I finally got around to reading the old geometry solution by CaptainEd.
It was quite confusing, but it seems our first two steps were exactly the same: solving for the red and blue areas.
However, while I calculate the result in just one more step, the old solution appears to involve at least half a dozen more constructions.
So it looks like I've found a much more trimmed-down proof.
That said, BMAD, why don't you mark this thread as solved?
The answer is clearly that the radius is √(π / (1 - √3 - π/3)), is it not?