Question 146
Consider the cyclic hexagon $ABCDEF$ and let $H_{1}$, $H_{2}$, $H_{3}$, $H_{4}$ be the orthocentres of the triangles $ABC$, $BCD$, $DEF$, $FAE$. Show that for any points $M$, $N$, $P$ and $Q$ in the plane satisfying the relation $\overrightarrow{H_{1}M} + \overrightarrow{H_{3}P} = \overrightarrow{H_{2}N} + \overrightarrow{H_{4}Q}$, $MNPQ$ is a parallelogram.