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Re: [EMHL] Re: PEDAL, ANTIPEDAL CIRCLES

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  • Antreas Hatzipolakis
    ...... and to any triangle center lying on the locus corresponds another triangle center, namely the point of the contact of the two circles. I have the
    Message 1 of 17 , Mar 14, 2013
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      ...... and to any triangle center lying on the locus
      corresponds another triangle center, namely the point
      of the contact of the two circles.

      I have the impression that the locus contains of no simple centers and
      probably no one
      in ETC!

      aph


      On Fri, Mar 15, 2013 at 12:07 AM, Francisco Javier
      <garciacapitan@...>wrote:

      > **
      >
      >
      > I get the cubic
      >
      > S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0.
      >
      > It is shaped like K016, but not the same at all.
      >
      >
      > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...>
      > wrote:
      > >
      > > Are there real points P such that the pedal and antipedal circles of P
      > > are tangent?
      > >
      > > APH
      > >
      >
      >
      >

      --
      http://anopolis72000.blogspot.com/


      [Non-text portions of this message have been removed]
    • Angel
      ... In private mail Fancisco Javier has corroborated what is exposed here: Are there real points P such that the pedal and antipedal circles of P are tangent?
      Message 2 of 17 , Mar 15, 2013
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        --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
        >
        > Are there real points P such that the pedal and antipedal circles of P
        > are tangent?
        >
        > APH
        >


        In private mail Fancisco Javier has corroborated what is exposed here:

        Are there real points P such that the pedal and antipedal circles of P are tangent?


        Yes, if P is on an octic or on the cubic K191="circumcircle pedal cubic, nK(X6, X6,?)".

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

        If P is on the cubic K191, then the point of the contact of the two circles are on the circumcircle.

        (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
        ------

        K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,

        or equivalently

        K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.

        S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))



        Barycentric equation of the octic is too complicated to be written here.
        (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)

        Angel M.
      • Antreas Hatzipolakis
        Dear Angel, Francisco Thanks!!! Now we have two new geometrical properties of K191
        Message 3 of 17 , Mar 15, 2013
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          Dear Angel, Francisco

          Thanks!!!

          Now we have two new geometrical properties of K191

          http://anthrakitis.blogspot.gr/2013/03/mccay-cubic-circumcevian-triangle.html
          http://anthrakitis.blogspot.gr/2013/03/k191-pedal-and-antipedal-circles-tangent.html

          Bernard may include them in the cubic's page:
          http://bernard.gibert.pagesperso-orange.fr/Exemples/k191.html

          APH

          On Fri, Mar 15, 2013 at 8:47 PM, Angel <amontes1949@...> wrote:

          > **
          >
          >
          >
          >
          > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...>
          > wrote:
          > >
          > > Are there real points P such that the pedal and antipedal circles of P
          > > are tangent?
          > >
          > > APH
          > >
          >
          > In private mail Fancisco Javier has corroborated what is exposed here:
          >
          >
          > Are there real points P such that the pedal and antipedal circles of P are
          > tangent?
          >
          > Yes, if P is on an octic or on the cubic K191="circumcircle pedal cubic,
          > nK(X6, X6,?)".
          >
          > ---------------
          >
          > If P is on the cubic K191, then the point of the contact of the two
          > circles are on the circumcircle.
          >
          > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
          > ------
          >
          > K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,
          >
          > or equivalently
          >
          > K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
          >
          > S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))
          >
          > Barycentric equation of the octic is too complicated to be written here.
          > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)
          >
          > Angel M.
          >
          > _
          >


          [Non-text portions of this message have been removed]
        • Bernard Gibert
          Dear Antreas, Angel, Francisco and all Hyacintists, ... It will be my pleasure to do so. Thanks to you all. By the way, I encourage (and thank) each member of
          Message 4 of 17 , Mar 15, 2013
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            Dear Antreas, Angel, Francisco and all Hyacintists,

            > Now we have two new geometrical properties of K191
            >
            > http://anthrakitis.blogspot.gr/2013/03/mccay-cubic-circumcevian-triangle.html
            > http://anthrakitis.blogspot.gr/2013/03/k191-pedal-and-antipedal-circles-tangent.html
            >
            > Bernard may include them in the cubic's page:
            > http://bernard.gibert.pagesperso-orange.fr/Exemples/k191.html

            It will be my pleasure to do so. Thanks to you all.

            By the way, I encourage (and thank) each member of Hyacinthos to contribute to CTC since it has become too heavy to me to verify and check all what's happening on the list.

            I think Clark has encountered the same problem with ETC and feel what I'm saying.

            So, when you find some interesting geometric properties of a catalogued cubic, any reference I could have forgotten, and of course any new interesting cubic, please do write a short note with any relevant references and send it to me.

            I'll be very happy to add it to CTC.

            With my warmest regards

            Bernard




            [Non-text portions of this message have been removed]
          • Francisco Javier
            As Bernard Gibert pointed out to us, we were wrong, this is a different cubic, not K191.
            Message 5 of 17 , Mar 16, 2013
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              As Bernard Gibert pointed out to us, we were wrong, this is a different cubic, not K191.



              --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@...> wrote:
              >
              >
              >
              > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
              > >
              > > Are there real points P such that the pedal and antipedal circles of P
              > > are tangent?
              > >
              > > APH
              > >
              >
              >
              > In private mail Fancisco Javier has corroborated what is exposed here:
              >
              > Are there real points P such that the pedal and antipedal circles of P are tangent?
              >
              >
              > Yes, if P is on an octic or on the cubic K191="circumcircle pedal cubic, nK(X6, X6,?)".
              >
              > ---------------
              >
              > If P is on the cubic K191, then the point of the contact of the two circles are on the circumcircle.
              >
              > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
              > ------
              >
              > K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,
              >
              > or equivalently
              >
              > K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
              >
              > S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))
              >
              >
              >
              > Barycentric equation of the octic is too complicated to be written here.
              > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)
              >
              > Angel M.
              >
            • Bernard Gibert
              Dear friends, ... indeed, although it is a cubic of the pencil generated by K024 and K191, since it is the locus of P whose pedal circle is orthogonal to the
              Message 6 of 17 , Mar 16, 2013
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                Dear friends,

                > As Bernard Gibert pointed out to us, we were wrong, this is a different cubic, not K191.

                indeed, although it is a cubic of the pencil generated by K024 and K191,

                since it is the locus of P whose pedal circle is orthogonal to the orthoptic circle of the Steiner in-ellipse.

                best regards

                Bernard

                [Non-text portions of this message have been removed]
              • Antreas
                Is K191 in http://tech.groups.yahoo.com/group/Hyacinthos/message/21743 ? APH
                Message 7 of 17 , Mar 16, 2013
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                  Is K191 in

                  http://tech.groups.yahoo.com/group/Hyacinthos/message/21743

                  ?

