Given any linear space $L$ over an ordered field $F$, consider the *equiproportion relation* $${\sim}=\{((x,y,z),(x',y',z'))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\quad(y-x=t(z-x)\wedge y'-x'=t(z'-x'))\},$$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 $y$. 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 $\sim$? 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.