Prove that Möbius transformation map every straight line to a line or circle, and map every circle to a line or circle

Question

I found this statement in wiki :

These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle

Talking about Möbius transformation , I found a proof for the invariance of angles but nothing for the rest does anyone know how can we prove it ?

Answer 1

The result follows from two facts:

1. Möbius transformations preserve the cross-ratio. That is, if $T$ is a Möbius transformation, then $(Tz_1,Tz_2;Tz_3,Tz_4) = (z_1,z_2;z_3,z_4)$ for any points $z_1,z_2,z_3,z_4$ in $\mathbb{C} \cup \{\infty\}$.

2. Four points lie on the same line or circle if and only if their cross-ratio is real. (Hint for proof: consider the angles in a quadrilateral formed by the four points.) Therefore, if $z_1,z_2,z_3$ are fixed, the set of points $z$ such that $(z_1,z_2;z_3,z)$ is real is a line or circle.