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Intersection of circumcircles of MNP and BOC

Source: Serbian National Olympiad 2013, Problem 3

April 8, 2013
geometrycircumcircletrigonometryratiosymmetrygeometric transformationreflection

Problem Statement

Let MM, NN and PP be midpoints of sides BC,ACBC, AC and ABAB, respectively, and let OO be circumcenter of acute-angled triangle ABCABC. Circumcircles of triangles BOCBOC and MNPMNP intersect at two different points XX and YY inside of triangle ABCABC. Prove that ∠BAX=∠CAY.\angle BAX=\angle CAY.
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