Kitap 12
2
Let there be inscribed, also, in the circle ABCD the polygon AOBPCQDR similar to the polygon EKFLGMHN; therefore, as the square on BD is to the square on FH, so is the polygon AOBPCQDR to the polygon EKFLGMHN. [XII. 1]
2
But, as the square on BD is to the square on FH, so also is the circle ABCD to the area S; therefore also, as the circle ABCD is to the area S, so is the polygon AOBPCQDR to the polygon EKFLGMHN; [V. 11] therefore, alternately, as the circle ABCD is to the polygon inscribed in it, so is the area S to the polygon EKFLGMHN. [V. 16]
2
But the circle ABCD is greater than the polygon inscribed in it; therefore the area S is also greater than the polygon EKFLGMHN.
2
But it is also less: which is impossible.
2
Therefore, as the square on BD is to the square on FH, so is not the circle ABCD to any area less than the circle EFGH.
2
Similarly we can prove that neither is the circle EFGH to any area less than the circle ABCD as the square on FH is to the square on BD.
2
I say next that neither is the circle ABCD to any area greater than the circle EFGH as the square on BD is to the square on FH.
2
For, if possible, let it be in that ratio to a greater area S.
2
Therefore, inversely, as the square on FH is to the square on DB, so is the area S to the circle ABCD.
