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Circle with center on another circle

Source: Sharygin contest 2008. The correspondence round. Problem 7

September 3, 2008
geometry proposedgeometry

Problem Statement

(A.Zaslavsky, 8--9) Given a circle and a point O O on it. Another circle with center O O meets the first one at points P P and Q Q. The point C C lies on the first circle, and the lines CP CP, CQ CQ meet the second circle for the second time at points A A and B B. Prove that AB\equal{}PQ.
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