# POINTS ON CUBICS

This continuation of Points on Cubics has two sections, accessible by scrolling down or clicking:

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

## C(U,L): Cross Conjugate Collinearity Cubics

### defined by

L(wy + vz)x2 + M(uz + wx)y2 + N(vx + uy)z2 - (Lu + Mv + Nw)xyz = 0

Locus: The cross conjugate collinearity cubic C(U,L) is the locus of a point X = x : y : z such that the U-cross conjugate of X, but not the point U itself, is on the line Lα + Mβ + Nγ = 0.

Notes:

1.   C(U,L) is also given by

(Mwy2 + Nvz2)x + (Nuz2 + Lwx2)y + (Lvx2 + Muy2)z - (Lu + Mv +Nw)xyz = 0.

2.   The cubic C(P -1U -1, U -1) is given by

up(qy2 + rz2)x + vq(rz2 + px2)y + wr(px2 + qy2)z - Kxyz = 0,

where K = (v2w2qr + w2u2rp + u2v2pq)/uvw.

Thus, if K = 0, the cubic C(P -1*U -1, U -1) is the cubic ZP(U,P).
Equivalently, the cubic C(U,L) is the cubic ZP(P -1, P*U -1).

For all K, the cubic C(P -1*U -1, U -1) is an nK-cubic as defined in Ehrmann and Gibert, "Special Isocubics in the Triangle Plane," downloadable from Cubics in the Triangle Plane.

3.   C(U,L) passes through the points A, B, C and these points:

0 : N : - M,    - N : 0 : L,    M : - L : 0.

4.   If U is on the line Lα + Mβ + Nγ = 0, then the left-hand side of the defining equation is satisfied by every point in the extended plane of triangle ABC.

5.   The discriminant d of C(U,P), obtained by solving the defining equation for x in terms of y and z, is given by

d2 = J2 + K2 - 2(3Lu + Mv + Nw)(Mwy3z + Nvyz3),

where J = Mwy2 + Nvz2   and    K = (Lu - Mv - Nw)yz.

It follows that if J2 = 2(3Lu + Mv + Nw)(Mwy3z + Nvyz3), then there are four solutions for z as multiples of y, and the discriminant simplifies so that

d = (Lu - Mv - Nw)yz.

For each of the four z, there are two resulting x, yielding eight points x : y : z on C(U,P). Each of the eight begets two others by cyclic permutations of L,M,N; u,v,w; x,y,z, for a total of 24 points on C(U,P). Of the eight points, four are written here using E as an abbreviation for (- MNvw)1/2:

uE : v(Nw + E) : w(Mv + E)    and     uE : v(- Nw + E) : w(Mv - E);

(Mv + Nw)E : Lv(Nw + E) : Lw(- Mv + E)    and     (Mv + Nw)E : Lv(- Nw + E) : Lw(Mv + E).

6.   In the list of cubics C(U,L(I,J)), the notation L(I,J) denotes the line X(I)X(J).

C(X(25),L(1,2)) passes through these centers:
4, 101, 1113, 1114, 37p1113, 37p1114

Collinear triples
 4 1113 1114 4 37p1113 37p1114 101 1113 37p1113 101 1114 37p1114

C(X(56),L(1,2)) passes through these centers:
57, 100, 1381, 1382, 2p1381, 2p1382, 1280p1617

Collinear triples
 57 100 1280p1617 57 1381 1382 57 2p1381 2p1382 100 1381 2p1381 100 1382 2p1382

C(X(764),L(1,2)) passes through these centers:
513, 514, 900, 1022

Collinear triple
 513 514 900

C(X(1470),L(1,2)) passes through these centers:
100, 104, 109, 2p997, 2p1381, 2p1382, 2p1470, 997p1381, 997p1382

Collinear triples
 100 104 2p997 100 109 997p1381 100 2p997 997p1382 104 109 2p1470 2p1381 2p1382 2p1470 2p1470 997p1381 997p1382

C(X(255),L(1,3)) passes through these centers:
63, 1822, 1823, 3p109, 1214p1822, 1214p1823

Collinear triples
 63 1822 1823 63 1214p1822 1214p1823 1822 3p109 1214p1822 1823 3p109 1214p1823

C(X(3),L(1,4)) passes through these centers:
2, 109, 1113, 1114, 226p1822, 226p1823

Collinear triples
 2 1113 1114 2 226p1822 226p1823 109 1113 226p1822 109 1114 1214p1823

C(X(56),L(1,4)) passes through these centers:
57, 108, 1381, 1382, 19p2338, 278p1381, 278p1382, 658p1456

Collinear triples
 57 108 19p2338 57 1381 1382 57 278p1381 278p1382 108 1381 278p1381 108 1382 278p1382

C(X(255),L(1,4)) passes through these centers:
57, 108, 1381, 1382, 19p2338, 278p1381, 278p1382, 658p1456

Collinear triples
 63 1822 1823 63 226p1822 226p1823 109 1822 278p1822 109 1823 226p1823

C(X(56),L(1,5)) passes through these centers:
57, 1381, 1382, 2222, 1381p2006, 1382p2006

Collinear triples
 57 1381 1382 57 1381p2006 1382p2006 1381 2222 1381p2006 1382 2222 1382p2006

C(X(255),L(1,5)) passes through these centers:
63, 1822, 1823, 3p2222

Collinear triple
 63 1822 1823

C(X(41),L(1,6)) passes through these centers:
55, 101, 1381, 1382, 220p1381, 220p1382

Collinear triples
 55 1381 1382 55 220p1381 220p1382 101 1381 220p1381 101 1382 220p1382

C(X(48),L(1,6)) passes through these centers:
3, 101, 1381, 1382, 55p2338, 219p1381, 219p1382

Collinear triples
 3 101 55p2338 3 1381 1382 3 219p1381 219p1382 101 1381 219p1381 101 1382 219p1382

C(X(56),L(1,6)) passes through these centers:
1, 57, 1381, 1382

Collinear triple
 57 1381 1382

C(X(73),L(1,6)) passes through these centers:
101, 1214, 1381, 1382, 72p1381, 72p1382

Collinear triples
 101 1381 72p1381 101 1382 72p1382 1214 1381 1382 1214 72p1381 72p1382

C(X(172),L(1,6)) passes through these centers:
101, 171, 1381, 1382, 1381p2329, 1382p2329

Collinear triples
 101 1381 1381p2329 101 1382 1382p2329 171 1381 1382 171 1381p2329 1381p2329

C(X(198),L(1,6)) passes through these centers:
40, 101, 1381, 1382, 1381p2324, 1382p2324

Collinear triples
 40 1381 1382 40 1381p2324 1382p2324 101 1381 1381p2324 101 1382 1382p2324

C(X(255),L(1,6)) passes through these centers:
63, 906, 1822, 1823, 72p1822, 72p1823

Collinear triples
 63 1822 1823 63 72p1822 72p1823 906 1822 72p1822 906 1823 72p1823

C(X(604),L(1,6)) passes through these centers:
56, 101, 1381, 1382, 6p1381, 6p1382, 218p1477, 1279p1292

Collinear triples
 56 101 218p1477 56 1381 1382 56 6p1381 6p1382 101 1381 6p1381 101 1382 6p1382

C(X(56),L(1,7)) passes through these centers:
6, 651, 1020, 1400, 2p1381, 2p1382, 6p2283, 10p1462, 1381p1400, 1382p1400

Collinear triples
 57 1381 1382 57 279p1381 279p1382 934 1381 279p1381 934 1382 279p1382

C(X(213),L(1,7)) passes through these centers:
6, 651, 1020, 1400, 2p1381, 2p1382, 6p2283, 10p1462, 1381p1400, 1382p1400

Collinear triples
 6 651 10p1462 6 1400 6p2282 651 1020 6p2282 651 2p1381 1381p1400 651 2p1382 1382p1400 1020 1400 10p1462 1400 2p1381 2p1382 1400 1381p1400 1382p1400

C(X(220),L(1,7)) passes through these centers:
9, 651, 1815, 2p1381, 2p1382, 9p1381, 9p1382, 100p1886

Collinear triples
 9 65 1815 9 2p1381 2p1382 9 9p1381 9p1382 651 2p1381 9p1381 651 2p1382 9p1382

C(X(255),L(1,7)) passes through these centers:
63, 1822, 1823, 3p934

Collinear triple
 63 1822 1823

C(X(603),L(2,3)) passes through these centers:
58, 162, 222, 651, 2p104, 3p1381, 3p1382, 27p1381, 27p1382, 109p859

Collinear triples
 58 162 2p104 58 222 109p859 162 651 109p839 162 3p1381 27p1381 162 3p1382 27p1382 222 651 2p104 222 3p1381 3p1382 222 27p1381 27p1382

C(X(1464),L(2,3)) passes through these centers:
1, 81, 278, 2p1983, 36p653, 100p1835

Collinear triples
 1 812 2p1983 1 278 100p1835 81 278 36p653 2p1983 36p653 100p1835

C(X(25),L(2,6)) passes through these centers:
4, 110, 1113, 1114

Collinear triple
 4 1113 1114

C(X(51),L(2,6)) passes through these centers:
5, 110, 1113, 1114, 5p1822, 5p1823

Collinear triples
 5 1113 1114 5 5p1822 5p1823 110 1113 5p1822 110 1114 5p1823

C(X(154),L(2,6)) passes through these centers:
20, 110, 1113, 1114, 20p1822, 20p1823

Collinear triples
 20 1113 1114 20 20p1822 20p1823 110 1113 20p1822 110 1114 20p1823

C(X(159),L(2,6)) passes through these centers:
110, 1113, 1114, 1370, 1370p1822, 1370p1823

Collinear triples
 110 1113 1370p1822 110 1114 1370p1823 1113 1114 1114 1114 1370p1822 1370p1823

C(X(161),L(2,6)) passes through these centers:
110, 1113, 1114, 75p161

Collinear triple
 1113 1114 75p161
(This is the Euler line.)

