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Florentin Smarandache
Compiled and Solved Problems
in Geometry and Trigonometry
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255 Compiled and Solved Problems in Geometry and Trigonometry
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FLORENTIN SMARANDACHE
255 Compiled and Solved Problems
in Geometry and Trigonometry
(from Romanian Textbooks)
Educational Publisher
2015
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Florentin Smarandache
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Peer reviewers:
Prof. Rajesh Singh, School of Statistics, DAVV, Indore (M.P.), India.
Dr. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences,
Beijing 100190, P. R. China.
Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000,
Pakistan
Prof. Stefan Vladutescu, University of Craiova, Romania.
Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik,
Casablanca B. P. 7951, Morocco.
E-publishing, Translation & Editing:
Dana Petras, Nikos Vasiliou
AdSumus Scientific and Cultural Society, Cantemir 13, Oradea, Romania
Copyright:
Florentin Smarandache 1998-2015
Educational Publisher, Chicago, USA
ISBN: 978-1-59973-299-2
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255 Compiled and Solved Problems in Geometry and Trigonometry
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Table of Content
Explanatory Note ..................................................................................................................................................... 4
Problems in Geometry (9th grade) ................................................................................................................... 5
Solutions ............................................................................................................................................................... 11
Problems in Geometry and Trigonometry ................................................................................................. 38
Solutions ............................................................................................................................................................... 42
Other Problems in Geometry and Trigonometry (10th grade) .......................................................... 60
Solutions ............................................................................................................................................................... 67
Various Problems ................................................................................................................................................... 96
Solutions ............................................................................................................................................................... 99
Problems in Spatial Geometry ...................................................................................................................... 108
Solutions ............................................................................................................................................................ 114
Lines and Planes ................................................................................................................................................. 140
Solutions ............................................................................................................................................................ 143
Projections ............................................................................................................................................................. 155
Solutions ............................................................................................................................................................ 159
Review Problems ................................................................................................................................................. 174
Solutions ............................................................................................................................................................ 182
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Explanatory Note
This book is a translation from Romanian of "Probleme Compilate ลi Rezolvate de
Geometrie ลi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and
includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and
solved from the Romanian Textbooks for 9th and 10th grade students, in the period
1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National
College in Balcesti, Valcea (Romania), Lycรฉe Sidi El Hassan Lyoussi in Sefrou (Morocco),
then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General
School (Romania), but also I did intensive private tutoring for students preparing their
university entrance examination. After that, I have escaped in Turkey in September 1988
and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I
immigrated to United States. The degree of difficulties of the problems is from easy
and medium to hard. The solutions of the problems are at the end of each chapter.
One can navigate back and forth from the text of the problem to its solution using
bookmarks. The book is especially a didactical material for the mathematical students
and instructors.
The Author
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Problems in Geometry (9th grade)
1. The measure of a regular polygonโs interior angle is four times bigger than
the measure of its external angle. How many sides does the polygon have?
Solution to Problem 1
2. How many sides does a convex polygon have if all its external angles are
obtuse?
Solution to Problem 2
3. Show that in a convex quadrilateral the bisector of two consecutive angles
forms an angle whose measure is equal to half the sum of the measures of
the other two angles.
Solution to Problem 3
4. Show that the surface of a convex pentagon can be decomposed into two
quadrilateral surfaces.
Solution to Problem 4
5. What is the minimum number of quadrilateral surfaces in which a convex
polygon with 9, 10, 11 vertices can be decomposed?
Solution to Problem 5
6. If (๐ด๐ต๐ถ) โก (๐ดโฒ๐ตโฒ๐ถโฒ) , then โ bijective function ๐ = (๐ด๐ต๐ถ) โ (๐ดโฒ๐ตโฒ๐ถโฒ) such
that for โ 2 points ๐, ๐ โ (๐ด๐ต๐ถ) , โ๐๐โ = โ๐(๐)โ, โ๐(๐)โ, and vice versa.
Solution to Problem 6
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7. If โ๐ด๐ต๐ถ โก โ๐ดโฒ๐ตโฒ๐ถโฒ then โ bijective function ๐ = ๐ด๐ต๐ถ โ ๐ดโฒ๐ตโฒ๐ถโฒ such that
(โ) 2 points ๐, ๐ โ ๐ด๐ต๐ถ, โ๐๐โ = โ๐(๐)โ, โ๐(๐)โ, and vice versa.
Solution to Problem 7
8. Show that if โ๐ด๐ต๐ถ~โ๐ดโฒ๐ตโฒ๐ถโฒ, then [๐ด๐ต๐ถ]~[๐ดโฒ๐ตโฒ๐ถโฒ].
Solution to Problem 8
9. Show that any two rays are congruent sets. The same property for lines.
Solution to Problem 9
10. Show that two disks with the same radius are congruent sets.
Solution to Problem 10
11. If the function ๐:๐ โ ๐โฒ is isometric, then the inverse function ๐โ1:๐ โ ๐โฒ
is as well isometric.
Solution to Problem 11
12. If the convex polygons ๐ฟ = ๐1, ๐2, โฆ , ๐๐ and ๐ฟโฒ = ๐1โฒ, ๐2
โฒ , โฆ , ๐๐โฒ have |๐๐ , ๐๐+1| โก
|๐๐โฒ, ๐๐+1
โฒ | for ๐ = 1, 2, โฆ , ๐ โ 1, and ๐๐๐๐+1๐๐+2 โก๐๐โฒ๐๐+1โฒ ๐๐+2
โฒ , (โ) ๐ = 1, 2, โฆ , ๐ โ
2, then ๐ฟ โก ๐ฟโฒ and [๐ฟ] โก [๐ฟโฒ].
Solution to Problem 12
13. Prove that the ratio of the perimeters of two similar polygons is equal to
their similarity ratio.
Solution to Problem 13
14. The parallelogram ๐ด๐ต๐ถ๐ท has โ๐ด๐ตโ = 6, โ๐ด๐ถโ = 7 and ๐(๐ด๐ถ) = 2. Find
๐(๐ท, ๐ด๐ต).
Solution to Problem 14
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255 Compiled and Solved Problems in Geometry and Trigonometry
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15. Of triangles ๐ด๐ต๐ถ with โ๐ต๐ถโ = ๐ and โ๐ถ๐ดโ = ๐, ๐ and ๐ being given
numbers, find a triangle with maximum area.
Solution to Problem 15
16. Consider a square ๐ด๐ต๐ถ๐ท and points ๐ธ, ๐น, ๐บ, ๐ป, ๐ผ, ๐พ, ๐ฟ,๐ that divide each side
in three congruent segments. Show that ๐๐๐ ๐ is a square and its area is
equal to 2
9๐[๐ด๐ต๐ถ๐ท].
Solution to Problem 16
17. The diagonals of the trapezoid ๐ด๐ต๐ถ๐ท (๐ด๐ต||๐ท๐ถ) cut at ๐.
a. Show that the triangles ๐ด๐๐ท and ๐ต๐๐ถ have the same area;
b. The parallel through ๐ to ๐ด๐ต cuts ๐ด๐ท and ๐ต๐ถ in ๐ and ๐. Show that
||๐๐|| = ||๐๐||.
Solution to Problem 17
18. ๐ธ being the midpoint of the non-parallel side [๐ด๐ท] of the trapezoid ๐ด๐ต๐ถ๐ท,
show that ๐[๐ด๐ต๐ถ๐ท] = 2๐[๐ต๐ถ๐ธ].
Solution to Problem 18
19. There are given an angle (๐ต๐ด๐ถ) and a point ๐ท inside the angle. A line
through ๐ท cuts the sides of the angle in ๐ and ๐. Determine the line ๐๐
such that the area โ๐ด๐๐ to be minimal.
Solution to Problem 19
20. Construct a point ๐ inside the triangle ๐ด๐ต๐ถ, such that the triangles ๐๐ด๐ต,
๐๐ต๐ถ, ๐๐ถ๐ด have equal areas.
Solution to Problem 20
21. Decompose a triangular surface in three surfaces with the same area by
parallels to one side of the triangle.
Solution to Problem 21
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22. Solve the analogous problem for a trapezoid.
Solution to Problem 22
23. We extend the radii drawn to the peaks of an equilateral triangle inscribed
in a circle ๐ฟ(๐, ๐), until the intersection with the circle passing through the
peaks of a square circumscribed to the circle ๐ฟ(๐, ๐). Show that the points
thus obtained are the peaks of a triangle with the same area as the
hexagon inscribed in ๐ฟ(๐, ๐).
Solution to Problem 23
24. Prove the leg theorem with the help of areas.
Solution to Problem 24
25. Consider an equilateral โ๐ด๐ต๐ถ with โ๐ด๐ตโ = 2๐. The area of the shaded
surface determined by circles ๐ฟ(๐ด, ๐), ๐ฟ(๐ต, ๐), ๐ฟ(๐ด, 3๐) is equal to the area of
the circle sector determined by the minor arc (๐ธ๐น) of the circle ๐ฟ(๐ถ, ๐).
Solution to Problem 25
26. Show that the area of the annulus between circles ๐ฟ(๐, ๐2) and ๐ฟ(๐, ๐2) is
equal to the area of a disk having as diameter the tangent segment to
circle ๐ฟ(๐, ๐1) with endpoints on the circle ๐ฟ(๐, ๐2).
Solution to Problem 26
27. Let [๐๐ด], [๐๐ต] two โฅ radii of a circle centered at [๐]. Take the points ๐ถ and
๐ท on the minor arc ๐ด๐ต๏ฟฝ๏ฟฝ such that ๐ด๏ฟฝ๏ฟฝโก๐ต๏ฟฝ๏ฟฝ and let ๐ธ, ๐น be the projections of
๐ถ๐ท onto ๐๐ต. Show that the area of the surface bounded by [๐ท๐น], [๐น๐ธ[๐ธ๐ถ]]
and arc ๐ถ๏ฟฝ๏ฟฝ is equal to the area of the sector determined by arc ๐ถ๏ฟฝ๏ฟฝ of the
circle ๐ถ(๐, โ๐๐ดโ).
Solution to Problem 27
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28. Find the area of the regular octagon inscribed in a circle of radius ๐.
Solution to Problem 28
29. Using areas, show that the sum of the distances of a variable point inside
the equilateral triangle ๐ด๐ต๐ถ to its sides is constant.
Solution to Problem 29
30. Consider a given triangle ๐ด๐ต๐ถ and a variable point ๐ โ |๐ต๐ถ|. Prove that
between the distances ๐ฅ = ๐(๐, ๐ด๐ต) and ๐ฆ = ๐(๐, ๐ด๐ถ) is a relation of ๐๐ฅ +
๐๐ฆ = 1 type, where ๐ and ๐ are constant.
Solution to Problem 30
31. Let ๐ and ๐ be the midpoints of sides [๐ต๐ถ] and [๐ด๐ท] of the convex
quadrilateral ๐ด๐ต๐ถ๐ท and {๐} = ๐ด๐ โฉ ๐ต๐ and {๐} = ๐ถ๐ โฉ ๐๐ท. Prove that the
area of the quadrilateral ๐๐๐๐ is equal to the sum of the areas of triangles
๐ด๐ต๐ and ๐ถ๐ท๐.
Solution to Problem 31
32. Construct a triangle having the same area as a given pentagon.
Solution to Problem 32
33. Construct a line that divides a convex quadrilateral surface in two parts
with equal areas.
Solution to Problem 33
34. In a square of side ๐, the middle of each side is connected with the ends of
the opposite side. Find the area of the interior convex octagon formed in
this way.
Solution to Problem 34
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35. The diagonal [๐ต๐ท] of parallelogram ๐ด๐ต๐ถ๐ท is divided by points ๐, ๐, in 3
segments. Prove that ๐ด๐๐ถ๐ is a parallelogram and find the ratio between
๐[๐ด๐๐ถ๐] and ๐[๐ด๐ต๐ถ๐ท].
Solution to Problem 35
36. There are given the points ๐ด, ๐ต, ๐ถ, ๐ท, such that ๐ด๐ต โฉ ๐ถ๐ท = {๐}. Find the
locus of point ๐ such that ๐[๐ด๐ต๐] = ๐[๐ถ๐ท๐].
Solution to Problem 36
37. Analogous problem for ๐ด๐ต||๐ถ๐ท.
Solution to Problem 37
38. Let ๐ด๐ต๐ถ๐ท be a convex quadrilateral. Find the locus of point ๐ฅ1 inside ๐ด๐ต๐ถ๐ท
such that ๐[๐ด๐ต๐] + ๐[๐ถ๐ท๐] = ๐, ๐ โ a constant. For which values of ๐ the
desired geometrical locus is not the empty set?
Solution to Problem 38
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Solutions
Solution to Problem 1.
180 (๐ โ 2)
๐= 4
180
5 ๐ = 10
Solution to Problem 2.
Let ๐ = 3๐ฅ1, ๐ฅ2, ๐ฅ3โข ext
โน
๐ฅ1 > 900
๐ฅ2 > 900
๐ฅ3 > 900
}โน ๐ฅ1 + ๐ฅ2 + ๐ฅ3 > 2700, so ๐ = 3 is possible.
Let ๐ = 4๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4โข ext
โน๐ฅ1 > 90
0
โฎ๐ฅ3 > 90
0} โน ๐ฅ1 + ๐ฅ2 + ๐ฅ3 + ๐ฅ4 > 360
0, so ๐ = 4 is impossible.
Therefore, ๐ = 3.
Solution to Problem 3.
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m(๐ด๐ธ๏ฟฝ๏ฟฝ) = m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ)
2
m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ) = 360ยฐ
m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ)
2 = 180ยฐ โ
m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ)
2
m(๐ด๐ธ๏ฟฝ๏ฟฝ) = 180ยฐ โ m(๏ฟฝ๏ฟฝ)
2โm(๏ฟฝ๏ฟฝ)
2 =
= 180ยฐ โ 180ยฐ +m(๏ฟฝ๏ฟฝ) + m(๏ฟฝ๏ฟฝ)
2 =m(๏ฟฝ๏ฟฝ) + (๏ฟฝ๏ฟฝ)
2
Solution to Problem 4.
Let ๐ธ๐ท๏ฟฝ๏ฟฝ โ ๐ด, ๐ต โ int. ๐ธ๐ท๏ฟฝ๏ฟฝ. Let ๐ โ |๐ด๐ต| โ ๐ โ int. ๐ธ๐ท๏ฟฝ๏ฟฝ โ |๐ท๐ โ int. ๐ธ๐ท๏ฟฝ๏ฟฝ, |๐ธ๐ด| โฉ
|๐ท๐ = โ โ ๐ท๐ธ๐ด๐ quadrilateral. The same for ๐ท๐ถ๐ต๐.
Solution to Problem 5.
9 vertices; 10 vertices; 11 vertices;
4 quadrilaterals. 4 quadrilaterals. 5 quadrilaterals.
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Solution to Problem 6.
We assume that ABC โก AโBโCโ . We construct a function ๐: ๐ด๐ต๐ถ โ ๐ดโฒ๐ตโฒ๐ถโฒ such that
{๐ (B) = Bโฒ
if P โ |BA, ๐(P) โ BโฒAโฒ
๐ โ |๐ต๐ถ, ๐(๐) โ ๐ตโฒ๐ถโฒ such that โ๐ต๐โ = โ๐ตโฒ๐โฒโ where ๐โฒ = ๐(๐น).
The so constructed function is bijective, since for different arguments there are
different corresponding values and โ point from ๐ดโฒ๐ตโฒ๐ถโฒ is the image of a single
point from ๐ด๐ต๏ฟฝ๏ฟฝ (from the axiom of segment construction).
If ๐, ๐ โ this ray,
โBPโ = โBโฒPโฒโ
โBQโ = โBโฒQโฒโ} โน โPQโ = โBQโ โ โBPโ = โBโฒQโฒโ โ โBโฒPโฒโ = โPโฒQโฒโ = โ๐(P), ๐(Q)โ.
If ๐, ๐ โ a different ray,
โBPโ = โBโฒPโฒโโBQโ = โBโฒQโฒโ
PBQ โก PโฒBโฒQโฒ
} โน โPBQ = โPโฒBโฒQโฒ โPQโ = โPโฒQโฒโ = โ๐ (P), ๐ (Q)โ.
Vice versa.
Let ๐ โถ ABC โ AโฒBโฒCโฒ such that ๐ bijective and โPQโ = โ๐(P), ๐(Q)โ.
Let ๐, ๐ โ |๐ต๐ด and ๐ ๐ โ |๐ต๐ถ.
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โPQโ = โPโฒQโฒโโPSโ = โPโฒSโฒโโQSโ = โQโฒSโฒโ
} โน โPQS โก โPโฒQโฒSโฒโน QPS โก QโฒPโฒSโฒ โนBPS โก BโฒPโฒSโฒ (1);
โ๐๐โ = โ๐โฒ๐โฒโโ๐ ๐โ = โ๐ โฒ๐โฒโโ๐๐โ = โ๐โฒ๐โฒโ
} โน โPRS โก โPโฒRโฒSโฒโฒ โน PSB โก PโฒSโฒBโฒ (2).
From (1) and (2) โน PBC โก PโฒBโฒSโฒ (as diff. at 180ยฐ) i.e. ABC โก AโฒBโฒCโฒ .
Solution to Problem 7.
Let โ๐ด๐ต๐ถ โก โ๐ดโฒ๐ตโฒ๐ถโฒ.
We construct a function ๐ โถ ๐ด๐ต๐ถ โ ๐ดโฒ๐ตโฒ๐ถโฒ such that ๐(๐ด) = ๐ดโฒ, ๐(๐ต) = ๐ตโฒ, ๐(๐ถ) = ๐ถโฒ
and so ๐ โ |๐ด๐ต| โ ๐โฒ = ๐(๐) โ |๐ดโฒ๐ตโฒ| such that ||๐ด๐|| = ||๐ดโฒ๐โฒ||;
๐ โ |๐ต๐ถ| โ ๐โฒ = ๐(๐) โ |๐ตโฒ๐ถโฒ| such that ||๐ต๐|| = ||๐ตโฒ๐โฒ||;
๐ โ |๐ถ๐ด| โ ๐โฒ = ๐(๐) โ |๐ถโฒ๐ดโฒ| such that ||๐ถ๐|| = ||๐ถโฒ๐โฒ||.
The so constructed function is bijective.
Let ๐ โ |๐ด๐ต| and ๐ โ |๐ถ๐ด| โน ๐โฒ โ |๐ดโฒ๐ตโฒ| and ๐โฒ โ |๐ถโฒ๐ดโฒ|.
โAPโ = โAโฒPโฒโ
โCQโ = โCโฒQโฒโ
โCAโ = โCโฒAโฒโ} โน โAQโ = โAโฒQโฒโ; A โก Aโฒโน โAPQ โก โAโฒPโฒQโฒ โน โPQโ = โPโฒQโฒโ.
Similar reasoning for (โ) point ๐ and ๐.
Vice versa.
We assume that โ a bijective function ๐ โถ ๐ด๐ต๐ถ โ ๐ดโฒ๐ตโฒ๐ถโฒ with the stated
properties.
We denote ๐(A) = Aโฒโฒ, ๐(B) = Bโฒโฒ, ๐(C) = Cโฒโฒ
โน โABโ = โAโฒโฒBโฒโฒโ, โBCโ = โBโฒโฒCโฒโฒโ, โACโ = โAโฒโฒCโฒโฒโโABC = โAโฒโฒBโฒโฒCโฒโฒ.
Because ๐(ABC) = ๐([AB] โช [BC] โช [CA]) = ๐([AB]) โช ๐([BC]) โช ๐([CA])
= [AโฒโฒBโฒโฒ] โช [BโฒโฒCโฒโฒ][CโฒโฒAโฒโฒ] = AโฒโฒBโฒโฒCโฒโฒ.
But by the hypothesis ๐(๐ด๐ต๐ถ) = ๐(๐ดโ๐ตโ๐ถโ), therefore
AโฒโฒBโฒโฒCโฒโฒ = โAโฒBโฒCโฒโน โABC โก โAโฒBโฒCโฒ.
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Solution to Problem 8.
If โ๐ด๐ต๐ถ~โ๐ดโ๐ตโ๐ถโ then (โ) ๐: ๐ด๐ต๐ถ โ ๐ดโ๐ตโ๐ถโ and ๐ > 0 such that:
||๐๐|| = ๐ ||๐(๐), ๐(๐)||, ๐, ๐ โ ๐ด๐ต๐ถ;
โ๐ด๐ต๐ถ~โ๐ดโฒ๐ตโฒ๐ถโฒ โน
โABโ
โAโฒBโฒโ=
โBCโ
โBโฒCโฒโ=
โCAโ
โCโฒAโฒโ= ๐
A โก Aโฒ; B โก Bโฒ; C โก Cโฒ} โน
โABโ = ๐โAโฒBโฒโ
โBCโ = ๐โBโฒCโฒโโCAโ = ๐โCโฒAโฒโ
.
We construct a function ๐: ๐ด๐ต๐ถ โ ๐ดโฒ๐ตโฒ๐ถโฒ such that ๐(๐ด) = ๐ดโฒ, ๐(๐ต) = ๐ตโฒ, ๐(๐ถ) = ๐ถโฒ;
if ๐ โ |๐ต๐ถ| โ ๐ โ |๐ตโฒ๐ถโฒ| such that ||๐ต๐|| = ๐||๐ตโฒ๐โฒ||;
if ๐ โ |๐ถ๐ด| โ ๐ โ |๐ถโฒ๐ดโฒ| such that ||๐ถ๐|| = ๐||๐ถโฒ๐โฒ||; ๐ โ similarity constant.
Let ๐, ๐ โ ๐ด๐ต such that ๐ โ |๐ต๐ถ|, ๐ โ |๐ด๐ถ| โน ๐โฒ โ |๐ตโฒ๐ถโฒ| and ||๐ต๐|| = ๐||๐ตโฒ๐โฒ||
Qโฒ โ |AโฒCโฒ| and โCQโ = ๐โCโฒQโฒโ (1);
As โBCโ = ๐โBโฒCโฒโ โน โPCโ = โBCโ โ โBPโ = ๐โBโฒCโฒโ โ ๐โBโฒPโฒโ =
= ๐(โBโฒCโฒโ โ โBโฒPโฒโ) = ๐โPโฒCโฒโ (2);
C โก Cโฒ (3).
From (1), (2), and (3) โน โPCQ~โPโฒCโฒQโฒ โน โPQโ = ๐โPโฒQโฒโ .
Similar reasoning for ๐, ๐ โ ๐ด๐ต๐ถ.
We also extend the bijective function previously constructed to the interiors of
the two triangles in the following way:
Let ๐ โ int. ๐ด๐ต๐ถ and we construct ๐โ โ int. ๐ดโ๐ตโ๐ถโ such that ||๐ด๐|| = ๐||๐ดโ๐โ|| (1).
Let ๐ โ int. ๐ด๐ต๐ถ โ ๐โฒ โ int. ๐ดโ๐ตโ๐ถโ such that BAQ โก BโAโQโ and ||๐ด๐|| = ๐||๐ดโ๐โ|| (2).
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From (1) and (2),
AP
AโฒPโฒ=AQ
AโฒQโฒ= ๐, PAQ โก PโฒAโฒQโฒ โนโAPQ ~ โAโฒPโฒQโฒโน โPQโ = ๐โPโฒQโฒโ,
but ๐, ๐ โ [๐ด๐ต๐ถ], so [๐ด๐ต๐ถ] ~[๐ดโฒ๐ตโฒ๐ถโฒ].
Solution to Problem 9.
a. Let |๐๐ด and |๐โฒ๐ดโฒ be two rays:
Let ๐: |๐๐ด โ |๐โฒ๐ดโฒ such that ๐(๐) = ๐โฒ and ๐(๐) = ๐โฒ with ||๐๐|| = ||๐โฒ๐โฒ|.
The so constructed point ๐โฒ is unique and so if ๐ โ ๐ โน ||๐๐|| โ ||๐๐|| โน
||๐โฒ๐โฒ|| โ ||๐โฒ๐โฒ|| โน ๐โฒ โ ๐โฒ and (โ)๐โฒ โ |๐โฒ๐ดโฒ (โ) a single point ๐ โ |๐๐ด such that
||๐๐|| = ||๐โฒ๐โฒ||.
The constructed function is bijective.
If ๐, ๐ โ |๐๐ด, ๐ โ |๐๐| โ ๐โฒ๐โฒ โ |๐โฒ๐ดโฒ such that โOPโ = โOโฒPโฒโ; โOQโ = โOโฒQโฒโ โน
โPQโ = โOQโ โ โOPโ = โOโฒQโฒโ โ โOโฒPโฒโ = โPโฒQโฒโ(โ)P;๐ โ |OA
โน the two rays are congruent.
b. Let ๐ and ๐โฒ be two lines.
Let ๐ โ ๐ and ๐โฒ โ ๐โฒ. We construct a function ๐: ๐ โ ๐โฒ such that ๐(๐) = ๐โฒ and
๐ (|๐๐ด) = |๐โฒ๐ดโฒ and ๐ (|๐๐ต) = |๐โฒ๐ตโฒ as at the previous point.
It is proved in the same way that ๐ is bijective and that ||๐๐|| = ||๐โฒ๐โฒ|| when ๐
and ๐ belong to the same ray.
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If ๐, ๐ belong to different rays:
โ๐๐โ = โ๐โฒ๐โฒโโ๐๐โ = โ๐โฒ๐โฒโ
} โน โ๐๐โ = โ๐๐โ + โ๐๐โ = โ๐โฒ๐โฒโ + โ๐โฒ๐โฒโ = โ๐โฒ๐โฒโ
and so the two rays are congruent.
Solution to Problem 10.
We construct a function ๐:๐ท โ ๐ทโฒ such that ๐(๐) = ๐โฒ, ๐(๐ด) = ๐ดโฒ and a point
(โ) ๐ โ ๐ท โ ๐โฒ โ ๐ทโฒ which are considered to be positive.
From the axiom of segment and angle construction โน that the so constructed
function is bijective, establishing a biunivocal correspondence between the elements
of the two sets.
Let ๐ โ ๐ท โ ๐โฒ โ ๐ทโฒ such that ||๐๐โฒ|| = ||๐๐||; ๐ด๐๏ฟฝ๏ฟฝ โก ๐ดโฒ๐โฒ๐โฒ .
As:
โ๐๐โ = โ๐โฒ๐โฒโโ๐๐โ = โ๐โฒ๐โฒโ
๐๐๏ฟฝ๏ฟฝ โก ๐โฒ๐โฒ๐โฒ}โนโ๐๐๐ โก ๐โฒ๐โฒ๐โฒ โนโ๐๐โ = โ๐โฒ๐โฒโ, (โ) ๐, ๐ โ ๐ท โน ๐ท โก ๐ทโฒ.
Solution to Problem 11.
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๐:๐ โ ๐โฒ is an isometry โน ๐ is bijective and (โ) ๐, ๐ โ ๐ we have ||๐๐|| =
||๐(๐), ๐(๐)||, ๐ โ bijective โน ๐ โ invertible and ๐โ1 โ bijective.
โ๐โฒ๐โฒโ = โ๐(๐); ๐(๐)โ = โ๐๐โ
โ๐โ1(๐โฒ); ๐โ1(๐โฒ)โ = โ๐โ1(๐(๐)), ๐โ1(๐(๐))โ = โ๐๐โ} โน
โ๐โฒ๐โฒโ = โ๐โ1(๐(๐โฒ)), ๐โ1(๐(๐โฒ))โ, (โ)๐โฒ, ๐โฒ โ ๐,
therefore ๐โ1:๐โฒ โ ๐ is an isometry.
Solution to Problem 12.
We construct a function ๐ such that ๐(๐๐) = ๐๐โฒ, ๐ = 1, 2, โฆ , ๐, and if ๐ โ
|๐๐ , ๐๐+1|.
The previously constructed function is also extended inside the polygon as
follows: Let ๐ โ int. ๐ฟ โ ๐โฒ โ int. ๐ฟโฒ such that ๐๐๐๐๐+1 โก ๐โฒ๐โฒ๐๐โฒ๐+1 and โ๐๐๐โ =
โ๐โฒ๐โฒ๐โ. We connect these points with the vertices of the polygon. It can be easily
proved that the triangles thus obtained are congruent.
We construct the function ๐: [๐ฟ] โ [๐ฟโฒ] such that
๐(๐) = {
๐(๐), if ๐ โ ๐ฟ
๐โฒ, if ๐ = ๐ ๐โฒ, if ๐ โ [๐๐๐๐๐+1] such that
๐๐๐๏ฟฝ๏ฟฝ โก ๐๐โฒ๐โฒ๐โฒ (โ)๐ = 1, 2, โฆ , ๐ โ 1
The so constructed function is bijective (โ) ๐, ๐ โ [๐ฟ]. It can be proved by the
congruence of the triangles ๐๐๐ and ๐โฒ๐โฒ๐โฒ that ||๐๐|| = ||๐โฒ๐โฒ||, so [๐ฟ] = [๐ฟโฒ]
โน if two convex polygons are decomposed
in the same number of triangles
respectively congruent,
they are congruent.
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Solution to Problem 13.
๐ฟ = ๐1๐2โฆ , ๐๐; ๐ฟโฒ = ๐1
โฒ๐2โฒ โฆ , ๐๐
โฒ
๐ฟ~๐ฟโฒ โน (โ)๐พ > 0 and ๐: ๐ฟ โ ๐ฟโฒ such that โ๐๐โ = ๐โ๐(๐)๐(๐)โ (โ)๐, ๐ โ ๐ฟ,
and ๐๐ผโฒ = ๐(๐๐).
Taking consecutively the peaks in the role of ๐ and ๐, we obtain:
โ๐1๐2โ = ๐โ๐1โฒ๐2โฒโ โ
โ๐1๐2โ
โ๐1โฒ๐2โฒโ= ๐
โ๐2๐3โ = ๐โ๐2โฒ๐3โฒโ โ
โ๐2๐3โ
โ๐2โฒ๐3โฒโ= ๐
โฎ
โ๐๐โ1๐๐โ = ๐โ๐๐โ1โฒ ๐๐
โฒโ โโ๐๐โ1๐๐โ
โ๐๐โ1โฒ ๐๐โฒโ
= ๐
โ๐๐๐1โ = ๐โ๐๐โฒ๐1โฒโ โ
โ๐๐๐1โ
โ๐๐โฒ๐1โฒโ= ๐
}
โน
โน ๐ =โ๐1๐2โ
โ๐1โฒ๐2โฒโ=โ๐2๐3โ
โ๐2โฒ๐3โฒโ= โฏ =
โ๐1๐2โ + โ๐2๐3โ +โฏ+ โ๐๐โ1๐๐โ + โ๐๐๐1โ
โ๐1โฒ๐2โฒโ + โ๐2
โฒ๐3โฒโ +โฏ+ โ๐๐โ1
โฒ ๐๐โฒโ + โ๐๐โฒ๐1โฒโ=๐
๐โฒ .
Solution to Problem 14.
๐[๐ด๐ท๐ถ] =2โ7
2= 7; ๐[๐ด๐ต๐ถ๐ท] = 2 โ 7 = 14 = 6โ๐ท๐นโ โน โ๐ท๐นโ =
14
6=7
3 .
Solution to Problem 15.
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โ = ๐ โ sin๐ถ โค ๐;
๐[๐ด๐ต๐ถ] =๐โโ
2 is max. when โ is max.
max. โ = ๐ when sin๐ถ = 1
โ ๐(๐ถ) = 90 โ ๐ด๐ต๐ถ has a right angle at ๐ถ.
Solution to Problem 16.
โ๐๐ทโ = โ๐ท๐ผโ โน ๐๐ท๐ผ โ an isosceles triangle.
โน๐(๐ท๐๏ฟฝ๏ฟฝ) = ๐(๐๐ผ๏ฟฝ๏ฟฝ) = 450 ;
The same way, ๐(๐น๐ฟ๏ฟฝ๏ฟฝ) = ๐(๐ด๐น๏ฟฝ๏ฟฝ) = ๐(๐ต๐ธ๏ฟฝ๏ฟฝ) = ๐(๐ธ๐ป๏ฟฝ๏ฟฝ).
โ๐ ๐พโโน โ๐๐โ = โ๐๐โ = โ๐๐ โ = โ๐ ๐โ โน ๐๐ ๐๐ is a square.
โ๐ด๐ตโ = ๐, โ๐ด๐ธโ =2๐
3, โ๐๐ผโ = โ
4๐2
9+4๐2
9=2๐โ2
3 ;
2โ๐ ๐ผโ2 =๐2
9โน โ๐ ๐ผโ2 =
๐2
18โน โ๐ ๐ผโ =
๐
3โ2=๐โ2
6 ;
โ๐๐ โ =2๐โ2
3โ 2
๐โ2
6=๐โ2
3 ;
๐[๐๐ ๐๐] =2๐2
9=2
9๐[๐ด๐ต๐ถ๐ท].
Solution to Problem 17.
๐[๐ด๐ถ๐ท] =โ๐ท๐ถโ โ โ๐ด๐ธโ
2
๐[๐ต๐ถ๐ท] =โ๐ท๐ถโ โ โ๐ต๐นโ
2โ๐ด๐ธโ = โ๐ต๐นโ }
โน ๐[๐ด๐ถ๐ท] = ๐[๐ต๐ถ๐ท]
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๐[๐ด๐๐ท] = ๐[๐ด๐๐] + ๐[๐๐๐ท]
๐[๐ด๐๐] = ๐[๐๐๐] =โ๐๐โ โ โ๐๐โ
2
๐[๐๐๐ท] = ๐[๐๐๐] =โ๐๐โ โ โ๐๐โ
2 }
โน ๐[๐ด๐๐ท] =โ๐๐โ(โ๐๐โ + โ๐๐โ)
2=โ๐๐โ โ โ
2
The same way,
๐[๐ต๐๐ถ] =โ๐๐โ โ โ
2 .
Therefore,
๐[๐ด๐๐ท] = ๐[๐ต๐๐ถ] โนโ๐๐โ โ โ
2=โ๐๐โ โ โ
2โน โ๐๐โ = โ๐๐โ.
Solution to Problem 18.
โ๐ด๐ธโ = โ๐ธ๐ทโ ;
We draw ๐๐ โฅ ๐ด๐ต;๐ท๐ถ;
โ๐ธ๐โ = โ๐ธ๐โ =โ
2;
๐[๐ต๐ธ๐ถ] =(โ๐ด๐ตโ + โ๐ท๐ถโ) โ โ
2โโ๐ด๐ตโ โ โ
4โโ๐ท๐ถโ โ โ
4=(โ๐ด๐ตโ + โ๐ท๐ถโ) โ โ
4=1
2๐[๐ด๐ต๐ถ๐ท];
Therefore, [๐ด๐ต๐ถ๐ท] = 2๐[๐ต๐ธ๐ถ] .
