HYACINTHOS 29235

[Antreas P. Hatzipolakis]:

Let ABC be a triangle.

Denote:

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

A' = AI /\ NbNc, B' = BI /\ NcNa, C' = CI /\ NaNb

A', B', C' are collinear (Hyacinthos 28099 )

La, Lb, Lc =: the reflections of the line A'B'C' in AI, BI, CI, resp.

A*B*C* = the triangle bounded by La, Lb, Lc,

Then
ABC, A*B*C* are parallelogic.

Parallelogic centers?


[Peter Moses]:


Hi Antreas,

ABC / A*B*C* parallelogic at X(109).

A*B*C* / ABC and A*B*C* / anticomplementary triangle parallelogic at:

MIDPOINT X(109) AND X(1807)  =

= a*(2*a^6 - a^5*b - 3*a^4*b^2 + 2*a^3*b^3 - a*b^5 + b^6 - a^5*c + 2*a^4*b*c + a^3*b^2*c - 2*a^2*b^3*c - 3*a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 - 2*a^2*b*c^3 + a*b^2*c^3 - b^2*c^4 - a*c^5 + c^6) : :

= lies on these lines: {1, 3}, {37, 1983}, {109, 1807}, {201, 22937}, {651, 12738}, {952, 11700}, {1104, 6788}, {1411, 6797}, {1455, 28204}, {1777, 31828}, {5081, 27529}, {5840, 15252}, {6198, 7012}, {6796, 32047}, {18340, 28160}, {24433, 31445}

= midpoint of X(109) and X(1807)
= crossdifference of every pair of points on line {650, 7297}


Anticomplementary triangle / A*B*C* parallelogic at:

 

X(2)X(109)∩X(4)X(151) =

 

= a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c + 3*a^4*b*c - 3*a*b^4*c + b^5*c - a^4*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 + b^4*c^2 + 2*a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - 3*a*b*c^4 + b^2*c^4 + a*c^5 + b*c^5 - c^6 : :
= 3 X[2] - 4 X[124],9 X[2] - 8 X[6718],3 X[4] - 2 X[10740],3 X[109] - 4 X[6718],X[109] - 3 X[10716],4 X[117] - 5 X[3091],3 X[124] - 2 X[6718],2 X[124] - 3 X[10716],3 X[151] - 4 X[10740],X[151] - 4 X[10747],7 X[3523] - 8 X[6711],3 X[3543] - 2 X[10726],5 X[3616] - 4 X[11700],7 X[3622] - 8 X[11734],3 X[3839] - 2 X[10709],3 X[5731] - 4 X[11713],4 X[6718] - 9 X[10716],X[10740] - 3 X[10747],8 X[29008] - 9 X[32558]  

= lies on the circumcircle of the anticomplementary triangle and these lines: {2, 109}, {4, 151}, {8, 153}, {20, 102}, {63, 147}, {92, 1836}, {117, 3091}, {144, 31091}, {145, 10703}, {146, 2779}, {148, 2785}, {149, 3738}, {150, 928}, {152, 2807}, {189, 9812}, {193, 10764}, {221, 17555}, {329, 2835}, {388, 1361}, {497, 1364}, {651, 23541}, {908, 4645}, {914, 3685}, {938, 12016}, {962, 2817}, {1330, 3869}, {1331, 27542}, {1456, 17923}, {1795, 3086}, {1845, 4295}, {2550, 3042}, {2551, 3040}, {2773, 3448}, {2841, 3436}, {2852, 14360}, {2853, 13219}, {3146, 10732}, {3523, 6711}, {3543, 10726}, {3616, 11700}, {3622, 11734}, {3839, 10709}, {4468, 14732}, {5081, 6001}, {5174, 12688}, {5698, 6350}, {5731, 11713}, {8048, 10446}, {8677, 21293}, {9364, 26010}, {9532, 12384}, {11109, 20306}, {11681, 15050}, {12699, 20220}, {14594, 26611}, {17615, 32850}, {18662, 33100}, {20557, 20562}, {29008, 32558}

