Barycenter Exists and is Well Defined

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Theorem

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 $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$.


Then the barycentre of $p_1, \ldots, p_n$ with weights $\lambda_1, \ldots, \lambda_n$ exists and is unique.


Proof

Let $r$ be any point in $\mathcal E$.

Set:

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

We are required to prove that for any other point $m \in \mathcal E$:

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


So:

\(\displaystyle m + \sum_{i \mathop = 1}^n \lambda_i \vec{m p_i}\) \(=\) \(\displaystyle m + \sum_{i \mathop = 1}^n \lambda_i \left({\vec{m r} + \vec{r p_i} }\right)\) Chasles' Relation
\(\displaystyle \) \(=\) \(\displaystyle m + \left(\sum_{i \mathop = 1}^n \lambda_i\right) \vec{m r} + \sum_{i \mathop = 1}^n \lambda_i \vec{r p_i}\)
\(\displaystyle \) \(=\) \(\displaystyle m + \vec{m r} + \sum_{i \mathop = 1}^n \lambda_i \vec{r p_i}\) by the assumption $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$
\(\displaystyle \) \(=\) \(\displaystyle r + \sum_{i \mathop = 1}^n \lambda_i \vec{r p_i}\) Axiom $(1)$ for an affine space
\(\displaystyle \) \(=\) \(\displaystyle q\) Definition of $q$

Hence the result.

$\blacksquare$