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Symmetric Tangents Concur on CD

Source: ISL 2022/G3

July 9, 2023
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Problem Statement

Let ABCDABCD be a cyclic quadrilateral. Assume that the points Q,A,B,PQ, A, B, P are collinear in this order, in such a way that the line ACAC is tangent to the circle ADQADQ, and the line BDBD is tangent to the circle BCPBCP. Let MM and NN be the midpoints of segments BCBC and ADAD, respectively. Prove that the following three lines are concurrent: line CDCD, the tangent of circle ANQANQ at point AA, and the tangent to circle BMPBMP at point BB.
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