Ive been thinking about the following question.
Let $p \in H^2,$ and $L$ be a complete geodesic in $H^2.$ Prove that there is a unique comlete geodesic $L'$ through $p$ and orthogonal to $L$ at some point $q \in L.$ Moreover, the line segment from $p$ to $q$ minimizes the distance from $p$ to any point on $L.$
The question is asking to build two complete geodesics that are orthogonal to each other at some point.
In my textbook, a complete geodesic is said to be an "open semi-circle" centered on the x-axis and delimited by two points on the x-axis.
Now, this seems curious. How can to semi-circles, both centered and delimited by the x-axis, intersect orthogonally?
Do angles work differently in hyperbolic space? Does a right-angle in hyperbolic space look different than a right angle in Euclidean space?