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pK60+

The pK with pole W = (p : q : r) and pivot P = (u : v : w) is a pK60+ if and only if P lies on the Neuberg cubic K001. Its pole W lies on the cubic K095 and its isopivot Q lies on K060. The locus of the common point X of the asymptotes is the bicircular quartic Q004.

The correspondences between W and P are given by the formulas :

Remark 1 : when P is one of the Fermat points X(13) or X(14), the pK60+ decomposes into the cevian lines of P. The corresponding poles are the barycentric squares X(11080), X(11085) of X(13), X(14) respectively.

Remark 2 : the hessian cubic of a pK60+ is always a focal cubic with singular focus the common point X of the three asymptotes.

The table below shows a selection of these cubics and other related points. Recall that W is the barycentric product P x Q.

W on K095

P on K001

Q on K060

X on Q004

cubic

hessian or centers on the hessian

X(6)

X(3)

X(4)

X(2)

McCay K003

McCay hessian cubic K048

X(53)

X(4)

X(5)

X(51)

McCay orthic K049

McCay orthic hessian cubic K957

X(395)

X(617)

X(34295)

X(619)

Fermat K046b

X(533), X(619)

X(396)

X(616)

X(34296)

X(618)

Fermat K046a

X(532), X(618)

X(1989)

X(30)

X(5627)

X(476)

Tixier K037

Tixier hessian cubic K958

X(1990)

X(5667)

X(34297)

X(14847)

K543

X(30), X(14847)

X(2160)

X(1)

X(79)

X(14844)

K097

X(14844)

X(2161)

X(3465)

X(34299)

X(34309)

 

 

X(11063)

X(399)

X(14451)

X(34306)

 

 

X(11069)

X(484)

X(14452)

X(34307)

 

 

X(11070)

X(1138)

X(30)

X(3258)

Fermat K515

X(3258)

X(11071)

X(1263)

X(1141)

X(34308)

 

 

X(11072)

X(1276)

?

?

 

 

X(11073)

X(1277)

?

?

 

 

X(11074)

X(2132)

?

?

 

 

X(11075)

X(3065)

X(80)

X(34311)

 

 

X(11076)

X(3464)

?

?

 

 

X(11077)

X(3484)

?

?

 

 

X(11079)

X(74)

X(265)

X(34310)

 

 

X(11080)

X(13)

X(13)

X(13)

see Remark 1

 

X(11081)

X(5668)

?

?

 

 

X(11082)

X(8175)

?

?

 

 

X(11083)

X(15)

X(11581)

?

 

X(61)

X(11084)

X(1337)

X(14373)

?

 

 

X(11085)

X(14)

X(14)

X(14)

see Remark 1

 

X(11086)

X(5669)

?

?

 

 

X(11087)

X(8174)

?

?

 

 

X(11088)

X(16)

X(11582)

?

 

X(62)

X(11089)

X(1338)

X(14372)

?

 

 

?

X(2133)

X(34298)

?

 

 

?

X(3440)

X(621)

?

 

 

?

X(3441)

X(622)

?

 

 

?

X(8487)

X(1117)

?

 

 

?

X(8431)

X(6761)

?

 

 

?

X(8494)

X(11584)

?

 

 

?

X(8446)

X(11600)

?

 

 

?

X(8456)

X(11601)

?

 

 

?

X(3466)

X(34300)

?

 

 

?

X(7164)

X(34301)

?

 

 

?

X(1157)

X(34302)

?

 

 

?

X(3483)

X(34303)

?

 

 

?

X(8439)

X(34304)

?

 

 

?

X(7165)

X(34305)

?

 

 

 

Other stelloids

The following table gathers together all listed stelloids with their respective types and their listed isogonal transforms (if any) in the last column.

The orange (resp. light blue) cells correspond to stelloids having the same asymptotic directions as K003 (resp. K024). Their isogonal transforms are CircumNormal (resp. CircumTangential) cubics as in Table 25.

The "orange" cubics are called McCay stelloids when they are circum-cubics and then, they are spK(X3, Q) for some Q. The asymptotes concur at X such that QX = 1/3 QH (vectors). See here for further properties.

Notations : o, c, n denote a circumscribed, central, nodal cubic respectively.

stelloid

X = X(i)

o

c

n

type

X(i) on the curve for i =

stelloid*

K003

2

o

 

 

pK

1, 3, 4, 1075, 1745, 3362, 13855, 39641, 39642, 46357, 46358

K003

K024

2

o

 

 

nK0

 

K024

K026

5

o

c

 

psK, spK

3, 4, 5, 5403, 5404, 8798, 14363

K361

K028

381

o

 

n

psK, spK

3, 4, 8, 76, 847, 3557, 3558, 3730, 8743, 10571, 14246, 14247, 14248, 14249, 14250, 14251, 14252, 14253, 14254, 14255, 14256, 14257, 14258, 14259, 14260, 14261, 14262, 14263, 14264, 14265, 14266, 14267, 14268, 25043, 34243, 34756, 38539, 38936, 38937, 38938, 38939, 39263, 39264, 39265, 39267, 39268, 39269, 39270

K009

K037

476

o

 

 

pK

30, 5627

K374

K046a

618

o

c

 

pK

13, 616, 618, 34296

 

K046b

619

o

c

 

pK

14, 617, 619, 34295

 

K049

51

o

 

