(

*Mathematics - Affine Geometry*)

__Centroid of a triangle__
The

centroid of a

triangle is the

concurrent point of the three

medians, or lines that pass through a

vertex and the

midpoint of the side opposite of that vertex.
The

centroid *G* of a

triangle *ABC* is the point

*G* = (1/3)(

*A* +

*B* +

*C*), which is equal to

*G* = (1/3)(

*A*) + (2/3)(1/2)(

*B* +

*C*), which shows that

*G* lies on the median that passes

*A* and the midpoint

*A'* of

*B* and

*C*, at a point in between

*A* and

*A'* that is closer to

*A'* than it is to

*A* by half the distance. Because of the symmetry of the equation,

*G* lies on all three medians.

__Centroid of a finite set of points__
To further generalise the definition, for a finite set of points

*A*_{1},

*A*_{2}, …

*A*_{n},
the

centroid is defined as (1/n)(

*A*_{1} +

*A*_{2} + …

*A*_{n}).
In fact, a

midpoint is the

centroid of two points.
An interesting fact is that the

centroid of the

roots of a

complex polynomial of degree greater than 1 is the same as the

centroid of the

roots of the

derivative function.

__Centroid of weighted points__
If points

*A*_{1},

*A*_{2}, …

*A*_{n} had weights

*w*_{1},

*w*_{2}, …

*w*_{n},
the

centroid *G* of the set of weighted points is:

*G* =
(1/

*w*_{total})(

*w*_{1}*A*_{1} +

*w*_{2}*A*_{2} + …

*w*_{n}*A*_{n})
where

*w*_{total}
=

*w*_{1} +

*w*_{2} + …

*w*_{n}.

We can

partition the set of weighted points {

*A*_{1}, …

*A*_{n}} into two sets,

*α* and

*β*.
Suppose that sets

*A*,

*α*,

*β* have total weights

*A*_{w},

*α*_{w},

*β*_{w} and

centroids
*A*_{G},

*α*_{G},

*β*_{G}.
Then it turns out that

*A*_{G} =
(1/

*A*_{w})
(

*α*_{w} *

*α*_{G}
+

*β*_{w} *

*β*_{G}
)
.
G lies on the line that passes

*α*_{G} and

*β*_{G}.
If we consider a triangle centroid to be the centroid of three points with weights 1, then by the above equation it is clear that the centroid lies on the medians at a location that leans closer towards the

midpoints.

__Other terms for centroid__
Another term for the centroid of weighted points is

affine combination.

When the weights of an

affine combination are

nonnegative, then the combination is called a

convex combination.

Another term for the centroid of non-weighted points on the Real line is

arithmetic mean.

In physics, centroid is also known as

center of mass.