# Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio

## Theorem

In the words of Euclid:

*If the square on a straight line be five times the square on a segment of it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line.*

(*The Elements*: Book $\text{XIII}$: Proposition $2$)

### Lemma

In the words of Euclid:

*That the double of $AC$ is greater than $BC$ is to be proved thus.*

(*The Elements*: Book $\text{XIII}$: Proposition $2$ : Lemma)

## Proof

Let the square on $AB$ be five times the square on $AC$.

Let $CD = 2 \cdot AC$.

It is to be demonstrated that when $CD$ is cut in extreme and mean ratio, the greater segment is $CB$.

Let the squares $AF$ and $CG$ be described on $AB$ and $CF$ respectively.

Let the figure $AF$ be drawn.

Let $BE$ be produced from $FB$.

We have that:

- $BA^2 = 5 \cdot AC^2$

That is:

- $AF = 5 \cdot AH$

Therefore the gnomon $MNO$ is $4$ times $AH$.

We have that:

- $DC = 2 \cdot CA$

Therefore:

- $DC^2 = 4 \cdot CA^2$

that is:

- $CG = 4 \cdot AH$

But:

- $MNO = 4 \cdot AH$

Therefore:

- $MNO = CG$

We have that:

- $DC = CK$

and:

- $AC = CH$

Therefore from Proposition $1$ of Book $\text{VI} $: Areas of Triangles and Parallelograms Proportional to Base:

- $KB = 2 \cdot BH$

But we also have:

- $LH + HB = 2 \cdot BH$

Therefore:

- $KB = LH + HB$

But:

- $MNO = CG$

Therefore:

- $HF = BG$

and as $CD = DG$:

- $BG = CD \cdot DB$

Also:

- $HF = CB^2$

Therefore:

- $CD \cdot DB = CB^2$

Therefore:

- $DC : CB = CB : BD$

But:

- $DC > CB$

therefore:

- $CB > BD$

Therefore, when $CD$ is cut in extreme and mean ratio, the greater segment is $CB$.

$\blacksquare$

## Historical Note

This proof is Proposition $2$ of Book $\text{XIII}$ of Euclid's *The Elements*.

It is the converse of Proposition $1$: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XIII}$. Propositions