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A cubic is said to be central when it is invariant under a symmetry with respect to a point N called the center of the cubic. Such center is necessarily an inflexion point on the curve.

Let A', B', C' be the reflections of A, B, C about N and let M be another point distinct from the seven previous points. Let M' be the reflection of M about N. A variable line L passing through M is reflected about N giving the line L'. The isoconjugation f that swaps M, N transforms L into the circum-conic L*. L' and L* meet at two points that lie on the central circum-cubic with center N, passing through A', B', C', M, M'. This cubic is therefore a kind of spK as in CL055 but the isogonal conjugation is replaced with the isoconjugation f mentioned above.

This central cubic meets the line at infinity at the same points as pK(X x Y, Y) and the circum-conic with perspector X x Y at the same points as pK(X x Y, Y'). When X and Y are isogonal conjugates, these pKs are isogonal pKs as in CL055. When M is giN, the inverse in (O) of the isogonal conjugate of N, the central cubic is a focal cubic, see below.

***

More specifically, a study of central isocubics can be found in Special Isocubics §3 where many examples are provided. Here is a short summary of the results.

 

Main theorem for non-isotomic central pKs

Denote by W (different of G) the pole of the isoconjugation. When W = G, the cubic is the union of the medians.

Theorem 1 : for a given W, there is only one non-isotomic non-degenerate central pK.

Its center is N = G-Ceva conjugate of W and its pivot is P homothetic of N under h(G,4) i.e. N is the midpoint of PN*. The asymptotes are the lines through N and the midpoints of ABC. The inflexional tangent at N is the line NW.

This central pK is closely related to the isotomic pK with pivot Q, the anticomplement of W i.e. the isotomic conjugate of N*. For M on this isotomic pK, denote by Ma, Mb, Mc the vertices of the cevian triangle of M. The parallels through Ma, Mb, Mc to the corresponding cevian lines of the isotomic conjugate of Q concur at Z which is a point on the central cubic. Furthermore, two isotomic conjugates M and M' on the isotomic pK correspond to two points Z and Z' on the central pK which are symmetric with respect to the center N.

The barycentric equations of a central pK are :

with W = p:q:r (pole) : [p(2q+2r-3p)+(q-r)^2] x (ry^2-qz^2) + cyclic = 0

with P = p:q:r (pivot) : (q+r)(q+r+2p)(qz-ry)yz + cyclic = 0

with N = p:q:r (center) : (q+r-3p) x [r(p+q-r)y^2-q(p-q+r)z^2] + cyclic = 0

with N* = p:q:r (isocenter) : p(q+r)[(p+q-r)y-(p-q+r)z]yz + cyclic = 0

The most remarkable central pKs are the Darboux cubic and the Fermat cubics.

The table below shows a selection of remarkable central pKs with center N, pole W, pivot P. N* is the isoconjugate of the center N. It is the reflection of P in N and also the anticomplement of N. K' is the isotomic related pK.

N

W

P

N*

cubic

other centers on the cubic

K'

X(3)

X(6)

X(20)

X(4)

K004 Darboux cubic

P* = X(64), see the page

K007

X(10)

X(37)

X(8)

X(1)

K033 Spieker central

P* = X(65), P/N = X(72), P/N* = X(40), X(3176)

K034

E(406)

X(187)

X(11061)

X(67)

K042 Droussent central

X(2930)

K008

X(5)

X(216)

X(4)

X(3)

K044 Euler central

P/N = X(52), P/N* = X(155)

K045

X(618)

X(396)

X(616)

X(13)

K046a Fermat 1

 

K264a

X(619)

X(395)

X(617)

X(14)

K046b Fermat 2

 

K264b

X(141)

X(39)

X(69)

X(6)

K140

P* = X(1843), P/N* = X(159)

K141

X(206)

X(32)

?

X(66)

K161

X(6), X(159), P=(b^4-c^4)^2+a^4(2b^4+2c^4-3a^4)

pK(X2, X315)

X(1)

X(9)

X(145)

X(8)

K201

P* = a (b+c-a) / (b+c-3a) : : , X(188), X(2136)

pK(X2, X7)

X(9)

X(1)

X(144)

X(7)

K202

P* = a / [(b-c)^2 + a(2b+2c-3a)] : :

K200

X(1125)

X(1213)

X(1)

X(10)

pK(X1213, X1)

X(596), X(3159)

pK(X2, X86)

X(140)

X(233)

X(3)

X(5)

K413

 

pK(X2, X95)

X(142)

X(1212)

X(7)

X(9)

pK(X1212, X7)

X(3059), X(3174)

pK(X2, X85)

X(6)

X(3)

X(193)

X(69)

K707

 

K170

X(11)

X(650)

X(149)

X(100)

pK(X650, X149)

 

pK(X2, X693)

X(113)

X(3003)

X(146)

X(74)

K255

X(265), X(399), X(1986), X(2935)

K279

X(114)

X(230)

X(147)

X(98)

pK(X230, X147)

 

pK(X2, X325)

X(115)

X(523)

X(148)

X(99)

pK(X523, X148)

 

pK(X2, X523)

X(946)

?

