  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.
