POINTS ON CUBICS

This continuation of Points on Cubics is devoted to one class of cubics. Access to related material is provided at the bottom of this page.

ZP(U,P): Z-Plus Cubics,

defined by

upx(qy2 + rz2) + vqy(rz2 + px2) + wrz(px2 + qy2) = 0

Notes about ZP Cubics:   

1.   ZP(U,P) is also given by

(vqy + wrz)px2 + (wrz + upx)qy2 + (upx + vqy)rz2 = 0.

2.   In Jean-Pierre Ehrmann and Bernard Gibert, "Special Isocubics in the Triangle Plane," downloadable from Cubics in the Triangle Plane, the cubics ZP-cubics are denoted by nK0.

3.   A subclass of cubics ZP(U,P) are mentioned in TCCT (1998, p. 241) as self-isogonal conjugate cubics. There, for example, ZP(X(513),X(1)) is denoted by Z+(L), where L denotes the line X(1)X(6). In order to reconcile the present notation with the earlier, note that X(513) is the point whose trilinears, b - c : c - a : a - b, are the coefficients of the line X(1)X(6); i.e., X(513) is the crossdifference of X(1) and X(6).

4.   As suggested in Note 3, it is useful to regard each U in the list of cubics ZP(U,P) as representative of the central line, any two of whose points have crossdifference U. The following list gives, for selected centers U, the corresponding line X(I)X(J):

U = X(1) represents the line X(44)X(513)
U = X(6) represents the line X(30)X(511), the line at infinity
U = X(244) represents the line X(100)X(101)
U = X(512) represents the line X(2)X(6)
U = X(513) represents the line X(1)X(6)
U = X(520) represents the line X(1)X(6)
U = X(523) represents the line X(3)X(6), the Brocard axis
U = X(647) represents the line X(2)X(3), the Euler line
U = X(647) represents the line X(2)X(3)
U = X(649) represents the line X(1)X(2)
U = X(650) represents the line X(1)X(3)
U = X(652) represents the line X(1)X(4)

5.   Z(U,P) is the set of points X for which a collinearity-determinant is zero; ZP(U,P) is the set of points X for which the corresponding permanent is zero. (The permanent is sometimes called "plus-determinant", as it results from changing the minus signs in the definition of determinant to plus signs.)

6.   These three points are on ZP(U,P):

vwqr(v2q - w2r) : - wurp(v2q + w2r) : uvpq(v2q + w2r),

vwqr(w2r + u2p) : wurp(w2r - u2p) : - uvpq(w2r + u2p),

- vwqr(u2p + v2q) : wurp(u2p + v2q) : uvpq(u2p - v2q).

7.   The discriminant d of ZP(U,P), obtained by solving the defining equation for x in terms of y and z, under the assumption that X does not satisfy vqy + wrz = 0, is given by

d2 = u2p2(qy2 + rz2)2 - 4pqryz(wy + vz)(vqy + wrz).

The discriminant can be used to show that the following six points are on ZP(U,P):

0 : v : - w      - u : 0 : w       u : - v : 0,

u(v2q + w2r) : v(w2r - v2q) : - w(w2r - v2q)

- u(u2q - w2r) : v(w2r + u2p) : w(u2p - w2r)

u(v2q - u2p) : - v(v2r - u2p) : w(u2p + v2q)

8.   Suppose that F = f : g : h is a triangle center. The collineation X --> F*X that carries each point x : y : z to the trilinear product fx : gy : hz also carries the cubic ZP(U,P) onto the the cubic ZP(F*U,P*F -2).

