Centre (geometry)

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Centre of a circle

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.

1 Circles
2 Symmetric objects
3 Triangles
4 See also
5 References

[edit] 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.

[edit] 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 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.

[edit] 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^hf(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.^[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 Encyclopedia of Triangle Centers lists over 3,000 different triangle centres.