Prove that a triangle is equilateral if and only if its centroid and circumcenter are the same point.
One way is pretty easy. As for the other way, we assume that the centroid and circumcenter are the same point, say, G. Let D be the midpoint of BC. Since D and G are two common points of the median and perpendicular bisector corresponding to BC, they must be the same line. Can you complete the proof ?