Proposition: 3.27: Angles on Equal Arcs are Equal

(Proposition 27 from Book 3 of Euclid's “Elements”)

In equal circles, angles standing upon equal circumferences are equal to one another, whether they are standing at the center or at the circumference.


Modern Formulation

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

This is the converse of the Prop. 3.26.

Proofs: 1

Proofs: 1 2 3 4 5 6 7
Propositions: 8

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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