C(X(184),L(2,6)) passes through these centers:
3, 110, 1113, 1114, 1370, 3p1822, 3p1823

Collinear triples
 3 1113 1370p1822 3 3p1822 3p1823 110 1113 3p1822 110 1114 3p1823

C(X(206),L(2,6)) passes through these centers:
22, 110, 1113, 1114, 1370, 22p1822, 22p1823, 255p1297

Collinear triples
 22 110 255p1297 22 1113 1114 22 22p1822 22p1823 110 1113 22p1822 110 1114 22p1823

C(X(232),L(2,6)) passes through these centers:
22, 110, 1113, 1114, 1370, 22p1822, 22p1823, 255p1297

Collinear triples
 110 1113 297p1822 110 1114 297p1823 297 1113 1114 297 297p1822 297p1823

C(X(311),L(2,6)) passes through these centers:
76, 300, 301, 662

Collinear triple
 300 301 5p662

C(X(25),L(2,7)) passes through these centers:
4, 109, 1113, 1114, 65p1113, 65p1114

Collinear triples
 4 1113 1114 4 65p1113 65p1114 109 1113 65p1113 109 1114 65p1114

C(X(31),L(2,7)) passes through these centers:
1, 109, 1822, 1823, 65p1113, 65p1114, 104p1470

Collinear triples
 1 109 104p1470 1 1822 1823 1 65p1113 65p1114 109 1822 65p1113 109 1823 65p1114

C(X(55),L(2,7)) passes through these centers:
1, 100, 1381, 1382, 8p1381, 8p1382, 999p1320

Collinear triples
 1 100 999p1320 1 1381 1382 1 8p1381 8p1382 100 1381 8p1381 100 1382 8p1382

C(X(222),L(2,7)) passes through these centers:
77, 651, 2338, 7p1381, 7p1382, 63p1381, 63p1382

Collinear triples
 77 651 2338 77 7p1381 7p1382 77 63p1381 63p1382 651 7p1381 63p1381 651 7p1382 63p1382

C(X(5),L(3,6)) passes through these centers:
2, 13, 14, 1625

Collinear triple
 13 14 1625

C(X(71),L(3,6)) passes through these centers:
1, 906, 1822, 1823, 37p1822, 37p1823

Collinear triples
 1 1822 1823 1 37p1822 37p1823 906 1822 37p1822 906 1823 37p1823

C(X(2140),L(3,6)) passes through these centers:
7, 86, 673, 692p2140

Collinear triple
 7 673 692p2140

C(X(3),L(4,6)) passes through these centers:
2, 112, 1113, 1114

Collinear triple
 2 1113 1114

C(X(15),L(4,6)) passes through these centers:
112, 470, 1113, 1114

Collinear triple
 470 1113 1114

C(X(16),L(4,6)) passes through these centers:
112, 471, 1113, 1114

Collinear triple
 471 1113 1114

C(X(32),L(4,6)) passes through these centers:
25, 112, 1113, 1114, 923p1995, 1113p1973, 1114p1973

Collinear triples
 25 112 923p1995 25 1113 1114 25 1113p1973 1114p1973 112 1113 1113p1973 112 1114 1114p1973

C(X(39),L(4,6)) passes through these centers:
112, 427, 1113, 1114

Collinear triple
 427 1113 1114

C(X(50),L(4,6)) passes through these centers:
112, 186, 1113, 1114

Collinear triple
 186 1113 1114

C(X(52),L(4,6)) passes through these centers:
112, 467, 1113, 1114

Collinear triple
 467 1113 1114

C(X(58),L(4,6)) passes through these centers:
27, 112, 1113, 1114, 28p1113, 28p1114

Collinear triples
 27 1113 1114 27 28p1113 28p1114 112 1113 28p1113 112 1114 28p1114

C(X(61),L(4,6)) passes through these centers:
112, 473, 1113, 1114

Collinear triple
 473 1113 1114

C(X(62),L(4,6)) passes through these centers:
112, 472, 1113, 1114

Collinear triple
 472 1113 1114

C(X(182),L(4,6)) passes through these centers:
112, 458, 1113, 1114

Collinear triple
 458 1113 1114

C(X(187),L(4,6)) passes through these centers:
112, 468, 1113, 1114

Collinear triple
 468 1113 1114

C(X(216),L(4,6)) passes through these centers:
5, 112, 1113, 1114, 53p1822, 53p1823

Collinear triples
 5 1113 1114 5 53p1882 53p1823 112 1113 53p1822 112 1114 53p1823

C(X(284),L(4,6)) passes through these centers:
29, 112, 1113, 1114, 1113p1172, 1114p1172

Collinear triples
 29 1113 1114 29 1113p1172 1114p1172 112 1113 1113p1172 112 1114 1114p1172

C(X(2193),L(4,6)) passes through these centers:
21, 112, 1113, 1114, 284p1113, 284p1114

Collinear triples
 21 1113 1114 21 284p1113 284p1114 112 1113 284p1113 112 1114 284p1114

## ZC(U,L): ZC-Cubics

L(wy - vz)x2 + M(uz - wx)y2 + N(vx - uy)z2 = 0

Locus:   The cubic ZC(U,L) is the locus of a point X = x : y : z such that the point

(wy - vz)x2 : (uz - wx)y2 : (vx - uy)z2

lies on the line L given by Lα + Mβ + Nγ = 0.

Notes:

1.   The bicentrics of the U-cross conjugate of a point X are the points

- v/y + w/z + u/x : - w/z + u/x + v/y : - u/x + v/y + w/z   and   - w/z + u/x + v/y : - u/x + v/y + w/z : - v/y + w/z + u/x.

It is easy to check that these bicentrics lie, respectively, on the cubics

L(wy - vz)x2 + M(uz - wx)y2 + N(vx - uy)z2 + (Lu + Mv + Nw)xyz = 0

and

L(wy - vz)x2 + M(uz - wx)y2 + N(vx - uy)z2 - (Lu + Mv + Nw)xyz = 0.

These equal ZC(U,L) if and only if U is on L; i.e., Lu + Mv + Nw = 0;
thus, in this case, the two bicentrics lie on ZC(U,L).

2.   ZC(U,L) is also given by

(Mwy2 - Nvz2)x + (Nuz2 - Lwx2)y + (Lvx2 - Muy2)z = 0.

3.   The cubic ZC(P -1*U -1), U -1) is the cubic Z(U,P); c.f., Note 2 in the section on the class of C-cubics. Conversely, the cubic ZC(U,L) is the cubic Z(L -1,L*U -1); here, the coefficients L, M, N are regarded as the point L having trilinears L : M : N, so that L -1 denotes the trilinear pole of the line Lα + Mβ + Nγ = 0.
Example: ZC(X(32),L(30,511)) = Z(X(2),X(75)).

4.   The point U is on ZC(U,L).

5.   The discriminant d of ZC(U,L), obtained by solving the defining equation for x in terms of y and z, is given by

d2 = (Mwy2 - Nvz2)2 - 4Luyz(My - Nz)(wy -vz).

6.   Suppose 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 ZC(U,L) onto the the cubic ZC(F*U,L[F]), where L[F} denotes the line Lghα + Mhfβ + Nfgγ = 0, this being the line consisting of the points F*X as X ranges through the line Lα + Mβ + Nγ = 0.

ZC(X(1),L(1,2)) passes through these centers:
1, 190, 649, 2p1016

Collinear triple
 190 649 2p1016

ZC(X(2),L(1,2)) passes through these centers:
2, 190, 514, 1016, 1016p1054

Collinear triples
 2 1016 1016p1054 190 514 1016

ZC(X(1),L(1,3)) passes through these centers:
1, 650, 651, 2p1262

Collinear triple
 650 651 2p1262

ZC(X(1),L(1,4)) passes through these centers:
1, 652, 653, 393p1275

Collinear triple
 652 653 653p1275

ZC(X(1),L(1,5)) passes through these centers:
1, 654, 655, 655p655

Collinear triple
 654 655 655p655

ZC(X(1),L(1,6)) passes through these centers:
1, 100, 513, 765, 765p1054

Collinear triples
 1 764 765p1054 100 513 764

ZC(X(6),L(1,6)) passes through these centers:
1, 100, 667, 1016

Collinear triple
 100 667 1016

ZC(X(279),L(1,7)) passes through these centers:
279, 658, 1275, 244p658, 1054p1275, 1987p1987

Collinear triples
 279 1275 1054p1275 658 1275 244p658

ZC(X(2),L(2,3)) passes through these centers:
2, 525, 648, 92p250

Collinear triple
 525 648 92p250

ZC(X(524),L(2,6)) passes through these centers:
99, 524, 1649, 662p892, 1379p1577, 1380p1577

Collinear triples
 99 1649 662p892 524 1379p1577 1380p1577

ZC(X(100),L(2,11)) passes through these centers:
100, 190, 664, 666, 2284, 2p919, 6p874, 8p919, 105p660, 666p673

Collinear triples
 100 190 6p874 100 2p919 8p919 190 664 2284 190 666 2p919 664 666 8p919 666 2284 666p673 666 6p874 105p660 2284 2p919 105p660 6p874 8p919 666p673

ZC(X(110),L(2,11)) passes through these centers:
99, 110, 648, 666, 919m 48p666

Collinear triples
 99 666 919 110 919 48p666 648 666 48p666

ZC(X(658),L(2,11)) passes through these centers:
651, 658, 664, 666, 883, 927, 2p927, 105p927, 165p927, 190p1456, 664p1376, 932p1447

Collinear triples
 651 658 190p1456 651 664 883 651 666 2p927 651 664p1376 932p1447 658 883 664p1376 658 927 165p927 664 666 927 666 883 105p927 927 2p927 664p1376 927 105p927 190p1456

ZC(X(665),L(2,11)) passes through these centers:
1, 101, 514, 665, 666, 1026, 1027, 2p1279, 6p1280, 10p1438, 21p241

Collinear triples
 1 101 10p1438 1 514 1027 1 1026 2p1279 101 514 666 514 665 2p1279 665 1027 10p1438 666 1026 1027 666 2p1279 6p1280 666 10p1438 21p241

ZC(X(1638),L(2,11)) passes through these centers:
2, 514, 527, 651, 666, 673, 918, 1638, 2p1027, 2p1279, 2p1308, 100p527, 105p527, 190p1456, 513p1121, 527p885, 666p1155

Collinear triples
 2 527 190p1456 2 666 105p527 2 673 666p1155 2 918 1638 2 2p1027 527p885 2 2p1279 100p527 514 527 918 514 666 666p1155 514 1638 2p1279 514 2p1027 105p527 527 666 673 651 666 527p885 651 673 105p527 651 918 100p527 651 1638 190p1456 666 2p1027 100p527 673 1628 2p1027

ZC(X(1639),L(2,11)) passes through these centers:
2, 519, 522, 644, 666, 918, 1639, 2p294, 2p2342, 7p1023, 8p910, 8p1022, 8p1024, 8p2222, 44p666, 105p2325, 190p2348, 214p885, 666p678, 666p1647, 673p1145

Collinear triples
 2 519 190p2348 2 666 105p2325 2 918 1639 2 2p294 44p666 2 7p1023 8p910 2 8p1024 666p1647 519 522 918 519 666 2p294 519 105p2325 666p678 519 214p885 666p1647 522 666 44p666 522 1639 8p910 522 8p1024 105p2325 644 666 666p16476 644 1639 7p1023 644 2p294 190p2348 666 2p2342 673p1145 666 7p1023 8p1024 666 8p1022 666p678 666 8p2222 214p885 1639 2p294 8p1024 1639 44p666 666p678 1639 105p2325 214p885 1639 666p1647 673p1145 2p294 8p1022 214p885 2p294 8p2222 673p1145 2p2342 8p1024 214p885 7p1023 44p666 214p885 7p1023 105p2325 673p1145 8p1024 8p2222 666p678

ZC(X(3),L(3,6)) passes through these centers:
3, 110, 250, 520

Collinear triple
 110 250 520

ZC(X(6),L(3,6)) passes through these centers:
6, 110, 249, 512, 148p1101

Collinear triples
 6 249 148p1101 110 249 512

ZC(X(101),L(3,6)) passes through these centers:
101, 110, 662, 692, 31p932, 43p662

Collinear triples
 101 662 43p662 101 31p932 932p1613 110 662 692 110 31p932 43p662

ZC(X(190),L(3,6)) passes through these centers:
100, 110, 190, 662, 932, 43p662

Collinear triples
 100 110 662 110 932 43p662 190 662 43p662

ZC(X(512),L(3,6)) passes through these centers:
100, 110, 190, 662, 932, 43p662

Collinear triples
 6 1379 1380 6 25p1822 25p1823 6 163p1312 523p1822 6 163p1313 523p1823 110 1379 661p1380 110 1380 661p1379 110 25p1822 523p1823 110 25p1823 523p1822 110 163p1312 163p1313 512 25p1822 163p1312 512 25p1823 163p1313 512 523p1822 523p1823 512 661p1379 661p1380

ZC(X(513),L(3,6)) passes through these centers:
110, 513, 942, 1175, 1381, 1382, 28p1822, 28p1823, 523p1822, 523p1823

