PERSPECTIVITY [Antipedal Triangles of P,P*] --1--

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

Which is the locus of P such that the triangles:

1. A"B"C", Triangle bounded by (AA',BB',CC')

2. A'B'C', Triangle bounded by (AA",BB",CC")

3. ABC, Triangle A*B*C* bounded by (A'A",B'B",C'C") are perspective ?

Antreas P. Hatzipolakis, Hyacinthos #21782

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The loci in 1. and 2 are the same:

line at infinity + circumcircle + K004[=Darboux cubic] + three cubics, each one relative to one of the vertices.

The cubic relative to A has equation

2 b^2 c^2 x^2 y + a^2 c^2 x y^2 + b^2 c^2 x y^2 + c^4 x y^2 + 2 b^2 c^2 x^2 z + 4 b^2 c^2 x y z + 2 a^2 c^2 y^2 z + a^2 b^2 x z^2 + b^4 x z^2 + b^2 c^2 x z^2 + 2 a^2 b^2 y z^2 =0.

It intersects the line at infinity at the infinite points of the bisectors of angle A and at the infinite point of the A-altitude.

It is its isogonal conjugate, then it intersects the circumcircle at the intersections of the circumcircle and the bisectors of angle A and at the antipode of A.

With respect to 3., the triangle bounded by (A'A", B' B ",C'C") is ALWAYS perspective with ABC.

If P=(x:y:z), then the perspector S is complicated: the coordinates of its isotomic conjugate, that is infinite when P lies on K004 [= Darboux cubic], are:

{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, -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, -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 #21782