Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio

(Proposition 9 from Book 13 of Euclid's “Elements”)

If the side of a hexagon and of a decagon inscribed in the same circle are added together then the whole straight line has been cut in extreme and mean ratio (at the junction point), and its greater piece is the side of the hexagon.


Modern Formulation

If the circle is of unit radius then the side of the hexagon is 1, whereas the side of the decagon is \[\frac {\sqrt{5}-1}2.\]

Proofs: 1

Proofs: 1 2

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