# 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.

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.

$FE : EG = FD : DG$

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.