ZP(X(1),X(244)) passes through these centers:
100, 813, 1023, 1025, 2p901, 6p874, 8p919, 100p1931, 101p447, 190p2054, 545p901

Collinear triple
1023 2p901 545p901

ZP(X(2),X(292)) passes through these centers:
1, 190, 239, 659, 2p1023, 88p659

Collinear triples
1 239 2p1023
190 659 2p1023

ZP(X(3),X(651)) passes through these centers:
1, 6, 522, 650, 8p2183, 104p513, 523p1822, 523p1823, 650p1113, 65pp1114

Collinear triples
1 6 8p2183
1 522 104p513
6 650 104p513
6 523p1822 650p1113
6 523p1823 650p1114
522 650 8p2183
522 523p1822 523p1823
522650p1113650p1114

ZP(X(3),X(662)) passes through these centers:
6, 325, 523, 19p878, 523p1822, 523p1823, 661p1113, 661p1114

Collinear triples
6 523 19p878
6 523p1822 661p1113
6 523p1823 661p1114
523 523p1822 523p1823
523661p1113 661p1114

ZP(X(4),X(296)) passes through these centers:
1, 243, 653, 108p527, 243p652

Collinear triples
1 243 108p527
653108p527 243p652

ZP(X(5),X(655)) passes through these centers:
1, 523. 654. 36p294

ZP(X(6),X(244)) passes through these centers:
100, 660, 898, 31p874, 100p899, 100p2107, 190p2106, 422p1331, 537p901

Collinear triples
660100p899 537p901
898 31p874 537p901

ZP(X(6),X(291)) passes through these centers:
1, 100, 238, 659, 100p899, 739p812

Collinear triples
1238 100p899
100 659 100p899

ZP(X(6),X(658)) passes through these centers:
55, 241, 650, 200p884, 650p1381, 650p1382

Collinear triples
55650 200p884
650 650p1381 650p1382

ZP(X(9),X(11)) passes through these centers:
100, 109, 677, 59p676, 100p1155, 109p1156, 144p1461

Collinear triples
100109 100p1155
100 677 144p1461
109 59p676 144p1461

ZP(X(9),X(101)) passes through these centers:
1, 514, 1027, 2p518, 514p1381, 514p1382, 522p1381, 522p1382

Collinear triples
1514 1027
1 514p1381 522p1381
1 514p1382 522p1382
514 514p1381 514p1381
514 522p1381 522p1382

ZP(X(9),X(105)) passes through these centers:
1, 100, 518, 2254, 88p1642, 100p1155, 518p1381, 518p1382, 519p840, 522p1381, 522p1382, 665p1121

Collinear triples
1100 88p1642
1 518 100p1155
1 2254 519p840
1 518p1381 522p1381
1 518p1382 522p1382
100518 519p840
100 2254 100p1155
518 2254 88p1642
518 518p1381 518p1382
518 522p1382 522p1382

ZP(X(9),X(106)) passes through these centers:
1, 519, 1026, 105p900, 519p1381, 519p1382, 522p1381, 522p1382

Collinear triples
1519 1026
1 519p1381 522p1381
1 519p1382 522p1382
519 519p1381 519p1382
519 522p1381 522p1382

ZP(X(19),X(293)) passes through these centers:
1, 162, 240, 4p684, 30p842, 110p468, 240p1822, 240p1823, 523p1113, 523p1114, 691p868

Collinear triples
1240 110p468
1 4p684 30p842
1 240p1822 523p1113
1 240p1823 523p1114
162 240 30p842
162 4p684 110p468
240 240p1822 240p1823
240 523p1113 523p1114

ZP(X(21),X(662)) passes through these centers:
1, 6, 523, 661, 10p2245, 513p759, 523p1379, 523p1380, 523p1822, 523p1823, 661p1113, 661,1114

Collinear triples
16 10p2245
1 523 513p759
6 661 513p759
6 523p1822 661p1113
6 523p1823 661p1114
523 661 10p2245
523 523p1822 523p1823
523 661p1113 661p1114
661 523p1379 523p1380

ZP(X(25),X(336)) passes through these centers:
6, 112, 232, 19p684, 112p896, 232p1822, 232p1823, 661p1113, 661p1114, 842p2173

Collinear triples
6232 112p896
6 19p684 842p2173
6 232p1822 661p1113
6 232p1823 661p1114
112 232 842p2173
112 19p684 112p896
232 232p1822 232p1823
232 661p1113 661p1114

