KöMaL problem A, 2018

Geometry Difficulty 8.0 Shortlist Prove it Hungary

Let $ABC$ be a scalene triangle with circumcenter $O$ and incenter $I$ . The $A$ -excircle, $B$ -excircle, and $C$ -excircle of triangle $ABC$ touch $BC$ , $CA$ , and $AB$ at points $A_1$ , $B_1$ , and $C_1$ , respectively. Let $P$ be the orthocenter of $AB_1C_1$ and $H$ be the orthocenter of $ABC$ . Show that if $M$ is the midpoint of $PA_1$ , then lines $HM$ and $OI$ are parallel.

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