A pentagon is inscribed inside a circle of fixed radius. Show that for the area of the pentagon to be maximum, all sides of it must be equal.
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Have you put any thought into this? Ideas, strategies, etc.? If so, what has come to mind? – Clayton Jan 05 '16 at 18:42
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1Possible duplicate of Given a polygon of n-sides, why does the regular one (i.e. all sides equal) enclose the greatest area given a constant perimeter? – Dietrich Burde Jan 05 '16 at 18:45
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See also this MSE qiestion. – Dietrich Burde Jan 05 '16 at 18:51
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OP replaced the text of the question with a completely unrelated question, and I've reverted it (not least because it orphans the answers already given). To ask another question, post it separately. – Travis Willse Jan 05 '16 at 18:53
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Here's a way to start. Imagine you draw any cyclic pentagon, and fix all but one vertex. Now slide the remaining vertex along the circle. Can you argue why the area of the pentagon is maximized when the vertex is equidistant (both in ambient space and along the circle) from its two neighbors?
Now consider what it means for this condition to hold at every vertex.
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