POINTS ON CUBICS:

Introduction and Z-Cubics

The title Points on Cubics covers several URLs devoted to the subject of cubic curves (henceforth, simply cubics) in the plane of an arbitrary triangle ABC. Most of the material consists of lists of triangle centers on selected cubics and tables of collinear triples of triangle centers.

The lists and tables should be of interest to those who seek to discover new properties of cubics, including on-cubic triangle centers, especially polynomial centers, more especially those of low degree, and most especially those of low degree with all coefficients in the set of integers.

Further properties for investigation include on-cubic bicentric pairs, on-cubic vertices of central triangles, and on-cubic vertices of bicentric triangles, as well as collinear triples among on-cubic points, asymptotes, related conics, pivot-properties, tangency, degeneracy, and transformations that preserve various properties.

The cubics are classed according to forms of equations using trilinear coordinates. The first edition of Points on Cubics, (December 13, 2003) contains ten classes:

Z(U,P): Pivotal Self-Isoconjugate Cubics (begins just below)

Names of the other nine classes are as follows. (Click for transportation.)

ZP(U,P): Z-Plus Cubics

H(U,P): Hirst Cubics
HP(U,P): H-Plus Cubics

C(U,P): Cross Conjugate Collinearity Cubics
ZC(U,P): ZC-Cubics

B(U,P): Bicentrics Collinearity Cubics
BP(U,P): B-Plus Cubics

D(U,P): Cross-Bicentrics Collinearity Cubics
DP(U,P): D-Plus Cubics

The cubics are defined in terms of points P = p : q : r and U = u : v : w, or else a point U and line given by trilinear coefficients L, M, N; these are arbitrary except for occasional tacitly understood cases, as when P or U lies on a sideline of triangle ABC.

Single numbers 1, 2, 3, . . . , up to 2364, refer to triangle centers X(1), X(2), X(3), . . . as listed in the Encyclopedia of Triangle Centers - ETC, and expressions of the form IpJ refer to trilinear products; specifically, IpJ abbreviates the trilinear product of centers X(I) and X(J).

Definitions of terms (e.g., isoconjugate, Hirst inverse, cross conjugate, bicentric points) are given in the Glossary (atop ETC).

A General Form: tricentral cubics.   The cubics covered by Points on Cubics fit a certain general form. In order to define it, let

F = f1x3 + f2y3 + f3z3,    H = xyz,

G = g1x2y + g2y2z + g3z2x - g4x2z - g5y2x - g6z2y,     G+ = g1x2y + g2y2z + g3z2x + g4x2z + g5y2x + g6z2y.

Suppose that the following conditions hold:

(1)   Either f1 = f2 = f3 = 0 or else f1 : f2 : f3 is a triangle center; that is, f1 is a nonzero function of a,b,c, homogeneous in a,b,c, such that

f2(a,b,c) = f1(b,c,a),    f3(a,b,c) = f1(c,a,b),    |f1(a,c,b)| = |f1(a,b,c)|.

(2)   One of the following holds:

g1 = g2 = g3 = g4 = g5 = g6 = 0;

g1 : g2 : g3 and g4 : g5 : g6 are equal and are a triangle center;

g1 : g2 : g3 and g4 : g5 : g6 are a bicentric pair.

(3)   s is a function of a,b,c symmetric in a,b,c.

(4)   At least one of the functions F, G, s is not identically zero, and if two or three of them are nonzero, then they all have the same degree of homogeneity in a,b,c.

A curve in the extended plane of triangle ABC (including the line at infinity) is a tricentral cubic if it consists of all the points x : y : z satisfying F + G + sH = 0 or F + G+ + sH = 0, where F, G, H, and G+ are as just described.

The collection of tricentral cubics is closed under various transformations: reflections, inversions, conjugations, products, quotients, collineations, intersections, etc., in much that same way that collections of triangle centers, central lines, and central conics are closed under such transformations.

With a computer algebra system, each of the equations

F + G + sH = 0   and   F + G+ + sH = 0

can be solved for x in terms of y and z, with solutions of the form

x = J(K + d)   and   x = J(K - d).

For some of the classes discussed below, the discriminant d is useful for finding and confirming that certain points are on certain cubics.

Acknowledgments.    I thank Amanda Singer and Brandi Warren for transcribing collinearity tables and the University of Evansville Alumni Association for financial support.



Z(U,P): Pivotal Self-Isoconjugate Cubics,

defined by

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

Locus:   The cubic Z(U,P) is the locus of a point X = x : y : z such that the P-isoconjugate of X is on the line UX.

Notes:

1.   Z(U,P) is also given by

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

2.   A list of cubics Z(U,X(1)) is given in TCCT (1998), and the more general class Z(U,P) is defined in publications dating from 2001. A rich discussion of these cubics, using barycentric coordinates, with notation pK, is given by Jean-Pierre Ehrmann and Bernard Gibert: "Special Isocubic in the Triangle Plane," downloadable from Gibert's magnificent site, Cubics in the Triangle Plane, which includes sketches of Z and ZP cubics.

3.   The descriptor self-isoconjugate indicates that if X is on Z(U,P), then the P-isoconjugate of X is on Z(U,P). The point U, also on Z(U,P), is the pivot of Z(U,P), and the three points U, X, P-isoconjugate of X are collinear.

4.   If X is on Z(U,P), then the U-Ceva conjugate of X is also on Z(U,P). This is proved here: let

x1 = x( - x/u + y/v + z/w),    y1 = y(x/u - y/v + z/w),    z1 = z(x/u + y/v - z/w);

t1 = upx1(qy12 - rz12),     t2 = vqy1(rz12 - px12),     t3 = wrz1(px12 - qy12).

Then x1 : y1 : z1 is the U-Ceva conjugate of X, and the equation that results from replacing X by U-Ceva conjugate of X in the equation that defines Z(U,P) is this: t1 + t2 + t3 = 0. Next, let

F1 = - u -3v -3w -3,     F2 = xvw - yuw - zuv,     F3 = ywu - zvu - xvw,     F4 = zuv - xwv - ywu,

F5 = upx(qy2 - rz2) + vqy(rz2 - px2) + wrz(px2 - qy2).

Then t1 + t2 + t3 factors as F1F2F3F4F5. Now, if X is on Z(U,P), then F5 = 0, so that t1 + t2 + t3 = 0, as asserted.

5.   Other points on Z(U,P) are the vertices A, B, C; the vertices of the cevian triangle of U, namely 0 : v : w,   u : 0 : w,   and u : v : 0; and the four points invariant under P-isoconjugation, namely P -1/2 and the vertices of the anticevian triangle of P -1/2.

6.   The discriminant d of Z(U,P), obtained by solving the defining equation for x in terms of y and z, under the assumption that X is not the P-isoconjugate of X, is given by

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

7.   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 Z(U,P) onto the the cubic Z(F*U,P*F -2). For example, if F = X(6), then Z(X(1),X(2)) is carried onto Z(X(6),X(76)).

8.   The trilinear square P -1*P -1 is on Z(X(1),P).


Z(X(1),X(2)) passes through these centers:
1, 6, 55, 57, 365, 1419, 2067, 4p2066, 6p1489, 9p2362, 57p2066, 364p365, 1419p2125

Collinear triples
1 55 57
1 2067 4p2066
1 9p2362 57p2066
6 57 1419
6 365 364p365
6 2067 57p2066
55 1419 1419p2125
55 4p2066 9p2362

Z(X(1),X(3)) passes through these centers:
1, 4, 1148


Z(X(1),X(4)) passes through these centers:
1, 3, 90, 3p46

Collinear triple
1 90 3p46

Z(X(1),X(6)) passes through these centers:
1, 2, 87, 192, 366

Collinear triple
1 87 192

Z(X(1),X(7)) passes through these centers:
1, 6, 9, 55, 259, 6p236, 9p289, 9p1743, 523p1293

Collinear triples
1 6 9
1 6p236 9p289
1 9p1743 513p1293
6 259 9p289
9 55 9p1743
55 259 6p236

Z(X(1),X(8)) passes through these centers:
1, 56, 84, 221, 266, 2067, 2362, 6p557, 6p558, 56p175, 56p176, 57p2066, 164p266, 225p1806, 266p505

Collinear triples
1 84 221
1 2067 225p1806
1 2362 57p2066
1 6p557 6p558
1 164p266 266p505
56 266 164p266
56 2067 57p2066
56 56p175 56p176
221 266 266p505
221 2362 225p1806
266 2067 6p558
266 6p557 57p2066
2067 2362 56p176
2362 6p558 164p266
6p557164p266225p1806
56p17557p2066225p1806

Z(X(1),X(9)) passes through these centers:
1, 57, 509, 1419


Z(X(1),X(10)) passes through these centers:
1, 58, 267, 501

Collinear triple
1 267 501

Z(X(1),X(11)) passes through these centers:
1, 59, 100p266


Z(X(1),X(12)) passes through these centers:
1, 21, 58, 60, 501, 21p266, 21p267, 21p1046, 21p2306, 21p2307, 58p1247, 284p554, 284p559

Collinear triples
1 21 58
1 501 21p267
1 21p1046 58p1247
1 21p2306 21p2307
1 284p554 284p559
21 60 21p1046
58 60 501
58 21p2307 284p559
501 21p2306 284p554

Z(X(1),X(21)) passes through these centers:
1, 4, 65, 73, 1148

Collinear triples
1 4 73
4 65 1148

Z(X(1),X(27)) passes through these centers:
1, 6, 55, 71, 72, 1214, 1751, 3p209, 72p1724, 1214p1754

Collinear triples
1 6 72
1 55 1214
1 1751 3p209
6 55 71
6 1751 1214p1754
55 1751 72p1724
71 72 72p1724
71 1214 1214p1754
72 1214 3p209

Z(X(1),X(29)) passes through these centers:
1, 3, 65, 73, 2067, 3p46, 10p2067, 12p1806, 46p921, 57p2066, 65p90, 485p2067

Collinear triples
1 3 65
1 2067 12p1806
1 3p64 65p90
1 10p2067 57p2066
3 73 3p46
65 2067 485p2067
65 3p64 46p921
65 10p2067 12p1806
73 2067 57p2066
3p46 10p2067 485p2067

Z(X(1),X(31)) passes through these centers:
1, 2, 75, 192, 330, 2p194

Collinear triples
1 192 330
2 75 192
2 330 2p194

Z(X(1),X(33)) passes through these centers:
1, 77, 2p2067, 7p2066, 77p1721

Collinear triple
1 2p2067 7p2066

Z(X(1),X(34)) passes through these centers:
1, 78, 2p2066, 8p2067, 78p1722

Collinear triple
1 2p2066 8p2067

Z(X(1),X(37)) passes through these centers:
1, 81, 2p2248, 58p1654

Collinear triple
1 2p2248 58p1654

Z(X(1),X(55)) passes through these centers:
1, 7, 174, 7p1742, 174p503

Collinear triple
7 174 174p503

Z(X(1),X(56)) passes through these centers:
1, 8, 188, 979, 8p978, 188p361

Collinear triples
1 979 8p978
8 188 188p361

Z(X(1),X(57)) passes through these centers:
1, 9, 9p509, 9p1743, 514p1293

Collinear triple
1 9p1743 514p1293

Z(X(1),X(59)) passes through these centers:
1, 11, 523, 9p1019, 11p1381, 11p1382, 174p650, 522p1381, 522p1382

Collinear triples
1 523 9p1019
1 11p1381 522p1382
1 11p1382 522p1381
11 11p1381 11p1382
523 522p1381 522p1382

Z(X(1),X(63)) passes through these centers:
1, 19, 204, 2184

Collinear triple
1 204 2184

Z(X(1),X(65)) passes through these centers:
1, 21, 1247, 2136, 21p1046, 21p2137

Collinear triples
1 1247 21p1046
1 2136 21p2137

Z(X(1),X(75)) [ K175] passes through these centers:
1, 6, 19, 31, 48, 55, 56, 204, 221, 2192, 3p64, 64p1498, 207p268, 1034p2199

Collinear triples
1 19 48
1 55 56
1 204 3p64
1 221 2192
1 207p268 1034p2199
6 19 221
6 31 55
6 48 56
6 204 2192
6 3p64 207p268
6 64p1498 1034p2199
19 31 204
19 56 207p268
31 56 221
31 2192 207p268
31 3p64 64p1498
48 55 2192
48 221 1034p2199
55 204 1034p2199
55 221 3p64
56 2192 64p1498

Z(X(1),X(76)) passes through these centers:
1, 32, 6p365, 31p1631


Z(X(1),X(77)) passes through these centers:
1, 33, 282, 2331


Z(X(1),X(78)) passes through these centers:
1, 34, 207, 56p1034

Collinear triple
1 207 56p1034

Z(X(1),X(79)) passes through these centers:
1, 35, 1129, 35p481, 35p482, 35p1127

Collinear triples
1 1129 35p1127
35 35p481 35p482

Z(X(1),X(80)) passes through these centers:
1, 15, 16, 36, 58, 106, 202, 203, 214, 501, 758, 1130, 13p36, 14p36, 36p484, 36p502, 36p1128, 38p202, 80p203

