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Re: [EMHL] Union of cubics [correction]

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  • Floor en Lyanne van Lamoen
    ... This one gives a quartic. I mistakenly sent that one for the one with the nicer result, and the result in accordance with the subject line: For which P are
    Message 1 of 8 , May 4, 2002
      > Dear all,
      >
      > Let A' be the intersection of the line through P perpendicular to BC and
      > the Cevian line of its isogonal conjugate P*.
      > Similarly B'C'.
      >
      > For which P are the lines through A'B'C' parallel to the corresponding
      > sies of ABC concurrent (ABC and A'B'C' paraspective)?

      This one gives a quartic. I mistakenly sent that one for the one with
      the nicer result, and the result in accordance with the subject line:

      For which P are the lines through A' //AP, through B' //AP and through
      C' //AC concurrent?

      >
      > Kind regards,
      > Sincerely,
      > Floor.
    • fredlangch
      ... BC and ... corresponding ... Dear Floor, this quartic is circular (mono), passes through X1,3,36,54, ABC,excenters and is invariant by the inversion
      Message 2 of 8 , May 4, 2002
        --- In Hyacinthos@y..., Floor en Lyanne van Lamoen <f.v.lamoen@w...>
        wrote:
        > > Dear all,
        > >
        > > Let A' be the intersection of the line through P perpendicular to
        BC and
        > > the Cevian line of its isogonal conjugate P*.
        > > Similarly B'C'.
        > >
        > > For which P are the lines through A'B'C' parallel to the
        corresponding
        > > sies of ABC concurrent (ABC and A'B'C' paraspective)?
        >
        > This one gives a quartic. ...

        Dear Floor,
        this quartic is circular (mono), passes through X1,3,36,54,
        ABC,excenters
        and is invariant by the inversion relative to the circumcircle.

        I give the trilinear equation for Edward Brisse:

        Regards Fred.


        -y*z^3*a^7 + y^3*z*a^7 + c*x*y^3*a^6 -
        b*x*z^3*a^6 - c*x*y*z^2*a^6 + b*x*y^2*z*a^6 +
        3*b^2*y*z^3*a^5 + 2*c^2*y*z^3*a^5 +
        2*b*c*x^2*y^2*a^5 - 2*b*c*x^2*z^2*a^5 -
        2*b^2*y^3*z*a^5 - 3*c^2*y^3*z*a^5 -
        b^2*x^2*y*z*a^5 + c^2*x^2*y*z*a^5 -
        3*c^3*x*y^3*a^4 + 3*b^3*x*z^3*a^4 +
        2*c^3*x*y*z^2*a^4 - b^2*c*x*y*z^2*a^4 +
        c^3*x^3*y*a^4 + b^2*c*x^3*y*a^4 -
        b^3*x^3*z*a^4 - b*c^2*x^3*z*a^4 -
        2*b^3*x*y^2*z*a^4 + b*c^2*x*y^2*z*a^4 -
        3*b^4*y*z^3*a^3 - c^4*y*z^3*a^3 -
        2*b^2*c^2*y*z^3*a^3 - 2*b*c^3*x^2*y^2*a^3 +
        2*b^3*c*x^2*z^2*a^3 + 2*b*c^3*y^2*z^2*a^3 -
        2*b^3*c*y^2*z^2*a^3 + b^4*y^3*z*a^3 +
        3*c^4*y^3*z*a^3 + 2*b^2*c^2*y^3*z*a^3 +
        2*b^4*x^2*y*z*a^3 - 2*c^4*x^2*y*z*a^3 +
        3*c^5*x*y^3*a^2 - 2*b^2*c^3*x*y^3*a^2 -
        b^4*c*x*y^3*a^2 - 3*b^5*x*z^3*a^2 +
        b*c^4*x*z^3*a^2 + 2*b^3*c^2*x*z^3*a^2 -
        c^5*x*y*z^2*a^2 + b^4*c*x*y*z^2*a^2 -
        2*c^5*x^3*y*a^2 + 2*b^2*c^3*x^3*y*a^2 +
        2*b^5*x^3*z*a^2 - 2*b^3*c^2*x^3*z*a^2 +
        b^5*x*y^2*z*a^2 - b*c^4*x*y^2*z*a^2 +
        b^6*y*z^3*a - b^2*c^4*y*z^3*a +
        2*b^3*c^3*x^2*y^2*a - 2*b^5*c*x^2*y^2*a +
        2*b*c^5*x^2*z^2*a - 2*b^3*c^3*x^2*z^2*a -
        2*b*c^5*y^2*z^2*a + 2*b^5*c*y^2*z^2*a -
        c^6*y^3*z*a + b^4*c^2*y^3*z*a - b^6*x^2*y*z*a +
        c^6*x^2*y*z*a + b^2*c^4*x^2*y*z*a -
        b^4*c^2*x^2*y*z*a - c^7*x*y^3 +
        2*b^2*c^5*x*y^3 - b^4*c^3*x*y^3 + b^7*x*z^3 +
        b^3*c^4*x*z^3 - 2*b^5*c^2*x*z^3 +
        b^2*c^5*x*y*z^2 - 2*b^4*c^3*x*y*z^2 +
        b^6*c*x*y*z^2 + c^7*x^3*y - 3*b^2*c^5*x^3*y +
        3*b^4*c^3*x^3*y - b^6*c*x^3*y - b^7*x^3*z +
        b*c^6*x^3*z - 3*b^3*c^4*x^3*z +
        3*b^5*c^2*x^3*z - b*c^6*x*y^2*z +
        2*b^3*c^4*x*y^2*z - b^5*c^2*x*y^2*z
      • Edward Brisse
        ... Except errors, you can add the following points : Inverse of X54, Isogonal from X1113 and X1114 (at the infinity) and the three second circumperp (TCCT
        Message 3 of 8 , May 4, 2002
          At 21:32 5/4/02 -0000, you wrote:
          >--- In Hyacinthos@y..., Floor en Lyanne van Lamoen <f.v.lamoen@w...>
          >wrote:
          >> > Dear all,
          >> >
          >> > Let A' be the intersection of the line through P perpendicular to
          >BC and
          >> > the Cevian line of its isogonal conjugate P*.
          >> > Similarly B'C'.
          >> >
          >> > For which P are the lines through A'B'C' parallel to the
          >corresponding
          >> > sies of ABC concurrent (ABC and A'B'C' paraspective)?
          >>
          >> This one gives a quartic. ...
          >
          >Dear Floor,
          >this quartic is circular (mono), passes through X1,3,36,54,
          >ABC,excenters
          >and is invariant by the inversion relative to the circumcircle.

