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Z lies on circle of diameter XY, orthogonal to circumcircle of cyclic ABCD

Source: Oliforum Contest III 2012 p5 https://artofproblemsolving.com/community/c2487525_oliforum_contest

September 28, 2021
geometrycyclic quadrilateralorthogonal

Problem Statement

Consider a cyclic quadrilateral ABCDABCD and define points X=AB∩CDX = AB \cap CD, Y=AD∩BCY = AD \cap BC, and suppose that there exists a circle with center ZZ inscribed in ABCDABCD. Show that the ZZ belongs to the circle with diameter XYXY , which is orthogonal to circumcircle of ABCDABCD.
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