Cut out a heptagonal ‘bite’ from a circle. The bite is four sides of a regular heptagon. The remaining arc length should be constant, and you wish to maximise the area. Below are two figures towards the extremes: a small circle with a small heptagon bite, or a far larger circle, but where the heptagon takes most of that area. In between these two figures there will be a point where the area is at a maximum, but how to construct it?

I had previously solved a far simpler version of this, where the bite was formed by two edges of a square. The optimum in that situation was for the inner vertex of the square to lie on the centre of the circle. In other words the resulting figure would be a three-quarter-circle. It turned out on further examination that the fact that the two lines of the square met at right angles was not important, but the fact that the lines were equal WAS. So if the bite was formed of two identical lines meeting at ANY angle, the maximum area for that angle is achieved when the vertex lies on the centre of the circle:

How can you use this information to construct the maximum area in the case of the heptagon?