Proposition: 3.14: Equal Chords in Circle

(Proposition 14 from Book 3 of Euclid's “Elements”)

In a circle, equal straight lines are equally far from the center, and (straight lines) which are equally far from the center are equal to one another. * Let $ABDC$1 be a circle, and let $AB$ and $CD$ be equal straight lines within it. * I say that $AB$ and $CD$ are equally far from the center.


Modern Formulation

Two chords in a circle are equal in length if and only if they are equally far from the center of the circle.

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. The Greek text has "$ABCD$", which is obviously a mistake (translator's note).