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

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