= anticomplement of X(109).
= reflection of X(i) in X(j) for these {i,j}: {2, 10716}, {4, 10747}, {8, 13532}, {20, 102}, {109, 124}, {145, 10703}, {149, 10777}, {151, 4}, {193, 10764}, {3146, 10732}
= orthoptic circle of the Steiner circumellipse inverse of X(1311)
= anticomplement of the isogonal of X(522)
= X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 522}, {2, 693}, {4, 521}, {6, 17496}, {7, 3900}, {8, 513}, {9, 514}, {11, 149}, {21, 523}, {29, 7253}, {33, 25259}, {41, 21225}, {55, 17494}, {57, 4025}, {75, 21302}, {78, 20294}, {80, 3738}, {81, 4467}, {84, 8058}, {87, 4449}, {88, 4453}, {101, 4552}, {104, 2804}, {108, 1897}, {162, 14544}, {189, 4131}, {190, 21272}, {200, 4468}, {210, 31290}, {256, 3907}, {261, 17166}, {274, 4374}, {277, 24002}, {278, 17896}, {280, 4397}, {281, 4391}, {282, 6332}, {284, 4560}, {291, 2254}, {294, 918}, {312, 20295}, {314, 512}, {318, 20293}, {329, 20297}, {333, 7192}, {346, 4462}, {513, 145}, {514, 7}, {521, 20}, {522, 8}, {523, 2475}, {525, 2897}, {643, 4427}, {644, 190}, {646, 668}, {647, 18667}, {649, 3210}, {650, 2}, {651, 664}, {652, 6360}, {653, 4566}, {656, 3152}, {657, 3177}, {661, 17778}, {662, 17136}, {663, 192}, {666, 883}, {668, 3888}, {693, 3434}, {885, 518}, {905, 347}, {941, 23880}, {983, 3810}, {1018, 3882}, {1019, 3875}, {1021, 63}, {1022, 1266}, {1024, 239}, {1039, 23874}, {1156, 6366}, {1169, 15420}, {1172, 525}, {1320, 900}, {1392, 4926}, {1577, 2893}, {1635, 30577}, {1639, 30578}, {1783, 651}, {1812, 6563}, {1896, 520}, {1929, 4458}, {1946, 3164}, {2170, 4440}, {2185, 17161}, {2194, 31296}, {2298, 3910}, {2299, 17498}, {2316, 21222}, {2319, 649}, {2320, 4777}, {2335, 23882}, {2344, 824}, {2346, 6362}, {2481, 926}, {2648, 2785}, {2766, 4242}, {2804, 153}, {2997, 8676}, {3063, 194}, {3064, 5905}, {3223, 23655}, {3239, 329}, {3254, 3887}, {3261, 21285}, {3271, 9263}, {3307, 3308}, {3308, 3307}, {3309, 7674}, {3596, 21301}, {3669, 4452}, {3680, 3667}, {3683, 14779}, {3699, 3952}, {3700, 2895}, {3709, 1655}, {3716, 17794}, {3737, 1}, {3738, 6224}, {3835, 20537}, {3900, 144}, {3952, 3909}, {4041, 1654}, {4086, 1330}, {4130, 30695}, {4391, 69}, {4397, 3436}, {4516, 148}, {4521, 8055}, {4560, 75}, {4581, 65}, {4594, 3903}, {4608, 20292}, {4612, 99}, {4631, 4576}, {4636, 6758}, {4768, 21290}, {4814, 17488}, {4858, 150}, {4866, 4778}, {4876, 812}, {4895, 17487}, {4900, 6006}, {4985, 2891}, {4997, 21297}, {5548, 2397}, {6332, 4329}, {6557, 4106}, {6591, 30699}, {6598, 6003}, {6601, 3309}, {6728, 7057}, {6729, 16018}, {6730, 16017}, {7105, 656}, {7110, 1577}, {7155, 4083}, {7192, 3873}, {7199, 20244}, {7220, 4724}, {7252, 17147}, {7253, 3869}, {7281, 4088}, {7649, 12649}, {7658, 31527}, {8047, 30613}, {8056, 3676}, {8058, 6223}, {8611, 3151}, {8641, 21218}, {8702, 12849}, {8851, 659}, {9282, 21105}, {9368, 6615}, {9375, 6129}, {9445, 6608}, {9503, 2400}, {10266, 8702}, {10309, 30201}, {10492, 4146}, {10495, 188}, {10566, 20247}, {11604, 8674}, {11609, 2787}, {12641, 2827}, {13426, 6365}, {13454, 6364}, {13486, 6742}, {14224, 2771}, {14298, 20211}, {14304, 151}, {14837, 5932}, {16612, 18632}, {17096, 17158}, {17420, 5484}, {17926, 92}, {18101, 25048}, {18108, 3891}, {18155, 17135}, {18191, 17154}, {18344, 193}, {19605, 3239}, {20906, 20350}, {21044, 21221}, {23189, 20222}, {23696, 3100}, {23836, 3880}, {23838, 519}, {23893, 527}, {24002, 6604}, {27527, 17149}, {28659, 21304}, {28660, 17217}, {30479, 8678}, {30513, 9001}, {30571, 4804}, {30610, 4554}, {31343, 3699}, {31623, 850}, {31628, 4998}, {32635, 4977}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {109, 124, 2}, {109, 10716, 124}

Best regards,
Peter Moses.