leftri rightri

This is PART 34: Centers X(66001) - X(68000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)



X(66001) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND-MOSES-MIYAMOTO-APOLLONIUS TRIANGLE WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^3-a^2*(b+c)+(b-c)^2*(b+c)-a*(b^2+c^2))*(a^4*(b+c)+a^2*b*c*(b+c)-2*a^3*(b^2+c^2)+2*a*(b-c)^2*(b^2+b*c+c^2)-(b-c)^2*(b+c)*(b^2+b*c+c^2))-2*a*(a^2-b^2+b*c-c^2)*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2))*S : :

X(66001) lies on these lines: {7, 80}, {104, 7133}, {1387, 30346}, {1768, 30400}, {2771, 63284}, {2800, 52808}, {5083, 30341}, {6203, 6326}, {6224, 52813}, {6264, 30319}, {6265, 30385}, {9946, 30276}, {9952, 30288}, {10265, 30380}, {11571, 30425}, {12515, 30296}, {12611, 30306}, {12619, 30313}, {12691, 30324}, {12758, 30333}, {12767, 30354}, {12770, 30360}, {12771, 30368}, {12772, 30418}, {12774, 30406}, {17638, 30375}, {18254, 30412}, {18460, 35774}, {49241, 52810}

X(66001) = reflection of X(i) in X(j) for these {i,j}: {66000, 11570}
X(66001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2801, 11570, 66000}

X(66002) = ORTHOLOGY CENTER OF THESE TRIANGLES: GEMINI 29 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^8*(b+c)-2*a^6*(b-c)^2*(b+c)-(b-c)^4*(b+c)^3*(b^2+c^2)-a^7*(2*b^2+b*c+2*c^2)-6*a^4*b*c*(b^3+c^3)+a*(b^2-c^2)^2*(2*b^4-5*b^3*c+3*b^2*c^2-5*b*c^3+2*c^4)+a^5*(6*b^4-3*b^3*c+5*b^2*c^2-3*b*c^3+6*c^4)+a^3*(-6*b^6+9*b^5*c-2*b^4*c^2+2*b^3*c^3-2*b^2*c^4+9*b*c^5-6*c^6)+2*a^2*(b^7+b^6*c-2*b^5*c^2-2*b^2*c^5+b*c^6+c^7)) : :
X(66002) = -3*X[2]+2*X[12665], -2*X[72]+3*X[38693], -2*X[1537]+3*X[3873], -3*X[3681]+4*X[64193], -5*X[3876]+6*X[21154], -5*X[3889]+4*X[64192], -3*X[5603]+2*X[66044], -3*X[5731]+2*X[64139], -4*X[5777]+5*X[31272], -3*X[5927]+4*X[58587], -6*X[10202]+5*X[64008]

X(66002) lies on these lines: {1, 66024}, {2, 12665}, {7, 80}, {11, 12528}, {57, 66061}, {72, 38693}, {100, 1071}, {104, 912}, {145, 2800}, {149, 63962}, {518, 64189}, {758, 64145}, {942, 17661}, {952, 14923}, {971, 10724}, {1320, 6001}, {1484, 38038}, {1537, 3873}, {1768, 3811}, {1858, 12740}, {2771, 7984}, {2802, 15071}, {2829, 3868}, {2950, 3870}, {3045, 47371}, {3218, 64188}, {3681, 64193}, {3869, 64191}, {3874, 34789}, {3876, 21154}, {3889, 64192}, {4996, 18446}, {5083, 14986}, {5603, 66044}, {5693, 11715}, {5731, 64139}, {5777, 31272}, {5840, 64358}, {5856, 12669}, {5884, 12751}, {5904, 46684}, {5927, 58587}, {6326, 35262}, {7080, 46685}, {10202, 64008}, {10711, 66047}, {10728, 24474}, {10742, 24475}, {11219, 47320}, {11220, 24466}, {12515, 62236}, {12531, 17654}, {13243, 18238}, {13278, 65998}, {13369, 34474}, {14266, 52409}, {14872, 59415}, {16173, 31803}, {16174, 61705}, {17100, 63399}, {18444, 51506}, {18861, 37700}, {34772, 48695}, {36845, 66060}, {40263, 59391}, {40266, 64742}, {64056, 66019}, {66021, 66045}

X(66002) = reflection of X(i) in X(j) for these {i,j}: {100, 1071}, {153, 11570}, {3869, 64191}, {5693, 11715}, {5904, 46684}, {10728, 24474}, {10742, 24475}, {12528, 11}, {12531, 17654}, {12532, 104}, {12665, 15528}, {12751, 5884}, {17661, 942}, {34789, 3874}, {40266, 64742}, {64056, 66019}, {66024, 1}
X(66002) = anticomplement of X(12665)
X(66002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {104, 912, 12532}, {2801, 11570, 153}, {10202, 66049, 64008}, {11570, 12736, 18419}

X(66003) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT INVERSE-OF-1ST-PAVLOV

Barycentrics    a*(a^9+11*a^7*b*c-3*a^8*(b+c)+(b-c)^4*(b+c)^3*(b^2-3*b*c+c^2)-4*a^2*b*(b-c)^2*c*(2*b^3-3*b^2*c-3*b*c^2+2*c^3)+2*a^6*(4*b^3-5*b^2*c-5*b*c^2+4*c^3)-a^4*(b-c)^2*(6*b^3-13*b^2*c-13*b*c^2+6*c^3)-a*(b^2-c^2)^2*(3*b^4-12*b^3*c+16*b^2*c^2-12*b*c^3+3*c^4)-2*a^5*(3*b^4+5*b^3*c-13*b^2*c^2+5*b*c^3+3*c^4)+a^3*(b-c)^2*(8*b^4+3*b^3*c-18*b^2*c^2+3*b*c^3+8*c^4)) : :
X(66003) = -3*X[3576]+X[64276], -3*X[5886]+2*X[64273]

X(66003) lies on these lines: {1, 6831}, {3, 51111}, {65, 104}, {79, 2829}, {355, 64274}, {515, 11263}, {517, 64268}, {523, 37628}, {546, 6261}, {944, 64345}, {997, 64294}, {999, 64284}, {1385, 64269}, {1476, 58595}, {1537, 65995}, {2646, 64173}, {2800, 47319}, {3244, 12616}, {3576, 64276}, {4511, 64270}, {4861, 14110}, {5450, 64044}, {5880, 31657}, {5884, 12114}, {5886, 64273}, {6001, 17637}, {6256, 64271}, {6264, 66006}, {6265, 63963}, {6583, 48694}, {6915, 37837}, {7354, 7702}, {10222, 66009}, {11500, 30147}, {11715, 66046}, {12650, 41865}, {12672, 66013}, {13375, 22766}, {17605, 21740}, {18493, 40257}, {22775, 31870}, {26066, 64275}, {35979, 64280}, {36975, 37468}, {39542, 64119}, {50371, 64201}, {64191, 65994}

X(66003) = midpoint of X(i) and X(j) for these {i,j}: {1, 64281}
X(66003) = reflection of X(i) in X(j) for these {i,j}: {355, 64274}, {6256, 64271}, {11500, 64286}, {64265, 63980}, {64266, 64293}, {64269, 1385}, {64298, 37837}

X(66004) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND-ANTI-PAVLOV WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a+b)*(a+c)*(a^7-3*a^6*(b+c)+(b-c)^2*(b+c)^3*(b^2+c^2)-a^5*(b^2-7*b*c+c^2)+a*(b^2-c^2)^2*(b^2-b*c+c^2)-a^2*(b-c)^2*(5*b^3+3*b^2*c+3*b*c^2+5*c^3)+a^4*(7*b^3-5*b^2*c-5*b*c^2+7*c^3)-a^3*(b^4+6*b^3*c-10*b^2*c^2+6*b*c^3+c^4)) : :

X(66004) lies on these lines: {11, 64382}, {21, 2800}, {58, 1768}, {81, 104}, {100, 64376}, {119, 5235}, {153, 333}, {952, 64720}, {1317, 64414}, {3193, 48694}, {4184, 12332}, {4221, 12515}, {4225, 22775}, {4653, 13253}, {5333, 6713}, {6224, 7415}, {9913, 64395}, {10058, 64420}, {10074, 64421}, {10698, 64415}, {10711, 64424}, {10742, 64405}, {11715, 64377}, {12138, 64378}, {12199, 64381}, {12248, 64384}, {12462, 64396}, {12463, 64397}, {12499, 64398}, {12751, 64401}, {12752, 64402}, {12753, 64403}, {12754, 64404}, {12761, 64406}, {12762, 64407}, {12763, 64408}, {12764, 64409}, {12767, 52680}, {12773, 64419}, {12775, 64422}, {12776, 64423}, {13913, 64417}, {13977, 64418}, {16704, 64009}, {16948, 57736}, {17551, 38133}, {17553, 50908}, {19081, 64385}, {19082, 64386}, {22799, 64399}, {35856, 64412}, {35857, 64413}, {37402, 46684}, {38602, 64393}, {38756, 64383}, {48464, 64379}, {48465, 64380}, {48684, 64387}, {48685, 64388}, {48686, 64389}, {48687, 64390}, {48692, 64391}, {48693, 64392}, {48695, 64394}, {48700, 64410}, {48701, 64411}, {64008, 64425}

X(66004) = reflection of X(i) in X(j) for these {i,j}: {66005, 64720}
X(66004) = X(104) of 2nd anti-Pavlov triangle
X(66004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 64720, 66005}

X(66005) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2ND-ANTI-PAVLOV WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a+b)*(a+c)*(a^3+b^3-3*b^2*c-3*b*c^2+c^3-3*a^2*(b+c)+a*(b^2+7*b*c+c^2)) : :

X(66005) lies on circumconic {{A, B, C, X(291), X(13143)}} and on these lines: {11, 5235}, {21, 2802}, {42, 81}, {58, 5541}, {80, 64401}, {104, 64376}, {119, 64400}, {149, 333}, {214, 64377}, {528, 4921}, {952, 64720}, {1043, 64743}, {1317, 64382}, {1320, 64415}, {1862, 64378}, {3035, 5333}, {3193, 48713}, {4184, 13205}, {4225, 22560}, {4653, 12653}, {4658, 15015}, {4720, 64056}, {6174, 42025}, {6224, 56018}, {9024, 41610}, {10087, 64420}, {10090, 64421}, {10707, 64424}, {10738, 64405}, {12331, 64419}, {13194, 64381}, {13199, 64384}, {13222, 64395}, {13228, 64396}, {13230, 64397}, {13235, 64398}, {13268, 64402}, {13269, 64403}, {13270, 64404}, {13271, 64406}, {13272, 64407}, {13273, 64408}, {13274, 64409}, {13278, 64422}, {13279, 64423}, {13922, 64417}, {13991, 64418}, {16173, 17557}, {16704, 20095}, {17553, 50891}, {19112, 64385}, {19113, 64386}, {22938, 64399}, {25438, 64394}, {31272, 64425}, {33814, 64393}, {35882, 64412}, {35883, 64413}, {38325, 53412}, {38484, 63917}, {48533, 64379}, {48534, 64380}, {48680, 64383}, {48703, 64387}, {48704, 64388}, {48705, 64389}, {48706, 64390}, {48711, 64391}, {48712, 64392}, {48714, 64410}, {48715, 64411}

X(66005) = reflection of X(i) in X(j) for these {i,j}: {66004, 64720}
X(66005) = X(100) of 2nd anti-Pavlov triangle
X(66005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 64720, 66004}

X(66006) = PERSPECTOR OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN AND ANTICEVIAN-OF-X(9)

Barycentrics    a*(a^6-2*a^5*(b+c)+(b^2-c^2)^2*(b^2+c^2)-a^2*(b+c)^2*(b^2-7*b*c+c^2)-a^4*(b^2+5*b*c+c^2)+a^3*(4*b^3+7*b^2*c+7*b*c^2+4*c^3)-a*(2*b^5+5*b^4*c+b^3*c^2+b^2*c^3+5*b*c^4+2*c^5)) : :

X(66006) lies on these lines: {1, 3925}, {9, 943}, {40, 3868}, {55, 191}, {65, 5541}, {78, 3646}, {79, 13146}, {149, 946}, {952, 65990}, {1058, 22836}, {1490, 5842}, {1855, 6198}, {1998, 31423}, {2136, 11529}, {2894, 5249}, {2949, 10902}, {2950, 65998}, {2951, 5762}, {3059, 44783}, {3174, 5880}, {3189, 3487}, {3333, 61033}, {3555, 7688}, {3970, 16550}, {4511, 40270}, {4654, 7702}, {5044, 5259}, {5528, 30424}, {5531, 65992}, {5905, 20066}, {6154, 65988}, {6264, 66003}, {7957, 41853}, {7982, 64316}, {13144, 64766}, {16558, 61763}, {31938, 63269}, {37625, 64276}, {37700, 40273}, {40263, 66020}, {41864, 52050}, {56176, 66009}, {64199, 66046}

X(66006) = reflection of X(i) in X(j) for these {i,j}: {64369, 943}
X(66006) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2894, 2949}, {5249, 9}, {63146, 40}

X(66007) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(7) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^9-a^8*(b+c)-(b-c)^6*(b+c)^3+a^7*(-4*b^2+9*b*c-4*c^2)+a*(b-c)^4*(b+c)^2*(b^2-5*b*c+c^2)+4*a^2*(b-c)^2*(b+c)^3*(b^2-b*c+c^2)+4*a^6*(b^3-3*b^2*c-3*b*c^2+c^3)-a^3*(b-c)^2*(4*b^4+5*b^3*c+10*b^2*c^2+5*b*c^3+4*c^4)+a^5*(6*b^4-5*b^3*c+30*b^2*c^2-5*b*c^3+6*c^4)-2*a^4*(3*b^5-5*b^4*c+6*b^3*c^2+6*b^2*c^3-5*b*c^4+3*c^5) : :
X(66007) = -2*X[80]+3*X[38149], -2*X[1484]+3*X[38107], -2*X[3254]+3*X[59386], -4*X[6594]+3*X[21168], -X[7993]+3*X[59372], -3*X[8236]+4*X[19907], -X[9803]+3*X[59412], -2*X[10265]+3*X[38052], -2*X[10707]+3*X[38073], -2*X[10738]+3*X[59385], -3*X[11038]+2*X[12737]

X(66007) lies on these lines: {7, 952}, {80, 38149}, {100, 329}, {104, 10427}, {119, 1156}, {149, 5805}, {153, 971}, {390, 6265}, {516, 5528}, {518, 66008}, {527, 48363}, {528, 10698}, {1484, 38107}, {2550, 2801}, {2800, 35514}, {3254, 59386}, {3488, 18801}, {4312, 5531}, {5083, 64155}, {5218, 5660}, {5220, 5657}, {5542, 6264}, {5728, 45043}, {5732, 12248}, {5762, 12331}, {5779, 11698}, {5856, 38665}, {6594, 21168}, {7972, 60924}, {7993, 59372}, {8236, 19907}, {9803, 59412}, {10265, 38052}, {10707, 38073}, {10738, 59385}, {10742, 36991}, {11038, 12737}, {11372, 21635}, {11729, 53055}, {12619, 40333}, {12730, 60926}, {12773, 31657}, {12775, 42843}, {15017, 51768}, {16116, 17857}, {17768, 66011}, {18230, 38752}, {19914, 59413}, {21630, 38036}, {33814, 59418}, {34122, 60959}, {38108, 66045}, {38137, 61601}, {38152, 66065}, {38755, 60901}, {57298, 60996}, {59381, 61562}, {61595, 66063}, {64008, 64738}

X(66007) = midpoint of X(i) and X(j) for these {i,j}: {4312, 5531}, {9809, 64696}
X(66007) = reflection of X(i) in X(j) for these {i,j}: {104, 10427}, {149, 5805}, {390, 6265}, {1156, 119}, {5759, 100}, {5779, 11698}, {6264, 5542}, {11372, 21635}, {12247, 2550}, {12248, 5732}, {12773, 31657}, {13199, 5528}, {36991, 10742}, {64264, 10265}, {66023, 37725}
X(66007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {104, 10427, 21151}, {119, 1156, 5817}, {2550, 2801, 12247}, {5851, 37725, 66023}, {38052, 64264, 10265}

X(66008) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(8) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^7-3*a^6*(b+c)-(b-c)^4*(b+c)^3+3*a*(b^2-c^2)^2*(b^2-3*b*c+c^2)+a^5*(b^2+15*b*c+c^2)-a^2*(b-c)^2*(b^3-15*b^2*c-15*b*c^2+c^3)+5*a^4*(b^3-3*b^2*c-3*b*c^2+c^3)-a^3*(5*b^4+6*b^3*c-30*b^2*c^2+6*b*c^3+5*c^4) : :
X(66008) = -3*X[2]+2*X[12737], -4*X[11]+5*X[5818], -4*X[214]+3*X[7967], -3*X[376]+2*X[64145], -5*X[631]+4*X[11715], -2*X[1484]+3*X[5790], -2*X[1537]+3*X[10711], -10*X[1656]+9*X[32558], -7*X[3090]+6*X[16173], -3*X[3241]+4*X[19907], -2*X[3254]+3*X[38149]

X(66008) lies on these lines: {2, 12737}, {3, 8}, {4, 2802}, {9, 64278}, {10, 6264}, {11, 5818}, {40, 12248}, {80, 497}, {119, 1320}, {145, 6265}, {149, 355}, {153, 517}, {214, 7967}, {376, 64145}, {388, 12749}, {392, 37162}, {515, 2950}, {518, 66007}, {519, 1512}, {528, 16112}, {631, 11715}, {758, 66017}, {938, 12735}, {946, 12653}, {962, 10742}, {1056, 12736}, {1317, 18391}, {1387, 31479}, {1482, 11698}, {1484, 5790}, {1537, 10711}, {1656, 32558}, {1768, 11362}, {1788, 10074}, {2096, 18802}, {2800, 5904}, {2801, 35514}, {2829, 6361}, {3086, 20586}, {3090, 16173}, {3241, 19907}, {3254, 38149}, {3421, 64139}, {3427, 56119}, {3476, 10090}, {3486, 10087}, {3487, 10956}, {3524, 50841}, {3545, 16174}, {3616, 38752}, {3617, 12619}, {3621, 12738}, {3632, 5531}, {3679, 7993}, {3873, 66047}, {3911, 41684}, {3913, 54134}, {4295, 12763}, {4677, 13146}, {5067, 32557}, {5176, 41389}, {5252, 17636}, {5533, 54361}, {5559, 20117}, {5587, 21630}, {5658, 64317}, {5660, 8166}, {5844, 48667}, {5853, 66010}, {5854, 10698}, {5881, 7701}, {5882, 15015}, {5886, 66045}, {6001, 44784}, {6713, 64141}, {6829, 63270}, {6842, 64201}, {6905, 22560}, {6906, 13205}, {6920, 45081}, {6941, 10912}, {6949, 22837}, {7972, 10573}, {7982, 21635}, {9778, 38753}, {9780, 57298}, {9802, 10738}, {9812, 22799}, {9897, 10572}, {9956, 66063}, {10057, 10629}, {10246, 61562}, {10595, 64137}, {10707, 38074}, {10724, 18499}, {10805, 64745}, {10806, 15863}, {11256, 34625}, {11500, 36972}, {11929, 64138}, {12115, 39776}, {12515, 59417}, {12747, 37705}, {12764, 30305}, {13253, 28234}, {13464, 15017}, {13607, 44848}, {16116, 25413}, {16202, 63917}, {17613, 28204}, {17660, 41687}, {18357, 51517}, {18446, 66062}, {19877, 34126}, {20075, 20085}, {22791, 38755}, {22935, 37727}, {24864, 32486}, {25415, 66012}, {28174, 38756}, {31162, 50906}, {31190, 33812}, {33337, 61296}, {33598, 64116}, {33709, 54447}, {33898, 66060}, {34611, 50798}, {34631, 50908}, {37002, 63133}, {37726, 59415}, {37736, 64163}, {37740, 41541}, {38138, 61601}, {38156, 66065}, {38762, 54445}, {39898, 66030}, {41701, 54366}, {44669, 66011}, {50810, 64189}, {50818, 64011}, {58659, 64734}, {63143, 64129}, {64322, 65948}

X(66008) = midpoint of X(i) and X(j) for these {i,j}: {153, 64743}, {3632, 5531}
X(66008) = reflection of X(i) in X(j) for these {i,j}: {4, 12751}, {8, 64140}, {104, 1145}, {145, 6265}, {149, 355}, {944, 100}, {962, 10742}, {1320, 119}, {1482, 11698}, {1768, 11362}, {6224, 12331}, {6264, 10}, {6361, 64136}, {7982, 21635}, {7993, 10265}, {9802, 10738}, {9803, 19914}, {9897, 47745}, {10698, 37725}, {12245, 64056}, {12247, 8}, {12248, 40}, {12653, 946}, {12747, 37705}, {12773, 5690}, {13199, 5541}, {26726, 25485}, {31162, 50906}, {34627, 50907}, {34631, 50908}, {37727, 22935}, {39898, 66030}, {49176, 15863}, {50810, 64746}, {50818, 64011}, {61296, 33337}, {64009, 12515}, {64136, 13996}, {66060, 33898}
X(66008) = anticomplement of X(12737)
X(66008) = X(1539) of 2nd-Conway triangle
X(66008) = X(6264) of outer-Garcia triangle
X(66008) = pole of line {2827, 39534} with respect to the polar circle
X(66008) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {153, 21290, 64743}
X(66008) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(4), X(63162)}}, {{A, B, C, X(517), X(2932)}}, {{A, B, C, X(1000), X(56757)}}, {{A, B, C, X(1809), X(12641)}}, {{A, B, C, X(34234), X(64290)}}
X(66008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 952, 12247}, {8, 9803, 19914}, {100, 952, 944}, {104, 1145, 5657}, {119, 1320, 5603}, {153, 64743, 517}, {355, 10284, 13729}, {528, 50907, 34627}, {952, 1145, 104}, {952, 12331, 6224}, {952, 19914, 9803}, {952, 5690, 12773}, {952, 64140, 8}, {2800, 64056, 12245}, {2802, 12751, 4}, {2829, 13996, 64136}, {2829, 64136, 6361}, {3679, 7993, 10265}, {5660, 26726, 25485}, {5854, 37725, 10698}, {38752, 64742, 3616}, {59417, 64009, 12515}

X(66009) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT ANTIPEDAL-OF-X(9)

Barycentrics    a*(-b^5+b^4*c+b*c^4-c^5+a^4*(b+c)-4*a^2*b*c*(b+c)-2*a^3*(b^2+c^2)+2*a*(b^4+b^3*c-2*b^2*c^2+b*c^3+c^4)) : :
X(66009) = -2*X[3678]+3*X[60986], -X[3869]+3*X[47357], X[3885]+3*X[7672], 3*X[3894]+X[60905], X[3901]+3*X[50836], -X[5696]+3*X[6173], -3*X[10202]+2*X[64113], -X[34784]+3*X[38057], -3*X[38054]+4*X[58607], -3*X[38150]+2*X[65466]

X(66009) lies on these lines: {1, 6}, {7, 7702}, {10, 61030}, {35, 60989}, {65, 528}, {79, 3254}, {142, 3841}, {354, 2886}, {390, 64043}, {497, 3873}, {516, 5884}, {527, 3874}, {912, 54370}, {942, 5880}, {946, 2801}, {971, 16127}, {1071, 5735}, {1210, 41570}, {1376, 1998}, {1445, 37579}, {1479, 61011}, {1818, 21346}, {1858, 11520}, {2078, 41539}, {2550, 5178}, {3035, 61660}, {3059, 3826}, {3338, 56583}, {3485, 3889}, {3678, 60986}, {3742, 5231}, {3811, 8257}, {3812, 64370}, {3868, 5698}, {3869, 47357}, {3870, 33925}, {3885, 7672}, {3892, 64110}, {3893, 41575}, {3894, 60905}, {3901, 50836}, {4654, 11235}, {5045, 28628}, {5083, 25558}, {5439, 41859}, {5536, 10167}, {5542, 11263}, {5686, 10587}, {5696, 6173}, {5705, 58634}, {5709, 11495}, {5732, 12704}, {5851, 66020}, {5852, 15007}, {5853, 30329}, {5887, 62860}, {5927, 41858}, {7675, 26357}, {8543, 63159}, {8545, 62861}, {8581, 41857}, {10198, 16216}, {10202, 64113}, {10222, 66003}, {10391, 54408}, {10527, 11025}, {10529, 11038}, {10569, 41866}, {10902, 65405}, {10943, 20330}, {11012, 65426}, {11020, 24477}, {11281, 17609}, {11510, 41712}, {12005, 43177}, {12109, 52359}, {12564, 24391}, {12669, 55109}, {12672, 65990}, {12675, 41854}, {12711, 41864}, {13369, 43178}, {14100, 16142}, {17660, 66065}, {17781, 49736}, {18406, 49176}, {18499, 52682}, {21617, 26481}, {21620, 31936}, {21746, 24476}, {24299, 52769}, {24386, 58626}, {24473, 28534}, {24475, 65998}, {26363, 58564}, {34195, 53055}, {34772, 64154}, {34784, 38057}, {37615, 54203}, {37787, 41538}, {38054, 58607}, {38150, 65466}, {41865, 58563}, {45230, 62837}, {56176, 66006}, {60968, 65129}, {62859, 66013}, {64155, 65994}, {64197, 64669}, {64264, 65992}

X(66009) = midpoint of X(i) and X(j) for these {i,j}: {2550, 30628}, {3868, 5698}, {5728, 15185}
X(66009) = reflection of X(i) in X(j) for these {i,j}: {72, 15254}, {142, 20116}, {1001, 5572}, {3059, 3826}, {5542, 61033}, {5784, 25557}, {5880, 942}, {25558, 5083}, {43177, 12005}, {43178, 13369}
X(66009) = X(5728) of anti-inner-Yff triangle
X(66009) = pole of line {55, 1004} with respect to the Feuerbach hyperbola
X(66009) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(2911)}}, {{A, B, C, X(79), X(5526)}}, {{A, B, C, X(220), X(43740)}}, {{A, B, C, X(3254), X(52405)}}
X(66009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 10177, 15254}, {354, 5784, 25557}, {518, 15254, 72}, {518, 5572, 1001}, {942, 15733, 5880}, {5696, 18398, 6173}, {5728, 15185, 518}, {11020, 24477, 58578}, {11025, 41228, 38053}

X(66010) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(9) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^8-2*a^7*(b+c)+a^6*(-2*b^2+b*c-2*c^2)-(b-c)^4*(b+c)^2*(b^2+c^2)-4*a^4*b*c*(5*b^2+4*b*c+5*c^2)+a^5*(6*b^3+13*b^2*c+13*b*c^2+6*c^3)+a^2*(b-c)^2*(2*b^4+21*b^3*c+26*b^2*c^2+21*b*c^3+2*c^4)-6*a^3*(b^5-3*b^3*c^2-3*b^2*c^3+c^5)+a*(b-c)^2*(2*b^5-7*b^4*c-15*b^3*c^2-15*b^2*c^3-7*b*c^4+2*c^5)) : :
X(66010) = -2*X[1484]+3*X[38108], -3*X[5686]+X[9803], -2*X[10265]+3*X[38057], -2*X[10707]+3*X[38075], -2*X[10738]+3*X[59389], -2*X[11715]+3*X[64154], -2*X[12737]+3*X[38316], -2*X[18482]+3*X[38755], -2*X[19914]+3*X[59414]

X(66010) lies on circumconic {{A, B, C, X(36101), X(64330)}} and on these lines: {9, 952}, {40, 5851}, {63, 100}, {80, 15298}, {84, 66056}, {104, 6594}, {119, 3254}, {149, 63970}, {153, 516}, {390, 9897}, {518, 6326}, {527, 48363}, {528, 11372}, {971, 2950}, {1001, 6264}, {1484, 38108}, {2800, 64319}, {2802, 43166}, {3243, 6265}, {3895, 37712}, {4321, 10090}, {4326, 10087}, {5686, 9803}, {5728, 37736}, {5731, 60912}, {5735, 5856}, {5805, 11698}, {5853, 66008}, {7972, 15299}, {8545, 20119}, {10265, 38057}, {10269, 22935}, {10698, 54159}, {10707, 38075}, {10738, 59389}, {10742, 52835}, {11715, 64154}, {12247, 24393}, {12248, 63413}, {12737, 38316}, {12738, 52026}, {12773, 31658}, {15296, 64278}, {15297, 61296}, {17660, 41712}, {17768, 66017}, {18482, 38755}, {19914, 59414}, {20095, 36991}, {20195, 38752}, {20400, 38205}, {21630, 38037}, {30500, 34894}, {37587, 41689}, {38122, 61562}, {38139, 61601}, {38159, 66065}, {38665, 64197}, {43175, 51082}, {59388, 61004}, {59418, 64009}

X(66010) = midpoint of X(i) and X(j) for these {i,j}: {5223, 5531}, {20095, 36991}, {38665, 66023}
X(66010) = reflection of X(i) in X(j) for these {i,j}: {84, 66056}, {104, 6594}, {149, 63970}, {3243, 6265}, {3254, 119}, {5528, 12331}, {5732, 100}, {5805, 11698}, {6264, 1001}, {12247, 24393}, {12248, 63413}, {12773, 31658}, {43166, 64765}, {52835, 10742}, {64197, 66023}
X(66010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 2801, 5732}, {104, 6594, 21153}, {119, 3254, 38150}, {971, 12331, 5528}, {2802, 64765, 43166}, {5223, 5531, 2801}

X(66011) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(21) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^9-2*a^8*(b+c)+b*(b-c)^4*c*(b+c)^3-2*a^7*(b^2-b*c+c^2)-a^5*b*c*(7*b^2+b*c+7*c^2)+6*a^6*(b^3+b^2*c+b*c^2+c^3)-a*(b^2-c^2)^2*(b^4+3*b^3*c-b^2*c^2+3*b*c^3+c^4)+a^4*(-6*b^5-5*b^4*c+b^3*c^2+b^2*c^3-5*b*c^4-6*c^5)+2*a^2*(b-c)^2*(b^5+2*b^4*c+2*b*c^4+c^5)+2*a^3*(b^6+4*b^5*c-2*b^3*c^3+4*b*c^5+c^6)) : :
X(66011) = -3*X[5426]+X[7993], -2*X[10738]+3*X[52269], -5*X[31254]+6*X[38752], -X[33557]+4*X[51525], -4*X[46028]+3*X[51517]

X(66011) lies on these lines: {21, 952}, {30, 153}, {72, 74}, {79, 66012}, {104, 15931}, {119, 11604}, {149, 6841}, {191, 5531}, {758, 5535}, {1006, 9803}, {2475, 10942}, {2800, 64280}, {2950, 13146}, {3065, 41166}, {3871, 13743}, {5253, 12009}, {5426, 7993}, {5428, 12773}, {6264, 35016}, {6265, 6583}, {6830, 42843}, {6920, 62354}, {6940, 22935}, {6950, 64313}, {7701, 8715}, {10087, 46816}, {10122, 37736}, {10202, 39778}, {10308, 51897}, {10698, 54161}, {10738, 52269}, {10742, 52841}, {10915, 12751}, {11499, 14450}, {11698, 37230}, {12247, 21677}, {12248, 44238}, {13465, 32141}, {15676, 16202}, {17484, 18524}, {17660, 41697}, {17768, 66007}, {20095, 37433}, {21635, 49177}, {26878, 58692}, {28461, 50907}, {31254, 38752}, {33557, 51525}, {33593, 60782}, {33667, 41701}, {33857, 41541}, {33860, 47032}, {37718, 63288}, {44258, 48680}, {44669, 66008}, {46028, 51517}

X(66011) = midpoint of X(i) and X(j) for these {i,j}: {191, 5531}, {20095, 37433}
X(66011) = reflection of X(i) in X(j) for these {i,j}: {104, 35204}, {149, 6841}, {3651, 100}, {6264, 35016}, {10308, 51897}, {11604, 119}, {12247, 21677}, {12248, 44238}, {12773, 5428}, {33858, 22935}, {34195, 6265}, {37230, 11698}, {48680, 44258}, {49177, 21635}, {52841, 10742}
X(66011) = pole of line {22765, 51420} with respect to the Stammler hyperbola
X(66011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 2771, 3651}, {104, 35204, 21161}

X(66012) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(35) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    (a^3+a^2*(b+c)-(b-c)^2*(b+c)-a*(b^2+b*c+c^2))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2)) : :
X(66012) = -2*X[1387]+3*X[4870]

X(66012) lies on these lines: {1, 153}, {5, 13751}, {10, 11571}, {11, 5045}, {12, 2771}, {30, 41541}, {55, 16128}, {65, 11698}, {79, 66011}, {80, 226}, {100, 1770}, {119, 912}, {214, 535}, {484, 17484}, {495, 17638}, {498, 1768}, {499, 15017}, {651, 56417}, {938, 37718}, {946, 7972}, {952, 11011}, {1145, 44663}, {1155, 61562}, {1317, 12611}, {1387, 4870}, {1478, 6326}, {1479, 37736}, {1484, 17605}, {1519, 25485}, {1836, 12331}, {1837, 38755}, {2800, 10039}, {2801, 8068}, {2829, 33597}, {3035, 3916}, {3085, 9809}, {3241, 18393}, {3452, 64012}, {3584, 29007}, {3585, 34772}, {3612, 12248}, {3614, 12009}, {3649, 6797}, {3822, 47320}, {3925, 58659}, {4187, 58591}, {4299, 15015}, {5080, 39778}, {5083, 39692}, {5219, 66059}, {5249, 6702}, {5252, 48667}, {5270, 45764}, {5531, 9612}, {5541, 34619}, {5570, 58613}, {5660, 10090}, {6224, 31053}, {6260, 10087}, {6265, 12763}, {7354, 22935}, {7951, 10265}, {8232, 51768}, {9803, 10590}, {10052, 66018}, {10058, 63259}, {10523, 17661}, {10572, 10742}, {10698, 12608}, {10738, 41701}, {10827, 12247}, {10895, 62354}, {10956, 12758}, {11375, 12773}, {11551, 12736}, {11813, 33812}, {12609, 59415}, {12647, 13253}, {12738, 13273}, {12743, 22799}, {12750, 59391}, {12767, 31434}, {13405, 63281}, {13601, 37725}, {16173, 21620}, {17100, 59719}, {17636, 39542}, {18480, 33594}, {19925, 53616}, {20118, 61580}, {25415, 66008}, {31272, 51706}, {37707, 64762}, {37730, 61605}, {37731, 46816}, {39991, 61231}, {53537, 56416}, {56419, 63334}, {60988, 66045}, {60992, 66023}

X(66012) = midpoint of X(i) and X(j) for these {i,j}: {3585, 41689}
X(66012) = reflection of X(i) in X(j) for these {i,j}: {3916, 3035}
X(66012) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2990, 19302}, {3065, 36052}, {21739, 32655}
X(66012) = X(i)-Dao conjugate of X(j) for these {i, j}: {119, 3065}
X(66012) = intersection, other than A, B, C, of circumconics {{A, B, C, X(484), X(18838)}}, {{A, B, C, X(1737), X(39991)}}, {{A, B, C, X(17484), X(64115)}}
X(66012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {80, 226, 33593}, {119, 11570, 1737}, {119, 12831, 11570}, {1317, 12611, 30384}, {6265, 12763, 45287}, {11570, 66021, 66014}

X(66013) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT PEDAL-OF-X(46)

Barycentrics    a*(a^8*(b+c)-2*a^7*(b^2+c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)+2*a*(b-c)^4*(b+c)^2*(b^2+b*c+c^2)-2*a^6*(b^3+b^2*c+b*c^2+c^3)+2*a^4*b*c*(b^3+b^2*c+b*c^2+c^3)+a^5*(6*b^4-2*b^3*c+4*b^2*c^2-2*b*c^3+6*c^4)+2*a^2*(b-c)^2*(b^5+b^4*c+b*c^4+c^5)+a^3*(-6*b^6+4*b^5*c+2*b^4*c^2-8*b^3*c^3+2*b^2*c^4+4*b*c^5-6*c^6)) : :

X(66013) lies on these lines: {1, 90}, {65, 5840}, {224, 22766}, {354, 10948}, {405, 14454}, {920, 14054}, {942, 7702}, {946, 5083}, {950, 5884}, {1071, 1479}, {1210, 41540}, {1737, 41559}, {1864, 10523}, {3870, 11508}, {5248, 18232}, {5719, 61559}, {5722, 41688}, {5728, 5880}, {8069, 11517}, {8071, 10391}, {9581, 41703}, {10073, 65994}, {10122, 11019}, {10175, 10395}, {10202, 41552}, {10394, 10629}, {10399, 56583}, {11507, 63437}, {11570, 65998}, {12672, 66003}, {12831, 40263}, {13369, 17437}, {13411, 58415}, {16465, 22836}, {17660, 65991}, {17700, 64341}, {30274, 41865}, {32760, 41538}, {33594, 33667}, {34772, 45393}, {37736, 66018}, {43740, 62864}, {53615, 65134}, {62859, 66009}

X(66013) = midpoint of X(i) and X(j) for these {i,j}: {90, 41685}
X(66013) = reflection of X(i) in X(j) for these {i,j}: {7702, 942}
X(66013) = pole of line {15313, 59973} with respect to the incircle
X(66013) = pole of line {3, 7702} with respect to the Feuerbach hyperbola
X(66013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {90, 41685, 912}

X(66014) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(46) WRT EXTOUCH-OF-FUHRMANN

