Here we introduce two pencils of cubics, each cubic is determined by a point on the Brocard Axis.
The first pencil is similar to, and indeed related to, the Euler Pencil given in Table 27 of CTC.
Each pencil contains a family of circumcevian invariant cubics, as also does the Euler Pencil.
The pencil of cubics of type pK appear also in the euler-brocard-jerabek net
The Special Centres S(n)
These are triangle centres which arise in this theory, but are not yet in ETC.
|1||a2(2a2-b2-c2)/(b2-c2)||circumcevian invariant, note 1|
|2||a2(3a4+ 6b2c2 - (a2+b2+c2)2)/(b2-c2)||circumcevian invariant, note 2|
S(2) is the inverse of S(1) in the Circumcircle.
S(2) is the intersection other than X3 of the Brocard Axis and the circle with diameter X3X110.
S(2) is circumcevian invariant as the inverse of S(1).
S(3) is the intersection of the Lemoine Axis and the line X2X6.
S(3) is the perspector of the circumconic which passes through X6 and X99.
It is this circumconic which is important in the theory of the nK0(W,X6) above.
Each nK0 contains A,B,C, the Apollonian points, and three fixed points on the Circumconic.