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

Expand Messages
• 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
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 -
> 72893343019705565184 p^56 r^10 R^10 -
> 3112184691970476605440 p^54 r^12 R^10 -
> 34778713913324856147968 p^52 r^14 R^10 -
> 194678995631713453867008 p^50 r^16 R^10 -
> 651897942709537425326080 p^48 r^18 R^10 -
> 1382677709812877123125248 p^46 r^20 R^10 -
> 1826916364132265909813248 p^44 r^22 R^10 -
> 1331477627334979179511808 p^42 r^24 R^10 -
> 505368121039486602706944 p^40 r^26 R^10 +
> 29153594519260818235719680 p^38 r^28 R^10 -
> 649891177268701670198476800 p^36 r^30 R^10 +
> 3594718538394933335540367360 p^34 r^32 R^10 -
> 10580630481120180413545840640 p^32 r^34 R^10 +
> 21634721605279312059849244672 p^30 r^36 R^10 -
> 23676771433351560491480121344 p^28 r^38 R^10 +
> 6981410500307845167232057344 p^26 r^40 R^10 +
> 1417575384900994078154424320 p^24 r^42 R^10 +
> 1324972206308853948369338368 p^22 r^44 R^10 +
> 1443013310955262695771734016 p^20 r^46 R^10 +
> 1525815879492663079795163136 p^18 r^48 R^10 +
> 1406535178062534160941056000 p^16 r^50 R^10 +
> 1074965006161160057191923712 p^14 r^52 R^10 +
> 662465341527183231260033024 p^12 r^54 R^10 +
> 322004127922513601871478784 p^10 r^56 R^10 +
> 120346496574408950777118720 p^8 r^58 R^10 +
> 33341177400778887436173312 p^6 r^60 R^10 +
> 6448537558939622411599872 p^4 r^62 R^10 +
> 777420591587070826250240 p^2 r^64 R^10 +
> 43997548962153448341504 r^66 R^10 +
> 3122503446693870043136 p^54 r^11 R^11 +
> 45125580691003955740672 p^52 r^13 R^11 +
> 281253968782101305098240 p^50 r^15 R^11 +
> 985367702805855850201088 p^48 r^17 R^11 +
> 2097441117330692833804288 p^46 r^19 R^11 +
> 2687310347642347515281408 p^44 r^21 R^11 +
> 1845380520299036875423744 p^42 r^23 R^11 +
> 435489803245050457489408 p^40 r^25 R^11 +
> 3779402583610605309001728 p^38 r^27 R^11 -
> 254219621442176392240300032 p^36 r^29 R^11 +
> 2547596932438529693515251712 p^34 r^31 R^11 -
> 10298772709505209142224093184 p^32 r^33 R^11 +
> 26558614271787053062712983552 p^30 r^35 R^11 -
> 35709740989491685492156006400 p^28 r^37 R^11 +
> 12772328994999642707715948544 p^26 r^39 R^11 +
> 2754097756873397576115683328 p^24 r^41 R^11 +
> 2987248231745152579134291968 p^22 r^43 R^11 +
> 3790035431735956572228026368 p^20 r^45 R^11 +
> 4728264051327986321976197120 p^18 r^47 R^11 +
> 5144680630990106993897766912 p^16 r^49 R^11 +
> 4609434692730113722897399808 p^14 r^51 R^11 +
> 3298719847823630049382236160 p^12 r^53 R^11 +
> 1843710853674216876687753216 p^10 r^55 R^11 +
> 785004820540588494296186880 p^8 r^57 R^11 +
> 245670843292551413319598080 p^6 r^59 R^11 +
> 53269440049688346674331648 p^4 r^61 R^11 +
> 7151296045216993527201792 p^2 r^63 R^11 +
> 447975043978289655840768 r^65 R^11 -
> 21301353436034384265216 p^52 r^12 R^12 -
> 225288225946847290589184 p^50 r^14 R^12 -
> 1006024842604404927365120 p^48 r^16 R^12 -
> 2417329166300755338985472 p^46 r^18 R^12 -
> 3251635732324684587335680 p^44 r^20 R^12 -
> 2225459664889876928528384 p^42 r^22 R^12 -
> 501353717293063604273152 p^40 r^24 R^12 +
> 258039765574763619024896 p^38 r^26 R^12 -
> 53554902929903282861113344 p^36 r^28 R^12 +
> 1230949428631904760469913600 p^34 r^30 R^12 -
> 7321525133069303464746549248 p^32 r^32 R^12 +
> 24676733911412131321436176384 p^30 r^34 R^12 -
> 42605104899908082373543067648 p^28 r^36 R^12 +
> 19095922438389745164109217792 p^26 r^38 R^12 +
> 4444989504888960957970120704 p^24 r^40 R^12 +
> 5658407409809468761044942848 p^22 r^42 R^12 +
> 8382535727647257047064379392 p^20 r^44 R^12 +
> 12386427245547852115275153408 p^18 r^46 R^12 +
> 16005517568600135326211506176 p^16 r^48 R^12 +
> 16927187497973802306711322624 p^14 r^50 R^12 +
> 14160254487780164383249793024 p^12 r^52 R^12 +
> 9154346603861932392765718528 p^10 r^54 R^12 +
> 4463094395773109054849679360 p^8 r^56 R^12 +
> 1584727036152792417703034880 p^6 r^58 R^12 +
> 386675306338096113222942720 p^4 r^60 R^12 +
> 57989428432590583583735808 p^2 r^62 R^12 +
> 4031775395804606902566912 r^64 R^12 +
> 83050694578231555653632 p^50 r^13 R^13 +
> 650691838473226627842048 p^48 r^15 R^13 +
> 2039333862882567765622784 p^46 r^17 R^13 +
> 3143220302857577286336512 p^44 r^19 R^13 +
> 2272524771183309363347456 p^42 r^21 R^13 +
> 488073050661848902270976 p^40 r^23 R^13 -
> 103573658491688335704064 p^38 r^25 R^13 -
> 4569912615177564639461376 p^36 r^27 R^13 +
> 373180600112307745299890176 p^34 r^29 R^13 -
> 3806757591785317805003374592 p^32 r^31 R^13 +
> 17263242693327851709236510720 p^30 r^33 R^13 -
> 39827351382434940759702503424 p^28 r^35 R^13 +
> 23179929590159857072579018752 p^26 r^37 R^13 +
> 5956814344421255049962848256 p^24 r^39 R^13 +
> 9024219152117621209490784256 p^22 r^41 R^13 +
> 15655273506408729623010476032 p^20 r^43 R^13 +
> 27493800088785396390852296704 p^18 r^45 R^13 +
> 42451007361465839649967570944 p^16 r^47 R^13 +
> 53385686587894961231242461184 p^14 r^49 R^13 +
> 52583521471457570838835888128 p^12 r^51 R^13 +
> 39577956688036911677555343360 p^10 r^53 R^13 +
> 22220039022283585134526464000 p^8 r^55 R^13 +
> 8994561963713721440812400640 p^6 r^57 R^13 +
> 2479624578187649025363148800 p^4 r^59 R^13 +
> 416815484663585596401254400 p^2 r^61 R^13 +
> 32254203166436855220535296 r^63 R^13 -
> 205333552593441281540096 p^48 r^14 R^14 -
> 1137258463029141983723520 p^46 r^16 R^14 -
> 2292756363333132540182528 p^44 r^18 R^14 -
> 1884404239481787945844736 p^42 r^20 R^14 -
> 377375099561615730671616 p^40 r^22 R^14 +
> 158176376139348619296768 p^38 r^24 R^14 +
> 141984030865591237607424 p^36 r^26 R^14 +
> 57331633900223757562150912 p^34 r^28 R^14 -
> 1393636013950784642756902912 p^32 r^30 R^14 +
