## SIMSON LINES Loci

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• 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:
Message 1 of 6 , Mar 6, 2013
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.

-------------

2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
(sharing the same cevian 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?

G,H are on the locus.

APH
• Antreas, 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)
Message 2 of 6 , Mar 6, 2013
Antreas,

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?

Randy

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
>
> 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.
>
> -------------
>
> 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
> and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
> (sharing the same cevian 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?
>
> G,H are on the locus.
>
>
> APH
>
• Dear Randy I was almost sure it is an isogonal cubic, but I expected a simpler one... (the simple is for the cubic itself, not for its pivot) Thanks APH
Message 3 of 6 , Mar 6, 2013
Dear Randy

I was almost sure it is an isogonal cubic, but I expected
a "simpler" one... (the "simple" is for the cubic itself, not for its pivot)

Thanks

APH

On Wed, Mar 6, 2013 at 9:54 PM, rhutson2 <rhutson2@...> wrote:

> **
>
>
> Antreas,
>
> 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?
>
> Randy
>
>
> --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
> >
> > 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.
> >
> > -------------
> >
> > 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
> > and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
> > (sharing the same cevian 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?
> >
> > G,H are on the locus.
> >
> >
> > APH
>

[Non-text portions of this message have been removed]
• And for (2.), I expected the locus to be the Lucas cubic, which is self-cyclocevian, as the trivial case (X(7),X(7)) - X(65). However, the pair (X(8),X(189)
Message 4 of 6 , Mar 6, 2013
And for (2.), I expected the locus to be the Lucas cubic, which is self-cyclocevian, as the trivial case (X(7),X(7)) -> X(65). However, the pair (X(8),X(189) does not lie on this locus.

Randy

--- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
>
> Dear Randy
>
> I was almost sure it is an isogonal cubic, but I expected
> a "simpler" one... (the "simple" is for the cubic itself, not for its pivot)
>
> Thanks
>
> APH
>
> On Wed, Mar 6, 2013 at 9:54 PM, rhutson2 <rhutson2@...> wrote:
>
> > **
> >
> >
> > Antreas,
> >
> > 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?
> >
> > Randy
> >
> >
> > --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
> > >
> > > 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.
> > >
> > > -------------
> > >
> > > 2. Let ABC be a triangle, P,P* two cyclocevian conjugate points,
> > > and A'B'C', A"B"C" the cevian triangles of P, P*, resp.
> > > (sharing the same cevian 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?
> > >
> > > G,H are on the locus.
> > >
> > >
> > > APH
> >
>
>
> [Non-text portions of this message have been removed]
>
• [Antreas] ... [Randy] ... 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
Message 5 of 6 , Mar 7, 2013
[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
• Dear Antreas For (1.), the locus is the McCay cubic and the circumcircle For (2. Message#321670), the locus is the sextic: CiclicSum[
Message 6 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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