The Pythagorean Theorem

The Pythagorean Theorem is one of the oldest, best known, and most useful theorems in all of mathematics, and it has also surely been proved in more different ways than any other. Euclid gave two proofs of it in the Elements, as Proposition I,47, and also as Proposition VI,31, a more general but less well-known formulation concerning arbitrary ‘figures’ described on the sides of a right triangle. The first of those demonstrations is based on a comparison of areas and the second on similarity theory, a basic distinction that can be used as a first step in classifying many other proofs of the theorem as well.