# 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

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 522 650p1113 650p1114

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 523 661p1113 661p1114

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

Collinear triples
 1 243 108p527 653 108p527 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
 660 100p899 537p901 898 31p874 537p901

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

Collinear triples
 1 238 100p899 100 659 100p899

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

Collinear triples
 55 650 200p884 650 650p1381 650p1382

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

Collinear triples
 100 109 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
 1 514 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
 1 100 88p1642 1 518 100p1155 1 2254 519p840 1 518p1381 522p1381 1 518p1382 522p1382 100 518 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
 1 519 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
 1 240 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
 1 6 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
 6 232 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
 1 37 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
 100 513 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
 6 513 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
 1 241 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
 7 651 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
 1 109 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
 1 294 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
 1 101 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
 1381 1382 11p36

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

Collinear triple
 1381 1382 11p1458

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

Collinear triple
 2 1113 1114

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

Collinear triple
 1 1113 1114

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

Collinear triples
 2 110 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
 1 6 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
 1 2 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
 1 19 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
 1 57 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
 2 4 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
 2 75 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
 2 99 111p2234 6 512 111p2234

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

Collinear triples
 2 648 852p2349 3 520 852p2349

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

Collinear triples
 3 647 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 187 248 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 88 239 537p739 88 241 528p2291 88 294 527p840

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 57 15p2306 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
 1 9 100p2246 1 100 527p2316 1 650 44p1156 9 100 44p1156 9 650 527p2316 100 650 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
 2 9 527 2 283p1832 1793p1832 2 283p1833 1793p1833 9 283p1832 1793p1833 9 1793p1832 1793p1833

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

Collinear triples
 1 19 6p1325 1 102 208 1 282 4p2182 19 102 282 19 208 4p2182

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

Collinear triples
 6 73 2183 6 281 2p2250 10 73 2p2250 10 281 2183

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
 2 6 524 2 13 16 2 14 15 6 13 14 6 15 6 13 15 530p2153 14 16 531p2154 111 530p2153 531p2154

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
 286 524 542p2349 297 671 542p2349 524 671 543p897 94p1094 94p1095 542p2349

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

Collinear triple
 4 6 1990

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

Collinear triple
 371 372 47p94

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

Collinear triple
 3 6 50

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

Collinear triple
 99 523 543p662

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

Collinear triple
 2 3 30

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

Collinear triples
 30 98 542p897 325 1494 542p897

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

Collinear triple
 2 4 30

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

Collinear triples
 2 25 30 2 1073 4p2312 2 1297 4p204 25 1073 1297 25 4p204 4p2312

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

Collinear triples
 522 44p666 100p528 522 88p918 693p840 664 44p666 693p840 664 88p918 100p528

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

Collinear triples
 666 7p1023 513p1121 666 8p1022 100p527 918 7p1023 100p527 918 8p1022 513p1121

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

Collinear triples
 513 100p537 335p1023 668 88p812 100p537

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

Collinear triples
 812 1022 693p739 812 44p668 100p536 1022 2p660 100p536 2p660 44p668 693p739

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

Collinear triple
 3 6 110p1755

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
 74 511 111p2247 74 1976 842p896 511 1495 842p896 1495 1976 111p2247

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

Collinear triple
 371 372 511

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

Collinear triples
 238 2p739 106p537 291 899 106p537 511 1495 842p896 1495 1976 111p2247

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

Collinear triples
 1 10 899 1 726 37p2162 1 2p192 10p727 10 726 2p192 10 10p727 37p2162

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

Collinear triples
 106 1458 2246p2291 106 2195 840p1155 902 1458 840p1155 902 2195 2246p2291

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
 105 1155 80p2246 105 1156 519p840 518 1155 519p840 518 1156 80p2246 80p202 80p203 80p2246

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

Collinear triples
 1 55 1155 1 13p1250 1833p2193 55 15p1251 1833p2193

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

Collinear triples
 1 56 1155 1 103 57p1419 56 1456 57p1419

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
 55 56 1936

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
 1910 2173 11p542 1910 2349 524p842 1959 2173 524p842 1959 2349 11p542

