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Re: [EMHL] Re: LOCUS problems

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  • 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 1 of 6 , Mar 11, 2013
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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 2 of 6 , Mar 11, 2013
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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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