Τρίτη, 19 Μαρτίου 2013

PERSPECTIVITY [Antipedal Triangles of P,P*] --2--

Let P,P* be two isogonal conjugate points and A'B'C',A"B"C" the antipedal triangles of P,P*, resp.

Denote: Abc = A'B' /\ A"C", Acb = A'C' /\ A"B"

Bca = B'C' /\ B"A", Bac = B'A' /\ B"C"

Cab = C'A' /\ C"B", Cba = C'B' /\ C"A"

Are the triangles AbcBcaCab, AcbBacCba perspective ?

Antreas P. Hatzipolakis, Hyacinthos #21788

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They are again ALWAYS perspective. The coordinates of the perspector are shown below and it is infinite when P lies on K003 [= McCay cubic].

{a^2 (-2 a^4 b^2 c^2 x^2 y + 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y - 4 a^2 b^2 c^4 x^2 y - 4 b^4 c^4 x^2 y + 6 b^2 c^6 x^2 y - a^6 c^2 x y^2 + a^4 b^2 c^2 x y^2 + a^2 b^4 c^2 x y^2 - b^6 c^2 x y^2 + a^4 c^4 x y^2 - 10 a^2 b^2 c^4 x y^2 + b^4 c^4 x y^2 + a^2 c^6 x y^2 + b^2 c^6 x y^2 - c^8 x y^2 + 2 a^4 b^2 c^2 x^2 z + 4 a^2 b^4 c^2 x^2 z - 6 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z + 2 b^2 c^6 x^2 z - 2 a^6 c^2 y^2 z - 4 a^4 b^2 c^2 y^2 z + 6 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z - 2 a^2 c^6 y^2 z + a^6 b^2 x z^2 - a^4 b^4 x z^2 - a^2 b^6 x z^2 + b^8 x z^2 - a^4 b^2 c^2 x z^2 + 10 a^2 b^4 c^2 x z^2 - b^6 c^2 x z^2 - a^2 b^2 c^4 x z^2 - b^4 c^4 x z^2 + b^2 c^6 x z^2 + 2 a^6 b^2 y z^2 - 4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 + 4 a^4 b^2 c^2 y z^2 + 4 a^2 b^4 c^2 y z^2 - 6 a^2 b^2 c^4 y z^2), -b^2 (-a^6 c^2 x^2 y + a^4 b^2 c^2 x^2 y + a^2 b^4 c^2 x^2 y - b^6 c^2 x^2 y + a^4 c^4 x^2 y - 10 a^2 b^2 c^4 x^2 y + b^4 c^4 x^2 y + a^2 c^6 x^2 y + b^2 c^6 x^2 y - c^8 x^2 y - 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 - 2 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 - 4 a^2 b^2 c^4 x y^2 + 6 a^2 c^6 x y^2 + 6 a^4 b^2 c^2 x^2 z - 4 a^2 b^4 c^2 x^2 z - 2 b^6 c^2 x^2 z - 4 a^2 b^2 c^4 x^2 z + 4 b^4 c^4 x^2 z - 2 b^2 c^6 x^2 z - 6 a^6 c^2 y^2 z + 4 a^4 b^2 c^2 y^2 z + 2 a^2 b^4 c^2 y^2 z + 4 a^4 c^4 y^2 z - 4 a^2 b^2 c^4 y^2 z + 2 a^2 c^6 y^2 z + 2 a^6 b^2 x z^2 - 4 a^4 b^4 x z^2 + 2 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 + 4 a^2 b^4 c^2 x z^2 - 6 a^2 b^2 c^4 x z^2 + a^8 y z^2 - a^6 b^2 y z^2 - a^4 b^4 y z^2 + a^2 b^6 y z^2 - a^6 c^2 y z^2 + 10 a^4 b^2 c^2 y z^2 - a^2 b^4 c^2 y z^2 - a^4 c^4 y z^2 - a^2 b^2 c^4 y z^2 + a^2 c^6 y z^2), c^2 (6 a^4 b^2 c^2 x^2 y - 4 a^2 b^4 c^2 x^2 y - 2 b^6 c^2 x^2 y - 4 a^2 b^2 c^4 x^2 y + 4 b^4 c^4 x^2 y - 2 b^2 c^6 x^2 y + 2 a^6 c^2 x y^2 + 4 a^4 b^2 c^2 x y^2 - 6 a^2 b^4 c^2 x y^2 - 4 a^4 c^4 x y^2 + 4 a^2 b^2 c^4 x y^2 + 2 a^2 c^6 x y^2 - a^6 b^2 x^2 z + a^4 b^4 x^2 z + a^2 b^6 x^2 z - b^8 x^2 z + a^4 b^2 c^2 x^2 z - 10 a^2 b^4 c^2 x^2 z + b^6 c^2 x^2 z + a^2 b^2 c^4 x^2 z + b^4 c^4 x^2 z - b^2 c^6 x^2 z + a^8 y^2 z - a^6 b^2 y^2 z - a^4 b^4 y^2 z + a^2 b^6 y^2 z - a^6 c^2 y^2 z + 10 a^4 b^2 c^2 y^2 z - a^2 b^4 c^2 y^2 z - a^4 c^4 y^2 z - a^2 b^2 c^4 y^2 z + a^2 c^6 y^2 z - 2 a^6 b^2 x z^2 - 4 a^4 b^4 x z^2 + 6 a^2 b^6 x z^2 + 4 a^4 b^2 c^2 x z^2 - 4 a^2 b^4 c^2 x z^2 - 2 a^2 b^2 c^4 x z^2 - 6 a^6 b^2 y z^2 + 4 a^4 b^4 y z^2 + 2 a^2 b^6 y z^2 + 4 a^4 b^2 c^2 y z^2 - 4 a^2 b^4 c^2 y z^2 + 2 a^2 b^2 c^4 y z^2)}

Francisco Javier, Hyacinthos #21789

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If P lies on Darboux cubic the perspector of the triangles AbcBcaCab, AcbBacCba coincides with the isotomic conjugate of the perspector of ABC and triangle bounded by (A'A", B'B ",C'C").

Does this happen only if P is in the Darboux cubic?

Some pairs of isogonal conjugate points (on Darboux cubic), and their corresponding perspector of the triangles AbcBcaCab, AcbBacCba:

(X3,X4): X512; (X20,X64): X647; (X40, X84): X663; .....

Angel Montesdeoca, Hyacinthos #21790