By a Tarski plane I understand a set $X$ endowed with a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying all Tarski axioms except for the Continuity Axiom and the Axiom of Euclid.
Let $X$ be a Tarski plane. Given three points $x,y,z\in X$ we write $Mxyz$ and say that $y$ is a midpoints between $x$ and $z$ if $Bxyz$ (i.e., $y$ is between $x$ and $z$) and $xy\equiv yz$.
Im am interested in the equivalence of the following weak versions of the Euclid parallel postulate:
(ML) $\forall a,b,c,x,y,z\in X\;((Maxb\;\wedge\; Mbyc\;\wedge\;Mcza)\;\Rightarrow\;xy\equiv az)$;
(CE) $\forall a,b,c\in X\;\exists p,q,x,y\in X\;(Mapc\,\wedge\,Mbqc\,\wedge\,Baxq\,\wedge\,Mpxy\,\wedge\, Mxyb)$.
The property (ML) says the length of the middle line of a triagnle is the half of the length of the base of the triangle.
The property (CE) says that the centroid divides the medians of a triangle in ratio $1:2$.
Question 1. Is (ML) equivalent to (CE)?
Question 2. Does (ML) imply (CE)?
Remark. I can prove that (ML) implies (CE) under the Archimedes Axiom:
$\forall a,b,c\in X\;\exists x_0,x_1,\dots,x_n\in X$ $(x_0=a\;\wedge\;x_1=b\;\wedge\;(\forall i\; Mx_{i-1}x_ix_{i+1})\;\wedge\;(\exists y\in X\;(Bayx_n\;\wedge\;ay\equiv ac)))$.
So, the problem is to prove the implication (ML)$\Rightarrow$(CE) without the Archimedes Axiom, or present an example of a Tarski space in which (ML) is satisfied but (CE) does not.
