Ultraparallel theorem: Difference between revisions

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In [[hyperbolic geometry]], the '''ultraparallel theorem''' states that every pair of [[ultraparallel line]]s in(lines thethat are not intersecting and not [[hyperbolic geometry#Models_of_the_hyperbolic_plane|hyperboliclimiting planeparallel]]) has a unique common [[perpendicular]] hyperbolic line.

==Proof in the Poincaré half-plane model==

==Hilberts construction==

Let r and s be two non-intersecting lines.

From any two points A and C on s draw AB and CB' perpendicular to r. (B and B' on r)

If it happens that AB = CB' the desired common perpendicular joins the midpoints AC and BB'(by the symmetry of the isocleses birectangle ACB'B ).

.

If not, suppose AB < CB'.

Take A' on CB' so that A'B' = AB.

Trough A' draw a line s', making the same angle with A'B' that s makes with AB.

Then s meets s' in an ordinary point D.

Take a point D' on ray AC so that AD' = A'D.

Then the perpendicular bisector of DD' is also perpendicular to r.<ref>{{cite book|last1=coxeter|title=non euclidean geometry|isbn=987-0-88385-522-5|pages=190-192}}</ref>

==Proof in the Poincaré half-plane model ==

[[Image:Ultraparallel theorem.svg|400px|right]]