Thomson's cubic

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• Hi ! - What is the G-Ceva of Thomson s cubic ? - What is the Anticomplement of Thomson s cubic ? That s all ! EB
Message 1 of 4 , Oct 31 2:29 AM
Hi !
- What is the G-Ceva of Thomson's cubic ?
- What is the Anticomplement of Thomson's cubic ?
That's all !
EB
• Dear Edward, ... Itself ! More generally, any pivotal isocubic with pivot P is invariant under P-Ceva conjugation. ... Lucas. Best regards Bernard
Message 2 of 4 , Oct 31 7:27 AM
Dear Edward,

> - What is the G-Ceva of Thomson's cubic ?

Itself !
More generally, any pivotal isocubic with pivot P is invariant under P-Ceva
conjugation.

> - What is the Anticomplement of Thomson's cubic ?

Lucas.

Best regards

Bernard
• ... Dear Bernard, 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
Message 3 of 4 , Oct 31 12:04 PM
[Edward Brisse]:
>> - What is the Anticomplement of Thomson's cubic ?

[Bernard Gibert]:
>Lucas.

Dear Bernard,

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 of ABC
is an isotomic cubic of ABC? (as in the above case)

Antreas
• Dear Antreas, Bernard, Edward and other Hyacinthists, [APH] ... of ABC ... If you note that an pivotal isogonal cubic wrt the antimedial triangle goes through
Message 4 of 4 , Oct 31 10:56 PM
Dear Antreas, Bernard, Edward and other Hyacinthists,

[APH]
> 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
of ABC
> 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
Then
M, f(M), P lie on a same line iff
M, g(M), f(P) lie on a same line
Friendly. Jean-Pierre
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