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This is PART 9: Centers X(16001) - X(18000)

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)


X(16001) =  X(3)X(13)∩X(4)X(542)

Barycentrics    (3*SA-2*SW)*S^2-SB*SC*(SW+4*sqrt(3)*S) : :
X(16001) = X(3)-3*X(13), 5*X(3)-3*X(5473), 2*X(3)-3*X(6771), X(3)+3*X(13103), 5*X(13)-X(5473), 2*X(140)-3*X(5459), 2*X(546)-3*X(5478), 3*X(616)-7*X(3090), 3*X(618)-4*X(3628), 5*X(632)-6*X(6669), 5*X(1656)-3*X(5463), 5*X(3091)-3*X(5617), 2*X(5473)-5*X(6771), X(5473)+5*X(13103), X(6771)+2*X(13103)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27108.

Let Na be the reflection of X(5) in the A-altitude, and define Nb and Nc cyclically. Then X(16001) = X(15)-of-NaNbNc. (Randy Hutson, March 14, 2018)

X(16001) lies on these lines:
{3, 13}, {4, 542}, {5, 530}, {18, 9115}, {61, 5472}, {62, 115}, {140, 5459}, {182, 5335}, {397, 575}, {511, 5318}, {546, 5478}, {616, 3090}, {618, 3628}, {632, 6669}, {1656, 5463}, {3091, 5617}, {3146, 6770}, {3303, 10062}, {3304, 10078}, {3746, 13076}, {5097, 5321}, {5334, 15520}, {5339, 11482}, {6321, 6778}, {7982, 9901}, {10594, 12142}, {11306, 12155}, {11542, 13350}, {11705, 15178}

X(16001) = midpoint of X(i) and X(j) for these {i,j}: {13, 13103}, {6321, 6778}
X(16001) = reflection of X(i) in X(j) for these (i,j): (6771, 13), (13350, 11542)


X(16002) =  X(3)X(14)∩X(4)X(542)

Barycentrics    (3*SA-2*SW)*S^2-SB*SC*(SW-4*sqrt(3)*S) : :
X(16002) = X(3)-3*X(14), 5*X(3)-3*X(5474), 2*X(3)-3*X(6774), X(3)+3*X(13102), 5*X(14)-X(5474), 2*X(140)-3*X(5460), 2*X(546)-3*X(5479), 3*X(617)-7*X(3090), 3*X(619)-4*X(3628), 5*X(632)-6*X(6670), 5*X(1656)-3*X(5464), 5*X(3091)-3*X(5613), 2*X(5474)-5*X(6774), X(5474)+5*X(13102), X(6774)+2*X(13102)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27108.

Let NaNbNc be as at X(16001). Then X(16002) = X(16)-of-NaNbNc. (Randy Hutson, March 14, 2018)

X(16002) lies on these lines:
{3, 14}, {4, 542}, {5, 531}, {17, 9117}, {61, 115}, {62, 5471}, {140, 5460}, {182, 5334}, {398, 575}, {511, 5321}, {546, 5479}, {617, 3090}, {619, 3628}, {632, 6670}, {1656, 5464}, {3091, 5613}, {3146, 6773}, {3303, 10061}, {3304, 10077}, {3746, 13075}, {5097, 5318}, {5335, 15520}, {5340, 11482}, {6321, 6777}, {7982, 9900}, {10594, 12141}, {11305, 12154}, {11543, 13349}, {11706, 15178}

X(16002) = midpoint of X(i) and X(j) for these {i,j}: {14, 13102}, {6321, 6777}
X(16002) = reflection of X(i) in X(j) for these (i,j): (6774, 14), (13349, 11543)


X(16003) =  X(3)X(67)∩X(5)X(113)

Barycentrics    2*(b^2+c^2)*a^8-(7*b^4-6*b^2* c^2+7*c^4)*a^6+(b^2+c^2)*(9*b^ 4-16*b^2*c^2+9*c^4)*a^4-(b^2- c^2)^2*(5*b^4+8*b^2*c^2+5*c^4) *a^2+(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (-18*R^2+3*SA+2*SW)*S^2+(18*R^ 2-SW)*SB*SC : :
X(16003) = X(4)-3*X(9140), 5*X(4)-7*X(15044), 4*X(5)-3*X(113), 2*X(5)-3*X(125), X(5)-3*X(10264), 11*X(5)-12*X(15088), 3*X(67)-X(15069), X(113)-4*X(10264), 3*X(113)-2*X(15063), 11*X(113)-16*X(15088), 3*X(125)-X(15063), 11*X(125)-8*X(15088), 15*X(9140)-7*X(15044), 3*X(9140)+X(15054), 6*X(10264)-X(15063), 11*X(10264)-4*X(15088), X(14981)-3*X(15357), 7*X(15044)+5*X(15054)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27108.

X(16003) lies on the Steiner circle and these lines:
{2, 14094}, {3, 67}, {4, 541}, {5, 113}, {20, 68}, {30, 6070}, {52, 2781}, {64, 265}, {110, 631}, {140, 5609}, {146, 3832}, {155, 15106}, {184, 15132}, {343, 14855}, {376, 15021}, {381, 15027}, {399, 3526}, {427, 13148}, {539, 2071}, {548, 12041}, {549, 11693}, {569, 5622}, {575, 15303}, {858, 13754}, {1112, 1907}, {1205, 10625}, {1209, 12827}, {1503, 8262}, {1511, 3530}, {1539, 3861}, {1656, 5655}, {1906, 12133}, {1986, 11806}, {2771, 12665}, {2931, 9715}, {2948, 9588}, {3028, 15888}, {3031, 9568}, {3043, 9706}, {3047, 9705}, {3091, 10706}, {3292, 15122}, {3520, 10116}, {3523, 9143}, {3524, 15020}, {3528, 12383}, {3545, 15025}, {3564, 10564}, {3580, 14915}, {3843, 7687}, {3853, 10113}, {3855, 15081}, {4301, 13605}, {4309, 10065}, {4317, 10081}, {4330, 12896}, {5067, 12900}, {5070, 6053}, {5071, 15029}, {5169, 5890}, {5449, 6241}, {5462, 12824}, {5504, 9936}, {5576, 13382}, {5734, 7978}, {5881, 13211}, {6000, 11799}, {6102, 14448}, {7722, 15100}, {8550, 12506}, {9589, 9904}, {9624, 11723}, {9643, 12888}, {9644, 10118}, {9656, 12373}, {9657, 12903}, {9670, 12904}, {9671, 12374}, {9680, 10819}, {9693, 10817}, {9714, 10117}, {9970, 15118}, {10111, 13293}, {10574, 14789}, {10575, 12359}, {10733, 12244}, {11579, 13352}, {11623, 11656}, {11694, 12108}, {12105, 15361}, {12121, 15041}, {12163, 15133}, {12219, 12284}, {12236, 13417}, {12301, 12302}, {13336, 15462}, {14683, 15035}, {15023, 15698}, {15039, 15720}

X(16003) = complement of X(14094)
X(16003) = Steiner-circle-antipode of X(15063)
X(16003) = X(15054)-of-Euler-triangle
X(16003) = X(15063)-of-Johnson-triangle
X(16003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 15063, 113), (125, 15063, 5), (140, 5609, 5642), (399, 15061, 5972), (631, 15057, 6699), (3523, 9143, 15034), (6053, 6723, 14643), (7689, 11457, 11750), (9140, 15054, 4), (12308, 14643, 6053)


X(16004) =  X(3)X(10624)∩X(5)X(516)

Barycentrics    2 a^7-a^6 b-4 a^5 b^2+a^4 b^3+2 a^3 b^4+a^2 b^5-b^7-a^6 c+8 a^5 b c+7 a^4 b^2 c-8 a^3 b^3 c-7 a^2 b^4 c+b^6 c-4 a^5 c^2+7 a^4 b c^2-12 a^3 b^2 c^2+6 a^2 b^3 c^2+3 b^5 c^2+a^4 c^3-8 a^3 b c^3+6 a^2 b^2 c^3-3 b^4 c^3+2 a^3 c^4-7 a^2 b c^4-3 b^3 c^4+a^2 c^5+3 b^2 c^5+b c^6-c^7 : :
X(16004) = 3 X[40] + X[1770], 3 X[3] - X[10624], 3 X[3654] - X[12527]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16004) lies on these lines:
{3,10624}, {5,516}, {30,5795}, {40,1478}, {165,6891}, {443,3587}, {517,4298}, {1788,3359}, {2771,6743}, {3654,12527}, {4292,12702}, {5082,7171}, {5766,6908}, {5853,13369}, {6738,13145}, {6847,9778}, {10310,11375}, {11278,12577}, {12575,13624}

X(16004) = midpoint of X(4292) and X(12702)
X(16004) = reflection of X(i) in X(j) for these {i,j}: {6738, 13145}, {11278, 12577}, {12575, 13624}


X(16005) =  X(9)X(3652)∩X(30)X(3680)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16005) lies on the Feuerbach hyperbola and these lines:
{9,3652}, {30,3680}, {2771,12641}, {2827,14224}, {2829,13143}, {3062,7681}, {5559,6001}, {7091,11373}, {12114,15180}

X(16005) = X(3)-vertex conjugate of X(5559)


X(16006) =  X(7)X(496)∩X(30)X(145)

Barycentrics    (2 a^3+a^2 b-2 a b^2-b^3+a^2 c-2 a b c+b^2 c-2 a c^2+b c^2-c^3) (2 a^4+a^3 b-5 a^2 b^2-a b^3+3 b^4+a^3 c+10 a^2 b c+a b^2 c-5 a^2 c^2+a b c^2-6 b^2 c^2-a c^3+3 c^4) : :
X(16006) = 4 X[3579] - 3 X[3650]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16006) lies on these lines:
{7,496}, {30,145}, {79,5586}, {442,10940}, {1317,10624}, {3305,3652}, {3579,3650}, {3647,6700}, {5815,11684}, {6001,13375}, {6831,10044}

X(16006) = reflection of X(10308) in X(11544)


X(16007) =  (name pending)

Barycentrics    a (3 a^8 b-6 a^7 b^2-6 a^6 b^3+18 a^5 b^4-18 a^3 b^6+6 a^2 b^7+6 a b^8-3 b^9+3 a^8 c+6 a^7 b c-2 a^6 b^2 c-18 a^5 b^3 c-12 a^4 b^4 c+18 a^3 b^5 c+18 a^2 b^6 c-6 a b^7 c-7 b^8 c-6 a^7 c^2-2 a^6 b c^2+56 a^5 b^2 c^2-12 a^4 b^3 c^2-14 a^3 b^4 c^2-2 a^2 b^5 c^2-36 a b^6 c^2+16 b^7 c^2-6 a^6 c^3-18 a^5 b c^3-12 a^4 b^2 c^3+28 a^3 b^3 c^3-22 a^2 b^4 c^3+6 a b^5 c^3+24 b^6 c^3+18 a^5 c^4-12 a^4 b c^4-14 a^3 b^2 c^4-22 a^2 b^3 c^4+60 a b^4 c^4-30 b^5 c^4+18 a^3 b c^5-2 a^2 b^2 c^5+6 a b^3 c^5-30 b^4 c^5-18 a^3 c^6+18 a^2 b c^6-36 a b^2 c^6+24 b^3 c^6+6 a^2 c^7-6 a b c^7+16 b^2 c^7+6 a c^8-7 b c^8-3 c^9) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16007) lies on this line: {971,3826}


X(16008) =  (name pending)

Barycentrics    (2 a^4+a^3 b-5 a^2 b^2-a b^3+3 b^4+a^3 c+10 a^2 b c+a b^2 c-5 a^2 c^2+a b c^2-6 b^2 c^2-a c^3+3 c^4) (4 a^6-3 a^5 b-9 a^4 b^2+6 a^3 b^3+6 a^2 b^4-3 a b^5-b^6-3 a^5 c+10 a^4 b c+10 a^3 b^2 c-12 a^2 b^3 c-7 a b^4 c+2 b^5 c-9 a^4 c^2+10 a^3 b c^2-12 a^2 b^2 c^2+10 a b^3 c^2+b^4 c^2+6 a^3 c^3-12 a^2 b c^3+10 a b^2 c^3-4 b^3 c^3+6 a^2 c^4-7 a b c^4+b^2 c^4-3 a c^5+2 b c^5-c^6) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16008) lies on the Mandart hyperbola and this line: {9,3652}


X(16009) =  X(1)X(10308)∩X(9)X(3652)

Barycentrics    a (a^9-a^8 b-4 a^7 b^2+4 a^6 b^3+6 a^5 b^4-6 a^4 b^5-4 a^3 b^6+4 a^2 b^7+a b^8-b^9-a^8 c+9 a^7 b c+8 a^6 b^2 c-29 a^5 b^3 c-16 a^4 b^4 c+31 a^3 b^5 c+12 a^2 b^6 c-11 a b^7 c-3 b^8 c-4 a^7 c^2+8 a^6 b c^2+42 a^5 b^2 c^2+10 a^4 b^3 c^2-20 a^3 b^4 c^2-24 a^2 b^5 c^2-18 a b^6 c^2+6 b^7 c^2+4 a^6 c^3-29 a^5 b c^3+10 a^4 b^2 c^3-14 a^3 b^3 c^3+8 a^2 b^4 c^3+11 a b^5 c^3+10 b^6 c^3+6 a^5 c^4-16 a^4 b c^4-20 a^3 b^2 c^4+8 a^2 b^3 c^4+34 a b^4 c^4-12 b^5 c^4-6 a^4 c^5+31 a^3 b c^5-24 a^2 b^2 c^5+11 a b^3 c^5-12 b^4 c^5-4 a^3 c^6+12 a^2 b c^6-18 a b^2 c^6+10 b^3 c^6+4 a^2 c^7-11 a b c^7+6 b^2 c^7+a c^8-3 b c^8-c^9) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27110.

X(16009) lies on the Jerabek hyperbola of the excentral triangle and on these lines:
{1,10308}, {9,3652}, {30,2136}, {40,3650}, {631,5506}, {2800,13144}, {2950,6223}, {3646,7701}, {5541,6361}

X(16009) = X(3650)-zayin conjugate of X(40)


X(16010) =  X(3)X(67)∩X(6)X(5663)

Barycentrics    a^2 (a^10-a^8 b^2-2 a^6 b^4+2 a^4 b^6+a^2 b^8-b^10-a^8 c^2+13 a^6 b^2 c^2-10 a^4 b^4 c^2+5 a^2 b^6 c^2-7 b^8 c^2-2 a^6 c^4-10 a^4 b^2 c^4-4 a^2 b^4 c^4+8 b^6 c^4+2 a^4 c^6+5 a^2 b^2 c^6+8 b^4 c^6+a^2 c^8-7 b^2 c^8-c^10) : :
X(16010) = 2 X(110) - 3 X(5085) = 2 X(3) - 3 X(5621) = X(2930) - 3 X(5621) = 3 X(5622) - 2 X(6593) = 3 X(6) - 2 X(9970) = 4 X(125) - 3 X(10516) = 4 X(5609) - 7 X(10541) = 3 X(5102) - 2 X(10752) = X(9970) - 3 X(11579) = 3 X(5050) - X(12308) = 3 X(2930) - 4 X(12584) = 9 X(5621) - 4 X(12584) = 3 X(3) - 2 X(12584) = 3 X(5622) - X(14094) = 3 X(10516) - 2 X(14982) = 2 X(3098) - 3 X(15041) = X(11477) + 2 X(15054) = 5 X(3763) - 6 X(15061) = 7 X(10541) - 6 X(15462) = 2 X(5609) - 3 X(15462)

X(16010) lies on these lines:
{3,67}, {6,5663}, {23,1503}, {40,2836}, {64,895}, {74,1296}, {110,5085}, {125,10516}, {146,5480}, {182,399}, {511,10620}, {524,7464}, {575,12162}, {613,7727}, {1177,1498}, {1181,5622}, {1351,9976}, {1352,10264}, {1593,5095}, {1995,9140}, {2393,15138}, {2892,6247}, {3098,15041}, {3292,15106}, {3564,12302}, {3763,15061}, {5050,12308}, {5102,10752}, {5609,10541}, {5941,11646}, {6776,8546}, {7496,9143}, {7527,8550}, {8584,13596}, {10065,10387}, {10510,13754}, {12163,14984}, {15063,15118}

X(16010) = midpoint of X(i) and X(j) for these {i,j}: {895, 15054}, {6776, 12317}
X(16010) = reflection of X(i) in X(j) for these {i,j}: {{6, 11579}, {146, 5480}, {399, 182}, {1350, 74}, {1351, 9976}, {1352, 10264}, {1498, 1177}, {2892, 6247}, {2930, 3}, {11061, 8550}, {11477, 895}, {14094, 6593}, {14982, 125}, {15063, 15118}, {15069, 67}
X(16010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 14982, 10516), (2930, 5621, 3), (5622, 14094, 6593)
X(16010) = crossdifference of every pair of points on line X(2492) X(9003)


X(16011) =  CROSSSUM OF X(1) AND X(188)

Trilinears    csc(B/2) + csc(C/2) : :

X(16011) lies on the conic {{A,B,C,X(1),X(6)}} and these lines:
{1, 188}, {6, 259}, {56, 266}, {58, 6727}, {106, 1130}, {269, 2091}, {10493, 10503}

X(16011) = X(1)-Ceva conjugate of X(15997)
X(16011) = crosspoint of X(1) and X(266)
X(16011) = crosssum of X(1) and X(188)
X(16011) = X(260)-isoconjugate of X(7057)
X(16011) = trilinear pole of line X(649)X(6729)
X(16011) = barycentric product X(i)*X(j) for these {i,j}: {9, 2091}, {174, 15997}, {177, 258}, {178, 289}, {266, 2090}, {514, 3659}, {1488, 7707}, {7028, 10490}
X(16011) = barycentric quotient X(i)/X(j) for these {i,j}: {2091, 85}, {3659, 190}, {15997, 556}
X(16011) = {X(1),X(361)}-harmonic conjugate of X(188)


X(16012) =  CROSSSUM OF X(1) AND X(174)

Trilinears    sec(B/2) + sec(C/2) : :

X(16012) lies on these lines:
{1, 167}, {33, 8122}, {55, 259}, {103, 13444}, {200, 6731}, {220, 6726}, {260, 1130}, {7589, 10498}, {7707, 10502}, {10490, 10500}, {10493, 10503}

X(16012) = X(1)-Ceva conjugate of X(7707)
X(16012) = X(i)-isoconjugate of X(j) for these (i,j): {7, 260}, {651, 10492}
X(16012) = crosspoint of X(i) and X(j) for these (i,j): {1, 259}, {7707, 15997}
X(16012) = crosssum of X(1) and X(174)
X(16012) = barycentric product X(i)*X(j) for these {i,j}: {9, 177}, {178, 259}, {188, 7707}, {200, 14596}, {234, 6726}, {236, 15997}, {3239, 13444}, {6731, 10490}
X(16012) = barycentric quotient X(i)/X(j) for these {i,j}: {41, 260}, {177, 85}, {663, 10492}, {7707, 4146}, {10490, 555}, {13444, 658}, {14596, 1088}
X(16012) = {X(1),X(503)}-harmonic conjugate of X(174)


X(16013) =  X(3)X(12278)∩X(4)X(13289)

Barycentrics    (SB+SC)*((5*R^2-2*SW)*SA^2-2*( 35*R^2*(2*R^2-SW)+4*SW^2)*SA+( 7*R^2-2*SW)*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27117.

X(16013) lies on these lines:
{3, 12278}, {4, 13289}, {74, 11250}, {184, 3357}, {186, 11704}, {378, 5895}, {1147, 12281}, {1614, 15062}, {2071, 7689}, {3448, 12118}, {4550, 10539}, {5622, 8537}, {11457, 12254}, {14059, 14385}


X(16014) =  X(65)X(267)∩X(517)X(11524)

Trilinears    (16*q*p^4+(4*q^2-3)*p*(4*p^2-3)+4*q*(4*q^2-9)*p^2-3*q*(4*q^2-5))*p : : , where p = sin(A/2), q = cos(B/2 - C/2)
Barycentrics    a*((b+c)*a^5+(2*b^2+b*c+2*c^2)*a^4+(b^2-c^2)*(b-c)*a^3-(b^4+b^2*c^2+c^4)*a^2-2*(b^3-c^3)*(b^2-c^2)*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27119.

