# Condition for Commensurability of Roots of Quadratic Equation/Lemma

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## Theorem

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)

## Proof

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 $CB$.

$\blacksquare$

## Historical Note

This proof is Proposition $17$ 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