Here is one approach.
First find a line $L$ that passes through two points of your set and has all points on $L$ or to one side.
Then reflect your point set over $L$. Finally, fit a full circle to the doubled set of points.

You might need a more careful approach, depending on how noisy is your data.
You could check the result against the center of gravity of your points.
I assume your points are from the semicircle boundary (as opposed to points from a half-disk).
Then I calculate that the center of gravity is $2 r/\pi$ to one side of the diameter.
If your fit circle's radius $r$ does not closely match this distance to the c.g.,
then adjustment of $L$ is indicated.