One of the oldest and simplest proofs of the Pythagorean theorem is found in the ancient...

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.

(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.