CGMO 2018 Problem 2

Points $D,E$ lie on segments $AB,AC$ of $\triangle ABC$ such that $DE\parallel BC$ . Let $O_1,O_2$ be the circumcenters of $\triangle ABE, \triangle ACD$ respectively. Line $O_1O _2$ meets $AC$ at $P$ , and $AB$ at $Q$ . Let $O$ be the circumcenter of $\triangle APQ$ , and $M$ be the intersection of $AO$ extended and $BC$ . Prove that $M$ is the midpoint of $BC$ .