My proof of the Pythagorean Theorem.

Start with a right triangle, labeling the short side (a), the long side (b), and the hypotenuse (c). Take four such triangles of identical dimensions, and arrange them so that their hypotenuses form the perimeter of a square. In this arrangement, the layout of the triangles would appear identical if the square were rotated 90, 180, or 270 degrees.

The area of the square is c². There is also an inner square, each side of which is the section of the (b) side of a triangle not shared by the (a) side of a contiguous triangle. The area of this inner square is (b - a)². The area of each triangle is ab/2, so that the area occupied by the four triangles is 2ab.

Because the area of the outer square is the sum of the areas of the triangles and the inner square, it can be expressed as (b - a)² + 2ab, which simplifies to b² + a².

Thus, c² = b² + a²; or, more as it's generally expressed, a² + b² = c².