geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a centre (or center; ) of an

object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...

is a

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

in some sense in the middle of the object. According to the specific definition of center taken into consideration, an object might have no center. 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 (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...

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

Circles, spheres, and segments

The center of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

is the point

equidistant A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal. In two-dimensional Euclidean geometry, the locus of points equidistant from two given (different) points is the ...

from the points on the edge. Similarly the center of a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

is the point equidistant from the points on the surface, and the center 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''-dimen ...

of the two ends.

Symmetric objects

For objects with several

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

, the

center of symmetry A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space. For an object, any unique centre and, more ...

is the point left unchanged by the symmetric actions. So the center of a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, rectangle,

rhombus In plane Euclidean geometry, a rhombus (plural 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 ...

or parallelogram is where the diagonals intersect, this is (among other properties) the fixed point of rotational symmetries. Similarly the center of an ellipse or a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...

is where the axes intersect.

Triangles

Several special points of a triangle are often described as

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...

s: *the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

, which is the center 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 surface of the figure. The same definition extends to any ...

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

, the point on which the triangle would balance if it had uniform density; *the

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

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

orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

, the intersection of the triangle's three

altitudes Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

; and *the

nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circle t ...

, the center of the circle that passes through nine key points of the triangle. For an

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

, 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 center is a point whose

trilinear coordinate In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

s 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 center 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 center 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 point In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

s (which are interchanged by a mirror-image reflection). As of 2020, the

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...

lists over 39,000 different triangle centers.

Tangential polygons and cyclic polygons

tangential polygon In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...

has each of its sides

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

to a particular circle, called the

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

or inscribed circle. The center of the incircle, called the incenter, can be considered a center of the polygon. A

cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...

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

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

or circumscribed circle. The center of the circumcircle, called the circumcenter, can be considered a center of the polygon. If a polygon is both tangential and cyclic, it is called bicentric. (All triangles are bicentric, for example.) The incenter and circumcenter of a bicentric polygon are not in general the same point.

General polygons

The center of a general

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

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 center, called just the

(center 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, ...

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 conic The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field. The usual d ...

s from a projectivity that is not a perspectivity. A symmetry of the projective plane with a given conic relates every point or

pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...

to a line called its

polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...

. The concept of center 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 In mathematics, a real-valued function u(x,y) defined on a connected open set \Omega \subset \R^2 is said to have a conjugate (function) v(x,y) if and only if they are respectively the real and imaginary parts of a holomorphic function f(z) of ...

of a point at infinity with respect to the end points of a finite sect is the 'center' of that sect. * The pole of the straight at infinity with respect to a certain conic is the 'center' of the conic. * The polar of any figurative point is on the center of the conic and is called a 'diameter'. * The center 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 center of a parabola is the contact point of the figurative straight. * The center of a hyperbola lies without the curve, since the figurative straight crosses the curve. The tangents from the center to the hyperbola are called 'asymptotes'. Their contact points are the two points at infinity on the curve.

References

{{DEFAULTSORT:Center (geometry) Elementary geometry *