Here is a solution that seems to match what you need (and @Okidoki figured out?). I never looked at your original blend file so you might need to tweak the interface. It was an interesting math problem.
My solution was to use the normal of each edge and scale it for distance of the circle from center, then use one of the edge vertices to find the required radius.
Here is a 3.6 blender file for you.
[circulate.blend|attachment]
(upload://yrq4QKShKsbnVqgwizTL8YZKNgn.blend) (960.9 KB)
@Okidoki : Compare and contrast! how did you work yours out?