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}





leftri   Points on Terzić hyperbolas: :X(X(66092)-X(660109)  rightri

Contributed by Clark Kimberling and Peter Moses, November 7, 2024.

Early in November, 2024, Predrag Terzić contributed notes on three hyperbolas, and Peter Moses found equations and pass-through points for the hyperbolas.

Points X(66092)-X(66097), along with the points X(i) for i = 5, 13, 14, 15, 16, 549, lie on the 1st Terzić hyperbola, given by the following barycentric equation:

(b - c)^2*(b + c)^2*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4)*x^2 - (a - c)*(a + c)*(-b + c)*(b + c)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 5*a^4*c^4 - 5*b^4*c^4 + 6*a^2*c^6 + 6*b^2*c^6 - 2*c^8)*x*y + (a - c)^2*(a + c)^2*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4)*y^2 - (a - b)*(a + b)*(b - c)*(b + c)*(a^8 - 5*a^4*b^4 + 6*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 6*b^6*c^2 + 6*a^4*c^4 - 5*b^4*c^4 - 4*a^2*c^6 + c^8)*x*z - (a - b)*(a + b)*(a - c)*(a + c)*(2*a^8 - 6*a^6*b^2 + 5*a^4*b^4 - b^8 - 6*a^6*c^2 + 4*b^6*c^2 + 5*a^4*c^4 - 6*b^4*c^4 + 4*b^2*c^6 - c^8)*y*z + (a - b)^2*(a + b)^2*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^4 - 3*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*z^2 = 0.

The center of this hyperbola is X(3055).

Points X(66098)-X(66102), along with the points X(i) for i = 1, 4, 9, 13, 14, 321, lie on the 2nd Terzić hyperbola, given by the following barycentric equation:

a*(b - c)^2*(b + c)*(a^2 - b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*x^2 - (a - c)*(b - c)*c*(3*a^6*b + 3*a^5*b^2 - 6*a^4*b^3 - 6*a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 + 2*a^6*c + 6*a^5*b*c - 2*a^4*b^2*c - 12*a^3*b^3*c - 2*a^2*b^4*c + 6*a*b^5*c + 2*b^6*c + 2*a^5*c^2 + 2*a^4*b*c^2 - 4*a^3*b^2*c^2 - 4*a^2*b^3*c^2 + 2*a*b^4*c^2 + 2*b^5*c^2 - a^4*c^3 + 4*a^2*b^2*c^3 - b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 + 2*a*b^2*c^4 - b^3*c^4 - a^2*c^5 - b^2*c^5 - a*c^6 - b*c^6)*x*y - b*(a - c)^2*(a + c)*(a^2 - b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*y^2 + (a - b)*b*(b - c)*(2*a^6*b + 2*a^5*b^2 - a^4*b^3 - a^3*b^4 - a^2*b^5 - a*b^6 + 3*a^6*c + 6*a^5*b*c + 2*a^4*b^2*c + 2*a^2*b^4*c - b^6*c + 3*a^5*c^2 - 2*a^4*b*c^2 - 4*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 2*a*b^4*c^2 - b^5*c^2 - 6*a^4*c^3 - 12*a^3*b*c^3 - 4*a^2*b^2*c^3 - b^4*c^3 - 6*a^3*c^4 - 2*a^2*b*c^4 + 2*a*b^2*c^4 - b^3*c^4 + 3*a^2*c^5 + 6*a*b*c^5 + 2*b^2*c^5 + 3*a*c^6 + 2*b*c^6)*x*z + a*(a - b)*(a - c)*(a^6*b + a^5*b^2 + a^4*b^3 + a^3*b^4 - 2*a^2*b^5 - 2*a*b^6 + a^6*c - 2*a^4*b^2*c - 2*a^2*b^4*c - 6*a*b^5*c - 3*b^6*c + a^5*c^2 - 2*a^4*b*c^2 - 4*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 2*a*b^4*c^2 - 3*b^5*c^2 + a^4*c^3 + 4*a^2*b^2*c^3 + 12*a*b^3*c^3 + 6*b^4*c^3 + a^3*c^4 - 2*a^2*b*c^4 + 2*a*b^2*c^4 + 6*b^3*c^4 - 2*a^2*c^5 - 6*a*b*c^5 - 3*b^2*c^5 - 2*a*c^6 - 3*b*c^6)*y*z - (a - b)^2*(a + b)*c*(a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*z^2 = 0

Points X(66103)-X(66109), along with the points X(i) for i = 3, 4, 10, 13, 14, 386, lie on the 3rd Terzić hyperbola, given by the following barycentric equation:

(b - c)^2*(b + c)*(a^2 - b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*x^2 + (a - c)*(b - c)*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 - 4*a^5*c^2 + 3*a^4*b*c^2 + a^3*b^2*c^2 + a^2*b^3*c^2 + 3*a*b^4*c^2 - 4*b^5*c^2 - a^4*c^3 + 2*a^2*b^2*c^3 - b^4*c^3 + 2*a^3*c^4 - 3*a^2*b*c^4 - 3*a*b^2*c^4 + 2*b^3*c^4 - a^2*c^5 - b^2*c^5 + a*c^6 + b*c^6 + 2*c^7)*x*y - (a - c)^2*(a + c)*(a^2 - b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*y^2 - (a - b)*(b - c)*(a^7 - 4*a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - a^2*b^5 + a*b^6 + 2*b^7 - a^6*c + 3*a^4*b^2*c - 3*a^2*b^4*c + b^6*c - 3*a^5*c^2 + a^3*b^2*c^2 + 2*a^2*b^3*c^2 - 3*a*b^4*c^2 - b^5*c^2 + 3*a^4*c^3 + a^2*b^2*c^3 + 2*b^4*c^3 + 3*a^3*c^4 + 3*a*b^2*c^4 - b^3*c^4 - 3*a^2*c^5 - 4*b^2*c^5 - a*c^6 + c^7)*x*z + (a - b)*(a - c)*(2*a^7 + a^6*b - a^5*b^2 + 2*a^4*b^3 - a^3*b^4 - 4*a^2*b^5 + b^7 + a^6*c - 3*a^4*b^2*c + 3*a^2*b^4*c - b^6*c - a^5*c^2 - 3*a^4*b*c^2 + 2*a^3*b^2*c^2 + a^2*b^3*c^2 - 3*b^5*c^2 + 2*a^4*c^3 + a^2*b^2*c^3 + 3*b^4*c^3 - a^3*c^4 + 3*a^2*b*c^4 + 3*b^3*c^4 - 4*a^2*c^5 - 3*b^2*c^5 - b*c^6 + c^7)*y*z - (a - b)^2*(a + b)*(a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*z^2= 0

underbar



X(66092) = X(13)X(53458)∩X(14)X(53469)

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 the 1st Terzić hyperbola and 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 the 1st Terzić hyperbola and 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 the 1st Terzić hyperbola and 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 the 1st Terzić hyperbola and 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 the 1st Terzić hyperbola and 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 the 1st Terzić hyperbola and 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 the 2nd Terzić hyperbola and 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 the 2nd Terzić hyperbola and 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 the 2nd Terzić hyperbola and 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 the 2nd Terzić hyperbola and 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 the 2nd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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 the 3rd Terzić hyperbola and 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}





leftri   Points on the Lester circle: X(66110)-X(66114)  rightri

Contributed by Clark Kimberling and Peter Moses, November 7, 2024.

Indices i < 40000 such that X(i) lies on the Lester circle:

3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365

Indices i > 40000 such that X(i) lies on the Lester circle: 66110, 66111, 66112, 66113, 66114

underbar



X(66110) = 1ST LESTER-MOSES POINT

Barycentrics    a^12*b^2 - 2*a^10*b^4 + 2*a^6*b^8 - a^4*b^10 + a^12*c^2 - 4*a^10*b^2*c^2 + 6*a^8*b^4*c^2 - 6*a^6*b^6*c^2 + 2*a^4*b^8*c^2 - a^2*b^10*c^2 - b^12*c^2 - 2*a^10*c^4 + 6*a^8*b^2*c^4 + 4*a^2*b^8*c^4 + 3*b^10*c^4 - 6*a^6*b^2*c^6 - 6*a^2*b^6*c^6 - 2*b^8*c^6 + 2*a^6*c^8 + 2*a^4*b^2*c^8 + 4*a^2*b^4*c^8 - 2*b^6*c^8 - a^4*c^10 - a^2*b^2*c^10 + 3*b^4*c^10 - b^2*c^12 : :

X(66110) lies on the Lester circle and these lines: {3, 59251}, {5, 39}, {6, 15542}, {111, 53876}, {187, 46633}, {542, 15544}, {804, 15543}, {1989, 34365}, {2088, 11646}, {6034, 15540}, {10413, 15535}, {11632, 15546}, {12188, 44533}, {14579, 15554}, {15550, 56401}, {23969, 43654}

X(66110) = Moses-circle-inverse of X(44468)
X(66110) = {X(115),X(1569)}-harmonic conjugate of X(44468)

X(66111) = 2ND LESTER-MOSES POINT

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(2*a^12 - 4*a^10*b^2 + 2*a^6*b^6 + 3*a^4*b^8 - 4*a^2*b^10 + b^12 - 4*a^10*c^2 + 12*a^8*b^2*c^2 - 6*a^6*b^4*c^2 - 14*a^4*b^6*c^2 + 12*a^2*b^8*c^2 - 5*b^10*c^2 - 6*a^6*b^2*c^4 + 24*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 11*b^8*c^4 + 2*a^6*c^6 - 14*a^4*b^2*c^6 - 8*a^2*b^4*c^6 - 14*b^6*c^6 + 3*a^4*c^8 + 12*a^2*b^2*c^8 + 11*b^4*c^8 - 4*a^2*c^10 - 5*b^2*c^10 + c^12) : :
X(66111) = 3 X[249] - X[14480], X[477] - 3 X[38702], X[842] - 3 X[38700], X[9158] - 3 X[26613], X[20957] - 3 X[57307]

X(66111) lies on the Lester circle and these lines: {3, 14995}, {30, 15535}, {187, 46633}, {249, 14480}, {476, 691}, {477, 38702}, {511, 46632}, {512, 7471}, {523, 9181}, {842, 38700}, {1316, 9169}, {3111, 15536}, {3258, 40544}, {5099, 22104}, {7472, 62489}, {9158, 26613}, {9179, 47327}, {11537, 61472}, {11549, 61474}, {15538, 25641}, {16168, 38611}, {16181, 45879}, {16182, 45880}, {20957, 57307}, {36180, 62508}, {46998, 53726}, {47502, 62510}

X(66111) = midpoint of X(476) and X(691)
X(66111) = reflection of X(i) in X(j) for these {i,j}: {3258, 40544}, {5099, 22104}
X(66111) = reflection of X(53735) in the Euler line

X(66112) = 3RD LESTER-MOSES POINT

Barycentrics    a^14 - 4*a^12*b^2 + 4*a^10*b^4 + a^8*b^6 - 3*a^6*b^8 + 2*a^4*b^10 - 2*a^2*b^12 + b^14 - 4*a^12*c^2 + 11*a^10*b^2*c^2 - 6*a^8*b^4*c^2 + 4*a^6*b^6*c^2 + 5*a^4*b^8*c^2 - a^2*b^10*c^2 - 3*b^12*c^2 + 4*a^10*c^4 - 6*a^8*b^2*c^4 - 12*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + 10*a^2*b^8*c^4 + 3*b^10*c^4 + a^8*c^6 + 4*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 14*a^2*b^6*c^6 - b^8*c^6 - 3*a^6*c^8 + 5*a^4*b^2*c^8 + 10*a^2*b^4*c^8 - b^6*c^8 + 2*a^4*c^10 - a^2*b^2*c^10 + 3*b^4*c^10 - 2*a^2*c^12 - 3*b^2*c^12 + c^14 : :

X(66112) lies on the Lester circle and these lines: {5, 32}, {6, 15545}, {381, 15546}, {5476, 15544}, {10413, 15539}, {11182, 61743}, {14356, 15550}, {14854, 56395}, {15542, 53504}, {15543, 59893}

X(66112) = orthocentroidal circle inverse of X(15546)

X(66113) = 4TH LESTER-MOSES POINT

Barycentrics    a^14 - 3*a^12*b^2 + a^10*b^4 - a^8*b^6 + 4*a^4*b^10 - 2*a^2*b^12 - 3*a^12*c^2 + 10*a^10*b^2*c^2 - 3*a^8*b^4*c^2 + 11*a^6*b^6*c^2 - 17*a^4*b^8*c^2 + 12*a^2*b^10*c^2 - b^12*c^2 + a^10*c^4 - 3*a^8*b^2*c^4 - 25*a^6*b^4*c^4 + 16*a^4*b^6*c^4 - 33*a^2*b^8*c^4 + 5*b^10*c^4 - a^8*c^6 + 11*a^6*b^2*c^6 + 16*a^4*b^4*c^6 + 44*a^2*b^6*c^6 - 4*b^8*c^6 - 17*a^4*b^2*c^8 - 33*a^2*b^4*c^8 - 4*b^6*c^8 + 4*a^4*c^10 + 12*a^2*b^2*c^10 + 5*b^4*c^10 - 2*a^2*c^12 - b^2*c^12 : :

X(66113) lies on the Lester circle and these lines: {3, 543}, {111, 53876}, {542, 15539}, {671, 57616}, {2782, 52036}, {2793, 14662}, {2854, 15536}, {11646, 15538}, {15342, 15544}

X(66113) = reflection of X(34010) in X(53726)

X(66114) = 5TH LESTER-MOSES POINT

Barycentrics    a^16 - 7*a^14*b^2 + 20*a^12*b^4 - 31*a^10*b^6 + 30*a^8*b^8 - 21*a^6*b^10 + 12*a^4*b^12 - 5*a^2*b^14 + b^16 - 7*a^14*c^2 + 24*a^12*b^2*c^2 - 37*a^10*b^4*c^2 + 28*a^8*b^6*c^2 + 3*a^6*b^8*c^2 - 26*a^4*b^10*c^2 + 21*a^2*b^12*c^2 - 6*b^14*c^2 + 20*a^12*c^4 - 37*a^10*b^2*c^4 + 33*a^8*b^4*c^4 - 23*a^6*b^6*c^4 + 39*a^4*b^8*c^4 - 33*a^2*b^10*c^4 + 16*b^12*c^4 - 31*a^10*c^6 + 28*a^8*b^2*c^6 - 23*a^6*b^4*c^6 - 32*a^4*b^6*c^6 + 17*a^2*b^8*c^6 - 26*b^10*c^6 + 30*a^8*c^8 + 3*a^6*b^2*c^8 + 39*a^4*b^4*c^8 + 17*a^2*b^6*c^8 + 30*b^8*c^8 - 21*a^6*c^10 - 26*a^4*b^2*c^10 - 33*a^2*b^4*c^10 - 26*b^6*c^10 + 12*a^4*c^12 + 21*a^2*b^2*c^12 + 16*b^4*c^12 - 5*a^2*c^14 - 6*b^2*c^14 + c^16 : :

X(66114) lies on the Lester circle and these lines: {2, 15544}, {5, 44386}, {114, 399}, {3258, 40544}, {5461, 15546}, {13582, 15553}, {15550, 18883}

X(66115) = X(2)X(2966)∩X(99)X(7471)

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

X(66115) lies on the cubic K1379 and these lines: {2, 2966}, {99, 7471}, {110, 685}, {250, 47259}, {316, 691}, {340, 687}, {450, 44146}, {476, 3268}, {892, 5466}, {3260, 56430}, {4240, 16077}, {4563, 14221}, {5651, 44155}, {5999, 26276}, {9140, 9141}, {9514, 30476}, {9979, 17708}, {10411, 18878}, {23357, 31072}, {30528, 57822}, {31174, 40866}, {32717, 34087}, {32729, 53365}, {52916, 65268}

X(66115) = X(36131)-anticomplementary conjugate of X(39356)
X(66115) = crosspoint of X(892) and X(16077)
X(66115) = crosssum of X(351) and X(9409)
X(66115) = barycentric product X(i)*X(j) for these {i,j}: {99, 48540}, {34537, 53327}, {43187, 56962}
X(66115) = barycentric quotient X(i)/X(j) for these {i,j}: {48540, 523}, {53327, 3124}, {56962, 3569}
X(66115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 850, 65713}, {110, 65713, 54108}, {476, 3268, 65768}, {850, 18020, 54108}, {7471, 30474, 99}, {18020, 65713, 110}, {53379, 65872, 892}

X(66116) = X(2)X(647)∩X(3)X(47258)

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

X(66116) lies on the cubic K1379 and these lines: {2, 647}, {3, 47258}, {5, 47248}, {110, 879}, {111, 2373}, {183, 55974}, {378, 46984}, {476, 1304}, {523, 1995}, {525, 15066}, {2394, 2986}, {2433, 3580}, {4580, 10130}, {4993, 15412}, {5466, 15398}, {5468, 17708}, {6563, 65612}, {7493, 47263}, {9147, 14270}, {9168, 11638}, {14592, 18883}, {18117, 62949}, {30739, 47256}, {30744, 59742}, {33752, 47250}, {34767, 40384}, {43957, 47260}, {44210, 47261}, {44212, 47175}, {46425, 65972}, {47001, 47596}, {52743, 63036}

X(66116) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9139, 21294}, {36034, 14360}, {36142, 146}, {65263, 34518}
X(66116) = crosspoint of X(i) and X(j) for these (i,j): {671, 46456}, {892, 40832}
X(66116) = crossdifference of every pair of points on line {237, 47414}
X(66116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31296, 62307}, {647, 18312, 2}, {2395, 53173, 46786}

X(66117) = X(2)X(525)∩X(110)X(476)

Barycentrics    (b^2 - c^2)*(-4*a^10 + 9*a^8*b^2 - 5*a^6*b^4 + a^4*b^6 - 3*a^2*b^8 + 2*b^10 + 9*a^8*c^2 - 20*a^6*b^2*c^2 + 10*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 3*b^8*c^2 - 5*a^6*c^4 + 10*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + b^6*c^4 + a^4*c^6 + 4*a^2*b^2*c^6 + b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + 2*c^10) : :
X(66117) = 4 X[14566] - 3 X[65614], 3 X[7426] - 4 X[58349], 3 X[14697] - 2 X[58349]

X(66117) lies on the cubic K1379 and these lines: {2, 525}, {110, 476}, {146, 1499}, {323, 65977}, {520, 45237}, {524, 9141}, {647, 34834}, {690, 858}, {850, 36789}, {1637, 3580}, {2799, 40112}, {2986, 10754}, {3124, 62572}, {3268, 11064}, {5466, 51405}, {7426, 14697}, {11646, 65609}, {23870, 41888}, {23871, 41887}, {44427, 62628}, {47258, 53725}

X(66117) = reflection of X(i) in X(j) for these {i,j}: {3268, 11064}, {3580, 1637}, {7426, 14697}, {51227, 65782}
X(66117) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {923, 62639}, {9214, 21294}, {9406, 39356}, {24001, 34518}, {32729, 18668}, {36060, 45289}, {36142, 30}
X(66117) = X(56395)-Dao conjugate of X(14559)
X(66117) = crosspoint of X(i) and X(j) for these (i,j): {671, 39290}, {892, 31621}
X(66117) = crosssum of X(i) and X(j) for these (i,j): {187, 52743}, {351, 9408}
X(66117) = crossdifference of every pair of points on line {1495, 2088}

X(66118) = X(2)X(523)∩X(110)X(525)

Barycentrics    (b^2 - c^2)*(4*a^10 - 7*a^8*b^2 + a^6*b^4 + 5*a^4*b^6 - 5*a^2*b^8 + 2*b^10 - 7*a^8*c^2 + 16*a^6*b^2*c^2 - 10*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - b^8*c^2 + a^6*c^4 - 10*a^4*b^2*c^4 + 8*a^2*b^4*c^4 - b^6*c^4 + 5*a^4*c^6 + 2*a^2*b^2*c^6 - b^4*c^6 - 5*a^2*c^8 - b^2*c^8 + 2*c^10) : :
X(66118) = 3 X[9168] - X[9213], X[47258] - 4 X[47627], 3 X[403] - 2 X[44203], X[6563] + 2 X[47216], X[33294] - 4 X[47217], X[41298] + 2 X[47175]

X(66118) lies on the cubic K1379 and these lines: {2, 523}, {30, 3268}, {110, 525}, {403, 44203}, {468, 9979}, {476, 65772}, {647, 60510}, {842, 2373}, {850, 34336}, {858, 14417}, {2799, 7426}, {2972, 37987}, {3258, 65978}, {3265, 14360}, {3580, 9033}, {6054, 30474}, {7665, 33294}, {9185, 47190}, {9529, 62288}, {10718, 34312}, {14611, 65776}, {17986, 34767}, {36904, 65782}, {41298, 47175}, {47325, 55135}

X(66118) = reflection of X(i) in X(j) for these {i,j}: {858, 14417}, {7426, 47219}, {9185, 47190}, {9979, 468}, {62629, 46986}
X(66118) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1304, 17491}, {16077, 21298}, {32715, 17497}, {36034, 858}, {36131, 524}, {65263, 316}
X(66118) = crosspoint of X(16077) and X(57539)
X(66118) = crosssum of X(9409) and X(39689)

X(66119) = X(2)X(476)∩X(4)X(110)

Barycentrics    a^10*b^2 - 4*a^6*b^6 + 2*a^4*b^8 + 3*a^2*b^10 - 2*b^12 + a^10*c^2 - 6*a^8*b^2*c^2 + 7*a^6*b^4*c^2 + a^4*b^6*c^2 - 10*a^2*b^8*c^2 + 7*b^10*c^2 + 7*a^6*b^2*c^4 - 8*a^4*b^4*c^4 + 7*a^2*b^6*c^4 - 10*b^8*c^4 - 4*a^6*c^6 + a^4*b^2*c^6 + 7*a^2*b^4*c^6 + 10*b^6*c^6 + 2*a^4*c^8 - 10*a^2*b^2*c^8 - 10*b^4*c^8 + 3*a^2*c^10 + 7*b^2*c^10 - 2*c^12 : :

X(66119) lies on the cubic K1379 and these lines: {2, 476}, {4, 110}, {5, 33927}, {264, 850}, {316, 5468}, {323, 18867}, {381, 9717}, {427, 65718}, {858, 5968}, {1007, 1272}, {1138, 3545}, {1553, 3543}, {3091, 59370}, {3268, 65775}, {3580, 35235}, {4240, 14165}, {6054, 31105}, {9140, 9214}, {9979, 53156}, {13448, 62551}, {14254, 20304}, {14355, 63036}, {15081, 51835}, {17511, 52772}, {18121, 37648}, {18301, 31857}, {38794, 58733}, {41724, 53351}, {47049, 51360}, {47050, 65086}, {47324, 57603}, {53346, 59422}, {61743, 64634}

X(66119) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {897, 3153}, {17983, 63642}, {36085, 65972}, {36128, 37779}, {52414, 14360}, {52668, 6360}
X(66119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14731, 65770}, {2, 52472, 476}, {3258, 14356, 2}, {32112, 58263, 65714}

X(66120) = X(2)X(17708)∩X(69)X(110)

Barycentrics    a^10*b^2 - 4*a^8*b^4 + 4*a^6*b^6 + 2*a^4*b^8 - 5*a^2*b^10 + 2*b^12 + a^10*c^2 + 2*a^8*b^2*c^2 - a^6*b^4*c^2 - 7*a^4*b^6*c^2 + 6*a^2*b^8*c^2 - b^10*c^2 - 4*a^8*c^4 - a^6*b^2*c^4 + 8*a^4*b^4*c^4 - a^2*b^6*c^4 - 2*b^8*c^4 + 4*a^6*c^6 - 7*a^4*b^2*c^6 - a^2*b^4*c^6 + 2*b^6*c^6 + 2*a^4*c^8 + 6*a^2*b^2*c^8 - 2*b^4*c^8 - 5*a^2*c^10 - b^2*c^10 + 2*c^12 : :

X(66120) lies on the cubic K1379 and these lines: {2, 17708}, {69, 110}, {76, 850}, {340, 4240}, {343, 65719}, {476, 65771}, {3580, 35910}, {5468, 37804}, {9140, 36890}, {14364, 52713}, {53348, 65715}, {60498, 65608}

X(66120) = anticomplement of X(60496)
X(66120) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {74, 17482}, {1494, 21274}, {2159, 14712}, {2349, 5189}, {16568, 146}, {65263, 9517}

X(66121) = X(2)X(35909)∩X(23)X(690)

Barycentrics    (b^2 - c^2)*(2*a^16 - 6*a^14*b^2 + 5*a^12*b^4 + a^10*b^6 - 4*a^8*b^8 + 4*a^6*b^10 - 3*a^4*b^12 + a^2*b^14 - 6*a^14*c^2 + 16*a^12*b^2*c^2 - 13*a^10*b^4*c^2 - 3*a^8*b^6*c^2 + 10*a^6*b^8*c^2 - 2*a^4*b^10*c^2 - 3*a^2*b^12*c^2 + b^14*c^2 + 5*a^12*c^4 - 13*a^10*b^2*c^4 + 20*a^8*b^4*c^4 - 12*a^6*b^6*c^4 - 9*a^4*b^8*c^4 + 11*a^2*b^10*c^4 - 2*b^12*c^4 + a^10*c^6 - 3*a^8*b^2*c^6 - 12*a^6*b^4*c^6 + 26*a^4*b^6*c^6 - 9*a^2*b^8*c^6 - b^10*c^6 - 4*a^8*c^8 + 10*a^6*b^2*c^8 - 9*a^4*b^4*c^8 - 9*a^2*b^6*c^8 + 4*b^8*c^8 + 4*a^6*c^10 - 2*a^4*b^2*c^10 + 11*a^2*b^4*c^10 - b^6*c^10 - 3*a^4*c^12 - 3*a^2*b^2*c^12 - 2*b^4*c^12 + a^2*c^14 + b^2*c^14) : :

X(66121) lies on the cubic K1379 and these lines: {2, 35909}, {23, 690}, {74, 2373}, {110, 62307}, {323, 9033}, {476, 2799}, {526, 3580}, {542, 850}, {858, 9517}, {895, 2986}, {3268, 65770}, {4240, 44427}, {47258, 62516}

X(66121) = reflection of X(47258) in X(62516)
X(66121) = X(48540)-anticomplementary conjugate of X(21294)

X(66122) = X(2)X(647)∩X(3)X(47004)

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

X(66122) lies on the cubic K1380 and these lines: {2, 647}, {3, 47004}, {5, 47002}, {125, 41167}, {126, 1560}, {523, 30739}, {525, 37648}, {549, 46984}, {858, 33752}, {879, 54012}, {1637, 5664}, {1648, 52628}, {1650, 3258}, {2394, 62927}, {2525, 65612}, {3265, 59766}, {3266, 50942}, {3267, 11059}, {5649, 6331}, {6676, 47262}, {6677, 47261}, {7493, 47255}, {7495, 47259}, {8371, 23105}, {8703, 46995}, {9148, 21731}, {10717, 54853}, {11064, 52743}, {14096, 42660}, {14618, 52147}, {18309, 53365}, {22112, 40550}, {30745, 57127}, {37439, 59742}, {40879, 47229}, {41357, 46371}, {43957, 47175}, {44210, 47252}, {44273, 47003}, {44814, 51479}, {44818, 47205}, {46336, 47254}, {47264, 52300}, {53327, 62489}, {57482, 65758}, {58900, 63084}

X(66122) = complement of X(66116)
X(66122) = X(43084)-Ceva conjugate of X(52628)
X(66122) = X(i)-isoconjugate of X(j) for these (i,j): {74, 36142}, {111, 36034}, {163, 9139}, {691, 2159}, {895, 36131}, {897, 32640}, {923, 44769}, {1304, 36060}, {2349, 32729}, {14908, 65263}, {36085, 40352}
X(66122) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 9139}, {1560, 1304}, {1637, 9213}, {1648, 9717}, {1649, 2433}, {2482, 44769}, {3163, 691}, {3258, 111}, {3284, 51478}, {6593, 32640}, {23992, 74}, {35582, 48451}, {38988, 40352}, {39008, 895}, {48317, 8749}, {57295, 10097}, {62569, 65321}, {62577, 2394}, {62594, 14919}, {62598, 671}, {65757, 14977}, {65763, 8430}
X(66122) = crosssum of X(i) and X(j) for these (i,j): {2433, 60499}, {10097, 60498}
X(66122) = crossdifference of every pair of points on line {237, 14908}
X(66122) = barycentric product X(i)*X(j) for these {i,j}: {30, 35522}, {468, 66073}, {524, 41079}, {670, 2682}, {690, 3260}, {850, 5642}, {1637, 3266}, {1990, 45807}, {2407, 52628}, {2642, 46234}, {4235, 65753}, {5468, 58261}, {5664, 43084}, {6148, 51479}, {9033, 44146}, {9214, 52629}, {13857, 65008}, {14210, 36035}, {14254, 45808}, {14417, 46106}, {36890, 58263}, {37778, 41077}, {52145, 65754}
X(66122) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 691}, {187, 32640}, {351, 40352}, {468, 1304}, {523, 9139}, {524, 44769}, {690, 74}, {896, 36034}, {1495, 32729}, {1511, 51478}, {1637, 111}, {1648, 2433}, {1649, 9717}, {2173, 36142}, {2631, 36060}, {2642, 2159}, {2682, 512}, {3258, 9213}, {3260, 892}, {5642, 110}, {9033, 895}, {9214, 34574}, {9409, 14908}, {11064, 65321}, {13857, 32583}, {14206, 36085}, {14273, 8749}, {14398, 32740}, {14417, 14919}, {14424, 46147}, {14559, 15395}, {32225, 65316}, {35522, 1494}, {36035, 897}, {36298, 9207}, {36299, 9206}, {37778, 15459}, {41079, 671}, {41586, 36831}, {43084, 39290}, {44102, 32715}, {44146, 16077}, {44814, 14385}, {45662, 51262}, {46106, 65350}, {51360, 36827}, {51429, 32112}, {51457, 35191}, {51479, 5627}, {52628, 2394}, {52629, 36890}, {52743, 52668}, {55265, 60498}, {58261, 5466}, {58263, 9214}, {58347, 2420}, {58349, 1495}, {60428, 32695}, {65753, 14977}, {65754, 5968}, {65755, 8430}, {66073, 30786}
X(66122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 850, 18312}, {2, 62307, 647}, {5664, 65757, 52624}

X(66123) = X(2)X(2966)∩X(112)X(57587)

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

X(66123) lies on the cubic K1380 and these lines: {2, 2966}, {112, 57587}, {115, 3154}, {125, 647}, {187, 468}, {1513, 1561}, {1637, 3258}, {1648, 1649}, {1650, 14401}, {2682, 58349}, {3284, 11064}, {6587, 65613}, {6781, 47351}, {10413, 18334}, {10418, 36166}, {14417, 62594}, {44892, 64603}

X(66123) = complement of X(66115)
X(66123) = isotomic conjugate of the polar conjugate of X(2682)
X(66123) = X(468)-Ceva conjugate of X(58349)
X(66123) = X(i)-isoconjugate of X(j) for these (i,j): {691, 65263}, {892, 36131}, {1304, 36085}, {16077, 36142}, {36034, 65350}, {36060, 42308}
X(66123) = X(i)-Dao conjugate of X(j) for these (i,j): {1560, 42308}, {1649, 16080}, {3258, 65350}, {14401, 30786}, {21905, 8749}, {23992, 16077}, {38988, 1304}, {38999, 65321}, {39008, 892}, {48317, 15459}, {57295, 671}, {62569, 52940}, {62598, 59762}, {65757, 18023}
X(66123) = crosspoint of X(1637) and X(60496)
X(66123) = crossdifference of every pair of points on line {691, 1304}
X(66123) = barycentric product X(i)*X(j) for these {i,j}: {69, 2682}, {125, 5642}, {187, 65753}, {351, 66073}, {468, 1650}, {690, 9033}, {1637, 14417}, {1648, 11064}, {3284, 52628}, {3292, 58261}, {9409, 35522}, {14273, 41077}, {14398, 45807}, {34767, 58349}, {35282, 65759}, {35912, 51429}, {43084, 47414}, {60496, 62594}
X(66123) = barycentric quotient X(i)/X(j) for these {i,j}: {351, 1304}, {468, 42308}, {690, 16077}, {1636, 65321}, {1637, 65350}, {1648, 16080}, {1650, 30786}, {2631, 36085}, {2642, 65263}, {2682, 4}, {5642, 18020}, {9033, 892}, {9409, 691}, {11064, 52940}, {14273, 15459}, {14443, 52475}, {20975, 9139}, {21906, 8749}, {33919, 18808}, {41079, 59762}, {58261, 46111}, {58349, 4240}, {65753, 18023}, {66073, 53080}
X(66123) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 647, 65724}, {1637, 3258, 65755}, {3154, 9209, 115}

X(66124) = X(2)X(523)∩X(23)X(53318)

Barycentrics    (b - c)*(b + c)*(a^2 + b^2 - 2*c^2)*(-a^2 + 2*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) : :
X(66124) = X[5466] + X[9213], 2 X[12075] + X[47173], 4 X[14341] - X[47216]

X(66124) lies on the cubic K1380 and these lines: {2, 523}, {23, 53318}, {30, 1637}, {107, 691}, {111, 2697}, {125, 525}, {468, 44564}, {512, 41670}, {647, 65729}, {671, 42738}, {858, 9979}, {895, 9007}, {1503, 14697}, {1560, 2501}, {2799, 47097}, {3233, 65776}, {3258, 65758}, {3265, 30786}, {3566, 13291}, {4846, 10097}, {5159, 14417}, {6055, 9209}, {9033, 11064}, {9140, 66117}, {10556, 15351}, {10561, 59652}, {12075, 47173}, {12079, 65978}, {14341, 47216}, {14401, 35912}, {14582, 51847}, {16177, 65759}, {24855, 47138}, {25644, 37969}, {31125, 33294}, {31621, 65973}, {37980, 53265}, {42736, 44569}, {46115, 47004}, {46982, 52450}, {46995, 62510}, {47159, 55122}, {52464, 52485}

X(66124) = midpoint of X(i) and X(j) for these {i,j}: {858, 9979}, {9140, 66117}, {16092, 62629}
X(66124) = reflection of X(i) in X(j) for these {i,j}: {468, 44564}, {14417, 5159}, {44569, 42736}, {47001, 9209}
X(66124) = complement of X(66118)
X(66124) = X(i)-isoconjugate of X(j) for these (i,j): {162, 9717}, {187, 65263}, {468, 36034}, {524, 36131}, {896, 1304}, {922, 16077}, {1101, 52475}, {2159, 4235}, {2349, 61207}, {5467, 36119}, {8749, 23889}, {14210, 32715}, {24039, 40354}, {32676, 36890}
X(66124) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 9717}, {523, 52475}, {1511, 5467}, {1650, 5642}, {3163, 4235}, {3258, 468}, {8552, 45808}, {14401, 14417}, {15477, 32715}, {15526, 36890}, {15899, 1304}, {38999, 3292}, {39008, 524}, {39061, 16077}, {39170, 14559}, {57295, 690}, {62569, 5468}, {62598, 44146}, {65757, 35522}
X(66124) = crosssum of X(690) and X(12828)
X(66124) = crossdifference of every pair of points on line {187, 9717}
X(66124) = barycentric product X(i)*X(j) for these {i,j}: {30, 14977}, {111, 66073}, {525, 9214}, {671, 9033}, {691, 65753}, {895, 41079}, {1636, 46111}, {1637, 30786}, {1650, 65350}, {2407, 51258}, {2631, 46277}, {3260, 10097}, {3284, 52632}, {5466, 11064}, {5968, 65778}, {9139, 52624}, {9213, 57482}, {9409, 18023}, {17983, 41077}, {35912, 62629}, {58261, 65321}
X(66124) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 4235}, {111, 1304}, {115, 52475}, {525, 36890}, {647, 9717}, {671, 16077}, {895, 44769}, {897, 65263}, {923, 36131}, {1495, 61207}, {1636, 3292}, {1637, 468}, {1650, 14417}, {2631, 896}, {2682, 58780}, {3284, 5467}, {5466, 16080}, {8430, 35908}, {8753, 32695}, {9033, 524}, {9139, 34568}, {9178, 8749}, {9213, 57487}, {9214, 648}, {9409, 187}, {10097, 74}, {11064, 5468}, {14391, 41586}, {14398, 44102}, {14401, 5642}, {14908, 32640}, {14977, 1494}, {17983, 15459}, {23894, 36119}, {32740, 32715}, {36060, 36034}, {41077, 6390}, {41079, 44146}, {44203, 37855}, {47414, 44814}, {51258, 2394}, {55265, 12828}, {56399, 14559}, {60496, 60503}, {64258, 18808}, {65350, 42308}, {65753, 35522}, {65778, 52145}, {66073, 3266}

X(66125) = X(2)X(476)∩X(3)X(125)

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

X(66125) lies on the cubic K1380 and these lines: {2, 476}, {3, 125}, {94, 7607}, {140, 14254}, {146, 59368}, {184, 65750}, {187, 1648}, {216, 647}, {328, 62698}, {373, 15000}, {376, 55319}, {381, 52056}, {468, 9176}, {549, 34209}, {631, 51835}, {1637, 60340}, {1656, 58733}, {1989, 10418}, {3523, 59428}, {3524, 5627}, {3530, 53137}, {5054, 14993}, {5055, 51345}, {5094, 53771}, {5467, 41586}, {5642, 9717}, {5651, 14560}, {5972, 33927}, {6053, 14264}, {6055, 47596}, {6070, 52772}, {6344, 52147}, {6639, 53168}, {6676, 57482}, {7493, 53768}, {7495, 43087}, {7499, 43089}, {7542, 58725}, {7664, 52145}, {8371, 15475}, {8553, 56404}, {9155, 14357}, {10272, 14670}, {11064, 16186}, {11078, 46825}, {11092, 46824}, {14389, 18114}, {14687, 61743}, {15329, 32223}, {18384, 52292}, {20125, 39239}, {24975, 47146}, {29012, 46602}, {30739, 43090}, {34577, 58926}, {35222, 44889}, {37779, 52603}, {43088, 65610}, {44210, 65620}, {44814, 51479}, {46127, 46155}, {47200, 66075}, {47327, 57603}, {50676, 59771}, {58723, 58729}, {59370, 64101}

X(66125) = complement of X(66119)
X(66125) = isotomic conjugate of the polar conjugate of X(56395)
X(66125) = isogonal conjugate of the polar conjugate of X(43084)
X(66125) = X(43084)-Ceva conjugate of X(56395)
X(66125) = X(i)-isoconjugate of X(j) for these (i,j): {92, 52668}, {111, 52414}, {162, 9213}, {186, 897}, {323, 36128}, {340, 923}, {2624, 65350}, {6149, 17983}, {9139, 35201}, {14165, 36060}, {14590, 23894}, {24006, 51478}, {34397, 46277}, {36085, 47230}, {36142, 44427}
X(66125) = X(i)-Dao conjugate of X(j) for these (i,j): {125, 9213}, {1560, 14165}, {1649, 35235}, {2482, 340}, {6593, 186}, {14993, 17983}, {15295, 8753}, {22391, 52668}, {23992, 44427}, {38988, 47230}, {39170, 9214}, {52881, 7799}, {62594, 3268}
X(66125) = cevapoint of X(5642) and X(41586)
X(66125) = crossdifference of every pair of points on line {186, 9126}
X(66125) = barycentric product X(i)*X(j) for these {i,j}: {3, 43084}, {69, 56395}, {94, 3292}, {187, 328}, {265, 524}, {476, 14417}, {525, 14559}, {690, 60053}, {1989, 6390}, {3266, 52153}, {4235, 43083}, {4558, 51479}, {5467, 14592}, {5468, 14582}, {9717, 57482}, {14560, 45807}, {20573, 23200}, {32662, 35522}, {36890, 56399}, {40709, 52040}, {40710, 52039}, {41586, 65326}, {44146, 50433}
X(66125) = barycentric quotient X(i)/X(j) for these {i,j}: {94, 46111}, {184, 52668}, {187, 186}, {265, 671}, {328, 18023}, {351, 47230}, {468, 14165}, {476, 65350}, {524, 340}, {647, 9213}, {690, 44427}, {896, 52414}, {1648, 35235}, {1989, 17983}, {3292, 323}, {5467, 14590}, {5642, 14920}, {6390, 7799}, {9717, 57487}, {11060, 8753}, {11079, 9139}, {14417, 3268}, {14559, 648}, {14567, 34397}, {14582, 5466}, {14592, 52632}, {23200, 50}, {23968, 53155}, {32661, 51478}, {32662, 691}, {35139, 59762}, {36061, 36085}, {41586, 14918}, {43083, 14977}, {43084, 264}, {44102, 52418}, {50433, 895}, {51479, 14618}, {52039, 471}, {52040, 470}, {52153, 111}, {56395, 4}, {56399, 9214}, {59209, 52750}, {59210, 52751}, {60053, 892}, {61207, 53176}
X(66125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 476, 14356}, {2, 65770, 3258}, {3, 39170, 51254}, {476, 14356, 14583}, {52039, 52040, 56395}, {59209, 59210, 56399}

X(66126) = X(2)X(525)∩X(125)X(523)

Barycentrics    (b - c)*(b + c)*(-2*a^2 + b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(-a^4 - a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 - c^4) : :
X(66126) = X[2394] - 3 X[65614], 3 X[468] - 2 X[58349]

X(66126) lies on the cubic K1380 and these lines: {2, 525}, {74, 1499}, {125, 523}, {468, 690}, {524, 14417}, {647, 65734}, {1494, 18823}, {1503, 47219}, {1513, 9191}, {1637, 47296}, {1648, 50942}, {1649, 5967}, {2501, 14223}, {2799, 44569}, {3265, 50567}, {3266, 45807}, {3268, 3580}, {3906, 22264}, {4143, 4563}, {5486, 14380}, {6563, 62722}, {7471, 65316}, {9140, 66118}, {9164, 12036}, {9168, 36875}, {9204, 52039}, {9205, 52040}, {11005, 17986}, {11064, 66083}, {16103, 33921}, {16243, 44451}, {30476, 58416}, {34150, 62507}, {46808, 64919}, {51823, 65611}, {57539, 62629}

X(66126) = midpoint of X(i) and X(j) for these {i,j}: {3268, 3580}, {9140, 66118}, {51227, 65973}
X(66126) = reflection of X(1637) in X(47296)
X(66126) = complement of X(66117)
X(66126) = isotomic conjugate of the polar conjugate of X(52475)
X(66126) = X(i)-isoconjugate of X(j) for these (i,j): {30, 36142}, {163, 9214}, {691, 2173}, {892, 9406}, {895, 56829}, {897, 2420}, {923, 2407}, {1495, 36085}, {4240, 36060}, {14206, 32729}, {14908, 24001}, {35266, 36045}
X(66126) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 9214}, {1560, 4240}, {1648, 5642}, {1649, 1637}, {2482, 2407}, {6593, 2420}, {9410, 892}, {21905, 14398}, {23992, 30}, {31654, 35266}, {36896, 691}, {38988, 1495}, {48317, 1990}, {56792, 60498}, {62577, 41079}, {62594, 11064}, {62606, 65321}
X(66126) = cevapoint of X(9204) and X(9205)
X(66126) = crossdifference of every pair of points on line {1495, 2420}
X(66126) = barycentric product X(i)*X(j) for these {i,j}: {69, 52475}, {74, 35522}, {468, 34767}, {523, 36890}, {524, 2394}, {690, 1494}, {850, 9717}, {2433, 3266}, {2642, 33805}, {5468, 12079}, {5627, 45808}, {5967, 65973}, {6390, 18808}, {8749, 45807}, {9139, 52629}, {9204, 36308}, {9205, 36311}, {14380, 44146}, {14417, 16080}, {32112, 52145}, {37778, 62665}, {44769, 52628}, {50942, 51227}
X(66126) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 691}, {187, 2420}, {351, 1495}, {468, 4240}, {523, 9214}, {524, 2407}, {690, 30}, {1494, 892}, {1648, 1637}, {1649, 5642}, {2159, 36142}, {2349, 36085}, {2394, 671}, {2433, 111}, {2642, 2173}, {2682, 58346}, {4750, 18653}, {5642, 3233}, {5967, 65776}, {9125, 35266}, {9139, 34574}, {9204, 41887}, {9205, 41888}, {9717, 110}, {11183, 51430}, {12079, 5466}, {14273, 1990}, {14380, 895}, {14385, 51478}, {14417, 11064}, {14419, 51420}, {14424, 51360}, {14432, 51382}, {14443, 2682}, {14919, 65321}, {16080, 65350}, {17986, 53155}, {18808, 17983}, {21906, 14398}, {32112, 5968}, {34767, 30786}, {35522, 3260}, {36875, 52035}, {36890, 99}, {40352, 32729}, {42713, 42716}, {44102, 23347}, {44814, 1511}, {45662, 64607}, {45808, 6148}, {46147, 36827}, {50567, 66074}, {50942, 51228}, {51227, 50941}, {51429, 65754}, {51479, 14254}, {52038, 35906}, {52475, 4}, {52628, 41079}, {56395, 41392}, {56792, 9213}, {58331, 58337}, {58349, 3081}, {65756, 62629}
X(66126) = {X(2394),X(63856)}-harmonic conjugate of X(1640)

X(66127) = X(2)X(9141)∩X(30)X(1637)

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

X(66127) lies on the cubic K1380 and these lines: {2, 9141}, {30, 1637}, {125, 23967}, {187, 1648}, {230, 6128}, {441, 44569}, {511, 58900}, {524, 14417}, {542, 647}, {1650, 3284}, {2482, 62594}, {3163, 3258}, {9140, 36904}, {10991, 57425}, {11645, 65489}, {51360, 57465}

X(66127) = complement of X(9141)
X(66127) = complement of the isogonal conjugate of X(9142)
X(66127) = complement of the isotomic conjugate of X(9140)
X(66127) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 5642}, {9140, 2887}, {9142, 10}
X(66127) = X(2)-Ceva conjugate of X(5642)
X(66127) = X(5642)-Dao conjugate of X(2)
X(66127) = crosspoint of X(i) and X(j) for these (i,j): {2, 9140}, {30, 524}
X(66127) = crosssum of X(74) and X(111)
X(66127) = crossdifference of every pair of points on line {9213, 9717}
X(66127) = barycentric product X(5642)*X(9140)
X(66127) = barycentric quotient X(i)/X(j) for these {i,j}: {5642, 9141}, {9142, 9139}

X(66128) = X(2)X(2986)∩X(115)X(647)

Barycentrics    (b - c)^2*(b + c)^2*(-2*a^2 + b^2 + c^2)*(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(66128) lies on the cubic K1380 and these lines: {2, 2986}, {115, 647}, {125, 61216}, {1637, 65764}, {3124, 14401}, {5642, 56395}, {6055, 51456}, {6388, 58900}, {8029, 20975}, {10418, 66078}, {14417, 52628}

X(66128) = X(i)-isoconjugate of X(j) for these (i,j): {15329, 36085}, {24041, 60498}, {36142, 61188}
X(66128) = X(i)-Dao conjugate of X(j) for these (i,j): {1649, 3580}, {3005, 60498}, {21905, 3003}, {23992, 61188}, {38988, 15329}, {48317, 16237}
X(66128) = cevapoint of X(2682) and X(21906)
X(66128) = crosspoint of X(43084) and X(52475)
X(66128) = crossdifference of every pair of points on line {15329, 21731}
X(66128) = barycentric product X(i)*X(j) for these {i,j}: {690, 15328}, {1648, 2986}, {2682, 40423}, {14273, 15421}, {14910, 52628}, {15470, 51479}, {18878, 33919}, {21906, 40832}
X(66128) = barycentric quotient X(i)/X(j) for these {i,j}: {351, 15329}, {690, 61188}, {1648, 3580}, {2682, 113}, {2986, 52940}, {3124, 60498}, {10420, 45773}, {14273, 16237}, {15328, 892}, {18878, 64460}, {21906, 3003}, {33919, 55121}, {61216, 65321}

X(66129) = X(2)X(112)∩X(122)X(35594)

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

X(66129) lies on the cubic K1380 and these lines: {2, 112}, {122, 35594}, {125, 65735}, {647, 15526}, {1637, 65753}, {1649, 2972}, {1650, 9409}, {3258, 65759}, {3269, 23616}, {10423, 57587}

X(66129) = X(i)-isoconjugate of X(j) for these (i,j): {24000, 60499}, {36131, 61181}, {46592, 65263}
X(66129) = X(i)-Dao conjugate of X(j) for these (i,j): {14401, 858}, {38999, 61198}, {39008, 61181}, {57295, 5523}
X(66129) = crossdifference of every pair of points on line {42665, 46592}
X(66129) = barycentric product X(i)*X(j) for these {i,j}: {1650, 2373}, {18876, 65753}, {41077, 60040}
X(66129) = barycentric quotient X(i)/X(j) for these {i,j}: {1636, 61198}, {1650, 858}, {2373, 42308}, {3269, 60499}, {9033, 61181}, {9409, 46592}, {60040, 15459}

X(66130) = X(2)X(35909)∩X(468)X(690)

Barycentrics    (b - c)*(b + c)*(-2*a^2 + 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^8 + a^6*b^2 + a^4*b^4 - a^2*b^6 + a^6*c^2 - 3*a^4*b^2*c^2 + a^2*b^4*c^2 + 3*b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 3*b^2*c^6) : :
X(66130) = 3 X[2] + X[66121]

X(66130) lies on the cubic K1380 and these lines: {2, 35909}, {468, 690}, {526, 47296}, {542, 30476}, {1637, 60496}, {1650, 9409}, {2781, 47252}, {2799, 22104}, {5159, 9517}, {6130, 9003}, {6699, 24284}, {9033, 11064}, {13202, 58344}, {47214, 55121}

X(66130) = complement of the isotomic conjugate of X(66115)
X(66130) = X(i)-complementary conjugate of X(j) for these (i,j): {1101, 44814}, {48540, 21253}, {53327, 24040}, {66115, 2887}
X(66130) = crosspoint of X(i) and X(j) for these (i,j): {2, 66115}, {690, 9033}
X(66130) = crosssum of X(691) and X(1304)

X(66131) = X(2)X(65780)∩X(125)X(18312)

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

X(66131) lies on the cubic K1380 and these lines: {2, 65780}, {125, 18312}, {468, 9176}, {647, 65732}, {1637, 65753}, {3150, 14566}, {3258, 65757}, {11064, 60496}, {14417, 52628}, {62577, 62594}

X(66131) = complement of the isotomic conjugate of X(66116)
X(66131) = X(i)-complementary conjugate of X(j) for these (i,j): {661, 12827}, {66116, 2887}
X(66131) = crosspoint of X(2) and X(66116)

X(66132) = X(2)X(35909)∩X(113)X(1560)

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

X(66132) lies on the cubics K1379 and K1380 and these lines: {2, 35909}, {113, 1560}, {125, 18312}, {265, 2433}, {468, 9517}, {526, 11064}, {542, 647}, {684, 1649}, {690, 858}, {1637, 14356}, {1648, 47138}, {1650, 6334}, {2799, 3258}, {3268, 66120}, {3580, 9033}, {5972, 60352}, {6333, 52629}, {9140, 66116}, {9979, 53156}, {14270, 15329}, {14314, 15354}, {16230, 58263}, {35235, 41079}, {47249, 62516}, {55121, 65709}

X(66132) = reflection of X(62516) in X(47249)
X(66132) = isotomic conjugate of X(66115)
X(66132) = complement of X(66121)
X(66132) = X(i)-isoconjugate of X(j) for these (i,j): {31, 66115}, {163, 48540}, {24041, 53327}, {36084, 56962}
X(66132) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 66115}, {115, 48540}, {3005, 53327}, {38987, 56962}
X(66132) = cevapoint of X(i) and X(j) for these (i,j): {686, 42665}, {690, 9033}, {1648, 65709}, {39474, 60340}
X(66132) = trilinear pole of line {23992, 39008}
X(66132) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 66115}, {523, 48540}, {3124, 53327}, {3569, 56962}

X(66133) = X(2)X(65780)∩X(125)X(65736)

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

X(66133) lies on the cubic K1380 and these lines: {2, 65780}, {125, 65736}, {232, 1560}, {237, 47414}, {1637, 65762}, {3003, 16186}, {3289, 47405}, {15462, 23357}, {36212, 62569}, {41270, 58267}

X(66133) = X(162)-isoconjugate of X(66116)
X(66133) = X(125)-Dao conjugate of X(66116)
X(66133) = barycentric quotient X(647)/X(66116)

X(66134) = X(4)X(523)∩X(30)X(15412)

Barycentrics    a^2*(b^2 - c^2)*(a^12 - a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 + a^4*b^8 - 5*a^2*b^10 + 2*b^12 - a^10*c^2 + 4*a^8*b^2*c^2 - a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - 5*b^10*c^2 - 4*a^8*c^4 - a^6*b^2*c^4 + 15*a^4*b^4*c^4 - 9*a^2*b^6*c^4 - b^8*c^4 + 6*a^6*c^6 - 11*a^4*b^2*c^6 - 9*a^2*b^4*c^6 + 8*b^6*c^6 + a^4*c^8 + 14*a^2*b^2*c^8 - b^4*c^8 - 5*a^2*c^10 - 5*b^2*c^10 + 2*c^12) : :
X(66134) = 3 X[37941] - 4 X[63830]

X(66134) lies on these lines: {4, 523}, {30, 15412}, {186, 15451}, {512, 14157}, {1510, 12112}, {3288, 7712}, {8718, 30210}, {9147, 47248}, {37941, 63830}, {64890, 65403}

X(66134) = reflection of X(186) in X(15451)
X(66134) = crossdifference of every pair of points on line {3284, 61691}

X(66135) = X(4)X(32)∩X(127)X(2896)

Barycentrics    a^2*(a^12 - 2*a^10*b^2 + a^8*b^4 + a^6*b^6 - 2*a^4*b^8 + a^2*b^10 - 2*a^10*c^2 + a^6*b^4*c^2 + a^4*b^6*c^2 - a^2*b^8*c^2 + b^10*c^2 + a^8*c^4 + a^6*b^2*c^4 - a^4*b^4*c^4 - b^8*c^4 + a^6*c^6 + a^4*b^2*c^6 - 2*a^4*c^8 - a^2*b^2*c^8 - b^4*c^8 + a^2*c^10 + b^2*c^10) : :

X(66135) lies on these lines: {4, 32}, {127, 2896}, {147, 10316}, {827, 64647}, {1297, 9821}, {1971, 11674}, {2076, 2781}, {2782, 10313}, {2799, 4580}, {3094, 10766}, {3098, 38717}, {3269, 62341}, {3455, 53026}, {5976, 28724}, {6720, 10583}, {7811, 10718}, {8743, 9861}, {9301, 53795}, {9941, 10705}, {9985, 58058}, {10357, 34841}, {10547, 14885}, {10749, 32151}, {12503, 38689}, {13116, 65127}, {13280, 49561}, {13313, 26318}, {13314, 26317}, {18503, 19163}, {18876, 35952}, {19114, 44605}, {19115, 44604}, {19165, 38525}, {20968, 39575}, {23128, 39836}, {26316, 38699}, {34217, 46283}, {39643, 39837}, {57304, 66096}

X(66135) = polar-circle inverse of X(27371)
X(66135) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 13236, 112}, {11610, 53767, 112}

X(66136) = X(4)X(542)∩X(25)X(39837)

Barycentrics    a^2*(a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 21*a^6*b^6 + 18*a^4*b^8 - 9*a^2*b^10 + 2*b^12 - 6*a^10*c^2 + 12*a^8*b^2*c^2 - 5*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - b^10*c^2 + 15*a^8*c^4 - 5*a^6*b^2*c^4 - 5*a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 11*b^8*c^4 - 21*a^6*c^6 - 5*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 20*b^6*c^6 + 18*a^4*c^8 + 5*a^2*b^2*c^8 - 11*b^4*c^8 - 9*a^2*c^10 - b^2*c^10 + 2*c^12) : :

X(66136) lies on these lines: {4, 542}, {25, 39837}, {74, 8588}, {110, 37457}, {378, 39849}, {690, 15412}, {1614, 3455}, {5621, 38520}, {5622, 10485}, {5655, 66097}, {11623, 43602}, {11674, 32438}, {12083, 39836}, {12834, 61576}, {20398, 43600}, {35473, 57011}, {39846, 52294}

X(66137) = X(1)X(4)∩X(35)X(2169)

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

X(66137) lies on these lines: {1, 4}, {35, 2169}, {48, 1324}, {522, 15412}, {820, 5399}, {916, 52407}, {5396, 62266}, {32613, 52430}

X(66138) = X(3)X(13509)∩X(6)X(186)

Barycentrics    a^2*(4*a^8 - 8*a^6*b^2 + 2*a^4*b^4 + 4*a^2*b^6 - 2*b^8 - 8*a^6*c^2 + 11*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 + 2*a^4*c^4 - 4*a^2*b^2*c^4 + 2*b^4*c^4 + 4*a^2*c^6 + b^2*c^6 - 2*c^8) : :

X(66138) lies on these lines: {3, 13509}, {6, 186}, {74, 8588}, {112, 11202}, {160, 39201}, {187, 11464}, {352, 15655}, {574, 62341}, {577, 15035}, {1249, 48361}, {1511, 18472}, {1614, 5206}, {1970, 44879}, {1971, 35473}, {3098, 38717}, {3164, 15412}, {3288, 7712}, {3484, 10979}, {5023, 9707}, {5092, 49124}, {5210, 11456}, {6241, 15513}, {7749, 12289}, {10282, 41367}, {10298, 32661}, {10986, 11430}, {12096, 22052}, {14585, 21844}, {15577, 21397}, {17506, 39643}, {17821, 41376}, {23128, 38448}, {39565, 40242}

X(66138) = circumcircle-inverse of X(13509)

X(66139) = X(4)X(9)∩X(389)X(23621)

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

X(66139) lies on these lines: {4, 9}, {389, 23621}, {514, 15412}, {2305, 8607}, {2361, 17798}, {2818, 42669}, {4300, 52425}, {54058, 54081}

X(66140) = X(3)X(39098)∩X(4)X(147)

Barycentrics    a^12 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + 2*a^2*b^10 - b^12 - 10*a^8*b^2*c^2 + 10*a^6*b^4*c^2 - 2*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + 2*b^10*c^2 - 4*a^8*c^4 + 10*a^6*b^2*c^4 - 3*a^4*b^4*c^4 - 4*a^2*b^6*c^4 + 6*a^6*c^6 - 2*a^4*b^2*c^6 - 4*a^2*b^4*c^6 - 2*b^6*c^6 - 4*a^4*c^8 + 2*a^2*b^2*c^8 + 2*a^2*c^10 + 2*b^2*c^10 - c^12 : :

X(66140) lies on these lines: {3, 39098}, {4, 147}, {98, 7749}, {99, 32152}, {114, 52034}, {115, 32467}, {315, 13172}, {804, 15412}, {1352, 5152}, {2548, 62356}, {2896, 33813}, {6298, 36776}, {6299, 61634}, {7709, 43449}, {8289, 15561}, {8721, 9862}, {9744, 14651}, {9890, 34623}, {11676, 35464}, {12188, 37446}, {14639, 43457}, {37334, 51872}, {48657, 55008}, {52128, 62341}

X(66140) = reflection of X(52034) in X(114)

X(66141) = X(384)-(B)LINE CONJUGATE OF X(1)

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

X(66141) lies on these lines: {1, 335}, {2, 4475}, {244, 17023}, {337, 24578}, {514, 661}, {1930, 62553}, {2809, 27919}, {3061, 36796}, {4554, 7146}, {14839, 40217}, {17284, 30846}, {17316, 17777}, {24255, 26590}, {27248, 27281}, {29960, 30000}

X(66142) = X(384)-(B)LINE CONJUGATE OF X(75)

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

X(66142) lies on these lines: {44, 513}, {75, 384}, {1740, 54406}, {9596, 26042}, {27633, 28264}

X(66143) = X(384)-(B)LINE CONJUGATE OF X(6)

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

X(66143) lies on these lines: {6, 194}, {325, 523}, {670, 59567}, {3003, 5976}, {3095, 58846}, {7750, 54334}, {7778, 30777}, {7792, 9465}, {14603, 40073}, {16084, 16098}, {19599, 40888}, {21531, 40074}

X(66143) = reflection of X(670) in X(59567)
X(66143) = crosspoint of X(99) and X(57988)
X(66143) = crossdifference of every pair of points on line {32, 3221}
X(66143) = {X(3001),X(35549)}-harmonic conjugate of X(325)

X(66144) = X(384)-(B)LINE CONJUGATE OF X(141)

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

X(66144) lies on these lines: {23, 385}, {141, 384}, {194, 35707}, {2916, 52637}, {7792, 26257}, {9019, 16985}, {9229, 46288}, {10997, 30736}, {19596, 40858}

X(66145) = X(384)-(B)LINE CONJUGATE OF X(523)

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

X(66145) lies on these lines: {2, 6}, {39, 4590}, {83, 31998}, {384, 523}, {892, 7804}, {2854, 4027}, {6680, 45212}, {7770, 36207}, {7783, 14588}, {7797, 54104}, {7816, 33799}, {7828, 23991}, {8289, 42007}, {8705, 10997}, {10583, 35511}, {14977, 46778}, {19686, 62508}, {36432, 41579}, {39565, 40429}, {46900, 53379}

X(66146) = X(1)-(B)LINE CONJUGATE OF X(385)

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

X(66146) lies on these lines: {1, 257}, {141, 523}, {1581, 4475}, {24348, 35552}

crossdifference of every pair of points on line {1691, 45882}

X(66147) = X(385)-(B)LINE CONJUGATE OF X(1)

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

X(66147) lies on these lines: {1, 257}, {514, 661}, {1089, 17760}, {1330, 1655}, {1930, 30077}, {2292, 49476}, {3061, 18140}, {4475, 27241}, {4876, 49753}, {20890, 30042}, {24632, 40773}, {29960, 45196}, {39044, 49755}, {52538, 59509}

X(66147) = crossdifference of every pair of points on line {31, 45882}

X(66148) = X(75)-(B)LINE CONJUGATE OF X(385)

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

X(66148) lies on these lines: {75, 385}, {141, 523}, {35551, 36227}

X(66149) = X(385)-(B)LINE CONJUGATE OF X(75)

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

X(66149) lies on these lines: {37, 8844}, {44, 513}, {75, 385}, {1333, 33295}, {1740, 3863}, {1914, 33891}, {2076, 8301}, {3509, 3862}, {4016, 4093}

X(66150) = X(385)-(B)LINE CONJUGATE OF X(76)

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

X(661) lies on these lines: {3, 33786}, {32, 76}, {39, 32748}, {187, 237}, {538, 51322}, {737, 805}, {2080, 35399}, {2387, 56978}, {5026, 33875}, {5970, 39632}, {6310, 13335}, {7804, 56442}, {8023, 8024}, {50665, 56428}

X(66150) = isogonal conjugate of X(57935)
X(66150) = isogonal conjugate of the isotomic conjugate of X(706)
X(66150) = X(707)-Ceva conjugate of X(6)
X(66150) = X(i)-isoconjugate of X(j) for these (i,j): {1, 57935}, {75, 707}
X(66150) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 57935}, {206, 707}, {706, 35528}
X(66150) = crosspoint of X(6) and X(707)
X(66150) = crosssum of X(2) and X(706)
X(66150) = crossdifference of every pair of points on line {2, 17415}
X(66150) = barycentric product X(i)*X(j) for these {i,j}: {6, 706}, {32, 35528}
X(66150) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 57935}, {32, 707}, {706, 76}, {35528, 1502}
X(66150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 3229, 21444}, {21444, 57016, 3229}

X(66151) = X(894)-(B)LINE CONJUGATE OF X(514)

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

X(66151) lies on these lines: {1, 2}, {6, 35957}, {37, 1016}, {75, 24281}, {86, 6631}, {99, 21888}, {190, 35103}, {335, 4482}, {514, 894}, {952, 26582}, {993, 62312}, {996, 31317}, {1145, 26629}, {1482, 26687}, {2345, 30225}, {2802, 4366}, {3125, 18047}, {3589, 45213}, {3754, 6645}, {3884, 16918}, {3918, 16917}, {4555, 4670}, {5697, 16916}, {6547, 16706}, {6630, 63053}, {7807, 8256}, {7824, 51111}, {7983, 20716}, {10944, 17670}, {11010, 17692}, {17116, 49751}, {17261, 32094}, {17289, 36230}, {17302, 54102}, {17351, 32028}, {18082, 29298}, {20172, 40587}, {24170, 62650}, {24358, 61187}, {33841, 37710}, {36234, 37756}, {36236, 46897}, {60480, 62324}

X(66151) = {X(86),X(6631)}-harmonic conjugate of X(36226)

X(66152) = X(894)-(B)LINE CONJUGATE OF X(10)

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

X(66152) lies on these lines: {8, 20102}, {10, 894}, {58, 257}, {63, 41232}, {191, 1655}, {239, 514}, {385, 758}, {484, 17759}, {519, 3099}, {740, 5184}, {1759, 17033}, {3125, 33295}, {3496, 16574}, {3509, 40859}, {3570, 21839}, {3721, 30168}, {3875, 41319}, {5282, 30114}, {5692, 16997}, {5883, 17000}, {5902, 16998}, {6763, 21226}, {7793, 22836}, {8682, 17731}, {9278, 17930}, {10176, 16999}, {11611, 17929}, {16611, 20142}, {17023, 24627}, {17030, 54382}, {17735, 49753}, {17768, 47286}, {20065, 49168}, {20372, 41240}, {21764, 30111}, {24514, 49500}, {25264, 56288}, {26085, 30165}, {26099, 30150}, {27081, 29610}, {27091, 54406}, {35101, 50252}, {36531, 40860}

X(66152) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {53686, 69}, {57682, 20243}, {60043, 21293}
X(66152) = crosssum of X(5277) and X(5291)
X(66152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1046, 17739, 17499}, {57017, 57029, 239}

X(66153) = X(32)-LINE CONJUGATE OF X(384)

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

X(66153) lies on these lines: {32, 76}, {647, 3221}

X(66153) = crossdifference of every pair of points on line {384, 17415}
X(66153) = X(i)-line conjugate of X(j) for these (i,j): {32, 384}, {647, 17415}

X(66154) = X(32)-LINE CONJUGATE OF X(385)

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

X(66154) lies on these lines: {32, 76}, {39, 512}, {695, 3493}, {2782, 51322}, {3095, 58212}

X(66154) = crossdifference of every pair of points on line {385, 17415}
X(66154) = X(i)-line conjugate of X(j) for these (i,j): {32, 385}, {39, 17415}

X(66155) = X(335)-LINE CONJUGATE OF X(1)

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

X(66155) lies on these lines: {1, 335}, {44, 513}, {190, 25800}, {1086, 25806}, {3758, 24722}, {4422, 25823}, {4440, 25805}, {9780, 26058}, {27627, 28264}

X(66155) = X(335)-line conjugate of X(1)
X(66155) = {X(1911),X(17738)}-harmonic conjugate of X(20356)

X(66156) = X(2)-LINE CONJUGATE OF X(423)

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

X(66156) lies on these lines: {2, 3}, {647, 22080}, {20966, 41172}

X(66156) = crossdifference of every pair of points on line {423, 647}
X(66156) = X(i)-line conjugate of X(j) for these (i,j): {2, 423}, {22080, 647}

X(66157) = X(1)-LINE CONJUGATE OF X(385)

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

X(66157) lies on these lines: {1, 257}, {39, 512}, {256, 36294}, {1934, 20356}, {52205, 59480}

X(66157) = crossdifference of every pair of points on line {385, 45882}
X(66157) = X(i)-line conjugate of X(j) for these (i,j): {1, 385}, {39, 45882}

X(66158) = X(257)-LINE CONJUGATE OF X(1)

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

X(66158) lies on these lines: {1, 257}, {44, 513}, {4396, 20356}, {4689, 8844}, {24628, 53541}

X(66158) = crossdifference of every pair of points on line {1, 45882}
X(66158) = X(i)-line conjugate of X(j) for these (i,j): {44, 45882}, {257, 1}
X(66158) = barycentric product X(350)*X(1908)
X(66158) = barycentric quotient X(1908)/X(291)

X(66159) = X(32)-LINE CONJUGATE OF X(76)

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

X(66159) lies on these lines: {32, 76}, {669, 688}, {3231, 41331}

X(66159) = crossdifference of every pair of points on line {76, 17415}
X(66159) = X(i)-line conjugate of X(j) for these (i,j): {32, 76}, {669, 17415}

X(66160) = X(669)X(1501)∩X(1084)X(1974)

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

X(66160) lies on these lines: {669, 1501}, {1084, 1974}, {1976, 41273}, {2715, 44116}, {35906, 52945}

X(66160) = X(i)-isoconjugate of X(j) for these (i,j): {325, 33805}, {561, 35910}, {799, 65973}, {1494, 46238}, {4602, 32112}, {24037, 65756}, {35908, 40364}
X(66160) = X(i)-Dao conjugate of X(j) for these (i,j): {512, 65756}, {38996, 65973}, {40368, 35910}
X(66160) = barycentric product X(i)*X(j) for these {i,j}: {30, 14601}, {32, 35906}, {98, 9407}, {248, 14581}, {669, 65776}, {878, 23347}, {1495, 1976}, {1501, 60869}, {1910, 9406}, {1974, 35912}, {1990, 14600}, {2420, 2422}, {2715, 14398}, {3284, 57260}, {9409, 32696}
X(66160) = barycentric quotient X(i)/X(j) for these {i,j}: {669, 65973}, {1084, 65756}, {1501, 35910}, {9406, 46238}, {9407, 325}, {9426, 32112}, {14581, 44132}, {14601, 1494}, {35906, 1502}, {35912, 40050}, {44162, 35908}, {58260, 65974}, {60869, 40362}, {65776, 4609}

X(66161) = X(2)X(46425)∩X(4)X(2881)

Barycentrics    b^2*(b^2 - c^2)*c^2*(-2*a^6 + a^4*b^2 + b^6 + a^4*c^2 - b^4*c^2 - b^2*c^4 + c^6) : :
X(66161) = 4 X[18312] - X[35522], 2 X[18312] + X[41079], X[35522] + 2 X[41079], 2 X[47138] + X[65612]

X(66161) lies on the cubic K1378 and these liines: {2, 46425}, {4, 2881}, {25, 47205}, {76, 52459}, {98, 804}, {115, 127}, {132, 50938}, {230, 65778}, {297, 65780}, {403, 523}, {427, 42665}, {525, 23285}, {850, 6587}, {1304, 65356}, {1503, 39073}, {1636, 23292}, {1989, 14592}, {2079, 54089}, {2394, 46105}, {2485, 18314}, {2508, 3767}, {3267, 14638}, {3569, 60527}, {6330, 16081}, {9148, 59742}, {9478, 63894}, {9979, 65972}, {13567, 52744}, {14977, 51967}, {15352, 65181}, {15595, 39473}, {16040, 31296}, {18311, 65757}, {22456, 59024}, {23105, 55122}, {23881, 60597}, {34129, 53173}, {44817, 62307}, {47206, 53318}, {52624, 62577}, {60516, 65980}

X(66161) = reflection of X(i) in X(j) for these {i,j}: {2485, 52585}, {53265, 6130}, {62307, 44817}
X(66161) = polar conjugate of X(44770)
X(66161) = X(1973)-complementary conjugate of X(39000)
X(66161) = X(i)-Ceva conjugate of X(j) for these (i,j): {16081, 338}, {22456, 52641}
X(66161) = X(i)-cross conjugate of X(j) for these (i,j): {33504, 4}, {55275, 523}
X(66161) = X(i)-isoconjugate of X(j) for these (i,j): {3, 36046}, {48, 44770}, {63, 32649}, {163, 1297}, {255, 32687}, {577, 36092}, {1101, 34212}, {2172, 46967}, {4575, 43717}, {8767, 32661}, {23995, 43673}, {32676, 64975}, {34072, 46164}, {36034, 51937}, {52430, 65265}
X(66161) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 1297}, {136, 43717}, {232, 4230}, {339, 64974}, {441, 2421}, {523, 34212}, {647, 2435}, {1249, 44770}, {3162, 32649}, {3258, 51937}, {6523, 32687}, {15449, 46164}, {15526, 64975}, {15595, 4558}, {18314, 43673}, {23285, 2419}, {23976, 110}, {33504, 3}, {35078, 51343}, {36103, 36046}, {36901, 35140}, {38970, 39265}, {39071, 32661}, {39073, 14966}, {50938, 112}, {57606, 52058}, {60341, 58796}, {65726, 43754}, {65757, 66077}
X(66161) = crosspoint of X(i) and X(j) for these (i,j): {76, 22456}, {98, 1289}, {2409, 21458}, {14618, 43665}
X(66161) = crosssum of X(i) and X(j) for these (i,j): {32, 39469}, {511, 8673}, {2435, 46164}, {14966, 32661}
X(66161) = crossdifference of every pair of points on line {160, 206}
X(66161) = barycentric product X(i)*X(j) for these {i,j}: {107, 58258}, {338, 34211}, {339, 2409}, {441, 14618}, {523, 30737}, {525, 60516}, {850, 1503}, {2052, 39473}, {2312, 20948}, {2799, 57490}, {3267, 16318}, {3268, 43089}, {3569, 51257}, {6333, 52641}, {15595, 43665}, {17879, 24024}, {18018, 55129}, {21458, 23285}, {23977, 36793}, {35282, 52632}, {41079, 63856}, {42671, 44173}, {43187, 57430}, {55275, 57799}, {57426, 65269}, {60506, 62431}, {65778, 65980}
X(66161) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 44770}, {19, 36046}, {25, 32649}, {66, 46967}, {115, 34212}, {125, 2435}, {132, 4230}, {158, 36092}, {338, 43673}, {339, 2419}, {393, 32687}, {441, 4558}, {523, 1297}, {525, 64975}, {804, 51343}, {826, 46164}, {850, 35140}, {879, 15407}, {1503, 110}, {1637, 51937}, {2052, 65265}, {2312, 163}, {2409, 250}, {2445, 57655}, {2501, 43717}, {6793, 2420}, {8766, 4575}, {8779, 32661}, {9475, 14966}, {14618, 6330}, {15595, 2421}, {16230, 39265}, {16318, 112}, {17994, 51822}, {21458, 827}, {23285, 64974}, {23977, 23964}, {24006, 8767}, {24024, 24000}, {30737, 99}, {34156, 43754}, {34211, 249}, {35282, 5467}, {36894, 65321}, {39473, 394}, {42671, 1576}, {43045, 4565}, {43089, 476}, {43665, 9476}, {51257, 43187}, {51363, 1625}, {51434, 35325}, {51437, 61206}, {51960, 65305}, {51963, 2715}, {52641, 685}, {53568, 15329}, {55129, 22}, {55275, 232}, {56572, 10425}, {57296, 58796}, {57426, 9517}, {57430, 3569}, {57490, 2966}, {57799, 55274}, {58258, 3265}, {60506, 57742}, {60516, 648}, {62612, 52058}, {63856, 44769}, {65753, 66077}
X(66161) = {X(18312),X(41079)}-harmonic conjugate of X(35522)

X(66162) = X(115)X(525)∩X(230)X(297)

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

X(66162) lies on the orthic asymptotic hyperbola, the cubic K1378, and these lines: {2, 36891}, {5, 34157}, {6, 56572}, {30, 61446}, {115, 525}, {141, 52091}, {230, 297}, {325, 52515}, {338, 3267}, {403, 935}, {523, 8754}, {524, 1989}, {671, 7799}, {868, 879}, {1503, 2065}, {2394, 60338}, {4064, 21043}, {4580, 34294}, {5139, 42399}, {6388, 55152}, {8773, 20337}, {11585, 53787}, {14120, 62489}, {14592, 52628}, {14977, 62551}, {15421, 62563}, {15526, 61339}, {15980, 51455}, {30476, 34988}, {34369, 65765}, {35078, 35132}, {42065, 44665}, {43705, 44377}, {44389, 57829}, {48982, 53419}

X(66162) = X(i)-Ceva conjugate of X(j) for these (i,j): {8781, 62645}, {35142, 35364}, {40428, 523}
X(66162) = X(i)-isoconjugate of X(j) for these (i,j): {162, 56389}, {163, 4226}, {230, 1101}, {249, 8772}, {662, 61213}, {1692, 24041}, {1733, 23357}, {17462, 57742}, {23995, 51481}, {23997, 60504}, {44099, 62719}
X(66162) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 4226}, {125, 56389}, {523, 230}, {647, 3564}, {1084, 61213}, {1649, 5477}, {3005, 1692}, {18314, 51481}, {35443, 6782}, {35444, 6783}, {41167, 47406}, {55267, 114}, {62562, 60504}
X(66162) = cevapoint of X(115) and X(868)
X(66162) = crosspoint of X(8781) and X(62645)
X(66162) = crosssum of X(1692) and X(61213)
X(66162) = trilinear pole of line {125, 8029}
X(66162) = crossdifference of every pair of points on line {56389, 61213}
X(66162) = barycentric product X(i)*X(j) for these {i,j}: {115, 8781}, {125, 35142}, {338, 2987}, {339, 3563}, {523, 62645}, {525, 60338}, {850, 35364}, {868, 40428}, {1109, 8773}, {2065, 62431}, {2394, 65758}, {2970, 43705}, {8029, 65277}, {8754, 57872}, {10425, 23105}, {12079, 36891}, {23962, 32654}, {23994, 36051}, {65756, 65781}
X(66162) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 230}, {125, 3564}, {338, 51481}, {512, 61213}, {523, 4226}, {647, 56389}, {868, 114}, {1109, 1733}, {1648, 5477}, {2065, 57742}, {2395, 60504}, {2643, 8772}, {2970, 44145}, {2971, 44099}, {2987, 249}, {3124, 1692}, {3563, 250}, {5466, 52035}, {8029, 55122}, {8754, 460}, {8773, 24041}, {8781, 4590}, {10425, 59152}, {12079, 36875}, {20975, 52144}, {22260, 42663}, {30465, 6782}, {30468, 6783}, {32654, 23357}, {32697, 47443}, {34246, 54965}, {35142, 18020}, {35364, 110}, {36051, 1101}, {40428, 57991}, {41172, 47406}, {42065, 47390}, {44114, 51335}, {51404, 65726}, {51441, 51820}, {57872, 47389}, {60338, 648}, {62645, 99}, {64258, 52450}, {65277, 31614}, {65354, 55270}, {65758, 2407}

X(66163) = X(2)X(99)∩X(230)X(297)

Barycentrics    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(66163) lies on the cubic K1378, and these lines: {2, 99}, {4, 35278}, {98, 868}, {107, 36191}, {125, 14651}, {187, 40885}, {230, 297}, {275, 14586}, {287, 11646}, {323, 7809}, {340, 16310}, {381, 51430}, {403, 1300}, {468, 39663}, {523, 17983}, {648, 3018}, {1316, 14639}, {1494, 1989}, {1632, 34981}, {1637, 2394}, {1993, 7926}, {2396, 8781}, {2450, 43460}, {2986, 18879}, {3545, 5642}, {3580, 14568}, {4226, 10723}, {5112, 38227}, {5191, 10722}, {5475, 52247}, {5477, 40867}, {5972, 23514}, {6036, 35922}, {6330, 16081}, {6529, 11547}, {6723, 38735}, {6781, 40853}, {6791, 37643}, {7473, 46982}, {7779, 60524}, {7831, 41237}, {7925, 36212}, {7934, 15066}, {8884, 52534}, {9142, 38393}, {9155, 64089}, {9410, 44576}, {10733, 64607}, {11007, 38224}, {11064, 33228}, {11176, 65488}, {11331, 13881}, {12066, 44877}, {14041, 51372}, {14590, 14910}, {16303, 37765}, {16312, 62237}, {16316, 36898}, {20218, 46208}, {30789, 53346}, {31998, 65730}, {32740, 65719}, {34473, 36163}, {38229, 57588}, {39563, 44575}, {40814, 52251}, {40884, 53419}, {43291, 44216}, {44533, 48871}, {44578, 63543}, {46453, 62955}, {52289, 63534}, {53266, 66082}, {53383, 65775}, {53485, 54105}, {53577, 57583}, {54837, 59091}, {65768, 66080}

X(66163) = isotomic conjugate of X(66083)
X(66163) = X(i)-isoconjugate of X(j) for these (i,j): {31, 66083}, {163, 65766}, {1755, 65783}
X(66163) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 66083}, {115, 65766}, {36899, 65783}, {65755, 65754}
X(66163) = crosspoint of X(15459) and X(60179)
X(66163) = crosssum of X(1636) and X(41172)
X(66163) = trilinear pole of line {1550, 52472}
X(66163) = barycentric product X(i)*X(j) for these {i,j}: {523, 65768}, {1494, 52472}, {1550, 5641}, {2966, 65977}, {35142, 52473}
X(66163) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 66083}, {98, 65783}, {523, 65766}, {1550, 542}, {52472, 30}, {52473, 3564}, {65763, 65754}, {65768, 99}, {65977, 2799}, {66080, 51389}
X(66163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 115, 41254}, {2, 148, 65722}, {2, 54395, 99}, {4, 47200, 35278}, {115, 48982, 671}, {1989, 62551, 48540}, {48540, 62551, 1494}

X(66164) = X(98)X(230)∩X(115)X(523)

Barycentrics    (b^2 - c^2)^2*(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(66164) = 3 X[115] - X[65755], X[65613] + 2 X[65724], 3 X[65613] - 2 X[65755], 3 X[65724] + X[65755], 3 X[671] + X[65768], 5 X[14061] - X[65713]

X(66164) lies on the cubic K1378, and these lines: {2, 36894}, {98, 230}, {115, 523}, {125, 9209}, {403, 1989}, {647, 3154}, {671, 65768}, {868, 2395}, {1637, 12079}, {2394, 62551}, {2501, 6070}, {14061, 65713}, {18121, 63534}, {36204, 65620}, {48981, 53419}, {65767, 65774}, {65782, 65978}

X(66164) = midpoint of X(i) and X(j) for these {i,j}: {115, 65724}, {11646, 34369}
X(66164) = reflection of X(65613) in X(115)
X(66164) = X(i)-isoconjugate of X(j) for these (i,j): {163, 66084}, {1101, 65765}
X(66164) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 66084}, {523, 65765}, {65754, 51389}, {65782, 325}, {65978, 2407}
X(66164) = crosspoint of X(i) and X(j) for these (i,j): {98, 2394}, {6344, 43665}
X(66164) = crosssum of X(i) and X(j) for these (i,j): {511, 2420}, {14966, 22115}
X(66164) = crossdifference of every pair of points on line {5467, 41167}
X(66164) = barycentric product X(i)*X(j) for these {i,j}: {98, 65978}, {523, 53383}, {2394, 65782}
X(66164) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 65765}, {523, 66084}, {53383, 99}, {65782, 2407}, {65978, 325}, {66081, 66074}
X(66164) = {X(98),X(34366)}-harmonic conjugate of X(230)

X(66165) = X(115)X(647)∩X(125)X(520)

Barycentrics    (b^2 - c^2)^2*(-a^2 + b^2 + c^2)*(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(66165) lies on the cubic K1378, and these lines: {2, 65618}, {4, 51895}, {5, 15454}, {30, 39986}, {98, 858}, {115, 647}, {125, 520}, {136, 16178}, {230, 8791}, {265, 2072}, {339, 3265}, {403, 1300}, {427, 16933}, {523, 2970}, {1312, 53384}, {1313, 53385}, {1368, 65729}, {1594, 38936}, {2351, 36192}, {3134, 12079}, {3150, 15421}, {3154, 8901}, {5576, 58924}, {7471, 13558}, {10024, 59288}, {11585, 16934}, {11799, 58942}, {13160, 58731}, {14611, 15928}, {14911, 47096}, {16089, 65267}, {16186, 43083}, {16221, 53993}, {30786, 57829}, {34209, 39375}, {37938, 39371}, {37987, 58353}, {42665, 51441}, {43090, 65765}, {47195, 62490}, {51404, 61216}

X(66165) = midpoint of X(39986) and X(60035)
X(66165) = X(i)-Ceva conjugate of X(j) for these (i,j): {1300, 15328}, {2986, 61216}, {10419, 523}, {57829, 15421}
X(66165) = X(1650)-cross conjugate of X(125)
X(66165) = X(i)-isoconjugate of X(j) for these (i,j): {162, 15329}, {163, 16237}, {250, 1725}, {403, 1101}, {662, 61209}, {2315, 23582}, {13754, 24000}, {23995, 44138}, {24041, 44084}, {32676, 61188}
X(66165) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 16237}, {125, 15329}, {523, 403}, {525, 62338}, {647, 3580}, {1084, 61209}, {1649, 12828}, {3005, 44084}, {14401, 62569}, {15526, 61188}, {18314, 44138}, {57295, 113}, {60342, 1986}
X(66165) = cevapoint of X(125) and X(16186)
X(66165) = crosspoint of X(i) and X(j) for these (i,j): {328, 2394}, {1300, 15328}, {15421, 57829}
X(66165) = crosssum of X(i) and X(j) for these (i,j): {110, 15472}, {2420, 34397}, {13754, 15329}, {44084, 61209}
X(66165) = crossdifference of every pair of points on line {15329, 61209}
X(66165) = barycentric product X(i)*X(j) for these {i,j}: {115, 57829}, {125, 2986}, {338, 5504}, {339, 14910}, {523, 15421}, {525, 15328}, {687, 5489}, {850, 61216}, {1300, 15526}, {3269, 65267}, {10419, 65753}, {12028, 62551}, {14592, 15470}, {16186, 40427}, {20902, 36053}, {20975, 40832}, {23105, 43755}, {30786, 66128}, {34767, 65615}, {35361, 62428}, {51456, 65727}, {53576, 60035}
X(66165) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 403}, {125, 3580}, {338, 44138}, {512, 61209}, {523, 16237}, {525, 61188}, {647, 15329}, {1300, 23582}, {1648, 12828}, {1650, 62569}, {2088, 1986}, {2986, 18020}, {3124, 44084}, {3269, 13754}, {3708, 1725}, {5489, 6334}, {5504, 249}, {8029, 47236}, {10420, 47443}, {12028, 39295}, {14582, 41512}, {14910, 250}, {15328, 648}, {15421, 99}, {15470, 14590}, {15526, 62338}, {16186, 34834}, {18210, 18609}, {18878, 55270}, {20975, 3003}, {35361, 35360}, {43755, 59152}, {47421, 52000}, {51404, 52451}, {57829, 4590}, {61216, 110}, {65615, 4240}, {65762, 4230}, {66128, 468}
X(66165) = {X(34978),X(37985)}-harmonic conjugate of X(125)

X(66166) = X(2)X(525)∩X(115)X(647)

Barycentrics    (b^2 - c^2)*(2*a^12 - 5*a^10*b^2 + 6*a^8*b^4 - 8*a^6*b^6 + 8*a^4*b^8 - 3*a^2*b^10 - 5*a^10*c^2 + 6*a^8*b^2*c^2 + a^6*b^4*c^2 - 5*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + b^10*c^2 + 6*a^8*c^4 + a^6*b^2*c^4 - 2*a^4*b^4*c^4 + a^2*b^6*c^4 - 4*b^8*c^4 - 8*a^6*c^6 - 5*a^4*b^2*c^6 + a^2*b^4*c^6 + 6*b^6*c^6 + 8*a^4*c^8 + 2*a^2*b^2*c^8 - 4*b^4*c^8 - 3*a^2*c^10 + b^2*c^10) : :

X(66166) lies on the cubic K1378, and these lines: {2, 525}, {115, 647}, {230, 3569}, {351, 50707}, {403, 47230}, {523, 1989}, {526, 61656}, {804, 2450}, {1138, 9213}, {1636, 44665}, {3288, 5915}, {6137, 61371}, {6138, 61370}, {14611, 41392}, {16171, 53416}, {16280, 22264}, {32120, 47200}, {35906, 56962}, {52743, 56395}

X(66166) = X(14356)-Dao conjugate of X(66075)
X(66166) = crosspoint of X(98) and X(39290)
X(66166) = crosssum of X(511) and X(52743)
X(66166) = crossdifference of every pair of points on line {1495, 15329}
X(66166) = barycentric product X(523)*X(65770)
X(66166) = barycentric quotient X(65770)/X(99)

X(66167) = X(4)X(523)∩X(6)X(13)

Barycentrics    2*a^12 - 3*a^10*b^2 - a^8*b^4 + 3*a^6*b^6 - 3*a^4*b^8 + 4*a^2*b^10 - 2*b^12 - 3*a^10*c^2 + 8*a^8*b^2*c^2 - 4*a^6*b^4*c^2 + 2*a^4*b^6*c^2 - 9*a^2*b^8*c^2 + 6*b^10*c^2 - a^8*c^4 - 4*a^6*b^2*c^4 + 2*a^4*b^4*c^4 + 5*a^2*b^6*c^4 - 6*b^8*c^4 + 3*a^6*c^6 + 2*a^4*b^2*c^6 + 5*a^2*b^4*c^6 + 4*b^6*c^6 - 3*a^4*c^8 - 9*a^2*b^2*c^8 - 6*b^4*c^8 + 4*a^2*c^10 + 6*b^2*c^10 - 2*c^12 : :

X(66167) = 3 X[4] + X[36875], 3 X[4] - X[52472], 3 X[381] - 2 X[14356], 3 X[381] - X[34810], 4 X[7687] - 3 X[65617], 4 X[14356] - 3 X[14995], 3 X[14995] - 2 X[34810], 3 X[3839] - X[9214]

X(66167) lies on the cubic K1378, and these lines: {2, 3233}, {3, 30715}, {4, 523}, {5, 52772}, {6, 13}, {25, 16221}, {30, 53274}, {147, 5968}, {157, 378}, {230, 54380}, {403, 23347}, {427, 16933}, {804, 23350}, {868, 1503}, {1550, 52451}, {1634, 66078}, {1637, 52469}, {1651, 47296}, {1995, 45030}, {2794, 56967}, {3258, 9717}, {3839, 9214}, {4235, 47000}, {5094, 9756}, {5467, 57603}, {5502, 16319}, {7418, 43460}, {7577, 34845}, {8371, 63768}, {11005, 42738}, {13448, 46818}, {14611, 66119}, {15069, 36207}, {16303, 44228}, {17511, 33927}, {18494, 35372}, {30549, 44438}, {34212, 47105}, {42854, 63535}, {47146, 53319}, {47354, 57618}, {51431, 53568}, {53246, 56962}, {57598, 65728}

X(66167) = midpoint of X(36875) and X(52472)
X(66167) = reflection of X(i) in X(j) for these {i,j}: {5467, 57603}, {14559, 113}, {14995, 381}, {34810, 14356}, {52772, 5}, {53267, 868}
X(66167) = X(65776)-Ceva conjugate of X(523)
X(66167) = X(65756)-Dao conjugate of X(65973)
X(66167) = crosspoint of X(98) and X(5627)
X(66167) = crosssum of X(i) and X(j) for these (i,j): {511, 1511}, {18334, 39469}, {41172, 58345}
X(66167) = crossdifference of every pair of points on line {526, 3284}
X(66167) = barycentric product X(523)*X(65773)
X(66167) = barycentric quotient X(65773)/X(99)
X(66167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 36875, 52472}, {4, 52488, 52219}, {381, 18440, 15928}, {381, 34810, 14356}, {14356, 34810, 14995}, {56395, 57464, 6}

X(66168) = X(2)X(65624)∩X(4)X(32)

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

X(66168) lies on the cubic K1378, and these lines: {2, 65624}, {4, 32}, {230, 2409}, {297, 65771}, {393, 523}, {868, 16318}, {1990, 62551}, {3545, 36435}, {5523, 7422}, {35088, 56601}, {37987, 51937}, {43291, 57608}
X(66168) = X(65759)-Dao conjugate of X(66077)

X(66169) = X(2)X(66083)∩X(98)X(858)

Barycentrics    2*a^14*b^2 - 5*a^12*b^4 + a^10*b^6 + 8*a^8*b^8 - 12*a^6*b^10 + 11*a^4*b^12 - 7*a^2*b^14 + 2*b^16 + 2*a^14*c^2 - 14*a^12*b^2*c^2 + 27*a^10*b^4*c^2 - 29*a^8*b^6*c^2 + 28*a^6*b^8*c^2 - 28*a^4*b^10*c^2 + 23*a^2*b^12*c^2 - 9*b^14*c^2 - 5*a^12*c^4 + 27*a^10*b^2*c^4 - 22*a^8*b^4*c^4 + 2*a^6*b^6*c^4 + 13*a^4*b^8*c^4 - 25*a^2*b^10*c^4 + 18*b^12*c^4 + a^10*c^6 - 29*a^8*b^2*c^6 + 2*a^6*b^4*c^6 + 9*a^2*b^8*c^6 - 23*b^10*c^6 + 8*a^8*c^8 + 28*a^6*b^2*c^8 + 13*a^4*b^4*c^8 + 9*a^2*b^6*c^8 + 24*b^8*c^8 - 12*a^6*c^10 - 28*a^4*b^2*c^10 - 25*a^2*b^4*c^10 - 23*b^6*c^10 + 11*a^4*c^12 + 23*a^2*b^2*c^12 + 18*b^4*c^12 - 7*a^2*c^14 - 9*b^2*c^14 + 2*c^16 : :

X(66167) lies on the cubic K1378, and these lines: {2, 66083}, {98, 858}, {115, 65622}, {523, 54395}, {524, 1989}, {868, 3564}, {1637, 3580}, {2394, 47286}, {2407, 16310}, {47159, 47348}, {62338, 62551}, {65719, 65765}, {65767, 65774}

X(66169) = reflection of X(i) in X(j) for these {i,j}: {2407, 16310}, {62338, 62551}
X(66169) = anticomplement of X(66083)
X(66169) = X(65768)-anticomplementary conjugate of X(17217)
X(66169) = X(65764)-Dao conjugate of X(66082)

X(66170) = X(3)X(7777)∩X(4)X(187)

Barycentrics    5*a^8 - 11*a^6*b^2 + 8*a^4*b^4 - a^2*b^6 - b^8 - 11*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + 3*b^6*c^2 + 8*a^4*c^4 + a^2*b^2*c^4 - 4*b^4*c^4 - a^2*c^6 + 3*b^2*c^6 - c^8 : :
2 X[3] + X[14712], 5 X[3] - 2 X[66092], 5 X[14712] + 4 X[66092], X[4] - 4 X[187], 2 X[15] + X[36995], 2 X[16] + X[36993], X[20] + 2 X[2080], 2 X[20] + X[43453], 4 X[2080] - X[43453], X[98] + 2 X[6781], 2 X[316] - 5 X[631], X[316] - 4 X[47113], 5 X[631] - 8 X[47113], X[376] + 2 X[51224], 2 X[385] + X[13172], 4 X[548] - X[47618], and many others

on lines {2, 38225}, {3, 7777}, {4, 187}, {13, 33388}, {14, 33389}, {15, 36995}, {16, 36993}, {20, 2080}, {30, 8859}, {98, 6781}, {99, 5965}, {141, 35950}, {315, 61126}, {316, 631}, {376, 511}, {378, 54091}, {381, 38230}, {385, 13172}, {420, 35282}, {548, 47618}, {550, 9301}, {621, 13349}, {622, 13350}, {625, 3525}, {691, 1141}, {754, 21166}, {944, 5184}, {1285, 1692}, {1352, 35951}, {1503, 2076}, {1691, 10788}, {2459, 3069}, {2460, 3068}, {2549, 10631}, {2782, 33265}, {3090, 13449}, {3091, 14693}, {3153, 57307}, {3398, 33260}, {3522, 35002}, {3524, 3849}, {3528, 18860}, {3534, 9755}, {3545, 26613}, {3564, 8598}, {3972, 38317}, {5023, 37446}, {5050, 35955}, {5067, 58448}, {5085, 60653}, {5104, 6776}, {5162, 36998}, {5189, 38611}, {5207, 14907}, {5215, 61899}, {5603, 38221}, {5667, 10295}, {6321, 63047}, {6658, 10104}, {7487, 58309}, {7612, 15682}, {7684, 44015}, {7685, 44016}, {7697, 19686}, {7779, 33813}, {7809, 38748}, {7812, 9734}, {8356, 38110}, {8588, 43461}, {9855, 12243}, {10150, 61868}, {10242, 34127}, {10359, 32965}, {12022, 54082}, {12176, 29317}, {13083, 21158}, {13084, 21159}, {14561, 57633}, {14830, 43532}, {15702, 31173}, {22521, 35006}, {22712, 47101}, {26869, 35941}, {31275, 61867}, {32447, 34604}, {32762, 35921}, {34623, 35927}, {35383, 62174}, {35937, 61690}, {35944, 62987}, {35945, 62986}, {38741, 40236}, {39561, 52691}, {39647, 46034}, {39656, 53023}, {39872, 48892}, {40246, 49102}, {46264, 52994}

X(66170) = reflection of X(i) in X(j) for these {i,j}: {2, 38225}, {4, 38227}, {381, 38230}, {3153, 57307}, {3545, 26613}, {5603, 38221}, {7809, 38748}, {10242, 34127}, {14651, 21445}, {14853, 1691}, {38227, 187}, {59397, 39555}, {59398, 39554}, {62174, 35383}
X(66170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 2080, 43453}, {316, 47113, 631}

X(66171) = X(5)X(147)∩X(76)X(65518)

Barycentrics    a^14 - 2*a^10*b^4 - a^8*b^6 + a^6*b^8 + 2*a^4*b^10 - b^14 - 2*a^10*b^2*c^2 + 5*a^8*b^4*c^2 - 6*a^6*b^6*c^2 - a^4*b^8*c^2 + 4*a^2*b^10*c^2 - 2*a^10*c^4 + 5*a^8*b^2*c^4 + 7*a^6*b^4*c^4 - a^4*b^6*c^4 + 3*a^2*b^8*c^4 + 6*b^10*c^4 - a^8*c^6 - 6*a^6*b^2*c^6 - a^4*b^4*c^6 - 14*a^2*b^6*c^6 - 5*b^8*c^6 + a^6*c^8 - a^4*b^2*c^8 + 3*a^2*b^4*c^8 - 5*b^6*c^8 + 2*a^4*c^10 + 4*a^2*b^2*c^10 + 6*b^4*c^10 - c^14 : :

X(66171) lies on these lines: {5, 147}, {76, 65518}, {1141, 7953}, {3153, 43453}, {3518, 34131}, {3818, 5890}, {5309, 50718}, {5667, 7576}, {14880, 58805}, {34864, 46226}, {37943, 42426}

X(66172) = X(30)X(568)∩X(53)X(403)

Barycentrics    a^14*b^2 - 5*a^12*b^4 + 10*a^10*b^6 - 10*a^8*b^8 + 5*a^6*b^10 - a^4*b^12 + a^14*c^2 - 5*a^12*b^2*c^2 + 8*a^10*b^4*c^2 - 2*a^8*b^6*c^2 - 6*a^6*b^8*c^2 + 4*a^4*b^10*c^2 + a^2*b^12*c^2 - b^14*c^2 - 5*a^12*c^4 + 8*a^10*b^2*c^4 + a^8*b^4*c^4 + a^6*b^6*c^4 - 8*a^4*b^8*c^4 - 3*a^2*b^10*c^4 + 6*b^12*c^4 + 10*a^10*c^6 - 2*a^8*b^2*c^6 + a^6*b^4*c^6 + 10*a^4*b^6*c^6 + 2*a^2*b^8*c^6 - 15*b^10*c^6 - 10*a^8*c^8 - 6*a^6*b^2*c^8 - 8*a^4*b^4*c^8 + 2*a^2*b^6*c^8 + 20*b^8*c^8 + 5*a^6*c^10 + 4*a^4*b^2*c^10 - 3*a^2*b^4*c^10 - 15*b^6*c^10 - a^4*c^12 + a^2*b^2*c^12 + 6*b^4*c^12 - b^2*c^14 : :
X(66172) = X[41202] - 4 X[51888], 5 X[3567] - 8 X[47153], 4 X[7575] - X[15112]

X(66172) lies on these lines: {30, 568}, {53, 403}, {54, 41202}, {112, 32439}, {186, 2052}, {476, 1141}, {2790, 14157}, {3153, 56302}, {3567, 47153}, {5667, 13619}, {7575, 15112}, {14644, 32428}, {14651, 62490}, {15912, 32339}, {16237, 62345}, {54927, 60130}

X(66172) = reflection of X(15111) in X(186)

X(66173) = X(30)X(14644)∩X(50)X(112)

Barycentrics    7*a^16 - 20*a^14*b^2 + 6*a^12*b^4 + 27*a^10*b^6 - 15*a^8*b^8 - 22*a^6*b^10 + 20*a^4*b^12 - a^2*b^14 - 2*b^16 - 20*a^14*c^2 + 66*a^12*b^2*c^2 - 59*a^10*b^4*c^2 - 34*a^8*b^6*c^2 + 87*a^6*b^8*c^2 - 37*a^4*b^10*c^2 - 12*a^2*b^12*c^2 + 9*b^14*c^2 + 6*a^12*c^4 - 59*a^10*b^2*c^4 + 117*a^8*b^4*c^4 - 65*a^6*b^6*c^4 - 27*a^4*b^8*c^4 + 42*a^2*b^10*c^4 - 14*b^12*c^4 + 27*a^10*c^6 - 34*a^8*b^2*c^6 - 65*a^6*b^4*c^6 + 88*a^4*b^6*c^6 - 29*a^2*b^8*c^6 + 7*b^10*c^6 - 15*a^8*c^8 + 87*a^6*b^2*c^8 - 27*a^4*b^4*c^8 - 29*a^2*b^6*c^8 - 22*a^6*c^10 - 37*a^4*b^2*c^10 + 42*a^2*b^4*c^10 + 7*b^6*c^10 + 20*a^4*c^12 - 12*a^2*b^2*c^12 - 14*b^4*c^12 - a^2*c^14 + 9*b^2*c^14 - 2*c^16 : :
X(66173) = X[477] - 4 X[10295], 4 X[7575] - X[44967], X[10721] - 4 X[47351], X[14989] - 4 X[47327], X[14989] + 2 X[56369], 2 X[47327] + X[56369]

X(66173) lies opn these lines: {30, 14644}, {50, 112}, {1157, 19651}, {3534, 9159}, {6761, 13619}, {7575, 44967}, {10721, 47351}, {14989, 16080}

X(66173) = {X(47327),X(56369)}-harmonic conjugate of X(14989)

X(66174) = X(5)X(38585)∩X(112)X(233)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(a^18 - 4*a^16*b^2 + 3*a^14*b^4 + 9*a^12*b^6 - 19*a^10*b^8 + 9*a^8*b^10 + 9*a^6*b^12 - 13*a^4*b^14 + 6*a^2*b^16 - b^18 - 4*a^16*c^2 + 8*a^14*b^2*c^2 + 7*a^12*b^4*c^2 - 26*a^10*b^6*c^2 + 20*a^8*b^8*c^2 - 16*a^6*b^10*c^2 + 27*a^4*b^12*c^2 - 22*a^2*b^14*c^2 + 6*b^16*c^2 + 3*a^14*c^4 + 7*a^12*b^2*c^4 - 21*a^10*b^4*c^4 + 7*a^8*b^6*c^4 + 5*a^6*b^8*c^4 - 15*a^4*b^10*c^4 + 29*a^2*b^12*c^4 - 15*b^14*c^4 + 9*a^12*c^6 - 26*a^10*b^2*c^6 + 7*a^8*b^4*c^6 + 4*a^6*b^6*c^6 + a^4*b^8*c^6 - 14*a^2*b^10*c^6 + 19*b^12*c^6 - 19*a^10*c^8 + 20*a^8*b^2*c^8 + 5*a^6*b^4*c^8 + a^4*b^6*c^8 + 2*a^2*b^8*c^8 - 9*b^10*c^8 + 9*a^8*c^10 - 16*a^6*b^2*c^10 - 15*a^4*b^4*c^10 - 14*a^2*b^6*c^10 - 9*b^8*c^10 + 9*a^6*c^12 + 27*a^4*b^2*c^12 + 29*a^2*b^4*c^12 + 19*b^6*c^12 - 13*a^4*c^14 - 22*a^2*b^2*c^14 - 15*b^4*c^14 + 6*a^2*c^16 + 6*b^2*c^16 - c^18) : :

X(66174) lies on the cubic K067 and these lines: {5, 38585}, {112, 233}, {1157, 37943}, {1173, 3574}, {3518, 14656}, {10745, 42441}
X(66174) = X(5)-Ceva conjugate of X(37943)

X(66175) = X(30)X(112)∩X(98)X(186)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^16 - 2*a^14*b^2 + a^10*b^6 + a^8*b^8 - 2*a^4*b^12 + a^2*b^14 - 2*a^14*c^2 + 2*a^12*b^2*c^2 + 3*a^10*b^4*c^2 - 2*a^8*b^6*c^2 - 3*a^6*b^8*c^2 + 3*a^4*b^10*c^2 - 2*a^2*b^12*c^2 + b^14*c^2 + 3*a^10*b^2*c^4 - 9*a^8*b^4*c^4 + 5*a^6*b^6*c^4 - a^4*b^8*c^4 + 4*a^2*b^10*c^4 - 2*b^12*c^4 + a^10*c^6 - 2*a^8*b^2*c^6 + 5*a^6*b^4*c^6 - 3*a^2*b^8*c^6 - b^10*c^6 + a^8*c^8 - 3*a^6*b^2*c^8 - a^4*b^4*c^8 - 3*a^2*b^6*c^8 + 4*b^8*c^8 + 3*a^4*b^2*c^10 + 4*a^2*b^4*c^10 - b^6*c^10 - 2*a^4*c^12 - 2*a^2*b^2*c^12 - 2*b^4*c^12 + a^2*c^14 + b^2*c^14) : :
X(66175) = X[935] + 2 X[41377], 2 X[37938] - 3 X[57346], 3 X[37943] - 2 X[42426], 4 X[44234] - 3 X[57319]

X(66175) lies on the cubic K937 and these lines: {4, 33695}, {23, 53769}, {30, 112}, {98, 186}, {403, 1503}, {1899, 36191}, {3153, 38971}, {5012, 57583}, {5890, 18338}, {5938, 37917}, {10151, 51940}, {10295, 47242}, {14591, 35912}, {14880, 39575}, {23293, 43389}, {37938, 57346}, {37943, 42426}, {40118, 53875}, {44234, 57319}, {54632, 60133}

X(66175) = midpoint of X(186) and X(41377)
X(66175) = reflection of X(i) in X(j) for these {i,j}: {935, 186}, {3153, 38971}, {51940, 10151}

X(66176) = X(2)X(112)∩X(403)X(44375)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12 - a^10*b^2 - 3*a^8*b^4 + 4*a^6*b^6 + a^4*b^8 - 3*a^2*b^10 + b^12 - a^10*c^2 - a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - b^10*c^2 - 3*a^8*c^4 + 4*a^6*b^2*c^4 + 2*a^4*b^4*c^4 - 2*a^2*b^6*c^4 - b^8*c^4 + 4*a^6*c^6 - 6*a^4*b^2*c^6 - 2*a^2*b^4*c^6 + 2*b^6*c^6 + a^4*c^8 + 5*a^2*b^2*c^8 - b^4*c^8 - 3*a^2*c^10 - b^2*c^10 + c^12) : :

X(66176) lies on the cubic K938 and these lines: {2, 112}, {403, 44375}, {1993, 61203}, {2967, 18531}, {3563, 18533}, {5207, 22151}, {14644, 14853}, {17035, 50718}, {41676, 65518}

X(66177) = X(3)X(112)∩X(132)X(52295)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^12*b^2 - 2*a^10*b^4 - a^8*b^6 + 4*a^6*b^8 - a^4*b^10 - 2*a^2*b^12 + b^14 + a^12*c^2 - a^10*b^2*c^2 - 3*a^8*b^4*c^2 + 5*a^6*b^6*c^2 - 2*a^4*b^8*c^2 - 2*a^10*c^4 - 3*a^8*b^2*c^4 + 5*a^6*b^4*c^4 - 3*a^4*b^6*c^4 + a^2*b^8*c^4 + 2*b^10*c^4 - a^8*c^6 + 5*a^6*b^2*c^6 - 3*a^4*b^4*c^6 + 2*a^2*b^6*c^6 - 3*b^8*c^6 + 4*a^6*c^8 - 2*a^4*b^2*c^8 + a^2*b^4*c^8 - 3*b^6*c^8 - a^4*c^10 + 2*b^4*c^10 - 2*a^2*c^12 + c^14) : :
X(66177) = X[112] + 2 X[53772]

X(66177) lies on the cubic K939 and these lines: {3, 112}, {132, 52295}, {262, 61451}, {1157, 19189}, {2781, 14644}, {2794, 18559}, {6761, 46450}, {10986, 66135}, {14983, 44288}, {43678, 54705}

X(66178) = X(3)X(252)∩X(137)X(52295)

Barycentrics    a^20*b^2 - 6*a^18*b^4 + 14*a^16*b^6 - 14*a^14*b^8 + 14*a^10*b^12 - 14*a^8*b^14 + 6*a^6*b^16 - a^4*b^18 + a^20*c^2 - 11*a^18*b^2*c^2 + 36*a^16*b^4*c^2 - 54*a^14*b^6*c^2 + 46*a^12*b^8*c^2 - 35*a^10*b^10*c^2 + 31*a^8*b^12*c^2 - 16*a^6*b^14*c^2 - a^4*b^16*c^2 + 4*a^2*b^18*c^2 - b^20*c^2 - 6*a^18*c^4 + 36*a^16*b^2*c^4 - 66*a^14*b^4*c^4 + 47*a^12*b^6*c^4 - 6*a^10*b^8*c^4 - 22*a^8*b^10*c^4 + 27*a^6*b^12*c^4 - 17*a^2*b^16*c^4 + 7*b^18*c^4 + 14*a^16*c^6 - 54*a^14*b^2*c^6 + 47*a^12*b^4*c^6 - 9*a^10*b^6*c^6 + 5*a^8*b^8*c^6 - 21*a^6*b^10*c^6 + 16*a^4*b^12*c^6 + 22*a^2*b^14*c^6 - 20*b^16*c^6 - 14*a^14*c^8 + 46*a^12*b^2*c^8 - 6*a^10*b^4*c^8 + 5*a^8*b^6*c^8 + 8*a^6*b^8*c^8 - 14*a^4*b^10*c^8 + a^2*b^12*c^8 + 28*b^14*c^8 - 35*a^10*b^2*c^10 - 22*a^8*b^4*c^10 - 21*a^6*b^6*c^10 - 14*a^4*b^8*c^10 - 20*a^2*b^10*c^10 - 14*b^12*c^10 + 14*a^10*c^12 + 31*a^8*b^2*c^12 + 27*a^6*b^4*c^12 + 16*a^4*b^6*c^12 + a^2*b^8*c^12 - 14*b^10*c^12 - 14*a^8*c^14 - 16*a^6*b^2*c^14 + 22*a^2*b^6*c^14 + 28*b^8*c^14 + 6*a^6*c^16 - a^4*b^2*c^16 - 17*a^2*b^4*c^16 - 20*b^6*c^16 - a^4*c^18 + 4*a^2*b^2*c^18 + 7*b^4*c^18 - b^2*c^20 : :

X(66178 lies on these lines: {3, 252}, {137, 52295}, {1157, 32428}, {5890, 32423}, {9381, 34418}, {11423, 27423}, {18353, 50718}, {25147, 39504}

X(66179) = X(2)X(53808)∩X(4)X(137)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^18 - 8*a^16*b^2 + 25*a^14*b^4 - 39*a^12*b^6 + 31*a^10*b^8 - 11*a^8*b^10 + 3*a^6*b^12 - 5*a^4*b^14 + 4*a^2*b^16 - b^18 - 8*a^16*c^2 + 36*a^14*b^2*c^2 - 61*a^12*b^4*c^2 + 48*a^10*b^6*c^2 - 18*a^8*b^8*c^2 + 11*a^4*b^12*c^2 - 12*a^2*b^14*c^2 + 4*b^16*c^2 + 25*a^14*c^4 - 61*a^12*b^2*c^4 + 43*a^10*b^4*c^4 - 7*a^8*b^6*c^4 + 3*a^6*b^8*c^4 - 7*a^4*b^10*c^4 + 9*a^2*b^12*c^4 - 5*b^14*c^4 - 39*a^12*c^6 + 48*a^10*b^2*c^6 - 7*a^8*b^4*c^6 - 12*a^6*b^6*c^6 + a^4*b^8*c^6 + 8*a^2*b^10*c^6 + b^12*c^6 + 31*a^10*c^8 - 18*a^8*b^2*c^8 + 3*a^6*b^4*c^8 + a^4*b^6*c^8 - 18*a^2*b^8*c^8 + b^10*c^8 - 11*a^8*c^10 - 7*a^4*b^4*c^10 + 8*a^2*b^6*c^10 + b^8*c^10 + 3*a^6*c^12 + 11*a^4*b^2*c^12 + 9*a^2*b^4*c^12 + b^6*c^12 - 5*a^4*c^14 - 12*a^2*b^2*c^14 - 5*b^4*c^14 + 4*a^2*c^16 + 4*b^2*c^16 - c^18) : :
X(66179) = 3 X[57317] - X[57369], X[4] + 2 X[933], X[4] - 4 X[18402], 5 X[4] - 2 X[44977], X[933] + 2 X[18402], 5 X[933] + X[44977], 10 X[18402] - X[44977], 2 X[5] + X[38585], X[20] - 4 X[38616], 5 X[631] - 2 X[18401], 7 X[3090] - 4 X[20625]

X(66179) lies on these lines: {2, 53808}, {3, 3462}, {4, 137}, {5, 38585}, {20, 38616}, {631, 18401}, {3090, 20625}, {3518, 54067}, {5890, 10628}, {13599, 64256}, {14940, 64257}, {39849, 61203}

X(66179) = reflection of X(2) in X(57317)
X(66179) = anticomplement of X(57369)
X(66179) = {X(933),X(18402)}-harmonic conjugate of X(4)

X(66180) = CENTER OF THE 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Trilinears    sqrt(3)*cos(A) - 2*(cos((A-2*Pi)/3) + 2*cos((B-2*Pi)/3)*cos((C - 2*Pi)/3))*(cos(2*A/3 - 7*Pi/6) + cos(2*B/3 - 7*Pi/6) + cos(2*C/3 - 7*Pi/6)) : :

See César Lozada, euclid 7201.

X(66180) lies on these lines: {3, 3276}, {4, 65155}

X(66180) = reflection of X(3276) in X(41109)
X(66180) = Cundy-Parry-Psi-transform of the anticomplement of X(41111)

X(66181) = CENTER OF THE 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Trilinears    sqrt(3)*cos(A) - 2*(cos((A-4*Pi)/3) + 2*cos((B-4*Pi)/3)*cos((C - 4*Pi)/3))*(cos(2*A/3 - 15*Pi/6) + cos(2*B/3 - 15*Pi/6) + cos(2*C/3 - 15*Pi/6)) : :

See César Lozada, euclid 7201.

X(66181) lies on this line: {3, 3277}

X(66181) = reflection of X(3277) in X(41110)
X(66181) = Cundy-Parry-Psi-transform of the anticomplement of X(41109)

X(66182) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 2nd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Trilinears    cos(A)+(cos((A-2*Pi)/3)+2*cos((B-2*Pi)/3)*cos((C-2*Pi)/3))*(1-2*cos((2*A-3*Pi)/3)-2*cos((2*B-3*Pi)/3)-2*cos((2*C-3*Pi)/3)) : :

See César Lozada, euclid 7201.

X(66182) lies on this line: {3, 3276}

X(66183) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 3rd HATZIPOLAKIS-VAN TIENHOVEN EQUILATERAL TRIANGLE

Trilinears    cos(A)+(cos((A-4*Pi)/3)+2*cos((B-4*Pi)/3)*cos((C-4*Pi)/3))*(1-2*cos((2*A-7*Pi)/3)-2*cos((2*B-7*Pi)/3)-2*cos((2*C-7*Pi)/3)) : :

See César Lozada, euclid 7201.

X(66183) lies on this line: {3, 3277}

X(66184) = X(115)X(826)∩X(230)X(9482)

Barycentrics    (b - c)^2*(b + c)^2*(b^2 + c^2)*(a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2) : :
X(66184) = 2 X[115] + X[62417], 4 X[230] - X[9482]

X(66184) lies on the cubic K1381 and these lines: {115, 826}, {230, 9482}, {262, 59273}, {688, 6784}, {732, 7840}, {1971, 39095}, {3124, 7668}, {4577, 13519}, {9019, 11673}, {9300, 11205}, {14416, 14424}, {20582, 45672}

X(66184) = X(i)-Ceva conjugate of X(j) for these (i,j): {262, 3005}, {59258, 23285}
X(66184) = X(4599)-isoconjugate of X(43357)
X(66184) = X(i)-Dao conjugate of X(j) for these (i,j): {3124, 43357}, {54263, 24273}
X(66184) = crosssum of X(i) and X(j) for these (i,j): {827, 41295}, {43357, 59262}
X(66184) = crossdifference of every pair of points on line {827, 43357}
X(66184) = barycentric product X(i)*X(j) for these {i,j}: {115, 10007}, {262, 55051}, {3329, 39691}, {14318, 23285}, {15449, 60860}, {59249, 62417}
X(66184) = barycentric quotient X(i)/X(j) for these {i,j}: {3005, 43357}, {10007, 4590}, {14318, 827}, {39691, 42006}, {55051, 183}, {60860, 57545}, {62417, 59262}

X(66185) = X(115)X(3906)∩X(141)X(12036)

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

X(66185) lies on the cubic K1381 and these lines: {115, 3906}, {141, 12036}, {1648, 6784}, {3815, 5642}, {6786, 17430}, {7777, 58854}, {8288, 17416}, {8704, 12494}

X(66185) = tripolar centroid of X(34246)
X(66185) = X(i)-Ceva conjugate of X(j) for these (i,j): {60240, 62568}, {64973, 8704}
X(66185) = X(i)-Dao conjugate of X(j) for these (i,j): {8704, 64973}, {17413, 6233}, {17436, 11167}
X(66185) = crosspoint of X(8704) and X(64973)
X(66185) = crossdifference of every pair of points on line {6233, 11636}
X(66185) = barycentric product X(i)*X(j) for these {i,j}: {115, 64942}, {3906, 8704}, {8288, 11163}, {17416, 64973}
X(66185) = barycentric quotient X(i)/X(j) for these {i,j}: {8288, 11167}, {8704, 35138}, {11186, 11636}, {17414, 6233}, {64942, 4590}

X(66186) = X(115)X(46462)∩X(523)X(13722)

Barycentrics    (b - c)^2*(b + c)^2*(-4*a^12 + 9*a^10*b^2 - 3*a^8*b^4 - 2*a^6*b^6 - 3*a^2*b^10 + 2*b^12 + 9*a^10*c^2 - 24*a^8*b^2*c^2 + 12*a^6*b^4*c^2 + 12*a^2*b^8*c^2 - 6*b^10*c^2 - 3*a^8*c^4 + 12*a^6*b^2*c^4 - 12*a^2*b^6*c^4 + 3*b^8*c^4 - 2*a^6*c^6 - 12*a^2*b^4*c^6 + 4*b^6*c^6 + 12*a^2*b^2*c^8 + 3*b^4*c^8 - 3*a^2*c^10 - 6*b^2*c^10 + 2*c^12 + Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(4*a^10 - 7*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 + 2*a^2*b^8 - 2*b^10 - 7*a^8*c^2 + 20*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 10*a^2*b^6*c^2 + 5*b^8*c^2 - 2*a^6*c^4 - 6*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 10*a^2*b^2*c^6 - 2*b^4*c^6 + 2*a^2*c^8 + 5*b^2*c^8 - 2*c^10)) : :
X(66186) = 3 X[13722] - 2 X[39022], 4 X[39022] - 3 X[46463]

X(66186) lies on the cubic K1381 and these lines: {115, 46462}, {523, 13722}, {671, 3413}, {1648, 8029}, {1989, 5638}, {13636, 64258}

X(66186) = reflection of X(46463) in X(13722)
X(66186) = tripolar centroid of X(13636)
X(66186) = X(i)-Ceva conjugate of X(j) for these (i,j): {3413, 13636}, {13722, 115}
X(66186) = X(i)-isoconjugate of X(j) for these (i,j): {1101, 6190}, {1379, 24041}
X(66186) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 6190}, {3005, 1379}, {13636, 57576}, {13722, 99}, {39023, 4590}, {39068, 249}, {62561, 31614}
X(66186) = crosspoint of X(i) and X(j) for these (i,j): {115, 13722}, {523, 39023}, {3413, 13636}, {30508, 62640}
X(66186) = crossdifference of every pair of points on line {249, 1380}
X(66186) = barycentric product X(i)*X(j) for these {i,j}: {115, 3413}, {338, 5638}, {523, 13636}, {1380, 23105}, {5466, 46462}, {6189, 8029}, {13722, 39023}, {52722, 64258}
X(66186) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 6190}, {1380, 59152}, {3124, 1379}, {3413, 4590}, {5638, 249}, {6189, 31614}, {8029, 3414}, {8754, 57014}, {13636, 99}, {13722, 57576}, {22260, 5639}, {33919, 52723}, {42344, 46463}, {46462, 5468}, {57013, 55270}, {61339, 13722}

X(66187) = X(115)X(46463)∩X(523)X(13636)

Barycentrics    (b - c)^2*(b + c)^2*(-4*a^12 + 9*a^10*b^2 - 3*a^8*b^4 - 2*a^6*b^6 - 3*a^2*b^10 + 2*b^12 + 9*a^10*c^2 - 24*a^8*b^2*c^2 + 12*a^6*b^4*c^2 + 12*a^2*b^8*c^2 - 6*b^10*c^2 - 3*a^8*c^4 + 12*a^6*b^2*c^4 - 12*a^2*b^6*c^4 + 3*b^8*c^4 - 2*a^6*c^6 - 12*a^2*b^4*c^6 + 4*b^6*c^6 + 12*a^2*b^2*c^8 + 3*b^4*c^8 - 3*a^2*c^10 - 6*b^2*c^10 + 2*c^12 - Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*(4*a^10 - 7*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 + 2*a^2*b^8 - 2*b^10 - 7*a^8*c^2 + 20*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 10*a^2*b^6*c^2 + 5*b^8*c^2 - 2*a^6*c^4 - 6*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 10*a^2*b^2*c^6 - 2*b^4*c^6 + 2*a^2*c^8 + 5*b^2*c^8 - 2*c^10)) : :
X(66187) = 3 X[13636] - 2 X[39023], 4 X[39023] - 3 X[46462]

X(66187) lies on the cubic K1381 and these lines: {115, 46463}, {523, 13636}, {671, 3414}, {1648, 8029}, {1989, 5639}, {13722, 64258}

X(66187) = reflection of X(46462) in X(13636)
X(66187) = tripolar centroid of X(13722)
X(66187) = X(i)-Ceva conjugate of X(j) for these (i,j): {3414, 13722}, {13636, 115}
X(66187) = X(i)-isoconjugate of X(j) for these (i,j): {1101, 6189}, {1380, 24041}
X(66187) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 6189}, {3005, 1380}, {13636, 99}, {13722, 57575}, {39022, 4590}, {39067, 249}, {62560, 31614}
X(66187) = crosspoint of X(i) and X(j) for these (i,j): {115, 13636}, {523, 39022}, {3414, 13722}, {30509, 62641}
X(66187) = crossdifference of every pair of points on line {249, 1379}
X(66187) = barycentric product X(i)*X(j) for these {i,j}: {115, 3414}, {338, 5639}, {523, 13722}, {1379, 23105}, {5466, 46463}, {6190, 8029}, {13636, 39022}, {52723, 64258}
X(66187) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 6189}, {1379, 59152}, {3124, 1380}, {3414, 4590}, {5639, 249}, {6190, 31614}, {8029, 3413}, {8754, 57013}, {13636, 57575}, {13722, 99}, {22260, 5638}, {33919, 52722}, {42344, 46462}, {46463, 5468}, {57014, 55270}, {61339, 13636}

X(66188) = X(101)X(40750)∩X(115)X(513)

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

X(66188) lies on the cubic K1381 and these lines: {101, 40750}, {115, 513}, {116, 38963}, {244, 665}, {838, 6784}, {3833, 61708}, {4893, 20982}, {5540, 24512}

X(66188) = tripolar centroid of X(60043)
X(66188) = X(59265)-Ceva conjugate of X(513)
X(66188) = X(765)-isoconjugate of X(59265)
X(66188) = X(513)-Dao conjugate of X(59265)
X(66188) = crosspoint of X(513) and X(59265)
X(66188) = crosssum of X(100) and X(59235)
X(66188) = barycentric product X(i)*X(j) for these {i,j}: {244, 51285}, {1086, 59235}
X(66188) = barycentric quotient X(i)/X(j) for these {i,j}: {1015, 59265}, {51285, 7035}, {59235, 1016}

X(66189) = X(11)X(244)∩X(115)X(514)

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

X(66189) lies on the cubic K1381 and these lines: {11, 244}, {115, 514}, {594, 3807}, {834, 6784}, {1213, 25383}, {4415, 27493}, {6627, 45661}, {7277, 24712}, {10707, 17726}, {14041, 25434}, {14568, 25432}, {17395, 17722}, {22110, 28530}, {27081, 30566}, {30997, 48632}, {33228, 35101}

X(66189) = tripolar centroid of X(60042)
X(66189) = X(59267)-Ceva conjugate of X(514)
X(66189) = X(1110)-isoconjugate of X(59267)
X(66189) = X(i)-Dao conjugate of X(j) for these (i,j): {514, 59267}, {54256, 24342}
X(66189) = crosspoint of X(514) and X(59267)
X(66189) = crosssum of X(101) and X(59238)
X(66189) = barycentric product X(i)*X(j) for these {i,j}: {1086, 51353}, {1111, 51294}, {23989, 59238}
X(66189) = barycentric quotient X(i)/X(j) for these {i,j}: {1086, 59267}, {51294, 765}, {51353, 1016}, {59238, 1252}

X(66190) = X(32)X(10694)∩X(115)X(804)

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

X(66190) lies on the cubic K1381 and these lines: {32, 10694}, {115, 804}, {597, 732}, {2086, 11183}, {3329, 60707}, {9468, 22735}

X(66190) = tripolar centroid of X(58784)
X(66190) = X(39685)-Ceva conjugate of X(14318)
X(66190) = X(37134)-isoconjugate of X(43357)
X(66190) = X(62649)-Dao conjugate of X(60667)
X(66190) = crossdifference of every pair of points on line {805, 1634}
X(66190) = barycentric product X(i)*X(j) for these {i,j}: {115, 64947}, {2086, 60707}, {2679, 39685}, {14295, 14318}, {41178, 59249}, {56976, 66184}
X(66190) = barycentric quotient X(i)/X(j) for these {i,j}: {2086, 60667}, {3329, 39292}, {5027, 43357}, {14318, 805}, {41178, 59262}, {64947, 4590}, {66184, 56977}

X(66191) = X(115)X(116)∩X(28602)X(35080)

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

X(66191) lies on the cubic K1381 and these lines: {115, 116}, {28602, 35080}, {60708, 60710}

X(66191) = tripolar centroid of X(4608)
X(66191) = X(37135)-isoconjugate of X(59080)
X(66191) = X(27929)-Dao conjugate of X(60669)
X(66191) = crossdifference of every pair of points on line {2702, 35327}
X(66191) = barycentric quotient X(5029)/X(59080)

X(66192) = X(115)X(127)∩X(381)X(5111)

Barycentrics    (b - c)^2*(b + c)^2*(-a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2)*(-(a^2*b^2) + b^4 - a^2*c^2 + c^4) : :
X(66192) = X[41172] + 2 X[62431]

X(66192) lies on the cubic K1381 and these lines: {115, 127}, {183, 458}, {327, 3314}, {381, 511}, {868, 35088}, {2373, 62512}, {6784, 45321}, {7778, 52251}, {7913, 34349}, {11168, 34094}, {22110, 45330}, {34765, 46245}, {36212, 65975}

X(66192) = tripolar centroid of X(850)
X(66192) = X(i)-Ceva conjugate of X(j) for these (i,j): {327, 41167}, {46806, 23878}, {46807, 2799}
X(66192) = X(i)-isoconjugate of X(j) for these (i,j): {110, 36132}, {163, 6037}, {662, 32716}, {2186, 57742}, {2715, 65252}, {3402, 57991}, {26714, 36084}, {36104, 65310}
X(66192) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 6037}, {244, 36132}, {1084, 32716}, {2799, 46807}, {23878, 46806}, {33569, 182}, {35088, 65271}, {36901, 53196}, {38970, 65349}, {38987, 26714}, {38997, 2715}, {39000, 65310}, {39009, 110}, {41167, 43718}, {41172, 63741}, {51580, 57991}, {54267, 5999}, {55267, 262}, {62596, 14966}
X(66192) = crosspoint of X(i) and X(j) for these (i,j): {2799, 46807}, {23878, 46806}, {44144, 63746}
X(66192) = crosssum of X(i) and X(j) for these (i,j): {2715, 51542}, {26714, 51543}
X(66192) = crossdifference of every pair of points on line {1576, 2715}
X(66192) = barycentric product X(i)*X(j) for these {i,j}: {115, 51373}, {182, 62431}, {183, 868}, {327, 62596}, {2799, 23878}, {9420, 44173}, {20023, 44114}, {34765, 45321}, {35088, 46806}, {41167, 63746}, {41172, 44144}, {51372, 65756}
X(66192) = barycentric quotient X(i)/X(j) for these {i,j}: {182, 57742}, {183, 57991}, {458, 60179}, {512, 32716}, {523, 6037}, {661, 36132}, {684, 65310}, {850, 53196}, {868, 262}, {2799, 65271}, {3288, 2715}, {3569, 26714}, {6784, 1976}, {9420, 1576}, {16230, 65349}, {23878, 2966}, {33569, 14966}, {35088, 46807}, {41167, 63741}, {41172, 43718}, {44114, 263}, {44144, 41174}, {45321, 34761}, {46806, 57562}, {51373, 4590}, {58260, 46319}, {59804, 51542}, {59805, 51543}, {62431, 327}, {62596, 182}



leftri

Points releated to the 1st Van-Khea-Pavlov triangle: X(66193)-X(66259)

rightri

This preamble and centers X(66193)-X(66259) were contributed by Ivan Pavlov on Nov 13, 2024.

Let PaPbPc be the intouch triangle. Let Ab and Ac be the reflections of Pa in the midpoints of BPb and CPc. Let AbAc intersect PbPc at point Ta, and similarly define Tb and Tc. TaTbTc is homothetic to the excenters-midpoints triangle with center X(55) and ratio r/R. It is bilogic to the following triangles: ABC, Garcia-reflection, 1st Pavlov, extouch-of-Fuhrmann.
We will call TaTbTc the 1st Van-Khea-Pavlov triangle. For more information see this Euclid thread.
Some of the properties below refer to CTR-triangles. More info on these series is in this catalog.

