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Circular pK(W, H)

Let P be the pivot of a pK. When P is different of H, there is one and only one circular such pK but when P = H, there are infinitely many such cubics.

These cubics have their singular focus F on the nine point circle and their pole W on the orthic axis. The isoconjugate H* of H is the point at infinity of the cubic.

pK(W, H) is also the locus of M whose orthotransversal has a fixed direction, that of any perpendicular to the direction given by H*.

The intersection with the real asymptote is X, the antipode of F on the nine point circle and this asymptote envelopes the Steiner deltoid H3. The "last" point on the circumcircle is E orthoassociate of X.

These cubics are isogonal pivotal circular cubics with respect to the orthic triangle.

They are also invariant under orthoassociation i.e. inversion in the polar circle and under three other inversions with poles A, B, C swapping H and the feet Ha, Hb, Hc of the relative altitude. They are also antigonal cubics i.e. cubics invariant under antigonal conjugation. They are the isogonal transforms of the inversible cubics.

All circular pK(W, H) form a pencil of circular cubics which contains three focal cubics : the pole must be a common point Wa, Wb, Wc of the orthic axis and a symmedian of ABC. For example pK(Wa, H) has singular focus Ha, meets its real asymptote at Xa antipode of Ha on the nine point circle. This asymptote is parallel to OA.

See also the related quartics at Q107.

The following table gives a selection of circular pK(W, H) with corresponding X and E.

H*

cubic

centers on the curve

W

F

X30

K059

X4, X13, X14, X30, X113, X1300

X1990

X125

X511

K337

X4, X114, X371, X372, X511, X2009, X2010, X3563

X232

X115

X512

X4, X112, X115, X512

X2489

X114

X513

X4, X11, X108, X513

X6591

X119

X514

X4, X116, X514, X26705

X7649

X118

X515

X4, X117, X515, X32706

X8755

X124

X516

X4, X118, X516, X917

X1886

X116

X517

K334

X1, X4, X46, X80, X119, X517, X915, X1785, X1845

X14571

X11

X518

X4, X120, X518, X15344

X5089

X5511

X519

X4, X121, X519, X40101

X8756

X5510

X520

X4, X122, X520, X1301

X647

X133

X521

X4, X123, X521, X40097

X650

X25640

X522

X4, X124, X522, X26704

X3064

X117

X523

X4, X107, X125, X523

X2501

X113

X524

K209

X2, X4, X126, X193, X468, X524, X671, X2374

X468

X5512

X525

X4, X127, X525, X1289

X523

X132

X526

X4, X526, X1304, X3258

?

X25641

X690

X4, X690, X935, X5099

X14273

X16188

X698

X4, X76, X698, X1916

?

?

X740

X4, X10, X242, X740

?

?

X900

X4, X900, X1309, X3259

?

X31841

X924

X4, X110, X136, X924

X6753

X131

X926

X4, X926, X1566, X40116

?

X33331

X1154

K050

X4, X5, X15, X16, X52, X128, X186, X1154, X1263, X2383, X2902, X2903, X2914, the Neuberg cubic of the orthic triangle

X11062

X137

X1503

X4, X98, X132, X1503

X16318

X127

X1510

X4, X137, X933, X1510

?

X128

X2390

X4, X106, X2390, X20619

?

?

X2393

K475

X4, X6, X25, X67, X111, X858, X1560, X2393

X14580

X14672

X13754

K339

X3, X4, X131, X155, X265, X403, X1299, X1986, X13754

X3003

X136

Remark : the centers Oa, Ob, Oc of the osculating circles at A, B, C to any pK(W, H) above lie on three perpendiculars to the real asymptote. In other words, the triangles ABC and OaObOc are perspective with perspector at infinity. OaObOc is also perspective to HaHbHc with perspector on a complicated sextic.

 

Other remarkable pK(W, H)

Let K = pK(W, H) be a pivotal cubic with isopivot H* not lying on the line at infinity. Recall that K is an isogonal pivotal cubic with pivot H* with respect to the orthic triangle.

• K is a pK+ i.e. has concurring asymptotes if and only if W lies on K208. In this case, H* (blue cells in the table) lies on K071 and the asymptotes concur on K412.

• When the pole W (green cells in the table) lies on the Brocard axis of the orthic triangle T, H* lies on the Euler line of T and the cubic is a member of the Euler pencil of T. It contains X(5) and X(52). This Brocard axis passes through X(i) for i = 5, 53, 216, 233, 3199, 6116, 6117, 6750, 8887, 10003, 10600, 11062, 14576, 14577, 14640, 27358, 27359, 27371, 32428, 35719, 36412, 39530, 39569.

• pK(W, H) contains its pole W if and only if W lies on the Euler line and then H* lies on the line GK (pink cells in the table). These cubics belong to a same pencil and always contain G and X(193).

