When is the area of a pentagon inscribed inside a fixed circle maximum?

Question

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.

Answer 1

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.