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Excenters and concyclic

Source: Bulgarian TST 2020 P6

January 23, 2021
geometry

Problem Statement

In triangle △ABC\triangle ABC, BC>ACBC>AC, IBI_B is the BB-excenter, the line through CC parallel to ABAB meets BIBBI_B at FF. MM is the midpoint of AIBAI_B and the AA-excircle touches side ABAB at DD. Point EE satisfies ∠BAC=∠BDE,DE=BC\angle BAC=\angle BDE, DE=BC, and lies on the same side as CC of ABAB. Let ECEC intersect AB,FMAB,FM at P,QP,Q respectively. Prove that P,A,M,QP,A,M,Q are concyclic.
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