Solution to Problem 19.
๐[๐ด๐ธ๐ท๐โฒ] is ct. because ๐ด, ๐ธ, ๐ท,๐โฒ are fixed points.
Let a line through ๐ท, and we draw โฅ to sides ๐๐ท and ๐ท๐ธ.
No matter how we draw a line through ๐ท, ๐[๐๐๐ด] is formed of: ๐[๐ด๐ธ๐ท๐] +
๐[๐๐๐] + ๐[๐ท๐ธ๐].
We have ๐[๐ด๐ธ๐ท๐] constant in all triangles ๐๐ด๐.
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Letโs analyse:
๐[๐๐โฒ๐ท] + ๐[๐ท๐ธ๐] =โ๐โฒ๐ทโ โ โ1
2+โ๐ธ๐โฒโ โ โ2
2=โ๐โฒ๐ทโ
2(โ1 +
โ๐ธ๐โ
โ๐๐ทโโ โ2)
=โ๐โฒ๐ทโ
2โ (โ1 +
โ2โ1โ โ2) =
โ๐โฒ๐ทโ
2โ1[(โ1 โ โ2)
2 + 2โ1โ2].
โ๐ด๐๐ is minimal when โ1 = โ2โน๐ท is in the middle of |๐๐|. The construction is
thus: โ๐ด๐๐ where ๐๐ || ๐ธ๐โฒ. In this case we have |๐๐ท| โก |๐ท๐|.
Solution to Problem 20.
๐[๐ด๐ต๐ถ] =โ๐ต๐ถโ โ โ๐ด๐ดโฒโ
2
Let the median be |๐ด๐ธ|, and ๐ be the centroid of the triangle.
Let ๐๐ท โฅ ๐ต๐ถ. ๐[๐ต๐๐ถ] =โ๐ต๐ถโโโ๐๐ทโ
2 .
๐ด๐ดโฒ โฅ ๐ต๐ถ๐๐ท โฅ ๐ต๐ถ
} โ ๐ด๐ดโฒ โฅ ๐๐ท โ โ๐๐ท๐ธ~โ๐ด๐ดโฒ๐ธ โโ๐๐ทโ
โ๐ด๐ดโฒโ=โ๐๐ธโ
โ๐ด๐ธโ=1
3โ โ๐๐ทโ =
โ๐ด๐ดโฒโ
3โ ๐[๐ต๐๐ถ]
=โ๐ต๐ถโ โ
โ๐ด๐ดโฒโ3
2=1
3
โ๐ต๐ถโ โ โ๐ด๐ดโฒโ
2=1
3๐[๐ด๐ต๐ถ].
We prove in the same way that ๐[๐๐ด๐ถ] = ๐[๐๐ด๐ต] =1
3๐[๐ด๐ต๐ถ], so the specific
point is the centroid.
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Solution to Problem 21.
Let ๐,๐ โ ๐ด๐ต such that ๐ โ |๐ด๐|.
We take ๐๐โฒ โฅ ๐ต๐ถ, ๐๐โฒ โฅ ๐ต๐ถ.
โ๐ด๐๐โฒ~โ๐ด๐ต๐ถ โ๐[๐ด๐๐โฒ]
๐[๐ด๐ต๐ถ]= (๐ด๐
๐ด๐ต)
๐[๐ด๐๐โฒ] =1
3๐, (
โ๐ด๐โ
โ๐ด๐ตโ)
2
=1
3 , โ๐ด๐โ =
โ๐ด๐ตโ
โ3; โ๐ด๐๐โฒ~โ๐ด๐ต๐ถ โน
๐[๐ด๐๐โฒ]
๐[๐ด๐ต๐ถ]= (
โ๐ด๐โ
โ๐ด๐ตโ)
2
๐[๐ด๐๐โฒ] =2
3๐[๐ด๐ต๐ถ]
}
โน (โ๐ด๐โ
โ๐ด๐ตโ)
2
=2
3โน โ๐ด๐โ = โ
2
3โ๐ด๐ตโ .
Solution to Problem 22.
โ๐๐ทโ = ๐, โ๐๐ดโ = ๐ ;
๐[โ๐ถ๐โฒ๐] = ๐[๐๐โฒ๐โฒ๐] = ๐[๐๐โฒ๐ต๐ด] =1
3๐[๐ด๐ต๐ถ๐ท] ;
โ๐๐ท๐ถ~โ๐๐ด๐ต โน๐[๐๐ท๐ถ]
๐[๐๐ด๐ต]=โ๐๐ทโ2
โ๐๐ดโ2=๐
๐ โน
๐[๐๐ท๐ถ]
๐[๐๐ด๐ต] โ ๐[๐๐ท๐ถ]=
๐2
๐2 โ ๐2โน
๐[๐๐ท๐ถ]
๐[๐ด๐ต๐ถ๐ท]
=๐2
๐2 โ ๐2 (1)
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โ๐๐ท๐ถ~โ๐๐๐โฒโน๐[๐๐ท๐ถ]
๐[๐๐๐โฒ]= (
โ๐๐ทโ
โ๐๐โ)
2
โน๐[๐๐ท๐ถ]
๐[๐๐๐โฒ] โ ๐[๐๐ท๐ถ]=
๐2
โ๐๐โ2
โน๐[๐๐ท๐ถ]
๐[๐ท๐ถ๐๐โฒ]=
๐2
โ๐๐โ2 โ ๐2โน
๐[๐๐ท๐ถ]
13๐[๐ด๐ต๐ถ๐ท]
=๐2
โ๐๐โ2 โ ๐2 (2)
โ๐๐๐โฒ~โ๐๐ท๐ถ โน๐[๐๐ท๐ถ]
๐[๐ท๐๐โฒ]=โ๐๐ทโ
โ๐๐โ=
๐2
โ๐๐โโน
๐[๐ผ๐ท๐ถ]
๐[๐๐๐โฒ] โ ๐[๐๐ท๐ถ]=
๐2
โ๐๐โ2
โน๐[๐๐ท๐ถ]
๐[๐ท๐ถ๐๐โฒ]=
๐2
โ๐๐โ2 โ ๐2โน
๐[๐๐ท๐ถ]
23๐[๐ด๐ต๐ถ๐ท]
=๐2
โ๐๐โ2 โ ๐2 (3)
We divide (1) by (3):
2
3=โ๐๐โ2 โ ๐2
๐2 โ ๐2 โน 3โ๐๐โ2 โ 3๐2 = 2๐2 โ 2๐2โน โ๐๐โ2 =
๐2 + 2๐2
3 .
Solution to Problem 23.
โ๐๐ดโ = ๐ โ โ๐ท๐ธโ = 2๐; ๐hexagon =3๐2โ2
2 (1)
๐ท๐ธ๐น๐ a square inscribed in the circle with radius ๐ โน
โน ๐4 = ๐ โ2 = โ๐ท๐ธโ โน ๐โ2 = 2๐ โน ๐ = ๐โ2
โ๐๐โ = ๐ = ๐โ2
๐[๐๐๐] =โ๐๐โ โ โ๐๐โ sin 120
2=๐โ2 โ ๐โ2 โ
โ32
2=๐2โ3
2
๐[๐๐๐] = 3๐[๐๐๐] = 3๐2โ3
2=3๐2โ2
2 (2)
From (1) and (2) โน ๐[๐๐๐] = ๐hexagon.
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Solution to Problem 24.
โ๐ด๐ตโ2 = โ๐ต๐ถโ โ โ๐ต๐ดโฒโ
We construct the squares ๐ต๐ถ๐ธ๐ท on the hypotenuse and ๐ด๐ต๐น๐บ on the leg.
We draw ๐ด๐ดโฒ โฅ ๐ต๐ถ.
๐[๐ด๐ต๐น๐บ] = โ๐ด๐ตโ2 ๐[๐ดโฒ๐ต๐ท๐ป] = โ๐ต๐ทโ โ โ๐ต๐ดโฒโ = โ๐ต๐ถโ โ โ๐ต๐ดโฒโโฆ
Solution to Problem 25.
๐(๐ 1) = ๐[๐ด๐ต๐ถ] โ 3๐[sect. ๐ด๐ท๐ป]
๐[๐ด๐ต๐ถ] =๐2โ3
4=(2๐)2โ3
4= ๐2โ3
๐[sect. ๐ด๐ท๐ป] =๐2
2๐(๐ท๏ฟฝ๏ฟฝ)
๐(๐ท๏ฟฝ๏ฟฝ) =๐
180๐(๐ท๏ฟฝ๏ฟฝ) =
๐
180โ 600 =
๐
3
๐[sect. ๐ด๐ท๐ป] =๐2
2โ๐
3=๐๐2
2
๐(๐ 1) = ๐2โ3 โ 3
๐๐2
6= ๐2โ3 โ
๐๐2
2 (1)
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26
๐(๐ 2) = ๐[sect. ๐ด๐ธ๐บ] โ ๐[sect. ๐ด๐ต๐ถ] โ ๐[sect. ๐ธ๐ถ๐น] โ ๐[sect. ๐บ๐ต๐น]
=(3๐)2
2โ๐
180โ 60 โ ๐2โ3 โ
๐2
2โ๐
180โ 120 =
9๐2
2โ๐
3โ ๐2โ3 โ
๐๐2
3โ๐๐2
3
=3๐๐2
2โ2๐๐2
3โ ๐2โ3 (2)
From (1) and (2)
โน ๐(๐ 1) + ๐(๐ 2) =3๐๐2
2โ2๐๐2
3โ๐๐2
2=2๐๐2
6=๐๐2
3 .
Solution to Problem 26.
โ๐ด๐ทโ2 = ๐22 โ ๐1
2
๐[๐ฟ(๐, ๐1)] = ๐๐12
๐[๐ฟ(๐, ๐2)] = ๐๐22
๐[annulus] = ๐๐22 โ ๐๐1
2 = ๐(๐22 โ ๐1
2) (1)
๐[disk diameterโ๐ด๐ตโ] = ๐โ๐ด๐ทโ2 = ๐(๐22 โ ๐1
2) (2)
From (1) and (2) โน ๐[annulus] = ๐[disk diameter].
Solution to Problem 27.
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๐[๐ถ๐ท๐ธ๐น] =๐[๐ถ๐ท๐ทโฒ๐ถโฒ]
2
๐[๐ถ๐ท๐ทโฒ๐ถโฒ] = ๐seg[๐ถ๐ท๐ต๐ทโฒ๐ถโฒ] โ ๐seg[๐ท๐ต๐ทโฒ]
We denote ๐(๐ด๏ฟฝ๏ฟฝ) = ๐(๐ต๏ฟฝ๏ฟฝ) = ๐ โน ๐(๐ถ๏ฟฝ๏ฟฝ) =๐
2โ 2๐
๐sect. =๐2
2(๐ผ โ sin๐ผ)
๐[๐ถ๐ท๐ต๐ทโฒ๐ถโฒ] =๐2
2[๐ โ 2๐ผ โ sin(๐ โ 2๐ผ)] = ๐
๐2
2[๐ โ 2๐ผ โ sin(๐ โ 2๐ผ)]
๐๐ด =๐2
2(2๐ผ โ sin 2๐ผ)
๐[๐ถ๐ท๐ทโฒ๐ถโฒ] = ๐[๐ถ๐ท๐ต๐ทโฒ๐ถโฒ] โ ๐[๐ท๐ต๐ทโฒ] =๐2
2(๐ โ 2๐ผ โ sin2๐ผ) =
๐2
2(2๐ผ โ sin2๐ผ)
=๐2
2(๐ โ 2๐ผ โ sin 2๐ผ โ 2๐ผ + sin 2๐ผ) =
๐2
2(๐ โ 4๐ผ)
โน ๐[๐ถ๐ท๐ธ๐น] =๐[๐ถ๐ท๐ทโฒ๐ถโฒ]
2=๐2
4(๐ โ 4๐ผ) =
๐2
2(๐
2โ 2๐ผ) (1)
๐[sect. ๐ถ๐๐ท] =๐2
2๐(๐ถ๏ฟฝ๏ฟฝ) =
๐2
2(๐
2โ 2๐ผ) (2)
From (1) and (2) โน ๐[๐ถ๐ท๐ธ๐น] = ๐[sect. ๐ถ๐๐ท].
โ๐1๐นโ = โ๐๐ธโ
๐[square] = โ๐ท๐ธโ2 = โ๐๐ดโ2 =๐๐
2= ๐[๐ด๐ต๐ถ]
Solution to Problem 28.
๐(๐ด๐๏ฟฝ๏ฟฝ) =๐
4
๐[๐ด๐๐ต] =๐2 sin
๐4
2=๐2โ222
=๐2โ2
4
๐[orthogon] = 8 โ๐2โ2
4= 2โ2๐2
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Solution to Problem 29.
๐[๐ด๐ต๐ถ] = ๐[๐ด๐๐ต] + ๐[๐ด๐๐ถ] + ๐[๐๐ต๐ถ] โน ๐โ๐ = ๐๐3 + ๐๐2 + ๐๐1
๐1 + ๐2 + ๐3 = โ๐ (๐ is the side of equilateral triangle)
โน ๐1 + ๐2 + ๐3 =๐โ3
2 (because โ๐ =
๐โ3
2 ).
Solution to Problem 30.
๐ด๐ต๐ถ โ given โ โน ๐, ๐, ๐, โ โ constant
๐[๐ด๐ต๐ถ] =๐โ
2
๐[๐ด๐ต๐ถ] = ๐[๐ด๐๐ต] + ๐[๐ด๐๐ถ]
โน๐โ
2=๐๐ฅ
2+๐๐ฆ
2โน ๐๐ฅ + ๐๐ฆ = ๐๐ โน
๐
๐โ๐ฅ +
๐
๐โ๐ฆ = 1 โน ๐๐ฅ + ๐๐ฆ = 1,
where ๐ =๐
๐โ and ๐ =
๐
๐โ .
Solution to Problem 31.
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We draw ๐ด๐ดโฒ โฅ ๐ต๐ถ;๐๐โฒ โฅ ๐ต๐ถ;๐ท๐ทโฒ โฅ ๐ต๐ถ โ๐ด๐ดโฒ โฅ ๐๐โฒ โฅ ๐ท๐ทโฒ
โ๐ด๐โ = โ๐๐ทโ} โน ๐๐โฒmedian line in
the trapezoid ๐ด๐ดโฒ๐ทโฒ๐ท โน โ๐๐โฒโ =โ๐ด๐ดโฒโ+โ๐ท๐ทโฒโ
2, ๐[๐ต๐ถ๐] =
โ๐ต๐ถโ+โ๐๐โฒโ
2.
Solution to Problem 32.
First, we construct a quadrilateral with the same area as the given pentagon. We
draw through C a parallel to BD and extend |AB| until it intersects the parallel at M.
๐[๐ด๐ต๐ถ๐ท๐ธ] = ๐[๐ด๐ต๐ท๐ธ] + ๐[๐ต๐ถ๐ท],
๐[๐ต๐ถ๐ท] = ๐[๐ต๐ท๐] (have the vertices on a parallel at the base).
Therefore, ๐[๐ด๐ต๐ถ๐ท๐ธ] = ๐[๐ด๐๐ท๐ธ].
Then, we consider a triangle with the same area as the quadrilateral ๐ด๐๐ท๐ธ.
We draw a parallel to ๐ด๐ท,๐ is an element of the intersection with the same
parallel.
๐[๐ด๐๐ท๐ธ] = ๐[๐ด๐ท๐ธ] + ๐[๐ด๐ท๐ธ] = ๐[๐ด๐ท๐ธ] + ๐[๐ด๐ท๐] = ๐[๐ธ๐ท๐].
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Solution to Problem 33.
โ๐ด๐ธโ = โ๐ธ๐ถโ
โ๐ธ๐นโ๐ต๐ท โน ๐[๐ต๐ท๐น] = ๐[๐ต๐ท๐ธ]
๐[๐ด๐ต๐น๐ท] = ๐[๐ด๐ต๐ธ๐ท] (1)
๐[๐ด๐ท๐ธ] = ๐[๐ท๐ธ๐ถ] equal bases and the same height;
๐[๐ด๐ท๐ธ] = ๐[๐ท๐ธ๐ถ]
๐[๐ด๐ต๐ธ๐ท] = ๐[๐ต๐ธ๐ท๐ถ] (2)
๐[๐ท๐ธ๐น] = ๐[๐ต๐ธ๐น] the same base and the vertices on parallel lines at the base;
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Solution to Problem 34.
It is proved in the same way that:
โน๐๐๐๐ rhombus with right angleโน๐๐๐๐ is a square.
It is proved in the same way that all the peaks of the octagon are elements of the
axis of symmetry of the square, thus the octagon is regular.
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Consider the square separately.
Solution to Problem 35.
โ๐๐โ = โ๐๐โ = โ๐๐ตโ โ๐ท๐ถโ?โน โ๐๐๐ถ = โ๐๐ต๐ด โน โ๐๐ถโ = โ๐ด๐โ
It is proved in the same way that โ๐ท๐ด๐ = โ๐ต๐ถ๐ โน โ๐๐ถโ = โ๐๐ถโ.
Thus ๐ด๐๐ถ๐ is a parallelogram.
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Solution to Problem 36.
To determine the angle ฮฑ:
We write
thus we have established the positions of the lines of the locus.
constant for ๐ด, ๐ต, ๐ถ, ๐ท โ fixed points.
We must find the geometrical locus of points ๐ such that the ratio of the
distances from this point to two concurrent lines to be constant.
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โ๐๐ธโ
โ๐๐นโ= ๐. Let M' be another point with the same property, namely
โ๐โฒ๐ธโฒโ
โ๐โฒ๐นโฒโ= ๐.
๐,๐,๐โฒ collinear โน the locus is a line that passes through ๐.
When the points are in โข๐ถ๐๐ต we obtain one more line that passes through ๐.
Thus the locus is formed by two concurrent lines through ๐, from which we
eliminate point ๐, because the distances from ๐ to both lines are 0 and their ratio is
indefinite.
Vice versa, if points ๐ and ๐โฒ are on the same line passing through ๐, the ratio
of their distances to lines ๐ด๐ต and ๐ถ๐ท is constant.
Solution to Problem 37.
We show in the same way as in the previous problem that:
โ๐๐ธโ
โ๐๐นโ= ๐ โน
โ๐๐ธโ
โ๐๐นโ + โ๐๐ธโ=
๐
1 โ ๐โน โ๐๐ธโ =
๐๐
1 + ๐ ,
and the locus of the points which are located at a constant distance from a given
line is a parallel to the respective line, located between the two parallels.
If ||๐ด๐ต|| > ||๐ถ๐ท|| โน ๐(๐๐ด๐ธ) < ๐(๐๐ถ๐ท).
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Then, if
๐๐ธ
๐๐น= ๐ โน
๐๐ธ
๐๐น โ๐๐ธ=
๐
1 โ ๐โน๐๐ธ
๐=
๐
1 โ ๐โน ๐๐ธ =
๐๐
1 โ ๐ ,
thus we obtain one more parallel to ๐ด๐ต.
Solution to Problem 38.
Solution no. 1
We suppose that ABCD is not a parallelogram. Let {๐ผ} = ๐ด๐ต โฉ ๐ถ๐ท. We build ๐ธ โ
(๐ผ๐ด such that ๐ผ๐ธ = ๐ด๐ต and ๐น โ (๐ผ๐ถ such that ๐ผ๐น = ๐ถ๐ท. If ๐ a point that verifies
๐[๐ด๐ต๐] + ๐[๐ถ๐ท๐] = 1 (1), then, because ๐[๐ด๐ต๐] = ๐[๐๐ผ๐ธ] and ๐[๐ถ๐ท๐] = ๐[๐๐ผ๐น], it
results that ๐[๐๐ผ๐ธ] + ๐[๐๐ผ๐น] = ๐ (2).
We obtain that ๐[๐๐ธ๐ผ๐น] = ๐.
On the other hand, the points ๐ธ, ๐น are fixed, therefore ๐[๐ผ๐ธ๐น] = ๐โฒ = const. That
is, ๐[๐๐ธ๐น] = ๐ โ ๐โฒ = const.
Because ๐ธ๐น = const., we have ๐(๐, ๐ธ๐น) =2(๐โ๐โฒ)
๐ธ๐น= const., which shows that ๐
belongs to a line that is parallel to ๐ธ๐น, taken at the distance 2(๐โ๐โฒ)
๐ธ๐น .
Therefore, the locus points are those on the line parallel to ๐ธ๐น, located inside the
quadrilateral ๐ด๐ต๐ถ๐ท. They belong to the segment [๐ธโฒ๐นโฒ] in Fig. 1.
Reciprocally, if ๐ โ [๐ธโฒ๐นโฒ], then ๐[๐๐ด๐ต] + ๐[๐๐ถ๐ท] = ๐[๐๐ผ๐ธ] + ๐[๐๐ผ๐น] =
๐[๐๐ธ๐ผ๐น] = ๐[๐ผ๐ธ๐น] + ๐[๐๐ธ๐] = ๐โฒ +๐ธ๐นโ2(๐โ๐โฒ)
2โ๐ธ๐น= ๐.
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In conclusion, the locus of points ๐ inside the quadrilateral ๐ด๐ต๐ถ๐ท which occurs
for relation (1) where ๐ is a positive constant smaller than ๐ = ๐[๐ด๐ต๐ถ๐ท] is a line
segment.
If ๐ด๐ต๐ถ๐ท is a trapeze having ๐ด๐ต and ๐ถ๐ท as bases, then we reconstruct the
reasoning as ๐ด๐ท โฉ ๐ต๐ถ = {๐ผ} and ๐[๐๐ด๐ท] + ๐[๐๐ต๐ถ] = ๐ โ ๐ = const.
If ๐ด๐ต๐ถ๐ท is a parallelogram, one shows without difficulty that the locus is a
segment parallel to ๐ด๐ต.
Solution no. 2 (Ion Patrascu)
We prove that the locus of points ๐ which verify the relationship ๐[๐๐ด๐ต] +
๐[๐๐ถ๐ท] = ๐ (1) from inside the convex quadrilateral ๐ด๐ต๐ถ๐ท of area ๐ (๐ โ ๐ ) is a line
segment.
Letโs suppose that ๐ด๐ต โฉ ๐ถ๐ท = {๐ผ}, see Fig. 2. There is a point ๐ of the locus which
belongs to the line ๐ถ๐ท. Therefore, we have (๐; ๐ด๐ต) =2๐
๐ด๐ต . Also, there is the point
๐ โ ๐ด๐ต such that ๐(๐; ๐ถ๐ท) =2๐
๐ถ๐ท .
Now, we prove that the points from inside the quadrilateral ๐ด๐ต๐ถ๐ท that are on
the segment [๐๐] belong to the locus.
Let ๐ โ int[๐ด๐ต๐ถ๐ท] โฉ [๐๐]. We denote ๐1 and ๐2 the projections of ๐ on ๐ด๐ต and
๐ถ๐ท respectively. Also, let ๐1 be the projection of ๐ on ๐ด๐ต and ๐1 the projection of
๐ on ๐ถ๐ท. The triangles ๐๐๐1 and ๐๐๐2 are alike, which means that
๐๐2๐๐1
=๐๐
๐๐ (2),
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and the triangles ๐๐1๐ and ๐๐1๐ are alike, which means that
๐๐1๐๐1
=๐๐
๐๐ (3).
By adding member by member the relations (2) and (3), we obtain
๐๐2๐๐1
+๐๐1๐๐1
=๐๐ +๐๐
๐๐= 1 (4).
Substituting in (4), ๐๐1 =2๐
๐ถ๐ท and ๐1 =
2๐
๐ด๐ต , we get ๐ด๐ต โ ๐๐1 + ๐ถ๐ท โ ๐๐2 = 2๐, that
is ๐[๐๐ด๐ต] + ๐[๐๐ถ๐ท] = ๐.
We prove now by reductio ad absurdum that there is no point inside the
quadrilateral ๐ด๐ต๐ถ๐ท that is not situated on the segment [๐๐], built as shown, to
verify the relation (1).
Let a point ๐โฒ inside the quadrilateral ๐ด๐ต๐ถ๐ท that verifies the relation (1), ๐โฒ โ
[๐๐]. We build ๐โฒ๐ โฉ ๐ด๐ต, ๐โฒ๐ โฅ ๐ถ๐ท, where ๐ and ๐ are situated on [๐๐], see Fig. 3.
We denote ๐1โฒ , ๐1, ๐1 the projections of ๐1, ๐, ๐ on ๐ด๐ต and ๐2
โฒ , ๐2, ๐2 the
projections of the same points on ๐ถ๐ท.
We have the relations:
๐โฒ๐1โฒ โ ๐ด๐ต +๐โฒ๐2
โฒ โ ๐ถ๐ท = 2๐ (5),
๐๐1 โ ๐ด๐ต + ๐๐2 โ ๐ถ๐ท = 2๐ (6).
Because ๐โฒ๐1โฒ = ๐๐1 and ๐โฒ๐2
โฒ = ๐๐2, substituting in (5), we get:
๐๐1 โ ๐ด๐ต + ๐๐2 โ ๐ถ๐ท = 2๐ (7).
From (6) and (7), we get that ๐๐2 = ๐๐2, which drives us to ๐๐ โฅ ๐ถ๐ท, false!
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38
Problems in Geometry and Trigonometry
39. Find the locus of the points such that the sum of the distances to two
concurrent lines to be constant and equal to ๐.
Solution to Problem 39
40. Show that in any triangle ๐ด๐ต๐ถ we have:
a. ๐ ๐๐๐ ๐ถ + ๐ ๐๐๐ ๐ต = ๐; b. ๐ ๐๐๐ ๐ต + ๐ ๐๐๐ ๐ถ = ๐ ๐๐๐ (๐ต โ ๐ถ).
Solution to Problem 40
41. Show that among the angles of the triangle ๐ด๐ต๐ถ we have:
a. ๐ ๐๐๐ ๐ถ โ ๐ ๐๐๐ ๐ต =๐2โ๐2
๐;
b. 2(๐๐ ๐๐๐ ๐ด + ๐๐ ๐๐๐ ๐ต + ๐๐ ๐๐๐ ๐ถ = ๐2 + ๐2 + ๐2.
Solution to Problem 41
42. Using the law of cosines prove that 4๐2
๐= 2(๐2 + ๐2) โ ๐2, where ๐๐ is the
length of the median corresponding to the side of ๐ length.
Solution to Problem 42
43. Show that the triangle ๐ด๐ต๐ถ where ๐+๐
๐= cot
๐ต
2 is right-angled.
Solution to Problem 43
44. Show that, if in the triangle ๐ด๐ต๐ถ we have cot ๐ด + cot๐ต = 2 cot ๐ถ โ ๐2 +
๐2 = 2๐2.
Solution to Problem 44
45. Determine the unknown elements of the triangle ๐ด๐ต๐ถ, given:
a. ๐ด, ๐ต and ๐;
b. ๐ + ๐ = ๐, ๐ด and ๐ต;
c. ๐, ๐ด; ๐ โ ๐ = ๐.
Solution to Problem 45
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46. Show that in any triangle ๐ด๐ต๐ถ we have tan๐ดโ๐ต
2tan
๐ถ
2=๐โ๐
๐+๐ (tangents
theorem).
Solution to Problem 46
47. In triangle ๐ด๐ต๐ถ it is given ๏ฟฝ๏ฟฝ = 60ยฐ and ๐
๐= 2 + โ3. Find tan
๐ตโ๐ถ
2 and angles
๐ต and ๐ถ.
Solution to Problem 47
48. In a convex quadrilateral ๐ด๐ต๐ถ๐ท, there are given โ๐ด๐ทโ = 7(โ6 โ โ2), โ๐ถ๐ทโ =
13, โ๐ต๐ถโ = 15, ๐ถ = arccos33
65, and ๐ท =
๐
4+ arccos
5
13 . The other angles of the
quadrilateral and โ๐ด๐ตโ are required.
Solution to Problem 48
49. Find the area of โ๐ด๐ต๐ถ when:
a. ๐ = 17, ๐ต = arcsin 24
25, ๐ถ = arcsin
12
13;
b. ๐ = 2, ๏ฟฝ๏ฟฝ โ 135ยฐ, ๏ฟฝ๏ฟฝ โ 30ยฐ;
c. ๐ = 7, ๐ = 5, ๐ = 6;
d. ๏ฟฝ๏ฟฝ โ 18ยฐ, ๐ = 4, ๐ = 6.
Solution to Problem 49
50. How many distinct triangles from the point of view of symmetry are there
such that ๐ = 15, ๐ = 13, ๐ = 24?
Solution to Problem 50
51. Find the area of โ๐ด๐ต๐ถ if ๐ = โ6, ๏ฟฝ๏ฟฝ โ 60ยฐ, ๐ + ๐ = 3 + โ3.
Solution to Problem 51
52. Find the area of the quadrilateral from problem 48.
Solution to Problem 52
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53. If ๐๐ is the area of the regular polygon with ๐ sides, find:
๐3, ๐4, ๐6, ๐8, ๐12, ๐20 in relation to ๐ , the radius of the circle inscribed in the
polygon.
Solution to Problem 53
54. Find the area of the regular polygon ๐ด๐ต๐ถ๐ทโฆ๐ inscribed in the circle with
radius ๐ , knowing that: 1
โ๐ด๐ตโ=
1
โ๐ด๐ถโ+
1
โ๐ด๐ทโ .
Solution to Problem 54
55. Prove that in any triangle ABC we have:
a. ๐ = (๐ โ ๐) tan๐ด
2;
b. ๐ = (๐ โ ๐) tan๐ด
2;
c. ๐ = 4๐ cos๐ด
2cos
๐ต
2cos
๐ถ
2;
d. ๐ โ ๐ = 4๐ cos๐ด
2cos
๐ต
2cos
๐ถ
2;
e. ๐๐2 = ๐ 2(sin2 ๐ด + 4 cos๐ด sin ๐ต sin ๐ถ;
f. โ๐ = 2๐ sin๐ต sin ๐ถ.
Solution to Problem 55
56. If ๐ is the center of the circle inscribed in triangle ๐ด๐ต๐ถ show that โ๐ด๐ผโ =
4๐ sin๐ต
2sin
๐ถ
2 .
Solution to Problem 56
57. Prove the law of sine using the analytic method.
Solution to Problem 57
58. Using the law of sine, show that in a triangle the larger side lies opposite
to the larger angle.
Solution to Problem 58
59. Show that in any triangle ๐ด๐ต๐ถ we have:
a.๐ cos ๐ถ โ ๐ cos๐ต
๐ cos๐ต โ ๐ cos๐ด+ cos๐ถ = 0, ๐ โ ๐;
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b.sin(๐ด โ ๐ต) sin๐ถ
1 + cos(๐ด โ ๐ต) cos๐ถ=๐2 โ ๐2
๐2 + ๐2;
c. (๐ + ๐) cos๐ต
4+ ๐ cos(๐ด +
3๐ต
4) = 2๐ cos
๐ต
2cos๐ต
4 .
Solution to Problem 59
60. In a triangle ๐ด๐ต๐ถ, ๐ด โ 45ยฐ, โ๐ด๐ตโ = ๐, โ๐ด๐ถโ =2โ2
3๐. Show that tan๐ต = 2.
Solution to Problem 60
61. Let ๐ดโฒ, ๐ตโฒ, ๐ถโฒ be tangent points of the circle inscribed in a triangle ๐ด๐ต๐ถ with
its sides. Show that ๐[๐ดโฒ๐ตโฒ๐ถโฒ]
๐[๐ด๐ต๐ถ]=
๐
2๐ .
Solution to Problem 61
62. Show that in any triangle ๐ด๐ต๐ถ sin๐ด
2โค
๐
2โ๐๐ .
Solution to Problem 62
63. Solve the triangle ๐ด๐ต๐ถ, knowing its elements ๐ด, ๐ต and area ๐.
Solution to Problem 63
64. Solve the triangle ๐ด๐ต๐ถ, knowing ๐ = 13, arc cos4
5 , and the corresponding
median for side ๐,๐๐ =1
2โ15โ3 .
Solution to Problem 64
65. Find the angles of the triangle ๐ด๐ต๐ถ, knowing that ๐ต โ ๐ถ =2๐
3 and ๐ = 8๐,
where ๐ and ๐ are the radii of the circles circumscribed and inscribed in the
triangle.
Solution to Problem 65
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42
Solutions
Solution to Problem 39.
Let ๐1 and ๐2 be the two concurrent lines. We draw 2 parallel lines to ๐1 located
on its both sides at distance ๐. These intersect on ๐2 at ๐ท and ๐ต, which will be
points of the locus to be found, because the sum of the distances ๐(๐ต, ๐1) +
๐(๐ต, ๐2) = ๐ + 0 verifies the condition from the statement.
We draw two parallel lines with ๐2 located at distance ๐ from it, which cut ๐1 in ๐ด
and ๐ถ, which are as well points of the locus to be found. The equidistant parallel
lines determine on ๐2 congruent segments โน|๐ท๐| โก |๐๐ต||๐ด๐| โก |๐๐ถ|
, in the same way ๐ด๐ต๐ถ๐ท
is a parallelogram.
โ๐ต๐๐ถ,โ๐ถ๐ถโฒโ = ๐(๐ถ, ๐2)
โ๐ต๐ตโฒโ = ๐(๐ต, ๐1)} โน โ๐ถ๐ถโฒโ = โ๐ต๐ตโฒโ
โน โ๐ต๐๐ถ is isosceles.
โน ||๐๐ถ|| = ||๐๐ต|| โน ๐ด๐ต๐ถ๐ท is a rectangle. Any point ๐ we take on the sides of this
rectangle, we have ||๐ 1, ๐1|| + ||๐, ๐2|| = ๐, using the propriety according to which
the sum of the distances from a point on the base of an isosceles triangle at the
sides is constant and equal to the height that starts from one vertex of the base,
namely ๐. Thus the desired locus is rectangle ๐ด๐ต๐ถ๐ท.
Solution to Problem 40.
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Solution to Problem 41.
Solution to Problem 42.
๐๐2 = ๐2 +
๐2
4โ 2
๐
2๐ cos๐ต ;
4๐๐2 = 4๐2 + ๐2 โ 4๐๐ cos๐ต = 4๐2 + ๐2 โ 4๐๐
๐2 + ๐2 โ ๐2
2๐๐= 4๐2 + ๐2 โ 2๐ โ 2๐2 + 2๐2
= 2๐2 + 2๐2 โ ๐2 = 2(๐2 + ๐2) โ ๐2 .
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44
Solution to Problem 43.
