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.