                  APH

                  --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
                  >
                  > As Bernard Gibert pointed out to us, we were wrong, this is a >different cubic, not K191.
                • Francisco Javier
                  No, it is the new cubic S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0 that was wrongly identified as K191.
                  Message 8 of 17 , Mar 16, 2013
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                    No, it is "the new cubic"

                    S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0

                    that was wrongly identified as K191.

                    --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
                    >
                    > Is K191 in
                    >
                    > http://tech.groups.yahoo.com/group/Hyacinthos/message/21743
                    >
                    > ?
                    >
                    > APH
                    >
                    > --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@> wrote:
                    > >
                    > > As Bernard Gibert pointed out to us, we were wrong, this is a >different cubic, not K191.
                    >
                  • Bernard Gibert
                    Dear friends ... this will be K634, the orthoptic pedal cubic. I will modify the pages K003, K191, K192 and I ask you to carefully check if this corresponds to
                    Message 9 of 17 , Mar 16, 2013
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                      Dear friends

                      > No, it is "the new cubic"
                      >
                      > S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0

                      this will be K634, the orthoptic pedal cubic.

                      I will modify the pages K003, K191, K192 and I ask you to carefully check if this corresponds to your findings.

                      many thanks

                      Bernard

                      [Non-text portions of this message have been removed]
                    • Angel
                      ... It was my mistake and I have confused to Francisco Javier. Are there real points P such that the pedal and antipedal circles of P are tangent?
                      Message 10 of 17 , Mar 16, 2013
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                        --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
                        >
                        > As Bernard Gibert pointed out to us, we were wrong, this is a different cubic, not K191.
                        >
                        >
                        >
                        > --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@> wrote:
                        > >
                        > >
                        > >
                        > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@> wrote:
                        > > >
                        > > > Are there real points P such that the pedal and antipedal circles of P
                        > > > are tangent?
                        > > >
                        > > > APH
                        > > >
                        > >
                        > >
                        > > In private mail Fancisco Javier has corroborated what is exposed here:
                        > >
                        > > Are there real points P such that the pedal and antipedal circles of P are tangent?
                        > >
                        > >
                        > > Yes, if P is on an octic or on the cubic K191="circumcircle pedal cubic, nK(X6, X6,?)".
                        > >
                        > > ---------------
                        > >
                        > > If P is on the cubic K191, then the point of the contact of the two circles are on the circumcircle.
                        > >
                        > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
                        > > ------
                        > >
                        > > K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,
                        > >
                        > > or equivalently
                        > >
                        > > K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
                        > >
                        > > S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))
                        > >
                        > >
                        > >
                        > > Barycentric equation of the octic is too complicated to be written here.
                        > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)
                        > >
                        > > Angel M.
                        > >
                        >


                        It was my mistake and I have confused to Francisco Javier.



                        Are there real points P such that the pedal and antipedal circles of P are tangent? (http://tech.groups.yahoo.com/group/Hyacinthos/message/21746)


                        The correct answer is:

                        Yes, if P is on an octic or on the cubic K634 (To appear, http://tech.groups.yahoo.com/group/Hyacinthos/message/21765).


                        K634: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.

                        If P is on the cubic K634, then the point of the contact of the two circles are on the circumcircle.

                        (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749Acorrected.eps)


                        Equation of the octic:

