Let ABC be a triangle with circumcenter O.
Denote:
Ga, Gb, Gc = the centroids of OBC, OCA, OAB, resp..
N' = the NPC center of GaGbGc.
Na, Nb, Nc = the reflections of N' in GbGc, GcGa, GaGb, resp.
Prove: ANa, BNb, CNc concur at a point X.
[Ercole Suppa]
X = X(4)X(2889) ∩ X(5)X(14483) =
= (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-3 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : : (barys)
= (5 S^2+SC^2) (SB+SC-SW) (-4 S^2+SB SC-SB SW+SC^2-SC SW) : : (barys)
= lies on the Jerabek circumhyperbola and these lines: {4,2889}, {5,14483}, {6,3411}, {20,11738}, {49,1176}, {54,549}, {64,3534}, {74,548}, {185,13623}, {265,1216}, {382,14490}, {1173,3628}, {3426,17800}, {3431,15717}, {3519,3917}, {3521,5562}, {3527,5055}, {3856,14487}, {4846,18436}, {5072,11850}, {7486,14491}, {10303,13472}, {10304,11270}, {10627,15108}, {11559,12121}, {13754,14861}, {15644,16620}, {15704,16659}, {15749,18531}, {15750,18532}
= isogonal conjugate of X*
= (6-9-13) search numbers [5.5986005659038030040, 3.2559808617312889554, -1.1974456066313454063]
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X* = isogonal conjugate of X = EULER LINE INTERCEPT OF X(6)X(15580) =
= a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-3 b^2 c^2+c^4) : : (barys)
= SB SC (SB+SC) (-4 S^2-SB SC+SB SW+SC SW-SW^2) : : (barys)
As a point on the Euler X* has Shinagawa coefficients {-4 f, 5 e + 4 f}
= lies on these lines: {2,3}, {6,15580}, {32,33885}, {51,1199}, {54,1495}, {64,11738}, {74,13474}, {93,32085}, {107,13597}, {110,5446}, {143,10540}, {155,15110}, {156,1994}, {184,9781}, {185,12112}, {232,5041}, {323,10263}, {389,14157}, {569,26881}, {578,26882}, {1056,10046}, {1058,10037}, {1112,2914}, {1173,13366}, {1179,6344}, {1204,11455}, {1216,15107}, {1629,11816}, {1829,33179}, {1831,6198}, {1843,5097}, {1968,10986}, {3060,10539}, {3085,9673}, {3086,9658}, {3199,5008}, {3431,17821}, {3527,26864}, {3563,7953}, {3567,6759}, {3817,9626}, {5102,7716}, {5603,8185}, {5890,26883}, {6242,22750}, {6403,11470}, {7592,17810}, {7687,32340}, {7689,11439}, {7713,16200}, {7967,11365}, {8718,9729}, {8884,11815}, {9590,18483}, {9591,10175}, {9609,31404}, {9625,19925}, {9700,31415}, {9707,10982}, {9713,31418}, {9777,14530}, {9798,10595}, {10095,11817}, {10117,15081}, {10282,15033}, {10575,15053}, {10596,26309}, {10597,26308}, {10984,15024}, {11002,12161}, {11270,14490}, {11278,31948}, {11423,15004}, {11424,11464}, {11438,12290}, {11440,16194}, {11451,13336}, {11456,31860}, {11491,20988}, {11550,26917}, {11572,14644}, {12022,15873}, {12241,12254}, {12310,20125}, {12325,31831}, {13339,32205}, {13353,13364}, {13419,25739}, {13451,14627}, {13472,17809}, {13567,16659}, {13568,32111}, {14683,32358}, {14853,20987},{15052,18436}, {15062,32110}, {15513,33880}, {15749,18532}, {16534,25714}, {16655,26879}, {18912,31383}
= isogonal conjugate of X
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7517,12088}, {3,4,13596}, {4,24,3520}, {4,25,3518}, {4,186,14865}, {4,3517,17506}, {4,3518,186}, {4,3542,7577}, {4,6143,15559}, {4,7487,18559}, {4,13619,1885}, {4,14940,427}, {4,21844,1593}, {5,23,7512}, {5,7512,7550}, {5,18378,23}, {20,14002,7506}, {22,7529,3090}, {24,378,15750}, {24,1593,21844}, {24,1598,4}, {24,3520,186}, {24,10594,1598}, {25,1598,24}, {25,5198,3517}, {25,10301,23}, {25,10594,4}, {51,1614,1199}, {186,26863,4}, {235,7576,4}, {235,7715,7576}, {378,5198,4}, {378,15750,23040}, {378,23040,3520}, {382,12106,22467}, {382,22467,7464}, {403,6756,4}, {428,1594,4}, {428,21841,1594}, {468,15559,6143}, {546,2070,14118}, {1495,10110,54}, {1593,21844,3520}, {1596,6240,4}, {1656,17714,6636}, {1658,3843,7527}, {1906,18560,4}, {1995,7387,631}, {3199,10312,8744}, {3199,10985,10312}, {3517,5198,378}, {3517,15750,24}, {3518,3520,24}, {3518,10594,26863}, {3518,26863,14865}, {3520,17506,23040}, {3542,6995,4}, {3567,6759,15032}, {3628,13564,15246}, {3855,7556,7503}, {3861,7575,14130}, {5020,10323,3525}, {5899,18369,140}, {7486,7492,7516}, {7503,9714,7556}, {7506,7530,20}, {7517,13861,2}, {7545,18378,5}, {10263,18350,323}, {11799,31830,34007}, {13564,21308,3628}, {15750,23040,17506}, {17928,18534,3529}
= (6-9-13) search numbers [-1.2041210079187199799, -2.0704644291937978833, 5.6298110903887253207]
Best regards,
Ercole Suppa
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