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Table 19-1 gives a list of triangles whose vertices lie on the Neuberg cubic K001 (P* is the isogonal conjugate of P). Other points are mentioned in Table 10 (points related with X370) and Table 18 (Ua, Ub, Uc and related points).

Table 19-2 gives the perspectors of these triangles which also lie on the Neuberg cubic (P/Q denotes a cevian quotient).

Table 19-3 gives the sixth common point S of the circumcircle of a triangle with the Neuberg cubic. See also Table 27.

Table 19-4 gives the tangentials of centers listed in ETC.

See the related paper Perspective Triangles inscribed in the Neuberg Cubic.

 

Table 19-1

#

A-vertex

triangle or definition

1

A

reference triangle

2

Ia

excentral triangle

3

A'

reflection triangle

4

A'*

A'* = X3 A /\ X30 A'

5

Ae

outer equilateral triangle

6

Ai

inner equilateral triangle

7

Ae*

8

Ai*

9

Ao

cevian triangle of X30

10

A1

A1 = X1 A' /\ X3 Ia

11

A2

A2 = X74 Ao /\ Bo Co

12

A3

A3 = X3 Ao /\ X74 A'

13

A4

A4 = X3 A' /\ B' C'

#

A-vertex

triangle or definition

14

A5

A5 = X1 Ao /\ X74 Ia

15

A1*

16

A2*

17

A3*

18

A4*

19

A5*

20

Ae'

Ae' = X15 Ae /\ Be Ce

21

Ai'

Ai' = X16 Ai /\ Bi Ci

22

Ae'*

23

Ai'*

24

A6

A6 = X4 Ao /\ X74 A'*

25

A6*

 

Table 19-2 : the cells in yellow contain centers listed in ETC, those in green were added in June 2014 (centers X5667-X5685), those in pink and light blue were added in October 2015 (centers above X7000). The remaining white cells were filled up in November 2015 with a good help from Peter Moses.

