IMO 1966 LL ROM39

In a plane, a circle with center O and radius R and two points

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IMO 1966 LL ROM39

Origin: ROM

Problem

In a plane, a circle with center O and radius R and two points A, B are given. (a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P, say P1, P2, and find the distance between them if OA = a, OB = b, AB = d.