Integer triangle cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles. Conditions are known in terms of elliptic curves for an integer triangle to have an integer ratio "N" of the circumradius to the inradius. The smallest case, that of the equilateral triangle, has
Triangle center if the trilinear coordinates of P can be expressed in the form f(A) : f(B) : f(C) where f(A) is a function of the angle A alone and does not depend on the other angles or on the side lengths. A triangle center P is called a "transcendental triangle center" if P has no trilinear representation using only algebraic functions of a, b and c. Let "f" be a triangle center function. If two sides of a triangle are equal (say "a" = "b") then so two components of the associated triangle center are always equal. Therefore all triangle centers