table sorted by increasing SEARCH numbers

#

#

SEARCH #

center

16

20

-263,42417606410

X8513

8

18

-105,97117045170

X8475

10

13

-31,79303507337

X8480

2

5

-24,75747987218

X1276

8

10

7

19

-22,77449322373

X8468

15

21

-20,38529954404

X8507

13

21

-13,20931700400

X1337

1

22

1

16

-12,72744887665

X2132 = X30/X74

4

17

2

19

3

25

8

21

-12,09074802946

X8477

5

19

-9,50649691661

X8449

14

24

-8,56804872862

X7328 = X5680*

9

22

-8,08438463657

X5681 = X30/X3479

17

25

-7,99810301418

X5676 = X30/X2133

9

16

11

23

-7,71921435140

X8492 = X5675*

20

23

-7,65530824139

X8528

7

24

-7,56731825822

X8471 = X8174*

3

14

-7,12562286909

X3466 = X3465*

11

15

2

24

15

20

-7,09894877848

X8506

16

22

-7,01644428708

X8515

11

13

-6,95294706960

X8487 = X5671*

13

20

-6,93913337660

X1338

1

23

8

11

-6,92073793301

X8452 = X5669*

5

24

3

7

-6,90974081789

X8172

6

18

8

20

6

10

-6,85080689138

X8454

2

3

-6,84088208278

X484

1

15

14

17

10

18

19

24

4

12

-6,80436889544

X1263 = X1157*

1

13

9

23

-6,79718756277

X5682 = X30/X3480

3

5

-6,78421149584

X16

1

8

6

21

7

17

-6,77364835600

X8466

18

21

-6,75459634942

X8522

10

21

-6,52868086149

X8482

9

15

-6,48255444489

X5677 = X30/X3065

2

17

3

19

9

13

-5,83828820861

X3484 = X30/X1157

22

23

12

25

16

23

-5,72437527035

X8516

-5,67697834270

X8535 = X5678*

1

3

-5,67661941093

X4

14

15

12

18

9

24

11

17

7

23

-5,67649655567

X8470

17

20

-5,67441835647

X8519

6

12

-5,67418845702

X8456 = X8173*

10

12

-5,59478415600

X7329 = X5685*

13

14

2

23

-5,57434076541

X8436

8

14

-5,53893303570

X7059 = X1277*

14

23

-5,53873868123

X8502

8

9

-5,53774499786

X5624 = X30/X14

5

17

7

15

-5,51158980238

X8464

16

19

-5,44518658306

X8512

19

22

-5,41596108283

X8525

3

20

-5,29312743553

X8437

3

11

-5,25494849240

X8431 = X5667*

1

24

4

6

-5,14020505969

X8175

7

13

5

23

22

24

-5,04101304602

X3440 = X616*

12

21

5

16

-4,96908901433

X8448

8

25

7

8

-4,96723694599

X399 = X30/X3

3

9

2

15

1

17

4

18

16

24

5

22

-4,91515929874

X8451

13

25

-4,83047253859

X5683 = X30/X3481

2

16

-4,24677722398

X8432

9

19

15

25

2

22

-4,16465456137

X8435

4

10

-4,05327953741

X3483

2

13

19

23

-3,90043696246

X8526

20

24

-3,84056015706

X8529

11

24

-3,74518947176

X8493

2

8

-3,65127826481

X5673 = X30/X1277

5

15

15

22

-3,52862295733

X8508

17

21

-3,36197974134

X8502

15

16

-3,29274094678

X8503

17

19

7

16

-3,19867129409

X8465

4

8

-3,08546739009

X5668 = X30/X15

5

9

7

22

23

25

-2,91181827916

X8533

4

13

-2,79179433014

X3482

2

9

-1,68618254416

X3464 = X30/X1

4

15

14

16

10

17

1

19

5

13

-1,66290685126

X8447

14

22

-1,54460939818

X8501

13

22

-1,39875648625

X8497

6

25

-1,35199329604

X8463

13

16

-1,13437049968

X8443

4

25

5

8

-0,85875716656

X617

4

21

3

22

8

22

-0,74241177525

X8478

3

16

-0,50752991831

X5670 = X30/X1138

9

17

15

19

18

25

8

16

-0,48139719780

X8473

17

18

-0,45303039991

X8518

3

21

-0,30176748118

X8438

7

9

-0,07593740528

X5623 = X30/X13

6

17

1

5

0,07594515180

X13

8

12

4

23

0,30189011567

X8442

12

13

0,45330765162

X8494 = X5684*

6

11

0,46735718657

X8455 = X5624*

6

20

0,70960433660

X8460

