We have a $\mathrm{\u25b3}ABC$ and a $\mathrm{\u25b3}{A}_{1}{B}_{1}{C}_{1}$. The segments $CL$ and ${C}_{1}{L}_{1}$ are angle bisectors. If $\mathrm{\u25b3}ALC\sim \mathrm{\u25b3}{A}_{1}{L}_{1}{C}_{1}$, I should show that $\mathrm{\u25b3}ABC\sim \mathrm{\u25b3}{A}_{1}{B}_{1}{C}_{1}$.

From the similarity, we have $\frac{AL}{{A}_{1}{L}_{1}}}={\displaystyle \frac{CL}{{C}_{1}{L}_{1}}}={\displaystyle \frac{AC}{{A}_{1}{C}_{1}}$. The only way I see from here is to show that $\mathrm{\u25b3}LBC\sim \mathrm{\u25b3}{L}_{1}{B}_{1}{C}_{1}$. Is this necessary for the solution?

From the similarity, we have $\frac{AL}{{A}_{1}{L}_{1}}}={\displaystyle \frac{CL}{{C}_{1}{L}_{1}}}={\displaystyle \frac{AC}{{A}_{1}{C}_{1}}$. The only way I see from here is to show that $\mathrm{\u25b3}LBC\sim \mathrm{\u25b3}{L}_{1}{B}_{1}{C}_{1}$. Is this necessary for the solution?