POINTS ON CUBICS

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

B(U,P): Bicentrics Collinearity Cubics
ZC(U,P): B-Plus Cubics

D(U,P): Cross-Bicentrics Collinearity Cubics
DP(U,P): D-Plus Cubics

Access to related material is provided at the end of this page.


B(U,P): Bicentrics Collinearity Cubics,

defined by

ux(qryz - p2x2) + vy(rpzx - q2y2) + wz(pqxy - r2z2) = 0

Locus: The bicentrics collinearity cubic B(U,P) is the locus of a point X = x : y : z such that the trilinear product U*X is on the line of the bicentric pair

qy : rz : px    and    rz : px : qy.

Notes:

1.   B(U,P) is also given by

up2x3 + vq2y3 + wr2z3 - (uqr + vrp + wpq)xyz = 0.

2.   B(U,P) meets the sidelines BC, CA, AB in these points:

0 : w1/3r2/3 : - v1/3q2/3,    - w1/3r2/3 : 0 : u1/3p2/3,    v1/3q2/3 : - u1/3p2/3 : 0

3.   If P = U, then defining equation yields

(px + qy + rz)(p2x2 + q2y2 + r2z2 - qryz - rpzx - pqxy) = 0,

so that B(U,U) is the union of a line and a conic.

4.   The point P -1 is on B(U,P).

5.   Problem: find general conditions on (U,P) and (U',P') such that B(U,P) = B(U',P'). Examples:

B(X(1),X(8)) = B(X(200),X(2)),     B(X(2),X(110)) = B(X(32),X(99)),     B(X(3),X(4)) = B(X(32),X(1492)).


B(X(1),X(8)) passes through these centers:
6, 56, 649, 1458, 513p1381, 513p1382

Collinear triples
6 561458
1458513p1381513p1382

B(X(1),X(58)) passes through these centers:
10, 37, 661, 740

Collinear triple
10 37740

B(X(1),X(513)) passes through these centers:
513, 514, 650, 812, 2254, 43p514, 513p1575, 514p1742

Collinear triples
513 514812
5136502254
514 43p514513p1575
65081243p514
2254 43p514514p1742

B(X(2),X(43)) passes through these centers:
1, 87, 649, 1463p2053

Collinear triple
1 871463p2053

B(X(2),X(81)) passes through these centers:
37, 42, 512, 2238

Collinear triple
37 422238

B(X(2),X(100)) passes through these centers:
513, 649, 663, 665, 43p513, 513p1742, 649p1575

Collinear triples
513 649659
51343p513649p1575
649 663665
65966343p513
665 43p513513p1742

B(X(2),X(107)) passes through these centers:
520, 647, 684, 20p810, 193p822, 230p822, 441p810

Collinear triples
520 647684
520193p822230p822
647 20p810441p810
68420p810193p822

B(X(2),X(110)) passes through these centers:
512, 523, 647, 804, 19p684, 194p661, 401p661, 512p2227

Collinear triples
512 523804
51264719p684
523 647401p661
523194p661512p2227
647 804194p661

B(X(3),X(4)) passes through these centers:
3, 6, 511, 647, 523p1822, 523p1823

Collinear triples
3 6511
511523p1822523p1823

B(X(3),X(29)) passes through these centers:
65, 73, 647, 851

Collinear triple
65 647851

B(X(3),X(78)) passes through these centers:
19, 34, 1876, 4p649

Collinear triple
19 341876

B(X(4),X(20)) passes through these centers:
6, 64, 647, 441p2155

Collinear triple
6 64441p2155

B(X(4),X(21)) passes through these centers:
65, 73, 647, 851

Collinear triple
65 73851

B(X(4),X(28)) passes through these centers:
71, 72, 647, 3p740

Collinear triple
71 723p740

B(X(4),X(76)) passes through these centers:
32, 184, 237, 3p798, 512p1822, 512p1823

Collinear triples
32 184237
237512p1822512p1823

B(X(4),X(112)) passes through these centers:
520, 525, 647, 684, 63p804, 194p656, 401p656, 647p2227

Collinear triples
520 525401p656
520647681
520 63p804194p656
52564763p804
525194p656647p2227

B(X(5),X(3)) passes through these centers:
4, 6, 647, 4p1955

Collinear triple
4 64p1955

B(X(6),X(9)) passes through these centers:
1, 57, 241, 513, 514p1381, 514p1382

Collinear triples
1 57241
241514p1381514p1382

B(X(6),X(69)) passes through these centers:
6, 25, 232, 512, 661p1113, 661p1114

Collinear triples
6 25232
232661p1113661p1114

B(X(7),X(75)) passes through these centers:
31, 41, 2223, 6p663, 663p1381, 663p1382

Collinear triples
31 412223
2223663p1381663p1382

B(X(8),X(85)) passes through these centers:
31, 41, 2223, 6p663, 663p1381, 663p1382

Collinear triples
31 412223
2223663p1381663p1382

B(X(9),X(7)) passes through these centers:
6, 55, 663, 672, 650p1381, 650p1382

Collinear triples
6 55672
672650p1381650p1382

B(X(10),X(1)) passes through these centers:
1, 6, 238, 649

Collinear triple
1 6238

B(X(10),X(76)) passes through these centers:
31, 32, 1919, 2210

Collinear triple
31 322110

B(X(10),X(78)) passes through these centers:
34, 56, 649, 1430

Collinear triple
34 561430

B(X(10),X(99)) passes through these centers:
512, 649, 798, 42p659, 423p810, 512p2106, 649p846, 649p1045

Collinear triples
512 649423p810
51279842p659
512 512p2106649p1045
649798512p2106
798423p810649p846
42p659649p846649p1045

B(X(10),X(101)) passes through these centers:
514, 649, 1459, 28p684, 81p804, 194p513, 401p513, 649p2227

Collinear triples
514 64981p804
5141459401p513
514 194p513649p2227
649145928p684
145981p804194p513

B(X(19),X(75)) passes through these centers:
31, 48, 810, 1755, 647p1113, 647p1114

Collinear triples
31 481755
1755647p1113647p1114

B(X(25),X(2)) passes through these centers:
3, 6, 511, 647, 523p1822, 523p1823

Collinear triples
3 6511
511523p1822523p1823

B(X(25),X(27)) passes through these centers:
71, 72, 647, 3p740

Collinear triple
71 723p740

B(X(28),X(29)) passes through these centers:
71, 73, 647, 65p1818, 656p1381, 656p1382

Collinear triples
71 7365p1818
65p1818656p1381656p1382

B(X(31),X(55)) passes through these centers:
71, 73, 647, 65p1818, 656p1381, 656p1382

