A nice discussion, started by Terry Tao, about different proofs of the Pythagorean theorem. Can you see why the two proofs he gives are essentially the same?

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This comment, by Gowers, is also nice:
There is a beautiful discussion of this proof in Polya’s Mathematics and Plausible Reasoning, where he shows not just the proof but how one might think of it. Roughly his account goes like this. You are trying to show that a^2+b^2=c^2. The obvious geometric interpretation of this is to put squares on the sides of the triangle, as one often sees. But a standard move in mathematics, one that Polya strongly advocates, is to generalize. In this instance, if you’re in a generalizing frame of mind, you will notice that you could use any shape: there’s no particular reason to use squares. But another standard move in mathematics is to specialize. One also observes that it is sufficient to prove the result for just one shape. Is there a good shape to pick? Well, a pretty good candidate is a triangle of the shape you start with, since if you put one on the hypotenuse it will have the same area as the triangle itself. It is then easy to see that the other two triangles are congruent to the smaller triangles in your diagram. I hope I’ve explained that clearly: it’s one of my alltime favourite howtothinkofthisproof discussions.