KöMaL 2015 B4693

$K$ is a point on side $AC$ of triangle $ABC$ such that $AK=2KC$ and $\angle ABK = 2 \angle KBC$ . Let $F$ denote the midpoint of side $AC$ , and let $L$ be the orthogonal projection of point $A$ onto the line segment $BK$ . Prove that the lines $FL$ and $BC$ are perpendicular.