Question 54
Let $\triangle ABC$ be a triangle with $\angle B \lt \angle C$. Let $O$ be the circle tangent to the line $AC$ at the point $C$ and passing through the point $B$. The lines $AB$ and $CO$ meet the circle $O$ at the points $D (\neq B)$ and $P (\neq C)$, respectively. Let $E$ be the intersection of $AC$ and the line passing through $P$ and parallel to $AO$. The line $EB$ and the circle $O$ meet at the point $L (\neq B)$. Let $F$ be the intersection of the perpendicular bisector of the line segment $BD$ and the line $AC$, and $K$ be the intersection of $LF$ and $CD$. Show that $EK$ and $CL$ are parallel.