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Re: conjugation

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  • Jeff Brooks
    Dear Alexey, ... and CD, AC and BD, AD and BC, P - arbitrary point distinct from X, Y, Z. Then the polars of P wrt all conics passing through A, B, C, D have
    Message 1 of 6 , May 12, 2005
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      Dear Alexey,

      In message 11252, you wrote:

      > Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
      and CD, AC and BD, AD and BC, P - arbitrary point distinct from X, Y,
      Z. Then the polars of P wrt all conics passing through A, B, C, D
      have the common point P'. If A, B, C, D are orthocentric then P' is
      isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
      is the centroid of three other, then P' is isotomic conjugated. There
      is an interesting corollary. Let U, U' and V, V' are two pairs of
      conjugated points. Then the points UV^U'V' and U'V^UV' are conjugated.
      >

      I believe a direct proof of your statements can be given without
      referring to well known geometric theorems. Personally, I would love
      to see a proof given using complex coordinates...Any ideas?

      Sincerely,

      Jeff
    • Jeff Brooks
      Alexey, Of course, I meant message 11251. Sorry, Jeff ... Y, ... There ... conjugated. ... love
      Message 2 of 6 , May 12, 2005
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        Alexey,

        Of course, I meant message 11251.

        Sorry,

        Jeff


        >
        > In message 11252, you wrote:
        >
        > > Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
        > and CD, AC and BD, AD and BC, P - arbitrary point distinct from X,
        Y,
        > Z. Then the polars of P wrt all conics passing through A, B, C, D
        > have the common point P'. If A, B, C, D are orthocentric then P' is
        > isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
        > is the centroid of three other, then P' is isotomic conjugated.
        There
        > is an interesting corollary. Let U, U' and V, V' are two pairs of
        > conjugated points. Then the points UV^U'V' and U'V^UV' are
        conjugated.
        > >
        >
        > I believe a direct proof of your statements can be given without
        > referring to well known geometric theorems. Personally, I would
        love
        > to see a proof given using complex coordinates...Any ideas?
        >
        > Sincerely,
        >
        > Jeff
      • Jeff Brooks
        Alexey, Doesn t look like we ll have anyone taking up this proposition anytime soon. How about we limit our observations to orthocentric configurations or
        Message 3 of 6 , May 20, 2005
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          Alexey,

          Doesn't look like we'll have anyone taking up this proposition
          anytime soon. How about we limit our observations to orthocentric
          configurations or maybe generalized circumconics just to get the ball
          rolling.

          Jeff



          > Dear Alexey,
          >
          > In message 11252, you wrote:
          >
          > > Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
          > and CD, AC and BD, AD and BC, P - arbitrary point distinct from X,
          Y,
          > Z. Then the polars of P wrt all conics passing through A, B, C, D
          > have the common point P'. If A, B, C, D are orthocentric then P' is
          > isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
          > is the centroid of three other, then P' is isotomic conjugated.
          There
          > is an interesting corollary. Let U, U' and V, V' are two pairs of
          > conjugated points. Then the points UV^U'V' and U'V^UV' are
          conjugated.
          > >
          >
          > I believe a direct proof of your statements can be given without
          > referring to well known geometric theorems. Personally, I would
          love
          > to see a proof given using complex coordinates...Any ideas?
          >
          > Sincerely,
          >
          > Jeff
        • Jeff Brooks
          Dear Keith Dean and Floor van Lamoen, Hope one of you are listening. There is a ball headed in your general direction with the word reciprocal conjugation or
          Message 4 of 6 , May 22, 2005
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            Dear Keith Dean and Floor van Lamoen,

            Hope one of you are listening. There is a ball headed in your general
            direction with the word "reciprocal conjugation" or "isoconjugation"
            written on it. I would really appreciate any input you might have
            regarding possible solutions to this problem (especially solutions
            using complex coordinates).