X(66193) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND ANTI-OUTER-YFF

Barycentrics    a*(a-b-c)*(a^11-a^10*(b+c)+(b-c)^6*(b+c)^5+a^9*(-5*b^2+2*b*c-5*c^2)-a^2*(b-c)^4*(b+c)^3*(5*b^2+6*b*c+5*c^2)+a^8*(5*b^3+7*b^2*c+7*b*c^2+5*c^3)-a*(b-c)^4*(b+c)^2*(b^4-4*b^3*c-2*b^2*c^2-4*b*c^3+c^4)+2*a^7*(5*b^4-6*b^3*c-2*b^2*c^2-6*b*c^3+5*c^4)-2*a^6*(5*b^5+7*b^4*c+4*b^3*c^2+4*b^2*c^3+7*b*c^4+5*c^5)+2*a^5*(-5*b^6+12*b^5*c+9*b^4*c^2+16*b^3*c^3+9*b^2*c^4+12*b*c^5-5*c^6)+a^3*(b-c)^2*(5*b^6-10*b^5*c-29*b^4*c^2-92*b^3*c^3-29*b^2*c^4-10*b*c^5+5*c^6)+2*a^4*(5*b^7+5*b^6*c-5*b^5*c^2-21*b^4*c^3-21*b^3*c^4-5*b^2*c^5+5*b*c^6+5*c^7)) : :

X(66193) lies on these lines: {1, 37514}, {40, 65987}, {55, 49171}, {950, 49169}, {1697, 12751}, {3359, 45639}, {6256, 31397}, {7162, 39692}, {10388, 66201}, {10629, 49163}, {10965, 49184}, {14100, 26358}, {60896, 60961}

X(66194) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND HEXYL

Barycentrics    a*(a-b-c)*(a^8-4*a^6*(b-c)^2+16*a^3*b^2*c^2*(b+c)+16*a*b^2*(b-c)^2*c^2*(b+c)-4*a^2*(b-c)^4*(b+c)^2+(b^2-c^2)^4+2*a^4*(3*b^4-8*b^3*c+18*b^2*c^2-8*b*c^3+3*c^4)) : :

X(66194) lies on these lines: {9, 2343}, {21, 66213}, {40, 950}, {56, 5732}, {84, 497}, {1479, 10042}, {1697, 20588}, {3057, 12629}, {3486, 7966}, {3601, 64154}, {5698, 10384}, {6264, 15558}, {6284, 10860}, {6762, 66197}, {7160, 66199}, {9898, 41229}, {11372, 12053}, {12705, 64320}, {12710, 64328}, {13996, 66206}, {15299, 66198}, {15803, 41853}, {16132, 61762}, {16141, 54302}, {16572, 66234}, {31435, 62333}, {41869, 45633}, {51785, 64740}, {63430, 66248}

X(66194) = pole of line {936, 1466} with respect to the Feuerbach hyperbola

X(66195) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND INCIRCLE-CIRCLES

Barycentrics    a*(a-b-c)*(b^5+8*a^3*b*c-2*a*b*(b-c)^2*c-b^3*c^2-b^2*c^3+c^5+a^4*(b+c)+a^2*(-2*b^3+5*b^2*c+5*b*c^2-2*c^3)) : :
X(66195) = -X[72]+3*X[17525], X[3868]+3*X[15678], -2*X[4662]+3*X[18253], -2*X[5044]+3*X[15673], -3*X[6675]+2*X[58658]

X(66195) lies on circumconic {{A, B, C, X(2287), X(10308)}} and on these lines: {1, 10308}, {9, 21}, {11, 6701}, {30, 553}, {35, 54190}, {55, 3647}, {57, 33557}, {72, 17525}, {79, 497}, {191, 9898}, {226, 37447}, {354, 51118}, {376, 10399}, {390, 3648}, {442, 9843}, {495, 22798}, {496, 49107}, {515, 66211}, {517, 66242}, {758, 3057}, {971, 63274}, {1058, 16116}, {1071, 64323}, {1210, 37401}, {1479, 11045}, {1697, 11684}, {1717, 63340}, {1725, 63356}, {1864, 15670}, {2475, 9776}, {2771, 12735}, {2801, 37080}, {3058, 3881}, {3065, 63288}, {3295, 3652}, {3303, 16140}, {3333, 63267}, {3486, 4302}, {3523, 61718}, {3529, 5902}, {3583, 58566}, {3649, 4890}, {3651, 10382}, {3743, 53524}, {3868, 15678}, {3873, 41864}, {3889, 60933}, {3982, 5045}, {4015, 4995}, {4294, 16113}, {4304, 15556}, {4313, 15677}, {4662, 18253}, {5044, 15673}, {5083, 63999}, {5218, 63286}, {5722, 47032}, {5883, 50239}, {5884, 6938}, {6001, 64282}, {6175, 9581}, {6284, 18977}, {6675, 58658}, {6767, 48668}, {6951, 37702}, {7354, 12564}, {7686, 66253}, {7701, 41546}, {9579, 11020}, {9965, 15680}, {10106, 12710}, {10176, 19526}, {10384, 16133}, {10385, 63278}, {10395, 44256}, {11034, 16118}, {11046, 16153}, {11047, 16154}, {11048, 16155}, {11220, 11518}, {11263, 66214}, {11281, 16120}, {12432, 15338}, {12512, 61663}, {12675, 64162}, {12680, 21628}, {12853, 41551}, {13145, 37730}, {13411, 16617}, {14450, 60926}, {14749, 50189}, {16117, 37545}, {16119, 16541}, {16132, 61762}, {16143, 66198}, {16148, 44623}, {16149, 44624}, {17768, 66210}, {18593, 48897}, {18839, 66207}, {20116, 52783}, {20117, 37571}, {24929, 31649}, {30384, 66254}, {33857, 34471}, {34195, 66197}, {37618, 66201}, {37724, 66019}, {44669, 66205}, {45230, 46816}, {45636, 49177}, {45637, 49178}, {50195, 66247}, {58380, 63332}, {59337, 63967}, {60911, 64342}, {60961, 63972}, {64160, 66248}, {64745, 66206}

X(66195) = midpoint of X(i) and X(j) for these {i,j}: {3555, 3650}, {6284, 18977}, {10543, 17637}, {15680, 39772}
X(66195) = reflection of X(i) in X(j) for these {i,j}: {11544, 5045}, {16120, 11281}, {40661, 21}
X(66195) = X(i)-Ceva conjugate of X(j) for these {i, j}: {63782, 650}
X(66195) = pole of line {3737, 4041} with respect to the incircle
X(66195) = pole of line {553, 1125} with respect to the Feuerbach hyperbola
X(66195) = pole of line {35057, 50346} with respect to the Suppa-Cucoanes circle
X(66195) = X(3519)-of-incircle-circles triangle triangle
X(66195) = X(3647)-of-Mandart-incircle triangle triangle
X(66195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1479, 16152, 16125}, {10543, 17637, 758}

X(66196) = TRIPOLE OF PERSPECTIVITY AXIS OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND INTOUCH

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

X(66196) lies on cubic K1082 and on these lines: {2, 60832}, {7, 3174}, {69, 35160}, {85, 344}, {144, 43762}, {279, 1445}, {347, 56783}, {480, 1358}, {1434, 41610}, {2369, 53888}, {6600, 40154}, {6601, 40615}, {8732, 17093}, {10509, 12848}, {17089, 56310}, {18230, 27818}, {23618, 60934}, {30379, 60831}, {41857, 57826}, {62782, 62784}

X(66196) = isotomic conjugate of X(56937)
X(66196) = trilinear pole of line {3309, 3676}
X(66196) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 3174}, {9, 21002}, {31, 56937}, {33, 22153}, {41, 36845}, {55, 16572}, {56, 24771}, {109, 59979}, {1253, 8732}, {2175, 20946}, {2194, 21096}, {3063, 65200}, {41573, 59141}
X(66196) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 24771}, {2, 56937}, {9, 3174}, {11, 59979}, {223, 16572}, {478, 21002}, {1214, 21096}, {3160, 36845}, {10001, 65200}, {17113, 8732}, {40593, 20946}
X(66196) = X(i)-Ceva conjugate of X(j) for these {i, j}: {63897, 2}
X(66196) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {63897, 6327}
X(66196) = X(i)-cross conjugate of X(j) for these {i, j}: {9, 7}, {277, 2}, {41790, 189}
X(66196) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(344)}}, {{A, B, C, X(4), X(43971)}}, {{A, B, C, X(7), X(85)}}, {{A, B, C, X(8), X(7674)}}, {{A, B, C, X(9), X(277)}}, {{A, B, C, X(57), X(8232)}}, {{A, B, C, X(142), X(12848)}}, {{A, B, C, X(144), X(30379)}}, {{A, B, C, X(189), X(673)}}, {{A, B, C, X(278), X(21446)}}, {{A, B, C, X(333), X(8051)}}, {{A, B, C, X(346), X(56322)}}, {{A, B, C, X(347), X(62786)}}, {{A, B, C, X(366), X(56707)}}, {{A, B, C, X(480), X(650)}}, {{A, B, C, X(514), X(6601)}}, {{A, B, C, X(1029), X(55937)}}, {{A, B, C, X(1156), X(42483)}}, {{A, B, C, X(1223), X(9311)}}, {{A, B, C, X(1440), X(30705)}}, {{A, B, C, X(3062), X(34578)}}, {{A, B, C, X(4373), X(24002)}}, {{A, B, C, X(5435), X(18230)}}, {{A, B, C, X(14189), X(41356)}}, {{A, B, C, X(15474), X(36101)}}, {{A, B, C, X(21454), X(41857)}}, {{A, B, C, X(21617), X(60939)}}, {{A, B, C, X(30275), X(52819)}}, {{A, B, C, X(37787), X(61019)}}, {{A, B, C, X(41563), X(60988)}}, {{A, B, C, X(41572), X(62778)}}, {{A, B, C, X(42309), X(62782)}}, {{A, B, C, X(42470), X(60832)}}, {{A, B, C, X(52803), X(58817)}}, {{A, B, C, X(60934), X(60992)}}, {{A, B, C, X(60941), X(60996)}}, {{A, B, C, X(60943), X(60948)}}, {{A, B, C, X(60955), X(60967)}}, {{A, B, C, X(61015), X(64142)}}, {{A, B, C, X(63178), X(63185)}}

X(66197) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND 5TH-MIXTILINEAR

Barycentrics    a*(a-b-c)*(a^5-3*a^4*(b+c)+a*(b-c)^2*(b^2-22*b*c+c^2)-2*a^3*(b^2-4*b*c+c^2)+6*a^2*(b^3-3*b^2*c-3*b*c^2+c^3)-(b-c)^2*(3*b^3+b^2*c+b*c^2+3*c^3)) : :

X(66197) lies on these lines: {1, 1407}, {9, 66216}, {11, 11530}, {55, 22754}, {100, 1697}, {145, 329}, {200, 2136}, {390, 60961}, {497, 3680}, {518, 66198}, {944, 12575}, {952, 56038}, {960, 9898}, {1058, 4342}, {1482, 8000}, {1490, 7966}, {2098, 3243}, {3247, 14749}, {3340, 10580}, {4345, 11518}, {4853, 10866}, {5048, 64263}, {5687, 9819}, {5732, 20789}, {6762, 66194}, {7971, 64897}, {7972, 66061}, {8163, 62823}, {9778, 61630}, {10384, 36846}, {12260, 64328}, {17614, 53053}, {17622, 25893}, {23764, 48338}, {25011, 50443}, {30323, 64766}, {31393, 63986}, {31435, 66200}, {34195, 66195}, {57279, 66201}, {62333, 66223}, {66206, 66222}, {66207, 66221}, {66210, 66215}, {66219, 66220}

X(66197) = reflection of X(i) in X(j) for these {i,j}: {7091, 1}
X(66197) = inverse of X(11530) in Feuerbach hyperbola
X(66197) = pole of line {3304, 3698} with respect to the Feuerbach hyperbola

X(66198) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND 6TH-MIXTILINEAR

Barycentrics    a*(a-b-c)*(a^6+a^4*(-7*b^2+6*b*c-7*c^2)+a^2*(b-c)^2*(3*b^2-38*b*c+3*c^2)+(b-c)^4*(3*b^2+2*b*c+3*c^2)-8*a*(b-c)^2*(b^3-3*b^2*c-3*b*c^2+c^3)+8*a^3*(b^3+b^2*c+b*c^2+c^3)) : :

X(66198) lies on these lines: {57, 2951}, {100, 4326}, {144, 36845}, {390, 66205}, {497, 3062}, {516, 41824}, {518, 66197}, {950, 5759}, {1479, 10045}, {1743, 66234}, {3057, 5223}, {3339, 66227}, {3486, 7990}, {5768, 7992}, {7993, 15558}, {11379, 12053}, {15299, 66194}, {16143, 66195}, {18222, 60937}, {63277, 66207}

X(66198) = pole of line {8580, 60937} with respect to the Feuerbach hyperbola

X(66199) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND 2ND-SCHIFFLER

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

X(66199) lies on these lines: {1, 104}, {2, 11}, {3, 1387}, {4, 11508}, {7, 33925}, {8, 4571}, {9, 14740}, {10, 25438}, {12, 12764}, {20, 11510}, {21, 643}, {33, 3749}, {35, 6940}, {36, 28194}, {40, 12736}, {56, 38693}, {57, 18240}, {65, 64189}, {80, 943}, {90, 66200}, {108, 52167}, {119, 3085}, {145, 10965}, {153, 10956}, {200, 46694}, {212, 8616}, {214, 3601}, {226, 11218}, {243, 37790}, {294, 38347}, {388, 2829}, {404, 11376}, {405, 1145}, {411, 12701}, {495, 10742}, {496, 11849}, {498, 6975}, {499, 65119}, {516, 2078}, {518, 1776}, {611, 10759}, {885, 15914}, {900, 53308}, {901, 14115}, {902, 1936}, {938, 12832}, {942, 12515}, {946, 64188}, {952, 3295}, {954, 13257}, {956, 25416}, {958, 5854}, {962, 37579}, {993, 7962}, {999, 38602}, {1000, 55966}, {1006, 5119}, {1012, 3476}, {1056, 12248}, {1058, 6977}, {1124, 19081}, {1155, 7677}, {1156, 2346}, {1259, 64068}, {1260, 38211}, {1279, 9371}, {1317, 3303}, {1319, 6909}, {1335, 19082}, {1385, 17622}, {1421, 24025}, {1478, 10728}, {1479, 6941}, {1484, 15172}, {1486, 54065}, {1497, 37529}, {1537, 3485}, {1612, 66249}, {1617, 3474}, {1618, 34949}, {1633, 20999}, {1697, 2802}, {1737, 65144}, {1788, 10306}, {1837, 3871}, {1858, 12532}, {1862, 7071}, {2066, 19113}, {2077, 44675}, {2098, 2975}, {2099, 62873}, {2293, 64710}, {2298, 14749}, {2310, 3722}, {2551, 55016}, {2646, 10179}, {2654, 5255}, {2771, 12711}, {2801, 10389}, {2831, 3744}, {2932, 34123}, {3036, 3913}, {3056, 10755}, {3065, 63288}, {3086, 6713}, {3091, 11501}, {3149, 38038}, {3218, 18839}, {3254, 66210}, {3256, 11019}, {3297, 48701}, {3298, 48700}, {3315, 53525}, {3487, 54441}, {3488, 12247}, {3579, 58587}, {3586, 6246}, {3616, 17100}, {3622, 22768}, {3651, 14798}, {3660, 17613}, {3681, 7082}, {3748, 10391}, {3750, 14547}, {3811, 18254}, {3870, 30223}, {3911, 5537}, {3938, 24430}, {3961, 7069}, {4031, 5563}, {4188, 18220}, {4189, 10966}, {4293, 38761}, {4294, 5840}, {4301, 37583}, {4302, 37430}, {4304, 12119}, {4305, 16202}, {4309, 6937}, {4313, 6224}, {4318, 8758}, {4326, 60964}, {4512, 10388}, {4551, 64013}, {4640, 17642}, {4996, 9785}, {5047, 64141}, {5048, 44663}, {5225, 11500}, {5229, 52836}, {5250, 64139}, {5251, 64056}, {5252, 6912}, {5258, 26726}, {5414, 19112}, {5531, 10382}, {5541, 53053}, {5603, 8069}, {5687, 34122}, {5722, 12619}, {5727, 15863}, {5734, 26437}, {5744, 42842}, {5851, 60934}, {5853, 58328}, {5901, 38722}, {5919, 20586}, {6264, 31393}, {6265, 24929}, {6284, 10724}, {6361, 7742}, {6596, 66219}, {6600, 64738}, {6702, 8715}, {6767, 12735}, {6796, 9614}, {6842, 10738}, {6888, 10957}, {6892, 37726}, {6897, 13199}, {6905, 30384}, {6914, 64742}, {6924, 38044}, {6946, 23708}, {6950, 22767}, {6981, 10591}, {6986, 37568}, {7080, 8668}, {7160, 66194}, {7162, 66201}, {7288, 10310}, {7489, 64140}, {7589, 8104}, {7676, 34879}, {7972, 28461}, {8071, 18861}, {8076, 13267}, {8543, 17718}, {9024, 10387}, {9654, 22799}, {9668, 22938}, {9669, 32141}, {9778, 37578}, {9809, 10578}, {9819, 12653}, {9848, 41541}, {9897, 50907}, {9898, 12868}, {9957, 12737}, {10056, 10711}, {10057, 10572}, {10106, 64145}, {10265, 63999}, {10321, 10531}, {10384, 64346}, {10386, 37438}, {10543, 63269}, {10595, 22766}, {10624, 10902}, {10679, 18391}, {10778, 46687}, {10950, 12531}, {11219, 64162}, {11256, 66256}, {11373, 26285}, {11374, 12611}, {11509, 14986}, {11604, 66207}, {11609, 66224}, {11688, 21333}, {11798, 49207}, {11997, 66067}, {12019, 12331}, {12332, 20418}, {12575, 21630}, {12641, 66205}, {12738, 63271}, {12739, 17638}, {12751, 31397}, {12763, 15888}, {12864, 35204}, {13143, 66242}, {13226, 41556}, {13266, 53523}, {13272, 15843}, {13405, 21635}, {14882, 37722}, {14935, 14947}, {15175, 24297}, {15179, 56036}, {15254, 58663}, {15298, 66023}, {15325, 35000}, {15446, 56040}, {15931, 64155}, {16371, 38026}, {16858, 64746}, {16865, 64743}, {17018, 61398}, {17127, 61397}, {17452, 38871}, {17603, 42819}, {17719, 35015}, {17724, 38357}, {18340, 24222}, {18990, 38753}, {19860, 39776}, {19914, 37730}, {21669, 45287}, {22775, 64192}, {23845, 53302}, {24840, 36237}, {25440, 32557}, {26476, 27529}, {28174, 41345}, {30117, 45269}, {31231, 65388}, {31479, 61580}, {32198, 66257}, {32635, 56121}, {33814, 64951}, {34772, 64042}, {35258, 54408}, {35445, 52769}, {36741, 38050}, {36868, 63268}, {37163, 63273}, {37403, 37618}, {37541, 42884}, {38759, 64074}, {40779, 66234}, {41546, 51897}, {42843, 63168}, {42886, 64151}, {43135, 47511}, {43974, 62306}, {44858, 61225}, {54318, 64745}, {56181, 64409}, {56288, 64046}, {58595, 64118}, {58604, 64670}, {59329, 64124}, {63208, 63983}, {64041, 66024}, {64290, 66211}

X(66199) = midpoint of X(i) and X(j) for these {i,j}: {37736, 64372}
X(66199) = reflection of X(i) in X(j) for these {i,j}: {51506, 5248}
X(66199) = inverse of X(64154) in Feuerbach hyperbola
X(66199) = isogonal conjugate of X(43947)
X(66199) = perspector of circumconic {{A, B, C, X(666), X(37136)}}
X(66199) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 43947}, {109, 43974}
X(66199) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 43947}, {11, 43974}, {64445, 40166}
X(66199) = X(i)-Ceva conjugate of X(j) for these {i, j}: {31615, 650}
X(66199) = X(i)-cross conjugate of X(j) for these {i, j}: {55334, 651}, {55379, 644}
X(66199) = pole of line {659, 53305} with respect to the circumcircle
X(66199) = pole of line {3738, 53523} with respect to the incircle
X(66199) = pole of line {59, 518} with respect to the Feuerbach hyperbola
X(66199) = pole of line {1319, 3286} with respect to the Stammler hyperbola
X(66199) = pole of line {30941, 43947} with respect to the Wallace hyperbola
X(66199) = pole of line {4435, 46393} with respect to the Hofstadter ellipse
X(66199) = pole of line {3008, 64115} with respect to the dual conic of Yff parabola
X(66199) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(52456)}}, {{A, B, C, X(9), X(56850)}}, {{A, B, C, X(11), X(64440)}}, {{A, B, C, X(100), X(54110)}}, {{A, B, C, X(104), X(14942)}}, {{A, B, C, X(105), X(45393)}}, {{A, B, C, X(109), X(1618)}}, {{A, B, C, X(497), X(14947)}}, {{A, B, C, X(514), X(14740)}}, {{A, B, C, X(522), X(5083)}}, {{A, B, C, X(528), X(42552)}}, {{A, B, C, X(650), X(3035)}}, {{A, B, C, X(673), X(34051)}}, {{A, B, C, X(1320), X(13576)}}, {{A, B, C, X(2342), X(28071)}}, {{A, B, C, X(7004), X(34949)}}, {{A, B, C, X(62715), X(64154)}}
X(66199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10058, 104}, {1, 12758, 10698}, {1, 1768, 5083}, {1, 63281, 10058}, {11, 13274, 10707}, {11, 3058, 13274}, {11, 55, 100}, {11, 6667, 10589}, {35, 16173, 10090}, {55, 1001, 5218}, {57, 64676, 18240}, {80, 10087, 38665}, {80, 3746, 10087}, {104, 12775, 66055}, {498, 39692, 64008}, {1001, 13205, 3035}, {1479, 8068, 59391}, {5083, 41166, 1768}, {6284, 13273, 10724}, {6284, 63270, 13273}, {6767, 12773, 12735}, {10389, 64372, 37736}, {10965, 22760, 145}, {13243, 14151, 17660}, {17638, 37080, 12739}, {18240, 46684, 57}, {24646, 24647, 497}, {26358, 62333, 8}, {26358, 66206, 13278}, {30384, 32760, 6905}, {37736, 64372, 2801}, {47744, 64008, 39692}

X(66200) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND INNER-YFF

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

X(66200) lies on these lines: {10, 50399}, {40, 64155}, {46, 18223}, {90, 66199}, {497, 7162}, {950, 6976}, {1478, 52860}, {1697, 47033}, {1709, 11508}, {3057, 9708}, {3295, 5779}, {3632, 66206}, {3872, 66219}, {5506, 51785}, {10058, 10085}, {10965, 66214}, {15558, 30323}, {21620, 60923}, {31435, 66197}, {41229, 62333}

X(66201) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND OUTER-YFF

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

X(66201) lies on these lines: {1, 11920}, {3, 14100}, {55, 58645}, {90, 497}, {191, 30223}, {946, 60924}, {950, 1728}, {1479, 1709}, {1697, 64056}, {2478, 42012}, {3057, 7082}, {3338, 18224}, {3586, 64292}, {5248, 66203}, {5250, 66205}, {6838, 62839}, {7162, 66199}, {7330, 37726}, {10075, 11508}, {10085, 18237}, {10382, 14798}, {10388, 66193}, {15845, 17437}, {16153, 37447}, {37618, 66195}, {43177, 62836}, {51090, 60949}, {57279, 66197}, {59316, 66239}

X(66202) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND INNER-YFF TANGENTS

Barycentrics    a*(a-b-c)*(a^11-a^10*(b+c)-a*(b-c)^8*(b+c)^2+(b-c)^6*(b+c)^5-a^9*(5*b^2+2*b*c+5*c^2)+a^8*(5*b^3+11*b^2*c+11*b*c^2+5*c^3)+2*a^7*(5*b^4-8*b^2*c^2+5*c^4)-2*a^6*(5*b^5+13*b^4*c+12*b^3*c^2+12*b^2*c^3+13*b*c^4+5*c^5)+2*a^5*(-5*b^6+6*b^5*c+17*b^4*c^2+36*b^3*c^3+17*b^2*c^4+6*b*c^5-5*c^6)+2*a^4*(5*b^7+11*b^6*c+5*b^5*c^2-45*b^4*c^3-45*b^3*c^4+5*b^2*c^5+11*b*c^6+5*c^7)-a^2*(b-c)^2*(5*b^7+15*b^6*c+17*b^5*c^2-21*b^4*c^3-21*b^3*c^4+17*b^2*c^5+15*b*c^6+5*c^7)+a^3*(5*b^8-16*b^7*c-144*b^5*c^3+310*b^4*c^4-144*b^3*c^5-16*b*c^7+5*c^8)) : :

X(66202) lies on these lines: {119, 7160}, {950, 12648}, {3295, 12686}, {5119, 65996}, {10587, 10940}, {10965, 14100}, {11047, 59333}, {12000, 12872}, {15558, 19860}, {60925, 60961}

X(66203) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND 1ST-ZANIAH

Barycentrics    a*(a-b-c)*(a^5*(b+c)+a^4*(-3*b^2+2*b*c-3*c^2)+(b-c)^4*(b^2+b*c+c^2)+a^2*(b-c)^2*(2*b^2-7*b*c+2*c^2)-3*a*(b-c)^2*(b^3-2*b^2*c-2*b*c^2+c^3)+a^3*(2*b^3-b^2*c-b*c^2+2*c^3)) : :
X(66203) = -X[5784]+3*X[38060], -X[11570]+3*X[41861], -X[25722]+5*X[31272], -3*X[52653]+X[64139]

X(66203) lies on circumconic {{A, B, C, X(34056), X(34894)}} and on these lines: {1, 651}, {7, 18240}, {9, 14740}, {11, 142}, {55, 6594}, {80, 5809}, {100, 4326}, {214, 7675}, {390, 2802}, {497, 3254}, {516, 12736}, {518, 15558}, {527, 18839}, {528, 950}, {946, 38055}, {952, 63972}, {971, 1387}, {1445, 46684}, {1461, 61762}, {1479, 11023}, {1768, 30330}, {2771, 15008}, {2800, 5728}, {2829, 12573}, {3035, 58608}, {3057, 3271}, {3059, 46694}, {3601, 64154}, {3660, 15726}, {3878, 6172}, {5083, 5572}, {5248, 66201}, {5250, 9898}, {5528, 60782}, {5537, 37787}, {5660, 60943}, {5784, 38060}, {5853, 66206}, {5856, 66210}, {5903, 12848}, {6667, 15587}, {6736, 38211}, {6745, 15733}, {9844, 12855}, {9951, 10384}, {10058, 15299}, {10531, 45655}, {10889, 38484}, {11024, 45043}, {11281, 16120}, {11372, 64334}, {11544, 58576}, {11570, 41861}, {11715, 42884}, {12758, 18412}, {17620, 64699}, {18254, 45395}, {25722, 31272}, {30628, 46685}, {37541, 41166}, {42014, 62333}, {52653, 64139}, {55432, 66234}, {60937, 64676}, {60995, 66021}, {61019, 65388}, {61030, 66204}

X(66203) = midpoint of X(i) and X(j) for these {i,j}: {11, 14100}, {10427, 36868}, {12758, 18412}, {30628, 46685}
X(66203) = reflection of X(i) in X(j) for these {i,j}: {7, 18240}, {3035, 58608}, {3059, 46694}, {5083, 5572}, {14740, 9}, {15587, 6667}
X(66203) = inverse of X(4845) in incircle
X(66203) = inverse of X(10427) in Feuerbach hyperbola
X(66203) = pole of line {3887, 4845} with respect to the incircle
X(66203) = pole of line {527, 1155} with respect to the Feuerbach hyperbola
X(66203) = pole of line {30379, 43065} with respect to the dual conic of Yff parabola
X(66203) = X(895)-of-inverse-in-incircle triangle
X(66203) = X(5181)-of-intouch triangle
X(66203) = X(5972)-of-Honsberger triangle
X(66203) = X(6594)-of-Mandart-incircle triangle
X(66203) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {11, 3022, 14100}
X(66203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 51768, 64765}, {5572, 5851, 5083}, {10177, 36868, 10427}, {51768, 64264, 1156}

X(66204) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND 1ST-PAVLOV

Barycentrics    a*(a-b-c)*(2*a^6-5*a^5*(b+c)+(b-c)^4*(b^2+4*b*c+c^2)+a^4*(b^2+12*b*c+c^2)-2*a^2*(b-c)^2*(2*b^2+3*b*c+2*c^2)+a^3*(6*b^3-9*b^2*c-9*b*c^2+6*c^3)-a*(b^5-b^3*c^2-b^2*c^3+c^5)) : :
X(66204) = X[1156]+3*X[2346]

X(66204) lies on these lines: {11, 6594}, {45, 66234}, {55, 5528}, {210, 34894}, {497, 55920}, {516, 63270}, {528, 24987}, {950, 45081}, {954, 17638}, {971, 63281}, {1155, 30379}, {1156, 2346}, {2801, 37080}, {3057, 5260}, {3295, 51768}, {3579, 64155}, {3683, 6068}, {5220, 62333}, {5851, 63265}, {8581, 10058}, {10389, 64264}, {16173, 31658}, {32636, 38055}, {36868, 61004}, {41166, 60961}, {61030, 66203}

X(66204) = inverse of X(6594) in Feuerbach hyperbola
X(66204) = pole of line {2078, 3935} with respect to the Feuerbach hyperbola

X(66205) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT EXCENTERS-MIDPOINTS

Barycentrics    (a-b-c)*(2*a^3-a^2*(b+c)+(b-c)^2*(b+c)-2*a*(b^2-4*b*c+c^2)) : :
X(66205) = -3*X[553]+4*X[10107], -3*X[3753]+2*X[66230], -4*X[3812]+3*X[66228], -3*X[3877]+4*X[18250], -X[3962]+3*X[34689]

X(66205) lies on these lines: {1, 2}, {9, 45193}, {11, 64205}, {55, 12640}, {72, 28234}, {75, 18811}, {165, 63133}, {226, 32049}, {329, 11531}, {355, 63989}, {376, 63138}, {390, 66198}, {392, 46677}, {452, 9819}, {497, 3680}, {515, 10914}, {516, 14923}, {517, 12527}, {518, 13601}, {528, 66247}, {553, 10107}, {944, 37560}, {946, 64087}, {950, 3880}, {952, 13369}, {956, 5450}, {958, 64744}, {960, 5854}, {1000, 31435}, {1145, 6684}, {1222, 9364}, {1259, 25439}, {1319, 63990}, {1320, 41012}, {1329, 33895}, {1376, 41426}, {1466, 24391}, {1479, 64203}, {1482, 21075}, {1697, 20588}, {1706, 3476}, {1837, 21627}, {2098, 3452}, {2122, 56942}, {2136, 3486}, {2321, 55432}, {2478, 4342}, {2550, 37709}, {2551, 7962}, {2802, 10624}, {2886, 32537}, {2975, 43174}, {3057, 5795}, {3295, 64768}, {3419, 6260}, {3421, 7982}, {3436, 4301}, {3660, 33956}, {3664, 20895}, {3686, 14735}, {3689, 37734}, {3753, 66230}, {3812, 66228}, {3877, 18250}, {3879, 63151}, {3884, 51379}, {3885, 12575}, {3893, 5853}, {3895, 4314}, {3911, 8256}, {3962, 34689}, {4030, 9371}, {4073, 49527}, {4187, 64703}, {4294, 64202}, {4297, 63130}, {4308, 64112}, {4311, 54286}, {4315, 36977}, {4345, 8165}, {4513, 41006}, {4534, 52528}, {4692, 23528}, {4696, 24026}, {4848, 12513}, {4863, 66251}, {4901, 63598}, {5048, 21031}, {5082, 5881}, {5123, 33559}, {5128, 34610}, {5175, 37712}, {5176, 19925}, {5178, 12531}, {5218, 64204}, {5250, 66201}, {5252, 36972}, {5258, 10058}, {5433, 37829}, {5440, 13607}, {5687, 5882}, {5697, 12572}, {5727, 64068}, {5745, 44784}, {5794, 66240}, {5836, 10106}, {5837, 62333}, {5844, 13600}, {5850, 64047}, {5855, 64171}, {5901, 51362}, {6691, 44848}, {6692, 20323}, {8666, 40293}, {10246, 59587}, {10306, 22758}, {10912, 12053}, {10944, 17612}, {11041, 41863}, {11220, 28236}, {11530, 26040}, {11682, 21060}, {12059, 12672}, {12245, 57279}, {12437, 37740}, {12512, 63136}, {12607, 64160}, {12641, 66199}, {13370, 54391}, {13462, 26062}, {13464, 17757}, {13996, 37568}, {15862, 58415}, {15888, 51416}, {17355, 23617}, {18802, 46684}, {20789, 58650}, {23659, 63977}, {25405, 47742}, {25568, 64964}, {28228, 64002}, {30620, 45275}, {30806, 58816}, {31509, 34919}, {32426, 64162}, {34471, 59584}, {37526, 61296}, {37618, 59675}, {37725, 64200}, {42012, 63135}, {44669, 66195}, {47746, 64897}, {50810, 54290}, {51380, 58679}, {51423, 56880}, {55016, 64137}, {59417, 62824}, {59572, 63208}, {59591, 64953}, {63971, 64697}, {66239, 66245}

X(66205) = midpoint of X(i) and X(j) for these {i,j}: {3893, 10950}
X(66205) = reflection of X(i) in X(j) for these {i,j}: {145, 6738}, {3057, 5795}, {3885, 12575}, {5697, 12572}, {6737, 8}, {10106, 5836}, {10944, 57284}, {66256, 66257}, {66258, 66256}
X(66205) = X(i)-isoconjugate-of-X(j) for these {i, j}: {604, 42339}
X(66205) = X(i)-Dao conjugate of X(j) for these {i, j}: {3161, 42339}, {6692, 3663}
X(66205) = X(i)-Ceva conjugate of X(j) for these {i, j}: {25268, 650}
X(66205) = X(i)-complementary conjugate of X(j) for these {i, j}: {39628, 513}
X(66205) = pole of line {3667, 46004} with respect to the Spieker circle
X(66205) = pole of line {3057, 3452} with respect to the Feuerbach hyperbola
X(66205) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(20323)}}, {{A, B, C, X(2), X(6692)}}, {{A, B, C, X(8), X(18811)}}, {{A, B, C, X(9), X(36846)}}, {{A, B, C, X(596), X(49169)}}, {{A, B, C, X(936), X(4900)}}, {{A, B, C, X(1000), X(14986)}}, {{A, B, C, X(1210), X(5559)}}, {{A, B, C, X(1222), X(6736)}}, {{A, B, C, X(3621), X(56200)}}, {{A, B, C, X(3623), X(34919)}}, {{A, B, C, X(3680), X(19861)}}, {{A, B, C, X(4866), X(12629)}}, {{A, B, C, X(4882), X(56094)}}, {{A, B, C, X(10200), X(42285)}}, {{A, B, C, X(12641), X(24982)}}, {{A, B, C, X(14942), X(20103)}}, {{A, B, C, X(25005), X(55076)}}, {{A, B, C, X(31434), X(64793)}}
X(66205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6736, 6745}, {1, 8, 6736}, {8, 145, 200}, {8, 4853, 4847}, {8, 4861, 6735}, {8, 519, 6737}, {8, 6734, 3626}, {499, 3679, 10}, {519, 6738, 145}, {1479, 64203, 64767}, {2975, 51433, 43174}, {3057, 5795, 40998}, {3632, 4915, 8}, {3893, 10950, 5853}, {4861, 6735, 1125}, {5836, 38455, 10106}, {5881, 11525, 5082}, {8256, 11260, 3911}, {11682, 56879, 21060}, {32426, 66256, 66258}, {32426, 66257, 66256}, {66256, 66257, 64162}

X(66206) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT FUHRMANN

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

X(66206) lies on these lines: {1, 5}, {8, 4571}, {9, 4534}, {46, 38761}, {55, 1145}, {56, 64188}, {57, 64145}, {65, 2829}, {100, 3486}, {104, 1470}, {140, 30538}, {153, 12831}, {390, 64743}, {480, 38211}, {497, 1320}, {499, 38032}, {515, 12736}, {517, 65516}, {519, 15558}, {528, 12743}, {908, 5048}, {950, 2802}, {1155, 38759}, {1159, 38756}, {1210, 11715}, {1319, 1512}, {1479, 64138}, {1519, 65948}, {1537, 2099}, {1697, 64056}, {1737, 6713}, {1788, 38693}, {1836, 3577}, {1858, 2800}, {1864, 12691}, {1877, 12138}, {2077, 40663}, {2098, 25416}, {2646, 3035}, {2801, 41558}, {3036, 3689}, {3057, 5854}, {3256, 9952}, {3295, 64140}, {3340, 34789}, {3359, 24466}, {3488, 66008}, {3586, 14217}, {3612, 38760}, {3632, 66200}, {4294, 64136}, {4295, 10728}, {4305, 34474}, {4848, 46684}, {5083, 6738}, {5193, 44425}, {5218, 64141}, {5552, 59415}, {5690, 65119}, {5795, 14740}, {5840, 10572}, {5853, 66203}, {6224, 60782}, {6246, 12608}, {6256, 13273}, {6596, 34918}, {6667, 17606}, {6737, 46694}, {6797, 34339}, {7354, 24465}, {7962, 26726}, {10058, 10573}, {10090, 10269}, {10106, 18240}, {10200, 34123}, {10385, 64746}, {10531, 10698}, {10543, 35204}, {10609, 22768}, {10679, 19914}, {10896, 38038}, {10915, 15863}, {10965, 49169}, {11011, 64192}, {11019, 41554}, {11239, 50890}, {11509, 48695}, {11871, 45628}, {11872, 45627}, {12053, 64137}, {12115, 64324}, {12119, 59333}, {12247, 12775}, {12531, 12648}, {12575, 66242}, {12611, 50194}, {12619, 55297}, {12665, 64041}, {12703, 30223}, {12709, 17661}, {12758, 23340}, {13384, 64012}, {13996, 66194}, {15096, 27778}, {15381, 40437}, {17652, 66226}, {20085, 45043}, {20118, 20418}, {20119, 60925}, {21154, 24914}, {22799, 39542}, {26364, 34122}, {28204, 58587}, {31272, 54361}, {34434, 58475}, {36279, 38753}, {37567, 64076}, {37606, 38762}, {41575, 46685}, {41684, 63281}, {41687, 49163}, {64078, 64189}, {64745, 66195}, {66012, 66052}, {66197, 66222}

X(66206) = midpoint of X(i) and X(j) for these {i,j}: {11, 10950}, {12743, 17636}, {41575, 46685}
X(66206) = reflection of X(i) in X(j) for these {i,j}: {3035, 66257}, {5083, 6738}, {6737, 46694}, {7354, 24465}, {10106, 18240}, {12735, 12433}, {14740, 5795}, {34434, 58475}
X(66206) = inverse of X(12740) in incircle
X(66206) = inverse of X(119) in Feuerbach hyperbola
X(66206) = perspector of circumconic {{A, B, C, X(655), X(46605)}}
X(66206) = X(i)-Ceva conjugate of X(j) for these {i, j}: {2397, 650}
X(66206) = pole of line {900, 12740} with respect to the incircle
X(66206) = pole of line {119, 517} with respect to the Feuerbach hyperbola
X(66206) = pole of line {52663, 61214} with respect to the Orthic inconic
X(66206) = X(974)-of-Ursa-minor triangle
X(66206) = X(1145)-of-Mandart-incircle triangle
X(66206) = intersection, other than A, B, C, of circumconics {{A, B, C, X(119), X(40437)}}, {{A, B, C, X(1411), X(45393)}}, {{A, B, C, X(2006), X(12641)}}, {{A, B, C, X(14584), X(30513)}}, {{A, B, C, X(15381), X(34586)}}, {{A, B, C, X(26482), X(56143)}}
X(66206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12751, 10956}, {1, 1837, 26476}, {1, 355, 26482}, {1, 80, 119}, {1, 9897, 12749}, {11, 10950, 952}, {11, 10959, 5533}, {11, 1317, 12740}, {104, 18391, 12832}, {952, 12433, 12735}, {2099, 12764, 1537}, {9581, 16173, 11}, {12736, 15528, 18838}, {12743, 17636, 528}, {13278, 66199, 26358}, {23477, 23517, 10523}

X(66207) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT 2ND-FUHRMANN

Barycentrics    (a-b-c)*(2*a^6-4*a^3*b*c*(b+c)-2*a*b*(b-c)^2*c*(b+c)+(b-c)^4*(b+c)^2-a^4*(3*b^2+4*b*c+3*c^2)) : :
X(66207) = -2*X[6743]+3*X[40661]

X(66207) lies on these lines: {1, 30}, {9, 1837}, {11, 5259}, {12, 18406}, {21, 497}, {55, 442}, {56, 44238}, {191, 30223}, {283, 54399}, {354, 64003}, {388, 52841}, {390, 2475}, {496, 5428}, {499, 28465}, {517, 66211}, {519, 66242}, {528, 24987}, {758, 950}, {1001, 44256}, {1058, 26437}, {1479, 6841}, {1697, 47033}, {1839, 1841}, {1884, 64753}, {2330, 51747}, {2646, 5249}, {2771, 31795}, {3057, 31938}, {3065, 12750}, {3086, 21161}, {3295, 18499}, {3303, 26332}, {3304, 64075}, {3486, 5905}, {3583, 63288}, {3601, 26725}, {3647, 10916}, {3648, 28610}, {3651, 4294}, {3683, 6734}, {3962, 17781}, {4309, 10267}, {4314, 11263}, {4857, 16617}, {5048, 33961}, {5218, 31254}, {5221, 37428}, {5225, 52269}, {5229, 61027}, {5274, 15674}, {5426, 51785}, {5427, 11012}, {5432, 41859}, {5499, 10386}, {5693, 37290}, {5698, 11684}, {5709, 41697}, {5715, 6253}, {5722, 16139}, {5840, 24299}, {5842, 37080}, {5904, 10950}, {6175, 10385}, {6743, 40661}, {6826, 64342}, {6865, 64341}, {9670, 37447}, {10072, 44255}, {10122, 63999}, {10399, 31789}, {10529, 15677}, {10572, 40263}, {10624, 64721}, {10902, 63273}, {10943, 31649}, {10966, 57002}, {11019, 41547}, {11113, 49168}, {11238, 15670}, {11240, 15678}, {11276, 34871}, {11604, 66199}, {11827, 37724}, {12053, 35016}, {12116, 21669}, {12575, 15558}, {12704, 16113}, {13743, 18543}, {14100, 16142}, {14794, 15325}, {15338, 37583}, {16141, 54302}, {16202, 47032}, {17525, 45700}, {18527, 22937}, {18839, 66195}, {20084, 60984}, {24541, 49736}, {24929, 33592}, {27529, 31660}, {28146, 58586}, {28460, 35252}, {33557, 33925}, {37563, 64275}, {37726, 46816}, {37740, 41863}, {41861, 60883}, {61663, 64004}, {63277, 66198}, {66197, 66221}

X(66207) = midpoint of X(i) and X(j) for these {i,j}: {3649, 6284}, {16142, 17637}
X(66207) = reflection of X(i) in X(j) for these {i,j}: {10122, 63999}, {15174, 15172}
X(66207) = X(i)-Ceva conjugate of X(j) for these {i, j}: {65205, 650}
X(66207) = pole of line {442, 942} with respect to the Feuerbach hyperbola
X(66207) = pole of line {2911, 8818} with respect to the Kiepert hyperbola
X(66207) = pole of line {35193, 37579} with respect to the Stammler hyperbola
X(66207) = pole of line {6741, 40622} with respect to the dual conic of Wallace hyperbola
X(66207) = X(442)-of-Mandart-incircle triangle
X(66207) = X(973)-of-Ursa-minor triangle
X(66207) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6598), X(52374)}}, {{A, B, C, X(43740), X(52382)}}
X(66207) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 49177, 3649}, {30, 15172, 15174}, {79, 41864, 10543}, {497, 26357, 26475}, {3649, 10543, 33857}, {3649, 6284, 30}, {16142, 17637, 17768}, {17637, 64046, 39772}

X(66208) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST MOSES-MIYAMOTO-APOLLONIUS TRIANGLE WRT 1ST-VAN-KHEA-PAVLOV

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

X(66208) lies on these lines: {7, 145}, {8, 14121}, {519, 52805}, {528, 45719}, {2136, 6204}, {2802, 66000}, {3244, 30342}, {3632, 30432}, {3633, 30426}, {3813, 30314}, {3893, 30376}, {3913, 30386}, {5836, 30347}, {11519, 30355}, {12437, 30277}, {12448, 30289}, {12513, 30297}, {12607, 30307}, {12625, 30325}, {12629, 30401}, {12642, 30361}, {12643, 30369}, {12644, 30419}, {12646, 30407}, {21627, 30381}, {44669, 63283}, {52811, 66243}

X(66208) = reflection of X(i) in X(j) for these {i,j}: {8, 60902}, {66209, 145}
X(66208) = intersection, other than A, B, C, of circumconics {{A, B, C, X(14121), X(27818)}}, {{A, B, C, X(19604), X(42013)}}
X(66208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 30334, 30413}, {145, 5853, 66209}

X(66209) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND MOSES-MIYAMOTO-APOLLONIUS TRIANGLE WRT 1ST-VAN-KHEA-PAVLOV

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

X(66209) lies on these lines: {7, 145}, {8, 7090}, {519, 52808}, {528, 45720}, {2136, 6203}, {2802, 66001}, {3241, 60902}, {3244, 30341}, {3632, 30431}, {3633, 30425}, {3813, 30313}, {3893, 30375}, {3913, 30385}, {5836, 30346}, {11519, 30354}, {11532, 31551}, {12437, 30276}, {12448, 30288}, {12513, 30296}, {12607, 30306}, {12625, 30324}, {12629, 30400}, {12642, 30360}, {12643, 30368}, {12644, 30418}, {12646, 30406}, {21627, 30380}, {44669, 63284}, {52813, 66243}

X(66209) = reflection of X(i) in X(j) for these {i,j}: {66208, 145}
X(66209) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7090), X(27818)}}, {{A, B, C, X(7133), X(19604)}}
X(66209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 30333, 30412}, {145, 5853, 66208}

X(66210) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT AGUILERA

Barycentrics    (a-b-c)*(2*a^4+a^2*(b-c)^2+(b-c)^4-3*a^3*(b+c)-a*(b-c)^2*(b+c)) : :
X(66210) = -3*X[354]+2*X[60945], -3*X[553]+4*X[58563], -3*X[5686]+4*X[18250], -3*X[7671]+X[41572], -5*X[11025]+3*X[60932], X[25722]+3*X[34611], -3*X[38036]+2*X[64001], -3*X[49736]+2*X[58608]

X(66210) lies on these lines: {1, 7}, {8, 9898}, {9, 497}, {10, 50399}, {11, 6594}, {38, 65671}, {55, 142}, {56, 63413}, {144, 36845}, {149, 60969}, {165, 8732}, {200, 7674}, {219, 66234}, {354, 60945}, {388, 52835}, {480, 3452}, {495, 18482}, {496, 31658}, {517, 29957}, {518, 950}, {519, 21084}, {527, 3058}, {528, 9951}, {553, 58563}, {673, 1223}, {946, 954}, {971, 15171}, {1001, 12053}, {1058, 5759}, {1253, 3008}, {1445, 11019}, {1479, 15298}, {1617, 11495}, {1697, 2550}, {1699, 8232}, {1837, 24393}, {2170, 42449}, {2257, 41325}, {2269, 18785}, {2346, 11218}, {3057, 3059}, {3062, 60934}, {3085, 38150}, {3086, 21153}, {3174, 10388}, {3243, 3486}, {3254, 66199}, {3295, 5805}, {3428, 42884}, {3474, 60955}, {3601, 38053}, {3748, 63258}, {3817, 60943}, {3911, 65405}, {3946, 41339}, {4357, 14942}, {4419, 4907}, {4995, 60999}, {5173, 5572}, {5218, 20195}, {5220, 10392}, {5223, 5809}, {5225, 59389}, {5274, 18230}, {5281, 60996}, {5432, 58433}, {5541, 45043}, {5686, 18250}, {5698, 10384}, {5728, 63999}, {5745, 6067}, {5762, 15172}, {5766, 19843}, {5850, 10394}, {5856, 66203}, {6173, 10385}, {6284, 8581}, {6600, 6745}, {6738, 7672}, {6767, 31671}, {7671, 41572}, {7676, 15931}, {7677, 59320}, {7678, 61015}, {8545, 51783}, {9580, 60937}, {9581, 38057}, {9614, 38037}, {9668, 31672}, {9669, 38108}, {9670, 60909}, {9848, 64723}, {10164, 61019}, {10171, 61017}, {10382, 61010}, {10386, 31657}, {10387, 47595}, {10580, 60939}, {10593, 38318}, {10947, 61004}, {11025, 60932}, {11238, 60986}, {11373, 38031}, {12706, 15071}, {12848, 30330}, {13464, 64286}, {14151, 64145}, {14746, 14749}, {14986, 59418}, {15726, 60961}, {15733, 61002}, {15841, 60938}, {17333, 63600}, {17768, 66195}, {18698, 42446}, {18839, 61033}, {19854, 38059}, {20330, 24929}, {21630, 51506}, {24466, 38055}, {25722, 34611}, {26015, 60970}, {29007, 64699}, {30223, 61005}, {30620, 41006}, {30621, 43035}, {30628, 60979}, {31393, 64316}, {34625, 50836}, {38036, 64001}, {38052, 53053}, {38122, 64951}, {40292, 44675}, {41573, 54408}, {42356, 64737}, {49736, 58608}, {50093, 63597}, {50865, 60967}, {51090, 60949}, {51099, 66229}, {52653, 64081}, {60910, 60942}, {60987, 64674}, {60990, 66239}, {63989, 64156}, {64163, 66211}, {66197, 66215}, {66216, 66219}

X(66210) = midpoint of X(i) and X(j) for these {i,j}: {6284, 8581}, {14100, 60919}, {30628, 60979}
X(66210) = reflection of X(i) in X(j) for these {i,j}: {390, 12575}, {4292, 5542}, {5223, 12572}, {5728, 63999}, {7672, 6738}, {12573, 1}, {14100, 15006}, {52819, 5572}, {60972, 49736}, {63972, 15172}
X(66210) = X(i)-Dao conjugate of X(j) for these {i, j}: {52542, 4847}
X(66210) = X(i)-Ceva conjugate of X(j) for these {i, j}: {65195, 650}
X(66210) = pole of line {514, 58322} with respect to the incircle
X(66210) = pole of line {142, 354} with respect to the Feuerbach hyperbola
X(66210) = pole of line {7, 218} with respect to the dual conic of Yff parabola
X(66210) = X(142)-of-Mandart-incircle triangle
X(66210) = X(3313)-of-intouch triangle
X(66210) = X(9969)-of-Ursa-minor triangle
X(66210) = intersection, other than A, B, C, of circumconics {{A, B, C, X(9), X(4350)}}, {{A, B, C, X(269), X(40505)}}, {{A, B, C, X(279), X(6601)}}, {{A, B, C, X(1223), X(62786)}}, {{A, B, C, X(10481), X(15909)}}, {{A, B, C, X(12573), X(14942)}}
X(66210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 516, 12573}, {7, 390, 4326}, {9, 6601, 4847}, {11, 15837, 6666}, {390, 11038, 4313}, {390, 7675, 4314}, {516, 12575, 390}, {516, 5542, 4292}, {1479, 15298, 63970}, {2346, 21617, 13405}, {3058, 14100, 15006}, {3058, 60919, 14100}, {4314, 5542, 7675}, {5572, 38454, 52819}, {5762, 15172, 63972}, {7676, 30379, 43151}, {13405, 65452, 21617}, {14100, 17642, 15185}, {14100, 60919, 527}, {52819, 64162, 5572}

X(66211) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT INVERSE-OF-1ST-PAVLOV

Barycentrics    2*a^7-4*a^6*(b+c)-2*a^2*(b-c)^4*(b+c)-(b-c)^4*(b+c)^3-a^5*(b^2-4*b*c+c^2)+a*(b^2-c^2)^2*(3*b^2-4*b*c+3*c^2)+a^4*(7*b^3-3*b^2*c-3*b*c^2+7*c^3)-4*a^3*(b^4-5*b^2*c^2+c^4) : :
X(66211) = -X[10944]+3*X[63287]

X(66211) lies on these lines: {1, 6831}, {9, 3632}, {55, 64275}, {497, 1389}, {515, 66195}, {517, 66207}, {519, 66219}, {952, 15174}, {1479, 64754}, {2346, 64270}, {3057, 5844}, {3295, 7489}, {3486, 11508}, {3746, 44669}, {4915, 41709}, {5794, 56583}, {5903, 38454}, {8069, 64269}, {8071, 64268}, {10039, 64294}, {10523, 64273}, {10543, 37621}, {10572, 13375}, {10944, 63287}, {11011, 12433}, {13750, 64284}, {15558, 63999}, {19920, 61286}, {24390, 37702}, {30323, 64766}, {37721, 64200}, {64163, 66210}, {64290, 66199}

X(66211) = midpoint of X(i) and X(j) for these {i,j}: {10572, 13375}, {10950, 45081}
X(66211) = pole of line {10039, 13375} with respect to the Feuerbach hyperbola

X(66212) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND ANTI-PAVLOV WRT 1ST-VAN-KHEA-PAVLOV

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

X(66212) lies on circumconic {{A, B, C, X(5559), X(59760)}} and on these lines: {1, 5235}, {8, 81}, {10, 5333}, {21, 519}, {58, 3632}, {86, 3617}, {100, 59302}, {145, 333}, {229, 17568}, {239, 26562}, {314, 4696}, {355, 64400}, {387, 32782}, {518, 41718}, {524, 2475}, {740, 11684}, {944, 64376}, {952, 64720}, {1014, 4848}, {1043, 3621}, {1046, 64010}, {1150, 19278}, {1408, 36920}, {1482, 64405}, {1724, 50638}, {1778, 17299}, {1817, 24391}, {1834, 2895}, {2098, 64409}, {2099, 64408}, {2287, 5839}, {2303, 17362}, {2975, 32853}, {3193, 13746}, {3241, 11110}, {3305, 35629}, {3578, 26117}, {3616, 64425}, {3626, 25526}, {3633, 4653}, {3679, 4658}, {3813, 14008}, {3828, 28618}, {3869, 17156}, {3913, 4184}, {3951, 42044}, {4101, 33133}, {4225, 12513}, {4255, 5372}, {4276, 5288}, {4278, 48696}, {4420, 18465}, {4642, 50016}, {4649, 59307}, {4677, 51669}, {4678, 8025}, {4803, 4816}, {4954, 8715}, {5016, 17363}, {5051, 31143}, {5174, 56014}, {5175, 56020}, {5178, 5847}, {5192, 10449}, {5253, 32919}, {5255, 39673}, {5260, 32864}, {5278, 56990}, {5284, 35633}, {5323, 41687}, {5361, 19765}, {5690, 64393}, {5846, 41610}, {6542, 16047}, {6765, 54356}, {8148, 64383}, {9534, 37633}, {10573, 64421}, {10912, 64406}, {10944, 64382}, {10950, 64414}, {11115, 31145}, {11362, 37402}, {12135, 64378}, {12195, 64381}, {12245, 64384}, {12410, 64395}, {12454, 64396}, {12455, 64397}, {12495, 64398}, {12626, 64402}, {12627, 64403}, {12628, 64404}, {12635, 64407}, {12645, 64419}, {12647, 64420}, {12648, 64422}, {13911, 64417}, {13973, 64418}, {14007, 53620}, {14552, 20019}, {14956, 64068}, {16050, 50079}, {16284, 16749}, {16454, 48850}, {16700, 21896}, {16859, 19723}, {17135, 62804}, {17151, 58786}, {17185, 63135}, {17553, 51093}, {17589, 42028}, {17697, 63060}, {17751, 27644}, {18206, 63130}, {19065, 64385}, {19066, 64386}, {19273, 19767}, {19875, 28620}, {20051, 29767}, {20054, 52352}, {20086, 49745}, {21997, 40891}, {22791, 64399}, {24632, 49770}, {24883, 30831}, {25446, 63344}, {25507, 46933}, {25650, 31204}, {26051, 42045}, {26064, 49718}, {26643, 29617}, {27368, 34195}, {27714, 42334}, {27754, 31446}, {28530, 31888}, {30939, 44720}, {33297, 33955}, {35842, 64412}, {35843, 64413}, {36846, 46877}, {37442, 62837}, {37652, 56989}, {40773, 49495}, {48493, 64379}, {48494, 64380}, {48746, 64389}, {48747, 64390}, {49060, 64391}, {49061, 64392}, {49169, 64394}, {49232, 64410}, {49233, 64411}, {49329, 64387}, {49330, 64388}, {50106, 54422}, {50625, 62848}, {53426, 63537}, {54335, 63333}

X(66212) = reflection of X(i) in X(j) for these {i,j}: {21, 64072}, {34195, 27368}
X(66212) = pole of line {5563, 16466} with respect to the Stammler hyperbola
X(66212) = pole of line {4346, 7321} with respect to the Wallace hyperbola
X(66212) = X(8)-of-2nd-anti-Pavlov triangle
X(66212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 64401, 5235}, {8, 56018, 81}, {10, 28619, 17551}, {10, 64377, 5333}, {21, 64072, 4921}, {58, 3632, 4720}, {145, 333, 64415}, {519, 64072, 21}, {1043, 16704, 16948}, {3679, 4658, 14005}, {4658, 14005, 42025}, {17551, 64377, 28619}, {24883, 41014, 30831}, {32919, 59303, 5253}, {49718, 64167, 26064}

X(66213) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND X(9)-CIRCUMCONCEVIAN-OF-X(7)

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

X(66213) lies on these lines: {7, 354}, {21, 66194}, {100, 4326}, {390, 3885}, {944, 63972}, {971, 17624}, {3243, 10384}, {4345, 9848}, {5274, 17668}, {12711, 18221}, {15845, 60988}, {17642, 60957}, {20075, 61009}, {27282, 63600}, {30330, 64129}, {30628, 60966}, {34784, 60910}, {40269, 66226}, {53055, 60964}, {58608, 59572}

X(66214) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND EXTOUCH-OF-FUHRMANN

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

X(66214) lies on these lines: {1, 1898}, {9, 26358}, {11, 41540}, {55, 58645}, {78, 62333}, {354, 10531}, {1479, 7702}, {3244, 15558}, {5880, 10940}, {9844, 10955}, {10965, 66200}, {11238, 13373}, {11248, 61653}, {11263, 66195}, {11376, 41871}, {18838, 41869}, {23340, 64766}

X(66214) = inverse of X(41540) in Feuerbach hyperbola
X(66214) = pole of line {46, 5552} with respect to the Feuerbach hyperbola

X(66215) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(7) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(3*a^5-4*a^3*b*c-5*a^4*(b+c)+(b-c)^4*(b+c)-a*(b-c)^2*(3*b^2+2*b*c+3*c^2)+4*a^2*(b^3+c^3)) : :
X(66215) = -5*X[3616]+4*X[64443], -2*X[3826]+3*X[56177], -3*X[5817]+4*X[42843], 3*X[8236]+X[12536], -2*X[24393]+3*X[47375]

X(66215) lies on circumconic {{A, B, C, X(2191), X(53623)}} and on these lines: {1, 142}, {7, 34195}, {8, 6600}, {9, 3486}, {56, 8730}, {100, 1617}, {145, 7672}, {390, 3877}, {480, 10950}, {497, 2900}, {516, 7971}, {518, 944}, {519, 7966}, {527, 34628}, {528, 10698}, {952, 3427}, {1001, 3488}, {1420, 41573}, {1445, 41575}, {1998, 64747}, {2886, 33993}, {3158, 4847}, {3243, 3476}, {3244, 8000}, {3616, 64443}, {3826, 56177}, {3869, 12706}, {3880, 11041}, {3913, 5657}, {4297, 60990}, {5086, 60943}, {5173, 17784}, {5441, 5698}, {5531, 25568}, {5817, 42843}, {5856, 6224}, {6067, 34471}, {6666, 66251}, {6765, 64111}, {7673, 20075}, {7677, 12649}, {7990, 61294}, {8236, 12536}, {12625, 24389}, {12632, 18221}, {12635, 61010}, {15931, 24477}, {19843, 56176}, {20007, 40659}, {20013, 34784}, {24393, 47375}, {25252, 51190}, {34607, 41338}, {49168, 52769}, {54193, 64696}, {59340, 60974}, {59413, 63260}, {63168, 64737}, {66197, 66210}

X(66215) = midpoint of X(i) and X(j) for these {i,j}: {145, 7674}
X(66215) = reflection of X(i) in X(j) for these {i,j}: {8, 6600}, {3174, 12437}, {6601, 1}, {12625, 24389}, {49168, 52769}, {60990, 4297}, {61010, 12635}, {66251, 6666}
X(66215) = pole of line {26641, 31603} with respect to the Steiner circumellipse
X(66215) = X(8730)-of-2nd-anti-circumperp-tangential triangle
X(66215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5853, 6601}, {5853, 12437, 3174}

X(66216) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CIRCUMCEVIAN-OF-X(8)

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

X(66216) lies on these lines: {1, 84}, {8, 210}, {9, 66197}, {10, 15845}, {11, 5836}, {33, 37542}, {55, 4855}, {56, 63985}, {65, 6890}, {72, 7962}, {145, 1864}, {200, 66245}, {354, 4323}, {388, 12679}, {390, 3890}, {392, 936}, {496, 517}, {518, 2098}, {519, 64131}, {944, 18239}, {950, 952}, {962, 64106}, {971, 20789}, {1040, 1616}, {1201, 9371}, {1319, 9943}, {1387, 34339}, {1388, 58567}, {1476, 20323}, {1482, 11432}, {1708, 8158}, {1858, 5048}, {1898, 3486}, {2057, 3913}, {2646, 10179}, {2800, 50196}, {2932, 17614}, {2943, 9364}, {3295, 45770}, {3476, 12688}, {3600, 17634}, {3622, 17603}, {3623, 10394}, {3660, 66019}, {3680, 17658}, {3698, 10589}, {3748, 45230}, {3753, 50443}, {3812, 11376}, {3868, 4345}, {3869, 17642}, {3878, 4342}, {3884, 12575}, {3898, 4314}, {3911, 31798}, {4308, 63995}, {4640, 10966}, {5045, 26200}, {5123, 26476}, {5205, 9435}, {5274, 14923}, {5691, 30294}, {5697, 51785}, {5722, 23340}, {5727, 41389}, {5728, 64964}, {5782, 54359}, {5882, 66248}, {5887, 64897}, {5927, 37709}, {6049, 11220}, {6767, 10393}, {7686, 30384}, {8236, 14100}, {9581, 10914}, {9612, 39779}, {9856, 10106}, {10167, 63208}, {10178, 37605}, {10284, 18527}, {10382, 37556}, {10384, 64197}, {10388, 15829}, {10523, 22835}, {10624, 31786}, {10679, 55298}, {10950, 17615}, {10959, 26015}, {11019, 13601}, {11240, 44663}, {11260, 22760}, {11373, 37562}, {11508, 37837}, {11510, 65404}, {11531, 41539}, {12513, 30223}, {12640, 51380}, {12647, 58631}, {13369, 25405}, {13464, 50195}, {13600, 64163}, {14110, 30305}, {16189, 18412}, {16483, 54295}, {17638, 20586}, {17857, 31393}, {18908, 37711}, {22767, 64118}, {24928, 64132}, {26358, 56176}, {31788, 44675}, {31792, 32900}, {34471, 62856}, {37588, 51361}, {38271, 56038}, {41426, 64129}, {55921, 56029}, {66210, 66219}

X(66216) = midpoint of X(i) and X(j) for these {i,j}: {1837, 3057}, {1898, 37738}, {2098, 64042}, {17638, 20586}
X(66216) = reflection of X(i) in X(j) for these {i,j}: {50196, 64703}, {59691, 58679}, {63987, 20789}, {64132, 24928}
X(66216) = inverse of X(45080) in Feuerbach hyperbola
X(66216) = perspector of circumconic {{A, B, C, X(646), X(37141)}}
X(66216) = pole of line {8, 56} with respect to the Feuerbach hyperbola
X(66216) = X(5836)-of-2nd-Johnson-Yff triangle
X(66216) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(1413)}}, {{A, B, C, X(84), X(341)}}, {{A, B, C, X(222), X(55112)}}, {{A, B, C, X(312), X(1422)}}, {{A, B, C, X(1265), X(1433)}}, {{A, B, C, X(2192), X(5423)}}, {{A, B, C, X(3057), X(9435)}}, {{A, B, C, X(3701), X(52384)}}, {{A, B, C, X(4723), X(45824)}}, {{A, B, C, X(9363), X(9368)}}
X(66216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12672, 66250}, {950, 15558, 9957}, {971, 20789, 63987}, {1837, 3057, 3880}, {1858, 5048, 34791}, {1898, 5919, 37738}, {2098, 64042, 518}, {2800, 64703, 50196}, {3057, 10866, 497}, {3057, 17604, 3893}, {3877, 9785, 3057}, {5919, 9848, 3486}, {17622, 66226, 1}

X(66217) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(8) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(3*a^6-6*a^5*(b+c)+(b-c)^4*(b+c)^2+a^4*(-5*b^2+34*b*c-5*c^2)+4*a^3*(3*b^3-7*b^2*c-7*b*c^2+3*c^3)-2*a*(b-c)^2*(3*b^3-7*b^2*c-7*b*c^2+3*c^3)+a^2*(b^4-40*b^3*c+94*b^2*c^2-40*b*c^3+c^4)) : :
X(66217) = -5*X[3616]+4*X[63644]

X(66217) lies on these lines: {1, 6692}, {8, 15347}, {100, 6049}, {390, 3885}, {497, 3680}, {519, 7971}, {944, 3880}, {952, 10309}, {2802, 64076}, {3189, 7972}, {3243, 45194}, {3616, 63644}, {3816, 10912}, {4421, 64173}, {5658, 32426}, {5853, 64697}, {5854, 10698}, {7966, 37560}, {7990, 34607}, {8058, 14812}, {8256, 33994}, {10629, 64203}, {11041, 58609}, {12437, 66231}, {31788, 47746}, {34711, 59326}, {37711, 64068}, {64129, 66245}

X(66217) = reflection of X(i) in X(j) for these {i,j}: {8, 15347}, {56089, 1}

X(66218) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(9) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a-b-c)*(a^7+3*a*(b-c)^6-3*a^6*(b+c)-(b-c)^6*(b+c)-a^2*(b+c)*(b^2-6*b*c+c^2)^2+a^5*(b^2+6*b*c+c^2)-a^3*(b+c)^2*(5*b^2-22*b*c+5*c^2)+a^4*(5*b^3-13*b^2*c-13*b*c^2+5*c^3)) : :

X(66218) lies on these lines: {1, 6600}, {9, 66197}, {145, 3174}, {390, 3872}, {518, 7971}, {519, 64319}, {944, 5732}, {952, 56273}, {1467, 2136}, {3880, 7966}, {4853, 6601}, {6764, 64150}, {7674, 36846}, {7990, 15347}, {8000, 42871}, {8726, 64173}, {10698, 54159}, {11526, 34195}, {15733, 18452}, {47375, 62835}

X(66218) = reflection of X(i) in X(j) for these {i,j}: {2136, 8730}, {42470, 1}

X(66219) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT ANTIPEDAL-OF-X(21)

Barycentrics    a*(a-b-c)*(-4*a^3*b*c-4*a*b^2*c^2+a^4*(b+c)+(b-c)^2*(b^3+c^3)-a^2*(2*b^3+3*b^2*c+3*b*c^2+2*c^3)) : :
X(66219) = -3*X[392]+X[442], -3*X[28465]+X[37562]

X(66219) lies on these lines: {1, 21}, {30, 9856}, {80, 58636}, {210, 66251}, {392, 442}, {405, 31806}, {452, 5692}, {497, 6598}, {517, 6675}, {519, 66211}, {950, 960}, {1479, 5175}, {1699, 46870}, {2475, 9812}, {2478, 10176}, {2800, 17009}, {3057, 4847}, {3145, 54180}, {3428, 37308}, {3452, 10958}, {3486, 66221}, {3649, 64106}, {3651, 63986}, {3872, 66200}, {3880, 58638}, {4301, 44256}, {4540, 59415}, {4853, 9898}, {5044, 12019}, {5086, 58699}, {5173, 8261}, {5693, 11111}, {5694, 50241}, {5697, 19843}, {5706, 16430}, {5745, 64043}, {5795, 14740}, {5883, 6910}, {5884, 16370}, {6362, 65442}, {6596, 66199}, {6857, 37625}, {6872, 31803}, {7173, 25917}, {7483, 31870}, {8582, 64107}, {10543, 14100}, {11019, 41574}, {11113, 20117}, {11114, 31871}, {11281, 58679}, {11344, 22836}, {11545, 58640}, {11729, 31838}, {12573, 17768}, {12635, 13615}, {12672, 44238}, {12758, 35204}, {14749, 40937}, {15071, 17576}, {15558, 18253}, {15680, 43161}, {15931, 51717}, {16139, 22770}, {17183, 18698}, {18250, 51379}, {19524, 54192}, {19861, 35979}, {25639, 41012}, {27086, 59320}, {28465, 37562}, {33858, 37292}, {35258, 66019}, {37286, 40257}, {44663, 58568}, {45955, 58479}, {61272, 64853}, {66197, 66220}, {66210, 66216}

X(66219) = midpoint of X(i) and X(j) for these {i,j}: {3057, 21677}, {3869, 39772}, {3878, 35016}, {10543, 44782}, {12672, 44238}, {12758, 35204}
X(66219) = reflection of X(i) in X(j) for these {i,j}: {10122, 35016}, {11281, 58679}, {40661, 960}
X(66219) = pole of line {2646, 5745} with respect to the Feuerbach hyperbola
X(66219) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {758, 35016, 10122}, {960, 44669, 40661}, {3869, 39772, 758}, {3877, 5250, 3878}

X(66220) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(21) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a-b-c)*(a^8-a^7*(b+c)-b*(b-c)^4*c*(b+c)^2-a^6*(3*b^2+b*c+3*c^2)+a^5*(3*b^3-b^2*c-b*c^2+3*c^3)-a^3*(b-c)^2*(3*b^3+b^2*c+b*c^2+3*c^3)+a^4*(3*b^4+b^3*c+16*b^2*c^2+b*c^3+3*c^4)+a*(b-c)^2*(b^5-b^4*c+8*b^3*c^2+8*b^2*c^3-b*c^4+c^5)-a^2*(b^6-b^5*c+7*b^4*c^2+2*b^3*c^3+7*b^2*c^4-b*c^5+c^6)) : :

X(66220) lies on these lines: {21, 390}, {56, 8730}, {145, 35979}, {404, 18221}, {519, 64280}, {758, 7971}, {944, 3428}, {3189, 59317}, {3880, 64173}, {3913, 11041}, {4847, 6598}, {5173, 34195}, {5853, 59320}, {10698, 54161}, {16143, 64697}, {21161, 34625}, {40292, 64068}, {46870, 64737}, {66197, 66219}

X(66221) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(79) WRT 1ST-VAN-KHEA-PAVLOV

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

X(66221) lies on these lines: {1, 442}, {21, 5837}, {30, 7971}, {100, 4848}, {145, 39772}, {517, 64287}, {519, 10902}, {631, 49168}, {758, 944}, {952, 6596}, {1420, 41574}, {2136, 37550}, {2475, 4323}, {3158, 5690}, {3189, 11041}, {3243, 37738}, {3486, 66219}, {3555, 7972}, {3649, 64263}, {3680, 8000}, {3880, 64766}, {3913, 36152}, {4301, 10698}, {6675, 37739}, {6762, 7966}, {6830, 22836}, {6904, 18221}, {6934, 37625}, {6987, 31806}, {7990, 34716}, {10106, 34195}, {10543, 66239}, {11523, 11827}, {12536, 33110}, {17768, 64697}, {21677, 37740}, {24987, 63260}, {46870, 64160}, {66197, 66207}

X(66221) = reflection of X(i) in X(j) for these {i,j}: {6598, 1}
X(66221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 44669, 6598}

X(66222) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(80) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(3*a^6-5*a^5*(b+c)+(b-c)^4*(b+c)^2-5*a^4*(b^2-5*b*c+c^2)-a*(b-c)^2*(5*b^3-9*b^2*c-9*b*c^2+5*c^3)+2*a^3*(5*b^3-8*b^2*c-8*b*c^2+5*c^3)+a^2*(b^4-25*b^3*c+52*b^2*c^2-25*b*c^3+c^4)) : :
X(66222) = -3*X[3158]+4*X[12735], -3*X[59415]+4*X[64205]

X(66222) lies on these lines: {1, 1145}, {80, 10912}, {100, 5193}, {145, 5083}, {390, 60940}, {519, 1519}, {528, 64697}, {944, 2802}, {952, 3680}, {1317, 2136}, {1320, 12053}, {1420, 18802}, {2804, 14812}, {2950, 6762}, {3158, 12735}, {3359, 3655}, {3586, 12653}, {3632, 39692}, {3880, 7972}, {3892, 11041}, {10073, 41702}, {10956, 64263}, {12531, 21627}, {12625, 13257}, {13271, 33956}, {13996, 45036}, {16173, 17619}, {16205, 64192}, {19907, 64768}, {25438, 34474}, {28234, 48695}, {34789, 38455}, {59415, 64205}, {66197, 66206}

X(66222) = reflection of X(i) in X(j) for these {i,j}: {80, 10912}, {2136, 1317}, {12531, 21627}, {12641, 1}, {64768, 19907}
X(66222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5854, 12641}

X(66223) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(90) WRT 1ST-VAN-KHEA-PAVLOV

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

X(66223) lies on these lines: {1, 11517}, {90, 956}, {100, 34489}, {145, 224}, {390, 4861}, {912, 6762}, {944, 12520}, {1056, 41540}, {3680, 7966}, {3872, 43740}, {5840, 12650}, {6261, 64317}, {7972, 56583}, {10395, 64081}, {10698, 66018}, {12629, 64287}, {37531, 66068}, {62333, 66197}

X(66223) = reflection of X(i) in X(j) for these {i,j}: {56278, 1}

X(66224) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT PEDAL-OF-X(171)

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

X(66224) lies on these lines: {1, 256}, {9, 23638}, {11, 51571}, {55, 2092}, {238, 1682}, {314, 497}, {740, 950}, {2269, 3747}, {2328, 7083}, {3057, 3883}, {3688, 4073}, {3819, 41886}, {4116, 20753}, {4357, 21334}, {4443, 11574}, {6007, 14100}, {8681, 24437}, {10480, 50295}, {11609, 66199}, {23868, 42671}

X(66224) = inverse of X(51571) in Feuerbach hyperbola
X(66224) = X(i)-Ceva conjugate of X(j) for these {i, j}: {53332, 650}
X(66224) = pole of line {512, 62749} with respect to the incircle
X(66224) = pole of line {1211, 2092} with respect to the Feuerbach hyperbola
X(66224) = X(2092)-of-Mandart-incircle triangle
X(66224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21746, 50621, 3056}

X(66225) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(241) WRT 1ST-VAN-KHEA-PAVLOV

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

X(66225) lies on these lines: {1, 2}, {9, 30619}, {1897, 5236}, {2389, 16465}, {3434, 9312}, {3879, 30628}, {3900, 4025}, {3977, 58327}, {4513, 7123}, {9436, 43736}, {9503, 14942}, {25019, 30620}, {41789, 56382}

X(66225) = perspector of circumconic {{A, B, C, X(190), X(30705)}}
X(66225) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {269, 152}, {911, 30695}, {24016, 513}, {32668, 514}, {36101, 54113}, {43736, 3436}, {52156, 21286}, {65245, 20295}, {65294, 21301}
X(66225) = pole of line {4057, 63177} with respect to the circumcircle
X(66225) = pole of line {3667, 6180} with respect to the incircle
X(66225) = pole of line {1863, 7649} with respect to the polar circle
X(66225) = pole of line {3057, 50441} with respect to the Feuerbach hyperbola
X(66225) = pole of line {279, 514} with respect to the Steiner circumellipse
X(66225) = pole of line {644, 44448} with respect to the Hutson-Moses hyperbola
X(66225) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(614), X(43932)}}, {{A, B, C, X(9503), X(25930)}}, {{A, B, C, X(26531), X(56353)}}

X(66226) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT X(1)-CIRCUMCONCEVIAN-OF-X(8)

Barycentrics    a*(a-b-c)*(b^5-b^4*c+4*a*b*(b-c)^2*c-b*c^4+c^5+a^4*(b+c)-2*a^2*(b^3-2*b^2*c-2*b*c^2+c^3)) : :
X(66226) = -5*X[3616]+4*X[58623], -4*X[3812]+5*X[31249], -5*X[3876]+4*X[58696], -4*X[20103]+5*X[25917], -3*X[61718]+X[64736]

X(66226) lies on these lines: {1, 84}, {8, 64131}, {11, 3753}, {55, 392}, {56, 64129}, {65, 4301}, {72, 519}, {78, 52804}, {200, 960}, {388, 6259}, {390, 3877}, {496, 37562}, {497, 517}, {516, 64106}, {518, 7962}, {551, 17603}, {758, 4342}, {912, 64897}, {938, 13601}, {942, 3656}, {944, 66248}, {956, 30223}, {971, 3476}, {995, 9371}, {1040, 16483}, {1060, 1480}, {1191, 54295}, {1259, 5250}, {1317, 2801}, {1319, 10167}, {1387, 10202}, {1420, 9943}, {1470, 17613}, {1478, 39779}, {1617, 64150}, {1699, 30294}, {1737, 15845}, {1788, 31798}, {1837, 10914}, {1858, 2098}, {1898, 10944}, {2078, 65404}, {2099, 5728}, {2800, 41556}, {3086, 31788}, {3100, 62848}, {3241, 10394}, {3295, 37700}, {3303, 10393}, {3486, 5887}, {3601, 4428}, {3616, 58623}, {3744, 45272}, {3752, 45269}, {3812, 31249}, {3869, 9785}, {3873, 4345}, {3876, 58696}, {3880, 5727}, {3884, 4314}, {3890, 4313}, {3895, 51379}, {3916, 10966}, {4002, 17606}, {4018, 64046}, {4294, 31786}, {4298, 17634}, {4308, 9961}, {4315, 63995}, {5048, 62822}, {5119, 64107}, {5252, 5927}, {5439, 11376}, {5603, 50195}, {5692, 9819}, {5697, 37721}, {5734, 62864}, {5836, 9581}, {5884, 64703}, {5903, 51785}, {6265, 24929}, {6735, 18236}, {6762, 66194}, {7288, 31787}, {7686, 9614}, {7982, 44547}, {8581, 66227}, {10106, 12688}, {10177, 53055}, {10179, 13384}, {10382, 31393}, {10480, 39594}, {10595, 16193}, {10624, 14110}, {10703, 14523}, {11224, 18412}, {11373, 34339}, {11508, 33597}, {12640, 46677}, {12647, 18908}, {12648, 17615}, {12680, 63987}, {12701, 64721}, {16465, 62826}, {17619, 26476}, {17626, 18838}, {17637, 33176}, {17652, 66206}, {18839, 24473}, {20103, 25917}, {20323, 64704}, {23340, 37730}, {31146, 44663}, {31397, 37725}, {31838, 64951}, {34434, 43213}, {34790, 64768}, {37539, 51476}, {37566, 66019}, {37610, 51361}, {40269, 66213}, {50196, 64021}, {58567, 63208}, {61718, 64736}, {61762, 64132}

X(66226) = midpoint of X(i) and X(j) for these {i,j}: {1864, 3057}, {3869, 36845}
X(66226) = reflection of X(i) in X(j) for these {i,j}: {65, 11019}, {200, 960}, {12711, 66239}, {17625, 1}, {17642, 4342}, {63995, 4315}, {64130, 9856}
X(66226) = perspector of circumconic {{A, B, C, X(37141), X(56248)}}
X(66226) = pole of line {521, 4895} with respect to the incircle
X(66226) = pole of line {10, 56} with respect to the Feuerbach hyperbola
X(66226) = X(997)-of-Mandart-incircle triangle
X(66226) = intersection, other than A, B, C, of circumconics {{A, B, C, X(84), X(44040)}}, {{A, B, C, X(1413), X(53089)}}
X(66226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 12672, 12709}, {1, 6001, 17625}, {1, 66216, 17622}, {65, 10866, 12053}, {758, 4342, 17642}, {1858, 2098, 3555}, {1864, 3057, 519}, {3057, 64042, 72}, {3057, 9848, 950}, {3878, 12575, 3057}, {5919, 17638, 64041}, {6001, 66239, 12711}, {12711, 17622, 1}, {66227, 66228, 66229}

X(66227) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT X(2)-CIRCUMCONCEVIAN-OF-X(8)

Barycentrics    (a-b-c)*(2*a^5+a^4*(b+c)-2*a^2*(b-c)^2*(b+c)+(b-c)^4*(b+c)+2*a*(b^2-c^2)^2-4*a^3*(b^2-4*b*c+c^2)) : :
X(66227) = -3*X[8236]+X[60936]

X(66227) lies on these lines: {1, 7955}, {7, 738}, {9, 7080}, {10, 60910}, {11, 38207}, {12, 64699}, {40, 61014}, {55, 21060}, {56, 43182}, {57, 64696}, {65, 516}, {142, 60925}, {144, 1697}, {226, 11372}, {388, 3062}, {390, 527}, {497, 553}, {499, 38123}, {946, 60923}, {971, 10106}, {1319, 43176}, {1386, 45275}, {1706, 61009}, {2310, 64174}, {2550, 10392}, {3057, 5850}, {3339, 66198}, {3476, 64697}, {3485, 24644}, {3486, 11531}, {3601, 52653}, {3911, 15299}, {4298, 9848}, {4307, 4907}, {4313, 60979}, {4326, 5698}, {4848, 10398}, {5083, 10391}, {5128, 60941}, {5759, 7994}, {5762, 10624}, {5779, 31397}, {5843, 9957}, {5851, 15558}, {5853, 10394}, {7221, 50294}, {7288, 64698}, {7675, 56387}, {7743, 61509}, {8236, 60936}, {8543, 63265}, {8581, 66226}, {9581, 59412}, {9614, 59386}, {9785, 20059}, {10593, 38172}, {10866, 12577}, {10896, 38151}, {11041, 28194}, {11373, 59380}, {11376, 38054}, {12573, 15726}, {12575, 60919}, {13257, 13405}, {15017, 51768}, {15587, 64131}, {16870, 52428}, {17768, 66195}, {21168, 61763}, {25722, 57284}, {30332, 41572}, {31393, 41705}, {31657, 44675}, {41339, 64017}, {42884, 43177}, {49476, 65952}, {50443, 62778}, {60896, 60993}, {60924, 63993}, {60926, 60962}, {60992, 63971}, {61012, 63990}, {63979, 65671}, {63998, 66248}

X(66227) = midpoint of X(i) and X(j) for these {i,j}: {30332, 41572}
X(66227) = reflection of X(i) in X(j) for these {i,j}: {950, 14100}, {25722, 57284}, {31391, 4298}, {60919, 12575}, {60961, 1}, {61003, 5698}
X(66227) = pole of line {226, 38054} with respect to the Feuerbach hyperbola
X(66227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 10384, 12053}, {516, 14100, 950}, {10398, 35514, 4848}, {66226, 66229, 66228}

X(66228) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT X(8)-CIRCUMCONCEVIAN-OF-X(8)

Barycentrics    2*a^4-a^3*(b+c)+a*(b-c)^2*(b+c)-(b^2-c^2)^2-a^2*(b^2-14*b*c+c^2) : :
X(66228) = X[3057]+2*X[4298], -5*X[3616]+2*X[5795], 5*X[3623]+X[57287], -2*X[3679]+3*X[46916], -4*X[3812]+X[66205], X[4292]+2*X[9957], -4*X[5045]+X[64163], X[6737]+2*X[34791], 2*X[6738]+X[10944], -4*X[6744]+X[10950], X[7354]+2*X[12575], X[10624]+2*X[18990] and many others

X(66228) lies on these lines: {1, 4}, {7, 7962}, {8, 5437}, {10, 3304}, {11, 51782}, {12, 10171}, {20, 37556}, {55, 4315}, {56, 10164}, {57, 59417}, {65, 10569}, {142, 3872}, {145, 9776}, {210, 34749}, {354, 519}, {377, 21627}, {390, 51779}, {443, 12629}, {495, 11230}, {496, 38140}, {516, 5434}, {517, 553}, {518, 60972}, {527, 3877}, {529, 10179}, {551, 17718}, {938, 37709}, {942, 5844}, {952, 5049}, {993, 33925}, {999, 3911}, {1000, 2093}, {1125, 15888}, {1210, 5790}, {1317, 38055}, {1319, 13405}, {1320, 60980}, {1387, 66052}, {1420, 54445}, {1697, 3600}, {1737, 37602}, {1836, 4342}, {1837, 21625}, {2098, 3671}, {2099, 5542}, {2136, 6904}, {2646, 63287}, {3057, 4298}, {3058, 28164}, {3085, 61762}, {3086, 54447}, {3241, 5853}, {3243, 60987}, {3244, 56997}, {3295, 4311}, {3303, 4297}, {3306, 12648}, {3333, 4848}, {3338, 11362}, {3340, 11037}, {3421, 5316}, {3474, 9819}, {3582, 10172}, {3601, 4308}, {3616, 5795}, {3622, 5748}, {3623, 57287}, {3663, 7223}, {3679, 46916}, {3742, 38455}, {3812, 66205}, {3817, 11237}, {3828, 61649}, {3879, 20037}, {3946, 9317}, {3947, 11376}, {3982, 64897}, {4292, 9957}, {4293, 31393}, {4301, 10404}, {4304, 6767}, {4317, 31730}, {4390, 5750}, {4512, 34610}, {4915, 26040}, {5045, 64163}, {5083, 50195}, {5218, 13462}, {5249, 38460}, {5252, 11019}, {5253, 63990}, {5261, 50443}, {5298, 58441}, {5442, 5563}, {5558, 18221}, {5586, 58245}, {5703, 63208}, {5719, 25405}, {5726, 10589}, {5727, 10580}, {5731, 10389}, {5745, 54391}, {5836, 32426}, {5837, 62874}, {5850, 31165}, {5902, 28234}, {6692, 6735}, {6736, 25524}, {6737, 34791}, {6738, 10944}, {6744, 10950}, {7288, 51784}, {7320, 20070}, {7354, 12575}, {7682, 64352}, {7966, 50701}, {8162, 30331}, {8236, 60967}, {8581, 66226}, {8582, 32049}, {9310, 61651}, {9578, 14986}, {9579, 9785}, {9581, 54448}, {9657, 51118}, {9843, 64087}, {9850, 12711}, {10039, 64124}, {10056, 10165}, {10072, 10175}, {10520, 43037}, {10578, 13384}, {10590, 37704}, {10624, 18990}, {10711, 16173}, {10914, 12436}, {11011, 12563}, {11035, 66250}, {11036, 64964}, {11224, 59372}, {11239, 35262}, {11240, 24386}, {11551, 63210}, {12527, 58679}, {12541, 56999}, {12573, 38454}, {12647, 51816}, {12943, 51783}, {13411, 24928}, {15170, 28160}, {15171, 28168}, {15172, 28190}, {15178, 63282}, {15558, 60961}, {16137, 33179}, {17614, 59722}, {17706, 50190}, {18527, 64839}, {19860, 51723}, {19925, 37722}, {22837, 51706}, {24391, 62832}, {24987, 62837}, {26062, 64204}, {27003, 51433}, {31735, 65454}, {31766, 65398}, {32636, 43174}, {33956, 58560}, {34605, 62835}, {34689, 61686}, {34716, 38316}, {36977, 54392}, {37717, 53618}, {40869, 61706}, {41558, 46681}, {41575, 62854}, {46943, 63126}, {50397, 51071}, {51103, 51112}, {51705, 59337}, {51714, 59719}, {52819, 64106}, {63430, 64322}, {64312, 64338}, {64377, 64582}

X(66228) = midpoint of X(i) and X(j) for these {i,j}: {210, 34749}, {3057, 11246}, {5434, 5919}
X(66228) = reflection of X(i) in X(j) for these {i,j}: {11246, 4298}, {40998, 10179}
X(66228) = pole of line {522, 21222} with respect to the incircle
X(66228) = pole of line {65, 4342} with respect to the Feuerbach hyperbola
X(66228) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(1065), X(63993)}}, {{A, B, C, X(51565), X(64162)}}
X(66228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10106, 950}, {1, 1056, 226}, {1, 10572, 40270}, {1, 13407, 13464}, {1, 388, 12053}, {1, 9613, 1058}, {495, 51788, 44675}, {529, 10179, 40998}, {999, 31397, 3911}, {3057, 11246, 28228}, {4298, 28228, 11246}, {5252, 61717, 38155}, {5434, 5919, 516}, {5558, 20050, 18221}, {6692, 6735, 44848}, {10944, 17609, 6738}, {11019, 38155, 61717}, {11239, 35262, 59584}, {15888, 20323, 1125}, {32636, 45081, 43174}, {66226, 66229, 66227}

X(66229) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT X(10)-CIRCUMCONCEVIAN-OF-X(8)

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

X(66229) lies on these lines: {1, 10309}, {226, 12678}, {354, 950}, {388, 10864}, {515, 10391}, {516, 2099}, {1056, 66239}, {1125, 22760}, {1837, 12436}, {2646, 12572}, {3486, 4292}, {3601, 12527}, {3872, 60925}, {4293, 10382}, {4297, 10393}, {4304, 37569}, {4305, 64004}, {5048, 10543}, {5434, 14100}, {6700, 22768}, {6738, 18838}, {7675, 41570}, {8581, 66226}, {10106, 12680}, {10404, 12053}, {10572, 30274}, {10624, 16200}, {13384, 40998}, {37469, 51375}, {51099, 66210}, {63265, 64009}

X(66229) = pole of line {8058, 17418} with respect to the incircle
X(66229) = pole of line {3671, 37566} with respect to the Feuerbach hyperbola
X(66229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66227, 66228, 66226}

X(66230) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT X(10)-ANTIPEDAL-OF-X(8)

Barycentrics    2*a^4-a^3*(b+c)+a*(b-c)^2*(b+c)-(b^2-c^2)^2-a^2*(b^2-10*b*c+c^2) : :
X(66230) = 3*X[354]+X[10944], -3*X[392]+X[12527], -3*X[553]+X[5903], -3*X[3753]+X[66205], -5*X[3889]+X[41575], 5*X[3890]+3*X[34605], -7*X[3983]+3*X[34689], -3*X[5049]+2*X[6744], 3*X[5919]+X[7354], -3*X[10179]+X[57288], -X[10950]+5*X[17609], X[37707]+7*X[50190]

X(66230) lies on these lines: {1, 4}, {2, 5828}, {3, 4315}, {5, 51782}, {7, 7982}, {8, 3306}, {10, 999}, {12, 20323}, {20, 31393}, {30, 10105}, {40, 3600}, {55, 4311}, {56, 6684}, {57, 11362}, {65, 28234}, {84, 64322}, {145, 8000}, {354, 10944}, {355, 7373}, {376, 53053}, {377, 36846}, {382, 51783}, {392, 12527}, {443, 4853}, {474, 6736}, {495, 1125}, {496, 19925}, {499, 10172}, {516, 9957}, {517, 4298}, {519, 942}, {527, 3878}, {529, 12572}, {548, 51787}, {551, 11236}, {553, 5903}, {631, 13462}, {938, 5881}, {952, 5045}, {1000, 7991}, {1059, 10570}, {1060, 30145}, {1071, 9850}, {1155, 45081}, {1210, 3304}, {1319, 13411}, {1385, 13405}, {1387, 38757}, {1388, 17718}, {1420, 3085}, {1467, 64733}, {1482, 3671}, {1512, 45977}, {1697, 4293}, {1770, 28232}, {1788, 38127}, {2098, 10404}, {2550, 12629}, {2800, 66250}, {2829, 20789}, {3057, 4292}, {3086, 9578}, {3090, 5726}, {3091, 37704}, {3157, 62828}, {3241, 11036}, {3244, 4780}, {3295, 4297}, {3296, 3633}, {3303, 4304}, {3338, 4848}, {3339, 12245}, {3361, 5657}, {3421, 8583}, {3524, 64350}, {3528, 31508}, {3545, 50444}, {3555, 6737}, {3576, 4308}, {3601, 51705}, {3616, 25522}, {3622, 26129}, {3623, 41870}, {3624, 8164}, {3632, 10980}, {3634, 15325}, {3635, 12563}, {3636, 5087}, {3649, 5048}, {3654, 37545}, {3746, 21578}, {3753, 66205}, {3812, 38455}, {3817, 9654}, {3889, 41575}, {3890, 34605}, {3895, 4190}, {3911, 5445}, {3947, 5886}, {3982, 63210}, {3983, 34689}, {4294, 37556}, {4295, 7962}, {4301, 57282}, {4305, 10389}, {4313, 50811}, {4314, 6767}, {4317, 5119}, {4342, 12699}, {4349, 64572}, {4355, 11531}, {4658, 64582}, {4855, 11239}, {4860, 41687}, {4861, 5249}, {5049, 6744}, {5083, 13750}, {5179, 9327}, {5226, 9624}, {5250, 20076}, {5253, 6735}, {5261, 8227}, {5265, 31423}, {5266, 11700}, {5274, 18492}, {5436, 34716}, {5438, 34619}, {5587, 14986}, {5703, 64953}, {5719, 15178}, {5722, 21625}, {5728, 17644}, {5745, 8666}, {5837, 62858}, {5844, 31794}, {5854, 10107}, {5884, 17625}, {5919, 7354}, {6049, 64952}, {6147, 10222}, {6284, 28172}, {6361, 9819}, {6700, 12607}, {6734, 62837}, {6745, 17614}, {6897, 60992}, {6904, 63137}, {6940, 13370}, {7091, 37560}, {7288, 31434}, {7686, 12915}, {7743, 12571}, {7989, 47743}, {8581, 12672}, {8582, 64087}, {8726, 12855}, {9363, 37469}, {9579, 30305}, {9581, 50796}, {9655, 51118}, {9657, 12701}, {9785, 41869}, {9799, 9845}, {9948, 63430}, {9949, 12684}, {10056, 37618}, {10072, 10827}, {10171, 10592}, {10179, 57288}, {10528, 35262}, {10569, 10914}, {10573, 51816}, {10590, 50443}, {10915, 63990}, {10950, 17609}, {11009, 11551}, {11023, 11518}, {11038, 61291}, {11237, 11376}, {11260, 25466}, {12005, 50195}, {12433, 28204}, {12447, 34790}, {12573, 14110}, {12609, 22837}, {12640, 54286}, {12676, 54198}, {12735, 16137}, {13374, 16215}, {13375, 53615}, {13600, 31775}, {13883, 35768}, {13936, 35769}, {15006, 43179}, {15171, 28164}, {15172, 28160}, {15934, 37727}, {18220, 38021}, {18391, 37709}, {19860, 36977}, {19861, 21075}, {19862, 31479}, {20060, 41012}, {20449, 64133}, {21096, 24247}, {21842, 63259}, {22759, 33925}, {23675, 49487}, {24929, 64706}, {24987, 54391}, {25440, 49626}, {28174, 31776}, {28186, 31795}, {30148, 37697}, {30337, 64005}, {30960, 43223}, {31410, 51789}, {31424, 34610}, {31445, 64109}, {31734, 31766}, {31735, 31767}, {31788, 63994}, {31806, 64106}, {31870, 50196}, {34640, 51071}, {37582, 43174}, {37602, 37710}, {37707, 50190}, {37734, 44840}, {37828, 40726}, {44669, 58609}, {47299, 53994}, {51362, 52264}, {51723, 54318}, {53597, 56928}, {55174, 65398}, {59691, 59722}, {63258, 64283}

X(66230) = midpoint of X(i) and X(j) for these {i,j}: {1, 10106}, {145, 63146}, {3057, 4292}, {3244, 17647}, {3555, 6737}, {7354, 10624}, {9957, 18990}, {10944, 64163}, {13600, 31775}, {31734, 31766}, {31735, 31767}
X(66230) = reflection of X(i) in X(j) for these {i,j}: {942, 12577}, {950, 40270}, {5795, 1125}, {5836, 12436}, {6738, 5045}, {12572, 58679}, {12575, 31792}, {15006, 43179}, {34790, 12447}, {37730, 6744}, {64163, 17706}
X(66230) = pole of line {522, 48282} with respect to the incircle
X(66230) = pole of line {65, 63993} with respect to the Feuerbach hyperbola
X(66230) = pole of line {14837, 47796} with respect to the Steiner inellipse
X(66230) = pole of line {57, 4415} with respect to the dual conic of Yff parabola
X(66230) = X(185)-of-incircle-circles triangle
X(66230) = X(6684)-of-2nd-anti-circumperp-tangential triangle
X(66230) = intersection, other than A, B, C, of circumconics {{A, B, C, X(29), X(63993)}}, {{A, B, C, X(33), X(56038)}}, {{A, B, C, X(1058), X(10570)}}, {{A, B, C, X(1059), X(10571)}}, {{A, B, C, X(1065), X(12053)}}, {{A, B, C, X(4900), X(65128)}}, {{A, B, C, X(13464), X(60041)}}, {{A, B, C, X(51565), X(63999)}}
X(66230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10106, 515}, {1, 1056, 21620}, {1, 1478, 12053}, {1, 226, 13464}, {1, 3476, 5882}, {1, 388, 946}, {1, 5270, 30384}, {1, 5290, 5603}, {1, 5691, 1058}, {1, 950, 40270}, {1, 9613, 497}, {10, 999, 64124}, {12, 20323, 44675}, {30, 31792, 12575}, {145, 11037, 11529}, {354, 10944, 64163}, {354, 64163, 17706}, {355, 7373, 11019}, {495, 24928, 1125}, {497, 9613, 31673}, {515, 40270, 950}, {519, 12436, 5836}, {519, 12577, 942}, {952, 5045, 6738}, {1319, 15888, 13411}, {1478, 12053, 18483}, {1697, 4293, 31730}, {3057, 4292, 28194}, {3057, 5434, 4292}, {3086, 9578, 10175}, {3244, 17647, 5853}, {3338, 12647, 4848}, {3890, 34605, 64002}, {5049, 37730, 6744}, {5563, 10039, 3911}, {5919, 7354, 10624}, {6744, 28236, 37730}, {6767, 18481, 4314}, {7354, 10624, 28150}, {9654, 11373, 3817}, {9957, 18990, 516}, {18391, 37709, 47745}, {25405, 37737, 3636}, {25524, 32049, 10}, {46681, 58566, 5045}, {57282, 64897, 4301}

X(66231) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(2)-ANTIPEDAL-OF-X(8) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a^3-3*b^3+19*b^2*c+19*b*c^2-3*c^3+3*a^2*(b+c)-a*(b^2+46*b*c+c^2)) : :
X(66231) = -3*X[165]+4*X[7966], -4*X[1001]+3*X[4915], -4*X[3577]+5*X[16189], -7*X[30389]+8*X[64735], -9*X[30392]+8*X[64733], -3*X[37712]+4*X[64322]

X(66231) lies on these lines: {1, 3689}, {40, 7990}, {100, 13462}, {145, 3339}, {165, 7966}, {390, 519}, {517, 64697}, {944, 3633}, {952, 3062}, {956, 11519}, {1000, 3632}, {1001, 4915}, {1743, 4752}, {2093, 6154}, {2136, 3361}, {2802, 30353}, {3243, 3880}, {3244, 18221}, {3577, 16189}, {3679, 36835}, {3680, 64263}, {3900, 14812}, {3913, 45036}, {4677, 30393}, {4882, 66256}, {5290, 12541}, {5531, 10698}, {7971, 11531}, {7987, 64173}, {7993, 64320}, {8000, 39779}, {9851, 61296}, {10980, 11041}, {11407, 64323}, {12437, 66217}, {12526, 20014}, {12629, 25439}, {12632, 56090}, {30389, 64735}, {30392, 64733}, {36867, 64766}, {37712, 64322}, {41702, 64731}, {48696, 53058}, {51781, 61158}, {58221, 61154}, {63260, 64199}, {64142, 66233}

X(66231) = midpoint of X(i) and X(j) for these {i,j}: {12632, 56090}
X(66231) = reflection of X(i) in X(j) for these {i,j}: {3632, 1000}, {4900, 1}, {16236, 145}
X(66231) = pole of line {7962, 61686} with respect to the Feuerbach hyperbola
X(66231) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2136, 12127, 3361}, {3632, 30337, 4866}

X(66232) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(7)-ANTIPEDAL-OF-X(8) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a^4-b^4-14*b^2*c^2-c^4+2*a^3*(b+c)+a^2*(-8*b^2+4*b*c-8*c^2)+2*a*(3*b^3+5*b^2*c+5*b*c^2+3*c^3)) : :

X(66232) lies on these lines: {1, 728}, {57, 14839}, {145, 10025}, {518, 7962}, {644, 7290}, {664, 49446}, {1697, 8844}, {3241, 3685}, {3243, 4919}, {3872, 7174}, {3875, 65953}, {16834, 32926}

X(66232) = reflection of X(i) in X(j) for these {i,j}: {39959, 1}

X(66233) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(8)-CIRCUMCONCEVIAN-OF-X(8) WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(7*a^3+17*a^2*(b+c)-(b-c)^2*(b+c)+a*(9*b^2-50*b*c+9*c^2)) : :
X(66233) = -16*X[3913]+7*X[20057], -55*X[5550]+64*X[64123], -4*X[12541]+13*X[19877]

X(66233) lies on these lines: {8, 9}, {519, 62095}, {3241, 64112}, {3880, 38314}, {3913, 20057}, {5550, 64123}, {5854, 10031}, {12541, 19877}, {12630, 13996}, {51786, 64151}, {64142, 66231}

X(66234) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND UNARY-COFACTOR-TRIANGLE-OF-MANDART-INCIRCLE

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

X(66234) lies on these lines: {1, 56913}, {6, 4319}, {45, 66204}, {55, 1438}, {218, 950}, {219, 66210}, {220, 2082}, {294, 497}, {650, 949}, {1212, 62333}, {1743, 66198}, {3663, 6180}, {4513, 41006}, {16572, 66194}, {40779, 66199}, {51190, 62799}, {55432, 66203}

X(66234) = X(i)-Ceva conjugate of X(j) for these {i, j}: {390, 55}
X(66234) = pole of line {28043, 37580} with respect to the Feuerbach hyperbola

X(66235) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY-COFACTOR-TRIANGLE-OF-ANTI-CONWAY WRT 1ST-VAN-KHEA-PAVLOV

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

X(66235) lies on these lines: {1, 6}, {8, 343}, {34, 14872}, {52, 1482}, {55, 44706}, {68, 952}, {145, 6515}, {161, 9798}, {221, 912}, {227, 5534}, {515, 64037}, {517, 1854}, {519, 64060}, {569, 10246}, {774, 3938}, {920, 3052}, {944, 6146}, {973, 7979}, {975, 16193}, {1038, 12675}, {1040, 63976}, {1060, 34046}, {1062, 7074}, {1069, 66036}, {1071, 8270}, {1072, 1837}, {1209, 5790}, {1385, 37476}, {1390, 51496}, {1465, 17857}, {1483, 13292}, {1725, 11508}, {1735, 5687}, {1737, 17054}, {1807, 11249}, {1858, 64449}, {1870, 9370}, {2801, 4347}, {2807, 6293}, {2917, 32371}, {3086, 17597}, {3176, 41361}, {3241, 61658}, {3465, 64077}, {3616, 37649}, {3622, 63085}, {3623, 63012}, {3811, 17102}, {3874, 41344}, {3920, 62864}, {4318, 12528}, {5266, 62810}, {5603, 45089}, {5693, 34040}, {5709, 51361}, {5711, 18389}, {5777, 34036}, {5844, 64066}, {7004, 10310}, {7718, 39898}, {7986, 50193}, {9630, 61397}, {9643, 41339}, {9817, 13374}, {10247, 37493}, {10573, 64172}, {11396, 16980}, {11496, 24430}, {11500, 37591}, {12410, 37488}, {12702, 37478}, {14986, 62814}, {17809, 31811}, {18391, 37549}, {18474, 18525}, {18477, 41711}, {19372, 58631}, {22770, 45272}, {24391, 51375}, {30142, 62852}, {34048, 63967}, {36565, 62873}, {36745, 41538}, {36746, 64349}, {37674, 54401}, {40836, 56137}, {46974, 62858}, {49542, 64085}, {60689, 64021}, {60786, 64132}, {61086, 66248}

X(66235) = reflection of X(i) in X(j) for these {i,j}: {64022, 9798}, {64057, 4347}
X(66235) = pole of line {55, 7395} with respect to the Feuerbach hyperbola
X(66235) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(1002), X(64722)}}, {{A, B, C, X(1280), X(64069)}}, {{A, B, C, X(1386), X(51496)}}, {{A, B, C, X(7078), X(56137)}}
X(66235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5904, 7078}, {2801, 4347, 64057}

X(66236) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY-COFACTOR-TRIANGLE-OF-2ND BROCARD WRT 1ST-VAN-KHEA-PAVLOV

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

X(66236) lies on these lines: {1, 7664}, {2, 7983}, {8, 111}, {110, 51192}, {145, 7665}, {952, 63719}, {2502, 5846}, {3124, 49524}, {3416, 40915}, {5147, 32848}, {10554, 51001}, {28538, 58854}

X(66237) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY-COFACTOR-TRIANGLE-OF-INNER TRI-EQUILATERAL WRT 1ST-VAN-KHEA-PAVLOV

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

X(66237) lies on these lines: {1, 6}, {3, 44659}, {8, 302}, {14, 7975}, {16, 51689}, {55, 65571}, {145, 62983}, {515, 41038}, {519, 9761}, {528, 37833}, {952, 63731}, {1082, 1376}, {1482, 5615}, {2809, 36940}, {3106, 14839}, {3241, 37785}, {3639, 5880}, {5698, 30339}, {11235, 51750}, {11295, 50849}, {11296, 50854}, {11480, 51688}, {11485, 11707}, {11705, 42974}, {17768, 37830}, {19781, 38221}, {25557, 30345}, {50254, 62197}

X(66237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5240, 1001}

X(66238) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY-COFACTOR-TRIANGLE-OF-OUTER TRI-EQUILATERAL WRT 1ST-VAN-KHEA-PAVLOV

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

X(66238) lies on these lines: {1, 6}, {3, 44660}, {8, 303}, {13, 7974}, {15, 51691}, {55, 65572}, {145, 62984}, {515, 41039}, {519, 9763}, {528, 37830}, {559, 1376}, {952, 63732}, {1482, 5611}, {2809, 36941}, {3107, 14839}, {3241, 37786}, {3638, 5880}, {5698, 30338}, {11235, 51749}, {11295, 50857}, {11296, 50852}, {11481, 51690}, {11486, 11708}, {11706, 42975}, {17768, 37833}, {19780, 38221}, {25557, 30344}, {50254, 62198}

X(66238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5239, 1001}

X(66239) = PERSPECTOR OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV AND CTR12-1.8

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

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(66239) lies on these lines: {1, 84}, {9, 55}, {11, 5437}, {19, 44695}, {31, 2257}, {33, 1395}, {35, 61122}, {40, 950}, {46, 37428}, {56, 5918}, {57, 497}, {63, 390}, {65, 12651}, {90, 3746}, {165, 15299}, {212, 16572}, {226, 11372}, {281, 7008}, {354, 12560}, {388, 21628}, {495, 18540}, {496, 37534}, {518, 10388}, {519, 1697}, {553, 3333}, {610, 20991}, {612, 2310}, {738, 30623}, {758, 7962}, {920, 4309}, {936, 64131}, {938, 63985}, {962, 62836}, {968, 2293}, {997, 3601}, {999, 7171}, {1001, 10383}, {1038, 35658}, {1040, 7290}, {1056, 66229}, {1058, 14646}, {1158, 63999}, {1190, 51418}, {1200, 28070}, {1210, 37560}, {1420, 12520}, {1445, 9778}, {1449, 61398}, {1453, 54295}, {1454, 9670}, {1467, 9943}, {1479, 59335}, {1490, 66248}, {1617, 5732}, {1621, 7675}, {1706, 1837}, {1711, 8616}, {1728, 61763}, {1743, 7074}, {1768, 41556}, {1858, 11523}, {2082, 28124}, {2136, 10950}, {2195, 23601}, {2328, 40979}, {2801, 10389}, {2999, 9371}, {3052, 8557}, {3057, 6762}, {3058, 3928}, {3086, 37526}, {3100, 62834}, {3219, 20015}, {3220, 16541}, {3295, 7330}, {3304, 7091}, {3305, 5281}, {3306, 5274}, {3338, 51785}, {3358, 63972}, {3359, 5722}, {3475, 60937}, {3583, 17699}, {3586, 5842}, {3600, 9800}, {3677, 7004}, {3870, 10394}, {4293, 58808}, {4307, 40960}, {4313, 5250}, {4321, 63995}, {4336, 62845}, {4413, 17604}, {4421, 15297}, {4666, 53055}, {4857, 17700}, {4863, 66252}, {5173, 43166}, {5218, 7308}, {5227, 10387}, {5249, 60925}, {5282, 54359}, {5432, 51780}, {5436, 62333}, {5441, 59342}, {5660, 51768}, {5687, 9844}, {5709, 15171}, {5727, 63137}, {5809, 17784}, {6244, 64157}, {6284, 37550}, {6769, 44547}, {6796, 66254}, {7069, 7322}, {7174, 24430}, {7221, 17469}, {7284, 37602}, {7701, 41546}, {7966, 10050}, {7994, 10398}, {8069, 52026}, {8545, 10578}, {8583, 18251}, {9578, 12617}, {9785, 28610}, {9845, 63987}, {10106, 10864}, {10167, 42884}, {10321, 63966}, {10386, 26921}, {10543, 66221}, {10582, 17603}, {10624, 62810}, {11220, 12706}, {11246, 60955}, {11518, 12564}, {12529, 19861}, {12575, 62858}, {12609, 50443}, {13405, 54370}, {14547, 37553}, {15006, 60974}, {15172, 24467}, {15298, 52665}, {15558, 66059}, {15829, 64042}, {16141, 63276}, {16670, 61397}, {16688, 23207}, {17594, 50616}, {17642, 62823}, {27542, 56519}, {30284, 62856}, {30503, 57278}, {31393, 61291}, {31795, 59318}, {32926, 65952}, {37730, 49163}, {38271, 58631}, {41229, 53053}, {41864, 54290}, {45633, 45637}, {55871, 63145}, {59316, 66201}, {60911, 64346}, {60990, 66210}, {61086, 62811}, {62776, 64108}, {62873, 64150}, {63969, 65671}, {66205, 66245}

X(66239) = midpoint of X(i) and X(j) for these {i,j}: {4294, 18391}, {12705, 63430}, {12711, 66226}
X(66239) = reflection of X(i) in X(j) for these {i,j}: {57, 62839}, {997, 5248}
X(66239) = isogonal conjugate of X(8829)
X(66239) = perspector of circumconic {{A, B, C, X(644), X(37141)}}
X(66239) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 8829}, {2, 8828}, {57, 56230}
X(66239) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 8829}, {5452, 56230}, {14986, 63151}, {32664, 8828}
X(66239) = pole of line {4394, 23224} with respect to the circumcircle
X(66239) = pole of line {9, 56} with respect to the Feuerbach hyperbola
X(66239) = pole of line {1014, 8829} with respect to the Stammler hyperbola
X(66239) = pole of line {8829, 57785} with respect to the Wallace hyperbola
X(66239) = pole of line {4162, 14298} with respect to the Hofstadter ellipse
X(66239) = pole of line {948, 24181} with respect to the dual conic of Yff parabola
X(66239) = intersection, other than A, B, C, of circumconics {{A, B, C, X(9), X(1422)}}, {{A, B, C, X(55), X(1413)}}, {{A, B, C, X(84), X(200)}}, {{A, B, C, X(210), X(52384)}}, {{A, B, C, X(222), X(55111)}}, {{A, B, C, X(480), X(2192)}}, {{A, B, C, X(1260), X(1433)}}, {{A, B, C, X(3694), X(52037)}}
X(66239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 30304, 17625}, {1, 7992, 66250}, {1, 7995, 12709}, {31, 4319, 7070}, {33, 52428, 5269}, {55, 14100, 10382}, {55, 1864, 200}, {55, 60910, 210}, {57, 10384, 497}, {4314, 12514, 1697}, {4326, 4512, 55}, {4512, 42012, 9}, {4907, 5269, 33}, {7994, 10398, 41539}, {11019, 64129, 57}, {11496, 12710, 1}, {12705, 63430, 6001}

X(66240) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-8 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    3*a^4-4*a^3*(b+c)-2*(b^2-c^2)^2-a^2*(b^2-12*b*c+c^2)+a*(4*b^3-6*b^2*c-6*b*c^2+4*c^3) : :
X(66240) = -4*X[40]+3*X[34620], -2*X[944]+3*X[4421], -2*X[962]+3*X[34739], -3*X[3679]+2*X[11260], -X[3680]+3*X[37712], -2*X[3813]+3*X[59388], -3*X[5587]+2*X[33895], -4*X[5690]+3*X[11194], -3*X[5731]+4*X[32157], -3*X[5790]+2*X[22837], -3*X[7967]+4*X[64123], -2*X[8666]+3*X[59503] and many others

Let A'B'C' be the cevian triangle of X(8). CTR1-8 is triangle formed by the Aubert lines of the following quadrilaterals AB'PC', BC'PA', CA'PB'.

X(66240) lies on these lines: {1, 5123}, {4, 5854}, {8, 56}, {10, 35272}, {11, 145}, {40, 34620}, {355, 381}, {515, 64744}, {517, 64725}, {518, 3632}, {528, 64000}, {529, 11826}, {944, 4421}, {952, 3913}, {956, 59334}, {958, 11508}, {962, 34739}, {1317, 5552}, {1320, 10896}, {1483, 26492}, {1709, 2136}, {2098, 5176}, {2802, 18525}, {3036, 3086}, {3057, 17615}, {3189, 27870}, {3244, 11373}, {3336, 4677}, {3434, 3621}, {3436, 10947}, {3623, 10584}, {3625, 17647}, {3633, 10826}, {3679, 11260}, {3680, 37712}, {3811, 46920}, {3812, 17624}, {3813, 59388}, {3880, 5881}, {4428, 45081}, {4701, 24391}, {5057, 63209}, {5289, 64087}, {5330, 31141}, {5587, 33895}, {5687, 37707}, {5690, 11194}, {5727, 17622}, {5731, 32157}, {5790, 22837}, {5794, 66205}, {5836, 17625}, {5844, 10525}, {5853, 16112}, {5927, 12448}, {6735, 37738}, {6890, 54177}, {6891, 54176}, {7967, 64123}, {8666, 59503}, {8715, 18526}, {9041, 24834}, {9053, 12586}, {9897, 17652}, {9948, 12640}, {10528, 37734}, {10598, 38156}, {10785, 34619}, {10915, 37727}, {10948, 17757}, {10950, 10965}, {11256, 15863}, {11280, 11523}, {11499, 22560}, {11500, 40255}, {11865, 12455}, {11866, 12454}, {12700, 28234}, {12737, 22836}, {13463, 59387}, {13895, 44635}, {13952, 44636}, {14450, 15679}, {17617, 35634}, {17618, 37714}, {17626, 58609}, {17765, 36280}, {18236, 58679}, {18516, 37705}, {20035, 36576}, {20050, 25568}, {21290, 59598}, {21669, 44669}, {30852, 33176}, {31140, 64201}, {31145, 34605}, {32426, 64068}, {33812, 51577}, {34629, 34697}, {35262, 37829}, {36920, 62874}, {37739, 49626}, {37828, 63987}, {38155, 64205}, {42871, 64163}, {44784, 63130}, {45700, 61510}, {56176, 61296}, {59719, 61287}, {61244, 64768}, {63324, 63415}

X(66240) = midpoint of X(i) and X(j) for these {i,j}: {61244, 64768}
X(66240) = reflection of X(i) in X(j) for these {i,j}: {1, 32537}, {145, 12607}, {3913, 49169}, {10912, 355}, {11235, 34717}, {11256, 15863}, {12513, 8}, {12635, 32049}, {18526, 8715}, {22560, 64140}, {24391, 4701}, {34710, 11236}, {37727, 10915}, {47746, 49600}, {61296, 56176}
X(66240) = pole of line {11238, 66216} with respect to the Feuerbach hyperbola
X(66240) = X(145)-of-inner-Johnson triangle
X(66240) = X(2883)-of-Ursa-major triangle
X(66240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 10944, 1376}, {8, 3476, 8256}, {8, 36977, 40663}, {8, 38455, 12513}, {355, 10912, 11235}, {355, 1482, 10893}, {355, 47746, 49600}, {355, 519, 10912}, {519, 11236, 34710}, {519, 32049, 12635}, {519, 49600, 47746}, {952, 49169, 3913}, {3632, 37708, 10914}, {3913, 12114, 13205}, {10912, 34717, 355}, {10915, 37727, 56177}, {32537, 33956, 1}

X(66241) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR5-8.8

Barycentrics    2*a^4-a^3*(b+c)+a*(b-c)^2*(b+c)-(b^2-c^2)^2-a^2*(b^2+26*b*c+c^2) : :
X(66241) = 3*X[553]+4*X[12575], 3*X[3058]+4*X[12577], 5*X[3623]+2*X[5795], -8*X[3636]+X[63146], 3*X[3742]+4*X[66259], X[4292]+6*X[15170], 6*X[5049]+X[10624], 5*X[5734]+2*X[64706], 3*X[5919]+4*X[6744], -9*X[38314]+2*X[57284], 3*X[40998]+4*X[58609]

Let XYZ be the anticevian triangle of X(8). Denote with X' the Kimberling-Pavlov X-conjugate of X(8) and X(8), and similarly define Y' and Z'. CTR5-8.8 is the triangle X'Y'Z'.

X(66241) lies on these lines: {1, 4}, {10, 8162}, {145, 62218}, {519, 3983}, {527, 62854}, {553, 12575}, {938, 51779}, {1125, 3689}, {1420, 8236}, {2646, 43179}, {3058, 12577}, {3303, 3911}, {3304, 30331}, {3474, 30343}, {3623, 5795}, {3636, 63146}, {3672, 63578}, {3742, 66259}, {3982, 12701}, {4114, 9589}, {4292, 15170}, {4666, 21627}, {4848, 10580}, {5049, 10624}, {5442, 64124}, {5734, 64706}, {5919, 6744}, {7308, 9797}, {7320, 64736}, {9785, 44841}, {10586, 59584}, {10587, 24386}, {15172, 28182}, {28194, 50190}, {38314, 57284}, {40998, 58609}

X(66241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40270, 950}, {3303, 21625, 3911}, {10580, 37556, 4848}, {12575, 17609, 553}

X(66242) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT MID-TRIANGLE-OF-MEDIAL-AND-INTOUCH

Barycentrics    a*(a-b-c)*(-16*a^3*b*c+10*a*b*(b-c)^2*c+3*a^4*(b+c)+a^2*(-6*b^3+3*b^2*c+3*b*c^2-6*c^3)+3*(b-c)^2*(b^3-2*b^2*c-2*b*c^2+c^3)) : :
X(66242) = -3*X[9957]+X[44685]

X(66242) lies on these lines: {1, 1389}, {9, 3885}, {497, 5559}, {517, 66195}, {519, 66207}, {950, 5844}, {1697, 64201}, {1837, 15862}, {3057, 3625}, {3486, 64766}, {3880, 58638}, {4342, 10958}, {5225, 64291}, {5493, 15338}, {7173, 12053}, {7962, 64199}, {9819, 11524}, {9957, 44685}, {12019, 15558}, {12575, 66206}, {13143, 66199}, {14100, 39777}, {20050, 61030}

X(66242) = pole of line {3626, 11011} with respect to the Feuerbach hyperbola

X(66243) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR10-8 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(a^3-5*a^2*(b+c)+(b-c)^2*(b+c)+a*(-5*b^2+18*b*c-5*c^2)) : :
X(66243) = -3*X[2]+2*X[3680], -5*X[3091]+4*X[64767], -6*X[3158]+5*X[3623], -5*X[3616]+4*X[10912], -7*X[4678]+6*X[24392], -3*X[7967]+2*X[47746], -9*X[9779]+8*X[13463], -3*X[9802]+4*X[13272], -3*X[9812]+4*X[32049], -8*X[11260]+9*X[64108], -2*X[12513]+3*X[34711], -2*X[12629]+3*X[59417] and many others

Let A'B'C' be the Gemini 29 trianlge. Denote with Pa the trace of AA' upon the circumconic with perspector X(8). Similarly define Pb and Pc. CTR10-8 is the triangle PaPbPc.

X(66243) lies on these lines: {1, 56090}, {2, 3680}, {4, 64768}, {7, 13601}, {8, 210}, {20, 519}, {57, 145}, {63, 66245}, {100, 6049}, {144, 3621}, {149, 7319}, {153, 962}, {390, 66198}, {404, 3241}, {443, 9874}, {517, 6223}, {952, 12246}, {1145, 5704}, {1320, 27383}, {1697, 11106}, {1788, 13996}, {2098, 64083}, {2475, 12648}, {3030, 36805}, {3085, 64203}, {3091, 64767}, {3149, 38665}, {3158, 3623}, {3189, 5854}, {3616, 10912}, {3632, 10624}, {3633, 4311}, {3648, 20053}, {3699, 8834}, {3895, 4313}, {4342, 8165}, {4345, 7080}, {4678, 24392}, {4848, 61630}, {4853, 5273}, {5328, 6736}, {5435, 36846}, {5697, 5815}, {5734, 6953}, {5768, 6764}, {5828, 30384}, {6556, 62297}, {6743, 8275}, {6766, 28234}, {7674, 12630}, {7963, 37743}, {7967, 47746}, {9623, 17554}, {9778, 32426}, {9779, 13463}, {9802, 13272}, {9812, 32049}, {9957, 17559}, {10580, 66256}, {11114, 34689}, {11260, 64108}, {12513, 34711}, {12629, 59417}, {12649, 64743}, {15347, 64114}, {15933, 16410}, {17460, 28016}, {20008, 64736}, {20014, 63145}, {20057, 56176}, {22837, 54445}, {28370, 61222}, {31509, 59414}, {31789, 34745}, {33895, 38314}, {34716, 50693}, {37709, 60961}, {38496, 64442}, {47444, 65966}, {49169, 59387}, {52804, 62837}, {52811, 66208}, {52813, 66209}

X(66243) = reflection of X(i) in X(j) for these {i,j}: {4, 64768}, {20, 64202}, {145, 2136}, {149, 12641}, {3680, 12640}, {6764, 12245}, {12536, 12632}, {12541, 8}, {12630, 7674}, {20050, 3189}, {64068, 64744}
X(66243) = anticomplement of X(3680)
X(66243) = X(i)-Ceva conjugate of X(j) for these {i, j}: {39126, 2}
X(66243) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {56, 3621}, {57, 21296}, {58, 11682}, {59, 3699}, {109, 3667}, {145, 3436}, {604, 17490}, {651, 4106}, {1252, 27834}, {1407, 4373}, {1412, 17151}, {1420, 8}, {1461, 3676}, {1743, 329}, {2149, 25268}, {3052, 144}, {3161, 54113}, {3451, 30567}, {3667, 33650}, {4394, 37781}, {4565, 4897}, {4848, 1330}, {4855, 52366}, {5435, 69}, {6049, 42020}, {8643, 39351}, {16948, 3869}, {18743, 21286}, {20818, 56943}, {30719, 150}, {32735, 53523}, {33628, 63}, {39126, 6327}, {40151, 33800}, {41629, 20245}, {51656, 149}, {57192, 4462}, {58858, 34548}, {62787, 3434}, {64736, 21291}
X(66243) = pole of line {4462, 4521} with respect to the Steiner circumellipse
X(66243) = pole of line {25268, 27834} with respect to the Yff parabola
X(66243) = pole of line {5328, 24175} with respect to the dual conic of Yff parabola
X(66243) = X(5895)-of-2nd-Conway triangle
X(66243) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(42020)}}, {{A, B, C, X(8), X(2137)}}, {{A, B, C, X(57), X(8055)}}, {{A, B, C, X(312), X(8051)}}, {{A, B, C, X(341), X(6553)}}, {{A, B, C, X(5423), X(56089)}}, {{A, B, C, X(7320), X(44301)}}
X(66243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 3057, 18228}, {8, 3880, 12541}, {8, 3885, 9785}, {57, 66258, 145}, {145, 2136, 64146}, {145, 63130, 4308}, {519, 12632, 12536}, {519, 64202, 20}, {3189, 5854, 20050}, {3680, 12640, 2}, {3680, 64204, 64205}, {3880, 64744, 64068}, {12640, 64205, 64204}, {36846, 63133, 5435}, {42020, 64563, 8055}, {64068, 64744, 8}

X(66244) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-7 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a^5-4*a^4*(b+c)+a^3*(4*b^2+6*b*c+4*c^2)+2*a^2*(b^3+c^3)+2*(b-c)^2*(b^3+c^3)+a*(-5*b^4+2*b^3*c-2*b^2*c^2+2*b*c^3-5*c^4)) : :
X(66244) = -3*X[38031]+2*X[60994]

Let A'B'C' be the intouch triangle. CTR1-7 is triangle formed by the Aubert lines of the following quadrilaterals AB'PC', BC'PA', CA'PB'.

X(66244) lies on these lines: {1, 6}, {7, 62830}, {8, 7679}, {11, 5748}, {78, 11526}, {390, 62826}, {517, 64312}, {519, 64731}, {527, 3655}, {528, 10698}, {944, 61010}, {1320, 12630}, {1376, 5173}, {1385, 60974}, {1389, 6601}, {1445, 56387}, {1482, 5805}, {1617, 5083}, {2099, 2550}, {3059, 11011}, {3174, 7982}, {3428, 56177}, {3811, 22753}, {3826, 5855}, {3869, 30284}, {3870, 17642}, {3897, 61024}, {3940, 24393}, {4018, 60968}, {4421, 41338}, {4511, 7672}, {4861, 34784}, {5330, 8236}, {5542, 62822}, {5780, 5886}, {5818, 12607}, {6265, 66054}, {6600, 22770}, {7373, 61033}, {7675, 11682}, {8730, 24474}, {10247, 61030}, {11038, 34195}, {11495, 14110}, {12573, 61021}, {13464, 24389}, {15934, 64734}, {15950, 64081}, {17614, 60985}, {17768, 36996}, {25893, 61663}, {30144, 30329}, {34588, 38288}, {38031, 60994}, {41712, 64154}, {44663, 65426}, {49168, 64294}

X(66244) = midpoint of X(i) and X(j) for these {i,j}: {944, 61010}, {3174, 7982}
X(66244) = reflection of X(i) in X(j) for these {i,j}: {6600, 22836}, {24389, 13464}, {49168, 64443}, {60974, 1385}
X(66244) = pole of line {1376, 25006} with respect to the dual conic of Moses-Feuerbach circumconic
X(66244) = intersection, other than A, B, C, of circumconics {{A, B, C, X(218), X(1389)}}, {{A, B, C, X(16601), X(56027)}}

X(66245) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-1.8 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a-b-c)*(a^5+a^4*(b+c)+(b-c)^4*(b+c)-2*a^3*(b^2+c^2)-2*a^2*(b^3-5*b^2*c-5*b*c^2+c^3)+a*(b^4+8*b^3*c-34*b^2*c^2+8*b*c^3+c^4)) : :
X(66245) = -3*X[3576]+4*X[8668]

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(66245) lies on these lines: {1, 11505}, {8, 9}, {40, 3880}, {57, 1476}, {63, 66243}, {84, 519}, {200, 66216}, {936, 3913}, {1210, 63137}, {1376, 12448}, {1706, 6692}, {2802, 66058}, {3158, 19861}, {3333, 10107}, {3576, 8668}, {4882, 58696}, {5435, 12541}, {5437, 64205}, {5854, 66059}, {6762, 63985}, {6765, 12672}, {7308, 64204}, {7320, 64146}, {7330, 64768}, {7701, 12703}, {7966, 64117}, {7991, 60990}, {7995, 15733}, {10864, 38455}, {11372, 32049}, {12536, 51786}, {12641, 64372}, {24392, 24982}, {34862, 49163}, {59335, 64203}, {64129, 66217}, {66205, 66239}

X(66245) = pole of line {1339, 4498} with respect to the Bevan circle
X(66245) = pole of line {4468, 60482} with respect to the Steiner circumellipse
X(66245) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(57), X(12640)}}, {{A, B, C, X(1476), X(3161)}}, {{A, B, C, X(2347), X(16945)}}
X(66245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 2136, 12640}, {3895, 12632, 2136}

X(66246) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-7.2 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a+b-c)*(a-b+c)*(5*a^5-11*a^4*(b+c)+(b-c)^4*(b+c)+2*a^3*(b^2+22*b*c+c^2)-a*(b-c)^2*(7*b^2+18*b*c+7*c^2)+2*a^2*(5*b^3-13*b^2*c-13*b*c^2+5*c^3)) : :

Let QaQbQc be the cevian triangle of X(2). CTR12-7.2 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(7)-circumconic.

X(66246) lies on these lines: {1, 7}, {8, 36620}, {144, 64980}, {145, 31527}, {479, 2098}, {517, 56870}, {518, 15913}, {934, 8158}, {3057, 3599}, {3598, 61630}, {3623, 56309}, {3680, 56275}, {4460, 33673}, {7320, 9446}, {9533, 11531}, {9778, 45228}, {15511, 37714}, {16284, 25718}, {28610, 43044}, {32003, 34060}, {40133, 60941}, {52819, 56043}

X(66246) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(43182)}}, {{A, B, C, X(2951), X(3680)}}, {{A, B, C, X(3160), X(56026)}}, {{A, B, C, X(56275), X(62787)}}

X(66247) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR12-8.8

Barycentrics    6*a^4-3*a^3*(b+c)+3*a*(b-c)^2*(b+c)+a^2*(-3*b^2+2*b*c-3*c^2)-3*(b^2-c^2)^2 : :
X(66247) = -3*X[553]+4*X[6738], -3*X[5434]+4*X[6744], -2*X[6743]+3*X[34606], -28*X[9780]+27*X[46916], -6*X[12433]+5*X[50191], -3*X[12743]+X[62617]

Let QaQbQc be the cevian triangle of X(8). CTR12-8.8 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(8)-circumconic.

X(66247) lies on these lines: {1, 4}, {2, 7319}, {8, 3929}, {10, 5217}, {20, 4848}, {30, 41551}, {63, 66251}, {65, 28164}, {80, 6684}, {144, 3621}, {145, 9580}, {149, 64205}, {355, 4304}, {382, 37739}, {390, 37709}, {498, 50796}, {499, 51705}, {516, 10950}, {517, 41562}, {519, 3962}, {527, 41575}, {528, 66205}, {551, 10896}, {553, 6738}, {942, 28186}, {952, 10624}, {1125, 7173}, {1145, 3626}, {1210, 18481}, {1385, 10593}, {1737, 5442}, {1770, 28172}, {1837, 3911}, {1898, 3878}, {2093, 3529}, {2098, 51783}, {2099, 51118}, {2646, 3614}, {2801, 64043}, {2829, 41558}, {2840, 18732}, {3057, 28236}, {3086, 50811}, {3091, 13384}, {3146, 3340}, {3244, 12701}, {3436, 12437}, {3600, 37723}, {3601, 59387}, {3612, 10175}, {3617, 5273}, {3625, 50242}, {3627, 50194}, {3634, 7483}, {3655, 9669}, {3671, 12943}, {3817, 34471}, {3828, 52793}, {4292, 28160}, {4294, 5881}, {4299, 37721}, {4301, 12953}, {4302, 11362}, {4305, 5587}, {4309, 37708}, {4311, 5722}, {4313, 9578}, {4314, 5252}, {4323, 17578}, {4330, 9897}, {4342, 9670}, {4551, 65670}, {5086, 5745}, {5119, 47745}, {5218, 37714}, {5274, 63208}, {5434, 6744}, {5441, 10039}, {5443, 17501}, {5493, 41687}, {5542, 9657}, {5657, 43734}, {5731, 9581}, {5818, 30282}, {5837, 6872}, {5903, 28150}, {6224, 41012}, {6700, 10609}, {6713, 12019}, {6735, 11015}, {6737, 57288}, {6743, 34606}, {6936, 64315}, {7080, 34701}, {7700, 51724}, {7987, 54361}, {9668, 37727}, {9778, 64895}, {9780, 46916}, {9812, 64964}, {9844, 64106}, {9957, 28224}, {10165, 10826}, {10392, 43161}, {10543, 13405}, {10573, 31730}, {10588, 53054}, {10589, 30389}, {10591, 64953}, {10592, 13411}, {10895, 34648}, {10944, 12575}, {11011, 65632}, {11041, 33703}, {11502, 63983}, {11545, 31663}, {12433, 50191}, {12512, 40663}, {12527, 44669}, {12571, 15950}, {12640, 20075}, {12736, 13369}, {12743, 62617}, {12764, 33337}, {13601, 15726}, {15171, 28204}, {15338, 43174}, {16193, 16616}, {16948, 64582}, {17632, 41539}, {17895, 18650}, {18220, 64849}, {18357, 61520}, {18525, 31397}, {18990, 28208}, {20007, 63916}, {20066, 51433}, {20070, 64736}, {20085, 64372}, {21578, 37702}, {22793, 37728}, {25440, 44848}, {26066, 63754}, {28194, 65134}, {28234, 37706}, {29353, 64580}, {30827, 63915}, {33697, 39542}, {34628, 53057}, {34773, 44675}, {37001, 54227}, {37080, 51782}, {37540, 44039}, {37606, 61261}, {37828, 63753}, {38127, 59316}, {38455, 66258}, {39892, 49505}, {40267, 41561}, {43759, 65822}, {49515, 64567}, {50195, 66195}, {50240, 64732}, {50693, 63207}, {59388, 61763}, {64087, 64117}

X(66247) = reflection of X(i) in X(j) for these {i,j}: {950, 10572}, {4292, 37730}, {6737, 57288}, {7354, 6738}, {10106, 950}, {10944, 12575}, {57287, 5795}
X(66247) = pole of line {65, 3817} with respect to the Feuerbach hyperbola
X(66247) = intersection, other than A, B, C, of circumconics {{A, B, C, X(34), X(7285)}}, {{A, B, C, X(5229), X(10570)}}
X(66247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 5229, 226}, {1, 5691, 5229}, {20, 5727, 4848}, {515, 10572, 950}, {515, 950, 10106}, {944, 3586, 12053}, {1837, 4297, 3911}, {3486, 5229, 1}, {5441, 37006, 10039}, {6738, 7354, 553}, {12943, 37724, 3671}, {12953, 37740, 4301}, {21578, 37702, 64124}, {28160, 37730, 4292}, {37568, 62616, 3626}

X(66248) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR12-9.21

Barycentrics    a*(a^5*(b+c)-(b-c)^2*(b+c)^4-a^4*(b^2-4*b*c+c^2)+2*a^2*(b+c)^2*(b^2-3*b*c+c^2)-2*a^3*(b^3+c^3)+a*(b^5-b^4*c-b*c^4+c^5)) : :
X(66248) = -5*X[3616]+3*X[17616], -7*X[3624]+6*X[10855]

Let QaQbQc be the cevian triangle of X(21). CTR12-9.21 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(9)-circumconic.

X(66248) lies on these lines: {1, 971}, {3, 30223}, {4, 12711}, {11, 9940}, {35, 3683}, {36, 31805}, {40, 1864}, {46, 64157}, {52, 517}, {55, 1898}, {56, 50528}, {65, 3586}, {72, 3189}, {90, 31445}, {100, 58649}, {210, 61763}, {226, 12710}, {354, 9614}, {380, 1903}, {390, 12528}, {392, 4305}, {405, 18251}, {443, 17668}, {452, 12529}, {496, 3660}, {497, 1071}, {498, 10157}, {499, 11227}, {516, 12432}, {518, 10624}, {912, 15171}, {938, 9961}, {942, 1479}, {944, 66226}, {946, 10391}, {950, 6001}, {960, 4304}, {962, 10394}, {1058, 17625}, {1210, 9943}, {1319, 16132}, {1385, 62333}, {1420, 63432}, {1467, 10092}, {1490, 66239}, {1617, 41854}, {1697, 14872}, {1728, 5584}, {1737, 31787}, {1770, 37544}, {1837, 31788}, {2310, 4300}, {2771, 31795}, {2801, 12575}, {3057, 5693}, {3085, 5927}, {3086, 10167}, {3295, 40263}, {3333, 63995}, {3465, 5266}, {3486, 12672}, {3488, 12709}, {3555, 30305}, {3583, 5806}, {3616, 17616}, {3624, 10855}, {3649, 5045}, {3871, 17615}, {3874, 51783}, {3962, 5697}, {4292, 15726}, {4295, 5728}, {4302, 31793}, {4314, 31803}, {4319, 7078}, {4857, 5570}, {5119, 34790}, {5128, 61718}, {5173, 12699}, {5259, 51768}, {5439, 10591}, {5441, 44782}, {5710, 36985}, {5711, 65128}, {5768, 17649}, {5784, 31435}, {5809, 12706}, {5882, 66216}, {5904, 64723}, {5918, 15803}, {6259, 10629}, {6361, 41539}, {6835, 60925}, {7082, 37601}, {7671, 11036}, {7743, 13373}, {7957, 18397}, {7965, 11018}, {7967, 17622}, {8069, 64804}, {8071, 34862}, {8193, 64121}, {8227, 17603}, {8715, 51380}, {9371, 37732}, {9589, 18412}, {9668, 24474}, {9669, 10202}, {9670, 64046}, {9812, 62864}, {9844, 18391}, {9942, 63989}, {9947, 10039}, {9957, 37738}, {10050, 11249}, {10058, 22935}, {10382, 12664}, {10393, 11496}, {10396, 12565}, {10523, 64813}, {10573, 31798}, {10966, 45632}, {11019, 64132}, {11220, 14986}, {11415, 16465}, {11529, 17634}, {11551, 15008}, {12053, 12675}, {12136, 34231}, {12514, 64171}, {12520, 57278}, {12669, 54228}, {12671, 63992}, {12758, 62617}, {13405, 31871}, {13407, 16201}, {13600, 37740}, {13601, 37730}, {15310, 37613}, {17637, 49177}, {17646, 54318}, {18236, 59591}, {18480, 64086}, {19541, 59335}, {24929, 31937}, {25466, 41871}, {25917, 30282}, {29207, 49542}, {31786, 64042}, {31822, 65632}, {33575, 59325}, {37411, 37550}, {37568, 58643}, {37722, 58576}, {38850, 58337}, {40262, 59334}, {41229, 42014}, {41339, 54301}, {41340, 64537}, {42450, 44670}, {44675, 58567}, {45120, 51090}, {51787, 56762}, {59316, 61709}, {61086, 66235}, {62810, 64077}, {63430, 66194}, {63998, 66227}, {63999, 66250}, {64160, 66195}

X(66248) = midpoint of X(i) and X(j) for these {i,j}: {1858, 6284}, {10624, 41562}
X(66248) = reflection of X(i) in X(j) for these {i,j}: {1770, 37544}, {13601, 37730}, {63999, 66254}, {66250, 63999}
X(66248) = pole of line {3900, 48302} with respect to the incircle
X(66248) = pole of line {5, 57} with respect to the Feuerbach hyperbola
X(66248) = pole of line {3900, 48307} with respect to the Suppa-Cucoanes circle
X(66248) = X(5777)-of-Mandart-incircle triangle
X(66248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 64741, 64057}, {4, 12711, 50195}, {55, 1898, 5777}, {496, 13369, 3660}, {497, 1071, 50196}, {946, 10391, 16193}, {1058, 64358, 17625}, {1858, 6284, 517}, {3583, 13750, 5806}, {5584, 60910, 1728}, {10624, 41562, 518}, {12688, 14100, 1}

X(66249) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-9.29 WRT 1ST-VAN-KHEA-PAVLOV

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

Let QaQbQc be the cevian triangle of X(29). CTR12-9.29 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(9)-circumconic.

X(66249) lies on these lines: {1, 3}, {4, 7078}, {6, 950}, {8, 29}, {10, 7074}, {11, 1714}, {19, 43213}, {20, 222}, {30, 3157}, {33, 72}, {73, 7580}, {77, 63141}, {78, 56178}, {81, 4313}, {109, 41402}, {145, 7538}, {201, 64750}, {212, 405}, {221, 516}, {255, 1012}, {278, 412}, {347, 20070}, {355, 7524}, {382, 8757}, {387, 497}, {388, 3332}, {390, 54358}, {394, 57287}, {452, 55432}, {474, 22072}, {515, 1498}, {518, 15954}, {519, 2192}, {580, 57278}, {603, 37022}, {651, 3146}, {758, 1854}, {912, 64054}, {916, 6285}, {938, 52424}, {946, 37695}, {958, 2328}, {990, 66250}, {1013, 3869}, {1191, 12053}, {1210, 36745}, {1249, 22124}, {1253, 59305}, {1364, 37482}, {1396, 6060}, {1468, 65670}, {1479, 5721}, {1612, 66199}, {1657, 23070}, {1745, 37411}, {1780, 22760}, {1785, 5812}, {1794, 64840}, {1816, 3871}, {1826, 10367}, {1837, 16471}, {1838, 12699}, {1891, 22132}, {1935, 22117}, {1944, 52346}, {2184, 11523}, {2292, 4336}, {2323, 12625}, {2361, 62333}, {3074, 6913}, {3100, 3868}, {3149, 22350}, {3190, 3913}, {3194, 44695}, {3346, 15501}, {3419, 40950}, {3434, 37235}, {3486, 62843}, {3522, 17074}, {3560, 52408}, {3586, 54301}, {3616, 7572}, {3682, 5687}, {3811, 51361}, {3901, 9576}, {3927, 24430}, {4194, 41883}, {4200, 36949}, {4295, 55010}, {4297, 34046}, {4301, 59645}, {4302, 63339}, {4303, 37426}, {4304, 36746}, {4314, 62805}, {4319, 12711}, {4383, 9581}, {4646, 54369}, {4847, 34831}, {5044, 9817}, {5250, 40937}, {5315, 51785}, {5438, 25934}, {5603, 7567}, {5691, 9370}, {5722, 36754}, {5752, 40944}, {5758, 7952}, {5763, 15252}, {5777, 65128}, {5806, 19372}, {6056, 56831}, {6180, 9579}, {6284, 64020}, {6361, 30268}, {6848, 52659}, {6872, 55400}, {6890, 43043}, {7355, 15951}, {7412, 63436}, {7531, 12245}, {9555, 49653}, {9785, 62804}, {10373, 11471}, {10535, 42463}, {10571, 64077}, {10624, 64449}, {10914, 37393}, {11429, 56960}, {12329, 22299}, {12575, 62828}, {12635, 45272}, {12649, 55399}, {12672, 57276}, {13346, 36059}, {14872, 36985}, {14923, 37253}, {15811, 63998}, {15852, 45126}, {16370, 22361}, {16389, 40953}, {17018, 35981}, {17811, 57284}, {19541, 37694}, {19843, 25490}, {23144, 64707}, {24391, 55405}, {26091, 64081}, {26932, 27505}, {30265, 54400}, {31837, 37696}, {33137, 37370}, {34043, 64005}, {36986, 51490}, {37046, 40152}, {37498, 56293}, {37723, 52423}, {37730, 44414}, {44661, 64022}, {45729, 57288}, {54386, 64131}, {60803, 61229}, {60925, 63341}, {63130, 64082}

X(66249) = reflection of X(i) in X(j) for these {i,j}: {64057, 3157}
X(66249) = perspector of circumconic {{A, B, C, X(651), X(36797)}}
X(66249) = X(i)-Ceva conjugate of X(j) for these {i, j}: {18655, 11347}, {44699, 108}
X(66249) = pole of line {7178, 44426} with respect to the polar circle
X(66249) = pole of line {1, 7535} with respect to the Feuerbach hyperbola
X(66249) = pole of line {14331, 36054} with respect to the MacBeath circumconic
X(66249) = pole of line {21, 222} with respect to the Stammler hyperbola
X(66249) = pole of line {1897, 57192} with respect to the Hutson-Moses hyperbola
X(66249) = pole of line {314, 348} with respect to the Wallace hyperbola
X(66249) = pole of line {52355, 57168} with respect to the dual conic of incircle
X(66249) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(2322)}}, {{A, B, C, X(3), X(2287)}}, {{A, B, C, X(8), X(1214)}}, {{A, B, C, X(29), X(57)}}, {{A, B, C, X(40), X(56146)}}, {{A, B, C, X(56), X(1172)}}, {{A, B, C, X(65), X(281)}}, {{A, B, C, X(219), X(22341)}}, {{A, B, C, X(517), X(3346)}}, {{A, B, C, X(607), X(1402)}}, {{A, B, C, X(942), X(64840)}}, {{A, B, C, X(1000), X(37528)}}, {{A, B, C, X(1429), X(14024)}}, {{A, B, C, X(2192), X(2352)}}, {{A, B, C, X(3362), X(15803)}}, {{A, B, C, X(7017), X(20618)}}, {{A, B, C, X(8758), X(43695)}}, {{A, B, C, X(14942), X(24310)}}, {{A, B, C, X(37582), X(55917)}}, {{A, B, C, X(41344), X(56261)}}
X(66249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1754, 56}, {1, 40, 1214}, {4, 7078, 34048}, {20, 3562, 222}, {30, 3157, 64057}, {212, 2654, 405}, {382, 23071, 8757}, {1498, 64069, 3173}, {4319, 54421, 12711}

X(66250) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR12-9.81

Barycentrics    a*(a+b-c)*(a-b+c)*(a^3*(b+c)-a*(b-c)^2*(b+c)-a^2*(b^2+c^2)+(b+c)^2*(b^2+c^2)) : :
X(66250) = -3*X[354]+X[1858], -3*X[553]+X[15556], -4*X[11035]+3*X[66228]

Let QaQbQc be the cevian triangle of X(81). CTR12-9.81 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(9)-circumconic.

