Kepler and Pythagoras

This work is a rearrangement proof of the Pythagorean theorem using Kepler triangles. The Kepler triangles in the corners have bases proportional to one, heights proportional to √φ, and hypotenuses proportional to φ; the edges form the geometric progression 1 : √φ : φ while the squares of the edges form the geometric progression 1 : φ : φ².

The central void bounded by the hypotenuses of the Kepler triangles has an area of φ². Rearranging the triangles such that triangles on opposing corners are joined creates two squares, with areas equal to 1 and φ. Because the area of the void is constant, the rearrangement proves that φ² = φ + 1.

The astronomer, mathematician, and polymath  Johannes Kepler, for whom the triangles are named, said:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.

This work mounts Kepler's jewel in Pythagoras' gold.