Euclidean geometry/Pythagorean theorem proofs

If you go back to Chapter 2 here, you'll see that in the introduction, we offered the Pythagorean Theorem as an example of a mathematical theorem that can be proven.

Well, here's a page that's going to show you just how true that is. Here are some hand-selected proofs (with additional commentary to facilitate learning) on proving the Pythagorean Theorem.

The Theorem Itself States...

For any right triangle with two legs a and b, and hypotenuse [the longest side] c, there is a relationship between the lengths of the three sides, such that

$a^{2}+b^{2}=c^{2}$

The length of side a times itself, plus the length of side b times itself, equals the length of side c, times itself.

Proof One:

As neither the size of the triangles nor the box they're contained in changes, the amount of empty space can't change either.