What is the mathematical definition of a circle?

Answer 1

A circle is the set of all points in a plane that are a given distance (or a fixed distance) from another point.

Take a plane, and then pick any point on it. Then pick a distance from that point you picked and connect all the points that are that distance from your fixed point. That will describe the circumference of a circle.

Another idea is to take a plane, pick any point, and then take a compass and put the point of that instrument on that point you picked. Then set a distance on the compass and draw a circle. That circle will be that set of all points a fixed distance from the original point you picked. (A string will work as well as a compass if that string is not stretched to draw the circle.)