
Pythagoras' Theorem: The "Aha!" Proof

This proof is certainly not original with me! It's at least
several hundred years old. None the less, no collection of visual
proofs seems complete without it. (But see also the
Aha! Proof, version 2.)
Besides
the picture, the only thing needed to complete the proof is the
observation that the sum of the area of the pink square, and the areas
of the four blue triangles, is (A
^{2} + B
^{2}  2AB) + 4*(AB/2) = A
^{2} + B
^{2}.
Actually this is not one of my favorite proofs. It's very nice but for some reason I've never felt like I could really
see the "reason" the crossterms cancel leaving A
^{2} + B
^{2}
standing alone. One thing that helps a bit is the further
observation that when A=0, the B side flops down along edge C, and then
it's obvious. And if we go the other way, and imagine A growing
until A=B, then we can see that the pink square will shrink away to
nothing, leaving four triangles, each of which has area AB/2 ( ... = A
^{2}/2 = B
^{2}/2, in that case); each triangle is half of a square which is half the area of the outer square and once again it's obvious.