Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane.
Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. $\{A,B,C\}$ are within the circle $\Gamma$ and $\overline{AB} \in K_1, \ \overline{BC} \in K_2, \ \overline{CA} \in K_3$ where $K_j \quad j= 1,2,3$ are normal circles on $\Gamma$.
For instance: any normal circle $K$ going through $p \in \overline{AB}$ cuts either $\overline{BC}$ or $\overline{CA}$ since $p$ is an element of the segment and not the vertices.