## NPC. locus.

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• Let ABC be a triangle and P a point. Which is the locus of P such that the NPC center of PBC lies on the line AP ? Ceva triangle variation: Let ABC be a
Message 1 of 11 , Apr 16, 2013
Let ABC be a triangle and P a point.

Which is the locus of P such that the NPC center of PBC
lies on the line AP ?

Ceva triangle variation:

Let ABC be a triangle, P a point and A'B'C'
the cevian triangle of P.
Which is the locus of P such that the NPC center
of PB'C' lies on the line APA' ?
(The I is on the locus)

APH
• A CIRCLE: Let ABC be a triangle, A B C the cevian triangle of I and N1, N2, N3 the NPC centers of IB C , IC A , IA B , resp. The points I, N1,N2,N3 are
Message 2 of 11 , Apr 17, 2013
A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
N2, N3
the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?

Antreas

On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@...> wrote:

> **
>
>
> Let ABC be a triangle and P a point.
>
> Which is the locus of P such that the NPC center of PBC
> lies on the line AP ?
>
> Ceva triangle variation:
>
> Let ABC be a triangle, P a point and A'B'C'
> the cevian triangle of P.
> Which is the locus of P such that the NPC center
> of PB'C' lies on the line APA' ?
> (The I is on the locus)
>
> APH
>
>
>
>
>

http://anopolis72000.blogspot.com/

[Non-text portions of this message have been removed]
• Yes, they are concyclic. The center X of the circle has trilinears: 2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : : ETC search: 2.387773069046934..,
Message 3 of 11 , Apr 17, 2013
Yes, they are concyclic.

The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :

ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..

X=Midpoint of X(I),X(J) for these (I,J):

(1,500)

X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (30,79), (58,5428), (79,500),
(140,3216),

(186,2906), (237,1896), (285,1082), (358,1882), (386,549), (500,554),
(511,1385),

(550,991), (554,1081), (1036,4022), (1081,1464), (1154,2646), (1464,1717),

(1675,4280), (1717,1836), (1836,3058), (2071,4309), (2072,3746),
(2771,3743),

(2943,4341), (3058,3649), (3108,3784), (3649,3655), (3655,3656),
(3656,3782),

(3704,3990), (3782,4654), (4654,4854), (4854,5160), (5160,5434),
(5434,5441)

Regards

_____

De: Hyacinthos@yahoogroups.com [mailto:Hyacinthos@yahoogroups.com] En nombre
de Antreas Hatzipolakis
Enviado el: Miércoles, 17 de Abril de 2013 05:34 a.m.
Para: Hyacinthos
Asunto: Re: [EMHL] NPC. locus.

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
N2, N3
the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?

Antreas

On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@...
<mailto:anopolis72%40gmail.com> > wrote:

> **
>
>
> Let ABC be a triangle and P a point.
>
> Which is the locus of P such that the NPC center of PBC
> lies on the line AP ?
>
> Ceva triangle variation:
>
> Let ABC be a triangle, P a point and A'B'C'
> the cevian triangle of P.
> Which is the locus of P such that the NPC center
> of PB'C' lies on the line APA' ?
> (The I is on the locus)
>
> APH
>
>
>
>
>

http://anopolis72000.blogspot.com/

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• Correction: X lies on line X(I),X(J) for these (I,J): (1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906), (386,549), (511,1385), (550,991),
Message 4 of 11 , Apr 17, 2013
Correction:

X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906),

(386,549), (511,1385), (550,991), (1154,2646), (2771,3743)

_____

De: Hyacinthos@yahoogroups.com [mailto:Hyacinthos@yahoogroups.com] En nombre
Enviado el: Miércoles, 17 de Abril de 2013 08:35 a.m.
Para: Hyacinthos@yahoogroups.com
Asunto: RE: [EMHL] NPC. locus.

Yes, they are concyclic.

The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :

ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..

