We start with a right triangle with sides a, b, and c. We then build a big square, out of four copies of our triangle. We end up with a square, in the middle, with sides c (we can easily show that this is a square).
We now construct a second big square, with identical triangles which are arranged as in the lower part of the diagram. This square has the same area as the square above it.
We now sum up the parts of the two big squares:
Area=2ab + c²
Area=2ab + a² + b²
These two areas are equal:
2ab + c²=2ab + a² + b²
c²=a² + b²