Collinear triples
1 15 13p36
1 16 14p36
1 58 758
1 106 214
1 202 80p203
1 203 80p202
1 501 36p502
1 1130 36p1128
15 16 58
15 36 202
15 214 80p202
16 36 203
16 214 80p203
36 58 501
36 214 758
58 106 36p484
106 202 203
202 758 14p36
203 758 13p36
501 13p36 14p36
758 36p484 36p502
13p36 36p484 80p203
14p36 36p484 80p202

Z(X(1),X(81)) passes through these centers:
1, 2, 37, 42, 192, 37p2162

Collinear triples
1 2 42
1 192 37p2162
2 37 192

Z(X(1),X(82)) passes through these centers:
1, 38, 75, 1964, 2p194

Collinear triples
1 75 1964
38 75 2p194

Z(X(1),X(85)) passes through these centers:
1, 41, 2067, 33p2066, 41p169, 41p508, 57p2066, 220p2362

Collinear triples
1 2067 33p2066
1 57p2066 220p2362
41 2067 57p2066
33p2066 41p169 220p2362

Z(X(1),X(86)) passes through these centers:
1, 6, 33, 37, 42, 55, 65, 73, 2331, 3p1903, 20p2357, 40p64, 64p1490

Collinear triples
1 6 37
1 33 73
1 55 65
1 2331 3p1903
1 20p2357 40p64
6 33 20p2357
6 42 55
6 65 2331
6 3p1903 64p1490
33 37 55
33 42 2331
33 65 64p1490
37 73 3p1903
37 2331 40p64
42 65 73
42 3p1903 20p2357
42 40p64 64p1490
55 73 40p64

Z(X(1),X(88)) passes through these centers:
1, 44, 88, 678, 13p1250

Collinear triple
1 88 678

Z(X(1),X(92)) passes through these centers:
1, 31, 48, 63, 2066, 2067, 2p184, 3p193, 3p371, 3p372, 6p485, 6p486, 9p2067, 57p2066, 371p493, 372p494

Collinear triples
1 31 63
1 2066 2067
1 3p371 6p485
1 3p372 6p486
1 9p2067 57p2066
31 2066 9p2067
31 3p371 371p493
31 3p372 372p494
31 6p485 6p486
48 63 3p193
48 2067 57p2066
48 3p371 3p372
63 2p184 3p193
2066 3p371 57p2066
2067 2p184 57p2066
2067 3p372 9p2067
2p184 3p371 3p372
3p193 6p485 372p494
3p193 6p486 371p493

Z(X(1),X(98)) passes through these centers:
1, 511, 511p1756


Z(X(1),X(99)) passes through these centers:
1, 512, 1015, 1018

Collinear triple
1 1015 1018

Z(X(1),X(100)) passes through these centers:
1, 100, 244, 513, 100p1054

Collinear triples
1 100 244
100 513 100p1054

Z(X(1),X(101)) passes through these centers:
1, 514, 190p2350, 1086p1621

Collinear triple
1 190p2350 1086p1621

Z(X(1),X(104)) passes through these centers:
1, 80, 517, 3p1845

Collinear triple
1 80 3p1845

Z(X(1),X(105)) passes through these centers:
1, 291, 518, 238p518, 291p2108

Collinear triples
1 291 238p518
291 518 291p2108

Z(X(1),X(107)) passes through these centers:
1, 520, 40p1364, 282p1020

Collinear triple
1 40p1364 282p1020

Z(X(1),X(158)) passes through these centers:
1, 3, 255, 921, 1069, 1124, 1335, 3p46, 3p155, 3p371, 3p372, 3p485, 3p486, 3p487, 3p488, 6p493, 6p494

Collinear triples
1 921 3p155
1 1069 3p46
1 1124 1335
1 3p371 3p485
1 3p372 3p486
1 3p487 6p494
1 3p488 6p493
3 255 3p46
3 1069 3p155
3 1124 3p371
3 1335 3p372
255 3p371 3p372
255 3p487 3p488
1124 3p46 3p486
1335 3p46 3p485
3p155 3p485 3p486
3p155 6p493 6p494
3p371 3p486 3p487
3p372 3p485 3p488

Z(X(1),X(226)) passes through these centers:
1, 21, 31, 48, 1172, 2194, 21p1046, 31p1247, 1762p2194

Collinear triples
1 21 31
1 48 1172
1 21p1046 31p1247
21 1172 1762p2194
21 2194 21p1046
31 48 2194
31 31p1247 1762p2194

Z(X(1),X(273)) passes through these centers:
1, 3, 6, 55, 212, 219, 3p46, 9p2164, 21p2174, 55p224, 71p79, 71p1780

Collinear triples
1 3 55
1 6 219
1 3p46 9p2164
1 21p2174 71p79
3 6 21p2174
3 212 3p46
3 219 71p1780
6 55 212
55 9p2164 71p1780
212 219 55p224
219 3p46 71p79
9p2164 21p2174 55p224

Z(X(1),X(279)) passes through these centers:
1, 8, 9, 37, 42, 210, 8p978, 9p1743, 42p979, 523p1293

Collinear triples
1 8 42
1 9 37
1 8p978 42p979
1 9p1743 523p1293
8 210 8p978
9 210 9p1743
37 42 210

Z(X(1),X(310)) passes through these centers:
1, 6, 55, 213, 1402, 1918, 1402p1764

Collinear triples
1 6 213
1 55 1402
6 55 1918
1402 1918 1402p1764

Z(X(1),X(313)) passes through these centers:
1, 31, 48, 58, 501, 1474, 2206, 31p267

Collinear triples
1 31 58
1 48 1474
1 501 31p267
31 48 2206
58 5012206

Z(X(1),X(321)) passes through these centers:
1, 28, 31, 48, 81, 1333, 6p2248, 58p199, 58p1654

Collinear triples
1 28 48
1 31 81
1 6p2248 58p1654
28 81 58p199
31 48 1333
31 6p2248 58p199
81 1333 58p1654

Z(X(1),X(326)) passes through these centers:
1, 19, 33, 34, 204, 207, 1096, 2331, 4p64, 4p1033, 4p1436, 25p1032, 25p1034

Collinear triples
1 33 34
1 204 4p64
1 207 25p1034
1 2331 4p1436
1 4p1033 25p1032
19 34 2331
19 204 1096
19 207 4p1436
19 4p64 4p1033
33 204 4p1436
33 207 4p64
33 1096 2331
33 4p1033 25p1034
34 207 1096
34 4p1033 4p1436
204 2331 25p1034
207 2331 25p1032

Z(X(1),X(346)) passes through these centers:
1, 56, 57, 221, 1407, 1419, 1422, 56p366

Collinear triples
1 56 57
1 221 1422
56 221 1407
57 1407 1419

Z(X(1),X(561)) passes through these centers:
1, 31, 48, 560, 1973, 2156, 6p206, 6p1676, 6p1677, 25p159, 32p1670, 32p1671

Collinear triples
1 48 1973
1 2156 6p206
1 6p1676 32p1671
1 6p1677 32p1670
31 48 560
31 1973 6p206
31 2156 25p159
560 1973 25p159
560 32p1670 32p1671
2p206 6p1676 6p1677

Z(X(1),X(673)) passes through these centers:
1, 6, 55, 241, 292, 518, 672, 673, 55p1362, 238p518, 291p2110

Collinear triples
1 6 518
1 55 241
1 292 238p518
1 673 55p1362
6 55 672
6 292 55p1362
55 673 238p518
241 518 55p1362
292 672 291p2110
518 672 238p518
518 673 291p2110

Z(X(1),X(739)) passes through these centers:
1, 2, 192, 536, 899, 10p715, 75p739, 87p899, 899p899

Collinear triples
1 2 899
1 192 87p899
1 75p739 899p899
2 192 536
192 10p715 899p899
536 899 899p899

Z(X(1),X(903)) passes through these centers:
1, 6, 44, 55, 106, 678, 902, 1319, 2161, 2342, 3p1846, 6p214

Collinear triples
1 6 44
1 55 1319
1 106 678
1 2161 6p214
1 2342 3p1846
6 55 902
6 106 6p214
6 2161 3p1846
44 678 902
44 1319 6p214
55 678 2161
55 2342 6p214
902 1319 3p1846

Z(X(1),X(961)) passes through these centers:
1, 8, 21, 960, 1193, 2292, 8p978, 21p1046, 43p256, 256p846, 979p1193, 1247p2292, 1999p2269

Collinear triples
1 8 1193
1 21 2292
1 8p978 979p1193
1 21p1046 1247p2292
8 21 1999p2269
8 960 8p978
8 2292 43p256
21 960 21p1046
21 1193 256p846
960 1193 2292
960 43p256 256p846
1193 979p1193 1999p2269
2292 1247p2292 1999p2269
8p978 256p846 1247p2292
21p1046 43p256 979p1193

Z(X(1),X(1043)) passes through these centers:
1, 34, 56, 64, 65, 73, 207, 221, 1042, 7p2357, 20p1042, 1034p1410

Collinear triples
1 34 73
1 56 65
1 64 20p1042
1 207 1034p1410
1 221 7p2357
34 56 20p1042
34 65 221
34 207 1042
56 207 7p2357
56 221 1042
64 65 207
64 73 221
65 73 1042
65 7p2357 20p1042
73 20p1042 1034p1410

Z(X(1),X(1219)) passes through these centers:
1, 56, 221, 1191, 1697, 2334, 84p1697

Collinear triples
1 56 1697
1 221 84p1697
56 221 1191

Z(X(1),X(1220)) passes through these centers:
1, 58, 501, 1193, 2067, 2292, 57p2066, 267p2292, 429p1805, 429p1806, 501p2127

Collinear triples
1 58 2292
1 501 267p2292
1 2067 429p1806
1 57p2066 429p1805
58 501 1193
501 2292 501p2127
1193 2067 57p2066
2292 429p1805 429p1806

Z(X(1),X(1259)) passes through these centers:
1, 4, 34, 207, 1118, 1148, 4p266, 34p1034

Collinear triples
1 4 34
1 207 34p1034
4 1118 1148
34 207 1118

Z(X(1),X(1268)) passes through these centers:
1, 6, 55, 58, 501, 1100, 1962, 2160, 2308, 6p553, 35p1100, 267p1962

Collinear triples
1 6 1100
1 55 6p553
1 58 1962
1 501 267p1962
1 2160 35p1100
6 55 2308
6 58 35p1100
55 1962 2160
58 501 2308
501 1100 2160
1100 1962 2308
1100 6p553 35p1100
1962 35p1100 267p1962

Z(X(1),X(1804)) passes through these centers:
1, 4, 33, 1148, 1857, 2331, 4p259, 4p282

Collinear triples
1 4 33
1 2331 4p282
4 1148 1857
33 1857 2331

Z(X(1),X(1821)) passes through these centers:
1, 31, 48, 240, 1755, 1821, 1959, 1967, 3p1987, 6p2009, 6p2010, 55p1355, 232p401, 325p1691, 511p1687, 511p1688

Collinear triples
1 31 1959
1 48 240
1 1821 55p1355
1 1967 325p1691
1 3p1987 232p401
1 6p2009 511p1687
1 6p2010 511p1688
31 48 1755
31 1821 232p401
31 1967 55p1355
48 1821 325p1691
48 3p1987 55p1355
240 755 232p401
240 1959 55p1355
1755 1959 325p1691
1755511p1687511p1688
6p2009 6p2010 55p1355

Z(X(1),X(1911)) passes through these centers:
1, 2, 86, 192, 239, 257, 335, 350, 385, 740, 2p294, 2p2068, 2p2069, 75p727, 81p1655, 87p239, 238p239, 238p726, 239p241, 274p2107, 740p2106, 1281p2113, 1654p1929

Collinear triples
1 2 239
1 86 740
1 192 87p239
1 257 385
1 335 238p239
1 2p294 239p241
1 2p2068 2p20691
1 75p727 238p726
1 274p2107 740p2106
2 86 385
2 192 350
2 335 238p726
2 2p294 238p239
86 335 740p2106
86 350 81p1655
86 238p239 1654p1929
192 257 740
192 335 239p241
192 75p727 238p239
239 257 81p1655
239 350 740p2106
239 385 239p241
239 740 238p239
257 2p294 740p2106
257 238p726 1654p1929
350 385 238p239
350 740 238p726
385 87p239 238p726
81p1655 238p239 274p2107
238p726 239p241 1281p2113

Z(X(1),X(1927)) passes through these centers:
1, 75, 336, 350, 1909, 1926, 1934, 1934, 1966, 2p83, 2p194, 2p732, 43p1909, 76p695, 76p699, 87p350, 240p1966, 384p385, 385p385, 385p698

Collinear triples
1 75 1966
1 336 240p1966
1 350 1909
1 1934 385p385
1 2p83 2p732
1 43p1909 87p350
1 76p695 384p385
1 76p699 385p698
75 336 385p698
75 350 87p350
75 1926 43p1909
75 1934 385p698
75 2p83 385p385
350 87p350 385p698
1909 2p194 87p350
1909 2p732 43p1909
1926 2p732 385p698
1926 384p385 385p385
1934 2p194 240p1966
1966 2p732 385p385
1966 240p1966 384p385
2p194 2p732 76p695
2p194 76p699 385p385