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

X(66014) lies on these lines: {1, 18254}, {4, 80}, {78, 51506}, {90, 12775}, {100, 920}, {119, 912}, {498, 46694}, {499, 5083}, {519, 64139}, {942, 38182}, {1210, 8068}, {1420, 6326}, {1445, 2801}, {1770, 41560}, {3811, 45393}, {5533, 49627}, {5570, 17533}, {5840, 41538}, {6594, 35204}, {6702, 18389}, {10057, 10941}, {10072, 18412}, {10073, 12649}, {10087, 14740}, {10391, 58666}, {12532, 18391}, {12736, 31164}, {12758, 23340}, {13750, 34122}, {17660, 66051}, {35976, 54432}, {39776, 41684}, {41562, 46684}, {50195, 58659}, {63399, 64188}

X(66014) = midpoint of X(i) and X(j) for these {i,j}: {10073, 41686}
X(66014) = reflection of X(i) in X(j) for these {i,j}: {11570, 12832}, {12758, 64042}
X(66014) = X(i)-Ceva conjugate of X(j) for these {i, j}: {4564, 61239}
X(66014) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(11570)}}, {{A, B, C, X(80), X(912)}}, {{A, B, C, X(18838), X(32760)}}, {{A, B, C, X(41552), X(55126)}}
X(66014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 66016, 11570}, {11570, 66021, 66012}

X(66015) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(57) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a+b-c)*(a-b+c)*(a^2+b^2+b*c+c^2-2*a*(b+c))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2)) : :

X(66015) lies on these lines: {2, 5083}, {11, 5173}, {56, 12738}, {57, 2801}, {65, 546}, {100, 1708}, {119, 912}, {518, 41556}, {528, 41539}, {952, 64106}, {1617, 41701}, {1830, 10772}, {2078, 3935}, {2800, 18391}, {2802, 64736}, {3256, 41166}, {3681, 14151}, {3940, 12739}, {4511, 41554}, {7672, 10707}, {9809, 45043}, {10090, 12757}, {11219, 18412}, {12691, 12736}, {12755, 54366}, {14740, 37736}, {15733, 25606}, {17660, 61653}, {18254, 54318}, {41558, 64139}, {65986, 66046}

X(66015) = reflection of X(i) in X(j) for these {i,j}: {11, 64157}
X(66015) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3254, 36052}, {37143, 61214}, {42064, 63190}
X(66015) = X(i)-Dao conjugate of X(j) for these {i, j}: {119, 3254}
X(66015) = intersection, other than A, B, C, of circumconics {{A, B, C, X(912), X(3887)}}, {{A, B, C, X(1737), X(3935)}}, {{A, B, C, X(2078), X(18838)}}, {{A, B, C, X(6594), X(12831)}}, {{A, B, C, X(11570), X(52456)}}, {{A, B, C, X(12832), X(41553)}}, {{A, B, C, X(37787), X(64115)}}, {{A, B, C, X(55126), X(61030)}}
X(66015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11570, 66014, 12665}, {11570, 66021, 12831}, {12832, 66016, 11570}

X(66016) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(65) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a+b-c)*(a-b+c)*(a^3+b^3+b^2*c+b*c^2+c^3-a^2*(b+c)-a*(b^2+b*c+c^2))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2)) : :
X(66016) = -2*X[12736]+3*X[61663]

X(66016) lies on these lines: {11, 12691}, {12, 8261}, {56, 6326}, {65, 79}, {72, 12739}, {100, 7098}, {119, 912}, {201, 64710}, {226, 47320}, {517, 12743}, {518, 1317}, {758, 41558}, {942, 8068}, {952, 13292}, {1071, 64188}, {1125, 5083}, {1388, 5692}, {1768, 11509}, {1858, 2800}, {1864, 12764}, {1898, 34789}, {2099, 13253}, {2801, 52819}, {3555, 20586}, {5172, 33667}, {5221, 15096}, {5432, 58666}, {5433, 58591}, {5531, 37550}, {5660, 37566}, {5727, 52860}, {5777, 15094}, {5904, 11510}, {6001, 52836}, {7702, 12528}, {9964, 60782}, {10073, 24474}, {10265, 18389}, {10698, 64042}, {10956, 24987}, {12432, 41551}, {12532, 29007}, {12619, 13750}, {12736, 61663}, {12763, 14872}, {13751, 37701}, {14882, 17637}, {17636, 64278}, {17661, 18961}, {18962, 45638}, {37579, 41685}, {37583, 41689}, {40269, 64009}, {41389, 41554}, {41537, 45393}, {54065, 64040}

X(66016) = midpoint of X(i) and X(j) for these {i,j}: {100, 64715}
X(66016) = reflection of X(i) in X(j) for these {i,j}: {11, 44547}, {15094, 5777}
X(66016) = X(i)-isoconjugate-of-X(j) for these {i, j}: {11604, 36052}, {61214, 65238}
X(66016) = X(i)-Dao conjugate of X(j) for these {i, j}: {119, 11604}
X(66016) = pole of line {1532, 3583} with respect to the Feuerbach hyperbola
X(66016) = intersection, other than A, B, C, of circumconics {{A, B, C, X(79), X(11570)}}, {{A, B, C, X(265), X(912)}}, {{A, B, C, X(1737), X(2166)}}, {{A, B, C, X(5172), X(18838)}}, {{A, B, C, X(12832), X(41541)}}
X(66016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1737, 11570, 66047}, {11570, 12665, 12831}, {11570, 12832, 18838}, {11570, 66014, 119}, {11570, 66015, 12832}

X(66017) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(79) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^10-a^9*(b+c)-(b-c)^6*(b+c)^4+a^8*(-3*b^2+8*b*c-3*c^2)+a*(b-c)^4*(b+c)^3*(b^2-5*b*c+c^2)+2*a^7*(b^3-4*b^2*c-4*b*c^2+c^3)+a^5*b*c*(13*b^3-6*b^2*c-6*b*c^2+13*c^3)+3*a^2*(b^2-c^2)^2*(b^4+b^3*c-2*b^2*c^2+b*c^3+c^4)+a^6*(4*b^4-9*b^3*c+19*b^2*c^2-9*b*c^3+4*c^4)-a^3*(b-c)^2*(2*b^5+2*b^4*c-9*b^3*c^2-9*b^2*c^3+2*b*c^4+2*c^5)-a^4*(4*b^6+4*b^5*c+7*b^4*c^2-18*b^3*c^3+7*b^2*c^4+4*b*c^5+4*c^6) : :
X(66017) = -3*X[6175]+2*X[10265], -5*X[15017]+4*X[16617], -2*X[22798]+3*X[38755], -2*X[33856]+3*X[38752]

X(66017) lies on these lines: {30, 5538}, {79, 952}, {80, 16152}, {100, 16113}, {104, 51569}, {119, 3065}, {149, 16125}, {758, 66008}, {1768, 37401}, {2475, 5884}, {2771, 12751}, {2886, 66059}, {3649, 6264}, {3652, 11698}, {4301, 10698}, {5441, 6265}, {5531, 16118}, {5660, 38722}, {5690, 12767}, {6175, 10265}, {6923, 15096}, {7972, 16153}, {9809, 31806}, {11604, 12757}, {11827, 63267}, {12773, 49107}, {15017, 16617}, {16139, 37725}, {17768, 66010}, {21635, 21669}, {22798, 38755}, {33856, 38752}, {49176, 56790}, {50834, 66023}, {53616, 65994}

X(66017) = midpoint of X(i) and X(j) for these {i,j}: {5531, 16118}
X(66017) = reflection of X(i) in X(j) for these {i,j}: {104, 51569}, {149, 16125}, {1768, 37401}, {3065, 119}, {3652, 11698}, {5441, 6265}, {6264, 3649}, {12773, 49107}, {16113, 100}, {21669, 21635}, {49176, 56790}, {49177, 47034}

X(66018) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(90) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^12-2*a^11*(b+c)+a^10*(-4*b^2+5*b*c-4*c^2)-(b-c)^6*(b+c)^4*(b^2+c^2)+a^9*(10*b^3+9*b^2*c+9*b*c^2+10*c^3)+a*(b-c)^4*(b+c)^3*(2*b^4-9*b^3*c+2*b^2*c^2-9*b*c^3+2*c^4)+a^8*(5*b^4-30*b^3*c-2*b^2*c^2-30*b*c^3+5*c^4)+2*a^6*b*c*(29*b^4-8*b^3*c+8*b^2*c^2-8*b*c^3+29*c^4)-4*a^7*(5*b^5+b^4*c-3*b^3*c^2-3*b^2*c^3+b*c^4+5*c^5)+a^2*(b^2-c^2)^2*(4*b^6+9*b^5*c-28*b^4*c^2+22*b^3*c^3-28*b^2*c^4+9*b*c^5+4*c^6)-2*a^3*(b-c)^2*(5*b^7-5*b^6*c-25*b^5*c^2-17*b^4*c^3-17*b^3*c^4-25*b^2*c^5-5*b*c^6+5*c^7)+2*a^5*(10*b^7-11*b^6*c-23*b^5*c^2+6*b^4*c^3+6*b^3*c^4-23*b^2*c^5-11*b*c^6+10*c^7)+a^4*(-5*b^8-44*b^7*c+56*b^6*c^2+16*b^5*c^3-30*b^4*c^4+16*b^3*c^5+56*b^2*c^6-44*b*c^7-5*c^8)) : :

X(66018) lies on these lines: {36, 912}, {90, 952}, {100, 1158}, {153, 63136}, {944, 18232}, {1768, 63752}, {3927, 15094}, {5541, 5691}, {6261, 12532}, {6264, 62333}, {10052, 66012}, {11698, 41688}, {37736, 66013}

X(66018) = reflection of X(i) in X(j) for these {i,j}: {6264, 62333}, {41688, 11698}

X(66019) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT ANTICEVIAN-OF-X(104)

Barycentrics    a*(a^5*(b+c)-a^4*(b^2+c^2)-(b^2-c^2)^2*(b^2-b*c+c^2)+a^2*(b+c)^2*(2*b^2-5*b*c+2*c^2)+a^3*(-2*b^3+3*b^2*c+3*b*c^2-2*c^3)+a*(b-c)^2*(b^3-2*b^2*c-2*b*c^2+c^3)) : :
X(66019) = -3*X[10]+2*X[5777], -3*X[165]+X[3869], -3*X[551]+4*X[9940], -2*X[946]+3*X[5883], -2*X[960]+3*X[10164], -X[962]+3*X[5902], -4*X[1385]+3*X[3898], -2*X[1482]+3*X[3892], -X[3057]+3*X[10167], -3*X[3576]+2*X[3884], -2*X[3678]+3*X[5657], -3*X[3679]+X[12528]

X(66019) lies on these lines: {1, 1106}, {3, 214}, {4, 3754}, {5, 13145}, {8, 2801}, {10, 5777}, {20, 5903}, {30, 35004}, {40, 758}, {46, 64150}, {65, 516}, {72, 43174}, {73, 45269}, {80, 37437}, {103, 65364}, {104, 11014}, {165, 3869}, {191, 12767}, {355, 47032}, {388, 60896}, {411, 484}, {515, 37562}, {517, 550}, {518, 12640}, {519, 1071}, {551, 9940}, {581, 4868}, {601, 63292}, {912, 11362}, {942, 4301}, {944, 2802}, {946, 5883}, {960, 10164}, {962, 5902}, {971, 5836}, {991, 37598}, {993, 1158}, {997, 7971}, {1012, 30147}, {1125, 12672}, {1329, 21635}, {1385, 3898}, {1388, 15558}, {1479, 12736}, {1482, 3892}, {1490, 54286}, {1519, 3825}, {1709, 19860}, {1768, 2975}, {1858, 4848}, {2077, 21740}, {2093, 12432}, {2098, 5083}, {2099, 37022}, {2646, 17613}, {2771, 5690}, {2809, 43163}, {2886, 33899}, {2951, 7672}, {3057, 10167}, {3256, 45230}, {3340, 10860}, {3359, 6261}, {3474, 64075}, {3486, 64076}, {3576, 3884}, {3579, 14988}, {3626, 14872}, {3671, 50195}, {3678, 5657}, {3679, 12528}, {3680, 9845}, {3698, 5927}, {3740, 31821}, {3746, 18444}, {3753, 12688}, {3812, 3817}, {3814, 12608}, {3833, 8227}, {3868, 3895}, {3872, 10085}, {3873, 11531}, {3876, 9588}, {3877, 7987}, {3881, 7982}, {3889, 11224}, {3890, 30389}, {3899, 16192}, {3901, 63468}, {3911, 64042}, {3918, 5587}, {3919, 7686}, {3968, 5818}, {4018, 7957}, {4067, 63976}, {4134, 58643}, {4300, 4424}, {4315, 64132}, {4342, 50196}, {4511, 59326}, {4691, 18908}, {4757, 6361}, {4853, 30304}, {4973, 11249}, {5046, 34789}, {5119, 10884}, {5123, 64813}, {5250, 52769}, {5261, 30290}, {5267, 64118}, {5330, 13253}, {5433, 17638}, {5443, 6972}, {5445, 6960}, {5450, 51111}, {5534, 63132}, {5537, 34772}, {5538, 62830}, {5584, 11517}, {5603, 15016}, {5691, 9961}, {5694, 61524}, {5697, 5731}, {5734, 50190}, {5881, 64358}, {5885, 22791}, {5887, 6684}, {5904, 59417}, {6244, 12635}, {6256, 64745}, {6702, 6941}, {6735, 12059}, {6736, 41561}, {6769, 12559}, {6842, 64763}, {6882, 64762}, {6922, 11813}, {6925, 10573}, {6932, 18395}, {6943, 18393}, {7330, 64733}, {7580, 15556}, {7680, 11263}, {7992, 9623}, {7995, 54370}, {8666, 63399}, {9778, 20612}, {9785, 18419}, {9949, 63970}, {9957, 58567}, {10087, 11010}, {10106, 64704}, {10107, 15726}, {10165, 40296}, {10175, 31937}, {10202, 13464}, {10269, 51714}, {10273, 28150}, {10283, 26200}, {10310, 22836}, {10391, 13601}, {10571, 24025}, {10624, 64045}, {10866, 17626}, {10912, 30283}, {10914, 12680}, {11220, 14923}, {11227, 58679}, {11496, 30143}, {11529, 12564}, {11826, 17654}, {11849, 33858}, {12053, 18838}, {12247, 12255}, {12512, 14110}, {12514, 30503}, {12616, 25639}, {12664, 17646}, {12678, 64087}, {12684, 40587}, {12699, 31870}, {12705, 54318}, {12709, 13405}, {12758, 21842}, {13257, 21031}, {13528, 33597}, {13600, 51071}, {13607, 23340}, {13752, 31849}, {14647, 26363}, {15049, 58487}, {15528, 64137}, {15623, 53252}, {16189, 62854}, {16209, 35262}, {17170, 59813}, {17649, 40290}, {17768, 31799}, {18221, 41861}, {18357, 31828}, {18397, 37421}, {18412, 64696}, {18417, 37402}, {18421, 62864}, {18481, 25413}, {20070, 43161}, {20116, 43166}, {20117, 26446}, {21616, 54198}, {22793, 61541}, {24474, 28194}, {24728, 31785}, {25439, 49163}, {25917, 58441}, {26492, 32557}, {29057, 44039}, {31777, 44669}, {31793, 44663}, {31835, 50821}, {35000, 37733}, {35242, 63915}, {36279, 64077}, {37529, 63354}, {37531, 62822}, {37568, 45288}, {37569, 62860}, {37725, 41543}, {38112, 56762}, {41389, 59587}, {47319, 64044}, {50031, 51409}, {54295, 54400}, {56288, 59320}, {59333, 63986}, {64056, 66002}

X(66019) = midpoint of X(i) and X(j) for these {i,j}: {8, 15071}, {20, 5903}, {40, 64021}, {2951, 7672}, {3868, 7991}, {4018, 7957}, {4084, 5493}, {5691, 9961}, {5881, 64358}, {6361, 37625}, {10914, 12680}, {11571, 64189}, {18412, 64696}, {18481, 25413}, {64056, 66002}
X(66019) = reflection of X(i) in X(j) for these {i,j}: {4, 3754}, {5, 13145}, {10, 31788}, {72, 43174}, {946, 34339}, {960, 31787}, {1482, 12005}, {1483, 26201}, {3244, 12675}, {3874, 5884}, {3878, 3}, {4067, 63976}, {4297, 9943}, {4301, 942}, {5693, 3678}, {5694, 61524}, {5882, 13369}, {5887, 6684}, {7982, 3881}, {9856, 3812}, {9957, 58567}, {12672, 1125}, {12688, 19925}, {12699, 31870}, {14110, 12512}, {14872, 3626}, {22791, 5885}, {22793, 61541}, {23340, 13607}, {31803, 10}, {31806, 3579}, {31828, 18357}, {31849, 13752}, {31871, 3918}, {37625, 4757}, {40266, 20117}, {43166, 20116}, {45776, 9940}, {51118, 7686}, {61705, 3968}, {63967, 5690}, {64137, 15528}
X(66019) = X(3754) of anti-Euler triangle
X(66019) = X(4297) of inner-Garcia triangle
X(66019) = pole of line {226, 20323} with respect to the Feuerbach hyperbola
X(66019) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {8, 15071, 38507}
X(66019) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6909), X(44040)}}, {{A, B, C, X(63983), X(65952)}}
X(66019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 64129, 63983}, {3, 2800, 3878}, {3, 40257, 214}, {8, 15071, 2801}, {10, 31803, 15064}, {10, 6001, 31803}, {40, 16132, 11491}, {40, 18446, 8715}, {40, 64021, 758}, {65, 12711, 6738}, {517, 12675, 3244}, {517, 13369, 5882}, {517, 26201, 1483}, {517, 5884, 3874}, {517, 9943, 4297}, {946, 34339, 5883}, {960, 31787, 10164}, {1482, 12005, 3892}, {2771, 5690, 63967}, {3359, 6261, 25440}, {3579, 14988, 31806}, {3753, 12688, 19925}, {3812, 9856, 3817}, {3918, 31871, 5587}, {3919, 51118, 7686}, {4084, 5493, 517}, {5603, 15016, 58565}, {6001, 31788, 10}, {7995, 64673, 54370}, {9940, 45776, 551}, {10914, 12680, 28236}, {26446, 40266, 20117}, {46684, 51717, 3}

X(66020) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT PEDAL-OF-X(165)

Barycentrics    a*(a^7*(b+c)+a^5*(b-c)^2*(b+c)+a^6*(-3*b^2+2*b*c-3*c^2)-(b^2-c^2)^4+a*(b-c)^2*(b+c)^3*(3*b^2-10*b*c+3*c^2)-a^3*(b-c)^2*(5*b^3+3*b^2*c+3*b*c^2+5*c^3)-a^2*(b-c)^2*(b^4-8*b^3*c-10*b^2*c^2-8*b*c^3+c^4)+a^4*(5*b^4-12*b^3*c+22*b^2*c^2-12*b*c^3+5*c^4)) : :
X(66020) = -5*X[5439]+4*X[64830], -3*X[5728]+2*X[5884], -3*X[5817]+2*X[15587], -3*X[6172]+2*X[63976], -3*X[7671]+2*X[12675], -3*X[21151]+4*X[58608], -2*X[31786]+3*X[50836]

X(66020) lies on these lines: {1, 971}, {4, 3812}, {7, 10309}, {9, 10310}, {40, 54135}, {55, 52684}, {210, 5537}, {382, 31788}, {516, 11827}, {517, 60905}, {946, 38055}, {997, 1012}, {1071, 6744}, {1156, 66055}, {1158, 5729}, {1466, 3358}, {1519, 25557}, {1532, 64113}, {1699, 3660}, {1709, 1864}, {1768, 61660}, {2801, 3244}, {2951, 37411}, {3059, 5779}, {3149, 43178}, {4312, 52860}, {5439, 64830}, {5572, 36996}, {5696, 5777}, {5698, 14110}, {5704, 30287}, {5728, 5884}, {5732, 37252}, {5805, 7702}, {5817, 15587}, {5851, 66009}, {5918, 19541}, {5927, 6745}, {6001, 10394}, {6172, 63976}, {6223, 12710}, {6700, 64699}, {6765, 12705}, {6769, 36973}, {6831, 10427}, {6906, 15254}, {7080, 25722}, {7330, 42014}, {7671, 12675}, {7681, 30379}, {8544, 22753}, {8545, 11496}, {8727, 17603}, {9843, 43182}, {9844, 9948}, {9940, 38107}, {10157, 41866}, {10177, 11263}, {10241, 11227}, {10391, 64130}, {10863, 17612}, {12666, 45776}, {12667, 36991}, {12671, 37434}, {12848, 64190}, {13600, 40266}, {13601, 18412}, {17650, 64163}, {17668, 63970}, {17768, 54145}, {21151, 58608}, {21628, 63257}, {21669, 64765}, {28160, 64332}, {31658, 59326}, {31786, 50836}, {31870, 66050}, {34789, 65986}, {37787, 64118}, {38036, 58576}, {40263, 66006}, {41560, 63266}, {60953, 64669}, {60992, 64658}, {63432, 63992}, {64155, 65987}

X(66020) = reflection of X(i) in X(j) for these {i,j}: {2951, 51489}, {3059, 5779}, {5696, 5777}, {5784, 54370}, {14110, 5698}, {14872, 64197}, {17668, 63970}, {31391, 5805}, {36996, 5572}
X(66020) = X(8548) of Ursa-minor triangle
X(66020) = pole of line {3900, 30235} with respect to the incircle
X(66020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15733, 64197, 14872}, {43177, 63973, 63989}

X(66021) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(165) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^4+7*a^2*b*c-2*a^3*(b+c)-(b-c)^2*(b^2+3*b*c+c^2)+2*a*(b^3-2*b^2*c-2*b*c^2+c^3))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2)) : :
X(66021) = X[72]+2*X[38757], X[153]+2*X[18254], -X[1071]+4*X[20400], -X[1145]+4*X[58687], X[2950]+2*X[34293], -2*X[5083]+5*X[15017], 2*X[9856]+X[13996], -4*X[9947]+X[62616], -4*X[10156]+5*X[31235], -X[12515]+4*X[58674], -X[12757]+4*X[66051], -2*X[13369]+5*X[38763] and many others

X(66021) lies on these lines: {2, 2801}, {11, 10157}, {72, 38757}, {100, 1709}, {119, 912}, {153, 18254}, {329, 14740}, {517, 50842}, {528, 5927}, {952, 5919}, {971, 6174}, {1071, 20400}, {1145, 58687}, {1768, 8580}, {2800, 3679}, {2802, 59387}, {2829, 64107}, {2950, 34293}, {3035, 10167}, {3219, 46684}, {3560, 12738}, {3678, 37437}, {4847, 21635}, {4915, 13253}, {5080, 64139}, {5083, 15017}, {5226, 18240}, {5531, 10382}, {5537, 60935}, {5777, 10039}, {5851, 61028}, {6326, 13384}, {6842, 56762}, {6893, 49176}, {6940, 64693}, {6941, 63967}, {9856, 13996}, {9947, 62616}, {10072, 61718}, {10156, 31235}, {10590, 12736}, {11678, 41338}, {12515, 58674}, {12647, 12751}, {12757, 66051}, {13257, 38211}, {13369, 38763}, {15528, 64008}, {21165, 64188}, {37712, 50907}, {41704, 59591}, {60782, 66023}, {64745, 66024}, {66002, 66045}

X(66021) = midpoint of X(i) and X(j) for these {i,j}: {10167, 17661}, {15104, 34789}
X(66021) = reflection of X(i) in X(j) for these {i,j}: {11, 10157}, {10167, 3035}, {15104, 14740}
X(66021) = intersection, other than A, B, C, of circumconics {{A, B, C, X(5537), X(18838)}}, {{A, B, C, X(60935), X(64115)}}
X(66021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 12665, 11570}, {119, 66049, 12665}, {5531, 30326, 51768}, {13227, 46694, 1768}

X(66022) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT PEDAL-OF-X(171)

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

X(66022) lies on these lines: {1, 256}, {65, 2783}, {79, 4014}, {181, 5143}, {946, 23821}, {1401, 33097}, {3736, 20470}, {5880, 6007}, {5884, 29057}, {11263, 49676}, {15488, 24851}, {17637, 29301}, {20864, 50595}, {24210, 50362}, {50605, 51575}, {64119, 64122}, {64751, 64753}

X(66022) = pole of line {45902, 48005} with respect to the Brocard inellipse
X(66022) = pole of line {16696, 30097} with respect to the dual conic of Yff parabola

X(66023) = ORTHOLOGY CENTER OF THESE TRIANGLES: X(1)-CIRCUMCONCEVIAN-OF-X(9) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^8-a^7*(b+c)+a^6*(-5*b^2+3*b*c-5*c^2)-(b-c)^4*(b+c)^2*(2*b^2+3*b*c+2*c^2)+a^5*(7*b^3+9*b^2*c+9*b*c^2+7*c^3)+5*a^4*(b^4-5*b^3*c-5*b*c^3+c^4)+a^2*(b-c)^2*(b^4+23*b^3*c+28*b^2*c^2+23*b*c^3+c^4)+a^3*(-11*b^5+5*b^4*c+14*b^3*c^2+14*b^2*c^3+5*b*c^4-11*c^5)+a*(b-c)^2*(5*b^5-3*b^4*c-18*b^3*c^2-18*b^2*c^3-3*b*c^4+5*c^5)) : :
X(66023) = -2*X[11]+3*X[5817], -4*X[142]+5*X[64008], -4*X[3035]+3*X[21151], -7*X[3090]+6*X[38205], -2*X[3254]+3*X[59391], -4*X[6713]+5*X[18230], -3*X[11038]+4*X[11729], -X[12248]+3*X[21168], -X[12653]+3*X[24644]

X(66023) lies on circumconic {{A, B, C, X(909), X(53911)}} and on these lines: {4, 5856}, {7, 119}, {9, 48}, {11, 5817}, {100, 971}, {142, 64008}, {144, 153}, {355, 20119}, {390, 952}, {392, 38669}, {480, 12332}, {516, 10728}, {517, 56551}, {518, 10698}, {527, 1512}, {528, 16112}, {912, 12755}, {1317, 60910}, {2096, 25606}, {2800, 5223}, {2802, 11372}, {2829, 5759}, {2950, 2951}, {3035, 21151}, {3062, 5541}, {3090, 38205}, {3254, 59391}, {3868, 66054}, {3911, 5660}, {4326, 66062}, {5083, 10398}, {5220, 5657}, {5316, 11219}, {5531, 41166}, {5732, 6594}, {5762, 10742}, {5785, 46694}, {5840, 36991}, {5843, 11698}, {5845, 66030}, {5850, 21635}, {6713, 18230}, {7993, 15558}, {8232, 38055}, {10031, 31156}, {10427, 36996}, {10724, 31672}, {10738, 60901}, {11038, 11729}, {12115, 60940}, {12248, 21168}, {12331, 60884}, {12653, 24644}, {12736, 60937}, {12737, 29007}, {12763, 60883}, {12764, 60919}, {12773, 51516}, {12775, 34894}, {12831, 54366}, {14151, 19907}, {14217, 63973}, {14872, 64173}, {15015, 64697}, {15017, 59372}, {15587, 58687}, {21630, 64699}, {22758, 60944}, {22799, 31671}, {30330, 46681}, {31272, 38108}, {31657, 38752}, {31658, 38693}, {38107, 61580}, {38124, 58421}, {38602, 59381}, {38665, 64197}, {38755, 60922}, {38761, 59418}, {39692, 60924}, {41389, 60935}, {43166, 54135}, {44848, 64830}, {45043, 60934}, {50834, 66017}, {52684, 64150}, {57298, 61511}, {60782, 66021}, {60957, 66052}, {60961, 64155}, {60966, 64139}, {60992, 66012}, {61006, 64009}, {62778, 66045}, {63346, 63384}

X(66023) = midpoint of X(i) and X(j) for these {i,j}: {144, 153}, {3062, 5541}, {12331, 60884}, {64197, 66010}
X(66023) = reflection of X(i) in X(j) for these {i,j}: {7, 119}, {104, 9}, {1156, 5779}, {2096, 25606}, {3254, 63970}, {3868, 66054}, {5732, 6594}, {5759, 6068}, {10698, 64765}, {10724, 31672}, {10738, 60901}, {14217, 63973}, {15587, 58687}, {20119, 355}, {21630, 64699}, {31671, 22799}, {35514, 1145}, {36996, 10427}, {38665, 66010}, {66007, 37725}
X(66023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 2801, 104}, {952, 5779, 1156}, {2829, 6068, 5759}, {3254, 63970, 59391}, {5732, 6594, 34474}, {5851, 37725, 66007}, {10759, 66057, 10698}, {38124, 58421, 60996}

X(66024) = ORTHOLOGY CENTER OF THESE TRIANGLES: X(9)-CIRCUMCONCEVIAN-OF-X(8) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^8*(b+c)-(b-c)^4*(b+c)^3*(b^2+c^2)-a^7*(2*b^2+3*b*c+2*c^2)-2*a^6*(b^3-3*b^2*c-3*b*c^2+c^3)-2*a^4*b*c*(7*b^3-5*b^2*c-5*b*c^2+7*c^3)+a*(b^2-c^2)^2*(2*b^4-7*b^3*c+7*b^2*c^2-7*b*c^3+2*c^4)+a^5*(6*b^4-b^3*c-9*b^2*c^2-b*c^3+6*c^4)+2*a^2*(b-c)^2*(b^5+5*b^4*c+5*b*c^4+c^5)+a^3*(-6*b^6+11*b^5*c+8*b^4*c^2-22*b^3*c^3+8*b^2*c^4+11*b*c^5-6*c^6)) : :
X(66024) = -4*X[960]+3*X[38693], -5*X[3091]+4*X[12736], -5*X[3616]+4*X[15528], -3*X[3873]+4*X[64192], -5*X[3876]+4*X[64193], -3*X[3877]+2*X[64191], -3*X[5692]+2*X[46684], -3*X[5927]+2*X[6797], -2*X[6246]+3*X[61705]

X(66024) lies on circumconic {{A, B, C, X(2745), X(46435)}} and on these lines: {1, 66002}, {4, 66044}, {8, 153}, {20, 64139}, {21, 104}, {72, 64189}, {78, 2950}, {80, 31803}, {100, 2745}, {119, 25005}, {214, 15071}, {390, 2801}, {517, 10728}, {758, 34789}, {912, 10698}, {952, 3885}, {960, 38693}, {997, 1768}, {1158, 4855}, {1320, 12672}, {1537, 3868}, {1697, 66061}, {1737, 11571}, {2829, 3869}, {3086, 11570}, {3091, 12736}, {3616, 15528}, {3873, 64192}, {3876, 64193}, {3877, 64191}, {3878, 64145}, {4511, 48695}, {4996, 6261}, {5086, 12761}, {5660, 18231}, {5692, 46684}, {5694, 12515}, {5777, 17654}, {5811, 12691}, {5927, 6797}, {6224, 64120}, {6246, 61705}, {6907, 11698}, {9588, 58698}, {9803, 10073}, {9961, 24466}, {10724, 12688}, {10742, 14988}, {12332, 38901}, {12531, 14872}, {12617, 33593}, {12755, 64765}, {12764, 45288}, {12775, 34772}, {14110, 63280}, {14740, 59417}, {17638, 20586}, {18446, 65739}, {18861, 45770}, {22799, 64044}, {27131, 32554}, {31788, 64141}, {31937, 59391}, {34339, 64008}, {37562, 66049}, {56288, 64188}, {64745, 66021}

X(66024) = reflection of X(i) in X(j) for these {i,j}: {4, 66044}, {8, 12665}, {20, 64139}, {80, 31803}, {104, 5887}, {1320, 12672}, {3868, 1537}, {9961, 24466}, {10724, 12688}, {11571, 21635}, {12515, 5694}, {12531, 14872}, {12532, 5693}, {12755, 64765}, {15071, 214}, {17654, 5777}, {37562, 66049}, {38669, 17638}, {64021, 119}, {64044, 22799}, {64145, 3878}, {64189, 72}, {66002, 1}
X(66024) = pole of line {38693, 61649} with respect to the Feuerbach hyperbola
X(66024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1158, 6326, 17100}, {2771, 5887, 104}, {2800, 12665, 8}, {2800, 5693, 12532}

X(66025) = PARALLELOGIC CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT ANTICEVIAN-OF-X(11)

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

X(66025) lies on these lines: {4, 2826}, {514, 11247}, {523, 3743}, {659, 20831}, {918, 3874}, {942, 52305}, {2804, 3913}, {12607, 55133}, {15171, 53578}, {16126, 49276}, {23887, 63800}, {30591, 31936}, {45061, 55137}

X(66025) = pole of line {2775, 5540} with respect to the Moses-Feuerbach circumconic

X(66026) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTICEVIAN-OF-X(11) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a-b-c)*(b-c)*(a^6-2*a^5*(b+c)-2*a^3*b*c*(b+c)+a^4*(b^2+4*b*c+c^2)+2*a*(b-c)^2*(b^3+c^3)-(b-c)^2*(b^4+c^4)-a^2*(b^4-2*b^3*c+b^2*c^2-2*b*c^3+c^4)) : :
X(66026) = -2*X[11]+3*X[11193], -3*X[16173]+4*X[32195]

X(66026) lies on these lines: {11, 11193}, {80, 11247}, {100, 650}, {149, 40166}, {513, 17660}, {528, 15914}, {654, 38325}, {3309, 34789}, {3738, 53523}, {11927, 13271}, {11934, 13274}, {16173, 32195}

X(66026) = reflection of X(i) in X(j) for these {i,j}: {11, 66064}, {80, 11247}, {42552, 11}
X(66026) = X(i)-Dao conjugate of X(j) for these {i, j}: {1252, 31615}
X(66026) = X(i)-Ceva conjugate of X(j) for these {i, j}: {149, 11}, {40166, 650}
X(66026) = pole of line {1090, 5532} with respect to the Feuerbach hyperbola
X(66026) = pole of line {21201, 63793} with respect to the Moses-Feuerbach circumconic
X(66026) = intersection, other than A, B, C, of circumconics {{A, B, C, X(11), X(5375)}}, {{A, B, C, X(1252), X(42552)}}
X(66026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 66064, 11193}, {11193, 42552, 11}

X(66027) = PARALLELOGIC CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT X(37)-ANTIPEDAL-OF-X(100)

Barycentrics    a*(4*a^2*b*c*(b+c)-b*(b-c)^2*c*(b+c)+2*a^3*(b^2+b*c+c^2)-a*(2*b^4+b^3*c-4*b^2*c^2+b*c^3+2*c^4)) : :
X(66027) = -2*X[3678]+3*X[49731], -X[3869]+3*X[49740], X[3901]+3*X[50296], -X[4067]+3*X[50297], -X[5904]+3*X[17330], -3*X[17392]+5*X[18398], -3*X[49738]+4*X[58565], 3*X[49746]+X[64047]

X(66027) lies on these lines: {1, 2245}, {10, 9054}, {65, 528}, {511, 13476}, {518, 3686}, {524, 3874}, {594, 38485}, {674, 942}, {3664, 9047}, {3678, 49731}, {3779, 3826}, {3869, 49740}, {3901, 50296}, {4067, 50297}, {4259, 25557}, {4260, 64524}, {5904, 17330}, {16678, 63393}, {17061, 40952}, {17392, 18398}, {17768, 21746}, {20718, 39543}, {49738, 58565}, {49746, 64047}, {64751, 66071}

X(66028) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ANTI-CONWAY WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^13-a^12*(b+c)+a^11*(-4*b^2+5*b*c-4*c^2)-(b-c)^6*(b+c)^5*(b^2+c^2)+4*a^10*(b^3+c^3)+a*(b^2-c^2)^4*(b^4-4*b^3*c+2*b^2*c^2-4*b*c^3+c^4)+4*a^2*(b-c)^4*(b+c)^3*(b^4+b^3*c+b^2*c^2+b*c^3+c^4)+a^9*(7*b^4-12*b^3*c+14*b^2*c^2-12*b*c^3+7*c^4)-a^4*(b-c)^2*(b+c)^3*(7*b^4-4*b^3*c+10*b^2*c^2-4*b*c^3+7*c^4)-a^8*(7*b^5-3*b^4*c+4*b^3*c^2+4*b^2*c^3-3*b*c^4+7*c^5)-a^3*(b^2-c^2)^2*(4*b^6-9*b^5*c-6*b^3*c^3-9*b*c^5+4*c^6)-2*a^7*(4*b^6-5*b^5*c+6*b^4*c^2-6*b^3*c^3+6*b^2*c^4-5*b*c^5+4*c^6)+4*a^6*(2*b^7+b^5*c^2+b^4*c^3+b^3*c^4+b^2*c^5+2*c^7)+a^5*(7*b^8-8*b^7*c-4*b^6*c^2-6*b^4*c^4-4*b^2*c^6-8*b*c^7+7*c^8) : :

X(66028) lies on these lines: {6, 119}, {52, 10742}, {68, 952}, {100, 6146}, {104, 343}, {153, 6515}, {569, 38752}, {1209, 57298}, {2829, 17834}, {3035, 37476}, {5840, 64037}, {9913, 37488}, {10711, 61658}, {10738, 18474}, {11698, 13292}, {37478, 38753}, {37493, 38755}, {37513, 38762}, {37649, 64008}, {49162, 64069}, {63085, 66045}

X(66028) = reflection of X(i) in X(j) for these {i,j}: {66029, 119}, {66035, 68}
X(66028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {68, 952, 66035}