 

pK

4, 5, 52, 847, 39641, 39642

K373

K054

13364

o

 

n

spK

4, 5, 143, 14254, 14374, 14375, 14584, 14632, 14633, 22100, 27359, 27375, 34356, 34434

 

K071

5891

o

 

 

psK, spK

4, 5, 20, 76, 5562, 15318

 

K077

376

 

 

 

 

1, 3, 20, 170, 194, 7991

 

K078

3524

 

 

 

 

1, 2, 3, 165, 5373, 6194, 21214, 32524

 

K080

3

o

c

 

spK

3, 4, 20, 1670, 1671, 15318

K405

K094

599

o

 

 

nK

 

 

K097

14844

o

 

 

pK

1, 79

 

K100

3

 

c

 

 

1, 3, 40, 1670, 1671, 17749

 

K115

3060

o

 

 

spK0

4, 6243, 25043, 39641, 39642

 

K139

5627

o

c

 

nK

30, 5627

 

K204

34365

o

 

 

nK0

 

 

K205

14846

o

 

 

nK0

 

 

K213

2

o

c

 

nK

2, 11058

 

K230

14629

o

 

n

cK

80, 2222, 23838, 37630

 

K258

549

 

 

n

 

1, 3, 5, 39, 2140, 14823, 15345

 

K268

3819

o

 

 

spK0

4, 20, 140, 15318, 15644, 39641, 39642

 

K309

5054

o

 

 

spK

3, 4, 376, 1340, 1341, 39158, 39159, 39160, 39161

 

K358

3545

o

 

 

spK

3, 4, 381, 39162, 39163, 39164, 39165

 

K412

14845

o

 

 

spK

2, 4, 5, 51, 262, 14249

 

K513

14846

o

 

 

 

6, 15, 16, 74, 265, 3016

 

K514

6781

o

 

 

spK

4, 15, 16, 39, 6781

 

K515

3258

o

 

 

pK

30, 1138, 31378

 

K516

262

o

 

 

spK0

4, 3095, 14251, 39641, 39642

 

K525

4

o

c

 

spK

3, 4, 382

 

K543

14847

o

 

 

pK

5667, 34297

 

K580

568

o

 

 

spK

4, 847, 5889

 

K581

5055

o

 

 

spK

2, 3, 4, 262

 

K582

14848

o

 

 

spK

2, 4, 6, 262, 4846, 14246, 20423, 31861

 

K594

5603

o

 

n

spK

1, 4, 1482, 14260, 17753

 

K595

14849

o

 

n

 

74, 98, 265, 290, 671, 9140

 

K596

14850

o

 

n

 

74, 99, 265, 290, 34174

 

K597

14851

o

 

n

 

30, 74, 265, 477, 43707

 

K598

6

 

 

 

 

 

 

K613

14643

o

 

 

nK, spK

4, 110, 1113, 1114, 38936, 41512

 

K643

14561

o

 

 

spK0

4, 6, 4846, 8743, 39641, 39642

 

K665

549

o

 

 

spK

3, 4, 39, 550

K664

K669

14644

o

 

 

nK, spK

4, 74, 265, 14254

 

K670

14852

o

 

 

psK, spK

4, 26, 64, 68, 847

 

K708

14853

o

 

 

spK0

4, 1344, 1345, 1351, 14248, 18906, 39641, 39642

 

K714

14640

o

 

 

spK

4, 14374, 14375, 30493

 

K724

14854

o

 

n

psK

74, 265, 5961, 6344, 11060, 39373, 39374, 39375, 39376

 

K827

14855

o

 

 

spK

4, 20, 550, 15318

 

K833

2

 

c

 

 

2, 3, 381, 39162, 39163, 39164, 39165, 39641, 39642

 

K852

376

o

 

 

spK

3, 4, 1657, 42411, 42412

 

K910

140

 

 

 

 

1, 3, 546, 1385

 

K911

5054

 

 

 

 

1, 3, 381, 1340, 1341, 3576, 39162, 39163, 39164, 39165

 

K928

10606

o

 

 

spK

4, 20, 64, 68, 12084, 15318, 20427, 38937

 

K929

15061

o

 

 

 

74, 265, 1113, 1114, 15328

 

K930

15362

o

 

 

spK

2, 4, 23, 74, 262, 265, 15360, 15363

K931

K1063

5667

 

 

 

 

1, 3, 8, 40, 21214, 21227, 21228, 21229, 21306, 21307

 

K1064

21445

 

 

 

 

3, 15, 16, 98, 385, 21444

 

K1098

7709

 

 

 

 

3, 194, 39641, 39642

 

K1101

32447

o

 

 

spK

4, 194, 14251

 

K1136

34290

o

 

 

nK0

 

 

K1137

381

o

 

 

nK

 

 

K1139

5640

o

 

 

spK

4, 568, 14254, 39641, 39642

 

K1140

5654

o

 

 

psK, spK

4, 155, 34756, 34757

 

K1161

5085

 

 

 

 

3, 6, 40122

 

K1261

3

 

c

 

 

2, 3, 376, 1670, 1671, 7757, 33706

 

Additional remarks :

• See the related Table 54 for green cells. These are the spK(X3, Q on the Euler line).

• the circum-cubics highlighted in yellow are those of Table 51 : they are spK(X3, Q on the Brocard axis). They form a pencil of K0s passing through the infinite points of K003, X(4) and the imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola. The asymptotes concur at X on the line X(2), X(51), X(262), X(263), X(373), X(511), X(2979), X(3060), etc.