Using the sine theorem, ๐ = ๐sin๐ด.
๐ด โ ๐ถ
2= โ
๐ต
2โน ๐ด โ ๐ต = ๐ถ or ๐ด โ ๐ถ = ๐ต โน
๐ด = ๐ต + ๐ถ 2๐ด = 1800 ๐ด = 900
or โน or โน or๐ด + ๐ต = ๐ถ 2๐ถ = 1800 ๐ถ = 900
Solution to Problem 44.
By substitution:
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45
Solution to Problem 45.
a. Using the law of sine,
b.
c.
Therefore,
We solve the system, and find ๐ต and ๐ถ. Then we find ๐ =๐ sin๐ต
sin๐ด and ๐ = ๐ โ ๐.
Solution to Problem 46.
๐
sin๐ด=
๐
sin๐ต=
๐
sin๐ถ= ๐โน
๐ = ๐sin๐ด๐ = ๐sin๐ต
;
๐ โ ๐
๐ + ๐=๐sin๐ด โ๐sin๐ต
๐ sin๐ด +๐sin๐ต=sin๐ด โ sin๐ต
sin๐ด + sin๐ต=2 sin
๐ด โ ๐ต2 cos
๐ด + ๐ต2
2 sin๐ด + ๐ต2 cos
๐ด โ ๐ต2
= tan๐ด โ ๐ต
2
sin๐ถ2
cos๐ถ2
= tan๐ด โ ๐ต
2tan๐ถ
2 .
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Solution to Problem 47.
Using tangentsโ theorem,
So
Solution to Problem 48.
In โ๐ต๐ท๐ถ we have
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In โ๐ด๐ท๐ต,
In โ๐ด๐ท๐ต we apply sineโs theorem:
Or we find ๐(๐ท๐ต๏ฟฝ๏ฟฝ) and we add it to ๐
6 .
Solution to Problem 49.
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48
sin๐ด =โ1+โ5
4, because ๐(๐ด) < 1800 and sin๐ด > 0.
Solution to Problem 50.
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Solution to Problem 51.
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Solution to Problem 52.
At problem 9 weโve found that
With Heronโs formula, we find the area of each triangle and we add them up.
Solution to Problem 53.
The formula for the area of a regular polygon:
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Solution to Problem 54.
In โ๐ต๐๐:
In โ๐๐๐ถ:
In โ๐๐๐ท:
Substituting (1), (2), (3) in the given relation:
or
which is impossible.
๐ =๐(complete circle)
๐(๐ด๏ฟฝ๏ฟฝ)=2๐
2๐7
= 7.
Thus the polygon has 7 sides.
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Solution to Problem 55.
Solution to Problem 56.
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We apply the law of sine in โ๐ด๐ต๐ผ:
The law of sine applied in โ๐ด๐ต๐ถ:
Solution to Problem 57.
In โ๐ด๐ถ๐ถโฒ: sin(1800 โ ๐ด) =โ๐ถ๐ถโฒโ
๐โน โ๐ถ๐ถโฒโ = ๐ sin๐ด ; cos(1800 โ ๐ด) = ๐ cos๐ด.
So the coordinates of ๐ถ are (โ๐ cos๐ด , ๐ sin๐ด).
The center of the inscribed circle is at the intersection of the perpendicular lines
drawn through the midpoints of sides ๐ด๐ต and ๐ด๐ถ.
The equation of the line ๐ธ๐:
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If we redo the calculus for the same draw, we have the following result:
(๐ cos๐ด , ๐ sin๐ด).
using the law of cosine.
Solution to Problem 58.
๐
sin๐ด=
๐
sin๐ต=
๐
sin๐ถ= 2๐ .
We suppose that ๐ > ๐. Letโs prove that ๐ด > ๐ต.
๐
sin๐ด=
๐
sin๐ตโ๐
๐=sin๐ด
sin๐ต
๐ > ๐ โ๐
๐> 1
}โนsin๐ด
sin๐ต> 1 โน ๐ด,๐ต, ๐ถ โ (0, ๐) โน sin๐ต > 0 โน sin๐ด > sin๐ต
โน sin๐ด โ sin๐ต > 0 โน 2sin๐ด โ ๐ต
2cos๐ด + ๐ต
2> 0 โน
๐ด+ ๐ต
2=1800 โ ๐ถ
2
= 900 โ๐ถ
2 .
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cos๐ด+๐ต
2= cos (90circ โ
๐ถ
2) = sin
๐ถ
2> 0, therefore
๐ดโ๐ต
2> 0โน ๐ด > ๐ต;
(โ๐
2<๐ด โ ๐ต
2<๐
2).
Solution to Problem 59.
b. We transform the product into a sum:
From (1) and (2) โน
We consider the last two terms:
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Solution to Problem 60.
We apply the law of cosines in triangle ABC:
Solution to Problem 61.
โ๐ผ๐ดโ = โ๐ผ๐ตโ = โ๐ผ๐ถโ = ๐ ๐ผ๐ถโฒ โฅ ๐ด๐ต๐ผ๐ดโฒ โฅ ๐ต๐ถ
} โ ๐ผ๐ดโฒ๐ตโฒ๐ถโฒ inscribable quadrilateral
๐(๐ดโฒ๐ผ๐ถโฒ) = 180 โ ๏ฟฝ๏ฟฝ โน sin(๐ดโฒ๐ผ๐ถโฒ) = sin ๏ฟฝ๏ฟฝ
Similarly, ๐ดโฒ๐ผ๐ตโฒ = sin๐ถ and ๐ถโฒ๐ผ๐ตโฒ = sin๐ด.
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In the same way,
Solution to Problem 62.
0 < ๐ด โ<๐ด
2<๐
2โ sin
๐ด
2> 0 ;
sin๐ด
2=โ
1 โ cos๐ด
2โน
sin2๐ด
2=1 โ cos๐ด
2
cos๐ด =๐2 + ๐2 โ ๐2
2๐๐
}
โน sin2๐ด
2=1 โ
๐2 โ ๐2 + ๐2
2๐๐2
=๐2 โ (๐ โ ๐)2
4๐๐โค๐2
4๐๐โน sin
๐ด
2โค
๐
2โ๐๐ .
Solution to Problem 63.
๐ถ = ๐ โ (๐ด + ๐ต)
๐ =๐2 sin๐ต sin๐ถ
2 sin๐ด
๐ด, ๐ต, ๐ถ are known} โน We find ๐.
๐
sin๐=
๐
sin๐โน ๐ =
๐ sin๐ต
sin๐ด. In the same way, we find ๐.
Solution to Problem 64.
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๐2 + ๐2 = 841๐๐ = 420
} โน {๐ = 21๐ = 20
or {๐ = 20๐ = 21
We find ๐ต.
We find the sum.
Or
Solution to Problem 65.
We already know that
๐
๐ = 4 sin
๐ด
2sin๐ต
2sin๐ถ
2
We write sin๐ด
2= ๐ก. We have
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255 Compiled and Solved Problems in Geometry and Trigonometry
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From this system we find ๐ต and ๐ถ.
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Other Problems in Geometry
and Trigonometry (10th grade)
66. Show that a convex polygon canโt have more than three acute angles.
Solution to Problem 66
67. Let ๐ด๐ต๐ถ be a triangle. Find the locus of points ๐ โ (๐ด๐ต๐ถ), for which
๐[๐ด๐ต๐] = ๐[๐ด๐ถ๐].
Solution to Problem 67
68. A convex quadrilateral ๐ด๐ต๐ถ๐ท is given. Find the locus of points ๐ โ
๐๐๐ก. ๐ด๐ต๐ถ๐ท, for which ๐[๐๐ต๐ถ๐ท] = ๐[๐๐ต๐ด๐ท].
Solution to Problem 68
69. Determine a line ๐๐, parallel to the bases of a trapezoid ๐ด๐ต๐ถ๐ท (๐ โ
|๐ด๐ท|, ๐ โ |๐ต๐ถ|) such that the difference of the areas of [๐ด๐ต๐๐] and [๐๐๐ถ๐ท]
to be equal to a given number.
Solution to Problem 69
70. On the sides of โ๐ด๐ต๐ถ we take the points ๐ท, ๐ธ, ๐น such that ๐ต๐ท
๐ท๐ถ=๐ถ๐ธ
๐ธ๐ด=๐ด๐น
๐น๐ต= 2.
Find the ratio of the areas of triangles ๐ท๐ธ๐น and ๐ด๐ต๐ถ.
Solution to Problem 70
71. Consider the equilateral triangle ๐ด๐ต๐ถ and the disk [๐ถ (๐,๐
3)], where ๐ is the
orthocenter of the triangle and ๐ = โ๐ด๐ตโ. Determine the area [๐ด๐ต๐ถ] โ
[๐ถ (๐,๐
3)].
Solution to Problem 71
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255 Compiled and Solved Problems in Geometry and Trigonometry
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72. Show that in any triangle ๐ด๐ต๐ถ we have:
a. 1 + cos๐ด cos(๐ต โ ๐ถ) =๐2+๐2
4๐ 2;
b. (๐2 + ๐2 = ๐2) tan๐ด = 4๐;
c. ๐+๐
2๐ cos๐ด
2
=sin(
๐ด
2+๐ถ)
sin(๐ด+๐ต);
d. ๐ = ๐ (cot๐ด
2+ cot
๐ต
2+ cot ๐ถ2);
e. cot๐ด
2+cot
๐ต
2+ cot
๐ถ
2=๐
๐ .
Solution to Problem 72
73. If ๐ป is the orthocenter of triangle ๐ด๐ต๐ถ, show that:
a. โ๐ด๐ปโ = 2๐ cos ๐ด;
b. ๐โ๐ด๐ปโ + ๐โ๐ต๐ปโ + ๐โ๐ถ๐ปโ = 4๐.
Solution to Problem 73
74. If ๐ is the orthocenter of the circumscribed circle of triangle ๐ด๐ต๐ถ and ๐ผ is
the center of the inscribed circle, show that โ๐๐ผโ2 = ๐ (๐ โ 2๐).
Solution to Problem 74
75. Show that in any triangle ๐ด๐ต๐ถ we have: cos2๐ตโ๐ถ
2โฅ2๐
๐ .
Solution to Problem 75
76. Find ๐ง๐ +1
๐ง๐ knowing that ๐ง +
1
๐ง= 2 sin ๐ผ.
Solution to Problem 76
77. Solve the equation: (๐ง + 1)๐ โ (๐ง โ 1)๐ = 0.
Solution to Problem 77
78. Prove that if ๐ง <1
2 then |(1 + ๐)๐ง3 + ๐๐ง| โค
3
4 .
Solution to Problem 78
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79. One gives the lines ๐ and ๐โฒ. Show that through each point in the space
passes a perpendicular line to ๐ and ๐โฒ.
Solution to Problem 79
80. There are given the lines ๐ and ๐โฒ, which are not in the same plane, and
the points ๐ด โ ๐, ๐ต โ ๐โฒ. Find the locus of points ๐ for which pr๐๐ = ๐ด and
pr๐โฒ๐ = ๐ต.
Solution to Problem 80
81. Find the locus of the points inside a trihedral angle ๐๐๏ฟฝ๏ฟฝ equally distant
from the edges of ๐, ๐, ๐.
Solution to Problem 81
82. Construct a line which intersects two given lines and which is perpendicular
to another given line.
Solution to Problem 82
83. One gives the points ๐ด and ๐ต located on the same side of a plane; find in
this plane the point for which the sum of its distances to ๐ด and ๐ต is
minimal.
Solution to Problem 83
84. Through a line draw a plane onto which the projections of two lines to be
parallel.
Solution to Problem 84
85. Consider a tetrahedron [๐ด๐ต๐ถ๐ท] and centroids ๐ฟ,๐,๐ of triangles
๐ต๐ถ๐ท, ๐ถ๐ด๐ท, ๐ด๐ต๐ท.
a. Show that (๐ด๐ต๐ถ) โฅ (๐ฟ๐๐);
b. Find the ratio ๐[๐ด๐ต๐ถ]
๐[๐ฟ๐๐] .
Solution to Problem 85
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86. Consider a cube [๐ด๐ต๐ถ๐ท๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ]. The point ๐ด is projected onto ๐ดโฒ๐ต, ๐ดโฒ๐ถ, ๐ดโฒ๐ท
respectively in ๐ด1, ๐ด2, ๐ด3. Show that:
a. ๐ดโฒ๐ถ โฅ (๐ด1๐ด2๐ด3);
b. ๐ด๐ด1 โฅ ๐ด1๐ด2, ๐ด๐ด3 โฅ ๐ด3๐ด2;
c. ๐ด๐ด1๐ด2๐ด3 is an inscribable quadrilateral.
Solution to Problem 86
87. Consider the right triangles ๐ต๐ด๐ถ and ๐ด๐ต๐ท (๐(๐ต๐ด๏ฟฝ๏ฟฝ)) = ๐((๐ด๐ต๐ท) = 900)
located on perpendicular planes ๐ and ๐, being midpoints of segments
[๐ด๐ต], [๐ถ๐ท]. Show that ๐๐ โฅ ๐ถ๐ท.
Solution to Problem 87
88. Prove that the bisector half-plane of a dihedral angle inside a tetrahedron
divides the opposite edge in proportional segments with the areas of the
adjacent faces.
Solution to Problem 88
89. Let ๐ด be a vertex of a regular tetrahedron and ๐, ๐ two points on its
surface. Show that ๐(๐๐ด๏ฟฝ๏ฟฝ) โค 600.
Solution to Problem 89
90. Show that the sum of the measures of the dihedral angles of a tetrahedron
is bigger than 360ยฐ.
Solution to Problem 90
91. Consider lines ๐1, ๐2 contained in a plane ๐ผ and a line ๐ด๐ต which intersects
plane ๐ผ at point ๐ถ. A variable line, included in ๐ผ and passing through ๐ถ all
๐1, ๐2 respectively at ๐๐. Find the locus of the intersection ๐ด๐ โฉ ๐ต๐. In
which case is the locus an empty set?
Solution to Problem 91
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64
92. A plane ๐ผ intersects sides [๐ด๐ต], [๐ต๐ถ], [๐ถ๐ท], [๐ท๐ด] of a tetrahedron [๐ด๐ต๐ถ๐ท] at
points ๐ฟ,๐,๐, ๐. Prove that โ๐ด๐ฟโ โ โ๐ต๐โ โ โ๐ถ๐โ โ โ๐๐ทโ = โ๐ต๐ฟโ โ โ๐ถ๐โ โ
โ๐ท๐โ โ โ๐ด๐โ.
Solution to Problem 92
93. From a point ๐ด located outside a plane ๐ผ, we draw the perpendicular line
๐ด๐, ๐ โ ๐ผ, and we take ๐ต, ๐ถ โ ๐ผ. Let ๐ป,๐ป1 be the orthocenters of triangles
๐ด๐ต๐ถ, ๐๐ต๐ถ; ๐ด๐ท and ๐ต๐ธ heights in triangle ๐ด๐ต๐ถ; and ๐ต๐ธ1 height in triangle
๐๐ต๐ถ. Show that:
a. ๐ป๐ป1 โฅ (๐ด๐ต๐ถ);
b. โ๐๐ด
๐ด๐ทโ โ โ
๐ท๐ป1
๐ป1๐ตโ โ โ
๐ต๐ธ
๐ธ๐ธ1โ = 1.
Solution to Problem 93
94. Being given a tetrahedron [๐ด๐ต๐ถ๐ท] where ๐ด๐ต โฅ ๐ถ๐ท and ๐ด๐ถ โฅ ๐ต๐ท, show that:
a. โ๐ด๐ตโ2 + โ๐ถ๐ทโ2 = โ๐ต๐ถโ2 + โ๐ด๐ทโ2 = โ๐ถ๐ดโ2 + โ๐ต๐ทโ2;
b. The midpoints of the 6 edges are located on a sphere.
Solution to Problem 94
95. It is given a triangular prism [๐ด๐ต๐ถ๐ดโฒ๐ตโฒ๐ถโฒ] which has square lateral faces. Let
๐ be a mobile point [๐ด๐ตโฒ], ๐ the projection of ๐ onto (๐ต๐ถ๐ถโฒ) and ๐ดสบ the
midpoint of [๐ตโฒ๐ถสบ]. Show that ๐ดโฒ๐ and ๐๐ดสบ intersect in a point ๐ and find
the locus of ๐.
Solution to Problem 95
96. We have the tetrahedron [๐ด๐ต๐ถ๐ท] and let ๐บ be the centroid of triangle
๐ต๐ถ๐ท. Show that if ๐ โ ๐ด๐บ then ๐[๐๐บ๐ต๐ถ] = ๐[๐๐บ๐ถ๐ท] = ๐[๐๐บ๐ท๐ต].
Solution to Problem 96
97. Consider point ๐ โ the interior of a trirectangular tetrahedron with its
vertex in ๐. Draw through ๐ a plane which intersects the edges of the
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65
respective tetrahedron in points ๐ด, ๐ต, ๐ถ so that ๐ is the orthocenter of
โ๐ด๐ต๐ถ.
Solution to Problem 97
98. A pile of sand has as bases two rectangles located in parallel planes and
trapezoid side faces. Find the volume of the pile, knowing the dimensions
๐โฒ, ๐โฒ of the small base, ๐, ๐ of the larger base, and โ the distance between
the two bases.
Solution to Problem 98
99. A pyramid frustum is given, with its height โ and the areas of the bases ๐ต
and ๐. Unite any point ๐ of the larger base with the vertices ๐ด, ๐ต, ๐ดโฒ, ๐ตโฒ of a
side face. Show that ๐[๐๐ดโฒ๐ตโฒ๐ด] =โ6
โ๐ต๐[๐๐ด๐ต๐ตโฒ].
Solution to Problem 99
100. A triangular prism is circumscribed to a circle of radius ๐ . Find the area
and the volume of the prism.
Solution to Problem 100
101. A right triangle, with its legs ๐ and ๐ and the hypotenuse ๐, revolves by
turns around the hypotenuse and the two legs, ๐1, ๐2, ๐3; ๐1, ๐2, ๐3 being the
volumes, respectively the lateral areas of the three formed shapes, show
that:
a. 1
๐12 =
1
๐22 =
1
๐32;
b. ๐2
๐3+๐3
๐2=๐2+๐3
๐1 .
Solution to Problem 101
102. A factory chimney has the shape of a cone frustum and 10m height, the
bases of the cone frustum have external lengths of 3,14m and 1,57m, and
the wall is 18cm thick. Calculate the volume of the chimney.
Solution to Problem 102
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103. A regular pyramid, with its base a square and the angle from the peak of
a side face of measure ๐ผ is inscribed in a sphere of radius ๐ . Find:
a. the volume of the inscribed pyramid;
b. the lateral and total area of the pyramid;
c. the value ๐ผ when the height of the pyramid is equal to the radius of
the sphere.
Solution to Problem 103
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Solutions
Solution to Problem 66.
Let ๐ด1, ๐ด2โฆ๐ด๐ the vertices of the convex polygon. Letโs assume that it has four
acute angles. The vertices of these angles form a convex quadrilateral ๐ด๐๐ด๐๐ด๐๐ด๐.
Due to the fact that the polygon is convex, the segments |๐ด๐๐ด๐|, |๐ด๐๐ด๐|, |๐ด๐๐ด๐|,
|๐ด๐๐ด๐| are inside the initial polygon. We find that the angles of the quadrilateral are
acute, which is absurd, because their sum is 360ยฐ.
Another solution: We assume that ๐ด๐๐ด๐๐ด๐๐ด๐ is a convex polygon with all its
angles acute โน the sum of the external angles is bigger than 360ยฐ, which is absurd
(the sum of the measures of the external angles of a convex polygon is 360ยฐ).
Solution to Problem 67.
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Let |๐ด๐ดโฒ| be the median from ๐ด and ๐ถ๐ โฅ ๐ด๐ดโฒ, ๐ต๐ โฅ ๐ด๐ดโฒ.
โ๐ต๐ดโฒ๐ โก ๐ถ๐ดโฒ๐ because:
{๐๐ต๏ฟฝ๏ฟฝ โก ๐ต๐ถ๏ฟฝ๏ฟฝ alternate interior
๐๐ดโฒ๏ฟฝ๏ฟฝ โก ๐ถ๐ดโฒ๏ฟฝ๏ฟฝ vertical angles
๐ต๐ดโฒ โก ๐ดโฒ๐ถ
โน ||๐ต๐|| = ||๐๐ถ|| and by its construction ๐ต๐ โฅ ๐ด๐ดโฒ, ๐ถ๐ โฅ ๐ด๐ดโฒ.
The desired locus is median |๐ด๐ดโฒ|. Indeed, for any ๐ โ |๐ด๐ดโฒ| we have ๐[๐ด๐ต๐] =
๐[๐ด๐ถ๐], because triangles ๐ด๐ต๐ and ๐ด๐ถ๐ have a common side |๐ด๐| and its
corresponding height equal ||๐ต๐|| = ||๐๐ถ||.
Vice-versa. If ๐[๐ด๐ต๐] = ๐[๐ด๐ถโฒ๐], letโs prove that ๐ โ |๐ด๐ดโฒ|.
Indeed: ๐[๐ด๐ต๐] ๐[๐ด๐ถ๐] โ ๐ (๐ต, ๐ด๐) = ๐(๐ถ, ๐ด๐), because |๐ด๐| is a common
side, ๐ (๐ต, ๐ด๐) = ||๐ต๐|| and ๐(๐ถ, ๐ด๐) = ||๐ถ๐|| and both are perpendicular to ๐ด๐ โน
๐๐ต๐๐ถ is a parallelogram, the points ๐,๐, ๐ are collinear (๐, ๐ the feet of the
perpendicular lines from ๐ต and ๐ถ to ๐ด๐).
In parallelogram ๐๐ต๐๐ถ we have |๐๐| and |๐ต๐ถ| diagonals โน ๐ด๐ passes through
the middle of |๐ต๐ถ|, so ๐ โ |๐ด๐ดโฒ|, the median from ๐ด.
Solution to Problem 68.
Let ๐ be the midpoint of diagonal |๐ด๐ถ| โน โ๐ด๐โ = โ๐๐ถโ.
๐[๐ด๐๐ท] = ๐[๐ถ๐๐ท] (1)
Because {โ๐ด๐โ = โ๐๐ถโ
โ๐๐ทโฒโ common height
๐[๐ด๐๐ต] = ๐[๐ถ๐๐ต] (2)
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the same reasons; we add up (1) and (2) โน
๐[๐ด๐ท๐๐ต] = ๐[๐ท๐ถ๐ต๐] (3),
so ๐ is a point of the desired locus.
We construct through ๐ a parallel to ๐ต๐ท until it cuts sides |๐ต๐ถ| and |๐ท๐ถ| at ๐
respectively ๐. The desired locus is |๐๐|.
Indeed (โ)๐ โ |๐๐| we have:
๐[๐ต๐ท๐] = ๐[๐ต๐ท๐] because ๐ and ๐ belongs to a parallel to ๐ต๐ท.
๐ต,๐ท โ a parallel to ๐๐.
So ๐[๐๐ต๐ด๐ท] = ๐[๐ด๐ต๐๐ท]
and ๐[๐ต๐ถ๐ท๐] = ๐[๐๐ต๐ถ๐ท]
and from (3)} โน ๐[๐๐ต๐ด๐ท] = ๐[๐ต๐ถ๐ท๐].
Vice-versa: If ๐[๐๐ต๐ถ๐ท] = ๐[๐๐ต๐ด๐ท], letโs prove that ๐ โ parallel line through ๐ to
๐ต๐ท. Indeed:
๐[๐ต๐ถ๐ท๐] = ๐[๐๐ต๐ด๐ท]
and because ๐[๐ต๐ถ๐ท๐] + ๐[๐๐ต๐ด๐ท] = ๐[๐ด๐ต๐ถ๐ท] } โน ๐[๐๐ต๐ถ๐ท] = ๐[๐๐ต๐ด๐ท] =
๐[๐ด๐ต๐ถ๐ท]
2 (2).
So, from (1) and (2) โน ๐[๐๐ต๐ด๐ท] = ๐[๐ด๐ต๐๐ท] โน ๐[๐ด๐ต๐ท] + ๐[๐ต๐ท๐] = ๐[๐ด๐ต๐ท] +
๐[๐ต๐ท๐] โน ๐[๐ต๐ท๐] โน ๐ and ๐ are on a parallel to ๐ต๐ท.
Solution to Problem 69.
We write ||๐ธ๐ด|| = ๐ and ||๐ธ๐ท|| = ๐, ||๐ธ๐|| = ๐ฅ.
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We subtract (2) from (3)
We subtract (2) from (4)
From the relation (3), by writing [๐ด๐ต๐ถ๐ท] โ ๐ โน ๐[๐ธ๐ถ๐ท] =๐๐2
๐2โ๐2 .
We substitute this in the relation of ๐ฅยฒ and we obtain:
and taking into consideration that ||๐ธ๐|| = ||๐ท๐|| + ๐, we have
so we have the position of point ๐ on the segment |๐ท๐ด| (but it was sufficient to
find the distance ||๐ธ๐||).
Solution to Problem 70.
We remark from its construction that ๐ธ๐||๐ด๐ต||๐ ๐ท, more than that, they are
equidistant parallel lines. Similarly, ๐ธ๐, ๐๐ท, ๐ด๐ถ and ๐ด๐ต, ๐ธ๐, ๐ ๐ท are also equidistant
parallel lines.
from the hypothesis
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We write ๐[๐ต๐น๐] = ๐.
Based on the following properties:
two triangles have equal areas if they have equal bases and the same
height;
two triangles have equal areas if they have the same base and the third
peak on a parallel line to the base,
we have:
by additionโ
So ๐[๐ท๐ธ๐น]
๐[๐ด๐ต๐ถ]=3๐
9๐=1
3 .
(If necessary the areas ๐ can be arranged).
Solution to Problem 71.
โ๐๐ตโ =๐โ3
6 (๐ต๐ตโฒ median)
In โ๐๐๐ตโฒ:
So (๐๐๏ฟฝ๏ฟฝ) =๐
3 .
We mark with ๐ด the disk surface bordered by a side of the triangle outside the
triangle.
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๐[๐ด] = ๐[circle sector ๐๐๐] โ ๐[๐๐๐]
=๐๐2
9 โ 6โ๐2
9 โ 2sin 600 =
๐๐2
9 โ 6โ๐2โ3
4 โ 9=๐2
18โ (๐
3โโ3
2).
If through the disk area we subtract three times ๐[๐ด], we will find the area of the
disk fraction from the interior of ๐ด๐ต๐ถ. So the area of the disk surface inside ๐ด๐ต๐ถ is:
The desired area is obtained by subtracting the calculated area form ๐[๐ด๐ต๐ถ].
So:
Solution to Problem 72.
a. 1 + cos๐ด โ cos(๐ต โ ๐ถ) =๐2+๐2
4๐ 2
b. We prove that tan๐ด =4๐
๐2+๐2โ๐2 .
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โน All terms reduce.
d. Letโs prove that cot๐ด
2+ cot
๐ต
2+ cot ๐ถ2 =
๐
๐ .
Indeed
We now have to prove that:
Solution to Problem 73.
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a. In triangle ๐ด๐ต๐ตโฒ: โ๐ด๐ตโฒโ = ๐ cos๐ด
In triangle ๐ด๐ป๐ตโฒ:
We used:
Solution to Problem 74.
Using the power of point ๐ผ in relation to circle ๐ถ(๐, ๐ )
Taking into consideration (1), we have โ๐ผ๐ดโ โ โ๐ผ๐ทโ = ๐ 2 โ โ๐๐ผโยฒ.
We now find the distances ||IA|| and ||ID||
In triangle โ๐ผ๐ด๐,
We also find โ๐ผ๐ทโ: ๐(๐ต๐ผ๏ฟฝ๏ฟฝ) = ๐(๐ท๐ต๐ผ ) have the same measure, more exactly:
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255 Compiled and Solved Problems in Geometry and Trigonometry
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In โ๐ด๐ต๐ท according to the law of sine, we have:
So taking into consideration (3),
Returning to the relation โ๐ผ๐ดโ โ โ๐ผ๐ทโ = ๐ 2 โ โ๐๐ผโยฒ. with (2) and (4) we have:
Solution to Problem 75.
Note. We will have to show that
Indeed:
(by Heronโs formula).
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Solution to Problem 76.
So:
We calculate for ๐ง1 and ๐ง2:
so ๐ง๐ +1
๐ง๐ takes the same value for ๐ง1 and for ๐ง2 and it is enough if we
calculate it for ๐ง1.
Analogously:
Solution to Problem 77.
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(we substitute โ1 with ๐ยฒ at denominator)
Solution to Problem 78.
Solution to Problem 79.
We construct ๐ผ โฅ ๐ and ๐ด โ ๐ผ. The so constructed plane is unique. Similarly we
construct ๐ฝ โฅ ๐โฒ and ๐ด โฅ ๐ฝ, ๐ผ โฉ ๐ฝ = ๐ โ ๐ด.
From ๐ผ โฅ ๐ โ ๐ โฅ ๐๐ฝ โฅ ๐โฒ โ ๐โฒ โฅ ๐
} โน ๐ is a line which passes through ๐ด and is perpendicular
to ๐ and ๐โฒ. The line ๐ is unique, because ๐ผ and ๐ฝ constructed as above are unique.
Solution to Problem 80.
We construct plane ๐ผ such that ๐ด โ ๐ผ and ๐ โฅ ๐ผ. We construct plane ๐ฝ such that
๐ต โ ๐ฝ and ๐โฒ โฅ ๐ฝ.
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The so constructed planes ๐ผ and ๐ฝ are unique.
Let ๐ = ๐ผ โฉ ๐ฝ โน ๐ โ ๐ผ so (โ) ๐ โ ๐ has the property pr๐๐ = ๐ด.
๐ผ โ ๐ฝ โน (โ) ๐ โ ๐ has the property ๐๐๐โฒ๐ = ๐ต.
Vice-versa. If there is a point ๐ in space such that pr๐๐ = ๐ด and ๐๐๐โฒ๐ = ๐ต โน
๐ โ ๐ and ๐ โ ๐ฝ โน๐ โ ๐ผ โฉ ๐ฝ โน ๐ โ ๐ (๐ผ and ๐ฝ previously constructed).
Solution to Problem 81.
Let ๐ด โ ๐, ๐ต โ ๐, ๐ถ โ ๐ such that โ๐๐ดโ = โ๐๐ตโ = โ๐๐ถโ. Triangles ๐๐ด๐ต,๐๐ต๐ถ, ๐๐ด๐ถ
are isosceles. The mediator planes of segments โ๐ด๐ตโ, โ๐ด๐ถโ, โ๐ต๐ถ โ pass through ๐
and ๐โฒ (the center of the circumscribed circle of triangle ๐ด๐ต๐ถ). Ray |๐๐โฒ| is the
desired locus.
Indeed (โ) ๐ โ |๐๐โฒ| โน ๐ โ mediator plane of segments |๐ด๐ต|, |๐ด๐ถ| and |๐ต๐ถ| โน
๐ is equally distant from ๐, ๐ and ๐.
Vice-versa: (โ) ๐ with the property: ๐(๐, ๐) = ๐(๐, ๐) = ๐(๐, ๐) โน ๐ โ mediator
plan, mediator planes of segments |๐ด๐ต|, |๐ด๐ถ| and |๐ต๐ถ| โน ๐ โ the intersection of
these planes โน๐ โ |๐๐โฒ|.
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Solution to Problem 82.
Let ๐, ๐, ๐ be the 3 lines in space.
I. We assume ๐ โฅ ๐ and ๐ โฅ ๐. Let ๐ผ be a plane such that:
The construction is possible because โฅ ๐ and ๐ โฅ ๐. Line ๐ด๐ต meets ๐ on ๐ and it is
perpendicular to ๐, because ๐ด๐ต โ ๐ผ and ๐ โฅ ๐ผ.
II. If ๐ โฅ ๐ or ๐ โฅ ๐, the construction is not always possible, only if plane ๐(๐, ๐)
is perpendicular to ๐.
III. If ๐ โฅ ๐ and ๐ โฅ ๐, we construct plane ๐ โฅ ๐ so that ๐ โ ๐ผ and ๐ โ ๐ผ โ โ . Any
point on line a connected with point ๐ โฉ ๐ผ is a desired line.
Solution to Problem 83.
We construct ๐ดโฒ the symmetrical point of ๐ด in relation to ๐ผ. ๐ดโฒ and ๐ต are on
different half-spaces, ๐ผ โฉ |๐ดโฒ๐ต| = ๐.
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๐ is the desired point, because ||๐๐ด|| + ||๐๐ต|| = ||๐๐ดโฒ|| + ||๐๐ต|| is minimal when
๐ โ |๐ดโฒ๐ต|, thus the desired point is ๐ = |๐ดโฒ๐ต| โฉ ๐ผ.
Solution to Problem 84.
Let ๐, ๐, ๐ be the 3 given lines and through ๐ we construct a plane in which ๐ and
๐ to be projected after parallel lines.
Let ๐ด be an arbitrary point on ๐. Through ๐ด we construct line ๐โฒ||๐. It results from
the figure ๐||๐ผ, ๐ผ = ๐(๐, ๐โฒ).
Let ๐ฝ such that ๐ โ ๐ฝ and ๐ฝ โฅ ๐ผ.
Lines ๐ and ๐โฒ are projected onto ๐ฝ after the same line ๐. Line ๐ is projected onto
๐ฝ after ๐1 and ๐1 โฅ ๐.
If ๐1 โฆ ๐,
absurd because ๐||๐ผ(๐||๐โฒ).
Solution to Problem 85.
๐ is the centroid in โ๐ด๐ถ๐ท โน
โน|๐๐ท|
|๐๐|= 2 (1)
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๐ is the centroid in โ๐ด๐ต๐ท โน
โน|๐๐ท|
|๐๐|= 2 (2)
๐ฟ is the centroid in โ๐ต๐ถ๐ท โน
โน|๐ฟ๐ท|
|๐ฟ๐|= 2 (3)
From 1 and 2,
and from 2 and 3
because:
So
Solution to Problem 86.
๐ต๐ท โฅ (๐ด๐ดโฒ๐ถ) from the hypothesis ๐ด๐ต๐ถ๐ท๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ cube (1).