0,93647210260

X8536 = X5679*

11

18

1,05985239030

X8439 = X3484*

3

24

3

4

1,13460438833

X30

5

7

6

8

1

9

10

15

11

16

12

17

13

18

14

19

20

22

21

23

24

25

1,27750126341

X370

18

20

1,28759991341

X8521

19

20

1,54802792230

X8523

1

2

1,69030850946

X1

3

10

9

14

12

15

11

19

2,08851089410

X370*

10

11

2,76314916251

X7327 = X5677*

12

14

3

18

2,80491649699

X5684 = X30/X3482

13

19

2,90916090233

X8485

10

25

21

24

2,92641639951

X8531

10

20

2,93309468208

X8481

3

6

3,10244402066

X15

1

7

5

20

5

11

3,21737310304

X8445 = X5623*

12

23

3,38344985821

X8496

16

25

3,77478964470

X8517

22

25

3,87263509427

X8532

8

19

3,96860638197

X8476

2

20

4,20701449325

X8433

2

11

4,29232349359

X7164 = X3464*

1

14

10

24

1

6

4,37343177201

X14

7

12

11

21

4,51975351723

X8490 = X5682*

1

18

4,61238076845

X1157 = X1263*

20

21

17

24

18

24

4,91058840478

X3481

7

20

5,00313786821

X8469

7

11

5,06270010066

X8462 = X5668*

6

24

1

10

5,23935593651

X3065 = X484*

2

12

4

14

15

23

5,29892807515

X8509

4

16

5,39111800938

X8440

9

25

4

22

5,43749424451

X8441

14

20

5,59419084956

X8499

11

14

5,63387199414

X8488

13

23

5,70925332465

X8498

5

12

5,76852629586

X8446 = X8172*

19

21

5,77192904717

X8524

4

7

5,80658311911

X5669 = X30/X16

6

9

8

23

1

21

5,86086297690

X3480 = X1338*

3

17

5,89586797378

X5671 = X30/X1263

9

18

8

17

6,10158941370

X8474

5

21

6,12827772813

X8450

2

4

6,15617358155

X3465 = X30/X484

9

10

13

15

12

19

14

25

4

9

6,16733538680

X5667 = X30/X4

12

16

13

17

10

19

1

25

12

22

6,19319521369

X8495

8

15

6,22591347882

X8472

15

17

6,38042757299

X8504

18

19

2

21

6,40581344896

X8434

6

19

6,44395052045

X8459

4

5

6,45797717165

X8174

8

13

6

22

5

10

6,46987301615

X8444

6

16

6,48657246249

X8458

7

25

18

22

6,53002144149

X5675 = X30/X1338

9

21

16

18

6,55991154588

X8511

6,64573209130

X8534 = X5676*

11

20

6,67852698580

X8489 = X5681*

3

8

6,70447414925

X8173

5

18

7

21

17

23

6,75048438897

X5678 = X30/X616*

20

25

1

4

6,78236289420

X3

5

6

2

10

9

12

3

13

11

25

10

22

6,80744200217

X8483

10

16

6,84565964096

X5680 = X30/X3466

4

19

2

25

3

15

7,08747856659

X5685 = X30/X3483

2

18

14

21

7,12885537346

X8500

2

6

7,19264423626

X1277

7

10

5

25

7,38581312867

X8453

17

22

7,48346819761

X5679 = X30/X617*

21

25

21

22

7,48395751441

X8530

16

21

7,55431061064

X8514

7

18

7,60157180331

X8467

8

24

7,65513888417

X8479 = X8175*

16

17

7,67010162214

X8510

6

7

7,73621607386

X616

4

20

3

23

4

11

7,79134466748

X1138 = X399*

1

12

10

14

13

24

1

11

7,85490562353

X2133 = X2132*

12

24

1

20

7,94636831360

X3479 = X1337*

11

12

8,06377960586

X8486 = X5670*

12

20

8,16674846519

X3441 = X617*

23

24

14

18

8,23665223338

X7165 = X3483*

15

24

6

14

8,33847477390

X7325 = X5673*

19

25

8,45117750091

X8527

7

14

9,41145337340

X7060 = X1276*

6

23

12,01788663654

X8461

9

11

12,65716281429

X74

3

12

2

14

4

24

9

20

13,14063785425

X5674 = X30/X1337

18

23

10

23

20,32666877404

X8484

5

14

22,71706812312

X7326 = X5672*

2

7

24,70077699181

X5672 = X30/X1276

6

15

15

18

31,73783490329

X8505

6

13

105,91794508580

X8457

