Problem

Source: Kazakhstan national olympiad 2022, grade 9, p1

Tags: geometry, ratio, Kazakhstan



$CH$ is an altitude in a right triangle $ABC$ $(\angle C = 90^{\circ})$. Points $P$ and $Q$ lie on $AC$ and $BC$ respectively such that $HP \perp AC$ and $HQ \perp BC$. Let $M$ be an arbitrary point on $PQ$. A line passing through $M$ and perpendicular to $MH$ intersects lines $AC$ and $BC$ at points $R$ and $S$ respectively. Let $M_1$ be another point on $PQ$ distinct from $M$. Points $R_1$ and $S_1$ are determined similarly for $M_1$. Prove that the ratio $\frac{RR_1}{SS_1}$ is constant.