Manipulate[
 Module[{p1, p2, p3},
  p1 = a {Cos[\[Theta]], Sin[\[Theta]]};
  p2 = a {Cos[\[Theta]], 0};
  p3 = p2 + b {Cos[\[Theta]], Sin[\[Theta]]};
  Show[
   Graphics[{
     Thickness[0.005],
     {Darker@Red, Circle[{0, 0}, a]},
     {Darker@Green, Circle[p2, b]},
     Line[{{0, 0}, p1, p2, p3}],
     PointSize[Medium],
     Point[{{0, 0}}],
     {Darker@Red, Point[p1]},
     {Darker@Green, Point[p2]},
     {RGBColor[0.01, 0.4, 0.5], Point[p3]}
     }, PlotRange -> (a + b), ImagePadding -> 10],
   ParametricPlot[
    {(a + b) Cos[\[Theta]t], b Sin[\[Theta]t]}, {\[Theta]t, 
     If[\[Theta] > 2 Pi, \[Theta] - 2 Pi, 0], 
     If[\[Theta] > 2 Pi, 2 Pi + 0.001, Max[\[Theta], 0.001]]},
    PlotStyle -> Directive[Thick, RGBColor[0.01, 0.4, 0.5]]
    ]
   ]
  ],
 {\[Theta], 0, 4 Pi}]