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 (A² + B² - 2AB) + 4*(AB/2) = A² + B².

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 cross-terms cancel leaving A² + B² 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 ( ... = A²/2 = B²/2, 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.