Barycenter Exists and is Well Defined
Jump to navigation
Jump to search
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.
Proof
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:
\(\ds m + \sum_{i \mathop = 1}^n \lambda_i \vec{m p_i}\) | \(=\) | \(\ds m + \sum_{i \mathop = 1}^n \lambda_i \paren {\vec{m r} + \vec{r p_i} }\) | Chasles' Relation | |||||||||||
\(\ds \) | \(=\) | \(\ds m + \paren {\sum_{i \mathop = 1}^n \lambda_i} \vec {m r} + \sum_{i \mathop = 1}^n \lambda_i \vec {r p_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m + \vec {m r} + \sum_{i \mathop = 1}^n \lambda_i \vec{r p_i}\) | 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.
$\blacksquare$