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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 $r \in \EE$

$\ds q = r + \sum_{i \mathop = 1}^n \lambda_i \vec {r p_i}$


It is conventional to write:

$q = \lambda_1 p_1 + \cdots + \lambda_n p_n$

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

Also known as

UK orthography encodes this as barycentre.

Also see