Collinear triples
 110 942 1175 110 28p1822 523p1823 110 28p1823 523p1822 513 523p1822 523p1823 942 1381 1382 942 28p1822 28p1823

ZC(X(521),L(3,6)) passes through these centers:
110, 521, 960, 1798, 21p1822, 21p1823, 523p1822, 523p1823

Collinear triples
 110 960 1178 110 21p1822 523p1823 110 21p1823 523p1822 521 523p1822 523p1823 960 21p1822 21p1823

ZC(X(523),L(3,6)) passes through these centers:
1, 5, 54, 110, 523, 1113, 1114, 9p1019, 523p1822, 523p1823

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

ZC(X(32),L(3,512)) passes through these centers:
6, 25, 32, 511, 31p2065, 1755p2065, 1959p2065

Collinear triples
 6 32 511 6 1755p2065 1959p2065 511 31p2065 1959p2065

ZC(X(39),L(3,512)) passes through these centers:
2, 6, 39, 511, 38p2065, 1959p2065

Collinear triples
 6 39 511 511 38p2065 1959p2065

ZC(X(2088),L(3,512)) passes through these centers:
6, 323, 511, 525, 526, 2088, 74p661, 255p1300, 1959p2065

Collinear triples
 6 511 2088 511 525 526

ZC(X(6),L(3,513)) passes through these centers:
1, 6, 57, 517, 2323, 3p915, 57p2077, 63p915, 78p913, 915p1795

Collinear triples
 1 6 2323 1 57 517 1 3p915 63p915 6 57 57p2077 6 3p915 78p913 57 63p915 78p913 517 63p915 915p1795 2323 78p913 915p1795 3p915 57p2077 v

ZC(X(37),L(3,513)) passes through these centers:
1, 2, 37, 2323, 63p915, 71p915, 72p915

Collinear triples
 1 37 2323 1 63p915 72p915 2 63p915 71p915 37 71p915 72p915

ZC(X(109),L(3,517)) passes through these centers:
101, 109, 651, 934, 100p2362, 109p366, 109p1489, 144p1461, 651p2067, 653p2066, 934p2066

Collinear triples
 101 651 934 101 100p2362 653p2066 109 934 144p1461 109 651p2067 934p2066 651 100p2362 934p2066 651 651p2067 653p2066

ZC(X(112),L(3,517)) passes through these centers:
108, 110, 112, 651, 63p1301, 101p229, 108p610

Collinear triples
 108 110 651 108 112 108p610 110 112 101p229 651 63p1301 108p610

ZC(X(163),L(3,517)) passes through these centers:
162, 163, 651, 662, 1415, 3p109, 99p2305, 163p1762, 1247p1415

Collinear triples
 162 651 3p109 162 662 163p1762 163 662 99p2305 163 1415 3p109 651 662 1415 651 99p2305 1247p1415 1415 163p1762 1247p1415

ZC(X(222),L(3,517)) passes through these centers:
222, 651, 905, 1262, 1262p1768

Collinear triples
 222 1262 1262p1768 651 905 1262

ZC(X(513),L(3,517)) passes through these centers:
6, 7, 509, 513, 651, 100p2350

Collinear triple
 6 7 651

ZC(X(514),L(3,517)) passes through these centers:
81, 174, 226, 514, 651, 2p1381, 2p1382, 514p1381, 514p1382

Collinear triples
 81 226 651 226 2p1381 2p1382 514 514p1381 514p1382 651 2p1381 514p1382 651 2p1382 514p1381

ZC(X(654),L(3,517)) passes through these centers:
101, 104, 109, 651, 654, 2316, 2p654, 3p1845, 7p654, 44p1443, 57p102, 515p2323

Collinear triples
 101 104 515p2323 101 109 654 101 651 7p654 101 2316 3p1845 104 109 44p1443 104 651 3p1845 109 651 2p654 109 3p1845 57p102 651 2316 44p1443 651 57p102 515p2323 654 2p654 515p2323 654 7p654 44p1443 2p654 3p1845 7p654

ZC(X(665),L(3,517)) passes through these centers:
101, 105, 109, 651, 665, 1362, 2254, 6p1280, 7p665, 7p911, 59p1566, 241p1279, 241p1381, 241p1382, 513p1381, 513p1382, 514p1381, 514p1382, 518p1381, 518p1382, 650p1381, 650p1382, 672p1381, 672p1382

Collinear triples
 101 105 59p1566 101 109 665 101 651 7p655 101 1362 6p1280 101 518p1381 650p1382 101 518p1381 650p1381 105 109 241p1279 105 651 1362 109 651 2254 109 1362 7p911 109 241p1381 513p1382 109 241p1382 513p1381 651 6p1280 241p1279 651 7p911 58p1566 651 241p1381 650p1382 651 241p1382 650p1381 651 513p1381 518p1382 651 513p1382 518p1381 651 514p1381 672p1382 651 514p1382 672p1381 665 59p1566 59p1566 665 241p1279 241p1279 665 650p1382 650p1382 665 650p1381 650p1381 665 514p1382 514p1382 1362 2254 7p665 1362 241p1381 518p1382 1362 241p1382 518p1381 1362 672p1381 672p1382 2254 650p1381 650p1382 7p665 513p1382 513p1382 59p1566 241p1382 241p1382 241p1279 518p1382 518p1382 241p1381 514p1381 650p1381 241p1382 514p1382 650p1382 241p1381 518p1381 518p1381 241p1381 672p1381 672p1381 241p1382 514p1382 518p1382 241p1382 650p1382 672p1382

ZC(X(668),L(3,517)) passes through these centers:
190, 651, 664, 668, 2p932, 43p664, 99p171, 100p508, 190p978, 662p1999, 664p979

Collinear triples
 190 651 664 190 668 190p978 651 2p932 43p664 651 190p978 664p999 664 668 43p664 668 99p171 662p1999 2p932 99p171 190p978 43p664 662p1999 664p979

ZC(X(812),L(3,517)) passes through these centers:
190, 651, 673, 812, 56p812, 190p978, 238p241

Collinear triples
 190 651 56p812 190 812 190p978 651 673 238p241 812 56p812 238p241

ZC(X(905),L(3,517)) passes through these centers:
2, 222, 651, 905, 63p1381, 63p1382, 226p1822, 226p1823, 514p1381, 514p1382, 514p1822, 572p693

Collinear triples
 2 222 651 2 905 572p693 2 226p1822 226p1823 222 63p1381 63p1382 651 63p1381 514p1382 651 63p1382 514p1381 651 226p1823 514p1822 905 514p1381 514p1382

ZC(X(906),L(3,517)) passes through these centers:
101, 109, 651, 906, 1331, 1813, 35p110, 46p1813, 90p101, 101p224, 906p1708

Collinear triples
 101 109 906 101 651 1813 101 90p101 906p1708 109 651 1331 109 1813 35p110 651 46p1813 90p101 906 1331 101p224 906 1813 46p1813 1331 1813 906p1708 35p110 90p101 101p224

ZC(X(928),L(3,517)) passes through these centers:
101, 109, 651, 928, 2p928, 7p928

Collinear triples
 101 109 906 101 651 7p928 109 651 2p928

ZC(X(1018),L(3,517)) passes through these centers:
100, 190, 651, 1018, 9p1020, 10p1415, 100p1745, 190p978, 226p1293

Collinear triples
 100 651 9p1020 100 1018 100p1743 190 651 10p1415 190 1018 190p978 651 100p1743 226p1293 1018 9p1020 10p1415

ZC(X(1022),L(3,517)) passes through these centers:
57, 88, 651, 1022, 2p901, 7p649

Collinear triples
 57 88 651 57 1022 7p649 651 2p901 7p649

ZC(X(1024),L(3,517)) passes through these centers:
6, 513, 651, 673, 1024, 2p919

Collinear triples
 6 513 1024 6 651 673 513 651 2p919

ZC(X(1026),L(3,517)) passes through these centers:
100, 651, 660, 664, 1025, 1026, 2284, 8p919, 43p664, 87p2284, 100p1743, 190p1477, 238p1025, 241p1025, 1025p2348

Collinear triples
 100 651 1025 100 1026 100p1743 100 8p919 1025p2348 651 660 238p1025 651 664 2284 651 8p919 241p1025 651 43p664 87p2284 651 190p1477 1025p2348 660 43p664 241p1025 664 1026 43p664 664 8p919 238p1025 1025 1026 238p1025 1025 2284 241p1025 1026 2284 1025p2348 100p1743 190p1477 241p1025

ZC(X(1415),L(3,517)) passes through these centers:
101, 108, 109, 651, 1415, 1461, 3p109, 40p1461, 64p1813, 84p101, 108p610

Collinear triples
 101 109 1415 101 651 1461 101 3p109 84p101 101 40p1461 64p1813 108 109 40p1461 108 651 3p109 108 1415 108p610 109 1461 3p109 109 84p101 108p610 651 40p1461 84p101 651 64p1813 108p610 1415 1461 40p1461

ZC(X(1643),L(3,517)) passes through these centers:
1, 6, 57, 513, 650, 651, 1643, 2246, 7p649, 7p2246, 44p2222, 56p528, 100p2246, 109p528, 514p840, 516p919, 517p2222, 528p1461, 1155p1308

Collinear triples
 1 6 100p2246 1 57 109p528 1 650 1643 1 651 56p528 1 2246 516p919 1 7p2246 517p2222 6 57 528p1461 6 513 1643 6 651 7p2246 6 2246 1155p1308 6 44p2222 56p528 57 651 2246 57 1643 7p649 513 650 2246 513 651 109p528 513 7p649 56p528 513 517p2222 528p1461 650 651 528p1461 650 7p649 7p2246 650 44p2222 109p528 651 7p649 100p2246 1643 44p2222 517p2222 7p649 109p528 1155p1308 7p649 516p919 528p1461

ZC(X(2283),L(3,517)) passes through these centers:
101, 109, 651, 927, 1025, 2283, 7p2283, 56p660, 101p1362, 238p1025

Collinear triples
 101 109 2283 101 651 7p2283 101 927 238p1025 109 651 1025 109 56p660 101p1362 651 927 101p1362 651 56p660 238p1025 1025 2283 238p1025 1025 7p2283 101p1362

ZC(X(6),L(3,520)) passes through these centers:
3, 6, 74, 1073, 3p2173, 3p2349, 20p2159, 74p2349

Collinear triples
 3 6 3p2173 3 74 3p2349 6 74 20p2159 1073 3p2349 20p2159 3p2173 3p2349 74p2349

ZC(X(6),L(3,523)) passes through these centers:
2, 4, 6, 30, 323, 19p2071, 48p1300, 63p1300, 255p1300

Collinear triples
 2 4 30 2 6 323 2 48p1300 63p1300 4 6 19p2071 4 63p1300 255p1300 6 48p1300 255p1300

ZC(X(30),L(3,523)) passes through these centers:
6, 13, 14, 30, 63p1300

Collinear triple
 6 13 14

ZC(X(190),L(4,6)) passes through these centers:
100, 107, 190, 823, 932, 43p823, 100p1714

Collinear triples
 100 107 823 100 190 100p1714 107 932 43p823 190 823 43p823

ZC(X(512),L(4,6)) passes through these centers:
19, 107, 512, 800, 1379, 1380, 6p821, 19p1020, 19p1021

Collinear triples
 19 512 19p1021 19 800 19p1020 107 800 6p821 107 19p1020 19p1021 800 1379 1380

ZC(X(523),L(4,6)) passes through these centers:
4, 107, 523, 1113, 1114, 162p1312