X(962)

X(40)

 

X(1), X(4)

K133

X(216)

X(5)

X(3164)

X(264)

 

 

K146

?

X(1108)

?

X(84)

 

X(4), X(1490)

K154

?

X(800)

?

X(64)

 

X(4), X(185), X(1498), X(2883)

K235

X(960)

X(2092)

?

X(65)

 

X(1), X(72), X(1829), X(2292)

K254

X(119)

?

X(153)

X(104)

 

X(80), X(2950)

 

X(214)

X(44)

?

X(80)

K510

X(104), X(1145), X(1317), X(1320)

K311

?

X(511)

?

X(290)

 

X(1987)

K355

?

X(1100)

?

X(79)

 

X(21), X(191)

K455

Remark : when N lies on the nine-point circle, W on the orthic axis, P on the circumcircle of the antimedial triangle i.e. C(H, 2R), N* on the circumcircle of ABC and the pivot of K' on the de Longchamps axis.

 

Main theorems for central nKs

Theorem 2 : for a given pole W, there are infinitely many non-degenerate central nK.

The center N lies on the circum-conic with perspector W. The root is the complement of the isotomic conjugate of the trilinear pole of NN*.

Theorem 3 : for a given center N, there are infinitely many non-degenerate central nK.

The pole W lies on the trilinear polar of N, the root lies on the trilinear polar of the isotomic conjugate of the anticomplement of N.

Theorem 4 : for a given root P, all the non-degenerate central nK form a pencil of cubics having a common real asymptote.

 

Consequence of theorem 2 : a central nK is isogonal if and only if its center N lies on the circum-circle and isotomic if and only if its center N lies on the Steiner circum-ellipse. Hence, the only point N for which there are simultaneously two such central cubics is the Steiner point X(99) : these are K084 and K087 (when N is a vertex of ABC, the cubic degenerates).

CL001 and CL002 are the classes of central isogonal and isotomic nKs respectively.

CL012 and CL013 are the classes of central nKs with center G and O respectively.

See also the class CL044 of nK0+ and nK0++.

Here is a selection of central nKs with pole W, root R, center N :

cubic

W

R

N

notes

K036

X(115)

?

X(476)

nK

K068

X(523)

X(524)

X(2)

nK0++, see CL012

K069

X(647)

X(30) x X69)

X(3)

nK++, see CL013

K084

X(6)

?

X(99)

nK++, see CL001

K087

X(2)

?

X(99)

nK++, see CL002

K139

X(1989)

?

X(1989)÷X(30)

nK60++

K398

X(690)

X(543)

X(2)

nK0++, see CL012

K399

X(900)

X(545)

X(2)

nK0++, see CL012

K400a

infinite points of axes of Steiner ellipses

X(2)

nK0++, see CL012

K400b

X(2)

nK0++, see CL012

K608

X(468) x X(523)

X(6)

X(468)

nK0++, see CL012

 

 

 

 

 

 

Other central circum-cubics and generalization

The following table gathers together other central cubics which are not isocubics.

cubic

N

notes

K026

X(5)

psK(X51, X2, X3), spK(X3, X140), central stelloid or K60++

K047

X(3)

spK(X2, X376)

K065

X(2)

O(X524), focal cubic

K080

X(3)

spK(X3, X550), central stelloid or K60++

K213

X(2)

central stelloid or K60++

K287

X(2)

spK(X6, X2)

K310

X(4)

 

K314

X(6)

spK(X2, X6), TC(X6) in CL040

K315

X(2)

TC(X376) in CL040

K425

X(4)

 

K426

X(3)

psK(X154, X2, X3), spK(X3146, X4)

K435

X(187)

focal cubic

K443

X(3)

psK(X3, X69, X3), spK(X4, X3)

K465

X(5)

focal cubic

K525

X(4)

spK(X3, X4)

K526

X(5)

spK(X54, X5)

K530

X(4)

focal cubic

K562

X(5)

psK(X5, X264, X3)

K566

X(3)

spK(X5, X548)

K573

h(X1,-1/2)(X1000)

psK(?, X7, X1000)

K706

X(3)

spK(X3543, X2)

K805

X(141)

 

K814

X(3)

spK(X3529, X1657)

K821

X(140)

 

K830

X(1)

psK(X1, X7, X1)

 

 

 

Notes :

• the cubics highlighted in green are those with center O, members of the pencil generated by the Darboux cubic K004 and the union of the circumcircle and the Euler line.

• the cubics highlighted in yellow are those with center G. This net of cubics contains one focal cubic K065, one equilateral cubic K213 and a pencil of nKs detailed in CL012.

