Major Straight Line is Divisible Uniquely

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In the words of Euclid:

A major straight line is divided at one and the same point only.

(The Elements: Book $\text{X}$: Proposition $45$)


Let $AB$ be a major straight line.

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 rational
$AC$ and $CB$ contain a medial 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:

$AB^2 = \left({AC + CB}\right)^2 = AC^2 + CB^2 + 2 \cdot AC \cdot CB$


$AB^2 = \left({AD + DB}\right)^2 = AD^2 + DB^2 + 2 \cdot AD \cdot DB$

and so:

$\left({AC^2 + CB^2}\right) - \left({AD^2 + DB^2}\right) = 2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$

But $AB^2 + CB^2$ and $AD^2 + DB^2$ are rational.

So $\left({AC^2 + CB^2}\right) - \left({AD^2 + DB^2}\right)$ is rational.

Therefore $2 \cdot AD \cdot DB - 2 \cdot AC \cdot CB$ is rational.

But $2 \cdot AD \cdot DB$ and $2 \cdot AC \cdot CB$ 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 $45$ of Book $\text{X}$ of Euclid's The Elements.