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The '''center''' of an object (in [[Geometry]]) is a point in some sense in the middle of the object. If geometry is regarded as the [[study]] of [http://en.wikipedia.org/wiki/Isometry_group isometry groups] then the centre is a fixed point of the isometries.

The '''center''' of an object (in [[Geometry]]) is a point in some sense in the middle of the object. If geometry is regarded as the [[study]] of [https://en.wikipedia.org/wiki/Isometry_group isometry groups] then the centre is a fixed point of the isometries.

==Circles==

==Circles==

The center 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.

The center 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.

==Symmetric objects==

==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 [http://en.wikipedia.org/wiki/Square_(geometry) square], [http://en.wikipedia.org/wiki/Rectangle rectangle], [http://en.wikipedia.org/wiki/Rhombus rhombus] or [http://en.wikipedia.org/wiki/Parallelogram 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.

For objects with several symmetries, the centre of [[symmetry]] is the point left unchanged by the symmetric [[actions]]. So the centre of a [https://en.wikipedia.org/wiki/Square_(geometry) square], [https://en.wikipedia.org/wiki/Rectangle rectangle], [https://en.wikipedia.org/wiki/Rhombus rhombus] or [https://en.wikipedia.org/wiki/Parallelogram 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==

==Triangles==

Several special points of a triangle are often described as triangle centres: the [http://en.wikipedia.org/wiki/Circumcentre circumcentre], centroid or centre of mass, incentre, excentres, orthocentre, nine-point centre. For an [http://en.wikipedia.org/wiki/Equilateral_triangle equilateral triangle], these (except for the excentres) are the same point.

Several special points of a triangle are often described as triangle centres: the [https://en.wikipedia.org/wiki/Circumcentre circumcentre], centroid or centre of mass, incentre, excentres, orthocentre, nine-point centre. For an [https://en.wikipedia.org/wiki/Equilateral_triangle 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:

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:

:2. 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.[1]

:2. 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.[1]

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 [http://en.wikipedia.org/wiki/Encyclopedia_of_Triangle_Centers Encyclopedia of Triangle Centers] lists over 3,000 different triangle centres.

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 [https://en.wikipedia.org/wiki/Encyclopedia_of_Triangle_Centers Encyclopedia of Triangle Centers] lists over 3,000 different triangle centres.

[[Category: Mathematics]]

[[Category: Mathematics]]

[[Category: General Reference]]

[[Category: General Reference]]