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LOCUS problems

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  • Antreas
    Let ABC be a triangle and A B C the circumcevian triangle of point P. The parallels to BC,CA,AB through A ,B ,C intersect the circumcircle again at
    Message 1 of 6 , Mar 10 12:20 PM
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      Let ABC be a triangle and A'B'C' the circumcevian
      triangle of point P.
      The parallels to BC,CA,AB through A',B',C'
      intersect the circumcircle again at A1,B1,C1,resp.
      The perpendiculars to AA', BB', CC' at A',B',C',
      intersect the circumcircle at A2,B2,C2, resp.

      Which is the locus of P such that:

      1. ABC, A1B1C1

      2. ABC, A2B2C2

      3. ABC, triangle bounded by (A1A2, B1B2, C1C2)

      are perspective?

      APH
    • Francisco Javier
      We have perspectivity in the three cases for any P: 1. The perspector is the isogonal conjugate of P 2. The perspector is the circumcenter. 3. The perspector
      Message 2 of 6 , Mar 10 2:22 PM
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        We have perspectivity in the three cases for any P:
        1. The perspector is the isogonal conjugate of P
        2. The perspector is the circumcenter.
        3. The perspector is the isogonal conjugate of the reflection of P on the orthocenter.

        Best regards,

        Francisco Javier.

        --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > Let ABC be a triangle and A'B'C' the circumcevian
        > triangle of point P.
        > The parallels to BC,CA,AB through A',B',C'
        > intersect the circumcircle again at A1,B1,C1,resp.
        > The perpendiculars to AA', BB', CC' at A',B',C',
        > intersect the circumcircle at A2,B2,C2, resp.
        >
        > Which is the locus of P such that:
        >
        > 1. ABC, A1B1C1
        >
        > 2. ABC, A2B2C2
        >
        > 3. ABC, triangle bounded by (A1A2, B1B2, C1C2)
        >
        > are perspective?
        >
        > APH
        >
      • Antreas Hatzipolakis
        So only the third is somehow interesting... How about to make it a little complicated? 4. Let A3, B3, C3 be the intersections with circumcircle of the
        Message 3 of 6 , Mar 10 2:40 PM
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          So only the third is somehow interesting...

          How about to make it a little complicated?

          4. Let A3, B3, C3 be the intersections with circumcircle of the reflections
          in AA', BB', CC' of the parallels to BC, CA, AB through A',B',C', resp.

          Which is the locus of P such that
          1. ABC, A3B3C3
          2. ABC, triangle bounded by (A'A3, B'B3, C'C3)

          are perspectivs?

          APH
          On Sun, Mar 10, 2013 at 11:22 PM, Francisco Javier
          <garciacapitan@...>wrote:

          > **
          >
          >
          > We have perspectivity in the three cases for any P:
          > 1. The perspector is the isogonal conjugate of P
          > 2. The perspector is the circumcenter.
          > 3. The perspector is the isogonal conjugate of the reflection of P on the
          > orthocenter.
          >
          > Best regards,
          >
          > Francisco Javier.
          >
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
          > >
          > > Let ABC be a triangle and A'B'C' the circumcevian
          > > triangle of point P.
          > > The parallels to BC,CA,AB through A',B',C'
          > > intersect the circumcircle again at A1,B1,C1,resp.
          > > The perpendiculars to AA', BB', CC' at A',B',C',
          > > intersect the circumcircle at A2,B2,C2, resp.
          > >
          > > Which is the locus of P such that:
          > >
          > > 1. ABC, A1B1C1
          > >
          > > 2. ABC, A2B2C2
          > >
          > > 3. ABC, triangle bounded by (A1A2, B1B2, C1C2)
          > >
          > > are perspective?
          > >
          > > APH
          >


          [Non-text portions of this message have been removed]
        • Francisco Javier
          In this case, 1. The locus is Ehrmann strophoid (K025) 2. The calculation takes a long time. I couldn t get the answer.
          Message 4 of 6 , Mar 11 12:56 AM
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            In this case,

            1. The locus is Ehrmann strophoid (K025)
            2. The calculation takes a long time. I couldn't get the answer.

            --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...> wrote:
            >
            > So only the third is somehow interesting...
            >
            > How about to make it a little complicated?
            >
            > 4. Let A3, B3, C3 be the intersections with circumcircle of the reflections
            > in AA', BB', CC' of the parallels to BC, CA, AB through A',B',C', resp.
            >
            > Which is the locus of P such that
            > 1. ABC, A3B3C3
            > 2. ABC, triangle bounded by (A'A3, B'B3, C'C3)
            >
            > are perspectivs?
            >
            > APH
            > On Sun, Mar 10, 2013 at 11:22 PM, Francisco Javier
            > <garciacapitan@...>wrote:
            >
            > > **
            > >
            > >
            > > We have perspectivity in the three cases for any P:
            > > 1. The perspector is the isogonal conjugate of P
            > > 2. The perspector is the circumcenter.
            > > 3. The perspector is the isogonal conjugate of the reflection of P on the
            > > orthocenter.
            > >
            > > Best regards,
            > >
            > > Francisco Javier.
            > >
            > >
            > > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
            > > >
            > > > Let ABC be a triangle and A'B'C' the circumcevian
            > > > triangle of point P.
            > > > The parallels to BC,CA,AB through A',B',C'
            > > > intersect the circumcircle again at A1,B1,C1,resp.
            > > > The perpendiculars to AA', BB', CC' at A',B',C',
            > > > intersect the circumcircle at A2,B2,C2, resp.
            > > >
            > > > Which is the locus of P such that:
            > > >
            > > > 1. ABC, A1B1C1
            > > >
            > > > 2. ABC, A2B2C2
            > > >
            > > > 3. ABC, triangle bounded by (A1A2, B1B2, C1C2)
            > > >
            > > > are perspective?
            > > >
            > > > APH
            > >
            >
            >
            > [Non-text portions of this message have been removed]
            >
          • Antreas Hatzipolakis
            Nice to meet old friend Jean-Pierre.... ! Now, I will rewrite the locus problem and cc Bernard, in the case it is not listed in the properties of the cubic.
            Message 5 of 6 , Mar 11 1:22 AM
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              Nice to "meet" old friend Jean-Pierre.... !

