Centre (geometry)

Encyclopedia

In geometry

, the

s then the centre is a fixed point of the isometries.

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.

, 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.

Several special points of a triangle are often described as triangle 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, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex.

A strict definition of a triangle centre is a point whose trilinear coordinates are

This strict definition exclude the excentres, and also excludes pairs of bicentric points such as the Brocard point

s (which are interchanged by a mirror-image reflection). The Encyclopedia of Triangle Centers

lists over 3,000 different triangle centres.

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the

**centre**(or**center**) of an object is a point in some sense in the middle of the object. If geometry is regarded as the study of isometry groupIsometry group

In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...

s then the centre is a fixed point of the isometries.

## Circles

The centre of a circleCircle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

is the point equidistant from the points on the edge. Similarly the centre of a sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

is the point equidistant from the points on the surface, and the centre of a line segment is the midpoint of the two ends.

## Symmetric objects

For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions. So the centre of a squareSquare (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

, rectangle

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

, rhombus

Rhombus

In Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...

or parallelogram

Parallelogram

In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...

is where the diagonals intersect, this being (amongst other properties) the fixed point of rotational symmetries. Similarly the centre of an ellipse

Ellipse

In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

is where the axes intersect.

## Triangles

Main article : Triangle centreSeveral special points of a triangle are often described as triangle centres

Triangle center

In geometry a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of...

: the circumcentre, centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

or centre of mass, incentre, excentres, orthocentre, nine-point centre. For an equilateral triangle, these (except for the excentres) are the same point, which lies at the intersection of the three axes of symmetry of the triangle, one third of the distance from its base to its apex.

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.

This strict definition exclude the excentres, and also excludes pairs of bicentric points such as the Brocard point

Brocard point

In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard , a French mathematician.-Definition:...

s (which are interchanged by a mirror-image reflection). The Encyclopedia of Triangle Centers

Encyclopedia of Triangle Centers

The Encyclopedia of Triangle Centers is an on-line list of more than 3,000 points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville....

lists over 3,000 different triangle centres.

## See also

- CentroidCentroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
- CentrepointCenterpoint (geometry)In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space...
- Centre of massCenter of massIn physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
- Chebyshev centreChebyshev centerIn geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or, alternatively, the center of largest inscribed ball of Q ....
- Fixed points of isometry groups in Euclidean spaceFixed points of isometry groups in Euclidean spaceA fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space....