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AX, BY, CZ concurrent iff AP, BQ, CR concurrent , 3 circles related

Source: 2017 XX All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p2

May 6, 2021
geometryconcurrencyconcurrentUkrainian TYM

Problem Statement

Points P,Q,RP, Q, R were marked on the sides BC,CA,ABBC, CA, AB, respectively. Let aa be tangent at point AA to the circumcircle of triangle AQRAQR, bb be tangent at point BB to the circumcircle of the triangle BPR, cc be tangent at point CC to the circumscribed circle triangle CPQCPQ. Let XX be the point of intersection of the lines bb and c,Yc, Y be the point the intersection of lines cc and a,Za, Z is the point of intersection of lines aa and bb. Prove that the lines AX,BY,CZAX, BY, CZ intersect at one point if and only if the lines AP,BQ,CRAP, BQ, CR intersect at one point.
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