Let $ABC$ be a triangle, and $A_1$, $B_1$, $C_1$ points on the sides $BC$, $CA$, $AB$, respectively, such that the triangle $A_1B_1C_1$ is equilateral. Let $I_1$ and $\omega_1$ be the incenter and the incircle of $AB_1C_1$. Define $I_2$, $\omega_2$ and $I_3$, $\omega_3$ similarly, with respect to the triangles $BA_1C_1$ and $CA_1B_1$, respectively. Let $l_1 \neq BC$ be the external tangent line to $\omega_2$ and $\omega_3$. Define $l_2$ and $l_3$ similarly, with respect to the pairs $\omega_1$, $\omega_3$ and $\omega_1$, $\omega_2$. Knowing that $A_1I_2 = A_1I_3$, show that the lines $l_1$, $l_2$, $l_3$ are concurrent.