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 $AC, CB$.
(The Elements: Book $\text{X}$: Proposition $54$ : Lemma)
Proof
We have that $DB = BF$ and $BE = BG$.
Thus $DE = FG$.
But from Proposition $34$ of Book $\text{I} $: Opposite Sides and Angles of Parallelogram are Equal:
- $DE = AH = KC$
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:
- $FB : BG = AB : DG$
and from Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:
- $DB : BE = DG : BC$
it follows from Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:
- $AB : DG = DG : BC$
Therefore $DG$ is a mean proportional between $AB$ and $BC$.
We also have that:
- $AD : DK = KG : GC$
and so from Proposition $1$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded:
- $AK : KD = KC : CG$
while:
- $AK : KD = AC : CD$
and from Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:
- $KC : CG = DC : CB$
it follows from Proposition $11$ of Book $\text{V} $: Equality of Ratios is Transitive:
- $AC : DC = DC : BC$
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.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions