Prove that a triangle is equilateral if and only if the circumcenter and incenter coincide.

Definitions:
- Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle.
- The circumcenter of a triangle is the point where the three perpendicular bisectors meet. This point is the same distance from each of the three vertices of the triangles.