Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points: $A_1$ and $A_2$, $B_1$ and $B_2$, and $C_1$ and $C_2$, where points $A_2,B_2,C_2$ lie inside the triangle $A_1B_1C_1$. Prove that \[A_1B_2 \cdot B_1C_2 \cdot C_1A_2 = A_1C_2 \cdot C_1B_2 \cdot B_1A_2 .\]