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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.

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

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 $z$. 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.

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.

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? 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.

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 $z$. 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.