X(66250) lies on these lines: {1, 84}, {2, 37566}, {3, 64132}, {4, 18239}, {5, 226}, {7, 8}, {9, 1467}, {12, 3812}, {20, 3057}, {21, 1319}, {27, 18178}, {34, 6180}, {37, 37523}, {38, 1042}, {46, 63976}, {55, 9943}, {56, 63}, {57, 72}, {73, 3666}, {78, 1466}, {109, 5266}, {201, 241}, {210, 1788}, {224, 11509}, {225, 3782}, {227, 986}, {354, 1858}, {390, 9961}, {392, 1420}, {404, 51379}, {405, 34489}, {442, 64115}, {443, 65000}, {452, 60934}, {495, 34339}, {496, 31937}, {497, 9799}, {517, 4292}, {519, 13601}, {553, 15556}, {603, 37539}, {651, 5262}, {758, 4298}, {938, 1864}, {944, 12671}, {946, 50196}, {950, 971}, {962, 17634}, {964, 28968}, {976, 9316}, {990, 66249}, {999, 5887}, {1004, 11501}, {1038, 1407}, {1056, 64021}, {1104, 1935}, {1108, 15656}, {1125, 3660}, {1155, 58637}, {1214, 4306}, {1259, 1470}, {1317, 47008}, {1376, 2057}, {1386, 23144}, {1387, 2771}, {1388, 10179}, {1406, 64349}, {1427, 37591}, {1445, 3951}, {1451, 4641}, {1452, 41611}, {1457, 11031}, {1458, 2292}, {1465, 3670}, {1478, 7686}, {1617, 12514}, {1697, 5732}, {1699, 30290}, {1706, 17658}, {1708, 3927}, {1737, 58631}, {1829, 23154}, {1836, 55109}, {1876, 1883}, {2082, 5781}, {2099, 11520}, {2285, 5782}, {2646, 18444}, {2800, 66230}, {2801, 6738}, {2975, 15823}, {3157, 64722}, {3218, 57283}, {3296, 55964}, {3303, 7675}, {3304, 62836}, {3333, 5693}, {3339, 5904}, {3340, 3555}, {3361, 5692}, {3474, 7957}, {3486, 9960}, {3487, 6833}, {3488, 64358}, {3585, 16616}, {3600, 3869}, {3601, 10167}, {3649, 10957}, {3663, 5930}, {3665, 34855}, {3671, 3874}, {3698, 4208}, {3740, 24914}, {3742, 11375}, {3752, 37694}, {3753, 9578}, {3754, 51782}, {3811, 37541}, {3838, 26481}, {3876, 5435}, {3877, 4308}, {3878, 4315}, {3880, 10944}, {3889, 4323}, {3890, 17576}, {3897, 18467}, {3901, 4355}, {3911, 5044}, {3916, 19525}, {3947, 5883}, {4032, 15443}, {4198, 37516}, {4293, 14110}, {4303, 37528}, {4304, 9957}, {4305, 63432}, {4311, 31786}, {4313, 5919}, {4321, 12526}, {4327, 54421}, {4640, 37579}, {4654, 14054}, {4662, 40663}, {4723, 56173}, {4847, 18251}, {4848, 34790}, {4870, 13751}, {4880, 15932}, {5047, 29007}, {5119, 37426}, {5126, 31838}, {5192, 28997}, {5217, 10178}, {5219, 5439}, {5221, 41538}, {5226, 6931}, {5261, 18419}, {5269, 35672}, {5270, 53615}, {5273, 7288}, {5290, 5902}, {5293, 9364}, {5433, 54357}, {5434, 34742}, {5438, 17612}, {5555, 41871}, {5563, 54432}, {5570, 12047}, {5665, 60953}, {5703, 6966}, {5708, 5780}, {5710, 54400}, {5714, 6968}, {5722, 40263}, {5728, 11518}, {5735, 9579}, {5768, 12664}, {5815, 41824}, {5884, 21620}, {5927, 9581}, {6284, 15726}, {6734, 57285}, {6735, 45080}, {6864, 11023}, {6871, 64715}, {6923, 24474}, {6958, 10202}, {6993, 61663}, {7091, 15829}, {7269, 64377}, {7289, 64022}, {7330, 57278}, {7354, 64003}, {7373, 40266}, {7411, 37568}, {7702, 64086}, {8069, 64118}, {8071, 37837}, {8165, 11678}, {8544, 63141}, {8545, 54392}, {8582, 12059}, {8679, 44545}, {8829, 44692}, {9119, 54405}, {9370, 54418}, {9785, 10430}, {9844, 37723}, {9856, 12053}, {9859, 12536}, {9940, 13411}, {9942, 18446}, {9964, 17660}, {10394, 60998}, {10399, 38271}, {10431, 12701}, {10444, 10480}, {10461, 10475}, {10506, 11888}, {10571, 37592}, {10914, 37709}, {10916, 64127}, {11011, 58609}, {11018, 63274}, {11019, 31803}, {11020, 17609}, {11035, 66228}, {11281, 58578}, {11508, 37287}, {11510, 20835}, {11551, 62859}, {11570, 13407}, {12005, 16193}, {12527, 61002}, {12529, 36845}, {12560, 15185}, {12563, 62852}, {12609, 64737}, {12669, 14100}, {13161, 51421}, {13243, 17638}, {13369, 24929}, {13373, 37737}, {14872, 18391}, {14923, 37435}, {15325, 58573}, {15528, 18260}, {15558, 20789}, {15605, 31794}, {15803, 64107}, {15845, 63989}, {15934, 37234}, {17054, 19372}, {17531, 37789}, {17614, 41389}, {17626, 50443}, {17637, 41695}, {18191, 37113}, {18221, 40269}, {18398, 23708}, {18527, 31828}, {18593, 63396}, {20117, 64124}, {20118, 58683}, {20323, 62873}, {21578, 44238}, {21871, 57286}, {22759, 37228}, {22767, 37302}, {24159, 37695}, {24391, 64171}, {24928, 32153}, {25066, 56546}, {26011, 34831}, {26357, 65404}, {26651, 27410}, {28610, 31165}, {30384, 37447}, {31231, 31446}, {31397, 31788}, {31821, 58577}, {31835, 34753}, {31837, 37582}, {34035, 54292}, {34050, 34937}, {34880, 37300}, {37080, 62800}, {37105, 63211}, {37106, 37605}, {37224, 60964}, {37468, 45287}, {37722, 65465}, {37738, 66256}, {39791, 41003}, {41554, 51529}, {43177, 66019}, {44675, 58576}, {46878, 51399}, {51380, 63990}, {54408, 64077}, {55921, 64344}, {58405, 62357}, {60936, 64002}, {63999, 66248}, {64006, 64700}

X(66250) = midpoint of X(i) and X(j) for these {i,j}: {1829, 23154}, {7354, 64043}, {45288, 64721}
X(66250) = reflection of X(i) in X(j) for these {i,j}: {15556, 37544}, {44547, 942}, {66248, 63999}
X(66250) = perspector of circumconic {{A, B, C, X(4554), X(37141)}}
X(66250) = pole of line {521, 3669} with respect to the incircle
X(66250) = pole of line {18344, 57089} with respect to the polar circle
X(66250) = pole of line {34948, 48330} with respect to the DeLongchamps ellipse
X(66250) = pole of line {20, 56} with respect to the Feuerbach hyperbola
X(66250) = pole of line {53761, 65205} with respect to the Kiepert parabola
X(66250) = pole of line {2194, 3057} with respect to the Stammler hyperbola
X(66250) = pole of line {4885, 17924} with respect to the Steiner inellipse
X(66250) = pole of line {21, 20895} with respect to the Wallace hyperbola
X(66250) = pole of line {521, 4905} with respect to the Suppa-Cucoanes circle
X(66250) = pole of line {14298, 21348} with respect to the Hofstadter ellipse
X(66250) = pole of line {650, 9364} with respect to the dual conic of DeLongchamps circle
X(66250) = pole of line {1214, 3663} with respect to the dual conic of Yff parabola
X(66250) = pole of line {2, 1466} with respect to the dual conic of Moses-Feuerbach circumconic
X(66250) = X(960)-of-2nd-anti-circumperp-tangential triangle
X(66250) = X(6146)-of-intouch triangle
X(66250) = X(9943)-of-Mandart-incircle triangle
X(66250) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(322)}}, {{A, B, C, X(7), X(1413)}}, {{A, B, C, X(8), X(2192)}}, {{A, B, C, X(21), X(20895)}}, {{A, B, C, X(69), X(1433)}}, {{A, B, C, X(75), X(84)}}, {{A, B, C, X(85), X(1422)}}, {{A, B, C, X(221), X(55015)}}, {{A, B, C, X(1122), X(1408)}}, {{A, B, C, X(1231), X(52037)}}, {{A, B, C, X(1394), X(8829)}}, {{A, B, C, X(1441), X(1476)}}, {{A, B, C, X(5555), X(56927)}}, {{A, B, C, X(11496), X(60158)}}, {{A, B, C, X(42696), X(55964)}}, {{A, B, C, X(44692), X(66239)}}
X(66250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1071, 10391}, {1, 12672, 66216}, {1, 15071, 12711}, {1, 7992, 66239}, {7, 3868, 65}, {7, 52385, 24471}, {12, 18838, 3812}, {55, 64704, 9943}, {56, 64041, 960}, {65, 5252, 5836}, {65, 8581, 388}, {553, 15556, 37544}, {912, 942, 44547}, {938, 12528, 1864}, {942, 5777, 1210}, {960, 63994, 56}, {1012, 1071, 18238}, {1071, 12672, 84}, {1478, 64045, 7686}, {1829, 23154, 34371}, {3057, 63995, 20}, {3057, 9850, 3476}, {3339, 5904, 41539}, {3600, 3869, 64106}, {3671, 3874, 5173}, {5083, 64160, 5045}, {5434, 45288, 64721}, {5570, 12047, 13374}, {5884, 21620, 50195}, {6147, 24475, 942}, {6180, 37549, 34}, {11036, 62864, 354}, {11520, 16465, 34791}, {11570, 13407, 13750}, {12005, 64110, 16193}, {12709, 17625, 1}, {17634, 17642, 962}, {18238, 45776, 1012}, {45288, 64721, 44663}

X(66251) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-10.10 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    (a-b-c)*(4*a^3-a^2*(b+c)+3*(b-c)^2*(b+c)-2*a*(b^2+c^2)) : :
X(66251) = -3*X[3158]+5*X[3617], -X[3189]+3*X[3679], -4*X[3634]+3*X[56177], -3*X[5790]+2*X[59722], -X[6765]+3*X[59388], -2*X[8715]+3*X[38127], -3*X[10175]+2*X[22836], -X[11523]+3*X[59387], X[12541]+3*X[31145], -2*X[12607]+3*X[38155], -2*X[13607]+3*X[45700], -3*X[25568]+5*X[37714] and many others

Let QaQbQc be the cevian triangle of X(10). CTR12-10.10 is the triangle with vertices at the inversion poles of QbQc, QaQc, and QaQb wrt to the X(10)-circumconic.

X(66251) lies on these lines: {1, 6856}, {8, 9}, {10, 6675}, {63, 66247}, {65, 17668}, {80, 21075}, {142, 5794}, {145, 5226}, {191, 10572}, {210, 66219}, {226, 5086}, {355, 381}, {495, 3244}, {515, 5709}, {527, 5691}, {528, 31799}, {758, 31673}, {952, 64804}, {1210, 17614}, {1479, 3421}, {1706, 62836}, {1837, 3452}, {2900, 19860}, {3158, 3617}, {3189, 3679}, {3243, 20008}, {3340, 5175}, {3419, 64163}, {3445, 51615}, {3486, 5745}, {3621, 3680}, {3625, 3878}, {3626, 3913}, {3634, 56177}, {3684, 63595}, {3753, 10122}, {3812, 9858}, {3893, 17658}, {3897, 6734}, {3919, 47319}, {4067, 47320}, {4082, 42378}, {4301, 5855}, {4701, 64744}, {4847, 10950}, {4848, 57287}, {4863, 66205}, {5325, 21677}, {5330, 12053}, {5534, 40257}, {5690, 64117}, {5790, 59722}, {5836, 15733}, {5844, 64272}, {5882, 10916}, {6173, 18221}, {6601, 37712}, {6666, 66215}, {6765, 59388}, {6872, 66253}, {8666, 35252}, {8715, 38127}, {9581, 26129}, {10106, 12649}, {10175, 22836}, {10573, 63146}, {11260, 51717}, {11523, 59387}, {11529, 41865}, {11530, 59413}, {12513, 28236}, {12515, 31730}, {12541, 31145}, {12607, 38155}, {12609, 14563}, {12629, 63986}, {12647, 41709}, {12684, 28164}, {13607, 45700}, {17016, 56317}, {18391, 57284}, {18395, 59587}, {20085, 66068}, {21095, 35104}, {21285, 52563}, {24174, 53614}, {24987, 62870}, {25568, 37714}, {26015, 63987}, {31399, 59719}, {32537, 38158}, {34625, 61296}, {34744, 64005}, {36845, 37709}, {44663, 51118}, {45036, 64114}, {49169, 49184}, {51071, 63282}, {51515, 64768}, {51978, 64582}, {54154, 63989}, {56088, 63169}, {57288, 60942}, {59311, 64739}, {59340, 60994}

X(66251) = midpoint of X(i) and X(j) for these {i,j}: {8, 12625}, {3621, 3680}, {3632, 64068}, {20085, 66068}
X(66251) = reflection of X(i) in X(j) for these {i,j}: {145, 64205}, {3244, 3813}, {3913, 3626}, {5882, 10916}, {12437, 10}, {12635, 19925}, {12640, 8}, {24391, 49168}, {64117, 5690}, {64744, 4701}, {66215, 6666}
X(66251) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {17097, 42020}
X(66251) = pole of line {3667, 5794} with respect to the Fuhrmann circle
X(66251) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(3161), X(7319)}}, {{A, B, C, X(10005), X(63169)}}, {{A, B, C, X(55337), X(62178)}}
X(66251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 12625, 5853}, {8, 5727, 5795}, {8, 5795, 24393}, {8, 5853, 12640}, {8, 950, 5837}, {10, 12437, 59584}, {10, 44669, 12437}, {145, 24392, 64205}, {515, 49168, 24391}, {519, 19925, 12635}, {1837, 6737, 3452}, {3617, 12536, 3158}, {4678, 64146, 64204}, {5086, 41575, 226}, {5794, 6738, 142}

X(66252) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR13-1.9 WRT 1ST-VAN-KHEA-PAVLOV

Barycentrics    a*(a-b-c)*(a^4-8*a*b*c*(b+c)-2*a^2*(b^2-8*b*c+c^2)+(b-c)^2*(b^2+10*b*c+c^2)) : :
X(66252) = -4*X[1001]+3*X[3158], -4*X[3813]+3*X[38052], -3*X[8236]+2*X[12437], -3*X[11038]+4*X[64205], -6*X[24386]+5*X[40333], -3*X[34701]+4*X[43175], -3*X[38036]+4*X[49600], -6*X[38057]+5*X[64204]

Let A'B'C' be the X(1)-circumconcevian triangle of X(9). CTR13-1.9 is the tangential triangle of A'B'C' wrt X(1)-circumconic.

X(66252) lies on these lines: {1, 15587}, {7, 21627}, {8, 9}, {144, 12541}, {145, 60937}, {200, 17604}, {516, 6762}, {518, 3062}, {519, 11372}, {528, 1768}, {673, 30567}, {971, 12629}, {1001, 3158}, {2099, 3243}, {2550, 5437}, {2951, 12513}, {3059, 10866}, {3174, 8583}, {3189, 12447}, {3340, 30628}, {3621, 60966}, {3813, 38052}, {3880, 5223}, {3893, 60910}, {4853, 14100}, {4863, 66239}, {5438, 42884}, {5732, 30283}, {5785, 9957}, {5836, 30330}, {6601, 7091}, {6765, 9947}, {7308, 64146}, {7962, 41228}, {8236, 12437}, {8545, 12630}, {8582, 24389}, {9623, 63972}, {9841, 35514}, {9845, 43182}, {9846, 25722}, {9856, 11523}, {10398, 10914}, {11038, 64205}, {12536, 37556}, {15008, 40587}, {15299, 30286}, {17144, 64695}, {18227, 62218}, {24386, 40333}, {30290, 41863}, {32922, 65957}, {33576, 42470}, {34701, 43175}, {38036, 49600}, {38057, 64204}, {39126, 42309}, {60990, 63984}, {61012, 63142}

X(66252) = midpoint of X(i) and X(j) for these {i,j}: {144, 12541}, {3062, 11519}
X(66252) = reflection of X(i) in X(j) for these {i,j}: {7, 21627}, {2136, 9}, {2951, 12513}, {3189, 30331}, {11523, 43166}
X(66252) = intersection, other than A, B, C, of circumconics {{A, B, C, X(673), X(2136)}}, {{A, B, C, X(1697), X(17107)}}, {{A, B, C, X(3062), X(3161)}}, {{A, B, C, X(7091), X(55337)}}, {{A, B, C, X(33576), X(56937)}}
X(66252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 10384, 9}, {9, 5853, 2136}, {3062, 11519, 518}, {11519, 12448, 3680}

X(66253) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR13-8.8

Barycentrics    10*a^4-5*a^3*(b+c)+5*a*(b-c)^2*(b+c)-5*(b^2-c^2)^2-a^2*(5*b^2+2*b*c+5*c^2) : :
X(66253) = -7*X[4678]+10*X[5795], -X[10483]+4*X[17706], -5*X[17647]+8*X[19878], -13*X[19877]+10*X[57284], -11*X[46933]+5*X[57287]

Let A'B'C' be the X(8)-circumconcevian triangle of X(8). CTR13-8.8 is the tangential triangle of A'B'C' wrt X(8)-circumconic.

X(66253) lies on these lines: {1, 4}, {10, 19526}, {55, 38155}, {65, 28158}, {553, 28164}, {942, 28190}, {1837, 10164}, {2646, 10171}, {3058, 28236}, {3244, 9670}, {3626, 63273}, {3828, 6174}, {3911, 61717}, {3982, 12943}, {4292, 28168}, {4294, 63143}, {4297, 17728}, {4304, 26446}, {4309, 47745}, {4313, 54448}, {4345, 51791}, {4678, 5795}, {4701, 44784}, {4848, 9778}, {5219, 64836}, {5328, 63913}, {5441, 6684}, {5727, 59417}, {5844, 10624}, {5853, 6172}, {5902, 28172}, {6284, 28228}, {6738, 11246}, {6872, 66251}, {7686, 66195}, {9581, 54445}, {10385, 37712}, {10483, 17706}, {10543, 19925}, {11015, 63990}, {12563, 65631}, {13411, 38140}, {17647, 19878}, {17718, 34648}, {19877, 57284}, {28146, 37730}, {28174, 64163}, {30332, 64736}, {31730, 37721}, {37724, 51118}, {37740, 51783}, {46933, 57287}, {50796, 59337}

X(66253) = reflection of X(i) in X(j) for these {i,j}: {11246, 6738}

X(66254) = ORTHOLOGY CENTER OF THESE TRIANGLES: 1ST-VAN-KHEA-PAVLOV WRT CTR13-9.21

Barycentrics    a*(a^5*(b+c)-a^4*(b^2-6*b*c+c^2)-(b^2-c^2)^2*(b^2+3*b*c+c^2)-a^3*(2*b^3+b^2*c+b*c^2+2*c^3)+a^2*(2*b^4-3*b^3*c-14*b^2*c^2-3*b*c^3+2*c^4)+a*(b^5-b^3*c^2-b^2*c^3+c^5)) : :
X(66254) = 3*X[3058]+X[41562]

Let A'B'C' be the X(9)-circumconcevian triangle of X(21). CTR13-9.21 is the tangential triangle of A'B'C' wrt X(9)-circumconic.

X(66254) lies on these lines: {55, 64693}, {390, 63967}, {497, 12005}, {758, 31795}, {943, 51768}, {946, 12671}, {950, 2800}, {971, 40270}, {2801, 15172}, {3058, 41562}, {3646, 25722}, {3678, 10386}, {4015, 51787}, {4314, 20117}, {5882, 9848}, {6796, 66239}, {9670, 18389}, {9844, 11362}, {12564, 22793}, {12710, 18483}, {12711, 31870}, {30329, 48661}, {30331, 40263}, {30384, 66195}, {40273, 58626}, {63999, 66248}

X(66254) = midpoint of X(i) and X(j) for these {i,j}: {63999, 66248}

X(66255) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR13-32.6 WRT 1ST-VAN-KHEA-PAVLOV

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

Let A'B'C' be the X(32)-circumconcevian triangle of X(6). CTR13-32.6 is the tangential triangle of A'B'C' wrt X(32)-circumconic.

X(66255) lies on these lines: {1, 19126}, {6, 12410}, {8, 1974}, {10, 19137}, {145, 19121}, {182, 517}, {184, 51192}, {206, 5846}, {952, 64052}, {962, 19124}, {1482, 19131}, {3416, 9306}, {5138, 37547}, {5157, 38315}, {5250, 26924}, {5844, 19154}, {5847, 52016}, {7983, 41274}, {8148, 19129}, {8193, 11574}, {9822, 11365}, {11511, 37546}, {12245, 19128}, {19127, 51147}, {19136, 49524}, {26923, 62874}, {37491, 64069}, {37515, 38029}, {56918, 59407}

X(66255) = midpoint of X(i) and X(j) for these {i,j}: {6, 12410}, {37491, 64069}
X(66255) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {6, 5091, 12410}

X(66256) = X(3) OF 1ST-VAN-KHEA-PAVLOV TRIANGLE

Barycentrics    a*(-b^3-10*a*b*c+3*b^2*c+3*b*c^2-c^3+a^2*(b+c)) : :
X(66256) = -3*X[2]+X[3893], -2*X[8]+3*X[3740], -2*X[10]+3*X[10179], -X[65]+3*X[3241], -3*X[210]+X[3621], -3*X[354]+5*X[3623], -3*X[392]+X[3632], -X[1071]+3*X[61291], -3*X[3555]+X[3901], -7*X[3622]+5*X[3698], -X[3625]+3*X[3898], -3*X[3656]+2*X[16616] and many others

X(66256) lies on these lines: {1, 474}, {2, 3893}, {5, 49626}, {8, 3740}, {9, 11519}, {10, 10179}, {37, 4051}, {43, 45219}, {55, 11260}, {56, 3895}, {65, 3241}, {72, 3633}, {100, 20323}, {144, 145}, {210, 3621}, {226, 13463}, {354, 3623}, {392, 3632}, {495, 49600}, {496, 5123}, {497, 32049}, {515, 13600}, {517, 550}, {519, 960}, {528, 10106}, {529, 10624}, {912, 10284}, {936, 8168}, {942, 2802}, {950, 38455}, {952, 31937}, {958, 12629}, {962, 12678}, {1001, 4853}, {1071, 61291}, {1155, 62837}, {1193, 17460}, {1222, 3685}, {1319, 3871}, {1320, 11011}, {1329, 63993}, {1385, 25439}, {1386, 37542}, {1387, 59719}, {1420, 4421}, {1482, 6261}, {1697, 4640}, {1722, 16486}, {1837, 12648}, {2098, 3870}, {2550, 12541}, {2605, 57209}, {2646, 38460}, {2886, 21627}, {2951, 3243}, {3059, 12630}, {3208, 40133}, {3303, 3872}, {3304, 63130}, {3333, 64202}, {3340, 4321}, {3476, 56936}, {3485, 34640}, {3555, 3901}, {3579, 62825}, {3582, 38411}, {3616, 31233}, {3622, 3698}, {3625, 3898}, {3656, 16616}, {3681, 20014}, {3696, 30090}, {3697, 4677}, {3754, 5049}, {3811, 64897}, {3813, 31397}, {3816, 6736}, {3825, 51362}, {3838, 15888}, {3876, 20053}, {3877, 20050}, {3878, 4525}, {3881, 50193}, {3892, 31794}, {3896, 20041}, {3911, 32157}, {3916, 37563}, {3918, 51103}, {3919, 50192}, {3922, 64149}, {3931, 50637}, {4002, 25055}, {4004, 50190}, {4050, 44798}, {4067, 51096}, {4360, 24471}, {4413, 63142}, {4511, 64199}, {4679, 56879}, {4731, 5550}, {4738, 59582}, {4861, 37080}, {4882, 66231}, {5045, 51071}, {5048, 34772}, {5087, 12053}, {5267, 51787}, {5289, 6765}, {5690, 49627}, {5722, 49169}, {5727, 17622}, {5784, 12536}, {5794, 64068}, {5844, 63976}, {5853, 12448}, {5854, 64163}, {5904, 34747}, {6001, 23340}, {6600, 15347}, {6734, 45081}, {6735, 37722}, {6762, 9819}, {6767, 51715}, {7373, 54286}, {7686, 10222}, {7962, 12635}, {7967, 58567}, {7972, 17652}, {7982, 50528}, {7987, 61153}, {8148, 12559}, {8162, 54392}, {8236, 58608}, {8256, 11019}, {8544, 11520}, {8715, 24928}, {9578, 11235}, {9614, 11236}, {9785, 24703}, {9856, 28236}, {9955, 11698}, {10178, 31798}, {10202, 61284}, {10247, 13374}, {10391, 37734}, {10459, 15569}, {10528, 11376}, {10580, 66243}, {10942, 22835}, {10944, 34699}, {10950, 17615}, {11194, 61763}, {11239, 11375}, {11240, 24914}, {11256, 66199}, {11278, 62822}, {11373, 45701}, {11682, 41711}, {12245, 58637}, {12531, 58683}, {12546, 35628}, {12575, 57288}, {12631, 40587}, {12645, 58631}, {12672, 61296}, {12710, 37728}, {12737, 33596}, {13278, 20586}, {13373, 61283}, {13405, 64205}, {13601, 63987}, {13607, 31788}, {14759, 14839}, {14986, 37828}, {15016, 61285}, {15481, 63135}, {15558, 64131}, {17318, 34371}, {17609, 20057}, {17636, 58611}, {18258, 65454}, {18391, 64744}, {19860, 42819}, {19861, 51786}, {19907, 33179}, {19925, 66065}, {20052, 63961}, {20691, 62370}, {21214, 21896}, {21620, 64767}, {22837, 24929}, {24600, 59616}, {24926, 33595}, {25440, 51788}, {25681, 34619}, {25716, 34855}, {26066, 34625}, {28234, 31786}, {28534, 34749}, {31145, 58629}, {32426, 64162}, {32636, 63136}, {32937, 64563}, {33815, 50191}, {34339, 61286}, {34434, 49471}, {34710, 66009}, {34748, 40266}, {36638, 39126}, {36867, 37585}, {37562, 61287}, {37567, 62832}, {37568, 54391}, {37614, 49465}, {37738, 66250}, {39776, 58591}, {40296, 50824}, {40883, 49466}, {44675, 64123}, {44720, 59506}, {49163, 64128}, {49450, 58693}, {51779, 64673}, {59507, 62697}, {59722, 64703}, {62861, 64963}

X(66256) = midpoint of X(i) and X(j) for these {i,j}: {65, 3885}, {72, 3633}, {145, 3057}, {2136, 17648}, {3059, 12630}, {3555, 5697}, {7972, 17652}, {12672, 61296}, {23340, 37727}, {66205, 66258}
X(66256) = reflection of X(i) in X(j) for these {i,j}: {8, 58679}, {10, 31792}, {65, 58609}, {942, 3635}, {960, 9957}, {3625, 5044}, {3632, 4662}, {3740, 5919}, {4711, 3898}, {5836, 1}, {7686, 10222}, {8256, 20789}, {9943, 5882}, {10914, 3812}, {12245, 58637}, {12531, 58683}, {12645, 58631}, {12675, 1483}, {13369, 32900}, {14923, 10107}, {17636, 58611}, {18258, 65454}, {31145, 58629}, {31788, 13607}, {34339, 61286}, {34790, 3884}, {34791, 3244}, {39776, 58591}, {49450, 58693}, {50193, 3881}, {57288, 12575}, {66205, 66257}, {66257, 66259}
X(66256) = complement of X(3893)
X(66256) = perspector of circumconic {{A, B, C, X(27834), X(30610)}}
X(66256) = pole of line {4162, 30198} with respect to the incircle
X(66256) = pole of line {4491, 48302} with respect to the DeLongchamps ellipse
X(66256) = pole of line {2, 2098} with respect to the Feuerbach hyperbola
X(66256) = pole of line {3669, 31287} with respect to the Steiner inellipse
X(66256) = pole of line {3452, 31197} with respect to the dual conic of Yff parabola
X(66256) = pole of line {5265, 12513} with respect to the dual conic of Moses-Feuerbach circumconic
X(66256) = X(5893)-of-Ursa-minor triangle
X(66256) = X(5894)-of-intouch triangle
X(66256) = X(5895)-of-inverse-in-incircle triangle
X(66256) = X(6225)-of-2nd-Zaniah triangle
X(66256) = intersection, other than A, B, C, of circumconics {{A, B, C, X(145), X(1376)}}, {{A, B, C, X(3445), X(7320)}}, {{A, B, C, X(3680), X(39702)}}, {{A, B, C, X(5437), X(18743)}}, {{A, B, C, X(8056), X(9311)}}
X(66256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10914, 3812}, {1, 2136, 1376}, {1, 3880, 5836}, {1, 3913, 59691}, {1, 48696, 17614}, {1, 5836, 3742}, {1, 63137, 25524}, {1, 64175, 4719}, {1, 64176, 52541}, {8, 58679, 3740}, {8, 5919, 58679}, {10, 31792, 10179}, {65, 3241, 58609}, {145, 3057, 518}, {354, 14923, 10107}, {392, 3632, 4662}, {517, 1483, 12675}, {517, 3244, 34791}, {517, 32900, 13369}, {517, 5882, 9943}, {519, 3884, 34790}, {1697, 12513, 4640}, {1837, 12648, 32537}, {2136, 17648, 3880}, {3241, 3885, 65}, {3555, 5697, 44663}, {3621, 3890, 210}, {3622, 3698, 3848}, {3623, 14923, 354}, {3625, 3898, 5044}, {3625, 5044, 4711}, {3812, 3880, 10914}, {3884, 34790, 960}, {4853, 37556, 1001}, {5697, 51093, 3555}, {7990, 11531, 2951}, {9819, 12127, 6762}, {9957, 34790, 3884}, {11019, 12640, 8256}, {11519, 30337, 9}, {12629, 31393, 958}, {13601, 63987, 63994}, {23340, 37727, 6001}, {32426, 66257, 66205}, {32426, 66259, 66257}, {66205, 66258, 32426}, {66257, 66259, 64162}

X(66257) = X(5) OF 1ST-VAN-KHEA-PAVLOV TRIANGLE

Barycentrics    2*a^4-2*a^3*(b+c)-(b^2-c^2)^2-a^2*(b^2+c^2)+2*a*(b^3-2*b^2*c-2*b*c^2+c^3) : :
X(66257) = 3*X[2]+X[10950], 3*X[51]+X[64580], 3*X[210]+X[41575], -X[1770]+5*X[4004], -X[3057]+3*X[49736], 3*X[3058]+X[14923], -5*X[3616]+X[10944], 7*X[3624]+X[37706], -5*X[3698]+3*X[49732], -3*X[3740]+X[6737], -3*X[3742]+X[10106], 3*X[3753]+X[10572] and many others

X(66257) lies on circumconic {{A, B, C, X(34918), X(56118)}} and on these lines: {1, 1329}, {2, 10950}, {5, 30147}, {8, 344}, {10, 6675}, {12, 11281}, {21, 14882}, {30, 3754}, {51, 64580}, {55, 5554}, {65, 17768}, {78, 9711}, {80, 442}, {100, 10543}, {140, 26287}, {145, 26105}, {210, 41575}, {214, 52264}, {355, 6881}, {388, 25557}, {405, 10573}, {484, 57002}, {495, 30143}, {497, 13463}, {515, 3812}, {516, 10107}, {517, 5462}, {518, 5795}, {519, 4015}, {524, 25371}, {528, 950}, {529, 942}, {535, 24470}, {938, 12513}, {944, 6946}, {952, 1125}, {958, 18391}, {960, 5855}, {997, 37739}, {1058, 10912}, {1145, 3746}, {1146, 41239}, {1213, 46823}, {1220, 17947}, {1376, 3486}, {1385, 6691}, {1389, 6902}, {1737, 4999}, {1770, 4004}, {1834, 60353}, {1837, 2886}, {2099, 2478}, {2292, 24433}, {2320, 17566}, {2329, 21049}, {2551, 12635}, {2646, 3035}, {2802, 15172}, {2829, 34339}, {3036, 10039}, {3057, 49736}, {3058, 14923}, {3304, 17051}, {3340, 24703}, {3419, 9710}, {3487, 11236}, {3488, 3913}, {3601, 37828}, {3616, 10944}, {3624, 37706}, {3649, 5080}, {3679, 5436}, {3698, 49732}, {3711, 20013}, {3740, 6737}, {3742, 10106}, {3753, 10572}, {3813, 5722}, {3814, 37737}, {3820, 22836}, {3822, 18357}, {3825, 5901}, {3826, 5727}, {3829, 9581}, {3833, 28224}, {3838, 19925}, {3847, 5886}, {3848, 28236}, {3868, 34606}, {3880, 63999}, {3884, 5844}, {3897, 5433}, {3924, 17061}, {3925, 5086}, {4193, 15950}, {4313, 4421}, {4640, 4848}, {4642, 64158}, {4679, 11682}, {4853, 37723}, {4860, 20076}, {4861, 37722}, {5084, 5289}, {5087, 64160}, {5123, 13411}, {5176, 15888}, {5248, 5690}, {5250, 15297}, {5251, 18253}, {5252, 54392}, {5259, 41684}, {5260, 21677}, {5330, 26127}, {5432, 25005}, {5439, 45287}, {5443, 17533}, {5445, 37298}, {5587, 28628}, {5691, 5880}, {5724, 41877}, {5784, 17632}, {5790, 10198}, {5837, 15254}, {5841, 61541}, {5852, 12527}, {5853, 58608}, {5854, 9957}, {5882, 9843}, {5883, 18990}, {5903, 11113}, {5919, 46677}, {5943, 34434}, {6224, 17531}, {6326, 64283}, {6692, 40262}, {6735, 37080}, {6744, 58609}, {6767, 49169}, {6840, 64754}, {6872, 37567}, {6920, 12247}, {7483, 18395}, {7681, 61146}, {8162, 36972}, {8261, 33961}, {8363, 30159}, {8582, 59691}, {9708, 49168}, {9946, 12675}, {10200, 10246}, {10385, 63133}, {10527, 61717}, {10582, 37709}, {10915, 33559}, {11011, 41012}, {11019, 11260}, {11415, 64963}, {11684, 63290}, {11729, 33281}, {12019, 25639}, {12104, 61524}, {12447, 58451}, {12572, 44663}, {12577, 58560}, {12609, 18480}, {12640, 30331}, {13205, 45080}, {13624, 58405}, {13747, 37525}, {14584, 54356}, {15325, 51111}, {17045, 21237}, {17070, 21935}, {17527, 30144}, {17540, 30124}, {17606, 24541}, {18243, 18516}, {18527, 49600}, {19861, 37740}, {20147, 54120}, {20292, 65631}, {20718, 58554}, {21031, 34772}, {21258, 24249}, {21616, 50194}, {24390, 37702}, {24926, 34123}, {24954, 56387}, {25055, 37707}, {25525, 37714}, {26007, 26532}, {28629, 59387}, {30389, 31190}, {31249, 63208}, {31284, 40483}, {31393, 64744}, {31397, 51715}, {31799, 38454}, {32198, 66199}, {32426, 64162}, {32537, 42819}, {33895, 63993}, {34640, 51785}, {34697, 64358}, {34749, 62854}, {35004, 37290}, {35010, 64145}, {35023, 63990}, {37162, 62826}, {37370, 51870}, {37733, 64282}, {40267, 60896}, {41711, 56879}, {49609, 59583}, {52835, 62178}, {56191, 63360}, {56426, 63318}, {56880, 63159}, {58565, 58570}, {59507, 64702}, {61286, 61551}, {62674, 62684}, {63136, 63273}

X(66257) = midpoint of X(i) and X(j) for these {i,j}: {10, 37730}, {65, 57288}, {950, 5836}, {960, 64163}, {3035, 66206}, {5795, 6738}, {35004, 37290}, {66205, 66256}
X(66257) = reflection of X(i) in X(j) for these {i,j}: {24470, 33815}, {58609, 6744}, {66256, 66259}
X(66257) = pole of line {200, 8256} with respect to the dual conic of Moses-Feuerbach circumconic
X(66257) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 24929, 64123}, {10, 37730, 44669}, {55, 5554, 8256}, {55, 8256, 32157}, {65, 57288, 17768}, {355, 54318, 25466}, {535, 33815, 24470}, {950, 5836, 528}, {960, 64163, 5855}, {1125, 6702, 3628}, {1125, 9956, 6668}, {2646, 24982, 3035}, {3698, 57287, 49732}, {5727, 64673, 5794}, {5794, 64673, 3826}, {5795, 6738, 518}, {6675, 11545, 10}, {18480, 64732, 12609}, {32426, 66259, 66256}, {64162, 66256, 66259}, {66205, 66256, 32426}

X(66258) = X(20) OF 1ST-VAN-KHEA-PAVLOV TRIANGLE

Barycentrics    2*a^4-5*a^3*(b+c)-(b^2-c^2)^2-a^2*(b^2-30*b*c+c^2)+a*(5*b^3-13*b^2*c-13*b*c^2+5*c^3) : :
X(66258) = -3*X[553]+2*X[14923], -3*X[3885]+X[64002], -3*X[10914]+4*X[12436]

X(66258) lies on these lines: {8, 5316}, {57, 145}, {72, 519}, {144, 12630}, {226, 3680}, {553, 14923}, {944, 3633}, {1058, 3632}, {1210, 64768}, {3241, 5438}, {3244, 3304}, {3621, 18228}, {3626, 50038}, {3880, 10106}, {3885, 64002}, {3911, 12640}, {3913, 41426}, {5844, 31793}, {5853, 25722}, {5854, 41558}, {7308, 7320}, {9797, 61630}, {10531, 47745}, {10912, 64160}, {10914, 12436}, {12648, 21627}, {13464, 41702}, {20050, 56936}, {21620, 64203}, {26726, 38665}, {31397, 31493}, {32426, 64162}, {37567, 64117}, {38455, 66247}

X(66258) = reflection of X(i) in X(j) for these {i,j}: {66205, 66256}
X(66258) = pole of line {42312, 58858} with respect to the incircle
X(66258) = pole of line {20196, 40688} with respect to the dual conic of Yff parabola
X(66258) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2137), X(57666)}}, {{A, B, C, X(6553), X(44040)}}
X(66258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {145, 2136, 63987}, {145, 51786, 12437}, {145, 66243, 57}, {12640, 36846, 3911}, {32426, 66256, 66205}, {66205, 66256, 64162}

X(66259) = X(140) OF 1ST-VAN-KHEA-PAVLOV TRIANGLE

Barycentrics    2*a^4+2*a*b*c*(b+c)-(b^2-c^2)^2-a^2*(b^2+20*b*c+c^2) : :
X(66259) = 5*X[145]+3*X[34689], 3*X[3058]+5*X[3623], 3*X[3241]+X[57288], X[3244]+3*X[15170], 5*X[3616]+3*X[34699], -7*X[3622]+3*X[49732], -3*X[3742]+7*X[66241], -11*X[5550]+3*X[34720], X[6284]+7*X[20057], -X[6743]+3*X[58679], X[15171]+3*X[51071]

X(66259) lies on these lines: {1, 528}, {145, 34689}, {390, 8163}, {519, 4547}, {529, 3635}, {1058, 12607}, {2829, 61286}, {2886, 10587}, {3058, 3623}, {3241, 57288}, {3244, 15170}, {3303, 6690}, {3616, 34699}, {3622, 49732}, {3742, 66241}, {3813, 6767}, {3816, 7080}, {3826, 64068}, {3871, 35023}, {3880, 40270}, {4301, 15570}, {5048, 33961}, {5220, 9797}, {5550, 34720}, {5794, 51779}, {5840, 33658}, {5842, 33179}, {5852, 34791}, {5854, 12433}, {5855, 9957}, {6284, 20057}, {6667, 27529}, {6691, 25439}, {6743, 58679}, {7680, 18543}, {9785, 42871}, {11019, 32157}, {11260, 30331}, {12575, 17768}, {15171, 51071}, {15888, 66065}, {17051, 63130}, {18530, 49169}, {21627, 42819}, {21630, 63282}, {31792, 44669}, {32426, 64162}, {38455, 63999}, {43179, 64205}

X(66259) = midpoint of X(i) and X(j) for these {i,j}: {3635, 15172}, {12575, 58609}, {66256, 66257}
X(66259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {12575, 58609, 17768}, {64162, 66256, 66257}, {66256, 66257, 32426}

X(66260) = TRIPOLAR CENTROID OF X(20578)

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

X(66260) lies on the cubic K1381 and these lines: {6, 11082}, {62, 36313}, {115, 30452}, {533, 5459}, {11136, 62199}, {15610, 58847}, {17403, 22511}, {18776, 42004}, {22738, 32037}, {35330, 61370}, {43229, 52868}, {44114, 66184}

X(66260) = tripolar centroid of X(20578)
X(66260) = X(i)-Ceva conjugate of X(j) for these (i,j): {14, 55223}, {17, 35444}
X(66260) = X(24041)-isoconjugate of X(34322)
X(66260) = X(3005)-Dao conjugate of X(34322)
X(66260) = crosssum of X(17402) and X(33527)
X(66260) = crossdifference of every pair of points on line {10410, 16807}
X(66260) = barycentric product X(i)*X(j) for these {i,j}: {14, 15610}, {115, 6672}, {533, 43967}
X(66260) = barycentric quotient X(i)/X(j) for these {i,j}: {3124, 34322}, {6672, 4590}, {15610, 299}, {43967, 11118}, {55223, 10410}

X(66261) = TRIPOLAR CENTROID OF X(20579)

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

X(66261) lies on the cubic K1381 and these lines: {6, 11087}, {61, 36312}, {115, 30453}, {532, 5460}, {11135, 62200}, {15609, 58848}, {17402, 22510}, {18777, 42003}, {22739, 32036}, {35329, 61371}, {43228, 52867}, {44114, 66184}

X(66261) = tripolar centroid of X(20579)
X(66261) = X(i)-Ceva conjugate of X(j) for these (i,j): {13, 55221}, {18, 35443}
X(66261) = X(24041)-isoconjugate of X(34321)
X(66261) = X(3005)-Dao conjugate of X(34321)
X(66261) = crosssum of X(17403) and X(33526)
X(66261) = crossdifference of every pair of points on line {10409, 16806}
X(66261) = barycentric product X(i)*X(j) for these {i,j}: {13, 15609}, {115, 6671}, {532, 43968}
X(66261) = barycentric quotient X(i)/X(j) for these {i,j}: {3124, 34321}, {6671, 4590}, {15609, 298}, {43968, 11117}, {55221, 10409}

X(66262) = TRIPOLAR CENTROID OF X(62631)

Barycentrics    (-a^2 + b^2 + c^2 + 2*Sqrt[3]*S)*(Sqrt[3]*(b^2 - c^2)^2*(-a^2 + b^2 + c^2) - 2*(b^2 - c^2)^2*S) : :

X(66262) lies on the cubic K1381 and these lines: {2, 17403}, {115, 23871}, {125, 526}, {141, 16536}, {233, 19294}, {302, 8838}, {471, 11081}, {532, 5459}, {624, 13162}, {2993, 11087}, {9205, 30468}, {15609, 46652}, {15610, 25178}, {19779, 34540}

X(66262) = midpoint of X(11126) and X(52220)
X(66262) = complement of X(17403)
X(66262) = complement of the isogonal conjugate of X(20579)
X(66262) = tripolar centroid of X(62631)
X(66262) = X(i)-complementary conjugate of X(j) for these (i,j): {14, 4369}, {301, 42327}, {661, 619}, {798, 40696}, {1109, 46651}, {2151, 8562}, {2154, 523}, {2643, 43962}, {3458, 14838}, {5994, 16598}, {8738, 8062}, {20579, 10}, {23896, 21254}, {30453, 8287}
X(66262) = X(i)-Ceva conjugate of X(j) for these (i,j): {338, 43962}, {2993, 523}, {13582, 23870}, {16771, 23872}, {19779, 23871}
X(66262) = X(1101)-isoconjugate of X(11087)
X(66262) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 11087}, {647, 52203}, {11130, 249}, {15610, 52930}, {23871, 19779}, {23872, 16771}, {35443, 11600}, {35444, 17}, {38994, 16806}, {43962, 32036}, {47899, 65346}, {60342, 8603}
X(66262) = crosspoint of X(i) and X(j) for these (i,j): {16771, 23872}, {19779, 23871}
X(66262) = crosssum of X(i) and X(j) for these (i,j): {5994, 11141}, {16806, 51890}
X(66262) = crossdifference of every pair of points on line {1625, 5994}
X(66262) = barycentric product X(i)*X(j) for these {i,j}: {115, 11132}, {302, 30468}, {338, 11126}, {339, 10632}, {8838, 62551}, {11128, 43968}, {11135, 23962}, {16771, 43962}, {23871, 23872}, {23994, 35199}, {30465, 52220}
X(66262) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 11087}, {125, 52203}, {2088, 8603}, {6138, 16806}, {8838, 39295}, {10632, 250}, {11126, 249}, {11132, 4590}, {11135, 23357}, {16771, 57580}, {23283, 60051}, {23871, 32036}, {23872, 23896}, {30465, 11600}, {30468, 17}, {35199, 1101}, {43962, 19779}, {43968, 11085}, {52343, 37848}, {55221, 5994}, {64465, 47390}
X(66262) = {X(30468),X(62551)}-harmonic conjugate of X(43962)

X(66263) = TRIPOLAR CENTROID OF X(62632)

Barycentrics    (-a^2 + b^2 + c^2 - 2*Sqrt[3]*S)*(Sqrt[3]*(b^2 - c^2)^2*(-a^2 + b^2 + c^2) + 2*(b^2 - c^2)^2*S) : :

X(66263) lies on the cubic K1381 and these lines: {2, 17402}, {115, 23870}, {125, 526}, {141, 16537}, {233, 19295}, {303, 8836}, {470, 11086}, {533, 5460}, {623, 13162}, {2992, 11082}, {9204, 30465}, {15609, 25173}, {15610, 46653}, {19778, 34541}

X(66263) = midpoint of X(11127) and X(52221)
X(66263) = complement of X(17402)
X(66263) = complement of the isogonal conjugate of X(20578)
X(66263) = tripolar centroid of X(62632)
X(66263) = X(i)-complementary conjugate of X(j) for these (i,j): {13, 4369}, {300, 42327}, {661, 618}, {798, 40695}, {1109, 46650}, {2152, 8562}, {2153, 523}, {2643, 43961}, {3457, 14838}, {5995, 16598}, {8737, 8062}, {20578, 10}, {23895, 21254}, {30452, 8287}
X(66263) = X(i)-Ceva conjugate of X(j) for these (i,j): {338, 43961}, {2992, 523}, {13582, 23871}, {16770, 23873}, {19778, 23870}
X(66263) = X(1101)-isoconjugate of X(11082)
X(66263) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 11082}, {647, 52204}, {11131, 249}, {15609, 52929}, {23870, 19778}, {23873, 16770}, {35443, 18}, {35444, 11601}, {38993, 16807}, {43961, 32037}, {47898, 65347}, {60342, 8604}
X(66263) = crosspoint of X(i) and X(j) for these (i,j): {16770, 23873}, {19778, 23870}
X(66263) = crosssum of X(i) and X(j) for these (i,j): {5995, 11142}, {16807, 51891}
X(66263) = crossdifference of every pair of points on line {1625, 5995}
X(66263) = barycentric product X(i)*X(j) for these {i,j}: {115, 11133}, {303, 30465}, {338, 11127}, {339, 10633}, {8836, 62551}, {11129, 43967}, {11136, 23962}, {16770, 43961}, {23870, 23873}, {23994, 35198}, {30468, 52221}
X(66263) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 11082}, {125, 52204}, {2088, 8604}, {6137, 16807}, {8836, 39295}, {10633, 250}, {11127, 249}, {11133, 4590}, {11136, 23357}, {16770, 57579}, {23284, 60052}, {23870, 32037}, {23873, 23895}, {30465, 18}, {30468, 11601}, {35198, 1101}, {43961, 19778}, {43967, 11080}, {52342, 37850}, {55223, 5995}, {64464, 47390}
X(66263) = {X(30465),X(62551)}-harmonic conjugate of X(43961)

X(66264) = TRIPOLAR CENTROID OF X(62645)

Barycentrics    (b^2 - c^2)^2*(-a^2 + b^2 + c^2)*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 2*b^2*c^2 + 3*c^4) : :
X(66264) = 4 X[6721] - X[52170], 5 X[31274] - 2 X[32661]

X(66264) lies on the cubic K1381 and these lines: {68, 7888}, {115, 525}, {122, 125}, {520, 6784}, {599, 45311}, {2482, 17702}, {3124, 13302}, {3564, 6055}, {5449, 7794}, {6388, 44564}, {6721, 52170}, {7687, 34360}, {7801, 14852}, {7821, 12359}, {7863, 9927}, {7873, 44158}, {31274, 32661}, {34767, 65756}, {34897, 38724}, {44569, 45312}

X(66264) = tripolar centroid of X(62645)
X(66264) = X(i)-complementary conjugate of X(j) for these (i,j): {40801, 4369}, {55972, 42327}, {64983, 21259}
X(66264) = X(i)-Ceva conjugate of X(j) for these (i,j): {37174, 64919}, {56267, 525}, {59257, 6368}
X(66264) = X(i)-isoconjugate of X(j) for these (i,j): {1101, 47735}, {23995, 42298}
X(66264) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 47735}, {525, 56267}, {647, 7612}, {18314, 42298}, {54259, 5921}, {64919, 37174}
X(66264) = crosspoint of X(i) and X(j) for these (i,j): {525, 56267}, {37174, 64919}
X(66264) = crosssum of X(112) and X(59229)
X(66264) = crossdifference of every pair of points on line {112, 61213}
X(66264) = barycentric product X(i)*X(j) for these {i,j}: {115, 10008}, {125, 1007}, {338, 59211}, {339, 1351}, {525, 64919}, {15526, 37174}, {17879, 51288}, {36793, 59229}
X(66264) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 47735}, {125, 7612}, {338, 42298}, {1007, 18020}, {1351, 250}, {10008, 4590}, {15526, 56267}, {37174, 23582}, {51288, 24000}, {59211, 249}, {59229, 23964}, {64919, 648}

X(66265) = TRIPOLAR CENTROID OF X(62672)

Barycentrics    (2*a^2 - b^2 - c^2)*(4*a^8 - 8*a^6*b^2 + 15*a^4*b^4 - 11*a^2*b^6 - 2*b^8 - 8*a^6*c^2 - 6*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + 19*b^6*c^2 + 15*a^4*c^4 + 3*a^2*b^2*c^4 - 30*b^4*c^4 - 11*a^2*c^6 + 19*b^2*c^6 - 2*c^8) : :
X(66265) = 2 X[1648] - 3 X[14971], 3 X[9166] - X[45291], 3 X[9167] - 4 X[11053], 2 X[14444] - 5 X[31274], X[15300] - 4 X[38239]

X(66265) lies on the cubic K1381 and these lines: {115, 524}, {351, 690}, {543, 5468}, {599, 5465}, {620, 51226}, {671, 61190}, {1648, 14971}, {9144, 14916}, {9166, 45291}, {9167, 11053}, {10717, 15342}, {14444, 31274}, {15300, 38239}

X(66265) = reflection of X(i) in X(j) for these {i,j}: {2482, 1641}, {51226, 620}
X(66265) = tripolar centroid of X(62672)

X(66266) = TRIPOLAR CENTROID OF X(65716)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(4*a^12 - 8*a^10*b^2 - a^8*b^4 + 13*a^6*b^6 - 13*a^4*b^8 + 7*a^2*b^10 - 2*b^12 - 8*a^10*c^2 + 26*a^8*b^2*c^2 - 21*a^6*b^4*c^2 + 15*a^4*b^6*c^2 - 13*a^2*b^8*c^2 + b^10*c^2 - a^8*c^4 - 21*a^6*b^2*c^4 + 6*a^2*b^6*c^4 + 14*b^8*c^4 + 13*a^6*c^6 + 15*a^4*b^2*c^6 + 6*a^2*b^4*c^6 - 26*b^6*c^6 - 13*a^4*c^8 - 13*a^2*b^2*c^8 + 14*b^4*c^8 + 7*a^2*c^10 + b^2*c^10 - 2*c^12) : :

X(66266) lies on the cubic K1381 and these lines: {30, 115}, {1636, 1637}, {2482, 46229}, {4240, 6103}, {6128, 45311}, {44564, 45331}, {44569, 45312}

X(66266) = tripolar centroid of X(65716)
X(66266) = crossdifference of every pair of points on line {74, 34291}

X(66267) = TRILINEAR POLE OF X(115)X(826)

Barycentrics    (b^2 - c^2)*(b^2 - a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2) : :
X(66267) = X[2395] + 3 X[34290], 3 X[8029] + X[62555]

X(66267) lies on the X-parabola of ABC (see X(12065)) and these lines: {2, 881}, {141, 523}, {427, 2501}, {476, 805}, {512, 7804}, {685, 4230}, {694, 804}, {733, 9076}, {850, 2528}, {876, 4581}, {892, 9178}, {1031, 17500}, {1502, 23285}, {1648, 35366}, {1916, 5466}, {2489, 51988}, {2492, 9468}, {2799, 23596}, {4024, 15523}, {5996, 8371}, {8430, 51494}, {8842, 45689}, {9479, 14316}, {11123, 65031}, {14970, 43098}, {15328, 36214}, {17941, 41209}, {17980, 47206}, {18105, 35222}, {18858, 53691}, {18896, 35522}, {19130, 32473}, {20027, 53331}, {23350, 36897}, {25423, 45329}, {30229, 64479}, {31065, 61418}, {37134, 60055}, {38393, 64258}, {40708, 62645}, {44768, 65327}, {58112, 59026}

X(66267) = isogonal conjugate of X(56980)
X(66267) = isotomic conjugate of X(17941)
X(66267) = isotomic conjugate of the isogonal conjugate of X(882)
X(66267) = isogonal conjugate of the isotomic conjugate of X(56981)
X(66267) = X(9477)-anticomplementary conjugate of X(21294)
X(66267) = X(i)-Ceva conjugate of X(j) for these (i,j): {18829, 1916}, {18858, 36897}, {39291, 47734}, {59026, 14970}, {65351, 694}
X(66267) = X(i)-cross conjugate of X(j) for these (i,j): {2799, 523}, {43665, 60036}
X(66267) = X(i)-isoconjugate of X(j) for these (i,j): {1, 56980}, {6, 56982}, {31, 17941}, {99, 1933}, {110, 1580}, {163, 385}, {419, 4575}, {560, 880}, {662, 1691}, {732, 34072}, {799, 14602}, {804, 1101}, {805, 51903}, {827, 2236}, {1576, 1966}, {1634, 56971}, {1926, 14574}, {1967, 46294}, {4164, 4570}, {4558, 56828}, {4579, 5009}, {4592, 44089}, {4593, 56915}, {4599, 8623}, {4602, 18902}, {5026, 36142}, {5027, 24041}, {9468, 46295}, {12215, 32676}, {14295, 23995}, {23997, 40820}, {24019, 58354}, {36034, 51430}, {36084, 36213}, {36134, 63736}, {37134, 51318}, {43754, 56679}
X(66267) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 17941}, {3, 56980}, {9, 56982}, {115, 385}, {136, 419}, {137, 63736}, {244, 1580}, {339, 35540}, {523, 804}, {647, 24284}, {1084, 1691}, {1649, 11183}, {3005, 5027}, {3124, 8623}, {3258, 51430}, {4858, 1966}, {4988, 4107}, {5139, 44089}, {6374, 880}, {8290, 46294}, {9467, 1576}, {15449, 732}, {15526, 12215}, {18314, 14295}, {23992, 5026}, {35071, 58354}, {35078, 4027}, {35088, 5976}, {35971, 16985}, {36901, 3978}, {38970, 39931}, {38986, 1933}, {38987, 36213}, {38996, 14602}, {39044, 46295}, {39092, 110}, {40810, 41337}, {41172, 46888}, {46669, 19576}, {47648, 2421}, {50330, 4164}, {55043, 2236}, {55050, 56915}, {55065, 4039}, {55152, 12829}, {60342, 39495}, {62562, 40820}
X(66267) = cevapoint of X(i) and X(j) for these (i,j): {523, 9479}, {868, 8029}, {3005, 3569}
X(66267) = crosspoint of X(i) and X(j) for these (i,j): {98, 65278}, {1916, 18829}, {14970, 59026}, {18858, 36897}
X(66267) = crosssum of X(i) and X(j) for these (i,j): {511, 5113}, {732, 24284}, {804, 63736}, {1691, 5027}, {8623, 62454}
X(66267) = trilinear pole of line {115, 826}
X(66267) = crossdifference of every pair of points on line {1691, 8623}
X(66267) = barycentric product X(i)*X(j) for these {i,j}: {6, 56981}, {76, 882}, {115, 18829}, {125, 65351}, {257, 35352}, {327, 39680}, {338, 805}, {512, 18896}, {523, 1916}, {661, 1934}, {669, 44160}, {694, 850}, {733, 23285}, {826, 14970}, {868, 39291}, {881, 1502}, {1109, 37134}, {1577, 1581}, {1967, 20948}, {2501, 40708}, {2799, 36897}, {2970, 65327}, {3267, 17980}, {8029, 39292}, {9468, 44173}, {9477, 9479}, {14295, 41517}, {14618, 36214}, {15449, 59026}, {17938, 23962}, {18858, 35088}, {18872, 52632}, {38947, 46245}, {39691, 41209}, {40810, 43665}, {43763, 62418}, {47734, 62645}, {52618, 56978}, {56977, 58784}, {60245, 60577}
X(66267) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 56982}, {2, 17941}, {6, 56980}, {76, 880}, {115, 804}, {125, 24284}, {148, 46291}, {262, 39681}, {338, 14295}, {385, 46294}, {512, 1691}, {520, 58354}, {523, 385}, {525, 12215}, {661, 1580}, {669, 14602}, {688, 56915}, {690, 5026}, {694, 110}, {733, 827}, {798, 1933}, {804, 4027}, {805, 249}, {826, 732}, {850, 3978}, {881, 32}, {882, 6}, {1577, 1966}, {1581, 662}, {1637, 51430}, {1648, 11183}, {1916, 99}, {1934, 799}, {1966, 46295}, {1967, 163}, {2088, 39495}, {2395, 40820}, {2489, 44089}, {2501, 419}, {2533, 27982}, {2799, 5976}, {3005, 8623}, {3120, 4107}, {3124, 5027}, {3125, 4164}, {3569, 36213}, {4010, 53681}, {4024, 4039}, {4444, 17103}, {5027, 51318}, {5466, 60863}, {8061, 2236}, {8789, 14574}, {9293, 46290}, {9426, 18902}, {9468, 1576}, {9477, 65278}, {9479, 8290}, {12077, 63736}, {14223, 57452}, {14251, 14966}, {14316, 19571}, {14618, 17984}, {14970, 4577}, {15391, 43754}, {16230, 39931}, {16732, 14296}, {17938, 23357}, {17970, 32661}, {17980, 112}, {17994, 51324}, {18105, 56975}, {18829, 4590}, {18858, 57562}, {18872, 5467}, {18896, 670}, {20948, 1926}, {22260, 2086}, {23285, 35540}, {23596, 56696}, {30671, 40731}, {34212, 51343}, {34238, 2715}, {35352, 894}, {36214, 4558}, {36897, 2966}, {37134, 24041}, {38947, 40866}, {39291, 57991}, {39292, 31614}, {39680, 182}, {40708, 4563}, {40810, 2421}, {41167, 46888}, {41517, 805}, {43534, 18047}, {43665, 14382}, {43763, 4599}, {44160, 4609}, {44173, 14603}, {46040, 16069}, {46292, 9218}, {47648, 41337}, {47734, 4226}, {52618, 56979}, {52651, 3573}, {52700, 9181}, {55122, 12829}, {55240, 56971}, {56977, 4576}, {56978, 1634}, {56981, 76}, {58784, 56976}, {59026, 57545}, {60028, 51510}, {60226, 47646}, {60338, 47736}, {60577, 27958}, {62417, 62454}, {65351, 18020}

X(66268) = ISOGONAL CONJUGATE OF X(5502)

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

X(66268) lies on the X-parabola (see X(12065)) and these lines: {2, 57295}, {20, 523}, {525, 33702}, {685, 53351}, {850, 14615}, {1249, 2501}, {1294, 5896}, {1650, 12079}, {2395, 34570}, {4024, 8804}, {4036, 52345}, {5466, 44877}, {8057, 33893}, {9033, 10152}, {14249, 18504}, {33897, 55127}, {42399, 52452}

X(66268) = isogonal conjugate of X(5502)
X(66268) = anticomplement of X(57295)
X(66268) = X(14345)-cross conjugate of X(525)
X(66268) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5502}, {162, 21663}, {163, 47296}, {662, 40135}, {1576, 18699}, {4575, 10151}, {13202, 36034}, {32676, 40996}
X(66268) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 5502}, {115, 47296}, {125, 21663}, {136, 10151}, {1084, 40135}, {3258, 13202}, {4858, 18699}, {15526, 40996}, {16177, 11598}
X(66268) = cevapoint of X(i) and X(j) for these (i,j): {30, 5972}, {512, 46425}, {523, 9033}
X(66268) = crosspoint of X(16077) and X(46206)
X(66268) = crosssum of X(9409) and X(34569)
X(66268) = trilinear pole of line {115, 6587}
X(66268) = crossdifference of every pair of points on line {21663, 40135}
X(66268) = barycentric product X(i)*X(j) for these {i,j}: {523, 44877}, {850, 34570}
X(66268) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5502}, {512, 40135}, {523, 47296}, {525, 40996}, {647, 21663}, {1577, 18699}, {1637, 13202}, {2501, 10151}, {5896, 46639}, {14345, 52874}, {34570, 110}, {44877, 99}, {46425, 11598}, {59652, 51998}

X(66269) = X(83)X(523)∩X(308)X(850)

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

X(66269) lies on the X-parabola (see X(12065)) and these lines: {83, 523}, {308, 850}, {476, 58112}, {755, 39427}, {2501, 32085}, {4024, 18082}, {4036, 56186}, {5466, 18010}, {14970, 43098}, {31065, 40425}, {52395, 58784}

X(66269) = X(i)-isoconjugate of X(j) for these (i,j): {163, 52906}, {1101, 33907}, {1634, 2244}, {4020, 46543}, {14403, 24037}, {24041, 62456}, {52958, 55239}
X(66269) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 52906}, {512, 14403}, {523, 33907}, {3005, 62456}
X(66269) = trilinear pole of line {115, 58784}
X(66269) = barycentric product X(i)*X(j) for these {i,j}: {338, 58112}, {755, 52618}, {43098, 58784}
X(66269) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 33907}, {523, 52906}, {755, 1634}, {1084, 14403}, {3124, 62456}, {18105, 8627}, {32085, 46543}, {34294, 14420}, {43098, 4576}, {51906, 14428}, {52618, 35549}, {55240, 2244}, {58112, 249}, {58784, 754}

X(66270) = X(42)X(2501)∩X(71)X(523)

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

X(66270) lies on the X-parabola (see X(12065)) and these lines: {42, 2501}, {71, 523}, {306, 850}, {476, 35182}, {1796, 4608}, {2259, 14775}, {2359, 4581}, {2989, 60042}, {3690, 4024}, {3949, 4036}

X(66270) = X(i)-isoconjugate of X(j) for these (i,j): {81, 4243}, {757, 56742}, {1101, 55125}, {1736, 4556}, {8608, 52935}
X(66270) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 55125}, {40586, 4243}, {40607, 56742}, {55065, 48381}
X(66270) = trilinear pole of line {115, 55230}
X(66270) = barycentric product X(i)*X(j) for these {i,j}: {12, 60569}, {338, 35182}, {339, 32699}, {917, 4064}, {2989, 4024}, {20902, 36107}, {55230, 57997}
X(66270) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 4243}, {115, 55125}, {1500, 56742}, {2989, 4610}, {4024, 48381}, {4079, 8608}, {4705, 1736}, {32699, 250}, {35182, 249}, {55230, 916}, {57997, 55229}, {60569, 261}

X(66271) = X(6)X(58784)∩X(39)X(523)

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

X(66271) lies on the X-parabola (see X(12065)) and these lines: {6, 58784}, {39, 523}, {141, 850}, {688, 27375}, {755, 39427}, {826, 41440}, {1843, 2501}, {2395, 51869}, {3954, 4036}, {4024, 21035}, {5466, 46154}, {8599, 30489}, {31065, 52554}, {33666, 53495}, {56978, 66267}

X(66271) = X(888)-cross conjugate of X(523)
X(66271) = X(i)-isoconjugate of X(j) for these (i,j): {662, 5201}, {4575, 46511}, {36133, 38998}
X(66271) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 46511}, {1084, 5201}, {39010, 38998}
X(66271) = cevapoint of X(i) and X(j) for these (i,j): {826, 9148}, {8029, 52625}
X(66271) = trilinear pole of line {115, 3005}
X(66271) = barycentric product X(i)*X(j) for these {i,j}: {523, 60111}, {826, 39427}
X(66271) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 5201}, {888, 38998}, {2501, 46511}, {39427, 4577}, {60111, 99}

X(66272) = X(12)X(523)∩X(80)X(60055)

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

X(66272) lies on the X-parabola (see X(12065)) and these lines: {12, 523}, {80, 60029}, {476, 2222}, {655, 60055}, {685, 65329}, {850, 34388}, {892, 35174}, {1769, 52383}, {1825, 24006}, {2006, 60043}, {2501, 8736}, {4036, 21682}, {4581, 60074}, {5466, 60091}, {15328, 52391}, {18815, 60042}, {44768, 65299}, {52356, 56321}, {55250, 55253}

X(66272) = X(35174)-Ceva conjugate of X(60091)
X(66272) = X(i)-isoconjugate of X(j) for these (i,j): {36, 4636}, {215, 65283}, {249, 654}, {662, 4282}, {1101, 3738}, {1983, 2185}, {2150, 4585}, {2323, 4556}, {2361, 52935}, {3615, 52603}, {3904, 23357}, {4575, 17515}, {4610, 52426}, {4612, 7113}, {4996, 36069}, {6369, 14587}, {8648, 24041}, {34544, 37140}, {44428, 47390}, {52407, 52914}, {55237, 57657}
X(66272) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 17515}, {523, 3738}, {1084, 4282}, {3005, 8648}, {15898, 4636}, {38982, 4996}, {55065, 4511}, {56325, 4585}, {62570, 55237}
X(66272) = crosspoint of X(35174) and X(60091)
X(66272) = crosssum of X(4282) and X(8648)
X(66272) = trilinear pole of line {115, 55197}
X(66272) = crossdifference of every pair of points on line {4282, 34544}
X(66272) = barycentric product X(i)*X(j) for these {i,j}: {12, 60074}, {115, 35174}, {125, 65329}, {338, 2222}, {523, 60091}, {655, 1109}, {1091, 60571}, {1365, 36804}, {1411, 52623}, {1441, 55238}, {1577, 52383}, {1825, 14592}, {2006, 4036}, {2610, 57645}, {2643, 46405}, {2970, 65299}, {4024, 18815}, {6354, 52356}, {6370, 34535}, {7178, 15065}, {8901, 62735}, {10412, 16577}, {14616, 55197}, {14618, 52391}, {20566, 57185}, {23994, 32675}
X(66272) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 4585}, {80, 4612}, {115, 3738}, {181, 1983}, {512, 4282}, {655, 24041}, {1109, 3904}, {1365, 3960}, {1411, 4556}, {1441, 55237}, {1825, 14590}, {2006, 52935}, {2161, 4636}, {2222, 249}, {2501, 17515}, {2610, 4996}, {2643, 654}, {3124, 8648}, {4024, 4511}, {4036, 32851}, {4079, 2361}, {4705, 2323}, {8736, 4242}, {8754, 65104}, {14582, 1789}, {14616, 55196}, {15065, 645}, {16577, 10411}, {18815, 4610}, {20566, 4631}, {21131, 53525}, {21741, 52603}, {21833, 53562}, {30572, 17191}, {32675, 1101}, {34535, 65283}, {34857, 5546}, {35174, 4590}, {36804, 6064}, {41221, 2600}, {42666, 34544}, {46405, 24037}, {50487, 52426}, {52356, 7058}, {52383, 662}, {52391, 4558}, {55197, 758}, {55234, 52407}, {55238, 21}, {56285, 65162}, {57185, 36}, {60074, 261}, {60091, 99}, {61052, 21758}, {63750, 37140}, {64835, 52914}, {65329, 18020}

X(66273) = X(37)X(2501)∩X(72)X(523)

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

X(66273) lies on the X-parabola (see X(12065)) and these lines: {37, 2501}, {72, 523}, {476, 6099}, {685, 65344}, {850, 20336}, {915, 43659}, {943, 14775}, {1791, 4581}, {1807, 2804}, {2990, 60043}, {3695, 4036}, {3949, 4024}, {12532, 55126}, {45393, 60029}, {60055, 65248}

X(66273) = X(i)-isoconjugate of X(j) for these (i,j): {58, 3658}, {60, 61231}, {270, 56410}, {593, 61239}, {849, 56881}, {1101, 55126}, {4556, 8609}, {4636, 18838}, {11570, 36069}, {51649, 52914}
X(66273) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 3658}, {523, 55126}, {4075, 56881}, {38982, 11570}, {55065, 1737}
X(66273) = trilinear pole of line {115, 55232}
X(66273) = barycentric product X(i)*X(j) for these {i,j}: {125, 65344}, {321, 3657}, {338, 6099}, {339, 32698}, {1109, 65248}, {2990, 4036}, {4064, 37203}, {20902, 36106}, {34388, 61214}, {36052, 52623}, {46133, 55232}
X(66273) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 3658}, {115, 55126}, {594, 56881}, {756, 61239}, {2171, 61231}, {2197, 56410}, {2610, 11570}, {2990, 52935}, {3657, 81}, {4024, 1737}, {4036, 48380}, {4064, 914}, {4705, 8609}, {6099, 249}, {32698, 250}, {36052, 4556}, {45393, 4612}, {46133, 55231}, {55230, 2252}, {55232, 912}, {55234, 51649}, {57185, 18838}, {61214, 60}, {65248, 24041}, {65344, 18020}

X(66274) = X(101)X(523)∩X(190)X(850)

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

X(66274) lies on the X-parabola (see X(12065)) and these lines: {101, 523}, {190, 850}, {1018, 4036}, {2501, 8750}, {4024, 4557}, {4608, 4629}, {4628, 58784}, {5134, 39993}, {10412, 56742}, {12079, 17747}, {18808, 41321}

X(66274) = isogonal conjugate of X(42744)
X(66274) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42744}, {81, 2774}, {905, 2073}, {1019, 56808}
X(66274) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 42744}, {40586, 2774}
X(66274) = trilinear pole of line {42, 115}
X(66274) = barycentric product X(i)*X(j) for these {i,j}: {10, 2690}, {1897, 38535}
X(66274) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 42744}, {42, 2774}, {2690, 86}, {4557, 56808}, {8750, 2073}, {38535, 4025}, {39993, 57214}

X(66275) = X(12)X(4036)∩X(65)X(523)

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

X(66275) lies on the X-parabola (see X(12065)) and these lines: {12, 4036}, {65, 523}, {104, 60029}, {476, 2720}, {685, 60568}, {850, 1441}, {892, 54953}, {961, 2401}, {1254, 21134}, {1411, 30725}, {1880, 2501}, {2171, 4024}, {14775, 43933}, {17097, 43728}, {34051, 60043}, {37136, 60055}, {48276, 61238}

X(66275) = X(i)-isoconjugate of X(j) for these (i,j): {249, 46393}, {283, 4246}, {284, 64828}, {517, 4636}, {643, 859}, {1098, 23981}, {1101, 2804}, {2150, 2397}, {2183, 4612}, {2185, 2427}, {7054, 24029}, {17139, 65375}, {22350, 52914}, {24041, 53549}, {36069, 64139}, {55258, 57657}
X(66275) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 2804}, {3005, 53549}, {15267, 23981}, {38982, 64139}, {40590, 64828}, {40622, 17139}, {55060, 859}, {55065, 6735}, {56325, 2397}, {62570, 55258}
X(66275) = trilinear pole of line {115, 57185}
X(66275) = barycentric product X(i)*X(j) for these {i,j}: {12, 2401}, {115, 54953}, {125, 65331}, {338, 2720}, {339, 32702}, {1109, 37136}, {1365, 13136}, {1441, 55259}, {2250, 4077}, {2423, 34388}, {4036, 34051}, {6354, 43728}, {7178, 38955}, {7180, 57984}, {18816, 57185}, {20902, 36110}, {21054, 47317}, {21134, 39294}, {23994, 32669}, {26942, 43933}, {36123, 57243}
X(66275) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 2397}, {65, 64828}, {104, 4612}, {115, 2804}, {181, 2427}, {909, 4636}, {1254, 24029}, {1365, 10015}, {1441, 55258}, {1880, 4246}, {2250, 643}, {2401, 261}, {2423, 60}, {2610, 64139}, {2643, 46393}, {2720, 249}, {3124, 53549}, {4024, 6735}, {7178, 17139}, {7180, 859}, {8736, 53151}, {13136, 6064}, {18816, 4631}, {20975, 52307}, {21131, 35015}, {32669, 1101}, {32702, 250}, {34051, 52935}, {37136, 24041}, {38955, 645}, {43728, 7058}, {43933, 46103}, {51663, 16586}, {53545, 23788}, {54953, 4590}, {55195, 14010}, {55197, 17757}, {55232, 51379}, {55234, 22350}, {55259, 21}, {57185, 517}, {57984, 62534}, {61052, 3310}, {61238, 1098}, {65331, 18020}

X(66276) = X(103)X(476)∩X(125)X(4024)

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

X(66276) lies on the X-parabola (see X(12065)) and these lines: {103, 476}, {125, 4024}, {423, 685}, {516, 47107}, {523, 4466}, {892, 18025}, {1815, 44768}, {2395, 55257}, {2400, 60042}, {2501, 3120}, {4036, 20902}, {4608, 15634}, {24315, 53133}, {36101, 60055}

X(66276) = X(52781)-Ceva conjugate of X(55257)
X(66276) = X(i)-isoconjugate of X(j) for these (i,j): {249, 910}, {516, 1101}, {2426, 52935}, {4241, 4575}, {23357, 30807}, {23995, 35517}
X(66276) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 4241}, {523, 516}, {647, 26006}, {4988, 14953}, {18314, 35517}, {36901, 55256}, {55065, 2398}
X(66276) = trilinear pole of line {115, 21134}
X(66276) = barycentric product X(i)*X(j) for these {i,j}: {103, 338}, {115, 18025}, {125, 52781}, {594, 15634}, {850, 55257}, {911, 23994}, {1109, 36101}, {1815, 2970}, {2400, 4024}, {2424, 52623}, {2643, 57996}, {4064, 53150}, {4092, 52156}, {20902, 36122}, {21131, 57928}, {57243, 60583}
X(66276) = barycentric quotient X(i)/X(j) for these {i,j}: {103, 249}, {115, 516}, {125, 26006}, {338, 35517}, {850, 55256}, {911, 1101}, {1109, 30807}, {1365, 43035}, {2400, 4610}, {2424, 4556}, {2501, 4241}, {2643, 910}, {3120, 14953}, {4024, 2398}, {4036, 42719}, {4079, 2426}, {4092, 40869}, {8754, 1886}, {15634, 1509}, {18025, 4590}, {21043, 17747}, {21046, 51366}, {21131, 676}, {21134, 39470}, {32657, 47390}, {36101, 24041}, {52156, 7340}, {52781, 18020}, {55257, 110}, {57996, 24037}

X(66277) = X(1)X(2501)∩X(63)X(523)

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

X(66277) lies on the X-parabola (see X(12065)) and these lines: {1, 2501}, {63, 523}, {72, 4024}, {92, 57083}, {293, 2395}, {304, 850}, {306, 4036}, {1214, 47887}, {1956, 62519}, {2349, 18808}, {2582, 39240}, {2583, 39241}, {6366, 9719}, {34055, 58784}

X(66277) = X(14414)-cross conjugate of X(522)
X(66277) = X(i)-isoconjugate of X(j) for these (i,j): {109, 1776}, {112, 64888}, {32676, 51608}
X(66277) = X(i)-Dao conjugate of X(j) for these (i,j): {11, 1776}, {15526, 51608}, {34591, 64888}
X(66277) = cevapoint of X(523) and X(6366)
X(66277) = trilinear pole of line {115, 656}
X(66277) = barycentric quotient X(i)/X(j) for these {i,j}: {525, 51608}, {650, 1776}, {656, 64888}, {14414, 52880}

X(66278) = X(76)X(523)∩X(264)X(2501)

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

X(66278) lies on the X-parabola (see X(12065)) and these lines: {76, 523}, {264, 2501}, {290, 2395}, {300, 20578}, {301, 20579}, {308, 53347}, {313, 4024}, {476, 9150}, {670, 34290}, {685, 32717}, {729, 2367}, {850, 1502}, {880, 886}, {882, 57993}, {3114, 46778}, {3978, 9178}, {4036, 27801}, {4581, 40827}, {5466, 14295}, {8599, 40826}, {10412, 20573}, {15328, 40832}, {18896, 35522}, {33919, 56981}, {34385, 55253}, {34389, 55199}, {34390, 55201}, {37132, 37219}, {50946, 57903}, {52632, 64258}, {53153, 57544}, {53154, 57543}

X(66278) = isotomic conjugate of X(5118)
X(66278) = isotomic conjugate of the isogonal conjugate of X(60028)
X(66278) = X(886)-Ceva conjugate of X(34087)
X(66278) = X(35366)-cross conjugate of X(60028)
X(66278) = X(i)-isoconjugate of X(j) for these (i,j): {31, 5118}, {163, 3231}, {560, 23342}, {662, 33875}, {887, 24041}, {888, 1101}, {1576, 2234}, {1917, 63747}, {4575, 46522}, {9148, 23995}, {24037, 65497}, {24039, 41294}, {34072, 52961}
X(66278) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 5118}, {115, 3231}, {136, 46522}, {512, 65497}, {523, 888}, {1084, 33875}, {3005, 887}, {4858, 2234}, {6374, 23342}, {15449, 52961}, {18314, 9148}, {35088, 6786}, {36901, 538}, {55065, 52894}
X(66278) = crosspoint of X(886) and X(34087)
X(66278) = crosssum of X(887) and X(33875)
X(66278) = trilinear pole of line {115, 850}
X(66278) = barycentric product X(i)*X(j) for these {i,j}: {76, 60028}, {115, 886}, {308, 35366}, {338, 9150}, {523, 34087}, {729, 44173}, {850, 3228}, {1502, 63749}, {3124, 57993}, {14608, 52632}, {20948, 37132}, {23962, 32717}, {23994, 36133}, {51510, 56981}
X(66278) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5118}, {76, 23342}, {115, 888}, {338, 9148}, {512, 33875}, {523, 3231}, {729, 1576}, {826, 52961}, {850, 538}, {886, 4590}, {1084, 65497}, {1502, 63747}, {1577, 2234}, {2501, 46522}, {2799, 6786}, {3124, 887}, {3228, 110}, {4024, 52894}, {4036, 52893}, {5466, 14609}, {8029, 52625}, {9148, 52067}, {9150, 249}, {14608, 5467}, {22260, 1645}, {32717, 23357}, {34087, 99}, {35366, 39}, {35522, 45672}, {36133, 1101}, {37132, 163}, {40495, 30938}, {43665, 36822}, {44173, 30736}, {51510, 56980}, {52632, 52756}, {52752, 2420}, {52762, 9181}, {52765, 14966}, {57459, 41412}, {57540, 32717}, {57993, 34537}, {60028, 6}, {63749, 32}

X(66279) = X(8)X(523)∩X(281)X(2501)

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

X(66279) lies on the X-parabola (see X(12065)) and these lines: {8, 523}, {281, 2501}, {476, 6083}, {850, 3596}, {2321, 4024}, {2395, 15628}, {3701, 4036}, {4404, 6757}, {4581, 50351}, {5466, 60251}, {6370, 6740}, {15328, 56103}, {53341, 60055}

X(66279) = X(i)-isoconjugate of X(j) for these (i,j): {163, 35466}, {1101, 6089}, {1884, 4575}
X(66279) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 35466}, {136, 1884}, {523, 6089}, {6741, 44669}
X(66279) = cevapoint of X(i) and X(j) for these (i,j): {523, 6370}, {758, 16598}
X(66279) = trilinear pole of line {115, 3700}
X(66279) = barycentric product X(i)*X(j) for these {i,j}: {75, 35354}, {338, 6083}, {523, 60251}
X(66279) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 6089}, {523, 35466}, {2501, 1884}, {2533, 27970}, {3700, 44669}, {6083, 249}, {17058, 65463}, {35354, 1}, {60251, 99}

X(66280) = X(80)X(758)∩X(100)X(523)

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

X(66280) lies on the X-parabola (see X(12065)) and these lines: {2, 38982}, {8, 58076}, {80, 758}, {100, 523}, {291, 21907}, {526, 64688}, {666, 9979}, {668, 850}, {901, 6089}, {1018, 4024}, {1783, 2501}, {3952, 4036}, {4427, 56321}, {4581, 53349}, {4596, 4608}, {4674, 5620}, {5379, 14775}, {5380, 5466}, {6370, 51562}, {10412, 56881}, {12079, 17757}, {18808, 53151}, {21956, 64258}

X(66280) = isogonal conjugate of X(42741)
X(66280) = isotomic conjugate of X(65669)
X(66280) = anticomplement of X(38982)
X(66280) = isotomic conjugate of the anticomplement of X(2610)
X(66280) = X(i)-cross conjugate of X(j) for these (i,j): {2610, 2}, {55238, 4080}
X(66280) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42741}, {21, 51646}, {31, 65669}, {58, 8674}, {86, 42670}, {513, 5127}, {514, 19622}, {649, 37783}, {1019, 17796}, {1459, 2074}, {1790, 47235}, {3737, 5172}, {32849, 57129}, {36034, 57447}, {36069, 38982}
X(66280) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 65669}, {3, 42741}, {10, 8674}, {3258, 57447}, {5375, 37783}, {39026, 5127}, {40600, 42670}, {40611, 51646}
X(66280) = cevapoint of X(i) and X(j) for these (i,j): {10, 6370}, {513, 50757}, {523, 758}, {3738, 4999}
X(66280) = crosspoint of X(35156) and X(65238)
X(66280) = trilinear pole of line {37, 115}
X(66280) = barycentric product X(i)*X(j) for these {i,j}: {10, 65238}, {37, 35156}, {190, 5620}, {321, 1290}, {3952, 21907}, {4552, 11604}
X(66280) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 65669}, {6, 42741}, {37, 8674}, {100, 37783}, {101, 5127}, {213, 42670}, {692, 19622}, {1290, 81}, {1400, 51646}, {1637, 57447}, {1783, 2074}, {1824, 47235}, {2610, 38982}, {3952, 32849}, {4557, 17796}, {4559, 5172}, {5620, 514}, {11604, 4560}, {21907, 7192}, {35156, 274}, {61170, 41542}, {61171, 41541}, {61178, 37799}, {65238, 86}

X(66281) = X(10)X(850)∩X(42)X(523)

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

X(66281) lies on the X-parabola (see X(12065)) and these lines: {10, 850}, {42, 523}, {291, 4453}, {476, 32682}, {659, 6187}, {675, 28482}, {756, 4036}, {1126, 4608}, {1500, 4024}, {2224, 60043}, {2333, 2501}, {4581, 60573}, {5466, 60135}, {23887, 65660}, {36087, 60055}, {53361, 56321}

X(66281) = X(i)-isoconjugate of X(j) for these (i,j): {593, 42723}, {662, 14964}, {674, 52935}, {1101, 23887}, {1444, 4249}, {2225, 4610}, {4556, 57015}, {4612, 43039}, {4623, 8618}, {24041, 65703}
X(66281) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 23887}, {1084, 14964}, {3005, 65703}, {55065, 3006}
X(66281) = crosssum of X(14964) and X(65703)
X(66281) = trilinear pole of line {115, 4079}
X(66281) = barycentric product X(i)*X(j) for these {i,j}: {12, 60573}, {338, 32682}, {523, 60135}, {675, 4024}, {1109, 36087}, {2224, 4036}, {4079, 43093}, {4705, 37130}
X(66281) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 23887}, {512, 14964}, {675, 4610}, {756, 42723}, {2224, 52935}, {2333, 4249}, {3124, 65703}, {4024, 3006}, {4079, 674}, {4705, 57015}, {32682, 249}, {36087, 24041}, {37130, 4623}, {43093, 52612}, {50487, 2225}, {53581, 8618}, {60135, 99}, {60573, 261}

X(66282) = X(37)X(523)∩X(105)X(53686)

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

X(66282) lies on the X-parabola (see X(12065)) and these lines: {37, 523}, {105, 53686}, {294, 60029}, {321, 850}, {335, 918}, {476, 919}, {594, 4036}, {650, 6690}, {666, 892}, {673, 60042}, {685, 65333}, {756, 4024}, {885, 2298}, {1024, 2161}, {1255, 4608}, {1824, 2501}, {2171, 4079}, {2284, 53358}, {2395, 56853}, {5466, 13576}, {6354, 57185}, {16600, 21201}, {18098, 58784}, {21132, 21808}, {28132, 40500}, {36086, 60055}

X(66282) = X(i)-Ceva conjugate of X(j) for these (i,j): {666, 13576}, {65333, 56853}
X(66282) = X(4155)-cross conjugate of X(523)
X(66282) = X(i)-isoconjugate of X(j) for these (i,j): {60, 1025}, {81, 54353}, {110, 18206}, {163, 30941}, {241, 4636}, {249, 2254}, {518, 4556}, {593, 1026}, {662, 3286}, {665, 24041}, {672, 52935}, {757, 2284}, {849, 42720}, {883, 2150}, {918, 1101}, {1458, 4612}, {1509, 54325}, {1576, 18157}, {1790, 4238}, {2185, 2283}, {2206, 55260}, {2223, 4610}, {4558, 54407}, {4575, 15149}, {4623, 9454}, {7054, 41353}, {7340, 46388}, {9455, 52612}, {23225, 46254}
X(66282) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 30941}, {136, 15149}, {244, 18206}, {523, 918}, {1084, 3286}, {3005, 665}, {4075, 42720}, {4858, 18157}, {4988, 23829}, {33675, 4623}, {40586, 54353}, {40603, 55260}, {40607, 2284}, {55065, 3912}, {56325, 883}, {62554, 52935}, {62599, 4610}
X(66282) = crosspoint of X(i) and X(j) for these (i,j): {666, 13576}, {4444, 60574}
X(66282) = crosssum of X(665) and X(3286)
X(66282) = trilinear pole of line {115, 4705}
X(66282) = barycentric product X(i)*X(j) for these {i,j}: {12, 885}, {105, 4036}, {115, 666}, {125, 65333}, {321, 55261}, {338, 919}, {523, 13576}, {594, 62635}, {673, 4024}, {850, 56853}, {884, 34388}, {927, 4092}, {1024, 6358}, {1027, 1089}, {1109, 36086}, {1365, 36802}, {1438, 52623}, {1577, 18785}, {2481, 4705}, {2643, 51560}, {3124, 36803}, {4064, 36124}, {4079, 18031}, {6057, 43930}, {6354, 28132}, {10099, 41013}, {23696, 56285}, {23994, 32666}, {28654, 43929}, {36796, 57185}, {54235, 55232}
X(66282) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 883}, {42, 54353}, {105, 52935}, {115, 918}, {181, 2283}, {294, 4612}, {321, 55260}, {512, 3286}, {523, 30941}, {594, 42720}, {661, 18206}, {666, 4590}, {673, 4610}, {756, 1026}, {872, 54325}, {884, 60}, {885, 261}, {919, 249}, {927, 7340}, {1024, 2185}, {1027, 757}, {1254, 41353}, {1365, 43042}, {1438, 4556}, {1500, 2284}, {1577, 18157}, {1824, 4238}, {2171, 1025}, {2195, 4636}, {2481, 4623}, {2501, 15149}, {2643, 2254}, {3120, 23829}, {3124, 665}, {4024, 3912}, {4036, 3263}, {4079, 672}, {4092, 50333}, {4155, 8299}, {4705, 518}, {6367, 4966}, {7063, 8638}, {10099, 1444}, {13576, 99}, {18031, 52612}, {18785, 662}, {20975, 53550}, {21043, 4088}, {21725, 53553}, {21833, 24290}, {24290, 16728}, {28132, 7058}, {32666, 1101}, {36086, 24041}, {36796, 4631}, {36802, 6064}, {36803, 34537}, {43929, 593}, {43930, 552}, {50487, 2223}, {51560, 24037}, {52030, 36066}, {52209, 65258}, {53581, 9454}, {54235, 55231}, {55230, 1818}, {55232, 25083}, {55261, 81}, {56853, 110}, {57185, 241}, {58289, 20683}, {61052, 53539}, {62635, 1509}, {65333, 18020}, {65751, 23225}

X(66283) = X(190)X(523)∩X(335)X(740)

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

X(66283) lies on the X-parabola (see X(12065)) and these lines: {99, 41076}, {190, 523}, {335, 740}, {660, 4155}, {690, 4555}, {835, 2702}, {850, 1978}, {874, 4639}, {1016, 6367}, {1897, 2501}, {1929, 17763}, {2054, 4039}, {2395, 2398}, {3943, 6543}, {3952, 4024}, {4033, 4036}, {4062, 4080}, {4427, 4600}, {4562, 18009}, {4581, 17940}, {5466, 17780}, {6758, 54118}, {9278, 41683}, {17791, 18032}, {17934, 60042}, {23354, 66267}, {27804, 40725}, {34246, 56797}, {36238, 65873}, {39921, 57040}, {47318, 53341}, {64071, 64236}

X(66283) = X(i)-cross conjugate of X(j) for these (i,j): {18001, 9278}, {18004, 10}, {18014, 11599}, {35352, 13576}
X(66283) = X(i)-isoconjugate of X(j) for these (i,j): {58, 9508}, {81, 5029}, {244, 17943}, {423, 22383}, {513, 1326}, {514, 64215}, {649, 1931}, {667, 17731}, {741, 38348}, {757, 17990}, {849, 18004}, {1019, 17735}, {1333, 2786}, {1757, 3733}, {1919, 52137}, {3248, 17934}, {6542, 57129}, {7192, 18266}, {17976, 57200}, {18268, 27929}
X(66283) = X(i)-Dao conjugate of X(j) for these (i,j): {10, 9508}, {37, 2786}, {4075, 18004}, {5375, 1931}, {6631, 17731}, {8299, 38348}, {9296, 52137}, {35068, 27929}, {39026, 1326}, {40586, 5029}, {40607, 17990}, {52872, 28602}
X(66283) = cevapoint of X(i) and X(j) for these (i,j): {10, 18004}, {37, 4155}, {523, 740}, {812, 17045}, {4427, 62644}, {9278, 18001}, {11599, 18014}
X(66283) = trilinear pole of line {10, 115}
X(66283) = barycentric product X(i)*X(j) for these {i,j}: {10, 35148}, {99, 6543}, {190, 11599}, {313, 2702}, {321, 37135}, {594, 17930}, {668, 9278}, {1016, 18014}, {1018, 18032}, {1897, 57848}, {1929, 4033}, {1978, 2054}, {3952, 6650}, {17940, 28654}, {17962, 27808}, {17982, 52609}, {18001, 31625}, {18004, 57560}
X(66283) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 2786}, {37, 9508}, {42, 5029}, {100, 1931}, {101, 1326}, {190, 17731}, {594, 18004}, {668, 52137}, {692, 64215}, {740, 27929}, {1016, 17934}, {1018, 1757}, {1252, 17943}, {1500, 17990}, {1897, 423}, {1929, 1019}, {2054, 649}, {2238, 38348}, {2702, 58}, {3943, 28602}, {3952, 6542}, {4033, 20947}, {4103, 6541}, {4557, 17735}, {4574, 17976}, {6367, 57461}, {6543, 523}, {6650, 7192}, {9278, 513}, {11599, 514}, {17780, 31059}, {17930, 1509}, {17940, 593}, {17962, 3733}, {17972, 7254}, {17982, 17925}, {18001, 1015}, {18004, 35080}, {18014, 1086}, {18032, 7199}, {35148, 86}, {37135, 81}, {40521, 20693}, {40767, 50456}, {57560, 17930}, {57681, 1459}, {57848, 4025}

X(66284) = X(1)X(523)∩X(10)X(522)

Barycentrics    (b - c)*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :
X(66284) = 3 X[21132] + X[39771], 2 X[30725] - 3 X[53314], X[30725] - 3 X[53522], 3 X[11125] - X[30572], 3 X[11125] - 2 X[59837]

X(66284) lies on the X-parabola (see X(12065)) and these lines: {1, 523}, {6, 55195}, {10, 522}, {19, 2501}, {35, 46611}, {36, 46610}, {37, 650}, {65, 513}, {75, 850}, {80, 900}, {82, 58784}, {190, 53359}, {225, 7649}, {476, 36069}, {512, 994}, {514, 4667}, {525, 56136}, {655, 885}, {656, 41501}, {659, 6187}, {676, 2006}, {759, 6089}, {826, 56149}, {892, 65283}, {897, 5466}, {1024, 2161}, {1168, 6550}, {1393, 4017}, {1411, 30725}, {1581, 66267}, {1647, 42754}, {1769, 52383}, {1807, 2804}, {1866, 54244}, {1910, 2395}, {2153, 20578}, {2154, 20579}, {2166, 10412}, {2168, 55253}, {2216, 50946}, {2222, 23981}, {2363, 4581}, {2397, 17780}, {2588, 39240}, {2589, 39241}, {2652, 9508}, {2826, 56426}, {3667, 31673}, {3668, 3676}, {3700, 17281}, {3738, 21112}, {4132, 34434}, {4151, 42285}, {4453, 63217}, {4608, 40438}, {4802, 21105}, {4926, 56174}, {4977, 14812}, {5425, 57130}, {5620, 10265}, {5903, 61637}, {6129, 61039}, {6370, 6740}, {6788, 38938}, {7252, 12077}, {8043, 62566}, {8599, 55927}, {8674, 17636}, {8677, 34242}, {8773, 57985}, {9268, 36236}, {10260, 39478}, {11125, 30572}, {14315, 56419}, {14616, 18827}, {15328, 36053}, {15475, 50344}, {16118, 28217}, {18011, 23352}, {18359, 41683}, {18808, 36119}, {20220, 39702}, {21186, 23604}, {23894, 64258}, {23987, 36110}, {26546, 26665}, {27529, 48204}, {28151, 30573}, {28161, 56221}, {28205, 56237}, {28221, 56135}, {29144, 40747}, {30591, 34920}, {34860, 65099}, {35055, 47054}, {35174, 53208}, {36801, 36804}, {37140, 60055}, {40172, 57051}, {40430, 56321}, {40437, 43728}, {42027, 64857}, {42337, 56259}, {42757, 52212}, {42763, 60845}, {47318, 53341}, {50333, 52351}, {51648, 52384}, {52371, 53523}, {52380, 60029}, {56284, 58322}, {56691, 65854}

X(66284) = reflection of X(i) in X(j) for these {i,j}: {24457, 21201}, {30572, 59837}, {48292, 44409}, {50350, 21186}, {50351, 62323}, {53314, 53522}, {53527, 21180}, {62329, 50574}, {62566, 8043}
X(66284) = polar conjugate of X(65162)
X(66284) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1168, 3448}, {32671, 30579}
X(66284) = X(i)-Ceva conjugate of X(j) for these (i,j): {476, 759}, {655, 2161}, {2222, 52383}, {23592, 64445}, {34535, 2170}, {40437, 11}, {51562, 80}, {65283, 24624}
X(66284) = X(i)-cross conjugate of X(j) for these (i,j): {244, 1168}, {1635, 514}, {1637, 7178}, {1647, 14584}, {1769, 513}, {2170, 34535}, {6089, 523}, {42666, 661}, {42759, 4}, {52338, 1086}, {55238, 60074}, {64445, 23592}
X(66284) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1983}, {3, 4242}, {6, 4585}, {36, 100}, {48, 65162}, {50, 15455}, {59, 3738}, {99, 3724}, {101, 3218}, {109, 4511}, {110, 758}, {163, 3936}, {190, 7113}, {213, 55237}, {214, 901}, {215, 35174}, {249, 2610}, {320, 692}, {643, 1464}, {651, 2323}, {654, 4564}, {655, 34544}, {662, 2245}, {664, 2361}, {668, 52434}, {765, 53314}, {813, 27950}, {860, 4575}, {906, 17923}, {934, 58328}, {1016, 21758}, {1023, 40215}, {1101, 6370}, {1110, 4453}, {1227, 32719}, {1252, 3960}, {1290, 35204}, {1293, 4881}, {1331, 1870}, {1332, 52413}, {1415, 32851}, {1443, 3939}, {1492, 3792}, {1576, 35550}, {1783, 22128}, {1897, 52407}, {2149, 3904}, {2222, 4996}, {2720, 64139}, {3257, 17455}, {3699, 52440}, {4053, 4556}, {4282, 4552}, {4554, 52426}, {4567, 21828}, {4570, 53527}, {4588, 4867}, {4591, 40988}, {4592, 44113}, {4736, 36069}, {4880, 8652}, {4973, 8701}, {4998, 8648}, {5081, 36059}, {5377, 53555}, {5546, 18593}, {6011, 27086}, {6099, 11570}, {6149, 6742}, {6516, 52427}, {6733, 63779}, {6739, 36034}, {6757, 52603}, {7045, 53285}, {9268, 53535}, {13589, 39166}, {15742, 22379}, {16586, 32641}, {16944, 17780}, {17515, 23067}, {20924, 32739}, {23344, 52553}, {23703, 62703}, {23981, 56757}, {24041, 42666}, {27757, 34073}, {32665, 51583}, {34586, 36037}, {35069, 37140}, {36804, 52059}, {39149, 57119}, {39778, 65881}, {41804, 65375}, {44717, 65104}, {53546, 59149}
X(66284) = X(i)-Dao conjugate of X(j) for these (i,j): {9, 4585}, {11, 4511}, {115, 3936}, {136, 860}, {244, 758}, {513, 53314}, {514, 4453}, {523, 6370}, {650, 3904}, {661, 3960}, {1015, 3218}, {1084, 2245}, {1086, 320}, {1146, 32851}, {1249, 65162}, {3005, 42666}, {3258, 6739}, {3259, 34586}, {4858, 35550}, {4988, 4707}, {5139, 44113}, {5190, 17923}, {5520, 52368}, {5521, 1870}, {6615, 3738}, {6626, 55237}, {8054, 36}, {14714, 58328}, {14838, 3268}, {14993, 6742}, {15898, 100}, {17115, 53285}, {20620, 5081}, {32664, 1983}, {34467, 52407}, {35092, 51583}, {36103, 4242}, {36909, 3699}, {38979, 214}, {38981, 64139}, {38982, 4736}, {38984, 4996}, {38986, 3724}, {38991, 2323}, {38995, 3792}, {39006, 22128}, {39025, 2361}, {40615, 17078}, {40617, 1443}, {40619, 20924}, {40622, 41804}, {40623, 27950}, {40627, 21828}, {50330, 53527}, {55045, 4867}, {55053, 7113}, {55055, 17455}, {55060, 1464}, {56416, 17780}, {61073, 27757}
X(66284) = cevapoint of X(i) and X(j) for these (i,j): {244, 6550}, {512, 3310}, {513, 59837}, {514, 45674}, {523, 900}, {661, 42666}, {1647, 21132}, {2310, 52316}, {3724, 21742}
X(66284) = crosspoint of X(i) and X(j) for these (i,j): {80, 51562}, {476, 2166}, {655, 18815}, {759, 2222}, {903, 65238}, {2401, 6548}, {4555, 60251}, {14616, 36804}, {24624, 65283}
X(66284) = crosssum of X(i) and X(j) for these (i,j): {1, 53406}, {36, 53314}, {55, 27780}, {523, 8068}, {526, 6149}, {654, 2361}, {758, 3738}, {2245, 42666}, {2323, 53285}, {2427, 23344}, {3724, 21758}
X(66284) = trilinear pole of line {115, 661}
X(66284) = crossdifference of every pair of points on line {36, 2245}
X(66284) = X(46610)-line conjugate of X(36)
X(66284) = barycentric product X(i)*X(j) for these {i,j}: {1, 60074}, {11, 655}, {12, 60571}, {57, 52356}, {80, 514}, {86, 55238}, {94, 2605}, {115, 65283}, {244, 36804}, {265, 65100}, {338, 36069}, {476, 8287}, {513, 18359}, {522, 2006}, {523, 24624}, {649, 20566}, {650, 18815}, {654, 57645}, {661, 14616}, {693, 2161}, {759, 1577}, {850, 34079}, {1019, 15065}, {1022, 51975}, {1086, 51562}, {1109, 37140}, {1168, 3762}, {1411, 4391}, {1635, 57788}, {1807, 17924}, {1989, 4467}, {2166, 14838}, {2170, 35174}, {2222, 4858}, {2341, 4077}, {2394, 56645}, {2401, 56416}, {2501, 57985}, {2611, 32680}, {3064, 52392}, {3120, 47318}, {3219, 43082}, {3261, 6187}, {3271, 46405}, {3669, 52409}, {3676, 36910}, {3737, 60091}, {3738, 34535}, {3904, 63750}, {4025, 64835}, {4049, 56950}, {4444, 36815}, {4560, 52383}, {6740, 7178}, {7004, 65329}, {7199, 34857}, {7649, 52351}, {9273, 23105}, {10015, 40437}, {10412, 40214}, {12077, 39277}, {14584, 60480}, {14618, 57736}, {14628, 23838}, {15475, 34016}, {17886, 32678}, {18070, 46160}, {20982, 35139}, {22094, 46456}, {23994, 32671}, {24002, 52371}, {30725, 36590}, {32675, 34387}, {34232, 60485}, {35015, 53811}, {37203, 61039}, {40166, 52377}, {42666, 57555}, {43728, 52212}, {45926, 56320}, {46107, 52431}, {52391, 57215}, {52780, 61041}
X(66284) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4585}, {4, 65162}, {11, 3904}, {19, 4242}, {31, 1983}, {80, 190}, {86, 55237}, {115, 6370}, {244, 3960}, {512, 2245}, {513, 3218}, {514, 320}, {522, 32851}, {523, 3936}, {649, 36}, {650, 4511}, {654, 4996}, {655, 4998}, {657, 58328}, {659, 27950}, {661, 758}, {663, 2323}, {667, 7113}, {693, 20924}, {759, 662}, {764, 53546}, {798, 3724}, {900, 51583}, {1015, 53314}, {1022, 52553}, {1086, 4453}, {1168, 3257}, {1411, 651}, {1459, 22128}, {1577, 35550}, {1635, 214}, {1637, 6739}, {1769, 16586}, {1807, 1332}, {1919, 52434}, {1960, 17455}, {1989, 6742}, {2006, 664}, {2087, 53535}, {2161, 100}, {2166, 15455}, {2170, 3738}, {2222, 4564}, {2341, 643}, {2489, 44113}, {2501, 860}, {2605, 323}, {2610, 4736}, {2611, 32679}, {2643, 2610}, {3063, 2361}, {3064, 5081}, {3120, 4707}, {3122, 21828}, {3124, 42666}, {3125, 53527}, {3248, 21758}, {3250, 3792}, {3261, 40075}, {3271, 654}, {3310, 34586}, {3669, 1443}, {3676, 17078}, {3762, 1227}, {4017, 18593}, {4036, 61410}, {4394, 4881}, {4467, 7799}, {4705, 4053}, {4730, 40988}, {4777, 27757}, {4813, 4880}, {4893, 4867}, {4979, 4973}, {6187, 101}, {6545, 4089}, {6591, 1870}, {6729, 63779}, {6740, 645}, {7178, 41804}, {7180, 1464}, {7649, 17923}, {8287, 3268}, {8648, 34544}, {8735, 44428}, {9273, 59152}, {14582, 52388}, {14584, 62669}, {14616, 799}, {14936, 53285}, {15065, 4033}, {15475, 8818}, {18359, 668}, {18815, 4554}, {20566, 1978}, {20982, 526}, {22094, 8552}, {22383, 52407}, {23345, 40215}, {24624, 99}, {30725, 41801}, {32671, 1101}, {32675, 59}, {34079, 110}, {34535, 35174}, {34857, 1018}, {35015, 53045}, {36069, 249}, {36590, 4582}, {36804, 7035}, {36815, 3570}, {36910, 3699}, {37140, 24041}, {40172, 1023}, {40214, 10411}, {40437, 13136}, {42657, 26744}, {42666, 35069}, {43052, 36589}, {43082, 30690}, {45926, 65205}, {46393, 64139}, {47227, 52368}, {47318, 4600}, {51562, 1016}, {51975, 24004}, {52316, 57434}, {52338, 51402}, {52351, 4561}, {52356, 312}, {52371, 644}, {52377, 31615}, {52380, 4612}, {52383, 4552}, {52391, 65233}, {52392, 65164}, {52409, 646}, {52431, 1331}, {54244, 52414}, {55208, 1835}, {55238, 10}, {56416, 2397}, {56645, 2407}, {57099, 42701}, {57181, 52440}, {57645, 46405}, {57736, 4558}, {57985, 4563}, {59283, 42718}, {59837, 40612}, {60074, 75}, {60571, 261}, {61039, 914}, {61238, 56757}, {63750, 655}, {64835, 1897}, {65100, 340}, {65283, 4590}
X(66284) = {X(11125),X(30572)}-harmonic conjugate of X(59837)

X(66285) = X(10)X(523)∩X(80)X(900)

Barycentrics    (a + b - 2*c)*(b - c)*(a - 2*b + c)*(b + c)^2 : :
X(66285) = 3 X[4049] - X[55244], 9 X[23598] - X[23838], X[23345] + 3 X[60480], 3 X[21112] + X[53533], X[53533] - 3 X[53565]

X(66285) lies on the X-parabola (see X(12065)) and these lines: {10, 523}, {80, 900}, {88, 60043}, {106, 2372}, {313, 850}, {476, 901}, {513, 3754}, {514, 4472}, {594, 4024}, {685, 32719}, {892, 4555}, {903, 35162}, {1022, 1224}, {1089, 4036}, {1220, 4581}, {1268, 3004}, {1320, 60029}, {1826, 2501}, {3257, 60055}, {4013, 6370}, {4080, 5466}, {4642, 24457}, {4732, 4777}, {4802, 19947}, {4806, 8599}, {4841, 55263}, {6542, 62626}, {7649, 12135}, {17390, 21200}, {17953, 52747}, {18082, 58784}, {21051, 45095}, {21112, 53533}, {21119, 40086}, {23352, 28183}, {31946, 51870}, {50342, 62732}

X(66285) = midpoint of X(i) and X(j) for these {i,j}: {21112, 53565}, {21119, 40086}
X(66285) = X(4555)-Ceva conjugate of X(4080)
X(66285) = X(6370)-cross conjugate of X(523)
X(66285) = X(i)-isoconjugate of X(j) for these (i,j): {44, 4556}, {60, 23703}, {101, 30576}, {110, 52680}, {163, 16704}, {214, 36069}, {249, 1635}, {250, 53532}, {593, 1023}, {662, 3285}, {757, 23344}, {849, 17780}, {900, 1101}, {902, 52935}, {1319, 4636}, {1404, 4612}, {1415, 30606}, {1437, 46541}, {1576, 30939}, {1960, 24041}, {2150, 62669}, {2185, 61210}, {2206, 55243}, {2251, 4610}, {3762, 23357}, {4575, 37168}, {4623, 9459}, {17455, 37140}, {23995, 65867}, {32671, 51583}
X(66285) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 16704}, {136, 37168}, {244, 52680}, {523, 900}, {1015, 30576}, {1084, 3285}, {1146, 30606}, {3005, 1960}, {4075, 17780}, {4858, 30939}, {9460, 4610}, {18314, 65867}, {38982, 214}, {40594, 52935}, {40595, 4556}, {40603, 55243}, {40607, 23344}, {55065, 519}, {56325, 62669}
X(66285) = cevapoint of X(2610) and X(4705)
X(66285) = crosspoint of X(4080) and X(4555)
X(66285) = crosssum of X(1960) and X(3285)
X(66285) = trilinear pole of line {115, 4024}
X(66285) = crossdifference of every pair of points on line {3285, 17455}
X(66285) = barycentric product X(i)*X(j) for these {i,j}: {10, 4049}, {12, 60480}, {88, 4036}, {106, 52623}, {115, 4555}, {125, 65336}, {313, 55263}, {321, 55244}, {338, 901}, {514, 4013}, {523, 4080}, {594, 6548}, {903, 4024}, {1022, 1089}, {1109, 3257}, {1365, 4582}, {1441, 61179}, {1577, 4674}, {2610, 57788}, {4064, 6336}, {4079, 57995}, {4103, 6549}, {4615, 21043}, {4634, 21833}, {4705, 20568}, {6358, 23838}, {21131, 62536}, {23345, 28654}, {23962, 32719}, {23994, 32665}
X(66285) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 62669}, {88, 52935}, {106, 4556}, {115, 900}, {181, 61210}, {313, 55262}, {321, 55243}, {338, 65867}, {512, 3285}, {513, 30576}, {522, 30606}, {523, 16704}, {594, 17780}, {661, 52680}, {756, 1023}, {901, 249}, {903, 4610}, {1022, 757}, {1089, 24004}, {1109, 3762}, {1168, 37140}, {1320, 4612}, {1365, 30725}, {1500, 23344}, {1577, 30939}, {1826, 46541}, {2171, 23703}, {2316, 4636}, {2501, 37168}, {2610, 214}, {2643, 1635}, {3124, 1960}, {3257, 24041}, {3708, 53532}, {4013, 190}, {4024, 519}, {4036, 4358}, {4049, 86}, {4064, 3977}, {4079, 902}, {4080, 99}, {4092, 1639}, {4555, 4590}, {4582, 6064}, {4674, 662}, {4705, 44}, {6057, 30731}, {6367, 4969}, {6370, 51583}, {6535, 4169}, {6548, 1509}, {17998, 5170}, {18004, 31059}, {20568, 4623}, {20975, 22086}, {21043, 4120}, {21046, 14429}, {21131, 1647}, {21833, 4730}, {23345, 593}, {23838, 2185}, {30575, 4622}, {32665, 1101}, {32719, 23357}, {42666, 17455}, {50487, 2251}, {52623, 3264}, {53527, 17191}, {53581, 9459}, {55197, 40663}, {55230, 22356}, {55232, 5440}, {55238, 56950}, {55244, 81}, {55263, 58}, {57185, 1319}, {57995, 52612}, {58289, 52963}, {60480, 261}, {61179, 21}, {65336, 18020}
X(66285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 4049, 18011}, {18004, 18005, 4080}

X(66286) = X(75)X(523)∩X(92)X(2501)

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

X(66286) lies on the X-parabola (see X(12065)) and these lines: {75, 523}, {92, 2501}, {274, 50351}, {291, 4453}, {313, 3261}, {321, 693}, {335, 918}, {476, 36066}, {514, 3572}, {561, 850}, {660, 883}, {661, 25759}, {813, 2860}, {826, 52619}, {874, 4639}, {875, 3112}, {892, 65285}, {1441, 20504}, {1821, 2395}, {1934, 66267}, {2997, 65099}, {3004, 51868}, {3113, 4367}, {4049, 6381}, {4369, 24631}, {4406, 29144}, {4411, 29204}, {4458, 14296}, {4467, 17155}, {4562, 35171}, {4583, 6548}, {5466, 40017}, {10412, 63759}, {14616, 18827}, {15328, 57738}, {16708, 20511}, {18895, 65867}, {20908, 23596}, {23807, 23877}, {34284, 49303}, {48326, 65101}, {52716, 62634}, {53361, 53377}, {57987, 62645}, {60055, 65258}, {63223, 65869}

X(66286) = reflection of X(27855) in X(20518)
X(66286) = isotomic conjugate of X(3573)
X(66286) = isotomic conjugate of the isogonal conjugate of X(876)
X(66286) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {741, 39353}, {919, 39367}, {2311, 14732}, {4584, 20344}, {4589, 20552}, {51866, 148}, {52030, 21221}, {52209, 3448}
X(66286) = X(i)-Ceva conjugate of X(j) for these (i,j): {4583, 334}, {65285, 40017}
X(66286) = X(i)-cross conjugate of X(j) for these (i,j): {20908, 3261}, {35352, 4444}, {48326, 514}
X(66286) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3573}, {32, 3570}, {100, 2210}, {101, 1914}, {110, 3747}, {163, 2238}, {190, 14599}, {238, 692}, {239, 32739}, {242, 32656}, {249, 46390}, {560, 874}, {659, 1110}, {662, 41333}, {668, 18892}, {740, 1576}, {812, 23990}, {813, 51328}, {825, 16514}, {827, 4093}, {862, 4575}, {906, 2201}, {1101, 4155}, {1252, 8632}, {1284, 65375}, {1331, 57654}, {1415, 3684}, {1428, 3939}, {1501, 27853}, {1933, 3903}, {1978, 18894}, {2149, 4435}, {3783, 34069}, {4148, 23979}, {4154, 17938}, {4432, 32719}, {4455, 4570}, {4557, 5009}, {4579, 61385}, {5384, 58864}, {6066, 43041}, {7193, 8750}, {8299, 32666}, {8300, 34067}, {14574, 35544}, {14602, 27805}, {32642, 51435}, {40729, 56982}, {51329, 52927}
X(66286) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 3573}, {115, 2238}, {136, 862}, {244, 3747}, {514, 659}, {523, 4155}, {650, 4435}, {661, 8632}, {1015, 1914}, {1084, 41333}, {1086, 238}, {1146, 3684}, {1577, 3716}, {2968, 58327}, {4369, 5027}, {4858, 740}, {4988, 21832}, {5190, 2201}, {5521, 57654}, {6374, 874}, {6376, 3570}, {6741, 4433}, {8054, 2210}, {9470, 692}, {14838, 53563}, {16592, 1580}, {26932, 7193}, {27929, 38348}, {35080, 8298}, {35088, 50440}, {35094, 8299}, {35119, 8300}, {36901, 3948}, {36906, 101}, {40615, 1429}, {40617, 1428}, {40618, 20769}, {40619, 239}, {40622, 1284}, {40623, 51328}, {40624, 3685}, {46398, 15507}, {50330, 4455}, {52656, 2284}, {55043, 4093}, {55053, 14599}, {61065, 3783}, {62557, 100}
X(66286) = cevapoint of X(i) and X(j) for these (i,j): {514, 4458}, {523, 918}, {4444, 60577}, {4858, 52305}, {6545, 21140}
X(66286) = crosspoint of X(i) and X(j) for these (i,j): {334, 4583}, {40017, 65285}
X(66286) = trilinear pole of line {115, 1111}
X(66286) = crossdifference of every pair of points on line {2210, 14599}
X(66286) = barycentric product X(i)*X(j) for these {i,j}: {75, 4444}, {76, 876}, {85, 60577}, {115, 65285}, {274, 35352}, {291, 3261}, {292, 40495}, {334, 514}, {335, 693}, {337, 17924}, {338, 36066}, {513, 18895}, {523, 40017}, {561, 3572}, {649, 44172}, {660, 23989}, {667, 44170}, {741, 20948}, {850, 37128}, {870, 23596}, {871, 30671}, {875, 1502}, {1086, 4583}, {1109, 65258}, {1111, 4562}, {1365, 36806}, {1577, 18827}, {1916, 4374}, {1934, 4369}, {2501, 57987}, {3120, 4639}, {3766, 40098}, {3801, 40834}, {4077, 36800}, {4367, 18896}, {4391, 7233}, {4458, 63895}, {4518, 24002}, {4584, 21207}, {4589, 16732}, {4817, 63228}, {4876, 52621}, {5378, 23100}, {7199, 43534}, {8033, 66267}, {14208, 65352}, {14618, 57738}, {14621, 63219}, {18268, 44173}, {23285, 39276}, {30663, 65101}, {44160, 56242}
X(66286) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3573}, {11, 4435}, {75, 3570}, {76, 874}, {115, 4155}, {244, 8632}, {291, 101}, {292, 692}, {295, 906}, {334, 190}, {335, 100}, {337, 1332}, {512, 41333}, {513, 1914}, {514, 238}, {522, 3684}, {523, 2238}, {561, 27853}, {649, 2210}, {659, 51328}, {660, 1252}, {661, 3747}, {667, 14599}, {693, 239}, {741, 163}, {812, 8300}, {813, 1110}, {824, 3783}, {850, 3948}, {875, 32}, {876, 6}, {882, 40729}, {905, 7193}, {918, 8299}, {1019, 5009}, {1086, 659}, {1111, 812}, {1491, 16514}, {1577, 740}, {1911, 32739}, {1916, 3903}, {1919, 18892}, {1934, 27805}, {1980, 18894}, {2196, 32656}, {2311, 65375}, {2501, 862}, {2643, 46390}, {2786, 8298}, {2799, 50440}, {3120, 21832}, {3125, 4455}, {3239, 58327}, {3252, 54325}, {3261, 350}, {3572, 31}, {3669, 1428}, {3676, 1429}, {3700, 4433}, {3762, 4432}, {3766, 4366}, {3776, 56805}, {3801, 18904}, {3837, 17475}, {3942, 22384}, {4025, 20769}, {4036, 4037}, {4077, 16609}, {4086, 3985}, {4367, 1691}, {4369, 1580}, {4374, 385}, {4391, 3685}, {4411, 4396}, {4444, 1}, {4453, 27950}, {4458, 19557}, {4466, 53556}, {4486, 3802}, {4500, 4489}, {4518, 644}, {4562, 765}, {4583, 1016}, {4584, 4570}, {4589, 4567}, {4639, 4600}, {4791, 4693}, {4815, 4771}, {4823, 4716}, {4824, 16369}, {4858, 3716}, {4876, 3939}, {4957, 4800}, {4978, 4974}, {5378, 59149}, {6545, 27846}, {6591, 57654}, {7178, 1284}, {7199, 33295}, {7200, 4164}, {7233, 651}, {7649, 2201}, {8033, 17941}, {8061, 4093}, {8287, 53563}, {9505, 2702}, {10015, 15507}, {14296, 53681}, {16592, 5027}, {16732, 4010}, {17103, 56982}, {17205, 50456}, {17924, 242}, {18111, 56971}, {18268, 1576}, {18827, 662}, {18895, 668}, {18896, 56241}, {20518, 27916}, {20908, 17793}, {20948, 35544}, {20981, 1933}, {21053, 20681}, {21140, 62558}, {21202, 27943}, {22116, 2284}, {23596, 984}, {23989, 3766}, {24002, 1447}, {24026, 4148}, {27855, 6652}, {30663, 813}, {30669, 4579}, {30671, 869}, {34067, 23990}, {35352, 37}, {35519, 3975}, {36038, 51381}, {36066, 249}, {36800, 643}, {36806, 6064}, {37128, 110}, {37207, 5384}, {39276, 827}, {40017, 99}, {40098, 660}, {40166, 4124}, {40217, 1026}, {40495, 1921}, {40848, 52923}, {43042, 34253}, {43534, 1018}, {43931, 51321}, {44170, 6386}, {44172, 1978}, {51866, 32666}, {52030, 919}, {52205, 34067}, {52209, 36086}, {52619, 30940}, {52621, 10030}, {52633, 38367}, {53239, 35338}, {53544, 51329}, {54229, 56828}, {56154, 5546}, {56242, 14602}, {57215, 14024}, {57554, 36066}, {57566, 65363}, {57738, 4558}, {57987, 4563}, {59941, 62785}, {60074, 36815}, {60577, 9}, {62415, 3797}, {62429, 62552}, {63219, 3661}, {63228, 3807}, {63234, 3799}, {63241, 4505}, {63895, 51614}, {63896, 37135}, {65101, 39044}, {65258, 24041}, {65285, 4590}, {65352, 162}, {66267, 52651}

X(66287) = X(65)X(513)∩X(109)X(476)

Barycentrics    (b - c)*(-a + b - c)*(a + b - c)*(b + c)^2 : :
X(66287) = X[39771] + 8 X[43052], X[21106] - 4 X[59750], 3 X[53356] + X[56321], 2 X[656] - 3 X[30574], 3 X[2457] - 2 X[53527], 4 X[7178] - X[30572], 3 X[30574] - X[62566], 8 X[59973] - 9 X[62579], 2 X[2605] - 3 X[11125], 4 X[8062] - 3 X[14432], 3 X[21052] - 2 X[52355], 3 X[23615] - 4 X[44426], 3 X[30573] - 4 X[48283]

X(66287) lies on the X-parabola (see X(12065)) and these lines: {1, 60029}, {7, 60042}, {57, 60043}, {65, 513}, {73, 15328}, {109, 476}, {225, 18808}, {226, 5466}, {307, 62645}, {514, 4581}, {522, 17950}, {523, 656}, {651, 60055}, {653, 685}, {661, 2501}, {663, 7649}, {664, 892}, {850, 4077}, {900, 21111}, {1214, 47887}, {1254, 21134}, {1400, 2395}, {1409, 20980}, {1441, 20504}, {1813, 44768}, {1880, 65103}, {1882, 16228}, {2171, 4079}, {2254, 57252}, {2517, 48278}, {2605, 11125}, {2610, 4024}, {2785, 7253}, {3668, 35347}, {3669, 4802}, {3676, 4608}, {3737, 21180}, {3907, 65099}, {4036, 4064}, {4105, 8058}, {4139, 53558}, {4160, 21109}, {4397, 23877}, {4449, 57241}, {4474, 23874}, {4626, 65559}, {4642, 23775}, {4705, 51663}, {4778, 58858}, {4913, 31603}, {4977, 21112}, {4988, 7180}, {6614, 65539}, {8062, 14432}, {8611, 47124}, {8672, 23755}, {8678, 21108}, {10015, 17420}, {12079, 21054}, {15228, 62499}, {15932, 21203}, {17072, 20294}, {17094, 47934}, {20360, 53501}, {21052, 52355}, {21103, 53314}, {21105, 37558}, {21142, 53538}, {21179, 48307}, {21185, 42312}, {21957, 47134}, {23615, 40149}, {23758, 50350}, {23943, 53545}, {28161, 49300}, {28175, 30725}, {28179, 30724}, {28191, 30719}, {28195, 51656}, {28473, 44409}, {30573, 48283}, {35352, 66267}, {38469, 51643}, {47701, 51650}, {47800, 59929}, {48293, 57198}, {50349, 57139}, {50354, 55126}, {50522, 57181}, {54194, 54244}, {55210, 55236}, {56816, 57224}

X(66287) = midpoint of X(23758) and X(50350)
X(66287) = reflection of X(i) in X(j) for these {i,j}: {663, 7649}, {3737, 21180}, {4017, 7178}, {4064, 4036}, {4088, 4086}, {17418, 21186}, {17420, 10015}, {20294, 17072}, {21103, 53314}, {21105, 48281}, {21106, 21173}, {21132, 21102}, {21173, 59750}, {30572, 4017}, {42312, 21185}, {48278, 2517}, {48307, 21179}, {55282, 23752}, {62566, 656}
X(66287) = isogonal conjugate of X(4636)
X(66287) = polar conjugate of the isotomic conjugate of X(57243)
X(66287) = polar conjugate of the isogonal conjugate of X(55234)
X(66287) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1247, 33650}, {53633, 3869}
X(66287) = X(i)-Ceva conjugate of X(j) for these (i,j): {12, 1365}, {65, 3120}, {653, 1400}, {664, 226}, {1441, 53545}, {4605, 2171}, {6354, 115}, {7178, 57185}, {36127, 225}, {40149, 21044}, {56285, 1109}
X(66287) = X(i)-cross conjugate of X(j) for these (i,j): {115, 6354}, {1365, 12}, {2643, 2171}, {4092, 8736}, {4705, 4024}, {21134, 1109}, {21944, 10}, {55234, 57243}
X(66287) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4636}, {3, 52914}, {6, 4612}, {9, 4556}, {21, 110}, {29, 4575}, {32, 4631}, {41, 4610}, {55, 52935}, {58, 643}, {59, 65575}, {60, 100}, {81, 5546}, {86, 65375}, {99, 2194}, {101, 2185}, {109, 1098}, {112, 1812}, {162, 283}, {163, 333}, {190, 2150}, {213, 55196}, {249, 650}, {250, 521}, {255, 52921}, {261, 692}, {270, 1331}, {284, 662}, {314, 1576}, {332, 32676}, {522, 1101}, {593, 644}, {645, 1333}, {648, 2193}, {651, 7054}, {663, 24041}, {667, 6064}, {757, 3939}, {799, 57657}, {849, 3699}, {906, 46103}, {931, 54417}, {934, 6061}, {960, 58982}, {1021, 52378}, {1172, 4558}, {1259, 52920}, {1332, 2189}, {1408, 7256}, {1412, 7259}, {1414, 2328}, {1415, 7058}, {1437, 36797}, {1783, 65568}, {1790, 65201}, {1813, 2326}, {1946, 18020}, {2175, 4623}, {2204, 4563}, {2206, 7257}, {2269, 65255}, {2287, 4565}, {2289, 52919}, {2299, 4592}, {2323, 37140}, {2327, 65232}, {2361, 65283}, {2617, 35196}, {3063, 4590}, {3683, 6578}, {3737, 4570}, {4282, 47318}, {4391, 23357}, {4511, 36069}, {4516, 59152}, {4566, 23609}, {4567, 7252}, {4578, 7341}, {5379, 23189}, {5548, 30576}, {6514, 24019}, {7258, 16947}, {7305, 40499}, {7340, 8641}, {9247, 55233}, {9447, 52612}, {13486, 35193}, {14574, 40072}, {14599, 36806}, {20967, 65281}, {23582, 36054}, {23995, 35519}, {24000, 57241}, {27083, 30238}, {30606, 32665}, {31623, 32661}, {32656, 57779}, {32671, 32851}, {32739, 52379}, {35518, 57655}, {36034, 51382}, {36059, 59482}, {44426, 47390}, {44769, 52949}, {47443, 53560}, {52425, 55231}, {56000, 65254}
X(66287) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 4636}, {9, 4612}, {10, 643}, {11, 1098}, {37, 645}, {115, 333}, {125, 283}, {136, 29}, {223, 52935}, {226, 4592}, {244, 21}, {478, 4556}, {523, 522}, {525, 52616}, {647, 6332}, {1015, 2185}, {1084, 284}, {1086, 261}, {1146, 7058}, {1214, 99}, {1365, 24161}, {1649, 14432}, {3005, 663}, {3125, 17185}, {3160, 4610}, {3258, 51382}, {4075, 3699}, {4858, 314}, {4988, 4560}, {5139, 2299}, {5190, 46103}, {5521, 270}, {6376, 4631}, {6523, 52921}, {6615, 65575}, {6626, 55196}, {6631, 6064}, {6741, 1043}, {8054, 60}, {10001, 4590}, {14714, 6061}, {15267, 109}, {15526, 332}, {18314, 35519}, {20620, 59482}, {21709, 4046}, {34591, 1812}, {35071, 6514}, {35092, 30606}, {36103, 52914}, {36901, 28660}, {36908, 1414}, {38982, 4511}, {38986, 2194}, {38991, 7054}, {38996, 57657}, {39006, 65568}, {39053, 18020}, {39060, 46254}, {40586, 5546}, {40590, 662}, {40593, 4623}, {40599, 7259}, {40600, 65375}, {40603, 7257}, {40607, 3939}, {40608, 2328}, {40611, 110}, {40615, 1509}, {40617, 757}, {40619, 52379}, {40622, 86}, {40627, 7252}, {47345, 648}, {50330, 3737}, {52119, 6734}, {55053, 2150}, {55060, 58}, {55064, 2287}, {55065, 8}, {55066, 2193}, {56325, 190}, {59577, 7256}, {59608, 4573}, {62565, 4563}, {62566, 7253}, {62570, 799}, {62576, 55233}, {62602, 55231}, {62614, 55207}
X(66287) = cevapoint of X(i) and X(j) for these (i,j): {2643, 21131}, {4705, 57185}
X(66287) = crosspoint of X(i) and X(j) for these (i,j): {225, 36127}, {226, 664}, {523, 24006}, {653, 57809}, {4077, 7178}
X(66287) = crosssum of X(i) and X(j) for these (i,j): {21, 65575}, {110, 4575}, {283, 57241}, {284, 663}, {3737, 54356}, {5546, 65375}, {46877, 57081}
X(66287) = trilinear pole of line {115, 1365}
X(66287) = crossdifference of every pair of points on line {60, 283}
X(66287) = barycentric product X(i)*X(j) for these {i,j}: {4, 57243}, {7, 4024}, {10, 7178}, {11, 4605}, {12, 514}, {37, 4077}, {56, 52623}, {57, 4036}, {65, 1577}, {73, 14618}, {75, 57185}, {85, 4705}, {86, 55197}, {108, 20902}, {109, 338}, {115, 664}, {125, 653}, {181, 3261}, {190, 1365}, {201, 17924}, {225, 525}, {226, 523}, {264, 55234}, {273, 55232}, {278, 4064}, {307, 2501}, {313, 7180}, {321, 4017}, {331, 55230}, {339, 32674}, {349, 512}, {513, 6358}, {522, 6354}, {594, 3676}, {647, 57809}, {649, 34388}, {651, 1109}, {656, 40149}, {658, 4092}, {661, 1441}, {693, 2171}, {756, 24002}, {810, 52575}, {826, 18097}, {850, 1400}, {905, 56285}, {1089, 3669}, {1111, 21859}, {1214, 24006}, {1254, 4391}, {1358, 4103}, {1402, 20948}, {1415, 23994}, {1425, 46110}, {1427, 4086}, {1446, 4041}, {1500, 52621}, {1813, 2970}, {1826, 17094}, {1880, 14208}, {1978, 61052}, {2006, 6370}, {2197, 46107}, {2533, 60245}, {2610, 18815}, {2632, 54240}, {2643, 4554}, {3064, 6356}, {3120, 4552}, {3124, 4572}, {3239, 6046}, {3267, 57652}, {3269, 52938}, {3649, 31010}, {3668, 3700}, {3701, 7216}, {3708, 18026}, {3952, 53545}, {4013, 30725}, {4025, 8736}, {4033, 53540}, {4049, 40663}, {4079, 6063}, {4080, 30572}, {4397, 7147}, {4444, 7235}, {4466, 61178}, {4551, 16732}, {4559, 21207}, {4566, 21044}, {4573, 21043}, {4620, 8029}, {4625, 21833}, {4707, 52383}, {4998, 21131}, {5930, 58759}, {6057, 58817}, {6535, 17096}, {6538, 30724}, {6545, 65958}, {6591, 57807}, {7143, 52622}, {7212, 43534}, {7250, 30713}, {7265, 52382}, {7649, 26942}, {8058, 13853}, {8754, 65164}, {11608, 18006}, {15526, 36127}, {16609, 35352}, {17422, 64990}, {18210, 65207}, {18359, 51663}, {20336, 55208}, {20567, 50487}, {20975, 46404}, {21054, 38340}, {21106, 31612}, {21124, 60086}, {21134, 46102}, {21824, 65292}, {23105, 52378}, {23752, 60188}, {27801, 51641}, {28654, 43924}, {37755, 44426}, {40086, 56326}, {40999, 55236}, {41013, 51664}, {41283, 53581}, {41804, 55238}, {43682, 57099}, {43683, 57107}, {43923, 52369}, {45196, 57162}, {50457, 60321}, {52565, 58757}, {53527, 60091}, {55242, 57810}, {55282, 60229}, {58005, 65796}
X(66287) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4612}, {6, 4636}, {7, 4610}, {10, 645}, {12, 190}, {19, 52914}, {37, 643}, {42, 5546}, {56, 4556}, {57, 52935}, {65, 662}, {73, 4558}, {75, 4631}, {85, 4623}, {86, 55196}, {109, 249}, {115, 522}, {125, 6332}, {181, 101}, {190, 6064}, {201, 1332}, {210, 7259}, {213, 65375}, {225, 648}, {226, 99}, {264, 55233}, {273, 55231}, {307, 4563}, {313, 62534}, {321, 7257}, {331, 55229}, {334, 36806}, {338, 35519}, {349, 670}, {393, 52921}, {512, 284}, {513, 2185}, {514, 261}, {520, 6514}, {522, 7058}, {523, 333}, {525, 332}, {594, 3699}, {647, 283}, {649, 60}, {650, 1098}, {651, 24041}, {653, 18020}, {656, 1812}, {657, 6061}, {658, 7340}, {661, 21}, {663, 7054}, {664, 4590}, {667, 2150}, {669, 57657}, {693, 52379}, {756, 644}, {762, 4069}, {798, 2194}, {810, 2193}, {850, 28660}, {900, 30606}, {961, 65255}, {1042, 4565}, {1089, 646}, {1109, 4391}, {1118, 52919}, {1214, 4592}, {1231, 55202}, {1254, 651}, {1356, 1919}, {1365, 514}, {1367, 30805}, {1400, 110}, {1402, 163}, {1409, 4575}, {1411, 37140}, {1415, 1101}, {1425, 1813}, {1426, 65232}, {1427, 1414}, {1441, 799}, {1446, 4625}, {1459, 65568}, {1500, 3939}, {1577, 314}, {1637, 51382}, {1648, 14432}, {1824, 65201}, {1826, 36797}, {1880, 162}, {2006, 65283}, {2170, 65575}, {2171, 100}, {2197, 1331}, {2321, 7256}, {2489, 2299}, {2501, 29}, {2533, 27958}, {2610, 4511}, {2623, 35196}, {2643, 650}, {2970, 46110}, {3064, 59482}, {3120, 4560}, {3122, 7252}, {3124, 663}, {3125, 3737}, {3261, 18021}, {3269, 57241}, {3668, 4573}, {3669, 757}, {3676, 1509}, {3690, 4587}, {3700, 1043}, {3701, 7258}, {3708, 521}, {3709, 2328}, {3949, 4571}, {4013, 4582}, {4017, 81}, {4024, 8}, {4036, 312}, {4041, 2287}, {4064, 345}, {4077, 274}, {4079, 55}, {4092, 3239}, {4103, 4076}, {4155, 3684}, {4171, 56182}, {4516, 1021}, {4551, 4567}, {4552, 4600}, {4554, 24037}, {4559, 4570}, {4566, 4620}, {4572, 34537}, {4605, 4998}, {4620, 31614}, {4705, 9}, {4838, 64401}, {4931, 4720}, {5930, 36841}, {6046, 658}, {6057, 6558}, {6058, 4103}, {6063, 52612}, {6354, 664}, {6356, 65164}, {6358, 668}, {6367, 3686}, {6370, 32851}, {6516, 62719}, {6535, 30730}, {6591, 270}, {7140, 65160}, {7143, 1461}, {7147, 934}, {7178, 86}, {7180, 58}, {7203, 763}, {7211, 18047}, {7212, 33295}, {7216, 1014}, {7233, 65258}, {7235, 3570}, {7250, 1412}, {7314, 4605}, {7363, 65290}, {7649, 46103}, {8013, 30729}, {8029, 21044}, {8611, 1792}, {8736, 1897}, {8754, 3064}, {9391, 6518}, {11608, 17931}, {13853, 53642}, {14321, 52352}, {14618, 44130}, {15526, 52616}, {16732, 18155}, {17094, 17206}, {17096, 6628}, {17924, 57779}, {17992, 5060}, {18006, 40882}, {18026, 46254}, {18097, 4577}, {18344, 2326}, {20336, 55207}, {20902, 35518}, {20948, 40072}, {20975, 652}, {21043, 3700}, {21044, 7253}, {21046, 52355}, {21054, 57066}, {21131, 11}, {21132, 26856}, {21134, 26932}, {21675, 65197}, {21725, 3287}, {21810, 61223}, {21824, 35057}, {21833, 4041}, {21834, 56181}, {21859, 765}, {24002, 873}, {24006, 31623}, {26942, 4561}, {27691, 57060}, {30572, 16704}, {30724, 30593}, {32660, 47390}, {32674, 250}, {34388, 1978}, {35352, 36800}, {36127, 23582}, {36197, 58329}, {37755, 6516}, {39691, 48278}, {40149, 811}, {40160, 54951}, {40999, 55235}, {41804, 55237}, {42661, 2269}, {42666, 2323}, {43924, 593}, {48005, 4877}, {50330, 17185}, {50487, 41}, {50538, 3691}, {51640, 18604}, {51641, 1333}, {51663, 3218}, {51664, 1444}, {52378, 59152}, {52383, 47318}, {52567, 3882}, {52575, 57968}, {52623, 3596}, {53321, 52378}, {53528, 30576}, {53540, 1019}, {53545, 7192}, {53551, 18206}, {53560, 57081}, {53581, 2175}, {54240, 23999}, {55197, 10}, {55206, 4183}, {55208, 28}, {55210, 35193}, {55214, 3193}, {55230, 219}, {55232, 78}, {55234, 3}, {55236, 3615}, {55238, 6740}, {55242, 285}, {55282, 16713}, {56285, 6335}, {57099, 56440}, {57107, 56439}, {57109, 3719}, {57181, 849}, {57185, 1}, {57243, 69}, {57652, 112}, {57809, 6331}, {57810, 55241}, {58289, 1334}, {58304, 52405}, {58757, 8748}, {58759, 5931}, {58817, 552}, {60229, 55281}, {60245, 4594}, {60321, 65230}, {61052, 649}, {61058, 4091}, {61364, 32739}, {64984, 65281}, {65164, 47389}, {65796, 950}, {65958, 6632}
X(66287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {30574, 62566, 656}, {55197, 57185, 4024}

X(66288) = X(106)X(476)∩X(115)X(4024)

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

X(66288) lies on the X-parabola (see X(12065)) and these lines: {88, 60055}, {106, 476}, {115, 4024}, {519, 17953}, {523, 3120}, {685, 6336}, {850, 21207}, {892, 903}, {1022, 60043}, {1109, 4036}, {1647, 42754}, {1797, 44768}, {2395, 55263}, {2501, 21950}, {4049, 5466}, {4062, 4080}, {4581, 43922}, {4608, 6549}, {4674, 5620}, {6548, 60042}, {23838, 60029}, {50755, 62732}, {52753, 65716}, {52759, 62672}, {61707, 63851}

X(66288) = X(i)-Ceva conjugate of X(j) for these (i,j): {903, 4049}, {6336, 55263}
X(66288) = X(i)-isoconjugate of X(j) for these (i,j): {44, 249}, {250, 5440}, {519, 1101}, {902, 24041}, {1023, 4556}, {1252, 30576}, {1576, 55243}, {2149, 30606}, {2251, 4590}, {3264, 23995}, {3285, 4567}, {4358, 23357}, {4570, 52680}, {4575, 46541}, {4612, 61210}, {4636, 23703}, {4730, 59152}, {9273, 40988}, {9459, 24037}, {18020, 23202}, {23344, 52935}, {38462, 47390}
X(66288) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 46541}, {512, 9459}, {523, 519}, {647, 3977}, {650, 30606}, {661, 30576}, {3005, 902}, {4858, 55243}, {4988, 16704}, {9460, 4590}, {17436, 4141}, {18314, 3264}, {36901, 55262}, {40594, 24041}, {40595, 249}, {40627, 3285}, {50330, 52680}, {55065, 17780}, {62582, 6064}
X(66288) = crosspoint of X(903) and X(4049)
X(66288) = trilinear pole of line {115, 21131}
X(66288) = barycentric product X(i)*X(j) for these {i,j}: {12, 60578}, {88, 1109}, {106, 338}, {115, 903}, {125, 6336}, {339, 8752}, {523, 4049}, {594, 6549}, {850, 55263}, {1022, 4036}, {1086, 4013}, {1365, 4997}, {1577, 55244}, {1797, 2970}, {2643, 20568}, {3120, 4080}, {3124, 57995}, {4024, 6548}, {4077, 61179}, {4555, 21131}, {4591, 23105}, {4615, 8029}, {4674, 16732}, {9456, 23994}, {12079, 52753}, {20902, 36125}, {21134, 65336}, {23345, 52623}, {28654, 43922}, {52759, 64258}
X(66288) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 30606}, {88, 24041}, {106, 249}, {115, 519}, {125, 3977}, {244, 30576}, {338, 3264}, {850, 55262}, {903, 4590}, {1022, 52935}, {1084, 9459}, {1109, 4358}, {1365, 3911}, {1577, 55243}, {2501, 46541}, {2643, 44}, {2970, 46109}, {3120, 16704}, {3122, 3285}, {3124, 902}, {3125, 52680}, {3708, 5440}, {4013, 1016}, {4024, 17780}, {4036, 24004}, {4049, 99}, {4079, 23344}, {4080, 4600}, {4092, 2325}, {4591, 59152}, {4615, 31614}, {4674, 4567}, {4705, 1023}, {4997, 6064}, {6336, 18020}, {6548, 4610}, {6549, 1509}, {8029, 4120}, {8288, 4141}, {8752, 250}, {8754, 8756}, {9456, 1101}, {16732, 30939}, {20568, 24037}, {20975, 22356}, {21043, 3943}, {21131, 900}, {21833, 21805}, {22260, 14407}, {23345, 4556}, {23838, 4612}, {32659, 47390}, {43922, 593}, {55244, 662}, {55263, 110}, {57185, 23703}, {57995, 34537}, {60578, 261}, {61052, 1404}, {61179, 643}, {64258, 52747}