ZP(X(35),X(651)) passes through these centers:
1, 37, 650, 2323, 9p1019, 11p2222

Collinear triples
137 2323
1 9p1019 11p2222
37 650 11p2222
650 2323 9p1019

ZP(X(44),X(1)) passes through these centers:
1, 100, 513, 100p537, 101p545, 109p528, 522p840, 527p919, 536p813, 665p1121, 739p812

Collinear triples
100513 101p545
100 100p537 739p812
100 109p528 665p1121
100 522p840 527p919
513 100p537 536p813
513 109p528 527p919
513 522p840 665p1121

ZP(X(55),X(190)) passes through these centers:
6, 513, 518, 6p1027, 513p1381, 513p1382, 650p1381, 650p1382

Collinear triples
6513 6p1027
6 513p1381 650p1381
6 513p1382 650p1382
513 513p1381 513p1382
513 650p1381 650p1382

ZP(X(57),X(294)) passes through these centers:
1, 241, 651, 2254, 44p651, 88p2254, 514p1381, 514p1382, 518p1381, 518p1382, 527p840

Collinear triples
1241 44p651
1 2254 527p840
1 514p1381 518p1381
1 514p1382 518p1382
241 651 527p840
241 514p1381 514p1382
241 518p1381 513p1382
651 2254 44p651

ZP(X(100),X(657)) passes through these centers:
7, 651, 883, 1462, 2p1381, 2p1382, 7p1381, 7p1382

Collinear triples
7651 1462
7 2p1381 2p1382
7 7p1381 7p1382
651 2p1381 7p1381
651 2p1382 7p1382

ZP(X(101),X(522)) passes through these centers:
1, 109, 1381, 1382, 104p999

Collinear triples
1109 104p999
1 1381 1382

ZP(X(105),X(241)) passes through these centers:
1, 294, 650, 2p919, 9p1022, 9p1381, 9p1382, 105p1023, 528p2291, 885p1381, 885p1382

Collinear triples
1294 105p1023
1 650 528p2291
1 9p1381 9p1382
1 885p1381 885p1382
294 2p919 528p2291
294 9p1381 885p1381
294 9p1382 885p1382
650 2p919 105p1023

ZP(X(109),X(514)) passes through these centers:
1, 101, 1381, 1382, 7p2283, 9p1381, 9p1382, 9p2195

Collinear triples
1101 9p2195
1 1381 1382
1 9p1381 9p1382
101 1381 9p1381
101 1382 9p1382

ZP(X(109),X(655)) passes through these centers:
101, 1381, 1382, 11p36

Collinear triple
13811382 11p36

ZP(X(109),X(666)) passes through these centers:
101, 1381, 1382, 11p1458

Collinear triple
13811382 11p1458

ZP(X(110),X(656)) passes through these centers:
2, 112, 1113, 1114

Collinear triple
21113 1114

ZP(X(112),X(656)) passes through these centers:
4, 110, 1113, 1114, 1302p1725

Collinear triple
11113 1114

ZP(X(112),X(661)) passes through these centers:
2, 110, 1113, 1114, 2p1822, 2p1823, 3p293, 19p877

Collinear triples
2110 3p293
2 1113 1114
2 2p1822 2p1823
110 1113 2p1822
110 1114 2p1823

ZP(X(238),X(2)) passes through these centers:
1, 6, 100, 649, 101p537, 106p899, 519p739

Collinear triples
16 101p537
1 100 106p899
1 649 519p739
6 100 519p739
6 649 106p899
100 649 101p637

ZP(X(239),X(6)) passes through these centers:
1, 2, 190, 513, 88p899, 100p537, 889p1960

Collinear triples
12 100p537
1 190 889p1960
1 513 88p899
2 190 88p899
2 513 889p1960
190 513 100p537

ZP(X(240),X(63)) passes through these centers:
1, 19, 162, 661, 30p111, 74p468, 110p1312, 110p1313, 112p542, 112p1312, 112p1313, 523p842

Collinear triples
119 112p542
1 162 74p468
1 661 30p111
19 162 30p111
19 661 74p468
162 661 112p542
162 110p1312 110p1313
162 112p1312 112p1313
661 110p1312 112p1312
661 110p1313 112p1313

