Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere/Lemma

Proof

 * Euclid-XIV-8-Lemma.png

Let $AB$ be cut at $C$ in extreme and mean ratio where $AC$ is the greater segment.

Let $DE$ be cut at $F$ in extreme and mean ratio where $DF$ is the greater segment.

It is to be demonstrated that:
 * $AB : AC = DE : DF$

Thus: