Collinear points
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between two point there is exactly one line between three points there is exactly one plane
No. The tiniest piece of a plane contains an infinite number of points. But if you give us just three points, then we know exactly what plane you're talking about, and it can't be any other plane.
yes
yes if it is a square of recatgle
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.