The four hyperbolic motions that produced <math>z</math> above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius <math>\sqrt{z}</math> to yield the unique hyperbolic line perpendicular to both ultraparallels <math>p</math> and <math>q</math>.
==Proof in the Klein model==
In the [[Klein model]] of the hyperbolic plane, two ultraparallel lines correspond to two non-intersecting chords. The ''[[Projective Harmonic Conjugates#Projective conics|poles'']] of these two lines are the respective intersections of the tangent lines to the unit circle at the endpoints of the chords. Lines ''perpendicular'' to line A are modeled by chords whose extension passes through the pole of A. Hence we draw the unique line between the poles of the two given lines, and intersect it with the unit disk; the chord of intersection will be the desired common perpendicular of the ultraparallel lines. If one of the chords happens to be a diameter, we do not have a pole, but in this case any chord perpendicular to the diameter is perpendicular as well in the hyperbolic plane, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the common perpendicular.
The proof is completed by showing this construction is always possible. If both
