HYACINTHOS 28568

[Antreas P. Hatzipolakis]:

 

 

Let ABC be a triangle and P a point.

Denote:

Na, Nb, Nc = the NPC centers of PBC, PCA, PAB, resp.

The reflections of NbNc, NcNa, NaNb in AP, BP, CP, resp. bound a triangle A*B*C*.

For P = I:

ABC, A*B*C* are orthologic. 

Orthologic centers ?

 

Locus?

 

 

[Peter Moses]:

Hi Antreas,

(ABC,  A*B*C*): X(36).
(A*B*C*,  ABC) : X(1385).

Locus: Q003 among other nasties.

P = X(2):

(ABC, A*B*C*) =

= MIDPOINT OF X(187) AND  X(5215) 

 

= 7 a^4-4 a^2 b^2+b^4-4 a^2 c^2-b^2 c^2+c^4 : : 

 

= X[2] + 2 X[187], 4 X[2] - X[316], 8 X[187] + X[316], 7 X[316] - 16 X[625], 7 X[2] - 4 X[625], 7 X[187] + 2 X[625], 4 X[230] - X[671], X[1992] - 4 X[2030], 2 X[549] + X[2080], X[385] + 2 X[2482], 2 X[597] + X[5104], 2 X[551] + X[5184], X[316] - 8 X[5215], 2 X[625] - 7 X[5215], 2 X[5461] + X[6781], X[691] + 2 X[7426], 4 X[2021] - X[7757], 4 X[620] - X[7840], 2 X[395] + X[8594], 2 X[396] + X[8595], 4 X[5461] - X[8597], 2 X[6781] + X[8597], 2 X[230] + X[8598], X[671] + 2 X[8598], X[843] + 2 X[9127], 2 X[115] + X[9855], 2 X[8997] + X[9893], 5 X[316] - 16 X[10150], 5 X[625] - 7 X[10150], 5 X[2] - 4 X[10150], 5 X[5215] - 2 X[10150], 5 X[187] + 2 X[10150], 2 X[99] + X[11054], 2 X[3111] + X[11673], 2 X[6055] + X[11676], 5 X[5071] - 2 X[13449], X[13677] + 2 X[13908], X[9891] + 2 X[13989], 2 X[8352] - 5 X[14061], 5 X[2482] - 2 X[14148], 5 X[385] + 4 X[14148], 2 X[13586] + X[14568], X[381] - 4 X[14693], 10 X[187] - X[14712], 5 X[2] + X[14712], 10 X[5215] + X[14712], 4 X[10150] + X[14712], 5 X[316] + 4 X[14712], 2 X[9181] + X[15360], X[9301] + 5 X[15693], X[8593] + 2 X[15993], X[842] - 4 X[18579], 5 X[15692] - 2 X[18860], 5 X[7925] - 8 X[22247], X[11054] - 4 X[22329], X[99] + 2 X[22329], 2 X[39] + X[22564].

 

= lies on these lines: {2,187}, {3,7827}, {30,9166}, {32,7622}, {39,22564}, {99,9136}, {115,9855}, {230,671}, {249,524}, {381,14693}, {385,2482}, {395,8594}, {396,8595}, {511,3524}, {512,15724}, {530,16267}, {531,16268}, {542,21445}, {543,5152}, {549,2080}, {551,5184}, {597,5104}, {599,7835}, {620,7840}, {691,7426}, {754,9167}, {842,18579}, {843,9127}, {1003,7610}, {1078,8369}, {1384,11163}, {1692,5032}, {1992,2030}, {2021,7618}, {3053,7769}, {3096,8366}, {3111,11673}, {3523,7878}, {3788,9939}, {5023,7841}, {5071,13449}, {5077,15655}, {5206,7828}, {5210,7790}, {5461,6781}, {6055,11676}, {7617,11361}, {7619,7753}, {7768,7870}, {7775,7907}, {7793,7801}, {7802,11318}, {7806,8588}, {7807,7883}, {7810,7832}, {7811,11288}, {7817,7847}, {7859,8359}, {7925,22247}, {7944,8365}, {8352,14061}, {8370,15597}, {8553,21395}, {8593,15993}, {8860,11159}, {8997,9893}, {9181,15360}, {9301,15693}, {9741,11055}, {9761,19781}, {9763,19780}, {9891,13989}, {11165,14614}, {11185,23055}, {13677,13908}, {14041,14971}, {15692,18860}.
= midpoint of X(i) and X(j) for these {i,j}: {187, 5215}, {8859, 13586}.
= reflection of X(i) in X(j) for these {i,j}: {2, 5215}, {5032, 1692}, {14041, 14971}, {14568, 8859}.
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 22329, 11054), (230, 8598, 671), (5461, 6781, 8597).
= X(661)-isoconjugate of X(9124).
= crossdifference of every pair of points on line {17414, 22260}.
= barycentric product X(99)X(9123).
= barycentric quotient X(i)/X(j) for these {i,j}: {110, 9124}, {9123, 523}.


