RU-ALLRUSSIAN 1986 Problem 11

ABC is a triangle with $\mathrm{AB} \neq \mathrm{AC}$ . Show that for each line through A, there is at most one point X on the line (excluding A, B, C) with $\angle \mathrm{ABX} = \angle \mathrm{ACX}$ . Which lines contain no such points X?