X(16014) lies on these lines: {65, 267}, {517, 11524}, {5697, 13744}


X(16015) =  CROSSPOINT OF X(2) AND X(174)

Trilinears    (sin B/2 + sin C/2) sec A/2 : :
Barycentrics    csc B/2 + csc C/2 : :

X(16015) lies on the concic {{A,B,C,X(1), X(2)}}, the cubic K748, and these lines:
{1, 188}, {2, 556}, {28, 8119}, {57, 173}, {81, 8125}, {88, 8126}, {105, 3659}, {164, 8351}, {177, 10490}, {234, 14596}, {236, 8056}, {279, 555}, {959, 8094}, {961, 7588}, {1002, 11033}, {1128, 1130}, {5430, 6553}, {6585, 9836}, {6724, 8422}, {6732, 11234}, {8078, 11924}, {8080, 8114}

X(16015) = complement X(556)
X(16015) = cevapoint of X(7707) and X(15997)
X(16015) = crosspoint of X(2) and X(174)
X(16015) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 2090}, {174, 2887}, {259, 1329}, {266, 141}, {604, 178}, {4146, 626}, {6729, 124}, {6733, 3835}, {7370, 2886}
X(16015) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 2090}, {1488, 15997}
X(16015) = X(i)-cross conjugate of X(j) for these (i,j): {7707, 177}, {10500, 7}, {16011, 2091}
X(16015) = X(173)-isoconjugate of X(260)
X(16015) = trilinear pole of line {513, 6728}
X(16015) = crosssum of X(6) and X(259)
X(16015) = barycentric product X(i)*X(j) for these {i,j}: {8, 2091}, {75, 16011}, {174, 2090}, {177, 7048}, {178, 1488}, {234, 7028}, {693, 3659}, {4146, 15997}
X(16015) = barycentric quotient X(i)/X(j) for these {i,j}: {177, 7057}, {2090, 556}, {2091, 7}, {3659, 100}, {7707, 236}, {10490, 2089}, {15997, 188}, {16011, 1}
X(16015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (174, 1488, 258), (174, 15495, 57)


X(16016) =  CROSSPOINT OF X(2) AND X(188)

Trilinears    (cos B/2 + cos C/2) csc A/2 : :
Barycentrics    sec B/2 + sec C/2 : :

X(16016) lies on the cubic K746 and these lines:
{2, 4146}, {9, 173}, {168, 12879}, {178, 10489}, {200, 6731}, {346, 5430}, {557, 15891}, {558, 5451}

X(16016) = complement X(4146)
X(16016) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 178}, {41, 2090}, {188, 2887}, {259, 141}, {266, 2886}, {556, 626}, {6726, 1329}, {6727, 3741}, {6729, 116}
X(16016) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 178}, {236, 7707}
X(16016) = X(16012)-cross conjugate of X(177)
X(16016) = X(i)-isoconjugate of X(j) for these (i,j): {57, 260}, {109, 10492}
X(16016) = crosspoint of X(i) and X(j) for these (i,j): {2, 188}, {178, 2090}
X(16016) = trilinear pole of line {3900, 6730}
X(16016) = crosssum of X(6) and X(266)
X(16016) = barycentric product X(i)*X(j) for these {i,j}: {8, 177}, {75, 16012}, {178, 188}, {234, 6731}, {236, 2090}, {346, 14596}, {556, 7707}, {4397, 13444}, {7027, 10490}
X(16016) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 260}, {177, 7}, {178, 4146}, {234, 555}, {650, 10492}, {7707, 174}, {10490, 7371}, {10502, 177}, {13444, 934}, {14596, 279}, {15997, 1488}, {16012, 1}
X(16016) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 178}, {41, 2090}, {188, 2887}, {259, 141}, {266, 2886}, {556, 626}, {6726, 1329}, {6727, 3741}, {6729, 116}
X(16016) = {X(188),X(236)}-harmonic conjugate of X(9)


X(16017) =  ANTICOMPLEMENT OF X(174)

Barycentrics    sin B/2 + sin C/2 - sin A/2 : :
X(16017) = 3 X(2) - 4 X(2090)

X(16017) lies on these lines:
{2, 174}, {8, 8372}, {145, 8241}, {329, 556}, {1143, 13387}, {1274, 13386}, {6731, 11686}

X(16017) = reflection of X(i) in X(j) for these {i,j}: {145, 8241}, {174, 2090}
X(16017) = anticomplement X(174)
X(16017) = X(556)-Ceva conjugate of X(2)
X(16017) = X(15495)-cross conjugate of X(2)
X(16017) = anticomplement of the isogonal conjugate of X(259)
X(16017) = anticomplement of the isotomic conjugate of X(556)
X(16017) = isotomic conjugate of the anticomplement X(15495)
X(16017) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6, 7057}, {174, 3434}, {188, 69}, {259, 8}, {266, 7}, {556, 6327}, {6724, 2893}, {6725, 1330}, {6726, 329}, {6727, 75}, {6728, 150}, {6729, 149}, {6731, 3436}, {6733, 693}, {7371, 6604}, {7591, 2897}
X(16017) = X(i)-isoconjugate of X(j) for these (i,j): {6, 505}
X(16017) = barycentric product X(i)*X(j) for these {i,j}: {75, 164}, {556, 15495}
X(16017) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 505}, {164, 1}, {15495, 174}
X(16017) = {X(174),X(2090)}-harmonic conjugate of X(2)


X(16018) =  ANTICOMPLEMENT OF X(556)

Barycentrics    csc B/2 + csc C/2 - csc A/2 : :

X(16018) lies on these lines:
{2, 556}, {145, 8094}, {174, 3210}, {192, 8125}, {258, 1999}

X(16018) = anticomplement X(556)
X(16018) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {174, 6327}, {259, 3436}, {266, 69}, {604, 7057}, {4146, 315}, {7370, 3434}
X(16018) = X(174)-Ceva conjugate of X(2)
X(16018) = barycentric product X(75)*X(361)
X(16018) = barycentric quotient X(361)/X(1)


X(16019) =  ANTICOMPLEMENT OF X(4146)

Barycentrics    sec B/2 + sec C/2 - sec A/2 : :

X(16019) lies on these lines:
{2, 4146}, {144, 7670}, {188, 3177}, {192, 11690}

X(16019) = anticomplement X(4146)
X(16019) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 7057}, {188, 6327}, {259, 69}, {266, 3434}, {556, 315}, {6726, 3436}, {6729, 150}, {7370, 6604}
X(16019) = X(188)-Ceva conjugate of X(2)
X(16019) = barycentric product X(75)*X(503)
X(16019) = barycentric quotient X(503)/X(1)


X(16020) =  X(1)X(2)∩X(3)X(105)

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

See Le Viet An and César Lozada, Hyacinthos 27123.

X(16020) lies on these lines:
{1, 2}, {3, 105}, {7, 238}, {9, 4310}, {31, 9776}, {56, 948}, {120, 3813}, {142, 4307}, {218, 3487}, {241, 7288}, {244, 5744}, {279, 1447}, {294, 7124}, {329, 748}, {390, 1738}, {443, 4339}, {516, 4859}, {527, 15601}, {537, 15590}, {726, 3161}, {740, 4402}, {885, 905}, {896, 2094}, {962, 9441}, {982, 5273}, {1001, 4000}, {1086, 5698}, {1212, 2275}, {1279, 2550}, {1386, 4648}, {1449, 4989}, {1475, 3333}, {1743, 5542}, {2263, 8732}, {3246, 5880}, {3361, 3598}, {3475, 4383}, {3485, 5228}, {3523, 11512}, {3576, 7390}, {3731, 4353}, {3751, 11038}, {3973, 5850}, {4220, 8273}, {4327, 8232}, {4419, 15254}, {4869, 5847}, {4966, 5839}, {5129, 13161}, {5247, 11037}, {5255, 11024}, {5573, 5745}, {5731, 7385}, {6361, 13635}, {6666, 7174}, {7407, 8227}, {8056, 10164}, {8616, 9778}, {10165, 11200}

X(16020) = reflection of X(7613) in X(4859)
X(16020) = center of orthoptic-circle-of-Steiner-inellipse-inverse-of-incircle
X(16020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (142, 7290, 4307), (3616, 5222, 1)


X(16021) =  X(15)X(1337)∩X(61)X(1994)

Barycentrics    ((SA+SW)*sqrt(3)+4*S)*(SA+ sqrt(3)*S)*(SB+SC) : :

See Le Viet An and César Lozada, Hyacinthos 27132.

X(16021) lies on these lines: {15, 1337}, {30, 11555}, {61, 1994}, {154, 3129}


X(16022) =  X(16)X(1338)∩X(62)X(1994)

Barycentrics    ((SA+SW)*sqrt(3)-4*S)*(SA- sqrt(3)*S)*(SB+SC) : :

See Le Viet An and César Lozada, Hyacinthos 27132.

X(16022) lies on these lines: {16, 1338}, {62, 1994}, {154, 3130}


X(16023) =  X(323)X(2981)∩X(396)X(11063)

Barycentrics    ((45*R^2+8*SA+6*SW)*S^2+4* sqrt(3)*(3*(SA+SW)*R^2+2*SA^2+ 2*S^2)*S+3*(3*R^2+2*SW)*SA^2)* (SB+SC) : :

See Le Viet An and César Lozada, Hyacinthos 27132.

X(16023) lies on these lines: {323, 2981}, {396, 11063}


X(16024) =  X(323)X(6151)∩X(395)X(11063)

Barycentrics    ((45*R^2+8*SA+6*SW)*S^2-4* sqrt(3)*(3*(SA+SW)*R^2+2*SA^2+ 2*S^2)*S+3*(3*R^2+2*SW)*SA^2)* (SB+SC) : :

See Le Viet An and César Lozada, Hyacinthos 27132.

X(16024) lies on these lines: {323, 6151}, {395, 11063}


X(16025) =  1ST HUNG-LOZADA-EULER POINT

Barycentrics    (S^2+sqrt(3)*(SA-3*SW+8*R^2)*S-3*(SB+SC)*R^2+SA^2+SB*SC-SW^2)*( sqrt(3)*SB*SC+S*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27134.

X(16025) lies on this line: {2,3}


X(16026) =  2ND HUNG-LOZADA-EULER POINT

Barycentrics    (S^2-sqrt(3)*(SA-3*SW+8*R^2)*S-3*(SB+SC)*R^2+SA^2+SB*SC-SW^2)*(sqrt(3)*SB*SC-S*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27134.

X(16026) lies on this line: {2,3}


X(16027) =  X(5)X(19)∩︀X(235)X(6212)

Trilinears    tan A + cos(B - C) : :

X(16027) lies on these lines:
{5, 19}, {235, 6212}, {427, 6213}, {485, 608}, {486, 607}

X(16027) = {X(5),X(19)}-harmonic conjugate of X(16033)


X(16028) =  X(6)X(63)∩︀X(9)X(1592)

Trilinears    cot A + cos(B - C) : :

X(16028) lies on these lines:
{5, 63}, {9, 1592}, {57, 1591}, {1267, 13387}, {3218, 15234}, {3219, 15233}, {3305, 15235}, {3306, 15236}


X(16029) =  X(6)X(24)&capsX(97)X(1578)

Trilinears    sin A + sec(B - C) : :

X(16029) lies on these lines:
{6, 24}, {97, 1578}, {275, 3093}, {485, 8901}, {3092, 8884}

X(16029) = barycentric product X(54)X(1591)
X(16029) = barycentric quotient X(1591)/X(311)
X(16029) = {X(6),X(54)}-harmonic conjugate of X(16034)


X(16030) =  CROSSSUM OF X(5) AND X(51)

Trilinears    cos A + sec(B - C) : :

X(16030) lies on these lines:
{2, 8901}, {3, 54}, {25, 262}, {95, 183}, {96, 7393}, {1593, 8884}, {1598, 4994}, {2623, 10329}, {3135, 11427}, {3933, 14096}, {4993, 5020}, {7485, 9755}, {9777, 9792}

X(16030) = X(826)-cross conjugate of X(1634)
X(16030) = crosspoint of X(54) and X(95)
X(16030) = crosssum of X(5) and X(51)
X(16030) = X(54)-waw conjugate of X(6)
X(16030) = X(i)-isoconjugate of X(j) for these (i,j): {5, 82}, {51, 3112}, {83, 1953}, {251, 14213}, {308, 2179}, {827, 2618}, {1799, 2181}, {4599, 12077}
X(16030) = barycentric product X(i)*X(j) for these {i,j}: {38, 2167}, {39, 95}, {54, 141}, {97, 427}, {275, 3917}, {933, 2525}, {1235, 14533}, {1634, 15412}, {1930, 2148}, {2623, 4576}, {3933, 8882}
X(16030) = barycentric quotient X(i)/X(j) for these {i,j}: {38, 14213}, {39, 5}, {54, 83}, {95, 308}, {97, 1799}, {141, 311}, {427, 324}, {1634, 14570}, {1843, 53}, {1923, 2179}, {1964, 1953}, {2148, 82}, {2167, 3112}, {3005, 12077}, {3051, 51}, {3917, 343}, {8061, 2618}, {14533, 1176}, {14586, 827}
X(16030) = isogonal conjugate of X(17500)
X(16030) = {X(3),X(54)}-harmonic conjugate of X(16035)


X(16031) =  X(19)X(54)∩X(275)X(6213)

Trilinears    tan A + sec(B - C) : :

X(16031) lies on these lines:
{19, 54}, {275, 6213}, {6212, 8884}

X(16031) = {X(2),X(54)}-harmonic conjugate of X(16036)


X(16032) =  X(2)X(54)∩X(95)X(491)

Trilinears    csc A + sec(B - C) : :

X(16032) lies on these lines:
{2, 54}, {95, 491}, {97, 1589}, {275, 485}, {590, 14533}, {1585, 8884}, {1591, 8901}, {3068, 8882}, {6413, 8795}

X(16032) = cevapoint of X(i) and X(j) for these (i,j): {485, 6413}, {5409, 11090}
X(16032) = X(i)-isoconjugate of X(j) for these (i,j): {371, 1953}, {486, 2180}, {492, 2179}, {2181, 5408}
X(16032) = barycentric product X(i)*X(j) for these {i,j}: {95, 485}, {96, 491}, {275, 11090}, {276, 6413}
X(16032) = barycentric quotient X(i)/X(j) for these {i,j}: {54, 371}, {95, 492}, {96, 486}, {97, 5408}, {275, 1585}, {372, 52}, {485, 5}, {1586, 467}, {5412, 14576}, {6413, 216}, {8577, 51}, {8882, 5413}, {11090, 343}, {13455, 7069}, {14533, 8911}

X(16032) = {X(2),X(54)}-harmonic conjugate of X(16037)


X(16033) =  X(5)X(19)∩X(235)X(6213)

Trilinears    tan A - cos(B - C) : :

X(16033) lies on these lines:
{5, 19}, {235, 6213}, {427, 6212}, {485, 607}, {486, 608}

X(16033) = {X(5),X(19)}-harmonic conjugate of X(16027)


X(16034) =  X(6)X(24)∩X(97)X(1579)

Trilinears    sin A - sec(B - C) : :

X(16034) lies on these lines:
{6, 24}, {97, 1579}, {275, 3092}, {486, 8901}, {3093, 8884}

X(16034) = barycentric product X(54)*X(1592)
X(16034) = barycentric quotient X(1592)/X(311)
X(16034) = {X(6),X(54)}-harmonic conjugate of X(16029)


X(16035) =  X(3)X(54)∩X(4)X(8901)

Trilinears    cos A - sec(B - C) : :

X(16035) lies on these lines:
{3, 54}, {4, 8901}, {6, 15653}, {25, 1093}, {96, 6642}, {275, 1593}, {378, 13381}, {1141, 11815}, {1597, 4994}, {1609, 14533}, {3515, 9307}, {4993, 11479}, {8573, 8882}

X(16035) = crosspoint of X(54) and X(8884)
X(16035) = crosssum of X(5) and X(5562)
X(16035) = X(i)-isoconjugate of X(j) for these (i,j): {5, 775}, {801, 1953}, {821, 5562}
X(16035) = barycentric product X(i)*X(j) for these {i,j}: {54, 13567}, {95, 800}, {97, 235}, {185, 275}, {417, 8794}, {774, 2167}, {1624, 15412}, {2190, 6508}, {6509, 8884}
X(16035) = barycentric quotient X(i)/X(j) for these {i,j}: {54, 801}, {185, 343}, {235, 324}, {774, 14213}, {800, 5}, {1624, 14570}, {2148, 775}, {8882, 1105}, {13567, 311}
X(16035) = {X(3),X(54)}-harmonic conjugate of X(16030)


X(16036) =  X(19)X(54)∩X(275)X(6212)

Trilinears    tan A - sec(B - C) : :

X(16036) lies on these lines: {19, 54}, {275, 6212}, {6213, 8884}

(16036) = {X(2),X(54)}-harmonic conjugate of X(16031)


X(16037) =  X(2)X(54)∩X(95)X(492)

Trilinears    csc A - sec(B - C) : :

X(16037) lies on these lines:
{2, 54}, {95, 492}, {97, 1590}, {275, 486}, {615, 14533}, {1586, 8884}, {1592, 8901}, {3069, 8882}, {6414, 8795}

X(16037) = cevapoint of X(i) and X(j) for these (i,j): {486, 6414}, {5408, 11091}
X(16037) = barycentric product X(i)*X(j) for these {i,j}: {95, 486}, {96, 492}, {275, 11091}, {276, 6414}
X(16037) = barycentric quotient X(i)/X(j) for these {i,j}: {54, 372}, {95, 491}, {96, 485}, {97, 5409}, {275, 1586}, {371, 52}, {486, 5}, {1585, 467}, {5413, 14576}, {6414, 216}, {8576, 51}, {8882, 5412}, {11091, 343}
X(16037) = X(i)-isoconjugate of X(j) for these (i,j): {372, 1953}, {485, 2180}, {491, 2179}, {2181, 5409}

X(16037) = {X(2),X(54)}-harmonic conjugate of X(16032)


X(16038) =  CENTER OF 1st DAO EQUILATERAL TRIANGLE

Barycentrics    a^3 + 4*a^2*b - 3*a*b^2 - 2*b^3 + 4*a^2*c + 6*a*b*c + 2*b^2*c - 3*a*c^2 + 2*b*c^2 - 2*c^3 + 6*Sqrt[3]*a*S : :      (Peter Moses, February 5, 2018)
X(16038) = ((a + b + c)^2 - 4 (b c + c a + a b) - 6 sqrt(3) S)*X(1) + 2 ((a + b + c)^2 - 4 (b c + c a + a b))*X(7) X(16038) = X(1) + 2 X(3638) = 4 X(1323) - X(10651) = 4 X(3638) - X(10652) = 2 X(1) + X(10652)      (Peter Moses, February 5, 2018)

Let D be the circle with center X(1) and radius 2*3-1/2r. Let {AB, AC} = D∩AB, where |AAB| < |BAB|. Define BC and CA cyclically, and define BA and CB cyclically. Let OA be the circle {{X(1), CA, CB}}, and define OB and OC cyclically.
Let TA = OB∩OC, TB = OC∩OA, TC = OA∩OB. The triangle TATBTC is here named the 1st Dao equilateral triangle.

See Dao Thanh Oai, ADGEOM 2197.

The A-vertex of the 1st Dao equilateral triangle is given by the following barycentrics:

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

Substituting - S for S yields another equilateral triangle, T*, congruent to TATBTC. The triangle T*, here named the 2nd Dao equilateral triangle, is perspective to ABC, with perspector X(3639).      (Peter Moses, February 5, 2018)

X(16038) lies on these lines: {1, 7}, {1277, 3338}, {5240, 5852}

X(16038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3638, 10652)


X(16039) =  TRILINEAR POLAR OF THE LINE X(3)X(161)

Barycentrics    (SA-SB)*(SA-SC)*(3*SB^2-8*R^2* SB+3*S^2-4*SC*SA)*(3*SC^2-8*R^ 2*SC+3*S^2-4*SA*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27147.

X(16039) lies on the MacBeath circumconic and this line: {895, 6145}

X(16039) = isogonal conjugate of X(16040)
X(16039) = trilinear pole of the line {3, 161}
X(16039) = barycentric product X(99)*X(6145)
X(16039) = barycentric quotient X(110)/X(7488)
X(16039) = trilinear product X(662)*X(6145)
X(16039) = trilinear quotient X(662)/X(7488)


X(16040) =  ISOGONAL CONJUGATE OF X(16039)

Barycentrics    (SB^2-SC^2)*(3*SA^2-8*R^2*SA+ 3*S^2-4*SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27147.

