Given any linear space $L$ over an ordered field $F$, consider the *equiproportion relation* $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-}a))\},$$called the *standard equiproportion relation* on $L$.
This relation seems to describe the affine geometry of $L$.


Observe that for any points $x,y,z\in L$ with $(x,y,z)\sim (x,y,z)$ the point $y$ lies *between* the points $x$ and $z$. Therefore the betweenness relation is encoded in the equiproportion relation.



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> **Problem.** Is there any characterization of the affine spaces $\mathbb{R}^n$ in terms of the equiproportion relation? More precisely, given a set $X$ endowed with a relation ${\sim}\subseteq X^3\times X^3$ I would like to have a reasonably short list of axioms guaranteeing that the relation algebra $(X,\sim)$ is isomorphic to the space $\mathbb R^n$ endowed with the standard equiproportion relation.
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Is the affine geometry a geometry of proportions?
The midpoint of $a$ and $b$ can be defined from equiproportion as the unique $m$ for which $(a,m,b)$ and $(b,m,a)$ are equiproportional.


Similarly, the collinearity of $a,b,c$ can be defined as one of $(a,b,c)$, $(b,c,a)$ or $(c,a,b)$ being equiproportional to itself.


Using these, $\mathbb{R}^n$ should be characterized by axioms for midpoints and collinearity plus the second-order axiom of continuity suggested in the comments.


For axioms of affine geometry from midpoint and collinearity, see Patrick [Suppes](https://www.jstor.org/stable/20117084), “Quantifier-Free Axioms for Constructive Affine Plane Geometry”, and the book referred to there: Wanda Szmielew, *From Affine to Euclidean Geometry: An Axiomatic Approach*.


