Least Squares Circumcenter of Polygons

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 $({x}_{0},{y}_{0})$ 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 $({x}_{1},{y}_{1})$, Q$({x}_{2},{y}_{2})$, R$({x}_{3},{y}_{3})$, and S $({x}_{4},{y}_{4})$. 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 ${A}^{\mathsf{T}}A\hat{x}={A}^{\mathsf{T}}b$, although any methods are welcome.

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 $({x}_{0},{y}_{0})$ 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 $({x}_{1},{y}_{1})$, Q$({x}_{2},{y}_{2})$, R$({x}_{3},{y}_{3})$, and S $({x}_{4},{y}_{4})$. 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 ${A}^{\mathsf{T}}A\hat{x}={A}^{\mathsf{T}}b$, although any methods are welcome.