Here’s another cool proof of the same fact. I remember reading somewhere that it may be due to Albert Einstein, but who knows if that’s true? Anyway, it’s a really nice proof that takes a quite different approach.
It has to do with a special property of right triangles: for any right triangle, if you draw a line perpendicular to the hypotenuse that goes through the right angle (an “altitude”), it splits the triangle into identical copies of itself. You can see this clearly in the special case of an isosceles right triangle, and it’s easy to prove for any right triangle. To prove it, imagine a right triangle as shown in figure 1, with vertices X, Y, and Z, where Z is the right angle. Let W be the point at which the altitude intersects the hypotenuse. We want to show that triangles XWZ and YWZ are similar to each other and to XYZ. Starting with triangle XWZ, note that ∠XWZ = 90, and so ∠XZW = 180-90-∠X. But 180-90-∠X is exactly what ∠Y is, so triangle XWZ must be similar (all angles equal) to triangle XYZ. The same exercise can be carried out for triangle YWZ.
Incidentally, the word for a little theorem that you have to prove in order to carry out the proof of a big theorem is a “lemma”. So we’ve just proven the lemma that right angles can be split into two smaller copies of themselves.
Alright so now we’ve got that out of the way, let’s get to the proof. A geometric interpretation of the Pythagorean theorem is a statement about areas. It says that for a configuration of squares like in figure 2, the areas of the smaller squares are equal to the area of the big square. Now how to prove it?
We start by using that lemma from above to create a picture like figure 3. I’ll name the newly made triangles with capital letters, with A being the new triangle that has hypotenuse a, B with hypotenuse b, and C the triangle we had in the first place with hypotenuse c.
Essentially by definition, we have A + B = C. File that away somewhere, we’ll need it later.
Let’s get a closer look at these house shapes.
Each of the triangles A, B, and C are similar, so these house shapes are just scaled versions of each other. That means that whatever the ratio of the areas A and a² is, it is the same as the ratio of the areas B and b², which is the same as the ratio of the areas C and c². Let’s give that ratio a name: r. So we have A/a² = B/b² = C/c² = r.
Flipping this equation around a bit, we see that we can write A = ra², B = rb², and C = rc². Can you see where this is going?
Remember that equation we tucked away for later: A + B = C. Plugging these new identities into this equation, we get ra² + rb² = rc². Dividing both sides of the equation by r gives the Pythagorean theorem in all its glory.
