That which produces Medial Whole with Rational Area is Irrational

Proof

 * Euclid-X-73.png

Let $AB$ be a straight line.

Let a straight line $BC$ such that:
 * $BC$ is incommensurable in square with $AB$
 * $AB^2 + BC^2$ is medial
 * the rectangle contained by $AB$ and $BC$ is rational

be cut off from $AB$.

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

while:
 * $2 \cdot AB \cdot BC$ is rational.

Therefore $AB^2 + BC^2$ is incommensurable with $2 \cdot AB \cdot BC$.

From:

and:

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

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

Therefore $AC^2$ is irrational.

Therefore $AC$ is irrational.

Such a straight line is known as that which produces with a rational area a medial whole.