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cyclic wanted, 3 incircles related

Source: Belarus TST 2015 2.3

June 9, 2020
geometryincircleConcyclic

Problem Statement

Let the incircle of the triangle ABCABC touch the side ABAB at point QQ. The incircles of the triangles QACQAC and QBCQBC touch AQ,ACAQ,AC and BQ,BCBQ,BC at points P,TP,T and D,FD,F respectively. Prove that PDFTPDFT is a cyclic quadrilateral.
I.Gorodnin
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