Circle Circumscribing Pentagon of Dodecahedron and Triangle of Icosahedron in Same Sphere

Theorem

In the words of Hypsicles of Alexandria:

The same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere.

(The Elements: Book $\text{XIV}$: Proposition $2$)

Lemma

In the words of Hypsicles of Alexandria:

If an equilateral and equiangular pentagon be inscribed in a circle, the sum of the squares on the straight line subtending two sides and on the side of the pentagon is five times the square on the radius.

(The Elements: Book $\text{XIV}$: Proposition $2$ : Lemma)

Proof

Let $AB$ be the diameter of the given sphere.

Let a regular dodecahedron and a regular icosahedron be inscribed within.

Let $CDEFG$ be one pentagonal face of the dodecahedron.

Let $KLH$ be one triangular face of the icosahedron.

It is to be demonstrated that the radii of the circles circumscribing $CDEFG$ and $KLH$ are equal.

Let $DG$ be joined.

From Proposition $17$ of Book $\text{XIII} $: Construction of Regular Dodecahedron within Given Sphere:

$DG$ is the side of a cube which has been inscribed within the given sphere.

Let $MN$ be a straight line such that:

$AB^2 = 5 \cdot MN^2$

From Porism to Proposition $16$ of Book $\text{XIII} $: Construction of Regular Icosahedron within Given Sphere:

the square on the diameter of the given sphere is $5$ times the square on the radius of the circle in which is inscribed the pentagonal base of the pyramidal tip of a regular icosahedron which is inscribed within that sphere.

Therefore $MN$ equals the radius of the circle in which that pentagon is inscribed.

Let $MN$ be cut in extreme and mean ratio at $O$ such that $MO$ is the greater segment.

Then from:

Proposition $9$ of Book $\text{XIII} $: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio

and the converse of:

Proposition $5$ of Book $\text{XIII} $: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment

it follows that:

$MO$ is the side of a regular decagon which has been inscribed in the circle whose radius is $MN$.

From Proposition $15$ of Book $\text{XIII} $: Construction of Cube within Given Sphere:

$5 \cdot MN^2 = AB^2 = 3 \cdot DG^2$

Let $DG$ be cut in extreme and mean ratio such that $CG$ is the greater segment.

From the Lemma to Proposition $8$ of Book $\text{XIV} $: Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere:

if two straight lines have been cut in extreme and mean ratio, their segments are in the same ratio.

So:

$3 \cdot DG^2 : 3 \cdot CG^2 = 5 \cdot MN^2 : 5 \cdot MO^2$

From Proposition $16$ of Book $\text{XIII} $: Construction of Regular Icosahedron within Given Sphere:

$MO$ is the side of a regular decagon which has been inscribed

and

$KL$ is the side of a regular pentagon which has been inscribed

in the circle whose radius is $MN$.

From Porism to Proposition $15$ of Book $\text{IV} $: Inscribing Regular Hexagon in Circle

$MN$ is the side of a regular hexagon which has also been inscribed in that same circle.

Thus from Proposition $10$ of Book $\text{XIII} $: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in same Circle:

$5 \cdot MO^2 + 5 \cdot MN^2 = 5 \cdot KL^2$

Therefore:

$5 \cdot KL^2 = 3 \cdot CG^2 + 3 \cdot DG^2$

But from Proposition $12$ of Book $\text{XIII} $: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle:

$5 \cdot KL^2$ equals the square on $15$ times the radius of the circle around $KLH$.

From Lemma to Proposition $2$ of Book $\text{XIV} $: Circle Circumscribing Pentagon of Dodecahedron and Triangle of Icosahedron in Same Sphere:

$3 \cdot DG^2 + 3 \cdot CG^2$ equals the square on $15$ times the radius of the circle around $CDEFG$.

Therefore the two radii are equal.

$\blacksquare$

Historical Note

This proof is Proposition $2$ of Book $\text{XIV}$ of Euclid's The Elements.
This was proved by Aristaeus the Elder.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): The So-Called Book $\text{XIV}$, by Hypsicles