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One can make Euclidean space an affine space by letting the points in the space be the points $p$ in your affine space $\Bbb A$ and letting addition by a vector $\mathbf v\in V$ (i.e. $p+\mathbf v$) represent a translation of $p$ a distance $\|\mathbf v\|$ in the direction of $\mathbf {\hat v}$. This is a nice way of getting to terms with what an affine space is.

But affine spaces can also be defined as a set of points closed under affine combinations. I.e. where the fundamental operation on points is affine combinations rather than addition of a point and a vector. I'm trying to figure out how to make Euclidean space an affine space in this sense, but the only way I can think to make sense of some affine combination $$q = a_1p_1 + a_2p_2 + \cdots + a_kp_k, \quad a_1 + \cdots + a_k = 1$$ is by converting back to point + vector form: $$q = p_1 + a_2(p_2-p_1) + \cdots + a_k(p_k-p_1)$$ Is there some intuitive way of geometrically defining an affine combination of points in a Euclidean space without having to go back to the other definition of an affine space?

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This is easier to see if you introduce a third view of affine spaces: an affine space is closed under binary affine combinations $(x,y)\mapsto (1-t)x+ty$ for $t\in \mathbb{R}$. A binary affine combination has a very simple geometric description: $(1-t)x+ty$ is the point on the line from $x$ to $y$ that is "$t$ of the way from $x$ to $y$" (that is, its signed distance from $x$ is $t$ times the distance from $x$ to $y$).

General $n$-ary affine combinations can then be constructed by taking repeated binary affine combinations. For instance, to get $a_1p_1+a_2p_2+a_3p_3$, you can first take a binary combination $p_{12}=\frac{a_1}{a_1+a_2}p_1+\frac{a_2}{a_1+a_2}p_2$ of $p_1$ and $p_2$ and then take a binary combination $(a_1+a_2)p_{12}+a_3p_3$ of $p_{12}$ and $p_3$.

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