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Q: Infinite number of points equidistant from a given point?

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a circle, centered at the given point.

That's a sphere whose radius is the constant equal distance.

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The points are collinear, and there is an infinite number of planes that contain a given line. A plane containing the line can be rotated about the line by any number of degrees to form an unlimited number of other planes.If, on the other hand, the points are not collinear, then the plane has no wriggle room: it is stuck fast in one place - there can be only one plane containing all the points. Provided they are non-colinear, three points will define a plane.

A circle.

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A locus of points is just the set of points satisfying a given condition. The locus of points equidistant from a point is a circle, since a circle is just a set of points which are all the same distance away from the center

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