Squares on the Sides

In a right triangle, the side facing the right angle is the hypotenuse, the longest side. The other two are the legs. Build a square on each side: the two squares on the legs together have exactly the area of the square on the hypotenuse.

a^2 + b^2 = c^2

This is the Pythagorean theorem, and a, b are the legs, c the hypotenuse. Why it holds: put four copies of the triangle inside a big square of side a + b, one in each corner. The space left in the middle is a square on c. Now slide the triangles together in pairs, into two rectangles in opposite corners: the space left is a square on a plus a square on b. Same big square, same four triangles, so the leftover areas are equal.

It works only for right triangles. Without the right angle, the squares on the two shorter sides add up to more or to less than the third.