The centre of a circle is equal distance from all points of its circumference
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This is the definition of a circle.
That's a "circle". The given distance is the circle's radius, and the given point is the circle's center.
A circle is the set of all the points that are the same (arbitrary) distance (called the radius) from some (arbitrary) point (called the center).
It is a circle or a sphere that fits the given description
The radius is always at a fixed, unchanging distance from the center of a circle to all the surrounding points.