Plane Figure with Maximum Area for given Perimeter is Circle
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Theorem
Let $F$ be a plane figure.
Let $P$ be the length of the perimeter of $F$.
Let the area of $F$ be the largest of all the plane figures whose perimeters are of length $P$.
Then $F$ is a circle.
Proof
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Historical Note
The problem of determining the plane figure which has the maximum area for a given perimeter was first solved by Jacob Bernoulli.
He also went ahead with a generalisation of the problem.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VIII}$: Nature or Nurture?