Proposition: 2.06: Square of Sum with One Halved Summand

(Proposition 6 from Book 2 of Euclid's “Elements”)

If a straight line is cut in half, and any straight line added to it straight-on, then the rectangle contained by the whole (straight line) with the (straight line) having being added, and the (straight line) having being added, plus the square on half (of the original straight line), is equal to the square on the sum of half (of the original straight line) and the (straight line) having been added.


Modern Formulation

Algebraically, with \(a:=AB\) and \(b:=BD\), the proposition states that \[\left(\frac a2\right)^2+(a+b)b=\left(\frac a2+b\right)^2.\]

Proofs: 1

Proofs: 1 2 3 4

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"