Kepler triangle

A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is ${\displaystyle {\sqrt {\varphi }}}$ where ${\displaystyle \varphi =(1+{\sqrt {5}})/2}$ is the golden ratio, and the progression can be written: ${\displaystyle 1:{\sqrt {\varphi }}:\varphi }$, or approximately ${\displaystyle 1:1.272:1.618}$. Squares on the edges of this triangle have areas in another geometric progression, ${\displaystyle 1:\varphi :\varphi ^{2}}$. Alternative definitions of the same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles.

This triangle is named after Johannes Kepler, but can be found in earlier sources. Although some sources claim that ancient Egyptian pyramids had proportions based on a Kepler triangle, most scholars believe that the golden ratio was not known to Egyptian mathematics and architecture.