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

Access to introductory and other material is provided at the end of this page.

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 1001280p1617
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 514900

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

Collinear triples
100 1042p997
100 109 997p1381
100 2p997 997p1382
104 109 2p1470
2p1381 2p1382 2p1470
2p1470997p1381997p1382

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

Collinear triples
63 18221823
631214p18221214p1823
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 11131114
2226p1822226p1823
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 10819p2338
5713811382
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 18221823
63226p1822226p1823
1091822 278p1822
109 1823 226p1823

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

Collinear triples
57 13811382
571381p20061382p2006
13812222 1381p2006
1382 2222 1382p2006

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

Collinear triple
63 18221823

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

Collinear triples
55 13811382
55220p1381220p1382
1011381 220p1381
101 1382 220p1382

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

Collinear triples
3 10155p2338
313811382
3219p1381 219p1382
101 1381 219p1381
101 1382 219p1382

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

Collinear triple
57 13811382

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

Collinear triples
101 138172p1381
101138272p1382
121413811382
121472p138172p1382

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

Collinear triples
101 13811381p2329
10113821382p2329
17113811382
1711381p23291381p2329

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

Collinear triples
40 13811382
401381p23241382p2324
10113811381p2324
10113821382p2324

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

Collinear triples
63 18221823
6372p182272p1823
906182272p1822
906182372p1823

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

Collinear triples
56 101218p1477
5613811382
566p13816p1382
10113816p1381
10113826p1382

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

Collinear triples
57 13811382
57279p1381279p1382
9341381279p1381
9341382279p1382

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

Collinear triples
6 65110p1462
614006p2282
65110206p2282
6512p13811381p1400
6512p13821382p1400
1020140010p1462
14002p13812p1382
14001381p14001382p1400

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

Collinear triples
9 651815
92p13812p1382
99p13819p1382
6512p13819p1381
6512p13829p1382

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

Collinear triple
63 18221823

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

Collinear triples
58 1622p104
58222109p859
162651109p839
1623p138127p1381
1623p138227p1382
2226512p104
2223p13813p1382
22227p138127p1382

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

Collinear triples
1 8122p1983
1278100p1835
8127836p653
2p198336p653100p1835

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

Collinear triple
4 11131114

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

Collinear triples
5 11131114
55p18225p1823
11011135p1822
11011145p1823

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

Collinear triples
20 11131114
2020p182220p1823
110111320p1822
110111420p1823

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

Collinear triples
110 11131370p1822
11011141370p1823
111311141114
11141370p18221370p1823

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

Collinear triple
1113 111475p161
(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 11131370p1822
33p18223p1823
11011133p1822
11011143p1823

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

Collinear triples
22 110255p1297
2211131114
2222p182222p1823
110111322p1822
110111422p1823

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

Collinear triples
110 1113297p1822
1101114297p1823
29711131114
297297p1822297p1823

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

Collinear triple
300 3015p662

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

Collinear triples
4 11131114
465p111365p1114
109111365p1113
109111465p1114

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

Collinear triples
1 109104p1470
118221823
165p111365p1114
109182265p1113
109182365p1114

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

Collinear triples
1 100999p1320
113811382
18p13818p1382
10013818p1381
10013828p1382

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

Collinear triples
77 6512338
777p13817p1382
7763p138163p1382
6517p138163p1381
6517p138263p1382

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

Collinear triple
13 141625

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

Collinear triples
1 18221823
137p182237p1823
906182237p1822
906182337p1823

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

Collinear triple
7 673692p2140

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

Collinear triple
2 11131114

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

Collinear triple
470 11131114

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

Collinear triple
471 11131114

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

Collinear triples
25 112923p1995
2511131114
251113p19731114p1973
11211131113p1973
11211141114p1973

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

Collinear triple
427 11131114

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

Collinear triple
186 11131114

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

Collinear triple
467 11131114

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

Collinear triples
27 11131114
2728p111328p1114
112111328p1113
112111428p1114

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

Collinear triple
473 11131114

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

Collinear triple
472 11131114

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

Collinear triple
458 11131114

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

Collinear triple
468 11131114

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

Collinear triples
5 11131114
553p188253p1823
112111353p1822
112111453p1823

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

Collinear triples
29 11131114
291113p11721114p1172
11211131113p1172
11211141114p1172

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

Collinear triples
21 11131114
21284p1113284p1114
1121113284p1113
1121114284p1114


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 6492p1016

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

Collinear triples
2 10161016p1054
190 5141016

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

Collinear triple
650 6512p1262

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

Collinear triple
652 653653p1275

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

Collinear triple
654 655655p655

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

Collinear triples
1 764765p1054
100 513764

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

Collinear triple
100 6671016

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

Collinear triples
279 12751054p1275
658 1275244p658

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

Collinear triple
525 64892p250

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

Collinear triples
99 1649662p892
524 1379p15771380p1577

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

Collinear triples
100 1906p874
100 2p9198p919
190 6642284
190 6662p919
664 6668p919
666 2284666p673
666 6p874105p660
2284 2p919105p660
6p874 8p919666p673

