Pythagoras' theorem

The Method of Two Squares.

Draw a square, and then draw a second smaller square obliquely inside it, so that the corners meet its edges.

The area of the system can be expressed in two ways. First as the area of the larger square, and secondly as the area of the smaller square plus four triangles.

area = (a + b)²

area = c² + 4.(a.b)/2

so

a² + 2.a.b + b² = c² + 2.a.b

and