For any pivot P, the cubic K = pK(O x P, P) has isopivot O = X(3) hence its tangents at A, B, C, P concur at O. The isogonal transform K* of K is a member of CL019 with pivot H. If P = O, the cubic splits into the cevian lines of O. K passes through O, P, the P-Ceva conjugate P/O of O (which is the tangential of O and the tertiary pivot of K) and the crossconjugate O©P of O and P. K meets the line at infinity (L∞) and the circumcircle (O) at the same points as two isogonal pKs with respective pivots P1, P2. We have P1 = a(ctP ÷ H) and P2 = agP. See CL073 for a generalization. pK(O x P, P) passes through a given point M ≠ O if and only if P lies on the circum-conic C(M) with perspector the barycentric quotient of M^2 and the trilinear pole of the line OM. In this case, the cubic must contain O©M and these cubics form a pencil. K is circular if and only if P2 = agP is on (L∞) hence P is on (O). In this case, K is an inversible cubic and its singular focus lies on the circumcircle of the tangential triangle. K is equilateral (i.e. is a pK60) if and only if P1 = O hence P = X(15318). K is a pK+ if and only if P lies on a circum-cubic passing through X(3), X(5), X(20), PM1 = {3,6}/\{217,1154}, PM2 = {3,3164}/\{5,324}. This cubic meets (L∞) at the same points as pK(X6, X34148) and (O) at the same points as pK(X6, PM3) where PM3 = {20,2979}/\{30,49}. These points PM1, PM2, PM3 are now X(41480), X(41481), X(41482) in ETC. The locus of the intersection of the asymptotes is then an equilateral cubic passing through X(2), X(3), X(154), X(1075), X(14059), the cevians of O, the infinite points of K003. *** The following table (contributed by Peter Moses) shows a large selection of these cubics. Notes • When P lies on the Kiepert hyperbola, the pole Ω = O x P of K lies on the Jerabek hyperbola. These cubics form a pencil and pass through X(485), X(486). The third point of K on the line through X(485), X(486) is the P-Ceva conjugate P/O of O. See green cells in the table. • When P lies on the Feuerbach hyperbola, the pole Ω = O x P of K lies on the circum-hyperbola with perspector X(1946) passing through X(6), X(48), etc. These cubics form another pencil and pass through X(1), X(90). The third point of K on the line through X(1), X(90) is the P-Ceva conjugate P/O of O. See blue cells in the table. • When P lies on the circum-hyperbola with perspector X(525), K passes through X(2) and X(69) and Ω lies on the circum-hyperbola with perspector X(520). this gives another pencil corresponding to the purple cells in the table. • The grey cells show the cubics passing through X(6) and X(8770) for P on the circum-conic with perspector X(512). The pink cells show the cubics passing through X(4) and X(254) for P on the circum-conic with perspector X(2501). • The red cells show cubics that belong to at least two of these pencils.
 P K or X(i) on K for i = remark K* 1 K1061 K691 2 K168 K233 4 K006 K006 6 K167 K181 25 K171 K170 32 K411 54 K373 K049 69 K099 K445 74 K114 circular inversible K059 95 K646 K350 98 K336 circular inversible K337 104 K436 circular inversible K334 264 K045 K176 393 K163 1141 K112 circular inversible K050 1976 K781 K718 2373 K113 circular inversible K475 8884 K919 K044 15318 K956 pK60 5 3, 5, 195 pK+ 7 1, 3, 7, 57, 63, 77, 90, 224, 3173, 15474, 40214 K1186 8 1, 3, 8, 40, 78, 90, 271 13 3, 13, 15, 62, 485, 486, 10217, 10661 14 3, 14, 16, 61, 370, 485, 486, 10218, 10662 20 3, 20, 1498 pK+ 21 1, 3, 21, 90, 283, 1800, 1805, 1806, 30556, 30557 42 3, 6, 42, 71, 8770, 15377, 18755 59 3, 59, 100, 1381, 1382, 1822, 1823, 2975, 36059 64 3, 64 K329 76 3, 76, 485, 486, 1670, 1671, 23128 81 3, 58, 63, 77, 81, 272, 284, 15376, 15393 84 1, 3, 84, 90, 1433 K807 96 3, 54, 68, 96, 485, 486 K415 105 3, 105, 3513, 3514, 15382, 34381, 40568, 40569 circular inversible 111 3, 6, 111, 187, 895, 6091, 8681, 8770, 15387, 15390, 15398 circular inversible K209 249 