Use the entire curve if possible. Well, all of it that you have. Not just a couple of isolated points.
Extract points from the entire accessible arc, as (x,y) pairs. At this point, it becomes a pure circle fitting problem.
Some years ago, I wrote a circle fitting tool. It tries to produce a fit that does as well as possible, even allowing the use of various solvers like backslash, pinv, or robustfit as alternatives. However, there are many algorithms one might postulate. I'll admit, I'm not in love with the one listed in the FAQ, but it is not that bad either. In some quick tests, it was comparable to the one I wrote. So if your data truly comes from a circle, you probably won't do too much better. And your data should not be too noisy since that arc is quite clear.
Can I do better than that though? Were I to try to do a bit better, I might look to using a maximum likelihood estimation, assuming noise in both x and y on each point. But your data is not that noisy, not that crappy that it is worth the effort.
So where will any "problems" arise? They will come from deviations from pure circularity. Thus, is that arc a perfectly circular arc, with a perfectly fixed radius? I doubt it. In fact, I'll bet that the largest amount of error in your circle parameter estimates come from those deviations from circularity, thus, lack of fit to a true circle, rather than from noise in the fit.
If that is true, then what does it mean to try to do better, fitting a circle perfectly to data that is not perfectly circular? There is a limit beyond which the fit becomes an exercise in absurdity. How can you improve things to best effect? Do a better job of extracting those points on the perimeter of the circle. Improving your data is always a good idea, because that means the algorithms meant to post-process the data will have a far easier time of it.
I would look carefully at the data you have. Compare it to the fit you estimate. Does it have systematic patterns in the residuals between your data and the circle fit. If you see patterns, that means your largest problem lies in lack of fit. And if my conjecture is true, then you won't improve things by making a better circle fit.