Trivial parmenides51 wrote: In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent. $\textit{WLOG}$ let $H$ be outside $\Delta ABC$ (configuration doesn't matter) . Clearly if we let $T=AE\cap{CD}$ , then $AHCT$ is a parallelogram. So we take the midpoint $N$ of $AC \Leftrightarrow{HT}$ , so $\overline{T-M-N-H}$ are collinear and this line passes through $AD\cap{CE}$.