Since linear fractional transformations are compositions of translations, mutliplications by a constant and inversion, I tried proving that an LTF would transform circles into circles by writing the equation of a circle as $z_0 + r\cdot\exp ( i \cdot \theta )$ , $0 \le \theta \le 2\pi$ and then looking at the effect of translations, multiplications by a constant and inversion.
The first two clearly leave me with a circle, as does inversion of a circle centered around the origin, but I can't get this to work for a circle centered at an arbitrary point $z_0$.
On the other hand, if I write down the general equation of a circle: $$A(x^2 + y^2) + Bx +Cy +D = 0$$
...and write $w = u+iv = \frac{1}{z} = \frac{1}{(x+iy)}$ and make the appropriate substitutions, I get the equation of circle in terms of $u$ and $v$. Is it possible to write $$\frac{1}{z_0 + re^{i\theta}}$$ where $0 \le \theta \le 2\pi$, in the form $$w_0 + r'e^{i\phi}$$ with $0 \le \phi \le 2\pi$?