incircle pattern

\[Theta]1[t_] := If[t < 1/3, 0, If[t < 2/3, (t - 1/3) Pi, Pi/3]];
\[Theta]2[t_] := If[t < 2/3, 2 Pi /3, 2 Pi/3 + (t - 2/3) Pi];
\[Theta]3[t_] := 4 Pi/3 +  Pi If[t < 1/3, t, 1/3];
ptsF[t_] := If[0 <= t < 1,
   pts = ({Sin[#], Cos[#]} &) /@ {\[Theta]1[t], \[Theta]2[
       t], \[Theta]3[t]},
   If[1 <= t < 2,
    pts = ({Sin[# + Pi + 4 Pi/3],
         Cos[# + Pi + 4 Pi/3]} &) /@ {\[Theta]3[t - 1], \[Theta]1[
        t - 1], \[Theta]2[t - 1]},
    If[2 <= t < 3,
     pts = ({Sin[# + 2 Pi/3], Cos[# + 2 Pi/3]} &) /@ {\[Theta]3[
         t - 2], \[Theta]1[t - 2], \[Theta]2[t - 2]},
     If[3 <= t < 4,
      pts = ({Sin[# + Pi + 0 Pi/3],
           Cos[# + Pi + 0 Pi/3]} &) /@ {\[Theta]3[t - 3], \[Theta]1[
          t - 3], \[Theta]2[t - 3]},
      If[4 <= t < 5,
       pts = ({Sin[# + 4 Pi/3], Cos[# + 4 Pi/3]} &) /@ {\[Theta]3[
           t - 4], \[Theta]1[t - 4], \[Theta]2[t - 4]},
       pts = ({Sin[# + Pi + 2 Pi/3],
            Cos[# + Pi + 2 Pi/3 ]} &) /@ {\[Theta]3[t - 5], \[Theta]1[
           t - 5], \[Theta]2[t - 5]}
       ]
      ]
     ]
    ]
   ];
cr[t_] := Module[{pts, lengths, c, r},
   pts = ptsF[t];
   lengths = Norm /@ (pts - RotateRight[pts]);
   perimeter = Total[lengths];
   c = {(First /@ pts).RotateRight[lengths]/
      perimeter, (Last /@ pts).RotateRight[lengths]/perimeter};
   r = 0.5 Sqrt[
      Times @@ (lengths + RotateRight[lengths] - RotateLeft[lengths])/
       perimeter];
   {c, r}
   ];
Manipulate[
 Module[{c, r, pts},
  pts = ptsF[Mod[t, 6, 0]];
  {c, r} = cr[Mod[t, 6, 0]];
  Show[
   Graphics[{
     {Darker@Red, Thick, Circle[{0, 0}, 1]},
     FaceForm[], EdgeForm[Black], Polygon[pts],
     PointSize[Medium],
     EdgeForm[Darker@Blue], FaceForm[RGBColor[0.7, 0.9, 1.0]],
     Disk[c, r],
     Darker@Red, Point[c], {Darker@Blue, Point[pts]},
     }, PlotRange -> 1, ImagePadding -> 10],
   ParametricPlot[
    cr[\[Tau]][[1]], {\[Tau], If[t < 6, 0, t - 6],
     If[t < 6, t, 6.001]}]
   ]
  ],
 {t, 0.001, 12}]