Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines

(Proposition 29 from Book 3 of Euclid's “Elements”)

In equal circles, equal straight lines subtend equal circumferences. * Let $ABC$ and $DEF$ be equal circles, and within them let the equal circumferences $BGC$ and $EHF$ have been cut off. * And let the straight lines $BC$ and $EF$ have been joined. * I say that $BC$ is equal to $EF$.


Modern Formulation

The segments ($\overline{BC}$, $\overline{EF}$) have equal lengths, if they connect the endpoints of two arcs being congruent in congruent circles.

Proofs: 1

Proofs: 1 2 3

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