A more motivated proof of the Pythagorean theorem

So far every proof I’ve known of the Pythagorean theorem has adhered to a narrative along the lines of

  • Notice, purely by accident, that in known right triangles it appears that the square on the hypotenuse is always equal to the sum of the squares on the other two sides.
  • Conjecture that this holds in general.
  • Draw a right triangle and a square on each side.
  • Figure out some ingenious geometric decomposition reassembling the two smaller squares into a copy of the bigger one.

This is fairly unsatisfying, because it only tells us that the theorem is true; it doesn’t do much to tell us why it’s true, or give us much intuition for what kind of information it does or does not encode.

Today I wondered if there was a better explanation, and I came across this:

Pythagoras’s theorem | What’s New

Terry Tao writes:

it is perhaps the most intuitive proof of the theorem that I have seen yet

The proof just comes down to examining the (obviously useful) construction where a right triangle is split into two smaller right triangles, both of which are similar to the big one.

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