Question

two distinct points in a plane determine a blank line

Answer 1

Three points that are not all on the same line (in other words, not colinear) define a plane. This is why a table with three legs (reasonably constructed) doesn't fall over and why a tripod supports a camera. The plane is the floor in either case.

If you have four points and they are not on the same plane (not coplanar), then they define a space. So the simplest Platonic solid is a tetrahedron (a four-sided die), which has four corners.

So how many planes contain two given points? An infinite number. How many contain three given (non-colinear) points? One. How many planes contain four (non-coplanar) points? Zero.