Euclid I.47: the windmill
Euclid, Elements, Book I, Proposition 47 (c. 300 BC); the proof shown as shears and a turn, redrawn
Garfield's trapezoid
James A. Garfield, New-England Journal of Education 3 (1876), p. 161; redrawn
More proofs as puzzles
Perigal, Bhaskara, Thabit ibn Qurra, an-Nayrizi, Liu Hui, Leonardo: cut and rebuild them on the dissection board.