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
 240 896 30p842 240 897 74p542 293 896 74p542 293 897 30p842 896 897 74p742 13p16 14p15 74p542

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

Collinear triples
 1 31 896 1 2151 2154 1 2152 2153 31 2151 2152 31 2153 2154

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

Collinear triples
 672 1055 44p840 672 2291 106p2246 1055 1438 106p2246 1438 2291 44p840 106p2246 202p2161 203p2161

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

Collinear triples
 55 840 88p1055 55 6p2246 44p2291 56 840 44p2291 56 6p2246 88p1055

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

Collinear triples
 6 37 6p899 6 192 37p727 6 1575 42p2162 37 192 1575 37 37p727 42p2162

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
 6 141 6p2234 6 38p699 75p194 141 698 75p194

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

Collinear triples
 6 55 101p2246 6 101 1055p1320 6 663 44p2291 55 101 44p2291 55 663 1055p1320 101 663 101p2246

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
 1 537 889p1960 75 537 88p536

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
 2 42 239 2 2p875 81p514 42 649 2p875 239 649 81p514

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

Collinear triples
 1 10 239 1 876 1019 1 514p1381 522p1381 1 514p1382 522p1382 10 514 876 239 514 1019 514 514p1381 514p1382 514 522p1381 522p1382

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

Collinear triple
 661 662 110p543

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

Collinear triples
 101 649 545p692 101 59p1643 665p1156 101 101p537 659p739 101 650p840 919p1155 649 59p1643 919p1155 649 101p537 813p899 649 650p840 665p1156

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

Collinear triples
 7 528 88p527 7 44p1121 75p840 8 528 44p1121 8 75p840 88p527

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

Collinear triples
 1 2 200 1 55 57 2 9 57 2 55 200 9 55 200

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

Collinear triples
 3 2066 57p2066 3 2067 9p2067 6 2066 9p2067 6 2067 57p2066

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

Collinear triples
 1 6 518 1 514 1027 6 513 1027 513 514 518

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

Collinear triples
 109 44p919 513p840 109 88p665 101p2246 663 44p919 101p2246 663 88p665 513p840

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

Collinear triples
 513 522 1638 513 676 514p1419 522 11p677 514p1419

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

Collinear triples
 650 88p2254 514p840 650 100p2246 105p1023 651 88p2254 100p2246 651 105p1023 514p840

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

Collinear triples
 1 75 2p524 1 2p13 2p14 1 2p15 2p16 75 2p13 2p16 75 2p14 2p15 897 13p530 14p531 2p13 2p15 13p530 2p14 2p16 14p531

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

Collinear triples
 1 662 691p888 1 798 524p729 31 662 524p729 31 798 691p888

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

Collinear triples
 1 100 690p901 1 661 44p897 37 100 44p897 37 661 690p901

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

Collinear triple
 1 100 690p901

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

Collinear triples
 6 31 537p692 6 667 44p739 31 101 44p739 101 667 537p692

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

Collinear triples
 1 513 88p896 1 662 691p900 81 513 691p900 81 662 88p896

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

Collinear triples
 1 63 110p542 1 656 74p524 1 662 30p895 63 656 30p895 63 662 74p524 656 662 110p542 662 110p1312 110p1313

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

Collinear triples
 81 1931 1962 81 514p1171 514p2054 661 1931 514p1171 661 1962 514p2054

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

Collinear triples
 1 799 351p886 1 6p523 111p538 75 799 111p538 75 6p523 351p886

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

Collinear triples
 6 667 739p2234 81 99 739p2234

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

Collinear triples
 162 656 1304p1651 162 110p542 691p868 656 523p842 691p868

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
 9 2246 44p1156 9 2p840 88p1155 57 2246 88p1155 57 2p840 44p1156

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

Collinear triples
 1 6 220 1 41 2246 1 55 56 6 41 56 6 2246 654p1308 41 55 220 55 55p840 654p1308

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

Collinear triple
 11 100 1001