Symbolic formula from GeometricScene?

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?

• There's currently no way to express l in terms of r through GeometricScene. Under the hood it finds instances of a scene through numerical methods. – Chip Hurst Jun 16 at 19:06
• @Chip Hurst ah okay, that's a little disappointing but maybe it's something the developers are working on. At the very least it'd be nice to get some kind of numeric function object out of GeometricScene for things like this so that passing it to one of the optimization functions could be easier/more efficient. – ccosm Jun 16 at 23:04
• seems like a good idea. I’ll pass this along to the development team. – Chip Hurst Jun 17 at 1:02