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.