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Two circles tangential to sides of a cyclic quadrilateral

Source: Germany 2017, Problem 4

May 4, 2017
geometrygeometry proposedcyclic quadrilateralCyclic Quadrilateralstangent circlesGermany

Problem Statement

Let ABCDABCD be a cyclic quadrilateral. The point PP is chosen on the line ABAB such that the circle passing through C,DC,D and PP touches the line ABAB. Similarly, the point QQ is chosen on the line CDCD such that the circle passing through A,BA,B and QQ touches the line CDCD.
Prove that the distance between PP and the line CDCD equals the distance between QQ and ABAB.