In the Poincare disk model I have two asymptotic lines (lines going towards the same ideal point) and I want to construct their angle bisector (or the line equidistant to both, or the line in who's reflection they change place )
How can I construct this line?
The tricks I know for a normal angle bisector don't work in this case.
For the diehards: the same question for diverging lines.
In the complex plane you can take an ideal point $p$ and invert it to infinity using a circular inversion $z\to p+r^2/(\bar z-\bar p)$. Asymptotic lines trough $p$ get mapped into parallel lines. Given a pair of these lines take the parallel line midway between them and invert back to get the angle bisector.
A different approach for any two hyperbolic lines (circular arcs orthogonal to the bounding circle) in the Poincare disk model which are not tangent is to complete the arcs into two circles $C_1$ and $C_2$ and construct the line between the two centers which will intersect the bounding circle in two ideal points $P_1$ and $P_2$. These two points determine two orthogonal family of circles. See the Wikipedia article Apollonian circles for an image of this.
Each circle in the family of circles not passing through $P_1$ and $P_2$ is determined as the locus of all points $Q$ such that the division ratio $P_1Q/P_2Q$ is constant. The division ratio constant of the middle circle is the geometric mean of the division ratio constants for $C_1$ and $C_2$.