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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.
==Construction of ultraparallel lines in the Klein model==
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 line, 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 polar of the other line intersecting the diameter at right angles to get the ultraparallel line.
==Proof in the Poincaré half-plane model==
Let
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