> 9007151197521420252763652096 p^30 r^32 R^14 -
> 28736680212880729092674027520 p^28 r^34 R^14 +
> 22586439484619506065738825728 p^26 r^36 R^14 +
> 6619046303235511859751682048 p^24 r^38 R^14 +
> 12124115442990313958482640896 p^22 r^40 R^14 +
> 24730022831800648279047274496 p^20 r^42 R^14 +
> 51767839856632313029198872576 p^18 r^44 R^14 +
> 96071509277876105046820651008 p^16 r^46 R^14 +
> 144784733279293708398395129856 p^14 r^48 R^14 +
> 169256190106254003458268987392 p^12 r^50 R^14 +
> 149401147715152224820985331712 p^10 r^52 R^14 +
> 97197796438114184531458129920 p^8 r^54 R^14 +
> 45092032343600228044237701120 p^6 r^56 R^14 +
> 14106298289739212943922298880 p^4 r^58 R^14 +
> 2667304761999502896581836800 p^2 r^60 R^14 +
> 230387165474548965860966400 r^62 R^14 +
> 317648757950544441507840 p^46 r^15 R^15 +
> 1126954969306610437980160 p^44 r^17 R^15 +
> 1183864014629469438345216 p^42 r^19 R^15 +
> 193638205372906428432384 p^40 r^21 R^15 -
> 230465381595854573404160 p^38 r^23 R^15 -
> 99035888762354361434112 p^36 r^25 R^15 +
> 1182607724879421158457344 p^34 r^27 R^15 -
> 315871032686289996094636032 p^32 r^29 R^15 +
> 3433312484233871582851760128 p^30 r^31 R^15 -
> 15648727621025535713462976512 p^28 r^33 R^15 +
> 17347857251079592551389331456 p^26 r^35 R^15 +
> 6090976931839821452975013888 p^24 r^37 R^15 +
> 13707392267884296799910887424 p^22 r^39 R^15 +
> 33066518218739653401596919808 p^20 r^41 R^15 +
> 82709556810553777903206662144 p^18 r^43 R^15 +
> 185469621100729400422964396032 p^16 r^45 R^15 +
> 337639972751642937448172879872 p^14 r^47 R^15 +
> 472540963384705489979859533824 p^12 r^49 R^15 +
> 493142205905382425062228885504 p^10 r^51 R^15 +
> 374401374604496055338249748480 p^8 r^53 R^15 +
> 200232658036985671456244367360 p^6 r^55 R^15 +
> 71421018483570901567458508800 p^4 r^57 R^15 +
> 15248738437736374143968870400 p^2 r^59 R^15 +
> 1474477859037113381510184960 r^61 R^15 -
> 280465037219638783508480 p^44 r^16 R^16 -
> 494556792269376388595712 p^42 r^18 R^16 -
> 17191016496691616415744 p^40 r^20 R^16 +
> 283089166317685219262464 p^38 r^22 R^16 +
> 174349500925130187997184 p^36 r^24 R^16 -
> 584671106742643494748160 p^34 r^26 R^16 -
> 26947879393149831662796800 p^32 r^28 R^16 +
> 895510267945729056877576192 p^30 r^30 R^16 -
> 6204637218649423784582316032 p^28 r^32 R^16 +
> 10196564021146560323565649920 p^26 r^34 R^16 +
> 4645364998179375192922390528 p^24 r^36 R^16 +
> 13006424620209803724960301056 p^22 r^38 R^16 +
> 37415312326052203837073129472 p^20 r^40 R^16 +
> 112101205896650376309848932352 p^18 r^42 R^16 +
> 305089674313785356802525757440 p^16 r^44 R^16 +
> 676216276767185459458076573696 p^14 r^46 R^16 +
> 1143622436960844450296653414400 p^12 r^48 R^16 +
> 1423800714960422680267111530496 p^10 r^50 R^16 +
> 1271468512677401329856937984000 p^8 r^52 R^16 +
> 789057687735016852206523514880 p^6 r^54 R^16 +
> 322599575243308168419842457600 p^4 r^56 R^16 +
> 78088318417911594197534638080 p^2 r^58 R^16 +
> 8478247689463401943683563520 r^60 R^16 +
> 99778918132891965718528 p^42 r^17 R^17 -
> 63687312694063214362624 p^40 r^19 R^17 -
> 272106970475830005727232 p^38 r^21 R^17 -
> 188785673698795658936320 p^36 r^23 R^17 -
> 87702267161965236322304 p^34 r^25 R^17 +
> 4276920078749616541007872 p^32 r^27 R^17 +
> 121122055075712275532742656 p^30 r^29 R^17 -
> 1664915641242225403727183872 p^28 r^31 R^17 +
> 4345952167244991258127499264 p^26 r^33 R^17 +
> 2951617280400649222925320192 p^24 r^35 R^17 +
> 10312723033806399378762498048 p^22 r^37 R^17 +
> 35775417872830083416153128960 p^20 r^39 R^17 +
> 128800842605501888764393291776 p^18 r^41 R^17 +
> 426879150070518259555615375360 p^16 r^43 R^17 +
> 1160513672185818754463643467776 p^14 r^45 R^17 +
> 2395072287361357912684137283584 p^12 r^47 R^17 +
> 3592878295650668392328313438208 p^10 r^49 R^17 +
> 3807594669087204100615459307520 p^8 r^51 R^17 +
> 2762337588069747578951832698880 p^6 r^53 R^17 +
> 1302103366345242134173282467840 p^4 r^55 R^17 +
> 358916443541935210705421598720 p^2 r^57 R^17 +
> 43887399804281139473185505280 r^59 R^17 +
> 40360605530473176760320 p^40 r^18 R^18 +
> 181346249666282951540736 p^38 r^20 R^18 +
> 156418592426988371181568 p^36 r^22 R^18 +
> 14645334711719455358976 p^34 r^24 R^18 +
> 1010960196711099712667648 p^32 r^26 R^18 -
> 9248963671277030130843648 p^30 r^28 R^18 -
> 232995146480147520934641664 p^28 r^30 R^18 +
> 1179175288255939351817486336 p^26 r^32 R^18 +
> 1580888400773303847484391424 p^24 r^34 R^18 +
> 6791640658892916851606552576 p^22 r^36 R^18 +
> 28821994443192896070348701696 p^20 r^38 R^18 +
> 125295974310253953704483880960 p^18 r^40 R^18 +
> 506989992556444474243603759104 p^16 r^42 R^18 +
> 1701480900267543838235197177856 p^14 r^44 R^18 +
> 4328283414675658271205095374848 p^12 r^46 R^18 +
> 7909160901728761058598259261440 p^10 r^48 R^18 +
> 10047037507177093766884192419840 p^8 r^50 R^18 +
> 8592876606616663575482911948800 p^6 r^52 R^18 +
> 4701044529494885140300981862400 p^4 r^54 R^18 +
> 1482718833348804931427814604800 p^2 r^56 R^18 +
> 204807865753311984208199024640 r^58 R^18 -
> 62709320033773562101760 p^38 r^19 R^19 -
> 87667091801164467929088 p^36 r^21 R^19 +
> 2588664144398515175424 p^34 r^23 R^19 +
> 81521238756497089888256 p^32 r^25 R^19 -
> 6610499792760813505216512 p^30 r^27 R^19 +
> 21537495840320526827388928 p^28 r^29 R^19 +
> 96994516532649044261994496 p^26 r^31 R^19 +
> 725208794481378341049860096 p^24 r^33 R^19 +
> 3686064570481940176467656704 p^22 r^35 R^19 +
> 19473650419221517482703454208 p^20 r^37 R^19 +
> 102972764155654163451722334208 p^18 r^39 R^19 +
> 509931308211518986614040690688 p^16 r^41 R^19 +
> 2123436239453848294729097150464 p^14 r^43 R^19 +
> 6723551761450187169910459727872 p^12 r^45 R^19 +