                        16 b^2 c^6 x^5 y^3+16 a^2 c^6 x^4 y^4+16 b^2 c^6 x^4 y^4-16 c^8 x^4
                        y^4+16 a^2 c^6 x^3 y^5-4 a^4 b^2 c^2 x^5 y^2 z+8 a^2 b^4 c^2 x^5 y^2
                        z-4 b^6 c^2 x^5 y^2 z-8 a^2 b^2 c^4 x^5 y^2 z+40 b^4 c^4 x^5 y^2 z+12
                        b^2 c^6 x^5 y^2 z-4 a^6 c^2 x^4 y^3 z+12 a^2 b^4 c^2 x^4 y^3 z-8 b^6
                        c^2 x^4 y^3 z-8 a^4 c^4 x^4 y^3 z+72 a^2 b^2 c^4 x^4 y^3 z+32 b^4 c^4
                        x^4 y^3 z+28 a^2 c^6 x^4 y^3 z-8 b^2 c^6 x^4 y^3 z-16 c^8 x^4 y^3 z-8
                        a^6 c^2 x^3 y^4 z+12 a^4 b^2 c^2 x^3 y^4 z-4 b^6 c^2 x^3 y^4 z+32 a^4
                        c^4 x^3 y^4 z+72 a^2 b^2 c^4 x^3 y^4 z-8 b^4 c^4 x^3 y^4 z-8 a^2 c^6
                        x^3 y^4 z+28 b^2 c^6 x^3 y^4 z-16 c^8 x^3 y^4 z-4 a^6 c^2 x^2 y^5 z+8
                        a^4 b^2 c^2 x^2 y^5 z-4 a^2 b^4 c^2 x^2 y^5 z+40 a^4 c^4 x^2 y^5 z-8
                        a^2 b^2 c^4 x^2 y^5 z+12 a^2 c^6 x^2 y^5 z-4 a^4 b^2 c^2 x^5 y z^2-8
                        a^2 b^4 c^2 x^5 y z^2+12 b^6 c^2 x^5 y z^2+8 a^2 b^2 c^4 x^5 y z^2+40
                        b^4 c^4 x^5 y z^2-4 b^2 c^6 x^5 y z^2+a^8 x^4 y^2 z^2-4 a^6 b^2 x^4
                        y^2 z^2+6 a^4 b^4 x^4 y^2 z^2-4 a^2 b^6 x^4 y^2 z^2+b^8 x^4 y^2 z^2-4
                        a^6 c^2 x^4 y^2 z^2-60 a^4 b^2 c^2 x^4 y^2 z^2+100 a^2 b^4 c^2 x^4 y^2
                        z^2-36 b^6 c^2 x^4 y^2 z^2+6 a^4 c^4 x^4 y^2 z^2+100 a^2 b^2 c^4 x^4
                        y^2 z^2+70 b^4 c^4 x^4 y^2 z^2-4 a^2 c^6 x^4 y^2 z^2-36 b^2 c^6 x^4
                        y^2 z^2+c^8 x^4 y^2 z^2+2 a^8 x^3 y^3 z^2-8 a^6 b^2 x^3 y^3 z^2+12 a^4
                        b^4 x^3 y^3 z^2-8 a^2 b^6 x^3 y^3 z^2+2 b^8 x^3 y^3 z^2-52 a^6 c^2 x^3
                        y^3 z^2+52 a^4 b^2 c^2 x^3 y^3 z^2+52 a^2 b^4 c^2 x^3 y^3 z^2-52 b^6
                        c^2 x^3 y^3 z^2+84 a^4 c^4 x^3 y^3 z^2+88 a^2 b^2 c^4 x^3 y^3 z^2+84
                        b^4 c^4 x^3 y^3 z^2-20 a^2 c^6 x^3 y^3 z^2-20 b^2 c^6 x^3 y^3 z^2-14
                        c^8 x^3 y^3 z^2+a^8 x^2 y^4 z^2-4 a^6 b^2 x^2 y^4 z^2+6 a^4 b^4 x^2
                        y^4 z^2-4 a^2 b^6 x^2 y^4 z^2+b^8 x^2 y^4 z^2-36 a^6 c^2 x^2 y^4
                        z^2+100 a^4 b^2 c^2 x^2 y^4 z^2-60 a^2 b^4 c^2 x^2 y^4 z^2-4 b^6 c^2
                        x^2 y^4 z^2+70 a^4 c^4 x^2 y^4 z^2+100 a^2 b^2 c^4 x^2 y^4 z^2+6 b^4
                        c^4 x^2 y^4 z^2-36 a^2 c^6 x^2 y^4 z^2-4 b^2 c^6 x^2 y^4 z^2+c^8 x^2
                        y^4 z^2+12 a^6 c^2 x y^5 z^2-8 a^4 b^2 c^2 x y^5 z^2-4 a^2 b^4 c^2 x
                        y^5 z^2+40 a^4 c^4 x y^5 z^2+8 a^2 b^2 c^4 x y^5 z^2-4 a^2 c^6 x y^5
                        z^2+16 b^6 c^2 x^5 z^3-4 a^6 b^2 x^4 y z^3-8 a^4 b^4 x^4 y z^3+28 a^2
                        b^6 x^4 y z^3-16 b^8 x^4 y z^3+72 a^2 b^4 c^2 x^4 y z^3-8 b^6 c^2 x^4
                        y z^3+12 a^2 b^2 c^4 x^4 y z^3+32 b^4 c^4 x^4 y z^3-8 b^2 c^6 x^4 y
                        z^3+2 a^8 x^3 y^2 z^3-52 a^6 b^2 x^3 y^2 z^3+84 a^4 b^4 x^3 y^2 z^3-20
                        a^2 b^6 x^3 y^2 z^3-14 b^8 x^3 y^2 z^3-8 a^6 c^2 x^3 y^2 z^3+52 a^4
                        b^2 c^2 x^3 y^2 z^3+88 a^2 b^4 c^2 x^3 y^2 z^3-20 b^6 c^2 x^3 y^2
                        z^3+12 a^4 c^4 x^3 y^2 z^3+52 a^2 b^2 c^4 x^3 y^2 z^3+84 b^4 c^4 x^3
                        y^2 z^3-8 a^2 c^6 x^3 y^2 z^3-52 b^2 c^6 x^3 y^2 z^3+2 c^8 x^3 y^2
                        z^3-14 a^8 x^2 y^3 z^3-20 a^6 b^2 x^2 y^3 z^3+84 a^4 b^4 x^2 y^3
                        z^3-52 a^2 b^6 x^2 y^3 z^3+2 b^8 x^2 y^3 z^3-20 a^6 c^2 x^2 y^3 z^3+88
                        a^4 b^2 c^2 x^2 y^3 z^3+52 a^2 b^4 c^2 x^2 y^3 z^3-8 b^6 c^2 x^2 y^3
                        z^3+84 a^4 c^4 x^2 y^3 z^3+52 a^2 b^2 c^4 x^2 y^3 z^3+12 b^4 c^4 x^2
                        y^3 z^3-52 a^2 c^6 x^2 y^3 z^3-8 b^2 c^6 x^2 y^3 z^3+2 c^8 x^2 y^3
                        z^3-16 a^8 x y^4 z^3+28 a^6 b^2 x y^4 z^3-8 a^4 b^4 x y^4 z^3-4 a^2
                        b^6 x y^4 z^3-8 a^6 c^2 x y^4 z^3+72 a^4 b^2 c^2 x y^4 z^3+32 a^4 c^4
                        x y^4 z^3+12 a^2 b^2 c^4 x y^4 z^3-8 a^2 c^6 x y^4 z^3+16 a^6 c^2 y^5
                        z^3+16 a^2 b^6 x^4 z^4-16 b^8 x^4 z^4+16 b^6 c^2 x^4 z^4-8 a^6 b^2 x^3
                        y z^4+32 a^4 b^4 x^3 y z^4-8 a^2 b^6 x^3 y z^4-16 b^8 x^3 y z^4+12 a^4
                        b^2 c^2 x^3 y z^4+72 a^2 b^4 c^2 x^3 y z^4+28 b^6 c^2 x^3 y z^4-8 b^4
                        c^4 x^3 y z^4-4 b^2 c^6 x^3 y z^4+a^8 x^2 y^2 z^4-36 a^6 b^2 x^2 y^2
                        z^4+70 a^4 b^4 x^2 y^2 z^4-36 a^2 b^6 x^2 y^2 z^4+b^8 x^2 y^2 z^4-4
                        a^6 c^2 x^2 y^2 z^4+100 a^4 b^2 c^2 x^2 y^2 z^4+100 a^2 b^4 c^2 x^2
                        y^2 z^4-4 b^6 c^2 x^2 y^2 z^4+6 a^4 c^4 x^2 y^2 z^4-60 a^2 b^2 c^4 x^2
                        y^2 z^4+6 b^4 c^4 x^2 y^2 z^4-4 a^2 c^6 x^2 y^2 z^4-4 b^2 c^6 x^2 y^2
                        z^4+c^8 x^2 y^2 z^4-16 a^8 x y^3 z^4-8 a^6 b^2 x y^3 z^4+32 a^4 b^4 x
                        y^3 z^4-8 a^2 b^6 x y^3 z^4+28 a^6 c^2 x y^3 z^4+72 a^4 b^2 c^2 x y^3
                        z^4+12 a^2 b^4 c^2 x y^3 z^4-8 a^4 c^4 x y^3 z^4-4 a^2 c^6 x y^3
                        z^4-16 a^8 y^4 z^4+16 a^6 b^2 y^4 z^4+16 a^6 c^2 y^4 z^4+16 a^2 b^6
                        x^3 z^5-4 a^6 b^2 x^2 y z^5+40 a^4 b^4 x^2 y z^5+12 a^2 b^6 x^2 y
                        z^5+8 a^4 b^2 c^2 x^2 y z^5-8 a^2 b^4 c^2 x^2 y z^5-4 a^2 b^2 c^4 x^2
                        y z^5+12 a^6 b^2 x y^2 z^5+40 a^4 b^4 x y^2 z^5-4 a^2 b^6 x y^2 z^5-8
                        a^4 b^2 c^2 x y^2 z^5+8 a^2 b^4 c^2 x y^2 z^5-4 a^2 b^2 c^4 x y^2
                        z^5+16 a^6 b^2 y^3 z^5=0


