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,