Collinear triples
 4 1113 1114 523 1113 162p1312

ZC(X(684),L(4,6)) passes through these centers:
107, 110, 132, 684, 1297, 110p920, 158p684

Collinear triples
 107 110 158p684 107 132 1297 110 684 110p920 132 684 158p684

ZC(X(112),L(4,8)) passes through these centers:
100, 107, 112, 648, 1783, 2p1897, 112p502, 191p648

Collinear triples
 100 107 2p1897 100 112 1783 112 648 191p648 648 1783 2p1897 2p1897 112p502 191p648

ZC(X(162),L(4,8)) passes through these centers:
100, 162, 811, 823, 1783, 2p1897, 112p1330, 648p1654, 1897p2248

Collinear triples
 100 162 1783 100 823 2p1897 162 811 648p1654 811 823 112p1330 811 1783 2p1897 1783 112p1330 1897p2248 2p1897 648p1654 1897p2248

ZC(X(112),L(4,9)) passes through these centers:
101, 107, 112, 648, 1897, 4p692, 112p1761, 162p1654, 1783p2248

Collinear triples
 101 107 1897 101 112 4p692 107 648 112p1761 112 648 162p1654 648 1897 4p692 1897 162p1654 1783p2248 4p692 112p1761 1783p2248

ZC(X(1459),L(4,9)) passes through these centers:
1, 109, 522, 1459, 1897, 27p836, 71p837, 109p1158

Collinear triples
 1 109 27p836 1 522 1459 1 71p837 109p1158 109 522 1897 109 1459 109p1158 1897 27p836 71p837

ZC(X(112),L(4,511)) passes through these centers:
99, 107, 112, 648, 2p811, 22p162, 66p811, 159, 823, 162p1670, 162p1671, 811p1676, 811p1677

Collinear triples
 99 107 2p811 99 112 648 107 112 159p823 107 648 22p162 112 162p1670 162p1671 648 66p811 159p823 2p811 22p162 66p811 2p811 162p1670 811p1677 2p811 162p1670 811p1676 22p162 811p1677 1783p2248

ZC(X(647),L(4,511)) passes through these centers:
2, 110, 647, 850, 1899, 2p811, 157p662

Collinear triples
 2 110 1899 2 647 850 110 647 157p662 110 850 2p811

ZC(X(4),L(4,512)) passes through these centers:
4, 98, 232, 264, 2p158, 4p1821, 4p1910, 4p1966, 92p1687, 92p1688, 98p1956, 158p401

Collinear triples
 4 98 4p1910 4 264 4p1966 4 2p158 158p401 4 92p1687 92p1688 98 232 98p1956 98 2p158 4p1821 232 264 2p158 264 4p1821 4p1910 4p1821 98p1956 158p401

ZC(X(39),L(4,512)) passes through these centers:
2, 4, 6, 39, 232, 248, 2p1733, 4p1821, 19p1235, 24p293, 38p98, 38p847, 311p1821, 427p1821, 427p1910

Collinear triples
 2 39 2p1733 2 232 19p1235 2 4p1821 427p1910 2 24p293 38p98 4 39 232 4 248 427p1910 4 2p1733 38p847 4 4p1821 38p98 6 248 211p1821 6 4p1821 427p1821 39 248 24p293 39 38p98 211p1821 39 427p1821 427p1910 248 4p1821 19p1235 4p1821 24p293 38p847 19p1235 38p98 427p1821 38p847 211p1821 427p1910

ZC(X(4),L(4,514)) passes through these centers:
4, 281, 1861, 1886, 92p103, 103p273

Collinear triples
 1 2 3 2 5 6

ZC(X(10),L(4,514)) passes through these centers:
2, 10, 75, 281, 1861, 2p1736, 92p103, 103p1441, 103p1826

Collinear triples
 2 10 2p1736 10 281 1861 75 92p103 103p1826 281 92p103 103p1441

ZC(X(516),L(4,514)) passes through these centers:
7, 281, 516, 673, 1861, 92p103

Collinear triples
 7 281 92p103 281 516 1861 673 1861 92p103

ZC(X(112),L(4,515)) passes through these centers:
107, 109, 112, 648, 653, 4p1415, 112p1762, 648p1046

Collinear triples
 107 109 653 107 648 112p1762 109 112 4p1415 112 648 648p1046 648 653 4p1415

ZC(X(163),L(4,515)) passes through these centers:
109, 162, 163, 653, 662, 4p1415, 162p1781

Collinear triples
 109 162 653 109 163 4p1415 162 163 162p1781 653 662 4p1415

ZC(X(112),L(4,516)) passes through these centers:
101, 107, 112, 648, 1897, 4p692, 112p1761, 162p1654, 1783p2248

Collinear triples
 101 107 1897 101 112 4p692 107 648 112p1761 112 648 162p1654 648 1897 4p692 1897 162p1654 1783p2248 4p692 112p1761 1783p2248

ZC(X(1459),L(4,516)) passes through these centers:
1, 109, 522, 1459, 1897, 27p836, 71p837, 109p1158

Collinear triples
 1 109 27p836 1 522 1459 1 71p837 109p1158 109 522 1897 109 1459 109p1158 1897 27p836 71p837

ZC(X(3),L(4,523)) passes through these centers:
3, 470, 471, 485, 486, 4p2349, 371p2349, 372p2349

Collinear triples
 3 470 471 3 371p2349 372p2349 485 4p2349 371p2349 486 4p2349 372p2349

ZC(X(4),L(4,523)) passes through these centers:
4, 470, 471, 1990, 4p2349

Collinear triple
 4 470 471

ZC(X(6),L(4,523)) passes through these centers:
2, 4, 6, 74, 186, 1990, 2p1725, 4p2159, 4p2349, 74p2166, 74p2349, 1138p2173, 1300p2159

Collinear triples
 2 4 186 2 6 2p1725 2 4p2159 4p2349 4 6 1990 4 74 4p2349 6 74 4p2159 74 186 74p2349 74 2p1725 74p2166 186 1990 1138p2173 186 4p2159 1300p2159 186 4p2349 74p2166 1990 4p2159 74p2166 1990 4p2349 74p2349 2p1725 4p2349 1300p2159

ZC(X(30),L(4,523)) passes through these centers:
13, 14, 30, 470, 471, 4p2349

Collinear triples
 13 470 4p2349 14 471 4p2349 30 470 471

ZC(X(140),L(4,523)) passes through these centers:
17, 18, 140, 470, 471, 4p2349

Collinear triple
 140 470 471

ZC(X(440),L(4,523)) passes through these centers:
226, 440, 470, 471, 1751, 4p2349, 950p2349

Collinear triples
 226 4p2349 950p2349 440 470 471

ZC(X(1865),L(4,523)) passes through these centers:
4, 281, 445, 860, 1865, 1990, 2p225, 4p2349, 1859p2349, 1865p2349

Collinear triples
 4 445 860 4 1865 1990 4 4p2349 1865p2349 281 860 1865 1865 1859p2349 1865p2349 2p225 4p2349 1859p2349

ZC(X(512),L(5,6)) passes through these centers:
216, 512, 925, 1379, 1380, 4p2168

Collinear triples
 216 925 4p2168 216 1379 1380

ZC(X(523),L(5,6)) passes through these centers:
3, 523, 847, 925, 1113, 1114

Collinear triples
 3 847 925 3 1113 1114

ZC(X(1),L(6,7)) passes through these centers:
651, 658, 664, 927, 2283, 2p919, 6p927, 56p874, 292p927, 673p927

Collinear triples
 651 664 56p874 651 2p919 6p927 658 664 2283 658 927 2p919 664 927 6p927 927 2283 673p927 927 56p874 292p927 2283 6p927 292p927 2p919 56p874 673p927

ZC(X(30),L(6,7)) passes through these centers:
30, 514, 516, 927

Collinear triple
 30 514 516

ZC(X(190),L(6,7)) passes through these centers:
100, 190, 664, 666, 927, 932, 2p927, 7p1025, 56p874, 85p1293, 190p1279, 192p927, 291p666, 294p666, 658p1376, 666p1743, 927p1280

Collinear triples
 100 190 190p1279 100 664 7p1025 100 927 2p927 190 664 56p874 190 666 666p1743 190 2p927 192p927 190 7p1025 658p1376 664 666 927 666 2p927 658p1376 666 190p1279 294p666 927 932 192p927 927 7p1025 294p666 927 56p874 291p666 927 85p1293 666p1743 927 190p1279 927p1280 932 56p874 658p1376 2p927 56p874 294p666 7p1025 192p927 291p666 7p1025 666p1743 927p1280 85p1293 190p1279 658p1376

ZC(X(511),L(6,7)) passes through these centers:
511, 514, 516, 927

Collinear triple
 511 514 516

ZC(X(512),L(6,7)) passes through these centers:
512, 514, 516, 927, 1379, 1380, 65p919

Collinear triples
 512 514 516 514 927 65p919

ZC(X(513),L(6,7)) passes through these centers:
241, 513, 514, 516, 927, 1381, 1382, 2p919, 2p1381, 2p1382, 6p927, 7p513, 7p1279, 7p1381, 7p1382, 105p673, 105p1280, 105p1280, 885p1381, 885p1382

Collinear triples
 241 514 7p513 241 927 105p673 241 1381 1382 241 2p919 105p1280 241 2p1321 7p1382 241 2p1382 7p1381 513 514 516 513 2p919 6p927 513 7p513 7p1279 514 927 6p927 514 885p1381 885p1382 516 2p919 105p673 927 2p919 7p513 927 7p1279 105p1280 927 7p1381 885p1382 927 7p1382 885p1381 2p919 2p1381 885p1382 2p919 2p1382 885p1381 2p1381 2p1382 7p1279 6p927 7p1279 105p673

ZC(X(522),L(6,7)) passes through these centers:
514, 516, 522, 666, 927, 666p1743, 1111p1293

Collinear triples
 514 516 522 514 666 927 522 666 666p1743 927 666p1743 1111p1293

ZC(X(523),L(6,7)) passes through these centers:
514, 516, 523, 693, 857, 927, 1113, 1114, 37p927, 42p927, 226p1113, 226p1114

Collinear triples
 514 516 523 514 693 857 514 927 37p927 523 37p927 42p927 693 927 42p927 857 1113 1114

ZC(X(526),L(6,7)) passes through these centers:
74, 110, 514, 516, 526, 927, 1442, 35p885, 105p758, 500p673, 658p2245

Collinear triples
 110 105p758 500p673 514 516 526 514 35p885 105p758 516 1442 658p2245 526 35p885 500p673 927 1442 105p758 927 35p885 658p2245

ZC(X(651),L(6,7)) passes through these centers:
651, 658, 664, 927, 2283, 2p919, 6p927, 56p874, 292p927, 673p927

Collinear triples
 651 664 56p874 651 2p919 6p927 658 664 2283 658 927 2p919 664 927 6p927 927 2283 673p927 927 56p874 292p927 2283 6p927 292p927 2p919 56p874 673p927

ZC(X(659),L(6,7)) passes through these centers:
57, 100, 105, 659, 927, 1447, 2p1027, 58p693, 59p812, 105p239, 190p1279, 812p1462

Collinear triples
 57 927 105p239 57 1447 190p1279 57 2p1027 812p1462 100 105 105p239 100 659 190p1279 100 927 812p1462 105 659 2p1027 105 927 1447 659 1447 812p1462 927 2p1027 59p812 1447 59p812 105p239 2p1027 58p693 105p239

