ISL 1966 Problem 39

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle.

a.) Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle.

b.) Show that generally, one gets two possible points $P$ ( $P_{1}$ and $P_{2}$ ) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.