KöMaL 2019 B5013

The excircle of triangle $ABC$ opposite to vertex $A$ touches line $AC$ at point $B_1$. The line segment $BB_1$ intersects the excircle at $B_2$, and the tangent drawn to the excircle at $B_2$ intersects side $BC$ at $B_3$. Similarly, the inscribed circle of the triangle touches side $AB$ at point $C_1$, line segment $CC_1$ intersects the incircle at $C_2$, and the tangent drawn to the incircle at $C_2$ intersects side $BC$ at $C_3$. Prove that $B_2B_3=C_2C_3$.