Problem 421

Upon inversion with the center at a given point $A$, the desired circle will transform into a line perpendicular to the images of the given circles $S_{1}$ and $S_{2}$, i.e., into a line connecting the centers $S_{1}^{*}$ and $S_{2}^{*}$. Thus, the desired circle is the image under this inversion of any line passing through the centers $S_{1}^{*}$ and $\text{2 2 *}$.