๐ด1 midpoint of segment |๐ต๐ดโฒ|
(๐ด๐ต๐ดโฒ) isosceles and ๐ด๐ด1 โฅ ๐ต๐ดโฒ
๐ด3 midpoint of |๐ดโฒ๐ท|
} |๐ด1๐ด3| mid-side in โ๐ดโฒ๐ต๐ท โน ๐ด1๐ด3 โฅ ๐ต๐ท (2)
From (1) and (2) โน ๐ด1๐ด3 โฅ (๐ด๐ดโฒ๐ถ) โน ๐ดโฒ๐ถ โฅ ๐ด1๐ด3 (3)
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From โ๐ด๐ถ๐ดโฒ:
From โ๐ด๐ต๐ดโฒ:
Similarly
In โ๐ด๐ถ๐ดโฒ:
and
๐ด1๐ด2๐ดโฒ right with ๐(๐ดโฒ๐ด2๐ด1) = 90 because
From (4) and (3) โน ๐ดโฒ๐ถ โฅ (๐ด1๐ด2๐ด3).
As ๐ดโฒ๐ถ โฅ (๐ด1๐ด2๐ด3)
๐ดโฒ๐ถ โฅ ๐ด2๐ด (by construction)} โน ๐ด1๐ด2๐ด3๐ด coplanar โน ๐ด1๐ด2๐ด3๐ด quadrilateral with
opposite angles ๐ด1 and ๐ด3 right โน ๐ด1๐ด2๐ด3๐ด inscribable quadrilateral.
Solution to Problem 87.
The conclusion is true only if ||๐ต๐ท|| = ||๐ด๐ถ|| that is ๐ = ๐.
๐๐ โฅ ๐ท๐ถ if
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Solution to Problem 88.
bisector plane
(๐ bisector half-plane)
In triangle ๐ท๐ท1๐ทโฒ:
But
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From 1, 2, 3 โน๐[๐ด๐ต๐ท]
๐[๐ด๐ต๐ถ]=โ๐ท๐ธโ
โ๐ธ๐ถโ q.e.d.
Solution to Problem 89.
Because the tetrahedron is regular ๐ด๐ต = โฆ =
we increase the denominator
If one of the points ๐ or ๐ is on face ๐ถ๐ต๐ท the problem is explicit.
Solution to Problem 90.
We consider tetrahedron ๐๐ฅ๐ฆ๐ง, and prove that the sum of the measures of the
dihedral angles of this trihedron is bigger than 360ยฐ. Indeed: let 100โฒ be the internal
bisector of trihedron ๐๐ฅ๐ฆ๐ง (1000โฒ the intersection of the bisector planes of the 3
dihedral angles) of the trihedron in ๐ด, ๐ต, ๐ถ.
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The size of each dihedron with edges ๐๐ฅ, ๐๐ฆ, ๐๐ง is bigger than the size of the
corresponding angles of ๐ด๐ต๐ถ, the sum of the measures of the dihedral angles of
trihedron ๐๐ฅ๐ฆ๐ง is bigger than 180ยฐ.
Let (๐, ๐) be a plane โฅ to ๐๐ง at ๐ถ; ๐ โฅ ๐๐ง, ๐ โฅ ๐๐ง, but |๐ถ๐ด and |๐ถ๐ต are on the same
half-space in relation to (๐๐) โ ๐(๏ฟฝ๏ฟฝ) < ๐(๐๏ฟฝ๏ฟฝ).
In tetrahedron ๐ด๐ต๐ถ๐ท, let ๐1, ๐2, ๐3, ๐4, ๐5 and ๐6 be the 6 dihedral angles formed
by the faces of the tetrahedron.
according to the inequality previously established.
Solution to Problem 91.
We mark with a the intersection of planes (๐ด, ๐1) and (๐ต, ๐2). So
Let ๐ be a variable line that passes through ๐ถ and contained in ๐ผ, which cuts ๐1
and ๐2 at ๐ respectively ๐. We have: ๐๐ด โ (๐ด, ๐1),๐๐ด โฉ ๐๐ต = ๐(๐๐ด and ๐๐ต
intersect because they are contained in the plane determined by (๐ด๐, ๐)).
Thus ๐ โ (๐ด, ๐1) and ๐ โ (๐ต, ๐2), โน๐ โ ๐, so ๐ describes line a the intersection of
planes (๐ด, ๐1) and (๐ต, ๐2).
Vice-versa: let ๐ โ ๐.
In the plane (๐ด, ๐1): ๐๐ด โฉ ๐1 = ๐โฒ
In the plane (๐ต, ๐2): ๐๐ต โฉ ๐2 = ๐โฒ
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Lines ๐โฒ๐โฒ and ๐ด๐ต are coplanar (both are on plane (๐, ๐ด, ๐ต)). But because ๐โฒ๐โฒ โ
๐ผ and ๐ด๐ต has only point ๐ถ in common with ๐ผ โน ๐โฒ๐โฒ โฉ ๐ด๐ต = ๐ถ.
So ๐โฒ๐โฒ passes through ๐ถ. If planes (๐ด, ๐1) and (๐ต, ๐2) are parallel, the locus is the
empty set.
Solution to Problem 92.
Remember the theorem: If a plane ๐พ intersects two planes ๐ผ and ๐ฝ such that
๐||๐ผ โน (๐พ โฉ ๐ผ)||(๐พ โฉ ๐ฝ). If plane (๐ฟ๐๐๐)||๐ต๐ท we have:
If (๐ฟ๐๐๐)||๐ด๐ถ we have:
โน relation ๐.
Solution:
Let ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ the projections of points ๐ด, ๐ต, ๐ถ, ๐ท onto plane (๐๐๐๐ฟ).
For ex. points ๐ตโฒ, ๐ฟ, ๐ดโฒ are collinear on plane (๐ฟ๐๐๐) because they are on the
projection of line ๐ด๐ต onto this plane.
Similarly we obtain:
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By multiplying the 4 relations,
โน relation (๐) from ๐.
Solution to Problem 93.
๐ โ Midpoint of |๐ต๐ถ|.
Solution to Problem 94.
(1) ๐ด๐ต โฅ ๐ถ๐ท (hypothesis)
From 1 and 2
height in โ๐ด๐ต๐ถ a
From 3 and 4 ๐ด๐ถ โฅ (๐ต๐ท๐ป) โน ๐ด๐ถ โฅ ๐ต๐ป โน ๐ต๐ป height in โ๐ด๐ต๐ถ b
From a and b โน๐ป orthocenter โ๐ด๐ต๐ถ. Let ๐ถ1 be the diametrical opposite point to
๐ถ in circle ๐ถ(๐ด๐ต๐ถ)๐ถ๐ถ1 diameter ๐(๐ถ1๐ต๏ฟฝ๏ฟฝ) = 900 but ๐ด๐ป โฅ ๐ต๐ถ โน ๐ด๐ป โฅ ๐ต๐ถ1. Similarly
๐ต๐ป||๐ถ1๐ด, so ๐ด๐ป๐ต๐ถ1 parallelogram, we have:
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similarly
diametrical opposite to B
but
by substituting above, we have:
Let ๐,๐,๐, ๐, ๐, ๐ midpoints of the edges of the quadrilateral ๐๐๐๐ because:
๐๐||๐ถ๐ท||๐๐ (median lines), ๐๐||๐ด๐ต||๐๐ (median lines), but ๐ถ๐ท โฅ ๐ด๐ต โ ๐๐๐๐
rectangle |๐๐| โฉ |๐๐| = {0}.
Similarly ๐๐๐๐ rectangle with |๐๐| common diagonal with a, the first rectangle,
so the 6 points are equally distant from โ๐โ the midpoint of diagonals in the two
rectangles โน the 6 points are on a sphere.
Solution to Problem 95.
๐ arbitrary point on |๐ด๐ตโฒ|
๐ดสบ midpoint of segment [๐ตโฒ๐ถโฒ]
When ๐ = ๐ตโฒ, point ๐ is in the position ๐ตโฒ.
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255 Compiled and Solved Problems in Geometry and Trigonometry
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When ๐ = ๐ด, point ๐ is in the position {๐1} = [๐ดโฒ๐ด1] โฉ [๐ด๐ดโฒโฒ]
(๐ดโฒ๐ดโฒโฒ๐ด1๐ด rectangle, so ๐1 is the intersection of the diagonals of the rectangle)
[The locus is [๐ตโฒ๐1]].
Let ๐ be arbitrary point ๐ โ |๐ด๐ตโฒ|.
because:
By the way it was constructed
โน (โ) plane that contains ๐ด๐ด1 is perpendicular to (๐ตโฒ๐ถ๐ถโฒโฒ), particularly to (๐ตโฒ๐ด๐ด1) โฅ
(๐ตโฒ๐ถโฒ๐ถ).
because ๐ตโฒ, ๐ โ (๐ตโฒ๐ด๐ดโฒโฒ)
from this reason ๐ตโฒ, ๐1 โ (๐ดโฒ๐ตโฒ๐ด1).
From 1 and 2
Let
So (โ) ๐ โ |๐ตโฒ๐ด|
and we have
Vice-versa. Let ๐ arbitrary point, ๐ โ |๐ตโฒ๐1| and
In plane
In plane
Indeed: ๐ดโฒ๐ดโฒโฒ || (๐ตโฒ๐ด๐ด1) thus any plane which passes through ๐ดโฒ๐ดโฒโฒ will intersect
(๐ตโฒ๐ด๐ด1) after a parallel line to ๐ดโฒ๐ดโฒโฒ. Deci ๐๐||๐ดโฒ๐ดโฒโฒ or ๐๐||๐ด๐ด1 as ๐ โ (๐ตโฒ๐ด๐ด1) โน
๐๐ โฅ (๐ตโฒ๐ถ๐ถโฒโฒ).
Weโve proved
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and
we have
describes |๐ตโฒ๐1| and vice-versa,
there is ๐|๐ตโฒ๐ด| and ๐|๐ตโฒ๐ด1| such that
and ๐ is the intersection of the diagonals of the quadrilateral ๐ดโฒ๐๐๐ดโฒโฒ.
Solution to Problem 96.
known result
From 1 and 3
Solution to Problem 97.
From the hypothesis:
๐๐ด โฅ ๐๐ต โฅ ๐๐ถ โฅ ๐๐ด
We assume the problem is solved.
Let ๐ be the orthocenter of triangle ๐ด๐ต๐ถ.
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But ๐ถ๐ถโฒ โฅ ๐ด๐ต โน๐ด๐ต โฅ (๐๐ถ๐ถโฒ)
๐๐ โฅ (๐ถ๐๐ถโฒ)} โน ๐ด๐ต โฅ ๐๐ โน๐๐ โฅ ๐ด๐ต (1)
๐ด๐ โฅ (๐ถ๐๐ต) โน ๐ด๐ โฅ ๐ต๐ถ, but ๐ด๐ดโฒ โฅ ๐ต๐ถ โน๐ต๐ถ โฅ (๐ด๐๐ดโฒ)
๐๐ โ (๐ด๐๐ดโฒ)} โน ๐ต๐ถ โฅ ๐๐ โน ๐๐ โฅ ๐ต๐ถ (2)
From (1) and (2) โน๐๐ โฅ (๐ด๐ต๐ถ)
So the plane (๐ด๐ต๐ถ) that needs to be drawn must be perpendicular to ๐๐ at ๐.
Solution to Problem 98.
๐ดโฒ๐ โฅ ๐ด๐ท, ๐ตโฒ๐ โฅ ๐ต๐ถ
โ๐ต๐โ =๐โ๐โฒ
2, โ๐๐โ =
๐โ๐โฒ
2
๐ฃ[๐ต๐๐๐๐ตโฒ] =๐ โ ๐โฒ
2โ๐ โ ๐โฒ
2โโ
3
๐ฃ[๐๐๐๐ ๐ดโฒ๐ตโฒ] =๐[๐๐๐ตโฒ] โ โ๐ตโฒ๐ดโฒโ
3=๐ โ ๐โฒ
2โโ
2โ ๐โฒ
๐ฃ[๐ตโฒ๐ดโฒ๐๐๐ถโฒ๐ทโฒ๐ท1๐ถ1] =(๐ + ๐โฒ)โ
2โ ๐โฒ
๐ฃ[๐ด๐ต๐ดโฒ๐ตโฒ๐ถ๐ท๐ถโฒ๐ทโฒ] = 2 [2 โ๐โ๐โฒ
2โ๐โ๐โฒ
2โโ
3+๐โ๐โฒ
2โโ
2โ ๐โฒ] + (
(๐+๐โฒ)โ
2) โ ๐โฒ =
โ
6(2๐๐ โ 2๐๐โฒ โ 2๐โฒ๐โฒ + 3๐๐โฒ โ 3๐โฒ๐โฒ + 3๐โฒ๐ + 3๐โฒ๐โฒ) =
โ
6[๐๐ + ๐โฒ๐โฒ +
(๐ + ๐โฒ)(๐ + ๐โฒ)].
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Solution to Problem 99.
๐ฃ[๐๐ด๐ต๐ตโฒ] =๐ต โ โ
2
๐ฃ[๐๐ดโฒ๐ตโฒ๐ด] = ๐ฃ[๐ด๐ต๐๐โฒ๐ดโฒ๐ตโฒ] โ ๐ฃ[๐ด๐ต๐ตโฒ๐] โ ๐ฃ[๐ดโฒ๐ตโฒ๐โฒ๐] =โ
3(๐ต + ๐ + โ๐ต๐) โ
๐ตโ
3โ๐โ
3=
โ
3โ๐๐ต.
So:
๐ฃ[๐๐ดโฒ๐ตโฒ๐ด]
๐ฃ[๐ท๐ด๐ต๐ตโฒ]=
โ3โ๐ต๐ โ โ๐ต๐
๐ตโ3 โ ๐ต
=โ๐
โ๐ตโน ๐ฃ[๐๐ดโฒ๐ตโฒ๐ด] =
โ๐
โ๐ตโ ๐ฃ[๐๐ด๐ต๐ตโฒ]
For the relation above, determine the formula of the volume of the pyramid
frustum.
Solution to Problem 100.
๐(๐บ๐บโฒ) = โ = 2๐
Let ๐ = โ๐ด๐ถโ โน โ๐ด๐ทโ =๐โ3
2โน โ๐บ๐ทโ =
๐โ3
6
Figure ๐บ๐ท๐๐ rectangle โน โ๐บ๐ทโ = โ๐๐โ โน๐โ3
6= ๐ = 2โ3๐
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255 Compiled and Solved Problems in Geometry and Trigonometry
93
So, the lateral area is ๐๐ = 3 โ 2โ3๐ โ ๐ = 12โ3๐ 2.
๐ฃ[๐ด๐ต๐ถ๐ดโฒ๐ตโฒ๐ถโฒ] = ๐[๐ด๐ต๐ถ] โ 2๐ = 2โ3๐ โ2โ3๐ โ3
4โ 2๐ = 6โ3๐ 2.
The total area:
๐๐ก = ๐๐ + 2๐[๐ด๐ต๐ถ] = 12โ3๐ 2 + 2 โ 3๐ โ3๐ 2 = 18โ3๐ 2
Solution to Problem 101.
Let ๐1 and ๐1 be the volume, respectively the area obtained revolving around ๐.
๐2 and ๐2 be the volume, respectively the area obtained after revolving around ๐.
๐3 and ๐3 be the volume, respectively the area obtained after revolving around c.
So:
๐1 =๐ โ ๐2(โ๐ถ๐ทโ + โ๐ท๐ตโ)
3=๐ โ ๐2 โ ๐
3
๐1 = ๐ โ ๐ โ ๐ + ๐ โ ๐ โ ๐ = ๐ โ ๐ โ (๐ + ๐)
๐2 =๐๐2๐
3=๐๐2๐2
3๐=๐๐2๐2๐
3๐2
๐2 = ๐ โ ๐ โ ๐
๐3 =๐๐2๐
3=๐๐2๐2
3๐=๐๐2๐2
3๐
๐3 = ๐ โ ๐ โ ๐
Therefore:
1
๐12 =
1
๐22 +
1
๐32โบ
9๐2
(๐๐2๐2)2=
9๐2
(๐๐2๐2)2+
9๐2
(๐๐2๐2)2
๐2๐3+๐3๐2=๐2 + ๐3๐1
โบ๐
๐+๐
๐=๐๐(๐ + ๐)
๐๐(๐ + ๐)โบ๐2 + ๐2
๐ โ ๐=๐
๐
But ๐ โ ๐ = ๐ โ ๐ โน๐2+๐2
๐๐=๐2
๐๐ , โ๐ด๐ทโ = ๐.
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Solution to Problem 102.
๐ = โ๐๐ดโ = 25 ๐๐
๐ = โ๐โฒ๐ตโ = 50 ๐๐
2๐๐ = 1,57 โน ๐ = 0,25 ๐
2๐๐ = 3,14 โน ๐ = 0,50 ๐
โ๐ถ๐โ = 18 ๐๐ = 0,18 ๐
โ๐ดโฒ๐ตโ = 25 ๐๐
โ๐ด๐ตโ = โ100 + 0,0625 = 10,003125
โ๐ดโฒ๐โ =โ๐ด๐ดโฒโ โ โ๐ดโฒ๐ตโ
โ๐ด๐ตโ=10 โ 0,25
10,003125โ 0,25
โ๐ถ๐โ
โ๐ดโฒ๐โ=โ๐ถ๐ตโ
โ๐ต๐ดโฒโโน0,18
0,25=โ๐ถ๐ตโ
0,25โน โ๐ถ๐ตโ = 0,18
โ๐โฒ๐ถโ = ๐ โฒ = 0,50 โ 0,18 = 0,32
โ๐๐โ = ๐โฒ = 0,25 โ 0,18 = 0,07
๐ =๐1
3(๐ 2 + ๐2 + ๐ ๐)
๐ =๐10
3(0,502 + 0,252 + 0,50 โ 0,25 โ 0,322 โ 0,072 โ 0,32 โ 0,07)
=๐10
3(0,4375 โ 0,1297) = 1,026๐๐3
Solution to Problem 103.
โ๐๐โ =๐
2 sin๐ผ2
cos๐ผ
2
In โ๐๐ด๐: โ๐๐ดโ =๐ด
2 sin๐ผ
2
.
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95
In โ๐๐ด๐โฒ: โ๐๐โฒโ2 =๐2
4 sin2๐ผ
2
โ๐2
2 .
โ๐๐โฒโ =๐โcos๐ผ
2 sin๐ผ2
In โ๐๐๐โฒ: โ๐๐โฒโ =๐โcos๐ผ
2 sin๐ผ
2
โ ๐ .
๐ 2 =๐2
2+ (
๐โcos๐ผ
2 sin๐ผ2
โ ๐ )
2
โน๐2
2+๐2 cos ๐ผ
4 sin2๐ผ2
โ2๐๐ โcos๐ผ
2 sin๐ผ2
โน ๐ =4๐ โcos๐ผ sin2
๐ผ2
sin๐ผ2(2 cos2
๐ผ2+ cos๐ผ)
โน ๐ = 4๐ โcos๐ผ โ sin๐ผ
2
๐ด๐ = 4๐2 cos
๐ผ2
2 โ 2 sin๐ผ2
=๐2 cos
๐ผ2
sin๐ผ2
= 16๐ 2 cos ๐ผ sin2๐ผ
2โcos๐ผ2
sin๐ผ2
= 8๐ 2 cos๐ผ sin๐ผ = 4๐ 2 sin 2๐ผ
๐ด๐ก = ๐ด๐ + ๐2 = 4๐ 2 sin 2๐ผ + 16๐ 2 cos๐ผ sin2
๐ผ
2
โ๐๐โฒโ = ๐ โน ๐ =๐โcos๐ผ
2 sin๐2
โน ๐ = 4๐ โcos๐ผ sin๐ผ
2โโcos๐ผ
2 sin๐ผ2
โน 2cos๐ผ = 1 โน ๐ผ = 600.
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Various Problems
104. Determine the set of points in the plane, with affine coordinates ๐ง that
satisfy:
a. |๐ง| = 1;
b. ๐ < arg ๐ง โค3๐
2; ๐ง โ 0;
c. arg ๐ง >4๐
3, ๐ง โ 0;
d. |๐ง + ๐| โค 2 .
Solution to Problem 104
105. Prove that the ๐ roots of the unit are equal to the power of the particular
root ํ1.
Solution to Problem 105
106. Knowing that complex number ๐ง verifies the equation ๐ง๐ = ๐, show that
numbers 2, โ๐๐ง and ๐๐ง verify this equation.
Application: Find (1 โ 2๐)4 and deduct the roots of order 4 of the number
โ7 + 24๐.
Solution to Problem 106
107. Show that if natural numbers ๐ and ๐ are coprime, then the equations
๐ง๐ โ 1 = 0 and ๐ง๐ โ 1 = 0 have a single common root.
Solution to Problem 107
108. Solve the following binomial equation: (2 โ 3๐)๐ง6 + 1 + 5๐ = 0.
Solution to Problem 108
109. Solve the equations:
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255 Compiled and Solved Problems in Geometry and Trigonometry
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Solution to Problem 109
110. Solve the equation ๐ง = ๐ง๐โ1, ๐ โ ๐, where ๐ง the conjugate of ๐ง.
Solution to Problem 110
111. The midpoints of the sides of a quadrilateral are the vertices of a
parallelogram.
Solution to Problem 111
112. Let ๐1๐2๐3๐4 and ๐1๐2๐3๐4 two parallelograms and ๐๐ the midpoints of
segments [๐๐๐๐], ๐ โ {1, 2, 3, 4}. Show that ๐1๐2๐3๐4 is a parallelogram or a
degenerate parallelogram.
Solution to Problem 112
113. Let the function ๐: ๐ถ โ ๐ถ, ๐(๐ง) = ๐๐ง + ๐; (๐, ๐, ๐ โ ๐ถ, ๐ โ 0). If ๐1 and
๐2 are of affixes ๐ง1 and ๐ง2, and ๐1โฒ and ๐2
โฒ are of affixes ๐(๐ง1), ๐(๐ง2), show
that โ๐1โฒ๐2
โฒโ = |๐| โ โ๐1๐2โ. We have โ๐1โฒ๐2
โฒโ = โ๐1๐2โ โ |๐| = 1.
Solution to Problem 113
114. Prove that the function z โ z, z โ C defines an isometry.
Solution to Problem 114
115. Let M1M2 be of affixes ๐ง1, ๐ง2 โ 0 and z2 = ฮฑz1. Show that rays |OM1, |OM2
coincide (respectively are opposed) โบ ฮฑ > 0 (respectively ฮฑ < 0).
Solution to Problem 115
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98
116. Consider the points M1M2M3 of affixes ๐ง1๐ง2๐ง3 and ๐1 โ ๐2. Show that:
a. M3 โ |M1M2โบz3โz1
z2โz1> 0;
b. M3 โ M1M2โบz3โz1
z2โz1โ R .
Solution to Problem 116
117. Prove Pompeiuโs theorem. If the point ๐ from the plane of the equilateral
triangle ๐1๐2๐3 โ the circumscribed circle โ ๐1๐2๐3 there exists a
triangle having sides of length โ๐๐1โ, โ๐๐2โ, โ๐๐3โ.
Solution to Problem 117
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255 Compiled and Solved Problems in Geometry and Trigonometry
99
Solutions
Solution to Problem 104.
a. |๐ง| = 1
|๐ง| = โ๐ฅ2 + ๐ฆ2} โน ๐ฅ2 + ๐ฆ2 = 1, so the desired set is the circle ๐ถ(0,1).
b. ๐ < arg๐ง โค3๐
2 .
The desired set is given by all the points of quadrant III, to which ray |๐๐ฆ is added,
so all the points with ๐ฅ < 0, ๐ฆ < 0.
c. arg๐ง >
4๐
3 , ๐ง โ 0
arg๐ง โ [0, 2๐]} โน
4๐
2< arg๐ง < 2๐
The desired set is that of the internal points of the angle with its sides positive
semi-axis and ray |๐๐ต.
d. |๐ง + ๐| โค 2; ๐ง = ๐ฅ + ๐ฆ๐, its geometric image ๐.
where ๐โฒ(0,โ1).
Thus, the desired set is the disk centered at ๐(0,โ1)โฒ and radius 2.
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100
Solution to Problem 105.
Solution to Problem 106.
Let the equation ๐ง4 = ๐. If ๐ง4 = 4 (๐ง is the solution) then: (โ๐ง)4 = (โ1)4๐ง4 = 1 โ ๐ =
๐, so โ ๐ง is also a solution.
โน ๐๐ง is the solution;
โนโ๐๐ง is the solution;
โน is the solution of the equation ๐ง4 = โ7 + 24๐.
The solutions of this equation are:
but based on the first part, if ๐ง โ 1 โ 2๐ is a root, then
are solutions of the given equation.
Solution to Problem 107.
If there exist ๐ and ๐โฒ with ๐ง๐ = ๐ง๐โฒ, then
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255 Compiled and Solved Problems in Geometry and Trigonometry
101
because (๐, ๐) = 1. Because ๐โฒ < ๐, ๐ < ๐, we have ๐โฒ = 0, ๐ = 0.
Thus the common root is ๐ง0.
Solution to Problem 108.
Solution to Problem 109.
Solution to Problem 110.
As:
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102
From:
positive
The given equation becomes
Solution to Problem 111.
We find the sum of the abscissa of the opposite points:
โน๐๐๐๐ a parallelogram.
Solution to Problem 112.
In the quadrilateral ๐1๐3๐3๐1 by connecting the midpoints we obtain the
parallelogram ๐โฒ๐1๐โฒโฒ๐3, with its diagonals intersecting at ๐, the midpoint of |๐โฒ๐โฒโฒ|
and thus |๐1๐| โก |๐๐3|. (1)
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In the quadrilateral ๐4๐2๐2๐4 by connecting the midpoints of the sides we obtain
the parallelogram ๐โฒ๐2๐โฒโฒ๐4 with its diagonals intersecting in ๐, the midpoint of
|๐โฒ๐โฒโฒ| and thus |๐2๐| โก |๐๐4|. (2)
From (1) and (2) ๐1๐2๐3๐4 a parallelogram.
Solution to Problem 113.
If:
If:
Solution to Problem 114.
Let ๐1 and ๐2 be of affixes ๐ง1 and ๐ง2. Their images through the given function
๐1โฒ and ๐2
โฒ with affixes ๐ง1 and ๐ง2, so
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From (1) and (2) โน โ๐1๐2โ = โ๐1โฒ๐2
โฒโ or โ๐1โฒ๐2
โฒโ = |๐ง2๐ง1| = |โ๐ง2 โ ๐ง1| =
|๐ง2 โ ๐ง1| = โ๐1๐2โ.
So ๐: ๐ถ โ ๐ถ, ๐(๐ง) = ๐ง defines an isometry because it preserves the distance
between the points.
Solution to Problem 115.
We know that the argument (๐๐ง1) = arg๐ง1 + arg๐ง๐ผ โ 2๐๐, where ๐ = 0 or ๐ = 1.
Because arg๐ง2 = arg(๐๐ง1), arg๐ง2 = arg๐ง1 + arg๐ง๐ผ โ 2๐๐.
a. We assume that
Vice versa,
b. Let |๐๐1 and |๐๐2 be opposed โน arg๐ง2 = arg๐ง1 + ๐
โ to the negative ray |๐๐ฅโฒโน ๐ผ < 0. Vice versa,
๐ = 0 or ๐ = 1 โน
arg๐ง2 = arg๐ง1 + ๐or
arg๐ง2 = arg๐ง1 โ ๐} โน |๐๐1 and |๐๐2 are opposed.
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Solution to Problem 116.
If n and ๐โฒ are the geometric images of complex numbers ๐ง and ๐งโฒ, then the image
of the difference ๐งโ ๐งโฒ is constructed on |OM1| and |๐โฒ๐| as sides.
We assume that ๐3 โ |๐1๐2
We construct the geometric image of ๐ง2โ ๐ง1 . It is the fourth vertex of the
parallelogram ๐๐1๐2๐1. The geometric image of ๐ง3โ ๐ง1 is ๐2, the fourth vertex of the
parallelogram ๐๐1๐3๐2.
๐๐1 โฅ ๐1๐2๐๐2 โฅ ๐1๐3
๐1๐2๐3collinear } โน ๐1, ๐2, ๐3 collinear โน
Vice versa, we assume that
If ๐3 and ๐2 โ the opposite ray to ๐, then ๐ง3โ ๐ง1 = ๐ผ(๐ง2โ ๐ง1) with ๐ผ < 0.
We repeat the reasoning from the previous point for the same case.
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Thus, when ๐3 โ ๐1๐2๐3 +๐2 we obtain for the respective ratio positive,
negative or having ๐3 = ๐1, so z3โz1
z2โz1โ R.
Solution to Problem 117.
The images of the roots of order 3 of the unit are the peaks of the equilateral
triangle.
But ษ1 = ษ22, so if we write ษ2 = ํ, then ษ1 = ษ2.
Thus ๐1(1),๐2(ํ),๐3(ํ2).
We use the equality:
adequate (โ)๐ง โ โ.
But
Therefore,
By substitution:
but
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thus
Therefore โ๐๐1โ, โ๐๐2โ, โ๐๐3โ sides of a โ.
Then we use โ๐ฅ| โ |๐ฆโ โค |๐ฅ โ ๐ฆ| and obtain the other inequality.
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Problems in Spatial Geometry
118. Show that if a line ๐ is not contained in plane ๐ผ, then ๐ โฉ ๐ผ is โ or it is
formed of a single point.
Solution to Problem 118
119. Show that (โ) ๐ผ, (โ) at least one point which is not situated in ๐ผ.
Solution to Problem 119
120. The same; there are two lines with no point in common.
Solution to Problem 120
121. Show that if there is a line ๐ (โ) at least two planes that contain line ๐.
Solution to Problem 121
122. Consider lines ๐, ๐โฒ, ๐โฒโฒ, such that, taken two by two, to intersect. Show
that, in this case, the 3 lines have a common point and are located on the
same plane.
Solution to Problem 122
123. Let ๐ด, ๐ต, ๐ถ be three non-collinear points and ๐ท a point located on the
plane (๐ด๐ต๐ถ). Show that:
a. The points ๐ท, ๐ด, ๐ต are not collinear, and neither are ๐ท, ๐ต, ๐ถ; ๐ท, ๐ถ, ๐ด.
b. The intersection of planes (๐ท๐ด๐ต), (๐ท๐ต๐ถ), (๐ท๐ถ๐ด) is formed of a single
point.
Solution to Problem 123
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124. Using the notes from the previous exercise, take the points ๐ธ, ๐น, ๐บ distinct
from ๐ด, ๐ต, ๐ถ, ๐ท, such that ๐ธ โ ๐ด๐ท, ๐น โ ๐ต๐ท, ๐บ โ ๐ถ๐ท. Let ๐ต๐ถ โฉ ๐น๐บ = {๐}, ๐บ๐ธ โฉ
๐ถ๐ด = {๐}, ๐ธ๐น โฉ ๐ด๐ต = {๐ }. Show that ๐, ๐, ๐ are collinear (T. Desarques).
Solution to Problem 124
125. Consider the lines ๐ and ๐โฒ which are not located on the same plane and
the distinct points ๐ด, ๐ต, ๐ถ โ ๐ and ๐ท, ๐ธ โ ๐โฒ. How many planes can we draw
such that each of them contains 3 non-collinear points of the given points?
Generalization.
Solution to Problem 125
126. Show that there exist infinite planes that contain a given line ๐.
Solution to Problem 126
127. Consider points ๐ด, ๐ต, ๐ถ, ๐ท which are not located on the same plane.
a. How many of the lines ๐ด๐ต, ๐ด๐ถ, ๐ด๐ท, ๐ต๐ถ, ๐ต๐ท, ๐ถ๐ท can be intersected by a
line that doesnโt pass through ๐ด, ๐ต, ๐ถ, ๐ท?
b. Or by a plane that doesnโt pass through ๐ด, ๐ต, ๐ถ, ๐ท?
Solution to Problem 127
128. The points ๐ผ and ๐ฝ are given, ๐ด, ๐ต โ ๐ผ. Construct a point ๐ โ ๐ผ at an
equal distance from ๐ด and ๐ต, that โ also to plan ๐ฝ.
Solution to Problem 128
129. Determine the intersection of three distinct planes ๐ผ, ๐ฝ, ๐พ.
Solution to Problem 129
130. Given: plane ๐ผ, lines ๐1, ๐2 and points ๐ด, ๐ต โ ๐ผ โช ๐1 โช ๐2. Find a point ๐ โ
๐ผ such that the lines ๐๐ด,๐๐ต intersect ๐1 and ๐2.
Solution to Problem 130
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131. There are given the plane ๐ผ, the line ๐ โ ๐ผ, the points ๐ด, ๐ต โ ๐ผ โช ๐, and
๐ถ โ ๐ผ. Let ๐ โ ๐ and ๐ดโฒ, ๐ตโฒ the points of intersection of the lines ๐๐ด,๐๐ต
with plane ๐ผ (if they exist). Determine the point ๐ such that the points
๐ถ, ๐ดโฒ, ๐ตโฒ to be collinear.
Solution to Problem 131
132. If points ๐ด and ๐ต of an open half-space ๐, then [๐ด๐ต] โ ๐. The property is
as well adherent for a closed half-space.
Solution to Problem 132
133. If point ๐ด is not situated on plane ๐ผ and ๐ต โ ๐ผ then |๐ต๐ด โ |๐ผ๐ด.
Solution to Problem 133
134. Show that the intersection of a line ๐ with a half-space is either line ๐ or
a ray or an empty set.
Solution to Problem 134
135. Show that if a plane ๐ผ and the margin of a half-space ๐ are secant
planes, then the intersection ๐ โฉ ๐ผ is a half-plane.
Solution to Problem 135
136. The intersection of a plane ๐ผ with a half-space is either the plane ๐ผ or a
half-plane, or an empty set.
Solution to Problem 136
137. Let ๐ด, ๐ต, ๐ถ, ๐ท four non coplanar points and ๐ผ a plane that doesnโt pass
through one of the given points, but it passes trough a point of the line
|๐ด๐ต|. How many of the segments |๐ด๐ต|, |๐ด๐ถ|, |๐ด๐ท|, |๐ต๐ถ|, |๐ต๐ท|, |๐ถ๐ท| can be
intersected by plane ๐ผ?
Solution to Problem 137
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138. Let ๐ be a line and ๐ผ, ๐ฝ two planes such that ๐ โฉ ๐ฝ = โ and ๐ผ โฉ ๐ฝ = โ .
Show that if ๐ด โ ๐ and ๐ต โ ๐ผ, then ๐ โ |๐ฝ๐ด and ๐ผ โ |๐ฝ๐ต.
Solution to Problem 138
139. Let |๐ผ๐ด and |๐ฝ๐ต be two half-spaces such that ๐ผ โ ๐ฝ and |๐ผ๐ด โ |๐ฝ๐ต or
|๐ผ๐ด โฉ |๐ฝ๐ต = โ . Show that ๐ผ โฉ ๐ฝ = โ .
