How to manipulate a circle in GeoGebra style?

This is what I find more intuitive:

circle[] := DynamicModule[{a = {0, 0}, b = {1, 0}, r = 1, w},
  {
   Dynamic@Circle[a, r],
   Locator[Dynamic[a, {(w = b - a) &, (a = #; b = a + w) &, None}]],
   Locator[Dynamic[b, (b = #; r = Norm[b - a]) &]]

   }]

Graphics[circle[], Frame -> True, PlotRange -> 2]

---

And this is what fits well OP's example:

circle2[] := DynamicModule[{a = {0, 0}, b = {1, 0}, r = 1, s, p, hand},
  {Thick,
   EventHandler[
    Dynamic@hand@Circle[a, r],
    {"MouseDown" :> {s = {p[], a, b}; },
     "MouseDragged" :> {{a, b} = (p[] - s[[1]] + #) & /@ Rest[s]}

     }],
   Locator[Dynamic[a, (a = #; r = Norm[b - a]) &]], 
   Locator[Dynamic[b, (b = #; r = Norm[b - a]) &]]}

  ,
  Initialization :> (
    p[] := MousePosition["Graphics"];
    hand = MouseAppearance[#, "LinkHand"] &;
    )]

Graphics[circle2[], Frame -> True, PlotRange -> 2]

You can improve styling ofc.

I do nor know how to implement what you want to do in a Manipulate expression using locators, because I don't know how to handle mouse events in a Manipulate expression. However, if you are willing to accept an answer using EventHandler, the behavior you ask for isn't very difficult to implement.

With[{δ = .2}, 
  DynamicModule[{p1 = {0, 0}, p2 = {2, 2}, mouse, action, dp},
    EventHandler[
      Dynamic @ Graphics[{
          {PointSize[Large], Point[{p1, p2}]},
          {Thick, Circle[p1, Norm[p2 - p1]]}},
        Frame -> True,
        PlotRange -> 5],
      {"MouseDown" :>
        (mouse = MousePosition["Graphics"];
         action =
          Which[
           Norm[p1 - mouse] < δ , "p1",
           Norm[p2 - mouse] < δ , "p2",
           Abs[Norm[mouse - p1] - Norm[p2 - p1]] < δ , "circle"]),
       "MouseDragged" :>
        (mouse = MousePosition["Graphics"];
         Switch[action,
          "p1", p1 = mouse,
          "p2", p2 = mouse,
          "circle", dp = p2 - p1; p1 = mouse; p2 = p1 + dp])}]]]

The plan behind this code is

• A mouse-down event detects what visual object the mouse is near. This in turn sets what action should be done during the drag.

• A mouse-dragged event carries out the selected action.