# Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters

## Theorem

In the words of Euclid:

*Similar polygons inscribed in circles are to one another as the squares on the diameters.*

(*The Elements*: Book $\text{XII}$: Proposition $1$)

## Proof

Let $ABC$ and $FGH$ be circles.

Let $ABCDE$ and $FGHKL$ be similar polygons inscribed in $ABC$ and $FGH$ respectively.

Let $BM$ and $GN$ be diameters of $ABC$ and $FGH$ respectively.

It is to be demonstrated that the ratio of the square on $BM$ to the square on $GN$ equals the ratio of the area of $ABCDE$ to the area of $FGHKL$.

Let $BE, AM, GL, FN$ be joined.

We have that $ABCDE$ is similar to $FGHKL$.

Thus:

- $\angle BAE = \angle GFL$

and so from Book $\text{VI}$ Definition $1$: Similar Rectilineal Figures:

- $BA : AE = GF : FL$

- $\triangle ABE$ is equiangular with $\triangle GFL$.

Therefore:

- $\angle AEB = \angle FLG$

But from Proposition $27$ of Book $\text{III} $: Angles on Equal Arcs are Equal:

- $\angle AEB = \angle AMB$

and:

- $\angle FLG = \angle FNG$

Therefore:

- $\angle AMB = \angle FNG$

But from Proposition $31$ of Book $\text{III} $: Relative Sizes of Angles in Segments:

- the right angle $\angle BAM$ equals the right angle $\angle GFN$.

Therefore from Proposition $32$ of Book $\text{I} $: Sum of Angles of Triangle equals Two Right Angles:

- $\angle ABM = \angle FGN$

Therefore $\triangle ABM$ is equiangular with $\triangle FGN$.

Therefore from Proposition $4$ of Book $\text{VI} $: Equiangular Triangles are Similar:

- $BM : GN = BA : GF$

But the ratio of the square on $BM$ to the square on $GN$ is the duplicate ratio of $BM$ to $GN$.

From Proposition $20$ of Book $\text{VI} $: Similar Polygons are composed of Similar Triangles:

- $ABCDE : FGHKL = BA^2 : GF^2$

Hence:

- $ABCDE : FGHKL = BM^2 : GN^2$

Hence the result.

$\blacksquare$

## Historical Note

This proof is Proposition $1$ of Book $\text{XII}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XII}$. Propositions