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11588Re: [EMHL] Re: supplementaire

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  • Steve Sigur
    Oct 1, 2005
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      Following Wison's lead and using his notation, I think I am getting a
      handle on this "supplement" operation. I am interested in this
      because the classical geometers used it and because it is not
      mentioned in ETC, and thereby presumably by modern geometers, the
      general principle here being that finding out what is not said is
      often as interesting as what is said. I am also interested in points
      whose trilinears coordinates are half-angles of A, B,C because they
      do not relate well to the great mass of points and because they have
      unique extraversion properties.

      I have combined Wison's and my tables below for reference.

      Some geometry from the supplementaire's.

      Consider P and its conjugate (alway isogonal) gP. Considering both
      enforces what is called anallagmatic symmetry. More symmetry is better.

      sP can be constructed as the the cevian quotient gP/Io by connecting
      the ex-incenters to the cevian traces of gP.

      Similarly for sgP.

      Now get aP and agP as harmonic conjugate (as Wison explains below).

      I am ignoring the cross conjugate operation because it seems too
      complicated for me to understand it as other than a mathematical
      operation. I do notice that it is an involution (to what?) and that
      may make it interesting later.

      The interesting part is that the isogonals of sP and sgP are colinear
      to P and gP!

      We now have 4 points in the P�gP line. Their tripolars meet at the
      perspector of the circumconic isogonal to this line with P, gP, sP,
      and sgP being on this conic . For example if P = H, the orthocenter,
      this conic is the isogonal of the Euler line, the Jerabek conic. From
      our lists below, we find out that X(65) and X(73) are on this conic.

      The Jerabek conic is unique in that one can do this same game in
      barycentrics using isotomic conjugation for that conic, finding
      different points.

      So... It took Wison and I a couple of days to find out that this is
      an interesting operation. it is easily constructable. The resulting
      coordinates are simple. It relates points related by nothing else.
      Why has this operation been ignored for the last hundred years?

      Friendly from the US,,

      Steve



      > [WS]
      > Can I perhaps add a few more sets of points,
      > and fill a few gaps.
      >
      > Let me write
      > sP for the supplement of P,
      > aP for the antisupplement of P,
      > gP for the isogonal conjugate of P
      >
      > Function P sP aP

      [I have combined my and Wilson's lists using his notation]
      >
      >> cos A 3, 65, 46
      >
      > sin A 6 37 9
      > csc A 2 42 43
      >
      > tan A 19 48 610
      > cot A 63 31 17

      >> sec B 4 73 1745

      >
      > cotA/2 9 6 1743
      > tanA/2 57 55 165

      >> sin B/2 266 , --, 164
      >> 258 on this line
      >> sec B/2 174 --, 503
      >> csc B/2 188 --, 361
      >> cos B/2 259 ---- 173

      >
      > These give new and simple forms for the trilinears
      > of X610, X1707, X1743, and alternative versions for
      > those of X6, X31, X37, X42.
      >
      > I cannot find much geometry, but do observe that
      >
      > aP = gP-Ceva conjugate of I
      > gP = sp-cross conjugate of I.
      >
      > Also, (P, sP, aP, I) is harmonic,
      > as is (P, aP, sP, X), where
      > X is the intersection of IP with the antiorthic axis.



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