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incenter and midpoints of arcs of <B,<C collinear iff AC + BC = 3AB

Source: 2014 Sharygin Geometry Olympiad Final Round 9.6

August 3, 2018
geometryincenter

Problem Statement

Let II be the incenter of triangle ABCABC, and M,NM, N be the midpoints of arcs ABCABC and BACBAC of its circumcircle. Prove that points M,I,NM, I, N are collinear if and only ifAC+BC=3AB AC + BC = 3AB.
(A. Polyansky)
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