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Concyclic iff collinear

Source: China Mathematical Olympiad 2017 Q2

November 23, 2016
geometry

Problem Statement

In acute triangle ABCABC, let ⊙O\odot O be its circumcircle, ⊙I\odot I be its incircle. Tangents at B,CB,C to ⊙O\odot O meet at LL, ⊙I\odot I touches BCBC at DD. AYAY is perpendicular to BCBC at YY, AOAO meets BCBC at XX, and OIOI meets ⊙O\odot O at P,QP,Q. Prove that P,Q,X,YP,Q,X,Y are concyclic if and only if A,D,LA,D,L are collinear.
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