ZC(X(676),L(6,7)) passes through these centers:
2, 105, 108, 676, 927, 2p1027, 7p513, 7p516, 7p1279, 190p1456, 516p1462, 666p910

Collinear triples
 2 105 666p910 2 927 516p1462 2 7p1279 190p1456 105 108 516p1462 105 676 2p1027 105 927 7p516 676 7p513 7p1279 676 190p1456 666p910 927 2p1027 190p1456 927 7p513 666p910 2p1027 7p513 516p1462 7p516 190p1456 516p1462

ZC(X(684),L(6,7)) passes through these centers:
110, 684, 693, 927, 1214, 1297, 105p1959, 108p1959, 520p673, 1814p2294

Collinear triples
 110 105p1959 1814p2294 684 520p673 1814p2294 693 105p1959 520p673 927 1214 105p1959 927 108p1959 520p673

ZC(X(804),L(6,7)) passes through these centers:
98, 99, 514, 516, 804, 927, 7p171, 105p740, 171p885, 239p1020, 740p961

Collinear triples
 514 516 804 514 105p740 171p885 516 7p171 239p1020 927 7p171 105p740 927 171p885 239p1020

ZC(X(900),L(6,7)) passes through these centers:
7, 100, 104, 514, 516, 655, 673, 885, 900, 927, 2p1022, 2p1319, 7p2254, 44p658, 44p673, 44p927, 190p1279, 678p927, 927p1647

Collinear triples
 7 516 44p658 7 673 44p927 7 885 927p1647 7 900 7p2254 7 927 44p673 7 2p1319 190p1279 100 673 44p673 100 900 190p1279 100 927 927p1647 100 7p2254 44p673 514 516 900 514 885 44p673 514 927 44p927 514 2p1319 7p2254 655 885 678p927 673 885 900 673 927 2p1319 885 927 44p658 900 44p927 678p927 927 2p1022 678p927 2p1319 44p673 678p927

ZC(X(918),L(6,7)) passes through these centers:
508, 514, 516, 664, 918, 927, 7p876, 56p874, 85p103

Collinear triples
 514 516 918 514 664 927 516 927 85p103 664 918 56p874 927 2p876 56p874

ZC(X(926),L(6,7)) passes through these centers:
1, 101, 103, 514, 516, 926, 927, 7p2283, 8p884, 105p1212, 241p2346

Collinear triples
 1 101 105p1212 1 514 8p884 1 516 7p2283 101 514 927 103 516 927 514 516 926 926 8p884 105p1212 927 7p2283 8p884 927 105p1212 241p2346

ZC(X(1633),L(6,7)) passes through these centers:
100, 653, 927, 934, 1633, 2p919, 2p1633, 6p927, 7p1292, 7p1633, 112p857, 190p1279, 516p934, 614p883, 614p927, 666p1438, 666p1473, 666p2082

Collinear triples
 100 927 614p927 100 934 614p883 100 1633 190p1279 653 927 666p1473 653 1633 112p857 653 2p919 666p2082 653 2p1633 516p934 653 7p1292 614p883 927 934 666p2082 927 2p919 7p1633 927 2p1633 6p927 927 614p883 666p1438 934 1633 516p934 934 6p927 666p1473 934 7p1633 112p857 1633 2p919 6p927 1633 614p927 666p2082 2p1633 7p1633 614p883 6p927 7p1292 614p927 7p1292 7p1633 190p1299 7p1633 614p927 666p1473 190p1279 666p1438 666p2082 516p934 614p927 666p1438

ZC(X(2284),L(6,7)) passes through these centers:
101, 664, 813, 927, 2284, 56p874, 100p509

Collinear triples
 101 664 927 664 2284 56p874 813 927 56p874

ZC(X(2284),L(6,8)) passes through these centers:
101, 190, 813, 2284, 99p2295, 100p1220, 572p1018, 1026p1220, 2284p2298

Collinear triples
 101 2284 572p1018 101 100p1220 1026p1220 190 2284 99p2295 190 100p1220 2284p2298

ZC(X(101),L(6,10)) passes through these centers:
101, 190, 835, 1897, 31p835, 48p835, 100p1011, 835p2282

Collinear triples
 101 31p835 48p835 190 835 31p835 190 1897 100p1011 835 1897 48p835 835 100p1011 835p2282

ZC(X(190),L(6,10)) passes through these centers:
101, 190, 835, 932, 2p835, 100p1698, 192p835

Collinear triples
 100 190 100p1698 100 835 2p835 190 2p835 192p835 835 932 192p835

ZC(X(190),L(6,11)) passes through these centers:
100, 190, 929, 932, 2p929, 192p929

Collinear triples
 100 929 2p929 190 2p929 192p929 929 932 192p929

ZC(X(351),L(6,13)) passes through these centers:
110, 111, 351, 476, 110p1749, 351p2166

Collinear triples
 110 351 110p1749 110 476 351p2166

ZC(X(511),L(6,13)) passes through these centers:
30, 476, 511, 523

Collinear triple
 30 511 523

ZC(X(512),L(6,13)) passes through these centers:
30, 476, 512, 523, 1379, 1380, 4p661, 4p2315, 31p476, 48p476, 393p2314, 1989p2159

Collinear triples
 30 476 1989p2159 30 512 523 476 523 31p476 476 4p661 48p476 512 4p661 393p2314 512 31p476 48p476 523 4p661 4p2315 1379 1380 4p2315 4p2315 31p476 1989p2159

ZC(X(523),L(6,13)) passes through these centers:
30, 476, 523, 1113, 1114, 74p2166

Collinear triples
 30 476 74p2166 30 1113 1114

ZC(X(526),L(6,13)) ( = Z(X(476),X(1)), [Tixier cubic, K130] passes through these centers:
1, 30, 74, 110, 476, 523, 526, 110p1749, 1109p1291 See Z(X(476),X(1)) for collinearities.

Collinear triples
 30 74 476 30 523 526 110 476 523 110 526 110p1749 476 110p1749 1109p1291

ZC(X(690),L(6,13)) passes through these centers:
13, 14, 30, 476, 523, 690, 13p896, 14p896, 111p1577, 395p662, 396p662, 476p896

Collinear triples
 13 14 476p896 13 476 14p896 13 13p896 395p662 14 476 13p896 14 14p896 396p662 30 523 690 476 523 476p896 690 395p662 396p662

ZC(X(804),L(6,13)) passes through these centers:
30, 98, 99, 476, 523, 804, 79p171, 80p238, 476p1580

Collinear triples
 30 523 804 476 523 476p1580 476 79p171 80p238

ZC(X(826),L(6,13)) passes through these centers:
30, 476, 523, 826, 850, 38p476, 476p1964

Collinear triples
 30 523 826 476 523 38p476 476 850 476p1964 826 38p476 476p1964

ZC(X(900),L(6,13)) passes through these centers:
30, 79, 100, 104, 476, 523, 900, 476p1647, 514p2341, 519p759, 553p2341

Collinear triples
 30 523 900 79 476 519p759 79 476p1647 514p2341 100 476 476p1647 100 519p759 553p2341 523 514p2341 519p759 900 514p2341 553p2341

ZC(X(926),L(6,13)) passes through these centers:
30, 101, 103, 476, 523, 926, 9p2160, 79p2283, 241p2341, 650p2341, 942p2341

Collinear triples
 30 523 926 30 9p2160 79p2283 101 241p2341 942p2341 476 9p2160 241p2341 476 79p2283 650p2341 523 241p2341 650p2341 926 650p2341 942p2341

ZC(X(1),L(30,511)) ( = Z(X(2),X(6)) passes through these centers:
1, 2, 7, 9, 366, 1489, 2p1419, 2p2067, 7p2066, 8p2362, 92p2066, 173p1489, 364p366
See Z(X(2),X(6)) for collinearities.

ZC(X(3),L(30,511)) ( = Z(X(2),X(19)) passes through these centers:
2, 3, 6, 69, 485, 486, 2p2066, 2p2067, 7p2066, 8p2067, 48p491, 48p492
See Z(X(2),X(19)) for collinearities.

ZC(X(6),L(30,511)) passes through these centers:
1, 2, 3, 4, 6, 9, 57, 223, 282, 1073, 1249, 1249, 84p1490, 204p1032, 221p1034, 1073p1712, the Thomson cubic.
See Z(X(2),X(1)) for collinearities

ZC(X(9),L(30,511)) ( = Z(X(2),X(56)) passes through these centers:
1, 2, 8, 9, 188, 236, 8p289, 8p1743, 514p1293
See Z(X(2),X(56)) for collinearities.

ZC(X(10),L(30,511)) passes through these centers:
2, 10, 37, 75, 2p192, 10p2162

Collinear triples
 2 37 75 2 2p192 10p2162 10 75 2p192

ZC(X(11),L(30,511)) passes through these centers:
2, 11, 650, 693, 8p2350, 508p650, 1111p1621

Collinear triples
 2 650 693 2 8p2350 1111p1621 11 693 1111p1621

ZC(X(15),L(30,511)) passes through these centers:
2, 3, 6, 13, 15, 18, 298, 470, 1082, 2p1094,2p1250, 15p1081, 16p2349, 61p2167, 283p1832, 303p2151, 533p2154, 621p2151

Collinear triples
 2 3 470 2 6 298 2 13 2p1094 2 18 303p2151 2 1082 283p1832 2 2p1250 15p1081 3 6 15 3 18 533p2154 3 1082 15p1081 3 2p1094 16p2349 3 61p2167 303p2151 6 2p1250 283p1832 13 18 621p2151 15 2p1094 303p2151 298 470 621p2151 298 2p1094 533p2154 1082 2p1250 303p2151 2p1094 15p1081 283p1832

ZC(X(16),L(30,511)) passes through these centers:
2, 3, 6, 14, 16, 17, 299, 471, 559, 2p1095, 9p559, 15p2349, 16p554, 62p2167, 283p1833, 302p2152, 532p2153, 617p2154, 622p2152

Collinear triples
 2 3 471 2 6 299 2 14 2p1095 2 17 302p2152 2 559 283p1833 2 9p559 16p554 3 6 16 3 17 532p2153 3 559 16p554 3 2p1095 15p2349 3 62p2167 302p2152 6 9p559 283p1833 14 16 617p2154 14 17 622p2152 16 2p1095 302p2152 299 471 622p2152 299 2p1095 532p2153 299 15p2349 617p2154 559 9p559 302p2152 2p1095 16p554 283p1833

ZC(X(23),L(30,511)) passes through these centers:
2, 23, 111, 251, 23p1930, 316p896

Collinear triples
 111 111 316p896 251 251 23p1930 23 23p1930 316p896

ZC(X(30),L(30,511)) passes through these centers:
2, 13, 14, 30, 1494, 6p1099, 298p2173, 299p2173, 617p2154

Collinear triples
 2 13 298p2173 2 14 299p2173 2 1494 6p1099 13 14 6p1099 14 30 617p2154 30 298p2173 299p2173 1494 298p2173 617p2154

ZC(X(32),L(30,511)) ( = Z(X(2),X(75)) passes through these centers:
2, 3, 6, 25, 32, 66, 206, 1676, 1677, 19p159, 31p1670, 31p1671
See Z(X(2),X(75)) for collinearities.

ZC(X(37),L(30,511)) ( = Z(X(2),X(58)) passes through these centers:
1, 2, 9, 10, 37, 226, 281, 1214, 2p2331, 10p1433, 20p1903, 40p2184, 1490p2184
See Z(X(2),X(58)) for collinearities.

