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Re: Two points

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  • jpehrmfr
    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
    • garciacapitan
      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
      • Bernard Gibert
        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]
        • garciacapitan
          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
            Dear Bernard and Peter, thank you for your answers,

            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]
            >
          • Bernard Gibert
            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?

              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 obtain a circle through A.

              That's all.



              Best regards

              Bernard



              [Non-text portions of this message have been removed]
            • fqces
              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
              • Francisco Javier
                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
                  >
                • Antreas Hatzipolakis
                  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]
                  • Francisco Javier
                    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]
                      >
                    • Bernard Gibert
                      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]
                      • Randy Hutson
                        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]
                        • Chandan Banerjee
                          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]
                          • Francisco Javier
                            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
                              > >
                              > >
                              > > 
                              > >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]
                              >
                            • CESAR
                              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
                                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
                                > > >
                                > > >
                                > > > 
                                > > >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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