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It is the circumcentre of the triangle formed by the three points. Draw the perpendicular bisectors of two of the lines joining the three points. They will meet at the point that is equidistant from the three points.
A triangle? Three lines that intersect in three points.
A plane is named by three points in the plane that is not on the same line.
Three points can determine a plane but not 3-d space.
No. A trinagle does not require four points, three are sufficient. And any three points, if they are not colinear, must be coplanar.