                        Best regards
                        Angel Montesdeoca
                      • Antreas Hatzipolakis
                        Dear Angel, Bernard, Francisco I think that, since for P lying on K634 the point of contact of the antipedal and pedal circles of P is lying on the
                        Message 11 of 17 , Mar 16, 2013
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                          Dear Angel, Bernard, Francisco

                          I think that, since for P lying on K634 the point of contact of the
                          antipedal and pedal circles of P
                          is lying on the circumcircle, the K634 is part of a more general locus:

                          Which is the locus of P such that the pedal circle of P, the antipedal
                          circle of P and the circumcircle,
                          are concurrent?

                          The locus should be K634 (for pedal and antipedal circles tangent) +
                          ???????????????????

                          Antreas


                          On Sat, Mar 16, 2013 at 3:18 PM, Angel <amontes1949@...> wrote:

                          > **
                          >
                          >
                          >
                          >
                          > --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...>
                          > wrote:
                          > >
                          > > As Bernard Gibert pointed out to us, we were wrong, this is a different
                          > cubic, not K191.
                          > >
                          > >
                          > >
                          > > --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@> wrote:
                          > > >
                          > > >
                          > > >
                          > > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@>
                          > wrote:
                          > > > >
                          > > > > Are there real points P such that the pedal and antipedal circles of
                          > P
                          > > > > are tangent?
                          > > > >
                          > > > > APH
                          > > > >
                          > > >
                          > > >
                          > > > In private mail Fancisco Javier has corroborated what is exposed here:
                          > > >
                          > > > Are there real points P such that the pedal and antipedal circles of P
                          > are tangent?
                          > > >
                          > > >
                          > > > Yes, if P is on an octic or on the cubic K191="circumcircle pedal
                          > cubic, nK(X6, X6,?)".
                          > > >
                          > > > ---------------
                          > > >
                          > > > If P is on the cubic K191, then the point of the contact of the two
                          > circles are on the circumcircle.
                          > > >
                          > > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
                          > > > ------
                          > > >
                          > > > K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,
                          > > >
                          > > > or equivalently
                          > > >
                          > > > K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
                          > > >
                          > > > S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))
                          > > >
                          > > >
                          > > >
                          > > > Barycentric equation of the octic is too complicated to be written
                          > here.
                          > > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)
                          > > >
                          > > > Angel M.
                          > > >
                          > >
                          >
                          > It was my mistake and I have confused to Francisco Javier.
                          >
                          > Are there real points P such that the pedal and antipedal circles of P are
                          > tangent? (http://tech.groups.yahoo.com/group/Hyacinthos/message/21746)
                          >
                          > The correct answer is:
                          >
                          > Yes, if P is on an octic or on the cubic K634 (To appear,
                          > http://tech.groups.yahoo.com/group/Hyacinthos/message/21765).
                          >
                          > K634: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
                          >
                          > If P is on the cubic K634, then the point of the contact of the two
                          > circles are on the circumcircle.
                          >
                          > (
                          > http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749Acorrected.eps
                          > )
                          >
                          > Equation of the octic:
                          >
                          > 16 b^2 c^6 x^5 y^3+16 a^2 c^6 x^4 y^4+16 b^2 c^6 x^4 y^4-16 c^8 x^4
                          > y^4+16 a^2 c^6 x^3 y^5-4 a^4 b^2 c^2 x^5 y^2 z+8 a^2 b^4 c^2 x^5 y^2
                          > z-4 b^6 c^2 x^5 y^2 z-8 a^2 b^2 c^4 x^5 y^2 z+40 b^4 c^4 x^5 y^2 z+12
                          > b^2 c^6 x^5 y^2 z-4 a^6 c^2 x^4 y^3 z+12 a^2 b^4 c^2 x^4 y^3 z-8 b^6
