   Any pK with pivot P=u:v:w, isopivot K (Lemoine point) and pole W = P x K (barycentric product) meets the circumcircle at A, B, C (where the tangents are the symmedians) and three other points Q1, Q2, Q3 (not necessarily all real) where the tangents are concurrent at a point X. The first barycentric coordinate of X is : a^2 [2 u v w (–13 b^2 c^2 u + a^2 c^2 v + a^2 b^2 w) + a^4 v^2 w^2 – 3 u^2 (c^4 v^2 + b^4 w^2)] This type of pK is the isogonal transform of a pK with pivot G and pole the isogonal conjugate of P. The pK always contains P, K, P/K (cevian quotient) and the crossconjugate of K and P (the isoconjugate of P/K). Locus property : Let S be the P–Ceva conjugate of K i.e. the perspector of the cevian triangle of P and the tangential triangle. The locus of M such that the anticevian triangle of M and the circumcevian triangle of S are perspective (at Q) is the pK above. The locus of Q is another pK with same pole and pivot the barycentric product of the S–Ceva conjugate of K and tgP (isotomic of isogonal of P). A simple example : with P = X(2) hence S = X(3), the two pKs above are both isogonal cubics, namely K002 and K004 respectively.    Special pK(P x K, P)  Circular cubic The only circular cubic is obtained with P = X(671) and the pole is W = X(111). This is K273. The two isotropic tangents meet at the singular focus F which lies on the tangent at X(111). F is X(11258), the reflection of O in X(111). *** Equilateral cubic The only pK60 is K375. It is obtained when P is the isotomic conjugate of the intersection of the lines X(3)X(523) and X(315)X(524). The three tangents at Q1, Q2, Q3 make 60° angles with one another. *** pK+ The cubic has three concurring asymptotes if and only if P lies on a circumcubic passing through K (the cubic decomposes into the union of the symmedians) and the points X(141), X(193), X(2998). This gives the cubics pK(X39, X141) and pK(X3053, X193). *** Cubics passing through the pole of isoconjugation All the pK(P x K, P) with P on the Kiepert hyperbola contain the pole W = P x K and also G and H. This is the case of the Thomson cubic K002 and several other cubics shown in the following table.  pivot centers on the cubic cubic X2 X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249 K002 X4 X2, X4, X6, X25, X193, X371, X372, X2362 K233 X10 X2, X4, X6, X10, X42, X71, X199, X1654 X13 X2, X4, X6, X13, X15, X62, X1251, X2306, X3129, X3180 X14 X2, X4, X6, X14, X16, X61, X3130, X3181 X76 X2, X4, X6, X22, X69, X76, X1670, X1671 K141 X83 X2, X4, X6, X83, X251, X1176 K644 X94 X2, X4, X6, X94, X265, X1989, X2070 X98 X2, X4, X6, X98, X237, X248, X385, X1687, X1688, X1976 K380 X226 X2, X4, X6, X73, X226, X1400 X262 X2, X4, X6, X262, X263, X1689, X1690, X3148 K791 X275 X2, X4, X6, X54, X275, X1993 X321 X2, X4, X6, X37, X72, X321, X2895, X2915 X598 X2, X4, X6, X598, X1383, X1992, X1995 K283 X671 X2, X4, X6, X23, X111, X524, X671, X895 K273 X801 X2, X4, X6, X20, X394, X801 X1446 X2, X4, X6, X1427, X1439, X1446 X1916 X2, X4, X6, X39, X256, X291, X511, X694, X1432, X1916 K354 X2052 X2, X4, X6, X24, X393, X847, X2052 K621 X2394 X2, X4, X6, X2394, X2433 X2592 X2, X4, X6, X2574, X2592 X2593 X2, X4, X6, X2575, X2593 X2986 X2, X4, X6, X30, X323, X2986 X(14534) X2, X4, X6, X21, X58, X81, X572, X961, X1169, X1220, X1798, X2298 K379   *** Cubics with pivot on the circumcircle All the pK(P x K, P) with P on the circumcircle contain two other imaginary points Q1, Q2 on the circumcircle and on the Lemoine axis (the trilinear polar of K). The tangents at Q1, Q2 meet on the line KP. In this case, P/K is the third point of the cubic on the Lemoine axis and its isoconjugate (P/K)* lies on the isogonal transform of the Steiner inellipse. The pole lies on the circumconic with perspector X(32). Here is a selection of these cubics where P/K is in red and (P/K)* is in blue.  pivot centers on the cubic cubic X74 X6, X15, X16, X74, X1495 X98 X2, X4, X6, X98, X237, X248, X385, X1687, X1688, X1976 K380 X100 X6, X100, X667, X1016 X101 X6, X101, X649, X1252 X103 X6, X55, X103, X672 X105 X1, X6, X57, X105, X238, X1438, X2195, X2223 X106 X6, X106, X902, X2226 X109 X6, X109, X663, X1262 X110 X6, X110, X249, X512 