> 15142850433138089943311929311232 p^10 r^47 R^19 +
> 23318326864717337836723812433920 p^8 r^49 R^19 +
> 23737513465873563008079400796160 p^6 r^51 R^19 +
> 15186180863053753729193827368960 p^4 r^53 R^19 +
> 5509938550738211508235901337600 p^2 r^55 R^19 +
> 862348908434997828245048524800 r^57 R^19 +
> 24236477627665630625792 p^36 r^20 R^20 -
> 11897757650381172637696 p^34 r^22 R^20 -
> 17357758034462367547392 p^32 r^24 R^20 -
> 907341673990346864852992 p^30 r^26 R^20 +
> 19629582266779739673853952 p^28 r^28 R^20 -
> 71487058120130415288123392 p^26 r^30 R^20 +
> 287369487008542789493325824 p^24 r^32 R^20 +
> 1632753651799766337467711488 p^22 r^34 R^20 +
> 10963891695563550634757062656 p^20 r^36 R^20 +
> 71235991434320377271268933632 p^18 r^38 R^20 +
> 433205585043047094672104095744 p^16 r^40 R^20 +
> 2246751098858846703852707643392 p^14 r^42 R^20 +
> 8935024153282716343692997689344 p^12 r^44 R^20 +
> 25111735559321047926749333028864 p^10 r^46 R^20 +
> 47467867697286255022559314575360 p^8 r^48 R^20 +
> 58147616990054804927245367377920 p^6 r^50 R^20 +
> 43879224378237800860205375815680 p^4 r^52 R^20 +
> 18424535133988164736413125836800 p^2 r^54 R^20 +
> 3276925852052991747331184394240 r^56 R^20 +
> 8205062922799893446656 p^34 r^21 R^21 +
> 11836544669270371467264 p^32 r^23 R^21 -
> 41206429538230357458944 p^30 r^25 R^21 +
> 4491674303356120483233792 p^28 r^27 R^21 -
> 37197127251046281773056000 p^26 r^29 R^21 +
> 96943616629618539384274944 p^24 r^31 R^21 +
> 583177627078794320881385472 p^22 r^33 R^21 +
> 5102650562769422107740733440 p^20 r^35 R^21 +
> 41258671271617174507664441344 p^18 r^37 R^21 +
> 309748756837517987013761957888 p^16 r^39 R^21 +
> 2007017467428334306959214772224 p^14 r^41 R^21 +
> 10101939182295255822830393098240 p^12 r^43 R^21 +
> 35879103507033452946239139086336 p^10 r^45 R^21 +
> 84415598460098333381815613521920 p^8 r^47 R^21 +
> 126004763318589725262468454809600 p^6 r^49 R^21 +
> 113285553614369456104852810629120 p^4 r^51 R^21 +
> 55427837013647279211140370923520 p^2 r^53 R^21 +
> 11235174349895971705135489351680 r^55 R^21 -
> 5014764676593697685504 p^32 r^22 R^22 -
> 1367931513591839064064 p^30 r^24 R^22 +
> 455234406138817594523648 p^28 r^26 R^22 -
> 9438583012851058560991232 p^26 r^28 R^22 +
> 26784944263367577584009216 p^24 r^30 R^22 +
> 165197719654797656308318208 p^22 r^32 R^22 +
> 1944458407625709948248260608 p^20 r^34 R^22 +
> 19874045207401120699437285376 p^18 r^36 R^22 +
> 185401880722310085264774529024 p^16 r^38 R^22 +
> 1506735581056166494729653977088 p^14 r^40 R^22 +
> 9657773797015750420398084194304 p^12 r^42 R^22 +
> 43884462401597927212204978864128 p^10 r^44 R^22 +
> 130468960796818084802353441013760 p^8 r^46 R^22 +
> 240716561062239685793612090572800 p^6 r^48 R^22 +
> 260864025194550810462557589995520 p^4 r^50 R^22 +
> 149912591757147312386453935226880 p^2 r^52 R^22 +
> 34726902536042094361327876177920 r^54 R^22 +
> 1372535036835941842944 p^30 r^23 R^23 +
> 18143400180158337908736 p^28 r^25 R^23 -
> 1411575084399478823714816 p^26 r^27 R^23 +
> 5780384520300990272897024 p^24 r^29 R^23 +
> 36185846628767700019576832 p^22 r^31 R^23 +
> 598723230668610468705206272 p^20 r^33 R^23 +
> 7918759422001196721078534144 p^18 r^35 R^23 +
> 92161396457870905431507861504 p^16 r^37 R^23 +
> 945067653736289346107041906688 p^14 r^39 R^23 +
> 7756595779258315414988482674688 p^12 r^41 R^23 +
> 45605775576117827040756472217600 p^10 r^43 R^23 +
> 174103367452193534902301192355840 p^8 r^45 R^23 +
> 403534119518397159107855552348160 p^6 r^47 R^23 +
> 534346045338989204665267755417600 p^4 r^49 R^23 +
> 364073349743065150317205546598400 p^2 r^51 R^23 +
> 96631380969856262570651481538560 r^53 R^23 -
> 1131475363181308674048 p^28 r^24 R^24 -
> 111590321045475436789760 p^26 r^26 R^24 +
> 926454006029351827537920 p^24 r^28 R^24 +
> 5931009999971132203073536 p^22 r^30 R^24 +
> 144970004748283398387138560 p^20 r^32 R^24 +
> 2609524952587826541499514880 p^18 r^34 R^24 +
> 37714068234620863057460461568 p^16 r^36 R^24 +
> 490730307479544072844829786112 p^14 r^38 R^24 +
> 5195539605798843020935977500672 p^12 r^40 R^24 +
> 39927769193522669139082914824192 p^10 r^42 R^24 +
> 198992890960222943654616603033600 p^8 r^44 R^24 +
> 590075142015419189214776540528640 p^6 r^46 R^24 +
> 970080427110220250219829453127680 p^4 r^48 R^24 +
> 792468135565563671352817011916800 p^2 r^50 R^24 +
> 241578452424640656426628703846400 r^52 R^24 -
> 2842298286027185324032 p^26 r^25 R^25 +
> 103391577596542511480832 p^24 r^27 R^25 +
> 713399639020649196814336 p^22 r^29 R^25 +
> 25852596071578209747992576 p^20 r^31 R^25 +
> 713420068449689058276278272 p^18 r^33 R^25 +
> 12702199466840310209184792576 p^16 r^35 R^25 +
> 207778175584004205065350938624 p^14 r^37 R^25 +
> 2874813650484473688865323876352 p^12 r^39 R^25 +
> 29173339340110925411263692406784 p^10 r^41 R^25 +
> 192942895896306350337541011406848 p^8 r^43 R^25 +
> 746948290277781643244497727913984 p^6 r^45 R^25 +
> 1553301335112359308363388172632064 p^4 r^47 R^25 +
> 1542116964091014693098516958412800 p^2 r^49 R^25 +
> 541135733431195070395648296615936 r^51 R^25 +
> 5684596572054370648064 p^24 r^26 R^26 +
> 70283274863935762726912 p^22 r^28 R^26 +
> 2929121507904199734591488 p^20 r^30 R^26 +
> 158182133962942750478303232 p^18 r^32 R^26 +
> 3629099061682083243973672960 p^16 r^34 R^26 +
> 70314125440648999364122902528 p^14 r^36 R^26 +
> 1292579317918978399232543686656 p^12 r^38 R^26 +
> 17600395713744815569153946550272 p^10 r^40 R^26 +
> 156928360163216881066960899538944 p^8 r^42 R^26 +
> 810801560142951990960465805049856 p^6 r^44 R^26 +
> 2179801065268808357215986574688256 p^4 r^46 R^26 +
> 2673819785409728553095649797603328 p^2 r^48 R^26 +
> 1082271466862390140791296593231872 r^50 R^26 +