X(66288) = X(1)X(850)∩X(31)X(523)

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

X(66288) lies on the X-parabola (see X(12065)) and these lines: {1, 850}, {31, 523}, {42, 4036}, {213, 4024}, {923, 5466}, {1967, 66267}, {1973, 2501}, {36051, 62645}, {37132, 37219}, {46289, 58784}

X(66288) = X(662)-isoconjugate of X(14963)
X(66288) = X(1084)-Dao conjugate of X(14963)
X(66288) = trilinear pole of line {115, 798}
X(66288) = barycentric product X(i)*X(j) for these {i,j}: {512, 37219}, {523, 60134}
X(66288) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 14963}, {37219, 670}, {60134, 99}

X(66289) = X(11)X(523)∩X(80)X(758)

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

X(66289) lies on the X-parabola (see X(12065)) and these lines: {11, 523}, {80, 758}, {476, 759}, {685, 36120}, {740, 51562}, {850, 17886}, {892, 14616}, {1365, 57423}, {1621, 36815}, {1807, 12081}, {2222, 3724}, {2292, 56416}, {2501, 8735}, {2650, 14584}, {2677, 4092}, {3992, 15065}, {4024, 21044}, {4036, 21054}, {4647, 51975}, {5466, 60074}, {6757, 37735}, {12077, 64445}, {13576, 34857}, {14628, 17874}, {18101, 58784}, {20566, 35544}, {24624, 60055}, {35016, 56950}, {35550, 57788}, {37702, 38938}, {56425, 63354}

X(66289) = X(i)-Ceva conjugate of X(j) for these (i,j): {14616, 60074}, {34535, 661}, {57788, 1577}, {60091, 55238}
X(66289) = X(i)-cross conjugate of X(j) for these (i,j): {10413, 7332}, {42759, 3120}
X(66289) = X(i)-isoconjugate of X(j) for these (i,j): {36, 4570}, {163, 4585}, {249, 2245}, {662, 1983}, {758, 1101}, {860, 47390}, {2323, 52378}, {3724, 24041}, {3936, 23357}, {4242, 4575}, {4282, 4564}, {4567, 7113}, {4600, 52434}, {4620, 52426}, {4736, 9274}, {5379, 52407}, {6742, 52603}, {9273, 35069}, {23995, 35550}, {32661, 65162}, {32739, 55237}, {42666, 59152}
X(66289) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 4585}, {136, 4242}, {523, 758}, {1084, 1983}, {3005, 3724}, {4988, 3218}, {6544, 17191}, {8287, 10411}, {15898, 4570}, {18314, 35550}, {40619, 55237}, {40627, 7113}, {50330, 36}, {50497, 52434}, {62566, 4511}, {62567, 4881}
X(66289) = crosspoint of X(i) and X(j) for these (i,j): {2166, 10412}, {14616, 60074}
X(66289) = crosssum of X(i) and X(j) for these (i,j): {1983, 3724}, {6149, 52603}
X(66289) = trilinear pole of line {115, 55195}
X(66289) = barycentric product X(i)*X(j) for these {i,j}: {11, 60091}, {80, 16732}, {94, 2611}, {115, 14616}, {338, 759}, {523, 60074}, {693, 55238}, {1086, 15065}, {1109, 24624}, {1989, 17886}, {2161, 21207}, {2166, 8287}, {3120, 18359}, {3125, 20566}, {4858, 52383}, {7178, 52356}, {7265, 43082}, {10412, 14838}, {14592, 54244}, {15475, 18160}, {18815, 21044}, {20982, 63759}, {23105, 37140}, {23989, 34857}, {23994, 34079}, {35174, 55195}, {52409, 53545}
X(66289) = barycentric quotient X(i)/X(j) for these {i,j}: {80, 4567}, {115, 758}, {338, 35550}, {512, 1983}, {523, 4585}, {693, 55237}, {759, 249}, {1109, 3936}, {1365, 18593}, {1411, 52378}, {1647, 17191}, {2161, 4570}, {2501, 4242}, {2611, 323}, {2643, 2245}, {3120, 3218}, {3121, 52434}, {3122, 7113}, {3124, 3724}, {3125, 36}, {3271, 4282}, {4516, 2323}, {8029, 2610}, {8034, 21758}, {8735, 17515}, {10412, 15455}, {14616, 4590}, {14838, 10411}, {15065, 1016}, {16732, 320}, {17886, 7799}, {18210, 22128}, {18359, 4600}, {18815, 4620}, {20566, 4601}, {20982, 6149}, {21043, 4053}, {21044, 4511}, {21054, 42701}, {21131, 53527}, {21207, 20924}, {21950, 4881}, {24006, 65162}, {24624, 24041}, {34079, 1101}, {34857, 1252}, {35174, 55194}, {36197, 58328}, {37140, 59152}, {42759, 16586}, {52356, 645}, {52383, 4564}, {52391, 44717}, {53545, 1443}, {54244, 14590}, {55195, 3738}, {55238, 100}, {57985, 62719}, {60074, 99}, {60091, 4998}, {63462, 8648}, {64835, 5379}

X(66290) = X(105)X(476)∩X(338)X(4036)

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

X(66290) lies on the X-parabola (see X(12065)) and these lines: {105, 476}, {338, 4036}, {518, 20556}, {523, 2486}, {673, 60055}, {685, 54235}, {885, 60029}, {892, 2481}, {1109, 4024}, {1814, 44768}, {2395, 55261}, {4581, 43921}, {10099, 15328}, {60043, 62635}

X(66290) = X(54235)-Ceva conjugate of X(55261)
X(66290) = X(i)-isoconjugate of X(j) for these (i,j): {110, 54353}, {249, 672}, {250, 1818}, {518, 1101}, {1861, 47390}, {2223, 24041}, {2283, 4636}, {2284, 4556}, {3263, 23995}, {3286, 4570}, {3912, 23357}, {4238, 4575}, {4590, 9454}, {9455, 24037}, {52935, 54325}
X(66290) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 4238}, {244, 54353}, {512, 9455}, {523, 518}, {647, 25083}, {3005, 2223}, {4988, 18206}, {18314, 3263}, {33675, 4590}, {36901, 55260}, {50330, 3286}, {55065, 1026}, {62554, 249}, {62599, 24041}
X(66290) = barycentric product X(i)*X(j) for these {i,j}: {105, 338}, {115, 2481}, {125, 54235}, {339, 8751}, {673, 1109}, {850, 55261}, {1027, 52623}, {1365, 36796}, {1438, 23994}, {1814, 2970}, {2643, 18031}, {4036, 62635}, {4092, 34018}, {10099, 14618}, {13576, 16732}, {18785, 21207}, {20902, 36124}, {21131, 51560}, {23962, 64216}, {28654, 43921}
X(66290) = barycentric quotient X(i)/X(j) for these {i,j}: {105, 249}, {115, 518}, {125, 25083}, {338, 3263}, {661, 54353}, {673, 24041}, {850, 55260}, {885, 4612}, {1024, 4636}, {1027, 4556}, {1084, 9455}, {1109, 3912}, {1365, 241}, {1438, 1101}, {2481, 4590}, {2501, 4238}, {2643, 672}, {2970, 46108}, {3120, 18206}, {3124, 2223}, {3125, 3286}, {3708, 1818}, {4024, 1026}, {4036, 42720}, {4079, 54325}, {4092, 3693}, {4705, 2284}, {8029, 24290}, {8751, 250}, {8754, 5089}, {10099, 4558}, {13576, 4567}, {16732, 30941}, {18031, 24037}, {18785, 4570}, {20975, 20752}, {21043, 3930}, {21131, 2254}, {21207, 18157}, {21833, 20683}, {31637, 62719}, {32658, 47390}, {34018, 7340}, {36796, 6064}, {43921, 593}, {54235, 18020}, {55261, 110}, {57185, 2283}, {61052, 52635}, {62635, 52935}, {64216, 23357}

X(66291) = X(112)X(685)∩X(263)X(512)

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

X(66291) lies on the X-parabola (see X(12065)) and these lines: {112, 685}, {262, 5466}, {263, 512}, {476, 26714}, {523, 3569}, {647, 4108}, {850, 2525}, {892, 65271}, {1637, 8599}, {2485, 50946}, {2501, 3005}, {3288, 54267}, {8288, 12079}, {10412, 62384}, {11182, 34246}, {14998, 47229}, {15328, 43718}, {22240, 33569}, {22734, 36900}, {42313, 62645}, {44768, 65310}, {53196, 53230}, {54262, 64919}, {55267, 66267}, {55275, 62519}, {60042, 60679}, {60055, 65252}

X(66291) = reflection of X(3288) in X(54267)
X(66291) = isotomic conjugate of the isogonal conjugate of X(52631)
X(66291) = X(i)-Ceva conjugate of X(j) for these (i,j): {65271, 262}, {65349, 263}
X(66291) = X(50549)-cross conjugate of X(850)
X(66291) = X(i)-isoconjugate of X(j) for these (i,j): {110, 52134}, {163, 183}, {182, 662}, {458, 4575}, {799, 34396}, {1101, 23878}, {1576, 3403}, {3288, 24041}, {4556, 60723}, {4558, 60685}, {4592, 10311}, {4599, 14096}, {5546, 60716}, {14994, 34072}, {23997, 46806}, {36034, 51372}, {36134, 59197}, {52935, 60726}
X(66291) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 183}, {136, 458}, {137, 59197}, {244, 52134}, {523, 23878}, {1084, 182}, {3005, 3288}, {3124, 14096}, {3258, 51372}, {4858, 3403}, {5139, 10311}, {15449, 14994}, {35088, 51373}, {36901, 20023}, {38996, 34396}, {55065, 60737}, {62562, 46806}, {63463, 59208}
X(66291) = crosspoint of X(262) and X(65271)
X(66291) = crosssum of X(i) and X(j) for these (i,j): {182, 3288}, {5052, 50550}, {23878, 59197}
X(66291) = trilinear pole of line {115, 44114}
X(66291) = crossdifference of every pair of points on line {182, 14096}
X(66291) = barycentric product X(i)*X(j) for these {i,j}: {76, 52631}, {115, 65271}, {125, 65349}, {262, 523}, {263, 850}, {327, 512}, {338, 26714}, {826, 42299}, {868, 6037}, {1109, 65252}, {1577, 2186}, {2395, 46807}, {2501, 42313}, {2970, 65310}, {3402, 20948}, {4024, 60679}, {10412, 57268}, {12077, 42300}, {14618, 43718}, {23285, 42288}, {23290, 51444}, {30735, 40803}, {32716, 62431}, {39682, 60036}, {43665, 51543}, {44114, 53196}, {44173, 46319}, {58757, 59257}
X(66291) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 23878}, {262, 99}, {263, 110}, {327, 670}, {512, 182}, {523, 183}, {661, 52134}, {669, 34396}, {826, 14994}, {850, 20023}, {1577, 3403}, {1637, 51372}, {2186, 662}, {2395, 46806}, {2422, 51542}, {2489, 10311}, {2501, 458}, {2799, 51373}, {3005, 14096}, {3124, 3288}, {3402, 163}, {4017, 60716}, {4024, 60737}, {4036, 42711}, {4079, 60726}, {4705, 60723}, {6037, 57991}, {12077, 59197}, {14618, 44144}, {15475, 56401}, {22260, 6784}, {26714, 249}, {32716, 57742}, {40803, 35575}, {42288, 827}, {42299, 4577}, {42313, 4563}, {43718, 4558}, {46319, 1576}, {46807, 2396}, {50549, 52658}, {51428, 45321}, {51513, 39530}, {51543, 2421}, {52631, 6}, {55219, 59208}, {57268, 10411}, {58260, 9420}, {58757, 33971}, {60679, 4610}, {61359, 35278}, {65252, 24041}, {65271, 4590}, {65349, 18020}, {66267, 8842}

X(66292) = X(162)X(685)∩X(256)X(60029)

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

X(66292) lies on the X-parabola (see X(12065)) and these lines: {162, 685}, {256, 60029}, {257, 26545}, {476, 29055}, {661, 2395}, {892, 65289}, {1431, 9013}, {1432, 27469}, {4017, 4369}, {4608, 29116}, {4804, 56321}, {5466, 27710}, {7249, 60042}, {16609, 58784}, {37137, 60055}

X(66292) = X(i)-Ceva conjugate of X(j) for these (i,j): {65289, 60245}, {65332, 65011}
X(66292) = X(i)-isoconjugate of X(j) for these (i,j): {60, 4579}, {163, 27958}, {171, 4636}, {172, 4612}, {249, 3287}, {1101, 3907}, {2150, 18047}, {2311, 56982}, {2329, 4556}, {2330, 52935}, {3955, 52914}, {4575, 14006}, {6064, 56242}, {14602, 36806}, {17103, 65375}, {18235, 58982}, {40608, 59152}, {56154, 56980}
X(66292) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 27958}, {136, 14006}, {523, 3907}, {16591, 17941}, {40622, 17103}, {55065, 7081}, {56325, 18047}
X(66292) = crosspoint of X(60245) and X(65289)
X(66292) = barycentric product X(i)*X(j) for these {i,j}: {115, 65289}, {125, 65332}, {338, 29055}, {523, 60245}, {850, 65011}, {1109, 37137}, {1365, 27805}, {1431, 52623}, {1432, 4036}, {4024, 7249}, {4077, 52651}, {7018, 57185}, {16609, 66267}, {32010, 55197}
X(66292) = barycentric quotient X(i)/X(j) for these {i,j}: {12, 18047}, {115, 3907}, {256, 4612}, {523, 27958}, {882, 2311}, {893, 4636}, {1284, 56982}, {1365, 4369}, {1431, 4556}, {1432, 52935}, {1934, 36806}, {2171, 4579}, {2501, 14006}, {2643, 3287}, {4024, 7081}, {4036, 17787}, {4077, 8033}, {4079, 2330}, {4092, 4529}, {4705, 2329}, {6354, 6649}, {7018, 4631}, {7178, 17103}, {7249, 4610}, {16609, 17941}, {21043, 4140}, {21131, 4459}, {27805, 6064}, {29055, 249}, {32010, 55196}, {37137, 24041}, {40729, 65375}, {52651, 643}, {53540, 18200}, {53545, 17212}, {55197, 1215}, {55234, 3955}, {57185, 171}, {60245, 99}, {61052, 20981}, {65011, 110}, {65289, 4590}, {65332, 18020}, {66267, 36800}

X(66293) = X(111)X(385)∩X(115)X(850)

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

X(66293) lies on the X-parabola (see X(12065)) and these lines: {111, 385}, {115, 850}, {476, 729}, {523, 3124}, {685, 57260}, {1084, 4108}, {1648, 35366}, {1916, 3266}, {2028, 30509}, {2029, 30508}, {2395, 15630}, {2501, 2971}, {3291, 46156}, {4036, 21833}, {5466, 60028}, {5640, 52765}, {5996, 9151}, {7804, 14608}, {14498, 65767}, {37132, 60055}, {41309, 52752}, {51906, 58784}

X(66293) = X(i)-Ceva conjugate of X(j) for these (i,j): {3228, 60028}, {34087, 35366}
X(66293) = X(i)-isoconjugate of X(j) for these (i,j): {163, 23342}, {249, 2234}, {538, 1101}, {662, 5118}, {3231, 24041}, {23995, 30736}, {24037, 33875}, {46522, 62719}
X(66293) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 23342}, {512, 33875}, {523, 538}, {1084, 5118}, {1649, 45672}, {3005, 3231}, {18314, 30736}, {36901, 63747}
X(66293) = crosspoint of X(3228) and X(60028)
X(66293) = crosssum of X(3231) and X(5118)
X(66293) = trilinear pole of line {115, 22260}
X(66293) = crossdifference of every pair of points on line {5118, 38366}
X(66293) = barycentric product X(i)*X(j) for these {i,j}: {115, 3228}, {338, 729}, {523, 60028}, {850, 63749}, {886, 22260}, {1109, 37132}, {3124, 34087}, {8029, 9150}, {12079, 52752}, {14608, 64258}, {23099, 57993}, {23105, 32717}, {35366, 58784}
X(66293) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 538}, {338, 30736}, {512, 5118}, {523, 23342}, {729, 249}, {850, 63747}, {1084, 33875}, {1648, 45672}, {2643, 2234}, {2971, 46522}, {3124, 3231}, {3228, 4590}, {8029, 9148}, {9150, 31614}, {16732, 30938}, {21833, 52893}, {22260, 888}, {23099, 887}, {23610, 65497}, {32717, 59152}, {34087, 34537}, {35366, 4576}, {37132, 24041}, {44114, 6786}, {51441, 36822}, {52625, 52067}, {60028, 99}, {63749, 110}, {64258, 52756}

X(66294) = X(104)X(476)∩X(125)X(4036)

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

X(66294) lies on the X-parabola (see X(12065)) and these lines: {104, 476}, {125, 4036}, {422, 685}, {517, 38952}, {523, 18210}, {892, 18816}, {1365, 2970}, {2395, 55259}, {2401, 60043}, {2501, 3125}, {3708, 4024}, {4581, 15635}, {5885, 14266}, {34234, 60055}, {42703, 57847}, {43728, 60029}, {44768, 65302}

X(66294) = X(16082)-Ceva conjugate of X(55259)
X(66294) = X(i)-isoconjugate of X(j) for these (i,j): {163, 64828}, {249, 2183}, {250, 22350}, {517, 1101}, {859, 4570}, {908, 23357}, {1785, 47390}, {2427, 4556}, {3262, 23995}, {4246, 4575}, {4636, 23981}
X(66294) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 64828}, {136, 4246}, {523, 517}, {18314, 3262}, {36901, 55258}, {50330, 859}
X(66294) = barycentric product X(i)*X(j) for these {i,j}: {104, 338}, {115, 18816}, {125, 16082}, {850, 55259}, {909, 23994}, {1109, 34234}, {1365, 36795}, {2250, 21207}, {2401, 4036}, {2970, 65302}, {3125, 57984}, {15635, 28654}, {16732, 38955}, {20902, 36123}, {21134, 65223}, {23962, 34858}
X(66294) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 249}, {115, 517}, {338, 3262}, {523, 64828}, {850, 55258}, {909, 1101}, {1109, 908}, {1365, 1465}, {2250, 4570}, {2401, 52935}, {2501, 4246}, {2643, 2183}, {3125, 859}, {3708, 22350}, {4036, 2397}, {4705, 2427}, {8754, 14571}, {14578, 47390}, {15635, 593}, {16082, 18020}, {16732, 17139}, {18816, 4590}, {21043, 21801}, {21131, 1769}, {21833, 51377}, {34234, 24041}, {34858, 23357}, {36795, 6064}, {38955, 4567}, {43728, 4612}, {55259, 110}, {56761, 30576}, {57185, 23981}, {57984, 4601}, {61238, 4636}

X(66295) = X(193)X(523)∩X(850)X(57518)

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

X(66295) lies on the X-parabola (see X(12065)) and these lines: {193, 523}, {850, 57518}, {2501, 6353}, {4024, 4028}, {4226, 44768}, {5466, 45687}, {8029, 58766}, {14977, 64217}, {40819, 55267}, {44554, 58784}, {53374, 65484}, {55122, 62645}

X(66295) = X(i)-cross conjugate of X(j) for these (i,j): {53374, 5466}, {65484, 2501}, {66162, 98}
X(66295) = X(i)-isoconjugate of X(j) for these (i,j): {163, 44377}, {662, 1570}, {24041, 63733}
X(66295) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 44377}, {1084, 1570}, {3005, 63733}
X(66295) = cevapoint of X(i) and X(j) for these (i,j): {512, 6132}, {523, 55122}
X(66295) = crosssum of X(1570) and X(63733)
X(66295) = trilinear pole of line {115, 3566}
X(66295) = barycentric product X(523)*X(60073)
X(66295) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 1570}, {523, 44377}, {3124, 63733}, {60073, 99}

X(66296) = X(99)X(685)∩X(183)X(47194)

Barycentrics    (b^2 - c^2)*(a^4 + 3*b^4 - 2*a^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 + 3*c^4) : :
X(66296) = 4 X[7804] - 3 X[59991]

X(66296) lies on the X-parabola (see X(12065)) and these lines: {99, 685}, {183, 47194}, {476, 9146}, {523, 4143}, {525, 2395}, {2419, 64983}, {2501, 2799}, {3268, 8599}, {5466, 30474}, {6563, 58784}, {7804, 59991}, {15414, 57069}, {18808, 55972}, {54259, 54267}, {54262, 64919}

X(66296) = isotomic conjugate of X(35278)
X(66296) = X(i)-isoconjugate of X(j) for these (i,j): {31, 35278}, {163, 7735}, {662, 40825}, {1576, 4008}, {4575, 6620}, {6776, 32676}, {23995, 30735}
X(66296) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 35278}, {115, 7735}, {136, 6620}, {525, 47194}, {1084, 40825}, {4858, 4008}, {15526, 6776}, {18314, 30735}, {35088, 1513}, {36901, 40814}, {62573, 37188}
X(66296) = cevapoint of X(525) and X(54260)
X(66296) = crosspoint of X(60093) and X(65276)
X(66296) = barycentric product X(i)*X(j) for these {i,j}: {338, 35575}, {523, 40824}, {525, 55972}, {850, 40802}, {3265, 64983}, {3267, 40801}, {40799, 44173}
X(66296) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 35278}, {338, 30735}, {512, 40825}, {523, 7735}, {525, 6776}, {850, 40814}, {1577, 4008}, {2501, 6620}, {2799, 1513}, {3265, 37188}, {3267, 62698}, {14618, 43976}, {15526, 47194}, {23878, 9755}, {30476, 56372}, {35575, 249}, {40799, 1576}, {40801, 112}, {40802, 110}, {40803, 26714}, {40823, 14574}, {40824, 99}, {41074, 60179}, {44173, 40822}, {55972, 648}, {60597, 42353}, {64919, 9752}, {64983, 107}

X(66297) = X(4)X(30200)∩X(108)X(476)

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

X(66297) lies on the X-parabola (see X(12065)) and these lines: {4, 30200}, {65, 15328}, {108, 476}, {225, 7649}, {273, 60042}, {278, 60043}, {521, 38949}, {523, 24006}, {651, 44768}, {653, 60055}, {685, 54240}, {892, 18026}, {1365, 2970}, {1441, 62645}, {1880, 2395}, {2501, 57185}, {4581, 17924}, {5466, 40149}, {14775, 18344}, {16231, 39579}, {44426, 56321}

X(66297) = polar conjugate of X(4612)
X(66297) = polar conjugate of the isogonal conjugate of X(57185)
X(66297) = X(i)-Ceva conjugate of X(j) for these (i,j): {18026, 40149}, {54240, 1880}
X(66297) = X(8754)-cross conjugate of X(8736)
X(66297) = X(i)-isoconjugate of X(j) for these (i,j): {3, 4636}, {21, 4575}, {48, 4612}, {60, 1331}, {101, 65568}, {110, 283}, {112, 6514}, {163, 1812}, {212, 52935}, {219, 4556}, {249, 652}, {250, 57241}, {255, 52914}, {261, 32656}, {284, 4558}, {332, 1576}, {333, 32661}, {521, 1101}, {522, 47390}, {593, 4587}, {643, 1437}, {662, 2193}, {849, 4571}, {906, 2185}, {1092, 52921}, {1098, 36059}, {1332, 2150}, {1444, 65375}, {1790, 5546}, {1813, 7054}, {1946, 24041}, {2194, 4592}, {2200, 55196}, {2327, 4565}, {3063, 62719}, {4563, 57657}, {4570, 23189}, {4610, 52425}, {4631, 9247}, {6332, 23357}, {7058, 32660}, {14585, 55233}, {18604, 65201}, {22074, 65255}, {23090, 52378}, {23181, 35196}, {23995, 35518}, {52616, 57655}, {55229, 62257}
X(66297) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 1812}, {136, 21}, {244, 283}, {523, 521}, {1015, 65568}, {1084, 2193}, {1214, 4592}, {1249, 4612}, {3005, 1946}, {4075, 4571}, {4858, 332}, {5139, 2194}, {5190, 2185}, {5521, 60}, {6523, 52914}, {6741, 1792}, {10001, 62719}, {15267, 36059}, {18314, 35518}, {20620, 1098}, {34591, 6514}, {36103, 4636}, {38966, 6061}, {39053, 24041}, {39060, 4590}, {40590, 4558}, {40611, 4575}, {40622, 1444}, {40837, 52935}, {47345, 662}, {50330, 23189}, {55060, 1437}, {55064, 2327}, {55065, 78}, {56325, 1332}, {62566, 57081}, {62570, 4563}, {62576, 4631}, {62602, 4610}
X(66297) = crosspoint of X(18026) and X(40149)
X(66297) = crosssum of X(1946) and X(2193)
X(66297) = crossdifference of every pair of points on line {2193, 22074}
X(66297) = barycentric product X(i)*X(j) for these {i,j}: {12, 17924}, {34, 52623}, {65, 14618}, {108, 338}, {115, 18026}, {125, 54240}, {158, 57243}, {225, 1577}, {226, 24006}, {264, 57185}, {273, 4024}, {278, 4036}, {286, 55197}, {313, 55208}, {331, 4705}, {512, 52575}, {514, 56285}, {523, 40149}, {651, 2970}, {653, 1109}, {661, 57809}, {693, 8736}, {850, 1880}, {1231, 58757}, {1254, 46110}, {1365, 6335}, {1441, 2501}, {1826, 4077}, {2171, 46107}, {2643, 46404}, {2973, 21859}, {3120, 65207}, {3669, 7141}, {3708, 52938}, {4079, 57787}, {4092, 13149}, {4554, 8754}, {6354, 44426}, {6358, 7649}, {6591, 34388}, {7140, 24002}, {7178, 41013}, {16732, 61178}, {20902, 36127}, {20948, 57652}, {23994, 32674}, {28654, 43923}, {55234, 57806}
X(66297) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 4612}, {12, 1332}, {19, 4636}, {34, 4556}, {65, 4558}, {108, 249}, {115, 521}, {181, 906}, {225, 662}, {226, 4592}, {264, 4631}, {273, 4610}, {278, 52935}, {286, 55196}, {313, 55207}, {331, 4623}, {338, 35518}, {349, 55202}, {393, 52914}, {512, 2193}, {513, 65568}, {523, 1812}, {594, 4571}, {653, 24041}, {656, 6514}, {661, 283}, {664, 62719}, {756, 4587}, {1109, 6332}, {1254, 1813}, {1365, 905}, {1400, 4575}, {1402, 32661}, {1415, 47390}, {1426, 4565}, {1441, 4563}, {1577, 332}, {1824, 5546}, {1826, 643}, {1880, 110}, {2171, 1331}, {2333, 65375}, {2489, 2194}, {2501, 21}, {2643, 652}, {2970, 4391}, {2971, 3063}, {3064, 1098}, {3124, 1946}, {3125, 23189}, {3700, 1792}, {3708, 57241}, {4017, 1790}, {4024, 78}, {4036, 345}, {4041, 2327}, {4064, 3719}, {4077, 17206}, {4079, 212}, {4092, 57055}, {4516, 23090}, {4554, 47389}, {4705, 219}, {6046, 65296}, {6335, 6064}, {6354, 6516}, {6358, 4561}, {6520, 52921}, {6591, 60}, {7140, 644}, {7141, 646}, {7178, 1444}, {7180, 1437}, {7649, 2185}, {8029, 53560}, {8735, 65575}, {8736, 100}, {8754, 650}, {13149, 7340}, {14618, 314}, {17924, 261}, {18026, 4590}, {18344, 7054}, {20902, 52616}, {20975, 36054}, {21043, 8611}, {21044, 57081}, {21131, 7004}, {24006, 333}, {32674, 1101}, {36197, 58338}, {37755, 6517}, {40149, 99}, {41013, 645}, {42661, 22074}, {43923, 593}, {44426, 7058}, {46107, 52379}, {46404, 24037}, {50487, 52425}, {51663, 22128}, {52575, 670}, {52623, 3718}, {52938, 46254}, {53008, 7259}, {53540, 7254}, {54240, 18020}, {55197, 72}, {55206, 2328}, {55208, 58}, {55212, 1819}, {55214, 1800}, {55230, 2289}, {55232, 1259}, {55234, 255}, {55236, 1789}, {55238, 1793}, {56285, 190}, {57185, 3}, {57243, 326}, {57652, 163}, {57787, 52612}, {57806, 55233}, {57809, 799}, {58289, 52370}, {58757, 1172}, {61052, 22383}, {61178, 4567}, {65103, 6061}, {65207, 4600}

X(66298) = X(187)X(523)∩X(524)X(850)

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

X(66298) lies on the X-parabola (see X(12065)) and these lines: {6, 5466}, {187, 523}, {249, 892}, {512, 64258}, {524, 850}, {843, 39450}, {2407, 62672}, {2501, 44102}, {4036, 21839}, {10412, 56395}, {12079, 44398}, {18872, 66267}, {34246, 51927}, {40879, 62645}

X(66298) = reflection of X(44398) in X(47229)
X(66298) = X(i)-isoconjugate of X(j) for these (i,j): {662, 46127}, {23889, 65320}
X(66298) = X(1084)-Dao conjugate of X(46127)
X(66298) = trilinear pole of line {115, 351}
X(66298) = barycentric product X(690)*X(39450)
X(66298) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 46127}, {8029, 15359}, {9178, 65320}, {39450, 892}

X(66299) = X(107)X(476)∩X(403)X(523)

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

X(66299) lies on the X-parabola (see X(12065)) and these lines: {4, 924}, {92, 57083}, {107, 476}, {264, 62645}, {393, 2395}, {403, 523}, {520, 16229}, {648, 44768}, {685, 15352}, {770, 2501}, {823, 60055}, {850, 6368}, {879, 57684}, {892, 6528}, {1093, 18808}, {1300, 53924}, {1896, 60029}, {2052, 5466}, {2970, 12079}, {6753, 15422}, {7650, 46110}, {8057, 39533}, {10412, 13450}, {14165, 47221}, {14249, 18504}, {16172, 57065}, {16868, 60342}, {18039, 62172}, {34334, 36169}, {44427, 59744}, {44732, 64935}, {46106, 47348}, {46151, 65183}, {59424, 63705}

X(66299) = midpoint of X(4) and X(57120)
X(66299) = reflection of X(14618) in X(23290)
X(66299) = polar conjugate of X(4558)
X(66299) = isotomic conjugate of the isogonal conjugate of X(58757)
X(66299) = polar conjugate of the isotomic conjugate of X(14618)
X(66299) = polar conjugate of the isogonal conjugate of X(2501)
X(66299) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {921, 34186}, {39416, 18664}, {60779, 21220}
X(66299) = X(i)-Ceva conjugate of X(j) for these (i,j): {107, 13450}, {1093, 2970}, {6528, 2052}, {15352, 393}
X(66299) = X(i)-cross conjugate of X(j) for these (i,j): {125, 6526}, {136, 4}, {2501, 14618}, {2970, 1093}, {8754, 393}, {23105, 2970}, {51513, 2501}, {65472, 58757}
X(66299) = X(i)-isoconjugate of X(j) for these (i,j): {3, 4575}, {48, 4558}, {63, 32661}, {99, 52430}, {101, 18604}, {110, 255}, {112, 6507}, {162, 1092}, {163, 394}, {184, 4592}, {249, 822}, {283, 36059}, {326, 1576}, {520, 1101}, {563, 65309}, {577, 662}, {643, 7335}, {648, 4100}, {656, 47390}, {799, 14585}, {811, 23606}, {820, 59039}, {906, 1790}, {1102, 61206}, {1331, 1437}, {1414, 6056}, {1415, 6514}, {1444, 32656}, {1804, 65375}, {1812, 32660}, {1813, 2193}, {2169, 23181}, {2194, 6517}, {2289, 4565}, {2315, 43755}, {2617, 19210}, {3049, 62719}, {3265, 23995}, {3964, 32676}, {3990, 4556}, {4020, 65307}, {4055, 52935}, {4563, 9247}, {4570, 23224}, {4602, 61361}, {4625, 62257}, {4636, 22341}, {5546, 7125}, {5562, 36134}, {7257, 62258}, {14575, 55202}, {15958, 44706}, {17974, 23997}, {22115, 36061}, {23357, 24018}, {24037, 58310}, {24041, 39201}, {36034, 51394}, {36054, 52378}, {36433, 57973}, {37754, 47443}, {44174, 63832}
X(66299) = X(i)-Dao conjugate of X(j) for these (i,j): {115, 394}, {122, 35602}, {125, 1092}, {135, 1147}, {136, 3}, {137, 5562}, {139, 52032}, {244, 255}, {338, 52347}, {512, 58310}, {523, 520}, {647, 52613}, {1015, 18604}, {1084, 577}, {1146, 6514}, {1214, 6517}, {1249, 4558}, {2501, 52584}, {2970, 12359}, {3005, 39201}, {3162, 32661}, {3258, 51394}, {4858, 326}, {4988, 4091}, {5099, 58357}, {5139, 184}, {5190, 1790}, {5521, 1437}, {6523, 110}, {6741, 1259}, {14363, 23181}, {15259, 1576}, {15525, 10607}, {15526, 3964}, {16178, 13754}, {16221, 22115}, {17423, 23606}, {18314, 3265}, {20620, 283}, {23285, 4143}, {34591, 6507}, {35078, 58354}, {35088, 51386}, {36103, 4575}, {36901, 3926}, {38970, 36212}, {38986, 52430}, {38996, 14585}, {40596, 47390}, {40608, 6056}, {40622, 1804}, {47345, 1813}, {47898, 44718}, {47899, 44719}, {48317, 3292}, {50330, 23224}, {53983, 3917}, {53986, 49}, {53989, 50461}, {55060, 7335}, {55064, 2289}, {55065, 3682}, {55066, 4100}, {56792, 53785}, {62562, 17974}, {62566, 57241}, {62576, 4563}, {62605, 4592}, {63463, 418}
X(66299) = cevapoint of X(i) and X(j) for these (i,j): {512, 6753}, {523, 65694}, {2501, 58757}, {2970, 23105}, {38359, 57154}, {58865, 58867}
X(66299) = crosspoint of X(i) and X(j) for these (i,j): {107, 8884}, {264, 30450}, {2052, 6528}
X(66299) = crosssum of X(i) and X(j) for these (i,j): {184, 30451}, {520, 5562}, {577, 39201}, {23606, 32320}
X(66299) = trilinear pole of line {115, 135}
X(66299) = crossdifference of every pair of points on line {577, 1092}
X(66299) = barycentric product X(i)*X(j) for these {i,j}: {4, 14618}, {76, 58757}, {92, 24006}, {107, 338}, {115, 6528}, {125, 15352}, {136, 30450}, {158, 1577}, {225, 46110}, {264, 2501}, {275, 23290}, {276, 51513}, {308, 65472}, {311, 15422}, {339, 6529}, {393, 850}, {512, 18027}, {523, 2052}, {525, 1093}, {648, 2970}, {656, 6521}, {661, 57806}, {823, 1109}, {847, 57065}, {1096, 20948}, {1826, 46107}, {2207, 44173}, {2394, 52661}, {2489, 18022}, {2623, 62275}, {2643, 57973}, {3064, 57809}, {3267, 6524}, {4563, 62524}, {5466, 37778}, {5489, 34538}, {6331, 8754}, {6344, 44427}, {6368, 8794}, {6520, 14208}, {6530, 43665}, {6753, 55553}, {7141, 17925}, {8747, 52623}, {8795, 12077}, {8884, 18314}, {8901, 65183}, {9290, 62521}, {9291, 62520}, {10412, 14165}, {13450, 15412}, {14222, 57486}, {14249, 58759}, {14273, 46111}, {15415, 61362}, {15459, 58261}, {16081, 16230}, {16089, 62519}, {17924, 41013}, {17994, 60199}, {18344, 52575}, {18808, 46106}, {18817, 47230}, {20031, 62431}, {20902, 36126}, {21044, 52938}, {21666, 52607}, {23105, 23582}, {23962, 32713}, {23994, 24019}, {27376, 52618}, {30735, 64983}, {32002, 55251}, {35235, 46456}, {36434, 52617}, {39183, 44732}, {39240, 46815}, {39241, 46812}, {40149, 44426}, {41221, 42405}, {43678, 59932}, {44132, 53149}, {44145, 60338}, {44161, 57204}, {44705, 52581}, {47236, 65267}, {52582, 57070}, {52632, 60428}, {55206, 57787}, {55219, 57844}, {56285, 57215}, {57868, 58812}, {58756, 62274}, {59745, 60841}
X(66299) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 4558}, {19, 4575}, {25, 32661}, {53, 23181}, {92, 4592}, {107, 249}, {112, 47390}, {115, 520}, {125, 52613}, {136, 52584}, {158, 662}, {225, 1813}, {226, 6517}, {264, 4563}, {338, 3265}, {339, 4143}, {393, 110}, {436, 62522}, {460, 56389}, {512, 577}, {513, 18604}, {522, 6514}, {523, 394}, {525, 3964}, {647, 1092}, {656, 6507}, {661, 255}, {669, 14585}, {798, 52430}, {804, 58354}, {810, 4100}, {811, 62719}, {823, 24041}, {847, 65309}, {850, 3926}, {1084, 58310}, {1093, 648}, {1096, 163}, {1109, 24018}, {1118, 4565}, {1300, 43755}, {1577, 326}, {1637, 51394}, {1824, 906}, {1826, 1331}, {1857, 5546}, {1880, 36059}, {1896, 4612}, {1969, 55202}, {2052, 99}, {2207, 1576}, {2333, 32656}, {2395, 17974}, {2489, 184}, {2492, 58357}, {2501, 3}, {2508, 58358}, {2623, 19210}, {2643, 822}, {2799, 51386}, {2969, 7254}, {2970, 525}, {2971, 3049}, {2973, 15419}, {3049, 23606}, {3064, 283}, {3120, 4091}, {3124, 39201}, {3125, 23224}, {3267, 4176}, {3566, 10607}, {3700, 1259}, {3709, 6056}, {4017, 7125}, {4024, 3682}, {4036, 3998}, {4041, 2289}, {4077, 7183}, {4079, 4055}, {4086, 3719}, {4516, 36054}, {4705, 3990}, {6331, 47389}, {6344, 60053}, {6520, 162}, {6521, 811}, {6524, 112}, {6526, 46639}, {6528, 4590}, {6529, 250}, {6530, 2421}, {6531, 43754}, {6587, 35602}, {6591, 1437}, {6753, 1147}, {7140, 4574}, {7141, 52609}, {7178, 1804}, {7180, 7335}, {7649, 1790}, {8029, 3269}, {8735, 23189}, {8736, 23067}, {8737, 38414}, {8738, 38413}, {8747, 4556}, {8748, 4636}, {8754, 647}, {8794, 18831}, {8882, 15958}, {8884, 18315}, {9426, 61361}, {12075, 22401}, {12077, 5562}, {12079, 62665}, {13400, 1181}, {13450, 14570}, {14165, 10411}, {14208, 1102}, {14249, 36841}, {14273, 3292}, {14569, 1625}, {14618, 69}, {15352, 18020}, {15422, 54}, {15475, 50433}, {16081, 17932}, {16230, 36212}, {16732, 4131}, {17924, 1444}, {17983, 65321}, {17994, 3289}, {18022, 52608}, {18027, 670}, {18314, 52347}, {18344, 2193}, {18384, 32662}, {18808, 14919}, {20031, 57742}, {20975, 32320}, {21044, 57241}, {21207, 30805}, {21447, 57216}, {21666, 15411}, {23105, 15526}, {23290, 343}, {23582, 59152}, {23962, 52617}, {24006, 63}, {24019, 1101}, {27376, 1634}, {30450, 57763}, {30735, 37188}, {32085, 65307}, {32230, 47443}, {32713, 23357}, {34208, 65311}, {34294, 58353}, {34854, 14966}, {35235, 8552}, {36127, 52378}, {36197, 58340}, {36417, 14574}, {36426, 15631}, {36434, 32713}, {37778, 5468}, {39240, 46814}, {39241, 46811}, {39416, 57638}, {40149, 6516}, {41013, 1332}, {41204, 62523}, {41221, 17434}, {42069, 23090}, {42455, 16731}, {43665, 6394}, {44426, 1812}, {44427, 52437}, {44705, 15905}, {46107, 17206}, {46110, 332}, {47230, 22115}, {47236, 13754}, {51513, 216}, {52335, 57057}, {52418, 52603}, {52439, 61206}, {52448, 4611}, {52623, 52396}, {52661, 2407}, {52938, 4620}, {53008, 4587}, {53149, 248}, {53569, 58359}, {55195, 1364}, {55197, 7066}, {55206, 212}, {55208, 603}, {55219, 418}, {55248, 60794}, {55251, 3519}, {55276, 40948}, {56285, 65233}, {57065, 9723}, {57070, 59155}, {57071, 3167}, {57094, 41608}, {57185, 22341}, {57204, 14575}, {57211, 63805}, {57652, 32660}, {57787, 55205}, {57806, 799}, {57809, 65164}, {57844, 55218}, {57973, 24037}, {58261, 41077}, {58310, 36433}, {58756, 14533}, {58757, 6}, {58759, 15394}, {58784, 28724}, {58812, 454}, {58865, 5408}, {58867, 5409}, {58882, 6461}, {59139, 55227}, {59932, 20806}, {60338, 43705}, {60428, 5467}, {61178, 44717}, {61362, 14586}, {62519, 14941}, {62520, 57686}, {62521, 56290}, {62524, 2501}, {63462, 61054}, {64983, 35575}, {65176, 44174}, {65472, 39}, {65478, 12096}, {65609, 51253}, {65694, 6503}

X(66300) = X(186)X(523)∩X(250)X(476)

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

X(66300) lies on the X-parabola (see X(12065)) and these lines: {4, 1510}, {54, 15328}, {95, 62645}, {96, 58727}, {186, 523}, {250, 476}, {275, 5466}, {340, 520}, {393, 55219}, {421, 2501}, {526, 562}, {685, 16813}, {687, 15958}, {688, 53149}, {879, 8795}, {892, 18831}, {924, 5962}, {1141, 32710}, {1825, 24006}, {2395, 8882}, {2713, 52779}, {3147, 47193}, {6368, 61440}, {7577, 34967}, {8739, 20579}, {8740, 20578}, {8884, 18808}, {8901, 12079}, {12077, 46088}, {15414, 57069}, {16230, 39182}, {18315, 44768}, {19128, 50946}, {23295, 53266}, {25044, 43088}, {36188, 43768}, {39177, 47844}, {43083, 59275}, {46138, 53346}, {60055, 65221}, {61181, 65716}, {62172, 64935}

X(66300) = reflection of X(i) in X(j) for these {i,j}: {4, 23290}, {15412, 23286}
X(66300) = isogonal conjugate of X(23181)
X(66300) = polar conjugate of X(14570)
X(66300) = anticomplement of the isogonal conjugate of X(65348)
X(66300) = isogonal conjugate of the anticomplement of X(53577)
X(66300) = isotomic conjugate of the anticomplement of X(47421)
X(66300) = isotomic conjugate of the isogonal conjugate of X(58756)
X(66300) = isotomic conjugate of the polar conjugate of X(15422)
X(66300) = polar conjugate of the isotomic conjugate of X(15412)
X(66300) = polar conjugate of the isogonal conjugate of X(2623)
X(66300) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2168, 39352}, {32692, 6360}, {65221, 40697}, {65273, 4329}, {65348, 8}
X(66300) = X(i)-Ceva conjugate of X(j) for these (i,j): {107, 51887}, {933, 4}, {8884, 8901}, {16813, 8882}, {18831, 275}, {65348, 54}
X(66300) = X(i)-cross conjugate of X(j) for these (i,j): {512, 2623}, {2623, 15412}, {2970, 4}, {8901, 8884}, {20975, 393}, {34338, 254}, {47421, 2}, {58308, 55253}, {58756, 15422}
X(66300) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23181}, {3, 2617}, {5, 4575}, {48, 14570}, {51, 4592}, {63, 1625}, {99, 62266}, {100, 44709}, {101, 16697}, {110, 44706}, {162, 5562}, {163, 343}, {216, 662}, {217, 799}, {255, 35360}, {304, 61194}, {326, 52604}, {418, 811}, {563, 65845}, {643, 30493}, {906, 17167}, {925, 63801}, {1087, 15958}, {1101, 6368}, {1154, 36061}, {1331, 18180}, {1414, 44707}, {1568, 36034}, {1576, 18695}, {1953, 4558}, {2179, 4563}, {2180, 65309}, {2290, 60053}, {2618, 47390}, {3737, 44710}, {4100, 65183}, {5546, 44708}, {6507, 61193}, {14213, 32661}, {15451, 24041}, {23997, 53174}, {23999, 58305}, {24037, 65485}, {32676, 52347}, {34055, 35319}, {35307, 65568}, {36084, 44716}, {36145, 52032}, {36148, 63805}, {40981, 55202}, {42293, 46254}, {44088, 57968}, {55219, 62719}
X(66300) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 23181}, {115, 343}, {125, 5562}, {135, 52}, {136, 5}, {244, 44706}, {512, 65485}, {523, 6368}, {647, 60597}, {1015, 16697}, {1084, 216}, {1249, 14570}, {2501, 63829}, {3005, 15451}, {3162, 1625}, {3258, 1568}, {4858, 18695}, {5139, 51}, {5190, 17167}, {5521, 18180}, {6523, 35360}, {8054, 44709}, {8901, 1216}, {15241, 63734}, {15259, 52604}, {15526, 52347}, {16178, 63735}, {16221, 1154}, {17423, 418}, {36103, 2617}, {36901, 28706}, {38970, 60524}, {38986, 62266}, {38987, 44716}, {38993, 44711}, {38994, 44712}, {38996, 217}, {39013, 52032}, {39018, 63805}, {40608, 44707}, {47898, 33529}, {47899, 33530}, {48317, 41586}, {53986, 143}, {53993, 5891}, {55060, 30493}, {62562, 53174}, {62603, 4563}, {63463, 61378}
X(66300) = cevapoint of X(i) and X(j) for these (i,j): {512, 2501}, {523, 924}, {2623, 58756}
X(66300) = crosspoint of X(i) and X(j) for these (i,j): {95, 65273}, {275, 18831}, {8795, 16813}, {46134, 60241}
X(66300) = crosssum of X(i) and X(j) for these (i,j): {51, 52317}, {216, 15451}, {418, 17434}, {647, 23195}, {684, 46094}, {20803, 23189}
X(66300) = trilinear pole of line {115, 136}
X(66300) = crossdifference of every pair of points on line {216, 217}
X(66300) = barycentric product X(i)*X(j) for these {i,j}: {4, 15412}, {54, 14618}, {69, 15422}, {76, 58756}, {92, 2616}, {95, 2501}, {96, 57065}, {107, 53576}, {115, 18831}, {125, 16813}, {136, 65273}, {264, 2623}, {275, 523}, {276, 512}, {317, 55253}, {338, 933}, {393, 62428}, {427, 39182}, {520, 8794}, {525, 8884}, {526, 65360}, {562, 2413}, {647, 8795}, {648, 8901}, {661, 40440}, {669, 57790}, {850, 8882}, {1109, 65221}, {1141, 44427}, {1577, 2190}, {2052, 23286}, {2167, 24006}, {2433, 43752}, {2489, 34384}, {2970, 18315}, {2971, 55218}, {3049, 57844}, {3267, 61362}, {3269, 52779}, {4580, 19174}, {6524, 15414}, {6591, 56189}, {6753, 34385}, {7649, 56246}, {16032, 58865}, {16037, 58867}, {17924, 56254}, {18027, 58308}, {18808, 43768}, {19189, 43665}, {20948, 62268}, {20975, 42405}, {34386, 58757}, {34980, 42401}, {35235, 64516}, {38808, 58759}, {39177, 56285}, {39181, 44732}, {39286, 55280}, {41221, 52939}, {44173, 62271}, {46138, 47230}, {55251, 63172}
X(66300) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 14570}, {6, 23181}, {19, 2617}, {25, 1625}, {54, 4558}, {95, 4563}, {96, 65309}, {115, 6368}, {125, 60597}, {136, 63829}, {275, 99}, {276, 670}, {317, 55252}, {393, 35360}, {512, 216}, {513, 16697}, {523, 343}, {525, 52347}, {647, 5562}, {649, 44709}, {661, 44706}, {669, 217}, {798, 62266}, {847, 65845}, {850, 28706}, {924, 52032}, {933, 249}, {1084, 65485}, {1093, 65183}, {1141, 60053}, {1510, 63805}, {1577, 18695}, {1637, 1568}, {1843, 35319}, {1974, 61194}, {2148, 4575}, {2167, 4592}, {2190, 662}, {2207, 52604}, {2395, 53174}, {2413, 63761}, {2433, 44715}, {2489, 51}, {2501, 5}, {2616, 63}, {2623, 3}, {2970, 18314}, {2971, 55219}, {3049, 418}, {3124, 15451}, {3569, 44716}, {3709, 44707}, {4017, 44708}, {4036, 42698}, {4559, 44710}, {6137, 44711}, {6138, 44712}, {6524, 61193}, {6591, 18180}, {6753, 52}, {7180, 30493}, {7649, 17167}, {8749, 36831}, {8754, 12077}, {8794, 6528}, {8795, 6331}, {8882, 110}, {8884, 648}, {8901, 525}, {14273, 41586}, {14586, 47390}, {14618, 311}, {15412, 69}, {15414, 4176}, {15422, 4}, {16230, 60524}, {16813, 18020}, {18808, 62722}, {18831, 4590}, {19174, 41676}, {19189, 2421}, {20578, 44713}, {20579, 44714}, {20975, 17434}, {23286, 394}, {23290, 45793}, {24006, 14213}, {32692, 44174}, {34384, 52608}, {35235, 41078}, {38808, 36841}, {39182, 1799}, {39286, 55279}, {40440, 799}, {41221, 57195}, {44427, 1273}, {44705, 42459}, {46088, 1092}, {47230, 1154}, {47236, 63735}, {51513, 36412}, {53149, 60517}, {53576, 3265}, {54034, 32661}, {55206, 7069}, {55208, 1393}, {55216, 63801}, {55219, 61378}, {55251, 25043}, {55253, 68}, {56246, 4561}, {56254, 1332}, {57065, 39113}, {57071, 41588}, {57204, 40981}, {57790, 4609}, {58306, 14966}, {58308, 577}, {58756, 6}, {58757, 53}, {58760, 3133}, {61193, 65959}, {61362, 112}, {62268, 163}, {62271, 1576}, {62276, 55202}, {62428, 3926}, {63634, 61195}, {65221, 24041}, {65273, 57763}, {65360, 35139}, {65472, 27371}, {65485, 46394}, {65751, 42293}
X(66300) = {X(2501),X(2623)}-harmonic conjugate of X(15422)

X(66301) = EULER LINE INTERCEPT OF X(86)X(55646)

Barycentrics    6*a^6 + a^5*b - 2*a^3*b^3 - 6*a^2*b^4 + a*b^5 + a^5*c + a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + b^5*c - 2*a^3*b*c^2 - 12*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - 6*a^2*c^4 + a*b*c^4 + a*c^5 + b*c^5 : :
X(66301) = 2 X[3] + X[36489], 4 X[3] - X[37416], X[20] + 2 X[7377], 5 X[631] - 2 X[36530], 2 X[36489] + X[37416]

X(66301) lies on these lines: {2, 3}, {86, 55646}, {165, 48854}, {511, 63108}, {1350, 46922}, {1447, 30282}, {1766, 51064}, {3098, 17379}, {5092, 17349}, {5731, 48849}, {9441, 48856}, {9746, 58221}, {10186, 28885}, {11179, 50074}, {12017, 63050}, {15668, 55656}, {16830, 35242}, {17238, 46264}, {17271, 43273}, {17277, 55676}, {17307, 48905}, {17343, 48906}, {17346, 51737}, {17378, 54169}, {17381, 48881}, {18755, 63006}, {25055, 48932}, {33863, 63024}, {33878, 37677}, {50133, 54173}, {50310, 51705}, {50967, 63052}

X(66301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13634, 2}, {3, 36477, 36699}, {3, 36489, 37416}, {5004, 5005, 16373}

X(66302) = EULER LINE INTERCEPT OF X(86)X(55676)

Barycentrics    6*a^6 - a^5*b + 2*a^3*b^3 - 6*a^2*b^4 - a*b^5 - a^5*c - a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c - a*b^4*c - b^5*c + 2*a^3*b*c^2 - 12*a^2*b^2*c^2 + 2*a*b^3*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - 6*a^2*c^4 - a*b*c^4 - a*c^5 - b*c^5 : :
X(66302) = 2 X[3] + X[36697], X[20] + 2 X[36652]

X(66302) lies on these lines: {2, 3}, {86, 55676}, {182, 63108}, {990, 51064}, {3098, 17349}, {5085, 46922}, {5092, 17379}, {7987, 48854}, {9441, 34632}, {11179, 50133}, {12017, 37677}, {16823, 35242}, {17232, 46264}, {17259, 55656}, {17277, 55646}, {17283, 48905}, {17297, 43273}, {17346, 54169}, {17352, 48881}, {17375, 48906}, {17378, 51737}, {17502, 44430}, {18755, 63024}, {24257, 51054}, {30282, 61018}, {33863, 63006}, {33878, 63050}, {50074, 54173}, {50286, 51705}, {50967, 63049}, {54174, 63086}

X(66302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13635, 2}, {3, 36699, 37416}

X(66303) = EULER LINE INTERCEPT OF X(86)X(55653)

Barycentrics    9*a^6 + a^5*b - 2*a^3*b^3 - 9*a^2*b^4 + a*b^5 + a^5*c + a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + b^5*c - 2*a^3*b*c^2 - 18*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - 9*a^2*c^4 + a*b*c^4 + a*c^5 + b*c^5 : :

X(66303) lies on these lines: {2, 3}, {86, 55653}, {3098, 46922}, {10164, 24808}, {17277, 55672}, {17307, 48892}, {17349, 55678}, {17379, 55639}, {33878, 63108}, {35242, 48854}, {37677, 55604}, {48932, 51109}

X(66303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13634, 13635}, {13634, 13635, 6998}

X(66304) = EULER LINE INTERCEPT OF X(86)X(55672)

Barycentrics    9*a^6 - a^5*b + 2*a^3*b^3 - 9*a^2*b^4 - a*b^5 - a^5*c - a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c - a*b^4*c - b^5*c + 2*a^3*b*c^2 - 18*a^2*b^2*c^2 + 2*a*b^3*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - 9*a^2*c^4 - a*b*c^4 - a*c^5 - b*c^5 : :

X(66304) lies on these lines: {2, 3}, {86, 55672}, {5092, 46922}, {12017, 63108}, {17277, 55653}, {17283, 48892}, {17349, 55639}, {17379, 55678}, {44430, 58221}, {55604, 63050}

X(66304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 13635, 13634}, {13634, 13635, 21554}

X(66305) = EULER LINE INTERCEPT OF X(86)X(55656)

Barycentrics    12*a^6 + a^5*b - 2*a^3*b^3 - 12*a^2*b^4 + a*b^5 + a^5*c + a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + b^5*c - 2*a^3*b*c^2 - 24*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - 12*a^2*c^4 + a*b*c^4 + a*c^5 + b*c^5 : :
X(66305) = 2 X[3] + X[36705], X[20] + 2 X[36651]

X(66305) lies on these lines: {2, 3}, {86, 55656}, {1350, 63108}, {3098, 37677}, {5092, 63050}, {16192, 48854}, {17349, 55676}, {17379, 55646}, {31884, 46922}, {50074, 51737}, {50133, 54169}

X(66305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 3524, 13632}

X(66306) = EULER LINE INTERCEPT OF X(3098)X(63050)

Barycentrics    12*a^6 - a^5*b + 2*a^3*b^3 - 12*a^2*b^4 - a*b^5 - a^5*c - a^4*b*c + 2*a^3*b^2*c + 2*a^2*b^3*c - a*b^4*c - b^5*c + 2*a^3*b*c^2 - 24*a^2*b^2*c^2 + 2*a*b^3*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - 12*a^2*c^4 - a*b*c^4 - a*c^5 - b*c^5 : :
X(66306) = 2 X[3] + X[36699], X[20] + 2 X[36473]

X(66306) lies on these lines: {2, 3}, {3098, 63050}, {5085, 63108}, {5092, 37677}, {17277, 55656}, {17349, 55646}, {17379, 55676}, {39586, 58217}, {46922, 53094}, {48849, 64108}, {48854, 58221}, {50074, 54169}, {50133, 51737}

X(66306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 3524, 13633}, {5004, 5005, 16421}

X(66307) = EULER LINE INTERCEPT OF X(86)X(33878)

Barycentrics    3*a^6 + 2*a^5*b - 4*a^3*b^3 - 3*a^2*b^4 + 2*a*b^5 + 2*a^5*c + 2*a^4*b*c - 4*a^3*b^2*c - 4*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - 4*a^3*b*c^2 - 6*a^2*b^2*c^2 - 4*a*b^3*c^2 - 4*a^3*c^3 - 4*a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 - 3*a^2*c^4 + 2*a*b*c^4 + 2*a*c^5 + 2*b*c^5 : :
X(66307) = X[3] + 2 X[36477], 4 X[140] - X[36474], 5 X[1656] - 2 X[36663], X[48853] - 3 X[49631], 3 X[9746] + X[48854]

X(66307) lies on these lines: {2, 3}, {86, 33878}, {230, 48837}, {515, 48853}, {517, 9746}, {519, 8667}, {536, 46475}, {540, 9766}, {542, 17251}, {952, 48849}, {966, 48906}, {1213, 46264}, {1351, 46922}, {1447, 15934}, {1654, 39899}, {1766, 51038}, {2271, 5306}, {3098, 15668}, {3579, 39586}, {3584, 37576}, {3654, 50291}, {3655, 50305}, {3818, 17327}, {5021, 9300}, {5092, 17259}, {5093, 63108}, {5224, 18440}, {5268, 18506}, {5790, 29081}, {6707, 48881}, {7179, 18541}, {7735, 48847}, {7778, 48835}, {8556, 48862}, {10056, 37580}, {10246, 29331}, {11179, 17330}, {11237, 17798}, {12017, 17277}, {12702, 16830}, {15271, 48863}, {17271, 50955}, {17313, 50977}, {17349, 55705}, {17379, 44456}, {17392, 54173}, {17398, 31670}, {18755, 48842}, {19758, 50178}, {19761, 50182}, {21850, 63055}, {24257, 50096}, {28150, 64302}, {28178, 44431}, {28194, 48900}, {28198, 48944}, {28204, 48851}, {28216, 64308}, {31394, 48805}, {34718, 50286}, {34773, 39581}, {37607, 48828}, {37654, 50979}, {48852, 54388}, {48861, 63006}, {49731, 51737}, {49738, 54169}, {50962, 63052}, {50967, 63110}

X(66307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13634, 3}, {6998, 13634, 2}, {13632, 36490, 36721}, {21869, 21898, 22267}

X(66308) = EULER LINE INTERCEPT OF X(86)X(12017)

Barycentrics    3*a^6 - 2*a^5*b + 4*a^3*b^3 - 3*a^2*b^4 - 2*a*b^5 - 2*a^5*c - 2*a^4*b*c + 4*a^3*b^2*c + 4*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + 4*a^3*b*c^2 - 6*a^2*b^2*c^2 + 4*a*b^3*c^2 + 4*a^3*c^3 + 4*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 3*a^2*c^4 - 2*a*b*c^4 - 2*a*c^5 - 2*b*c^5 : :
X(66308) = 4 X[140] - X[36674], 5 X[1656] - 2 X[36661]

X(66308) lies on these lines: {2, 3}, {86, 12017}, {542, 17313}, {990, 51038}, {1385, 48854}, {2271, 9300}, {3098, 17259}, {3582, 37576}, {3654, 50305}, {3655, 50291}, {3818, 17265}, {4648, 48906}, {4755, 46475}, {5021, 5306}, {5050, 46922}, {5092, 15668}, {5690, 48849}, {6211, 31178}, {6684, 48853}, {9441, 31162}, {10072, 37580}, {10246, 44430}, {11179, 17392}, {12702, 16823}, {13624, 39586}, {15934, 61018}, {16020, 22791}, {17234, 18440}, {17245, 46264}, {17251, 50977}, {17277, 33878}, {17297, 50955}, {17300, 39899}, {17330, 54173}, {17337, 31670}, {17349, 44456}, {17379, 55705}, {18358, 53665}, {21850, 37650}, {24257, 50111}, {26241, 35000}, {29085, 38107}, {29369, 59381}, {34718, 50310}, {37705, 39570}, {38034, 44431}, {39581, 61524}, {48851, 50821}, {48856, 50824}, {49731, 54169}, {49738, 51737}, {50962, 63049}, {50979, 63054}, {53091, 63108}

X(66308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13635, 3}, {13633, 36730, 36731}, {13635, 21554, 2}

X(66309) = EULER LINE INTERCEPT OF X(86)X(44456)

Barycentrics    3*a^6 + 4*a^5*b - 8*a^3*b^3 - 3*a^2*b^4 + 4*a*b^5 + 4*a^5*c + 4*a^4*b*c - 8*a^3*b^2*c - 8*a^2*b^3*c + 4*a*b^4*c + 4*b^5*c - 8*a^3*b*c^2 - 6*a^2*b^2*c^2 - 8*a*b^3*c^2 - 8*a^3*c^3 - 8*a^2*b*c^3 - 8*a*b^2*c^3 - 8*b^3*c^3 - 3*a^2*c^4 + 4*a*b*c^4 + 4*a*c^5 + 4*b*c^5 : :
X(66309) = 4 X[140] - X[36706], 5 X[1656] - 2 X[36659], 7 X[3851] - 4 X[36686]

X(66309) lies on these lines: {2, 3}, {86, 44456}, {355, 48853}, {966, 39899}, {990, 51049}, {1213, 18440}, {1482, 48854}, {2271, 3017}, {3584, 37580}, {3828, 48932}, {4688, 46475}, {5093, 46922}, {5275, 45923}, {6707, 31670}, {8148, 16830}, {11179, 49731}, {12017, 17259}, {12645, 48849}, {12702, 39586}, {15668, 33878}, {17251, 50955}, {17277, 55705}, {18481, 39580}, {18526, 39581}, {21850, 63014}, {26446, 28849}, {34718, 50291}, {34748, 50310}, {39605, 48661}, {48851, 50798}, {48856, 50805}, {49738, 54173}, {50962, 63054}

X(66309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 5054, 13632}

X(66310) = EULER LINE INTERCEPT OF X(86)X(55705)

Barycentrics    3*a^6 - 4*a^5*b + 8*a^3*b^3 - 3*a^2*b^4 - 4*a*b^5 - 4*a^5*c - 4*a^4*b*c + 8*a^3*b^2*c + 8*a^2*b^3*c - 4*a*b^4*c - 4*b^5*c + 8*a^3*b*c^2 - 6*a^2*b^2*c^2 + 8*a*b^3*c^2 + 8*a^3*c^3 + 8*a^2*b*c^3 + 8*a*b^2*c^3 + 8*b^3*c^3 - 3*a^2*c^4 - 4*a*b*c^4 - 4*a*c^5 - 4*b*c^5 : :
X(66310) = 4 X[140] - X[36698], 5 X[631] + X[7406], 5 X[1656] - 2 X[36526]

X(66310) lies on these lines: {2, 3}, {86, 55705}, {1766, 51049}, {3582, 37580}, {4648, 39899}, {8148, 16823}, {9441, 38021}, {10246, 48854}, {10247, 44430}, {11179, 49738}, {12017, 15668}, {17245, 18440}, {17259, 33878}, {17277, 44456}, {17313, 50955}, {18530, 24239}, {26446, 48853}, {34718, 50305}, {34748, 50286}, {37654, 50962}, {38066, 48851}, {46922, 53091}, {48849, 59503}, {49731, 54173}, {50979, 63110}

X(66310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {381, 5054, 13633}

X(66311) = EULER LINE INTERCEPT OF X(86)X(37517)

Barycentrics    3*a^6 + 5*a^5*b - 10*a^3*b^3 - 3*a^2*b^4 + 5*a*b^5 + 5*a^5*c + 5*a^4*b*c - 10*a^3*b^2*c - 10*a^2*b^3*c + 5*a*b^4*c + 5*b^5*c - 10*a^3*b*c^2 - 6*a^2*b^2*c^2 - 10*a*b^3*c^2 - 10*a^3*c^3 - 10*a^2*b*c^3 - 10*a*b^2*c^3 - 10*b^3*c^3 - 3*a^2*c^4 + 5*a*b*c^4 + 5*a*c^5 + 5*b*c^5 : :
X(66311) = X[3] + 5 X[36527], 17 X[3533] - 5 X[36543], 11 X[5056] - 5 X[7384]

X(66311) lies on these lines: {2, 3}, {86, 37517}, {3818, 31248}, {5097, 46922}, {9441, 38068}, {11278, 16830}, {15668, 55582}, {16200, 48854}, {17259, 55699}, {17277, 50664}, {28858, 49631}, {38155, 48853}, {44430, 63468}, {51214, 63110}, {53018, 54447}

X(66311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6998, 13634}, {2, 13634, 21554}

X(66312) = EULER LINE INTERCEPT OF X(86)X(50664)

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

X(66312) lies on these lines: {2, 3}, {86, 50664}, {11278, 16823}, {15668, 55699}, {16200, 44430}, {17259, 55582}, {17277, 37517}, {24808, 38155}, {39561, 46922}, {39586, 64954}, {48854, 64952}

X(66312) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13635, 6998}, {2, 21554, 13635}

X(66313) = EULER LINE INTERCEPT OF X(86)X(1350)

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

X(66313) lies on these lines: {1, 1447}, {2, 3}, {8, 20769}, {40, 16830}, {86, 1350}, {98, 43359}, {165, 39586}, {182, 17349}, {183, 1043}, {192, 46475}, {262, 48939}, {385, 20018}, {388, 17798}, {511, 17379}, {572, 48878}, {576, 63108}, {962, 28862}, {966, 25406}, {991, 18792}, {1213, 44882}, {1281, 8235}, {1351, 37677}, {1352, 17238}, {1503, 5224}, {1654, 6776}, {2271, 5304}, {3085, 37576}, {3564, 17343}, {3576, 16823}, {3579, 44430}, {3598, 11036}, {3815, 64159}, {3920, 37529}, {3923, 8245}, {3945, 62174}, {4292, 7179}, {4297, 49631}, {4300, 32462}, {4352, 19758}, {5021, 37665}, {5050, 63050}, {5085, 17277}, {5232, 5921}, {5275, 37537}, {5314, 27287}, {5480, 17381}, {5731, 39581}, {5882, 50310}, {6626, 7710}, {7735, 18755}, {7736, 33863}, {7774, 20077}, {7778, 59625}, {7991, 48854}, {8550, 17346}, {9534, 54388}, {10516, 17307}, {10519, 17300}, {11362, 50286}, {11477, 46922}, {14853, 63053}, {14912, 62989}, {15069, 17271}, {15668, 31884}, {15803, 61018}, {16020, 54445}, {17000, 37474}, {17206, 37668}, {17245, 21167}, {17251, 64080}, {17259, 53094}, {17327, 36990}, {17375, 48876}, {17398, 29181}, {19851, 33891}, {20090, 63428}, {20731, 37523}, {22712, 48894}, {23151, 54398}, {23863, 28265}, {24239, 37608}, {24320, 26059}, {24342, 24728}, {24467, 56512}, {25521, 51687}, {26277, 65659}, {26921, 56513}, {26939, 27547}, {26971, 31394}, {27401, 41828}, {30761, 64711}, {31089, 46483}, {31144, 43273}, {31730, 39605}, {33748, 62985}, {37527, 37683}, {39587, 59417}, {43174, 50291}, {44434, 48936}, {44698, 45141}, {48925, 64005}, {48944, 64308}, {50074, 63722}, {51212, 63055}, {55104, 56517}, {56511, 63399}

X(66313) = anticomplement of X(7380)
X(66313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 7379}, {2, 22, 37103}, {2, 3146, 7407}, {2, 7390, 7385}, {3, 4, 37416}, {3, 5, 36697}, {3, 6998, 2}, {3, 13727, 20}, {3, 36477, 4}, {20, 36693, 4}, {631, 36705, 3}, {1010, 5999, 7379}, {1010, 37053, 2}, {4220, 56774, 411}, {5002, 5003, 7560}, {5004, 5005, 1011}, {6998, 13634, 3}, {7397, 36660, 36692}, {13725, 37182, 7379}, {14784, 14785, 36663}, {16060, 56562, 2}, {16062, 56731, 2}, {19310, 37099, 37442}, {37039, 60651, 7379}, {37149, 56774, 2}

X(66314) = EULER LINE INTERCEPT OF X(86)X(5085)

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

X(66314) lies on these lines: {1, 61018}, {2, 3}, {8, 24591}, {40, 16823}, {86, 5085}, {98, 60236}, {147, 25650}, {182, 17379}, {391, 62174}, {511, 17349}, {516, 15485}, {575, 63108}, {962, 9441}, {1213, 21167}, {1350, 17277}, {1351, 63050}, {1352, 17232}, {1385, 44430}, {1447, 15803}, {1503, 17234}, {1654, 10519}, {2271, 37665}, {3086, 37576}, {3564, 17375}, {3576, 16830}, {3624, 48932}, {3705, 63146}, {3757, 10476}, {4298, 37608}, {4314, 24239}, {4648, 25406}, {4869, 5921}, {5021, 5304}, {5050, 37677}, {5268, 64679}, {5272, 12651}, {5480, 17352}, {5882, 50286}, {6194, 16552}, {6211, 24349}, {6776, 17300}, {7081, 57279}, {7179, 13411}, {7191, 37529}, {7288, 17798}, {7293, 27287}, {7735, 33863}, {7736, 18755}, {7987, 39586}, {8550, 17378}, {9588, 48851}, {9746, 16192}, {10165, 39605}, {10310, 26241}, {10444, 21153}, {10446, 13329}, {10516, 17283}, {11037, 37607}, {11362, 50310}, {11499, 31073}, {14853, 63051}, {14912, 20090}, {14986, 37580}, {15069, 17297}, {15271, 59625}, {15589, 17206}, {15668, 53094}, {16783, 54388}, {16825, 18788}, {17245, 44882}, {17259, 31884}, {17265, 36990}, {17313, 64080}, {17337, 29181}, {17343, 48876}, {18525, 24808}, {19782, 20018}, {20731, 37694}, {20769, 27383}, {23863, 28271}, {24467, 56513}, {26470, 31084}, {26921, 56512}, {27268, 46475}, {30389, 48854}, {33748, 62997}, {37521, 37683}, {37650, 51212}, {37800, 62314}, {43174, 50305}, {44431, 48900}, {46922, 53093}, {50133, 63722}, {55104, 56511}, {56517, 63399}, {62989, 63428}

X(66314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 7385}, {2, 3522, 7390}, {2, 50699, 37443}, {3, 5, 36489}, {3, 4229, 10304}, {3, 6996, 20}, {3, 21554, 2}, {3, 36477, 36705}, {3, 36697, 37416}, {3, 49129, 376}, {20, 36692, 4}, {140, 7380, 2}, {631, 36699, 3}, {3090, 36705, 36477}, {5004, 5005, 4191}, {5999, 17682, 7385}, {7413, 16434, 2}, {7474, 40916, 2}, {13635, 21554, 3}, {14784, 14785, 36661}, {16061, 56563, 2}, {19512, 49131, 36652}, {19649, 56775, 6986}

X(66315) = EULER LINE INTERCEPT OF X(86)X(55639)

Barycentrics    9*a^6 + 2*a^5*b - 4*a^3*b^3 - 9*a^2*b^4 + 2*a*b^5 + 2*a^5*c + 2*a^4*b*c - 4*a^3*b^2*c - 4*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - 4*a^3*b*c^2 - 18*a^2*b^2*c^2 - 4*a*b^3*c^2 - 4*a^3*c^3 - 4*a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 - 9*a^2*c^4 + 2*a*b*c^4 + 2*a*c^5 + 2*b*c^5 : :

X(66315) lies on these lines: {2, 3}, {86, 55639}, {3579, 48854}, {9746, 17502}, {15668, 55653}, {17259, 55672}, {17277, 55678}, {17327, 48892}, {17379, 55604}, {33878, 46922}, {34773, 48849}, {44456, 63108}, {48932, 51108}

X(66316) = EULER LINE INTERCEPT OF X(86)X(55678)

Barycentrics    9*a^6 - 2*a^5*b + 4*a^3*b^3 - 9*a^2*b^4 - 2*a*b^5 - 2*a^5*c - 2*a^4*b*c + 4*a^3*b^2*c + 4*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + 4*a^3*b*c^2 - 18*a^2*b^2*c^2 + 4*a*b^3*c^2 + 4*a^3*c^3 + 4*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 9*a^2*c^4 - 2*a*b*c^4 - 2*a*c^5 - 2*b*c^5 : :

X(66316) lies on these lines: {2, 3}, {86, 55678}, {12017, 46922}, {13624, 48854}, {15668, 55672}, {17259, 55653}, {17265, 48892}, {17277, 55639}, {17349, 55604}, {48849, 61524}, {55705, 63108}

X(66316) = midpoint of X(376) and X(36670)

X(66317) = EULER LINE INTERCEPT OF X(148)X(7884)

Barycentrics    7*a^4 + a^2*b^2 + b^4 + a^2*c^2 + 7*b^2*c^2 + c^4 : :
X(66317) = X[2] + 2 X[384], 4 X[2] - X[6655], 7 X[2] - 4 X[6656], 5 X[2] + X[6658], X[2] - 4 X[6661], 5 X[2] - 8 X[7819], 5 X[2] - 2 X[7924], 13 X[2] - 10 X[7948], 23 X[2] - 8 X[8357], 19 X[2] - 16 X[8364], 2 X[2] + X[19686], 11 X[2] + 4 X[19687], 2 X[2] - 5 X[19689], 11 X[2] - 5 X[19690], 13 X[2] - X[19691], X[2] - 7 X[19692], 7 X[2] + 5 X[19693], 11 X[2] - 14 X[19694], 25 X[2] - 4 X[19695], 19 X[2] + 2 X[19696], X[2] - 16 X[19697], 13 X[2] - 28 X[19702], 17 X[2] - 2 X[33256], X[376] - 4 X[44224], X[381] - 4 X[44237], 8 X[384] + X[6655], 7 X[384] + 2 X[6656], 10 X[384] - X[6658], X[384] + 2 X[6661], 5 X[384] + 4 X[7819], and manyu others

X(66317) lies on these lines: {2, 3}, {148, 7884}, {698, 59373}, {1992, 42421}, {2896, 47005}, {3241, 51710}, {3329, 59634}, {3734, 19570}, {3972, 63044}, {5306, 17128}, {5309, 10583}, {7739, 63020}, {7753, 7836}, {7766, 32836}, {7785, 7880}, {7787, 32833}, {7788, 20088}, {7799, 7804}, {7809, 7820}, {7811, 46226}, {7822, 11057}, {7832, 14537}, {7846, 11648}, {7865, 14712}, {7875, 20094}, {9143, 64602}, {10353, 64090}, {14907, 60728}, {20081, 63006}, {32874, 63048}, {34604, 63939}

X(66317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 33021}, {2, 384, 19686}, {2, 3543, 7933}, {2, 6658, 7924}, {2, 6661, 19689}, {2, 14031, 3543}, {2, 19686, 6655}, {2, 19692, 6661}, {2, 32981, 33263}, {2, 33019, 33223}, {2, 33201, 33266}, {2, 33246, 33259}, {2, 33266, 33004}, {2, 61944, 33277}, {2, 61985, 33283}, {2, 62063, 33258}, {376, 16898, 2}, {381, 7892, 2}, {384, 6656, 19693}, {384, 6661, 2}, {384, 7819, 6658}, {384, 19689, 6655}, {384, 19692, 19689}, {384, 19694, 19687}, {384, 19697, 19692}, {384, 19702, 19691}, {384, 44230, 14031}, {547, 33245, 2}, {7770, 33246, 2}, {7819, 7924, 2}, {8368, 33013, 2}, {11286, 14036, 2}, {11361, 33237, 2}, {14033, 33196, 52942}, {16911, 50202, 2}, {16924, 33224, 2}, {19686, 19689, 2}, {19687, 19694, 19690}, {33005, 33197, 2}

X(66318) = EULER LINE INTERCEPT OF X(83)X(59634)

Barycentrics    10*a^4 + a^2*b^2 + b^4 + a^2*c^2 + 10*b^2*c^2 + c^4 : :
X(66318) = X[2] + 3 X[384], 11 X[2] - 3 X[6655], 5 X[2] - 3 X[6656], 13 X[2] + 3 X[6658], X[2] - 3 X[6661], 2 X[2] - 3 X[7819], 7 X[2] - 3 X[7924], 19 X[2] - 15 X[7948], 8 X[2] - 3 X[8357], 7 X[2] - 6 X[8364], 5 X[2] + 3 X[19686], 7 X[2] + 3 X[19687], 7 X[2] - 15 X[19689], 31 X[2] - 15 X[19690], 35 X[2] - 3 X[19691], 5 X[2] - 21 X[19692], 17 X[2] + 15 X[19693], 17 X[2] - 21 X[19694], 17 X[2] - 3 X[19695], 25 X[2] + 3 X[19696], X[2] - 6 X[19697], 11 X[2] - 21 X[19702], 23 X[2] - 3 X[33256], 11 X[384] + X[6655], 5 X[384] + X[6656], 13 X[384] - X[6658], 2 X[384] + X[7819], 7 X[384] + X[7924], 19 X[384] + 5 X[7948], and many others

X(66318) lies on these lines: {2, 3}, {83, 59634}, {698, 63124}, {3734, 5306}, {3793, 3972}, {6390, 7804}, {6781, 34573}, {7745, 7880}, {7750, 47005}, {7753, 7789}, {7788, 18907}, {7794, 63944}, {7820, 14537}, {7884, 32819}, {8556, 37809}, {8584, 42421}, {8667, 19661}, {13188, 51732}, {14614, 59780}, {18358, 55007}, {21358, 47102}, {30435, 32836}, {32892, 63954}, {47287, 63020}, {51071, 51710}, {51122, 59373}

X(66318) = midpoint of X(i) and X(j) for these {i,j}: {384, 6661}, {6656, 19686}, {7924, 19687}
X(66318) = reflection of X(i) in X(j) for these {i,j}: {6661, 19697}, {7819, 6661}, {7924, 8364}
X(66318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1003, 8703}, {2, 3830, 33184}, {2, 8370, 5066}, {2, 8598, 8358}, {2, 8703, 8359}, {2, 11001, 11287}, {2, 14033, 3830}, {2, 15640, 33190}, {2, 32983, 61920}, {2, 32985, 15693}, {2, 33216, 61843}, {2, 35297, 11812}, {2, 41099, 11318}, {2, 44543, 61910}, {2, 61989, 33285}, {384, 19689, 19687}, {384, 19692, 6656}, {384, 19694, 19693}, {384, 19697, 7819}, {3830, 33237, 2}, {5055, 33242, 33224}, {5066, 8368, 2}, {6656, 19687, 19691}, {7770, 33255, 549}, {8363, 14034, 3853}, {8364, 19689, 7819}, {8367, 11812, 2}, {8370, 14036, 8368}, {11285, 33266, 17504}, {11286, 14039, 8369}, {14031, 33217, 3627}, {14033, 33237, 33184}, {14035, 33219, 15687}, {15687, 33185, 33219}, {19687, 19689, 8364}, {19691, 19692, 19689}, {19693, 19694, 19695}, {32971, 33224, 5055}, {33197, 61932, 2}, {33231, 61915, 2}

X(66319) = EULER LINE INTERCEPT OF X(6)X(47287)

Barycentrics    8*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 8*b^2*c^2 - c^4 : :
X(66319) = X[2] - 3 X[384], 7 X[2] - 3 X[6655], 4 X[2] - 3 X[6656], 5 X[2] + 3 X[6658], 2 X[2] - 3 X[6661], 5 X[2] - 6 X[7819], 5 X[2] - 3 X[7924], 17 X[2] - 15 X[7948], 11 X[2] - 6 X[8357], 13 X[2] - 12 X[8364], X[2] + 3 X[19686], 2 X[2] + 3 X[19687], 11 X[2] - 15 X[19689], 23 X[2] - 15 X[19690], 19 X[2] - 3 X[19691], 13 X[2] - 21 X[19692], X[2] + 15 X[19693], 19 X[2] - 21 X[19694], 10 X[2] - 3 X[19695], 11 X[2] + 3 X[19696], 7 X[2] - 12 X[19697], 16 X[2] - 21 X[19702], 13 X[2] - 3 X[33256], 7 X[384] - X[6655], 4 X[384] - X[6656], 5 X[384] + X[6658], 5 X[384] - 2 X[7819], 5 X[384] - X[7924], 17 X[384] - 5 X[7948], 11 X[384] - 2 X[8357], and many others

X(66319) lies on these lines: {2, 3}, {6, 47287}, {99, 9300}, {141, 11057}, {325, 14537}, {543, 39593}, {698, 8584}, {736, 14711}, {3314, 19569}, {3734, 37671}, {3972, 5306}, {5182, 42421}, {5309, 32819}, {6645, 15170}, {6680, 39563}, {6781, 40344}, {7737, 7788}, {7745, 7799}, {7747, 7880}, {7753, 7816}, {7762, 32833}, {7789, 7809}, {7792, 11648}, {7794, 63943}, {7802, 47005}, {7828, 63543}, {7835, 48913}, {7837, 18907}, {7868, 43618}, {13172, 18583}, {17130, 63952}, {20094, 63633}, {26686, 65140}, {31859, 63024}, {32815, 63006}, {40706, 53429}, {40707, 53441}, {51103, 51710}, {51123, 63028}, {52229, 63038}, {53428, 62877}, {53440, 62876}

X(66319) = midpoint of X(i) and X(j) for these {i,j}: {384, 19686}, {6658, 7924}, {6661, 19687}
X(66319) = reflection of X(i) in X(j) for these {i,j}: {6656, 6661}, {6661, 384}, {7924, 7819}, {19687, 19686}, {19695, 7924}
X(66319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 8356}, {2, 3845, 33228}, {2, 9855, 8354}, {2, 11361, 3845}, {2, 13586, 12100}, {2, 15682, 7841}, {2, 33005, 61908}, {2, 33007, 3534}, {2, 33013, 10109}, {2, 33016, 19709}, {2, 33274, 15713}, {2, 35287, 15719}, {2, 35927, 19708}, {2, 61966, 32984}, {2, 62007, 16041}, {2, 62051, 33210}, {2, 62094, 33215}, {2, 62160, 32986}, {381, 33255, 7807}, {382, 14037, 8363}, {384, 6655, 19697}, {384, 6658, 7819}, {384, 19687, 6656}, {384, 19696, 19689}, {384, 33256, 19692}, {1003, 8370, 35297}, {1003, 14033, 8370}, {1003, 44543, 32985}, {3534, 11286, 2}, {3543, 14001, 33219}, {3543, 33219, 33229}, {3552, 33015, 33227}, {3839, 33201, 33224}, {3839, 33224, 7887}, {3845, 8369, 2}, {6658, 7819, 19695}, {7753, 7816, 59634}, {7770, 32981, 33250}, {7819, 19695, 6656}, {7866, 15684, 33278}, {8369, 11361, 33228}, {11159, 35954, 8352}, {11286, 33007, 8356}, {11288, 19709, 2}, {14035, 33255, 381}, {14038, 33019, 33185}, {14039, 15682, 2}, {16924, 33266, 5054}, {19687, 19695, 6658}, {19689, 19696, 8357}, {19692, 33256, 8364}, {33191, 41106, 2}, {33278, 33280, 15684}, {35942, 35943, 6661}

X(66320) = EULER LINE INTERCEPT OF X(698)X(5032)

Barycentrics    11*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 11*b^2*c^2 - c^4 : :
X(66320) = X[2] - 4 X[384], 5 X[2] - 2 X[6655], 11 X[2] - 8 X[6656], 2 X[2] + X[6658], 5 X[2] - 8 X[6661], 13 X[2] - 16 X[7819], 7 X[2] - 4 X[7924], 23 X[2] - 20 X[7948], 31 X[2] - 16 X[8357], 35 X[2] - 32 X[8364], X[2] + 2 X[19686], 7 X[2] + 8 X[19687], 7 X[2] - 10 X[19689], 8 X[2] - 5 X[19690], 7 X[2] - X[19691], 4 X[2] - 7 X[19692], X[2] + 5 X[19693], 25 X[2] - 28 X[19694], 29 X[2] - 8 X[19695], 17 X[2] + 4 X[19696], 17 X[2] - 32 X[19697], 41 X[2] - 56 X[19702], 19 X[2] - 4 X[33256], 10 X[384] - X[6655], 11 X[384] - 2 X[6656], 8 X[384] + X[6658], 5 X[384] - 2 X[6661], 13 X[384] - 4 X[7819], 7 X[384] - X[7924], 23 X[384] - 5 X[7948], and many others

X(66320) lies on these lines: {2, 3}, {698, 5032}, {3972, 19570}, {7739, 20094}, {7795, 19569}, {7836, 14537}, {8591, 10353}, {10583, 11648}, {11057, 46226}, {16984, 63543}, {16989, 35369}, {20088, 32833}, {32836, 50248}, {42421, 63127}, {59634, 63018}

X(66320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19686, 6658}, {2, 19691, 7924}, {2, 19693, 19686}, {2, 62005, 33290}, {381, 33225, 2}, {384, 6658, 19692}, {384, 19686, 2}, {384, 19687, 19689}, {384, 19693, 6658}, {384, 19696, 19697}, {549, 33020, 2}, {3543, 14037, 2}, {6655, 6661, 2}, {6655, 19689, 8364}, {6658, 19692, 19690}, {7924, 19689, 2}, {11286, 33265, 2}, {15692, 33269, 2}, {15721, 33261, 2}, {16044, 33246, 2}, {16913, 31156, 2}, {19687, 19689, 19691}, {19687, 19691, 6658}, {32966, 33224, 2}, {32971, 33266, 2}, {33198, 33263, 2}, {33262, 61912, 2}

X(66321) = EULER LINE INTERCEPT OF X(698)X(20583)

Barycentrics    14*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 14*b^2*c^2 - c^4 : :
X(66321) = X[2] - 5 X[384], 13 X[2] - 5 X[6655], 7 X[2] - 5 X[6656], 11 X[2] + 5 X[6658], 3 X[2] - 5 X[6661], 4 X[2] - 5 X[7819], 9 X[2] - 5 X[7924], 29 X[2] - 25 X[7948], 11 X[2] - 10 X[8364], 3 X[2] + 5 X[19686], 17 X[2] - 25 X[19689], 41 X[2] - 25 X[19690], 37 X[2] - 5 X[19691], 19 X[2] - 35 X[19692], 7 X[2] + 25 X[19693], 31 X[2] - 35 X[19694], 19 X[2] - 5 X[19695], 23 X[2] + 5 X[19696], 5 X[2] - 7 X[19702], 5 X[2] - X[33256], 13 X[384] - X[6655], 7 X[384] - X[6656], 11 X[384] + X[6658], 3 X[384] - X[6661], 4 X[384] - X[7819], 9 X[384] - X[7924], 29 X[384] - 5 X[7948], 10 X[384] - X[8357], 11 X[384] - 2 X[8364], 3 X[384] + X[19686], and many others

X(66321) lies on these lines: {2, 3}, {698, 20583}, {1285, 32869}, {1384, 46951}, {3734, 3793}, {5026, 6329}, {6390, 7753}, {6680, 63543}, {7789, 14537}, {7816, 9300}, {12150, 52229}, {15484, 32837}, {18907, 32833}, {19661, 63955}, {32893, 46453}

X(66321) = midpoint of X(i) and X(j) for these {i,j}: {2, 19687}, {6661, 19686}
X(66321) = reflection of X(i) in X(j) for these {i,j}: {2, 19697}, {8357, 2}
X(66321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33187, 33254}, {2, 33235, 17504}, {2, 33239, 15688}, {2, 62003, 33292}, {384, 19686, 6661}, {384, 19687, 19697}, {384, 19693, 6656}, {3545, 33236, 2}, {3845, 33220, 8361}, {7770, 33187, 8703}, {8357, 19697, 7819}, {8368, 11361, 37350}, {8370, 33246, 547}, {11159, 14039, 33184}, {11361, 35954, 8368}, {14035, 33220, 3845}, {14269, 33242, 2}, {15684, 33237, 33223}, {19687, 19697, 8357}, {19687, 19702, 33256}, {33185, 33699, 33251}

X(66322) = EULER LINE INTERCEPT OF X(32)X(47005)

Barycentrics    5*a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 5*b^2*c^2 + 2*c^4 : :
X(66322) =2 X[2] + X[384], 7 X[2] - X[6655], 5 X[2] - 2 X[6656], 11 X[2] + X[6658], X[2] + 2 X[6661], X[2] - 4 X[7819], 4 X[2] - X[7924], 8 X[2] - 5 X[7948], 19 X[2] - 4 X[8357], 11 X[2] - 8 X[8364], 5 X[2] + X[19686], 13 X[2] + 2 X[19687], X[2] + 5 X[19689], 17 X[2] - 5 X[19690], 25 X[2] - X[19691], 5 X[2] + 7 X[19692], 19 X[2] + 5 X[19693], 4 X[2] - 7 X[19694], 23 X[2] - 2 X[19695], 20 X[2] + X[19696], 7 X[2] + 8 X[19697], X[2] + 14 X[19702], 16 X[2] - X[33256], X[376] + 2 X[44230], X[381] + 2 X[44224], 7 X[384] + 2 X[6655], 5 X[384] + 4 X[6656], 11 X[384] - 2 X[6658], X[384] - 4 X[6661], X[384] + 8 X[7819], 2 X[384] + X[7924], and many others

X(66322) lies on these lines: {2, 3}, {32, 47005}, {83, 7880}, {99, 16987}, {187, 16988}, {599, 42421}, {698, 47352}, {736, 8859}, {3329, 7799}, {3589, 59634}, {3679, 51710}, {3734, 7884}, {3972, 7865}, {5306, 10583}, {5309, 7846}, {7739, 7875}, {7753, 7832}, {7783, 7889}, {7784, 19569}, {7787, 7788}, {7792, 19570}, {7795, 7837}, {7804, 7809}, {7811, 7822}, {7836, 9300}, {7839, 32833}, {7867, 48913}, {7883, 63943}, {7885, 7915}, {7906, 63024}, {7919, 39563}, {7923, 11648}, {7925, 60855}, {7928, 11057}, {7942, 18362}, {9140, 64602}, {10159, 35007}, {10333, 12150}, {10353, 48657}, {15513, 31268}, {16989, 32836}, {31652, 55767}, {32820, 51860}, {34604, 63944}, {37671, 46226}, {43527, 53096}, {48310, 52695}

X(66322) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 7876}, {2, 381, 7901}, {2, 384, 7924}, {2, 6661, 384}, {2, 7924, 7948}, {2, 8369, 33273}, {2, 8370, 14046}, {2, 11286, 14041}, {2, 14001, 33246}, {2, 14035, 33223}, {2, 14036, 13586}, {2, 14037, 376}, {2, 14039, 7833}, {2, 16913, 44217}, {2, 19686, 6656}, {2, 19689, 6661}, {2, 19692, 19686}, {2, 33020, 547}, {2, 33193, 33230}, {2, 33224, 7907}, {2, 33225, 549}, {2, 33246, 7824}, {2, 33261, 61895}, {2, 33262, 61859}, {2, 33263, 32956}, {2, 33266, 16043}, {2, 33269, 5071}, {2, 61936, 33248}, {381, 33217, 2}, {384, 6656, 19696}, {384, 7819, 19694}, {384, 7948, 33256}, {384, 19694, 7948}, {6655, 19697, 384}, {6656, 19692, 384}, {6661, 7819, 2}, {7770, 14043, 32967}, {7819, 19689, 384}, {7819, 19702, 19689}, {7819, 44224, 33217}, {7924, 19694, 2}, {8703, 37340, 11299}, {8703, 37341, 11300}, {14001, 16895, 7824}, {14037, 44230, 384}, {16044, 33185, 14047}, {16045, 33224, 2}, {16895, 33246, 2}, {33222, 61895, 2}, {33225, 44237, 384}

X(66323) = EULER LINE INTERCEPT OF X(385)X(47005)

Barycentrics    7*a^4 + 4*a^2*b^2 + 4*b^4 + 4*a^2*c^2 + 7*b^2*c^2 + 4*c^4 : :
X(66323) = 4 X[2] + X[384], 11 X[2] - X[6655], 7 X[2] - 2 X[6656], 19 X[2] + X[6658], 3 X[2] + 2 X[6661], X[2] + 4 X[7819], 6 X[2] - X[7924], 29 X[2] - 4 X[8357], 13 X[2] - 8 X[8364], 9 X[2] + X[19686], 23 X[2] + 2 X[19687], 5 X[2] - X[19690], 41 X[2] - X[19691], 13 X[2] + 7 X[19692], 7 X[2] + X[19693], 2 X[2] - 7 X[19694], 37 X[2] - 2 X[19695], 34 X[2] + X[19696], 17 X[2] + 8 X[19697], 11 X[2] + 14 X[19702], 26 X[2] - X[33256], 11 X[384] + 4 X[6655], 7 X[384] + 8 X[6656], 19 X[384] - 4 X[6658], 3 X[384] - 8 X[6661], X[384] - 16 X[7819], 3 X[384] + 2 X[7924], X[384] + 2 X[7948], 29 X[384] + 16 X[8357], 13 X[384] + 32 X[8364], and many others

X(66323) lies on these lines: {2, 3}, {385, 47005}, {3329, 7880}, {5306, 46226}, {5346, 7846}, {7753, 7931}, {7799, 7889}, {7809, 7915}, {7820, 16987}, {7822, 63952}, {7875, 32833}, {7883, 63947}, {7884, 17128}, {7914, 11057}, {7920, 32836}, {7943, 11648}, {7944, 14537}, {7947, 9300}, {10583, 37671}, {12150, 63942}, {19875, 51710}, {21358, 42421}, {44562, 55778}

X(66323) = midpoint of X(2) and X(19689)
X(66323) = reflection of X(7948) in X(2)
X(66323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3545, 14065}, {2, 6661, 7924}, {2, 33187, 32956}, {2, 33198, 33251}, {2, 33220, 7824}, {2, 33237, 13586}, {2, 33255, 7876}, {2, 33278, 33221}, {6661, 7924, 384}, {7819, 19694, 384}, {7948, 19689, 384}, {8364, 19692, 33256}, {14001, 16897, 33276}, {14038, 32956, 33267}, {14067, 16045, 16922}, {19692, 33256, 384}, {19696, 19697, 384}

X(66324) = EULER LINE INTERCEPT OF X(385)X(7865)

Barycentrics    a^4 + 4*a^2*b^2 + 4*b^4 + 4*a^2*c^2 + b^2*c^2 + 4*c^4 : :
X(66324) = 4 X[2] - X[384], 5 X[2] + X[6655], X[2] + 2 X[6656], 13 X[2] - X[6658], 5 X[2] - 2 X[6661], 7 X[2] - 4 X[7819], 2 X[2] + X[7924], 2 X[2] - 5 X[7948], 11 X[2] + 4 X[8357], 5 X[2] - 8 X[8364], 7 X[2] - X[19686], 17 X[2] - 2 X[19687], 11 X[2] - 5 X[19689], 7 X[2] + 5 X[19690], 23 X[2] + X[19691], 19 X[2] - 7 X[19692], 29 X[2] - 5 X[19693], 10 X[2] - 7 X[19694], 19 X[2] + 2 X[19695], 22 X[2] - X[19696], 23 X[2] - 8 X[19697], 29 X[2] - 14 X[19702], 14 X[2] + X[33256], 2 X[381] + X[7470], 5 X[384] + 4 X[6655], X[384] + 8 X[6656], 13 X[384] - 4 X[6658], 5 X[384] - 8 X[6661], 7 X[384] - 16 X[7819], X[384] + 2 X[7924], and many others

X(66324) lies on these lines: {2, 3}, {76, 54748}, {115, 16988}, {141, 19570}, {316, 16987}, {385, 7865}, {698, 21358}, {2896, 5306}, {3096, 5309}, {3314, 7739}, {3329, 7809}, {4045, 7799}, {7753, 7859}, {7783, 7880}, {7788, 7839}, {7797, 37671}, {7803, 7837}, {7808, 48913}, {7811, 7834}, {7817, 31168}, {7831, 16984}, {7849, 39593}, {7852, 40344}, {7856, 63952}, {7864, 32833}, {7883, 63038}, {7893, 63006}, {7911, 14537}, {7914, 7918}, {7920, 63093}, {7935, 7943}, {7941, 9300}, {10356, 14458}, {12150, 63943}, {31268, 39565}, {34573, 63543}, {36811, 52088}, {60728, 64093}

X(66324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 7892}, {2, 549, 33245}, {2, 3543, 16898}, {2, 6655, 6661}, {2, 6656, 7924}, {2, 6661, 19694}, {2, 7791, 33246}, {2, 7924, 384}, {2, 7933, 381}, {2, 11287, 13586}, {2, 19686, 7819}, {2, 19690, 19686}, {2, 32965, 33224}, {2, 32986, 14036}, {2, 33021, 549}, {2, 33184, 33013}, {2, 33190, 11361}, {2, 33223, 5025}, {2, 33224, 14067}, {2, 33246, 14043}, {2, 33258, 15702}, {2, 33263, 14001}, {2, 33264, 33237}, {2, 33266, 14069}, {2, 33277, 61895}, {2, 33283, 5071}, {2, 44217, 16911}, {376, 33221, 2}, {549, 8363, 2}, {3096, 7923, 17129}, {3845, 37351, 11304}, {3845, 37352, 11303}, {5025, 32956, 16897}, {6655, 8364, 19694}, {6655, 19694, 384}, {6656, 7819, 19690}, {6656, 7948, 384}, {6656, 8363, 37243}, {6656, 8364, 6655}, {6661, 8364, 2}, {7791, 14043, 33276}, {7819, 19690, 33256}, {7819, 33256, 384}, {7824, 7866, 14047}, {7865, 7884, 385}, {7865, 7913, 7884}, {7884, 7937, 7865}, {7913, 7937, 385}, {7914, 7918, 17128}, {7914, 11648, 47005}, {7918, 47005, 11648}, {7924, 7948, 2}, {7948, 19694, 8364}, {8357, 19689, 19696}, {8363, 33021, 33245}, {11648, 47005, 17128}, {14036, 32986, 9855}, {14065, 16043, 16923}, {15694, 33218, 2}, {16895, 32974, 14042}, {16921, 33180, 33284}, {19689, 19696, 384}, {32956, 33223, 2}, {32965, 33194, 14067}, {33013, 33184, 33291}, {33194, 33224, 2}, {33245, 37243, 384}

X(66325) = EULER LINE INTERCEPT OF X(698)X(48310)

Barycentrics    8*a^4 + 5*a^2*b^2 + 5*b^4 + 5*a^2*c^2 + 8*b^2*c^2 + 5*c^4 : :
X(66325) = 5 X[2] + X[384], 13 X[2] - X[6655], 4 X[2] - X[6656], 23 X[2] + X[6658], 2 X[2] + X[6661], X[2] + 2 X[7819], 7 X[2] - X[7924], 11 X[2] - 5 X[7948], 17 X[2] - 2 X[8357], 7 X[2] - 4 X[8364], 11 X[2] + X[19686], 14 X[2] + X[19687], 7 X[2] + 5 X[19689], 29 X[2] - 5 X[19690], 49 X[2] - X[19691], 17 X[2] + 7 X[19692], 43 X[2] + 5 X[19693], X[2] - 7 X[19694], 22 X[2] - X[19695], 41 X[2] + X[19696], 11 X[2] + 4 X[19697], 8 X[2] + 7 X[19702], 31 X[2] - X[33256], 2 X[381] + X[44251], 13 X[384] + 5 X[6655], 4 X[384] + 5 X[6656], 23 X[384] - 5 X[6658], 2 X[384] - 5 X[6661], X[384] - 10 X[7819], 7 X[384] + 5 X[7924]

X(66325) lies on these lines: {2, 3}, {698, 48310}, {3589, 7799}, {3828, 51710}, {5306, 7846}, {6390, 16987}, {7753, 7915}, {7771, 51128}, {7792, 17131}, {7809, 53489}, {7820, 59634}, {7822, 37671}, {7832, 9300}, {7835, 51126}, {7880, 7889}, {7881, 63024}, {7884, 47286}, {7919, 63543}, {9606, 43527}, {12150, 63944}, {20582, 42421}, {45311, 64602}

X(66325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 381, 8363}, {2, 3543, 33221}, {2, 5071, 33218}, {2, 6661, 6656}, {2, 7819, 6661}, {2, 7892, 549}, {2, 7924, 8364}, {2, 16898, 381}, {2, 16911, 50395}, {2, 19686, 7948}, {2, 19689, 7924}, {2, 33013, 33213}, {2, 33198, 33223}, {2, 33224, 11285}, {2, 33237, 8356}, {2, 33245, 10124}, {2, 33246, 8362}, {6656, 7819, 19702}, {7819, 8364, 19689}, {7846, 47005, 5306}, {7948, 19695, 6656}, {7948, 19697, 19695}, {8363, 44251, 6656}, {8364, 19687, 6656}, {8364, 19689, 19687}, {16895, 33185, 32992}

X(66326) = EULER LINE INTERCEPT OF X(698)X(9466)

Barycentrics    2*a^4 + 5*a^2*b^2 + 5*b^4 + 5*a^2*c^2 + 2*b^2*c^2 + 5*c^4 : :
X(66326) = 5 X[2] - X[384], 7 X[2] + X[6655], 17 X[2] - X[6658], 3 X[2] + X[7924], X[2] - 5 X[7948], 4 X[2] + X[8357], 9 X[2] - X[19686], 11 X[2] - X[19687], 13 X[2] - 5 X[19689], 11 X[2] + 5 X[19690], 31 X[2] + X[19691], 23 X[2] - 7 X[19692], 37 X[2] - 5 X[19693], 11 X[2] - 7 X[19694], 13 X[2] + X[19695], 29 X[2] - X[19696], 7 X[2] - 2 X[19697], 17 X[2] - 7 X[19702], 19 X[2] + X[33256], 7 X[384] + 5 X[6655], X[384] + 5 X[6656], 17 X[384] - 5 X[6658], 3 X[384] - 5 X[6661], 2 X[384] - 5 X[7819], 3 X[384] + 5 X[7924], X[384] - 25 X[7948], 4 X[384] + 5 X[8357], X[384] - 10 X[8364], 9 X[384] - 5 X[19686], 11 X[384] - 5 X[19687], and many others

X(66326) lies on these lines: {2, 3}, {115, 34573}, {141, 5309}, {597, 7818}, {626, 9300}, {698, 9466}, {736, 22110}, {3096, 5305}, {3314, 63633}, {3589, 7753}, {3619, 46951}, {3631, 5355}, {3763, 64093}, {3793, 7792}, {3933, 7739}, {4045, 6390}, {4995, 30104}, {5007, 63944}, {5024, 32837}, {5254, 7914}, {5298, 30103}, {5306, 7767}, {5319, 63951}, {5475, 51126}, {6329, 7845}, {6680, 40344}, {7750, 7943}, {7776, 63024}, {7788, 7803}, {7790, 47005}, {7794, 39593}, {7799, 7944}, {7809, 7859}, {7822, 11648}, {7829, 63939}, {7832, 59634}, {7837, 7938}, {7846, 11057}, {7861, 63543}, {7868, 15048}, {7869, 9607}, {7879, 63093}, {7883, 63940}, {7889, 14537}, {7915, 63548}, {7919, 43291}, {7923, 19570}, {12150, 63945}, {14535, 32827}, {14929, 16989}, {15170, 26590}, {15484, 63119}, {16987, 53489}, {18358, 39882}, {18840, 32874}, {19883, 51710}, {21356, 63954}, {21358, 63955}, {22165, 41748}, {22329, 31168}, {41750, 63124}, {42421, 48310}

X(66326) = midpoint of X(i) and X(j) for these {i,j}: {2, 6656}, {6661, 7924}, {7794, 39593}, {8352, 10997}, {37351, 37352}
X(66326) = reflection of X(i) in X(j) for these {i,j}: {2, 8364}, {7819, 2}
X(66326) = complement of X(6661)
X(66326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3524, 32954}, {2, 3839, 16045}, {2, 7791, 33220}, {2, 7887, 15699}, {2, 7924, 6661}, {2, 8356, 8368}, {2, 10304, 14069}, {2, 11285, 11539}, {2, 11287, 8369}, {2, 14064, 5055}, {2, 15705, 33183}, {2, 15708, 33189}, {2, 16043, 5054}, {2, 32969, 61887}, {2, 32976, 61882}, {2, 32978, 61864}, {2, 32986, 33237}, {2, 33008, 8366}, {2, 33180, 3545}, {2, 33190, 11286}, {2, 33196, 11318}, {2, 33199, 61899}, {2, 33202, 3524}, {2, 33219, 5}, {2, 33220, 33185}, {2, 33223, 381}, {2, 33228, 8367}, {2, 33230, 11287}, {2, 33249, 47599}, {2, 33251, 7770}, {2, 33255, 33217}, {2, 33278, 16898}, {2, 35297, 8365}, {2, 61844, 32959}, {2, 61924, 32957}, {2, 66099, 17540}, {381, 33223, 33184}, {1003, 33263, 15686}, {3096, 7884, 37671}, {3363, 33184, 16041}, {3524, 33194, 2}, {5306, 7865, 7767}, {6656, 6661, 7924}, {6656, 7819, 8357}, {6656, 7948, 8364}, {6656, 8364, 7819}, {6656, 19687, 19690}, {7770, 33251, 3845}, {7791, 33220, 8703}, {7834, 7865, 5306}, {7866, 8362, 8361}, {7866, 32956, 8362}, {7876, 8363, 140}, {7884, 37671, 5305}, {8356, 8368, 27088}, {8356, 33246, 34200}, {8358, 8365, 35297}, {8368, 34200, 33246}, {8369, 11287, 8354}, {8703, 33185, 33220}, {11539, 33186, 2}, {15709, 32953, 2}, {19690, 19694, 19687}, {31693, 31694, 3860}, {32827, 63120, 14535}, {33187, 33234, 19710}, {33194, 33202, 32954}, {33246, 34200, 27088}, {37170, 37171, 3839}

X(66327) = EULER LINE INTERCEPT OF X(698)X(51185)

Barycentrics    11*a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 11*b^2*c^2 + 2*c^4 : :
X(66327) = 2 X[2] + 3 X[384], 13 X[2] - 3 X[6655], 11 X[2] - 6 X[6656], 17 X[2] + 3 X[6658], X[2] - 6 X[6661], 7 X[2] - 12 X[7819], 8 X[2] - 3 X[7924], 4 X[2] - 3 X[7948], 37 X[2] - 12 X[8357], 29 X[2] - 24 X[8364], 7 X[2] + 3 X[19686], 19 X[2] + 6 X[19687], X[2] - 3 X[19689], 7 X[2] - 3 X[19690], 43 X[2] - 3 X[19691], X[2] - 21 X[19692], 5 X[2] + 3 X[19693], 16 X[2] - 21 X[19694], 41 X[2] - 6 X[19695], 32 X[2] + 3 X[19696], X[2] + 24 X[19697], 17 X[2] - 42 X[19702], 28 X[2] - 3 X[33256], 13 X[384] + 2 X[6655], 11 X[384] + 4 X[6656], 17 X[384] - 2 X[6658], X[384] + 4 X[6661], 7 X[384] + 8 X[7819], 4 X[384] + X[7924], 2 X[384] + X[7948], and many others

X(66327) lies on these lines: {2, 3}, {698, 51185}, {5346, 17128}, {7753, 7947}, {7868, 19569}, {7880, 7941}, {7931, 14537}, {15534, 42421}, {16988, 40344}, {32892, 63065}, {51093, 51710}

X(66327) = midpoint of X(19686) and X(19690)
X(66327) = reflection of X(i) in X(j) for these {i,j}: {7924, 7948}, {19689, 6661}
X(66327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3845, 14046}, {2, 11317, 33288}, {2, 14030, 8597}, {384, 7819, 33256}, {384, 19689, 7948}, {384, 19694, 19696}, {7948, 33256, 19690}, {8366, 19709, 2}, {14038, 33198, 7824}, {19689, 19690, 7819}, {19692, 19697, 384}

X(66328) = EULER LINE INTERCEPT OF X(698)X(15534)

Barycentrics    7*a^4 - 2*a^2*b^2 - 2*b^4 - 2*a^2*c^2 + 7*b^2*c^2 - 2*c^4 : :
X(66328) = 2 X[2] - 3 X[384], 5 X[2] - 3 X[6655], 7 X[2] - 6 X[6656], X[2] + 3 X[6658], 5 X[2] - 6 X[6661], 11 X[2] - 12 X[7819], 4 X[2] - 3 X[7924], 16 X[2] - 15 X[7948], 17 X[2] - 12 X[8357], 25 X[2] - 24 X[8364], X[2] - 3 X[19686], X[2] - 6 X[19687], 13 X[2] - 15 X[19689], 19 X[2] - 15 X[19690], 11 X[2] - 3 X[19691], 17 X[2] - 21 X[19692], 7 X[2] - 15 X[19693], 20 X[2] - 21 X[19694], 13 X[2] - 6 X[19695], 4 X[2] + 3 X[19696], 19 X[2] - 24 X[19697], 37 X[2] - 42 X[19702], 8 X[2] - 3 X[33256], 5 X[384] - 2 X[6655], 7 X[384] - 4 X[6656], X[384] + 2 X[6658], 5 X[384] - 4 X[6661], 11 X[384] - 8 X[7819], 8 X[384] - 5 X[7948], and many others

X(66328) lies on these lines: {2, 3}, {99, 14537}, {148, 5306}, {543, 63038}, {599, 14976}, {698, 15534}, {3314, 43618}, {3734, 11057}, {3972, 11648}, {7737, 7837}, {7747, 7799}, {7753, 7783}, {7785, 59634}, {7788, 19569}, {7809, 7816}, {7811, 17128}, {7823, 32833}, {7875, 43619}, {7880, 7885}, {7893, 32836}, {7923, 65633}, {7925, 48913}, {7928, 47005}, {8591, 41624}, {8859, 18546}, {9766, 11164}, {11152, 12156}, {12117, 44422}, {12150, 39593}, {14712, 37671}, {16529, 36366}, {16530, 36368}, {17005, 32456}, {18907, 20094}, {19570, 32819}, {32815, 63093}, {39141, 54131}, {42421, 51185}, {51105, 51710}

X(66328) = midpoint of X(i) and X(j) for these {i,j}: {6658, 19686}, {7924, 19696}
X(66328) = reflection of X(i) in X(j) for these {i,j}: {384, 19686}, {6655, 6661}, {7924, 384}, {19686, 19687}, {33256, 7924}
X(66328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3830, 14041}, {2, 8352, 33291}, {2, 8703, 33273}, {2, 11001, 7833}, {2, 15640, 33017}, {2, 15697, 33008}, {2, 32994, 61901}, {2, 33193, 11001}, {2, 33208, 15698}, {2, 33265, 8703}, {2, 61989, 33006}, {2, 62168, 33272}, {4, 33187, 33246}, {384, 6655, 19694}, {384, 6658, 19696}, {384, 19696, 33256}, {384, 33256, 7948}, {1003, 3830, 2}, {3529, 14031, 7876}, {3543, 32981, 33255}, {3543, 33255, 5025}, {3545, 33239, 33266}, {3545, 33266, 7907}, {3552, 14042, 32967}, {3627, 33225, 14045}, {5066, 35297, 2}, {6656, 19693, 384}, {6658, 19687, 384}, {8370, 8703, 2}, {8370, 33265, 33273}, {11001, 14033, 2}, {11361, 13586, 33013}, {11361, 33007, 13586}, {11361, 33274, 33016}, {14001, 62042, 33278}, {14033, 33193, 7833}, {14035, 33214, 32968}, {14035, 33257, 7824}, {14068, 33239, 7907}, {14068, 33266, 3545}, {15684, 33219, 33019}, {15693, 44543, 2}, {15698, 32983, 2}, {16044, 33250, 33276}, {32979, 33254, 33015}, {32981, 33280, 5025}, {32985, 41099, 2}, {33007, 33016, 35927}, {33016, 35927, 33274}, {33018, 33235, 16923}, {33201, 33279, 14065}, {33216, 61932, 2}, {33224, 62017, 14063}, {33255, 33280, 3543}, {33274, 35927, 13586}

X(66329) = 1st HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR

Barycentrics    (-8*Cos[A/3]^3*Cos[B/3]^2 + Cos[B/3]*Cos[C/3] + 4*Cos[A/3]^2*Cos[B/3]*Cos[C/3] - 4*Cos[B/3]^3*Cos[C/3] - 8*Cos[A/3]^3*Cos[C/3]^2 + 8*Cos[A/3]*Cos[B/3]^2*Cos[C/3]^2 + 32*Cos[A/3]^3*Cos[B/3]^2*Cos[C/3]^2 - 4*Cos[B/3]*Cos[C/3]^3)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7236.

X(66329) lies on these lines: {356, 357}, {1135, 3605}

X(66330) = 2nd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR

Barycentrics    (-8*Cos[A/3-2Pi/3]^3*Cos[B/3-2Pi/3]^2 + Cos[B/3-2Pi/3]*Cos[C/3-2Pi/3] + 4*Cos[A/3-2Pi/3]^2*Cos[B/3-2Pi/3]*Cos[C/3-2Pi/3] - 4*Cos[B/3-2Pi/3]^3*Cos[C/3-2P/i3] - 8*Cos[A/3-2Pi/3]^3*Cos[C/3-2Pi/3]^2 + 8*Cos[A/3-2Pi/3]*Cos[B/3-2Pi/3]^2*Cos[C/3-2Pi/3]^2 + 32*Cos[A/3-2Pi/3]^3*Cos[B/3-2Pi/3]^2*Cos[C/3-2Pi/3]^2 - 4*Cos[B/3-2Pi/3]*Cos[C/3-2Pi/3]^3)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7236.

X(66330) lies on these lines: {358,3606}, {1136,1137}

X(66331) = 3rd HATZIPOLAKIS-MOSES TRISECTOR TRIANGLES PERSPECTOR

Barycentrics    (-8*Cos[A/3+2Pi/3]^3*Cos[B/3+2Pi/3]^2 + Cos[B/3+2Pi/3]*Cos[C/3+2Pi/3] + 4*Cos[A/3+2Pi/3]^2*Cos[B/3+2Pi/3]*Cos[C/3+2Pi/3] - 4*Cos[B/3+2Pi/3]^3*Cos[C/3+2P/i3] - 8*Cos[A/3+2Pi/3]^3*Cos[C/3+2Pi/3]^2 + 8*Cos[A/3+2Pi/3]*Cos[B/3+2Pi/3]^2*Cos[C/3+2Pi/3]^2 + 32*Cos[A/3+2Pi/3]^3*Cos[B/3+2Pi/3]^2*Cos[C/3+2Pi/3]^2 - 4*Cos[B/3+2Pi/3]*Cos[C/3+2Pi/3]^3)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7236.

X(66331) lies on these lines: {356,1134}, {1137,3607}

X(66332) = CENTER OF THE 1st MORLEY CONIC

Barycentrics    (1 + 2*Cos[A/3])^2*Cos[B/3]*Cos[C/3]*(-(Cos[A/3]*Cos[B/3]) - Cos[A/3]*Cos[C/3] + Cos[B/3]*Cos[C/3] - 4*Cos[A/3]*Cos[B/3]*Cos[C/3] + 4*Cos[A/3]^2*Cos[B/3]*Cos[C/3] - 4*Cos[A/3]*Cos[B/3]^2*Cos[C/3] - 4*Cos[A/3]*Cos[B/3]*Cos[C/3]^2) : :

See Peter Moses, euclid 7248.

X(66332) lies on these lines: { }

X(66333) = PERSPECTOR OF THE 1st MORLEY CONIC

Barycentrics    (4*Cos[A/3]*Cos[B/3] + Cos[C/3] + 4*Cos[A/3]^2*Cos[C/3] + 4*Cos[B/3]^2*Cos[C/3] + 16*Cos[A/3]^2*Cos[B/3]^2*Cos[C/3] + 16*Cos[A/3]*Cos[B/3]*Cos[C/3]^2)*(Cos[B/3] + 4*Cos[A/3]^2*Cos[B/3] + 4*Cos[A/3]*Cos[C/3] + 16*Cos[A/3]*Cos[B/3]^2*Cos[C/3] + 4*Cos[B/3]*Cos[C/3]^2 + 16*Cos[A/3]^2*Cos[B/3]*Cos[C/3]^2) : :

See Peter Moses, euclid 7248.

X(66333) lies on these lines: { }

X(66334) = EULER LINE INTERCEPT OF X(5305)X(7865)

Barycentrics    2*a^4 + 11*a^2*b^2 + 11*b^4 + 11*a^2*c^2 + 2*b^2*c^2 + 11*c^4 : :
X(66334) = 11 X[2] - 3 X[384], 13 X[2] + 3 X[6655], X[2] + 3 X[6656], 35 X[2] - 3 X[6658], 7 X[2] - 3 X[6661], 5 X[2] - 3 X[7819], 5 X[2] + 3 X[7924], 7 X[2] - 15 X[7948], 7 X[2] + 3 X[8357], 2 X[2] - 3 X[8364], 19 X[2] - 3 X[19686], 23 X[2] - 3 X[19687], 31 X[2] - 15 X[19689], 17 X[2] + 15 X[19690], 61 X[2] + 3 X[19691], 53 X[2] - 21 X[19692], and many others

X(66334) lies on these lines: {2, 3}, {698, 51143}, {5305, 7865}, {5306, 7913}, {7767, 7884}, {7788, 63633}, {7829, 63944}, {7853, 9300}, {7937, 37671}, {7944, 59634}, {14929, 63006}, {48310, 63956}

X(66334) = midpoint of X(i) and X(j) for these {i,j}: {6655, 66321}, {6656, 66326}, {6661, 8357}, {7819, 7924}
X(66334) = reflection of X(8364) in X(66326)
X(66334) = complement of X(66318)
X(66334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5066, 8367}, {2, 6655, 66327}, {2, 7924, 66319}, {2, 8359, 11812}, {2, 8703, 8368}, {2, 11001, 33237}, {2, 11287, 8703}, {2, 11318, 61910}, {2, 33184, 5066}, {2, 33190, 3830}, {2, 33285, 61920}, {2, 61796, 33231}, {2, 62059, 33197}, {2, 66319, 7819}, {2, 66327, 66325}, {6655, 66325, 66321}, {6656, 7948, 8357}, {6656, 66324, 66326}, {7819, 8357, 6658}, {7866, 32990, 33212}, {7924, 66317, 19695}, {8362, 33219, 547}, {11812, 33213, 2}, {33211, 45759, 33224}

X(66335) = EULER LINE INTERCEPT OF X(5305)X(7811)

Barycentrics    2*a^4 - 7*a^2*b^2 - 7*b^4 - 7*a^2*c^2 + 2*b^2*c^2 - 7*c^4 : :
X(66335) = 7 X[2] - 3 X[384], 5 X[2] + 3 X[6655], X[2] - 3 X[6656], 19 X[2] - 3 X[6658], 5 X[2] - 3 X[6661], 4 X[2] - 3 X[7819], X[2] + 3 X[7924], 11 X[2] - 15 X[7948], 2 X[2] + 3 X[8357], 5 X[2] - 6 X[8364], 11 X[2] - 3 X[19686], 13 X[2] - 3 X[19687], 23 X[2] - 15 X[19689], X[2] + 15 X[19690], 29 X[2] + 3 X[19691], 37 X[2] - 21 X[19692], 47 X[2] - 15 X[19693], 25 X[2] - 21 X[19694], 11 X[2] + 3 X[19695], 31 X[2] - 3 X[19696], 11 X[2] - 6 X[19697], 31 X[2] - 21 X[19702], 17 X[2] + 3 X[33256], 17 X[2] - 9 X[66317], 25 X[2] - 9 X[66320], 8 X[2] - 3 X[66321], 13 X[2] - 9 X[66322], 19 X[2] - 15 X[66323], 5 X[2] - 9 X[66324], and many others

X(66335) lies on these lines: {2, 3}, {141, 11648}, {230, 40344}, {524, 39593}, {698, 14711}, {3589, 14537}, {3793, 5306}, {3934, 63543}, {4045, 9300}, {5254, 7865}, {5305, 7811}, {5309, 7767}, {6292, 39563}, {6390, 7853}, {7739, 7784}, {7750, 7884}, {7788, 15048}, {7790, 37671}, {7792, 11057}, {7827, 63940}, {7829, 63943}, {7831, 43291}, {7837, 63633}, {7847, 59634}, {7873, 63944}, {7875, 19569}, {7880, 63548}, {7902, 63952}, {7928, 19570}, {14929, 63093}, {15170, 26561}, {19661, 47102}, {21356, 32892}, {32819, 47005}, {32893, 55732}, {32896, 51122}, {44678, 47352}, {51109, 51710}, {51126, 62203}

X(66335) = midpoint of X(i) and X(j) for these {i,j}: {6655, 6661}, {6656, 7924}, {8357, 66326}, {19686, 19695}
X(66335) = reflection of X(i) in X(j) for these {i,j}: {6661, 8364}, {7819, 66326}, {8357, 7924}, {19686, 19697}, {66318, 2}, {66321, 7819}, {66326, 6656}
X(66335) = complement of X(66319)
X(66335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 8369}, {2, 6655, 66328}, {2, 7841, 3845}, {2, 8354, 27088}, {2, 8356, 12100}, {2, 15682, 11286}, {2, 16041, 19709}, {2, 19708, 11288}, {2, 32984, 61908}, {2, 32986, 3534}, {2, 33210, 15682}, {2, 33215, 15701}, {2, 33228, 10109}, {2, 62094, 33191}, {2, 62160, 14039}, {2, 66318, 7819}, {2, 66328, 6661}, {549, 33219, 8361}, {6655, 6656, 8364}, {6655, 66324, 6661}, {6656, 6661, 66324}, {6656, 8357, 7819}, {6656, 19695, 7948}, {6661, 66324, 8364}, {7770, 33278, 15687}, {7791, 33219, 549}, {7924, 66324, 6655}, {7948, 19686, 66325}, {7948, 19695, 19697}, {8360, 12100, 2}, {8364, 66324, 66326}, {11287, 33184, 8359}, {11287, 33190, 33184}, {15682, 33230, 2}, {15686, 33185, 33255}, {15701, 33240, 2}, {19686, 66325, 19697}, {19694, 66320, 6661}, {19695, 66325, 19686}, {19708, 33196, 2}, {33023, 33224, 15688}, {33025, 33232, 7866}, {33210, 33230, 11286}, {33220, 33263, 550}, {33234, 33255, 15686}, {66318, 66326, 2}, {66324, 66328, 2}

X(66336) = EULER LINE INTERCEPT OF X(5306)X(7928)

Barycentrics    a^4 + 7*a^2*b^2 + 7*b^4 + 7*a^2*c^2 + b^2*c^2 + 7*c^4 : :
X(66336) = 7 X[2] - 2 X[384], 4 X[2] + X[6655], X[2] + 4 X[6656], 11 X[2] - X[6658], 9 X[2] - 4 X[6661], 13 X[2] - 8 X[7819], 3 X[2] + 2 X[7924], 17 X[2] + 8 X[8357], 11 X[2] - 16 X[8364], 6 X[2] - X[19686], 29 X[2] - 4 X[19687], 19 X[2] + X[19691], 17 X[2] - 7 X[19692], 5 X[2] - X[19693], 19 X[2] - 14 X[19694], 31 X[2] + 4 X[19695], 37 X[2] - 2 X[19696], 41 X[2] - 16 X[19697], 53 X[2] - 28 X[19702], 23 X[2] + 2 X[33256], 8 X[2] - 3 X[66317], 23 X[2] - 8 X[66318], 19 X[2] - 4 X[66319], 13 X[2] - 3 X[66320], 33 X[2] - 8 X[66321], 11 X[2] - 6 X[66322], X[2] - 6 X[66324], 17 X[2] - 12 X[66325], 3 X[2] - 8 X[66326], 5 X[2] - 2 X[66327], 17 X[2] - 2 X[66328], and many others

X(66336) lies on these lines: {2, 3}, {148, 47005}, {2896, 5346}, {3096, 19570}, {5306, 7928}, {5309, 7937}, {7739, 7938}, {7797, 7865}, {7811, 7913}, {7853, 63018}, {7923, 37671}, {7929, 63006}, {10583, 11057}, {11648, 46226}, {34604, 63947}, {50570, 55738}

X(66336) = midpoint of X(i) and X(j) for these {i,j}: {2, 19690}, {7924, 66323}
X(66336) = reflection of X(i) in X(j) for these {i,j}: {2, 7948}, {19689, 2}, {19693, 66327}
X(66336) = anticomplement of X(66323)
X(66336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6655, 66317}, {2, 6658, 66322}, {2, 7924, 19686}, {2, 19692, 66325}, {2, 33025, 33187}, {2, 33251, 16044}, {2, 33260, 33220}, {2, 33287, 61924}, {2, 66320, 7819}, {5054, 14065, 2}, {6656, 7948, 19690}, {6656, 66324, 2}, {6656, 66326, 7924}, {6658, 66321, 19686}, {7876, 33219, 2}, {7924, 7948, 66323}, {7924, 19686, 6655}, {7924, 66324, 66326}, {7924, 66326, 2}, {7948, 19690, 19689}, {8357, 66325, 66328}, {8364, 66322, 2}, {8366, 14093, 33246}, {19689, 19690, 6655}, {32956, 33251, 2}, {33221, 33255, 2}, {66321, 66326, 8364}, {66325, 66328, 19692}

X(66337) = EULER LINE INTERCEPT OF X(5306)X(7923)

Barycentrics    a^4 - 5*a^2*b^2 - 5*b^4 - 5*a^2*c^2 + b^2*c^2 - 5*c^4 : :
X(66337) = 5 X[2] - 2 X[384], 2 X[2] + X[6655], X[2] - 4 X[6656], 7 X[2] - X[6658], 7 X[2] - 4 X[6661], 11 X[2] - 8 X[7819], X[2] + 2 X[7924], 7 X[2] - 10 X[7948], 7 X[2] + 8 X[8357], 13 X[2] - 16 X[8364], 4 X[2] - X[19686], 19 X[2] - 4 X[19687], 8 X[2] - 5 X[19689], X[2] + 5 X[19690], 11 X[2] + X[19691], 13 X[2] - 7 X[19692], 17 X[2] - 5 X[19693], 17 X[2] - 14 X[19694], 17 X[2] + 4 X[19695], 23 X[2] - 2 X[19696], 31 X[2] - 16 X[19697], 43 X[2] - 28 X[19702], 13 X[2] + 2 X[33256], 17 X[2] - 8 X[66318], 13 X[2] - 4 X[66319], 23 X[2] - 8 X[66321], 13 X[2] - 10 X[66323], 5 X[2] - 4 X[66325], 5 X[2] - 8 X[66326], 19 X[2] - 10 X[66327], and many others

X(66337) lies on these lines: {2, 3}, {148, 7937}, {698, 21356}, {736, 41136}, {2896, 5309}, {3096, 11648}, {4045, 7809}, {5306, 7923}, {7739, 7779}, {7748, 47005}, {7753, 7911}, {7761, 7884}, {7768, 39593}, {7784, 7837}, {7788, 7864}, {7790, 7865}, {7797, 7811}, {7799, 7853}, {7827, 63939}, {7828, 40344}, {7834, 11057}, {7847, 7880}, {7859, 14537}, {7860, 51860}, {7868, 20094}, {7872, 46226}, {7885, 9300}, {7898, 63020}, {7900, 63024}, {7910, 10583}, {7913, 14712}, {7928, 37671}, {7929, 63093}, {7931, 59634}, {7936, 63952}, {7938, 32833}, {11185, 60728}, {16988, 53419}, {34604, 63943}, {42421, 63109}, {63038, 63944}

X(66337) = midpoint of X(i) and X(j) for these {i,j}: {6655, 66317}, {7924, 66324}
X(66337) = reflection of X(i) in X(j) for these {i,j}: {2, 66324}, {384, 66325}, {19686, 66317}, {66317, 2}, {66320, 66322}, {66324, 6656}, {66325, 66326}
X(66337) = complement of X(66320)
X(66337) = anticomplement of X(66322)
X(66337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 33225}, {2, 381, 33020}, {2, 6655, 19686}, {2, 6658, 6661}, {2, 7924, 6655}, {2, 15683, 14037}, {2, 19686, 19689}, {2, 19690, 7924}, {2, 19692, 66323}, {2, 32986, 33265}, {2, 33014, 33224}, {2, 33023, 33266}, {2, 33025, 33263}, {2, 33260, 33246}, {2, 33263, 3552}, {2, 33290, 61936}, {2, 61806, 33262}, {2, 61944, 33261}, {2, 61985, 33269}, {2, 66320, 66322}, {381, 7876, 2}, {384, 66326, 2}, {549, 7901, 2}, {6656, 7924, 2}, {6656, 8357, 7948}, {6656, 19690, 6655}, {6658, 8357, 6655}, {6661, 7948, 2}, {7790, 7865, 19570}, {7791, 32951, 33022}, {7791, 33223, 2}, {7865, 19570, 63044}, {7866, 33246, 2}, {7948, 8357, 6658}, {8359, 14046, 2}, {8360, 33273, 2}, {8364, 33256, 19692}, {8364, 66319, 66323}, {8364, 66323, 2}, {15702, 33248, 2}, {19694, 19695, 19693}, {33008, 33196, 2}, {33017, 33230, 2}, {33256, 66323, 66319}, {66319, 66323, 19692}, {66320, 66322, 66317}

X(66338) = EULER LINE INTERCEPT OF X(5309)X(7928)

Barycentrics    a^4 - 8*a^2*b^2 - 8*b^4 - 8*a^2*c^2 + b^2*c^2 - 8*c^4 : :
X(66338) = 8 X[2] - 3 X[384], 7 X[2] + 3 X[6655], X[2] - 6 X[6656], 23 X[2] - 3 X[6658], 11 X[2] - 6 X[6661], 17 X[2] - 12 X[7819], 2 X[2] + 3 X[7924], 2 X[2] - 3 X[7948], 13 X[2] + 12 X[8357], 19 X[2] - 24 X[8364], 13 X[2] - 3 X[19686], 31 X[2] - 6 X[19687], 5 X[2] - 3 X[19689], X[2] + 3 X[19690], 37 X[2] + 3 X[19691], 41 X[2] - 21 X[19692], 11 X[2] - 3 X[19693], 26 X[2] - 21 X[19694], 29 X[2] + 6 X[19695], 38 X[2] - 3 X[19696], 49 X[2] - 24 X[19697], 67 X[2] - 42 X[19702], 22 X[2] + 3 X[33256], 19 X[2] - 9 X[66317], 9 X[2] - 4 X[66318], 7 X[2] - 2 X[66319], 29 X[2] - 9 X[66320], 37 X[2] - 12 X[66321], and many others

X(66338) lies on these lines: {2, 3}, {698, 50993}, {5309, 7928}, {5346, 7811}, {7739, 7939}, {7865, 7918}, {7872, 47005}, {7883, 39593}, {7884, 7935}, {7913, 11057}, {7919, 40344}, {7937, 11648}, {31173, 55778}, {63038, 63942}

X(66338) = midpoint of X(7924) and X(7948)
X(66338) = reflection of X(i) in X(j) for these {i,j}: {384, 66323}, {7924, 19690}, {19693, 6661}, {66323, 7948}, {66327, 2}
X(66338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6655, 66319}, {2, 11001, 14036}, {2, 66319, 66322}, {2, 66327, 66323}, {6655, 66326, 66322}, {6656, 7924, 66324}, {6656, 19690, 7948}, {7924, 66322, 6655}, {7924, 66324, 384}, {7948, 66327, 2}, {16897, 32974, 14044}, {66319, 66326, 2}, {66323, 66324, 7948}

X(66339) = EULER LINE INTERCEPT OF X(5309)X(7914)

Barycentrics    4*a^4 + 7*a^2*b^2 + 7*b^4 + 7*a^2*c^2 + 4*b^2*c^2 + 7*c^4 : :
X(66339) = 7 X[2] - X[384], 11 X[2] + X[6655], 2 X[2] + X[6656], 25 X[2] - X[6658], 4 X[2] - X[6661], 5 X[2] - 2 X[7819], 5 X[2] + X[7924], X[2] + 5 X[7948], 13 X[2] + 2 X[8357], X[2] - 4 X[8364], 13 X[2] - X[19686], 16 X[2] - X[19687], 17 X[2] - 5 X[19689], 19 X[2] + 5 X[19690], 47 X[2] + X[19691], 31 X[2] - 7 X[19692], 53 X[2] - 5 X[19693], 13 X[2] - 7 X[19694], 20 X[2] + X[19695], 43 X[2] - X[19696], 19 X[2] - 4 X[19697], 22 X[2] - 7 X[19702], 29 X[2] + X[33256], 5 X[2] - X[66317], 11 X[2] - 2 X[66318], 10 X[2] - X[66319], 9 X[2] - X[66320], 17 X[2] - 2 X[66321], 11 X[2] - 5 X[66323], X[2] + 2 X[66326], 23 X[2] - 5 X[66327], and many others

X(66339) lies on these lines: {2, 3}, {141, 7884}, {736, 41133}, {3096, 5306}, {3589, 7809}, {4045, 59634}, {5254, 47005}, {5309, 7914}, {7739, 7868}, {7792, 7865}, {7811, 7943}, {7834, 37671}, {7853, 53489}, {7859, 9300}, {7879, 63006}, {7883, 63944}, {7913, 47286}, {7919, 34573}, {7934, 51126}, {14568, 20582}, {16988, 43291}

X(66339) = midpoint of X(i) and X(j) for these {i,j}: {2, 66324}, {6656, 66325}, {7924, 66317}
X(66339) = reflection of X(i) in X(j) for these {i,j}: {6656, 66324}, {6661, 66325}, {66317, 7819}, {66319, 66317}, {66324, 66326}, {66325, 2}
X(66339) = complement of X(66322)
X(66339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 33217}, {2, 6655, 66323}, {2, 6656, 6661}, {2, 7876, 549}, {2, 7901, 547}, {2, 7924, 7819}, {2, 7948, 66326}, {2, 14046, 8367}, {2, 15721, 33222}, {2, 19686, 19694}, {2, 33196, 44543}, {2, 33202, 33224}, {2, 33223, 7770}, {2, 33230, 1003}, {2, 33246, 33185}, {2, 33248, 15703}, {2, 33273, 8365}, {2, 66326, 6656}, {6655, 66323, 66318}, {6656, 7819, 19695}, {6656, 19702, 6655}, {6656, 66319, 7924}, {6661, 19695, 66319}, {7819, 7924, 66319}, {7819, 66319, 6661}, {7924, 66319, 19695}, {7948, 8364, 6656}, {8362, 33212, 33015}, {8364, 66326, 2}, {19702, 66318, 6661}, {66318, 66323, 19702}

X(66340) = EULER LINE INTERCEPT OF X(5306)X(7822)

Barycentrics    10*a^4 + 7*a^2*b^2 + 7*b^4 + 7*a^2*c^2 + 10*b^2*c^2 + 7*c^4 : :
X(66340) = 7 X[2] + X[384], 17 X[2] - X[6655], 5 X[2] - X[6656], 31 X[2] + X[6658], 3 X[2] + X[6661], 9 X[2] - X[7924], 13 X[2] - 5 X[7948], 11 X[2] - X[8357], 15 X[2] + X[19686], 19 X[2] + X[19687], 11 X[2] + 5 X[19689], 37 X[2] - 5 X[19690], 65 X[2] - X[19691], 25 X[2] + 7 X[19692], 59 X[2] + 5 X[19693], X[2] + 7 X[19694], 29 X[2] - X[19695], 55 X[2] + X[19696], 4 X[2] + X[19697], 13 X[2] + 7 X[19702], 41 X[2] - X[33256], 13 X[2] + 3 X[66317], 5 X[2] + X[66318], 11 X[2] + X[66319], 29 X[2] + 3 X[66320], 9 X[2] + X[66321], 5 X[2] + 3 X[66322], 3 X[2] + 5 X[66323], 11 X[2] - 3 X[66324], X[2] + 3 X[66325], 19 X[2] + 5 X[66327], and many others

X(66340) lies on these lines: {2, 3}, {141, 63952}, {620, 51127}, {698, 44562}, {736, 44401}, {3589, 7880}, {5306, 7822}, {7792, 47005}, {7846, 37671}, {7849, 63944}, {7859, 59634}, {7889, 9300}, {32833, 63633}, {32837, 63119}

X(66340) = midpoint of X(i) and X(j) for these {i,j}: {2, 7819}, {6656, 66318}, {6661, 66326}, {7924, 66321}, {8357, 66319}
X(66340) = reflection of X(8364) in X(2)
X(66340) = complement of X(66326)
X(66340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3839, 33194}, {2, 5055, 33186}, {2, 6661, 66326}, {2, 14069, 5054}, {2, 16045, 5055}, {2, 16898, 33219}, {2, 19689, 66324}, {2, 19694, 66325}, {2, 32954, 11539}, {2, 32957, 61887}, {2, 32959, 61871}, {2, 33183, 15709}, {2, 33189, 61864}, {2, 33220, 8362}, {2, 61924, 32953}, {2, 66317, 7948}, {2, 66322, 6656}, {2, 66323, 6661}, {2, 66325, 7819}, {6656, 6661, 19686}, {6656, 19696, 8357}, {6656, 66322, 66318}, {6661, 7924, 66321}, {6661, 19686, 66318}, {6661, 66323, 7819}, {6661, 66325, 66323}, {7819, 8357, 19689}, {7819, 8364, 19697}, {7819, 66318, 66322}, {7819, 66326, 6661}, {7948, 19691, 6656}, {8359, 33246, 14891}, {8362, 33220, 12100}, {11286, 33223, 15687}, {11301, 11302, 3524}, {15699, 33211, 2}, {19686, 66322, 6661}, {19689, 66324, 66319}, {66319, 66324, 8357}, {66321, 66326, 7924}

X(66341) = EULER LINE INTERCEPT OF X(141)X(7856)

Barycentrics    4*a^4 + 5*a^2*b^2 + 5*b^4 + 5*a^2*c^2 + 4*b^2*c^2 + 5*c^4 : :
X(66341) = 15 X[2] - X[384], 27 X[2] + X[6655], 6 X[2] + X[6656], 57 X[2] - X[6658], 8 X[2] - X[6661], 9 X[2] - 2 X[7819], 13 X[2] + X[7924], 9 X[2] + 5 X[7948], 33 X[2] + 2 X[8357], 3 X[2] + 4 X[8364], 29 X[2] - X[19686], 36 X[2] - X[19687], 33 X[2] - 5 X[19689], 51 X[2] + 5 X[19690], 111 X[2] + X[19691], 9 X[2] - X[19692], 117 X[2] - 5 X[19693], 48 X[2] + X[19695], 99 X[2] - X[19696], 39 X[2] - 4 X[19697], 6 X[2] - X[19702], 69 X[2] + X[33256], 31 X[2] - 3 X[66317], 23 X[2] - 2 X[66318], 22 X[2] - X[66319], 59 X[2] - 3 X[66320], 37 X[2] - 2 X[66321], 17 X[2] - 3 X[66322], 19 X[2] - 5 X[66323], 11 X[2] + 3 X[66324], and many others

