# 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

## 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 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 : 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 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 : 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 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 : 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 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