Symbolic formula from GeometricScene?

Question

I'm trying to solve the following problem with GeometricScene

What is the shortest circular arc of which the altitude above its chord is one?

I have the following code:

GeometricScene[
    {
        {a -> {0, 0}, b -> {0, r}, c -> {0, r - 1}, d}, 
        {r, l}
    },
    {
        GeometricAssertion[{Line[{b, c}], Line[{c, d}]}, "Perpendicular"],
        GeometricAssertion[{Circle[a, r], Line[{c, d}]}, {"Concurrent", d}],
        l == 2 r PlanarAngle[a -> {c, d}]
    }
]

from which I get random instances like this

Is there any way to get a formula for l in terms of r? I know that I can substitute values of r into the scene and then use l /. RandomInstance[scene]["Quantities"] to get a numeric answer, but can this be done symbolically?

Obviously one can do the geometry by hand, but referring specifically to GeometricScene, does it have these capabilities?

Answer 1

ResourceFunction["GeometricSolve"][
 GeometricScene[{{a -> {0, 0}, b -> {0, r}, c -> {0, r - 1}, d}, {r, 
    l}}, {GeometricAssertion[{Line[{b, c}], Line[{c, d}]}, 
    "Perpendicular"], 
   GeometricAssertion[{Circle[a, r], Line[{c, d}]}, {"Concurrent", 
     d}]}], 2 r PlanarAngle[a -> {c, d}]]

Answer 2

This is a much simpler problem than for GeometricScene.

Here my solution with Manipulate.

Manipulate[
 With[{alp := ArcSin[(d - 1)/d]}, 
  Graphics[{{White, Circle[{0, 0}, 5]}, Circle[{0, 0}, d], 
    Point[{0, d Sin[alp]}], 
    Point[{0, d}], {EdgeForm[Thick], White, 
     Triangle[{{0, 0}, {-d Cos[alp], d Sin[alp]}, {d Cos[alp], 
        d Sin[alp]}}]}, Line[{{0, d Sin[alp]}, {0, d}}], 
    Text[Style[1, 16], {-0.15, d - 0.5}], {EdgeForm[Thick], Red, 
     Circle[{0, 0}, 
      d, {alp - \[Pi]/2 + \[Pi]/2, -alp + \[Pi]}]}}]], {d, 1, 5}]

There is only one parameter-free. The restriction makes this two-dimensional appearing problem one-dimensional.

Every equivalent to Manipulate creates the video or gif or animation.