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