IMO 1983 LL GBR29

Origin: GBR

Problem

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.