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

Collinear triples
99 666919
110 91948p666
648 66648p666

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 658190p1456
651 664883
651 6662p927
651 664p1376932p1447
658 883664p1376
658 927165p927
664 666927
666 883105p927
927 2p927664p1376
927 105p927190p1456

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

Collinear triples
1 10110p1438
1 5141027
1 10262p1279
101 514666
514 6652p1279
665 102710p1438
666 10261027
666 2p12796p1280
666 10p143821p241

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 527190p1456
2 666105p527
2 673666p1155
2 9181638
2 2p1027527p885
2 2p1279100p527
514 527918
514 666666p1155
514 16382p1279
514 2p1027105p527
527 666673
651 666527p885
651 673105p527
651 918100p527
651 1638190p1456
666 2p1027100p527
673 16282p1027

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 519190p2348
2 666105p2325
2 9181639
2 2p29444p666
2 7p10238p910
2 8p1024666p1647
519 522918
519 6662p294
519 105p2325666p678
519 214p885666p1647
522 66644p666
522 16398p910
522 8p1024105p2325
644 666666p16476
644 16397p1023
644 2p294190p2348
666 2p2342673p1145
666 7p10238p1024
666 8p1022666p678
666 8p2222214p885
1639 2p2948p1024
1639 44p666666p678
1639 105p2325214p885
1639 666p1647673p1145
2p294 8p1022214p885
2p294 8p2222673p1145
2p2342 8p1024214p885
7p1023 44p666214p885
7p1023105p2325673p1145
8p1024 8p2222666p678

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

Collinear triple
110 250520

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

Collinear triples
6 249148p1101
110 249512

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

Collinear triples
101 66243p662
101 31p932932p1613
110 662692
110 31p93243p662

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

Collinear triples
100 110662
110 93243p662
190 66243p662

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

Collinear triples
6 13791380
6 25p182225p1823
6 163p1312523p1822
6 163p1313523p1823
110 1379661p1380
110 1380661p1379
110 25p1822523p1823
110 25p1823523p1822
110 163p1312163p1313
512 25p1822163p1312
512 25p1823163p1313
512 523p1822523p1823
512 661p1379661p1380

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

Collinear triples
110 9421175
110 28p1822523p1823
110 28p1823523p1822
513 523p1822523p1823
942 13811382
942 28p182228p1823

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

Collinear triples
110 9601178
110 21p1822523p1823
110 21p1823523p1822
521 523p1822523p1823
960 21p182221p1823

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

Collinear triples
1 59p1020
1 5239p1019
5 54110
5 11131114
110 1113523p1823
110 1114523p1823
110 9p10199p1020
523 523p1822523p1823

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

Collinear triples
6 32511
6 1755p20651959p2065
511 31p20651959p2065

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

Collinear triples
6 39511
511 38p20651959p2065

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

Collinear triples
6 5112088
511 525526

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

Collinear triples
1 62323
1 57517
1 3p91563p915
6 5757p2077
6 3p91578p913
57 63p91578p913
517 63p915915p1795
2323 78p913915p1795
3p915 57p2077v

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

Collinear triples
1 372323
1 63p91572p915
2 63p91571p915
37 71p91572p915

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

Collinear triples
101 651934
101 100p2362653p2066
109 934144p1461
109 651p2067934p2066
651 100p2362934p2066
651 651p2067653p2066

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

Collinear triples
108 110651
108 112108p610
110 112101p229
651 63p1301108p610

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

Collinear triples
162 6513p109
162 662163p1762
163 66299p2305
163 14153p109
651 6621415
651 99p23051247p1415
1415 163p17621247p1415

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

Collinear triples
222 12621262p1768
651 9051262

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

Collinear triple
6 7651

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

Collinear triples
81 226651
226 2p13812p1382
514 514p1381514p1382
651 2p1381514p1382
651 2p1382514p1381

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 104515p2323
101 109654
101 6517p654
101 23163p1845
104 10944p1443
104 6513p1845
109 6512p654
109 3p184557p102
651 231644p1443
651 57p102515p2323
654 2p654515p2323
654 7p65444p1443
2p654 3p18457p654

