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A Circle.
All points on the circumference of a circle drawn on a plane are equidistant from the single point on the plane which is the center of the circle.
If your asking "What is ...?" Then, the answer is a circle. This is the definition of a circle!
A circle is the set of all points in a plane that are equidistant from a given point, called the center.
They are called equidistant points and form points on a sphere for a solid or a circle on a plane figure.