NL-NWO 2012 Problem 4

We are given an acute triangle $ABC$ and points $D$ on $BC$ and $E$ on $AC$ such that $AD$ is perpendicular to $BC$ and $BE$ is perpendicular to $AC$ . The intersection of $AD$ and $BE$ is called $H$ . A line through $H$ intersects line segment $BC$ in $P$ , and intersects line segment $AC$ in $Q$ . Furthermore, $K$ is a point on $BE$ such that $PK$ is perpendicular to $BE$ , and $L$ is a point on $AD$ such that $QL$ is perpendicular to $AD$ . Prove that $DK$ and $EL$ are parallel.