Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P.
However, there's one line M which is "special" in that the closest that it gets to L is exactly P.
One way to construct this line is to draw the perpendicular from P down to L, then obtain M by drawing the perpendicular through L at P.
Another way, in the Klein model, is to get the unique chord M going through P which is "parallel" to L if L and M are both extended outside of the disk to the rest of the Euclidean plane.
I believe this line should also be the "bisector" of the angle made by the asymptotic limiting parallels to L which go through P.
Does this line have a name in the hyperbolic geometry literature? It is, in a certain sense, the "most parallel" line to L going through P.