  We define and study a variant of the center of mass of a polygon and, more
generally, of a simplicial polytope which we call the Circumcenter of Mass
(CCM). The Circumcenter of Mass is an affine combination of the circumcenters
of the simplices in a triangulation of a polytope, weighted by their volumes.
For an inscribed polytope, CCM coincides with the circumcenter.
  Our motivation comes from the study of completely integrable discrete
dynamical systems, where the CCM is an invariant of the discrete bicycle
(Darboux) transformation and of recuttings of polygons.
  We show that the CCM satisfies an analog of Archimedes' Lemma, a familiar
property of the center of mass. We define and study a generalized Euler line
associated to any simplicial polytope, extending the previously studied Euler
line associated to the quadrilateral. We show that the generalized Euler line
for polygons consists of all centers satisfying natural continuity and
homogeneity assumptions and Archimedes' Lemma.
  Finally, we show that CCM can also be defined in the spherical and hyperbolic
settings.

Metric Geometry