> 14802899629602592784384 p^22 r^27 R^27 +
> 97943708235301243584512 p^20 r^29 R^27 +
> 25646745319133977375670272 p^18 r^31 R^27 +
> 937999471332890944965967872 p^16 r^33 R^27 +
> 19015128889123603518391844864 p^14 r^35 R^27 +
> 456199764737146843150724104192 p^12 r^37 R^27 +
> 8645569177941535720416556875776 p^10 r^39 R^27 +
> 105700679739340200112556685131776 p^8 r^41 R^27 +
> 745919084750607446906405121949696 p^6 r^43 R^27 +
> 2659221489859258452182316175327232 p^4 r^45 R^27 +
> 4112503190683216359626976748634112 p^2 r^47 R^27 +
> 1924038163310915805851193943523328 r^49 R^27 -
> 42276622767616547618816 p^20 r^28 R^28 +
> 2651003208111011041116160 p^18 r^30 R^28 +
> 215664258486118441614835712 p^16 r^32 R^28 +
> 4647818159006569187487252480 p^14 r^34 R^28 +
> 116855989060939087632253583360 p^12 r^36 R^28 +
> 3369501445376414868384186892288 p^10 r^38 R^28 +
> 58121132666460782907284086849536 p^8 r^40 R^28 +
> 573286643770874513145680501932032 p^6 r^42 R^28 +
> 2790895526900322513685507683123200 p^4 r^44 R^28 +
> 5578618730591958264090782953635840 p^2 r^46 R^28 +
> 3023488542345724837766161911250944 r^48 R^28 +
> 250710851627291122860032 p^18 r^29 R^29 +
> 33938018257837378372632576 p^16 r^31 R^29 +
> 1373598424762343931397537792 p^14 r^33 R^29 +
> 17865693951336343910226067456 p^12 r^35 R^29 +
> 971575628163997742436548870144 p^10 r^37 R^29 +
> 25682514634144290635733620752384 p^8 r^39 R^29 +
> 361656075078774027731354664304640 p^6 r^41 R^29 +
> 2486568222758292078378918418055168 p^4 r^43 R^29 +
> 6623513572233271906756707252961280 p^2 r^45 R^29 +
> 4170329023925137707263671601725440 r^47 R^29 +
> 2051703369031702066233344 p^16 r^30 R^30 +
> 441761338110954191645048832 p^14 r^32 R^30 +
> 772150129141336647850262528 p^12 r^34 R^30 +
> 154990626026205160410512883712 p^10 r^36 R^30 +
> 8961318804489297763430490439680 p^8 r^38 R^30 +
> 183238489627573261128815127560192 p^6 r^40 R^30 +
> 1848917721811483557833682890784768 p^4 r^42 R^30 +
> 6813977819683415608685264675799040 p^2 r^44 R^30 +
> 5004394828710165248716405922070528 r^46 R^30 +
> 86069344231320126652153856 p^14 r^31 R^31 +
> 22733785843183725121110016 p^12 r^33 R^31 -
> 21276945347855241791087312896 p^10 r^35 R^31 +
> 2417491612907686651228632121344 p^8 r^37 R^31 +
> 72526982125708035007321009553408 p^6 r^39 R^31 +
> 1122456132217084850877562724810752 p^4 r^41 R^31 +
> 5991594173292925723905546669522944 p^2 r^43 R^31 +
> 5165826919958880256739515790524416 r^45 R^31 +
> 49967378201987789715668992 p^12 r^32 R^32 -
> 20615421373635766442799398912 p^10 r^34 R^32 +
> 489353378405164055498466525184 p^8 r^36 R^32 +
> 21571980145495141093520491675648 p^6 r^38 R^32 +
> 540801656138259791298685082533888 p^4 r^40 R^32 +
> 4419348224682312148599691873878016 p^2 r^42 R^32 +
> 4520098554964020224647076316708864 r^44 R^32 -
> 4349941182268803154857426944 p^10 r^33 R^33 +
> 69865345931146192877174915072 p^8 r^35 R^33 +
> 4515860038737802794067570982912 p^6 r^37 R^33 +
> 199269407855838365654879840501760 p^4 r^39 R^33 +
> 2662350450080648843787879223132160 p^2 r^41 R^33 +
> 3287344403610196527016055503060992 r^43 R^33 +
> 5576806977003948932377608192 p^8 r^34 R^34 +
> 578966225681854961540235526144 p^6 r^36 R^34 +
> 53403771721195400848761455378432 p^4 r^38 R^34 +
> 1258976309111435186846370491269120 p^2 r^40 R^34 +
> 1933732002123645015891797354741760 r^42 R^34 +
> 31242753998865355900566110208 p^6 r^35 R^35 +
> 9600045899464528234808688508928 p^4 r^37 R^35 +
> 438424010829544066590308260904960 p^2 r^39 R^35 +
> 883991772399380578693393076453376 r^41 R^35 +
> 929469686571277543884248317952 p^4 r^36 R^36 +
> 99973483235211993844017893736448 p^2 r^38 R^36 +
> 294663924133126859564464358817792 r^40 R^36 +
> 11195651926810990882177629028352 p^2 r^37 R^37 +
> 63711118731486888554478780284928 r^39 R^37 +
> 6706433550682830374155661082624 r^38 R^38)/(372952245 p^76 +
> 15980705934 p^74 r^2 + 336173294795 p^72 r^4 +
> 4631786402212 p^70 r^6 + 47064014247235 p^68 r^8 +
> 376686432120542 p^66 r^10 + 2478431189385661 p^64 r^12 +
> 13826654967408224 p^62 r^14 + 67036384589031188 p^60 r^16 +
> 288639616825182040 p^58 r^18 + 1126903604016879468 p^56 r^20 +
> 4069813441547044080 p^54 r^22 + 13509874641123561276 p^52 r^24 +
> 36268543044616026936 p^50 r^26 +
> 612301273451214945348 p^48 r^28 +
> 1878060975958315120800 p^46 r^30 -
> 21343504649330854044874 p^44 r^32 +
> 436647802539362719236 p^42 r^34 +
> 133856974745153633981002 p^40 r^36 -
> 231776926901194657508136 p^38 r^38 +
> 133856974745153633981002 p^36 r^40 +
> 436647802539362719236 p^34 r^42 -
> 21343504649330854044874 p^32 r^44 +
> 1878060975958315120800 p^30 r^46 +
> 612301273451214945348 p^28 r^48 +
> 36268543044616026936 p^26 r^50 + 13509874641123561276 p^24 r^52 +
> 4069813441547044080 p^22 r^54 + 1126903604016879468 p^20 r^56 +
> 288639616825182040 p^18 r^58 + 67036384589031188 p^16 r^60 +
> 13826654967408224 p^14 r^62 + 2478431189385661 p^12 r^64 +
> 376686432120542 p^10 r^66 + 47064014247235 p^8 r^68 +
> 4631786402212 p^6 r^70 + 336173294795 p^4 r^72 +
> 15980705934 p^2 r^74 + 372952245 r^76 - 56688741240 p^74 r R -
> 2244532913256 p^72 r^3 R - 43475035178864 p^70 r^5 R -
> 549393624304272 p^68 r^7 R - 5098288085363928 p^66 r^9 R -
> 37090592312136776 p^64 r^11 R - 220666251383971968 p^62 r^13 R -
> 1106692625487217536 p^60 r^15 R -
> 4792509218347527008 p^58 r^17 R -
> 18298660127667036192 p^56 r^19 R -
> 62838399361548285504 p^54 r^21 R -
> 197629660897462535616 p^52 r^23 R -
> 558923319005456895840 p^50 r^25 R -
> 1181966747136112397088 p^48 r^27 R -
> 23310084190912485416832 p^46 r^29 R -
> 36787867039753598448768 p^44 r^31 R +
> 549031978623694095525744 p^42 r^33 R -
> 556018343464323899033520 p^40 r^35 R -
