d = 6 Pi/99;
Manipulate[
 Graphics[{
   {
    If[4 Pi >= s > 2 Pi,
     {
      Darker@Red,
      Table[
       Block[{p}, p = {Sin[t], Cos[t]}; Circle[p, Norm[{0, 1} - p]]]
       , {t, 2 Pi, Max[2 Pi + 0.01, s], d}]
      }
     ,
     If[4 Pi < s,
      {
       Darker@Red,
       Table[
        Block[{p}, p = {Sin[t], Cos[t]}; Circle[p, Norm[{0, 1} - p]]]
        , {t, s, 6 Pi, d}]
       }
      ,
      {}
      ]
     ],
    Thickness[0.01],
    Circle[{0, 0}, 1],
    Circle[{Sin[s], Cos[s]}, Norm[{0, 1} - {Sin[s], Cos[s]}]]
    },
   Darker@Red, PointSize[Large], Point[{Sin[s], Cos[s]}], 
   Line[{{0, 1}, {Sin[s], Cos[s]}}]
   }, PlotRange -> 3.1, ImageSize -> 220]
 , {s, 0, 6 Pi, d}]