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No, A plane can be drawn through any 3 points. If the 3 points are collinear then they make a line and a plane can contain a line. If the points are noncollinear then they can be used to form the corners of a triangle; all points of a triangle are in the same plane.
points
line or a segment
If 2 points determine a line, then a line contains infinitely many planes.
Yes, if no endpoints are defined.