  A set of n non-collinear points in the Euclidean plane defines at least n
different lines. Chen and Chv\'atal in 2008 conjectured that the same results
is true in metric spaces for an adequate definition of line. More recently,
this conjecture was studied in the context of quasi-metric spaces. In this work
we prove that there is a quasi-metric space on four points a, b, c and d whose
betweenness is B={(c,a,b),(a,b,c),(d,b,a),(b,a,d)}. Then, this space has only
three lines none of which has four points. Moreover, we show that the
betweenness of any quasi-metric space on four points with this property is
isomorphic to B. Since B is not metric, we get that Chen and Chv\'atal's
conjecture is valid for any metric space on four points.