ZP(X(241),X(9)) passes through these centers:
1, 57, 513, 651, 88p1155, 109p528, 514p840, 1121p1404

Collinear triples
157 109p528
1 513 88p1155
1 651 1121p1404
57 513 1121p1404
57 651 88p1155
513 651 109p528

ZP(X(297),X(48)) passes through these centers:
2, 4, 523, 648, 30p897, 162p542, 162p1312, 468p2349, 662p1312, 842p1577

Collinear triples
24 162p542
2 523 30p897
2 648 468p2349
4 523 468p2349
4 648 30p897
523 648 162p542
523 162p1312 662p1312

ZP(X(350),X(42)) passes through these centers:
2, 75, 514, 668, 88p536, 190p537, 889p1635

Collinear triples
275 190p537
2 514 88p536
2 668 889p1635
75 514 889p1635
75 668 88p536
514 668 190p537

ZP(X(385),X(1)) [Brocard (first) cubic, K017] passes through these centers:
2, 6, 99, 512, 111p2234

Collinear triples
299 111p2234
6 512 111p2234

ZP(X(401),X(19)) passes through these centers:
2, 3, 520, 648, 852p2349

Collinear triples
2648 852p2349
3 520 852p2349

ZP(X(450),X(1)) passes through these centers:
3, 4, 647, 648, 852p897

Collinear triples
3647 852p897
4 648 852p897

ZP(X(511),X(92)) passes through these centers:
3, 6, 110, 647, 74p896, 110p2247, 163p1312, 163p1313, 656p842, 895p2173

Collinear triples
3 6 110p2247
3 110 74p896
3 647 895p2173
6 110 895p2173
6 647 74p896
110 647 110p2247
110 163p1312 163p1313

ZP(X(512),X(31)) passes through these centers:
2, 385, 538, 1916, 10p2106, 75p729, 274p2107, 297p1957 (no collinearities found)

ZP(X(512),X(75)) passes through these centers:
6, 111, 187, 232, 248, 385, 2p1927, 6p1757, 6p1758, 6p1929, 13p2152, 14p2151, 19p450, 48p1942, 71p422, 74p2247, 82p2076, 421p1820, 425p1400, 543p923, 842p2173, 843p896, 2080p2186

Collinear triples
111 187 543p923
111 232 74p2247
111 248 842p2173
187 232 842p2173
187248 74p2247
13p2152 14p2151 74p2247

ZP(X(513),X(1)) [ K137] passes through these centers:
1, 44, 88, 239, 241, 292, 294, 1931, 2p2054, 6p447, 106p545, 527p840, 528p2291, 537p739, 1280p1743

Collinear triples
44 88 106p545
44 241 527p840
44 292 537p739
44 294 528p2291
88239 537p739
88 241 528p2291
88294527p840

ZP(X(513),X(8)) passes through these centers:
6, 57, 2291, 6p1323, 13p2307, 15p2306, 554p2152, 559p2154

Collinear triples
6 57 6p1323
6 13p2307 559p2154
6 15p2306 554p2152
57 13p2307 554p2152
5715p2306 559p2154

ZP(X(514),X(31)) passes through these centers:
2, 291, 350, 447, 519, 2p241, 2p294, 2p1932, 10p1929, 88p545, 145p1280, 528p1156

Collinear triples
519 903 88p545
519 2p294 528p1156
903 2p241 528p1156

ZP(X(514),X(55)) passes through these centers:
1, 7, 1156, 1323, 13p1082, 14p559, 15p1081, 16p554

Collinear triples
1 7 1323
1 13p1082 14p559
1 15p1081 16p554
7 13p1082 16p554
7 14p559 15p1081

ZP(X(514),X(81)) passes through these centers:
2, 42, 55, 226, 519, 672, 10p105, 37p106

Collinear triples
2 42 519
2 55 10p105
2 226 672
42 55 672
42 226 10p105

ZP(X(518),X(57)) passes through these centers:
1, 9, 100, 650, 44p1156, 100p2246, 522p840, 527p2316

