# Prove that all hyperbolic straight lines are congruent to $x$-axis

I have the notes on the proof but I cannot fully understand the proof.

Let $C$ be a hyperbolic straight line through $z_o\in \mathbb{D}$ and $z^*_o$ the point symmetric to $z_o$ wrt the unit circle $S^1$. Then we have $z^*_o$ is outside $S^1$.

Since C is perpendicular to $S^1$, we have $z^*_0$ lies on $C$.

I can't understand that how the last statement will happen. Thanks.

$z_o^*$ is the inverse of $z_o$. The inverse of $C$ is $C$ itself, since circle inversion is conformal and knowing the two points of intersection with $S^1$ already uniquely defines the single orthogonal circle through these two points. Inversion preserves incidence, so if $z_o$ lies on $C$, then $z_o^*$ lies on the image of $C$, i.e. on $C$ itself as well.