FKMO 2009 Problem 4

$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$ ) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$ . $O$ meets the segment $AB$ at $D$ . $CO$ meets the circle $(O)$ again at $P$ , a line, which passes $P$ and parallel to $AO$ , meets $AC$ at $E$ , and $EB$ meets the circle $(O)$ again at $L$ . A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$ . Prove that two lines $EK$ and $CL$ are parallel.