# Centre (geometry)

﻿
Centre (geometry)

In geometry, the centre (or center, in American English) of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groups then the centre is a fixed point of the isometries.

Circles

The centre of a circle is the point equidistant from the points on the edge. Similarly the centre of a sphere is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends.

ymmetric objects

For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions. So the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this being (amongst other properties) the fixed point of rotational symmetries. Similarly the centre of an ellipse is where the axes intersect.

Triangles

Several special points of a triangle are often described as centres: the circumcentre, centroid or centre of mass, incentre, excentres, orthocentre, nine-point centre. For an equilateral triangle, these (except for the excentres) are the same point.

A strict definition of a triangle centre is a point whose trilinear coordinates are "f"("a","b","c") : "f"("b","c","a") : "f"("c","a","b") where "f" is a function of the lengths of the three sides of the triangle, "a", "b", "c" such that:

# "f" is homogenous in "a", "b", "c" i.e. "f"("ta","tb","tc")="t""h""f"("a","b","c") for some real power "h"; thus the position of a centre is independent of scale.
# "f" is symmetric in its last two arguments i.e. "f"("a","b","c")= "f"("a","c","b"); thus position of a centre in a mirror-image triangle is the mirror-image of its position in the original triangle. [ [http://faculty.evansville.edu/ck6/tcenters/roads.html Algebraic Highways in Triangle Geometry] ]

This strict definition exclude the excentres, and also excludes pairs of bicentric points such as the Brocard points (which are interchanged by a mirror-image reflection). The Encyclopedia of Triangle Centers lists over 3,000 different triangle centres.

ee also

* Fixed points of isometry groups in Euclidean space
* Centroid
* Centerpoint (geometry)

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Geometry & Topology — (ISSN 1364 0380 online, 1465 3060 printed) is a peer refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published… …   Wikipedia

• Centre of Mathematics and Design — Centre of Mathematics Design (MAyDI), the research Centre of Mathematics Design MAyDI was created at the Faculty of Architecture, Design and Urban Planning, University of Buenos Aires, Argentina, in 1995, under the direction of Dr. Vera W. de… …   Wikipedia

• geometry — /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… …   Universalium

• centre — /ˈsɛntə / (say sentuh) noun 1. Geometry the middle point, as the point within a circle or sphere equidistant from all points of the circumference or surface, or the point within a regular polygon equidistant from the vertices. 2. a point, pivot,… …   Australian English dictionary

• Centre de Recherches Mathématiques — The Centre de Recherches Mathématiques (CRM) [http://www.crm.umontreal.ca/en/] is the first mathematical research institute in Canada. It was founded in 1969, and was recognized as a national research center for the mathematical sciences in… …   Wikipedia

• Euclidean geometry — geometry based upon the postulates of Euclid, esp. the postulate that only one line may be drawn through a given point parallel to a given line. [1860 65] * * * Study of points, lines, angles, surfaces, and solids based on Euclid s axioms. Its… …   Universalium

• non-Euclidean geometry — geometry based upon one or more postulates that differ from those of Euclid, esp. from the postulate that only one line may be drawn through a given point parallel to a given line. [1870 75; NON + EUCLIDEAN] * * * Any theory of the nature of… …   Universalium

• Musgrave non-dead-centre engine — Park Street Mill engine in the Bolton Steam Museum …   Wikipedia

• Octahedral molecular geometry — Idealized structure of a compound with octahedral coordination geometry. In chemistry, octahedral molecular geometry describes the shape of compounds where in six atoms or groups of atoms or ligands are symmetrically arranged around a central… …   Wikipedia

• Euclidean geometry — A Greek mathematician performing a geometric construction with a compass, from The School of Athens by Raphael. Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his… …   Wikipedia