Use the Origin Lemma to show that, given two d-lines meeting at right angles, there is a hyperbolic reflection mapping them to two perpendicular diameters of $\mathscr D$.
my work:
case 1 : two d-lines are diameters, then the hyperbolic reflection in any of the d-lines would do
case 2: two d-lines are arcs
case 3: one of the d-lines is arc, the other one is diameter.
I am not sure how origin lemma is relating to this problem. Thank you for your response and any feedback are appreciated.
Origin Lemma:
Let A be a point of $\mathscr D$ other than the origin O. Then there exists a d-line l such that hyperbolic reflection in l maps A to O.