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:
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.