These are some Cabri observations. Most have been proved by Maple calculations.
notation
TC is the Tucker Cubic.
SE is the Steiner Ellipse.
SI is the Steiner Inellipse.
G is the centroid.
C(P,Q) is the circumconic through points P and Q.
P(P,Q) is the perspector of C(P,Q).
tP is the isotomic conjugate of a point P.
cP is the complement of a point P.
familiar facts
The following are equivalent :
(A) P is on TC.
(B) C(P,Q) is a parabola, where Q = tP, cP or ctP.
(C) PtP is a tangent to SE.
P and Q must then be the intersections of C(P,Q) with its isotome.
They are therefore uniquely defined by the circumparabola.
We shall refer to them as the Tucker Points of the conic.
If we intersect TC with a circumconic, we should get three
intersections other than the vertices of the triangle. Two may
be complex, but we always will have a real one.
For a parabola all are real.
observations
Suppose that P is on TC and that Q = tP.
Let R be the contact point of PQ and SE.
(1) C(P,Q) is the isotome of PQ.
(2) P(P,Q) is cR.
(3) C(P,R) and C(Q,R) are parabolas.
(4) P(P,R) and P(Q,R) are antipodes on SI.
(5) cPcQ is tangent to SI at P(P,Q).
(6) the mid-point of P(P,Q) and P(P,R) lies on QG.
(7) the mid-point of P(P,Q) and P(Q,R) lies on PG.
(8) the tangents to C(P,R) at P,and C(Q,R) at Q are parallel.
(9) the tangents in (8) are also tangent to SE
their contact points with SE are collinear with P(P,R) and P(Q,R).
(10) the tangents in (8) are the isotomes of C(P,R), C(Q,R).
Let D be the circumconic with centre P(P,Q).
Let S be the fourth intersection of D and C(P,Q),
let T be the fourth intersection of C(P,Q) and SE,
and T' the fourth intersection of D and SE.
(11) S is on TC - so is the third intersection of TC,C(P,Q).
We call it the third Tucker point of C(P,Q).
(12) P is the third Tucker point of C(P,R),
Q is the third Tucker point of C(Q,R).
(13) the tangent to C(P,Q) at S is parallel to PQ.
and is tangent to SE at T'.
(14) the line P(P,R)P(Q,R) touches D at G.
it is the isotome of D.
As S is on C(P,Q), its isotomic conjugate V is on PQ.
As S is on TC, V is on TC.
(15) SV is tangent to SE at T.
(16) TT' is tangent to D at T'.
(17) TC, PQ and the tripolar of R concur at V.
(18) the tangent to D at S meets TT' at a point of C(P,Q).
and is concurrent with cPcQ and P(P,R)P(Q,R).
The point of concurrence is on C(S,V).
(19) the lines P(P,Q)P(Q,R) and P(P,Q)P(P,R) are the asymptotes of D,
they are parallel to the axes of C(P,R) and C(Q,R).
(20) the tangents to C(P,Q) at cP,cQ meet on the tripolar of P(P,Q).