X(16040) lies on these lines: {6, 14346}, {230, 231}, {1510, 2623}, {2165, 10412}

X(16040) = midpoint of X(647) and X(6753)
X(16040) = isogonal conjugate of X(16039)
X(16040) = barycentric product X(523)*X(7488)
X(16040) = barycentric quotient X(512)/X(6145)
X(16040) = trilinear product X(661)*X(7488)
X(16040) = trilinear quotient X(661)/X(6145)
X(16040) = crossdifference of every pair of points on line X(3)X(161)


X(16041) =  X(2)X(3)∩X(69)X(115)

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

X(16041) lies on these lines:
{2, 3}, {69, 115}, {315, 14568}, {316, 2031}, {625, 1007}, {754, 3767}, {1285, 7806}, {1570, 1992}, {1916, 5485}, {2548, 7861}, {2996, 3933}, {3618, 5475}, {3619, 7853}, {3785, 13468}, {5254, 9766}, {5286, 7773}, {5319, 7843}, {6337, 7748}, {6392, 7776}, {7615, 9466}, {7736, 7790}, {7737, 7844}, {7738, 7752}, {7739, 7775}, {7757, 9770}, {7811, 9166}, {7845, 11008}, {7856, 12156}, {7934, 11185}, {8182, 14971}, {9167, 11147}, {14061, 14907}


X(16042) = X(2)X(3)∩X(39)X(111)

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

X(16042) lies on these lines:
{2, 3}, {6, 9716}, {32, 8585}, {39, 111}, {76, 5971}, {110, 373}, {182, 10546}, {194, 9870}, {323, 576}, {352, 13330}, {511, 10545}, {597, 2930}, {748, 5363}, {750, 7301}, {1383, 3053}, {1506, 10418}, {1994, 3292}, {2502, 5038}, {3054, 11063}, {3066, 11002}, {3291, 5007}, {3329, 9149}, {3746, 5297}, {5012, 6688}, {5017, 8617}, {5085, 7712}, {5158, 15355}, {5182, 9966}, {5459, 13859}, {5460, 13858}, {5480, 7693}, {5505, 12039}, {5563, 7292}, {5609, 13363}, {5650, 15107}, {5888, 14810}, {5913, 7745}, {6090, 11004}, {6248, 9775}, {6800, 10541}, {7605, 14389}, {7664, 7769}, {7772, 9465}, {8550, 9143}, {9166, 13233}, {9225, 13410}, {9306, 11422}, {9544, 10601}, {9730, 14094}, {10162, 14682}, {10539, 11465}, {11430, 15020}, {15030, 15054}


X(16043) = X(2)X(3)∩X(39)X(69)

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

X(16043) lies on these lines:
{2, 3}, {6, 3785}, {32, 3618}, {39, 69}, {76, 7738}, {83, 14907}, {141, 3926}, {160, 15435}, {183, 5286}, {193, 7767}, {315, 7736}, {574, 3619}, {620, 7914}, {626, 1007}, {988, 3912}, {1078, 7735}, {1285, 7787}, {1352, 13334}, {1506, 7935}, {1992, 7772}, {2021, 5207}, {2548, 6683}, {2549, 3934}, {2896, 7774}, {3053, 3589}, {3096, 7763}, {3329, 7904}, {3620, 3933}, {3763, 7789}, {3767, 4045}, {3815, 7784}, {4176, 8041}, {5007, 15810}, {5171, 14561}, {5182, 10991}, {5206, 7889}, {5254, 15271}, {5319, 7780}, {6392, 15048}, {7737, 7808}, {7739, 7751}, {7749, 7913}, {7750, 11174}, {7754, 15589}, {7764, 7865}, {7769, 7937}, {7771, 7859}, {7777, 7928}, {7818, 9698}, {7820, 15515}, {7826, 11008}, {7839, 14482}, {7847, 11185}, {7883, 9770}, {9306, 14133}, {9606, 9766}, {9862, 10352}, {10350, 10359}


X(16044) =  X(2)X(3)∩X(83)X(115)

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

X(16044 lies on these lines:
{2, 3}, {11, 6645}, {12, 4366}, {39, 148}, {51, 6310}, {69, 7900}, {76, 5475}, {83, 115}, {99, 1506}, {141, 7885}, {147, 6248}, {183, 7823}, {192, 9596}, {194, 2548}, {316, 2896}, {330, 9599}, {385, 7745}, {538, 7858}, {543, 9698}, {598, 6179}, {625, 7832}, {671, 7765}, {695, 3124}, {1078, 7747}, {1975, 7777}, {3096, 7825}, {3314, 7773}, {3329, 5254}, {3398, 14651}, {3411, 12155}, {3412, 12154}, {3589, 7923}, {3734, 7752}, {3767, 7787}, {3815, 7783}, {3933, 7941}, {3972, 7746}, {5007, 14568}, {5304, 5395}, {5309, 7878}, {5319, 7615}, {6292, 7911}, {6321, 11272}, {6680, 14061}, {6683, 7847}, {7603, 7769}, {7608, 10992}, {7737, 7793}, {7748, 7786}, {7751, 7812}, {7753, 7760}, {7754, 7921}, {7755, 12150}, {7768, 7843}, {7775, 7796}, {7780, 14537}, {7789, 7925}, {7790, 7808}, {7794, 7809}, {7795, 7912}, {7800, 7898}, {7801, 7814}, {7802, 7815}, {7804, 7828}, {7806, 13881}, {7820, 7899}, {7822, 7934}, {7831, 7842}, {7835, 7862}, {7844, 7846}, {7851, 7875}, {7854, 7860}, {7859, 7861}, {7864, 11174}, {7870, 8176}, {7889, 7919}, {7904, 15271}, {8596, 9606}, {10352, 14639}

X(16044) = anticomplement of X(7824)


X(16045) =  X(2)X(3)∩X(69)X(83)

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

X(16045) lies on these lines:
{2, 3}, {69, 83}, {76, 3618}, {194, 14482}, {315, 3619}, {1007, 7832}, {1235, 1249}, {1285, 3785}, {1992, 7878}, {2548, 7821}, {3589, 5286}, {3620, 7762}, {3673, 5749}, {3734, 6704}, {3763, 7745}, {3767, 7889}, {3926, 11174}, {3934, 7735}, {4385, 5222}, {5319, 9466}, {5485, 7827}, {6292, 7737}, {6337, 7786}, {7736, 7764}, {7800, 7804}, {7812, 10159}, {7859, 11185}, {10359, 14912}


X(16046) =  X(2)X(3)∩X(81)X(99)

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

X(16046) lies on these lines:
{2, 3}, {81, 99}, {172, 3175}, {536, 1333}, {1931, 4921}, {2303, 4664}, {5337, 7816}


X(16047) =  X(2)X(3)∩X(45)X(86)

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

X(16047) lies on these lines:
{2, 3}, {45, 86}, {3662, 4877}


X(16048) =  X(2)X(3)∩X(8)X(105)

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

X(16048) lies on these lines:
{2, 3}, {8, 105}, {34, 7131}, {120, 6284}, {218, 3868}, {614, 3976}, {1001, 4687}, {1486, 4429}, {2975, 16020}, {3290, 4426}, {3836, 7295}, {3889, 7191}, {4645, 7083}


X(16049) =  X(2)X(3)∩X(46)X(58)

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

X(16049) lies on these lines:
{1, 1719}, {2, 3}, {40, 283}, {46, 58}, {65, 81}, {75, 1444}, {86, 4329}, {100, 1792}, {104, 925}, {105, 3565}, {110, 1295}, {197, 3436}, {205, 1766}, {229, 2646}, {284, 2285}, {321, 1791}, {347, 1014}, {476, 2694}, {517, 1437}, {610, 2327}, {759, 4278}, {915, 13398}, {1043, 7169}, {1155, 5324}, {1610, 1812}, {1717, 3612}, {1722, 5358}, {1774, 1780}, {1778, 2245}, {1800, 2360}, {1806, 2362}, {1819, 6282}, {1880, 1950}, {2182, 2287}, {2252, 4269}, {2687, 10420}, {4259, 12220}, {4267, 11509}, {4276, 5530}, {4324, 9591}, {4511, 14868}


X(16050) =  X(2)X(3)∩X(37)X(86)

Barycentrics    (a + b)*(a + 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) : :

X(16050) lies on these lines:
{2, 3}, {37, 86}, {58, 3912}, {69, 1778}, {81, 7123}, {239, 1043}, {306, 5247}, {314, 5336}, {333, 3661}, {344, 2303}, {3662, 8822}, {3695, 6542}, {4357, 4877}, {4676, 5327}


X(16051) =  X(2)X(3)∩X(69)X(125)

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

Let A"B"C" be as at X(376). Then A"B"C" is homothetic to the medial triangle, and the center of homothety is X(16051). (Randy Hutson, March 14, 2018)

X(16051) lies on these lines:
{2, 3}, {69, 125}, {122, 126}, {305, 6340}, {339, 9464}, {575, 11427}, {576, 11433}, {577, 13611}, {1007, 3260}, {1038, 10588}, {1040, 10589}, {1060, 5297}, {1062, 7292}, {1249, 1560}, {1495, 14927}, {1506, 15880}, {1853, 14826}, {1899, 3292}, {1992, 13857}, {2548, 15820}, {2892, 6593}, {2986, 7612}, {3098, 6723}, {3284, 7735}, {3618, 15812}, {3619, 5650}, {4549, 6699}, {5158, 7736}, {5921, 6090}, {6334, 9191}, {6515, 8538}, {6776, 11064}, {7998, 9967}, {8717, 12900}, {9463, 14965}, {9465, 14961}, {11422, 14912}, {11477, 13567}

X(16051) = isotomic conjugate of X(10603)
X(16051) = complement of X(4232)


X(16052) =  X(2)X(3)∩X(115)X(121)

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

X(16052) lies on these lines:
{2, 3}, {10, 3967}, {115, 121}, {519, 1834}, {524, 3017}, {1211, 3679}, {3175, 3695}, {3241, 3936}, {3820, 4429}, {3822, 4026}


X(16053) = X(2)X(3)∩X(9)X(86)

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

X(16053) lies on these lines:
{2, 3}, {9, 86}, {81, 218}, {142, 4877}, {226, 1434}, {284, 6666}, {329, 5333}, {333, 1174}, {1014, 8232}, {1043, 4384}, {1213, 2893}, {1778, 4648}, {1901, 6707}, {3008, 4653}, {3786, 5728}, {5327, 15254}


X(16054) = X(2)X(3)∩X(57)X(85)

Barycentrics    (a + b)*(a + 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) : :

X(16054) lies on these lines:
{2, 3}, {7, 2287}, {9, 8822}, {57, 85}, {58, 3008}, {81, 277}, {86, 142}, {239, 942}, {610, 10436}, {653, 1441}, {1014, 8732}, {1043, 3912}, {1781, 11683}, {2271, 5712}, {2303, 4000}, {2328, 9441}, {2999, 4281}, {3187, 3889}, {3794, 10855}, {4273, 4675}, {4393, 15934}, {4877, 6666}, {5235, 5744}, {5327, 5880}, {5745, 6626}


X(16055) =  X(2)X(3)∩X(76)X(111)

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

X(16055) lies on these lines:
{2, 3}, {76, 111}, {126, 7756}, {194, 5971}, {626, 10418}, {3266, 7781}, {3291, 7780}, {3788, 7664}, {5913, 7750}, {6719, 7830}, {7665, 7836}, {7760, 9465}, {7773, 9745}, {7793, 11580}, {7810, 9172}, {7815, 8585}, {9775, 11257}


X(16056) =  X(2)X(3)∩X(43)X(57)

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

X(16056) lies on these lines:
{2, 3}, {42, 942}, {43, 57}, {141, 1376}, {171, 1582}, {228, 5249}, {511, 1730}, {579, 2238}, {1402, 1738}, {1715, 5907}, {1754, 9306}, {1764, 3819}, {1836, 15507}, {3185, 5880}, {3240, 5708}, {4362, 4372}, {6685, 12436}


X(16057) =  X(2)X(3)∩X(42)X(106)

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

X(16057) lies on these lines:
{2, 3}, {42, 106}, {1222, 4651}


X(16058) = X(2)X(3)∩X(43)X(55)

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

X(16058) lies on these lines:
{2, 3}, {42, 3295}, {43, 55}, {182, 2328}, {228, 3305}, {572, 9306}, {573, 5943}, {846, 1403}, {956, 10453}, {958, 3741}, {991, 3819}, {993, 3840}, {999, 3720}, {1376, 8053}, {1790, 5651}, {2223, 5268}, {2238, 4254}, {2267, 3955}, {3185, 15254}, {3740, 15624}, {3781, 14547}, {3913, 4685}, {3941, 4682}, {4428, 15621}, {5248, 6685}, {5544, 6244}


X(16059) =  X(2)X(3)∩X(43)X(56)

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

X(16059) lies on these lines:
{2, 3}, {42, 999}, {43, 56}, {228, 3306}, {573, 3819}, {991, 5943}, {1054, 1403}, {1376, 3741}, {1575, 2178}, {2183, 3784}, {2223, 5272}, {2238, 5120}, {3295, 3720}, {3474, 15507}, {3742, 15624}, {4685, 12513}, {5687, 10453}, {6686, 15654}, {8053, 8167}, {9306, 13329}


X(16060) =  X(2)X(3)∩X(32)X(86)

Barycentrics    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 - a*c^3 - b*c^3 : :

X(16060) lies on these lines:
{2, 3}, {32, 86}, {39, 2669}, {69, 2271}, {333, 980}, {966, 3926}, {988, 4384}, {1213, 7789}, {1654, 3933}, {3053, 15668}, {3618, 5021}, {3785, 4648}, {4352, 7754}, {5224, 7795}


X(16061) =  X(2)X(3)∩X(39)X(86)

Barycentrics    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 + a*c^3 + b*c^3 : :

X(16061) lies on these lines:
{2, 3}, {39, 86}, {69, 5021}, {239, 5266}, {333, 5337}, {966, 3785}, {1654, 7767}, {2271, 3618}, {3926, 4648}, {5013, 15668}, {5224, 7800}, {6292, 6626}


X(16062) =  X(2)X(3)∩X(10)X(75)

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

X(16062) lies on these lines:
{1, 977}, {2, 3}, {6, 1330}, {8, 3891}, {10, 75}, {12, 1403}, {69, 387}, {86, 315}, {141, 1834}, {192, 3695}, {239, 5814}, {316, 1326}, {333, 1714}, {386, 3454}, {942, 3662}, {956, 5484}, {966, 5286}, {978, 3846}, {1046, 4655}, {1211, 9534}, {1213, 5254}, {1220, 1478}, {1654, 7754}, {3216, 5233}, {3661, 5295}, {3714, 3844}, {3920, 5300}, {3927, 6646}, {3933, 4352}, {4257, 6693}, {4260, 10381}, {4645, 5711}, {4650, 8258}, {4658, 7768}, {4660, 5255}, {4911, 10436}, {5016, 5262}, {5290, 7247}, {5292, 14829}, {5799, 10446}, {7784, 15668}


X(16063) =  X(2)X(3)∩X(67)X(69)

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

X(16063) lies on these lines:
{2, 3}, {50, 7735}, {67, 69}, {74, 4549}, {113, 8717}, {125, 3098}, {149, 11677}, {305, 7768}, {315, 3266}, {316, 11059}, {323, 6776}, {566, 7736}, {577, 6103}, {612, 5270}, {614, 4857}, {1216, 11457}, {1350, 3580}, {1352, 7998}, {1478, 5297}, {1479, 7292}, {1503, 15066}, {1899, 2979}, {1992, 10510}, {1993, 8550}, {2548, 15302}, {2549, 9465}, {3001, 7774}, {3291, 7748}, {3574, 13347}, {3818, 5650}, {3819, 11550}, {3917, 11442}, {5085, 14389}, {5095, 11511}, {5286, 5354}, {5800, 14996}, {5986, 10991}, {5987, 9862}, {6800, 11064}, {7607, 13579}, {7612, 13582}, {7712, 13203}, {7771, 11056}, {11004, 14912}, {11179, 11422}, {11206, 15139}, {11444, 14216}, {12220, 15812}, {14683, 15106}, {14853, 15018}, {15431, 15435}


X(16064) =  X(2)X(3)∩X(38)X(55)

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

X(16064) lies on these lines:
{2, 3}, {35, 3961}, {38, 55}, {51, 13329}, {56, 4332}, {184, 991}, {228, 3220}, {1324, 5010}, {2223, 5322}, {2328, 3917}, {2916, 8053}, {3286, 5347}


X(16065) =  X(2)X(3)∩X(38)X(86)

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

X(16065) lies on these lines:
{2, 3}, {38, 86}, {333, 15523}


X(16066) =  X(2)X(3)∩X(34)X(87)

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

X(16066) lies on these lines:
{2, 3}, {34, 87}, {281, 7119}, {318, 11363}, {1148, 1870}, {1395, 9364}, {1788, 3215}, {1834, 10192}, {7058, 9306}


X(16067) =  X(2)X(3)∩X(11)X(75)

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

X(16067) lies on these lines:
{2, 3}, {11, 75}, {343, 3794}, {496, 5211}, {1329, 5205}, {3825, 5121}


X(16068) =  MIDPOINT OF X(805) AND X(1916)

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

,P. X(16068) lies on the cubics K019 and K756 and on these lines:
{39, 512}, {98, 385}, {694, 3291}, {1567, 2782}, {1692, 8789}, {2021, 9468}, {2023, 2679}

X(16068) = reflection of X(2679) in X(2023)
X(16068) = isogonal conjugate of X(16069)
X(16068) = orthogonal projection of X(98) on line PU(1)
X(16068) = X(1966)-isoconjugate of X(2698)
X(16068) = X(882)-Hirst inverse of X(14251)
X(16068) = X(i)-line conjugate of X(j) for these (i,j): {98, 385}, {511, 385}, {805, 385}, {1916, 385}, {5999, 385}, {9467, 385}, {10754, 385}, {13137, 385}, {13207, 385}, {13225, 385}, {14510, 385}, {15630, 385}
X(16068) = barycentric product X(694)*X(2782)
X(16068) = barycentric quotient X(i)/X(j) for these {i,j}: {2782, 3978}, {9468, 2698}


X(16069) =  X(99)X(511)∩X(287)X(694)

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

X(16069) lies on the cubics K019 and K757 and on these lines:
{99, 511}, {287, 694}, {804, 14382}, {880, 5976}

X(16069) = isogonal conjugate of X(16068)
X(16069) = antipode of X(98) in circle {{X(98),PU(1)}}
X(16069) = intersection of tangents at X(98) and PU(1) to Brocard (third) cubic K019
X(16069) = X(1967)-isoconjugate of X(2782)
X(16069) = barycentric product X(2698)*X(3978)
X(16069) = barycentric quotient X(i)/X(j) for these {i,j}: {385, 2782}, {2086, 6071}, {2698, 694}
X(16069) = intersection, other than X(98) and PU(1), of circle {{X(98),PU(1)}} and Brocard (third) cubic K019


X(16070) =  ISOGONAL CONJUGATE OF X(13414)

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

X(16070) lies on the circle {{X(4),PU(1)}}, the cubics K019, K289, K501, and these lines:
{98, 648}, {232, 1113}, {325, 1313}, {511, 1114}, {1312, 6530}, {1344, 5968}, {1346, 14356}, {2574, 9513}, {8427, 8430}

X(16070) = reflection of X(i) in X(j) for these {i,j}: {1113, 15167}, {15165, 1313}
X(16070) = isogonal conjugate of X(13414)
X(16070) = antigonal image of X(15165)
X(16070) = symgonal image of X(15167)
X(16070) = trilinear pole of line {2575, 3569}


X(16071) =  ISOGONAL CONJUGATE OF X(13415)

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

X(16071) lies on the circle {{X(4),PU(1)}}, the cubics K019, K289, K501, and these lines:
{98, 648}, {232, 1114}, {325, 1312}, {511, 1113}, {1313, 6530}, {1345, 5968}, {1347, 14356}, {2575, 9513}, {8426, 8430}

X(16071) = reflection of X(i) in X(j) for these {i,j}: {1114, 15166}, {15164, 1312}
X(16071) = isogonal conjugate of X(13415)
X(16071) = antigonal image of X(15164)
X(16071) = symgonal image of X(15166)
X(16071) = trilinear pole of line {2574, 3569}


X(16072) =  3RD HUNG-LOZADA-EULER POINT

Barycentrics    a^10-4*(b^4+c^4)*a^6+2*(b^2+c^ 2)^3*a^4+3*(b^2-c^2)^4*a^2-2*( b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (8*R^2-SW)*S^2+3*(4*R^2-SW)* SB*SC : :
X(16072) = 4*X(5) - X(25) = X(4) + 2*X(1368) = 4*X(5) - X(25) = 2*X(1352) + X(10602)

As a point on the Euler line, this center has Shinagawa coefficients (E-F, -3*F)

See Tran Quang Hung and César Lozada, Hyacinthos 27151.

X(16072) lies on these lines:
{2, 3}, {6, 1568}, {1352, 10602}, {1853, 15030}, {2393, 10516}, {3167, 12022}, {5651, 13851}, {5654, 11402}, {5891, 14852}, {6761, 9308}, {7809, 14615}, {9140, 12825}, {9627, 11238}, {11178, 14913}, {11180, 13562}, {14644, 14984}

X(16072) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6816, 7395), (5, 12362, 3542), (235, 6643, 11414), (376, 3542, 10154), (376, 10154, 9715), (1370, 3091, 1596), (2043, 2044, 235), (2072, 9818, 5094), (3091, 6804, 7399), (3542, 12362, 9715), (3830, 7529, 13490), (10154, 12362, 376)


X(16073) =  X(30)X(5459)∩X(402)X(5972)

Barycentrics    ((12*R^2-SA-2*SW)*S^2+sqrt( 3)*(-SW^2-4*SB*SC+6*SA^2+5*S^ 2+4*(SW-6*SA)*R^2)*S+(6*(3*SA- 5*SW)*R^2-2*SA^2+2*SB*SC+5*SW^ 2)*SA)*(S^2-3*SB*SC) : :

See Seiichi Kirikami and César Lozada, Hyacinthos 27155.

X(16073) lies on these lines: {30, 5459}, {402, 5972}


X(16074) =  X(30)X(5460)∩X(402)X(5972)

Barycentrics    ((12*R^2-SA-2*SW)*S^2-sqrt( 3)*(-SW^2-4*SB*SC+6*SA^2+5*S^ 2+4*(SW-6*SA)*R^2)*S+(6*(3*SA- 5*SW)*R^2-2*SA^2+2*SB*SC+5*SW^ 2)*SA)*(S^2-3*SB*SC) : :

See Seiichi Kirikami and César Lozada, Hyacinthos 27155.

