Center

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 isometry groups then the centre is a fixed point of the isometries.

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.

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.

Triangles

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.

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:

1. f is homogenous in a, b, c i.e. f(ta,tb,tc)=thf(a,b,c) for some real power h; thus the position of a centre is independent of scale.

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 Encyclopedia of Triangle Centers lists over 3,000 different triangle centres.