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
Malfatti circles corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces. The difference in area for an equilateral triangle is small, just over 1%, but as pointed out, for an isosceles triangle with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles. The problem of constructing three circles tangent to each other within a triangle was posed by the 18th-century Japanese mathematician Ajima Naonobu prior to the work of Malfatti,