From Wikipedia's triangle center article: "Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The Encyclopedia of Triangle Centers is an ever-expanding list of interesting ones."
Hyperbolic Barycentric Coordinates are used in: http://ajmaa.org/searchroot/files/pdf/v6n1/v6i1p18.pdf
What makes a triangle center interesting, and why would hyperbolic triangle centers be interesting ? Wouldn't they just be copies of the euclidean triangle centers ?