Side of Rational Plus Medial Area is Divisible Uniquely
Theorem
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 $46$)
Proof
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:
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:
- $AB^2 = \paren {AC + CB}^2 = AC^2 + CB^2 + 2 \cdot AC \cdot CB$
and:
- $AB^2 = \paren {AD + DB}^2 = AD^2 + DB^2 + 2 \cdot AD \cdot DB$
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 $D$.
$\blacksquare$
Historical Note
This proof is Proposition $46$ of Book $\text{X}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions