Dear Antreas, Bernard, Edward and other Hyacinthists,
> There should be a general theorem saying what cubic of ABC
> is a given (pivotal) isogonal cubic of its antimedial triangle.
> Is it always true that an isogonal cubic of the antimedial triangle
> is an isotomic cubic of ABC? (as in the above case)
If you note that an pivotal isogonal cubic wrt the antimedial
triangle goes through ABC iff the pivot is the centroid, the above
case is the only possible one.
A kind of generalization could be the following :
f is an isoconjugation wrt the precevian triangle of P
g is the isoconjugation wrt ABC with P as fixed point
M, f(M), P lie on a same line iff
M, g(M), f(P) lie on a same line