<< Graphics`ImplicitPlot`

f[{a_, b_, c_}] := 
        Solve[{
(((a[[1]] - α))^2) + (((a[[2]] + β))^2) == 
              (r^2), 
(((b[[1]] - α))^2) + (((b[[2]] + β))^2) == 
              (r^2), 
(((c[[1]] - α))^2) + (((c[[2]] + β))^2) == (r^2)}, {α, β, 
            r}];

track[s_] := {s, -Sin[s]}

A[s_, t_] := {track[s + t], track[s], track[s - t]};

g1 = 
      ParametricPlot[track[s], {s, 0, π}, PlotRange -> {-1, 1}, 
        Compiled -> False, AspectRatio -> Automatic]

s = π/2;

k = 1; dk =  .01;

t1 = 
      Table[ImplicitPlot[(x - α /. f[
                        A[s, t]][[1]])^2 +  (y + β /. f[
                        A[s, t]][[1]])^2 == (((r /. f[A[s, t]][[1]]))^2), {x, 0, π}, AspectRatio -> Automatic, 
          AxesOrigin -> {0, 0}], {t, dk, k, dk}]

p1 = 
      Table[ListPlot[A[s, t], AspectRatio -> Automatic, 
          AxesOrigin -> {0, 0}, PlotStyle -> PointSize[ .02]], {t, 
          dk, k, dk}]

Table[
      Show[g1, t1[[k/dk - (r/dk)]], p1[[k/dk - (r/dk)]], 
        AspectRatio -> Automatic, Axes -> False], {r, 0, k - dk, 
        dk}]

Export["OSCsine.GIF", %]