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Plane Bisector: Quick Illustrator

The locus of points in space that are equidistant from any two points is a plane that passes through the midpoint of the segment connecting these two points. This plane is also perpendicular to this segment.
Note this point on the plane bisector of the segment (connecting these two points) also serves as the center of a sphere that passes through these two points. The cross section of this sphere and plane is a circle that slices the sphere in half.