HYACINTHOS 28967

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

Denote:

(Na), (Nb), (Nc) = the NPCs of IBC, ICA, IAB, resp.

Ra = the radical axis of (Nb), (Nc)
Rb = the radical axis of (Nc), (Na)
Rc = the radical axis of (Na), (Nb)
(concurrent at X(11))

La, Lb, Lc = the Euler lines of  IBC, ICA, IAB, resp.
(concurrent at X(21))

A* = Ra /\ La
B* = Rb /\ Lb
C* = Rc /\ Lc

ABC, A*B*C* are orthologic.
The orthologic center (ABC, A*B*C*) is X(21)
The other one (A*B*C*, ABC) ?


[Peter Moses]:

Hi Antreas,

X(1125).

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A*B*C* is perspective to the third Euler triangle at

 X(4)X(1001)∩X(5)X(6261)


= a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7+2 a^5 b c-3 a^4 b^2 c-6 a^3 b^3 c+2 a^2 b^4 c+4 a b^5 c+b^6 c+a^5 c^2-3 a^4 b c^2-4 a^3 b^2 c^2-8 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3-6 a^3 b c^3-8 a^2 b^2 c^3-8 a b^3 c^3-3 b^4 c^3-2 a^3 c^4+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5+4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :

 = lies on these lines: {4,1001}, {5,6261} ,{8,12}, {11,11281}, {21,12615}, {142,9948}, {411,6690}, {431,1848}, {442,960}, {497,6871}, {946,3822}, {1125,6841}, {1329,6856}, {1858,5249}, {3816,6828}, {3826,11024}, {3829,15933}, {3847,6873}, {3869,3925}, {3957,5086}, {4197,11415}, {5141,10958}, {5880,8728}, {5887,6881}, {6668,6853}, {6691,6852}, {6824,25524}, {6825,7680}, {6867,18242}, {6870,26105}, {6985,10198}, {6991,28629}, {10590,15843}

= {X(2476),X(3485)}-harmonic conjugate of X(2886)

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A*B*C* orthologic to the extouch triangle at:

 X(1)X(1898)∩X(3)X(1709) 

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

= 5 X[3] - 3 X[5918],3 X[3] - 5 X[25917],3 X[4] + X[3869],3 X[5] - 2 X[3812],X[65] - 3 X[381],3 X[210] - X[12702],3 X[354] - 5 X[18493],X[355] - 3 X[5927],3 X[355] - X[10914],3 X[392] - X[18481],5 X[631] - X[9961],3 X[946] - X[3874],X[1071] - 3 X[5886],9 X[1699] - X[3901],3 X[1699] + X[5693],3 X[1699] - X[24474],X[1770] - 3 X[28452],4 X[3530] - 3 X[10178],X[3555] - 3 X[3656],3 X[3654] - 5 X[3697],3 X[3817] - X[5884],3 X[3845] - 2 X[16616],X[3869] - 3 X[5887],X[3874] + 3 X[31803],5 X[3876] - X[6361],X[3884] + 3 X[31871],5 X[3889] - 9 X[5603],5 X[3889] + 3 X[12528],X[3901] + 3 X[5693],X[3901] - 3 X[24474],3 X[5603] + X[12528],X[5696] + 3 X[11372],3 X[5886] - 2 X[13373],X[5903] - 5 X[18492],X[5904] + 3 X[31162],3 X[5918] + 5 X[12688],9 X[5918] - 25 X[25917],3 X[5919] - X[18526],9 X[5927] - X[10914],3 X[5927] + X[12672],9 X[7988] - 5 X[15016],5 X[8227] - 3 X[10202],5 X[8227] - X[15071],2 X[9940] - 3 X[11230],2 X[9956] - 3 X[10157],3 X[10157] - X[31788],3 X[10176] - X[31730],3 X[10202] - X[15071],3 X[10246] - X[12680],X[10914] + 3 X[12672],3 X[11231] - 2 X[31787],X[11362] - 3 X[15064],3 X[12688] + 5 X[25917],3 X[15049] - X[31728],3 X[15178] - 2 X[26089],3 X[17502] - 2 X[31805],X[22793] + 2 X[31821] 

