Construction of Regular Dodecahedron within Given Sphere/Porism

Porism to Construction of Regular Dodecahedron within Given Sphere

In the words of Euclid:

From this it is manifest that, when the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron.

(The Elements: Book $\text{XIII}$: Proposition $17$ : Porism)

Proof

Apparent from the construction.

$\blacksquare$

Historical Note

This proof is Proposition $17$ of Book $\text{XIII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XIII}$. Propositions