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Re: [EMHL] Sequence of NPC centers [Circumcenter of N1N2N3 Re: Radical axes, NPCs, Os]

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  • Antreas Hatzipolakis
    Thanks!!!!! I would like the n-th ratio be something like this: [f(n)/g(n)] * F(a,b,c) with f(n), g(n) integer sequences, which I would send to seqfan
    Message 1 of 18 , Apr 14, 2013
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      Thanks!!!!!

      I would like the n-th ratio be something like this: [f(n)/g(n)] * F(a,b,c)
      with f(n), g(n) integer sequences, which I would send to seqfan list......
      :-)

      APH


      On Sun, Apr 14, 2013 at 10:17 PM, Francisco Javier
      <garciacapitan@...>wrote:

      > **
      >
      >
      > I calculated the ratios NO1:O1O y NO2:O2O. The expression for the first
      > one is quite long, and that for the second one is enormous:
      >
      > If p stands for the semiperimeter, we have
      >
      > NO1:O1O = (2 p^4 - 12 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 8 p^2 R^2
      > + 40 r^2 R^2 + 32 r R^3 + 9 R^4)/(2 p^4 + 20 p^2 r^2 + 2 r^4 - 16 p^2 r R +
      > 16 r^3 R - 16 p^2 R^2 + 48 r^2 R^2 + 64 r R^3 + 23 R^4)
      >
      > NO2:O2O = -(88758909 p^76 + 3798796350 p^74 r^2 + 79806242051 p^72 r^4 +
      > 1097924922436 p^70 r^6 + 11137581114427 p^68 r^8 +
      > 88982021250830 p^66 r^10 + 584403495879877 p^64 r^12 +
      > 3255333525339488 p^62 r^14 + 15776328653191220 p^60 r^16 +
      > 68115591934376728 p^58 r^18 + 269093311225860684 p^56 r^20 +
      > 1011575212060493424 p^54 r^22 + 3919400530626845340 p^52 r^24 +
      > 26307388875386659320 p^50 r^26 -
      > 104586738964268538652 p^48 r^28 -
      > 896768742292988544608 p^46 r^30 +
      > 9725386151452194478246 p^44 r^32 -
      > 43918348723830627877980 p^42 r^34 +
      > 103657064418461238590682 p^40 r^36 -
      > 137207590879950726501224 p^38 r^38 +
      > 103657064418461238590682 p^36 r^40 -
      > 43918348723830627877980 p^34 r^42 +
      > 9725386151452194478246 p^32 r^44 -
      > 896768742292988544608 p^30 r^46 -
      > 104586738964268538652 p^28 r^48 +
      > 26307388875386659320 p^26 r^50 + 3919400530626845340 p^24 r^52 +
      > 1011575212060493424 p^22 r^54 + 269093311225860684 p^20 r^56 +
      > 68115591934376728 p^18 r^58 + 15776328653191220 p^16 r^60 +
      > 3255333525339488 p^14 r^62 + 584403495879877 p^12 r^64 +
      > 88982021250830 p^10 r^66 + 11137581114427 p^8 r^68 +
      > 1097924922436 p^6 r^70 + 79806242051 p^4 r^72 +
      > 3798796350 p^2 r^74 + 88758909 r^76 - 13491354168 p^74 r R -
      > 533535320232 p^72 r^3 R - 10320113598704 p^70 r^5 R -
      > 130214276473104 p^68 r^7 R - 1206295457258136 p^66 r^9 R -
      > 8759690922834824 p^64 r^11 R - 52018272648672384 p^62 r^13 R -
      > 260493745783918464 p^60 r^15 R -
      > 1127823175696908896 p^58 r^17 R -
      > 4321424179053229344 p^56 r^19 R -
      > 15053294229208739904 p^54 r^21 R -
      > 49687519342662839232 p^52 r^23 R -
      > 169798910372440499808 p^50 r^25 R -
      > 1092955755646119147552 p^48 r^27 R +
      > 5276425314216860693632 p^46 r^29 R +
      > 23420174439158772733824 p^44 r^31 R -
      > 256829442074011440211728 p^42 r^33 R +
      > 959523021655301486259408 p^40 r^35 R -
      > 1788779537002991394984864 p^38 r^37 R +
      > 1788779537002991394984864 p^36 r^39 R -
      > 959523021655301486259408 p^34 r^41 R +
      > 256829442074011440211728 p^32 r^43 R -
      > 23420174439158772733824 p^30 r^45 R -
      > 5276425314216860693632 p^28 r^47 R +
      > 1092955755646119147552 p^26 r^49 R +
      > 169798910372440499808 p^24 r^51 R +
      > 49687519342662839232 p^22 r^53 R +
      > 15053294229208739904 p^20 r^55 R +
      > 4321424179053229344 p^18 r^57 R +
      > 1127823175696908896 p^16 r^59 R +
      > 260493745783918464 p^14 r^61 R + 52018272648672384 p^12 r^63 R +
      > 8759690922834824 p^10 r^65 R + 1206295457258136 p^8 r^67 R +
      > 130214276473104 p^6 r^69 R + 10320113598704 p^4 r^71 R +
      > 533535320232 p^2 r^73 R + 13491354168 r^75 R +
      > 934911705712 p^72 r^2 R^2 + 33961662171136 p^70 r^4 R^2 +
      > 600860599804384 p^68 r^6 R^2 + 6901666686733184 p^66 r^8 R^2 +
      > 57896032527784624 p^64 r^10 R^2 +
      > 378424562062522368 p^62 r^12 R^2 +
      > 2008989630631125248 p^60 r^14 R^2 +
      > 8923740143784088576 p^58 r^16 R^2 +
      > 33960413509965697216 p^56 r^18 R^2 +
      > 113165359238973171712 p^54 r^20 R^2 +
      > 338469987441761757312 p^52 r^22 R^2 +
      > 943347078380146897408 p^50 r^24 R^2 +
      > 2442074571426910193344 p^48 r^26 R^2 +
      > 22240110752039997585408 p^46 r^28 R^2 -
      > 89999476996366730227968 p^44 r^30 R^2 -
      > 403777959903803392267264 p^42 r^32 R^2 +
      > 3531458465043963499978272 p^40 r^34 R^2 -
      > 10008278685321486967244800 p^38 r^36 R^2 +
      > 13585837759327469757214528 p^36 r^38 R^2 -
      > 9288596594975772417534720 p^34 r^40 R^2 +
