Side of Rational plus Medial Area is Irrational

Theorem

In the words of Euclid:

If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole straight line is irrational; and let it be called the side of a rational plus a medial area.

Proof

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

So:

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

$\blacksquare$

Historical Note

This proof is Proposition $40$ of Book $\text{X}$ of Euclid's The Elements.