Collinear triples
19 100p2246
1100 527p2316
1650 44p1156
9100 44p1156
9650 527p2316
100650 100p2246

ZP(X(520),X(31)) passes through these centers:
2, 401, 1972, 63p450, 92p852, 92p1942, 226p416, 325p1958

ZP(X(520),X(92)) passes through these centers:
3, 6, 74, 3p2173

Collinear triple
3 6 3p2173

ZP(X(521),X(1)) passes through these centers:
9, 57, 1944, 1945

Collinear triple
9 57 1944

ZP(X(521),X(4)) passes through these centers:
1, 3, 2338, 63p1456

Collinear triple
1 3 63p1456

ZP(X(521),X(31)) passes through these centers:
2, 448, 1944, 1952, 63p425, 92p416

ZP(X(521),X(81)) passes through these centers:
9, 65, 2182, 2331, 10p102, 10p1433

Collinear triples
9 2182 10p1433
9 2331 10p1102
65 2182 2331
65 10p102 10p1433

ZP(X(521),X(92)) passes through these centers:
3, 6, 221, 268, 2182, 3p102, 48p1325

Collinear triples
3 6 48p1325
3 221 3p102
3 268 2182
6 221 2182
6 268 3p102

ZP(X(522),X(31)) passes through these centers:
2, 527, 673, 1121, 1948, 2p296, 2p518, 63p415, 88p528, 694p1281

Collinear triples
527 673 88p528
1121 2p518 88p528

ZP(X(522),X(55)) passes through these centers:
2, 57, 527, 2p103, 2p1456, 7p1419, 7p2291

Collinear triples
2 57 527
2 2p103 7p1419
57 2p1456 7p1419

ZP(X(522),X(56)) passes through these centers:
2, 9, 527, 8p2291, 283p1832, 283p1833, 1793p1832, 1793p1833

Collinear triples
29527
2283p18321793p1832
2283p18331793p1833
9283p18321793p1833
91793p18321793p1833

ZP(X(522),X(63)) passes through these centers:
1, 19, 102, 208, 282, 4p2182, 6p1325

Collinear triples
1196p1325
1102208
12824p2182
19102282
192084p2182

ZP(X(522),X(81)) passes through these centers:
1, 19, 102, 208, 282, 4p2182, 6p1325

Collinear triples
6732183
62812p2250
10732p2250
102812183

ZP(X(523),X(1)) [a Brocard cubic, K018] passes through these centers:
2, 6, 13, 14, 15, 16, 111, 368, 524, 530p2153, 531p2154

Collinear triples
26524
21316
21415
61314
6156
1315530p2153
1416531p2154
111530p2153531p2154

ZP(X(523),X(31)) [ K185] passes through these centers:
2, 287, 297, 524, 671, 694, 2p1757, 2p1758, 2p1929, 2p1966, 63p1942, 92p450, 94p1094, 94p1095, 226p425, 306p422, 542p2349, 543p897

Collinear triples
286524542p2349
297671542p2349
524671543p897
94p109494p1095542p2349

ZP(X(523),X(63)) passes through these centers:
4, 6, 74, 1990

Collinear triple
461990

ZP(X(523),X(91)) passes through these centers:
371, 372, 47p94

Collinear triple
37137247p94

ZP(X(523),X(92)) passes through these centers:
3, 6, 50, 265

Collinear triple
3650

ZP(X(524),X(31)) passes through these centers:
2, 99, 523, 162p542, 543p662, 843p1577

Collinear triple
99523543p662

ZP(X(525),X(19)) passes through these centers:
2, 3, 30, 3p2349

Collinear triple
2330

ZP(X(525),X(31)) passes through these centers:
2, 30, 98, 325, 1494, 3p1956, 72p447, 92p401, 226p448, 441p2184, 542p897, 1297p1895

Collinear triples
3098542p897
3251494542p897

ZP(X(525),X(48)) passes through these centers:
2, 4, 30, 4p2349

Collinear triple
2430

ZP(X(525),X(63)) passes through these centers:
2, 25, 30, 1073, 1297, 4p204, 4p2159, 4p2312

