Euclid's Definition of Circle

The traditional definition of Circle, derived from Euclid, is 'a set of points equidistant from a given point'. What is plainly inadequate about this definition is that from a given point, there are an infinite number of sets of equidistant points, i. e. a given point can be the center of an infinite number of circles. Spinoza's genetic variation on the definition does not escape the difficulty. For, a line pivoting on one of its ends does not describe merely one circle, i. e. the path of the free end of the line, as Spinoza intends--in the process, the motion of every point on the line describes a circle, as well. The problem with any such attempt to define Circle in terms of Center seems to be that it infers that because the points of a circle defines a unique center, a center defines a unique circle.