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
Equilateral triangle the location of "P". Pompeiu's theorem states that, if "P" is an arbitrary point in an equilateral triangle "ABC", then there exists a triangle with sides of lengths "PA", "PB", and "PC". That is, "PA", "PB", and "PC" satisfy the triangle inequality that any two of them sum to at least as great as the third. By Euler's inequality, the equilateral triangle has the smallest ratio "R"/"r" of the circumradius to the inradius of any triangle: specifically, "R"/"r" = 2. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest