## Re: Two points

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• Dear Juan Carlos [JCS] ... P1 is the isogonal conjugate of X(144)= anticomplement of the Gergonne point and P2 is the isogonal conjugate of X(165)= centroid of
Message 1 of 15 , Jun 30, 2004
Dear Juan Carlos
[JCS]
> Let ABC be triangle and external tangent circles
> (O1),(O2),(O3) to circumcircle (O) at J,K,L and
> (O1),(O2),(O3) touch the lines (AB,CA),(AB,BC),(BC,CA)
> at points (H,I),(F,G),(E,D) respectively.
> The tangents to circumcircle (O) at tangency points
> J,K,L perform the triangle UVW and the lines FG,DE,HI
> perform the triangle XYZ.
> Then:
> 1)ABC and UVW are perspective with center of
> perpectivity P1
> 2)ABC and XYZ are perspective with center of
> perspectivity P2
> Are P1 and P2 well-know points?
P1 is the isogonal conjugate of X(144)= anticomplement of the
Gergonne point and P2 is the isogonal conjugate of X(165)= centroid
of the excentral triangle.
I don't think that these points are in the current edition of ETC.
Friendly. Jean-Pierre
• Dear all, There are two points U1 and U2 of the form {0,v,w} (barycentric coordinates) on sideline BC satisfying b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w =
Message 2 of 15 , Sep 3, 2006
Dear all,

There are two points U1 and U2 of the form {0,v,w} (barycentric
coordinates) on sideline BC satisfying

b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w = 0

The midpoint of U1 and U2 is

U={0, 2a^2 + b^2 - c^2, 2a^2 - b^2 + c^2}

How can read geometrically U, or better U1 and U2?

Best regards,

Francisco Javier García Capitán,
http://garciacapitan.auna.com
• Dear Francisco, ... U1, U2 are the (real or not) intersections of BC and the circle through A, midpoints of AB, AC. Best regards Bernard [Non-text portions of
Message 3 of 15 , Sep 5, 2006
Dear Francisco,

> There are two points U1 and U2 of the form {0,v,w} (barycentric
> coordinates) on sideline BC satisfying
>
> b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w = 0
>
> The midpoint of U1 and U2 is
>
> U={0, 2a^2 + b^2 - c^2, 2a^2 - b^2 + c^2}
>
> How can read geometrically U, or better U1 and U2?

U1, U2 are the (real or not) intersections of BC and the circle
through A, midpoints of AB, AC.

Best regards

Bernard

[Non-text portions of this message have been removed]
• Dear Bernard and Peter, thank you for your answers, Bernard, yours is really simple. Can we follow a method to arrive it? Best regards,
Message 4 of 15 , Sep 5, 2006

Bernard, yours is really simple. Can we follow a 'method' to arrive
it?

Best regards,

--- In Hyacinthos@yahoogroups.com, Bernard Gibert <bg42@...> wrote:
>
> Dear Francisco,
>
> > There are two points U1 and U2 of the form {0,v,w} (barycentric
> > coordinates) on sideline BC satisfying
> >
> > b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w = 0
> >
> > The midpoint of U1 and U2 is
> >
> > U={0, 2a^2 + b^2 - c^2, 2a^2 - b^2 + c^2}
> >
> > How can read geometrically U, or better U1 and U2?
>
>
> U1, U2 are the (real or not) intersections of BC and the circle
> through A, midpoints of AB, AC.
>
>
>
> Best regards
>
> Bernard
>
>
>
> [Non-text portions of this message have been removed]
>
• Dear Francisco, ... I have added terms in your equation to obtain p u^2 + q u v+r u w +b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w = 0 and found p, q r to
Message 5 of 15 , Sep 5, 2006
Dear Francisco,

> Can we follow a 'method' to arrive
> it?

p u^2 + q u v+r u w +b^2 w^2 + c^2 v^2 + (b^2 + c^2 - 2*a^2)*v*w = 0

and found p, q r to obtain a circle through A.

That's all.

Best regards

Bernard

[Non-text portions of this message have been removed]
• Dear friends, Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line
Message 6 of 15 , May 8, 2012
Dear friends,

Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?

Greetings
Quim Castellsaguer
• I did a search and they weren t found.
Message 7 of 15 , May 8, 2012
I did a search and they weren't found.

--- In Hyacinthos@yahoogroups.com, "fqces" <qcastell@...> wrote:
>
> Dear friends,
>
> Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?
>
> Greetings
> Quim Castellsaguer
>
• Are these triangle **centers** in order to be included in ETC? APH ... [Non-text portions of this message have been removed]
Message 8 of 15 , May 8, 2012
Are these triangle **centers** in order to be included in ETC?

