Apotome is Irrational

Theorem
Every apotome is irrational, i.e.:


 * $\ds \forall a, b \in \set {x \in \R_{>0} : x^2 \in \Q}: \paren {\frac a b \notin \Q \land \paren {\frac a b}^2 \in \Q} \implies \paren {\paren {a - b} \notin \Q \land \paren {a - b}^2 \notin \Q}$

Proof

 * Euclid-X-73.png

Let $AB$ be a rational straight line.

Let a rational straight line $BC$ which is commensurable in square only with $AB$ be cut off from $AB$.

We have that $AB$ is incommensurable in length with $BC$.

We also have:
 * $AB : BC = AB^2 : AB \cdot AC$

Therefore from :
 * $AB^2$ is incommensurable with $AB \cdot AC$.

But from :
 * $AB^2 + BC^2$ is commensurable with $AB^2$.

and from :
 * $2 \cdot AB \cdot AC$ is commensurable with $AB \cdot AC$.

From :
 * $AB^2 + BC^2 = 2 \cdot AB \cdot AC + CA^2$

and so from:

and:

it follows that:
 * $AB^2 + BC^2$ is incommensurable with $AC^2$.

But $AB$ and $BC$ are rational.

Therefore $AC$ is irrational.

Such a straight line is known as an apotome.