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 10559p1566
101 109665
101 6517p655
101 13626p1280
101 518p1381650p1382
101 518p1381650p1381
105 109241p1279
105 6511362
109 6512254
109 13627p911
109 241p1381513p1382
109 241p1382513p1381
651 6p1280241p1279
651 7p91158p1566
651 241p1381650p1382
651 241p1382650p1381
651 513p1381518p1382
651 513p1382518p1381
651 514p1381672p1382
651 514p1382672p1381
665 59p156659p1566
665 241p1279241p1279
665 650p1382650p1382
665 650p1381650p1381
665 514p1382514p1382
1362 22547p665
1362 241p1381518p1382
1362 241p1382518p1381
1362 672p1381672p1382
2254 650p1381650p1382
7p665 513p1382513p1382
59p1566 241p1382241p1382
241p1279 518p1382518p1382
241p1381 514p1381650p1381
241p1382 514p1382650p1382
241p1381 518p1381518p1381
241p1381 672p1381672p1381
241p1382 514p1382518p1382
241p1382 650p1382672p1382

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

Collinear triples
190 651664
190 668190p978
651 2p93243p664
651 190p978664p999
664 66843p664
668 99p171662p1999
2p932 99p171190p978
43p664 662p1999664p979

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

Collinear triples
190 65156p812
190 812190p978
651 673238p241
812 56p812238p241

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 222651
2 905572p693
2 226p1822226p1823
222 63p138163p1382
651 63p1381514p1382
651 63p1382514p1381
651 226p1823514p1822
905 514p1381514p1382

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

Collinear triples
101 109906
101 6511813
101 90p101906p1708
109 6511331
109 181335p110
651 46p181390p101
906 1331101p224
906 181346p1813
1331 1813906p1708
35p110 90p101101p224

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

Collinear triples
101 109906
101 6517p928
109 6512p928

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

Collinear triples
100 6519p1020
100 1018100p1743
190 65110p1415
190 1018190p978
651 100p1743226p1293
1018 9p102010p1415

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

Collinear triples
57 88651
57 10227p649
651 2p9017p649

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

Collinear triples
6 5131024
6 651673
513 6512p919

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 6511025
100 1026100p1743
100 8p9191025p2348
651 660238p1025
651 6642284
651 8p919241p1025
651 43p66487p2284
651 190p14771025p2348
660 43p664241p1025
664 102643p664
664 8p919238p1025
1025 1026238p1025
1025 2284241p1025
1026 22841025p2348
100p1743 190p1477241p1025

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

Collinear triples
101 1091415
101 6511461
101 3p10984p101
101 40p146164p1813
108 10940p1461
108 6513p109
108 1415108p610
109 14613p109
109 84p101108p610
651 40p146184p101
651 64p1813108p610
1415 146140p1461

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 6100p2246
1 57109p528
1 6501643
1 65156p528
1 2246516p919
1 7p2246517p2222
6 57528p1461
6 5131643
6 6517p2246
6 22461155p1308
6 44p222256p528
57 6512246
57 16437p649
513 6502246
513 651109p528
513 7p64956p528
513517p2222528p1461
650 651528p1461
650 7p6497p2246
650 44p2222109p528
651 7p649100p2246
1643 44p2222517p2222
7p649 109p5281155p1308
7p649 516p919528p1461

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

Collinear triples
101 1092283
101 6517p2283
101 927238p1025
109 6511025
109 56p660101p1362
651 927101p1362
651 56p660238p1025
1025 2283238p1025
1025 7p2283101p1362

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

Collinear triples
3 63p2173
3 743p2349
6 7420p2159
1073 3p234920p2159
3p21733p234974p2349

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

Collinear triples
2 430
2 6323
2 48p130063p1300
4 619p2071
4 63p1300255p1300
6 48p1300255p1300

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

Collinear triple
6 1314

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

Collinear triples
100 107823
100 190100p1714
107 93243p823
190 82343p823

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

Collinear triples
19 51219p1021
19 80019p1020
107 8006p821
107 19p102019p1021
800 13791380

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

Collinear triples
4 11131114
523 1113162p1312

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

Collinear triples
107 110158p684
107 1321297
110 684110p920
132 684158p684

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

Collinear triples
100 1072p1897
100 1121783
112 648191p648
648 17832p1897
2p1897 112p502191p648

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

Collinear triples
100 1621783
100 8232p1897
162 811648p1654
811 823112p1330
811 17832p1897
1783 112p13301897p2248
2p1897 648p16541897p2248

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

Collinear triples
101 1071897
101 1124p692
107 648112p1761
112 648162p1654
648 18974p692
1897 162p16541783p2248
4p692 112p17611783p2248

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

Collinear triples
1 10927p836
1 5221459
1 71p837109p1158
109 5221897
109 1459109p1158
1897 27p83671p837

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 1072p811
99 112648
107 112159p823
107 64822p162
112 162p1670162p1671
648 66p811159p823
2p811 22p16266p811
2p811 162p1670811p1677
2p811 162p1670811p1676
22p162 811p16771783p2248

