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The circumcevian and circumanticevian triangles of any point M are perspective at Q. See Table 6.

Now, if P is a fixed point, the points P, M, Q are collinear if and only if M lies on the pivotal cubic with pole X(32) and pivot P.

All pivotal cubics with pole X(32) contain the Lemoine point K and the vertices of the tangential triangle KaKbKc.

The isogonal transform K* of K = pK(X32, P) is pK(X2, tgP), the isotomic pivotal pK with pivot tgP (the isotomic conjugate of the isogonal conjugate of P). See the related Table 35 (for cells highlighted in orange, P on the Euler line) and CL048 for locus properties.

Any cubic K = pK(X32, P) is the barycentric product by X(1) of a cubic K' = pK(X6, P') where P' = P ÷ X(1) = P x X(75). In other words, the trilinear equation of K is the barycentric equation of K'. These three cubics K, K*, K' are equivalent.

The following table shows a large selection of these pKs with at least eight ETC centers and also several special cubics. See notes below table.

P

Centers on the cubic K

K

K* or tgP

K' or P'

X1

X1, X6, X19, X31, X48, X55, X56, X204, X221, X2192

K175

K034

K002

X2

X2, X3, X6, X25, X32, X66, X206, X1676, X1677, X3162

K177

K141

K968

X3

X3, X6, X25, X55, X56, X64, X154, X198, X1033, X1035, X1436

K172

K007

K343

X4

X3, X4, X6, X25, X155, X184, X571, X2165

K176

K045

X92

X19

X6, X19, X48, X2164, X2178

 

X92

K006

X20

X3, X6, X20, X25, X393, X577, X1498, X1660, X1661

K236

K235

X18750

X21

X1, X3, X6, X21, X25, X31, X37, X1333, X1402, X2217, X3185

K430

K254

X333

X22

X3, X6, X22, X25, X159, X2353

K174

X315

X1760

X23

X3, X6, X23, X25, X111, X187, X1177, X2393, X2930

K108

K008

X16568

X25

X3, X6, X25

K171

K170

K1039

X28

X3, X6, X19, X25, X28, X48, X65, X228, X2194, X2218, X2352

K431

K610

K109

X30

X3, X6, X25, X30, X50, X399, X1989

K495

K279

X14206

X35

X6, X35, X42, X55, X56, X58, X1030, X3444, X6186

K1056

K455

K1055

X36

X6, X36, X55, X56, X106, X902, X909, X2183, X3196

K312

K311

K717

X40

X6, X34, X40, X55, X56, X212, X2208, X3197

K179

K154

X329

X48

X6, X19, X48

 

X63

K003

X63

X1, X6, X31, X63, X220, X610, X1407, X1973, X2155

K1043

K605

K169

X69

X6, X69, X159, X1974

K178

X305

X304

X84

X6, X33, X84, X198, X221, X603, X963, X1436, X2187, X2192

K180

K133

X189

X96

X5, X6, X24, X96, X571, X2165, X2351, X3135

 

?

?

X110

X6, X110, X512, X1379, X1380, X2574, X2575

K1067

K242

X662

X163

X6, X163, X661, X1953, X2148, X2576, X2577, X2578, X2579

K1005

K1004

K316

X172

X6, X37, X172, X893, X1333, X2162, X2176, X2248

 

X894

X171

X186

X3, X6, X25, X74, X186, X1495, X2931, X3003

 

K611

?

X206

X6, X66, X206

K160

X22

X2172

X237

X3, X6, X25, X98, X237, X694, X1691, X1971, X1987

 

K355

X1755

X241

X6, X55, X56, X220, X241, X910, X911, X1279, X1407

 

?

X9436

X297

X3, X6, X25, X230, X297, X394, X1503, X2207

 

?

?

X468

X3, X6, X25, X67, X468

K478

?

?

X610

X6, X19, X48, X198, X610, X1436, X2155, X3197

 

X18750

K004

X1495

X4, X74, X1495

 

K860

X2173

X1580

X1, X6, X31, X75, X560, X1403, X1580, X1755, X1910, X1967, X2053

K432

K985

K128

X1582

X6, X19, X48, X75, X82, X560, X1582, X1740, X1964

K999

K998

K020

X1725

X1, X6, X31, X1406, X1725, X1820, X2159, X2173

 

?

X3580

X1953

X6, X19, X48, X1953, X2148

 

X14213

K005

X2173

X6, X19, X48, X2151, X2152, X2153, X2154, X2159, X2173

K1042

X14206

K001

X2223

X6, X55, X56, X105, X292, X1914, X2110, X2223

 

X518

X672

X2303

X6, X19, X37, X48, X1333, X2214, X2281, X2303

 

?

X1010

X2328

X1, X6, X31, X64, X71, X154, X1042, X1474, X2328

 

X1043

X2287

X2360

X6, X19, X48, X64, X73, X154, X2299, X2357, X2360

 

X8822

X1817

X3129

X3, X6, X25, X3129, X3438, X3440, X3490, X11142, X11243, X19305

K1054a

K1053a

 

X3130

X3, X6, X25, X3130, X3439, X3441, X3489, X11141, X11244, X19304

K1054b

K1053b

 

X5596

X6, X66, X159, X206, X5596

K161

 

 

X6660

X3, X6, X25, X2076, X5989, X6660, X8852, X10329, X14370, X17798

K1001

K1000

K968

X7712

X6, X7712, X14479

K922

K092

 

X17798

X6, X55, X56, X2248, X3286, X8301, X8424, X8852, X9500, X17735, X17798, X17962

K1003

K1002

K1025

X18882

X6, X18882

K428

a pK60

 

Notes

• when P lies on the line {1, 19}, K contains X(19), X(48) and K' is an Euler cubic of Table 27. See green cells.

• when P lies on the line {1, 3}, K contains X(55), X(56). See blue cells.