# Root of Area contained by Rational Straight Line and First Binomial/Lemma

## Lemma to Root of Area contained by Rational Straight Line and First Binomial

In the words of Euclid:

Let there be two squares $AB, BC$, and let them be placed so that $DB$ is in a straight line with $BE$; therefore $FB$ is also in a straight line with $BG$.
Let the parallelogram $AC$ be completed; I say that $AC$ is a square, that $DG$ is a mean proportional between $AB, BC$, and further, that $DC$ is a mean proportional between $AC, CB$.

## Proof

We have that $DB = BF$ and $BE = BG$.

Thus $DE = FG$.

$DE = AH = KC$

and:

$FG = AK = HC$.

Therefore:

$AH = AK = HC = KC$

Therefore the parallelogram $AKCH$ is equilateral.

The parallelogram $AKCH$ is also rectangular.

Therefore by definition $AKCH$ is a square.

We have that:

$FB : BG = DB : BE$

and:

$FB : BG = AB : DG$
$DB : BE = DG : BC$
$AB : DG = DG : BC$

Therefore $DG$ is a mean proportional between $AB$ and $BC$.

We also have that:

$AD : DK = KG : GC$
$AK : KD = KC : CG$

while:

$AK : KD = AC : CD$
$KC : CG = DC : CB$
$AC : DC = DC : BC$

Therefore $DC$ is a mean proportional between $AC$ and $BC$.

$\blacksquare$

## Historical Note

This proof is Proposition $54$ of Book $\text{X}$ of Euclid's The Elements.
It can be argued that this lemma is not original to Euclid.