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A general study of spK cubics is to be found in CL055.

In this page, we consider cubics spK(P, Q) with P, Q on the Euler line such that P ≠ X(30). These cubics form a net and all the cubics pass through A, B, C, X(3), X(4).

Naturally, when P = Q, we find the pKs of the Euler pencil of Table 27. This net contains other pencils of remarkable cubics as detailed below. Each pencil is characterized by four other points, not necessarily distinct from the previous five points in which case the tangents at the "repeated" points are the same for all cubics of the pencil.

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Each line of the following table corresponds to a given point Q and all the cubics in this line must pass through the foci of the inconic with center Q. In other words, the cubics in each line are in a same pencil which can be generated by K187 and a cubic pK(X6, Q) of the Euler pencil. Recall that K187 is the locus of foci of inconics with center on the Euler line.

If Q is the point with abscissa t in (X3, X2), any cubic of the pencil related with Q is K(k) = pK(X6, Q) + k K187 where k is a real number or infinity (giving K187).

K(k) is spK(Pk, Qk) where Pk, Qk are the points with abscissa t – k, t + k respectively.

Since pK(X6, Q) and K187 are both self-isogonal cubics (K187 is a nK), the pencil K(k) is invariant under isogonal conjugation and K(k)* = K(– k).

See the comments below to examine various special cubics of a pencil associated with special values of k.

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On the other hand, each column contains cubics related with a certain point P either fixed on the Euler line or simply related with Q. Each column also contains the cubics of a same pencil. These cubics pass through four additional points mentioned at the bottom of the table.

 

Notations :

• aQ, cQ, gQ, tQ are the anticomplement, complement, isogonal, isotomic of Q respectively,

• [Xn] is the reflection of an ETC center Xn about Q. The cubic spK([Xn], Q) is the isogonal transform of spK(Xn, Q) : this is shown by columns of same colour in the table.

D denotes a decomposed cubic into the line at infinity and the Jerabek hyperbola. D* denotes a decomposed cubic into the circumcircle and the Euler line.

• When a cubic is not listed in CTC, the corresponding point P is given when listed in ETC.

t

Q

P=Q

P=aQ

P=aaQ

P=X3

P=[X3]

P=X20

P=[X20]

P=X2

P=[X2]

P=S

P=S'

notes

cubic type –>

pK

psK1

psK2

stelloid

CN

 

central

 

 

 

 

X30

K001

K447

K446

D

D*

D

D*

D

D*

K811

K854

note 1

1

X2

K002

K002

K002

K358

X381

K847

K706

K002

K002

K812

 

note 2

0

X3

K003

K443

K376

K003

K003

K376

K443

K851

K810

K810

K851

note 4

–3

X20

K004

X3146

X5059

K852

X1657

K004

K004

 

?

?

 

 

3/2

X5

K005

K028

K009

K028

K009

K846

K850

K762

K759

K762

K759

note 3

3

X4

K006

K841

K426

K525

X382

K841

K426

 

X3543

K006

K006

 

– 1

X376

K243

X3543

?

 

X3534

K615

K047

K047

K615

?

 

 

3/4

X140

 

K361

K026

K026

K361

 

X3627

 

X549

K026

K361

 

2

X381

 

X376

X3543

 

X3830

 

?

 

K804

X3545

 

 

– 3/4

X548

 

X3627

X1657

K665

K664

K848

K566

 

?

X3534

 

 

1/2

X549

 

X381

X376

K581

X2

 

X3830

 

K581

?

 

 

– 3/2

X550

 

X382

X3529

K080

K405

K405

K080

 

?

K405

K080

 

6/5

X1656

 

K813

X3091

 

X3843

 

?

 

X5071

K813

 

 

– 6

X1657

 

?

?

 

?

 

K814

 

?

K814

 

 

9/8

X3628

 

K849

K569

 

X546

 

?

 

X547

X549

 

 

– 1/2

X8703

 

X3830

?

K309

X376

 

X381

 

X3534

?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

other cubics

 

 

 

 

 

K820

 

 

 

 

 

 

 

 

 

 

 

K844

 

 

 

 

 

 

four points

X1

excenters

X4

A, B, C

X3

midpoints

X4

∞K003

X3

CN triangle

X64

∞K004

X20

antipodes

of A,B,C

X6

∞K002

X2

Thomson

triangle

X3-OAP

isog.

X3-OAP

 

Comments :

• psK1 = spK(aQ, Q) = psK(gaQ, taQ, X3) obtained with k = 3(t – 1).

• psK2 = spK(aaQ, Q) = psK(gtaQ, X2, X3) obtained with k = – 3(t – 1). See Table 50.

• spK(X3, Q) is a stelloid with asymptotes parallel to those of the McCay cubic K003 obtained with k = t. The radial center X is the homothetic of Q under h(X4, 2/3).

• spK([X3], Q) is a CircumNormal cubic obtained with k = – t. See Table 25.

• spK(X20, Q) is a cubic with three real asymptotes parallel to the altitudes of ABC obtained with k = t + 3. See Table 58 for further details and complements.

• spK([X20], Q) is a central cubic with center X3 obtained with k = – (t + 3).

• spK(X2, Q) is a cubic with three real asymptotes parallel to those of the Thomson cubic K002 obtained with k = t – 1.

• spK([X2], Q) is a cubic passing through the vertices of the Thomson triangle obtained with k = 1 – t.

• spK(S, Q) is a cubic passing through the four X3-OAP points. S is the homothetic of Q under h(X4, 4/3). It is obtained with k = (3 – t) / 7. See also Table 53.

• spK(S', Q) is the isogonal transform of spK(S, Q), hence a cubic passing through the isogonal conjugates of the four X3-OAP points. It is obtained with k = – (3 – t) / 7. S' is the homothetic of Q under h(X4, 2/3) hence it is the radial center X of the corresponding stelloid spK(X3, Q).

***

Notes :

• note 1 : these cubics are circular cubics passing through X(30), X(74) but they are not proper spKs. They form a pencil generated by two decomposed cubics namely the union of the line at infinity and the Jerabek hyperbola, the union of the circumcircle of ABC and the Euler line. This pencil also contains the focal cubic K187 and the axial cubic K448.

• note 2 : any spK(P, X2) passes through the foci of the Steiner inellipse.

• note 3 : any spK(P, X5) passes through the four foci of the MacBeath inconic : X(3), X(4) and two imaginary. The tangents at X(3), X(4) pass through X(74) when they are defined i.e. when the cubic is not a nodal cubic such as K009 and K028.

• note 4 : any spK(P, X3) passes through the foci of the inellipse with center X3.