Inscribing Regular 15-gon in Circle

Theorem
In a given circle, it is possible to inscribe a regular 15-gon.

Construction

 * Euclid-IV-16.png

Let $ABCD$ be the given circle.

In $ABCD$ we inscribe an equilateral triangle, one of whose vertices is at $A$.

In $ABCD$ we also inscribe a regular pentagon, one of whose vertices is at $A$.

Thus we have that $AC$ is one of the sides of the equilateral triangle, and $AB$ is one of the sides of the regular pentagon.

Let $BC$ be bisected by a line which passes through the center of the circle.

Let this line intersect the circumference of the circle at $E$.

We fit as many copies of the straight line $BE$ around the circumference of $ABCD$, starting each one at the point the previous one ends.

The resulting polygon is the required regular 15-gon.

Proof
Consider the circumference of circle $ABCD$ as divided into $15$ equal arcs.

Of these, there will be $5$ in the shorter arc $AC$, and $3$ in the shorter arc $AB$.

So there are $2$ in the shorter arc $BC$.

Once this has been bisected by the construction which produces $E$, we see that each of these parts is a copy of these $15$ equal arcs.

Then $BE$ and $EC$ are $\dfrac 1 {15}$ of the length of the circumference of $ABCD$.

Hence the result.

Corollary
In the same way as for the regular pentagon, we can draw tangents to the circle at the vertices of the regular 15-gon.

This will draw a regular 15-gon which has been circumscribed about the circle.

Further, in a similar way to methods used for the regular pentagon, a circle can be inscribed in a regular 15-gon and circumscribed about a regular 15-gon.