circular geometry robust constructions proofs chapter 4
TRANSCRIPT
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Circular GeometryRobust Constructions
Proofs
Chapter 4
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Axiom Systems:Ancient and Modern Approaches
• Euclid’s definitions A point is that which has no part A line is breadthless length A straight line is a line which lies evenly with
the points on itself etc.
• We need clarification
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Axiom Systems:Ancient and Modern Approaches
• David Hilbert redefined, clarified Cleaned up ambiguities
• Basic objects of geometry point line considered undefined terms plane
• Many geometry texts use Hilbert’s axioms
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Language of Circles
• Definition: Set of points Fixed distance from point A Distance called the radius A called the center
• Interior:
• Exterior:
: ( , )P d A B r
: ( , )P d A B r
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Language of Circles
• Chord of a circle:Line segment joiningtwo points on the circle
• Diameter: a chordcontaining the center
• Tangent: a line containing exactly one point of the circle Will be perpendicular to radius at that point
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• Circumference:Length of theperimeter
• Sector:Pie shaped portionbounded by arc andtwo radii
Language of Circles
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Language of Circles
• Segment:Region bounded byarc and chord
• Central angleCAB, center is theangle vertex
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Language of Circles
• Inscribed angleCDB Vertex is on thecircle Also called an angle
subtended by chord CB
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Inscribed Angles
• Recall results of recent activity 1.4, 1.5 …
• Note fixed relationship between central and inscribed angle subtending same arc
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Language of Circles
• What would the conjecture of Activity 1.7 have to do with this figure from Activity 1.9
• What conjecture would you make here?
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Language of Circles
• Recall conjecture made in Activity 1.8
• This also is a consequence of what we saw in Activity 1.7
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Language of Circles
• Any triangle will be cyclic (vertices lie on a circle)
• Is this true for any four non collinear points?
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Language of Circles
• Some quadrilaterals will be cyclic
• Again, note the properties of such a quadrilateral
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Language of Circles
• Using the results of this activity
• Construct a line through a point exterior to a circle and tangent to a circle
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Robust Constructions
Developing visual proof• Distinction between
“drawing” and “construction”
• In Sketchpad and Goegebra Allowable constructions based on Euclid’s
postulates
• Constructions develop visual proof Guide us in making step by step proofs
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Step-by-Step Proofs
• Each line of the proof presents A new idea or concept
• Together with previous steps Produces new result
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Allowable Argument Justifications
• Site the given conditions
• Base argument on Definitions Postulates and axioms
• Constructions implicitly linked to axioms, postulates
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Allowable Argument Justifications
• Any previously proved theorem
• Previous step in current proof
• “Common notions” Properties of equality, congruence Arithmetic, algebraic computations Rules of logic
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Methods of Proof
1. Start by being sure of what is given
2. Clearly state the conjecture or theorema) P Q
b) If hypotheses then conclusion
3. Note the steps of Geogebra constructiona) Steps of proof may well follow similar order
4. Proof should stand up to questioning of colleagues
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Direct Proof
• Start with given, work step by step towards conclusion
• Goal is to show P Q using modus ponens Based on P, show sufficient conditions to
conclude Q
• Use syllogism: P R, R S, S Qthen P Q
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Indirect Proof
• Use logic role of modus tollens P Q is equivalent to Q P
• We assume Q
• Then work step by step to show that P cannot be true That is P
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Indirect Proof
• Alternatively we use this fact
• Begin by assuming P and not Q• Use logical reasoning to look for contradiction• This gives us• Which means that P Q must be true
P Q P Q
P Q
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Counter Examples
• Consider a conjecture you make P Q You create a Geogebra diagram to illustrate
your conjecture
• Then you discover a specific example where all the requirements of P hold true But Q is definitely not true
• This is a counter example to show that
P Q
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If-And-Only-If Proofs
• This means that
• Also written
• Proof must proceed in both directions Assume P, show Q is true Assume Q, show P is true
P Q and Q P P Q
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Proofs
• Constructed diagrams provide visual proof demonstration of geometric theorems
• Consider this diagram• How might it help
us prove that thenon adjacent angles ofa cyclic quadrilateralare supplementary.
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Proof of Theorem 4.3
• Assume ABCD cyclic
• Consider pair of non-adjacentangles
• Let arc BAD be arc subtendingangle a, BCD be arc subtending b
• We know a + b = 360 and
• Also ½ a = , ½ b =
• So
• And they are supplementary
,BAD BCD
1( ) 1802a b
BAD BCD
180BAD BCD
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Incircles and Excircles
• Consider concurrency of angle bisectors of exterior angles
• PerpendicularPJ gives radiusfor excircle
• Note the otherexterior anglesare congruent
• How to show tangency points M and N?
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Incircles and Excircles
• Proof : Drop perpendiculars from P to lines XY andXZ
• Look for congruentright triangles
• Finish the proof
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Families of Circles
• Orthogonal circles:Tangents are perpendicular at points of intersection
• Describe how youconstructed these in Activity 8.
• How would you construct more circles orthogonal to circle A?
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Orthogonal Circles
• Describe what happens when point Q approaches infinity.
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Families of Circles
• Circles that share a common chord
• Note centers are collinear Use this to construct more circles with chord AB
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The Arbelos and Salinon
• Figures bounded by semicircular arcs
• What did you discover about the arbelos in Activity 7?
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The Arbelos and Salinon
• Note the two areas – the arbelos and the circle with diameter RP
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The Arbelos
Algebraic proof
• Calculate areas of allthe semicircles
• Calculate the are of circlewith diameter RP
• Show equality
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Power of a Point
• We are familiar with the concept of a function Examples:
• Actually Geogebra commands are functions that take points and/or lines as parameters
2 3
2
( ) 6 9
( , ) 3 7 4
f x x x
g x y x xy y
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Power of a Point
• Consider a mathematical function involving distances with a point and a circle
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Power of a Point
• This calculation clams to be another way to calculate Power (P, C)
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Power of a Point
• Requirements for calculating Power (P, C) Given radius, r, of circle O and distance d
(length of PO) use
or …
Given line intersecting circle O at Q and R with collinear point P use
2 2d r
PQ PR
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Point at Infinity
• Recall stipulation Pcannot be at O
• As P approachesO, P’ gets infinitelyfar away
• This is a “point at infinity” (denoted by )
• Thus Inversion(O, C) =
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The Radical Axis
• Consider Power as a measure of Distance d from P to given circle
• If radius = 0, Power is d2
• Consider two circles, centered at A, B Point P has a power for each There will be some points P where Power is
equal for both circles Set of such points called radical axis
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The Radical Axis
• Set of points P with Power equal will be bisector of segment AB
• Construction when circles do not intersect?
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The Radical Axis
• Suppose three circles are given for which the centers are not collinear. Each pair of circles determines a radical axis, These three radical axes are concurrent.
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Nine-Point Circle (2nd Pass)
• Recall circle which intersects feet of altitudes (Activity 2.8)
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Nine-Point Circle
• Note all the points which lie on this circle
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Nine-Point Circle
• Additional phenomena Nine point circle tangent to incircle and
excircles
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Circular GeometryRobust Constructions
Proofs
Chapter 4