Question 154
In triangle $ABC$ consider the points $M \in (AB)$, $N \in (BC)$, $P \in (CA)$ such that $AM = BN = CP$. If $G_1, G_2, G_3$ are the centroids of triangles $AMP$, $BMN$ and $CNP$, respectively, show that triangles $ABC$ and $G_1G_2G_3$ have the same centroid if and only if triangle $ABC$ is equilateral.