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Ingeometry, acentre (British English) orcenter (American English) (from Ancient Greek κέντρον (kéntron) 'pointy object') of anobject is apoint 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 ofisometry groups, then a centre is a fixed point of all the isometries that move the object onto itself.
The centre of acircle is the pointequidistant 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 themidpoint of the two ends.
For objects with severalsymmetries, thecentre of symmetry is the point left unchanged by the symmetric actions. So the centre of asquare,rectangle,rhombus orparallelogram is where the diagonals intersect, this is (among other properties) the fixed point of rotational symmetries. Similarly the centre of anellipse or ahyperbola is where the axes intersect.
Several special points of a triangle are often described astriangle centres:
For anequilateral 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 whosetrilinear coordinates aref(a,b,c) :f(b,c,a) :f(c,a,b) wheref 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 theBrocard points (which are interchanged by a mirror-image reflection). As of 2020, theEncyclopedia of Triangle Centers lists over 39,000 different triangle centres.[2]
Atangential polygon has each of its sidestangent to a particular circle, called theincircle or inscribed circle. The centre of the incircle, called the incentre, can be considered a centre of the polygon.
Acyclic polygon has each of its vertices on a particular circle, called thecircumcircle 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 calledbicentric. (All triangles are bicentric, for example.) The incentre and circumcentre of a bicentric polygon are not in general the same point.
The centre of a generalpolygon can be defined in several different ways. The "vertex centroid" comes from considering the polygon as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just thecentroid (centre of area) comes from considering the surface of the polygon as having constant density. These three points are in general not all the same point.
Inprojective geometry every line has apoint 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 calledconics in projective geometry and may be constructed asSteiner conics from a projectivity that is not a perspectivity. A symmetry of the projective plane with a given conic relates every point orpole to a line called itspolar. The concept of centre in projective geometry uses this relation. The following assertions are fromG. B. Halsted.[3]