Transitivity on pointed hyperbolic lines
= Transitivity on pointed hyperbolic lines
The group $PSL_2(\mathbb R)$ acts transitively on pairs $(\ell,P)$ consisting of a hyperbolic line and a point on it. Send the two ideal endpoints of $\ell$ to $0$ and $\infty$, then use a positive dilation to move the resulting point on the imaginary axis to $i$.