Problem (2018KoMaLA719): 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.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; i = incenter a b c; ia = excenter a b c; a1 = foot ia b c; ib = excenter b c a; b1 = foot ib c a; ic = excenter c a b; c1 = foot ic a b; p = orthocenter a b1 c1; h = orthocenter a b c; m = midpoint p a1 ? para h m o i
HAGeo formalization: A B C = triangle; O = circumcenter A B C; I = incenter A B C; Ia = excenter A B C; Ib = excenter B C A; Ic = excenter C A B; BC = line B C; A1 = foot Ia BC; AC = line A C; B1 = foot Ib AC; AB = line A B; C1 = foot Ic AB; P = orthocenter A B1 C1; H = orthocenter A B C; M = midpoint P A1; HM = line H M; IO = line I O; Prove: parallel HM IO
Difficulty score: 4.7