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A point is on a fixed line

Source: Vietnam Mathematical Olympiad 2016, Day 1, Problem 3

January 6, 2016
geometry proposedgeometry

Problem Statement

Let ABCABC be an acute triange with B,CB,C fixed. Let DD be the midpoint of BCBC and E,FE,F be the foot of the perpendiculars from DD to AB,ACAB,AC, respectively. a) Let OO be the circumcenter of triangle ABCABC and M=EF∩AO,N=EF∩BCM=EF\cap AO, N=EF\cap BC. Prove that the circumcircle of triangle AMNAMN passes through a fixed point; b) Assume that tangents of the circumcircle of triangle AEFAEF at E,FE,F are intersecting at TT. Prove that TT is on a fixed line.
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