= Solution
The <Poincare disc model> is
$$
\mathbb D=\{z=x+iy:|z|<1\},
\qquad
ds^2=\frac{4(dx^2+dy^2)}{(1-|z|^2)^2}.
$$
Its geodesics through the origin are the Euclidean diameters. To see directly that the radial segment from $0$ to $re^{i\theta_0}$ minimizes length, write an arbitrary joining curve as $z(t)=r(t)e^{i\theta(t)}$. Its <hyperbolic length> satisfies
$$
\begin{aligned}
L
&=\int\frac{2\sqrt{\dot r^{\,2}+r^2\dot\theta^{\,2}}}{1-r^2}\,dt\\
&\geq\int\frac{2|\dot r|}{1-r^2}\,dt
\geq2\operatorname{artanh}r.
\end{aligned}
$$
The radial segment has constant $\theta$ and monotone $r$, so equality holds. Hence
$$
\boxed{d_{\mathbb D}(0,z)=2\operatorname{artanh}|z|}
$$
and every radial diameter is length minimizing. Rotational symmetry and uniqueness for the <geodesic equation> show that these are all geodesics through $0$.
Given any geodesic and a point $p$ on it, a disc-preserving map
$$
T(z)=e^{i\varphi}\frac{z-p}{1-\overline pz}
$$
takes $p$ to the origin. By part (a), $T$ is a hyperbolic isometry and commutes with $J(z)=1/\overline z$. The transformed geodesic is a diameter, hence a generalized circle invariant under $J$. Its inverse image is therefore also a <Generalized circle under a Möbius transformation> invariant under $J$. Thus every hyperbolic geodesic is the part in $\mathbb D$ of a Euclidean line or circle preserved by reflection in the unit circle; equivalently, it is a diameter or a circle orthogonal to the unit circle. This is the <Geodesics of the Poincare disc> description.
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