On the outside of triangle ABC construct equilateral triangles $ABC_1,BCA_1, CAB_1$, and inside of ABC, construct equilateral triangles $ABC_2,BCA_2, CAB_2$. Let $G_1,G_2,G_3$ $G_3,G_4,G_6$be respectively the centroids of triangles $ABC_1,BCA_1, CAB_1$, $ABC_2,BCA_2, CAB_2$.
Prove that the centroids of triangle $G_1G_2G_3$ and of triangle $G_4G_5G_6$ coincide
The easiest way to get at the result is to apply an affine transformation to the original triangle so as to make it equilateral. The constructed "interior" triangles will all then coincide with the original triangle, and the conclusion can be easily drawn using basic geometry.