notes on the family cK(#F,R)

The cubic cK(#F,R) is the conico-pivotal isocubic with node F = [f,g,h] and root R = [r,s,t]. It has equation

(1)             rx(hy-gz)2+sy(fz-hx)2+tz(gx-xy)2 = 0.

For brevity, we often use the notation ΣF(x,y,z,f,g,h,r,s,t) to mean the sum over all values of F with (x,y,z),
(f,g,h), and (r,s,t) simultaneously cyclically permuted. Thus, (1) becomes Σrx(hy-gz)2 = 0. Likewise, we
abbreviate the point K to [f], R to [r] and so on.

The cubic is invariant under F-isoconjugation, the map iF taking [x] to [f2/x].

Proposition 1
The point X lies on cK(#F,R) if and only if the circumconic C through X and iF(X)
has either of the properties
(1) the pole of Σx/f = 0 with respect to C lies on Σrx/f2.
(2) the perspector of C lies on C(F,R) : Σ(r/f)(x/f)2-(s/g+t/h)(y/g)(z/h) = 0.

Proof
If X = [x], the conic C has perspector P = [f2x(h2y2-g2z2)].
The pole of Σx/f is then [x(hy-gz)2], and the first result follows.
The second is just the condition that the conic with perspector [x] has the property.
This is derived from the matrix condition aTM-1b = 0 for the pole of line b.x = 0
to lie on a.x = 0.

The conic C(F,R) contains the vertices of the cevian triangles of F and iF(R).
When F = R, C(F,R) is the inconic with perspector F.

Proposition 2
The point X lies on cK(#F,R) if and only if the line L through X and iF(X)
touches the conic dual to C*(F,R) = Σrfx2-(sh+tg)yz = 0.

Proof
The line is the dual of L' = [x(y2h2-z2g2)]. This is [p/f2], where [p] is the perspector in Proposition 1.
By Proposition 1, the condition is equivalent to L' being on C*(F,R), i.e. L being tangent to the dual.

The dual conic is called the pivotal conic of cK(#F,R).

The conic C*(F,R) contains the vertices of the isotomes [1/f] and [1/r].
When F = R, it is the inconic with perspector [1/f].
Note also that C*(F,R) = C*(R,F), so the cubics cK(#F,R), cK(#R,F) have the same pivotal conic.

In some respects the appearance of duality in Proposition 2 is unnatural. It refers to "duality with
respect to the complex conic Σx2 = 0" which is not a projective notion. The conic C(F,R) has
an unexpected geometrical connection with the problem - see simson lines.

Instead, we introduce the idea of F-duality - duality with respect to the conic F Σ(p/k)2 = 0
the F-dual of point [u] is the line Σ(u/f2)x = 0,
the F-dual of the line Σux = 0 is the point [f2u].

The F-dual of the conic xTMx = 0 is the conic xTDM-1DX = 0, where D = diag(1/f2,1/g2,1/h2).

Observe that the conic F is invariant under F-duality.

Now the line L in Proposition 2 is the F-dual of the perspector P of Proposition 1.
Then P is on C(F,R) is equivalent to L touches the F-dual. This is the pivotal conic.

Note that the conics C(F,R) and C*(F,R) are identical only when F = G.

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