Determining the position of a polygon inside a circle from only the angle of opposing sides/edges.

I have a simple convex irregular polygon (octagon in example image) inside a circle (circle and polygon are not always concentric and never touching or intersecting) and I need to determine the polygon's position relative to the circle.

Unfortunately I only know three basic things...

1. I know the exact angle between each of the polygon's opposing sides. No sides are parallel.
2. When extended to the circle, the polygon's edges/sides form inscribed angles. Since one angle accounts for two polygon sides, there are always half as many inscribed angles as polygon sides. In the example there are four inscribed angles because it's an octagon.
3. The vertices of each inscribed angle are evenly spaced around the circle.

My worry is that this kind of calculation might involve an infinite number of iterations and/or multiple solutions.

If I knew the relative angles (labeled 'Unknown' in drawing) of the inscribed angles, that should be enough to determine everything else, right? Even if I could approximate the 'centre' of the irregular polygon somehow...

*The bottom right edges in the example meet with such an obtuse angle that they appear to be one line, sorry. It is an octagon with each inscribed angle creating two sides of the polygon.

Here are two diagrams, both satisfying your conditions and similar to your octagon in your diagram. The only difference between the two is the orientation of the $9.04°$ angle. I hope you can see the differences.  