11

22

263,37110789000

X8491 = X5674*

table sorted by increasing triangles numbers

#

#

SEARCH #

center

1

2

1,69030850946

X1

1

3

-5,67661941093

X4

1

4

6,78236289420

X3

1

5

0,07594515180

X13

1

6

4,37343177201

X14

1

7

3,10244402066

X15

1

8

-6,78421149584

X16

1

9

1,13460438833

X30

1

10

5,23935593651

X3065 = X484*

1

11

7,85490562353

X2133 = X2132*

1

12

7,79134466748

X1138 = X399*

1

13

-6,80436889544

X1263 = X1157*

1

14

4,29232349359

X7164 = X3464*

1

15

-6,84088208278

X484

1

16

-12,72744887665

X2132 = X30/X74

1

17

-4,96723694599

X399 = X30/X3

1

18

4,61238076845

X1157 = X1263*

1

19

-1,68618254416

X3464 = X30/X1

1

20

7,94636831360

X3479 = X1337*

1

21

5,86086297690

X3480 = X1338*

1

22

-13,20931700400

X1337

1

23

-6,93913337660

X1338

1

24

-5,25494849240

X8431 = [X30/X4]*

1

25

6,16733538680

X5667 = X30/X4

2

3

-6,84088208278

X484

2

4

6,15617358155

X3465 = X30/X484

2

5

-24,75747987218

X1276

2

6

7,19264423626

X1277

2

7

24,70077699181

X5672 = X30/X1276

2

8

-3,65127826481

X5673 = X30/X1277

2

9

-1,68618254416

X3464 = X30/X1

2

10

6,78236289420

X3

2

11

4,29232349359

X7164 = X3464*

2

12

5,23935593651

X3065 = X484*

2

13

-4,05327953741

X3483

2

14

12,65716281429

X74

2

15

-4,96723694599

X399 = X30/X3

2

16

-4,24677722398

X8432

2

17

-6,48255444489

X5677 = X30/X484*

2

18

7,08747856659

X5685

2

19

-12,72744887665

X2132 = X30/X74

2

20

4,20701449325

X8433

2

21

6,40581344896

X8434

2

22

-4,16465456137

X8435

2

23

-5,57434076541

X8436

2

24

-7,12562286909

X3466 = X3465*

2

25

6,84565964096

X5680

3

4

1,13460438833

X30

3

5

-6,78421149584

X16

3

6

3,10244402066

X15

3

7

-6,90974081789

X8172

3

8

6,70447414925

X8173

3

9

-4,96723694599

X399 = X30/X3

3

10

1,69030850946

X1

3

11

-5,25494849240

X8431 = [X30/X4]*

3

12

12,65716281429

X74

3

13

6,78236289420

X3

3

14

-7,12562286909

X3466 = X3465*

3

15

7,08747856659

X5685

3

16

-0,50752991831

X5670 = X30/X1138

3

17

5,89586797378

X5671 = X30/X1263

3

18

2,80491649699

X5684

3

19

-6,48255444489

X5677 = X30/X484*

3

20

-5,29312743553

X8437

3

21

-0,30176748118

X8438

3

22

-0,85875716656

X617

3

23

7,73621607386

X616

3

24

1,05985239030

X8439 = X3484*

3

25

-12,72744887665

X2132 = X30/X74

4

5

6,45797717165

X8174

4

6

-5,14020505969

X8175

4

7

5,80658311911

X5669 = X30/X16

4

8

-3,08546739009

X5668 = X30/X15

4

9

6,16733538680

X5667 = X30/X4

4

10

-4,05327953741

X3483

4

11

7,79134466748

X1138 = X399*

4

12

-6,80436889544

X1263 = X1157*

4

13

-2,79179433014

X3482

4

14

5,23935593651

X3065 = X484*

4

15

-1,68618254416

X3464 = X30/X1

4

16

5,39111800938

X8440

4

17

-12,72744887665

X2132 = X30/X74

4

18

-4,96723694599

X399 = X30/X3

4

19

6,84565964096

X5680

4

20

7,73621607386

X616

4

21

-0,85875716656

X617

4

22

5,43749424451

X8441

4

23

0,30189011567

X8442

4

24

12,65716281429

X74

4

25

-1,13437049968

X8443

5

6

6,78236289420

X3

5

7

1,13460438833

X30

5

8

-0,85875716656

X617

5

9

-3,08546739009

X5668 = X30/X15

5

10

6,46987301615

X8444

5

11

3,21737310304

X8445 = X5623*

5

12

5,76852629586

X8446

5

13

-1,66290685126

X8447

5

14

22,71706812312

X7326 = [X30/X1276]*

5

15

-3,65127826481

