This is a slight variation on the classic

"Aha!" proof. I find this version a little clearer. The picture says most of it; we'll explain the details below.

The pink inner square has side length

*c*, and hence has area

*c*^{2}. The outer square, which contains it, has side length

*a+b*, and hence has total area

*(***a**+**b**)^{2}, which, multiplied out, is equal to

**a**^{2}*+***b**^{2} + 2

**a****b**.

From the picture, we see that the four blue triangles each have area

**ab**/2, and the total area of the four blue triangles together is 2

**ab**. So, adding the blue triangles and the pink inner square, the outer square must also have area

*c*^{2} + 2

**a****b.**And so, we have

**a**^{2}*+***b**^{2} + 2

**a****b** =

*c*^{2} + 2

**a****b**. Canceling the "cross term" we're left with

**a**^{2}*+***b**^{2} =

*c*^{2}.

Page created on 2/16/2008