HYACINTHOS 29455

[Antreas P. Hatzipolakis]:

 

VARIATIONS of Hyacinthos 29402 

Let ABC be a triangle and P a point on the (O) [ = circumcircle]

Denote:

Ga, Gb, Gc = the G's (centroids) of PBC, PCA, PAB, resp.

The (O) [ = circumcircle] of GaGbGc passes through G of ABC.

 

 

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Two related centers:

 

 (name pending) =

 

= 2 a^10 - 9 a^8 (b^2 + c^2) + 2 a^6 (5 b^4 + 9 b^2 c^2 + 5 c^4) + a^4 (4 b^6 - 9 b^4 c^2 - 9 b^2 c^4 + 4 c^6) - 3 a^2 (b^2 - c^2)^2 (4 b^4 + 3 b^2 c^2 + 4 c^4) + 5 (b^2 - c^2)^4 (b^2 + c^2) : :

= lies on this line: {2,3}

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EULER LINE INTERCEPT OF X(36)X(15170) =

 = 14 a^4- 13 a^2 (b^2 + c^2) - (b^2 - c^2)^2  : :

= lies on these lines: {2,3}, {36,15170}, {143,17704}, {146,15042}, {165,3655}, {511,20583}, {519,8688}, {524,14810}, {541,11694}, {542,3631}, {551,17502}, {597,17508}, {962,32533}, {1125,28202}, {1992,33750}, {2549,5585}, {3058,7280}, {3098,3629}, {3163,22052}, {3241,4935}, {3244,3579}, {3576,28212}, {3582,15338}, {3584,15326}, {3626,28204}, {3632,3654}, {3636,13624}, {3653,22791}, {3656,7987}, {3679,28224}, {3828,28160}, {3982,5719}, {4031,24929}, {4995,18990}, {5010,5434}, {5023,7739}, {5092,6329}, {5204,15172}, {5206,5306}, {5210,15048}, {5298,15171}, {5305,15513}, {5309,8588}, {5447,31834}, {5493,31666}, {5655,13392}, {5657,17063}, {5690,16192}, {5892,13451}, {5901,12512}, {5907,11592}, {6154,7688}, {6390,7811}, {6449,19054}, {6450,19053}, {6452,9541}, {6455,19117}, {6456,19116}, {6459,6497}, {6460,6496}, {6500,9543}, {6684,28208}, {7753,8589}, {7880,32459}, {8182,8716}, {9143,15041}, {9167,22505}, {9729,14449}, {9778,10283}, {9880,26614}, {10164,28186}, {10165,28178}, {10168,29181}, {10178,14988}, {10263,16226}, {10990,11693}, {11178,21167}, {11179,31884}, {11230,28182}, {11231,28190}, {11645,33751}, {12006,21849}, {12041,24981}, {12244,22251}, {12702,20057}, {13340,20791}, {13348,13630}, {13353,13482}, {13391,16836}, {13421,15012}, {14128,14641}, {14830,21166}, {14855,15067}, {15040,22250}, {15808,31730}, {16881,21969}, {18583,19924}, {19883,22793}, {20582,29012}, {28216,31162}, {28228,31662}, {31805,31835}, {32006,32887}, {32448,33706}, {32450,32516}  

= midpoint of X(i) and X(j) for these {i,j}: {2, 550}, {3, 8703}, {4, 19710}, {5, 3534}, {20, 3845}, {140, 15690}, {376, 549}, {381, 15686}, {547, 15691}, {548, 12100}, {632, 15697}, {1657, 33699}, {3522, 15711}, {3594, 17827}, {3627, 11001}, {3830, 15704}, {5066, 12103}, {5655, 14677}, {9778, 10283}, {11539, 15689}, {14093, 15714}, {14855, 15067}, {15681, 15687}, {15688, 17504}, {15695, 15712}, {15696, 15713}, {15759, 33923}, {16190, 29318}, {23745, 550}, {32448, 33706}

= reflection of X(i) in X(j) for these {i,j}: {2, 3530}, {3, 15759}, {4, 10109}, {5, 11812}, {140, 12100}, {381, 10124}, {546, 2}, {547, 549}, {548, 8703}, {549, 14891}, {3545, 14890}, {3627, 3860}, {3830, 3850}, {3845, 3628}, {3850, 11540}, {3853, 5066}, {3860, 16239}, {5066, 140}, {5655, 13392}, {8703, 33923}, {10109, 12108}, {12100, 3}, {12101, 5}, {12103, 15690}, {13451, 5892}, {14892, 5054}, {14893, 547}, {15682, 12102}, {15687, 11737}, {15690, 548}, {15691, 376}, {21849, 12006}, {21969, 16881}, {25338, 18579}, {33591, 18324}, {33699, 3861}

Angel Montesdeoca: Centro de cónica sobre la recta de Euler