X(66341) lies on these lines: {2, 3}, {141, 7856}, {597, 7922}, {698, 51128}, {3096, 63928}, {3589, 7858}, {6329, 7917}, {7752, 51126}, {7758, 7868}, {7792, 7854}, {7822, 7902}, {7828, 34573}, {7832, 59546}, {7834, 17131}, {7843, 7889}, {7846, 7936}, {7859, 7909}, {7863, 7915}, {7875, 7946}, {13468, 55738}, {16989, 63936}, {19878, 51710}, {31268, 58446}, {42421, 51127}, {47005, 63923}

X(66341) = midpoint of X(6656) and X(19702)
X(66341) = reflection of X(i) in X(j) for these {i,j}: {19692, 7819}, {19702, 19694}
X(66341) = complement of X(19694)
X(66341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7876, 33185}, {2, 7907, 33211}, {2, 7948, 7819}, {2, 8363, 32992}, {2, 8364, 6656}, {2, 14047, 3628}, {2, 16897, 140}, {2, 32956, 33217}, {2, 33194, 7887}, {2, 33221, 7770}, {2, 66326, 66325}, {384, 66326, 6656}, {6656, 6661, 19695}, {6656, 7819, 19687}, {6656, 66319, 8357}, {6656, 66325, 384}, {7819, 7948, 6656}, {7819, 8364, 7948}, {7819, 19687, 6661}, {7819, 19692, 19702}, {7876, 33185, 35297}, {7887, 33194, 8363}, {7948, 19696, 66324}, {8357, 19689, 66319}, {8357, 66324, 6656}, {8362, 33211, 7907}, {19689, 66324, 8357}, {19692, 19694, 7819}, {19696, 66319, 19687}, {32956, 33217, 8356}

X(66342) = EULER LINE INTERCEPT OF X(141)X(6179)

Barycentrics    4*a^4 + 3*a^2*b^2 + 3*b^4 + 3*a^2*c^2 + 4*b^2*c^2 + 3*c^4 : :
X(66342) = 9 X[2] + X[384], 21 X[2] - X[6655], 6 X[2] - X[6656], 39 X[2] + X[6658], 4 X[2] + X[6661], 3 X[2] + 2 X[7819], 11 X[2] - X[7924], 27 X[2] - 2 X[8357], 9 X[2] - 4 X[8364], 19 X[2] + X[19686], 24 X[2] + X[19687], 3 X[2] + X[19689], 9 X[2] - X[19690], 81 X[2] - X[19691], 33 X[2] + 7 X[19692], 15 X[2] + X[19693], 3 X[2] + 7 X[19694], 36 X[2] - X[19695], 69 X[2] + X[19696], 21 X[2] + 4 X[19697], 18 X[2] + 7 X[19702], 51 X[2] - X[33256], 17 X[2] + 3 X[66317], 13 X[2] + 2 X[66318], 14 X[2] + X[66319], 37 X[2] + 3 X[66320], 23 X[2] + 2 X[66321], 7 X[2] + 3 X[66322], 13 X[2] - 3 X[66324], 2 X[2] + 3 X[66325], and many others

X(66342) lies on these lines: {2, 3}, {141, 6179}, {325, 7889}, {597, 7796}, {620, 39784}, {626, 53489}, {698, 7786}, {736, 31239}, {1078, 34573}, {3589, 7832}, {3618, 7881}, {3634, 51710}, {3815, 7930}, {3933, 7875}, {5254, 7943}, {5305, 46226}, {5346, 7751}, {6329, 7905}, {6704, 7874}, {6723, 64602}, {7745, 7944}, {7750, 7914}, {7762, 7868}, {7763, 47355}, {7767, 10583}, {7789, 7859}, {7834, 17130}, {7836, 16987}, {7849, 63942}, {7852, 59635}, {7856, 47005}, {7864, 47287}, {7869, 41624}, {7870, 9606}, {7884, 63923}, {7888, 63101}, {7909, 9300}, {7913, 32819}, {7932, 64093}, {7938, 18907}, {7940, 15491}, {7945, 31406}, {18841, 62988}, {31400, 63120}, {39142, 55752}, {53033, 63119}, {55730, 55738}, {55732, 55735}, {55733, 55734}, {55744, 55829}, {55745, 55824}, {55746, 55820}, {55747, 55818}, {55749, 55801}, {55751, 55798}, {55753, 55796}, {55755, 55793}, {55757, 55788}, {55762, 55785}, {55767, 55778}, {55770, 55774}, {55771, 55773}, {56791, 61550}

X(66342) = midpoint of X(i) and X(j) for these {i,j}: {2, 66323}, {384, 19690}, {7948, 19689}
X(66342) = reflection of X(i) in X(j) for these {i,j}: {6656, 7948}, {19689, 7819}
X(66342) = complement of X(7948)
X(66342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 384, 8364}, {2, 7770, 8363}, {2, 7819, 6656}, {2, 7892, 8362}, {2, 14043, 140}, {2, 14069, 11285}, {2, 16045, 7887}, {2, 16895, 5}, {2, 16898, 7866}, {2, 16921, 33186}, {2, 17540, 33034}, {2, 19689, 7948}, {2, 19694, 7819}, {2, 32968, 33218}, {2, 32971, 33194}, {2, 32987, 32953}, {2, 33020, 14047}, {2, 33183, 32978}, {2, 33198, 33221}, {2, 33217, 7807}, {2, 33225, 16897}, {2, 66322, 66326}, {2, 66325, 6661}, {384, 6656, 19695}, {384, 7819, 19702}, {384, 7948, 19690}, {384, 8364, 6656}, {384, 19695, 19687}, {384, 19702, 6661}, {631, 8366, 7807}, {632, 1656, 40336}, {6655, 19697, 66319}, {6655, 66322, 19697}, {6655, 66326, 6656}, {6656, 6661, 19687}, {6656, 7819, 6661}, {6656, 19702, 384}, {6656, 66319, 6655}, {6656, 66325, 7819}, {6661, 19695, 384}, {7770, 8363, 33228}, {7819, 8364, 384}, {7819, 19694, 66325}, {7819, 19697, 66322}, {7819, 66326, 19697}, {7866, 16898, 8370}, {7889, 7915, 325}, {7892, 8362, 35297}, {7948, 19694, 66323}, {7948, 66323, 19689}, {8364, 19702, 19695}, {11285, 14069, 7807}, {11285, 33217, 14069}, {11287, 14037, 33250}, {11307, 11308, 3530}, {14001, 33202, 33235}, {16045, 32953, 32987}, {16897, 33225, 8359}, {19689, 19690, 384}, {19689, 19693, 66327}, {19689, 66323, 7819}, {19697, 66326, 6655}, {32953, 32987, 7887}, {32971, 33194, 33219}, {33198, 33221, 7841}, {33202, 33235, 8356}, {39387, 39388, 15720}, {66319, 66322, 6661}, {66322, 66326, 66319}

X(66343) = EULER LINE INTERCEPT OF X(141)X(5346)

Barycentrics    6*a^4 + 7*a^2*b^2 + 7*b^4 + 7*a^2*c^2 + 6*b^2*c^2 + 7*c^4 : :
X(66343) = 21 X[2] - X[384], 39 X[2] + X[6655], 9 X[2] + X[6656], 81 X[2] - X[6658], 11 X[2] - X[6661], 6 X[2] - X[7819], 19 X[2] + X[7924], 3 X[2] + X[7948], 24 X[2] + X[8357], 3 X[2] + 2 X[8364], 41 X[2] - X[19686], 51 X[2] - X[19687], 9 X[2] - X[19689], 15 X[2] + X[19690], 159 X[2] + X[19691], 87 X[2] - 7 X[19692], 33 X[2] - X[19693], 27 X[2] - 7 X[19694], 69 X[2] + X[19695], 141 X[2] - X[19696], 27 X[2] - 2 X[19697], 57 X[2] - 7 X[19702], 99 X[2] + X[33256], 43 X[2] - 3 X[66317], 16 X[2] - X[66318], 31 X[2] - X[66319], 83 X[2] - 3 X[66320], 26 X[2] - X[66321], 23 X[2] - 3 X[66322], 5 X[2] - X[66323], and many others

X(66343) lies on these lines: {2, 3}, {141, 5346}, {698, 31239}, {1506, 51127}, {3096, 3793}, {3589, 7821}, {5305, 7943}, {6390, 7915}, {7746, 51128}, {7755, 20582}, {7767, 7914}, {7822, 63923}, {7852, 34573}, {7867, 51126}, {39784, 44377}

X(66343) = midpoint of X(6656) and X(19689)
X(66343) = reflection of X(i) in X(j) for these {i,j}: {7948, 8364}, {66321, 66327}
X(66343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8364, 7819}, {2, 11285, 33211}, {6656, 6661, 33256}, {6656, 19694, 19697}, {7819, 8357, 66318}, {7819, 8364, 66326}, {7819, 66326, 8357}, {7948, 19689, 6656}, {7948, 66323, 19690}, {15712, 33185, 8366}, {19694, 19697, 7819}, {32490, 32491, 61900}

X(66344) = EULER LINE INTERCEPT OF X(5305)X(7822)

Barycentrics    6*a^4 + 5*a^2*b^2 + 5*b^4 + 5*a^2*c^2 + 6*b^2*c^2 + 5*c^4 : :
X(66344) = 15 X[2] + X[384], 33 X[2] - X[6655], 9 X[2] - X[6656], 63 X[2] + X[6658], 7 X[2] + X[6661], 3 X[2] + X[7819], 17 X[2] - X[7924], 21 X[2] - 5 X[7948], 21 X[2] - X[8357], 31 X[2] + X[19686], 39 X[2] + X[19687], 27 X[2] + 5 X[19689], 69 X[2] - 5 X[19690], 129 X[2] - X[19691], 57 X[2] + 7 X[19692], 123 X[2] + 5 X[19693], 9 X[2] + 7 X[19694], 57 X[2] - X[19695], 111 X[2] + X[19696], 9 X[2] + X[19697], 33 X[2] + 7 X[19702], 81 X[2] - X[33256], 29 X[2] + 3 X[66317], 11 X[2] + X[66318], 23 X[2] + X[66319], 61 X[2] + 3 X[66320], 19 X[2] + X[66321], 13 X[2] + 3 X[66322], 11 X[2] + 5 X[66323], 19 X[2] - 3 X[66324], and many others

X(66344) lies on these lines: {2, 3}, {597, 7869}, {698, 6683}, {736, 44381}, {3589, 7764}, {3631, 63929}, {3788, 51126}, {3793, 10583}, {5305, 7822}, {6329, 7895}, {6390, 7859}, {6680, 34573}, {6704, 44377}, {7767, 7846}, {7780, 20582}, {7795, 63633}, {7815, 42421}, {7821, 7889}, {7834, 63923}, {7849, 63940}, {7852, 43291}, {7914, 63935}, {10159, 22329}, {31406, 47355}, {32825, 63109}, {43527, 63101}, {50991, 63927}, {51073, 51710}

X(66344) = midpoint of X(i) and X(j) for these {i,j}: {6656, 19697}, {7819, 8364}
X(66344) = complement of X(8364)
X(66344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1656, 33212}, {2, 7819, 8364}, {2, 16895, 8363}, {2, 16896, 32992}, {2, 19694, 6656}, {2, 32952, 3526}, {2, 33185, 140}, {2, 33195, 46219}, {2, 33217, 8362}, {2, 66325, 66326}, {140, 33185, 8365}, {384, 66325, 7819}, {632, 37466, 140}, {6655, 19702, 66318}, {6655, 66323, 19702}, {6656, 6658, 8357}, {6656, 6661, 6658}, {6656, 7819, 19697}, {6656, 19694, 7819}, {6658, 7948, 6656}, {6661, 7948, 8357}, {7770, 8360, 3850}, {7770, 33283, 3363}, {7819, 8357, 6661}, {7819, 66318, 19702}, {7819, 66326, 384}, {8361, 8367, 35018}, {8362, 8368, 3530}, {8362, 33217, 8368}, {8364, 19697, 6656}, {11315, 11316, 10303}, {19692, 19695, 66321}, {19692, 66324, 19695}, {19702, 66323, 7819}, {32956, 33237, 550}, {33202, 33242, 8703}

X(66345) = EULER LINE INTERCEPT OF X(141)X(7923)

Barycentrics    a^4 + 3*a^2*b^2 + 3*b^4 + 3*a^2*c^2 + b^2*c^2 + 3*c^4 : :
X(66345) = 9 X[2] - 2 X[384], 6 X[2] + X[6655], 3 X[2] + 4 X[6656], 15 X[2] - X[6658], 11 X[2] - 4 X[6661], 15 X[2] - 8 X[7819], 5 X[2] + 2 X[7924], 3 X[2] - 10 X[7948], 27 X[2] + 8 X[8357], 9 X[2] - 16 X[8364], 8 X[2] - X[19686], 39 X[2] - 4 X[19687], 12 X[2] - 5 X[19689], 9 X[2] + 5 X[19690], 27 X[2] + X[19691], 33 X[2] - 5 X[19693], 45 X[2] + 4 X[19695], 51 X[2] - 2 X[19696], 51 X[2] - 16 X[19697], 9 X[2] - 4 X[19702], 33 X[2] + 2 X[33256], 10 X[2] - 3 X[66317], 29 X[2] - 8 X[66318], 25 X[2] - 4 X[66319], 17 X[2] - 3 X[66320], 43 X[2] - 8 X[66321], 13 X[2] - 6 X[66322], 17 X[2] - 10 X[66323], X[2] + 6 X[66324], and many others

X(66345) lies on these lines: {2, 3}, {76, 14125}, {141, 7923}, {148, 7822}, {315, 63020}, {626, 55085}, {698, 3619}, {2896, 6179}, {3096, 7751}, {3589, 7885}, {3618, 7900}, {4045, 7836}, {5007, 63946}, {5319, 44367}, {5355, 32027}, {5550, 51710}, {6292, 7919}, {7745, 16987}, {7755, 31168}, {7759, 51860}, {7761, 7943}, {7779, 7803}, {7784, 7875}, {7785, 7853}, {7790, 7914}, {7792, 7928}, {7800, 7932}, {7827, 7849}, {7829, 7883}, {7831, 7852}, {7846, 7935}, {7847, 7915}, {7851, 16986}, {7854, 7884}, {7856, 7865}, {7864, 7868}, {7873, 34604}, {7879, 7920}, {7889, 7911}, {7902, 19570}, {7922, 13571}, {7929, 16989}, {10159, 63924}, {16988, 59635}, {31276, 60728}, {42421, 63119}

X(66345) = reflection of X(i) in X(j) for these {i,j}: {384, 19702}, {19692, 19694}
X(66345) = complement of X(19692)
X(66345) = anticomplement of X(19694)
X(66345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5025, 33020}, {2, 6655, 19689}, {2, 6656, 6655}, {2, 6658, 7819}, {2, 7791, 33225}, {2, 7924, 66317}, {2, 7933, 16044}, {2, 19690, 384}, {2, 19692, 19694}, {2, 33010, 32957}, {2, 33014, 14069}, {2, 33018, 16045}, {2, 33019, 16898}, {2, 33021, 33259}, {2, 33025, 14037}, {2, 33180, 32966}, {2, 33200, 33269}, {2, 33202, 33004}, {2, 33260, 7892}, {2, 33283, 33002}, {2, 33287, 32987}, {2, 66320, 66323}, {5, 16897, 2}, {140, 14047, 2}, {384, 6656, 19690}, {384, 7924, 19695}, {384, 7948, 8364}, {384, 8357, 19691}, {384, 8364, 2}, {384, 19690, 6655}, {384, 19694, 19702}, {384, 19695, 6658}, {384, 19702, 19692}, {2896, 7834, 63019}, {3096, 7797, 63044}, {3096, 7913, 7797}, {4045, 7944, 7836}, {6655, 19689, 19686}, {6655, 66317, 6658}, {6656, 7819, 7924}, {6656, 7948, 2}, {6656, 8363, 11356}, {6656, 8364, 384}, {6656, 66326, 7948}, {6658, 7819, 66317}, {6658, 7924, 6655}, {6661, 33256, 19693}, {7761, 7943, 10583}, {7784, 7875, 20088}, {7790, 7914, 46226}, {7791, 14069, 33014}, {7791, 33221, 2}, {7800, 7932, 63047}, {7803, 7938, 7779}, {7819, 7924, 6658}, {7819, 19695, 384}, {7819, 66317, 19689}, {7822, 7918, 148}, {7824, 8363, 2}, {7834, 7937, 2896}, {7846, 7935, 14712}, {7853, 7859, 7785}, {7866, 7876, 2}, {7866, 11285, 14065}, {7876, 14065, 11285}, {7879, 7920, 50248}, {7892, 11287, 33260}, {7901, 8362, 2}, {7948, 66324, 6656}, {8357, 19691, 6655}, {10997, 32974, 6655}, {11285, 14065, 2}, {14037, 33025, 33264}, {14069, 33014, 33225}, {16043, 32953, 33000}, {16045, 33251, 33018}, {16898, 33190, 33019}, {16923, 33186, 2}, {16925, 33194, 2}, {19690, 19691, 8357}, {19696, 19697, 66320}, {19696, 66323, 19697}, {32953, 33000, 2}, {32960, 33248, 2}, {32987, 33180, 33287}, {32987, 33287, 32966}, {33004, 33202, 33021}, {33015, 33218, 2}, {33221, 33230, 7791}, {66324, 66326, 2}

X(66346) = EULER LINE INTERCEPT OF X(141)X(7902)

Barycentrics    2*a^4 + 7*a^2*b^2 + 7*b^4 + 7*a^2*c^2 + 2*b^2*c^2 + 7*c^4 : :
X(66346) = 21 X[2] - 5 X[384], 27 X[2] + 5 X[6655], 3 X[2] + 5 X[6656], 69 X[2] - 5 X[6658], 13 X[2] - 5 X[6661], 9 X[2] - 5 X[7819], 11 X[2] + 5 X[7924], 9 X[2] - 25 X[7948], 3 X[2] + X[8357], 3 X[2] - 5 X[8364], 37 X[2] - 5 X[19686], 9 X[2] - X[19687], 57 X[2] - 25 X[19689], 39 X[2] + 25 X[19690], 123 X[2] + 5 X[19691], 99 X[2] - 35 X[19692], 153 X[2] - 25 X[19693], 51 X[2] - 35 X[19694], 51 X[2] + 5 X[19695], 117 X[2] - 5 X[19696], 15 X[2] - 7 X[19702], 15 X[2] + X[33256], 47 X[2] - 15 X[66317], 17 X[2] - 5 X[66318], 29 X[2] - 5 X[66319], 79 X[2] - 15 X[66320], 5 X[2] - X[66321], 31 X[2] - 15 X[66322], 41 X[2] - 25 X[66323], and many others

X(66346) lies on these lines: {2, 3}, {141, 7902}, {3589, 7843}, {3793, 7928}, {4045, 59546}, {5305, 7854}, {6292, 43291}, {6390, 7944}, {7758, 63633}, {7767, 7856}, {7792, 7936}, {7825, 51126}, {7829, 63940}, {7834, 63928}, {7861, 34573}, {19662, 38627}, {20582, 63924}, {50991, 63925}

X(66346) = midpoint of X(i) and X(j) for these {i,j}: {6656, 8364}, {8357, 19697}
X(66346) = complement of X(19697)
X(66346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6656, 8357}, {2, 8357, 19697}, {2, 19687, 7819}, {2, 33232, 382}, {2, 33241, 5}, {2, 33256, 19702}, {3, 33194, 33211}, {140, 7866, 33213}, {5025, 8367, 12811}, {6656, 6661, 19690}, {6656, 7948, 7819}, {6656, 66326, 8364}, {7791, 8368, 33923}, {7807, 8358, 61792}, {7819, 7948, 8364}, {7819, 8357, 19687}, {7819, 19687, 19697}, {7819, 66326, 7948}, {7866, 16043, 33186}, {8357, 66321, 33256}, {8360, 8362, 3628}, {8364, 19697, 2}, {11287, 33185, 548}, {11287, 33221, 33185}, {16043, 33186, 140}, {19694, 19695, 66318}, {19702, 33256, 66321}, {19702, 66321, 19697}, {32960, 33240, 55856}, {33025, 33237, 15704}, {33194, 33236, 2}, {33221, 33226, 2}

X(66347) = EULER LINE INTERCEPT OF X(141)X(7872)

Barycentrics    2*a^4 - 5*a^2*b^2 - 5*b^4 - 5*a^2*c^2 + 2*b^2*c^2 - 5*c^4 : :
X(66347) = 15 X[2] - 7 X[384], 9 X[2] + 7 X[6655], 3 X[2] - 7 X[6656], 39 X[2] - 7 X[6658], 11 X[2] - 7 X[6661], 9 X[2] - 7 X[7819], X[2] + 7 X[7924], 27 X[2] - 35 X[7948], 3 X[2] + 7 X[8357], 6 X[2] - 7 X[8364], 23 X[2] - 7 X[19686], 27 X[2] - 7 X[19687], 51 X[2] - 35 X[19689], 3 X[2] - 35 X[19690], 57 X[2] + 7 X[19691], 81 X[2] - 49 X[19692], 99 X[2] - 35 X[19693], 57 X[2] - 49 X[19694], 3 X[2] + X[19695], 9 X[2] - X[19696], 12 X[2] - 7 X[19697], 69 X[2] - 49 X[19702], 33 X[2] + 7 X[33256], 37 X[2] - 21 X[66317], 13 X[2] - 7 X[66318], 19 X[2] - 7 X[66319], 53 X[2] - 21 X[66320], 17 X[2] - 7 X[66321], 29 X[2] - 21 X[66322], ande many others

X(66347) lies on these lines: {2, 3}, {141, 7872}, {315, 63633}, {597, 63931}, {626, 59546}, {3589, 7842}, {3793, 7797}, {4045, 7843}, {5007, 63945}, {5254, 7854}, {5286, 14929}, {5305, 7761}, {6144, 10542}, {6292, 53419}, {6390, 7847}, {7750, 7856}, {7758, 7784}, {7767, 7790}, {7792, 7910}, {7794, 52229}, {7818, 9607}, {7829, 63941}, {7853, 7863}, {7858, 7911}, {7861, 43291}, {7864, 7946}, {7865, 63923}, {7873, 63940}, {7928, 47286}, {7937, 32819}, {15172, 26561}, {22110, 31652}, {32455, 44499}

X(66347) = midpoint of X(i) and X(j) for these {i,j}: {6655, 7819}, {6656, 8357}
X(66347) = reflection of X(i) in X(j) for these {i,j}: {8364, 6656}, {19697, 8364}
X(66347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6655, 19696}, {3, 33180, 33186}, {20, 33194, 33242}, {140, 7791, 8358}, {384, 6656, 66326}, {546, 8362, 8367}, {550, 7866, 8368}, {550, 33211, 32973}, {3363, 32996, 546}, {3628, 5025, 8355}, {5025, 8359, 3628}, {5077, 14001, 15704}, {6655, 6656, 7819}, {6655, 7948, 19687}, {6655, 19693, 33256}, {6655, 19696, 19695}, {6656, 7924, 8357}, {6656, 19687, 7948}, {6656, 19695, 2}, {7761, 7902, 63928}, {7791, 33184, 140}, {7807, 8354, 33923}, {7819, 8357, 6655}, {7841, 8362, 546}, {7866, 32973, 33211}, {7866, 32986, 550}, {7902, 63928, 5305}, {7924, 19690, 6656}, {7933, 8356, 8361}, {7948, 19687, 7819}, {8356, 8361, 3530}, {8369, 33234, 12103}, {11286, 33238, 62036}, {11287, 32974, 5}, {11288, 33226, 46853}, {11318, 32990, 632}, {19691, 19694, 66319}, {27088, 33260, 62087}, {32954, 33023, 8703}, {32956, 33210, 382}, {32973, 33211, 8368}, {32992, 37350, 12811}, {33023, 33223, 32954}, {33025, 33190, 3}, {33180, 33186, 8360}, {33194, 33242, 33185}, {33212, 46853, 11288}, {33213, 33923, 7807}

X(66348) = EULER LINE INTERCEPT OF X(148)X(7914)

Barycentrics    a^4 + 5*a^2*b^2 + 5*b^4 + 5*a^2*c^2 + b^2*c^2 + 5*c^4 : :
X(66348) = 15 X[2] - 4 X[384], 9 X[2] + 2 X[6655], 3 X[2] + 8 X[6656], 12 X[2] - X[6658], 19 X[2] - 8 X[6661], 27 X[2] - 16 X[7819], 7 X[2] + 4 X[7924], 9 X[2] - 20 X[7948], 39 X[2] + 16 X[8357], 21 X[2] - 32 X[8364], 13 X[2] - 2 X[19686], 63 X[2] - 8 X[19687], 21 X[2] - 10 X[19689], 6 X[2] + 5 X[19690], 21 X[2] + X[19691], 18 X[2] - 7 X[19692], 27 X[2] - 5 X[19693], 39 X[2] - 28 X[19694], 69 X[2] + 8 X[19695], 81 X[2] - 4 X[19696], 87 X[2] - 32 X[19697], 111 X[2] - 56 X[19702], 51 X[2] + 4 X[33256], 17 X[2] - 6 X[66317], 49 X[2] - 16 X[66318], 41 X[2] - 8 X[66319], 14 X[2] - 3 X[66320], 71 X[2] - 16 X[66321], and many others

X(66348) lies on these lines: {2, 3}, {148, 7914}, {2896, 7856}, {3096, 7902}, {4045, 7909}, {7758, 7938}, {7784, 63020}, {7797, 7854}, {7803, 7946}, {7818, 51860}, {7834, 7936}, {7843, 7859}, {7853, 7858}, {7863, 7944}, {7918, 46226}, {7920, 63936}, {7923, 63044}, {7928, 63019}, {7931, 59546}, {7935, 10583}, {7943, 14712}, {43527, 63956}

X(66348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6655, 19692}, {2, 6656, 19690}, {2, 7924, 66320}, {2, 7933, 32993}, {2, 19690, 6658}, {2, 19691, 19689}, {2, 19693, 7819}, {2, 33287, 33010}, {384, 66325, 19689}, {3552, 33221, 2}, {6655, 7819, 19693}, {6655, 7948, 2}, {6655, 19687, 19691}, {6655, 19689, 19687}, {6655, 19692, 6658}, {6656, 7948, 6655}, {6656, 8364, 7924}, {6656, 66326, 384}, {7819, 19693, 19692}, {7866, 33021, 2}, {7924, 8364, 19689}, {7924, 19687, 6655}, {7924, 19689, 19691}, {7933, 32956, 2}, {8357, 19694, 19686}, {8363, 33259, 2}, {8364, 19689, 2}, {16897, 33020, 2}, {16897, 33184, 33020}, {19689, 19691, 66320}, {19690, 19692, 6655}, {19691, 66320, 6658}, {33182, 33258, 2}

X(66349) = EULER LINE INTERCEPT OF X(115)X(40344)

Barycentrics    4*a^4 - 5*a^2*b^2 - 5*b^4 - 5*a^2*c^2 + 4*b^2*c^2 - 5*c^4 : :
X(66349) = 5 X[2] - 3 X[384], X[2] + 3 X[6655], 2 X[2] - 3 X[6656], 11 X[2] - 3 X[6658], 4 X[2] - 3 X[6661], 7 X[2] - 6 X[7819], X[2] - 3 X[7924], 13 X[2] - 15 X[7948], X[2] - 6 X[8357], 11 X[2] - 12 X[8364], 7 X[2] - 3 X[19686], 8 X[2] - 3 X[19687], 19 X[2] - 15 X[19689], 7 X[2] - 15 X[19690], 13 X[2] + 3 X[19691], 29 X[2] - 21 X[19692], 31 X[2] - 15 X[19693], 23 X[2] - 21 X[19694], 4 X[2] + 3 X[19695], 17 X[2] - 3 X[19696], 17 X[2] - 12 X[19697], 26 X[2] - 21 X[19702], 7 X[2] + 3 X[33256], 13 X[2] - 9 X[66317], 17 X[2] - 9 X[66320], 11 X[2] - 6 X[66321], 11 X[2] - 9 X[66322], 17 X[2] - 15 X[66323], 7 X[2] - 9 X[66324], 10 X[2] - 9 X[66325], and many others

X(66349) lies on these lines: {2, 3}, {115, 40344}, {316, 9300}, {598, 62893}, {626, 59634}, {671, 60217}, {698, 22165}, {754, 39593}, {2549, 7788}, {3314, 47287}, {3815, 48913}, {4045, 14537}, {5254, 7811}, {5306, 7790}, {5309, 7750}, {7739, 7762}, {7748, 7865}, {7753, 7842}, {7756, 7880}, {7760, 63944}, {7761, 11648}, {7765, 63939}, {7767, 19570}, {7784, 32833}, {7799, 7911}, {7802, 7884}, {7809, 7847}, {7827, 63941}, {7831, 53419}, {7837, 7898}, {7860, 9607}, {7868, 43619}, {7879, 32836}, {7935, 32819}, {7936, 63923}, {12156, 63124}, {13468, 55164}, {18362, 37688}, {18907, 19569}, {32480, 51123}, {32885, 63533}, {33458, 53428}, {33459, 53440}, {39563, 59635}, {51108, 51710}, {63006, 64018}, {63038, 63945}, {63101, 63956}

X(66349) = midpoint of X(i) and X(j) for these {i,j}: {6655, 7924}, {6661, 19695}, {19686, 33256}
X(66349) = reflection of X(i) in X(j) for these {i,j}: {384, 66326}, {6656, 7924}, {6658, 66321}, {6661, 6656}, {7924, 8357}, {19686, 7819}, {19687, 6661}, {66319, 2}, {66321, 8364}, {66328, 66318}
X(66349) = complement of X(66328)
X(66349) = anticomplement of X(66318)
X(66349) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3830, 8370}, {2, 5077, 8353}, {2, 7833, 8703}, {2, 8353, 8598}, {2, 8703, 35297}, {2, 11001, 1003}, {2, 14041, 5066}, {2, 15640, 14033}, {2, 15697, 32985}, {2, 19686, 66327}, {2, 33006, 61920}, {2, 33008, 15693}, {2, 33017, 3830}, {2, 33272, 11001}, {2, 33273, 11812}, {2, 33291, 8355}, {2, 41099, 44543}, {2, 61989, 32983}, {2, 62059, 33216}, {2, 66319, 6661}, {2, 66327, 7819}, {2, 66328, 66318}, {20, 33223, 33220}, {376, 32974, 33219}, {376, 33219, 7807}, {381, 33278, 33229}, {384, 66325, 6661}, {384, 66326, 66325}, {3830, 11287, 2}, {5066, 8359, 2}, {6655, 6656, 19695}, {6655, 8357, 6656}, {6655, 19690, 33256}, {6656, 19695, 19687}, {6656, 19702, 7948}, {6656, 66319, 2}, {6656, 66325, 66326}, {6658, 66322, 66321}, {7761, 11648, 37671}, {7790, 11057, 5306}, {7791, 33229, 32992}, {7791, 33278, 381}, {7819, 19690, 6656}, {7833, 33184, 35297}, {7841, 8356, 33228}, {7841, 32986, 8356}, {7866, 15681, 33255}, {7866, 32997, 33250}, {7924, 33256, 66324}, {7924, 66324, 19690}, {8358, 37350, 2}, {8364, 66321, 66322}, {8703, 33184, 2}, {11001, 33190, 2}, {11287, 33017, 8370}, {11318, 15693, 2}, {11648, 37671, 47286}, {15681, 33255, 33250}, {15689, 32954, 33266}, {15698, 33285, 2}, {19686, 19690, 66324}, {19686, 66324, 7819}, {19690, 33256, 7819}, {19696, 66323, 66320}, {19702, 66317, 6661}, {32974, 33234, 7807}, {32986, 33210, 7841}, {32997, 33255, 15681}, {33025, 33238, 7770}, {33180, 33247, 33235}, {33180, 62120, 33224}, {33190, 33272, 1003}, {33196, 35927, 8366}, {33200, 33226, 33233}, {33219, 33234, 376}, {33220, 33223, 8363}, {33224, 33247, 62120}, {33224, 62120, 33235}, {33251, 33263, 3}, {33253, 33266, 15689}, {33256, 66324, 19686}, {66318, 66328, 66319}, {66320, 66323, 19697}, {66324, 66327, 2}

X(66350) = X(115)X(511)∩X(684)X(2491)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^12*b^4 - 4*a^10*b^6 + 6*a^8*b^8 - 4*a^6*b^10 + a^4*b^12 + 2*a^12*b^2*c^2 - 4*a^10*b^4*c^2 + 5*a^8*b^6*c^2 - a^6*b^8*c^2 - a^4*b^10*c^2 - a^2*b^12*c^2 + a^12*c^4 - 4*a^10*b^2*c^4 + 2*a^8*b^4*c^4 - 3*a^6*b^6*c^4 - a^4*b^8*c^4 + 5*a^2*b^10*c^4 - 2*b^12*c^4 - 4*a^10*c^6 + 5*a^8*b^2*c^6 - 3*a^6*b^4*c^6 + 6*a^4*b^6*c^6 - 4*a^2*b^8*c^6 + 8*b^10*c^6 + 6*a^8*c^8 - a^6*b^2*c^8 - a^4*b^4*c^8 - 4*a^2*b^6*c^8 - 12*b^8*c^8 - 4*a^6*c^10 - a^4*b^2*c^10 + 5*a^2*b^4*c^10 + 8*b^6*c^10 + a^4*c^12 - a^2*b^2*c^12 - 2*b^4*c^12) : :

X(66350) lies on the cubic K1381 and these lines: {115, 511}, {684, 2491}, {3016, 14966}, {6784, 56392}, {46303, 60517}

X(66351) = X(11)X(1146)∩X(115)X(522)

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

X(66351) lies on the cubic K1381 and these lines: {11, 1146}, {115, 522}, {3738, 66188}, {3910, 66189}, {6784, 8676}

X(66351) = barycentric product X(24026)*X(51305)
X(66351) = barycentric quotient X(51305)/X(7045)

X(66352) = TRIPOLAR CENTROID OF X(476)

Barycentrics    (2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(2*a^12 - 4*a^10*b^2 + 4*a^8*b^4 - 7*a^6*b^6 + 7*a^4*b^8 - a^2*b^10 - b^12 - 4*a^10*c^2 + 4*a^8*b^2*c^2 + 3*a^6*b^4*c^2 - 6*a^4*b^6*c^2 - 2*a^2*b^8*c^2 + 5*b^10*c^2 + 4*a^8*c^4 + 3*a^6*b^2*c^4 + 3*a^2*b^6*c^4 - 11*b^8*c^4 - 7*a^6*c^6 - 6*a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 14*b^6*c^6 + 7*a^4*c^8 - 2*a^2*b^2*c^8 - 11*b^4*c^8 - a^2*c^10 + 5*b^2*c^10 - c^12) : :
X(66352) = 2 X[23968] + X[57464], X[23968] + 2 X[63788], X[57464] - 4 X[63788]

X(66352) lies on the cubic K1381 and these lines: {6, 13}, {1640, 6041}, {34761, 35906}, {40138, 53155}, {46048, 57598}

X(66352) = tripolar centroid of X(476)
X(66352) = crossdifference of every pair of points on line {526, 842}
X(66352) = {X(23968),X(63788)}-harmonic conjugate of X(57464)

X(66353) = X(50)X(230)∩X(115)X(523)

Barycentrics    (b^2 - c^2)^2*(3*a^4 - 3*a^2*b^2 + 2*b^4 - 3*a^2*c^2 - b^2*c^2 + 2*c^4) : :
X(66353) = 5 X[115] + X[23992], 2 X[115] + X[44398], 3 X[115] + X[45212], 3 X[115] - X[61339], 4 X[115] - X[64258], 5 X[23991] - X[23992], 2 X[23991] + X[31644], 3 X[23991] - X[45212], X[23991] + 2 X[57515], 3 X[23991] + X[61339], 4 X[23991] + X[64258], 2 X[23992] + 5 X[31644], 2 X[23992] - 5 X[44398], 3 X[23992] - 5 X[45212], X[23992] + 10 X[57515], 3 X[23992] + 5 X[61339], 4 X[23992] + 5 X[64258], 3 X[31644] + 2 X[45212], X[31644] - 4 X[57515], 3 X[31644] - 2 X[61339], 3 X[44398] - 2 X[45212], X[44398] + 4 X[57515], 3 X[44398] + 2 X[61339], 2 X[44398] + X[64258], X[45212] + 6 X[57515], 4 X[45212] + 3 X[64258], 6 X[57515] - X[61339], 8 X[57515] - X[64258], 4 X[61339] - 3 X[64258], 9 X[9166] - X[31998], 9 X[9166] - 5 X[40429], X[31998] - 5 X[40429], 3 X[671] + X[33799], X[2482] - 4 X[9165], X[4590] - 5 X[14061], X[4590] - 4 X[40511], 5 X[14061] - 4 X[40511], 4 X[5461] - X[44397], X[31372] + 3 X[44373]

X(66353) lies on these lines: {2, 14588}, {50, 230}, {115, 523}, {524, 5103}, {671, 33799}, {1510, 14113}, {1648, 45294}, {2482, 9165}, {3124, 62572}, {4580, 34294}, {4590, 14061}, {5254, 18122}, {5461, 44397}, {9182, 54104}, {10026, 17162}, {13881, 40879}, {21906, 38393}, {22110, 62311}, {24345, 62322}, {28213, 41180}, {31372, 44373}, {32740, 65719}, {37804, 44377}, {40350, 47171}, {41254, 65774}, {47243, 47349}, {62508, 63543}

X(66353) = midpoint of X(i) and X(j) for these {i,j}: {115, 23991}, {9182, 54104}, {31644, 44398}, {45212, 61339}
X(66353) = reflection of X(i) in X(j) for these {i,j}: {115, 57515}, {14588, 36953}, {31644, 115}, {44398, 23991}, {64258, 31644}
X(66353) = complement of X(14588)
X(66353) = anticomplement of X(36953)
X(66353) = anticomplement of the isogonal conjugate of X(39024)
X(66353) = anticomplement of the isotomic conjugate of X(14061)
X(66353) = complement of the isotomic conjugate of X(42345)
X(66353) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {14060, 4329}, {14061, 6327}, {33799, 17217}, {33803, 7192}, {33809, 44445}, {39024, 8}, {62663, 21294}
X(66353) = X(i)-complementary conjugate of X(j) for these (i,j): {40429, 42327}, {42345, 2887}, {57728, 4369}
X(66353) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 523}, {14061, 2}
X(66353) = X(1101)-isoconjugate of X(40511)
X(66353) = X(i)-Dao conjugate of X(j) for these (i,j): {523, 40511}, {8029, 115}, {12076, 40469}
X(66353) = crosspoint of X(2) and X(42345)
X(66353) = barycentric product X(99)*X(12076)
X(66353) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 40511}, {12076, 523}
X(66353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14588, 36953}, {115, 44398, 64258}, {115, 45212, 61339}, {9166, 31998, 40429}, {23991, 57515, 31644}, {23991, 61339, 45212}

X(66354) = TRIPOLAR CENTROID OF X(110)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :
X(66354) = 5 X[14966] + X[14967], X[14966] + 2 X[34349], 2 X[14966] + X[41172], X[14967] - 10 X[34349], 2 X[14967] - 5 X[41172], 4 X[34349] - X[41172]

X(66354) lies on these lines: {2, 66192}, {3, 6}, {99, 50436}, {115, 57603}, {232, 4230}, {237, 44114}, {248, 57742}, {684, 2491}, {877, 40138}, {1640, 6041}, {1989, 34365}, {2421, 35910}, {2482, 2799}, {2493, 5106}, {3148, 44127}, {3455, 50433}, {3815, 11007}, {4226, 35906}, {5649, 46787}, {6103, 60502}, {7736, 35922}, {7820, 62431}, {9171, 55143}, {9513, 43718}, {10311, 36176}, {14995, 18487}, {22240, 37918}, {23976, 35067}, {33927, 61198}, {37183, 56980}, {39469, 47405}, {41273, 44533}, {42671, 61213}, {46807, 63028}, {47412, 47433}, {48451, 51262}, {53346, 60517}, {61067, 65905}, {61194, 61679}, {65906, 65918}, {65908, 65923}

X(66354) = reflection of X(66192) in X(2)
X(66354) = isogonal conjugate of the polar conjugate of X(54380)
X(66354) = tripolar centroid of X(i) for these i: {110, 36885}
X(66354) = X(i)-Ceva conjugate of X(j) for these (i,j): {5649, 41167}, {40083, 237}, {46786, 542}, {46787, 511}, {53695, 512}, {54439, 47079}, {61446, 5191}
X(66354) = X(i)-isoconjugate of X(j) for these (i,j): {842, 1821}, {1577, 53691}, {1910, 5641}, {2349, 53866}, {14223, 36084}, {14998, 36036}, {36120, 65308}
X(66354) = X(i)-Dao conjugate of X(j) for these (i,j): {511, 46787}, {542, 46786}, {2679, 14998}, {8623, 57452}, {11672, 5641}, {23967, 290}, {38987, 14223}, {40601, 842}, {41167, 65727}, {41172, 34765}, {42426, 16081}, {46094, 65308}, {65728, 43665}, {65730, 57799}
X(66354) = crosspoint of X(i) and X(j) for these (i,j): {248, 65736}, {511, 46787}, {542, 46786}, {5649, 57742}
X(66354) = crosssum of X(i) and X(j) for these (i,j): {6, 7418}, {98, 34369}, {297, 41253}, {842, 52199}, {868, 1640}, {5641, 57452}
X(66354) = crossdifference of every pair of points on line {98, 523}
X(66354) = barycentric product X(i)*X(j) for these {i,j}: {3, 54380}, {74, 57431}, {114, 61446}, {232, 65722}, {325, 5191}, {511, 542}, {523, 42743}, {684, 7473}, {1640, 2421}, {1959, 2247}, {2396, 6041}, {3289, 60502}, {3569, 14999}, {4230, 65723}, {5968, 45662}, {6103, 36212}, {9155, 16092}, {11672, 46786}, {14966, 18312}, {16188, 40083}, {23967, 46787}, {32112, 64607}, {33752, 53232}, {34369, 36790}, {34761, 41167}, {45321, 63741}, {48451, 51389}, {51262, 65754}, {52491, 65748}, {52492, 65750}
X(66354) = barycentric quotient X(i)/X(j) for these {i,j}: {237, 842}, {511, 5641}, {542, 290}, {1495, 53866}, {1576, 53691}, {1640, 43665}, {2247, 1821}, {2421, 6035}, {2491, 14998}, {3289, 65308}, {3569, 14223}, {5191, 98}, {6041, 2395}, {6103, 16081}, {7473, 22456}, {9155, 52094}, {9419, 52199}, {11672, 46787}, {14966, 5649}, {14999, 43187}, {23967, 46786}, {34369, 34536}, {36213, 57452}, {36885, 53196}, {39469, 35909}, {41167, 34765}, {41172, 65727}, {42743, 99}, {45321, 63746}, {45662, 52145}, {46786, 57541}, {46787, 57547}, {51335, 34174}, {54380, 264}, {57431, 3260}, {58262, 23350}, {60502, 60199}, {61446, 40428}, {65722, 57799}
X(66354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 2420, 187}, {6, 3003, 21906}, {6, 5467, 3284}, {39, 187, 2088}, {9475, 47406, 11672}, {14966, 34349, 41172}, {41196, 41197, 41172}

X(66355) = TRIPOLAR CENTROID OF X(675)

Barycentrics    (b - c)*(-2*a^4 + 2*a^3*b - a*b^3 + b^4 + 2*a^3*c - 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4) : :
X(66355) = 2 X[6] + X[21133], X[6] + 2 X[21202], X[21133] - 4 X[21202]

X(66355) lies on the cubic K1382 and these lines: {6, 514}, {11, 244}, {37, 905}, {522, 17301}, {3672, 63251}, {4000, 60479}, {4025, 4363}, {4850, 27486}, {7178, 47935}, {7277, 23730}, {17354, 25259}, {17366, 42462}, {17369, 53583}, {17395, 23757}, {17720, 47787}, {21188, 23810}, {29212, 50313}, {31139, 44551}, {35093, 61066}, {40138, 53150}

X(66355) = tripolar centroid of X(675)
X(66355) = X(i)-isoconjugate of X(j) for these (i,j): {100, 38884}, {692, 57893}
X(66355) = X(i)-Dao conjugate of X(j) for these (i,j): {1086, 57893}, {8054, 38884}
X(66355) = crosssum of X(101) and X(52986)
X(66355) = crossdifference of every pair of points on line {101, 674}
X(66355) = barycentric product X(i)*X(j) for these {i,j}: {514, 544}, {23989, 52986}
X(66355) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 57893}, {544, 190}, {649, 38884}, {52986, 1252}
X(66355) = {X(6),X(21202)}-harmonic conjugate of X(21133)

X(66356) = TRIPOLAR CENTROID OF X(842)

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

X(66356) lies on the cubic K1382 and these lines: {6, 526}, {2088, 16186}, {2420, 52603}, {2436, 34210}, {5890, 19902}, {40112, 45681}, {40138, 53158}

X(66356) = tripolar centroid of X(842)
X(66356) = crossdifference of every pair of points on line {476, 542}

X(66357) = TRIPOLAR CENTROID OF X(1113)

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

X(66357) lies on the cubic K1382 and these lines: {6, 1345}, {647, 15167}, {1636, 1637}, {2593, 43530}, {8115, 65308}, {22340, 60872}, {32663, 57026}, {40138, 53154}

X(66357) = tripolar centroid of X(1113)
X(66357) = X(11064)-Ceva conjugate of X(14500)
X(66357) = X(i)-isoconjugate of X(j) for these (i,j): {74, 2581}, {1114, 2349}, {1304, 2582}, {1494, 2577}, {1823, 16080}, {2159, 15165}, {2574, 65263}, {2578, 16077}, {2584, 15459}, {2587, 14919}, {2588, 44769}, {2592, 36034}, {8116, 36119}, {22339, 36131}, {33805, 44124}, {35200, 46812}
X(66357) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 46812}, {1312, 16080}, {1511, 8116}, {3163, 15165}, {3258, 2592}, {15167, 1494}, {38999, 46814}, {39008, 22339}, {62569, 46810}
X(66357) = crosssum of X(44068) and X(57025)
X(66357) = crossdifference of every pair of points on line {74, 1114}
X(66357) = barycentric product X(i)*X(j) for these {i,j}: {30, 2575}, {1113, 9033}, {1114, 14500}, {1495, 22340}, {1636, 46815}, {1637, 8115}, {1784, 2585}, {1822, 36035}, {1990, 46811}, {2173, 2583}, {2579, 14206}, {2580, 2631}, {2593, 3284}, {3260, 42667}, {8106, 11064}, {9409, 15164}, {14398, 46813}, {41079, 57026}, {44123, 66073}
X(66357) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 15165}, {1113, 16077}, {1495, 1114}, {1636, 46814}, {1637, 2592}, {1990, 46812}, {2173, 2581}, {2420, 39299}, {2575, 1494}, {2576, 65263}, {2579, 2349}, {2583, 33805}, {2631, 2582}, {3284, 8116}, {8106, 16080}, {9033, 22339}, {9406, 2577}, {9407, 44124}, {9409, 2574}, {11064, 46810}, {14398, 8105}, {14500, 22340}, {42667, 74}, {44123, 1304}, {57026, 44769}

X(66358) = TRIPOLAR CENTROID OF X(1114)

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

X(66358) lies on the cubic K1382 and these lines: {6, 1344}, {647, 15166}, {1636, 1637}, {2592, 43530}, {8116, 65308}, {22339, 60872}, {32663, 57025}, {40138, 53153}

X(66358) = tripolar centroid of X(1114)
X(66358) = X(11064)-Ceva conjugate of X(14499)
X(66358) = X(i)-isoconjugate of X(j) for these (i,j): {74, 2580}, {1113, 2349}, {1304, 2583}, {1494, 2576}, {1822, 16080}, {2159, 15164}, {2575, 65263}, {2579, 16077}, {2585, 15459}, {2586, 14919}, {2589, 44769}, {2593, 36034}, {8115, 36119}, {22340, 36131}, {33805, 44123}, {35200, 46815}
X(66358) = X(i)-Dao conjugate of X(j) for these (i,j): {133, 46815}, {1313, 16080}, {1511, 8115}, {3163, 15164}, {3258, 2593}, {15166, 1494}, {38999, 46811}, {39008, 22340}, {62569, 46813}
X(66358) = crosssum of X(44067) and X(57026)
X(66358) = crossdifference of every pair of points on line {74, 1113}
X(66358) = barycentric product X(i)*X(j) for these {i,j}: {30, 2574}, {1113, 14499}, {1114, 9033}, {1495, 22339}, {1636, 46812}, {1637, 8116}, {1784, 2584}, {1823, 36035}, {1990, 46814}, {2173, 2582}, {2578, 14206}, {2581, 2631}, {2592, 3284}, {3260, 42668}, {8105, 11064}, {9409, 15165}, {14398, 46810}, {41079, 57025}, {44124, 66073}
X(66358) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 15164}, {1114, 16077}, {1495, 1113}, {1636, 46811}, {1637, 2593}, {1990, 46815}, {2173, 2580}, {2420, 39298}, {2574, 1494}, {2577, 65263}, {2578, 2349}, {2582, 33805}, {2631, 2583}, {3284, 8115}, {8105, 16080}, {9033, 22340}, {9406, 2576}, {9407, 44123}, {9409, 2575}, {11064, 46813}, {14398, 8106}, {14499, 22339}, {42668, 74}, {44124, 1304}, {57025, 44769}

X(66359) = TRIPOLAR CENTROID OF X(1297)

Barycentrics    a^2*(b^2 - c^2)*(a^2 - b^2 - c^2)^2*(2*a^12 - 2*a^10*b^2 - a^8*b^4 - 4*a^6*b^6 + 8*a^4*b^8 - 2*a^2*b^10 - b^12 - 2*a^10*c^2 + 4*a^8*b^2*c^2 + 4*a^6*b^4*c^2 - 8*a^4*b^6*c^2 - 2*a^2*b^8*c^2 + 4*b^10*c^2 - a^8*c^4 + 4*a^6*b^2*c^4 + 4*a^2*b^6*c^4 - 7*b^8*c^4 - 4*a^6*c^6 - 8*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 8*b^6*c^6 + 8*a^4*c^8 - 2*a^2*b^2*c^8 - 7*b^4*c^8 - 2*a^2*c^10 + 4*b^2*c^10 - c^12) : :

X(66359) lies on these lines: {3, 58796}, {6, 520}, {184, 2430}, {217, 32320}, {1073, 52584}, {1636, 2972}, {6587, 41369}, {7729, 9242}, {39473, 41145}, {40138, 43701}

X(66359) = tripolar centroid of X(1297)
X(66359) = X(823)-isoconjugate of X(53914)
X(66359) = crossdifference of every pair of points on line {107, 1503}
X(66359) = barycentric product X(520)*X(9530)
X(66359) = barycentric quotient X(i)/X(j) for these {i,j}: {9530, 6528}, {39201, 53914}

X(66360) = TRIPOLAR CENTROID OF X(1302)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^10 - 11*a^6*b^4 + 13*a^4*b^6 - 3*a^2*b^8 - b^10 + 16*a^6*b^2*c^2 - 11*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 - 11*a^6*c^4 - 11*a^4*b^2*c^4 + 22*a^2*b^4*c^4 - 2*b^6*c^4 + 13*a^4*c^6 - 8*a^2*b^2*c^6 - 2*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 - c^10) : :

X(66360) lies on these lines: {6, 30}, {184, 3081}, {577, 16190}, {1636, 1637}, {1640, 55141}, {1650, 5158}, {1990, 11251}, {3284, 12113}, {3580, 58875}, {4240, 40138}, {6749, 18507}, {8749, 9140}, {18554, 56399}, {20126, 51544}, {23967, 65911}, {40135, 56395}, {40385, 46233}, {45331, 45681}

X(66360) = tripolar centroid of X(1302)
X(66360) = X(i)-isoconjugate of X(j) for these (i,j): {841, 2349}, {2159, 57892}
X(66360) = X(i)-Dao conjugate of X(j) for these (i,j): {3163, 57892}, {53984, 16080}
X(66360) = crosssum of X(74) and X(52976)
X(66360) = crossdifference of every pair of points on line {74, 841}
X(66360) = barycentric product X(i)*X(j) for these {i,j}: {30, 541}, {36789, 52976}
X(66360) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 57892}, {541, 1494}, {1495, 841}, {52976, 40384}

X(66361) = TRIPOLAR CENTROID OF X(1311)

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

X(66361) lies on the cubic K1382 and these lines: {6, 522}, {11, 1146}, {45, 59998}, {905, 3752}, {1212, 57055}, {8058, 17281}, {40138, 53152}

X(66361) = tripolar centroid of X(1311)
X(66361) = crossdifference of every pair of points on line {109, 8679}

X(66362) = TRIPOLAR CENTROID OF X(2373)

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

X(66362) lies on the cubic K1382 and these lines: {6, 525}, {122, 125}, {216, 52613}, {520, 40673}, {523, 23327}, {599, 8057}, {2435, 5486}, {40138, 43673}, {41614, 57069}

X(66362) = tripolar centroid of X(2373)
X(66362) = crossdifference of every pair of points on line {112, 2393}

X(66363) = TRIPOLAR CENTROID OF X(9058)

Barycentrics    a*(a^2*b - b^3 + a^2*c - 2*a*b*c + b^2*c + b*c^2 - c^3)*(2*a^7 - 2*a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + a*b^6 - b^7 - 2*a^6*c + 12*a^5*b*c - 7*a^4*b^2*c - 6*a^3*b^3*c + 8*a^2*b^4*c - 6*a*b^5*c + b^6*c - 3*a^5*c^2 - 7*a^4*b*c^2 + 16*a^3*b^2*c^2 - 8*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 6*a^3*b*c^3 - 8*a^2*b^2*c^3 + 12*a*b^3*c^3 - 3*b^4*c^3 + 8*a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 - 6*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7) : :

X(66363) lies on the cubic K1382 and these lines: {6, 517}, {45, 34345}, {1643, 35013}, {1769, 3310}, {5158, 35014}, {19297, 34346}, {40138, 53151}, {61066, 65926}

X(66363) = tripolar centroid of X(9058)
X(66363) = crossdifference of every pair of points on line {104, 9001}

X(66364) = TRIPOLAR CENTROID OF X(9059)

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

X(66364) lies on the cubic K1382 and these lines: {6, 519}, {45, 62630}, {900, 1635}, {1647, 17369}, {4363, 62621}, {17354, 62620}, {17780, 54389}

X(66364) = tripolar centroid of X(9059)
X(66364) = crossdifference of every pair of points on line {106, 9002}

X(66365) = X(356)-CEVA CONJUGATE OF X(357)

Barycentrics    Cos[B/3]*Cos[C/3]*(Cos[A/3]^2*Cos[B/3]^2 + 4*Cos[A/3]^3*Cos[B/3]*Cos[C/3] + Cos[A/3]^2*Cos[C/3]^2 - Cos[B/3]^2*Cos[C/3]^2 + 4*Cos[A/3]^2*Cos[B/3]^2*Cos[C/3]^2)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7254.

X(66365) lies on the cubic K029 and this line: {357, 3605}

X(66365) = X(356)-Ceva conjugate of X(357)

X(66366) = X(3276)-CEVA CONJUGATE OF X(1136)

Barycentrics    Cos[B/3-2*Pi/3]*Cos[C/3-2*Pi/3]*(Cos[A/3-2*Pi/3]^2*Cos[B/3-2*Pi/3]^2 + 4*Cos[A/3-2*Pi/3]^3*Cos[B/3-2*Pi/3]*Cos[C/3-2*Pi/3] + Cos[A/3-2*Pi/3]^2*Cos[C/3-2*Pi/3]^2 - Cos[B/3-2*Pi/3]^2*Cos[C/3-2*Pi/3]^2 + 4*Cos[A/3-2*Pi/3]^2*Cos[B/3-2*Pi/3]^2*Cos[C/3-2*Pi/3]^2)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7254.

X(66366) lies on the cubic K031 and this line: {1136, 3606}

X(66366) = X(3276)-Ceva conjugate of X(1136)

X(66367) = X(3277)-CEVA CONJUGATE OF X(1134)

Barycentrics    Cos[B/3+2*Pi/3]*Cos[C/3+2*Pi/3]*(Cos[A/3+2*Pi/3]^2*Cos[B/3+2*Pi/3]^2 + 4*Cos[A/3+2*Pi/3]^3*Cos[B/3+2*Pi/3]*Cos[C/3+2*Pi/3] + Cos[A/3+2*Pi/3]^2*Cos[C/3+2*Pi/3]^2 - Cos[B/3+2*Pi/3]^2*Cos[C/3+2*Pi/3]^2 + 4*Cos[A/3+2*Pi/3]^2*Cos[B/3+2*Pi/3]^2*Cos[C/3+2*Pi/3]^2)*Sin[A] : :

See Antreas Hatzipolakis and Peter Moses, euclid 7254.

X(66367) lies on the cubic K030 and this line: {1134, 3607}

X(66367) = X(3277)-Ceva conjugate of X(1134)

X(66368) = EULER LINE INTERCEPT OF X(69)X(5648)

Barycentrics   7*a^6 - a^4*b^2 - 7*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 7*a^2*c^4 - b^2*c^4 + c^6 : :
X(66368) = X[2] + 2 X[22], 7 X[2] - 4 X[427], 5 X[2] - 8 X[6676], 4 X[2] - X[7391], 5 X[2] + X[20062], 5 X[2] - 2 X[31133], 13 X[2] - 10 X[31236], X[2] - 4 X[44210], 19 X[2] - 16 X[64852], X[4] + 8 X[7555], X[4] - 4 X[44262], X[20] + 8 X[16618], X[20] - 4 X[44261], 7 X[22] + 2 X[427], 5 X[22] + 4 X[6676], 8 X[22] + X[7391], and many others

X(66368) lies on these lines: {2, 3}, {69, 5648}, {110, 54173}, {111, 21843}, {323, 50967}, {353, 63043}, {524, 6800}, {542, 35268}, {1350, 40112}, {1383, 7736}, {1495, 50977}, {1992, 11003}, {2781, 33884}, {3098, 5642}, {3241, 51692}, {3448, 64014}, {3580, 43273}, {3618, 48912}, {4549, 10706}, {5012, 44490}, {5210, 5913}, {5304, 14836}, {5640, 38064}, {5651, 32267}, {5987, 64090}, {6032, 43618}, {6776, 44555}, {7664, 14907}, {7693, 63119}, {8588, 10418}, {9019, 11002}, {9140, 46264}, {9544, 33522}, {9870, 37667}, {10168, 34417}, {10519, 35265}, {11057, 37804}, {11064, 50965}, {11160, 16789}, {11179, 15080}, {11645, 61644}, {12117, 62298}, {12824, 54334}, {14389, 54131}, {15066, 35266}, {15107, 20423}, {15448, 21766}, {16962, 54363}, {16963, 54362}, {18361, 52898}, {18911, 32225}, {20192, 50983}, {26233, 32833}, {26276, 56435}, {32269, 51737}, {36427, 52058}, {37645, 54170}, {37779, 50974}, {41896, 57822}, {44569, 44882}, {44822, 45317}, {45311, 48892}, {45331, 59227}, {45794, 64802}, {46818, 50955}, {47296, 50971}, {47582, 50979}, {51028, 63082}, {54174, 64058}, {54674, 60255}

X(66368) = midpoint of X(22) and X(47596)
X(66368) = reflection of X(i) in X(j) for these {i,j}: {2, 47596}, {7391, 31105}, {10304, 44837}, {31105, 2}, {47596, 44210}
X(66368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 10989}, {2, 376, 16063}, {2, 3543, 5169}, {2, 7492, 376}, {2, 7519, 381}, {2, 7533, 5071}, {2, 15677, 31106}, {2, 15683, 31099}, {2, 15692, 7496}, {2, 20062, 31133}, {2, 33263, 31107}, {2, 37901, 4}, {2, 37909, 26255}, {2, 61936, 7570}, {3, 7426, 2}, {22, 1995, 12083}, {22, 6676, 20062}, {22, 7493, 44440}, {22, 7495, 44831}, {22, 7502, 7492}, {22, 16387, 20}, {22, 25337, 7519}, {22, 44210, 2}, {22, 44260, 37913}, {22, 44262, 37901}, {376, 7493, 2}, {381, 7495, 2}, {381, 47313, 7519}, {549, 1995, 2}, {549, 37904, 1995}, {3524, 26255, 2}, {4232, 15692, 2}, {5004, 5005, 31861}, {5094, 15681, 47314}, {6676, 31133, 2}, {7394, 16063, 18531}, {7426, 47313, 16619}, {7485, 44212, 2}, {7492, 7493, 16063}, {7492, 10298, 6636}, {7494, 37913, 7394}, {7495, 47313, 381}, {7555, 37969, 22}, {9832, 45662, 2}, {10201, 34006, 376}, {10298, 10565, 7493}, {10989, 52300, 2}, {11179, 15360, 37644}, {15080, 15360, 11179}, {16063, 44440, 7391}, {26257, 33246, 2}, {30775, 62130, 1370}, {35266, 54169, 15066}, {37900, 53843, 3830}, {37969, 44210, 44262}, {44210, 44261, 16387}, {56966, 62344, 3543}

X(66369) = EULER LINE INTERCEPT OF X(69)X(12367)

Barycentrics    5*a^6 + a^4*b^2 - 5*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 5*a^2*c^4 + b^2*c^4 - c^6 : :
X(66369) = 5 X[2] - 4 X[427], 7 X[2] - 8 X[6676], 4 X[2] - 3 X[31105], 11 X[2] - 10 X[31236], 3 X[2] - 4 X[44210], 5 X[2] - 6 X[47596], 17 X[2] - 16 X[64852], X[20] + 2 X[12082], 5 X[22] - 2 X[427], 7 X[22] - 4 X[6676], 4 X[22] - X[7391], 2 X[22] + X[20062], 8 X[22] - 3 X[31105], 3 X[22] - X[31133], 11 X[22] - 5 X[31236], 3 X[22] - 2 X[44210], 5 X[22] - 3 X[47596], and many others

X(66369) lies on these lines: {2, 3}, {69, 12367}, {98, 54782}, {110, 48873}, {125, 48896}, {146, 41465}, {154, 40112}, {184, 19924}, {251, 7739}, {542, 45794}, {543, 5986}, {1495, 48880}, {1627, 19220}, {1899, 15360}, {1992, 9019}, {1994, 54132}, {2781, 9143}, {3060, 11179}, {3163, 36414}, {3410, 33522}, {3448, 14927}, {3580, 48905}, {3796, 54131}, {5012, 20423}, {5370, 65134}, {5422, 51737}, {5476, 22352}, {5651, 48885}, {5987, 13172}, {6515, 64014}, {6800, 29181}, {7302, 10483}, {7712, 37645}, {7811, 16276}, {8193, 34668}, {8267, 63093}, {8878, 19569}, {10385, 29815}, {11002, 25406}, {11003, 51212}, {11442, 11645}, {13337, 63024}, {13338, 63006}, {14389, 48910}, {14683, 63428}, {14912, 16981}, {15066, 48881}, {15073, 54384}, {15080, 31670}, {15107, 37644}, {16165, 37669}, {18353, 62992}, {18911, 48898}, {19127, 59373}, {20099, 34106}, {22112, 33751}, {24981, 55587}, {29317, 35268}, {29323, 61644}, {31166, 41715}, {31383, 54173}, {32237, 48920}, {33534, 50434}, {33586, 43273}, {33878, 46818}, {34417, 48892}, {36967, 54363}, {36968, 54362}, {37636, 47353}, {37775, 42091}, {37776, 42090}, {37779, 39874}, {38314, 51692}, {48870, 54341}, {48879, 51360}, {48912, 63084}

X(66369) = midpoint of X(i) and X(j) for these {i,j}: {2, 20062}, {3534, 44457}
X(66369) = reflection of X(i) in X(j) for these {i,j}: {2, 22}, {378, 44261}, {7391, 2}, {15640, 35480}, {31133, 44210}, {31723, 44262}, {35481, 3534}, {44287, 7555}
X(66369) = anticomplement of X(31133)
X(66369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3839, 37353}, {2, 7391, 31105}, {2, 7500, 62963}, {2, 10304, 15246}, {2, 33187, 16949}, {2, 33278, 63797}, {2, 35929, 33255}, {2, 37349, 3545}, {2, 37909, 6353}, {2, 62032, 7409}, {2, 62963, 7394}, {3, 7519, 62937}, {3, 37900, 7519}, {20, 23, 16063}, {22, 20062, 7391}, {22, 31133, 44210}, {378, 44261, 10304}, {427, 47596, 2}, {548, 10301, 40916}, {550, 37899, 1995}, {1995, 43957, 2}, {3524, 6997, 2}, {3529, 7493, 5189}, {3534, 47313, 2}, {5004, 5005, 7530}, {5054, 37990, 2}, {6636, 7500, 7394}, {6636, 62963, 2}, {6995, 10304, 2}, {7386, 26255, 2}, {7426, 31152, 2}, {7426, 52397, 31152}, {7492, 20063, 4}, {7500, 59343, 6636}, {7667, 47312, 44212}, {9909, 15681, 31152}, {9909, 31152, 7426}, {9909, 52397, 2}, {11414, 34726, 38323}, {12083, 44831, 44440}, {12103, 37910, 30739}, {15080, 31670, 63036}, {15107, 46264, 37644}, {15158, 15159, 10989}, {15681, 31152, 52397}, {15686, 44212, 7667}, {31101, 37907, 2}, {31133, 44210, 2}, {31723, 44262, 3545}, {34726, 38323, 31304}, {44218, 44837, 15692}, {52399, 52400, 7527}

X(66370) = EULER LINE INTERCEPT OF X(154)X(54173)

Barycentrics    10*a^6 - a^4*b^2 - 10*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 10*a^2*c^4 - b^2*c^4 + c^6 : :
X(66370) = X[2] + 3 X[22], 5 X[2] - 3 X[427], 2 X[2] - 3 X[6676], 11 X[2] - 3 X[7391], 13 X[2] + 3 X[20062], 17 X[2] - 9 X[31105], 7 X[2] - 3 X[31133], 19 X[2] - 15 X[31236], X[2] - 3 X[44210], 5 X[2] - 9 X[47596], 7 X[2] - 6 X[64852], 5 X[22] + X[427], 2 X[22] + X[6676], 11 X[22] + X[7391], 13 X[22] - X[20062], 17 X[22] + 3 X[31105], 7 X[22] + X[31133], and many others

X(66370) lies on these lines: {2, 3}, {154, 54173}, {3058, 7298}, {3167, 50967}, {3564, 35268}, {3796, 50979}, {3819, 32267}, {3917, 35266}, {5310, 15170}, {5345, 5434}, {5370, 18990}, {5943, 50983}, {6030, 11245}, {6800, 34380}, {7302, 15171}, {8584, 19127}, {8854, 52045}, {8855, 52046}, {9019, 21849}, {9306, 54169}, {10117, 61610}, {10192, 50965}, {11003, 61624}, {11179, 41588}, {11206, 50955}, {14677, 32227}, {14810, 15448}, {15080, 47582}, {15533, 16789}, {16165, 64062}, {17810, 38064}, {19924, 23292}, {20192, 43650}, {33651, 59634}, {35260, 55610}, {42912, 54363}, {42913, 54362}, {45298, 51737}, {47296, 48892}, {50992, 61771}, {51071, 51692}, {51108, 51718}, {55584, 64058}, {55593, 64177}, {55649, 61507}, {61345, 65006}

X(66370) = midpoint of X(i) and X(j) for these {i,j}: {22, 44210}, {12083, 44218}
X(66370) = reflection of X(i) in X(j) for these {i,j}: {6676, 44210}, {31133, 64852}, {64474, 549}
X(66370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 428, 5066}, {2, 8703, 10691}, {2, 11001, 34609}, {2, 15640, 62975}, {2, 34608, 3830}, {2, 47313, 428}, {2, 52397, 47311}, {427, 44210, 47596}, {468, 548, 10300}, {468, 7492, 548}, {550, 7493, 5159}, {3530, 47630, 1995}, {6636, 7426, 43957}, {6636, 43957, 34200}, {6677, 34200, 43957}, {7426, 43957, 6677}, {7495, 37899, 546}, {8703, 10154, 2}, {8703, 18579, 12100}, {10128, 11812, 2}, {15698, 62979, 2}, {33923, 47316, 30739}, {37454, 37900, 3853}, {37911, 44245, 16063}

X(66371) = EULER LINE INTERCEPT OF X(51)X(51737)

Barycentrics    8*a^6 + a^4*b^2 - 8*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 8*a^2*c^4 + b^2*c^4 - c^6 : :
X(66371) = X[2] - 3 X[22], 4 X[2] - 3 X[427], 5 X[2] - 6 X[6676], 7 X[2] - 3 X[7391], 5 X[2] + 3 X[20062], 13 X[2] - 9 X[31105], 5 X[2] - 3 X[31133], 17 X[2] - 15 X[31236], 2 X[2] - 3 X[44210], 7 X[2] - 9 X[47596], 13 X[2] - 12 X[64852], 4 X[22] - X[427], 5 X[22] - 2 X[6676], 7 X[22] - X[7391], 5 X[22] + X[20062], 13 X[22] - 3 X[31105], 5 X[22] - X[31133], 17 X[22] - 5 X[31236], and many others

X(66371) lies on these lines: {2, 3}, {51, 51737}, {110, 48874}, {343, 11645}, {597, 22352}, {599, 31383}, {1495, 48881}, {1829, 34642}, {3058, 5322}, {3060, 50979}, {3796, 20423}, {3917, 50965}, {5310, 5434}, {5370, 6284}, {5972, 48920}, {7302, 7354}, {8584, 9019}, {9798, 34656}, {10192, 13857}, {11057, 45201}, {11064, 48880}, {11179, 33586}, {11180, 33522}, {11206, 50967}, {11245, 43273}, {11402, 54132}, {12135, 34712}, {12410, 34667}, {13394, 29317}, {14836, 33872}, {15080, 21850}, {15107, 48906}, {16276, 37671}, {16789, 22165}, {19127, 63124}, {25406, 61657}, {26276, 59766}, {26881, 40112}, {29181, 35268}, {29323, 45303}, {32223, 48891}, {32237, 48885}, {32269, 48898}, {34634, 49553}, {35283, 55649}, {37648, 48892}, {41586, 64196}, {42942, 54363}, {42943, 54362}, {43653, 47353}, {46128, 58347}, {46264, 47582}, {47298, 65633}, {51103, 51692}, {51110, 51718}, {54013, 55629}, {54174, 63174}, {59411, 61506}

X(66371) = midpoint of X(i) and X(j) for these {i,j}: {376, 12082}, {20062, 31133}
X(66371) = reflection of X(i) in X(j) for these {i,j}: {381, 16618}, {427, 44210}, {549, 7555}, {31133, 6676}, {44210, 22}, {44218, 7502}, {44285, 44261}, {61690, 35268}
X(66371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 7667}, {2, 9909, 37904}, {2, 15682, 5064}, {2, 34603, 3845}, {2, 37904, 62978}, {2, 62160, 44442}, {3, 37899, 10301}, {22, 6636, 7555}, {22, 7500, 16618}, {22, 12082, 25}, {22, 20062, 6676}, {23, 550, 30739}, {25, 376, 43957}, {428, 21213, 62978}, {548, 37910, 1995}, {1657, 7493, 46517}, {3529, 5094, 47095}, {3534, 9909, 2}, {3534, 44265, 8703}, {7492, 37900, 5}, {7493, 46517, 52293}, {7495, 20063, 3627}, {7667, 9909, 62978}, {7667, 37904, 2}, {7714, 19708, 2}, {8703, 44265, 44268}, {12103, 37897, 16063}, {43957, 47312, 25}, {44265, 47313, 37904}, {47630, 62123, 10300}

X(66372) = EULER LINE INTERCEPT OF X(323)X(54170)

Barycentrics    11*a^6 + a^4*b^2 - 11*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 11*a^2*c^4 + b^2*c^4 - c^6 : :
X(66372) = X[2] - 4 X[22], 11 X[2] - 8 X[427], 13 X[2] - 16 X[6676], 5 X[2] - 2 X[7391], 2 X[2] + X[20062], 7 X[2] - 4 X[31133], 23 X[2] - 20 X[31236], 5 X[2] - 8 X[44210], 3 X[2] - 4 X[47596], 35 X[2] - 32 X[64852], 11 X[22] - 2 X[427], 13 X[22] - 4 X[6676], 10 X[22] - X[7391], 8 X[22] + X[20062], 6 X[22] - X[31105], 7 X[22] - X[31133], 23 X[22] - 5 X[31236], and many others

X(66372) lies on these lines: {2, 3}, {323, 54170}, {1992, 8547}, {2781, 64059}, {5032, 9019}, {5987, 8591}, {6031, 32815}, {9143, 50967}, {11003, 54132}, {11004, 51028}, {11160, 14683}, {11179, 15107}, {13857, 48880}, {14389, 51024}, {14836, 63005}, {15066, 50965}, {15080, 20423}, {15360, 46264}, {19127, 63127}, {19924, 35268}, {20099, 37667}, {32225, 48898}, {32267, 48885}, {35266, 48881}, {36415, 36427}, {37644, 43273}, {37648, 50971}, {41424, 50968}, {44300, 55675}, {44555, 64014}, {54131, 63036}, {55656, 59776}

X(66372) = reflection of X(31105) in X(47596)
X(66372) = anticomplement of X(31105)
X(66372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15683, 5189}, {2, 20063, 3543}, {2, 37901, 7519}, {22, 12083, 23}, {23, 376, 2}, {23, 6636, 6644}, {549, 62937, 2}, {3534, 7426, 16063}, {3534, 47335, 376}, {7391, 44210, 2}, {7426, 16063, 2}, {7492, 37901, 2}, {7493, 10989, 2}, {7493, 11001, 10989}, {8703, 47312, 1995}, {9909, 15689, 47597}, {20063, 44831, 20062}, {31105, 47596, 2}, {33532, 44265, 376}, {35481, 37969, 4232}, {36445, 36463, 7514}, {46860, 46861, 20}, {52301, 62063, 2}

X(66373) = EULER LINE INTERCEPT OF X(524)X(35268)

Barycentrics    16*a^6 - a^4*b^2 - 16*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 16*a^2*c^4 - b^2*c^4 + c^6 : :
X(66373) = X[2] + 5 X[22], 8 X[2] - 5 X[427], 7 X[2] - 10 X[6676], 17 X[2] - 5 X[7391], 19 X[2] + 5 X[20062], 9 X[2] - 5 X[31105], 11 X[2] - 5 X[31133], 31 X[2] - 25 X[31236], 2 X[2] - 5 X[44210], 3 X[2] - 5 X[47596], 23 X[2] - 20 X[64852], 8 X[22] + X[427], 7 X[22] + 2 X[6676], 17 X[22] + X[7391], 19 X[22] - X[20062], 9 X[22] + X[31105], 11 X[22] + X[31133], and many others

X(66373) lies on these lines: {2, 3}, {524, 35268}, {1495, 54169}, {3058, 7302}, {3098, 35266}, {5092, 20192}, {5210, 16317}, {5370, 5434}, {5642, 50965}, {9143, 50978}, {11179, 47582}, {13394, 19924}, {13857, 48881}, {14810, 32267}, {15018, 50987}, {15080, 50979}, {15360, 48906}, {16789, 24981}, {19127, 20583}, {26864, 50967}, {32225, 44882}, {34417, 50983}, {36427, 45141}, {37643, 50975}, {44109, 51132}, {50968, 59767}

X(66373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15681, 46517}, {2, 37900, 15687}, {2, 47312, 10301}, {2, 62042, 62977}, {3534, 7493, 47097}, {7426, 7492, 8703}, {7426, 8703, 30739}, {7495, 37901, 3845}, {10304, 47597, 43957}, {34200, 37897, 2}

X(66374) = EULER LINE INTERCEPT OF X(3058)X(5345)

Barycentrics    14*a^6 + a^4*b^2 - 14*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 14*a^2*c^4 + b^2*c^4 - c^6 : :
X(66374) = X[2] - 5 X[22], 7 X[2] - 5 X[427], 4 X[2] - 5 X[6676], 13 X[2] - 5 X[7391], 11 X[2] + 5 X[20062], 23 X[2] - 15 X[31105], 9 X[2] - 5 X[31133], 29 X[2] - 25 X[31236], 3 X[2] - 5 X[44210], 11 X[2] - 15 X[47596], 11 X[2] - 10 X[64852], 7 X[22] - X[427], 4 X[22] - X[6676], 13 X[22] - X[7391], 11 X[22] + X[20062], 23 X[22] - 3 X[31105], 9 X[22] - X[31133], and many others

X(66374) lies on these lines: {2, 3}, {3058, 5345}, {3167, 54170}, {5322, 15170}, {5370, 15171}, {5434, 7298}, {7302, 18990}, {9019, 20583}, {9306, 50965}, {9591, 34634}, {15074, 21969}, {15448, 48885}, {32267, 53415}, {33522, 50955}, {33586, 50979}, {41588, 43273}, {47296, 48891}, {55593, 64059}

X(66374) = midpoint of X(i) and X(j) for these {i,j}: {12082, 44285}, {12083, 44261}
X(66374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 62976, 38071}, {22, 12083, 44260}, {376, 9909, 44212}, {376, 44212, 10691}, {550, 37897, 10300}, {7492, 37899, 140}, {7493, 15704, 47315}, {7499, 62963, 5066}, {10154, 15686, 31152}, {15688, 20850, 2}, {20063, 37454, 62026}, {44245, 47630, 30739}, {47316, 62123, 16063}

X(66375) = EULER LINE INTERCEPT OF X(3796)X(9140)

Barycentrics    7*a^6 - 4*a^4*b^2 - 7*a^2*b^4 + 4*b^6 - 4*a^4*c^2 - 4*b^4*c^2 - 7*a^2*c^4 - 4*b^2*c^4 + 4*c^6 : :
X(66375) = 4 X[2] + X[22], 7 X[2] - 2 X[427], X[2] + 4 X[6676], 11 X[2] - X[7391], 19 X[2] + X[20062], 13 X[2] - 3 X[31105], 6 X[2] - X[31133], 3 X[2] + 2 X[44210], 2 X[2] + 3 X[47596], 13 X[2] - 8 X[64852], 7 X[22] + 8 X[427], X[22] - 16 X[6676], 11 X[22] + 4 X[7391], 19 X[22] - 4 X[20062], 13 X[22] + 12 X[31105], 3 X[22] + 2 X[31133], X[22] + 2 X[31236], and many others

X(66375) lies on these lines: {2, 3}, {3796, 9140}, {5422, 32225}, {5965, 61644}, {9019, 11451}, {9544, 50955}, {10168, 61645}, {11178, 35264}, {11402, 44555}, {15059, 53094}, {16789, 59373}, {19127, 21358}, {19875, 51692}, {23293, 43273}, {25561, 44082}, {26881, 47353}, {40112, 43653}

X(66375) = reflection of X(31236) in X(2)
X(66375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6676, 47596}, {2, 13595, 5055}, {2, 15246, 32216}, {2, 26255, 37990}, {2, 31105, 64852}, {2, 37907, 5020}, {2, 37909, 37353}, {2, 44210, 31133}, {2, 47596, 22}, {1656, 37760, 1995}, {3545, 44261, 52842}, {5054, 37453, 2}, {5054, 44262, 378}, {5094, 15696, 60455}, {31133, 44210, 22}, {31133, 47596, 44210}

X(66376) = EULER LINE INTERCEPT OF X(6)X(6032)

Barycentrics    a^6 - 4*a^4*b^2 - a^2*b^4 + 4*b^6 - 4*a^4*c^2 - 4*b^4*c^2 - a^2*c^4 - 4*b^2*c^4 + 4*c^6 : :
X(66376) = 4 X[2] - X[22], X[2] + 2 X[427], 7 X[2] - 4 X[6676], 5 X[2] + X[7391], 13 X[2] - X[20062], 2 X[2] + X[31133], 2 X[2] - 5 X[31236], 5 X[2] - 2 X[44210], 5 X[2] - 8 X[64852], X[4] + 2 X[44218], X[22] + 8 X[427], 7 X[22] - 16 X[6676], 5 X[22] + 4 X[7391], 13 X[22] - 4 X[20062], X[22] + 4 X[31105], X[22] + 2 X[31133], X[22] - 10 X[31236], and many others

X(66376) lies on these lines: {2, 3}, {6, 6032}, {98, 54803}, {110, 47353}, {115, 9745}, {125, 5476}, {230, 41394}, {232, 36430}, {262, 15363}, {323, 50955}, {524, 45303}, {542, 11187}, {597, 18911}, {599, 41721}, {1007, 62299}, {1351, 44555}, {1352, 40112}, {1992, 51744}, {1993, 64802}, {2453, 34312}, {2781, 5640}, {3066, 15059}, {3241, 51718}, {3291, 18362}, {3580, 20423}, {3818, 5642}, {5306, 47298}, {5480, 44569}, {5651, 25561}, {5913, 43620}, {5968, 11184}, {5996, 53266}, {6054, 15928}, {6800, 11645}, {7693, 40920}, {7699, 10706}, {7736, 14836}, {7739, 15880}, {7811, 11056}, {7837, 19577}, {7840, 36207}, {7880, 30747}, {7998, 9019}, {8585, 39601}, {8724, 62298}, {9143, 18440}, {9166, 58046}, {9209, 39491}, {9759, 44420}, {10415, 51926}, {10418, 18424}, {10546, 16165}, {11064, 47354}, {11163, 14995}, {11174, 23297}, {11178, 13857}, {11179, 14389}, {11180, 37645}, {11477, 38397}, {14848, 26869}, {15107, 51024}, {15360, 37638}, {15431, 64058}, {15533, 23061}, {15534, 41724}, {19130, 45311}, {19924, 61644}, {20192, 47296}, {20481, 39602}, {30474, 65754}, {30718, 47284}, {30785, 47005}, {31125, 31859}, {31174, 33752}, {31176, 44823}, {32110, 51993}, {35908, 46808}, {37779, 50962}, {39490, 59969}, {39492, 59982}, {39493, 48182}, {41428, 64094}, {42972, 54362}, {42973, 54363}, {46818, 51023}, {46983, 66116}, {48310, 64730}, {50974, 63082}, {50977, 51360}, {50979, 63036}, {53136, 66119}, {54384, 58470}

X(66376) = midpoint of X(i) and X(j) for these {i,j}: {2, 31105}, {31133, 47596}
X(66376) = reflection of X(i) in X(j) for these {i,j}: {22, 47596}, {31105, 427}, {31133, 31105}, {44837, 5054}, {47596, 2}
X(66376) = orthocentroidal-circle-inverse of X(7426)
X(66376) = orthoptic-circle-of-Steiner-inellipse-inverse of X(44265)
X(66376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7426}, {2, 376, 7495}, {2, 381, 1995}, {2, 427, 31133}, {2, 3543, 7493}, {2, 3839, 26255}, {2, 5169, 381}, {2, 7391, 44210}, {2, 7394, 44212}, {2, 7496, 15694}, {2, 7570, 15703}, {2, 10989, 3}, {2, 16063, 549}, {2, 31074, 31152}, {2, 31099, 376}, {2, 31100, 44217}, {2, 31106, 15670}, {2, 31133, 22}, {2, 31152, 7485}, {2, 31857, 10989}, {2, 37901, 52300}, {2, 53161, 9832}, {2, 61985, 4232}, {2, 62975, 34603}, {5, 47097, 2}, {376, 31099, 47314}, {378, 1995, 22}, {381, 5094, 2}, {381, 7579, 39484}, {381, 44287, 378}, {381, 56966, 4}, {381, 56967, 11317}, {427, 5094, 378}, {427, 31236, 22}, {427, 37454, 31723}, {427, 39504, 5169}, {427, 52262, 31099}, {427, 64852, 7391}, {547, 30739, 2}, {549, 37454, 2}, {549, 47311, 16063}, {858, 53843, 2}, {1344, 1345, 23}, {1346, 1347, 858}, {1995, 7485, 6644}, {3524, 3839, 38320}, {3543, 7493, 47313}, {3545, 8889, 30775}, {3545, 30775, 2}, {3843, 52292, 14002}, {5055, 32216, 2}, {5071, 16051, 2}, {5094, 5169, 1995}, {5133, 8889, 30744}, {5169, 7577, 5133}, {5169, 44287, 31133}, {7391, 7533, 44263}, {7495, 47314, 376}, {7495, 52842, 22}, {7577, 8889, 5094}, {11178, 13857, 15066}, {15687, 37904, 7519}, {30769, 61936, 2}, {31133, 31236, 2}, {31861, 39484, 381}, {37454, 47311, 549}, {37638, 54131, 15360}, {39504, 44287, 381}, {44210, 64852, 2}, {44212, 62958, 2}, {44218, 56966, 378}, {47296, 50959, 20192}, {52267, 52268, 381}

X(66377) = EULER LINE INTERCEPT OF X(69)X(61655)

Barycentrics    3*a^6 - 2*a^4*b^2 - 3*a^2*b^4 + 2*b^6 - 2*a^4*c^2 - 2*b^4*c^2 - 3*a^2*c^4 - 2*b^2*c^4 + 2*c^6 : :
X(66377) = 6 X[2] + X[22], 9 X[2] - 2 X[427], 3 X[2] + 4 X[6676], 15 X[2] - X[7391], 27 X[2] + X[20062], 17 X[2] - 3 X[31105], 8 X[2] - X[31133], 12 X[2] - 5 X[31236], 5 X[2] + 2 X[44210], 4 X[2] + 3 X[47596], 15 X[2] - 8 X[64852], 6 X[3] + X[35480], 5 X[3] + 2 X[44263], 3 X[3] + 4 X[46029], 4 X[5] + 3 X[44837], 8 X[5] - X[52842], 3 X[22] + 4 X[427], and many others

X(66377) lies on these lines: {2, 3}, {69, 61655}, {1209, 9707}, {1698, 51692}, {1853, 15080}, {1993, 58447}, {2781, 44299}, {3410, 26864}, {3618, 16789}, {3763, 19127}, {3796, 23293}, {3819, 54384}, {3917, 58480}, {5012, 37638}, {5085, 26913}, {5422, 61646}, {5550, 51718}, {6030, 7703}, {6800, 21243}, {9019, 47355}, {11422, 64060}, {11442, 13394}, {11605, 58428}, {12270, 15151}, {15059, 16165}, {17005, 43980}, {17809, 41724}, {31267, 34177}, {33651, 64982}, {34507, 64064}, {37513, 61701}, {39576, 63611}, {41588, 63036}, {45794, 61690}, {51744, 63119}

X(66377) = crossdifference of every pair of points on line {647, 39481}
X(66377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 30744}, {2, 22, 31236}, {2, 1995, 7571}, {2, 6353, 37990}, {2, 6636, 5094}, {2, 6676, 22}, {2, 7391, 64852}, {2, 7394, 37454}, {2, 7493, 5133}, {2, 7494, 858}, {2, 7495, 7485}, {2, 7496, 31255}, {2, 7499, 40916}, {2, 13595, 7539}, {2, 15246, 30771}, {2, 16063, 62958}, {2, 31101, 52298}, {2, 47596, 31133}, {2, 52300, 25}, {2, 62937, 11548}, {3, 46029, 35480}, {3, 52298, 31101}, {5, 44837, 52842}, {22, 6676, 47596}, {22, 31236, 31133}, {26, 7569, 7566}, {140, 52297, 2}, {3549, 7503, 63657}, {6353, 37990, 1995}, {6639, 7568, 7509}, {6676, 64852, 44210}, {7391, 44210, 22}, {7542, 7558, 17928}, {10154, 37454, 7394}, {11548, 62978, 62937}, {16419, 52292, 2}, {31101, 52298, 30744}, {31236, 47596, 22}, {44210, 64852, 7391}, {58447, 61644, 1993}

X(66378) = EULER LINE INTERCEPT OF X(69)X(9544)

Barycentrics    3*a^6 - a^4*b^2 - 3*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6 : :
X(66378) = 3 X[2] + 2 X[22], 9 X[2] - 4 X[427], 3 X[2] - 8 X[6676], 6 X[2] - X[7391], 9 X[2] + X[20062], 8 X[2] - 3 X[31105], 7 X[2] - 2 X[31133], X[2] + 4 X[44210], X[2] - 6 X[47596], 21 X[2] - 16 X[64852], X[3] + 4 X[25337], 6 X[3] - X[35481], 4 X[3] + X[44440], X[4] + 4 X[7502], 3 X[4] - 8 X[46029], 4 X[5] + X[44831], X[20] + 4 X[15760], 3 X[20] + 2 X[35480], and many others

X(66378) lies on these lines: {2, 3}, {8, 51692}, {69, 9544}, {110, 43653}, {141, 35264}, {154, 37636}, {184, 5965}, {193, 16789}, {343, 6800}, {1194, 5346}, {1352, 26881}, {1369, 6031}, {1799, 14247}, {1899, 15080}, {1993, 13394}, {2979, 54384}, {3060, 58480}, {3410, 7712}, {3448, 16165}, {3580, 3796}, {3618, 9019}, {5012, 37644}, {5092, 61645}, {5422, 32269}, {5596, 34177}, {6030, 23293}, {6515, 11003}, {7998, 59543}, {9627, 29815}, {9973, 41578}, {10192, 15066}, {11002, 63085}, {11008, 13622}, {11427, 44439}, {11442, 61644}, {11456, 44201}, {14389, 33586}, {14826, 35265}, {16960, 54363}, {16961, 54362}, {16990, 21458}, {17809, 41628}, {18911, 22352}, {19126, 64724}, {19220, 21843}, {21243, 35268}, {21766, 53415}, {24206, 44082}, {26233, 34254}, {26883, 32348}, {32223, 43650}, {33522, 37645}, {33884, 37669}, {34507, 44110}, {37494, 61619}, {37517, 61659}, {38317, 44106}, {40897, 63021}, {46934, 51718}, {51707, 54445}

X(66378) = midpoint of X(22) and X(31236)
X(66378) = anticomplement of X(31236)
X(66378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 31074}, {2, 22, 7391}, {2, 23, 7394}, {2, 25, 62937}, {2, 5189, 8889}, {2, 6636, 16063}, {2, 6995, 37353}, {2, 7492, 1370}, {2, 7500, 5169}, {2, 7519, 5133}, {2, 10565, 23}, {2, 14002, 7392}, {2, 20062, 427}, {2, 37913, 4}, {2, 59343, 31099}, {2, 59344, 31106}, {22, 427, 20062}, {22, 6676, 2}, {22, 47596, 6676}, {23, 37353, 6995}, {25, 7495, 2}, {26, 7558, 7544}, {427, 20062, 7391}, {468, 7485, 2}, {631, 7493, 37760}, {1995, 7499, 2}, {3410, 7712, 11206}, {3522, 30745, 16063}, {3523, 62973, 2}, {3549, 7512, 37444}, {5004, 5005, 7526}, {5054, 44457, 44236}, {5133, 9909, 7519}, {5899, 60763, 4}, {6030, 23293, 46264}, {6636, 52300, 2}, {6639, 7525, 47528}, {6676, 44210, 22}, {6677, 40916, 2}, {6995, 37353, 7394}, {7492, 60455, 17538}, {7493, 7494, 2}, {7499, 10154, 1995}, {7502, 37932, 38435}, {9715, 13160, 31304}, {15760, 44837, 20}, {22352, 61646, 18911}, {33522, 37645, 62188}, {38282, 46336, 2}, {44210, 47596, 2}

X(66379) = EULER LINE INTERCEPT OF X(145)X(51692)

Barycentrics    5*a^6 - a^4*b^2 - 5*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 5*a^2*c^4 - b^2*c^4 + c^6 : :
X(66379) = 3 X[2] + 4 X[22], 15 X[2] - 8 X[427], 9 X[2] - 16 X[6676], 9 X[2] - 2 X[7391], 6 X[2] + X[20062], 13 X[2] - 6 X[31105], 11 X[2] - 4 X[31133], 27 X[2] - 20 X[31236], X[2] - 8 X[44210], 5 X[2] - 12 X[47596], 39 X[2] - 32 X[64852], X[4] - 8 X[25337], X[20] - 8 X[7502], 5 X[22] + 2 X[427], 3 X[22] + 4 X[6676], 6 X[22] + X[7391], 8 X[22] - X[20062], and many others

X(66379) lies on these lines: {2, 3}, {145, 51692}, {193, 19127}, {323, 33522}, {1899, 6030}, {3796, 37644}, {6800, 45794}, {9019, 51171}, {11002, 58480}, {11442, 35268}, {14683, 16165}, {15066, 59699}, {16789, 20080}, {16981, 63030}, {20079, 34177}, {26881, 43653}, {33586, 63036}, {33878, 61655}, {41736, 58439}, {52987, 64064}, {54384, 62188}

X(66379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 22, 20062}, {2, 20063, 7378}, {2, 37913, 7519}, {2, 59343, 5189}, {22, 6676, 7391}, {22, 47596, 427}, {23, 6636, 15818}, {23, 7494, 2}, {1370, 52300, 2}, {3547, 38435, 31304}, {6353, 15246, 2}, {6636, 7493, 2}, {6676, 7391, 2}, {7394, 7495, 2}, {7495, 9909, 7394}, {7499, 62937, 2}, {13564, 37119, 20}, {16618, 44837, 44440}, {44440, 44837, 3522}

X(66380) = EULER LINE INTERCEPT OF X(51)X(51732)

Barycentrics    6*a^6 - a^4*b^2 - 6*a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 - 6*a^2*c^4 - b^2*c^4 + c^6 : :
X(66380) = 3 X[2] + 5 X[22], 9 X[2] - 5 X[427], 3 X[2] - 5 X[6676], 21 X[2] - 5 X[7391], 27 X[2] + 5 X[20062], 31 X[2] - 15 X[31105], 13 X[2] - 5 X[31133], 33 X[2] - 25 X[31236], X[2] - 5 X[44210], 7 X[2] - 15 X[47596], 6 X[2] - 5 X[64852], 3 X[22] + X[427], 7 X[22] + X[7391], 9 X[22] - X[20062], 31 X[22] + 9 X[31105], 13 X[22] + 3 X[31133], 11 X[22] + 5 X[31236], and many others

X(66380) lies on these lines: {2, 3}, {51, 51732}, {154, 48876}, {184, 34380}, {343, 35268}, {1350, 59553}, {3098, 10192}, {3167, 33522}, {3244, 51692}, {3580, 6030}, {3629, 19127}, {3631, 15585}, {3796, 41588}, {3819, 15448}, {5012, 47582}, {5206, 40326}, {5310, 15172}, {5345, 18990}, {5907, 15152}, {6329, 9019}, {6390, 33651}, {7298, 15171}, {8770, 21843}, {8780, 10519}, {8854, 35255}, {8855, 35256}, {11245, 15080}, {11402, 61624}, {14810, 53415}, {15153, 44829}, {15808, 51718}, {16165, 24981}, {16621, 32348}, {16789, 40341}, {17040, 63026}, {17809, 64067}, {17810, 38110}, {18289, 42216}, {18290, 42215}, {19126, 41585}, {22352, 32269}, {23332, 48898}, {29181, 58447}, {31884, 59543}, {35260, 62217}, {37669, 55610}, {43653, 61545}, {44110, 64062}, {44882, 61646}, {46728, 61607}, {54169, 59699}, {55584, 63092}, {55606, 61681}, {55614, 59551}

X(66380) = midpoint of X(i) and X(j) for these {i,j}: {22, 6676}, {7502, 16618}, {7555, 25337}, {12083, 64474}
X(66380) = reflection of X(64852) in X(6676)
X(66380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 37900, 52285}, {3, 6677, 7734}, {3, 10154, 6677}, {3, 10565, 10154}, {22, 44210, 6676}, {22, 47596, 7391}, {25, 140, 10128}, {26, 16197, 9825}, {428, 7495, 11548}, {428, 11548, 3850}, {428, 37913, 37910}, {468, 6636, 10691}, {548, 7493, 37911}, {1595, 47525, 140}, {2937, 34002, 6756}, {5159, 7492, 44245}, {6636, 10691, 33923}, {6677, 10154, 47316}, {7494, 9909, 5}, {7495, 37910, 3850}, {7495, 37913, 428}, {7734, 47316, 6677}, {10128, 47630, 25}, {11548, 37910, 428}, {16531, 34200, 3530}, {22352, 32269, 45298}, {34609, 59343, 15704}, {44210, 44260, 25337}, {52300, 52397, 62958}, {52397, 62958, 47315}

X(66381) = EULER LINE INTERCEPT OF X(51)X(44882)

Barycentrics    4*a^6 + a^4*b^2 - 4*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 4*a^2*c^4 + b^2*c^4 - c^6 : :
X(66381) = 3 X[2] - 5 X[22], 6 X[2] - 5 X[427], 9 X[2] - 10 X[6676], 9 X[2] - 5 X[7391], 3 X[2] + 5 X[20062], 19 X[2] - 15 X[31105], 7 X[2] - 5 X[31133], 27 X[2] - 25 X[31236], 4 X[2] - 5 X[44210], 13 X[2] - 15 X[47596], 21 X[2] - 20 X[64852], 3 X[22] - 2 X[6676], 3 X[22] - X[7391], 19 X[22] - 9 X[31105], 7 X[22] - 3 X[31133], 9 X[22] - 5 X[31236], 4 X[22] - 3 X[44210], and many others

X(66381) lies on these lines: {2, 3}, {51, 44882}, {154, 48872}, {184, 29181}, {251, 15048}, {343, 29012}, {394, 48873}, {612, 15338}, {614, 15326}, {1180, 18907}, {1196, 6781}, {1350, 31383}, {1353, 62187}, {1495, 59699}, {1799, 32819}, {1899, 47582}, {2781, 24981}, {2979, 48874}, {3060, 48906}, {3629, 6467}, {3631, 16789}, {3636, 51692}, {3796, 31670}, {3819, 48885}, {3917, 48881}, {5012, 21850}, {5310, 7354}, {5322, 6284}, {5345, 65134}, {5480, 22352}, {5943, 48892}, {6030, 14389}, {6329, 19127}, {7298, 10483}, {7712, 61655}, {7750, 16276}, {7802, 45201}, {8550, 21969}, {9306, 48880}, {9777, 25406}, {9924, 40341}, {10192, 51360}, {10313, 42459}, {10386, 29815}, {11064, 48879}, {11206, 64716}, {11245, 33586}, {11402, 51212}, {12220, 46444}, {12290, 33523}, {15107, 41588}, {16194, 35254}, {16655, 46728}, {17810, 59411}, {18289, 42276}, {18290, 42275}, {19924, 34986}, {21243, 29323}, {23292, 35268}, {26881, 59553}, {29317, 61690}, {31406, 38862}, {31802, 52525}, {31804, 64051}, {32269, 48896}, {36990, 43653}, {37636, 39884}, {37648, 48891}, {37649, 48901}, {40904, 47287}, {41628, 54036}, {41715, 64719}, {42087, 54363}, {42088, 54362}, {43291, 63538}, {43726, 51744}, {44082, 53415}, {44762, 45187}, {46818, 62188}, {50979, 53863}, {52987, 64062}, {53100, 54636}, {54426, 64159}, {61044, 63174}

X(66381) = midpoint of X(i) and X(j) for these {i,j}: {22, 20062}, {12082, 44831}
X(66381) = reflection of X(i) in X(j) for these {i,j}: {427, 22}, {7391, 6676}, {31723, 16618}
X(66381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 382, 52285}, {3, 428, 37439}, {3, 7500, 428}, {20, 25, 7667}, {20, 34608, 25}, {20, 37899, 30739}, {20, 37945, 52069}, {20, 39568, 1885}, {22, 427, 44210}, {22, 7391, 6676}, {23, 1368, 62978}, {23, 52397, 1368}, {25, 7667, 30739}, {25, 34608, 37899}, {376, 6995, 7484}, {550, 37900, 10301}, {858, 10154, 52297}, {858, 37913, 10154}, {1368, 15704, 52397}, {1370, 9909, 468}, {1657, 9909, 1370}, {3146, 7494, 5064}, {3534, 44454, 49669}, {5059, 10565, 44442}, {5064, 7494, 37454}, {6636, 20063, 34603}, {6636, 34603, 5}, {6676, 7391, 427}, {7387, 18531, 47093}, {7396, 37453, 47097}, {7493, 17800, 47095}, {7493, 34609, 62958}, {7667, 37899, 25}, {10565, 44442, 5094}, {11414, 31305, 3575}, {18531, 47093, 235}, {31133, 64852, 427}, {31304, 33524, 31829}, {33586, 46264, 11245}, {34614, 37931, 21312}, {37910, 62144, 16063}, {47095, 62958, 34609}

X(66382) = EULER LINE INTERCEPT OF X(154)X(48873)

Barycentrics    6*a^6 + a^4*b^2 - 6*a^2*b^4 - b^6 + a^4*c^2 + b^4*c^2 - 6*a^2*c^4 + b^2*c^4 - c^6 : :
X(66382) = 3 X[2] - 7 X[22], 9 X[2] - 7 X[427], 6 X[2] - 7 X[6676], 15 X[2] - 7 X[7391], 9 X[2] + 7 X[20062], 29 X[2] - 21 X[31105], 11 X[2] - 7 X[31133], 39 X[2] - 35 X[31236], 5 X[2] - 7 X[44210], 17 X[2] - 21 X[47596], 15 X[2] - 14 X[64852], 3 X[22] - X[427], 5 X[22] - X[7391], 3 X[22] + X[20062], 29 X[22] - 9 X[31105], 11 X[22] - 3 X[31133], 13 X[22] - 5 X[31236], and many others

X(66382) lies on these lines: {2, 3}, {154, 48873}, {251, 63633}, {394, 48874}, {3796, 21850}, {5310, 18990}, {5322, 15171}, {5345, 6284}, {6688, 33751}, {7298, 7354}, {7767, 16276}, {8280, 42272}, {8281, 42271}, {9019, 32366}, {9306, 48881}, {9641, 29815}, {10313, 59649}, {10625, 44544}, {11206, 33878}, {11245, 15107}, {13567, 48898}, {14826, 55610}, {15448, 48920}, {18289, 42264}, {18290, 42263}, {18439, 33523}, {18440, 33522}, {18583, 22352}, {23292, 29317}, {31383, 48876}, {32237, 53415}, {33586, 48906}, {39884, 43653}, {41588, 46264}, {41724, 54036}, {42122, 54363}, {42123, 54362}, {44106, 64730}, {44882, 45298}, {48896, 61646}, {51392, 61606}, {55584, 63174}, {61624, 62187}

X(66382) = midpoint of X(i) and X(j) for these {i,j}: {427, 20062}, {12082, 44239}, {44249, 44457}
X(66382) = reflection of X(i) in X(j) for these {i,j}: {6676, 22}, {7391, 64852}, {52262, 7555}, {64474, 7502}
X(66382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 9909, 1368}, {22, 7391, 44210}, {22, 20062, 427}, {23, 7667, 6677}, {23, 12103, 10300}, {25, 550, 10691}, {428, 6636, 140}, {1368, 9909, 37897}, {1370, 10154, 5159}, {3522, 7714, 16419}, {3529, 10565, 34609}, {3534, 20850, 7386}, {6636, 37900, 428}, {6677, 7667, 10300}, {6677, 12103, 7667}, {7386, 20850, 44212}, {7391, 44210, 64852}, {7485, 10301, 10128}, {7492, 34603, 7499}, {7499, 34603, 546}, {7553, 13564, 16197}, {10128, 33923, 7485}, {10154, 15704, 1370}, {10691, 37910, 25}, {11414, 65376, 31829}, {34608, 59343, 3}, {37913, 52397, 468}, {44210, 64852, 6676}, {47630, 62136, 16063}

X(66383) = EULER LINE INTERCEPT OF X(110)X(48872)

Barycentrics    7*a^6 + 2*a^4*b^2 - 7*a^2*b^4 - 2*b^6 + 2*a^4*c^2 + 2*b^4*c^2 - 7*a^2*c^4 + 2*b^2*c^4 - 2*c^6 : :
X(66383) = 2 X[2] - 3 X[22], 7 X[2] - 6 X[427], 11 X[2] - 12 X[6676], 5 X[2] - 3 X[7391], X[2] + 3 X[20062], 11 X[2] - 9 X[31105], 4 X[2] - 3 X[31133], 16 X[2] - 15 X[31236], 5 X[2] - 6 X[44210], 8 X[2] - 9 X[47596], 25 X[2] - 24 X[64852], 7 X[22] - 4 X[427], 11 X[22] - 8 X[6676], 5 X[22] - 2 X[7391], X[22] + 2 X[20062], 11 X[22] - 6 X[31105], 8 X[22] - 5 X[31236], and many others

X(66383) lies on these lines: {2, 3}, {110, 48872}, {1495, 48879}, {1993, 19924}, {3060, 43273}, {3796, 51024}, {5012, 54131}, {5640, 59411}, {5651, 48920}, {6800, 29317}, {7605, 55682}, {9019, 15531}, {11057, 16276}, {14389, 43621}, {14683, 55584}, {15066, 48880}, {15080, 48910}, {15107, 48905}, {16789, 50990}, {19127, 51185}, {21766, 48885}, {29323, 61700}, {34417, 48891}, {34633, 37557}, {34712, 64039}, {34796, 44750}, {45968, 64014}, {51105, 51692}

X(66383) = reflection of X(i) in X(j) for these {i,j}: {7391, 44210}, {31133, 22)
X(66383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11001, 52397}, {2, 15640, 62964}, {2, 34608, 47313}, {20, 37900, 1995}, {22, 31133, 47596}, {428, 8703, 2}, {550, 7519, 40916}, {10154, 47311, 2}, {11001, 34608, 2}, {15696, 62968, 7496}, {15704, 37899, 16063}, {31133, 47596, 31236}, {47313, 52397, 2}

X(66384) = X(1)X(3)∩X(632)X(5123)

Barycentrics    a (8 a^6-11 a^5 b-13 a^4 b^2+22 a^3 b^3+2 a^2 b^4-11 a b^5+3 b^6-11 a^5 c+36 a^4 b c-18 a^3 b^2 c-30 a^2 b^3 c+29 a b^4 c-6 b^5 c-13 a^4 c^2-18 a^3 b c^2+52 a^2 b^2 c^2-18 a b^3 c^2-3 b^4 c^2+22 a^3 c^3-30 a^2 b c^3-18 a b^2 c^3+12 b^3 c^3+2 a^2 c^4+29 a b c^4-3 b^2 c^4-11 a c^5-6 b c^5+3 c^6) : :

See David Nguyen and Francisco Javier García Capitán, euclid 7255.

X(66384) lies on these lines:{1, 3}, {632, 5123}, {1532, 28208}, {3525, 5176}, {3627, 22835}, {3653, 6947}, {3655, 6880}, {4881, 38665}, {5087, 38028}, {6938, 51709}, {10598, 18481}, {12737, 35271}, {20418, 28204}, {21578, 38032}

X(66385) = ISOGONAL CONJUGATE OF X(1507)

Barycentrics    (1 + 2*Cos[A/3] + 2*Cos[B/3] - 2*Cos[C/3])*(1 + 2*Cos[A/3] - 2*Cos[B/3] + 2*Cos[C/3])*Sin[A] : :

X(66385) lies on the cubic K029 and these lines: {356, 1507}, {1135, 1508}

X(66385) = isogonal conjugate of X(1507)

X(66386) = ISOGONAL CONJUGATE OF X(1508)

Barycentrics    (2 + Sec[A/3] + Sec[B/3] - Sec[C/3])*(2 + Sec[A/3] - Sec[B/3] + Sec[C/3])*Sin[A] : :

X(66386) lies on the cubic K029 and these lines: {356, 1508}

X(66386) = isogonal conjugate of X(1508).

X(66387) = EULER LINE INTERCEPT OF X(83)X(44519)

Barycentrics    7*a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 + 6*b^2*c^2 - 2*c^4 : :
X(66387) = 2 X[2] - 3 X[1003], 4 X[2] - 3 X[7841], 13 X[2] - 12 X[8360], 23 X[2] - 24 X[8365], 14 X[2] - 15 X[8366], 11 X[2] - 12 X[8368], 5 X[2] - 6 X[8369], X[2] - 3 X[33007], 5 X[2] - 3 X[33017], 7 X[2] - 6 X[33184], 5 X[2] - 9 X[33187], 7 X[2] - 3 X[33192], X[2] + 3 X[33193], 25 X[2] - 24 X[33213], 10 X[2] - 9 X[33219], 8 X[2] - 9 X[33220], 11 X[2] - 9 X[33251], and many others

X(66387) lies on these lines: {2, 3}, {83, 44519}, {99, 7926}, {148, 1384}, {183, 6781}, {187, 18546}, {193, 47287}, {325, 43618}, {543, 14614}, {599, 11057}, {671, 62898}, {754, 1975}, {1351, 13172}, {2482, 63956}, {3053, 14568}, {3849, 7788}, {3972, 44526}, {5182, 54131}, {5858, 8595}, {5859, 8594}, {5969, 8593}, {7737, 31859}, {7753, 34504}, {7757, 12156}, {7774, 51123}, {7781, 41750}, {7782, 65630}, {7792, 43619}, {7802, 7879}, {7812, 8716}, {7816, 7818}, {7837, 8591}, {7840, 19569}, {7851, 65633}, {8667, 51224}, {10131, 18501}, {11163, 14537}, {11185, 13468}, {11648, 32479}, {11742, 60855}, {12154, 41100}, {12155, 41101}, {13846, 54507}, {13847, 54503}, {15031, 44535}, {15655, 17004}, {17503, 60073}, {20094, 22253}, {22486, 43273}, {23698, 39656}, {32456, 62203}, {32819, 63955}, {32821, 63931}, {32833, 63941}, {37671, 47102}, {41134, 48913}, {45103, 60178}, {47286, 63034}, {52229, 63093}

X(66387) = midpoint of X(33007) and X(33193)
X(66387) = reflection of X(i) in X(j) for these {i,j}: {1003, 33007}, {7818, 7816}, {7841, 1003}, {33017, 8369}, {33192, 33184}, {33219, 33187}
X(66387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 35955}, {2, 9855, 3534}, {2, 11001, 8353}, {2, 14030, 11286}, {2, 15682, 8352}, {2, 19686, 14030}, {2, 33288, 33240}, {2, 52942, 3845}, {2, 62094, 47061}, {2, 66328, 11159}, {3, 11361, 44543}, {4, 439, 33249}, {4, 33235, 33233}, {4, 33250, 33235}, {4, 35927, 35297}, {20, 14033, 8356}, {20, 19687, 7770}, {382, 3552, 7887}, {382, 11288, 14041}, {384, 1657, 33234}, {384, 33264, 11287}, {550, 14035, 11285}, {1003, 7841, 33220}, {1003, 8366, 33255}, {1003, 33219, 8369}, {1657, 11287, 33264}, {3146, 33239, 7807}, {3363, 12100, 2}, {3529, 14039, 33272}, {3529, 32981, 6656}, {3534, 11159, 2}, {3543, 32985, 33228}, {3552, 14041, 11288}, {3552, 19696, 382}, {3845, 27088, 2}, {3850, 33227, 33000}, {5059, 14001, 19695}, {6658, 33257, 3}, {6658, 33265, 11361}, {7819, 62155, 32997}, {7833, 14030, 2}, {7833, 19686, 11286}, {7837, 8591, 51122}, {7866, 49137, 33256}, {8353, 66319, 2}, {8356, 14033, 7770}, {8356, 19687, 14033}, {8357, 62159, 33271}, {8359, 15686, 33207}, {8361, 62041, 33279}, {8362, 62144, 33253}, {8366, 33192, 7841}, {8369, 33017, 33219}, {8369, 33187, 1003}, {9855, 11159, 35955}, {9855, 66328, 2}, {11286, 15681, 7833}, {11287, 33264, 33234}, {11288, 14041, 7887}, {11361, 33257, 33265}, {11361, 33265, 3}, {14031, 33253, 8362}, {14037, 33271, 8357}, {14039, 33272, 6656}, {14042, 33014, 1656}, {14066, 33259, 3851}, {14068, 33254, 140}, {16044, 33268, 3}, {16924, 33214, 548}, {32954, 49136, 33019}, {32973, 33229, 33218}, {32973, 33703, 33229}, {32981, 33272, 14039}, {33001, 33252, 33923}, {33007, 33017, 33187}, {33016, 33208, 549}, {33017, 33187, 8369}, {33017, 33219, 7841}, {33018, 33276, 3526}, {33184, 33255, 8366}, {33192, 33255, 33184}, {33198, 62152, 33247}, {33201, 33238, 8363}, {33201, 49140, 33238}, {33244, 33280, 5}, {33250, 35297, 35927}, {35297, 35927, 33235}, {35954, 66349, 2}, {44903, 66321, 33263}

X(66388) = EULER LINE INTERCEPT OF X(6)X(12156)

Barycentrics    5*a^4 - 3*a^2*b^2 - 4*b^4 - 3*a^2*c^2 + 6*b^2*c^2 - 4*c^4 : :
X(66388) = 4 X[2] - 3 X[1003], 2 X[2] - 3 X[7841], 11 X[2] - 12 X[8360], 25 X[2] - 24 X[8365], 16 X[2] - 15 X[8366], 13 X[2] - 12 X[8368], 7 X[2] - 6 X[8369], 5 X[2] - 3 X[33007], X[2] - 3 X[33017], 5 X[2] - 6 X[33184], 13 X[2] - 9 X[33187], X[2] + 3 X[33192], 7 X[2] - 3 X[33193], 23 X[2] - 24 X[33213], 8 X[2] - 9 X[33219], 10 X[2] - 9 X[33220], 7 X[2] - 9 X[33251], and many others

X(66388) lies on these lines: {2, 3}, {6, 12156}, {115, 47101}, {183, 18546}, {316, 9766}, {325, 43619}, {385, 14976}, {543, 7788}, {598, 54905}, {671, 8667}, {754, 7748}, {1350, 10723}, {1975, 7818}, {2549, 41624}, {3849, 11648}, {5210, 14061}, {5969, 11161}, {7750, 63955}, {7752, 44519}, {7756, 7773}, {7792, 43618}, {7802, 14568}, {7809, 8716}, {7810, 63957}, {7811, 34505}, {7840, 51122}, {7847, 65630}, {7879, 32819}, {7910, 31168}, {8556, 55164}, {8860, 18362}, {9879, 62188}, {9939, 63954}, {10722, 55177}, {11054, 63951}, {11163, 63956}, {11174, 62203}, {11184, 48913}, {13449, 63424}, {13468, 14907}, {15514, 15534}, {17503, 60101}, {19569, 63038}, {19570, 63950}, {22329, 47102}, {22486, 51024}, {37668, 47287}, {41748, 63943}, {41750, 63931}, {43448, 63034}, {44969, 59231}, {45103, 60096}, {47286, 64018}, {60228, 60280}, {63093, 63945}

X(66388) = midpoint of X(i) and X(j) for these {i,j}: {7818, 65633}, {33017, 33192}
X(66388) = reflection of X(i) in X(j) for these {i,j}: {1003, 7841}, {1975, 7818}, {7818, 7842}, {7841, 33017}, {14614, 11648}, {33007, 33184}, {33193, 8369}, {33220, 33278}
X(66388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3830, 11317}, {2, 8353, 35955}, {2, 8597, 3830}, {2, 11001, 8598}, {2, 33291, 11318}, {4, 8356, 44543}, {4, 19695, 33234}, {4, 33023, 32992}, {4, 33234, 11285}, {4, 33272, 8356}, {20, 16041, 35297}, {20, 33229, 7887}, {316, 44526, 31859}, {382, 6655, 7770}, {382, 11287, 11361}, {550, 14063, 33233}, {671, 11057, 8667}, {1003, 7841, 33219}, {1003, 33219, 8366}, {1657, 5025, 33235}, {1657, 11288, 33265}, {2549, 44678, 41624}, {3146, 33210, 14033}, {3146, 33238, 6656}, {3363, 8358, 2}, {3529, 32982, 7807}, {3529, 33285, 35927}, {3543, 32986, 8370}, {3830, 5077, 2}, {3845, 8354, 2}, {5025, 19691, 1657}, {5025, 33265, 11288}, {5059, 14064, 33250}, {5077, 8597, 11317}, {6655, 11361, 11287}, {7819, 62041, 33280}, {7841, 33220, 33184}, {7842, 65633, 1975}, {7866, 49136, 6658}, {8352, 8353, 2}, {8356, 19695, 33272}, {8356, 33272, 33234}, {8356, 44543, 11285}, {8357, 62036, 14035}, {8358, 12101, 3363}, {8359, 15687, 33016}, {8361, 62155, 33244}, {8362, 62026, 14068}, {8703, 37350, 2}, {11287, 11361, 7770}, {11288, 33265, 33235}, {11318, 15681, 13586}, {13586, 33291, 2}, {14033, 33210, 6656}, {14033, 33238, 33210}, {14041, 33256, 33264}, {14041, 33264, 3}, {14044, 33004, 3851}, {14062, 33260, 1656}, {14063, 33271, 550}, {16041, 35297, 7887}, {18362, 46893, 8860}, {19687, 32974, 33217}, {32954, 49137, 33257}, {32961, 33209, 548}, {32966, 33267, 3}, {32974, 33703, 19687}, {32982, 35927, 33285}, {32993, 33275, 3526}, {32996, 33253, 140}, {32997, 33279, 5}, {33000, 33243, 33923}, {33006, 33207, 549}, {33007, 33017, 33278}, {33007, 33184, 33220}, {33007, 33220, 1003}, {33007, 33278, 33184}, {33016, 33263, 8359}, {33019, 33256, 3}, {33019, 33264, 14041}, {33184, 33278, 7841}, {33193, 33251, 8369}, {33200, 49140, 33239}, {33227, 62136, 33252}, {33229, 35297, 16041}, {33234, 44543, 8356}, {33285, 35927, 7807}, {33292, 62147, 439}, {33824, 50239, 33035}

X(66389) = EULER LINE INTERCEPT OF X(69)X(14976)

Barycentrics    13*a^4 - 6*a^2*b^2 - 5*b^4 - 6*a^2*c^2 + 12*b^2*c^2 - 5*c^4 : :
X(66389) = 5 X[2] - 6 X[1003], 7 X[2] - 6 X[7841], 25 X[2] - 24 X[8360], 47 X[2] - 48 X[8365], 29 X[2] - 30 X[8366], 23 X[2] - 24 X[8368], 11 X[2] - 12 X[8369], 2 X[2] - 3 X[33007], 4 X[2] - 3 X[33017], 13 X[2] - 12 X[33184], 7 X[2] - 9 X[33187], 5 X[2] - 3 X[33192], X[2] - 3 X[33193], 49 X[2] - 48 X[33213], 19 X[2] - 18 X[33219], 17 X[2] - 18 X[33220], and many others

X(66389) lies on these lines: {2, 3}, {69, 14976}, {99, 44678}, {148, 63034}, {543, 63093}, {5969, 63064}, {6781, 17008}, {7737, 12156}, {7774, 43618}, {8591, 19569}, {11185, 47101}, {12154, 46334}, {12155, 46335}, {16989, 43619}, {32479, 63065}, {32480, 63024}, {32532, 60136}, {62203, 63083}

X(66389) = reflection of X(i) in X(j) for these {i,j}: {33007, 33193}, {33017, 33007}, {33192, 1003}
X(66389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15640, 8597}, {2, 52943, 8353}, {4, 33014, 32998}, {4, 33254, 33000}, {20, 11361, 33008}, {20, 19696, 33280}, {20, 32979, 33275}, {20, 33280, 16924}, {382, 33244, 32961}, {384, 5059, 33271}, {439, 50691, 14062}, {550, 14068, 33001}, {1003, 33192, 33251}, {1657, 14035, 33253}, {3146, 33257, 16925}, {3146, 35927, 14041}, {3522, 14042, 32999}, {3528, 33018, 33003}, {3529, 6658, 7791}, {3529, 14033, 33264}, {3543, 13586, 33006}, {3552, 33279, 33248}, {3552, 33703, 33279}, {3830, 8598, 2}, {5073, 33250, 14063}, {5077, 66319, 2}, {6658, 33264, 14033}, {7770, 62155, 33209}, {8353, 11159, 2}, {8353, 15685, 52943}, {8370, 15681, 33207}, {8703, 11317, 2}, {11159, 15685, 8353}, {11285, 62144, 33243}, {11361, 33008, 16924}, {11541, 33239, 33019}, {14001, 62171, 19691}, {14033, 33264, 7791}, {14036, 33256, 33210}, {14041, 33257, 35927}, {14041, 35927, 16925}, {17800, 19687, 32997}, {19687, 32997, 16898}, {32971, 62152, 33267}, {32981, 33210, 14036}, {32981, 49140, 33256}, {32989, 50690, 14044}, {33007, 33017, 33255}, {33007, 33251, 1003}, {33008, 33280, 11361}, {33192, 33251, 33017}, {33235, 62036, 32996}, {33246, 33291, 2}

X(66390) = EULER LINE INTERCEPT OF X(148)X(14976)

Barycentrics    11*a^4 - 6*a^2*b^2 - 7*b^4 - 6*a^2*c^2 + 12*b^2*c^2 - 7*c^4 : :
X(66390) = 7 X[2] - 6 X[1003], 5 X[2] - 6 X[7841], 23 X[2] - 24 X[8360], 49 X[2] - 48 X[8365], 31 X[2] - 30 X[8366], 25 X[2] - 24 X[8368], 13 X[2] - 12 X[8369], 4 X[2] - 3 X[33007], 2 X[2] - 3 X[33017], 11 X[2] - 12 X[33184], 11 X[2] - 9 X[33187], X[2] - 3 X[33192], 5 X[2] - 3 X[33193], 47 X[2] - 48 X[33213], 17 X[2] - 18 X[33219], 19 X[2] - 18 X[33220], and many others

X(66390) lies on these lines: {2, 3}, {148, 14976}, {671, 47102}, {754, 65633}, {1992, 19569}, {3849, 63093}, {5969, 50992}, {7618, 48913}, {7739, 12156}, {7774, 43619}, {7802, 63955}, {11648, 63065}, {14712, 63034}, {14907, 18546}, {16989, 43618}, {17008, 47101}, {41624, 44526}

X(66390) = reflection of X(i) in X(j) for these {i,j}: {33007, 33017}, {33017, 33192}, {33193, 7841}
X(66390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15682, 52942}, {2, 40246, 15682}, {2, 52943, 3534}, {2, 62160, 9855}, {4, 19691, 33271}, {4, 33008, 33005}, {4, 33253, 33001}, {4, 33260, 32999}, {4, 33264, 33008}, {4, 33271, 33253}, {20, 32980, 33276}, {20, 33279, 32961}, {20, 54097, 7907}, {382, 32997, 16924}, {550, 32996, 33000}, {1657, 14063, 33254}, {3146, 33256, 7791}, {3146, 33272, 11361}, {3522, 14062, 32998}, {3529, 16041, 33265}, {3529, 33019, 16925}, {3534, 8352, 2}, {3543, 7833, 33016}, {3830, 8353, 2}, {3832, 33275, 33003}, {3845, 35955, 2}, {5073, 19695, 14035}, {6655, 33280, 16898}, {6655, 33703, 33280}, {7841, 33193, 33255}, {7887, 62155, 33214}, {7924, 14030, 2}, {8354, 11317, 2}, {8354, 33699, 11317}, {11159, 66349, 2}, {11361, 33256, 33272}, {11361, 33272, 7791}, {11541, 33238, 6658}, {15681, 33228, 33208}, {16041, 33265, 16925}, {17800, 33229, 33244}, {32972, 62152, 33268}, {32974, 50692, 19696}, {32982, 49140, 33257}, {32990, 50690, 14066}, {33005, 33008, 33001}, {33005, 33253, 33008}, {33007, 33017, 33251}, {33008, 33264, 33253}, {33008, 33271, 33264}, {33017, 33255, 7841}, {33019, 33265, 16041}, {33023, 50691, 14042}, {33193, 33255, 33007}, {33226, 62021, 33018}, {33229, 33244, 33248}, {33233, 62144, 33252}, {33234, 62036, 14068}, {33246, 33288, 2}, {33247, 62028, 16044}, {41106, 47061, 2}

X(66391) = EULER LINE INTERCEPT OF X(99)X(12156)

Barycentrics    8*a^4 - 3*a^2*b^2 - b^4 - 3*a^2*c^2 + 6*b^2*c^2 - c^4 : :
X(66391) = X[2] - 3 X[1003], 5 X[2] - 3 X[7841], 7 X[2] - 6 X[8360], 11 X[2] - 12 X[8365], 13 X[2] - 15 X[8366], 5 X[2] - 6 X[8368], 2 X[2] - 3 X[8369], X[2] + 3 X[33007], 7 X[2] - 3 X[33017], 4 X[2] - 3 X[33184], X[2] - 9 X[33187], 11 X[2] - 3 X[33192], 5 X[2] + 3 X[33193], 13 X[2] - 12 X[33213], 11 X[2] - 9 X[33219], 7 X[2] - 9 X[33220], 13 X[2] - 9 X[33251], and many others

X(66391) lies on these lines: {2, 3}, {99, 12156}, {141, 6781}, {187, 13468}, {230, 18546}, {543, 5306}, {598, 12040}, {599, 47102}, {620, 53418}, {754, 3933}, {1285, 22253}, {1353, 13188}, {1383, 62299}, {1384, 32815}, {1569, 5052}, {1992, 51122}, {2482, 14537}, {3053, 63955}, {3314, 14976}, {3734, 47101}, {3815, 32456}, {3972, 15048}, {4558, 18373}, {5034, 63124}, {5215, 20112}, {5475, 32459}, {5480, 38738}, {6390, 7737}, {6645, 10386}, {7747, 59545}, {7750, 32027}, {7766, 47287}, {7778, 43618}, {7782, 31406}, {7788, 63945}, {7789, 7818}, {7801, 63941}, {7812, 59634}, {8182, 8556}, {8588, 58446}, {8589, 15491}, {8591, 63038}, {9890, 44532}, {10546, 43964}, {11164, 14614}, {11168, 46893}, {12154, 35692}, {12155, 35696}, {14568, 32819}, {14712, 14929}, {15655, 34229}, {16509, 26613}, {18362, 44401}, {18424, 44381}, {22110, 63956}, {22486, 50979}, {32833, 63940}, {32836, 63950}, {33458, 52022}, {33459, 52021}, {35007, 63923}, {37671, 51224}, {38741, 39884}, {41133, 48913}, {42052, 65030}, {44377, 62203}, {45103, 56064}, {49843, 49844}, {53142, 63024}

X(66391) = midpoint of X(i) and X(j) for these {i,j}: {1003, 33007}, {7841, 33193}
X(66391) = reflection of X(i) in X(j) for these {i,j}: {7818, 7789}, {7841, 8368}, {8369, 1003}, {33017, 8360}, {33184, 8369}
X(66391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 8354}, {2, 3830, 37350}, {2, 5077, 66335}, {2, 8598, 8703}, {2, 9855, 8353}, {2, 11001, 5077}, {2, 11317, 5066}, {2, 35955, 8358}, {20, 14039, 11287}, {99, 41624, 51123}, {140, 33235, 33227}, {376, 11286, 8359}, {382, 32973, 8361}, {384, 550, 8362}, {384, 33250, 550}, {384, 33265, 8356}, {1003, 7841, 33255}, {1003, 33193, 8368}, {1657, 14001, 8357}, {3529, 33201, 7866}, {3543, 33191, 11318}, {3552, 11361, 35297}, {3552, 19687, 5}, {6656, 33257, 15704}, {6658, 7807, 3627}, {7770, 33244, 548}, {7841, 33255, 8368}, {7887, 33280, 3853}, {8353, 9855, 19710}, {8353, 35954, 2}, {8354, 66318, 2}, {8356, 33250, 33265}, {8356, 33265, 550}, {8358, 15690, 35955}, {8359, 66321, 11286}, {8364, 62144, 33234}, {8366, 33251, 33213}, {8368, 33255, 8369}, {8370, 13586, 549}, {8598, 66319, 2}, {11159, 27088, 3363}, {11285, 33254, 33923}, {11287, 14039, 7819}, {11361, 35297, 5}, {12103, 19697, 7791}, {13586, 19686, 8370}, {14031, 33254, 11285}, {14033, 33239, 35927}, {14033, 35927, 3}, {14034, 33014, 32992}, {14035, 33235, 140}, {14036, 33257, 33264}, {14036, 33264, 6656}, {14037, 33234, 8364}, {14042, 33249, 3858}, {14068, 33233, 3850}, {15681, 33237, 32986}, {17800, 33242, 32974}, {18907, 51123, 41624}, {19687, 35297, 11361}, {19696, 33225, 33229}, {19696, 33229, 62041}, {32973, 54097, 33222}, {32981, 33239, 3}, {32981, 35927, 14033}, {32983, 35287, 5054}, {32986, 33237, 66326}, {32992, 33014, 15712}, {32997, 33217, 66347}, {33007, 33187, 1003}, {33007, 33255, 33193}, {33017, 33220, 8360}, {33185, 62155, 6655}, {33186, 62041, 33229}, {33193, 33255, 7841}, {33225, 33229, 33186}, {62139, 66340, 33263}, {62151, 66347, 32997}

X(66392) = EULER LINE INTERCEPT OF X(6)X(44678)

Barycentrics    4*a^4 - 3*a^2*b^2 - 5*b^4 - 3*a^2*c^2 + 6*b^2*c^2 - 5*c^4 : :
X(66392) = 5 X[2] - 3 X[1003], X[2] - 3 X[7841], 5 X[2] - 6 X[8360], 13 X[2] - 12 X[8365], 17 X[2] - 15 X[8366], 7 X[2] - 6 X[8368], 4 X[2] - 3 X[8369], 7 X[2] - 3 X[33007], X[2] + 3 X[33017], 2 X[2] - 3 X[33184], 17 X[2] - 9 X[33187], 5 X[2] + 3 X[33192], 11 X[2] - 3 X[33193], 11 X[2] - 12 X[33213], 7 X[2] - 9 X[33219], 11 X[2] - 9 X[33220], 5 X[2] - 9 X[33251], and many others

X(66392) lies on these lines: {2, 3}, {6, 44678}, {115, 13468}, {230, 47101}, {316, 15048}, {325, 51123}, {524, 11648}, {597, 14537}, {598, 54773}, {671, 37671}, {754, 5254}, {1570, 8584}, {2031, 3849}, {2549, 9766}, {3589, 62203}, {3793, 63034}, {3933, 7748}, {4045, 53418}, {5024, 32827}, {5309, 63941}, {5461, 46893}, {5969, 14711}, {6321, 48876}, {6390, 44526}, {7615, 8556}, {7745, 7872}, {7750, 14568}, {7761, 18546}, {7765, 41750}, {7767, 44518}, {7778, 43619}, {7788, 52229}, {7789, 65633}, {7790, 18907}, {7810, 39563}, {7825, 63548}, {7827, 12156}, {7830, 63534}, {7843, 9607}, {7847, 31406}, {7865, 63957}, {7873, 63923}, {7897, 47287}, {7898, 14929}, {7910, 59635}, {7911, 32819}, {7998, 20326}, {8588, 44381}, {9300, 63956}, {11057, 22329}, {11168, 40344}, {14614, 63945}, {14907, 43291}, {15491, 43457}, {15655, 63104}, {15810, 20112}, {16509, 55164}, {17503, 60099}, {18424, 58446}, {23334, 63024}, {32532, 60259}, {32892, 50990}, {39764, 63124}, {39838, 44882}, {41748, 63944}, {44415, 63094}, {45103, 62894}, {48913, 52691}

X(66392) = midpoint of X(i) and X(j) for these {i,j}: {1003, 33192}, {7748, 7818}, {7841, 33017}
X(66392) = reflection of X(i) in X(j) for these {i,j}: {1003, 8360}, {3933, 7818}, {8369, 33184}, {33007, 8368}, {33184, 7841}
X(66392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3534, 27088}, {2, 3845, 3363}, {2, 5077, 8354}, {2, 8352, 3845}, {2, 8353, 8703}, {2, 11159, 66318}, {2, 14030, 6661}, {2, 15682, 11159}, {2, 35955, 12100}, {2, 40246, 66328}, {2, 47061, 15701}, {2, 66328, 35954}, {4, 8357, 8362}, {4, 33210, 11287}, {20, 33285, 11288}, {381, 32986, 8359}, {382, 32974, 7819}, {1003, 7841, 33251}, {1003, 33251, 8360}, {3529, 33200, 32954}, {3543, 33190, 11286}, {3853, 66347, 7770}, {5025, 19695, 550}, {5025, 33264, 35297}, {5066, 8358, 2}, {6655, 14041, 8356}, {6655, 33229, 5}, {6656, 33019, 3627}, {7761, 53419, 64093}, {7770, 33279, 3853}, {7807, 33256, 15704}, {7833, 33228, 549}, {7841, 33192, 8360}, {7887, 32997, 548}, {7898, 47286, 14929}, {7901, 19691, 33250}, {7933, 19687, 33185}, {8352, 66349, 2}, {8354, 37350, 2}, {8355, 12100, 2}, {8356, 14041, 5}, {8356, 33229, 14041}, {8360, 33251, 33184}, {8364, 62026, 14035}, {8367, 14893, 33016}, {11285, 32996, 3850}, {11286, 33190, 66326}, {11287, 33210, 8357}, {11288, 33285, 8361}, {13586, 33288, 2}, {14045, 33260, 33249}, {14046, 33256, 33265}, {14046, 33265, 7807}, {14062, 32992, 3858}, {14063, 33234, 140}, {15681, 33240, 32985}, {15684, 33223, 66321}, {16041, 33238, 33272}, {16041, 33272, 3}, {16043, 54097, 3843}, {17800, 33241, 32973}, {19691, 33250, 62159}, {19695, 35297, 33264}, {32972, 33247, 3}, {32980, 33226, 3526}, {32982, 33238, 3}, {32982, 33272, 16041}, {33007, 33219, 8368}, {33017, 33251, 33192}, {33017, 33278, 7841}, {33185, 62041, 19687}, {33186, 62155, 3552}, {33192, 33251, 1003}, {33233, 33253, 33923}, {33235, 33271, 62144}, {33249, 33260, 15712}, {33253, 33290, 33233}, {33264, 35297, 550}, {33271, 33283, 33235}

X(66393) = EULER LINE INTERCEPT OF X(6)X(51123)

Barycentrics    10*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 + 6*b^2*c^2 + c^4 : :
X(66393) =X[2] + 3 X[1003], 7 X[2] - 3 X[7841], 4 X[2] - 3 X[8360], 5 X[2] - 6 X[8365], 11 X[2] - 15 X[8366], 2 X[2] - 3 X[8368], X[2] - 3 X[8369], 5 X[2] + 3 X[33007], 11 X[2] - 3 X[33017], 5 X[2] - 3 X[33184], 7 X[2] + 9 X[33187], 19 X[2] - 3 X[33192], 13 X[2] + 3 X[33193], 7 X[2] - 6 X[33213], 13 X[2] - 9 X[33219], 5 X[2] - 9 X[33220], 17 X[2] - 9 X[33251], X[2] - 9 X[33255], and many others

X(6693) lies on these lines: {2, 3}, {6, 51123}, {99, 63633}, {141, 47101}, {524, 41413}, {754, 7789}, {2482, 9300}, {3589, 32456}, {3734, 13468}, {3972, 6390}, {5039, 8584}, {5305, 7816}, {5306, 52229}, {5969, 36521}, {7778, 44678}, {7799, 12156}, {7801, 63940}, {7804, 32459}, {7863, 41750}, {7880, 63941}, {8667, 37809}, {9766, 18907}, {11165, 63024}, {11544, 30123}, {12150, 59634}, {14148, 32455}, {14537, 22110}, {14614, 19661}, {15300, 39593}, {18546, 43291}, {18583, 33813}, {20582, 40344}, {21309, 32817}, {22331, 63926}, {32896, 63064}, {39141, 61624}, {47287, 63019}, {51122, 63006}

X(66393) =midpoint of X(i) and X(j) for these {i,j}: {1003, 8369}, {33007, 33184}
X(66393) =reflection of X(i) in X(j) for these {i,j}: {7841, 33213}, {8360, 8368}, {8368, 8369}, {33184, 8365}
X(66393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3363, 10109}, {2, 3845, 8355}, {2, 8353, 66335}, {2, 8598, 8354}, {2, 8703, 8358}, {2, 9855, 66349}, {2, 11159, 3845}, {2, 14030, 8370}, {2, 27088, 12100}, {2, 35954, 66318}, {2, 66328, 8352}, {20, 33185, 66347}, {20, 33242, 33185}, {382, 33181, 33186}, {549, 11286, 8367}, {550, 14001, 8364}, {1003, 7841, 33187}, {1003, 33220, 33007}, {1003, 33255, 8369}, {3543, 33197, 33240}, {3552, 7819, 548}, {3552, 14036, 8356}, {6661, 13586, 8359}, {7866, 33239, 15704}, {7892, 33250, 8357}, {8354, 8598, 15690}, {8356, 14036, 7819}, {8357, 33250, 62144}, {8359, 13586, 34200}, {8361, 19687, 3853}, {8362, 33235, 33923}, {8365, 33220, 8368}, {8369, 33007, 8365}, {8369, 33184, 33220}, {8369, 33187, 33213}, {11285, 33227, 61792}, {11286, 32985, 549}, {11287, 35927, 550}, {11288, 14033, 5}, {14001, 35927, 11287}, {14030, 33246, 2}, {14033, 32973, 11288}, {14037, 33235, 8362}, {19687, 33225, 8361}, {27088, 66318, 2}, {32954, 32981, 3627}, {33007, 33220, 33184}, {33183, 33703, 33241}, {33184, 33220, 8365}, {33211, 62155, 32974}, {33246, 66321, 547}, {44245, 66344, 7791}

X(66394) = EULER LINE INTERCEPT OF X(141)X(18546)

Barycentrics    2*a^4 - 3*a^2*b^2 - 7*b^4 - 3*a^2*c^2 + 6*b^2*c^2 - 7*c^4 : :
X(66394) = 7 X[2] - 3 X[1003], X[2] + 3 X[7841], 2 X[2] - 3 X[8360], 7 X[2] - 6 X[8365], 19 X[2] - 15 X[8366], 4 X[2] - 3 X[8368], 5 X[2] - 3 X[8369], 11 X[2] - 3 X[33007], 5 X[2] + 3 X[33017], X[2] - 3 X[33184], 25 X[2] - 9 X[33187], 13 X[2] + 3 X[33192], 19 X[2] - 3 X[33193], 5 X[2] - 6 X[33213], 5 X[2] - 9 X[33219], 13 X[2] - 9 X[33220], X[2] - 9 X[33251], and many others

X(66394) lies on these lines: {2, 3}, {141, 18546}, {316, 12156}, {597, 63956}, {599, 42023}, {754, 5305}, {2549, 51123}, {3631, 32457}, {3793, 7898}, {5254, 7818}, {5306, 63945}, {5309, 63940}, {5461, 40344}, {5969, 19662}, {6390, 7934}, {7761, 13468}, {7767, 7911}, {7784, 63955}, {7790, 7926}, {7806, 14976}, {7817, 63941}, {7844, 47101}, {7853, 53419}, {7913, 53418}, {7935, 63534}, {8556, 16509}, {9300, 31173}, {9466, 63543}, {9766, 15048}, {11168, 18362}, {11544, 30119}, {11648, 52229}, {18907, 44678}, {31168, 59635}, {37672, 44415}, {39524, 63094}, {44401, 46893}, {48913, 63101}

X(66394) = midpoint of X(i) and X(j) for these {i,j}: {5254, 7818}, {7841, 33184}, {8369, 33017}
X(66394) = reflection of X(i) in X(j) for these {i,j}: {1003, 8365}, {8360, 33184}, {8368, 8360}, {8369, 33213}
X(66394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5077, 8703}, {2, 8353, 27088}, {2, 8354, 12100}, {2, 8597, 66319}, {2, 33291, 33228}, {2, 37350, 5066}, {2, 66349, 8354}, {5, 32974, 66347}, {20, 33241, 33186}, {382, 33180, 33185}, {1003, 7841, 33278}, {3543, 33196, 33237}, {3627, 7866, 19697}, {3861, 66346, 7770}, {5025, 8357, 140}, {6655, 8361, 548}, {6655, 14046, 35297}, {7819, 33229, 3853}, {7841, 33219, 33017}, {7841, 33251, 33184}, {7866, 32982, 3627}, {7924, 33228, 8359}, {7924, 33291, 2}, {7933, 33229, 7819}, {8353, 27088, 15690}, {8355, 8358, 2}, {8359, 33228, 547}, {8362, 14063, 3850}, {8369, 33184, 33219}, {8369, 33219, 33213}, {11287, 16041, 5}, {11288, 33272, 550}, {11318, 32986, 549}, {14045, 19690, 32992}, {14046, 35297, 8361}, {14064, 33272, 11288}, {15687, 33223, 66340}, {16041, 32974, 11287}, {32954, 33238, 15704}, {33017, 33184, 33213}, {33017, 33219, 8369}, {33025, 33292, 1656}, {33182, 33703, 33242}, {33184, 33278, 8365}, {33200, 33210, 33285}, {33210, 33285, 3}, {33212, 62155, 32973}, {33213, 33219, 8360}, {33227, 33253, 41981}, {37350, 66335, 2}

X(66395) = EULER LINE INTERCEPT OF X(32)X(32479)

Barycentrics    11*a^4 - 5*a^2*b^2 - 4*b^4 - 5*a^2*c^2 + 10*b^2*c^2 - 4*c^4 : :
X(66395) = 4 X[2] - 5 X[1003], 6 X[2] - 5 X[7841], 21 X[2] - 20 X[8360], 39 X[2] - 40 X[8365], 24 X[2] - 25 X[8366], 19 X[2] - 20 X[8368], 9 X[2] - 10 X[8369], 3 X[2] - 5 X[33007], 7 X[2] - 5 X[33017], 11 X[2] - 10 X[33184], 11 X[2] - 15 X[33187], 9 X[2] - 5 X[33192], X[2] - 5 X[33193], 41 X[2] - 40 X[33213], 16 X[2] - 15 X[33219], 14 X[2] - 15 X[33220], and many others

X(66395) lies on these lines: {2, 3}, {32, 32479}, {543, 7754}, {598, 5013}, {599, 7802}, {671, 3053}, {1975, 3849}, {2482, 7773}, {5182, 51024}, {5206, 8860}, {5969, 10488}, {6337, 23334}, {7617, 15513}, {7622, 39590}, {7747, 11163}, {7759, 15300}, {7782, 11184}, {7785, 11165}, {7793, 40727}, {7801, 11164}, {7812, 31859}, {7817, 65633}, {7823, 8591}, {7827, 44526}, {7847, 47352}, {7936, 50993}, {8182, 59635}, {8593, 11477}, {9605, 32480}, {11152, 48673}, {11160, 32822}, {11456, 35706}, {12154, 42158}, {12155, 42157}, {12191, 38905}, {12355, 58765}, {13108, 22564}, {13881, 26613}, {14712, 63950}, {14907, 15598}, {20065, 52229}, {32826, 63029}, {33698, 62880}, {34505, 51224}, {34511, 43618}, {39785, 63931}, {44519, 52691}, {44678, 59634}, {53105, 60103}, {53109, 60211}, {54494, 60198}

X(66395) = reflection of X(i) in X(j) for these {i,j}: {7841, 33007}, {33192, 8369}, {65633, 7817}
X(66395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14038, 33237}, {2, 14045, 11318}, {20, 8370, 35955}, {382, 33257, 33235}, {1003, 7841, 8366}, {1657, 6658, 7770}, {1657, 11159, 7833}, {3146, 32985, 8352}, {3146, 33250, 7887}, {3529, 19687, 33234}, {3552, 8597, 11318}, {3627, 27088, 33006}, {3627, 33244, 33233}, {5073, 11318, 8597}, {5206, 47617, 8860}, {6658, 7833, 11159}, {7747, 34504, 11163}, {7819, 62162, 33271}, {7833, 11159, 7770}, {7833, 14034, 2}, {7841, 8366, 33219}, {7841, 33007, 1003}, {7841, 33220, 8360}, {7866, 62170, 19691}, {8352, 32985, 7887}, {8352, 33250, 32985}, {8360, 33017, 7841}, {8362, 58203, 33209}, {8369, 33192, 7841}, {8370, 35955, 11285}, {11286, 15685, 33264}, {12102, 33227, 32963}, {14033, 15683, 8353}, {15682, 35927, 33228}, {19695, 32981, 33217}, {19695, 35954, 33190}, {27088, 33006, 33233}, {32981, 33190, 35954}, {32981, 49138, 19695}, {33006, 33244, 27088}, {33007, 33192, 8369}, {33190, 35954, 33217}, {33208, 52942, 5}, {33239, 49135, 33229}, {33264, 66328, 11286}

X(66396) = EULER LINE INTERCEPT OF X(148)X(63954)

Barycentrics    9*a^4 - 5*a^2*b^2 - 6*b^4 - 5*a^2*c^2 + 10*b^2*c^2 - 6*c^4 : :
X(66396) = 6 X[2] - 5 X[1003], 4 X[2] - 5 X[7841], 19 X[2] - 20 X[8360], 41 X[2] - 40 X[8365], 26 X[2] - 25 X[8366], 21 X[2] - 20 X[8368], 11 X[2] - 10 X[8369], 7 X[2] - 5 X[33007], 3 X[2] - 5 X[33017], 9 X[2] - 10 X[33184], 19 X[2] - 15 X[33187], X[2] - 5 X[33192], 9 X[2] - 5 X[33193], 39 X[2] - 40 X[33213], 14 X[2] - 15 X[33219], 16 X[2] - 15 X[33220], and many others

X(66396) lies on these lines: {2, 3}, {148, 63954}, {183, 63957}, {316, 8716}, {538, 65633}, {3849, 41748}, {5182, 48905}, {5969, 40341}, {7748, 14614}, {7754, 63941}, {7756, 63956}, {7757, 44526}, {7802, 8667}, {7818, 32479}, {7842, 7881}, {7880, 11164}, {11055, 63932}, {11057, 34505}, {11152, 38744}, {11185, 15598}, {12355, 34734}, {14976, 63950}, {22486, 48910}, {31859, 43619}, {33698, 60248}, {44562, 62203}, {53105, 62892}, {54494, 62922}

X(66396) = reflection of X(i) in X(j) for these {i,j}: {1003, 33017}, {14614, 7748}, {33193, 33184}
X(66396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14034, 11286}, {2, 33284, 33240}, {381, 33264, 35955}, {382, 33256, 33234}, {1003, 33017, 7841}, {1003, 33219, 8368}, {1657, 33019, 7887}, {3146, 19695, 7770}, {3529, 33229, 33235}, {3543, 8356, 11317}, {3627, 8354, 33016}, {3627, 32997, 11285}, {3830, 7833, 44543}, {5077, 15684, 11361}, {7866, 49133, 19696}, {8354, 33016, 11285}, {8357, 62044, 33280}, {8368, 33007, 1003}, {8597, 33264, 381}, {11288, 62158, 9855}, {11318, 15685, 33265}, {11361, 40246, 15684}, {13586, 14045, 2}, {15682, 33272, 8370}, {15683, 16041, 8598}, {17538, 54097, 33249}, {17578, 33247, 32992}, {32982, 33250, 33218}, {32982, 49138, 33250}, {32997, 33016, 8354}, {33017, 33193, 33184}, {33184, 33193, 1003}, {33238, 49135, 19687}

X(66397) = EULER LINE INTERCEPT OF X(148)X(63950)

Barycentrics    13*a^4 - 7*a^2*b^2 - 8*b^4 - 7*a^2*c^2 + 14*b^2*c^2 - 8*c^4 : :
X(66397) =8 X[2] - 7 X[1003], 6 X[2] - 7 X[7841], 27 X[2] - 28 X[8360], 57 X[2] - 56 X[8365], 36 X[2] - 35 X[8366], 29 X[2] - 28 X[8368], 15 X[2] - 14 X[8369], 9 X[2] - 7 X[33007], 5 X[2] - 7 X[33017], 13 X[2] - 14 X[33184], 25 X[2] - 21 X[33187], 3 X[2] - 7 X[33192], 11 X[2] - 7 X[33193], 55 X[2] - 56 X[33213], 20 X[2] - 21 X[33219], 22 X[2] - 21 X[33220], and many others

X(66397) lies on these lines: {2, 3}, {148, 63950}, {543, 7855}, {626, 11164}, {1975, 32479}, {3849, 7754}, {5023, 9166}, {5585, 51237}, {7756, 11163}, {7773, 34504}, {7776, 8591}, {7802, 34505}, {7812, 44526}, {7893, 8596}, {7910, 21358}, {7926, 31859}, {11054, 63938}, {11161, 53097}, {14976, 63954}, {15300, 32821}, {44518, 51224}, {52691, 65630}, {53106, 60220}, {53107, 62895}, {54646, 62881}

X(66397) =reflection of X(7841) in X(33192)
X(66397) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14040, 33237}, {2, 33289, 11318}, {382, 7833, 11317}, {3543, 8353, 44543}, {5073, 33256, 7770}, {7833, 11317, 11285}, {7833, 14066, 2}, {7833, 40246, 382}, {7841, 8369, 33219}, {7841, 33007, 8366}, {8359, 62036, 52942}, {8366, 33007, 1003}, {8369, 33017, 7841}, {9855, 11318, 33235}, {9855, 33019, 11318}, {11318, 17800, 9855}, {15704, 33279, 33233}, {15704, 37350, 33208}, {17800, 33019, 33235}, {19691, 40246, 7833}, {32997, 52942, 8359}, {33006, 52943, 550}, {33017, 33193, 33213}, {33208, 33279, 37350}, {33208, 37350, 33233}

X(66398) = EULER LINE INTERCEPT OF X(543)X(20065)

Barycentrics    17*a^4 - 8*a^2*b^2 - 7*b^4 - 8*a^2*c^2 + 16*b^2*c^2 - 7*c^4 : :
X(66398) = 7 X[2] - 8 X[1003], 9 X[2] - 8 X[7841], 33 X[2] - 32 X[8360], 63 X[2] - 64 X[8365], 39 X[2] - 40 X[8366], 31 X[2] - 32 X[8368], 15 X[2] - 16 X[8369], 3 X[2] - 4 X[33007], 5 X[2] - 4 X[33017], 17 X[2] - 16 X[33184], 5 X[2] - 6 X[33187], 65 X[2] - 64 X[33213], 25 X[2] - 24 X[33219], 23 X[2] - 24 X[33220], 13 X[2] - 12 X[33251], 11 X[2] - 12 X[33255], and many others

X(66398) lies on these lines: {2, 3}, {543, 20065}, {3849, 7855}, {5969, 20105}, {6392, 8596}, {7620, 7793}, {7785, 53142}, {7812, 43618}, {7827, 43619}, {7926, 34511}, {8587, 38259}, {9939, 32815}, {10484, 18845}, {12154, 43633}, {12155, 43632}, {14976, 32836}, {15300, 63931}, {32816, 52695}, {35369, 63042}, {44519, 63101}, {44526, 63045}, {60113, 62904}

X(66398) = reflection of X(i) in X(j) for these {i,j}: {2, 33193}, {33192, 33007}
X(66398) = anticomplement of X(33192)
X(66398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33209, 7833}, {2, 33214, 33208}, {4, 9855, 33208}, {4, 33208, 2}, {4, 33268, 33206}, {4, 33276, 33270}, {382, 8598, 33006}, {1003, 7841, 8365}, {1003, 33278, 2}, {1657, 33280, 32965}, {3146, 33244, 32996}, {3528, 14066, 33009}, {3529, 7833, 52943}, {3529, 14035, 33209}, {3529, 19696, 14035}, {3543, 33265, 2}, {3627, 33254, 32963}, {5059, 6658, 32997}, {6658, 32997, 14031}, {7833, 14035, 2}, {7833, 52943, 33209}, {8359, 33269, 2}, {8369, 33007, 33187}, {8369, 33213, 8366}, {8370, 32965, 2}, {8597, 32985, 14063}, {8597, 33257, 32985}, {8598, 33006, 32964}, {9855, 33208, 33214}, {9855, 37461, 33265}, {11001, 11361, 33207}, {11361, 33207, 2}, {14033, 33263, 2}, {14035, 52943, 7833}, {14037, 33190, 2}, {14041, 33266, 2}, {14042, 17538, 33012}, {14063, 32985, 2}, {15704, 16924, 33243}, {15708, 32994, 2}, {19686, 33272, 2}, {19687, 49137, 33271}, {32964, 33006, 2}, {32985, 33703, 8597}, {33007, 33017, 8369}, {33007, 33192, 2}, {33017, 33187, 2}, {33192, 33193, 33007}, {33250, 49136, 33279}, {33257, 33703, 14063}

X(66399) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES EQUILATERAL AND CIRCUMTANGENTIAL

Barycentrics    a*(b*c*((b*cos(B/3)-cos(A/3)*(b*cos(C/3)+c*cos(B/3))+c*cos(C/3))*(S^2+SB*SC)-a^3*SA*(cos(A/3)-cos(B/3)*cos(C/3)))+a*SA*S^2) : :

César Lozada, Nov 19, 2024.
The equilateral Hatzipolakis-Moses triangle was introduced in Euclid 6964.

