[Tran Quang Hung]:
Let ABC be a triangle.
L is its de Longchamps point.
H is its orthocenter.
La, Lb, Lc are de Longchamps points of LBC, LCA, LAB resp.
Then ALa, BLb, CLc are concurrent at H. This means ABC and LaLbLc are orthologic, which is other orthologic center?
Let Lha, Lhb, Lhc be de Longchamps points of HBC, HCA, HAB resp.
Let La', Lb', Lc' be de Longchamps points of HLbLc, HLcLa, HLaLb resp.
Then La'Lha, Lb'Lhb, Lc'Lhc are concurrent at L. This means La'Lb'Lc' and LhaLhbLhc are orthologic, which is other orthologic center?
Also ABC and LhaLhbLhc are orthologic, which are the orthologic centers?
L is orthocenter of the triangle LhaLhbLhc. Let Lha', Lhb', Lhc' be de Longchamps points of LLhbLhc, LLhcLha, LLhaLhb, resp.
Then ABC and Lha', Lhb', Lhc' are also orthologic. Which are the orthologic centers?
[Ercole Suppa]:
Hi Tran Quang Hung,
We have:
U1 = orthology center(LaLbLc, ABC) = H (orthocenter of ABC)
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U2 = orthology center(ABC, LaLbLc)
= X(3)X(31361) ∩ X(4)X(8567)
= X(3)X(31361) ∩ X(4)X(8567)
= (a^2+b^2-c^2) (a^2-b^2+c^2) (5 a^4-10 a^2 b^2+5 b^4+2 a^2 c^2+2 b^2 c^2-7 c^4) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4) (5 a^4+2 a^2 b^2-7 b^4-10 a^2 c^2+2 b^2 c^2+5 c^4) : : (barys)
= (384 R^4+16 R^2 SB+16 R^2 SC-SB SC-192 R^2 SW-4 SB SW-4 SC SW+24 SW^2)S^2 -1152 R^4 SB SC+528 R^2 SB SC SW-60 SB SC SW^2 : : (barys)
= 3*X[376]-2*X[16251]
= lies on these lines: {3,31361}, {4,8567}, {30,253}, {376,16251}, {1294,11001}, {3079,10152}, {3346,3529}, {5925,22049}, {11541,15319}
= barycentric quotient X(1249)/X(3543)
= trilinear quotient X(1895)/X(3543)
= (6-9-13) search numbers: [-94.9800273850920979472, -98.2037062581429574990, 115.4647814537412410396]
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V1 = orthology center(La'Lb'Lc', LhaLhbLhc) = L (de Longchamps point of ABC)
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V2 = orthology center(LhaLhbLhc,La'Lb'Lc')
= 507 a^28-2158 a^26 b^2-275 a^24 b^4+16900 a^22 b^6-34517 a^20 b^8+11294 a^18 b^10+54069 a^16 b^12-93192 a^14 b^14+65193 a^12 b^16-14418 a^10 b^18-8801 a^8 b^20+7364 a^6 b^22-2383 a^4 b^24+482 a^2 b^26-65 b^28-2158 a^26 c^2+15478 a^24 b^2 c^2-29924 a^22 b^4 c^2-23932 a^20 b^6 c^2+165542 a^18 b^8 c^2-225598 a^16 b^10 c^2+81640 a^14 b^12 c^2+91800 a^12 b^14 c^2-115522 a^10 b^16 c^2+58778 a^8 b^18 c^2-24004 a^6 b^20 c^2+9892 a^4 b^22 c^2-1846 a^2 b^24 c^2-146 b^26 c^2-275 a^24 c^4-29924 a^22 b^2 c^4+139938 a^20 b^4 c^4-180932 a^18 b^6 c^4-113125 a^16 b^8 c^4+498552 a^14 b^10 c^4-444356 a^12 b^12 c^4+107288 a^10 b^14 c^4+19827 a^8 b^16 c^4+24300 a^6 b^18 c^4-21182 a^4 b^20 c^4-3540 a^2 b^22 c^4+3429 b^24 c^4+16900 a^22 c^6-23932 a^20 b^2 c^6-180932 a^18 b^4 c^6+569308 a^16 b^6 c^6-487000 a^14 b^8 c^6-198360 a^12 b^10 