AUSTRIAN-POLISH 1992 Problem 5

Given a circle $k$ with center $M$ and radius $r$ , let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$ . Denote by $t$ the tangent of $k$ at $A$ . For any chord $CD$ through $K$ other than $AB$ , denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$ , respectively. Prove that $AP\cdot AQ$ does not depend on $CD$ .