Two Brocard Pencils

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

 k P(k) W(k) φ Q(φ),Q(-φ) pK(X6,P(k)) pK(W(k),X3) nK0(W(k),X6) E(k) P'(k) ∞ X30 X50 π/3 X15,X16 K001 K073 K148 X186 X74 1/3 X2 X32 0 X6,X6 K002 K172 nK0(X32,X6) X25 X6 -1 X20 X577 π/2 X3,X3 K004 pK(X577,X3) nK0(X577,X6) X3 X3 1/2 X5 X2965 π/6 X61,X62 K005 K349 nK0(X2965,X6) ? X1173 1 X4 X571 π/4 X371,X372 K006 pK(X571,X3) nK0(X571,X6) X24 X4 0 X3 X6 complex K003 K003 K024 X4 X54 -5 ? ? X1151,X1152 K156 ?

The Special Centres S(n)
These are triangle centres which arise in this theory, but are not yet in ETC.

 n first barycentric notes 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 3 a2(2b2c2-a2b2-a2c2) note 3

Note 1
S(1) is the intersection of the Brocard Axis and the line X110X351.
The latter line is the tangent at X110 to the Circumcincle.
S(1) also lies on the circumconic with perspector X187, which passes through X6 and X110.
S(1) is circumcevian invariant, and lies on K148 = nK0(X50,X6). It is the third intersection
along with X15 and X16. Since K148 is circumcevian invariant, S(1) must be.

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).

Note 3
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.