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Re: [EMHL] Inverting Kiepert triangles

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  • Darij Grinberg
    Dear John Conway, I am very glad to see you posting at Hyacinthos again! Thanks for the reply. ... A little historical digression. I think this is what you and
    Message 1 of 6 , Jun 12, 2003
      Dear John Conway,

      I am very glad to see you posting at Hyacinthos again!
      Thanks for the reply.

      >> Yes, at least to me. It's much more general than
      >> the Kiepert situation. Erect what I call
      >> (alpha,beta,gamma)-Napoleons on the edges, namely
      >> triangles whose base angles are alpha at A, beta at B,
      >> gamma at C, then their apices form a triangle that's
      >> in perspective with ABC at a point P(alpha,beta,gamma).

      A little historical digression. I think this is what you
      and Antreas call "Traditional Theorem" or "Isogonal Theorem"
      or "de Villiers Theorem". In fact, it is very old - for
      instance, see

      William Allen Whitworth, "Trilinear Coordinates",
      Cambridge 1986.

      (This book is accessible through
      http://library5.library.cornell.edu/math_W.html
      .)

      On page 57, the exercise (41) reads:

      "On the three sides of a triangle ABC triangles PBC,
      QCA, RAB are described so that the angles QAC, RAB
      are equal, the angles RBA, PBC are equal, and the
      angles PCB, QCA are equal; prove that the straight
      lines, AP, BQ, CR pass through one point."

      In one paper, Armin Saam calls this fact "Jacobi
      Theorem", however he doesn't say which Jacobi this is
      and where the fact was published first.

      >> The inverses of the apices (in the circumcircle)
      >> also form a triangle in perspective with ABC,

      My sketches don't confirm that! Have we confused some
      points?

      Sincerely,
      Darij Grinberg
    • Darij Grinberg
      ... ^^^^ 1886. With my apologies for the typo, Sincerely, Darij Grinberg
      Message 2 of 6 , Jun 13, 2003
        I wrote:

        >> A little historical digression. I think this is what you
        >> and Antreas call "Traditional Theorem" or "Isogonal Theorem"
        >> or "de Villiers Theorem". In fact, it is very old - for
        >> instance, see
        >>
        >> William Allen Whitworth, "Trilinear Coordinates",
        >> Cambridge 1986.
        ^^^^
        1886.

        With my apologies for the typo,
        Sincerely,
        Darij Grinberg
      • John Conway
        ... I should explain my long absence. It was caused by the large backlog that built up, initially because I went away from Princeton for two weeks sometime
        Message 3 of 6 , Jun 13, 2003
          On Thu, 12 Jun 2003, Darij Grinberg wrote:

          > Dear John Conway,
          >
          > I am very glad to see you posting at Hyacinthos again!
          > Thanks for the reply.

          I should explain my long absence. It was caused by the
          large backlog that built up, initially because I went away
          from Princeton for two weeks sometime before Christmas, and
          then got larger and larger because I couldn't face handling
          it.

          A few weeks ago, I solved this problem by just "throwing away"
          more than 14,000 messages, and starting again. [I actually archived
          those messages, and plan to go through them "some day".] However,
          I find I'm now getting between 100 and 200 every day (most of
          which are junk), and so am still having difficulty keeping up
          to date.

          At the moment, the way I'm doing this is just by deleting
          most of them without reading, even when I know I might be
          interested in them, as with the Hyacinthos ones. But I plan
          to find some automatic way of getting rid of a goodly number
          and will them start taking an interest in Hyacinthos again.

          > A little historical digression. I think this is what you
          > and Antreas call "Traditional Theorem" or "Isogonal Theorem"
          > or "de Villiers Theorem". In fact, it is very old - for
          > instance, see
          >
          > William Allen Whitworth, "Trilinear Coordinates",
          > Cambridge 1986.

          I have never used any of those names, but knew it was at
          least a century old. Somehow I doubt "Jacobi", though.

          I think I said a few things wrong. In our book, Steve
          and I use this theorem to define the generalized pivot points
          P^(alpha,beta,gamma) and P_(alpha,beta,gamma), which become
          ordinary pivot points when alpha + beta + gamma = pi.
          perhaps some of the assertions only hold in that case.

          I regard the theorem as a fundamental one in view of
          they importance of these concepts. I think it's foolish
          to try to name it after someone (especially a recent someone)
          for the usual reasons - any such attribution will almost
          certainly be wrong, and in any case it wouldn't help to
          recall the theorem. So let's call it something like
          "the generalized pivot theorem".

          Actually, there are two, corresponding to the two types
          of pivot. The base-angle (al,be,ga)-Napoleons defined like this:


          ___________
          ga / \al
          / B \
          / \
          \be / \ be/
          \ / C A \ /
          ------------
          \al ga/
          \ /

          have apices in perspective with ABC, the perspector being the
          first type of generalized pivot. Apex-angle (al,be,ga)-Naps
          defined like this

          ___________________
          \al /\ ga/
          \ / \ /
          \ / \ /
          \ / \ /
          ---------
          \ /
          \ /
          \be/

          have circumcircles whose radical center is the other type.
          The two types are mutually isogonal. If al+be+ga = pi, then
          the circumcircles just mentioned concur in the appropriate pivot.