 
= lies on these lines: {1,1898}, {3,1709}, {4,8}, {5,3812}, {11,113}, {21,4881}, {30,960}, {36,7701}, {40,18491}, {56,90}, {65,381}, {84,10269}, {140,9943}, {210,12702}, {224,1012}, {354,18493}, {382,14110}, {392,6872}, {411,17613}, {500,6051}, {515,3884}, {516,20117}, {518,21850}, {546,7686}, {550,15726}, {631,9961}, {758,18483}, {912,946}, {971,1001}, {1071,5886}, {1125,13369}, {1156,15179}, {1158,6911}, {1319,26321}, {1376,3579}, {1456,18447}, {1482,14872}, {1490,10267}, {1519,26470}, {1539,2778}, {1697,18528}, {1699,3901}, {1770,28452}, {2392,31751}, {2476,7703}, {2646,13743}, {2800,19925}, {2801,13464}, {3057,9668}, {3058,9957}, {3333,30290}, {3359,7995}, {3485,5045}, {3486,17622}, {3487,16216}, {3530,10178}, {3555,3656}, {3556,9818}, {3583,16155}, {3612,28444}, {3652,3916}, {3654,3697}, {3666,5492}, {3678,28194}, {3753,6871}, {3817,5884}, {3818,3827}, {3830,31165}, {3845,16616}, {3876,6361}, {3878,31673}, {3889,5603}, {3925,6842}, {4295,6849}, {4297,31838}, {4511,21669}, {4870,17637}, {5119,18518}, {5252,10043}, {5259,13151}, {5439,10584}, {5450,18857}, {5453,15569}, {5696,11372}, {5698,6869}, {5719,12710}, {5720,11248}, {5722,12709}, {5779,22770}, {5780,6244}, {5806,10893}, {5836,18357}, {5881,23340}, {5885,6828}, {5901,12675}, {5903,18492}, {5904,31162}, {5919,18526}, {6000,9895}, {6583,10883}, {6705,6713}, {6767,9848}, {6824,9940}, {6825,11231}, {6838,26446}, {6857,17612}, {6866,31794}, {6870,10598}, {6873,10129}, {6883,12520}, {6900,20292}, {6912,15178}, {6913,12664}, {6923,12679}, {6944,14647}, {7171,8583}, {7330,11249}, {7373,8581}, {7491,28160}, {7548,17654}, {7728,10693}, {7741,18838}, {7743,10948}, {7988,15016}, {7991,18529}, {8227,10202}, {8543,17620}, {10057,10742}, {10085,16203}, {10176,31730}, {10246,12680}, {10394,15008}, {10532,10941}, {10679,17857}, {10827,18542}, {10912,11278}, {10915,27870}, {10949,30384}, {11114,28208}, {11362,15064}, {11374,12711}, {11699,12889}, {11826,28146}, {12446,16004}, {12571,31870}, {12737,17661}, {12773,20323}, {13374,24475}, {13600,26200}, {13750,17605}, {15049,31728}, {15842,21616}, {17502,31805}, {18243,25466}, {22753,24467}, {22760,24928}, {28174,31835}

= midpoint of X(i) and X(j) for these {i,j}: {3, 12688}, {4, 5887}, {72, 12699}, {355, 12672}, {382, 14110}, {946, 31803}, {1385, 31828}, {1482, 14872}, {3057, 18525}, {3830, 31165}, {3878, 31673}, {5693, 24474}, {5694, 22793}, {5777, 9856}, {5881, 23340}, {7728, 10693}, {10742, 17638}, {12737, 17661}, {16138, 17653}

= reflection of X(i) and X(j) for these {i,j}: {942, 9955}, {1071, 13373}, {3579, 5044}, {4297, 31838}, {5694, 31821}, {5836, 18357}, {7686, 546}, {9943, 140}, {12675, 5901}, {13369, 1125}, {13600, 26200}, {24475, 13374}, {31788, 9956}, {31837, 20117}, {31870, 12571}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 18540, 18761}, {4, 11415, 12699}, {355, 12699, 3434}, {355, 18516, 18480}, {1071, 5886, 13373}, {1699, 5693, 24474}, {1858, 12047, 942}, {3560, 6261, 1385}, {5259, 16132, 13151}, {5720, 12705, 11248}, {5927, 12672, 355}, {6841, 12047, 9955}, {6985, 12514, 3579}, {8227, 15071, 10202}, {10157, 31788, 9956}, {11373, 17625, 5045}, {12608, 12617, 5}, {17614, 17653, 17616}, {18480, 22793, 18407}

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extouch triangle orthologic to A*B*C* at

MIDPOINT OF X(79) AND X(5904) 

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

= 3 X[21] - 5 X[3876],3 X[210] - X[17637],3 X[210] - 2 X[18253],3 X[392] - 2 X[15174],3 X[442] - 2 X[942],3 X[3681] - X[11684],X[3868] - 3 X[6175],4 X[4662] - 3 X[21677],4 X[5044] - 3 X[15670],X[5441] - 3 X[5692],6 X[11281] - 5 X[17609]

= lies on the Mandart hyperbola and these lines: {2,10122}, {8,79}, {9,21}, {10,3580}, {30,72}, {35,3219}, {40,3681}, {63,3651}, {69,23581}, {144,3648}, {191,200}, {210,17637}, {319,340}, {392,15174}, {442,942}, {445,1844}, {500,16585}, {518,3649}, {527,10123}, {908,6841}, {960,10543}, {1071,5771}, {1145,2771}, {3059,17768}, {3146,31803}, {3555,16137}, {3652,11248}, {3868,4654}, {3872,8000}, {3874,6701}, {3927,16117}, {3940,13743}, {3984,21669}, {4511,5259}, {4662,6735}, {4847,11263}, {4853,16126}, {4855,21161}, {5044,15670}, {5086,18406}, {5178,16125}, {5223,16143}, {5428,5440}, {5441,5692}, {5687,16139}, {5693,6925}, {6675,27385}, {6912,20117}, {10176,16865}, {10527,26725}, {11281,17609}, {12691,31837}, {14740,15227}, {15674,27383}, {15680,20007}, {17653,17658}, {18259,31660}, {20084,20214}, {31649,31835}

= anticomplement of X(10122)
= midpoint of X(79) and X(5904)
= reflection of X(i) and X(j) for these {i,j}: {3555, 16137}, {3647, 3678}, {3874, 6701}, {10543, 960}, {14450, 16120}, {17637, 18253}, {31649, 31835}
= X(8)-Ceva conjugate of X(6734)
= X(2160)-isoconjugate of X(2982)
= crosspoint of X(8) and X(4420)
= barycentric product X(i) X(j) for these {i,j}: {8, 16585}, {78, 445}, {312, 500}, {345, 1844}, {3219, 6734}, {4420, 5249}
= barycentric quotient X(i) / X(j) for these {i,j}: {35, 2982}, {445, 273}, {500, 57}, {1844, 278}, {6734, 30690}, {14547, 2160}, {16585, 7}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {210, 17637, 18253}, {2475, 5905, 79}, {3681, 12528, 3951}

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Best regards,
Peter Moses.