The bride's chair

2026-05-11

Draw squares on every side of a right triangle. The two small squares have a job: between them, they have to fill the big one. Watch them do it.

This is Proposition I.47 of Euclid's Elements, the proof of Pythagoras' theorem the western world has been teaching for 2,300 years. The figure is sometimes called the bride's chair, sometimes the windmill - the shape the three squares cut around the central triangle.

The hidden line

Drop a perpendicular from the right-angle vertex straight down onto the hypotenuse. Where it lands - call that point F - splits the hypotenuse into two pieces. The left piece has length a²/c, the right piece b²/c. (Add them: a²/c + b²/c = (a² + b²)/c. By the theorem we are about to prove, this is c²/c = c. The pieces fit.)

Now extend that perpendicular through the hypotenuse-square. It divides the big square into two rectangles - one for each piece of the hypotenuse. The left rectangle has width a²/c and height c: area a². The right rectangle has width b²/c and height c: area b².

So the hypotenuse-square has already split itself into a piece of area a² and a piece of area b². The theorem is sitting there, before either of the leg-squares has moved.

The proof Euclid wrote

Euclid does not lerp polygons. He proves the equality with a chain of congruent triangles. Take the blue square (on side a): cut it in half along a diagonal. Take the left rectangle inside the hypotenuse-square: cut it in half along a diagonal too. Show that one half of the blue square is congruent to one half of the rectangle - two sides equal, the angle between them equal. Same half-areas, therefore same full-areas.

The two key ingredients Euclid leans on are: (a) a triangle sharing a base and a height with a parallelogram has half the parallelogram's area, and (b) sliding a triangle along a line parallel to its base doesn't change its area. Two facts about area, four congruent half-triangles, no algebra. The proof works identically for the red square on the other side.

Why this proof felt definitive

Pythagoras' theorem was almost certainly known to the Babylonians a thousand years before Euclid. What Euclid did was build it inside a formal system of definitions and axioms - the first time a mathematical claim was nailed to a foundation strong enough that you could not, in principle, doubt it. The windmill diagram is the capstone of Book I of the Elements. Everything before it is there so that this can be proved.

The big square was already a sum of two rectangles. The leg-squares just had to walk over and prove it.

Other proofs of a² + b² = c²: Bhāskara's rearrangement, Garfield's trapezoid.