Given a triangle $ABC$, $A_1$ divides the length of the path $CAB$ into two equal parts, and define $B_1$ and $C_1$ analogously. Let $l_A$, $l_B$, $l_C$ be the lines passing through $A_1$, $B_1$ and $C_1$ and being parallel to the bisectors of $\angle A$, $\angle B$, and $\angle C$. Show that $l_A$, $l_B$, $l_C$ are concurrent.