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Concyclic with circumcenters

Source: China TST 4 2018 Day 1 Q3

March 27, 2018
geometrycircumcircle

Problem Statement

In isosceles △ABC\triangle ABC, AB=ACAB=AC, points D,E,FD,E,F lie on segments BC,AC,ABBC,AC,AB such that DE∥ABDE\parallel AB, DF∥ACDF\parallel AC. The circumcircle of △ABC\triangle ABC ω1\omega_1 and the circumcircle of △AEF\triangle AEF ω2\omega_2 intersect at A,GA,G. Let DEDE meet ω2\omega_2 at K≠EK\neq E. Points L,ML,M lie on ω1,ω2\omega_1,\omega_2 respectively such that LG⊥KG,MG⊥CGLG\perp KG, MG\perp CG. Let P,QP,Q be the circumcenters of △DGL\triangle DGL and △DGM\triangle DGM respectively. Prove that A,G,P,QA,G,P,Q are concyclic.
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