Problem (2023G7): Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$. Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear.
AlphaGeometry formalization: a b c = triangle; h = orthocenter a b c; b1 = reflect b c h; c1 = reflect c b h; c2 = reflect c a h; a2 = reflect a c h; a3 = reflect a b h; b3 = reflect b a h; p = on_line b1 c1, on_line c2 a2; q = on_line c2 a2, on_line a3 b3; r = on_line a3 b3, on_line b1 c1; h1 = orthocenter p q r; o1 = circumcenter p q r ? coll h1 o1 h
HAGeo formalization: A B C = acute_triangle; H = orthocenter A B C; CH = line C H; B1 = reflect_point_wrt_line B CH; BH = line B H; C1 = reflect_point_wrt_line C BH; la = line B1 C1; A1 = reflect A BH; AH = line A H; B2 = reflect B AH; lc = line A1 B2; A2 = reflect A CH; C2 = reflect C AH; lb = line A2 C2; X = intersection lb lc; Y = intersection la lc; Z = intersection la lb; H1 = orthocenter X Y Z; O1 = circumcenter X Y Z; Prove: collinear H H1 O1
Difficulty score: 5.0