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

Proof

 * Euclid-XII-1.png

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 :
 * $BA : AE = GF : FL$

Thus from :
 * $\triangle ABE$ is equiangular with $\triangle GFL$.

Therefore:
 * $\angle AEB = \angle FLG$

But from :
 * $\angle AEB = \angle AMB$

and:
 * $\angle FLG = \angle FNG$

Therefore:
 * $\angle AMB = \angle FNG$

But from :
 * the right angle $\angle BAM$ equals the right angle $\angle GFN$.

Therefore from :
 * $\angle ABM = \angle FGN$

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

Therefore from :
 * $BM : GM = BA : GF$

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

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

Hence:
 * $ABCDE : FGHKL = BM^2 : GN^2$

Hence the result.