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
Isodynamic point point is the intersection of three circles that intersect the circumcircle to form lenses with apex angle π/3. The angles formed by the first isodynamic point with the triangle vertices satisfy the equations formula_29, formula_30, and formula_31. Analogously, the angles formed by the second isodynamic point satisfy the equationsformula_32, formula_33, and formula_34. The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from formula_1 to each of the three sides of triangle formula_3, is equilateral, as is the triangle formed by reflecting formula_1 across each side of the triangle. Among all the equilateral triangles inscribed in triangle