
> 
> **Problem.** Is there a subset $X$ in the Euclidean plane such that $X$ is not contained in a line and for any points $a,b,c,d\in X$ with $a\ne b$ and $c\ne d$, the intersection $X\cap\overline{ab}$ is infinite and the intersection $X\cap\overline{ab}\cap\overline{cd}$ is not empty.
> 
> 
> 


Here $\overline {uv}$ denotes the (unique) line containing distinct points $u,v$ in the Euclidean plane.



A projective plane in the Euclidean plane
Let $\ P^2(\mathbb Q)\ $ and $\ P^2(\mathbb R)\ $ be the projective planes over rationals and reals. Let $\,\ L\subseteq P^2(\mathbb R)\,\ $ be a straight line in the real plane such that


$$ L\cap P^2(\mathbb Q)\,\ =\,\ \emptyset, $$


and let projective map $\,\ F: P^2(\mathbb R)\to P^2(\mathbb R)\,\ $ map $\ L\ $ onto the ***infinite*** line. Then


$$ X\,\ :=\,\ F(P^2(\mathbb Q)) $$


is the required set (see the ***OP Question***).