Solution to Problem 139
140. Show that the intersection of a dihedral angle with a plane ๐ผ can be: a
right angle, the union of two lines, a line, an empty set or a closed half-
plane and cannot be any other type of set.
Solution to Problem 140
141. Let ๐ be the edge of a proper dihedron โ ๐ผโฒ๐ฝโฒ, ๐ด โ ๐ผโฒโ ๐, ๐ โ ๐ฝโฒโ ๐ and
๐ โ int.โ ๐ผโฒ๐ฝโฒ. Show that:
a. (๐๐) โฉ int.โ ๐ผโฒ๐ฝโฒ = |๐๐;
b. If ๐ โ ๐, int.โ ๐ด๐๐ต = int. ๐ผโฒ๐ฝโฒ โฉ (๐ด๐๐ต).
Solution to Problem 141
142. Consider the notes from the previous problem. Show that:
a. The points ๐ด and ๐ต are on different sides of the plane (๐๐);
b. The segment |๐ด๐ต| and the half-plane |๐๐ have a common point.
Solution to Problem 142
143. If โ ๐๐๐ is a trihedral angle, ๐ โ int.โ ๐๐๐ and ๐ด, ๐ต, ๐ถ are points on edges
๐, ๐, ๐, different from ๐, then the ray |๐๐ and int. ๐ด๐ต๐ถ have a common
point.
Solution to Problem 143
144. Show that any intersection of convex sets is a convex set.
Solution to Problem 144
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145. Show that the following sets are convex planes, half-planes, any open or
closed half-space and the interior of a dihedral angle.
Solution to Problem 145
146. Can a dihedral angle be a convex set?
Solution to Problem 146
147. Which of the following sets are convex:
a. a trihedral angle;
b. its interior;
c. the union of its faces;
d. the union of its interior with all its faces?
Solution to Problem 147
148. Let ๐ be an open half-space bordered by plane ๐ผ and ๐ a closed convex
set in plane ๐ผ. Show that the set ๐ โฉ ๐ is convex.
Solution to Problem 148
149. Show that the intersection of sphere ๐(๐, ๐) with a plane which passes
through ๐, is a circle.
Solution to Problem 149
150. Prove that the int. ๐(๐, ๐) is a convex set.
Solution to Problem 150
151. Show that, by unifying the midpoints of the opposite edges of a
tetrahedron, we obtain concurrent lines.
Solution to Problem 151
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152. Show that the lines connecting the vertices of a tetrahedron with the
centroids of the opposite sides are concurrent in the same point as the
three lines from the previous example.
Solution to Problem 152
153. Let ๐ด๐ต๐ถ๐ท be a tetrahedron. We consider the trihedral angles which have
as edges [๐ด๐ต, [๐ด๐ธ, [๐ด๐ท, [๐ต๐ด, [๐ต๐ถ, [๐ต๐ท, [๐ถ๐ด, [๐ถ๐ต, [๐ถ๐ท, [๐ท๐ด, [๐ท๐ต, [๐ท๐ถ. Show that
the intersection of the interiors of these 4 trihedral angles coincides with
the interior of tetrahedron [๐ด๐ต๐ถ๐ท].
Solution to Problem 153
154. Show that (โ) ๐ โ int. [๐ด๐ต๐ถ๐ท] (โ) ๐ โ |๐ด๐ต| and ๐ โ |๐ถ๐ท| such that ๐ โ
โ๐๐.
Solution to Problem 154
155. The interior of tetrahedron [๐ด๐ต๐ถ๐ท] coincides with the union of segments
|๐๐| with ๐ โ |๐ด๐ต| and ๐ โ |๐ถ๐ท|, and tetrahedron [๐ด๐ต๐ถ๐ท] is equal to the
union of the closed segments [๐๐], when ๐ โ [๐ด๐ต] and ๐ โ [๐ถ๐ท].
Solution to Problem 155
156. The tetrahedron is a convex set.
Solution to Problem 156
157. Let ๐1 and ๐2 convex sets. Show that by connecting segments [๐๐], for
which ๐ โ ๐1 and ๐ โ ๐2 we obtain a convex set.
Solution to Problem 157
158. Show that the interior of a tetrahedron coincides with the intersection of
the open half-spaces determined by the planes of the faces and the
opposite peak. Define the tetrahedron as an intersection of half-spaces.
Solution to Problem 158
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Solutions
Solution to Problem 118.
We assume that ๐ โฉ ๐ผ = {๐ด, ๐ต} โ ๐ โ ๐ผ.
It contradicts the hypothesis โน ๐ โฉ ๐ผ = {๐ด} or ๐ โฉ ๐ผ = โ .
Solution to Problem 119.
We assume that all the points belong to the plane ๐ผ โน (โ) for the points that
are not situated in the same plane. False!
Solution to Problem 120.
โ ๐ด, ๐ต, ๐ถ, ๐ท, which are not in the same plane. We assume that ๐ด๐ต โฉ ๐ถ๐ท = {0} โน
๐ด๐ต and ๐ถ๐ท are contained in the same plane and thus ๐ด, ๐ต, ๐ถ, ๐ท are in the same
plane. False, it contradicts the hypothesis โน๐ด๐ต โฉ ๐ถ๐ท = โ โน (โ) lines with no
point in common.
Solution to Problem 121.
(โ) ๐ด โ ๐ (if all the points would โ ๐, the existence of the plane and space would
be negated). Let ๐ผ = (๐๐ด), (โ)๐ต โ ๐ผ (otherwise the space wouldnโt exist). Let ๐ฝ =
(๐ต๐), ๐ผ โ ๐ฝ and both contain line ๐.
Solution to Problem 122.
We show that ๐ โ ๐โฒ โ ๐โฒโฒ โ ๐.
Let ๐ โฉ ๐โฒ = {๐ด} = (๐, ๐โฒ) โน {๐ โ ๐ผ๐โฒ โ ๐ผ
๐ โฉ ๐โฒ = {๐ต}๐ต โ ๐ด
โน {๐ต โ ๐๐ โโ
โน ๐ต โ ๐ , ๐ต โ ๐โฒ
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๐โฒโฒ โฉ ๐โฒ = {๐ถ}๐ถ โ ๐ต๐ถ โ ๐ด
โน ๐ถ โ ๐โฒ
๐โฒ โ โโน ๐ถ โ ๐ผ, ๐ถ โ ๐โฒโฒ
โน๐ = ๐โฒ
or๐ = ๐โฒโฒ
โน ๐โฒโฒ โ ๐ผ, so the lines are located on the same plane ฮฑ.
If ๐ โฉ ๐โฒ = {๐ด} โน ๐ด โ ๐โฒ
๐โฒโฒ โฉ ๐ = {๐ด} โน ๐ด โ ๐โฒโฒ} โน ๐โฒ โฉ ๐โฒโฒ = {๐ด}, and the three lines have a point in
common.
Solution to Problem 123.
a. ๐ท โ (๐ด๐ต๐ถ).
We assume that ๐ท, ๐ด, ๐ต collinear โน (โ)๐ such that ๐ท โ ๐, ๐ด โ ๐, ๐ต โ ๐๐ด โ (๐ด๐ต๐ถ), ๐ต โ (๐ด๐ต๐ถ)
}๐2โ ๐ โ
(๐ด๐ต๐ถ) โน ๐ท โ (๐ด๐ต๐ถ) โ false. Therefore, the points ๐ท, ๐ด, ๐ต are not collinear.
b. Let (๐ท๐ด๐ต) โฉ (๐ต๐ถ๐ท) โฉ (๐ท๐ถ๐ด) = ๐ธ.
As the planes are distinct, their intersections are:
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(๐ท๐ด๐ต) โฉ (๐ท๐ต๐ถ) = ๐ท๐ต(๐ท๐ด๐ต) โฉ (๐ท๐ถ๐ด) = ๐ท๐ด(๐ท๐ต๐ถ) โฉ (๐ท๐ถ๐ด) = ๐ท๐ถ
}โน If (๐ท๐ด๐ต) = (๐ท๐ต๐ถ) โน ๐ด, ๐ต, ๐ถ, ๐ท coplanar, contrary to the hypothesis.
We suppose that (โ)๐ โ ๐ธ,๐ โ ๐ท โน๐ โ ๐ท๐ต๐ โ ๐ท๐ด
} โน๐ต โ ๐๐ท๐ด โ ๐๐ท
}โน ๐ด, ๐ต, ๐ท are collinear
(false, contrary to point a.). Therefore, set ๐ธ has a single point ๐ธ = {๐ท}.
Solution to Problem 124.
We showed at the previous exercise that if ๐ท โ (๐ด๐ต๐ถ), (๐ท๐ด๐ต) โ (๐ท๐ต๐ถ). We show
that ๐ธ, ๐น, ๐บ are not collinear. We assume the opposite. Then,
Having three common points ๐ท,๐ต and ๐บ โน false. So ๐ธ, ๐น, ๐บ are not collinear and
determine a plane (๐ธ๐น๐บ).
โน๐,๐, ๐ are collinear because โ to the line of intersection of the two planes.
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Solution to Problem 125.
The planes are (๐ด, ๐โฒ); (๐ต, ๐โฒ); (๐ถ, ๐โฒ).
Generalization: The number of planes corresponds to the number of points on
line ๐ because ๐โฒ contains only 2 points.
Solution to Problem 126.
Let line ๐ be given, and ๐ด any point such that ๐ด โ ๐.
We obtain the plane ๐ผ = (๐ด, ๐), and let ๐ โ ๐ผ. The line ๐โฒ = ๐ด๐, ๐โฒ โ ๐ผ is not
thus contained in the same plane with ๐. The desired planes are those of type
(๐๐),๐ โ ๐โฒ, that is an infinity of planes.
Solution to Problem 127.
a. (โ) 3 points determine a plane. Let plane (๐ด๐ต๐ท). We choose in this plane ๐ โ
|๐ด๐ท| and ๐ โ |๐ด๐ต| such that ๐ โ |๐ต๐|, then the line ๐๐ separates the points ๐ด
and ๐ท, but does not separate ๐ด and ๐ต, so it separates ๐ and ๐ท โ ๐๐ โฉ |๐ต๐ท| = ๐ ,
where ๐ โ |๐ต๐ท|.
Thus, the line ๐๐ meets 3 of the given lines. Letโs see if it can meet more.
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Florentin Smarandache
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We assume that
it has two points in common with the plane.
โน๐ด,๐ต, ๐ถ, ๐ท coplanar โ false.
Thus,
false.
We show in the same way that ๐๐ does not cut ๐ด๐ถ or ๐ท๐ถ, so a line meets at
most three of the given lines.
b. We consider points ๐ธ, ๐น, ๐บ such that ๐ธ โ |๐ต๐ถ|, ๐ด โ |๐ท๐น|, ๐ท โ |๐ต๐บ|. These points
determine plane (๐ธ๐น๐บ) which obviously cuts the lines ๐ต๐ถ, ๐ต๐ท and ๐ต๐ท. ๐น๐บ does not
separate ๐ด and ๐ท or ๐ต๐ท โน it does not separate ๐ด or ๐ต โน ๐ด โ |๐ต๐ |.
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Letโs show that (๐ธ๐น๐บ) meets as well the lines ๐ด๐ต, ๐ถ๐ท, ๐ด๐ถ. In the plane (๐ด๐ต๐ท) we
consider the triangle ๐น๐ท๐บ and the line ๐ด๐ต.
As this line cuts side |๐น๐ท|, but it does not cut |๐ท๐บ|, it must cut side |๐น๐บ|, so ๐ด๐ต โฉ
|๐น๐บ| = {๐ } โน ๐ โ |๐น๐บ| โ (๐ธ๐น๐บ), so ๐ด๐ต โฉ (๐ธ๐น๐บ) = {๐ }. In the plane (๐ต๐ถ๐ท), the line
๐ธ๐บ cuts |๐ต๐ถ| and does not cut |๐ต๐ท|, so ๐ธ๐บ cuts the side |CD|, ๐ธ๐บ โฉ |๐ถ๐ท| = {๐} โน
๐ โ ๐ธ๐บ โ (๐ธ๐น๐บ) โน ๐ถ๐ท โฉ (๐ธ๐น๐บ) = {๐}.
๐ โ (๐ธ๐น๐บ), ๐ does not separate ๐ด and ๐ต๐ธ separates ๐ต and ๐ถ
} โน ๐ โโฉ |๐ด๐ถ| = ๐ โน ๐ โ ๐ ๐ธ โน ๐
โ (๐ธ๐น๐บ) โฉ ๐ด๐ถ = {๐}.
Solution to Problem 128.
We assume problem is solved, if ๐ โ ๐ผ๐ โ ๐ฝ
} โน ๐ผ โฉ ๐ฝ โ โ ,โน ๐ผ โฉ ๐ฝ = ๐.
As ||๐๐ด|| = ||๐๐ต|| โน ๐ โ the bisecting line of the segment [๐ด๐ต].
So, to find ๐, we proceed as follows:
1. We look for the line of intersection of planes ๐ผ and ๐ฝ, d. If ๐ผ โฅ ๐ฝ, the
problem hasnโt got any solution.
2. We construct the bisecting line ๐โฒ of the segment [๐ด๐ต] in the plane ๐ผ.
3. We look for the point of intersection of lines ๐ and ๐โฒ. If ๐ โฅ ๐โฒ, the problem
hasnโt got any solution.
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Solution to Problem 129.
If ๐ผ โฉ ๐ฝ = โ โน ๐ผ โฉ ๐ฝ โฉ ๐พ = โ . If ๐ผ โฉ ๐ฝ = ๐, the desired intersection is ๐ โฉ ๐พ, which
can be a point (the 3 planes are concurrent), the empty set (the line of intersection
of two planes is || with the third) or line ๐ (the 3 planes which pass through ๐ are
secant).
Solution to Problem 130.
To determine ๐, we proceed as follows:
1. We construct plane (๐ด๐1) and we look for the line of intersection with ๐ผ1, ๐1.
If ๐1 (/โ), โ neither does ๐.
2. We construct plane (๐ต๐2) and we look for the line of intersection with ๐ผ, ๐2โฒ.
If ๐2โฒ does not exist, neither does ๐.
3. We look for the point of intersection of lines ๐1โฒ and ๐2โฒ. The problem has
only one solution if the lines are concurrent, an infinity if they are coinciding
lines and no solution if they are parallel.
Solution to Problem 131.
We assume the problem is solved.
a. First we assume that ๐ด. ๐ต, ๐ถ are collinear. As ๐ด๐ดโฒ and ๐ต๐ตโฒ are concurrent lines,
they determine a plane ๐ฝ, that intersects ๐ผ after line ๐ดโฒ๐ตโฒ.
As
and points ๐ถ, ๐ด, ๐ตโฒ are collinear (โ)๐ โ ๐.
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b. We assume that ๐ด, ๐ต, ๐ถ are not collinear.
We notice that: (๐ด๐ดโฒ, ๐ต๐ตโฒ) = ๐ฝ (plane determined by 2 concurrent lines).
๐ฝ โช ๐ผ = ๐โฒ and ๐ถ โ ๐โฒ.
To determine ๐ we proceed as follows:
1) We determine plane (๐ด๐ต๐ถ);
2) We look for the point of intersection of this plane with line ๐, so ๐ โฉ
(๐ด๐ต๐ถ) = {๐} is the desired point.
Then (๐ด๐ต๐ถ) โฉ ๐ผ = ๐โฒ.
โน ๐ดโฒ,๐ตโฒ, ๐ถโฒ are collinear.
Solution to Problem 132.
๐ด โ ๐ and ๐ต โ ๐ โน [๐ด๐ต] โฉ ๐ผ โ โ .
Let ๐ = |๐ผ๐ด = |๐ผ๐ต.
Let ๐ โ |๐ด๐ต| and we must show that ๐ โ ๐(โ)๐ inside the segment.
We assume the contrary that ๐ โ ๐ โน (โ)๐ such that [๐ด๐] โฉ ๐ = {๐} โน ๐ โ
[๐ด๐] โน ๐ โ [๐ด๐ต] โน [๐ด๐ต] โฉ ๐ผ โ โ false.
๐ โ ๐ผ, so ๐ โ ๐. The property is also maintained for the closed half-space.
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Compared to the previous case there can appear the situation when one of the
points ๐ด and ๐ต โ ๐ผ or when both belong to ๐ผ.
If ๐ด โ ๐ผ, ๐ต โ ๐, |๐ด๐ต| โฉ ๐ผ โ โ and we show as we did above that:
If:
Solution to Problem 133.
Let
So
Solution to Problem 134.
Let ๐ผ be a plane and ๐1, ๐2 the two half-spaces that it determines. We consider
half-space ๐1.
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๐ determines on ๐ two rays, |๐๐ด and |๐๐ต where
๐ด and ๐ต are in different half-spaces.
We assume
Solution to Problem 135.
Let ๐ be an open half-space and ๐ its margin and let ๐ = ๐ผ โฉ ๐ฝ.
We choose points ๐ด and ๐ต โ ๐ผ โ ๐, on both sides of line ๐ โน
โน ๐ด,๐ต are on one side and on the other side of ๐ฝ and it means that only one of
them is on ๐.
We assume that ๐ด โ ๐ โน ๐ต โ ๐. We now prove ๐ผ โฉ ๐ = |๐๐ด.
๐ผ โฉ ๐ โ |๐๐ด
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Let
[๐๐ต] โ ๐ โ โ โน ๐ and ๐ต are on one side and on the other side of line ๐ โน ๐ is
on the same side of line ๐ with ๐ด โน ๐ โ |๐๐ด
โน๐ โ ๐ โฉ ๐, so |๐๐ด โ ๐ โฉ ๐.
Solution to Problem 136.
Let ฯ be the considered half-space and ฮฒ its margin. There are more possible
cases:
In this case it is possible that:
Let
is a half-plane according to a previous problem.
Solution to Problem 137.
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The intersection of two planes is a line and it cuts only two sides of a triangle.
There are more possible cases:
1. ๐ cuts |๐ด๐ต| and |๐ต๐ถ|
๐โฒ cuts |๐ด๐ต| and |๐ด๐ท|, ๐ผ cuts |๐ด๐ท| so it has a point in common with (๐ด๐ท๐ถ) and let
(๐ด๐ท๐ถ) โฉ ๐ผ = ๐โฒโฒ.
๐โฒโฒ cuts |๐ด๐ท| and does not cut |๐ด๐ถ| โน ๐โฒโฒ cuts |๐ท๐ถ|
๐ผ cuts |๐ท๐ถ| and |๐ต๐ถ| โน it does not cut |๐ต๐ท|. In this case ๐ผ cuts 4 of the 6
segments (the underlined ones).
2. ๐ cuts |๐ด๐ต| and |๐ด๐ถ|, it does not cut |๐ต๐ท|
๐โฒ cuts |๐ด๐ต| an |๐ด๐ท|, it does not cut |๐ต๐ท|
๐โฒโฒ cuts |๐ด๐ท| an |๐ด๐ถ|, it does not cut |๐ท๐ถ|
โน ๐ผ does not intersect plane (๐ต๐ถ๐ท). In this case ๐ผ intersects only 3 of the 6
segments.
3. ๐ cuts |๐ด๐ต| and |๐ต๐ถ|, it does not cut |๐ด๐ถ|
๐โฒ cuts |๐ด๐ต| an |๐ต๐ท|, it does not cut |๐ท๐ถ|
๐ผ intersects |๐ต๐ท| and |๐ต๐ถ|, so it does not cut |๐ท๐ถ|
In โ๐ต๐ท๐ถ โน ๐ผ does not intersect plane (๐ด๐ท๐ถ)
In this case ๐ผ intersects only three segments.
4. ๐ cuts |๐ด๐ต| and |๐ด๐ถ|, it does not cut |๐ต๐ถ|
๐โฒ cuts |๐ด๐ต| an |๐ต๐ท|, it does not cut |๐ด๐ท|
๐โฒโฒ cuts |๐ด๐ถ| an |๐ท๐ถ|
๐ผ does not cut |๐ต๐ถ| in triangle ๐ต๐ท๐ถ. So ๐ผ intersects 4 or 3 segments.
Solution to Problem 138.
Let
Let
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Solution to Problem 139.
We first assume that ๐ผ โ ๐ฝ and |๐ผ๐ด โ |๐ฝ๐ต.
As
The hypothesis can then be written as ๐ผ โ ๐ฝ and |๐ผ๐ด โ |๐ฝ๐ต. Letโs show that ๐ผ โฉ
๐ฝ = โ . By reductio ad absurdum, we assume that ๐ผ โฉ ๐ฝ โ โ โน (โ)๐ = ๐ผ โฉ ๐ฝ and let
๐ โ ๐, so ๐ โ ๐ผ and ๐ โ ๐ฝ. We draw through ๐ด and ๐ a plane ๐, such that ๐ โ ๐, so
the three planes ๐ผ, ๐ฝ and ๐ do not pass through this line. As ๐ has the common
point ๐ with ๐ผ and ๐ฝ, it is going to intersect these planes.
which is a common point of the 3 planes. Lines ๐ฟ and ๐ฟโฒ determine 4 angles in
plane ๐, having ๐ as a common peak, ๐ด โ the interior of one of them, let ๐ด โ int.
โ๏ฟฝ๏ฟฝ. We consider ๐ถ โ int. โ๏ฟฝ๏ฟฝ.
Then ๐ถ is on the same side with ๐ด in relation to ๐ฟโฒ, so ๐ถ is on the same side with
๐ด in relation to ๐ผ โน ๐ถ โ |๐ผ๐ด.
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But ๐ถ is on the opposite side of ๐ด in relation to ๐ฟ, so ๐ถ is on the opposite side of
๐ด in relation to ๐ฝ โน ๐ถ โ |๐ฝ๐ด. So |๐ผ๐ด โ |๐ฝ๐ด โ false โ it contradicts the hypothesis โน
So ๐ผ โฉ ๐ฝ = โ .
Solution to Problem 140.
Let ๐ be the edge of the given dihedral angle. Depending on the position of a line
in relation to a plane, there can be identified the following situations:
The ray with its origin in ๐, so ๐ผ โ ๐ฝโฒ๐พโฒ = ๐โฒ๐โฒโฒ thus an angle.
Indeed, if we assumed that ๐โฒ โฉ ๐โฒโฒ โ โ โน (โ)๐ โ ๐โฒ โฉ ๐โฒโฒ.
false โ it contradicts the hypothesis.
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Florentin Smarandache
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Or
in this case ๐ผ โฉ ๐ฝโฒ๐พ โฒ = ๐โฒโฒ - a line.
Then ๐ผ โฉ ๐ฝโฒ๐พ โฒ = โ .
๐ โฉ ๐ผ = ๐, but ๐ผ โ ๐ฝ, ๐ผ โ ๐พ
๐ผ โฉ ๐ฝโฒ๐พ โฒ = ๐ thus the intersection is a line.
In this case the intersection is a closed half-plane.
Solution to Problem 141.
is a half-plane
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From (*) and (**),
Solution to Problem 142.
โน points ๐ด and ๐ต are on different sides of (๐๐).
Solution to Problem 143.
is a half-plane
so they are secant planes
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Let rays
As ๐ is interior to the dihedron formed by any half-plane passing through ๐ of the
trihedral, so
So
๐ and ๐ in the same half-plane det. ๐๐ด โ ๐ and ๐ on the same side of ๐๐ด (1)
โน ๐ and ๐ด are on the same side of (๐๐ต๐ถ) โฉ ๐พโฒ โน ๐ and ๐ด are on the same side of
๐๐ (2).
From (1) and (2) โน
Solution to Problem 144.
Let ๐ and ๐โฒโฒbe two convex sets and ๐ โฉ๐โฒ their intersection. Let
so the intersection is convex.
Solution to Problem 145.
a. Let ๐, ๐ โ ๐ผ; ๐ โ ๐ โน |๐๐ = ๐๐ (the line is a convex set)
๐๐ โ ๐ผ, so |๐๐| โ ๐ผ, so the plane is a convex set.
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b. Half-planes: Let ๐ = |๐๐ด and ๐, ๐ โ ๐ โน |๐๐| โฉ ๐ = โ . Let ๐ โ |๐๐| โน |๐๐| โ
|๐๐| โน |๐๐| โฉ ๐ = โ โน ๐ and ๐ are in the same half-plane โน๐ โ ๐. So
|๐๐| โ ๐ and ๐ is a convex set.
Let ๐โฒ = [๐๐ด. There are three situations:
1) ๐, ๐ โ |๐๐ด โ previously discussed;
2) ๐, ๐ โ ๐ โน |๐๐| โ ๐ โ ๐โฒ;
3) ๐ โ ๐, ๐ โ ๐ โน |๐๐| โ |๐๐ โน |๐๐| โ |๐๐ด โ [๐๐ด so [๐๐ด is a convex set.
c. Half-spaces: Let ๐ = |๐ผ๐ด and let ๐, ๐ โ ๐ โน |๐๐| โฉ ๐ผ = โ .
Let ๐ โ |๐๐| โน |๐๐| โ |๐๐| โน |๐๐| โฉ ๐ผ = โ .
Let ๐โฒ = [๐ผ๐ด. There are three situations:
1) ๐, ๐ โ |๐ผ๐ด previously discussed;
2) ๐, ๐ โ ๐ผ โน |๐๐| โ ๐ผ โ ๐โฒ;
3) ๐ โ ๐ผ, ๐ โ ๐ผ.
and so ๐โฒ is a convex set.
d. the interior of a dihedral angle:
๐๐๐ก. ๐ผโฒ๐ฝโฒ = |๐ผ๐ด โฉ |๐ฝ๐ต and as each half-space is a convex set and their intersection
is the convex set.
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Solution to Problem 146.
No. The dihedral angle is not a convex set, because if we consider it as in the
previous figure ๐ด โ ๐ฝโฒ and ๐ต โ ๐ผโฒ.
Only in the case of the null or straight angle, when the dihedral angle
becomes a plane or closed half-plane, is a convex set.
Solution to Problem 147.
a. No. The trihedral angle is not the convex set, because, if we take ๐ด โ ๐ and ๐ โ
the int. ๐๏ฟฝ๏ฟฝ determined by ๐ โ the int. ๐๐๏ฟฝ๏ฟฝ, (โ)๐ such that |๐๐ โฉ the int. ๐ด๐ต๐ถ =
{๐ }, ๐ โ |๐ด๐|, ๐ โ ๐๐๏ฟฝ๏ฟฝ.
So ๐ด, ๐ โ ๐๐๏ฟฝ๏ฟฝ, but |๐ด๐| โ ๐๐๐.
b. ๐ฃ = (๐๐ถ๐ด), ๐พ = (๐๐ด๐ต) is a convex set as an intersection of convex sets.
C) It is the same set from a. and it is not convex.
D) The respective set is [๐ผ๐ด โฉ [๐ฝ๐ต โฉ [๐พ๐ถ, intersection of convex sets and, thus, it
is convex.
Solution to Problem 148.
Let ๐ = |๐ผ๐ด and ๐ โ ๐ผ. Let ๐, ๐ โ ๐ โฉ ๐.
We have the following situations:
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Solution to Problem 149.
Solution to Problem 150.
Let
In plane (๐๐๐), let ๐ โ (๐๐).
Solution to Problem 151.
Let: ๐ midpoint of |๐ด๐ต|
๐ midpoint of |๐ต๐ถ|
๐ midpoint of |๐ท๐ถ|
๐ midpoint of |๐ด๐ท|
๐ midpoint of |๐ต๐ท|
๐ midpoint of |๐ด๐ถ|
or
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In triangle ABC:
In triangle DAC:
โน parallelogram โน |๐๐| and |๐๐ | intersect at their midpoint ๐.
โน ๐๐๐ ๐ parallelogram.
โน |๐๐| passes through midpoint ๐ of |๐๐ |.
Thus the three lines ๐๐ , ๐๐ , ๐๐ are concurrent in ๐.
Solution to Problem 152.
Let tetrahedron ๐ด๐ต๐ถ๐ท and ๐ธ be the midpoint of |๐ถ๐ท|. The centroid ๐บ of the face
๐ด๐ถ๐ท is on |๐ด๐ธ| at a third from the base. The centroid ๐บโฒ of the face ๐ต๐ถ๐ท is on |๐ต๐ธ|
at a third from the base |๐ถ๐ท|.
We separately consider โ๐ด๐ธ๐ต. Let ๐น be the midpoint of ๐ด๐ต, so ๐ธ๐น is median in
this triangle and, in the previous problem, it was one of the 3 concurrent segments
in a point located in the middle of each.
Let ๐ be the midpoint of |๐ธ๐น|. We write ๐ด๐ โฉ ๐ธ๐ต = {๐บโฒ} and ๐ต๐บ โฉ ๐ธ๐ด = {๐บ}.
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From (1) and (2)
โน ๐บโฒ is exactly the centroid of face ๐ต๐ถ๐ท, because it is situated on median |๐ธ๐ต| at
a third from ๐ธ. We show in the same way that ๐บ is exactly the centroid of face ๐ด๐ถ๐ท.
Weโve thus shown that ๐ต๐บ and ๐ด๐บโฒ pass through point ๐ from the previous
problem.
We choose faces ๐ด๐ถ๐ท and ๐ด๐ถ๐ต and mark by ๐บโฒโฒ the centroid of face ๐ด๐ถ๐ต, we
show in the same way that ๐ต๐บ and ๐ท๐บโฒโฒ pass through the middle of the segment
|๐๐| (|๐ด๐| โก |๐๐ถ|, |๐ต๐| โก |๐๐ท|) thus also through point ๐, etc.
Solution to Problem 153.
We mark planes (๐ด๐ต๐ถ) = ๐ผ, (๐ด๐ท๐ถ) = ๐ฝ, (๐ต๐ท๐ถ) = ๐พ, (๐ด๐ต๐) = ๐ฟ.
Let ๐ be the intersection of the interiors of the 4 trihedral angles.
We show that:
๐ = int. [๐ด๐ต๐ถ๐ท], by double inclusion.
1. ๐ โ ๐ โน ๐ โ int. ๐๐๏ฟฝ๏ฟฝ โฉ int. ๐๐๏ฟฝ๏ฟฝ โฉ int. ๐๐๏ฟฝ๏ฟฝ โฉ int. ๐๐๏ฟฝ๏ฟฝ โน ๐ โ |๐ผ๐ท โฉ |๐พ๐ถ โฉ ๐ฝ๐ต and
๐ โ |๐ฟ๐ด โฉ |๐พ๐ถ โฉ |๐ฝ๐ต โน ๐ โ |๐ผ๐ท and ๐ โ |๐ฝ๐ต and ๐ โ |๐พ๐ถ and ๐ โ |๐ฟ๐ถ โน ๐ โ
|๐ผ๐ท โฉ |๐พ๐ถ โฉ ๐ฝ๐ต โฉ ๐ฟ๐ด โน ๐ โ int. [๐ด๐ต๐ถ๐ท]. So ๐ โ [๐ด๐ต๐ถ๐ท].
2. Following the inverse reasoning we show that [๐ด๐ต๐ถ๐ท] โ ๐ from where the
equality.
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Solution to Problem 154.
such that
such that ๐ โ |๐ถ๐|, (๐ด๐ท๐ต) โฉ (๐ท๐๐ถ) = ๐ท๐.
From ๐ โ |๐ถ๐| and โ |๐ท๐| ๐๐๐๐๐โ int. ๐ท๐๐ถ โน ๐ โ int. ๐ท๐๐ถ โน (โ)๐ โ |๐ท๐ถ|.
So we showed that (โ)๐ โ |๐ด๐ต| and ๐ โ |๐ท๐ถ| such that ๐ โ |๐๐|.
Solution to Problem 155.
Let โณ be the union of the open segments |๐๐|. We must prove that: int. [๐ด๐ต๐ถ๐ท] =
โณ through double inclusion.
1. Let ๐ โ int. [๐ด๐ต๐ถ๐ท] โน (โ)๐ โ |๐ด๐ต and ๐ โ |๐ถ๐ท such that ๐ โ |๐๐| โน ๐ โ โณ so
int. [๐ด๐ต๐ถ๐ท] โ โณ.
2. Let ๐ โโณ โน (โ)๐ โ |๐ด๐ต| and ๐ โ |๐ถ๐ท| such that ๐ โ |๐๐|. Points ๐ท, ๐ถ and ๐
determine plane (๐๐ท๐ถ) and (๐๐ท๐ถ) โฉ (๐ด๐ถ๐ต) = ๐๐ถ, (๐๐๐ถ) โฉ (๐ด๐ท๐ต) = ๐๐ท.
As (โ)๐ โ |๐ถ๐ท| such that ๐ โ |๐๐| โน ๐ โ [๐๐ถ๐ท] โน |(โ)๐ โ |๐๐ถ| such that ๐ โ |๐ท๐ |.
If ๐ โ |๐ด๐ต| and ๐ โ |๐๐ถ| โน ๐ โ int. ๐ด๐ถ๐ต such that ๐ โ |๐ท๐ | โน ๐ int. [๐ด๐ต๐ถ๐ท] โน
โณ โ int. [๐ด๐ต๐ถ๐ท].
Working with closed segments we obtain that (โ)๐ โ [๐ด๐ถ๐ต] such that ๐ โ [๐ท๐ ],
thus obtaining tetrahedron [๐ด๐ต๐ถ๐ท].
Solution to Problem 156.
Let ๐ โ [๐ด๐ต๐ถ๐ท] โน (โ) ๐ โ [๐ด๐ต๐ถ] such that ๐ โ [๐ท๐].
Let ๐ โ [๐ด๐ต๐ถ๐ท] โน (โ) ๐ โ [๐ด๐ต๐ถ] such that ๐ โ [๐ท๐].
The concurrent lines ๐ท๐ and ๐ท๐ determine angle ๐ท๐๐.
The surface of triangle ๐ท๐๐ is a convex set.
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Let
such that ๐ โ [๐ท๐ ]. But [๐๐] โ [๐ด๐ต๐ถ] because ๐ โ [๐ด๐ต๐ถ] โฉ ๐ โ [๐ด๐ต๐ถ] and the
surface of the triangle is convex. So (โ)๐ โ [๐ด๐ต๐ถ] such that
and the tetrahedron is a convex set.
Note: The tetrahedron can be regarded as the intersection of four closed half-
spaces which are convex sets.
Solution to Problem 157.
Let โณ be the union of the segments [๐๐] with ๐ โ โณ1 and ๐ โ โณ2.
Let ๐ฅ, ๐ฅโ โ โณ โน (โ)๐ โ โณ1 and ๐ โ โณ2 such that ๐ฅ โ [๐๐];
(โ)๐โฒ โ โณ1 and ๐โฒ โ โณ2 such that ๐ฅโฒ โ [๐โฒ๐โฒ].
