First Apotome of Medial is Irrational

Proof

 * Euclid-X-73.png

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 $2 \cdot AB \cdot BC$.

From :
 * if $AB$ is incommensurable with either $AC$ or $CB$, $AC$ and $CB$ are incommensurable with each other.

Therefore by :
 * $2 \cdot AB \cdot BC$ 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.