Question 113
Consider the quadrilateral $ABCD$ with non-perpendicular diagonals and non-parallel sides $AB$ and $CD$. Let $O$ be the intersection point of the diagonals, $H_{1}$ the orthocentre of the triangle $AOB$ and $H_{2}$ the orthocentre of the triangle $COD$. Denote by $M$ the midpoint of the side $[AB]$ and by $N$ the midpoint of the side $[CD]$. Show that $H_{1}H_{2}$ is parallel to $MN$ if and only if $AC = BD$.