ISL 1996 C4

Determine whether or nor there exist two disjoint infinite sets $A$ and $B$ of points in the plane satisfying the following conditions:

a.) No three points in $A \cup B$ are collinear, and the distance between any two points in $A \cup B$ is at least 1.

b.) There is a point of $A$ in any triangle whose vertices are in $B,$ and there is a point of $B$ in any triangle whose vertices are in $A.$