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''of a set of points in Euclidean space''

Consider  $  m $
points  $  a _ {i} $, 
$  i = 1 \dots m $, 
in  $  \mathbf R  ^ {n} $, 
with positive weights  $  t _ {i} $
attached to them. Then the centroid of this system is the point

$$ 
c = {
\frac{\sum _ {i = 1 } ^ { m }  t _ {i} a _ {i} }{\sum _ {i = j } ^ { m }  t _ {i} }
 } .
$$

This is also the centre of gravity, or barycentre, or centre of mass (in the sense of mechanics) of these mass points. The concept belongs to [[Affine geometry|affine geometry]] rather than to the vector space (Euclidean space)  $  \mathbf R  ^ {n} $, 
in the sense that the vector

$$ 
x + \left ( \sum _ {i = 1 } ^ { m }  t _ {i} \right ) ^ {- 1 } \sum _ {i = 1 } ^ { m }  t _ {i} ( a _ {i} - x )
$$

is obviously independent of  $  x $.

If  $  t _ {i} = 1 $, 
$  i = 1 \dots m $, 
the term equi-barycentre is sometimes used. The concept can be extended to the case of possibly negative  $  t _ {i} $(
electric charges), see [[#References|[a1]]].

A related concept is that of the centre of a compact set  $  K $
with non-empty interior in  $  \mathbf R  ^ {n} $, 
which is defined as:

$$ 
c = \mu ( K ) ^ {- 1 } \int\limits _ { K } x  {d \mu } ,
$$

where  $  \mu $
is [[Lebesgue measure|Lebesgue measure]]. This is the centre of mass, in the sense of mechanics, of the body  $  K $
with uniform mass density. Here, a weight function (mass distribution) could be introduced.

For a triangle, the centroid of the three corners, each with weight unity, is the intersection point of the three medians of the triangle (see [[Plane trigonometry|Plane trigonometry]]; [[Median (of a triangle)|Median (of a triangle)]]). This is also the centre of the convex hull of the three corners (Archimedes' theorem). This is no longer necessarily true for four points in the plane.

The centroid of  $  a _ {1} \dots a _ {m} $
with weights  $  t _ {i} $, 
$  i = 1 \dots m $, 
minimizes the weighted sum of the squared distances  $  \sum _ {i = 1 }   ^ {m} t _ {i} \| {x - a _ {i} } \|  ^ {2} $.

See also [[Chebyshev centre|Chebyshev centre]] of a bounded set.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''I''' , Springer  (1987)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1969)  (Edition: Second)</TD></TR></table>