VTMO 2026 - Grade 9 - Final - SET 12 - Self-study Question 4

Let $G$ be an interior point of triangle $ABC$, and let $M,N,P$ be the intersections of $AG,BG,CG$ with $BC,CA,AB$, respectively. Suppose
$$\frac{AG}{GM}+\frac{BG}{GN}+\frac{CG}{GP}=6.$$
Prove that $G$ is the centroid of $ABC$.