c^6+515448 a^10 b^12 c^6-162248 a^8 b^14 c^6-92908 a^6 b^16 c^6+20500 a^4 b^18 c^6+35980 a^2 b^20 c^6-12756 b^22 c^6-34517 a^20 c^8+165542 a^18 b^2 c^8-113125 a^16 b^4 c^8-487000 a^14 b^6 c^8+971446 a^12 b^8 c^8-492796 a^10 b^10 c^8-249874 a^8 b^12 c^8+262504 a^6 b^14 c^8+56255 a^4 b^16 c^8-95930 a^2 b^18 c^8+17495 b^20 c^8+11294 a^18 c^10-225598 a^16 b^2 c^10+498552 a^14 b^4 c^10-198360 a^12 b^6 c^10-492796 a^10 b^8 c^10+684636 a^8 b^10 c^10-177256 a^6 b^12 c^10-232120 a^4 b^14 c^10+124878 a^2 b^16 c^10+6770 b^18 c^10+54069 a^16 c^12+81640 a^14 b^2 c^12-444356 a^12 b^4 c^12+515448 a^10 b^6 c^12-249874 a^8 b^8 c^12-177256 a^6 b^10 c^12+338076 a^4 b^12 c^12-60024 a^2 b^14 c^12-57723 b^16 c^12-93192 a^14 c^14+91800 a^12 b^2 c^14+107288 a^10 b^4 c^14-162248 a^8 b^6 c^14+262504 a^6 b^8 c^14-232120 a^4 b^10 c^14-60024 a^2 b^12 c^14+85992 b^14 c^14+65193 a^12 c^16-115522 a^10 b^2 c^16+19827 a^8 b^4 c^16-92908 a^6 b^6 c^16+56255 a^4 b^8 c^16+124878 a^2 b^10 c^16-57723 b^12 c^16-14418 a^10 c^18+58778 a^8 b^2 c^18+24300 a^6 b^4 c^18+20500 a^4 b^6 c^18-95930 a^2 b^8 c^18+6770 b^10 c^18-8801 a^8 c^20-24004 a^6 b^2 c^20-21182 a^4 b^4 c^20+35980 a^2 b^6 c^20+17495 b^8 c^20+7364 a^6 c^22+9892 a^4 b^2 c^22-3540 a^2 b^4 c^22-12756 b^6 c^22-2383 a^4 c^24-1846 a^2 b^2 c^24+3429 b^4 c^24+482 a^2 c^26-146 b^2 c^26-65 c^28 : : (barys)
= 9 S^6 + (2816 R^4-144 R^2 SB-144 R^2 SC-18 SB SC-1120 R^2 SW+36 SB SW+36 SC SW+104 SW^2)S^4 + (147456 R^8-30720 R^6 SB-30720 R^6 SC-5248 R^4 SB SC-102400 R^6 SW+20736 R^4 SB SW+20736 R^4 SC SW+2336 R^2 SB SC SW+23808 R^4 SW^2-4608 R^2 SB SW^2-4608 R^2 SC SW^2-256 SB SC SW^2-1920 R^2 SW^3+336 SB SW^3+336 SC SW^3+16 SW^4)S^2 -147456 R^8 SB SC+124928 R^6 SB SC SW-38400 R^4 SB SC SW^2+4992 R^2 SB SC SW^3-224 SB SC SW^4: : (barys)
= (6-9-13) search numbers: [7.6297601206671894399, 5.5540768141183889343, -3.7258933681747468237]
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W1 = orthology center(ABC, Lha'Lhb'Lhc') = H (orthocenter of ABC)
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W2 = orthology center(Lha'Lhb'Lhc',ABC) = ANTICOMPLEMENT OF X(1657)
= 9 a^4-4 a^2 b^2-5 b^4-4 a^2 c^2+10 b^2 c^2-5 c^4 : : (barys)
= 2 S^2 - 7 SB SC : : (barys)
= 15*X[2]-14*X[3], 7*X[40]-8*X[4691], 2*X[52]-X[12279], 3*X[146]-2*X[23236], 9*X[165]-10*X[31399], 7*X[944]-8*X[3635], 10*X[946]-9*X[30392], 5*X[962]-4*X[11278], 4*X[1216]-5*X[11439], 3*X[3060]-2*X[10575], 25*X[3616]-24*X[31662], 6*X[3630]-7*X[15069], 6*X[4297]-7*X[9624], 5*X[4668]-7*X[5691], 16*X[5097]-15*X[14912], 6*X[5102]-5*X[6776], 4*X[5446]-3*X[15072], 9*X[5485]-8*X[7751], 2*X[5562]-3*X[11455], 3*X[5656]-2*X[17845], 3*X[5657]-4*X[31673], 3*X[5667]-2*X[23241], 3*X[5731]-4*X[22793], 10*X[5734]-9*X[7967], 15*X[5818]-14*X[9588], 6*X[5886]-7*X[10248], 3*X[5890]-4*X[13598], 2*X[5894]-3*X[18405], 