Problem (2012CTSTp2): Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
AlphaGeometry formalization: a b c = triangle; i = incenter a b c; d = foot i b c; e = foot i a c; f = foot i a b; l = reflect d e f; m = reflect e f d; n = reflect f d e; p = on_line a l, on_line b c; q = on_line b m, on_line a c; r = on_line c n, on_line a b ? coll p q r
HAGeo formalization: A B C = triangle; I = incenter A B C; BC = line B C; D = foot I BC; AC = line A C; E = foot I AC; AB = line A B; F = foot I AB; EF = line E F; L = reflect_point_wrt_line D EF; DF = line D F; M = reflect_point_wrt_line E DF; DE = line D E; N = reflect F DE; AL = line A L; P = intersection AL BC; BM = line B M; Q = intersection BM AC; CN = line C N; R = intersection CN AB; Prove: collinear P Q R
Difficulty score: 2.5