KöMaL 2019 B5005

The feet of the altitudes of an acute-angled triangle $ABC$ drawn to the sides $BC$, $CA$, $AB$ are $D$, $E$, $F$, respectively. The orthocentre of triangle $ABC$ is $M$. Let $k_1$ denote the circle of diameter $AB$, and let $k_2$ be the circumscribed circle of triangle $DEM$. Let $P$ be an interior point of the arc $EM$ of $k_2$ that does not contain point $D$. Let line $DP$ intersect circle $k_1$ again at point $Q$, and let the midpoint of line segment $PQ$ be $R$. Show that lines $AQ$, $MP$, $FR$ are concurrent.