My question is about the barycenter. Given an affine space $(A,V)$, being $A$ a set and $V$ a $\mathbb{R}$-vector space, for $\{p_1,\dots,p_k\}$ and weights $\alpha_1,\dots,\alpha_k \in \mathbb{R}$, you define the barycenter as the element $g \in A$ such that $$\alpha_1\vec{gp_1}+\cdots+\alpha_k\vec{gp_k} = \vec{0}.$$
Now, in the real affine plane $(\mathbb{R}^2,\mathbb{R}^2)$, if you want to draw the barycenter of $\{a,b,c\}$, vertices of an equilateral triangle, with weights $\{-1,2,3\}$, what you have?
My try:
Let $g'$ the barycenter of $\{b,c\}$ with weights $2,3$, so $2\vec{g'b}+3\vec{g'c} = \vec{0}$. Then $2\vec{g'b}=3\vec{cg'}$.
Now, $g$ the barycenter of $\{g',a\}$ with weights $\{3+2,-1\}$ satisfies $-\vec{ga}+5\vec{gg'}=\vec{0}$. Therefore $\vec{ga}=5\vec{gg'}$.
I think the next picture is wrong (because the orientation of vectors?):
So, how you can draw this barycenter?
What a negative weight means (even physic interpretation)?
Is there exist a method to draw/calculate negative weights?
Thank you very much in advance!
We can think to the negative weight $-w_i$ applied at $\vec p_i$ as a positive weight $w_i$ applied at $-\vec p_i$.