- [Antreas]
> > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,

[Randy]

> > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.

> > (sharing the same pedal circle)

> >

> > Denote:

> >

> > sA' = the Simson line of A' wrt A"B"C"

> > sB' = the Simson line of B' wrt A"B"C"

> > sC' = the Simson line of C' wrt A"B"C"

> >

> > Which is the locus of P such the lines sA',sB',sC' are concurrent?

> >

> > O,H are on the locus.

> For (1.), the locus appears to be the McCay cubic.

3. Let P,P* be two isogonal conjugate points,

> The concurrence points are, for (P,P*):

>

> (X(1),X(1)) -> X(65)

> (X(3),X(4)) -> X(389)

> (X(1075),X*(1075)) -> non-ETC 0.540058752563797

> (X(1745),X(3362)) -> non-ETC 1.986126910721852

>

> What is the locus of the points of concurrence?

and A'B'C', A"B"C" the circumcevian triangles

of P,P*, resp.

Denote:

sA' = the Simson line of A' wrt A"B"C"

sB' = the Simson line of B' wrt A"B"C"

sC' = the Simson line of C' wrt A"B"C"

Which is the locus of P such the lines sA',sB',sC'

are concurrent?

The McCay cubic ??

Antreas - Dear Antreas

For (1.), the locus is the McCay cubic and the circumcircle

For (2. Message#321670), the locus is the sextic:

CiclicSum[ a^2(b^2-c^2)y^3z^3-b^2c^2x^3y(y-z)z) ] =0,

The vertices of ABC are singular points of multiplicity 3; the vertices of the anticomplementary triangle are inflexion points with tangents the medians. The sextic contains the centroid.

For (3.), the locus is the McCay cubic, the cubic Kjp = K024 and the circumcircle

Can be a lucus propertie of K024 not cited in Bernard Gibert Cataloge, CTC?

Best regards

Angek Montesdeoca

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:

>

> [Antreas]

> > > 1. Let ABC be a triangle, P,P* two isogonal conjugate points,

> > > and A'B'C', A"B"C" the pedal triangles of P, P*, resp.

> > > (sharing the same pedal circle)

> > >

> > > Denote:

> > >

> > > sA' = the Simson line of A' wrt A"B"C"

> > > sB' = the Simson line of B' wrt A"B"C"

> > > sC' = the Simson line of C' wrt A"B"C"

> > >

> > > Which is the locus of P such the lines sA',sB',sC' are concurrent?

> > >

> > > O,H are on the locus.

>

> [Randy]

> > For (1.), the locus appears to be the McCay cubic.

> > The concurrence points are, for (P,P*):

> >

> > (X(1),X(1)) -> X(65)

> > (X(3),X(4)) -> X(389)

> > (X(1075),X*(1075)) -> non-ETC 0.540058752563797

> > (X(1745),X(3362)) -> non-ETC 1.986126910721852

> >

> > What is the locus of the points of concurrence?

>

> 3. Let P,P* be two isogonal conjugate points,

> and A'B'C', A"B"C" the circumcevian triangles

> of P,P*, resp.

> Denote:

>

> sA' = the Simson line of A' wrt A"B"C"

> sB' = the Simson line of B' wrt A"B"C"

> sC' = the Simson line of C' wrt A"B"C"

>

> Which is the locus of P such the lines sA',sB',sC'

> are concurrent?

>

> The McCay cubic ??

>

> Antreas

>