          John Conway
        • Darij Grinberg
          Dear John Conway, ... archived ... I wish you good luck with following up the emails. Let me tell you how I joined Hyacinthos in December 2002 and had to read
          Message 4 of 6 , Jun 13, 2003
            Dear John Conway,

            Many thanks again. You wrote:

            >> I should explain my long absence. It was caused by the
            >> large backlog that built up, initially because I went away
            >> from Princeton for two weeks sometime before Christmas, and
            >> then got larger and larger because I couldn't face handling
            >> it.
            >>
            >> A few weeks ago, I solved this problem by just "throwing away"
            >> more than 14,000 messages, and starting again. [I actually
            archived
            >> those messages, and plan to go through them "some day".] However,
            >> I find I'm now getting between 100 and 200 every day (most of
            >> which are junk), and so am still having difficulty keeping up
            >> to date.
            >>
            >> At the moment, the way I'm doing this is just by deleting
            >> most of them without reading, even when I know I might be
            >> interested in them, as with the Hyacinthos ones. But I plan
            >> to find some automatic way of getting rid of a goodly number
            >> and will them start taking an interest in Hyacinthos again.

            I wish you good luck with following up the emails.

            Let me tell you how I joined Hyacinthos in December 2002
            and had to read through all messages from #1 to #6122. First,
            I had to download all them in HTML files partitioned with 15
            messages in each file. However, copying took five days, but
            reading took some months. I also tried to reprove some of the
            theorems I found in Hyacinthos, what was a quite good way of
            appreciating them.

            >> > A little historical digression. I think this is what you
            >> > and Antreas call "Traditional Theorem" or "Isogonal Theorem"
            >> > or "de Villiers Theorem". In fact, it is very old - for
            >> > instance, see
            >> >
            >> > William Allen Whitworth, "Trilinear Coordinates",
            >> > Cambridge 1986.
            >>
            >> I have never used any of those names, but knew it was at
            >> least a century old. Somehow I doubt "Jacobi", though.

            Another source, may be more reliable:

            Peter Baptist, "Die Entwicklung der neueren
            Dreiecksgeometrie", Mannheim-Leipzig-Wien-Zürich 1992.

            This book also manifests Jacobi as the first discoverer (alas,
            I have only a copy of the pages with the geometry and don't
            know the initials of Jacobi). Jacobi presented his theorem in
            his schoolbook.

            Of course, this is not a reason to name it after Jacobi, if
            there is a good alternative term.

            >> I think I said a few things wrong. In our book, Steve
            >> and I use this theorem to define the generalized pivot points
            >> P^(alpha,beta,gamma) and P_(alpha,beta,gamma), which become
            >> ordinary pivot points when alpha + beta + gamma = pi.
            >> perhaps some of the assertions only hold in that case.

            Hope you won't mind me digressing again. Let triangles
            BA'C, CB'A and AC'B are constructed on BC, CA, AB so that

            angle A'BC = angle BCA' = alpha;
            angle B'CA = angle CAB' = beta;
            angle C'AB = angle ABC' = gamma.

            Here, I consider DIRECTED ANGLES MODULO 180°. (These are
            THE angles of circle geometry.)

            Now, I usually call triangle A'B'C' the
            (alpha, beta, gamma)-Jacobi triangle. If

            alpha + beta + gamma = 0°,

            then, I also call triangle A'B'C' a special Jacobi
            triangle. (I formerly called it special Schaal triangle -
            forget this nonsense name.) While for any Jacobi triangle,
            the lines AA', BB', CC' concur, for any special Jacobi
            triangle, the point of concurrence also lies on the
            circles A'BC, B'CA and C'AB.

            However, neither for all Jacobi triangles nor for all
            special Jacobi triangles the inverses in the circumcircle
            form a triangle perspective to ABC. Actually, they do for
            all Kiepert triangles, which is due to the fact that the
            vertices of a Kiepert triangle lie on the perpendicular
            bisectors of the sides of ABC.

            >> I regard the theorem as a fundamental one in view of
            >> they importance of these concepts. I think it's foolish
            >> to try to name it after someone (especially a recent someone)
            >> for the usual reasons - any such attribution will almost
            >> certainly be wrong, and in any case it wouldn't help to
            >> recall the theorem. So let's call it something like
            >> "the generalized pivot theorem".
            >>
            >> Actually, there are two, corresponding to the two types
            >> of pivot. The base-angle (al,be,ga)-Napoleons defined like this:
            >>
            >>
            >> ___________
            >> ga / \al
            >> / B \
            >> / \
            >> \be / \ be/
            >> \ / C A \ /
            >> ------------
            >> \al ga/
            >> \ /
            >>
            >> have apices in perspective with ABC, the perspector being the
            >> first type of generalized pivot.

            I think such triangles are not perspective with ABC. In fact,
            if they were perspective, we would have

            cot A + cot be cot B + cot ga cot C + cot al
            -------------- * -------------- * -------------- = 1,
            cot A + cot ga cot B + cot al cot C + cot be

            what doesn't seem to be correct.

            Perhaps the angles must be positioned another way round:

            ___________
            be / \be
            / B \
            / \
            \ga / \ al/
            \ / C A \ /
            ------------
            \ga al/
            \ /

            This is the "Jacobi" case.

            >> Apex-angle (al,be,ga)-Naps defined like this
            >>
            >> ___________________
            >> \al /\ ga/
            >> \ / \ /
            >> \ / \ /
            >> \ / \ /
            >> ---------
            >> \ /
            >> \ /
            >> \be/
            >>
            >> have circumcircles whose radical center is the other type.

            But they are not defined by one angle! The circumcircles
            are uniquely defined, but not the triangles themselves.

            >> The two types are mutually isogonal. If al+be+ga = pi, then
            >> the circumcircles just mentioned concur in the appropriate pivot.

            Now it seems that we are speaking of different things, if not
            something is wrong.

            Sincerely,
            Darij Grinberg
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