Let $H=\{z=x+i y \in \mathbb{C}: y>0\}$ be the hyperbolic half-plane with the metric $g_{H}=\left(d x^{2}+d y^{2}\right) / y^{2}$. Define the length of a continuously differentiable curve in $H$ with respect to $g_{H}$.

What are the hyperbolic lines in $H$ ? Show that for any two distinct points $z, w$ in $H$, the infimum $\rho(z, w)$ of the lengths (with respect to $g_{H}$ ) of curves from $z$ to $w$ is attained by the segment $[z, w]$ of the hyperbolic line with an appropriate parameterisation.

The 'hyperbolic Pythagoras theorem' asserts that if a hyperbolic triangle $A B C$ has angle $\pi / 2$ at $C$ then

$$\cosh c=\cosh a \cosh b,$$

where $a, b, c$ are the lengths of the sides $B C, A C, A B$, respectively.

Let $l$ and $m$ be two hyperbolic lines in $H$ such that

$$\inf \{\rho(z, w): z \in l, w \in m\}=d>0$$

Prove that the distance $d$ is attained by the points of intersection with a hyperbolic line $h$ that meets each of $l, m$ orthogonally. Give an example of two hyperbolic lines $l$ and $m$ such that the infimum of $\rho(z, w)$ is not attained by any $z \in l, w \in m$.

[You may assume that every Möbius transformation that maps H onto itself is an isometry of $\left.g_{H} \cdot\right]$