Z(X(1),X(1934)) passes through these centers:
1, 31, 48, 82, 172, 1428, 1580, 1910, 1914, 1927, 1933, 2330, 4p1691, 31p1281, 31p2236, 32p695, 147p1976, 194p699, 237p385, 384p385, 385p385, 385p2076

Collinear triples
1 31 1580
1 48 4p1691
1 82 31p2236
1 172 1914
1 1428 2330
1 1910 237p285
1 19275 385p385
1 32p695 384p385
31 48 1933
31 82 384p385
31 1914 2330
31 1927 237p385
31 194p699 385p385
48 172 1428
48 1910 385p385
48 31p2236 32p695
82 1927 385p2076
172 1927 31p1281
172 2330 31p2236
1428 1580 31p1281
1428 1914 237p385
1580 1933 385p2076
1580 31p2236 385p385
1910 1933 147p1976
1910 2330 31p1281
1914 1933 31p1281
1927 4p1691 147p1976
1933 31p2236 237p385
1933 384p385 385p385
4p1691 237p385 384p385
32p695 194p699 385p2076

Z(X(1),X(2221)) passes through these centers:
1, 2, 192, 612, 1219, 2345, 87p612, 1191p2345

Collinear triples
1 2 612
1 192 87p612
1 1219 1191p2345
2 192 2345
612 2345 1191p2345

Z(X(1),X(2287)) passes through these centers:
1, 57, 65, 73, 278, 1419, 1427

Collinear triples
1 57 65
1 73 278
57 1419 1427
65 73 1427

Z(X(2),X(1)) [Thomson cubic, K002] passes through these centers:
1, 2, 3, 4, 6, 9, 57, 223, 282, 1073, 1249, 84p1490, 204p1032, 221p1034, 1073p1712

Collinear triples
1 3 57
1 4 223
1 6 9
1 282 1249
1 1073 84p1490
1 221p1034 1073p1712
2 3 4
2 9 57
2 223 282
2 1073 1249
2 84p1490 221p1034
2 204p1032 1073p1712
3 9 2823
3 223 221p1034
3 1249 204p1032
4 6 1249
4 57 84p1490
6 57 223
6 282 84p1490
6 1073 1073p1712
9 223 1073
9 1249 221p1034
9 84p1490 204p1032
571073p17121073p1712

Z(X(2),X(2)) passes through these centers:
2, 31, 365, 6p1631, 365p510

Collinear triple
31 365 365p510

Z(X(2),X(3)) passes through these centers:
2, 19, 19p1763


Z(X(2),X(4)) passes through these centers:
2, 48, 48p1726


Z(X(2),X(5)) passes through these centers:
2, 2148


Z(X(2),X(6)) passes through these centers:
1, 2, 7, 9, 366, 1489, 2p1419, 2p2067, 7p2066, 8p2362, 92p2066, 173p1489, 364p366

Collinear triples
172p1419
1366364p366
12p20677p2066
279
220p206792p2066
27p20668p2362
98p236292p2066
14892p2067173p1489

Z(X(2),X(7)) passes through these centers:
2, 41, 259, 55p1626, 259p362

Collinear triple
41259259p362

Z(X(2),X(8)) passes through these centers:
2, 266, 604, 266p504

Collinear triple
266604266p504

Z(X(2),X(9)) passes through these centers:
2, 56, 478, 509, 2362, 2p2067, 7p2066, 225p1806

Collinear triples
223627p2066
22p2067225p1806
562p20677p2066
4782362225p1806

Z(X(2),X(10)) passes through these centers:
2, 3, 6, 28, 81, 1333, 2p501, 6p267

Collinear triples
2328
2681
22p5016p267
361333
8113332p501

Z(X(2),X(19)) passes through these centers:
2, 3, 6, 69, 485, 486, 2p2066, 2p2067, 7p2066, 8p2067, 48p491, 48p492

Collinear triples
2669
248548p492
248648p491
22p20662p2067
27p20668p2067
36963p193
32p20677p2066
348p49148p492
6485486
62p20668p2067
2p20667p206648p492
2p20678p206748p491

Z(X(2),X(29)) passes through these centers:
2, 63, 1400, 1409, 45p63, 65p2164, 1400p1764

Collinear triples
2631400
245p6365p2164
63140945p63
140014091400p1764

Z(X(2),X(32)) passes through these centers:
2, 75, 330, 2p192, 2p366

Collinear triple
23302p192

Z(X(2),X(33)) passes through these centers:
2, 57, 63, 222, 223, 2p1433, 46p63, 222p1158, 912p2006

Collinear triples
25763
22232p1433
57222223
6322246p63
632p1433222p1158

Z(X(2),X(34)) passes through these centers:
1, 2, 9, 63, 78, 219, 8p2164, 9p224, 12p1789, 21p35, 46p63, 72p1780

Collinear triples
1278
19219
16321p35
2963
28p216446p63
212p178921p35
98p216472p1780
637872p1780
6321946p63
782199p224
7812p178946p63
8p21649p22421p35

Z(X(2),X(37)) passes through these centers:
2, 3, 6, 27, 58, 86, 2248, 58p1761, 81p1654

Collinear triples
2 3 27
2 6 86
2 2248 81p1654
3 6 58
58 86 21p1654

Z(X(2),X(55)) passes through these centers:
2, 57, 174, 189, 223, 557, 558, 1659, 2p2067, 7p2066, 57p175, 57p176, 164p174, 174p505, 273p2066

Collinear triples
2 189 223
2 557 558
2 1659 7p2066
2 2p2067 273p2066
2 164p174 174p505
57 174 164p174
57 2p2067 7p2066
57 57p175 57p176
174 223 174p505
174 557 7p2066
174 558 2p2067
223 1659 273p2066
557 164p174 293p2066
558 1659 164p174
1659 2p2067 57p176
7p206657p175273p2066

Z(X(2),X(56)) passes through these centers:
1, 2, 8, 9, 188, 236, 8p289, 8p1743, 514p1293

Collinear triples
128
11888p289
22368p289
28p1743514p1293
898p1743
9188236

Z(X(2),X(57)) passes through these centers:
2, 55, 2p2067, 7p2066, 9p509, 9p1486, 200p2362, 281p2066

Collinear triples
22p2067281p2066
27p2066200p2362
552p20677p2066
9p1486200p2362281p2066

Z(X(2),X(58)) passes through these centers:
1, 2, 9, 10, 37, 226, 281, 1214, 2p2331, 10p1433, 20p1903, 40p2184, 1490p2184

Collinear triples
1 2 10
1 9 37
1 226 2p2331
1 281 20p1903
1 10p1433 1490p2184
2 9 226
2 281 1214
2 2p2331 10p1433
2 20p1903 40p2184
9 10 281
9 1214 40p2184
10 1214 10p1433
10 2p2331 40p2184
37 226 1214
37 281 2331
3710p143320p1903
3740p21841490p2184
2262811490p2184

Z(X(2),X(59)) passes through these centers:
2, 2170, 174p650, 513p1222, 514p2347

Collinear triple
2513p1222514p2347

Z(X(2),X(64)) passes through these centers:
2, 57, 223, 3p1249, 8p610, 20p282

Collinear triples
2578p610
222320p282
572233p1249

Z(X(2),X(65)) passes through these centers:
2, 3, 6, 29, 284, 333, 6p1247, 284p1762, 314p2305

Collinear triples
2329
26333
26p1247314p2305
36284
66p1247284p1762
29333284p1762
284333314p2305

Z(X(2),X(73)) passes through these centers:
2, 4, 21, 1172, 1249, 9p229, 21p2184

Collinear triples
2421
2124921p2184
411721249
2111729p229

Z(X(2),X(75)) [ K177] passes through these centers:
2, 3, 6, 25, 32, 66, 206, 1676, 1677, 19p159, 31p1670, 31p1671

Collinear triples
2 3 25
2 66 206
2 1676 31p1671
2 1677 31p1670
3 6 32
6 25 206
66619p159
25 32 19p159
32 31p1670 31p1671
20616761697

Z(X(2),X(82)) passes through these centers:
2, 3, 6, 39, 141, 427, 38p1342, 38p1343, 63p66, 427p2172, 1370p2156

Collinear triples
236
26141
263p66427p2172
3639
314164p66
64271370p2156
39427427p2172
3938p134238p1343
3963p661370p2156

Z(X(2),X(86)) passes through these centers:
1, 2, 9, 42, 213, 1400, 1400p1764

Collinear triples
1242
19213
291400
21314001400p1764

Z(X(2),X(87)) passes through these centers:
1, 2, 9, 43, 1423, 2176, 37p904

Collinear triples
1243
192176
291423
94337p904

Z(X(2),X(91)) passes through these centers:
2, 3, 6, 24, 70, 571, 1993, 26p47, 49p2190, 161p1748

Collinear triples
2 3 24
2 6 1993
2 70 26p47
3 6 571
3 1993 49p2190
24 1993 26p47

Z(X(2),X(92)) passes through these centers:
2, 184, 31p485, 31p486, 31p1670, 31p1671, 48p157, 48p491, 48p492, 63p1676, 63p1677, 75p1485

Collinear triples
2 31p485 48p492
2 31p486 48p491
2 31p1670 63p1677
2 31p1671 63p1676
2 48p157 75p1485
184 31p1670 31p1671
184 48p491 48p492
31p485 31p486 48p157
48p157 63p1676 63p1677

Z(X(2),X(99)) passes through these centers:
2, 798, 10p932, 192p1977

Collinear triple
2 10p932 192p1977

Z(X(2),X(100)) passes through these centers:
2, 649, 100p596, 244p595

Collinear triple
2 100p596 244p595

Z(X(2),X(101)) passes through these centers:
2, 513, 668, 1015, 1978p1979

Collinear triples
2 668 1015
513 668 1978p1979

Z(X(2),X(103)) passes through these centers:
2, 57, 105, 223, 910, 8p910, 271p1886, 516p518

Collinear triples
2 57 8p910
2 105 516p518
2 223 271p1886
57 223 910
910 8p910 516p518

Z(X(2),X(106)) passes through these centers:
1, 2, 9, 44, 80, 88, 214, 519, 2p678, 2p1319, 2p2342, 57p1145

Collinear triples
1 2 519
1 9 44
1 80 57p1145
1 88 214
2 9 2p1319
2 80 214
2 88 2p678
2 2p2342 57p1145
9 80 2p678
9 214 2p2342
44 519 2p678
44 2p1319 57p1145
214 519 2p1319

Z(X(2),X(109)) passes through these centers:
2, 11, 100, 650, 101p149

Collinear triples
2 11 100
100 650 101p149

Z(X(2),X(110)) passes through these centers:
2, 244, 661, 2p1018

Collinear triple
2 244 2p1018

Z(X(3),X(1)) [McCay cubic, K003]] passes through these centers:
1, 3, 4, 1075, 1745

Collinear triple
1 4 1745

Z(X(3),X(2)) passes through these centers:
3, 19, 55, 57, 84, 198, 365

Collinear triples
3 55 57
3 84 198
19 55 198

Z(X(3),X(7)) passes through these centers:
1, 3, 33, 55, 198, 259, 282, 1745

Collinear triples
1 3 55
1 33 1745
3 198 282
33 55 198

Z(X(3),X(8)) passes through these centers:
1, 3, 34, 56, 266, 1035, 1745, 56p1034

Collinear triples
1 3 56
1 34 1745
3 1035 56p1034
34 56 1035

Z(X(3),X(75)) [ K172] passes through these centers:
3, 6, 25, 55, 56, 64, 154, 198, 1033, 1035, 1436, 31p1032, 31p1034

Collinear triples
3 55 56
3 64 154
3 198 1436
3 1033 31p1032
3 1035 31p1034
6 25 154
6 56 198
6 64 1033
6 1035 1046
25 55 198
25 56 1035
55 64 1035
55 154 1436
55 1033 31p1034
56 1033 1436
154 198 31p1034
198 1035 31p1032

Z(X(4),X(1)) [Orthocubic, K006] passes through these centers:
1, 3, 4, 46, 90, 155, 254, 371, 372, 485, 486, 487, 488, 3p1123, 3p1336, 19p493, 19p494

Collinear triples
1 3 46
1 90 55
1 371 3p1336
1 372 3p1123
3 371 372
3 487 488
4 46 20
4 155 254
4 371 485
4 372 486
4 487 19p494
4 488 19p493
4 3p1123 3p1336
46 485 3p1123
46 486 3p1336
155 485 486
155 19p493 19p494
371 486 487
372 485 488

Z(X(4),X(31)) [ K170] passes through these centers:
2, 4, 69, 193, 487, 488, 2p2128, 2p2129, 19p1267, 63p1123, 91p1599, 91p1600, 92p493, 92p494

Collinear triples
2 69 193
2 487 91p1599
2 488 91p1600
4 487 92p494
4 488 92p493
4 2p2128 2p2129
4 19p1267 63p1123
4 91p1599 91p1600
69 487 488
193 91p1599 92p493
193 91p1600 92p494
2p2128 92p493 92p494

Z(X(4),X(63)) passes through these centers:
2, 4, 6, 25, 193, 371, 372, 2362, 4p2066, 9p2362, 19p485, 19p486, 193p2129, 225p1806

