# Proof

(related to Corollary: Barycentric Coordinates, Barycenter)

Let $$\mathcal A=(A,V_A,v)$$ be an affine space with $$V_A$$ as the associated vector space over a field $$F$$ . Take any $$Q\in \mathcal A$$. By definition of an affine space there is a unique vector $$q\in V_A$$ defined by $$q:=\overrightarrow{QP_0}$$, which is equivalent to $$Q=P_0 + q$$.

By hypothesis, $$P_0,P_1,P_2\ldots,P_n$$ is an affine basis of $$\mathcal A$$, which by definition means that the vectors $$x_1=\overrightarrow{P_0P_1},\,x_2=\overrightarrow{P_0P_2},\,\ldots,\,x_n=\overrightarrow{P_0P_n}$$ form a basis of the corresponding vector space $$V_A$$.

By definition of a basis of a vector space it means that there exist unique elements $$\lambda_1, \ldots, \lambda_n\in F$$ such that $q=\lambda_1x_1+ \cdots +\lambda_nx_n.$

Then it follows

$\begin{array}{rcl} Q&=&P_0+q\\ &=&P_0+\lambda_1x_1+\cdots+\lambda_nx_n\\ &=&P_0+\lambda_1\overrightarrow{P_1P_0}+\cdots+\lambda_n\overrightarrow{P_nP_0}\\ &=&P_0+\lambda_1(P_1-P_0)+\cdots+\lambda_n(P_n-P_0)\\ &=&P_0+\lambda_1P_1+\cdots+\lambda_nP_n -\lambda_1P_0-\cdots-\lambda_nP_0\\ &=&(1-(\lambda_1+\cdots+\lambda_n))P_0+\lambda_1P_1+\cdots+\lambda_nP_n\\ &=&\lambda_0P_0+\lambda_1P_1+\cdots+\lambda_nP_n \end{array}$ with $$\lambda_0:=1-(\lambda_1+\cdots+\lambda_n)\in F$$.

Because all $$\lambda_0, \ldots, \lambda_n\in F$$ form a unique $$(n+1)$$-tuple, it follows

$Q=\lambda _{0}P_{0}+\cdots +\lambda_{n}P_{n},\quad\quad\lambda_0+\cdots+\lambda_n=1.$

The following figure demonstrates the ideas of this proof for the special case of an affine plane, in which any point $$Q$$ can be represented in barycentric coordinates using the affine basis of points of a triangle.

(c) bookofproofs own work

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994