X(66029) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-ANTI-CONWAY WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^2*(a^11-a^10*(b+c)+a^9*(-5*b^2+3*b*c-5*c^2)+(b-c)^4*(b+c)^3*(b^2+c^2)^2+a^8*(5*b^3+3*b^2*c+3*b*c^2+5*c^3)-a^2*(b-c)^2*(b+c)^3*(5*b^4-8*b^3*c+14*b^2*c^2-8*b*c^3+5*c^4)+2*a^7*(5*b^4-6*b^3*c+6*b^2*c^2-6*b*c^3+5*c^4)-2*a^5*(b-c)^2*(5*b^4+b^3*c+2*b^2*c^2+b*c^3+5*c^4)-2*a^6*(5*b^5+b^4*c+2*b^3*c^2+2*b^2*c^3+b*c^4+5*c^5)-a*(b^2-c^2)^2*(b^6-3*b^5*c+3*b^4*c^2-10*b^3*c^3+3*b^2*c^4-3*b*c^5+c^6)+2*a^4*(5*b^7-b^6*c-b^5*c^2+5*b^4*c^3+5*b^3*c^4-b^2*c^5-b*c^6+5*c^7)+a^3*(5*b^8-12*b^7*c+4*b^6*c^2+4*b^5*c^3-18*b^4*c^4+4*b^3*c^5+4*b^2*c^6-12*b*c^7+5*c^8)) : :
X(66029) =

X(66029) lies on these lines: {6, 119}, {11, 17814}, {100, 1181}, {104, 394}, {149, 11441}, {153, 1993}, {155, 952}, {323, 64009}, {511, 9913}, {576, 58543}, {1191, 6265}, {1484, 15068}, {1498, 5840}, {2323, 66058}, {2771, 17847}, {2783, 39849}, {2787, 39820}, {2829, 37498}, {3035, 37514}, {3045, 19357}, {5020, 58508}, {5422, 66045}, {5531, 56535}, {6713, 17811}, {6759, 13222}, {8674, 17838}, {9024, 19149}, {10601, 64008}, {10711, 63094}, {10738, 18451}, {10742, 36747}, {11432, 58504}, {11456, 13199}, {11698, 12161}, {12331, 18445}, {12515, 62245}, {15017, 16472}, {15811, 64186}, {17825, 58421}, {17834, 54065}, {20095, 43605}, {22758, 63346}, {22799, 44413}, {36749, 38755}, {36752, 38752}, {37483, 38753}

X(66029) = reflection of X(i) in X(j) for these {i,j}: {13222, 6759}, {17834, 54065}, {66028, 119}, {66036, 155}
X(66029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {155, 952, 66036}

X(66030) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ANTI-HONSBERGER WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^9-a^8*(b+c)+2*a^6*(b-c)^2*(b+c)+a^7*(-2*b^2+5*b*c-2*c^2)+2*a^2*(b-c)^2*(b+c)^3*(b^2+c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)+a*(b^2-c^2)^2*(b^4-4*b^3*c+2*b^2*c^2-4*b*c^3+c^4)+2*a^5*(b^4-b^3*c+4*b^2*c^2-b*c^3+c^4)-2*a^4*(b^5+3*b^3*c^2+3*b^2*c^3+c^5)+a^3*(-2*b^6+b^5*c+2*b^4*c^2+6*b^3*c^3+2*b^2*c^4+b*c^5-2*c^6) : :
X(66030) = -2*X[11]+3*X[10516], -2*X[182]+3*X[38752], -X[1351]+3*X[38755], -4*X[3035]+3*X[5085], -4*X[3589]+5*X[64008], -5*X[3618]+7*X[66045], -5*X[3620]+X[64009], -5*X[3763]+4*X[6713], -4*X[5092]+5*X[38762]

X(66030) lies on these lines: {4, 9024}, {6, 119}, {11, 10516}, {67, 2771}, {69, 153}, {100, 1503}, {104, 141}, {182, 38752}, {376, 51158}, {511, 10742}, {518, 12751}, {524, 10711}, {528, 47353}, {613, 39692}, {742, 66057}, {952, 1352}, {1317, 12589}, {1350, 2829}, {1351, 38755}, {1469, 12763}, {1484, 18358}, {2783, 11646}, {2800, 3416}, {2801, 47595}, {2802, 64085}, {2830, 36883}, {3035, 5085}, {3056, 12764}, {3098, 38753}, {3564, 11698}, {3589, 64008}, {3618, 66045}, {3620, 64009}, {3763, 6713}, {3818, 10738}, {5092, 38762}, {5227, 66058}, {5480, 10755}, {5820, 38144}, {5840, 36990}, {5845, 66023}, {5846, 10698}, {5847, 21635}, {5848, 12587}, {6174, 43273}, {6326, 39885}, {6776, 51157}, {8674, 14982}, {9041, 50907}, {9913, 37485}, {10519, 12248}, {10541, 38763}, {10707, 47354}, {10728, 29181}, {10778, 32274}, {11477, 38757}, {11729, 38315}, {12199, 42534}, {12331, 18440}, {14561, 61580}, {14810, 38754}, {15017, 16475}, {20400, 53093}, {22799, 31670}, {24206, 57298}, {24466, 48905}, {25485, 49681}, {28538, 50908}, {31884, 38761}, {32233, 53743}, {33814, 46264}, {33878, 38756}, {34380, 61605}, {34474, 44882}, {38119, 47355}, {38531, 51390}, {38758, 55711}, {38759, 55646}, {38760, 53094}, {39898, 66008}, {40341, 66052}, {48906, 61562}, {48910, 52836}, {59415, 63470}

X(66030) = midpoint of X(i) and X(j) for these {i,j}: {69, 153}, {6326, 39885}, {12331, 18440}, {33878, 38756}, {39898, 66008}
X(66030) = reflection of X(i) in X(j) for these {i,j}: {6, 119}, {104, 141}, {376, 51158}, {1350, 51007}, {1484, 18358}, {6776, 51157}, {10707, 47354}, {10738, 3818}, {10755, 5480}, {10778, 32274}, {31670, 22799}, {32233, 53743}, {38531, 51390}, {38753, 3098}, {43273, 6174}, {46264, 33814}, {48905, 24466}, {48906, 61562}, {48910, 52836}, {49681, 25485}, {66031, 6}, {66037, 1352}
X(66030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 1352, 66037}, {2829, 51007, 1350}, {38119, 58421, 47355}

X(66031) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-EHRMANN WRT EXTOUCH-OF-FUHRMANN

Barycentrics    3*a^9-3*a^8*(b+c)+a^7*(-8*b^2+11*b*c-8*c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)+4*a^2*(b-c)^2*(b+c)^3*(b^2-b*c+c^2)+2*a^5*(b-c)^2*(4*b^2+b*c+4*c^2)+8*a^6*(b^3+c^3)+a*(b^2-c^2)^2*(b^4-4*b^3*c+2*b^2*c^2-4*b*c^3+c^4)-2*a^4*(4*b^5-b^4*c+b^3*c^2+b^2*c^3-b*c^4+4*c^5)+a^3*(-4*b^6+7*b^5*c+4*b^4*c^2-6*b^3*c^3+4*b^2*c^4+7*b*c^5-4*c^6) : :
X(66031) = -3*X[6]+2*X[119], -X[153]+3*X[1992], -6*X[182]+5*X[38762], -4*X[575]+3*X[38752], -6*X[597]+5*X[64008], -3*X[599]+4*X[6713], -3*X[1350]+4*X[38759], -3*X[1351]+X[38756], -3*X[1352]+4*X[60759], -4*X[3035]+5*X[53093], -5*X[3763]+6*X[38119], -3*X[5085]+2*X[51007] and many others

X(66031) lies on these lines: {6, 119}, {11, 15069}, {100, 8550}, {104, 524}, {153, 1992}, {182, 38762}, {193, 48692}, {511, 38753}, {542, 10738}, {575, 38752}, {576, 10742}, {597, 64008}, {599, 6713}, {952, 63722}, {1350, 38759}, {1351, 38756}, {1352, 60759}, {1484, 3564}, {1503, 10724}, {2771, 64104}, {2783, 64092}, {2787, 64091}, {2829, 11477}, {3035, 53093}, {3045, 64061}, {3629, 10759}, {3763, 38119}, {4663, 12751}, {5085, 51007}, {5840, 64080}, {6776, 9024}, {8540, 12764}, {8584, 10711}, {8674, 64103}, {10541, 38760}, {11179, 33814}, {11482, 38755}, {12763, 19369}, {14912, 51157}, {20423, 22799}, {24466, 43273}, {25485, 47356}, {29959, 58508}, {34507, 57298}, {38069, 50993}, {38754, 52987}, {38757, 53858}, {38761, 53097}, {39897, 63270}, {47352, 58421}, {47353, 65948}, {50979, 61562}, {52836, 54131}, {59373, 66045}

X(66031) = reflection of X(i) in X(j) for these {i,j}: {100, 8550}, {10711, 8584}, {10742, 576}, {10759, 3629}, {12751, 4663}, {15069, 11}, {53097, 38761}, {66030, 6}, {66039, 63722}
X(66031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 63722, 66039}

X(66032) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 1ST-KENMOTU-DIAGONALS WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^9-2*a^8*(b+c)+a^7*(-5*b^2+8*b*c-5*c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)+a*(b-c)^2*(b+c)^2*(b^2+c^2)*(b^2-4*b*c+c^2)+a^2*(b-c)^2*(b+c)^3*(3*b^2-2*b*c+3*c^2)+a^5*(b^2+c^2)*(5*b^2-8*b*c+5*c^2)+a^6*(b+c)*(5*b^2-6*b*c+5*c^2)-a^4*(b+c)*(b^2+c^2)*(5*b^2-6*b*c+5*c^2)+a^3*(-3*b^6+4*b^5*c+3*b^4*c^2+3*b^2*c^4+4*b*c^5-3*c^6)+2*(-2*a*b*c+a^2*(b+c)-(b-c)^2*(b+c))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2))*S : :

X(66032) lies on these lines: {6, 119}, {11, 6289}, {100, 45406}, {104, 492}, {153, 62987}, {591, 48684}, {952, 45713}, {2783, 50719}, {2800, 49347}, {2829, 9733}, {3035, 43119}, {5840, 13748}, {6713, 45472}, {10711, 45421}, {10738, 45375}, {10742, 45488}, {10956, 45490}, {11729, 45398}, {12305, 38761}, {12751, 45426}, {13991, 39679}, {37725, 49317}, {37726, 45496}, {38752, 45411}, {44392, 48701}, {45438, 65948}

X(66032) = midpoint of X(i) and X(j) for these {i,j}: {13748, 48703}
X(66032) = reflection of X(i) in X(j) for these {i,j}: {66033, 119}, {66040, 49355}
X(66032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 49355, 66040}, {13748, 48703, 5840}

X(66033) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-KENMOTU DIAGONALS WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^9-2*a^8*(b+c)+a^7*(-5*b^2+8*b*c-5*c^2)-(b-c)^4*(b+c)^3*(b^2+c^2)+a*(b-c)^2*(b+c)^2*(b^2+c^2)*(b^2-4*b*c+c^2)+a^2*(b-c)^2*(b+c)^3*(3*b^2-2*b*c+3*c^2)+a^5*(b^2+c^2)*(5*b^2-8*b*c+5*c^2)+a^6*(b+c)*(5*b^2-6*b*c+5*c^2)-a^4*(b+c)*(b^2+c^2)*(5*b^2-6*b*c+5*c^2)+a^3*(-3*b^6+4*b^5*c+3*b^4*c^2+3*b^2*c^4+4*b*c^5-3*c^6)-2*(-2*a*b*c+a^2*(b+c)-(b-c)^2*(b+c))*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2))*S : :

X(66033) lies on these lines: {6, 119}, {11, 6290}, {100, 45407}, {104, 491}, {153, 62986}, {952, 45714}, {1991, 48685}, {2783, 50720}, {2800, 49348}, {2829, 9732}, {3035, 43118}, {5840, 13749}, {6713, 45473}, {10711, 45420}, {10738, 45376}, {10742, 45489}, {10956, 45491}, {11729, 45399}, {12306, 38761}, {12751, 45427}, {13922, 39648}, {37725, 49318}, {37726, 45497}, {38752, 45410}, {44394, 48700}, {45439, 65948}

X(66033) = midpoint of X(i) and X(j) for these {i,j}: {13749, 48704}
X(66033) = reflection of X(i) in X(j) for these {i,j}: {66032, 119}, {66041, 49356}
X(66033) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 49356, 66041}, {13749, 48704, 5840}

X(66034) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ORTHOCENTROIDAL-ISOGONIC WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(2*a^7*b*c+2*a^8*(b+c)+2*a^3*b*c*(b^2-c^2)^2-a^5*b*c*(b^2+c^2)-3*a*b*c*(b^2-c^2)^2*(b^2+c^2)+a^2*(b-c)^2*(b+c)^3*(b^2-b*c+c^2)-5*a^6*(b^3+c^3)-(b-c)^2*(b+c)^3*(b^4+b^2*c^2+c^4)+a^4*(3*b^5-b^4*c+b^3*c^2+b^2*c^3-b*c^4+3*c^5)) : :

X(66034) lies on these lines: {30, 62305}, {72, 74}, {104, 112}, {214, 22054}, {376, 25252}, {900, 9409}, {2783, 9862}, {2828, 5667}, {3569, 9980}, {5191, 9978}, {5260, 11259}, {12775, 13265}, {40948, 44243}, {41191, 42662}, {44427, 55126}, {53252, 53282}

X(66034) = reflection of X(i) in X(j) for these {i,j}: {66042, 9409}
X(66034) = pole of line {53248, 53762} with respect to the circumcircle
X(66034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {900, 9409, 66042}

X(66035) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ANTI-CONWAY WRT EXTOUCH-OF-FUHRMANN

Barycentrics    (a-b-c)*(a^8+a^5*b*c*(b+c)-a^3*b*(b-c)^2*c*(b+c)+a^6*(-2*b^2+b*c-2*c^2)+(b-c)^4*(b+c)^2*(b^2+c^2)+a^4*(b-c)^2*(2*b^2+b*c+2*c^2)-2*a^2*(b-c)^2*(b^4+c^4)) : :

X(66035) lies on these lines: {6, 11}, {52, 10738}, {68, 952}, {80, 7686}, {100, 343}, {104, 6146}, {149, 6515}, {161, 54065}, {528, 64060}, {569, 57298}, {1209, 38752}, {1484, 13292}, {2829, 64037}, {5840, 17834}, {6713, 37476}, {10071, 64069}, {10707, 61658}, {10742, 18474}, {11750, 38753}, {13222, 37488}, {21293, 38357}, {31272, 37649}, {37493, 51517}, {45089, 59391}, {63085, 66063}

X(66035) = reflection of X(i) in X(j) for these {i,j}: {66028, 68}, {66036, 11}
X(66035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 5848, 66036}, {68, 952, 66028}

X(66036) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-ANTI-CONWAY WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^2*(a-b-c)*(a^6+a^3*b*c*(b+c)-a*b*(b-c)^2*c*(b+c)-3*a^4*(b^2-b*c+c^2)+a^2*(b-c)^2*(3*b^2+b*c+3*c^2)-(b^3-b^2*c+b*c^2-c^3)^2) : :
X(66036) = -3*X[154]+2*X[54065]

X(66036) lies on these lines: {1, 34976}, {6, 11}, {55, 53324}, {100, 394}, {104, 1181}, {119, 17814}, {149, 1993}, {153, 9370}, {154, 54065}, {155, 952}, {221, 2800}, {222, 1768}, {227, 66058}, {323, 20095}, {399, 18340}, {511, 13222}, {528, 37672}, {576, 58539}, {651, 9809}, {692, 1364}, {1191, 12740}, {1413, 66055}, {1484, 12161}, {1498, 2829}, {1854, 2771}, {2003, 64372}, {2192, 2801}, {2323, 66068}, {2783, 39820}, {2787, 39849}, {2807, 36059}, {3035, 17811}, {3562, 9803}, {4585, 27542}, {5020, 58504}, {5422, 66063}, {5531, 51361}, {5840, 37498}, {6326, 7078}, {6667, 17825}, {6713, 37514}, {6759, 9913}, {6797, 44414}, {8674, 17847}, {8679, 10535}, {8757, 16128}, {9371, 22128}, {9817, 58683}, {10058, 36746}, {10090, 36745}, {10265, 41344}, {10601, 31272}, {10707, 63094}, {10738, 36747}, {10742, 18451}, {10982, 59391}, {11432, 58508}, {11456, 12248}, {11698, 15068}, {12758, 64449}, {12767, 34043}, {12773, 18445}, {13243, 23144}, {13253, 34040}, {15805, 34126}, {15811, 52836}, {16473, 37718}, {17638, 64020}, {17660, 19354}, {19357, 58056}, {19372, 58613}, {21635, 34048}, {22938, 44413}, {23071, 45272}, {36749, 51517}, {36752, 57298}, {43605, 64009}, {53295, 53554}, {60691, 62354}

X(66036) = reflection of X(i) in X(j) for these {i,j}: {9913, 6759}, {66029, 155}, {66035, 11}
X(66036) = pole of line {5172, 44670} with respect to the Feuerbach hyperbola
X(66036) = pole of line {15252, 65808} with respect to the MacBeath circumconic
X(66036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 5848, 66035}, {155, 952, 66029}

X(66037) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ANTI-HONSBERGER WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^5-b^5+a^3*b*c+b^4*c+b*c^4-c^5-a^4*(b+c)+a*(b-c)^2*(b^2+c^2) : :
X(66037) = -3*X[2]+2*X[51157], -2*X[119]+3*X[10516], -2*X[182]+3*X[57298], -2*X[597]+3*X[59377], -X[1351]+3*X[51517], -2*X[1386]+3*X[16173], -4*X[3035]+5*X[3763], -4*X[3036]+3*X[59407], -4*X[3589]+5*X[31272], -5*X[3618]+7*X[66063], -5*X[3620]+X[20095], -X[3751]+3*X[37718] and many others

X(66037) lies on these lines: {2, 51157}, {6, 11}, {67, 8674}, {69, 149}, {80, 518}, {100, 141}, {104, 1503}, {119, 10516}, {182, 57298}, {511, 10738}, {524, 10707}, {528, 599}, {597, 59377}, {611, 8068}, {613, 5533}, {732, 32454}, {742, 66067}, {952, 1352}, {1086, 18343}, {1156, 5845}, {1317, 12588}, {1320, 5846}, {1350, 5840}, {1351, 51517}, {1386, 16173}, {1387, 5820}, {1469, 13273}, {1484, 3564}, {2771, 14982}, {2781, 10767}, {2787, 11646}, {2800, 64085}, {2802, 3416}, {2805, 36883}, {2810, 10770}, {2829, 36990}, {2854, 10778}, {3035, 3763}, {3036, 59407}, {3056, 13274}, {3315, 3448}, {3410, 62814}, {3583, 9037}, {3589, 31272}, {3618, 66063}, {3620, 20095}, {3675, 24713}, {3751, 37718}, {3818, 10742}, {3827, 17638}, {4265, 10058}, {4585, 31126}, {5085, 6713}, {5096, 10090}, {5227, 66068}, {5480, 10759}, {5847, 21630}, {5856, 50995}, {5969, 10769}, {6174, 21358}, {6264, 39885}, {6667, 47355}, {6702, 38047}, {7232, 21280}, {7289, 64372}, {7972, 49465}, {8679, 12764}, {9021, 12532}, {9041, 50890}, {9053, 12531}, {9897, 16496}, {10519, 13199}, {10711, 47354}, {10724, 29181}, {11442, 17597}, {11698, 18358}, {12019, 64070}, {12247, 39898}, {12587, 62616}, {12595, 15069}, {12773, 18440}, {13194, 42534}, {13222, 37485}, {14561, 60759}, {15863, 49688}, {16174, 38035}, {16686, 26932}, {20418, 64080}, {20987, 54065}, {21154, 53094}, {21356, 51158}, {22769, 39892}, {22938, 31670}, {24206, 38752}, {24466, 31884}, {25416, 49679}, {28538, 50891}, {29012, 38753}, {31523, 32298}, {32233, 53753}, {33709, 38049}, {33878, 48680}, {34378, 47320}, {34380, 61601}, {37998, 46158}, {38090, 51185}, {38119, 53093}, {38602, 46264}, {38693, 44882}, {38754, 48898}, {38759, 59411}, {38761, 48905}, {39692, 45729}, {40341, 66065}, {45310, 47352}, {48906, 61566}, {48910, 64186}, {49524, 59415}, {49681, 64137}, {50949, 64746}, {51003, 64011}, {53023, 65948}

X(66037) = midpoint of X(i) and X(j) for these {i,j}: {69, 149}, {6264, 39885}, {9897, 16496}, {12247, 39898}, {12773, 18440}, {33878, 48680}
X(66037) = reflection of X(i) in X(j) for these {i,j}: {6, 11}, {100, 141}, {7972, 49465}, {10711, 47354}, {10742, 3818}, {10759, 5480}, {11698, 18358}, {31670, 22938}, {32233, 53753}, {32298, 31523}, {46264, 38602}, {48905, 38761}, {48906, 61566}, {48910, 64186}, {49679, 25416}, {49681, 64137}, {49688, 15863}, {51008, 45310}, {64011, 51003}, {64746, 50949}, {66030, 1352}, {66039, 6}
X(66037) = anticomplement of X(51157)
X(66037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5848, 66039}, {11, 5848, 6}, {69, 149, 9024}, {952, 1352, 66030}, {10759, 59391, 5480}, {45310, 51008, 47352}

X(66038) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-BROCARD WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(2*a^8-2*a^6*(b-c)^2-2*a^7*(b+c)+3*a^2*(b-c)^4*(b+c)^2+2*a^5*(b^3+c^3)-a^4*(b^4+3*b^3*c-4*b^2*c^2+3*b*c^3+c^4)+a^3*(b^5+3*b^4*c-2*b^3*c^2-2*b^2*c^3+3*b*c^4+c^5)+b*c*(b^6+c^6)+a*(-3*b^7+b^6*c+7*b^5*c^2-7*b^4*c^3-7*b^3*c^4+7*b^2*c^5+b*c^6-3*c^7)) : :

X(66038) lies on these lines: {2, 10769}, {11, 7664}, {23, 667}, {37, 100}, {110, 10755}, {149, 7665}, {952, 63719}, {2502, 9024}, {2783, 7417}, {3124, 51157}, {40915, 51007}, {46131, 53743}

X(66039) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-EHRMANN WRT EXTOUCH-OF-FUHRMANN

Barycentrics    3*a^5-b^5+b^4*c+b*c^4-c^5-3*a^4*(b+c)+2*a^2*(b-c)^2*(b+c)+a^3*(-2*b^2+7*b*c-2*c^2)+a*(b-c)^2*(b^2+c^2) : :
X(66039) = -3*X[6]+2*X[11], -X[149]+3*X[1992], -4*X[575]+3*X[57298], -6*X[597]+5*X[31272], -3*X[599]+4*X[3035], -3*X[1351]+X[48680], -3*X[1352]+4*X[61580], -3*X[3242]+4*X[12735], -3*X[3751]+X[9897], -8*X[6667]+9*X[47352], -4*X[6713]+5*X[53093], -3*X[6776]+X[12248]

X(66039) lies on circumconic {{A, B, C, X(36902), X(60362)}} and on these lines: {6, 11}, {69, 51157}, {80, 4663}, {100, 524}, {104, 8550}, {119, 15069}, {149, 1992}, {193, 9024}, {518, 7972}, {528, 15534}, {542, 10742}, {575, 57298}, {576, 10738}, {597, 31272}, {599, 3035}, {952, 63722}, {1351, 48680}, {1352, 61580}, {1503, 10728}, {2771, 64103}, {2783, 64091}, {2787, 64092}, {2829, 64080}, {2836, 11571}, {3242, 12735}, {3564, 11698}, {3629, 10755}, {3751, 9897}, {4316, 9037}, {5840, 11477}, {6174, 15533}, {6667, 47352}, {6713, 53093}, {6776, 12248}, {8540, 13274}, {8584, 10707}, {8674, 64104}, {9004, 17660}, {10541, 21154}, {11160, 51158}, {11179, 38602}, {11482, 51517}, {13273, 19369}, {15863, 47359}, {20423, 22938}, {21358, 31235}, {24466, 53097}, {25416, 51000}, {28538, 64056}, {29959, 58504}, {34507, 38752}, {35023, 40341}, {38119, 55711}, {38761, 43273}, {38762, 40107}, {45310, 51185}, {47356, 64137}, {50979, 61566}, {54131, 64186}, {58056, 64061}, {59373, 66063}, {59377, 63124}

X(66039) = reflection of X(i) in X(j) for these {i,j}: {6, 51198}, {69, 51157}, {80, 4663}, {104, 8550}, {599, 51008}, {10707, 8584}, {10738, 576}, {10755, 3629}, {11160, 51158}, {15069, 119}, {15533, 6174}, {40341, 51007}, {53097, 24466}, {66031, 63722}, {66037, 6}
X(66039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5848, 66037}, {952, 63722, 66031}, {5848, 51198, 6}

X(66040) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 1ST-KENMOTU-DIAGONALS WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^5-2*a^4*(b+c)+a^2*(b-c)^2*(b+c)+a*(b-c)^2*(b^2+c^2)-(b-c)^2*(b+c)*(b^2+c^2)-a^3*(b^2-4*b*c+c^2)+2*(a-b-c)*(b-c)^2*S : :

X(66040) lies on these lines: {6, 11}, {80, 45426}, {100, 492}, {104, 45406}, {119, 6289}, {149, 62987}, {528, 591}, {952, 45713}, {1145, 45444}, {1317, 45476}, {1387, 45398}, {2787, 50719}, {2802, 49347}, {2829, 13748}, {3035, 45472}, {5840, 9733}, {5851, 60888}, {5854, 49329}, {6713, 43119}, {10707, 45421}, {10738, 45488}, {10742, 45375}, {10956, 45458}, {12305, 24466}, {12959, 19048}, {13977, 39679}, {37725, 45456}, {37726, 45422}, {44392, 48715}, {45411, 57298}, {45428, 54065}, {45440, 65948}

X(66040) = midpoint of X(i) and X(j) for these {i,j}: {13748, 48684}, {45713, 49337}
X(66040) = reflection of X(i) in X(j) for these {i,j}: {66032, 49355}, {66041, 11}
X(66040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 5848, 66041}, {952, 49355, 66032}, {13748, 48684, 2829}, {45713, 49337, 952}

X(66041) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF 2ND-KENMOTU DIAGONALS WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^5-2*a^4*(b+c)+a^2*(b-c)^2*(b+c)+a*(b-c)^2*(b^2+c^2)-(b-c)^2*(b+c)*(b^2+c^2)-a^3*(b^2-4*b*c+c^2)-2*(a-b-c)*(b-c)^2*S : :

X(66041) lies on these lines: {6, 11}, {80, 45427}, {100, 491}, {104, 45407}, {119, 6290}, {149, 62986}, {528, 1991}, {952, 45714}, {1145, 45445}, {1317, 45477}, {1387, 45399}, {2787, 50720}, {2802, 49348}, {2829, 13749}, {3035, 45473}, {5840, 9732}, {5851, 60889}, {5854, 49330}, {6713, 43118}, {10707, 45420}, {10738, 45489}, {10742, 45376}, {10956, 45459}, {12306, 24466}, {12958, 19047}, {13913, 39648}, {37725, 45457}, {37726, 45423}, {44394, 48714}, {45410, 57298}, {45429, 54065}, {45441, 65948}

X(66041) = midpoint of X(i) and X(j) for these {i,j}: {13749, 48685}, {45714, 49338}
X(66041) = reflection of X(i) in X(j) for these {i,j}: {66033, 49356}, {66040, 11}
X(66041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 5848, 66040}, {952, 49356, 66033}, {13749, 48685, 2829}, {45714, 49338, 952}

X(66042) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR TRIANGLE OF ORTHOCENTROIDAL-ISOGONIC WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(b-c)*(2*a^10*(b+c)-a^3*(b^2-c^2)^4-a^2*(b-c)^4*(b+c)^3*(b^2+c^2)-2*a^9*(b^2+3*b*c+c^2)+a^8*(-5*b^3+2*b^2*c+2*b*c^2-5*c^3)+b*(b-c)^2*c*(b+c)^3*(b^4+b^2*c^2+c^4)+a^7*(5*b^4+8*b^3*c+2*b^2*c^2+8*b*c^3+5*c^4)+a^6*(3*b^5-7*b^4*c-7*b*c^4+3*c^5)+a*(b^2-c^2)^2*(b^6-2*b^5*c-b^4*c^2-2*b^3*c^3-b^2*c^4-2*b*c^5+c^6)-a^5*(3*b^6+b^4*c^2+10*b^3*c^3+b^2*c^4+3*c^6)+a^4*(b^7+b^6*c-5*b^5*c^2+8*b^4*c^3+8*b^3*c^4-5*b^2*c^5+b*c^6+c^7)) : :

X(66042) lies on these lines: {74, 104}, {100, 112}, {214, 40613}, {900, 9409}, {2787, 9862}, {2803, 5667}, {2804, 44427}, {3569, 9978}, {5191, 9980}, {53248, 53762}

X(66042) = reflection of X(i) in X(j) for these {i,j}: {66034, 9409}
X(66042) = pole of line {53252, 53282} with respect to the circumcircle
X(66042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {900, 9409, 66034}

X(66043) = PERSPECTOR OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN AND INVERSE-OF-X(37)-CIRCUMCONCEVIAN-OF-X(1))

Barycentrics    (b+c)*(-2*a^3-3*a^2*(b+c)+3*b*c*(b+c)-a*(b^2+4*b*c+c^2)) : :
X(66043) = X[65]+3*X[3159], 3*X[2901]+5*X[3697], 9*X[3175]+7*X[4002], X[3874]+3*X[64426], -9*X[3971]+X[4067], 5*X[18398]+3*X[24068], X[35633]+3*X[59718]

X(66043) lies on circumconic {{A, B, C, X(43972), X(64071)}} and on these lines: {10, 3995}, {65, 3159}, {502, 3178}, {519, 960}, {537, 64428}, {740, 4540}, {756, 3626}, {1089, 1125}, {1215, 3636}, {2901, 3697}, {3175, 4002}, {3244, 3952}, {3293, 14752}, {3634, 3666}, {3842, 4681}, {3874, 64426}, {3971, 4067}, {3992, 4065}, {4015, 58395}, {4125, 56221}, {5045, 59717}, {6534, 58565}, {6540, 32004}, {18398, 24068}, {21021, 24051}, {21864, 24067}, {27538, 50588}, {35633, 59718}, {51562, 56950}

X(66043) = midpoint of X(i) and X(j) for these {i,j}: {4075, 63800}
X(66043) = pole of line {47793, 48085} with respect to the Steiner inellipse
X(66043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4075, 63800, 519}

X(66044) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-8 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^8*(b+c)-2*a^7*(b^2+b*c+c^2)-(b-c)^4*(b+c)^3*(b^2+b*c+c^2)+a^4*b*c*(-13*b^3+9*b^2*c+9*b*c^2-13*c^3)+a^6*(-2*b^3+5*b^2*c+5*b*c^2-2*c^3)+2*a*(b^2-c^2)^2*(b^4-3*b^3*c+2*b^2*c^2-3*b*c^3+c^4)+a^5*(6*b^4-2*b^3*c-6*b^2*c^2-2*b*c^3+6*c^4)+a^2*(b-c)^2*(2*b^5+11*b^4*c+3*b^3*c^2+3*b^2*c^3+11*b*c^4+2*c^5)+a^3*(-6*b^6+10*b^5*c+8*b^4*c^2-20*b^3*c^3+8*b^2*c^4+10*b*c^5-6*c^6)) : :
X(66044) = -3*X[381]+2*X[12736], -3*X[5603]+X[66002], -3*X[5657]+4*X[58674], -3*X[5886]+2*X[15528], -2*X[9943]+3*X[38760], -X[9961]+3*X[34474], -2*X[13369]+3*X[34123], -4*X[18240]+5*X[18493], -5*X[31235]+4*X[40296], -3*X[38128]+4*X[58631]

Triangle CTR1-8 is defined by the Aubert (Steiner) lines of quadrilaterals ABPC, BCPA, CAPB, where P=X(8).

X(66044) lies on these lines: {4, 66024}, {11, 113}, {30, 64139}, {80, 18516}, {104, 55961}, {119, 5123}, {355, 2800}, {381, 12736}, {517, 12665}, {912, 1537}, {952, 12672}, {960, 38761}, {1071, 11729}, {1145, 17615}, {1376, 12515}, {1387, 17625}, {1709, 6326}, {2801, 10247}, {2802, 18525}, {2829, 5887}, {2950, 5720}, {3434, 12532}, {3869, 10728}, {5083, 11373}, {5541, 18528}, {5603, 66002}, {5657, 58674}, {5692, 35249}, {5693, 10525}, {5694, 11826}, {5777, 45080}, {5840, 12688}, {5886, 15528}, {5927, 9952}, {6246, 31871}, {6265, 12114}, {6923, 46435}, {9856, 64138}, {9943, 38760}, {9961, 34474}, {10698, 12528}, {10826, 11571}, {10914, 46685}, {10944, 12758}, {11715, 26321}, {12616, 21635}, {12699, 13271}, {12702, 14740}, {12735, 17622}, {12738, 13205}, {12773, 41554}, {12775, 37700}, {13369, 34123}, {14988, 22799}, {15071, 26492}, {15906, 44013}, {16138, 66048}, {17613, 33814}, {17614, 38602}, {17617, 35638}, {17618, 60759}, {17619, 61580}, {17660, 65991}, {18240, 18493}, {18542, 64745}, {20117, 46684}, {31235, 40296}, {31828, 64000}, {33898, 34293}, {38128, 58631}, {45764, 51897}, {45770, 48695}, {49171, 55298}, {51515, 63967}, {64197, 64267}

X(66044) = midpoint of X(i) and X(j) for these {i,j}: {4, 66024}, {3869, 10728}, {5693, 34789}, {10698, 12528}, {10742, 40266}, {12672, 17661}
X(66044) = reflection of X(i) in X(j) for these {i,j}: {11, 31937}, {1071, 11729}, {1145, 66049}, {6246, 31871}, {11570, 12611}, {12515, 18254}, {12702, 14740}, {33898, 34293}, {38761, 960}, {46684, 20117}, {64138, 9856}
X(66044) = X(1511) of Ursa-major triangle
X(66044) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {355, 16128, 12761}, {2771, 12611, 11570}, {10742, 40266, 2800}, {12672, 17661, 952}, {18519, 48667, 12737}

X(66045) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR5-2.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^7-a^6*(b+c)+3*(b-c)^4*(b+c)^3-a*(b^2-c^2)^2*(3*b^2-11*b*c+3*c^2)-a^5*(5*b^2+b*c+5*c^2)+a^4*(5*b^3+7*b^2*c+7*b*c^2+5*c^3)-a^2*(b-c)^2*(7*b^3+17*b^2*c+17*b*c^2+7*c^3)+a^3*(7*b^4-10*b^3*c-6*b^2*c^2-10*b*c^3+7*c^4) : :
X(66045) = -9*X[2]+2*X[104], 3*X[4]+4*X[33814], -8*X[5]+X[149], 3*X[8]+4*X[25485], 6*X[10]+X[13253], -4*X[11]+11*X[5056], -X[20]+8*X[3035], 2*X[100]+5*X[3091], -8*X[140]+X[12248], 3*X[210]+4*X[58613], 4*X[214]+3*X[59387], 3*X[354]+4*X[58687] and many others

CTR5-2.2 is the triangle homothetic to ABC with center X(2) and ratio 2/7.

