Prove the following statement in Mobius geometry

Question

Let $C$ be a cline, $z$ and $z^{*}$ be two distinct symmetric points w.r.t. $C$.

Then

1. Any cline $C'$ that is orthogonal to $C$ and passing through $z$, must also pass through $z^{*}$.

And its converse

2. Any cline passing through $z$ and $z^{*}$, is orthogonal to $C$.

I have an idea, since all clines are congruent in hyperbolic geometry, and all Mobius transformations preserve angles and pairs of symmetric points. We can simply consider the case with $C$ being a straight line, but I am not sure how to continue...