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

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 $\mathcal E$ and set
 * $\displaystyle q = r + \sum_{i = 1}^n \lambda_i \vec{r p_i}$

We are required to prove that for any other point $m \in \mathcal E$, we have
 * $\displaystyle q = m + \sum_{i = 1}^n \lambda_i \vec{m p_i}$

Indeed, we find that

This concludes the proof.