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Asian Pacific Mathematical Olympiad 2010 Problem 4

Source:

May 7, 2010
geometrycircumcircleEulerincentergeometric transformationreflectiontrapezoid

Problem Statement

Let ABCABC be an acute angled triangle satisfying the conditions AB>BCAB>BC and AC>BCAC>BC. Denote by OO and HH the circumcentre and orthocentre, respectively, of the triangle ABC.ABC. Suppose that the circumcircle of the triangle AHCAHC intersects the line ABAB at MM different from AA, and the circumcircle of the triangle AHBAHB intersects the line ACAC at NN different from A.A. Prove that the circumcentre of the triangle MNHMNH lies on the line OHOH.
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