VMO 1974 Problem 3

Let $ABC$ be a triangle with $A = 90^o, AH$ the altitude, $P,Q$ the feet of the perpendiculars from $H$ to $AB,AC$ respectively. Let $M$ be a variable point on the line $PQ$ . The line through $M$ perpendicular to $MH$ meets the lines $AB,AC$ at $R, S$ respectively. i) Prove that circumcircle of $ARS$ always passes the fixed point $H$ . ii) Let $M_1$ be another position of $M$ with corresponding points $R_1, S_1$ . Prove that the ratio $RR_1/SS_1$ is constant. iii) The point $K$ is symmetric to $H$ with respect to $M$ . The line through $K$ perpendicular to the line $PQ$ meets the line $RS$ at $D$ . Prove that $\angle BHR = \angle DHR, \angle DHS = \angle CHS$ .