Namespaces
Variants
Actions

Difference between revisions of "Triangle centre"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
Given a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301801.png" />, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:
+
{{TEX|done}}
 +
Given a triangle $A_1A_2A_3$, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:
  
 
the centroid (i.e. the centre of mass), the common intersection point of the three medians (see [[Median (of a triangle)|Median (of a triangle)]]);
 
the centroid (i.e. the centre of mass), the common intersection point of the three medians (see [[Median (of a triangle)|Median (of a triangle)]]);
Line 11: Line 12:
 
the [[Gergonne point|Gergonne point]], the common intersection point of the lines joining the vertices with the opposite tangent points of the incircle;
 
the [[Gergonne point|Gergonne point]], the common intersection point of the lines joining the vertices with the opposite tangent points of the incircle;
  
the Fermat point (also called the Torricelli point or first isogonic centre), the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301802.png" /> that minimizes the sum of the distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301803.png" />;
+
the Fermat point (also called the Torricelli point or first isogonic centre), the point $X$ that minimizes the sum of the distances $|A_1X|+|A_2X|+|A_3X|$;
  
the Grebe point (also called the Lemoine point or symmedean point), the common intersection point of the three symmedeans (the symmedean through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301804.png" /> is the isogonal line of the median through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301805.png" />, see [[Isogonal|Isogonal]]);
+
the Grebe point (also called the Lemoine point or symmedean point), the common intersection point of the three symmedeans (the symmedean through $A_i$ is the isogonal line of the median through $A_i$, see [[Isogonal|Isogonal]]);
  
 
the [[Nagel point|Nagel point]], the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see [[Plane trigonometry|Plane trigonometry]]).
 
the [[Nagel point|Nagel point]], the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see [[Plane trigonometry|Plane trigonometry]]).
  
In [[#References|[a1]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130180/t1301806.png" /> different triangle centres are described.
+
In [[#References|[a1]]], $400$ different triangle centres are described.
  
 
The Nagel point is the isotomic conjugate of the Gergonne point, and the symmedean point is the isogonal conjugate of the centroid (see [[Isogonal|Isogonal]] for both notions of  "conjugacy" ).
 
The Nagel point is the isotomic conjugate of the Gergonne point, and the symmedean point is the isogonal conjugate of the centroid (see [[Isogonal|Isogonal]] for both notions of  "conjugacy" ).

Revision as of 16:24, 9 April 2014

Given a triangle $A_1A_2A_3$, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:

the centroid (i.e. the centre of mass), the common intersection point of the three medians (see Median (of a triangle));

the incentre, the common intersection point of the three bisectrices (see Bisectrix) and hence the centre of the incircle (see Plane trigonometry);

the circumcentre, the centre of the circumcircle (see Plane trigonometry);

the orthocentre, the common intersection point of the three altitude lines (see Plane trigonometry);

the Gergonne point, the common intersection point of the lines joining the vertices with the opposite tangent points of the incircle;

the Fermat point (also called the Torricelli point or first isogonic centre), the point $X$ that minimizes the sum of the distances $|A_1X|+|A_2X|+|A_3X|$;

the Grebe point (also called the Lemoine point or symmedean point), the common intersection point of the three symmedeans (the symmedean through $A_i$ is the isogonal line of the median through $A_i$, see Isogonal);

the Nagel point, the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see Plane trigonometry).

In [a1], $400$ different triangle centres are described.

The Nagel point is the isotomic conjugate of the Gergonne point, and the symmedean point is the isogonal conjugate of the centroid (see Isogonal for both notions of "conjugacy" ).

References

[a1] C. Kimberling, "Triangle centres and central triangles" Congr. Numer. , 129 (1998) pp. 1–285
[a2] R.A. Johnson, "Modern geometry" , Houghton–Mifflin (1929)
How to Cite This Entry:
Triangle centre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Triangle_centre&oldid=31470
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article