# Square on Medial Straight Line/Lemma

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## Lemma for Square on Medial Straight Line

In the words of Euclid:

*If there be two straight lines, then, as the first is to the second, so is the square on the first to the rectangle contained by the two straight lines.*

(*The Elements*: Book $\text{X}$: Proposition $22$ : Lemma)

Algebraically:

- $a : b = a^2 : a b$

## Proof

Let $FE$ and $EG$ be straight lines.

Let the square $DF$ be described on $FE$.

Let the rectangle $GD$ be completed.

From Areas of Triangles and Parallelograms Proportional to Base:

- $FE : EG = FD : DG$

We have that $DG$ is the rectangle contained by $DE$ and $EG$.

But as $DF$ is a square, then $DE = FE$.

Thus $DG$ is the rectangle contained by $FE$ and $EG$.

So as $FE$ is to $EG$, so is the square on $EF$ to the rectangle contained by $FE$ and $EG$.

$\blacksquare$

## Historical Note

This proof is Proposition $22$ of Book $\text{X}$ of Euclid's *The Elements*.

## Sources

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