(related to Definition: Affine Basis, Affine Coordinate System)
Let \(\mathcal A=(A,V_A,v)\) be an affine space with \(V_A\) as the associated vector space over a field \(F\) . Let \[P_0,P_1,P_2\ldots,P_n\] be an affine basis of \(\mathcal A\). Then every point \(Q\in\mathcal A\) can be uniquely represented by the \((n+1)\)-tuple of field elements \(\lambda_0, \ldots, \lambda_n\in F \) such that \[\lambda_0+ \cdots +\lambda_n=1\] and \[Q=\lambda _{0}P_{0}+\cdots +\lambda_{n}P_{n}.\]
The elements \(\lambda_0, \ldots, \lambda_n\in F \) are called the barycentric coordinates. If \(\lambda_0= \cdots =\lambda_n\), then such a point is called the barycenter of \(\mathcal A\).
Proofs: 1