Barycenter Exists and is Well Defined

Theorem

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

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

Let $r$ be any point in $\EE$.

Set:

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

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

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

So:

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

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

by the assumption $\ds \sum_{i \mathop = 1}^n \lambda_i = 1$

$\ds $

$=$

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

Axiom $(1)$ for an affine space

$\ds $

$=$

$\ds q$

Definition of $q$

Hence the result.

$\blacksquare$