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

Collinear triples
2 1101899
2 647850
110 647157p662
110 8502p811

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 984p1910
4 2644p1966
4 2p158158p401
4 92p168792p1688
98 23298p1956
98 2p1584p1821
232 2642p158
264 4p18214p1910
4p1821 98p1956158p401

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 392p1733
2 23219p1235
2 4p1821427p1910
2 24p29338p98
4 39232
4 248427p1910
4 2p173338p847
4 4p182138p98
6 248211p1821
6 4p1821427p1821
39 24824p293
39 38p98211p1821
39 427p1821427p1910
248 4p182119p1235
4p1821 24p29338p847
19p1235 38p98427p1821
38p847 211p1821427p1910

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

Collinear triples
1 23
2 56

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

Collinear triples
2 102p1736
10 2811861
75 92p103103p1826
281 92p103103p1441

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

Collinear triples
7 28192p103
281 5161861
673 186192p103

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

Collinear triples
107 109653
107 648112p1762
109 1124p1415
112 648648p1046
648 6534p1415

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

Collinear triples
109 162653
109 1634p1415
162 163162p1781
653 6624p1415

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

Collinear triples
101 1071897
101 1124p692
107 648112p1761
112 648162p1654
648 18974p692
1897 162p16541783p2248
4p692 112p17611783p2248

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

Collinear triples
1 10927p836
1 5221459
1 71p837109p1158
109 5221897
109 1459109p1158
1897 27p83671p837

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

Collinear triples
3 470471
3 371p2349372p2349
485 4p2349371p2349
486 4p2349372p2349

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

Collinear triple
4 470471

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 4186
2 62p1725
2 4p21594p2349
4 61990
4 744p2349
6 744p2159
74 18674p2349
74 2p172574p2166
186 19901138p2173
186 4p21591300p2159
186 4p234974p2166
1990 4p215974p2166
1990 4p234974p2349
2p1725 4p23491300p2159

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

Collinear triples
13 4704p2349
14 4714p2349
30 470471

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

Collinear triple
140 470471

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

Collinear triples
226 4p2349950p2349
440 470471

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

Collinear triples
4 445860
4 18651990
4 4p23491865p2349
281 8601865
1865 1859p23491865p2349
2p225 4p23491859p2349

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

Collinear triples
216 9254p2168
216 13791380

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

Collinear triples
3 847925
3 11131114

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

Collinear triples
651 66456p874
651 2p9196p927
658 6642283
658 9272p919
664 9276p927
927 2283673p927
927 56p874292p927
2283 6p927292p927
2p919 56p874673p927

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

Collinear triple
30 514516

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 190190p1279
100 6647p1025
100 9272p927
190 66456p874
190 666666p1743
190 2p927192p927
190 7p1025658p1376
664 666927
666 2p927658p1376
666 190p1279294p666
927 932192p927
927 7p1025294p666
927 56p874291p666
927 85p1293666p1743
927 190p1279927p1280
932 56p874658p1376
2p927 56p874294p666
7p1025 192p927291p666
7p1025 666p1743927p1280
85p1293 190p1279658p1376

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

Collinear triple
511 514516

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

Collinear triples
512 514516
514 92765p919

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 5147p513
241 927105p673
241 13811382
241 2p919105p1280
241 2p13217p1382
241 2p13827p1381
513 514516
513 2p9196p927
513 7p5137p1279
514 9276p927
514 885p1381885p1382
516 2p919105p673
927 2p9197p513
927 7p1279105p1280
927 7p1381885p1382
927 7p1382885p1381
2p919 2p1381885p1382
2p919 2p1382885p1381
2p1381 2p13827p1279
6p927 7p1279105p673

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

Collinear triples
514 516522
514 666927
522 666666p1743
927 666p17431111p1293

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 516523
514 693857
514 92737p927
523 37p92742p927
693 92742p927
857 11131114

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

Collinear triples
110 105p758500p673
514 516526
514 35p885105p758
516 1442658p2245
526 35p885500p673
927 1442105p758
927 35p885658p2245

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

Collinear triples
651 66456p874
651 2p9196p927
658 6642283
658 9272p919
664 9276p927
927 2283673p927
927 56p874292p927
2283 6p927292p927
2p919 56p874673p927

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 927105p239
57 1447190p1279
57 2p1027812p1462
100 105105p239
100 659190p1279
100 927812p1462
105 6592p1027
105 9271447
659 1447812p1462
927 2p102759p812
1447 59p812105p239
2p1027 58p693105p239

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 105666p910
2 927516p1462
2 7p1279190p1456
105 108516p1462
105 6762p1027
105 9277p516
676 7p5137p1279
676 190p1456666p910
927 2p1027190p1456
927 7p513666p910
2p1027 7p513516p1462
7p516 190p1456516p1462

