Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$. The foot of perpendicular from $A$ to $BC$ is $O$, and denote $K,L$ by the midpoints of $AB, AC$, respectively. For a point $D(\neq O,B,C)$ on segment $BC$, let $E,F$ be the orthocenters of triangles $ABD, ACD$, respectively, and denote $M,N$ by the midpoints of $DE,DF$. The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$. If $Q$ is the midpoint of $EF$, and $S$ is the orthocenter of triangle $HPQ$, then prove that as $D$ varies on $BC$, the ratio $\frac{OS}{OH}$, $\frac{OQ}{OP}$ remains constant.