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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 CL048 for related 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 รท 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.

P

Centers on the cubic

K

K*

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

X75

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

 

 

K006

X20

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

K236

K235

 

X21

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

K430

K254

X333

X22

X3, X6, X22, X25, X159, X2353

K174

pK(X2, X315)

X1760

X23

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

K108

K008

 

X25

X3, X6, X25

K171

K170

X19

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

 

X36

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

K312

K311

X3218

X40

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

K179

K154

X329

X48

X6, X19, X48

 

 

K003

X63

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

 

 

K169

X69

X6, X69, X159, X1974

K178

pK(X2, 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

 

 

 

X163

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

 

 

K316

X172

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

 

 

X171

X186

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

 

 

 

X206

X6, X66, X206

K160

pK(X2, X22)

X2172

X237

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

 

K355

X1755

X241

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

 

 

 

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

 

 

K004

X1580

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

K432

pK(X2, X1966)

K128

X1582

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

 

 

K020

X1725

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

 

 

X3580

X1953

X6, X19, X48, X1953, X2148

 

 

K005

X2173

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

 

 

K001

X2223

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

 

 

X672

X2303

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

 

 

X1986

X2328

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

 

 

X2287

X2360

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

 

 

X1817

X5596

X6, X69, X159, X206, X5596

K161

 

 

P428

X6

K428

a pK60

 

 

 

 

 

 

Notes :

P428, intersection of the lines X(20)X(49) and X(23)X(206), SEARCH = 6.97881772221443