Collinear triples
2 4 25
2 6 193
4 371 19p485
4 372 19p486
4 2362 4p2066
4 9p2362 225p1806
6 371 372
6 2362 225p1806
25 193 193p2129
25 4p2066 9p2362
25 19p485 19p486
371 4p2066 225p1806
372 2362 9p2362

Z(X(4),X(75)) [ K176] passes through these centers:
3, 4, 6, 25, 155, 184, 571, 2165, 25p921

Collinear triples
3 4 25
3 6 571
3 155 184
4 155 25p921
4 571 2165
6 25 184
6 155 2165
25 571 25p921

Z(X(4),X(77)) passes through these centers:
1, 4, 9, 19, 33, 46, 55, 9p1723, 10p2160, 29p2174, 33p90, 37p1780

Collinear triples
1 4 33
1 9 37p1780
1 19 29p2174
1 46 55
4 9 19
4 46 33p90
4 10p2160 29p2174
9 46 10p2160
9 55 9p1723
19 33 55
33 33p90 37p1780
9p1723 29p2174 33p90

Z(X(4),X(78)) passes through these centers:
1,4,34,46, 56, 84, 208, 34p90, 36p915, 56p1158, 80p1455, 84p1720, 102p1870, 1411p1737

Collinear triples
1 4 34
1 46 56
1 84 56p1158
1 208 102p1870
4 46 34p90
4 84 208
4 36p915 1411p1737
4 80p1455 102p1870
34 46 36p915
34 56 208
34 34p90 56p1158
46 84 80p1455
56 84 84p1720
56 80p1455 1411p1737
208 34p90 1411p1737
34p9084p1720102p1870
36p915 56p1158 102p1870

Z(X(5),X(1)) [Feuerbach cubic, K005] passes through these centers:
1, 3, 4, 5, 17, 18, 54, 61, 62, 195, 627, 628, 2120, 2121, 74p1749

Collinear triples
3 4 5
3 54 195
3 61 62
5 17 61
5 18 62
5 2120 2122
17 18 195
17 62 627
18 61 628
54 627 628

Z(X(6),X(1)) [Grebe cubic, K102] passes through these centers:
1, 2, 6, 43, 87, 194

Collinear triples
1 2 43
1 87 194
6 43 87

Z(X(6),X(6)) passes through these centers:
6, 75, 366


Z(X(6),X(57)) passes through these centers:
1, 6, 8, 9, 43, 979, 2319, 9p509, 9p978, 10p893, 21p171, 284p1999, 846p1247

Collinear triples
1 6 9
1 8 43
1 284p1999 846p1247
6 43 2319
6 979 9p978
6 10p893 21p171
8 9 9p978
8 21p171 284p1999
9 43 10p893
43 979 284p1999
2319 9p978 21p171
9p978 10p893 846p1247

Z(X(7),X(1)) passes through these centers:
1, 7, 9, 55, 57, 218, 277

Collinear triples
1 9 218
1 55 57
7 9 57
7 218 277

Z(X(6),X(33)) passes through these centers:
1, 3, 7, 57, 63, 77, 90, 224, 3p1708, 21p2003, 46p77, 79p1214

Collinear triples
1 3 57
1 7 77
1 90 3p1708
3 63 224
3 77 46p77
7 57 63
7 90 46p77
7 21p2003 79p1214
57 77 21p2003
63 77 3p1708
63 46p77 79p1214
90 224 21p2003

Z(X(7),X(55)) passes through these centers:
1, 2, 7, 57, 145, 174, 1488, 2089, 145p2137

Collinear triples
1 2 145
1 174 2089
2 7 57
7 1488 2089
57 145 145p2137
57 174 1488

Z(X(8),X(1)) passes through these centers:
1, 8, 40, 56, 84, 2122, 2123

Collinear triples
1 40 56
1 84 2122
8 40 84
8 2122 2123

Z(X(8),X(6)) passes through these centers:
1, 2, 8, 40, 57, 144, 189, 366

Collinear triples
1 2 8
1 40 57
2 57 144
8 40 189

Z(X(8),X(31)) passes through these centers:
2, 7, 8, 144, 175, 176, 1143, 1274, 2p364, 2p2124, 2p2125, 19p1267, 63p1123

Collinear triples
2 7 144
2 175 19p1267
2 176 63p1123
7 175 176
8 1143 1274
8 2p2124 2p2125
8 19p1267 63p1123

Z(X(8),X(34)) passes through these centers:
1, 3, 8, 40, 78, 90, 271, 3p1158, 10p1800, 80p912, 271p1720, 515p1807

Collinear triples
1 3 40
1 8 78
1 90 3p1158
3 78 10p1800
3 271 271p1720
3 80p912 515p1807
8 40 271
8 90 10p1800
40 90 80p912
78 271 3p1158
271 10p1800 515p1807

Z(X(8),X(56)) [ K199] passes through these centers:
1, 8, 40, 175, 176, 188, 280, 483, 2p2066, 8p2067, 8p2362, 9p557, 92p2066, 164p188, 188p505

Collinear triples
1 175 176
1 188 164p188
1 2p2066 8p2067
8 40 280
8 483 9p557
8 2p2066 8p2362
8 8p2067 92p2066
8 164p188 188p505
40 188 188p505
40 8p2362 92p2066
175 2p2066 92p2066
176 8p2067 8p2362
188 483 8p2067
188 2p2066 9p557
483 8p2362 164p188
9p557 92p2066 164p188

Z(X(8),X(58)) [Spieker central cubic, K033] passes through these centers:
1, 4, 8, 10, 40, 65, 72, 2p1903, 4p1490, 8p64, 10p1394, 73p1034, 1032p2331

Collinear triples
1 4 10p1394
1 8 10
1 40 65
1 2p1903 4p1490
4 8 72
4 10 40
4 65 4p1490
8 40 2p1903
8 4p1490 73p1034
8 8p64 10p1394
10 65 72
10 2p1903 10p1394
10 4p1490 8p64
40 72 8p64
40 10p1394 1032p2331
72 4p1490 1032p2331
72 10p1394 73p1034

Z(X(8),X(106)) passes through these centers:
1, 8, 40, 104, 519, 1145, 1319, 1339, 44p189

Collinear triples
1 8 519
1 40 1319
8 40 44p189
8 104 1145
519 1145 1319
1145 1339 44p189

Z(X(9),X(1)) passes through these centers:
1, 9, 57, 165, 364, 2124, 2125, 3p1123, 6p175, 6p176, 6p1143, 6p1274

Collinear triples
1 57 165
1 3p1123 6p176
1 3p1336 6p175
9 2124 2125
9 3p1123 3p1336
9 6p1143 6p1274
57 6p175 6p176

Z(X(9),X(2)) passes through these centers:
1, 6, 9, 56, 84, 165, 198, 365, 34p271

Collinear triples
1 6 9
1 56 165
6 56 1985
9 84 198

Z(X(9),X(81)) passes through these centers:
1, 4, 9, 37, 65, 71, 165

Collinear triples
1 9 37
1 65 165
4 9 71
37 65 71

Z(X(9),X(86)) passes through these centers:
6, 9, 19, 37, 71, 198, 1400, 1903, 8p2155, 9p207, 20p1400, 1034p1409, 1498p1903

Collinear triples
6 9 37
6 19 20p1400
6 198 1400
6 903 9p207
6 8p2155 1498p1903
9 19 71
9 198 1903
9 8p2155 20p1400
9 9p207 1034p1409
19 37 198
19 1400 9p207
37 71 1400
37 1903 20p1400
37 8p2155 9p207
371034p14091498p1903
71 198 8p2155
71 20p1400 1032p2331
1400 1903 1498p1903

Z(X(9),X(105)) passes through these centers:
1, 2, 9, 105, 165, 241, 518, 672, 2p103, 2p1742, 8p1362, 43p291, 59p1566, 291p1282, 518p1376, 1280p1743

Collinear triples
1 2 518p1376
1 9 518
1 105 59p1566
1 165 241
1 8p1362 1280p1743
2 9 672
2 241 2p1742
2 8p1362 43p291
9 105 8p1362
9 2p103 59p1566
105 672 291p1282
165 2p103 8p1362
241 672 59p1566
241 8p1362 518p1376
24143p291291p1282
518 672 8p1362
2p103291p1282518p1376

Z(X(10),X(1)) passes through these centers:
1, 10, 58, 191, 267, 2126, 2127, 3p1123, 3p1336

Collinear triples
1 58 191
1 267 2126
10 191 267
10 2126 2127
10 3p1123 3p1336

Z(X(10),X(6)) passes through these centers:
1, 2, 10, 81, 191, 366, 1029, 1654, 2p2248, 1255p1961

Collinear triples
1 2 10
1 81 191
2 81 1654
10 191 1029
10 1654 2p2248
191 2p2248 1255p1961

Z(X(10),X(56)) passes through these centers:
1, 8, 10, 21, 188, 191, 1247, 2p1250, 2p1251, 8p267, 8p1046, 9p554, 9p559, 484p1320

Collinear triples
1 8 10
1 21 191
1 2p1250 9p559
8 21 8p1046
10 191 8p267
10 1247 8p1046
10 2p1250 2p1251
10 9p554 9p559
191 2p1251 9p554
8p267 8p1046 484p1320

Z(X(10),X(81)) passes through these centers:
2, 4, 6, 10, 42, 71, 199, 1654, 37p2248

Collinear triples
2 4 199
2 6 1654
2 10 42
4 10 71
6 42 71
10 1654 37p2248
42 199 37p2248

Z(X(13),X(1)) passes through these centers:
1, 13, 15, 18, 62, 13p202

Collinear triples
1 15 13p202
13 18 62

Z(X(14),X(1)) passes through these centers:
1, 14, 16, 17, 61, 14p203

Collinear triples
1 16 14p203
14 17 61

Z(X(19),X(1)) passes through these centers:
1, 19, 63, 1707, 2128, 2129, 3p1123, 3p1336, 4p493, 4p494, 6p487, 6p488, 371p486, 372p485

Collinear triples
1 63 1707
1 6p487 371p486
1 6p488 372p485
19 2128 2129
19 3p1123 3p1336
19 4p493 6p488
19 4p494 6p487
19 371p486 372p485
63 6p487 6p488
1707 4p493 371p486
1707 4p494 372p485
2128 4p493 4p494

Z(X(19),X(19)) passes through these centers:
19, 326, 2p493, 2p494, 6p487, 6p488

Collinear triples
19 2p493 6p488
19 2p494 6p487
326 6p487 6p488

Z(X(20),X(1)) [Darboux cubic, K004] passes through these centers:
1, 3, 4, 20, 40, 64, 84, 1490, 1498, 2130, 2131, 19p1032, 56p1034

Collinear triples
1 3 40
1 4 1490
1 84 1498
3 4 20
3 64 1498
3 84 1490
4 2130 19p1032
20 40 84
20 1490 56p1034
20 1498 19p1032
40 64 1490
40 1498 56p1034
84 2130 56p1034

Z(X(20),X(6)) passes through these centers:
20, 40, 189, 366, 1490, 2184, 57p1034

Collinear triples
20 40 189
20 1490 57p1034
40 1490 2184

Z(X(20),X(75)) passes through these centers:
3, 6, 20, 25, 393, 577, 1498, 1661, 6p2155, 1032p1973, 1973p2063

Collinear triples
3 6 577
3 20 25
3 1498 6p2155
6 25 1973p2063
6 393 1498
6 1661 6p2155
20 393 577
20 1498 1032p1973
25 393 1661
577 1661 1032p1973
577 6p2155 1973p2063

Z(X(21),X(10)) passes through these centers:
1, 3, 21, 28, 56, 58, 84, 1394, 2360, 21p64, 21p207, 84p1498, 603p1034

Collinear triples
1 3 56
1 21 58
1 28 2360
1 84 1394
1 21p64 21p207
1 84p1498 603p1034
3 21 28
3 1394 603p1034
3 2360 21p64
21 84 2360
21 1394 21p64
21 21p207 603p1084
28 56 21p207
28 58 1394
56 58 2360
56 84 84p1498
58 84 21p207
58 21p64 84p1498

Z(X(21),X(75)) passes through these centers:
1, 3, 6, 21, 25, 31, 37, 1333, 1402, 2217, 6p573, 31p1764, 37p572, 58p197, 1400p1610

Collinear triples
1 3 1402
1 6 37
1 21 31
1 2217 58p197
3 6 1333
3 21 25
3 2217 37p572
6 25 58p197
6 31 37p572
6 1402 6p573
6 2217 1400p1610
21 37 1333
21 2217 6p573
25 31 1400p1610
25 37 6p573
31 1333 6p573
31 1402 31p1764
37 1402 37p572
37 2217 31p1764
1333 1402 58p197

Z(X(23),X(75)) [ K108] passes through these centers:
3, 6, 23, 25, 111, 187, 1177, 6p2157, 31p858

Collinear triples
3 6 187
3 23 25
6 25 31p858
23 111 187
23 1177 31p858
187 6p2157 31p858

Z(X(27),X(1)) [ K109] passes through these centers:
1, 3, 4, 19, 27, 63, 71, 226, 284, 579, 1751, 1780, 3p1713, 63p1714, 72p1612, 226p580, 377p2219

