The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at points $A_1$, $B_1$, and $C_1$. The line through the midpoints of segments $AB_1$ and $AC_1$ intersects the tangent at $A$ to the circumcircle of triangle $ABC$ at point $A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that points $A_2$, $B_2$, and $C_2$ lie on a line.