      > 2972072380643917794384544 p^32 r^42 R^2 -
      > 237882965151856088045568 p^30 r^44 R^2 -
      > 117215766407943490070784 p^28 r^46 R^2 +
      > 21602952052787711052800 p^26 r^48 R^2 +
      > 3471424591303306097600 p^24 r^50 R^2 +
      > 1152155592838123419648 p^22 r^52 R^2 +
      > 398646644554123985024 p^20 r^54 R^2 +
      > 130253705294057017856 p^18 r^56 R^2 +
      > 38444833235486085056 p^16 r^58 R^2 +
      > 9973722675429124096 p^14 r^60 R^2 +
      > 2222489070371096832 p^12 r^62 R^2 +
      > 415127277305295872 p^10 r^64 R^2 +
      > 63063469834238704 p^8 r^66 R^2 + 7472424541528064 p^6 r^68 R^2 +
      > 647165835343328 p^4 r^70 R^2 + 36412560710528 p^2 r^72 R^2 +
      > 998360208432 r^74 R^2 - 39038499623936 p^70 r^3 R^3 -
      > 1293644515622656 p^68 r^5 R^3 - 20771094521846272 p^66 r^7 R^3 -
      > 215255727681242880 p^64 r^9 R^3 -
      > 1618260615263305728 p^62 r^11 R^3 -
      > 9405732493204150272 p^60 r^13 R^3 -
      > 43995696382930284544 p^58 r^15 R^3 -
      > 170286876359619360768 p^56 r^17 R^3 -
      > 556941007802720112640 p^54 r^19 R^3 -
      > 1566238866195774100480 p^52 r^21 R^3 -
      > 3848559023743470442496 p^50 r^23 R^3 -
      > 8357108809849657349120 p^48 r^25 R^3 -
      > 6433204985224296431616 p^46 r^27 R^3 -
      > 324201437887497135556608 p^44 r^29 R^3 +
      > 838523185478967383519232 p^42 r^31 R^3 +
      > 4690983051030825395296256 p^40 r^33 R^3 -
      > 28532047959087471899514880 p^38 r^35 R^3 +
      > 57181691192636824779314688 p^36 r^37 R^3 -
      > 51298990974560661877873664 p^34 r^39 R^3 +
      > 19520132389007557791712768 p^32 r^41 R^3 -
      > 858845455565215994167296 p^30 r^43 R^3 -
      > 1558455966076296026892288 p^28 r^45 R^3 +
      > 270042879838727751266304 p^26 r^47 R^3 +
      > 44592015388841825351680 p^24 r^49 R^3 +
      > 16786533911874255646720 p^22 r^51 R^3 +
      > 6650810158116756048896 p^20 r^53 R^3 +
      > 2482366930880015230976 p^18 r^55 R^3 +
      > 831747470989058214912 p^16 r^57 R^3 +
      > 243244673992116142080 p^14 r^59 R^3 +
      > 60687929982491363328 p^12 r^61 R^3 +
      > 12611944293765353472 p^10 r^63 R^3 +
      > 2119445116876744704 p^8 r^65 R^3 +
      > 276369175719185408 p^6 r^67 R^3 +
      > 26216899924128000 p^4 r^69 R^3 + 1608787920463360 p^2 r^71 R^3 +
      > 47921290004736 r^73 R^3 + 1092823189112320 p^68 r^4 R^4 +
      > 32761182035786240 p^66 r^6 R^4 +
      > 472835032487885568 p^64 r^8 R^4 +
      > 4372059196604125184 p^62 r^10 R^4 +
      > 29071275068040159232 p^60 r^12 R^4 +
      > 147887213661159096320 p^58 r^14 R^4 +
      > 597637754164684040192 p^56 r^16 R^4 +
      > 1965480535379458265088 p^54 r^18 R^4 +
      > 5339498721623052900352 p^52 r^20 R^4 +
      > 12051771603976330864640 p^50 r^22 R^4 +
      > 22301520720804800689152 p^48 r^24 R^4 +
      > 32569132162341419732992 p^46 r^26 R^4 -
      > 235425802008777590800384 p^44 r^28 R^4 +
      > 3400978616673249642733568 p^42 r^30 R^4 -
      > 4726652097417898513229824 p^40 r^32 R^4 -
      > 35610675650658451268612096 p^38 r^34 R^4 +
      > 137097809753072658028123136 p^36 r^36 R^4 -
      > 172505810502481528845030400 p^34 r^38 R^4 +
      > 77288660324758901838389760 p^32 r^40 R^4 +
      > 4976413637199941356531712 p^30 r^42 R^4 -
      > 14062430505530323999055872 p^28 r^44 R^4 +
      > 2392189795873423920381952 p^26 r^46 R^4 +
      > 404171275470019334019072 p^24 r^48 R^4 +
      > 172548327102189597634560 p^22 r^50 R^4 +
      > 78475590368046233925632 p^20 r^52 R^4 +
      > 33583839212088035051520 p^18 r^54 R^4 +
      > 12824815299404783547392 p^16 r^56 R^4 +
      > 4244124350688201535488 p^14 r^58 R^4 +
      > 1189750352335237758976 p^12 r^60 R^4 +
      > 275977629928628707328 p^10 r^62 R^4 +
      > 51455437085403949056 p^8 r^64 R^4 +
      > 7403563347714928640 p^6 r^66 R^4 +
      > 771126451821532672 p^4 r^68 R^4 +
      > 51723653888453120 p^2 r^70 R^4 + 1677245150165760 r^72 R^4 -
      > 21553555724797952 p^66 r^5 R^5 -
      > 578498869769111552 p^64 r^7 R^5 -
      > 7414237087227805696 p^62 r^9 R^5 -
      > 60285365604881334272 p^60 r^11 R^5 -
      > 348320065047484907520 p^58 r^13 R^5 -
      > 1516741492955807727616 p^56 r^15 R^5 -
      > 5144106113001019244544 p^54 r^17 R^5 -
      > 13810449082232855789568 p^52 r^19 R^5 -
      > 29338956925242013868032 p^50 r^21 R^5 -
      > 47832076531708039127040 p^48 r^23 R^5 -
      > 51995767644709101469696 p^46 r^25 R^5 -
      > 96374125568224486293504 p^44 r^27 R^5 +
      > 4087945917163617221230592 p^42 r^29 R^5 -
      > 25202259836786374042468352 p^40 r^31 R^5 +
      > 17883120958093732544544768 p^38 r^33 R^5 +
      > 154238149257637020433711104 p^36 r^35 R^5 -
      > 335968289044342424388743168 p^34 r^37 R^5 +
      > 167140462955903711719444480 p^32 r^39 R^5 +
      > 80850656951482862342733824 p^30 r^41 R^5 -
      > 92068644828655608689524736 p^28 r^43 R^5 +
      > 15942594123582405915918336 p^26 r^45 R^5 +
      > 2750936998979706192510976 p^24 r^47 R^5 +
