Areas of Circles are as Squares on Diameters/Lemma

Proof

 * Euclid-XII-2.png

Let it be contrived that:
 * $S : ABCD = EFGH : T$

for some area $T$.

It is to be demonstrated that $T$ is less than the circle $ABCD$.

From the relationship:
 * $S : ABCD = EFGH : T$

it follows from that:
 * $S : EFGH = ABCD : T$

But:
 * $S > EFGH$

Therefore:
 * $ABCD > T$

Hence the result.