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Firstly, please note that the related question can also be found at mathoverflow.

The question is stated as following:

In Euclidean Geometry, we know that from a given point there is an unique line perpendicular to a given line. In constant negative curvature spaces (hyperbolic geometry), from the answers over mathoverflow, I know that there also exists an unique (geodesic) line which pass a given point and perpendicular to a given line.

After a second thought of my question, I also found a method to prove it. Since we know that there are several models for hyperbolic geometry, in the projective disk model, the proof should be almost apparent. But in the conformal disk model, the proof is not so easy to reach.

Now since the answer is positive, I asked to exactly work it out in conformal disk model. More exactly,

diagram goes here As the above figure shows, disk A is the conformal model disk. J is the given point, and CD is the given line; I asked how to get the line GF, such that J is on it and GF perpendicular to CD, ie., $\angle M=\pi/2$.

P.S.The step should only be completed by compass and straightedge.

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