Geometric Proof

Orthocenter = intersection of the three altitudes
Circumcenter = intersection of the perpendicular bisectors of the three sides
Centroid = intersection of the three medians
Incenter = intersection of the three angle bisectors

In an equilateral triangle, the altitude, perpendicular bisector, median, and angle bisector (relative to a given side and its opposite angle) are all the same line segment (this is what you need to prove). Therefore, the intersections of these triplets are all the same point.

You can show that these line segments are colinear by rotating the equilateral triangle around its centroid (which by the way is the intersection of these line segments)---

its a classic case of circular logic.

Rap

Thank You
I needed this information to complete my question. Thank you all for your great reply.

Janet