JBMO-SL 2014 G6

Let $A B C D$ be a quadrilateral whose sides $A B$ and $C D$ are not parallel, and let $O$ be the intersection of its diagonals. Denote with $H_{1}$ and $H_{2}$ the orthocenters of the triangles $O A B$ and OCD, respectively. If $M$ and $N$ are the midpoints of the segments $\overline{A B}$ and $\overline{C D}$, respectively, prove that the lines $M N$ and $\mathrm{H}_{1} \mathrm{H}_{2}$ are parallel if and only if $\overline{A C}=\overline{B D}$.