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 to the points where the angle bisectors of ∠"APB", ∠"BPC", and ∠"CPA" cross the sides ("A", "B", and "C" being the vertices). For any point P in the plane, with distances "p", "q", and "t" from the vertices A, B, and C respectively, For any point P on the inscribed circle of an equilateral triangle, with distances "p", "q", and "t" from the vertices, and For any point P on the minor arc BC of the circumcircle, with distances "p", "q", and "t" from A, B, and C respectively, and moreover, if point D on side BC divides PA into