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
Isosceles triangle an equilateral triangle. The first solutions of the problem with unbounded oscillations were found in the isosceles three-body problem. Long before isosceles triangles were studied by the ancient Greek mathematicians, the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area. Problems of this type are included in the Moscow Mathematical Papyrus and Rhind Mathematical Papyrus. The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid. This result has been called the "pons asinorum" (the bridge of asses) or the isosceles triangle theorem. Rival explanations for this name