HYACINTHOS 29637

[Kadir Aktintas]

Let ABC be a triangle.

Denote:

Na, Nb. Nc = the NPC centers of IBC, ICA, IAB, resp.

G' = the centroid of NaNbNc

Ga, Gb, Gc = the reflections of of G' in NbNc, NcNa, NaNb, resp.

 

ABC, GaGbGc are perspective.

 

 

[Ercole Suppa]

 

The perspector of ABC, GaGbGc is the point

 

X = X(8)X(31870) ∩ X(9)X(5886)

 

= (a^5+2 a^4 b-3 a^3 b^2-3 a^2 b^3+2 a b^4+b^5-a^4 c-3 a^3 b c+4 a^2 b^2 c-3 a b^3 c-b^4 c-2 a^3 c^2-3 a^2 b c^2-3 a b^2 c^2-2 b^3 c^2+2 a^2 c^3+3 a b c^3+2 b^2 c^3+a c^4+b c^4-c^5) (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5+2 a^4 c-3 a^3 b c-3 a^2 b^2 c+3 a b^3 c+b^4 c-3 a^3 c^2+4 a^2 b c^2-3 a b^2 c^2+2 b^3 c^2-3 a^2 c^3-3 a b c^3-2 b^2 c^3+2 a c^4-b c^4+c^5) : : (barys)

 

= lies on the Feuerbach circumhyperbola and these lines: {8,31870}, {9,5886}, {21,13464}, {79,12675}, {90,11522}, {946,1156}, {1537,3065}, {4866,30315}, {5551,10806}, {5665,18990}, {5715,33576}, {5882,17097}, {6596,19907}, {7317,10597}, {7319,26332}, {11496,15446}, {11604,12757}

 

= isogonal conjugate of X*

 

= (6-9-13) search numbers [-0.7839722700258233846, -0.8713748285930834830, 4.6057573340222784612]

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X* = ISOGONAL CONJUGATE OF X = X(1)X(3) ∩ X(21)X(5882)

 

= a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c-3 a^3 b c+3 a^2 b^2 c+3 a b^3 c-2 b^4 c-2 a^3 c^2+3 a^2 b c^2-4 a b^2 c^2+3 b^3 c^2+2 a^2 c^3+3 a b c^3+3 b^2 c^3+a c^4-2 b c^4-c^5) : : (barys)

 

= lies on these lines: {1,3}, {21,5882}, {100,10165}, {104,33812}, {140,6174}, {355,5259}, {392,6326}, {405,5881}, {411,13464}, {498,6978}, {515,1621}, {519,1006}, {551,6905}, {581,3915}, {631,8715}, {632,10943}, {944,5248}, {952,5251}, {993,7967}, {1001,5587}, {1012,4428}, {1125,6946}, {1283,30285}, {1479,6982}, {1483,5288}, {1953,5011}, {2267,2323}, {2302,22356}, {2772,14094}, {2800,18444}, {2975,13607}, {3058,6907}, {3090,3825}, {3146,10587}, {3149,9624}, {3523,11240}, {3525,10806}, {3529,10532}, {3584,6882}, {3616,6796}, {3624,11499}, {3628,26470}, {3651,4301}, {3655,6914}, {3679,6883}, {3871,6684}, {3884,21740}, {4187,20400}, {4304,12119}, {4309,6850}, {4853,11517}, {4857,6842}, {4863,26446}, {5047,24987}, {5079,18544}, {5231,5687}, {5250,5693}, {5270,7491}, {5284,10175}, {5315,5396}, {5398,16474}, {5657,25439}, {6419,26458}, {6420,26464}, {6827,10056}, {6830,10197}, {6853,24387}, {6875,8666}, {6891,31452}, {6911,25055}, {6916,10385}, {6954,10072}, {6985,11522}, {6986,11362}, {6992,11239}, {7411,28194}, {7412,23710}, {7420,18613}, {7489,28204}, {7580,31162}, {7701,12680}, {7741,26487}, {7988,18491}, {7989,18518}, {8227,11500}, {9956,25542}, {10303,10527}, {10386,11826}, {10541,12595}, {10597,17538}, {11024,17572}, {11230,18524}, {11231,12331}, {12672,16132}, {13218,15020}, {14217,33593}, {14869,32214}, {15172,15908}, {15254,18908}, {15325,21155}, {15888,31789}, {16842,17619}, {17531,24541}, {17536,31399}, {17857,31435}, {19546,29640}, {21628,21669}

 

= isogonal conjugate of X

 

= reflection of X(7688) in X(15931)

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,10267,10902}, {1,10268,12704}, {1,10902,11012}, {1,16208,5709}, {3,3303,7982}, {3,15178,5563}, {55,3576,2077}, {3303,11510,33925}, {3303,33925,1}, {3428,6767,16200}, {3576,12703,30503}, {5010,30392,10269}, {10246,32613,36}, {10267,16202,1}, {10267,24299,14798}

 

= (6-9-13) search numbers [3.9823977416477836882, 3.5829465061622586991, -0.6778666723500078523]

 

 

Best regards,

Ercole Suppa