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HM is parallel to OI

Source: KöMaL A. 719

April 13, 2018
geometry

Problem Statement

Let ABCABC be a scalene triangle with circumcenter OO and incenter II. The AA-excircle, BB-excircle, and CC-excircle of triangle ABCABC touch BCBC, CACA, and ABAB at points A1A_1, B1B_1, and C1C_1, respectively. Let PP be the orthocenter of AB1C1AB_1C_1 and HH be the orthocenter of ABCABC. Show that if MM is the midpoint of PA1PA_1, then lines HMHM and OIOI are parallel.
Michael Ren
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