Question 14
Given a quadrilateral $ABCD$, isosceles triangles $ABK$, $BCL$, $CDM$ and $DAN$ are constructed, whose bases are the sides $AB$, $BC$, $CD$ and $DA$, and such that $K$, $L$, $M$ and $N$ are different points and no three of them are aligned. The perpendicular to the line $KL$ drawn by $B$ cuts the perpendicular to the line $LM$ drawn by $C$ at the point $P$; the perpendicular to the line $MN$ drawn by $D$ cuts the perpendicular to the line $NK$ drawn by $A$ at the point $Q$. Show that, if $P$ and $Q$ are different points, then $PQ$ is perpendicular to $KM$.