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. 

>**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 guaranteeng that the relation algebra $(X,\sim)$ is isomorphic to the space $\mathbb R^n$ endowed with the standard equiproportion relation.