Given that $l_1$ and $l_2$ are hyperbolic lines and they are ultraparallel, prove that there exists a line perpendicular to both $l_1$ and $l_2$.
My progress: In the half-plane model, I can prove that the situation when $l_1$ is a Euclidean vertical line and $l_2$ is a Euclidean semi-circle. However, I cannot prove the statement when $l_1$ and $l_2$ are both Euclidean circles. I let the radius of the two circles containing $l_1$ and $l_2$ be $r_1$ and $r_2$. I let the radius of the circle containing the perpendicular line be $r$. I let the center of the circles containing $l_1$, $l_2$ and the perpendicular line to be $c_1$, $c_2$ and $c$. Then I have the equation: $$ (c-c_1)^2 = r_1^2+r^2$$ $$ (c-c_2)^2 = r_2^2+r^2$$ I want to show that $r>0$ but I couldn't.