ZC(X(39),L(30,511)) ( - Z(X(2),X(82)) passes through these centers:
2, 3, 6, 39, 141, 427, 38p1342, 38p1343, 63p66, 427p2172, 1370p2156
See Z(X(2),X(82)) for collinearities.

ZC(X(44),L(30,511)) ( = Z(X(2),X(106)) passes through these centers:
1, 2, 9, 44, 80, 88, 214, 519, 2p678, 2p1319, 2p2342, 57p1145
See Z(X(2),X(106)) for collinearities.

ZC(X(45),L(30,511)) passes through these centers:
1, 2, 9, 45, 1000, 2p45, 2p2099, 44p80, 80p517

Collinear triples
 1 2 2p45 1 9 45 1 2p2099 80p517 2 9 2p2099 9 2p45 44p80 45 44p80 80p517

ZC(X(50),L(30,511)) passes through these centers:
2, 3, 6, 50, 186, 323, 323p1749, 661p1291

Collinear triples
 2 3 186 2 6 323 2 323p1749 661p1291 3 6 50 50 323 323p1749

ZC(X(52),L(30,511)) passes through these centers:
2, 3, 6, 52, 467, 47p311

Collinear triples
 2 3 467 2 6 47p311 3 6 52

ZC(X(53),L(30,511)) passes through these centers:
2, 4, 5, 53, 216, 1249, 2p158, 5p2184, 92p1075

Collinear triples
 2 4 5 2 216 2p158 2 1249 5p2184 4 53 1249 5 53 216 53 2p158 92p1075

ZC(X(56),L(30,511)) ( = Z(X(2),X(9)) passes through these centers:
2, 56, 478, 509, 2362, 2p2067, 7p2066, 225p1806
See Z(X(2),X(9) for collinearities.

ZC(X(57),L(30,511)) ( = Z(X(2),X(55)) passes through these centers:
2, 57, 174, 189, 223, 557, 558, 1659, 2p2067, 7p2066, 57p175, 57p176, 164p174, 174p505, 273p2066
See Z(X(2),X(55)) for collinearities.

ZC(X(58),L(30,511)) ( = Z(X(2),X(37)) passes through these centers:
2, 3, 6, 27, 58, 86, 2248, 81p1654
See Z(X(2),X(55)) for collinearities.

ZC(X(61),L(30,511)) passes through these centers:
2, 3, 6, 61, 302, 473

Collinear triples
 2 3 473 2 6 302 3 6 61

ZC(X(62),L(30,511)) passes through these centers:
2, 3, 6, 62, 303, 472

Collinear triples
 2 3 472 2 6 303 3 6 62

ZC(X(72),L(30,511)) passes through these centers:
1, 2, 9, 72, 306, 307, 2p1751, 63p209, 306p1724, 307p1754

Collinear triples
 1 2 306 1 9 72 1 2p1751 307p1754 2 9 307 2 2p1751 63p209 9 2p1751 306p1724 72 306 306p1724 72 307 307p1754 306 307 63p209

ZC(X(115),L(30,511)) passes through these centers:
2, 115, 523, 1312, 1313, 338p1822, 338p1823, 661p1113, 661p1114, 1109p1379, 1109p1380, 1379p1577, 1380p1577

Collinear triples
 2 1312 1313 2 338p1822 661p1114 2 338p1823 661p1113 2 1109p1379 1380p1577 2 1109p1380 1379p1577 115 1312 661p1113 115 1313 661p1114 115 338p1822 338p1823 115 1109p1379 1109p1380 523 1312 338p1822 523 1313 338p1823 523 661p1113 661p1114 523 1379p1577 1380p1577

ZC(X(173),L(30,511)) passes through these centers:
2, 173, 174, 188, 236, 2089, 164p174, 168p2089, 177p1128, 236p505

Collinear triples
 2 174 236 2 188 2089 2 164p174 236p505 173 174 164p174 173 188 236 173 2089 168p2089 2089 164p174 177p1128

ZC(X(173),L(30,511)) passes through these centers:
2, 3, 6, 182, 183, 458, 98p1966, 290, 1955

Collinear triples
 2 3 458 2 6 183 3 6 182 3 183 98p1966 6 458 290p1955 182 98p1966 290p1955

ZC(X(187),L(30,511)) [Droussent medial cubic, K043] passes through these centers:
2, 3, 6, 67, 111, 187, 468, 524, 1560, 9p1366, 23p896, 63p1177

Collinear triples
 2 3 468 2 6 524 2 67 23p896 2 111 9p1366 2 1560 63p1366 3 6 187 3 67 9p1366 3 23p896 63p1177 6 67 1560 6 111 23p896 187 468 1560 187 524 9p1366 468 524 23p896

ZC(X(213),L(30,511)) ( = Z(X(2),X(86)) passes through these centers:
1, 2, 9, 42, 213, 1400, 1400p1764
See Z(X(2),X(86)) for collinearities.

ZC(X(216),L(30,511)) passes through these centers:
2, 3, 5, 6, 216, 343, 2165, 47p343, 52p921, 68p920

Collinear triples
 2 3 5 2 6 343 2 2165 47p343 2 52p921 68p920 3 6 216 3 343 68p920 5 6 2165 5 47p343 52p921 216 343 47p343 216 2165 68p920

ZC(X(219),L(30,511)) ( = Z(X(2),X(34)) passes through these centers:
1, 2, 9, 63, 78, 219, 8p2164, 9p224, 12p1789, 21p35, 46p63, 72p1780
See Z(X(2),X(34)) for collinearities.

ZC(X(238),L(30,511)) passes through these centers:
1, 2, 9, 86, 238, 239, 292, 673, 893, 1447, 1929, 1966, 2238, 6p1281, 81p1654, 238p350, 238p518, 239p1757, 335p2112, 366p2068, 366p2069, 673p1282, 740p2248

Collinear triples
 1 2 239 1 9 238 1 86 1966 1 292 238p518 1 1929 6p1281 2 9 1447 2 86 2238 2 292 238p350 2 673 238p518 2 893 1966 2 1929 239p1757 2 81p1654 740p2248 2 366p2068 366p2069 9 673 238p250 9 893 2238 86 238 81p1654 86 292 239p1757 86 673 6p1281 238 239 239p1757 238 292 335p2112 238 673 673p1282 238 1966 238p250 238 2238 238p518 239 673 335p2112 239 893 81p1654 239 1447 6p1281 239 2238 238p250 292 893 6p1281 292 1447 673p1282 1447 1966 238p518 1929 2238 335p2112 1929 81p1654 238p250 1966 239p1757 740p2248 2238 6p1281 740p2248

ZC(X(269),L(30,511)) passes through these centers:
2, 7, 57, 223, 269, 1440, 2p1419, 7p365

Collinear triples
 2 7 57 2 223 1440 7 269 2p1419 57 223 269

ZC(X(284),L(30,511)) ( - Z(X(2),X(65)) passes through these centers:
2, 3, 6, 29, 284, 333, 6p1247, 284p1762, 314p2305
See Z(X(2),X(65) for collinearities.

ZC(X(371),L(30,511)) passes through these centers:
2, 3, 6, 371, 492, 1585

Collinear triples
 2 3 1585 2 6 492 3 6 371

ZC(X(372),L(30,511)) passes through these centers:
2, 3, 6, 372, 1586

Collinear triples
 2 3 1586 3 6 372

ZC(X(393),L(30,511)) passes through these centers:
2, 4, 278, 281, 393, 1249, 2p207, 2p2331, 4p84, 4p1712, 4p2184, 19p1032, 19p1034

Collinear triples
 2 278 281 2 1249 4p2184 2 2p207 19p1034 2 2p2331 4p84 2 4p1712 19p1032 4 278 2p2331 4 393 1249 4 2p207 4p84 4 4p1712 4p2184 278 393 2p207 278 4p84 4p1712 281 393 2p2331 281 1249 4p84 281 2p207 4p2184 281 4p1712 19p1034 1249 2p2331 19p1034 2p207 2p2331 19p1032

ZC(X(397),L(30,511)) passes through these centers:
2, 4, 17, 397, 465, 1249, 465p2184

Collinear triples
 2 4 465 2 1249 465p2184 4 397 1249

ZC(X(398),L(30,511)) passes through these centers:
2, 4, 18, 398, 466, 1249, 466p2184

Collinear triples
 2 4 465 2 1249 466p2184 4 398 1249

ZC(X(440),L(30,511)) passes through these centers:
2, 226, 306, 440, 1214, 1751, 2p950, 2p1104, 21p440, 306p1724

Collinear triples
 2 226 21p440 2 306 2p1104 2 1214 2p950 226 440 1214 226 1751 2p1104 306 440 306p1724 1751 2p950 306p1724 2p950 2p1104 21p440

ZC(X(511),L(30,511)) passes through these centers:
2, 3, 6, 290, 297, 325, 511, 694, 2009, 2010, 3p1956, 9p1355, 237p1926, 240p401, 1687p1959, 1688p1959

Collinear triples
 2 3 297 2 6 325 2 290 9p1355 2 694 237p1926 2 2009 1687p1959 2 2010 1688p1959 2 3p1956 240p401 3 6 511 3 290 237p1926 3 3p1956 2p1355 6 290 240p401 6 694 9p1355 297 325 9p1355 297 511 240p401 325 511 237p1926 511 1687p1959 1688p1959 2009 2010 9p1355

ZC(X(513),L(30,511)) passes through these centers:
2, 513, 668, 1015, 1978p1979

Collinear triples
 2 668 1015 513 668 1978p1979

ZC(X(514),L(30,511)) passes through these centers:
2, 190, 514, 1086, 190p1054

Collinear triples
 2 190 1086 190 514 190p1054

ZC(X(516),L(30,511)) passes through these centers:
2, 7, 516, 673, 2p1419, 8p910, 9p1360, 75p103, 516p518, 673p1282

Collinear triples
 2 7 8p910 2 673 516p518 2 9p1360 75p103 7 516 2p1419 7 673 9p1360 516 673 673p1282 516 8p910 516p518 2p1419 75p103 516p518 8p910 75p103 673p1282

ZC(X(518),L(30,511)) passes through these centers:
1, 2, 9, 291, 518, 2p241, 2p518, 2p673, 9p1362, 239p518, 335p2110

Collinear triples
 1 2 2p518 1 9 518 1 291 9p1362 2 9 2p241 2 291 239p518 2 2p673 9p1362 9 2p673 239p518 291 518 335p2110 518 2p518 239p518 2p241 2p518 9p1362 2p578 2p673 335p2110

ZC(X(523),L(30,511)) passes through these centers:
2, 99, 115, 523, 148p662

Collinear triples
 2 99 115 99 523 148p662

ZC(X(532),L(30,511)) passes through these centers:
2, 299, 396, 532, 618, 94p1095

Collinear triples
 2 299 396 2 618 94p1095 396 532 618

ZC(X(533),L(30,511)) passes through these centers:
2, 298, 395, 533, 619, 94p1094

Collinear triples
 2 298 395 2 619 94p1094 395 533 619

ZC(X(570),L(30,511)) passes through these centers:
2, 3, 6, 54, 570, 1209, 1594, 19p1238, 48p93

Collinear triples
 2 3 1594 2 6 19p1238 2 54 1209 3 6 570 6 1209 48p93 1209 1594 19p1238

ZC(X(571),L(30,511)) passes through these centers:
2, 3, 6, 24, 70, 571, 1993, 26p47, 49p2190

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

ZC(X(577),L(30,511)) passes through these centers:
2, 3, 6, 68, 394, 577, 1147, 9p921, 63p155, 547p1820

