RIOPLATENSE 2019 Problem 5

Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$ . Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$ . Let $P$ be the foot of perpendicular of $A$ to the line $BC$ , and the point $Q$ is the intersection of $GP$ and $\omega$ ( $Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$ . If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$ , $G$ and $K$ are collinears.