# Construction of Regular Heptadecagon

## Theorem

It is possible to construct a regular hepadecagon (that is, a regular polygon with $17$ sides) using a compass and straightedge construction.

## Construction

The construction will inscribe a regular hepadecagon inside any arbitrary circle.

By Proposition $11$ of Book $\text{I} $: Construction of Perpendicular Line:

- construct $OC$ perpendicular to $OA$.

By Proposition $10$ of Book $\text{I} $: Bisection of Straight Line twice:

- join $OA$.

By Proposition $9$ of Book $\text{I} $: Bisection of Angle twice:

- construct $\angle ODE$ to be $\dfrac 1 4$ the angle $\angle ODA$.

By Proposition $11$ of Book $\text{I} $: Construction of Perpendicular Line and Proposition $9$ of Book $\text{I} $: Bisection of Angle:

- construct $\angle EDF$ to be half a right angle.

Using Proposition $10$ of Book $\text{I} $: Bisection of Straight Line and Euclid's Third Postulate:

- construct a semicircle on $AF$ intersecting $OC$ at $G$.

- construct a semicircle with center $E$ and radius $EG$, intersecting $AB$ at $H$ and $K$.

By Proposition $11$ of Book $\text{I} $: Construction of Perpendicular Line:

- construct $HL$ and $KM$ perpendicular to $OA$, intersecting the circle $ACB$ at $L$ and $M$.

By Proposition $9$ of Book $\text{I} $: Bisection of Angle:

- join $NM$.

$NM$ is one of the sides of a regular hepadecagon which has been inscribed inside circle $ACB$.

## Proof

It remains to be demonstrated that the line segment $NM$ is the side of a regular hepadecagon inscribed in circle $ACB$.

This will be done by demonstrating that $\angle LOM$ is equal to $\dfrac {2 \pi} {17}$ radians, that is, $\dfrac 1 {17}$ of the full circle $ACB$.

For convenience, let the radius $OA$ be equal to $4 a$.

By Pythagoras's Theorem, $AD = a \sqrt {17}$.

By definition of tangent, $OE = a \arctan \left({\dfrac {\angle ODA} 4}\right)$.

By construction, $\angle EDF = \dfrac \pi 4$ radians.

Thus:

\(\displaystyle \frac {\tan \angle ODE + \tan \angle ODF} {1 - \tan \angle ODE \tan \angle ODF}\) | \(=\) | \(\displaystyle \tan \angle EDF\) | $\quad$ Tangent of Sum | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) | $\quad$ Tangent of $\dfrac \pi 4$ | $\quad$ |

## Also see

## Historical Note

The existence of the construction of the regular heptadecagon was first demonstrated by Carl Friedrich Gauss on $30$th March $1796$, at the age of $19$.

Some sources suggest that it was this discovery that led him to consider mathematics as a career option.

The construction given here is the one given by Herbert William Richmond in $1893$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $17$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $17$