[Kadir Altintas]:
A conjecture on conics (van Lamoen like)
Let ABC be a triangle, P a point and DEF the circumcevian triangle of P.
The six circumcenters of the triangles PAE, PEC, PPCD, PDB, PBF, PFA lie on a conic.
Open problem: For which point P these circumcenters lie on a circle? (van Lamoen circle like)
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[Ercole Suppa]
Let P=X(i) and W=W(X(i)) the center of conic through circumcenters of triangles PAE, PEC, PPCD, PDB, PBF, PFA
*** Some pairs {X(i),X(j)=W(X(i))}: {1, 1385}, {2, 549}, {3, 3}, {4, 5}, {5, 140}, {6, 182}, {8, 5690}, {10, 6684}, {11, 6713}
*** Some points:
W(X(7)) = COMPLEMENT OF X(5779)
= 2 a^5 b-3 a^4 b^2-2 a^3 b^3+4 a^2 b^4-b^6+2 a^5 c+4 a^4 b c-2 a^3 b^2 c-6 a^2 b^3 c+2 b^5 c-3 a^4 c^2-2 a^3 b c^2+4 a^2 b^2 c^2+b^4 c^2-2 a^3 c^3-6 a^2 b c^3-4 b^3 c^3+4 a^2 c^4+b^2 c^4+2 b c^5-c^6 : : (barys)
= 3*X[2]-X[5779], X[3]+X[7], X[5]-2*X[142], X[9]-2*X[140], X[144]-5*X[631], 2*X[548]+X[5735], 5*X[632]-4*X[6666], X[1482]-3*X[11038], 5*X[1656]-3*X[5817], X[2951]+X[12699], X[3062]-5*X[8227], 7*X[3523]+X[20059], 7*X[3526]-5*X[18230], 4*X[3530]-3*X[21153], 3*X[3576]+X[4312], X[3627]-2*X[18482], 4*X[3628]-5*X[20195], 3*X[5054]-X[6172], X[5223]-3*X[26446], X[5572]-2*X[13373], X[6916]+X[15934], X[6987]+X[18541], 7*X[9624]-3*X[24644], X[12675]+X[15587], 5*X[15016]-X[18412], X[15937]+X[15970], X[20430]-3*X[27475]
= lies on these lines on lines: {2,5779}, {3,7}, {5,142}, {9,140}, {30,5732}, {40,5586}, {55,24465}, {144,631}, {182,5845}, {226,11227}, {390,6948}, {442,10861}, {495,8581}, {496,14100}, {516,550}, {517,5542}, {518,5690}, {527,549}, {548,5735}, {632,6666}, {952,2550}, {990,4675}, {991,1086}, {1001,6914}, {1071,8728}, {1482,11038}, {1483,5853}, {1656,5817}, {2096,16418}, {2346,11849}, {2801,3826}, {2808,20328}, {2951,12699}, {3062,8227}, {3243,5844}, {3452,10156}, {3475,6244}, {3517,7717}, {3523,20059}, {3526,18230}, {3530,21153}, {3576,4312}, {3627,18482}, {3628,20195}, {3649,7987}, {3742,7956}, {3834,12618}, {4326,15172}, {4654,10857}, {5054,6172}, {5219,11407}, {5220,26487}, {5223,26446}, {5249,8727}, {5428,17768}, {5572,13373}, {5696,6067}, {5708,6908}, {5728,6907}, {5729,6863}, {5768,17528}, {5785,5791}, {5811,16853}, {5850,6684}, {5851,6713}, {5886,7171}, {6825,8732}, {6842,10394}, {6846,12684}, {6850,12433}, {6881,12669}, {6891,8232}, {6916,15934}, {6922,21617}, {6954,12848}, {6955,9945}, {6987,18541}, {7263,29016}, {7411,26842}, {8158,11037}, {8226,11220}, {9624,24644}, {9776,19541}, {10004,28344}, {10267,11495}, {10384,11373}, {10884,20420}, {10943,17668}, {11019,15008}, {11112,18444}, {11246,15931}, {12528,17529}, {12675,15587}, {13329,17365}, {13727,26806}, {15016,18412}, {15299,15325}, {15726,27869}, {15937,15970}, {16112,26492}, {18857,28534}, {20430,27475}, {24390,25722}
