Euclid I.47: the windmill
Euclid's proof of the Pythagorean theorem, shown as shears and turns that keep area.
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Euclid I.47: the windmill
- A right triangle ABC, right angle at C, with the squares on its sides: a² on BC, b² on AC, and c² on the hypotenuse AB.
- Draw the line from C perpendicular to AB, through D, across the square on AB. It cuts that square into two rectangles.
- Shear the square on AC: keep its side AF and slide the opposite side along its own line until C reaches B. Same base, same height, so the parallelogram still has area b².
- Turn the parallelogram a quarter turn about A. Nothing changes size. (This is Euclid's step: the triangles FAB and CAK are congruent, side-angle-side.)
- Shear it again, keeping its side on AK and sliding the far side down the line through C and D. It becomes the rectangle under AD, so that rectangle has area b².
- Do the same with the square on BC: shear it, keeping its outer side, until C reaches A.
- Turn it a quarter turn about B.
- Shear it onto the rectangle under DB: that rectangle has area a².
- The two rectangles make the square on AB: a² + b² = c².
Euclid, Elements, Book I, Proposition 47 (c. 300 BC); the proof shown as shears and a turn, redrawn
Where this proof comes from
Euclid taught in Alexandria, Egypt, around 300 BC. His book, the Elements, was used to teach geometry for more than 2,000 years, and this theorem is Book I, Proposition 47. Euclid’s own proof uses pairs of equal triangles; the slides and turns here show the same steps as moving shapes. Loomis’s collection of proofs (1927) lists it as proof 15.
Read Euclid’s proof in David Joyce’s free online Elements: Book I, Proposition 47