Proposition: 6.13: Construction of Mean Proportional

(Proposition 13 from Book 6 of Euclid's “Elements”)1

To find the (straight line) in mean proportion3 to two given straight lines. * Let $AB$ and $BC$ be the two given straight lines. * So it is required to find the (straight line) in mean proportion3 to $AB$ and $BC$.


Modern Formulation

It is possible to construct a segment with a length being the geometric mean3 of the lengths of two given other segments.

Proofs: 1

Corollaries: 1
Proofs: 2 3 4 5
Propositions: 6 7
Solutions: 8

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. In other words, to find the geometric mean of two given straight lines (translator's note).