NL-NWO 1979 Problem 4

Given is the non-equilateral triangle $A_1A_2A_3$ . $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$ . Prove that lines $B_{12}B_{21}$ , $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.