= complement of X(5779)
= midpoint of X(i) in X(j) for these {i,j}: {3,7}, {5732,5805}, {6916,15934}, {6987,18541}, {12675,15587}, {15937,15970}
= reflection of X(i) in X(j) for these {i,j}: {5,142}, {9,140}, {3627,18482}, {5572,13373}, {20330,25557}, {22791,20330}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,6147,5763}, {7,21151,3}, {3523,20059,21168}, {3824,6245,5}, {5249,10167,8727}, {5732,6173,5805}, {11220,27186,8226}
= (6-9-13) search numbers [3.76326193660129327, 3.34436131883569677, -0.411552709564007864]
W(X(9)) = COMPLEMENT OF X(5805)
= a (2 a^5-3 a^4 b-2 a^3 b^2+4 a^2 b^3-b^5-3 a^4 c-2 a^3 b c+4 a^2 b^2 c+2 a b^3 c-b^4 c-2 a^3 c^2+4 a^2 b c^2-4 a b^2 c^2+2 b^3 c^2+4 a^2 c^3+2 a b c^3+2 b^2 c^3-b c^4-c^5) : : (barys)
= 3*X[2]+X[5759], X[3]+X[9], X[4]-5*X[18230], X[7]-5*X[631], X[20]+3*X[5817], 2*X[140]-X[142], X[144]+7*X[3523], X[390]+3*X[5657], X[944]+3*X[5686], X[3062]+7*X[16192], X[3243]-3*X[10246], 3*X[3524]+X[6172], 7*X[3526]-X[5735], X[3587]+X[6913], 3*X[5054]-X[6173], X[5220]+2*X[13624], X[6068]+3*X[21154], 3*X[8236]+X[12245]
= lies on these lines: {1,15837}, {2,5759}, {3,9}, {4,18230}, {5,516}, {7,631}, {20,5817}, {35,14100}, {36,8581}, {37,13329}, {40,5806}, {44,991}, {45,990}, {55,15299}, {56,15298}, {57,10156}, {63,11227}, {72,6986}, {140,142}, {144,3523}, {165,3683}, {182,518}, {210,5531}, {390,5657}, {495,12573}, {517,1001}, {527,549}, {528,12619}, {942,954}, {944,5686}, {952,6594}, {962,17552}, {1006,5728}, {1071,26878}, {1125,5763}, {1155,4312}, {1214,2954}, {1538,4679}, {1699,7964}, {1708,11018}, {2550,6827}, {2646,18412}, {2801,15481}, {2808,28345}, {3035,3452}, {3059,10902}, {3062,16192}, {3088,7717}, {3219,10167}, {3243,10246}, {3305,7580}, {3419,6992}, {3524,6172}, {3526,5735}, {3530,5843}, {3576,3940}, {3587,6913}, {3601,10398}, {3748,15104}, {3824,5812}, {3927,8726}, {3929,10857}, {4188,10861}, {4297,5302}, {4304,10392}, {4422,12618}, {4512,6244}, {5054,6173}, {5122,8545}, {5220,13624}, {5259,7957}, {5542,5719}, {5584,9856}, {5690,5853}, {5698,6825}, {5729,7675}, {5791,6865}, {5832,6958}, {5856,6713}, {5901,14150}, {5927,7411}, {6068,21154}, {6282,16418}, {6600,10267}, {6875,10394}, {6887,12699}, {6940,29007}, {6985,11495}, {7677,24928}, {8071,15518}, {8128,8389}, {8236,12245}, {9581,9588}, {9709,10268}, {9940,26921}, {10225,28534}, {10386,15006}, {10884,15650}, {11230,25379}, {11362,12433}, {12260,20790}, {12572,22792}, {13727,17260}, {15296,26286}, {15297,26285}, {15587,18233}, {15935,28234}, {17348,29016}, {17768,22937}
