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another conjugation

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  • rhutson2
    Dear friends, Consider the mapping F(P), defined equivalently as: 1) F(P) = pole, wrt polar circle, of the trilinear polar of P 2) F(P) = trilinear pole of the
    Message 1 of 4 , Nov 9, 2012
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      Dear friends,

      Consider the mapping F(P), defined equivalently as:

      1) F(P) = pole, wrt polar circle, of the trilinear polar of P
      2) F(P) = trilinear pole of the polar of P wrt polar circle

      This is a conjugate mapping, i.e. F(F(P)) = P.

      Some F-conjugate pairs X(I), X(J): (1,92), (2,4), (3,2052), (5,275), (6,264), (7,281), (8,278), (9,273), (10,27), (13,470), (14,471), (19,75), (20,459), (25,76), (28,321), (29,226), (31,1969), (33,85), (34,312), (37,286), (51,276), (53,95), (54,324), (55,331), (57,318), (63,158), (69,393), (98,297), (99,2501), (108,4391), (112,850).

      F(Euler line) = Kiepert hyperbola
      F(orthic axis) = Steiner circumellipse
      F(Brocard axis) = conic {A,B,C,X(264),X(2052)}
      F(circumcircle) = line X(297)X(525)
      F(Jerabek hyperbola) = line X(2)X(216)
      F(Feuerbach hyperbola) = line X(2)X(92)
      F(MacBeath circumconic) = line X(403)X(523)

      Is this a known conjugation? If not, it seems 'polar conjugate' might be a fitting name.

      Is there another construction for this conjugation?

      Best regards,
      Randy Hutson
    • rhutson2
      Question: Is there a construction of this mapping that does not involve the polar circle, which is only defined for obtuse triangles? Randy
      Message 2 of 4 , Nov 15, 2012
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        Question: Is there a construction of this mapping that does not involve the polar circle, which is only defined for obtuse triangles?

        Randy

        --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
        >
        > Dear friends,
        >
        > Consider the mapping F(P), defined equivalently as:
        >
        > 1) F(P) = pole, wrt polar circle, of the trilinear polar of P
        > 2) F(P) = trilinear pole of the polar of P wrt polar circle
        >
        > This is a conjugate mapping, i.e. F(F(P)) = P.
        >
        > Some F-conjugate pairs X(I), X(J): (1,92), (2,4), (3,2052), (5,275), (6,264), (7,281), (8,278), (9,273), (10,27), (13,470), (14,471), (19,75), (20,459), (25,76), (28,321), (29,226), (31,1969), (33,85), (34,312), (37,286), (51,276), (53,95), (54,324), (55,331), (57,318), (63,158), (69,393), (98,297), (99,2501), (108,4391), (112,850).
        >
        > F(Euler line) = Kiepert hyperbola
        > F(orthic axis) = Steiner circumellipse
        > F(Brocard axis) = conic {A,B,C,X(264),X(2052)}
        > F(circumcircle) = line X(297)X(525)
        > F(Jerabek hyperbola) = line X(2)X(216)
        > F(Feuerbach hyperbola) = line X(2)X(92)
        > F(MacBeath circumconic) = line X(403)X(523)
        >
        > Is this a known conjugation? If not, it seems 'polar conjugate' might be a fitting name.
        >
        > Is there another construction for this conjugation?
        >
        > Best regards,
        > Randy Hutson
        >
      • Bernard Gibert
        Dear Randy, ... your F(P) is the H-isoconjugate of P i.e. the image of P under the isoconjugation that swaps G and H. best regards Bernard [Non-text portions
        Message 3 of 4 , Nov 16, 2012
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          Dear Randy,

          > Question: Is there a construction of this mapping that does not involve the polar circle, which is only defined for obtuse triangles?

          your F(P) is the H-isoconjugate of P i.e. the image of P under the isoconjugation that swaps G and H.

          best regards

          Bernard

          [Non-text portions of this message have been removed]
        • rhutson2
          Dear Bernard, After looking at it more closely, it looks like F(P) is the X(48)-isoconjugate of P, or equivalently, the trilinear product of X(92) and the
          Message 4 of 4 , Nov 20, 2012
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            Dear Bernard,

            After looking at it more closely, it looks like F(P) is the X(48)-isoconjugate of P, or equivalently, the trilinear product of X(92) and the isogonal conjugate of P.

            Best regards,
            Randy

            --- In Hyacinthos@yahoogroups.com, Bernard Gibert <bg42@...> wrote:
            >
            > Dear Randy,
            >
            > > Question: Is there a construction of this mapping that does not involve the polar circle, which is only defined for obtuse triangles?
            >
            > your F(P) is the H-isoconjugate of P i.e. the image of P under the isoconjugation that swaps G and H.
            >
            > best regards
            >
            > Bernard
            >
            > [Non-text portions of this message have been removed]
            >
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