3, 99, 249, 1078, 1379, 1380, 32661 250 3, 110, 250, 1113, 1114, 15460, 15461 K238 251 3, 6, 32, 251, 1176, 1799, 8770, 15371 K517 252 3, 17, 18, 54, 96, 252, 3519, 5449 K416 253 2, 3, 64, 69, 253, 1073, 3146, 15400, 37672 262 3, 262, 485, 486, 1689, 1690, 19139 265 3, 5, 30, 265, 1807, 7100, 10217, 10218, 15392, 20123 287 2, 3, 69, 248, 287, 401, 3289, 12215, 15391, 17974 305 2, 3, 69, 305, 1370, 3926, 20806 306 2, 3, 69, 71, 306, 3151, 3682, 15377, 22133 307 2, 3, 69, 73, 307, 3152, 40152 328 2, 3, 69, 265, 328, 3153, 15392 593 3, 58, 593, 1333, 1444, 1790, 15376, 15408 604 3, 48, 56, 604, 1403, 15373, 15375 847 3, 4, 68, 254, 847, 7505, 34853 909 3, 48, 104, 909, 2252, 15373, 15381 941 1, 3, 6, 90, 941, 5227, 8770, 17594, 34259, 36744 943 1, 3, 35, 72, 90, 943, 1175, 1794, 11517 K1184 961 3, 56, 65, 961, 1791, 1798, 15375 1014 3, 56, 57, 1014, 1444, 1790, 15375 1073 3, 219, 1073, 38292 K709 1169 3, 6, 1169, 1333, 1791, 1798, 8770, 15408 1171 3, 6, 58, 1171, 1796, 8770, 15376 1172 1, 3, 90, 284, 1172, 1754, 15393 1300 3, 4, 186, 254, 1300, 5504, 12028, 15478 circular inversible K339 1441 2, 3, 65, 69, 1214, 1441, 2475 1444 3, 21, 63, 77, 283, 1444, 1790, 1804, 1817 1494 2, 3, 30, 69, 74, 323, 1494, 10419, 14919, 20123 K496 1792 3, 21, 78, 271, 283, 1259, 1792, 1819, 13614 1799 2, 3, 22, 69, 1176, 1799, 28724 1814 3, 63, 77, 1814, 20752, 20769, 36057 1826 3, 4, 71, 254, 1826, 4213, 15377 1896 1, 3, 29, 90, 1715, 1896, 3559 1916 3, 39, 485, 486, 511, 1916, 15372 1937 1, 3, 73, 90, 243, 296, 1758, 1937, 1940, 37142 K1185 1972 2, 3, 69, 216, 511, 1972, 1987, 9291, 14941, 16089, 40853 1974 3, 25, 31, 184, 1974, 11325, 15369, 15370, 15389 1989 3, 6, 265, 1989, 8770, 11063, 15392 2346 1, 3, 55, 90, 2346, 15374, 40443 2359 3, 48, 71, 1791, 1798, 2359, 3955, 15373, 15377 3399 3, 485, 486, 3102, 3103, 3399, 7594 3407 3, 32, 182, 485, 486, 3407, 15371 3964 3, 394, 1259, 1804, 3964, 6617, 15394, 35602 5392 3, 264, 485, 486, 5392, 11090, 11091 5627 3, 74, 265, 3470, 5627, 10419, 15392, 39170 K1157 6330 2, 3, 69, 297, 1297, 6330, 15407 6344 3, 4, 254, 265, 6344, 15392, 37943 6531 3, 4, 248, 254, 419, 6531, 15391 7054 3, 21, 283, 284, 2193, 7054, 15393 8587 3, 187, 485, 486, 575, 8587, 15390 8749 3, 6, 74, 3003, 8749, 8770, 10419 8791 3, 6, 67, 427, 468, 8770, 8791 8882 3, 6, 54, 96, 571, 8770, 8882 K1087 9139 3, 74, 895, 9139, 9717, 10419, 15398 14534 3, 58, 485, 486, 572, 14534, 15376 14910 3, 6, 50, 5504, 8770, 12028, 14910 14998 3, 6, 647, 842, 8770, 14998, 35909 15395 3, 74, 110, 250, 476, 10419, 15395 16080 3, 470, 471, 485, 486, 16080, 39377, 39378 16277 3, 66, 485, 486, 1176, 1799, 2353, 16277 17983 3, 4, 254, 468, 895, 15398, 17983 18018 2, 3, 66, 69, 7391, 14376, 18018 18019 2, 3, 67, 69, 5189, 18019, 34897 18316 3, 265, 485, 486, 3431, 15392, 18316 18532 3, 24, 378, 3422, 7163, 14517, 18532 20336 2, 3, 69, 72, 3998, 20336, 23130 22455 3, 74, 186, 378, 3431, 10419, 22455 30786 2, 3, 69, 858, 895, 6390, 15398, 22151, 30786 32085 3, 4, 25, 254, 1176, 1799, 15369, 32085 34405 3, 394, 489, 490, 491, 492, 8946, 8948, 15394, 34405 35510 2, 3, 69, 3532, 5059, 35510, 36609 36101 3, 63, 77, 103, 2338, 15380, 36101 36121 1, 3, 90, 102, 1735, 15379, 36121 39645 3, 6, 230, 5254, 6530, 8770, 39645 40032 2, 3, 69, 15466, 15740, 37201, 40032, 40680 40395 3, 284, 485, 486, 580, 15393, 40395 40410 2, 3, 5, 69, 1173, 1994, 31626, 40410 40412 2, 3, 21, 69, 81, 272, 283, 1175, 40412 40413 2, 3, 25, 69, 193, 15369, 40413 K1165 40454 1, 3, 90, 1791, 1798, 3435, 40454 40708 2, 3, 69, 3917, 36212, 36214, 40708 40709 2, 3, 69, 10217, 19772, 36296, 40709 40710 2, 3, 69, 10218, 19773, 36297, 40710 40802 3, 6, 183, 394, 1350, 1975, 8770, 15394, 40801, 40802 40824 3, 485, 486, 3926, 10008, 40801, 40824 41013 3, 4, 72, 254, 451, 39131, 41013 41480 3, 41480 pK+ 41481 3, 41481 pK+ 41483 3, 54, 76, 83, 96, 276, 1078, 34386, 41483 K1088 41484 3, 6, 64, 8770, 41484 K182