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International Zhautykov olympiad 2014 problem 1

Source:

January 14, 2014
geometrycircumcirclegeometry proposedeasy

Problem Statement

Points MM, NN, KK lie on the sides BCBC, CACA, ABAB of a triangle ABCABC, respectively, and are different from its vertices. The triangle MNKMNK is called beautiful if ∠BAC=∠KMN\angle BAC=\angle KMN and ∠ABC=∠KNM\angle ABC=\angle KNM. If in the triangle ABCABC there are two beautiful triangles with a common vertex, prove that the triangle ABCABC is right-angled.
Proposed by Nairi M. Sedrakyan, Armenia
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