Nine-Point Circle of an Orthocentric System
Points A, B, C, and D are an ORTHOCENTRIC SYSTEM - meaning any of the four points is the orthocenter of the remaining three. The Nine-Point Circle of triangle ABC is also the Nine-Point circle of any of the other three triangles formed by any three points in the othocentric system.
Each of the four triangles in the orthocentric system has an incircle and three excircles, making for 16 distinct circles - 4 incircles and 12 excircles - in the orthocentric system. The Nine-Point Circle is tangent to each of these 16 circles.