APH

On Tue, May 8, 2012 at 1:19 PM, Francisco Javier <garciacapitan@...>wrote:

> **
>
>
> I did a search and they weren't found.
>
>
> --- In Hyacinthos@yahoogroups.com, "fqces" <qcastell@...> wrote:
> >
> > Dear friends,
> >
> > Some years ago Paul Yiu posted at Hyacinthos several results about two
> interesting points, namely the intersections of the De Longchamps line with
> the circumcircle. I found these points in the lists of E. Brisse as the
> "extra-centers" E1036 and E1038, but I don't know if they have been, since
> then, included in ETC. Does anybody knows abou them?
> >
> > Greetings
> > Quim Castellsaguer
> >
>
>
>

[Non-text portions of this message have been removed]
• They are, that is, a cyclic permutation of one of the points gives the same point.
Message 9 of 15 , May 8, 2012
They are, that is, a cyclic permutation of one of the points gives the same point.

--- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
>
> Are these triangle **centers** in order to be included in ETC?
>
> APH
>
>
> On Tue, May 8, 2012 at 1:19 PM, Francisco Javier <garciacapitan@...>wrote:
>
> > **
> >
> >
> > I did a search and they weren't found.
> >
> >
> > --- In Hyacinthos@yahoogroups.com, "fqces" <qcastell@> wrote:
> > >
> > > Dear friends,
> > >
> > > Some years ago Paul Yiu posted at Hyacinthos several results about two
> > interesting points, namely the intersections of the De Longchamps line with
> > the circumcircle. I found these points in the lists of E. Brisse as the
> > "extra-centers" E1036 and E1038, but I don't know if they have been, since
> > then, included in ETC. Does anybody knows abou them?
> > >
> > > Greetings
> > > Quim Castellsaguer
> > >
> >
> >
> >
>
>
> [Non-text portions of this message have been removed]
>
• Dear Quim ... these points are real only when ABC is obtuseangle so I expect this is the end of the story... Best regards Bernard [Non-text portions of this
Message 10 of 15 , May 8, 2012
Dear Quim

> [QC] Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?

these points are real only when ABC is obtuseangle so I expect this is the end of the story...

Best regards

Bernard

[Non-text portions of this message have been removed]
• There are other centers in ETC that are real only for certain triangles, for exampleX(5000) = WALSMITH POINT, which is complex when ABC is obtuse, and the
Message 11 of 15 , May 8, 2012
There are other centers in ETC that are real only for certain triangles, for exampleX(5000) = WALSMITH POINT, which is complex when ABC is obtuse, and the bicentric pair PU(4) = 1st and 2nd GRINBERG INTERSECTIONS, the intersections of the circumcircle and nine-point circle, which are real if and only if ABC is not acute.

Do we have coordinates for these 2 points (intersection of circumcircle and de Longchamps line)?

Best regards,
Randy

>________________________________
> From: Bernard Gibert <bg42@...>
>To: Hyacinthos@yahoogroups.com
>Sent: Tuesday, May 8, 2012 5:58 AM
>Subject: Re: [EMHL] Two points
>
>

>Dear Quim
>
>> [QC] Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?
>
>these points are real only when ABC is obtuseangle so I expect this is the end of the story...
>
>Best regards
>
>Bernard
>
>[Non-text portions of this message have been removed]
>
>
>
>
>

[Non-text portions of this message have been removed]
• So it seems that the two points which are being discussed in this post are anti-complements of the 1st and 2nd Grinberg Intersections. ... -- CHANDAN [Non-text
Message 12 of 15 , May 8, 2012
So it seems that the two points which are being discussed in this post are
anti-complements of the 1st and 2nd Grinberg Intersections.

On Tue, May 8, 2012 at 7:15 PM, Randy Hutson <rhutson2@...> wrote:

> **
>
>
> There are other centers in ETC that are real only for certain triangles,
> for exampleX(5000) = WALSMITH POINT, which is complex when ABC is obtuse,
> and the bicentric pair PU(4) = 1st and 2nd GRINBERG INTERSECTIONS, the
> intersections of the circumcircle and nine-point circle, which are real if
> and only if ABC is not acute.
>
> Do we have coordinates for these 2 points (intersection of circumcircle
> and de Longchamps line)?
>
> Best regards,
> Randy
>
> >________________________________
> > From: Bernard Gibert <bg42@...>
> >To: Hyacinthos@yahoogroups.com
> >Sent: Tuesday, May 8, 2012 5:58 AM
> >Subject: Re: [EMHL] Two points
>
> >
> >
> >
> >Dear Quim
> >
> >> [QC] Some years ago Paul Yiu posted at Hyacinthos several results about
> two interesting points, namely the intersections of the De Longchamps line
> with the circumcircle. I found these points in the lists of E. Brisse as
> the "extra-centers" E1036 and E1038, but I don't know if they have been,
> since then, included in ETC. Does anybody knows abou them?
> >
> >these points are real only when ABC is obtuseangle so I expect this is
> the end of the story...
> >
> >Best regards
> >
> >Bernard
> >
> >[Non-text portions of this message have been removed]
> >
> >
> >
> >
> >
>
> [Non-text portions of this message have been removed]
>
>
>

