MEMO 2009 Problem 10

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