KöMaL 2021 A796

Let $ABCD$ be a cyclic quadrilateral. Let lines $AB$ and $CD$ intersect in $P$, and lines $BC$ and $DA$ intersect in $Q$. The feet of the perpendiculars from $P$ to $BC$ and $DA$ are $K$ and $L$, and the feet of the perpendiculars from $Q$ to $AB$ and $CD$ are $M$ and $N$. The midpoint of diagonal $AC$ is $F$. Prove that the circumcircles of triangles $FKN$ and $FLM$, and the line $PQ$ are concurrent.