Collinear triples
1 3 226p580
1 4 226
1 19 284
1 63 1780
1 71 579
1 1751 72p1612
3 4 27
3 63 72p1612
3 71 3p1713
3 284 579
4 19 71
4 284 377p2219
4 1751 1780
19 27 63
19 1751 226p580
27 226 284
27 579 1751
63 71 63p1714
63 226 579
63 226p580 377p2219
71 226 226p580
71 284 1780
226 1751 3p1713
284 1751 63p1714

Z(X(27),X(71)) passes through these centers:
1, 2, 7, 27, 28, 579, 1172, 2p1724, 4p272, 7p1754

Collinear triples
1 2 2p1724
1 7 7p1754
1 28 1172
2 7 579
2 27 28
7 27 1172
27 579 4p272
28 4p272 7p1754
1172 2p1724 4p272

Z(X(28),X(63)) passes through these centers:
1, 2, 19, 25, 28, 37, 1724, 469p1245, 2214p2303

Collinear triples
1 19 28
1 25 469p1245
1 37 1724
2 25 28
2 1724 2214p2303
19 25 2214p2303

Z(X(28),X(75)) passes through these centers:
3, 6, 19, 25, 28, 48, 65, 228, 2194, 2218, 2352, 3p1714, 6p1780, 48p1713, 65p580, 71p1612

Collinear triples
3 6 6p1780
3 25 28
3 48 71p1612
3 65 2352
3 228 3p1714
6 19 65
6 25 2194
6 48 65p580
6 228 2352
6 2218 71p1612
19 25 228
19 28 48
19 2218 6p1780
25 2218 65p580
28 65 2194
28 2218 2352
48 228 48p1713
48 2194 2352
65 228 65p580
65 2218 48p1713
228 2194 6p1780
219422183p1714

Z(X(30),X(1)) [Neuberg cubic, K001] passes through these centers:
1, 3, 4, 13, 14, 15, 16, 30, 74, 399, 484, 617, 1138, 1157, 1263, 1276, 1277, 1337, 1338, 2132, 2133

Collinear triples
1 3 484
1 15 1276
1 16 1277
3 4 30
3 15 16
3 74 399
4 399 1263
4 617 1338
4 1138 2132
4 1276 1277
13 14 399
13 15 30
13 484 1277
14 15 617
14 16 30
14 484 1276
30 399 1138
30 1157 1263
30 2132 2133
1157 1337 1338

Z(X(30),X(75)) passes through these centers:
3, 6, 25, 30, 50, 399, 1989, 6p2159, 31p146, 31p1138, 31p2071

Collinear triples
3 6 50
3 25 30
3 399 6p2159
6 25 31p2071
6 399 1989
6 6p2159 31p146
25 1989 31p146
30 50 1989
30 399 31p1138
50 6p2159 31p2071
50 31p146 31p1138

Z(X(35),X(2)) passes through these centers:
35, 55, 57, 267, 365, 1030, 2160

Collinear triples
35 55 57
35 267 1030
55 1030 2160

Z(X(35),X(75)) passes through these centers:
6, 35, 42, 55, 56, 58, 1030, 6p267, 6p2160

Collinear triples
6 42 55
6 58 1030
35 42 58
35 55 56
35 1030 6p267
55 1030 6p2160

Z(X(36),X(75)) passes through these centers:
6, 36, 55, 56, 106, 902, 909, 2183, 6p2161

Collinear triples
6 55 902
6 56 2183
36 55 56
36 106 902
36 909 2183
902 2183 6p2161

Z(X(37),X(2)) passes through these centers:
1, 6, 37, 58, 267, 365, 846, 1030, 2248, 1126p1961

Collinear triples
1 6 37
1 58 846
6 58 1030
37 267 1030
37 846 2248
1030 2248 1126p1961

Z(X(37),X(7)) passes through these centers:
6, 9, 37, 259, 284, 1030, 1250, 1251, 6p1247, 8p2305, 9p267, 35p1251, 55p554, 484p2316

Collinear triples
6 9 37
6 284 1030
6 1250 35p1251
9 284 8p2305
37 1030 9p267
37 1250 1251
37 6p1247 8p2305
37 35p1251 55p554
1030 1251 55p554
8p2305 9p267 484p2316

Z(X(37),X(57)) passes through these centers:
1, 9, 21, 37, 846, 893, 1247, 2p2329, 8p2248, 8p2305, 9p509, 9p1999, 43p979, 978p2319

Collinear triples
1 9 37
1 21 846
1 9p1999 43p979
9 21 8p2305
9 846 893
9 2p2329 978p2319
21 2p2329 9p1999
37 846 8p2248
37 893 2p2329
37 1247 8p2305
37 43p979 978p2319
846 1247 9p1999
893 8p2305 43p979
2p2329 8p2248 8p2305

Z(X(40),X(2)) passes through these centers:
40, 55, 57, 365, 1436, 2066, 2362, 6p1490, 9p2067, 56p1034, 225p1806

Collinear triples
40 55 57
40 2066 2362
40 6p1490 56p1034
40 9p2067 225p1806
55 1436 6p1490
55 2066 9p2067
2362 6p1490 225p1806

Z(X(40),X(75)) [ K179] passes through these centers:
6, 34, 40, 55, 56, 212, 2208, 6p1490, 212p1767, 604p1034

Collinear triples
6 34 6p1490
6 55 212
6 56 212p1767
34 40 212
40 55 56
40 6p1490 604p1034
55 2208 6p1490
55 2208 212p1767

Z(X(40),X(86)) passes through these centers:
19, 40, 55, 64, 65, 71, 2357, 6p1490, 10p154, 65p1498, 607p1032, 1034p1400

Collinear triples
19 40 71
19 55 10p154
19 65 6p1490
40 55 65
40 64 10p154
40 6p1490 1034p1400
40 65p1498 607p1032
55 64 65p1498
55 2357 6p1490
64 71 6p1490
65 2357 65p1498
71 2357 10p154
71 65p1498 1034p1400
6p149010p154607p1032

Z(X(42),X(6)) passes through these centers:
1, 2, 42, 274, 366, 1045

Collinear triples
1 2 42
1 274 1045

Z(X(56),X(2)) passes through these centers:
9, 55, 56, 57, 365, 1423, 2053, 2137, 6p2136

Collinear triples
9 55 6p2136
9 57 1423
55 56 57
56 1423 2053
56 2137 6p2136

Z(X(56),X(9)) passes through these centers:
1, 2, 56, 57, 87, 509, 978, 1423, 10p1431, 57p979, 58p894, 58p1999

Collinear triples
1 2 978
1 56 57
1 1423 10p1431
2 57 1423
2 58p894 58p1999
56 87 1423
56 978 57p979
56 10p1431 58p894
87 978 58p894
1423 57p979 58p1999

Z(X(57),X(1)) passes through these centers:
1, 9, 57, 173, 258, 1743, 2136, 2137, 514p1293

Collinear triples
1 9 1743
1 2136 514p1293
57 173 258
57 1743 514p1293
57 2136 2137

Z(X(57),X(6)) passes through these centers:
7, 8, 9, 57, 366, 2136, 2319, 2p1423, 2p2137

Collinear triples
7 8 2p1423
7 9 57
8 9 2136
57 2136 2p2137
57 2319 2p2137

Z(X(57),X(8)) passes through these centers:
1, 6, 56, 57, 266, 289, 1743, 6p2089, 1743p2137

Collinear triples
1 6 1743
1 56 57
6 266 6p2089
56 266 289
56 1743 1743p2137
57 289 6p2089

Z(X(57),X(21)) passes through these centers:
1, 2, 37, 57, 65, 1400, 1743, 2p978, 979p1400

Collinear triples
1 37 1743
1 57 65
2 37 2p978
2 57 1400
37 65 1400
57 2p978 979p1400

Z(X(57),X(80)) passes through these centers:
1, 36, 57, 284, 1743, 1795, 1845, 2316, 2323, 2p1464, 44p1443, 102p223, 173p1130, 258p1130

Collinear triples
1 36 57
1 1743 2323
1 1795 44p1443
1 1845 102p223
36 1845 2p1464
57 284 2p1464
57 1795 1845
57 2316 44p1443
1743 1845 2316
2323 2p1464 44p1443
2p1464 173p1130 258p1130

Z(X(57),X(86)) passes through these centers:
9, 55, 57, 65, 1334, 1400, 2136, 42p2137

Collinear triples
9 57 1400
9 1334 2136
55 57 65
57 2136 42p2137
65 1334 1400

Z(X(63),X(1)) passes through these centers:
1, 9, 19, 40, 57, 63, 84, 610, 1712, 2184, 2p1035, 6p1032, 6p1034

Collinear triples
1 19 610
1 40 57
1 84 2p1035
1 1712 2184
9 19 40
9 57 63
9 84 610
9 1712 6p1034
9 2184 2p1035
19 57 2p1035
40 63 84
40 610 6p1034
40 2p1035 6p1032
63 610 1712
63 1712 2184
632p10356p1032

Z(X(63),X(6)) passes through these centers:
4, 7, 9, 40, 63, 189, 366

Collinear triples
4 9 40
7 9 63
40 63 189

Z(X(63),X(56)) passes through these centers:
2, 9, 40, 63, 188, 280, 281, 2p1745, 164p236

Collinear triples
2 9 63
2 281 2p1745
9 40 281
9 188 164p236
40 63 280

Z(X(63),X(75)) passes through these centers:
1, 6, 31, 63, 220, 610, 1407, 1973, 2155, 6p159, 6p1763, 6p2138, 6p2139, 1801p1824

Collinear triples
1 6 220
1 31 63
1 610 1973
6 31 1801p1824
6 610 1407
6 1973 6p1763
31 1407 6p1763
31 1973 6p159
31 2155 6p2138
63 220 1407
63 610 2155
63 6p2138 6p2139
220 1973 1801p1824
220 2155 6p1763
610 6p159 6p2139

Z(X(65),X(8)) passes through these centers:
1, 56, 58, 65, 266, 267, 1046, 2306, 2307, 6p554, 6p559, 56p191, 56p1247, 106p484

Collinear triples
1 56 65
1 58 1046
56 58 56p191
56 2307 6p559
65 267 56p191
65 1046 56p1247
65 2306 2307
65 6p554 6p559
267 1046 106p484
2306 6p554 56p191

Z(X(65),X(9)) passes through these centers:
1, 57, 65, 81, 509, 894, 1046, 1431, 1999, 2p2248, 56p1654, 57p1247, 87p978, 979p1423

Collinear triples
1 57 65
1 81 1046
1 894 87p978
1 1431 56p1654
57 81 56p1954
57 1999 979p1423
65 894 1431
65 1046 57p1247
65 2p2248 56p1654
65 87p978 979p1423
81 894 1999
894 1046 2p2248
1046 1431 979p1423
1999 56p1654 57p1247

Z(X(65),X(31)) passes through these centers:
2, 7, 8, 65, 257, 314, 894, 1999, 7p1045

Collinear triples
2 7 894
2 8 1999
2 257 7p1045
7 8 65
7 314 7p1045
65 257 894
314 894 1999

Z(X(65),X(75)) passes through these centers:
6, 55, 56, 65, 2160, 2174, 2194, 6p1781, 41p1029, 56p191, 1717p2164, 1723p2164

Collinear triples
6 56 2174
6 2160 56p191
6 2194 6p1781
55 56 65
55 2160 6p1781
55 2174 1723p2164
56 2160 1717p2164
56 2194 56p191
65 2160 2174
65 41p1029 56p191
2194 6p1781 41p1029
2194 1717p2164 1723p2164

Z(X(69),X(1)) [ K169] passes through these centers:
1, 2, 6, 20, 25, 64, 69, 159, 200, 269, 1763, 2138, 2139, 1801p1826

Collinear triples
1 2 200
1 6 1801p1826
1 20 269
1 25 1763
2 6 69
2 20 25
6 25 159
6 64 2138
6 269 1763
20 64 69
20 159 2139
25 200 1801p1826
64 200 1726
69 200 269
69 2138 2139

Z(X(69),X(31)) [Lucas cubic, K007] passes through these centers:
2, 4, 7, 8, 20, 69, 189, 253, 329, 1032, 1034, 2p1712, 7p1490

Collinear triples
2 4 20
2 7 329
2 189 7p1490
2 253 2p1712
4 7 7p1490
4 8 329
7 8 69
7 189 2p1712
8 20 189
8 253 7p1490
8 1034 2p1712
20 69 253
20 329 1034
69 189 329
69 1032 2p1712
69 1034 7p1490
329 1032 7p1490

Z(X(69),X(75)) [ K178] passes through these centers:
6, 69, 159, 1974

Collinear triple
6 159 1974

Z(X(69),X(82)) [ K140] passes through these centers:
6, 66, 69, 141, 159, 1843, 22p38

Collinear triples
6 69 141
6 159 1843
66 69 22p38
66 141 159
141 1843 22p38

Z(X(75),X(6)) passes through these centers:
6, 75, 366, 1631, 366p510

Collinear triples
1 366 366p510

Z(X(75),X(31)) [Spieker perspector cubic, K034] passes through these centers:
1, 2, 7, 8, 63, 75, 92, 280, 347, 1895, 2p1073, 189p1490, 223p1034, 253p1498, 1032p1249

