1. What Should Adorn Next Year’s Math Contest T-shirt? By Kevin Ferland

2. Last Year’s T-shirt total area = area of inner square + area of 4 triangles b a a b a b ab c c c c

3. Proof Pythagoras 585-500 B.C.E. The t-shirt proof is believed to be the type used by the Pythagoreans.

4. Pythagorean Theorem Given a right triangle we have b c a

5. James Garfield 1876 total area = area of inner half-square + area of 2 triangles b a b a c c

6. Euclid 323-285 B.C.E. (modernized) proof from Elements I II I ← c → ab xc-x

7. The result was known by Babylonian mathematicians circa 1800 B.C.E.

8. The oldest known proof is found in a Chinese text circa 600 B.C.E.

9. January 2010

10. Last Year’s T-shirt total area = area of inner square + area of 4 triangles b a a b a b ab c c c c

11. Generalizing the t-shirt total area = area of inner hexagon + area of 6 triangles ba b a b a ba b a b a c c c c c c 120°

12. 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

13. Proof

14. Result Given a 120°-triangle we have 120° a b c

15. b a ba b c c a c

16. STOP?

17. Clearly, this argument extends to any regular 2k-gon for k ≥ 2. DON’T STOP

18. Result (n = 2k) Given a θ-triangle we have θ a b c

19. θnEquation c 2 = 90°4 120°6 135°8 144°10 150°12 ׃׃׃

20. FACT : The area of a regular n-gon with side length s is s s s s s s s s θ/2 θ θ θ

21. h s/2

22. FACT : The area of a θ-triangle with (short) sides a and b is θ a b c h

23. Trig Identity : Proof

24. Generalized Pythagorean Theorem total n-gon area = area of inner n-gon + area of n θ-triangles b a ba b c c a c

25. Proof

26. Result Given a θ-triangle we have θ a b c The LAW OF COSINES in these cases.

27. Pythagorean Triples A triple (a, b, c) of positive integers such that a 2 + b 2 = c 2 is called a Pythagorean triple. It is called primitive if a, b, and c are relatively prime.

28. E.g. Whereas, (6, 8, 10) is a Pythagorean triple that is not primitive. 3 5 4

29. Theorem : All primitive solutions to a 2 + b 2 = c 2 (satisfying a even and b consequently odd) are given by where

30. stabc 21435 3212513 4181513 4324725 52202129 5440941 61123537 65601161 72284553 74563365 76841385 ׃׃׃׃׃

31. E.g. n = 6, θ = 120° What is the characterization of all such primitive triples? 120° 5 3 7

32. Other 120°-triples (7, 8, 13), (5, 16, 19), … There does exist a characterization of these. What can you find?

33. Next Year’s t-shirt ba b a b a ba b a b a c c c c c c 120°