The Lucas cubic and a generalisation

First a basic result which is useful.

Theorem 1
Suppose that p = p:q:r, Q = u:v:w. Then C(P,Q) is degenerate
if and only if pqruvw(pv+qu)(pw+ru)(rv+qw) = 0.
This is equivalent to P or Q on a sideline of ΔABC, or
Q on the anticevian triangle of P (or vice versa).

A conic degenerates if and only if its matrix is singular.
The result is now a simple Maple computation.

Notes on the Lucas Cubic : K007 = pK(X2,X69).
We know that, if P has a cevian triangle with a non-degenerate circumcircle C having
centre X, then PX contains the cyclocevian conjugate of P when P is on the Lucas Cubic.
Further, we have

Suppose that P is on K007.
(1) Q, the cyclocevian conjugate of P is on K007.
(2) C(P,Q) is a circle, with centre X on PQ.
(3) PQ meets the cubic again in L = X20.

These can be rephrased in a variety of ways, for example
(4) ct(P),ct(Q) are isogonal conjugates (on K002) - this is (1).
(5) P,Q,X,L are collinear,
(6) PL meets the cubic again in Q.
(7) X1032, the cyclocevian conjugate of L is the tangential of L (see (3)).

The real foci F1, F2 of the Steiner Ellipse are on K007. Their isotomic conjugates t(F1), t(F2)
are related to f1, f2, the real foci of the Steiner Inellipse, as ct(t(Fi)) = c(Fi) = fi.

t(F1), t(F2) are cyclocevian conjugates,
L is on t(F1)-t(F2),
the centre of C(t(F1),t(F2)) is on this line.
the cyclocevian conjugate of Fi is the third meet of the cubic with LFi.
The circumcircle of the cevian triangle of Fi is on LFi.

Now K007 has three conjugations :
(a) isoconjugation - in this case isotomic conjugation
the X2-isoconjugate of P is the third meet with PX69 (X69 the pivot).
It follows that the tangential of X69 is its isotomic conjugate H = X4.
(b) ceva conjugation - here X69-Ceva conjugation
the X69-Ceva conjugate of P is the third meet with PH (H the secondary pivot).
It follows that the tangential of H is its X69-Ceva conjugate (= tX1032).
(c) cyclocevian conjugation
the cyclocevian conjugate of P is the third meet with PL (see above).
It follows that the tangential of L is its cyclocevian conjugate X1032.

Note that the locus { P : P, tagct(P) passes through X20 } consists of :

the sidelines of ΔABC,
the sidelines of the antimedial triangle,
pK(X2,X69) = K007,
the isotomic conjugate of the anticomplement of an nK with pole K = X6, root H = X4.

The third conjugation comes from isogonal conjugation on the complement K002.
It maps P to tagct(P), where g(X) denotes the isogonal conjugate of X.

A simpler conjugation maps P to agc(P) - the anticomplementary conjugate of ETC.
Now the conjugate of P is the third meet with PG.
The tangential of G is therefore its image under this map X69.

The cubic K007 is the locus { P : P-agc(P) passes through X2 }.

Notes on CTC-K007
(1) tF1, tF2 are cyclocevian conjugates,
(2) the points tY1 and tY2 are the wrong way round. In other words, tY2 is on L-F1,
     and tY1 on L-F2. Indeed {F1,tY2} and {F2,tY2} are cyclocevian conjugate pairs.
(3) Y1 = X69-Ceva conjugate of F1 is also agctF2, Y2 = X69-Ceva conjugate of F2
      is also agctF1 (from the collinearities with G,H and tF2, tF1).


Above, we looked at the circumconics of a cevian triangle homothetic with the Circumcircle.
We can replace the Circumcircle by the circumconic with perspector S. Then the cyclocevian
conjugate is replaced by the map taking M to tasct(M), where s denotes S-isoconjugation.

Note. When a(S) is on the Steiner Ellipse, S is on the Steiner Inellipse. Then a calculation
shows that the maps in (b) and (c) have the same effect on the cubic pK(G,a(S)). This is
reflected in the table below since we then have ta(S) = ata(S) (K242).

The Lucas Cubic is replaced by pK(G,a(S)), with complement pK(S,G).

Again we have four conjugations on pK(G,a(S)).
(a) isotomic conjugation.
This takes M to the third meet of the cubic with M-a(S).
Then the tangential of a(S) is its isotomic conjugate ta(S).
(b) a(S)-Ceva conjugation.
This takes M to the third meet of the cubic with M-ta(S).
Then the tangential of ta(S) is its a(S)-Ceva conjugate asctata(S).
(c) the analogue of cyclocevian conjugation, mapping M to tasct(M).
This takes M to the third meet of the cubic with M-ata(S).
Then the tangential of ata(S) is its image tasctata(S).
(d) the map taking M to asc(M).
This takes M to the third meet of the cubic with M-G.
Then the tangential of G is its image a(S).

(a), (b) and (d) can be used to define the cubic pK(G,a(S)) as a locus.
The map in (d) is a version of X-anticomplementary conjugation in ETC.

Here is a table showing the various points associated with cubics pK(X2,a(S))
listed in CTC. The final column shows the related pK(S,X2). An asterisk shows
that CTC does not state that the cubics are related by complementation. If an
isotomic cubic in CTC does not appear, it means that S is not listed in ETC.

CTC     S   a(S) ta(S) ata(S) complement
K007 X6 X69 X4 X20 K002
K008 X187 X316 X67 aX67 K043
K034 X37 X75 X1 X8 K345*
K045 X216 X264 X3 X4 pK(X216,X2)
K141 X39 X76 X6 X69 pK(X39,X2)
K146 X5 X3 X264 aX264 pK(X5,X2)
K154 X1108 X322 X84 aX84 pK(X1108,X2)
K170 X3 X4 X69 X193 K168*
K200 X1 X8 X7 X144 pK(X1,X2)
K242 X115 X99 X523 X523 K237*
K264a X396 X298 X13 X616 pK(X396,X2)
K264b X395 X299 X14 X617 pK(X395,X2)
K355 X511 X511 X290 aX290 K357