The Euler-Brocard-Jerabek Net

Let K1 be the McCay Cubic : F1(x,y,z) = Σa2(b2+c2-a2)x(c2y2-b2z2) = 0.
Define two degenerate cubics
K2, the union of the X3-Cevians, F2(x,y,z) = Π(b2(c2+a2-b2)z-c2(a2+b2-c2)y) = 0, and
K3, the union of the Circumcircle and the Brocard Line : F3(x,y,z) = (Σa2yz)(Σb2c2(b2-c2)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
ka2b2c2F1(x,y,z)+lF2(x,y,z)+m(Σa4-2Σb2c2)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.

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 t2 = 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 cos2(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