Question 88
Let $H,O$ be the orthocenter and circumcenter of an acute triangle $ABC$. Let $D,E$ be the feet of the altitudes from $A,B$, respectively. Set $K=OD\cap BE$ and $L=OE\cap AD$. Let $X$ be the second intersection of the circumcircles of $HKD,HLE$, and $M$ the midpoint of $AB$. Prove that $K,L,M$ are collinear if and only if $X$ is the circumcenter of triangle $EOD$.