Are perpendicular bisectors of an arbitrary $n$-gon concurrent ? If so, how to prove it?

Question

Are perpendicular bisectors of an arbitrary $n$-gon concurrent ? If so, how to prove it ?

Answer 1

Hint What does it mean for all perpendicular bisectors of an $n-$gon $A_1A_2...A_n$ to be concurrent at $P$?

Well, it means, that all segments $PA_1, PA_2, ..., PA_n$ are equal in length. Why?

Observe that the triangles $\triangle A_1A_2P, \triangle A_2, A_3P...$ are all isosceles since $P$ lies on their perpendicular bisector. All of them being isosceles means that $$A_1P=A_2P\quad A_2P=A_3P\quad A_3P=A_4P...\implies A_1P=A_2P=A_3P=A_4P=...$$

$P$ is thus the circumcentre of the $n-$ gon.

But this is only possible if you polygon is $\color{blue}{cyclic}$