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Concept
Hyperbolic triangle
Summary
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices.
Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces.
A hyperbolic triangle consists of three non-collinear points and the three segments between them.
Hyperbolic triangles have some properties that are analogous to those of triangles 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.
Hyperbolic triangles have some properties that are analogous to those of triangles in 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.
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.
Official source
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