X=Midpoint of X(I),X(J) for these (I,J):

(1,500)

X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (30,79), (58,5428), (79,500),
(140,3216),

(186,2906), (237,1896), (285,1082), (358,1882), (386,549), (500,554),
(511,1385),

(550,991), (554,1081), (1036,4022), (1081,1464), (1154,2646), (1464,1717),

(1675,4280), (1717,1836), (1836,3058), (2071,4309), (2072,3746),
(2771,3743),

(2943,4341), (3058,3649), (3108,3784), (3649,3655), (3655,3656),
(3656,3782),

(3704,3990), (3782,4654), (4654,4854), (4854,5160), (5160,5434),
(5434,5441)

Regards

_____

De: Hyacinthos@yahoogroups.com <mailto:Hyacinthos%40yahoogroups.com>
[mailto:Hyacinthos@yahoogroups.com <mailto:Hyacinthos%40yahoogroups.com> ]
En nombre
de Antreas Hatzipolakis
Enviado el: Miércoles, 17 de Abril de 2013 05:34 a.m.
Para: Hyacinthos
Asunto: Re: [EMHL] NPC. locus.

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
N2, N3
the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?

Antreas

On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@...
<mailto:anopolis72%40gmail.com>
<mailto:anopolis72%40gmail.com> > wrote:

> **
>
>
> Let ABC be a triangle and P a point.
>
> Which is the locus of P such that the NPC center of PBC
> lies on the line AP ?
>
> Ceva triangle variation:
>
> Let ABC be a triangle, P a point and A'B'C'
> the cevian triangle of P.
> Which is the locus of P such that the NPC center
> of PB'C' lies on the line APA' ?
> (The I is on the locus)
>
> APH
>
>
>
>
>

http://anopolis72000.blogspot.com/

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• ... The most interesting line is the first one (X1,X30): If I0 is the center of the circle, then the line II0 is parallel to Euler line of ABC. There are three
Message 5 of 11 , Apr 17, 2013
[APH]:
> A CIRCLE:
>
> Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
> the NPC centers of IB'C', IC'A', IA'B', resp.
>
> The points I, N1,N2,N3 are concyclic.
>
> Center of the circle?
>
> LOCUS:
>
> Let ABC be a triangle, P a point, A'B'C' the cevian triangle of
> P and N1,
> N2, N3
> the NPC centers of PB'C', PC'A', PA'B', resp.
>
> Which is the locus of P such that the points P, N1,N2,N3
> are concyclic ?

> Yes, they are concyclic.
>
> The center X of the circle has trilinears:
>
> 2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :
>
> ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..
>
> X=Midpoint of X(I),X(J) for these (I,J):
>
> (1,500)
>

> X lies on line X(I),X(J) for these (I,J):
>
> (1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906),
>
> (386,549), (511,1385), (550,991), (1154,2646), (2771,3743)

The most interesting line is the first one (X1,X30):
If I0 is the center of the circle, then the line II0
is parallel to Euler line of ABC.

There are three more centers of circles (but not triangle centrers):
I1, I2, I3 = the extraversions of I0.

ie:

Let Ia, Ib, Ic be the excenters of ABC.

Denote:

A1B1C1, A2B2C2, A3B3C3 = the cevian triangles of Ia, Ib, Ic, resp.

I1 = the center of the circle (Ia, N11, N12, N13), where
N11 = the NPC center of IaB1C1,
N12 = the NPC center of IaC1A1,
N13 = the NPC center of IaA1B1.

Similarly I2, I3.

Antreas
• A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle
Message 6 of 11 , Apr 17, 2013
A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.

Randy

--- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
>
> A CIRCLE:
>
> Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
> the NPC centers of IB'C', IC'A', IA'B', resp.
>
> The points I, N1,N2,N3 are concyclic.
>
> Center of the circle?
>
> LOCUS:
>
> Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
> N2, N3
> the NPC centers of PB'C', PC'A', PA'B', resp.
>
> Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?
>
> Antreas
>
>
> On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@...> wrote:
>
> > **
> >
> >
> > Let ABC be a triangle and P a point.
> >
> > Which is the locus of P such that the NPC center of PBC
> > lies on the line AP ?
> >
> > Ceva triangle variation:
> >
> > Let ABC be a triangle, P a point and A'B'C'
> > the cevian triangle of P.
> > Which is the locus of P such that the NPC center
> > of PB'C' lies on the line APA' ?
> > (The I is on the locus)
> >
> > APH
> >
> >
> >
> >
> >
>
> http://anopolis72000.blogspot.com/
>
>
> [Non-text portions of this message have been removed]
>
• This is the tricircular sextic Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z) where S=twice the area of ABC. (tricircular = the circular points
Message 7 of 11 , Apr 17, 2013
This is the tricircular sextic

Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z)

where S=twice the area of ABC.

(tricircular = the circular points are triple points of the curve).

Francisco Javier.

--- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
>
> A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.
>
> Randy
>
> --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
> >
> > A CIRCLE:
> >
> > Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
> > the NPC centers of IB'C', IC'A', IA'B', resp.
> >
> > The points I, N1,N2,N3 are concyclic.
> >
> > Center of the circle?
> >
> > LOCUS:
> >
> > Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
> > N2, N3
> > the NPC centers of PB'C', PC'A', PA'B', resp.
> >
> > Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?
> >
> > Antreas
> >
> >
> > On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@> wrote:
> >
> > > **
> > >
> > >
> > > Let ABC be a triangle and P a point.
> > >
> > > Which is the locus of P such that the NPC center of PBC
> > > lies on the line AP ?
> > >
> > > Ceva triangle variation:
> > >
> > > Let ABC be a triangle, P a point and A'B'C'
> > > the cevian triangle of P.
> > > Which is the locus of P such that the NPC center
> > > of PB'C' lies on the line APA' ?
> > > (The I is on the locus)
> > >
> > > APH
> > >
> > >
> > >
> > >
> > >
> >
> > http://anopolis72000.blogspot.com/
> >
> >
> > [Non-text portions of this message have been removed]
> >
>
• Francisco Javier, While Q014 mentions PU(5), it does not contain them, nor X(13) and X(14), which are on this curve. Also, X(80) which is on Q014, is not on
Message 8 of 11 , Apr 18, 2013
Francisco Javier,

While Q014 mentions PU(5), it does not contain them, nor X(13) and X(14), which are on this curve. Also, X(80) which is on Q014, is not on this curve (X(80) and the NPCs of BCX(80), CAX(80), BCX(80) are not concyclic).

By the way, the point Qi on this curve is the point which is the incenter of its anticevian triangle, and has ETC search value 1.999434154060428. Coordinates? Its isogonal conjugate Qi* is the point which is the nine-point center of its pedal triangle (ETC search 1.142779079509848). The line QiQi* passes through X(5).

Randy

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
>
> This is the tricircular sextic
>
> Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z)
>
> where S=twice the area of ABC.
>
> (tricircular = the circular points are triple points of the curve).
>
> Francisco Javier.
>
> --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
> >
> > A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.
> >
> > Randy
> >
> > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
> > >
> > > A CIRCLE:
> > >
> > > Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
> > > the NPC centers of IB'C', IC'A', IA'B', resp.
> > >
> > > The points I, N1,N2,N3 are concyclic.
> > >
> > > Center of the circle?
> > >
> > > LOCUS:
> > >
> > > Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
> > > N2, N3
> > > the NPC centers of PB'C', PC'A', PA'B', resp.
> > >
> > > Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?
> > >
> > > Antreas
> > >
> > >
> > > On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@> wrote:
> > >
> > > > **
> > > >
> > > >
> > > > Let ABC be a triangle and P a point.
> > > >
> > > > Which is the locus of P such that the NPC center of PBC
> > > > lies on the line AP ?
> > > >
> > > > Ceva triangle variation:
> > > >
> > > > Let ABC be a triangle, P a point and A'B'C'
> > > > the cevian triangle of P.
> > > > Which is the locus of P such that the NPC center
> > > > of PB'C' lies on the line APA' ?
> > > > (The I is on the locus)
> > > >
> > > > APH
> > > >
> > > >
> > > >
> > > >
> > > >
> > >
> > > http://anopolis72000.blogspot.com/
> > >
> > >
> > > [Non-text portions of this message have been removed]
> > >
> >
>
• My apologies. I misread Francisco Javier s post below. As he explained privately, he meant the difference: LOCUS = SEXTIC(Q014) - SEXTIC(S 4 S^2 x y z (x + y
Message 9 of 11 , Apr 18, 2013
My apologies. I misread Francisco Javier's post below. As he explained privately, he meant the difference:

LOCUS = SEXTIC(Q014) - SEXTIC(S 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z) )

Question: does this locus contain any ETC centers or bicentric pairs besides X(13), X(14), and PU(5)? Are the circumcircle intercepts of line X(5)X(523) triangle centers or another bicentric pair?