X(16074) lies on these lines: {30, 5460}, {402, 5972}


X(16075) =  CEVAPOINT OF X(30) AND X(1651)

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

X(16075) lies on the cubic K953 and these: {2, 9033}, {30, 648}, {1494, 1650}, {3163, 4240}

X(16075) = reflection of X(i) in X(j) for these {i,j}: {1494, 1650}, {4240, 3163}
X(16075) = isotomic conjugate of X(16076)
X(16075) = X(1651)-cross conjugate of X(30)
X(16075) = cevapoint of X(30) and X(1651)
X(16075) = trilinear pole of line {30, 14401}
X(16075) = antitomic conjugate of X(4240)
X(16075) = barycentric quotient X(3163)/X(1651)


X(16076) =  ISOTOMIC CONJUGATE OF X(16075)

Barycentrics    (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)*(2*a^8 - 2*a^6*b^2 - 3*a^4*b^4 + 4*a^2*b^6 - b^8 - 2*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 + 6*b^4*c^4 + 4*a^2*c^6 - 2*b^2*c^6 - c^8) : :
X(16076) = X(1494) + 3 X(9410)

X(16076) lies on the cubic K953 and these lines: {2, 525}, {30, 340}

X(16076) = isotomic conjugate of X(16075)
X(16076) = barycentric quotient X(1651)/X(3163)


X(16077) =  ISOGONAL CONJUGATE OF X(9409)

Barycentrics    (a - b)*(a + b)*(a - c)*(a + c)*(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 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :
Barycentrics    1/[(tan B - tan C)(tan B + tan C - 2 tan A)] : :
Barycentrics    1/[(sin 2B - sin 2C)(sin 2B + sin 2C - 2 sin 2A)] : :
Barycentrics    1/[sec B csc(A - C) + sec C csc(A - B)] : :
X(16077) = X(1494) - 3 X(9410)

X(16077) lies on the Steiner circumellipse and thesse lines:
{4, 5641}, {30, 340}, {74, 290}, {99, 1304}, {297, 671}, {316, 10152}, {325, 1552}, {401, 14919}, {447, 903}, {525, 648}, {685, 690}, {850, 6528}, {877, 892}, {2394, 2966}, {3228, 8749}, {11093, 11118}, {11094, 11117}

X(16077) = isogonal conjugate of X(9409)
X(16077) = isotomic conjugate of X(9033)
X(16077) = cevapoint of X(i) and X(j) for these (i,j): {2, 9033}, {30, 525}, {74, 2394}, {107, 2404}, {110, 14590}, {648, 4240}, {6368, 14918}
X(16077) = trilinear pole of line {2, 648}
X(16077) = polar conjugate of X(1637)
X(16077) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 9409}, {43, 2631}
X(16077) = X(i)-cross conjugate of X(j) for these (i,j): {476, 687}, {1304, 15459}, {3268, 264}, {3580, 4590}, {4240, 648}, {9033, 2}, {15107, 250}
X(16077) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9409}, {6, 2631}, {19, 1636}, {30, 810}, {31, 9033}, {48, 1637}, {63, 14398}, {71, 14399}, {228, 11125}, {525, 9406}, {647, 2173}, {656, 1495}, {661, 3284}, {798, 11064}, {822, 1990}, {1400, 14395}, {1409, 14400}, {1820, 14397}, {2148, 14391}, {2155, 14345}, {2156, 14396}, {2159, 14401}, {2420, 3708}, {3049, 14206}, {9407, 14208}
X(16077) = X(3)-vertex conjugate of X(685)
X(16077) = barycentric product X(i)*X(j) for these {i,j}: {69, 15459}, {74, 6331}, {76, 1304}, {648, 1494}, {670, 8749}, {811, 2349}, {6528, 14919}
X(16077) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2631}, {2, 9033}, {3, 1636}, {4, 1637}, {5, 14391}, {6, 9409}, {20, 14345}, {21, 14395}, {22, 14396}, {24, 14397}, {25, 14398}, {27, 11125}, {28, 14399}, {29, 14400}, {30, 14401}, {74, 647}, {99, 11064}, {107, 1990}, {110, 3284}, {112, 1495}, {162, 2173}, {250, 2420}, {340, 5664}, {525, 1650}, {648, 30}, {687, 15454}, {811, 14206}, {823, 1784}, {1304, 6}, {1494, 525}, {2159, 810}, {2349, 656}, {2394, 125}, {2404, 133}, {2409, 6793}, {4235, 5642}, {4240, 3163}, {5627, 14582}, {6331, 3260}, {8749, 512}, {9139, 10097}, {10152, 6587}, {14264, 686}, {14273, 2682}, {14380, 3269}, {14570, 1568}, {14590, 1511}, {14919, 520}, {15404, 2430}, {15459, 4}


X(16078) = ISOTOMIC CONJUGATE OF X(15519)

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

In the plane of a triangle ABC, let P be a point and A'B'C' = intouch triangle. Let A'' be the point other than A' in which the line PA' intersects the incircle, and define B'' and C'' cyclically. Then ABC and A"B"C" are perspective. In particular, for P=X(2) the perspector is X(16078). (César Lozada, February 08, 2018)

X(16078) lies on the the circumconic {{A,B,C,X(2),X(7)}} and this line: {86,16079}

X(16078) = isotomic conjugate of X(15519)
X(16078) = barycentric quotient X(i)/X(j) for these (i,j): (279, 6049), (514, 4943), (3680, 4936), (4373, 3161), (6557, 6555), (8056, 3158), (10029, 4899)
X(16078) = trilinear quotient X(i)/X(j) for these (i,j): (693, 4943), (1088, 6049), (4373, 3158), (6557, 4936)


X(16079) = ISOGONAL CONJUGATE OF X(15519)

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

X(16079) lies on the circumconic {{A,B,C,X(1),X(6)}} and these lines:
{6,9050}, {86,16078}, {1120,1358}

X(16079) = isogonal conjugate of X(15519)
X(16079) = barycentric quotient X(i)/X(j) for these (i,j): (649, 4943), (1407, 6049), (3445, 3161), (16078, 76)
X(16079) = trilinear quotient X(i)/X(j) for these (i,j): (269, 6049), (513, 4943), (3445, 3158), (3680, 6555), (8056, 3161), (16078, 75)


X(16080) = POLAR CONJUGATE OF X(30)

Barycentrics    (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 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4) : :
Barycentrics    (sec A)/(cos A - 2 cos B cos C) : :
Barycentrics    (sec A)/(3 cos A - 2 sin B sin C) : :
Barycentrics    1/{(b^2 + c^2 - a^2)[(b^2 - c^2)^2 + a^2(b^2 + c^2 - 2a^2)]} : :
Barycentrics    1/((S^2 - 3*SB*SC)*SA) : :

X(16080) lies on the circumconic centered at X(1249) (conic {A,B,C,X(107),X(648)}). (Randy Hutson, March 14, 2018)

X(16080) lies on the Kiepert hyperbola, the cubics K490 and K564, and on these lines:
{2, 648}, {4, 74}, {10, 1897}, {13, 470}, {14, 471}, {25, 14458}, {76, 6331}, {96, 10018}, {98, 468}, {132, 14484}, {186, 5627}, {226, 653}, {262, 5094}, {275, 6749}, {297, 671}, {321, 6335}, {338, 2052}, {340, 687}, {343, 801}, {420, 9302}, {427, 14492}, {458, 598}, {459, 11547}, {472, 12817}, {473, 12816}, {1327, 1585}, {1328, 1586}, {1446, 13149}, {1637, 2394}, {1648, 6531}, {1650, 10714}, {1656, 13599}, {1990, 14165}, {2501, 14223}, {3424, 4232}, {3429, 4248}, {3470, 14940}, {3535, 14241}, {3536, 14226}, {4049, 6336}, {4240, 9140}, {5392, 15466}, {5702, 6793}, {6130, 14380}, {6330, 9979}, {7607, 9717}, {10295, 11657}, {11433, 15291}, {13582, 14918}

X(16080) = isogonal conjugate of X(3284)
X(16080) = isotomic conjugate of X(11064)
X(16080) = X(i)-Ceva conjugate of X(j) for these (i,j): {1494, 10152}, {15459, 2394}
X(16080) = X(i)-cross conjugate of X(j) for these (i,j): {6, 5627}, {74, 1494}, {403, 264}, {1637, 107}, {1989, 1300}, {1990, 4}, {2394, 15459}, {2433, 1304}, {14165, 275}, {15311, 253}
X(16080) = cevapoint of X(i) and X(j) for these (i,j): {2, 3580}, {4, 1990}, {6, 186}, {74, 8749}, {125, 1637}, {281, 860}, {470, 471}, {2433, 12079}
X(16080) = crossdifference of every pair of points on line {1636, 9409}
X(16080) = trilinear pole of line {4, 523}
X(16080) = polar conjugate of X(30)
X(16080) = polar-circle inverse of X(14847)
X(16080) = pole wrt polar circle of trilinear polar of X(30) (line X(1636)X(1637))
X(16080) = perspector of ABC and orthoanticevian triangle of X(1494)
X(16080) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3284}, {1745, 2173}, {3737, 2631}
X(16080) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3284}, {3, 2173}, {30, 48}, {31, 11064}, {63, 1495}, {69, 9406}, {109, 14395}, {110, 2631}, {162, 1636}, {163, 9033}, {184, 14206}, {212, 6357}, {255, 1990}, {304, 9407}, {326, 14581}, {577, 1784}, {603, 7359}, {610, 11589}, {656, 2420}, {662, 9409}, {810, 2407}, {822, 4240}, {906, 11125}, {1331, 14399}, {1568, 2148}, {1637, 4575}, {2315, 15454}, {3260, 9247}, {4592, 14398}
X(16080) = barycentric product X(i)*X(j) for these {i,j}: {4, 1494}, {74, 264}, {76, 8749}, {92, 2349}, {253, 10152}, {340, 5627}, {525, 15459}, {648, 2394}, {850, 1304}, {1969, 2159}, {2052, 14919}, {2433, 6331}, {6528, 14380}
X(16080) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11064}, {4, 30}, {5, 1568}, {6, 3284}, {19, 2173}, {25, 1495}, {64, 11589}, {74, 3}, {92, 14206}, {107, 4240}, {112, 2420}, {125, 1650}, {158, 1784}, {186, 1511}, {264, 3260}, {278, 6357}, {281, 7359}, {340, 6148}, {378, 10564}, {381, 1531}, {393, 1990}, {403, 113}, {468, 5642}, {512, 9409}, {523, 9033}, {647, 1636}, {648, 2407}, {650, 14395}, {661, 2631}, {860, 6739}, {1300, 15454}, {1304, 110}, {1312, 14500}, {1313, 14499}, {1494, 69}, {1552, 2777}, {1596, 1533}, {1637, 14401}, {1784, 1099}, {1973, 9406}, {1974, 9407}, {1990, 3163}, {2159, 48}, {2207, 14581}, {2349, 63}, {2394, 525}, {2433, 647}, {2485, 14396}, {2489, 14398}, {2501, 1637}, {2777, 12113}, {3064, 14400}, {4240, 3233}, {5094, 13857}, {5627, 265}, {5667, 15774}, {6070, 13212}, {6344, 14254}, {6587, 14345}, {6591, 14399}, {6623, 1514}, {6753, 14397}, {7649, 11125}, {8749, 6}, {9139, 895}, {9717, 3292}, {10151, 13202}, {10152, 20}, {10419, 5504}, {10421, 12383}, {11251, 1553}, {12077, 14391}, {12079, 125}, {14165, 14920}, {14264, 13754}, {14380, 520}, {14581, 9408}, {14919, 394}, {14989, 12121}, {15291, 15905}, {15311, 3184}, {15459, 648} X(16080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 14847, 7687), (186, 5627, 10421), (5667, 14847, 107)


X(16081) = POLAR CONJUGATE OF X(511)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4 + b^4 - a^2*c^2 - b^2*c^2)*(-a^4 + a^2*b^2 + b^2*c^2 - c^4) : :
Barycentrics    sec A sec(A + ω) : :
Barycentrics    (csc 2A)/(b^4 + c^4 - a^2b^2 - a^2c^2) : :
Barycentrics    1/[a^2(b^4 + c^4 - a^2b^2 - a^2c^2)(b^2 + c^2 - a^2)] : :

X(16081) lis on the circumconic {{A,B,C,X(2),X(6)} and on these lines:
{2, 6331}, {4, 263}, {6, 264}, {25, 98}, {37, 6335}, {39, 276}, {42, 1897}, {115, 6528}, {248, 8795}, {251, 324}, {297, 694}, {393, 15352}, {419, 685}, {653, 1400}, {687, 2966}, {1427, 13149}, {5254, 9291}, {8749, 15459}, {8770, 15466}, {8791, 14165}, {11547, 13854}, {14618, 14998}

X(16081) = isogonal conjugate of X(3289)
X(16081) = isotomic of the isogonal of X(6531)
X(16081) = polar conjugate of X(30)
X(16081) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3289}, {1745, 1755}
X(16081) = X(i)-cross conjugate of X(j) for these (i,j): {98, 290}, {232, 4}, {2395, 685}
X(16081) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3289}, {3, 1755}, {48, 511}, {63, 237}, {69, 9417}, {163, 684}, {184, 1959}, {232, 255}, {240, 577}, {293, 11672}, {304, 9418}, {325, 9247}, {326, 2211}, {336, 9419}, {560, 6393}, {656, 14966}, {810, 2421}, {822, 4230}, {1790, 5360}, {2491, 4592}, {3569, 4575}, {4100, 6530}
X(16081) = X(264)-Hirst inverse of X(290)
X(16081) = cevapoint of X(i) and X(j) for these (i,j): {4, 232}, {98, 6531}
X(16081) = trilinear pole of line {4, 512}
X(16081) = pole wrt polar circle of trilinear polar of X(511) (line X(684)X(2491))
X(16081) = barycentric product X(i)*X(j) for these {i,j}: {4, 290}, {76, 6531}, {92, 1821}, {98, 264}, {158, 336}, {287, 2052}, {685, 850}, {879, 6528}, {1093, 6394}, {1910, 1969}, {2395, 6331}, {2966, 14618}
X(16081) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 511}, {6, 3289}, {19, 1755}, {25, 237}, {76, 6393}, {92, 1959}, {98, 3}, {107, 4230}, {112, 14966}, {158, 240}, {232, 11672}, {248, 577}, {264, 325}, {287, 394}, {290, 69}, {293, 255}, {336, 326}, {393, 232}, {468, 9155}, {523, 684}, {648, 2421}, {685, 110}, {850, 6333}, {877, 15631}, {879, 520}, {1093, 6530}, {1821, 63}, {1824, 5360}, {1910, 48}, {1973, 9417}, {1974, 9418}, {1976, 184}, {2052, 297}, {2207, 2211}, {2211, 9419}, {2395, 647}, {2422, 3049}, {2489, 2491}, {2501, 3569}, {2966, 4558}, {2970, 868}, {3404, 4020}, {5967, 3292}, {6331, 2396}, {6344, 14356}, {6394, 3964}, {6528, 877}, {6530, 2967}, {6531, 6}, {9154, 895}, {11610, 10316}, {12131, 446}, {14265, 3564}, {14382, 12215}, {14600, 14585}, {14601, 14575}, {14618, 2799}


X(16082) = POLAR CONJUGATE OF X(517)

Barycentrics    b*c*(-a^2 + b^2 - c^2)*(a^2 + b^2 - 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) : :
Barycentrics    (sec A)/(cos B + cos C - 1) : :

X(16082) lies on the circumconics {{A,B,C,X(1),X(2)}} and {{A,B,C,X(107),X(648)}} and on these lines:
{1, 318}, {2, 6335}, {4, 957}, {28, 104}, {57, 92}, {81, 648}, {105, 243}, {274, 6331}, {278, 2052}, {279, 331}, {422, 685}, {1022, 6336}, {1257, 4696}, {2250, 2282}, {2990, 13136}, {15466, 15474}

X(16082) = X(1745)-zayin conjugate of X(2183)
X(16082) = polar conjugate X(517)
X(16082) = X(i)-cross conjugate of X(j) for these (i,j): {1455, 2995}, {1870, 286}, {14571, 4}
X(16082) = cevapoint of X(i) and X(j) for these (i,j): {4, 14571}, {1146, 14312}
X(16082) = trilinear pole of line {4, 513}
X(16082) = pole wrt polar circle of trilinear polar of X(517) (line X(1769)X(3310))
X(16082) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2183}, {48, 517}, {71, 859}, {101, 8677}, {184, 908}, {212, 1465}, {219, 1457}, {255, 14571}, {577, 1785}, {822, 4246}, {906, 1769}, {1331, 3310}, {1459, 2427}, {1875, 2289}, {2196, 15507}, {3262, 9247}
X(16082) = barycentric product X(i)*X(j) for these {i,j}: {104, 264}, {693, 1309}, {909, 1969}, {2401, 6335}
X(16082) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 517}, {19, 2183}, {28, 859}, {34, 1457}, {92, 908}, {104, 3}, {107, 4246}, {158, 1785}, {242, 15507}, {264, 3262}, {278, 1465}, {318, 6735}, {393, 14571}, {513, 8677}, {909, 48}, {1118, 1875}, {1309, 100}, {1783, 2427}, {1795, 255}, {1809, 1259}, {1875, 1361}, {2250, 71}, {2342, 212}, {2401, 905}, {5146, 15906}, {6335, 2397}, {6591, 3310}, {7649, 1769}, {13136, 1332}, {14266, 912}, {14578, 577}, {14776, 692}, {15501, 7078}, {15635, 3937}
X(16082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (243, 14198, 1309)


X(16083) =  X(2)X(647)∩X(30)X(290)

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

X(16083) lies on the cubic K953 and these lines:
{2, 647}, {30, 290}, {325, 14941}, {2966, 10317}, {3398, 14382}


X(16084) =  X(2)X(39)∩X(30)X(670)

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

X(16084) lies on the cubic K953 and these lines:
{2, 39}, {30, 670}, {99, 2868}, {886, 1494}, {2367, 2858}, {2549, 6374}, {2782, 14603}, {3734, 9230}

X(16084) = cevapoint of X(865) and X(9035)
X(16084) = X(865)-cross conjugate of X(9035)
X(16084) = X(798)-isoconjugate of X(9091)
X(16084) = X(2)-Hirst inverse of X(305)
X(16084) = antitomic image of X(15014)
X(16084) = barycentric product X(i)*X(j) for these {i,j}: {305, 15014}, {670, 9035} X(16084) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 9091}, {865, 1084}, {9035, 512}, {15014, 25}


X(16085) =  X(2)X(37)∩X(30)X(668)

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

X(16085) lies on the cubic K953 and these lines: {2, 37}, {30, 668}, {889, 1494}

X(16085) = barycentric quotient X(868)/X(1015)


X(16086) =  X(1)X(2)∩X(30)X(190)

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

X(16086) lies on the cubic K953 and these lines:
{1, 2}, {4, 1265}, {30, 190}, {72, 1330}, {80, 3992}, {100, 1324}, {150, 3263}, {312, 3419}, {325, 4561}, {341, 355}, {344, 3488}, {392, 4514}, {515, 3717}, {758, 4645}, {953, 8706}, {960, 5015}, {1016, 5379}, {1043, 3695}, {1264, 10446}, {1494, 4555}, {2273, 2345}, {2975, 4218}, {3057, 5100}, {3701, 5086}, {3702, 5178}, {3710, 7283}, {3869, 5300}, {3876, 5016}, {3877, 5014}, {3940, 4417}, {3952, 5080}, {3969, 4720}, {4115, 5134}, {4385, 5794}, {4388, 4680}, {4389, 11359}, {4723, 5176}, {4737, 5252}, {5687, 11334}

X(16086) = antitomic image of X(447)
X(16086) = X(2)-Hirst inverse of X(306)
X(16086) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (72, 7270, 1330), (4680, 5692, 4388)
X(16086) = barycentric product X(i)*X(j) for these {i,j}: {306, 447}, {867, 1016}
X(16086) = barycentric quotient X(i)/X(j) for these {i,j}: {447, 27}, {867, 1086}


X(16087) =  X(2)X(650)∩X(30)X(2481)

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

X(16087) lies on the cubic K953 and these lines:
{2, 650}, {30, 2481}, {98, 927}, {105, 2864}, {4872, 10030}


X(16088) =  X(2)X(514)∩X(30)X(903)

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

X(16088) lies on the cubic K953 and these lines:
{2, 514}, {30, 903}, {519, 3007}, {857, 4945}, {1443, 6549}, {1494, 4555}


X(16089) =  X(2)X(216)∩X(30)X(6528)

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

X(16089) lies on the cubics K780 and K953 and on these lines:
on lines {2, 216}, {3, 9291}, {30, 6528}, {140, 276}, {230, 16081}, {290, 297}, {317, 1899}, {325, 6331}, {327, 11331}, {340, 520}, {648, 3289}

X(16089) = isotomic conjugate of X(14941)
X(16089) = crosspoint of X(1972) and X(9290)
X(16089) = crosssum of X(1970) and X(1971)
X(16089) = X(2)-daleth conjugate of X(14767)
X(16089) = X(i)-Ceva conjugate of X(j) for these (i,j): {290, 264}, {1972, 9291}
X(16089) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14941}, {48, 1987}, {184, 1956}, {1972, 9247}
X(16089) = X(2)-Hirst inverse of X(264)
X(16089) = antitomic image of X(401)
X(16089) = barycentric product X(i)*X(j) for these {i,j}: {264, 401}, {1955, 1969}, {6130, 6331}
X(16089) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14941}, {4, 1987}, {92, 1956}, {264, 1972}, {275, 1298}, {401, 3}, {1955, 48}, {1971, 184}, {6130, 647}


X(16090) =  X(2)X(92)∩X(3)X(331)

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

X(16090) lies on the cubic K953 and these lines:
{2, 92}, {3, 331}, {98, 927}, {286, 6356}, {325, 4554}, {521, 693}, {2973, 6905}, {4872, 10446}, {6604, 6851}

X(16090) = antitomic image of X(448)
X(16090) = X(2)-Hirst inverse of X(1441)
X(16090) = barycentric product X(448)*X(1441)
X(16090) = barycentric quotient X(448)/X(21)


X(16091) =  X(2)X(7)∩X(30)X(664)

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

X(16091) lies on the cubic K953 and these lines:
on lines {2, 7}, {30, 664}, {69, 7046}, {75, 10400}, {150, 912}, {152, 971}, {388, 760}, {637, 13436}, {638, 13453}, {1231, 1330}, {1275, 5379}, {2893, 7282}, {4566, 5080}, {6356, 8822}, {7009, 10446}, {7183, 7330}, {9312, 9579}

X(16091) = anticomplement X(8558)
X(16091) = X(2)-Hirst inverse of X(307)
X(16091) = barycentric product X(1231)*X(14192)
X(16091) = barycentric quotient X(14192)/X(1172)


X(16092) =  X(2)X(523)∩X(30)X(98)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(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(16092) = X(23) - 3 X(8859) = 3 X(9166) - 2 X(14120)

X(16092) lies on the cubics K394, K492, K741, K953 and on these lines:
{2, 523}, {5, 14246}, {23, 8859}, {30, 98}, {67, 524}, {111, 230}, {115, 5912}, {325, 892}, {385, 10989}, {403, 8753}, {468, 10416}, {512, 5465}, {525, 11006}, {542, 1550}, {543, 7472}, {1499, 9144}, {1503, 14833}, {2452, 11163}, {3767, 14263}, {5099, 5461}, {5133, 8877}, {5309, 14609}, {6103, 7473}, {7468, 9149}, {7610, 9832}, {8705, 11673}, {9166, 14120}

X(16092) = midpoint of X(i) and X(j) for these {i,j}: {385, 10989}, {671, 691}
X(16092) = reflection of X(i) in X(j) for these {i,j}: {5099, 5461}, {7426, 230}
X(16092) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9214, 5968), (10415, 15899, 858), (10416, 15398, 468)
X(16092) = orthoptic-circle-of-Steiner-inellipe-inverse of X(8371)
X(16092) = antitomic image of X(7473)
X(16092) = X(i)-isoconjugate of X(j) for these (i,j): {842, 896}, {922, 5641}, {2642, 5649}
X(16092) = X(16092) = X(i)-Hirst inverse of X(j) for these (i,j): {2, 14977}, {5466, 9214}
X(16092) = X(23)-vertex conjugate of X(5466)
X(16092) = trilinear pole of line {542, 1640}
X(16092) = barycentric product X(i)*X(j) for these {i,j}: {542, 671}, {892, 1640}, {5466, 14999}, {7473, 14977}
X(16092) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 842}, {542, 524}, {671, 5641}, {691, 5649}, {892, 6035}, {1640, 690}, {2247, 896}, {5191, 187}, {5466, 14223}, {6041, 351}, {6103, 468}, {7473, 4235}, {9178, 14998}, {14999, 5468}


X(16093) =  X(2)X(99)∩X(30)X(892)

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

X(16093) lies on the cubic K953 and these lines:
{2, 99}, {30, 892}, {69, 14833}, {11643, 13233}

{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 148, 14832), (111, 148, 671), (7664, 8591, 99)


X(16094) =  X(2)X(881)∩X(30)X(14970)

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

X(16094) lies on the cubic K953 and these lines:
{2, 881}, {30, 14970}, {98, 783}, {733, 1691}


X(16095) =  X(2)X(32)∩X(30)X(4577)

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

X(16095) lies on the cubic K953 and these lines:
{2, 32}, {30, 4577}, {316, 827}, {826, 11416}, {2794, 8928}, {6655, 14885}, {7802, 14247}

X16095) = X(2)-Hirst inverse of X(1799)


X(16096) =  ISOTOMIC CONJUGATE OF X(14944)

Barycentrics    (a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^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(16095) lies on the cubics K776 and K953 and on these lines:
on lines {2, 253}, {3079, 6527}, {3265, 8057}, {14944, 15312}

X(16096) = isotomic conjugate of X(14944)
X(16096) = crosssum of X(8778) and X(8779)
X(16096) = antitomic image of X(441)
X(16096) = X(253)-daleth conjugate of X(1073)
X(16096) = X(1503)-cross conjugate of X(441)
X(16096) = X(i)-isoconjugate of X(j) for these (i,j): {31, 14944}, {154, 8767}, {204, 1297}
X(16096) = X(2)-Hirst inverse of X(253)
X(16096) = barycentric product X(i)*X(j) for these {i,j}: {253, 441}, {2409, 14638}
X(16096) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14944}, {253, 6330}, {441, 20}, {1073, 1297}, {1503, 1249}, {2184, 8767}, {2312, 204}, {8766, 610}, {8779, 154}, {14638, 2419}


X(16097) =  X(2)X(1235)∩X(3267)X(8673)

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

X(16097) lies on the cubic K953 and these lines: {2, 1235}, {3267, 8673}

X(16097) = antitomic image of X(15013)
X(16097) = barycentric quotient X(15013)/X(22)


X(16098) =  X(6)X(2882)∩X(25)X(1084)

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

X(16098) lies on the circumconic {{A,B,C,X(2),X(6)}}, the cubic K953, and on these lines:
{6, 2882}, {25, 1084}, {30, 3228}, {111, 9091}, {148, 2998}, {393, 2971}, {670, 1368}, {694, 2393}, {888, 2433}, {2987, 11416}

X(16098) = reflection of X(i) in X(j) for these {i,j}: {25, 1084}, {670, 1368}
X(16098) = antitomic image of X(25)
X(16098) = X(i)-cross conjugate of X(j) for these (i,j): {865, 512}, {2386, 25}
X(16098) = X(i)-isoconjugate of X(j) for these (i,j): {63, 15014}, {662, 9035}
X(16098) = cevapoint of X(512) and X(865)
X(16098) = barycentric product X(523)*X(9091)
X(16098) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 15014}, {512, 9035}, {1084, 865}, {9091, 99}


X(16099) =  MIDPOINT OF X(3151) AND X(4440)

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

X(16099) lies on the circumconic {{A,B, C, X(2), X(7)}}, the cubic K953, and on these lines:
{2, 1762}, {27, 1086}, {30, 903}, {75, 150}, {86, 1565}, {190, 440}, {335, 8680}, {1234, 1240}, {3151, 4440}, {6548, 11125}

X(16099) = midpoint of X(3151) and X(4440)
X(16099) = reflection of X(i) in X(j) for these {i,j}: {27, 1086}, {190, 440}
X(16099) = cevapoint of X(i) and X(j) for these (i,j): {514, 867}, {3937, 3960}
X(16099) = antitomic image of X(27)
X(867)-cross conjugate of X(514)
X(16099) = X(i)-isoconjugate of X(j) for these (i,j): {228, 447}, {867, 1110}
X(16099) = barycentric quotient X(i)/X(j) for these {i,j}: {27, 447}, {1086, 867}


X(16100) =  REFLECTION OF X(28) IN X(1015)

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

X(16100) lies on the circumconic {{A, B, C, X(1), X(2)}}, the cubic K953, and on these lines:
{28, 1015}, {30, 3227}, {81, 3937}, {274, 1565}

X(16100) = reflection of X(28) in X(1015)
X(16100) = antitomic image of X(28)
X(16100) = cevapoint of X(513) and X(866)
X(16100) = X(866)-cross conjugate of X(513)
X(16100) = X(765)-isoconjugate of X(866)
X(16100) = barycentric quotient X(1015)/X(866)


X(16101) =  ISOTOMIC CONJUGATE OF X(3505)

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

X(16101) lies on the cubics K739 and K953, and on this line: {1502, 5989}

X(16101) = isotomic conjugate of X(3505)
X(16101) = antitomic image of X(384)
X(16101) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3505}, {5207, 9236}, {6660, 9288}
X(16101) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3505}, {384, 6660}, {9230, 5207}


X(16102) =  ANTITOMIC IMAGE OF X(427)

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

X(16102) lies on the cubic K953 and these lines:
{2, 9512}, {30, 14718}, {427, 15449}, {4577, 6676}

X(16102) = reflection of X(i) in X(j) for these {i,j}: {427, 15449}, {4577, 6676}
X(16102) = antitomic image of X(427)


X(16103) =  ANTITOMIC IMAGE OF X(468)

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

X(16103) lies on the cubic K953 and these lines:
{2, 2452}, {523, 14832}, {892, 5159}, {4590, 6390}

X(16103) = reflection of X(892) in X(5159)
X(16103) = antitomic image of X(468)


X(16104) =  X(26)X(15454)∩X(14254)X(15761)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

X(16104) lies on these lines: {26,15454}, {14254,15761}


X(16105) =  MIDPOINT OF X(1986) AND X(10721)

Barycentrics    a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-8 a^10 b^2 c^2+8 a^8 b^4 c^2+8 a^6 b^6 c^2-7 a^4 b^8 c^2-8 a^2 b^10 c^2+6 b^12 c^2-4 a^10 c^4+8 a^8 b^2 c^4-28 a^6 b^4 c^4+12 a^4 b^6 c^4+24 a^2 b^8 c^4-12 b^10 c^4+5 a^8 c^6+8 a^6 b^2 c^6+12 a^4 b^4 c^6-40 a^2 b^6 c^6+7 b^8
X(16105) = 4 X(389) - 3 X(974) = { 2 X(389) - 3 X(1112) = 3 X(1539) - X(5876) = 3 X(1986) - X(6241) = X(6243) + 3 X(7728) = 3 X(74) - 7 X(9781) = 3 X(113) - X(10625) = X(6241) + 3 X(10721) = 3 X(51) - X(10990) = 7 X(9781) - 6 X(11746) = X(974) - 4 X(11807) = X(389) - 3 X(11807) = 4 X(10110) - 3 X(12099) = X(20) - 3 X(12824) = X(11381) - 3 X(13202) = 3 X(5972) - 2 X(13348) = X(11381) + 3 X(13417) = 3 X(13417) - X(14448) = 3 X(13202) + X(14448) = 3 X(11557) - X(14641) = 9 X(5640) - 5 X(15021) = 3 X(12041) - 5 X(15026) = 9 X(7998) - 13 X(15029) = 13 X(15028) - 9 X(15055) = 3 X(125) - 4 X(15465)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27169

X(16105) lies on these lines:
{4,67}, {20,12824}, {51,10990}, {52,3627}, {74,9781}, {113,10625}, {125,1595}, {389,974}, {511,1514}, {541,5446}, {542,13598}, {973,13488}, {1192,2935}, {1539,5876}, {1986,5895}, {2854,10752}, {3542,15131}, {5198,15106}, {5622,10982}, {5640,15021}, {5972,13348}, {6000,13148}, {6243,7728}, {6593,12082}, {7530,15132}, {7731,12292}, {7998,15029}, {9707,15463}, {9919,11426}, {10110,12099}, {10117,11425}, {10628,11576}, {11414,15462}, {11557,14641}, {12041,15026}, {12244,15151}, {14984,15063}, {15028,15055}

X(16105) = midpoint of X(i) and X(j) for these {i,j}: {1986, 10721}, {7731, 12292}, {11381, 14448}, {13202, 13417}
X(16105) = reflection of X(i) in X(j) for these {i,j}: {74, 11746}, {974, 1112}, {1112, 11807}, {12244, 15151}, {15738, 4}
X(16105) = crosssum of X(3) and X(16003)
X(16105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11381, 13417, 14448), (13202, 14448, 11381)


X(16106) =  (name pending)

Trilinears    (8*cos(2*A)-8*cos(4*A)-15)*cos(B-C)+6*cos(3*A)*cos(2*(B-C))-cos(3*(B-C))+10*cos(3*A) : :
Barycentrics    5*S^4+(64*R^4-6*R^2*(SA+5*SW)+4*SA^2-7*SB*SC+SW^2)*S^2-(32*R^2*(R^2-SW)+7*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27170

X(16106) lies on this line: {523, 973}


X(16107) =  (name pending)

Trilinears    (34*cos(2*A)+3*cos(4*A)+5*cos(6*A)-21/2)*cos(B-C)+(-13*cos(A)+3*cos(3*A)-6*cos(5*A)+cos(7*A))*cos(2*(B-C))+(6*cos(2*A)+5*cos(4*A)-cos(6*A)-3/2)*cos(3*(B-C))-cos(3*A)*cos(4*(B-C))-cos(7*A)-19*cos(A)+6*cos(3*A)-10*cos(5*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27170

X(16107) lies on this line: {973,1510}


X(16108) =  (name pending)

Trilinears    (72*cos(2*A)+8*cos(4*A)+65)* cos(B-C)+(-16*cos(A)-10*cos(3* A))*cos(2*(B-C))-cos(3*(B-C))- 22*cos(3*A)-96*cos(A) : :
Barycentrics    11*S^4+(192*R^4-2*R^2*(29* SA+17*SW)+12*SA^2-9*SB*SC-SW^ 2)*S^2-3*(32*R^2*(3*R^2-SW)+3* SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27171

X(16108) lies on this line: {523, 974}


X(16109) =  (name pending)

Trilinears    (164*cos(2*A)+59*cos(4*A)+5* cos(6*A)+217/2)*cos(B-C)-(103* cos(A)+43*cos(3*A)+8*cos(5*A)- cos(7*A))*cos(2*(B-C))+(12* cos(2*A)+5*cos(4*A)-cos(6*A)+ 15/2)*cos(3*(B-C))-cos(3*A)* cos(4*(B-C))-cos(7*A)-113*cos( A)-76*cos(3*A)-16*cos(5*A) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27171

X(16109) lies on this line: {974, 1510}


X(16110) =  REFLECTION OF X(80) IN X(15906)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27177

X(16110) lies on this line: {80, 15906}

X16110) = reflection of X(80) in X(15906)


X(16111) =  MIDPOINT OF X(20) AND X(74)

Barycentrics    4 a^10-6 a^8 b^2-5 a^6 b^4+11 a^4 b^6-3 a^2 b^8-b^10-6 a^8 c^2+22 a^6 b^2 c^2-13 a^4 b^4 c^2-6 a^2 b^6 c^2+3 b^8 c^2-5 a^6 c^4-13 a^4 b^2 c^4+18 a^2 b^4 c^4-2 b^6 c^4+11 a^4 c^6-6 a^2 b^2 c^6-2 b^4 c^6-3 a^2 c^8+3 b^2 c^8-c^10 : :
X16111) = X(110) - 3 X(376) = 3 X(74) - X(3448) = 3 X(20) + X(3448) = X(146) - 5 X(3522) = 3 X(113) - 4 X(5972) = 3 X(3) - 2 X(5972) = 3 X(381) - 4 X(6723) = 3 X(113) - 2 X(7728) = 3 X(3) - X(7728) = 2 X(1112) - 3 X(9730) = 3 X(125) - 2 X(10113) = 3 X(3534) + X(10620) = 2 X(550) + X(10990) = 3 X(3576) - 2 X(11723) = 5 X(10113) - 6 X(11801) = 5 X(125) - 4 X(11801) = 2 X(11801) - 5 X(12041) = X(10113) - 3 X(12041) = 3 X(3534) - X(12121) = 3 X(376) + X(12244) = 8 X(11801) - 5 X(12295) = 4 X(10113) - 3 X(12295) = 4 X(12041) - X(12295) = 3 X(165) - X(12368) = 5 X(631) - 4 X(12900) = 10 X(5972) - 9 X(14643) = 5 X(7728) - 9 X(14643) = 5 X(113) - 6 X(14643) = 5 X(3) - 3 X(14643) = X(3146) - 3 X(14644) = X(11562) - 3 X(14855) = X(10733) - 5 X(15021) = X(3529) + 5 X(15021) = X(146) - 3 X(15035) = 5 X(3522) - 3 X(15035) = X(265) - 3 X(15041) = X(1657) + 3 X(15041) = 7 X(3528) - 5 X(15051) = X(4) - 3 X(15055) = 2 X(6699) - 3 X(15055) = 3 X(4) - 5 X(15059) = 6 X(6699) - 5 X(15059) = 9 X(15055) - 5 X(15059) = X(382) - 3 X(15061) = 2 X(7687) - 3 X(15061) = 4 X(548) - X(15063) = 3 X(3845) - 4 X(15088) = X(399) - 5 X(15696) = 2 X(3448) - 3 X(16003) = 2 X(20) + X(16003)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27179

X(16111) lies on thesse lines:
{2,10721}, {3,113}, {4,6699}, {5,13202}, {20,68}, {22,12893}, {30,125}, {52,974}, {110,376}, {140,1539}, {146,3522}, {165,12368}, {247,7422}, {265,1657}, {378,7706}, {381,6723}, {382,7687}, {399,15696}, {516,11709}, {542,1350}, {548,1511}, {550,5562}, {569,15472}, {631,12900}, {1060,10118}, {1112,9730}, {1209,11598}, {1216,12825}, {1531,15122}, {1533,7575}, {1593,15473}, {2778,9943}, {2781,9967}, {2931,11414}, {2972,12113}, {3024,15326}, {3028,15338}, {3070,8994}, {3071,13969}, {3098,5181}, {3146,14644}, {3528,15051}, {3529,10733}, {3576,11723}, {3845,15088}, {4299,10065}, {4302,10081}, {4324,12896} et al

X16111) = midpoint of X(i) and X(j) for these {i,j}: {20,74}, {110,12244}, {265,1657}, {550,14677}, {3529,10733}, {5925,11744}, {6241,12219}, {6361,7984}, {7722,13201}, {9140,11001}, {10264,15704}, {10620,12121}, {12383,15054}, {13445,13619}
X16111) = reflection of X(i) in X(j) for these {i,j}: {4,6699}, {52,974}, {113,3}, {125,12041}, {382,7687}, {1511,548}, {1531,15122}, {1533,7575}, {1539,140}, {5181,3098}, {5642,8703}, {7728,5972}, {10990,14677}, {11693,15688}, {11807,9729}, {12162,12358}, {12295,125}, {12699,11735}, {12825,1216}, {13202,5}, {13417,14708}, {15063,1511}, {16003,74}, {16105,9826}
X16111) = complement X(10721)
X16111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7728, 5972), (4, 15055, 6699), (146, 3522, 15035), (376, 12244, 110), (382, 15061, 7687), (1657, 15041, 265), (3534, 10620, 12121), (5972, 7728, 113)






leftri  2nd Fuhrmann triangle and related centers: X(16112) - X(16162)  rightri

This preamble and centers X(16112)-X(16162) were contributed by César Eliud Lozada, February 13, 2018.

Let ABC be a triangle and let A1 be the midpoint of the arc BC of its circumcircle not containing A. Define B' and C' cyclically. The triangle A'B'C' is called the Fuhrmann triangle of ABC. (See Fuhrmann triangle from MathWorld). The Furhrman triangle is the reflection of the 2nd circumperp triangle in the sidelines of ABC.

Let ABC be a triangle and let A'' be the midpoint of the arc BC containing A, and define B" and C" cyclically.. The triangle A"B"C" is here named the 2nd Fuhrmann triangle of ABC. This triangle is the reflection of the 1st circumperp triangle in the sidelines of ABC. Its A-vertex has barycentric coordinates:

 A" = -a^2 : b*c+a^2-c^2 : b*c+a^2-b^2

A"B"C" has area S*(2*r+3*R)/(4*r), where r, R and S are the inradius, circumradius and double-area of ABC, respectively.

The 2nd Fuhrmann triangle is perspective to triangles in the following list with perspector X(3): anti-Hutson intouch, anti-incircle-circles, 6th anti-mixtilinear, Ara, Ascella, 1st Brocard, 1st circumperp, 2nd circumperp, 1st Ehrmann, 2nd Euler, Fuhrmann, Johnson, inner-Johnson, outer-Johnson, Kosnita, McCay, medial, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, tangential, Trinh, inner-Vecten, outer-Vecten. Also, A"B"C" is perspective to the inner- and outer-Johnson triangles with perspectors X(16112) and X(12635), respectively.

The appearance of (T, i, j) in the following list means that triangles A"B"C" and T are orthologic with centers X(i) and X(j):
(ABC, 3, 79), (ABC-X3 reflections, 3, 16113), (anti-Aquila, 3, 3649), (anti-Ara, 3, 16114), (5th anti-Brocard, 3, 16115), (anti-Euler, 3, 16116), (anti-Mandart-incircle, 3, 16117), (anticomplementary, 3, 3648), (Aquila, 3, 16118), (Ara, 3, 16119), (Ascella, 4, 6675), (Atik, 4, 16120), (1st Auriga, 3, 16121), (2nd Auriga, 3, 16122), (5th Brocard, 3, 16123), (2nd circumperp tangential, 3, 13743), (1st circumperp, 4, 3651), (2nd circumperp, 4, 21), (inner-Conway, 4, 11684), (Conway, 4, 21), (2nd Conway, 4, 14450), (3rd Conway, 4, 16124), (Euler, 3, 16125), (3rd Euler, 4, 6841), (4th Euler, 4, 442), (excenters-midpoints, 12635, 442), (excenters-reflections, 4, 16126), (excentral, 4, 191), (extouch, 16127, 3650), (2nd extouch, 4, 442), (Feuerbach, 5, 442), (Fuhrmann, 6326, 191), (inner-Garcia, 16128, 3648), (outer-Garcia, 3, 11684), (Garcia-reflection, 12635, 21), (Gossard, 3, 16129), (inner-Grebe, 3, 16130), (outer-Grebe, 3, 16131), (hexyl, 4, 16132), (Honsberger, 4, 16133), (Hutson extouch, 16134, 442), (inner-Hutson, 4, 16135), (Hutson intouch, 4, 10543), (outer-Hutson, 4, 16136), (incircle-circles, 4, 16137), (intouch, 4, 3649), (inverse-in-incircle, 4, 10122), (Johnson, 3, 3652), (inner-Johnson, 3, 16138), (outer-Johnson, 3, 16139), (1st Johnson-Yff, 3, 16140), (2nd Johnson-Yff, 3, 16141), (Lucas homothetic, 3, 16161), (Lucas(-1) homothetic, 3, 16162), (Mandart-incircle, 3, 16142), (medial, 3, 3647), (5th mixtilinear, 3, 5441), (6th mixtilinear, 4, 16143), (2nd Pamfilos-Zhou, 4, 16144), (1st Schiffler, 16145, 10266), (2nd Schiffler, 12635, 11604), (1st Sharygin, 4, 21), (tangential-midarc, 4, 16146), (2nd tangential-midarc, 4, 16147), (3rd tri-squares-central, 3, 16148), (4th tri-squares-central, 3, 16149), (X3-ABC reflections, 3, 16150), (Yff central, 4, 16151), (inner-Yff, 3, 16152), (outer-Yff, 3, 16153), (inner-Yff tangents, 3, 16154), (outer-Yff tangents, 3, 16155)

Note: if T* is a triangle homothetic to ABC, then T* and the 2nd Fuhrmann triangles are orthologic with centers X(3)-of-ABC and X(79)-of-T*.

The appearance of (T, i, j) in the following list means that triangles A"B"C" and T are parallelogic with centers X(i) and X(j): (1st Parry, 3, 16156), (2nd Parry, 3, 16157), (2nd Sharygin, 4, 16158)

The appearance of (T, i, j) in the following list means that triangles A"B"C" and T are cyclologic with centers X(i) and X(j): (ABC, 4, 5127), (excentral, 399, 6326), (Fuhrmann, 3448, 4).