      > 1331848498568252382740480 p^22 r^49 R^5 +
      > 696945362385637725634560 p^20 r^51 R^5 +
      > 343283183995967591129088 p^18 r^53 R^5 +
      > 150027950964713217859584 p^16 r^55 R^5 +
      > 56408272554137008046080 p^14 r^57 R^5 +
      > 17833715146563534258176 p^12 r^59 R^5 +
      > 4633170211802385334272 p^10 r^61 R^5 +
      > 961380201025088143360 p^8 r^63 R^5 +
      > 153060560890642137088 p^6 r^65 R^5 +
      > 17548730693934579712 p^4 r^67 R^5 +
      > 1289618618784348160 p^2 r^69 R^5 + 45621068084508672 r^71 R^5 +
      > 305355215278092288 p^64 r^6 R^6 +
      > 7236955642523164672 p^62 r^8 R^6 +
      > 80998019570024398848 p^60 r^10 R^6 +
      > 567273581337377947648 p^58 r^12 R^6 +
      > 2773466066478922145792 p^56 r^14 R^6 +
      > 9976324744019306143744 p^54 r^16 R^6 +
      > 26985585713960639561728 p^52 r^18 R^6 +
      > 54549619578201829900288 p^50 r^20 R^6 +
      > 77754030650635237662720 p^48 r^22 R^6 +
      > 59744691201375143206912 p^46 r^24 R^6 -
      > 43356988578497367326720 p^44 r^26 R^6 +
      > 1931475899799566980956160 p^42 r^28 R^6 -
      > 37368709869658687929114624 p^40 r^30 R^6 +
      > 135549465316777486799331328 p^38 r^32 R^6 -
      > 79163643213750803320209408 p^36 r^34 R^6 -
      > 229312655301523237547212800 p^34 r^36 R^6 +
      > 30853044863374223670247424 p^32 r^38 R^6 +
      > 519181777377456858825187328 p^30 r^40 R^6 -
      > 455113947878699062604152832 p^28 r^42 R^6 +
      > 82811225651577031800209408 p^26 r^44 R^6 +
      > 14604472884763164431769600 p^24 r^46 R^6 +
      > 8021747961690420426547200 p^22 r^48 R^6 +
      > 4840494820344785938710528 p^20 r^50 R^6 +
      > 2754877411509856951697408 p^18 r^52 R^6 +
      > 1383965982785680479739904 p^16 r^54 R^6 +
      > 593731977771817744359424 p^14 r^56 R^6 +
      > 212543575042097613062144 p^12 r^58 R^6 +
      > 62068894396541671882752 p^10 r^60 R^6 +
      > 14380356196106834763776 p^8 r^62 R^6 +
      > 2540837633285637251072 p^6 r^64 R^6 +
      > 321524801039704915968 p^4 r^66 R^6 +
      > 25949664123636678656 p^2 r^68 R^6 +
      > 1003663497859190784 r^70 R^6 - 3086000322010644480 p^62 r^7 R^7 -
      > 63254444018589368320 p^60 r^9 R^7 -
      > 602096083551165546496 p^58 r^11 R^7 -
      > 3506522805081767280640 p^56 r^13 R^7 -
      > 13804901524573771694080 p^54 r^15 R^7 -
      > 38004986682042514931712 p^52 r^17 R^7 -
      > 71572024937280078479360 p^50 r^19 R^7 -
      > 78864459701899622612992 p^48 r^21 R^7 -
      > 355208993878347317248 p^46 r^23 R^7 +
      > 164254144242293920628736 p^44 r^25 R^7 +
      > 738727797734975116345344 p^42 r^27 R^7 -
      > 26844557821203945533341696 p^40 r^29 R^7 +
      > 227368210799746208900153344 p^38 r^31 R^7 -
      > 573970743130970319144615936 p^36 r^33 R^7 +
      > 624462478502301295921856512 p^34 r^35 R^7 -
      > 1054738493670002805536980992 p^32 r^37 R^7 +
      > 2129476073229459119993094144 p^30 r^39 R^7 -
      > 1741199963468989574975717376 p^28 r^41 R^7 +
      > 342779356632131024956096512 p^26 r^43 R^7 +
      > 61982103646378812675588096 p^24 r^45 R^7 +
      > 38672068824535086359937024 p^22 r^47 R^7 +
      > 26965705356154440133115904 p^20 r^49 R^7 +
      > 17804796357091335565606912 p^18 r^51 R^7 +
      > 10329662146062291520192512 p^16 r^53 R^7 +
      > 5079704267903251635863552 p^14 r^55 R^7 +
      > 2067780077630768119939072 p^12 r^57 R^7 +
      > 681391316730095240806400 p^10 r^59 R^7 +
      > 176879538305835773853696 p^8 r^61 R^7 +
      > 34791717950383418540032 p^6 r^63 R^7 +
      > 4872858489528198889472 p^4 r^65 R^7 +
      > 433010881028197187584 p^2 r^67 R^7 +
      > 18352703960853774336 r^69 R^7 +
      > 21157461200939778048 p^60 r^8 R^8 +
      > 360182970303358173184 p^58 r^10 R^8 +
      > 2751858566304020037632 p^56 r^12 R^8 +
      > 12172854610892077924352 p^54 r^14 R^8 +
      > 32727913042765889667072 p^52 r^16 R^8 +
      > 45866686885097012985856 p^50 r^18 R^8 -
      > 14430309661224207122432 p^48 r^20 R^8 -
      > 216536312958720295829504 p^46 r^22 R^8 -
      > 475042582939514331201536 p^44 r^24 R^8 -
      > 425847805833839038431232 p^42 r^26 R^8 -
      > 9972172940151551751618560 p^40 r^28 R^8 +
      > 205403553317534037266661376 p^38 r^30 R^8 -
      > 1000889976384192969714171904 p^36 r^32 R^8 +
      > 2179936708676258798685913088 p^34 r^34 R^8 -
      > 3829098623875418719299829760 p^32 r^36 R^8 +
      > 6199886976246065477320704000 p^30 r^38 R^8 -
      > 5235572772386089217723203584 p^28 r^40 R^8 +
      > 1146701269456164114437242880 p^26 r^42 R^8 +
      > 213748796751548032159842304 p^24 r^44 R^8 +
      > 151864861169836678496649216 p^22 r^46 R^8 +
      > 122617848579714306055864320 p^20 r^48 R^8 +
      > 94318878419593329776787456 p^18 r^50 R^8 +
      > 63508263335597357654605824 p^16 r^52 R^8 +
      > 35976405290230771482099712 p^14 r^54 R^8 +
      > 16729634850212066083995648 p^12 r^56 R^8 +
      > 6246616401621674442358784 p^10 r^58 R^8 +
