The two most famous isotomic pivotal cubics K007 = Lucas cubic and K008 = Droussent cubic generate a pencil of cubics (K) we call the Lucas - Droussent pencil. Each member of this pencil is (K), an isotomic pK with pivot P on the line X4, X69 and isopivot on the Jerabek hyperbola. These cubics are : – the isogonal conjugates of pKs with pole X32, pivot on the Euler line. See Table 34. – the complements (K') of pKs with pole cP (the complement of P) on the Brocard axis, pivot G i.e. the cubics of the pencil generated by K002 = Thomson cubic and K043 = Droussent medial cubic. The nine base points of the pencil are A, B, C, G, H, X69, Ga, Gb, Gc (vertices of the antimedial triangle). The following tables (contributed by Peter Moses) presents a selection of these cubics according to the pivot P or the isopivot P* or the third point Q on the line X2, X69. The base points are not repeated.
 P X(i) on (K) for i = 2, 4, 69 and (K) (K') 4 193, 487, 488, 2996, 13386, 13387, 13428, 13439, 19583, 24243, 24244 K170 K168 69 7, 8, 20, 189, 253, 329, 1032, 1034, 5932, 14361, 14362, 14365, 34162, 39158, 39159, 39160, 39161, 41080 K007 K002 76 6, 22, 76, 1670, 1671, 18018, 19613 K141 K836 264 3, 254, 264, 1993, 5392, 40697, 40698 K045 K612 286 21, 63, 72, 92, 286, 1441, 2997, 3868, 40571 K610 --- 311 54, 311, 1994, 2888, 3459, 7488, 11140 --- --- 314 1, 65, 75, 81, 314, 321, 1764, 2995, 3869, 16049, 17751, 20028 K254 K253 315 66, 315, 1370, 5596, 13575 --- K177 316 67, 316, 524, 671, 858, 2373, 11061, 13574, 14360, 14364, 34163, 34164, 34165, 34166, 39157 K008 K043 317 68, 317, 6193, 6504, 6515 --- --- 340 30, 265, 340, 1494, 2986, 3580, 12383 K611 --- 511 290, 385, 401, 511, 1916, 1972, 25332, 39355 K355 K357 621 616, 621, 628, 2992, 3180, 11121, 16770, 19713, 19772, 19774, 19776, 19778, 36929 K1053a K341a 622 617, 622, 627, 2993, 3181, 11122, 16771, 19712, 19773, 19775, 19777, 19779, 36928 K1053b K341b 877 877, 879, 4226, 34211 --- --- 1232 1173, 1232, 2889, 34545 --- --- 1234 1175, 1234 --- --- 1235 1176, 1235, 6636 --- --- 1236 1177, 1236, 2892, 22151, 37929 --- --- 1330 1330, 1654, 3151, 6625, 8044 --- --- 1352 1352, 7774 --- --- 1843 384, 1843, 9229, 10340 --- --- 2893 2893, 3152, 17778 --- --- 3186 3186, 3552 --- --- 3260 74, 94, 146, 323, 1138, 1272, 2071, 3260 K279 K489 3818 3818, 7837 --- --- 5207 147, 1031, 2896, 4388, 4645, 5207, 7224, 7261, 7779, 9473, 11606, 20021, 20022, 34214 K1000 K252 5562 5562, 8613, 8795, 9290 --- --- 7768 83, 141, 1369, 7768, 15321 --- --- 7850 598, 599, 7850 --- --- 7860 7860, 40341 --- --- 9230 695, 3051, 9230, 37889, 39953, 40016 --- --- 9291 418, 9291 --- --- 10446 