Let $D$ be the unit disc model of the hyperbolic plane, with metric

$$\frac{4|d \zeta|^{2}}{\left(1-|\zeta|^{2}\right)^{2}}$$

(i) Show that the group of Möbius transformations mapping $D$ to itself is the group of transformations

$$\zeta \mapsto \omega \frac{\zeta-\lambda}{\bar{\lambda} \zeta-1},$$

where $|\lambda|<1$ and $|\omega|=1$.

(ii) Assuming that the transformations in (i) are isometries of $D$, show that any hyperbolic circle in $D$ is a Euclidean circle.

(iii) Let $P$ and $Q$ be points on the unit circle with $\angle P O Q=2 \alpha$. Show that the hyperbolic distance from $O$ to the hyperbolic line $P Q$ is given by

$$2 \tanh ^{-1}\left(\frac{1-\sin \alpha}{\cos \alpha}\right)$$

(iv) Deduce that if $a>2 \tanh ^{-1}(2-\sqrt{3})$ then no hyperbolic open disc of radius $a$ is contained in a hyperbolic triangle.