POINTS ON 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
6p557	164p266	225p1806
56p175	57p2066	225p1806

---

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	501	2206

---

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
1755	511p1687	511p1688
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
57	1073p1712	1073p1712

---

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

1	7	2p1419
1	366	364p366
1	2p2067	7p2066
2	7	9
2	20p2067	92p2066
2	7p2066	8p2362
9	8p2362	92p2066
1489	2p2067	173p1489

---

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

Collinear triple

41

259

259p362

---

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

Collinear triple

266

604

266p504

---

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

Collinear triples

2	2362	7p2066
2	2p2067	225p1806
56	2p2067	7p2066
478	2362	225p1806

---

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

Collinear triples

2	3	28
2	6	81
2	2p501	6p267
3	6	1333
81	1333	2p501

---

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

Collinear triples

2	6	69
2	485	48p492
2	486	48p491
2	2p2066	2p2067
2	7p2066	8p2067
3	69	63p193
3	2p2067	7p2066
3	48p491	48p492
6	485	486
6	2p2066	8p2067
2p2066	7p2066	48p492
2p2067	8p2067	48p491

---

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

Collinear triples

2	63	1400
2	45p63	65p2164
63	1409	45p63
1400	1409	1400p1764

---

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

Collinear triple

2

330

2p192

---

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

Collinear triples

2	57	63
2	223	2p1433
57	222	223
63	222	46p63
63	2p1433	222p1158

---

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

Collinear triples

1	2	78
1	9	219
1	63	21p35
2	9	63
2	8p2164	46p63
2	12p1789	21p35
9	8p2164	72p1780
63	78	72p1780
63	219	46p63
78	219	9p224
78	12p1789	46p63
8p2164	9p224	21p35

---

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
7p2066	57p175	273p2066

---

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

Collinear triples

1	2	8
1	188	8p289
2	236	8p289
2	8p1743	514p1293
8	9	8p1743
9	188	236

---

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

Collinear triples

2	2p2067	281p2066
2	7p2066	200p2362
55	2p2067	7p2066
9p1486	200p2362	281p2066

---

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
37	10p1433	20p1903
37	40p2184	1490p2184
226	281	1490p2184

---

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

Collinear triple

2

513p1222

514p2347

---

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

Collinear triples

2	57	8p610
2	223	20p282
57	223	3p1249

---

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

Collinear triples

2	3	29
2	6	333
2	6p1247	314p2305
3	6	284
6	6p1247	284p1762
29	333	284p1762
284	333	314p2305

---

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

Collinear triples

2	4	21
2	1249	21p2184
4	1172	1249
21	1172	9p229

---

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
6	66	19p159
25	32	19p159
32	31p1670	31p1671
206	1676	1697

---

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

Collinear triples

2	3	6
2	6	141
2	63p66	427p2172
3	6	39
3	141	64p66
6	427	1370p2156
39	427	427p2172
39	38p1342	38p1343
39	63p66	1370p2156

---

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

Collinear triples

1	2	42
1	9	213
2	9	1400
213	1400	1400p1764

---

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

Collinear triples

1	2	43
1	9	2176
2	9	1423
9	43	37p904

---

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
34p90	84p1720	102p1870
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
37	1034p1409	1498p1903
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
241	43p291	291p1282
518	672	8p1362
2p103	291p1282	518p1376

---

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
2194	2218	3p1714

---

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
6p1490	10p154	607p1032

---

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
63	2p1035	6p1032

---

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
512	1379p1577	1380p1577

---

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
2p1423	2p2112	75p727
2p1931	2p2110	2p2113
2p1931	2p2115	2p2116
2p2054	2p2107	2p2144
2p2106	2p2111	2p2112
2p2106	2p2114	2p2117
2p2107	2p2112	1655p2248
2p2108	2p2110	2p2145
2p2108	2p2118	2p2147
2p2110	2p2115	694p1281
2p2111	2p2118	2p2146

---

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
1	1096	4p401
19	63	2p419
19	1096	2312
19	1910	25p147
63	240	1096
63	2312	64p193
63	4p1297	25p147
240	1967	2p419
240	2312	4p1297
240	2p1692	297p2065
1096	1967	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
241	2114	2115

---

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
9	57	1447
9	241	269
9	2114	7p103
57	105	2114
57	269	910
105	910	1279
241	292	1447
241	910	7p103
241	1279	2p1477
241	2114	7p2115
269	292	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
55	56	241
55	910	19p218
56	1279	3p2191
220	241	1407
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
3	4	415
3	158	243
158	415	1047p1247
415	1937	4p416

---

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
20	64	253
64	154	1073
253	1073	1249
253	7p221	8p2192

---

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
2	2p91	69p920
3	69	69p920
4	69	264
4	254	1993
254	264	69p920
254	1993	2p91

---

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
5	30	186
5	54	265
30	74	265
265	35p80	36p79

---

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
1	57	6p1708
1	2p1406	65p79
1	267p1781	1770p2160
6	19	34
6	57	2p1406
6	65p79	1770p2160
6	267p1717	267p1781
19	57	278
19	4p2164	6p1708
19	27p2174	267p1781
34	57	27p2174
34	4p2164	1770p2160
57	65p79	267p1717
278	2p1406	4p2164
278	27p2174	65p79
1723	4p2164	27p2174

