Scattered through these pages are various results about such cubics, sometimes as special cases
of more general results. Here, we collect these for future reference.
Result 1
The cubic nK0(F2,R) is a cK0 precisely when one of the following equivalent conditions holds:
(1) R is on T(F),
(2) F is on C(R),
(3) the barycentric quotient R/F is at infinity.
From now on, we shall assume that these conditions are met.
Notation
F = [f,g,h].
R = [r,s,t].
K = cK0(#F,R).
For a point U,
C(U) is the circumconic with perspector U,
T(U) is the tripolar of U,
F(U) is the F2-isoconjugate of U.
Results
First, some information about an important line
Result 2
The line L(F,R) may be defined in any of the following equivalent ways:
(1) the tripolar of the isoconjugate of R, i.e. T(F(R)),
(2) the tangent to C(R) at F,
(3) the polar of R with respect to C(F).
Now two characterizations of K which do not apply to general cK(#F,R).
Result 3
K is the locus of points X such that XF(X) is cut harmonically by T(F) and L(F,R).
Result 4
K is the F-Hirst inverse of C(R).
If C(R) contains known centres, this gives centres on K.
If we take the ETC definition of Hirst inversion, where the pole is the perspector of the
conic, then only a cK0 can be the Hirst inverse of a conic. When we allow an arbitrary
pole, then we get the entire family of cK(#F,R).
Result 5
Other than the points on the sidelines, K contains:
(1) F, the node,
(2) F1, the fourth intersection of C(F) and C(R), see Result 4,
(3) F2, the intersection of T(F) and L(F,R), see Result 3,
(4) the intersections of T(F) and C(F), which are complex.
Note that F1 and F2 are isoconjugates as are the points in (4).
We can describe the nodal tangents in three quite different ways.
Result 6
The nodal tangents are the tripolars of the intersections of C(F) and L(F,R).
Result 7
The nodal tangents are the tangents from F to the diagonal conic x2/rf+y2/sg+z2/th = 0.
The contact points with the conic lie on T(R).
Result 8
The nodal tangents are the lines FU, FV, where U,V are the intersections of T(F) and C(R).
Notes
Result 6 holds for general cK(#F,R) provided we interpret L(F,R) as the polar of R for C(F).
Result 7 holds for general cK(#F,R). For a cK0(#F,R), we can describe the conic geometrically.
Result 9
The diagonal conic in result 7 touches T(F) at R and is
(1) an inconic of the cevian triangle of F,
(2) a circumconic of the anticevian triangle of R.
From either point of view, the perspector (relative to the triangle) is the F-Ceva conjugate of R.
Some further facts about cK0:
Result 10 (may14.mws)
The pivotal conic is the dual of C(R) with respect to C(F).
The tangent T3 to C(F) at F1 is therefore a tangent to the pivotal conic.
In fact it touches it at r/(t/h-s/g) - a point of C(R).
Also T3 passes through r(t/h-s/g) - the intersection of T(F) and T(R)
Note
For a general cK, the tangent to C(F) at F1 is a tangent to the pivotal conic.
The contact point is f(s/g+t/h+2r/f)/(t/h-s/g).
Bernard Gibert's List
This currently includes six cK0(#F,R) - K040, K137, K147, K185, K217, K218.
Of these, only K040 and K185 are noted as being Hirst inverses.
| cubic | circumconic | inverting conic | L(F,R) |
| K040 | Feuerbach Hyperbola | C(X(1)) - in TCCT | OI |
| K137 | C(X(513)) - in TCCT | C(X(1)) - in TCCT | IK |
| K147 | Circumcircle | C(X(110)) | X(110)X(351) |
| K185 | Kiepert Hyperbola | Steiner Ellipse | GK |
| K217 | C(X(1640)) - see note | Kiepert Hyperbola | X(523)X(868) |
| K218 | C(X(1648)) - see note | Kiepert Hyperbola | GX(523) |
Note. The circumconics for K217, K218 have tripolar centroids as their perspectors.
This means that they have explicit asymptotes, and contain six easily defined centres.