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11596Re: supplementaire and Jerabek hyperbola

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  • Wilson Stothers
    Oct 3, 2005
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      --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@c...> wrote:
      > Wilson, and now Dick,
      >
      > I first owe a bit of an apology. I have been rereading my old
      > French books and the "supplementaire" of a point is not :z+x: in
      > trilinears. It is that point referred to the incentral triangle.
      >
      > So I am going to call our operation (which is an entirely valid one)
      > the supplement, and use the French word for the now no longer in use
      > idea.
      >

      I have looked at your translation of Mineur. From this, I think that
      your apology is not needed. The two notions coincide.

      If you look at a result in section 26 (in your translation):

      When a triangle is inscribed in ABC is in perspective to ABC, it is
      again in perspective to the antisupplementary triangle, and the second
      center of perspective is the antisupplementary to the inverse of the
      first.

      This translates into more modern language as follows :

      The perspector of the anticevian triangle of I (the antisupplementary
      triangle) and the cevian triangle of P (= the P-ceva conjugate of I)
      is the antisupplementaire of the isogonal conjugate of P,

      In our discussion, we found that :

      the P-ceva conjugate of I is agP, the antisupplement of the isogonal
      conjugate of P.

      Thus our antisupplement is the same as Mineur's antisupplementaire.

      There is confirmation later (section 31, subsection 2)

      certain conics are found to meet at the antisupplementaire of O.
      A Cabri sketch confirms that they meet at X(46).
      Clark gives this as the H-ceva conjugate of I,
      which we interpret as the antisupplement of O = gH.

      Note that your translation describes the point as the reflection
      of the incentre in the circumcentre, that is X(40), so there is
      something amiss here.

      Good news?

      Wilson

      PS (re your other message)
      You might like to check Clark's Glossary, the cevapoint
      operation is not an involution.
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