POINTS ON CUBICS: B-, BP-, D-, and DP- Cubics

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 : w^1/3r^2/3 : - v^1/3q^2/3,    - w^1/3r^2/3 : 0 : u^1/3p^2/3,    v^1/3q^2/3 : - u^1/3p^2/3 : 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 56 1458

1458 513p1381 513p1382

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

10 37 740

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

513 514 812

513 650 2254

514 43p514 513p1575

650 812 43p514

2254 43p514 514p1742

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

1 87 1463p2053

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

37 42 2238

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

513 649 659

513 43p513 649p1575

649 663 665

659 663 43p513

665 43p513 513p1742

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

520 647 684

520 193p822 230p822

647 20p810 441p810

684 20p810 193p822

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

512 523 804

512 647 19p684

523 647 401p661

523 194p661 512p2227

647 804 194p661

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

3 6 511

511 523p1822 523p1823

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

65 647 851

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

19 34 1876

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

6 64 441p2155

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

65 73 851

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

71 72 3p740

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

32 184 237

237 512p1822 512p1823

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

520 525 401p656

520 647 681

520 63p804 194p656

525 647 63p804

525 194p656 647p2227
B(X(5),X(3)) passes through these centers:
4, 6, 647, 4p1955
Collinear triple

4 6 4p1955

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

1 57 241

241 514p1381 514p1382

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

6 25 232

232 661p1113 661p1114

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

31 41 2223

2223 663p1381 663p1382

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

31 41 2223

2223 663p1381 663p1382

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

6 55 672

672 650p1381 650p1382

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

1 6 238

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

31 32 2110

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

34 56 1430

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

512 649 423p810

512 798 42p659

512 512p2106 649p1045

649 798 512p2106

798 423p810 649p846

42p659 649p846 649p1045

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

514 649 81p804

514 1459 401p513

514 194p513 649p2227

649 1459 28p684

1459 81p804 194p513

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

31 48 1755

1755 647p1113 647p1114

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

3 6 511

511 523p1822 523p1823

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

71 72 3p740

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

71 73 65p1818

65p1818 656p1381 656p1382

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

2 7 2p241

2p241 693p1381 693p1382

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

1 19 240

240 523p1113 523p1114

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

513 514 812

513 650 2254

514 43p514 513p1575

650 812 43p514

2254 43p514 514p1742

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

2 4 297

297 1113p1577 1114p1577

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

512 523 19p684

512 647 19p684

523 647 401p661

523 194p661 512p2227

647 19p684 194p661

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

514 522 918

514 693 2p812

522 2p812 43p693

693 43p693 513p726

918 43p693 693p1742

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

514 523 423p656

514 661 523p2106

523 661 10p659

523 513p1655 523p2106

661 423p656 513p1654

10p659 513p1654 513p1655

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

1 9 518

518 522p1381 522p1382

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

514 522 918

514 9p676 144p513

522 145p650 514p2348

918 144p513 145p650

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

1 9 518

518 522p1381 522p1382

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

514 522 918

514 9p676 144p513

522 145p650 514p2348

918 144p513 145p650

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

6 55 672

672 650p1381 650p1382

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

513 650 2254

513 55p676 144p649

650 145p663 513p2348

2254 144p649 145p663

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

514 523 42p656

514 661 523p2106

523 661 10p659

523 513p1655 523p2106

661 423p656 513p1654

10p659 513p1654 513p1655

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

512 513 661p2106

512 661 37p659

512 423p647 513p846

513 661 423p647

661 513p1045 661p2106

37p659 513p846 513p1045

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

513 650 2254

513 109p1566 513p1282

513 238p2254 514p2110

