Inscribing a Regular 15-gon in a Circle
Contents |
Theorem
In a given circle, it is possible to inscribe a regular 15-gon.
Construction
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.
$\blacksquare$
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.
Historical Note
This is Proposition 16 of Book IV of Euclid's The Elements.