Collinear triples
2 72p241
2p241693p1381693p1382

B(X(31),X(63)) passes through these centers:
1, 19, 240, 661, 523p1113, 523p1114

Collinear triples
1 19240
240523p1113523p1114

B(X(31),X(100)) passes through these centers:
513, 514, 650, 812, 2254, 43p514, 513p1575, 514p1742

Collinear triples
513 514812
5136502254
514 43p514513p1575
65081243p514
225443p514514p1742

B(X(32),X(3)) passes through these centers:
2, 4, 297, 523, 1113p1577, 1114p1577

Collinear triples
2 4297
2971113p15771114p1577

B(X(32),X(99)) passes through these centers:
512, 523, 647, 804, 19p684, 194p661, 401p661, 512p2227

Collinear triples
512 52319p684
51264719p684
523 647401p661
523194p661512p2227
64719p684194p661

B(X(32),X(101)) passes through these centers:
514, 522, 693, 918, 2p812, 43p693, 513p726, 693p1742

Collinear triples
514 522918
5146932p812
522 2p81243p693
69343p693513p726
91843p693693p1742

B(X(42),X(110)) passes through these centers:
514, 523, 661, 10p659, 423p656, 513p1654, 513p1655, 523p2106

Collinear triples
514 523423p656
514661523p2106
523 66110p659
523513p1655523p2106
661423p656513p1654
10p659513p1654513p1655

B(X(55),X(57)) passes through these centers:
1, 9, 518, 650, 522p1381, 522p1382

Collinear triples
1 9518
518522p1381522p1382

B(X(55),X(109)) passes through these centers:
514, 522, 918, 9p676, 144p513, 145p650, 514p2348

Collinear triples
514 522918
5149p676144p513
522 145p650514p2348
918144p513145p650

B(X(56),X(1)) passes through these centers:
1, 9, 518, 650, 522p1381, 522p1382

Collinear triples
1 9518
518522p1381522p1382

B(X(56),X(101)) passes through these centers:
514, 522, 918, 9p676, 144p513, 145p650, 514p2348

Collinear triples
514 522918
5149p676144p513
522 145p650514p2348
918144p513145p650

B(X(57),X(2)) passes through these centers:
6, 55, 663, 672, 650p1381, 650p1382

Collinear triples
6 55672
672650p1381650p1382

B(X(57),X(100)) passes through these centers:
513, 650, 2254, 55p676, 144p649, 145p663, 513p2348

Collinear triples
513 6502254
51355p676144p649
650 145p663513p2348
2254144p649145p663

B(X(58),X(101)) passes through these centers:
514, 523, 661, 10p659, 423p656, 513p1654, 513p1655, 523p2106

Collinear triples
514 52342p656
514661523p2106
523 66110p659
523513p1655523p2106
661 423p656513p1654
10p659513p1654513p1655

B(X(81),X(100)) passes through these centers:
514, 523, 661, 10p659, 423p656, 513p1654, 513p1655, 523p2106

Collinear triples
512 513661p2106
51266137p659
512 423p647513p846
513661423p647
661 513p1045661p2106
37p659513p846513p1045

B(X(105),X(100)) passes through these centers:
513, 650, 665, 2254, 109p1566, 238p2254, 513p1282, 514p2110

Collinear triples
513 6502254
513109p1566513p1282
513 238p2254514p2110
650238p2254513p1282
665 2254238p2254

B(X(109),X(100)) passes through these centers:
513, 650, 2170, 2254, 11p1381, 11p1382

Collinear triples
513 6502254
225411p138111p1382

B(X(110),X(107)) passes through these centers:
520, 647, 684, 41p1367, 125p1822, 125p1823

Collinear triples
520 647684
681125p1822125p1823

B(X(114),X(98)) passes through these centers:
3, 6, 511, 19p684

Collinear triple
3 6511

B(X(125),X(110)) passes through these centers:
6, 523, 1316, 19p684

Collinear triple
6 5231316

B(X(132),X(98)) passes through these centers:
3, 6, 511, 511p656

Collinear triple
3 6511

B(X(141),X(6)) passes through these centers:
2, 6, 385, 512

Collinear triple
2 6385

B(X(184),X(2)) passes through these centers:
6, 523, 1316, 19p684

Collinear triple
6 5231316

B(X(184),X(25)) passes through these centers:
2, 69, 325, 525, 850p1822, 850,1823

Collinear triples
2 69325
325850p1822850p1823

B(X(184),X(107)) passes through these centers:
520, 525, 647, 684, 63p804, 194p656, 401p656, 647p2227

Collinear triples
520 525401p656
520647684
52063p804194p646
52564762p804
525194p656647p2227

B(X(200),X(2)) passes through these centers:
6, 56, 649, 1458, 513p1381, 513p1382

Collinear triples
6 561458
1458513p1381513p1382

B(X(213),X(1)) passes through these centers:
1, 2, 239, 513

Collinear triple
1 2239

B(X(213),X(32)) passes through these centers:
75, 76, 693, 1921

Collinear triple
75 761921

B(X(213),X(72)) passes through these centers:
28, 81, 513, 27p511, 514p1113, 514p1114

Collinear triples
28 8127p511
27p511514p1113514p1114

B(X(217),X(3)) passes through these centers:
4, 470, 471, 458p661

Collinear triple
4 470471

B(X(217),X(4)) passes through these centers:
2, 3, 401, 520

Collinear triple
2 3401

B(X(220),X(1)) passes through these centers:
1, 57, 241, 513, 514p1381, 514p1382

Collinear triples
1 57241
241514p1381514p1382

B(X(225),X(1)) passes through these centers:
1, 3, 652, 1936

Collinear triple
1 31936

B(X(226),X(99)) passes through these centers:
512, 663, 810, 243p810, 415p810, 416p512, 650p2305, 663p1047

Collinear triples
512 663415p810
512810243p810
512 416p512663p1047
663810416p572
810 415p810650p2305
243p810650p2305663p1047

B(X(226),X(109)) passes through these centers:
522, 652, 663, 21p804, 194p650, 232p521, 401p650, 663p2227

Collinear triples
522 652401p650
52266321p804
522 194p650663p2227
652663232p521
652 21p804194p650

B(X(235),X(3)) passes through these centers:
3, 4, 450, 647

Collinear triple
3 4450

B(X(244),X(11)) passes through these centers:
59, 101, 1252, 59p518, 100p1381, 100p1382

Collinear triples
59 125259p518
59p518100p1381100p1382

B(X(244),X(88)) passes through these centers:
44, 1023, 1635, 1023p1052, 1023p1054

Collinear triples
44 1023p10521023p1054
102316351023p1054

B(X(255),X(1)) passes through these centers:
1, 19, 240, 661, 523p1113, 523p1114

Collinear triples
1 19240
240523p1113523p1114

B(X(284),X(109)) passes through these centers:
522, 523, 656, 243p656, 415p656, 416p523, 522p1046, 522p1047

Collinear triples
522 523415p656
522656416p523
523 656243p656
523416p523522p1047
656 415p656522p1046
243p656522p1046522p1047

B(X(287),X(107)) passes through these centers:
520, 647, 684, 31p684, 132p822, 147p810, 684p1580

Collinear triples
520 647684
520147p810684p1580
647 132p822147p810
68431p684684p1580

B(X(291),X(100)) passes through these centers:
513, 649, 659, 37p659, 513p2108, 659p1575, 659p1757

Collinear triples
513 649659
513513p2108659p1575
649 513p2108659p1757
65937p659659p1757

B(X(291),X(101)) passes through these centers:
513, 514, 812, 37p659, 238p2254, 514p2110

Collinear triples
513 514812
513238p2254514p2110

B(X(292),X(101)) passes through these centers:
513, 514, 812, 10p659, 514p2108, 659p726, 812p1757

Collinear triples
513 514812
513514p2108812p1757
514 514p2108659p726
81210p659812p1757

B(X(326),X(75)) passes through these centers:
31, 798, 1973, 2p2211, 512p1113, 512p1114

Collinear triples
31 19732p2211
2p2211512p1113512p1114

B(X(335),X(100)) passes through these centers:
513, 649, 659, 42p659, 238p665, 513p2110

Collinear triples
513 649659
649238p665513p2110

B(X(341),X(76)) passes through these centers:
32, 1397, 1919, 31p1458, 667p1381, 667p1382

Collinear triples
32 139731p1458
31p1458667p1381667p1382

B(X(346),X(75)) passes through these centers:
31, 604, 667, 6p1458, 649p1381, 649p1382

Collinear triples
31 6046p1458
6p1458649p1381649p1382

B(X(394),X(2)) passes through these centers:
6, 25, 232, 512, 661p1113, 661p1114

Collinear triples
6 25232
232661p1113661p1114

B(X(512),X(2)) passes through these centers:
6, 15, 16, 6p2234

Collinear triple
6 1516

B(X(512),X(3)) passes through these centers:
2, 4, 193, 230, 297, 1249, 1503

Collinear triples
2 4297
2 193230
4 12491503
193 2971249

B(X(512),X(10)) passes through these centers:
58, 81, 284, 21p1423, 21p1458, 58p1575, 81p1742

Collinear triples
58 8121p1423
58 28421p1458
81 21p142315p1575
284 21p142321p1423
21p1423 21p145881p1742

B(X(512),X(15)) passes through these centers:
2, 13, 30, 395, 94p1095

Collinear triple
2 1394p1095

B(X(512),X(16)) passes through these centers:
2, 14, 30, 396, 94p1094, 617p2154

Collinear triples
2 1494p1094
14 30617p2154

B(X(512),X(32)) passes through these centers:
2, 69, 76, 325, 698, 2p1966, 75p194, 75p401

Collinear triples
2 69325
2 762p1966
69 7675p401
69 2p196675p194
76 69875p194

B(X(512),X(58)) passes through these centers:
2, 10, 37, 740, 1654, 1655, 2p1757, 10p2106

Collinear triples
2 102p1757
2 3710p1757
10 37750
10 165510p1757
37 16542p1757
740 16541655

B(X(512),X(76)) passes through these centers:
6, 32, 184, 237, 1613, 1691, 1971, 32p2227

Collinear triples
6 321691
6 1841971
6 161332p2227
32 184237
184 16131691

B(X(513),X(6)) passes through these centers:
1, 2, 9, 192, 239, 518, 1575, 2p1742

Collinear triples
1 2239
1 9518
2 1921575
9 192239
192 5182p1742

B(X(513),X(9)) passes through these centers:
1, 57, 241, 910, 1279, 1419, 1743

Collinear triples
1 57241
1 2411743
57 9101419
241 14191743

B(X(513),X(37)) passes through these centers:
1, 6, 81, 238, 1931, 2106, 58p1654, 58p1655

Collinear triples
1 6238
1 811931
1 210658p1655
6 812106
6 193158p1654
238 58p165458p1655

B(X(513),X(75)) passes through these centers:
6, 31, 41, 1914, 2176, 2223, 6p1742, 31p1575

Collinear triples
6 311914
6 217631p1575
31 412223
41 19142176
2176 22236p1742

B(X(514),X(1)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 6p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

B(X(514),X(8)) passes through these centers:
6, 56, 1458, 6p910, 6p1279, 6p1419, 6p1743

Collinear triples
6 561458
6 6p12796p1743
56 6p9106p1419
1458 6p14196p1743

B(X(514),X(10)) passes through these centers:
6, 31, 58, 1326, 1914, 6p2106, 58p846, 58p1045

Collinear triples
6 311914
6 581326
6 6p210658p1045
31 586p2106
31 132658p846
1914 58p84658p1045

B(X(514),X(58)) passes through these centers:
10, 42, 71, 10p1580, 10p1740, 10p1755, 10p1955, 42p2227

Collinear triples
10 4210p1580
10 7110p1955
10 10p174042p2227
42 7110p1755
71 10p158010p1740

B(X(514),X(76)) passes through these centers:
31, 32, 2175, 2209, 2210, 6p2223, 31p1742, 32p1575

Collinear triples
31 322210
31 220932p1575
32 21756p2223
2175 22092210
2209 6p222331p1742

B(X(520),X(6)) passes through these centers:
2, 4, 193, 230, 297, 1249, 1503

Collinear triples
2 4297
2 193230
4 12491503
193 2971249

B(X(522),X(7)) passes through these centers:
6, 55, 672, 6p2348, 41p145, 41p516

Collinear triples
6 55672
55 6p234841p145

B(X(522),X(9)) passes through these centers:
6, 56, 57, 1423, 1429, 1458, 6p1463, 57p1742

Collinear triples
6 561458
6 14231429
56 571429
57 14236p1463
1423 145857p1742

B(X(522),X(63)) passes through these centers:
6, 19, 2178, 2202, 6p1148

Collinear triples
6 192202
2178 22026p1148

B(X(523),X(3)) passes through these centers:
6, 19, 2178, 2202, 6p1148

Collinear triple
371 3724p2315

B(X(523),X(6)) passes through these centers:
2, 3, 6, 194, 385, 401, 511, 6p2227

Collinear triples
2 3401
2 6385
2 1946p2227
3 6511
3 194385

B(X(523),X(30)) passes through these centers:
6, 15, 16, 74, 617p2159

Collinear triples
6 1516
16 74617p2159

B(X(523),X(37)) passes through these centers:
58, 81, 284, 21p1423, 21p1429, 21p1458, 58p1575, 81p1742