3*X[5918]-4*X[16616], 3*X[6031]-4*X[31824], 5*X[8567]-6*X[23324], 3*X[9778]-4*X[18480], 9*X[9779]-8*X[13624], 21*X[9780]-20*X[31447], 6*X[9812]-5*X[10595], X[9862]-2*X[10723], 3*X[10167]-4*X[31822], 5*X[10574]-4*X[14641], 2*X[10625]-3*X[15305], 2*X[10721]-X[12383], 2*X[10722]-X[13172], 2*X[10724]-X[12248], 2*X[10728]-X[13199], 2*X[10733]-X[12244], 2*X[10735]-X[12253], 9*X[11002]-8*X[13630], 3*X[11206]-4*X[22802], 2*X[11381]-X[11412], 5*X[11444]-6*X[16194], 3*X[11459]-4*X[13474], 17*X[11465]-16*X[17704], 8*X[11793]-9*X[16261], 4*X[12121]-5*X[20125], 4*X[12512]-5*X[18492], 6*X[14855]-7*X[15043], 14*X[15057]-15*X[15081], 5*X[15058]-4*X[15644], 2*X[18525]-X[20070], 3*X[20127]-4*X[20379], 2*X[20427]-3*X[32064], 3*X[21168]-4*X[31672], 3*X[23039]-4*X[32137]
= as a point on the Euler line, W2 has Shinagawa coefficients (2,-7)
= lies on these lines: {1,28172}, {2,3}, {8,28146}, {40,4691}, {52,12279}, {55,31410}, {68,11738}, {69,29317}, {74,15749}, {99,32823}, {145,28186}, {146,23236}, {155,12112}, {165,31399}, {262,18844}, {315,32822}, {316,32818}, {325,32876}, {355,28154}, {371,23269}, {372,23275}, {388,4309}, {390,9655}, {393,18849}, {497,4317}, {511,12290}, {515,3633}, {516,3625}, {517,20053}, {574,31417}, {578,8718}, {944,3635}, {946,30392}, {950,4114}, {958,31420}, {962,11278}, {1056,6284}, {1058,7354}, {1075,15005}, {1093,18847}, {1151,23253}, {1152,23263}, {1204,18918}, {1216,11439}, {1285,5254}, {1478,4330}, {1479,4325}, {1503,6144}, {1587,6431}, {1588,6432}, {1620,15752}, {1788,4333}, {1870,9643}, {1975,32875}, {2549,5041}, {2777,12317}, {3060,10575}, {3085,9656}, {3086,9671}, {3183,6761}, {3488,9579}, {3585,8164}, {3600,9668}, {3616,31662}, {3619,7910}, {3621,28212}, {3630,15069}, {3632,28232}, {4045,18841}, {4293,12953}, {4294,12943}, {4297,9624}, {4299,5225}, {4302,5229}, {4304,5714}, {4316,7288}, {4324,5218}, {4338,10572},{4668,5691}, {5008,5319}, {5013,31407}, {5097,14912}, {5102,6776}, {5261,31480}, {5270,10385}, {5343,22238}, {5344,22236}, {5446,15072}, {5475,31450}, {5485,7751}, {5562,11455}, {5656,17845}, {5657,31673}, {5667,23241}, {5731,22793}, {5734,7967}, {5818,9588}, {5886,10248}, {5889,14915}, {5890,13598}, {5894,18405}, {5918,16616}, {6000,14531}, {6031,31824}, {6225,11271}, {6337,7814}, {6429,9541}, {6430,13939}, {6437,23249}, {6438,23259}, {6459,23267}, {6460,23273}, {6484,31412}, {6560,7582}, {6561,7581}, {6759,9705}, {7689,15107}, {7736,7756}, {7737,7765}, {7738,7747}, {7750,32826}, {7759,32479}, {7761,18840}, {7796,32006}, {8567,23324}, {8884,18851}, {8981,9692}, {9539,32047}, {9542,10137}, {9543,13903}, {9607,14482}, {9706,13352}, {9778,18480}, {9779,13624}, {9780,31447}, {9812,10595}, {9862,10723}, {9936,17702}, {10138,13993}, {10167,31822}, {10574,14641}, {10580,31776}, {10588,18513}, {10589,18514}, {10590,15338}, {10591,15326}