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Thomson's cubic

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  • Edward Brisse
    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
    • Bernard Gibert
      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
      • xpolakis@otenet.gr
        ... 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
        • jean-pierre.ehrmann@wanadoo.fr
          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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