Collinear triples
58 8121p1429
58 28421p1458
81 21p142858p1575
284 21p142821p1429
21p1423 21p145881p1742

B(X(523),X(69)) passes through these centers:
6, 25, 232, 1692, 6p204, 6p1707, 6p2312

Collinear triples
6 25232
6 16926p1707
25 6p2046p2312
232 6p2046p1707

B(X(523),X(86)) passes through these centers:
1, 42, 213, 3p862, 6p846, 6p1045, 6p1757, 42p2106

Collinear triples
6 426p1757
6 21342p2106
42 2133p862
42 6p104542p2106
213 6p8466p1757
3p862 6p1045

B(X(525),X(3)) passes through these centers:
4, 6, 25, 232, 419, 4p1740, 4p1955, 25p2227

Collinear triples
4 64p1955
4 25419
4 4p174025p2227
6 25232
6 4194p1740

B(X(525),X(4)) passes through these centers:
3, 6, 154, 511, 3p1707, 3p2312, 48p230

Collinear triples
3 6511
3 3p170748p230
6 1543p2312
154 5113p1707

B(X(525),X(72)) passes through these centers:
28, 1474, 2299, 27p1914, 27p2176, 27p2223, 28p1742, 726p2203

Collinear triples
28 147427p1914
28 27p2176726p2203
1474 229927p2223
2299 27p191427p2172
27p2176 27p222328p1742

B(X(526),X(50)) passes through these centers:
94, 265, 1989, 94p1580, 94p1740, 94p1755, 94p1955, 1989p2227

Collinear triples
94 26594p1955
94 198994p1580
94 94p17401989p2227
265 198994p1755
265 94p158094p1740

B(X(560),X(48)) passes through these centers:
75, 92, 1577, 2p297, 850p1113, 850p1114

Collinear triples
75 922p297
2p297 850p1113850p1114

B(X(647),X(25)) passes through these centers:
2, 20, 69, 325, 441, 3p1733, 63p193

Collinear triples
2 20441
2 69325
20 32563p193
69 3p173363p193

B(X(649),X(2)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 6p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

B(X(649),X(31)) passes through these centers:
2, 8, 75, 350, 726, 2p192, 2p518, 75p1742

Collinear triples
2 82p518
2 75350
8 3502p192
75 7262p192
2p192 2p51875p1742

B(X(649),X(42)) passes through these centers:
1, 2, 86, 239, 2p1931, 2p2106, 81p1654, 81p1655

Collinear triples
1 2239
1 862p2106
1 2p193181p1654
2 862p1931
2 2p210681p1655
239 81p165481p1655

B(X(649),X(55)) passes through these centers:
2, 7, 145, 516, 2p241, 2p1279, 2p1419

Collinear triples
2 72p241
2 1452p1279
7 5162p1419
145 2p2412p1419

B(X(649),X(60)) passes through these centers:
10, 12, 10p851, 10p1046, 10p1047, 10p1758, 12p416, 37p1214

Collinear triples
10 1210p1758
10 12p41637p1214
12 10p85137p1214
12 10p104712p416
10p85110p104610p1047
10p104610p175837p1214

B(X(650),X(3)) passes through these centers:
1, 4, 46, 243, 1148

Collinear triples
1 4243
46 2431148

B(X(650),X(55)) passes through these centers:
1, 7, 57, 241, 1447, 1463, 2p1423, 7p1742

Collinear triples
1 57241
1 14472p1423
7 571447
7 14632p1423
241 2p14237p1742

B(X(650),X(57)) passes through these centers:
1, 9, 165, 518, 2348, 9p910, 9p1743

Collinear triples
1 9518
1 1659p910
9 23489p1743
165 5189p1743

B(X(650),X(65)) passes through these centers:
1, 3, 21, 416, 1936, 3p415, 21p1046, 21p1047

Collinear triples
1 31936
1 213p415
1 41621p1047
3 21416
3 3p41521p1046
1936 21p104621p1047

B(X(651),X(59)) passes through these centers:
11, 2170, 2310, 11p43, 11p238, 22p672, 11p1742, 1463p2310

Collinear triples
11 217011p238
11 11p431463p2310
2170 231011p672
2310 11p4311p238
11p43 11p67211p1742

B(X(652),X(73)) passes through these centers:
1, 4, 29, 243, 415, 4p416, 29p1046, 29p1047

Collinear triples
1 4243
1 294p416
1 41529p1046
4 29415
4 4p41629p1047
243 29p104629p1047

B(X(656),X(19)) passes through these centers:
1, 63, 610, 1959, 3p193, 3p230, 3p1503

Collinear triples
1 631959
1 6103p1503
63 3p1933p230
610 19593p193

B(X(656),X(48)) passes through these centers:
1, 19, 92, 240, 2p419, 4p194, 4p401, 19p2227

Collinear triples
1 19240
1 924p401
1 2p4194p194
19 922p419
92 4p19419p2227

B(X(656),X(71)) passes through these centers:
27, 28, 1172, 21p1876, 27p43, 27p238, 27p1742, 28p1575

Collinear triples
27 2827p238
27 27p4328p1575
28 117221p1876
1172 27p4327p238
21p1876 27p4327p1742

B(X(657),X(6)) passes through these centers:
2, 7, 145, 516, 2p241, 2p1279, 2p1419

Collinear triples
2 72p241
2 1452p1279
7 5162p1419
145 2p2412p1419

B(X(657),X(7)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 6p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

B(X(657),X(42)) passes through these centers:
7, 77, 86, 416, 7p1936, 77p415, 86p1046, 86p1047

Collinear triples
7 777p1936
7 8677p415
7 7p41686p1047
77 867p416
77 77p41586p1046
7p1936 86p104686p1047

B(X(661),X(21)) passes through these centers:
1, 65, 73, 851, 1046, 1047, 1758, 65p416

Collinear triples
1 651758
1 7365p416
65 73851
65 104765p416
73 10461758
851 10461047

B(X(661),X(42)) passes through these centers:
21, 81, 86, 21p241, 21p1447, 21p1463, 43p86, 86p1742

Collinear triples
21 8121p241
21 21p144743p86
81 8621p1447
86 21p146343p86
21p241 43p8686p1742

B(X(661),X(63)) passes through these centers:
1, 19, 204, 240, 1707, 2312, 2p1692

Collinear triples
1 19240
1 17072p1692
19 2042312
204 2401707

B(X(661),X(81)) passes through these centers:
1, 37, 42, 846, 1045, 1757, 2238, 37p2106

Collinear triples
1 371757
1 4237p2106
37 422238
37 104537p2106
42 8461757
846 10452238

B(X(663),X(8)) passes through these centers:
6, 56, 57, 1423, 1429, 1458, 6p1463, 57p1742