, {10625,15305}, {10721,12383}, {10722,13172}, {10724,12248}, {10728,13199}, {10733,12244}, {10735,12253}, {11002,13630}, {11037,31795}, {11206,22802}, {11270,32110}, {11381,11412}, {11431,12241}, {11444,16194}, {11459,13474}, {11465,17704}, {11793,16261}, {12121,20125}, {12289,29012}, {12512,18492}, {12645,28216}, {12702,28182}, {13346,14157}, {13391,18439}, {14483,15740}, {14855,15043}, {15057,15081}, {15058,15644}, {15602,31457}, {16263,18853}, {18525,20070}, {18540,26878}, {20127,20379}, {20427,32064}, {21168,31672}, {23039,32137}, {31404,31492}, {32815,32877}, {32819,32878}, {32827,32889}
= anticomplement of X(1657)
= reflection of X(i) in X(j) for these {i,j}: {4,3146}, {3146,5073}, {11001,3543}, {12383,10721}, {13199,10728}, {15161,10736}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,20,548}, {2,1657,17538}, {2,3627,4}, {2,10304,15718}, {2,14093,3524}, {2,17538,21735}, {2,33271,33247}, {3,4,3545}, {3,26,13620}, {3,382,3853}, {3,1656,11812}, {3,1657,15686}, {3,3533,15719}, {3,3543,4}, {3,3545,3533}, {3,3832,5067}, {3,3845,5056}, {3,3850,2}, {3,3851,15723}, {3,3853,3832}, {3,5056,15702}, {3,5059,11001}, {3,5067,631}, {3,11539,3523}, {3,15690,3522}, {3,15719,10299}, {4,20,631}, {4,376,3090}, {4,631,3855}, {4,1657,21735}, {4,3146,15682}, {4,3524,3091}, {4,3525,381}, {4,3528,5}, {4,3529,376}, {4,3544,3839}, {4,5067,3832}, {4,5071,546}, {4,6899,6965}, {4,6916,6900}, {4,11001,3}, {4,13619,3147}, {4,15704,15698}, {4,17538,2}, {4,19708,3544}, {5,20,3528}, {5,382,17578}, {5,3528,631}, {5,15696,15717}, {5,17578,4}, {5,17800,20}, {20,382,4}, {20,548,17538}, {20,631,376}, {20,3091,21734}, {20,3146,382}, {20,3543,3832}, {20,3832,3}, {20,3853,5067}, {20,3861,3524}, {20,7486,3522}, {20,15717,15696}, {20,17578,5}, {20,21734,550}, {23,12084,21844}, {140,3839,3544},{140,5076,3839}, {376,3090,10299}, {376,3533,3}, {376,3545,15719}, {376,3855,631}, {381,3522,3525}, {381,3530,7486}, {381,15690,15708}, {381,15704,3522}, {382,1657,3843}, {382,3526,3830}, {382,3529,3855}, {382,3843,3627}, {382,3853,3543}, {382,5059,5067}, {382,15685,3856}, {382,17800,5}, {384,33192,33238}, {384,33238,33190}, {439,33228,32959}, {546,3523,5071}, {546,3534,3523}, {547,550,3}, {547,3853,3861}, {547,15686,14093}, {548,1657,20}, {548,3627,3843}, {548,3843,2}, {548,3853,3850}, {548,12812,3530}, {548,23046,3526}, {550,3091,3524}, {550,3526,21734}, {550,3627,23046}, {550,3830,3091}, {550,3861,3526}, {550,12108,14093}, {550,23046,12108}, {631,3529,20}, {631,3545,5067}, {631,3855,3090}, {631,5067,3533}, {631,15698,3530}, {1003,32982,32951}, {1593,12082,7512}, {1656,12103,10304}, {1656,15685,12103}, {1657,3627,2}, {1657,3830,12108}, {1657,3843,548}, {1657,5072,15689}, {1657,5076,15706}, {1657,15684,3627}, {1657,15718,12103}, {2041,2042,3146}, {2043,2044,15692}, {3090,10299,15709}, {3090,15