Centroid

Definition

Let $k$ be a positive integer, and let $K$ be a field in which $k \neq 0$. Let $A$ be an affine space over a $K$-vector space, and let $p_1, \dots, p_k$ be points of $A$. The centroid of $p_1, \dots, p_k$ is the affine combination with all weights equal to $\frac{1}{k}$: $$G = \frac{1}{k} p_1 + \frac{1}{k} p_2 + \dots + \frac{1}{k} p_k.$$ The centroid of a triangle $ABC$ is the centroid $G = \frac{1}{3}(A + B + C)$ of its vertices.

Remarks

In $\mathbb{R}^n$, the coordinates of the centroid are the means of the coordinates of the points. In $\mathbb{R}^n$, the centroid is the point $x$ that makes the sum $\sum_i \|x - p_i\|^2$ smallest. The centroid of two points is their midpoint. The centroid of a triangle is also the centre of mass of a uniform triangular plate.

Explanations

L1 human

The centroid is the average of the points. For the triangle with the vertices $(0, 0)$, $(6, 0)$ and $(0, 3)$, the centroid is $\left(\frac{0 + 6 + 0}{3}, \frac{0 + 0 + 3}{3}\right) = (2, 1)$. A cardboard triangle balances on a pin at this point.