Collinear triples
22530
210734p2312
212974p204
2510731297
254p2044p2312

ZP(X(527),X(31)) passes through these centers:
2, 522, 664, 44p666, 88p918, 100p528, 693p840

Collinear triples
52244p666100p528
52288p918693p840
66444p666693p840
66488p918100p528

ZP(X(528),X(31)) passes through these centers:
2, 666, 918, 7p1023, 8p1022, 100p527, 513p1121

Collinear triples
6667p1023513p1121
6668p1022100p527
9187p1023100p527
9188p1022513p1121

ZP(X(536),X(31)) passes through these centers:
2, 513, 668, 88p812, 100p537, 335p1023

Collinear triples
513100p537335p1023
66888p812100p537

ZP(X(537),X(31)) passes through these centers:
2, 812, 1022, 2p660, 44p668, 100p536, 693p739

Collinear triples
8121022693p739
81244p668100p536
10222p660100p536
2p66044p668693p739

ZP(X(542),X(92)) passes through these centers:
3, 6, 879, 110p1755

Collinear triple
36110p1755

ZP(X(647),X(75)) passes through these centers:
6, 74, 511, 1495, 1976, 4p1955, 48p1987, 111p2247, 204p1297, 228p447, 441p2155, 448p1400, 765p866, 842p896

Collinear triples
74511111p2247
741976842p896
5111495842p896
14951976111p2247

ZP(X(647),X(91)) passes through these centers:
371, 372, 511, 24p293

Collinear triple
371372511

ZP(X(649),X(1)) passes through these centers:
1, 238, 291, 899, 2107, 2p739, 2p2106, 3p422, 105p1983, 106p537, 1463p2053

Collinear triples
2382p739106p537
291899106p537
5111495842p896
14951976111p2247

ZP(X(649),X(58)) passes through these centers:
1, 10, 726, 899, 2p192, 10p727, 37p2162, 321p739

Collinear triples
110899
172637p2162
12p19210p727
107262p192
1010p72737p2162

ZP(X(649),X(75)) passes through these centers:
6, 106, 238, 902, 1326, 1458, 1911, 2054, 2195, 31p447, 840p1155, 2246p2291

Collinear triples
10614582246p2291
1062195840p1155
9021458840p1155
90221952246p2291

ZP(X(650),X(1)) [Pelletier strophoic, K040] passes through these centers:
1, 105, 243, 296, 518, 1155, 1156, 3p415, 80p202, 80p203, 88p2246, 519p840, 1281p1967

Collinear triples
105115580p2246
1051156519p840
5181155519p840
518115680p2246
80p20280p20380p2246

ZP(X(650),X(7)) passes through these centers:
1, 55, 1155, 9p2291, 13p1250, 15p1251, 1833p2193

Collinear triples
1551155
113p12501833p2193
5515p12511833p2193

ZP(X(650),X(8)) passes through these centers:
1, 56, 103, 1155, 1456, 56p1156, 57p1419

Collinear triples
1561155
110357p1419
56145657p1419

ZP(X(652),X(1)) passes through these centers:
1, 1936, 1937, 3p425, 4p416, 6p448

ZP(X(652),X(75)) passes through these centers:
55, 56, 1936, 6p1945

Collinear triple
55561936

ZP(X(656),X(1)) passes through these centers:
1, 1910, 1959, 2173, 2349, 3p1987, 4p401, 64p441, 65p448, 71p447, 111p542, 449p2213, 524p842, 866p1016, 1249p1297

Collinear triples
1910217311p542
19102349524p842
19592173524p842
1959234911p542

ZP(X(659),X(1)) passes through these centers:
1, 2, 6, 88p899, 519p739, 730p743, 731p742

ZP(X(661),X(1)) passes through these centers:
1, 240, 293, 896, 897, 1757, 1758, 1929, 1967, 3p1942, 4p450, 13p16, 14p15, 30p842, 65p425, 68p421, 72p422, 74p542, 83p2076, 99p804, 111p543, 262p2080, 524p843, 1687p2010, 1688p2009

Collinear triples
24089630p842
24089774p542
29389674p542
29389730p842
89689774p742
13p1614p1574p542

ZP(X(661),X(75)) passes through these centers:
1, 31, 896, 923, 2151, 2152, 2153, 2154

Collinear triples
131896
121512154
121522153
3121512152
3121532154

ZP(X(663),X(75)) passes through these centers:
6, 672, 1055, 1438, 1949, 2202, 2291, 44p840, 48p415, 103p1419, 106p2246, 202p2161, 203p2161

Collinear triples
672105544p840
6722291106p2246
10551438106p2246
1438229144p840
106p2246202p2161203p2161

ZP(X(665),X(75)) passes through these centers:
6, 55, 56, 840, 1083, 6p2246, 44p2291, 88p1055, 761p2239

Collinear triples
5584088p1055
556p224644p2291
5684044p2291
566p224688p1055

ZP(X(667),X(86)) passes through these centers:
6, 37, 192, 1575, 6p899, 10p739, 37p727, 42p2162

Collinear triples
6376p899
619237p727
6157542p2162
371921575
3737p72742p2162

ZP(X(669),X(75)) passes through these centers:
6, 694, 729, 1691, 6p2234, 42p2106, 81p2107, 232p1957

ZP(X(669),X(82)) passes through these centers:
6, 141, 698, 6p2234, 38p699, 75p194, 729p1930

Collinear triples
61416p2234
638p69975p194
14169875p194

ZP(X(672),X(85)) passes through these centers:
6, 55, 101, 663, 44p2291, 101p2246, 650, 840, 1055p1320

Collinear triples
655101p2246
61011055p1320
666344p2291
5510144p2291
556631055p1320
101663101p2246

ZP(X(798),X(1)) passes through these centers:
1, 1580, 1581, 2234, 2p729, 37p2106, 86p2107, 240p1957   (no collinear triples)

ZP(X(812),X(31)) passes through these centers:
1, 2, 75, 537, 88p536, 889p1969

Collinear triples
1537889p1960
7553788p536

ZP(X(822),X(1)) passes through these centers:
1, 1955, 1956, 3p450, 4p852, 4p1942, 65p416, 511p1975   (no collinear triples)

ZP(X(894),X(100)) passes through these centers:
2, 42, 239, 649, 2p875, 81p514

Collinear triples
242239
22p87581p514
426492p875
23964981p514

ZP(X(894),X(101)) passes through these centers:
1, 10, 239, 514, 876, 1019, 514p1381, 514p1382, 522p1381, 522p1382

Collinear triples
110239
18761019
1514p1381522p1381
1514p1382522p1382
10514876
2395141019
514514p1381514p1382
514522p1381522p1382

ZP(X(896),X(1)) passes through these centers:
1, 661, 662, 110p543, 112p542, 523p843, 525p842, 538p805, 729p804, 1022p1757, 1023p1929

Collinear triple
661662110p543

ZP(X(902),X(75)) passes through these centers:
6, 101, 649, 59p1643, 101p537, 545p692, 650p840, 659p739, 665p1156, 813p899, 919p1155

Collinear triples
101649545p692
10159p1643665p1156
101101p537659p739
101650p840919p1155
64959p1643919p1155
649101p537813p899
649650p840665p1156

ZP(X(918),X(31)) passes through these centers:
2, 7, 8, 528, 44p1121, 75p840, 75p1083, 88p527

Collinear triples
752888p527
744p112175p840
852844p1121
875p84088p527

ZP(X(918),X(57)) passes through these centers:
1, 2, 9, 55, 57, 200, 528, 9p840, 149p1110

Collinear triples
12200
15557
2957
255200
955200

ZP(X(928),X(92)) passes through these centers:
3, 6, 2066, 2067, 9p2067, 57p2066

Collinear triples
3206657p2066
320679p2067
620669p2067
6206757p2066

ZP(X(1001),X(100)) passes through these centers:
1, 6, 513, 514, 518, 1027

Collinear triples
16518
15141027
65131027
513514518

ZP(X(1055),X(75)) passes through these centers:
6, 109, 663, 44p919, 88p665, 101p2246, 513p840

