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.


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

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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