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
Automedian triangle sharing no common factor) as with formula_4 and formula_5 coprime, formula_6 odd, and to satisfy the triangle inequality formula_7 (if the quantity inside the absolute value signs is negative) or formula_8 (if that quantity is positive). Then this triangle’s medians formula_9 are found by using the above expressions for its sides in the general formula for medians: where the second equation in each case reflects the automedian feature formula_11 From this can be seen the similarity relationships There is a primitive integer-sided automedian triangle that is not generated from a right triangle: namely, the equilateral triangle with sides of unit