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Any three points will determine a plane, provided they are not collinear. If you pick any two points, you can draw a line to connect them.
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There can be any number of points on a plane, or even on a line - and any number of lines on a plane.
A minimum of three points are required to define a plne (if they are not collinear). And in projective geometry you can have a plane with only 3 points. Boring, but true. In normal circumstances, a plane will have infinitely many points. Not only that, there are infinitely many in the tiniest portion of the plane.