Problem #WSP-5303

Problem

Let $ABC$ be a triangle with midpoints $D$ on the side $BC$, $E$ on the side $AC$ and $F$ on the side $AB$. Let $M$ be the point of intersection of all medians of the triangle $ABC$ and let $H$ be the point of intersection of the heights $AJ$, $BI$ and $CK$. Consider the Euler circle of the triangle $ABC$, which is the one that contains the points $D,J,I,E,F,K$. This circle intersects the segments $AH$, $BH$ and $CH$ at the points $O$, $P$ and $Q$ respectively. Prove that $O$, $P$ and $Q$ are the midpoints of the segments $AH$, $BH$ and $CH$.