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}]