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Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What shape will the string form when this closed figure has the maximum possible area?

If A and B are the same point, then it is well-known that the shape is a circle. A and B are moveable, then it has been shown elsewhere that the maximal-area shape is a semicircle: find maximum area .

My instinct says that when A and B are fixed (as in this question) the shape will be an arc of some circle with AB as a chord, but I would like there to be some sort of proof.

So far, I can argue that the maximal-area shape must be convex, since if it wasn't you could turn the inwards-facing bit outwards to make a bigger area without changing the length L.

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    $\begingroup$ See Isoperimetric problem for some useful references in this vein $\endgroup$
    – Semiclassical
    Commented Jul 22, 2014 at 22:29
  • $\begingroup$ Also, I'm not sure I see a question in here? Though I do believe you're correct about the line segment will be serve as a chord of some circle. $\endgroup$
    – Semiclassical
    Commented Jul 22, 2014 at 22:37
  • $\begingroup$ The question is "what shape will the string form when the closed figure has the maximum possible area?" $\endgroup$
    – DavidButlerUofA
    Commented Jul 22, 2014 at 22:38
  • $\begingroup$ Ok. Then I believe your solution is correct, though I don't recall the manner of proof. $\endgroup$
    – Semiclassical
    Commented Jul 22, 2014 at 22:40
  • $\begingroup$ This older question looks applicable $\endgroup$
    – Semiclassical
    Commented Jul 23, 2014 at 1:33

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