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Circumcentres of 6 circles are concyclic

Source: Chinese TST 1 2013 Day 2 Q2

April 1, 2013
geometrycircumcirclegeometric transformationhomothetytrigonometryEulergeometry proposed

Problem Statement

Let PP be a given point inside the triangle ABCABC. Suppose L,M,NL,M,N are the midpoints of BC,CA,ABBC, CA, AB respectively and PL:PM:PN=BC:CA:AB.PL: PM: PN= BC: CA: AB. The extensions of AP,BP,CPAP, BP, CP meet the circumcircle of ABCABC at D,E,FD,E,F respectively. Prove that the circumcentres of APF,APE,BPF,BPD,CPD,CPEAPF, APE, BPF, BPD, CPD, CPE are concyclic.
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