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in the Euclidean plane

Three points (the vertices) and the straight line segments (the sides) with ends at these points. Sometimes the definition of a triangle refers to the convex part of the plane that is bounded by the sides of the triangle (the solid triangle).

The notion of a triangle can be introduced in manifolds different from the Euclidean plane. A triangle is usually defined as three points and three geodesic segments with ends at these points. Such are e.g. spherical triangles in spherical geometry, and triangles in the hyperbolic or Lobachevskii plane (see Non-Euclidean geometries).


[1] H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975)
[2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969)
[3] S.I. Zetel', "A new geometry of triangles" , Moscow (1962) (In Russian)
[4] J. Hadamard, "Leçons de géométrie élémentaire. Géométrie plane" , J. Gabay, reprint (1990) pp. Chapt. 1
[5] D. Efremov, "A new geometry of triangles" , Odessa (1902) (In Russian)


For relations between angles and sides of a triangle see Plane trigonometry.


[a1] M. Berger, "Geometry" , 1–2 , Springer (1987) pp. Chapt. 9 (Translated from French)
How to Cite This Entry:
Triangle. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098