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Let Ω = p : q : r be a point and K(Ω) the pseudo-pivotal cubic psK(Ω, X2, X3). See Pseudo-Pivotal Cubics and Poristic Triangles, specially Tables 3 and 4.

The equation of K(Ω) is : p b^2 c^2 y z ((b^2 - c^2) x + a^2 (y - z)) = 0, showing that all these cubics form a net which can be generated by three independent members. It is convenient to choose K(A), K(B), K(C) since these are very simple. Indeed, K(A) is the decomposed cubic, the union of the sidelines AB, AC and the perpendicular bisector of BC.

Each cubic K(Ω) contains seven fixed points namely A, B, C, X3 and the midpoints of ABC. Recall that the tangents at A, B, C concur at the pseudo-isopivot Ω.

K(Ω) is invariant under two simple transformations, one being the inverse of the other :

T1 : M = x : y : z -> p y z (x + y - z) (x - y + z) : : ,

T2 : M = x : y : z -> p y z (- p y z + q x z + r x y) : : .

***

Other points on K(Ω) :

• K(Ω) meets the line at infinity at the same points as pK(X6, atgΩ). The six remaining common points lie on the circum-conic with perspector the X32-isoconjugate of the tripole of the line Ω-X6. This latter conic passes through gtgΩ.

• K(Ω) meets the circumcircle at the same points as pK(X6, tgΩ). The three remaining common points lie on the line which is the isogonal transform of the conic above and one of them is the pivot tgΩ.

Note that the three cubics coincide into K002 when Ω = X6.

Now, if mΩ denotes the midpoint of atgΩ, tgΩ (which is actually ctgΩ and also the isogonal conjugate of the cevapoint of X6 and Ω), it can be easily verified that K(Ω) is spK(atgΩ, mΩ) as in CL055 with all the consequences mentioned there.

 

Special cubics K(Ω)

Pivotal cubics

K(Ω) is a pK if and only if it contains X2 hence Ω must lie on the Brocard axis. In this case, these cubics form a pencil and each cubic passes through the seven fixed points above and also X2, X6. Obviously, K(Ω) contains the (real or not) square roots of Ω on the Stammler rectangular hyperbola.

gΩ lies on the Kiepert hyperbola and K(Ω) also contains the foci of the inconic with perspector gΩ and center ctgΩ.

The following table gives a list of all CTC cubics and also a selection of other cubics that pass through at least seven ETC centers (2, 3, 6 are not repeated) and/or some other interesting points.

Ω

pK or X(i) on the pK for i =

3

K168 / foci of the orthic inconic

6

K002

15

K341a

16

K341b

32

K177

39

K836

187

K043 / circular cubic

216

K612

511

K357

574

K284

800

K924

1691

K252

2092

K253

3003

K489

3094

K1012

5019

K321

50

50, 186, 323, 11597, 14579

58

27, 58, 86, 2248, 6626, 8044

61

61, 302, 473, 3391, 3392, 10640

62

62, 303, 472, 3366, 3367, 10639

371

371, 492, 1585, 3387, 3388, 10962

372

372, 491, 1586, 3373, 3374, 10960

570

54, 570, 1209, 1594, 2963

577

68, 394, 577, 1147, 6503

1692

114, 230, 460, 1692, 1976

2088

2088, 2433, 5664, 13636, 13722, 15328

3284

30, 265, 1511, 3163, 3284, 11064, 14910, 14919

4258

391, 461, 3635, 4258, 15519

4266

8, 995, 3161, 4266, 5233

5007

251, 428, 3589, 5007, 6292, 15321

5008

597, 1383, 5008, 10301, 15810

5065

1593, 5065, 14379, 14390, 14457, 17811

18591

72, 226, 442, 942, 1214, 18591

 

Circular cubics

K(Ω) is circular if and only if Ω lies on the Lemoine axis. In this case, these cubics also form a pencil and each cubic meets the line at infinity again at a real point M = tgΩ and the circumcircle again at gM = gtgΩ.

The real asymptote is the Simson line of the antipode of gM on (O) hence it envelopes the Steiner deltoid H3 and it meets K(Ω) again on Q011.

The singular focus F of K(Ω) lies on C(X3, R/2) since it is the midpoint of X3-gM.

The only pK of the pencil is K043 = K(X187) with focus X(14650).

There are three focal cubics in this pencil and one of them is always real. They are obtained when Ω is the barycentric product of X6 and one of the infinite points of the Napoleon cubic K005 or, equivalently, the common points of the Lemoine axis and pK(X1501, X51). The singular foci are the midpoints of the segments joining X3 and the common points of (O) and K005. These three focal cubics are actually central cubics, symmetric about their singular focus.

CL068circ CL068foc

Circular K(Ω)

Focal K(Ω)

The following table (computed by Peter Moses) gives a list of all CTC circular cubics and also a selection of other cubics together with the points M, F mentioned above. 0 in the second column means Ω is unlisted in the current edition of ETC (2018-05-18).

