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

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## Theorem

In the words of Euclid:

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

## 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:

$\angle BAE = \angle GFL$
$BA : AE = GF : FL$
$\triangle ABE$ is equiangular with $\triangle GFL$.

Therefore:

$\angle AEB = \angle FLG$
$\angle AEB = \angle AMB$

and:

$\angle FLG = \angle FNG$

Therefore:

$\angle AMB = \angle FNG$
the right angle $\angle BAM$ equals the right angle $\angle GFN$.
$\angle ABM = \angle FGN$

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

$BM : GN = BA : GF$

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

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

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.