more results on the tucker cubic

To some extent, this is the start of an attemp to clean up previous pages.

Notation

TC = Tucker Nodal Cubic
SE = Steiner Ellipse
SI = Steiner Inellipse
G = centrioid of reference triangle
tP = isotomic conjugate of P
cP = complement of P
C(P,Q) = circumconic through P and Q

characterizations of TC

Theorem 1
For a triangle centre other than G, the following are equivalent:
(a) P is on TC
(b) C(P,tP) is a parabola
(c) PtP is tangent to SE
(d) the tripolars of P, tP meet on SE
(e) the tripolar of P is parallel to PG.

related matters

Let us write U*V for the barycentric product of U and V.

Lemma
If C is SE or SI and U,V are antipodal points on C, then U*V is on C.

Proof
Any point on SE is tW, W at infinity. Any point on SI is W*W, W at infinity.
Let W = [u,v,w] be at infinity, and W' = [v-w,w-u,u-v], also at infinity
Observe that W*W' is at infinity.
For SE, if U = [1/u,1/v,1/w], then V is [1/(v-w),1/(w-u),1/(u-v)].
For SI, if U = W*W, then V is W'*W'.
In either case, we are done.

Theorem 2
If U,V are antipodal points on SI, and W = U*V, then the circumconics
CU,CV,CW with respective perspectors U,V,W meet in the three points
P = CUnCW, Q = cVnCW, R = CUnCV.
(a) CU, CV, CW are parabolas,
(b) P,Q,R are collinear, and their line is tangent to SE,
(c) R is on SE, and is the anticomplement of W,
(d) P,Q are isotomic conjugates, and on TC,
(e) P is the third Tucker point of CU, Q that of CV.

Proof
(a) is immediate after the Lemma, since all three perspectors are on SI.
For the rest, suppose U = [u2], with u+v+w = 0. Then Q = [(v-w)2], W = [u2(v-w)2].
Now it is easy to verify that P = [u/(v-w)], Q = [(v-w)/u], R = [1/u(v-w)], and then
that PQ is the line whose equation has as coefficients the coordinates of W.
This is the tangent to SE at R. Clearly the complement of R is W.
Clearly P, Q are isotomic conjugates.
It is a routine calculation to show that P, and hence Q, is on TC.
A circumparabola has three intersections with TC, two are isotomes.
P,Q define CW, so Q is not on CU. Thus p is the third Tucker point of CU.
Similarly Q is the third Tucker point of CV.

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