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

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

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

Proof

 * Euclid-X-60-Lemma.png

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

So:
 * $AC \cdot CB < AD^2$

and so:
 * $2 \cdot AC \cdot CB < 2 \cdot AD^2$

But from :
 * $AC^2 + CB^2 = 2 \cdot AD^2 + DC^2$

Therefore:
 * $AC^2 + CB^2 > 2 \cdot AC \cdot CB$