Equilateral Triangle - Characterizations

Characterizations

A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii r_a, r_b, r_c (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following eight categories is true. These are also properties of an equilateral triangle.