(
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
A1,
A2, …
An,
the
centroid is defined as (1/n)(
A1 +
A2 + …
An).
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
A1,
A2, …
An had weights
w1,
w2, …
wn,
the
centroid G of the set of weighted points is:
G =
(1/
wtotal)(
w1A1 +
w2A2 + …
wnAn)
where
wtotal
=
w1 +
w2 + …
wn.
We can
partition the set of weighted points {
A1, …
An} into two sets,
α and
β.
Suppose that sets
A,
α,
β have total weights
Aw,
αw,
βw and
centroids
AG,
αG,
βG.
Then it turns out that
AG =
(1/
Aw)
(
αw *
αG
+
βw *
βG
)
.
G lies on the line that passes
αG and
βG.
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.