SHARYGIN 2015 Problem 19

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$ . Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$ . The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$ . Prove that $Q$ lies on the medial line of triangle $LKP$ .