650 238p2254 513p1282

665 2254 238p2254

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

513 650 2254

2254 11p1381 11p1382

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

520 647 684

681 125p1822 125p1823

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

3 6 511

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

6 523 1316

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

3 6 511

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

2 6 385

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

6 523 1316

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

2 69 325

325 850p1822 850p1823

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

520 525 401p656

520 647 684

520 63p804 194p646

525 647 62p804

525 194p656 647p2227

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

6 56 1458

1458 513p1381 513p1382

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

1 2 239

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

75 76 1921

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

28 81 27p511

27p511 514p1113 514p1114

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

4 470 471

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

2 3 401

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

1 57 241

241 514p1381 514p1382

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

1 3 1936

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

512 663 415p810

512 810 243p810

512 416p512 663p1047

663 810 416p572

810 415p810 650p2305

243p810 650p2305 663p1047

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

522 652 401p650

522 663 21p804

522 194p650 663p2227

652 663 232p521

652 21p804 194p650

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

3 4 450

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

59 1252 59p518

59p518 100p1381 100p1382

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

44 1023p1052 1023p1054

1023 1635 1023p1054

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

1 19 240

240 523p1113 523p1114

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

522 523 415p656

522 656 416p523

523 656 243p656

523 416p523 522p1047

656 415p656 522p1046

243p656 522p1046 522p1047

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

520 647 684

520 147p810 684p1580

647 132p822 147p810

684 31p684 684p1580

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

513 649 659

513 513p2108 659p1575

649 513p2108 659p1757

659 37p659 659p1757

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

513 514 812

513 238p2254 514p2110

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

513 514 812

513 514p2108 812p1757

514 514p2108 659p726

812 10p659 812p1757

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

31 1973 2p2211

2p2211 512p1113 512p1114

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

513 649 659

649 238p665 513p2110

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

32 1397 31p1458

31p1458 667p1381 667p1382

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

31 604 6p1458

6p1458 649p1381 649p1382

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

6 25 232

232 661p1113 661p1114

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

6 15 16

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

2 4 297

2 193 230

4 1249 1503

193 297 1249

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

58 81 21p1423

58 284 21p1458

81 21p1423 15p1575

284 21p1423 21p1423

21p1423 21p1458 81p1742

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

2 13 94p1095

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

2 14 94p1094

14 30 617p2154

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

2 69 325

2 76 2p1966

69 76 75p401

69 2p1966 75p194

76 698 75p194

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

2 10 2p1757

2 37 10p1757

10 37 750

10 1655 10p1757

37 1654 2p1757

740 1654 1655

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

6 32 1691

6 184 1971

6 1613 32p2227

32 184 237

184 1613 1691

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

1 2 239

1 9 518

2 192 1575

9 192 239

192 518 2p1742

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

1 57 241

1 241 1743

57 910 1419

241 1419 1743

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

1 6 238

1 81 1931

1 2106 58p1655

6 81 2106

6 1931 58p1654

238 58p1654 58p1655

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

6 31 1914

6 2176 31p1575

31 41 2223

41 1914 2176

2176 2223 6p1742

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

6 56 1458

6 6p1279 6p1743

56 6p910 6p1419

1458 6p1419 6p1743

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

6 31 1914

6 58 1326

6 6p2106 58p1045

31 58 6p2106

31 1326 58p846

1914 58p846 58p1045

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

10 42 10p1580

10 71 10p1955

10 10p1740 42p2227

42 71 10p1755

71 10p1580 10p1740

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

31 32 2210

31 2209 32p1575

32 2175 6p2223

2175 2209 2210

2209 6p2223 31p1742

