Editing Hyperbolic triangle (section)

==Triangles with ideal vertices==

[[File:Ideal circles.svg|thumb|right|200px|Three ideal triangles in the [[Poincaré disk model]]]]

The definition of a triangle can be generalized, permitting vertices on the [[ideal point|ideal boundary]] of the plane while keeping the sides within the plane. If a pair of sides is ''[[limiting parallel]]'' (i.e. the distance between them approaches zero as they tend to the [[ideal point]], but they do not intersect), then they end at an '''ideal vertex''' represented as an ''[[ideal point|omega point]]''.

Such a pair of sides may also be said to form an angle of [[zero]].

A triangle with a zero angle is impossible in [[Euclidean geometry]] for [[line (geometry)|straight]] sides lying on distinct lines. However, such zero angles are possible with [[tangent circles]].

A triangle with one ideal vertex is called an '''omega triangle'''.

Special Triangles with ideal vertices are:

===Triangle of parallelism===
A triangle where one vertex is an ideal point, one angle is right: the third angle is the [[angle of parallelism]] for the length of the side between the right and the third angle.

===Schweikart triangle===
The triangle where two vertices are ideal points and the remaining angle is [[right angle|right]], one of the first hyperbolic triangles (1818) described by [[Ferdinand Karl Schweikart]].

===Ideal triangle===
{{Main|Ideal triangle}}
The triangle where all vertices are ideal points, an [[ideal triangle]] is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles.