table35


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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).

***

Geometric properties

(K) = pK(X2, P) meets the line at infinity and the circum-circle at the same points as two isogonal pKs (K1), (K2) with pivots P1 and P2 respectively.

P1 lies on the line {2, 6} ad P2 lies on the line {66, 69}. See figure 1.

(K) and (K1) meet again at six finite points on a circum-conic (C), namely A, B, C, G, P1 and Q1 on Q066. Note that the line P1Q1 is tangent to the Stammler hyperbola.

(K) and (K2) meet at A, B, C, T1, T2, T3 on (O) and at three points on a line (L) passing through X(6) and gP1.

Note that (L) and (C) meet on the Grebe cubic K102.

When P traverses the line {4, 69},

• the line {P, P1} envelopes an ellipse (E), with center X(115), passing through {2, 315, 671, 46707, 54097}. (E) is inscribed in the antimedial triangle, also in the anticevian triangle of X(99). It is the anticomplement of the ellipse with center X(620) inscribed in ABC.

• the line {P, P2} envelopes a hyperbola (H), with center X(110) x X(636), passing through {76, 2979}. This center lies on the lines {39, 695}, {99, 512}, {110, 827}.

• the line {P1, P2} envelopes a parabola (P), passing through {6, 2854, 12220}, with directrix the line {110, 1296}, focus F on the lines {67, 524}, {110, 1499}.

Remarks :

(E) and (H) are tangent to the line {4, 69} at X(315), X(76) respectively.
(E) and (P) are tangent to the line {2, 6} at X(2), X(6) respectively.
(H) and (P) are tangent to the line {66, 69} at X(2979), X(12220) respectively.

Hence, when one the three points P, P1, P2 is given, it is easy to find the others. See figure 2.

• the sidelines of the triangle (T) = T1T2T3 envelope a parabola (π), with focus X(2373), directrix the line {66, 69}, passing through X(30209) at infinity. This parabola is tangent to the lines {297, 525}, {325, 523}, {125, 339}. See figure 3.

• the conic inscribed in ABC and (T) is also tangent to the line {125, 339} and has its center on the line {141, 1368}. See figure 3.

• the isogonal conjugation with respect to (T) transforms (K) into a central cubic (K") with center the isogonal conjugate X in (T) of P, passing through X(20), T1, T2, T3 and the infinite points of the altitudes of ABC. (K") meets (O) again at R1, R2, R3 which are the antipodes of the points S1, S2, S3 (apart A, B, C) where pK(X6, P1) meets (O). X lies on the line {69, 74, 99, 376, 542, etc}. See figure 4.


Figure 1 : (K1) and (K2)	Figure 2 : (E), (H) and (P)


Figure 3 : (T), (C) and (π)	Figure 4 : (K) and (K")

Additional remarks :

• Figure 1 : (K1) and (K2) meet at A, B, C, X(1), Ia, Ib, Ic and two other points M1, M2 on the line {P1, P2} and on the circum-conic which is its isogonal transform. This line is parallel to (L). These points M1, M2 lie on an isogonal circular circum-sextic passing through X(1), X(2), X(6), X(1113), X(1114), X(2574), X(2575), excenters, infinite points of the conic ABCGK and their isogonal conjugates.

• Figure 4 : Y2, Y4, Y69 are the isogonal conjugates with respect to (T) of X(2), X(4), X(69). These points lie on the lines {20, 64, 69, ...}, {22, 69, 159, ...}, {2, 3, 4, 20, ...} (Euler line).

The following tables (contributed by Peter Moses) present 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	K1365	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	---