          Except errors, you can add the following points :
          Inverse of X54, Isogonal from X1113 and X1114 (at the infinity)
          and the three second circumperp (TCCT 6.22)
          Best Regards, Edward Brisse


          >
          >I give the trilinear equation for Edward Brisse:
          >
          >Regards Fred.
          >
          >
          >-y*z^3*a^7 + y^3*z*a^7 + c*x*y^3*a^6 -
          > b*x*z^3*a^6 - c*x*y*z^2*a^6 + b*x*y^2*z*a^6 +
          > 3*b^2*y*z^3*a^5 + 2*c^2*y*z^3*a^5 +
          > 2*b*c*x^2*y^2*a^5 - 2*b*c*x^2*z^2*a^5 -
          > 2*b^2*y^3*z*a^5 - 3*c^2*y^3*z*a^5 -
          > b^2*x^2*y*z*a^5 + c^2*x^2*y*z*a^5 -
          > 3*c^3*x*y^3*a^4 + 3*b^3*x*z^3*a^4 +
          > 2*c^3*x*y*z^2*a^4 - b^2*c*x*y*z^2*a^4 +
          > c^3*x^3*y*a^4 + b^2*c*x^3*y*a^4 -
          > b^3*x^3*z*a^4 - b*c^2*x^3*z*a^4 -
          > 2*b^3*x*y^2*z*a^4 + b*c^2*x*y^2*z*a^4 -
          > 3*b^4*y*z^3*a^3 - c^4*y*z^3*a^3 -
          > 2*b^2*c^2*y*z^3*a^3 - 2*b*c^3*x^2*y^2*a^3 +
          > 2*b^3*c*x^2*z^2*a^3 + 2*b*c^3*y^2*z^2*a^3 -
          > 2*b^3*c*y^2*z^2*a^3 + b^4*y^3*z*a^3 +
          > 3*c^4*y^3*z*a^3 + 2*b^2*c^2*y^3*z*a^3 +
          > 2*b^4*x^2*y*z*a^3 - 2*c^4*x^2*y*z*a^3 +
          > 3*c^5*x*y^3*a^2 - 2*b^2*c^3*x*y^3*a^2 -
          > b^4*c*x*y^3*a^2 - 3*b^5*x*z^3*a^2 +
          > b*c^4*x*z^3*a^2 + 2*b^3*c^2*x*z^3*a^2 -
          > c^5*x*y*z^2*a^2 + b^4*c*x*y*z^2*a^2 -
          > 2*c^5*x^3*y*a^2 + 2*b^2*c^3*x^3*y*a^2 +
          > 2*b^5*x^3*z*a^2 - 2*b^3*c^2*x^3*z*a^2 +
          > b^5*x*y^2*z*a^2 - b*c^4*x*y^2*z*a^2 +
          > b^6*y*z^3*a - b^2*c^4*y*z^3*a +
          > 2*b^3*c^3*x^2*y^2*a - 2*b^5*c*x^2*y^2*a +
          > 2*b*c^5*x^2*z^2*a - 2*b^3*c^3*x^2*z^2*a -
          > 2*b*c^5*y^2*z^2*a + 2*b^5*c*y^2*z^2*a -
          > c^6*y^3*z*a + b^4*c^2*y^3*z*a - b^6*x^2*y*z*a +
          > c^6*x^2*y*z*a + b^2*c^4*x^2*y*z*a -
          > b^4*c^2*x^2*y*z*a - c^7*x*y^3 +
          > 2*b^2*c^5*x*y^3 - b^4*c^3*x*y^3 + b^7*x*z^3 +
          > b^3*c^4*x*z^3 - 2*b^5*c^2*x*z^3 +
          > b^2*c^5*x*y*z^2 - 2*b^4*c^3*x*y*z^2 +
          > b^6*c*x*y*z^2 + c^7*x^3*y - 3*b^2*c^5*x^3*y +
          > 3*b^4*c^3*x^3*y - b^6*c*x^3*y - b^7*x^3*z +
          > b*c^6*x^3*z - 3*b^3*c^4*x^3*z +
          > 3*b^5*c^2*x^3*z - b*c^6*x*y^2*z +
          > 2*b^3*c^4*x*y^2*z - b^5*c^2*x*y^2*z
          >
          >
          >
          >
          >
          >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
          >
          >
          >
        • Antreas P. Hatzipolakis
