MK-MO 2015 Problem 4

Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$ . A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$ . Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$ , respectively. Let $P$ be the intersection of the lines $AM$ and $CD$ . The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$ . The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$ . Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$ . Prove that $MNPQ$ is a parallelogram.