Napoleon's theorem is easy enough to prove using a geometric approach. What I am unable to do is complete a proof using vector methods. For the purposes of discussion, let OAB be an arbitrary triangle. Let H, F and G be the centroids of the equilaterial triangles OAC, ABE and OBD constructed on the sides OA, AB and OB respectively. One must prove that triangle HFG is equilateral.

I have pursued an approach involving scalar (dot) products but seem to run into a loop and a fair bit of messy algebra. Any thoughts welcome.