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Re: The nonpivotal isocubics nK(W,G*W)

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  • Angel
    Dear Antreas Suppose we are given a triangle ABC and two other points W=(p:q:r) and P=(x:y:z), not on its sidelines, and WaWbWc, PaPbPc the cevian triangles
    Message 1 of 4 , Mar 22, 2013
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      Dear Antreas

      Suppose we are given a triangle ABC and two other points W=(p:q:r) and P=(x:y:z), not on its sidelines, and WaWbWc, PaPbPc the cevian triangles of W,P, resp.

      A' = (Parallel through Pb to WaWb) /\ (Parallel through Pc to WaWc),
      B' = (Parallel through Pc to WbWc) /\ (Parallel through Pa to WbWa),
      C' = (Parallel through Pa to WcWa) /\ (Parallel through Pb to WcWb).

      When W=H, which is the locus of P such that:

      1. ABC, A'B'C'

      2. HaHbHc, A'B'C'

      are perspective?


      1. The triangles ABC and A'B'C' are perspective if and only if P lies on the circumconic of perspector the crosspoint of P and G , p(q+r)yz+ q(r+p)zx+ r(p+q)xy=0, or lies on the quartic p^2(q^2- r^2)y^2z^2+ q^2(r^2- p^2)z^2x^2+ r^2(p^2- q^2)x^2y^2=0.

      Points of the quartic: W, the vertices de ABC (are double points), and the vertices of the antimedial triangle, Ga, Gb and Gc.
      The tangents to quartic at Ga,Gb, Gc determine a triangle perspective with ABC of the perspector
      1/( p^2(q^2- r^2)):1/( q^2(r^2- p^2)): 1/( r^2(p^2- q^2)).

      In particular, if W=H
      The triangles ABC and A'B'C' are perspective if and only if P lies on the circumcircle or lies on the quartic Q066 ( Stammler quartic, http://bernard.gibert.pagesperso-orange.fr/curves/q066.html)

      If P lies on circumcircle the perspector Q of the triangles ABC and A'B'C' is the isogonal conjugate of P.
      If P lies on the Stammler quartic the perspector Q of the triangles ABC and A'B'C' lies on the JERABEK HYPERBOLA

      ---------------------------------------------------------
      Another geometry property of the quartic Q066:
      (EPS file: http://f1.grp.yahoofs.com/v1/wBlNUXTk_prMpAQuRSkfBeorkY6PtiaKnCxn-V4o6o9X1W2FkiOwaDALdn3PHq-ILiA4gk809bSkFT7ISn2Ot_Ts4e_Z1Y7w/Hyacinthos21808A.eps)

      Let P be a point and PaPbPc its cevian triangle, HaHbHc
      the orthic triangle (cevian triangle of H).
      The parallel through the point Pb to the sideline HaHb and
      the parallel through the point Pc to the sideline HaHc
      intersect at A'. The points B' and C' defined likewise.

      The triangles ABC and A'B'C' are perspective if and
      only if P lies on the Stammler quartic (together with the circumcircle).
      ----------------------------------------------------------------


      2. The triangles WaWbWc and A'B'C' are perspective if and only if P lies on the cevians of W.

      Best regards
      Angel Montesdeoca



      --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
      >
      >
      >
      > --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@> wrote:
      > >
      > > -------------------------------------------------
      > > Another geometry property of K211:
      > >
      > > Let P be a point and PaPbPc its cevian triangle, HaHbHc
      > > the orthic triangle (cevian triangle of H).
      > > The parallel through the point Pb to the sideline HaHb and
      > > the parallel through the point Pc to the sideline HaHc
      > > intersect at A'. The points B' and C' defined likewise.
      > >
      > > The triangles PaPbPc and A'B'C' are perspective if and
      > > only if P lies on on K211 (together with the three
      > > circm-rectangular hyperbolas passing through Ga, Gb, Gc).
      >
      > Naturally one may ask which are the rest two loci we get by combinations
      > of the triangles in the configuration, ie
      >
      > Which is the locus of P such that:
      >
      > 1. ABC, A'B'C'
      >
      > 2. HaHbHc, A'B'C'
      >
      > are perspective?
      >
      > Furthermore, we can define more points and ask for loci.
      >
      > For example:
      >
      > Denote A* = PbHc /\ PcHb and similarly B*,C*.
      >
      > Which is the locus of P such that:
      >
      > A*B*C* is perspective with:
      >
      > 1. ABC 2. HaHbHc. 3. PaPbPc. 4. A'B'C' ?
      >
      >
      > APH
      >
    • Antreas Hatzipolakis
      Dear Angel Nice results! This configuration is rich in locus problems, but I am not sure if we get such nice results for all! Now, we can replace the parallels
      Message 2 of 4 , Mar 23, 2013
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        Dear Angel

        Nice results!

        This configuration is rich in locus problems, but I am not sure if we get
        such nice results for all!

        Now, we can replace the parallels with perpendiculars.

        For the case of W = H, we have to draw perpendiculars through the
        feet of H (not of P, since we get the ABC)

        That is:

        Suppose we are given a triangle ABC and two other points W=(p:q:r) and
        P=(x:y:z), not on its sidelines,
        and WaWbWc, PaPbPc the cevian triangles of W,P, resp.

