Plane Geometry - Bruce E. Shapiro

198 SECTION 38. THE PYTHAGOREAN THEOREMProof. (Third Proof of the Pythagorean Theorem 7 ) Let △ABC bea right triangle with vertices A, B, C opposite sides a, b, c, and right angleat C. Label the vertices so that a ≤ b. See figure 38.6.On the left hand side of figure 38.6, the large square of area c 2 is equalto four times the area of triangle △ABC plus the area of the small squarewith sides of length b − a. HenceÅ ã abc 2 = (4) + (b − a) 2 (38.8)2= 2ab + b 2 − 2ab + a 2 (38.9)= a 2 + b 2Proof. (Fourth Proof of the Pythagorean Theorem.) This is a varianton the previous proof. Rearrange the triangles as shown on the right handside of figure 38.6. The four triangles plus the small square are rearranged7 Attributed to the Indian mathematician Bhaskara Acharya (1114-1185), one of thegreatest mathematicians of all time. Bhaskara is also attributed with the first proofsof the general solutions of the quadratic, cubic, and quartic equations; as well as thesolution of several Diophantine equations of the second order that were later posed byFermat, not knowing that they had been solved centuries earlier; and the developmentof the differential calculus and theory of infinitesimals, some 500 years prior to Newton.Essentially the same proof is also given in the Zhoubi Suanjing, published in Chinaapproximately 1000 years prior to that, the the logic by which (38.8) is reached isslightly different.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012