The Pythagorean theorem states that, for a right triangle with side lengths $a$, $b$ and hypotenuse $c$:

There are many proofs, but my favourite is a proof by diagram. Arrange four copies of the triangle in a square with side length $a+b$ in two different ways. In the first the leftover space forms two squares of area $a^2$ and $b^2$ respectively. In the other, the leftover space is a single square of area $c^2$.

This proof makes for a neat design that I used for my latest watercolour-on-wood experiment.