Problem (2012USAMOp5): Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.
AlphaGeometry formalization: a b c = triangle; p = free; t = free; a1 = reflect a p t; a2 = on_line b c, on_line p a1; b1 = reflect b p t; b2 = on_line a c, on_line p b1; c1 = reflect c p t; c2 = on_line a b, on_line p c1 ? coll a2 b2 c2
HAGeo formalization: A B C = triangle; P = point; T = point; PT = line P T; A1 = reflect_point_wrt_line A PT; B1 = reflect_point_wrt_line B PT; C1 = reflect_point_wrt_line C PT; A1P = line A1 P; BC = line B C; A2 = intersection A1P BC; B1P = line B1 P; AC = line A C; B2 = intersection B1P AC; C1P = line C1 P; AB = line A B; C2 = intersection C1P AB; Prove: collinear A2 B2 C2
Difficulty score: 1.7