[Kadir Altintas]
Let ABC be a triangle, O, H, the circumcenter, orthocenter, resp. and P a point.
Denote:
DEF = the pedal triangle of P
A'B'C' = the circumcevian triangle of O.
A'' = the reflection of A' in D. Define B'', C'' cyclically
Prove that A'', B'', C'', H are concyclic.
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[Ercole Suppa]
Let Q(X(i)) be the center of fircle (A''B''C'') with respect to point P=X(i). Some points:
Q(X(2)) = X(3830)
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Q(X(3)) = X(4)
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Q(X(4)) = X(382)
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Q(X(5))= X(3627)
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Q(X(6)) = ANTICOMPLEMENT OF X(3098)
= a^6+3 a^4 b^2-3 a^2 b^4-b^6+3 a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-3 a^2 c^4+b^2 c^4-c^6 : : (barys)
= (SB+SC-SW)S^2 + 3 SB SC SW : : (barys)
= 3*X[2]-2*X[3098], 3*X[3]-4*X[3589], 3*X[4]-X[69], 2*X[5]-X[1350], X[20]-2*X[182], X[67]-2*X[10113], 2*X[141]-3*X[381], 3*X[376]-5*X[3618], X[382]+X[1351], 4*X[546]-3*X[10516], 2*X[550]-3*X[5085], 4*X[575]-X[3529], 2*X[576]+X[3146], 2*X[597]-X[3534], X[599]-2*X[3845], 5*X[631]-4*X[14810], X[895]+X[10721], 2*X[1353]-3*X[5102], 2*X[1386]-X[18481], X[1657]-3*X[5050], X[2892]-2*X[19506], X[3242]-2*X[22791], X[3416]-2*X[18480], 5*X[3522]-6*X[17508], 7*X[3526]-6*X[21167], 9*X[3545]-7*X[3619], 5*X[3620]-9*X[3839], 4*X[3631]-9*X[14269], 4*X[3853]-X[15069], 3*X[5032]+X[15640], 4*X[5066]-3*X[21358], X[5073]+3*X[5093], 5*X[5076]-X[11898], 4*X[5097]-3*X[14912], 3*X[5621]-2*X[14677], 3*X[6034]-2*X[12042], 8*X[6329]-9*X[14848], 2*X[6593]-X[12121], 2*X[9880]-X[19905], 4*X[10168]-3*X[10304], 7*X[10541]-4*X[12103], X[10620]-2*X[25328], X[10723]+X[10753], X[10724]+X[10759], X[10725]+X[10758], X[10726]+X[10764], X[10727]+X[10756], X[10728]+X[10755], X[10732]+X[10757], 3*X[11180]-X[20080], 5*X[11482]-4*X[12007], X[11541]+8*X[22330], X[11646]-2*X[22515], 4*X[12101]-X[15533], 3*X[12177]-2*X[14928], X[12244]-3*X[25320], X[12254]-2*X[19150], X[12383]-2*X[19140], 2*X[15118]-X[16111], 5*X[17538]-8*X[20190], 5*X[19709]-4*X[20582], 3*X[21356]-4*X[25561], X[22677]-2*X[22682], 3*X[23049]-2*X[23300]
= lies on these lines: {2,3098}, {3,3589}, {4,69}, {5,1350}, {6,30}, {20,182}, {22,14389}, {25,11064}, {51,1370}, {52,14216}, {66,265}, {67,10113}, {68,10263}, {110,7519}, {141,381}, {146,148}, {159,18534}, {184,7500}, {206,13352}, {263,14957}, {303,383}, {343,5064}, {376,3618}, {382,1351}, {394,428}, {495,10387}, {518,12699}, {524,3830}, {546,10516}, {550,5085}, {575,3529}, {576,3146}, {597,3534}, {599,3845}, {611,6284}, {613,7354}, {631,14810}, {698,7758}, {895,10721}, {1176,15033}, {1353,5102}, {1368,17810}, {1386,18481}, {1428,4299}, {1469