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

2 4 297

2 193 230

4 1249 1503

193 297 1249

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

6 55 672

55 6p2348 41p145

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

6 56 1458

6 1423 1429

56 57 1429

57 1423 6p1463

1423 1458 57p1742

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

6 19 2202

2178 2202 6p1148

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

371 372 4p2315

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

2 3 401

2 6 385

2 194 6p2227

3 6 511

3 194 385

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

6 15 16

16 74 617p2159

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

58 81 21p1429

58 284 21p1458

81 21p1428 58p1575

284 21p1428 21p1429

21p1423 21p1458 81p1742

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

6 25 232

6 1692 6p1707

25 6p204 6p2312

232 6p204 6p1707

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

6 42 6p1757

6 213 42p2106

42 213 3p862

42 6p1045 42p2106

213 6p846 6p1757

3p862 6p1045

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

4 6 4p1955

4 25 419

4 4p1740 25p2227

6 25 232

6 419 4p1740

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

3 6 511

3 3p1707 48p230

6 154 3p2312

154 511 3p1707

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

28 1474 27p1914

28 27p2176 726p2203

1474 2299 27p2223

2299 27p1914 27p2172

27p2176 27p2223 28p1742

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

94 265 94p1955

94 1989 94p1580

94 94p1740 1989p2227

265 1989 94p1755

265 94p1580 94p1740

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

75 92 2p297

2p297 850p1113 850p1114

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

2 20 441

2 69 325

20 325 63p193

69 3p1733 63p193

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

2 8 2p518

2 75 350

8 350 2p192

75 726 2p192

2p192 2p518 75p1742

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

1 2 239

1 86 2p2106

1 2p1931 81p1654

2 86 2p1931

2 2p2106 81p1655

239 81p1654 81p1655

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

2 7 2p241

2 145 2p1279

7 516 2p1419

145 2p241 2p1419

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

10 12 10p1758

10 12p416 37p1214

12 10p851 37p1214

12 10p1047 12p416

10p851 10p1046 10p1047

10p1046 10p1758 37p1214

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

1 4 243

46 243 1148

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

1 57 241

1 1447 2p1423

7 57 1447

7 1463 2p1423

241 2p1423 7p1742

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

1 9 518

1 165 9p910

9 2348 9p1743

165 518 9p1743

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

1 3 1936

1 21 3p415

1 416 21p1047

3 21 416

3 3p415 21p1046

1936 21p1046 21p1047

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

11 2170 11p238

11 11p43 1463p2310

2170 2310 11p672

2310 11p43 11p238

11p43 11p672 11p1742

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

1 4 243

1 29 4p416

1 415 29p1046

4 29 415

4 4p416 29p1047

243 29p1046 29p1047

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

1 63 1959

1 610 3p1503

63 3p193 3p230

610 1959 3p193

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

1 19 240

1 92 4p401

1 2p419 4p194

19 92 2p419

92 4p194 19p2227

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

27 28 27p238

27 27p43 28p1575

28 1172 21p1876

1172 27p43 27p238

21p1876 27p43 27p1742

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

2 7 2p241

2 145 2p1279

7 516 2p1419

145 2p241 2p1419

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

7 77 7p1936

7 86 77p415

7 7p416 86p1047

77 86 7p416

77 77p415 86p1046

7p1936 86p1046 86p1047

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

1 65 1758

1 73 65p416

65 73 851

65 1047 65p416

73 1046 1758

851 1046 1047

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

21 81 21p241

21 21p1447 43p86

81 86 21p1447

86 21p1463 43p86

21p241 43p86 86p1742

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

1 19 240

1 1707 2p1692

19 204 2312

204 240 1707

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

1 37 1757

1 42 37p2106

37 42 2238

37 1045 37p2106

42 846 1757

846 1045 2238

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

6 56 1548

6 1423 1429

56 57 1429

57 1423 6p1463

1423 1458 57p1742

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

2 7 2p241

2 2p1447 7p192

7 85 2p1447

85 2p1463 7p192

2p241 7p192 85p1742

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

2 8 2p518

2 144 8p910

8 2p2348 8p1743

144 2p518 8p1743

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

12 226 12p1931

12 2171 10p1284

12 12p2106 65p1655

226 2171 12p2106

2171 12p1931 65p1654

10p1284 65p1654 65p1655

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

1 2 239

1 9 518

2 192 1575

9 192 239

192 518 2p1742

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

75 76 1921

75 312 69p1861

76 2p726 43p76

312 1921 43p76

43p76 69p1861 76p1742

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

