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 \(CH\).