Square on Medial Straight Line/Lemma

Lemma for Square on Medial Straight Line
Algebraically:
 * $a : b = a^2 : a b$

Proof

 * Euclid-X-22-Lemma.png

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 $EG$.