Consider a closed curve of finite length.
There is at least one straight line that can bisect both the perimeter and the area of the curve.
Why is this statment true?
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It isn't true. Here's one example of a closed curve that can't be bisected in this way:
r3mainer
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I guess I missed it. Wouldn't a vertical line through the center of the top circle successfully bisect it both? --granted, the perimeter (and the area too) will be broken into several pieces, but the part on the left and the part on the right will both have the same total length. – Dan Uznanski Aug 13 '14 at 13:00
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i wont consider this as a counter example ( even though it's clear that the closed curve you posted could be divided easily ) , this is strong statement that i post in the question , i read its proof once but i was too young to understand why its correct , however now a day it sounds like i dont find it anywhere , I will appreciate it if any one have information or proof about it . – Bswan Aug 14 '14 at 09:31
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@IkramIbrahim The statement in your question is false because this property is only true for convex closed curves. You can find the proof in the linked question above. – r3mainer Aug 14 '14 at 13:44
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thank you all ,i will see it . – Bswan Aug 14 '14 at 16:57