## another conjugation

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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
Message 1 of 4 , Nov 9, 2012
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
• 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
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
>
• 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
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]
• 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
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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