Collinear triples
 2 6 394 2 68 1147 2 6p921 63p155 3 6 577 3 68 63p155 3 394 1147 6 68 547p1820 6 1147 6p921 394 577 63p155

ZC(X(594),L(30,511)) passes through these centers:
2, 10, 37, 321, 594, 596, 595p1089

Collinear triples
 2 37 321 2 596 595p1089 10 37 594 10 321 595p1089

ZC(X(698),L(30,511)) passes through these centers:
2, 76, 698, 1916, 6p2227, 75p194, 75p699, 385p2227, 698p2227

Collinear triples
 2 76 6p2227 2 1916 385p2227 2 75p699 698p2227 76 698 75p194 76 1916 698p2227 698 6p2227 385p2227 75p194 75p699 385p2227

ZC(X(726),L(30,511)) passes through these centers:
2, 75, 335, 726, 1575, 2p192, 75p727, 87p1575, 238p726, 335p2108, 726p1575

Collinear triples
 2 75 1575 2 335 238p726 2 2p192 87p1575 2 75p727 726p1575 75 335 726p1575 75 626 2p192 335 726 335p2108 726 1575 238p726 1575 75p727 335p2108 1575 87p1575 726p1575 2p192 75p727 238p726

ZC(X(732),L(30,511)) passes through these centers:
2, 39, 141, 385, 732, 2p1966, 75p733, 732p2236, 1966p2076

Collinear triples
 2 39 2p1966 2 141 385 2 75p733 732p2236 39 141 732 39 75p733 1966p2076 385 732 1966p2076 385 2p1966 732p2236

ZC(X(740),L(30,511)) passes through these centers:
2, 10, 37, 239, 350, 740, 2p2107, 10p1929, 28p337, 239p1757, 740p2238

Collinear triples
 2 10 239 2 37 350 2 10p1929 239p1757 2 28p337 740p2238 10 37 740 10 10p1929 239p1757 37 2p2107 740p2238 37 28p337 239p1757 239 350 740p2238 239 740 239p1757

ZC(X(960),L(30,511)) passes through these centers:
1, 2, 9, 314, 960, 2092, 2p960, 7p960, 8p893, 69p444, 256p1045, 960p1999

Collinear triples
 1 2 2p960 1 9 960 1 314 69p444 2 9 7p960 2 314 2092 2 8p893 69p444 9 314 960p1999 9 2092 8p893 314 7p960 256p1045 960 2p960 960p1999 960 7p960 69p444 960 8p893 256p1045 2092 2p960 7p960

ZC(X(984),L(30,511)) passes through these centers:
1, 2, 9, 75, 984, 2276, 2p192, 2p984, 2p1002, 7p984, 10p893, 87p2276, 291p518, 291p1575, 984p1001

Collinear triples
 1 2 2p984 1 9 984 1 2276 291p518 2 9 7p984 2 75 2276 2 2p192 87p2276 2 2p1002 984p1001 9 2276 10p983 9 87p2276 291p1575 75 984 2p192 75 2p984 291p1575 984 2276 284p1001 984 291p518 291p1575 2p192 2p984 10p893 2p192 7p984 291p518

ZC(X(984),L(30,511)) passes through these centers:
1, 2, 9, 79, 81, 553, 110, 1125, 1213, 2p501, 35p1125, 267p1213

Collinear triples
 1 2 1125 1 9 1100 1 81 35p1125 2 9 553 2 79 35p1125 2 81 1213 2 2p501 267p1213 9 79 1213 79 1125 2p501 81 1100 2p501 553 1125 35p1125 1100 1125 1213 1213 35p1125 267p1213

ZC(X(1104),L(30,511)) passes through these centers:
1, 2, 9, 28, 440, 1104, 2p1104, 7p1104

Collinear triples
 1 2 2p1104 1 9 1104 2 9 7p1104 2 28 440 440 1104 2p1104

ZC(X(1107),L(30,511)) passes through these centers:
1, 2, 9, 256, 274, 1107, 2p1107, 7p1107, 10p1197, 86p1655, 894p1107

Collinear triples
 1 2 2p1107 1 9 1107 1 274 894p1107 2 9 7p1107 2 256 894p1107 2 274 10p1197 9 256 10p1197 256 2p1107 86p1655 274 1107 86p1655 1107 2p1107 10p1197 2p1107 7p1107 894p1107

ZC(X(1108),L(30,511)) passes through these centers:
1, 2, 9, 84, 278, 1108, 1210, 2p207, 7p1108, 40p1210, 63p1864

Collinear triples
 1 2 1210 1 9 1108 1 278 40p1210 2 9 7p1108 2 84 40p1210 2 278 63p1864 9 84 63p1864 84 1210 2p207 278 1108 2p207 1108 1210 63p1864 1210 7p1108 40p1210

ZC(X(1147),L(30,511)) passes through these centers:
2, 317, 371, 372, 577, 1147, 48p491, 48p492

Collinear triples
 2 317 577 2 371 48p491 2 372 48p492 371 372 577 1147 48p491 48p492

ZC(X(1212),L(30,511)) passes through these centers:
1, 2, 9, 142, 277, 1212, 2p1212, 344p2293

Collinear triples
 1 2 2p1212 1 9 1212 1 142 277 2 9 142 2 277 344p2293 1212 2p1212 344p2293

ZC(X(1333),L(30,511)) 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

ZC(X(1418),L(30,511)) passes through these centers:
2, 57, 142, 223, 279, 1212, 1418, 7p2124, 84p142, 354p2125

Collinear triples
 2 57 142 2 223 84p142 2 279 1212 2 7p2124 354p2125 57 223 1418 142 1212 1418 279 1418 7p2124

ZC(X(1427),L(30,511)) passes through these centers:
2, 57, 223, 226, 278, 1214, 1427, 2184, 2p207, 7p1903, 20p1427, 208p1032

Collinear triples
 2 57 226 2 223 7p1903 2 278 1214 2 2184 20p1427 57 223 1427 57 278 20p1427 57 2p207 7p1903 223 226 278 223 1214 2184 223 20p1427 208p1032 226 1214 1427 226 2184 2p207 226 7p1903 20p1427 278 1427 2p207 1214 2p207 208p1032

ZC(X(1465),L(30,511)) passes through these centers:
2, 57, 223, 908, 1465, 2006, 2p102, 36p908, 84p908, 515p1465

Collinear triples
 2 57 908 2 223 84p908 2 2006 36p908 2 2p102 515p1465 57 223 1465 57 2006 515p1465 223 2p102 36p908 908 1465 36p908 908 84p908 515p1465

ZC(X(1503),L(30,511)) passes through these centers:
2, 4, 98, 441, 1249, 1503, 63p132, 75p1297, 147p1910, 441p2184, 1503p2312

Collinear triples
 2 4 441 2 98 63p132 2 1249 441p2184 2 75p1297 1503p2312 4 98 1503p2312 4 1249 1503 98 1503 147p1910 441 1503 63p132 441 75p1297 147p1910 441 441p2184 1503p2312 1249 63p132 75p1297

ZC(X(1513),L(30,511)) passes through these centers:
2, 98, 132, 232, 262, 325, 1513, 147p1910, 237p1926, 336p1513, 1513p1910, 1513p1959

Collinear triples
 2 98 1513p1959 2 232 336p1513 2 325 1513p1910 98 132 1513p1910 98 1513 147p1910 98 237p1926 336p1513 132 232 1513 232 325 1513p1959 325 1513 237p1926

ZC(X(1691),L(30,511)) passes through these centers:
2, 3, 6, 83, 98, 171, 238, 385, 419, 1429, 1691, 2329, 2p1927, 6p1281, 6p2236, 31p695, 147p1910, 237p1966, 384p1966, 385p1966, 1966p2076

Collinear triples
 2 3 419 2 6 385 2 83 6p2236 2 98 237p1966 2 171 238 2 1429 2329 2 2p1927 385p1966 2 31p695 384p1966 3 6 1691 3 98 385p1966 3 171 1429 3 6p2236 31p695 6 83 384p1966 6 238 2329 6 2p1927 237p1966 83 2p1927 1966p2076 98 1691 147p1910 98 2329 6p1281 171 2329 6p2236 171 2p1927 6p1281 238 1429 237p1966 238 1691 6p1281 385 1429 6p1281 385 1691 1966p2076 385 6p2236 385p1966 419 2p1927 147p1910 419 237p1966 384p1966 1691 6p2236 237p1966 1691 384p1966 385p1966

ZC(X(1692),L(30,511)) passes through these centers:
2, 3, 6, 114, 230, 460, 1692, 1976

Collinear triples
 2 3 460 2 6 230 2 114 1967 3 6 1692 114 230 1692

ZC(X(1834),L(30,511)) passes through these centers:
2, 4, 10, 37, 440, 1249, 1834, 2p1104, 2p1427, 7p950, 8p2264, 440p2184

Collinear triples
 2 4 440 2 10 2p1104 2 37 7p950 2 1249 440p2184 2 2p1427 8p2264 4 10 8p2264 4 2249 1834 10 37 1834 37 440 2p1427 440 1834 2p1104 440 8p2264 440p2184 1249 2p1104 2p1427 2p1104 7p950 8p2264

ZC(X(1901),L(30,511)) passes through these centers:
2, 4, 226, 1214, 1249, 1901, 2p1903, 21p1901, 29p1901, 63p1901, 1817p1901

Collinear triples
 2 4 63p1901 2 226 21p1901 2 1214 29p1901 2 2p1903 1817p1901 4 226 1817p1901 4 1249 1901 226 1214 1901 1214 2p1903 63p1901 1249 2p1903 21p1901 1901 21p1901 63p1901 21p1901 29p1901 1817p1901

ZC(X(1921),L(30,511)) passes through these centers:
2, 75, 274, 334, 350, 1921, 1966, 2p192, 2p257, 2p740, 8p673, 75p2068, 75p2069, 76p727, 86p1655, 87p350, 238p350, 239p726, 241p350, 310p2107

Collinear triples
 2 75 350 2 274 2p740 2 334 238p350 2 1966 2p257 2 2p192 87p350 2 8p673 241p350 2 75p2068 75p2069 2 76p727 239p726 75 274 1966 75 334 239p726 75 1921 2p192 75 8p673 238p350 274 1921 86p1655 334 2p192 241p350 350 1966 241p350 350 2p257 86p1655 350 2p740 238p350 1921 1966 238p350 1921 2p740 239p726 1966 87p350 239p726 2p192 2p257 7p740 2p192 76p727 238p350 86p1655 238p350 310p2107

ZC(X(1990),L(30,511)) passes through these centers:
2, 4, 30, 133, 1249, 1294, 1990, 4p2349, 6p1099, 30p2184

Collinear triples
 2 4 30 2 133 1249 2 1249 30p2184 2 4p2349 6p1099 4 133 4p2349 4 1249 1990 30 133 30p2184 30 1990 6p1099 1249 1249 6p1099

ZC(X(2088),L(30,511)) passes through these centers:
2, 3, 6, 2088, 74p661, 75p2088, 92p2088, 129p371, 526p1725, 656p1300, 1109p1379, 1109p1380, 1511p1577

Collinear triples
 2 3 92p2088 2 6 75p2088 2 74p661 1511p1577 2 526p1725 656p1300 3 6 2088 3 656p1300 1511p1577 6 74p661 526p1725 2088 75p2088 1511p1577 2088 1109p1379 1109p1380 75p2088 92p2088 526p1725

ZC(X(2092),L(30,511)) passes through these centers:
2, 3, 6, 10, 37, 65, 429, 960, 1193, 1211, 2092, 2p1193, 65p573, 197p1848, 572p2292

