Möbius transformations map circles and lines to circles and lines

Statement

A generalized circle in the Riemann sphere $\widehat{\mathbb{C}}$ is either a circle in $\mathbb{C}$, or a set $L \cup \{\infty\}$ where $L$ is a line in $\mathbb{C}$. Let $f$ be a Möbius transformation. Then for each generalized circle $C$, the image $f(C)$ is a generalized circle.

Remarks

A circle can go to a line: the map $z \mapsto \frac{1}{z}$ sends the circle $|z - 1| = 1$, which contains $0$, to the line $\operatorname{Re} w = \frac{1}{2}$ together with $\infty$. The image of a generalized circle is a line together with $\infty$ exactly when the generalized circle contains the point that $f$ sends to $\infty$. The proof checks the result for $z \mapsto z + b$, $z \mapsto az$ and $z \mapsto \frac{1}{z}$, because each Möbius transformation is a composition of such maps.

Explanations

L1 human

For a Möbius map, a line is only a circle through the point $\infty$. So the theorem says that circles go to circles, if we count lines as circles. For example, $z \mapsto 2z + 3$ sends each circle to a circle with twice the radius.