One of the oldest and simplest proofs of the Pythagorean

Chapter 1, Problem 33

(choose chapter or problem)

One of the oldest and simplest proofs of the Pythagorean theorem is found in the ancient Chinese text Chou Pei Suan Ching. This text was written during the Han period (206 B.C.A.D. 222), but portions of it may date back to 600 B.C. The proof in Chou Pei Suan Ching is based on this diagram from the text. In this exercise we explain the details of the proof. A diagram accompanying a proof of the Pythagorean theorem in the ancient Chinese text Chou Pei Suan Ching [from Science and Civilisation in China, vol. 3, by Joseph Needham (Cambridge, England: Cambridge University Press, 1959)]. (a) Starting with the right triangle in Figure A, we make four replicas of this triangle and arrange them to form the pattern shown in Figure B. Explain why the outer quadrilateral in Figure B is a square. Figure A Figure B c b a A B C D c-y b a x y c (b) The unshaded region in the center of Figure B is a square. What is the length of each side? (c) The area of the outer square in Figure B is (side)2 c2 . This area can also be computed by adding up the areas of the four right triangles and the inner square. Compute the area in this fashion. After simplifying, you should obtain a2 b2 . Now conclude that a2 b2 c2 , since both expressions represent the same area. C