Möbius transformations lines and circles I am looking for a basic outline of a proof
I know that all MT's are of the form $\frac{ax+b}{cx+d}$
For $c=0$, I know that lines/circles are preserved because translations and dilations do not change a line/circle from being a line/circle
But I am not sure how to prove this for all cases
My exam is actually tomorrow, so it would be great if someone could help me today :)

To begin with, if $c\neq0,$ then put $f_1(z)=z+d/c,$ $f_2(z)=1/z,$ $f_3(z)=\frac{bc-ad}{c^2}z,$ and $f_4(z)=z+a/c.$ Then $$(f_4\circ f_3\circ f_2\circ f_1)(z)=\frac{az+b}{cz+d}.$$ So, it suffices to show that each of $f_1,f_2,f_3,f_4$ maps generalized circles to generalized circles. The fact that $ad-bc\ne0$ will be essential here. The only somewhat tricky part is showing that this is true for $f_2.$ The idea is to show that lines are mapped to generalized circles through the origin and vice versa, while circles that don't pass through the origin are again mapped to circles that don't pass through the origin.
The $c=0$ case is straightforward.
