Let
K1 be the McCay Cubic : F1(x,y,z) = Σa^{2}(b^{2}+c^{2}-a^{2})x(c^{2}y^{2}-b^{2}z^{2}) = 0.

Define two degenerate cubics

K2, the union of the X3-Cevians, F2(x,y,z) = Π(b^{2}(c^{2}+a^{2}-b^{2})z-c^{2}(a^{2}+b^{2}-c^{2})y) = 0, and

K3, the union of the Circumcircle and the Brocard Line : F3(x,y,z) = (Σa^{2}yz)(Σb^{2}c^{2}(b^{2}-c^{2})x) = 0.

These generate a *net* of cubics, which we will call the Euler-Brocard-Jerabek Net.

The equation of each member has the form

ka^{2}b^{2}c^{2}F1(x,y,z)+lF2(x,y,z)+m(Σa^{4}-2Σb^{2}c^{2})F3(x,y,z) = 0,

where k,l,m are symmetric in a,b,c, and of the same degree. The case where they are real numbers

is particularly important. We have the

**Theorem**

If k,l,m are real, the cubic is circumcevian invariant.

See circumcevian points and circumcevian invariant cubics

The net contains three pencils, each consisting of cubics of type pK.

**The Euler Pencil**

This is discussed in Table 27. The cubics are of the form pK(X6,P), with P on the Euler Line.

This arises when m = l. The pivot is P, where X3P:PX4 = l:k-2l.

The common points are A,B,C,X3,X4 and the members of the extraversion class {X1}.

Each member meets the Euler Line in X3,X4 and P.

Each member meets the Brocard Line in X3, and at points harmonic with X3,X6 (see below).

Each member meets the Jerabek Hyperbola in A,B,C,X3,X4 and the isoconjugate of P.

Obviously, K1 belongs to this pencil.

**Note.**

The Euler Pencil is quite remarkable. Each member **K** with real values of k,l has the property

if **K** contains a point X, then it also contains

the circumcevian points of X, and

the pedal and antipedal points of X.

**The Brocard Pencil**

The cubics have the form pK(W,X3), with W on the Brocard Line.

This arises when m = 0.

Each contains A,B,C,X3, and the feet of the X3-Cevians; the tangent at X3 is the Euler Line.

Observe that the seven fixed points on the cubic mean that it is of the form pK(W,X3) for

*some* point W. This is CL021.The condition on the tangent means W is on the Brocard Line.

Each member meets the Euler Line in X3 (twice) and at E', the isoconjugate of X3.

Each member meets the Brocard Line in X3, and at points harmonic with X3,X6 (see below).

Each member meets the Jerabek Hyperbola in A,B,C,X3 and two further points.

The isogonal conjugates of these cut X3X4 in the ratio 1:±t, where t^{2} = k/l.

Note that K2 belongs to this pencil, with W = X577, the barycentric square of X3.

**The Jerabek Pencil**

The cubics have the form pK(P'<->X3,P'), with P' on the Jerabek Hyperbola.

This arises when m=½k. The isogonal conjugate of the pole is P", with X3P":P"X4 = k-2l:k.

Each contains A,B,C and X3; the tangents at A,B,C are the X3-Cevians, and the tangent at X3

is the tangent to the Jerabek Hyperbola at that point. The last is the line X3X184.

The four fixed points, and the tangents at A,B,C mean that the cubic is of type pK(P'<->X3,P')

for some point P'. The condition on the tangent at X3 means that P' is on the Jerabek Hyperbola.

All members of the pencil are invariant under the Hirst Inversion with pole X3 and conic the

Jerabek hyperbola. Thus, every line through X3 meets the cubic in points M,N, and the hyperbola

in the M,N- harmonic of X3. Of course, as X3 = P'*, M,N are P'-Ceva conjugate.

For example, as X4, X6 are on the hyperbola,

Each member meets the Euler Line at X3, and at two points harmonic with X3,X4.

These may be complex - this depends on the shape of the triangle as well as k,l.

Each member meets the Brocard Line at X3, and at two points harmonic with X3,X6 (see below).

Each member mets the Jerabek Hyperbola at A,B,C,X3(twice) and P'.

**The Brocard Line**

Each member of the net cuts the Brocard Line at X3, and at two points Q,R harmonic with X3,X6.

These are real provided 4 ≥k/l ≥ 0.

The points Q,R have first barycentrics sin(A)sin(A±φ), with cos^{2}(φ) = k/4l.

If we know Q, R, then the cubics of type pK have one of the forms

pK(X6,P), with P the anticomplement of the crosssum of Q and R,

pK(W,X3), with W the barycentric product of Q and R,

pK(P'<->X3,P'), with P' the cevapoint of Q and R.

The three cubics appear in a row of the table below. They have the common points

A, B, C,

X3, Q, R on the Brocard Axis,

three points on the Circumcircle.

This follows since they belong to a set with the same value of k/l, and differ by multiples of **K3**.

row | k | l | P | W | P' | E' | Q,R | pK(X6,P) | pK(W,X3) | pK(P'<->X3,P') | notes |

1 | 1 | 1 | X30 | X50 | X74 | X186 | X15,X16 | K001 | K073 | K114 | |

2 | 4 | 1 | X2 | X32 | X6 | X25 | X6,X6 | K002 | K172 | K167 | |

3 | 1 | 0 | X3 | X6 | X54 | X4 | complex | K003 | K003 | pK(X54<->X3,X54) | |

4 | 0 | 1 | X20 | X577 | X3 | X3 | X3,X3 | K004 | X3-cevians | X3-cevians | |

5 | 3 | 1 | X5 | X2965 | X1173 | ? | X61,X62 | K005 | K349 | pK(X1173<->X3,X1173) | |

6 | 2 | 1 | X4 | X571 | X4 | X24 | X371,X372 | K006 | pK(X571,X3) | K006 | |

7 | 4 | 5 | X1151,X1152 | K156 |