2 3 401

2 6 385

2 194 6p2227

3 6 511

3p76 194 385

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

86 333 86p518

274 43p274 81p726

43p274 86p518 274p1742

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

94 300 301

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

2 10 2p1757

2 37 10p2106

10 37 740

10 1655 10p2106

37 1654 2p1757

740 1654 1655

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

6 31 1914

6 2176 31p1575

31 41 2223

41 1914 2176

2176 2223 6p1742

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

37 213 10p1691

37 228 10p1971

37 10p1613 213p2227

213 228 10p237

228 10p1613 10p1691

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

21 81 21p241

21 21p1447 43p86

81 86 21p1447

86 21p1463 43p86

21p241 43p86 86p1742

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

1 31 1580

1 48 1955

1 1740 31p2227

31 48 1755

48 1580 1740

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

1 37 1757

1 42 37p2106

37 42 2238

37 1045 37p2106

42 846 1757

846 1045 2238

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

27 28 27p238

27 27p43 28p1575

28 1172 21p1872

1172 27p43 27p238

21p1872 27p43 27p1742

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

1 65 1758

1 73 65p416

65 73 851

65 1047 65p416

73 1046 1758

851 1046 1047

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

75 92 2p297

75 1733 2p193

92 1895 2p1503

1895 2p193 2p297

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

1 31 1580

1 1740 31p2227

1 1995 34p2289

31 1755 34p2289

1580 1740 34p2289

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

83 251 82p385

83 1176 82p401

83 82p194 251p2227

251 1176 82p511

1176 82p194 82p385

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

6 32 1691

6 184 1971

6 1613 32p2227

32 184 237

184 1613 1691

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

1 2 239

1 2p2110 238p518

2 2p2110 740p2106

239 2238 740p2106

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

1 6 238

1 2108 238p1575

6 2108 238p1757

238 2238 238p1757

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

2 75 350

2 75p2108 350p1757

75 75p2108 239p726

350 2p740 350p1757

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

1 6 238

1 2108 238p1575

6 2108 238p1557

238 2238 238p1557

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

1 2 239

1 2p2110 238p518

2 2p2110 740p2106

239 2238 740p2106

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

31 32 2210

31 31p2108 1575p2210

32 31p2108 1757p2210

2210 31p2238 1757p2210

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

2 4 297

2 19p147 232p1966

4 132 19p147

232 297 232p1966

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

6 25 232

6 147p1973 232p1580

25 31p132 147p1973

232 2211 232p1580

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

2 7 2p241

2 2p2114 239p241

7 2p2114 241p516

7 85p2110 239p241

241 2p241 239p241

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

6 56 1458

6 6p2114 238p1458

56 6p2114 672p1456

56 57p2110 238p1458

1458 6p1458 238p1458

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

88 106 88p238

88 43p88 106p1575

106 2316 88p672

2316 43p88 88p238

43p88 88p672 88p1742

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

19 34 1876

19 19p1279 19p1743

34 6p1886 19p1419

1876 19p1419 19p1743

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

1 37 1757

1 42 37p2106

37 42 2238

37 1045 37p2106

42 846 1757

846 1045 2238

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

10 37 740

10 10p43 37p1575

37 210 10p672

210 740 10p43

10p43 10p672 10p1742

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

1 65 1758

1 73 65p416

65 73 851

65 1047 65p416

73 1046 1758

851 1046 1047

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

1 57 241

1 2114 238p241

57 2114 241p910

57 7p2110 238p241

241 1458 238p241

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

1 9 518

1 1282 59p1566

1 2p2110 238p518

9 1282 238p518

518 672 238p518

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

19 25 2201

19 4p2176 25p1575

25 607 2356

607 2201 4p2176

2356 4p2176 19p1742

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

4 19 242

4 27 423

4 4p2106 27p1045

19 27 4p2106

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

1 31 1580

1 48 1955

1 1740 31p2227

31 48 1755

48 1580 1740

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

1 244 1054

513 1052 1054

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

2 8 2p518

2 75 350

8 350 2p192

75 726 2p192

2p192 2p518 75p1742

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

1 63 1959

1 75 83p732

63 75 2p401

63 2p164 83p732

75 2227 2p194

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

1 9 518

1 1282 59p1566

1 2p2110 238p518

9 1282 238p518

518 672 238p518

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 : w^1/3r^2/3 : - v^1/3q^2/3,    - w^1/3r^2/3 : 0 : u^1/3p^2/3,    v^1/3q^2/3 : - u^1/3p^2/3 : 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 15p798 16p798