            Sincerely,
            Jeff Brooks

            PS
            I am a long-time admirer of your works,
            Re: http://forumgeom.fau.edu/FG2001volume1/FG200116.pdf



            > Alexey,
            >
            > Doesn't look like we'll have anyone taking up this proposition
            > anytime soon. How about we limit our observations to orthocentric
            > configurations or maybe generalized circumconics just to get the
            ball
            > rolling.
            >
            > Jeff
            >
            >
            >
            > > Dear Alexey,
            > >
            > > In message 11252, you wrote:
            > >
            > > > Let give 4 points A, B, C, D. X, Y, Z are the common points of
            AB
            > > and CD, AC and BD, AD and BC, P - arbitrary point distinct from
            X,
            > Y,
            > > Z. Then the polars of P wrt all conics passing through A, B, C, D
            > > have the common point P'. If A, B, C, D are orthocentric then P'
            is
            > > isogonally conjugated to P wrt XYZ, and if one of points A, B, C,
            D
            > > is the centroid of three other, then P' is isotomic conjugated.
            > There
            > > is an interesting corollary. Let U, U' and V, V' are two pairs of
            > > conjugated points. Then the points UV^U'V' and U'V^UV' are
            > conjugated.
            > > >
            > >
            > > I believe a direct proof of your statements can be given without
            > > referring to well known geometric theorems. Personally, I would
            > love
            > > to see a proof given using complex coordinates...Any ideas?
            > >
            > > Sincerely,
            > >
            > > Jeff
          • Floor en Lyanne van Lamoen
            Dear Jeff, The statement with A, B, C and D very much sound to me as P-perpendicular generalization of isogonal conjugacy. All statements for the
            Message 5 of 6 , May 23, 2005
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              Dear Jeff,

              The statement with A, B, C and D very much sound to me as "P-perpendicular"
              generalization of isogonal conjugacy. All statements for the orthocentric
              configuration smoothly generalize to general A,B,C,D by
              "P-perpendicularity".

              See http://forumgeom.fau.edu/FG2001volume1/FG200122.pdf

              Kind regards,
              Floor van Lamoen.

              > Dear Keith Dean and Floor van Lamoen,
              >
              > Hope one of you are listening. There is a ball headed in your general
              > direction with the word "reciprocal conjugation" or "isoconjugation"
              > written on it. I would really appreciate any input you might have
              > regarding possible solutions to this problem (especially solutions
              > using complex coordinates).
              >
              > Sincerely,
              > Jeff Brooks
              >
              > PS
              > I am a long-time admirer of your works,
              > Re: http://forumgeom.fau.edu/FG2001volume1/FG200116.pdf
              >
              >
              >
              > > Alexey,
              > >
              > > Doesn't look like we'll have anyone taking up this proposition
              > > anytime soon. How about we limit our observations to orthocentric
              > > configurations or maybe generalized circumconics just to get the
              > ball
              > > rolling.
              > >
              > > Jeff
              > >
              > >
              > >
              > > > Dear Alexey,
              > > >
              > > > In message 11252, you wrote:
              > > >
              > > > > Let give 4 points A, B, C, D. X, Y, Z are the common points of
              > AB
              > > > and CD, AC and BD, AD and BC, P - arbitrary point distinct from
              > X,
              > > Y,
              > > > Z. Then the polars of P wrt all conics passing through A, B, C, D
              > > > have the common point P'. If A, B, C, D are orthocentric then P'
              > is
              > > > isogonally conjugated to P wrt XYZ, and if one of points A, B, C,
              > D
              > > > is the centroid of three other, then P' is isotomic conjugated.
              > > There
              > > > is an interesting corollary. Let U, U' and V, V' are two pairs of
              > > > conjugated points. Then the points UV^U'V' and U'V^UV' are
              > > conjugated.
              > > > >
              > > >
              > > > I believe a direct proof of your statements can be given without
              > > > referring to well known geometric theorems. Personally, I would
              > > love
              > > > to see a proof given using complex coordinates...Any ideas?
              > > >
              > > > Sincerely,
              > > >
              > > > Jeff
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