Each of the red triangles shown below have the same area.
|DA || CB. Slide C to B.||Rotate ΔDAB to ΔCAF||AF || CG. Slide C to E.|
The area of ΔDAC is half the area of the upper left square, and the area of ΔAEF is half the area of the rectangle AEGF. Thus the upper left square has the same area as rectangle AEGF. By a similar argument, the upper right square has the same area as the right hand rectangle. Combining these results gives that the sum of the areas on the two sides of the right triangle ABC is the area of the square on the hypotenuse.