***

Let N = p:q:r be a point not lying on the line at infinity or on the sidelines of ABC. Let A1, B1, C1 be the reflections of A, B, C about N.

The union of the lines AN, BC1 and CB1 can be considered as a degenerate central cubic Ka with center N. Two other cubics Kb and Kc are defined similarly.

Any central cubic K with center N can be written under the form : K = u Ka + v Kb + w Kc where P = u:v:w is a point.

The equation of K is : u(ry-qz)[(p-q+r)x+2py][(p+q-r)x+2pz]=0.

In particular, when N = G, this equation becomes : u(y-z)(x+2y)(x+2z)=0 which is the equation of a central K0 (without term in xyz).

 

Central focal circum-cubics

There is one and only one central focal with given center (and focus) N not lying on the line at infinity or on a sideline of ABC and distinct of O, all these special cases giving decomposed cubics.

The real asymptote (A) passes through cgN (the complement of the isogonal conjugate of N).

The inflexional tangent (T) at N is the tangent at N to the circum-conic through N and giN (the isogonal conjugate of the inverse in (O) of N). This point giN and its symmetric sgiN in N are two points on the cubic.

Any circle through giN and sgiN meets the cubic again at two points lying on a parallel to the asymptote. This gives A1, B1, C1 on the parallels passing through A, B, C and A2, B2, C2 on the parallels passing through the reflections A', B', C' of A, B, C in N.

The polar conic of the real infinite point is the rectangular hyperbola (H) with center N passing through the midpoints of AA1, BB1, CC1, A'A2, B'B2, C'C2.

Any circle passing through N with center M on the normal (N) at N to the cubic also meets the cubic again at two points lying on a parallel to the asymptote. This parallel passes through the intersection of (N) and the polar line of M in (H). This gives a construction of the cubic and, in particular, the last common point S of the cubic and the circumcircle of ABC when the circle passes through sgiN. Obviously, its reflection S' in N is on the cubic.

The centers of anallagmaty E1, E2 lie on (H) and on the bisectors of (A) and (T). Only two are real.

centralfocal1

The table below shows a selection of central focal cubics according to their center N.

N

cubic

centers on the cubic

remarks

X(1)

K831

X(1), X(80), X(519), X(2718)

 

X(2)

K065

X(2), X(13), X(14), X(67), X(524), X(2770)

orthopivotal cubic

X(4)

K530

X(4), X(30), X(265), X(477)

 

X(5)

K465

X(3), X(4), X(5), X(1154), X(1157), X(1263)

 

X(6)

 

X(6), X(524), X(671), X(843)

 

X(99)

K084

X(99), X(512)

isogonal nK, see CL001

X(187)

K435

X(2), X(15), X(16), X(187), X(352), X(524), X(843)

 

X(468)

 

X(69), X(468), X(524), X(2770)

 

X(1511)

K614

X(30), X(74), X(399), X(1138), X(1511)

 

 

 

 

 

 

 

 

 

Remark : a central focal cubic with center N passes through a given infinite point P if and only if X lies on psK(X6 x P, X2, X3). For example, with P = X(524), the cubic is K043 = pK(X187, X2).

 

Central equilateral circum-cubics and points Na, Nb, Nc

Let us consider again the net of central circum-cubics with (radial) center N defined as above by K = u Ka + v Kb + w Kc, where P = u:v:w is a point.

K is an equilateral cubic, hence a central stelloid or a K60++, if and only if P lies on three lines L1, L2, L3 which are generally distinct and concurrent at a point Y.

It follows that, in general, there is one and only one central circum-stelloid with center N. For instance, with N = X(2), X(3), X(5) we obtain K213, K080, K026 respectively.

In particular, these lines are parallel (i.e. Y is a point on the line at infinity) if and only if P lies on a conic (C) which contains X(1989).

But if two of these lines coincide then all three coincide in a line L. This occurs when N lies on three circular quartics adding up to the conic (C) and the line at infinity (twice).

Taking N on (C), a quick computation shows that one can find three points Na, Nb, Nc on (C) such that L1, L2, L3 coincide therefore, for each of these points, one can find a pencil of central circum-stelloids with center Ni when P is a point on the corresponding line Li.

These points also lie on the circle C(H, R) and they are the base points, apart X(5), of the pencil of rectangular hyperbolas generated by those passing through X(4), X(5), X(265), X(2574), X(2575), X(2888) and X(5), X(13), X(14), X(3413), X(3414). These are clearly homothetic to the Jerabek and Kiepert hyperbolas respectively.

It follows that these points Na, Nb, Nc are the complements of the points Ua, Ub, Uc we meet in the page K001 and therefore they must lie on the cubics K026, K485, K607. They are also the reflections about X(5) of the intersections (apart A, B, C) of K005 and the circumcircle.

K060 is another cubic containing Na, Nb, Nc.

At last, recall that the Hessian of a central circum-stelloid is a focal central cubic with singular focus the center of the stelloid.