      > 1823553907872154814251008 p^8 r^60 R^8 +
      > 400626991009365011202048 p^6 r^62 R^8 +
      > 62286941836222551818240 p^4 r^64 R^8 +
      > 6110195501050340311040 p^2 r^66 R^8 +
      > 284466911393233502208 r^68 R^8 -
      > 80303512707748855808 p^58 r^9 R^9 -
      > 970410565041953701888 p^56 r^11 R^9 -
      > 4244264538712111579136 p^54 r^13 R^9 -
      > 2803692602922429841408 p^52 r^15 R^9 +
      > 51854130438864292020224 p^50 r^17 R^9 +
      > 266243999419618037858304 p^48 r^19 R^9 +
      > 681175498579659842387968 p^46 r^21 R^9 +
      > 1030866995016179937968128 p^44 r^23 R^9 +
      > 855489992893501405134848 p^42 r^25 R^9 -
      > 1501476066112463538487296 p^40 r^27 R^9 +
      > 105685525935459059268321280 p^38 r^29 R^9 -
      > 1013212795832229811948879872 p^36 r^31 R^9 +
      > 3492955971362623539002736640 p^34 r^33 R^9 -
      > 7754792082920360008554119168 p^32 r^35 R^9 +
      > 13348697389531372107417518080 p^30 r^37 R^9 -
      > 12481448672884232938609180672 p^28 r^39 R^9 +
      > 3126959004859565160459665408 p^26 r^41 R^9 +
      > 605393382522430267133526016 p^24 r^43 R^9 +
      > 491817076861626529883357184 p^22 r^45 R^9 +
      > 460735312243689423638626304 p^20 r^47 R^9 +
      > 414632062700555521946025984 p^18 r^49 R^9 +
      > 325745901645704401956896768 p^16 r^51 R^9 +
      > 213712167035471587639820288 p^14 r^53 R^9 +
      > 114094734113179585197637632 p^12 r^55 R^9 +
      > 48488118468674673303355392 p^10 r^57 R^9 +
      > 15981837141146112180092928 p^8 r^59 R^9 +
      > 3935463401850600246214656 p^6 r^61 R^9 +
      > 681318864409164102238208 p^4 r^63 R^9 +
      > 73986002842923072225280 p^2 r^65 R^9 +
      > 3792892151909780029440 r^67 R^9 -
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      > 3960842325507507954888212480 p^28 r^30 R^18 -
      > 6227297285136569292036243456 p^26 r^32 R^18 -
      > 4851814980738665962549542912 p^24 r^34 R^18 -
      > 5271747349465723121837277184 p^22 r^36 R^18 +
      > 11939478522117128617856073728 p^20 r^38 R^18 +
      > 150871882362794159165090562048 p^18 r^40 R^18 +
      > 861117039806189425062397870080 p^16 r^42 R^18 +
      > 3446568358583418742751300157440 p^14 r^44 R^18 +
      > 9917973899939331118854473515008 p^12 r^46 R^18 +
      > 20164375708836368101019083603968 p^10 r^48 R^18 +
      > 28349368308168771690636446269440 p^8 r^50 R^18 +
      > 26787345978836905877181227335680 p^6 r^52 R^18 +
      > 16182946893734427408613559500800 p^4 r^54 R^18 +
      > 5637219100018390760565610905600 p^2 r^56 R^18 +
      > 860573368768608013150008115200 r^58 R^18 +
      > 86903163784711364411392 p^38 r^19 R^19 +
      > 203438072906108079439872 p^36 r^21 R^19 +
      > 112012426906292038139904 p^34 r^23 R^19 -
      > 2374551496330974330880 p^32 r^25 R^19 -
      > 11095504634645442704965632 p^30 r^27 R^19 +
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      > 2192578317063412230811287552 p^26 r^31 R^19 -
      > 2028555403651001861464391680 p^24 r^33 R^19 -
      > 3887715095501705646241218560 p^22 r^35 R^19 +
      > 2483070202753722172116566016 p^20 r^37 R^19 +
      > 98164245344292635763005194240 p^18 r^39 R^19 +
      > 758781114493719667324548546560 p^16 r^41 R^19 +
      > 3887188920667275322702270824448 p^14 r^43 R^19 +
      > 14094683366260191626719657984000 p^12 r^45 R^19 +
      > 35640897842774363339225760268288 p^10 r^47 R^19 +
      > 61383978714221305854757982699520 p^8 r^49 R^19 +
      > 69940629320434458746986891837440 p^6 r^51 R^19 +
      > 50194228014588169059798267985920 p^4 r^53 R^19 +
      > 20493177229331369061876813004800 p^2 r^55 R^19 +
      > 3623466815867823213263192064000 r^57 R^19 -
      > 74195345706200351113216 p^36 r^20 R^20 -
      > 75626182657161394388992 p^34 r^22 R^20 -
      > 8558166383846970359808 p^32 r^24 R^20 +
      > 469409007942290938789888 p^30 r^26 R^20 +
      > 63206377361837984057393152 p^28 r^28 R^20 -
      > 511959551426105566757388288 p^26 r^30 R^20 -
      > 650364972533117098148233216 p^24 r^32 R^20 -
      > 2143251502708157496569102336 p^22 r^34 R^20 -
      > 1734327637466493820952641536 p^20 r^36 R^20 +
      > 49384894688058377559628316672 p^18 r^38 R^20 +
      > 552824265991921733263830286336 p^16 r^40 R^20 +
      > 3691724982551474681290705338368 p^14 r^42 R^20 +
      > 17057099136341611627053716602880 p^12 r^44 R^20 +
      > 54250553620054320396239527477248 p^10 r^46 R^20 +
      > 115766688712057206592268843089920 p^8 r^48 R^20 +
      > 160752529797325953373936193372160 p^6 r^50 R^20 +
      > 138354243641429071365378114846720 p^4 r^52 R^20 +
      > 66755549822975288609491428311040 p^2 r^54 R^20 +
      > 13769173900297728210400129843200 r^56 R^20 +
      > 27341004387959997202432 p^34 r^21 R^21 +
      > 24717769060311367680 p^32 r^23 R^21 +
      > 28953189990736693559296 p^30 r^25 R^21 -
      > 488722558226595466706944 p^28 r^27 R^21 -
      > 62282211155722082956869632 p^26 r^29 R^21 -
      > 163169328993966088054112256 p^24 r^31 R^21 -
