Problem (2024VietnamTSTp3): Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.
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; x = foot a b c; y = foot b a c; z = foot c a b; a1 = reflect x e f; b1 = reflect y f d; c1 = reflect z d e ? simtri a b c a1 b1 c1
HAGeo formalization: A B C = triangle; I = incenter A B C; BC = line B C; AC = line A C; AB = line A B; D = foot I BC; E = foot I AC; F = foot I AB; X = foot A BC; Y = foot B AC; Z = foot C AB; EF = line E F; FD = line F D; DE = line D E; A1 = reflect_point_wrt_line X EF; B1 = reflect_point_wrt_line Y FD; C1 = reflect_point_wrt_line Z DE; Prove: similar A B C A1 B1 C1
Difficulty score: 5.7