  Hermann Schwarz, while studying complex analysis, introduced the geometric
interpretation for the Poisson kernel in 1890. We shall see here that the
geometric interpretation can be useful to develop a new approach to some old
classical problems as well as to obtain several new results, mostly related to
hyperbolic geometry. For example, we obtain One Radius Theorem saying that any
two radial eigenfunctions of a Hyperbolic Laplacian assuming the value 1 at the
origin can not assume any other common value within some interval [0, p], where
the length of this interval depends only on the location of the eigenvalues on
the complex plane and does not depend on the distance between them.
