VMO 1970 Problem 4

$AB$ and $CD$ are perpendicular diameters of a circle. $L$ is the tangent to the circle at $A$ . $M$ is a variable point on the minor arc $AC$ . The ray $BM, DM$ meet the line $L$ at $P$ and $Q$ respectively. Show that $AP\cdot AQ = AB\cdot PQ$ . Show how to construct the point $M$ which gives $BQ$ parallel to $DP$ . If the lines $OP$ and $BQ$ meet at $N$ find the locus of $N$ . The lines $BP$ and $BQ$ meet the tangent at $D$ at $P'$ and $Q'$ respectively. Find the relation between $P'$ and $Q$ '. The lines $D$ P and $DQ$ meet the line $BC$ at $P"$ and $Q"$ respectively. Find the relation between $P"$ and $Q"$ .