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 to 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