X5673 = X30/X1277

5

16

-4,96908901433

X8448

5

17

-5,53774499786

X5624 = X30/X14

5

18

6,70447414925

X8173

5

19

-9,50649691661

X8449

5

20

3,10244402066

X15

5

21

6,12827772813

X8450

5

22

-4,91515929874

X8451

5

23

-5,14020505969

X8175

5

24

-6,92073793301

X8452 = [X30/X16]*

5

25

7,38581312867

X8453

6

7

7,73621607386

X616

6

8

1,13460438833

X30

6

9

5,80658311911

X5669 = X30/X16

6

10

-6,85080689138

X8454

6

11

0,46735718657

X8455 = X5624*

6

12

-5,67418845702

X8456

6

13

105,91794508580

X8457

6

14

8,33847477390

X7325

6

15

24,70077699181

X5672 = X30/X1276

6

16

6,48657246249

X8458

6

17

-0,07593740528

X5623 = X30/X13

6

18

-6,90974081789

X8172

6

19

6,44395052045

X8459

6

20

0,70960433660

X8460

6

21

-6,78421149584

X16

6

22

6,45797717165

X8174

6

23

12,01788663654

X8461

6

24

5,06270010066

X8462 = [X30/X15]*

6

25

-1,35199329604

X8463

7

8

-4,96723694599

X399 = X30/X3

7

9

-0,07593740528

X5623 = X30/X13

7

10

7,19264423626

X1277

7

11

5,06270010066

X8462 = [X30/X15]*

7

12

4,37343177201

X14

7

13

-5,14020505969

X8175

7

14

9,41145337340

X7060 = X1276*

7

15

-5,51158980238

X8464

7

16

-3,19867129409

X8465

7

17

-6,77364835600

X8466

7

18

7,60157180331

X8467

7

19

-22,77449322373

X8468

7

20

5,00313786821

X8469

7

21

6,70447414925

X8173

7

22

-3,08546739009

X5668 = X30/X15

7

23

-5,67649655567

X8470

7

24

-7,56731825822

X8471

7

25

6,48657246249

X8458

8

9

-5,53774499786

X5624 = X30/X14

8

10

-24,75747987218

X1276

8

11

-6,92073793301

X8452 = [X30/X16]*

8

12

0,07594515180

X13

8

13

6,45797717165

X8174

8

14

-5,53893303570

X7059 = X1277*

8

15

6,22591347882

X8472

8

16

-0,48139719780

X8473

8

17

6,10158941370

X8474

8

18

-105,97117045170

X8475

8

19

3,96860638197

X8476

8

20

-6,90974081789

X8172

8

21

-12,09074802946

X8477

8

22

-0,74241177525

X8478

8

23

5,80658311911

X5669 = X30/X16

8

24

7,65513888417

X8479

8

25

-4,96908901433

X8448

9

10

6,15617358155

X3465 = X30/X484

9

11

12,65716281429

X74

9

12

6,78236289420

X3

9

13

-5,83828820861

X3484 = X30/X1157

9

14

1,69030850946

X1

9

15

-6,48255444489

X5677 = X30/X484*

9

16

-7,99810301418

X5676 = X30/X2133

9

17

-0,50752991831

X5670 = X30/X1138

9

18

5,89586797378

X5671 = X30/X1263

9

19

-4,24677722398

X8432

9

20

13,14063785425

X5674 = X30/X1337

9

21

6,53002144149

X5675 = X30/X1338

9

22

-8,08438463657

X5681

9

23

-6,79718756277

X5682

9

24

-5,67661941093

X4

9

25

5,39111800938

X8440

10

11

2,76314916251

X7327 = [X30/X484*]*

10

12

-5,59478415600

X7329

10

13

-31,79303507337

X8480

10

14

7,79134466748

X1138 = X399*

10

15

1,13460438833

X30

10

16

6,84565964096

X5680

10

17

-1,68618254416

X3464 = X30/X1

10

18

-6,84088208278

X484

10

19

6,16733538680

X5667 = X30/X4

10

20

2,93309468208

X8481

10

21

-6,52868086149

X8482

10

22

6,80744200217

X8483

10

23

20,32666877404

X8484

10

24

4,29232349359

X7164 = X3464*

10

25

2,90916090233

X8485

11

12

8,06377960586

X8486 = [X30/X1138]*

11

13

-6,95294706960

X8487 = [X30/X1263]*

11

14

5,63387199414

X8488

11

15

-7,12562286909

X3466 = X3465*

11

16

1,13460438833

X30

11

17

-5.67661941093

X4

11

18

1,05985239030

X8439 = X3484*

11

19