X(66045) lies on circumconic {{A, B, C, X(6713), X(57769)}} and on these lines: {2, 104}, {4, 33814}, {5, 149}, {8, 25485}, {10, 13253}, {11, 5056}, {20, 3035}, {100, 3091}, {140, 12248}, {145, 6981}, {210, 58613}, {214, 59387}, {354, 58687}, {376, 22799}, {381, 13199}, {498, 63281}, {528, 61936}, {549, 38756}, {631, 10742}, {632, 61605}, {952, 3090}, {1145, 8166}, {1317, 10589}, {1329, 6960}, {1387, 8164}, {1484, 5055}, {1537, 6969}, {1656, 11698}, {1698, 12767}, {1768, 3634}, {1862, 6622}, {2800, 9780}, {2801, 60996}, {2829, 3523}, {2932, 6912}, {3085, 39692}, {3146, 34474}, {3305, 66058}, {3522, 10728}, {3524, 38753}, {3525, 38602}, {3543, 24466}, {3544, 51525}, {3545, 10738}, {3577, 30852}, {3616, 12751}, {3617, 10698}, {3618, 66030}, {3620, 10759}, {3628, 12773}, {3814, 6840}, {3817, 5541}, {3832, 5840}, {3839, 6174}, {3850, 48680}, {3854, 10993}, {3855, 22938}, {3917, 58543}, {4666, 66062}, {4699, 66057}, {5067, 57298}, {5068, 20095}, {5070, 61566}, {5071, 60759}, {5076, 38636}, {5083, 5704}, {5187, 65739}, {5218, 12764}, {5226, 12736}, {5260, 22775}, {5422, 66029}, {5531, 59419}, {5550, 11715}, {5587, 6224}, {5603, 64743}, {5657, 12611}, {5660, 6702}, {5720, 39778}, {5731, 64012}, {5734, 64056}, {5748, 64139}, {5818, 6265}, {5886, 66008}, {5889, 58504}, {6264, 32558}, {6326, 10175}, {6594, 59385}, {6667, 38669}, {6856, 9952}, {6858, 66051}, {6859, 20085}, {6860, 10609}, {6879, 54448}, {6908, 32554}, {6933, 59415}, {6959, 20060}, {6979, 11681}, {6982, 61156}, {6993, 60782}, {7288, 12763}, {7485, 9913}, {7486, 10585}, {7988, 21630}, {7993, 33709}, {8068, 45043}, {8889, 12138}, {8972, 19082}, {9779, 14217}, {9802, 16174}, {9809, 19877}, {9956, 12247}, {10087, 10591}, {10090, 10590}, {10265, 54447}, {10299, 38754}, {10303, 31235}, {10595, 64140}, {10707, 61924}, {10956, 14986}, {11002, 58522}, {11231, 16128}, {11451, 58508}, {11500, 63917}, {12735, 47743}, {12738, 38182}, {12739, 54361}, {12747, 61259}, {13729, 27529}, {13941, 19081}, {15015, 19925}, {15022, 23513}, {15692, 38759}, {15717, 38761}, {17572, 18861}, {19907, 59388}, {20418, 46935}, {21154, 55864}, {22935, 61261}, {26364, 37437}, {27355, 58539}, {31412, 48715}, {32785, 48700}, {32786, 48701}, {33337, 37714}, {33812, 37712}, {34126, 61886}, {35023, 59390}, {35882, 42274}, {35883, 42277}, {37106, 64188}, {37126, 54065}, {37163, 63964}, {37718, 63259}, {37726, 61914}, {38042, 48667}, {38077, 61938}, {38084, 61913}, {38108, 66007}, {38141, 61945}, {38637, 61850}, {40333, 64765}, {42561, 48714}, {45310, 61906}, {50689, 64186}, {51529, 60781}, {54445, 58453}, {59373, 66031}, {59377, 61912}, {60988, 66012}, {62778, 66023}, {63085, 66028}, {66002, 66021}

X(66045) = reflection of X(i) in X(j) for these {i,j}: {66063, 3090}
X(66045) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 64009, 6713}, {104, 58421, 2}, {104, 64008, 58421}, {119, 58421, 104}, {119, 6713, 10711}, {140, 38755, 12248}, {1537, 64141, 59417}, {5068, 20095, 59391}, {5660, 6702, 9803}, {6713, 10711, 64009}, {10711, 64009, 153}, {10728, 38760, 3522}, {22799, 38762, 376}, {31235, 38693, 10303}, {31235, 38757, 38693}, {38752, 61580, 4}, {58453, 64145, 54445}

X(66046) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT CTR7-2.7

Barycentrics    a*(a^8*(b+c)-(b-c)^4*(b+c)^3*(b^2-3*b*c+c^2)-2*a^7*(b^2+3*b*c+c^2)-2*a^6*(b^3-4*b^2*c-4*b*c^2+c^3)-3*a^4*b*c*(5*b^3-6*b^2*c-6*b*c^2+5*c^3)+a*(b^2-c^2)^2*(2*b^4-9*b^3*c+13*b^2*c^2-9*b*c^3+2*c^4)-a^3*(b-c)^2*(6*b^4-19*b^2*c^2+6*c^4)+a^5*(6*b^4+3*b^3*c-20*b^2*c^2+3*b*c^3+6*c^4)+a^2*(b-c)^2*(2*b^5+6*b^4*c-15*b^3*c^2-15*b^2*c^3+6*b*c^4+2*c^5)) : :

Triangle CTR7-2.7 vertices are the barycentric sums of the corresponding vertices of the cevian triangles of X(2) and X(7).

X(66046) lies on these lines: {1, 1389}, {79, 2800}, {515, 17637}, {517, 5499}, {952, 47319}, {2771, 65999}, {2829, 66048}, {3244, 6583}, {3754, 22765}, {5559, 61105}, {5880, 64044}, {5884, 18990}, {6246, 65995}, {11715, 66003}, {31806, 64275}, {45081, 64345}, {64199, 66006}, {65986, 66015}

X(66046) = reflection of X(i) in X(j) for these {i,j}: {1389, 31870}, {31806, 64275}

X(66047) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR7-2.7 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2))*(a^5-a^4*(b+c)+2*a^2*(b-c)^2*(b+c)-(b-c)^4*(b+c)+a^3*(-2*b^2+3*b*c-2*c^2)+a*(b^4-3*b^3*c+6*b^2*c^2-3*b*c^3+c^4)) : :
X(66047) = -X[72]+3*X[38752], -X[104]+3*X[10202], -3*X[354]+X[12737], -3*X[3753]+X[19914], 3*X[3873]+X[66008], -5*X[5439]+3*X[57298], -X[5693]+5*X[15017], -3*X[5886]+X[17638]

Triangle CTR7-2.7 vertices are the barycentric sums of the corresponding vertices of the cevian triangles of X(2) and X(7).

X(66047) lies on circumconic {{A, B, C, X(18838), X(22765)}} and on these lines: {5, 2771}, {11, 13750}, {65, 6265}, {72, 38752}, {100, 24474}, {104, 10202}, {119, 912}, {214, 517}, {354, 12737}, {355, 17660}, {942, 952}, {971, 22799}, {1071, 10742}, {1125, 2800}, {1317, 5570}, {1385, 58591}, {1387, 50195}, {1484, 11019}, {1537, 37374}, {1768, 3560}, {1858, 39692}, {2801, 60980}, {2829, 13369}, {2932, 37533}, {3035, 31837}, {3244, 6583}, {3306, 17654}, {3530, 31788}, {3555, 64140}, {3753, 19914}, {3812, 12619}, {3870, 12331}, {3873, 66008}, {3911, 14988}, {5045, 64742}, {5439, 57298}, {5535, 35204}, {5693, 15017}, {5777, 61580}, {5806, 22938}, {5886, 17638}, {5887, 11571}, {5902, 6326}, {6001, 12611}, {6264, 18398}, {6745, 61562}, {6826, 9803}, {6831, 33594}, {6893, 9809}, {6905, 39778}, {6917, 10044}, {6924, 22836}, {6929, 16128}, {6940, 10698}, {9940, 38602}, {10073, 65994}, {10087, 64046}, {10165, 13145}, {10167, 38753}, {10175, 47320}, {10247, 17652}, {10427, 66054}, {10711, 66002}, {11231, 58666}, {11698, 24475}, {11715, 13373}, {12532, 64008}, {12735, 50196}, {12739, 64045}, {15015, 37625}, {15556, 61530}, {15904, 56423}, {17100, 33596}, {17661, 38755}, {18254, 58421}, {18443, 66058}, {19920, 35597}, {24929, 38722}, {25413, 35262}, {27778, 38156}, {31838, 34123}, {31849, 53743}, {34474, 37585}, {36167, 46044}, {38042, 58659}, {38182, 58683}, {38762, 64107}, {40296, 46684}, {52005, 53537}, {56387, 64044}

X(66047) = midpoint of X(i) and X(j) for these {i,j}: {65, 6265}, {100, 24474}, {119, 11570}, {355, 17660}, {1071, 10742}, {3555, 64140}, {5884, 21635}, {5887, 11571}, {9946, 12736}, {10427, 66054}, {10698, 37562}, {11698, 24475}, {17654, 48667}
X(66047) = reflection of X(i) in X(j) for these {i,j}: {1385, 58591}, {1387, 58604}, {1484, 58587}, {5777, 61580}, {6797, 61541}, {11715, 13373}, {12611, 58613}, {12619, 3812}, {18254, 58421}, {22938, 5806}, {31837, 3035}, {38602, 9940}, {46684, 40296}, {64742, 5045}, {66049, 119}
X(66047) = X(i)-isoconjugate-of-X(j) for these {i, j}: {36052, 64290}
X(66047) = X(i)-Dao conjugate of X(j) for these {i, j}: {119, 64290}
X(66047) = pole of line {23087, 39200} with respect to the DeLongchamps ellipse
X(66047) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {113, 119, 11570}, {18341, 31849, 46044}
X(66047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 11570, 912}, {119, 912, 66049}, {952, 61541, 6797}, {1737, 11570, 66016}, {5884, 21635, 2771}, {6001, 58613, 12611}, {9946, 12736, 952}

X(66048) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT CTR7-2.8

Barycentrics    a*(a^8*(b+c)-2*a^7*(b^2-b*c+c^2)-(b-c)^4*(b+c)^3*(b^2+b*c+c^2)-2*a^6*(b^3+c^3)-3*a^4*b*c*(b^3-2*b^2*c-2*b*c^2+c^3)-3*a^3*(b-c)^2*(2*b^4-b^2*c^2+2*c^4)+a*(b^2-c^2)^2*(2*b^4-5*b^3*c-3*b^2*c^2-5*b*c^3+2*c^4)+3*a^5*(2*b^4-3*b^3*c+4*b^2*c^2-3*b*c^3+2*c^4)+a^2*(b-c)^2*(2*b^5+6*b^4*c+b^3*c^2+b^2*c^3+6*b*c^4+2*c^5)) : :
X(66048) = -3*X[3652]+2*X[3678], -4*X[3918]+3*X[47032], -3*X[16116]+5*X[18398]

Triangle CTR7-2.8 vertices are the barycentric sums of the corresponding vertices of the cevian triangles of X(2) and X(8).

X(66048) lies on these lines: {1, 10308}, {78, 7701}, {946, 58595}, {2771, 3244}, {2800, 64766}, {2829, 66046}, {3652, 3678}, {3742, 9955}, {3918, 47032}, {5880, 31672}, {6246, 65996}, {6831, 64345}, {7702, 16125}, {13145, 31673}, {16116, 18398}, {16138, 66044}, {31870, 65988}, {35982, 63267}, {41540, 51569}, {41865, 52269}

X(66049) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR7-2.8 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^3*(b+c)-a*(b-c)^2*(b+c)+(b^2-c^2)^2-a^2*(b^2+c^2))*(a^5-a^4*(b+c)-(b-c)^2*(b+c)^3+a^3*(-2*b^2+3*b*c-2*c^2)+2*a^2*(b^3+b^2*c+b*c^2+c^3)+a*(b^4-3*b^3*c-2*b^2*c^2-3*b*c^3+c^4)) : :
X(66049) = -3*X[5]+2*X[58587], -3*X[210]+X[12515], -3*X[355]+X[17636], -X[1071]+3*X[38752], -5*X[3876]+X[12248], X[5541]+3*X[61705], -3*X[5790]+X[17654], -3*X[5927]+X[10738], -3*X[10157]+2*X[60759], -3*X[10167]+5*X[38762]

Triangle CTR7-2.8 vertices are the barycentric sums of the corresponding vertices of the cevian triangles of X(2) and X(8).

X(66049) lies on circumconic {{A, B, C, X(18838), X(35000)}} and on these lines: {3, 17661}, {5, 58587}, {10, 2771}, {72, 10742}, {80, 64041}, {100, 40263}, {119, 912}, {210, 12515}, {355, 17636}, {499, 17660}, {517, 22799}, {518, 12611}, {942, 61580}, {950, 952}, {971, 6594}, {1071, 38752}, {1145, 17615}, {1898, 10087}, {2800, 3626}, {2801, 6666}, {2802, 31937}, {2829, 31837}, {3035, 13369}, {3579, 58663}, {3872, 48667}, {3876, 12248}, {5044, 38602}, {5450, 22935}, {5541, 61705}, {5790, 17654}, {5887, 12751}, {5927, 10738}, {6001, 58687}, {6265, 14872}, {6326, 22758}, {6797, 18357}, {10058, 12738}, {10157, 60759}, {10167, 38762}, {10202, 64008}, {10265, 15064}, {10711, 12532}, {10728, 37585}, {11230, 58595}, {11729, 46681}, {12059, 37406}, {12619, 58631}, {12647, 17638}, {12672, 64140}, {12735, 64131}, {12749, 64042}, {12773, 19861}, {12775, 41560}, {15528, 58421}, {18443, 66061}, {18908, 19914}, {19919, 58640}, {21635, 63967}, {31838, 64191}, {34293, 40659}, {37562, 66024}, {38753, 64107}, {39991, 56881}, {46684, 58630}, {58573, 65388}, {58674, 64193}

X(66049) = midpoint of X(i) and X(j) for these {i,j}: {3, 17661}, {72, 10742}, {100, 40263}, {119, 12665}, {1145, 66044}, {5887, 12751}, {6265, 14872}, {10728, 37585}, {12672, 64140}, {21635, 63967}, {37562, 66024}
X(66049) = reflection of X(i) in X(j) for these {i,j}: {942, 61580}, {3579, 58663}, {6797, 18357}, {12619, 58631}, {13369, 3035}, {15528, 58421}, {38602, 5044}, {46684, 58630}, {64191, 31838}, {64193, 58674}, {66047, 119}
X(66049) = pole of line {35460, 40663} with respect to the Feuerbach hyperbola
X(66049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {119, 12665, 912}, {119, 912, 66047}, {12665, 66021, 119}

X(66050) = ORTHOLOGY CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT CTR9-2.11

Barycentrics    8*a^4+17*a^3*(b+c)-17*a*(b-c)^2*(b+c)+a^2*(-3*b^2+14*b*c-3*c^2)-5*(b^2-c^2)^2 : :
X(66050) = -6*X[3828]+5*X[51572]

Triangle CTR9-2.11 vertices are the barycentric sums of the corresponding vertices of the cevian triangle of X(2) and the anticevian triangle of X(11).

X(66050) lies on these lines: {1, 376}, {65, 27778}, {946, 13226}, {1125, 37545}, {3244, 24470}, {3306, 3646}, {3452, 11263}, {3671, 5122}, {3828, 51572}, {4640, 14150}, {4691, 5850}, {5183, 21620}, {5905, 11024}, {9948, 64119}, {10404, 11362}, {18480, 30424}, {31870, 66020}, {41869, 65383}

X(66051) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-2.11 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^7-4*a^6*(b+c)+(b-c)^4*(b+c)^3+a^5*(-3*b^2+4*b*c-3*c^2)+4*a^3*b*c*(-2*b^2+b*c-2*c^2)+a*(b^2-c^2)^2*(b^2+4*b*c+c^2)-2*a^2*(b-c)^2*(3*b^3+4*b^2*c+4*b*c^2+3*c^3)+a^4*(9*b^3+b^2*c+b*c^2+9*c^3) : :
X(66051) = 3*X[4]+X[9963], -5*X[100]+X[6361], -3*X[381]+X[12690], -5*X[631]+X[13243], -X[1768]+3*X[38760], X[5541]+3*X[50908], -3*X[6174]+X[12515], 3*X[6905]+X[17484], -X[9803]+5*X[64008], X[9809]+3*X[34474], -2*X[9956]+3*X[38758]

Triangle CTR9-2.11 vertices are the barycentric sums of the corresponding vertices of the cevian triangle of X(2) and the anticevian triangle of X(11).

X(66051) lies on these lines: {1, 5}, {3, 13257}, {4, 9963}, {7, 6911}, {9, 549}, {30, 908}, {78, 37406}, {100, 6361}, {104, 6883}, {140, 1071}, {149, 6849}, {153, 6827}, {214, 3452}, {224, 37356}, {381, 12690}, {515, 66052}, {528, 12611}, {550, 6259}, {631, 13243}, {912, 3911}, {936, 44222}, {942, 64475}, {1145, 3940}, {1512, 5844}, {1537, 12331}, {1768, 38760}, {1862, 15763}, {2771, 3035}, {2800, 31837}, {2801, 6666}, {2829, 22935}, {2932, 45393}, {3579, 35023}, {3652, 52793}, {3913, 18491}, {4304, 37290}, {4999, 56762}, {5433, 27778}, {5541, 50908}, {5692, 61524}, {5748, 6224}, {5761, 9802}, {5763, 51525}, {5770, 31188}, {5780, 38752}, {5806, 64192}, {5840, 21635}, {5843, 50573}, {5851, 31658}, {6001, 66053}, {6154, 12699}, {6174, 12515}, {6700, 13369}, {6702, 58463}, {6825, 64141}, {6848, 10698}, {6856, 59415}, {6858, 66045}, {6905, 17484}, {6970, 64142}, {7308, 64012}, {7682, 25485}, {8167, 38028}, {8257, 25558}, {8703, 31142}, {9803, 64008}, {9809, 34474}, {9844, 64476}, {9955, 66065}, {9956, 38758}, {9964, 61539}, {10090, 12831}, {10265, 58421}, {10609, 10742}, {10993, 34789}, {11108, 12773}, {11495, 12332}, {12619, 20400}, {12757, 66021}, {13411, 16617}, {15015, 38761}, {16128, 24466}, {17660, 66014}, {18228, 28466}, {18397, 34753}, {18516, 56177}, {18524, 51409}, {20117, 31659}, {25011, 64853}, {27065, 28465}, {27131, 28459}, {27385, 40263}, {28174, 44425}, {28452, 31053}, {29243, 34461}, {31835, 52265}, {31937, 59719}, {36922, 50823}, {38032, 38669}, {38042, 64335}, {38112, 61628}, {38602, 51506}, {44286, 64186}, {50205, 61566}, {52638, 61533}, {61562, 64193}

X(66051) = midpoint of X(i) and X(j) for these {i,j}: {3, 13257}, {11, 12738}, {119, 6326}, {1145, 48667}, {1537, 12331}, {5531, 37726}, {6154, 12699}, {6265, 37725}, {10609, 10742}, {10993, 34789}, {16128, 24466}, {18524, 51409}, {38665, 64138}
X(66051) = reflection of X(i) in X(j) for these {i,j}: {3579, 35023}, {10265, 58421}, {12619, 20400}, {13226, 140}, {64193, 61562}, {66065, 9955}
X(66051) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 13257, 18342}
X(66051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 12738, 952}, {5660, 6326, 119}, {5720, 37713, 5}

X(66052) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-2.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^7-2*a^6*(b+c)-3*(b-c)^4*(b+c)^3-a^5*(b^2-16*b*c+c^2)+a*(b^2-c^2)^2*(3*b^2-14*b*c+3*c^2)+a^4*(b^3-13*b^2*c-13*b*c^2+c^3)+4*a^2*(b-c)^2*(b^3+5*b^2*c+5*b*c^2+c^3)-2*a^3*(2*b^4+b^3*c-12*b^2*c^2+b*c^3+2*c^4) : :
X(66052) = -9*X[2]+5*X[104], -5*X[11]+7*X[3851], -5*X[100]+X[3529], -2*X[140]+3*X[38758], -5*X[149]+13*X[61982], -5*X[1484]+9*X[38071], -5*X[3035]+4*X[3530], -7*X[3528]+5*X[38761], -17*X[3544]+15*X[23513], -X[3632]+5*X[12751], -4*X[3636]+5*X[11729], -X[3644]+5*X[66057] and many others

CTR9-2.2 is the triangle homothetic to ABC with center X(2) and ratio 5/4.

X(66052) lies on these lines: {2, 104}, {11, 3851}, {80, 11529}, {100, 3529}, {140, 38758}, {149, 61982}, {355, 7700}, {382, 5840}, {515, 66051}, {528, 15687}, {546, 946}, {550, 2829}, {1145, 16128}, {1478, 64341}, {1484, 38071}, {2771, 9947}, {2800, 3626}, {2801, 60980}, {3035, 3530}, {3036, 47320}, {3528, 38761}, {3544, 23513}, {3632, 12751}, {3636, 11729}, {3644, 66057}, {3855, 10597}, {3870, 12690}, {3982, 12736}, {5079, 12773}, {5083, 12019}, {5818, 13243}, {5841, 44425}, {5884, 18357}, {6174, 15688}, {6326, 18528}, {6667, 51529}, {6745, 9945}, {6929, 64735}, {9956, 13226}, {10299, 12248}, {10698, 20050}, {10707, 61967}, {10724, 62017}, {10728, 10993}, {10738, 14269}, {10759, 11008}, {11019, 38140}, {11715, 15808}, {11737, 60759}, {12138, 52285}, {12619, 50238}, {13199, 62042}, {14869, 20400}, {15017, 38032}, {15681, 24466}, {15700, 38762}, {15720, 38752}, {16205, 34747}, {20418, 35018}, {21154, 55863}, {28186, 54192}, {31235, 61850}, {34200, 38759}, {34474, 62097}, {34641, 50906}, {38665, 50688}, {38693, 38763}, {38754, 62074}, {40341, 66030}, {45310, 47478}, {51525, 62044}, {57298, 61905}, {59377, 61928}, {60957, 66023}, {61566, 61894}

X(66052) = midpoint of X(i) and X(j) for these {i,j}: {119, 153}, {355, 13257}, {382, 6154}, {1145, 16128}, {10728, 10993}, {10742, 37725}, {12331, 52836}, {24466, 38756}, {38665, 64186}
X(66052) = reflection of X(i) in X(j) for these {i,j}: {104, 58421}, {550, 35023}, {6713, 119}, {13226, 9956}, {20418, 61580}, {38602, 20400}, {38757, 61605}, {38759, 61562}, {51529, 6667}, {66065, 546}
X(66052) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {24466, 38756, 63407}
X(66052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {104, 119, 58421}, {104, 58421, 6713}, {153, 10711, 119}, {546, 952, 66065}, {952, 61605, 38757}, {2829, 35023, 550}, {10742, 12331, 52836}, {10742, 37725, 5840}, {20418, 61580, 38319}, {37725, 52836, 12331}

X(66053) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR9-2.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^10-2*a^9*(b+c)+(b-c)^6*(b+c)^4+a^8*(-9*b^2+8*b*c-9*c^2)-2*a*(b-c)^4*(b+c)^3*(2*b^2-3*b*c+2*c^2)-4*a^2*b*c*(b^2-c^2)^2*(5*b^2-8*b*c+5*c^2)+2*a^7*(5*b^3+7*b^2*c+7*b*c^2+5*c^3)-2*a^4*(b-c)^2*(4*b^4-19*b^3*c-36*b^2*c^2-19*b*c^3+4*c^4)+2*a^6*(7*b^4-20*b^3*c-4*b^2*c^2-20*b*c^3+7*c^4)-6*a^5*(3*b^5+2*b^4*c-7*b^3*c^2-7*b^2*c^3+2*b*c^4+3*c^5)+2*a^3*(b-c)^2*(7*b^5+9*b^4*c-18*b^3*c^2-18*b^2*c^3+9*b*c^4+7*c^5) : :
X(66053) = -5*X[100]+X[64144], -3*X[38758]+2*X[64813], -5*X[64008]+X[66060]

CTR9-2.2 is the triangle homothetic to ABC with center X(2) and ratio 5/4.

X(66053) lies on these lines: {1, 13226}, {9, 119}, {100, 64144}, {952, 3913}, {1125, 2800}, {1145, 6244}, {1158, 10942}, {1387, 17626}, {1537, 36279}, {1768, 10956}, {2077, 9945}, {2829, 3579}, {3035, 31787}, {3256, 9952}, {5128, 34789}, {5552, 13257}, {5787, 6154}, {6001, 66051}, {6256, 61524}, {6735, 17613}, {10269, 64109}, {10528, 13243}, {10679, 14647}, {10915, 34862}, {11231, 12608}, {12019, 63266}, {12690, 64078}, {12751, 24466}, {18542, 64190}, {31794, 64192}, {35023, 64804}, {35445, 64145}, {38758, 64813}, {55297, 55301}, {64008, 66060}, {64191, 64951}

X(66053) = midpoint of X(i) and X(j) for these {i,j}: {119, 2950}, {5787, 6154}, {16128, 52116}
X(66053) = reflection of X(i) in X(j) for these {i,j}: {64804, 35023}

X(66054) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-7 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^10*(b+c)-(b-c)^6*(b+c)^3*(b^2-b*c+c^2)+2*a*(b-c)^6*(b+c)^2*(2*b^2+b*c+2*c^2)-2*a^9*(2*b^2+3*b*c+2*c^2)+3*a^8*(b^3+4*b^2*c+4*b*c^2+c^3)+2*a^7*(4*b^4-4*b^3*c-7*b^2*c^2-4*b*c^3+4*c^4)+4*a^5*b*c*(5*b^4-2*b^3*c-3*b^2*c^2-2*b*c^3+5*c^4)+2*a^4*(b-c)^2*(7*b^5+2*b^4*c-5*b^3*c^2-5*b^2*c^3+2*b*c^4+7*c^5)-2*a^6*(7*b^5+3*b^4*c-6*b^3*c^2-6*b^2*c^3+3*b*c^4+7*c^5)-2*a^3*(b-c)^2*(4*b^6+4*b^5*c-3*b^4*c^2-8*b^3*c^3-3*b^2*c^4+4*b*c^5+4*c^6)-a^2*(b-c)^2*(3*b^7-7*b^6*c-b^5*c^2+13*b^4*c^3+13*b^3*c^4-b^2*c^5-7*b*c^6+3*c^7)) : :
X(66054) = -3*X[5817]+X[12532], -2*X[18254]+3*X[38108], -3*X[38032]+4*X[58564], -3*X[38053]+4*X[58604]

Triangle CTR1-7 is defined by the Aubert (Steiner) lines of quadrilaterals ABPC, BCPA, CAPB, where P=X(7).

X(66054) lies on these lines: {4, 12755}, {11, 5173}, {119, 518}, {517, 25606}, {952, 15185}, {971, 11570}, {1159, 2800}, {2801, 5805}, {3868, 66023}, {5817, 12532}, {5856, 24474}, {6594, 25485}, {7672, 10698}, {10427, 66047}, {10728, 12669}, {11372, 11571}, {11715, 20116}, {12738, 22753}, {18254, 38108}, {18861, 60948}, {38032, 58564}, {38053, 58604}, {57278, 66056}

X(66054) = midpoint of X(i) and X(j) for these {i,j}: {4, 12755}, {3868, 66023}, {7672, 10698}, {10728, 12669}, {11372, 11571}
X(66054) = reflection of X(i) in X(j) for these {i,j}: {10427, 66047}, {11715, 20116}

X(66055) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR4-100 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^9-2*a^8*(b+c)-b*(b-c)^4*c*(b+c)^3+a^5*b*c*(-17*b^2+18*b*c-17*c^2)+a^7*(-2*b^2+9*b*c-2*c^2)+a^2*(b-c)^4*(2*b^3+3*b^2*c+3*b*c^2+2*c^3)+a^6*(6*b^3-3*b^2*c-3*b*c^2+6*c^3)-a*(b^2-c^2)^2*(b^4-b^3*c+4*b^2*c^2-b*c^3+c^4)+a^3*(b-c)^2*(2*b^4+11*b^3*c+6*b^2*c^2+11*b*c^3+2*c^4)-a^4*(6*b^5-11*b^4*c+b^3*c^2+b^2*c^3-11*b*c^4+6*c^5)) : :
X(66055) = -2*X[6256]+3*X[10711], -4*X[6705]+3*X[11219], -3*X[10707]+4*X[63980]

Triangle CTR4-100 is defined as follows. Let DEF be cevian triangle of X(100). AD intersects the circle (AEF) at A1 different from A. Define B1, C1 cyclically, then CTR4-100 is the triangle A1B1C1. It is similar to ABC.

X(66055) lies on these lines: {1, 104}, {3, 1633}, {4, 55966}, {11, 1466}, {20, 100}, {21, 54442}, {78, 12666}, {80, 59329}, {84, 2801}, {119, 6850}, {145, 12114}, {214, 12520}, {404, 64119}, {515, 5537}, {516, 48713}, {528, 64074}, {912, 56941}, {946, 64155}, {952, 3189}, {962, 13279}, {971, 12738}, {997, 12686}, {1012, 14647}, {1156, 66020}, {1376, 38757}, {1413, 66036}, {1537, 4295}, {1770, 10090}, {2077, 6745}, {2096, 8069}, {2475, 12761}, {2771, 17649}, {2932, 13257}, {3035, 6908}, {3065, 44861}, {3651, 5660}, {4189, 22775}, {4305, 64191}, {4511, 6001}, {5010, 5924}, {5440, 48697}, {5840, 6851}, {6174, 37427}, {6245, 49176}, {6256, 10711}, {6261, 37403}, {6700, 21635}, {6705, 11219}, {6713, 6892}, {6736, 12751}, {6845, 59391}, {6888, 31272}, {6895, 10724}, {6937, 64008}, {6940, 12608}, {6950, 14646}, {7971, 63983}, {9809, 10309}, {10087, 64145}, {10609, 12330}, {10707, 63980}, {11248, 34619}, {11496, 14986}, {11698, 33898}, {12515, 31788}, {12616, 21669}, {12675, 14151}, {12680, 41701}, {12737, 13600}, {13243, 18238}, {13601, 17654}, {18237, 51636}, {18419, 62873}, {18861, 40293}, {26333, 47744}, {31730, 64280}, {34772, 65998}, {35238, 64148}, {37401, 38752}, {37434, 45043}, {37560, 46684}, {37561, 50908}, {38697, 61221}, {43178, 52026}, {54052, 63168}, {56288, 64189}

X(66055) = reflection of X(i) in X(j) for these {i,j}: {100, 12332}, {104, 48695}, {12667, 37725}, {33898, 11698}, {38669, 12114}, {46435, 21635}, {48694, 5450}, {48697, 5440}, {49176, 6245}, {64267, 11715}, {66058, 46684}
X(66055) = pole of line {6366, 53305} with respect to the circumcircle
X(66055) = X(84) of anti-inner-Garcia triangle
X(66055) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(1795), X(55966)}}, {{A, B, C, X(6001), X(37725)}}, {{A, B, C, X(15501), X(34894)}}
X(66055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {78, 49171, 12666}, {153, 7080, 37725}, {1768, 10058, 104}, {2800, 11715, 64267}, {2800, 5450, 48694}, {2829, 37725, 12667}, {10310, 37725, 100}, {48694, 48695, 5450}

X(66056) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(1)-CIRCUMCONCEVIAN-OF-X(7)) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^11-3*a^10*(b+c)-a^9*(b^2-9*b*c+c^2)-(b-c)^6*(b+c)^3*(b^2+b*c+c^2)+a^8*(11*b^3-b^2*c-b*c^2+11*c^3)+a*(b-c)^4*(b+c)^2*(3*b^4-2*b^3*c+2*b^2*c^2-2*b*c^3+3*c^4)-a^7*(6*b^4+23*b^3*c+8*b^2*c^2+23*b*c^3+6*c^4)+a^6*(-14*b^5+23*b^4*c+25*b^3*c^2+25*b^2*c^3+23*b*c^4-14*c^5)+a^4*(b-c)^2*(6*b^5-17*b^4*c-53*b^3*c^2-53*b^2*c^3-17*b*c^4+6*c^5)-a^3*(b-c)^2*(11*b^6+11*b^5*c-5*b^4*c^2-26*b^3*c^3-5*b^2*c^4+11*b*c^5+11*c^6)+a^5*(14*b^6+11*b^5*c-10*b^4*c^2-54*b^3*c^3-10*b^2*c^4+11*b*c^5+14*c^6)+a^2*(b-c)^2*(b^7+10*b^6*c+6*b^5*c^2-b^4*c^3-b^3*c^4+6*b^2*c^5+10*b*c^6+c^7)) : :
X(66056) = -5*X[18230]+X[66060]

X(66056) lies on these lines: {9, 119}, {84, 66010}, {100, 971}, {104, 2346}, {480, 12665}, {518, 48695}, {952, 3358}, {1001, 2800}, {1158, 5851}, {1445, 1537}, {1768, 15298}, {2801, 6600}, {5728, 12775}, {6594, 64156}, {11372, 59390}, {17654, 53055}, {18230, 66060}, {24466, 58808}, {41166, 64338}, {57278, 66054}, {60970, 64189}, {64188, 65405}

X(66056) = midpoint of X(i) and X(j) for these {i,j}: {9, 2950}, {84, 66010}
X(66056) = reflection of X(i) in X(j) for these {i,j}: {64156, 6594}, {64188, 65405}

X(66057) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(10)-CIRCUMCONCEVIAN-OF-X(37)) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^6*b*c+a^5*(b^3+2*b^2*c+2*b*c^2+c^3)-(b^2-c^2)^2*(b^4-b^2*c^2+c^4)-a^4*(b^4+6*b^3*c+b^2*c^2+6*b*c^3+c^4)+a^2*(b-c)^2*(2*b^4+9*b^3*c+10*b^2*c^2+9*b*c^3+2*c^4)+a*(b-c)^2*(b^5-2*b^4*c-7*b^3*c^2-7*b^2*c^3-2*b*c^4+c^5)-2*a^3*(b^5-b^4*c-2*b^3*c^2-2*b^2*c^3-b*c^4+c^5)) : :
X(66057) = X[3644]+4*X[66052], -4*X[3739]+5*X[64008], -5*X[4687]+4*X[6713], -5*X[4699]+7*X[66045], -5*X[4704]+X[64009], -7*X[4751]+8*X[58421], -2*X[30271]+3*X[34474], -3*X[38752]+2*X[64728], -3*X[57298]+4*X[61522]

X(66057) lies on these lines: {37, 104}, {75, 119}, {153, 192}, {518, 10698}, {536, 10711}, {537, 50908}, {726, 21635}, {740, 12751}, {742, 66030}, {952, 20430}, {984, 2800}, {2801, 51058}, {2805, 38665}, {2829, 30273}, {3644, 66052}, {3739, 64008}, {4687, 6713}, {4699, 66045}, {4704, 64009}, {4751, 58421}, {5840, 51063}, {6174, 51044}, {7201, 12736}, {10707, 51038}, {10742, 29010}, {12332, 34247}, {13253, 49448}, {25485, 49490}, {29054, 34789}, {30271, 34474}, {38752, 64728}, {57298, 61522}

X(66057) = midpoint of X(i) and X(j) for these {i,j}: {153, 192}, {13253, 49448}
X(66057) = reflection of X(i) in X(j) for these {i,j}: {75, 119}, {104, 37}, {10707, 51038}, {30273, 51062}, {49490, 25485}, {51044, 6174}, {66067, 20430}
X(66057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 20430, 66067}, {2829, 51062, 30273}, {10698, 66023, 10759}

X(66058) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-1.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^9-a^8*(b+c)+4*a^2*(b-c)^4*(b+c)^3-(b-c)^6*(b+c)^3+a^7*(-4*b^2+b*c-4*c^2)+4*a^6*(b^3+b^2*c+b*c^2+c^3)-2*a^4*(b-c)^2*(3*b^3+7*b^2*c+7*b*c^2+3*c^3)+a*(b^2-c^2)^2*(b^4-5*b^3*c+12*b^2*c^2-5*b*c^3+c^4)+a^5*(6*b^4-7*b^3*c-6*b^2*c^2-7*b*c^3+6*c^4)+a^3*(-4*b^6+11*b^5*c-14*b^3*c^3+11*b*c^5-4*c^6)) : :
X(66058) = -3*X[165]+2*X[12332]

Let MaMbMc be the medial triangle. CTR12-1.2 is the triangle with vertices at the inversion poles of MbMc, MaMc, and MaMb wrt to the X(1)-circumconic.

X(66058) lies on circumconic {{A, B, C, X(36100), X(46435)}} and on these lines: {1, 22775}, {4, 64372}, {9, 119}, {11, 12858}, {40, 78}, {46, 80}, {57, 104}, {63, 153}, {144, 64148}, {165, 12332}, {191, 18242}, {227, 66036}, {484, 6001}, {515, 3218}, {517, 64267}, {518, 66062}, {908, 66060}, {912, 66061}, {952, 5709}, {1001, 58613}, {1158, 2475}, {1317, 7966}, {1445, 10265}, {1490, 2771}, {1697, 10698}, {1709, 12761}, {1727, 41698}, {2093, 17654}, {2323, 66029}, {2787, 24469}, {2801, 60990}, {2932, 6282}, {3035, 61122}, {3220, 9913}, {3305, 66045}, {3333, 11715}, {3336, 12114}, {3576, 64359}, {3587, 33814}, {3929, 10711}, {5119, 13253}, {5128, 12691}, {5220, 58687}, {5227, 66030}, {5251, 64118}, {5437, 6713}, {5531, 5904}, {5536, 7993}, {5541, 41338}, {5720, 40266}, {5727, 12248}, {5903, 59366}, {6260, 9809}, {6264, 12704}, {6596, 37531}, {6769, 13205}, {7171, 38753}, {7308, 64008}, {7330, 10742}, {7686, 15932}, {7951, 64119}, {7972, 65129}, {8068, 12705}, {8580, 58666}, {9841, 38761}, {9897, 49170}, {10058, 59335}, {10175, 61012}, {10980, 58595}, {11698, 26921}, {12331, 37584}, {12514, 21635}, {12616, 18406}, {12650, 12773}, {12751, 57279}, {12762, 41229}, {12764, 30223}, {13528, 58328}, {15015, 59340}, {15737, 64761}, {17638, 63992}, {17699, 63281}, {18237, 37567}, {18397, 56889}, {18443, 66047}, {18491, 31828}, {18540, 22799}, {18802, 63137}, {20420, 64265}, {24468, 64743}, {25485, 31393}, {31018, 40256}, {34256, 55931}, {34474, 37551}, {36922, 63132}, {37526, 38693}, {37534, 38602}, {37560, 46684}, {38036, 63254}, {48695, 59333}, {51768, 65948}, {51780, 58421}, {62354, 64261}, {63430, 64145}

X(66058) = reflection of X(i) in X(j) for these {i,j}: {1, 22775}, {84, 1768}, {2950, 12515}, {5531, 11500}, {6264, 48694}, {6326, 64188}, {6769, 13205}, {9809, 6260}, {12650, 12773}, {64261, 62354}, {66055, 46684}, {66068, 5709}
X(66058) = inverse of X(102) in the Bevan circle
X(66058) = X(2931) of excentral triangle
X(66058) = pole of line {102, 104} with respect to the Bevan circle
X(66058) = pole of line {16548, 66068} with respect to the Gheorghe circle
X(66058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 5709, 66068}, {1768, 2829, 84}, {2800, 64188, 6326}, {2950, 46435, 12686}, {6326, 64188, 52026}

X(66059) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-1.8 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^6-3*a^3*b*c*(b+c)+3*a*b*(b-c)^2*c*(b+c)-(b-c)^2*(b+c)^4+a^4*(-3*b^2+7*b*c-3*c^2)+a^2*(3*b^4-5*b^3*c+8*b^2*c^2-5*b*c^3+3*c^4)) : :
X(66059) = -2*X[119]+3*X[11219], -3*X[165]+2*X[12331], -4*X[1387]+3*X[50908], -4*X[1484]+3*X[1699], -5*X[1698]+4*X[11698], -3*X[2077]+2*X[3689], -7*X[3624]+8*X[61566], -5*X[5071]+4*X[50909], -3*X[5131]+2*X[18524], -3*X[5660]+4*X[6713], -3*X[6909]+X[62236], -5*X[7987]+4*X[22935] and many others

Let QaQbQc be the cevian triangle of X(8). CTR12-1.8 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(1)-circumconic.

