(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.