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

Let *ABC* and *FGH* be circles, let *ABCDE* and *FGHKL* be similar polygons inscribed in them, and let *BM* and *GN* be diameters of the circles.

I say that the square on *BM* is to the square on *GN* as the polygon *ABCDE* is to the polygon *FGHKL.*

Join *BE, AM, GL,* and *FN.*

Now, since the polygon *ABCDE* is similar to the polygon *FGHKL,* therefore the angle *BAE* equals the angle *GFL,* and *BA* is to *AE* as *GF* is to *FL.*

Thus *BAE* and *GFL* are two triangles which have one angle equal to one angle, namely the angle *BAE* equal to the angle *GFL,* and the sides about the equal angles proportional, therefore the triangle *ABE* is equiangular with the triangle *FGL.* Therefore the angle *AEB* equals the angle *FLG.*

But the angle *AEB* equals the angle *AMB,* for they stand on the same circumference, and the angle *FLG* equals the angle *FNG,* therefore the angle *AMB* also equals the angle *FNG.*

But the right angle *BAM* also equals the right angle *GFN,* therefore the remaining angle equals the remaining angle. Therefore the triangle *ABM* is equiangular with the triangle *FGN.*

Therefore, proportionally *BM* is to *GN* as *BA* is to *GF.*

But the ratio of the square on *BM* to the square on *GN* is duplicate of the ratio of *BM* to *GN,* and the ratio of the polygon *ABCDE* to the polygon *FGHKL* is duplicate of the ratio of *BA* to *GF.*

Therefore the square on *BM* is to the square on *GN* as the polygon *ABCDE* is to the polygon *FGHKL.*

Therefore, *similar polygons inscribed in circles are to one another as the squares on their diameters.*

Q.E.D.

This proposition is in preparation for the next in which it is shown that circles are proportional to the squares on their diameters. The connection is that the circles can be arbitrarily closely approximated by polygons, so that if the polygons are proportional to the squares, then so will the circles be proportional to the squares. The difficulty in that proof coming up is to make that argument rigorous.