# 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 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