There are an infinite number of any kind of points in any plane. But once you have three ( 3 ) non-collinear points, you know exactly which plane they're in, because
there's no other plane that contains the same three non-collinear points.
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no
Only one plane can pass through 3 non-collinear points.
exactly one and only one.
If you're asking a question, then the answer is 'no'. If you're making a statement, then the statement is false. I can always lay a single plane down on any three points you choose. If your points are in the same straight line, then there an infinite number of other planes that your points all lie in. If they're not all in the same straight line, then there's only one plane.
The set of all points in the plane equidistant from one point in the plane is named a parabola.