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2021 EGMO P3: E, F, N, M lie on a circle

Source: 2021 EGMO P3

April 13, 2021
EGMO 2021geometryTriangleEGMO

Problem Statement

Let ABCABC be a triangle with an obtuse angle at AA. Let EE and FF be the intersections of the external bisector of angle AA with the altitudes of ABCABC through BB and CC respectively. Let MM and NN be the points on the segments ECEC and FBFB respectively such that EMA=BCA\angle EMA = \angle BCA and ANF=ABC\angle ANF = \angle ABC. Prove that the points E,F,N,ME, F, N, M lie on a circle.