Randy

--- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
>
> Francisco Javier,
>
> While Q014 mentions PU(5), it does not contain them, nor X(13) and X(14), which are on this curve. Also, X(80) which is on Q014, is not on this curve (X(80) and the NPCs of BCX(80), CAX(80), BCX(80) are not concyclic).
>
> By the way, the point Qi on this curve is the point which is the incenter of its anticevian triangle, and has ETC search value 1.999434154060428. Coordinates? Its isogonal conjugate Qi* is the point which is the nine-point center of its pedal triangle (ETC search 1.142779079509848). The line QiQi* passes through X(5).
>
> Randy
>
> --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@> wrote:
> >
> > This is the tricircular sextic
> >
> > Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z)
> >
> > where S=twice the area of ABC.
> >
> > (tricircular = the circular points are triple points of the curve).
> >
> > Francisco Javier.
> >
> > --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@> wrote:
> > >
> > > A related locus: Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.
> > >
> > > Randy
> > >
> > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
> > > >
> > > > A CIRCLE:
> > > >
> > > > Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
> > > > the NPC centers of IB'C', IC'A', IA'B', resp.
> > > >
> > > > The points I, N1,N2,N3 are concyclic.
> > > >
> > > > Center of the circle?
> > > >
> > > > LOCUS:
> > > >
> > > > Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1,
> > > > N2, N3
> > > > the NPC centers of PB'C', PC'A', PA'B', resp.
> > > >
> > > > Which is the locus of P such that the points P, N1,N2,N3 are concyclic ?
> > > >
> > > > Antreas
> > > >
> > > >
> > > > On Wed, Apr 17, 2013 at 1:43 AM, Antreas <anopolis72@> wrote:
> > > >
> > > > > **
> > > > >
> > > > >
> > > > > Let ABC be a triangle and P a point.
> > > > >
> > > > > Which is the locus of P such that the NPC center of PBC
> > > > > lies on the line AP ?
> > > > >
> > > > > Ceva triangle variation:
> > > > >
> > > > > Let ABC be a triangle, P a point and A'B'C'
> > > > > the cevian triangle of P.
> > > > > Which is the locus of P such that the NPC center
> > > > > of PB'C' lies on the line APA' ?
> > > > > (The I is on the locus)
> > > > >
> > > > > APH
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > >
> > > > http://anopolis72000.blogspot.com/
> > > >
> > > >
> > > > [Non-text portions of this message have been removed]
> > > >
> > >
> >
>
• [APH Hyacinthos 21971 ]: A CIRCLE: Let ABC be a triangle, A B C the cevian
Message 10 of 11 , Feb 12
[APH Hyacinthos 21971]:

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

[

Yes, they are concyclic.

The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :

[....]

****************************

1. Point in ETC X5453

2. Stathis Koutras proved it synthetically in the FB group "Romantics of Geometry" #265

[APH  Hyacinthos 21968]
Let Ia, Ib, Ic be the excenters of ABC.

Denote:

A1B1C1, A2B2C2, A3B3C3 = the cevian triangles of Ia, Ib, Ic, resp.

I1 = the center of the circle (Ia, N11, N12, N13), where
N11 = the NPC center of IaB1C1,
N12 = the NPC center of IaC1A1,
N13 = the NPC center of IaA1B1.

Similarly I2, I3.

************

What properties have the circles (I1), (I2), (I3) ?

Are the triangles IaIbIc, I1I2I3 perspective ?
(Are IaI1, IbI2, IcI3 parallels?)

APH

• [APH Hyacinthos 21971 ]: A CIRCLE: Let ABC be a triangle, A B C the cevian
Message 11 of 11 , Feb 12

[APH Hyacinthos 21971]:

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3
the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

Center of the circle?

[

Yes, they are concyclic.

The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2- C/2)+ cos(B-C)+2 : :

[....]

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1. Point in ETC X5453

2. Stathis Koutras proved it synthetically in the FB group "Romantics of Geometry" #265

[APH  Hyacinthos 21968]
Let Ia, Ib, Ic be the excenters of ABC.

Denote:

A1B1C1, A2B2C2, A3B3C3 = the cevian triangles of Ia, Ib, Ic, resp.

I1 = the center of the circle (Ia, N11, N12, N13), where
N11 = the NPC center of IaB1C1,
N12 = the NPC center of IaC1A1,
N13 = the NPC center of IaA1B1.

Similarly I2, I3.

************

What properties have the circles (I1), (I2), (I3) ?

Are the triangles IaIbIc, I1I2I3 perspective ?

(Are IaI1, IbI2, IcI3 parallels to II0 = X1X5453  ?
If so then they concur at the infinity point of X1X5453 )

APH

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