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China TST 2010, Problem 1

Source:

August 28, 2010
geometrycircumcircletrigonometryparallelogramgeometric transformationgeometry unsolved

Problem Statement

Given acute triangle ABCABC with AB>ACAB>AC, let MM be the midpoint of BCBC. PP is a point in triangle AMCAMC such that ∠MAB=∠PAC\angle MAB=\angle PAC. Let O,O1,O2O,O_1,O_2 be the circumcenters of △ABC,△ABP,△ACP\triangle ABC,\triangle ABP,\triangle ACP respectively. Prove that line AOAO passes through the midpoint of O1O2O_1 O_2.
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