Problem 1069

28.38. If circles $R_{1}$ and $R_{2}$ intersect or touch, then the inversion with the center at their point of intersection will transform circles $S_{1}, S_{2}, \ldots, S_{n}$ into circles touching a pair of lines and each other at points $A_{1}^{*}, A_{2}^{*}, \ldots, A_{n-1}^{*}$, lying on the bisector of the angle formed by lines $R_{1}^{*}$ and $R_{2}^{*}$, if $R_{1}^{*}$ and $R_{2}^{*}$ intersect, and on a line parallel to $R_{1}^{*}$ and $R_{2}^{*}$, if these lines do not intersect. Applying the inversion again, we obtain that points $A_{1}^{*}, A_{2}^{*}, \ldots, A_{n-1}^{*}$ lie on one circle.

If circles $R_{1}$ and $R_{2}$ do not intersect, then according to problem 28.6, there exists an inversion that transforms them into a pair of concentric circles. In this case, points $A_{1}^{*}, A_{2}^{*}, \ldots, A_{n-1}^{*}$ lie on a circle concentric with $R_{1}^{*}$ and $R_{2}^{*}$, which means that points $A_{1}, A_{2}, \ldots, A_{n-1}$ lie on one circle.