I want to emulate the GeoGebra app. I want to get an effect like this in Mathematica.
I don't know how to move whole circle without changing the radius of the circle.
My sample code：
Manipulate[ Graphics[{Circle[p, Norm[p2 - p]]}, PlotRange -> 5, Frame -> 1], {{p, {0, 0}}, Locator}, {{p2, {2, 2}}, Locator}] 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.
circle. I think it detracts from your answer to leave it in. $\endgroup$ circle based version - which garnered my upvote. While it didn't replicate the GeoGebra behavior exactly, it certainly got the same idea across and with much simpler code. Furthermore, it's not clear from the question that an exact replication was necessary - at least not to me. $\endgroup$ 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.
x = {3, 3}; y = {5, 3}; LocatorPane[Dynamic[{x, y}], Dynamic@Graphics[{{Gray, Circle[x, Abs[y[[1]] - x[[1]]]]}, {Blue, PointSize[0.02], Point[{x, {y[[1]], x[[2]]}}]}}, Axes -> True, PlotRange -> {{-2, 8}, {-2, 8}}, AxesOrigin -> {0, 0}], Appearance -> None] 