X(66399) lies on these lines: {3, 65156}, {5, 3280}, {30, 13590}

X(66399) = (X(3), X(65156))-harmonic conjugate of X(66400)

X(66400) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES EQUILATERAL AND CIRCUMNORMAL

Barycentrics    a*(b*c*(-(b*cos(B/3)-cos(A/3)*(b*cos(C/3)+c*cos(B/3))+c*cos(C/3))*(3*S^2-SB*SC)-a^3*SA*(cos(A/3)-cos(B/3)*cos(C/3)))+a*SA*S^2) : :

César Lozada, Nov 19, 2024.
The equilateral Hatzipolakis-Moses triangle was introduced in Euclid 6964.

X(66400) lies on these lines: {3, 65156}, {140, 3281}

X(66400) = (X(3), X(65156))-harmonic conjugate of X(66399)

X(66401) = CENTER OF THE 2nd DAO PERSPECONIC OF THESE TRIANGLES: 2nd CONWAY TO ABC

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

Let ABC and A'B'C' be two triangles, neither inscribed in the other, such that AA', BB', CC' are concurrent. Let AB' meet BC at Ba, AC' meet BC at Ca, define Ab, Cb, Bc, Ac cyclically. Then six points Ba, Ca, Cb, Ab, Ac, Bc lie on a conic. (Dao Thanh Oai, Nov. 14, 2024)

(César Lozada, - Nov. 19, 2024): This conic is named here the 2nd Dao-perspeconic of ABC to A'B'C'. Of course, there exists a 2nd Dao-perspeconic of A'B'C' to ABC.

The appearance of (T, i, j) in the following list means that the centers of the 2nd Dao-perspeconic ABC to T and T to ABC are X(i) and X(j): (ABC-X3 reflections, 17807, 45188), (2nd Conway, 478, 66401), (Ehrmann-mid, 45191, 45192), (outer-Garcia, 3588, 45189), (Johnson, 31353, 45190), (5th mixtilinear, 45193, 45194), (orthic axes, 5702, 66402)

The appearance of (T, i, j) in the following list means that the perspectors of the 2nd Dao-perspeconic ABC to T and T to ABC are X(i) and X(j): (circumsymmedial, 6, 6), (2nd mixtilinear, 1, 1), (orthic axes, 4, 4).

X(66401) lies on these lines: {2, 2140}, {8, 22278}

X(66402) = CENTER OF THE 2nd DAO PERSPECONIC OF THESE TRIANGLES: ORTHIC AXES TO ABC

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(14*a^4-15*(b^2+c^2)*a^2+(b^2-c^2)^2) : :

César Lozada, - Nov. 19, 2024.
The center of the reciprocal 2nd Dao perspeconic of these triangles is X(5702).

X(66402) lies on these lines: {4, 6}, {216, 34200}, {264, 20583}, {577, 44682}, {3284, 12108}, {5158, 62104}, {5421, 16328}, {15689, 15851}, {15693, 15905}, {15705, 36748}, {15860, 58203}, {27377, 63124}, {32001, 51185}, {36751, 58188}, {38292, 55863}, {43981, 63062}, {48154, 52704}, {52703, 62078}, {59657, 61942}

X(66402) = polar conjugate of the isotomic conjugate of X(61792)
X(66402) = pole of the line {3580, 3861} with respect to the Dou circles radical circle
X(66402) = pole of the line {44436, 61792} with respect to the Moses circles radical circle
X(66402) = barycentric product X(4)*X(61792)
X(66402) = trilinear product X(19)*X(61792)
X(66402) = trilinear quotient X(61792)/X(63)
X(66402) = (X(393), X(6749))-harmonic conjugate of X(6748)

X(66403) = X(11)X(1385)∩X(10260)X(10269)

Barycentrics    a*(4*a^12 - 10*a^11*b - 4*a^10*b^2 + 32*a^9*b^3 - 22*a^8*b^4 - 28*a^7*b^5 + 48*a^6*b^6 - 8*a^5*b^7 - 32*a^4*b^8 + 22*a^3*b^9 + 4*a^2*b^10 - 8*a*b^11 + 2*b^12 - 10*a^11*c + 38*a^10*b*c - 33*a^9*b^2*c - 55*a^8*b^3*c + 128*a^7*b^4*c - 56*a^6*b^5*c - 86*a^5*b^6*c + 118*a^4*b^7*c - 30*a^3*b^8*c - 38*a^2*b^9*c + 31*a*b^10*c - 7*b^11*c - 4*a^10*c^2 - 33*a^9*b*c^2 + 130*a^8*b^2*c^2 - 99*a^7*b^3*c^2 - 137*a^6*b^4*c^2 + 264*a^5*b^5*c^2 - 98*a^4*b^6*c^2 - 99*a^3*b^7*c^2 + 107*a^2*b^8*c^2 - 33*a*b^9*c^2 + 2*b^10*c^2 + 32*a^9*c^3 - 55*a^8*b*c^3 - 99*a^7*b^2*c^3 + 304*a^6*b^3*c^3 - 170*a^5*b^4*c^3 - 184*a^4*b^5*c^3 + 273*a^3*b^6*c^3 - 86*a^2*b^7*c^3 - 36*a*b^8*c^3 + 21*b^9*c^3 - 22*a^8*c^4 + 128*a^7*b*c^4 - 137*a^6*b^2*c^4 - 170*a^5*b^3*c^4 + 388*a^4*b^4*c^4 - 166*a^3*b^5*c^4 - 111*a^2*b^6*c^4 + 116*a*b^7*c^4 - 26*b^8*c^4 - 28*a^7*c^5 - 56*a^6*b*c^5 + 264*a^5*b^2*c^5 - 184*a^4*b^3*c^5 - 166*a^3*b^4*c^5 + 248*a^2*b^5*c^5 - 70*a*b^6*c^5 - 14*b^7*c^5 + 48*a^6*c^6 - 86*a^5*b*c^6 - 98*a^4*b^2*c^6 + 273*a^3*b^3*c^6 - 111*a^2*b^4*c^6 - 70*a*b^5*c^6 + 44*b^6*c^6 - 8*a^5*c^7 + 118*a^4*b*c^7 - 99*a^3*b^2*c^7 - 86*a^2*b^3*c^7 + 116*a*b^4*c^7 - 14*b^5*c^7 - 32*a^4*c^8 - 30*a^3*b*c^8 + 107*a^2*b^2*c^8 - 36*a*b^3*c^8 - 26*b^4*c^8 + 22*a^3*c^9 - 38*a^2*b*c^9 - 33*a*b^2*c^9 + 21*b^3*c^9 + 4*a^2*c^10 + 31*a*b*c^10 + 2*b^2*c^10 - 8*a*c^11 - 7*b*c^11 + 2*c^12) : :

See David Nguyen and Peter Moses, euclid 7262.

X(66403) lies on these lines: {11, 1385}, {10260, 10269}

X(66404) = MIDPOINT OF X(11) AND X(66403)

Barycentrics    4*a^16 - 14*a^15*b + 56*a^13*b^3 - 58*a^12*b^4 - 66*a^11*b^5 + 148*a^10*b^6 - 20*a^9*b^7 - 150*a^8*b^8 + 110*a^7*b^9 + 56*a^6*b^10 - 96*a^5*b^11 + 10*a^4*b^12 + 34*a^3*b^13 - 12*a^2*b^14 - 4*a*b^15 + 2*b^16 - 14*a^15*c + 74*a^14*b*c - 97*a^13*b^2*c - 128*a^12*b^3*c + 430*a^11*b^4*c - 214*a^10*b^5*c - 455*a^9*b^6*c + 650*a^8*b^7*c - 50*a^7*b^8*c - 490*a^6*b^9*c + 329*a^5*b^10*c + 76*a^4*b^11*c - 158*a^3*b^12*c + 38*a^2*b^13*c + 15*a*b^14*c - 6*b^15*c - 97*a^13*b*c^2 + 382*a^12*b^2*c^2 - 340*a^11*b^3*c^2 - 593*a^10*b^4*c^2 + 1359*a^9*b^5*c^2 - 486*a^8*b^6*c^2 - 1062*a^7*b^7*c^2 + 1200*a^6*b^8*c^2 - 113*a^5*b^9*c^2 - 484*a^4*b^10*c^2 + 258*a^3*b^11*c^2 - 15*a^2*b^12*c^2 - 5*a*b^13*c^2 - 4*b^14*c^2 + 56*a^13*c^3 - 128*a^12*b*c^3 - 340*a^11*b^2*c^3 + 1304*a^10*b^3*c^3 - 897*a^9*b^4*c^3 - 1369*a^8*b^5*c^3 + 2434*a^7*b^6*c^3 - 680*a^6*b^7*c^3 - 1147*a^5*b^8*c^3 + 921*a^4*b^9*c^3 - 56*a^3*b^10*c^3 - 78*a^2*b^11*c^3 - 50*a*b^12*c^3 + 30*b^13*c^3 - 58*a^12*c^4 + 430*a^11*b*c^4 - 593*a^10*b^2*c^4 - 897*a^9*b^3*c^4 + 2720*a^8*b^4*c^4 - 1428*a^7*b^5*c^4 - 1718*a^6*b^6*c^4 + 2273*a^5*b^7*c^4 - 452*a^4*b^8*c^4 - 448*a^3*b^9*c^4 + 117*a^2*b^10*c^4 + 70*a*b^11*c^4 - 16*b^12*c^4 - 66*a^11*c^5 - 214*a^10*b*c^5 + 1359*a^9*b^2*c^5 - 1369*a^8*b^3*c^5 - 1428*a^7*b^4*c^5 + 3260*a^6*b^5*c^5 - 1246*a^5*b^6*c^5 - 997*a^4*b^7*c^5 + 756*a^3*b^8*c^5 - 30*a^2*b^9*c^5 + 29*a*b^10*c^5 - 54*b^11*c^5 + 148*a^10*c^6 - 455*a^9*b*c^6 - 486*a^8*b^2*c^6 + 2434*a^7*b^3*c^6 - 1718*a^6*b^4*c^6 - 1246*a^5*b^5*c^6 + 1852*a^4*b^6*c^6 - 386*a^3*b^7*c^6 - 90*a^2*b^8*c^6 - 135*a*b^9*c^6 + 68*b^10*c^6 - 20*a^9*c^7 + 650*a^8*b*c^7 - 1062*a^7*b^2*c^7 - 680*a^6*b^3*c^7 + 2273*a^5*b^4*c^7 - 997*a^4*b^5*c^7 - 386*a^3*b^6*c^7 + 140*a^2*b^7*c^7 + 80*a*b^8*c^7 + 30*b^9*c^7 - 150*a^8*c^8 - 50*a^7*b*c^8 + 1200*a^6*b^2*c^8 - 1147*a^5*b^3*c^8 - 452*a^4*b^4*c^8 + 756*a^3*b^5*c^8 - 90*a^2*b^6*c^8 + 80*a*b^7*c^8 - 100*b^8*c^8 + 110*a^7*c^9 - 490*a^6*b*c^9 - 113*a^5*b^2*c^9 + 921*a^4*b^3*c^9 - 448*a^3*b^4*c^9 - 30*a^2*b^5*c^9 - 135*a*b^6*c^9 + 30*b^7*c^9 + 56*a^6*c^10 + 329*a^5*b*c^10 - 484*a^4*b^2*c^10 - 56*a^3*b^3*c^10 + 117*a^2*b^4*c^10 + 29*a*b^5*c^10 + 68*b^6*c^10 - 96*a^5*c^11 + 76*a^4*b*c^11 + 258*a^3*b^2*c^11 - 78*a^2*b^3*c^11 + 70*a*b^4*c^11 - 54*b^5*c^11 + 10*a^4*c^12 - 158*a^3*b*c^12 - 15*a^2*b^2*c^12 - 50*a*b^3*c^12 - 16*b^4*c^12 + 34*a^3*c^13 + 38*a^2*b*c^13 - 5*a*b^2*c^13 + 30*b^3*c^13 - 12*a^2*c^14 + 15*a*b*c^14 - 4*b^2*c^14 - 4*a*c^15 - 6*b*c^15 + 2*c^16 : :

See David Nguyen and Peter Moses, euclid 7262.

X(66404) lies on this line: {11, 1385}

X(66404) = midpoint of X(11) and X(66403)

X(66405) = EULER LINE INTERCEPT OF X(99)X(63956)

Barycentrics    6*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 7*b^2*c^2 - 3*c^4 : :
X(66405) = 4 X[2] - 3 X[7833], 5 X[2] - 4 X[8354], 7 X[2] - 6 X[8356], 9 X[2] - 8 X[8358], 13 X[2] - 12 X[8359], 23 X[2] - 24 X[8367], 5 X[2] - 6 X[8370], 2 X[2] - 3 X[11361], 5 X[2] - 3 X[33264], 4 X[3845] - 3 X[55008], 9 X[7833] - 8 X[8353], 15 X[7833] - 16 X[8354], 7 X[7833] - 8 X[8356], 27 X[7833] - 32 X[8358], 13 X[7833] - 16 X[8359], and many others

X(66405) lies on these lines: {2, 3}, {99, 63956}, {148, 14614}, {316, 7908}, {385, 43618}, {524, 19569}, {538, 7823}, {543, 7837}, {2482, 48913}, {3329, 43619}, {3849, 14711}, {5182, 48901}, {6321, 58765}, {6781, 17004}, {7737, 63038}, {7747, 7757}, {7748, 7920}, {7777, 62203}, {7785, 8716}, {7802, 9466}, {7840, 44678}, {7864, 65633}, {7893, 32819}, {8591, 9766}, {8667, 14712}, {9300, 32480}, {11152, 23698}, {11185, 47102}, {14458, 54750}, {14537, 32479}, {14976, 37671}, {17129, 32826}, {17503, 60104}, {18362, 26613}, {18546, 51224}, {20081, 63940}, {20094, 51122}, {22253, 35369}, {22486, 29012}, {39141, 51163}, {45103, 60233}, {54540, 54839}

X(66405) = reflection of X(i) in X(j) for these {i,j}: {7757, 7747}, {7802, 9466}, {7833, 11361}, {14976, 37671}, {33264, 8370}
X(66405) = anticomplement of X(8353)
X(66405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 11159, 14030}, {2, 15640, 66390}, {2, 15682, 8597}, {2, 33264, 8354}, {2, 40246, 66388}, {2, 66389, 9855}, {4, 19696, 33257}, {4, 33193, 13586}, {4, 33244, 32967}, {4, 33257, 7907}, {20, 14042, 16921}, {20, 33016, 33273}, {381, 33265, 33274}, {381, 66395, 33265}, {382, 6658, 5025}, {550, 33018, 33015}, {1003, 33240, 33225}, {1657, 16044, 33275}, {3146, 33280, 384}, {3529, 14068, 7824}, {3529, 32983, 33207}, {3534, 11317, 2}, {3543, 33007, 14041}, {3552, 3627, 14062}, {3830, 66387, 2}, {3832, 33254, 16923}, {3845, 8598, 2}, {3853, 33250, 32966}, {5059, 16924, 33267}, {5073, 11286, 66396}, {5076, 33235, 32993}, {6655, 14034, 16895}, {7770, 49136, 19691}, {7841, 19686, 14036}, {7866, 19693, 14040}, {8352, 66391, 2}, {8354, 8370, 2}, {8597, 66328, 2}, {11159, 62040, 66388}, {11159, 66388, 2}, {11286, 66396, 6655}, {11287, 62045, 66397}, {13586, 19696, 33193}, {13586, 32967, 33216}, {13586, 33193, 33257}, {14031, 33238, 7948}, {14033, 33192, 7924}, {14033, 62042, 33192}, {14035, 33256, 7876}, {14035, 33703, 33256}, {14041, 33007, 33246}, {14042, 33273, 33016}, {14068, 33207, 32983}, {16925, 17578, 14044}, {19687, 33019, 7892}, {19687, 62036, 33019}, {32968, 62171, 33209}, {32971, 50692, 33271}, {32979, 49140, 33253}, {32981, 33279, 7901}, {32981, 50691, 33279}, {32983, 33207, 7824}, {32996, 33239, 33245}, {33016, 33273, 16921}, {33216, 33244, 13586}, {33239, 62021, 32996}, {33699, 66391, 8352}, {35480, 40890, 62954}, {35927, 50687, 33006}, {35954, 66394, 2}, {52942, 66389, 2}, {62040, 66388, 40246}, {66319, 66392, 2}

X(66406) = EULER LINE INTERCEPT OF X(148)X(8667)

Barycentrics    6*a^4 - 4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 5*b^2*c^2 - 3*c^4 : :
X(66406) = 2 X[2] - 3 X[7833], 3 X[2] - 4 X[8354], 5 X[2] - 6 X[8356], 7 X[2] - 8 X[8358], 11 X[2] - 12 X[8359], 25 X[2] - 24 X[8367], 7 X[2] - 6 X[8370], 4 X[2] - 3 X[11361], X[2] - 3 X[33264], 4 X[3534] - 3 X[60651], 2 X[3830] - 3 X[55008], 3 X[7833] - 4 X[8353], 9 X[7833] - 8 X[8354], 5 X[7833] - 4 X[8356], 21 X[7833] - 16 X[8358], 11 X[7833] - 8 X[8359], and many others

X(66406) lies on these lines: {2, 3}, {148, 8667}, {194, 63941}, {385, 43619}, {524, 14976}, {538, 7802}, {543, 11057}, {671, 47101}, {754, 11055}, {1078, 63957}, {2549, 63038}, {2794, 11152}, {3329, 43618}, {3849, 7837}, {5182, 48898}, {6781, 7806}, {7756, 7757}, {7777, 63956}, {7779, 51122}, {7785, 44519}, {7788, 8591}, {7809, 34504}, {7811, 14711}, {7842, 7891}, {7864, 12150}, {7877, 63947}, {7904, 9466}, {7906, 8716}, {7921, 63548}, {11648, 51224}, {14537, 52691}, {14614, 14712}, {15300, 32458}, {17004, 46893}, {19569, 32480}, {22165, 51374}, {22486, 29317}, {23698, 33706}, {33610, 49848}, {33611, 49847}, {38741, 58765}

X(66406) = reflection of X(i) in X(j) for these {i,j}: {2, 8353}, {7757, 7756}, {7823, 7757}, {7833, 33264}, {11361, 7833}, {19569, 41624}
X(66406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8353, 7833}, {2, 11001, 9855}, {2, 15640, 52942}, {2, 33264, 8353}, {2, 40246, 3830}, {2, 52943, 11001}, {2, 62160, 66389}, {2, 66328, 14030}, {2, 66389, 66328}, {2, 66390, 8597}, {4, 33207, 33273}, {4, 33209, 33267}, {4, 33267, 33275}, {4, 33275, 33015}, {20, 5025, 33268}, {20, 32982, 33254}, {20, 33017, 13586}, {20, 33256, 5025}, {20, 33271, 33256}, {376, 14041, 33274}, {376, 33192, 14041}, {382, 33260, 16921}, {550, 33019, 7907}, {1657, 6655, 33257}, {3146, 7824, 14066}, {3146, 33253, 7824}, {3522, 33279, 32967}, {3528, 32996, 16923}, {3529, 32986, 33193}, {3529, 32997, 384}, {3534, 66388, 2}, {3543, 33008, 33013}, {3830, 35955, 2}, {5059, 7791, 19696}, {5077, 15685, 66387}, {5077, 66387, 2}, {6655, 33257, 7892}, {6658, 7876, 14032}, {6658, 33234, 7876}, {7791, 19696, 14034}, {7841, 15681, 33265}, {7841, 33265, 33246}, {7924, 33007, 14036}, {7948, 32981, 14040}, {8352, 8703, 2}, {8358, 8370, 2}, {8598, 66392, 2}, {11287, 62158, 66395}, {11287, 66395, 19686}, {11541, 33226, 14068}, {12103, 33229, 33014}, {13586, 33017, 5025}, {13586, 33256, 33017}, {14033, 62161, 66398}, {14063, 17538, 33276}, {14064, 62146, 33214}, {15683, 33272, 33007}, {15704, 19695, 3552}, {16041, 62130, 33208}, {17800, 33234, 6658}, {19569, 32480, 41624}, {19710, 66392, 8598}, {32965, 33703, 14042}, {32970, 62133, 33252}, {32982, 33245, 5025}, {32982, 33254, 33245}, {32985, 33278, 14046}, {32986, 33193, 384}, {32997, 33193, 32986}, {33007, 33272, 7924}, {33023, 49140, 33280}, {33207, 33273, 33275}, {33238, 33244, 7901}, {33238, 62147, 33244}, {33247, 49138, 14035}, {33263, 66398, 14033}, {33267, 33273, 33207}, {35954, 66335, 2}, {41099, 47061, 2}, {54097, 62097, 33000}, {66349, 66391, 2}

X(66407) = EULER LINE INTERCEPT OF X(543)X(19569)

Barycentrics    12*a^4 - 5*a^2*b^2 - 6*b^4 - 5*a^2*c^2 + 13*b^2*c^2 - 6*c^4 : :
X(66407) = 7 X[2] - 6 X[7833], 5 X[2] - 4 X[8353], 9 X[2] - 8 X[8354], 13 X[2] - 12 X[8356], 17 X[2] - 16 X[8358], 25 X[2] - 24 X[8359], 47 X[2] - 48 X[8367], 11 X[2] - 12 X[8370], 5 X[2] - 6 X[11361], 4 X[2] - 3 X[33264], 15 X[7833] - 14 X[8353], 27 X[7833] - 28 X[8354], 13 X[7833] - 14 X[8356], 51 X[7833] - 56 X[8358], 25 X[7833] - 28 X[8359], 47 X[7833] - 56 X[8367], and many others

X(66407) lies on these lines: {2, 3}, {543, 19569}, {5182, 48904}, {7766, 43618}, {7793, 63957}, {8591, 44678}, {8596, 63093}, {9939, 14711}, {10723, 58765}, {12150, 65633}, {14537, 32480}, {20081, 63941}, {22486, 29323}, {33623, 49938}, {33625, 49937}, {43619, 62994}, {63021, 63956}

X(66407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15640, 40246}, {2, 52944, 15640}, {2, 62168, 52943}, {3543, 66398, 33265}, {3830, 9855, 2}, {3853, 33268, 33011}, {5073, 19696, 33019}, {8353, 11361, 2}, {8597, 66387, 2}, {11001, 52942, 2}, {11541, 33280, 19691}, {14030, 66349, 2}, {14066, 15704, 33022}, {15682, 66389, 2}, {15684, 66395, 14041}, {62040, 66387, 8597}, {66328, 66388, 2}

X(66408) = EULER LINE INTERCEPT OF X(32)X(63957)

Barycentrics    6*a^4 - a^2*b^2 - 3*b^4 - a^2*c^2 + 8*b^2*c^2 - 3*c^4 : :
X(66408) = 5 X[2] - 3 X[7833], 4 X[2] - 3 X[8356], 5 X[2] - 4 X[8358], 7 X[2] - 6 X[8359], 11 X[2] - 12 X[8367], 2 X[2] - 3 X[8370], X[2] - 3 X[11361], 7 X[2] - 3 X[33264], 4 X[5066] - 3 X[37345], 6 X[7833] - 5 X[8353], 9 X[7833] - 10 X[8354], 4 X[7833] - 5 X[8356], 3 X[7833] - 4 X[8358], 7 X[7833] - 10 X[8359], 11 X[7833] - 20 X[8367], 2 X[7833] - 5 X[8370], and many others

X(66408) lies on these lines: {2, 3}, {32, 63957}, {76, 63941}, {99, 53418}, {148, 18907}, {183, 43618}, {325, 62203}, {538, 7747}, {543, 14537}, {598, 9300}, {671, 5306}, {754, 14711}, {1503, 22486}, {2549, 53489}, {3849, 37671}, {3972, 53419}, {5182, 5480}, {5210, 53127}, {5254, 12150}, {6781, 37688}, {7620, 63034}, {7737, 14614}, {7745, 7757}, {7750, 9466}, {7754, 32826}, {7756, 44562}, {7774, 47287}, {7775, 59634}, {7788, 44678}, {7812, 11055}, {7823, 63940}, {7826, 63947}, {7837, 52229}, {7843, 32820}, {8556, 14907}, {8584, 8593}, {8591, 51123}, {8594, 33458}, {8595, 33459}, {8667, 11185}, {8716, 65630}, {8781, 45103}, {9777, 32463}, {11174, 43619}, {12154, 41107}, {12155, 41108}, {13468, 51224}, {14160, 38748}, {14458, 54751}, {14712, 64093}, {17131, 63948}, {17503, 60093}, {18513, 26629}, {18514, 26686}, {18546, 22329}, {18842, 54889}, {19569, 63945}, {19661, 41135}, {20065, 63954}, {20112, 26613}, {21969, 55005}, {22110, 48913}, {23698, 44422}, {32456, 37647}, {32479, 63101}, {32532, 62930}, {36521, 50280}, {60260, 60281}

X(66408) = reflection of X(i) in X(j) for these {i,j}: {7750, 9466}, {7756, 44562}, {7757, 7745}, {8353, 2}, {8356, 8370}, {8370, 11361}, {33264, 8359}, {41624, 14537}
X(66408) = anticomplement of X(8354)
X(66408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3830, 8352}, {2, 7833, 8358}, {2, 8353, 8356}, {2, 8597, 66392}, {2, 9855, 8703}, {2, 11001, 35955}, {2, 11159, 66319}, {2, 14030, 66318}, {2, 15682, 66388}, {2, 15697, 47061}, {2, 52942, 3830}, {2, 66319, 35954}, {2, 66328, 66391}, {2, 66387, 8598}, {2, 66388, 66349}, {2, 66389, 3534}, {2, 66390, 5077}, {4, 1003, 33228}, {4, 19687, 7807}, {4, 32981, 7887}, {5, 6658, 33250}, {5, 33227, 16923}, {381, 33007, 35297}, {382, 11286, 33017}, {382, 14035, 6656}, {384, 3627, 33229}, {384, 33229, 8363}, {546, 3552, 33249}, {1003, 7887, 33191}, {1003, 33228, 7807}, {3146, 7770, 19695}, {3146, 32986, 66396}, {3363, 8703, 2}, {3529, 32979, 11285}, {3543, 14033, 7841}, {3552, 14066, 546}, {3830, 11159, 2}, {3832, 33239, 33233}, {3845, 66391, 2}, {5066, 27088, 2}, {5077, 62040, 66390}, {6658, 14042, 5}, {7747, 32819, 7762}, {7770, 66396, 32986}, {7819, 62026, 33019}, {7841, 14033, 6661}, {7866, 62023, 33279}, {7933, 14032, 19697}, {8352, 11159, 35954}, {8352, 66319, 2}, {8353, 8370, 2}, {8360, 66321, 14036}, {8361, 12102, 14062}, {8362, 62041, 33256}, {8369, 15687, 14041}, {11159, 52942, 8352}, {11286, 33017, 6656}, {11287, 15684, 33192}, {11288, 14269, 33006}, {11317, 66387, 2}, {12101, 66393, 37350}, {14031, 33279, 7866}, {14034, 33019, 7819}, {14035, 33017, 11286}, {14041, 19686, 8369}, {14068, 33193, 33016}, {14068, 33280, 3}, {16044, 19696, 550}, {19687, 33228, 1003}, {32456, 43457, 37647}, {32954, 62008, 32996}, {32971, 33703, 33234}, {32981, 33191, 1003}, {32986, 66396, 19695}, {32995, 33254, 3526}, {32999, 33214, 3}, {33002, 33268, 3530}, {33003, 33252, 3}, {33005, 33208, 5054}, {33006, 33187, 11288}, {33008, 66398, 15681}, {33013, 33265, 549}, {33016, 33193, 3}, {33016, 33280, 33193}, {33018, 33257, 140}, {33024, 33276, 632}, {33272, 62042, 66397}, {33699, 66392, 8597}, {37350, 66393, 2}, {44543, 66395, 376}, {66318, 66394, 2}

X(66409) = EULER LINE INTERCEPT OF X(76)X(63940)

Barycentrics    6*a^4 + a^2*b^2 - 3*b^4 + a^2*c^2 + 10*b^2*c^2 - 3*c^4 : :
X(66409) = 7 X[2] - 3 X[7833], 5 X[2] - 3 X[8356], 4 X[2] - 3 X[8359], 5 X[2] - 6 X[8367], X[2] - 3 X[8370], X[2] + 3 X[11361], 11 X[2] - 3 X[33264], 9 X[7833] - 7 X[8353], 6 X[7833] - 7 X[8354], 5 X[7833] - 7 X[8356], 9 X[7833] - 14 X[8358], 4 X[7833] - 7 X[8359], 5 X[7833] - 14 X[8367], X[7833] - 7 X[8370], X[7833] + 7 X[11361], 11 X[7833] - 7 X[33264], 2 X[8353] - 3 X[8354], and many others

X(66409) lies on these lines: {2, 3}, {76, 63940}, {141, 62203}, {148, 53489}, {524, 14537}, {538, 7745}, {543, 9300}, {597, 11648}, {598, 11055}, {599, 44678}, {671, 62900}, {2548, 8716}, {2996, 43136}, {3055, 32456}, {3564, 22486}, {3734, 53418}, {3793, 7737}, {3933, 65630}, {3972, 43291}, {4027, 61600}, {5182, 6321}, {5254, 63957}, {5305, 12150}, {5306, 18546}, {5475, 6390}, {6781, 58446}, {7603, 32459}, {7615, 19661}, {7620, 63006}, {7747, 7767}, {7757, 32819}, {7788, 59780}, {7789, 39590}, {7804, 53419}, {7817, 63543}, {8556, 47102}, {9605, 32826}, {11054, 12156}, {11163, 51123}, {11164, 12040}, {11168, 47101}, {11185, 14614}, {11544, 30139}, {12154, 35693}, {12155, 35697}, {13669, 49261}, {13789, 49262}, {15271, 43618}, {15484, 32815}, {17503, 60215}, {18362, 20112}, {18800, 39593}, {21849, 55005}, {32892, 50992}, {37671, 63945}, {39601, 44381}, {40727, 63034}, {41748, 63923}, {43457, 44377}, {44562, 63548}, {45103, 60213}, {46951, 63950}, {47286, 63038}, {47287, 63018}, {49794, 49795}, {60201, 60281}

X(66409) = midpoint of X(i) and X(j) for these {i,j}: {7747, 9466}, {7757, 32819}, {8370, 11361}
X(66409) = reflection of X(i) in X(j) for these {i,j}: {7767, 9466}, {8353, 8358}, {8354, 2}, {8356, 8367}, {63548, 44562}
X(66409) = complement of X(8353)
X(66409) = anticomplement of X(8358)
X(66409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3830, 66392}, {2, 3845, 37350}, {2, 8352, 66394}, {2, 8353, 8358}, {2, 8354, 8359}, {2, 8597, 66349}, {2, 8598, 12100}, {2, 11159, 66391}, {2, 11317, 3845}, {2, 14030, 35954}, {2, 15682, 5077}, {2, 52942, 66388}, {2, 66319, 66393}, {2, 66328, 8598}, {2, 66387, 8703}, {2, 66389, 35955}, {2, 66391, 27088}, {2, 66392, 66335}, {4, 11286, 33184}, {148, 53489, 63633}, {381, 14033, 8369}, {382, 32971, 8362}, {384, 546, 8361}, {384, 33228, 8368}, {546, 8368, 33228}, {1003, 33016, 5}, {3363, 11159, 27088}, {3363, 66391, 2}, {3526, 33239, 33227}, {3627, 7770, 8357}, {3839, 14039, 11318}, {3861, 19697, 5025}, {5066, 66393, 2}, {6656, 14042, 3853}, {6658, 32992, 548}, {6661, 14041, 8360}, {7737, 64093, 3793}, {7770, 14068, 3627}, {7807, 33018, 3850}, {8353, 8358, 8354}, {8360, 14893, 14041}, {8364, 12102, 33229}, {8368, 33228, 8361}, {8369, 14033, 66321}, {11285, 33280, 15704}, {11286, 33184, 7819}, {12101, 66394, 8352}, {14034, 33018, 7807}, {14035, 33016, 1003}, {14066, 33229, 12102}, {14269, 33237, 16041}, {16044, 19687, 140}, {16921, 33250, 3530}, {19686, 33013, 35297}, {32962, 33235, 632}, {32991, 33239, 3526}, {32995, 33233, 5}, {33007, 44543, 549}, {33008, 66395, 15686}, {33013, 35297, 547}, {33185, 61988, 14063}, {33242, 61970, 32972}, {33291, 66327, 2}, {35955, 66389, 19710}, {37350, 66318, 2}, {52942, 66388, 33699}, {62013, 66347, 33019}

X(66410) = EULER LINE INTERCEPT OF X(83)X(63957)

Barycentrics    6*a^4 + 2*a^2*b^2 - 3*b^4 + 2*a^2*c^2 + 11*b^2*c^2 - 3*c^4 : :
X(66410) = 8 X[2] - 3 X[7833], 7 X[2] - 2 X[8353], 9 X[2] - 4 X[8354], 11 X[2] - 6 X[8356], 13 X[2] - 8 X[8358], 17 X[2] - 12 X[8359], 19 X[2] - 24 X[8367], X[2] - 6 X[8370], 2 X[2] + 3 X[11361], 13 X[2] - 3 X[33264], 4 X[3534] - 9 X[60654], 8 X[5066] - 3 X[55008], 21 X[7833] - 16 X[8353], 27 X[7833] - 32 X[8354], 11 X[7833] - 16 X[8356], 39 X[7833] - 64 X[8358], and many others

X(66410) lies on these lines: {2, 3}, {83, 63957}, {538, 7921}, {598, 7837}, {3314, 63956}, {5346, 12150}, {5965, 22486}, {7753, 11055}, {7812, 14711}, {7823, 9466}, {8556, 14712}, {11057, 55730}, {11185, 63038}, {12154, 42520}, {12155, 42521}, {14537, 63942}, {16986, 62203}, {17503, 62891}, {20088, 63954}, {31276, 63941}, {51122, 63018}

X(66410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33264, 8358}, {3363, 66319, 2}, {3843, 19689, 5025}, {5066, 35954, 2}, {5076, 7770, 19690}, {13586, 32983, 16921}, {14033, 33013, 33246}, {14034, 16044, 7907}, {14035, 32983, 13586}, {14042, 32971, 7876}, {16044, 19693, 1656}, {19686, 44543, 33274}

X(66411) = EULER LINE INTERCEPT OF X(39)X(63947)

Barycentrics    6*a^4 - 8*a^2*b^2 - 3*b^4 - 8*a^2*c^2 + b^2*c^2 - 3*c^4 : :
X(66411) = 2 X[2] + 3 X[7833], 3 X[2] + 2 X[8353], X[2] + 4 X[8354], X[2] - 6 X[8356], 3 X[2] - 8 X[8358], 7 X[2] - 12 X[8359], 29 X[2] - 24 X[8367], 11 X[2] - 6 X[8370], 8 X[2] - 3 X[11361], 7 X[2] + 3 X[33264], 2 X[3534] + 3 X[55008], 9 X[7833] - 4 X[8353], 3 X[7833] - 8 X[8354], X[7833] + 4 X[8356], 9 X[7833] + 16 X[8358], 7 X[7833] + 8 X[8359], 29 X[7833] + 16 X[8367], and many others

X(66411) lies on these lines: {2, 3}, {39, 63947}, {148, 8556}, {538, 7904}, {543, 55730}, {2896, 8716}, {5346, 7864}, {7757, 7830}, {7790, 46893}, {7802, 44562}, {7837, 52691}, {7921, 63941}, {9300, 14976}, {11055, 55164}, {11057, 63028}, {14711, 40344}, {14907, 63038}, {19569, 63101}, {31168, 34504}, {32480, 37671}, {50991, 59548}

X(66411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 8354, 7833}, {2, 9855, 14030}, {3522, 7791, 7948}, {7791, 33275, 7892}, {7876, 33260, 33268}, {7876, 33268, 14038}, {7924, 33008, 33274}, {8353, 8356, 8358}, {8353, 8358, 2}, {8354, 8356, 2}, {8354, 8358, 8353}, {8366, 11287, 66336}, {11287, 14093, 8366}, {14093, 66336, 33246}, {16043, 33267, 14034}, {16897, 33244, 14040}, {19689, 33260, 15696}, {19693, 62131, 33257}, {32965, 32986, 33273}, {32986, 33273, 5025}, {33215, 33263, 14041}, {33247, 33258, 14042}, {62104, 66342, 3552}

X(66412) = EULER LINE INTERCEPT OF X(141)X(63956)

Barycentrics    6*a^4 + 5*a^2*b^2 - 3*b^4 + 5*a^2*c^2 + 14*b^2*c^2 - 3*c^4 : :
X(66412) = 11 X[2] - 3 X[7833], 5 X[2] - X[8353], 7 X[2] - 3 X[8356], 5 X[2] - 3 X[8359], 2 X[2] - 3 X[8367], X[2] + 3 X[8370], 5 X[2] + 3 X[11361], 19 X[2] - 3 X[33264], 15 X[7833] - 11 X[8353], 9 X[7833] - 11 X[8354], 7 X[7833] - 11 X[8356], 6 X[7833] - 11 X[8358], 5 X[7833] - 11 X[8359], 2 X[7833] - 11 X[8367], X[7833] + 11 X[8370], 5 X[7833] + 11 X[11361], 19 X[7833] - 11 X[33264], and many others

X(66412) lies on these lines: {2, 3}, {141, 63956}, {597, 18546}, {598, 37671}, {3934, 63941}, {5182, 51732}, {6329, 32457}, {7736, 51122}, {7737, 8556}, {7745, 7826}, {7753, 14711}, {7804, 43291}, {7808, 63957}, {8667, 18907}, {8716, 31406}, {9300, 52229}, {9766, 59780}, {10352, 61600}, {11185, 63633}, {12150, 59635}, {14535, 43448}, {14537, 63945}, {14614, 64093}, {15271, 47102}, {18841, 63536}, {22486, 34380}, {40727, 63006}, {46893, 58446}, {53489, 63038}, {55005, 58470}

X(66412) = midpoint of X(i) and X(j) for these {i,j}: {7745, 9466}, {8359, 11361}
X(66412) = reflection of X(8358) in X(2)
X(66412) = complement of X(8354)
X(66412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3363, 5066}, {2, 3845, 66394}, {2, 5066, 8355}, {2, 8352, 66335}, {2, 8353, 8359}, {2, 11159, 8703}, {2, 11317, 66392}, {2, 11361, 8353}, {2, 27088, 11812}, {2, 66319, 27088}, {2, 66391, 12100}, {2, 66394, 66334}, {5, 11286, 8368}, {546, 7770, 8364}, {3850, 33213, 33228}, {3851, 33198, 33186}, {3856, 66344, 5025}, {7770, 33016, 33184}, {7819, 16044, 3850}, {7819, 33228, 33213}, {8358, 8367, 2}, {8368, 11286, 19697}, {8369, 44543, 547}, {11286, 32983, 5}, {11317, 66392, 12101}, {32954, 32991, 5}, {32971, 32983, 11286}, {33016, 33184, 546}

X(66413) = EULER LINE INTERCEPT OF X(6)X(19570)

Barycentrics    2*a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 + 5*b^2*c^2 - c^4 : :
X(66413) = 4 X[2] - X[7833], 11 X[2] - 2 X[8353], 13 X[2] - 4 X[8354], 5 X[2] - 2 X[8356], 17 X[2] - 8 X[8358], 7 X[2] - 4 X[8359], 5 X[2] - 8 X[8367], X[2] + 2 X[8370], 2 X[2] + X[11361], 7 X[2] - X[33264], 2 X[4] + X[60651], 4 X[5] - X[55008], 5 X[5071] - 2 X[37345], 11 X[7833] - 8 X[8353], 13 X[7833] - 16 X[8354], 5 X[7833] - 8 X[8356], 17 X[7833] - 32 X[8358], and many others

X(66413) lies on these lines: {2, 3}, {6, 19570}, {76, 7753}, {83, 5309}, {115, 7875}, {148, 11174}, {193, 32874}, {194, 9300}, {316, 7865}, {532, 25167}, {533, 25157}, {538, 63028}, {597, 39141}, {598, 754}, {626, 47005}, {671, 10352}, {1506, 7891}, {2548, 7906}, {2896, 65630}, {3096, 39590}, {3314, 5475}, {3329, 7739}, {3589, 63543}, {3734, 7777}, {3815, 59634}, {3934, 7811}, {3972, 17004}, {4027, 11632}, {4366, 10056}, {5024, 20094}, {5182, 39515}, {5306, 7787}, {5395, 32834}, {5640, 55005}, {6034, 42534}, {6645, 10072}, {6694, 16631}, {6695, 16630}, {6704, 7918}, {7603, 7835}, {7745, 7893}, {7747, 7904}, {7750, 19569}, {7752, 7880}, {7766, 53489}, {7773, 46226}, {7774, 32836}, {7779, 15484}, {7785, 7788}, {7790, 39563}, {7802, 31239}, {7804, 7806}, {7808, 7864}, {7812, 9466}, {7827, 18546}, {7828, 18362}, {7831, 62203}, {7834, 15031}, {7839, 63024}, {7846, 39565}, {7858, 17130}, {7878, 63924}, {7883, 63956}, {7898, 53418}, {7919, 18424}, {7925, 31415}, {7932, 63534}, {7934, 43457}, {8591, 42849}, {8667, 34604}, {10333, 10358}, {10583, 13881}, {14639, 38317}, {14712, 15271}, {14762, 52691}, {16984, 43620}, {17008, 32885}, {17129, 46951}, {26752, 49719}, {31404, 32837}, {31407, 32824}, {41135, 47352}, {47286, 62994}, {52713, 63017}, {63038, 63955}

X(66413) = reflection of X(60653) in X(5054)
X(66413) = orthocentroidal-circle-inverse of X(7924)
X(66413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 7924}, {2, 376, 7824}, {2, 381, 5025}, {2, 384, 33246}, {2, 439, 15721}, {2, 549, 33015}, {2, 1003, 33274}, {2, 3543, 7791}, {2, 3552, 549}, {2, 3839, 33251}, {2, 5071, 32967}, {2, 6661, 7892}, {2, 7924, 7876}, {2, 7933, 66326}, {2, 8370, 11361}, {2, 11286, 14036}, {2, 11361, 7833}, {2, 14031, 33266}, {2, 14033, 13586}, {2, 14035, 376}, {2, 14037, 33224}, {2, 14063, 33223}, {2, 14068, 33263}, {2, 15677, 17684}, {2, 15683, 32990}, {2, 15692, 33001}, {2, 15721, 33003}, {2, 16044, 381}, {2, 16898, 66323}, {2, 16914, 15670}, {2, 19686, 3}, {2, 31156, 33047}, {2, 32962, 5071}, {2, 32964, 15702}, {2, 32979, 3543}, {2, 32981, 15692}, {2, 32983, 33013}, {2, 32991, 61936}, {2, 33002, 547}, {2, 33006, 14046}, {2, 33007, 33273}, {2, 33009, 61895}, {2, 33016, 14041}, {2, 33030, 44217}, {2, 33187, 3524}, {2, 33193, 33215}, {2, 33204, 61865}, {2, 33206, 61859}, {2, 33223, 7948}, {2, 33224, 33245}, {2, 33246, 7907}, {2, 33251, 66324}, {2, 33259, 15694}, {2, 33263, 16043}, {2, 33264, 8359}, {2, 33266, 631}, {2, 61912, 32998}, {2, 61927, 32988}, {2, 61936, 32961}, {2, 61944, 32972}, {2, 61972, 33200}, {2, 61985, 32974}, {2, 62005, 33025}, {2, 66317, 33220}, {5, 6661, 2}, {76, 7753, 7837}, {115, 60855, 7875}, {376, 14035, 66328}, {376, 32968, 2}, {376, 66328, 33257}, {381, 7770, 2}, {384, 16921, 7907}, {384, 16922, 16925}, {384, 16923, 32973}, {384, 16924, 16921}, {547, 7807, 2}, {547, 66318, 7807}, {549, 32992, 2}, {549, 66319, 3552}, {2548, 17128, 7906}, {3090, 14037, 33245}, {3090, 33224, 2}, {3091, 16898, 7901}, {3524, 14033, 33187}, {3524, 33187, 13586}, {3552, 32992, 33015}, {3839, 33251, 14041}, {3934, 14537, 7811}, {5025, 7770, 16895}, {5055, 11286, 33220}, {5055, 33220, 2}, {5071, 14001, 2}, {6175, 17541, 2}, {6656, 33018, 14062}, {6658, 11285, 33275}, {7745, 31276, 7893}, {7753, 7837, 7921}, {7770, 16044, 5025}, {7791, 32979, 14042}, {7811, 14537, 7823}, {7819, 32966, 14065}, {7824, 14035, 33257}, {7824, 66328, 376}, {7866, 32993, 5025}, {7887, 19689, 14067}, {7901, 66323, 2}, {7924, 60651, 7833}, {8356, 8367, 2}, {8362, 15687, 66349}, {8363, 66340, 2}, {8367, 37345, 32958}, {11286, 33220, 66317}, {11286, 44543, 2}, {14001, 32962, 32967}, {14030, 33274, 1003}, {14031, 33261, 631}, {14032, 33015, 3552}, {14035, 32968, 7824}, {14041, 66324, 33251}, {14044, 16897, 32974}, {14063, 16045, 7948}, {14068, 16043, 33256}, {14068, 33263, 15682}, {14893, 66335, 33229}, {15682, 16043, 33263}, {15682, 33263, 33256}, {15687, 66349, 33019}, {15702, 32975, 2}, {15703, 33233, 2}, {16045, 33223, 2}, {16045, 41099, 33223}, {16921, 33246, 2}, {16922, 32987, 16921}, {16924, 16925, 32987}, {16924, 32971, 384}, {16925, 32987, 16922}, {19686, 33020, 2}, {19687, 33004, 33268}, {19689, 33024, 7887}, {32961, 33198, 14043}, {32970, 61895, 2}, {32973, 32999, 16923}, {32977, 61888, 2}, {32981, 33001, 33276}, {32990, 33280, 33267}, {32991, 33198, 32961}, {32992, 66319, 549}, {33016, 33251, 3839}, {33033, 50202, 2}, {33181, 61912, 2}, {33198, 61936, 2}, {33220, 44543, 5055}, {33220, 66317, 14036}, {33221, 61964, 33290}, {33223, 41099, 14063}, {33261, 33266, 2}, {46951, 63093, 17129}, {47005, 48913, 626}, {53489, 64093, 7766}

X(66414) = EULER LINE INTERCEPT OF X(39)X(7811)

Barycentrics    2*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 - b^2*c^2 - c^4 : :
X(66414) = 2 X[2] + X[7833], 7 X[2] + 2 X[8353], 5 X[2] + 4 X[8354], X[2] + 2 X[8356], X[2] + 8 X[8358], X[2] - 4 X[8359], 11 X[2] - 8 X[8367], 5 X[2] - 2 X[8370], 4 X[2] - X[11361], 5 X[2] + X[33264], 2 X[3] + X[55008], 4 X[3] - X[60651], X[376] + 2 X[37345], 7 X[7833] - 4 X[8353], 5 X[7833] - 8 X[8354], X[7833] - 4 X[8356], X[7833] - 16 X[8358], and many others

X(66414) lies on these lines: {2, 3}, {39, 7811}, {99, 16986}, {141, 59634}, {147, 52771}, {148, 15271}, {183, 19570}, {187, 7875}, {194, 37671}, {316, 15482}, {385, 7739}, {532, 3107}, {533, 3106}, {538, 10335}, {574, 3314}, {597, 59232}, {599, 59236}, {620, 7937}, {754, 55164}, {1078, 5309}, {1384, 63020}, {1506, 7910}, {2896, 5013}, {2996, 32893}, {3096, 7880}, {3329, 14907}, {3785, 7839}, {3815, 7898}, {4027, 14830}, {4045, 7771}, {4995, 26561}, {5023, 10583}, {5024, 7779}, {5206, 7859}, {5298, 26590}, {5306, 7793}, {6034, 39652}, {6292, 7782}, {6683, 7802}, {6781, 60855}, {7738, 17129}, {7745, 19569}, {7749, 7918}, {7750, 7921}, {7753, 7786}, {7757, 7810}, {7760, 63952}, {7761, 7777}, {7763, 7928}, {7764, 7936}, {7768, 53096}, {7769, 7935}, {7780, 39593}, {7783, 7800}, {7790, 17004}, {7796, 31652}, {7801, 31168}, {7812, 44562}, {7815, 7847}, {7832, 15515}, {7834, 43459}, {7835, 8589}, {7836, 15815}, {7846, 15513}, {7860, 9698}, {7868, 9878}, {7872, 18362}, {7885, 31401}, {7900, 31406}, {7911, 31455}, {7941, 31400}, {7947, 32837}, {7998, 55005}, {9229, 57822}, {9863, 13334}, {9939, 41624}, {10333, 37479}, {10351, 12054}, {11174, 14712}, {12150, 47101}, {13571, 22332}, {16984, 21843}, {16990, 32836}, {20065, 63024}, {21358, 52695}, {24726, 25362}, {26801, 49719}, {31276, 63548}, {31859, 63044}, {32152, 61132}, {32885, 43448}, {39141, 51737}, {54393, 61104}, {55085, 63935}, {58446, 63543}, {63101, 63941}

X(66414) = midpoint of X(3524) and X(57633)
X(66414) = reflection of X(60654) in X(3524)
X(66414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 33246}, {2, 376, 384}, {2, 381, 16921}, {2, 549, 7907}, {2, 3543, 16924}, {2, 3552, 6661}, {2, 5071, 16922}, {2, 6175, 33045}, {2, 6655, 381}, {2, 6661, 16895}, {2, 7791, 7924}, {2, 7833, 11361}, {2, 7924, 5025}, {2, 8356, 7833}, {2, 10304, 33255}, {2, 13586, 14036}, {2, 14001, 66323}, {2, 14063, 5071}, {2, 15677, 16916}, {2, 15683, 32971}, {2, 15692, 16925}, {2, 15702, 16923}, {2, 15721, 33000}, {2, 17565, 44217}, {2, 19686, 7770}, {2, 31156, 16918}, {2, 32963, 61895}, {2, 32964, 33224}, {2, 32965, 376}, {2, 32966, 547}, {2, 32980, 61912}, {2, 32982, 61936}, {2, 32986, 14041}, {2, 33004, 549}, {2, 33008, 13586}, {2, 33012, 15702}, {2, 33017, 33013}, {2, 33023, 3543}, {2, 33063, 15670}, {2, 33188, 61859}, {2, 33190, 14046}, {2, 33192, 32983}, {2, 33207, 14033}, {2, 33208, 14039}, {2, 33210, 33006}, {2, 33215, 33273}, {2, 33223, 7901}, {2, 33224, 14043}, {2, 33246, 7892}, {2, 33255, 66322}, {2, 33260, 19686}, {2, 33263, 4}, {2, 33264, 8370}, {2, 33265, 11286}, {2, 33266, 14001}, {2, 33270, 61888}, {2, 33272, 33016}, {2, 33273, 33274}, {2, 33278, 3545}, {2, 33823, 6175}, {2, 44651, 41231}, {2, 50410, 16900}, {2, 61778, 33205}, {2, 61806, 32989}, {2, 61936, 32999}, {2, 61985, 32987}, {2, 62063, 32973}, {2, 62081, 33201}, {2, 66336, 7866}, {2, 66337, 33219}, {2, 66369, 16950}, {3, 7876, 7892}, {3, 33021, 7876}, {3, 55008, 60651}, {39, 7811, 7837}, {39, 7904, 7893}, {39, 40344, 7811}, {376, 16043, 2}, {381, 11285, 2}, {384, 32965, 33275}, {384, 33275, 33268}, {549, 6656, 2}, {574, 7831, 3314}, {574, 7865, 7799}, {631, 33223, 2}, {2896, 5013, 7906}, {3096, 37512, 7891}, {3534, 7770, 19686}, {3534, 19686, 33257}, {3545, 32986, 33278}, {3545, 33278, 14041}, {3552, 8362, 16895}, {3552, 16895, 14038}, {4045, 7771, 7806}, {5025, 7824, 33015}, {5054, 11287, 33219}, {5054, 33219, 2}, {5071, 32978, 2}, {6655, 11285, 16921}, {6655, 16921, 14062}, {6656, 7907, 14065}, {6656, 33004, 7907}, {6661, 8362, 2}, {6661, 8703, 3552}, {7753, 7830, 11057}, {7753, 11057, 7823}, {7770, 33257, 14032}, {7770, 33260, 33257}, {7786, 7830, 7823}, {7786, 11057, 7753}, {7791, 7824, 5025}, {7791, 32961, 33025}, {7791, 32990, 7824}, {7791, 33001, 32974}, {7799, 7831, 7865}, {7799, 7865, 3314}, {7807, 66326, 2}, {7811, 7837, 7893}, {7811, 40344, 7904}, {7824, 7924, 2}, {7824, 32967, 33001}, {7833, 33246, 60651}, {7837, 7904, 7811}, {7876, 33246, 2}, {7887, 15694, 2}, {7901, 66338, 33223}, {7948, 16925, 14067}, {8354, 8370, 33264}, {8354, 33264, 7833}, {8356, 8359, 2}, {8356, 8370, 8354}, {8358, 8359, 8356}, {8359, 37345, 16043}, {8362, 8703, 6661}, {10124, 33249, 2}, {10304, 33255, 13586}, {11286, 35955, 33265}, {11287, 33219, 66337}, {12100, 66326, 7807}, {13586, 66322, 33255}, {14001, 19708, 33266}, {14033, 33207, 9855}, {14035, 33226, 33267}, {14063, 32978, 16922}, {14064, 15702, 2}, {14064, 33012, 16923}, {15670, 17670, 2}, {15690, 66321, 33250}, {15692, 33202, 2}, {15698, 32956, 33224}, {15698, 33224, 32964}, {15721, 33180, 2}, {16043, 32965, 384}, {16045, 33244, 384}, {16897, 33276, 14001}, {16897, 66323, 2}, {16924, 33023, 33256}, {16924, 33256, 14066}, {16925, 33202, 7948}, {19686, 33260, 3534}, {19708, 33266, 33276}, {32956, 32964, 14043}, {32956, 33224, 2}, {32957, 33247, 14068}, {32960, 33226, 14035}, {32962, 33238, 14044}, {32965, 37345, 7833}, {32967, 32974, 5025}, {32968, 32997, 14042}, {32969, 61859, 2}, {32971, 33253, 19696}, {32974, 33001, 32967}, {32976, 61865, 2}, {33008, 33255, 10304}, {33016, 33272, 8597}, {33044, 50727, 2}, {33199, 61846, 2}, {33216, 33230, 2}, {33246, 55008, 11361}, {33255, 66322, 14036}, {33258, 33263, 2}, {33259, 66336, 2}

X(66415) = EULER LINE INTERCEPT OF X(6)X(63955)

Barycentrics    2*a^4 + 3*a^2*b^2 - b^4 + 3*a^2*c^2 + 6*b^2*c^2 - c^4 : :
X(66415) = 5 X[2] - X[7833], 7 X[2] - X[8353], 4 X[2] - X[8354], 5 X[2] - 2 X[8358], 3 X[2] + X[11361], 9 X[2] - X[33264], X[3543] + 3 X[60654], 3 X[3839] + X[60651], 3 X[5055] - X[37345], 5 X[5071] - X[55008], 7 X[7833] - 5 X[8353], 4 X[7833] - 5 X[8354], 3 X[7833] - 5 X[8356], 2 X[7833] - 5 X[8359], X[7833] - 10 X[8367], X[7833] + 5 X[8370], 3 X[7833] + 5 X[11361], and many others

X(66415) lies on these lines: {2, 3}, {6, 63955}, {32, 13468}, {69, 15484}, {76, 41624}, {83, 5305}, {115, 3589}, {141, 5475}, {183, 3793}, {187, 58446}, {230, 7804}, {316, 31168}, {385, 53489}, {524, 5052}, {538, 9300}, {543, 2023}, {574, 15491}, {597, 5034}, {598, 7811}, {620, 3055}, {671, 62894}, {754, 3934}, {1384, 34229}, {1506, 7789}, {1975, 31406}, {1992, 46951}, {2548, 3933}, {2882, 61676}, {3329, 47286}, {3564, 7697}, {3614, 30104}, {3618, 14535}, {3619, 32827}, {3629, 17131}, {3631, 7845}, {3734, 3815}, {3820, 20172}, {3972, 37688}, {4045, 53419}, {5008, 50774}, {5024, 32815}, {5031, 20582}, {5182, 11632}, {5254, 7808}, {5355, 6329}, {5544, 32463}, {5943, 55005}, {6033, 18358}, {6292, 39590}, {6337, 31467}, {6683, 63548}, {6704, 7861}, {7173, 30103}, {7603, 7820}, {7610, 19661}, {7615, 47352}, {7694, 10516}, {7737, 15271}, {7739, 34505}, {7747, 31239}, {7757, 52229}, {7761, 53418}, {7762, 31276}, {7772, 63923}, {7778, 31415}, {7781, 9606}, {7786, 32819}, {7792, 43291}, {7800, 44678}, {7810, 14537}, {7812, 37671}, {7815, 47101}, {7834, 63534}, {7835, 37647}, {7853, 34573}, {7859, 15031}, {7865, 63956}, {7880, 22110}, {7889, 39565}, {7904, 14976}, {7913, 18424}, {8584, 41748}, {8716, 42849}, {9698, 59546}, {11163, 32833}, {11168, 42535}, {11174, 11185}, {12150, 22329}, {13877, 49253}, {13930, 49252}, {14929, 16990}, {15171, 27020}, {15174, 30135}, {16137, 30139}, {18842, 32893}, {18990, 26959}, {20112, 48310}, {21309, 37667}, {22253, 37665}, {22682, 29181}, {22712, 34733}, {24273, 53484}, {24512, 50185}, {26687, 31419}, {29438, 49745}, {30111, 39544}, {30435, 32828}, {30886, 37691}, {31455, 59545}, {32837, 63025}, {32898, 39142}, {39141, 51732}, {40727, 59373}, {41134, 63647}, {47617, 63543}, {50280, 51143}, {51258, 52758}

X(66415) = midpoint of X(i) and X(j) for these {i,j}: {2, 8370}, {76, 41624}, {7753, 9466}, {7810, 14537}, {7812, 37671}, {8356, 11361}
X(66415) = reflection of X(i) in X(j) for these {i,j}: {2, 8367}, {7833, 8358}, {8354, 8359}, {8359, 2}
X(66415) = complement of X(8356)
X(66415) = orthocentroidal-circle-inverse of X(11287)
X(66415) = X(9069)-Ceva conjugate of X(523)
X(66415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 11287}, {2, 381, 33184}, {2, 384, 35297}, {2, 1003, 549}, {2, 3091, 33285}, {2, 3363, 37350}, {2, 3545, 11318}, {2, 3839, 33190}, {2, 3845, 66335}, {2, 6661, 8368}, {2, 11286, 8369}, {2, 11287, 8362}, {2, 11361, 8356}, {2, 14033, 3}, {2, 14035, 33008}, {2, 14036, 7807}, {2, 14039, 11288}, {2, 14041, 6656}, {2, 16044, 14041}, {2, 16924, 44543}, {2, 19686, 33273}, {2, 32971, 14033}, {2, 32979, 33272}, {2, 32983, 381}, {2, 32984, 33240}, {2, 32985, 5054}, {2, 33006, 33219}, {2, 33008, 11285}, {2, 33013, 33228}, {2, 33016, 7841}, {2, 33184, 66326}, {2, 33216, 15694}, {2, 33228, 8360}, {2, 33265, 7824}, {2, 33272, 16043}, {2, 33285, 7866}, {2, 35297, 140}, {2, 44543, 5}, {2, 66319, 12100}, {3, 14033, 66391}, {4, 8362, 8357}, {4, 11287, 66392}, {5, 7770, 7819}, {5, 7819, 8361}, {5, 33185, 7887}, {5, 33186, 32961}, {83, 59635, 5305}, {140, 66393, 35297}, {183, 18907, 3793}, {381, 32983, 3363}, {381, 33184, 37350}, {384, 32992, 140}, {384, 33020, 32992}, {384, 35297, 66393}, {546, 66394, 14041}, {547, 8368, 2}, {549, 1003, 27088}, {1992, 46951, 63954}, {3090, 33198, 32954}, {3091, 16045, 7866}, {3329, 47286, 63633}, {3363, 33184, 381}, {3628, 19697, 7807}, {3734, 3815, 6390}, {3850, 8364, 5025}, {3861, 66347, 33229}, {3934, 7745, 7767}, {5055, 33237, 2}, {5066, 8360, 33228}, {5070, 33242, 32970}, {6656, 14041, 66394}, {6656, 16044, 546}, {7603, 7820, 44377}, {7770, 7887, 16898}, {7770, 16924, 5}, {7770, 32962, 33185}, {7770, 44543, 2}, {7807, 16921, 3628}, {7824, 19687, 548}, {7841, 33016, 3845}, {7876, 33018, 33229}, {7876, 33229, 66347}, {7887, 16898, 33185}, {7887, 32962, 5}, {7892, 33002, 33249}, {8355, 66340, 33213}, {8356, 8370, 11361}, {8362, 66392, 11287}, {8363, 16895, 66344}, {8367, 8370, 8359}, {8369, 11286, 66318}, {8598, 33273, 34200}, {11159, 11287, 14532}, {11174, 11185, 15048}, {11285, 14035, 550}, {11285, 66387, 33008}, {11286, 11288, 14039}, {11287, 66392, 8357}, {11288, 14039, 8369}, {11317, 33017, 15687}, {11737, 33213, 8355}, {14001, 32987, 1656}, {14031, 33001, 33235}, {14033, 32968, 2}, {14034, 33004, 33250}, {14035, 33008, 66387}, {14036, 16921, 2}, {14037, 32999, 33233}, {14042, 19695, 62026}, {14042, 33021, 19695}, {14064, 32991, 3851}, {14068, 33234, 62036}, {16043, 32979, 382}, {16045, 33285, 2}, {16895, 32966, 8363}, {16898, 16924, 32962}, {16898, 32962, 7887}, {16898, 33185, 7819}, {16918, 33033, 50205}, {16924, 33269, 7770}, {19686, 33273, 8598}, {19709, 33240, 32984}, {27088, 66321, 1003}, {31693, 31694, 5066}, {32815, 63041, 5024}, {32961, 33217, 33186}, {32968, 32971, 3}, {32973, 32975, 3526}, {32978, 32981, 3}, {32980, 33221, 33241}, {32992, 35297, 2}, {32999, 33233, 55856}, {33001, 33235, 15712}, {33002, 33249, 35018}, {33004, 33250, 33923}, {33008, 66387, 550}, {33013, 33228, 5066}, {33018, 33229, 3861}, {33183, 61914, 32958}, {33193, 35955, 15686}, {33197, 61899, 2}, {33207, 66395, 19710}, {33213, 66340, 2}, {33231, 61895, 2}, {33241, 61953, 32980}, {33263, 52942, 66396}, {35948, 35949, 54993}, {37170, 37171, 61936}, {37348, 44543, 3363}, {37350, 66326, 33184}, {37351, 37352, 547}, {37665, 52713, 22253}, {52942, 66396, 35404}

X(66416) = EULER LINE INTERCEPT OF X(76)X(9300)

Barycentrics    2*a^4 + 5*a^2*b^2 - b^4 + 5*a^2*c^2 + 8*b^2*c^2 - c^4 : :
X(66416) = 7 X[2] - X[7833], 10 X[2] - X[8353], 11 X[2] - 2 X[8354], 4 X[2] - X[8356], 13 X[2] - 4 X[8358], 5 X[2] - 2 X[8359], X[2] - 4 X[8367], 2 X[2] + X[8370], 5 X[2] + X[11361], 13 X[2] - X[33264], 4 X[547] - X[37345], 7 X[3090] - X[55008], 5 X[3091] + X[60651], 10 X[7833] - 7 X[8353], 11 X[7833] - 14 X[8354], 4 X[7833] - 7 X[8356], 13 X[7833] - 28 X[8358], and many others

X(66416) lies on these lines: {2, 3}, {76, 9300}, {83, 5306}, {99, 15491}, {141, 7809}, {183, 53489}, {230, 60855}, {373, 55005}, {538, 14762}, {597, 12151}, {598, 63941}, {1506, 7880}, {2548, 7788}, {3055, 7835}, {3058, 27020}, {3096, 48913}, {3329, 19570}, {3589, 7884}, {3631, 7926}, {3734, 59634}, {3815, 7799}, {3934, 7753}, {3972, 58446}, {4045, 39563}, {5024, 47287}, {5309, 7808}, {5434, 26959}, {5475, 7865}, {6683, 32819}, {6704, 39565}, {7735, 32885}, {7736, 32836}, {7739, 11174}, {7745, 7811}, {7747, 40344}, {7750, 14537}, {7752, 47005}, {7754, 46951}, {7790, 63543}, {7804, 37688}, {7810, 63943}, {7812, 63944}, {7820, 37647}, {7831, 53418}, {7834, 18362}, {7837, 31276}, {7859, 63534}, {7868, 31415}, {7875, 43291}, {7881, 31404}, {7904, 19569}, {7919, 51126}, {7934, 34573}, {9166, 48310}, {9466, 41624}, {9698, 32820}, {10352, 11632}, {11842, 61618}, {12150, 13468}, {14535, 16989}, {14651, 38110}, {15484, 16990}, {17128, 31406}, {26590, 65140}, {27091, 49732}, {29438, 49744}, {31859, 63041}, {32828, 63006}, {32874, 37665}, {37678, 48848}, {38317, 39663}, {39593, 63924}, {55085, 63923}

X(66416) = midpoint of X(3839) and X(60654)
X(66416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 11285}, {2, 381, 6656}, {2, 384, 549}, {2, 547, 33249}, {2, 3091, 33223}, {2, 3543, 16043}, {2, 3545, 33219}, {2, 5025, 66326}, {2, 5071, 7887}, {2, 6175, 17670}, {2, 6661, 7807}, {2, 7770, 6661}, {2, 7907, 10124}, {2, 7924, 8362}, {2, 8370, 8356}, {2, 11286, 35297}, {2, 11361, 8359}, {2, 15692, 32978}, {2, 16044, 7924}, {2, 16895, 66340}, {2, 16916, 15670}, {2, 16918, 50202}, {2, 16921, 547}, {2, 16922, 61885}, {2, 16923, 61869}, {2, 16924, 381}, {2, 16925, 15694}, {2, 16950, 44210}, {2, 19686, 7824}, {2, 32971, 376}, {2, 32973, 15702}, {2, 32983, 7841}, {2, 32987, 5071}, {2, 32989, 61859}, {2, 32993, 66336}, {2, 32999, 15703}, {2, 33000, 15723}, {2, 33005, 11318}, {2, 33013, 33184}, {2, 33016, 11287}, {2, 33037, 50727}, {2, 33198, 33224}, {2, 33201, 15721}, {2, 33205, 61846}, {2, 33219, 66339}, {2, 33224, 33233}, {2, 33246, 140}, {2, 33255, 5054}, {2, 33266, 33001}, {2, 41231, 40884}, {2, 44543, 33228}, {2, 61912, 32969}, {2, 61927, 33199}, {2, 61936, 14064}, {2, 61944, 33180}, {2, 61985, 33202}, {2, 66323, 33185}, {2, 66336, 16897}, {140, 66318, 33246}, {376, 32957, 2}, {547, 7819, 2}, {3545, 33219, 33228}, {3845, 7924, 33229}, {3845, 8362, 7924}, {3934, 7753, 37671}, {5054, 11286, 33255}, {5054, 33255, 35297}, {5066, 66326, 5025}, {5071, 16045, 2}, {6661, 32992, 2}, {7753, 37671, 7762}, {7770, 32968, 32992}, {7770, 32992, 7807}, {7770, 33233, 33198}, {7819, 16921, 33249}, {7824, 19686, 8703}, {7887, 16045, 66342}, {7924, 16044, 3845}, {8353, 8359, 8356}, {8353, 8370, 11361}, {8359, 11361, 8353}, {8361, 66340, 2}, {8362, 16044, 33229}, {8370, 33249, 37345}, {8703, 19686, 33250}, {10109, 66340, 8361}, {11285, 32971, 19687}, {11286, 35297, 35954}, {11287, 14269, 33278}, {11287, 33016, 8352}, {12100, 66321, 3552}, {14069, 61895, 2}, {14269, 33278, 8352}, {15703, 32954, 2}, {16045, 32987, 7887}, {16895, 33002, 8361}, {16898, 33261, 1656}, {32957, 32971, 11285}, {32959, 61884, 2}, {32960, 32979, 33234}, {32967, 66323, 2}, {32975, 33198, 33233}, {32975, 33224, 2}, {33001, 33266, 15693}, {33016, 33278, 14269}, {33189, 61888, 2}, {33219, 44543, 3545}, {33228, 66339, 33219}, {46951, 63024, 7754}

X(66417) = EULER LINE INTERCEPT OF X(39)X(37671)

Barycentrics    2*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 - 4*b^2*c^2 - c^4 : :
X(66417) = 5 X[2] + X[7833], 8 X[2] + X[8353], 7 X[2] + 2 X[8354], 2 X[2] + X[8356], 5 X[2] + 4 X[8358], X[2] + 2 X[8359], 7 X[2] - 4 X[8367], 4 X[2] - X[8370], 7 X[2] - X[11361], 11 X[2] + X[33264], 2 X[549] + X[37345], 5 X[631] + X[55008], 7 X[3523] - X[60651], 8 X[7833] - 5 X[8353], 7 X[7833] - 10 X[8354], 2 X[7833] - 5 X[8356], X[7833] - 4 X[8358], X[7833] - 10 X[8359], and many others

X(66417) lies on these lines: {2, 3}, {39, 37671}, {141, 7799}, {183, 7739}, {230, 7884}, {316, 15491}, {325, 7865}, {524, 13331}, {574, 59634}, {597, 12212}, {754, 15810}, {1078, 5306}, {2896, 31406}, {3054, 7919}, {3055, 7934}, {3058, 26959}, {3582, 26590}, {3584, 26561}, {3589, 7771}, {3785, 63024}, {3793, 62994}, {3815, 7809}, {4045, 37688}, {5013, 32833}, {5024, 16990}, {5309, 7815}, {5434, 27020}, {5650, 55005}, {6292, 7880}, {6390, 16986}, {6683, 7750}, {6704, 15513}, {7738, 46951}, {7745, 11057}, {7762, 7786}, {7767, 7837}, {7768, 9606}, {7772, 63952}, {7788, 7800}, {7789, 47005}, {7790, 58446}, {7810, 41624}, {7827, 13468}, {7830, 14537}, {7835, 34573}, {7853, 37647}, {7869, 31457}, {7879, 31400}, {7881, 32837}, {7937, 44377}, {9605, 63093}, {10352, 14830}, {11168, 14568}, {11648, 59635}, {14907, 53489}, {14929, 63018}, {15048, 19570}, {15271, 47286}, {15602, 35022}, {17030, 49732}, {21445, 38110}, {26613, 48310}, {31239, 32819}, {31360, 57822}, {31450, 32821}, {31652, 32820}, {31859, 32836}, {37686, 48848}, {50652, 64802}, {55085, 63928}, {55164, 63941}, {63028, 63940}

X(66417) = midpoint of X(3545) and X(60653)
X(66417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 6661}, {2, 376, 7770}, {2, 381, 32992}, {2, 549, 7807}, {2, 3523, 33224}, {2, 3524, 33220}, {2, 3543, 32968}, {2, 5025, 547}, {2, 7791, 381}, {2, 7824, 549}, {2, 7876, 66326}, {2, 7892, 66340}, {2, 7924, 5}, {2, 8356, 8370}, {2, 8359, 8356}, {2, 11287, 33228}, {2, 11361, 8367}, {2, 15677, 17541}, {2, 15692, 14001}, {2, 15702, 33233}, {2, 15721, 32970}, {2, 17684, 15670}, {2, 31107, 53843}, {2, 32961, 15703}, {2, 32967, 61885}, {2, 32972, 61895}, {2, 32974, 5071}, {2, 32986, 44543}, {2, 32988, 61888}, {2, 32990, 376}, {2, 32998, 61883}, {2, 33001, 15694}, {2, 33003, 15723}, {2, 33004, 33246}, {2, 33008, 11286}, {2, 33015, 10124}, {2, 33021, 7924}, {2, 33025, 61936}, {2, 33047, 50202}, {2, 33200, 61912}, {2, 33202, 33223}, {2, 33215, 1003}, {2, 33216, 8366}, {2, 33220, 66325}, {2, 33223, 7887}, {2, 33224, 33217}, {2, 33225, 66323}, {2, 33246, 7819}, {2, 33251, 5055}, {2, 33263, 16924}, {2, 33266, 16898}, {2, 33273, 8369}, {2, 33274, 8368}, {2, 44217, 33033}, {2, 61806, 33181}, {2, 61825, 33203}, {2, 61846, 32977}, {2, 61936, 32975}, {2, 62063, 33198}, {2, 66326, 8363}, {2, 66336, 7901}, {140, 7876, 8363}, {140, 66326, 2}, {376, 7770, 66319}, {376, 32960, 2}, {376, 66319, 33250}, {381, 7791, 66349}, {381, 66349, 33229}, {547, 66335, 5025}, {549, 8362, 2}, {3524, 33220, 35297}, {3830, 33263, 19695}, {5055, 11287, 33251}, {5055, 33251, 33228}, {6683, 40344, 7753}, {7753, 40344, 7750}, {7786, 7811, 9300}, {7791, 32992, 33229}, {7807, 37345, 8370}, {7810, 44562, 41624}, {7811, 9300, 7762}, {7819, 12100, 33246}, {7824, 8362, 7807}, {7824, 16897, 33259}, {7833, 8358, 8356}, {7866, 15694, 2}, {7887, 11285, 32978}, {7887, 33202, 6656}, {8354, 8367, 11361}, {8356, 8370, 8353}, {8359, 8362, 37345}, {8361, 10124, 2}, {8362, 33185, 16897}, {10124, 66334, 8361}, {11285, 16043, 6656}, {11286, 15688, 33187}, {11286, 33008, 8598}, {15688, 33187, 8598}, {15702, 32956, 2}, {16043, 32978, 33202}, {16897, 33259, 33185}, {16923, 66345, 33186}, {16924, 33263, 3830}, {32951, 61859, 2}, {32955, 61865, 2}, {32960, 32990, 7770}, {32978, 33202, 7887}, {32978, 33223, 2}, {32986, 44543, 8352}, {32992, 66349, 381}, {33004, 33246, 12100}, {33008, 33187, 15688}, {33185, 33259, 7807}, {33260, 66328, 15686}, {34200, 66318, 3552}, {35297, 66325, 33220}

X(66418) = EULER LINE INTERCEPT OF X(141)X(7908)

Barycentrics    2*a^4 - 9*a^2*b^2 - b^4 - 9*a^2*c^2 - 6*b^2*c^2 - c^4 : :
X(66418) = 7 X[2] + X[7833], 11 X[2] + X[8353], 5 X[2] + X[8354], 3 X[2] + X[8356], 2 X[2] + X[8358], 5 X[2] - X[8370], 9 X[2] - X[11361], 15 X[2] + X[33264], 3 X[5054] + X[37345], 5 X[5071] + 3 X[60653], 11 X[7833] - 7 X[8353], 5 X[7833] - 7 X[8354], 3 X[7833] - 7 X[8356], 2 X[7833] - 7 X[8358], X[7833] - 7 X[8359], 2 X[7833] + 7 X[8367], 5 X[7833] + 7 X[8370], and many others

X(66418) lies on these lines: {2, 3}, {141, 7908}, {183, 63633}, {325, 31168}, {524, 10007}, {597, 5039}, {620, 34573}, {754, 6683}, {3054, 7913}, {3055, 7853}, {3329, 3793}, {3564, 40108}, {3589, 41413}, {3618, 44839}, {3815, 7818}, {4045, 43291}, {5013, 51123}, {5305, 7815}, {5309, 11168}, {5480, 52770}, {5569, 48310}, {7736, 14929}, {7739, 8556}, {7753, 15810}, {7761, 15491}, {7767, 7786}, {7798, 15598}, {7800, 9766}, {7808, 47101}, {7810, 9300}, {7811, 63101}, {7854, 9606}, {7880, 20582}, {7937, 37647}, {8584, 63952}, {8716, 59780}, {9466, 52229}, {12040, 21358}, {15048, 15271}, {15172, 26959}, {21843, 47355}, {22246, 63042}, {31239, 63548}, {40344, 63941}, {42850, 63954}, {63024, 63950}

X(66418) = midpoint of X(i) and X(j) for these {i,j}: {2, 8359}, {7767, 41624}, {7810, 9300}, {8354, 8370}, {8358, 8367}
X(66418) = reflection of X(i) in X(j) for these {i,j}: {8358, 8359}, {8367, 2}
X(66418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 549, 8368}, {2, 3524, 33237}, {2, 7791, 44543}, {2, 7824, 35297}, {2, 8368, 66340}, {2, 11287, 5}, {2, 11318, 15699}, {2, 14041, 32992}, {2, 15708, 33197}, {2, 15721, 33231}, {2, 16043, 11287}, {2, 32990, 14033}, {2, 33004, 14036}, {2, 33008, 7770}, {2, 33021, 14041}, {2, 33184, 547}, {2, 33190, 5055}, {2, 33202, 33285}, {2, 33215, 11286}, {2, 33230, 33240}, {2, 33258, 33008}, {2, 33272, 32968}, {2, 33273, 6661}, {2, 33285, 1656}, {2, 35297, 7819}, {2, 66326, 33213}, {2, 66335, 10109}, {5, 11287, 66394}, {140, 8362, 8364}, {547, 33184, 8355}, {3363, 33017, 14893}, {3526, 32956, 33186}, {4045, 58446, 43291}, {5077, 32983, 15687}, {6661, 33273, 27088}, {7770, 33008, 66391}, {7791, 44543, 66392}, {7819, 7824, 3530}, {7866, 32978, 632}, {7876, 8361, 66346}, {8356, 8370, 33264}, {8356, 33264, 8354}, {8357, 32992, 3850}, {8362, 11285, 140}, {10124, 33213, 2}, {10997, 35297, 27088}, {11286, 33215, 8703}, {11287, 66394, 66347}, {11301, 11302, 15702}, {12108, 66344, 7807}, {16239, 66346, 8361}, {27088, 33273, 14891}, {32992, 33021, 8357}, {33008, 66391, 548}, {33211, 61837, 32970}, {44543, 66392, 546}

X(66419) = EULER LINE INTERCEPT OF X(32)X(671)

Barycentrics    4*a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 5*b^2*c^2 - 2*c^4 : :
X(66419) = 7 X[2] - 4 X[8353], 11 X[2] - 8 X[8354], 5 X[2] - 4 X[8356], 19 X[2] - 16 X[8358], 9 X[2] - 8 X[8359], 15 X[2] - 16 X[8367], 3 X[2] - 4 X[8370], 4 X[5] - 3 X[57633], 5 X[3091] - 4 X[37345], 5 X[3522] - 6 X[60654], 7 X[3523] - 6 X[60653], 3 X[3839] - 2 X[55008], 7 X[7833] - 6 X[8353], 11 X[7833] - 12 X[8354], 5 X[7833] - 6 X[8356], 19 X[7833] - 24 X[8358], and many others

X(66419) lies on these lines: {2, 3}, {32, 671}, {39, 598}, {76, 3849}, {83, 54737}, {99, 7775}, {148, 7737}, {187, 47617}, {194, 543}, {316, 7801}, {385, 34505}, {524, 7823}, {576, 8593}, {597, 7864}, {599, 7929}, {627, 10808}, {628, 10809}, {736, 43688}, {754, 19569}, {1383, 31125}, {1975, 7840}, {1992, 8596}, {2482, 7752}, {2549, 62994}, {3053, 8859}, {3095, 10811}, {3329, 44526}, {3734, 7883}, {3767, 41135}, {3926, 23334}, {3934, 55164}, {3972, 7817}, {4027, 10723}, {4366, 12943}, {5023, 8860}, {5206, 7617}, {5210, 17006}, {5461, 7857}, {5475, 34504}, {6179, 63922}, {6321, 10788}, {6337, 8786}, {6645, 12953}, {6781, 34506}, {7697, 34510}, {7745, 63028}, {7746, 26613}, {7748, 7787}, {7755, 36523}, {7756, 52691}, {7757, 14537}, {7762, 20105}, {7763, 52695}, {7764, 15300}, {7769, 8176}, {7773, 11164}, {7774, 20094}, {7777, 53418}, {7779, 32815}, {7782, 39590}, {7783, 11163}, {7785, 8591}, {7793, 51224}, {7799, 63956}, {7802, 7810}, {7806, 53419}, {7811, 14976}, {7816, 7870}, {7825, 7945}, {7842, 7938}, {7843, 39785}, {7862, 64019}, {7891, 22110}, {7893, 63945}, {7928, 21358}, {7946, 63931}, {8182, 32832}, {8594, 34509}, {8595, 34508}, {9466, 11057}, {9737, 12117}, {9830, 13330}, {9993, 39809}, {10131, 18502}, {10810, 59363}, {11055, 41750}, {11177, 36998}, {11185, 14712}, {11606, 54752}, {11645, 22486}, {11648, 12150}, {12154, 16965}, {12155, 16964}, {13657, 54507}, {13777, 54503}, {14484, 54833}, {14568, 63957}, {14711, 63943}, {15098, 62295}, {15515, 55801}, {16118, 30139}, {17129, 63950}, {20065, 32826}, {22561, 52674}, {32833, 44678}, {33342, 35703}, {33343, 35702}, {35295, 61743}, {35706, 61752}, {39141, 48901}, {41133, 59545}, {41895, 62905}, {42849, 44519}, {43448, 63019}, {43620, 51238}, {46313, 52088}, {51581, 54494}, {53101, 60234}, {54476, 62932}, {55005, 62187}, {60113, 60263}, {63044, 64018}, {63101, 63548}, {63107, 63533}

X(66419) = reflection of X(i) in X(j) for these {i,j}: {2, 11361}, {194, 7812}, {7757, 14537}, {7802, 7810}, {7812, 7747}, {7833, 8370}, {9878, 671}, {9939, 76}, {11055, 41750}, {11057, 9466}, {14976, 7811}, {15683, 60651}, {32480, 598}, {33264, 2}
X(66419) = anticomplement of X(7833)
X(66419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3146, 33192}, {2, 6658, 33007}, {2, 7841, 7933}, {2, 15683, 33207}, {2, 19690, 33230}, {2, 32993, 32984}, {2, 33007, 3552}, {2, 33013, 33002}, {2, 33014, 33274}, {2, 33018, 33013}, {2, 33019, 7841}, {2, 33192, 6655}, {2, 33193, 33265}, {2, 33210, 66337}, {2, 33244, 35287}, {2, 33260, 33215}, {2, 35287, 33259}, {2, 66320, 14039}, {2, 66398, 20}, {3, 11317, 33013}, {3, 14042, 33018}, {3, 33013, 2}, {3, 33018, 33002}, {3, 66395, 9855}, {4, 3552, 32966}, {4, 6658, 3552}, {4, 16925, 32993}, {4, 32985, 33006}, {4, 33007, 2}, {4, 33239, 32961}, {4, 33280, 6658}, {4, 35951, 5}, {5, 8598, 33274}, {5, 33257, 33014}, {5, 33274, 2}, {20, 14068, 16044}, {20, 16044, 33004}, {148, 7737, 7766}, {376, 33016, 2}, {381, 13586, 2}, {381, 66387, 13586}, {382, 384, 33019}, {382, 7841, 8597}, {382, 11159, 7841}, {384, 7841, 2}, {384, 8597, 7841}, {384, 33019, 7933}, {439, 50689, 32963}, {546, 7907, 33011}, {546, 33250, 7907}, {550, 16921, 33022}, {1003, 3830, 14041}, {1003, 14041, 2}, {3091, 33244, 33259}, {3091, 35287, 2}, {3146, 14035, 6655}, {3146, 33192, 40246}, {3524, 33005, 2}, {3529, 16924, 33260}, {3534, 44543, 33273}, {3627, 8369, 8352}, {3627, 19687, 5025}, {3830, 66328, 2}, {3839, 35927, 2}, {3843, 33235, 32967}, {3853, 7807, 14062}, {5025, 8369, 2}, {5059, 32979, 32965}, {5073, 5077, 66397}, {5073, 7770, 33256}, {5076, 7887, 14044}, {5077, 66397, 33256}, {6655, 40246, 33192}, {7770, 66397, 5077}, {7785, 8591, 34511}, {7791, 33703, 19691}, {7816, 31173, 7870}, {7823, 32819, 20081}, {7833, 8370, 2}, {7833, 11361, 8370}, {7841, 8597, 33019}, {7841, 11159, 384}, {7870, 31173, 7912}, {7892, 8360, 2}, {7901, 8366, 2}, {7924, 11286, 2}, {8352, 8369, 5025}, {8352, 19687, 8369}, {8356, 8370, 8367}, {8360, 35954, 7892}, {8365, 14065, 2}, {8597, 11159, 2}, {8598, 33274, 33014}, {9855, 11317, 2}, {9855, 14042, 33013}, {9855, 19696, 66395}, {9855, 33013, 3}, {11001, 32983, 33008}, {11111, 33031, 2}, {11185, 43618, 14712}, {11285, 17800, 33267}, {11286, 15684, 66388}, {11286, 66388, 7924}, {11287, 62040, 66396}, {11317, 33013, 33018}, {11317, 66395, 3}, {11541, 16043, 33271}, {14001, 62028, 33279}, {14030, 66392, 2}, {14031, 32974, 19689}, {14033, 15682, 33017}, {14033, 33017, 2}, {14035, 33192, 2}, {14036, 33184, 2}, {14039, 33251, 2}, {14041, 66328, 1003}, {14042, 19696, 3}, {14042, 33013, 11317}, {14042, 66395, 2}, {14046, 33220, 2}, {14063, 32981, 33225}, {14066, 33257, 5}, {14068, 66398, 2}, {15687, 66391, 33228}, {15704, 32992, 33275}, {16041, 33255, 2}, {16898, 33230, 2}, {16898, 33238, 19690}, {16924, 33215, 2}, {16925, 32984, 2}, {17578, 32981, 14063}, {19696, 33013, 9855}, {32965, 32979, 33020}, {32968, 49138, 33253}, {32971, 32997, 33021}, {32971, 49135, 32997}, {32973, 50688, 32996}, {32983, 33008, 2}, {32985, 33006, 2}, {32986, 62042, 66390}, {32990, 49140, 33209}, {32991, 50693, 33012}, {32995, 33214, 3523}, {33006, 33007, 32985}, {33007, 52942, 4}, {33009, 33252, 15717}, {33014, 35951, 3552}, {33016, 66389, 376}, {33184, 66319, 14036}, {33201, 54097, 33283}, {33228, 33246, 2}, {33228, 66391, 33246}, {33229, 35954, 8360}, {33257, 33274, 8598}, {33273, 44543, 2}, {33280, 52942, 33007}

X(66420) = EULER LINE INTERCEPT OF X(76)X(14976)

Barycentrics    8*a^4 - 3*a^2*b^2 - 4*b^4 - 3*a^2*c^2 + 9*b^2*c^2 - 4*c^4 : :
X(66420) = 5 X[2] - 4 X[7833], 11 X[2] - 8 X[8353], 19 X[2] - 16 X[8354], 9 X[2] - 8 X[8356], 35 X[2] - 32 X[8358], 17 X[2] - 16 X[8359], 31 X[2] - 32 X[8367], 7 X[2] - 8 X[8370], 3 X[2] - 4 X[11361], 11 X[7833] - 10 X[8353], 19 X[7833] - 20 X[8354], 9 X[7833] - 10 X[8356], 7 X[7833] - 8 X[8358], 17 X[7833] - 20 X[8359], 31 X[7833] - 40 X[8367], 7 X[7833] - 10 X[8370], and many others

X(66420) lies on these lines: {2, 3}, {76, 14976}, {148, 43618}, {193, 35369}, {194, 41750}, {538, 19569}, {754, 20081}, {7752, 45017}, {7753, 32480}, {7757, 32479}, {7787, 65633}, {7793, 18546}, {7823, 20105}, {7900, 44678}, {14712, 63955}, {23698, 32469}, {39141, 48904}, {41895, 60136}, {43449, 54749}, {44526, 62994}, {51224, 63957}

X(66420) = reflection of X(i) in X(j) for these {i,j}: {14976, 76}, {33264, 11361}
X(66420) = anticomplement of X(33264)
X(66420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 19691, 33272}, {2, 33265, 33014}, {4, 33014, 33011}, {4, 33265, 2}, {4, 66389, 33265}, {20, 33018, 33022}, {382, 19696, 3552}, {382, 66387, 14041}, {549, 32994, 2}, {550, 14066, 33002}, {1003, 15684, 8597}, {1657, 14042, 33004}, {3146, 6658, 33019}, {3543, 33193, 2}, {3552, 14041, 2}, {3627, 33257, 32966}, {3830, 66395, 13586}, {3839, 33208, 2}, {5059, 14068, 33260}, {6655, 14033, 2}, {7933, 14036, 2}, {11159, 62045, 66396}, {11159, 66396, 7924}, {11286, 62046, 66397}, {11317, 15681, 33273}, {11361, 33264, 2}, {14033, 33703, 66390}, {14033, 66390, 6655}, {14035, 33272, 2}, {14035, 49135, 19691}, {14036, 66392, 7933}, {14041, 19696, 66387}, {14041, 66387, 3552}, {16044, 33008, 2}, {17578, 33244, 32993}, {19686, 33017, 2}, {19686, 40246, 33017}, {19687, 66392, 14036}, {32966, 35297, 2}, {33004, 44543, 2}, {33017, 62042, 40246}, {33190, 66317, 2}, {33225, 33285, 2}, {33235, 62023, 14044}, {33250, 62026, 14062}, {33280, 33703, 6655}, {33280, 66390, 14033}, {35948, 35949, 35951}

X(66421) = EULER LINE INTERCEPT OF X(76)X(32479)

Barycentrics    8*a^4 - 5*a^2*b^2 - 4*b^4 - 5*a^2*c^2 + 7*b^2*c^2 - 4*c^4 : :
X(66421) = 3 X[2] - 4 X[7833], 5 X[2] - 8 X[8353], 13 X[2] - 16 X[8354], 7 X[2] - 8 X[8356], 29 X[2] - 32 X[8358], 15 X[2] - 16 X[8359], 33 X[2] - 32 X[8367], 9 X[2] - 8 X[8370], 5 X[2] - 4 X[11361], 5 X[3091] - 6 X[57633], 7 X[3832] - 8 X[37345], 5 X[7833] - 6 X[8353], 13 X[7833] - 12 X[8354], 7 X[7833] - 6 X[8356], 29 X[7833] - 24 X[8358], 5 X[7833] - 4 X[8359], and many others

X(66421) lies on these lines: {2, 3}, {76, 32479}, {194, 3849}, {315, 8591}, {316, 34504}, {524, 20105}, {538, 14976}, {543, 7802}, {671, 7793}, {2482, 7912}, {2549, 34604}, {5206, 9166}, {5254, 62204}, {7617, 43459}, {7747, 52691}, {7748, 51224}, {7756, 7812}, {7757, 19569}, {7766, 44526}, {7771, 47617}, {7782, 31173}, {7784, 11164}, {7796, 15300}, {7801, 7898}, {7825, 41134}, {7842, 7870}, {7860, 39785}, {7893, 52229}, {7900, 34511}, {7941, 11165}, {8596, 9878}, {8859, 44518}, {11054, 63935}, {11055, 63943}, {11161, 52987}, {11163, 44519}, {14712, 43619}, {17005, 44541}, {19570, 47102}, {20094, 64018}, {31276, 55164}, {32006, 41136}, {35369, 63046}, {39141, 48896}, {63028, 63548}

X(66421) = reflection of X(i) in X(j) for these {i,j}: {2, 33264}, {7812, 7756}, {8596, 9878}, {9939, 7802}, {11361, 8353}, {19569, 7757}, {20081, 9939}
X(66421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5059, 66398}, {2, 19691, 33192}, {2, 33192, 33019}, {2, 66398, 6658}, {3, 66397, 8597}, {4, 33022, 33010}, {20, 19691, 33019}, {20, 32972, 33252}, {20, 33019, 33014}, {20, 33192, 2}, {382, 33267, 33004}, {382, 35955, 33013}, {550, 8352, 33274}, {1657, 7841, 9855}, {1657, 33256, 3552}, {3146, 33209, 33260}, {3146, 33260, 33018}, {3529, 33271, 6655}, {3534, 66396, 14041}, {3543, 33207, 2}, {3552, 7841, 2}, {3627, 33275, 33002}, {5059, 32997, 6658}, {5077, 17800, 66395}, {5077, 66395, 384}, {6655, 33007, 2}, {7756, 7812, 32480}, {7833, 11361, 8359}, {7841, 9855, 3552}, {7933, 8369, 2}, {8352, 33274, 32966}, {8353, 8359, 7833}, {9855, 33256, 7841}, {11001, 33017, 33265}, {14039, 66337, 2}, {14063, 35287, 2}, {15681, 66388, 13586}, {16044, 33215, 2}, {17538, 33279, 33259}, {19686, 32986, 2}, {19689, 33230, 2}, {19695, 33257, 7933}, {19695, 62155, 33257}, {32966, 33274, 2}, {32982, 62149, 33214}, {32984, 33259, 2}, {32986, 62161, 66389}, {32986, 66389, 19686}, {32997, 66398, 2}, {33004, 33013, 2}, {33013, 33267, 35955}, {33013, 35955, 33004}, {33017, 33265, 2}, {33187, 33210, 2}, {33192, 52943, 20}, {33193, 33272, 2}, {33215, 33703, 52942}, {33215, 52942, 16044}, {33229, 62144, 33268}, {33234, 49137, 19696}, {33247, 33280, 33021}, {33247, 62171, 33280}, {33253, 33703, 16044}, {33253, 52942, 33215}, {33272, 62160, 33193}, {33278, 35927, 2}

X(66422) = EULER LINE INTERCEPT OF X(543)X(7823)

Barycentrics    10*a^4 - 4*a^2*b^2 - 5*b^4 - 4*a^2*c^2 + 11*b^2*c^2 - 5*c^4 : :
X(66422) = 6 X[2] - 5 X[7833], 13 X[2] - 10 X[8353], 23 X[2] - 20 X[8354], 11 X[2] - 10 X[8356], 43 X[2] - 40 X[8358], 21 X[2] - 20 X[8359], 39 X[2] - 40 X[8367], 9 X[2] - 10 X[8370], 4 X[2] - 5 X[11361], 7 X[2] - 5 X[33264], 14 X[3528] - 15 X[60654], 16 X[3530] - 15 X[60653], 17 X[3544] - 15 X[57633], 11 X[3855] - 10 X[37345], 13 X[7833] - 12 X[8353], and many others

X(66422) lies on these lines: {2, 3}, {543, 7823}, {598, 7756}, {3053, 41135}, {3849, 7893}, {5569, 15031}, {7745, 32480}, {7747, 63028}, {7754, 8596}, {7773, 52695}, {7777, 34504}, {7812, 32450}, {7814, 36521}, {7827, 65633}, {7836, 11164}, {7891, 31173}, {7906, 8591}, {7929, 59780}, {7941, 23334}, {8587, 53105}, {9939, 32819}, {10484, 53109}, {14712, 34505}, {17004, 47617}, {20081, 63945}, {33698, 62904}, {37637, 51238}

X(66422) = reflection of X(9939) in X(32819)
X(66422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33192, 33238}, {2, 33276, 33274}, {2, 66395, 33257}, {4, 9855, 33274}, {4, 33214, 16923}, {4, 66398, 9855}, {20, 14066, 33015}, {20, 52942, 33013}, {382, 33257, 14062}, {382, 66395, 2}, {3146, 19696, 5025}, {3146, 33007, 8597}, {3529, 14042, 33275}, {3543, 66389, 13586}, {5073, 7841, 40246}, {6658, 40246, 7841}, {8359, 33264, 7833}, {8597, 19696, 33007}, {8597, 33007, 5025}, {9855, 33274, 33268}, {11159, 49136, 66397}, {11159, 66397, 6655}, {14068, 49138, 33267}, {14068, 52943, 33215}, {15682, 33193, 14041}, {33007, 33279, 2}, {33013, 52942, 14066}, {33017, 66328, 14036}, {33192, 52944, 33703}, {33215, 49138, 52943}, {33215, 52943, 33267}, {33244, 62028, 14044}, {33256, 33280, 14034}, {33280, 49135, 33256}

X(66423) = EULER LINE INTERCEPT OF X(543)X(41750)

Barycentrics    10*a^4 - 3*a^2*b^2 - 5*b^4 - 3*a^2*c^2 + 12*b^2*c^2 - 5*c^4 : :
X(66423) = 7 X[2] - 5 X[7833], 8 X[2] - 5 X[8353], 13 X[2] - 10 X[8354], 6 X[2] - 5 X[8356], 23 X[2] - 20 X[8358], 11 X[2] - 10 X[8359], 19 X[2] - 20 X[8367], 4 X[2] - 5 X[8370], 3 X[2] - 5 X[11361], 9 X[2] - 5 X[33264], 8 X[7833] - 7 X[8353], 13 X[7833] - 14 X[8354], 6 X[7833] - 7 X[8356], 23 X[7833] - 28 X[8358], 11 X[7833] - 14 X[8359], 19 X[7833] - 28 X[8367], and many others

X(66423) lies on these lines: {2, 3}, {543, 41750}, {754, 32819}, {1975, 44678}, {3629, 10754}, {7747, 32450}, {7753, 32479}, {7767, 14976}, {7785, 51123}, {19569, 63940}, {22329, 63957}, {33698, 60073}, {43618, 63955}, {44526, 53489}, {47101, 59635}, {53105, 62898}, {53482, 54507}, {53483, 54503}, {53491, 60195}, {54494, 60178}, {59634, 63956}

X(66423) = reflection of X(i) in X(j) for these {i,j}: {8353, 8370}, {8356, 11361}, {14976, 7767}, {41624, 7747}
X(66423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33235, 35297}, {2, 33243, 33008}, {2, 33265, 33276}, {2, 62042, 66396}, {4, 33250, 33249}, {4, 66387, 35297}, {381, 33193, 8598}, {382, 19687, 33229}, {382, 33280, 19687}, {1003, 3543, 8352}, {1657, 14068, 32992}, {3146, 14033, 66388}, {3363, 15686, 33273}, {3627, 6658, 7807}, {3627, 66391, 14041}, {3830, 33007, 33228}, {5073, 11287, 66390}, {5073, 14035, 19695}, {6658, 14041, 66391}, {8356, 11361, 8370}, {8362, 62047, 19691}, {8597, 19686, 33184}, {11159, 15684, 33017}, {11159, 33017, 6661}, {11286, 33192, 66349}, {11286, 62040, 33192}, {11287, 66390, 19695}, {14033, 33238, 2}, {14033, 66388, 6656}, {14035, 66390, 11287}, {14036, 33019, 66394}, {14036, 66394, 8363}, {14041, 66391, 7807}, {15640, 32986, 66397}, {15683, 32983, 35955}, {19686, 33184, 35954}, {33016, 66398, 3534}, {33184, 35404, 8597}, {33193, 52942, 381}, {35297, 66387, 33250}

X(66424) = EULER LINE INTERCEPT OF X(194)X(63945)

Barycentrics    10*a^4 - 7*a^2*b^2 - 5*b^4 - 7*a^2*c^2 + 8*b^2*c^2 - 5*c^4 : :
X(66424) = 3 X[2] - 5 X[7833], 2 X[2] - 5 X[8353], 7 X[2] - 10 X[8354], 4 X[2] - 5 X[8356], 17 X[2] - 20 X[8358], 9 X[2] - 10 X[8359], 21 X[2] - 20 X[8367], 6 X[2] - 5 X[8370], 7 X[2] - 5 X[11361], X[2] - 5 X[33264], 4 X[546] - 5 X[37345], 11 X[3855] - 15 X[57633], 2 X[7833] - 3 X[8353], 7 X[7833] - 6 X[8354], 4 X[7833] - 3 X[8356], 17 X[7833] - 12 X[8358], and many others

X(66424) lies on these lines: {2, 3}, {194, 63945}, {325, 34504}, {524, 7802}, {543, 7750}, {597, 7847}, {2482, 7842}, {2896, 59780}, {3629, 8586}, {3849, 7756}, {3933, 8591}, {5254, 51224}, {5461, 15513}, {6329, 10485}, {6781, 7817}, {7618, 7773}, {7745, 52691}, {7747, 63101}, {7748, 22329}, {7782, 22110}, {7795, 11164}, {7797, 19661}, {7810, 32479}, {7812, 63548}, {7821, 36521}, {7823, 32480}, {7825, 41133}, {7851, 37809}, {7885, 52695}, {7936, 50991}, {8176, 15515}, {8593, 64196}, {8596, 17129}, {9939, 52229}, {11054, 63928}, {11055, 63944}, {12154, 43632}, {12155, 43633}, {14907, 34505}, {14929, 20094}, {14976, 63940}, {15048, 34604}, {15300, 32820}, {15597, 43459}, {15814, 51581}, {17006, 51238}, {32006, 53142}, {32826, 42850}, {37688, 47617}, {41895, 55823}, {43618, 53489}, {43619, 47286}, {53101, 55794}, {53105, 60220}, {53109, 62895}, {54494, 62881}

X(66424) = reflection of X(i) in X(j) for these {i,j}: {7812, 63548}, {8353, 33264}, {8356, 8353}, {8370, 7833}, {11361, 8354}, {32819, 7810}
X(66424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3529, 66395}, {2, 33192, 33279}, {2, 33238, 7841}, {2, 66395, 19687}, {3, 33192, 8352}, {20, 7841, 8598}, {20, 19695, 7807}, {20, 33238, 33235}, {376, 66388, 33228}, {548, 33019, 33249}, {548, 37350, 33274}, {550, 33256, 33229}, {1003, 33272, 66349}, {1657, 5077, 33007}, {1657, 32997, 6656}, {3146, 33215, 11317}, {3529, 33234, 19687}, {3534, 11318, 33208}, {3534, 33017, 35297}, {5059, 33247, 7770}, {5077, 33007, 6656}, {6655, 9855, 8369}, {6655, 15704, 33250}, {6655, 33250, 8363}, {6656, 33007, 35954}, {7791, 66398, 11159}, {7833, 8370, 8356}, {7841, 8598, 7807}, {7841, 33235, 2}, {8353, 8370, 7833}, {8357, 62151, 33257}, {8361, 62136, 33268}, {8362, 62162, 19696}, {8367, 11361, 8370}, {8369, 9855, 33250}, {8369, 15704, 9855}, {8598, 19695, 7841}, {11001, 33272, 1003}, {11159, 17800, 66398}, {11286, 62158, 66389}, {11287, 15685, 33193}, {11287, 33193, 66319}, {11317, 33215, 32992}, {11318, 33208, 35297}, {15683, 32986, 66387}, {15686, 66392, 13586}, {19691, 33267, 5}, {19710, 33184, 33265}, {32954, 62142, 33214}, {32986, 66387, 6661}, {32997, 33007, 5077}, {32997, 52943, 33007}, {33007, 52943, 1657}, {33013, 40246, 3627}, {33017, 33208, 11318}, {33019, 33274, 37350}, {33192, 33243, 2}, {33207, 66390, 381}, {33209, 33271, 3}, {33234, 66395, 2}, {33243, 33279, 3}, {33260, 40246, 33013}, {33263, 66389, 11286}, {33274, 37350, 33249}, {35955, 66397, 4}

X(66425) = EULER LINE INTERCEPT OF X(543)X(7762)

Barycentrics    14*a^4 - 5*a^2*b^2 - 7*b^4 - 5*a^2*c^2 + 16*b^2*c^2 - 7*c^4 : :
X(66425) = 9 X[2] - 7 X[7833], 10 X[2] - 7 X[8353], 17 X[2] - 14 X[8354], 8 X[2] - 7 X[8356], 31 X[2] - 28 X[8358], 15 X[2] - 14 X[8359], 27 X[2] - 28 X[8367], 6 X[2] - 7 X[8370], 5 X[2] - 7 X[11361], 11 X[2] - 7 X[33264], 8 X[3850] - 7 X[37345], 10 X[7833] - 9 X[8353], 17 X[7833] - 18 X[8354], 8 X[7833] - 9 X[8356], 31 X[7833] - 36 X[8358], 5 X[7833] - 6 X[8359], and many others

X(66425) lies on these lines: {2, 3}, {543, 7762}, {598, 63548}, {3849, 7826}, {6781, 47617}, {7747, 32479}, {7756, 63101}, {7823, 52229}, {7843, 15300}, {7910, 20582}, {12154, 42431}, {12155, 42432}, {15031, 15597}, {26613, 63534}, {34504, 62203}, {35007, 36523}, {43618, 47286}, {43619, 53489}, {53106, 60103}, {53107, 60211}, {54493, 62880}, {54646, 60198}

X(66425) = reflection of X(8353) in X(11361)
X(66425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33703, 66397}, {2, 66397, 19695}, {4, 66395, 8598}, {382, 33007, 8352}, {3543, 66387, 33228}, {3627, 19696, 33250}, {3830, 33193, 35297}, {3853, 33257, 33249}, {5073, 11159, 33192}, {5073, 33280, 6656}, {6658, 8597, 8369}, {6658, 62036, 33229}, {7841, 19687, 35954}, {7841, 35954, 8363}, {8352, 33007, 7807}, {8353, 8370, 8359}, {8359, 11361, 8370}, {8369, 8597, 33229}, {8369, 62036, 8597}, {11159, 33192, 6656}, {11286, 62045, 66390}, {14033, 15640, 66396}, {14033, 66396, 66349}, {33192, 33280, 11159}, {52942, 66398, 3}



leftri

Points releated to the 2nd outer-Grebe triangle: X(66426)-X(66474)

rightri

This preamble and centers X(66426)-X(66474) were contributed by Ivan Pavlov on Nov 25, 2024.

On the sides of ABC, construct squares ABCbCa, BCAcAb, and CABaBc. The triangle formed by lines BaCa, AbCb, and AcBc is called here the 2nd outer-Grebe triangle.
It is homothetic to the Artzt triangle and the center of homothety is X(6811).

For more information about the 2nd outer-Grebe triangle see this Euclid thread.
Some of the properties below refer to CTR-triangles. More info on these series of triangles is available in this catalog.

