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.


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