Some of them (e.g. Apollonius point, Ajima-Malfatti points, Morley centers) arise from classical configurations. Others were found with the help of visualization software, such as Geometer's Sketchpad. Still others were found using number-crunching computer programs that looped through thousands of possibilities and outprinted exceptional cases. Finally, some of them were found by sheer geometric (and algebraic!) mental power, curiosity, persistence, and luck.
Here is a clickable list. You will find each center is described, sketched (by Geometer's Sketchpad), and "pedigreed" by trilinear coordinates.
