table37


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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