KöMaL 2011 B4368

Let $D$, $E$ and $F$, respectively, be points on sides $AB$, $BC$, $CA$ of a triangle ABC such that $AD$:$DB$=$BE$:$EC$=$CF$:$FA \ne$1. The lines $AE$, $BF$, $CD$ intersect one another at points $G$, $H$, $I$, respectively. Prove that the centroids of triangles ABC and GHI coincide. (Suggested by Sz. Miklós, Herceghalom)