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 15p661 16p661

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 1491 238p824

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 1491 238p824

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

686 371p656 372p656

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 15 16

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

21 1805 1806

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 : w^1/3q^1/3 : - v^1/3r^1/3,    - w^1/3p^1/3 : 0 : u^1/3r^1/3,    v^1/3p^1/3 : - u^1/3q^1/3 : 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 6 511

511 523p1822 523p1823

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 55 672

672 650p1381 650p1382

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

1 9 518

9 522p1381 522p1382

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

6 55 672

672 650p1381 650p1382

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

6 56 1458

1458 513p1381 513p1382

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

1 6 238

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

101 109 2283

1381 1382 2283

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

3 6 511

511 523p1822 523p1823

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

1 63 1959

1959 525p1113 525p1114

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

1 6 238

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

1 2 239

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

1 19 240

240 523p1113 523p1114

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

1 57 241

241 514p1381 514p1382

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

1 9 518

518 522p1381 522p1382

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

37 65 10p1458

10p1458 523p1381 523p1382

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

6 25 232

232 661p1113 661p1114

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

31 41 2223

6p663 663p1381 663p1382

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

31 48 1755

1755 647p1113 647p1114

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

1 57 241

241 514p1381 514p1382

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

1 19 240

240 523p1113 523p1114

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

6 25 232

232 661p1113 661p1114

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

6 56 1458

1458 513p1381 513p1382

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

1 9 518

1 165 9p910

9 2348 9p1743

165 518 9p1743

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

1 37 1757

1 42 37p2106

37 42 2238

37 1045 37p2106

42 846 1757

846 1045 2238

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

36 202 15p1251

36 203 1833p2193

55 15p1251 1833p2193

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

6 37 71p422

6 1030 48p424

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

2 69 325

325 850p1822 850p1823

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

1 2 239

1 9 518

2 192 1575

9 192 239

192 518 2p1742

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

1 37 1757

1 42 37p2106

37 42 2238

37 1045 37p2106

42 846 1757

846 1045 2238

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

1 3 1936

1745 1936 3p46

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

1 6 238

1 43 6p1575

6 55 672

43 55 238

43 672 1742

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

1 57 241

1 241 1743

57 910 1419

241 1419 1743

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

1 65 1758

1 73 65p416

65 73 851

65 1047 65p416

73 1046 1758

851 1046 1047

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

1 65 1758

1 73 65p416

65 73 851

65 1047 65p416

73 1046 1758

851 1046 1047

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

1 19 240

1 1707 2p1692

19 204 2312

204 240 1707

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

1 31 1580

1 48 1955

1 1740 31p2227

31 48 1755

48 1580 1740

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

1 57 241

1 1447 2p1423

7 57 1447

7 1463 2p1423

241 2p1423 7p1742

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

2 3 401

2 6 385

2 194 6p2227

3 6 511

3p76 194 385

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

1 3 1936

1 416 21p1047

3 21 416

1936 21p1046 21p1047

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

1 31 1580

1 48 1955

1 1740 31p2227

31 48 1755

48 1580 1740

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

1 63 1959

1 610 3p1503

63 3p193 3p230

610 1959 3p193

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

1 6 238

1 81 1931

1 2106 58p1655

6 81 2106

6 1931 58p1654

238 58p1654 58p1655

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

1 2 239

1 9 518

2 192 1575

9 192 239

192 518 2p1742

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

2 8 2p518

2 144 8p910

8 2p2348 8p1743

144 2p518 8p1743

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

2 10 2p1757

2 37 10p1757

10 37 750

10 1655 10p1757

37 1654 2p1757

740 1654 1655

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 : w^1/3q^1/3 : - v^1/3r^1/3,    - w^1/3p^1/3 : 0 : u^1/3r^1/3,    v^1/3p^1/3 : - u^1/3q^1/3 : 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 15p1251 1833p2193