> 514837454496905172814496 p^38 r^37 R +
> 514837454496905172814496 p^36 r^39 R +
> 556018343464323899033520 p^34 r^41 R -
> 549031978623694095525744 p^32 r^43 R +
> 36787867039753598448768 p^30 r^45 R +
> 23310084190912485416832 p^28 r^47 R +
> 1181966747136112397088 p^26 r^49 R +
> 558923319005456895840 p^24 r^51 R +
> 197629660897462535616 p^22 r^53 R +
> 62838399361548285504 p^20 r^55 R +
> 18298660127667036192 p^18 r^57 R +
> 4792509218347527008 p^16 r^59 R +
> 1106692625487217536 p^14 r^61 R +
> 220666251383971968 p^12 r^63 R + 37090592312136776 p^10 r^65 R +
> 5098288085363928 p^8 r^67 R + 549393624304272 p^6 r^69 R +
> 43475035178864 p^4 r^71 R + 2244532913256 p^2 r^73 R +
> 56688741240 r^75 R + 3963039321328 p^72 r^2 R^2 +
> 144191196884992 p^70 r^4 R^2 + 2555739238794976 p^68 r^6 R^2 +
> 29417401301014400 p^66 r^8 R^2 +
> 247360601399505712 p^64 r^10 R^2 +
> 1621107940579964928 p^62 r^12 R^2 +
> 8630557391821011200 p^60 r^14 R^2 +
> 38439319752734503936 p^58 r^16 R^2 +
> 146527136509356597952 p^56 r^18 R^2 +
> 487357010822781847552 p^54 r^20 R^2 +
> 1439571768278905001088 p^52 r^22 R^2 +
> 3837676303957809496576 p^50 r^24 R^2 +
> 9048150323207964887232 p^48 r^26 R^2 -
> 1815243583793406953472 p^46 r^28 R^2 +
> 369988458771349864920832 p^44 r^30 R^2 +
> 573515640249251277982720 p^42 r^32 R^2 -
> 6010158566595449521434336 p^40 r^34 R^2 +
> 8333193206667689418958848 p^38 r^36 R^2 -
> 7303518297673879073329856 p^36 r^38 R^2 +
> 9346268627381392915635456 p^34 r^40 R^2 -
> 6063835426486192233240160 p^32 r^42 R^2 +
> 145041679785200513361920 p^30 r^44 R^2 +
> 421396759020218144456448 p^28 r^46 R^2 +
> 16955118113786507848704 p^26 r^48 R^2 +
> 10694329990184069795264 p^24 r^50 R^2 +
> 4489318058644971067392 p^22 r^52 R^2 +
> 1649148567469859859584 p^20 r^54 R^2 +
> 548821948341990777344 p^18 r^56 R^2 +
> 162846067417861630400 p^16 r^58 R^2 +
> 42277303380590995456 p^14 r^60 R^2 +
> 9412204628618633472 p^12 r^62 R^2 +
> 1755531383455880192 p^10 r^64 R^2 +
> 266283307650093424 p^8 r^66 R^2 +
> 31506665468069888 p^6 r^68 R^2 + 2725161124920032 p^4 r^70 R^2 +
> 153154264606592 p^2 r^72 R^2 + 4194966851760 r^74 R^2 -
> 168888554624000 p^70 r^3 R^3 - 5611080220927744 p^68 r^5 R^3 -
> 90364279965035008 p^66 r^7 R^3 -
> 939760192070299392 p^64 r^9 R^3 -
> 7094116873562308608 p^62 r^11 R^3 -
> 41432243476841797632 p^60 r^13 R^3 -
> 194892517456360173568 p^58 r^15 R^3 -
> 759161670296761202688 p^56 r^17 R^3 -
> 2499733819721028849664 p^54 r^19 R^3 -
> 7070764458306129998848 p^52 r^21 R^3 -
> 17404828225650211837952 p^50 r^23 R^3 -
> 37885638521295034496000 p^48 r^25 R^3 -
> 74522508353325912260608 p^46 r^27 R^3 +
> 497989269327906056943616 p^44 r^29 R^3 -
> 3858774782387726085066752 p^42 r^31 R^3 -
> 6276562667060370758805504 p^40 r^33 R^3 +
> 41054680582661198249977856 p^38 r^35 R^3 -
> 62470549684908451380337152 p^36 r^37 R^3 +
> 64888488031508207340930048 p^34 r^39 R^3 -
> 36151590573896024171637248 p^32 r^41 R^3 -
> 2968791212029773607878656 p^30 r^43 R^3 +
> 4804999320120506789042176 p^28 r^45 R^3 +
> 136135133974919672655872 p^26 r^47 R^3 +
> 125377485676721827379200 p^24 r^49 R^3 +
> 63385644367311862755328 p^22 r^51 R^3 +
> 27080249838398950403072 p^20 r^53 R^3 +
> 10356508975509156669440 p^18 r^55 R^3 +
> 3498766375483407117312 p^16 r^57 R^3 +
> 1025757989189607948288 p^14 r^59 R^3 +
> 256000566717468100608 p^12 r^61 R^3 +
> 53175174481486712832 p^10 r^63 R^3 +
> 8929483219144648704 p^8 r^65 R^3 +
> 1163474708920404992 p^6 r^67 R^3 +
> 110289932460837120 p^4 r^69 R^3 + 6763590726161920 p^2 r^71 R^3 +
> 201358408884480 r^73 R^3 + 4902405953107456 p^68 r^4 R^4 +
> 147648901531380224 p^66 r^6 R^4 +
> 2142667443690603264 p^64 r^8 R^4 +
> 19941953313286885376 p^62 r^10 R^4 +
> 133653330442225328128 p^60 r^12 R^4 +
> 686542878068406124544 p^58 r^14 R^4 +
> 2808396100004105775104 p^56 r^16 R^4 +
> 9381133602406991007744 p^54 r^18 R^4 +
> 26017705097626973280256 p^52 r^20 R^4 +
> 60495918156694038259712 p^50 r^22 R^4 +
> 118176286393369560142848 p^48 r^24 R^4 +
> 196914928710399907057664 p^46 r^26 R^4 +
> 571165246915062629498880 p^44 r^28 R^4 -
> 8853252773411091289251840 p^42 r^30 R^4 +
> 30265161765536643391543296 p^40 r^32 R^4 +
> 31658856739794134386561024 p^38 r^34 R^4 -
> 168019259761589103369423872 p^36 r^36 R^4 +
> 219711386519285936477424640 p^34 r^38 R^4 -
> 109572423291453282917395968 p^32 r^40 R^4 -
> 50708632310274041924112384 p^30 r^42 R^4 +
> 38688063911709764665286656 p^28 r^44 R^4 +
> 602383542598091926093824 p^26 r^46 R^4 +
> 1004815328060474212866048 p^24 r^48 R^4 +
> 623436871546995076509696 p^22 r^50 R^4 +
> 312018456882407019345920 p^20 r^52 R^4 +
> 137946843611363106244608 p^18 r^54 R^4 +
> 53342562157956855747584 p^16 r^56 R^4 +
> 17744932749553889046528 p^14 r^58 R^4 +
> 4985860278022946701312 p^12 r^60 R^4 +
> 1157780159764272373760 p^10 r^62 R^4 +
> 215985668790557435904 p^8 r^64 R^4 +
> 31087122781307039744 p^6 r^66 R^4 +
> 3238724664948442624 p^4 r^68 R^4 +
> 217287394551503360 p^2 r^70 R^4 + 7047544310956800 r^72 R^4 -
> 102583302989170688 p^66 r^5 R^5 -
> 2777124580628252672 p^64 r^7 R^5 -
> 35963039703303553024 p^62 r^9 R^5 -
> 296151319453169156096 p^60 r^11 R^5 -
> 1738543223010614132736 p^58 r^13 R^5 -
> 7727058511332947378176 p^56 r^15 R^5 -
> 26934536085211330215936 p^54 r^17 R^5 -
> 75163084348708410556416 p^52 r^19 R^5 -
> 169480257619810096537600 p^50 r^21 R^5 -
> 307688306287650398724096 p^48 r^23 R^5 -
> 436464088743321079349248 p^46 r^25 R^5 -
> 438904457710989947502592 p^44 r^27 R^5 -
> 8115545909110603876384768 p^42 r^29 R^5 +
> 88700285123003474031788032 p^40 r^31 R^5 -
> 198062321046935290818560000 p^38 r^33 R^5 +
> 11485734035852492138151936 p^36 r^35 R^5 +
