Show that two non intersecting circles can always be mapped by a suitable Möbius transformation to two concentric circles.
I wanted to map the center of the first circle to the center of the second circle then using scaling the radius would give me the concentric circles. But having problem to execute.
Suggestions please,
Use one Möbius transform to turn one of the circles into a straight line $\ell$. This also transfroms the other circel: into a circle $C$ that does not intersect (nor touch) $\ell$. Let $O$ be the center of $C$ and $A$ the orthogonal projection of $O$ onto $\ell$. Then $A$ is outside $C$. One of th etwo tangent lines to $C$ through $A$ touches $C$ in a point $B$. Let $P$ be one of th eintersection points of the circle $C_1$ around $A$ through $B$ and the line $OA$. A Möbius transfrom that takes $P$ to infinity transforms $C_1$ and $OA$ to lines that are orthogonal to both transformed given circles. Therefore these are concentric.