# 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