Let $\mathbb{H}^{2}$ denote the hyperbolic plane, and $T \subset \mathbb{H}^{2}$ be a non-degenerate triangle, i.e. the bounded region enclosed by three finite-length geodesic arcs. Prove that the three angle bisectors of $T$ meet at a point.

Must the three vertices of $T$ lie on a hyperbolic circle? Justify your answer.