Sorry, an error occurred while loading the content.

## Re: [EMHL] Union of cubics [correction]

Expand Messages
• ... 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.
• ... 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
• ... 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/
>
>
>
• ... 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
• 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
• 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.
• 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
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
• ... 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
Your message has been successfully submitted and would be delivered to recipients shortly.