> 215537793518102426153635840 p^34 r^37 R^5 +
> 3872333435710933797392384 p^32 r^39 R^5 -
> 409486819041565905757700096 p^30 r^41 R^5 +
> 233282004018392947108937728 p^28 r^43 R^5 +
> 572968223785813968240640 p^26 r^45 R^5 +
> 5803573022939318906503168 p^24 r^47 R^5 +
> 4537446576201845877211136 p^22 r^49 R^5 +
> 2681572111165572841144320 p^20 r^51 R^5 +
> 1379286406206520331771904 p^18 r^53 R^5 +
> 613964767940576280256512 p^16 r^55 R^5 +
> 232934102227413499740160 p^14 r^57 R^5 +
> 74021625998470939738112 p^12 r^59 R^5 +
> 19295048913988818026496 p^10 r^61 R^5 +
> 4013528735181135167488 p^8 r^63 R^5 +
> 640246550170503577600 p^6 r^65 R^5 +
> 73528661295890366464 p^4 r^67 R^5 +
> 5411490376911173632 p^2 r^69 R^5 + 191693205258024960 r^71 R^5 +
> 1597211684703842304 p^64 r^6 R^6 +
> 38481513904474169344 p^62 r^8 R^6 +
> 439531868628907278336 p^60 r^10 R^6 +
> 3158496974214928973824 p^58 r^12 R^6 +
> 15970724368343959937024 p^56 r^14 R^6 +
> 60149236110063722487808 p^54 r^16 R^6 +
> 173925713361163110350848 p^52 r^18 R^6 +
> 391029100559009428111360 p^50 r^20 R^6 +
> 680164390678337766236160 p^48 r^22 R^6 +
> 884987776519754131578880 p^46 r^24 R^6 +
> 735752495618850336325632 p^44 r^26 R^6 -
> 2471567008735468800229376 p^42 r^28 R^6 +
> 101234619089666748192202752 p^40 r^30 R^6 -
> 622933271871308553815728128 p^38 r^32 R^6 +
> 1183319439170882206296506368 p^36 r^34 R^6 -
> 1206290907242587190572810240 p^34 r^36 R^6 +
> 1592921517202410358830202880 p^32 r^38 R^6 -
> 2165354184334904176930365440 p^30 r^40 R^6 +
> 1090463049842573732755456000 p^28 r^42 R^6 -
> 11655516417392673157988352 p^26 r^44 R^6 +
> 24706504281729604406501376 p^24 r^46 R^6 +
> 25338650160333083379081216 p^22 r^48 R^6 +
> 17840832905824776077475840 p^20 r^50 R^6 +
> 10748982760282074517176320 p^18 r^52 R^6 +
> 5541413195231952712097792 p^16 r^54 R^6 +
> 2410892020583563636793344 p^14 r^56 R^6 +
> 870758311476429120290816 p^12 r^58 R^6 +
> 255915553482700979748864 p^10 r^60 R^6 +
> 59589501987504609910784 p^8 r^62 R^6 +
> 10572849475905298472960 p^6 r^64 R^6 +
> 1342770243169288323072 p^4 r^66 R^6 +
> 108721248385224409088 p^2 r^68 R^6 +
> 4217250515676549120 r^70 R^6 -
> 18828904738971942912 p^62 r^7 R^7 -
> 398956441092335730688 p^60 r^9 R^7 -
> 3962815575285302493184 p^58 r^11 R^7 -
> 24427665390453727559680 p^56 r^13 R^7 -
> 104144553087344964173824 p^54 r^15 R^7 -
> 323376349335905920352256 p^52 r^17 R^7 -
> 747573247266248410923008 p^50 r^19 R^7 -
> 1284201825597322889003008 p^48 r^21 R^7 -
> 1584081189483208361869312 p^46 r^23 R^7 -
> 1266115080253475161767936 p^44 r^25 R^7 -
> 529804148417648123510784 p^42 r^27 R^7 +
> 58406923381360197711822848 p^40 r^29 R^7 -
> 844479577816128314865778688 p^38 r^31 R^7 +
> 3340785160448020340768505856 p^36 r^33 R^7 -
> 6007921853791232434595168256 p^34 r^35 R^7 +
> 7863758472094247821321961472 p^32 r^37 R^7 -
> 8247096547328768307217727488 p^30 r^39 R^7 +
> 4037869124531303269523914752 p^28 r^41 R^7 -
> 96151804815657732736352256 p^26 r^43 R^7 +
> 77109994127850389718564864 p^24 r^45 R^7 +
> 111097772643314525051584512 p^22 r^47 R^7 +
> 94141320588086236232810496 p^20 r^49 R^7 +
> 66922789166844614552977408 p^18 r^51 R^7 +
> 40216753149459420147351552 p^16 r^53 R^7 +
> 20182813798581001193357312 p^14 r^55 R^7 +
> 8328904386528650414915584 p^12 r^57 R^7 +
> 2772993942710130018418688 p^10 r^59 R^7 +
> 725850517160374767058944 p^8 r^61 R^7 +
> 143784576885030726959104 p^6 r^63 R^7 +
> 20262707412028498116608 p^4 r^65 R^7 +
> 1810481869135833726976 p^2 r^67 R^7 +
> 77115438000942612480 r^69 R^7 +
> 169370097660939730944 p^60 r^8 R^8 +
> 3110978076460700139520 p^58 r^10 R^8 +
> 26407733158451288735744 p^56 r^12 R^8 +
> 136636232006793226354688 p^54 r^14 R^8 +
> 477664235291633751687168 p^52 r^16 R^8 +
> 1177975819241442890481664 p^50 r^18 R^8 +
> 2064508770007982272675840 p^48 r^20 R^8 +
> 2495306583890947526361088 p^46 r^22 R^8 +
> 1888330556046315130716160 p^44 r^24 R^8 +
> 815034519917407845482496 p^42 r^26 R^8 +
> 16014163818092813983678464 p^40 r^28 R^8 -
> 638402011437450112196673536 p^38 r^30 R^8 +
> 4904189225122369696637386752 p^36 r^32 R^8 -
> 14012877999401857130277896192 p^34 r^34 R^8 +
> 22399611993595084522430136320 p^32 r^36 R^8 -
> 23624015683370056462902493184 p^30 r^38 R^8 +
> 12010074341986089615445458944 p^28 r^40 R^8 -
> 446889620220685167549480960 p^26 r^42 R^8 +
> 166548136168232546294824960 p^24 r^44 R^8 +
> 388111387464751845217665024 p^22 r^46 R^8 +
> 400502825953973581687554048 p^20 r^48 R^8 +
> 338536610631552636344598528 p^18 r^50 R^8 +
> 238790196048401323628691456 p^16 r^52 R^8 +
> 139104312058758855158923264 p^14 r^54 R^8 +
> 65962341505856871698792448 p^12 r^56 R^8 +
> 25004754440054009603293184 p^10 r^58 R^8 +
> 7391160913997325261078528 p^8 r^60 R^8 +
> 1641261657496733869670400 p^6 r^62 R^8 +
> 257580876744337817600000 p^4 r^64 R^8 +
> 25480737277724398714880 p^2 r^66 R^8 +
> 1195289289014610493440 r^68 R^8 -
> 1162714643358370758656 p^58 r^9 R^9 -
> 18187961426666498031616 p^56 r^11 R^9 -
> 129061706798349622968320 p^54 r^13 R^9 -
> 544909087501429566865408 p^52 r^15 R^9 -
> 1504297291422898994544640 p^50 r^17 R^9 -
> 2794365660869112738349056 p^48 r^19 R^9 -
> 3426365280115344483024896 p^46 r^21 R^9 -
> 2549651955029024769572864 p^44 r^23 R^9 -
> 945213588504139915067392 p^42 r^25 R^9 +
> 781277535610266217086976 p^40 r^27 R^9 -
> 278150961483101756961325056 p^38 r^29 R^9 +
> 4402001960181266736901783552 p^36 r^31 R^9 -
> 20569062204167712291342516224 p^34 r^33 R^9 +
> 43708255184311255160135352320 p^32 r^35 R^9 -
> 52057009710488398574590623744 p^30 r^37 R^9 +
> 28949290867961322095283011584 p^28 r^39 R^9 -
> 1494280421640850443129061376 p^26 r^41 R^9 +
