Let $B_1B_2\ldots B_n$ be non-cyclic polygon and $B_kB_{k+1}B_{k+2}B_{k+3}$ are four consecutive verteces which not lie on a circle. Making $B_kB_{k+1}B_{k+2}B_{k+3}$ a cyclic polygon and keeping all points excepting $B_{k+1}$ and $B_{k+2}$ at the same places you will get bigger area.

Let $B_1B_2\ldots B_n$ be non-cyclic polygon and $B_kB_{k+1}B_{k+2}B_{k+3}$ are four consecutive verteces which not lie on a circle. Making $B_kB_{k+1}B_{k+2}B_{k+3}$ a cyclic polygon and keeping all points excepting $B_{k+1}$ and $B_{k+2}$ at the same places you will get bigger area.

See also Maximum area of a flexible polygon.