geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a centre (

Commonwealth English The use of the English language in current and former Member states of the Commonwealth of Nations, countries of Commonwealth of Nations, the Commonwealth was largely inherited from British Empire, British colonisation, with some exceptions. Eng ...

) or center (

American English American English, sometimes called United States English or U.S. English, is the set of variety (linguistics), varieties of the English language native to the United States. English is the Languages of the United States, most widely spoken lang ...

) () 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 group In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...

s, then a centre is a fixed point of all the isometries that move the object onto itself.

Circles, spheres, and segments

The centre of a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

is the point equidistant from the points on the edge. Similarly the centre of a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

is the point equidistant from the points on the surface, and the centre of a line segment is the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

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 In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

is where the diagonals intersect, this is (among other properties) the fixed point of rotational symmetries. Similarly the centre of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

or a hyperbola is where the axes intersect.

Triangles

Several special points of a triangle are often described as triangle centres: *the circumcentre, which is the centre of the circle that passes through all three vertices; *the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

or centre of mass, the point on which the triangle would balance if it had uniform density; *the incentre, the centre of the circle that is internally tangent to all three sides of the triangle; *the orthocentre, the intersection of the triangle's three altitudes; and *the nine-point centre, the centre of the circle that passes through nine key points of the triangle. For an

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

, 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: # ''f'' is homogeneous in ''a'', ''b'', ''c''; i.e., ''f''(''ta'',''tb'',''tc'')=''t''^''h''''f''(''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. 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.

Tangential polygons and cyclic polygons

A tangential polygon has each of its sides

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 In geometry, a set (mathematics), set of point (geometry), points are said to be concyclic (or cocyclic) if they lie on a common circle. A polygon whose vertex (geometry), vertices are concyclic is called a cyclic polygon, and the circle is cal ...

has each of its vertices on a particular circle, called the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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.

General polygons

The centre of a general

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

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 the

(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.

Projective conics

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

every line has a

point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...

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. G. B. Halsted (1903) ''Synthetic Projective Geometry'', #130, #131, #132, #139 * The harmonic conjugate of a point at infinity with respect to the end points of a finite sect is the 'centre' of that sect. * The pole of the straight at infinity with respect to a certain conic is the 'centre' of the conic. * The polar of any figurative point is on the centre of the conic and is called a 'diameter'. * The centre of any ellipse is within it, for its polar does not meet the curve, and so there are no tangents from it to the curve. The centre of a parabola is the contact point of the figurative straight. * The centre of a hyperbola lies without the curve, since the figurative straight crosses the curve. The tangents from the centre to the hyperbola are called 'asymptotes'. Their contact points are the two points at infinity on the curve.

References

{{DEFAULTSORT:Center (geometry) Elementary geometry *