From ๐, ๐โฒ โ โณ1 โน [๐๐โฒ]โฒ โ โณ1 which is a convex set.
From ๐,๐โฒ โ โณ2โน [๐๐โฒ] โ โณ2 which is a convex set.
The union of all the segments [๐๐] with ๐ โ [๐๐โฒ] and ๐ โ [๐๐โฒ] is tetrahedron
[๐๐โฒ๐๐โฒ] โ โณ.
So from ๐ฅ, ๐ฅโฒ โ โณ โน |๐ฅ๐ฅโ| โ โณ, so set โณ is convex.
Solution to Problem 158.
The interior of the tetrahedron coincides with the union of segments |๐๐|, ๐ โ
|๐ด๐ต| and ๐ โ |๐ถ๐ท|, that is int. [๐ด๐ต๐ถ๐ท] = {|๐๐| โ ๐ โ |๐ด๐ต|, ๐ โ |๐ถ๐ท|}.
Letโs show that:
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1. Let
2. Let
If we assume ๐ โ |๐ท๐| โน |๐ท๐| โฉ (๐ด๐ต๐ถ) โ โ โน ๐ and ๐ท are in different half-
spaces in relation to (๐ด๐ต๐ถ) โน ๐ โ (๐ด๐ต๐ถ), ๐ท, false (it contradicts the hypothesis).
So
and the second inclusion is proved.
As regarding the tetrahedron: [๐ด๐ต๐ถ๐ท] = {[๐๐] โ ๐ โ [๐ด๐ต] and ๐ โ [๐ถ๐ท]}.
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If
๐ = ๐ด,๐ โ [๐ถ๐ท], [๐๐] describes face [๐ด๐ท๐ถ]
๐ = ๐ต,๐ โ [๐ถ๐ท], [๐๐] describes face [๐ต๐ท๐ถ]
๐ = ๐ถ, ๐ โ [๐ด๐ต], [๐๐] describes face [๐ด๐ต๐ถ].
Because the triangular surfaces are convex sets and along with their two points ๐,
๐, segment [๐๐] is included in the respective surface.
So, if we add these two situations to the equality from the previous case, we
obtain:
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Lines and Planes
159. Let ๐, ๐โฒ be two parallel lines. If the line ๐ is parallel to a plane ๐ผ, show
that ๐โฒ||๐ผ or ๐โฒ โ ๐ผ.
Solution to Problem 159
160. Consider a line ๐, parallel to the planes ๐ผ and ๐ฝ, which intersects after the
line ๐. Show that ๐โ๐.
Solution to Problem 160
161. Through a given line ๐, draw a parallel plane with another given line ๐โฒ.
Discuss the number of solutions.
Solution to Problem 161
162. Determine the union of the lines intersecting a given line ๐ and parallel
to another given line ๐โฒ (๐ โฆ ๐โฒ).
Solution to Problem 162
163. Construct a line that meets two given lines and that is parallel to a third
given line. Discuss.
Solution to Problem 163
164. If a plane ๐ผ intersects the secant planes after parallel lines, then ๐ผ is
parallel to line ๐ฝ โฉ ๐พ.
Solution to Problem 164
165. A variable plane cuts two parallel lines in points ๐ and ๐. Find the
geometrical locus of the middle of segment [๐๐].
Solution to Problem 165
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166. Two lines are given. Through a given point, draw a parallel plane with
both lines. Discuss.
Solution to Problem 166
167. Construct a line passing through a given point, which is parallel to a
given plane and intersects a given line. Discuss.
Solution to Problem 167
168. Show that if triangles ๐ด๐ต๐ถ and ๐ดโฒ๐ตโฒ๐ถโฒ, located in different planes, have
๐ด๐ต โฅ ๐ดโฒ๐ตโฒ, ๐ด๐ถ โฅ ๐ดโฒ๐ถโฒ and ๐ต๐ถ โฅ ๐ตโฒ๐ถโฒ, then lines ๐ด๐ดโฒ, ๐ต๐ตโฒ, ๐ถ๐ถโฒ are concurrent or
parallel.
Solution to Problem 168
169. Show that, if two planes are parallel, then a plane intersecting one of
them after a line cuts the other one too.
Solution to Problem 169
170. Through the parallel lines ๐ and ๐โฒ we draw the planes ๐ผ and ๐ผโฒ distinct
from (๐, ๐โฒ). Show that ๐ผ โฅ ๐ผโฒ or (๐ผ โฉ ๐ผโฒ) โฅ ๐.
Solution to Problem 170
171. Given a plane ๐ผ, a point ๐ด โ ๐ผ and a line ๐ โ ๐ผ.
a. Construct a line ๐โฒ such that ๐โฒ โ ๐ผ, ๐ด โ ๐โฒ and ๐โฒ โฅ ๐.
b. Construct a line through ๐ด included in ๐ผ, which forms with ๐ an angle
of a given measure ๐. How many solutions are there?
Solution to Problem 171
172. Show that relation ๐ผ โฅ ๐ฝ defined on the set of planes is an equivalence
relation. Define the equivalence classes.
Solution to Problem 172
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173. Consider on the set of all lines and planes the relation โ๐ฅ โฅ ๐ฆโ or ๐ฅ = ๐ฆ,
where ๐ฅ and ๐ฆ are lines or planes. Have we defined an equivalence
relation?
Solution to Problem 173
174. Show that two parallel segments between parallel planes are concurrent.
Solution to Problem 174
175. Show that through two lines that are not contained in the same plane, we
can draw parallel planes in a unique way. Study also the situation when the
two lines are coplanar.
Solution to Problem 175
176. Let ๐ผ and ๐ฝ be two parallel planes, ๐ด, ๐ต โ ๐ผ, and ๐ถ๐ท is a parallel line with
๐ผ and ๐ฝ. Lines ๐ถ๐ด, ๐ถ๐ต, ๐ท๐ต, ๐ท๐ด cut plane ๐ฝ respectively in ๐,๐, ๐, ๐. Show
that these points are the vertices of a parallelogram.
Solution to Problem 176
177. Find the locus of the midpoints of the segments that have their
extremities in two parallel planes.
Solution to Problem 177
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Solutions
Solution to Problem 159.
Let
๐ด โ ๐ผ๐ โฅ ๐ผ ๐โฒโฒ โฅ ๐
} โน ๐โฒโฒ โฅ ๐ผ
๐โฒ โฅ ๐๐โฒโฒ โฅ ๐
} โน ๐โฒ โฅ ๐โฒโฒ}
โน ๐โฒโฒ โฅ ๐ผ or ๐โฒ โ ๐ผ.
Solution to Problem 160.
Let ๐ด โ ๐ โ ๐ด โ ๐ผ โฉ ๐ด โ ๐ฝ. We draw through A, ๐โฒ โฅ ๐.
๐ด โ ๐ผ,๐ โฅ ๐ผ๐โฒ โฅ ๐
} โน ๐โฒ โ ๐ผ
๐ด โ ๐ฝ,๐ โฅ ๐ฝ
๐โฒ โฅ ๐} โน ๐โฒ โ ๐ฝ
}โน๐โฒ โ ๐ผ โ ๐ฝ๐ผ โฉ ๐ฝ = ๐
} โน๐โฒ = ๐๐โฒ โฅ ๐
} โน ๐ โฅ ๐
Solution to Problem 161.
a. If ๐ โฆ ๐โฒ there is only one solution and it can be obtained as it follows:
Let ๐ด โ ๐. In the plane (๐ด, ๐โฒโฒ) we draw ๐โฒโฒ โฅ ๐โฒ. The concurrent lines ๐ and ๐โฒโฒ
determine plane ๐. As ๐โฒโฒ โฅ ๐ โน ๐ โฅ ๐ผ, in the case of the non-coplanar lines.
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b. If ๐ โฅ ๐โฒ || d'', (โ) infinite solutions. Any plane passing through ๐ is parallel to
๐โฒโฒ, with the exception of plane (๐, ๐โฒ).
c. ๐ โฆ ๐โฒ, but they are coplanar (โ) solutions.
Solution to Problem 162.
Let ๐ด โ ๐, we draw through ๐ด, ๐1 โฅ ๐โฒ. We write ๐ผ = (๐, ๐1). As ๐1 โฅ ๐โฒ โน ๐โฒ โฅ ๐ผ.
Let ๐ โ ๐, arbitrary โน๐ โ ๐ผ.
We draw ๐ฟ โฅ ๐ฟโฒ,๐ โ ๐ฟ๐โฒ โฅ ๐
} โน ๐ฟ โ ๐ผ, so all the parallel lines to ๐โฒ intersecting ๐ are
contained in plane ๐ผ.
Let ๐พ โ ๐ผ, ๐พ โฅ ๐โฒโน ๐พ โฉ ๐ = ๐ต, so (โ) parallel to ๐โฒ from ๐ผ intersects ๐. Thus, the
plane ๐ผ represents the required union.
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Solution to Problem 163.
We draw ๐ through ๐ such that ๐ โฅ ๐โฒ๐โฒ โฅ ๐3
} โน ๐ โฅ ๐3. According to previous
problem: ๐ โฉ ๐1 = {๐}. Therefore,
a. If ๐3 โฅ ๐1, the plane ๐ผ is unique, and if ๐2 โฉ ๐ผ โ โ , the solution is unique.
b. If ๐1 โฅ ๐3, (โ) ๐ โฅ ๐1
๐ โฉ ๐1 โ โ , because it would mean that we can draw through a
point two parallel lines ๐, ๐1 to the same line ๐3. So there is no solution.
c. If ๐1 โฆ ๐3 and ๐2 โฉ ๐ผ โ โ , all the parallel lines to ๐2 cutting ๐1 are on the plane ๐ผ
and none of them can intersect ๐2, so the problem has no solution.
d. If ๐2 โ ๐ผ, ๐1 โฉ ๐2 โ โ , let ๐1 โฉ ๐2 = {๐}, and the required line is parallel to ๐3
drawn through ๐ โน one solution.
e. If ๐2 โ ๐ผ, ๐1 โฅ ๐2. The problem has infinite solutions, (โ) || to ๐3 which cuts ๐1,
also cuts ๐2.
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Solution to Problem 164.
Solution to Problem 165.
The problem is reduced to the geometrical locus of the midpoints of the
segments that have extremities on two parallel lines. ๐ is such a point |๐๐| = |๐๐|.
We draw ๐ด๐ต โฅ ๐1 โ ๐ด๐ต?๐๐๏ฟฝ๏ฟฝ = ๐ต๐๐ โน ๐ฅ๐๐ด๐ = ๐ฅ๐๐ต๐ โน |๐๐ด| โก |๐๐ต| โน ||๐ด๐||?โน
the geometrical locus is the parallel to ๐1 and ๐2 drawn on the mid-distance
between them. It can also be proved vice-versa.
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Solution to Problem 166.
Let ๐1 โฆ ๐2. In plane (๐,๐) we draw ๐1โฒ โฅ ๐1, ๐ โ ๐1
โฒ . In plane (๐2๐) we draw
๐2โฒ โฅ ๐2, ๐ โ ๐2
โฒ . We note ๐ผ = ๐1โฒ๐2โฒ the plane determined by two concurrent lines.
๐ โ ๐ผ the only solution.
Let ๐1 โฅ ๐2, ๐ โ ๐1,๐ โ ๐2.
๐1 = ๐1โฒ = ๐2
โฒ
In this case ๐1โฒ = ๐2
โฒ = ๐ and infinite planes pass through ๐;
๐2 โฅ ๐๐1 โฅ ๐
} โน ๐1, ๐2 are parallel lines with (โ) of the planes passing through ๐.
The problem has infinite solutions. But ๐ โ ๐1 or ๐ โ ๐2, the problem has no
solution because the plane canโt pass through a point of a line and be parallel to
that line.
Solution to Problem 167.
Let ๐ด be the given point, ๐ผ the given plane and ๐ the given line.
a. We assume that ๐ โฆ ๐ผ, ๐ โฉ ๐ผ = {๐}. Let plane (๐๐ด) which has a common point
๐ with ๐ผ โ (๐๐ด) โฉ ๐ผ = ๐โฒ.
We draw in plane (๐๐ด) through point ๐ด a parallel line to ๐โฒ.
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โน ๐ is the required line.
b. ๐ โฅ ๐ผ, (๐๐ด) โฉ ๐ผ โ โ .
Let (dA) โฉ ฮฑ = ฮฑโ
๐ โฅ ๐ผ} โ ๐โฒ โฅ ๐
All the lines passing through ๐ด and intersecting ๐ are contained in plane (๐๐ด). But
all these lines also cut ๐โฒ โฅ ๐, so they canโt be parallel to ๐ผ. There is no solution.
c. ๐ โฅ ๐ผ, (๐๐ด) โฉ ๐ผ = โ .
Let ๐ โ ๐ and line ๐ด๐ โ (๐๐ด); (๐๐ด) โฉ ๐ผ = โ โน ๐ด๐ โฉ ๐ผ = โ โน ๐ด๐ โฅ ๐ผ, (โ)๐ โ ๐.
The problem has infinite solutions.
Solution to Problem 168.
(๐ด๐ต๐ถ) and (๐ดโฒ๐ตโฒ๐ถโฒ) are distinct planes, thus the six points ๐ด, ๐ต, ๐ถ, ๐ดโฒ, ๐ตโฒ, ๐ถโฒ canโt be
coplanar.
๐ด๐ต โฅ ๐ดโฒ๐ตโ โน ๐ด,๐ต, ๐ดโฒ, ๐ตโฒ are coplanar.
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The points are coplanar four by four, that is (๐ด๐ต๐ตโฒ๐ด), (๐ด๐ถ๐ถโฒ๐ดโฒ), (๐ต๐ถ๐ถโฒ๐ตโฒ), and
determine four distinct planes. If we assumed that the planes coincide two by two,
it would result other 6 coplanar points and this is false.
In plane ๐ด๐ต๐ตโฒ๐ดโฒ, lines ๐ด๐ดโฒ, ๐ต๐ตโฒ can be parallel or concurrent.
First we assume that:
is a common point to the 3 distinct planes, but the intersection of 3 distinct
planes can be only a point, a line or โ . It canโt be a line because lines
โน are distinct if we assumed that two of them coincide, the 6 points would be
coplanar, thus there is no common line to all the three planes. There is one
possibility left, that is they have a common point ๐ and from
We assume ๐ โฉ ๐ฝ = โ โน ๐ โฅ ๐ฝ โน ๐ โ plane โฅ ๐ฝ drawn through ๐ด โน ๐ โ ๐ผ, false.
So ๐ โฉ ๐ฝ = {๐ต}.
Solution to Problem 169.
Hypothesis: ๐ผ โฅ ๐ฝ, ๐พ โฉ ๐ผ = ๐1.
Conclusion: ๐พ โฉ ๐ฝ = ๐2.
We assume that ๐พ โฉ ๐ฝ = โ โน ๐พ โฅ ๐ฝ.
Let
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because from a point we can draw only one parallel plane with the given plane.
But this result is false, it contradicts the hypothesis ๐พ โฉ ๐ผ = ๐1 so ๐พ โฉ ๐ฝ = ๐2.
Solution to Problem 170.
Hypothesis: ๐ โฅ ๐โฒ; ๐ โ ๐ผ; ๐โฒ โ ๐ผโฒ; ๐ผ, ๐โฒ โ (๐๐โฒ).
Conclusion: ๐ผ โฅ ๐ผโฒ or ๐โฒโฒ โฅ ๐.
As ๐ผ, ๐โฒ โ (๐๐โฒ) โน ๐ผ โ ๐โฒ.
If ๐ โฉ ๐โฒ = โ โน ๐ โฅ ๐โฒ.
If ๐ โฉ ๐โฒ = โ โน ๐ โฉ ๐โฒ = ๐โฒโฒ.
If ๐ โฅ ๐โฒโฒ
๐โฒ โ ๐โฒโน๐ โฅ ๐โฒ
๐ โ ๐} โน ๐โฒโฒโ๐.
Solution to Problem 171.
a. If ๐ด โ ๐, then ๐โฒ = ๐. If ๐ด โ ๐, we draw through ๐ด, ๐โฒ โฅ ๐.
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b. We draw ๐1 โ ๐ผ, ๐ด โ ๐, such that ๐(๐1๐โฒ) = ๐ and ๐ โ ๐ผ, ๐ด โ ๐2, such that
๐(๐2๐โฒ) = ๐, a line in each half-plane determined by ๐โฒ. So (โ) 2 solutions
excepting the situation ๐ = 0 or ๐ = 90 when (โ) only one solution.
Solution to Problem 172.
๐ผ โฅ ๐ฝ or ๐ผ = ๐ฝ โบ ๐ผ~๐ฝ
1. ๐ผ = ๐ผ โน ๐ผ~๐ผ, the relation is reflexive;
2. ๐ผ~๐ฝ โน ๐ฝ~ โ, the relation is symmetric.
๐ผ โฅ ๐ฝ or ๐ผ = ๐ฝ โน ๐ฝ โฅ ๐ผ or ๐ฝ = ๐ผ โน ๐ฝ~ โ;
3. ๐ผ~๐ฝ โฉ ๐ฝ~๐พ โน ๐ผ~๐พ.
If ๐ผ = ๐ฝ โฉ ๐ฝ~๐พ โน ๐ผ~๐พ.
If ๐ผ โ ๐ฝ and ๐ผ~๐ฝ โน ๐ผ โฅ ๐ฝ๐ฝ~๐พ โน ๐ฝ = ๐พ or ๐ฝ โฅ ๐พ
} โน ๐ผ โฅ ๐พ โน ๐ผ~๐พ.
The equivalence class determined by plane ๐ผ is constructed of planes ๐ผโฒ with ๐ผโฒ~๐ผ,
that is of ๐ผ and all the parallel planes with ๐ผ.
Solution to Problem 173.
No, it is an equivalence relation, because the transitive property is not true. For
example, ๐ฅ is a line, ๐ฆ a plane, ๐ง a line. From ๐ฅ || ๐ฆ and || ๐ง โ ๐ฅ || ๐ง, lines ๐ฅ and ๐ง
could be coplanar and concurrent or non-coplanar.
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Solution to Problem 174.
โน ๐ด๐ต๐ถ๐ท parallelogram.
So ||๐ด๐ถ|| = ||๐ต๐ท||.
Solution to Problem 175.
We consider ๐ด โ ๐ and draw through it ๐1 โฅ ๐โฒ. We consider ๐ต โ ๐โฒ and draw
๐2 โฅ ๐. Plane (๐1๐2) โฅ (๐๐1), because two concurrent lines from the first plane are
parallel with two concurrent lines from the second plane.
When ๐ and ๐โฒ are coplanar, the four lines ๐, ๐1, ๐2 and ๐โฒ are coplanar and the
two planes coincide with the plane of the lines ๐ and ๐โฒ.
Solution to Problem 176.
Let planes:
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From (1), (2), (3), (4) โน๐๐๐๐ parallelogram.
Solution to Problem 177.
Let [๐ด๐ต] and [๐ถ๐ท] be two segments, with ๐ด, ๐ถ โ ๐ผ and ๐ต,๐ท โ ๐ฝ such that |๐ด๐| =
|๐๐ต| and |๐ถ๐| = |๐๐ท|.
In plane (๐๐ถ๐ท) we draw through ๐,๐ธ๐น||๐ถ๐ท โน ๐ธ๐ถ||๐ท๐น โน ๐ธ๐น๐ท๐ถ parallelogram
โน |๐ธ๐น| โก |๐ถ๐ท|.
Concurrent lines ๐ด๐ต and ๐ธ๐น determine a plane which cuts planes ๐ผ after 2 parallel
lines โน ๐ธ๐ด||๐ต๐น.
In this plane, |๐ด๐| โก |๐ต๐|.
๐ธ๐๏ฟฝ๏ฟฝ โก ๐ต๐๐น (angles opposed at peak)
๐ธ๐ด๏ฟฝ๏ฟฝ โก ๐น๐ต๐(alternate interior angles)}
In parallelogram ๐ธ๐ถ๐ท๐น,
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So the segment connecting the midpoints of two of the segments with the
extremity in ๐ผ and ๐ฝ is parallel to these planes. We also consider [๐บ๐ป] with ๐บ โ
๐ผ,๐ป โ ๐ฝ and |๐บ๐| โก |๐๐ป| and we show in the same way that ๐๐||๐ผ and ๐๐||๐ฝ. (2)
From (1) and (2) โน๐,๐, ๐ are elements of a parallel plane to ๐ผ and ๐ฝ, marked by
๐พ.
Vice-versa, letโs show that any point from this plane is the midpoint of a segment,
with its extremities in ๐ผ and ๐ฝ.
Let segment [๐ด๐ต] with ๐ด โ ๐ผ and ๐ต โ ๐ฝ and |๐ด๐| = |๐ต๐|. Through ๐, we draw the
parallel plane with ๐ผ and ๐ฝ and in this plane we consider an arbitrary point ๐ โ ๐พ.
Through ๐ we draw a line such that ๐ โฉ ๐ผ = {๐ผ} and ๐ โฉ ๐ฝ = {๐ผ}.
In plane (๐๐ด๐ต) we draw ๐ดโฒ๐ตโฒ || ๐ด๐ต. Plane (๐ด๐ดโฒ๐ตโฒ๐ต) cuts the three parallel planes
after parallel lines โน
In plane (๐ดโฒ๐ตโฒ๐ผ) โน |๐ดโฒ๐| โก |๐๐ตโฒ| โน ๐ผ๐ดโฒ || ๐ตโฒ๐ผ and thus |๐ดโฒ๐| โก |๐๐ตโฒ|
๐ผ๐๐ดโฒ โก ๐ผ๐๐ตโฒ and ๐ผ๐ดโฒ๏ฟฝ๏ฟฝ โก
๐ผ๐ตโฒ๏ฟฝ๏ฟฝ โน
๐ is the midpoint of a segment with extremities in planes ๐ผ and ๐ฝ.
Thus the geometrical locus is plane ๐พ, parallel to ๐ผ and ๐ฝ and passing through the
mid-distance between ๐ผ and ๐ฝ.
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Projections
178. Show that if lines ๐ and ๐โฒ are parallel, then pr๐ผ๐ โฅ pr๐ผ๐โฒ or pr๐ผ๐ = pr๐ผ๐
โฒ.
What can we say about the projective planes of ๐ and ๐โฒ?
Solution to Problem 178
179. Show that the projection of a parallelogram on a plane is a parallelogram
or a segment.
Solution to Problem 179
180. Knowing that side [๐๐ด of the right angle ๐ด๐๐ต is parallel to a plane ๐ผ,
show that the projection of ๐ด๐๏ฟฝ๏ฟฝ onto the plane ๐ผ is a right angle.
Solution to Problem 180
181. Let ๐ดโฒ๐ตโฒ๐ถโฒ be the projection of โ๐ด๐ต๐ถ onto a plane ๐ผ. Show that the
centroid of โ๐ด๐ต๐ถ is projected onto the centroid of โ๐ดโ๐ตโ๐ถโ. Is an analogous
result true for the orthocenter?
Solution to Problem 181
182. Given the non-coplanar points ๐ด, ๐ต, ๐ถ, ๐ท, determine a plane on which the
points ๐ด, ๐ต, ๐ถ, ๐ท are projected onto the peaks of parallelogram.
Solution to Problem 182
183. Consider all triangles in space that are projected onto a plane ๐ผ after the
same triangle. Find the locus of the centroid.
Solution to Problem 183
184. Let ๐ด be a point that is not on line ๐. Determine a plane ๐ผ such that pr๐ผ๐
passes through pr๐ผ๐ด.
Solution to Problem 184
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185. Determine a plane onto which three given lines to be projected after
concurrent lines.
Solution to Problem 185
186. Let ๐ผ, ๐ฝ be planes that cut each other after a line ๐ and let ๐ be a
perpendicular line to ๐. Show that the projections of line ๐ onto ๐ผ, ๐ฝ are
concurrent.
Solution to Problem 186
187. Consider lines ๐๐ด,๐๐ต, ๐๐ถ โฅ two by two. We know that ||๐๐ด|| = ๐,
||๐๐ต|| = ๐, ||๐๐ถ|| = ๐. Find the measure of the angle of planes (๐ด๐ต๐ถ) and
(๐๐ด๐ต).
Solution to Problem 187
188. A line cuts two perpendicular planes ๐ผ and ๐ฝ at ๐ด and ๐ต. Let ๐ดโฒ and ๐ตโฒ be
the projections of points ๐ด and ๐ต onto line ๐ผ โฉ ๐ฝ.
a. Show that ||๐ด๐ต||ยฒ = ||๐ด๐ดโฒ||ยฒ + ||๐ดโฒ๐ตโฒ||ยฒ + ||๐ตโฒ๐ต||ยฒ;
b. If ๐, ๐, ๐ are the measures of the angles of line ๐ด๐ต with planes ๐ผ, ๐ฝ and
with ๐ผ โฉ ๐ฝ, then cos ๐โ๐ดโฒ๐ตโฒโ
โ๐ด๐ตโ and sin2 ๐ + sin2 ๐ = sin2 ๐.
Solution to Problem 188
189. Let ๐ด๐ต๐ถ be a triangle located in a plane ๐ผ, ๐ดโฒ๐ตโฒ๐ถโฒ the projection of
โ๐ดโฒ๐ตโฒ๐ถโฒ onto plane ๐ผ. We mark with ๐, ๐โฒ, ๐โฒโฒ the areas of โ๐ด๐ต๐ถ,
โ๐ดโฒ๐ตโฒ๐ถโฒ, โ๐ดสบ๐ตสบ๐ถสบ, show that ๐โฒ is proportional mean between ๐ and ๐โฒโฒ.
Solution to Problem 189
190. A trihedral [๐ด๐ต๐ถ๐ท] has |๐ด๐ถ| โก |๐ด๐ท| โก |๐ต๐ถ| โก |๐ต๐ท|. ๐,๐ are the midpoints
of edges [๐ด๐ต], [๐ถ๐ท], show that:
a. ๐๐ โฅ ๐ด๐ต,๐๐ โฅ ๐ถ๐ท, ๐ด๐ต โฅ ๐ถ๐ท
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b. If ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ are the feet of the perpendicular lines drawn to the peaks
๐ด, ๐ต, ๐ถ, ๐ท on the opposite faces of the tetrahedron, points ๐ต, ๐ดโฒ, ๐ are
collinear and so are ๐ด, ๐ตโฒ, ๐; ๐ท, ๐ถโฒ,๐; ๐ถ, ๐ทโฒ, ๐.
c. ๐ด๐ดโฒ, ๐ต๐ตโฒ,๐๐ and ๐ถ๐ถโฒ, ๐ท๐ทโฒ,๐๐ are groups of three concurrent lines.
Solution to Problem 190
191. If rays [๐๐ด and [๐๐ต with their origin in plane ๐ผ, ๐๐ด โฅ ๐ผ, then the two rays
form an acute or an obtuse angle, depending if they are or are not on the
same side of plane ๐ผ.
Solution to Problem 191
192. Show that the 6 mediator planes of the edges of a tetrahedron have a
common point. Through this point pass the perpendicular lines to the faces
of the tetrahedron, drawn through the centers of the circles of these faces.
Solution to Problem 192
193. Let ๐ and ๐โฒ be two non-coplanar lines. Show that (โ) unique points ๐ด โ
๐, ๐ดโฒ โ ๐โฒ such that ๐ด๐ดโฒ โฅ ๐ and ๐ด๐ดโฒ โฅ ๐โฒ. The line ๐ด๐ดโฒ is called the common
perpendicular of lines ๐ and ๐โฒ.
Solution to Problem 193
194. Consider the notations from the previous problem. Let ๐ โ ๐, ๐โฒ โ ๐โฒ.
Show that โ๐ด๐ดโฒโ โค โ๐๐โฒโ. The equality is possible only if ๐ = ๐ด, ๐โฒ = ๐ดโฒ.
Solution to Problem 194
195. Let ๐ด๐ดโฒ be the common โฅ of non-coplanar lines ๐, ๐โฒโฒ and ๐ โ ๐, ๐โฒ โ ๐โฒ
such that |๐ด๐| โก |๐ดโฒ๐โฒ|. Find the locus of the midpoint of segment [๐๐โฒ].
Solution to Problem 195
196. Consider a tetrahedron ๐๐ด๐ต๐ถ with the following properties. ๐ด๐ต๐ถ is an
equilateral triangle of side ๐, (๐ด๐ต๐ถ) โฅ (๐๐ต๐ถ), the planes (๐๐ด๐ถ) and (๐๐ด๐ต)
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form with plane (๐ด๐ต๐ถ) angles of 60ยฐ. Find the distance from point ๐ to
plane (๐ด๐ต๐ถ).
Solution to Problem 196
197. All the edges of a trihedral are of length a. Show that a peak is projected
onto the opposite face in its centroid. Find the measure of the dihedral
angles determined by two faces.
Solution to Problem 197
198. Let ๐ท๐ธ be a perpendicular line to the plane of the square ๐ด๐ต๐ถ๐ท. Knowing
that โ๐ต๐ธโ = ๐ and that the measure of the angle formed by [๐ต๐ธ and (๐ด๐ต๐ถ)
is ๐ฝ, determine the length of segment ๐ด๐ธ and the angle of [๐ด๐ธ with plane
(๐ด๐ต๐ถ).
Solution to Problem 198
199. Line ๐ถ๐ท โฅ plane of the equilateral โ๐ด๐ต๐ถ of side ๐, and [๐ด๐ท and [๐ต๐ท form
with plane (๐ด๐ต๐ถ) angles of measure ๐ฝ. Find the angle of planes (๐ด๐ต๐ถ) and
๐ด๐ต๐ท.
Solution to Problem 199
200. Given plane ๐ผ and โ๐ด๐ต๐ถ, โ๐ดโ๐ตโ๐ถโ that are not on this plane. Determine a
โ๐ท๐ธ๐น, located on ๐ผ such that on one side lines ๐ด๐ท, ๐ต๐ธ, ๐ถ๐น and on the
other side lines ๐ดโฒ๐ท, ๐ตโฒ๐ธ, ๐ถโฒ๐น are concurrent.
Solution to Problem 200
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Solutions
Solution to Problem 178.
Let ๐ โฅ ๐โฒ, ๐ฝ the projective plane of ๐.
We assume that ๐โฒ โ ๐ฝ, which means that is plane ๐, ๐โฒ โฅ ๐ผ, โน the projective
plane of ๐โฒ is ๐ฝโฒ. We want to show that pr๐๐ โฅ pr๐๐โฒ. We assume that pr๐๐ โฉ
pr๐๐โฒ = {๐} โ (โ) ๐ โ ๐ such that pr๐๐ = ๐ and (โ)๐โฒ โ ๐โฒ such that pr๐๐
โฒ = ๐.
โน๐๐ โฅ ๐ผ๐๐โฒ โฅ ๐ผ
} โน in the point ๐ on plane ๐ผ we can draw two distinct perpendicular
lines. False.
If ๐ฝ is the projective plane of ๐ and ๐ฝ of ๐โฒ, then ๐ฝ โฅ ๐ฝโฒ, because if they had a
common point their projections should be elements of pr๐๐ and pr๐๐โฒ, and thus
they wouldnโt be anymore parallel lines.
If ๐โฒ โ ๐ฝ or ๐ โ ๐ฝโฒ, that is (๐, ๐โฒ) โฅ ๐ผ โน ๐ and ๐โฒ have the same projective plane
โน pr๐๐ = pr๐๐โฒ.
Solution to Problem 179.
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We assume that ๐ด๐ต๐ถ๐ท โฅ ๐ผ. Let ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ be the projections of points ๐ด, ๐ต, ๐ถ, ๐ท.
๐ด๐ต โฅ ๐ท๐ถ๐๐1โ ๐ดโฒ๐ตโฒ โฅ ๐ทโฒ๐ถโฒ
๐ด๐ท โฅ ๐ต๐ถ โน ๐ดโฒ๐ทโฒ โฅ ๐ตโฒ๐ทโฒ} โน ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ parallelogram.
If (๐ด๐ต๐ถ๐ท) โฅ ๐ผ โ the projection ๐ดโฒ, ๐ตโฒ, ๐ถโฒ, ๐ทโฒ โ the line (๐ด๐ต๐ถ๐ท) โฉ ๐ผ โน the
projection of the parallelogram is a segment.
Solution to Problem 180.
If ๐๐ด||๐ผ โน proj๐ผ๐๐ด||๐๐ด โน ๐โฒ๐ดโฒ||๐๐ด because (โ) a plane which passes through
๐๐ด cuts the plane ๐ผ after a parallel to ๐๐ด.
โน๐โฒ๐ดโฒ โฅ ๐โฒ๐ตโฒ โน ๐ดโฒ๐โฒ๐ตโฒ is a right angle.
Solution to Problem 181.
In the trapezoid ๐ต๐ถ๐ถโฒ๐ตโฒ (๐ต๐ตโฒ||๐ถ๐ถโฒ),
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โน ๐ดโฒ๐โฒ is a median.
๐๐โฒ โฅ ๐ด๐ดโฒโน๐๐โฒ๐ดโฒ๐ด trapezoid
โ๐ด๐บโ
โ๐บ๐โ= 2, ๐บ๐บโฒ โฅ ๐ด๐ดโฒ
}โ๐ดโฒ๐บโ
โ๐บโฒ๐โฒโ=โ๐ด๐บโ
โ๐บ๐โ= 2
โน๐บโฒ is on median A'M' at 2/3 from the peak and 1/3 from the base.
Generally no, because the right angle ๐ด๐๐ถ should be projected after a
right angle. The same thing is true for another height. This is achieved if the
sides of the โ are parallel to the plane.
Solution to Problem 182.
Let ๐ด, ๐ต, ๐ถ, ๐ท be the 4 non-coplanar points and ๐,๐ midpoints of
segments |๐ด๐ต| and |๐ถ๐ท|.
๐ and ๐ determine a line and let a plane ๐ผ โฅ ๐๐, ๐ and ๐ are projected
in the same point ๐ onto ๐ผ.
โน ๐ดโฒ๐ตโฒ๐ถโฒ๐ทโฒ a parallelogram.
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Solution to Problem 183.
Let ๐ดโฒ๐ตโฒ๐ถโฒ, ๐ดโฒโฒ๐ตโฒโฒ๐ถโฒโฒ two triangles of this type, with the following property:
โน ๐บโฒ๐บ = ๐บ๐บโฒโฒโน ๐บโฒโฒ, ๐บโฒ, ๐บ are collinear.
Due to the fact that by projection the ratio is maintained, we show that ๐บโฒโฒ is the
centroid of ๐ด, ๐ต, ๐ถ.
Solution to Problem 184.
