  The symmedian point of a triangle enjoys several geometric and optimality
properties, which also serve to define it. We develop a new dynamical
coordinatization of the symmedian, which naturally generalizes to other ideal
hyperbolic polygons beyond triangles. We prove that in general this point still
satisfies analogous geometric and optimality properties to those of the
symmedian, making it into a hyperbolic barycenter. We initiate a study of
moduli spaces of ideal polygons with fixed hyperbolic barycenter, and of some
additional optimality properties of this point for harmonic (and sufficiently
regular) ideal polygons.
