Proposition: 2.09: Sum of Squares of Sum and Difference

(Proposition 9 from Book 2 of Euclid's “Elements”)

If a straight line is cut into equal and unequal (pieces) then the (sum of the) squares on the unequal pieces of the whole (straight line) is double the (sum of the) square on half (the straight line) and (the square) on the (difference) between the (equal and unequal) pieces.


Modern Formulation

Algebraically, with \(a=AC\) and \(b=CD\), this proposition states that


Proofs: 1

Proofs: 1

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"