,1479}, {1478,3056}, {1539,9973}, {1568,28419}, {1595,17834}, {1596,7716}, {1597,3867}, {1657,5050}, {1899,3060}, {1974,15462}, {1992,11645}, {2330,4302}, {2548,3094}, {2777,11579}, {2810,10741}, {2854,7728}, {2892,19506}, {2979,7394}, {3091,7938}, {3095,8721}, {3242,22791}, {3313,18420}, {3416,18480}, {3448,16981}, {3522,17508}, {3526,21167}, {3545,3619}, {3564,3627}, {3583,12589}, {3585,12588}, {3593,7374}, {3595,7000}, {3620,3839}, {3631,14269}, {3767,5017}, {3819,7392}, {3853,15069}, {3917,6997}, {4232,5972}, {4260,6851}, {5012,20062}, {5028,7747}, {5032,15640}, {5033,6781}, {5034,7756}, {5039,5286}, {5052,7748}, {5066,21358}, {5073,5093}, {5076,11898}, {5080,25304}, {5097,14912}, {5138,6869}, {5189,11002}, {5227,18540}, {5319,12212}, {5446,14790}, {5486,8705}, {5596,18400}, {5621,14677}, {5640,16063} ,{5654,7530}, {5722,24471}, {5728,24701}, {5846,18525}, {5921,5965}, {5943,7386}, {5969,6033}, {5999,7806}, {6034,12042}, {6036,9752}, {6193,13419}, {6201,21737}, {6329,14848}, {6393,7773}, {6515,11550}, {6593,12121}, {6643,10110}, {6803,13348}, {6995,9306}, {7378,21243}, {7401,15644}, {7408,14826}, {7470,7803}, {7487,13346}, {7517,15577}, {7528,10625}, {7553,9833}, {7576,20806}, {7667,10601}, {7694,22728}, {7703,15360}, {7731,12317}, {7800,9821}, {7931,13862}, {8148,9053}, {8177,9301}, {9019,18438}, {9024,10742}, {9037,12586}, {9047,12587}, {9308,16264}, {9739,21736}, {9822,18537}, {9873,20065}, {9880,19905}, {9909,23292}, {9969,15812}, {9970,17702}, {10168,10304}, {10192,20850}, {10210,16771}, {10541,12103}, {10620,25328}, {10691,17825}, {10723,10753}, {10724,10759}, {10725,10758}, {10726,10764}, {10727,10756}, {10728,10755}, {10732,10757}, {11003,20063}, {11114,15988}, {11180,20080}, {11433,21849}, {11482,12007}, {11487,15606}, {11541,22330}, {11646,22515}, {11663,15800}, {11800,13203}, {11807,12319}, {11818,13391}, {11819,12118}, {12101,15533}, {12160,16655}, {12164,16621}, {12177,14928}, {12244,25320}, {12254,19150}, {12383,19140}, {13634,17381}, {13635,17352}, {13857,26255}, {14965,15075}, {15118,16111}, {15311,15583}, {15435,18489}, {16625,18909}, {17532,26543}, {17538,20190}, {18390,21851}, {18396,26926}, {18559,22151}, {19709,20582}, {21356,25561}, {22677,22682}, {23049,23300}, {24248,29301}, {24695,29097}
= midpoint of X(895) and X(10721)
= reflection of X(i) in X(j) for these {i,j}: {3,5480}, {6,21850}, {20,182}, {66,18382}, {67,10113}, {69,3818}, {376,5476}, {550,18583}, {599,3845}, {1350,5}, {1352,4}, {2892,19506}, {3094,14881}, {3098,19130}, {3242,22791}, {3416,18480}, {3534,597}, {6776,576}, {9821,24256}, {10620,25328}, {11179,20423}, {11646,22515}, {12118,19139}, {12121,6593}, {12254,19150}, {12383,19140}, {16111,15118}, {18481,1386}, {19161,5446}, {19905,9880}, {22677,22682}
