The Life of a Mathematician

Okay, I'm not in this semester's Modern Geometry class, but I have become mildly obsessed with one of their homework problems. The statement is simple enough: Given that the circumcenter of a triangle is the same as the Gergonne point, prove that the triangle is equilateral. The circumcenter is the center of the circumcircle, which is the circle that circumscribes the triangle, i.e., intersects all three vertices. It is also the intersection point of the perpendicular bisectors of the sides. The Gergonne point is defined as the point of intersection of the lines connecting the vertices with the points of tangency for the incircle, where the incircle is the circle inscribed in the triangle. The approach I've been trying is to show that when the Gergonne point and the circumcenter are the same, they must also be the same as the incenter, which is the center of the incircle. But I haven't had success yet. Apparantly no one in the class, including the teacher, has been able to solve this.