X(66059) lies on the 2nd Evans circle and on these lines: {1, 399}, {3, 3711}, {9, 48}, {11, 3333}, {30, 51463}, {40, 550}, {46, 9897}, {57, 80}, {63, 6224}, {84, 1320}, {90, 56036}, {100, 4652}, {119, 11219}, {149, 41869}, {153, 3306}, {165, 12331}, {191, 34773}, {200, 2932}, {484, 28204}, {515, 3218}, {517, 7993}, {528, 58808}, {912, 4867}, {946, 9809}, {971, 64264}, {999, 60884}, {1012, 42871}, {1158, 3895}, {1317, 31393}, {1387, 50908}, {1484, 1699}, {1490, 22775}, {1537, 5851}, {1697, 7972}, {1698, 11698}, {1706, 15863}, {1709, 12737}, {2077, 3689}, {2093, 17636}, {2802, 6762}, {2829, 10864}, {2886, 66017}, {2950, 12703}, {2975, 16132}, {3036, 5794}, {3059, 7688}, {3219, 51705}, {3220, 9912}, {3336, 18525}, {3337, 18480}, {3338, 10742}, {3339, 6797}, {3340, 11571}, {3359, 19914}, {3464, 12407}, {3612, 41689}, {3624, 61566}, {3646, 34123}, {3881, 21669}, {3929, 64011}, {4654, 33593}, {5071, 50909}, {5131, 18524}, {5219, 66012}, {5258, 13369}, {5289, 5693}, {5437, 6702}, {5536, 28160}, {5563, 40263}, {5660, 6713}, {5691, 62354}, {5791, 13226}, {5881, 12247}, {5902, 18519}, {6001, 64267}, {6211, 56807}, {6265, 7330}, {6765, 13205}, {6909, 62236}, {7280, 35451}, {7308, 64012}, {7987, 22935}, {7989, 38755}, {8227, 21635}, {8580, 58659}, {8666, 64358}, {9355, 32486}, {9616, 35882}, {9802, 28194}, {9841, 38665}, {9845, 59347}, {9956, 35010}, {10057, 59335}, {10058, 37736}, {10074, 61762}, {10165, 35595}, {10389, 63281}, {10476, 13244}, {10529, 16127}, {10698, 12705}, {10860, 64189}, {10980, 58587}, {11010, 18526}, {11012, 12680}, {11014, 15071}, {11525, 39776}, {11529, 11570}, {11715, 64260}, {12248, 12625}, {12332, 52027}, {12514, 33337}, {12531, 63137}, {12611, 18540}, {12619, 37534}, {12687, 45632}, {12699, 64289}, {12738, 15015}, {12739, 27778}, {12740, 30223}, {12743, 54408}, {12747, 37532}, {13257, 20418}, {14217, 63974}, {14872, 37561}, {15017, 57298}, {15079, 18542}, {15096, 22758}, {17437, 53616}, {17857, 59332}, {18398, 18761}, {18518, 37524}, {18976, 37550}, {18991, 35856}, {18992, 35857}, {20095, 31730}, {21630, 31162}, {22560, 50528}, {22791, 64740}, {22936, 26089}, {23958, 50864}, {24390, 49178}, {25524, 58683}, {27003, 50796}, {27065, 50828}, {30282, 41541}, {31871, 45977}, {34628, 37584}, {34789, 37726}, {35638, 39552}, {37234, 50190}, {37612, 37714}, {37618, 45764}, {38617, 63911}, {38631, 64742}, {38753, 41338}, {45043, 60938}, {46681, 53055}, {47034, 57282}, {48713, 54441}, {50907, 51781}, {51780, 58453}, {54370, 61275}, {58609, 63266}, {60936, 63993}, {61261, 61605}, {63143, 64129}

X(66059) = midpoint of X(i) and X(j) for these {i,j}: {7993, 12767}, {9803, 64009}, {13243, 38669}
X(66059) = reflection of X(i) in X(j) for these {i,j}: {1, 12773}, {40, 1768}, {153, 10265}, {1490, 22775}, {5531, 3}, {5541, 12515}, {5691, 62354}, {5881, 12247}, {6264, 38669}, {6265, 51529}, {6326, 104}, {6765, 13205}, {7982, 6264}, {9809, 946}, {12738, 38602}, {13253, 12737}, {13257, 20418}, {16128, 1484}, {20095, 31730}, {34789, 37726}, {37725, 13226}, {38665, 46684}, {41869, 149}, {64278, 9803}, {64742, 38631}, {66068, 62858}
X(66059) = inverse of X(12515) in Bevan circle
X(66059) = X(399) of excentral triangle
X(66059) = X(3448) of hexyl triangle
X(66059) = X(6361) of anti-inner-Garcia triangle
X(66059) = pole of line {900, 12515} with respect to the Bevan circle
X(66059) = pole of line {8674, 14288} with respect to the Conway circle
X(66059) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(909), X(28193)}}, {{A, B, C, X(3065), X(52663)}}
X(66059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {104, 2801, 6326}, {104, 6326, 3576}, {153, 10265, 5587}, {515, 9803, 64278}, {952, 12515, 5541}, {1484, 16128, 1699}, {1768, 5541, 12515}, {2771, 12773, 1}, {2800, 38669, 6264}, {2800, 6264, 7982}, {5541, 12515, 40}, {9803, 64009, 515}, {10742, 37718, 18492}, {12737, 13253, 16200}, {12738, 38602, 15015}, {13243, 38669, 2800}, {18540, 51816, 38021}

X(66060) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-2.85 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^10+9*a^7*b*c*(b+c)-5*a*b*(b-c)^4*c*(b+c)^3-(b-c)^6*(b+c)^4+a^8*(-5*b^2+3*b*c-5*c^2)+a^5*b*c*(-23*b^3+27*b^2*c+27*b*c^2-23*c^3)+a^3*b*(b-c)^2*c*(19*b^3-3*b^2*c-3*b*c^2+19*c^3)+5*a^6*(2*b^4-3*b^3*c-2*b^2*c^2-3*b*c^3+2*c^4)+a^2*(b^2-c^2)^2*(5*b^4-13*b^3*c+8*b^2*c^2-13*b*c^3+5*c^4)-a^4*(b-c)^2*(10*b^4-3*b^3*c-30*b^2*c^2-3*b*c^3+10*c^4) : :
X(66060) = -3*X[2]+2*X[2950], -5*X[3616]+4*X[48695], -3*X[5658]+2*X[12331], -3*X[9778]+4*X[64188], -4*X[12761]+3*X[59387], -2*X[13199]+3*X[54051], -5*X[18230]+4*X[66056], -5*X[64008]+4*X[66053]

Let QaQbQc be the cevian triangle of X(85). CTR12-2.85 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the Steiner circumconic.

X(66060) lies on these lines: {2, 2950}, {4, 6797}, {7, 104}, {8, 153}, {80, 64130}, {149, 9799}, {214, 63971}, {329, 55016}, {515, 9802}, {908, 66058}, {952, 6223}, {962, 1320}, {1158, 3306}, {1490, 20095}, {1519, 37789}, {1737, 12767}, {1768, 3086}, {2476, 11024}, {3616, 48695}, {4295, 12736}, {4345, 12248}, {5082, 17661}, {5328, 12515}, {5531, 54227}, {5658, 12331}, {5703, 12775}, {5811, 38755}, {5853, 66061}, {6001, 9803}, {9778, 64188}, {9785, 64191}, {10580, 15528}, {11037, 64192}, {11415, 17100}, {12246, 12773}, {12743, 64321}, {12761, 59387}, {13199, 54051}, {14450, 64120}, {14986, 45655}, {18228, 64193}, {18230, 66056}, {22775, 64190}, {24466, 64696}, {25005, 54156}, {30305, 64145}, {33898, 66008}, {36845, 66002}, {38460, 64009}, {64008, 66053}

X(66060) = reflection of X(i) in X(j) for these {i,j}: {153, 46435}, {5531, 54227}, {9799, 149}, {9809, 63962}, {12246, 12773}, {20095, 1490}, {64009, 64267}, {64190, 22775}, {66008, 33898}
X(66060) = anticomplement of X(2950)
X(66060) = X(2950) of anticomplementary triangle
X(66060) = X(5504) of 2nd-Conway triangle
X(66060) = X(i)-Dao conjugate of X(j) for these {i, j}: {2950, 2950}
X(66060) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2800, 46435, 153}, {2800, 63962, 9809}

X(66061) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-9.1 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^9-5*a^8*(b+c)+(b-c)^4*(b+c)^3*(3*b^2+2*b*c+3*c^2)+a^7*(4*b^2+9*b*c+4*c^2)+4*a^6*(3*b^3-2*b^2*c-2*b*c^2+3*c^3)-3*a^5*(6*b^4+b^3*c+2*b^2*c^2+b*c^3+6*c^4)-a*(b^2-c^2)^2*(7*b^4-15*b^3*c+4*b^2*c^2-15*b*c^3+7*c^4)-6*a^4*(b^5-5*b^4*c-5*b*c^4+c^5)-4*a^2*(b-c)^2*(b^5+6*b^4*c+7*b^3*c^2+7*b^2*c^3+6*b*c^4+c^5)+a^3*(20*b^6-21*b^5*c-8*b^4*c^2+2*b^3*c^3-8*b^2*c^4-21*b*c^5+20*c^6)) : :
X(66061) = -2*X[2950]+3*X[3158], -3*X[3928]+4*X[64188]

Let QaQbQc be the cevian triangle of X(1). CTR12-9.1 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(9)-circumconic.

X(66061) lies on these lines: {1, 17661}, {9, 48}, {57, 66002}, {153, 5727}, {912, 66058}, {952, 3680}, {1490, 66068}, {1537, 3243}, {1697, 66024}, {2136, 2800}, {2771, 5534}, {2829, 11523}, {2932, 30304}, {2950, 3158}, {3928, 64188}, {5437, 15528}, {5531, 5687}, {5853, 66060}, {6001, 66062}, {7982, 10728}, {7992, 13205}, {9803, 64115}, {12528, 64372}, {12767, 48696}, {15829, 64191}, {18443, 66049}, {34789, 41863}, {62218, 64193}

X(66061) = reflection of X(i) in X(j) for these {i,j}: {7992, 13205}, {66068, 1490}

X(66062) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-9.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a^9+5*a^7*b*c-3*a^8*(b+c)+(b-c)^4*(b+c)^5-2*a^2*b*(b-c)^2*c*(3*b^3+17*b^2*c+17*b*c^2+3*c^3)+2*a^6*(4*b^3+b^2*c+b*c^2+4*c^3)-a*(b^2-c^2)^2*(3*b^4-b^3*c-12*b^2*c^2-b*c^3+3*c^4)-a^5*(6*b^4+9*b^3*c+22*b^2*c^2+9*b*c^3+6*c^4)-6*a^4*(b^5-b^4*c-4*b^3*c^2-4*b^2*c^3-b*c^4+c^5)+a^3*(8*b^6+3*b^5*c+4*b^4*c^2-46*b^3*c^3+4*b^2*c^4+3*b*c^5+8*c^6)) : :
X(66062) = -3*X[3158]+2*X[12332], -2*X[22560]+3*X[52026]

Let QaQbQc be the medial triangle. CTR12-9.2 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(9)-circumconic.

X(66062) lies on these lines: {1, 5}, {84, 13205}, {100, 10270}, {104, 200}, {153, 3870}, {518, 66058}, {519, 64267}, {528, 42470}, {912, 12767}, {936, 11715}, {1001, 58687}, {1145, 30503}, {1320, 16205}, {1490, 2802}, {1750, 14217}, {2057, 38669}, {2077, 3689}, {2771, 49163}, {2800, 6765}, {2801, 2950}, {2829, 6769}, {2900, 12641}, {2932, 63430}, {3158, 12332}, {3359, 12331}, {3935, 64009}, {4326, 66023}, {4666, 66045}, {5437, 58595}, {5840, 63981}, {6001, 66061}, {6282, 64145}, {6713, 8580}, {6735, 9803}, {6762, 22775}, {9913, 40910}, {10582, 64008}, {10679, 60884}, {10728, 12651}, {10738, 18528}, {11500, 66068}, {12565, 64136}, {12653, 63988}, {12705, 17661}, {14872, 64372}, {17654, 63137}, {18446, 66008}, {18529, 65948}, {22560, 52026}, {30350, 58604}, {30393, 58674}, {34474, 64679}, {37561, 64116}, {38752, 64668}, {42871, 58613}, {58663, 61122}, {58666, 62218}, {64150, 64743}

X(66062) = reflection of X(i) in X(j) for these {i,j}: {84, 13205}, {2950, 25438}, {5531, 5534}, {6762, 22775}, {66068, 11500}
X(66062) = X(5531) of anti-outer-Yff triangle
X(66062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {952, 5534, 5531}, {2801, 25438, 2950}, {6264, 6326, 12740}

X(66063) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR5-2.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a^3-a^2*(b+c)+3*(b-c)^2*(b+c)+a*(-3*b^2+7*b*c-3*c^2) : :
X(66063) = -2*X[1]+9*X[32558], 3*X[2]+4*X[11], -5*X[4]+12*X[38141], -8*X[5]+X[153], -X[8]+8*X[6702], 6*X[10]+X[12653], -X[20]+8*X[6713], 2*X[80]+5*X[3616], 2*X[104]+5*X[3091], -4*X[119]+11*X[5056], -8*X[140]+X[13199], -X[144]+8*X[64738] and many others

CTR5-2.2 is the triangle homothetic to ABC with center X(2) and ratio 2/7.

X(66063) lies on these lines: {1, 32558}, {2, 11}, {4, 38141}, {5, 153}, {8, 6702}, {10, 12653}, {20, 6713}, {80, 3616}, {88, 62221}, {104, 3091}, {119, 5056}, {140, 13199}, {144, 64738}, {145, 1387}, {210, 58611}, {214, 5550}, {354, 58683}, {376, 22938}, {377, 51636}, {381, 12248}, {404, 10593}, {499, 5046}, {549, 48680}, {551, 9897}, {631, 10738}, {632, 61601}, {952, 3090}, {956, 4193}, {962, 16174}, {1023, 26074}, {1125, 6224}, {1145, 46933}, {1156, 38205}, {1317, 10588}, {1320, 3617}, {1484, 1656}, {1537, 6956}, {1647, 33148}, {1698, 21630}, {1768, 3817}, {1862, 8889}, {2475, 7741}, {2486, 27342}, {2771, 61268}, {2802, 9780}, {2805, 4751}, {2829, 3832}, {2932, 17531}, {2975, 3847}, {3036, 3621}, {3085, 5533}, {3086, 5154}, {3146, 38693}, {3241, 15863}, {3243, 30852}, {3254, 18230}, {3305, 66068}, {3306, 64372}, {3315, 37691}, {3485, 20118}, {3522, 10724}, {3523, 5840}, {3525, 33814}, {3533, 38762}, {3543, 38761}, {3544, 51529}, {3545, 10742}, {3582, 5080}, {3583, 36004}, {3618, 66037}, {3619, 9024}, {3620, 10755}, {3623, 12531}, {3628, 12331}, {3634, 5541}, {3825, 5251}, {3839, 10728}, {3850, 38756}, {3855, 22799}, {3868, 58587}, {3877, 6797}, {3890, 17636}, {3917, 58539}, {3957, 64676}, {4188, 10058}, {4189, 10090}, {4430, 5748}, {4666, 5531}, {4678, 5854}, {4699, 66067}, {4857, 20107}, {4928, 38325}, {4996, 16865}, {5055, 11698}, {5057, 61649}, {5059, 38759}, {5067, 38752}, {5068, 20418}, {5070, 61562}, {5071, 38084}, {5076, 38637}, {5083, 5226}, {5087, 17484}, {5141, 39692}, {5219, 30318}, {5223, 27131}, {5225, 37307}, {5253, 7173}, {5260, 22560}, {5328, 46694}, {5422, 66036}, {5433, 15680}, {5528, 58433}, {5603, 12619}, {5704, 12736}, {5731, 6246}, {5775, 26129}, {5818, 12737}, {5848, 51171}, {5886, 12247}, {5889, 58508}, {6264, 10175}, {6622, 12138}, {6681, 65140}, {6856, 34123}, {6859, 11729}, {6879, 10698}, {6894, 63963}, {6904, 47744}, {6933, 12019}, {6952, 64792}, {6979, 18491}, {7288, 13273}, {7485, 13222}, {7486, 10587}, {7705, 11373}, {7972, 38314}, {7988, 21635}, {8047, 56365}, {8164, 12735}, {8166, 52682}, {8227, 10265}, {8972, 19113}, {9345, 17717}, {9669, 17566}, {9779, 34789}, {9802, 19877}, {9809, 11219}, {9812, 46684}, {9956, 66008}, {10006, 17494}, {10074, 10590}, {10171, 15017}, {10246, 61553}, {10303, 34474}, {10595, 19914}, {10711, 61924}, {10896, 37256}, {10993, 61856}, {11002, 58475}, {11230, 62354}, {11451, 58504}, {11604, 15674}, {11681, 63270}, {11715, 59387}, {12119, 54445}, {12747, 38028}, {13226, 38107}, {13595, 54065}, {13902, 49241}, {13941, 19112}, {13959, 49240}, {14217, 38133}, {15015, 19862}, {15022, 38669}, {15325, 20067}, {15677, 56790}, {15692, 38069}, {15717, 24466}, {16468, 29662}, {16859, 51506}, {17100, 17572}, {17483, 17728}, {17533, 54391}, {17570, 48713}, {17578, 59390}, {17605, 26842}, {17660, 64149}, {18240, 18412}, {18398, 47320}, {19632, 28222}, {20070, 64193}, {23343, 27290}, {24465, 64142}, {25005, 50443}, {25055, 33337}, {25416, 31145}, {25439, 27529}, {26102, 64710}, {26136, 58371}, {26492, 37437}, {27138, 37998}, {27186, 31249}, {27355, 58543}, {29688, 60688}, {29817, 37736}, {30143, 45764}, {30577, 44006}, {31276, 32454}, {31412, 48701}, {32785, 48714}, {32786, 48715}, {33703, 38754}, {35856, 42274}, {35857, 42277}, {37725, 61914}, {37758, 60459}, {38038, 64189}, {38077, 61985}, {38090, 63127}, {38099, 50894}, {38104, 50891}, {38161, 64145}, {38636, 61850}, {38665, 46936}, {38760, 55864}, {39778, 54392}, {41541, 62870}, {42561, 48700}, {48667, 61272}, {50689, 52836}, {51157, 63119}, {51198, 63123}, {51525, 60781}, {53620, 64056}, {59373, 66039}, {59388, 64742}, {59417, 64138}, {61595, 66007}, {63085, 66035}, {63975, 64155}

X(66063) = reflection of X(i) in X(j) for these {i,j}: {66045, 3090}
X(66063) = pole of line {918, 4409} with respect to the Steiner circumellipse
X(66063) = intersection, other than A, B, C, of circumconics {{A, B, C, X(105), X(24302)}}, {{A, B, C, X(149), X(56365)}}, {{A, B, C, X(3035), X(8047)}}, {{A, B, C, X(35023), X(43974)}}
X(66063) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 33709, 32558}, {2, 20095, 3035}, {11, 3035, 10707}, {11, 6667, 100}, {80, 32557, 3616}, {100, 6667, 2}, {104, 23513, 3091}, {381, 61566, 12248}, {952, 3090, 66045}, {1125, 37718, 6224}, {1156, 38205, 62778}, {1320, 34122, 3617}, {1387, 59415, 145}, {3035, 10707, 20095}, {3086, 5154, 20060}, {6713, 59391, 20}, {9669, 17566, 20066}, {10707, 20095, 149}, {10724, 21154, 3522}, {12737, 38182, 5818}, {15325, 37375, 20067}, {19914, 38044, 10595}, {26726, 38213, 8}, {33709, 59419, 1}, {37726, 38319, 64008}, {38141, 38753, 4}, {38319, 64008, 7486}, {38693, 65948, 3146}

X(66064) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR7-2.149 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a-b-c)*(b-c)*(a^6-2*a^5*(b+c)-2*a^3*b*c*(b+c)+a^4*(b^2+4*b*c+c^2)+a*(b-c)^2*(2*b^3-b^2*c-b*c^2+2*c^3)-(b-c)^2*(b^4-b^2*c^2+c^4)-a^2*(b^4-3*b^3*c+3*b^2*c^2-3*b*c^3+c^4)) : :
X(66064) = -X[11]+3*X[11193]

Triangle CTR7-2.149 vertices are the barycentric sums of the corresponding vertices of the cevian triangles of X(2) and X(149).

X(66064) lies on these lines: {11, 11193}, {100, 31628}, {149, 885}, {497, 42547}, {513, 5083}, {528, 64440}, {663, 64710}, {952, 11247}, {1387, 32195}, {1862, 18344}, {2520, 37998}, {3035, 10006}, {3900, 14740}, {8641, 65739}, {11927, 42863}, {11934, 15914}, {13274, 40166}, {38325, 65664}

X(66064) = midpoint of X(i) and X(j) for these {i,j}: {11, 66026}
X(66064) = reflection of X(i) in X(j) for these {i,j}: {1387, 32195}, {10006, 17115}
X(66064) = X(i)-Ceva conjugate of X(j) for these {i, j}: {31611, 650}
X(66064) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11193, 66026, 11}

X(66065) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR9-2.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^3-2*a^2*(b+c)-3*(b-c)^2*(b+c)+a*(3*b^2-4*b*c+3*c^2) : :
X(66065) = -3*X[2]+5*X[11], -5*X[80]+X[3632], -5*X[104]+X[3529], -5*X[119]+7*X[3851], -X[153]+3*X[59390], -5*X[214]+7*X[15808], -X[550]+5*X[1484], -X[1145]+3*X[37718], -5*X[1156]+X[60957], -5*X[1387]+4*X[3636], -7*X[3528]+5*X[24466], -4*X[3530]+5*X[6713] and many others

CTR9-2.2 is the triangle homothetic to ABC with center X(2) and ratio 5/4.

X(66065) lies on these lines: {1, 12690}, {2, 11}, {5, 25439}, {80, 3632}, {104, 3529}, {119, 3851}, {153, 59390}, {214, 15808}, {226, 15570}, {382, 2829}, {496, 17563}, {516, 13226}, {529, 3583}, {546, 946}, {550, 1484}, {900, 4458}, {956, 1479}, {960, 2802}, {1125, 9945}, {1145, 37718}, {1156, 60957}, {1279, 17070}, {1317, 3485}, {1320, 7319}, {1329, 9669}, {1387, 3636}, {1476, 11604}, {1537, 49176}, {1699, 3243}, {1837, 13463}, {1848, 1862}, {2805, 4739}, {2810, 38390}, {3039, 21090}, {3062, 3254}, {3436, 9671}, {3528, 24466}, {3530, 6713}, {3544, 38665}, {3616, 9963}, {3627, 62825}, {3629, 5848}, {3631, 9024}, {3644, 66067}, {3722, 37691}, {3742, 63972}, {3746, 6668}, {3756, 24715}, {3822, 15170}, {3838, 64162}, {3847, 5687}, {3855, 10599}, {3871, 7173}, {3873, 27778}, {3874, 31828}, {3913, 10591}, {3914, 59477}, {3982, 5083}, {4023, 21283}, {4031, 24465}, {4293, 34706}, {4512, 51791}, {4640, 24386}, {4649, 33106}, {4847, 15481}, {4857, 5251}, {4973, 28178}, {4996, 17574}, {4999, 15171}, {5057, 5852}, {5079, 12331}, {5082, 9711}, {5087, 5853}, {5176, 32426}, {5223, 24392}, {5225, 12513}, {5528, 38205}, {5533, 51636}, {5541, 31435}, {5572, 18240}, {5727, 34640}, {5794, 51785}, {5856, 24389}, {5880, 17051}, {6068, 60983}, {6261, 12737}, {6264, 63992}, {6326, 38038}, {6691, 37720}, {6982, 64735}, {7671, 33558}, {7681, 18491}, {7741, 64123}, {8256, 9581}, {8715, 10593}, {9041, 21093}, {9345, 33104}, {9668, 45700}, {9670, 10527}, {9802, 13996}, {9812, 13243}, {9897, 25416}, {9946, 13374}, {9955, 66051}, {10058, 19535}, {10090, 19537}, {10129, 37703}, {10299, 13199}, {10300, 18589}, {10529, 12953}, {10609, 16173}, {10711, 61967}, {10724, 49135}, {10728, 62017}, {10742, 14269}, {10755, 11008}, {10896, 12607}, {10993, 15720}, {11240, 12943}, {11698, 38071}, {11715, 65404}, {11737, 61580}, {11928, 18242}, {12248, 62042}, {12531, 20054}, {12630, 25568}, {12735, 50892}, {12915, 41871}, {13279, 50244}, {14740, 58683}, {14869, 33814}, {15172, 25639}, {15681, 38761}, {15687, 22938}, {15863, 34641}, {16468, 33141}, {16866, 51506}, {17533, 48696}, {17571, 48713}, {17606, 32157}, {17660, 66009}, {17719, 53534}, {17721, 66071}, {17757, 65140}, {17768, 26015}, {18483, 34791}, {18527, 64732}, {19641, 28162}, {20085, 62617}, {20850, 54065}, {22793, 49627}, {24477, 63975}, {26470, 64792}, {27065, 61032}, {30384, 44669}, {31936, 34503}, {32557, 51724}, {34126, 61853}, {34200, 61566}, {34474, 61814}, {35018, 60759}, {36835, 38200}, {37722, 52367}, {38069, 61829}, {38077, 61947}, {38140, 49626}, {38152, 66007}, {38156, 66008}, {38159, 66010}, {38319, 61562}, {38669, 50688}, {38693, 62097}, {38752, 61905}, {38753, 49139}, {38754, 62128}, {38760, 55863}, {38762, 61855}, {38763, 61892}, {39692, 65132}, {40341, 66037}, {42886, 64152}, {46816, 57002}, {51198, 62995}, {51525, 58421}, {51529, 62044}, {51768, 66068}, {52985, 64445}, {54391, 65632}, {61649, 63145}, {62354, 64138}, {62837, 65631}, {64140, 64335}

X(66065) = midpoint of X(i) and X(j) for these {i,j}: {1, 12690}, {11, 149}, {1320, 62616}, {1537, 49176}, {5057, 51463}, {9802, 13996}, {9897, 25416}, {10738, 37726}, {12773, 64186}, {20085, 62617}, {38669, 52836}, {38761, 48680}, {54391, 65632}, {62354, 64138}
X(66065) = reflection of X(i) in X(j) for these {i,j}: {100, 6667}, {3035, 11}, {3036, 12019}, {5083, 58611}, {9945, 1125}, {9946, 13374}, {12331, 20400}, {14740, 58683}, {20418, 1484}, {38757, 65948}, {38759, 20418}, {51525, 58421}, {66051, 9955}, {66052, 546}
X(66065) = complement of X(6154)
X(66065) = anticomplement of X(35023)
X(66065) = X(i)-Dao conjugate of X(j) for these {i, j}: {35023, 35023}
X(66065) = pole of line {659, 44807} with respect to the circumcircle
X(66065) = pole of line {17719, 53523} with respect to the incircle
X(66065) = pole of line {918, 27191} with respect to the Steiner inellipse
X(66065) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {11, 149, 10776}
X(66065) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(4428), X(14947)}}, {{A, B, C, X(20095), X(43974)}}
X(66065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {11, 100, 6667}, {11, 149, 528}, {11, 6154, 2}, {100, 6667, 3035}, {149, 10707, 11}, {497, 11235, 2886}, {528, 6667, 100}, {546, 952, 66052}, {952, 65948, 38757}, {1479, 3813, 57288}, {1484, 5840, 20418}, {2802, 12019, 3036}, {3058, 11680, 6690}, {3434, 11238, 3816}, {5057, 51463, 5852}, {5840, 20418, 38759}, {9802, 59415, 13996}, {9897, 50891, 25416}, {10738, 12773, 64186}, {10738, 37726, 2829}, {12331, 23513, 20400}, {15171, 24387, 4999}, {24386, 51783, 4640}, {24646, 24647, 4428}, {37726, 64186, 12773}

X(66066) = PARALLELOGIC CENTER OF THESE TRIANGLES: EXTOUCH-OF-FUHRMANN WRT CTR4-100

Barycentrics    a^9*(b+c)-7*a^5*b*(b-c)^2*c*(b+c)+a^8*(-3*b^2+2*b*c-3*c^2)+(b^2-c^2)^4*(b^2+c^2)-2*a^2*b*c*(b^2-c^2)^2*(b^2-4*b*c+c^2)-a*(b-c)^4*(b+c)^3*(b^2-b*c+c^2)+2*a^6*(b-c)^2*(4*b^2+5*b*c+4*c^2)+a^7*(-2*b^3+3*b^2*c+3*b*c^2-2*c^3)-2*a^4*(b-c)^2*(3*b^4+3*b^3*c+2*b^2*c^2+3*b*c^3+3*c^4)+a^3*(b-c)^2*(2*b^5+5*b^4*c-3*b^3*c^2-3*b^2*c^3+5*b*c^4+2*c^5) : :

Triangle CTR4-100 is defined as follows. Let DEF be cevian triangle of X(100). AD intersects the circle (AEF) at A1 different from A. Define B1, C1 cyclically, then CTR4-100 is the triangle A1B1C1. It is similar to ABC.

X(66066) lies on these lines: {1, 18339}, {10, 521}, {11, 12016}, {117, 24030}, {496, 942}, {502, 15232}, {900, 11798}, {1210, 31849}, {1385, 28347}, {1387, 25437}, {2695, 2720}, {10950, 15524}, {11373, 23869}, {12053, 24201}, {12608, 64512}, {13138, 50917}, {20264, 35580}, {37702, 56814}

X(66066) = midpoint of X(i) and X(j) for these {i,j}: {13138, 50917}
X(66066) = X(135) of Fuhrmann triangle

X(66067) = PARALLELOGIC CENTER OF THESE TRIANGLES: INVERSE-OF-X(10)-CIRCUMCONCEVIAN-OF-X(37)) WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(b^2-b*c+(a-c)*c)*(a*b-b^2-b*c+c^2) : :
X(66067) = -4*X[3035]+5*X[4687], X[3644]+4*X[66065], -2*X[3696]+3*X[59415], -4*X[3739]+5*X[31272], -2*X[4688]+3*X[59377], -5*X[4699]+7*X[66063], -5*X[4704]+X[20095], -7*X[4751]+8*X[6667], -2*X[6174]+3*X[51488], -2*X[10427]+3*X[27475], -3*X[16173]+2*X[24325]

X(66067) lies on these lines: {11, 75}, {37, 100}, {80, 740}, {148, 24500}, {149, 192}, {190, 4516}, {335, 876}, {518, 1156}, {528, 4664}, {536, 4956}, {537, 50891}, {726, 21630}, {742, 66037}, {903, 3675}, {952, 20430}, {984, 2802}, {1025, 62764}, {2087, 37129}, {2161, 3573}, {2170, 24482}, {2310, 25048}, {2397, 13576}, {2611, 64863}, {2829, 51063}, {3035, 4687}, {3254, 14947}, {3644, 66065}, {3696, 59415}, {3739, 31272}, {4043, 4451}, {4440, 17463}, {4475, 24338}, {4499, 7202}, {4518, 24004}, {4688, 59377}, {4699, 66063}, {4704, 20095}, {4751, 6667}, {4777, 27493}, {4919, 36278}, {4941, 20274}, {5083, 7201}, {5840, 30273}, {5848, 49496}, {5854, 49450}, {5856, 51052}, {6174, 51488}, {7972, 49471}, {9024, 49509}, {9897, 49469}, {10427, 27475}, {10711, 51038}, {10738, 29010}, {12531, 28581}, {12653, 49448}, {13205, 34247}, {13243, 54344}, {14217, 29054}, {15863, 49459}, {16173, 24325}, {17660, 64546}, {21887, 22209}, {21889, 52923}, {24516, 24715}, {27809, 37842}, {30271, 38693}, {31057, 47842}, {32557, 40328}, {37718, 49474}, {38752, 61522}, {49457, 64056}, {49490, 64137}, {50111, 64011}, {51034, 64746}, {57298, 64728}, {59391, 64088}

X(66067) = midpoint of X(i) and X(j) for these {i,j}: {149, 192}, {9897, 49469}, {12653, 49448}
X(66067) = reflection of X(i) in X(j) for these {i,j}: {75, 11}, {100, 37}, {7972, 49471}, {10711, 51038}, {17660, 64546}, {49459, 15863}, {49490, 64137}, {64011, 50111}, {64056, 49457}, {64746, 51034}, {66057, 20430}
X(66067) = pole of line {8540, 9025} with respect to the Feuerbach hyperbola
X(66067) = intersection, other than A, B, C, of circumconics {{A, B, C, X(111), X(30992)}}, {{A, B, C, X(11609), X(24490)}}
X(66067) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 2805, 100}, {952, 20430, 66057}, {1156, 1320, 10755}

X(66068) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR12-1.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    a*(a-b-c)*(a^4+(b-c)^4+a*b*c*(b+c)+a^2*(-2*b^2+b*c-2*c^2)) : :
X(66068) = -3*X[165]+2*X[13205], -2*X[13272]+3*X[37718], -4*X[21635]+3*X[28609]

Let MaMbMc be the medial triangle. CTR12-1.2 is the triangle with vertices at the inversion poles of MbMc, MaMc, and MaMb wrt to the X(1)-circumconic.

X(66068) lies on these lines: {1, 6596}, {9, 11}, {38, 56317}, {40, 104}, {46, 2136}, {57, 100}, {63, 149}, {80, 57279}, {84, 5840}, {165, 13205}, {190, 4939}, {191, 3813}, {214, 3333}, {244, 3939}, {484, 3880}, {518, 5531}, {519, 5535}, {527, 9809}, {528, 1768}, {952, 5709}, {1001, 58611}, {1054, 61222}, {1145, 1706}, {1317, 37550}, {1320, 1697}, {1331, 1421}, {1484, 26921}, {1490, 66061}, {1616, 10899}, {1709, 13271}, {1750, 17661}, {2323, 66036}, {2771, 54422}, {2783, 24469}, {2900, 17660}, {2932, 15803}, {3035, 5437}, {3218, 5853}, {3219, 24386}, {3220, 13222}, {3305, 66063}, {3336, 3913}, {3337, 56176}, {3338, 15015}, {3359, 3655}, {3587, 38602}, {3601, 4996}, {3646, 32557}, {3738, 13256}, {3882, 26141}, {3894, 34600}, {3929, 10707}, {4666, 63917}, {4853, 17636}, {4860, 6600}, {5119, 12653}, {5220, 58683}, {5227, 66037}, {5436, 51506}, {5438, 10090}, {5528, 60968}, {5759, 24477}, {6224, 62874}, {6326, 11523}, {6597, 24298}, {6667, 51780}, {6713, 61122}, {6763, 65134}, {6765, 12331}, {6769, 12332}, {6797, 9623}, {7091, 12641}, {7289, 9024}, {7308, 31272}, {7330, 10738}, {7993, 12513}, {8580, 58663}, {8668, 37572}, {9802, 21627}, {9803, 24391}, {9841, 24466}, {10087, 59335}, {10389, 65739}, {10390, 34894}, {10427, 60955}, {10912, 11010}, {10980, 58591}, {11034, 35023}, {11500, 66062}, {11520, 39778}, {12119, 63430}, {12248, 12625}, {12514, 21630}, {12526, 17638}, {12629, 59318}, {12705, 14217}, {12773, 37584}, {13199, 63399}, {13243, 60990}, {13272, 37718}, {13274, 30223}, {13277, 53400}, {13279, 15829}, {14740, 60782}, {15932, 34791}, {16173, 31435}, {17059, 33115}, {17154, 65206}, {18240, 64154}, {18540, 22938}, {18839, 58328}, {20588, 30827}, {21342, 56178}, {21635, 28609}, {22770, 64267}, {23958, 64146}, {25438, 59333}, {26877, 64117}, {27003, 59584}, {30578, 60368}, {31393, 64137}, {33814, 37534}, {33895, 37563}, {34474, 37526}, {35445, 64359}, {36975, 44669}, {37551, 38693}, {38316, 64676}, {45043, 55869}, {49168, 64278}, {50865, 51897}, {51768, 66065}, {57036, 65164}, {62819, 64710}, {63130, 64743}, {63137, 64056}

X(66068) = reflection of X(i) in X(j) for these {i,j}: {1, 22560}, {2136, 5541}, {6326, 48713}, {6765, 12331}, {6769, 12332}, {7993, 12513}, {9802, 21627}, {9803, 24391}, {11523, 6326}, {12641, 13996}, {64267, 22770}, {64278, 49168}, {66058, 5709}, {66059, 62858}, {66061, 1490}, {66062, 11500}
X(66068) = inverse of X(1293) in Bevan circle
X(66068) = X(3189) of anti-inner-Garcia triangle
X(66068) = pole of line {100, 1293} with respect to the Bevan circle
X(66068) = pole of line {7677, 61035} with respect to the dual conic of Moses-Feuerbach circumconic
X(66068) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1280), X(6596)}}, {{A, B, C, X(3254), X(43760)}}
X(66068) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 149, 64372}, {952, 5709, 66058}, {5541, 5854, 2136}

X(66069) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR12-8.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    (a-b-c)*(3*a^5-a^4*(b+c)-a^3*(2*b^2+5*b*c+2*c^2)-2*a^2*(b^3-4*b^2*c-4*b*c^2+c^3)+(b-c)^2*(3*b^3-7*b^2*c-7*b*c^2+3*c^3)-a*(b^4-15*b^3*c+32*b^2*c^2-15*b*c^3+c^4)) : :

Let QaQbQc be the medial triangle. CTR12-8.2 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(8)-circumconic.

