◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--2-geometric-algebra / Proposition: 2.04: Square of Sum

Proposition: 2.04: Square of Sum

(Proposition 4 from Book 2 of Euclid's “Elements”)

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

• For let the straight line $AB$ have been cut, at random, at (point) $C$.
• I say that the square on $AB$ is equal to the (sum of the) squares on $AC$ and $CB$, and twice the rectangle contained by $AC$ and $CB$.

Modern Formulation

With \(a=AC\) and \(b=CB\), this proposition is a geometric version of the algebraic identity: \[(a+b)^2 = a^2+2\,a\,b+b^2.\]

See also binomial theorem for $n=2.$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13
Sections: 14

Thank you to the contributors under CC BY-SA 4.0!

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non-Github:

@Calahan

@Casey

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References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
2. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Adapted from (Public Domain)

1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

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