If points f and g are contained in a plane, then fg is entirely contained in that plane

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