Collinear triples
1 2 8
1 7 347
1 92 1895
1 280 189p1490
1 2p1073 253p1498
2 7 63
2 92 347
2 280 1895
2 2p1073 189p1490
2 223p1034 253p1498
7 8 75
7 92 189p1490
7 280 253p1498
8 63 280
8 347 2p1073
8 1895 223p1034
8 189p1490 1032p1249
63 75 92
63 347 223p1034
63 1895 1032p1249
75 280 347
75 1895 2p1073
75 189p1490 223p1034
75 253p1498 1032p1249

Z(X(75),X(42)) passes through these centers:
1, 27, 58, 63, 75, 86, 267, 75p501

Collinear triples
1 58 63
1 75 86
27 63 75
58 86 75p501
75 267 75p501

Z(X(75),X(75)) passes through these centers:
6, 75, 560, 1631, 6p1670, 6p1671, 6p1676, 6p1677, 75p1672, 75p1673

Collinear triples
6 560 1631
6 6p1670 75p1673
6 6p1671 75p1672
75 6p1670 6p1677
75 6p1671 6p1676
75 6p1672 75p1673
560 6p1670 6p1671
1631 6p1676 75p1673
1631 6p1670 75p1672

Z(X(76),X(1)) passes through these centers:
1, 32, 76, 1670, 1671, 1676, 1677, 1759, 76p1672, 76p1673

Collinear triples
1 32 1759
1 1670 76p1673
1 1671 76p1672
32 1670 1671
76 1670 1677
76 1671 1676
76 76p1672 76p1673
1676 1759 76p1673
1677 1759 76p1672

Z(X(76),X(31)) [ K141] passes through these centers:
2, 4, 6, 22, 69, 76, 1670, 1671, 19p1370, 66p75, 75p1676, 75p1677

Collinear triples
2 4 22
2 6 69
2 19p1370 66p75
4 6 19p1370
4 69 76
6 1670 1671
22 76 66p75
22 75p1676 75p1677
76 1670 75p1677
76 1671 75p1676

Z(X(77),X(55)) passes through these centers:
1, 4, 7, 77, 174, 189, 223, 7p1745, 164p2089, 505p1488

Collinear triples
1 4 223
1 7 77
1 174 164p2089
4 7 7p1745
77 189 223
77 164p2089 505p1488
174 223 505p1488

Z(X(78),X(6)) passes through these centers:
1, 2, 78, 278, 366, 1490, 57p1034

Collinear triples
1 2 78
1 278 1490
78 1490 57p1034

Z(X(78),X(56)) passes through these centers:
1, 4, 8, 78, 188, 1034, 1490, 8p1745

Collinear triples
1 4 1490
1 8 78
4 8 8p1745
78 1034 1490

Z(X(80),X(36)) passes through these centers:
1, 10, 13, 14, 80, 484, 502, 519, 759, 1128, 1168, 13p35, 14p35, 80p202, 80p203, 80p501, 80p1130

Collinear triples
1 10 519
1 759 80p501
1 13p35 80p203
1 14p35 80p202
10 13 80p203
10 14 80p202
10 80 759
10 484 502
13 14 80p501
13 80 14p35
14 80 13p85
80 5022 80p501
80 519 1168
80 1128 80p1130
484 759 1168
759 13p35 14p35
1168 80p202 80p203

Z(X(81),X(10)) passes through these centers:
1, 6, 57, 58, 81, 222, 284, 1172, 1433, 4p2360, 20p1436, 64p1817

Collinear triples
1 58 81
1 222 1433
1 284 1172
1 4p2360 64p1817
6 57 222
6 58 284
6 1172 4p2360
6 1433 20p1436
57 58 4p2360
57 81 284
58 1172 20p1436
81 222 1172
81 1433 4p2360
81 20p1436 64p1817
222 284 64p1817

Z(X(83),X(38)) passes through these centers:
2, 4, 6, 83, 251, 1176, 1342, 1343, 4p2172, 1799p2156

Collinear triples
2 83 251
2 1176 1799p2156
4 6 4p2172
4 83 1176
6 251 1176
6 1342 1343
83 4p2172 1799p2156

Z(X(84),X(6)) passes through these centers:
40, 189, 329, 366, 6p189, 40p87, 43p189

Collinear triples
40 189 6p189
189 329 43p189
6p189 40p87 43p189

Z(X(84),X(75)) [ K180] passes through these centers:
6, 33, 84, 198, 221, 603, 963, 1436, 2187, 2192, 6p2270, 198p1622, 603p1753

Collinear triples
6 33 2192
6 198 603p1753
6 603 1436
6 963 198p1622
6 2187 6p2270
33 84 603
33 198 6p2270
84 198 1436
84 221 2192
84 963 6p2270
198 221 198p1622
221 603 6p2270
603 2187 603p1753
963 2192 603p1753
1436 2187 2192

Z(X(86),X(42)) passes through these centers:
1, 2, 7, 21, 29, 77, 81, 86, 2p1433, 4p1817, 20p84, 253p2360, 1034p1394

Collinear triples
1 7 77
1 21 81
1 29 4p1817
1 2p1433 20p84
2 21 29
2 77 2p1433
2 81 86
2 4p1817 253p3260
2 20p84 1034p1394
7 21 86
7 81 4p1817
21 77 253p2360
21 4p1817 1034p1394
29 77 86
29 81 20p84
86 2p1433 4p1817
86 20p84 253p2360

Z(X(87),X(2)) passes through these centers:
43, 87, 365, 1423, 2053, 2162, 2176

Collinear triples
43 87 2162
87 1423 2053
2053 2162 2176

Z(X(92),X(1)) passes through these centers:
1, 19, 47, 48, 63, 91, 92, 2p155, 6p254

Collinear triples
1 19 48
1 47 63
1 91 2p155
19 47 6p254
19 63 92
47 91 92
48 63 2p155
92 2p155 6p254

Z(X(96),X(75)) passes through these centers:
5, 6, 24, 96, 571, 2165, 2351, 5p2158, 6p2148, 6p2180, 26p2148

Collinear triples
5 6 2165
5 96 6p2148
6 24 26p2148
6 2351 6p2148
24 96 2351
96 571 2165
96 5p2158 26p2148
2165 2351 6p2180
6p2148 6p2180 26p2148

Z(X(98),X(1)) passes through these centers:
1, 98, 511, 1687, 1688, 1756, 2009, 2010, 98p2007, 98p2008

Collinear triples
1 511 1656
1 1687 98p2007
98 1687 2009
98 1688 2010
511 1687 1688
1756 2010 98p2007

Z(X(99),X(1)) [Steiner cubic, K035] passes through these centers:
1, 39, 83, 99, 512, 1018, 1019, 1379, 1380, 1379p1577, 1380p1577

Collinear triples
1 39 1018
1 512 1019
39 83 99
39 1379 1380
99 1018 1019
99 1379 1380p1577
99 1380 1379p1577
5121379p15771380p1577

Z(X(100),X(6)) passes through these centers:
100, 142, 366, 514, 2346, 2p1381, 2p1382, 522p1381, 522p1382

Collinear triples
100 142 2346
100 2p1381 522p1382
100 2p1382 522p1381
142 2p1381 2p1382
514 522p1381 522p1381

Z(X(101),X(2)) passes through these centers:
101, 354, 365, 513, 1174, 1381, 1382, 650p1381, 650p1382

Collinear triples
101 354 1174
101 1381 650p1382
101 1382 650p1381
354 1381 1382
513 650p1382 650p1382

Z(X(108),X(78)) passes through these centers:
4, 56, 108, 513, 1381, 1382, 65p1113, 65p1114, 513p1113, 513p1114

Collinear triples
4 56 108
4 65p1113 65p1114
56 1381 1382
108 65p1113 65p1114
108 65p1114 65p1113

Z(X(109),X(8)) passes through these centers:
58, 65, 109, 266, 513, 1381, 1382, 513p1381, 513p1382

Collinear triples
58 65 109
65 1381 1382
109 1381 513p1382
109 1382 513p1381
513 513p1381 513p1382

Z(X(110),X(1)) passes through these centers:
1, 5, 54, 110, 523, 1113, 1114, 9p1019, 9p1020, 523p1822, p23p1823

Collinear triples
1 5 9p1020
1 1523 19p1019
5 54 110
5 1113 1114
110 1113 523p1823
110 1114 523p1822
110 9p1019 9p1020
523 523p1822 9p1020

Z(X(165),X(75)) passes through these centers:
6,55,56, 165, 269, 1253, 1615, 56p2125

Collinear triples
6 55 1253
6 269 1615
55 56 165
165 269 1253
165 1615 56p2125

Z(X(171),X(1)) passes through these centers:
1, 10, 43, 58, 87, 171, 256, 846, 2p2248, 978p1247, 979p1046

Collinear triples
1 10 43
1 58 846
10 58 171
10 846 978p1247
43 58 979p1046
43 87 171
43 256 846
171 846 2p2248
171 978p1247 979p1046

Z(X(171),X(2)) [ K131] passes through these centers:
2, 31, 42, 43, 55, 57, 81, 171, 365, 846, 893, 2162, 2248, 6p1045, 1282p2111

Collinear triples
2 31 171
2 42 43
2 81 6p1045
31 42 55
31 81 846
42 81 171
43 171 2162
43 846 893
55 57 171
55 893 6p1045
171 846 2248

Z(X(171),X(75)) passes through these centers:
6, 55, 56, 86, 171, 904, 1918, 6p1045

Collinear triples
6 55 1918
6 86 6p1045
55 56 171
55 904 6p1045
86 171 1918

Z(X(172),X(75)) passes through these centers:
6, 37, 172, 893, 1333, 2162, 2176, 2248, 6p846, 979p2305

Collinear triples
6 37 2176
6 1333 6p846
37 172 1333
172 2162 2176
172 2248 6p846
893 2176 6p846
1333 2176 979p2305

Z(X(172),X(76)) passes through these centers:
1, 32, 41, 56, 58, 172, 213, 904, 2176, 6p365, 6p846, 6p2162, 6p2248, 31p1045, 727p1757, 1189p1972

Collinear triples
1 32 172
1 58 31p1045
1 213 2176
32 41 213
32 58 6p846
32 2176 727p1757
41 56 172
41 904 31p1045
58 172 213
172 2176 6p2162
172 6p846 6p2248
904 2176 6p846

Z(X(174),X(1)) passes through these centers:
1, 173, 174, 188, 258, 259, 266, 145p259

Collinear triples
1 173 259
1 258 266
173 174 258
174 188 266
188 259 145p259

Z(X(189),X(1)) passes through these centers:
1, 40, 84, 189, 198, 222, 223, 281, 282, 2270, 2p963, 40p1622, 222p1753

Collinear triples
1 40 222p1753
1 84 222
1 198 2270
1 281 282
1 2p963 40p1622
40 84 189
40 223 40p1622
40 281 2270
84 198 282
189 222 281
189 223 282
189 2270 2p963
198 222 222p1753
222 223 2270
282 2p963 222p1753

Z(X(190),X(106)) passes through these centers:
88, 100, 190, 644, 900, 1120, 1635, 2p678, 7p1635, 519p1149

Collinear triples
88 100 519p1149
88 190 2p678
100 190 900
100 644 1635
190 644 7p1635
190 1120 519p1149
644 1120 2p678
900 1635 2p678
900 7p1635 519p1149

Z(X(220),X(2)) passes through these centers:
1, 6, 170, 220, 269, 365, 1615, 57p2125

Collinear triples
1 6 220
1 170 269
6 269 1615
220 1615 57p2125

Z(X(220),X(7)) passes through these centers:
6, 9, 57, 220, 259, 1615, 9p1742, 259p844

Collinear triples
6 9 220
6 57 1615
9 57 9p1742
57 259 259p844
220 1615 2125

Z(X(222),X(8)) passes through these centers:
6, 19, 57, 84, 221, 222, 266, 57p1745, 289p505

Collinear triples
6 19 221
6 57 222
19 57 57p1745
84 221 222
221 266 289p505

Z(X(226),X(1)) passes through these centers:
1, 9, 35, 57, 79, 226, 284, 1781, 7p1030, 9p267, 90p1717, 90p1723

Collinear triples
1 35 57
1 79 7p1030
1 284 1781
9 35 90p1723
9 57 226
9 79 1781
35 79 226
35 1781 9p267
57 79 90p1717
57 284 7p1030
226 7p1030 9p267
284 90p1717 90p1723

Z(X(226),X(29)) passes through these centers:
1, 48, 63, 73, 226, 1400, 1781

Collinear triples
1 48 1781
1 73 226
48 73 1400
63 226 1400

Z(X(226),X(41)) passes through these centers:
2, 7, 86, 226, 508, 1432, 1909, 2p1046, 2p1999, 7p846, 7p1247, 75p2248, 330p978

Collinear triples
2 7 226
2 86 2p1046
2 1432 7p846
2 1909 330p978
7 86 7p846
86 1909 2p1999
226 1432 1909
226 2p1046 7p1247
226 7p846 75p2248
1909 2p1046 75p2248
2p1999 7p846 7p1247

Z(X(226),X(55)) [ K134] passes through these centers:
2, 57, 81, 174, 226, 554, 559, 1029, 1081, 1082, 2p1046, 7p1030, 57p1247, 88p484

Collinear triples
2 57 226
2 81 2p1046
57 81 2p1046
57 559 1082
226 554 559
226 1029 7p1030
226 1081 1082
226 2p1046 57p1247
554 1081 7p1030
1029 2p1046 88p484

Z(X(237),X(75)) passes through these centers:
3, 6, 25, 98, 237, 694, 1691, 1971

Collinear triples
3 6 1691
3 25 237
6 25 1971
98 1691 1971
237 649 1691

Z(X(238),X(1)) passes through these centers:
1, 238, 291, 2108, 2p2109

Collinear triples
1 3 4
2 4 5

Z(X(238),X(2)) (2nd equal areas cubic, [ K155]) passes through these centers:
1, 2, 6, 31, 105, 238, 292, 365, 672, 1423, 1931, 2053, 2054, 2106, 2107, 2108, 2109, 2110, 2111, 2112, 2113, 2114, 2115, 2116, 2117, 2118, 2119, 2144, 2145, 2146, 2147, 2p727, 6p1575, 1045p2248, 1281p1967

Collinear triples
1 2 6p1575
1 6 238
1 31 1931
1 105 2112
1 292 2108
1 1423 1281p1967
1 2111 2144
1 2119 2146
2 6 2106
2 31 238
2 105 2110
2 672 1423
2 1931 1045p2248
2 2054 2108
2 2113 2144
6 31 672
6 105 2116
6 292 2110
6 365 2118
6 2054 2112
6 2144 2144
6 1045p2248 1281p1967
31 105 2114
31 292 2112
31 2053 6p1575
31 2107 2110
31 2108 2p727
105 238 672
105 2106 1281p1967
105 2144 2p727
238 1423 2053
238 1931 2054
238 2106 2107
238 2108 2109
238 2110 2111
238 2112 2113
238 2114 2115
238 2116 2117
238 2118 2119
238 2144 2145
238 2146 2147
238 2p727 6p1575
292 365 2146
292 672 6p1575
292 1423 2114
292 1931 2106
365 2110 2119
365 2144 2147
672 2108 2113
672 2110 2117
672 2112 2115
1423 2107 2116
1423 2112 2p727
1931 2110 2113
1931 2115 2116
2053 2115 2144
2054 2107 2144
2054 6p1575 1281p1967
2106 2111 2112
2106 2114 21171
2107 2112 1045p2248
2108 2110 2145
2108 2118 2147
2109 2112 6p1575
2110 2115 1281p1967
2111 2116 6p1575
2111 2118 2146
2113 2114 6p1575

Z(X(238),X(7)) passes through these centers:
6, 9, 238, 259, 2110, 8p1911, 9p2111

Collinear triples
1 2 3
1 5 6
3 5 7

Z(X(238),X(9)) passes through these centers:
87, 238, 509, 1423, 2114, 7p292, 7p2115

Collinear triples
87 238 1423
238 2114 7p2115
1423 2114 7p292

Z(X(238),X(27)) passes through these centers:
6, 72, 238, 2110, 10p2196, 72p2111

Collinear triples
6 72 238
6 2110 10p2196
238 2108 306p2109

Z(X(238),X(57)) passes through these centers:
1, 9, 238, 2108, 8p292, 8p2109, 9p509

Collinear triples
1 9 238
1 2108 8p292
238 2108 8p2109

Z(X(238),X(239)) passes through these centers:
238, 672, 6p1575, 105p291, 291p292, 335p727

Collinear triples
238 672 238
238 6p1575 335p727
672 6p1575 291p292

Z(X(239),X(6)) passes through these centers:
1, 2, 6, 75, 239, 291, 366, 518, 673, 1575, 2319, 2p1423, 2p1931, 2p2054, 2p2106, 2p2107, 2p2108, 2p2109, 2p2110, 2p2111, 2p2112, 2p2112, 2p2113, 2p2114, 2p2115, 2p2116, 2p2117, 2p2118, 2p2119, 2p2144, 2p2145, 2p2146, 2p2147, 75p727, 694p1281, 1655p2248

Collinear triples
1 2 239
1 6 518
1 75 2p2106
1 291 2p2110
1 366 2p2118
1 673 2p2116
1 2p2054 2p2112
1 2p2109 2p2144
1 694p1281 1655p2248
2 6 2p1931
2 75 1575
2 291 2p2108
2 673 2p2112
2 2p1423 694p1281
2 2p2111 2p2144
2 2p2119 2p2146
6 75 239
6 291 2p2112
6 673 2p2114
6 1575 2319
6 2p2107 2p2110
6 2p2108 75p727
75 518 2p1423
75 673 2p2110
75 2p1931 1655p2248
75 2p2054 2p2108
75 2p2113 2p2144
239 518 673
239 1575 75p727
239 2319 2p1423
239 2p1931 2p2054
239 2p2106 2p2107
239 2p2108 2p2109
239 2p2110 2p2111
239 2p2112 2p2113
239 2p2114 2p2115
238 2p2116 2p2117
239 2p2118 2p2119
239 2p2144 2p2145
239 2p2146 2p2147
291 366 2p2146
291 518 1575
291 2p1423 2p2114
291 2p1931 2p2106
366 2p2110 2p2119
366 2p2144 2p2147
518 2p2108 2p2113
518 2p2110 2p2117
518 2p2112 2p2115
673 2p2106 694p1281
673 2p2144 75p727
1575 2p2054 694p1281
1575 2p2109 2p2112
1575 2p2111 2p2116
1575 2p2113 2p2114
2319 2p2115 2p2144
2p1423 2p2107 2p2116
2p14232p211275p727
2p19312p2110 2p2113
2p19312p21152p2116
2p20542p21072p2144
2p2106 2p2111 2p2112
2p2106 2p2114 2p2117
2p21072p21121655p2248
2p21082p21102p2145
2p21082p21182p2147
2p21102p2115694p1281
2p21112p21182p2146

Z(X(240),X(63)) passes through these centers:
1, 19, 63, 240, 1096, 1910, 1967, 2312, 2p419, 2p1692, 4p401, 4p1297, 25p147, 64p193, 297p2065

Collinear triples
1 19 240
1 63 2p1692
110964p401
1963 2p419
1910962312
19 191025p147
63 240 1096
63 2312 64p193
63 4p1297 25p147
240 1967 2p419
240 2312 4p1297
240 2p1692 297p2065
10961967 25p147
1910 2312 2p1692
1910 2p419 4p401

Z(X(241),X(2)) passes through these centers:
55, 57, 241, 365, 2114, 2115, 2195

Collinear triples
55 57 241
57 2114 2195
24121142115

Z(X(241),X(9)) passes through these centers:
1, 9, 57, 105, 241, 269, 292, 509, 910, 1279, 1447, 2114, 2p1477, 7p103, 7p2115, 87p1742, 355p1355, 1425p1427

Collinear triples
1 9 1279
1 57 241
9571447
9241 269
921147p103
57 1052114
57 269 910
105 910 1279
241 292 1447
241 910 7p103
241 1279 2p1477
241 2114 7p2115
269292 2114
269 1447 87p1742

Z(X(241),X(75)) passes through these centers:
6, 55, 56, 220, 241, 910, 911, 1279, 1407, 3p2191, 6p2195, 19p218, 25p1810

Collinear triples
6 220 1279
6 910 1407
5556241
55910 19p218
5612793p2191
220 2411407
241 910 911
241 1279 25p1810
241 3p2191 19p218
910 1279 6p2195

Z(X(243),X(3)) passes through these centers:
1, 3, 4, 158, 243, 415, 1937, 4p416, 90p1148, 1047p1247

Collinear triples
1 4 243
1 158 4p416
34415
3158 243
1584151047p1247
415 19374p416

Z(X(253),X(1)) passes through these centers:
1, 20, 64, 154, 253, 1073, 1249, 7p221, 8p2192

Collinear triples
1 64 7p221
1 1073 8p2192
2064253
64154 1073
25310731249
253 7p2218p2192

Z(X(264),X(31)) [Euler perspector cubic, K045] passes through these centers:
2, 3, 4, 69, 254, 264, 1993, 2p91, 69p920

Collinear triples
2 3 4
2 69 1993
22p9169p920
369 69p920
469264
4 2541993
25426469p920
254 19932p91

Z(X(265),X(1)) passes through these centers:
5, 30, 54, 74, 186, 265, 35p80, 36p79, 79p1458

Collinear triples
1 5 35p80
1 30 36p79
530186
554 265
3074265
265 35p8036p79

Z(X(278),X(78)) passes through these centers:
1, 6, 19, 34, 57, 278, 1723, 2p1406, 4p2164, 6p1708, 27p2174, 65p79, 267p1717, 267p1781, 1570p1987, 1770p2160

Collinear triples
1 6 1723
1 34 278
1576p1708
12p1406 65p79
1267p17811770p2160
6 1934
6 57 2p1406
6 65p79 1770p2160
6267p1717267p1781
1957 278
194p21646p1708
19 27p2174267p1781
34 57 27p2174
34 4p2164 1770p2160
5765p79267p1717
2782p1406 4p2164
27827p217465p79
1723 4p216427p2174

Z(X(385),X(1)) [ K128] passes through these centers:
1, 2, 6, 32, 76, 98, 385, 511, 694, 1423, 2319, 4p1740, 6p2227, 19p147, 75p699, 292p1281

Collinear triples
1 32 292p1281
1 511 1423
123196p2227
26 385
2766p2227
2 1423292p1281
6 32 511
32 76 385
329819p147
76511 4p1740
98385511
385 14232319
385 6p2227 75p699
511 694 6p2227
6944p174019p147
23194p1740 292p1281

Z(X(286),X(31)) passes through these centers:
2, 4, 21, 63, 69, 72, 92, 286, 1441, 2p579, 2p1751, 2p1780, 63p1713, 69p1714, 306p1612, 580p1441

Collinear triples
2 4 21
2 63 580p1441
2692p1780
272 2p579
2921441
2 2p1751306p1612
4 69 286
4 72 92
42p1751580p1441
2163 2p579
21722p1780
21 2861441
21 2p1751 69p1714
6369306p1612
6372 63p1713
6392286
69 7269p1714
6914412p579
721441 580p1441
922p17512p1780
286 2p5792p1751
1441 2p175163p1713

Z(X(291),X(2)) passes through these centers:
42, 81, 105, 291, 365, 672, 1914

Collinear triples
42 81 291
42 672 1914
105291672

Z(X(291),X(239)) [ K135] passes through these centers:
1, 6, 42, 57, 239, 291, 292, 672, 894, 1757, 1967, 2p741, 8p1911, 37p2248, 105p291, 238p2113, 241p2115, 291p1911, 292p1929, 741p1654, 1281p1911, 1463p2053, 1575p2109

Collinear triples
1 6 1757
1 42 239
1571281p1911
1291 292
18941463p2053
1 1967741p1654
6 42672
6 239894
6 292 8p1911
6 2p741 741p1654
422912p741
421967 8p1911
4237p22481281p1911
57 2918p1911
57 672894
239 291291p1911
239 1757 1575p2109
239 8p1911 105p291
239238p21131281p1911
239292p1929 741p1654
291672105p291
291 8941967
291 1757292p1929
291 37p2248741p1654
292 672291p1911
292 894 2p741
292 292p1929 1281p1911
6721757238p2113
672241p2115 1281p1911
894175737p2248
1757 2p741291p1911
1967 291p19111281p1911
2p741 105p2911281p1911
8p1911 291p19111463p2053
1281p1911 1463p20531575p2053

Z(X(292),X(238)) [ K136] passes through these centers:
1, 2, 37, 87, 171, 238, 241, 291, 292, 1575, 1581, 2p741, 37p2106, 43p291, 291p291, 291p294, 291p2068, 291p2069, 335p727, 518p2111, 741p1655

Collinear triples
1 2 37p2106
1 37 238
1171241
1291 292
11581741p1655
2 371575
2 171238
2 291 43p291
2 2p741 741p1655
372922p741
371581 43p291
871711575
87 29243p291
171 2912p741
171 2921581
238 291291p294
238 292 291p291
238 37p2106518p2111
238 43p291 335p727
241 292 291p294
24143p291291p291
2911575 291p291
2921575335p727
292 291p2068291p2069
1581 37p2106291p294
2p741 37p2106291p291

Z(X(297),X(48)) passes through these centers:
2, 4, 69, 98, 230, 297, 393, 694, 1503, 4p1966, 19p147, 92p401, 92p1297, 193p2184

Collinear triples
2 4 297
2 69 230
239392p401
469 4p1966
49819p147
4 3931503
69 297 393
69 1503 193p2184
6919p14792p1297
98230 1503
984p14792p401
297 6944p1966
297 1503 92p1297
393 694 19p147

Z(X(297),X(75)) passes through these centers:
3, 6, 25, 230, 297, 394, 1503, 2207, 31p248, 31p1297, 47p66, 91p206, 1755p2065

Collinear triples
3 25 297
3 1503 47p66
6230394
61503 2207
2523091p206
230 2971755p2065
230 1503 31p248
297 394 2207
297150331p1297
29747p66 91p206

Z(X(304),X(31)) passes through these centers:
1, 2, 19, 75, 279, 304, 346, 2184, 2p20, 2p159, 2p1763, 2p2138, 2p2139