> 186461667390105505736163328 p^24 r^43 R^9 +
> 1089643524200336324752834560 p^22 r^45 R^9 +
> 1389204303226421276240248832 p^20 r^47 R^9 +
> 1407870758676442388908474368 p^18 r^49 R^9 +
> 1174170708608547021419184128 p^16 r^51 R^9 +
> 799331895460934060645285888 p^14 r^53 R^9 +
> 438207453391077358593638400 p^12 r^55 R^9 +
> 190170593602992201293365248 p^10 r^57 R^9 +
> 63788078236265033829974016 p^8 r^59 R^9 +
> 15947394142253395416711168 p^6 r^61 R^9 +
> 2798086738039119068266496 p^4 r^63 R^9 +
> 307520080347766498263040 p^2 r^65 R^9 +
> 15937190520194806579200 r^67 R^9 +
> 6050678173276349399040 p^56 r^10 R^10 +
> 78825788227973713756160 p^54 r^12 R^10 +
> 454447757962277558419456 p^52 r^14 R^10 +
> 1507691328833731254288384 p^50 r^16 R^10 +
> 3118982558182767922774016 p^48 r^18 R^10 +
> 4039748890589000644952064 p^46 r^20 R^10 +
> 3073496402265307546124288 p^44 r^22 R^10 +
> 1203393835148033526857728 p^42 r^24 R^10 +
> 135682141197484966281216 p^40 r^26 R^10 -
> 63974015390707304871493632 p^38 r^28 R^10 +
> 2498534897594724800889094144 p^36 r^30 R^10 -
> 20320166027452005379722969088 p^34 r^32 R^10 +
> 61458290435787429962408001536 p^32 r^34 R^10 -
> 89246328140127766988768411648 p^30 r^36 R^10 +
> 56840039794143366108227305472 p^28 r^38 R^10 -
> 3887027513035504724429766656 p^26 r^40 R^10 -
> 229033868321824764137046016 p^24 r^42 R^10 +
> 2466730023755305075660029952 p^22 r^44 R^10 +
> 3958654207158420157721739264 p^20 r^46 R^10 +
> 4852738946337849323739414528 p^18 r^48 R^10 +
> 4822444406786271681588494336 p^16 r^50 R^10 +
> 3864137080713383165921591296 p^14 r^52 R^10 +
> 2465177629627292341564866560 p^12 r^54 R^10 +
> 1231982186884357293602766848 p^10 r^56 R^10 +
> 471393863305849001119580160 p^8 r^58 R^10 +
> 133304668124176787586416640 p^6 r^60 R^10 +
> 26257826812180030140973056 p^4 r^62 R^10 +
> 3218113483247233368326144 p^2 r^64 R^10 +
> 184871410034259756318720 r^66 R^10 -
> 23539561980203477499904 p^54 r^11 R^11 -
> 248190854013593128534016 p^52 r^13 R^11 -
> 1119678149916013038141440 p^50 r^15 R^11 -
> 2773141905622894460796928 p^48 r^17 R^11 -
> 3997578928574334117085184 p^46 r^19 R^11 -
> 3258714941300647043006464 p^44 r^21 R^11 -
> 1412931593888652616269824 p^42 r^23 R^11 -
> 440490013231393066188800 p^40 r^25 R^11 -
> 5750221510090936100585472 p^38 r^27 R^11 +
> 879083213640034962286575616 p^36 r^29 R^11 -
> 13699651754527602672729587712 p^34 r^31 R^11 +
> 63238163508038884309778563072 p^32 r^33 R^11 -
> 119467472661529242655311003648 p^30 r^35 R^11 +
> 91078983214726367171840049152 p^28 r^37 R^11 -
> 8177110935416458234396934144 p^26 r^39 R^11 -
> 1722583222997896155761737728 p^24 r^41 R^11 +
> 4489948099828789201193664512 p^22 r^43 R^11 +
> 9311604543845941380464508928 p^20 r^45 R^11 +
> 13940477063421981911838359552 p^18 r^47 R^11 +
> 16641206140679869794279751680 p^16 r^49 R^11 +
> 15815065597155999483240120320 p^14 r^51 R^11 +
> 11824027538366559159278632960 p^12 r^53 R^11 +
> 6848104368279956977675665408 p^10 r^55 R^11 +
> 3006008914432113825609154560 p^8 r^57 R^11 +
> 966374429780732201266053120 p^6 r^59 R^11 +
> 214655167603468959171477504 p^4 r^61 R^11 +
> 29454071587342091651383296 p^2 r^63 R^11 +
> 1882327083985190246154240 r^65 R^11 +
> 66990294212835595517952 p^52 r^12 R^12 +
> 550909650592661332033536 p^50 r^14 R^12 +
> 1851685070785564823781376 p^48 r^16 R^12 +
> 3206068408611131858354176 p^46 r^18 R^12 +
> 2977455944600219040088064 p^44 r^20 R^12 +
> 1517314510701748327284736 p^42 r^22 R^12 +
> 574251715048434342821888 p^40 r^24 R^12 +
> 40661419027692487966720 p^38 r^26 R^12 +
> 176494100527659116277006336 p^36 r^28 R^12 -
> 6222240631360813329926848512 p^34 r^30 R^12 +
> 47510630497681949305667059712 p^32 r^32 R^12 -
> 124505152362064703784531001344 p^30 r^34 R^12 +
> 118937896011702400566188572672 p^28 r^36 R^12 -
> 14216206403111058836937506816 p^26 r^38 R^12 -
> 4813952942916579214051246080 p^24 r^40 R^12 +
> 6490163773304530677190885376 p^22 r^42 R^12 +
> 18123625007723025645988478976 p^20 r^44 R^12 +
> 33493869326350079694446002176 p^18 r^46 R^12 +
> 48434868228426305664435879936 p^16 r^48 R^12 +
> 55035243851643855470363410432 p^14 r^50 R^12 +
> 48585630977432601740649168896 p^12 r^52 R^12 +
> 32834853139583088976312926208 p^10 r^54 R^12 +
> 16636069300411352429698744320 p^8 r^56 R^12 +
> 6113015546423661981503324160 p^6 r^58 R^12 +
> 1538591317079563369426452480 p^4 r^60 R^12 +
> 237381483574846606238810112 p^2 r^62 R^12 +
> 16940943755866712215388160 r^64 R^12 -
> 135397290214508965396480 p^50 r^13 R^13 -
> 829996907464682838687744 p^48 r^15 R^13 -
> 1967383926730356705722368 p^46 r^17 R^13 -
> 2273149513878827457052672 p^44 r^19 R^13 -
> 1456429249469202936365056 p^42 r^21 R^13 -
> 656701670100366568456192 p^40 r^23 R^13 -
> 89500688445094802489344 p^38 r^25 R^13 +
> 16806100774147714411659264 p^36 r^27 R^13 -
> 1829848003688919623052296192 p^34 r^29 R^13 +
> 25661597968317685464364482560 p^32 r^31 R^13 -
> 100069608047776596425632120832 p^30 r^33 R^13 +
> 125921818779579549303122690048 p^28 r^35 R^13 -
> 20624980352662279300661116928 p^26 r^37 R^13 -
> 9143558972442422236520906752 p^24 r^39 R^13 +
> 7209712365659318361052413952 p^22 r^41 R^13 +
> 29189220217369843012483416064 p^20 r^43 R^13 +
> 67435616927062976174944681984 p^18 r^45 R^13 +
> 119168660172629247111355957248 p^16 r^47 R^13 +
> 163267948098242126556254175232 p^14 r^49 R^13 +
> 171567532069665275824526327808 p^12 r^51 R^13 +
> 136298732879155141956587225088 p^10 r^53 R^13 +
> 80237400465130012852145356800 p^8 r^55 R^13 +
> 33898434535017685126867845120 p^6 r^57 R^13 +
> 9717938366236238701343539200 p^4 r^59 R^13 +
> 1693621471545112046390476800 p^2 r^61 R^13 +
