Problem

Source: Moldova TST 2023

Tags: geometry



Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$