              Now, I will rewrite the locus problem and cc Bernard, in the case it is not
              listed
              in the properties of the cubic.

              Let ABC be a triangle, P a point and A'B'C' the circumcevian triangle of P.
              Let La,Lb,Lc be the parallels to BC,CA,AB through A',B',C', resp. and
              Ma,Mb,Mc the reflections of La,Lb,Lc in AA',BB',CC', resp..

              Denote:
              A* = Ma /\ (circumcircle - A') ie the other than A' intersection of Ma with
              the circumcircle.
              Similarly B*,C*.

              As Francisco found (quoted below), the locus of P such that ABC, A*B*C* are
              perspective is Ehrmann strophoid (K025)

              APH

              On Mon, Mar 11, 2013 at 9:56 AM, Francisco Javier
              <garciacapitan@...>wrote:

              > **
              >
              >
              > In this case,
              >
              > 1. The locus is Ehrmann strophoid (K025)
              > 2. The calculation takes a long time. I couldn't get the answer.
              >
              >
              > --- In Hyacinthos@yahoogroups.com, Antreas Hatzipolakis <anopolis72@...>
              > wrote:
              > >
              > > So only the third is somehow interesting...
              > >
              > > How about to make it a little complicated?
              > >
              > > 4. Let A3, B3, C3 be the intersections with circumcircle of the
              > reflections
              > > in AA', BB', CC' of the parallels to BC, CA, AB through A',B',C', resp.
              > >
              > > Which is the locus of P such that
              > > 1. ABC, A3B3C3
              > > 2. ABC, triangle bounded by (A'A3, B'B3, C'C3)
              > >
              > > are perspectivs?
              > >
              > > APH
              > > On Sun, Mar 10, 2013 at 11:22 PM, Francisco Javier
              > > <garciacapitan@...>wrote:
              > >
              > > > **
              >
              > > >
              > > >
              > > > We have perspectivity in the three cases for any P:
              > > > 1. The perspector is the isogonal conjugate of P
              > > > 2. The perspector is the circumcenter.
              > > > 3. The perspector is the isogonal conjugate of the reflection of P on
              > the
              > > > orthocenter.
              > > >
              > > > Best regards,
              > > >
              > > > Francisco Javier.
              > > >
              > > >
              > > > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
              > > > >
              > > > > Let ABC be a triangle and A'B'C' the circumcevian
              > > > > triangle of point P.
              > > > > The parallels to BC,CA,AB through A',B',C'
              > > > > intersect the circumcircle again at A1,B1,C1,resp.
              > > > > The perpendiculars to AA', BB', CC' at A',B',C',
              > > > > intersect the circumcircle at A2,B2,C2, resp.
              > > > >
              > > > > Which is the locus of P such that:
              > > > >
              > > > > 1. ABC, A1B1C1
              > > > >
              > > > > 2. ABC, A2B2C2
              > > > >
              > > > > 3. ABC, triangle bounded by (A1A2, B1B2, C1C2)
              > > > >
              > > > > are perspective?
              > > > >
              > > > > APH
              >
              >
              >


              [Non-text portions of this message have been removed]
            • Francisco Javier
              In message #21715 I wrote: We have perspectivity in the three cases for any P: 1. The perspector is the isogonal conjugate of P 2. The perspector is the
              Message 6 of 6 , Mar 11 11:22 AM
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                In message #21715 I wrote:

                We have perspectivity in the three cases for any P:
                1. The perspector is the isogonal conjugate of P
                2. The perspector is the circumcenter.
                3. The perspector is the isogonal conjugate of the reflection of P on the
                orthocenter.

                There is a typo here because it should be:

                3. The perspector is the isogonal conjugate of the reflection of P on the
                CIRCUMCENTER.

                (thanks to Angel Montesdeoca for pointing it out)

                Best regards,

                Francisco Javier.


                --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
                >
                > We have perspectivity in the three cases for any P:
                > 1. The perspector is the isogonal conjugate of P
                > 2. The perspector is the circumcenter.
                > 3. The perspector is the isogonal conjugate of the reflection of P on the orthocenter.
                >
                > Best regards,
                >
                > Francisco Javier.
                >
                > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
                > >
                > > Let ABC be a triangle and A'B'C' the circumcevian
                > > triangle of point P.
                > > The parallels to BC,CA,AB through A',B',C'
                > > intersect the circumcircle again at A1,B1,C1,resp.
                > > The perpendiculars to AA', BB', CC' at A',B',C',
                > > intersect the circumcircle at A2,B2,C2, resp.
                > >
                > > Which is the locus of P such that:
                > >
                > > 1. ABC, A1B1C1
                > >
                > > 2. ABC, A2B2C2
                > >
                > > 3. ABC, triangle bounded by (A1A2, B1B2, C1C2)
                > >
                > > are perspective?
                > >
                > > APH
                > >
                >
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