ILL 1983 Problem 29

Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$ , where $A,C$ are the midpoints of $OB,OD$ , respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$