IMO Shortlist 1978 problem 12

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In a triangle ABC we have AB = AC. A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides AB, AC in the points P, respectively Q. Prove that the midpoint of PQ is the center of the inscribed circle of the triangle ABC.
Source: Međunarodna matematička olimpijada, shortlist 1978