If you have two circles that always have equal radius on a plane, and the circles have different center points, both on the x axis to make it simpler. Such that they intersect twice.
As the radius increases the distance between the points of intersection will increase, and they will be almost opposite each other on the circle, but never quite reach that point.
As the radius tends towards infinity the distance between the edges if the circles will get smaller and smaller, so it will tend towards zero, and so the circles will overlap and must be the same.
The reason I ask is because: where z is a complex number, $|z - a| = |z - b|$, where $a \neq b$ and a and b are real numbers (to make it simpler as above), is a locus, the perpendicular bisector of the line ab. (This is because it can be thought of as all the points where circles around the points a and b (each) intersect, in the same way you'd draw a perpendicular bisector with a compass)
So in addition to the perpendicular bisector, won't the locus also include a circle of infinite radius?