In this case we get the sidelines, the line at infinity, the circumcircle and a septic trhough the orthocenter, the incenter, the excenters and the feet of the altitudes

2 a^4 c^6 x^4 y^3 - 2 b^4 c^6 x^4 y^3 - 4 a^2 c^8 x^4 y^3 +

2 c^10 x^4 y^3 + 2 a^4 c^6 x^3 y^4 - 2 b^4 c^6 x^3 y^4 +

4 b^2 c^8 x^3 y^4 - 2 c^10 x^3 y^4 - a^6 b^2 c^2 x^4 y^2 z +

3 a^4 b^4 c^2 x^4 y^2 z - 3 a^2 b^6 c^2 x^4 y^2 z +

b^8 c^2 x^4 y^2 z + 3 a^4 b^2 c^4 x^4 y^2 z +

6 a^2 b^4 c^4 x^4 y^2 z - 9 b^6 c^4 x^4 y^2 z -

7 a^2 b^2 c^6 x^4 y^2 z + 3 b^4 c^6 x^4 y^2 z + 5 b^2 c^8 x^4 y^2 z -

a^8 c^2 x^3 y^3 z + 2 a^6 b^2 c^2 x^3 y^3 z -

2 a^2 b^6 c^2 x^3 y^3 z + b^8 c^2 x^3 y^3 z + 7 a^6 c^4 x^3 y^3 z +

3 a^4 b^2 c^4 x^3 y^3 z - 3 a^2 b^4 c^4 x^3 y^3 z -

7 b^6 c^4 x^3 y^3 z - 11 a^4 c^6 x^3 y^3 z + 11 b^4 c^6 x^3 y^3 z +

5 a^2 c^8 x^3 y^3 z - 5 b^2 c^8 x^3 y^3 z - a^8 c^2 x^2 y^4 z +

3 a^6 b^2 c^2 x^2 y^4 z - 3 a^4 b^4 c^2 x^2 y^4 z +

a^2 b^6 c^2 x^2 y^4 z + 9 a^6 c^4 x^2 y^4 z -

6 a^4 b^2 c^4 x^2 y^4 z - 3 a^2 b^4 c^4 x^2 y^4 z -

3 a^4 c^6 x^2 y^4 z + 7 a^2 b^2 c^6 x^2 y^4 z - 5 a^2 c^8 x^2 y^4 z +

a^6 b^2 c^2 x^4 y z^2 - 3 a^4 b^4 c^2 x^4 y z^2 +

7 a^2 b^6 c^2 x^4 y z^2 - 5 b^8 c^2 x^4 y z^2 -

3 a^4 b^2 c^4 x^4 y z^2 - 6 a^2 b^4 c^4 x^4 y z^2 -

3 b^6 c^4 x^4 y z^2 + 3 a^2 b^2 c^6 x^4 y z^2 + 9 b^4 c^6 x^4 y z^2 -

b^2 c^8 x^4 y z^2 + 12 a^4 b^4 c^2 x^3 y^2 z^2 -

12 a^2 b^6 c^2 x^3 y^2 z^2 - 12 a^4 b^2 c^4 x^3 y^2 z^2 +

12 a^2 b^2 c^6 x^3 y^2 z^2 + 12 a^6 b^2 c^2 x^2 y^3 z^2 -

12 a^4 b^4 c^2 x^2 y^3 z^2 + 12 a^2 b^4 c^4 x^2 y^3 z^2 -

12 a^2 b^2 c^6 x^2 y^3 z^2 + 5 a^8 c^2 x y^4 z^2 -

7 a^6 b^2 c^2 x y^4 z^2 + 3 a^4 b^4 c^2 x y^4 z^2 -

a^2 b^6 c^2 x y^4 z^2 + 3 a^6 c^4 x y^4 z^2 +

6 a^4 b^2 c^4 x y^4 z^2 + 3 a^2 b^4 c^4 x y^4 z^2 -

9 a^4 c^6 x y^4 z^2 - 3 a^2 b^2 c^6 x y^4 z^2 + a^2 c^8 x y^4 z^2 -

2 a^4 b^6 x^4 z^3 + 4 a^2 b^8 x^4 z^3 - 2 b^10 x^4 z^3 +

2 b^6 c^4 x^4 z^3 + a^8 b^2 x^3 y z^3 - 7 a^6 b^4 x^3 y z^3 +

11 a^4 b^6 x^3 y z^3 - 5 a^2 b^8 x^3 y z^3 -

2 a^6 b^2 c^2 x^3 y z^3 - 3 a^4 b^4 c^2 x^3 y z^3 +

5 b^8 c^2 x^3 y z^3 + 3 a^2 b^4 c^4 x^3 y z^3 -

11 b^6 c^4 x^3 y z^3 + 2 a^2 b^2 c^6 x^3 y z^3 +

7 b^4 c^6 x^3 y z^3 - b^2 c^8 x^3 y z^3 -

12 a^6 b^2 c^2 x^2 y^2 z^3 + 12 a^2 b^6 c^2 x^2 y^2 z^3 +

12 a^4 b^2 c^4 x^2 y^2 z^3 - 12 a^2 b^4 c^4 x^2 y^2 z^3 +

5 a^8 b^2 x y^3 z^3 - 11 a^6 b^4 x y^3 z^3 + 7 a^4 b^6 x y^3 z^3 -