= complement of X(5805)
= midpoint of X(i) in X(j) for these {i,j}: {3,9}, {3587,6913}, {5732,5779}, {5759,5805}
= reflection of X(i) in X(j) for these {i,j}: {5,6666}, {142,140}, {18482,5}, {20330,1125}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,5759,5805}, {3,5779,5732}, {9,5438,5785}, {9,5732,5779}, {9,21153,3}, {40,11108,5806}, {144,3523,21151}, {165,7308,19541}, {631,21168,7}, {954,1445,942}, {3305,7580,10157}, {5812,6989,3824}, {7411,27065,5927}
= (6-9-13) search numbers [5.17716779671315120, 4.06371693187506308, -1.56214006941211897]
W(X(12)) = COMPLEMENT OF X(26470)
= 2 a^7-2 a^6 b-5 a^5 b^2+5 a^4 b^3+4 a^3 b^4-4 a^2 b^5-a b^6+b^7-2 a^6 c+3 a^4 b^2 c-2 a^3 b^3 c+2 a b^5 c-b^6 c-5 a^5 c^2+3 a^4 b c^2+4 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+5 a^4 c^3-2 a^3 b c^3+4 a^2 b^2 c^3-4 a b^3 c^3+3 b^4 c^3+4 a^3 c^4+a b^2 c^4+3 b^3 c^4-4 a^2 c^5+2 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : : (barys)
= 2 R S^2 + (3 a SB-3 b SB-c SB+3 a SC-b SC-3 c SC-2 a SW)S + 2 R SB SC : : (barys)
= X[3]+X[12], 5*X[631]-X[2975], 7*X[3523]+X[20060], 3*X[3584]+X[11012], 3*X[4995]-X[11849]
= lies on these lines: {2,10267}, {3,12}, {5,5248}, {10,140}, {21,119}, {35,5840}, {55,6863}, {78,26446}, {100,6853}, {182,5849}, {355,7483}, {495,15865}, {499,10959}, {517,13411}, {529,549}, {631,2975}, {758,6684}, {993,10942}, {1001,6959}, {1006,27529}, {1621,6949}, {1737,24299}, {3085,6954}, {3523,20060}, {3526,19854}, {3530,23961}, {3584,11012}, {3652,13257}, {3884,11729}, {3898,5901}, {3911,13373}, {4309,11928}, {4995,11849}, {4996,6940}, {5217,6923}, {5218,6825}, {5433,10246}, {5554,17566}, {5690,5855}, {5790,24953}, {5885,15556}, {6284,6980}, {6675,9956}, {6745,9940}, {6824,18491}, {6862,11500}, {6868,10588}, {6875,11681}, {6882,10902}, {6883,26364}, {6889,11517}, {6892,18761}, {6907,26285}, {6910,10786}, {6911,10198}, {6914,18242}, {6924,25466}, {6934,10585}, {6998,26231}, {7491,7951}, {8070,14795}, {10056,10680}, {10225,11277}, {12639,22937}, {13226,26201}, {13747,19860}, {15178,15325}, {15867,16203}, {15888,22765}, {16617,19925}
= complement of X(26470)
= midpoint of X(i) in X(j) for these {i,j}: {3,12}, {35,6842}, {11491,26470}, {11849,15908}
= reflection of X(i) in X(j) for these {i,j}: {5,6668}, {4999,140}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,11491,26470}, {12,21155,3}, {140,1385,6713}, {3085,6954,11249}, {4995,15908,11849}, {5218,6825,11248}, {6910,10786,22758}
= (6-9-13) search numbers [4.05826340940835219, 3.47820072704388288, -0.640365287311394672]
Best regards
Ercole Suppa
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