Proposition: 2.03: Rectangle is Sum of Square and Rectangle

(Proposition 3 from Book 2 of Euclid's “Elements”)

If a straight line is cut at random then the rectangle contained by the whole (straight line), and one of the pieces (of the straight line), is equal to the rectangle contained by (both of) the pieces, and the square on the aforementioned piece. * For let the straight line $AB$ have been cut, at random, at (point) $C$. * I say that the rectangle contained by $AB$ and $BC$ is equal to the rectangle contained by $AC$ and $CB$, plus the square on $BC$.


Modern Formulation

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

Proofs: 1

Proofs: 1

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



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016