◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--2-geometric-algebra / Proposition: 2.07: Sum of Squares

Proposition: 2.07: Sum of Squares

(Proposition 7 from Book 2 of Euclid's “Elements”)

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

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

Modern Formulation

With \(a:=AB\) and \(b:=CB\), this proposition is a geometric version of the algebraic binomial formula: \[(a-b)^2=a^2-2ab+b^2.\]

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Sections: 19

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

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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"