First Apotome of Medial is Irrational

Theorem

In the words of Euclid:

If from a medial straight line there be subtracted a medial straight line commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a first apotome of a medial straight line.

Proof

Let $AB$ be a medial straight line.

Let a medial straight line $BC$ such that:

$BC$ is commensurable in square only with $AB$
the rectangle contained by $AB$ and $BC$ is rational

be cut off from $AB$.

We have that $AB$ and $BC$ are medial.

So by definition $AB^2$ and $BC^2$ are both medial.

But $2 \cdot AB \cdot BC$ is rational.

Therefore:

$AB^2$ and $BC^2$ are incommensurable with $2 \cdot AB \cdot BC$.
if $AB$ is incommensurable with either $AC$ or $CB$, $AC$ and $CB$ are incommensurable with each other.
$2 \cdot AB \cdot BC$ is incommensurable with $AC^2$.

Therefore $AC^2$ is irrational.

Therefore by definition $AC$ is irrational.

Such a straight line is known as a first apotome of a medial.

$\blacksquare$

Historical Note

This proof is Proposition $74$ of Book $\text{X}$ of Euclid's The Elements.