a^2 b^8 x y^3 z^3 - 5 a^8 c^2 x y^3 z^3 + 3 a^4 b^4 c^2 x y^3 z^3 +

2 a^2 b^6 c^2 x y^3 z^3 + 11 a^6 c^4 x y^3 z^3 -

3 a^4 b^2 c^4 x y^3 z^3 - 7 a^4 c^6 x y^3 z^3 -

2 a^2 b^2 c^6 x y^3 z^3 + a^2 c^8 x y^3 z^3 + 2 a^10 y^4 z^3 -

4 a^8 b^2 y^4 z^3 + 2 a^6 b^4 y^4 z^3 - 2 a^6 c^4 y^4 z^3 -

2 a^4 b^6 x^3 z^4 + 2 b^10 x^3 z^4 - 4 b^8 c^2 x^3 z^4 +

2 b^6 c^4 x^3 z^4 + a^8 b^2 x^2 y z^4 - 9 a^6 b^4 x^2 y z^4 +

3 a^4 b^6 x^2 y z^4 + 5 a^2 b^8 x^2 y z^4 -

3 a^6 b^2 c^2 x^2 y z^4 + 6 a^4 b^4 c^2 x^2 y z^4 -

7 a^2 b^6 c^2 x^2 y z^4 + 3 a^4 b^2 c^4 x^2 y z^4 +

3 a^2 b^4 c^4 x^2 y z^4 - a^2 b^2 c^6 x^2 y z^4 -

5 a^8 b^2 x y^2 z^4 - 3 a^6 b^4 x y^2 z^4 + 9 a^4 b^6 x y^2 z^4 -

a^2 b^8 x y^2 z^4 + 7 a^6 b^2 c^2 x y^2 z^4 -

6 a^4 b^4 c^2 x y^2 z^4 + 3 a^2 b^6 c^2 x y^2 z^4 -

3 a^4 b^2 c^4 x y^2 z^4 - 3 a^2 b^4 c^4 x y^2 z^4 +

a^2 b^2 c^6 x y^2 z^4 - 2 a^10 y^3 z^4 + 2 a^6 b^4 y^3 z^4 +

4 a^8 c^2 y^3 z^4 - 2 a^6 c^4 y^3 z^4

Best regards,

Francisco Javier.

--- In Hyacinthos@yahoogroups.com, "xpolakis" <xpolakis@...> wrote:

>

> Dear Francisco

>

> Thanks!

>

> Here is a similar problem:

>

> Let ABC be a triangle, P a point and A'B'C', A"B"C"

> its pedal, circumcevian triangles, resp.

>

> Let O' be the center of the pedal circle of P, and

> A*B*C* the circumcevian of O' wrt A'B'C' in the pedal

> circle of P. (ie A*,B*,C* are the antipodes of A',B',C'

> in the pedal circle).

>

> Which is the locus of P such that A"B"C", A*B*C* are

> perspective?

>

> Antreas

>

> [APH]

> > > Let ABC be a triangle, P,P* two isogonal conjugate points

> > > and A*B*C* the pedal triangle of P*.

> > >

> > > Denote:

> > >

> > > A'B'C' := the circumcevian triangle of P wrt ABC

> > > (in the circumcircle of ABC)

> > >

> > > A"B"C" := the circumcevian triangle of P* wrt A*B*C* in

> > > the pedal circle of P*

> > >

> > > (ie A',B',C' are the second intersections of AP,BP,CP

> > > with

> > > the circumcircle of ABC, and A",B",C" are the second

> > > intersections

> > > of P*A*,PB*,P*C* with the circumcircle of A*B*C*).

> > >

> > > Which is the locus of P such that A'B'C', A"B"C" are perspective?

> > >

> > > (For P = O or H the perspector is G)

>

> [Francisco Javier]

> > Yes, I get the McCay cubic, also with the line at infinity,

> > the circumcircle and the cubic with equation

> >

> > 4 S^2 xyz + CyclicSum(a^2 yz(c^2y+b^2z)) = 0

> >

> > Surely Bernard can give this cubic a name.

>