# Inscribing Circle in Regular Pentagon

## Theorem

In any given regular pentagon it is possible to inscribe a circle.

In the words of Euclid:

*In a given pentagon, which is equilateral and equiangular, to inscribe a circle.*

(*The Elements*: Book $\text{IV}$: Proposition $13$)

## Construction

Let $ABCDE$ be the given regular pentagon.

Let $\angle BCD$ and $\angle CDE$ be bisected by $CM$ and $DG$ respectively, and let these cross at $F$.

Then construct the circle whose center is $F$ and whose radius is $FM$ (or $FG$).

This circle is the one required.

## Proof

Join the straight lines $FB, FA, FE$.

We have that $BC = CD$, $CF$ is common and $\angle BCF = \angle DCF$.

So by Triangle Side-Angle-Side Equality $\triangle BCF = \triangle DCF$ and so $BF = DF$.

Thus $\angle CBF = \angle CDF$.

Since $\angle CDE = 2 \angle CDF$ and $\angle CDE = \angle CBF$, then $\angle CDF = \angle CBF$.

So $\angle ABF = \angle FBC$ and so $\angle ABC$ has been bisected by the straight line $BF$.

Similarly it can be shown that $\angle BAE, \angle AED$ have been bisected by the straight lines $FA, FE$ respectively.

Now join $FH, FK, FL$ from $F$ perpendicular to $BC, CD, DE$.

We have that:

- $\angle HCF = \angle KCF$
- $\angle FHC = \angle FKC$ (both are right angles)
- $FC$ is common and subtends one of the equal angles.

So from Triangle Side-Angle-Angle Equality:

- $\triangle FHC = \triangle FKC$

and so:

- $FH = FK$

Similarly it is shown that $FL = FM = FG = FH = FK$.

Therefore the circle whose center is $F$ and radius is $FM$ (or $FG$) passes through all of the points $G, H, K, L, M$.

We have that the angles at each of those points is a right angle.

So from Line at Right Angles to Diameter of Circle, the circle $GHKLM$ is tangent to each of lines $AB, BC, CD, DE, EA$.

Hence the result.

$\blacksquare$

## Historical Note

This theorem is Proposition $13$ of Book $\text{IV}$ of Euclid's *The Elements*.

## Sources

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