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 \(ABC\), let \(H\) be the point of intersection of the heights \(AJ\), \(BI\) and \(CK\). Consider the Euler circle of the triangle \(ABC\), the one that contains the points \(D,J,I,E,F,K\). This circle intersects the segments \(AH\), \(BH\), \(CH\) at points \(O\), \(P\), \(Q\) respectively. Prove that \(O\), \(P\), \(Q\) are the midpoints of the segments \(AH\), \(BH\), \(CH\).