what should adorn next year’s math contest t-shirt? by kevin ferland
TRANSCRIPT
What Should AdornNext Year’s MathContest T-shirt?
By
Kevin Ferland
Last Year’s T-shirt
total area = area of inner square+ area of 4 triangles
b a
a
b
a
b
a b
c
c
c
c
Proof
Pythagoras 585-500 B.C.E.The t-shirt proof is believed to be the type used by the Pythagoreans.
2 2 12( ) 4( )a b c ab
2 2 22 2a ab b c ab 2 2 2a b c
Pythagorean Theorem
Given a right triangle
we have222 bac
bc
a
James Garfield 1876
total area = area of inner half-square+ area of 2 triangles
)(2))(( 212
21
21 abcbaba
b
a
b
a
c
c
Euclid 323-285 B.C.E.
(modernized) proof from Elements
222
2
2
)(
cba
xccb
cxa
bc
xcb
ac
xa
I
II
I II
← c →
a b
x c-x
The result was known by Babylonian mathematicians circa 1800 B.C.E.
The oldest known proof is found in a Chinese text circa 600 B.C.E.
January 2010
Last Year’s T-shirt
total area = area of inner square+ area of 4 triangles
b a
a
b
a
b
a b
c
c
c
c
Generalizing the t-shirt
total area = area of inner hexagon
+ area of 6 triangles
b a
b
a
b
a
ba
b
a
b
a
c
c
c
c
c
c
120° 120°
120° 120°
120° 120°
• FACT: The area of a regular hexagon with side length s is
• FACT: The area of a 120°-triangle with (short) sides a and b is
22
33 s
ab43
Proof
)(6)( 432
2332
233 abcba
abcba 22)(
abcbaba 222 2222 cabba
Result
Given a 120°-triangle
we have
abbac 222
120°
a
b
c
8, 135n
b
a
b ab
c
c
a
c
STOP?
Clearly, this argument extends to any regular 2k-gon for k ≥ 2.
DON’T STOP
Result (n = 2k)
Given a θ-triangle
we have
180( 2)(for )nn
θ
a
b
c
2 2a b
2 2a b ab
2 2 2a b ab
2 2a b ab
2 2 3a b ab
θ n Equation c2 =
90° 4
120° 6
135° 8
144° 10
150° 12
׃ ׃ ׃
FACT: The area of a regular n-gon with side length s is2 180( 2)
4 2tan (where )n s nn
s
s
s
s
s
s
s
s
θ/2
θ/2
θ
θ
θ
2
2 2
12
4 2
2 2
tan
triangle area
tan
-gon area = tan
s
s
s
h
sh
n n
θ/2
h
s/2 s/2
FACT: The area of a θ-triangle with (short) sides a and b is12 sinab
1 12 2
sin(180 ) sin
area sin
h b b
ah ab
θ
a
b
ch
Trig Identity:
Proof
2
sintan
1 cos
22 2 2 2 2(1 cos ) tan (2cos ) tan 2cos sin sin
Generalized Pythagorean Theorem
total n-gon area = area of inner n-gon
+ area of n θ-triangles
b
a
b ab
c
c
a
c
Proof
)sin21(
2tan
4
2
2tan
4
2)( abnncban
1sin
2cos1sin
4
2
cos1sin
4
2)(
nabncban
)cos1(2)( 22 abcba
cos222 222 ababcbaba
222 cos2 cabba
Result
Given a θ-triangle
we have2 2 2 2 cosc a b ab
180(2 2)2(for , k 2)kk
θ
a
b
c
The LAW OF COSINES in these cases.
Pythagorean Triples
A triple (a, b, c) of positive integers such that a2 + b2 = c2
is called a Pythagorean triple.
It is called primitive if a, b, and c are relatively prime.
E.g.
Whereas, (6, 8, 10) is a Pythagorean triple that is not primitive.
35
4
Theorem: All primitive solutions to a2 + b2 = c2 (satisfying a even and b consequently odd)
are given by
where
2 2
2 2
a st
b s t
c s t
0
gcd( , ) 1
(mod 2)
s t
s t
s t
s t a b c
2 1 4 3 5
3 2 12 5 13
4 1 8 15 13
4 3 24 7 25
5 2 20 21 29
5 4 40 9 41
6 1 12 35 37
6 5 60 11 61
7 2 28 45 53
7 4 56 33 65
7 6 84 13 85
׃ ׃ ׃ ׃ ׃
E.g. n = 6, θ = 120°
What is the characterization of all such primitive triples?
120°
5
3
7
Other 120°-triples
(7, 8, 13), (5, 16, 19), …
There does exist a characterization of these.
What can you find?
Next Year’s t-shirtb a
b
a
b
a
ba
b
a
b
a
c
c
c
c
c
c
120° 120°
120° 120°
120° 120°
222 cabba