Proposition 1

Let $(x_i)_{i \in I}$ be a family of points in an affine space $E$ and $(\lambda_i)_{i \in L}$ a family of elements of $K$ of finite support such that $\sum_{i \in I} \lambda_i = 1$ (resp. $\sum_{i \in I} \lambda_i = 0$*). If $a$ is any point of $E$, the point $x \in E$ defined by

(resp. the free vector $\sum_{i \in I} \lambda_i (x_i - a)$) is independent of the point considered.

If $a'$ is another point of $E$, then

If $\sum_i \lambda_i = 1$, then $\sum_i \lambda_i (x_i - a) = (x - a) + (a - a') = x - a'$; if $\sum_i \lambda_i = 0$, then $\sum_i \lambda_i (x - a') = \sum_i \lambda_i (x - a)$; whence the proposition.

Under the conditions of Proposition 1, the point $x$ defined by

(resp. the free vector $\sum_{i \in I} \lambda_i (x_i - a)$) is denoted by $\sum_{i \in I} \lambda_i x_i$.

Thus in particular the notation $b - a$ introduced in no. 1 is recovered. When $\sum_i \lambda_i = 1$, the point $x = \sum_i \lambda_i x_i$ is called the barycentre of the points $x_i$ given the masses $\lambda_i$.

Given $m$ points $a_1, \ldots, a_m$ of $E$, whose number $m$ is not a multiple of the characteristic of $K$ (V, § 1), the point $g = \sum_{i=1}^m \frac{1}{m} a_i$ is called (by an abuse of

language) the barycentre of the points $a_i$ ($1 \leq i \leq m$) (for $m = 2$, we say "midpoint" instead of "barycentre"); it is characterized by the relation