      > 899064114581610913798815744 p^22 r^33 R^21 -
      > 2232034559026249303377051648 p^20 r^35 R^21 +
      > 17365971320002505861710741504 p^18 r^37 R^21 +
      > 326966105037951266276891426816 p^16 r^39 R^21 +
      > 2930859214576566448516646305792 p^14 r^41 R^21 +
      > 17490294624701432356244857815040 p^12 r^43 R^21 +
      > 70800572457472979092232625717248 p^10 r^45 R^21 +
      > 189468868908336764256665783500800 p^8 r^47 R^21 +
      > 324363116628513907064195682140160 p^6 r^49 R^21 +
      > 338296343721675022484936211824640 p^4 r^51 R^21 +
      > 194662273781143317620941243023360 p^2 r^53 R^21 +
      > 47208596229592211007086159462400 r^55 R^21 +
      > 3650293299706190626816 p^32 r^22 R^22 +
      > 13370485576546062958592 p^30 r^24 R^22 -
      > 547690344386816462815232 p^28 r^26 R^22 +
      > 1525135460840007396229120 p^26 r^28 R^22 -
      > 34938554381768745155559424 p^24 r^30 R^22 -
      > 286686578519783864063229952 p^22 r^32 R^22 -
      > 1361160070616839080514682880 p^20 r^34 R^22 +
      > 2730157156242413819181137920 p^18 r^36 R^22 +
      > 151739556945086099658788306944 p^16 r^38 R^22 +
      > 1922203427136435844005192794112 p^14 r^40 R^22 +
      > 15096878383824883009156301193216 p^12 r^42 R^22 +
      > 78798136901775476030533426741248 p^10 r^44 R^22 +
      > 267872871901259207024361477242880 p^8 r^46 R^22 +
      > 572523935102658300894177656832000 p^6 r^48 R^22 +
      > 731918664418950048279096176148480 p^4 r^50 R^22 +
      > 507334492822911069108985732792320 p^2 r^52 R^22 +
      > 145917479255103197658266311065600 r^54 R^22 -
      > 7909599967956764196864 p^30 r^23 R^23 -
      > 30474939381142209429504 p^28 r^25 R^23 +
      > 1614450340658054854344704 p^26 r^27 R^23 -
      > 7174867659643837971693568 p^24 r^29 R^23 -
      > 69066086152443533291683840 p^22 r^31 R^23 -
      > 557155272898970916239704064 p^20 r^33 R^23 -
      > 1176676493029807670805135360 p^18 r^35 R^23 +
      > 51483180170420045436432678912 p^16 r^37 R^23 +
      > 1020691366497614534906350665728 p^14 r^39 R^23 +
      > 10867490311791360718162496585728 p^12 r^41 R^23 +
      > 74299026789038800780375252533248 p^10 r^43 R^23 +
      > 325341967475872831280038061015040 p^8 r^45 R^23 +
      > 880010842381433390684471411343360 p^6 r^47 R^23 +
      > 1396464793987193987041261623705600 p^4 r^49 R^23 +
      > 1179037108958629820248028400844800 p^2 r^51 R^23 +
      > 406031246622895854353436691660800 r^53 R^23 +
      > 3700864518989096681472 p^28 r^24 R^24 +
      > 190173507094473468805120 p^26 r^26 R^24 -
      > 1331770845546286153728000 p^24 r^28 R^24 -
      > 12663961840907598964981760 p^22 r^30 R^24 -
      > 162471467459737530614677504 p^20 r^32 R^24 -
      > 1074959459488841462911598592 p^18 r^34 R^24 +
      > 10354852825522837050261241856 p^16 r^36 R^24 +
      > 423454173842126474989145161728 p^14 r^38 R^24 +
      > 6429899889403364174755486760960 p^12 r^40 R^24 +
      > 58851525233512293009664149291008 p^10 r^42 R^24 +
      > 337183004501089786817871725199360 p^8 r^44 R^24 +
      > 1171452490258177887661560000675840 p^6 r^46 R^24 +
      > 2339657873524029131702466993192960 p^4 r^48 R^24 +
      > 2435874899855361009160320083558400 p^2 r^50 R^24 +
      > 1015078116557239635883591729152000 r^52 R^24 +
      > 4962295753017808388096 p^26 r^25 R^25 -
      > 180916163279729530503168 p^24 r^27 R^25 -
      > 1813937317396153389547520 p^22 r^29 R^25 -
      > 34645412806122091004821504 p^20 r^31 R^25 -
      > 426747965470688813848199168 p^18 r^33 R^25 -
      > 305156232628922299598241792 p^16 r^35 R^25 +
      > 128136542576282044697684213760 p^14 r^37 R^25 +
      > 3052027082832976185528024039424 p^12 r^39 R^25 +
      > 38694945414443397269002498605056 p^10 r^41 R^25 +
      > 295772346910577475875158181806080 p^8 r^43 R^25 +
      > 1341632654269970207659115453349888 p^6 r^45 R^25 +
      > 3424152480435980745295411166576640 p^4 r^47 R^25 +
      > 4456374908226398311068088387239936 p^2 r^49 R^25 +
      > 2273774981088216784379245473300480 r^51 R^25 -
      > 9924591506035616776192 p^24 r^26 R^26 -
      > 218801992583441958305792 p^22 r^28 R^26 -
      > 5940687261666931980107776 p^20 r^30 R^26 -
      > 104079126289975277506265088 p^18 r^32 R^26 -
      > 10208008991505472<br/><br/>(Message over 64 KB, truncated)
    • Barry Wolk
      ... This was already answered, and is the whole plane. Another way of stating this result is:   The Poncelet points of the quads BCPP*, CAPP*, ABPP* form a
      Message 2 of 18 , Apr 15, 2013
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        > Let ABC be a triangle, P, P* two isogonala conjugate points.
        >
        > Denote:
        >
        > Ra = radical axis of (NPC_PBC), (NPC_P*BC)
        >
        > Rb = radical axis of (NPC_PCA), (NPC_P*CA)
        >
        > Rc = radical axis of (NPC_PAB), (NPC_P*AB)
        >
        > Which is the locus of P such that Ra,Rb,Rc are concurrent?
        > The entire plane?
        >
        > Antreas