Collinear triples
6 561548
6 14231429
56 571429
57 14236p1463
1423 145857p1742

B(X(663),X(41)) passes through these centers:
2, 7, 85, 2p241, 2p1447, 2p1463, 7p192, 85p1742

Collinear triples
2 72p241
2 2p14477p192
7 852p1447
85 2p14637p192
2p241 7p19285p1742

B(X(663),X(56)) passes through these centers:
2, 8, 144, 2p518, 2p2348, 8p910, 8p1743

Collinear triples
2 82p518
2 1448p910
8 2p23488p1743
144 2p5188p1743

B(X(663),X(60)) passes through these centers:
12, 226, 2171, 10p1284, 12p1931, 12p2106, 65p1654, 65p1655

Collinear triples
12 22612p1931
12 217110p1284
12 12p210665p1655
226 217112p2106
2171 12p193165p1654
10p1284 65p165465p1655

B(X(667),X(1)) passes through these centers:
1, 2, 9, 192, 239, 518, 1575, 2p1742

Collinear triples
1 2239
1 9518
2 1921575
9 192239
192 5182p1742

B(X(667),X(32)) passes through these centers:
75, 76, 312, 1921, 2p726, 43p76, 69p1861, 76p1742

Collinear triples
75 761921
75 31269p1861
76 2p72643p76
312 192143p76
43p76 69p186176p1742

B(X(669),X(2)) passes through these centers:
2, 3, 6, 194, 385, 401, 511, 6p2227

Collinear triples
2 3401
2 6385
2 1946p2227
3 6511
3p76 194385

B(X(669),X(42)) passes through these centers:
86, 274, 333, 43p274, 81p726, 86p518, 274p1742

Collinear triples
86 33386p518
274 43p27481p726
43p274 86p518274p1742

B(X(669),X(50)) passes through these centers:
94, 300, 301, 75p395, 75p396

Collinear triple
94 300301

B(X(669),X(81)) passes through these centers:
2, 10, 37, 740, 1654, 1655, 2p1757, 10p2106

Collinear triples
2 102p1757
2 3710p2106
10 37740
10 165510p2106
37 16542p1757
740 16541655

B(X(693),X(2)) passes through these centers:
6, 31, 41, 1914, 2176, 2223, 6p1742, 31p1575

Collinear triples
6 311914
6 217631p1575
31 412223
41 19142176
2176 22236p1742

B(X(693),X(81)) passes through these centers:
37, 213, 228, 10p237, 10p1613, 10p1691, 10p1971, 213p2227

Collinear triples
37 21310p1691
37 22810p1971
37 10p1613213p2227
213 22810p237
228 10p161310p1691

B(X(798),X(37)) passes through these centers:
21, 81, 86, 21p241, 21p1447, 21p1463, 43p86, 86p1742

Collinear triples
21 8121p241
21 21p144743p86
81 8621p1447
86 21p146343p86
21p241 43p8686p1742

B(X(798),X(37)) passes through these centers:
1, 31, 48, 1580, 1740, 1755, 1955, 31p2227

Collinear triples
1 311580
1 481955
1 174031p2227
31 481755
48 15801740

B(X(798),X(86)) passes through these centers:
1, 37, 42, 846, 1045, 1757, 2238, 37p2106

Collinear triples
1 371757
1 4237p2106
37 422238
37 104537p2106
42 8461757
846 10452238

B(X(810),X(72)) passes through these centers:
27, 28, 1172, 21p1876, 27p43, 27p238, 27p1742, 28p1575

Collinear triples
27 2827p238
27 27p4328p1575
28 117221p1872
1172 27p4327p238
21p1872 27p4327p1742

B(X(822),X(29)) passes through these centers:
1, 65, 73, 851, 1046, 1047, 1758, 65p416

Collinear triples
1 651758
1 7365p416
65 73851
65 104765p416
73 10461758
851 10461047

B(X(822),X(31)) passes through these centers:
75, 92, 1733, 1895, 2p193, 2p297, 2p1503

Collinear triples
75 922p297
75 17332p193
92 18952p1503
1895 2p1932p297

B(X(822),X(92)) passes through these centers:
1, 31, 1580, 1740, 1755, 1955, 31p2227, 34p2289

Collinear triples
1 311580
1 174031p2227
1 199534p2289
31 175534p2289
1580 174034p2289

B(X(826),X(39)) passes through these centers:
83, 251, 1176, 82p194, 82p385, 82p401, 82p511, 251p2227

Collinear triples
83 25182p385
83 117682p401
83 82p194251p2227
251 117682p511
1176 82p19482p385

B(X(850),X(2)) passes through these centers:
6, 32, 184, 237, 1613, 1691, 1971, 32p2227

Collinear triples
6 321691
6 1841971
6 161332p2227
32 184237
184 16131691

B(X(875),X(1)) passes through these centers:
1, 2, 239, 2238, 2p2110, 238p518, 740p2106

Collinear triples
1 2239
1 2p2110238p518
2 2p2110740p2106
239 2238740p2106

B(X(875),X(2)) passes through these centers:
1, 6, 238, 2108, 2238, 238p1575, 238p1757

Collinear triples
1 6238
1 2108238p1575
6 2108238p1757
238 2238238p1757

B(X(875),X(31)) passes through these centers:
2, 75, 350, 2p740, 75p2108, 239p726, 350p1757

Collinear triples
2 75350
2 75p2108350p1757
75 75p2108239p726
350 2p740350p1757

B(X(876),X(1)) passes through these centers:
1, 6, 238, 2108, 2238, 238p1575, 238p1757

Collinear triples
1 6238
1 2108238p1575
6 2108238p1557
238 2238238p1557

B(X(876),X(6)) passes through these centers:
1, 2, 239, 2238, 2p2110, 238p518, 740p2106

Collinear triples
1 2239
1 2p2110238p518
2 2p2110740p2106
239 2238740p2106

B(X(876),X(76)) passes through these centers:
31, 32, 2210, 31p2108, 31p2238, 1575p2210, 1757, 2210

Collinear triples
31 322210
31 31p21081575p2210
32 31p21081757p2210
2210 31p22381757p2210

B(X(878),X(3)) passes through these centers:
2, 4, 132, 232, 297, 19p147, 232p1966

Collinear triples
2 4297
2 19p147232p1966
4 13219p147
232 297232p1966

B(X(879),X(69)) passes through these centers:
6, 25, 232, 2211, 21p132, 147p1973, 232p1580

Collinear triples
6 25232
6 147p1973232p1580
25 31p132147p1973
232 2211232p1580

B(X(884),X(55)) passes through these centers:
2, 7, 241, 2p241, 2p2114, 85p2110, 239p241, 241p516

