Proof of the ultraparallel theorem in the Beltrami Klein model

Question

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct.

(the ultra parallel theorem is a theorem in hyperbolic geometry, and the Beltrami-Klein model an well known model of it)

My question is about the second part of the proof, and then especially the part when neither of the chords are diameters:

The proof is completed by showing this construction is always possible:

• If both chords are diameters, they intersect.(at the center of the boundary circle)
• If only one of the chords is a diameter, the other chord projects orthogonally down to a section of the first chord contained in its interior, and a line from the pole orthogonal to the diameter intersects both the diameter and the chord.
• If both lines are not diameters, then we may extend the tangents drawn from each pole to produce a quadrilateral with the unit circle inscribed within it. The poles are opposite vertices of this quadrilateral, and the chords are lines drawn between adjacent sides of the vertex, across opposite corners. Since the quadrilateral is convex, the line between the poles intersects both of the chords drawn across the corners, and the segment of the line between the chords defines the required chord perpendicular to the two other chords.

Not always will " the tangents drawn from each pole to produce a quadrilateral" (two tangents can be parallel lines , in the Euclidean sense)

Also not always will the the unit circle be inscribed within the quadrilateral. (mainly when the angle pole -center of circle - pole is less than $90^\circ$ )

Can this proof be improved /salvaged?

I don't doubt the theorem as such, but how can the proof be formulated to be correct?

Answer 1

I don't really understand the problem.

(1) The tangents are parallel lines only if one of the segments is a diameter. (2) As shown below, there is no quadrilateral if the chords lie on the same side of a diameter:

In this case the line connecting the intersection points of the tangent lines will intersect both of the cords. This is because the connecting (red) line lies inside the angles of the crossing tangents.