Definition:Barycenter

Definition
Let $\mathcal E$ be an affine space over a field $k$.

Let $p_1,\ldots,p_n \in \mathcal E$ be points.

Let $\lambda_1,\ldots,\lambda_n \in k$ such that $\sum_{i=1}^n \lambda_i = 1$.

The barycentre of $p_1,\ldots,p_n$ with weights $\lambda_1,\ldots,\lambda_n$ is the unique point $q$ of $\mathcal E$ such that for every point $r \in \mathcal E$
 * $\displaystyle q = r + \sum_{i=1}^n\lambda_i \vec{r p_i}$

Notation
It is conventional to write:
 * $r = \lambda_1 p_1 + \cdots + \lambda_np_n$

despite the fact that linear combinations are not defined in affine spaces.

Also see

 * Barycentre Exists and is Well Defined