Side of Rational Plus Medial Area is Divisible Uniquely

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In the words of Euclid:

The side of a rational plus a medial area is divided at one point only.

(The Elements: Book $\text{X}$: Proposition $46$)


Let $AB$ be a side of a rational plus a medial area.

Let $AB$ be divided at $C$ to create $AC$ and $CB$ such that:

$AC$ and $CB$ are incommensurable in square
$AC^2 + CB^2$ is medial
$AC$ and $CB$ contain a rational rectangle.

From Proposition $19$ of Book $\text{X} $: Product of Rationally Expressible Numbers is Rational:

$2 \cdot AC \cdot CB$ is a rational rectangle.

Let $AB$ be divided at $D$ such that $AD$ and $DB$ have the same properties as $AB$ and $CB$.

From Proposition $4$ of Book $\text{II} $: Square of Sum:

$AB^2 = \paren {AC + CB}^2 = AC^2 + CB^2 + 2 \cdot AC \cdot CB$


$AB^2 = \paren {AD + DB}^2 = AD^2 + DB^2 + 2 \cdot AD \cdot DB$

and so:

$\paren {AC^2 + CB^2} - \paren {AD^2 + DB^2} = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$

Since $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational:

$\paren {AC^2 + CB^2} - \paren {AD^2 + DB^2}$ is rational.

But $\paren {AC^2 + CB^2}$ and $\paren {AD^2 + DB^2}$ are medial.

From Proposition $26$ of Book $\text{X} $: Medial Area not greater than Medial Area by Rational Area this cannot be the case.

So there can be no such $D$.


Historical Note

This proof is Proposition $46$ of Book $\text{X}$ of Euclid's The Elements.