> 135527550046933697723105280 r^63 R^13 +
> 188493135616004111466496 p^48 r^14 R^14 +
> 828454851182595019898880 p^46 r^16 R^14 +
> 1375321234112532988297216 p^44 r^18 R^14 +
> 1199251467128579433693184 p^42 r^20 R^14 +
> 636925999921721247793152 p^40 r^22 R^14 -
> 6341837279477599567872 p^38 r^24 R^14 +
> 484576499580721615077376 p^36 r^26 R^14 -
> 320262499550059760675979264 p^34 r^28 R^14 +
> 9671399830185061886518624256 p^32 r^30 R^14 -
> 60994595766820432815247589376 p^30 r^32 R^14 +
> 107024846534844360451141140480 p^28 r^34 R^14 -
> 24946981962690172899513335808 p^26 r^36 R^14 -
> 13102910231700219143647133696 p^24 r^38 R^14 +
> 5597732352947944566349103104 p^22 r^40 R^14 +
> 38783991708651025683338756096 p^20 r^42 R^14 +
> 113839777262660353534281646080 p^18 r^44 R^14 +
> 248088812351891608722015780864 p^16 r^46 R^14 +
> 413423452637132606758692323328 p^14 r^48 R^14 +
> 521562696446099641372466216960 p^12 r^50 R^14 +
> 490961676993751311744350814208 p^10 r^52 R^14 +
> 338232345596298754993485250560 p^8 r^54 R^14 +
> 165340130308018532027203584000 p^6 r^56 R^14 +
> 54292372578619677005221724160 p^4 r^58 R^14 +
> 10741268233935050919798374400 p^2 r^60 R^14 +
> 968053928906669269450752000 r^62 R^14 -
> 181615303270788841340928 p^46 r^15 R^15 -
> 592698337079744087982080 p^44 r^17 R^15 -
> 787935985559457006354432 p^42 r^19 R^15 -
> 488385023717526158180352 p^40 r^21 R^15 +
> 110197357779507547734016 p^38 r^23 R^15 +
> 281798130842772103495680 p^36 r^25 R^15 -
> 28012303591446025624092672 p^34 r^27 R^15 +
> 2414658158210248150865674240 p^32 r^29 R^15 -
> 27465882569754396913863491584 p^30 r^31 R^15 +
> 71890538704204071100777234432 p^28 r^33 R^15 -
> 24892546454618599383359291392 p^26 r^35 R^15 -
> 14650687724781281346098036736 p^24 r^37 R^15 +
> 1864102878297275568339550208 p^22 r^39 R^15 +
> 42182720115038430058038427648 p^20 r^41 R^15 +
> 160998277625241011547772289024 p^18 r^43 R^15 +
> 436953614103046027636476215296 p^16 r^45 R^15 +
> 893646369718215538744326356992 p^14 r^47 R^15 +
> 1365723774671613669391571353600 p^12 r^49 R^15 +
> 1536387305540864876859105476608 p^10 r^51 R^15 +
> 1248317762448921510045580001280 p^8 r^53 R^15 +
> 710965759735592192257692794880 p^6 r^55 R^15 +
> 269042360472935630545417666560 p^4 r^57 R^15 +
> 60749517798373350285099663360 p^2 r^59 R^15 +
> 6195545145002683324484812800 r^61 R^15 +
> 137342342706655727190016 p^44 r^16 R^16 +
> 360940753722042520436736 p^42 r^18 R^16 +
> 258313845034407356792832 p^40 r^20 R^16 -
> 219617843181640667365376 p^38 r^22 R^16 -
> 293558630424681373499392 p^36 r^24 R^16 -
> 1053754849771710278795264 p^34 r^26 R^16 +
> 362291473417574007732961280 p^32 r^28 R^16 -
> 8778732810790026374413287424 p^30 r^30 R^16 +
> 37289673924650764177349017600 p^28 r^32 R^16 -
> 20111035293255076488702590976 p^26 r^34 R^16 -
> 12911883230811487628750225408 p^24 r^36 R^16 -
> 2325410812773289823356059648 p^22 r^38 R^16 +
> 36924007707350919782766477312 p^20 r^40 R^16 +
> 190254805636540906734532689920 p^18 r^42 R^16 +
> 650384013510062443638992928768 p^16 r^44 R^16 +
> 1647498027824898576508912664576 p^14 r^46 R^16 +
> 3078830938186067834177627095040 p^12 r^48 R^16 +
> 4177277504683269400120690475008 p^10 r^50 R^16 +
> 4036747708859365468186985103360 p^8 r^52 R^16 +
> 2698854325580282978648960532480 p^6 r^54 R^16 +
> 1184783161572612562076471132160 p^4 r^56 R^16 +
> 307109539359768187408471818240 p^2 r^58 R^16 +
> 35624384583765429115787673600 r^60 R^16 -
> 83819060844054198616064 p^42 r^17 R^17 -
> 63370348873176687050752 p^40 r^19 R^17 +
> 271430302321193902931968 p^38 r^21 R^17 +
> 330622946491212914229248 p^36 r^23 R^17 +
> 66163452392208219504640 p^34 r^25 R^17 +
> 26255512436671584617365504 p^32 r^27 R^17 -
> 1864862637996988046508032000 p^30 r^29 R^17 +
> 14437956894043583926052585472 p^28 r^31 R^17 -
> 12819123118821477649712414720 p^26 r^33 R^17 -
> 8957566535623023836467822592 p^24 r^35 R^17 -
> 4982683265523733343873531904 p^22 r^37 R^17 +
> 25056481138116817486618820608 p^20 r^39 R^17 +
> 186836579211584134502468812800 p^18 r^41 R^17 +
> 816375331015864361893463326720 p^16 r^43 R^17 +
> 2585870177505225463071539986432 p^14 r^45 R^17 +
> 5966822812937124540481793949696 p^12 r^47 R^17 +
> 9859315115932066232208612917248 p^10 r^49 R^17 +
> 11435563071747301358191824076800 p^8 r^51 R^17 +
> 9048689530371316579872339394560 p^6 r^53 R^17 +
> 4641593435443702137577915023360 p^4 r^55 R^17 +
> 1389954701046734118713378734080 p^2 r^57 R^17 +
> 184408579021844574246430310400 r^59 R^17 -
> 8952098876789574598656 p^40 r^18 R^18 -
> 215469185073990642696192 p^38 r^20 R^18 -
> 304536861491279716941824 p^36 r^22 R^18 -
> 118517810445598448418816 p^34 r^24 R^18 +
> 370036947071834005700608 p^32 r^26 R^18 -
> 231117049513531861872148480 p^30 r^28 R^18 +
> 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 +
> 701540231741039846755401728 p^28 r^29 R^19 -
> 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)
• ... 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
> 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

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
• 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
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.
• [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
[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.
>
• 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
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.
>
• ... 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
> [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

[Non-text portions of this message have been removed]
Your message has been successfully submitted and would be delivered to recipients shortly.