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ASU 428 All Soviet Union MO 1986 equal angles impossible

Source:

August 6, 2019
geometryequal anglesLine

Problem Statement

A line is drawn through the AA vertex of triangle ABCABC with ∣AB∣≠∣AC∣|AB|\ne|AC|. Prove that the line can not contain more than one point MM such, that MM is not a triangle vertex, and ∠ABM=∠ACM\angle ABM = \angle ACM. What lines do not contain such a point MM at all?
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