RS-SMO 2011 Problem 3

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$ . $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$ , respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$ , and $M$ is midpoint of $AB$ . Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$ .