Barbier's Theorem

Theorem

Let $K$ be a closed curve of constant diameter.


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Let the circumference of $K$ be $c$.

Let the diameter of $K$ be $d$.

Then:

$\dfrac c d = \pi$


Proof


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Source of Name

This entry was named for Joseph-Émile Barbier.


Sources

  • 1860: Joseph-Émile Barbier: Note sur le problème de l'aiguille et le jeu du joint couvert (J. Math. Pures Appl. Ser. 2 Vol. 5: pp. 273 – 286)
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
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