Collinear triples
2 72p241
2 2p2114239p241
7 2p2114241p516
7 85p2110239p241
241 2p241239p241

B(X(885),X(8)) passes through these centers:
6, 56, 1458, 6p1458, 57p2110, 238p1458, 672p1456

Collinear triples
6 561458
6 6p2114238p1458
56 6p2114672p1456
56 57p2110238p1458
1458 6p1458238p1458

B(X(900),X(44)) passes through these centers:
88, 106, 2316, 43p88, 88p238, 88p672, 88p1742, 106p1575

Collinear triples
88 10688p238
88 43p88106p1575
106 231688p672
2316 43p8888p238
43p88 88p67288p1742

B(X(905),X(78)) passes through these centers:
19, 34, 1876, 6p1886, 19p1279, 19p1419, 19p1743

Collinear triples
19 341876
19 19p127919p1743
34 6p188619p1419
1876 19p141919p1743

B(X(1019),X(1)) passes through these centers:
1, 37, 42, 846, 1045, 1757, 2238, 37p2106

Collinear triples
1 371757
1 4237p2106
37 422238
37 104537p2106
42 8461757
846 10452238

B(X(1019),X(58)) passes through these centers:
10, 37, 210, 740, 10p43, 10p672, 10p1742, 37p1575

Collinear triples
10 37740
10 10p4337p1575
37 21010p672
210 74010p43
10p43 10p67210p1742

B(X(1021),X(1)) passes through these centers:
1, 65, 73, 851, 1046, 1047, 1758, 65p416

Collinear triples
1 651758
1 7365p416
65 73851
65 104765p416
73 10461758
851 10461047

B(X(1024),X(9)) passes through these centers:
1, 57, 241, 1458, 2114, 7p2110, 238p241, 241p910

Collinear triples
1 57241
1 2114238p241
57 2114241p910
57 7p2110238p241
241 1458238p241

B(X(1027),X(1)) passes through these centers:
1, 9, 518, 672, 1282, 2p2110, 59p1566, 238p518

Collinear triples
1 9518
1 128259p1566
1 2p2110238p518
9 1282238p518
518 672238p518

B(X(1459),X(69)) passes through these centers:
19, 25, 607, 2201, 2356, 4p2176, 19p1742, 25p1575

Collinear triples
19 252201
19 4p217625p1575
25 6072356
607 22014p2176
2356 4p217619p1742

B(X(1459),X(71)) passes through these centers:
4, 19, 27, 242, 423, 4p2106, 27p1045

Collinear triples
4 19242
4 27423
4 4p210627p1045
19 274p2106

B(X(1577),X(71)) passes through these centers:
1, 31, 48, 1580, 1740, 1755, 1955, 31p2227

Collinear triples
1 311580
1 481955
1 174031p2227
31 481755
48 15801740

B(X(1635),X(100)) passes through these centers:
1, 244, 513, 1052, 1054

Collinear triples
1 2441054
513 10521054

B(X(1919),X(6)) passes through these centers:
2, 8, 75, 350, 726, 2p192, 2p518, 75p1742

Collinear triples
2 82p518
2 75350
8 3502p192
75 7262p192
2p192 2p51875p1742

B(X(1924),X(1)) passes through these centers:
1, 63, 75, 1959, 2227, 2p194, 2p401 83p732

Collinear triples
1 631959
1 7583p732
63 752p401
63 2p16483p732
75 22272p194

B(X(2254),X(105)) passes through these centers:
1, 9, 518, 672, 1282, 2p2110, 59p1566, 238p518

Collinear triples
1 9518
1 128259p1566
1 2p2110238p518
9 1282238p518
518 672238p518



BP(U,P): B-Plus Cubics,

defined by

ux(qryz + p2x2) + vy(rpzx + q2y2) + wz(pqxy + r2z2) = 0

Notes:

1.   BP(U,P) is also given by

up2x3 + vq2y3 + wr2z3 + (uqr + vrp + wpq)xyz = 0.

2.   B(U,P) is the set of points X for which a collinearity-determinant is zero; BP(U,P) is the set of points X for which the corresponding permanent is zero. (The permanent is sometimes called "plus-determinant", as it results from changing the minus signs in the definition of determinant to plus signs.)

3.   BP(U,P) meets the sidelines BC, CA, AB in these points:

0 : w1/3r2/3 : - v1/3q2/3,    - w1/3r2/3 : 0 : u1/3p2/3,    v1/3q2/3 : - u1/3p2/3 : 0

4.   Problem: find general conditions on (U,P) and (U',P') such that BP(U,P) = BP(U',P').
Example: BP(X(6),X(825)) = BP(X(32),X(1492)).


BP(X(1),X(88)) passes through these centers:
1, 44


BP(X(1),X(100)) passes through these centers:
513, 1635


BP(X(2),X(99)) passes through these centers:
513, 1635


BP(X(2),X(670)) passes through these centers:
669, 887, 15p798, 16p798

Collinear triple
669 15p79816p798

BP(X(2),X(671)) passes through these centers:
187, 351


BP(X(3),X(110)) passes through these centers:
523, 1637


BP(X(4),X(107)) passes through these centers:
520, 1636


BP(X(6),X(99)) passes through these centers:
512, 888, 15p661, 16p661

Collinear triple
512 15p66116p661

BP(X(6),X(100)) passes through these centers:
513, 891


BP(X(6),X(101)) passes through these centers:
514, 900


BP(X(6),X(106)) passes through these centers:
519,900


BP(X(6),X(110)) passes through these centers:
523, 690


BP(X(6),X(111)) passes through these centers:
524, 690


BP(X(6),X(825)) passes through these centers:
824, 1491, 238p824, 752p1491, 1491p2230

Collinear triple
824 1491238p824

BP(X(306),X(2)) passes through these centers:
6, 647


BP(X(32),X(1492)) passes through these centers:
824, 1491, 238p824, 752p1491, 1491p2230

Collinear triple
824 1491238p824

BP(X(184),X(648)) passes through these centers:
523, 686, 371p656, 372p656

Collinear triple
686 371p656372p656

BP(X(187),X(6)) passes through these centers:
2, 523


BP(X(512),X(2)) passes through these centers:
6, 15, 16, 6p2234

Collinear triple
6 1516

BP(X(512),X(226)) passes through these centers:
21, 1805, 1806, 9p859

Collinear triple
21 18051806



D(U,P): Cross Bicentrics Collinearity Cubics,

defined by

ux(p2yz - qrx2) + vy(q2zx - rpy2) + wz(r2xy - pqz2)

Definition:   The cross bicentrics of P and X are a bicentric pair of points here introduced by trilinears

ry : pz : qx    and    qz : rx : py

Locus:

(1)   The cross bicentrics collinearity cubic D(U,P) is the locus of a point X = x : y : z such that the trilinear product U*X is on the line of the cross bicentrics of P and X.