= isotomic conjugate of isogonal conjugate of X(20897)
= anticomplement of X(3098)
= barycentric product of X(i) and X(j) for these {i,j}: {76, 20897}
= barycentric quotient of X(i) and X(j) for these {i,j}: {20897, 6}
= trilinear product of X(i) and X(j) for these {i,j}: {75, 20897}, {75, 20897}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,5480,14561}, {4,69,3818}, {6,21850,20423}, {20,14853,182}, {69,3818,1352}, {376,3618,5092}, {550,18583,5085}, {1531,1843,3818}, {3060,7391,1899}, {3091,10519,24206}, {3098,19130,2}, {5092,5476,3618}, {5189,11002,18911}, {11550,21969,6515}
= (6-9-13) search numbers [-11.4660738100572121, -10.9367524526182682, 16.5042963999003549]
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Q(X(7)) = X(3)X(142) ∩ X(4)X(144)
= 3 a^6-4 a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4+2 a b^5-2 b^6-4 a^5 c-2 a^4 b c+2 a^3 b^2 c-2 a^2 b^3 c+2 a b^4 c+4 b^5 c-2 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2-4 a b^3 c^2+2 b^4 c^2+2 a^3 c^3-2 a^2 b c^3-4 a b^2 c^3-8 b^3 c^3+a^2 c^4+2 a b c^4+2 b^2 c^4+2 a c^5+4 b c^5-2 c^6 : : (barys)
= 3*X[3]-4*X[142], 3*X[4]-X[144], 2*X[5]-X[5759], 2*X[9]-3*X[381], 4*X[546]-3*X[5817], 2*X[550]-3*X[21151], X[673]-2*X[24827], X[1156]-2*X[22938], X[1657]-2*X[5732], 2*X[2550]-X[12702], 5*X[3091]-3*X[21168], 2*X[3243]-X[18526], 2*X[3254]-X[12773], 7*X[3526]-6*X[21153], X[3534]-2*X[6173], 2*X[3845]-X[6172], 9*X[5054]-10*X[20195], 9*X[5055]-8*X[6666], X[5223]-2*X[18480], 2*X[5542]-X[18481], 3*X[5686]-4*X[18357], 3*X[10246]-4*X[20330], X[12669]-2*X[24475]
= lies on these lines: {3,142}, {4,144}, {5,5759}, {7,30},{9,381}, {57,11238}, {382,971}, {390,6869}, {480,18491}, {517,3059}, {518,18345}, {527,3830}, {528,4930}, {546,5817}, {550,21151}, {673,24827}, {908,1260}, {942,4312}, {954,6985}, {962,5730}, {1156,22938}, {1537,9945}, {1596,7717}, {1657,5732}, {1699,3683}, {2095,10738}, {2550,12702}, {2951,18443} ,{3019,16884}, {3091,21168}, {3243,18526}, {3254,12773}, {3428,15909}, {3526,21153}, {3534,6173}, {3543,12690}, {3587,28198}, {3627,5843}, {3845,6172}, {5054,20195}, {5055,6666}, {5223,18480}, {5542,18481}, {5572,18530}, {5686,18357}, {5698,6841}, {5708,6851}, {5709,5789}, {5744,8727}, {5791,18483}, {5853,8148}, {5856,10742}, {6361,8728}, {6600,18524}, {6826,28174}, {7373,12573}, {8581,9655}, {9580,11018}, {9654,15298}, {9669,15299}, {10246,20330}, {12669,24475}, {12953,17637}, {13727,17236}, {15733,18499}, {24391,28646}
= reflection of X(i) in X(j) for these {i,j}: {3,5805}, {9,18482}, {390,22791}, {673,24827}, {1156,22938}, {1657,5732}, {3534,6173}, {5223,18480}, {5759,5}, {5779,4}, {6172,3845}, {11372,22793}, {12669,24475}, {12702,2550}, {12773,3254}, {18481,5542}, {18526,3243}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {9,18482,381}
