[ edited 9th april 2019]
I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.
If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?
(1) I choose an arbitrary point O (as first "center").
(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")
(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.
(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".
(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.
Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory
Mathematicians already have a symbol for a circle of radius $r$ centred at $(x_0 ,y_0 )$, it is $$B((x_0 ,y_0 ),r)=\{ (x,y) | (x-x_0 )^2 + (y - y_0 )^2 = r \}$$ Then, the set of all circles is $$C=\{B((x_0 ,y_0 ),r) | r>0, (x_0 ,y_0 ) \in Plane \}$$