Proposition: 3.26: Equal Angles and Arcs in Equal Circles

(Proposition 26 from Book 3 of Euclid's “Elements”)

In equal circles, equal angles stand upon equal circumferences whether they are standing at the center or at the circumference. * Let $ABC$ and $DEF$ be equal circles, and within them let $BGC$ and $EHF$ be equal angles at the center, and $BAC$ and $EDF$ (equal angles) at the circumference. * I say that circumference $BKC$ is equal to circumference $ELF$.


Modern Formulation

Let two given circles be congruent and let some of its inscribed angles be congruent ($\angle{BAC}=\angle{EDF}$) or some of its central angles be congruent ($\angle{BGC}=\angle{EHF}$) be congruent. Then the corresponding arcs are also congruent ($BKC=ELF$).

Proofs: 1

Proofs: 1 2 3 4 5
Propositions: 6

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