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Incentre of reflected triangle

Source: India Postals 2015 Set 2

November 7, 2015
geometryincenter

Problem Statement

OO is the centre of the circumcircle of triangle ABCABC, and MM is its orthocentre. Point AA is reflected in the perpendicular bisector of the side BCBC,B B is reflected in the perpendicular bisector of the side CACA, and finally CC is reflected in the perpendicular bisector of the side ABAB. The images are denoted by A1,B1,C1A_1, B_1, C_1 respectively. Let KK be the centre of the inscribed circle of triangle A1B1C1A_1B_1C_1. Prove that OO bisects the line segment MKMK.
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