Binomial is Irrational

Proof
Let $AB$ and $BC$ be two rational straight lines which are commensurable in square only.

By definition, $AB$ and $BC$ are incommensurable in length.

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

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

But by :
 * $2 AB \cdot BC$ is commensurable with $AB \cdot BC$.

We have that $AB$ and $BC$ are commensurable in square.

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

So from :
 * $2 AB \cdot BC$ is incommensurable with $AB^2 + BC^2$.

Thus from :
 * $AC^2 = \left({AB + BC}\right)^2 = 2 AB \cdot BC + AB^2 + BC^2$ is incommensurable with $AB^2 + BC^2$.

But $AB^2 + BC^2$ is rational.

Therefore $AC^2$ is irrational.

Thus by :
 * $AC$ is irrational.

Such a straight line is called binomial.