While I tried to prove a problem,          Suppose that a linear fractional transformation carries one pair of concentric circles into another pair of concentric circles. Prove that the ratios of the radii must be the same.      Following is my try     Suppose concentric circles $z_1=\alpha+Re^{i\theta}$ and $z_2=\alpha+t Re^{i\theta}$, where $t&gt;0,\ \theta\in [0,2\pi)$, through LFT $\frac{az+b}{cz+d}$ mapping to          $$     \omega_1=\frac{a Re^{i\theta}+a\alpha+b}{cRe^{i\theta}+c\alpha +d};\\   \omega_2=\frac{a tRe^{i\theta}+a\alpha+b}{ctRe^{i\theta}+c\alpha +d}.    $$      According to condition the mapping maps the concentric circles to concentric circles, so $\omega_1$ and $\omega_2$ are concentric. Suppose $\omega_1=\beta_1+R_1e^{i\theta}$ and $\omega_2=\beta_2+R_2e^{i\theta}$, solve the equations, we get the results as follows,         $$   \beta_1\to \frac{(a \alpha+b) (\alpha c+d)-a c R^2}{(\alpha  c+d)^2-c^2 R^2},R_1\to \frac{R (a d-b c)}{(\alpha  c+d)^2-c^2 R^2};    $$         $$   \beta_2\to \frac{(a \alpha+b) (\alpha c+d)-a c R^2 t^2}{(\alpha c+d)^2-c^2 R^2 t^2},R_2\to \frac{R t (a d-b c)}{(\alpha c+d)^2-c^2 R^2 t^2}.    $$           We simplify $\beta_1-\beta_2=0$, that is          $$     c(c\alpha+ d)=0.    $$   (i)$c=0$, $R_2/R_1=t$.      (ii)$c\alpha+d=0$, $R_2/R_1=1/t$.   Now my question is the first situation is normal and second one seems like the original proposition is not right, what's wrong?   Besides, I think my way to solve this question especially solving the equations are complicated, it there any other simpler solution? Thanks!
Given a Möbius transformation that maps one pair of concentric circles to another pair of concentric circles, why is the ratio of the radii preserved through the map?   I thought about how Möbius transformations are compositions of rotations, scaling, inversion, and translation, and that intuitively, these types of maps shouldn't change the ratio of radii between two circles.   Would it be correct to just say that if $\frac{r_1}{r_2}$ is the ratio of radii between the two circles, then   1) The radii are invariant under translation, $z \mapsto z+a$, so $\frac{r_1}{r_2}$ stays the same  2) Under scaling by a factor $z \mapsto az$, $\frac{ar_1}{ar_2} = \frac{r_1}{r_2}$   3) Under inversion, $z \mapsto \frac{1}{z}$, $\frac{1/r_1}{1/r_2} = \frac{r_2}{r_1}$  Or is there a different/better way to think about this problem?
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