          ... Dear Floor, Note that there is a cubic with a similar name: orthic curve . Brooks, C. E.: A note on the orthic cubic curve. Johns Hophins Univ. Circ.
          Message 4 of 8 , May 4, 2002
            [FvL]:
            > Union of Linf, McCay, iso-H (does this one have a name? I believe that I
            > have read it been called "orthocubic", but is there more reason for that
            > than the orthocenter as pivot only?).
            >

            Dear Floor,

            Note that there is a cubic with a similar name:
            "orthic curve".

            Brooks, C. E.: A note on the orthic cubic curve.
            Johns Hophins Univ. Circ. $1904_2$, 47-52.
            Eine ebene Kurve $C_1$ $n$-ter Ordnung heisst eine ``orthische'',
            wenn sie zum Paar der imaginären Kreispunkte apolar ist. Insbesondere
            ist eine orthische Kurve zweiter Ordnung $C_2$ eine gleichseitige
            Hyperbel, eine orthische Kurve dritter Ordnung $C_3$ eine solche,
            deren sämtliche Polarkurven zweiter Ordnung gleichseitige Hyperbeln
            sind. Es wird eine Reihe bemerkenswerter Eigenschaften der orthischen
            $C_3$ aufgestellt, unter Verwendung komplexer Koordinaten. So geht
            durch sechs Punkte eines Kreises eine und nur eine orthische $C_3$;
            die Bahn eines Punktes, der zu drei festen Punkten konstant orientiert
            ist, ist eine orthische $C_3$; jede Kurve dritter Ordnung kann in eine
            orthische projiziert werden. Zum Schlusse wird noch der allgemeinere
            Satz bewiesen, dass die Asymptoten eines Büschels orthischer $C_n$
            eine Hypozykloide der Ordnung $2n$ und der Klasse $2n-1$ umhüllen.

            Germanophones may inform us what cubic is.

            APH
          • Bernard Gibert
            Dear friends, [FvL] ... [FL] ... [EB] ... This is a very nice circular quartic Q with many interesting properties. a). O is a node with two tangents // to the
            Message 5 of 8 , May 5, 2002
              Dear friends,

              [FvL]
              > Let A' be the intersection of the line through P perpendicular to
              > BC and the Cevian line of its isogonal conjugate P*.
              > Similarly B'C'.
              > For which P are the lines through A'B'C' parallel to the
              > corresponding sides of ABC concurrent (ABC and A'B'C' paraspective)?

              [FL]
              > this quartic is circular (mono), passes through X1,3,36,54,
              > ABC,excenters
              > and is invariant by the inversion relative to the circumcircle.