--
CHANDAN

[Non-text portions of this message have been removed]
• A (non-symmetric) coordinates are {2 a^2 b^2 c^2, -c^2(a^4 + b^4 - c^4 + R), b^2(-a^4+b^4-c^4 + R )}, {2 a^2 b^2 c^2, c^2 (-a^4 - b^4 + c^4 + R), -b^2
Message 13 of 15 , May 8, 2012
A (non-symmetric) coordinates are

{2 a^2 b^2 c^2, -c^2(a^4 + b^4 - c^4 + R), b^2(-a^4+b^4-c^4 + R )},
{2 a^2 b^2 c^2, c^2 (-a^4 - b^4 + c^4 + R), -b^2 (a^4-b^4+c^4 + R}

where R is the square root of

(a^2-b^2-c^2)(a^2+b^2-c^2)(a^2-b^2+c^2)(a^2+b^2+c^2)

The midpoint of the two points is X858.

Best regards,

Francisco Javier.

--- In Hyacinthos@yahoogroups.com, Randy Hutson <rhutson2@...> wrote:
>
> There are other centers in ETC that are real only for certain triangles, for exampleX(5000) = WALSMITH POINT, which is complex when ABC is obtuse, and the bicentric pair PU(4) = 1st and 2nd GRINBERG INTERSECTIONS, the intersections of the circumcircle and nine-point circle, which are real if and only if ABC is not acute.
>
> Do we have coordinates for these 2 points (intersection of circumcircle and de Longchamps line)?
>
> Best regards,
> Randy
>
>
>
>
> >________________________________
> > From: Bernard Gibert <bg42@...>
> >To: Hyacinthos@yahoogroups.com
> >Sent: Tuesday, May 8, 2012 5:58 AM
> >Subject: Re: [EMHL] Two points
> >
> >
> >Â
> >Dear Quim
> >
> >> [QC] Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?
> >
> >these points are real only when ABC is obtuseangle so I expect this is the end of the story...
> >
> >Best regards
> >
> >Bernard
> >
> >[Non-text portions of this message have been removed]
> >
> >
> >
> >
> >
>
> [Non-text portions of this message have been removed]
>
• Barycentrics: u = f(SA, SB, SC) and f(SA, SB, SC) = 1/((SB+SC)*((SB+SC)*(SA^2-SB*SC)+(SB-SC)*(SA*(SB-SC)+`&+-`(2*sqrt(-SA*SB*SC*(SA+SB+SC)))))) There is + -
Message 14 of 15 , May 9, 2012
Barycentrics:

u = f(SA, SB, SC) and f(SA, SB, SC) = 1/((SB+SC)*((SB+SC)*(SA^2-SB*SC)+(SB-SC)*(SA*(SB-SC)+`&+-`(2*sqrt(-SA*SB*SC*(SA+SB+SC))))))

There is + - before the 2*sqrt(....)
Best regards

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
>
>
> A (non-symmetric) coordinates are
>
> {2 a^2 b^2 c^2, -c^2(a^4 + b^4 - c^4 + R), b^2(-a^4+b^4-c^4 + R )},
> {2 a^2 b^2 c^2, c^2 (-a^4 - b^4 + c^4 + R), -b^2 (a^4-b^4+c^4 + R}
>
> where R is the square root of
>
> (a^2-b^2-c^2)(a^2+b^2-c^2)(a^2-b^2+c^2)(a^2+b^2+c^2)
>
> The midpoint of the two points is X858.
>
> Best regards,
>
> Francisco Javier.
>
>
>
>
> --- In Hyacinthos@yahoogroups.com, Randy Hutson <rhutson2@> wrote:
> >
> > There are other centers in ETC that are real only for certain triangles, for exampleX(5000) = WALSMITH POINT, which is complex when ABC is obtuse, and the bicentric pair PU(4) = 1st and 2nd GRINBERG INTERSECTIONS, the intersections of the circumcircle and nine-point circle, which are real if and only if ABC is not acute.
> >
> > Do we have coordinates for these 2 points (intersection of circumcircle and de Longchamps line)?
> >
> > Best regards,
> > Randy
> >
> >
> >
> >
> > >________________________________
> > > From: Bernard Gibert <bg42@>
> > >To: Hyacinthos@yahoogroups.com
> > >Sent: Tuesday, May 8, 2012 5:58 AM
> > >Subject: Re: [EMHL] Two points
> > >
> > >
> > >Â
> > >Dear Quim
> > >
> > >> [QC] Some years ago Paul Yiu posted at Hyacinthos several results about two interesting points, namely the intersections of the De Longchamps line with the circumcircle. I found these points in the lists of E. Brisse as the "extra-centers" E1036 and E1038, but I don't know if they have been, since then, included in ETC. Does anybody knows abou them?
> > >
> > >these points are real only when ABC is obtuseangle so I expect this is the end of the story...
> > >
> > >Best regards
> > >
> > >Bernard
> > >
> > >[Non-text portions of this message have been removed]
> > >
> > >
> > >
> > >
> > >
> >
> > [Non-text portions of this message have been removed]
> >
>
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