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A difficult geometry from Iranian TST 2017

Source: Iranian TST 2017, first exam day 2, problem 5

April 6, 2017
geometryIranIranian TSTcircumcircle

Problem Statement

In triangle ABCABC, arbitrary points P,QP,Q lie on side BCBC such that BP=CQBP=CQ and PP lies between B,QB,Q.The circumcircle of triangle APQAPQ intersects sides AB,ACAB,AC at E,FE,F respectively.The point TT is the intersection of EP,FQEP,FQ.Two lines passing through the midpoint of BCBC and parallel to ABAB and ACAC, intersect EPEP and FQFQ at points X,YX,Y respectively. Prove that the circumcircle of triangle TXYTXY and triangle APQAPQ are tangent to each other.
Proposed by Iman Maghsoudi