(A*B*C*, ABC) :

 

= MIDPOINT OF X(3) AND X(9166)

 

= 10 a^8-21 a^6 b^2+25 a^4 b^4-18 a^2 b^6+4 b^8-21 a^6 c^2+4 a^4 b^2 c^2+7 a^2 b^4 c^2-15 b^6 c^2+25 a^4 c^4+7 a^2 b^2 c^4+22 b^4 c^4-18 a^2 c^6-15 b^2 c^6+4 c^8 : : 

 

= 7 X[2] - X[6033], X[549] + 2 X[6036], 7 X[3526] - X[6054], 2 X[140] + X[6055], X[3845] - 4 X[6722], 2 X[5461] + X[8703], 11 X[2] + X[9862], 11 X[6033] + 7 X[9862], X[114] - 4 X[10124], 11 X[3525] + X[11177], 5 X[631] + X[11632], X[2482] - 4 X[11812], X[11161] + 5 X[12017], 2 X[9862] - 11 X[12042], 2 X[2] + X[12042], 2 X[6033] + 7 X[12042], X[115] + 2 X[12100], X[3534] + 5 X[14061], X[10723] + 5 X[14093], 5 X[9862] - 11 X[14830], 5 X[12042] - 2 X[14830], 5 X[2] + X[14830], 5 X[6033] + 7 X[14830], 2 X[11623] + 7 X[14869], 10 X[140] - X[14981], 5 X[9167] - X[14981], 5 X[6055] + X[14981], X[3830] - 4 X[15092], X[6321] + 5 X[15692], X[671] + 5 X[15693], X[98] + 5 X[15694], X[12117] - 7 X[15700], X[99] - 7 X[15701], X[8724] - 7 X[15702], X[14651] + 3 X[15708], X[15561] - 3 X[15709], 2 X[620] - 5 X[15713], X[12355] + 11 X[15718], X[148] + 11 X[15719], X[12243] + 11 X[15721], X[10991] + 8 X[16239], 5 X[15712] + 4 X[20398], 3 X[15707] - X[21166], 4 X[547] - X[22505], 4 X[5461] - X[22515], 2 X[8703] + X[22515], 4 X[6033] - 7 X[22566], 4 X[2] - X[22566], 2 X[12042] + X[22566], 4 X[14830] + 5 X[22566], 4 X[9862] + 11 X[22566], 5 X[14971] - 3 X[23514].

 

= lies on these lines: {2,5191}, {3,9166}, {30,5215}, {98,15694}, {99,15701}, {114,10124}, {115,12100}, {140,6055}, {148,15719}, {542,11539}, {543,549}, {547,22505}, {620,15713}, {631,11632}, {671,15693}, {2482,11812}, {2782,5054}, {2794,15699}, {3525,11177}, {3526,6054}, {3534,14061}, {3830,15092}, {3845,6722}, {5461,8703}, {6321,15692}, {7610,13085}, {8724,15702}, {10723,14093}, {10991,16239}, {11161,12017}, {11623,14869}, {12117,15700}, {12243,15721}, {12355,15718}, {14639,15688}, {14651,15708}, {15561,15709},

{15707,21166}, {15712,20398}, {17504,23698}.
= midpoint of X(i) and X(j) for these {i,j}: {3, 9166}, {6055, 9167}, {14639, 15688}
= reflection of X(9167) in X(140).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12042, 22566), (5461, 8703, 22515).


Best regards,
Peter Moses.