X(66069) lies on these lines: {11, 3161}, {100, 6557}, {149, 8055}, {952, 8834}, {2827, 9809}, {2899, 9802}, {5423, 13274}, {6224, 28661}, {20095, 62297}, {34122, 39800}

X(66069) = pole of line {24036, 65818} with respect to the dual conic of incircle

X(66070) = PARALLELOGIC CENTER OF THESE TRIANGLES: CTR12-10.2 WRT EXTOUCH-OF-FUHRMANN

Barycentrics    (b+c)*(-a^5-a*b*(b-c)^2*c-a^2*b*c*(b+c)+b*(b-c)^2*c*(b+c)+a^3*(b^2+b*c+c^2)) : :

Let QaQbQc be the medial triangle. CTR12-10.2 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(10)-circumconic.

X(66070) lies on the Yff contact circle and on these lines: {11, 37}, {72, 952}, {98, 100}, {125, 21091}, {149, 3995}, {428, 528}, {740, 51377}, {908, 29010}, {1145, 5295}, {1867, 37725}, {1868, 12138}, {2801, 22001}, {2802, 2901}, {3035, 31993}, {3191, 6326}, {3198, 6154}, {4024, 24979}, {4552, 18210}, {4847, 22004}, {5057, 29073}, {6224, 56318}, {6358, 21319}, {6745, 29347}, {10006, 55210}, {13244, 21061}, {18359, 65313}, {20095, 62227}, {21635, 22000}, {22014, 34789}, {24269, 32931}, {26893, 54035}, {28850, 38389}, {29327, 63145}, {38665, 41013}, {43223, 58397}

X(66070) = X(i)-Dao conjugate of X(j) for these {i, j}: {21091, 150}
X(66070) = X(i)-Ceva conjugate of X(j) for these {i, j}: {44184, 10}

X(66071) = X(6) OF EXTOUCH-OF-FUHRMANN

Barycentrics    2*a^2*(b+c)+(b-c)^2*(b+c)+a*(b^2+c^2) : :
X(66071) = -5*X[3616]+3*X[48805], -5*X[3618]+X[24280], -7*X[3624]+3*X[50126], -4*X[3634]+3*X[17359], -X[3886]+5*X[17304], -2*X[4527]+3*X[50097], X[4780]+2*X[17235], -7*X[9780]+3*X[50107], -3*X[16475]+X[64016], 3*X[17274]+X[49495], -4*X[24295]+5*X[51126], -3*X[47358]+X[49451]

X(66071) lies on circumconic {{A, B, C, X(34578), X(60276)}} and on these lines: {1, 528}, {2, 3712}, {5, 2486}, {6, 17768}, {8, 4389}, {10, 536}, {11, 4850}, {37, 1738}, {39, 5701}, {42, 3782}, {43, 4415}, {55, 7465}, {65, 22464}, {69, 49486}, {75, 4026}, {79, 11076}, {81, 5196}, {100, 17602}, {120, 26242}, {141, 740}, {142, 4356}, {192, 3932}, {238, 17366}, {239, 24723}, {244, 17051}, {321, 26251}, {386, 63997}, {495, 4868}, {496, 53564}, {516, 1386}, {518, 3663}, {519, 4743}, {524, 4655}, {527, 4663}, {545, 32935}, {550, 29032}, {575, 53792}, {594, 32784}, {597, 2796}, {612, 49732}, {614, 49736}, {726, 4085}, {752, 49477}, {758, 48847}, {846, 33132}, {942, 44670}, {946, 4719}, {950, 45275}, {968, 24789}, {984, 17246}, {986, 1834}, {1001, 4000}, {1009, 4436}, {1100, 50307}, {1125, 17067}, {1211, 32776}, {1266, 49483}, {1281, 7792}, {1284, 5132}, {1449, 4312}, {1503, 24257}, {1621, 33150}, {1698, 16676}, {1714, 18253}, {1756, 4271}, {1757, 17334}, {1836, 5256}, {1999, 33068}, {2177, 17724}, {2321, 3844}, {2550, 3672}, {2792, 8550}, {2795, 15048}, {2805, 5883}, {2831, 5884}, {2886, 3666}, {2887, 4970}, {2999, 24703}, {3008, 15254}, {3011, 4689}, {3035, 17720}, {3058, 7191}, {3120, 5718}, {3122, 24443}, {3123, 4642}, {3187, 32950}, {3210, 32773}, {3240, 33151}, {3247, 38052}, {3329, 5992}, {3416, 3875}, {3434, 17599}, {3589, 3923}, {3616, 48805}, {3618, 24280}, {3624, 50126}, {3626, 4407}, {3627, 29113}, {3629, 17770}, {3634, 17359}, {3644, 3790}, {3649, 19767}, {3662, 4966}, {3670, 57022}, {3683, 26723}, {3685, 16706}, {3696, 4357}, {3703, 4972}, {3704, 16062}, {3706, 54311}, {3717, 49523}, {3720, 40688}, {3729, 28556}, {3739, 39580}, {3740, 4656}, {3742, 24177}, {3743, 8728}, {3750, 33147}, {3751, 5852}, {3752, 3816}, {3756, 24217}, {3757, 19796}, {3772, 6690}, {3773, 28522}, {3775, 4709}, {3813, 37592}, {3815, 5988}, {3823, 4078}, {3829, 24239}, {3836, 3993}, {3848, 24175}, {3886, 17304}, {3891, 4030}, {3896, 17184}, {3912, 49462}, {3920, 34612}, {3924, 64158}, {3925, 28606}, {3931, 23537}, {3936, 64161}, {3943, 29674}, {3944, 37662}, {3980, 6703}, {4003, 26015}, {4021, 64174}, {4023, 26580}, {4046, 32782}, {4133, 17229}, {4202, 64071}, {4205, 28612}, {4260, 20718}, {4310, 42871}, {4331, 5228}, {4346, 64165}, {4353, 5853}, {4360, 4645}, {4361, 50295}, {4362, 44419}, {4365, 32781}, {4392, 51463}, {4395, 16825}, {4398, 24349}, {4399, 50308}, {4414, 33128}, {4417, 4734}, {4419, 5220}, {4424, 64172}, {4425, 5743}, {4438, 59583}, {4450, 17150}, {4523, 9021}, {4527, 50097}, {4640, 40940}, {4643, 17224}, {4646, 12607}, {4647, 13728}, {4648, 7613}, {4649, 17365}, {4650, 61661}, {4657, 50314}, {4660, 5846}, {4667, 30424}, {4676, 17367}, {4684, 49475}, {4693, 29637}, {4715, 64073}, {4716, 17362}, {4733, 5224}, {4780, 17235}, {4852, 5847}, {4863, 62833}, {4884, 29673}, {4899, 49513}, {4906, 64162}, {4991, 28508}, {4995, 29665}, {5057, 17012}, {5091, 5135}, {5222, 5698}, {5249, 37593}, {5262, 6284}, {5263, 17302}, {5313, 51409}, {5432, 33133}, {5434, 17015}, {5480, 29057}, {5699, 37340}, {5700, 37341}, {5902, 11809}, {6057, 29679}, {6147, 59301}, {6650, 20132}, {6679, 59580}, {6685, 48643}, {6738, 64932}, {7263, 24325}, {7354, 17016}, {8543, 37771}, {8584, 28558}, {8692, 52653}, {9052, 64553}, {9053, 49455}, {9055, 49519}, {9780, 50107}, {9791, 17277}, {10327, 50071}, {11269, 17595}, {11281, 19765}, {12722, 58562}, {13747, 43135}, {14267, 52902}, {15172, 30148}, {15338, 62802}, {16475, 64016}, {16670, 60905}, {16823, 37756}, {16830, 17320}, {17011, 20292}, {17017, 33094}, {17018, 33146}, {17024, 34611}, {17045, 50302}, {17056, 17592}, {17231, 49461}, {17237, 49468}, {17258, 60731}, {17274, 49495}, {17330, 24697}, {17340, 33159}, {17345, 34379}, {17351, 28526}, {17355, 28557}, {17369, 29633}, {17388, 32846}, {17390, 50281}, {17398, 24342}, {17529, 27785}, {17591, 33141}, {17593, 33140}, {17596, 33135}, {17600, 33109}, {17601, 29658}, {17717, 62221}, {17721, 66065}, {17726, 33104}, {17764, 49482}, {17765, 49464}, {17766, 49472}, {17771, 49685}, {17772, 50304}, {18139, 27804}, {18343, 36154}, {19623, 35916}, {19637, 40432}, {19784, 50044}, {19786, 32932}, {20160, 29590}, {20872, 41230}, {21850, 29301}, {21956, 41269}, {22791, 50604}, {23536, 37548}, {23681, 37553}, {24231, 49478}, {24293, 35101}, {24295, 51126}, {24440, 24456}, {24476, 40965}, {24692, 62467}, {24728, 29181}, {24988, 31035}, {25453, 32934}, {26227, 50102}, {27186, 62840}, {28174, 62828}, {28297, 50313}, {28329, 50781}, {28333, 50283}, {28534, 50114}, {28538, 49630}, {28542, 49726}, {28566, 49684}, {28570, 51196}, {28582, 49529}, {29093, 39884}, {29097, 48906}, {29243, 47373}, {29631, 32845}, {29659, 49493}, {29667, 50106}, {29815, 49719}, {29821, 33095}, {29850, 32936}, {30768, 50104}, {31083, 54291}, {31151, 50113}, {31264, 48642}, {32774, 32929}, {32911, 33100}, {32915, 33125}, {32924, 32947}, {32928, 32948}, {32937, 62229}, {33087, 48632}, {33098, 61358}, {33101, 42043}, {33103, 42042}, {33136, 46901}, {33139, 62796}, {33148, 37703}, {33152, 60714}, {33165, 49445}, {34937, 56176}, {35652, 62673}, {37159, 44396}, {37312, 41811}, {39543, 64524}, {39586, 41312}, {40724, 56851}, {42356, 53599}, {42819, 63977}, {44669, 48837}, {45398, 52805}, {45399, 52808}, {47356, 64299}, {47358, 49451}, {48631, 49471}, {48822, 49733}, {49446, 49688}, {49458, 50285}, {49491, 53601}, {49508, 49701}, {49515, 49772}, {49518, 49531}, {49520, 49693}, {49747, 50282}, {50065, 54418}, {50441, 62697}, {51400, 64306}, {56177, 60751}, {56519, 59536}, {60896, 62183}, {61716, 63008}, {63334, 64345}, {64751, 66027}

X(66071) = midpoint of X(i) and X(j) for these {i,j}: {6, 24248}, {8, 49453}, {69, 49486}, {3416, 3875}, {3663, 3755}, {3751, 17276}, {4655, 49488}, {4660, 32921}, {4780, 49511}, {17301, 50080}, {24476, 40965}, {47356, 64299}, {48829, 50101}, {49446, 49688}, {49518, 49531}, {49630, 50109}, {49747, 50282}
X(66071) = reflection of X(i) in X(j) for these {i,j}: {141, 3821}, {1386, 3946}, {2321, 3844}, {3629, 49489}, {3923, 3589}, {4133, 17229}, {12722, 58562}, {48810, 17382}, {48821, 50091}, {49465, 4353}, {49484, 1125}, {49511, 17235}, {49524, 4085}, {51147, 49472}
X(66071) = complement of X(5695)
X(66071) = X(53) of Fuhrmann triangle
X(66071) = perspector of circumconic {{A, B, C, X(35177), X(37143)}}
X(66071) = pole of line {9037, 18839} with respect to the Feuerbach hyperbola
X(66071) = pole of line {5164, 5692} with respect to the Kiepert hyperbola
X(66071) = pole of line {48550, 48571} with respect to the Steiner circumellipse
X(66071) = pole of line {1638, 4776} with respect to the Steiner inellipse
X(66071) = pole of line {527, 4688} with respect to the dual conic of Yff parabola
X(66071) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {11, 115, 1358}
X(66071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1086, 25557}, {1, 33149, 1086}, {6, 24248, 17768}, {8, 50101, 49453}, {37, 1738, 3826}, {42, 33145, 3782}, {43, 33154, 4415}, {55, 19785, 17061}, {81, 33102, 11246}, {100, 33155, 17602}, {142, 4356, 15569}, {516, 3946, 1386}, {536, 50091, 48821}, {726, 4085, 49524}, {740, 3821, 141}, {1125, 28580, 49484}, {1125, 49484, 48810}, {3120, 46904, 5718}, {3589, 28530, 3923}, {3662, 49470, 4966}, {3663, 3755, 518}, {3751, 17276, 5852}, {3752, 24210, 3816}, {3823, 4681, 4078}, {3836, 3993, 17243}, {3844, 28484, 2321}, {4000, 64168, 1001}, {4353, 5853, 49465}, {4414, 33128, 35466}, {4646, 13161, 12607}, {4649, 32857, 17365}, {4655, 49488, 524}, {4660, 32921, 5846}, {4689, 50103, 3011}, {4780, 49511, 28581}, {17017, 33094, 63979}, {17235, 28581, 49511}, {17301, 50080, 528}, {17302, 62392, 5263}, {17382, 49484, 1125}, {17592, 17889, 17056}, {17596, 33135, 37646}, {17766, 49472, 51147}, {17770, 49489, 3629}, {25453, 32934, 44416}, {28606, 33131, 3925}, {29674, 49452, 3943}, {32776, 32860, 1211}, {32784, 49474, 594}, {48829, 49453, 8}, {48829, 50101, 28503}, {49630, 50109, 28538}, {49736, 59477, 614}

X(66072) = X(3)X(125)∩X(15059)X(39118)

Barycentrics    (a^2-b^2-c^2) (2 a^20-8 a^18 b^2+9 a^16 b^4+3 a^14 b^6-12 a^12 b^8+9 a^10 b^10-12 a^8 b^12+21 a^6 b^14-18 a^4 b^16+7 a^2 b^18-b^20-8 a^18 c^2+30 a^16 b^2 c^2-39 a^14 b^4 c^2+13 a^12 b^6 c^2+9 a^10 b^8 c^2+19 a^8 b^10 c^2-69 a^6 b^12 c^2+75 a^4 b^14 c^2-37 a^2 b^16 c^2+7 b^18 c^2+9 a^16 c^4-39 a^14 b^2 c^4+56 a^12 b^4 c^4-30 a^10 b^6 c^4-22 a^8 b^8 c^4+89 a^6 b^10 c^4-122 a^4 b^12 c^4+80 a^2 b^14 c^4-21 b^16 c^4+3 a^14 c^6+13 a^12 b^2 c^6-30 a^10 b^4 c^6+34 a^8 b^6 c^6-41 a^6 b^8 c^6+101 a^4 b^10 c^6-88 a^2 b^12 c^6+36 b^14 c^6-12 a^12 c^8+9 a^10 b^2 c^8-22 a^8 b^4 c^8-41 a^6 b^6 c^8-72 a^4 b^8 c^8+38 a^2 b^10 c^8-42 b^12 c^8+9 a^10 c^10+19 a^8 b^2 c^10+89 a^6 b^4 c^10+101 a^4 b^6 c^10+38 a^2 b^8 c^10+42 b^10 c^10-12 a^8 c^12-69 a^6 b^2 c^12-122 a^4 b^4 c^12-88 a^2 b^6 c^12-42 b^8 c^12+21 a^6 c^14+75 a^4 b^2 c^14+80 a^2 b^4 c^14+36 b^6 c^14-18 a^4 c^16-37 a^2 b^2 c^16-21 b^4 c^16+7 a^2 c^18+7 b^2 c^18-c^20) : :
X(66072) = X(3)+3*X(34310), X(13496)-3*X(38727), X(13558)+3*X(15061), 5*X(15059)-X(39118), 2*X(20397)-3*X(45309), X(22823)-3*X(23515)

See Antreas Hatzipolakis and Ercole Suppa, euclid 7152.

X(66072) lies on these lines: {3, 125}, {15059, 39118}, {22823, 23515}

X(66072) = midpoint of X(125) and X(5961)

X(66073) = BARYCENTRIC QUOTIENT X(30)/X(112)

Barycentrics    b^2*(b - c)*c^2*(b + c)*(-a^2 + b^2 + c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4) : :

X(66073) lies on the cubic K1377 and these lines: {2, 46425}, {3, 47205}, {69, 14220}, {76, 58257}, {99, 107}, {114, 34336}, {125, 339}, {325, 523}, {328, 34767}, {343, 52744}, {525, 686}, {1368, 42665}, {1370, 2881}, {1494, 54988}, {1531, 30209}, {1636, 11064}, {1637, 5664}, {2373, 34168}, {2419, 18019}, {2799, 47236}, {7630, 30476}, {8552, 14592}, {12384, 14360}, {13203, 53331}, {14618, 20580}, {18314, 31174}, {23105, 45688}, {30786, 57799}, {36255, 65710}, {47230, 62307}, {53266, 57829}

X(66073) = reflection of X(i) in X(j) for these {i,j}: {15421, 8552}, {41079, 65757}, {42665, 1368}
X(66073) = isogonal conjugate of X(32715)
X(66073) = isotomic conjugate of X(1304)
X(66073) = anticomplement of X(46425)
X(66073) = polar conjugate of X(32695)
X(66073) = anticomplement of the isogonal conjugate of X(48373)
X(66073) = isotomic conjugate of the anticomplement of X(16177)
X(66073) = isotomic conjugate of the isogonal conjugate of X(9033)
X(66073) = isotomic conjugate of the polar conjugate of X(41079)
X(66073) = polar conjugate of the isogonal conjugate of X(41077)
X(66073) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {162, 51968}, {11744, 21221}, {22239, 5905}, {48373, 8}, {51967, 21294}, {65263, 59434}
X(66073) = X(i)-Ceva conjugate of X(j) for these (i,j): {328, 339}, {3267, 52624}, {6331, 36789}, {35139, 69}, {40832, 338}, {57932, 394}
X(66073) = X(i)-cross conjugate of X(j) for these (i,j): {1650, 11064}, {9033, 41079}, {16177, 2}, {52624, 3267}
X(66073) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32715}, {6, 36131}, {19, 32640}, {25, 36034}, {31, 1304}, {32, 65263}, {48, 32695}, {74, 32676}, {112, 2159}, {162, 40352}, {163, 8749}, {560, 16077}, {662, 40354}, {799, 40351}, {1576, 36119}, {1973, 44769}, {2349, 61206}, {9247, 15459}, {9406, 34568}, {18808, 23995}, {18877, 24019}, {32713, 35200}, {36083, 44080}, {36114, 51821}, {36129, 61354}, {36831, 62268}, {40353, 56829}
X(66073) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 1304}, {3, 32715}, {6, 32640}, {9, 36131}, {30, 23347}, {115, 8749}, {125, 40352}, {133, 32713}, {525, 14380}, {647, 2433}, {1084, 40354}, {1249, 32695}, {1511, 1576}, {1637, 47230}, {1650, 1495}, {1990, 2442}, {3003, 61209}, {3163, 112}, {3258, 25}, {3284, 14591}, {4858, 36119}, {6337, 44769}, {6374, 16077}, {6376, 65263}, {6505, 36034}, {6587, 61215}, {8552, 526}, {9033, 9409}, {9410, 34568}, {11064, 15329}, {14401, 647}, {14918, 53176}, {15526, 74}, {18314, 18808}, {23285, 2394}, {34591, 2159}, {35071, 18877}, {35088, 35908}, {36901, 16080}, {38996, 40351}, {38999, 184}, {39005, 51821}, {39008, 6}, {39020, 15291}, {39170, 14560}, {44436, 46587}, {47296, 5502}, {52032, 36831}, {52869, 52604}, {52874, 57153}, {57295, 512}, {62551, 186}, {62569, 110}, {62572, 57487}, {62573, 14919}, {62576, 15459}, {62577, 52475}, {62594, 9717}, {62598, 4}, {62612, 15292}, {62613, 250}, {65730, 51262}, {65732, 17986}, {65753, 403}, {65757, 523}, {65760, 4230}, {65763, 17994}
X(66073) = cevapoint of X(i) and X(j) for these (i,j): {9033, 41077}, {9409, 14396}
X(66073) = crosspoint of X(99) and X(57829)
X(66073) = crosssum of X(i) and X(j) for these (i,j): {512, 44084}, {3049, 9407}, {14270, 61354}
X(66073) = trilinear pole of line {52624, 65753}
X(66073) = crossdifference of every pair of points on line {32, 40351}
X(66073) = barycentric product X(i)*X(j) for these {i,j}: {30, 3267}, {69, 41079}, {76, 9033}, {99, 65753}, {264, 41077}, {304, 36035}, {305, 1637}, {325, 65778}, {328, 5664}, {339, 2407}, {340, 18557}, {525, 3260}, {561, 2631}, {656, 46234}, {850, 11064}, {1494, 52624}, {1502, 9409}, {1636, 18022}, {1650, 6331}, {1990, 52617}, {3265, 46106}, {3268, 57482}, {3284, 44173}, {4143, 52661}, {4240, 36793}, {4563, 58261}, {6148, 14592}, {6333, 60869}, {6334, 52552}, {9214, 45807}, {11125, 40071}, {14206, 14208}, {14254, 45792}, {14345, 41530}, {14391, 34384}, {14396, 40421}, {14398, 40050}, {17879, 24001}, {23974, 58071}, {34767, 36789}, {43752, 60597}, {46229, 57819}, {57570, 58257}, {57799, 65754}, {57829, 65757}
X(66073) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36131}, {2, 1304}, {3, 32640}, {4, 32695}, {6, 32715}, {30, 112}, {63, 36034}, {69, 44769}, {75, 65263}, {76, 16077}, {113, 61209}, {122, 61215}, {125, 2433}, {133, 2442}, {264, 15459}, {328, 39290}, {338, 18808}, {339, 2394}, {343, 36831}, {477, 32712}, {512, 40354}, {520, 18877}, {523, 8749}, {525, 74}, {647, 40352}, {656, 2159}, {669, 40351}, {686, 51821}, {850, 16080}, {1099, 56829}, {1494, 34568}, {1495, 61206}, {1511, 14591}, {1568, 1625}, {1577, 36119}, {1636, 184}, {1637, 25}, {1650, 647}, {1784, 24019}, {1990, 32713}, {2173, 32676}, {2407, 250}, {2416, 15404}, {2420, 57655}, {2525, 46147}, {2631, 31}, {2697, 59108}, {2799, 35908}, {3163, 23347}, {3258, 47230}, {3260, 648}, {3265, 14919}, {3267, 1494}, {3268, 57487}, {3284, 1576}, {4240, 23964}, {4846, 32681}, {5642, 61207}, {5664, 186}, {6148, 14590}, {6331, 42308}, {6333, 35910}, {6334, 14264}, {6793, 2445}, {8057, 15291}, {8552, 14385}, {9033, 6}, {9409, 32}, {11064, 110}, {11125, 1474}, {14206, 162}, {14208, 2349}, {14345, 154}, {14380, 40353}, {14391, 51}, {14395, 2194}, {14396, 206}, {14397, 44077}, {14398, 1974}, {14399, 2203}, {14400, 2299}, {14401, 1495}, {14417, 9717}, {14499, 52132}, {14500, 52131}, {14582, 40355}, {14592, 5627}, {14920, 53176}, {14977, 9139}, {15328, 40388}, {15421, 10419}, {15454, 32708}, {15526, 14380}, {16163, 2420}, {16177, 46425}, {18312, 17986}, {18557, 265}, {18558, 52153}, {23347, 41937}, {24001, 24000}, {24018, 35200}, {34767, 40384}, {35906, 32696}, {35912, 2715}, {36035, 19}, {36102, 36117}, {36789, 4240}, {36793, 34767}, {36891, 32697}, {37638, 65316}, {39008, 9409}, {41077, 3}, {41079, 4}, {42716, 5379}, {43083, 11079}, {43752, 16813}, {43768, 933}, {44204, 33885}, {45807, 36890}, {46106, 107}, {46229, 378}, {46234, 811}, {46809, 58994}, {47414, 14270}, {51254, 32662}, {51349, 32711}, {51360, 35325}, {51389, 4230}, {51392, 61203}, {51393, 61208}, {51394, 32661}, {51403, 61204}, {51937, 32649}, {52355, 15627}, {52485, 32687}, {52552, 687}, {52624, 30}, {52628, 52475}, {52661, 6529}, {52743, 34397}, {52945, 52604}, {53235, 32663}, {55141, 47228}, {55265, 44084}, {56399, 14560}, {57295, 40135}, {57482, 476}, {57606, 15292}, {58071, 23590}, {58085, 32646}, {58257, 39008}, {58261, 2501}, {58263, 1990}, {58346, 14581}, {60053, 15395}, {60597, 44715}, {60869, 685}, {62172, 52418}, {62569, 15329}, {62583, 46587}, {62624, 41433}, {63171, 36064}, {64603, 46249}, {65325, 64774}, {65722, 51262}, {65723, 48451}, {65753, 523}, {65754, 232}, {65755, 17994}, {65757, 403}, {65758, 3563}, {65759, 34212}, {65778, 98}
X(66073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {850, 3268, 65972}, {850, 30474, 23285}, {3268, 65972, 35522}, {22339, 22340, 3265}, {30474, 57069, 3265}

X(66074) = BARYCENTRIC QUOTIENT X(30)/X(2395)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(66074) = 2 X[3003] - 3 X[35297]

X(66074) lies on the cubic K1377 and these lines: {2, 60498}, {30, 3260}, {76, 54600}, {99, 523}, {110, 2855}, {114, 325}, {316, 46988}, {877, 2396}, {2407, 2420}, {3003, 35297}, {3233, 51263}, {4576, 30474}, {5468, 57627}, {6563, 40049}, {14570, 64919}, {18020, 31510}, {18878, 53776}, {23342, 45808}, {36891, 52472}, {47207, 62310}, {51389, 65755}

X(66074) = midpoint of X(99) and X(14221)
X(66074) = X(65754)-cross conjugate of X(51389)
X(66074) = X(i)-isoconjugate of X(j) for these (i,j): {878, 36119}, {1910, 2433}, {2159, 2395}, {2349, 2422}, {35200, 53149}, {36034, 51441}, {36131, 51404}
X(66074) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 53149}, {1511, 878}, {3163, 2395}, {3258, 51441}, {3284, 60777}, {5976, 2394}, {11672, 2433}, {35088, 12079}, {39008, 51404}, {51389, 53266}, {62569, 879}, {62590, 14380}, {62595, 18808}, {62613, 98}, {65760, 523}, {65763, 8029}
X(66074) = cevapoint of X(51389) and X(65754)
X(66074) = crossdifference of every pair of points on line {2422, 21906}
X(66074) = barycentric product X(i)*X(j) for these {i,j}: {30, 2396}, {99, 51389}, {325, 2407}, {877, 11064}, {2421, 3260}, {4240, 6393}, {4590, 65754}, {6035, 57431}, {15631, 60869}, {23997, 46234}, {31614, 65755}, {32458, 65776}, {42716, 51369}
X(66074) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 2395}, {297, 18808}, {325, 2394}, {511, 2433}, {877, 16080}, {1495, 2422}, {1511, 60777}, {1637, 51441}, {1990, 53149}, {2396, 1494}, {2407, 98}, {2420, 1976}, {2421, 74}, {2799, 12079}, {3233, 35906}, {3260, 43665}, {3284, 878}, {4230, 8749}, {4240, 6531}, {5642, 52038}, {6393, 34767}, {9033, 51404}, {11064, 879}, {14398, 15630}, {14966, 40352}, {15631, 35910}, {23347, 57260}, {23997, 2159}, {24001, 36120}, {32458, 65973}, {36212, 14380}, {36790, 32112}, {42743, 48451}, {51386, 62665}, {51389, 523}, {57431, 1640}, {58343, 14398}, {62555, 65756}, {62720, 36119}, {64607, 34369}, {65754, 115}, {65755, 8029}, {65760, 53266}, {65776, 41932}
X(66074) = {X(99),X(31998)}-harmonic conjugate of X(65713)

X(66075) = BARYCENTRIC QUOTIENT X(30)/X(65779)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :

X(66075) lies on the cubic K1377 and these lines: {2, 94}, {99, 5649}, {110, 476}, {114, 5968}, {265, 2782}, {325, 34370}, {543, 56395}, {648, 47443}, {2421, 2799}, {4230, 16230}, {4558, 40173}, {5149, 11060}, {6054, 54554}, {6331, 46456}, {9149, 52153}, {16237, 47230}, {18384, 56390}, {32680, 37137}, {34368, 44114}, {35138, 54959}, {35139, 65271}, {35910, 51389}, {36166, 53768}, {36170, 51847}, {36173, 53771}, {39182, 64516}, {52056, 53793}, {53692, 53760}, {53725, 56397}

X(66075) = reflection of X(65975) in X(51389)
X(66075) = isogonal conjugate of X(60777)
X(66075) = X(i)-cross conjugate of X(j) for these (i,j): {2799, 65979}, {65754, 325}, {65762, 35908}
X(66075) = X(i)-isoconjugate of X(j) for these (i,j): {1, 60777}, {98, 2624}, {293, 47230}, {526, 1910}, {661, 14355}, {878, 52414}, {1821, 14270}, {1976, 32679}, {2088, 36084}, {2159, 65779}, {2395, 6149}, {16186, 36104}
X(66075) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 60777}, {132, 47230}, {3163, 65779}, {5976, 3268}, {8623, 39495}, {11672, 526}, {14993, 2395}, {15295, 2422}, {35088, 62551}, {36830, 14355}, {38970, 35235}, {38987, 2088}, {39000, 16186}, {39040, 32679}, {40601, 14270}, {55071, 18334}, {60596, 41078}, {62590, 8552}, {62595, 44427}, {65760, 5664}
X(66075) = cevapoint of X(2799) and X(51389)
X(66075) = crosssum of X(526) and X(39495)
X(66075) = trilinear pole of line {511, 868}
X(66075) = crossdifference of every pair of points on line {2088, 14270}
X(66075) = barycentric product X(i)*X(j) for these {i,j}: {94, 2421}, {99, 14356}, {265, 877}, {297, 60053}, {325, 476}, {328, 4230}, {511, 35139}, {1959, 32680}, {1989, 2396}, {2407, 65979}, {2799, 39295}, {14966, 20573}, {20022, 46155}, {23997, 63759}, {32662, 44132}, {32678, 46238}, {36061, 40703}, {36212, 46456}, {39290, 51389}, {58979, 62431}, {60524, 64516}
X(66075) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 60777}, {30, 65779}, {94, 43665}, {110, 14355}, {232, 47230}, {237, 14270}, {265, 879}, {297, 44427}, {325, 3268}, {476, 98}, {511, 526}, {684, 16186}, {877, 340}, {1755, 2624}, {1959, 32679}, {1989, 2395}, {2396, 7799}, {2421, 323}, {2799, 62551}, {3569, 2088}, {4230, 186}, {5968, 9213}, {6393, 45792}, {9155, 44814}, {11060, 2422}, {14356, 523}, {14559, 5967}, {14560, 1976}, {14582, 51404}, {14966, 50}, {15475, 51441}, {15631, 51383}, {16230, 35235}, {18384, 53149}, {23968, 34369}, {23997, 6149}, {32112, 56792}, {32662, 248}, {32678, 1910}, {32680, 1821}, {34370, 14998}, {35139, 290}, {36061, 293}, {36129, 36120}, {36212, 8552}, {36213, 39495}, {39295, 2966}, {39374, 35364}, {41392, 35906}, {41512, 52451}, {42717, 42701}, {44114, 65709}, {46155, 20021}, {46456, 16081}, {50567, 45808}, {51389, 5664}, {52153, 878}, {52449, 52076}, {56395, 52038}, {57482, 65778}, {58070, 52418}, {58979, 57742}, {60053, 287}, {60524, 41078}, {62720, 52414}, {63741, 57268}, {65317, 52190}, {65754, 3258}, {65979, 2394}
X(66075) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2493, 43084, 18883}, {23895, 23896, 14559}

X(66076) = BARYCENTRIC QUOTIENT X(132)/X(523)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*a^6 - a^4*b^2 - b^6 - a^4*c^2 + b^4*c^2 + b^2*c^4 - c^6) : :

X(66076) lies on the cubic K1377 and these lines: {2, 65624}, {20, 39265}, {30, 2967}, {99, 20580}, {112, 57065}, {147, 47105}, {232, 297}, {250, 523}, {401, 52058}, {441, 9475}, {525, 1625}, {1235, 44345}, {2409, 2445}, {3163, 40884}, {4230, 65754}, {4235, 5664}, {7482, 62510}, {14570, 57069}, {15595, 65980}, {16237, 18311}, {20577, 35318}, {35907, 40866}, {36891, 57493}, {41677, 57222}, {41678, 57071}, {44332, 50945}, {44333, 50944}, {53205, 65271}, {56601, 65771}

X(66076) = reflection of X(i) in X(j) for these {i,j}: {297, 232}, {30737, 441}
X(66076) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 4230}, {648, 34211}, {23582, 297}
X(66076) = X(55275)-cross conjugate of X(132)
X(66076) = X(i)-isoconjugate of X(j) for these (i,j): {293, 34212}, {661, 15407}, {798, 57761}, {810, 9476}, {1910, 2435}
X(66076) = X(i)-Dao conjugate of X(j) for these (i,j): {132, 34212}, {232, 523}, {441, 525}, {5976, 2419}, {11672, 2435}, {15595, 53173}, {23976, 879}, {31998, 57761}, {36830, 15407}, {39062, 9476}, {39073, 647}, {50938, 2395}, {62595, 43673}
X(66076) = cevapoint of X(132) and X(55275)
X(66076) = trilinear pole of line {132, 15595}
X(66076) = crossdifference of every pair of points on line {878, 41172}
X(66076) = barycentric product X(i)*X(j) for these {i,j}: {99, 132}, {162, 17875}, {297, 34211}, {325, 2409}, {648, 15595}, {877, 1503}, {2396, 16318}, {2407, 65980}, {2421, 60516}, {4230, 30737}, {4590, 55275}, {6331, 9475}, {6393, 23977}, {15631, 52641}, {55270, 57430}
X(66076) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 57761}, {110, 15407}, {132, 523}, {232, 34212}, {297, 43673}, {325, 2419}, {441, 53173}, {511, 2435}, {648, 9476}, {877, 35140}, {1503, 879}, {2409, 98}, {2421, 64975}, {2445, 1976}, {4230, 1297}, {4590, 55274}, {9475, 647}, {15595, 525}, {15639, 51963}, {16318, 2395}, {17875, 14208}, {23977, 6531}, {24024, 36120}, {34211, 287}, {42671, 878}, {44704, 61189}, {51437, 2422}, {55275, 115}, {58070, 43717}, {60506, 47388}, {60516, 43665}, {65754, 65759}, {65980, 2394}

X(66077) = BARYCENTRIC QUOTIENT X(30)/X(2409)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^2*c^4 + b^2*c^4 - 2*c^6)*(-a^6 - a^2*b^4 + 2*b^6 + a^4*c^2 - b^4*c^2 + a^2*c^4 - c^6) : :

X(66077) lies on the cubic K1377 and these lines: {2, 2419}, {30, 41077}, {99, 20580}, {114, 60590}, {127, 525}, {523, 1297}, {2435, 4846}, {2799, 16318}, {5664, 51937}, {8057, 65749}, {11064, 14345}, {15351, 39359}, {16177, 65759}, {16251, 53016}, {31510, 44770}, {35140, 53201}, {40512, 60597}, {46115, 52613}, {52485, 65754}, {64975, 65325}