X(66426) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND ABC-X3 REFLECTIONS

Barycentrics    12*a^8-45*a^6*(b^2+c^2)+a^2*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^2*(3*b^4-14*b^2*c^2+3*c^4)+a^4*(35*b^4+26*b^2*c^2+35*c^4)-6*(5*a^6-2*(b^2-c^2)^2*(b^2+c^2)-a^2*(3*b^4+2*b^2*c^2+3*c^4))*S : :

X(66426) lies on these lines: {3, 54874}, {30, 66438}, {376, 55041}, {1503, 13666}, {3316, 6250}, {3564, 66432}, {5420, 14234}, {6561, 39656}, {6811, 9757}, {11257, 66434}, {12117, 66431}, {12159, 66462}, {23698, 42024}, {32421, 53141}

X(66426) = reflection of X(i) in X(j) for these {i,j}: {54874, 3}

X(66427) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND ANTI-ARTZT

Barycentrics    17*a^4-b^4+10*b^2*c^2-c^4-8*a^2*(b^2+c^2)+6*(a^2+b^2+c^2)*S : :

X(66427) lies on these lines: {2, 3}, {99, 13637}, {492, 51224}, {543, 13638}, {597, 66472}, {598, 11158}, {599, 66471}, {1285, 13759}, {1992, 66430}, {3068, 53142}, {8593, 66431}, {8974, 53141}, {9741, 13639}, {11147, 33338}, {11160, 66473}, {11161, 66442}, {11165, 13644}, {12150, 12158}, {12155, 66433}, {12159, 66436}, {13757, 66429}, {13758, 37809}, {13761, 50719}, {13789, 66439}, {22486, 66434}, {49543, 66437}

X(66427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66428, 66438, 13637}, {66430, 66443, 1992}

X(66428) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND 1ST TRI-SQUARES

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

X(66428) lies on these lines: {2, 38423}, {3, 40286}, {4, 45574}, {30, 45484}, {98, 485}, {99, 13637}, {376, 3068}, {590, 21843}, {597, 15484}, {637, 19103}, {1587, 11825}, {2549, 13644}, {3311, 53491}, {5210, 13846}, {6564, 66464}, {7581, 35794}, {7583, 9732}, {7585, 58804}, {7790, 61389}, {8975, 23249}, {8982, 13886}, {9540, 12124}, {13639, 66430}, {13640, 66431}, {13646, 66433}, {13650, 66432}, {13651, 62986}, {13663, 66471}, {13664, 66472}, {13665, 13910}, {13712, 13920}, {13833, 66439}, {14242, 31412}, {18512, 33878}, {19102, 32489}, {22541, 32808}, {22722, 66434}, {31411, 53487}, {32419, 44594}, {38424, 51171}, {44656, 50721}, {49620, 66437}

X(66428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 66443, 66429}, {13637, 66427, 66438}

X(66429) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND 2ND TRI-SQUARES

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

X(66429) lies on circumconic {{A, B, C, X(9164), X(54627)}} and on these lines: {2, 38423}, {4, 13880}, {6, 41490}, {30, 50721}, {187, 13989}, {193, 13771}, {230, 32421}, {371, 45522}, {372, 6811}, {376, 3069}, {488, 45576}, {492, 41411}, {524, 620}, {574, 32788}, {591, 1384}, {615, 5475}, {639, 12968}, {641, 6423}, {754, 44390}, {1152, 45577}, {1692, 8997}, {1992, 13769}, {2459, 48726}, {3068, 55041}, {3830, 13988}, {5062, 64691}, {5092, 35256}, {5860, 46453}, {6398, 36733}, {6560, 66464}, {6566, 53498}, {6781, 53515}, {7584, 43141}, {11008, 13650}, {11315, 45574}, {12601, 13933}, {13665, 13692}, {13757, 66427}, {13759, 66430}, {13760, 66431}, {13763, 13847}, {13765, 66433}, {13770, 66432}, {13773, 43460}, {13782, 66439}, {13783, 66471}, {13784, 66472}, {13834, 58803}, {13849, 33457}, {13935, 45552}, {13966, 43121}, {13972, 21850}, {13993, 53492}, {21843, 41491}, {22723, 66434}, {26288, 37689}, {38425, 49786}, {39387, 45515}, {39679, 48734}, {41410, 62987}, {44391, 58448}, {49621, 66437}, {50723, 53497}

X(66429) = midpoint of X(i) and X(j) for these {i,j}: {187, 44392}, {6566, 53498}, {6781, 53515}
X(66429) = reflection of X(i) in X(j) for these {i,j}: {44391, 58448}
X(66429) = pole of line {9168, 38425} with respect to the Steiner inellipse
X(66429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 66443, 66428}, {187, 44392, 32419}

X(66430) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT ANTI-ARTZT

Barycentrics    7*a^4+b^4+14*b^2*c^2+c^4-16*a^2*(b^2+c^2)-12*(2*a^2-b^2-c^2)*S : :

X(66430) lies on these lines: {2, 13988}, {376, 524}, {538, 66434}, {543, 66431}, {1991, 5485}, {1992, 66427}, {5503, 14229}, {5860, 11165}, {5861, 52229}, {6811, 9770}, {11148, 13798}, {13639, 66428}, {13759, 66429}, {17132, 66437}, {32421, 53141}, {66464, 66466}

X(66430) = reflection of X(i) in X(j) for these {i,j}: {5485, 1991}, {5860, 11165}, {26288, 53142}, {66471, 66472}, {66473, 66471}
X(66430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 66473, 66471}, {524, 53142, 26288}, {524, 66471, 66473}, {1992, 66427, 66443}, {66471, 66472, 376}

X(66431) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 1ST ANTI-BROCARD

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

X(66431) lies on these lines: {69, 74}, {531, 66433}, {543, 66430}, {671, 6230}, {2482, 33430}, {2782, 66434}, {2796, 66437}, {5071, 50721}, {5477, 66443}, {6054, 6811}, {8593, 66427}, {12117, 66426}, {13640, 66428}, {13760, 66429}, {19109, 42602}, {23234, 50719}, {23235, 66439}, {26289, 66464}

X(66431) = reflection of X(i) in X(j) for these {i,j}: {671, 6230}, {33430, 2482}, {66442, 376}
X(66431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 542, 66442}

X(66432) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 3RD ANTI-TRI-SQUARES

Barycentrics    12*a^4-3*b^4+14*b^2*c^2-3*c^4-13*a^2*(b^2+c^2)-2*(5*a^2-4*(b^2+c^2))*S : :

X(66432) lies on these lines: {376, 5860}, {487, 1131}, {524, 66439}, {3564, 66426}, {6811, 9767}, {7615, 42602}, {12150, 12158}, {13650, 66428}, {13770, 66429}, {22485, 22592}

X(66432) = reflection of X(i) in X(j) for these {i,j}: {42024, 487}

X(66433) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT BANKOFF

Barycentrics    -11*a^4+b^4-10*b^2*c^2+c^4+14*a^2*(b^2+c^2)+sqrt(3)*(6*a^4-3*(b^2-c^2)^2-3*a^2*(b^2+c^2))+2*(6*a^2-3*(b^2+c^2)+sqrt(3)*(a^2+b^2+c^2))*S : :

X(66433) lies on these lines: {376, 530}, {531, 66431}, {543, 1991}, {5463, 23011}, {6306, 42035}, {6811, 9762}, {12155, 66427}, {13646, 66428}, {13765, 66429}, {41620, 66443}

X(66433) = reflection of X(i) in X(j) for these {i,j}: {42035, 6306}

X(66434) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 1ST BROCARD-REFLECTED

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

X(66434) lies on these lines: {39, 19063}, {76, 22727}, {262, 486}, {371, 10841}, {376, 511}, {538, 66430}, {726, 66437}, {2782, 66431}, {5052, 66443}, {5871, 22699}, {6194, 6462}, {9732, 9755}, {9738, 21445}, {10851, 11824}, {11257, 66426}, {19108, 35840}, {22486, 66427}, {22525, 44486}, {22682, 66464}, {22712, 66438}, {22722, 66428}, {22723, 66429}, {22728, 36733}, {43532, 60195}, {49326, 66442}

X(66434) = reflection of X(i) in X(j) for these {i,j}: {76, 22727}, {262, 3103}, {33434, 39}

X(66435) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT CIRCUMMEDIAL

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

X(66435) lies on these lines: {2, 3}, {69, 42226}, {141, 42276}, {524, 6560}, {543, 1991}, {590, 43619}, {591, 3849}, {597, 6561}, {599, 42264}, {615, 43618}, {1327, 13835}, {1328, 54628}, {1588, 8411}, {1992, 42216}, {2549, 13644}, {2782, 66431}, {3589, 42275}, {3618, 42225}, {5860, 43256}, {5861, 52229}, {6200, 13663}, {6398, 13757}, {6452, 32807}, {6564, 32479}, {6565, 13783}, {7737, 8376}, {8584, 9974}, {12123, 45862}, {12158, 20423}, {13637, 13665}, {13639, 23267}, {13669, 46264}, {13763, 13847}, {13828, 13850}, {13846, 44526}, {15048, 19054}, {15533, 32421}, {15534, 32419}, {18907, 19053}, {21356, 58803}, {22485, 22592}, {32810, 64018}, {32811, 32815}, {38072, 45545}, {41946, 50681}, {42215, 59373}, {42263, 47352}, {45544, 54131}, {54656, 60195}, {63059, 63633}

X(66435) = midpoint of X(i) and X(j) for these {i,j}: {599, 42264}, {1992, 58804}
X(66435) = reflection of X(i) in X(j) for these {i,j}: {1992, 42216}, {6561, 597}, {54131, 45544}
X(66435) = X(1991)-of-anti-Artzt
X(66435) = intersection, other than A, B, C, of circumconics {{A, B, C, X(54628), X(62957)}}, {{A, B, C, X(60224), X(62956)}}
X(66435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66438, 66439, 66472}

X(66436) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 1ST HALF-SQUARES

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

X(66436) lies on these lines: {2, 13880}, {193, 13771}, {376, 5861}, {485, 1991}, {491, 13834}, {524, 19145}, {543, 22502}, {641, 5860}, {6278, 6811}, {7775, 22484}, {10519, 41491}, {12159, 66427}, {12222, 51952}, {13651, 62986}, {14645, 55040}, {19103, 45420}, {22485, 22592}, {22642, 22645}, {26289, 53016}, {32419, 66464}, {36733, 43139}, {60224, 60270}

X(66436) = reflection of X(i) in X(j) for these {i,j}: {485, 1991}, {5860, 641}, {6278, 9768}, {9768, 35685}, {42023, 485}

X(66437) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 1ST JENKINS

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

X(66437) lies on these lines: {40, 376}, {551, 66438}, {726, 66434}, {2796, 66431}, {4052, 49625}, {4856, 66443}, {6811, 49554}, {17132, 66430}, {28329, 66472}, {49543, 66427}, {49620, 66428}, {49621, 66429}

X(66437) = reflection of X(i) in X(j) for these {i,j}: {4052, 49625}

X(66438) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 1ST TRI-SQUARES-CENTRAL

Barycentrics    8*a^4+5*b^4-2*b^2*c^2+5*c^4-17*a^2*(b^2+c^2)-18*a^2*S : :

X(66438) lies on these lines: {2, 13988}, {6, 12040}, {30, 66426}, {99, 13637}, {371, 9770}, {376, 13666}, {485, 543}, {486, 9771}, {491, 55164}, {524, 19145}, {551, 66437}, {590, 40727}, {597, 38426}, {1151, 63945}, {1327, 13835}, {1328, 8176}, {1504, 9167}, {1991, 8182}, {1992, 13769}, {3068, 9741}, {3849, 53130}, {5418, 7610}, {5463, 23011}, {5464, 23002}, {5861, 13701}, {6054, 6811}, {6561, 66466}, {7615, 42602}, {7619, 43255}, {8592, 33343}, {9168, 54029}, {9540, 9740}, {11147, 12159}, {11163, 61389}, {11165, 32787}, {13639, 66443}, {13663, 38425}, {13690, 13828}, {13720, 14482}, {13846, 52229}, {13847, 63647}, {15597, 43254}, {20112, 42277}, {22712, 66434}, {26289, 48778}, {26613, 45420}, {27088, 61388}, {33456, 66462}, {35822, 53142}, {41963, 63950}, {54628, 60240}

X(66438) = reflection of X(i) in X(j) for these {i,j}: {60223, 2}
X(66438) = inverse of X(13637) in Wallace hyperbola
X(66438) = X(9741)-of-3rd-tri-squares-central
X(66438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13637, 13669, 13662}, {13637, 66427, 66428}, {66435, 66472, 66439}

X(66439) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT 2ND TRI-SQUARES-CENTRAL

Barycentrics    44*a^4-13*b^4+34*b^2*c^2-13*c^4-35*a^2*(b^2+c^2)-6*(a^2-2*(b^2+c^2))*S : :

X(66439) lies on these lines: {376, 13786}, {524, 66432}, {1327, 13835}, {6811, 13801}, {8182, 42023}, {8703, 60223}, {11148, 13798}, {13782, 66429}, {13789, 66427}, {13833, 66428}, {23235, 66431}, {32479, 55040}, {47102, 66471}

X(66439) = reflection of X(i) in X(j) for these {i,j}: {60224, 13835}
X(66439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66435, 66472, 66438}

X(66440) = ORTHOLOGY CENTER OF THESE TRIANGLES: VIJAY POLAR EXCENTRAL WRT 2ND OUTER-GREBE

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

X(66440) lies on these lines: {1, 2}, {226, 42051}, {312, 4072}, {345, 59579}, {518, 10440}, {524, 62189}, {527, 10443}, {536, 2051}, {573, 3928}, {740, 3817}, {752, 50808}, {908, 42044}, {1682, 31165}, {2092, 50092}, {2321, 37662}, {3175, 22020}, {3452, 3950}, {3596, 20942}, {3663, 4417}, {3739, 56226}, {3752, 4035}, {3846, 4356}, {3913, 19517}, {3932, 59686}, {3936, 24177}, {3977, 63010}, {3986, 5743}, {4058, 44417}, {4082, 32848}, {4098, 5233}, {4361, 58463}, {4656, 5741}, {4851, 6692}, {4856, 37642}, {4869, 8056}, {5085, 5847}, {5226, 17151}, {5542, 42053}, {5717, 19276}, {5814, 19279}, {5846, 59584}, {8715, 16435}, {9535, 64143}, {9568, 24391}, {9569, 19542}, {10445, 17132}, {15828, 56078}, {17314, 30827}, {17355, 63089}, {17490, 63589}, {18134, 24175}, {18228, 59585}, {21060, 42054}, {24386, 28581}, {27739, 50068}, {28313, 42047}, {31034, 62240}, {33071, 63969}, {34454, 34899}, {38408, 61661}, {42034, 50100}, {51090, 59547}, {59583, 60942}

X(66440) = midpoint of X(i) and X(j) for these {i,j}: {28609, 42049}
X(66440) = reflection of X(i) in X(j) for these {i,j}: {4052, 66465}
X(66440) = inverse of X(52907) in excircles-radical circle
X(66440) = pole of line {2976, 3667} with respect to the excircles-radical circle
X(66440) = intersection, other than A, B, C, of circumconics {{A, B, C, X(145), X(2051)}}, {{A, B, C, X(3009), X(7660)}}, {{A, B, C, X(4052), X(17751)}}, {{A, B, C, X(19998), X(37865)}}, {{A, B, C, X(54355), X(54553)}}
X(66440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3752, 4035, 21255}, {28609, 42049, 17132}

X(66441) = ORTHOLOGY CENTER OF THESE TRIANGLES: GEMINI 63 WRT 2ND OUTER-GREBE

Barycentrics    2*a^2-b^2-5*b*c-c^2-2*a*(b+c) : :
X(66441) = 2*X[37]+X[50088], X[192]+2*X[50099], X[1278]+2*X[50090], 4*X[3686]+5*X[4699], 2*X[3696]+X[49746], 8*X[3739]+X[17363], 4*X[4399]+5*X[4687], 2*X[4416]+7*X[4772], -4*X[4698]+X[50123], 8*X[4739]+X[17347], 7*X[4751]+2*X[17362], 2*X[4755]+X[50085] and many others

X(66441) lies on these lines: {1, 2}, {37, 50088}, {75, 545}, {86, 50131}, {192, 50099}, {319, 17313}, {320, 31139}, {335, 50075}, {391, 17116}, {524, 27483}, {527, 17488}, {536, 16590}, {594, 17338}, {740, 31331}, {752, 24452}, {894, 37654}, {903, 4643}, {966, 17117}, {1213, 17396}, {1278, 50090}, {1573, 4850}, {1654, 17274}, {2796, 31310}, {3219, 21373}, {3618, 28635}, {3654, 6996}, {3662, 17271}, {3686, 4699}, {3696, 49746}, {3739, 17363}, {3758, 10022}, {3797, 50086}, {3829, 26019}, {3948, 4479}, {3997, 14997}, {4034, 17300}, {4044, 27772}, {4359, 33934}, {4360, 31332}, {4361, 17248}, {4370, 4665}, {4371, 17319}, {4395, 17250}, {4399, 4687}, {4402, 17324}, {4416, 4772}, {4421, 16367}, {4422, 62228}, {4478, 17241}, {4659, 17487}, {4664, 28309}, {4688, 4715}, {4690, 31138}, {4698, 50123}, {4725, 31306}, {4733, 48810}, {4739, 17347}, {4740, 28301}, {4751, 17362}, {4755, 50085}, {4785, 14433}, {4921, 26643}, {4967, 17349}, {4969, 41847}, {4971, 31322}, {5224, 17382}, {5233, 27747}, {5278, 11352}, {5564, 17242}, {5839, 63110}, {5936, 51171}, {6651, 50126}, {7263, 17328}, {7384, 31162}, {7406, 34632}, {10436, 63052}, {11194, 11329}, {14621, 55955}, {14839, 63961}, {17119, 17256}, {17120, 63086}, {17133, 27480}, {17227, 64712}, {17234, 50081}, {17251, 37756}, {17260, 42696}, {17261, 32087}, {17270, 48633}, {17277, 17281}, {17278, 32025}, {17301, 31144}, {17317, 62224}, {17337, 48630}, {17341, 48636}, {17343, 24199}, {17348, 17368}, {17360, 34824}, {17377, 31238}, {17381, 28633}, {17392, 50077}, {17755, 50096}, {17950, 36595}, {18146, 59212}, {19281, 19723}, {20132, 50283}, {20137, 49497}, {20138, 32941}, {20142, 50300}, {20152, 49680}, {20154, 48805}, {20533, 51102}, {24589, 64133}, {25057, 51583}, {25590, 62989}, {28329, 31319}, {28534, 60927}, {28562, 31329}, {28610, 50735}, {30044, 34282}, {30583, 47762}, {31151, 50308}, {31178, 50309}, {31347, 31352}, {32029, 47358}, {34627, 36698}, {35957, 64463}, {36588, 39721}, {40480, 48639}, {41842, 64299}, {41845, 50836}, {42026, 52755}, {43527, 65022}, {49725, 50289}, {50166, 50220}, {51381, 64906}, {59772, 63051}

X(66441) = midpoint of X(i) and X(j) for these {i,j}: {75, 66451}
X(66441) = reflection of X(i) in X(j) for these {i,j}: {17333, 66451}, {66451, 17330}
X(66441) = pole of line {1213, 17342} with respect to the Kiepert hyperbola
X(66441) = pole of line {514, 48183} with respect to the Steiner inellipse
X(66441) = pole of line {86, 4795} with respect to the Wallace hyperbola
X(66441) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(75), X(17310)}}, {{A, B, C, X(551), X(14621)}}, {{A, B, C, X(3241), X(39721)}}, {{A, B, C, X(3624), X(43527)}}, {{A, B, C, X(3661), X(55955)}}, {{A, B, C, X(3679), X(27483)}}, {{A, B, C, X(16826), X(39704)}}, {{A, B, C, X(17316), X(36588)}}, {{A, B, C, X(19875), X(57725)}}, {{A, B, C, X(29624), X(65081)}}, {{A, B, C, X(29834), X(57721)}}, {{A, B, C, X(36871), X(50016)}}, {{A, B, C, X(49769), X(60276)}}
X(66441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3679, 3661}, {2, 4393, 551}, {75, 66451, 545}, {545, 17330, 66451}, {545, 66451, 17333}, {966, 17117, 17247}, {3679, 4384, 2}, {3686, 4699, 17364}, {3686, 50116, 50074}, {3739, 50082, 17378}, {4688, 17346, 50128}, {4699, 50074, 50116}, {4751, 17362, 17391}, {4751, 50132, 49738}, {4755, 50085, 50121}, {5564, 17259, 17242}, {17277, 28634, 48628}, {17277, 48628, 17339}, {17278, 32025, 48634}, {17330, 17333, 17331}, {17362, 49738, 50132}, {17378, 50082, 17363}, {31139, 66454, 320}, {49731, 50098, 4664}

X(66442) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT ANTI-MCCAY

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

X(66442) lies on these lines: {4, 6568}, {69, 74}, {98, 485}, {115, 23249}, {1503, 58033}, {2794, 5870}, {6230, 34473}, {6560, 10722}, {7694, 45510}, {10991, 33431}, {11161, 66427}, {12188, 36733}, {12256, 13989}, {13773, 43460}, {35820, 54877}, {48905, 49367}, {49326, 66434}

X(66442) = reflection of X(i) in X(j) for these {i,j}: {10722, 50719}, {33431, 10991}, {66431, 376}
X(66442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 542, 66431}

X(66443) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND PEDAL-OF-X(6)

Barycentrics    (3*a^2+b^2-c^2)*(3*a^2-b^2+c^2)+12*a^2*S : :

X(66443) lies on circumconic {{A, B, C, X(52187), X(54627)}} and on these lines: {2, 38423}, {6, 376}, {20, 45515}, {30, 26456}, {115, 23249}, {187, 44594}, {372, 37665}, {597, 66471}, {599, 66473}, {1384, 19054}, {1587, 6423}, {1588, 5062}, {1992, 66427}, {2549, 43256}, {3068, 26288}, {3069, 15484}, {3146, 19102}, {3815, 13935}, {4856, 66437}, {5052, 66434}, {5477, 66431}, {5860, 13644}, {6221, 26462}, {6460, 15048}, {6560, 61322}, {7585, 41411}, {7586, 13770}, {7736, 8376}, {8974, 32421}, {9112, 33440}, {9113, 33442}, {9540, 12968}, {13639, 66438}, {15640, 19099}, {18907, 19053}, {19108, 26289}, {20583, 66472}, {21309, 26463}, {23253, 49221}, {23263, 62203}, {32787, 46453}, {35822, 37689}, {35944, 44502}, {35945, 44656}, {36733, 42216}, {41410, 63015}, {41620, 66433}, {43407, 44526}, {43511, 45512}, {44596, 62220}, {61389, 63058}

X(66443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66428, 66429, 2}

X(66444) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(10) WRT 2ND OUTER-GREBE

Barycentrics    4*a^3-3*a^2*(b+c)+(b+c)^3+a*(-6*b^2+4*b*c-6*c^2) : :
X(66444) = -3*X[9741]+X[51121]

X(66444) lies on these lines: {2, 2415}, {10, 11359}, {40, 376}, {57, 3950}, {99, 17222}, {142, 59583}, {190, 45204}, {345, 21255}, {527, 10443}, {538, 62189}, {543, 34899}, {545, 66465}, {551, 3743}, {726, 10164}, {1018, 59173}, {2796, 6054}, {2802, 61671}, {3175, 22003}, {3218, 50292}, {3241, 6553}, {3752, 59579}, {3817, 28526}, {4061, 36263}, {4847, 32845}, {4856, 62820}, {5435, 55998}, {5437, 59585}, {5463, 49595}, {5464, 49594}, {5542, 59547}, {5745, 53594}, {6692, 17262}, {7757, 50114}, {9741, 51121}, {11019, 32934}, {15828, 37679}, {16833, 28638}, {24068, 59675}, {24386, 28530}, {28562, 55177}, {28582, 59584}, {28610, 64700}, {29573, 65384}, {29594, 31168}, {29671, 30424}, {31191, 44416}, {33116, 63589}, {42045, 62240}, {47039, 51122}, {47040, 51071}, {49517, 59593}, {49730, 52229}, {50109, 61661}, {52907, 59599}, {59572, 59732}

X(66444) = midpoint of X(i) and X(j) for these {i,j}: {3928, 42049}
X(66444) = reflection of X(i) in X(j) for these {i,j}: {4052, 2}
X(66444) = inverse of X(41629) in Wallace hyperbola
X(66444) = pole of line {28296, 59969} with respect to the orthoptic circle of the Steiner Inellipse
X(66444) = pole of line {3667, 25020} with respect to the Steiner inellipse
X(66444) = pole of line {17132, 41629} with respect to the Wallace hyperbola
X(66444) = pole of line {8, 21949} with respect to the dual conic of Yff parabola
X(66444) = X(4052)-of-Gemini-107
X(66444) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(17132), X(41629)}}, {{A, B, C, X(17951), X(60172)}}, {{A, B, C, X(18743), X(28655)}}, {{A, B, C, X(39980), X(47636)}}
X(66444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17132, 4052}, {2, 8055, 28655}, {2, 8056, 66468}, {3928, 42049, 519}, {28655, 63621, 2}, {56078, 62300, 24175}

X(66445) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(61) WRT 2ND OUTER-GREBE

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

X(66445) lies on these lines: {2, 6}, {5, 49914}, {13, 41114}, {14, 36366}, {61, 532}, {99, 22488}, {376, 5865}, {381, 5873}, {383, 576}, {398, 22113}, {473, 648}, {530, 41101}, {531, 41107}, {533, 7760}, {598, 22487}, {616, 11485}, {618, 36386}, {621, 42974}, {622, 33626}, {628, 11302}, {633, 37352}, {634, 11297}, {671, 6778}, {1080, 63722}, {1351, 6770}, {1353, 6773}, {3105, 7757}, {3412, 50859}, {3534, 51484}, {3543, 5869}, {3758, 40713}, {3830, 33625}, {3845, 36319}, {5055, 51487}, {5097, 5613}, {5459, 42506}, {5463, 42532}, {5978, 44498}, {6179, 16962}, {7812, 22495}, {7858, 16268}, {8014, 23895}, {8594, 35692}, {8703, 51485}, {11055, 12155}, {11087, 11144}, {12154, 12156}, {13102, 61600}, {14537, 35693}, {14568, 22496}, {16267, 34508}, {22114, 42156}, {22491, 41119}, {22492, 41113}, {22573, 47866}, {22579, 36362}, {22580, 36383}, {22666, 51208}, {22855, 36368}, {33464, 42581}, {33613, 49961}, {33622, 42633}, {33623, 33699}, {33627, 49945}, {35696, 41745}, {35931, 42511}, {36396, 49955}, {36397, 49957}, {37171, 42999}, {40714, 62231}, {42533, 45880}, {42632, 51224}, {42976, 45879}, {46709, 66328}, {49855, 49911}

X(66445) = reflection of X(i) in X(j) for these {i,j}: {633, 37352}, {11299, 61}, {11303, 61719}
X(66445) = X(i)-Dao conjugate of X(j) for these {i, j}: {36830, 39636}
X(66445) = pole of line {2501, 14446} with respect to the polar circle
X(66445) = pole of line {99, 39636} with respect to the Kiepert parabola
X(66445) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(396), X(8738)}}, {{A, B, C, X(524), X(12816)}}, {{A, B, C, X(18842), X(63102)}}
X(66445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 193, 5862}, {2, 3180, 5859}, {2, 5859, 299}, {2, 5862, 298}, {6, 5859, 2}, {396, 3181, 302}, {396, 3629, 3181}, {533, 61719, 11303}

X(66446) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(62) WRT 2ND OUTER-GREBE

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

X(66446) lies on these lines: {2, 6}, {5, 49911}, {13, 36368}, {14, 41115}, {62, 533}, {99, 22487}, {376, 5864}, {381, 5872}, {383, 63722}, {397, 22114}, {472, 648}, {530, 41108}, {531, 41100}, {532, 7760}, {576, 1080}, {598, 22488}, {617, 11486}, {619, 36388}, {621, 33627}, {622, 42975}, {627, 11301}, {633, 11298}, {634, 37351}, {671, 6777}, {1351, 6773}, {1353, 6770}, {3104, 7757}, {3411, 50860}, {3534, 51485}, {3543, 5868}, {3758, 40714}, {3830, 33623}, {3845, 36344}, {5055, 51486}, {5097, 5617}, {5460, 42507}, {5464, 42533}, {5979, 44497}, {6179, 16963}, {7812, 22496}, {7858, 16267}, {8015, 23896}, {8595, 35696}, {8703, 51484}, {11055, 12154}, {11082, 11143}, {12155, 12156}, {13103, 61600}, {14537, 35697}, {14568, 22495}, {16268, 34509}, {22113, 42153}, {22491, 41112}, {22492, 41120}, {22574, 47865}, {22579, 36382}, {22580, 36363}, {22665, 51209}, {22901, 36366}, {33465, 42580}, {33612, 49962}, {33624, 42634}, {33625, 33699}, {33626, 49946}, {34508, 61719}, {35692, 41746}, {35932, 42510}, {36400, 49956}, {36401, 49958}, {37170, 42998}, {40713, 62231}, {42532, 45879}, {42631, 51224}, {42977, 45880}, {46708, 66328}, {49858, 49914}

X(66446) = reflection of X(i) in X(j) for these {i,j}: {634, 37351}, {11300, 62}
X(66446) = X(i)-isoconjugate-of-X(j) for these {i, j}: {661, 39637}
X(66446) = X(i)-Dao conjugate of X(j) for these {i, j}: {36830, 39637}
X(66446) = pole of line {2501, 14447} with respect to the polar circle
X(66446) = pole of line {99, 39637} with respect to the Kiepert parabola
X(66446) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(395), X(8737)}}, {{A, B, C, X(524), X(12817)}}, {{A, B, C, X(18842), X(63103)}}
X(66446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 193, 5863}, {2, 3181, 5858}, {2, 5858, 298}, {2, 5863, 299}, {6, 5858, 2}, {62, 533, 11300}, {395, 3180, 303}, {395, 3629, 3180}

X(66447) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(76) WRT 2ND OUTER-GREBE

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

X(66447) lies on these lines: {2, 59535}, {6, 66448}, {76, 11287}, {99, 1384}, {376, 538}, {524, 39882}, {543, 14458}, {698, 13468}, {732, 33976}, {2549, 14711}, {3102, 55041}, {3103, 55040}, {5463, 23009}, {5464, 23000}, {5503, 60095}, {5969, 6054}, {7757, 8369}, {7837, 8592}, {8354, 37671}, {8667, 55178}, {8782, 32451}, {9466, 33230}, {9741, 63006}, {12203, 63954}

X(66447) = reflection of X(i) in X(j) for these {i,j}: {11055, 51122}, {60180, 2}
X(66447) = inverse of X(14614) in Wallace hyperbola
X(66447) = pole of line {14614, 41413} with respect to the Wallace hyperbola

X(66448) = ORTHOLOGY CENTER OF THESE TRIANGLES: ANTIPEDAL-OF-X(83) WRT 2ND OUTER-GREBE

Barycentrics    3*a^8+5*b^6*c^2+6*b^4*c^4+5*b^2*c^6-a^6*(b^2+c^2)-a^4*(9*b^4+19*b^2*c^2+9*c^4)-a^2*(b^6+7*b^4*c^2+7*b^2*c^4+c^6) : :
X(66448) = -3*X[6308]+4*X[46893]

X(66448) lies on these lines: {2, 60181}, {6, 66447}, {83, 1975}, {99, 12156}, {371, 6274}, {372, 6275}, {376, 754}, {524, 55178}, {538, 32467}, {543, 14492}, {732, 5085}, {3734, 14482}, {5182, 8290}, {5463, 23010}, {5464, 23001}, {5503, 8592}, {6054, 9765}, {6308, 46893}, {7788, 11165}, {7799, 8359}, {8716, 35701}, {9741, 63024}, {9766, 48905}, {11179, 64243}

X(66448) = midpoint of X(i) and X(j) for these {i,j}: {8716, 35701}
X(66448) = reflection of X(i) in X(j) for these {i,j}: {60181, 2}
X(66448) = inverse of X(41624) in Wallace hyperbola
X(66448) = pole of line {41622, 41624} with respect to the Wallace hyperbola

X(66449) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(216) WRT 2ND OUTER-GREBE

Barycentrics    2*a^8-a^6*(b^2+c^2)+5*a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(-5*b^4+4*b^2*c^2-5*c^4)-(b^2-c^2)^2*(b^4+c^4) : :
X(66449) = 2*X[216]+X[27377], X[3164]+2*X[6748]

X(66449) lies on these lines: {2, 6}, {5, 648}, {30, 30258}, {53, 17035}, {216, 27377}, {297, 5158}, {381, 6530}, {401, 6749}, {671, 60121}, {1494, 42330}, {1513, 8541}, {1990, 52247}, {3087, 20477}, {3091, 15274}, {3164, 6748}, {3545, 41371}, {3759, 53821}, {3839, 42831}, {5007, 26205}, {7399, 7760}, {7750, 26216}, {7769, 36841}, {7772, 26155}, {7812, 34664}, {9744, 10602}, {14570, 32819}, {14912, 20792}, {15526, 52289}, {15851, 17907}, {15860, 23583}, {16813, 62603}, {17813, 64711}, {32002, 42459}, {34836, 56297}, {36794, 41005}, {37188, 62213}, {40065, 40680}, {42353, 60693}, {43461, 47277}, {44096, 44212}, {44285, 51224}, {52281, 64781}, {52766, 58875}, {62595, 64923}

X(66449) = midpoint of X(i) and X(j) for these {i,j}: {27377, 35937}
X(66449) = reflection of X(i) in X(j) for these {i,j}: {35937, 216}
X(66449) = X(i)-complementary conjugate of X(j) for these {i, j}: {54732, 2887}
X(66449) = pole of line {2501, 42731} with respect to the polar circle
X(66449) = pole of line {6467, 14461} with respect to the Jerabek hyperbola
X(66449) = pole of line {2, 54732} with respect to the Kiepert hyperbola
X(66449) = intersection, other than A, B, C, of circumconics {{A, B, C, X(524), X(60121)}}, {{A, B, C, X(11064), X(42330)}}
X(66449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {216, 64783, 35937}

X(66450) = ORTHOLOGY CENTER OF THESE TRIANGLES: PEDAL-OF-X(284) WRT 2ND OUTER-GREBE

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

X(66450) lies on these lines: {2, 6}, {21, 6172}, {27, 31164}, {29, 648}, {78, 3758}, {190, 34772}, {284, 527}, {528, 5327}, {662, 1434}, {671, 54526}, {1043, 50107}, {1817, 2094}, {2966, 53193}, {4234, 50127}, {4248, 11520}, {4273, 17301}, {5703, 54280}, {6173, 16054}, {6734, 62231}, {14543, 34195}, {14616, 32040}, {14953, 60984}, {16053, 60986}, {17188, 31146}, {30728, 42724}, {34393, 65276}, {34619, 62843}, {50129, 56019}, {54966, 65835}, {56948, 60951}, {58786, 60971}, {60014, 65271}

X(66450) = midpoint of X(i) and X(j) for these {i,j}: {35935, 56020}
X(66450) = reflection of X(i) in X(j) for these {i,j}: {8822, 35935}, {35935, 284}
X(66450) = X(i)-isoconjugate-of-X(j) for these {i, j}: {661, 28291}
X(66450) = X(i)-Dao conjugate of X(j) for these {i, j}: {36830, 28291}
X(66450) = pole of line {2501, 30574} with respect to the polar circle
X(66450) = pole of line {99, 28291} with respect to the Kiepert parabola
X(66450) = pole of line {2, 51121} with respect to the Wallace hyperbola
X(66450) = pole of line {3265, 53334} with respect to the dual conic of Orthic inconic
X(66450) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(12848)}}, {{A, B, C, X(69), X(1121)}}, {{A, B, C, X(183), X(60014)}}, {{A, B, C, X(325), X(53193)}}, {{A, B, C, X(394), X(60047)}}, {{A, B, C, X(524), X(28292)}}, {{A, B, C, X(4585), X(32040)}}, {{A, B, C, X(5232), X(58002)}}, {{A, B, C, X(14548), X(39704)}}, {{A, B, C, X(14552), X(55956)}}, {{A, B, C, X(34393), X(37668)}}, {{A, B, C, X(37658), X(47375)}}, {{A, B, C, X(41570), X(51384)}}
X(66450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {284, 56020, 8822}, {35935, 56020, 527}

X(66451) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(1)-ANTIPEDAL-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    4*a^2-2*b^2-b*c-2*c^2-4*a*(b+c) : :
X(66451) = 2*X[37]+X[50074], X[192]+2*X[50082], X[3644]+8*X[3686], 4*X[4416]+5*X[4687], -5*X[4704]+2*X[50123], 7*X[4751]+2*X[17347], -4*X[4755]+X[50133], -X[4764]+4*X[50099], X[17363]+2*X[50113], -X[17364]+4*X[49738], -X[17389]+4*X[49737], -X[49499]+4*X[50305] and many others

X(66451) lies on these lines: {1, 31332}, {2, 44}, {8, 66452}, {9, 17228}, {10, 24452}, {37, 50074}, {45, 17310}, {75, 545}, {190, 3679}, {192, 50082}, {239, 24441}, {329, 41846}, {333, 17196}, {391, 17258}, {513, 31992}, {519, 751}, {524, 51488}, {903, 4384}, {1150, 27776}, {1386, 1992}, {1654, 17281}, {1743, 17400}, {1762, 31153}, {2550, 5080}, {3241, 62231}, {3644, 3686}, {3661, 4370}, {3707, 4389}, {3731, 17386}, {3759, 17257}, {3789, 24482}, {3929, 64907}, {3973, 17307}, {4416, 4687}, {4422, 48639}, {4690, 4908}, {4704, 50123}, {4751, 17347}, {4755, 50133}, {4762, 55954}, {4764, 50099}, {4859, 17274}, {4945, 65052}, {5224, 50115}, {5296, 63110}, {10005, 50107}, {10022, 29576}, {15492, 17238}, {15533, 29575}, {15534, 29580}, {16468, 25055}, {16814, 17240}, {16815, 31139}, {16885, 17252}, {17241, 17344}, {17247, 50112}, {17249, 17349}, {17253, 17370}, {17260, 17313}, {17261, 50087}, {17272, 17341}, {17294, 36911}, {17315, 63001}, {17321, 63086}, {17337, 48637}, {17338, 48638}, {17339, 48640}, {17363, 50113}, {17364, 49738}, {17389, 49737}, {17393, 50131}, {17394, 63052}, {19875, 24342}, {20917, 39996}, {20973, 62796}, {21356, 61023}, {22165, 29582}, {24697, 50287}, {25728, 32025}, {28301, 49748}, {28309, 29617}, {28606, 39974}, {28840, 62634}, {31225, 41801}, {36522, 64712}, {36872, 65054}, {38098, 50118}, {41312, 63049}, {42030, 42044}, {42034, 49724}, {48829, 60731}, {49499, 50305}, {49731, 50128}, {50297, 51055}

X(66451) = midpoint of X(i) and X(j) for these {i,j}: {2, 17488}, {17333, 66441}
X(66451) = reflection of X(i) in X(j) for these {i,j}: {2, 16590}, {75, 66441}, {24452, 10}, {39704, 2}, {66441, 17330}
X(66451) = pole of line {4777, 47775} with respect to the Steiner circumellipse
X(66451) = pole of line {4777, 47778} with respect to the Steiner inellipse
X(66451) = pole of line {5235, 25057} with respect to the Wallace hyperbola
X(66451) = pole of line {551, 24452} with respect to the dual conic of Yff parabola
X(66451) = intersection, other than A, B, C, of circumconics {{A, B, C, X(80), X(4795)}}, {{A, B, C, X(89), X(751)}}, {{A, B, C, X(519), X(29895)}}, {{A, B, C, X(3679), X(4715)}}, {{A, B, C, X(4945), X(26738)}}, {{A, B, C, X(5235), X(25057)}}, {{A, B, C, X(28658), X(41416)}}, {{A, B, C, X(31138), X(57725)}}, {{A, B, C, X(35170), X(39704)}}, {{A, B, C, X(52901), X(65054)}}, {{A, B, C, X(63233), X(65052)}}
X(66451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16590, 41848}, {2, 17488, 4715}, {2, 20072, 4795}, {2, 24616, 63233}, {2, 25057, 30608}, {2, 30564, 25057}, {2, 31056, 27751}, {2, 4741, 31138}, {2, 4795, 41847}, {9, 17271, 17342}, {37, 50074, 50132}, {45, 66454, 17310}, {545, 17330, 66441}, {1654, 17336, 48630}, {4643, 17335, 17227}, {4664, 17346, 50077}, {16590, 17488, 39704}, {16885, 17252, 17371}, {17256, 54280, 3758}, {17257, 37654, 17320}, {17271, 17342, 17228}, {17277, 17329, 48629}, {17310, 66454, 17360}, {17320, 37654, 3759}, {17328, 17342, 17271}, {17330, 17332, 17333}, {17330, 17333, 75}, {17331, 17333, 17330}, {17333, 66441, 545}, {17346, 50093, 4664}, {17347, 63978, 4751}, {39704, 41848, 2}, {50088, 50090, 3644}

X(66452) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(8)-ANTIPEDAL-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    7*a^2+b^2+14*b*c+c^2-16*a*(b+c) : :
X(66452) = 8*X[3950]+X[60957], 5*X[18230]+4*X[55998], -4*X[29573]+X[60971], -4*X[37654]+7*X[60983], -16*X[59585]+7*X[60996]

X(66452) lies on circumconic {{A, B, C, X(3241), X(28301)}} and on these lines: {2, 1266}, {7, 545}, {8, 66451}, {190, 3241}, {192, 31349}, {346, 50090}, {390, 519}, {522, 31992}, {536, 61023}, {903, 29627}, {3161, 17352}, {3616, 31332}, {3950, 60957}, {4346, 41141}, {4353, 25055}, {4370, 5222}, {4419, 4908}, {4460, 25728}, {4488, 17378}, {4664, 15569}, {4733, 9791}, {4762, 63246}, {4901, 50093}, {5226, 36595}, {6707, 7229}, {17132, 59374}, {17160, 31722}, {17256, 51068}, {17261, 32087}, {17310, 20073}, {17316, 17487}, {17318, 36522}, {17333, 32099}, {17488, 50079}, {18228, 36916}, {18230, 55998}, {20090, 25269}, {24441, 29611}, {28503, 52653}, {28580, 59413}, {29573, 60971}, {31153, 64143}, {37654, 60983}, {37681, 50108}, {41140, 62706}, {59585, 60996}

X(66452) = reflection of X(i) in X(j) for these {i,j}: {2, 36911}, {36588, 2}
X(66452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 28301, 36588}, {28301, 36911, 2}, {62706, 66456, 41140}

X(66453) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(9)-ANTIPEDAL-OF-X(2) WRT 2ND OUTER-GREBE

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

X(66453) lies on these lines: {2, 6603}, {85, 64462}, {527, 31169}, {664, 6173}, {1121, 40719}, {2550, 3241}, {3872, 17297}, {3900, 64149}, {4762, 39704}, {10004, 17079}, {23058, 55082}, {32007, 42050}, {38948, 60876}

X(66453) = reflection of X(i) in X(j) for these {i,j}: {55954, 2}

X(66454) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(1)-CIRCUMCONCEVIAN-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    5*a^2-2*a*(b+c)-2*(2*b^2+b*c+2*c^2) : :
X(66454) = 2*X[4665]+X[64015], -2*X[4670]+3*X[19875], -4*X[4708]+3*X[25055], -5*X[4748]+3*X[38314], -3*X[35578]+7*X[51068], -3*X[41312]+2*X[51071]

X(66454) lies on these lines: {2, 6}, {8, 545}, {9, 50081}, {10, 4795}, {45, 17310}, {144, 4478}, {190, 17488}, {319, 17262}, {320, 31139}, {519, 4643}, {527, 4669}, {536, 4677}, {742, 50075}, {903, 4741}, {3241, 4364}, {3679, 4363}, {3686, 7232}, {3707, 41141}, {3828, 4667}, {3830, 48938}, {4034, 17345}, {4042, 31134}, {4346, 4405}, {4357, 50131}, {4361, 17274}, {4370, 17269}, {4384, 31138}, {4389, 40891}, {4416, 4445}, {4644, 10022}, {4665, 64015}, {4670, 19875}, {4708, 25055}, {4713, 31136}, {4725, 51093}, {4748, 38314}, {4908, 17294}, {4945, 31172}, {5220, 64906}, {5839, 17323}, {5845, 50949}, {6172, 36522}, {6646, 50088}, {11354, 63939}, {16590, 17374}, {16666, 25503}, {16675, 17373}, {16677, 17386}, {16777, 17328}, {16884, 17252}, {16885, 17287}, {17230, 41138}, {17250, 62212}, {17253, 17320}, {17254, 50077}, {17255, 17362}, {17257, 50113}, {17272, 17382}, {17275, 50116}, {17276, 50099}, {17290, 41140}, {17293, 50115}, {17299, 50090}, {17309, 17332}, {17311, 17331}, {17325, 62231}, {17387, 41848}, {17677, 63933}, {20072, 61321}, {24699, 50095}, {25057, 27757}, {27949, 43287}, {28301, 34641}, {28333, 51072}, {29069, 50798}, {29615, 49721}, {29617, 49747}, {33082, 48829}, {35578, 51068}, {41312, 51071}, {48817, 63944}, {49742, 50079}, {50076, 50093}, {50275, 64912}, {50276, 57006}, {50950, 51034}, {51678, 63938}, {64802, 66307}

X(66454) = midpoint of X(i) and X(j) for these {i,j}: {4419, 31145}
X(66454) = reflection of X(i) in X(j) for these {i,j}: {3241, 4364}, {3679, 4690}, {4363, 3679}, {4644, 10022}, {4667, 3828}, {4795, 10}, {10022, 64712}, {17318, 24441}, {24441, 4643}
X(66454) = pole of line {1125, 31139} with respect to the dual conic of Yff parabola
X(66454) = X(4795)-of-outer-Garcia
X(66454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {319, 17333, 50087}, {320, 66441, 31139}, {519, 24441, 17318}, {519, 4643, 24441}, {3679, 4715, 4363}, {4419, 31145, 28309}, {4644, 53620, 10022}, {4690, 4715, 3679}, {10022, 64712, 53620}, {17254, 50077, 50120}, {17310, 66451, 45}, {17332, 32099, 17309}, {17333, 50087, 17262}, {17344, 50082, 17274}, {17360, 66451, 17310}

X(66455) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(6)-CIRCUMCONCEVIAN-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    5*a^4-2*a^2*(b^2+c^2)-2*(2*b^4+b^2*c^2+2*c^4) : :
X(66455) = -X[7737]+3*X[21356], -3*X[7739]+X[63064], -2*X[7804]+3*X[21358], -3*X[11286]+5*X[50993], -3*X[11287]+X[15534], -3*X[14033]+7*X[50994], 3*X[32986]+X[50992]

X(66455) lies on circumconic {{A, B, C, X(83), X(13377)}} and on these lines: {2, 32}, {6, 63942}, {30, 22165}, {69, 543}, {76, 8597}, {141, 63945}, {183, 7617}, {325, 7622}, {376, 14981}, {524, 7761}, {538, 5077}, {574, 7840}, {599, 3734}, {620, 8182}, {625, 7610}, {671, 7898}, {1007, 7619}, {1992, 4045}, {2482, 7908}, {2549, 11160}, {2794, 54173}, {3314, 51224}, {3363, 63956}, {3631, 59780}, {3830, 6248}, {3933, 34504}, {5206, 7870}, {5254, 63953}, {5306, 63948}, {5461, 63029}, {5569, 22110}, {6054, 8722}, {6656, 63937}, {6722, 23055}, {7615, 15589}, {7618, 37668}, {7737, 21356}, {7739, 63064}, {7750, 7801}, {7751, 7841}, {7759, 8359}, {7764, 33215}, {7767, 7825}, {7768, 7781}, {7772, 7936}, {7779, 52691}, {7780, 11318}, {7784, 7817}, {7788, 35955}, {7790, 44367}, {7794, 33007}, {7804, 21358}, {7826, 7872}, {7827, 7893}, {7829, 33230}, {7830, 7916}, {7831, 63028}, {7842, 34505}, {7844, 22329}, {7845, 11163}, {7849, 33237}, {7854, 8370}, {7860, 33013}, {7863, 33208}, {7866, 63930}, {7869, 8369}, {7888, 33274}, {7897, 8588}, {7902, 14023}, {7903, 7904}, {7917, 15515}, {7919, 62204}, {7924, 41748}, {7934, 8859}, {7946, 53096}, {8176, 11168}, {8352, 18546}, {8355, 13468}, {8357, 63934}, {8360, 63928}, {8366, 35007}, {8584, 63940}, {9466, 11317}, {9737, 34510}, {9766, 40344}, {9855, 11057}, {10513, 14148}, {11007, 38239}, {11054, 11648}, {11286, 50993}, {11287, 15534}, {14033, 50994}, {14762, 15484}, {14971, 17008}, {15300, 32833}, {18907, 20582}, {21843, 22247}, {27088, 47101}, {32974, 63927}, {32986, 50992}, {33184, 63952}, {36523, 63955}, {42850, 66466}, {44543, 50280}, {47074, 47596}, {50991, 63941}, {51185, 63946}, {51186, 63947}, {55801, 63021}, {62203, 63044}, {63124, 63944}

X(66455) = midpoint of X(i) and X(j) for these {i,j}: {2549, 11160}, {5077, 15533}
X(66455) = reflection of X(i) in X(j) for these {i,j}: {599, 7848}, {1992, 4045}, {3734, 599}, {18907, 20582}, {59780, 3631}
X(66455) = pole of line {39, 353} with respect to the Stammler hyperbola
X(66455) = pole of line {826, 9191} with respect to the Steiner circumellipse
X(66455) = pole of line {141, 8598} with respect to the Wallace hyperbola
X(66455) = X(3098)-of-anti-Artzt
X(66455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {183, 31173, 7617}, {315, 7810, 7775}, {599, 3849, 3734}, {3849, 7848, 599}, {5077, 15533, 538}, {7775, 7810, 7815}, {7784, 63950, 7817}, {7840, 55164, 574}, {7845, 15810, 11163}, {7850, 55164, 7840}, {7883, 9939, 32}, {7929, 9939, 7883}, {11163, 15810, 15482}

X(66456) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(8)-CIRCUMCONCEVIAN-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    a^2+b^2-22*b*c+c^2+14*a*(b+c) : :
X(66456) = -5*X[3623]+4*X[4795], -3*X[4363]+4*X[66457], -2*X[4659]+3*X[38314], -5*X[4747]+8*X[17318], -5*X[24441]+4*X[64712], -3*X[35578]+4*X[51071]

X(66456) lies on these lines: {2, 37}, {145, 545}, {391, 50090}, {519, 64015}, {3241, 4454}, {3623, 4795}, {3875, 63086}, {4346, 17310}, {4363, 66457}, {4373, 17313}, {4659, 38314}, {4677, 28313}, {4715, 20049}, {4747, 17318}, {4779, 49453}, {17132, 51093}, {17262, 32105}, {20059, 50132}, {20073, 40891}, {24441, 64712}, {28297, 51092}, {29069, 50872}, {35578, 51071}, {41140, 62706}, {50108, 55998}, {50121, 60984}, {50123, 62999}

X(66456) = reflection of X(i) in X(j) for these {i,j}: {4454, 3241}, {31145, 4419}
X(66456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3241, 28301, 4454}, {41140, 66452, 62706}

X(66457) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(10)-CIRCUMCONCEVIAN-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    10*a^2+b^2-4*b*c+c^2+14*a*(b+c) : :
X(66457) = 3*X[4363]+X[66456], -X[4665]+3*X[25055], -5*X[24441]+X[64015]

X(66457) lies on these lines: {1, 545}, {2, 594}, {519, 4708}, {524, 49465}, {527, 51107}, {536, 51103}, {551, 4472}, {3241, 4364}, {3679, 25358}, {3723, 24199}, {4363, 66456}, {4395, 39260}, {4664, 36522}, {4665, 25055}, {4725, 51091}, {4745, 28329}, {6707, 50099}, {8584, 16973}, {9055, 50111}, {10022, 17318}, {16521, 29584}, {17133, 51108}, {17288, 17320}, {17330, 17393}, {17392, 36525}, {20582, 50013}, {24441, 64015}, {28297, 51104}, {28313, 41150}, {28337, 41312}, {28639, 50108}, {29580, 49733}, {31285, 41140}, {31332, 40891}, {46845, 50116}

X(66457) = midpoint of X(i) and X(j) for these {i,j}: {3241, 4364}, {10022, 17318}
X(66457) = reflection of X(i) in X(j) for these {i,j}: {3679, 25358}, {4472, 551}
X(66457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {551, 28309, 4472}, {17318, 38314, 10022}

X(66458) = ORTHOLOGY CENTER OF THESE TRIANGLES: INVERSE-OF-X(69)-CIRCUMCONCEVIAN-OF-X(2) WRT 2ND OUTER-GREBE

Barycentrics    a^4+b^4-22*b^2*c^2+c^4+14*a^2*(b^2+c^2) : :
X(66458) = -3*X[1992]+2*X[11159], -5*X[3618]+4*X[59780], -3*X[3734]+4*X[61046], -3*X[5032]+4*X[7798], -2*X[7737]+3*X[63027], -4*X[8584]+3*X[14033], -6*X[11287]+5*X[50990], -4*X[15048]+3*X[21356], -2*X[15533]+3*X[32986], -3*X[33272]+X[63118], -7*X[63109]+8*X[63633]

X(66458) lies on these lines: {2, 39}, {30, 55724}, {99, 11148}, {148, 23334}, {193, 543}, {385, 53142}, {524, 44526}, {549, 40925}, {671, 32827}, {754, 63116}, {1285, 11164}, {1992, 11159}, {2549, 11160}, {2782, 54132}, {2996, 7775}, {3618, 59780}, {3734, 61046}, {5032, 7798}, {5077, 50992}, {5485, 11163}, {6390, 63107}, {7615, 62988}, {7616, 15708}, {7617, 63077}, {7618, 37667}, {7620, 7774}, {7737, 63027}, {7754, 8598}, {7781, 35287}, {7812, 32826}, {7837, 52942}, {7840, 43448}, {8369, 32824}, {8584, 14033}, {8667, 47061}, {8724, 9752}, {9166, 63098}, {9741, 22329}, {9770, 47286}, {9855, 63093}, {11008, 63945}, {11147, 63654}, {11165, 23055}, {11287, 50990}, {11318, 32825}, {12040, 23053}, {15048, 21356}, {15300, 35927}, {15484, 63651}, {15533, 32986}, {15589, 52691}, {19569, 52944}, {27088, 51122}, {31859, 63029}, {32480, 63046}, {32515, 50967}, {32816, 37350}, {32820, 33197}, {33215, 63933}, {33272, 63118}, {35954, 63006}, {37689, 41134}, {40727, 63025}, {43619, 63942}, {51187, 63941}, {51224, 53141}, {52695, 63048}, {52713, 63101}, {53143, 62203}, {60118, 60228}, {63109, 63633}

X(66458) = reflection of X(i) in X(j) for these {i,j}: {1992, 22253}, {11160, 2549}, {32815, 1992}, {50992, 5077}
X(66458) = pole of line {512, 9189} with respect to the Steiner circumellipse
X(66458) = pole of line {6, 9741} with respect to the Wallace hyperbola
X(66458) = pole of line {9209, 23878} with respect to the dual conic of orthoptic circle of the Steiner Inellipse
X(66458) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(39236)}}, {{A, B, C, X(11059), X(60268)}}
X(66458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1992, 52229, 32815}, {22253, 52229, 1992}, {53141, 63042, 51224}

X(66459) = PARALLELOGIC CENTER OF THESE TRIANGLES: PEDAL-OF-X(237) WRT 2ND OUTER-GREBE

Barycentrics    (b-c)^2*(b+c)^2*(-a^4+2*b^2*c^2+a^2*(b^2+c^2)) : :
X(66459) = X[3001]+2*X[53474], X[14957]+2*X[63736]

X(66459) lies on these lines: {2, 2782}, {3, 65767}, {4, 47202}, {6, 18575}, {51, 22682}, {98, 1316}, {110, 12188}, {115, 125}, {127, 5522}, {137, 46665}, {148, 35922}, {182, 56401}, {230, 47526}, {262, 37988}, {338, 523}, {339, 2972}, {381, 5640}, {389, 54005}, {401, 21445}, {458, 9755}, {511, 56392}, {542, 57618}, {671, 36194}, {895, 36207}, {1312, 2593}, {1313, 2592}, {1503, 50707}, {1650, 46229}, {1899, 7694}, {2450, 39663}, {2452, 48540}, {2453, 9142}, {2794, 57598}, {2967, 57583}, {2970, 2971}, {3001, 53474}, {3014, 25328}, {3018, 15118}, {3060, 22728}, {3134, 9191}, {3143, 65870}, {3148, 9756}, {3154, 51258}, {3258, 17436}, {3288, 59804}, {3424, 6620}, {3448, 6033}, {3580, 15980}, {3734, 13233}, {3767, 14003}, {4226, 12042}, {5094, 8426}, {5099, 6070}, {5309, 61743}, {5466, 46040}, {5475, 13410}, {5650, 9466}, {5652, 56788}, {6036, 65722}, {6055, 45662}, {6194, 37190}, {6248, 37338}, {6321, 36163}, {6784, 45321}, {7612, 37188}, {7753, 61712}, {8599, 12079}, {8719, 41275}, {8842, 20023}, {8902, 53570}, {9475, 60517}, {9775, 11284}, {9832, 63719}, {10485, 30540}, {10991, 51431}, {11007, 54395}, {11197, 11245}, {11328, 48663}, {11579, 15928}, {11623, 15000}, {11792, 46654}, {13188, 54439}, {14096, 15819}, {14356, 20301}, {14957, 63736}, {15271, 33900}, {15449, 55152}, {16052, 34122}, {17511, 38953}, {18911, 37348}, {20021, 46124}, {20775, 61684}, {21531, 32515}, {22087, 64782}, {22681, 62949}, {23635, 41760}, {25317, 64882}, {26235, 30739}, {31127, 61576}, {32216, 40727}, {32274, 66167}, {35933, 38225}, {36181, 38741}, {37916, 58849}, {38361, 38393}, {39266, 56442}, {41221, 53569}, {44651, 66170}, {44774, 52658}, {46512, 51430}, {50188, 57586}

X(66459) = midpoint of X(i) and X(j) for these {i,j}: {2, 53346}, {20021, 46124}
X(66459) = reflection of X(i) in X(j) for these {i,j}: {9155, 2}, {51335, 46124}
X(66459) = inverse of X(3569) in Kiepert hyperbola
X(66459) = perspector of circumconic {{A, B, C, X(523), X(23878)}}
X(66459) = center of circumconic {{A, B, C, X(183), X(262)}}
X(66459) = X(i)-isoconjugate-of-X(j) for these {i, j}: {110, 65252}, {162, 65310}, {163, 65271}, {249, 2186}, {262, 1101}, {263, 24041}, {327, 23995}, {662, 26714}, {2421, 36132}, {3402, 4590}, {4575, 65349}, {6037, 23997}, {24000, 54032}, {24037, 46319}, {36084, 63741}
X(66459) = X(i)-Dao conjugate of X(j) for these {i, j}: {115, 65271}, {125, 65310}, {136, 65349}, {244, 65252}, {512, 46319}, {523, 262}, {525, 59257}, {647, 42313}, {1084, 26714}, {3005, 263}, {4988, 60679}, {18314, 327}, {23878, 183}, {35078, 39681}, {38987, 63741}, {38997, 110}, {39009, 2421}, {51580, 4590}, {54262, 11328}, {55051, 1634}, {55267, 46807}, {60342, 57268}, {62562, 6037}, {63463, 52926}, {65728, 36885}
X(66459) = X(i)-Ceva conjugate of X(j) for these {i, j}: {183, 23878}, {262, 523}, {458, 3288}
X(66459) = X(i)-complementary conjugate of X(j) for these {i, j}: {30535, 4369}, {60101, 42327}
X(66459) = pole of line {7669, 21525} with respect to the circumcircle
X(66459) = pole of line {868, 7668} with respect to the nine-point circle
X(66459) = pole of line {6, 526} with respect to the orthocentroidal circle
X(66459) = pole of line {98, 804} with respect to the orthoptic circle of the Steiner Inellipse
X(66459) = pole of line {648, 1634} with respect to the polar circle
X(66459) = pole of line {523, 3569} with respect to the Kiepert hyperbola
X(66459) = pole of line {868, 8754} with respect to the MacBeath inconic
X(66459) = pole of line {249, 2080} with respect to the Stammler hyperbola
X(66459) = pole of line {148, 59775} with respect to the Steiner circumellipse
X(66459) = pole of line {115, 46656} with respect to the Steiner inellipse
X(66459) = pole of line {47284, 56962} with respect to the Yff hyperbola
X(66459) = pole of line {4590, 39099} with respect to the Wallace hyperbola
X(66459) = pole of line {3124, 23962} with respect to the dual conic of circumcircle
X(66459) = pole of line {4563, 23181} with respect to the dual conic of polar circle
X(66459) = pole of line {5, 76} with respect to the dual conic of Stammler hyperbola
X(66459) = pole of line {2, 51} with respect to the dual conic of Wallace hyperbola
X(66459) = X(237)-of-orthocentroidal
X(66459) = X(9155)-of-Gemini-107
X(66459) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {2, 9140, 53346}, {4, 22265, 50401}, {23, 14957, 33873}, {98, 6785, 11005}, {671, 6787, 9144}, {20021, 46124, 46130}
X(66459) = intersection, other than A, B, C, of circumconics {{A, B, C, X(115), X(43532)}}, {{A, B, C, X(125), X(46806)}}, {{A, B, C, X(182), X(2088)}}, {{A, B, C, X(183), X(1648)}}, {{A, B, C, X(262), X(59804)}}, {{A, B, C, X(338), X(458)}}, {{A, B, C, X(523), X(3288)}}, {{A, B, C, X(690), X(23878)}}, {{A, B, C, X(1637), X(8599)}}, {{A, B, C, X(1640), X(45321)}}, {{A, B, C, X(2081), X(3268)}}, {{A, B, C, X(2970), X(39691)}}, {{A, B, C, X(3124), X(6784)}}, {{A, B, C, X(5466), X(63746)}}, {{A, B, C, X(7668), X(60497)}}, {{A, B, C, X(8288), X(12079)}}, {{A, B, C, X(9155), X(46142)}}, {{A, B, C, X(11182), X(34246)}}, {{A, B, C, X(14223), X(31953)}}, {{A, B, C, X(20975), X(34396)}}, {{A, B, C, X(39680), X(56748)}}
X(66459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2782, 9155}, {2, 53346, 2782}, {98, 1316, 5191}, {98, 41254, 1316}, {115, 125, 868}, {125, 16280, 53132}, {338, 7668, 20975}, {339, 3150, 2972}, {7668, 59739, 338}, {30465, 30468, 8288}, {57583, 60502, 2967}

X(66460) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR OF 2ND BROCARD WRT 2ND OUTER-GREBE

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

X(66460) lies on these lines: {2, 99}, {4, 9759}, {23, 51224}, {30, 63719}, {94, 60220}, {110, 1992}, {193, 10554}, {338, 11168}, {351, 523}, {519, 66236}, {524, 2502}, {528, 66038}, {542, 7417}, {597, 3124}, {599, 20998}, {648, 4232}, {804, 11631}, {1084, 9465}, {1634, 1995}, {1641, 5969}, {1648, 9830}, {2493, 45331}, {2770, 18823}, {2782, 14694}, {3363, 23297}, {4576, 12036}, {4590, 34539}, {5191, 37904}, {5640, 17430}, {5642, 46124}, {5648, 9129}, {5913, 8598}, {6054, 60066}, {6791, 18800}, {6792, 8593}, {7472, 52232}, {7493, 23055}, {7801, 16055}, {7812, 14002}, {8584, 39689}, {8860, 47596}, {9149, 46589}, {9169, 51798}, {9486, 16317}, {9745, 11317}, {9775, 64090}, {9829, 62411}, {9870, 11054}, {10552, 15534}, {10553, 63064}, {11172, 58268}, {11580, 26613}, {14568, 37907}, {14653, 57594}, {14666, 57620}, {14916, 35279}, {14932, 53374}, {15597, 53495}, {20583, 20976}, {23699, 57624}, {31173, 40350}, {32424, 57604}, {32479, 39602}, {33274, 39576}, {36168, 53136}, {37748, 46453}, {37775, 37785}, {37776, 37786}, {37855, 47187}, {37860, 62672}, {39024, 63127}, {40112, 57257}, {40283, 45662}, {48540, 63029}, {51438, 52231}, {55957, 60211}

X(66460) = reflection of X(i) in X(j) for these {i,j}: {2, 10418}, {58854, 2502}
X(66460) = inverse of X(10717) in Wallace hyperbola
X(66460) = perspector of circumconic {{A, B, C, X(598), X(892)}}
X(66460) = X(i)-isoconjugate-of-X(j) for these {i, j}: {163, 34206}, {2642, 53613}
X(66460) = X(i)-vertex conjugate of X(j) for these {i, j}: {1995, 9123}
X(66460) = X(i)-Dao conjugate of X(j) for these {i, j}: {115, 34206}
X(66460) = X(i)-Ceva conjugate of X(j) for these {i, j}: {598, 20381}
X(66460) = X(i)-cross conjugate of X(j) for these {i, j}: {20381, 598}
X(66460) = pole of line {1995, 9123} with respect to the circumcircle
X(66460) = pole of line {381, 2793} with respect to the orthoptic circle of the Steiner Inellipse
X(66460) = pole of line {5094, 14273} with respect to the polar circle
X(66460) = pole of line {524, 8288} with respect to the Kiepert hyperbola
X(66460) = pole of line {1499, 5468} with respect to the Kiepert parabola
X(66460) = pole of line {598, 65870} with respect to the Lemoine inellipse
X(66460) = pole of line {9134, 53418} with respect to the Orthic inconic
X(66460) = pole of line {187, 9027} with respect to the Stammler hyperbola
X(66460) = pole of line {690, 1992} with respect to the Steiner circumellipse
X(66460) = pole of line {597, 690} with respect to the Steiner inellipse
X(66460) = pole of line {53341, 65701} with respect to the Yff parabola
X(66460) = pole of line {524, 9146} with respect to the Wallace hyperbola
X(66460) = pole of line {7790, 9979} with respect to the dual conic of circumcircle
X(66460) = pole of line {44317, 50755} with respect to the dual conic of Yff parabola
X(66460) = pole of line {2, 1637} with respect to the dual conic of anti-Artzt circle
X(66460) = pole of line {1648, 3906} with respect to the dual conic of Wallace hyperbola
X(66460) = X(1641)-of-1st-anti-Brocard
X(66460) = X(7417)-of-2nd-Parry
X(66460) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3448, 45291, 62295}
X(66460) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(23287)}}, {{A, B, C, X(99), X(34205)}}, {{A, B, C, X(111), X(46001)}}, {{A, B, C, X(351), X(574)}}, {{A, B, C, X(523), X(42008)}}, {{A, B, C, X(524), X(10717)}}, {{A, B, C, X(543), X(2770)}}, {{A, B, C, X(671), X(8599)}}, {{A, B, C, X(9100), X(22329)}}, {{A, B, C, X(9185), X(11167)}}, {{A, B, C, X(37860), X(41134)}}
X(66460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 148, 42008}, {2, 8591, 10717}, {23, 62294, 51224}, {111, 7665, 7664}, {524, 2502, 58854}, {2482, 9172, 2}, {7426, 22329, 51541}, {7426, 62311, 22329}, {35279, 50639, 14916}

X(66461) = ORTHOLOGY CENTER OF THESE TRIANGLES: UNARY COFACTOR OF 1ST BROCARD-REFLECTED WRT 2ND OUTER-GREBE

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

X(66461) lies on these lines: {2, 39}, {524, 52658}, {574, 56442}, {702, 9462}, {3231, 7798}, {3849, 33873}, {5106, 31859}, {6379, 59373}, {7781, 37338}, {7804, 62301}, {8623, 14614}, {8716, 11328}, {8842, 11163}, {10335, 53375}, {11183, 23878}, {13586, 41278}, {14608, 43950}, {14957, 63956}, {15048, 59765}, {22253, 62712}, {22486, 25332}, {52637, 63557}, {62949, 63957}

X(66461) = pole of line {512, 22564} with respect to the Steiner circumellipse
X(66461) = pole of line {6, 41143} with respect to the Wallace hyperbola
X(66461) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(6), X(41143)}}, {{A, B, C, X(538), X(43950)}}, {{A, B, C, X(7757), X(60667)}}, {{A, B, C, X(9462), X(60707)}}
X(66461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20023, 9466}, {2, 7757, 3117}

X(66462) = ORTHOLOGY CENTER OF THESE TRIANGLES: 2ND OUTER-GREBE WRT UNARY COFACTOR OF LUCAS BROCARD

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

X(66462) lies on these lines: {2, 3}, {487, 53856}, {489, 50992}, {490, 50990}, {1991, 66472}, {3595, 42276}, {5860, 66471}, {5861, 43209}, {6459, 63124}, {6460, 8584}, {6560, 13639}, {6781, 13950}, {11148, 13798}, {12159, 66426}, {13783, 52666}, {13849, 33457}, {15534, 42259}, {33456, 66438}, {42258, 51185}, {43133, 63116}, {43134, 63117}, {61389, 63058}

X(66462) = reflection of X(i) in X(j) for these {i,j}: {52666, 13783}

X(66463) = PARALLELOGIC CENTER OF THESE TRIANGLES: UNARY COFACTOR OF ORTHOCENTROIDAL-ISOGONIC WRT 2ND OUTER-GREBE

Barycentrics    (b-c)*(b+c)*(-5*a^8+b^8-11*a^4*b^2*c^2-b^6*c^2-b^2*c^6+c^8+8*a^6*(b^2+c^2)-4*a^2*(b^2-c^2)^2*(b^2+c^2)) : :
X(66463) = X[20]+2*X[62438], -3*X[3524]+2*X[14417], -3*X[3545]+4*X[44564], -3*X[9191]+4*X[16235], -3*X[14644]+4*X[42736], -3*X[42731]+X[58346]

X(66463) lies on these lines: {2, 44202}, {3, 3268}, {4, 1637}, {20, 62438}, {30, 9979}, {74, 98}, {112, 30247}, {186, 42659}, {376, 2799}, {378, 53265}, {523, 9409}, {1499, 1513}, {2793, 9862}, {2826, 66034}, {2848, 5667}, {3524, 14417}, {3543, 44203}, {3545, 44564}, {3830, 44204}, {4235, 65776}, {5191, 9123}, {5664, 9517}, {5890, 39469}, {6130, 42733}, {6776, 9003}, {7612, 9180}, {9191, 16235}, {9479, 61776}, {9529, 11001}, {9744, 35909}, {10706, 14697}, {14582, 18316}, {14644, 42736}, {18556, 44810}, {25644, 35921}, {41377, 65107}, {42731, 58346}, {46229, 53345}, {47050, 66121}, {47333, 66118}

X(66463) = reflection of X(i) in X(j) for these {i,j}: {2, 44202}, {4, 1637}, {3268, 3}, {3543, 44203}, {3830, 44204}, {10706, 14697}, {18556, 44810}, {42733, 6130}, {66118, 47333}
X(66463) = pole of line {378, 53246} with respect to the circumcircle
X(66463) = pole of line {18361, 35481} with respect to the 2nd DrozFarny circle
X(66463) = pole of line {18568, 64923} with respect to the circumcircle of the Johnson triangle
X(66463) = pole of line {6, 67} with respect to the orthoptic circle of the Steiner Inellipse
X(66463) = pole of line {381, 64923} with respect to the polar circle
X(66463) = pole of line {6749, 47204} with respect to the Orthic inconic
X(66463) = pole of line {37645, 65767} with respect to the Steiner circumellipse
X(66463) = pole of line {9745, 9755} with respect to the Artzt circle
X(66463) = pole of line {31174, 52720} with respect to the dual conic of Wallace hyperbola
X(66463) = X(1637)-of-anti-Euler
X(66463) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {2, 11177, 62294}

X(66464) = PERSPECTOR OF THESE TRIANGLES: 2ND OUTER-GREBE AND CTR1-2

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

X(66464) lies on these lines: {2, 3}, {6, 13794}, {69, 45438}, {115, 23249}, {193, 45375}, {485, 13920}, {488, 45542}, {638, 46951}, {1327, 6250}, {1328, 45101}, {1503, 13674}, {1587, 5309}, {1588, 7753}, {5475, 23259}, {6033, 62986}, {6054, 33432}, {6289, 22806}, {6560, 66429}, {6564, 66428}, {6776, 35822}, {9738, 13798}, {12124, 13701}, {12256, 45862}, {12257, 13846}, {12297, 13678}, {12322, 22625}, {12602, 48677}, {13082, 13695}, {13665, 39874}, {13696, 18988}, {13748, 32787}, {13812, 32806}, {14227, 14241}, {14853, 35823}, {15484, 23273}, {19054, 45406}, {20112, 66471}, {22631, 41022}, {22633, 41023}, {22682, 66434}, {23267, 61322}, {26289, 66431}, {26330, 45407}, {31463, 42283}, {32419, 66436}, {32788, 45440}, {32827, 49016}, {40727, 66473}, {42284, 62202}, {48778, 55041}, {66430, 66466}

X(66464) = midpoint of X(i) and X(j) for these {i,j}: {12297, 13678}, {12602, 48677}
X(66464) = reflection of X(i) in X(j) for these {i,j}: {1327, 6250}, {6289, 22806}, {12124, 13701}, {12257, 13846}, {13846, 45861}, {32810, 6289}, {55041, 48778}
X(66464) = anticomplement of X(60655)
X(66464) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1585), X(54652)}}, {{A, B, C, X(14229), X(62956)}}, {{A, B, C, X(45101), X(62957)}}

X(66465) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-1 WRT 2ND OUTER-GREBE

Barycentrics    -4*a*b*c+3*a^2*(b+c)-3*(b-c)^2*(b+c) : :
X(66465) = 2*X[4]+X[12437], -4*X[5]+X[24391], -7*X[3090]+X[54422], 5*X[3091]+X[11523], -3*X[3653]+X[34740], X[3811]+2*X[18483], -7*X[3832]+X[12625], X[4301]+2*X[12607], -X[5493]+4*X[64123], X[5758]+5*X[63966], X[5763]+2*X[64813], -3*X[7988]+X[24477] and many others

X(66465) lies on these lines: {2, 7}, {4, 12437}, {5, 24391}, {10, 3838}, {12, 5837}, {30, 6260}, {65, 18236}, {72, 17530}, {145, 7319}, {210, 61031}, {312, 4035}, {355, 381}, {516, 4421}, {518, 3817}, {528, 21635}, {529, 551}, {535, 28459}, {536, 2051}, {537, 49554}, {545, 66444}, {758, 10175}, {936, 5714}, {948, 59610}, {960, 3947}, {1001, 8171}, {1086, 45204}, {1125, 11194}, {1210, 17533}, {1329, 3671}, {1427, 16578}, {1699, 5853}, {1738, 36634}, {1770, 59587}, {1836, 6745}, {1997, 17298}, {2321, 4417}, {2325, 56084}, {2478, 63274}, {2886, 21060}, {3090, 54422}, {3091, 11523}, {3158, 9812}, {3175, 22000}, {3241, 12053}, {3243, 5274}, {3436, 64160}, {3475, 64667}, {3485, 5795}, {3543, 34701}, {3614, 3962}, {3649, 8582}, {3653, 34740}, {3663, 37662}, {3679, 12047}, {3687, 42029}, {3755, 3944}, {3811, 18483}, {3816, 5542}, {3823, 59686}, {3828, 12609}, {3832, 12625}, {3848, 38054}, {3873, 61718}, {3912, 20942}, {3946, 63089}, {3951, 6933}, {3984, 6871}, {4054, 4980}, {4104, 25385}, {4138, 59511}, {4187, 64664}, {4292, 16371}, {4295, 63990}, {4298, 25681}, {4301, 12607}, {4304, 33595}, {4312, 59572}, {4387, 50753}, {4428, 13405}, {4641, 37691}, {4644, 39980}, {4656, 5718}, {4667, 39595}, {4677, 18393}, {4698, 56226}, {4847, 17605}, {4848, 11681}, {4849, 62221}, {4869, 6557}, {4870, 34606}, {4892, 62673}, {4902, 36603}, {4921, 17167}, {4945, 52753}, {5055, 55108}, {5087, 11019}, {5121, 33103}, {5187, 11520}, {5261, 15829}, {5289, 51782}, {5493, 64123}, {5758, 63966}, {5763, 64813}, {5836, 58696}, {5850, 10171}, {5855, 38155}, {5880, 20103}, {5882, 6928}, {6147, 9843}, {6690, 51090}, {6700, 16417}, {6737, 10895}, {6842, 11362}, {6893, 13464}, {6919, 11518}, {7988, 24477}, {8727, 59687}, {9579, 27383}, {9580, 63168}, {9612, 57284}, {9779, 24392}, {10129, 25006}, {10164, 17768}, {10440, 20718}, {10588, 12526}, {10589, 62823}, {10591, 41863}, {10863, 30291}, {11374, 12572}, {11375, 12527}, {11522, 64205}, {11813, 37728}, {12536, 50689}, {12608, 28194}, {12610, 17132}, {12699, 59722}, {13407, 25055}, {13411, 16370}, {13587, 27385}, {14526, 63278}, {14554, 65021}, {15185, 17604}, {15677, 41550}, {16580, 41310}, {16602, 63589}, {16833, 20257}, {17067, 23511}, {17097, 34918}, {17133, 42047}, {17182, 42028}, {17197, 41629}, {17549, 64002}, {17718, 40998}, {17775, 44307}, {18134, 62297}, {18250, 28628}, {18589, 41313}, {18908, 38039}, {19517, 24328}, {19872, 28647}, {19875, 34744}, {19883, 34646}, {20060, 63987}, {21024, 29594}, {21096, 24045}, {21557, 31540}, {21562, 31541}, {21630, 51096}, {22019, 50100}, {22793, 64117}, {22836, 31673}, {23681, 63126}, {23806, 44567}, {24175, 51415}, {24177, 37663}, {24210, 42042}, {24239, 33101}, {25101, 41878}, {25760, 53663}, {27739, 50048}, {27747, 49724}, {28164, 56177}, {28301, 42049}, {28534, 50808}, {28645, 51073}, {28646, 31253}, {28657, 59646}, {29600, 44664}, {29639, 42039}, {30384, 51093}, {30568, 30828}, {31162, 34619}, {31397, 51409}, {31730, 35251}, {32856, 42040}, {33105, 42041}, {34048, 37672}, {34607, 50865}, {34625, 38021}, {34716, 38314}, {37364, 43177}, {37374, 41561}, {37634, 62240}, {38123, 58441}, {38204, 58451}, {39570, 59599}, {39948, 63007}, {41539, 46694}, {41883, 63844}, {45334, 46396}, {49511, 50609}, {49599, 49636}, {50829, 64113}, {51118, 56176}, {51724, 63282}, {52374, 56234}, {56089, 62178}, {57287, 62969}, {57477, 65415}, {60071, 60267}, {61029, 61686}, {62189, 64912}, {64011, 66012}

X(66465) = midpoint of X(i) and X(j) for these {i,j}: {355, 4930}, {1699, 25568}, {3158, 9812}, {3543, 34701}, {4052, 66440}, {11236, 34647}, {31162, 34619}, {34607, 50865}
X(66465) = reflection of X(i) in X(j) for these {i,j}: {11194, 1125}, {11235, 50802}, {24386, 3817}
X(66465) = complement of X(3928)
X(66465) = X(i)-complementary conjugate of X(j) for these {i, j}: {6, 45036}, {7319, 141}, {41441, 10}, {65046, 2886}, {65047, 2887}
X(66465) = pole of line {23865, 39225} with respect to the circumcircle
X(66465) = pole of line {28292, 59912} with respect to the orthoptic circle of the Steiner Inellipse
X(66465) = pole of line {3663, 17056} with respect to the Kiepert hyperbola
X(66465) = pole of line {522, 21052} with respect to the Steiner inellipse
X(66465) = pole of line {1, 4004} with respect to the dual conic of Yff parabola
X(66465) = pole of line {20907, 57244} with respect to the dual conic of Hofstadter ellipse
X(66465) = X(154)-of-Wasat
X(66465) = X(3928)-of-medial
X(66465) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(65065)}}, {{A, B, C, X(312), X(3929)}}, {{A, B, C, X(2051), X(5435)}}, {{A, B, C, X(3219), X(56234)}}, {{A, B, C, X(3928), X(65047)}}, {{A, B, C, X(4052), X(52358)}}, {{A, B, C, X(5745), X(34918)}}, {{A, B, C, X(7319), X(64114)}}, {{A, B, C, X(8732), X(54689)}}, {{A, B, C, X(14554), X(31231)}}, {{A, B, C, X(21454), X(60071)}}, {{A, B, C, X(37797), X(60172)}}, {{A, B, C, X(54366), X(54928)}}
X(66465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 329, 3929}, {2, 3929, 5745}, {226, 3452, 142}, {226, 5316, 5249}, {226, 908, 3452}, {518, 3817, 24386}, {519, 50802, 11235}, {2886, 21060, 24393}, {3929, 5219, 2}, {4052, 66440, 536}, {11236, 34647, 519}, {12635, 19925, 66251}, {17533, 24473, 1210}, {18228, 25525, 6666}

X(66466) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR1-2 WRT 2ND OUTER-GREBE

Barycentrics    5*a^4-7*b^4+10*b^2*c^2-7*c^4+4*a^2*(b^2+c^2) : :
X(66466) = 8*X[546]+X[7758], -7*X[3090]+4*X[34506], 5*X[3091]+4*X[7843], X[3146]+2*X[34504], -3*X[3524]+4*X[7619], 7*X[3832]+2*X[7759], 2*X[3845]+X[9766], 8*X[3850]+X[63932], -7*X[3851]+X[63950], -17*X[3854]+2*X[63953], -11*X[3855]+2*X[7751], -10*X[3858]+X[63933] and many others

X(66466) lies on these lines: {2, 187}, {3, 9771}, {4, 543}, {5, 7610}, {6, 37350}, {30, 7618}, {32, 14971}, {69, 43457}, {99, 52942}, {114, 19911}, {115, 1992}, {193, 18424}, {315, 33013}, {325, 11317}, {376, 7622}, {381, 524}, {511, 57634}, {538, 3839}, {546, 7758}, {574, 63025}, {591, 1327}, {597, 15484}, {599, 3363}, {671, 7774}, {754, 3545}, {1003, 41133}, {1007, 2482}, {1285, 6722}, {1328, 1991}, {1384, 44401}, {1506, 33215}, {2548, 7841}, {2549, 8352}, {3090, 34506}, {3091, 7843}, {3146, 34504}, {3524, 7619}, {3543, 32479}, {3767, 7812}, {3815, 5077}, {3830, 11165}, {3832, 7759}, {3845, 9766}, {3850, 63932}, {3851, 63950}, {3854, 63953}, {3855, 7751}, {3858, 63933}, {3859, 63926}, {5007, 32980}, {5008, 61304}, {5032, 5309}, {5055, 15597}, {5056, 63935}, {5066, 8667}, {5068, 7780}, {5071, 10788}, {5072, 63928}, {5076, 59546}, {5355, 63022}, {5395, 7852}, {5461, 7735}, {5485, 14492}, {6033, 9830}, {6561, 66438}, {6781, 34803}, {7739, 14041}, {7745, 11318}, {7747, 9167}, {7752, 33007}, {7753, 16041}, {7763, 52695}, {7768, 32995}, {7769, 33208}, {7773, 7795}, {7777, 8597}, {7784, 8367}, {7785, 41135}, {7791, 31417}, {7798, 36523}, {7801, 32816}, {7808, 33230}, {7809, 33016}, {7810, 32006}, {7811, 33005}, {7814, 14068}, {7818, 21356}, {7821, 32979}, {7825, 33190}, {7827, 14063}, {7829, 33292}, {7833, 31401}, {7838, 63533}, {7840, 11185}, {7842, 31404}, {7845, 11160}, {7854, 32991}, {7858, 32996}, {7860, 32962}, {7870, 14035}, {7873, 32987}, {7878, 33290}, {7883, 16924}, {7913, 63109}, {7926, 11054}, {7936, 33261}, {8355, 18907}, {8369, 65630}, {8591, 63021}, {8598, 43618}, {8703, 63647}, {8716, 15687}, {9698, 33238}, {9743, 60658}, {9761, 10653}, {9763, 10654}, {9888, 58851}, {9890, 22566}, {9939, 32832}, {10297, 16279}, {10717, 56435}, {11148, 61989}, {11159, 22110}, {11167, 54826}, {11179, 15980}, {11632, 63722}, {12101, 51123}, {13377, 46645}, {13468, 19709}, {13608, 14666}, {14160, 64802}, {14269, 53143}, {14866, 34165}, {14881, 18768}, {15533, 64093}, {17131, 50992}, {18309, 23878}, {18362, 63034}, {19695, 31450}, {19924, 64942}, {22329, 43620}, {27088, 50571}, {31105, 42008}, {32457, 63091}, {32515, 40277}, {32815, 39785}, {32833, 41136}, {32966, 34604}, {32988, 35007}, {33008, 55801}, {33017, 52691}, {33184, 47352}, {33210, 44562}, {36733, 66472}, {36775, 36970}, {36882, 64613}, {37348, 54173}, {37667, 39601}, {38071, 63940}, {41750, 63027}, {41895, 60095}, {42850, 66455}, {47332, 60696}, {50687, 53141}, {51122, 61993}, {53144, 61967}, {54616, 62900}, {54659, 60240}, {54753, 62895}, {54833, 60103}, {54841, 54901}, {54915, 60268}, {55823, 61899}, {57618, 61506}, {61924, 63943}, {61930, 63947}, {61943, 63948}, {61944, 63952}, {61948, 63944}, {61954, 63939}, {61955, 63936}, {61956, 63951}, {61964, 63924}, {61970, 63923}, {61977, 63651}, {61982, 63922}, {61985, 63957}, {61997, 63654}, {66430, 66464}

X(66466) = midpoint of X(i) and X(j) for these {i,j}: {2, 23334}, {4, 9770}, {3543, 53142}, {3830, 11165}, {8176, 63956}, {8182, 44678}
X(66466) = reflection of X(i) in X(j) for these {i,j}: {2, 8176}, {3, 9771}, {376, 7622}, {5485, 18546}, {7610, 5}, {7615, 381}, {7618, 11184}, {8182, 2}, {8667, 16509}, {8703, 63647}, {9770, 7775}, {14023, 9740}, {16509, 5066}, {19911, 114}, {23334, 63956}, {34511, 9770}, {40727, 20112}, {44678, 23334}, {47101, 1153}, {47102, 8182}, {63029, 7617}, {63955, 7615}
X(66466) = anticomplement of X(5569)
X(66466) = X(i)-Dao conjugate of X(j) for these {i, j}: {5569, 5569}
X(66466) = pole of line {1499, 23288} with respect to the orthocentroidal circle
X(66466) = pole of line {8704, 9125} with respect to the orthoptic circle of the Steiner Inellipse
X(66466) = pole of line {597, 2549} with respect to the Kiepert hyperbola
X(66466) = pole of line {3906, 39905} with respect to the Steiner circumellipse
X(66466) = X(5569)-of-anticomplementary
X(66466) = X(7610)-of-Johnson
X(66466) = X(8182)-of-Gemini-107
X(66466) = X(9770)-of-Euler
X(66466) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {4, 6792, 9770}
X(66466) = intersection, other than A, B, C, of circumconics {{A, B, C, X(5503), X(64982)}}, {{A, B, C, X(14492), X(61345)}}, {{A, B, C, X(14907), X(36882)}}, {{A, B, C, X(46645), X(55164)}}, {{A, B, C, X(51224), X(64613)}}, {{A, B, C, X(54826), X(64973)}}
X(66466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 23334, 3849}, {2, 32827, 31173}, {2, 3849, 8182}, {2, 44678, 47102}, {2, 51224, 21843}, {2, 63956, 44678}, {2, 7737, 37809}, {4, 7775, 34511}, {4, 9770, 543}, {5, 63945, 7610}, {30, 11184, 7618}, {32, 14971, 63107}, {316, 8176, 49788}, {381, 40727, 20112}, {524, 20112, 40727}, {524, 7615, 63955}, {543, 7775, 9770}, {754, 7617, 63029}, {1153, 3849, 47101}, {3091, 7843, 14023}, {3543, 53142, 32479}, {3545, 63029, 7617}, {3849, 63956, 23334}, {3849, 8176, 2}, {7812, 33006, 3767}, {7812, 9166, 63065}, {8352, 11163, 2549}, {20112, 40727, 7615}, {22110, 53418, 11159}, {22491, 22492, 20423}, {31173, 50280, 5475}, {32984, 63107, 14971}, {33006, 63065, 9166}

X(66467) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-7.2 WRT 2ND OUTER-GREBE

Barycentrics    (a+b-c)*(a-b+c)*(a^4-8*a^3*(b+c)-16*a*(b-c)^2*(b+c)+(b-c)^2*(5*b^2+22*b*c+5*c^2)+2*a^2*(9*b^2-10*b*c+9*c^2)) : :

X(66467) lies on circumconic {{A, B, C, X(19605), X(31507)}} and on these lines: {2, 3160}, {7, 31507}, {479, 11238}, {516, 32079}, {519, 66246}, {527, 15913}, {1996, 56331}, {2898, 65384}, {3058, 3599}, {9533, 50865}, {10004, 50802}, {11019, 62788}, {15511, 30308}, {36640, 51364}, {36644, 42047}, {56310, 59374}

X(66467) = X(i)-Ceva conjugate of X(j) for these {i, j}: {36605, 7}
X(66467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31527, 36620, 3160}

X(66468) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-8.2 WRT 2ND OUTER-GREBE

Barycentrics    (a-b-c)*(a^2+5*b^2-22*b*c+5*c^2+6*a*(b+c)) : :

X(66468) lies on these lines: {2, 2415}, {8, 31509}, {519, 8834}, {528, 66069}, {1997, 4488}, {2899, 3241}, {5325, 31722}, {5328, 42033}, {5423, 11238}, {11679, 56075}, {16284, 20942}, {25567, 28530}, {28808, 56201}, {39570, 50802}, {46937, 54689}, {56084, 65384}, {56085, 59374}

X(66468) = reflection of X(i) in X(j) for these {i,j}: {2, 28655}
X(66468) = X(i)-Ceva conjugate of X(j) for these {i, j}: {36606, 8}
X(66468) = pole of line {514, 2490} with respect to the dual conic of incircle
X(66468) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8056), X(31509)}}, {{A, B, C, X(47636), X(54689)}}
X(66468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4052, 62403}, {2, 8056, 66444}, {3161, 4052, 6557}, {3161, 8055, 8056}, {4373, 8055, 62297}, {6557, 62297, 4052}, {6557, 8055, 3161}, {17132, 28655, 2}

X(66469) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR12-9.2 WRT 2ND OUTER-GREBE

Barycentrics    a*(3*a^2+3*b^2+2*b*c+3*c^2-6*a*(b+c)) : :
X(66469) = 2*X[2136]+X[11531], 2*X[3189]+X[5691], -2*X[3680]+5*X[16189], -8*X[3829]+9*X[7988], 2*X[4301]+X[12632], -2*X[6762]+5*X[7987], -X[6766]+4*X[64804], 2*X[7674]+X[63974], -X[7992]+4*X[10306], -7*X[9588]+4*X[24391], -3*X[10304]+2*X[34646], -5*X[11522]+2*X[64068] and many others

X(66469) lies on these lines: {1, 2}, {9, 4428}, {30, 5534}, {55, 3929}, {57, 3689}, {63, 31508}, {72, 53053}, {100, 53056}, {165, 518}, {171, 39959}, {210, 10389}, {326, 50132}, {354, 46917}, {373, 64685}, {390, 21060}, {405, 4866}, {480, 30330}, {516, 64143}, {527, 2951}, {528, 1750}, {529, 34628}, {536, 65957}, {537, 58035}, {758, 63468}, {956, 33595}, {968, 42041}, {1001, 30393}, {1054, 1280}, {1260, 10398}, {1376, 3243}, {1449, 44798}, {1490, 28194}, {1697, 31165}, {1699, 5853}, {1707, 4712}, {1708, 41553}, {1721, 17132}, {1743, 3693}, {1992, 56179}, {2093, 48696}, {2136, 11531}, {2177, 42039}, {3058, 10388}, {3189, 5691}, {3219, 64343}, {3304, 16411}, {3333, 16417}, {3339, 5687}, {3361, 3555}, {3419, 5726}, {3475, 38052}, {3543, 12651}, {3545, 64669}, {3550, 62820}, {3553, 50087}, {3576, 33575}, {3654, 30503}, {3656, 5720}, {3680, 16189}, {3681, 4512}, {3697, 17542}, {3711, 3748}, {3722, 62875}, {3731, 3750}, {3740, 38316}, {3744, 16469}, {3745, 39948}, {3746, 13615}, {3829, 7988}, {3830, 18528}, {3845, 18529}, {3869, 4917}, {3871, 12526}, {3873, 64112}, {3880, 11224}, {3886, 42034}, {3889, 36006}, {3893, 64964}, {3913, 7580}, {3940, 31393}, {3973, 8616}, {3996, 42029}, {4018, 63138}, {4301, 12632}, {4312, 17784}, {4314, 5815}, {4321, 65384}, {4326, 6172}, {4383, 16487}, {4413, 44841}, {4654, 34612}, {4661, 35258}, {4662, 5436}, {4702, 59597}, {4849, 7290}, {4863, 5219}, {4864, 5573}, {4921, 17194}, {4930, 7982}, {4936, 64579}, {4954, 50106}, {4980, 63131}, {4995, 10383}, {5054, 64668}, {5234, 16418}, {5264, 16398}, {5274, 12630}, {5290, 63146}, {5437, 30350}, {5438, 34791}, {5440, 13462}, {5537, 30304}, {5732, 28610}, {5785, 62800}, {5850, 9778}, {5881, 8727}, {5904, 61763}, {5927, 24644}, {6173, 41548}, {6174, 37736}, {6282, 50811}, {6600, 15931}, {6762, 7987}, {6766, 64804}, {7323, 41239}, {7411, 8715}, {7674, 63974}, {7675, 50835}, {7992, 10306}, {8056, 56009}, {8167, 36835}, {8226, 12607}, {9588, 24391}, {9589, 50696}, {9851, 37022}, {9909, 40910}, {9954, 14100}, {10025, 55998}, {10075, 10396}, {10304, 34646}, {10382, 10385}, {10860, 64697}, {10883, 61252}, {11235, 17618}, {11518, 37271}, {11522, 64068}, {11525, 50194}, {12565, 34632}, {12739, 50842}, {12767, 25438}, {13587, 62874}, {14022, 37721}, {14828, 25590}, {14942, 65047}, {15671, 64680}, {15699, 64670}, {15829, 30337}, {15909, 42470}, {16126, 35990}, {16192, 62858}, {16370, 57279}, {16496, 60714}, {16667, 17716}, {17151, 32920}, {17314, 40869}, {17549, 62824}, {17658, 18412}, {17715, 60846}, {17857, 31162}, {18228, 30331}, {18421, 63137}, {18443, 50821}, {18446, 50810}, {19346, 54327}, {19605, 36627}, {20173, 49469}, {21031, 37723}, {24216, 63621}, {24283, 49446}, {24477, 59584}, {25439, 53052}, {25524, 30343}, {25716, 31627}, {28204, 37531}, {28534, 41860}, {30291, 66252}, {30353, 60971}, {30392, 56177}, {31164, 49719}, {31231, 51463}, {32946, 52164}, {33092, 40609}, {34195, 63142}, {34631, 63986}, {34894, 64264}, {36002, 58245}, {37364, 37727}, {37533, 50798}, {37615, 38066}, {37703, 41867}, {37712, 44669}, {38031, 58688}, {38455, 61294}, {42040, 62695}, {42054, 65952}, {42819, 51780}, {47352, 64671}, {47375, 61030}, {50827, 64733}, {51576, 64951}, {51786, 62826}, {59374, 64672}, {59376, 64676}, {61154, 63214}, {62856, 63961}, {64005, 64117}

X(66469) = reflection of X(i) in X(j) for these {i,j}: {165, 3158}, {1699, 25568}, {3928, 4421}, {6762, 11194}, {7982, 4930}, {11194, 56176}, {24477, 59584}, {28610, 50808}, {34628, 34701}, {34632, 34639}, {50865, 28609}
X(66469) = X(i)-Ceva conjugate of X(j) for these {i, j}: {10405, 9}
X(66469) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(8), X(32003)}}, {{A, B, C, X(1280), X(3621)}}, {{A, B, C, X(3008), X(36603)}}, {{A, B, C, X(3617), X(39959)}}, {{A, B, C, X(3912), X(65047)}}, {{A, B, C, X(3957), X(34525)}}, {{A, B, C, X(4847), X(65952)}}, {{A, B, C, X(4853), X(56098)}}, {{A, B, C, X(4882), X(56140)}}, {{A, B, C, X(5222), X(39980)}}, {{A, B, C, X(6744), X(60078)}}, {{A, B, C, X(8580), X(56179)}}, {{A, B, C, X(9282), X(51615)}}, {{A, B, C, X(10582), X(56330)}}, {{A, B, C, X(11019), X(55993)}}, {{A, B, C, X(15909), X(36845)}}, {{A, B, C, X(17014), X(39948)}}, {{A, B, C, X(20008), X(62178)}}, {{A, B, C, X(20057), X(56030)}}, {{A, B, C, X(27304), X(36871)}}, {{A, B, C, X(29627), X(56722)}}, {{A, B, C, X(36627), X(64083)}}
X(66469) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {100, 62823, 53056}, {200, 3870, 1}, {518, 4421, 3928}, {528, 28609, 50865}, {529, 34701, 34628}, {1001, 62218, 30393}, {1376, 3243, 10980}, {2136, 12635, 11531}, {3158, 3928, 4421}, {3689, 41711, 57}, {3711, 3748, 7308}, {3870, 3935, 200}, {3873, 64135, 64112}, {3913, 11523, 7991}, {3928, 4421, 165}, {5437, 42871, 30350}, {5534, 6769, 63981}, {5687, 41863, 3339}, {5853, 25568, 1699}, {6762, 56176, 7987}, {8715, 54422, 63469}, {12607, 12625, 37714}, {59216, 63087, 3973}

X(66470) = ORTHOLOGY CENTER OF THESE TRIANGLES: CTR13-32.6 WRT 2ND OUTER-GREBE

Barycentrics    a^2*(3*a^6-2*b^2*c^2*(b^2+c^2)+a^2*(-3*b^4+4*b^2*c^2-3*c^4)) : :
X(66470) = 2*X[26]+X[576], 2*X[575]+X[7387], X[1351]+3*X[10245], -5*X[3618]+X[44442], -3*X[5085]+X[54992], -8*X[10226]+11*X[55675], 7*X[10244]+5*X[11482], 2*X[10282]+X[44492], -4*X[11250]+7*X[55681], X[11477]+5*X[16195], -2*X[12084]+5*X[55687], -X[12085]+4*X[20190] and many others

X(66470) lies on circumconic {{A, B, C, X(45819), X(60125)}} and on these lines: {2, 1974}, {6, 9909}, {20, 32300}, {22, 11511}, {23, 8541}, {25, 9813}, {26, 576}, {30, 182}, {49, 50962}, {68, 542}, {110, 11160}, {154, 8681}, {184, 1992}, {206, 524}, {376, 19128}, {381, 19131}, {511, 11202}, {519, 66255}, {541, 19138}, {543, 39840}, {569, 14848}, {575, 7387}, {578, 20423}, {599, 9306}, {671, 41274}, {1092, 50967}, {1147, 33591}, {1176, 34608}, {1351, 10245}, {1495, 41614}, {1576, 5171}, {1614, 50974}, {1658, 6593}, {2937, 8538}, {3098, 18324}, {3506, 64923}, {3543, 19124}, {3618, 44442}, {3830, 19129}, {5012, 63127}, {5027, 64925}, {5085, 54992}, {5157, 34609}, {5158, 6660}, {6403, 37939}, {6800, 40673}, {7488, 11470}, {8263, 15448}, {8584, 64028}, {9426, 64916}, {10168, 44441}, {10201, 11178}, {10226, 55675}, {10243, 64026}, {10244, 11482}, {10282, 44492}, {10539, 50955}, {10984, 34621}, {11003, 63000}, {11188, 44082}, {11250, 55681}, {11265, 44473}, {11266, 44474}, {11416, 37913}, {11477, 16195}, {11574, 19118}, {11645, 34775}, {12084, 55687}, {12085, 20190}, {12107, 55721}, {13346, 64061}, {13347, 50983}, {13383, 34507}, {14790, 25555}, {15331, 55637}, {15462, 37480}, {16199, 17809}, {17714, 22234}, {19125, 63094}, {19132, 37491}, {19143, 32419}, {19144, 32421}, {21637, 21969}, {22112, 30775}, {23327, 29012}, {25406, 61744}, {26881, 37784}, {26883, 51023}, {30558, 35287}, {32223, 63129}, {34117, 46730}, {34148, 51028}, {34351, 50977}, {34643, 38023}, {34713, 38087}, {34725, 38072}, {35264, 61667}, {35268, 52238}, {37478, 45016}, {37515, 38064}, {37897, 41585}, {37904, 44080}, {39561, 64599}, {39568, 53093}, {41719, 64883}, {43572, 63428}, {43650, 63109}, {43812, 51176}, {44077, 44210}, {44108, 61692}, {44489, 50979}, {47354, 63663}, {50978, 61753}

X(66470) = midpoint of X(i) and X(j) for these {i,j}: {6, 9909}, {37491, 37672}
X(66470) = reflection of X(i) in X(j) for these {i,j}: {3098, 18324}, {11178, 10201}, {44441, 10168}, {50977, 34351}
X(66470) = pole of line {3620, 7998} with respect to the Stammler hyperbola
X(66470) = X(9876)-of-1st-Brocard
X(66470) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {6, 1316, 9909}
X(66470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {597, 51745, 5476}, {1974, 19121, 19126}, {1974, 19126, 19137}, {19127, 19136, 182}, {19127, 32217, 19136}

X(66471) = X(3) OF 2ND OUTER-GREBE

Barycentrics    19*a^4-5*b^4+2*b^2*c^2-5*c^4-10*a^2*(b^2+c^2)+6*(2*a^2-b^2-c^2)*S : :

X(66471) lies on these lines: {376, 524}, {591, 3849}, {597, 66443}, {599, 66427}, {1991, 8182}, {5860, 66462}, {6811, 7610}, {13663, 66428}, {13783, 66429}, {20112, 66464}, {36733, 40727}, {47102, 66439}

X(66471) = midpoint of X(i) and X(j) for these {i,j}: {66430, 66473}
X(66471) = reflection of X(i) in X(j) for these {i,j}: {1991, 8182}, {66430, 66472}
X(66471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 66430, 66472}, {376, 66473, 66430}, {66430, 66473, 524}

X(66472) = X(5) OF 2ND OUTER-GREBE

Barycentrics    13*a^4-13*a^2*(b^2+c^2)-2*(b^4-4*b^2*c^2+c^4)-3*(2*a^2-b^2-c^2)*S : :

X(66472) lies on these lines: {376, 524}, {597, 66427}, {1327, 13835}, {1991, 66462}, {6811, 9771}, {13664, 66428}, {13784, 66429}, {13801, 45863}, {20583, 66443}, {28329, 66437}, {36733, 66466}

X(66472) = midpoint of X(i) and X(j) for these {i,j}: {66430, 66471}
X(66472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {376, 66430, 66471}, {66430, 66471, 524}, {66438, 66439, 66435}

X(66473) = X(20) OF 2ND OUTER-GREBE

Barycentrics    31*a^4-11*b^4-10*b^2*c^2-11*c^4-4*a^2*(b^2+c^2)+24*(2*a^2-b^2-c^2)*S : :

X(66473) lies on these lines: {2, 40286}, {376, 524}, {599, 66443}, {5485, 54652}, {5861, 13701}, {6811, 9740}, {11160, 66427}, {40727, 66464}

X(66473) = reflection of X(i) in X(j) for these {i,j}: {66430, 66471}
X(66473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {66430, 66471, 376}

X(66474) = X(69)X(74)∩X(114)X(489)

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

Let P be a point on line X(3)X(6). In this case the 2nd outer-Grebe triangle is orthologic to the pedal of P. The locus of the orthology center is a conic with center X(66474). The locus of the reciprocal orthology center is X(2)X(6).

X(66474) lies on these lines: {2, 50721}, {3, 42009}, {69, 74}, {114, 489}, {115, 590}, {147, 51952}, {148, 50722}, {187, 13989}, {487, 6230}, {491, 39809}, {543, 1991}, {620, 6200}, {637, 12974}, {639, 15885}, {671, 43536}, {2482, 13701}, {2794, 11824}, {5418, 60270}, {5477, 49267}, {5861, 49096}, {5981, 35748}, {6033, 36733}, {6036, 33341}, {6564, 63957}, {6721, 12322}, {7692, 62348}, {9894, 13835}, {13821, 13968}, {14061, 43374}, {14645, 49367}, {19108, 26289}, {21166, 33430}, {23235, 33431}, {23514, 45509}, {31274, 41945}, {43124, 43134}

X(66474) = midpoint of X(i) and X(j) for these {i,j}: {23235, 33431}
X(66474) = reflection of X(i) in X(j) for these {i,j}: {148, 50722}, {13968, 13821}, {50719, 620}
X(66474) = anticomplement of X(50721)
X(66474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {49266, 58033, 115}

X(66475) = 1st MIYAMOTO-MOSES EQUILATERAL TRIANGLE CENTER

Barycentrics    a^2*(2*Sqrt[3]*a^2*b^2*c^2 + (a^2*b^2 - b^4 + a^2*c^2 + 4*b^2*c^2 - c^4)*S) : :
X(66475) = X[396]+3X[11624] = 3X[396]+X[36978] = 9X[11624]-X[36978]

Let ABC be a triangle, and HAHBHC be the orthic triangle. Let SA be X(15) = 1st isodynamic point of AHBHC. Define SB, SC cyclically. Let AAABAC be the pedal triangle of SA with respect to AHBHC. Define BBBCBA, CCCACB cyclically. Let MA be the midpoint of ABAC. Define MB, MC cyclically. Then, MAMBMC is an equilateral triangle. This property holds when we replace X(15) with X(16). (Keita Miyamoto, Nov 28, 2024)

The center of MAMBMC is X(66475). For X(16), the center is X(66476). For X(15), barycentric coordinates of MA is
{a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 - 16*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6 - 2*Sqrt[3]*(a^2*b^2 - b^4 + a^2*c^2 + 6*b^2*c^2 - c^4)*S, b^2*(-a^2 + b^2 + c^2)*(a^2 - b^2 - 3*c^2 - 2*Sqrt[3]*S), c^2*(-a^2 + b^2 + c^2)*(a^2 - 3*b^2 - c^2 - 2*Sqrt[3]*S)}.
For the X(16) version, replace S with -S. (Peter Moses, Nov 29, 2024)

X(66475) lies on these lines: {6,1196}, {13,6000}, {17,11793}, {51,37640}, {52,42988}, {61,10110}, {62,11695}, {303,51386}, {373,37641}, {389,40693}, {395,6688}, {396,511}, {397,9729}, {1154,42496}, {3060,63032}, {3819,16644}, {3917,11488}, {5321,13570}, {5335,64100}, {5340,46850}, {5472,50387}, {5640,63079}, {5907,42156}, {9730,42974}, {10219,23303}, {10653,16836}, {11243,36757}, {11451,63080}, {11459,43542}, {11542,13754}, {12111,22235}, {13348,16772}, {13382,42992}, {13474,42162}, {13598,22236}, {14845,42975}, {14915,43416}, {15030,43403}, {15644,42152}, {16194,42128}, {16645,63632}, {17704,42148}, {21849,49947}, {21969,49813}, {23039,42817}, {40578,54472}, {42166,44870}, {42998,64854}, {43228,58470}

X(66475) = crossdifference of every pair of points on line {3566, 10676}

X(66476) = 2nd MIYAMOTO-MOSES EQUILATERAL TRIANGLE CENTER

Barycentrics    a^2*(2*Sqrt[3]*a^2*b^2*c^2 - (a^2*b^2 - b^4 + a^2*c^2 + 4*b^2*c^2 - c^4)*S) : :
X(66476) = X[395]+3X[11626] = 3X[395]+X[36980] = 9X[11626]-X[36980]

X(66476) lies on these lines: {6,1196}, {14,6000}, {18,11793}, {51,37641}, {52,42989}, {61,11695}, {62,10110}, {302,51386}, {373,37640}, {389,40694}, {395,511}, {396,6688}, {398,9729}, {1154,42497}, {3060,63033}, {3819,16645}, {3917,11489}, {5318,13570}, {5334,64100}, {5339,46850}, {5471,50387}, {5640,63080}, {5907,42153}, {9730,42975}, {10219,23302}, {10654,16836}, {11244,36758}, {11451,63079}, {11459,43543}, {11543,13754}, {12111,22237}, {13348,16773}, {13382,42993}, {13474,42159}, {13598,22238}, {14845,42974}, {14915,43417}, {15030,43404}, {15644,42149}, {16194,42125}, {16644,63632}, {17704,42147}, {21849,49948}, {21969,49812}, {23039,42818}, {40579,54473}, {42163,44870}, {42999,64854}, {43229,58470}

X(66476) = crossdifference of every pair of points on line {3566, 10675}

X(66477) = X(125)X(656)∩X(513)X(1364)

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

See David Nguyen and Juan José Isach Mayo, euclid 7317.

X(66477) lies on these lines: {11, 21172}, {125, 656}, {222, 61732}, {513, 1364}, {522, 4081}, {971, 1785}, {1071, 51889}, {1456, 15524}, {1464, 45272}, {1768, 55315}, {14584, 18340}, {16870, 60062}, {18838, 43909}, {35014, 57291}, {35015, 55359}

X(66477) = reflection of X(1456) in X(51616)
X(66477) = X(15252)-Dao conjugate of X(8)
X(66477) = crosspoint of X(7) and X(26932)
X(66477) = crosssum of X(55) and X(7115)
X(66477) = barycentric product X(15252)*X(26932)
X(66477) = barycentric quotient X(15252)/X(46102)

X(66478) = CENTER OF THE 1st EULER-ROUSSEL EQUILATERAL TRIANGLE

Trilinears    2*(2*S^2-SB*SC)*(cos(A/3)+2*cos(B/3)*cos(C/3))+a*(-a^2+b^2+c^2)*(b*(2*cos(A/3)*cos(C/3)+cos(B/3))+c*(2*cos(A/3)*cos(B/3)+cos(C/3))) : :
X(66478) = 2*X(3)+X(356)

See Antreas Hatzipolakis and César Lozada, euclid 7318.

X(66478) lies on these lines: {3, 356}, {376, 5455}

X(66479) = CENTER OF THE 2nd EULER-ROUSSEL EQUILATERAL TRIANGLE

Trilinears    -(S^2-2*SB*SC)*(cos(A/3)-cos(B/3)*cos(C/3))+a*(-a^2+b^2+c^2)*((cos(A/3)*cos(C/3)-cos(B/3))*b+(cos(A/3)*cos(B/3)-cos(C/3))*c) : :
X(66479) = 4*X(3)-X(3276) = 2*X(356)+X(5635)

See Antreas Hatzipolakis and César Lozada, euclid 7318.

X(66479) lies on these lines: {3, 3276}, {20, 65156}, {356, 5635}

X(66480) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EULER-ROUSSEL AND 1st MORLEY

Trilinears    σ*(cos(A/3)+2*cos(B/3)*cos(C/3))+sqrt(3)*(cos((B-C)/3)+sqrt(3)*sin((A+Pi)/3))+8*cos(A)*sin((A+Pi)/3)*sin((B+Pi)/3)*sin((C+Pi)/3) : :, where σ = CyclicSum(sqrt(3)*cos(2*(B-C)/3)+2*sin((2*A+Pi)/3)+sin(2*A/3)+sin(4*A/3))

See Antreas Hatzipolakis and César Lozada, euclid 7318.

X(66480) lies on these lines: {3, 356}, {1134, 66183}, {5454, 10258}

X(66480) = (X(8002), X(15857))-harmonic conjugate of X(356)

X(66481) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER-ROUSSEL AND 1st MORLEY

Trilinears    -2*sqrt(3)*(cos(A)-2*cos(B/3)*cos(C/3)*(-1+cos(2*B/3)+cos(2*C/3))-4*sin((B+Pi)/3)*sin((C+Pi)/3))+2*cos(A/3)*(sin(2*B/3)+sin(2*C/3))+2*(-3*cos(2*(B-C)/3)+2*cos((B-C)/3)*(-cos((A+Pi)/3)+2*sin((2*A+Pi)/6)))*cos((2*A+Pi)/6) : :

See Antreas Hatzipolakis and César Lozada, euclid 7318.

X(66481) lies on these lines: {356, 5635}, {357, 1134}, {3277, 41111}

X(66482) = X(3)X(6)∩X(394)X(454)

Barycentrics    a^4 (a^2-b^2-c^2) (a^10-5 a^8 b^2+10 a^6 b^4-10 a^4 b^6+5 a^2 b^8-b^10-5 a^8 c^2+12 a^6 b^2 c^2-6 a^4 b^4 c^2-4 a^2 b^6 c^2+3 b^8 c^2+10 a^6 c^4-6 a^4 b^2 c^4+6 a^2 b^4 c^4-2 b^6 c^4-10 a^4 c^6-4 a^2 b^2 c^6-2 b^4 c^6+5 a^2 c^8+3 b^2 c^8-c^10) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 7321.

X(66482) lies on these lines: {3, 6}, {4, 63835}, {20, 63762}, {26, 12095}, {30, 53171}, {97, 3547}, {394, 454}, {1092, 44405}, {1154, 59369}, {1628, 34756}, {4558, 11411}, {6759, 13557}, {7387, 39109}, {13398, 57697}, {44752, 47195}, {61748, 64035}

X(66483) = X(2)X(6)∩X(1357)X(1358)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66483) lies on these lines: {2, 6}, {1357, 1358}

X(66483) = perspector of the circumconic through X(99) X(17096)
X(66483) = pole of the line {4897, 53538} with respect to the incircle
X(66483) = pole of the line {1125, 4106} with respect to the circumhyperbola dual of Yff parabola
X(66483) = pole of the line {6, 6065} with respect to the Stammler hyperbola
X(66483) = pole of the line {2, 4076} with respect to the Steiner-Wallace hyperbola
X(66483) = barycentric product X(7192)*X(47884)
X(66483) = trilinear product X(1019)*X(47884)
X(66483) = trilinear quotient X(i)/X(j) for these (i,j): (7203, 59117), (47884, 1018)

X(66484) = X(7)X(55)∩X(11)X(116)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66484) lies on these lines: {7, 55}, {11, 116}, {5432, 56255}, {42454, 42462}, {45293, 62728}

X(66484) = cross-difference of every pair of points on the line X(10581)X(35326)
X(66484) = pole of the line {2310, 21104} with respect to the incircle
X(66484) = pole of the line {650, 58816} with respect to the circumhyperbola dual of Yff parabola
X(66484) = pole of the line {514, 5572} with respect to the Feuerbach circumhyperbola
X(66484) = barycentric product X(i)*X(j) for these {i,j}: {11, 62728}, {1638, 62725}, {6366, 56322}, {6606, 52334}, {21453, 33573}
X(66484) = trilinear product X(i)*X(j) for these {i,j}: {1170, 33573}, {1638, 62747}, {2170, 62728}, {6366, 58322}, {14392, 65552}, {14413, 62725}, {52334, 65222}, {56322, 65680}
X(66484) = trilinear quotient X(i)/X(j) for these (i,j): (1638, 63203), (4858, 62731), (6366, 35338), (33573, 1212), (52334, 21127), (56284, 35348), (56322, 37139), (58322, 14733), (62728, 4564), (65680, 35326)

X(66485) = X(8)X(56)∩X(11)X(1357)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66485) lies on these lines: {8, 56}, {11, 1357}, {4081, 61079}

X(66485) = pole of the line {3667, 66216} with respect to the Feuerbach circumhyperbola
X(66485) = barycentric product X(i)*X(j) for these {i,j}: {900, 60482}, {1647, 40420}, {3911, 40451}, {6613, 52338}, {30725, 56323}
X(66485) = trilinear product X(i)*X(j) for these {i,j}: {1319, 40451}, {1476, 1647}, {1635, 60482}, {2087, 40420}, {30725, 62748}, {40528, 62789}, {53528, 56323}
X(66485) = trilinear quotient X(i)/X(j) for these (i,j): (900, 61222), (1476, 9268), (1647, 3057), (2087, 2347), (3762, 25268), (6550, 6615), (30572, 61166), (30725, 21362), (40420, 5376), (40451, 1320), (53528, 23845), (60482, 3257), (62748, 5548)

X(66486) = X(2)X(7)∩X(11)X(1111)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66486) lies on these lines: {2, 7}, {11, 1111}, {1647, 3676}, {2006, 56783}, {2078, 37757}, {3254, 43762}, {3323, 14027}, {4904, 43960}, {4997, 35160}, {5091, 17718}, {10589, 40154}, {12019, 59490}, {15728, 60782}, {17181, 63574}, {30857, 43760}, {40629, 57435}, {53546, 57442}

X(66486) = perspector of the circumconic through X(664) X(24002)
X(66486) = pole of the line {1086, 3676} with respect to the incircle
X(66486) = pole of the line {105, 28292} with respect to the orthoptic circle of Steiner inellipse
X(66486) = pole of the line {3064, 56183} with respect to the polar circle
X(66486) = pole of the line {1, 3676} with respect to the circumhyperbola dual of Yff parabola
X(66486) = pole of the line {3309, 14100} with respect to the Feuerbach circumhyperbola
X(66486) = pole of the line {522, 4904} with respect to the Steiner inellipse
X(66486) = barycentric product X(i)*X(j) for these {i,j}: {11, 37757}, {693, 43050}, {1111, 37787}, {1358, 17264}, {2078, 23989}, {3676, 30565}, {3887, 24002}, {4858, 38459}, {22108, 52621}, {40629, 62723}, {52156, 57439}
X(66486) = trilinear product X(i)*X(j) for these {i,j}: {11, 38459}, {514, 43050}, {1086, 37787}, {1111, 2078}, {1358, 3935}, {2170, 37757}, {3669, 30565}, {3676, 3887}, {6549, 41553}, {8645, 52621}, {17264, 53538}, {22108, 24002}, {34051, 57435}, {34056, 40629}, {43736, 57439}
X(66486) = trilinear quotient X(i)/X(j) for these (i,j): (11, 42064), (693, 60488), (1111, 3254), (2078, 1110), (3676, 1308), (3887, 3939), (3935, 6065), (19624, 6066), (23100, 60489), (24002, 37143), (30565, 644), (37757, 4564), (37787, 1252), (38459, 59), (40629, 6603), (43050, 101), (47007, 19624), (52621, 35171), (57439, 41339)

X(66487) = X(662)X(2407)∩X(3960)X(4560)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66487) lies on these lines: {662, 2407}, {3960, 4560}, {21222, 30690}

X(66487) = pole of the line {3874, 6742} with respect to the Steiner circumellipse
X(66487) = barycentric product X(i)*X(j) for these {i,j}: {85, 62746}, {3025, 35139}, {3615, 4453}, {3904, 52393}, {7192, 63642}, {18155, 56844}
X(66487) = trilinear product X(i)*X(j) for these {i,j}: {7, 62746}, {1019, 63642}, {3025, 32680}, {3615, 3960}, {3738, 52393}, {3904, 52375}, {4560, 56844}
X(66487) = trilinear quotient X(i)/X(j) for these (i,j): (7, 63202), (3025, 2624), (3904, 3678), (3960, 2594), (4453, 16577), (4560, 56422), (7192, 65228), (7199, 63778), (18155, 41226), (32679, 7144), (32680, 46649), (52375, 32675), (52393, 2222), (53314, 21741), (53525, 55210), (53527, 21794), (56844, 4559), (62746, 55), (63642, 1018)

X(66488) = X(514)X(4581)∩X(693)X(3669)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66488) lies on these lines: {514, 4581}, {693, 3669}, {927, 8687}, {961, 29162}, {3676, 28094}, {4017, 6002}, {4444, 60086}, {4462, 25981}, {4555, 6648}, {4560, 7180}, {4608, 30725}, {6548, 30724}, {7178, 7192}, {8707, 59117}, {15309, 51659}, {17096, 52619}, {17925, 29126}, {21222, 41299}, {27469, 51664}, {30719, 58860}, {36098, 37143}

X(66488) = cross-difference of every pair of points on the line X(2269)X(20967)
X(66488) = perspector of the circumconic through X(31643) X(64984)
X(66488) = pole of the line {4298, 49598} with respect to the incircle
X(66488) = pole of the line {3704, 3965} with respect to the polar circle
X(66488) = pole of the line {65, 1999} with respect to the Steiner circumellipse
X(66488) = pole of the line {3812, 39595} with respect to the Steiner inellipse
X(66488) = barycentric product X(i)*X(j) for these {i,j}: {7, 4581}, {85, 62749}, {278, 15420}, {513, 31643}, {514, 64984}, {693, 961}, {1086, 6648}, {1111, 36098}, {1220, 3676}, {1240, 43924}, {1357, 65282}, {1358, 8707}, {1365, 65281}, {2298, 24002}, {2363, 4077}, {3668, 57161}, {3669, 30710}, {7178, 14534}, {7180, 40827}, {7192, 60086}
X(66488) = trilinear product X(i)*X(j) for these {i,j}: {7, 62749}, {34, 15420}, {57, 4581}, {244, 6648}, {513, 64984}, {514, 961}, {649, 31643}, {1019, 60086}, {1086, 36098}, {1111, 8687}, {1169, 4077}, {1220, 3669}, {1240, 57181}, {1357, 65229}, {1358, 36147}, {1365, 65255}, {1427, 57161}, {1434, 57162}, {2298, 3676}, {2363, 7178}
X(66488) = trilinear quotient X(i)/X(j) for these (i,j): (2, 61223), (7, 3882), (34, 61205), (57, 53280), (65, 61168), (85, 53332), (226, 61172), (244, 52326), (278, 61226), (513, 2269), (514, 960), (522, 3965), (523, 21033), (649, 20967), (661, 40966), (693, 3687), (961, 101), (1019, 4267), (1086, 17420), (1111, 3910)
X(66488) = (X(7178), X(57079))-harmonic conjugate of X(7192)

X(66489) = X(2)X(1577)∩X(4608)X(20578)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66489) lies on these lines: {2, 1577}, {4608, 20578}, {7192, 37772}, {62631, 66284}

X(66489) = trilinear product X(11073)*X(16755)
X(66489) = trilinear quotient X(i)/X(j) for these (i,j): (1019, 42623), (7192, 3179)

X(66490) = X(2)X(1577)∩X(4608)X(20579)

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

See Sriram Panchapakesan and César Lozada, euclid 7324.

X(66490) lies on these lines: {2, 1577}, {4608, 20579}, {7192, 37773}, {62632, 66284}

X(66490) = barycentric product X(i)*X(j) for these {i,j}: {7150, 7199}, {42624, 52619}
X(66490) = trilinear product X(i)*X(j) for these {i,j}: {7150, 7192}, {7199, 42624}, {11072, 16755}
X(66490) = trilinear quotient X(i)/X(j) for these (i,j): (7150, 4557), (7192, 41225)

X(66491) = X(30)X(974)∩X(3580)X(16237)

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^14-4*(b^2+c^2)*a^12+(5*b^4+7*b^2*c^2+5*c^4)*a^10-7*(b^2+c^2)*b^2*c^2*a^8-(5*b^8+5*c^8-2*(8*b^4-9*b^2*c^2+8*c^4)*b^2*c^2)*a^6+4*(b^4-c^4)*(b^2-c^2)*(b^4-3*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(b^8+c^8+(b^4-16*b^2*c^2+c^4)*b^2*c^2)*a^2+3*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 7325.

X(66491) lies on these lines: {30, 974}, {3003, 44436}, {3580, 16237}

X(66492) = X(64)X(155)∩X(1112)X(1301)

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

See Antreas Hatzipolakis and César Lozada, euclid 7325.

X(66492) lies on the cubic K500 and these lines: {64, 155}, {459, 56063}, {1073, 1993}, {1112, 1301}, {1539, 20123}, {6102, 14379}, {6793, 11064}, {13157, 64923}, {51491, 64505}

X(66492) = pole of the line {15291, 52948} with respect to the Stammler hyperbola
X(66492) = barycentric product X(i)*X(j) for these {i, j}: {64, 6148}, {253, 1511}, {340, 11589}, {1073, 14920}, {5664, 46639}, {19611, 35201}, {34403, 39176}, {44326, 52743}
X(66492) = trilinear product X(i)*X(j) for these {i, j}: {1073, 35201}, {1511, 2184}, {2155, 6148}, {11589, 52414}, {14920, 19614}, {19611, 39176}
X(66492) = trilinear quotient X(i)/X(j) for these (i, j): (1511, 610), (2155, 40355), (2184, 5627), (5664, 17898), (6148, 18750), (6149, 15291), (14920, 1895), (19614, 11079), (35201, 1249), (39176, 204), (52414, 10152)

X(66493) = X(2)X(6120)∩X(631)X(6123)

Barycentrics    2*a^4 + (b^2 - c^2)^2 - 3*a^2*(b^2 + c^2) + a^2*((-a^2 + b^2 + c^2)*Cos[2*A/3] + 2*S*Sin[2*A/3]) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7333.

X(66493) lies on these lines: {2, 6120}, {140, 358}, {631, 6123}

X(66494) = X(2)X(6122)∩X(631)X(6125)

Barycentrics    2*a^4 + (b^2 - c^2)^2 - 3*a^2*(b^2 + c^2) + a^2*((-a^2 + b^2 + c^2)*Cos[2*A/3 + 2*Pi/3] + 2*S*Sin[2*A/3 + 2*Pi/3]) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7333.

X(66494) lies on these lines: {2, 6122}, {140, 1137}, {631, 6125}

X(66495) = X(2)X(6121)∩X(631)X(6124)

Barycentrics    2*a^4 + (b^2 - c^2)^2 - 3*a^2*(b^2 + c^2) + a^2*((-a^2 + b^2 + c^2)*Cos[2*A/3 + 2*Pi/3] + 2*S*Sin[2*A/3 - 2*Pi/3]) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7333.

X(66495) lies on these lines: {2, 6121}, {140, 1135}, {631, 6124}

X(66496) = (name pending)

Barycentrics    2*a^4 + (b^2 - c^2)^2 - 3*a^2*(b^2 + c^2) + a^2*((-a^2 + b^2 + c^2)*Cos[2*A/5] + 2*S*Sin[2*A/5]) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7333.

X(66496) lies on these lines: { }

X(66497) = (name pending)

Barycentrics    2*a^4 + (b^2 - c^2)^2 - 3*a^2*(b^2 + c^2) + a^2*((-a^2 + b^2 + c^2)*Cos[4*A/5] + 2*S*Sin[4*A/5]) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7333.

X(66497) lies on these lines: { }

X(66498) = X(5)X(690)∩X(113)X(526)

Barycentrics    (b^2 - c^2)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - b^2*c^6) : :
X(66498) = 4 X[39509] - X[53567], 3 X[19912] - X[21731], 3 X[19912] + X[41079], 2 X[23301] - 3 X[34964], X[23301] - 3 X[59745]

See Tran Quang Hung and Peter Moses, euclid 7347.

X(66498) lies on these lines: {2, 53247}, {4, 53263}, {5, 690}, {12, 53563}, {30, 14270}, {113, 526}, {115, 804}, {140, 44826}, {235, 16230}, {247, 53577}, {351, 54395}, {403, 44427}, {512, 51548}, {523, 11799}, {550, 39477}, {858, 9185}, {1637, 15543}, {2450, 3566}, {2491, 5254}, {2797, 6132}, {6140, 62489}, {6334, 15760}, {6753, 16229}, {9189, 30739}, {9208, 21531}, {9409, 46985}, {11176, 51389}, {14271, 44882}, {14295, 59635}, {15367, 45147}, {43917, 55121}

X(66498) = midpoint of X(i) and X(j) for these {i,j}: {4, 53263}, {21731, 41079}
X(66498) = reflection of X(i) in X(j) for these {i,j}: {5, 39509}, {550, 39477}, {15543, 1637}, {34964, 59745}, {44826, 140}, {44882, 14271}, {53567, 5}
X(66498) = complement of X(53247)
X(66498) = X(2433)-Ceva conjugate of X(523)
X(66498) = crosspoint of X(2394) and X(52618)
X(66498) = crossdifference of every pair of points on line {1634, 5063}
X(66498) = {X(19912),X(41079)}-harmonic conjugate of X(21731)

X(66499) = TRILINEAR POLE OF X(30)X(12900)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(8*a^8 + a^6*b^2 - 18*a^4*b^4 + a^2*b^6 + 8*b^8 - 17*a^6*c^2 + 23*a^4*b^2*c^2 + 23*a^2*b^4*c^2 - 17*b^6*c^2 + 3*a^4*c^4 - 37*a^2*b^2*c^4 + 3*b^4*c^4 + 13*a^2*c^6 + 13*b^2*c^6 - 7*c^8)*(8*a^8 - 17*a^6*b^2 + 3*a^4*b^4 + 13*a^2*b^6 - 7*b^8 + a^6*c^2 + 23*a^4*b^2*c^2 - 37*a^2*b^4*c^2 + 13*b^6*c^2 - 18*a^4*c^4 + 23*a^2*b^2*c^4 + 3*b^4*c^4 + a^2*c^6 - 17*b^2*c^6 + 8*c^8) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7356.

X(66499) lies on this line: {9214, 50687}

X(66499) = trilinear pole of line {30, 12900}

X(66500) = TRILINEAR POLE OF X(30)X(9826)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^10*b^2 - 3*a^8*b^4 + 2*a^6*b^6 + 2*a^4*b^8 - 3*a^2*b^10 + b^12 - a^10*c^2 + 7*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 12*a^4*b^6*c^2 + 13*a^2*b^8*c^2 - 3*b^10*c^2 + 4*a^8*c^4 - 8*a^6*b^2*c^4 + 22*a^4*b^4*c^4 - 12*a^2*b^6*c^4 + 2*b^8*c^4 - 6*a^6*c^6 - 8*a^4*b^2*c^6 - 4*a^2*b^4*c^6 + 2*b^6*c^6 + 4*a^4*c^8 + 7*a^2*b^2*c^8 - 3*b^4*c^8 - a^2*c^10 + b^2*c^10)*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 - a^10*c^2 - 7*a^8*b^2*c^2 + 8*a^6*b^4*c^2 + 8*a^4*b^6*c^2 - 7*a^2*b^8*c^2 - b^10*c^2 + 3*a^8*c^4 + 4*a^6*b^2*c^4 - 22*a^4*b^4*c^4 + 4*a^2*b^6*c^4 + 3*b^8*c^4 - 2*a^6*c^6 + 12*a^4*b^2*c^6 + 12*a^2*b^4*c^6 - 2*b^6*c^6 - 2*a^4*c^8 - 13*a^2*b^2*c^8 - 2*b^4*c^8 + 3*a^2*c^10 + 3*b^2*c^10 - c^12) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7356.

X(66500) lies on this line: {2407, 61209}

X(66500) = trilinear pole of line {30, 9826}

X(66501) = TRILINEAR POLE OF X(517)X(58421)

Barycentrics    (a - b)*(a - c)*(3*a^5*b - 3*a^4*b^2 - 6*a^3*b^3 + 6*a^2*b^4 + 3*a*b^5 - 3*b^6 + 5*a^5*c - 13*a^4*b*c + 10*a^3*b^2*c + 10*a^2*b^3*c - 15*a*b^4*c + 3*b^5*c + 14*a^3*b*c^2 - 30*a^2*b^2*c^2 + 10*a*b^3*c^2 + 6*b^4*c^2 - 10*a^3*c^3 + 14*a^2*b*c^3 + 10*a*b^2*c^3 - 6*b^3*c^3 - 13*a*b*c^4 - 3*b^2*c^4 + 5*a*c^5 + 3*b*c^5)*(5*a^5*b - 10*a^3*b^3 + 5*a*b^5 + 3*a^5*c - 13*a^4*b*c + 14*a^3*b^2*c + 14*a^2*b^3*c - 13*a*b^4*c + 3*b^5*c - 3*a^4*c^2 + 10*a^3*b*c^2 - 30*a^2*b^2*c^2 + 10*a*b^3*c^2 - 3*b^4*c^2 - 6*a^3*c^3 + 10*a^2*b*c^3 + 10*a*b^2*c^3 - 6*b^3*c^3 + 6*a^2*c^4 - 15*a*b*c^4 + 6*b^2*c^4 + 3*a*c^5 + 3*b*c^5 - 3*c^6) : :

See Antreas Hatzipolakis and Peter Moses, euclid 7357.

X(66501) lies on these lines: { }

X(66501) = trilinear pole of line {517, 58421}

X(66502) = X(1)X(88)∩X(1960)X(2827)

Barycentrics    a*(4*a^6 - 9*a^5*b - 8*a^4*b^2 + 17*a^3*b^3 + 3*a^2*b^4 - 8*a*b^5 + b^6 - 9*a^5*c + 44*a^4*b*c - 23*a^3*b^2*c - 45*a^2*b^3*c + 29*a*b^4*c - 2*b^5*c - 8*a^4*c^2 - 23*a^3*b*c^2 + 72*a^2*b^2*c^2 - 14*a*b^3*c^2 - 2*b^4*c^2 + 17*a^3*c^3 - 45*a^2*b*c^3 - 14*a*b^2*c^3 + 2*b^3*c^3 + 3*a^2*c^4 + 29*a*b*c^4 - 2*b^2*c^4 - 8*a*c^5 - 2*b*c^5 + c^6) : :
X(66502) = X[13541] + 3 X[14193], X[6246] - 3 X[57300], X[10774] - 3 X[32557], X[11814] - 3 X[34123], X[21290] - 5 X[64012], 3 X[38695] - X[46684]

See Antreas Hatzipolakis, David Nguyen and Peter Moses, euclid 7358.

X(66502) lies on these lines: {1, 88}, {121, 58453}, {1960, 2827}, {2800, 38604}, {2801, 59783}, {6246, 57300}, {6702, 6715}, {10774, 32557}, {11814, 34123}, {21290, 64012}, {38695, 46684}

X(66502) = midpoint of X(106) and X(214)
X(66502) = reflection of X(i) in X(j) for these {i,j}: {121, 58453}, {6702, 6715}

X(66503) = X(7)X(4076)∩X(1357)X(16185)

Barycentrics    (b - c)^2 (a - b + c) (a + b - c) (-3 a + b + c) (2 a^3 + b^3 -3 b^2 c - 3 b c^2 + c^3 - 4 a^2 (b + c) - a (b^2 - 12 b c + c^2)) : :

See David Nguyen and Juan José Isach Mayo, euclid 7366.

X(66503) lies on these lines: {7, 4076}, {1357, 16185}, {3667, 40617}, {5048, 37743}, {14027, 14112}, {52907, 60058}

X(66503) = crosspoint of X(7) and X(40617)

X(66504) = X(7)X(46649)∩X(900)X(4542)

Barycentrics    -(2 a - b - c) (b - c)^2 (a^2 - b^2 + b c - c^2) (2 a^6 + 12 a^4 b c - 4 a^5 (b + c) + (b^2 - c^2)^2 (b^2 - 3 b c + c^2) - a (b - c)^2 (b^3 - 8 b^2 c - 8 b c^2 + c^3) + a^3 (5 b^3 - 9 b^2 c - 9 b c^2 + 5 c^3) -a^2 (3 b^4 + 6 b^3 c - 20 b^2 c^2 + 6 b c^3 + 3 c^4)) : :

See David Nguyen and Juan José Isach Mayo, euclid 7366.

X(66504) lies on these lines: {7, 46649}, {900, 4542}, {952, 43909}, {2827, 3025}

X(66505) = X(7)X(6065)∩X(527)X(60059)

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

See David Nguyen and Juan José Isach Mayo, euclid 7366.

X(66505) lies on these lines: {7, 6065}, {527, 60059}, {1358, 16184}, {3309, 40615}

X(66505) = crosspoint of X(7) and X(40615)

X(66506) = X(141)X(1369)∩X(512)X(5943)

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

See Juan José Isach Mayo, euclid 7390.

X(66506) lies on these lines: {2, 59266}, {83, 40163}, {141, 1369}, {512, 5943}, {5133, 42421}, {7745, 20022}, {7804, 40379}, {8878, 24273}, {10191, 29012}, {39691, 59180}

X(66506) = crosspoint of X(8) and X(1031)
X(66506) = crosssum of X(39) and X(10329)

X(66507) = (name pending)

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

As a point on the Euler line, X(66507) has Shinagawa coefficients {-(E + F)^3 - 2 e S^2 + 3 (E +F) S^2, 8 e S^2 -(E + F) ((E +F)^2 + 13 S^2)}

See Juan José Isach Mayo, euclid 7390.

X(66507) lies on this line: {2, 3}

X(66508) = COMPLEMENT OF X(3257)

Barycentrics    (b - c)^2*(-2*a + b + c)*(-a^2 + b^2 - b*c + c^2) : :
X(66508) = 3 X[1086] - 2 X[60578], 4 X[35023] - 3 X[46973]

X(66508) = lies on these lines: {2, 3257}, {7, 655}, {11, 513}, {44, 3911}, {121, 3836}, {320, 2245}, {495, 56750}, {514, 1086}, {518, 1145}, {523, 43909}, {527, 4370}, {679, 8046}, {900, 4542}, {908, 3834}, {1279, 11700}, {1387, 14190}, {1647, 42084}, {3328, 53578}, {3942, 45234}, {4089, 46398}, {4129, 8287}, {4293, 36944}, {4530, 21129}, {4675, 9318}, {4957, 28851}, {4977, 7336}, {5218, 56758}, {6544, 40629}, {16597, 34587}, {16732, 23755}, {17237, 24318}, {17301, 60692}, {17455, 41801}, {20317, 26932}, {21127, 38375}, {22102, 39154}, {23766, 42754}, {24870, 61730}, {28217, 55376}, {30379, 39063}, {35023, 46973}, {35175, 36804}, {36275, 37651}, {37691, 51908}, {40622, 51664}, {52556, 53665}

X(66508) = midpoint of X(i) and X(j) for these {i,j}: {7, 37131}, {320, 3218}, {679, 8046}
X(66508) = reflection of X(i) in X(j) for these {i,j}: {44, 3911}, {908, 3834}, {14190, 1387}, {39154, 22102}
X(66508) = complement of X(3257)
X(66508) = complement of the isogonal conjugate of X(1635)
X(66508) = complement of the isotomic conjugate of X(3762)
X(66508) = medial-isogonal conjugate of X(4928)
X(66508) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 4928}, {2, 53571}, {6, 900}, {31, 3960}, {32, 3310}, {42, 59737}, {44, 513}, {56, 44902}, {58, 45674}, {101, 62630}, {213, 21894}, {512, 3936}, {513, 3834}, {514, 21241}, {519, 3835}, {649, 519}, {650, 5123}, {667, 16610}, {900, 141}, {902, 514}, {1015, 1647}, {1017, 6544}, {1023, 24003}, {1252, 6550}, {1319, 4885}, {1333, 59837}, {1404, 522}, {1635, 10}, {1639, 1329}, {1647, 116}, {1877, 46396}, {1919, 8610}, {1960, 2}, {2087, 11}, {2226, 33922}, {2251, 650}, {2279, 45328}, {2316, 59997}, {2325, 59971}, {2384, 33920}, {2423, 1387}, {3251, 16594}, {3264, 21262}, {3285, 523}, {3310, 56416}, {3572, 25351}, {3689, 20317}, {3733, 4395}, {3762, 2887}, {3911, 17072}, {3943, 31946}, {4120, 3454}, {4358, 21260}, {4432, 27854}, {4448, 20333}, {4530, 124}, {4730, 1211}, {4768, 21244}, {4775, 27751}, {4895, 3452}, {6187, 21198}, {6544, 121}, {8661, 6547}, {8756, 20316}, {9459, 6586}, {14407, 1213}, {14408, 34832}, {14418, 34823}, {14425, 2885}, {14584, 46397}, {16704, 512}, {17780, 27076}, {21758, 52537}, {21805, 4129}, {22086, 3}, {22356, 20315}, {22383, 60415}, {23344, 4422}, {23703, 21232}, {25426, 45342}, {30572, 17052}, {30573, 31844}, {30576, 52601}, {30725, 2886}, {30731, 3038}, {30939, 42327}, {32641, 22102}, {35092, 3259}, {37168, 30476}, {43924, 17067}, {45144, 53573}, {47420, 10017}, {52338, 46100}, {52556, 6085}, {52680, 4369}, {52963, 661}, {53528, 142}, {53532, 18589}, {60665, 45340}, {60865, 6373}, {60873, 9461}, {61210, 3035}, {62789, 46399}, {65867, 626}
X(66508) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3960}, {7, 900}, {89, 23888}, {1111, 1647}, {7192, 6550}, {8046, 514}, {8047, 519}, {40218, 30725}, {41801, 53535}, {54452, 513}
X(66508) = X(66504)-cross conjugate of X(7)
X(66508) = X(i)-isoconjugate of X(j) for these (i,j): {1168, 1252}, {2149, 36590}, {2161, 9268}, {2222, 5548}, {2316, 52377}, {5376, 6187}, {23990, 57788}, {32665, 51562}, {32719, 36804}, {52925, 58955}
X(66508) = X(i)-Dao conjugate of X(j) for these (i,j): {44, 765}, {650, 36590}, {661, 1168}, {1639, 8}, {3310, 56416}, {3936, 1016}, {3960, 2}, {6544, 80}, {21198, 30578}, {35092, 51562}, {38984, 5548}, {40584, 9268}, {40612, 5376}
X(66508) = cevapoint of X(3259) and X(35092)
X(66508) = crosspoint of X(i) and X(j) for these (i,j): {2, 3762}, {514, 3911}
X(66508) = crosssum of X(i) and X(j) for these (i,j): {6, 32665}, {101, 2316}
X(66508) = crossdifference of every pair of points on line {2427, 23344}
X(66508) = barycentric product X(i)*X(j) for these {i,j}: {7, 51402}, {11, 41801}, {214, 1111}, {244, 1227}, {320, 1647}, {519, 4089}, {693, 53535}, {900, 4453}, {1086, 51583}, {2087, 20924}, {3762, 3960}, {3904, 30725}, {4358, 53546}, {4530, 17078}, {14425, 27836}, {16727, 40988}, {16732, 17191}, {17455, 23989}, {40218, 46398}, {53314, 65867}
X(66508) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 36590}, {36, 9268}, {214, 765}, {244, 1168}, {320, 62536}, {654, 5548}, {900, 51562}, {1111, 57788}, {1227, 7035}, {1319, 52377}, {1647, 80}, {2087, 2161}, {3025, 62703}, {3218, 5376}, {3259, 56416}, {3762, 36804}, {3904, 4582}, {3960, 3257}, {4089, 903}, {4453, 4555}, {4530, 36910}, {6550, 66284}, {14027, 14584}, {14584, 46649}, {17191, 4567}, {17455, 1252}, {21758, 32665}, {30725, 655}, {41801, 4998}, {42084, 40172}, {51402, 8}, {51583, 1016}, {53314, 901}, {53525, 1320}, {53528, 2222}, {53535, 100}, {53537, 65573}, {53546, 88}, {56761, 40437}, {57434, 51984}

X(66509) = COMPLEMENT OF X(3756)

Barycentrics    2*a^3 - 4*a^2*b - a*b^2 + b^3 - 4*a^2*c + 12*a*b*c - 3*b^2*c - a*c^2 - 3*b*c^2 + c^3 : :
X(66509) = 3 X[2] + X[3699], 9 X[2] - X[58371], X[3699] - 3 X[12035], 3 X[3699] + X[58371], X[3756] + 3 X[12035], 3 X[3756] - X[58371], 9 X[12035] + X[58371], X[1120] - 5 X[3616], 5 X[1698] - X[26727]

X(66509) lies on these lines: {1, 2885}, {2, 1280}, {10, 1387}, {11, 9458}, {100, 16594}, {121, 952}, {149, 30855}, {244, 58413}, {513, 3038}, {518, 50535}, {519, 11731}, {528, 11814}, {545, 1054}, {900, 3035}, {1120, 3616}, {1293, 38384}, {1358, 25605}, {1644, 3722}, {1698, 26727}, {2611, 65561}, {3952, 43055}, {4432, 35023}, {4997, 26073}, {5048, 60443}, {5121, 9053}, {5205, 5846}, {5853, 52907}, {6154, 24709}, {6557, 25567}, {6692, 59596}, {6788, 65742}, {9791, 62379}, {15325, 59669}, {17044, 27076}, {17259, 24669}, {17279, 65957}, {17719, 40480}, {20103, 62674}, {20315, 55317}, {20316, 36951}, {24988, 37691}, {26139, 43290}, {28530, 62297}, {30566, 44006}, {31235, 33115}, {33070, 37663}, {34824, 61158}, {35466, 37762}, {37828, 59704}, {52264, 59666}, {56176, 65993}, {58451, 62689}, {59506, 59583}, {59572, 59580}, {59599, 63621}

X(66509) = midpoint of X(i) and X(j) for these {i,j}: {1, 52871}, {2, 12035}, {121, 6789}, {1293, 38384}, {3699, 3756}, {5205, 51415}, {6788, 65742}
X(66509) = complement of X(3756)
X(66509) = X(i)-complementary conjugate of X(j) for these (i,j): {59, 12640}, {100, 5510}, {101, 40617}, {692, 40621}, {765, 2885}, {1110, 3161}, {1293, 11}, {2149, 63621}, {3680, 46100}, {5382, 141}, {15403, 3057}, {23990, 63622}, {27834, 116}, {31343, 124}, {32665, 62559}, {34080, 1086}, {36042, 1647}, {38266, 6547}, {38828, 4904}, {53647, 21252}, {59095, 24237}, {65173, 17059}
X(66509) = crossdifference of every pair of points on line {8659, 9259}
X(66509) = barycentric quotient X(66503)/X(40617)
X(66509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3699, 3756}, {3035, 24003, 4422}, {3756, 12035, 3699}, {4997, 26073, 62221}, {24003, 62630, 3035}, {26139, 43290, 53534}

X(66510) = X(5)X(275)∩X(140)X(233)

Barycentrics    (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^12-3 a^10 b^2+a^8 b^4+6 a^6 b^6-9 a^4 b^8+5 a^2 b^10-b^12-3 a^10 c^2+3 a^8 b^2 c^2+2 a^6 b^4 c^2+6 a^4 b^6 c^2-15 a^2 b^8 c^2+7 b^10 c^2+a^8 c^4+2 a^6 b^2 c^4+6 a^4 b^4 c^4+10 a^2 b^6 c^4-19 b^8 c^4+6 a^6 c^6+6 a^4 b^2 c^6+10 a^2 b^4 c^6+26 b^6 c^6-9 a^4 c^8-15 a^2 b^2 c^8-19 b^4 c^8+5 a^2 c^10+7 b^2 c^10-c^12) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 7406.

X(66510) lies on these lines: {5, 275}, {30, 58468}, {140, 233}, {381, 3462}, {546, 6750}, {3628, 58417}, {3850, 7687}, {4993, 15780}, {5462, 32438}, {8955, 18538}, {13147, 32078}, {14635, 35717}, {15712, 60171}, {31389, 35311}

X(66511) = X(2)X(187)∩X(5)X(543)

Barycentrics    2 a^4-11 a^2 b^2+8 b^4-11 a^2 c^2-14 b^2 c^2+8 c^4 : :

See Antreas Hatzipolakis, Antonio Roberto Martínez Fernández, and Francisco Javier García Capitán, euclid 7394.

X(66511) lies on these lines: {2, 187}, {5, 543}, {30, 7619}, {39, 9166}, {148, 64809}, {381, 7622}, {524, 547}, {538, 5055}, {597, 6722}, {620, 3363}, {671, 17005}, {754, 15597}, {1506, 7817}, {1656, 7610}, {2482, 37647}, {2548, 63107}, {2782, 14159}, {3055, 37350}, {3090, 7758}, {3091, 34504}, {3399, 5503}, {3545, 7618}, {3628, 7843}, {3815, 5461}, {4045, 8355}, {5056, 34511}, {5066, 63647}, {5071, 7615}, {5079, 34505}, {5459, 33474}, {5460, 33475}, {5485, 54645}, {6114, 33476}, {6115, 33477}, {6683, 11318}, {7486, 7759}, {7620, 61924}, {7746, 63065}, {7751, 61905}, {7764, 35018}, {7769, 52695}, {7781, 61919}, {7816, 33013}, {7844, 42849}, {7848, 11168}, {7849, 32999}, {7861, 32984}, {7870, 33002}, {7880, 41133}, {7883, 16922}, {7915, 8367}, {8370, 9167}, {8589, 8597}, {8598, 43457}, {8667, 61901}, {8716, 61925}, {9761, 22495}, {9763, 22496}, {9766, 61908}, {9830, 25561}, {10109, 52229}, {11159, 18584}, {11165, 18546}, {11317, 32456}, {12506, 14666}, {12812, 63924}, {13335, 26614}, {13468, 61898}, {14041, 55801}, {14061, 33694}, {14161, 64802}, {16509, 61910}, {19911, 64089}, {21358, 40332}, {31274, 35954}, {31455, 33006}, {31652, 33011}, {32457, 63083}, {32480, 39563}, {33274, 39590}, {39565, 41135}, {43620, 63025}, {47478, 53144}, {47599, 63941}, {51123, 61918}, {53141, 61930}, {53142, 61936}, {54750, 62895}, {55857, 63931}, {61886, 63935}, {61887, 63943}, {61889, 63947}, {61894, 63930}, {61897, 63946}, {61899, 63029}, {61912, 63955}

X(66512) = ORTHOLOGIC CONJUGATE OF X(4) WRT ABC AND EXCENTRAL

Barycentrics    (b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^3 + a*b^2 - a*b*c + b^2*c + a*c^2 + b*c^2) : :
X(66512) = 3X[4]-4X[16228] = 2X[16228]-3X[44426] = 5X[631]-4X[59973] = 9X[3545]-8X[44923]

In the following, ℓ(⊥AB, P, θ) denotes the line obtained by rotating ℓ(⊥AB, P) by angle θ about P, where ℓ(⊥AB, P) denotes the line through a point P perpendicular to a line AB, and -π/2 < θ ≤ π/2 (counter-clockwise is the positive direction).

Let A1B1C1 and A2B2C2 be two triangles.
If there exists θ = θ0 such that the three lines ℓ(⊥B2C2, A1, θ0), ℓ(⊥C2A2, B1, θ0), ℓ(⊥A2B2, C1, θ0) concur in a point P1, then, the three lines ℓ(⊥B1C1, A2, -θ0), ℓ(⊥C1A1, B2, -θ0), ℓ(⊥A1B1, C2, -θ0) also concur in a point P2. Here, we call A1B1C1 and A2B2C2 the θ0-isologic triangles. We call P1 the isologic center of A1B1C1 with respect to A2B2C2, and P2 the isologic center of A2B2C2 with respect to A1B1C1, and θ0 the isologic angle of A1B1C1 and A2B2C2.

Let P be a point in the plane. Let A' = B1C1 ∩ ℓ(⊥PA2, P1, θ0). Define B', C' cyclically. Similarly, let A'' = B2C2 ∩ ℓ(⊥PA1, P2, -θ0). Define B'', C'' cyclically. Then,

  1. The triangle bounded by A'A'', B'B'', C'C'' is perspective with both A1B1C1 and A2B2C2.
  2. A', B', C' lie on the same line ℓ1. Similarly, A'', B'', C'' lie on the same line ℓ2. We call ℓ1 the A1B1C1-isologic transversal of P with respect to A2B2C2, and ℓ2 the A2B2C2-isologic transversal of P with respect to A1B1C1.
  3. Let Q = ℓ1 ∩ ℓ2. Suppose that P1 ≢ P2. Let ℓX be the perpendicular bisector of P1P2. We call ℓX the isologic axis of A1B1C1 and A2B2C2. The line PQ passes through a fixed point U on ℓX, and angle ∡P2UP1 = 2*θ0. We call U the isologic pole of A1B1C1 and A2B2C2.
  4. The mapping f : P → Q is an involution, i.e., f(f(P)) = P. We call Q the isologic conjugate of P with respect to A1B1C1 and A2B2C2. If P coincides with U, then, ℓ1 coincides with ℓ2, and therefore Q is undefined. Let ℓU be the A1B1C1-isologic transversal of U with respect to A2B2C2 (or equivalently, A2B2C2-isologic transversal of U with respect to A1B1C1). We call ℓU the isologic polar of A1B1C1 and A2B2C2. Conversely, if P lies on the isologic polar, then Q coincides with the isologic pole U. ℓU is perpendicular to ℓX. The intersection point of A1B1C1-isologic transversal of P and A1B1C1-isologic transversal of Q lies on ℓU.
  5. Suppose that θ0 ≠ 0, and P1 ≢ P2. Let the circle (P1P2U) intersect ℓX in T different from U. Let T1 and T2 be the real intersection points of ℓU and (P1P2U). If P lies on the circle centered at T and whose radical axis with (P1P2U) is ℓU, then, Q coincides with P (except for T1 and T2), and the triangle bounded by A'A'', B'B'', C'C'' degenerates to a point. Q is also undefined if P coincides with T1 or T2.

Note that isologic centers are not necessarily uniquely determined. For example, let ABC be a triangle, and let A1B1C1 and A2B2C2 be the pedal triangles of X(15), X(16), respectively. Then, A1B1C1 and A2B2C2 have infinitely many isologic centers. This is because A1B1C1 and A2B2C2 are inversely similar. Also, if θ0 = 0 and P1 coincides with P2, then, the two isologic transversals of P always coincide, and the isologic conjugate of P is undefined.

If θ0 = 0, then, A1B1C1 and A2B2C2 are orthologic. In this case, we call Q the orthologic conjugate of P with respect to A1B1C1 and A2B2C2. For example, X(47805) is the orthologic conjugate of X(2) with respect to ABC and the excentral triangle. The orthologic conjugates of X(4), X(6), X(9) with respect to ABC and the excentral triangle were found by Peter Moses, Dec 14, 2024. The line P1P2 is the isologic polar.
If θ0 = π/2, then, A1B1C1 and A2B2C2 are parallelogic.

Based on notes by Keita Miyamoto, Dec 3, 2024, updated on Dec 23, 2024.

X(66512) lies on these lines: {4,513}, {25,47805}, {27,47763}, {28,57246}, {108,53702}, {186,523}, {242,514}, {273,57167}, {427,48164}, {451,48165}, {469,47759}, {522,45766}, {631,59973}, {649,17926}, {650,57166}, {1119,24002}, {1172,21007}, {2812,4086}, {3064,62748}, {3520,48390}, {3545,44923}, {3937,21666}, {4091,28623}, {4212,47824}, {4213,47821}, {4581,17924}, {4778,54239}, {4817,53150}, {4977,39534}, {6198,48307}, {6353,47804}, {6590,57043}, {6949,42769}, {7192,46107}, {7490,47762}, {7718,48324}, {8672,14618}, {8889,44429}, {13619,62492}, {16231,28225}, {17496,23187}, {20949,54314}, {26704,59073}, {35360,37966}, {38282,47803}, {47136,57224}, {47802,52299}, {48246,52252}, {48298,58313}

X(66512) = reflection of X(i) in X(j) for these {i,j}: {4, 44426}, {17496, 23187}, {59915, 7649}
X(66512) = polar conjugate of X(56188)
X(66512) = polar conjugate of the isotomic conjugate of X(17496)
X(66512) = X(108)-Ceva conjugate of X(4)
X(66512) = X(51662)-cross conjugate of X(21173)
X(66512) = X(i)-isoconjugate of X(j) for these (i,j): {3, 56194}, {48, 56188}, {71, 65260}, {72, 59006}, {184, 56252}, {228, 65275}, {906, 2051}, {1331, 34434}, {4574, 53083}, {4575, 51870}, {6516, 60817}, {22350, 53702}, {32656, 54121}
X(66512) = X(i)-Dao conjugate of X(j) for these (i,j): {136, 51870}, {1249, 56188}, {4391, 35518}, {5190, 2051}, {5521, 34434}, {21189, 57111}, {34589, 72}, {36103, 56194}, {53566, 22076}, {62605, 56252}
X(66512) = cevapoint of X(650) and X(21645).
X(66512) = crosspoint of X(i) and X(j) for these (i,j): {107, 57669}, {286, 653}, {8795, 54240}
X(66512) = crosssum of X(i) and X(j) for these (i,j): {228, 652}, {408, 520}, {418, 36054}, {1459, 22344}, {22368, 65102}, {22383, 23196}
X(66512) = trilinear pole of line {11998, 53566}
X(66512) = crossdifference of every pair of points on line {71, 216}
X(66512) = barycentric product X(i)*X(j) for these {i,j}: {4, 17496}, {19, 57244}, {92, 21173}, {108, 40624}, {278, 57091}, {514, 11109}, {572, 46107}, {648, 53566}, {653, 34589}, {1847, 58339}, {1897, 24237}, {2052, 23187}, {2973, 65203}, {2975, 17924}, {7649, 14829}, {11998, 18026}, {17074, 44426}, {17751, 17925}, {31623, 51662}, {36123, 64825}, {37558, 57215}, {38344, 52938}, {40149, 57125}
X(66512) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 56188}, {19, 56194}, {27, 65275}, {28, 65260}, {92, 56252}, {572, 1331}, {1474, 59006}, {2501, 51870}, {2975, 1332}, {6591, 34434}, {7649, 2051}, {11109, 190}, {11998, 521}, {14829, 4561}, {17074, 6516}, {17496, 69}, {17751, 52609}, {17924, 54121}, {17925, 20028}, {20986, 906}, {21173, 63}, {23187, 394}, {24237, 4025}, {34589, 6332}, {36123, 64824}, {37558, 65233}, {38344, 57241}, {40624, 35518}, {43925, 52150}, {46107, 57905}, {51662, 1214}, {52139, 4574}, {53566, 525}, {55323, 23067}, {55362, 23113}, {57091, 345}, {57125, 1812}, {57200, 53083}, {57244, 304}, {58339, 3692}

X(66513) = ORTHOLOGIC CONJUGATE OF X(6) WRT ABC, EXCENTRAL

Barycentrics    a^2*(b - c)*(a*(a - b - c) - 2*b*c) : :
X(66513) = 2X[6]-3X[3063] = 4X[6]-3X[20980] = X[6]-3X[21007] = 7X[6]-6X[39521] = 9X[1643]-8X[54250] = 7X[3063]-4X[39521] = X[20980]-4X[21007] = 7X[20980]-8X[39521] = 7X[21007]-2X[39521]

Contributed by Peter Moses, Dec 14, 2024.