              [EB]
              > Except errors, you can add the following points :
              > Inverse of X54, Isogonal from X1113 and X1114 (at the infinity)
              > and the three second circumperp (TCCT 6.22)

              This is a very nice circular quartic Q with many interesting properties.

              a). O is a node with two tangents // to the asymptotes of the Jerabek
              hyperbola.

              b). Q has two real asymptotes // to the asymptotes of the Jerabek hyperbola
              and intersecting at an unknown point [a^4 SA(4SA^2 - b^2c^2) : : ]

              c). The singular focus (not on the curve) is unknown as well but ugly.
              [ a^2(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + a^4*b^2*c^2 -
              a^2*b^4*c^2 + 2*b^6*c^2 - a^2*b^2*c^4 - 2*b^4*c^4 + 2*a^2*c^6
              + 2*b^2*c^6 - c^8) : : ]

              d). Q meets the circumcircle again at the vertices U,V,W of the circumnormal
              triangle (like McCay)

              e). the tangents at A,B,C,U,V,W all pass through O.

              f). the 3rd and 4th intersections with BC lie on the circle through A & O,
              centered at Oa = BC /\ perp. bisector OA.
              The points Oa, Ob, Oc are collinear on the tripolar of a very simple unknown
              point of GK with baryc. [R^2 - a^2 : : ] (R circumradius)

              I'm sure there are other things to discover.

              I hope I didn't make too many mistakes.

              Best regards

              Bernard
            • Floor en Lyanne van Lamoen
              Dear all, ... I have to send a rectification about the nicer result. I didn t realize that the quartic was so rich!! Thanks Bernard, Fred and Edward. The two
              Message 6 of 8 , May 5, 2002
                Dear all,

                [FvL]:
                > > Dear all,
                > >
                > > Let A' be the intersection of the line through P perpendicular to BC and
                > > the Cevian line of its isogonal conjugate P*.
                > > Similarly B'C'.
                > >
                > > For which P are the lines through A'B'C' parallel to the corresponding
                > > sies of ABC concurrent (ABC and A'B'C' paraspective)?
                >
                > This one gives a quartic. I mistakenly sent that one for the one with
                > the nicer result, and the result in accordance with the subject line:
                >
                > For which P are the lines through A' //AP, through B' //AP and through
                > C' //AC concurrent?

                I have to send a rectification about the "nicer" result. I didn't
                realize that the quartic was so rich!! Thanks Bernard, Fred and Edward.

                The two cubic locus of the intended problem, I thought was nice:

                Union of Linf, McCay, iso-H (does this one have a name? I believe that I
                have read it been called "orthocubic", but is there more reason for that
                than the orthocenter as pivot only?).

                Kind regards,
                Sincerely,
                Floor.
              • Gilles Boutte
                Dear all Hyacinthists, ... Are you sur of Kurve $C_1$ ? I mean that it is Kurve $C_n$ ... Are you sure of apolar ? I mean autopolar . ... Are you sur of
                Message 7 of 8 , May 5, 2002
                  Dear all Hyacinthists,

                  >[APH]
                  >
                  >Note that there is a cubic with a similar name:
                  >"orthic curve".
                  >
                  >Brooks, C. E.: A note on the orthic cubic curve.
                  >Johns Hophins Univ. Circ. $1904_2$, 47-52.
                  >Eine ebene Kurve $C_1$ $n$-ter Ordnung heisst eine ``orthische'',
                  >
                  Are you sur of "Kurve $C_1$ ? I mean that it is "Kurve $C_n$

                  >wenn sie zum Paar der imaginären Kreispunkte apolar ist. Insbesondere
                  >
                  Are you sure of "apolar" ? I mean "autopolar".

                  >ist eine orthische Kurve zweiter Ordnung $C_2$ eine gleichseitige
                  >Hyperbel, eine orthische Kurve dritter Ordnung $C_3$ eine solche,
                  >deren sämtliche Polarkurven zweiter Ordnung gleichseitige Hyperbeln
                  >sind. Es wird eine Reihe bemerkenswerter Eigenschaften der orthischen
                  >$C_3$ aufgestellt, unter Verwendung komplexer Koordinaten. So geht
                  >durch sechs Punkte eines Kreises eine und nur eine orthische $C_3$;
                  >die Bahn eines Punktes, der zu drei festen Punkten konstant orientiert
                  >
                  Are you sur of this sentence, I can't understand it, it seems some
                  words are omitted. (see the PS.)