= (6-9-13) search numbers [-11.7148213261166409, -11.6323434842767825, 17.1006659676913705]
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Q(X(8)) = X(18525)
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Q(X(9)) = MIDPOINT OF X(382) AND X(5779)
= a^6-3 a^5 b+3 a^4 b^2-3 a^2 b^4+3 a b^5-b^6-3 a^5 c-4 a^4 b c+2 a^2 b^3 c+3 a b^4 c+2 b^5 c+3 a^4 c^2+2 a^2 b^2 c^2-6 a b^3 c^2+b^4 c^2+2 a^2 b c^3-6 a b^2 c^3-4 b^3 c^3-3 a^2 c^4+3 a b c^4+b^2 c^4+3 a c^5+2 b c^5-c^6 : : (barys)
= 3*X[3]-4*X[6666], 3*X[4]-X[7], 2*X[5]-X[5732], X[20]-3*X[5817], 2*X[142]-3*X[381], 3*X[376]-5*X[18230], 2*X[550]-3*X[21153], 2*X[1001]-X[18481], X[1156]+X[10728], 3*X[1699]-2*X[20330], X[2550]-2*X[18480], X[2951]-3*X[5587], 5*X[3091]-3*X[21151], X[3146]+X[5759], X[3243]-2*X[22791], X[3254]-2*X[22938], 2*X[3845]-X[6173], 4*X[3853]-X[5735], X[5528]-2*X[11698], X[5542]-2*X[18483], 3*X[5686]-X[6361], X[6172]+X[15682], 2*X[11495]-3*X[26446]
= lies on these lines: {3,6666}, {4,7}, {5,5732}, {9,30}, {20,5817}, {33,6357}, {80,2093}, {142,381}, {144,3419}, {355,382}, {376,18230}, {495,4326}, {496,4321}, {515,6767}, {518,12699}, {527,3830}, {550,21153}, {990,17366}, {1001,18481}, {1156,10728},{1478,14100}, {1479,8581},{1490,5719}, {1697,5252}, {1699,20330}, {1728,3358}, {1750,5219}, {1836,18412}, {1837,4312}, {2550,18480}, {2801,10738}, {2951,5587}, {3091,21151}, {3146,5759}, {3174,18528}, {3243,22791}, {3254,22938}, {3627,5762}, {3820,18529}, {3845,6173}, {3853,5735}, {3911,19541}, {4292,10392}, {4302,15837}, {4335,5725}, {5046,10861}, {5080,25722}, {5528,11698}, {5542,18483}, {5561,15909}, {5686,6361}, {5853,18525}, {5927,10431}, {6172,15682}, {6923,15726}, {6930,28160}, {7354,15299}, {7675,11374}, {7678,18450}, {8232,24929}, {8257,28452}, {9579,10398}, {9613,10384}, {9655,12573}, {10430,11227}, {10947,17642}, {11495,26446}, {12618,17293}, {13727,17368}, {16152,18513}, {17668,18516}
= midpoint of X(i) and X(j) for these {i,j}: {382,5779}, {1156,10728}, {3146,5759}, {5691,11372}, {6172,15682}
= reflection of X(i) in X(j) for these {i,j}: {7,18482}, {2550,18480}, {3243,22791}, {3254,22938}, {5528,11698}, {5542,18483}, {5732,5}, {5805,4}, {6173,3845}, {12702,24393}, {18481,1001}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,7,18482}, {4,9799,5806}, {7,18482,5805}
= (6-9-13) search numbers [-8.88700960589292502, -10.1936322581980499, 14.7994912479951483]
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Q(X(10)) = MIDPOINT OF X(4) AND X(5691)
= 4 a^4-a^3 b-a^2 b^2+a b^3-3 b^4-a^3 c+2 a^2 b c-a b^2 c-a^2 c^2-a b c^2+6 b^2 c^2+a c^3-3 c^4 : : (barys)
