I have already solved that $\Phi(z)=z$ in the geodesic $g$, but I am stuck on this part of the problem:
Let $g$ be a complete geodesic of $H^2$, which is a semicircle of radius $R$ centered at $C=(x_0,0)$. Consider the map $\Phi (z)= x_0 + R^2\dfrac{z-x_0}{|z-x_0|^2}$
Consider a point $P\in H^2$. Let $h$ be the complete geodesic that contains $P$ and orthogonally meets $g$ at a point $Q$. Show that $\Phi$ sends $h$ to $h$. (use the fact that $\Phi$ respects angles). Conclude that $\Phi(P)$ is the point of the geodesic $h$ that is at the same hyperbolic distance from $Q$ as $P$, but is on the other side of $g$. This construction is the hyperbolic analogue of the Euclidean reflection across a straight line.
We were previously given the following: Let $g$ be a complete geodesic of $H^2$, which is a semicircle of radius $R$ centered at $C=(x_0,0)$. Consider the map $\Phi (z)= x_0 + R^2\dfrac{z-x_0}{|z-x_0|^2}$ and that $\Phi$ is the inversion across $g$.