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These are the pivotal cubics with pivot X(1), the incenter of ABC, and isopivot P*.

All these cubics contain A, B, C, X(1), the vertices of the incentral triangle.

For any P* on the cubic K702 = psK+(X1125, X75, X1), the cubic pK(X1 x P*, X1) has concurring asymptotes and is therefore a pK+. This is the case when P* is X(1) giving a decomposed cubic, X(8) giving K157, X(79) giving K097, X(274), X(1125).

The table gives cubics passing through at least seven centers of ETC and some special other cubics. The pole W is the barycentric product X(1) x P*.

P*

W

Cubic or centers on the cubic

X2

X1

K101

X6

X31

X1, X6, X55, X57, X365, X1419, X2067

X8

X9

K157

X31

X32

K175

X36

 

K206, the only circular cubic

X42

X213

K362

X56

X604

K632

X71

X228

X1, X6, X55, X71, X72, X1214, X1751

X79

X2160

K097, the only equilateral cubic

X210

X1334

X1, X8, X9, X37, X42, X210, X3158

X212

 

X1, X3, X6, X55, X212, X219, X3157

X255

X577

X1, X3, X255, X921, X1069, X1124, X1335, X3157

X350

X239

K770

X603

 

X1, X3, X56, X221, X603, X1433, X3157

X672

X2223

X1, X6, X55, X241, X292, X518, X672, X673

X851

 

X1, X65, X73, X243, X851, X1936, X2652, X2660

X902

X2251

X1, X6, X44, X55, X106, X678, X902, X1319, X2161, X2342

X922

 

X1, X31, X48, X896, X922, X923, X2157

X1042

 

X1, X34, X56, X64, X65, X73, X207, X221, X1042

X1096

X2207

X1, X19, X33, X34, X204, X207, X1096, X2331

X1109

X115

K672

X1407

X1106

X1, X56, X57, X221, X1407, X1419, X1422

X1418

 

X1, X57, X279, X354, X1418, X1419, X2293

X1427

X1042

X1, X57, X65, X73, X278, X1419, X1427

X1457

 

X1, X56, X102, X221, X517, X1411, X1457

X1464

 

X1, X36, X65, X73, X74, X1464, X1870

X1755

X237

X1, X31, X48, X240, X1755, X1821, X1959, X1967

X1914

X2210

K774

X1926

 

X1, X75, X336, X350, X1909, X1926, X1934, X1966, X3112

X1933

X14602

K861

X2151

 

X1, X31, X48, X1094, X1250, X2151, X2153, X2307

X2206

 

X1, X31, X48, X58, X501, X1474, X2206

X2238

 

X1, X37, X42, X238, X239, X2107, X2238

X2268

 

X1, X6, X55, X940, X958, X2268, X2339

X2276

X869

X1, X2, X6, X55, X192, X869, X984, X1002, X2276

X2293

 

X1, X6, X55, X354, X1212, X2191, X2293

X2308

 

X1, X6, X55, X58, X501, X1100, X1962, X2160, X2308

X2309

X1197

X1, X6, X55, X86, X893, X1107, X2309

X2594

 

X1, X35, X54, X65, X73, X2307, X2594, X2599

X3056

 

X1, X6, X55, X982, X2319, X3056, X3061

X3116

X3117

X1, X31, X48, X75, X1469, X2186, X2275, X2276, X3056, X3116