Formal definition of plane

Question

The formal definition of plane says that:

A plane is a set of points such that if any two points are taken on it, all the points lying on the line joining these two points also lie on the plane.

The definition is not very intuitive(to me). How does this definition specify a plane, i.e., why are we led to believe that a set of points possessing the above property is actually a plane?

Answer 1

There are infinitely many planes that intersect any given line. So we can not define a plane by two points alone, as you seem to be suggesting. Two points define/determine a line, since all points lying on the line connecting the two points necessarily also lie on that line. Period. So the "formal definition" you give is incorrect.

You need three non-collinear points to determine/define a plane.

Answer 2

Hmm...this looks more like the definition of a convex set. For example, let $D$ be the unit disk in the plane. Then for any two points in $D$ the line segment joining those two points are also in $D$.