Woah you got it! This is definitely a complex code, I wouldn've never been been able to figure it out haha. Thank you!

Here is an attempt:

pointP[s_] = {0, s};
Assuming[0 < s < 2,
  pointQ[s_] = SolveValues[{x, y} \[Element] Circle[{0, 0}, s] &&
      {x, y} \[Element] Circle[{1, 0}, 1] &&
      y > 0,
     {x, y}][[1]];
  pointR[s_] = SolveValues[{x, y} \[Element] InfiniteLine[{pointQ[s],
         pointP[s]}] &&
      y == 0,
     {x, y}][[1]]];
Manipulate[Graphics[{Circle[{0, 0}, r], Circle[{1, 0}, 1],
   PointSize[Large], Point[{pointQ[r], pointP[r], pointR[r]}], 
   InfiniteLine[{pointQ[r], pointP[r]}]},
  Axes -> True, PlotRange -> {{-2, 4.5}, {-2, 3}}],
 {{r, 3/2}, 0, 2}]