The problem is a space geometry problem.
Space geometry operates on axioms and theorems built on these axioms.
1. If G and F are on a plane say P, then the line l through G and F is also in plane P
2. Segment GF is a subset of points of l, which is itself a subset of points of the plane P
So GF⊆l⊆P, thus GF⊆P, so GF is contained entirely in P