In the following list, (i, j) means that X(i)-of-A"B"C" = X(j): (2, 1699), (3, 16159), (4, 7701), (5, 16160)

underbar

X(16112) = PERSPECTOR OF THESE TRIANGLES: 2nd FUHRMANN AND INNER-JOHNSON

Barycentrics    a*(a^4-(b+c)*a^3-(3*b^2-8*b*c+3*c^2)*a^2+5*(b^2-c^2)*(b-c)*a-2*(b^2+3*b*c+c^2)*(b-c)^2) : :
X(16112) = 5*X(7)-9*X(9779) = 5*X(9)-3*X(165) = 3*X(9)-X(2951) = 5*X(142)-6*X(10171) = 9*X(165)-5*X(2951) = 3*X(165)+5*X(3062) = 6*X(165)-5*X(11495) = 5*X(1001)-4*X(1385) = 5*X(1156)-X(13243) = X(2951)+3*X(3062) = 2*X(2951)-3*X(11495) = 2*X(3062)+X(11495) = 5*X(5220)-2*X(12702) = 5*X(5779)-X(12702) = X(7982)-5*X(11372)

X(16112) lies on these lines:
{3,16138}, {5,16127}, {6,9355}, {7,11}, {9,165}, {40,15481}, {44,1721}, {45,1742}, {142,10171}, {144,3434}, {220,5527}, {355,382}, {381,10265}, {390,10944}, {518,5693}, {527,11235}, {631,10308}, {958,12688}, {971,1001}, {1012,6326}, {1125,12684}, {1788,5825}, {2310,6180}, {2801,10247}, {3149,7701}, {3305,5918}, {3358,15297}, {3686,9950}, {3715,9778}, {3748,8545}, {3812,7992}, {3826,5817}, {3832,5221}, {3843,16125}, {3913,12705}, {4312,5729}, {4423,11220}, {5076,13465}, {5223,10914}, {5302,12565}, {5542,11373}, {5658,6690}, {5708,12571}, {5732,15254}, {5759,11826}, {5762,10525}, {5795,9949}, {5805,10893}, {5832,12679}, {5843,10943}, {5845,12586}, {5880,12616}, {5905,7965}, {6244,15064}, {6259,12617}, {6600,13205}, {6666,15346}, {7308,10178}, {8167,10167}, {8232,8255}, {9856,12513}, {12700,12857}

X(16112) = midpoint of X(9) and X(3062)
X(16112) = reflection of X(i) in X(j) for these (i,j): (40, 15481), (5220, 5779), (5732, 15254), (11495, 9)
X(16112) = X(7) of inner-Johnson triangle
X(16112) = X(3629) of excentral triangle
X(16112) = X(3631) of 6th mixtilinear triangle
X(16112) = Ursa-minor-to-Ursa-major similarity image of X(7)
X(16112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1709, 5927, 1376), (9779, 13243, 4860)


X(16113) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 2nd FUHRMANN

Barycentrics    3*a^7-2*(b+c)*a^6-(6*b^2+b*c+6*c^2)*a^5+3*(b^3+c^3)*a^4+(b^2+b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a^3+(b^2-c^2)*(b-c)*b*c*a^2-(b^2-c^2)^3*(b-c) : :
X(16113) = 3*X(3)-X(16150) = 3*X(21)-2*X(946) = 3*X(79)-2*X(16150) = 3*X(165)-X(16118) = 3*X(191)-X(5691) = 3*X(376)-X(16116) = 5*X(631)-4*X(6701) = X(962)-3*X(15677) = 3*X(3576)-2*X(3649) = 3*X(3651)-4*X(12512) = 2*X(5690)-3*X(16139) = 3*X(5886)-4*X(12104) = 3*X(6175)-4*X(6684) = 5*X(8227)-6*X(15670) = 3*X(10902)-2*X(13407)

X(16113) lies on these lines:
{2,16125}, {3,79}, {4,3647}, {20,3648}, {21,946}, {30,40}, {35,16152}, {36,16153}, {56,16142}, {102,930}, {165,16118}, {182,16115}, {376,16116}, {515,11684}, {517,4330}, {631,6701}, {758,944}, {962,15677}, {1593,16114}, {1749,7491}, {1770,7688}, {2077,3651}, {2771,12119}, {3065,5840}, {3098,16123}, {3428,13743}, {3576,3649}, {3579,3585}, {3587,4333}, {3650,12526}, {5428,16159}, {5432,9612}, {5732,16132}, {5886,12104}, {6175,6684}, {6256,9778}, {6282,16143}, {6284,16141}, {6928,15079}, {6987,15016}, {7280,7702}, {7354,16140}, {7982,10543}, {8227,15670}, {9540,16148}, {10310,16117}, {10902,13407}, {11248,16154}, {11249,16155}, {11414,16119}, {12120,12651}, {13935,16149}, {15908,16160}

X(16113) = midpoint of X(20) and X(3648)
X(16113) = reflection of X(i) in X(j) for these (i,j): (4, 3647), (79, 3), (7982, 10543), (16159, 5428)
X(16113) = anticomplement of X(16125)
X(16113) = X(79) of ABC-X3 reflections triangle
X(16113) = X(3647) of anti-Euler triangle
X(16113) = X(6152) of hexyl triangle
X(16113) = X(6242) of 1st circumperp triangle
X(16113) = X(12226) of 2nd circumperp triangle
X(16113) = X(12606) of excentral triangle


X(16114) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARA TO 2nd FUHRMANN

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

X(16114) lies on these lines:
{4,3648}, {25,79}, {30,1829}, {33,16142}, {235,16125}, {427,3647}, {468,6701}, {758,12135}, {1593,16113}, {1598,16150}, {2771,12137}, {3649,11363}, {5090,11684}, {5441,11396}, {7487,16116}, {7713,16118}, {11380,16115}, {11383,16117}, {11384,16121}, {11385,16122}, {11386,16123}, {11388,16130}, {11389,16131}, {11390,16138}, {11391,16139}, {11392,16140}, {11393,16141}, {11398,16152}, {11399,16153}, {11400,16154}, {11401,16155}, {11832,16129}, {13884,16148}, {13937,16149}

X(16114) = X(79) of anti-Ara triangle


X(16115) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD TO 2nd FUHRMANN

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

X(16115) lies on these lines:
{30,12194}, {32,79}, {83,3647}, {98,16125}, {182,16113}, {758,12195}, {1078,6701}, {2771,12198}, {3648,7787}, {3649,11364}, {3652,10796}, {5441,10800}, {10788,16116}, {10789,16118}, {10790,16119}, {10791,11684}, {10792,16130}, {10793,16131}, {10794,16138}, {10795,16139}, {10797,16140}, {10798,16141}, {10799,16142}, {10801,16152}, {10802,16153}, {10803,16154}, {10804,16155}, {11380,16114}, {11490,16117}, {11837,16121}, {11838,16122}, {11839,16129}, {11842,16150}, {13938,16149}

X(16115) = X(79) of 5th anti-Brocard triangle


X(16116) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-EULER TO 2nd FUHRMANN

Barycentrics    a^7+(b+c)*a^6-3*(b^2-b*c+c^2)*a^5-(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^3+3*(b^4-c^4)*(b-c)*a^2-(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :
X(16116) = 3*X(4)-4*X(16125) = 3*X(79)-2*X(16125) = 3*X(191)-4*X(6684) = 2*X(355)-3*X(2475) = 3*X(376)-2*X(16113) = 5*X(631)-4*X(3647) = X(962)-3*X(14450) = 4*X(1385)-3*X(15677) = 7*X(3090)-8*X(6701) = 4*X(3649)-3*X(5603) = 2*X(4297)-3*X(16132) = 4*X(5901)-3*X(13743) = X(10308)-4*X(11544) = X(10308)+2*X(16006) = 2*X(11544)+X(16006)

X(16116) lies on these lines:
{1,12248}, {2,3652}, {3,3648}, {4,79}, {5,9782}, {7,496}, {21,2096}, {24,16119}, {30,944}, {104,5606}, {153,355}, {191,6684}, {329,3650}, {376,16113}, {388,10043}, {442,5811}, {497,10052}, {515,16118}, {553,16009}, {631,3647}, {758,12115}, {912,5178}, {1158,14526}, {1385,15677}, {1490,10123}, {1749,6949}, {1768,6952}, {3085,16140}, {3086,16141}, {3090,6701}, {3255,10305}, {3296,16005}, {3304,3649}, {3651,5759}, {3871,5905}, {3889,12699}, {4294,16142}, {4295,10950}, {4297,16132}, {4466,6173}, {5057,13369}, {5441,7967}, {5499,13465}, {5657,10942}, {5693,6951}, {5714,10523}, {5768,10248}, {5818,6175}, {5885,13729}, {6757,12317}, {6965,15016}, {7487,16114}, {9862,16123}, {10122,10531}, {10785,16138}, {10786,16139}, {10788,16115}, {10805,16154}, {10806,16155}, {11491,16117}, {11844,16122}, {11845,16129}, {12005,13129}, {12540,12556}, {13886,16148}, {13939,16149}

X(16116) = reflection of X(i) in X(j) for these (i,j): (4, 79), (3648, 3), (7701, 11263), (13465, 5499)
X(16116) = anticomplement of X(3652)
X(16116) = X(79) of anti-Euler triangle
X(16116) = X(1493) of 2nd Conway triangle
X(16116) = X(3648) of ABC-X3 reflections triangle
X(16116) = X(12280) of 2nd circumperp triangle
X(16116) = X(12291) of 1st circumperp triangle
X(16116) = X(15801) of Fuhrmann triangle
X(16116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5885, 16128, 13729), (11544, 16006, 10308)


X(16117) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 2nd FUHRMANN

Barycentrics    a*(a^6-(b+c)*a^5-(b+2*c)*(2*b+c)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2+3*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+2*(b^2-c^2)^2*b*c) : :
X(16117) = 3*X(2)-4*X(11277) = 3*X(3)-2*X(21) = 5*X(3)-4*X(5428) = 11*X(3)-8*X(12104) = X(21)-3*X(3651) = 3*X(165)-X(7701) = 3*X(5426)-4*X(13624) = X(16126)-3*X(16132)

As a point on the Euler line, X(16117) has Shinagawa coefficients (5*R+2*r, -9*R-2*r)

X(16117) lies on these lines:
{2,3}, {35,16118}, {40,2771}, {55,79}, {56,5441}, {100,3648}, {165,7701}, {191,210}, {197,16119}, {355,12511}, {500,4658}, {516,16159}, {517,16126}, {758,3913}, {970,14855}, {999,10543}, {1001,6701}, {1030,8818}, {1376,3647}, {1836,14526}, {2795,12188}, {3295,3649}, {3652,5779}, {4324,5172}, {5426,13624}, {5584,5790}, {5687,11684}, {5708,10122}, {6011,12121}, {6259,6796}, {6361,14450}, {6767,16137}, {7373,15174}, {7742,9668}, {8666,12773}, {10310,16113}, {10679,12333}, {11246,13995}, {11263,12699}, {11383,16114}, {11490,16115}, {11491,16116}, {11494,16123}, {11496,16125}, {11500,13465}, {11501,16140}, {11502,16141}, {11507,16152}, {11508,16153}, {11509,16154}, {11510,16155}, {11848,16129}, {11849,16150}, {12515,12519}, {13887,16148}, {13940,16149}, {14882,15228}

X(16117) = midpoint of X(i) and X(j) for these {i,j}: {40, 16143}, {6361, 14450}
X(16117) = reflection of X(i) in X(j) for these (i,j): (3, 3651), (4, 5499), (191, 3579), (3830, 6175), (12699, 11263), (13465, 16139), (13743, 3), (15677, 8703), (16138, 3647), (16160, 11277)
X(16117) = anticomplement of X(16160)
X(16117) = X(79) of anti-Mandart-incircle triangle
X(16117) = X(195) of 1st circumperp triangle
X(16117) = X(3651) of X3-ABC reflections triangle
X(16117) = X(5499) of anti-Euler triangle
X(16117) = X(8254) of 6th mixtilinear triangle
X(16117) = X(12307) of 2nd circumperp triangle
X(16117) = X(13743) of ABC-X3 reflections triangle
X(16117) = Stammler-isogonal conjugate of X(1)
X(16117) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 382, 7489), (3, 3830, 405), (3, 3843, 6883), (3, 5073, 3560), (3, 6918, 15720), (5, 7411, 3), (21, 442, 11108), (404, 8703, 3), (411, 550, 3), (548, 6905, 3), (3522, 6924, 3), (4188, 15680, 21), (6097, 7430, 3), (11277, 16160, 2)


X(16118) = ORTHOLOGIC CENTER OF THESE TRIANGLES: AQUILA TO 2nd FUHRMANN

Barycentrics    3*a^4+(b+c)*a^3-(b^2-b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(16118) = 3*X(1)-4*X(3649) = 3*X(1)-2*X(5441) = 5*X(1)-4*X(10543) = 5*X(1)-8*X(11544) = 9*X(1)-8*X(15174) = 7*X(1)-8*X(16137) = 3*X(79)-2*X(3649) = 3*X(79)-X(5441) = 5*X(79)-2*X(10543) = 5*X(79)-4*X(11544) = 9*X(79)-4*X(15174) = 7*X(79)-4*X(16137) = 5*X(3649)-3*X(10543) = 5*X(3649)-6*X(11544) = 3*X(3649)-2*X(15174) = 7*X(3649)-6*X(16137) = 5*X(5441)-6*X(10543) = 5*X(5441)-12*X(11544) = 3*X(5441)-4*X(15174) = 7*X(5441)-12*X(16137) = 9*X(10543)-10*X(15174) = 7*X(10543)-10*X(16137) = 9*X(11544)-5*X(15174) = 7*X(11544)-5*X(16137) = 7*X(15174)-9*X(16137)

X(16118) lies on these lines:
{1,30}, {4,1768}, {5,5131}, {10,191}, {12,15228}, {20,14526}, {21,3624}, {35,16117}, {36,9955}, {46,1749}, {145,9802}, {165,16113}, {226,4324}, {382,5902}, {515,16116}, {516,5270}, {517,16150}, {758,3632}, {946,4325}, {1125,15677}, {1478,6361}, {1697,16142}, {1698,3647}, {1699,5450}, {2771,5903}, {3099,16123}, {3337,3583}, {3467,6839}, {3586,5586}, {3616,4299}, {3627,11246}, {3651,4333}, {3652,5587}, {3679,11684}, {3830,5221}, {3843,15079}, {4312,10394}, {4316,12047}, {4317,9812}, {4330,13407}, {4338,5691}, {4668,5223}, {4757,11571}, {5443,15326}, {5499,7951}, {5560,10308}, {5588,16131}, {5589,16130}, {5697,9655}, {6841,15803}, {6924,15017}, {7713,16114}, {7741,16160}, {8185,16119}, {8187,16122}, {8275,9613}, {9578,16140}, {9581,16141}, {10122,11220}, {10572,11552}, {10789,16115}, {10826,16138}, {10827,16139}, {11852,16129}, {13888,16148}, {13942,16149}

X(16118) = reflection of X(i) in X(j) for these (i,j): (1, 79), (191, 2475), (3648, 10), (3679, 15679), (4330, 13407), (5441, 3649), (10543, 11544), (15680, 11263), (16126, 14450)
X(16118) = X(79) of Aquila triangle
X(16118) = X(3648) of outer-Garcia triangle
X(16118) = X(6242) of excentral triangle
X(16118) = X(12316) of intouch triangle
X(16118) = X(12606) of 6th mixtilinear triangle
X(16118) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 3648, 191), (21, 6701, 3624), (46, 7701, 1749), (79, 5441, 3649), (79, 10483, 16132), (1770, 3585, 484), (1836, 10483, 1), (2475, 3648, 10), (3583, 4292, 3337), (3647, 6175, 1698), (3649, 5441, 1), (4333, 9612, 5010), (11263, 15680, 5426)


X(16119) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARA TO 2nd FUHRMANN

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

X(16119) lies on these lines:
{3,3647}, {22,3648}, {24,16116}, {25,79}, {30,9798}, {191,7085}, {197,16117}, {758,12410}, {1598,16125}, {2771,3556}, {3220,7701}, {3649,11365}, {5020,6701}, {5441,8192}, {7517,16150}, {8185,16118}, {8193,11684}, {10037,16152}, {10046,16153}, {10790,16115}, {10828,16123}, {10829,16138}, {10830,16139}, {10831,16140}, {10832,16141}, {10833,16142}, {10834,16154}, {10835,16155}, {11414,16113}, {11853,16129}, {13889,16148}, {13943,16149}

X(16119) = X(79) of Ara triangle


X(16120) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO 2nd FUHRMANN

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

X(16120) lies on these lines:
{8,79}, {21,3062}, {30,9856}, {142,1898}, {191,8580}, {2550,12059}, {2771,9947}, {3649,8581}, {5290,12529}, {5587,12666}, {5927,6260}, {10122,11019}, {10123,15587}, {10543,10866}, {10864,16132}, {10865,16133}, {11035,16137}, {11519,16126}, {11678,11684}, {11856,16135}, {11857,16136}, {11858,16146}, {11859,16147}, {11860,16151}

X(16120) = reflection of X(10122) in X(11263)
X(16120) = X(54) of Atik triangle


X(16121) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st AURIGA TO 2nd FUHRMANN

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*D-a^2*(a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(16121) lies on these lines:
{30,55}, {79,5597}, {758,12454}, {3647,5599}, {3649,11366}, {3652,8200}, {5441,5598}, {8196,16125}, {8197,11684}, {10543,11367}, {11384,16114}, {11493,13743}, {11837,16115}, {11861,16123}, {11869,16140}, {11871,16141}, {11873,16142}, {11877,16152}, {11879,16153}, {11881,16154}, {11883,16155}, {13890,16148}

X(16121) = reflection of X(16122) in X(55)
X(16121) = X(79) of 1st Auriga triangle
X(16121) = X(5441) of 2nd Auriga triangle


X(16122) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd AURIGA TO 2nd FUHRMANN

Barycentrics    -(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*D-a^2*(a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c)) : : , where D=4*S*sqrt(R*(4*R+r))

X(16122) lies on these lines:
{30,55}, {79,5598}, {3647,5600}, {3649,11367}, {5441,5597}, {8187,16118}, {8203,16125}, {8204,11684}, {8205,16130}, {10543,11366}, {11385,16114}, {11492,13743}, {11838,16115}, {11844,16116}, {11862,16123}, {11866,16138}, {11868,16139}, {11870,16140}, {11872,16141}, {11874,16142}, {11878,16152}, {11880,16153}, {11882,16154}

X(16122) = reflection of X(16121) in X(55)
X(16122) = X(79) of 2nd Auriga triangle
X(16122) = X(5441) of 1st Auriga triangle


X(16123) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th BROCARD TO 2nd FUHRMANN

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

X(16123) lies on these lines:
{30,9941}, {32,79}, {758,12495}, {2771,12498}, {2896,3648}, {3096,3647}, {3098,16113}, {3099,16118}, {3649,11368}, {3652,9996}, {5441,9997}, {9301,16150}, {9857,11684}, {9862,16116}, {9993,16125}, {9994,16130}, {9995,16131}, {10038,16152}, {10047,16153}, {10828,16119}, {10871,16138}, {10872,16139}, {10873,16140}, {10874,16141}, {10877,16142}, {10878,16154}, {10879,16155}, {11386,16114}, {11861,16121}, {11862,16122}, {11885,16129}, {13892,16148}, {13946,16149}

X(16123) = X(79) of 5th Brocard triangle


X(16124) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO 2nd FUHRMANN

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

X(16124) lies on these lines:
{1,30}, {21,10435}, {191,1764}, {442,10887}, {758,12435}, {1699,8235}, {2771,10441}, {3651,10434}, {6675,10856}, {6841,10886}, {7701,10476}, {10122,11021}, {10439,12547}, {10446,14450}, {10465,15680}, {10478,11263}, {10862,16120}, {10889,16133}, {11521,16126}, {11679,11684}, {11892,16135}, {11893,16136}, {11894,16146}, {11895,16147}, {11896,16151}

X(16124) = X(54) of 3rd Conway triangle


X(16125) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EULER TO 2nd FUHRMANN

Barycentrics    2*a^7-(b+c)*a^6-3*(b^2+c^2)*a^5-b*c*(b+c)*a^4-b*c*(b-c)^2*a^3+3*(b^3-c^3)*(b^2-c^2)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-2*(b^2-c^2)^3*(b-c) : :
X(16125) = 3*X(4)+X(16116) = 3*X(21)-5*X(8227) = X(40)-3*X(6175) = 3*X(79)-X(16116) = 3*X(191)-7*X(7989) = X(355)+3*X(16159) = 3*X(381)-X(3652) = 3*X(381)+X(16150) = 3*X(442)-2*X(6684) = X(962)+3*X(2475) = 3*X(1699)+X(16118) = 5*X(3091)-X(3648) = X(4297)-3*X(11263) = X(5441)-3*X(5603) = 3*X(5587)-X(11684)

X(16125) lies on these lines:
{2,16113}, {3,6701}, {4,79}, {5,3647}, {11,1354}, {12,16142}, {21,8227}, {30,551}, {40,6175}, {98,16115}, {117,137}, {191,7989}, {235,16114}, {355,758}, {371,16148}, {372,16149}, {381,3652}, {442,6684}, {515,3649}, {517,15862}, {962,2475}, {971,13159}, {1478,16153}, {1479,11045}, {1598,16119}, {1699,5450}, {2829,15911}, {3091,3648}, {3651,5715}, {3671,11544}, {3843,16112}, {3884,12699}, {5267,9955}, {5330,15679}, {5441,5603}, {5499,7680}, {5535,7548}, {5561,10308}, {5587,11684}, {5882,16137}, {6201,16131}, {6202,16130}, {6840,9782}, {6841,12571}, {7681,16160}, {8196,16121}, {8203,16122}, {9964,11604}, {9993,16123}, {10248,10430}, {10531,16154}, {10532,16155}, {10543,13464}, {10893,16138}, {10894,16139}, {10895,16140}, {10896,16141}, {11230,12104}, {11496,16117}, {11897,16129}

