This is an announcement for the paper "Fixed points of groups of biholomorphic transformations of operator balls using the midpoint property" by M.I. Ostrovskii, V.S. Shulman, and L. Turowska. Abstract: A new techniques for proving the existence of fixed points of groups of isometric transformations is developed. It is used to find simpler proofs and real-case versions of previous results of the authors. In particular, we use the obtained fixed point theorem to show that a bounded representation in a separable, real or complex, Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is orthogonalizable or unitarizable (equivalent to an orthogonal or unitary representation), respectively. Archive classification: math.MG math.OA Mathematics Subject Classification: 47H10; 47B50; 22D10; 54E35 The source file(s), midpoints10.tex: 42873 bytes, is(are) stored in gzipped form as 0902.1784.gz with size 14kb. The corresponding postcript file has gzipped size 109kb. Submitted from: ostrovsm@stjohns.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0902.1784 or http://arXiv.org/abs/0902.1784 or by email in unzipped form by transmitting an empty message with subject line uget 0902.1784 or in gzipped form by using subject line get 0902.1784 to: math@arXiv.org.