\begin{definition}[Definition:Barycenter]
Let $\EE$ be an affine space over a field $k$.
Let $p_1, \ldots, p_n \in \EE$ be points.
Let $\lambda_1, \ldots, \lambda_n \in k$ such that $\ds \sum_{i \mathop = 1}^n \lambda_i = 1$.
The '''barycenter''' of $p_1, \ldots, p_n$ with '''weights''' $\lambda_1, \ldots, \lambda_n$ is the unique point $q$ of $\EE$ such that for every point $r \in \EE$
:$\ds q = r + \sum_{i \mathop = 1}^n \lambda_i \vec {r p_i}$
\end{definition}