Shortest curve that divides circle into two regions of equal area

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! – Sgernesto Dec 20 '12 at 9:44

One has to add that all closed curves in the ball of radius $r$ not intersecting the circle have greater length than any diameter, since by the isoperimetric inequality they must have length at least $\sqrt{2} \pi r > 2 r$. – Thomas Dec 20 '12 at 13:42