Two Honest Proofs of the Pythagorean Theorem

1. The picture proof: one square, four triangles

Take four copies of one right triangle, with legs a and b and long side (hypotenuse) c. Lay them in a big square whose side is a + b, first one way and then another. Watch what is left uncovered.

How this picture stays honest. The triangles never stretch, shrink, fade or turn into anything else. Each one slides as a solid piece, and you can see every bit of its trip. The numbers under the picture are worked out from the shapes on the screen at every moment. Change a and b and it still works, because the argument works for every right triangle.

2. Euclid's proof: Elements Book I, Proposition 47

Here is the proof in Euclid's Elements. In our figure the right angle is at A, the legs are AB = a and AC = b, and the hypotenuse is BC = c. Squares stand on all three sides. The square on the legs ABFG is moved, one honest step at a time, onto one part of the big square, and the square ACKH onto the other part.

A word about shears. A shear slides one side of a shape along its own line while the opposite side stays put. The base and the height stay the same, so the area stays the same (Euclid I.35 and I.41). Every shear here is labelled. While a piece moves, its area is measured from the shape on the screen and shown under the picture. Euclid himself argues with the hatched triangles; the shears and the turn are a way to see his argument. While a piece turns, it may pass over the other piece or the triangle. Each piece is still one solid shape, so its area does not change.

A note on the name. The theorem is named for Pythagoras of Samos. The proof on this page is the one written down in Euclid's Elements, Book I, Proposition 47, which closes Book I. No writing by Pythagoras himself has come down to us.

3. Try it