No, two points define a line. It takes three points to define a plane.
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A plane has no endpoints. It is an endless two-dimensional field. ■
In Euclidean geometry, they can only intersect in 0, 1 or infinitely many points. If there are two points of intersection then the whole line lies in the plane.
A line segment would connect two points on a plane.
Any 3 geometric points, as long as they are all in different locations and not superimposed on each other, will define a plane. In other words, there is only one plane that can pass through 3 distinct points. If you had only two points, it would define a line, but not a plane. A plane can include 2 points but if there are only 2 that are specified, the plane can rotate around those 2 points, generating infinitely many planes.
Because a place can rotate around those two points as long as the line in on the plane. Two points defines a line. A plane requires 3 points.