Let ๐ โ ๐ and ๐ด โ ๐. The two points determine a line and let ๐ผ be a
perpendicular plane to this line, ๐ด๐ โฅ ๐ผ โน ๐ด and ๐ are projected onto ๐ผ in the
same point ๐ดโฒ through which also passes projฮฑ๐ = projฮฑ๐ด โ projฮฑ๐.
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Solution to Problem 185.
We determine a line which meets the three lines in the following way.
Let
Let now a plane
Solution to Problem 186.
Let ๐ผ โฉ ๐ฝ = ๐ and ๐ โ ๐. We project this point onto ๐ผ and ๐ฝ:
โน ๐ โฅ onto the projective plane of ๐ onto ๐ฝ.
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Let
โน๐๐โฒ โฉ ๐๐โฒโฒ = {๐}, so the two projections are concurrent.
Solution to Problem 187.
the angle of planes (๐ด๐ต๐ถ) and (๐๐ด๐ต) is ๐๐๏ฟฝ๏ฟฝ = ๐ผ.
Solution to Problem 188.
Let ๐ผ โฉ ๐ฝ = a and ๐ด๐ดโฒ โฅ ๐, ๐ต๐ตโฒ โฅ ๐.
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As
โนโข of line ๐ด๐ต with ๐ผ, ๐ต๐ด๐ตโฒ = ๐
โนโข of line ๐ด๐ต with ๐ฝ, ๐ด๐ต๐ดโฒ = ๐
In the plane ๐ฝ we draw through ๐ต a parallel line to ๐ and through ๐ดโฒ a parallel line
to ๐ต๐ตโฒ. Their intersection is ๐ถ, and ||๐ดโฒ๐ตโฒ|| = ||๐ต๐ถ||, ||๐ต๐ตโฒ|| = ||๐ดโฒ๐ถ||. The angle of line
๐ด๐ต with ๐ผ is ๐ด๐ต๏ฟฝ๏ฟฝ = ๐.
As ๐ด๐ดโ โฅ ๐ฝ โน ๐ด๐ดโฒ โฅ ๐ดโฒ๐ถ โน โ๐ด๐ถโ2 = โ๐ด๐ดโฒโ2 + โ๐ดโฒ๐ถโ2 = โ๐ด๐ดโฒโ2 + โ๐ตโฒ๐ตโ2(1)
๐ตโฒ๐ต๐ถ๐ด rectangle
โน โ๐ด๐ถ๐ต is right in ๐ถ.
We divide the relation (1) with ||๐ด๐ต||ยฒ:
Solution to Problem 189.
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Solution to Problem 190.
a. |๐ด๐ถ| โก |๐ต๐ถ| โน โ๐ด๐ถ๐ต isosceles
๐ถ๐ median } โน ๐ถ๐ โฅ ๐ด๐ต (1)
|๐ด๐ท| โก |๐ต๐ถ| โน โ๐ด๐ต๐ท isosceles๐ท๐ median
} โน ๐ท๐ โฅ ๐ด๐ต (2)
From (1) and (2) โน ๐ด๐ต โฅ (๐ท๐๐ถ) =๐ด๐ต โฅ ๐๐ ๐ด๐ต โฅ ๐ท๐ถ
}
|๐ต๐ถ| โก |๐ต๐ท|๐ต๐ median
} โน ๐ต๐ โฅ ๐ท๐ถ
|๐ด๐ท| โก |๐ด๐ถ|๐ด๐ median
} โน ๐ด๐ โฅ ๐ท๐ถ}โน ๐ท๐ถ โฅ (๐ด๐ต๐) โน ๐ท๐ถ โฅ ๐๐
b.
โน ๐ดโฒ โ ๐ต๐ โน ๐ต,๐ดโฒ, ๐ are collinear.
In the same way:
From (๐ด๐ท๐ถ) โฅ (๐ด๐ต๐) โน ๐ด,๐ตโฒ, ๐ collinear
(๐ด๐ต๐ถ) โฅ (๐ท๐๐ถ) โน ๐,๐ทโฒ, ๐ถ collinear
(๐ด๐ต๐ท) โฅ (๐ท๐๐ถ) โน ๐ท, ๐ถโฒ,๐ collinear
C. At point a. weโve shown that ๐๐ โฅ ๐ด๐ต
๐ด๐ดโฒ, ๐ต๐ตโฒ and ๐๐ are heights in โ๐ด๐ต๐, so they are concurrent lines.
In the same way, ๐ท๐ทโฒ, ๐ถ๐ถโฒ,๐๐ will be heights in โ๐ท๐๐ถ.
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Solution to Problem 191.
We assume that [๐๐ด, [๐๐ต are on the same side of plane ๐ผ.
We draw
(โ) plane (๐ด๐ท, ๐ต๐ตโฒ) = ๐ฝ โน |๐๐ด, |๐๐ต are in the same half-plane.
In plane ๐ฝ we have ๐(๐ด๐๏ฟฝ๏ฟฝ) = 900 = ๐(๐ต๐๏ฟฝ๏ฟฝโฒ) < 900โน ๐ด๐๏ฟฝ๏ฟฝ acute.
We assume that [๐๐ด and [๐๐ต are in different half-planes in relation to ๐ผ โน ๐ด and ๐ต are
in different half-planes in relation to ๐๐ตโฒ in plane ๐ฝ โน |๐๐ตโ โ int. ๐ต๐๏ฟฝ๏ฟฝ โน ๐(๐ด๐๏ฟฝ๏ฟฝ) =
900 +๐(๐ต๐๐ตโ) > 900โน ๐ด๐๐ต obtuse.
Solution to Problem 192.
We know the locus of the points in space equally distant from the peaks of โ๐ต๐ถ๐ท
is the perpendicular line ๐ to the pl. โ in the center of the circumscribed circle of
this โ, marked with ๐. We draw the mediator plane of side |๐ด๐ถ|, which intersects
this โฅ ๐ at point ๐. Then, point ๐ is equally distant from all the peaks of the
tetrahedron ||๐๐ด|| = ||๐๐ต|| = ||๐๐ถ|| = ||๐๐ท||. We connect ๐ with midpoint ๐ธ of side
|๐ด๐ต|. From |๐๐ด| โก |๐๐ต| โน โ๐๐ด๐ต isosceles โน๐๐ถ โฅ ๐ด๐ต (1).
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We project ๐ onto plane (๐ด๐ต๐ท) in point ๐2.
As |๐๐ด| โก |๐๐ต| โก |๐๐ท||๐)๐2 common side
} โน โ๐๐ด๐2 = โ๐๐ต๐2 = โ๐๐ท๐2
โน |๐2๐ด| โก |๐ต๐2| โก |๐ท๐2| โน
โน๐2 is the center of the circumscribed circle of โ๐ด๐ต๐ท. We show in the same
way that ๐ is also projected on the other faces onto the centers of the
circumscribed circles, thus through ๐ pass all the perpendicular lines to the faces of
the tetrahedron. These lines are drawn through the centers of the circumscribed
circles. So b. is proved.
From |๐2๐ด| โก |๐๐ต2| โน โ๐2๐ด๐ต isosceles ๐2๐ธ โฅ ๐ด๐ต (2)
From (1) and (2) โน ๐ด๐ต โฅ (๐ธ๐2๐)|๐ด๐ธ| โก |๐ธ๐ต|
} โน
โน (๐ธ๐2๐) is a mediator plane of side |๐ด๐ต| and passes through ๐ and the
intersection of the 3 mediator planes of sides |๐ต๐ถ|, |๐ถ๐ท|, |๐ต๐ท| belongs to line ๐, thus
O is the common point for the 6 mediator planes of the edges of a tetrahedron.
Solution to Problem 193.
Let ๐ โ ๐ and ๐ฟ||๐โฒ, ๐ โ ๐ฟ. Let = (๐, ๐ฟ) โน ๐โฒ||๐ผ.
Let
= {๐ด} otherwise ๐ and ๐โฒ would be parallel, thus coplanar. Let ๐ฝ be the projective
plane of line
In plane ๐ฝ we construct a perpendicular to ๐โฒโฒ in point ๐ด and
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Solution to Problem 194.
We draw
We can obtain the equality only when ๐ = ๐ด and ๐โฒ = ๐ดโฒ.
Solution to Problem 195.
Let ๐ โ ๐, ๐โฒ โ ๐โฒ such that |๐ด๐| โก |๐ดโฒ๐โฒ|. Let ๐โฒโฒ = pr๐ผ๐โฒ and ๐โฒ๐โฒโฒ โฅ ๐โฒโฒ โน
๐โฒ๐โฒโฒ โฅ ๐ผ โน ๐โฒ๐โฒโฒ โฅ ๐โฒโฒ๐.
โน โ๐ด๐๐โฒ isosceles.
Let ๐ be the midpoint of |๐๐โฒ| and ๐โฒ = pr๐ผ๐ โน ๐๐โฒ โฅ ๐โฒ๐โฒโฒ โน ๐โฒ is the midpoint
of ๐๐โฒโฒ, โ๐ด๐๐โฒโฒ isosceles โน [๐ด๐โฒ the bisector of ๐โฒ๐ด๏ฟฝ๏ฟฝ. (๐๐โฒ) is midline in
โ๐โฒ๐สบ๐ โน โ๐๐โฒโ =1
2โ๐โฒ๐โฒโฒ =
1
2โ๐ดโฒ๐ดโ = constant.
Thus, the point is at a constant distance from line ๐ด๐โฒ, thus on a parallel line to
this line, located in the โฅ plane ๐ผ, which passes through ๐ด๐โฒ.
When ๐ = ๐ด and ๐โฒ = ๐ดโฒ โน ||๐ด๐|| = ||๐โฒ๐ดโฒ = 0 โน ๐ = ๐ , where ๐ is the midpoint
of segment |๐ด๐ดโฒ|. So the locus passes through ๐ and because
โน ๐ ๐ is contained in the mediator plane of segment |๐ด๐ดโฒ|.
So ๐ ๐ is the intersection of the mediator plane of segment |๐ด๐ดโฒ| with the โฅ plane
to ๐ผ, passing through one of the bisectors of the angles determined by ๐ and ๐โฒ,
we obtain one more line contained by the mediator plane of [๐ด๐ดโฒ], the parallel line
with the other bisector of the angles determined by ๐ and ๐โฒโฒ.
So the locus will be formed by two perpendicular lines.
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Vice-versa, let ๐ โ ๐ ๐ a (โ) point on this line and ๐โ = pr๐ผ๐ โน ๐โฒ โ |๐ด๐โฒ bisector.
We draw ๐๐โฒโฒ โฅ ๐ด๐โฒ and because ๐ด๐โฒ is both bisector and height โน โ๐ด๐๐โฒโฒ
isosceles โน |๐ด๐โฒ| median โน |๐๐โฒ| โก |๐โฒ๐โฒโฒ|.
We draw
As
โน |๐โฒ๐| midline in โ๐๐โฒ๐โฒโฒโน๐,๐โฒ, ๐ collinear and |๐๐โฒ| โก |๐๐|.
Solution to Problem 196.
where ฮฑ = (ABC).
VD common side
coplanar
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Solution to Problem 197.
Let
โน๐ is the center of the circumscribed circle and as โ๐ด๐ต๐ถ is equilateral โน๐ is the
centroid โน
Solution to Problem 198.
In โ๐ด๐ธ๐ต, right in ๐ด:
||VO|| common
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Solution to Problem 199.
๐ถ๐ธ โฅ ๐ต๐ด๐ท๐ธ โฅ ๐ด๐ต
} โน โขpl. (๐ด๐ต๐ถ) and ๐ด๐ต๐ท are ๐(๐ท๐ธ๏ฟฝ๏ฟฝ).
๐ด๐ต๐ถ equilateral โน โ๐ถ๐ธโ =๐โ3
2 .
Solution to Problem 200.
We consider the problem solved and we take on plane ๐ผ, โ๐ท๐ธ๐น, then points ๐
and ๐โฒ which are not located on ๐ผ.
We also construct lines |๐ท๐, |๐น๐, |๐ธ๐ respectively |๐ท๐โฒ, |๐น๐โฒ, |๐ธ๐โฒ. On these rays we
take โ๐ด๐ต๐ถ and โ๐ดโฒ๐ตโฒ๐ถโฒ. Obviously, the way we have constructed the lines ๐ด๐ท, ๐ต๐ธ, ๐ถ๐น
shows that they intersect at ๐. We extend lines ๐ต๐ด, ๐ต๐ถ, ๐ถ๐ด until they intersect plane
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๐ผ at points ๐ต, ๐ถ respectively ๐ด. Then, we extend lines ๐ถโฒ๐ดโฒ, ๐ถโฒ๐ตโฒ, ๐ดโฒ๐ตโฒ until they
intersect plane ๐ผ at points ๐ด2, ๐ถ2 respectively ๐ต2.
Obviously, points ๐ด1, ๐ต1, ๐ถ1 are collinear (because โ ๐ผ โฉ (๐ด๐ต๐ถ)) and ๐ด2, ๐ต2, ๐ถ2 are as
well collinear (because โ ๐ผ โฉ (๐ดโฒ๐ตโฒ๐ถโฒ)).
On the other side, points ๐ท, ๐น, ๐ด1, ๐ด2 are collinear because:
thus collinear (1)
โน๐ท,๐น, ๐ด2 collinear (2)
From (1) and (2) โน๐ท,๐น, ๐ด1, ๐ด2 collinear. Similarly ๐ถ, ๐ธ, ๐น, ๐ถ2 collinear and
๐ต1, ๐ธ, ๐ท, ๐ท2 collinear.
Consequently, ๐ท๐ธ๐น is at the intersection of lines ๐ด1๐ด2, ๐ถ1๐ถ2 , ๐ต1๐ต2 on plane ๐ผ,
thus uniquely determined.
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Review Problems
201. Find the position of the third peak of the equilateral triangle, the affixes
of two peaks being ๐ง1 = 1, ๐ง2 = 2 + ๐.
Solution to Problem 201
202. Let ๐ง1, ๐ง2, ๐ง3 be three complex numbers, not equal to 0, + two by 2, and
of equal moduli. Prove that if ๐ง1 + ๐ง2๐ง3, ๐ง2 + ๐ง3๐ง1, ๐ง2 + ๐ง 1๐ง3 โ ๐ โ ๐ง1๐ง2๐ง3 = 1.
Solution to Problem 202
203. We mark by ๐บ the set of ๐ roots of the unit, ๐บ = {ํ0, ํ1, โฆ , ํ๐โ1}. Prove
that:
a. ํ๐ โ ํ๐ โ ๐บ, (โ) ๐, ๐ โ {0, 1, โฆ , ๐ โ 1};
b. ํ๐โ1 โ ๐บ, (โ ) ๐ โ {0, 1, โฆ , ๐ โ 1}.
Solution to Problem 203
204. Let the equation ๐๐งยฒ + ๐๐งยฒ + ๐ = 0, ๐, ๐, ๐ โ ๐ถ and arg๐ + arg๐ = 2arg๐, and
|๐| + |๐| = |๐|. Show that the given equation has at list one root of unity.
Solution to Problem 204
205. Let ๐ง1, ๐ง2, ๐ง3 be three complex numbers, not equal to 0, such that |๐ง1| =
|๐ง2| = |๐ง3|.
a. Prove that (โ) complex numbers ๐ผ and ๐ฝ such that ๐ง2 = ๐ผ๐ง1, ๐ง3 = ๐ฝ๐ง2
and |๐ผ| = |๐ฝ| = 1;
b. Solve the equation ๐ผยฒ + ๐ฝยฒโ๐ผ โ ๐ฝโ ๐ผโ ๐ฝ + 1 = 0 in relation to one of the
unknowns.
c. Possibly using the results from ๐. and ๐., prove that if ๐ง12 + ๐ง2
2 + ๐ง32 =
๐ง1๐ง2 + ๐ง2๐ง3 + ๐ง1๐ง3 , then we have ๐ง1 = ๐ง2 = ๐ง3 or the numbers ๐ง1, ๐ง2, ๐ง3
are affixes of the peaks of an equilateral โ.
Solution to Problem 205
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255 Compiled and Solved Problems in Geometry and Trigonometry
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206. Draw a plane through two given lines, such that their line of intersection
to be contained in a given plane.
Solution to Problem 206
207. Let ๐, ๐, ๐ be three lines with a common point and ๐ a point not located
on any of them. Show that planes (๐๐), (๐๐), (๐๐) contain a common line.
Solution to Problem 207
208. Let ๐ด, ๐ต, ๐ถ, ๐ท be points and ๐ผ a plane separating points ๐ด and ๐ต, ๐ด and ๐ถ,
๐ถ and ๐ท. Show that ๐ผ โฉ |๐ต๐ท| โ โ and ๐ผ โฉ |๐ด๐ท| = โ .
Solution to Problem 208
209. On edges ๐, ๐, ๐ of a trihedral angle with its peak ๐, take points ๐ด, ๐ต, ๐ถ; let
then ๐ท โ |๐ต๐ถ| and ๐ธ โ |๐ด๐ท|. Show that |๐๐ธ โ ๐๐๐ก. โ ๐๐๐.
Solution to Problem 209
210. Show that the following sets are convex: the interior of a trihedral angle,
a tetrahedron without an edge (without a face).
Solution to Problem 210
211. Let ๐ด, ๐ต, ๐ถ, ๐ท be four non-coplanar points and ๐ธ, ๐น, ๐บ, ๐ป the midpoints of
segments [๐ด๐ต], [๐ต๐ถ], [๐ถ๐ท], [๐ท๐ด]. Show that ๐ธ๐น || (๐ด๐ถ๐ท) and points ๐ธ, ๐น, ๐บ, ๐ป
are coplanar.
Solution to Problem 211
212. On lines ๐, ๐โฒ consider distinct points ๐ด, ๐ต, ๐ถ; ๐ดโฒ, ๐ตโฒ, ๐ถโฒ respectively. Show
that we can draw through lines ๐ด๐ดโฒ, ๐ต๐ตโฒ, ๐ถ๐ถโฒ three parallel planes if and only
if โ๐ด๐ตโ
โ๐ดโฒ๐ตโฒโ=
โ๐ต๐ถโ
โ๐ตโฒ๐ถโฒโ .
Solution to Problem 212
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Florentin Smarandache
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213. Let ๐,๐โฒ be each mobile points on the non-coplanar lines ๐, ๐โฒ. Find the
locus of points ๐ that divide segment |๐๐โฒ| in a given ratio.
Solution to Problem 213
214. Construct a line that meets three given lines, respectively in ๐,๐, ๐ and
for which โ๐๐โ
โ๐๐โ to be given ratio.
Solution to Problem 214
215. Find the locus of the peak ๐ of the triangle ๐,๐, ๐ if its sides remain
parallel to three fixed lines, the peak ๐ describes a given line ๐, and the
peak ๐ โ a given plane ๐ผ.
Solution to Problem 215
216. On the edges [๐๐ด, [๐๐ต, [๐๐ถ of a trihedral angle we consider points ๐,๐, ๐
such that โ๐๐โ = ๐โ๐๐ดโ, โ๐๐โ = ๐โ๐๐ตโ, โ๐๐โ = ๐โ๐๐ถโ, where ๐ is a
positive variable number. Show the locus of the centroid of triangle ๐๐๐.
Solution to Problem 216
217. ๐ด๐ต๐ถ๐ท and ๐ด1๐ต1๐ถ1๐ท1 are two parallelograms in space. We take the points
๐ด2, ๐ต2, ๐ถ2, ๐ท2 which divide segments [๐ด๐ด1], [๐ต๐ต1], [๐ถ๐ถ1], [๐ท๐ท1] in the same
ratio. Show that ๐ด2๐ต2๐ถ2๐ท2 is a parallelogram.
Solution to Problem 217
218. The lines ๐, ๐โฒ are given, which cut a given plane ๐ผ in ๐ด and ๐ดโฒ. Construct
the points ๐,๐โฒ on ๐, ๐โฒ such that ๐๐โฒ โฅ ๐ผ and segment [๐๐โฒ] to have a
given length ๐. Discuss.
Solution to Problem 218
219. Construct a line which passes through a given point ๐ด and that is
perpendicular to two given lines ๐ and ๐โฒ.
Solution to Problem 219
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255 Compiled and Solved Problems in Geometry and Trigonometry
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220. Show that there exist three lines with a common point, perpendicular two
by two.
Solution to Problem 220
221. Let ๐ ๐, ๐, ๐ four lines with a common point, ๐ is perpendicular to ๐ ๐, ๐.
Show that lines ๐, ๐, ๐ are coplanar.
Solution to Problem 221
222. Show that there do not exist four lines with a common point that are
perpendicular two by two.
Solution to Problem 222
223. Let ๐ โฅ ๐ผ and ๐โฒ โฅ ๐. Show that ๐โฒ โฅ ๐ผ.
Solution to Problem 223
224. Show that two distinct perpendicular lines on a plane are parallel.
Solution to Problem 224
225. Let ๐ โฅ ๐ผ and ๐โฒ[โฅ ๐ผ. Show that ๐โฒ โฅ ๐.
Solution to Problem 225
226. Show that two perpendicular planes on the same line are parallel with
each other.
Solution to Problem 226
227. Show that the locus of the points equally distant from two distinct points
๐ด and ๐ต is a perpendicular plane to ๐ด๐ต, passing through midpoint ๐ of the
segment [๐ด๐ต] (called mediator plane of [๐ด๐ต]).
Solution to Problem 227
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Florentin Smarandache
178
228. Find the locus of the points in space equally distant from the peaks of a
triangle ๐ด๐ต๐ถ.
Solution to Problem 228
229. The plane ๐ผ and the points ๐ด โ ๐ผ, ๐ต โ ๐ผ are given. A variable line ๐ passes
through ๐ด and it is contained in plane ๐ผ. Find the locus of the โฅ feet from
๐ต to ๐.
Solution to Problem 229
230. A line ๐ผ, and a point ๐ด โ ๐ผ are given. Find the locus of the feet of the
perpendicular lines from ๐ด to planes passing through ๐ผ.
Solution to Problem 230
231. Consider a plane ๐ผ that passes through the midpoint of segment [๐ด๐ต].
Show that points ๐ด and ๐ต are equally distant from plane ๐ผ.
Solution to Problem 231
232. Through a given point, draw a line that intersects a given line and is โฅ to
another given line.
Solution to Problem 232
233. Let ๐ผ and ๐ฝ be two distinct planes and the line ๐ their intersection. Let ๐
be a point that is not located on ๐ผ โช ๐ฝ. We draw the lines ๐๐1 and ๐๐2 โฅ
on ๐ผ and ๐ฝ. Show that the line ๐ is โฅ to (๐๐1๐๐2).
Solution to Problem 233
234. A plane ๐ผ and a point ๐ด, ๐ด โ ๐ผ are given. Find the locus of points ๐ โ ๐ผ
such that segment |๐ด๐| has a given length.
Solution to Problem 234
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255 Compiled and Solved Problems in Geometry and Trigonometry
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235. Let ๐, ๐ด, ๐ต, ๐ถ be four points such that ๐๐ด โฅ ๐๐ต โฅ ๐๐ถ โฅ ๐ท๐ด and we write
๐ = โ๐๐ดโ, ๐ = โ๐๐ตโ, ๐ = โ๐๐ถโ.
a. Find the length of the sides of โ๐ด๐ต๐ถ in relation to ๐, ๐, ๐;
b. Find ๐[๐ด๐ต๐ถ] and demonstrate the relation ๐[๐ด๐ต๐ถ]2 = ๐[๐ท๐ด๐ต]2 +
๐[๐๐ต๐ถ]2 + ๐[๐๐ถ๐ด]2;
c. Show that the orthogonal projection of point ๐ on plane (๐ด๐ต๐ถ) is the
orthocenter ๐ป of โ๐ด๐ต๐ถ;
d. Find the distance โ๐๐ปโ.
Solution to Problem 235
236. Consider non-coplanar points ๐ด, ๐ต, ๐ถ, ๐ท and lines ๐ด๐ดโฒ, ๐ต๐ตโฒ, ๐ถ๐ถโฒ, ๐ท๐ทโฒ
perpendicular to (๐ต๐ถ๐ท), (๐ด๐ถ๐ท), (๐ด๐ต๐ท). Show that if lines ๐ด๐ดโฒ and ๐ต๐ตโฒ are
concurrent, then lines ๐ถ๐ถโฒ, ๐ท๐ทโฒ are coplanar.
Solution to Problem 236
237. Let ๐ด, ๐ต, ๐ถ, ๐ท four non-coplanar points. Show that ๐ด๐ต โฅ ๐ถ๐ท and ๐ด๐ถ โฅ ๐ต๐ท
โน ๐ด๐ท โฅ ๐ต๐ถ.
Solution to Problem 237
238. On the edges of a triangle with its peak ๐, take the points ๐ด, ๐ต, ๐ถ such
that |๐๐ด| โก |๐๐ต| โก |๐๐ถ|. Show that the โฅ foot in ๐ to the plane (๐ด๐ต๐ถ)
coincides with the point of intersection of the bisectors โ๐ด๐ต๐ถ.
Solution to Problem 238
239. Let a peak ๐ด of the isosceles triangle ๐ด๐ต๐ถ (|๐ด๐ต| โก |๐ด๐ถ|) be the
orthogonal projection onto ๐ดโฒ on a plane ๐ผ which passes through ๐ต๐ถ. Show
that ๐ต๐ดโฒ๏ฟฝ๏ฟฝ > ๐ต๐ด๏ฟฝ๏ฟฝ.
Solution to Problem 239
240. With the notes of Theorem 1, let [๐ด๐ตโฒ be the opposite ray to [๐ด๐ตโฒโฒ. Show
that for any point ๐ โ ๐ผโ [๐ด๐ตโฒโฒ we have ๐ตโฒโฒ๐ด๏ฟฝ๏ฟฝ > ๐๐ด๏ฟฝ๏ฟฝ.
Solution to Problem 240
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Florentin Smarandache
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241. Let ๐ผ be a plane, ๐ด โ ๐ผ and ๐ต and ๐ถ two points on the same side of ๐ผ
such that ๐ด๐ถ โฅ ๐ผ. Show that ๐ถ๐ด๏ฟฝ๏ฟฝ is the complement of the angle formed
by [๐ด๐ต with ๐ผ.
Solution to Problem 241
242. Let ๐ผโฒ๐ฝโฒ be a trihedral angle with edge ๐ and ๐ด โ ๐. Show that of all the
rays with origin at ๐ด and contained in half-plane ๐ฝโฒ, the one that forms
with plane ๐ผ the biggest possible angle is that โฅ ๐ โ ๐ (its support is
called the line with the largest slope of ๐ฝ in relation to ๐ผ).
Solution to Problem 242
243. Let ๐ผ be a plane, ๐ a closed half-plane, bordered by ๐ผ, ๐ผโฒ a half-plane
contained in ๐ผ and ๐ a real number between 00 and 1800. Show that there
is only one half-space ๐ฝโฒ that has common border with ๐ผโฒ such that ๐ฝโฒ โ ๐
and ๐(๐ผโฒ๐ฝโฒ) = ๐.
Solution to Problem 243
244. Let (๐ผโฒ๐ฝโฒ) be a proper dihedral angle. Construct a half-plane ๐พโฒ such that
๐(๐ผโฒ๐ฝโฒ) = ๐(๐พโฒ๐ฝโฒ). Show that the problem has two solutions, one of which
is located in the int. ๐ผโฒ๐ฝโฒ (called bisector half-plane of ๐ผโฒ๐ฝโฒ).
Solution to Problem 244
245. Show that the locus of the points equally distant from two secant planes
๐ผ, ๐ฝ is formed by two โฅ planes, namely by the union of the bisector planes
of the dihedral angles ๐ผ and ๐ฝ.
Solution to Problem 245
246. If ๐ผ and ๐ฝ are two planes, ๐ โ ๐ฝ and ๐ โฅ through ๐ on ๐ผ. Show that ๐ โ
๐ฝ.
Solution to Problem 246
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255 Compiled and Solved Problems in Geometry and Trigonometry
181
247. Consider a line ๐ โ ๐ผ. Show that the union of the โฅ lines to ๐ผ, which
intersect line ๐, is a plane โฅ ๐ผ.
Solution to Problem 247
248. Find the locus of the points equally distant from two concurrent lines.
Solution to Problem 248
249. Show that a plane ๐ผ โฅ to two secant planes is โฅ to their intersection.
Solution to Problem 249
250. Let ๐ด be a point that is not on plane ๐ผ. Find the intersection of all the
planes that contain point ๐ด and are โฅ to plane ๐ผ.
Solution to Problem 250
251. From a given point draw a โฅ plane to two given planes.
Solution to Problem 251
252. Intersect a dihedral angle with a plane as the angle of sections is right.
Solution to Problem 252
253. Show that a line ๐ and a plane ๐ผ, which are perpendicular to another
plane, are parallel or line ๐ is contained in ๐ผ.
Solution to Problem 253
254. If three planes are โฅ to a plane, they intersect two by two after lines
๐, ๐, ๐. Show that ๐ โฅ ๐ โฅ ๐.
Solution to Problem 254
255. From a point ๐ด we draw perpendicular lines ๐ด๐ต and ๐ด๐ถ to the planes of
the faces of a dihedral angle ๐ผโฒ๐ฝโฒ. Show that ๐(๐ต๐ด๏ฟฝ๏ฟฝ) = ๐(๐ผโฒ๐ฝโฒ) or
๐(๐ต๐ด๏ฟฝ๏ฟฝ) = 1800 โ๐(๐ผโฒ๐ฝโฒ).
Solution to Problem 255
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Florentin Smarandache
182
Solutions
Solution to Problem 201.
๐1 โ ๐ง1 = 1
๐2 โ ๐ง1 = 2 + ๐
๐1 โ ๐ง1 = ๐ฅ + ๐ฆ๐
โ๐1๐2๐3 equilateral โน โ๐1๐2โ = โ๐1๐3โ = โ๐2๐3โ โ |๐ง2 โ ๐ง1| = |๐ง3 โ ๐ง2| =
|๐ง1 โ ๐ง3|
โ โ2 = โ(๐ฅ โ 2)2 + (๐ฆ โ 1)2 โ {(๐ฅ โ 2)2 + (๐ฆ โ 1)2 = 2
(1 โ ๐ฅ)2 + ๐ฆ2 = 2โ {
๐ฅ + ๐ฆ = 2
๐ฅ2 + ๐ฆ2 โ 2๐ฅ = 1
โ ๐ฆ = 2 โ ๐ฅ
๐ฅ2 + 4 + ๐ฅ2 โ 4๐ฅ โ 2๐ฅ = 1 โ ๐ฅ1,2 =3 ยฑ โ3
2โ
[ ๐ฆ1 =
1 โ โ3
2
๐ฆ21 + โ3
2
Thus: ๐3 (3+โ3
2,1โโ3
2) or ๐3 (
3โโ3
2,1+โ3
2).
There are two solutions!
Solution to Problem 202.
๐ง1 = ๐(cos ๐ก1 + ๐ sin ๐ก1)
๐ง2 = ๐(cos ๐ก2 + ๐ sin ๐ก2)
๐ง3 = ๐(cos ๐ก3 + ๐ sin ๐ก3)
๐ง1 โ ๐ง2 โ ๐ง3 โ ๐ก1 โ ๐ก2 โ ๐ก3
{
๐ง1 + ๐ง2๐ง3 โ โ โ sin ๐ก1 + ๐ sin(๐ก2 + ๐ก3) = 0
๐ง2 + ๐ง3๐ง1 โ โ โ sin ๐ก2 + ๐ sin(๐ก1 + ๐ก3) = 0
๐ง3 + ๐ง1๐ง2 โ โ โ sin ๐ก3 + ๐ sin(๐ก1 + ๐ก2) = 0
โน
{
sin ๐ก1(1 โ ๐ cos ๐ก) + ๐ sin ๐ก โ cos ๐ก1 = 0
sin ๐ก2(1 โ ๐ cos ๐ก) + ๐ sin ๐ก โ cos ๐ก2 = 0
sin ๐ก3(1 โ ๐ cos ๐ก) + ๐ sin ๐ก โ cos ๐ก3 = 0
๐ก1 โ ๐ก2 โ ๐ก3
These equalities are simultaneously true only if 1 โ ๐ โ cos ๐ก = 0 and ๐ โ
sin ๐ก = 0, as ๐ โ 0 โ sin ๐ก = 0 โ ๐ก = 0 โ cos ๐ก = 1 โ 1 โ ๐ = 0 โ ๐ = 1, so
๐ง1๐ง2๐ง3 = 1 โ (cos 0 + sin 0) = 1.
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255 Compiled and Solved Problems in Geometry and Trigonometry
183
Solution to Problem 203.
a. ํ๐ =2๐๐
๐+ ๐ sin
2๐๐
๐ , ๐ โ {0, 1, โฆ , ๐ โ 1}.
So ํ๐ = cos
2๐๐
๐+ ๐ sin
2๐๐
๐
ํ๐ = cos2๐๐
๐+ ๐ sin
2๐๐
๐
} โน ํ๐ํ๐ = cos2๐(๐+๐)
๐+ ๐ sin
2๐(๐+๐)
๐, ๐, ๐ โ {0, 1, โฆ , ๐ โ 1}.
1) ๐ + ๐ < ๐ โ 1 โน ๐ + ๐ = ๐ โ {0, 1, โฆ , ๐ โ 1} โน ํ๐ํ๐ = ํ๐ โ ๐บ;
2) ๐ + ๐ = ๐ โน ํ๐ํ๐ = cos 2๐ + ๐ sin 2๐ = 1 = ํ๐ โ ๐บ;
3) ๐ + ๐ > ๐ โน ๐ + ๐ = ๐ โ ๐ + ๐, 0 โค ๐ < ๐, ํ๐ํ๐ = cos2๐(๐โ๐+๐)
๐+ ๐ sin
2๐(๐โ๐+๐)
๐=
cos (2๐๐ +2๐๐
๐) + ๐ sin (2๐๐+
2๐๐
๐) = cos
2๐๐
๐+ ๐ sin
2๐๐
๐= ํ๐ โ ๐บ.
b. ํ๐ = cos2๐๐
๐+ ๐ sin
2๐๐
๐
ํ๐โ1 =
1
๐๐=
cos0+๐ sin0
cos2๐๐
๐+๐ sin
2๐๐
๐
= cos (โ2๐๐
๐) + ๐ sin (โ
2๐๐
๐) = cos (2๐ โ
2๐๐
๐) +
๐ sin (2๐ โ2๐๐
๐) = cos
2๐๐โ2๐๐
๐+ ๐ sin
2๐๐โ2๐๐
๐= cos
2๐(๐โ1)
๐+ ๐ sin
2๐(๐โ1)
๐ ,
๐ โ {0, 1,โฆ , ๐ โ 1}.