X(66077) = X(i)-isoconjugate of X(j) for these (i,j): {1304, 2312}, {1503, 36131}, {2159, 2409}, {2349, 2445}, {8766, 32695}, {16318, 36034}, {18877, 24024}, {23977, 35200}, {32676, 63856}, {42671, 65263}
X(66077) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 23977}, {1650, 6793}, {3163, 2409}, {3258, 16318}, {15526, 63856}, {35088, 65980}, {38999, 8779}, {39008, 1503}, {61505, 35908}, {62569, 34211}, {62598, 60516}, {65763, 55275}
X(66077) = trilinear pole of line {9033, 65759}
X(66077) = crossdifference of every pair of points on line {2445, 42671}
X(66077) = barycentric product X(i)*X(j) for these {i,j}: {30, 2419}, {99, 65759}, {2435, 3260}, {3265, 52485}, {3267, 51937}, {6330, 41077}, {9033, 35140}, {11064, 43673}, {41079, 64975}, {55274, 65755}, {57761, 65754}
X(66077) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 2409}, {525, 63856}, {1297, 1304}, {1495, 2445}, {1636, 8779}, {1637, 16318}, {1784, 24024}, {1990, 23977}, {2419, 1494}, {2435, 74}, {2631, 2312}, {2799, 65980}, {6330, 15459}, {6793, 15639}, {9033, 1503}, {9409, 42671}, {11064, 34211}, {14391, 51363}, {14398, 51437}, {14401, 6793}, {34212, 8749}, {35140, 16077}, {35912, 60506}, {41077, 441}, {41079, 60516}, {43673, 16080}, {43717, 32695}, {51937, 112}, {52485, 107}, {61189, 10152}, {61505, 15292}, {64975, 44769}, {65754, 132}, {65755, 55275}, {65759, 523}, {65778, 57490}

X(66078) = BARYCENTRIC QUOTIENT X(30)/X(65780)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6) : :

X(66078) lies on the cubic K1377 and these lines: {2, 65618}, {3, 523}, {6, 1511}, {22, 842}, {24, 250}, {25, 3233}, {26, 3447}, {99, 264}, {186, 2407}, {232, 14966}, {237, 56925}, {262, 1995}, {325, 7418}, {381, 3613}, {511, 21525}, {1485, 44259}, {2070, 39371}, {2799, 40083}, {3425, 52505}, {4230, 6530}, {7468, 52472}, {7503, 58731}, {7514, 12028}, {7526, 59288}, {8430, 47079}, {9139, 10419}, {9307, 15078}, {12084, 48379}, {14356, 51389}, {15329, 16319}, {15478, 41768}, {16303, 44221}, {17928, 46426}, {18575, 31861}, {18878, 46142}, {32112, 47049}, {33752, 34157}, {35901, 61216}, {37123, 52692}, {38610, 59231}, {39375, 56400}

X(66078) = isogonal conjugate of X(52451)
X(66078) = isogonal conjugate of the anticomplement of X(47049)
X(66078) = X(65754)-cross conjugate of X(4230)
X(66078) = X(i)-isoconjugate of X(j) for these (i,j): {1, 52451}, {98, 1725}, {293, 403}, {336, 44084}, {1821, 3003}, {1910, 3580}, {2159, 65780}, {2315, 16081}, {6334, 36104}, {13754, 36120}, {21731, 36036}, {36084, 55121}
X(66078) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 52451}, {132, 403}, {2679, 21731}, {3163, 65780}, {11672, 3580}, {35088, 65972}, {38987, 55121}, {39000, 6334}, {39073, 53568}, {40601, 3003}, {46094, 13754}, {55071, 60342}, {62590, 62338}, {62595, 44138}
X(66078) = trilinear pole of line {3289, 3569}
X(66078) = crossdifference of every pair of points on line {3003, 55121}
X(66078) = barycentric product X(i)*X(j) for these {i,j}: {99, 65762}, {232, 57829}, {237, 40832}, {297, 5504}, {325, 14910}, {511, 2986}, {684, 687}, {868, 18879}, {877, 61216}, {1300, 36212}, {1959, 36053}, {2421, 15328}, {2799, 10420}, {3289, 65267}, {3569, 18878}, {4230, 15421}, {6333, 32708}, {10419, 51389}, {15454, 35910}, {16230, 43755}, {39371, 65979}, {39469, 57932}, {43034, 56103}, {46787, 51456}
X(66078) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 52451}, {30, 65780}, {232, 403}, {237, 3003}, {297, 44138}, {511, 3580}, {684, 6334}, {687, 22456}, {1300, 16081}, {1755, 1725}, {2211, 44084}, {2421, 61188}, {2491, 21731}, {2799, 65972}, {2986, 290}, {3289, 13754}, {3569, 55121}, {4230, 16237}, {5504, 287}, {9475, 53568}, {10420, 2966}, {14356, 57486}, {14910, 98}, {14966, 15329}, {15328, 43665}, {15454, 60869}, {17994, 47236}, {18878, 43187}, {18879, 57991}, {32112, 65614}, {32708, 685}, {35361, 61196}, {35910, 65715}, {36053, 1821}, {36212, 62338}, {39469, 686}, {40832, 18024}, {43755, 17932}, {51456, 46786}, {51980, 60498}, {52505, 31635}, {52557, 14355}, {57829, 57799}, {57932, 65272}, {60035, 53245}, {61216, 879}, {65262, 36036}, {65267, 60199}, {65754, 65757}, {65762, 523}
X(66078) = {X(3),X(15454)}-harmonic conjugate of X(51895)

X(66079) = BARYCENTRIC QUOTIENT X(524)/X(9140)

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

X(66079) lies on the cubic K1377 and these lines: {99, 14995}, {325, 3233}, {1494, 51262}, {2482, 2799}, {5467, 52094}, {5642, 45808}, {14559, 36890}, {15303, 50567}, {34319, 36884}, {35522, 45662}

X(66079) = X(i)-isoconjugate of X(j) for these (i,j): {897, 9142}, {923, 9140}
X(66079) = X(i)-Dao conjugate of X(j) for these (i,j): {2482, 9140}, {6593, 9142}
X(66079) = cevapoint of X(2482) and X(5642)
X(66079) = trilinear pole of line {8030, 58347}
X(66079) = barycentric product X(524)*X(9141)
X(66079) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 9142}, {524, 9140}, {9141, 671}

X(66080) = BARYCENTRIC QUOTIENT X(30)/X(65783)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - a^6*b^2 + a^4*b^4 - 3*a^2*b^6 + 2*b^8 - a^6*c^2 - a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 5*b^6*c^2 + a^4*c^4 + 3*a^2*b^2*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 5*b^2*c^6 + 2*c^8) : :
X(66080) = 3 X[2] - 4 X[65765]

X(66080) lies on the cubic K1377 and these lines: {2, 65613}, {23, 12384}, {30, 51228}, {297, 2799}, {325, 34370}, {523, 54395}, {524, 39358}, {685, 10723}, {1990, 2407}, {3233, 14920}, {4230, 6530}, {24975, 59694}, {41079, 65780}, {46236, 62310}, {51389, 65755}, {53416, 62551}

X(66080) = reflection of X(i) in X(j) for these {i,j}: {2407, 1990}, {65774, 65765}
X(66080) = anticomplement of X(65774)
X(66080) = X(99)-Ceva conjugate of X(65754)
X(66080) = X(2159)-isoconjugate of X(65783)
X(66080) = X(i)-Dao conjugate of X(j) for these (i,j): {3163, 65783}, {65755, 523}
X(66080) = barycentric product X(i)*X(j) for these {i,j}: {99, 65763}, {325, 52472}, {2407, 65977}, {65754, 65768}
X(66080) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 65783}, {52472, 98}, {65754, 65766}, {65763, 523}, {65977, 2394}
X(66080) = {X(65765),X(65774)}-harmonic conjugate of X(2)

X(66081) = BARYCENTRIC QUOTIENT X(65782)/X(98)

Barycentrics    (b^2 - c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 + c^4)*(-2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4)*(3*a^8 - 5*a^6*b^2 + 3*a^4*b^4 - 3*a^2*b^6 + 2*b^8 - 5*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + 3*a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 - b^2*c^6 + 2*c^8) : :
X(66081) = 3 X[65714] - X[65775], 3 X[65754] - 2 X[65763]

X(66081) lies on the cubic K1377 and these lines: {2, 65623}, {30, 41077}, {99, 65714}, {146, 147}, {325, 6333}, {523, 65722}, {1272, 57009}, {2407, 9033}, {4226, 65871}, {4230, 16230}, {5664, 16163}, {51389, 65754}

X(66081) = midpoint of X(4226) and X(65871)
X(66081) = X(99)-Ceva conjugate of X(51389)
X(66081) = X(i)-Dao conjugate of X(j) for these (i,j): {65754, 523}, {65782, 2394}, {65978, 98}
X(66081) = crosspoint of X(325) and X(2407)
X(66081) = crosssum of X(1976) and X(2433)
X(66081) = barycentric product X(i)*X(j) for these {i,j}: {325, 65782}, {2407, 65978}, {51389, 53383}
X(66081) = barycentric quotient X(i)/X(j) for these {i,j}: {65754, 65765}, {65782, 98}, {65978, 2394}

X(66082) = BARYCENTRIC QUOTIENT X(523)/X(65767)

Barycentrics    (b^2 - c^2)*(a^4*b^4 - 2*a^2*b^6 + b^8 + 2*a^6*c^2 - a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 2*b^6*c^2 - 4*a^4*c^4 - a^2*b^2*c^4 + b^4*c^4 + 2*a^2*c^6)*(2*a^6*b^2 - 4*a^4*b^4 + 2*a^2*b^6 - a^4*b^2*c^2 - a^2*b^4*c^2 + a^4*c^4 + 3*a^2*b^2*c^4 + b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8) : :
X(66082) = 3 X[65610] - 4 X[65763]

X(66082) lies on the cubic K1377 and these lines: {2, 51480}, {30, 21731}, {98, 35364}, {99, 34291}, {113, 114}, {115, 65610}, {230, 3569}, {351, 3233}, {403, 44427}, {523, 54395}, {526, 2407}, {804, 23350}, {1989, 15328}, {2411, 39985}, {4226, 6132}, {4230, 53263}, {14273, 62172}, {16230, 57609}, {35522, 62555}, {41079, 51479}, {51389, 65766}, {55121, 62551}

X(66082) = reflection of X(4226) in X(6132)
X(66082) = X(65754)-cross conjugate of X(523)
X(66082) = X(i)-isoconjugate of X(j) for these (i,j): {163, 65767}, {1101, 53266}, {34810, 36034}, {36084, 47049}
X(66082) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 65767}, {523, 53266}, {3258, 34810}, {35088, 65975}, {38987, 47049}
X(66082) = cevapoint of X(i) and X(j) for these (i,j): {526, 6132}, {1637, 55122}, {3569, 21731}
X(66082) = trilinear pole of line {1648, 3258}
X(66082) = barycentric product X(99)*X(65764)
X(66082) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 53266}, {523, 65767}, {1637, 34810}, {2799, 65975}, {3569, 47049}, {65754, 65760}, {65764, 523}

X(66083) = BARYCENTRIC QUOTIENT X(30)/X(52472)

Barycentrics    (2*a^8 - 5*a^6*b^2 + 6*a^4*b^4 - 5*a^2*b^6 + 2*b^8 - 3*a^6*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 3*b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 + b^4*c^4 - a^2*c^6 - b^2*c^6 + c^8)*(2*a^8 - 3*a^6*b^2 + a^4*b^4 - a^2*b^6 + b^8 - 5*a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + 6*a^4*c^4 + 3*a^2*b^2*c^4 + b^4*c^4 - 5*a^2*c^6 - 3*b^2*c^6 + 2*c^8) : :

X(66083) lies on the cubic K1377 and these lines: {114, 468}, {523, 65722}, {524, 62590}, {2407, 56021}, {3564, 5967}, {5664, 6390}, {12079, 51456}, {16310, 24975}, {40429, 41254}, {51227, 60053}, {51228, 59634}, {51389, 65765}, {65730, 65734}

X(66083) = midpoint of X(2407) and X(62338)
X(66083) = reflection of X(16310) in X(24975)
X(66083) = X(65754)-cross conjugate of X(99)
X(66083) = X(i)-isoconjugate of X(j) for these (i,j): {798, 65768}, {2159, 52472}
X(66083) = X(i)-Dao conjugate of X(j) for these (i,j): {3163, 52472}, {23967, 1550}, {31998, 65768}, {35067, 52473}, {35088, 65977}
X(66083) = cevapoint of X(i) and X(j) for these (i,j): {325, 59634}, {684, 2088}
X(66083) = trilinear pole of line {690, 16163}
X(66083) = barycentric product X(i)*X(j) for these {i,j}: {99, 65766}, {325, 65783}
X(66083) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 52472}, {99, 65768}, {542, 1550}, {2799, 65977}, {3564, 52473}, {65754, 65763}, {65766, 523}, {65783, 98}
X(66083) = {X(65722),X(65760)}-harmonic conjugate of X(65774)

X(66084) = BARYCENTRIC QUOTIENT X(30)/X(65782)

Barycentrics   (a^2 - b^2)*(a^2 - c^2)*(2*a^8 - 3*a^6*b^2 + 3*a^4*b^4 - 5*a^2*b^6 + 3*b^8 - a^6*c^2 + a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 5*b^6*c^2 - 2*a^4*c^4 + a^2*b^2*c^4 + 3*b^4*c^4 - a^2*c^6 - 3*b^2*c^6 + 2*c^8)*(2*a^8 - a^6*b^2 - 2*a^4*b^4 - a^2*b^6 + 2*b^8 - 3*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 3*b^6*c^2 + 3*a^4*c^4 + 5*a^2*b^2*c^4 + 3*b^4*c^4 - 5*a^2*c^6 - 5*b^2*c^6 + 3*c^8) : :
X(66084) = 3 X[34211] - 4 X[65777]

X(66084) lies on the cubics K1371 and K1377 and these lines: {2, 65613}, {30, 36890}, {99, 65714}, {114, 52094}, {476, 34767}, {1494, 54527}, {2799, 34211}, {2966, 43673}, {3233, 5468}, {3268, 4240}, {4226, 34765}, {4235, 5664}, {6337, 58271}, {30737, 52145}, {34761, 62645}, {53383, 65768}

X(66084) = isotomic conjugate of X(53383)
X(66084) = isotomic conjugate of the anticomplement of X(65754)
X(66084) = X(65754)-cross conjugate of X(2)
X(66084) = X(i)-isoconjugate of X(j) for these (i,j): {31, 53383}, {2159, 65782}
X(66084) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 53383}, {3163, 65782}, {35088, 65978}
X(66084) = cevapoint of X(i) and X(j) for these (i,j): {30, 2799}, {441, 9033}, {511, 8552}
X(66084) = trilinear pole of line {524, 3163}
X(66084) = barycentric product X(99)*X(65765)
X(66084) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 53383}, {30, 65782}, {2799, 65978}, {65765, 523}

X(66085) = (name pending)

Barycentrics    (a^2 - b^2 - c^2)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)*(10*a^32 - 24*a^30*b^2 - 67*a^28*b^4 + 264*a^26*b^6 - 157*a^24*b^8 - 312*a^22*b^10 + 349*a^20*b^12 + 72*a^18*b^14 + 15*a^16*b^16 - 328*a^14*b^18 - 33*a^12*b^20 + 472*a^10*b^22 - 255*a^8*b^24 - 104*a^6*b^26 + 135*a^4*b^28 - 40*a^2*b^30 + 3*b^32 - 24*a^30*c^2 + 226*a^28*b^2*c^2 - 308*a^26*b^4*c^2 - 835*a^24*b^6*c^2 + 2053*a^22*b^8*c^2 - 513*a^20*b^10*c^2 - 997*a^18*b^12*c^2 - 1044*a^16*b^14*c^2 + 1834*a^14*b^16*c^2 + 896*a^12*b^18*c^2 - 1678*a^10*b^20*c^2 - 219*a^8*b^22*c^2 + 1017*a^6*b^24*c^2 - 529*a^4*b^26*c^2 + 151*a^2*b^28*c^2 - 30*b^30*c^2 - 67*a^28*c^4 - 308*a^26*b^2*c^4 + 2026*a^24*b^4*c^4 - 1745*a^22*b^6*c^4 - 3055*a^20*b^8*c^4 + 3031*a^18*b^10*c^4 + 3104*a^16*b^12*c^4 - 1010*a^14*b^14*c^4 - 4809*a^12*b^16*c^4 + 1442*a^10*b^18*c^4 + 2906*a^8*b^20*c^4 - 1661*a^6*b^22*c^4 + 299*a^4*b^24*c^4 - 261*a^2*b^26*c^4 + 108*b^28*c^4 + 264*a^26*c^6 - 835*a^24*b^2*c^6 - 1745*a^22*b^4*c^6 + 6438*a^20*b^6*c^6 - 2106*a^18*b^8*c^6 - 3452*a^16*b^10*c^6 - 4694*a^14*b^12*c^6 + 7560*a^12*b^14*c^6 + 2144*a^10*b^16*c^6 - 3375*a^8*b^18*c^6 - 1073*a^6*b^20*c^6 + 458*a^4*b^22*c^6 + 554*a^2*b^24*c^6 - 138*b^26*c^6 - 157*a^24*c^8 + 2053*a^22*b^2*c^8 - 3055*a^20*b^4*c^8 - 2106*a^18*b^6*c^8 + 2754*a^16*b^8*c^8 + 4198*a^14*b^10*c^8 - 470*a^12*b^12*c^8 - 5384*a^10*b^14*c^8 - 145*a^8*b^16*c^8 + 1965*a^6*b^18*c^8 + 1333*a^4*b^20*c^8 - 854*a^2*b^22*c^8 - 132*b^24*c^8 - 312*a^22*c^10 - 513*a^20*b^2*c^10 + 3031*a^18*b^4*c^10 - 3452*a^16*b^6*c^10 + 4198*a^14*b^8*c^10 - 6288*a^12*b^10*c^10 + 3004*a^10*b^12*c^10 - 886*a^8*b^14*c^10 + 4082*a^6*b^16*c^10 - 3023*a^4*b^18*c^10 - 435*a^2*b^20*c^10 + 594*b^22*c^10 + 349*a^20*c^12 - 997*a^18*b^2*c^12 + 3104*a^16*b^4*c^12 - 4694*a^14*b^6*c^12 - 470*a^12*b^8*c^12 + 3004*a^10*b^10*c^12 + 3948*a^8*b^12*c^12 - 4226*a^6*b^14*c^12 - 2279*a^4*b^16*c^12 + 2753*a^2*b^18*c^12 - 492*b^20*c^12 + 72*a^18*c^14 - 1044*a^16*b^2*c^14 - 1010*a^14*b^4*c^14 + 7560*a^12*b^6*c^14 - 5384*a^10*b^8*c^14 - 886*a^8*b^10*c^14 - 4226*a^6*b^12*c^14 + 7212*a^4*b^14*c^14 - 1868*a^2*b^16*c^14 - 426*b^18*c^14 + 15*a^16*c^16 + 1834*a^14*b^2*c^16 - 4809*a^12*b^4*c^16 + 2144*a^10*b^6*c^16 - 145*a^8*b^8*c^16 + 4082*a^6*b^10*c^16 - 2279*a^4*b^12*c^16 - 1868*a^2*b^14*c^16 + 1026*b^16*c^16 - 328*a^14*c^18 + 896*a^12*b^2*c^18 + 1442*a^10*b^4*c^18 - 3375*a^8*b^6*c^18 + 1965*a^6*b^8*c^18 - 3023*a^4*b^10*c^18 + 2753*a^2*b^12*c^18 - 426*b^14*c^18 - 33*a^12*c^20 - 1678*a^10*b^2*c^20 + 2906*a^8*b^4*c^20 - 1073*a^6*b^6*c^20 + 1333*a^4*b^8*c^20 - 435*a^2*b^10*c^20 - 492*b^12*c^20 + 472*a^10*c^22 - 219*a^8*b^2*c^22 - 1661*a^6*b^4*c^22 + 458*a^4*b^6*c^22 - 854*a^2*b^8*c^22 + 594*b^10*c^22 - 255*a^8*c^24 + 1017*a^6*b^2*c^24 + 299*a^4*b^4*c^24 + 554*a^2*b^6*c^24 - 132*b^8*c^24 - 104*a^6*c^26 - 529*a^4*b^2*c^26 - 261*a^2*b^4*c^26 - 138*b^6*c^26 + 135*a^4*c^28 + 151*a^2*b^2*c^28 + 108*b^4*c^28 - 40*a^2*c^30 - 30*b^2*c^30 + 3*c^32) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7158.

X(66085) lies on this line: {122, 154}

X(66085) = midpoint of X(122) and X(48448)

X(66086) = (name pending)

Barycentrics    2*a^32 - 8*a^30*b^2 + 3*a^28*b^4 + 27*a^26*b^6 - 33*a^24*b^8 - 24*a^22*b^10 + 49*a^20*b^12 + 11*a^18*b^14 - 27*a^16*b^16 - 32*a^14*b^18 + 29*a^12*b^20 + 33*a^10*b^22 - 39*a^8*b^24 + 15*a^4*b^28 - 7*a^2*b^30 + b^32 - 8*a^30*c^2 + 58*a^28*b^2*c^2 - 103*a^26*b^4*c^2 - 55*a^24*b^6*c^2 + 272*a^22*b^8*c^2 - 74*a^20*b^10*c^2 - 211*a^18*b^12*c^2 - 15*a^16*b^14*c^2 + 192*a^14*b^16*c^2 + 110*a^12*b^18*c^2 - 253*a^10*b^20*c^2 + 43*a^8*b^22*c^2 + 88*a^6*b^24*c^2 - 62*a^4*b^26*c^2 + 23*a^2*b^28*c^2 - 5*b^30*c^2 + 3*a^28*c^4 - 103*a^26*b^2*c^4 + 394*a^24*b^4*c^4 - 340*a^22*b^6*c^4 - 373*a^20*b^8*c^4 + 387*a^18*b^10*c^4 + 422*a^16*b^12*c^4 - 92*a^14*b^14*c^4 - 715*a^12*b^16*c^4 + 307*a^10*b^18*c^4 + 250*a^8*b^20*c^4 - 144*a^6*b^22*c^4 + 13*a^4*b^24*c^4 - 15*a^2*b^26*c^4 + 6*b^28*c^4 + 27*a^26*c^6 - 55*a^24*b^2*c^6 - 340*a^22*b^4*c^6 + 880*a^20*b^6*c^6 - 195*a^18*b^8*c^6 - 373*a^16*b^10*c^6 - 812*a^14*b^12*c^6 + 956*a^12*b^14*c^6 + 297*a^10*b^16*c^6 - 253*a^8*b^18*c^6 - 256*a^6*b^20*c^6 + 116*a^4*b^22*c^6 - a^2*b^24*c^6 + 9*b^26*c^6 - 33*a^24*c^8 + 272*a^22*b^2*c^8 - 373*a^20*b^4*c^8 - 195*a^18*b^6*c^8 - 14*a^16*b^8*c^8 + 744*a^14*b^10*c^8 + 254*a^12*b^12*c^8 - 698*a^10*b^14*c^8 - 321*a^8*b^16*c^8 + 232*a^6*b^18*c^8 + 191*a^4*b^20*c^8 - 35*a^2*b^22*c^8 - 24*b^24*c^8 - 24*a^22*c^10 - 74*a^20*b^2*c^10 + 387*a^18*b^4*c^10 - 373*a^16*b^6*c^10 + 744*a^14*b^8*c^10 - 1268*a^12*b^10*c^10 + 314*a^10*b^12*c^10 - 110*a^8*b^14*c^10 + 960*a^6*b^16*c^10 - 546*a^4*b^18*c^10 - 13*a^2*b^20*c^10 + 3*b^22*c^10 + 49*a^20*c^12 - 211*a^18*b^2*c^12 + 422*a^16*b^4*c^12 - 812*a^14*b^6*c^12 + 254*a^12*b^8*c^12 + 314*a^10*b^10*c^12 + 860*a^8*b^12*c^12 - 880*a^6*b^14*c^12 - 219*a^4*b^16*c^12 + 197*a^2*b^18*c^12 + 26*b^20*c^12 + 11*a^18*c^14 - 15*a^16*b^2*c^14 - 92*a^14*b^4*c^14 + 956*a^12*b^6*c^14 - 698*a^10*b^8*c^14 - 110*a^8*b^10*c^14 - 880*a^6*b^12*c^14 + 984*a^4*b^14*c^14 - 149*a^2*b^16*c^14 - 7*b^18*c^14 - 27*a^16*c^16 + 192*a^14*b^2*c^16 - 715*a^12*b^4*c^16 + 297*a^10*b^6*c^16 - 321*a^8*b^8*c^16 + 960*a^6*b^10*c^16 - 219*a^4*b^12*c^16 - 149*a^2*b^14*c^16 - 18*b^16*c^16 - 32*a^14*c^18 + 110*a^12*b^2*c^18 + 307*a^10*b^4*c^18 - 253*a^8*b^6*c^18 + 232*a^6*b^8*c^18 - 546*a^4*b^10*c^18 + 197*a^2*b^12*c^18 - 7*b^14*c^18 + 29*a^12*c^20 - 253*a^10*b^2*c^20 + 250*a^8*b^4*c^20 - 256*a^6*b^6*c^20 + 191*a^4*b^8*c^20 - 13*a^2*b^10*c^20 + 26*b^12*c^20 + 33*a^10*c^22 + 43*a^8*b^2*c^22 - 144*a^6*b^4*c^22 + 116*a^4*b^6*c^22 - 35*a^2*b^8*c^22 + 3*b^10*c^22 - 39*a^8*c^24 + 88*a^6*b^2*c^24 + 13*a^4*b^4*c^24 - a^2*b^6*c^24 - 24*b^8*c^24 - 62*a^4*b^2*c^26 - 15*a^2*b^4*c^26 + 9*b^6*c^26 + 15*a^4*c^28 + 23*a^2*b^2*c^28 + 6*b^4*c^28 - 7*a^2*c^30 - 5*b^2*c^30 + c^32 : :

See Antreas Hatzipolakis and Peter Moses, euclid 7158.

X(66086) lies on this line: {25, 125}

X(66086) = midpoint of X(125) and X(10229)

X(66087) = HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE CENTER

Trilinears    Sqrt[3] Cos[A] - 2 (Cos[A/3] + 2 Cos[B/3] Cos[C/3]) (Cos[2A/3 + Pi/6] + Cos[2B/3 + Pi/6] + Cos[2C/3 + Pi/6]) : :

See Antreas Hatzipolakis and Chris van Tienhoven, euclid 7157.

X(66087) lies on this line: {3, 356}

X(66088) = MIDPOINT OF X(115) AND X(265)

Barycentrics    2*a^14 - 6*a^12*b^2 + 7*a^10*b^4 - 5*a^8*b^6 + 6*a^6*b^8 - 10*a^4*b^10 + 9*a^2*b^12 - 3*b^14 - 6*a^12*c^2 + 14*a^10*b^2*c^2 - 11*a^8*b^4*c^2 - 3*a^6*b^6*c^2 + 20*a^4*b^8*c^2 - 27*a^2*b^10*c^2 + 13*b^12*c^2 + 7*a^10*c^4 - 11*a^8*b^2*c^4 + 12*a^6*b^4*c^4 - 12*a^4*b^6*c^4 + 35*a^2*b^8*c^4 - 21*b^10*c^4 - 5*a^8*c^6 - 3*a^6*b^2*c^6 - 12*a^4*b^4*c^6 - 34*a^2*b^6*c^6 + 11*b^8*c^6 + 6*a^6*c^8 + 20*a^4*b^2*c^8 + 35*a^2*b^4*c^8 + 11*b^6*c^8 - 10*a^4*c^10 - 27*a^2*b^2*c^10 - 21*b^4*c^10 + 9*a^2*c^12 + 13*b^2*c^12 - 3*c^14 : :
X(66088) = 3 X[115] - X[18332], 3 X[265] + X[18332], X[99] - 5 X[15081], X[110] - 3 X[23514], X[114] - 3 X[14644], 3 X[125] - X[53710], 3 X[671] + X[18331], 2 X[36253] + X[38734], X[3448] + 3 X[14639], X[6321] + 3 X[38724], X[15357] - 3 X[38724], 2 X[6721] - 3 X[23515], 3 X[23515] - X[53735], X[10991] - 3 X[14849], X[12121] - 3 X[38737], X[12383] - 5 X[14061], X[12407] + 3 X[38220], X[12902] + 3 X[38224], 3 X[14971] - X[64182], 11 X[15025] - 5 X[38751], 5 X[15059] - 3 X[38748], 3 X[15061] - X[38738], X[15545] + 3 X[38732], 2 X[33511] - 3 X[38735], 3 X[34127] - X[34153], 3 X[34953] - X[46981]

X(66088) lies on these lines: {5, 50711}, {6, 13}, {74, 39809}, {99, 15081}, {110, 23514}, {114, 14644}, {125, 23698}, {620, 20304}, {671, 18331}, {690, 24978}, {1511, 6722}, {2782, 11801}, {2794, 10113}, {3448, 14639}, {6036, 15359}, {6321, 15357}, {6699, 38736}, {6721, 23515}, {9140, 9880}, {10264, 22515}, {10272, 15092}, {10628, 39806}, {10733, 38749}, {10991, 14849}, {11557, 58518}, {12121, 38737}, {12295, 53709}, {12383, 14061}, {12407, 38220}, {12902, 38224}, {14971, 64182}, {15025, 38751}, {15059, 38748}, {15061, 38738}, {15545, 38732}, {19457, 39825}, {20398, 53725}, {32423, 61576}, {33511, 38735}, {34127, 34153}, {34953, 46981}, {45311, 65722}

X(66088) = midpoint of X(i) and X(j) for these {i,j}: {74, 39809}, {115, 265}, {6321, 15357}, {9140, 9880}, {10113, 15535}, {10264, 22515}, {10733, 38749}, {12295, 53709}
X(66088) = reflection of X(i) in X(j) for these {i,j}: {620, 20304}, {1511, 6722}, {6036, 15359}, {10272, 15092}, {11557, 58518}, {38736, 6699}, {53725, 20398}, {53735, 6721}
X(66088) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6321, 38724, 15357}, {23515, 53735, 6721}

X(66089) = PERSPECTOR OF THESE TRIANGLES: 1st MORLEY AND HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Trilinears    Cos[A] + (Cos[A/3] + 2 Cos[B/3] Cos[C/3]) (1 + 2 Sin[(2 A)/3 - Pi/6] + 2 Sin[(2 B)/3 - Pi/6] + 2 Sin[(2 C)/3 - Pi/6]) : :

See Antreas Hatzipolakis and Chris van Tienhoven, euclid 7157.

X(66089) lies on this line: {3, 356}

X(66090) = X(2)X(7)∩X(342)X(347)

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

X(66090) lies on the curve Q189 and these lines: {2, 7}, {4, 38290}, {342, 347}, {651, 36413}, {653, 8894}, {1119, 6848}, {1490, 5932}, {2060, 2062}, {3487, 52097}, {6356, 6908}, {6527, 18026}, {6617, 55114}, {7080, 55015}, {7282, 37434}, {8809, 16870}, {17037, 65355}, {33672, 46350}, {40837, 56943}, {46352, 47634}, {56873, 64156}

X(66090) = X(33672)-Ceva conjugate of X(5932)
X(66090) = X(1490)-cross conjugate of X(329)
X(66090) = X(i)-isoconjugate of X(j) for these (i,j): {84, 7037}, {282, 7152}, {650, 8064}, {1034, 2208}, {1433, 7007}, {1436, 47850}, {2188, 7149}, {2192, 3345}, {7118, 41514}, {7151, 57643}
X(66090) = X(i)-Dao conjugate of X(j) for these (i,j): {57, 3345}, {278, 40836}, {281, 40838}
X(66090) = barycentric product X(i)*X(j) for these {i,j}: {223, 33672}, {322, 47848}, {329, 5932}, {347, 56943}, {664, 8063}, {1490, 40702}, {40212, 47436}
X(66090) = barycentric quotient X(i)/X(j) for these {i,j}: {40, 47850}, {109, 8064}, {196, 7149}, {198, 7037}, {207, 7129}, {221, 7152}, {223, 3345}, {329, 1034}, {347, 41514}, {1035, 1436}, {1490, 282}, {2331, 7007}, {3176, 7003}, {3197, 2192}, {5932, 189}, {7952, 40838}, {8063, 522}, {13612, 5514}, {33672, 34404}, {40212, 3342}, {40702, 56596}, {40837, 40836}, {47848, 84}, {55015, 63877}, {56943, 280}, {57117, 40117}, {64082, 57643}, {64708, 8806}
X(66090) = {X(226),X(40212)}-harmonic conjugate of X(329)

X(66091) = X(2)X(271)∩X(92)X(280)

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

X(66091) lies on the curve Q189 and these lines: {2, 271}, {8, 57643}, {85, 46352}, {92, 280}, {189, 3345}, {312, 46350}, {1311, 8064}, {2060, 7111}, {7020, 40838}, {7149, 52780}, {8806, 50442}, {63877, 64081}

X(66091) = X(56596)-Ceva conjugate of X(189)
X(66091) = X(i)-cross conjugate of X(j) for these (i,j): {3345, 1034}, {40836, 280}, {40838, 41514}
X(66091) = X(i)-isoconjugate of X(j) for these (i,j): {40, 1035}, {198, 47848}, {207, 7078}, {221, 1490}, {223, 3197}, {1415, 8063}, {2187, 5932}, {2199, 56943}, {3176, 7114}, {47438, 55015}
X(66091) = X(i)-Dao conjugate of X(j) for these (i,j): {1146, 8063}, {3341, 1490}, {3351, 40212}, {6129, 13612}
X(66091) = barycentric product X(i)*X(j) for these {i,j}: {189, 1034}, {280, 41514}, {282, 56596}, {309, 47850}, {3345, 34404}, {7037, 44190}, {7129, 57782}, {7149, 44189}, {7152, 57793}, {8064, 35519}, {46355, 63877}, {57643, 64988}
X(66091) = barycentric quotient X(i)/X(j) for these {i,j}: {84, 47848}, {189, 5932}, {280, 56943}, {282, 1490}, {522, 8063}, {1034, 329}, {1436, 1035}, {2192, 3197}, {3342, 40212}, {3345, 223}, {5514, 13612}, {7003, 3176}, {7007, 2331}, {7037, 198}, {7129, 207}, {7149, 196}, {7152, 221}, {8064, 109}, {8806, 64708}, {34404, 33672}, {40117, 57117}, {40836, 40837}, {40838, 7952}, {41514, 347}, {47850, 40}, {56596, 40702}, {57643, 64082}, {63877, 55015}

X(66092) = X(13)X(53458)∩X(14)X(53459)

Barycentrics    a^6*b^2 + 2*a^4*b^4 - 4*a^2*b^6 + b^8 + a^6*c^2 + 4*a^4*b^2*c^2 - a^2*b^4*c^2 - 3*b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 + 4*b^4*c^4 - 4*a^2*c^6 - 3*b^2*c^6 + c^8 : :
X(66092) = 3 X[3] - X[14712], 3 X[5] - 4 X[625], 3 X[5103] - 2 X[19130], X[6033] - 3 X[7809], 4 X[140] - 3 X[38230], 2 X[2080] - 3 X[38230], 2 X[187] - 3 X[549], 3 X[381] - X[43453], 5 X[632] - 4 X[14693], 3 X[15980] - X[47286], 4 X[3530] - 3 X[38225], X[18322] - 3 X[33873], 4 X[3628] - 3 X[38227], X[3793] - 3 X[56370], 3 X[5207] - X[18440], X[18440] + 3 X[35458], 6 X[5215] - 7 X[61851], and many others

X(66092) lies on these lines: {2, 9301}, {3, 7777}, {4, 7897}, {5, 141}, {6, 43456}, {13, 53458}, {14, 53469}, {15, 53441}, {16, 53429}, {30, 99}, {83, 140}, {113, 46669}, {187, 549}, {262, 7937}, {315, 32151}, {376, 63021}, {381, 3314}, {385, 61560}, {524, 49006}, {550, 18860}, {632, 7889}, {754, 12042}, {1916, 15980}, {2021, 31406}, {2076, 2548}, {2782, 7813}, {3095, 7790}, {3098, 7775}, {3530, 38225}, {3580, 18322}, {3627, 13449}, {3628, 7944}, {3767, 15514}, {3793, 56370}, {3845, 31173}, {3849, 8703}, {3933, 39266}, {5025, 48673}, {5055, 16986}, {5066, 10302}, {5111, 5305}, {5149, 32459}, {5162, 7745}, {5184, 61524}, {5189, 38583}, {5207, 7776}, {5215, 61851}, {5965, 51523}, {5999, 9866}, {6034, 8586}, {6329, 35377}, {7470, 7941}, {7516, 54091}, {7575, 47570}, {7698, 15360}, {7752, 9821}, {7759, 14880}, {7773, 40279}, {7779, 12188}, {7812, 26316}, {7817, 55716}, {7818, 9996}, {7824, 42788}, {7832, 18502}, {7835, 34733}, {7840, 61102}, {7844, 37517}, {7845, 58849}, {7853, 44422}, {7858, 12054}, {7885, 37243}, {7892, 18501}, {9300, 54964}, {10150, 61890}, {10242, 10723}, {10264, 14962}, {10277, 65517}, {11318, 44456}, {11539, 15491}, {11676, 61561}, {11812, 26613}, {12100, 51224}, {13862, 22728}, {14485, 60213}, {15699, 31275}, {15712, 47113}, {15919, 44262}, {16188, 37938}, {18572, 46338}, {19924, 22566}, {20428, 41024}, {20429, 41025}, {21536, 51360}, {25338, 57311}, {29317, 38745}, {31415, 54173}, {32816, 35456}, {33330, 55051}, {34105, 37950}, {34209, 57272}, {36248, 36249}, {37466, 37690}, {38743, 40236}, {40927, 61545}, {41136, 48657}, {42010, 55009}, {42215, 53514}, {42216, 53511}, {44282, 47584}, {44289, 50858}, {46264, 47619}, {50855, 52649}, {53452, 60319}, {53463, 60318}, {54718, 60202}, {58309, 64474}

