One day last fall, I was making some notes before a lecture in my Geometry for Teachers class. My then-officemate, Todor Milanov, asked me what I would be talking about. When I said “the Pythagorean Theorem”, he asked me if I was going to prove it. When I said I was, he told me that I was going to prove it “the wrong way”.
Well, this was intriguing. I hadn’t told him which of the various proofs I knew was the one I intended to give. How was he so sure I would do it “the wrong way”? And what did that even mean?
I asked him (a bit skeptically at first) to show me “the right proof”, and now I will show you.
First, “the wrong proof”. Traditionally, the Pythagorean Theorem is phrased in terms of squares.
If you draw three squares, one based on each side of a right triangle, then the combined area of the smaller two squares equals the area of the largest square.
But we can give an even simpler visual proof of the Pythagorean Theorem, if we step back and understand better what it really says.
The crucial concept for us is the similarity ratio, which is something that most people understand on an intuitive level. Suppose that I decide to build, in my backyard, a larger-than-life statue of me. It will be exactly the same shape as me, but twice as tall. Then, without knowing anything about the precise shape of me, you know that its left arm will be twice as long as mine, its belt will be twice as long as mine, etc. But its footprint will be four times as large as mine, since that’s a two-dimensional measurement; likewise its shirt would take four times as much fabric as mine, etc. And it would take up eight times as much space as I do.
Suppose that the sides of a right triangle are in the proportion , and suppose that I take three similar versions of the same shape (it could be a square, but it could be any shape: a semicircle, a stop sign, the outline of my face…) in the same proportions. Then the areas of the three shapes will be in the proportions .
So the Pythagorean Theorem is really asserting the following.
If you draw three similar shapes, one based on each side of a right triangle, then the combined area of the smaller two shapes equals the area of the largest shapes.
Crucially, the statement is true for one shape if and only if it’s true for all shapes. The key is to use the triangle itself. Consider the following diagram.
Here an arbitrary right triangle is divided into two by an altitude. By standard angle-chasing, the two smaller triangles are similar to each other and to the large triangle. Furthermore, these three similar triangles are in the same proportions as the sides of the right triangle (look at the hypotenuses).
Bottom line: to prove the Pythagorean Theorem once and for all, all I have to do is show that the green triangle and the blue triangle together have the same area as the large triangle. But this is obvious.
The Pythagorean Theorem has nothing to do with squares and everything to do with similarity ratios.