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

Collinear triples
110105p19591814p2294
684 520p6731814p2294
693 105p1959520p673
927 1214105p1959
927 108p1959520p673

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

Collinear triples
514516804
514 105p740171p885
516 7p171239p1020
927 7p171105p740
927 171p885239p1020

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 51644p658
7 67344p927
7 885927p1647
7 9007p2254
7 92744p673
7 2p1319190p1279
100 67344p673
100 900190p1279
100 927927p1647
100 7p225444p673
514 516900
514 88544p673
514 92744p927
514 2p13197p2254
655 885678p927
673 885900
673 9272p1319
885 92744p658
900 44p927678p927
927 2p1022678p927
2p1319 44p673678p927

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

Collinear triples
514516918
514 664927
516 92785p103
664 91856p874
927 2p87656p874

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

Collinear triples
1101105p1212
1 5148p884
1 5167p2283
101 514927
103 516927
514 516926
926 8p884105p1212
927 7p22838p884
927 105p1212241p2346

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
100927614p927
100 934614p883
100 1633190p1279
653 927666p1473
653 1633112p857
653 2p919666p2082
653 2p1633516p934
653 7p1292614p883
927 934666p2082
927 2p9197p1633
927 2p16336p927
927 614p883666p1438
934 1633516p934
934 6p927666p1473
934 7p1633112p857
1633 2p9196p927
1633 614p927666p2082
2p1633 7p1633614p883
6p927 7p1292614p927
7p1292 7p1633190p1299
7p1633 614p927666p1473
190p1279 666p1438666p2082
516p934 614p927666p1438

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

Collinear triples
101664927
664 228456p874
813 92756p874

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

Collinear triples
1012284572p1018
101100p12201026p1220
190 228499p2295
190 100p12202284p2298

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

Collinear triples
10131p83548p835
19083531p835
190 1897100p1011
835 189748p835
835 100p1011835p2282

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

Collinear triples
100190100p1698
1008352p835
190 2p835192p835
835 932192p835

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

Collinear triples
1009292p929
190 2p929192p929
929 932192p929

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

Collinear triples
110351110p1749
110 476351p2166

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

Collinear triple
30511523

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

Collinear triples
304761989p2159
30512523
476 52331p476
476 4p66148p476
512 4p661393p2314
51231p47648p476
523 4p6614p2315
1379 13804p2315
4p2315 31p4761989p2159

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

Collinear triples
3047674p2166
3011131114

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
3074476
30523526
110476523
110526110p1749
476110p17491109p1291

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

Collinear triples
1314476p896
1347614p896
1313p896395p662
1447613p896
1414p896396p662
30523690
476523476p896
690395p662396p662

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

Collinear triples
30523804
476523476p1580
47679p17180p238

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

Collinear triples
30523826
47652338p476
476850476p1964
82638p476476p1964

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

Collinear triples
30523900
79476519p759
79476p1647514p2341
100476476p1647
100519p759553p2341
523514p2341519p759
900514p2341553p2341

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

Collinear triples
30523926
309p216079p2283
101241p2341942p2341
4769p2160241p2341
47679p2283650p2341
523241p2341650p2341
926650p2341942p2341

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
23775
22p19210p2162
10752p192

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

Collinear triples
2650693
28p23501111p1621
116931111p1621

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
23470
26298
2132p1094
218303p2151
21082283p1832
22p125015p1081
3615
318533p2154
3108215p1081
32p109416p2349
361p2167303p2151
62p1250283p1832
1318621p2151
152p1094303p2151
298470621p2151
2982p1094533p2154
10822p1250303p2151
2p109415p1081283p1832

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
23471
26299
2142p1095
217302p2152
2559283p1833
29p55916p554
3616
317532p2153
355916p554
32p109515p2349
362p2167302p2152
69p559283p1833
1416617p2154
1417622p2152
162p1095302p2152
299471622p2152
2992p1095532p2153
29915p2349617p2154
5599p559302p2152
2p109516p554283p1833

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

Collinear triples
111111316p896
25125123p1930
2323p1930316p896

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

Collinear triples
213298p2173
214299p2173
214946p1099
13146p1099
1430617p2154
30298p2173299p2173
1494298p2173617p2154

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
122p45
1945
12p209980p517
292p2099
92p4544p80
4544p8080p517

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

Collinear triples
23186
26323
2323p1749661p1291
3650
50323323p1749

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

Collinear triples
23467
2647p311
3652

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

Collinear triples
245
22162p158
212495p2184
4531249
553216
532p15892p1075

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
23473
26302
3661

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

Collinear triples
23472
26303
3662

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

Collinear triples
12306
1972
12p1751307p1754
29307
22p175163p209
92p1751306p1724
72306306p1724
72307307p1754
30630763p209

