JBMO-SL 2001 Problem 12

Consider the triangle $ABC$ with $\angle A= 90^{\circ}$ and $\angle B \not= \angle C$ . A circle $\mathcal{C}(O,R)$ passes through $B$ and $C$ and intersects the sides $AB$ and $AC$ at $D$ and $E$ , respectively. Let $S$ be the foot of the perpendicular from $A$ to $BC$ and let $K$ be the intersection point of $AS$ with the segment $DE$ . If $M$ is the midpoint of $BC$ , prove that $AKOM$ is a parallelogram.