Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles

(Proposition 28 from Book 3 of Euclid's “Elements”)

In equal circles, equal straight lines cut off equal circumferences, the greater (circumference being equal) to the greater, and the lesser to the lesser. * Let $ABC$ and $DEF$ be equal circles, and let $AB$ and $DE$ be equal straight lines in these circles, cutting off the greater circumferences $ACB$ and $DFE$, and the lesser (circumferences) $AGB$ and $DHE$ (respectively). * I say that the greater circumference $ACB$ is equal to the greater circumference $DFE$, and the lesser circumference $AGB$ to (the lesser) $DHE$.


Modern Formulation

The arcs are congruent (the longer ones ${ACB}\cong {DFE}$ and the shorter ones ${AGB}\cong {DHE}$) if segments with equal lengths ($|\overline{AB} | = | \overline{DE} |$) connect their endpoints in congruent circles.

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