, 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 center taken into consideration, an object might have no center. If geometry is regarded as the study of isometry group
s then a center is a fixed point of all the isometries which move the object onto itself.
Circles, spheres, and segments
The center of a circle
is the point equidistant
from the points on the edge. Similarly the center of a sphere
is the point equidistant from the points on the surface, and the center of a line segment is the midpoint
of the two ends.
For objects with several symmetries
, the center of symmetry
is the point left unchanged by the symmetric actions. So the center of a square
is where the diagonals intersect, this being (amongst other properties) the fixed point of rotational symmetries. Similarly the center of an ellipse
or a hyperbola
is where the axes intersect.
Several special points of a triangle are often described as triangle center
, which is the center of the circle that passes through all three vertices
or center of mass
, the point on which the triangle would balance if it had uniform density;
, the center of the circle that is internally tangent to all three sides of the triangle;
, the intersection of the triangle's three altitudes
*the nine-point center
, the center of the circle that passes through nine key points of the triangle.
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 center is a point whose trilinear coordinate
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
s (which are interchanged by a mirror-image reflection). As of 2020, the Encyclopedia of Triangle Centers
lists over 39,000 different triangle centers.
Tangential polygons and cyclic polygons
A tangential polygon
has each of its sides tangent
to a particular circle, called the incircle
or inscribed circle. The center of the incircle, called the incenter, can be considered a center of the polygon.
A cyclic polygon
has each of its vertices on a particular circle, called the circumcircle
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.
The center of a general 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 center, called just the centroid
(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.
* center of mass
* Chebyshev center
* Fixed points of isometry groups in Euclidean space