X(66513) lies on these lines: {1,54249}, {6,513}, {111,2711}, {187,237}, {514,4435}, {523,4501}, {525,4976}, {650,1734}, {652,10581}, {654,17425}, {661,6004}, {693,24285}, {739,14665}, {826,48277}, {832,48022}, {840,59049}, {1019,1429}, {1480,41162}, {1499,4773}, {2291,12032}, {2724,32726}, {3247,21348}, {3287,3667}, {3709,48340}, {3723,48302}, {3731,21390}, {3800,48276}, {4079,61036}, {4132,63785}, {4378,16971}, {4380,57148}, {4394,48331}, {4724,45755}, {4776,37680}, {4782,16514}, {4784,16782}, {4785,23597}, {4839,29302}, {4979,22383}, {6185,30804}, {6586,48306}, {6590,48305}, {7180,51652}, {7927,48275}, {7950,50482}, {8693,35280}, {13401,36054}, {14996,47763}, {14997,47759}, {16489,21786}, {16785,48324}, {17023,23828}, {17117,20906}, {17494,53335}, {20295,24601}, {20963,54253}, {21005,50500}, {21127,65102}, {21261,30836}, {22155,48616}, {23892,41436}, {28478,58773}, {36274,48352}, {37633,47762}, {37675,47804}, {46385,50539}, {48026,48586}, {49293,65097}

X(66513) = midpoint of X(17494) and X(53335)
X(66513) = reflection of X(i) in X(j) for these {i,j}: {649, 8659}, {693, 24285}, {3063, 21007}, {4378, 24286}, {20980, 3063}, {24290, 650}
X(66513) = isogonal conjugate of X(32041)
X(66513) = isogonal conjugate of the anticomplement of X(61076)
X(66513) = isogonal conjugate of the isotomic conjugate of X(4762)
X(66513) = X(i)-Ceva conjugate of X(j) for these (i,j): {100, 40732}, {3423, 3271}, {4784, 54279}, {8693, 6}, {35280, 37580}
X(66513) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32041}, {2, 37138}, {37, 51563}, {75, 8693}, {99, 60677}, {100, 27475}, {101, 59255}, {190, 1002}, {644, 62784}, {651, 60668}, {658, 59269}, {664, 40779}, {668, 2279}, {672, 53227}, {673, 63743}, {919, 63231}, {1461, 59260}, {3263, 36138}, {3699, 42290}, {3799, 63882}, {3939, 62946}, {3952, 42302}, {4033, 51443}, {4554, 60673}, {35338, 42310}, {36086, 62622}, {59193, 65195}
X(66513) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 32041}, {206, 8693}, {1015, 59255}, {2276, 4505}, {8054, 27475}, {32664, 37138}, {35508, 59260}, {38980, 63231}, {38986, 60677}, {38989, 62622}, {38991, 60668}, {39012, 3263}, {39025, 40779}, {40589, 51563}, {40617, 62946}, {55053, 1002}, {55059, 321}, {61076, 76}, {62554, 53227}
X(66513) = crosspoint of X(i) and X(j) for these (i,j): {6, 8693}, {100, 14621}, {651, 52013}, {54440, 60721}
X(66513) = crosssum of X(i) and X(j) for these (i,j): {2, 4762}, {37, 4824}, {390, 650}, {513, 2276}, {514, 29571}, {518, 33570}, {3700, 4733}, {3797, 62552}
X(66513) = crossdifference of every pair of points on line {2, 210}
X(66513) = barycentric product X(i)*X(j) for these {i,j}: {1, 4724}, {6, 4762}, {57, 45755}, {58, 4804}, {244, 54440}, {513, 1001}, {514, 2280}, {522, 1471}, {647, 31926}, {649, 4384}, {650, 5228}, {657, 42309}, {661, 60721}, {663, 40719}, {665, 63236}, {667, 4441}, {693, 60722}, {739, 45338}, {798, 60735}, {840, 45322}, {1019, 59207}, {1893, 23189}, {1919, 21615}, {2223, 63221}, {2310, 65187}, {3063, 60720}, {3669, 37658}, {3696, 3733}, {3737, 42289}, {3886, 43924}, {3900, 59242}, {4044, 57129}, {4702, 23345}, {6185, 33570}, {6591, 23151}, {8693, 61076}, {28809, 57181}, {54251, 56705}, {58322, 59217}
X(66513) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 32041}, {31, 37138}, {32, 8693}, {58, 51563}, {105, 53227}, {513, 59255}, {649, 27475}, {663, 60668}, {665, 62622}, {667, 1002}, {798, 60677}, {1001, 668}, {1471, 664}, {1919, 2279}, {2223, 63743}, {2254, 63231}, {2280, 190}, {3063, 40779}, {3669, 62946}, {3675, 63223}, {3696, 27808}, {3789, 4505}, {3900, 59260}, {4384, 1978}, {4441, 6386}, {4724, 75}, {4762, 76}, {4804, 313}, {5228, 4554}, {8641, 59269}, {31926, 6331}, {33570, 4437}, {37658, 646}, {40719, 4572}, {40732, 3799}, {42309, 46406}, {43924, 62784}, {45338, 35543}, {45755, 312}, {54440, 7035}, {57129, 42302}, {57181, 42290}, {59207, 4033}, {59242, 4569}, {60721, 799}, {60722, 100}, {60735, 4602}, {63236, 36803}
X(66513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 663, 665}, {649, 8632, 667}, {667, 8632, 8657}, {4790, 7252, 57181}

X(66514) = ORTHOLOGIC CONJUGATE OF X(9) WRT ABC, EXCENTRAL

Barycentrics    a*(b - c)*(a^3 - 3*a^2*b + 3*a*b^2 - b^3 - 3*a^2*c - b^2*c + 3*a*c^2 - b*c^2 - c^3) : :
X(66514) = X[885]-3X[47804] = 3X[1635]-X[45755] = 3X[30234]-2X[45695] = 5X[27013]-X[53357] = 2X[40551]-3X[47803]

Contributed by Peter Moses, Dec 14, 2024.

X(66514) lies on these lines: {2,30804}, {9,513}, {104,43079}, {241,514}, {392,3250}, {647,3800}, {649,3309}, {667,3900}, {812,54264}, {885,40127}, {1024,42884}, {1436,61040}, {1635,14077}, {2256,21007}, {2291,2725}, {2516,48344}, {2530,47966}, {2726,59049}, {2826,47766}, {3239,59590}, {3667,4130}, {3887,42322}, {4521,59979}, {4790,42325}, {5316,47760}, {5744,47762}, {8659,48329}, {8732,24002}, {9001,22108}, {24562,48060}, {27013,53357}, {40551,47803}, {48026,48591}, {57167,62775}

X(66514) = midpoint of X(i) and X(j) for these {i,j}: {4130, 17410}, {43042, 47890}
X(66514) = complement of X(30804)
X(66514) = complement of the isotomic conjugate of X(37223)
X(66514) = X(i)-complementary conjugate of X(j) for these (i,j): {37223, 2887}, {39959, 21252}, {52013, 17059}
X(66514) = crosspoint of X(i) and X(j) for these (i,j): {2, 37223}, {100, 39273}
X(66514) = crosssum of X(i) and X(j) for these (i,j): {513, 40131}, {650, 3242}
X(66514) = crossdifference of every pair of points on line {55, 614}
X(66514) = {X(650),X(665)}-harmonic conjugate of X(905)

X(66515) = X(1)X(6)∩X(2)X(165)

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

See Tran Viet Hung and Juan José Isach Mayo, euclid 7416.

X(66515) lies on these lines: {1, 6}, {2, 165}, {3, 2951}, {5, 10268}, {7, 1125}, {8, 25101}, {10,390}, {21, 3062}, {31, 17022}, {35, 16293}, {36, 9814}, {40,5806}, {46, 25542}, {55, 7308}, {56, 60937}, {57, 3683}, {63,5284}, {100, 36835}, {104, 46947}, {105, 14439}, {140, 10270}, {142,3624}, {144, 3616}, {145, 43179}, {191, 60974}, {200, 1621}, {210,10389}, {214, 1156}, {333, 35613}, {344, 3883}, {354, 3929}, {355,61511}, {386, 4343}, {406, 1890}, {442, 42356}, {443, 63413}, {452,5691}, {474, 11495}, {480, 3295}, {496, 6067}, {515, 5817}, {517,16857}, {519, 5686}, {527, 17561}, {528, 19875}, {551, 5850}, {612,62875}, {614, 46901}, {673, 4432}, {740, 16833}, {748, 968}, {846,5272}, {936, 4326}, {938, 18249}, {943, 42015}, {946, 5759}, {962,17554}, {971, 3576}, {978, 4335}, {993, 4321}, {997, 5785}, {999,36973}, {1011, 35289}, {1282, 28345}, {1319, 60909}, {1376,31508}, {1385, 5779}, {1387, 6068}, {1420, 8581}, {1445,3339}, {1456, 59215}, {1471, 4328}, {1479, 1698}, {1519,6878}, {1697, 15837}, {1707, 26102}, {1709, 10857}, {1721, 56775}, {1738, 31183}, {1750, 13615}, {1764, 16345}, {1768,11407}, {1836, 41867}, {2093, 8257}, {2293, 56809}, {2346,4866}, {2478, 7989}, {2551, 51784}, {2646, 60910}, {2771,18443}, {2801, 16858}, {2802, 9623}, {2975, 60966}, {3059,5044}, {3158, 3740}, {3174, 5506}, {3219, 4666}, {3306,62838}, {3333, 31445}, {3336, 60985}, {3337, 60968}, {3338,60990}, {3340, 41712}, {3358, 7992}, {3485, 52819}, {3486,10392}, {3487, 61003}, {3523, 43151}, {3579, 16853}, {3601,5696}, {3612, 5784}, {3622, 17120}, {3625, 12630}, {3632,24393}, {3634, 30332}, {3636, 60983}, {3647, 16133}, {3649,63277}, {3663,16020}, {3671, 60939}, {3678, 34784}, {3679,5853}, {3681, 62856}, {3685, 4384}, {3715, 3748}, {3720,62812}, {3729, 16823}, {3742, 3928}, {3744, 7322}, {3745,25430}, {3746, 6600}, {3753, 17542}, {3755, 37650}, {3757,30568}, {3795, 36634}, {3814, 7679}, {3826, 4187}, {3828,38201}, {3868, 20116}, {3869, 30329}, {3870, 27065}, {3874,11025}, {3876, 30628}, {3877, 11224}, {3878, 7672}, {3886,17277}, {3897, 17543}, {3916, 60955}, {3923, 25590}, {3925,9580}, {3984, 62870}, {4002, 63138}, {4189, 43178}, {4208,51118}, {4297, 11106}, {4307, 29571}, {4313, 12447}, {4349,5308}, {4356, 5222}, {4383, 37553}, {4385, 56085}, {4413, 35445}, {4414, 62695}, {4421, 58451}, {4640, 5437}, {4652,5550}, {4668, 41709}, {4669, 50839}, {4676, 10436}, {4677, 59414}, {4679, 5219}, {4684, 54280}, {4759, 60960}, {4853,5260}, {4855, 25722}, {4859, 24248}, {4888, 24695}, {4900,55920}, {4915, 9708}, {5018, 21446}, {5047, 5250}, {5085,35273}, {5131, 28534}, {5159, 47470}, {5204, 31391}, {5218, 5316}, {5231, 54357}, {5257, 5819}, {5267, 8544}, {5268, 8616}, {5269, 44307}, {5273, 11019}, {5281, 20103}, {5287, 17127}, {5290, 8232}, {5296, 19868}, {5325, 24477}, {5432, 20196}, {5438, 15587}, {5439, 54290}, {5493, 11024}, {5541,6594}, {5563, 60965}, {5587, 34746}, {5603, 21168}, {5657,38130}, {5726, 61015}, {5735, 41012}, {5745, 26105}, {5750,41325}, {5762, 5886}, {5766, 19843}, {5790, 38179}, {5805,6675}, {5832, 23708}, {5833, 26363}, {5843, 38028}, {5846, 41313}, {5847, 29573}, {5851, 34123}, {5852, 51110}, {5856,16173}, {5857, 37701}, {5880, 7483}, {5918, 10855}, {6173,15670}, {6210, 16850}, {6684, 17559}, {6690, 30827}, {6702,20119}, {6744, 54398}, {6763, 61005}, {6872, 24564}, {6883,30503}, {6910, 16209}, {6986, 12565}, {7162, 42470}, {7262,62820}, {7288, 60992}, {7489, 37611}, {7611, 38029}, {7671, 10176}, {7676, 25440}, {7678, 25639}, {7688, 50204}, {7982,16860}, {8056, 17596}, {8582, 9588}, {8728, 41869}, {9589,17552}, {9614, 19854}, {9624, 20330}, {9791, 17304}, {9843,62775}, {9955, 31671}, {10004, 17106}, {10058, 15015}, {10165, 21151}, {10175, 38149}, {10198, 60943}, {10200, 61019}, {10246,51516}, {10383, 30223}, {10434, 16058}, {10442, 10455}, {10578,21060}, {10624, 19855}, {10856, 25514}, {10882, 28383}, {10886,37370}, {10916, 31446}, {11113, 59389}, {11230, 38107}, {11231,38121}, {11375, 60883}, {11376, 60919}, {11496, 61122}, {11522,60959}, {11531, 16859}, {12437, 45085}, {12512, 17580}, {12526,54392}, {12669, 31803}, {12699, 50205}, {12717, 16849}, {12730,15863}, {13405, 18228}, {13745, 29181}, {15006, 63146}, {15726,16370}, {15733, 59337}, {15808, 30340}, {15908, 17527}, {15950,61007}, {16112, 17614}, {16189, 17544}, {16287, 61124}, {16357,62320}, {16367, 35291}, {16408, 35242}, {16593, 32784}, {16825, 17151}, {16834, 27484}, {16842, 63469}, {16863, 31663}, {16865,19861}, {17021, 30653}, {17123, 17594}, {17125, 54390}, {17140,25734}, {17185, 35621}, {17259, 49484}, {17261, 49446}, {17282,24723}, {17284, 50295}, {17308, 20533}, {17313, 28570}, {17349,49495}, {17355, 39581}, {17556, 61264}, {17582, 31730}, {17588,43169}, {17687, 48900}, {17718, 31142}, {18222, 37551}, {18393,60978}, {18398, 58564}, {18421, 37787}, {18481, 50243}, {19225,31996}, {19862, 60996}, {19863, 25513}, {19872, 61001}, {19876,31159}, {19883, 59374}, {20059, 43180}, {20835, 41860}, {21154,21164}, {21165, 59386}, {21616, 21617}, {21628, 37423}, {21677,37723}, {22793, 50726}, {24199, 24280}, {24325, 51052}, {24349,25728}, {24392, 49736}, {24541, 60895}, {24646, 52790}, {24647,52791}, {24697, 47595}, {24703, 25525}, {24929, 42014}, {24953,50443}, {24987, 37714}, {25557, 60933}, {26446, 38113}, {27475,50300}, {27549, 49466}, {28363, 45047}, {28503, 36911}, {28516,55998}, {29576, 41845}, {29598, 38187}, {29602, 50284}, {29628,62392}, {30144, 30284}, {30147, 60954}, {30315, 37162}, {30318,51111}, {30326, 54348}, {30343, 62874}, {30571, 39980}, {31156,34628}, {31162, 50202}, {31249, 59491}, {31672, 50241}, {33761,62833}, {35595, 61155}, {36277, 37633}, {37224, 59340}, {37244,59320}, {37270, 41853}, {37569, 54203}, {37617, 46943}, {37712,38154}, {38046, 41312}, {38047, 49740}, {38049, 59405}, {38101,53620}, {38126, 63143}, {38137, 61269}, {38158, 59387}, {38194,59406}, {38216, 59415}, {38217, 59416}, {38318, 54447}, {39586,49482}, {40719, 52511}, {40966, 63511}, {41228, 62829}, {41857,51706}, {41859, 50206}, {43173, 56769}, {44675, 60997}, {45043,59419}, {45834, 56203}, {48830, 59408}, {49451, 60731}, {49511,51190}, {49598, 58398}, {50111, 51053}, {50834, 51103}, {50835,51071}, {50837, 51108}, {50838, 51093}, {50840, 51098}, {50996,51005}, {50997, 51003}, {51006, 51191}, {51066, 51102}, {51099,51105}, {51409, 60982}, {51700, 61596}, {52050, 63264}, {52457,60923}, {52682, 61595}, {53052, 63137}, {53058, 60935}, {54422,61024}, {60949, 62858}, {63292, 63384}

X(66515) = midpoint of X(i) and X(j) for these {i,j}: {2, 52653}, {9, 38316}, {165, 24644}, {390, 59413}, {5603,21168}, {5686, 8236}, {5692, 41861}, {6172, 11038}, {10246,51516}, {10384, 46917}, {15015, 51768}, {30331, 38210}, {38052,50836}, {38054, 51090}, {38057, 47357}
X(66515) = reflection of X(i) in X(j) for these {i,j}: {1, 38316}, {2, 38059}, {7, 38054}, {8, 38210}, {165, 21153}, {1699,38037}, {3576, 38031}, {3679, 38057}, {4677, 59414}, {5587, 38108}, {5657, 38130}, {5790, 38179}, {5886, 38043}, {11038,551}, {16173, 38060}, {16475, 38048}, {21151, 10165}, {25055,38025}, {26446, 38113}, {37701, 38061}, {37712, 38154}, {38024,25055}, {38030, 38028}, {38036, 5886}, {38052, 2}, {38054,1125}, {38057, 60986}, {38107, 11230}, {38121, 11231}, {38137, 61269}, {38149, 10175}, {38151, 10171}, {38201, 3828}, {38316,1001}, {41861, 10177}, {45043, 59419}, {50836, 52653}, {53620, 38101}, {59372, 38053}, {59374, 19883}, {59385, 3817}, {59387,38158}, {59405, 38049}, {59406, 38194}, {59412, 38204}, {59413, 10}, {59415, 38216}, {59416, 38217}, {63143, 38126}

X(66516) = ORTHOLOGIC CONJUGATE OF X(7) WRT ABC, EXCENTRAL

Barycentrics    (b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - 3*a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3) : :
X(66516) = 2X[3126]-3X[47824] = 3X[4379]-2X[54264] = 4X[40551]-3X[47821]

Let T = (t1:t2:t3), P = (p1:p2:p3), Q(T, P) = orthologic conjugate of P wrt ABC and antipedal triangle of T. Then

Q(T, P) = c^2*(a^2 - b^2 + c^2)*p1*p2*t1 - b^2*(a^2 + b^2 - c^2)*p1*p3*t1 - 2*a^2*(b - c)*(b + c)*p2*p3*t1 - c^2*(a^2 + b^2 - c^2)*p1^2*t2 - 2*a^2*c^2*p1*p2*t2 - a^2*(a^2 + b^2 - c^2)*p1*p3*t2 - 2*a^4*p2*p3*t2 + b^2*(a^2 - b^2 + c^2)*p1^2*t3 + a^2*(a^2 - b^2 + c^2)*p1*p2*t3 + 2*a^2*b^2*p1*p3*t3 + 2*a^4*p2*p3*t3::

Contributed by Peter Moses, Dec 15, 2024.

X(66516) lies on these lines: {7,513}, {239,514}, {663,3676}, {693,3309}, {1444,57246}, {1459,17096}, {2320,14154}, {3126,47824}, {3887,47780}, {3900,17166}, {4040,20520}, {4367,8638}, {4374,57091}, {4379,54264}, {4435,21104}, {4453,45695}, {4897,6362}, {5744,47762}, {6003,46402}, {14330,28878}, {20295,42325}, {21390,38379}, {23828,35355}, {31019,47759}, {40551,47821}, {43041,48151}, {43349,54440}, {43932,59936}, {47694,50556}

X(66516) = midpoint of X(7192) and X(53357)
X(66516) = X(43349)-anticomplementary conjugate of X(69)
X(66516) = X(6183)-Ceva conjugate of X(7)
X(66516) = X(692)-isoconjugate of X(60227)
X(66516) = X(1086)-Dao conjugate of X(60227)
X(66516) = crosspoint of X(190) and X(42310)
X(66516) = crossdifference of every pair of points on line {42, 8012}
X(66516) = barycentric product X(i)*X(j) for these {i,j}: {514, 14828}, {693, 62797}, {4025, 37389}
X(66516) = barycentric quotient X(i)/X(j) for these {i,j}: {514, 60227}, {14828, 190}, {37389, 1897}, {62797, 100}

X(66517) = ORTHOLOGIC CONJUGATE OF X(8) WRT ABC, EXCENTRAL

Barycentrics    (b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 9*a^2*b*c - 5*a*b^2*c + b^3*c - a^2*c^2 - 5*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :
X(66517) = 5X[3616]-4X[59972] = 2X[6615]-3X[48243] = 2X[24457]-3X[48246]

Contributed by Peter Moses, Dec 15, 2024.

X(66517) lies on these lines: {8,513}, {522,4318}, {900,4057}, {2827,20293}, {3616,59972}, {3667,4063}, {4394,57168}, {4397,30198}, {4926,48330}, {4962,7253}, {6615,48243}, {8689,53343}, {23838,25005}, {24457,48246}, {28183,48304}, {34758,48390}, {43728,63163}, {56940,61040}

X(66517) = X(56145)-anticomplementary conjugate of X(33650)
X(66517) = crossdifference of every pair of points on line {2347, 17053}

X(66518) = ORTHOLOGIC CONJUGATE OF X(10) WRT ABC, EXCENTRAL

Barycentrics    (b - c)*(2*a^4 + a^3*b + a*b^3 + a^3*c + 2*a^2*b*c - 2*a*b^2*c + b^3*c - 2*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :
X(66518) = X[6615]-3X[47817] = 3X[47845]-X[48293]

Contributed by Peter Moses, Dec 15, 2024.

X(66518) lies on these lines: {10,513}, {514,4581}, {522,1324}, {523,48328}, {3667,6211}, {4063,57091}, {4448,59726}, {4807,6003}, {4962,48063}, {6006,50344}, {6615,47817}, {7081,47805}, {16824,48320}, {30115,48307}, {33138,47824}, {47845,48293}, {48283,49682}

X(66518) = midpoint of X(i) and X(j) for these {i,j}: {4063, 57091}, {4581, 50346}

X(66519) = ORTHOLOGIC CONJUGATE OF X(11) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66519) lies on these lines: {11,513}, {108,43933}, {514,11715}, {522,46684}, {659,14667}, {676,1421}, {900,1768}, {2804,13205}, {2826,22775}, {2829,42455}, {6264,6366}, {11700,21201}, {15252,24457}, {53298,53878}, {53321,64440}

X(66519) = crossdifference of every pair of points on line {2427, 13006}

X(66520) = ORTHOLOGIC CONJUGATE OF X(20) WRT ABC, EXCENTRAL

Barycentrics    (a - b - c)*(b - c)*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - a*b^4*c + b^5*c - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*a^3*c^3 - 2*b^3*c^3 + a^2*c^4 - a*b*c^4 + a*c^5 + b*c^5) : :
X(66520) = 3X[2]-4X[59973] = 5X[3091]-4X[16228] = 7X[3832]-8X[44923] = 2X[8062]-3X[14414] = 2X[14304]-3X[48243] = 2X[42455]-3X[48165]

Contributed by Peter Moses, Dec 15, 2024.

X(66520) lies on these lines: {2,44426}, {20,513}, {22,47805}, {100,44710}, {280,61040}, {347,24002}, {521,17496}, {522,663}, {523,2071}, {1331,41906}, {1370,48164}, {2804,65099}, {3091,16228}, {3151,47759}, {3153,62492}, {3832,44923}, {4025,8058}, {4296,48281}, {4391,57101}, {6360,45290}, {7361,63744}, {7396,44429}, {7488,48383}, {7560,47763}, {8062,14414}, {9538,48302}, {10565,47804}, {14304,48243}, {16049,57246}, {20222,64362}, {20298,47995}, {27086,44428}, {28623,57108}, {37437,42769}, {42455,48165}, {57072,57089}

X(66520) = midpoint of X(20294) and X(59926)
X(66520) = reflection of X(i) in X(j) for these {i,j}: {4391, 57101}, {7253, 57241}, {44426, 59973}
X(66520) = anticomplement of X(44426)
X(66520) = anticomplement of the isogonal conjugate of X(36059)
X(66520) = anticomplement of the isotomic conjugate of X(6516)
X(66520) = isotomic conjugate of the polar conjugate of X(57166)
X(66520) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3, 33650}, {48, 37781}, {59, 20293}, {73, 3448}, {77, 21293}, {108, 5906}, {109, 4}, {163, 92}, {184, 39351}, {222, 150}, {255, 34188}, {603, 149}, {651, 21270}, {653, 317}, {664, 11442}, {692, 5942}, {906, 329}, {1214, 21294}, {1262, 46400}, {1331, 3436}, {1332, 21286}, {1409, 21221}, {1414, 20242}, {1415, 5905}, {1461, 56927}, {1813, 69}, {2149, 4391}, {4558, 20245}, {4565, 17220}, {4575, 3869}, {4587, 54113}, {6516, 6327}, {6517, 1370}, {23067, 1330}, {24027, 521}, {32651, 52673}, {32656, 144}, {32660, 2}, {32661, 63}, {32669, 48380}, {32674, 6515}, {32739, 30694}, {36040, 5081}, {36059, 8}, {40152, 13219}, {44717, 20295}, {52378, 850}, {52411, 4440}, {52610, 2893}, {65164, 315}, {65179, 21279}, {65233, 21287}, {65296, 21285}, {65299, 21277}, {65300, 21276}
X(66520) = X(6516)-Ceva conjugate of X(2)
X(66520) = X(108)-isoconjugate of X(57672)
X(66520) = X(i)-Dao conjugate of X(j) for these (i,j): {38983, 57672}, {40626, 57838}
X(66520) = crosspoint of X(664) and X(31623)
X(66520) = crosssum of X(i) and X(j) for these (i,j): {513, 64522}, {663, 1409}
X(66520) = crossdifference of every pair of points on line {800, 1400}
X(66520) = barycentric product X(i)*X(j) for these {i,j}: {69, 57166}, {412, 6332}, {3562, 4391}, {35518, 38860}
X(66520) = barycentric quotient X(i)/X(j) for these {i,j}: {412, 653}, {652, 57672}, {3562, 651}, {6332, 57838}, {38860, 108}, {57166, 4}
X(66520) = {X(44426),X(59973)}-harmonic conjugate of X(2)

X(66521) = ORTHOLOGIC CONJUGATE OF X(25) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66521) lies on these lines: {2,44426}, {4,44429}, {25,513}, {230,231}, {427,16228}, {905,8760}, {1829,48332}, {3063,45786}, {3737,51644}, {4232,47805}, {4233,57246}, {6353,47804}, {6995,48164}, {6997,44923}, {7713,48335}, {11363,48327}, {16231,44432}, {20621,50933}, {39534,47799}, {44428,47798}, {47757,54239}, {47797,59915}

X(66521) = {X(16228),X(47802)}-harmonic conjugate of X(427)

X(66522) = ORTHOLOGIC CONJUGATE OF X(31) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66522) lies on these lines: {31,513}, {58,48320}, {81,48281}, {171,47824}, {238,47821}, {595,48352}, {649,834}, {661,22384}, {663,22160}, {748,47822}, {750,47823}, {985,4817}, {1451,43052}, {1468,4378}, {1621,48307}, {2254,6003}, {2280,3063}, {2605,65703}, {3915,4775}, {4017,18108}, {4581,48106}, {5276,21390}, {17124,48216}, {17125,48197}, {20949,33295}, {21761,47918}, {21791,57171}, {23752,48101}, {48283,62821}, {48302,62849}

X(66522) = X(100)-isoconjugate of X(45964)
X(66522) = X(8054)-Dao conjugate of X(45964)
X(66522) = crossdifference of every pair of points on line {10, 17451}
X(66522) = barycentric product X(i)*X(j) for these {i,j}: {514, 5135}, {649, 37670}
X(66522) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 45964}, {5135, 190}, {37670, 1978}

X(66523) = ORTHOLOGIC CONJUGATE OF X(37) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66523) lies on these lines: {37,513}, {649,50355}, {650,667}, {661,4367}, {665,830}, {798,38469}, {814,6590}, {1019,24290}, {3309,4790}, {3669,14349}, {3700,6002}, {3777,4813}, {4140,4581}, {4435,48322}, {4832,35057}, {16785,48324}, {21438,21613}, {23882,47678}, {28475,47881}, {29051,48276}, {29170,48269}, {29246,49293}, {30836,31250}, {32779,47762}, {32849,47763}, {48022,50353}, {50342,50541}

X(66523) = crosspoint of X(100) and X(40776)
X(66523) = crosssum of X(513) and X(40750)
X(66523) = crossdifference of every pair of points on line {238, 846}

X(66524) = ORTHOLOGIC CONJUGATE OF X(44) WRT ABC, EXCENTRAL

Barycentrics    a*(5*a - b - c)*(b - c) : :
X(66524) = 2X[649]+X[650] = 5X[649]+X[661] = 5X[649]+4X[2516] = X[649]+2X[4394] = 4X[649]-X[4790] = 11X[649]+X[4813] = 3X[649]+X[4893] = 7X[649]-X[4979] = 4X[649]+X[47777] = 17X[649]+X[48019] = 8X[649]+X[48026] = 7X[649]+X[48544], and many others

Contributed by Peter Moses, Dec 15, 2024.

X(66524) lies on these lines: {2,6008}, {44,513}, {55,667}, {57,1022}, {100,6017}, {105,2384}, {165,3309}, {354,4083}, {522,4773}, {523,47768}, {665,29350}, {812,45313}, {824,45679}, {891,54249}, {900,4944}, {901,52985}, {905,48011}, {918,4786}, {1002,43928}, {1019,47921}, {1639,3667}, {2291,2718}, {2487,48398}, {2490,48269}, {2527,6590}, {2529,48275}, {2786,47770}, {3004,48605}, {3063,21786}, {3700,4962}, {3748,48330}, {3798,47890}, {3835,45675}, {3887,42322}, {3900,58140}, {4025,48095}, {4041,58143}, {4106,4928}, {4369,48125}, {4380,4885}, {4705,58146}, {4729,58138}, {4730,58141}, {4750,30520}, {4752,6014}, {4762,47762}, {4763,4785}, {4765,28147}, {4776,44567}, {4777,14435}, {4778,47876}, {4820,28221}, {4822,58180}, {4834,6050}, {4841,28229}, {4860,14421}, {4895,58136}, {4897,11068}, {4905,53056}, {4926,4984}, {4932,47962}, {4940,26853}, {4976,28161}, {4977,47883}, {5273,20317}, {5338,18344}, {6002,48559}, {6006,52593}, {6009,21183}, {6084,47758}, {7192,47920}, {7234,22314}, {7653,47672}, {7658,23729}, {8027,9010}, {8678,58144}, {10196,28867}, {11051,61238}, {14422,48332}, {14425,28217}, {14838,48128}, {15931,39227}, {16892,48132}, {17069,47960}, {17494,48133}, {20295,31287}, {21196,49281}, {21301,26040}, {23813,24924}, {23880,48565}, {23882,48566}, {25577,35310}, {25666,48016}, {26777,48107}, {27115,48079}, {27486,28894}, {28175,45745}, {28195,47878}, {28205,47873}, {28213,49293}, {28250,28255}, {28840,48560}, {28846,47884}, {28882,45674}, {28898,47771}, {29078,48219}, {29150,48561}, {29178,45664}, {29328,47803}, {29370,48222}, {31150,47763}, {31207,48114}, {38347,61674}, {39386,47764}, {42316,59239}, {43067,48008}, {43079,53608}, {45326,47786}, {46919,47756}, {47755,47892}, {47785,47880}, {47882,48558}, {47914,48000}, {47915,48003}, {47919,48101}, {47965,48064}, {48099,58179}, {48111,63207}, {48189,48220}, {48322,58142}, {48327,58139}, {48329,63211}, {48557,53333}, {50500,50510}, {50501,50512}, {50504,50515}, {50507,58182}, {55163,59921}

X(66524) = isogonal conjugate of X(65235)
X(66524) = X(39960)-complementary conjugate of X(116)
X(66524) = X(i)-Ceva conjugate of X(j) for these (i,j): {4752, 16666}, {56150, 3248}, {65235, 1}
X(66524) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65235}, {2, 6014}, {6, 53659}, {100, 39963}, {101, 36588}, {109, 56075}, {190, 41436}, {651, 4900}, {662, 56159}, {692, 40029}, {901, 36915}, {4638, 36924}
X(66524) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65235}, {9, 53659}, {11, 56075}, {1015, 36588}, {1084, 56159}, {1086, 40029}, {8054, 39963}, {32664, 6014}, {38979, 36915}, {38991, 4900}, {52593, 693}, {55053, 41436}
X(66524) = crosspoint of X(i) and X(j) for these (i,j): {1, 65235}, {89, 100}
X(66524) = crosssum of X(i) and X(j) for these (i,j): {45, 513}, {514, 5316}, {650, 7962}
X(66524) = crossdifference of every pair of points on line {1, 3689}
X(66524) = barycentric product X(i)*X(j) for these {i,j}: {1, 6006}, {75, 8656}, {89, 52593}, {513, 3241}, {514, 16670}, {522, 13462}, {649, 30829}, {650, 64142}, {1019, 4029}, {3669, 62706}, {4982, 47947}, {7192, 21870}, {17924, 23073}
X(66524) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 53659}, {6, 65235}, {31, 6014}, {512, 56159}, {513, 36588}, {514, 40029}, {649, 39963}, {650, 56075}, {663, 4900}, {667, 41436}, {1635, 36915}, {3241, 668}, {3251, 36924}, {4029, 4033}, {4982, 65161}, {6006, 75}, {8656, 1}, {13462, 664}, {16670, 190}, {21870, 3952}, {23073, 1332}, {30829, 1978}, {52593, 4671}, {62706, 646}, {64142, 4554}
X(66524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 650, 4790}, {649, 4394, 650}, {650, 4790, 48026}, {654, 17410, 4790}, {661, 2516, 650}, {667, 50499, 4162}, {4106, 31286, 31250}, {4380, 27013, 4885}, {4834, 6050, 50508}, {4897, 11068, 48087}, {17069, 48060, 47960}, {26853, 31209, 4940}, {50501, 50512, 50517}, {50504, 58145, 50515}

X(66525) = ORTHOLOGIC CONJUGATE OF X(58) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66525) lies on these lines: {1,57246}, {58,513}, {86,23790}, {284,21007}, {512,1326}, {656,3737}, {757,57059}, {1019,4017}, {4040,65575}, {4653,48307}, {4658,48281}, {7180,7252}, {42741,48306}, {50346,57093}, {51646,57239}, {57125,57148}

X(66525) = isogonal conjugate of the isotomic conjugate of X(57248)
X(66525) = X(109)-Ceva conjugate of X(58)
X(66525) = X(i)-isoconjugate of X(j) for these (i,j): {1018, 55090}, {4551, 55091}
X(66525) = X(4560)-Dao conjugate of X(35519)
X(66525) = crosspoint of X(i) and X(j) for these (i,j): {109, 55101}, {4565, 52558}
X(66525) = crosssum of X(i) and X(j) for these (i,j): {522, 55091}, {523, 21674}, {3700, 8013}, {4171, 21704}
X(66525) = crossdifference of every pair of points on line {1213, 2294}
X(66525) = barycentric product X(i)*X(j) for these {i,j}: {1, 57189}, {6, 57248}, {57, 57093}, {81, 50346}, {109, 40625}, {110, 24224}, {651, 64416}, {1019, 5260}, {1509, 58302}, {3733, 55095}, {4560, 55101}, {7192, 55100}, {7252, 55096}
X(66525) = barycentric quotient X(i)/X(j) for these {i,j}: {3733, 55090}, {5260, 4033}, {7252, 55091}, {24224, 850}, {40625, 35519}, {50346, 321}, {55095, 27808}, {55100, 3952}, {55101, 4552}, {57093, 312}, {57189, 75}, {57248, 76}, {58302, 594}, {64416, 4391}
X(66525) = {X(57093),X(57189)}-harmonic conjugate of X(50346)

X(66526) = ORTHOLOGIC CONJUGATE OF X(59) WRT ABC, EXCENTRAL

Barycentrics    a^2*(a - b)*(a - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 3*a^2*b*c - a*b^2*c + 3*b^3*c - a^2*c^2 - a*b*c^2 - 6*b^2*c^2 + a*c^3 + 3*b*c^3) : :
X(66526) = 3X[59]-4X[692] = 2X[692]-3X[1618]

Contributed by Peter Moses, Dec 15, 2024.

X(66526) lies on these lines: {59,513}, {60,37019}, {100,2742}, {101,649}, {109,663}, {110,1019}, {517,7291}, {650,64616}, {840,59021}, {934,2720}, {999,1318}, {1155,11349}, {1290,58974}, {1292,6099}, {1878,57654}, {2078,8647}, {2222,53243}, {2283,57105}, {2340,5537}, {4511,6001}, {4588,53887}, {5057,6996}, {5091,5222}, {5126,26884}, {10426,37541}, {13589,14513}, {25268,39185}, {34921,58104}, {38674,38682}, {39026,48340}, {43344,65881}, {44858,61435}, {51682,52213}, {55380,64372}

X(66526) = reflection of X(59) in X(1618)
X(66526) = reflection of X(7291) in the anti-Orthic axis
X(66526) = isogonal conjugate of the anticomplement of X(52873)
X(66526) = crosspoint of X(840) and X(18771)
X(66526) = crosssum of X(i) and X(j) for these (i,j): {528, 3035}, {23757, 45884}, {62579, 65858}
X(66526) = crossdifference of every pair of points on line {1647, 17435}
X(66526) = barycentric product X(i)*X(j) for these {i,j}: {651, 53055}, {1275, 58370}
X(66526) = barycentric quotient X(i)/X(j) for these {i,j}: {53055, 4391}, {58370, 1146}

X(66527) = ORTHOLOGIC CONJUGATE OF X(63) WRT ABC, EXCENTRAL

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

Contributed by Peter Moses, Dec 15, 2024.

X(66527) lies on these lines: {63,513}, {522,693}, {649,6003}, {656,47785}, {1734,4560}, {2504,44409}, {3733,8646}, {4784,65401}, {4905,23887}, {8676,57184}, {13277,47823}, {21189,47798}, {44551,59753}, {50354,64917}, {53361,62811}

X(66527) = crossdifference of every pair of points on line {41, 3924}

X(66528) = ORTHOLOGIC CONJUGATE OF X(81) WRT ABC, EXCENTRAL

Barycentrics    a*(a + b)*(b - c)*(a + c)*(a^3 - 5*a*b^2 - 9*a*b*c - 3*b^2*c - 5*a*c^2 - 3*b*c^2) : :

Contributed by Peter Moses, Dec 15, 2024.

X(66528) lies on these lines: {21,48352}, {81,513}, {512,57189}, {661,1021}, {669,2106}, {1499,4560}, {1621,57246}, {2775,57227}, {4017,7203}, {4775,64415}, {4789,7253}, {4833,47827}, {5235,47821}, {5333,47824}, {47822,64425}, {48320,64377}, {51356,57059}

X(66528) = crossdifference of every pair of points on line {2650, 16589}

X(66529) = 99TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^6 + a^4*b^2 - 2*a^2*b^4 - b^6 + a^4*c^2 + 8*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 - c^6 : :
X(66529) = 3 X[2] - 4 X[13361], 3 X[2] + X[34603], 5 X[2] - 3 X[43957], 5 X[2] - X[52397], 5 X[2] + 3 X[62963], X[4] + 2 X[9825], 2 X[4] + X[31829], 3 X[4] + X[44458], 2 X[5] + X[6756], 5 X[5] + X[11819], 4 X[5] - X[12362], X[5] - 4 X[23411], 3 X[381] + X[38321], 3 X[428] + X[7667], 3 X[428] + 2 X[7734], X[428] + 2 X[10128], 2 X[428] + X[10691], 3 X[428] + 4 X[13361], 3 X[428] - X[34603], 5 X[428] + 3 X[43957], 5 X[428] + X[52397], 5 X[428] - 3 X[62963], 4 X[546] - X[13488], 2 X[546] + X[31833], 5 X[1656] + X[7553], and many others

See Antreas Hatzipolakis and Peter Moses, euclid 7426.

X(66529) lies on these lines: {2, 3}, {51, 3564}, {53, 10314}, {98, 54629}, {143, 31831}, {154, 14561}, {161, 61610}, {184, 18583}, {206, 597}, {230, 36412}, {262, 54496}, {264, 65063}, {275, 54709}, {343, 18358}, {394, 21850}, {519, 51719}, {524, 9969}, {528, 35652}, {539, 58545}, {542, 11746}, {612, 15171}, {614, 18990}, {801, 14492}, {1184, 18907}, {1196, 7745}, {1351, 14826}, {1352, 17810}, {1353, 9777}, {1495, 37649}, {1498, 9815}, {1503, 5943}, {1611, 7737}, {1899, 3066}, {1992, 6391}, {3060, 34380}, {3070, 8855}, {3071, 8854}, {3163, 15527}, {3167, 14853}, {3527, 6193}, {3796, 38110}, {3818, 13567}, {3819, 29181}, {3867, 19137}, {3917, 35283}, {3920, 15172}, {5012, 51732}, {5050, 11206}, {5093, 63174}, {5268, 6284}, {5272, 7354}, {5322, 15325}, {5345, 5433}, {5432, 7298}, {5462, 18914}, {5475, 34481}, {5480, 9306}, {5544, 14927}, {5640, 11245}, {5806, 31832}, {5907, 11745}, {6390, 16276}, {6688, 29012}, {7693, 34545}, {7767, 40022}, {8280, 42273}, {8281, 42270}, {8770, 65630}, {8780, 11427}, {9157, 57304}, {9300, 53420}, {9729, 16621}, {9827, 58484}, {10095, 13292}, {10110, 13142}, {10219, 29323}, {10311, 65809}, {10547, 42037}, {10601, 31383}, {11179, 64719}, {11402, 59399}, {11433, 18440}, {11542, 54362}, {11543, 54363}, {11550, 37648}, {11566, 12358}, {12134, 61713}, {13157, 61349}, {13394, 44082}, {13419, 64038}, {13568, 44870}, {14458, 37874}, {14615, 37671}, {15024, 16659}, {15045, 16658}, {15311, 46847}, {15435, 37491}, {15448, 58447}, {15466, 16264}, {16187, 51163}, {16654, 64100}, {16655, 64854}, {16656, 46850}, {17359, 49732}, {17811, 31670}, {17814, 31802}, {17825, 46264}, {18289, 42265}, {18290, 42262}, {18928, 62209}, {19125, 59373}, {19130, 23292}, {19139, 52077}, {19568, 52229}, {19583, 32826}, {19875, 34657}, {19883, 34633}, {20423, 37672}, {21969, 64062}, {23291, 51537}, {24981, 34565}, {25055, 34634}, {26864, 63085}, {27355, 61139}, {32269, 44106}, {32819, 57518}, {33586, 48876}, {33651, 59635}, {34612, 50126}, {34656, 53620}, {34668, 38314}, {35259, 38136}, {35264, 61690}, {35266, 64064}, {37636, 47582}, {39899, 63031}, {40179, 43136}, {40326, 53418}, {43588, 58531}, {43670, 60127}, {44683, 64095}, {44935, 54040}, {45089, 61607}, {45303, 61645}, {45968, 61657}, {47328, 63475}, {50675, 64781}, {51212, 62217}, {54836, 60125}, {59655, 63005}, {64919, 65393}

X(66529) = midpoint of X(i) and X(j) for these {i,j}: {2, 428}, {5, 13490}, {3575, 52069}, {7576, 34664}, {7667, 34603}, {12134, 61713}, {16654, 64100}, {21969, 64062}, {31833, 44804}, {38323, 62962}, {43957, 62963}, {44935, 54040}
X(66529) = reflection of X(i) in X(j) for these {i,j}: {2, 10128}, {6756, 13490}, {7667, 7734}, {7734, 13361}, {10127, 23410}, {10691, 2}, {13488, 44804}, {44804, 546}, {45298, 5943}
X(66529) = complement of X(7667)
X(66529) = anticomplement of X(7734)
X(66529) = orthocentroidal circle inverse of X(34609)
X(66529) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(46451)
X(66529) = pole of line {6, 34609} with respect to the Kiepert circumhyperbola
X(66529) = pole of line {185, 21850} with respect to the Jerabek circumhyperbola
X(66529) = pole of line {3574, 18583} with respect to the ABCHN
X(66529) = pole of line {1196, 10154} with respect to the ABCGK
X(66529) = pole of line {3589, 5907} with respect to the Jerabek circumhyperbola of the medial triangle
X(66529) = pole of line {12111, 44456} with respect to the Jerabek circumhyperbola of the anticomplementary triangle
X(66529) = pole of line {525, 55190} with respect to the Steiner inellipse
X(66529) = pole of line {523, 34609} with respect to the Yff hyperbola
X(66529) = pole of line {5650, 59553} with respect to the Thomson-Gibert-Moses hyperbola
X(66529) = pole of line {6, 34609} with respect to the BG KHO conic
X(66529) = pole of line {523, 34609} with respect to the orthocentroidal circle
X(66529) = pole of line {523, 46451} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(66529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 34609}, {2, 25, 10154}, {2, 3839, 62975}, {2, 6995, 34608}, {2, 7667, 7734}, {2, 7714, 9909}, {2, 10154, 6676}, {2, 21213, 34477}, {2, 26255, 62965}, {2, 34603, 7667}, {2, 34608, 3}, {2, 34609, 1368}, {2, 52397, 43957}, {2, 62963, 52397}, {2, 62964, 31152}, {2, 66371, 12100}, {4, 5020, 1368}, {4, 7398, 5020}, {4, 9825, 31829}, {5, 25, 6676}, {5, 6676, 11548}, {5, 6756, 12362}, {5, 7715, 3}, {5, 10154, 2}, {5, 13861, 21841}, {5, 21841, 63667}, {5, 37440, 140}, {5, 44233, 37942}, {5, 65376, 7395}, {22, 25, 37440}, {22, 37439, 140}, {22, 62937, 37439}, {23, 7499, 66380}, {23, 37990, 7499}, {25, 6676, 37897}, {25, 6995, 7715}, {25, 6997, 5}, {25, 7539, 7493}, {140, 10301, 37910}, {427, 1995, 6677}, {427, 6677, 5159}, {427, 7394, 546}, {428, 7667, 34603}, {428, 10128, 10691}, {428, 34658, 7540}, {428, 34664, 34659}, {468, 5133, 64852}, {468, 7533, 3850}, {546, 1995, 5159}, {546, 6677, 427}, {546, 31833, 13488}, {858, 37349, 52285}, {1352, 17810, 41588}, {1370, 62976, 3627}, {1595, 6642, 16196}, {1598, 7401, 6823}, {1656, 20850, 7494}, {1995, 7394, 427}, {2043, 2044, 11479}, {3545, 7576, 34664}, {3545, 62979, 2}, {3627, 11284, 10300}, {3628, 66380, 7499}, {3839, 38323, 62962}, {3843, 30771, 7378}, {3845, 10154, 15809}, {3845, 39487, 5066}, {3850, 13163, 31830}, {3850, 64852, 5133}, {3856, 37911, 5169}, {5004, 5005, 16661}, {5020, 34609, 2}, {5133, 13595, 468}, {6644, 64474, 16976}, {6995, 7392, 3}, {7378, 40132, 30771}, {7386, 7408, 382}, {7392, 7715, 6676}, {7392, 34608, 2}, {7395, 37122, 65376}, {7403, 7506, 140}, {7405, 7517, 16197}, {7484, 7500, 550}, {7484, 62968, 7500}, {7485, 7519, 66381}, {7485, 66381, 548}, {7494, 52301, 20850}, {7499, 37990, 3628}, {7528, 7529, 5}, {7533, 13595, 5133}, {7545, 37347, 37971}, {7667, 7734, 10691}, {7734, 10128, 13361}, {7734, 13361, 2}, {10110, 64035, 13142}, {10154, 15818, 66370}, {10301, 37439, 22}, {10301, 62937, 140}, {10601, 31383, 48906}, {11112, 11113, 48817}, {11284, 62976, 1370}, {11548, 37897, 6676}, {12106, 52262, 37935}, {12811, 47316, 37454}, {13361, 34603, 10691}, {13621, 50137, 7542}, {14002, 37454, 47316}, {15765, 18585, 1595}, {17928, 63666, 1907}, {18586, 18587, 6643}, {20405, 20406, 7426}, {34559, 34562, 50138}, {35018, 47630, 7495}, {37349, 52285, 3861}, {37439, 37440, 6676}, {47597, 62980, 2}, {63838, 65154, 5159}

X(66530) = X(1)X(442)∩X(2)X(25419)

Barycentrics    -2 a^3-8 a^2 (b+c)-a (3 b^2+8 b c+3 c^2)+(b-c)^2 (b+c) : :

See Juan José Isach Mayo, euclid 7425.

X(66530) lies on these lines: {1, 442}, {2, 25419}, {55, 35997}, {524, 10180}, {846, 63401}, {1962, 17768}, {3712, 8025}, {3742, 35104}, {4028, 28639}, {4046, 5333}, {4658, 18253}, {4682, 59584}, {4733, 25507}, {4854, 37635}, {4938, 8040}, {5625, 6703}, {6690, 37595}, {6707, 21085}, {15569, 40998}, {17390, 43223}, {17392, 17592}, {17770, 58381}, {20182, 25557}, {23812, 28530}, {27811, 42045}, {29580, 33126}, {42028, 59574}, {49564, 49728}, {49724, 53034}, {49734, 58399}, {49743, 58380}

X(66530) = midpoint of X(1962) and X(37631)

X(66531) = X(4)X(64)∩X(51)X(428)

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

See Juan José Isach Mayo, euclid 7425.

X(66531) lies on these lines: {2, 21167}, {4, 64}, {5, 46728}, {6, 6995}, {20, 17825}, {22, 3589}, {23, 37649}, {25, 5480}, {30, 5892}, {51, 428}, {53, 52448}, {125, 52285}, {141, 6997}, {154, 7714}, {159, 43726}, {161, 10594}, {184, 10301}, {185, 16656}, {343, 7394}, {373, 7667}, {389, 16621}, {427, 34417}, {468, 44106}, {524, 3060}, {546, 21243}, {578, 7715}, {597, 3796}, {1216, 23411}, {1350, 7392}, {1368,48901}, {1370, 3066}, {1495, 61659}, {1595, 23329}, {1598, 12233}, {1619, 15583}, {1629, 1990}, {1864, 1890}, {1885, 50709}, {1899, 62976}, {1906, 5893}, {1907, 6696}, {1995, 53415}, {2051, 33302}, {2393, 51745}, {2883, 5198}, {2979, 35283}, {3146, 18928}, {3167, 20423}, {3168, 16264}, {3527, 9833}, {3564, 21849}, {3567, 16655}, {3575, 44079}, {3580, 37349}, {3763, 33522}, {3818, 41588}, {3819, 10128}, {3845, 51993}, {3850, 13565}, {3867, 23327}, {3981, 7745}, {4186, 5799}, {5012, 6329}, {5020, 31670}, {5064, 23332}, {5085, 34608}, {5102, 63174}, {5133, 32269}, {5422, 7519}, {5640, 34603}, {5890,16654}, {5894, 11403}, {6030, 47313}, {6146, 9781}, {6201, 19219}, {6217, 6218}, {6353, 31860}, {6676, 19130}, {6688, 10691}, {6703, 35996}, {6756, 10110}, {7354, 63511}, {7378, 26958}, {7386, 48910}, {7391, 37648}, {7398, 17811}, {7408, 11433}, {7409, 37643}, {7484, 48881}, {7500, 10601}, {7533, 37636}, {7545, 51425}, {7576, 16657}, {7687, 11566}, {7693, 15246}, {7734, 63632}, {8550, 9777}, {9306, 21850}, {9815, 39568}, {9825, 13598}, {9909, 14561}, {9969, 15255}, {10095, 61299}, {10154, 38136}, {10169, 19136}, {10982, 34782}, {11002, 61658}, {11064, 13595}, {11427, 52301}, {11451, 52397}, {11477, 14826}, {11645, 32068}, {11746, 36201}, {11808, 44056}, {12007, 15004}, {13361, 15082}, {13391, 23410}, {13490, 44665}, {13621, 59648}, {13861, 59659}, {15045, 34613}, {15107, 34573}, {15274, 61348}, {15360, 50960}, {15559, 38848}, {16419, 48873}, {17845, 52518}, {18388, 61612}, {19468, 34484}, {20192, 31133}, {22352, 37899}, {25555, 37910}, {26005, 37456}, {26926, 52789}, {29012, 45298}, {34093, 62509}, {34796, 61989}, {35266, 51130}, {35919 ,63440}, {37353, 48912}, {37645, 59699}, {37897, 58447}, {37935, 46265}, {42874, 43462}, {43588, 58533}, {43621, 62209}, {44082, 61690}, {45440, 52291}, {45474, 45475}, {45862, 52286}, {45863, 52287}, {46818, 53863}, {47328, 63688}, {53094, 59343}, {58434, 61743}, {61680, 62979}, {61735, 62975}

X(66531) = midpoint of X(i) and X(j) for these {i,j}: {51, 428}, {3575, 61744}, {5890, 16654},{ 7576, 16657}
X(66531) = reflection of X(i) in X(j) for these {i,j}: {3819, 10128}, {10691, 6688}, {45298, 58470}
X(66531) = crosspoint of X(i) in X(j) for these {i,j}: {4, 57408}, {253, 10159}



leftri

Points related to the Kirikami-Steiner trifolium: X(66532)-X(66547)

rightri

This preamble and centers X(66532)-X(66547) were contributed by Ivan Pavlov on Dec 18, 2024.

Let P be a point not on the sides of ABC and G its centroid. Denote A' = Kirikami center of PBCQ, A'' = Kirikami center of PCBQ and similarly define B',B'',C',C''. The locus of points P for which A'B'C' and A''B''C'' is a circumquartic, which is called here the Kirikami-Steiner trifolium. It is tangent to the Steiner circumellipse at A, B, and C and has a triple point at G. The Kirikami-Steiner trifolium is the inverse image of K015 in the Steiner circumellipse.

In the following list (i,j) means that for P=X(i) the perspector of A'B'C' and A''B''C'' is X(j): (1,66543), (3,66544), (6,66545), (37,66546), (39,66547)

For more information see this Euclid post.

X(66532) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(3)

Barycentrics    (a-b)*(a+b)*(a-c)*(a+c)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(2*a^8-2*a^6*(b^2+c^2)+4*a^2*(b^2-c^2)^2*(b^2+c^2)+a^4*(-3*b^4+8*b^2*c^2-3*c^4)-(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)) : :
X(66532) = -X[15351]+4*X[46115]

X(66532) lies on the Kirikami-Steiner trifolium and on these lines: {2, 3}, {107, 39533}, {525, 648}, {685, 1499}, {15351, 46115}, {16076, 64923}, {23583, 39008}, {32662, 41679}, {42307, 42308}

X(66532) = midpoint of X(i) and X(j) for these {i,j}: {648, 16077}, {23582, 39062}
X(66532) = reflection of X(i) in X(j) for these {i,j}: {39008, 23583}
X(66532) = inverse of X(4240) in Steiner circumellipse
X(66532) = inverse of X(402) in Steiner inellipse
X(66532) = isotomic conjugate of X(62624)
X(66532) = trilinear pole of line {1651, 47204}
X(66532) = perspector of circumconic {{A, B, C, X(648), X(42308)}}
X(66532) = X(i)-isoconjugate-of-X(j) for these {i, j}: {31, 62624}, {810, 53201}, {2159, 47071}, {2631, 41433}
X(66532) = X(i)-Dao conjugate of X(j) for these {i, j}: {2, 62624}, {3163, 47071}, {39062, 53201}, {64923, 52720}
X(66532) = X(i)-cross conjugate of X(j) for these {i, j}: {52720, 64923}
X(66532) = pole of line {3, 1636} with respect to the Stammler hyperbola
X(66532) = pole of line {525, 4240} with respect to the Steiner circumellipse
X(66532) = pole of line {402, 525} with respect to the Steiner inellipse
X(66532) = pole of line {69, 41077} with respect to the Wallace hyperbola
X(66532) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(16076)}}, {{A, B, C, X(3), X(44769)}}, {{A, B, C, X(4), X(15459)}}, {{A, B, C, X(25), X(32695)}}, {{A, B, C, X(30), X(648)}}, {{A, B, C, X(99), X(40884)}}, {{A, B, C, X(376), X(2966)}}, {{A, B, C, X(381), X(53205)}}, {{A, B, C, X(402), X(39062)}}, {{A, B, C, X(468), X(47204)}}, {{A, B, C, X(476), X(46869)}}, {{A, B, C, X(525), X(1650)}}, {{A, B, C, X(868), X(42733)}}, {{A, B, C, X(3154), X(14223)}}, {{A, B, C, X(4240), X(23582)}}, {{A, B, C, X(6528), X(40885)}}, {{A, B, C, X(11050), X(53201)}}, {{A, B, C, X(15184), X(40512)}}, {{A, B, C, X(15351), X(45289)}}, {{A, B, C, X(18831), X(44651)}}, {{A, B, C, X(31152), X(53202)}}, {{A, B, C, X(31153), X(35169)}}, {{A, B, C, X(31154), X(53203)}}, {{A, B, C, X(31155), X(53206)}}, {{A, B, C, X(44216), X(53639)}}
X(66532) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {648, 39062, 16077}, {16077, 23582, 648}

X(66533) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(7)

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

X(66533) lies on the Kirikami-Steiner trifolium and on these lines: {2, 7}, {522, 664}, {651, 30181}, {927, 6006}, {17044, 35091}

X(66533) = midpoint of X(i) and X(j) for these {i,j}: {664, 35157}, {1275, 10001}
X(66533) = reflection of X(i) in X(j) for these {i,j}: {35091, 17044}
X(66533) = inverse of X(56543) in Steiner circumellipse
X(66533) = trilinear pole of line {14477, 64462}
X(66533) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3063, 53212}
X(66533) = X(i)-Dao conjugate of X(j) for these {i, j}: {10001, 53212}, {64462, 14476}
X(66533) = X(i)-cross conjugate of X(j) for these {i, j}: {14476, 64462}
X(66533) = pole of line {522, 56543} with respect to the Steiner circumellipse
X(66533) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(35157)}}, {{A, B, C, X(7), X(60487)}}, {{A, B, C, X(522), X(14476)}}, {{A, B, C, X(527), X(664)}}, {{A, B, C, X(666), X(6172)}}, {{A, B, C, X(1275), X(56543)}}, {{A, B, C, X(10001), X(36956)}}, {{A, B, C, X(31142), X(53208)}}, {{A, B, C, X(31164), X(53211)}}
X(66533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {664, 10001, 35157}, {1275, 35157, 664}

X(66534) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(11)

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

X(66534) lies on the Kirikami-Steiner trifolium and on these lines: {2, 11}, {666, 918}, {900, 34906}, {927, 6009}

X(66534) = inverse of X(63745) in Steiner circumellipse
X(66534) = pole of line {918, 63745} with respect to the Steiner circumellipse
X(66534) = intersection, other than A, B, C, of circumconics {{A, B, C, X(528), X(666)}}, {{A, B, C, X(10707), X(53214)}}, {{A, B, C, X(10712), X(53213)}}, {{A, B, C, X(57536), X(63745)}}

X(66535) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(37)

Barycentrics    (a-b)*b*(a-c)*c*(-2*b^2*c^2+2*a*b*c*(b+c)+a^2*(b^2-4*b*c+c^2)) : :
X(66535) = -X[9295]+4*X[36957]

X(66535) lies on the Kirikami-Steiner trifolium and on these lines: {2, 37}, {513, 668}, {1015, 40552}, {1978, 4411}, {3761, 35043}, {4568, 64867}, {4583, 4777}, {9295, 36957}, {27076, 39011}, {33908, 46796}

X(66535) = midpoint of X(i) and X(j) for these {i,j}: {668, 889}, {9296, 31625}
X(66535) = reflection of X(i) in X(j) for these {i,j}: {1015, 40552}, {39011, 27076}
X(66535) = inverse of X(41314) in Steiner circumellipse
X(66535) = trilinear pole of line {33908, 36847}
X(66535) = X(i)-isoconjugate-of-X(j) for these {i, j}: {649, 59053}
X(66535) = X(i)-Dao conjugate of X(j) for these {i, j}: {5375, 59053}, {33908, 14474}
X(66535) = X(i)-cross conjugate of X(j) for these {i, j}: {14474, 33908}
X(66535) = pole of line {513, 41314} with respect to the Steiner circumellipse
X(66535) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(889)}}, {{A, B, C, X(190), X(41142)}}, {{A, B, C, X(513), X(1646)}}, {{A, B, C, X(536), X(668)}}, {{A, B, C, X(4562), X(4664)}}, {{A, B, C, X(4688), X(53216)}}, {{A, B, C, X(4740), X(54985)}}, {{A, B, C, X(9296), X(36957)}}, {{A, B, C, X(18830), X(41144)}}, {{A, B, C, X(31625), X(41314)}}
X(66535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {668, 889, 513}, {668, 9296, 889}, {889, 31625, 668}

X(66536) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(39)

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

X(66536) lies on the Kirikami-Steiner trifolium and on these lines: {2, 39}, {99, 9489}, {512, 670}, {36950, 39010}

X(66536) = midpoint of X(i) and X(j) for these {i,j}: {670, 886}, {9428, 44168}
X(66536) = reflection of X(i) in X(j) for these {i,j}: {39010, 36950}
X(66536) = inverse of X(63747) in Steiner circumellipse
X(66536) = X(i)-isoconjugate-of-X(j) for these {i, j}: {798, 59051}
X(66536) = X(i)-Dao conjugate of X(j) for these {i, j}: {31998, 59051}
X(66536) = pole of line {32, 65497} with respect to the Stammler hyperbola
X(66536) = pole of line {512, 63747} with respect to the Steiner circumellipse
X(66536) = pole of line {6, 887} with respect to the Wallace hyperbola
X(66536) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(886)}}, {{A, B, C, X(76), X(57993)}}, {{A, B, C, X(99), X(41143)}}, {{A, B, C, X(512), X(1645)}}, {{A, B, C, X(538), X(670)}}, {{A, B, C, X(7757), X(18829)}}, {{A, B, C, X(44168), X(63747)}}
X(66536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {670, 886, 512}, {670, 9428, 886}, {886, 44168, 670}

X(66537) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(98)

Barycentrics    (a-b)*(a+b)*(a-c)*(a+c)*(a^4+b^4-a^2*c^2-b^2*c^2)*(a^4-a^2*b^2-b^2*c^2+c^4)*(2*a^12-4*a^10*(b^2+c^2)+4*a^8*(b^4+b^2*c^2+c^4)-4*a^6*(b^6+c^6)+a^4*(b^8+c^8)-(b^2-c^2)^2*(b^8+4*b^4*c^4+c^8)+2*a^2*(b^10-b^8*c^2-b^2*c^8+c^10)) : :

X(66537) lies on the Kirikami-Steiner trifolium and on these lines: {2, 98}, {685, 2793}, {690, 43113}, {2799, 2966}

X(66537) = inverse of X(34761) in Steiner circumellipse
X(66537) = pole of line {2799, 34761} with respect to the Steiner circumellipse
X(66537) = intersection, other than A, B, C, of circumconics {{A, B, C, X(125), X(42738)}}, {{A, B, C, X(542), X(2966)}}, {{A, B, C, X(648), X(6054)}}, {{A, B, C, X(9140), X(53229)}}, {{A, B, C, X(34761), X(57562)}}

X(66538) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(351)

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

X(66538) lies on the Kirikami-Steiner trifolium and on these lines: {2, 351}, {6, 47646}, {5939, 5970}, {5969, 35146}

X(66538) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(351), X(59051)}}, {{A, B, C, X(804), X(35146)}}

X(66539) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(512)

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

X(66539) lies on the Kirikami-Steiner trifolium and on these lines: {2, 512}, {187, 9150}, {385, 729}, {538, 886}, {1084, 66547}, {3972, 41309}, {7804, 51510}, {41143, 46156}

X(66539) = midpoint of X(i) and X(j) for these {i,j}: {886, 3228}
X(66539) = reflection of X(i) in X(j) for these {i,j}: {1084, 66547}
X(66539) = inverse of X(63749) in Steiner circumellipse
X(66539) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2234, 59051}
X(66539) = pole of line {538, 63749} with respect to the Steiner circumellipse
X(66539) = pole of line {23342, 52067} with respect to the Wallace hyperbola
X(66539) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(886)}}, {{A, B, C, X(512), X(3228)}}, {{A, B, C, X(538), X(62611)}}, {{A, B, C, X(1916), X(5996)}}, {{A, B, C, X(57540), X(63749)}}
X(66539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {886, 57540, 3228}

X(66540) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(514)

Barycentrics    (a+b-2*c)*(a-2*b+c)*(2*a^2-b^2+4*b*c-c^2-2*a*(b+c)) : :
X(66540) = -3*X[32106]+4*X[36522]

X(66540) lies on K296, Kirikami-Steiner trifolium and on these lines: {1, 52574}, {2, 514}, {88, 41140}, {106, 9089}, {239, 42026}, {291, 40833}, {320, 519}, {350, 20568}, {545, 6633}, {551, 27922}, {1086, 64463}, {1644, 34762}, {3912, 4945}, {4080, 17310}, {4480, 9326}, {4615, 6629}, {4997, 41141}, {17378, 49751}, {17487, 53582}, {26749, 50116}, {32106, 36522}, {39704, 42482}, {40891, 62732}

X(66540) = midpoint of X(i) and X(j) for these {i,j}: {903, 4555}, {9460, 54974}
X(66540) = reflection of X(i) in X(j) for these {i,j}: {903, 6549}, {1086, 66543}, {6633, 35121}, {17487, 53582}
X(66540) = inverse of X(6548) in Steiner circumellipse
X(66540) = trilinear pole of line {14475, 545}
X(66540) = perspector of circumconic {{A, B, C, X(903), X(34762)}}
X(66540) = X(i)-isoconjugate-of-X(j) for these {i, j}: {44, 2384}, {101, 52225}, {692, 34764}, {1017, 64459}, {2251, 35168}
X(66540) = X(i)-Dao conjugate of X(j) for these {i, j}: {545, 1644}, {1015, 52225}, {1086, 34764}, {9460, 35168}, {35121, 519}, {40595, 2384}
X(66540) = X(i)-cross conjugate of X(j) for these {i, j}: {1644, 545}, {33920, 6633}
X(66540) = pole of line {519, 6548} with respect to the Steiner circumellipse
X(66540) = pole of line {903, 1647} with respect to the dual conic of Yff parabola
X(66540) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(4555)}}, {{A, B, C, X(291), X(4792)}}, {{A, B, C, X(350), X(27921)}}, {{A, B, C, X(514), X(545)}}, {{A, B, C, X(519), X(1644)}}, {{A, B, C, X(679), X(1022)}}, {{A, B, C, X(1266), X(21129)}}, {{A, B, C, X(4379), X(4510)}}, {{A, B, C, X(6545), X(6549)}}, {{A, B, C, X(6548), X(54974)}}, {{A, B, C, X(14421), X(52745)}}, {{A, B, C, X(14475), X(35121)}}, {{A, B, C, X(23598), X(36594)}}, {{A, B, C, X(24841), X(24858)}}, {{A, B, C, X(31992), X(35168)}}, {{A, B, C, X(44009), X(62413)}}
X(66540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {545, 35121, 6633}, {903, 4555, 519}, {903, 54974, 6549}, {903, 9460, 4555}, {4555, 54974, 903}, {17953, 46795, 46790}, {36887, 52755, 52759}, {64463, 66543, 1086}

X(66541) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(649)

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

X(66541) lies on the Kirikami-Steiner trifolium, circumconic {{A, B, C, X(3226), X(4785)}}, and on these lines: {2, 649}, {726, 3226}, {3253, 49479}, {4759, 8709}

X(66541) = midpoint of X(i) and X(j) for these {i,j}: {33678, 57535}
X(66541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {33678, 57535, 726}

X(66542) = SECOND INTERSECTION OF KIRIKAMI-STEINER TRIFOLIUM AND LINE X(2)X(690)

Barycentrics    (a^4+b^4+2*b^2*c^2-2*c^4+a^2*(-4*b^2+2*c^2))*(a^4-2*b^4+2*b^2*c^2+c^4+2*a^2*(b^2-2*c^2))*(2*a^8-b^8-4*b^6*c^2+12*b^4*c^4-4*b^2*c^6-c^8-4*a^6*(b^2+c^2)-6*a^4*(b^4-4*b^2*c^2+c^4)+4*a^2*(2*b^6-3*b^4*c^2-3*b^2*c^4+2*c^6)) : :

X(66542) lies on the Kirikami-Steiner trifolium and on these lines: {2, 690}, {543, 18823}, {2482, 9170}, {31632, 36521}

X(66542) = inverse of X(34763) in Steiner circumellipse
X(66542) = pole of line {543, 34763} with respect to the Steiner circumellipse
X(66542) = intersection, other than A, B, C, of circumconics {{A, B, C, X(690), X(18823)}}, {{A, B, C, X(34763), X(57561)}}
X(66542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9180, 51226, 2}

X(66543) = X(2)X(52574)∩X(545)X(6549)

Barycentrics    2*a^4+5*b^4+12*a^2*b*c-22*b^3*c+36*b^2*c^2-22*b*c^3+5*c^4-4*a^3*(b+c)+2*a*(b^3-3*b^2*c-3*b*c^2+c^3) : :
X(66543) = 5*X[903]+3*X[6631], -5*X[4370]+3*X[32106], -3*X[6547]+X[35168]

X(66543) lies on these lines: {2, 52574}, {519, 7238}, {545, 6549}, {903, 6631}, {1086, 64463}, {4370, 32106}, {4795, 36234}, {6547, 35168}, {6550, 21204}, {24628, 41140}

X(66543) = midpoint of X(i) and X(j) for these {i,j}: {903, 35121}, {1086, 66540}

X(66544) = X(2)X(31621)∩X(1494)X(39062)

Barycentrics    2*a^16-4*a^14*(b^2+c^2)-4*a^12*(b^4-5*b^2*c^2+c^4)-4*a^6*(b^2-c^2)^2*(7*b^6-12*b^4*c^2-12*b^2*c^4+7*c^6)+2*a^10*(7*b^6-9*b^4*c^2-9*b^2*c^4+7*c^6)-2*a^2*(b^2-c^2)^4*(11*b^6+37*b^4*c^2+37*b^2*c^4+11*c^6)-a^8*(b^8+46*b^6*c^2-96*b^4*c^4+46*b^2*c^6+c^8)+(b^2-c^2)^4*(5*b^8+22*b^6*c^2+36*b^4*c^4+22*b^2*c^6+5*c^8)+2*a^4*(b^2-c^2)^2*(19*b^8+2*b^6*c^2-48*b^4*c^4+2*b^2*c^6+19*c^8) : :
X(66544) = X[2]+3*X[31621], 5*X[1494]+3*X[39062]

X(66544) lies on these lines: {2, 31621}, {1494, 39062}, {11049, 38240}, {15526, 16076}

X(66544) = midpoint of X(i) and X(j) for these {i,j}: {15526, 16076}

X(66545) = X(2)X(52551)∩X(115)X(17948)

Barycentrics    2*a^8+5*b^8+12*a^4*b^2*c^2-22*b^6*c^2+36*b^4*c^4-22*b^2*c^6+5*c^8-4*a^6*(b^2+c^2)+2*a^2*(b^6-3*b^4*c^2-3*b^2*c^4+c^6) : :
X(66545) = 5*X[671]+3*X[31998], X[892]+3*X[41135], 3*X[4590]+X[8596], -3*X[9166]+X[23992], -3*X[14971]+2*X[40486], -X[18823]+3*X[23991], 3*X[39061]+X[44373]

X(66545) lies on these lines: {2, 52551}, {115, 17948}, {523, 5461}, {524, 32457}, {543, 40553}, {671, 31998}, {892, 41135}, {4590, 8596}, {9166, 23992}, {9169, 51428}, {9172, 16092}, {10278, 33919}, {14971, 40486}, {18823, 23991}, {39061, 44373}, {44401, 47171}

X(66545) = midpoint of X(i) and X(j) for these {i,j}: {115, 17948}, {671, 35087}
X(66545) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {115, 17948, 46980}, {16092, 16278, 51258}
X(66545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {35087, 61339, 671}

X(66546) = X(2)X(57542)∩X(3227)X(9296)

Barycentrics    12*a^2*b^3*c^3+2*b^4*c^4-4*a*b^3*c^3*(b+c)+2*a^3*b*c*(b^3-3*b^2*c-3*b*c^2+c^3)+a^4*(5*b^4-22*b^3*c+36*b^2*c^2-22*b*c^3+5*c^4) : :
X(66546) = X[2]+3*X[57542], 5*X[3227]+3*X[9296]

X(66546) lies on these lines: {2, 57542}, {1015, 46796}, {3227, 9296}, {33908, 40552}, {33917, 38238}

X(66546) = midpoint of X(i) and X(j) for these {i,j}: {1015, 46796}

X(66547) = X(2)X(57540)∩X(3228)X(9428)

Barycentrics    12*a^4*b^6*c^6+2*b^8*c^8-4*a^2*b^6*c^6*(b^2+c^2)+2*a^6*b^2*c^2*(b^6-3*b^4*c^2-3*b^2*c^4+c^6)+a^8*(5*b^8-22*b^6*c^2+36*b^4*c^4-22*b^2*c^6+5*c^8) : :
X(66547) = X[2]+3*X[57540], 5*X[3228]+3*X[9428]

X(66547) lies on these lines: {2, 57540}, {1084, 66539}, {3228, 9428}, {33918, 38237}

X(66547) = midpoint of X(i) and X(j) for these {i,j}: {1084, 66539}

X(66548) = (name pending)

Barycentrics    a^2*(a^26 - 7*a^24*b^2 + 20*a^22*b^4 - 28*a^20*b^6 + 13*a^18*b^8 + 21*a^16*b^10 - 48*a^14*b^12 + 48*a^12*b^14 - 21*a^10*b^16 - 13*a^8*b^18 + 28*a^6*b^20 - 20*a^4*b^22 + 7*a^2*b^24 - b^26 - 7*a^24*c^2 + 42*a^22*b^2*c^2 - 103*a^20*b^4*c^2 + 128*a^18*b^6*c^2 - 76*a^16*b^8*c^2 + 8*a^14*b^10*c^2 + 14*a^12*b^12*c^2 - 28*a^10*b^14*c^2 + 77*a^8*b^16*c^2 - 114*a^6*b^18*c^2 + 89*a^4*b^20*c^2 - 36*a^2*b^22*c^2 + 6*b^24*c^2 + 20*a^22*c^4 - 103*a^20*b^2*c^4 + 216*a^18*b^4*c^4 - 231*a^16*b^6*c^4 + 123*a^14*b^8*c^4 - 22*a^12*b^10*c^4 + 22*a^10*b^12*c^4 - 101*a^8*b^14*c^4 + 171*a^6*b^16*c^4 - 153*a^4*b^18*c^4 + 72*a^2*b^20*c^4 - 14*b^22*c^4 - 28*a^20*c^6 + 128*a^18*b^2*c^6 - 231*a^16*b^4*c^6 + 212*a^14*b^6*c^6 - 99*a^12*b^8*c^6 - 3*a^10*b^10*c^6 + 68*a^8*b^12*c^6 - 115*a^6*b^14*c^6 + 120*a^4*b^16*c^6 - 66*a^2*b^18*c^6 + 14*b^20*c^6 + 13*a^18*c^8 - 76*a^16*b^2*c^8 + 123*a^14*b^4*c^8 - 99*a^12*b^6*c^8 + 61*a^10*b^8*c^8 - 28*a^8*b^10*c^8 + 17*a^6*b^12*c^8 - 32*a^4*b^14*c^8 + 21*a^2*b^16*c^8 + 21*a^16*c^10 + 8*a^14*b^2*c^10 - 22*a^12*b^4*c^10 - 3*a^10*b^6*c^10 - 28*a^8*b^8*c^10 + 26*a^6*b^10*c^10 - 4*a^4*b^12*c^10 + 6*a^2*b^14*c^10 - 13*b^16*c^10 - 48*a^14*c^12 + 14*a^12*b^2*c^12 + 22*a^10*b^4*c^12 + 68*a^8*b^6*c^12 + 17*a^6*b^8*c^12 - 4*a^4*b^10*c^12 - 8*a^2*b^12*c^12 + 8*b^14*c^12 + 48*a^12*c^14 - 28*a^10*b^2*c^14 - 101*a^8*b^4*c^14 - 115*a^6*b^6*c^14 - 32*a^4*b^8*c^14 + 6*a^2*b^10*c^14 + 8*b^12*c^14 - 21*a^10*c^16 + 77*a^8*b^2*c^16 + 171*a^6*b^4*c^16 + 120*a^4*b^6*c^16 + 21*a^2*b^8*c^16 - 13*b^10*c^16 - 13*a^8*c^18 - 114*a^6*b^2*c^18 - 153*a^4*b^4*c^18 - 66*a^2*b^6*c^18 + 28*a^6*c^20 + 89*a^4*b^2*c^20 + 72*a^2*b^4*c^20 + 14*b^6*c^20 - 20*a^4*c^22 - 36*a^2*b^2*c^22 - 14*b^4*c^22 + 7*a^2*c^24 + 6*b^2*c^24 - c^26) : :

See Antreas Hatzipolakis, Elias Hagos and Peter Moses, euclid 7446.

X(66548) lies on this line: {186, 2970}

X(66549) = X(1)X(6)∩X(391)X(551)

Barycentrics    a (7 a + 3 (b + c)) : :

See Juan José Isach Mayo, euclid 7450.

X(66549) lies on these lines: {1, 6}, {2, 4856}, {41, 30343}, {81, 18186}, {86, 16833}, {346, 3635}, {391, 551}, {572, 7991}, {573, 17474}, {604, 3339}, {902, 62842}, {940, 8056}, {966, 25055}, {988, 1384}, {1051, 5268}, {1419, 7274}, {1698, 4982}, {1766, 11224}, {2177, 62845}, {2262, 18398}, {2268, 9819}, {2269, 53054}, {2270, 51816}, {2276, 46189}, {2277, 9336}, {2280, 10980}, {2297, 41434}, {2321, 51093}, {2345, 3633}, {2999, 37633}, {3187, 19741}, {3217,9327}, {3241, 4898}, {3244, 5749}, {3337, 54420}, {3361, 4262}, {3618, 29573}, {3619, 3879}, {3620, 17023}, {3623, 3950}, {3624, 3686}, {3630, 4657}, {3631, 17306}, {3632, 5750}, {3636, 5296}, {3663, 60984}, {3664, 17014}, {3672, 60976}, {3679, 4545}, {3729, 29584}, {3746, 5120}, {3758, 55998}, {3759, 16832}, {3875, 46922}, {3945, 4859}, {3946, 4888}, {3986, 38314}, {4034, 17398}, {4058, 20050}, {4060, 26039}, {4254, 5563}, {4263, 63493}, {4357, 11008}, {4384, 31312}, {4393, 4821}, {4416, 63061}, {4512, 21747}, {4648, 60999}, {4667, 4862}, {4668, 17303}, {4677, 59772}, {4747, 53594}, {4851, 51126}, {4889, 61344}, {4902, 17301}, {4910, 7227}, {4969, 34595}, {5024, 37552}, {5256, 14996}, {5271, 19740}, {5272, 37675}, {5287, 14997}, {5393, 13941}, {5405, 8972}, {5540, 54385}, {6144, 41311}, {6173, 63401}, {7222, 50109}, {7271, 33633}, {7987, 37508}, {10199, 27524}, {11011, 38296}, {15048, 48827}, {15668, 50124}, {16192, 41456}, {16829, 20146}, {16831, 17121}, {16834, 17117}, {17011, 62812}, {17022, 37680}, {17120, 25269}, {17257, 63026}, {17272, 20080}, {17275, 61302}, {17284, 63119}, {17294, 63053}, {17296, 34573}, {17299, 34747}, {17304, 20090}, {17319, 63108}, {17321, 62996}, {17330, 51110}, {17349, 29597}, {17353, 29602}, {17363, 29603}, {17368, 29605}, {17393, 50127}, {17395, 60933}, {17396, 63052}, {18907, 48818}, {19743, 56082}, {19745, 31993}, {19876, 50082}, {20818, 44841}, {21309, 37592}, {24239, 37689}, {24512, 42043}, {25101, 63123}, {29574, 51171}, {29578, 31313}, {29580, 63050}, {30308, 32431}, {30337, 55100}, {32087, 49543}, {32455, 41312}, {33630, 34231}, {37654, 51105}, {37685, 62818}, {42042, 63066}, {46475, 55716}, {47355, 50125}, {48824, 63633}

X(66549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 6, 3731}, {1, 1449, 16667}, {1, 1743, 16673}, {1, 3973, 3247}, {1, 16475, 60846}, {1, 16667, 1743}, {6, 3247, 3973}, {6, 3723, 9}, {6, 3731,1743}, {6, 15492, 16670}, {6, 16674, 44}, {6, 16777, 15492}, {6,16884, 3723}, {9, 1449, 16666}, {9, 3247, 16677}, {9, 16884,1}, {1100, 1449, 1}, {1100, 16666, 16884}, {1100, 62212,1449}, {3241, 17355, 4898}, {3247, 3973, 3731}, {3723, 16666,6}, {3723, 16677, 3247}, {3731, 16667, 6}, {3945, 50114, 4859}, {3946, 63054, 4888}, {4034, 17398, 19875}, {5256, 14996, 62695}, {15492, 16668, 6}, {16666, 16884, 9}, {16668, 16777, 16670}, {16671, 16672, 9}, {16834, 17379, 25590}, {17398, 50131, 4034}, {29584, 37677, 3729}, {38314, 62985, 3986}

X(66550) = X(1)X(6)∩X(2)X(4898)

Barycentrics    a (5 a + 9 (b + c)) : :

See Juan José Isach Mayo, euclid 7450.

X(66550) lies on these lines: {1, 6}, {2, 4898}, {346, 3636}, {391, 51071}, {573, 16189}, {594, 34595}, {966, 51093}, {1213, 4677}, {1255, 2999}, {1766, 30392}, {2171, 13462}, {2321, 25055}, {3175, 19746}, {3241, 3986}, {3619, 29573}, {3620, 29574}, {3622, 3950}, {3624, 17314}, {3631, 41312}, {3633, 5257}, {3635, 5296}, {3663, 59375}, {3729, 29580}, {3945, 60971}, {4021, 29624}, {4034,34747}, {4058, 5550}, {4668, 17388}, {4686, 36834}, {4740, 17319}, {4764, 10436}, {4772, 16826}, {4788, 17116}, {4902, 17392}, {5287, 62695}, {7991, 37508}, {9331, 17053}, {10980, 41423}, {14996, 27789}, {15655, 37552}, {15828, 51104}, {16831, 17117}, {16833, 17393}, {17019, 23958}, {17022, 62851}, {17272, 29585}, {17299, 19875}, {17321, 29602}, {17330, 51097}, {17355, 38314}, {17398, 51110}, {19741, 56082}, {21871, 50190}, {25502, 60724}, {37499, 58245}, {37587, 54285}, {37602, 54322}, {37633, 62816}, {41456, 63468}, {46475, 55594}, {50093, 62996}, {50110, 63014}, {50113, 59772}, {50123, 51066}, {51105, 63055}, {51107, 63086}

X(66550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 37, 16667}, {1, 3247, 3731}, {1, 16673, 1743}, {9, 3247, 16674}, {37, 3973, 3731}, {37, 62212, 9}, {1449, 46845, 1}, {3247,3723, 1}, {3247, 3731, 16673}, {3723, 16777, 3247}, {3723, 16814, 46845}, {3731, 16667, 3973}, {16672, 46845, 1449}, {16826, 17151, 31312}, {17299, 62648, 19875}, {17319, 29597, 25590}

X(66551) = X(1)X(6)∩X(200)X(4695)

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

See Juan José Isach Mayo, euclid 7450.

X(66551) lies on these lines: {1, 6}, {78, 62695}, {200, 4695}, {902, 12526}, {3241, 4656}, {3339, 56010}, {3430, 7991}, {3940, 23511}, {3953, 7963}, {3961, 18421}, {3984, 8951}, {4257, 54422}, {4902, 11112}, {7322, 44840}, {11520, 37633}, {13462, 62865}, {16371, 33795}, {17022, 63159}, {17276, 34701}, {17597, 46943}, {54310, 62823}

X(66551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3973, 16485}, {1, 5692, 60846}, {72, 16485, 3973}, {10179, 15600, 1}

X(66552) = X(3)X(8745)∩X(24)X(394)

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

See Antreas Hatzipolakis and Juan José Isach Mayo, euclid 7477.

X(66552) lies on these lines: {3, 8745}, {4, 52350}, {24, 394}, {52, 8882}, {96, 56303}, {97, 7488}, {317, 3926}, {389, 17974}, {1073, 6642}, {1594, 52415}, {7401, 14376}, {7509, 63154}, {7576, 14111}, {14919, 44802}, {31626, 37126}, {46728, 54032}, {53173, 58734}

X(66552) = isogonal conjugate of X(6146)

X(66553) = EULER LINE INTERCEPT OF X(8029)X(31990)

Barycentrics    4 a^16-12 a^14 b^2+6 a^12 b^4+10 a^10 b^6-3 a^8 b^8-13 a^6 b^10+7 a^4 b^12+3 a^2 b^14-2 b^16-12 a^14 c^2+48 a^12 b^2 c^2-50 a^10 b^4 c^2-28 a^8 b^6 c^2+63 a^6 b^8 c^2-7 a^4 b^10 c^2-23 a^2 b^12 c^2+9 b^14 c^2+6 a^12 c^4-50 a^10 b^2 c^4+122 a^8 b^4 c^4-62 a^6 b^6 c^4-57 a^4 b^8 c^4+53 a^2 b^10 c^4-14 b^12 c^4+10 a^10 c^6-28 a^8 b^2 c^6-62 a^6 b^4 c^6+118 a^4 b^6 c^6-33 a^2 b^8 c^6+7 b^10 c^6-3 a^8 c^8+63 a^6 b^2 c^8-57 a^4 b^4 c^8-33 a^2 b^6 c^8-13 a^6 c^10-7 a^4 b^2 c^10+53 a^2 b^4 c^10+7 b^6 c^10+7 a^4 c^12-23 a^2 b^2 c^12-14 b^4 c^12+3 a^2 c^14+9 b^2 c^14-2 c^16 : :
Barycentrics    3*S^6-SB*SC*(10*R^2-SW)*SW^3+S^4*(-648*R^4-11*SB*SC+270*R^2*SW-30*SW^2)+S^2*(1728*R^4*SB*SC-738*R^2*SB*SC*SW+86*SB*SC*SW^2+6*R^2*SW^3-SW^4) : :
X(66553) = 3*X(8029)-4*X(31990), 9*X(8029)-8*X(32061), 3*X(31990)-2*X(32061), 5*X(31990)-4*X(32211), 5*X(32061)-6*X(32211)

See Tran Quang Hung and Ercole Suppa, euclid 7485.

X(66553) lies on these lines: {2, 3}, {8029, 31990}




leftri  Perspectors related to Morley triangles: X(66554) - X(66580)  rightri

This preamble and centers X(66554)-X(66580) were contributed by César Eliud Lozada, December 23, 2024.

Many thanks to Chris van Tienhoven for sharing his deep knowledge on this topic.

Two triangles related to the original Morley triangles are introduced here:

  1. The Morley-homothetic-inscribed triangle is the triangle inscribed in ABC and homothetic to the 1st-, 2nd- and 3rd- Morley triangles. It has center X(3272) and A-vertex with trilinear coordinates:
    A' = 0 : (c+2*b*x+2*a*y)*(2*c*(-y+2*x*z)+a*(-1+4*z^2)) : (a*(-1+4*y^2)+2*b*(2*x*y-z))*(b+2*c*x+2*a*z)

    where x = cos(A/3), y = cos(B/3), z = cos(C/3)

  2. The Morley-homothetic-circumscribed triangle is the triangle circumscribed to ABC and homothetic to the 1st-, 2nd- and 3rd- Morley triangles. It has center X(8011) and A-vertex with trilinear coordinates:
    A" = -1 : (a*(-1+4*z^2)-(2*(y-2*x*z))*c)/((-1+4*z^2)*b-(2*(x-2*y*z))*c): (a*(-1+4*y^2)-(2*(z-2*x*y))*b)/((-1+4*y^2)*c-(2*(x-2*y*z))*b)

    where x = cos(A/3), y = cos(B/3), z = cos(C/3)

underbar

X(66554) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY AND 1st MORLEY-MIDPOINT

Trilinears    cos(A/3)*(3+8*cos(A/3)*cos(B/3)*cos(C/3))+8*cos((B-C)/3)*cos((A-Pi)/3)*(cos((A-Pi)/3)+cos((B-Pi)/3)*cos((C-Pi)/3)) : :

X(66554) lies on these lines: {3, 356}, {357, 5456}, {358, 66557}, {1135, 66560}, {3273, 66568}, {3275, 3604}, {8065, 66575}, {66555, 66563}, {66556, 66564}

X(66555) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY AND 2nd MORLEY-MIDPOINT

Trilinears    sin((A-B)/3)*sin((A-C)/3)*(-((1+2*cos((2*B)/3))*cos((2*C+Pi)/6)*sin(C/3)*(sqrt(3)+2*sin((2*(A-Pi))/3)+2*sin((2*(B-Pi))/3)))+(1+2*cos((2*C)/3))*cos((2*B+Pi)/6)*sin(B/3)*(sqrt(3)+2*sin((2*(A-Pi))/3)+2*sin((2*(C-Pi))/3))) : :

X(66555) lies on these lines: {356, 41109}, {357, 8065}, {358, 66558}, {1135, 66561}, {3273, 66569}, {3274, 3602}, {3278, 31934}, {3279, 31931}, {66554, 66563}, {66556, 66565}

X(66556) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY AND 3rd MORLEY-MIDPOINT

Trilinears    2*sin(A/3)+cos(C)*sin(B/3)+cos(B)*sin(C/3)-sqrt(3)*(2*cos(A/3)+5*cos(B/3)*cos(C/3)+sin(B/3)*sin(C/3))+2*(cos(A/3)+2*cos(B/3)*cos(C/3))*(cos(B/3)*cos((2*B+Pi)/6)+cos(C/3)*cos((2*C+Pi)/6)-cos(A/3)*sin((A-Pi)/3)) : :

X(66556) lies on these lines: {2, 5456}, {356, 41110}, {357, 38417}, {358, 66559}, {1134, 15859}, {1135, 66562}, {3273, 3603}, {3275, 46642}, {3277, 66479}, {3278, 31935}, {3279, 31932}, {3283, 5635}, {8065, 66577}, {66554, 66564}, {66555, 66565}

X(66556) = (X(3273), X(66570))-harmonic conjugate of X(3603)

X(66557) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 1st MORLEY-MIDPOINT

Trilinears    (-1+4*cos(B/3)^2)*(csc(A)*csc((2*B-Pi)/6)*sec((2*A+Pi)/6)*sin(A/3)-csc(B)*csc((2*A-Pi)/6)*sec((2*B+Pi)/6)*sin(B/3))*sin((A-C)/3)-(-1+4*cos(C/3)^2)*sin((A-B)/3)*(csc(A)*csc((2*C-Pi)/6)*sec((2*A+Pi)/6)*sin(A/3)-csc(C)*csc((2*A-Pi)/6)*sec((2*C+Pi)/6)*sin(C/3)) : :

X(66557) lies on these lines: {357, 15857}, {358, 66554}, {1136, 38415}, {1137, 66560}, {3273, 46641}, {3274, 3602}, {3276, 41111}, {3279, 5636}, {3280, 31933}, {3281, 31930}, {8066, 66575}, {66558, 66563}, {66559, 66564}

X(66557) = barycentric product X(7309)*X(38415)
X(66557) = (X(3274), X(66568))-harmonic conjugate of X(3604)

X(66558) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 2nd MORLEY-MIDPOINT

Trilinears    cos(A)-4*(cos(A/3)-cos(B/3)*cos(C/3))*(-1+sin((4*A+Pi)/6)+sin((4*B+Pi)/6)+sin((4*C+Pi)/6)) : :

X(66558) lies on these lines: {3, 3276}, {358, 66555}, {1136, 38416}, {1137, 66561}, {3273, 3602}, {3274, 66569}, {8066, 66576}, {66557, 66563}, {66559, 66565}

X(66559) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND 3rd MORLEY-MIDPOINT

Trilinears    sin((A-B)/3)*sin((A-C)/3)*(-((sqrt(3)+2*sin((2*A)/3)+2*sin((2*B)/3))*sin(C/3)*sin((C-2*Pi)/3)*(1-2*sin((4*B+Pi)/6)))-sin(B/3)*(sqrt(3)+2*sin((2*A)/3)+2*sin((2*C)/3))*sin((B-2*Pi)/3)*(-1+2*sin((4*C+Pi)/6))) : :

X(66559) lies on these lines: {358, 66556}, {1136, 8066}, {1137, 66562}, {3272, 7309}, {3274, 66570}, {3275, 3602}, {3276, 41110}, {3280, 31935}, {3281, 31932}, {66557, 66564}, {66558, 66565}

X(66560) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 1st MORLEY-MIDPOINT

Trilinears    csc(A/3)*sin((A-B)/3)*sin((A-C)/3)*((1+2*cos((2*C)/3))*(4*cos((A-B)/3)+sqrt(3)*csc(C/3))*sin((B+Pi)/3)-(1+2*cos((2*B)/3))*(4*cos((A-C)/3)+sqrt(3)*csc(B/3))*sin((C+Pi)/3)) : :

X(66560) lies on these lines: {1134, 8067}, {1135, 66554}, {1137, 66557}, {3272, 5456}, {3273, 3603}, {3275, 66568}, {3277, 41111}, {3282, 31933}, {3283, 31930}, {66561, 66563}, {66562, 66564}

X(66561) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 2nd MORLEY-MIDPOINT

Trilinears    (-1+2*cos(C/3)*cos((C-Pi)/3))*((1+sqrt(3)*cot(C/3))*csc(C)*csc((2*A+Pi)/6)-(1+sqrt(3)*cot(A/3))*csc(A)*csc((2*C+Pi)/6))*sin((A-B)/3)-(-1+2*cos(B/3)*cos((B-Pi)/3))*((1+sqrt(3)*cot(B/3))*csc(B)*csc((2*A+Pi)/6)-(1+sqrt(3)*cot(A/3))*csc(A)*csc((2*B+Pi)/6))*sin((A-C)/3) : :

X(66561) lies on these lines: {2, 7309}, {1134, 38416}, {1135, 66555}, {1136, 15858}, {1137, 66558}, {3274, 46643}, {3275, 3602}, {3277, 41109}, {3281, 5634}, {3282, 31934}, {3283, 31931}, {8067, 66576}, {66560, 66563}, {66562, 66565}

X(66561) = barycentric product X(5456)*X(38416)
X(66561) = trilinear quotient X(5456)/X(7047)
X(66561) = (X(3275), X(66569))-harmonic conjugate of X(3602)

X(66562) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd MORLEY AND 3rd MORLEY-MIDPOINT

Trilinears    cos(A)+2*(1+cos((2*(A-Pi))/3)+cos((2*(B-Pi))/3)+cos((2*(C-Pi))/3))*(sin((2*A+Pi)/6)-2*sin((2*B+Pi)/6)*sin((2*C+Pi)/6)) : :

X(66562) lies on these lines: {3, 3277}, {1134, 7309}, {1135, 66556}, {1137, 66559}, {3274, 3603}, {3275, 66570}, {8067, 66577}, {66560, 66564}, {66561, 66565}

X(66563) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY-MIDPOINT AND 2nd MORLEY-MIDPOINT

Trilinears    sec((2*A+Pi)/6)*((csc(C)*csc((A-2*Pi)/3)-csc(A)*csc((C-2*Pi)/3))*sec((2*B+Pi)/6)*(-(csc(A)*sin(A/3))+csc(B)*sin(B/3))+(csc(B)*csc((A-2*Pi)/3)-csc(A)*csc((B-2*Pi)/3))*sec((2*C+Pi)/6)*(csc(A)*sin(A/3)-csc(C)*sin(C/3))) : :

X(66563) lies on these lines: {2, 3604}, {395, 66564}, {3602, 38415}, {5454, 5455}, {31930, 31931}, {31933, 31934}, {41109, 41111}, {66554, 66555}, {66557, 66558}, {66560, 66561}, {66568, 66569}, {66575, 66576}

X(66563) = (X(2), X(38416))-harmonic conjugate of X(23030)

X(66564) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY-MIDPOINT AND 3rd MORLEY-MIDPOINT

Trilinears    -(csc((A-2*Pi)/3)*(-((-(csc(A)*csc(C/3))+csc(A/3)*csc(C))*csc((B-2*Pi)/3)*(cos((2*B+Pi)/6)*csc(B)+csc(A)*sin((A-Pi)/3)))-(csc(A)*csc(B/3)-csc(A/3)*csc(B))*csc((C-2*Pi)/3)*(cos((2*C+Pi)/6)*csc(C)+csc(A)*sin((A-Pi)/3)))) : :

X(66564) lies on these lines: {2, 3603}, {395, 66563}, {3277, 65156}, {3604, 38417}, {7309, 65155}, {31930, 31932}, {31933, 31935}, {41110, 41111}, {66554, 66556}, {66557, 66559}, {66560, 66562}, {66568, 66570}, {66575, 66577}

X(66564) = (X(2), X(38415))-harmonic conjugate of X(23029)

X(66565) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY-MIDPOINT AND 3rd MORLEY-MIDPOINT

Trilinears    csc(A/3)*(csc(B/3)*(csc(A)*csc((C-Pi)/3)+csc(C)*sec((2*A+Pi)/6))*(csc(A)*sin((A+Pi)/3)-csc(B)*sin((B+Pi)/3))+csc(C/3)*(csc(B)*csc((A-Pi)/3)+csc(A)*sec((2*B+Pi)/6))*(csc(A)*sin((A+Pi)/3)-csc(C)*sin((C+Pi)/3))) : :

X(66565) lies on these lines: {2, 3602}, {357, 5455}, {395, 66563}, {3603, 38416}, {31931, 31932}, {31934, 31935}, {41109, 41110}, {66555, 66556}, {66558, 66559}, {66561, 66562}, {66569, 66570}, {66576, 66577}

X(66565) = (X(2), X(38417))-harmonic conjugate of X(23031)

X(66566) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st MORLEY AND ROUSSEL

Trilinears    csc((B-C)/3)*csc((2*A+Pi)/6)*(-2*cos(C/3-Pi/6)*csc((2*B+Pi)/6)*sin((B-Pi)/3)*(cos(B+Pi/6)-2*sin(B/3)+2*(sin((2*A)/3)+sin((2*C)/3))*sin(B/3-Pi/6))+2*cos(B/3-Pi/6)*csc((2*C+Pi)/6)*sin((C-Pi)/3)*(cos(C+Pi/6)-2*sin(C/3)+2*(sin((2*A)/3)+sin((2*B)/3))*sin(C/3-Pi/6))) : :
X(66566) = 3*X(356)+X(66583)

X(66566) lies on these lines: {356, 1134}, {358, 66567}, {3273, 66571}, {8065, 14166}

X(66567) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd MORLEY AND ROUSSEL

Trilinears    sec(A/3)*(-4*(-1/2*sqrt(3)+cos((4*A+Pi)/6)+cos((4*B+Pi)/6)+cos((4*C+Pi)/6))*sin((A-B)/3)*sin((B-C)/3)*sin((-A+C)/3)+(cos((2*A)/3)-cos((2*B)/3))*(cos((2*A)/3)-cos((2*C)/3))*(cos((2*B)/3)-cos((2*C)/3))*sec(B/3)*sec(C/3)*sin((2*A-Pi)/6)) : :

X(66567) lies on these lines: {357, 1137}, {358, 66566}, {3274, 66571}, {3276, 11228}, {3280, 3602}, {8066, 14166}

X(66568) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-INSCRIBED AND 1st MORLEY-MIDPOINT

Trilinears    2*(2*sin((A-5*Pi)/3)+cos(C)*sin((B-5*Pi)/3)+cos(B)*sin((C-5*Pi)/3))+cos((B-C)/3)*(3*sqrt(3)-2*cos((4*A+Pi)/6)-2*cos((4*B+Pi)/6)-2*cos((4*C+Pi)/6)-2*sin((2*A)/3)-2*sin((2*B)/3)-2*sin((2*C)/3)-4*sin((2*(A-Pi))/3)-4*sin((2*(B-Pi))/3)-4*sin((2*(C-Pi))/3)) : :

X(66568) lies on these lines: {3272, 41111}, {3273, 66554}, {3274, 3602}, {3275, 66560}, {3334, 31933}, {3335, 31930}, {66563, 66569}, {66564, 66570}

X(66568) = barycentric product X(3604)*X(23029)
X(66568) = (X(3604), X(66557))-harmonic conjugate of X(3274)

X(66569) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-INSCRIBED AND 2nd MORLEY-MIDPOINT

Trilinears    sqrt(3)*cos(A/3)+cos(C)*cos((2*B+Pi)/6)+cos(B)*cos((2*C+Pi)/6)-sin(A/3)+cos((B-C)/3)*((-3*sqrt(3))/2+sin((2*A)/3)+sin((2*B)/3)+sin((2*C)/3)) : :

X(66569) lies on these lines: {358, 66480}, {3272, 41109}, {3273, 66555}, {3274, 66558}, {3275, 3602}, {3334, 31934}, {3335, 31931}, {66563, 66568}, {66565, 66570}

X(66569) = barycentric product X(3602)*X(23030)
X(66569) = (X(3602), X(66561))-harmonic conjugate of X(3275)

X(66570) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-INSCRIBED AND 3rd MORLEY-MIDPOINT

Trilinears    2*csc(B/3)*csc(C/3)+csc(A/3)*(-3*sqrt(3)-3*sqrt(3)*cot(B/3)*cot(C/3)+cos(B)*csc(B/3)+cos(C)*csc(C/3))+4*cos((B-C)/3)*(cot(A/3)*cot(C/3)+cos((2*B+Pi)/6)*cot(B/3)*csc(A/3)*csc(C/3)) : :

X(66570) lies on these lines: {3272, 41110}, {3273, 3603}, {3274, 66559}, {3275, 66562}, {3334, 31935}, {3335, 31932}, {66564, 66568}, {66565, 66569}

X(66570) = barycentric product X(3603)*X(23031)
X(66570) = (X(3603), X(66556))-harmonic conjugate of X(3273)

X(66571) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-INSCRIBED AND ROUSSEL

Trilinears    sec(A/3)*(sqrt(3)*(-4*(-1/2*sqrt(3)+cos((4*A+Pi)/6)+cos((4*B+Pi)/6)+cos((4*C+Pi)/6))*sin((A-B)/3)*sin((B-C)/3)*sin((-A+C)/3)+(cos((2*A)/3)-cos((2*B)/3))*(cos((2*A)/3)-cos((2*C)/3))*(cos((2*B)/3)-cos((2*C)/3))*sec(B/3)*sec(C/3)*sin((2*A-Pi)/6))-2*cos((2*A+Pi)/6)*sec(B/3)*sec(C/3)*sin((A-B)/3)*sin((B-C)/3)*sin((-A+C)/3)*(3*cos((4*A+Pi)/6)+3*cos((4*B+Pi)/6)+3*cos((4*C+Pi)/6)+sin((2*A)/3)+sin((2*B)/3)+sin((2*C)/3)+2*sin((2*(A-Pi))/3)+2*sin((2*(B-Pi))/3)+2*sin((2*(C-Pi))/3))) : :

X(66571) lies on these lines: {3272, 11228}, {3273, 66566}, {3274, 66567}, {3275, 3604}

X(66572) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-INSCRIBED AND STAMMLER

Trilinears    cos(A)-4*cos(B/3)*cos(C/3)+2*cos((A-Pi)/3) : :

X(66572) lies on these lines: {3, 3272}, {6, 14146}, {3273, 8002}, {3274, 8003}, {3275, 8004}, {3526, 3609}

X(66572) = (X(3272), X(3335))-harmonic conjugate of X(3)

X(66573) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND CIRCUMNORMAL

Trilinears    (2*cos((2*(B-C))/3)*cos(C)*(-cos((B-C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-B)/3)+cos(A-B)-2*cos((2*(A-B))/3)*sin(C-Pi/6)))*((cos(B/3)-cos(B)+2*cos(A/3)*cos(C/3))*(sin((2*A+Pi)/6)-2*cos((B-C)/3)*sin((4*A+Pi)/6))-(cos(A/3)-cos(A)+2*cos(B/3)*cos(C/3))*(sin((2*B+Pi)/6)-2*cos((-A+C)/3)*sin((4*B+Pi)/6)))-(2*cos(B)*cos((2*(B-C))/3)*(-cos((B-C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-C)/3)+cos(A-C)-2*cos((2*(A-C))/3)*sin(B-Pi/6)))*((2*cos(A/3)*cos(B/3)+cos(C/3)-cos(C))*(sin((2*A+Pi)/6)-2*cos((B-C)/3)*sin((4*A+Pi)/6))-(cos(A/3)-cos(A)+2*cos(B/3)*cos(C/3))*(sin((2*C+Pi)/6)-2*cos((A-B)/3)*sin((4*C+Pi)/6))) : :

X(66573) lies on these lines: {3, 8011}, {3279, 8065}, {3281, 8066}, {3283, 8067}, {31930, 66575}, {31931, 66576}, {31932, 66577}

X(66573) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (3, 8011, 66574), (3, 66580, 8011)

X(66574) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND CIRCUMTANGENTIAL

Trilinears    (cos(C/3)*cos(A+Pi/6)-cos(A/3)*cos(C+Pi/6))*(2*cos(B)*cos((2*(B-C))/3)*(-cos((B-C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-C)/3)+cos(A-C)-2*cos((2*(A-C))/3)*sin(B-Pi/6)))-(cos(B/3)*cos(A+Pi/6)-cos(A/3)*cos(B+Pi/6))*(2*cos((2*(B-C))/3)*cos(C)*(-cos((B-C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-B)/3)+cos(A-B)-2*cos((2*(A-B))/3)*sin(C-Pi/6))) : :

X(66574) lies on these lines: {3, 8011}, {16, 358}, {31933, 66575}, {31934, 66576}, {31935, 66577}

X(66574) = (X(3), X(8011))-harmonic conjugate of X(66573)

X(66575) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND 1st MORLEY-MIDPOINT

Trilinears    (3+2*cos((2*A)/3)+2*cos((2*B)/3)+2*cos((2*C)/3))*csc(A)*csc((2*A+Pi)/6)*csc((2*B+Pi)/6)*csc((2*C+Pi)/6)*sin((A-B)/3)*sin((B-C)/3)*sin((-A+C)/3)+4*sin((A-2*Pi)/3)*(cos((2*C+Pi)/6)*(-(cot((2*A+Pi)/6)*csc(C))+cot((2*B+Pi)/6)*csc(C)+(-(csc(B)*csc((2*A+Pi)/6))+csc(A)*csc((2*B+Pi)/6))*(-sin((2*(A-Pi))/3)-sin((2*(B-Pi))/3)))+cos((2*B+Pi)/6)*(cot((2*A+Pi)/6)*csc(B)-cot((2*C+Pi)/6)*csc(B)+(csc(C)*csc((2*A+Pi)/6)-csc(A)*csc((2*C+Pi)/6))*(-sin((2*(A-Pi))/3)-sin((2*(C-Pi))/3)))+cos((2*A+Pi)/6)*(-(cot((2*B+Pi)/6)*csc(A))+cot((2*C+Pi)/6)*csc(A)+(csc(C)*csc((2*B+Pi)/6)-csc(B)*csc((2*C+Pi)/6))*(sin((2*(B-Pi))/3)+sin((2*(C-Pi))/3)))) : :

X(66575) lies on these lines: {2, 3273}, {1134, 8067}, {3604, 23029}, {8011, 41111}, {8065, 66554}, {8066, 66557}, {31930, 66573}, {31933, 66574}, {66563, 66576}, {66564, 66577}

X(66576) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND 2nd MORLEY-MIDPOINT

Trilinears    (cos((2*C+Pi)/6)*csc(A)-cos((2*A+Pi)/6)*csc(C))*(sec(B/3)*(sin((2*A+Pi)/6)-2*cos((B-C)/3)*sin((4*A+Pi)/6))-sec(A/3)*(sin((2*B+Pi)/6)-2*cos((-A+C)/3)*sin((4*B+Pi)/6)))-(cos((2*B+Pi)/6)*csc(A)-cos((2*A+Pi)/6)*csc(B))*(sec(C/3)*(sin((2*A+Pi)/6)-2*cos((B-C)/3)*sin((4*A+Pi)/6))-sec(A/3)*(sin((2*C+Pi)/6)-2*cos((A-B)/3)*sin((4*C+Pi)/6))) : :

X(66576) lies on these lines: {2, 3274}, {357, 8065}, {3602, 23030}, {8011, 41109}, {8066, 66558}, {8067, 66561}, {31931, 66573}, {31934, 66574}, {66563, 66575}, {66565, 66577}

X(66577) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND 3rd MORLEY-MIDPOINT

Trilinears    csc((2*A-Pi)/6)*(-2*csc(C/3-Pi/6)*sin(B/3)*sin((A-C)/3)*((sqrt(3)*cos((A-C)/3)-cos(C)*sin(A/3)-2*sin(B/3)-cos(A)*sin(C/3))*(cos(A)+2*cos((2*(-B+C))/3)*(cos((-B+C)/3)-sin(A-Pi/6)))-(sqrt(3)*cos((-B+C)/3)-2*sin(A/3)-cos(C)*sin(B/3)-cos(B)*sin(C/3))*(cos(B)+cos((A-C)/3)+cos(A-C)-2*cos((2*(A-C))/3)*sin(B-Pi/6)))*(1+sin((4*B+Pi)/6))+2*csc(B/3-Pi/6)*sin((A-B)/3)*sin(C/3)*((sqrt(3)*cos((A-B)/3)-cos(B)*sin(A/3)-cos(A)*sin(B/3)-2*sin(C/3))*(cos(A)+2*cos((2*(B-C))/3)*(cos((B-C)/3)-sin(A-Pi/6)))-(sqrt(3)*cos((B-C)/3)-2*sin(A/3)-cos(C)*sin(B/3)-cos(B)*sin(C/3))*(cos((A-B)/3)+cos(A-B)+cos(C)-2*cos((2*(A-B))/3)*sin(C-Pi/6)))*(1+sin((4*C+Pi)/6))) : :

X(66577) lies on these lines: {2, 3275}, {1136, 8066}, {3603, 23031}, {8011, 41110}, {8065, 66556}, {8067, 66562}, {31932, 66573}, {31935, 66574}, {66564, 66575}, {66565, 66576}

X(66578) = PERSPECTOR OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND 2nd MORLEY-ADJUNCT

Trilinears    csc((B-C)/3)*((sqrt(3)-cos((8*C+Pi)/6)-sin((2*A)/3)-sin((2*B)/3)-2*sin((2*C)/3))*(sqrt(3)/2-2*cos((4*A+Pi)/6)-2*cos((4*B+Pi)/6)-2*cos((4*C+Pi)/6)-sin((2*B)/3)+sin((4*B)/3)+sin((2*B+Pi)/3))-(sqrt(3)-cos((8*B+Pi)/6)-sin((2*A)/3)-2*sin((2*B)/3)-sin((2*C)/3))*(sqrt(3)/2-2*cos((4*A+Pi)/6)-2*cos((4*B+Pi)/6)-2*cos((4*C+Pi)/6)-sin((2*C)/3)+sin((4*C)/3)+sin((2*C+Pi)/3))) : :

X(66578) lies on these lines: {358, 5390}, {3602, 8065}, {8066, 66582}

X(66579) = PERSPECTOR OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND 3rd MORLEY-ADJUNCT

Trilinears    (csc(A/3)*(sin((A-2*Pi)/3)*sin((B-2*Pi)/3)+(cos((4*A+Pi)/6)+cos((4*B+Pi)/6))*sin((C-2*Pi)/3))-csc(C/3)*((cos((4*B+Pi)/6)+cos((4*C+Pi)/6))*sin((A-2*Pi)/3)+sin((B-2*Pi)/3)*sin((C-2*Pi)/3)))*(sin((2*B-Pi)/6)*(cos((2*(B-C))/3)*cos((A-Pi)/3)-sin((2*A)/3)*sin(A)-sin(B)*sin(C)+cos((B-C)/3)*sin((8*A+Pi)/6))-sin((2*A-Pi)/6)*(cos((2*(-A+C))/3)*cos((B-Pi)/3)-sin((2*B)/3)*sin(B)-sin(A)*sin(C)+cos((-A+C)/3)*sin((8*B+Pi)/6)))-(csc(A/3)*((cos((4*A+Pi)/6)+cos((4*C+Pi)/6))*sin((B-2*Pi)/3)+sin((A-2*Pi)/3)*sin((C-2*Pi)/3))-csc(B/3)*((cos((4*B+Pi)/6)+cos((4*C+Pi)/6))*sin((A-2*Pi)/3)+sin((B-2*Pi)/3)*sin((C-2*Pi)/3)))*(sin((2*C-Pi)/6)*(cos((2*(B-C))/3)*cos((A-Pi)/3)-sin((2*A)/3)*sin(A)-sin(B)*sin(C)+cos((B-C)/3)*sin((8*A+Pi)/6))-sin((2*A-Pi)/6)*(cos((2*(A-B))/3)*cos((C-Pi)/3)-sin(A)*sin(B)-sin((2*C)/3)*sin(C)+cos((A-B)/3)*sin((8*C+Pi)/6))) : :

X(66579) lies on these lines: {1137, 10258}, {3603, 8066}, {8067, 66583}

X(66580) = HOMOTHETIC CENTER OF THESE TRIANGLES: MORLEY-HOMOTHETIC-CIRCUMSCRIBED AND STAMMLER

Trilinears    (2*cos((2*(B-C))/3)*cos(C)*(-cos((B-C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-B)/3)+cos(A-B)-2*cos((2*(A-B))/3)*sin(C-Pi/6)))*(-((2*cos(B/3)-cos(B)+4*cos(A/3)*cos(C/3))*(cos((A-Pi)/3)-2*cos((B-C)/3)*sin((4*A+Pi)/6)))+(2*cos(A/3)-cos(A)+4*cos(B/3)*cos(C/3))*(cos((B-Pi)/3)-2*cos((A-C)/3)*sin((4*B+Pi)/6)))-(2*cos(B)*cos((2*(-B+C))/3)*(-cos((-B+C)/3)+sin(A-Pi/6))+cos(A)*(cos((A-C)/3)+cos(A-C)-2*cos((2*(A-C))/3)*sin(B-Pi/6)))*(-((4*cos(A/3)*cos(B/3)+2*cos(C/3)-cos(C))*(cos((A-Pi)/3)-2*cos((-B+C)/3)*sin((4*A+Pi)/6)))+(2*cos(A/3)-cos(A)+4*cos(B/3)*cos(C/3))*(cos((C-Pi)/3)-2*cos((A-B)/3)*sin((4*C+Pi)/6))) : :

X(66580) lies on these lines: {3, 8011}, {8002, 8065}, {8003, 8066}, {8004, 8067}

X(66580) = (X(8011), X(66573))-harmonic conjugate of X(3)

X(66581) = TRIPOLAR-PERSPECTOR (OR PERSPECTIVE CENTROID) OF THESE TRIANGLES: 1st MORLEY-ADJUNCT TO ABC

Trilinears    cos(A/3)+cos(B/3)*cos(C/3) : :

Chris van Tienhoven - Dec 10, 2024.
The reciprocal tripolar perspector of these triangles is X(3602).
Note: Tripolar perspectors, or perspective centroids, were introduced in the preamble of X(58747).

X(66581) lies on the curves Q104, Q174 and these lines: {356, 357}, {3273, 3603}, {3278, 6120}, {3279, 6123}, {3605, 5628}, {8065, 14166}

X(66581) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (356, 358, 357), (356, 3602, 58843), (357, 358, 3276), (358, 58843, 3602), (3273, 3604, 3603), (3602, 58843, 357)

X(66582) = TRIPOLAR-PERSPECTOR (OR PERSPECTIVE CENTROID) OF THESE TRIANGLES: 2nd MORLEY-ADJUNCT TO ABC

Trilinears    cos((A+Pi)/3)-cos((B+Pi)/3)*cos((C+Pi)/3) : :

Chris van Tienhoven - Dec 10, 2024.
The reciprocal tripolar perspector of these triangles is X(3603).
Note: Tripolar perspectors, or perspective centroids, were introduced in the preamble of X(58747).

X(66582) lies on these lines: {1136, 1137}, {3274, 3602}, {3280, 6122}, {3281, 6125}, {3606, 5630}, {8066, 66578}

X(66582) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (1136, 1137, 3277), (1137, 3276, 1136), (1137, 58844, 3603), (3274, 3602, 3604), (3276, 3603, 58844), (3603, 58844, 1136)

X(66583) = TRIPOLAR-PERSPECTOR (OR PERSPECTIVE CENTROID) OF THESE TRIANGLES: 3rd MORLEY-ADJUNCT TO ABC

Trilinears    cos(A/3)+cos((B-C)/3)-3*sin(A/3+Pi/6) : :
X(66583) = 3*X(356)-4*X(66566)

Chris van Tienhoven - Dec 10, 2024.
The reciprocal tripolar perspector of these triangles is X(3604).
Note: Tripolar perspectors, or perspective centroids, were introduced in the preamble of X(58747).

X(66583) lies on these lines: {356, 1134}, {3275, 3602}, {3282, 6121}, {3283, 6124}, {3607, 5632}, {8067, 66579}

X(66583) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (1134, 1135, 356), (1135, 3277, 1134), (1135, 58845, 3604), (3275, 3603, 3602), (3277, 3604, 58845), (3604, 58845, 1134)

X(66584) = EULER LINE INTERCEPT OF LINE X(69)X(1511)

Barycentrics    7 a^10-15 a^8 b^2+2 a^6 b^4+14 a^4 b^6-9 a^2 b^8+b^10-15 a^8 c^2+20 a^6 b^2 c^2-10 a^4 b^4 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-10 a^4 b^2 c^4+2 a^2 b^4 c^4+2 b^6 c^4+14 a^4 c^6+8 a^2 b^2 c^6+2 b^4 c^6-9 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics    S^2*(15*R^2-4*SW)-3*SB*SC*(3*R^2-SW) : :
X(66584) = X(2)+2*X(18324), X(2)-4*X(34477), 5*X(2)+4*X(48368), 5*X(3)+4*X(13383), X(3)+2*X(34351), 7*X(3)-X(34622), X(4)+8*X(15331), X(20)+8*X(10020), 2*X(26)+7*X(3523), X(376)+2*X(10201), X(376)+8*X(15330)

As a point on the Euler line, X(66584) has Shinagawa coefficients (E + 16*F, -3*E - 12*F)

See David Nguyen and Ercole Suppa, euclid 7494.

X(66584) lies on these lines: {2, 3}, {69, 1511}, {524, 47391}, {542, 11202}, {599, 44201}, {1992, 15361}, {2931, 35228}, {3018, 21843}, {3098, 48378}, {3431, 37644}, {3581, 37645}, {4549, 5972}, {5569, 64781}, {5642, 63425}, {5654, 10182}, {5663, 35260}, {5892, 38064}, {5944, 18909}, {5946, 59373}, {6225, 32210}, {6699, 46264}, {7622, 64783}, {7712, 20773}, {8182, 14649}, {9126, 64920}, {9826, 18438}, {9833, 20191}, {10264, 39874}, {11179, 18475}, {11411, 32171}, {11464, 18917}, {11472, 15448}, {11645, 23329}, {13367, 18951}, {14805, 63084}, {15136, 64061}, {15303, 15462}, {17821, 44158}, {18451, 35266}, {18931, 61752}, {20423, 64095}, {22151, 50967}, {25406, 34513}, {32223, 64096}, {32620, 61507}, {32837, 52149}, {35254, 59767}, {35268, 38727}, {39242, 61506}, {41465, 51391}, {43273, 62376}, {43394, 64048}, {58484, 64050}, {63649, 64802}

X(66584) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 376, 18531), (3, 381, 44285), (3, 468, 49669), (41465, 62708, 51391)

X(66585) = EULER LINE INTERCEPT OF LINE X(17986)X(47188)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^12-13 a^10 b^2+25 a^8 b^4-10 a^6 b^6-20 a^4 b^8+23 a^2 b^10-7 b^12-13 a^10 c^2+22 a^8 b^2 c^2-18 a^6 b^4 c^2+36 a^4 b^6 c^2-41 a^2 b^8 c^2+14 b^10 c^2+25 a^8 c^4-18 a^6 b^2 c^4-24 a^4 b^4 c^4+18 a^2 b^6 c^4+7 b^8 c^4-10 a^6 c^6+36 a^4 b^2 c^6+18 a^2 b^4 c^6-28 b^6 c^6-20 a^4 c^8-41 a^2 b^2 c^8+7 b^4 c^8+23 a^2 c^10+14 b^2 c^10-7 c^12) : :
Barycentrics    SB*SC*(108*R^4-36*R^2*SA-48*R^2*SW+9*SA*SW+5*SW^2) : :

As a point on the Euler line, X(66585) has Shinagawa coefficients (36*F^2,-E^2-8*E*F+20*F^2)

See David Nguyen and Ercole Suppa, euclid 7494.

X(66585) lies on these lines: {2, 3}, {17986, 47188}, {32274, 62375}

X(66585) = (X(378), X(403))-harmonic conjugate of X(47332)

X(66586) = MIDPOINT OF X(66584) AND X(66585)

Barycentrics    130 a^22-599 a^20 b^2+706 a^18 b^4+803 a^16 b^6-2452 a^14 b^8+1106 a^12 b^10+1820 a^10 b^12-2146 a^8 b^14+242 a^6 b^16+757 a^4 b^18-446 a^2 b^20+79 b^22-599 a^20 c^2+2512 a^18 b^2 c^2-3543 a^16 b^4 c^2+512 a^14 b^6 c^2+4586 a^12 b^8 c^2-6576 a^10 b^10 c^2+3458 a^8 b^12 c^2+1568 a^6 b^14 c^2-3507 a^4 b^16 c^2+1984 a^2 b^18 c^2-395 b^20 c^2+706 a^18 c^4-3543 a^16 b^2 c^4+6600 a^14 b^4 c^4-5548 a^12 b^6 c^4+1524 a^10 b^8 c^4+2406 a^8 b^10 c^4-5800 a^6 b^12 c^4+6132 a^4 b^14 c^4-3030 a^2 b^16 c^4+553 b^18 c^4+803 a^16 c^6+512 a^14 b^2 c^6-5548 a^12 b^4 c^6+6144 a^10 b^6 c^6-3750 a^8 b^8 c^6+5568 a^6 b^10 c^6-4924 a^4 b^12 c^6+800 a^2 b^14 c^6+395 b^16 c^6-2452 a^14 c^8+4586 a^12 b^2 c^8+1524 a^10 b^4 c^8-3750 a^8 b^6 c^8-3156 a^6 b^8 c^8+1542 a^4 b^10 c^8+3476 a^2 b^12 c^8-1738 b^14 c^8+1106 a^12 c^10-6576 a^10 b^2 c^10+2406 a^8 b^4 c^10+5568 a^6 b^6 c^10+1542 a^4 b^8 c^10-5568 a^2 b^10 c^10+1106 b^12 c^10+1820 a^10 c^12+3458 a^8 b^2 c^12-5800 a^6 b^4 c^12-4924 a^4 b^6 c^12+3476 a^2 b^8 c^12+1106 b^10 c^12-2146 a^8 c^14+1568 a^6 b^2 c^14+6132 a^4 b^4 c^14+800 a^2 b^6 c^14-1738 b^8 c^14+242 a^6 c^16-3507 a^4 b^2 c^16-3030 a^2 b^4 c^16+395 b^6 c^16+757 a^4 c^18+1984 a^2 b^2 c^18+553 b^4 c^18-446 a^2 c^20-395 b^2 c^20+79 c^22 : :
Barycentrics    S^2*(13284*R^6-10008*R^4*SW+2508*R^2*SW^2-209*SW^3)-3*SB*SC*(324*R^6-432*R^4*SW+156*R^2*SW^2-17*SW^3) : :

As a point on the Euler line, X(66586) has Shinagawa coefficients (E^3 + 24*E^2*F - 3344*F^3, -3*E^3 + 576*E*F^2 + 816*F^3)

See David Nguyen and Ercole Suppa, euclid 7494.

X(66586) lies on this line: {2, 3}

X(66586) = midpoint of X(66584) and X(66585)

X(66587) = X(4)X(524)∩X(6)X(598)

Barycentrics    7 a^4-a^2 (b^2+c^2)-8 b^4+20 b^2 c^2-8 c^4 : :
X(66587) = 3*X[3]-4*X[1153, 2*X[5]-X[7618], X[376]-2*X[15597, X[382]+4*X[53144], 4*X[546]-X[34511], 5*X[1656]-4*X[7619], X[1657]-4*X[34506], 5*X[3091]-X[53141], X[3529]-5*X[55823], X[3534]-2*X[5569], X[3543]+X[63029], 3*X[3545]-2*X[9771], 5*X[3843]-2*X[7775], 4*X[3845]-X[9766], 4*X[3860]-X[51123], 3*X[5055]-2*X[7622], 2*X[5066]-X[12040, 3*X[5085]-4*X[7606], 2*X[7751]+7*X[62008], X[7758]-10*X[61988], 2*X[7759]-11*X[61990], 4*X[7764]-13*X[61975], 4*X[7780]+5*X[62023], 2*X[7781]-11*X[61970], 4*X[7843]-13*X[61991], X[8719]-2*X[40248], X[9740]+3*X[50687], X[9741]-5*X[41099], 4*X[12101]-X[44678], 8*X[12102]+X[14023], 2*X[13468]+X[15682], X[15685]-4*X[46893], 2*X[33699]+X[47102], 2*X[47101]+X[62040], X[51122]-7*X[61974], 4*X[59546]-13*X[61964]

See Juan José Isach Mayo, euclid 7505.

X(66587) lies on these lines: {2, 11147}, {3, 1153}, {4, 524}, {5, 7618}, {6, 598}, {30, 7610}, {76, 33698}, {83, 60630}, {114, 381}, {115, 11159}, {126, 10355}, {148, 11163}, {183, 8597}, {316, 15533}, {376, 15597}, {382, 53144}, {405, 7621}, {538, 14269}, {542, 53017}, {546, 34511}, {597, 43448}, {599, 8352}, {754, 38335}, {1003, 9166}, {1656, 7619}, {1657, 34506}, {1992, 53418}, {1995, 42008}, {2453, 36196}, {2482, 18424}, {2549, 3363}, {2996, 54476}, {3053, 8859}, {3091, 53141}, {3096, 7841}, {3529, 55823}, {3534, 5569}, {3543, 63029}, {3545, 9771}, {3767, 19661}, {3830, 3849}, {3839, 9742}, {3843, 7775}, {3845, 9766}, {3860, 51123}, {5013, 32480}, {5032, 7745}, {5055, 7622}, {5066, 12040}, {5077, 15271}, {5085, 7606}, {5210, 8860}, {5215, 18362}, {5254, 59373}, {5463, 55950}, {5464, 55951}, {5476, 38734}, {5503, 54713}, {6329, 18842}, {7751, 62008}, {7758, 61988}, {7759, 61990}, {7764, 61975}, {7773, 41136}, {7777, 8596}, {7778, 37350}, {7780, 62023}, {7781, 61970}, {7784, 21356}, {7803, 8370}, {7843, 61991}, {7874, 11318}, {7907, 51238}, {8556, 55164}, {8598, 37637}, {8719, 40248}, {9169, 15638}, {9740, 50687}, {9741, 41099}, {9877, 13860}, {9878, 11361}, {10302, 17503}, {11167, 54718}, {11286, 39563}, {11288, 14971}, {11295, 31710}, {11296, 31709}, {12101, 44678}, {12102, 14023}, {13233, 33980}, {13449, 50955}, {13468, 15682}, {13881, 33007}, {14033, 63543}, {14066, 34604}, {15031, 15815}, {15534, 47286}, {15685, 46893}, {16808, 36775}, {22110, 32815}, {22165, 52713}, {22329, 52942}, {23234, 42011}, {27088, 43620}, {32532, 54616}, {32819, 33006}, {32826, 32984}, {32985, 41139}, {33016, 63101}, {33192, 59635}, {33208, 44535}, {33699, 47102}, {37809, 43291}, {38259, 54639}, {44541, 53127}, {44543, 52691}, {45103, 60626}, {47101, 62040}, {51122, 61974}, {51170, 53101}, {53105, 60238}, {53106, 60131}, {54646, 60228}, {54720, 60629}, {54871, 54913}, {59546, 61964}, {60113, 60628}, {62939, 62945}

X(66587) = midpoint of X(i) in X(j) for these {i,j}: {4, 7620}, {3543, 63029}, {3830, 40727}, {5485, 23334}, {34511, 53143}
X(66587) = reflection of X(i) in X(j) for these {i,j}: {2, 20112}, {3, 7617}, {376, 15597}, {3534, 5569}, {7610, 7615}, {7617, 47617}, {7618, 5}, {8182, 16509}, {8667, 40727}, {8716, 11184}, {8719, 40248}, {11165, 8176}, {11184, 381}, {12040, 5066}, {34504, 7619}, {34505, 7620}, {40727, 18546}, {53142, 9771}
X(66587) = {X (i), X (j)} -harmonic conjugate of X (k) for these {i, j, k} : {4, 5485, 23334}, {381, 11165, 8176}, {671, 11317, 6}, {2549, 3363, 42849}, {3545, 53142, 9771}, {3830, 18546, 8667}, {7615, 8182, 16509}, {7620, 23334, 5485}, {8176, 11165, 11184}, {8182, 16509, 7610}, {8352, 11185, 599}, {8860, 9855, 5210}, {22575, 22576, 381}

X(66588) = X(2)X(154)∩X(373)X(468)

Barycentrics    6 a^6-5 a^4 (b^2+c^2)-4 a^2 (b^4+c^4)+3 (b^2-c^2)^2 (b^2+c^2) : :
X(66588) = X[427]+X[35268], 11*X[3525]+X[11456], X[15760]+X[39242], X[44201]+2*X[61619]

See Juan José Isach Mayo, euclid 7519.

X(66588) lies on these lines: {2, 154}, {5, 15448}, {140, 5663}, {141, 6090}, {182, 47296}, {373, 468}, {427, 35268}, {441, 21163}, {466, 21158}, {511, 6676}, {524, 61644}, {546, 32237}, {597, 61506}, {631, 59767}, {632, 16187}, {852, 41328}, {1176, 15139}, {1368, 17508}, {1495, 37454}, {1624, 54004}, {3091, 41424}, {3292, 3631}, {3523, 62708}, {3525, 11456}, {3547, 37497}, {3549, 12241}, {3580, 12007}, {5050, 13567}, {5066, 32267}, {5092, 5159}, {5094, 44882}, {5102, 11427}, {5157, 5651}, {5480, 7493}, {5640, 37649}, {5642, 20582}, {5646, 10303}, {5650, 7499}, {5893, 10117}, {6677, 63632}, {6689, 11743}, {6696, 15072}, {6723, 20190}, {6748, 41203}, {7386, 55673}, {7494, 31884}, {7495, 7998}, {7542, 9730}, {7552, 16657}, {7568, 9820}, {7605, 37907}, {7789, 9155}, {8359, 35282}, {8550, 37638}, {10020, 13363}, {10154, 38136}, {10300, 55674}, {10691, 55670}, {11002, 14389}, {11188, 15585}, {11284, 35707}, {11746, 44479}, {11801, 40291}, {12024, 14852}, {12039, 47449}, {12045, 58445}, {12099, 44323}, {14915, 52262}, {15030, 16252}, {15055, 15131}, {15520, 41588}, {15760, 39242}, {16051, 53094}, {16511, 47457}, {17825, 38282}, {18583, 32223}, {19130, 37897}, {22352, 62958}, {23515, 37513}, {25328, 32227}, {26958, 55703}, {29181, 44210}, {30771, 55682}, {32225, 61657}, {32348, 61607}, {32455, 41586}, {34664, 36518}, {34828, 44888}, {37643, 53093}, {37904, 50959}, {37910, 48895}, {38317, 44212}, {39561, 61646}, {40132, 47355}, {40280, 62378}, {40550, 47249}, {43650, 52297}, {44158, 45956}, {44201, 61619}, {44891, 46127}, {45298, 55706}, {46847, 63679}, {47200, 58446}, {47311, 50971}, {47315, 48892}, {47597, 48310}, {48905, 52284}, {49731, 61694}, {52292, 54012}, {55166, 55292}, {55711, 63081}, {58437, 61676}

X(66588) = midpoint of X(i) in X(j) for these {i,j}: {2, 13394}, {5, 34513}, {427, 35268}, {6800, 45303}, {15760, 39242}, {44210, 61743}, {61644, 61690}
X(66588) = complement of X(45303)
X(66588) = center of bicevian conic X(2) and X(6800)
X(66588) = {X (i), X (j)} -harmonic conjugate of X (k) for these {i, j, k} : {2, 3796, 23332}, {2, 6800, 45303}, {2, 25406, 61735}, {2, 35260, 10516}, {2, 61680, 61507}, {140, 44516, 59659}, {3589, 32218, 25488}, {6676, 58447, 23292}, {13394, 45303, 6800}, {14389, 52300, 32269}

X(66589) = EULER LINE INTERCEPT OF LINE X(69)X(51393)

Barycentrics    7 a^10-15 a^8 b^2+2 a^6 b^4+14 a^4 b^6-9 a^2 b^8+b^10-15 a^8 c^2+12 a^6 b^2 c^2-6 a^4 b^4 c^2+12 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-6 a^4 b^2 c^4-6 a^2 b^4 c^4+2 b^6 c^4+14 a^4 c^6+12 a^2 b^2 c^6+2 b^4 c^6-9 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics    2*S^2*(7*R^2-2*SW)-3*SB*SC*(2*R^2-SW) : :
X(66589) = X(2)+2*X(14070), X(2)-4*X(34351), X(3)+2*X(10154), X(3)+8*X(44277), X(4)+8*X(1658), X(4)-4*X(10201), X(20)+8*X(13383), 4*X(26)+5*X(631), 5*X(26)+4*X(23336), X(26)+2*X(34477), 4*X(26)-X(34608), 8*X(26)+X(34938), 2*X(26)+X(44441), 4*X(140)+5*X(16195), 8*X(140)+X(31305), 4*X(140)-X(34609), X(376)-4*X(18324), X(376)+8*X(44213), X(6193)-10*X(17821), X(6193)+2*X(64060)

As a point on the Euler line, X(66589) has Shinagawa coefficients (E + 8*F, -3*E - 6*F)

See David Nguyen and Ercole Suppa, euclid 7509.

X(66589) lies on these lines: {2, 3}, {69, 51393}, {524, 23041}, {1154, 64177}, {1568, 41465}, {5944, 18951}, {6193, 17821}, {6515, 11464}, {8780, 44683}, {10282, 11411}, {11427, 64095}, {11433, 18475}, {12245, 51696}, {12317, 20773}, {13367, 64048}, {13754, 35260}, {14826, 44201}, {16111, 40196}, {18925, 61713}, {18928, 37513}, {19154, 22115}, {19357, 61658}, {20806, 50967}, {28708, 54170}, {34781, 44158}, {37478, 37669}, {44673, 46264}, {45011, 56071}, {46730, 61681}, {51392, 62708}, {63703, 64802}

X(66589) = midpoint of X(5054) and X(10245)
X(66589) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 20, 31180), (2, 376, 6643)

X(66590) = X(4)X(193)∩X(235)X(9752)

Barycentrics    (3*a^2-b^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^4-2*(b^2-c^2)^2-a^2*(b^2+c^2)) : :
Barycentrics    SB*SC*(2*SA-SW)*(4*SB*SC-SB*SW-SC*SW) : :

See David Nguyen and Ercole Suppa, euclid 7509.

X(66590) lies on these lines: {4, 193}, {25, 15591}, {51, 63544}, {132, 37197}, {235, 9752}, {382, 2971}, {460, 6525}, {2207, 41521}, {3053, 5139}, {5140,27373}, {8754, 17813}, {14593, 58878}, {32982, 63549}, {54097, 63545}

X(66590) = barycentric product of X(i) and X(j) for these (i,j): (4, 63611), (6353, 44518), (34481, 54412)
X(66590) = barycentric quotient of X(i) and X(j) for these {i,j}: {6353, 63182}, {19118, 56362}, {34481, 6391}, {44518, 6340}, {63611, 69}
X(66590) = trilinear product X(19)*X(63611)
X(66590) = trilinear quotient X(63611)/X(63)
X(66590) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (4, 6392, 5203), (4, 14248, 63535)

X(66591) = EULER LINE INTERCEPT OF LINE X(21663)X(44750)

Barycentrics    19 a^10 - 42 a^8 b^2 + 8 a^6 b^4 + 38 a^4 b^6 - 27 a^2 b^8 + 4 b^10 - 42 a^8 c^2 + 44 a^6 b^2 c^2 - 22 a^4 b^4 c^2 + 32 a^2 b^6 c^2 - 12 b^8 c^2 + 8 a^6 c^4 - 22 a^4 b^2 c^4 - 10 a^2 b^4 c^4 + 8 b^6 c^4 + 38 a^4 c^6 + 32 a^2 b^2 c^6 + 8 b^4 c^6 - 27 a^2 c^8 - 12 b^2 c^8 + 4 c^10, 4 a^10 - 27 a^8 b^2 + 38 a^6 b^4 + 8 a^4 b^6 - 42 a^2 b^8 + 19 b^10 - 12 a^8 c^2 + 32 a^6 b^2 c^2 - 22 a^4 b^4 c^2 + 44 a^2 b^6 c^2 - 42 b^8 c^2 + 8 a^6 c^4 - 10 a^4 b^2 c^4 - 22 a^2 b^4 c^4 + 8 b^6 c^4 + 8 a^4 c^6 + 32 a^2 b^2 c^6 + 38 b^4 c^6 - 12 a^2 c^8 - 27 b^2 c^8 + 4 c^10, 4 a^10 - 12 a^8 b^2 + 8 a^6 b^4 + 8 a^4 b^6 - 12 a^2 b^8 + 4 b^10 - 27 a^8 c^2 + 32 a^6 b^2 c^2 - 10 a^4 b^4 c^2 + 32 a^2 b^6 c^2 - 27 b^8 c^2 + 38 a^6 c^4 - 22 a^4 b^2 c^4 - 22 a^2 b^4 c^4 + 38 b^6 c^4 + 8 a^4 c^6 + 44 a^2 b^2 c^6 + 8 b^4 c^6 - 42 a^2 c^8 - 42 b^2 c^8 + 19 c^10 : :

See David Nguyen, euclid 7517.

X(66591) lies on these lines: {2, 3}, {21663, 44750}, {38793, 55610}, {43273, 44673}, {50955, 51393}

(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)