# Inscribing Regular 15-gon in Circle/Corollary

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## Corollary to Inscribing Regular 15-gon in Circle

In the words of Euclid:

*And, in like manner as in the case of the pentagon, if through the points of division on the circle we draw tangents to the circle, there will be circumscribed about the circle a fifteen-angled figure which is equilateral and equiangular.*

And further, by proofs similar to those in the case of the regular pentagon, we can both inscribe a circle in the given fifteen-angled figure and circumscribe one about it.

(*The Elements*: Book $\text{IV}$: Proposition $16$ : Corollary)

## Proof

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.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{IV}$. Propositions