Constructing bisectors for a triangle in Poincare Disk (wolfram mathematica)

Question

I'm having difficulties constructing angle bisector in Poincare Disc(Hyperbolic geometry), specifically with writing code in Wolfram Mathematica. I've managed to create a triangle, but now have no clue how to construct its bisectors.If someone can help me with the code - I would be very glad. here's my code

Answer 1

Disclaimer

In this answer, I am going to describe an Euclidean construction whose analytic version may be useful when writing the code. But I am not going to give the analytic version or the code in any programming language.

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The actual answer

The basic principle of this solution is that the Poincaré disc model uses Euclidean objects to represent hyperbolic objects. Namely, the hyperbolic straights in this model are represented by parts of Euclidean circles (inside the Poincare circle.) These circles of Euclidean nature are perpendicular to the Poincare circle. Also, the hyperbolic angles, in this model, are identical to the visible Euclidean angles.

The figure below depicts the Poincaré circle (white) and two hyperbolic straights (red and blue.) These hyperbolic straights cross each other at a point (black).

The angle between the red and the blue line equals the Euclidean angle of the two tangent lines (white). Now, the angle bisector can be constructed by the well known Euclidean method. The black Euclidean straight is the Euclidean angle bisector.

The actual hyperbolic angle bisector is part of an Euclidean circle, is inside the Poincare circle, and is tangent to the black line and is perpendicular to the white Poincaré circle. (purple)

Is this enough? Or you need the Euclidean construction of a circle tangent to a given line (inside a given circle) and perpendicular to that given circle...