(2)   D(U,P) is the locus of a point X = x : y : z such that the P-Hirst inverse of X is on the line

uxα + vyβ + wzγ = 0.

(3)   D(U,P) is the locus of X such that U*X is collinear with crossdifference(P,X) and crosssum(P,X).

Notes:

1.   D(U,P) is also given by

uqrx3 + vrpy3 + wpqz3 - (up2 + vq2 + wr2)xyz = 0.

2.   D(U,P) meets the sidelines BC, CA, AB in these points:

0 : w1/3q1/3 : - v1/3r1/3,    - w1/3p1/3 : 0 : u1/3r1/3,    v1/3p1/3 : - u1/3q1/3 : 0

3.   The point P is on D(U,P).

4.   D(U,P) = B(U*P,P -1). That is, every D-cubic is a B-cubic, and in particular, D(U,X(1)) = B(U*P,X(1)).


D(X(1),X(3)) passes through these centers:
3, 6, 511, 647, 523p1822, 523p1823

Collinear triples
3 6511
511 523p1822523p1823

D(X(1),X(5)) passes through these centers:
5, 523, 2081


D(X(2),X(55)) passes through these centers:
6, 55, 663, 672, 650p1381, 650p1382

Collinear triples
6 55672
672 650p1381650p1382

D(X(6),X(9)) passes through these centers:
1, 9, 518, 650, 522p1381, 522p1382

Collinear triples
1 9518
9 522p1381522p1382

D(X(7),X(6)) passes through these centers:
6, 55, 663, 672, 650p1381, 650p1382

Collinear triples
6 55 672
672 650p1381650p1382

D(X(8),X(56)) passes through these centers:
6, 56, 649, 1458, 513p1381, 513p1382

Collinear triples
6 561458
1458 513p1381513p1382

D(X(10),X(1)) passes through these centers:
1, 6, 238, 649

Collinear triple
1 6238

D(X(11),X(109)) passes through these centers:
6, 101, 109, 1381, 1382, 2283

Collinear triples
101 1092283
1381 13822283

D(X(19),X(6)) passes through these centers:
3, 6, 511, 647, 523p1822, 523p1823

Collinear triples
3 6511
511 523p1822523p1823

D(X(31),X(63)) passes through these centers:
1, 63, 656, 1959, 525p1113, 525p1114, 684p1316

Collinear triples
1 631959
1959 525p1113525p1114

D(X(37),X(6)) passes through these centers:
1, 6, 238, 649

Collinear triple
1 6238

D(X(42),X(2)) passes through these centers:
1, 6, 238, 649

Collinear triple
1 2239

D(X(48),X(19)) passes through these centers:
1, 19, 240, 4p647, 523p1113, 523p1114

Collinear triples
1 19240
240 523p1113523p1114

D(X(55),X(57)) passes through these centers:
1, 57, 241, 513, 514p1381, 514p1382

Collinear triples
1 57241
241 514p1381514p1382

D(X(56),X(1)) passes through these centers:
1, 9, 518, 650, 522p1381, 522p1382

Collinear triples
1 9518
518 522p1381522p1382

D(X(60),X(65)) passes through these centers:
37, 65, 661, 10p1458, 523p1381, 523p1382

Collinear triples
37 6510p1458
10p1458 523p1381523p1382

D(X(63),X(25)) passes through these centers:
6, 25, 232, 512, 661p1113, 661p1114

Collinear triples
6 25232
232 661p1113661p1114

D(X(76),X(41)) passes through these centers:
31, 41, 2223, 6p663, 663p1381, 663p1381

Collinear triples
31 412223
6p663 663p1381663p1382

D(X(92),X(31)) passes through these centers:
31, 48, 810, 1755, 647p1113, 647p1114

Collinear triples
31 481755
1755 647p1113647p1114

D(X(220),X(1)) passes through these centers:
1, 57, 241, 513, 514p1381, 514p1382

Collinear triples
1 57241
241 514p1381514p1382

D(X(255),X(1)) passes through these centers:
1, 19, 240, 661, 523p1113, 523p1114

Collinear triples
1 19240
240 523p1113523p1114

D(X(326),X(6)) passes through these centers:
6, 25, 232, 512, 661p1113, 661p1114

Collinear triples
6 25232
232 661p1113661p1114

D(X(346),X(6)) passes through these centers:
6, 56, 649, 1458, 513p1381, 513p1382

Collinear triples
6 561458
1458 513p1381513p1382

D(X(513),X(6)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 6p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

D(X(513),X(9)) passes through these centers:
1, 9, 165, 518, 2348, 9p910, 9p1743

Collinear triples
1 9518
1 1659p910
9 23489p1743
165 5189p1743

D(X(513),X(37)) passes through these centers:
1, 37, 42, 846, 1045, 1757, 2238, 37p2106

Collinear triples
1 371757
1 4237p2106
37 422238
37 104537p2106
42 8461757
846 10452238

D(X(514),X(1)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 5p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

D(X(514),X(36)) passes through these centers:
36, 55, 202, 203, 15p1251, 1833p2193

Collinear triples
36 20215p1251
36 2031833p2193
55 15p12511833p2193

D(X(514),X(37)) passes through these centers:
6, 37, 1030, 48p424, 71p422

Collinear triples
6 3771p422
6 103048p424

D(X(560),X(69)) passes through these centers:
2, 69, 325, 525, 850p1822, 850p1823

Collinear triples
2 69325
325 850p1822850p1823

D(X(649),X(2)) passes through these centers:
1, 2, 9, 192, 239, 518, 1575, 2p1742

Collinear triples
1 2239
1 9518
2 1921575
9 192239
192 5182p1742

D(X(649),X(42)) passes through these centers:
1, 37, 42, 846, 1045, 1757, 2238, 37p2106

Collinear triples
1 371757
1 4237p2106
37 422238
37 104537p2106
42 8461757
846 10452238

D(X(650),X(3)) passes through these centers:
1, 3, 1745, 1936, 3p46

Collinear triples
1 31936
1745 19363p46

D(X(650),X(55)) passes through these centers:
1, 6, 43, 55, 238, 672, 1742, 6p1575

Collinear triples
1 6238
1 436p1575
6 55672
43 55238
43 6721742

D(X(650),X(57)) passes through these centers:
1, 57, 241, 910, 1279, 1419, 1743

Collinear triples
1 57241
1 2411743
57 9101419
241 14191743

D(X(650),X(65)) passes through these centers:
1, 65, 73, 851, 1046, 1047, 1758, 65p416

Collinear triples
1 651758
1 7365p416
65 73851
65 104765p416
73 10461758
851 10461047

