This is a Problem 3 part b from Chapter 8 from the book Complex Analysis by Stein and Shakarchi.
Show that if $\phi : \mathbb{D} \to \mathbb{D}$ preserves the hyperbolic distance then $\phi$ or $\bar{\phi}$ is an automorphism of $\mathbb{D}$.
Apparently no assumption on $\phi$ is made in this part.
The hyperbolic distance between two points $z_1$ and $z_2$ in $\mathbb{D}$ is defined as
$d(z_1, z_2) = \text{inf}_{\gamma} \int_{0}^{1}||\gamma ' (t)||_{\gamma(t)} dt$ where $\gamma$ ranges over all smooth curves from $z_1$ to $z_2$ in $\mathbb{D}$.
Also, $||w||_{z} $ is defined as $\frac{w}{1 - |z|^2}$ for any $w \in {C}$ and $z \in \mathbb{D}$.
I can reduce the problem to $\phi(0) = 0 $ by using an automorphism of $\mathbb{D}$ and automorphisms are isometries have been shown.
There is an answer here but I do not understand the line where it is said 'since $f$ i an isometry it takes circles around the origin to circles'. It seems like the author of the answer is taliking about Euclidean circles since next they use this observation to compose with a rotation. I have not any knowledge of Hyperbolic geometry.
Update: I am able to see that the hyperbolic circles with centre origin are same as Euclidean circles (again using the rotation automorphism and fact that automorphisms are isometries). Now I only need to prove that any two hyperbolic circles intersect only at a single point. Can anyone help me to prove this from the definitions introduced in this problem?
Thank you.