Question 26
Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and let $K$, $L$, $M$ and $N$ denote the midpoints of the sides $AB$, $BC$, $CD$ and $DA$ respectively. Suppose that the lines $KM$ and $LN$ do not pass through $O$. Let $E := KM \cap LN$. Point $P$ is chosen on the segment $KM$ to satisfy $\angle KOE = \angle MOP$ and point $Q$ is chosen on the segment $LN$ to satisfy $\angle LOE = \angle NOQ$. Show that the points $O$, $P$ and $Q$ are collinear.
(Batzaya G.)
(Batzaya G.)