H*

W

Cubic or X(i) on the cubic

M for OAPs

Remarks

X3

X6

K006 Orthocubic

X3

isogonal pK

X5562

X216

K044 Euler central cubic

X185

the only central cubic, a pK+, the Darboux cubic of the orthic triangle

X5

X53

K049 McCay orthic cubic

X550

the only equilateral cubic, a pK+, the McCay cubic of the orthic triangle

X69

X2

K170

X6776

isotomic pK

X184

X32

K176

 

 

X2

X4

K181

X376

 

X37669

X20

K182

 

 

X6

X25

K233

X1350

 

X125

X115

K238

 

 

X20

X1249

K329

X4

a pK+

X51

X3199

K350

 

the Thomson cubic of the orthic triangle

X52

X14576

K415

 

the Orthocubic of the orthic triangle

X143

X14577

K416

 

the Napoleon cubic of the orthic triangle

X25

X2207

K445

 

 

X1495

X14581

K496

 

 

X141

X427

K517

 

 

X1

X19

K691

X40

 

X1249

X6525

K709

 

 

X325

X297

K718

 

 

X40

X2331

K807

X1

 

X343

X5

K1087

 

 

X27374

?

K1088

 

 

X1511

X39176

K1157

 

 

X6467

X1196

K1165

 

 

X942

X1841

K1184

 

 

X1936

X2202

K1185

 

 

X55

X607

K1186

 

 

X3690

X1500

K1187

 

 

X4

X393

union of the altitudes

X20

a pK+

X76

X264

X(4), X(76)

 

a pK+

?

X233

X(4), X(5), X(52), X(140)

 

 

X1568

?

X(4), X(5), X(30), X(52), X(113), X(1568)

 

 

X1213

X430

X(2), X(4), X(10), X(193), X(430), X(1213), X(1839), X(2901)

 

 

X1350

?

X(4), X(371), X(372), X(1350)

X6

 

X376

X40138

X(4), X(376)

X2

 

X550

?

X(4), X(550)

X5

 

X22

X8743

X(4), X(22)

X378

 

X378

?

X(4), X(378)

X22

 

 

 

 

 

 

M-OAP points

These points are studied in Table 53. We simply recall that any pK with pivot H, isopivot H* (not lying on the line at infinity) contains the four M-OAP points where M is the reflection of H* about the circumcenter O.

***

Additional cubics by Peter Moses

Ω

X(i) on pK(Ω, H) for i =

3

2, 3, 4, 155, 193, 394, 5374, 6391, 6504

39

3, 4, 69, 155, 427, 1843, 3917, 19583

235

2, 4, 185, 193, 235, 2052, 9307, 13567

325

4, 76, 114, 487, 488, 511, 6393, 8781

343

4, 264, 487, 488, 5562, 8905, 11090, 11091

430

2, 4, 10, 193, 430, 1213, 1839, 2901

571

3, 4, 24, 155, 1147, 2904, 15317, 34756

1184

4, 6, 25, 487, 488, 5286, 7386, 19459, 34817

1495

4, 371, 372, 3284, 6110, 6111, 36296, 36297

1824

4, 10, 19, 33, 37, 209, 226, 2901

3172

4, 6, 20, 25, 154, 1249, 3532, 32605

4185

1, 2, 4, 46, 193, 940, 4185, 5307

6620

2, 4, 193, 263, 393, 3424, 6525, 6620, 6776, 7735, 9292, 9752

7473

2, 4, 193, 542, 648, 7473, 14999, 16188

8735

4, 11, 513, 2588, 2589, 3307, 3308, 34434

8739

4, 13, 15, 18, 371, 372, 2902, 10633, 15441, 15445

8740

4, 14, 16, 17, 371, 372, 2903, 10632, 15442, 15444

8754

4, 115, 512, 2039, 2040, 3413, 3414, 14618, 27375

14165

4, 340, 470, 471, 2992, 2993, 38427, 38428

18487

4, 30, 113, 265, 381, 1531, 4846, 40909

34854

4, 6, 25, 232, 297, 1987, 6530, 16081, 17980, 39931

36417

4, 6, 25, 1974, 2207, 13854, 17409, 34207

39008

4, 30, 113, 122, 520, 1650, 9033, 16177, 38956

40938

4, 22, 315, 371, 372, 427, 1843, 3313

 

Bi-isogonal pK(W, H)

Recall that K= pK(W, H) is an isogonal pivotal cubic with pivot H* = W ÷ H with respect to the orthic triangle. The isopivot is the H-Ceva conjugate of H*. It is the tangential of H* in pK(W, H).

When W lies on K627 = pK(X3199, X393), this cubic is also an isogonal pK with respect to another triangle T and then the cubic is said to be a bi-isogonal pK with a second pivot Q.

See Bi-isogonal and Tri-isogonal Pivotal Cubics.

 

Points of pK(W, H) on (L∞) and (O)

Every cubic pK(W, H) meets the line at infinity (L∞) and the circumcircle (O) at the same points as two isogonal pKs with respective pivots P1, P2.

Obviously, when W = X(6), P1 = P2 = H corresponding to the Orthocubic K006.

When W ≠ X(6), P1 = a(Ω ÷ H) and P2 = at(cevapoint(Ω ÷ X6, H)). See CL073 for a generalization.

The isogonal transform of pK(W, H) is a pK with isopivot O, a member of CL074,