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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 E2 (the Eucidean plane) then there must exist a line through two points of A that intersects A in only finitely many points. (2) If A spans E3 (Euclidean three-space) then there must exist a line through exactly two points of A. (3) If A U B spans E2 then there must exist a line through at least two points of one of the sets that does not intersect the other set.