---

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
1	2319	6p2227
2	6	385
2	76	6p2227
2	1423	292p1281
6	32	511
32	76	385
32	98	19p147
76	511	4p1740
98	385	511
385	1423	2319
385	6p2227	75p699
511	694	6p2227
694	4p1740	19p147
2319	4p1740	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
2	69	2p1780
2	72	2p579
2	92	1441
2	2p1751	306p1612
4	69	286
4	72	92
4	2p1751	580p1441
21	63	2p579
21	72	2p1780
21	286	1441
21	2p1751	69p1714
63	69	306p1612
63	72	63p1713
63	92	286
69	72	69p1714
69	1441	2p579
72	1441	580p1441
92	2p1751	2p1780
286	2p579	2p1751
1441	2p1751	63p1713

---

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

Collinear triples

42	81	291
42	672	1914
105	291	672

---

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
1	57	1281p1911
1	291	292
1	894	1463p2053
1	1967	741p1654
6	42	672
6	239	894
6	292	8p1911
6	2p741	741p1654
42	291	2p741
42	1967	8p1911
42	37p2248	1281p1911
57	291	8p1911
57	672	894
239	291	291p1911
239	1757	1575p2109
239	8p1911	105p291
239	238p2113	1281p1911
239	292p1929	741p1654
291	672	105p291
291	894	1967
291	1757	292p1929
291	37p2248	741p1654
292	672	291p1911
292	894	2p741
292	292p1929	1281p1911
672	1757	238p2113
672	241p2115	1281p1911
894	1757	37p2248
1757	2p741	291p1911
1967	291p1911	1281p1911
2p741	105p291	1281p1911
8p1911	291p1911	1463p2053
1281p1911	1463p2053	1575p2053

---

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
1	171	241
1	291	292
1	1581	741p1655
2	37	1575
2	171	238
2	291	43p291
2	2p741	741p1655
37	292	2p741
37	1581	43p291
87	171	1575
87	292	43p291
171	291	2p741
171	292	1581
238	291	291p294
238	292	291p291
238	37p2106	518p2111
238	43p291	335p727
241	292	291p294
241	43p291	291p291
291	1575	291p291
292	1575	335p727
292	291p2068	291p2069
1581	37p2106	291p294
2p741	37p2106	291p291

---

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
2	393	92p401
4	69	4p1966
4	98	19p147
4	393	1503
69	297	393
69	1503	193p2184
69	19p147	92p1297
98	230	1503
98	4p147	92p401
297	694	4p1966
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
6	230	394
6	1503	2207
25	230	91p206
230	297	1755p2065
230	1503	31p248
297	394	2207
297	1503	31p1297
297	47p66	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
1	279	2p1763
1	2184	2p2138
2	19	2p1763
2	75	346
2	279	2p20
19	75	2p20
279	304	346
304	2184	2p20
304	2p2138	2p2139
346	2184	2p1763
2p20	2p159	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
2	280	318
2	329	77p1753
2	75p963	329p1622
40	77	77p1753
40	189	280
77	309	318
77	347	962
189	309	329
280	309	347
280	75p963	77p1753
309	962	75p963
318	329	962
329	347	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
1	65	1764
1	75	314
1	81	2p573
2	4	81p1766
2	65	2p573
2	69	81
2	75	321
2	75p2217	226p1610
4	69	314
4	321	2p573
65	69	75
65	81	81p1766
65	321	321p572
69	75p2217	321p572
75	75p2217	81p1766
81	314	321
314	2p573	75p2217
321	1767	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
4	69	316
67	524	858
316	524	671
316	858	75p1177

---

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
7	8	319
8	79	2p191
10	86	319
319	1029	2p191

---

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
7	8	320
80	519	908
320	519	903
320	908	2p104

---

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
2	273	2p1490
7	8	322
8	84	2p1490
78	84	57p2057
78	273	322
322	2p1490	57p1034

---

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
1	19	21p478
1	58	63
1	573	1400
1	2p2217	65p1610
2	6	333
2	63	1400
2	2p2217	21p478
6	19	65p1610
6	58	573
6	1400	6p1764
10	19	573
10	58	333
10	1400	10p572
10	2p2217	6p1764
19	63	333
58	1400	21p478
63	2p2217	10p572
333	573	2p2217

---

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
356	357	358
356	1134	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
366	7p365	9p365

---

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
32	76	384
39	76	194
39	83	384

---

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
4	6	393
254	393	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
2	19p617	621p2159
2	19p628	622p2148
6	16	62
14	16	395
14	62	19p617
16	18	19628
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
2	19p627	621p2158
6	15	61
13	15	386
15	17	19p627
17	61	396

---

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
110	476	523
110	526	110p1749
476	110p1749	1109p1291

---

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
4	69	511
290	385	401
385	511	1916
401	511	1972

---

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
3	394	48p230
6	25	3p2312
6	248	31p147
6	385	394
25	394	511
25	3p1967	31p147
248	385	401
248	3p2312	48p230
385	511	3p1967
394	1297	31p147
394	3p2312	1073p1707
401	511	48p1987
511	1297	3p2312
511	48p230	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
36	84	90p1720
40	80	90p1718
40	84	515
102	90p1718	90p1720

---

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
40	515	2357
40	1826	2183
48	73	2183
48	515	1826
48	2357	1528p2188
515	2183	2250
515	65p1295	1528p2188
2183	42p102	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
19	55	9p2182
19	219	517
44	517	2316
44	2342	9p2182
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
6p2109	6p2112	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
75	561	2227
75	2p1423	291p1281
561	1959	4p194
1581	1959	2227
1581	4p147	4p194
2p2319	4p147	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

---