to prove:the centroid and circumcenter are equal

Centeroid is the intersection of 3 medians of the triangle while circumcenter is the intersection of perpendicular bisectors of the 3 sides of a triangle meet. 

 

Now, since in an equilateral triangle, perpendicular bisectors and medians are essntially the same (can be proven either by congruency), so the centeroid and circumcenter's location is the same.