                          > c^2 x^4 y^3 z-8 a^4 c^4 x^4 y^3 z+72 a^2 b^2 c^4 x^4 y^3 z+32 b^4 c^4
                          > x^4 y^3 z+28 a^2 c^6 x^4 y^3 z-8 b^2 c^6 x^4 y^3 z-16 c^8 x^4 y^3 z-8
                          > a^6 c^2 x^3 y^4 z+12 a^4 b^2 c^2 x^3 y^4 z-4 b^6 c^2 x^3 y^4 z+32 a^4
                          > c^4 x^3 y^4 z+72 a^2 b^2 c^4 x^3 y^4 z-8 b^4 c^4 x^3 y^4 z-8 a^2 c^6
                          > x^3 y^4 z+28 b^2 c^6 x^3 y^4 z-16 c^8 x^3 y^4 z-4 a^6 c^2 x^2 y^5 z+8
                          > a^4 b^2 c^2 x^2 y^5 z-4 a^2 b^4 c^2 x^2 y^5 z+40 a^4 c^4 x^2 y^5 z-8
                          > a^2 b^2 c^4 x^2 y^5 z+12 a^2 c^6 x^2 y^5 z-4 a^4 b^2 c^2 x^5 y z^2-8
                          > a^2 b^4 c^2 x^5 y z^2+12 b^6 c^2 x^5 y z^2+8 a^2 b^2 c^4 x^5 y z^2+40
                          > b^4 c^4 x^5 y z^2-4 b^2 c^6 x^5 y z^2+a^8 x^4 y^2 z^2-4 a^6 b^2 x^4
                          > y^2 z^2+6 a^4 b^4 x^4 y^2 z^2-4 a^2 b^6 x^4 y^2 z^2+b^8 x^4 y^2 z^2-4
                          > a^6 c^2 x^4 y^2 z^2-60 a^4 b^2 c^2 x^4 y^2 z^2+100 a^2 b^4 c^2 x^4 y^2
                          > z^2-36 b^6 c^2 x^4 y^2 z^2+6 a^4 c^4 x^4 y^2 z^2+100 a^2 b^2 c^4 x^4
                          > y^2 z^2+70 b^4 c^4 x^4 y^2 z^2-4 a^2 c^6 x^4 y^2 z^2-36 b^2 c^6 x^4
                          > y^2 z^2+c^8 x^4 y^2 z^2+2 a^8 x^3 y^3 z^2-8 a^6 b^2 x^3 y^3 z^2+12 a^4
                          > b^4 x^3 y^3 z^2-8 a^2 b^6 x^3 y^3 z^2+2 b^8 x^3 y^3 z^2-52 a^6 c^2 x^3
                          > y^3 z^2+52 a^4 b^2 c^2 x^3 y^3 z^2+52 a^2 b^4 c^2 x^3 y^3 z^2-52 b^6
                          > c^2 x^3 y^3 z^2+84 a^4 c^4 x^3 y^3 z^2+88 a^2 b^2 c^4 x^3 y^3 z^2+84
                          > b^4 c^4 x^3 y^3 z^2-20 a^2 c^6 x^3 y^3 z^2-20 b^2 c^6 x^3 y^3 z^2-14
                          > c^8 x^3 y^3 z^2+a^8 x^2 y^4 z^2-4 a^6 b^2 x^2 y^4 z^2+6 a^4 b^4 x^2
                          > y^4 z^2-4 a^2 b^6 x^2 y^4 z^2+b^8 x^2 y^4 z^2-36 a^6 c^2 x^2 y^4
                          > z^2+100 a^4 b^2 c^2 x^2 y^4 z^2-60 a^2 b^4 c^2 x^2 y^4 z^2-4 b^6 c^2
                          > x^2 y^4 z^2+70 a^4 c^4 x^2 y^4 z^2+100 a^2 b^2 c^4 x^2 y^4 z^2+6 b^4
                          > c^4 x^2 y^4 z^2-36 a^2 c^6 x^2 y^4 z^2-4 b^2 c^6 x^2 y^4 z^2+c^8 x^2
                          > y^4 z^2+12 a^6 c^2 x y^5 z^2-8 a^4 b^2 c^2 x y^5 z^2-4 a^2 b^4 c^2 x
                          > y^5 z^2+40 a^4 c^4 x y^5 z^2+8 a^2 b^2 c^4 x y^5 z^2-4 a^2 c^6 x y^5
                          > z^2+16 b^6 c^2 x^5 z^3-4 a^6 b^2 x^4 y z^3-8 a^4 b^4 x^4 y z^3+28 a^2
                          > b^6 x^4 y z^3-16 b^8 x^4 y z^3+72 a^2 b^4 c^2 x^4 y z^3-8 b^6 c^2 x^4
                          > y z^3+12 a^2 b^2 c^4 x^4 y z^3+32 b^4 c^4 x^4 y z^3-8 b^2 c^6 x^4 y
                          > z^3+2 a^8 x^3 y^2 z^3-52 a^6 b^2 x^3 y^2 z^3+84 a^4 b^4 x^3 y^2 z^3-20
                          > a^2 b^6 x^3 y^2 z^3-14 b^8 x^3 y^2 z^3-8 a^6 c^2 x^3 y^2 z^3+52 a^4
                          > b^2 c^2 x^3 y^2 z^3+88 a^2 b^4 c^2 x^3 y^2 z^3-20 b^6 c^2 x^3 y^2
                          > z^3+12 a^4 c^4 x^3 y^2 z^3+52 a^2 b^2 c^4 x^3 y^2 z^3+84 b^4 c^4 x^3
                          > y^2 z^3-8 a^2 c^6 x^3 y^2 z^3-52 b^2 c^6 x^3 y^2 z^3+2 c^8 x^3 y^2
                          > z^3-14 a^8 x^2 y^3 z^3-20 a^6 b^2 x^2 y^3 z^3+84 a^4 b^4 x^2 y^3
                          > z^3-52 a^2 b^6 x^2 y^3 z^3+2 b^8 x^2 y^3 z^3-20 a^6 c^2 x^2 y^3 z^3+88
                          > a^4 b^2 c^2 x^2 y^3 z^3+52 a^2 b^4 c^2 x^2 y^3 z^3-8 b^6 c^2 x^2 y^3
                          > z^3+84 a^4 c^4 x^2 y^3 z^3+52 a^2 b^2 c^4 x^2 y^3 z^3+12 b^4 c^4 x^2
                          > y^3 z^3-52 a^2 c^6 x^2 y^3 z^3-8 b^2 c^6 x^2 y^3 z^3+2 c^8 x^2 y^3
                          > z^3-16 a^8 x y^4 z^3+28 a^6 b^2 x y^4 z^3-8 a^4 b^4 x y^4 z^3-4 a^2
                          > b^6 x y^4 z^3-8 a^6 c^2 x y^4 z^3+72 a^4 b^2 c^2 x y^4 z^3+32 a^4 c^4
                          > x y^4 z^3+12 a^2 b^2 c^4 x y^4 z^3-8 a^2 c^6 x y^4 z^3+16 a^6 c^2 y^5
                          > z^3+16 a^2 b^6 x^4 z^4-16 b^8 x^4 z^4+16 b^6 c^2 x^4 z^4-8 a^6 b^2 x^3
                          > y z^4+32 a^4 b^4 x^3 y z^4-8 a^2 b^6 x^3 y z^4-16 b^8 x^3 y z^4+12 a^4
                          > b^2 c^2 x^3 y z^4+72 a^2 b^4 c^2 x^3 y z^4+28 b^6 c^2 x^3 y z^4-8 b^4
                          > c^4 x^3 y z^4-4 b^2 c^6 x^3 y z^4+a^8 x^2 y^2 z^4-36 a^6 b^2 x^2 y^2
                          > z^4+70 a^4 b^4 x^2 y^2 z^4-36 a^2 b^6 x^2 y^2 z^4+b^8 x^2 y^2 z^4-4
                          > a^6 c^2 x^2 y^2 z^4+100 a^4 b^2 c^2 x^2 y^2 z^4+100 a^2 b^4 c^2 x^2
                          > y^2 z^4-4 b^6 c^2 x^2 y^2 z^4+6 a^4 c^4 x^2 y^2 z^4-60 a^2 b^2 c^4 x^2
                          > y^2 z^4+6 b^4 c^4 x^2 y^2 z^4-4 a^2 c^6 x^2 y^2 z^4-4 b^2 c^6 x^2 y^2
                          > z^4+c^8 x^2 y^2 z^4-16 a^8 x y^3 z^4-8 a^6 b^2 x y^3 z^4+32 a^4 b^4 x
                          > y^3 z^4-8 a^2 b^6 x y^3 z^4+28 a^6 c^2 x y^3 z^4+72 a^4 b^2 c^2 x y^3
                          > z^4+12 a^2 b^4 c^2 x y^3 z^4-8 a^4 c^4 x y^3 z^4-4 a^2 c^6 x y^3
                          > z^4-16 a^8 y^4 z^4+16 a^6 b^2 y^4 z^4+16 a^6 c^2 y^4 z^4+16 a^2 b^6
                          > x^3 z^5-4 a^6 b^2 x^2 y z^5+40 a^4 b^4 x^2 y z^5+12 a^2 b^6 x^2 y
                          > z^5+8 a^4 b^2 c^2 x^2 y z^5-8 a^2 b^4 c^2 x^2 y z^5-4 a^2 b^2 c^4 x^2
                          > y z^5+12 a^6 b^2 x y^2 z^5+40 a^4 b^4 x y^2 z^5-4 a^2 b^6 x y^2 z^5-8
                          > a^4 b^2 c^2 x y^2 z^5+8 a^2 b^4 c^2 x y^2 z^5-4 a^2 b^2 c^4 x y^2
                          > z^5+16 a^6 b^2 y^3 z^5=0
                          >
                          > Best regards
                          > Angel Montesdeoca
                          >
                          >
                          >



