Triangle centers with Triangle Center Functions of the form are called th Power points. The 0th power point is the Incenter, with Triangle Center Function .
See also Incenter, Triangle Center Function
References
Groenman, J. T. and Eddy, R. H. ``Problem 858 and Solution.'' Crux Math. 10, 306-307, 1984.
Kimberling, C. ``Problem 865.'' Crux Math. 10, 325-327, 1984.
Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.