- Dear Juan Carlos

[JCS]> Let ABC be triangle and external tangent circles

P1 is the isogonal conjugate of X(144)= anticomplement of the

> (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?

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 - 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

César Lozada

--- 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]

> >

>