## 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.

construct a circle with center $O$ and radius $OA$.
produce $OA$ to $B$, hence making $AB$ a diameter of this circle.
construct $OC$ perpendicular to $OA$.
construct $OD$ whose length is $\dfrac 1 4$ the length of $OC$.
join $OA$.
construct $\angle ODE$ to be $\dfrac 1 4$ the angle $\angle ODA$.
construct $\angle EDF$ to be half a right angle.
construct a semicircle on $AF$ intersecting $OC$ at $G$.
construct a semicircle with center $E$ and radius $EG$, intersecting $AB$ at $H$ and $K$.
construct $HL$ and $KM$ perpendicular to $OA$, intersecting the circle $ACB$ at $L$ and $M$.
bisect $\angle LOM$ to obtain angle $\angle NOM$.
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:

 $\displaystyle \frac {\tan \angle ODE + \tan \angle ODF} {1 - \tan \angle ODE \tan \angle ODF}$ $=$ $\displaystyle \tan \angle EDF$ Tangent of Sum $\displaystyle$ $=$ $\displaystyle 1$ Tangent of $\dfrac \pi 4$

## 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 $1893$.