Editing Hyperbolic triangle (section)

==Properties==
Hyperbolic triangles have some properties that are analogous to those of [[triangle]]s in [[Euclidean geometry]]:

*Each hyperbolic triangle has an [[inscribed circle]] but not every hyperbolic triangle has a [[circumscribed circle]] (see below). Its vertices can lie on a [[horocycle]] or [[hypercycle (geometry)|hypercycle]].

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Hyperbolic triangles have some properties that are analogous to those of triangles in [[spherical geometry|spherical]] or [[elliptic geometry]]:

*Two triangles with the same angle sum are equal in area.
*There is an upper bound for the area of triangles.
*There is an upper bound for radius of the [[inscribed circle]].
*Two triangles are congruent [[if and only if]] they correspond under a finite product of line reflections.
*Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).

Hyperbolic triangles have some properties that are the opposite of the properties of triangles in spherical or elliptic geometry:

*The angle sum of a triangle is less than 180°.
*The area of a triangle is proportional to the deficit of its angle sum from 180°.

Hyperbolic triangles also have some properties that are not found in other geometries:

*Some hyperbolic triangles have no [[circumscribed circle]], this is the case when at least one of its vertices is an [[ideal point]] or when all of its vertices lie on a [[horocycle]] or on a one sided [[hypercycle (geometry)|hypercycle]].
*[[δ-hyperbolic space|Hyperbolic triangles are thin]], there is a maximum distance δ from a point on an edge to one of the other two edges. This principle gave rise to [[δ-hyperbolic space]].