M

Ω

cubic

F

X(i) on the cubic for i =

30

1495

K446

12041

3, 4, 30, 74, 133, 1511, 11589, 14993

511

237

K570

12042

3, 32, 98, 132, 511, 5000, 5001, 5403, 5404, 5976, 9467

512

669

 

 

3, 99, 512, 5139

513

667

 

 

3, 100, 513, 5521

514

649

 

 

3, 101, 514, 5190

515

0

 

 

3, 102, 515, 10570

516

0

 

 

3, 103, 516, 14377, 20622

517

0

 

 

3, 56, 104, 517, 1145

518

2223

 

 

3, 105, 518, 3513, 3514, 20621

519

902

 

 

1, 3, 106, 214, 519, 1319, 15898, 20619

520

0

 

 

3, 107, 125, 520

521

1946

 

 

3, 11, 108, 521

522

663

 

 

3, 109, 522, 20620

523

512

K567

1511

3, 110, 136, 523, 6328, 15454

524

187

K043

14650

2, 3, 6, 67, 111, 187, 468, 524, 1560, 2482, 6593, 10354, 13608, 15899, 15900, 18876

525

647

 

 

3, 112, 115, 525

526

14270

 

 

3, 476, 526, 16221

527

1055

 

 

3, 9, 57, 527, 1155, 2291, 6594

532

0

 

 

3, 13, 16, 532, 618, 2380, 3479

533

0

 

 

3, 14, 15, 533, 619, 2381, 3480

539

0

 

 

3, 128, 539, 2383

542

5191

 

 

3, 542, 842, 14357

732

8623

 

 

3, 39, 732, 733, 1691, 8290

758

3724

 

 

3, 10, 36, 65, 758, 759, 10693

912

0

 

 

3, 119, 912, 915, 11517, 18838

916

0

 

 

3, 118, 916, 917

924

0

 

 

3, 135, 924, 925

971

0

 

 

3, 971, 972, 7367

1154

0

 

 

3, 5, 186, 1141, 1154, 11597, 18402

1503

0

 

 

3, 1297, 1503, 14376

2393

0

 

 

3, 25, 206, 858, 2373, 2393, 3455, 5181, 15477

2574

0

 

 

3, 1113, 1313, 2574

2575

0

 

 

3, 1114, 1312, 2575

2782

0

 

 

3, 2698, 2782, 14382

2850

0

 

 

3, 2766, 2850, 5520

3307

0

 

 

3, 1381, 2446, 3307

3308

0

 

 

3, 1382, 2447, 3308

3413

5638

 

 

3, 1379, 2028, 3413, 6178

3414

5639

 

 

3, 1380, 2029, 3414, 6177

3564

0

 

 

3, 114, 1692, 3563, 3564, 6337

3738

8648

 

 

3, 2222, 3738, 13999

5663

0

 

 

3, 477, 5663, 14385, 18809

6000

0

 

 

3, 1294, 6000, 14379

6368

15451

 

 

3, 137, 933, 6368

8057

0

 

 

3, 122, 1301, 8057

8673

0

 

 

3, 127, 1289, 8673

8677

0

 

 

3, 1309, 3259, 8677

8681

0

 

 

3, 126, 2374, 8681

9019

0

 

 

3, 23, 141, 427, 9019, 9076

9028

0

 

 

3, 5513, 9028, 9085

9033

9409

 

 

3, 1304, 3258, 9033

9517

0

 

 

3, 935, 5099, 9517

13754

0

 

 

3, 113, 403, 1147, 1300, 13754

15311

0

 

 

3, 64, 3184, 5897, 6523, 11744, 12096, 15311

15733

0

 

 

3, 55, 3660, 6600, 10427, 15728, 15733

17770

0

 

 

3, 58, 6626, 17770, 20666

 

Other remarkable cubics

K(Ω) is equilateral if and only if Ω = X(51) corresponding to K026.

K(Ω) is a psK+ if and only if Ω lies on a complicated cubic passing through X(51), X(154) corresponding to the cubics K026, K426 which are actually both central cubics.

K(Ω) is an unicursal cubic if and only if Ω lies on a very complicated sextic passing through X(55), X(184), X(8041) corresponding to the cubics K259, K009, K1068.

The table below presents other listed cubics and a selection of K(Ω) with at least seven ETC centers. Some additional points are given : (∞)Knnn and (O)Knnn denote the infinite points and the points on the circumcircle of Knnn.

Ω

K(Ω) or X(i) on the cubic for i =

Other remarkable points

2

K512

(∞)K410, (O)K184, vertices of 1st Brocard triangle

25

K376

(∞)K004, (O)K006

51

K026 / central stelloid

(∞)K003, (O)K005

55

K259 / nodal

(∞)pK(X6,X145), (O)K692

56

K1060

(∞)K1044, (O)K1059, foci of the Mandart inellipse

110

K559

(∞)pK(X6,X148), (O)K035

154

K426 / central

(∞)pK(X6,X3146), (O)K004

184

K009 / nodal

(∞)K006, (O)K003

8041

K1068 / nodal

(∞)pK(X6, X7760), (O)pK(X6, X7794)

13366

K569

(∞)K005, (O)pK(X6,X140)

33578

K1110

(∞)pK(X6, aX17810), (O)pK(X6, X17810)

902

1, 3, 106, 214, 519, 1319, 15898, 20619

 

1193

3, 9, 57, 58, 893, 2092, 3666, 4357, 6626

 

1400

3, 9, 10, 19, 57, 65, 226, 478, 1903, 13478

 

2260

3, 9, 27, 57, 442, 942, 2160, 5249

 

2361

1, 3, 21, 36, 80, 860, 4511, 6149

 

3124

3, 76, 115, 512, 2028, 2029, 3413, 3414, 14384

 

3271

3, 8, 11, 513, 2446, 2447, 3307, 3308

 

7113

3, 9, 36, 57, 79, 104, 1333, 2245, 3218, 19302

 

Remark : for any Ω on the line passing through X(6) and X(41), K(Ω) passes through X(9) and X(57) just like K002.