Barycenter Exists and is Well Defined

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:

Hence the result.