Areas of Circles are as Squares on Diameters/Lemma

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Theorem

In the words of Euclid:

I say that, the area $S$ being greater than the circle $EFGH$, as the area $S$ is to the circle $ABCD$, so is the circle $EFGH$ to some area less than the circle $ABCD$.

(The Elements: Book $\text{XII}$: Proposition $2$ : 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 Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately that:

$S : EFGH = ABCD : T$

But:

$S > EFGH$

Therefore:

$ABCD > T$

Hence the result.

$\blacksquare$


Historical Note

This theorem is Proposition $2$ of Book $\text{XII}$ of Euclid's The Elements.


Sources