Square on Medial Straight Line/Lemma

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