HYACINTHOS 6878

• There are lots of concyclicities related to this. Using the notation A*
  for reflection of vertex A in side BC, and A+ and A- as before, we have
  that AA+A-A* are concyclic (trivial). Such circles can be thought of as
  "lines" in the definition of a configuration, so one can find
  configurations of circles and lines, with 4 points on each circle. One sort
  of circle that might not be known is this: I,Ia,A*,P are concyclic, as are
  I,Ib,B*,P, and I,Ic,C*,P, where P is Parry's reflection point X(399).
  These circles have some analogy with the three Apollonian circles. In each
  case, all 4 points on the circle lie on Neuberg's cubic curve. I was
  surprised to see P show up in this, and expect many other well-known and
  newer points will occur in the concurrences of these circles.
  
  Larry Evans