Collinear triples
1 19 2p159
1 75 304
12792p1763
12184 2p2138
2192p1763
2 75346
2 279 2p20
19 75 2p20
279304346
3042184 2p20
3042p2138 2p2139
34621842p1763
2p202p159 2p2139

Z(X(309),X(31)) [ K133] passes through these centers:
2, 40, 77, 189, 280, 309, 318, 329, 347, 962, 75p963, 77p1753, 329p1622

Collinear triples
2 40 962
2 77 189
2280318
2329 77p1753
275p963329p1622
40 7777p1753
40 189 280
77 309 318
77347962
189309 329
280309 347
28075p96377p1753
309962 75p963
318329962
329347 329p1622

Z(X(314),X(31)) passes through these centers:
1, 2, 4, 65, 69, 75, 81, 314, 321, 1764, 2p573, 75p2217, 81p1766, 226p1610, 321p572

Collinear triples
1 2 321p572
1 4 226p1610
1651764
175 314
1812p573
2 481p1766
2 65 2p573
2 69 81
275321
275p2217 226p1610
469 314
43212p573
6569 75
658181p1766
65321 321p572
6975p2217 321p572
7575p221781p1766
81314 321
3142p57375p2217
3211767 75p2217

Z(X(316),X(31)) [Droussent cubic, K008] passes through these centers:
2, 4, 67, 69, 316, 524, 671, 858, 75p1177

Collinear triples
2 4 858
2 69 524
469316
67524 858
316524671
316 85875p1177

Z(X(319),X(31)) passes through these centers:
2, 7, 8, 10, 79, 86, 319, 1029, 2p191

Collinear triples
2 8 10
2 86 2p191
78319
879 2p191
1086319
319 10292p191

Z(X(320),X(31)) passes through these centers:
2, 7, 8, 80, 320, 519, 903, 908, 2p104

Collinear triples
2 7 908
2 8 519
78320
80519 908
320519903
320 9082p104

Z(X(322),X(31)) [ K154] passes through these centers:
2, 7, 8, 78, 84, 273, 322, 2p1490, 57p1034, 57p2057

Collinear triples
2 7 57p2057
2 8 78
22732p1490
78 322
8842p1490
78 8457p2057
78 273322
322 2p149057p1034

Z(X(333),X(1)) passes through these centers:
1, 2, 6, 10, 19, 58, 63, 333, 573, 1400, 2p2217, 6p1764, 10p572, 21p478, 65p1610

Collinear triples
1 2 10
1 6 10p572
11921p478
158 63
15731400
1 2p221765p1610
2 6333
2 631400
2 2p2217 21p478
61965p1610
658 573
614006p1764
10 19573
10 58333
10 140010p572
102p2217 6p1764
1963333
58 140021p478
63 2p221710p572
333 5732p2217

Z(X(356),X(1)) [a Morley cubic, K029] passes through these centers:
1, 356, 357, 358, 1134, 1137, 1507, 1508

Collinear triples
1 357 1508
1 358 1507
356357358
3561134 1137

Z(X(365),X(1)) passes through these centers:
1, 365, 366, 43p366, 87p365

Collinear triple
365 43p366 87p365

Z(X(366),X(1)) passes through these centers:
1, 364, 365, 366, 7p365, 9p365, 365p1489, 366p1419, 366p2067

Collinear triples
1 364 365
365 7p365 366p1419
3667p3659p365

Z(X(384),X(1)) [a Brocard cubic, K020] passes through these centers:
1, 3, 4, 32, 39, 76, 83, 194, 384, 695, 31p1031

Collinear triples
3 4 384
3 32 39
3276384
3976 194
3983384

Z(X(393),X(63)) [ K163] passes through these centers:
3, 4, 6, 254, 393, 459, 1609

Collinear triples
3 4 459
3 6 1609
46393
254393 1609

Z(X(395),X(1)) [ K129a] passes through these centers:
1, 2, 6, 14, 16, 18, 62, 395, 1653, 19p617, 19p628, 621p2159, 622p2148

Collinear triples
1 16 1653
2 6 395
219p617621p2159
219p628 622p2148
61662
14 16395
14 6219p617
16 1819628
18 62 395

Z(X(396),X(1)) [ K129b] passes through these centers:
1, 2, 6, 13, 15, 17, 61, 396, 1652, 19p627, 621p2148, 622p2159

Collinear triples
1 15 1652
2 6 396
219p627621p2158
615 61
1315386
15 1719p627
17 61396

Z(X(476),X(1)) [ K130] passes through these centers:
1, 30, 74, 110, 476, 523, 626, 110p1749, 1109p1291

Collinear triples
30 74 476
30 523 526
110476523
110526 110p1749
476110p17491109p1291

Z(X(511),X(31)) passes through these centers:
2, 4, 69, 290, 385, 401, 511, 1916, 1972

Collinear triples
2 4 401
2 69 385
469511
290385 401
3855111916
401 5111972

Z(X(511),X(92)) passes through these centers:
3, 6, 25, 248, 385, 394, 401, 511, 1297, 3p1967, 3p2312, 31p147, 48p230, 48p1987, 1073p1707, 1959p2065

Collinear triples
3 6 511
3 25 401
339448p230
625 3p2312
624831p147
6 385394
25 394 511
25 3p1967 31p147
248385401
2483p2312 48p230
3855113p1967
394 129731p147
394 3p2312 1073p1707
401 511 48p1987
51112973p2312
51148p230 1959p2065

Z(X(512),X(1)) [1st Equal Areas Cubic, K021] passes through these centers:
1, 99, 512, 2142, 2143

Collinear triple
512 2142 2143

Z(X(515),X(1)) passes through these centers:
1, 36, 40, 80, 84, 102, 515, 90p1718, 90p1720

Collinear triples
1 36 40
36 80 515
368490p1720
4080 90p1718
4084515
102 90p171890p1720

Z(X(515),X(86)) passes through these centers:
33, 40, 48, 73, 515, 1826, 2183, 2250, 2357, 42p102, 65p1295, 1528p2188

Collinear triples
33 73 515
33 1826 1528p2188
405152357
401826 2183
48732183
48 5151826
48 2357 1528p2188
515 2183 2250
51565p12951528p2188
218342p102 1528p2188

Z(X(517),X(7)) passes through these centers:
1, 19, 44, 55, 102, 219, 259, 517, 2316, 2342, 9p2182, 145p2192

Collinear triples
1 44 219
1 55 517
19559p2182
19219 517
445172316
44 23429p2182
102 517 9p2182
217 9p2182 145p2192

Z(X(518),X(57)) passes through these centers:
1, 9, 57, 200, 239, 294, 518, 1280, 1282, 2348, 2p103, 2p2115, 8p1911, 9p509, 9p910, 1742p2319

Collinear triples
1 9 518
1 57 9p910
1 200 239
1 294 1289
9 200 2348
57 200 5181
57 239 9p509
57 1282 8p1911
200 1282 2p103
239 518 8p1911
294 2348 9p910
518 1280 2348
518 1282 2p2115
518 2p103 9p910

Z(X(894),X(1)) passes through these centers:
1, 9, 57, 213, 274, 893, 894, 1045

Collinear triples
1 9 213
1 274 1045
9 57 894
9 893 1045
213 274 894

Z(X(894),X(6)) [ K132] passes through these centers:
6, 7, 9, 37, 75, 86, 87, 192, 256, 366, 894, 1045, 1654, 2p2248, 1575p1929

Collinear triples
6 9 37
6 75 894
6 86 1654
7 9 894
9 256 1045
37 75 192
37 86 894
75 86 1045
75 1654 1575p1929
87 192 894
192 256 1654
894 1654 2p2248

Z(X(894),X(31)) passes through these centers:
2, 81, 192, 257, 321, 330, 894, 1654, 75p2248

Collinear triples
2 81 1654
2 192 321
81 321 894
192 257 1654
192 330 894
894 1654 75p2248

Z(X(1465),X(8)) passes through these centers:
3, 6, 34, 57, 106, 266, 909, 1319, 1465, 2182, 57p102, 84p1743, 164p289

Collinear triples
3 34 1465
3 57 1319
3 2182 84p1743
6 34 2182
6 57 1465
106 1319 1465
266 1319 164p289
909 1319 2182
1465 2182 57p102

Z(X(1580),X(75)) ( = H(X(31),X(1)) ) passes through these centers:
1, 6, 31, 75, 560, 1403, 1580, 1755, 1910, 1967, 2053, 2p699, 4p1613, 25p147, 31p2227, 83p2076, 1281p1911

Collinear triples
1 31 1580
1 75 31p2227
1 560 83p2076
1 1403 1281p1911
6 560 1281p1911
6 1403 1755
6 2053 31p2227
31 560 1755
75 560 1580
75 1755 4p1613
560 1910 25p147
1403 1580 2053
1580 1755 1910
1580 2p699 31p2227
1755 1967 31p2227
1967 4p1613 25p147
2053 4p1613 1281p1911

Z(X(1770),X(1)) passes through these centers:
1, 35, 46, 79, 90, 191, 267, 1717, 1770, 1780

Collinear triples
1 35 46
1 79 1717
1 191 1780
35 79 1770
35 267 1717
46 79 191
46 90 1770
90 1717 1780
191 267 1770

Z(X(1909),X(32)) passes through these centers:
1, 8, 10, 76, 85, 257, 274, 330, 1655, 1909, 2p192, 2p366, 2p1654, 75p2248, 726p1919

Collinear triples
1 8 10
1 76 1909
1 274 2p1654
8 85 1909
8 257 1655
10 76 2p192
10 274 1909
76 274 1655
76 2p1654 729p1929
257 2p192 2p1654
330 1909 2p192
1909 2p1654 75p2248

Z(X(1914),X(76)) passes through these centers:
1, 6, 31, 32, 727, 1326, 1403, 1438, 1911, 1914, 2223, 6p365, 6p2053, 6p2054, 6p2106, 6p2107, 6p2108, 6p2109, 6p2110, 6p2111, 6p2112, 6p2113, 6p2114, 6p2115, 6p2116, 6p2117, 6p2118, 6p2119, 6p2144, 6p2145, 6p2146, 6p2147, 31p1575

Collinear triples
1 6 31p1575
1 31 6p2106
1 32 1914
1 1403 2223
1 1438 6p2110
1 6p2054 6p2108
1 2p2113 6p2144
6 31 1914
6 32 1326
6 1438 6p2112
6 1911 6p2108
6 6p2111 6p2144
6 6p2119 6p2146
31 32 2223
31 1438 6p2116
31 1911 6p2110
31 6p365 6p2118
31 6p2054 6p2112
31 6p2109 6p2144
32 727 6p2108
32 1438 6p2114
32 1911 6p2112
32 6p2053 31p1575
32 6p2107 6p2110
727 1403 6p2112
727 1438 6p2144
727 1914 31p1575
1326 1911 6p2106
1326 1914 6p2054
1326 6p2110 6p2113
1326 6p2115 6p2116
1403 1911 6p2114
1403 1914 6p2053
1403 6p2107 6p2116
1438 1914 2223
1911 2223 31p1575
1911 6p365 6p2146
1914 6p2106 6p2107
1914 6p2108 6p2109
1914 6p2110 6p2111
1914 6p2112 6p2113
1914 6p2114 6p2115
1914 6p2116 6p2117
1914 6p2118 6p2119
1914 6p2144 6p2145
1914 6p2146 6p2147
2223 6p2108 6p2113
2223 6p2110 6p2117
2223 6p2112 6p2115
6p365 6p2110 6p2119
6p365 6p2144 6p2147
6p2053 6p2115 6p2144
6p2054 6p2107 6p2144
6p2106 6p2111 6p2112
6p2106 6p2114 6p2117
6p2108 6p2110 6p2145
6p2108 6p2118 6p2147
6p21096p2112 31p1575
6p2111 6p2116 31p1575
6p2111 6p2118 6p2146
6p2113 6p2114 31p1575

Z(X(1935),X(9)) passes through these centers:
34, 58, 63, 87, 226, 509, 1046, 1423, 1745, 1935, 57p1247

Collinear triples
34 63 1935
34 226 1745
58 63 1046
58 226 1935
63 226 1425
87 1423 1935
1046 1935 57p1247

Z(X(1966),X(31)) ( = H(X(75),X(1)) ) passes through these centers:
1, 2, 31, 75, 561, 1581, 1821, 1959, 1966, 2227, 2p1423, 2p2319, 4p147, 4p194, 76p699, 291p1281, 308p2076

Collinear triples
1 31 1959
2 31 291p1281
2 1959 2p1423
2 2227 2p2319
31 75 308p2076
31 1821 4p147
75561 2227
752p1423 291p1281
561 1959 4p194
1581 1959 2227
1581 4p147 4p194
2p23194p147 291p1281

Z(X(2328),X(75)) passes through these centers:
1, 6, 31, 64, 71, 154, 1042, 1474, 2128

Collinear triples
1 31 2328
6 31 71
6 154 1474
31 154 1042
64 154 2328
71 1474 2328

Z(X(2360),X(75)) passes through these centers:
6, 19, 48, 64, 73, 154, 2299, 2357, 2360

Collinear triples
6 48 73
6 154 2299
19 48 2360
48 154 2357
64 154 2360
73 2299 2360


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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