That is also a visual proof of the Pythagorean theorem for the 45-45-90 special case.Socrates showed a shepherd boy how to make a square one half the size of another one...
...that was many centuries ago...but it still works
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Here is a visual proof of the Pythagorean theorem for arbitrary lengths of two sides. The two black squares have the same total area as the green square which is also on the hypotenuse of the right triangle formed by the two shorter lengths:
The pattern also works as a way to tile a room with two batches of square tiles of different sizes.