X(16125) = midpoint of X(i) and X(j) for these {i,j}: {4, 79}, {3652, 16150}
X(16125) = reflection of X(i) in X(j) for these (i,j): (3, 6701), (3647, 5), (5882, 16137), (10543, 13464)
X(16125) = complement of X(16113)
X(16125) = X(79) of Euler triangle
X(16125) = X(3647) of Johnson triangle
X(16125) = X(6242) of 3rd Euler triangle
X(16125) = X(6701) of X3-ABC reflections triangle
X(16125) = X(12226) of 4th Euler triangle
X(16125) = {X(381), X(16150)}-harmonic conjugate of X(3652)


X(16126) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO 2nd FUHRMANN

Barycentrics    a*(a^3-3*(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)*(3*b^2-4*b*c+3*c^2)) : :
X(16126) = 3*X(1)-2*X(21) = 7*X(1)-4*X(3647) = 4*X(1)-3*X(5426) = 5*X(1)-2*X(11684) = 4*X(21)-3*X(191) = 7*X(21)-6*X(3647) = 8*X(21)-9*X(5426) = 5*X(21)-3*X(11684) = 2*X(79)+X(3633) = 2*X(145)+X(16118) = 7*X(191)-8*X(3647) = 2*X(191)-3*X(5426) = 5*X(191)-4*X(11684) = 10*X(3647)-7*X(11684) = 15*X(5426)-8*X(11684)

X(16126) lies on these lines:
{1,21}, {8,11263}, {30,7982}, {35,4018}, {72,5425}, {79,3633}, {100,4757}, {145,9802}, {442,3679}, {474,5902}, {484,4084}, {517,16117}, {519,2475}, {551,15674}, {942,4867}, {952,16159}, {1392,3065}, {1482,2771}, {1698,11374}, {2646,4880}, {2802,13146}, {3241,4309}, {3243,5441}, {3304,4930}, {3336,4188}, {3337,4511}, {3340,3632}, {3555,11009}, {3576,16139}, {3586,16155}, {3624,11281}, {3651,7991}, {3654,11277}, {3656,16160}, {3913,5541}, {3919,4420}, {3962,5251}, {4127,5260}, {4295,12536}, {4668,6701}, {4677,6175}, {4861,13089}, {5506,5692}, {5538,5884}, {5730,8261}, {6598,7700}, {6675,11518}, {6737,11551}, {6841,11522}, {7962,10543}, {9897,11604}, {10247,13465}, {11224,12650}, {11519,16120}, {11521,16124}, {11526,16133}, {11527,16135}, {11528,16136}, {11531,16143}, {11534,16146}, {11535,16151}, {11899,16147}, {12647,14526}

X(16126) = midpoint of X(i) and X(j) for these {i,j}: {145, 14450}, {11531, 16143}
X(16126) = reflection of X(i) in X(j) for these (i,j): (8, 11263), (191, 1), (4677, 6175), (7991, 3651), (9897, 11604), (16118, 14450)
X(16126) = X(54) of excenters-reflections triangle
X(16126) = X(191) of 5th mixtilinear triangle
X(16126) = X(9920) of Hutson intouch triangle
X(16126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 191, 5426), (1, 3901, 6763), (145, 11280, 12653), (3892, 5330, 1)


X(16127) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FUHRMANN TO EXTOUCH

Barycentrics    a^7+(b+c)*a^6-3*(b-c)^2*a^5-(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^3+3*(b^4-c^4)*(b-c)*a^2-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(16127) = 3*X(84)-5*X(8227) = 2*X(355)-3*X(6256) = X(355)-3*X(6259) = X(962)+3*X(6223) = 3*X(1158)-4*X(6684) = 2*X(4297)-3*X(6261) = 3*X(5658)-2*X(6796) = 4*X(5901)-3*X(12114) = 3*X(6245)-4*X(12571) = 3*X(6260)-2*X(6684) = 7*X(7989)-3*X(7992) = 7*X(7989)-6*X(12616) = 5*X(8227)-6*X(12608) = 3*X(9799)-7*X(10248) = X(12245)-3*X(12667)

X(16127) lies on these lines:
{2,7701}, {4,79}, {5,16112}, {9,1158}, {20,6326}, {30,12635}, {84,5249}, {145,515}, {355,5836}, {516,5534}, {946,4654}, {1519,10085}, {1768,6834}, {2800,5904}, {3062,5715}, {3091,9782}, {3336,16009}, {3543,14450}, {3583,10052}, {4295,5727}, {4297,6261}, {5178,12528}, {5221,16006}, {5259,5450}, {5658,6796}, {5693,6925}, {5812,15726}, {5901,12114}, {6245,12571}, {6845,10308}, {6872,16132}, {6928,16128}, {6957,15016}, {6984,7989}, {9799,10248}, {10310,13257}, {12672,12678}

X(16127) = reflection of X(i) in X(j) for these (i,j): (84, 12608), (1158, 6260), (6256, 6259), (7992, 12616), (12246, 5450)


X(16128) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FUHRMANN TO INNER-GARCIA

Barycentrics    a^7-(2*b^2-3*b*c+2*c^2)*a^5-(b^3+c^3)*a^4+(b^4+c^4)*a^3+(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a^2-3*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(16128) = 3*X(4)-X(9803) = 2*X(104)-3*X(5886) = 4*X(140)-5*X(15017) = 5*X(355)-4*X(15863) = 3*X(381)-2*X(10265) = 2*X(550)-3*X(15015) = 2*X(1484)-3*X(1699) = 4*X(1537)-3*X(3656) = 3*X(3656)-2*X(12737) = 3*X(5587)-X(12767) = 3*X(5886)-4*X(12611) = X(9803)+3*X(9809) = 5*X(10742)-2*X(15863) = 3*X(12699)-2*X(14217)

X(16128) lies on these lines:
{4,2771}, {5,1768}, {9,119}, {11,3338}, {30,5538}, {40,11698}, {79,546}, {80,1836}, {104,5886}, {140,15017}, {149,3555}, {153,517}, {355,2800}, {381,10265}, {382,12635}, {516,12331}, {550,15015}, {908,2932}, {946,12773}, {952,3627}, {1385,12248}, {1484,1699}, {1537,3656}, {1837,11571}, {2801,10738}, {2827,14286}, {2829,6259}, {3419,12532}, {3652,6842}, {3654,10711}, {3655,12678}, {4295,6797}, {5057,6224}, {5587,12767}, {5722,11570}, {5805,5851}, {5812,5840}, {5880,6702}, {5885,13729}, {6831,16138}, {6928,16127}, {7972,12701}, {10058,11374}, {10074,11373}, {12758,12763}

X(16128) = midpoint of X(4) and X(9809)
X(16128) = reflection of X(i) in X(j) for these (i,j): (40, 11698), (104, 12611), (355, 10742), (1768, 5), (3654, 10711), (12248, 1385), (12515, 119), (12737, 1537), (12738, 13257), (12773, 946)
X(16128) = X(1768) of Johnson triangle
X(16128) = X(9809) of Euler triangle
X(16128) = X(10272) of 2nd Conway triangle
X(16128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (104, 12611, 5886), (1537, 12737, 3656), (11570, 12764, 5722), (13729, 16116, 5885)


X(16129) = ORTHOLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 2nd FUHRMANN

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^12+(b+c)*a^11-2*(2*b^2+b*c+2*c^2)*a^10-(b+c)*(2*b^2+b*c+2*c^2)*a^9-(3*b^4+3*c^4-2*b*c*(b^2+8*b*c+c^2))*a^8-(b+c)*(2*b^4+2*c^4-b*c*(b^2+9*b*c+c^2))*a^7+(12*b^6+12*c^6+(3*b^4+3*c^4-2*b*c*(7*b^2+4*b*c+7*c^2))*b*c)*a^6+(b+c)*(8*b^6+8*c^6-b^2*c^2*(9*b^2+b*c+9*c^2))*a^5-2*(b^2-c^2)^2*(4*b^4+4*c^4+b*c*(2*b^2+11*b*c+2*c^2))*a^4-(b^2-c^2)^2*(b+c)*(7*b^4+7*c^4-b*c*(b^2-15*b*c+c^2))*a^3+(b^2-c^2)^2*(b^4+c^4+2*b*c*(5*b^2+2*b*c+5*c^2))*b*c*a^2+(b^2-c^2)^2*(b+c)*(2*b^2+b*c+2*c^2)*(b^4+c^4-b*c*(b^2-3*b*c+c^2))*a+(b^4-c^4)^2*(b^2-c^2)^2) : :
X(16129) = 2*X(3649)-3*X(11831) = 4*X(6701)-5*X(15183) = 3*X(11845)-X(16116) = 3*X(11852)-X(16118) = 3*X(11897)-2*X(16125) = 3*X(11911)-X(16150)

X(16129) lies on these lines:
{30,40}, {79,402}, {758,12626}, {1650,3647}, {2771,12729}, {3648,4240}, {3649,11831}, {5441,11910}, {6701,15183}, {11684,11900}, {11832,16114}, {11839,16115}, {11845,16116}, {11848,16117}, {11852,16118}, {11853,16119}, {11885,16123}, {11897,16125}, {11905,16140}, {11906,16141}, {11909,16142}, {11911,16150}, {11912,16152}, {11913,16153}, {11914,16154}, {11915,16155}

X(16129) = midpoint of X(3648) and X(4240)
X(16129) = reflection of X(i) in X(j) for these (i,j): (79, 402), (1650, 3647)
X(16129) = X(79) of Gossard triangle


X(16130) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-GREBE TO 2nd FUHRMANN

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

X(16130) lies on these lines:
{6,79}, {30,3641}, {1271,3648}, {3647,5591}, {3649,11370}, {3652,6215}, {5441,5605}, {5589,16118}, {5689,11684}, {6202,16125}, {8205,16122}, {8974,16148}, {9994,16123}, {10040,16152}, {10048,16153}, {10783,16116}, {10792,16115}, {10919,16138}, {10923,16140}, {10925,16141}, {10927,16142}, {10929,16154}, {10931,16155}, {11388,16114}, {11916,16150}, {13949,16149}

X(16130) = reflection of X(16131) in X(79)
X(16130) = X(79) of inner-Grebe triangle


X(16131) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-GREBE TO 2nd FUHRMANN

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

X(16131) lies on these lines:
{6,79}, {30,3640}, {1270,3648}, {3647,5590}, {3649,11371}, {3652,6214}, {5441,5604}, {5588,16118}, {5688,11684}, {6201,16125}, {8975,16148}, {9995,16123}, {10041,16152}, {10049,16153}, {10793,16115}, {10920,16138}, {10922,16139}, {10924,16140}, {10926,16141}, {10928,16142}, {10930,16154}, {10932,16155}, {11389,16114}, {11917,16150}

X(16131) = reflection of X(16130) in X(79)
X(16131) = X(79) of outer-Grebe triangle


X(16132) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO 2nd FUHRMANN

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2+b*c+c^2)*a^4+(b+c)*(4*b^2-5*b*c+4*c^2)*a^3-(b^2-3*b*c+c^2)*(b+c)^2*a^2-(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a+(b^4-c^4)*(b^2-c^2)) : :
X(16132) = 3*X(3)-X(13465) = 2*X(21)-3*X(3576) = 3*X(165)-2*X(16139) = 3*X(191)-2*X(13465) = 4*X(442)-3*X(5587) = 4*X(1385)-3*X(5426) = 3*X(3576)-X(7701) = 2*X(3652)-5*X(7987) = 2*X(4297)+X(16116) = 3*X(5426)-2*X(13743) = 4*X(5428)-5*X(7987) = 3*X(5731)-X(15680) = 3*X(5886)-2*X(16160) = 4*X(8261)-5*X(15016) = 2*X(16117)+X(16126)

X(16132) lies on these lines:
{1,30}, {3,191}, {4,11263}, {20,14450}, {21,84}, {36,1858}, {40,758}, {72,7688}, {78,11684}, {104,6597}, {165,16139}, {355,5499}, {411,5535}, {442,1490}, {515,2475}, {517,16117}, {550,5538}, {944,6264}, {952,13146}, {997,3647}, {1385,5426}, {1420,10050}, {1478,14526}, {1749,7280}, {1789,11709}, {2077,9943}, {3065,15446}, {3149,8261}, {3333,10122}, {3579,3962}, {3601,10042}, {3612,7171}, {3648,4511}, {3652,5428}, {3683,13624}, {4295,10123}, {4297,16116}, {5259,13151}, {5427,16141}, {5531,5690}, {5553,6596}, {5731,15680}, {5732,16113}, {5886,16160}, {5887,15931}, {5901,7965}, {5902,6985}, {6001,10902}, {6675,8726}, {6841,8227}, {6872,16127}, {6960,10265}, {7590,16151}, {7675,16133}, {7966,7982}, {8081,16146}, {8082,16147}, {8111,16135}, {8112,16136}, {8234,16144}, {8583,15670}, {9624,11281}, {10165,15674}, {10393,11529}, {10860,11919}, {10864,16120}, {12739,15338}

X(16132) = midpoint of X(i) and X(j) for these {i,j}: {1, 16143}, {20, 14450}
X(16132) = reflection of X(i) in X(j) for these (i,j): (4, 11263), (40, 3651), (191, 3), (355, 5499), (3652, 5428), (7701, 21), (13743, 1385)
X(16132) = X(54) of hexyl triangle
X(16132) = X(191) of ABC-X3 reflections triangle
X(16132) = X(2888) of 2nd circumperp triangle
X(16132) = X(3651) of inner-Garcia triangle
X(16132) = X(6288) of excentral triangle
X(16132) = X(10115) of 2nd Conway triangle
X(16132) = X(10610) of 6th mixtilinear triangle
X(16132) = X(11263) of anti-Euler triangle
X(16132) = X(12254) of 1st circumperp triangle
X(16132) = X(16143) of anti-Aquila triangle
X(16132) = hexyl-isogonal conjugate of X(3)
X(16132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (79, 10483, 16118), (411, 5884, 5535), (944, 11014, 6264), (1385, 13743, 5426), (3576, 7701, 21), (6261, 10884, 3576)


X(16133) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO 2nd FUHRMANN

Barycentrics    a*(a^3-2*(b+c)*a^2+(b^2+3*b*c+c^2)*a+5*b*c*(b+c))*(a+b-c)*(a-b+c) : :
X(16133) = 3*X(21)-4*X(1001) = 5*X(2346)-4*X(3746) = 2*X(2550)-3*X(6175) = 8*X(5427)-9*X(7677) = 3*X(8236)-2*X(10543) = 4*X(10122)-5*X(11025) = 3*X(11038)-4*X(16137)

X(16133) lies on these lines:
{7,21}, {9,11684}, {30,390}, {65,15481}, {79,516}, {100,226}, {191,1445}, {354,13243}, {442,7679}, {528,15679}, {553,5284}, {651,4649}, {758,5223}, {954,3651}, {1156,2771}, {1443,15569}, {1456,7269}, {1476,3255}, {1621,4654}, {2475,7674}, {2550,6175}, {3671,5251}, {4312,5010}, {4326,16143}, {4413,5226}, {5045,16138}, {5542,10074}, {6675,8732}, {6700,11263}, {6841,7678}, {7671,11372}, {7675,16132}, {8236,10543}, {8237,16144}, {8385,16135}, {8387,16146}, {8388,16147}, {8389,16151}, {10122,11025}, {10865,16120}, {10889,16124}, {11038,16137}, {11526,16126}, {14151,15570}

X(16133) = reflection of X(i) in X(j) for these (i,j): (7, 3649), (4312, 13159), (11684, 9)
X(16133) = X(54) of Honsberger triangle
X(16133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 8543, 7677), (8545, 12560, 7672)


X(16134) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FUHRMANN TO HUTSON EXTOUCH

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

X(16134) lies on these lines:
{3,11281}, {9,946}, {355,12731}, {1697,12855}, {3832,5815}, {4301,6261}, {5719,12260}, {5726,9898}, {5920,12859}, {6838,7160}, {6867,12599}, {6936,12120}, {7982,12541}, {12439,12699}, {12854,12860}

X(16134) = reflection of X(i) in X(j) for these (i,j): (7160, 12612), (12516, 12864), (12731, 12856)


X(16135) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-HUTSON TO 2nd FUHRMANN

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : : , where
F(a,b,c) = 2*b*c*(b+c)*(2*a+b+c)*(a+b-c)*(a-b+c)
G(a,b,c) = -2*(a+c)*(a+b-c)*(-a+b+c)*(a^2+a*b+b^2-c^2)*c
H(a,b,c) = a^6-2*(b+c)*a^5-(b^2+5*b*c+c^2)*a^4+(b+c)*(4*b^2-5*b*c+4*c^2)*a^3-(b^4+c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a+(b^2-c^2)^2*(b+c)^2

X(16135) lies on these lines:
{21,8109}, {30,9836}, {191,363}, {442,5934}, {758,9805}, {2771,12488}, {3649,8113}, {6732,16147}, {8111,16132}, {8133,16146}, {8385,16133}, {8390,10543}, {9783,14450}, {10122,11026}, {11039,16137}, {11527,16126}, {11684,11685}, {11923,16151}, {16136,16143}

X(16135) = reflection of X(16136) in X(16143)
X(16135) = X(54) of inner-Hutson triangle


X(16136) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-HUTSON TO 2nd FUHRMANN

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : : , where
F(a,b,c) = 2*b*c*(b+c)*(2*a+b+c)*(a+b-c)*(a-b+c)
G(a,b,c) = -2*(a+c)*(a+b-c)*(-a+b+c)*(a^2+a*b+b^2-c^2)*c
H(a,b,c) = a^6-2*(b+c)*a^5-(b^2+5*b*c+c^2)*a^4+(b+c)*(4*b^2-5*b*c+4*c^2)*a^3-(b^4+c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a+(b^2-c^2)^2*(b+c)^2

X(16136) lies on these lines:
{21,8110}, {30,9837}, {79,1127}, {168,191}, {442,5935}, {758,9806}, {2771,12489}, {3649,8114}, {3651,8108}, {6675,11855}, {6841,8378}, {8112,16132}, {8135,16146}, {8138,16147}, {8140,16135}, {8386,16133}, {8392,10543}, {9787,14450}, {10122,11027}, {11040,16137}, {11528,16126}, {11684,11686}, {11857,16120}, {11893,16124}

X(16136) = reflection of X(16135) in X(16143)
X(16136) = X(54) of outer-Hutson triangle


X(16137) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 2nd FUHRMANN

Barycentrics    2*a^4-4*(b+c)*a^3-(3*b^2+4*b*c+3*c^2)*a^2+4*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(16137) = 3*X(1)+X(79) = 5*X(1)-X(5441) = 3*X(1)-X(10543) = 2*X(1)+X(11544) = 7*X(1)+X(16118) = X(79)-3*X(3649) = 5*X(79)+3*X(5441) = 2*X(79)-3*X(11544) = 2*X(79)+3*X(15174) = 7*X(79)-3*X(16118) = 5*X(3649)+X(5441) = 3*X(3649)+X(10543) = 2*X(3649)+X(15174) = 7*X(3649)-X(16118) = 3*X(5441)-5*X(10543) = 2*X(5441)+5*X(11544) = 2*X(5441)-5*X(15174) = 7*X(5441)+5*X(16118) = 2*X(10543)+3*X(11544) = 2*X(10543)-3*X(15174) = 7*X(10543)+3*X(16118) = 7*X(11544)-2*X(16118) = 7*X(15174)+2*X(16118)

X(16137) lies on these lines:
{1,30}, {8,442}, {12,5425}, {21,999}, {56,5428}, {57,16139}, {65,5719}, {140,5902}, {145,6175}, {191,3333}, {354,5887}, {484,11276}, {496,3485}, {515,15911}, {519,6701}, {548,11246}, {549,5221}, {551,3647}, {553,13624}, {758,942}, {938,10593}, {952,13407}, {1056,2475}, {1159,3085}, {1387,2771}, {1479,15935}, {1482,3475}, {1698,11374}, {2099,5499}, {2294,7359}, {2646,11551}, {3244,11263}, {3295,3651}, {3304,10283}, {3336,3530}, {3616,11684}, {3633,4863}, {3824,6737}, {3947,14563}, {4084,6690}, {5427,5563}, {5442,11812}, {5542,5625}, {5603,9799}, {5844,15888}, {5882,16125}, {5886,11518}, {5903,11277}, {6738,12019}, {6744,7743}, {6767,16117}, {7373,13743}, {8092,16151}, {8351,16147}, {8728,12635}, {9856,12675}, {10021,15950}, {10123,10624}, {11011,14526}, {11035,16120}, {11039,16135}, {11042,16144}, {11552,15338}, {12047,12433}, {12812,15079}

X(16137) = midpoint of X(i) and X(j) for these {i,j}: {1, 3649}, {79, 10543}, {5882, 16125}, {10123, 10624}, {11011, 14526}, {11544, 15174}
X(16137) = reflection of X(i) in X(j) for these (i,j): (6675, 11281), (10122, 5045), (11544, 3649), (15174, 1), (15673, 551)
X(16137) = X(54) of incircle-circles triangle
X(16137) = X(3649) of anti-Aquila triangle
X(16137) = X(6288) of inverse-in-incircle triangle
X(16137) = X(10610) of intouch triangle
X(16137) = X(11576) of 2nd circumperp triangle
X(16137) = X(15174) of 5th mixtilinear triangle
X(16137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 79, 10543), (1, 1464, 5453), (3296, 3622, 999), (3485, 15934, 496), (3616, 11684, 15670), (3622, 11036, 3296), (3649, 10543, 79)


X(16138) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-JOHNSON TO 2nd FUHRMANN

