How can I transform the circle $x^2+y^2+a x+b y+c=0$ into $ (x-A)^2+(y-B)^2+C=0$

Updated

sol = Solve[
  Thread[CoefficientList[x^2 + y^2 + a x + b y + c, {x, y}] == 
    CoefficientList[(x - A)^2 + (y - B)^2 + C, {x, y}]], {A, B, C}]

(* {{A -> -(a/2), B -> -(b/2), C -> 1/4 (-a^2 - b^2 + 4 c)}} *)

HoldForm[(x - A)^2 + (y - B)^2 + C==0] /. First@sol
% // ReleaseHold

Original

Solve[Thread[
  CoefficientList[x^2 + y^2 + a x + b y + c, {x, y}] == 
   CoefficientList[(x - A)^2 + (y - B)^2 + C, {x, y}]], {a, b, c}]

{{a -> -2 A, b -> -2 B, c -> A^2 + B^2 + C}}

You have the right idea, but go wrong on the details.

rules =
  Solve[
    CoefficientList[x^2 + y^2 + a x + b y + c, {x, y}] == 
    CoefficientList[(x - A)^2 + (y - B)^2 + C, {x, y}], 
    {A, B, C}]

{{A -> -(a/2), B -> -(b/2), C -> 1/4 (-a^2 - b^2 + 4 c)}}

To show that this is correct transformation:

((x - A)^2 + (y - B)^2 + C == 0 /. rules)[[1]]// Simplify

c + a x + x^2 + b y + y^2 == 0

which is the original equation except that the terms are reordered in the way Mathematica prefers them.

This is a job for SolveAlways, but it does not choose from the desired parameters and the extra parameters just the ones desired. So we use the equivalent Solve[!Eliminate[!eqns,vars]] form, but with the Solve variables specified.

std = r2 + (x - x1)^2 + (y - y1)^2;
Solve[! Eliminate[! (c + a x + x^2 + b y + y^2 == std), {x, y}], {x1, y1, r2}]
std == 0 /. First[%]
(*
  {{x1 -> -(a/2), y1 -> -(b/2), r2 -> 1/4 (-a^2 - b^2 + 4 c)}}
  1/4 (-a^2 - b^2 + 4 c) + (a/2 + x)^2 + (b/2 + y)^2 == 0
*)