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Diameter, tangent and chord

Source: Baltic Way 2004 Problem 16

November 19, 2004
projective geometrygeometry proposedgeometry

Problem Statement

Through a point PP exterior to a given circle pass a secant and a tangent to the circle. The secant intersects the circle at AA and BB, and the tangent touches the circle at CC on the same side of the diameter through PP as the points AA and BB. The projection of the point CC on the diameter is called QQ. Prove that the line QCQC bisects the angle ∠AQB\angle AQB.
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