X699 X6, X32, X699, X1691 X727 X6, X31, X727, X1914, X3009 X733 X6, X83, X251, X694, X733 X741 X6, X58, X81, X292, X741, X1326, X1911, X2106 X1297 X3, X6, X511, X1073, X1297 X1298 X6, X54, X275, X1298, X1987 X1477 X6, X56, X1458, X1477 X2249 X6, X284, X1172, X1945, X1949, X2249   *** pK passing through a given point Q distinct of K pK(K x P, P) contains the given point Q if and only if P lies on the circumconic (C) passing through Q and the cevapoint (or cevian product) of K and Q. All these cubics form a pencil of pKs tangent at A, B, C to the symmedians and passing through K, Q and the crossconjugate Q' of K and Q. Examples : if Q = G, we find Q' = H and (C) is the Kiepert hyperbola as seen above. if Q = I, we find Q' = X(57) and (C) is the circumconic through I and G. if Q = O, we find Q' = X(1073) and (C) is the circumconic through O and G. if Q = X(9), we find Q' = X(282) and (C) is the circumconic through X(9) and G. Obviously, the Thomson cubic contains all these points. Here is a selection of these cubics.  pivot centers on the cubic cubic cubics passing through X(9) and X(282) X2 X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249 K002 X9 X6, X9, X198, X259, X282, X2066 X200 X6, X9, X55, X200, X282 X281 X6, X9, X19, X281, X282, X2331 X346 X6, X9, X282, X346, X1604, X2324 X2287 X6, X9, X219, X282, X284, X610, X1172, X2287 X2297 X6, X9, X282, X1449, X2297 cubics passing through O and X(1073) X2 X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249 K002 X97 X3, X6, X54, X97, X275, X577, X1073 X394 X3, X6, X219, X222, X394, X1073, X1433, X1498 X1214 X3, X6, X65, X1073, X1214 X1297 X3, X6, X511, X1073, X1297 cubics passing through I and X(57) X1 X1, X6, X55, X57, X365, X1419, X2067 X2 X1, X2, X3, X4, X6, X9, X57, X223, X282, X1073, X1249 K002 X28 X1, X6, X28, X57, X1474, X1724, X2299, X2352 X57 X1, X6, X56, X57, X266, X289, X1743 X81 X1, X6, X57, X58, X81, X222, X284, X1172, X1433 K319 X88 X1, X6, X36, X44, X57, X88, X106, X1168, X1465, X2226, X2316 K454 X89 X1, X6, X57, X89, X999, X2163, X2364 X105 X1, X6, X57, X105, X238, X1438, X2195, X2223 X274 X1, X6, X57, X86, X274, X333 X277 X1, X6, X57, X277, X2191 X278 X1, X6, X19, X34, X57, X278, X1723 X279 X1, X6, X57, X269, X279, X1617 X291 X1, X6, X42, X57, X239, X291, X292, X672, X894, X1757, X1967 K135 X330 X1, X6, X57, X87, X330, X2319 X959 X1, X6, X57, X959, X1460, X2258 X961 X1, X6, X57, X961, X1402 X985 X1, X6, X31, X57, X985, X2280 X1002 X1, X6, X57, X1002, X2279 X1170 X1, X6, X57, X218, X1170, X1174 X1219 X1, X6, X57, X1219, X2297 X1255 X1, X6, X35, X37, X57, X1126, X1171, X1255 X1257 X1, X6, X57, X72, X1257, X2983 X1258 X1, X6, X57, X171, X213, X1258 X1280 X1, X6, X57, X518, X1280 X1390 X1, X6, X57, X984, X1390 X1422 X1, X6, X57, X1413, X1422, X1436 X1432 X1, X6, X57, X893, X1431, X1432 X2006 X1, X6, X57, X1411, X2006, X2161 X2982 X1, X6, X57, X65, X1175, X2003, X2259, X2982 X2990 X1, X6, X57, X517, X2323, X2990     Generalization CL043 is a sub-class of a more general type of pK intersecting the circumcircle at three points where the tangents are concurrent. If W = p:q:r and P=u:v:w are the pole and the pivot of the pK, we must have the following conditions : for a given pivot P, W lies on a quintic Q(P) passing through : A, B, C which are nodes, the tangents being the cevian lines of X(32) and the sidelines of the anticevian triangle of P x K P x K (barycentric product) P^2 (barycentric square) and the vertices of its cevian triangle for a given pole W, P lies on a quintic Q(W) passing through : A, B, C which are nodes the square roots of W W÷K (barycentric quotient) the common points of the circumcircle and the trilinear polar of W÷K (barycentric quotient) the common points of the circumcircle and the line passing through W÷K and the crossconjugate of K and W÷K the vertices of the cevian triangle of Z, isoconjugate of the crossconjugate of K and W÷K in the isoconjugation with fixed point W÷K for a given pole W, the isopivot S lies on a quartic Q(S) which is the isoconjugate of Q(W). Their equations are downloadable as a text file.      For example, with W = K, we obtain the circular quintic Q063 and its isogonal conjugate Q113.  