Side of Rational plus Medial Area is Irrational

Proof
From :
 * Let $AB$ and $BC$ be straight lines which are incommensurable in square which make the sum of the squares on them medial but the rectangle contained by them rational.

We have that:
 * $AB^2 + BC^2$ is medial

and from :
 * $2 \cdot AB \cdot BC$ is rational.

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

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

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

Therefore $AC^2$ is irrational.

Hence $AC$ is irrational.

Such a straight line is called the side of a rational plus a medial area.