Calabi triangle triangle or the Calabi triangle. Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square. The Calabi triangle is isosceles. The ratio of the base to either leg is which approximates to 1.55138752454. In terms of trigonometric functions, it is 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...°. Calabi triangle The Calabi triangle is a
Langley’s Adventitious Angles the same 80-80-20 triangle but different internal angles. A quadrilateral such as BCEF in which the angles formed by the diagonals with the sides are all rational (when measured in degrees) is called an adventitious quadrangle. Numerous constructions for other adventitious quadrangles beyond the one appearing in Langley's puzzle are known. They form several infinite families and an additional set of sporadic examples. Classifying the adventitious quadrangles (which need not be convex) turns out to be equivalent to classifying all triple intersections of diagonals in regular polygons. This was solved by Gerrit Bol in 1936 (Beantwoording van prijsvraag # 17,