MathNet 2018-part-04 — 181 problems (EN–VI) Question 160

Triangle $ABC$ is inscribed in the circle $\mathcal{C}(O, 1)$. Let $G_1, G_2, G_3$ be the centroids of triangles $OBC, OAC$ and $OAB$, respectively. Prove that triangle $ABC$ is equilateral if and only if $AG_1 + BG_2 + CG_3 = 4$.