10446, 37683 --- --- 10449 1246, 7560, 10449, 17379 --- --- 10477 10477, 16998 --- --- 11185 1992, 5485, 5486, 7493, 11185 --- K284 14221 99, 523, 14221, 14223, 14999 --- --- 14615 64, 394, 2052, 3346, 6225, 6527, 11413, 14615 K235 K924 14994 3329, 14994 --- --- 15164 1113, 2574, 2592, 8115, 15164, 22339 --- --- 15165 1114, 2575, 2593, 8116, 15165, 22340 --- --- 17139 4080, 16704, 17139, 38955 --- --- 17984 237, 17984, 18024, 36214 --- --- 18906 194, 263, 2998, 3212, 6194, 7155, 7766, 18906, 19222, 20023, 24349, 32937 K1037 K1012 20558 20536, 20558 --- --- 21287 1029, 2895, 21287 --- --- 22468 801, 13567, 22466, 22468, 22647 --- --- 32000 3522, 11469, 15740, 32000 --- --- 32001 3146, 15077, 32001, 35510 --- --- 32002 5, 95, 3519, 11271, 32002 --- --- 32006 7396, 16774, 20080, 32006, 38259 --- --- 33297 10, 86, 8049, 15320, 17135, 33297 --- --- 34282 940, 34258, 34282 --- --- 34507 7777, 34507 --- --- 38434 38433, 38434 --- --- 38437 38436, 38437 --- --- 38440 38439, 38440 --- --- 39266 39099, 39266 --- ---
 P* X(i) on (K) for i = 2, 4, 69 and (K) 3 3, 254, 264, 1993, 5392, 40697, 40698 K045 4 7, 8, 20, 189, 253, 329, 1032, 1034, 5932, 14361, 14362, 14365, 34162, 39158, 39159, 39160, 39161, 41080 K007 6 6, 22, 76, 1670, 1671, 18018, 19613 K141 54 54, 311, 1994, 2888, 3459, 7488, 11140 --- 64 64, 394, 2052, 3346, 6225, 6527, 11413, 14615 K235 65 1, 65, 75, 81, 314, 321, 1764, 2995, 3869, 16049, 17751, 20028 K254 66 66, 315, 1370, 5596, 13575 --- 67 67, 316, 524, 671, 858, 2373, 11061, 13574, 14360, 14364, 34163, 34164, 34165, 34166, 39157 K008 68 68, 317, 6193, 6504, 6515 --- 69 193, 487, 488, 2996, 13386, 13387, 13428, 13439, 19583, 24243, 24244 K170 70 70, 13579, 37444 --- 71 71, 4184, 17220, 17911 --- 72 21, 63, 72, 92, 286, 1441, 2997, 3868, 40571 K610 73 73, 4225 --- 74 74, 94, 146, 323, 1138, 1272, 2071, 3260 K279 248 248, 393, 1297, 2987, 3926, 30737, 34137, 37183 --- 265 30, 265, 340, 1494, 2986, 3580, 12383 K611 290 290, 385, 401, 511, 1916, 1972, 25332, 39355 K355 695 695, 3051, 9230, 37889, 39953, 40016 --- 879 877, 879, 4226, 34211 --- 895 23, 895, 18019, 37784 --- 1173 1173, 1232, 2889, 34545 --- 1175 1175, 1234 --- 1176 1176, 1235, 6636 --- 1177 1177, 1236, 2892, 22151, 37929 --- 1243 1243, 36029 --- 1246 1246, 7560, 10449, 17379 --- 1439 1439, 1817, 1895, 19611 --- 1903 1446, 1903, 2184, 2287, 18750 --- 1987 1987, 3289 --- 2435 2435, 4230 --- 2574 1113, 2574, 2592, 8115, 15164, 22339 --- 2575 1114, 2575, 2593, 8116, 15165, 22340 --- 2992 616, 621, 628, 2992, 3180, 11121, 16770, 19713, 19772, 19774, 19776, 19778, 36929 K1053a 2993 617, 622, 627, 2993, 3181, 11122, 16771, 19712, 