Square on Binomial Straight Line applied to Rational Straight Line/Lemma

Lemma to Square on Binomial Straight Line applied to Rational Straight Line

$\forall x, y \in \R: x \ne y \implies x^2 + y^2 > 2 x y$

In the words of Euclid:

If a straight line be cut into unequal parts, the squares on the unequal parts are greater than twice the rectangle contained by the unequal parts.

(The Elements: Book $\text{X}$: Proposition $60$ : Lemma)

Proof

Let $AB$ be a straight line.

Let $AB$ be cut into unequal parts at $C$ such that $AC > CB$.

Let $AB$ be bisected at $D$.

From Proposition $5$ of Book $\text{II} $: Difference of Two Squares:

$AC \cdot CB + CD^2 = AD^2$

So:

$AC \cdot CB < AD^2$

and so:

$2 \cdot AC \cdot CB < 2 \cdot AD^2$

But from Proposition $9$ of Book $\text{II} $: Sum of Squares of Sum and Difference:

$AC^2 + CB^2 = 2 \cdot AD^2 + DC^2$

Therefore:

$AC^2 + CB^2 > 2 \cdot AC \cdot CB$

$\blacksquare$

Historical Note

This proof is Proposition $60$ of Book $\text{X}$ of Euclid's The Elements.
It is practically certain that this lemma was not original to Euclid, as the result itself is used in Proposition $44$ of Book $\text{X} $: Second Bimedial Straight Line is Divisible Uniquely.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions