Bijaganita, 1150
Show another proofBehold
A square built on the hypotenuse can be cut into four copies of the original right triangle and one small square — and that dissection alone forces .
Behold.
- Build a square with side , the hypotenuse. Inside it, place four copies of the right triangle, each hypotenuse lying along one side of the square, each rotated a quarter turn from its neighbor.
- The four right-angle vertices land at the corners of a smaller square in the middle, with side .
- The big square's area is exactly the four triangles plus that small square: .
- Expand the right side: , and .
- Add them together and the terms cancel exactly: .
- So .
Therefore c² = a² + b². ∎
The twelfth-century mathematician Bhāskara II is said to have presented this dissection with a single word of commentary — “Behold” — trusting the picture to carry the whole argument. Swap the inner square for or and the algebra runs identically either way, since both square to the same thing.