The Perpendicular Bisector Construction in $n$-dimensional Euclidean and Non-euclidean Geometries

The "Perpendicular Bisectors Construction" is a natural way to seek a replacement for the circumcenter of a noncyclic quadrilateral in the plane. In this paper, we generalize this iterative construction to a construction on polytopes with $n$ vertices in $(n-2)$-dimensional Euclidean, Hyperbolic and Elliptic geometries. We then show that a number of nice properties concerning this iterative construction continue to hold in these geometries. We also introduce an analogue of the isoptic point of a quadrilateral, which is the limit point of the Perpendicular Bisectors Construction, in $\mathbb{R}^{n}$ and prove some of its properties.