Pyhtogorean Theorams of Traingles

Baudhayana (c. 800 B.c.), Pythagoras of Samos (c. 580 B.C.-c. 500 B.c.) Today, young children sometimes first hear of the famous Pythagorean theorem from the mouth of the Scarecrow, when he finally gets a brain in MGM's 1939 film version of The Wizard o{Oz. Alas, the Scarecrow's recitation of the famous theorem is completely wrong! The Pythagorean theorem states that for any right triangle, the square of the hypotenuse length c is equaI to the sum of the squares on the two (shorter) "leg" lengths a and b-which is written as a 2 + b2 = c2 The theorem has more published proofs than any other, and Elisha Scott Loomis's book Pythagorean Proposition contains 367 proofs. Pythagorean triangles (PTs) are right triangles with integer sides. The "3-4-5" PT-with legs oflengths 3 and 4, and a hypotenuse oflength 5- is the only PT with three sides as consecutive numbers and the only triangle with integer sides, the sum of whose sides (12) is equal to double its area (6). After the 3-4-5 PT, the next triangle with consecutive leg lengths is 21-20-29. The tenth such triangle is much larger: 27304197-

273041%-38613965. In 1643, French mathematician Pierre de Fermat (1601- 1665) asked for a PT, such that both the hypotenuse c and the sum (0 + b) had values that were square numbers. It was startling to find that the smallest three numbers satisfying these conditions are 4,565,486,027,761, 1,061,652,293,520, and 4,687,298,610,289. It turns out that the 

second such triangle would be so "large" that if its numbers were represented as feet, the triangle's legs would project from Earth to beyond the sun! Although Pythagoras is often credited with the formulation of the Pythagorean theorem, evidence suggests that the theorem was developed by the Hindu mathematician Baudhayana centuries earlier around 800 B.C. in his book Baudhayana Sulba Sutra. Pythagorean triangles were probably known even earlier to the Babylonians. See also Plimpton 322 (c. 1800 B C.), Pythagoras Founds Mathemaheal Brotherhood (c. 530 B.C.), Quadrature of the Lune (c. +W B.C.), Law or Cosines (c. 1427), and Viviani's Theorem (1659)