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Proposition: 2.08: Square of Sum with One Doubled Summand

(Proposition 8 from Book 2 of Euclid's “Elements”)

If a straight line is cut at random then four times the rectangle contained by the whole (straight line), and one of the pieces (of the straight line), plus the square on the remaining piece, is equal to the square described on the whole and the former piece, as on one (complete straight line).

• For let any straight line $AB$ have been cut, at random, at point $C$.
• I say that four times the rectangle contained by $AB$ and $BC$, plus the square on $AC$, is equal to the square described on $AB$ and $BC$, as on one (complete straight line).

Modern Formulation

With \(a:=AC\) and \(b:=CB\), this proposition is a geometric version of the algebraic identity: \[4\,(a+b)\,b+a^2 = (a+2b)^2.\]

Table of Contents

Proofs: 1

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References

Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

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"