FKMO 2019 Problem 4

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.