If ๐ = 0 โน ๐ โ ๐ = ๐ โน ํ0โ1 = ํ0 โ ๐.
If ๐ โ 0 โน ๐ โ ๐ โค ๐ โ 1 โน โ = ๐ โ ๐ โ {0, 1, โฆ , ๐ โ 1}โน ๐โ1 = cos2๐โ
๐+
๐ sin2๐โ
๐โ ๐บ.
Solution to Problem 204.
{
๐ = ๐1(cos ๐ก1 + ๐ sin ๐ก1)
๐ = ๐2(cos ๐ก2 + ๐ sin ๐ก2)
๐ = ๐3(cos ๐ก3 + ๐ sin ๐ก3)
arg๐ + arg๐ = 2arg๐ โน ๐ก1 + ๐ก3 = 2๐ก2
and |๐| + |๐| = |๐| โน ๐1 + ๐3 = ๐2
๐๐ง2 + ๐๐ง + ๐ = 0 โน ๐ง1,2 =โ๐ ยฑ โ๐2 โ 4๐๐
2๐
=โ๐2(cos ๐ก2 + ๐ sin ๐ก2) ยฑ โ๐2
2(cos2๐ก2 + ๐ sin2๐ก2) โ 4๐1๐3(cos(๐ก1 + ๐ก3) + ๐ sin(๐ก1 + ๐ก3))
2๐1(cos ๐ก1 + ๐ sin ๐ก1)
=โ๐2(cos ๐ก2 + ๐ sin ๐ก2) ยฑ โ(cos 2๐ก2 + ๐ sin 2๐ก2)(๐2
2 โ 4๐1๐3)
2๐1(cos ๐ก1 + ๐ sin ๐ก1)
But ๐1 + ๐3 = ๐2โน ๐22 = ๐1
2 + ๐12 + ๐3
2 + 2๐1๐3โน ๐22 โ 4๐1๐3 = ๐1
2 + ๐12 + ๐3
2 + 2๐1๐3 โ
4๐1๐3 = (๐1 โ ๐3)2.
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Therefore:
๐ง1,2 =โ๐2(cos ๐ก2 + ๐ sin ๐ก2) ยฑ (cos ๐ก2 + ๐ sin ๐ก2)(๐1 โ ๐3)
2๐1(cos ๐ก1 + ๐ sin ๐ก1)
We observe that:
๐ง2 =(cos ๐ก2+๐ sin ๐ก2)(โ2๐1)
2๐1(cos ๐ก1+๐ sin ๐ก1)= โ[cos(๐ก2 โ ๐ก1) + ๐ sin(๐ก2 โ ๐ก1)] = cos[๐ + (๐ก2 โ ๐ก1)] +
๐ sin[๐ + ๐ก2 โ ๐ก1] and ๐ก2 = 1.
Solution to Problem 205.
Let
Let
So ๐ผ is determined.
So ๐ฝ is determined.
If we work with reduced arguments, then ๐ก4 = ๐ก2 โ ๐ก1 or ๐ก4 = ๐ก2 โ ๐ก1 + 2๐, in the
same way ๐ก5.
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According to a. (โ) the complex numbers of modulus 1, ๐ผ and ๐ฝ such that ๐ง2 = ๐ผ๐ง1
and ๐ง3 = ๐ฝ๐ง1.
In the given relation, by substitution we obtain:
๐ผ = 1 and ๐ฝ = 1 verify this equality, so in this case ๐ง2 = ๐ง3 = ๐ง1.
According to point b.,
where ๐ฝ = ๐ฅ + ๐๐ฆ, when
We construct the system:
The initial solution leads us to ๐ง1 = ๐ง2 = ๐ง3.
and gives
By substituting,
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|๐ผ| = 2 does not comply with the condition |๐ผ| = 1.
But
so
If
then
and then
If
are on the circle with radius ๐ and the arguments are
they are the peaks of an equilateral triangle.
Solution to Problem 206.
a. We assume that ๐ โฉ ๐ผ โ โ and ๐โฒ โฉ ๐ผ = {๐ต}.
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Let ๐ โฉ ๐ผ = {๐ด} and ๐โฒ โฉ ๐ผ = {๐ต} and the planes determined by pairs of concurrent
lines (๐, ๐ด๐ต); (๐โฒ, ๐ด๐ต).
We remark that these are the required planes, because
b. We assume ๐ โฉ ๐ผ = {๐ด} and ๐โฒ||๐ผ.
We draw through ๐ด, in plane ๐ผ, line ๐โฒ||๐ and we consider planes (๐, ๐โฒโฒ) and
(๐โฒ, ๐โฒโฒ) and we remark that
c. We assume ๐ โฉ ๐ผ = โ and ๐โฒ โฉ ๐ผ = โ and ๐โฒ โ direction ๐.
Let ๐ด โ ๐ผ and ๐โฒโฒ||๐ โน ๐โฒโฒ||๐โฒ and the planes are (๐, ๐โฒโฒ) and (๐โฒ, ๐โฒโฒ). The reasoning
is the same as above.
Solution to Problem 207.
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Solution to Problem 208.
If ๐ผ separates points ๐ด and ๐ต, it means they are in different half-spaces and let
๐ = |๐ผ๐ด and ๐โฒ = |๐ผ๐ต.
Because ๐ผ separates ๐ด and ๐ถ โน ๐ถ โ ๐โฒ.
Because ๐ผ separates ๐ถ and ๐ท โน ๐ท โ ๐.
From ๐ต โ ๐โฒ and ๐ท โ ๐ โน ๐ผ separates points ๐ต and ๐ท
From ๐ด โ ๐ and ๐ท โ ๐ โน |๐ต๐ท| โฉ ๐ผ = โ .
Solution to Problem 209.
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From (1), (2), (3)
Solution to Problem 210.
a. int. (|๐๐ด, |๐๏ฟฝ๏ฟฝ, |๐๐ถ) = |(๐๐ด๐ต), ๐ถ โฉ |(๐๐ต๐ถ), ๐ด โฉ |(๐๐ด๐ถ), ๐ต is thus an intersection of
convex set and thus the interior of a trihedron is a convex set.
b. Tetrahedron [๐๐ด๐ต๐ถ] without edge [๐ด๐ถ]. We mark with โณ1 = [๐ด๐ต๐ถ]โ [๐ด๐ถ] = [๐ด๐ต, ๐ถ โฉ
[๐ต๐ถ, ๐ด โฉ |๐ด๐ถ, ๐ต is thus a convex set, being intersection of convex sets.
is a convex set.
In the same way
is a convex set, where
But [๐๐ด๐ต๐ถ] โ [๐ด๐ถ] =
and thus it is a convex set as intersection of convex sets.
c. Tetrahedron [๐๐ด๐ต๐ถ] without face [๐ด๐ต๐ถ]
๐ is thus intersection of convex sets โน is a convex set.
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Solution to Problem 211.
In plane (๐ต๐ด๐ถ) we have ๐ธ๐น||๐ด๐ถ. In plane (๐ท๐ด๐ถ) we have ๐ด๐ถ โ (๐ท๐ด๐ถ) โน ๐ธ๐น||(๐ท๐ด๐ถ).
In this plane we also have ๐ป๐บ||๐ด๐ถ. So ๐ธ๐น||๐ป๐บ โน ๐ธ, ๐น, ๐บ, ๐ป are coplanar and because
โ๐ธ๐นโ =โ๐ด๐ถโ
2= โ๐ป๐บโ โน๐ธ๐น๐บ๐ป is a parallelogram.
Solution to Problem 212.
We assume we have ๐ผ||๐ฝ||๐พ such that ๐ด๐ดโ โ ๐ผ, ๐ต๐ตโ โ , ๐ถ๐ถโ โ ๐พ.
We draw through ๐ดโฒ a parallel line with ๐: ๐โฒโฒ||๐. As ๐ intersects all the 3 planes
๐ดโฒ โ ๐โฒโฒ at ๐ด, ๐ต, ๐ถ โน and its || ๐โโ cuts them at ๐ดโฒ, ๐ตโฒโฒ, ๐ถโฒโฒ.
Because
Let plane (๐โฒ, ๐โฒโฒ ). Because this plane has in common with planes ๐ผ, ๐ฝ, ๐พ the points
๐ดโฒ, ๐ตโฒโฒ, ๐ถโฒโฒ and because ๐ผ || ๐ฝ || ๐พ โน it intersects them after the parallel lines
Taking into consideration (1) and (2)
The vice-versa can be similarly proved.
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Solution to Problem 213.
Let
such that
and
such that
So
according to problem 7, three planes can be drawn || ๐ฝ || ๐ผ || ๐พ such that
and ๐๐โฒ โ ๐ผ.
So by marking ๐ and letting ๐โฒ variable, ๐โฒ โ a parallel plane with the two lines,
which passes through ๐. It is known that this plane is unique, because by drawing
through ๐ parallel lines to ๐ and ๐โ in order to obtain this plane, it is well
determined by 2 concurrent lines.
Vice-versa: Let ๐ โ ๐ผ, that is the plane passing through ๐ and it is parallel to ๐
and ๐โ.
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(๐โฒโฒ, ๐) determines a plane, and (๐โฒโฒ, ๐โฒ) determines a plane โน the two planes,
which have a common point, intersect after a line(๐โฒโฒ, ๐)โฉ (๐โฒโฒ, ๐โฒ) = ๐๐โฒ where ๐ โ
๐ and ๐โฒ โ ๐โฒ.
Because
such that ๐๐ โ ๐ฝ, ๐ฝ||๐ผ.
Because
such that ๐โฒ๐โฒ โ ๐พ, ๐พ||๐ผ.
So the required locus is a parallel plane with ๐ and ๐โ.
Solution to Problem 214.
We consider the plane, which according to a previous problem, represents the
locus of the points dividing the segments with extremities on lines ๐1 and ๐3 in a
given ratio ๐. To obtain this plane, we take a point ๐ด โ ๐1, ๐ต โ ๐3 and point ๐ถ โ ๐ด๐ต
such that โ๐ด๐ถโ
โ๐ถ๐ตโ= ๐. Through this point ๐ถ we draw two parallel lines ๐1 and ๐3 which
determine the above mentioned plane ๐ผ.
Let ๐2 โฉ ๐ผ = {๐}. We must determine a segment that passes through ๐ and with
its extremities on ๐1 and ๐3, respectively at ๐ and ๐. As the required line passes
through ๐ and ๐
The same line must pass through ๐ and ๐ and because
๐,๐, ๐ collinear.
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From (1) and (2)
Then, according to previous problem 8:
and the required line is ๐๐.
Solution to Problem 215.
Let โ๐๐๐ such that
Let โ๐โฒ๐โฒ๐โฒ such that
Line ๐๐ generates a plane ๐ฝ, being parallel to a fixed direction ๐1 and it is based
on a given line ๐. In the same way, the line ๐๐ generates a plane ๐พ, parallel to a
fixed direction ๐2, and based on a given line ๐. As ๐ is contained by ๐พ โน ๐ is a
common point for ๐ผ and ๐พ โน ๐ผ โฉ ๐พ โ โ โน ๐ผ โฉ ๐พ = ๐โฒ, ๐ โ ๐โฒ.
(โ) the considered โ, so ๐ also describes a line ๐โฒ โ ๐ผ.
Because plane ๐พ is well determined by line ๐ and direction ๐2, is fixed, so ๐โฒ = ๐ผ โฉ
๐พ is fixed.
In the same way, ๐๐ will generate a plane ๐ฟ, moving parallel to the fixed direction
๐3 and being based on the given line ๐โ.
As
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(โ)๐ variable peak, ๐ โ ๐โฒโฒ.
Thus, in the given conditions, for any โ๐๐๐, peak ๐ โ ๐โฒโฒ.
Vice-versa, let ๐โฒ โ ๐โฒโฒ. On plane (๐โฒ, ๐โฒโฒ) we draw ๐โฒ,๐โฒ||๐๐ โน (๐โฒ๐โฒ๐โฒ)||(๐๐๐) โน
(๐๐โฒ) the intersection of two parallel planes after parallel lines ๐โฒ๐โฒ||๐๐ and the so
constructed โ๐โฒ๐โฒ๐โฒ has its sides parallel to the three fixed lines, has ๐โฒ โ ๐ and
๐โฒ โ ๐ผ, so it is one of the triangles given in the text.
So the locus is line ๐โฒโฒ. Weโve seen how it can be constructed and it passes
through ๐.
In the situation when ๐ท||๐ผ we obtain
In this case the locus is a parallel line with ๐.
Let ๐๐๐ and ๐โฒ๐โฒ๐โฒ such that
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We assume ๐ผ โฉ ๐ฝ = ๐ and let ๐ โฉ (๐๐๐) = {๐} and ๐ โฉ (๐โฒ๐โฒ๐โฒ) = {๐โฒ} โน
a plane cuts the parallel planes after parallel lines.
In the same way, ๐๐||๐โฒ๐โฒ and because
We use the property: Let ๐1 and ๐2 2 parallel planes and ๐ด, ๐ต, ๐ถ โ ๐1 and ๐ดโฒ๐ตโฒ๐ถโฒ โ
๐2, ๐ด๐ต โฅ ๐ดโฒ๐ตโฒ,
Letโs show that ๐ต๐ถ||๐ตโฒ๐ถโฒ. Indeed (๐ต๐ตโฒ๐ถโฒ) is a plane which intersects the 2 planes
after parallel lines.
Applying in (1) this property โน๐๐||๐โฒ๐โฒ. Maintaining ๐๐ fixed and letting ๐โ
variable, always ๐๐||๐โฒ๐โฒ.=, so ๐โฒ๐โฒ generates a plane which passes through ๐. We
assume ๐ฝ||๐ผ.
๐๐๐โฒ๐โฒ parallelogram
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Considering ๐โฒ fix and ๐ variable โน ๐๐โฒ||๐ผ and the set of parallel lines drawn
๐๐โฒ||๐ฝ to a plane through an exterior point is a parallel plane with the given plane.
So the locus is a parallel plane with ๐ผ and ๐ฝ.
Solution to Problem 216.
In plane ๐ท๐ด๐ถ we have:
In plane ๐ท๐ด๐ต we have:
In plane ๐๐ต๐ถ we have:
From ๐๐||๐ด๐ถ and ๐๐||๐ต๐ถ โน (๐๐๐ )||(๐ด๐ต๐ถ).
Let ๐ and ๐ท be midpoints of sides |๐๐| and |๐ด๐ต|.
are collinear.
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Concurrent lines ๐๐ท and ๐๐ถ determine a plane which cuts the parallel planes
are collinear.
So ๐บโ โ |๐๐บ โน the required locus is ray |๐๐บ.
Vice-versa: we take a point on |๐๐บ, ๐บโฒโฒ, and draw through it a parallel plane
to (๐ด๐ต๐ถ), plane (๐โฒโฒ, ๐โฒโฒ, ๐โฒโฒ), similar triangles are formed and the ratios from
the hypothesis appear.
Solution to Problem 217.
Let ๐ด2, ๐ต2, ๐ถ2, ๐ท2 such that
Mark on lines ๐ด๐ท1 and ๐ต๐ถ1 points ๐ and ๐ such that
From
Next is
The same,
As
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we obtain
From
is a parallelogram.
is parallelogram.
So
is a parallelogram.
Solution to Problem 218.
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We draw through ๐ดโฒ a line ๐โฒโฒ||๐. We draw two parallel planes with ๐ผ, which will
intersect the three lines in ๐ตโฒ, ๐ตโ, ๐ต and ๐ถโฒ, ๐ถโ, ๐ถ. Plane (๐, ๐โ) intersects planes ๐ผ,
(๐ตโฒ๐ตโ๐ต), (๐ถโฒ๐ถโ๐ถ) after parallel lines
= โ๐ต๐ตโโ = โ๐ถโฒ๐ถโโ.
Plane (๐โฒ, ๐โ) intersects parallel planes (๐ตโฒ๐ตโ๐ต), (๐ถโฒ๐ถโ๐ถ) after parallel lines
So (โ) parallel plane with ๐ผ we construct, the newly obtained triangle has a side of
๐ผ length and the corresponding angle to ๐ตโฒ๐ตโ๏ฟฝ๏ฟฝ is constant. We mark with a line
that position of the plane, for which the opposite length of the required angle is ๐.
With the compass spike at ๐ถ and with a radius equal with ๐, we trace a circle arc
that cuts segment | ๐ถโฒ๐ถโ| at ๐ or line ๐ถโฒ๐ถโ. Through ๐ we draw at (๐โฒ, ๐โ) a parallel
line to ๐โ which precisely meets ๐โฒ in a point ๐โฒ. Through ๐โฒ, we draw the || plane
to ๐ผ, which will intersect the three lines in ๐,๐โฒ,๐โ.
is a parallelogram.
โน ๐ถ๐๐โฒ๐ is a parallelogram.
and line ๐๐โฒ, located in a parallel plane to ๐ผ, is parallel to ๐ผ.
Discussion:
Assuming the plane (๐ถโฒ๐ถโ๐ถ) is variable, as | ๐ถ๐ถโ| and ๐ถ๐ถโฒ๐ถโ are constant, then
๐(๐ถโฒ๐ถโ๐ถ) = ๐ = also constant
If ๐ < ๐ we donโt have any solution.
If ๐ = ๐ (โ) a solution, the circle of radius ๐, is tangent to ๐ถโฒ๐ถโ.
If ๐ > ๐ (โ ) two solutions: circle of radius ๐, cuts ๐ถโฒ๐ถโ at two points ๐ and ๐.
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Solution to Problem 219.
We draw through ๐ด planes ๐ผ โฅ ๐ and ๐ผโฒ โฅ ๐โฒ.
As ๐ด is a common point
โน ๐ผ โฉ ๐ผโฒ = โโน ๐ด โ โ.
โน โ the required line
If ๐ผ โ ๐ผโฒ - we have only one solution.
If ๐ผ = ๐ผโฒ (โ) line from ๐ผ which passes through ๐ด corresponds to the problem, so
(โ) infinite solutions.
Solution to Problem 220.
Let ๐1 โฅ ๐2 two concurrent perpendicular lines, ๐1 โฉ ๐2 = {๐}. They determine a
plane ๐ผ = (๐1, ๐2) and ๐ โ ๐ผ. We construct on ๐ผ in ๐.
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Solution to Problem 221.
We use the reductio ad absurdum method.
Let ๐ โฅ ๐, ๐ โฅ ๐, ๐ โฅ ๐. We assume that these lines are not coplanar. Let ๐ผ =
(๐, ๐), ๐ผโฒ = (๐, ๐), ๐ผ โ ๐ผโฒ. Then ๐ โฅ ๐ผ, ๐ โฅ ๐ผโ.
Thus through point ๐, 2 perpendicular planes to ๐ can be drawn. False โน ๐, ๐, ๐ are
coplanar.
Solution to Problem 222.
By reductio ad absurdum:
Let ๐ โฉ ๐ โฉ ๐ โฉ ๐ = {๐} and they are perpendicular two by two. From ๐ โฅ ๐, ๐ โฅ
๐, ๐ โฅ ๐ โน ๐, ๐, ๐ are coplanar and ๐ โฅ ๐, ๐ โฅ ๐, so we can draw to point ๐ two
distinct perpendicular lines. False. So the 4 lines cannot be perpendicular two by
two.
Solution to Problem 223.
We assume that ๐ โฅ ๐ผ.
In ๐โฒ โฉ ๐ผ = {๐} we draw line ๐โฒโฒ โฅ ๐ผ. Lines ๐โฒ and ๐โฒโฒ are concurrent and determine a
plane ๐ฝ = (๐โฒ, ๐โฒโฒ) and as ๐โฒ โ ๐ฝ, ๐โฒ โ ๐ผ โน
From (1) and (2) โน in plane ๐ฝ, on line ๐, at point ๐โฒ two distinct perpendicular lines
had been drawn. False. So ๐โฒ||๐ผ.
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Solution to Problem 224.
Reductio ad absurdum. Let ๐ โฆ ๐โฒ. We draw ๐โฒโฒ||๐ through ๐โ.
โน at point ๐โฒ we can draw two perpendicular lines to plane ๐ผ. False.
So ๐||๐โฒ.
Solution to Problem 225.
Let ๐ โฅ ๐ผ and ๐ โฉ ๐ผ = {๐}. We draw through ๐ a parallel to ๐โฒ, which will be
contained in ๐ผ, then ๐||๐ผ.
Solution to Problem 226.
We assume ๐ฝ โน ๐ผ โฉ ๐ฝ โ โ and let ๐ด โ ๐ผ โฉ ๐ฝ โน through a point ๐ด there can be
drawn two distinct perpendicular planes on this line. False.
โน ๐ผ || ๐ฝ.
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Solution to Problem 227.
Let ๐ be a point in space with the property ||๐๐ด|| = ||๐๐ต||.
We connect ๐ with the midpoint of segment [๐ด๐ต], point ๐.
So ๐ is on a line drawn through ๐, perpendicular to ๐ด๐ต.
But the union of all perpendicular lines drawn through ๐ to ๐ด๐ต is the perpendicular
plane to ๐ด๐ต at point ๐, marked with ๐ผ, so ๐ โ ๐ผ.
Vice-versa: let ๐ โ ๐ผ,
Solution to Problem 228.
Let ๐ be a point in space with this property:
Let ๐ be the center of the circumscribed circle โ๐ด๐ต๐ถ โน ||๐๐ด|| = ๐๐ต|| = ||๐๐ถ||, so
๐ is also a point of the desired locus.
common side
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According to the previous problem the locus of the points in space equally distant
from ๐ด and ๐ต is in the mediator plane of segment [๐ด๐ต], which also contains ๐. We
mark with ๐ผ this plane. The locus of the points in space equally distant from ๐ต and
๐ถ is in the mediator plane of segment [๐ต๐ถ], marked ๐ฝ, which contains both ๐ and
๐. So ๐ผ โฉ ๐ฝ = ๐๐.
so ๐ โ the perpendicular line to plane (๐ด๐ต๐ถ) in the center of the circumscribed
circle โ๐ด๐ต๐ถ.
Vice-versa, let ๐ โ this perpendicular line
= ||๐ถ๐||, so ๐ has the property from the statement.
Solution to Problem 229.
We draw โฅ from ๐ต to the plane. Let ๐ be the foot of this perpendicular line.
Let
the circle of radius ๐๐ด. Vice-versa, let ๐ โ this circle
so ๐ represents the foot from ๐ต to ๐ด๐.
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Solution to Problem 230.
Let ๐ผ be a plane that passes through ๐ and let ๐ be the โฅ foot from ๐ด to ๐ผ โน
๐ด๐ โฅ ๐ผ.
From
so ๐ โ a perpendicular line to ๐ in ๐ดโฒ, thus it is an element of the perpendicular
plane to ๐ in ๐ดโฒ, which we mark as ๐ and which also contains ๐ด.
๐ โ the circle of radius ๐ด๐ดโฒ from plane ฯ.
*Vice-versa, let ๐ be a point on this circle of radius ๐ด๐ดโฒ from plane ๐.
โน๐ is the foot of a โฅ drawn from ๐ด to a plane that passes through ๐.
Solution to Problem 231.
Let ๐ดโฒ and ๐ตโฒ be the feet of the perpendicular lines from ๐ด and ๐ต to ๐ผ
(โ) a plane ๐ฝ = (๐ด๐ดโฒ, ๐ต๐ตโฒ) and ๐ด๐ต โ ๐ฝ
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are collinear.
In plane ๐ฝ we have
Solution to Problem 232.
Let ๐, ๐โฒ be given lines, ๐ด given point. We draw through ๐ด plane ๐ผ โฅ to ๐โฒ.
If ๐ โฉ ๐ผ = {๐ต}, then line ๐ด๐ต is the desired one, because it passes through ๐ด, meets ๐
and from ๐โฒ๐ผ ๐โฒ๐ด๐ต. If ๐ โฉ ๐ผ = โ there is no solution.
If ๐ โ ๐ผ, then any line determined by ๐ด and a point of ๐ represents solution to the
problem, so there are infinite solutions.
Solution to Problem 233.
right
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Solution to Problem 234.
Let ๐ be a point such that ||๐ด๐|| = ๐.
We draw ๐ด๐ดโฒ โฅ ๐ผ โ ๐ดโฒ fixed point and ๐ด๐ดโฒ โฅ ๐ดโฒ๐.
We write ||๐ด๐ดโฒ|| = ๐.
Then
๐ โ a circle centered at ๐ดโฒ and of radius โ๐2 โ ๐2, for ๐ > ๐.
For ๐ = ๐ we obtain 1 point.
For ๐ < ๐ empty set.
Vice-versa, let ๐ be a point on this circle โน
so ๐ has the property from the statement.
Solution to Problem 235.
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In โ๐๐ถ๐:
But
3. Let ๐ป be the projection of ๐ lcp. plane ๐ด๐ต๐ถ, so
๐ป โ corresponding heights of side ๐ด๐ต. We show in the same way that ๐ด๐ถ โฅ
๐ต๐ป and thus ๐ป is the point of intersection of the heights, thus orthocenter.
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Solution to Problem 236.
First we prove that if a line is โฅ to two concurrent planes โน the planes coincide.
Let
โ๐ด๐ต๐ has two right angles. False. We return to the given problem.
being concurrent, they determine a plane โน ๐ถ๐ท โฅ ๐ด๐ต.
๐ถ, ๐ท, ๐ถโฒ, ๐ทโฒ are coplanar โน ๐ถ๐ถโฒ and ๐ท๐ทโฒ are coplanar.
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Solution to Problem 237.
We draw
โน ๐ต๐ดโฒ height in โ๐ต๐ถ๐ท (1)
โน ๐ดโฒ๐ถ height in โ๐ด๐ต๐ถ (2)
From (1) and (2) โน ๐ดโฒ is the orthocenter โ๐ด๐ต๐ถ โน ๐๐ดโฒ โฅ ๐ต๐ถ.
Solution to Problem 238.
Let
โ๐ต๐๐โฒ and โ๐ถ๐๐โฒ are right at ๐โฒ.
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As |๐๐ด| โก |๐๐ต| โก |๐๐ถ||๐๐โฒ| common side
}
is the center of the circumscribed circle โ๐ด๐ต๐ถ.
Solution to Problem 239.
Let ๐ท be the midpoint of [๐ต๐ถ] and ๐ธ โ |๐ท๐ดโฒ such that ||๐ท๐ธ|| = ||๐ท๐ด||.
๐ด๐ท is median in the โ isosceles
being external for
Solution to Problem 240.
Let ๐ be a point in the plane and |๐ด๐โฒ the opposite ray to ๐ด๐.
According to theorem 1
common
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Solution to Problem 241.
We construct ๐ต on the plane
โน ๐ด๐ถ and ๐ต๐ตโฒ determine a plane ๐ฝ = (๐ด๐ถ, ๐ต๐ตโฒ) โน ๐ด๐ต โ ๐ฝ and on this plane
๐(๐ถ๐ด๏ฟฝ๏ฟฝ) = 900 โ๐(๐ต๐ด๐ตโฒ) .
Solution to Problem 242.
Let ray |๐ด๐ต โ ๐ฝโฒ such that ๐ด๐ต โฅ ๐. Let |๐ด๐ถ another ray such that |๐ด๐ถ โ ๐ฝโ. We draw
๐ต๐ตโฒ โฅ ๐ผ and ๐ถ๐ถโฒ โฅ ๐ผ to obtain the angle of the 2 rays with ๐ผ, namely ๐ต๐ด๐ตโฒ > ๐ถ๐ด๐ถโฒ.
We draw line |๐ด๐ดโ such that ๐ด๐ดโฒ โฅ ๐ผ and is on the same side of plane ๐ผ as well as
half-plane ๐ฝโ.
[๐ด๐ต is the projection of ray [๐ด๐ดโฒ on plane ๐ฝ
Solution to Problem 243.
Let ๐ be the border of ๐ผโฒ and ๐ด โ ๐. We draw a plane โฅ on ๐ in ๐ด, which we mark
as ๐พ.
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In this plane, there is only one ray ๐, with its origin in ๐ด, such that ๐(๐, ๏ฟฝ๏ฟฝ) = ๐.
The desired half-plane is determined by ๐ and ray ๐, because from
Solution to Problem 244.
Let ๐ be the edge of the dihedral angle and ๐ด โ ๐. We draw ๐ โฅ ๐, ๐ โ ๐ผโฒ and ๐ โฅ
๐, ๐ โ ๐ฝโ two rays with origin in ๐ด. It results ๐ โฅ (๐๐). We draw on plane (๐, ๐) ray ๐
such that ๐(๐๏ฟฝ๏ฟฝ) = ๐(๐๏ฟฝ๏ฟฝ) (1).
As ๐ โฅ (๐๐) โน ๐ โฅ ๐.
Half-plane ๐พโฒ = (๐, ๐) is the desired one, because
If we consider the opposite ray to ๐, ๐โฒ, half-plane ๐พโฒโฒ = (๐, ๐โฒ) also forms
concurrent angles with the two half-planes, being supplementary to the others.
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Solution to Problem 245.
Let ๐ be a point in space equally distant from the half-planes ๐ผโฒ, ๐ฝโฒ โน ||๐๐ด|| =
||๐๐ต||.
where ๐ = ๐ผ โฉ ๐ฝ.
Let
|๐๐ด| = |๐๐ต||๐๐| common side right triangle
} โน
โน๐ โ bisector of the angle.
๐ด๐ท๏ฟฝ๏ฟฝ โน ๐ โ bisector half-plane of the angle of half-planes ๐ผโฒ, ๐ฝโฒ.
If ๐โฒ is equally distant from half-planes ๐ฝโฒ and ๐ผโฒโฒ we will show in the same way
that ๐โฒ โ bisector half-plane of these half-planes. We assume that ๐ and ๐โฒ are on
this plane โฅ to ๐, we remark that ๐(๐๐๐โฒ ) = 900, so the two half-planes are โฅ.
Considering the two other dihedral angles, we obtain 2 perpendicular planes, the 2
bisector planes.
Vice-versa: we can easily show that a point on these planes is equally distant form
planes ๐ผ and ๐ฝ.
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Solution to Problem 246.
Let ๐ผ โฉ ๐ฝ = ๐ผ. In plane ๐ฝ we draw
As
but
so from a point it can be drawn only one perpendicular line to a plane,
Solution to Problem 247.
Let
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the line with the same direction. We know that the union of the lines with the same
direction and are based on a given line is a plane. As this plane contains a
perpendicular line to ๐ผ, it is perpendicular to ๐ผ.
Solution to Problem 248.
Let ๐ผ = (๐1, ๐2) the plane of the two concurrent lines and ๐ is a point with the
property ๐(๐, ๐1) = ๐(๐, ๐2). We draw
Let
a bisector of the angle formed by the two lines, and ๐ is on a line ๐ผ which
meets a bisector โ ๐ โ a plane โฅ ๐ผ and which intersects ๐ผ after a bisector. Thus
the locus will be formed by two planes โฅ ฮฑ and which intersects ๐ผ after the two
bisectors of the angle formed by ๐1, ๐2. The two planes are โฅ.
โนโ๐โฒ๐ดโ = โ๐โฒ๐ตโ
||MMโฒ|| common side} โน ๐๐ด โฅ ๐1
And in the same way ๐๐ต โฅ ๐2 โ ๐ has the property from the statement.
Solution to Problem 249.
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Let ๐ฝ โฉ ๐พ = ๐ and ๐ โ ๐ โน ๐ โ ๐ฝ,๐ โ ๐พ. We draw โฅ from ๐ to ๐ผ, line ๐โ.
According to a previous problem
Solution to Problem 250.
Let ๐ฝ and ๐พ be such planes, that is
From
are secant planes and โฅ to ๐ผ.
So their intersection is โฅ through ๐ด to plane ๐ผ.
Solution to Problem 251.
We construct the point on the two planes and the desired plane is determined by
the two perpendicular lines.
Solution to Problem 252.
Let ๐ผ โฉ ๐ฝ = ๐ and ๐ โ ๐. We consider a ray originating in ๐, ๐ โ ๐ผ and we
construct a โฅ plane to ๐ in ๐, plane ๐พ.
Because
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and let a ray originating in ๐, ๐ โ ๐ฝ โฉ ๐พ โ ๐ โ ๐ฝ , ๐ โ.๐พ As ๐ โฅ ๐พ โน ๐ โฅ ๐ and the
desired plane is that determined by rays (๐, ๐).
Solution to Problem 253.
Let ๐ผ โฉ ๐ฝ = ๐ and ๐ โฉ ๐ฝ = {๐ด}.
We suppose that ๐ด โ ๐. Let ๐ โ ๐, we build ๐ โฅ ๐ฝ, ๐ โ ๐ โน ๐ โ ๐ผ.
If
Solution to Problem 254.
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From (1), (2), (3) โน ๐ โฅ ๐ โฅ ๐.
Solution to Problem 255.
Let ๐ด โ int. (๐ผโฒ๐ฝโฒ), ๐ผ โฉ ๐ฝ = ๐.
Let ๐ด โ int. (๐ผโฒโฒ๐ฝโฒ). We show the same way that
๐(๐ต๐ด๏ฟฝ๏ฟฝ) = 1800 โ๐(๐ผโฒโฒ๐ฝโฒ)
๐(๐ผโฒโฒ๐ฝโฒ) = 1800 โ๐(๐ผโฒ๐ฝโฒ)} โน ๐(๐ต๐ด๏ฟฝ๏ฟฝ) = 1800 โ 1800 +๐(๐ผโฒ๐ฝโฒ) = ๐(๐ผโฒ๐ฝโฒ).
If ๐ด โ int. (๐ผโฒโฒ๐ฝโฒโฒ)โน๐(๐ต๐ด๏ฟฝ๏ฟฝ) = 1800 โ๐(๐ผโฒโฒ๐ฝโฒ).
If ๐ด โ int. (๐ผโฒ๐ฝโฒโฒ) โน ๐(๐ต๐ด๏ฟฝ๏ฟฝ) = 1800 โ๐(๐ผโฒ๐ฝโฒ).
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This book is a translation from Romanian of "Probleme Compilate ลi Rezolvate de Geometrie ลi
Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D
and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks
for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics
at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycรฉe Sidi El Hassan
Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and
Dragotesti General School (Romania), but also I did intensive private tutoring for students
preparing their university entrance examination. After that, I have escaped in Turkey in September
1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated
to United States. The degree of difficulties of the problems is from easy and medium to hard. The
solutions of the problems are at the end of each chapter. One can navigate back and forth from
the text of the problem to its solution using bookmarks. The book is especially a didactical
material for the mathematical students and instructors.
The Author