Center (geometry)

In geometry, a centre (or center) (from Greek κέντρον) of an object is a point in some sense in the middle of the object. According to the specific definition of centre taken into consideration, an object might have no centre. If geometry is regarded as the study of isometry groups then a centre is a fixed point of all the isometries which move the object onto itself.

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.

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 or a hyperbola is where the axes intersect.

Several special points of a triangle are often described as triangle centres:

For an equilateral triangle, these 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:

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