Ratio of Lengths of Sides of Cube and Regular Icosahedron in Same Sphere

Proof

 * Euclid-XIV-7.png

Let a regular dodecahedron, a regular icosahedron and a cube be inscribed in a given sphere.

From :
 * the circle which circumscribes the regular pentagon which is the face of the regular dodecahedron is the same size as the circle which circumscribes the equilateral triangle which is the face of the regular icosahedron.

Let $AHB$ be the circle which circumscribes both that regular pentagon and that equilateral triangle.

Let $C$ be the center of the circumscribing circle.

Let $CB$ be a radius of $AHB$.

Let $CB$ be divided at $D$ in extreme and mean ratio where $CD$ is the greater segment.

Then from:

and the converse of:

it follows that:
 * $CD$ is the side of a regular decagon which has been inscribed in $AHB$.

Let $E$ be the side of the equilateral triangle that is the face of the regular icosahedron.

Let $F$ be the side of the regular pentagon which is the face of the regular dodecahedron.

Let $G$ be the side of the square which is the face of the cube.

From :
 * if $G$ is divided in extreme and mean ratio, the greater segment is equal to $F$.

Thus:

Hence the result.