Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line

(Proposition 61 from Book 10 of Euclid's “Elements”)

The square on a first bimedial (straight line) applied to a rational (straight line) produces as breadth a second binomial (straight line).1 * Let $AB$ be a first bimedial (straight line) having been divided into its (component) medial (straight lines) at $C$, of which $AC$ (is) the greater. * And let the rational (straight line) $DE$ be laid down. * And let the parallelogram $DF$, equal to the (square) on $AB$, have been applied to $DE$, producing $DG$ as breadth. * I say that $DG$ is a second binomial (straight line).


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1

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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, the square of a first bimedial is a second binomial. See [Prop. 10.55] (translator's note)