Middle European Mathematical Olympiad 2018 Problem T-5

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let $D$ be the foot of its altitude from $A$. Points $B'$ and $C'$ lie on the rays $AB$ and $AC$, respectively, so that points $B'$, $C'$ and $D$ are collinear and points $B$, $C$, $B'$ and $C'$ lie on one circle with center $O$. Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC$, then $DHMO$ is a parallelogram.