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

2067 2183 57p2066

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

9 283p1832 283p1833

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

371 372 4p2315

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

1 15p1081 16p554

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

6 15 16

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

517 2p2067 7p2066

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

1 2 239

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

1 2p15 2p16

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

2 617 299p2173

30 298p2173 299p2173

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

1725 2p371 2p372

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

19 4p15 4p16

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

6 31 1914

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

42 10p2151 10p2152

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

1 6 238

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

8 298p1251 1812p1833

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

1 298 299

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

513	514	812
513	650	2254
514	43p514	513p1575
650	812	43p514
2254	43p514	514p1742

513	649	659
513	43p513	649p1575
649	663	665
659	663	43p513
665	43p513	513p1742

512	523	804
512	647	19p684
523	647	401p661
523	194p661	512p2227
647	804	194p661

520	525	401p656
520	647	681
520	63p804	194p656
525	647	63p804
525	194p656	647p2227

512	649	423p810
512	798	42p659
512	512p2106	649p1045
649	798	512p2106
798	423p810	649p846
42p659	649p846	649p1045

514	649	81p804
514	1459	401p513
514	194p513	649p2227
649	1459	28p684
1459	81p804	194p513

514	522	918
514	693	2p812
522	2p812	43p693
693	43p693	513p726
918	43p693	693p1742

514	523	423p656
514	661	523p2106
523	661	10p659
523	513p1655	523p2106
661	423p656	513p1654
10p659	513p1654	513p1655

513	650	2254
513	55p676	144p649
650	145p663	513p2348
2254	144p649	145p663

514	523	42p656
514	661	523p2106
523	661	10p659
523	513p1655	523p2106
661	423p656	513p1654
10p659	513p1654	513p1655

512	513	661p2106
512	661	37p659
512	423p647	513p846
513	661	423p647
661	513p1045	661p2106
37p659	513p846	513p1045

513	650	2254
513	109p1566	513p1282
513	238p2254	514p2110
650	238p2254	513p1282
665	2254	238p2254

520	525	401p656
520	647	684
520	63p804	194p646
525	647	62p804
525	194p656	647p2227

512	663	415p810
512	810	243p810
512	416p512	663p1047
663	810	416p572
810	415p810	650p2305
243p810	650p2305	663p1047

522	652	401p650
522	663	21p804
522	194p650	663p2227
652	663	232p521
652	21p804	194p650

522	523	415p656
522	656	416p523
523	656	243p656
523	416p523	522p1047
656	415p656	522p1046
243p656	522p1046	522p1047

520	647	684
520	147p810	684p1580
647	132p822	147p810
684	31p684	684p1580

513	649	659
513	513p2108	659p1575
649	513p2108	659p1757
659	37p659	659p1757

513	514	812
513	514p2108	812p1757
514	514p2108	659p726
812	10p659	812p1757

58	81	21p1423
58	284	21p1458
81	21p1423	15p1575
284	21p1423	21p1423
21p1423	21p1458	81p1742

2	10	2p1757
2	37	10p1757
10	37	750
10	1655	10p1757
37	1654	2p1757
740	1654	1655

1	6	238
1	81	1931
1	2106	58p1655
6	81	2106
6	1931	58p1654
238	58p1654	58p1655

6	31	1914
6	58	1326
6	6p2106	58p1045
31	58	6p2106
31	1326	58p846
1914	58p846	58p1045

10	42	10p1580
10	71	10p1955
10	10p1740	42p2227
42	71	10p1755
71	10p1580	10p1740

31	32	2210
31	2209	32p1575
32	2175	6p2223
2175	2209	2210
2209	6p2223	31p1742

58	81	21p1429
58	284	21p1458
81	21p1428	58p1575
284	21p1428	21p1429
21p1423	21p1458	81p1742

6	42	6p1757
6	213	42p2106
42	213	3p862
42	6p1045	42p2106
213	6p846	6p1757
3p862		6p1045

28	1474	27p1914
28	27p2176	726p2203
1474	2299	27p2223
2299	27p1914	27p2172
27p2176	27p2223	28p1742

94	265	94p1955
94	1989	94p1580
94	94p1740	1989p2227
265	1989	94p1755
265	94p1580	94p1740

1	2	239
1	86	2p2106
1	2p1931	81p1654
2	86	2p1931
2	2p2106	81p1655
239	81p1654	81p1655