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In [[hyperbolic geometry]], the '''ultraparallel theorem''' states that every pair of [[ultraparallel]] lines in the hyperbolic plane has a unique common [[perpendicular]] hyperbolic line.
==
In the [[Klein model]] of the hyperbolic plane, two ultraparallel lines correspond to two non-intersecting chords. The ''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 modelled by chords such that when extended, the extension passes through the pole of A, and vice-versa. 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 ultraparallel line. 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 space, and so we draw a line through the polw of the other line intersecting the diameter at right angles to get the ultraparallel line.
The proof is completed by showing this construction is always possible. If both
chords are diameters, they intersect. 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, the 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, accross 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 is the required chord perpendicular to the two other chords.
==Proof in the Poincaré half-plane model==
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