Collinear triples
10944p919513p840
10988p665101p2246
66344p919101p2246
66388p665513p840

ZP(X(1146),X(59)) passes through these centers:
513, 522, 676, 9p1638, 11p677, 513p1156, 514p1419

Collinear triples
5135221638
513676514p1419
52211p677514p1419

ZP(X(1155),X(1)) passes through these centers:
1, 650, 651, 88p2254, 100p2246, 105p1023, 514p840

Collinear triples
65088p2254514p840
650100p2246105p1023
65188p2254100p2246
651105p1023514p840

ZP(X(1577),X(31)) passes through these centers:
1, 75, 897, 2p13, 2p14, 2p15, 2p16, 2p524, 13p530, 14p531

Collinear triples
1752p524
12p132p14
12p152p16
752p132p16
752p142p15
89713p53014p531
2p132p1513p530
2p142p1614p531

ZP(X(1580),X(75)) passes through these centers:
1, 31, 662, 798, 524p729, 691p888

Collinear triples
1662691p888
1798524p729
31662524p729
31798691p888

ZP(X(1757),X(81)) passes through these centers:
1, 37, 100, 661, 44p897, 690p901

Collinear triples
1100690p901
166144p897
3710044p897
37661690p901

ZP(X(1797),X(44)) passes through these centers:
2, 1113, 1114

Collinear triple
1100690p901

ZP(X(1914),X(76)) passes through these centers:
6, 31, 101, 667, 44p739, 537p692

Collinear triples
631537p692
666744p739
3110144p739
101667537p692

ZP(X(1931),X(37)) passes through these centers:
1, 81, 513, 662, 88p896, 691p900

Collinear triples
151388p896
1662691p900
81513691p900
8166288p896

ZP(X(1959),X(19)) passes through these centers:
1, 63, 656, 662, 30p895, 74p524, 110p542, 110p1312, 110p1313, 525p842

Collinear triples
163110p542
165674p524
166230p895
6365630p895
6366274p524
656662110p542
662110p1312110p1313

ZP(X(1963),X(100)) passes through these centers:
81, 661, 1931, 1962, 514p1171, 514p2054

Collinear triples
8119311962
81514p1171514p2054
6611931514p1171
6611962514p2054

ZP(X(1966),X(31)) passes through these centers:
1, 75, 799, 6p523, 111p538, 351p886

Collinear triples
1799351p886
16p523111p538
75799111p538
756p523351p886

ZP(X(2106),X(10)) passes through these centers:
6, 81, 99, 667, 739p2234

Collinear triples
6667739p2234
8199739p2234

ZP(X(2173),X(1)) passes through these centers:
1, 162, 656, 110p542, 523p842, 691p868, 866p898, 867p901, 1304p1651

Collinear triples
1626561304p1651
162110p542691p868
656523p842691p868

ZP(X(2234),X(1)) passes through these centers:
1, 798, 799, 111p804, 524p805

ZP(X(2243),X(1)) passes through these centers:
1, 1491, 1492, 716p795, 717p794

ZP(X(2254),X(1)) passes through these centers:
1, 9, 57, 2246, 2p840, 2p1083, 44p1156, 88, 1155, 742p761, 743p760

Collinear triples
9224644p1156
92p84088p1155
57224688p1155
572p84044p1156

ZP(X(2254),X(85)) passes through these centers:
1, 6, 41, 55, 56, 220, 2246, 55p840, 654p1308

Collinear triples
16220
1412246
15556
64156
62246654p1308
4155220
5555p840654p1308

ZP(X(2284),X(109)) passes through these centers:
11, 100, 1001, 522p1002, 522p1381, 522p1382, 644p2350, 1111p1621

Collinear triple
111001001

POINTS ON CUBICS: INTRODUCTION and Z-CUBICS
POINTS ON CUBICS: ZP-CUBICS
POINTS ON CUBICS: H- and HP- CUBICS
POINTS ON CUBICS: C- and ZC- CUBICS
POINTS ON CUBICS: B-, BP-, D-, and DP- CUBICS

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