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 | a^{2}(2a^{2}-b^{2}-c^{2})/(b^{2}-c^{2}) |
circumcevian invariant, note 1 |

2 | a^{2}(3a^{4}+ 6b^{2}c^{2} - (a^{2}+b^{2}+c^{2})^{2})/(b^{2}-c^{2}) |
circumcevian invariant, note 2 |

3 | a^{2}(2b^{2}c^{2}-a^{2}b^{2}-a^{2}c^{2}) |
note 3 |

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.