For which circles $\Gamma$ does inversion in $\Gamma$ interchange 0 and $\infty$ ?

Let $\Gamma$ be a circle that lies entirely within the unit $\operatorname{disc} \mathbb{D}=\{z \in \mathbb{C}:|z|<1\} .$ Let $K$ be inversion in this circle $\Gamma$, let $J$ be inversion in the unit circle, and let $T$ be the Möbius transformation $K \circ J$. Show that, if $z_{0}$ is a fixed point of $T$, then

$$J\left(z_{0}\right)=K\left(z_{0}\right)$$

and this point is another fixed point of $T$.

By applying a suitable isometry of the hyperbolic plane $\mathbb{D}$, or otherwise, show that $\Gamma$ is the set of points at a fixed hyperbolic distance from some point of $\mathbb{D}$.