- 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 ofisometry group s 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 asphere 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 orparallelogram is where the diagonals intersect, this being (amongst other properties) the fixed point of rotational symmetries. Similarly the centre of anellipse is where the axes intersect.**Triangles**Several special points of a triangle are often described as centres: the

circumcentre ,centroid orcentre of mass ,incentre ,excentre s,orthocentre ,nine-point centre . For anequilateral triangle , these (except for the excentres) are the same point.A strict definition of a triangle centre is a point whose

trilinear coordinate s 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 point s (which are interchanged by a mirror-image reflection). TheEncyclopedia 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.*

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