Let $\phi = \dfrac{1}{r} ∘ T_{-O_C}$, $O_C$ the center of $C$, $T_{-O_C}$ is a translation.
I want to show that the inversion of a circle $C \in \mathbb{C}$ can be written as: $$\iota_C = \phi ∘ \iota ∘ \phi^{-1}$$
I have to show 3 things:
$\forall z \in C$ , $\iota_C (z) \in C$
$\iota_C$ exchanges the inside and the outside of C
$\iota_C$ globally preserves the circles and the lines perpendicular to $C$
The first properties are easy to show but for the third property, I don't see why after a homotheticity of $1/r$, and after a translation, a line $l$ perpendicular to $C$ remains the same