MCCAY CUBIC. Locus Problems (Parallels to cevians)

Let ABC be a triangle, P a point and A1B1C1, A2B2C2 the pedal triangles of P and its isogonal conjugate P*.

The parallels through A1,B1,C1 to AP,BP,CP bound a triangle A3B3C3.

The above parallels intersect the pedal circle of A1B1C1 (and A2B2C2)at A4,B4,C4.

Which is the locus of P such that:

1. A2B2C2, A3B3C3

2. A2B2C2, A4B4C4

3. A3B3C3, A4B4C4

4. A1B1C1, A3B3C3

5. ABC, A3B3C3

6. ABC, A4B4C4

are perspective?

Antreas P. Hatzipolakis, Hyacinthos #20745

1. A2B2C2, A3B3C3: line at infinity + circumcircle + McCay cubic.

2. A2B2C2, A4B4C4: The whole plane. The perspector R (1) is collinear with P and P* when P is on the McCay cubic.

3. A3B3C3, A4B4C4: line at infinity + McCay cubic + sextic

4. A1B1C1, A3B3C3: Same as previous.

5. ABC, A3B3C3: sides of anticomplementary triangle + circumcircle + McCay cubic

6. ABC, A4B4C4: line at infinity + circumcircle + McCay cubic.

(1) For P=(x:y:z) the perspector R is the point

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

Francisco Javier, Hyacinthos #20745