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Of all the curves that divide the circle into two regions with the same area, is the diameter the shortest one?

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    An elementary proof is prefered!2012-12-20

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It can be shown, using a variational argument, that the curve should intersect the circle perpendicularly and has constant curvature. It's not hard to show that only the diameter is the only curve of this type dividing the disk in two equal halfs.

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    @Thomas. It remains the case where the curve has its ends in the circle line.As it's written below, this is clear if the the endpoints are opposite respect the center of the circle.2012-12-20