Collinear triples
 2 3 429 2 6 1211 2 10 1193 2 37 2p1193 2 65 960 3 6 2092 3 65 2p1193 6 37 960 6 65 197p1848 6 1193 65p573 10 37 2092 10 65 1211 37 65 572p2292 37 429 65p573 65 2092 65p573 429 960 1211 429 2092 197p1848 960 1193 2p1193 1193 2092 572p2292 1211 2092 2p1193

ZC(X(2176),L(30,511)) ( = 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

ZC(X(2193),L(30,511)) passes through these centers:
2, 3, 6, 21, 1812, 2193, 9p229

Collinear triples
 2 3 21 2 6 1812 3 6 2193 21 2193 9p229

ZC(X(2207),L(30,511)) passes through these centers:
2, 4, 19, 25, 1249, 1435, 2207, 4p220, 4p2155, 19p159, 19p1763, 19p2138, 19p2139

Collinear triples
 2 4 25 2 1249 4p2155 2 1435 4p220 2 19p2138 19p2139 4 19 4p220 4 1249 2207 19 1249 1435 19 2207 19p1763 25 1435 19p1763 25 2209 19p159 25 4p2155 19p2138 1249 19p159 19p2139 4p220 4p2155 19p1763

ZC(X(2256),L(30,511)) passes through these centers:
1, 2, 9, 268, 936, 2256, 2p939, 2p1466, 196p936, 268p1767, 936p2257

Collinear triples
 1 2 936 1 9 2256 1 2p1466 268p1767 2 9 2p1466 2 268 196p936 2 2p939 936p2257 9 268 936 268 2256 268p1767 936 2256 936p2257 2p1466 196p936 936p2257

ZC(X(2284),L(30,511)) passes through these centers:
1, 2, 9, 1025, 1026, 2284

Collinear triples
 1 2 1026 1 9 2284 2 9 1025

ZC(X(2321),L(30,511)) passes through these centers:
2, 8, 10, 37, 312, 2321, 8p1743, 37p979, 312p978, 1293p1577

Collinear triples
 2 8 10 2 37 312 2 8p1743 1293p1577 2 37p979 312p978 8 2321 8p1743 10 37 2321 312 2321 312p978

ZC(X(851),L(44,513)) passes through these centers:
1, 65, 73, 243, 851, 1936, 2p2249, 851p851

Collinear triples
 1 65 1936 1 73 243 1 2p2249 851p851 65 73 851 73 2p2249 1758p1936 243 1936 851p851 851 1936 1758p1936

ZC(X(910),L(44,513)) passes through these centers:
1, 57, 105, 910, 1419, 2p103, 9p910, 55p1360, 59p1566, 105p1282

Collinear triples
 1 57 9p910 1 105 59p1566 1 2p103 55p1360 57 105 55p1360 57 910 1419 105 910 105p1282 910 9p910 59p1566 1419 2p103 59p1566 2p103 9p910 105p1282

ZC(X(922),L(44,513)) passes through these centers:
1, 31, 48, 896, 922, 923, 2157, 3p1177, 4p187, 6p1560, 23p187, 55p1366

Collinear triples
 1 31 896 1 48 4p187 1 923 55p1366 1 2157 23p187 1 3p1177 6p1560 31 48 922 31 923 23p187 31 2157 6p1560 48 2157 55p1366 48 3p1177 23p187 896 922 55p1366 896 4p187 23p187 922 4p187 6p1560

ZC(X(1418),L(44,513)) passes through these centers:
1, 57, 279, 354, 1418, 1419, 2293, 170p279

Collinear triples
 1 57 354 1 279 2293 57 1418 1419 279 1418 170p279 354 1418 2293

ZC(X(1457),L(44,513)) passes through these centers:
1, 56, 102, 221, 517, 1411, 1457, 3p1845, 84p517, 515p1457

Collinear triples
 1 56 517 1 102 515p1457 1 221 84p517 1 1411 3p1845 56 221 1457 56 1411 515p1457 102 221 3p1845 517 1457 3p1845 517 84p517 515p1457

ZC(X(1464),L(44,513)) passes through these centers:
1, 36, 65, 73, 74, 1464, 1870, 3p1832, 3p1833, 15p2306, 30p1464, 36p484, 554p2152

Collinear triples
 1 36 65 1 73 1870 1 74 30p1464 1 3p1832 554p2152 1 3p1833 15p2306 36 1464 36p484 36 1870 30p1464 65 73 1464 65 3p1832 3p1833 73 74 36p484 73 15p2306 554p2152

ZC(X(1575),L(44,513)) passes through these centers:
1, 2, 192, 291, 1575, 2p727, 6p1575, 238p1575, 291p2108, 1463p2053, 1575p1575

Collinear triples
 1 2 6p1575 1 192 1463p2053 1 291 238p1575 1 2p727 1575p1575 2 192 1575 2 291 1575p1575 192 2p727 238p1575 291 1575 291p2108 1575 6p1575 238p1575 2p727 6p1575 291p2108 6p1575 1463p2053 1575p1575

ZC(X(1864),L(44,513)) passes through these centers:
1, 4, 9, 1108, 1148, 1864, 3p1864, 9p1743

Collinear triples
 1 4 3p1864 1 9 1108 4 1148 1864 9 1864 9p1743 1108 1864 3p1864

ZC(X(1905),L(44,513)) passes through these centers:
1, 4, 1061, 1064, 1148, 1905, 1060p1905

Collinear triples
 1 4 1064 1 1061 1060p1905 4 1148 1905 1064 1905 1060p1905

ZC(X(1962),L(44,513)) passes through these centers:
1, 37, 42, 1100, 1125, 1962, 31p596, 1230p2220

Collinear triples
 1 37 1100 1 42 1125 1 31p596 1230p2220 37 42 1962 42 1100 31p596 1125 1962 1230p2220

ZC(X(2151),L(44,513)) passes through these centers:
1, 31, 48, 1094, 1250, 2151, 2153, 2307, 2p15, 4p15, 6p18, 15p62, 15p1251, 15p2306, 16p74, 54p61, 1250p1653

Collinear triples
 1 31 2p15 1 48 4p15 1 1094 2153 1 1250 15p2306 1 2307 15p1251 1 6p18 15p62 31 48 2151 31 1250 15p1251 48 1094 16p74 48 2307 15p2306 48 15p62 54p61 1094 2151 15p62 1094 15p2151 15p2306 1250 2151 1250p1653 1250 2307 15p62 2307 2p15 1250p1653 6p18 15p1251 1250p1653

ZC(X(2173),L(44,513)) passes through these centers:
1, 2153, 2154, 2173, 2349, 13p1511, 14p1511, 30p1495

Collinear triples
 1 2153 14p1511 1 2154 13p1511 1 2349 30p1495 2153 2154 30p1495 2173 13p1571 14p1511

ZC(X(2181),L(44,513)) passes through these centers:
1, 19, 158, 204, 1953, 2181, 2p217, 4p1075, 5p64

Collinear triples
 1 19 1953 1 158 2p217 1 204 5p64 19 204 2181 158 2181 4p1075 1953 2181 2p217

ZC(X(2227),L(44,513)) passes through these centers:
1, 75, 1581, 2227, 2p194, 2p699, 31p2227, 698p1691, 2227p2227

Collinear triples
 1 75 31p2227 1 1581 698p1691 1 2p699 2227p2227 75 1581 2227p2227 75 2227 2p194 2227 31p2227 698p1691 2p194 6p699 698p1691

ZC(X(2236),L(44,513)) passes through these centers:
1, 38, 1580, 1964, 2236, 2p733, 69p419, 385p2076, 2236p2236

Collinear triples
 1 38 1580 1 1964 69p419 1 2p733 2236p2236 38 1964 2236 1580 2236 385p2076 1580 69p419 2236p2236 1964 2p733 385p2076

ZC(X(2238),L(44,513)) passes through these centers:
1, 37, 42, 238, 239, 2107, 2238, 2p741, 2p2054, 238p1757, 740p2106, 2238p2238

Collinear triples
 1 37 238 1 42 239 1 2107 740p2106 1 2p741 2238p2238 1 2p2054 238p1757 37 42 2238 37 2p741 740p2106 37 2p2054 2238p2238 42 2107 2238p2238 42 2p741 238p1757 238 239 2238p2238 238 2238 238p1757 239 2238 740p2106

ZC(X(2268),L(44,513)) passes through these centers:
1, 6, 55, 940, 958, 2268, 2339, 21p894, 21p1999, 612p1468

Collinear triples
 1 6 958 1 55 940 1 2339 612p1468 6 55 2268 6 940 21p894 55 958 21p1999 940 958 612p1468 2268 21p894 21p1999

ZC(X(2276),L(44,513)) passes through these centers:
1, 2, 6, 55, 192, 869, 984, 1002, 2276, 2p1469, 10p904, 87p869, 291p672, 292p1575, 984p2280, 1282p2113, 2108p2113

Collinear triples
 1 2 869 1 6 984 1 55 2p1469 1 192 87p869 1 1002 984p2280 2 192 2276 2 984 292p1575 6 55 2276 6 869 291p672 55 869 10p904 55 87p869 292p1575 192 984 10p904 192 2p1469 291p672 869 2276 984p2280 984 291p672 2108p2113 2276 291p672 292p1575 2276 1282p2113 2108p2113 2p1469 292p1575 1282p2113

ZC(X(2293),L(44,513)) passes through these centers:
1, 6, 55, 354, 1212, 2293, 218p1212, 513p1292

Collinear triples
 1 6 1212 1 55 354 1 218p1212 513p1292 6 55 2293 6 354 513p1292 1212 2293 218p1212

ZC(X(2309),L(44,513)) passes through these centers:
1, 6, 55, 86, 893, 1107, 2309, 7p2309, 37p1197, 81p1655, 171p1107

Collinear triples
 1 6 1107 1 55 7p2309 1 86 37p1197 1 893 171p1107 6 55 2309 6 86 171p1107 55 893 37p1197 86 2309 81p1655 893 1107 81p1655 1107 2309 37p1197 1107 7p2309 171p1107

ZC(X(2312),L(44,513)) passes through these centers:
1, 19, 204, 1910, 2312, 2p1297, 3p132, 3p1503, 64p441, 147p1976, 2312p2312

Collinear triples
 1 19 3p1503 1 204 64p441 1 1910 3p132 1 2p1297 2312p2312 19 204 2312 19 1910 2312p2312 204 2p1297 3p132 1910 2312 147p1976 2312 3p132 3p1503 2p1297 3p1503 147p1976 3p1503 64p441 2312p2312

ZC(X(2313),L(44,513)) passes through these centers:
1, 1953, 1955, 2313, 2p217, 2p1298, 4p401, 2313p2313

Collinear triples
 1 1953 1955 1 2p217 4p401 1 2p1298 2313p2313 1953 2313 2p217 1955 4p401 2313p2313

ZC(X(2348),L(44,513)) passes through these centers:
1, 9, 294, 1279, 2348, 2p1477, 9p1743, 241p2348, 2348p2348

Collinear triples
 1 9 1279 1 294 241p2348 1 2p1477 2348p2348 9 294 2348p2348 9 2348 9p1743 1279 2348 241p2348 2p1477 9p1743 241p2348

ZC(X(2361),L(44,513)) passes through these centers:
1, 6, 36, 55, 2323, 2361, 36p484, 650p1290,

Collinear triples
 1 2 2323 1 6 55 1 55 2361 1 36p484 650p1290 1 2361 36p484