1,69030850946

X1

11

20

6,67852698580

X8489

11

21

4,51975351723

X8490

11

22

263,37110789000

X8491 = [X30/X1337]*

11

23

-7,71921435140

X8492 = [X30/X1338]*

11

24

-3,74518947176

X8493

11

25

6,78236289420

X3

12

13

0,45330765162

X8494

12

14

2,76314916251

X7327 = [X30/X484*]*

12

15

1,69030850946

X1

12

16

6,16733538680

X5667 = X30/X4

12

17

1,13460438833

X30

12

18

-5,67661941093

X4

12

19

6,15617358155

X3465 = X30/X484

12

20

8,16674846519

X3441 = X617*

12

21

-5,04101304602

X3440 = X616*

12

22

6,19319521369

X8495

12

23

3,38344985821

X8496

12

24

7,85490562353

X2133 = X2132*

12

25

-5,83828820861

X3484 = X30/X1157

13

14

-5,59478415600

X7329

13

15

6,15617358155

X3465 = X30/X484

13

16

-1,13437049968

X8443

13

17

6,16733538680

X5667 = X30/X4

13

18

1,13460438833

X30

13

19

2,90916090233

X8485

13

20

-6,93913337660

X1338

13

21

-13,20931700400

X1337

13

22

-1,39875648625

X8497

13

23

5,70925332465

X8498

13

24

7,79134466748

X1138 = X399*

13

25

-4,83047253859

X5683

14

15

-5,67661941093

X4

14

16

-1,68618254416

X3464 = X30/X1

14

17

-6,84088208278

X484

14

18

8,23665223338

X7165

14

19

1,13460438833

X30

14

20

5,59419084956

X8499

14

21

7,12885537346

X8500

14

22

-1,54460939818

X8501

14

23

-5,53873868123

X8502

14

24

-8,56804872862

X7328

14

25

6,15617358155

X3465 = X30/X484

15

16

-3,29274094678

X8503

15

17

6,38042757299

X8504

15

18

31,73783490329

X8505

15

19

-0,50752991831

X5670 = X30/X1138

15

20

-7,09894877848

X8506

15

21

-20,38529954404

X8507

15

22

-3,52862295733

X8508

15

23

5,29892807515

X8509

15

24

8,23665223338

X7165

15

25

-4,24677722398

X8432

16

17

7,67010162214

X8510

16

18

6,55991154588

X8511

16

19

-5,44518658306

X8512

16

20

-263,42417606410

X8513

16

21

7,55431061064

X8514

16

22

-7,01644428708

X8515

16

23

-5,72437527035

X8516

16

24

-4,96723694599

X399 = X30/X3

16

25

3,77478964470

X8517

17

18

-0,45303039991

X8518

17

19

-3,29274094678

X8503

17

20

-5,67441835647

X8519

17

21

-3,36197974134

X8520

17

22

7,48346819761

X5679 = X30/X617*

17

23

6,75048438897

X5678 = X30/X616*

17

24

4,61238076845

X1157 = X1263*

17

25

-7,99810301418

X5676 = X30/X2133

18

19

6,38042757299

X8504

18

20

1,28759991341

X8521

18

21

-6,75459634942

X8522

18

22

6,53002144149

X5675 = X30/X1338

18

23

13,14063785425

X5674 = X30/X1337

18

24

4,91058840478

X3481

18

25

-0,50752991831

X5670 = X30/X1138

19

20

1,54802792230

X8523

19

21

5,77192904717

X8524

19

22

-5,41596108283

X8525

19

23

-3,90043696246

X8526

19

24

-6,84088208278

X484

19

25

8,45117750091

X8527

20

21

4,61238076845

X1157 = X1263*

20

22

1,13460438833

X30

20

23

-7,65530824139

X8528

20

24

-3,84056015706

X8529

20

25

6,75048438897

X5678 = X30/X616*

21

22

7,48395751441

X8530

21

23

1,13460438833

X30

21

24

2,92641639951

X8531

21

25

7,48346819761

X5679 = X30/X617*

22

23

-5,83828820861

X3484 = X30/X1157

22

24

-5,04101304602

X3440 = X616*

22

25

3,87263509427

X8532

23

24

8,16674846519

X3441 = X617*

23

25

-2,91181827916

X8533

24

25

1,13460438833

X30

-5,67697834270

X8535 = X5678*

0,93647210260

X8536 = X5679*

1,27750126341

X370

2,08851089410

X370*

6,64573209130

X8534 = X5676*

 

