It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic.
I would like to extend the definition of an circumcenter for noncyclic polygons. Namely, let us define the least squares circumcenter as the point A such that the point A minimizes the sum of the squares of the residuals.
Let us consider the case for a noncyclic quadrilateral with vertices P , Q, R, and S . Let us also define the origin O(0,0).
How would we solve for the point A in this case? I was thinking of using matrices and solving , although any methods are welcome.