Re: SIMSON LINES Loci

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• Dear Antreas For (1.), the locus is the McCay cubic and the circumcircle For (2. Message#321670), the locus is the sextic: CiclicSum[
Message 1 of 6 , Mar 7, 2013
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
>
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