Someone showed me the following problem at the meeting of ATMIM (Association of Teachers of Mathematics in Massachusetts.) It's proof is a great non-routine use of the Pythagorean theorem. I think the proof is too easy to post, but if anyone can't figure it out, post a comment and I will make the solution available.
Let ABCD be a rectangle, and P any point in the interior. Prove that AP2 + PC2 = BP2 + PD2.