# 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 $FG$.

### 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

Proofs: 1

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick

### References

1. 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).