Calabi triangle

The Calabi triangle is a special triangle found by Eugenio Calabi. Consider the largest square that can be placed in an arbitrary triangle. It may be that such a square could be positioned in the triangle in more than one way. If the largest such square can be positioned in three different ways, then the triangle is either an equilateral triangle or the Calabi triangle.

The Calabi triangle is isosceles. The ratio of the base to either leg is

which approximates to 1.55138752454. It is the largest positive root of

and has continued fraction representation [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...].

The Calabi triangle is obtuse with base angles 39.1320261...° and third angle 101.7359477...°.