Table 19-3

#

SEARCH number of S

point S

1

12,65716281429

X74

2

-1,686182544157

X3464 = X30/X1

3

2,804916496991

X5684

4

-4,83047253859

X5683

5

-4.91515929874390

X8451

6

12.0178866365392

X8461

7

2.98484382343539

 

8

-7.87940401803044

 

9

-7,998103014183

X5676 = X30/X2133

10

-31,79303507337

X8480

11

5,986883645766

 

12

3,491609800358

 

13

-8,60752539145

 

14

5,633871994141

X8488

15

-19,06668342127

 

16

-6,106852470866

 

17

-5,139363821194

 

18

16,4825281142

 

19

-5,638307465751

 

20

-3,027822557307

 

21

-7,771368838429

 

22

-6,748919072806

 

23

-7,744974884784

 

24

6,527093481101

 

25

-3,026205180289

 

If UVW is one of the 25 triangles above, the tangents at its vertices to the Neuberg cubic concur at T hence the polar conic of T passes through T (counted twice), U, V, W and a 6th point M which is the isogonal conjugate of the perspector of the reference triangle ABC and UVW.

If U', V', W' are the third intersections of K001 with the sidelines of UVW then M is the common point of the lines UU', VV', WW'.

It follows that K001 is a pK with pivot M, isopivot T with respect to UVW.

The circumcircle of UVW meets K001 at a 6th point S which lies on the line through M and T*. S is also the common point of the lines UU'*, VV'*, WW'*.

Example : UVW = TR9,

M = X74, T = X2132, T* = X2133, S = X5676.

 

Table 19-4

Xi

tangential

SEARCH

 

Xi

tangential

SEARCH

X1

X30

1.134604388331988

X3483

 

-7.437477277252003

X3

X1157

4.612380768446445

X3484

 

-9.943057527163664

X4

X3484

-5.838288208608522

X5623

 

12.21802245042616

X13

X5674

13.14063785424721

X5624

 

-11.27517160352744

X14

X5675

6.53002144149218

X5667

 

-4.522496737391494

X15

X1337

-13.20931700385151

X5668

X8489

6.678526985801815

X16

X1338

-6.939133376622704

X5669

X8490

4.519753517225083

X30

X74

12.657162814291

X5670

 

-11.61589278228621

X74

X2132

-12.72744887664726

X5671

 

3.987349260570008

X399

X8487

-6.952947069597033

X5672

X8491

263.3711078878636

X484

X1263

-6.804368895444312

X5673

X8492

-7.719214351353764

X616

 

6.50070960067425

X5674

 

-13.74206648847509

X617

 

-10.96247585757698

X5675

 

-5.213815527501939

X1138

X8511

6.559911545875819

X5676

 

4.845429151702456

X1157

 

6.388333803719578

X5677

 

-7.787270372874163

X1263

 

-6.88663861404061

X5678

 

4.422609390446225

X1276

X3479

7.946368313606507

X5679

 

15.47486565777944

X1277

X3480

5.860862976915729

X5680

 

-6.683487917046746

X1337

 

6.71230642228289

X5681

 

4.996844537390252

X1338

 

-5.330018136329866

X5682

 

-6.904740406300832

X2132

X8534

6.64573209128735

X5683

 

5.258942411393891

X2133

 

-6.99757752679892

X5684

 

-16.54455741767844

X3065

X5671

5.895867973781977

X5685

 

5.66752684753887

X3440

 

-6.926501318064256

X7059

X5682

-6.797187562774348

X3441

 

10.88315097276115

X7060

X5681

-8.084384636568658

X3464

X2133

7.854905623524792

X7164

X5676

-7.998103014182646

X3465

X8439

1.059852390261943

X7165

 

7.237407996906251

X3466

X8443

-1.134370499681268

X7325

X8514

7.554310610636976

X3479

 

-7.036835325167965

X7326

X8513

-263.4241760641431

X3480

 

4.216403891227347

X7327

 

7.628526547647783

X3481

 

-7.134669987016505

X7328

 

5.543898071022024

X3482

 

6.858343185808408

X7329

 

-6.739192163704805