Brad Klee privately informs that http://mathworld.wolfram.com/RegularIcosahedron.html attributes this to Apollonius, given the "obvious" lemma: If a Platonic solid is scaled to have the same inradius as its dual, then they will also have the same circumradius. Moldy enough! —rwg On Sun, May 26, 2019 at 12:01 AM Bill Gosper <billgosper@gmail.com> wrote:

This just in: If a regular dodecahedron and icosahedron are inscribed in the same sphere, the ratios of their areas and volumes are both √(√5 GoldenRatio/3). (If true, does anyone have a moldy-oldie reference?) —rwg