        This was already answered, and is the whole plane.
        Another way of stating this result is:
         
        The Poncelet points of the quads BCPP*, CAPP*, ABPP*
        form a triangle which is perspective to the medial triangle.
        --
        Barry Wolk
      • Antreas
        I guess that for P = I (instead of N) the ratio should be simpler.... ie Let A1, B1, C1 be the NPC centers of IBC, ICA, IAB and O1 the circumcenter of A1B1C1.
        Message 3 of 18 , Apr 15, 2013
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          I guess that for P = I (instead of N) the ratio should be simpler....

          ie

          Let A1, B1, C1 be the NPC centers of IBC, ICA, IAB
          and O1 the circumcenter of A1B1C1.
          It lies on the Euler line.
          http://tech.groups.yahoo.com/group/Hyacinthos/message/21947
          Ratio O1O / O1N ?

          In general:

          Which is the locus of P such that the circumcenter
          of A1B1C1 is on the Euler line (where A1,B1,C1 are the
          NPC centers of PBC, PCA, PAB)?

          P = H (trivial case), I, N, .......

          APH

          --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@...> wrote:
          >
          > I calculated the ratios NO1:O1O y NO2:O2O. The expression for the first one is quite long, and that for the second one is enormous:
          >
          > If p stands for the semiperimeter, we have
          >
          > NO1:O1O = (2 p^4 - 12 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 8 p^2 R^2 + 40 r^2 R^2 + 32 r R^3 + 9 R^4)/(2 p^4 + 20 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 16 p^2 R^2 + 48 r^2 R^2 + 64 r R^3 + 23 R^4)
          >
          [...]
          >
          > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
          > >
          > > Let ABC be a triangle.
          > >
          > > Denote:
          > >
          > > A1, B1, C1 = The NPC centers of NBC, NCA, NAB, resp.
          > >
          > > A2, B2 ,C2 = The NPC centers of A1BC, B1CA, C1AB, resp.
          > >
          > > A3, B3, C3 = The NPC centers of A2BC, B2CA, C2AB
          > >
          > > An, Bn, Cn = The NPC centers of A_n-1BC, B_n-1CA, C_n-1AB.
          > >
          > > On = the circumcenter of the triangle AnBnCn.
          > >
          > > The points O1, O2,......, On, ..... lie on the Euler Line of ABC.
          > >
          > > Coordinates of On? Ratio of OnO / OnN ?
          > >
          > > APH
          > >
          > >
          > > [APH]
          > > > Let ABC be a triangle, P a point and N1,N2,N3 the NPC centers
          > > > of PBC, PCA, PAB, resp.
          > > >
          > > > Which is the circumcenter Op of N1N2N3 ?
          > > >
          > >
          > > > P = N. Op lies on the Euler line of ABC.
        • Francisco Javier
          [APH] Which is the locus of P such that the circumcenter of A1B1C1 is on the Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)? ... It is a
          Message 4 of 18 , Apr 15, 2013
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            [APH]

            Which is the locus of P such that the circumcenter of A1B1C1 is on the Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)?

            ----

            It is a circular quintic through I, H and N.
            The other three intersection points with Euler line are X30 (the infinite point) and the instersections with the circumcircle X1113 and X1114.

            Francisco Javier.

