Condition for Commensurability of Roots of Quadratic Equation/Lemma

Proof

 * Euclid-X-17-Lemma.png

Let $AB$ be a straight line.

Let the parallelogram $AD$ be applied to $AB$ which is deficient by the square $DB$.

As $DB$ is a square:
 * $DC = BC$

Thus $AD$ is the rectangle contained by $AC$ and $CB$.