        A* = (Perpendicular through Wb to PaPb) /\ (Perpendicular through Wc to
        PaPc),
        B* = (Perpendicular through Wc to PbPc) /\ (Perpendicular through Wa to
        PbPa),
        C* = (Perpendicular through Wa to PcPa) /\ (Perpendicular through Wb to
        PcPb).

        When W=H, which is the locus of P such that:

        1. ABC, A*'B*C*

        2. WaWbWc [= HaHbHc], A*B*C*

        are perspective?

        and also 3. A*B*C*, PaPbPc.

        But I see more..... and will come back later, when I have free time.....


        APH

        On Sat, Mar 23, 2013 at 5:03 AM, Angel <amontes1949@...> wrote:

        > **
        >
        >
        > Dear Antreas
        >
        >
        > Suppose we are given a triangle ABC and two other points W=(p:q:r) and
        > P=(x:y:z), not on its sidelines, and WaWbWc, PaPbPc the cevian triangles of
        > W,P, resp.
        >
        > A' = (Parallel through Pb to WaWb) /\ (Parallel through Pc to WaWc),
        > B' = (Parallel through Pc to WbWc) /\ (Parallel through Pa to WbWa),
        > C' = (Parallel through Pa to WcWa) /\ (Parallel through Pb to WcWb).
        >
        > When W=H, which is the locus of P such that:
        >
        >
        > 1. ABC, A'B'C'
        >
        > 2. HaHbHc, A'B'C'
        >
        > are perspective?
        >
        > 1. The triangles ABC and A'B'C' are perspective if and only if P lies on
        > the circumconic of perspector the crosspoint of P and G , p(q+r)yz+
        > q(r+p)zx+ r(p+q)xy=0, or lies on the quartic p^2(q^2- r^2)y^2z^2+ q^2(r^2-
        > p^2)z^2x^2+ r^2(p^2- q^2)x^2y^2=0.
        >
        > Points of the quartic: W, the vertices de ABC (are double points), and the
        > vertices of the antimedial triangle, Ga, Gb and Gc.
        > The tangents to quartic at Ga,Gb, Gc determine a triangle perspective with
        > ABC of the perspector
        > 1/( p^2(q^2- r^2)):1/( q^2(r^2- p^2)): 1/( r^2(p^2- q^2)).
        >
        > In particular, if W=H
        > The triangles ABC and A'B'C' are perspective if and only if P lies on the
        > circumcircle or lies on the quartic Q066 ( Stammler quartic,
        > http://bernard.gibert.pagesperso-orange.fr/curves/q066.html)
        >
        > If P lies on circumcircle the perspector Q of the triangles ABC and A'B'C'
        > is the isogonal conjugate of P.
        > If P lies on the Stammler quartic the perspector Q of the triangles ABC
        > and A'B'C' lies on the JERABEK HYPERBOLA
        >
        > ---------------------------------------------------------
        > Another geometry property of the quartic Q066:
        > (EPS file:
        > http://f1.grp.yahoofs.com/v1/wBlNUXTk_prMpAQuRSkfBeorkY6PtiaKnCxn-V4o6o9X1W2FkiOwaDALdn3PHq-ILiA4gk809bSkFT7ISn2Ot_Ts4e_Z1Y7w/Hyacinthos21808A.eps
        > )
        >
        >
        > Let P be a point and PaPbPc its cevian triangle, HaHbHc
        > the orthic triangle (cevian triangle of H).
        > The parallel through the point Pb to the sideline HaHb and
        > the parallel through the point Pc to the sideline HaHc
        > intersect at A'. The points B' and C' defined likewise.
        >
        > The triangles ABC and A'B'C' are perspective if and
        > only if P lies on the Stammler quartic (together with the circumcircle).
        > ----------------------------------------------------------
        >
        > 2. The triangles WaWbWc and A'B'C' are perspective if and only if P lies
        > on the cevians of W.
        >
        > Best regards
        > Angel Montesdeoca
        >
        > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
        >
        > >
        > >
        > >
        > > --- In Hyacinthos@yahoogroups.com, "Angel" <amontes1949@> wrote:
        > > >
        > > > -------------------------------------------------
        > > > Another geometry property of K211:
        > > >
        > > > Let P be a point and PaPbPc its cevian triangle, HaHbHc
        > > > the orthic triangle (cevian triangle of H).
        > > > The parallel through the point Pb to the sideline HaHb and
        > > > the parallel through the point Pc to the sideline HaHc
        > > > intersect at A'. The points B' and C' defined likewise.
        > > >
        > > > The triangles PaPbPc and A'B'C' are perspective if and
        > > > only if P lies on on K211 (together with the three
        > > > circm-rectangular hyperbolas passing through Ga, Gb, Gc).
        > >
        > > Naturally one may ask which are the rest two loci we get by combinations
        > > of the triangles in the configuration, ie
        > >
        > > Which is the locus of P such that:
        > >
        > > 1. ABC, A'B'C'
        > >
        > > 2. HaHbHc, A'B'C'
        > >
        > > are perspective?
        > >
        > > Furthermore, we can define more points and ask for loci.
        > >
        > > For example:
        > >
        > > Denote A* = PbHc /\ PcHb and similarly B*,C*.
        > >
        > > Which is the locus of P such that:
        > >
        > > A*B*C* is perspective with:
        > >
        > > 1. ABC 2. HaHbHc. 3. PaPbPc. 4. A'B'C' ?
        > >
        > >
        > > APH
        > >
        >
        > _
        >


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