KöMaL 2024 B5427

Let $P$ be an internal point in triangle $ABC$. Let lines $AP$, $BP$ and $CP$ intersect sides $BC$, $CA$ and $AB$ in points $L$, $M$ and $N$, respectively. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $LMN$.