Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles

Theorem

In the words of Euclid:

Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.

(The Elements: Book $\text{XII}$: Proposition $16$)

Proof

Let $ABCD$ and $EFGH$ be two concentric circles whose centers are at $K$.

Let $ABCD$ be the larger.

It is required that a regular polygon with an even number of sides be inscribed in $ABCD$ such that it does not touch $EFGH$.

Let the straight line $BKD$ be drawn through the center $K$.

From the point $G$ let $GA$ be drawn perpendicular to $BD$ and carried through to $ABCD$ at $C$.

From Porism to Proposition $16$ of Book $\text{III} $: Line at Right Angles to Diameter of Circle:

$AC$ is tangent to $EFGH$.

Let the arc $BAD$ of $ABCD$ be bisected repeatedly.

By Proposition $1$ of Book $\text{X} $: Existence of Fraction of Number Smaller than Given:

this can be done until an arc remains less than $AD$.

Let such remaining arc be $LD$.

Let $LM$ be drawn perpendicular to $BD$ and carried through to $ABCD$ at $N$.

Let $LD$ and $DN$ be joined.

From Proposition $3$ of Book $\text{III} $: Conditions for Diameter to be Perpendicular Bisector

$LD = DN$

We have that $LN \parallel AC$.

We also have that $AC$ is tangent to $EFGH$.

Therefore $LN$ does not touch $EFGH$.

Therefore $LD$ and $DN$ are far from touching $EFGH$.

We can then fit straight lines equal to $LD$ into $ABCD$ as chords going all the way round the circle.

Thus a regular polygon with an even number of sides has been inscribed in $ABCD$ such that it does not touch $EFGH$.

$\blacksquare$

Historical Note

This proof is Proposition $16$ of Book $\text{XII}$ 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 $\text{XII}$. Propositions