                          --
                          http://anopolis72000.blogspot.com/


                          [Non-text portions of this message have been removed]
                        • Bernard Gibert
                          Dear Antreas, ... + another circular octic ! Best regards Bernard [Non-text portions of this message have been removed]
                          Message 12 of 17 , Mar 16, 2013
                          • 0 Attachment
                            Dear Antreas,

                            > I think that, since for P lying on K634 the point of contact of the
                            > antipedal and pedal circles of P
                            > is lying on the circumcircle, the K634 is part of a more general locus:
                            >
                            > Which is the locus of P such that the pedal circle of P, the antipedal
                            > circle of P and the circumcircle,
                            > are concurrent?
                            >
                            > The locus should be K634 (for pedal and antipedal circles tangent) +

                            + another circular octic !

                            Best regards

                            Bernard




                            [Non-text portions of this message have been removed]
                          • Francisco Javier
                            It is another octic, tangent to the sides at the traces of the circumcenter.
                            Message 13 of 17 , Mar 16, 2013
                            • 0 Attachment
                              It is another octic, tangent to the sides at the traces of the circumcenter.

                              --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
                              >
                              > Dear Angel, Bernard, Francisco
                              >
                              > I think that, since for P lying on K634 the point of contact of the
                              > antipedal and pedal circles of P
                              > is lying on the circumcircle, the K634 is part of a more general locus:
                              >
                              > Which is the locus of P such that the pedal circle of P, the antipedal
                              > circle of P and the circumcircle,
                              > are concurrent?
                              >
                              > The locus should be K634 (for pedal and antipedal circles tangent) +
                              > ???????????????????
                              >
                              > Antreas
                              >
                              >
                              > On Sat, Mar 16, 2013 at 3:18 PM, Angel <amontes1949@...> wrote:
                              >
                              > > **
                              > >
                              > >
                              > >
                              > >
                              > > --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@>
                              > > wrote:
                              > > >
                              > > > As Bernard Gibert pointed out to us, we were wrong, this is a different
                              > > cubic, not K191.
                              > > >
                              > > >
                              > > >
                              > > > --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@> wrote:
                              > > > >
                              > > > >
                              > > > >
                              > > > > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@>
                              > > wrote:
                              > > > > >
                              > > > > > Are there real points P such that the pedal and antipedal circles of
                              > > P
                              > > > > > are tangent?
                              > > > > >
                              > > > > > APH
                              > > > > >
                              > > > >
                              > > > >
                              > > > > In private mail Fancisco Javier has corroborated what is exposed here:
                              > > > >
                              > > > > Are there real points P such that the pedal and antipedal circles of P
                              > > are tangent?
                              > > > >
                              > > > >
                              > > > > Yes, if P is on an octic or on the cubic K191="circumcircle pedal
                              > > cubic, nK(X6, X6,?)".
                              > > > >
                              > > > > ---------------
                              > > > >
                              > > > > If P is on the cubic K191, then the point of the contact of the two
                              > > circles are on the circumcircle.
                              > > > >
                              > > > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749A.eps)
                              > > > > ------
                              > > > >
                              > > > > K191: S^2 xyz + CyclicSum[ a^2 y z (c^2 y + b^2 z)] = 0,
                              > > > >
                              > > > > or equivalently
