Areas of Circles are as Squares on Diameters

Proof

 * Euclid-XII-2.png

Let $ABCD$ and $EFGH$ be circles.

Let $BD$ be a diameter of $ABCD$.

Let $FH$ be a diameter of $EFGH$.

It is to be demonstrated that the ratio of the area of $ABCD$ to the area of $EFGH$ equals the ratio of the square on $BD$ to the square on $FH$.

Suppose to the contrary that the square on $BD$ to the square on $FH$ does not equal the area of $ABCD$ to the area of $EFGH$.

Then:
 * $BD^2 : FH^2 = ABCD : S$

where $S \ne EFGH$.

Suppose WLOG that $S < EFGH$.