X(66092) = midpoint of X(i) and X(j) for these {i,j}: {4, 47618}, {316, 35002}, {5189, 38583}, {5207, 35458}, {7779, 12188}, {7845, 58849}
X(66092) = reflection of X(i) in X(j) for these {i,j}: {385, 61560}, {550, 18860}, {2080, 140}, {3627, 13449}, {3845, 31173}, {5184, 61524}, {7575, 47570}, {11676, 61561}, {43460, 61599}, {51224, 12100}, {51872, 325}
X(66092) = complement of X(9301)
X(66092) = reflection of X(51872) in the De Longchamps axis
X(66092) = complement of the isogonal conjugate of X(9302)
X(66092) = X(9302)-complementary conjugate of X(10)
X(66092) = crossdifference of every pair of points on line {3050, 6041}
X(66092) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {140, 2080, 38230}, {623, 624, 5103}, {626, 14881, 5}, {7818, 58851, 9996}, {7832, 18502, 44237}

X(66093) = X(15)X(5469)∩X(16)X(5470)

Barycentrics    4*a^8 - 9*a^6*b^2 + 16*a^4*b^4 - 18*a^2*b^6 + 7*b^8 - 9*a^6*c^2 - 8*a^4*b^2*c^2 + 13*a^2*b^4*c^2 - 27*b^6*c^2 + 16*a^4*c^4 + 13*a^2*b^2*c^4 + 40*b^4*c^4 - 18*a^2*c^6 - 27*b^2*c^6 + 7*c^8 : :
X(66093) = 7 X[2] - X[13188], 5 X[2] - 2 X[61561], 5 X[13188] - 14 X[61561], X[5] - 4 X[5461], 5 X[5] + 4 X[11623], X[5] + 8 X[20398], 5 X[5] - 2 X[22566], 13 X[5] - 4 X[38745], X[5] + 2 X[49102], 7 X[5] + 2 X[51523], 5 X[5461] + X[11623], X[5461] + 2 X[20398], 10 X[5461] - X[22566], 13 X[5461] - X[38745], 2 X[5461] + X[49102], 14 X[5461] + X[51523], X[11623] - 10 X[20398], 2 X[11623] + X[22566], and many others

X(66093) lies on these lines: {2, 13188}, {5, 542}, {15, 5469}, {16, 5470}, {30, 9166}, {98, 5066}, {99, 10124}, {114, 61910}, {115, 549}, {140, 671}, {147, 61920}, {148, 15694}, {381, 7806}, {543, 11539}, {546, 14830}, {547, 11632}, {550, 9880}, {620, 61869}, {631, 12355}, {632, 2482}, {1656, 12243}, {2782, 14971}, {2794, 23046}, {2796, 11231}, {3090, 48657}, {3524, 38732}, {3525, 8596}, {3526, 8591}, {3530, 12117}, {3628, 8724}, {3845, 6055}, {3857, 10991}, {5054, 38635}, {5055, 14651}, {5071, 12188}, {5465, 10264}, {5690, 12258}, {5969, 16509}, {5984, 61932}, {6033, 11737}, {6034, 8586}, {6036, 8703}, {6054, 10109}, {6321, 12100}, {6721, 61890}, {6722, 61885}, {8593, 51732}, {8981, 49215}, {9167, 61874}, {9830, 38110}, {9884, 51700}, {10054, 15325}, {10722, 61978}, {10723, 15690}, {10992, 61837}, {11006, 40685}, {11177, 19709}, {11540, 38750}, {11656, 20304}, {11812, 61600}, {12042, 15687}, {12101, 38741}, {12812, 52090}, {13172, 15701}, {13670, 42215}, {13790, 42216}, {13881, 42787}, {13908, 19117}, {13966, 49214}, {13968, 19116}, {14159, 63101}, {14891, 38730}, {14981, 61900}, {15092, 61942}, {15561, 47599}, {15686, 22515}, {15692, 38733}, {15703, 64090}, {15711, 38738}, {15712, 38734}, {15713, 33813}, {16239, 64019}, {17504, 23698}, {18583, 19905}, {20094, 61859}, {21166, 61827}, {22247, 51524}, {22505, 41148}, {23235, 48154}, {23514, 38071}, {34200, 38739}, {35018, 38664}, {35021, 61963}, {35404, 38749}, {36519, 61917}, {38064, 43620}, {38634, 62020}, {38731, 61782}, {38737, 45759}, {38743, 61924}, {38744, 41106}, {38747, 62154}, {38748, 61841}, {39809, 44903}, {41134, 47598}, {50881, 61272}, {52695, 61864}, {59378, 59384}, {59379, 59383}, {61575, 61916}, {61599, 61922}, {61896, 64089}

X(66093) = midpoint of X(i) and X(j) for these {i,j}: {3524, 38732}, {5054, 41135}, {5055, 14651}, {9166, 38224}, {11632, 23234}, {59378, 59384}, {59379, 59383}
X(66093) = reflection of X(i) in X(j) for these {i,j}: {11539, 34127}, {15561, 47599}, {15699, 14971}, {17504, 26614}, {21166, 61827}, {23234, 547}, {38071, 23514}, {38229, 9166}, {38731, 61782}, {41134, 47598}, {45759, 38737}, {51872, 23234}
X(66093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {547, 11632, 51872}, {5461, 20398, 49102}, {5461, 49102, 5}, {6055, 61576, 3845}, {11632, 14061, 547}, {22566, 49102, 11623}

X(66094) = X(76)X(140)∩X(182)X(524)

Barycentrics    2*a^8 - 9*a^6*b^2 + 12*a^4*b^4 - 6*a^2*b^6 + b^8 - 9*a^6*c^2 + 12*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 3*b^6*c^2 + 12*a^4*c^4 + 5*a^2*b^2*c^4 + 4*b^4*c^4 - 6*a^2*c^6 - 3*b^2*c^6 + c^8 : :

X(66094) lies on these lines: {2, 13188}, {3, 7777}, {4, 42788}, {5, 574}, {6, 38230}, {13, 44223}, {14, 52650}, {30, 43461}, {39, 14693}, {76, 140}, {141, 39498}, {182, 524}, {547, 52691}, {550, 9734}, {620, 24256}, {632, 7789}, {1352, 5116}, {2549, 38229}, {3094, 18583}, {3106, 61513}, {3107, 61514}, {3526, 7891}, {3530, 26316}, {3628, 7790}, {5066, 11669}, {6713, 51046}, {7619, 49102}, {7709, 61560}, {7844, 55856}, {8724, 55801}, {10277, 52036}, {10484, 11170}, {11539, 59780}, {11842, 33274}, {13108, 33015}, {13449, 37512}, {15464, 43084}, {15482, 51848}, {15920, 61548}, {17004, 32519}, {31406, 32134}, {38110, 59695}, {42215, 53498}, {42216, 53497}, {54482, 60233}, {61104, 62362}

X(66094) = midpoint of X(3) and X(7777)
X(66094) = reflection of X(i) in X(j) for these {i,j}: {5, 3055}, {37688, 140}

X(66095) = X(76)X(140)∩X(230)X(549)

Barycentrics    2*a^8 - a^6*b^2 - 2*a^2*b^6 + b^8 - a^6*c^2 - 8*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 3*b^6*c^2 - 3*a^2*b^2*c^4 + 4*b^4*c^4 - 2*a^2*c^6 - 3*b^2*c^6 + c^8 : :

X(66095) lies on these lines: {2, 7711}, {3, 7797}, {5, 182}, {6, 43456}, {15, 53440}, {16, 53428}, {30, 3972}, {76, 140}, {230, 549}, {316, 3398}, {381, 7875}, {550, 20576}, {625, 50664}, {632, 7822}, {2549, 44532}, {2782, 7820}, {3091, 48674}, {3094, 5305}, {3407, 11170}, {3526, 46226}, {3628, 7943}, {3767, 5116}, {4045, 12042}, {4846, 43721}, {5026, 10168}, {5050, 5207}, {5054, 17004}, {5066, 14458}, {5092, 7817}, {5254, 44224}, {5989, 11185}, {6033, 7919}, {6036, 40108}, {6656, 32151}, {7622, 15713}, {7709, 61561}, {7775, 55710}, {7807, 32516}, {7827, 35002}, {7828, 12054}, {7829, 14881}, {7846, 44237}, {7851, 40279}, {7856, 9821}, {7866, 39899}, {7913, 9996}, {7920, 48673}, {7923, 37243}, {9301, 63019}, {9734, 15712}, {10124, 47005}, {10272, 15920}, {11318, 55705}, {11539, 58446}, {11623, 58445}, {12100, 52691}, {13334, 58448}, {14389, 21531}, {14693, 21163}, {15921, 61572}, {24206, 51523}, {32515, 37450}, {35705, 38224}, {37348, 38229}, {38064, 43620}, {39499, 53567}, {42215, 53515}, {42216, 53512}, {44380, 50979}, {60115, 60215}, {60659, 63047}

X(66095) = midpoint of X(7790) and X(26316)
X(66095) = {X(7834),X(14880)}-harmonic conjugate of X(5)

X(66096) = X(5)X(32)∩X(140)X(262)

Barycentrics    2*a^8 - 5*a^6*b^2 + 2*a^4*b^4 + b^8 - 5*a^6*c^2 - 8*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - b^6*c^2 + 2*a^4*c^4 - 3*a^2*b^2*c^4 - b^2*c^6 + c^8 : :
X(66096) = X[7898] + 3 X[10788]

X(66096) lies on these lines: {2, 9301}, {3, 7875}, {5, 32}, {6, 51872}, {15, 44223}, {16, 52650}, {30, 3972}, {76, 44237}, {140, 262}, {381, 7806}, {495, 10047}, {496, 10038}, {546, 9873}, {547, 7811}, {549, 3098}, {576, 7908}, {590, 35783}, {598, 5066}, {615, 35782}, {952, 11368}, {1656, 2896}, {2076, 14561}, {2080, 7831}, {2782, 5355}, {3091, 18503}, {3094, 18583}, {3095, 7835}, {3096, 3628}, {3099, 5886}, {3104, 61537}, {3105, 61538}, {3398, 40239}, {3407, 44230}, {3526, 10357}, {3530, 35248}, {3850, 18500}, {5025, 18501}, {5055, 17004}, {5432, 65127}, {5901, 9941}, {6033, 12150}, {6680, 14881}, {7583, 44605}, {7584, 44604}, {7736, 37466}, {7753, 61575}, {7807, 40252}, {7819, 32521}, {7828, 18502}, {7865, 15699}, {7880, 55716}, {7892, 48673}, {7898, 10788}, {7907, 42788}, {7914, 55856}, {7919, 12110}, {8176, 61910}, {8254, 9985}, {8368, 22486}, {8782, 32447}, {9857, 38042}, {9923, 61544}, {9956, 49561}, {9981, 20253}, {9982, 20252}, {9983, 61550}, {9984, 61548}, {9997, 10283}, {10272, 13210}, {10277, 34845}, {10346, 37446}, {10347, 38227}, {10592, 10873}, {10593, 10874}, {10828, 13861}, {11272, 46283}, {11386, 21841}, {11623, 22681}, {11801, 12501}, {11842, 13862}, {12040, 42536}, {12042, 19130}, {12188, 63019}, {12495, 61510}, {12496, 61556}, {12497, 61524}, {12498, 61553}, {12499, 61566}, {12502, 61540}, {13235, 61562}, {13236, 61573}, {14389, 44215}, {14853, 35456}, {15092, 43457}, {15325, 18957}, {15806, 43854}, {16123, 61552}, {19011, 19117}, {19012, 19116}, {19686, 38733}, {22745, 61516}, {22746, 61515}, {24825, 61621}, {25555, 40108}, {30435, 43450}, {32268, 61543}, {32448, 63633}, {38229, 43449}, {54716, 62912}, {60900, 61509}

X(66096) = midpoint of X(i) and X(j) for these {i,j}: {9993, 26316}, {11842, 13862}
X(66096) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 32, 32151}, {140, 9821, 42787}, {381, 7806, 61560}, {7846, 9821, 140}

X(66097) = X(15)X(53446)∩X(16)X(53434)

Barycentrics    2*a^8 + 3*a^6*b^2 - 22*a^4*b^4 + 24*a^2*b^6 - 7*b^8 + 3*a^6*c^2 - 28*a^4*b^2*c^2 - 19*a^2*b^4*c^2 + 27*b^6*c^2 - 22*a^4*c^4 - 19*a^2*b^2*c^4 - 40*b^4*c^4 + 24*a^2*c^6 + 27*b^2*c^6 - 7*c^8 : :

X(66097) lies on these lines: {2, 38230}, {5, 524}, {15, 53446}, {16, 53434}, {30, 43461}, {140, 598}, {148, 381}, {262, 5066}, {547, 7811}, {549, 3055}, {3628, 7936}, {3972, 10124}, {5611, 51483}, {5615, 51482}, {7753, 41675}, {8860, 14161}, {10796, 15699}, {14159, 22329}, {25154, 44289}, {25164, 52649}, {31415, 54173}, {38735, 61046}, {43450, 54964}

X(66097) = midpoint of X(381) and X(7777)
X(66097) = reflection of X(i) in X(j) for these {i,j}: {549, 3055}, {37688, 547}

X(66098) = X(1)X(381)∩X(4)X(3017)

Barycentrics    2*a^7 - 5*a^6*b + 6*a^4*b^3 - 6*a^3*b^4 + 3*a^2*b^5 + 4*a*b^6 - 4*b^7 - 5*a^6*c - 2*a^5*b*c - a^4*b^2*c + a^3*b^3*c + 2*a^2*b^4*c + a*b^5*c + 4*b^6*c - a^4*b*c^2 + 10*a^3*b^2*c^2 - 5*a^2*b^3*c^2 - 4*a*b^4*c^2 + 12*b^5*c^2 + 6*a^4*c^3 + a^3*b*c^3 - 5*a^2*b^2*c^3 - 2*a*b^3*c^3 - 12*b^4*c^3 - 6*a^3*c^4 + 2*a^2*b*c^4 - 4*a*b^2*c^4 - 12*b^3*c^4 + 3*a^2*c^5 + a*b*c^5 + 12*b^2*c^5 + 4*a*c^6 + 4*b*c^6 - 4*c^7 : :

X(66098) lies on these lines: {1, 381}, {4, 3017}, {30, 35466}, {321, 54516}, {376, 24880}, {549, 24902}, {3019, 61983}, {3543, 24883}, {3839, 45924}, {3845, 56402}, {5721, 48861}, {6175, 48897}, {14269, 45923}, {15683, 24898}, {24936, 61936}, {37428, 41501}, {37718, 62491}, {48842, 63982}, {49744, 63318}

X(66099) = X(2)X(3)∩X(8)X(3058)

Barycentrics    (a - b - c)*(2*a^3 + 2*a^2*b + a*b^2 + b^3 + 2*a^2*c + a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(66099) = 5 X[2] - 2 X[11112], X[2] + 2 X[11113], 2 X[2] + X[11114], 4 X[2] - X[17579], X[4] + 2 X[28459], 4 X[549] - X[37430], 5 X[631] - 2 X[28458], 5 X[631] + 4 X[37290], 7 X[3090] + 2 X[7491], 5 X[3091] + 4 X[31789], 10 X[3091] - X[59355], X[3543] + 2 X[37428], 11 X[5056] - 2 X[37468], 13 X[5068] - 4 X[20420], 5 X[5071] - 2 X[28452], and many others

X(66099) lies on these lines: {2, 3}, {8, 3058}, {10, 41872}, {13, 54379}, {14, 54378}, {56, 26127}, {81, 48870}, {145, 15170}, {149, 9708}, {329, 15933}, {392, 28204}, {517, 5640}, {519, 3681}, {528, 38057}, {529, 3475}, {535, 25055}, {540, 3794}, {551, 3897}, {553, 64002}, {612, 48827}, {614, 48818}, {936, 11015}, {950, 3876}, {958, 11238}, {993, 3582}, {1001, 5080}, {1211, 48859}, {1329, 4995}, {1478, 5284}, {1479, 5260}, {1621, 10056}, {1655, 7837}, {1724, 3017}, {1737, 62838}, {2346, 11239}, {2551, 3871}, {2829, 54445}, {2975, 10072}, {3219, 5722}, {3241, 5330}, {3303, 56880}, {3305, 3586}, {3419, 27065}, {3488, 31018}, {3578, 10449}, {3583, 33108}, {3584, 5248}, {3615, 43531}, {3616, 5434}, {3617, 15171}, {3621, 15172}, {3634, 65134}, {3648, 5221}, {3654, 34629}, {3679, 5178}, {3697, 31795}, {3720, 48825}, {3753, 28198}, {3816, 5298}, {3826, 65632}, {3828, 7705}, {3841, 18514}, {3868, 10399}, {3885, 5795}, {3889, 12527}, {3920, 48824}, {3951, 37723}, {4383, 48842}, {4428, 31141}, {4511, 4679}, {4512, 19875}, {4654, 54392}, {4669, 34719}, {4720, 14555}, {4745, 34649}, {5057, 54318}, {5251, 11680}, {5262, 50068}, {5283, 7753}, {5287, 48828}, {5303, 10200}, {5325, 6734}, {5362, 10654}, {5367, 10653}, {5550, 7354}, {5554, 50810}, {5985, 11632}, {6284, 9780}, {6740, 48863}, {7191, 48819}, {7679, 64086}, {7737, 37675}, {7739, 33854}, {7811, 18140}, {8167, 12943}, {8582, 50808}, {8583, 34628}, {9668, 33110}, {9709, 20066}, {9711, 63273}, {10197, 31160}, {10327, 48798}, {10479, 49729}, {10483, 19862}, {10546, 51420}, {11180, 63070}, {14537, 16589}, {14997, 48847}, {15934, 17484}, {15988, 20423}, {16998, 19570}, {17024, 48820}, {17127, 37715}, {17182, 57722}, {17183, 17378}, {17185, 48839}, {17194, 48868}, {17757, 61155}, {18135, 37671}, {18253, 56203}, {18444, 37822}, {18990, 46934}, {19767, 49739}, {19784, 34657}, {19860, 31162}, {19861, 50811}, {20195, 51790}, {21077, 62870}, {24564, 31673}, {24929, 27131}, {24987, 50796}, {25005, 50821}, {25011, 31730}, {26062, 34630}, {26543, 47354}, {29814, 48823}, {30117, 33151}, {32836, 45962}, {32911, 48857}, {33090, 48804}, {33091, 48800}, {34617, 51709}, {34637, 51108}, {34690, 51103}, {34695, 64143}, {34720, 51072}, {36263, 53619}, {36889, 57818}, {37657, 48848}, {37680, 48837}, {38074, 59416}, {40663, 60954}, {41698, 52769}, {44663, 61663}, {47353, 63470}, {48861, 63074}, {48866, 51382}, {50865, 64673}, {50890, 66008}, {56879, 64199}, {57721, 60079}, {57822, 57830}

X(66099) = orthocentroidal-circle-inverse of X(6175)
X(66099) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 6175}, {2, 376, 404}, {2, 381, 2476}, {2, 452, 31156}, {2, 549, 17566}, {2, 2475, 44217}, {2, 3543, 377}, {2, 4189, 549}, {2, 5046, 381}, {2, 5071, 7504}, {2, 5154, 547}, {2, 5187, 5071}, {2, 6175, 4197}, {2, 6872, 376}, {2, 7924, 33840}, {2, 11111, 17549}, {2, 11113, 11114}, {2, 11114, 17579}, {2, 15677, 3}, {2, 15683, 6904}, {2, 15692, 6921}, {2, 16859, 50202}, {2, 16865, 15670}, {2, 16920, 6661}, {2, 17561, 15671}, {2, 17576, 15692}, {2, 17692, 33246}, {2, 19686, 16915}, {2, 26117, 50321}, {2, 31156, 21}, {2, 33824, 7924}, {2, 36004, 16417}, {2, 37291, 15694}, {2, 37299, 16371}, {2, 48817, 51669}, {2, 50407, 14005}, {2, 50430, 17553}, {2, 51678, 19336}, {2, 61936, 6933}, {2, 61985, 5177}, {2, 62005, 37161}, {2, 62048, 56999}, {2, 62969, 17528}, {3, 6965, 6945}, {4, 5047, 4197}, {4, 6992, 7411}, {5, 15670, 2}, {21, 2478, 4193}, {376, 5084, 2}, {376, 6872, 15678}, {377, 3543, 15679}, {377, 5129, 17536}, {381, 405, 2}, {404, 15678, 376}, {405, 5046, 2476}, {442, 50202, 2}, {452, 2478, 21}, {452, 6919, 11106}, {547, 7483, 2}, {547, 15673, 7483}, {547, 50243, 15673}, {549, 4187, 2}, {549, 17525, 4189}, {549, 50241, 17525}, {550, 17575, 17572}, {1006, 6929, 6932}, {1995, 56960, 1325}, {2478, 6910, 6919}, {2478, 31156, 2}, {3091, 31789, 59355}, {3543, 5129, 2}, {3545, 17561, 2}, {3560, 6902, 6943}, {3830, 11108, 44217}, {3830, 44217, 2475}, {3845, 50202, 442}, {4187, 4189, 17566}, {4187, 17525, 549}, {4187, 50241, 4189}, {4190, 17559, 17535}, {4205, 50323, 2}, {5047, 6175, 2}, {5071, 6857, 2}, {5073, 16855, 56997}, {5084, 6872, 404}, {5187, 6857, 7504}, {5192, 50321, 2}, {6175, 15678, 33557}, {6827, 6976, 6912}, {6840, 6913, 10883}, {6868, 6898, 6915}, {6871, 16845, 31254}, {6893, 6936, 411}, {6910, 11106, 21}, {6919, 11106, 6910}, {6920, 6928, 6828}, {6930, 6947, 6909}, {6957, 6987, 36002}, {7924, 16918, 2}, {11108, 44217, 2}, {11114, 17566, 37430}, {15670, 17525, 12104}, {15671, 16858, 17561}, {15677, 37162, 2}, {16408, 50242, 37256}, {16417, 57006, 36004}, {16418, 17556, 2}, {16857, 17532, 2}, {16858, 37375, 2}, {16861, 17577, 2}, {16916, 17685, 17550}, {16918, 33824, 33840}, {17527, 57002, 4188}, {17528, 17542, 2}, {17552, 50727, 2}, {17558, 61936, 2}, {17590, 50395, 2}, {18586, 18587, 64473}, {20846, 28466, 17549}, {33046, 33246, 2}, {34606, 49736, 3241}

X(66100) = X(1)X(2)∩X(30)X(333)

Barycentrics    a^4 + 3*a^3*b - 2*a^2*b^2 - 3*a*b^3 + b^4 + 3*a^3*c - 3*a^2*b*c - 9*a*b^2*c - 3*b^3*c - 2*a^2*c^2 - 9*a*b*c^2 - 8*b^2*c^2 - 3*a*c^3 - 3*b*c^3 + c^4 : :
X(66100) = X[8] + 2 X[54335]

X(66100) lies on these lines: {1, 2}, {30, 333}, {391, 3839}, {1010, 61661}, {1043, 15670}, {1330, 3578}, {1654, 19570}, {1834, 49730}, {2475, 50215}, {2891, 25466}, {3543, 43533}, {3545, 14555}, {3681, 61699}, {3695, 4102}, {4042, 11237}, {4405, 25455}, {4720, 15671}, {4921, 50171}, {5055, 5233}, {5123, 25679}, {5224, 51593}, {5295, 42033}, {5325, 7283}, {5737, 48842}, {5814, 42030}, {6757, 28612}, {11110, 49739}, {14534, 54786}, {15673, 52352}, {15682, 46976}, {16052, 41816}, {16267, 37834}, {16268, 37831}, {16418, 56946}, {17330, 56745}, {17346, 17532}, {17677, 49724}, {24597, 51591}, {25648, 64200}, {26051, 49744}, {26117, 49729}, {26131, 50256}, {32853, 48825}, {34258, 54677}, {37631, 56018}, {37652, 48870}, {41629, 50169}, {45923, 56440}, {48839, 54119}, {49735, 64424}, {50074, 56291}, {51668, 56974}, {54510, 60206}

X(66100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 9780, 27558}, {8, 25446, 25650}, {10, 3017, 2}, {3578, 6175, 1330}, {6175, 64401, 3578}

X(66101) = X(2)X(37)∩X(9)X(3017)

Barycentrics    a^5 + a^4*b + 4*a^3*b^2 + 4*a^2*b^3 + a*b^4 + b^5 + a^4*c + 9*a^3*b*c + 10*a^2*b^2*c + 3*a*b^3*c + b^4*c + 4*a^3*c^2 + 10*a^2*b*c^2 + 4*a*b^2*c^2 - 2*b^3*c^2 + 4*a^2*c^3 + 3*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5 : :

X(66101) lies on these lines: {2, 37}, {9, 3017}, {30, 2303}, {45, 24883}, {941, 1989}, {965, 48842}, {1333, 15677}, {1778, 61661}, {1901, 37631}, {2325, 25441}, {4021, 25651}, {4029, 25645}, {4873, 24931}, {6175, 53417}, {7739, 24275}, {16672, 24936}, {16673, 24937}, {16676, 24880}, {17330, 53427}, {17592, 61710}, {19570, 26110}, {19767, 62210}, {48818, 54385}, {48857, 52405}, {50066, 54405}, {61650, 61699}

X(66102) = X(2)X(759)∩X(9)X(80)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(2*a^3 + 2*a^2*b + a*b^2 + b^3 + 2*a^2*c + 2*a*b*c + a*c^2 + c^3) : :

X(66102) lies on these lines: {2, 759}, {9, 80}, {30, 35466}, {1168, 5180}, {1751, 54528}, {1793, 31156}, {2166, 14583}, {3017, 52380}, {4075, 34857}, {5248, 6187}, {6740, 48863}, {9143, 56405}, {11114, 24624}, {12699, 56426}, {17537, 52367}, {21363, 28459}, {34311, 37718}, {37702, 57263}, {51303, 56645}

X(66102) = barycentric product X(i)*X(j) for these {i,j}: {80, 29833}, {14616, 53037}
X(66102) = barycentric quotient X(i)/X(j) for these {i,j}: {29833, 320}, {53037, 758}

X(66103) = X(4)X(572)∩X(30)X(386)

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

X(66103) lies on these lines: {4, 572}, {10, 54544}, {30, 386}, {381, 46976}, {2049, 32431}, {2794, 49130}, {3543, 19766}, {16124, 24725}, {17777, 28661}, {34258, 64748}, {35203, 48839}, {37823, 49129}

X(66104) = X(4)X(6)∩X(10)X(45)

Barycentrics    a^4 - 2*a^3*b - a^2*b^2 - 2*b^4 - 2*a^3*c - 2*a^2*b*c - a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :

X(66104) lies on these lines: {1, 3838}, {2, 49734}, {3, 45939}, {4, 6}, {5, 4255}, {8, 4415}, {10, 45}, {20, 37646}, {21, 31187}, {30, 4252}, {31, 12953}, {40, 5036}, {42, 10895}, {55, 21935}, {58, 382}, {65, 1900}, {115, 2271}, {149, 37542}, {154, 37226}, {230, 7390}, {377, 37674}, {381, 386}, {443, 37682}, {497, 1616}, {546, 48847}, {595, 9668}, {599, 10449}, {938, 1086}, {940, 2475}, {950, 3772}, {966, 43533}, {995, 9669}, {1030, 37320}, {1104, 3586}, {1191, 1479}, {1193, 10896}, {1201, 11238}, {1203, 18514}, {1330, 40341}, {1468, 12943}, {1620, 37410}, {1656, 4256}, {1657, 4257}, {1714, 11113}, {1837, 1853}, {2047, 8253}, {2049, 5110}, {2334, 9656}, {2476, 19765}, {2478, 37679}, {2549, 5022}, {2650, 61716}, {3017, 3830}, {3052, 5230}, {3053, 49132}, {3086, 8572}, {3091, 37662}, {3146, 37642}, {3192, 37197}, {3214, 31141}, {3216, 17556}, {3242, 13161}, {3445, 37722}, {3485, 62221}, {3543, 61661}, {3583, 16466}, {3752, 9581}, {3755, 19925}, {3763, 16062}, {3767, 4258}, {3782, 12649}, {3815, 7407}, {3832, 63089}, {3845, 48857}, {3913, 37716}, {3915, 9670}, {3944, 12635}, {4190, 37634}, {4208, 17245}, {4214, 37538}, {4259, 15488}, {4294, 21000}, {4383, 5046}, {4385, 59407}, {4642, 7069}, {4646, 5587}, {4648, 37161}, {4857, 16483}, {5021, 7748}, {5064, 54426}, {5086, 33134}, {5096, 37415}, {5124, 37062}, {5129, 17337}, {5177, 17056}, {5187, 37663}, {5204, 29662}, {5275, 23903}, {5290, 49478}, {5295, 56541}, {5710, 52367}, {5717, 16884}, {5718, 6871}, {5722, 17054}, {5737, 26117}, {5793, 32773}, {5794, 24210}, {5814, 62224}, {6144, 56018}, {6703, 50408}, {6734, 50065}, {6840, 37537}, {6850, 37501}, {6872, 35466}, {6919, 51415}, {6923, 36746}, {6928, 36745}, {6998, 37637}, {7074, 10953}, {7297, 7713}, {7300, 54397}, {7354, 11269}, {7380, 31489}, {7773, 33296}, {7841, 17034}, {8252, 63810}, {8609, 15852}, {9598, 42316}, {9664, 14974}, {10448, 31245}, {10459, 31140}, {10479, 50056}, {10516, 50591}, {10525, 64449}, {10827, 64175}, {10894, 37529}, {11114, 24883}, {11236, 50581}, {11287, 29455}, {11354, 20083}, {11359, 50605}, {11679, 50050}, {12293, 56295}, {12433, 24159}, {12513, 33141}, {12572, 16885}, {13736, 62689}, {13740, 47355}, {13881, 18755}, {14893, 48861}, {15069, 37823}, {15668, 26051}, {15687, 48870}, {15955, 18525}, {16052, 48862}, {16394, 25441}, {16418, 24880}, {16644, 37144}, {16645, 37145}, {17276, 24391}, {17327, 37164}, {17334, 54398}, {17374, 35629}, {17392, 50736}, {17555, 26958}, {17577, 19767}, {17578, 37666}, {17676, 37660}, {17685, 20154}, {17720, 57287}, {17734, 64951}, {18961, 34046}, {19744, 37314}, {20131, 33030}, {20135, 33028}, {20155, 33031}, {20156, 33029}, {20157, 33026}, {21049, 62693}, {21949, 64673}, {23681, 37723}, {24443, 61717}, {25446, 48814}, {31295, 63078}, {31479, 33771}, {31884, 50425}, {33094, 37567}, {33137, 57288}, {33863, 44526}, {36695, 63534}, {37146, 43029}, {37147, 43028}, {37224, 41501}, {37234, 45926}, {37411, 54431}, {37424, 50677}, {37522, 50239}, {37540, 54355}, {37657, 63537}, {45219, 51785}, {48801, 50608}, {48841, 50740}, {48846, 50741}, {49168, 63997}, {50242, 52680}, {51118, 64016}, {51599, 64850}, {54698, 57720}, {56819, 64127}, {57282, 62223}, {63541, 63604}

X(66104) = reflection of X(4252) in X(5292)
X(66104) = crosspoint of X(4) and X(43533)
X(66104) = crosssum of X(3) and X(4252)
X(66104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1834, 6}, {5, 48837, 4255}, {1479, 64172, 1191}, {3146, 37642, 64159}, {5086, 33134, 37614}, {5230, 6284, 3052}, {5722, 23537, 17054}

X(66105) = X(3)X(142)∩X(5)X(28858)

Barycentrics    2*a^5 - 5*a^4*b - a^3*b^2 - 2*a^2*b^3 + 5*a*b^4 + b^5 - 5*a^4*c + a^2*b^2*c + 4*b^4*c - a^3*c^2 + a^2*b*c^2 - 10*a*b^2*c^2 - 5*b^3*c^2 - 2*a^2*c^3 - 5*b^2*c^3 + 5*a*c^4 + 4*b*c^4 + c^5 : :
X[12699] + 3 X[48900], 5 X[18493] + 3 X[48944], X[31730] - 3 X[48932]

X(66105) lies on these lines: {3, 142}, {5, 28858}, {10, 54657}, {546, 28877}, {1699, 17367}, {2784, 18480}, {3008, 50802}, {6625, 54668}, {9955, 28854}, {9956, 28889}, {17397, 50865}, {19925, 28909}, {28866, 40273}, {29628, 30308}

X(66106) = X(1)X(4)∩X(3)X(45)

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

X(66106) lies on these lines: {1, 4}, {3, 45}, {30, 4415}, {31, 61705}, {36, 7069}, {44, 580}, {58, 2341}, {65, 52371}, {88, 6915}, {381, 30117}, {411, 62796}, {500, 46976}, {595, 31937}, {612, 50528}, {756, 7688}, {936, 54389}, {971, 37469}, {975, 41854}, {976, 41869}, {990, 5720}, {1071, 37520}, {2173, 57281}, {2783, 12738}, {3072, 31803}, {3073, 31871}, {3120, 18406}, {3149, 17595}, {3811, 28580}, {3924, 18492}, {3938, 31162}, {3961, 28194}, {4080, 34772}, {4217, 19861}, {4306, 37696}, {4346, 50700}, {4420, 32932}, {4887, 64001}, {5293, 31730}, {5396, 29061}, {6796, 17601}, {6831, 37691}, {6841, 24160}, {6849, 24159}, {7986, 18491}, {8583, 51673}, {11362, 54997}, {11552, 56422}, {12528, 37530}, {15955, 18525}, {16132, 59305}, {17012, 37732}, {18357, 30449}, {18480, 56426}, {18540, 37817}, {20117, 37570}, {33597, 54387}, {34627, 49494}, {34648, 49682}, {35242, 36510}, {37522, 64358}, {41543, 49745}, {41562, 54339}, {49712, 63967}, {50796, 60353}, {52544, 62210}, {54310, 66059}

X(66107) = X(1)X(3838)∩X(3)X(9)

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

X(66107) lies on these lines: {1, 3838}, {3, 9}, {4, 11813}, {10, 16132}, {57, 64715}, {72, 484}, {73, 56317}, {78, 9579}, {80, 442}, {200, 3962}, {214, 6920}, {224, 5219}, {226, 2475}, {329, 37256}, {405, 37616}, {950, 6224}, {1376, 15071}, {1706, 17857}, {2099, 2900}, {2476, 9581}, {3419, 37707}, {3430, 16548}, {3452, 64707}, {4640, 16143}, {5172, 6597}, {5293, 35338}, {5436, 37525}, {5531, 5836}, {5714, 22836}, {5881, 6937}, {5903, 11523}, {5927, 59691}, {6326, 17647}, {6596, 13273}, {6598, 57285}, {6840, 63998}, {6913, 26287}, {6943, 30827}, {8583, 33576}, {10382, 34471}, {10483, 58798}, {12625, 36846}, {13089, 34871}, {14799, 37284}, {14800, 37249}, {15556, 35990}, {15829, 63988}, {17668, 56176}, {19925, 65990}, {30147, 50741}, {37163, 57284}, {37572, 54290}, {54305, 56824}

X(66108) = X(3)X(17281)∩X(4)X(519)

Barycentrics    2*a^7 + a^6*b + a^4*b^3 - 2*a^3*b^4 - a^2*b^5 - b^7 + a^6*c + 2*a^5*b*c - 3*a^4*b^2*c - 2*a^3*b^3*c + 5*a^2*b^4*c - 3*b^6*c - 3*a^4*b*c^2 + 4*a^2*b^3*c^2 - b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 + 4*a^2*b^2*c^3 + 5*b^4*c^3 - 2*a^3*c^4 + 5*a^2*b*c^4 + 5*b^3*c^4 - a^2*c^5 - b^2*c^5 - 3*b*c^6 - c^7 : :

X(66108) lies on these lines: {3, 17281}, {4, 519}, {10, 8235}, {84, 1766}, {321, 3429}, {386, 3553}, {511, 22036}, {946, 50589}, {2321, 3430}, {2345, 5438}, {3175, 13442}, {3971, 35099}, {5777, 50594}, {12528, 50633}, {12618, 50608}

X(66108) = midpoint of X(12528) and X(50633)
X(66108) = reflection of X(i) in X(j) for these {i,j}: {50589, 946}, {50594, 5777}

X(66109) = X(1)X(3)∩X(946)X(17012)

Barycentrics    a*(a^6 + 2*a^5*b - a^4*b^2 - 4*a^3*b^3 - a^2*b^4 + 2*a*b^5 + b^6 + 2*a^5*c - a^4*b*c - 7*a^3*b^2*c - a^2*b^3*c + 5*a*b^4*c + 2*b^5*c - a^4*c^2 - 7*a^3*b*c^2 - 7*a*b^3*c^2 - b^4*c^2 - 4*a^3*c^3 - a^2*b*c^3 - 7*a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 + 5*a*b*c^4 - b^2*c^4 + 2*a*c^5 + 2*b*c^5 + c^6) : :

X(66109) lies on these lines: {1, 3}, {946, 17012}, {962, 17013}, {1029, 31673}, {2999, 38021}, {3755, 18406}, {4646, 62210}, {5256, 31162}, {6684, 17021}, {7292, 39605}, {7592, 12705}, {16132, 59301}, {16474, 66059}, {17011, 28194}, {46976, 52524}, {48903, 56426}, {51599, 64673}

(Part 35 will be started in the future.)

This is the end of PART 34: Centers X(66001) - X(68000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)