= X[1]-3*X[4], 3*X[2]-5*X[18492], 3*X[3]-4*X[3634], 2*X[5]-X[4297], X[8]+3*X[3543], X[20]-3*X[5587], X[40]+X[3146], X[146]+X[12407], 3*X[165]-X[3529], X[355]+X[382], 3*X[376]-5*X[1698], 3*X[381]-2*X[1125], 2*X[546]-X[1385], 2*X[548]-3*X[11231], X[550]-2*X[9956], X[551]-2*X[3845], 5*X[631]-7*X[7989], X[942]-2*X[16616], 3*X[962]+X[3621], X[1482]-5*X[5076], X[1657]-2*X[12512], 7*X[3090]-5*X[7987], 5*X[3091]-3*X[3576], X[3244]-6*X[15687], X[3534]-2*X[3828], 9*X[3545]-7*X[3624], 5*X[3616]-9*X[3839], 4*X[3628]-3*X[17502], 2*X[3635]-3*X[3656], 4*X[3636]-3*X[3655], 3*X[3654]-4*X[4691], 3*X[3679]-X[6361], 7*X[3832]-3*X[5731], 5*X[3843]-3*X[5886], 4*X[3850]-3*X[11230], 7*X[3851]-6*X[10171], 11*X[3855]-9*X[7988], 2*X[3861]-X[5901], 5*X[4816]+3*X[9589], 9*X[5055]-8*X[19878], 4*X[5066]-3*X[19883], X[5073]+3*X[5790], X[5493]-2*X[5690], 2*X[5806]-X[12675], 3*X[5883]-2*X[13369], 3*X[5927]-X[14110], 2*X[7687]-X[11709], X[7957]-3*X[18908], X[7982]-3*X[9812], 7*X[9624]-9*X[9779], X[9864]+X[10723], 2*X[10113]-X[13605], X[10222]-4*X[12102], 9*X[10304]-13*X[19877], X[10721]+X[13211], X[10722]+X[13178], X[10724]+X[12751], X[10733]+X[12368], X[10735]+X[12784], X[11001]-3*X[19875], 3*X[11220]-5*X[15016], X[11599]-2*X[22515], 2*X[12005]-X[12680], 4*X[12009]-3*X[26201], X[12747]+X[16128], 7*X[16192]-5*X[17538], 5*X[19708]-7*X[19876], X[21630]-2*X[22938], X[21635]-2*X[22799], X[21636]-2*X[22505]
= lies on these lines: {1,4}, {2,18492}, {3,3634}, {5,4297}, {7,17706}, {8,3543}, {10,30}, {11,4311}, {12,4304}, {20,5587}, {35,21669}, {40,3146}, {44,10445}, {65,16006}, {79,16615}, {80,1770}, {84,7319}, {104,17501}, {140,28190}, {146,12407}, {165,3529}, {307,18661}, {355,382}, {376,1698}, {378,8185}, {381,1125}, {495,4314}, {496,4315}, {517,3625}, {519,3830}, {535,10916}, {546,1385}, {548,11231}, {550,9956}, {551,3845}, {631,7989}, {936,18529}, {942,16616}, {952,3853}, {962,3621}, {971,5884}, {993,6985}, {1012,5217}, {1155,12616}, {1158,5128}, {1159,6259}, {1210,7354}, {1420,10591}, {1482,5076}, {1503,4663}, {1532,7173}, {1597,9798}, {1657,12512}, {1737,10483}, {1826,2173}, {1837,4292}, {1872,2817}, {2349,2816}, {2475,18406}, {2771,4084}, {2784,6321}, {2800,12688}, {2801,24474}, {2807,13474}, {2829,6245}, {3090,7987}, {3091,3576}, {3149,5204}, {3244,15687}, {3419,12527}, {3452,17647}, {3520,9590}, {3534,3828}, {3545,3624}, {3555,12690}, {3577,5556}, {3601,10590}, {3614,6831}, {3616,3839}, {3628,17502}, {3635,3656}, {3636,3655}, {3651,5251}, {3654,4691}, {3671,11544}, {3679,6361}, {3755,29040}, {3811,18528}, {3822,6841}, {3832,5731}, {3843,5886}, {3850,11230}, {3851,10171}, {3855,7988}, {3861,5901}, {3911,4299}, {3947,24929}, {4293,9581}, {4294,9578}, {4295,5727}, {4298,5722}, {4302,10827}, {4305,5219}, {4325,12248}, {4420,5080}, {4667,13408}, {4669,28198}, {4816,9589}, {5055,19878}, {5066,19883}, {5073,5790}, {5086,11684}, {5126,10593}, {5195,25719}, {5252,10624}, {5260,7688}, {5316,6903}, {5493,5690}, {5542,12433}, {5708,5