            Equation:

            a^6 c^2 x^3 y^2 - 3 a^4 b^2 c^2 x^3 y^2 + 3 a^2 b^4 c^2 x^3 y^2 -
            b^6 c^2 x^3 y^2 - a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 -
            a^2 c^6 x^3 y^2 + c^8 x^3 y^2 + a^6 c^2 x^2 y^3 -
            3 a^4 b^2 c^2 x^2 y^3 + 3 a^2 b^4 c^2 x^2 y^3 - b^6 c^2 x^2 y^3 -
            a^2 b^2 c^4 x^2 y^3 + b^4 c^4 x^2 y^3 + b^2 c^6 x^2 y^3 -
            c^8 x^2 y^3 + 3 a^2 b^4 c^2 x^3 y z - 3 b^6 c^2 x^3 y z -
            3 a^2 b^2 c^4 x^3 y z + 3 b^2 c^6 x^3 y z + 2 a^6 c^2 x^2 y^2 z -
            2 b^6 c^2 x^2 y^2 z - 4 a^4 c^4 x^2 y^2 z + 4 b^4 c^4 x^2 y^2 z +
            2 a^2 c^6 x^2 y^2 z - 2 b^2 c^6 x^2 y^2 z + 3 a^6 c^2 x y^3 z -
            3 a^4 b^2 c^2 x y^3 z + 3 a^2 b^2 c^4 x y^3 z - 3 a^2 c^6 x y^3 z -
            a^6 b^2 x^3 z^2 + a^4 b^4 x^3 z^2 + a^2 b^6 x^3 z^2 - b^8 x^3 z^2 +
            3 a^4 b^2 c^2 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 - 3 a^2 b^2 c^4 x^3 z^2 +
            b^2 c^6 x^3 z^2 - 2 a^6 b^2 x^2 y z^2 + 4 a^4 b^4 x^2 y z^2 -
            2 a^2 b^6 x^2 y z^2 + 2 b^6 c^2 x^2 y z^2 - 4 b^4 c^4 x^2 y z^2 +
            2 b^2 c^6 x^2 y z^2 + 2 a^6 b^2 x y^2 z^2 - 4 a^4 b^4 x y^2 z^2 +
            2 a^2 b^6 x y^2 z^2 - 2 a^6 c^2 x y^2 z^2 + 4 a^4 c^4 x y^2 z^2 -
            2 a^2 c^6 x y^2 z^2 + a^8 y^3 z^2 - a^6 b^2 y^3 z^2 -
            a^4 b^4 y^3 z^2 + a^2 b^6 y^3 z^2 + a^4 b^2 c^2 y^3 z^2 -
            3 a^2 b^4 c^2 y^3 z^2 + 3 a^2 b^2 c^4 y^3 z^2 - a^2 c^6 y^3 z^2 -
            a^6 b^2 x^2 z^3 + b^8 x^2 z^3 + 3 a^4 b^2 c^2 x^2 z^3 +
            a^2 b^4 c^2 x^2 z^3 - b^6 c^2 x^2 z^3 - 3 a^2 b^2 c^4 x^2 z^3 -
            b^4 c^4 x^2 z^3 + b^2 c^6 x^2 z^3 - 3 a^6 b^2 x y z^3 +
            3 a^2 b^6 x y z^3 + 3 a^4 b^2 c^2 x y z^3 - 3 a^2 b^4 c^2 x y z^3 -
            a^8 y^2 z^3 + a^2 b^6 y^2 z^3 + a^6 c^2 y^2 z^3 -
            a^4 b^2 c^2 y^2 z^3 - 3 a^2 b^4 c^2 y^2 z^3 + a^4 c^4 y^2 z^3 +
            3 a^2 b^2 c^4 y^2 z^3 - a^2 c^6 y^2 z^3 = 0

            --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
            >
            > I guess that for P = I (instead of N) the ratio should be simpler....
            >
            > ie
            >
            > Let A1, B1, C1 be the NPC centers of IBC, ICA, IAB
            > and O1 the circumcenter of A1B1C1.
            > It lies on the Euler line.
            > http://tech.groups.yahoo.com/group/Hyacinthos/message/21947
            > Ratio O1O / O1N ?
            >
            > In general:
            >
            > Which is the locus of P such that the circumcenter
            > of A1B1C1 is on the Euler line (where A1,B1,C1 are the
            > NPC centers of PBC, PCA, PAB)?
            >
            > P = H (trivial case), I, N, .......
            >
            > APH
            >
            > --- In Hyacinthos@yahoogroups.com, "Francisco Javier" <garciacapitan@> wrote:
            > >
            > > I calculated the ratios NO1:O1O y NO2:O2O. The expression for the first one is quite long, and that for the second one is enormous:
            > >
            > > If p stands for the semiperimeter, we have
            > >
            > > NO1:O1O = (2 p^4 - 12 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 8 p^2 R^2 + 40 r^2 R^2 + 32 r R^3 + 9 R^4)/(2 p^4 + 20 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 16 p^2 R^2 + 48 r^2 R^2 + 64 r R^3 + 23 R^4)
            > >
            > [...]
            > >
            > > --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@> wrote:
            > > >
            > > > Let ABC be a triangle.
            > > >
            > > > Denote:
            > > >
            > > > A1, B1, C1 = The NPC centers of NBC, NCA, NAB, resp.
            > > >
            > > > A2, B2 ,C2 = The NPC centers of A1BC, B1CA, C1AB, resp.
            > > >
            > > > A3, B3, C3 = The NPC centers of A2BC, B2CA, C2AB
            > > >
            > > > An, Bn, Cn = The NPC centers of A_n-1BC, B_n-1CA, C_n-1AB.
            > > >
            > > > On = the circumcenter of the triangle AnBnCn.
            > > >
            > > > The points O1, O2,......, On, ..... lie on the Euler Line of ABC.
            > > >
            > > > Coordinates of On? Ratio of OnO / OnN ?
            > > >
            > > > APH
            > > >
            > > >
            > > > [APH]
            > > > > Let ABC be a triangle, P a point and N1,N2,N3 the NPC centers
            > > > > of PBC, PCA, PAB, resp.
            > > > >
            > > > > Which is the circumcenter Op of N1N2N3 ?
            > > > >
            > > >
            > > > > P = N. Op lies on the Euler line of ABC.
            >
          • rhutson2
            For P = O, Op = complement of X(157). In general, Op = complement of the nine-point-center of the antipedal triangle of P. Randy
            Message 5 of 18 , Apr 15, 2013
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              For P = O, Op = complement of X(157).

