Szyszkowski equation

The Calabi triangle is a very special triangle found by Eugenio Calabi. Consider the largest square that can be inscribed in an arbitrary triangle (all vertices of the square lie on sides of the triangle). It may be that such a square could be positioned in the triangle in more than one way. If the largest inscribed 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

$x={1 \over 3}+{(-23+3i{\sqrt {237}})^{1/3} \over 3\cdot 2^{2/3}}+{11 \over 3\cdot (2\cdot (-23+3i{\sqrt {237}}))^{1/3}}$

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

$2x^{3}-2x^{2}-3x+2=0$

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

See also

• Biggest Little Polygon