787}, {5745,6869}, {5794,12572}, {5806,12675}, {5842,21628}, {5883,13369}, {5927,14110}, {6845,7951}, {6912,10902}, {6920,15931}, {6928,9842}, {6996,29596}, {6999,16815}, {7384,29578}, {7406,29579}, {7491,12617}, {7687,11709}, {7741,21578}, {7957,18908}, {7982,9812}, {7991,28232}, {8582,11112}, {8666,18519}, {8715,18518}, {8727,10592}, {9579,18391}, {9624,9779}, {9626,14118}, {9654,13405}, {9656,17718}, {9668,12575}, {9864,10723}, {10113,13605}, {10151,11363}, {10222,12102}, {10304,19877}, {10721,13211}, {10722,13178}, {10724,12751}, {10733,12368}, {10735,12784}, {10742,12437}, {10863,13729}, {10895,13411}, {11001,19875}, {11019,18990}, {11114,24987}, {11220,15016}, {11365,18535}, {11372,29007}, {11456,16473}, {11552,16116}, {11599,22515}, {12005,12680}, {12009,26201}, {12114,19541}, {12135,13473}, {12514,18540}, {12618,29024}, {12640,18499}, {12747,16128}, {13746,18653}, {15326,17606},{16192,17538}, {16948,24624}, {17577,24541}, {17579,24982}, {17679,25967}, {18491,25440}, {18527,21625}, {19708,19876}, {21630,22938}, {21635,22799}, {21636,22505}
= midpoint of X(i) and X(j) for these {i,j}: {4,5691}, {40,3146}, {80,10728}, {146,12407}, {355,382}, {962,5881}, {3654,15684}, {3679,15682}, {9589,12245}, {9864,10723}, {10721,13211}, {10722,13178}, {10724,12751}, {10726,13532}, {10733,12368}, {10735,12784}, {12699,18525}, {12747,16128}
= reflection of X(i) in X(j) for these {i,j}: {1,18483}, {3,19925}, {10,18480}, {20,6684}, {550,9956}, {551,3845}, {942,16616}, {944,13464}, {946,4}, {1385,546}, {1657,12512}, {3244,22791}, {3452,18516}, {3534,3828}, {3579,18357}, {4297,5}, {4301,22793}, {5493,5690}, {5542,18482}, {5882,946}, {5884,7686}, {5901,3861}, {10265,6246}, {11362,355}, {11599,22515}, {11709,7687}, {12437,21077}, {12675,5806}, {12680,12005}, {12702,3626}, {13605,10113}, {14110,20117}, {18481,1125}, {21630,22938}, {21635,22799}, {21636,22505}, {22793,3853}
= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,4,18483}, {1,18483,946}, {3,19925,10175}, {4,944,1699}, {4,12667,26332}, {5,4297,10165}, {5,13624,19862}, {20,5587,6684}, {80,1770,4848}, {355,12702,3626}, {381,18481,1125}, {546,1385,3817}, {550,9956,10164}, {631,7989,10172}, {944,1699,13464}, {950,1478,21620}, {1657,26446,12512}, {1699,13464,946}, {1837,12943,4292}, {3529,5818,165}, {3579,18357,10}, {3579,18480,18357}, {3585,10572,226}, {3626,12702,11362}, {3655,18493,3636}, {3656,18526,3635}, {3830,18525,12699}, {3832,5731,8227}, {3843,5886,12571}, {4297,19862,13624}, {4299,10826,3911}, {5252,12953,10624}, {5722,9655,4298}, {5927,14110,20117}, {6985,18761,993}, {13624,19862,10165}
= (6-9-13) search numbers [-9.36008337111832110, -10.6419006574565712, 15.3281726468166151]
Best regards
Ercole Suppa
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