              In general, Op = complement of the nine-point-center of the antipedal triangle of P.

              Randy

              --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
              >
              > Let ABC be a triangle, P a point and N1,N2,N3 the NPC centers
              > of PBC, PCA, PAB, resp.
              >
              > Which is the circumcenter Op of N1N2N3 ?
              >
              > P = H. Op = N [N1 = N2 = N3 = N (N1N2N3 is degenerated)]
              >
              > P = I. Op = N
              >
              > P = O. Op lies on the line NU, where U is the Poncelet point of O
              > wrt ABC (ie the point where concur the NPCs of OBC,OCA,OAB and ABC)
              > Coordinates???
              >
              > P = N. Op lies on the Euler line of ABC.
              > Coordinates???
              >
              > If P describes a line (Euler line, etc) which is the locus of Op?
              >
              > If P describes the circumcircle, Op is the Poncelet point of P
              > wrt ABC. If P describes other curves?
              >
              > Figures:
              > http://anthrakitis.blogspot.gr/2013/04/circumcenter-of-n1n2n3.html
              >
              > APH
              >
              > [APH]
              > > Let ABC be a triangle.
              > >
              > > Denote:
              > >
              > > N1,N2,N3 = The NPC centers of IBC, ICA, IAB, resp.
              >
              > >
              > > 2. The Circumcenter of N1N2N3 is N, the NPC center of ABC.
              >
            • Antreas Hatzipolakis
              ... Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)? ... I think it is the same with Bernard s Q038:
              Message 6 of 18 , Apr 16, 2013
              • 0 Attachment
                > [APH]
                >
                > Which is the locus of P such that the circumcenter of A1B1C1 is on the
                Euler line (where A1,B1,C1 are the NPC centers of PBC, PCA, PAB)?

                [Francisco]:

                > It is a circular quintic through I, H and N.
                > The other three intersection points with Euler line are X30
                > (the infinite point) and the instersections with the
                > circumcircle X1113 and X1114.
                > Equation:
                >
                > a^6 c^2 x^3 y^2 - 3 a^4 b^2 c^2 x^3 y^2 + 3 a^2 b^4 c^2 x^3 y^2 -
                > b^6 c^2 x^3 y^2 - a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 -
                > a^2 c^6 x^3 y^2 + c^8 x^3 y^2 + a^6 c^2 x^2 y^3 -
                > 3 a^4 b^2 c^2 x^2 y^3 + 3 a^2 b^4 c^2 x^2 y^3 - b^6 c^2 x^2 y^3 -
                > a^2 b^2 c^4 x^2 y^3 + b^4 c^4 x^2 y^3 + b^2 c^6 x^2 y^3 -
                > c^8 x^2 y^3 + 3 a^2 b^4 c^2 x^3 y z - 3 b^6 c^2 x^3 y z -
                > 3 a^2 b^2 c^4 x^3 y z + 3 b^2 c^6 x^3 y z + 2 a^6 c^2 x^2 y^2 z -
                > 2 b^6 c^2 x^2 y^2 z - 4 a^4 c^4 x^2 y^2 z + 4 b^4 c^4 x^2 y^2 z +
                > 2 a^2 c^6 x^2 y^2 z - 2 b^2 c^6 x^2 y^2 z + 3 a^6 c^2 x y^3 z -
                > 3 a^4 b^2 c^2 x y^3 z + 3 a^2 b^2 c^4 x y^3 z - 3 a^2 c^6 x y^3 z -
                > a^6 b^2 x^3 z^2 + a^4 b^4 x^3 z^2 + a^2 b^6 x^3 z^2 - b^8 x^3 z^2 +
                > 3 a^4 b^2 c^2 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 - 3 a^2 b^2 c^4 x^3 z^2 +
                > b^2 c^6 x^3 z^2 - 2 a^6 b^2 x^2 y z^2 + 4 a^4 b^4 x^2 y z^2 -
                > 2 a^2 b^6 x^2 y z^2 + 2 b^6 c^2 x^2 y z^2 - 4 b^4 c^4 x^2 y z^2 +
                > 2 b^2 c^6 x^2 y z^2 + 2 a^6 b^2 x y^2 z^2 - 4 a^4 b^4 x y^2 z^2 +
                > 2 a^2 b^6 x y^2 z^2 - 2 a^6 c^2 x y^2 z^2 + 4 a^4 c^4 x y^2 z^2 -
                > 2 a^2 c^6 x y^2 z^2 + a^8 y^3 z^2 - a^6 b^2 y^3 z^2 -
                > a^4 b^4 y^3 z^2 + a^2 b^6 y^3 z^2 + a^4 b^2 c^2 y^3 z^2 -
                > 3 a^2 b^4 c^2 y^3 z^2 + 3 a^2 b^2 c^4 y^3 z^2 - a^2 c^6 y^3 z^2 -
                > a^6 b^2 x^2 z^3 + b^8 x^2 z^3 + 3 a^4 b^2 c^2 x^2 z^3 +
                > a^2 b^4 c^2 x^2 z^3 - b^6 c^2 x^2 z^3 - 3 a^2 b^2 c^4 x^2 z^3 -
                > b^4 c^4 x^2 z^3 + b^2 c^6 x^2 z^3 - 3 a^6 b^2 x y z^3 +
                > 3 a^2 b^6 x y z^3 + 3 a^4 b^2 c^2 x y z^3 - 3 a^2 b^4 c^2 x y z^3 -
                > a^8 y^2 z^3 + a^2 b^6 y^2 z^3 + a^6 c^2 y^2 z^3 -
                > a^4 b^2 c^2 y^2 z^3 - 3 a^2 b^4 c^2 y^2 z^3 + a^4 c^4 y^2 z^3 +
                > 3 a^2 b^2 c^4 y^2 z^3 - a^2 c^6 y^2 z^3 = 0


                I think it is the same with Bernard's Q038:
                http://bernard.gibert.pagesperso-orange.fr/curves/q038.html

                If yes, Bernard may add that property of the quintic.

                APH


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