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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 ($BKC=ELF$).

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5
Propositions: 6

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References

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", https://proofwiki.org/wiki/Main_Page, 2016