                              > > > >
                              > > > > K191: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
                              > > > >
                              > > > > S= 2*area(ABC) (In CTC of Bernad Gibert, S=area(ABC))
                              > > > >
                              > > > >
                              > > > >
                              > > > > Barycentric equation of the octic is too complicated to be written
                              > > here.
                              > > > > (http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749B.eps)
                              > > > >
                              > > > > Angel M.
                              > > > >
                              > > >
                              > >
                              > > It was my mistake and I have confused to Francisco Javier.
                              > >
                              > > Are there real points P such that the pedal and antipedal circles of P are
                              > > tangent? (http://tech.groups.yahoo.com/group/Hyacinthos/message/21746)
                              > >
                              > > The correct answer is:
                              > >
                              > > Yes, if P is on an octic or on the cubic K634 (To appear,
                              > > http://tech.groups.yahoo.com/group/Hyacinthos/message/21765).
                              > >
                              > > K634: S^2 xyz + CyclicSum[ a^2 x (c^2 y^2 + b^2 z^2)] = 0.
                              > >
                              > > If P is on the cubic K634, then the point of the contact of the two
                              > > circles are on the circumcircle.
                              > >
                              > > (
                              > > http://groups.yahoo.com/group/Hyacinthos/files/Hyacinthos21749Acorrected.eps
                              > > )
                              > >
                              > > Equation of the octic:
                              > >
                              > > 16 b^2 c^6 x^5 y^3+16 a^2 c^6 x^4 y^4+16 b^2 c^6 x^4 y^4-16 c^8 x^4
                              > > y^4+16 a^2 c^6 x^3 y^5-4 a^4 b^2 c^2 x^5 y^2 z+8 a^2 b^4 c^2 x^5 y^2
                              > > z-4 b^6 c^2 x^5 y^2 z-8 a^2 b^2 c^4 x^5 y^2 z+40 b^4 c^4 x^5 y^2 z+12
                              > > b^2 c^6 x^5 y^2 z-4 a^6 c^2 x^4 y^3 z+12 a^2 b^4 c^2 x^4 y^3 z-8 b^6
                              > > c^2 x^4 y^3 z-8 a^4 c^4 x^4 y^3 z+72 a^2 b^2 c^4 x^4 y^3 z+32 b^4 c^4
                              > > x^4 y^3 z+28 a^2 c^6 x^4 y^3 z-8 b^2 c^6 x^4 y^3 z-16 c^8 x^4 y^3 z-8
                              > > a^6 c^2 x^3 y^4 z+12 a^4 b^2 c^2 x^3 y^4 z-4 b^6 c^2 x^3 y^4 z+32 a^4
                              > > c^4 x^3 y^4 z+72 a^2 b^2 c^4 x^3 y^4 z-8 b^4 c^4 x^3 y^4 z-8 a^2 c^6
                              > > x^3 y^4 z+28 b^2 c^6 x^3 y^4 z-16 c^8 x^3 y^4 z-4 a^6 c^2 x^2 y^5 z+8
                              > > a^4 b^2 c^2 x^2 y^5 z-4 a^2 b^4 c^2 x^2 y^5 z+40 a^4 c^4 x^2 y^5 z-8
                              > > a^2 b^2 c^4 x^2 y^5 z+12 a^2 c^6 x^2 y^5 z-4 a^4 b^2 c^2 x^5 y z^2-8
                              > > a^2 b^4 c^2 x^5 y z^2+12 b^6 c^2 x^5 y z^2+8 a^2 b^2 c^4 x^5 y z^2+40
                              > > b^4 c^4 x^5 y z^2-4 b^2 c^6 x^5 y z^2+a^8 x^4 y^2 z^2-4 a^6 b^2 x^4
                              > > y^2 z^2+6 a^4 b^4 x^4 y^2 z^2-4 a^2 b^6 x^4 y^2 z^2+b^8 x^4 y^2 z^2-4
                              > > a^6 c^2 x^4 y^2 z^2-60 a^4 b^2 c^2 x^4 y^2 z^2+100 a^2 b^4 c^2 x^4 y^2
                              > > z^2-36 b^6 c^2 x^4 y^2 z^2+6 a^4 c^4 x^4 y^2 z^2+100 a^2 b^2 c^4 x^4
                              > > y^2 z^2+70 b^4 c^4 x^4 y^2 z^2-4 a^2 c^6 x^4 y^2 z^2-36 b^2 c^6 x^4
                              > > y^2 z^2+c^8 x^4 y^2 z^2+2 a^8 x^3 y^3 z^2-8 a^6 b^2 x^3 y^3 z^2+12 a^4
                              > > b^4 x^3 y^3 z^2-8 a^2 b^6 x^3 y^3 z^2+2 b^8 x^3 y^3 z^2-52 a^6 c^2 x^3
                              > > y^3 z^2+52 a^4 b^2 c^2 x^3 y^3 z^2+52 a^2 b^4 c^2 x^3 y^3 z^2-52 b^6
                              > > c^2 x^3 y^3 z^2+84 a^4 c^4 x^3 y^3 z^2+88 a^2 b^2 c^4 x^3 y^3 z^2+84
                              > > b^4 c^4 x^3 y^3 z^2-20 a^2 c^6 x^3 y^3 z^2-20 b^2 c^6 x^3 y^3 z^2-14
                              > > c^8 x^3 y^3 z^2+a^8 x^2 y^4 z^2-4 a^6 b^2 x^2 y^4 z^2+6 a^4 b^4 x^2
                              > > y^4 z^2-4 a^2 b^6 x^2 y^4 z^2+b^8 x^2 y^4 z^2-36 a^6 c^2 x^2 y^4
                              > > z^2+100 a^4 b^2 c^2 x^2 y^4 z^2-60 a^2 b^4 c^2 x^2 y^4 z^2-4 b^6 c^2
                              > > x^2 y^4 z^2+70 a^4 c^4 x^2 y^4 z^2+100 a^2 b^2 c^4 x^2 y^4 z^2+6 b^4
                              > > c^4 x^2 y^4 z^2-36 a^2 c^6 x^2 y^4 z^2-4 b^2 c^6 x^2 y^4 z^2+c^8 x^2
                              > > y^4 z^2+12 a^6 c^2 x y^5 z^2-8 a^4 b^2 c^2 x y^5 z^2-4 a^2 b^4 c^2 x
                              > > y^5 z^2+40 a^4 c^4 x y^5 z^2+8 a^2 b^2 c^4 x y^5 z^2-4 a^2 c^6 x y^5
                              > > z^2+16 b^6 c^2 x^5 z^3-4 a^6 b^2 x^4 y z^3-8 a^4 b^4 x^4 y z^3+28 a^2
                              > > b^6 x^4 y z^3-16 b^8 x^4 y z^3+72 a^2 b^4 c^2 x^4 y z^3-8 b^6 c^2 x^4
                              > > y z^3+12 a^2 b^2 c^4 x^4 y z^3+32 b^4 c^4 x^4 y z^3-8 b^2 c^6 x^4 y
                              > > z^3+2 a^8 x^3 y^2 z^3-52 a^6 b^2 x^3 y^2 z^3+84 a^4 b^4 x^3 y^2 z^3-20
