Pythagorean Cuts

Overview

During the third century BCE, Euclid of Alexandria made the Theorem of Pythagoras (and its converse) the climax of Book I of his Elements of Geometry. Euclid stated the theorem geometrically, as a theorem about three squares built on the sides of a right triangle. His proof consisted of cutting the square on the hypotenuse into two rectangles, each having area equal to that of one of the remaining squares on the sides. Can Euclid's "Pythagorean cut" be adapted to parallelograms, triangles, n-gons, and semicircles built on the sides of a right triangle? To find out, read on!