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Proposition: 2.02: Square is Sum of Two Rectangles

(Proposition 2 from Book 2 of Euclid's “Elements”)

(Proposition 2 from Book 2 of Euclid's “Elements”)

If a straight line is cut at random then the (sum of the) rectangle(s) contained by the whole (straight line), and each of the pieces (of the straight line), is equal to the square on the whole.

• For let the straight line $AB$ have been cut, at random, at point $C$.
• I say that the rectangle contained by $AB$ and $BC$, plus the rectangle contained by $BA$ and $AC$, is equal to the square on $AB$.

Modern Formulation

With \(a=AB\), \(b=AC\), \(C=CB\), we have $a=b+c$, in which case this proposition is a geometric version of the algebraic identity: \[a=b+c\Rightarrow a\,b+a\,c=a^2.\]

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016