Condition for Commensurability of Roots of Quadratic Equation/Lemma

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In the words of Euclid:

If to any straight line there be applied a parallelogram deficient by a square figure, the applied parallelogram is equal to the rectangle contained by the segments of the straight line resulting from the application.

(The Elements: Book $\text{X}$: Proposition $17$ : Lemma)



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 $CB$.


Historical Note

This proof is Proposition $17$ of Book $\text{X}$ of Euclid's The Elements.