Calabi triangle

The Calabi triangle is a special triangle found by Eugenio Calabi.[1] 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.[2][3]

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, ...].[2]

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

Biggest Little Polygon

References

Eugenio Calabi (3 Nov 1997). "Outline of Proof Regarding Squares Wedged in Triangle". Retrieved 15 November 2012.
Calabi's triangle at Mathworld
Conway, J.H.; Guy, R.K. (1996). The Book of Numbers. New York: Springer-Verlag. p. 206.

Mathematics Encyclopedia