Pythagoras Theorem 

There are literally hundreds of proofs of this theorem, including one found/created by James A. Garfield, who later became US president.
Albert Einstein also discovered a proof which is demonstrated at: http://demonstrations.wolfram.com/EinsteinsMostExcellentProof/
What is less commonly known is that this is an "if and only if."
Consider a triangle T with sides a, b and c, with c the longest. Stating both parts:
Consider a rightangled triangle. Draw the triangle on the complex plane with the hypotenuse running from the origin into the first quadrant, and the right angle on the Xaxis. The vertices of the triangle are now at 0, /a+0i/, and a+bi. Using Euler we can write $a+bi=ce^{i\theta}.$
Take the complex conjugate, and multiply. That gives us
$(a+bi)(abi)=(ce^{i\theta})(ce^{i\theta})$ 
$a^2+b^2=c^2$ 
Most of this is reversible, so there's very little to check for the other direction.