KURSCHAK 1990 Problem 2

The incenter of $\triangle A_1A_2A_3$ is $I$ , and the center of the $A_i$ -excircle is $J_i$ ( $i=1,2,3$ ). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ( $A_{i+3}:=A_i$ $\forall i$ ). Prove that the three lines $B_iJ_i$ are concurrent.