Sylvester's problem and Motzkin's theorem for countable and compact sets

The following three variations of Sylvester's Problem are established. Let $A$ and $B$ be compact, countable and disjoint sets of points.

(1) If $A$ spans ${E^2}$ (the Euclidean plane) then there must exist a line through two points of $A$ that intersects $A$ in only finitely many points.

(2) If $A$ spans ${E^3}$ (Euclidean three-space) then there must exist a line through exactly two points of $A$.

(3) If $A \cup B$ spans ${E^2}$ then there must exist a line through at least two points of one of the sets that does not intersect the other set.