Centre (geometry)

In geometry, a centre (British English) or center (American English) (from Ancient Greek κέντρον (kéntron) 'pointy object') 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 that move the object onto itself.

Circles, spheres, and segments[edit]

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.

Symmetric objects[edit]

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 is (among other properties) the fixed point of rotational symmetries. Similarly the centre of an ellipse or a hyperbola is where the axes intersect.

the , which is the centre of the circle that passes through all three vertices;

circumcentre

the or centre of mass, the point on which the triangle would balance if it had uniform density;

centroid

the , the centre of the circle that is internally tangent to all three sides of the triangle;

incentre

the , the intersection of the triangle's three altitudes; and

orthocentre

the , the centre of the circle that passes through nine key points of the triangle.

nine-point centre

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:

This strict definition excludes pairs of bicentric points such as the Brocard points (which are interchanged by a mirror-image reflection). As of 2020, the Encyclopedia of Triangle Centers lists over 39,000 different triangle centres.^[2]

Tangential polygons and cyclic polygons[edit]

A tangential polygon has each of its sides tangent to a particular circle, called the incircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon.

A cyclic polygon has each of its vertices on a particular circle, called the circumcircle or circumscribed circle. The centre of the circumcircle, called the circumcentre, can be considered a centre of the polygon.

If a polygon is both tangential and cyclic, it is called bicentric. (All triangles are bicentric, for example.) The incentre and circumcentre of a bicentric polygon are not in general the same point.

The of a point at infinity with respect to the end points of a finite sect is the 'centre' of that sect.

harmonic conjugate

In projective geometry every line has a point at infinity or "figurative point" where it crosses all the lines that are parallel to it. The ellipse, parabola, and hyperbola of Euclidean geometry are called conics in projective geometry and may be constructed as Steiner conics from a projectivity that is not a perspectivity. A symmetry of the projective plane with a given conic relates every point or pole to a line called its polar. The concept of centre in projective geometry uses this relation. The following assertions are from G. B. Halsted.^[3]