Gou-Gu Theorem

One of the basic theorems of geometry in both the East and West concerns the relationship between the sides of a right triangle and their squares, known in the West as the “Pythagorean” theorem, but understood in an equivalent form as the Gou-Gu theorem in China. The ancient Egyptians, Babylonians, and Chinese probably discovered this remarkable property of right triangles by empirically examining the simplest case of 3-4-5 triangles. Whether in its geometric form or more familiar algebraic expression, ${3}^2+{4}^2={5}^2$, the theorem concludes that the sum of the squares on either “side” of the right angle is equal to the square on the hypotenuse (Xian). In China, this was established for right triangles in general, i.e., not just for the 3-4-5 triangle, or for those with sides of integer lengths. The Greek made this discovery as well, but proved it rather differently in the argument presented at the end of the first book of Euclid’s Elements, Proposition I-47.

One of the great...