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{{short description|Theorem in hyperbolic geometry}}
[[File:Ultraparallel.png|thumb|200px|right|Poincaré disc model: The pink line is ultraparallel to the blue line and the green lines are [[limiting parallel]] to the blue line.]]
In [[hyperbolic geometry]], two lines are said to be '''ultraparallel''' if they do not intersect and are not [[limiting parallel]].
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==Hilbert's construction==
Let {{mvar|r}} and {{mvar|s}} be two ultraparallel lines.
From any two distinct points {{mvar|A}} and {{mvar|C}} on s draw {{mvar|AB}} and {{mvar|CB'}} perpendicular to {{mvar|r}} with {{mvar|B}} and {{mvar|B'}} on {{mvar|r}}.
If it happens that AB = CB', then the desired common perpendicular joins the midpoints of AC and BB' (by the symmetry of the [[Saccheri quadrilateral]] ACB'B).
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