Školjka

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 $AB$, $AC$ respectively, so that points $B'$, $C'$ and $D$ are collinear and points $B$, $C$, $B'$, $C'$ are concyclic with center $O$. Prove that if $M$ is the midpoint of $\overline{BC}$ and $H$ is the orthocenter of $ABC$, then $DHMO$ is a parallelogram.