I have a polygon with $n$-vertices and of fixed length, I need to find a condition for which the polygon gives the maximum area without using isoperimetric inequality.
I tried with the simple polygon "Triangle" ($n=3$) and find that it gives maximum area when it's 3 sides are equal. (See here for proof)
Next, I break one side of the triangle to form a Quadrilateral ($n=4$). And find that it gives maximum area when it is a Square. (See proof here)
My intuition tells me that if I increase $n$ then the corresponding polygon will give maximum if their sides are all equal, I assumed this is true for $n=k$ but I am unable to show this for $n=k+1$. Or is there is any other way to prove it? Any help will be appreciated. Thanks in advance.