Proposition: 3.14: Equal Chords in Circle

(Proposition 14 from Book 3 of Euclid's “Elements”)

In a circle, equal straight lines are equally far from the center, and (straight lines) which are equally far from the center are equal to one another.

Let $ABDC$¹ be a circle, and let $AB$ and $CD$ be equal straight lines within it.
I say that $AB$ and $CD$ are equally far from the center.

fig14e

Modern Formulation

Two chords in a circle are equal in length if and only if they are equally far from the center of the circle.

Proofs: 1

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References

Adapted from (subject to copyright, with kind permission)

Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Footnotes

The Greek text has "$ABCD$", which is obviously a mistake (translator's note). ↩

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Proposition: 3.14: Equal Chords in Circle

(Proposition 14 from Book 3 of Euclid's “Elements”)

Modern Formulation

Table of Contents

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References

Adapted from (subject to copyright, with kind permission)

Adapted from CC BY-SA 3.0 Sources:

Footnotes