719,3533}, {3091,3522,15713}, {3091,3830,4}, {3091,15683,550}, {3091,21734,3526}, {3146,5059,3543}, {3146,11541,3529}, {3146,15640,5073}, {3153,31725,4}, {3520,7387,7556}, {3522,3525,15698}, {3522,7486,3530}, {3522,15708,3}, {3524,3526,631}, {3525,3530,631}, {3526,3830,3861}, {3526,3861,3091}, {3526,21734,3524}, {3529,3543,3533}, {3529,15682,4}, {3530,7486,3525}, {3533,3545,3090}, {3534,5071,15710}, {3534,15710,376}, {3543,3832,3853}, {3543,5059,3}, {3543,11001,3545}, {3543,15683,547}, {3544,19708,140}, {3545,3832,3855}, {3545,11001,376}, {3545,15682,3543}, {3545,15710,11539}, {3552,16041,33189}, {3552,33279,16041}, {3627,15686,3850}, {3627,15704,12812}, {3627,15712,14893}, {3628,14269,3854}, {3830,14093,23046}, {3830,15683,3524}, {3832,3853,4}, {3832,5059,20}, {3832,5067,3545}, {3839,5076,4}, {3839,15681,19708}, {3843,3850,3832}, {3843,15684,382}, {3843,17538,631}, {3845,15702,3545}, {3850,12108,547}, {3850,15686,3}, {3850,21735,3533}, {3851,8703,10303}, {3853,11001,631}, {3853,16239,3845}, {3854,15692,3628}, {3858,5054,15022}, {3861,23046,3843}, {5025,33193,33239}, {5025,33239,33191}, {5055,15697,15715}, {5056,16239,5067}, {5059,11001,3529}, {5059,15708,15704}, {5067,15702,16239}, {5072,15689,15712}, {5072,15712,2}, {5073,11541,15682}, {5076,15681,140}, {6655,14033,32956}, {6655,33280,14033}, {6658,33017,14001}, {7391,20063,7493}, {7580,21669,6875}, {7819,33210,33232}, {7833,14068,32968}, {7841,32981,14069}, {8352,33235,32972}, {8356,32979,32957}, {8357,11159,33198}, {8357,33198,33230}, {8369,33200,32953}, {8370,33023,32960}, {8597,33257,14063}, {8703,12102,3851}, {9541,23251,13886}, {9541,31454,9693}, {9693,13886,31454}, {9812,18481,10595}, {9855,14062,32964}, {11114,31295,443}, {11361,32997,16043}, {11413,18534,3518}, {12103,15687,1656}, {13586,32996,32969}, {14035,32986,16045}, {14035,33256,32986}, {14039,32974,33194}, {14041,33244,32970}, {14042,32965,32983}, {14044,33268,2}, {14062,32964,32984}, {14063,32985,32955}, {14063,33257,32985}, {14066,33267,2}, {14093,23046,2}, {14269,19710,15692}, {14784,14785,3534}, {14892,15718,2}, {14893,15689,2}, {14893,15712,5072}, {14927,31670,14912}, {15644,32062,15058}, {15682,21735,3627}, {15683,21734,20}, {15684,15686,3543}, {15685,15687,10304}, {15691,19709,15705}, {15696,15717,3528}, {15704,15713,550}, {15722,19708,15698}, {16044,33253,33215}, {16924,33264,33226}, {17538,21735,376}, {17578,17800,3528}, {18586,18587,12101}, {19687,32974,14039}, {32966,33254,33216}, {32973,33229,33285}, {32973,33285,33195}, {32995,33243,33273}, {33006,33014,32977}, {33007,33019,14064}, {33008,33018,32975}, {33016,33260,32978}, {33184,33201,32952}
= (6-9-13) search numbers: [-55.5125486314180638661,-56.2356250493524613824,68.1941965767290356907]
Best regards,
Ercole Suppa
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