MathDB
congruent triangles, 3 circumcircles related

Source: 2012 Balkan Shortlist G3 BMO

April 3, 2020
geometrycongruent trianglescircumcircle

Problem Statement

Let ABCABC be a triangle with circumcircle cc and circumcenter OO, and let DD be a point on the side BCBC different from the vertices and the midpoint of BCBC. Let KK be the point where the circumcircle c1c_1 of the triangle BODBOD intersects cc for the second time and let ZZ be the point where c1c_1 meets the line ABAB. Let MM be the point where the circumcircle c2c_2 of the triangle CODCOD intersects cc for the second time and let EE be the point where c2c_2 meets the line ACAC. Finally let NN be the point where the circumcircle c3c_3 of the triangle AEZAEZ meets cc again. Prove that the triangles ABCABC and NKMNKM are congruent.
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