Proposition: 6.30: Construction of the Inverse Golden Section

(Proposition 30 from Book 6 of Euclid's “Elements”)

To cut1 a given finite straight line in extreme and mean ratio. * Let $AB$ be the given finite straight line. * So it is required to cut the straight line $AB$ in extreme and mean ratio. fig30e

Modern Formulation

It is possible to cut a given segment at a point being its inverse golden ratio.

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


  1. This method of cutting a straight line is sometimes called the Golden Section" - see " [Prop. 2.11] (translator's note).