19773, 19775, 19777, 19779, 36928 K1053b 3426 3426, 15066, 21312, 34289 --- 3431 3431, 10298, 11004, 18387 --- 3519 5, 95, 3519, 11271, 32002 --- 3521 550, 3521 --- 3527 3527, 5422, 11414 --- 3532 3532, 37672 --- 4846 376, 4846, 15454, 36889, 37645 --- 5486 1992, 5485, 5486, 7493, 11185 --- 5504 186, 328, 5504 --- 5505 5505, 37980 --- 6145 275, 343, 6145, 12225, 32354 --- 6391 25, 305, 6339, 6391, 6392, 40318 --- 8044 1330, 1654, 3151, 6625, 8044 --- 8795 5562, 8613, 8795, 9290 --- 9513 9513, 37918 --- 10097 10097, 11634 --- 10099 4236, 10099 --- 10261 5591, 10261 --- 10262 5590, 10262 --- 10293 10293, 40112 --- 10378 1896, 10378 --- 10693 10693, 37783 --- 11138 395, 11138, 13484, 40706 --- 11139 396, 11139, 13483, 40707 --- 11559 11559, 18859 --- 11564 10296, 11564 --- 11744 11064, 11744, 16080, 16386 --- 13472 13472, 38435 --- 13622 3629, 13622 --- 13623 8703, 13623 --- 14220 7471, 14220 --- 14380 14380, 15329 --- 14457 11433, 14457, 37201 --- 14483 14483, 15018 --- 14528 14528, 38444 --- 14542 11427, 14542 --- 14841 3843, 14841 --- 14861 548, 14861 --- 15002 2937, 15002 --- 15077 3146, 15077, 32001, 35510 --- 15232 226, 333, 15232, 20245 --- 15316 24, 15316, 20563 --- 15317 26, 15317, 20564 --- 15320 10, 86, 8049, 15320, 17135, 33297 --- 15321 83, 141, 1369, 7768, 15321 --- 15328 2394, 2407, 15328, 30508, 30509, 30512 --- 15453 15453, 40049 --- 15740 3522, 11469, 15740, 32000 --- 15749 5059, 15749 --- 16665 15331, 16665 --- 16774 7396, 16774, 20080, 32006, 38259 --- 16867 1658, 16867 --- 17040 10565, 17040 --- 17505 15704, 17505 --- 18123 2475, 18123 --- 18124 7391, 18124 --- 18125 5189, 18125 --- 18434 18434, 37638 --- 18550 3534, 18550 --- 19222 194, 263, 2998, 3212, 6194, 7155, 7766, 18906, 19222, 20023, 24349, 32937 K1037 20029 388, 3436, 5739, 8048, 20029, 30479 --- 20421 20421, 35493 --- 21400 1657, 21400 --- 22334 17811, 22334, 30698, 37874 --- 22336 597, 10302, 22336 --- 22466 801, 13567, 22466, 22468, 22647 --- 26861 3530, 26861 --- 28786 27, 306, 28786 --- 28787 28, 20336, 28787 --- 28788 29, 307, 28788 --- 30496 1613, 30496, 32747, 40162 --- 32533 3529, 32533 --- 32618 5002, 32618 --- 32619 5003, 32619 --- 33565 3153, 13582, 33565, 37779 --- 34135 34135, 41195, 41199 --- 34136 34136, 41194, 41198 --- 34207 20806, 34207 --- 34259 4189, 34259 --- 34440 22133, 34440 --- 34483 140, 34483, 40410 --- 34800 3651, 34800 --- 34801 378, 34801 --- 34802 7464, 34802 --- 34817 7485, 34817 --- 35364 2421, 35364 --- 35909 110, 850, 7468, 35909 --- 36214 237, 17984, 18024, 36214 --- 36296 34008, 36296 --- 36297 34009, 36297 --- 38260 7387, 38260 --- 38263 9909, 38263 --- 38433 38433, 38434 --- 