Difference between revisions of "Triangle"
From Encyclopedia of Mathematics
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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). | 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 [[ | + | 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]]). |
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====Comments==== | ====Comments==== | ||
| − | For relations between angles and sides of a triangle see [[ | + | For relations between angles and sides of a triangle, see [[Plane trigonometry]]. |
====References==== | ====References==== | ||
| − | * {{Ref|a1}} | + | * {{Ref|1}} H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975) {{MR|3155265}} |
| + | * {{Ref|2}} H.S.M. Coxeter, "Introduction to geometry" , Wiley (1969) | ||
| + | * {{Ref|3}} S.I. Zetel', "A new geometry of triangles" , Moscow (1962) (In Russian) | ||
| + | * {{Ref|4}} J. Hadamard, "Leçons de géométrie élémentaire. Géométrie plane" , J. Gabay, reprint (1990) pp. Chapt. 1 | ||
| + | * {{Ref|5}} D. Efremov, "A new geometry of triangles" , Odessa (1902) (In Russian) | ||
| + | * {{Ref|a1}} M. Berger, "Geometry" , '''1–2''' , Springer (1987) Chapt. 9 (Translated from French) | ||
Latest revision as of 18:23, 14 August 2023
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).
Comments
For relations between angles and sides of a triangle, see Plane trigonometry.
References
- [1] H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer. (1975) MR3155265
- [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)
- [a1] M. Berger, "Geometry" , 1–2 , Springer (1987) Chapt. 9 (Translated from French)
How to Cite This Entry:
Triangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle&oldid=54033
Triangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle&oldid=54033
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article