Proposition: 3.15: Relative Lengths of Chords of Circles

(Proposition 15 from Book 3 of Euclid's “Elements”)

In a circle, a diameter (is) the greatest (straight line), and for the others, a (straight line) nearer to the center is always greater than one further away. * Let $ABCD$ be a circle, and let $AD$ be its diameter, and $E$ (its) center. * And let $BC$ be nearer to the diameter $AD$,1 and $FG$ further away. * I say that $AD$ is the greatest (straight line), and $BC$ (is) greater than $FG$.


Modern Formulation

The longest chord in a circle is its diameter $\beta$. All chords in a circle have lengths $\gamma$ with $0 < \gamma \le \beta$. The chords are the longer the nearer they are to the center.

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


  1. Euclid should have said "to the center", rather than to the diameter $AD$, since $BC$, $AD$ and $FG$ are not necessarily parallel (translator's note).