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Re: Io--tIo line

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  • Wilson Stothers
    ... are ... on ... This ... organized ... one ... that ... Hi Seve I am afraid that there are no strong points on your line. Think of a line L as the tripolar
    Message 1 of 8 , Feb 17, 2006
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      --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:

      > There is often (usually a strong point on a weak line). Examples
      are
      > the symmedian point in the incenter-Mittenpunkt line, the centroid
      on
      > the Nagel line, the isotomic H point on the Gergonne-Nagel line.
      This
      > is so common that it is easy to think the weak points are
      organized
      > around strong ones in this way.
      >
      > I am wondering if the line between the incenter its isotomic
      > conjugate contains a strong point. I have not been able to find
      one
      > and I was wondering I anyone out there could find one.
      >
      > These are two very unique points (algebraically) and I suspect
      that
      > they contain no strong points.

      Hi Seve

      I am afraid that there are no strong points on your line.
      Think of a line L as the tripolar of a point P.
      If P is weak, it has extraversions. The tripolars of these
      give the "extraversions" of L. Now if L has a strong point,
      this point will also be on the extraversions of L. Here, L
      has four extraversions (including L), and they do not concur.
      In fact, this shows that a strong point on a weak line (with
      more than two versions) will be unusual. For a point P with,
      with just one extraversion Q, the tripolar will contain a
      strong point - the crosspoint of P and Q. For example, with
      P = X13, Q = X14, we get X30.

      Sorry to be negative

      regards

      Wilson
    • Wilson Stothers
      ... Dear all, Sorry, I got my signs confused at the end of the message ... NOT the crosspoint, but the perspector of the circumconic through P and Q For
      Message 2 of 8 , Feb 17, 2006
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        --- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@...> wrote:
        Dear all,

        Sorry, I got my signs confused at the end of the message

        > I am afraid that there are no strong points on your line.
        > Think of a line L as the tripolar of a point P.
        > If P is weak, it has extraversions. The tripolars of these
        > give the "extraversions" of L. Now if L has a strong point,
        > this point will also be on the extraversions of L. Here, L
        > has four extraversions (including L), and they do not concur.
        > In fact, this shows that a strong point on a weak line (with
        > more than two versions) will be unusual. For a point P with,
        > with just one extraversion Q, the tripolar will contain a
        > strong point - the crosspoint of P and Q.

        NOT the crosspoint, but the perspector of the circumconic
        through P and Q

        For example,
        with P = X13, Q = X14, we get X523, conic the Kiepert Hyperbola

        >
        >> regards
        >
        > Wilson
        >
      • Wilson Stothers
        Dear All, After the last two posts, we have a theorem - with hindsight, it s rather obvious! Suppose that P is a weak point. Then L, the tripolar of P,
        Message 3 of 8 , Feb 18, 2006
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          Dear All,

          After the last two posts, we have a theorem - with hindsight, it's
          rather obvious!

          Suppose that P is a weak point. Then L, the tripolar of P, contains
          a strong point Q if and only if P and all its extraversions lie on a
          circumconic C. In such a case, Q is the perspector of C.
          Proof
          Observe that a point Q is on the tripolar of P if and only if P is
          on the circumconic with perspector Q. Now Q will be strong if and
          only if it lies on the tripolars of all the extraversions of P. This
          requires that P is weak so that we have at least two tripolars.
          Thus, L contains a strong point if and only if the extraversions lie
          on a circumconic C. The strong point is then the perspector of C.

          Clearly, for a point P with more than two versions, it is unlikely
          that all versions lie on a single circumconic. Well, Steve, it now
          seems that the early evidence is misleading - weak lines will seldom
          contain strong points!

          Regards

          Wilson
        • Steve Sigur
          Hi Wilson, Good to hear from you and to be back in triangle geometry after an extended trip into the 4th dimension. This is very interesting. At first I
          Message 4 of 8 , Feb 18, 2006
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            Hi Wilson,

            Good to hear from you and to be back in triangle geometry after an
            extended trip into the 4th dimension.

            This is very interesting. At first I thought your result was wrong,
            but after some thought I think it is right.

            Here are my investigations that latch into your results:
            I had suspected that strong points on weak lines were rare, but what
            interests me is that they almost always happen for what I call "the
            classical weak points." These are the inceter Io, Spieker So,
            Feuerbach Fo, Nagel No, Gergonne Go, Mittenpunkt Mo, reflection of Io
            in O = oIo, and some others. I sometimes call these the "friends of
            the incenter" since they all depend on the incenter and/or incircles.
            These points (other than oIo) are all part of what I call the "sweep
            of the incenter."

            Connect most (perhaps all) of these and you will find a strong point
            on the line. I am making a chart of these strong points on weak lines
            linked in "Steve's Voodoo pages" on my web site.

            For fundamental reasons I want to add the point :1/(c-a): = isotomic
            of infinity point on the dual of Io (the dual = tripolar of the
            isotomic conjugate). Comment: In the projective world the tripolar is
            nicer, but in the affine world the dual is nicer.

