The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield).
What's the most elegant proof?
My favorite is this graphical one:
According to cut-the-knot:
Loomis (pp. 49-50) mentions that the proof "was devised by Maurice Laisnez, a high school boy, in the Junior-Senior High School of South Bend, Ind., and sent to me, May 16, 1939, by his class teacher, Wilson Thornton."
The proof has been published by Rufus Isaac in Mathematics Magazine, Vol. 48 (1975), p. 198.