Question 152
The incircle $\omega$ of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. Point $D$ is chosen on line $AA_1$ so that $AD = AC_1$, and point $A$ lies between $A_1$ and $D$. Lines $DB_1$ and $DC_1$ intersect $\omega$ at points $B_2 \neq B_1$ and $C_2 \neq C_1$, respectively. Prove that $B_2C_2$ is a diameter of $\omega$. (R. Zhenodarov)