Proposition: 3.15: Relative Lengths of Chords of Circles

(Proposition 15 from Book 3 of Euclid's “Elements”)

In a circle, a diameter (is) the greatest (straight line), and for the others, a (straight line) nearer to the center is always greater than one further away.

Let $ABCD$ be a circle, and let $AD$ be its diameter, and $E$ (its) center.
And let $BC$ be nearer to the diameter $AD$,¹ and $FG$ further away.
I say that $AD$ is the greatest (straight line), and $BC$ (is) greater than $FG$.

fig15e

Modern Formulation

The longest chord in a circle is its diameter $\beta$. All chords in a circle have lengths $\gamma$ with $0 < \gamma \le \beta$. The chords are the longer the nearer they are to the center.

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

Euclid should have said "to the center", rather than to the diameter $AD$, since $BC$, $AD$ and $FG$ are not necessarily parallel (translator's note). ↩

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