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
213121313
2338p1822661p1114
2338p1823661p1113
21109p13791380p1577
21109p13801379p1577
1151312661p1113
1151313661p1114
115338p1822338p1823
1151109p13791109p1380
5231312338p1822
5231313338p1823
523661p1113661p1114
5231379p15771380p1577

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

Collinear triples
2174236
21882089
2164p174236p505
173174164p174
173188236
1732089168p2089
2089164p174177p1128

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

Collinear triples
23458
26183
36182
318398p1966
6458290p1955
18298p1966290p1955

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
23468
26524
26723p896
21119p1366
2156063p1366
36187
3679p1366
323p89663p1177
6671560
611123p896
1874681560
1875249p1366
46852423p896

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
235
26343
2216547p343
252p92168p920
36216
334368p920
562165
547p34352p921
21634347p343
216216568p920

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
12239
19238
1861966
1292238p518
119296p1281
291447
2862238
2292238p350
2673238p518
28931966
21929239p1757
281p1654740p2248
2366p2068366p2069
9673238p250
98932238
8623881p1654
86292239p1757
866736p1281
238239239p1757
238292335p2112
238673673p1282
2381966238p250
2382238238p518
239673335p2112
23989381p1654
23914476p1281
2392238238p250
2928936p1281
2921447673p1282
14471966238p518
19292238335p2112
192981p1654238p250
1966239p1757740p2248
22386p1281740p2248

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

Collinear triples
2757
22231440
72692p1419
57223269

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
231585
26492
36371

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

Collinear triples
231586
36372

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
2278281
212494p2184
22p20719p1034
22p23314p84
24p171219p1032
42782p2331
43931249
42p2074p84
44p17124p2184
2783932p207
2784p844p1712
2813932p2331
28112494p84
2812p2074p2184
2814p171219p1034
12492p233119p1034
2p2072p233119p1032

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

Collinear triples
24465
21249465p2184
43971249

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

Collinear triples
24465
21249466p2184
43981249

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

Collinear triples
222621p440
23062p1104
212142p950
2264401214
22617512p1104
306440306p1724
17512p950306p1724
2p9502p110421p440

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
23297
26325
22909p1355
2694237p1926
220091687p1959
220101688p1959
23p1956240p401
36511
3290237p1926
33p19562p1355
6290240p401
66949p1355
2973259p1355
297511240p401
325511237p1926
5111687p19591688p1959
200920109p1355

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

Collinear triples
26681015
5136681978p1979

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

Collinear triples
21901086
190514190p1054

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

Collinear triples
278p910
2673516p518
29p136075p103
75162p1419
76739p1360
516673673p1282
5168p910516p518
2p141975p103516p518
8p91075p103673p1282

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

Collinear triples
122p518
19518
12919p1362
292p241
2291239p518
22p6739p1362
92p673239p518
291518335p2110
5182p518239p518
2p2412p5189p1362
2p5782p673335p2110

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

Collinear triples
299115
99523148p662

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

Collinear triples
2299396
261894p1095
396532618

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

Collinear triples
2298395
261994p1094
395533619

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

Collinear triples
231594
2619p1238
2541209
36570
6120948p93
1209159419p1238

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

Collinear triples
2324
261993
27026p47
36571
6199349p2190
24199326p47

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

Collinear triples
26394
2681147
26p92163p155
36577
36863p155
33941147
668547p1820
611476p921
39457763p155

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

Collinear triples
237321
2596595p1089
1037594
10321595p1089

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

Collinear triples
2766p2227
21916385p2227
275p699698p2227
7669875p194
761916698p2227
6986p2227385p2227
75p19475p699385p2227

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

Collinear triples
2751575
2335238p726
22p19287p1575
275p727726p1575
75335726p1575
756262p192
335726335p2108
7261575238p726
157575p727335p2108
157587p1575726p1575
2p19275p727238p726

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

Collinear triples
2392p1966
2141385
275p733732p2236
39141732
3975p7331966p2076
3857321966p2076
3852p1966732p2236

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

Collinear triples
210239
237350
210p1929239p1757
228p337740p2238
1037740
1010p1929239p1757
372p2107740p2238
3728p337239p1757
239350740p2238
239740239p1757

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

Collinear triples
122p960
19960
131469p444
297p960
23142092
28p89369p444
9314960p1999
920928p893
3147p960256p1045
9602p960960p1999
9607p96069p444
9608p893256p1045
20922p9607p960

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
122p984
19984
12276291p518
297p984
2752276
22p19287p2276
22p1002984p1001
9227610p983
987p2276291p1575
759842p192
752p984291p1575
9842276284p1001
984291p518291p1575
2p1922p98410p893
2p1927p984291p518

