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