            This point seems to me to be and extremely important point. It is to
            the weak points as the Steiner point is to the strong points.
            But...join it to the other friends of the incenter and you do not
            seem to get lines with strong points as often.

            This now connects me to your result. Here is what I have been
            thinking: that points whose coordinate have a factor of (c-a) behave
            differently than points that do not. I first notice this in some work
            by Darij and me on the Shiffler and coSpieker points (linked on my
            website).

            Now almost all the lines between classical weak points have this
            factor. For example the weak line G--Io = : c-a : which is the
            tripolar of the point :1/(c-a): .

            So my topic for investigation, based on your results, is: are points
            like :1/(c-a): and its extraversions on circumconics, and, if so,
            which ones.

            This could be fun since it combines all the threads of thought in my
            head at the moment.

            Friendly from Atlanta,

            Steve






            On Feb 18, 2006, at 3:48 AM, Wilson Stothers wrote:

            > After the last two posts, we have a theorem - with hindsight, it's
            > rather obvious!
            >
            > Suppose that P is a weak point. Then L, the tripolar of P, contains
            > a strong point Q if and only if P and all its extraversions lie on a
            > circumconic C. In such a case, Q is the perspector of C.
            > Proof
            > Observe that a point Q is on the tripolar of P if and only if P is
            > on the circumconic with perspector Q. Now Q will be strong if and
            > only if it lies on the tripolars of all the extraversions of P. This
            > requires that P is weak so that we have at least two tripolars.
            > Thus, L contains a strong point if and only if the extraversions lie
            > on a circumconic C. The strong point is then the perspector of C.
            >
            > Clearly, for a point P with more than two versions, it is unlikely
            > that all versions lie on a single circumconic. Well, Steve, it now
            > seems that the early evidence is misleading - weak lines will seldom
            > contain strong points!





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          • Steve Sigur
            ... Not at all, I both expect and want this. But I am wondering about another circumstance. Could it be true that there be a strong point on Io-- tIo where the
            Message 5 of 8 , Feb 18, 2006
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              On Feb 18, 2006, at 2:03 AM, Wilson Stothers wrote:

              >
              > Sorry to be negative

              Not at all,

              I both expect and want this. But I am wondering about another
              circumstance. Could it be true that there be a strong point on Io--
              tIo where the extraversions (which, unllike most points are the
              harmonic associate versions) go through the harmonic associates of a
              strong point.

              The reason I think this might be possible is because of a theorem I
              proved a long time ago. For each desmic situation with desmon P and
              harmon Q, there is one with harmon Q and desmon P.

              For example the mittens and comittens have desmon G and harmon Io. In
              this situation the Mx--gMx lines all go through G

              Hence there is one with harmon G, which means that the corresponding
              lines go.one at a time, through the harmonic assocaties of G (the
              vertices of the antimedial (or anticomplementary) triangle).

              Steve



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            • Wilson Stothers
              ... But I ve just shown that there are no such points
              Message 6 of 8 , Feb 18, 2006
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                --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:
                >
                >
                > On Feb 18, 2006, at 2:03 AM, Wilson Stothers wrote:
                >
                > >
                > > Sorry to be negative
                >
                > Not at all,
                >
                > I both expect and want this. But I am wondering about another
                > circumstance. Could it be true that there be a strong point on Io--

                > tIo where the extraversions .....

                But I've just shown that there are no such points
              • Steve Sigur
                ... This is very interesting. Some examples, The Spiekers are on the Kiepert hyperbola, so lets use that P = So So that L = ~t So = : 1/(c+a) : , where ~
                Message 7 of 8 , Feb 20, 2006
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                  On Feb 18, 2006, at 3:48 AM, Wilson Stothers wrote:

                  > Suppose that P is a weak point. Then L, the tripolar of P, contains
                  > a strong point Q if and only if P and all its extraversions lie on a
                  > circumconic C. In such a case, Q is the perspector of C.

                  This is very interesting.

                  Some examples,

                  The Spiekers are on the Kiepert hyperbola, so lets use that

                  P = So

                  So that L = ~t So = : 1/(c+a) : , where ~ indicates the dual.

                  Q = : cc - aa : the Kiepert perspector, the dual of which is GK, on
                  which all extraversions of t So lie.

                  Generalization (obviously true): if all P lie on a circumconic, then
                  all tP and all gP lie on lines.
                  ----

                  Now let's go the other direction.

                  tH is on the Go--No line : b(c-a)sb :

                  Hence the tH circumconic ( a favorite of Bernard's) contians all
                  the : 1/b(c-a)sb :

                  I'll have to check this out.

                  This one might be easier

                  K is on Mo--Io = : (c-a)/b :

                  The isotomics of which are known to be on the K circumconic, the
                  circumcircle.

                  Very interesting,

                  Steve







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