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.


Modern Formulation

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

Proofs: 1

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!



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"