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Vertex on radical axis

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 1.

August 5, 2016
geometrycircumcircleSharygin Geometry Olympiad

Problem Statement

A line parallel to the side BCBC of a triangle ABCABC meets the sides ABAB and ACAC at points PP and QQ, respectively. A point MM is chosen inside the triangle APQAPQ. The segments MBMB and MCMC meet the segment PQPQ at points EE and FF, respectively. Let NN be the second intersection point of the circumcircles of the triangles PMFPMF and QMEQME. Prove that the points A,M,NA,M,N are collinear.
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