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Difference between revisions of "Triangle"

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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 [[Spherical geometry|spherical geometry]], and triangles in the hyperbolic or Lobachevskii plane (see [[Non-Euclidean geometries|Non-Euclidean geometries]]).
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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]]).
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H.S.M. Coxeter,  S.L. Greitzer,  "Geometry revisited" , Math. Assoc. Amer.  (1975) {{MR|3155265}}</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.I. Zetel',  "A new geometry of triangles" , Moscow  (1962)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Hadamard,  "Leçons de géométrie élémentaire. Géométrie plane" , J. Gabay, reprint  (1990) pp. Chapt. 1</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Efremov,  "A new geometry of triangles" , Odessa  (1902)  (In Russian)</TD></TR></table>
 
 
 
 
 
  
 
====Comments====
 
====Comments====
For relations between angles and sides of a triangle see [[Plane trigonometry|Plane trigonometry]].
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For relations between angles and sides of a triangle, see [[Plane trigonometry]].
  
 
====References====
 
====References====
* {{Ref|a1}} M. Berger,   "Geometry" , '''1–2''' , Springer  (1987) Chapt. 9  (Translated from French)
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* {{Ref|1}} H.S.M. Coxeter, S.L. Greitzer, "Geometry revisited" , Math. Assoc. Amer.  (1975) {{MR|3155265}}
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* {{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
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article