# Hyperplane with convex / affine combination

I have a $(d-1)$-simplex with the $n$ vertices $x_1,...,x_n$ and look at points described by the convex combination $$\sum_{i=1}^n a_i x_i, \qquad \sum a_i = 1.$$ Now can I describe the slice hyperplane between $\frac{1}{2}(x_1+x_2)$ and $\frac{1}{2}(x_3+x_4)$ by just setting $$a_1 + a_2 = a_3+a_4$$ and between $\frac{1}{3}(x_1+x_2 + x_3)$ and $\frac{1}{3}(x_4+x_5 + x_6)$ with the equation $$a_1+a_2+a_3 = a_4+a_5+a_6 ~?$$

I have some problems imagining this. I know, it is true for $n=4$ and $a_1=a_2$ and should also be true for $a_1 + a_2 = a_3 + a_4$ - but here I

If there is already some literature about linear algebra / geometry with affine combinations? If so, please point me towards it.

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