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Let $C$ be the regular icosahedron, each of whose vertex exists at the centroid of the each surface of the regular dodecahedron $B$, each of whose vertex exists at the centroid of the each surface of a regular icosahedron $A$.

Also, let $R_A, R_B, R_C$ be the radius of the circumscribed sphere of $A,B,C$ respectively.

Then, here is my question.

Question : Can we explain the following relation in a geometrical aspect?

$$\begin{align}R_A : R_B\ =\ R_B : R_C\ =\ \tan60^{\circ} : \tan54^{\circ}\qquad(\star)\end{align}$$

Motivation : I've thought about the similarity ratio between $A$ and $C$, which is $$\frac 13 \tan^254^{\circ}=\frac{5+2\sqrt 5}{15}\approx 0.631475.$$

Then, I found $(\star)$. However, I can't find any 'nice' explanation. Can anyone help?

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