Middle European Mathematical Olympiad 2009 Problem T-6

Suppose that $ABCD$ is a cyclic quadrilateral and $CD = DA$. Points $E$ and $F$ belong to the segments $AB$ and $BC$ respectively, and $\measuredangle ADC = 2\measuredangle EDF$. Segments $DK$ and $DM$ are height and median of the triangle $DEF$, respectively. $L$ is the point symmetric to $K$ with respect to $M$. Prove that the lines $DM$ and $BL$ are parallel.