D(X(652),X(73)) passes through these centers:
1, 65, 73, 851, 1046, 1047, 1758, 65p416

Collinear triples
1 651758
1 7365p416
65 73851
65 104765p416
73 10461758
851 10461047

D(X(656),X(19)) passes through these centers:
1, 19, 204, 240, 1707, 2312, 2p1692

Collinear triples
1 19240
1 17072p1692
19 2042312
204 2401707

D(X(656),X(48)) passes through these centers:
1, 31, 48, 1580, 1740, 1755, 1955, 31p2227

Collinear triples
1 311580
1 481955
1 174031p2227
31 481755
48 15801740

D(X(657),X(7)) passes through these centers:
1, 7, 57, 241, 1447, 1463, 2p1423, 7p1742

Collinear triples
1 57241
1 14472p1423
7 571447
7 14632p1423
241 2p14237p1742

D(X(661),X(2)) passes through these centers:
2, 3, 6, 194, 385, 401, 511, 6p2227

Collinear triples
2 3401
2 6385
2 1946p2227
3 6511
3p76 194385

D(X(661),X(21)) passes through these centers:
1, 3, 21, 416, 1936, 21p1046, 21p1047

Collinear triples
1 31936
1 41621p1047
3 21416
1936 21p104621p1047

D(X(661),X(31)) passes through these centers:
1, 31, 48, 1580, 1740, 1755, 1955, 31p2227

Collinear triples
1 311580
1 481955
1 174031p2227
31 481755
48 15801740

D(X(661),X(63)) passes through these centers:
1, 63, 610, 1959, 3p193, 3p230, 3p1503

Collinear triples
1 631959
1 6103p1503
63 3p1933p230
610 19593p193

D(X(661),X(81)) passes through these centers:
1, 6, 81, 238, 1931, 2106, 58p1654, 58p1655

Collinear triples
1 6238
1 811931
1 210658p1655
6 812106
6 193158p1654
238 58p165458p1655

D(X(667),X(1)) passes through these centers:
1, 2, 9, 192, 239, 518, 1575, 2p1742

Collinear triples
1 2239
1 9518
2 1921575
9 192239
192 5182p1742

D(X(667),X(8)) passes through these centers:
2, 8, 144, 2p518, 2p2348, 8p910, 8p1743

Collinear triples
2 82p518
2 1448p910
8 2p23488p1743
144 2p5188p1743

D(X(667),X(10)) passes through these centers:
2, 10, 37, 740, 1654, 1655, 2p1757, 10p2106

Collinear triples
2 102p1757
2 3710p1757
10 37750
10 165510p1757
37 16542p1757
740 16541655



DP(U,P): D-Plus Cubics,

defined by

ux(p2yz + qrx2) + vy(q2zx + rpy2) + wz(r2xy + pqz2)

Locus:

DP(U,P) is the locus of a point X = x : y : z such that the Hirstpoint of P and X is on the line

uxα + vyβ + wzγ = 0.

Notes:

1.   D(U,P) is also given by

uqrx3 + vrpy3 + wpqz3 + (up2 + vq2 + wr2)xyz = 0.

2.   D(U,P) is the set of points for which a collinearity-determinant is zero; DP(U,P) is the set of points X for which the corresponding permanent is zero. (The permanent is sometimes called "plus-determinant", as it results from changing the minus signs in the definition of determinant to plus signs.)

3.   DP(U,P) meets the sidelines BC, CA, AB in these points:

0 : w1/3q1/3 : - v1/3r1/3,    - w1/3p1/3 : 0 : u1/3r1/3,    v1/3p1/3 : - u1/3q1/3 : 0

4.   DP(U,P) = BP(U*P,P -1). That is, every DP-cubic is a BP-cubic, and in particular, DP(U,X(1)) = BP(U*P,X(1)).


DP(X(1),X(88)) passes through these centers:
88, 44p1635


DP(X(1),X(100)) passes through these centers:
100, 2087


DP(X(31),X(99)) passes through these centers:
99, 1648


DP(X(44),X(1)) passes through these centers:
1, 513


DP(X(75),X(99)) passes through these centers:
99, 31p1645


DP(X(75),X(100)) passes through these centers:
100, 31p1646


DP(X(75),X(101)) passes through these centers:
101, 6p2087


DP(X(75),X(106)) passes through these centers:
106, 44p1960


DP(X(75),X(110)) passes through these centers:
110, 31p1648


DP(X(75),X(111)) passes through these centers:
111, 31p1649


DP(X(88),X(44)) passes through these centers:
44, 1635


DP(X(514),X(55)) passes through these centers:
55, 15p1251, 55p506, 1833p2193

Collinear triple
55 15p12511833p2193

DP(X(552),X(1)) passes through these centers:
1, 2067, 2183, 57p2066

Collinear triple
2067 218357p2066

DP(X(649),X(9)) passes through these centers:
9, 9p506, 283p1832, 283p1833

Collinear triple
9 283p1832283p1833

DP(X(656),X(4)) passes through these centers:
4, 371, 372, 4p2315

Collinear triple
371 3724p2315

DP(X(657),X(1)) passes through these centers:
1, 506, 15p1081, 16p554, 506p1155

Collinear triple
1 15p108116p554

DP(X(661),X(6)) passes through these centers:
6, 15, 16, 6p2234

Collinear triple
6 1516

DP(X(663),X(2)) passes through these centers:
2, 517, 2p2067, 7p2066

Collinear triple
517 2p20677p2066

DP(X(788),X(2)) passes through these centers:
1, 2, 239, 752, 2230

Collinear triple
1 2239

DP(X(798),X(1)) passes through these centers:
1, 2234, 2p15, 2p16

Collinear triple
1 2p152p16

DP(X(798),X(30)) passes through these centers:
2, 30, 617, 298p2173, 299p2173

Collinear triples
2 617299p2173
30 298p2173299p2173

DP(X(810),X(92)) passes through these centers:
1725, 2p371, 2p372, 17p473

Collinear triple
1725 2p3712p372

DP(X(822),X(19)) passes through these centers:
19, 4p15, 4p16, 19p2234

Collinear triple
19 4p154p16

DP(X(824),X(31)) passes through these centers:
6, 31, 1914, 6p2243, 31p2230

Collinear triple
6 311914

DP(X(1019),X(42)) passes through these centers:
42, 10p2151, 10p2152, 42p2234

Collinear triple
42 10p215110p2152

DP(X(1491),X(1)) passes through these centers:
1, 6, 238, 2243, 6p2230

Collinear triple
1 6238

DP(X(1919),X(8)) passes through these centers:
8, 8p506, 298p1251, 1812p1833

Collinear triple
8 298p12511812p1833

DP(X(1924),X(2)) passes through these centers:
2, 298, 299, 538

Collinear triple
1 298299


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

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