                  >ist, ist eine orthische $C_3$; jede Kurve dritter Ordnung kann in eine
                  >orthische projiziert werden. Zum Schlusse wird noch der allgemeinere
                  >Satz bewiesen, dass die Asymptoten eines Büschels orthischer $C_n$
                  >eine Hypozykloide der Ordnung $2n$ und der Klasse $2n-1$ umhüllen.
                  >
                  >Germanophones may inform us what cubic is.
                  >
                  Translation in French

                  Une courbe plane Cn d'ordre n est appelée "orthique" si elle est
                  autopolaire par rapport aux points cycliques.
                  En particulier une conique C2 orthique est une hyperbole équilatère.
                  Une cubique C3 orthique est telle que toutes ses coniques polaires sont
                  des hyperboles équilatères.
                  Nous établissons une liste de propriétés remarquables des cubiques
                  orthiques en utilisant des affixes complexes.
                  Ainsi, par six points d'un cercle, passe une cubique orthique et une
                  seule ; le lieu d'un point [qui est orienté de façon constante par
                  rapport à trois points fixes ??] est une cubique C3 orthique ; toute
                  cubique peut être projetée en une cubique orthique.
                  En conclusion nous prouverons le résultat général : les asymptotes d'un
                  faisceau de courbes Cn orthiques enveloppent une hypocycloïde d'ordre 2n
                  et de classe 2n-1.

                  Trial of English translation

                  A plane curve Cn of order n is called "orthic", if it is autopolar wrt
                  the cyclic points.
                  Particularly a conic C2 orthic is a rectangular hyperbola.
                  A cubic C3 orthic is such, that all its polar conic ara rectangular
                  hyperbolae.
                  It is established a schedule of remarkable properties of orthic cubics,
                  using complex numbers.
                  So one orthic cubic C3 and only one passes through six points lying on a
                  circle ; the locus of a point [constantly oriented wrt three fixed
                  points ??] is an orthic cubic C3 ; any cubic mauy be projected to an
                  orthic cubic.
                  In conclusion, the general proposition is proved : the asymptots of a
                  pencil of othic curves Cn envelop a hypocycloid of ordre 2n and class 2n-1.

                  English speakers may give a preferable translation.

                  Best regards,
                  Happy Eater to oriental Churches members,

                  Gilles Boutte

                  PS Dear Antreas, in a previous message, you gave the reference

                  Blanchard, R. - Thebault, V.: Sur la cubique de MacCay. Mathesis 60,
                  244-248 (1951).

                  ABC, A'B'C' seien zwei einem Kreis (O) einbeschriebene perspektive
                  Dreiecke
                  mit im Endlichen liegendem Perspektivitaetszentrum P; A_1 und A'_1, B_1
                  und B'_1, C_1 und C'_1 seien die Punkte, in denen die
                  Winkelgegengeraden von
                  AA' und A'A, BB' und B'B, CC' und C'C den Kreis (O) schneiden; ...

                  I translate :

                  ABC and A'B'C' are two perspective triangles inscribed in a circle
                  (O), whose perspector P not on the line at infinity ; let A1 and
                  A'1, B1 and B'1, C1 and C'1 be the points at which the isogonal
                  lines of AA' and A'A', BB' and B'B, CC' and C'C intersects the
                  circle (O) ; ...

                  It is incomprehensible!!!

                  I read the top copy of Blanchard and Thebaud. They wrote (I translate) :

                  ... let A1 and A'1, B1 and B'1, C1 and C'1 be the points at which
                  the isogonal lines of AA' wrt ABC and A'A wrt A'B'C', BB' wrt ABC
                  and B'B wrt A'B'C', CC' wrt ABC and C'C wrt A'B'C' intersects the
                  circle (O) ; ...


                  Friendly, yours
                  Gilles Boutte
                • Antreas P. Hatzipolakis
                  ... Dear Gilles, Thank you for your translations. I simply had copy&paste -ed the german texts from the JFM database. Antreas
                  Message 8 of 8 , May 6, 2002
                    On Sunday, May 5, 2002, at 06:37 PM, Gilles Boutte wrote:

                    > Dear all Hyacinthists,
                    >
                    >> [APH]
                    >>
                    >> Note that there is a cubic with a similar name:
                    >> "orthic curve".
                    >>
                    >> Brooks, C. E.: A note on the orthic cubic curve.
                    >> Johns Hophins Univ. Circ. $1904_2$, 47-52.
                    >> Eine ebene Kurve $C_1$ $n$-ter Ordnung heisst eine ``orthische'',
                    >>
                    > Are you sure of "Kurve $C_1$ ? I mean that it is "Kurve $C_n$

                    Dear Gilles,

                    Thank you for your translations.

                    I simply had "copy&paste"-ed the german texts from the
                    JFM database.

                    Antreas
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