Pythagoras' Theorem

This is with good reason the most famous theorem in all of Mathematics!
It is the key to understanding how angles and distances are related, and it allows us to find the distances between distant points in space, as well as showing us the existence of lengths which cannot be related by whole number ratios. (An example of such an "irrational" length occurs if we take a and b of length 1 in the picture, for then c squared is 2 and this is not possible for any ratio of whole numbers. This was very upsetting to Pythagoras and his followers as their philosophy idealized such ratios. Ironically it is the theorem bearing Pythagoras' own name which was the downfall of much that he believed in.)

There are many ways of proving this important theorem and no-one's education can be said to be complete without understanding at least one.

Here are some links to proofs of the theorem that you can find on the Web:

This award winning presentation of an animated proof of Pythagoras' Theorem by UBC's Jim Morrey was an early example of the use of JAVA to help with the understanding of mathematical ideas.

Here's another presentation of the same proof (by the IES group in Japan)

(This latter site also illustrates a number of other proofs of the theorem, and allows you to see how some apparently different arguments are really just different aspects of the same basic principle.)