A metric invariant of Mobius transformations

The complex unit disk D = {z is an element of C: vertical bar z vertical bar < 1} is endowed with Mobius addition circle plus(M) defined by w circle plus(M) z = w+z/1+<(w)over bar>z. We prove that the metric d(T) defined on D by d(T)(w, z) = tan(-1) vertical bar - w circle plus(M) z vertical bar is an invariant of Mobius transformations carrying D onto itself. We also prove that (D, d(T)) and (D, d(P)), where d(P) denotes the Poincare metric, have the same isometry group and then classify the isometrics of (D, d(T)).