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Two points are reflections across diagonal

Source: China Team Selection Test 2016 Test 3 Day 1 Q3

March 25, 2016
geometrycyclic quadrilateralincenter

Problem Statement

In cyclic quadrilateral ABCDABCD, AB>BCAB>BC, AD>DCAD>DC, I,JI,J are the incenters of △ABC\triangle ABC,△ADC\triangle ADC respectively. The circle with diameter ACAC meets segment IBIB at XX, and the extension of JDJD at YY. Prove that if the four points B,I,J,DB,I,J,D are concyclic, then X,YX,Y are the reflections of each other across ACAC.
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