Question 175
Let $ABCD$ be a convex quadrilateral whose diagonals are not perpendicular and such that the lines $(AB)$ and $(CD)$ are not parallel. Let $O$ be the intersection point of $[AC]$ and $[BD]$. Let $H_1$ and $H_2$ be the orthocenters of triangles $AOB$ and $COD$ respectively. Let $M$ and $N$ be the midpoints of $[AB]$ and $[CD]$ respectively.
Prove that the lines $(H_1 H_2)$ and $(MN)$ are parallel if and only if $AC = BD$.
Prove that the lines $(H_1 H_2)$ and $(MN)$ are parallel if and only if $AC = BD$.