38436 38436, 38437 --- 38439 38439, 38440 --- 38535 5196, 38535 --- 38955 4080, 16704, 17139, 38955 --- 40441 7512, 40441 --- Q X(i) on (K) for i = 2, 4, 69 and (K) 6 6, 22, 76, 1670, 1671, 18018, 19613 K141 81 1, 65, 75, 81, 314, 321, 1764, 2995, 3869, 16049, 17751, 20028 K254 86 10, 86, 8049, 15320, 17135, 33297 --- 141 83, 141, 1369, 7768, 15321 --- 183 183, 262 --- 193 193, 487, 488, 2996, 13386, 13387, 13428, 13439, 19583, 24243, 24244 K170 230 230, 8781 --- 298 13, 298 --- 299 14, 299 --- 302 17, 302 --- 303 18, 303 --- 323 74, 94, 146, 323, 1138, 1272, 2071, 3260 K279 325 98, 325 --- 333 226, 333, 15232, 20245 --- 343 275, 343, 6145, 12225, 32354 --- 385 290, 385, 401, 511, 1916, 1972, 25332, 39355 K355 394 64, 394, 2052, 3346, 6225, 6527, 11413, 14615 K235 395 395, 11138, 13484, 40706 --- 396 396, 11139, 13483, 40707 --- 491 486, 491 --- 492 485, 492 --- 524 67, 316, 524, 671, 858, 2373, 11061, 13574, 14360, 14364, 34163, 34164, 34165, 34166, 39157 K008 597 597, 10302, 22336 --- 599 598, 599, 7850 --- 940 940, 34258, 34282 --- 966 966, 17732 --- 1007 1007, 7612 --- 1184 1184, 40831 --- 1211 1211, 14534 --- 1213 1213, 32014 --- 1270 1131, 1270 --- 1271 1132, 1271 --- 1613 1613, 30496, 32747, 40162 --- 1654 1330, 1654, 3151, 6625, 8044 --- 1812 1812, 40149, 40457 --- 1992 1992, 5485, 5486, 7493, 11185 --- 1993 3, 254, 264, 1993, 5392, 40697, 40698 K045 1994 54, 311, 1994, 2888, 3459, 7488, 11140 --- 2238 2238, 9510, 40017 --- 2287 1446, 1903, 2184, 2287, 18750 --- 2407 2394, 2407, 15328, 30508, 30509, 30512 --- 2421 2421, 35364 --- 2895 1029, 2895, 21287 --- 3051 695, 3051, 9230, 37889, 39953, 40016 --- 3068 3068, 5490 --- 3069 3069, 5491 --- 3180 616, 621, 628, 2992, 3180, 11121, 16770, 19713, 19772, 19774, 19776, 19778, 36929 K1053a 3181 617, 622, 627, 2993, 3181, 11122, 16771, 19712, 19773, 19775, 19777, 19779, 36928 K1053b 3231 3231, 34087, 39361 --- 3289 1987, 3289 --- 3314 3314, 3407 --- 3329 3329, 14994 --- 3570 3570, 4444 --- 3580 30, 265, 340, 1494, 2986, 3580, 12383 K611 3589 3589, 10159, 40002 --- 3593 3590, 3593 --- 3595 3591, 3595 --- 3618 3618, 18840, 39978 --- 3619 3619, 18841 --- 3620 3620, 5395 --- 3629 3629, 13622 --- 3936 3936, 24624 --- 4383 4383, 40012 --- 4417 4417, 13478 --- 4648 4648, 32022 --- 5235 5235, 30588 --- 5422 3527, 5422, 11414 --- 5468 5466, 5468 --- 5590 5590, 10262 --- 5591 5591, 10261 --- 5739 388, 3436, 5739, 8048, 20029, 30479 --- 6144 6144, 30744 --- 6189 3414, 6189 --- 6190 3413, 6190 --- 6515 68, 317, 6193, 6504, 6515 --- 7735 7735, 40824 --- 7766 194, 263, 2998, 3212, 6194, 7155, 7766, 18906, 19222, 20023, 24349, 32937 K1037 7774 1352, 7774 --- 7777 7777, 34507 --- 7779 147, 1031, 2896, 4388, 4645, 5207, 7224, 7261, 7779, 9473, 11606, 20021, 20022, 34214 K1000 7788 7788, 14458 --- 7837 3818, 7837 --- 8025 6539, 8025 --- 8115 1113, 2574, 2592, 8115, 15164, 22339 --- 8116 1114, 2575, 2593, 8116, 15165, 22340 --- 9182 9180, 9182 --- 9770 9770, 11172 --- 10601 10601, 33524 --- 11004 3431, 10298, 11004, 18387 --- 11064 11064, 11744, 16080, 16386 --- 11163 11163, 11167 --- 11427 11427, 14542 --- 11433 11433, 14457, 37201 --- 13567 801, 13567, 22466, 22468, 22647 --- 14607 512, 670, 14607 --- 14829 2051, 14829 --- 14997 14997, 40021 --- 14999 99, 523, 14221, 14223, 14999 --- 15018 14483, 15018 --- 15066 3426, 15066, 21312, 34289 --- 15108 11538, 15108 --- 15533 15533, 17503 --- 15589 14484, 15589 --- 16704 4080, 16704, 17139, 38955 --- 16998 10477, 16998 --- 17277 17277, 17758, 40007 --- 17379 1246, 7560, 10449, 17379 --- 17731 11599, 17731, 20351 --- 17778 2893, 3152, 17778 --- 17811 17811, 22334, 30698, 37874 --- 18134 1751, 18134 --- 19623 11611, 19623 --- 20080 7396, 16774, 20080, 32006, 38259 --- 20536 20536, 20558 --- 20806 20806, 34207 --- 20965 20965, 31630 --- 21356 18842, 21356 --- 22133 22133, 34440 --- 22151 1177, 1236, 2892, 22151, 37929 --- 22329 5503, 22329 --- 24512 24512, 40024 --- 25508 4651, 25508, 39734 --- 26860 26860, 27797 --- 27398 8808, 27398 --- 27644 3223, 17149, 27644 --- 29767 29767, 40515 --- 30941 13576, 30941 --- 30966 30966, 40718 --- 32805 3316, 32805 --- 32806 3317, 32806 --- 32807 10195, 32807 --- 32808 1327, 32808 --- 32809 1328, 32809 --- 32810 14241, 32810 --- 32811 14226, 32811 --- 32812 32812, 34089 --- 32813 32813, 34091 --- 32911 18133, 32911, 35998, 39748, 40013 --- 34211 877, 879, 4226, 34211 --- 34229 14494, 34229 --- 34545 1173, 1232, 2889, 34545 --- 37636 37636, 40393 --- 37638 18434, 37638 --- 37645 376, 4846, 15454, 36889, 37645 --- 37652 7538, 37652 --- 37657 37657, 40030 --- 37668 3424, 37668 --- 37669 459, 30552, 37669 --- 37671 14492, 37671 --- 37672 3532, 37672 --- 37673 37673, 40031 --- 37680 37680, 39994, 39996 --- 37683 10446, 37683 --- 37684 145, 4373, 37684 --- 37685 7520, 37685 --- 37688 7608, 37688 --- 37779 3153, 13582, 33565, 37779 --- 37783 10693, 37783 --- 37784 23, 895, 18019, 37784 --- 37792 34899, 37792 --- 37894 37892, 37894 --- 39099 39099, 39266 --- 39113 96, 39113 --- 40112 10293, 40112 --- 40123 40123, 40178 --- 40316 1368, 40316, 40413 --- 40318 25, 305, 6339, 6391, 6392, 40318 --- 40341 7860, 40341 --- 40571 21, 63, 72, 92, 286, 1441, 2997, 3868, 40571 K610 40882 11608, 40882 --- 41133 10153, 41133 --- 41198 34136, 41194, 41198 --- 41199 34135, 41195, 41199 ---