◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--3-circles / Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles

Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles

(Proposition 28 from Book 3 of Euclid's “Elements”)

In equal circles, equal straight lines cut off equal circumferences, the greater (circumference being equal) to the greater, and the lesser to the lesser.

• Let $ABC$ and $DEF$ be equal circles, and let $AB$ and $DE$ be equal straight lines in these circles, cutting off the greater circumferences $ACB$ and $DFE$, and the lesser (circumferences) $AGB$ and $DHE$ (respectively).
• I say that the greater circumference $ACB$ is equal to the greater circumference $DFE$, and the lesser circumference $AGB$ to (the lesser) $DHE$.

Modern Formulation

The arcs are congruent (the longer ones ${ACB}\cong {DFE}$ and the shorter ones ${AGB}\cong {DHE}$) if segments with equal lengths ($|\overline{AB} | = | \overline{DE} |$) connect their endpoints in congruent circles.

Table of Contents

Proofs: 1

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Proofs: 1 2

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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