                              > > a^2 b^6 x^3 y^2 z^3-14 b^8 x^3 y^2 z^3-8 a^6 c^2 x^3 y^2 z^3+52 a^4
                              > > b^2 c^2 x^3 y^2 z^3+88 a^2 b^4 c^2 x^3 y^2 z^3-20 b^6 c^2 x^3 y^2
                              > > z^3+12 a^4 c^4 x^3 y^2 z^3+52 a^2 b^2 c^4 x^3 y^2 z^3+84 b^4 c^4 x^3
                              > > y^2 z^3-8 a^2 c^6 x^3 y^2 z^3-52 b^2 c^6 x^3 y^2 z^3+2 c^8 x^3 y^2
                              > > z^3-14 a^8 x^2 y^3 z^3-20 a^6 b^2 x^2 y^3 z^3+84 a^4 b^4 x^2 y^3
                              > > z^3-52 a^2 b^6 x^2 y^3 z^3+2 b^8 x^2 y^3 z^3-20 a^6 c^2 x^2 y^3 z^3+88
                              > > a^4 b^2 c^2 x^2 y^3 z^3+52 a^2 b^4 c^2 x^2 y^3 z^3-8 b^6 c^2 x^2 y^3
                              > > z^3+84 a^4 c^4 x^2 y^3 z^3+52 a^2 b^2 c^4 x^2 y^3 z^3+12 b^4 c^4 x^2
                              > > y^3 z^3-52 a^2 c^6 x^2 y^3 z^3-8 b^2 c^6 x^2 y^3 z^3+2 c^8 x^2 y^3
                              > > z^3-16 a^8 x y^4 z^3+28 a^6 b^2 x y^4 z^3-8 a^4 b^4 x y^4 z^3-4 a^2
                              > > b^6 x y^4 z^3-8 a^6 c^2 x y^4 z^3+72 a^4 b^2 c^2 x y^4 z^3+32 a^4 c^4
                              > > x y^4 z^3+12 a^2 b^2 c^4 x y^4 z^3-8 a^2 c^6 x y^4 z^3+16 a^6 c^2 y^5
                              > > z^3+16 a^2 b^6 x^4 z^4-16 b^8 x^4 z^4+16 b^6 c^2 x^4 z^4-8 a^6 b^2 x^3
                              > > y z^4+32 a^4 b^4 x^3 y z^4-8 a^2 b^6 x^3 y z^4-16 b^8 x^3 y z^4+12 a^4
                              > > b^2 c^2 x^3 y z^4+72 a^2 b^4 c^2 x^3 y z^4+28 b^6 c^2 x^3 y z^4-8 b^4
                              > > c^4 x^3 y z^4-4 b^2 c^6 x^3 y z^4+a^8 x^2 y^2 z^4-36 a^6 b^2 x^2 y^2
                              > > z^4+70 a^4 b^4 x^2 y^2 z^4-36 a^2 b^6 x^2 y^2 z^4+b^8 x^2 y^2 z^4-4
                              > > a^6 c^2 x^2 y^2 z^4+100 a^4 b^2 c^2 x^2 y^2 z^4+100 a^2 b^4 c^2 x^2
                              > > y^2 z^4-4 b^6 c^2 x^2 y^2 z^4+6 a^4 c^4 x^2 y^2 z^4-60 a^2 b^2 c^4 x^2
                              > > y^2 z^4+6 b^4 c^4 x^2 y^2 z^4-4 a^2 c^6 x^2 y^2 z^4-4 b^2 c^6 x^2 y^2
                              > > z^4+c^8 x^2 y^2 z^4-16 a^8 x y^3 z^4-8 a^6 b^2 x y^3 z^4+32 a^4 b^4 x
                              > > y^3 z^4-8 a^2 b^6 x y^3 z^4+28 a^6 c^2 x y^3 z^4+72 a^4 b^2 c^2 x y^3
                              > > z^4+12 a^2 b^4 c^2 x y^3 z^4-8 a^4 c^4 x y^3 z^4-4 a^2 c^6 x y^3
                              > > z^4-16 a^8 y^4 z^4+16 a^6 b^2 y^4 z^4+16 a^6 c^2 y^4 z^4+16 a^2 b^6
                              > > x^3 z^5-4 a^6 b^2 x^2 y z^5+40 a^4 b^4 x^2 y z^5+12 a^2 b^6 x^2 y
                              > > z^5+8 a^4 b^2 c^2 x^2 y z^5-8 a^2 b^4 c^2 x^2 y z^5-4 a^2 b^2 c^4 x^2
                              > > y z^5+12 a^6 b^2 x y^2 z^5+40 a^4 b^4 x y^2 z^5-4 a^2 b^6 x y^2 z^5-8
                              > > a^4 b^2 c^2 x y^2 z^5+8 a^2 b^4 c^2 x y^2 z^5-4 a^2 b^2 c^4 x y^2
                              > > z^5+16 a^6 b^2 y^3 z^5=0
                              > >
                              > > Best regards
                              > > Angel Montesdeoca
                              > >
                              > >
                              > >
                              >
                              >
                              >
                              > --
                              > http://anopolis72000.blogspot.com/
                              >
                              >
                              > [Non-text portions of this message have been removed]
                              >
                            • Antreas Hatzipolakis
                              We can naturally ask the same question for the NPCs of the same triangles ie Which is the locus of P such that the NPCs of the pedal triangle of P and the NPC
                              Message 14 of 17 , Mar 17, 2013
                              • 0 Attachment
                                We can naturally ask the same question for the NPCs of the same triangles ie

                                Which is the locus of P such that the NPCs of the pedal triangle of P and
                                the NPC of the antipedal triangle of P are tangent

                                but probably the locus, if exists, is too complicated!!

                                More simple becomes the locus, if we replace the pedal triangle's NPC with
                                the NPC of the antipedal triangle of the isogonal conjugate of P,

                                that is:

                                Let ABC be a triangle and P, P* two isogonal conjugate points.
                                Which is the locus of P such that the NPCs of the antipedal
                                triangles of P and P* are tangent.

                                I guess that the locus is McCay cubic +???????

                                And if we replace the tangency of the circles with the orthogonal intersection,
                                the locus will be Kjp + ???? ????

                                Just some midnight thoughts!!! ..... :-)

                                APH

                                On Thu, Mar 14, 2013 at 11:13 PM, Antreas Hatzipolakis
                                <anopolis72@...> wrote:
                                > Are there real points P such that the pedal and antipedal circles of P
                                > are tangent?
                                >
                                > APH
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