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

Collinear triples
121125
191100
18135p1125
29553
27935p1125
2811213
22p501267p1213
9791213
7911252p501
8111002p501
553112535p1125
110011251213
121335p1125267p1213

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

Collinear triples
122p1104
191104
297p1104
228440
44011042p1104

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

Collinear triples
122p1107
191107
1274894p1107
297p1107
2256894p1107
227410p1197
925610p1197
2562p110786p1655
274110786p1655
11072p110710p1197
2p11077p1107894p1107

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

Collinear triples
121210
191108
127840p1210
297p1108
28440p1210
227863p1864
98463p1864
8412102p207
27811082p207
1108121063p1864
12107p110840p1210

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

Collinear triples
2317577
237148p491
237248p492
371372577
114748p49148p492

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

Collinear triples
122p1212
191212
1142277
29142
2277344p2293
12122p1212344p2293

ZC(X(1333),L(30,511)) passes through these centers:
2, 3, 6, 28, 81, 1333, 2p501, 6p267

Collinear triples
2328
2681
22p5016p267
361333
8113332p501

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

Collinear triples
257142
222384p142
22791212
27p2124354p2125
572231418
14212121418
27914187p2124

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

Collinear triples
257226
22237p1903
22781214
2218420p1427
572231427
5727820p1427
572p2077p1903
223226278
22312142184
22320p1427208p1032
22612141427
22621842p207
2267p190320p1427
27814272p207
12142p207208p1032

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

Collinear triples
257908
222384p908
2200636p908
22p102515p1465
572231465
572006515p1465
2232p10236p908
908146536p908
90884p908515p1465

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

Collinear triples
24441
29863p132
21249441p2184
275p12971503p2312
4981503p2312
412491503
981503147p1910
441150363p132
44175p1297147p1910
441441p21841503p2312
124963p13275p1297

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

Collinear triples
2981513p1959
2232336p1513
23251513p1910
981321513p1910
981513147p1910
98237p1926336p1513
1322321513
2323251513p1959
3251513237p1926

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
23419
26385
2836p2236
298237p1966
2171238
214292329
22p1927385p1966
231p695384p1966
361691
398385p1966
31711429
36p223631p695
683384p1966
62382329
62p1927237p1966
832p19271966p2076
981691147p1910
9823296p1281
17123296p2236
1712p19276p1281
2381429237p1966
23816916p1281
38514296p1281
38516911966p2076
3856p2236385p1966
4192p1927147p1910
419237p1966384p1966
16916p2236237p1966
1691384p1966385p1966

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

Collinear triples
23460
26230
21141967
361692
1142301692

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

Collinear triples
24440
2102p1104
2377p950
21249440p2184
22p14278p2264
4108p2264
422491834
10371834
374402p1427
44018342p1104
4408p2264440p2184
12492p11042p1427
2p11047p9508p2264

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

Collinear triples
2463p1901
222621p1901
2121429p1901
22p19031817p1901
42261817p1901
412491901
22612141901
12142p190363p1901
12492p190321p1901
190121p190163p1901
21p190129p19011817p1901

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
275350
22742p740
2334238p350
219662p257
22p19287p350
28p673241p350
275p206875p2069
276p727239p726
752741966
75334239p726
7519212p192
758p673238p350
274192186p1655
3342p192241p350
3501966241p350
3502p25786p1655
3502p740238p350
19211966238p350
19212p740239p726
196687p350239p726
2p1922p2577p740
2p19276p727238p350
86p1655238p350310p2107

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

Collinear triples
2430
21331249
2124930p2184
24p23496p1099
41334p2349
412491990
3013330p2184
3019906p1099
124912496p1099

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
2392p2088
2675p2088
274p6611511p1577
2526p1725656p1300
362088
3656p13001511p1577
674p661526p1725
208875p20881511p1577
20881109p13791109p1380
75p208892p2088526p1725

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
23429
261211
2101193
2372p1193
265960
362092
3652p1193
637960
665197p1848
6119365p573
10372092
10651211
3765572p2292
3742965p573
65209265p573
4299601211
4292092197p1848
96011932p1193
11932092572p2292
121120922p1193

ZC(X(2176),L(30,511)) ( = Z(X(2),X(87)) passes through these centers:
1, 2, 9, 43, 1423, 2176, 37p904

Collinear triples
1243
192176
291423
94337p904

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

Collinear triples
2321
261812
362193
2121939p229

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
2425
212494p2155
214354p220
219p213819p2139
4194p220
412492207
1912491435
19220719p1763
25143519p1763
25220919p159
254p215519p2138
124919p15919p2139
4p2204p215519p1763

