Staring with an equilateral triangle $\Delta$, inscribe a circle, then in the gaps, inscribe other circles, ad infinitum. Similarly, inside the circumscribed circle but outside $\Delta$, continue to fill in every gap with a touching circle:
Q. Is some inscribed radius equal to some circumscribed radius, among all the gap-filling circles? Is ever a blue circle congruent to a red circle? Or do they each generate distinct infinite series of radii that never share a radius?
I would especially appreciate a resolution without resorting to explicit calculation of every radius.
