Euclid's proof of Pythagoras' Theorem

This proof is essentially Euclid's own proof of Proposition I.47.

Like many proofs, it partitions the square on the hypotenuse by dropping a perpendicular from the right angle through it. Then it performs a sequence of shears and rotations to show corresponding areas are equal. It does not, however, transform the areas directly concerned, but rather looks at triangles obtained by halving them.

Click on a figure to start an animation, to pause it in the middle of an animation, or to reset it to its initial configuration if it is finished.

The Jim Morey's first Java proof of Pythagoras' Theorem (which won Grand Prize in the first Java contest) follows a variant of this argument.


In Euclid's diagram, three of the lines drawn seem to meet in a single point around the middle of the triangle. Do they? Prove your assertion.

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