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

Collinear triples
12936
192256
12p1466268p1767
292p1466
2268196p936
22p939936p2257
9268936
2682256268p1767
9362256936p2257
2p1466196p936936p2257

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

Collinear triples
121026
192284
291025

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

Collinear triples
2810
237312
28p17431293p1577
237p979312p978
823218p1743
10372321
3122321312p978

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

Collinear triples
1651936
173243
12p2249851p851
6573851
732p22491758p1936
2431936851p851
85119361758p1936

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

Collinear triples
1579p910
110559p1566
12p10355p1360
5710555p1360
579101419
105910105p1282
9109p91059p1566
14192p10359p1566
2p1039p910105p1282

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

Collinear triples
131896
1484p187
192355p1366
1215723p187
13p11776p1560
3148922
3192323p187
3121576p1560
48215755p1366
483p117723p187
89692255p1366
8964p18723p187
9224p1876p1560

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

Collinear triples
157354
12792293
5714181419
2791418170p279
35414182293

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

Collinear triples
156517
1102515p1457
122184p517
114113p1845
562211457
561411515p1457
1022213p1845
51714573p1845
51784p517515p1457

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
13665
1731870
17430p1464
13p1832554p2152
13p183315p2306
36146436p484
36187030p1464
65731464
653p18323p1833
737436p484
7315p2306554p2152

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

Collinear triples
126p1575
11921463p2053
1291238p1575
12p7271575p1575
21921575
22911575p1575
1922p727238p1575
2911575291p2108
15756p1575238p1575
2p7276p1575291p2108
6p15751463p20531575p1575

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

Collinear triples
143p1864
191108
411481864
918649p1743
110818643p1864

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

Collinear triples
141064
110611060p1905
411481905
106419051060p1905

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

Collinear triples
1371100
1421125
131p5961230p2220
37421962
42110031p596
112519621230p2220

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
1312p15
1484p15
110942153
1125015p2306
1230715p1251
16p1815p62
31482151
31125015p1251
48109416p74
48230715p2306
4815p6254p61
1094215115p62
109415p215115p2306
125021511250p1653
1250230715p62
23072p151250p1653
6p1815p12511250p1653

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

Collinear triples
1215314p1511
1215413p1511
1234930p1495
2153215430p1495
217313p157114p1511

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

Collinear triples
1191953
11582p217
12045p64
192042181
15821814p1075
195321812p217

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

Collinear triples
17531p2227
11581698p1691
12p6992227p2227
7515812227p2227
7522272p194
222731p2227698p1691
2p1946p699698p1691

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

Collinear triples
1381580
1196469p419
12p7332236p2236
3819642236
15802236385p2076
158069p4192236p2236
19642p733385p2076

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

Collinear triples
137238
142239
12107740p2106
12p7412238p2238
12p2054238p1757
37422238
372p741740p2106
372p20542238p2238
4221072238p2238
422p741238p1757
2382392238p2238
2382238238p1757
2392238740p2106

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

Collinear triples
16958
155940
12339612p1468
6552268
694021p894
5595821p1999
940958612p1468
226821p89421p1999

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
12869
16984
1552p1469
119287p869
11002984p2280
21922276
2984292p1575
6552276
6869291p672
5586910p904
5587p869292p1575
19298410p904
1922p1469291p672
8692276984p2280
984291p6722108p2113
2276291p672292p1575
22761282p21132108p2113
2p1469292p15751282p2113

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

Collinear triples
161212
155354
1218p1212513p1292
6552293
6354513p1292
12122293218p1212

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

Collinear triples
161107
1557p2309
18637p1197
1893171p1107
6552309
686171p1107
5589337p1197
86230981p1655
893110781p1655
1107230937p1197
11077p2309171p1107

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

Collinear triples
1193p1503
120464p441
119103p132
12p12972312p2312
192042312
1919102312p2312
2042p12973p132
19102312147p1976
23123p1323p1503
2p12973p1503147p1976
3p150364p4412312p2312

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

Collinear triples
119531955
12p2174p401
12p12982313p2313
195323132p217
19554p4012313p2313

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

Collinear triples
191279
1294241p2348
12p14772348p2348
92942348p2348
923489p1743
12792348241p2348
2p14779p1743241p2348

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

Collinear triples
122323
1655
1552361
136p484650p1290
1236136p484

POINTS ON CUBICS: INTRODUCTION and Z-CUBICS
POINTS ON CUBICS: ZP-CUBICS
POINTS ON CUBICS: H- and HP- CUBICS
POINTS ON CUBICS: C- and ZC- CUBICS
POINTS ON CUBICS: B-, BP-, D-, and DP- CUBICS

Return to Front Cover of ETC
Last update: 01/07/2004 15:48:03