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