Barycentrics    a*(a^6-(3*b^2-5*b*c+3*c^2)*a^4+b*c*(b+c)*a^3+(3*b^4+3*c^4-2*b*c*(b^2+c^2))*a^2-(b^2-c^2)*(b-c)*b*c*a-(b^2+3*b*c+c^2)*(b^2-c^2)^2) : :
X(16138) = 5*X(21)-4*X(13624) = 3*X(40)-5*X(191) = 2*X(40)-5*X(3652) = X(40)-5*X(7701) = 4*X(40)-5*X(16139) = 2*X(191)-3*X(3652) = X(191)-3*X(7701) = 4*X(191)-3*X(16139) = 4*X(7701)-X(16139) = 3*X(10246)-5*X(13743) = 5*X(10308)+4*X(13624)

X(16138) lies on these lines:
{3,16112}, {11,79}, {12,16154}, {21,4881}, {30,40}, {84,3255}, {546,1768}, {758,8148}, {1376,3647}, {2771,7984}, {3336,3845}, {3434,3648}, {3579,3921}, {3646,7171}, {3649,11373}, {3651,5927}, {3813,12699}, {3853,5535}, {5045,16133}, {5428,16143}, {5441,10386}, {5506,12100}, {6583,13243}, {6831,16128}, {6841,12676}, {10246,12114}, {10523,16152}, {10728,12515}, {10785,16116}, {10794,16115}, {10826,16118}, {10829,16119}, {10871,16123}, {10893,16125}, {10914,11684}, {10919,16130}, {10920,16131}, {10947,16142}, {10948,16153}, {10949,16155}, {11390,16114}, {11866,16122}, {11928,16150}, {12616,12761}, {13895,16148}, {13952,16149}

X(16138) = midpoint of X(21) and X(10308)
X(16138) = reflection of X(i) in X(j) for these (i,j): (79, 16160), (3652, 7701), (16117, 3647), (16139, 3652), (16143, 5428)
X(16138) = X(79) of inner-Johnson triangle
X(16138) = X(16154) of outer-Johnson triangle
X(16138) = {X(79), X(3065)}-harmonic conjugate of X(16141)


X(16139) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON TO 2nd FUHRMANN

Barycentrics    a*(a^6-3*(b^2+b*c+c^2)*a^4+b*c*(b+c)*a^3+(3*b^4+3*c^4+2*b*c*(b^2+c^2))*a^2-(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)*(b-c)*(b^3+c^3)) : :
X(16139) = 2*X(40)+X(3652) = 3*X(40)+X(7701) = 4*X(40)+X(16138) = 3*X(165)-X(16132) = 3*X(191)-X(7701) = 4*X(191)-X(16138) = X(2475)-3*X(5657) = 3*X(3576)-X(16126) = 2*X(3579)+X(11684) = 2*X(3647)+X(12702) = 3*X(3652)-2*X(7701) = 3*X(5426)-X(7982) = 3*X(5426)-4*X(12104) = 2*X(5690)+X(16113) = 4*X(7701)-3*X(16138) = X(7982)-4*X(12104)

X(16139) lies on these lines:
{1,5424}, {3,758}, {11,16155}, {12,79}, {21,517}, {30,40}, {46,3649}, {57,16137}, {72,74}, {140,5535}, {165,16132}, {411,5694}, {442,5812}, {500,1046}, {547,5506}, {548,1768}, {549,3336}, {582,986}, {958,3647}, {1482,4428}, {1697,15174}, {1727,15338}, {1749,5441}, {2475,5657}, {2886,5791}, {2949,6598}, {3218,13624}, {3295,10122}, {3338,3653}, {3436,3648}, {3576,16126}, {3656,5250}, {5119,10543}, {5128,11544}, {5426,7982}, {5536,5901}, {5603,15674}, {5659,16160}, {5709,5886}, {5719,15932}, {5885,6986}, {6265,11012}, {6684,11263}, {6762,7966}, {10523,16153}, {10786,16116}, {10795,16115}, {10827,16118}, {10830,16119}, {10872,16123}, {10894,16125}, {10922,16131}, {10953,16142}, {10954,16152}, {10955,16154}, {10993,12695}, {11260,12737}, {11391,16114}, {11500,13465}, {11604,12619}, {11868,16122}, {11929,16150}, {13896,16148}, {13953,16149}

X(16139) = midpoint of X(i) and X(j) for these {i,j}: {40, 191}, {3651, 11684}, {12702, 13743}, {13465, 16117}
X(16139) = reflection of X(i) in X(j) for these (i,j): (1, 5428), (79, 5499), (3651, 3579), (3652, 191), (3656, 15670), (11263, 6684), (11604, 12619), (12699, 6841), (13743, 3647), (16138, 3652), (16159, 442)
X(16139) = X(79) of outer-Johnson triangle
X(16139) = X(5428) of Aquila triangle
X(16139) = X(6288) of 1st circumperp triangle
X(16139) = X(10610) of excentral triangle
X(16139) = X(13565) of 6th mixtilinear triangle
X(16139) = X(16155) of inner-Johnson triangle
X(16139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1749, 5441, 16141), (1749, 11010, 5441), (5690, 11827, 355)


X(16140) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF TO 2nd FUHRMANN

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

X(16140) lies on these lines:
{1,3652}, {4,16142}, {5,16153}, {12,79}, {21,1319}, {30,5119}, {46,11544}, {56,3647}, {57,191}, {65,3219}, {388,3648}, {495,16152}, {632,3336}, {758,956}, {846,1464}, {1046,11553}, {1317,3065}, {1727,5719}, {1749,15950}, {2771,10058}, {3085,16116}, {3218,4870}, {3476,15677}, {3584,12515}, {3650,10404}, {3743,8614}, {3748,12710}, {3833,5221}, {5226,14450}, {5427,13462}, {5441,10386}, {7354,16113}, {7701,10543}, {8545,15296}, {9654,16150}, {10797,16115}, {10831,16119}, {10873,16123}, {10895,16125}, {10923,16130}, {10924,16131}, {10956,16154}, {10957,16155}, {11392,16114}, {11501,16117}, {11869,16121}, {11870,16122}, {13465,15934}, {13897,16148}, {13954,16149}

X(16140) = reflection of X(16152) in X(495)
X(16140) = X(79) of 1st Johnson-Yff triangle
X(16140) = X(3652) of inner-Yff triangle
X(16140) = X(16141) of inner-Yff tangents triangle
X(16140) = {X(1), X(3652)}-harmonic conjugate of X(16141)


X(16141) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF TO 2nd FUHRMANN

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

X(16141) lies on these lines:
{1,3652}, {5,16152}, {11,79}, {21,60}, {30,46}, {55,3647}, {90,11375}, {191,1697}, {496,16153}, {497,3648}, {758,2098}, {1210,12764}, {1749,5441}, {1864,3651}, {2771,10074}, {3057,11684}, {3058,6763}, {3086,16116}, {3255,6597}, {3333,3649}, {3336,3627}, {3338,11544}, {3486,15677}, {3612,12104}, {5427,16132}, {5499,10958}, {6284,16113}, {7082,15670}, {7681,12679}, {9669,16150}, {10798,16115}, {10832,16119}, {10874,16123}, {10896,16125}, {10925,16130}, {10959,16155}, {11393,16114}, {11502,16117}, {11871,16121}, {11872,16122}, {11906,16129}, {13955,16149}

X(16141) = reflection of X(16153) in X(496)
X(16141) = X(79) of 2nd Johnson-Yff triangle
X(16141) = X(3652) of outer-Yff triangle
X(16141) = X(16140) of outer-Yff tangents triangle
X(16141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3652, 16140), (79, 3065, 16138), (497, 3648, 16142), (1749, 5441, 16139)


X(16142) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE TO 2nd FUHRMANN

Barycentrics    (-a+b+c)*(2*a^6+(b+c)*a^5-3*(b^2+c^2)*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3-b*c*(b-c)^2*a^2+(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b-c)^2) : :
X(16142) = 3*X(3649)-2*X(4292) = 3*X(10543)-4*X(12575)

X(16142) lies on these lines:
{3,16153}, {4,16140}, {11,3647}, {12,16125}, {30,3057}, {33,16114}, {55,79}, {56,16113}, {191,7082}, {497,3648}, {516,15888}, {758,10950}, {1317,10624}, {1364,7159}, {1479,3652}, {1697,16118}, {1770,5719}, {1837,11684}, {2098,5441}, {2646,3649}, {2771,12743}, {3058,3881}, {3065,13274}, {3295,16150}, {4092,6068}, {4294,16116}, {4313,14450}, {5048,10543}, {5432,6701}, {5789,9671}, {10799,16115}, {10833,16119}, {10877,16123}, {10927,16130}, {10928,16131}, {10947,16138}, {10953,16139}, {10965,16154}, {10966,13743}, {11374,16159}, {11873,16121}, {11874,16122}, {11909,16129}, {13901,16148}, {13958,16149}

X(16142) = reflection of X(1770) in X(11544)
X(16142) = X(79) of Mandart-incircle triangle
X(16142) = X(6242) of Hutson intouch triangle
X(16142) = X(12226) of intouch triangle
X(16142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497, 3648, 16141), (3295, 16150, 16152)


X(16143) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO 2nd FUHRMANN

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2+5*b*c+c^2)*a^4+(b+c)*(4*b^2-5*b*c+4*c^2)*a^3-(b^4+c^4-b*c*(3*b^2+4*b*c+3*c^2))*a^2-(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a+(b^2-c^2)^2*(b+c)^2) : :
X(16143) = 4*X(21)-5*X(7987) = 3*X(165)-2*X(191) = 3*X(165)-4*X(3651) = 8*X(442)-7*X(7989) = 3*X(1699)-4*X(11263) = 3*X(3576)-2*X(13743) = 4*X(5499)-3*X(5587) = 8*X(6841)-9*X(7988) = 5*X(8227)-4*X(16160)

X(16143) lies on these lines:
{1,30}, {3,7701}, {20,5538}, {21,3062}, {40,2771}, {74,2940}, {78,3648}, {84,15910}, {90,3065}, {165,191}, {200,11684}, {411,1768}, {442,1750}, {516,14450}, {550,6326}, {758,6765}, {936,3647}, {1699,10884}, {2475,5691}, {2941,3430}, {3336,6985}, {3337,13369}, {3576,13743}, {3579,4005}, {3652,5720}, {4297,15680}, {4312,10123}, {4326,16133}, {5428,16138}, {5499,5587}, {6282,16113}, {6675,10857}, {6841,7988}, {6922,15017}, {7580,15071}, {7990,11224}, {8089,16146}, {8090,16147}, {8140,16135}, {8227,16160}, {8244,16144}, {8423,16151}, {9612,14526}, {10092,13462}, {10122,10980}, {11531,16126}, {12511,12528}, {12556,12769}, {12688,15931}

X(16143) = reflection of X(i) in X(j) for these (i,j): (1, 16132), (40, 16117), (11531, 16126)
X(16143) = X(54) of 6th mixtilinear triangle
X(16143) = X(195) of hexyl triangle
X(16143) = X(2888) of excentral triangle
X(16143) = X(7701) of ABC-X3 reflections triangle
X(16143) = X(12325) of 2nd circumperp triangle
X(16143) = X(16132) of Aquila triangle
X(16143) = {X(191), X(3651)}-harmonic conjugate of X(165)


X(16144) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO 2nd FUHRMANN

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

X(16144) lies on these lines:
{30,7596}, {79,7133}, {191,8231}, {442,8230}, {758,9808}, {2771,12490}, {3649,8243}, {3651,8224}, {6675,10858}, {6841,8228}, {8234,16132}, {8237,16133}, {8239,10543}, {8244,16143}, {8247,16146}, {8248,16147}, {9789,14450}, {10122,11030}, {11042,16137}, {11263,12610}, {11684,11687}, {11922,16135}, {11996,16151}

X(16144) = X(54) of 2nd Pamfilos-Zhou triangle


X(16145) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd FUHRMANN TO 1st SCHIFFLER

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

X(16145) lies on these lines:
{9,12519}, {355,12745}, {6261,12524}, {6595,14450}, {10266,12615}, {12877,12947}, {12913,12957}

X(16145) = reflection of X(10266) in X(12615)


X(16146) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO 2nd FUHRMANN

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c) : : , where
F(a,b,c) = -2*b*c*(b+c)*(2*a+b+c)
G(a,b,c) = 2*c*(a+c)*(a^2+a*b+b^2-c^2)
H(a,b,c) = (a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))

X(16146) lies on these lines:
{1,16147}, {21,177}, {30,8091}, {79,15997}, {191,8078}, {442,8079}, {758,8093}, {2089,3649}, {2771,8099}, {3651,8075}, {6675,8733}, {6841,8085}, {8081,16132}, {8089,16143}, {8095,11192}, {8097,11013}, {8133,16135}, {8135,16136}, {8241,10543}, {8247,16144}, {8387,16133}, {9793,14450}, {10122,11032}, {11044,16137}, {11534,16126}, {11684,11690}, {11858,16120}, {11894,16124}

X(16146) = X(54) of tangential-midarc triangle
X(16146) = X(12341) of Hutson intouch triangle
X(16146) = X(16147) of 5th mixtilinear triangle


X(16147) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO 2nd FUHRMANN

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-H(a,b,c) : : , where
F(a,b,c) = -2*b*c*(b+c)*(2*a+b+c)
G(a,b,c) = 2*c*(a+c)*(a^2+a*b+b^2-c^2)
H(a,b,c) = (a+b+c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))

X(16147) lies on these lines:
{1,16146}, {21,7588}, {30,8092}, {174,3649}, {191,258}, {442,8080}, {758,8094}, {2771,8100}, {3651,8076}, {6675,8734}, {6732,16135}, {6841,8086}, {8082,16132}, {8090,16143}, {8096,11217}, {8125,11684}, {8138,16136}, {8242,10543}, {8248,16144}, {8351,16137}, {8388,16133}, {9795,14450}, {10122,11033}, {11859,16120}, {11895,16124}, {11899,16126}

X(16147) = X(54) of 2nd tangential-midarc triangle
X(16147) = X(12341) of intouch triangle
X(16147) = X(16146) of 5th mixtilinear triangle
X(16147) = {X(174), X(3649)}-harmonic conjugate of X(16151)


X(16148) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd TRI-SQUARES-CENTRAL TO 2nd FUHRMANN

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

X(16148) lies on these lines:
{6,6701}, {30,8983}, {79,3068}, {371,16125}, {442,13936}, {590,3647}, {758,13911}, {2771,8988}, {3648,8972}, {3649,13883}, {3652,8976}, {5441,13902}, {8974,16130}, {8975,16131}, {9540,16113}, {11684,13893}, {13884,16114}, {13885,16115}, {13886,16116}, {13887,16117}, {13888,16118}, {13889,16119}, {13890,16121}, {13891,16122}, {13892,16123}, {13895,16138}, {13896,16139}, {13897,16140}, {13898,16141}, {13901,16142}, {13903,16150}, {13904,16152}, {13905,16153}, {13906,16154}, {13907,16155}

X(16148) = X(79) of 3rd tri-squares central triangle


X(16149) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th TRI-SQUARES-CENTRAL TO 2nd FUHRMANN

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

X(16149) lies on these lines:
{6,6701}, {30,13971}, {79,3069}, {372,16125}, {442,13883}, {758,13973}, {2771,13976}, {3648,13941}, {3649,13936}, {3652,13951}, {5441,13959}, {11684,13947}, {13935,16113}, {13937,16114}, {13938,16115}, {13939,16116}, {13940,16117}, {13942,16118}, {13943,16119}, {13946,16123}, {13949,16130}, {13952,16138}, {13953,16139}, {13954,16140}, {13955,16141}, {13958,16142}, {13961,16150}, {13962,16152}, {13963,16153}, {13964,16154}, {13965,16155}

X(16149) = X(79) of 4th tri-squares central triangle


X(16150) = ORTHOLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO 2nd FUHRMANN

Barycentrics    3*a^7-(b+c)*a^6-(6*b^2-b*c+6*c^2)*a^5+(b^2+b*c+c^2)*(3*b^2-4*b*c+3*c^2)*a^3+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-2*(b^2-c^2)^3*(b-c) : :
X(16150) = 3*X(3)-2*X(16113) = 3*X(79)-X(16113) = 3*X(191)-4*X(9956) = 3*X(381)-2*X(3652) = 3*X(381)-4*X(16125) = X(944)-3*X(14450) = 4*X(946)-3*X(13743) = 2*X(946)-3*X(16159) = 5*X(1656)-4*X(3647) = 3*X(2475)-2*X(5690) = 7*X(3526)-8*X(6701) = 4*X(3649)-3*X(10246) = 2*X(5441)-3*X(10247) = 3*X(5790)-2*X(11684) = 4*X(5901)-3*X(15677)

X(16150) lies on these lines:
{3,79}, {5,3648}, {30,944}, {191,9956}, {381,3652}, {517,16118}, {758,12645}, {946,12600}, {999,16153}, {1478,12702}, {1598,16114}, {1656,3647}, {1749,15079}, {2475,5690}, {2771,5691}, {3295,16142}, {3485,11544}, {3526,6701}, {3627,9803}, {3649,4317}, {5441,10247}, {5779,13465}, {5790,11684}, {5812,10123}, {5901,15677}, {7517,16119}, {9301,16123}, {9654,16140}, {9669,16141}, {10595,15680}, {10738,13243}, {11842,16115}, {11849,16117}, {11911,16129}, {11916,16130}, {11917,16131}, {11928,16138}, {11929,16139}, {12000,16154}, {12001,16155}, {12699,12773}, {13903,16148}, {13961,16149}

X(16150) = reflection of X(3) in X(79)
X(16150) = X(79) of X3-ABC reflections triangle
X(16150) = X(3648) of Johnson triangle
X(16150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3652, 16125, 381), (16142, 16152, 3295)


X(16151) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO 2nd FUHRMANN

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

X(16151) lies on these lines:
{21,177}, {30,8351}, {79,1127}, {173,191}, {174,3649}, {442,7593}, {758,12445}, {2771,12491}, {3651,7589}, {6675,8729}, {6841,8379}, {7590,16132}, {8083,10122}, {8092,16137}, {8126,11684}, {8389,16133}, {8423,16143}, {10543,11924}, {11195,12685}, {11860,16120}, {11891,14450}, {11896,16124}, {11923,16135}, {11996,16144}

X(16151) = X(54) of Yff-central triangle
X(16151) = {X(174), X(3649)}-harmonic conjugate of X(16147)


X(16152) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 2nd FUHRMANN

Barycentrics    a^7-(2*b^2-3*b*c+2*c^2)*a^5-(b+c)*(b^2+b*c+c^2)*a^4+(b^2+b*c+c^2)*(b^2-4*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a^2-(b^2-c^2)^3*(b-c) : :
X(16152) = 3*X(79)-2*X(1836) = 2*X(1836)+3*X(16154)

X(16152) lies on these lines:
{1,30}, {4,10044}, {5,16141}, {12,1727}, {21,14526}, {35,16113}, {226,10058}, {354,10738}, {388,10043}, {442,1728}, {495,16140}, {498,3647}, {499,6701}, {758,12647}, {1479,11045}, {1709,7680}, {1737,6175}, {1749,3820}, {2771,10057}, {3065,8068}, {3085,3648}, {3295,16142}, {3336,6907}, {3582,3838}, {3583,10391}, {3584,4640}, {4679,15673}, {5057,15678}, {5270,6001}, {5499,14883}, {5728,10073}, {5902,6923}, {9612,10042}, {10037,16119}, {10038,16123}, {10039,11684}, {10040,16130}, {10523,16138}, {10801,16115}, {10954,16139}, {11398,16114}, {11507,16117}, {11877,16121}, {11878,16122}, {11912,16129}, {13904,16148}, {13962,16149}

X(16152) = X(79) of inner-Yff triangle
X(16152) = X(3652) of 1st Johnson-Yff triangle
X(16152) = X(16153) of inner-Yff tangents triangle
X(16152) = X(16154) of outer-Yff triangle
X(16152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 79, 16153), (79, 16155, 16159), (3295, 16150, 16142), (10543, 16159, 16155)


X(16153) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 2nd FUHRMANN

Barycentrics    a^7-(2*b^2+b*c+2*c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+(b^2+c^2)*(b^2+b*c+c^2)*a^3+(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^2-(b^2-c^2)^3*(b-c) : :
X(16153) = 3*X(79)+2*X(12701) = 2*X(12701)-3*X(16155)

X(16153) lies on these lines:
{1,30}, {3,16142}, {4,10051}, {5,16140}, {11,3652}, {36,16113}, {496,16141}, {497,10052}, {498,6701}, {499,3647}, {758,3436}, {938,14450}, {999,16150}, {1478,16125}, {1737,11684}, {1898,2771}, {3065,5533}, {3086,3648}, {3336,6922}, {3582,3916}, {3585,12709}, {3612,7702}, {5902,6928}, {6175,10039}, {9614,10050}, {10046,16119}, {10047,16123}, {10048,16130}, {10049,16131}, {10057,10914}, {10074,12053}, {10523,16139}, {10572,10728}, {10802,16115}, {10948,16138}, {11263,13411}, {11399,16114}, {11508,16117}, {11879,16121}, {11880,16122}, {11913,16129}, {13905,16148}, {13963,16149}

X(16153) = X(79) of outer-Yff triangle
X(16153) = X(3652) of 2nd Johnson-Yff triangle
X(16153) = X(16152) of outer-Yff tangents triangle
X(16153) = X(16155) of inner-Yff triangle
X(16153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 79, 16152), (1836, 11544, 79), (3649, 16159, 79)


X(16154) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 2nd FUHRMANN

Barycentrics    a^7-(2*b^2-7*b*c+2*c^2)*a^5-(b+c)*(b^2+b*c+c^2)*a^4+(b^4+c^4-b*c*(5*b^2+2*b*c+5*c^2))*a^3+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a^2-2*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(16154) = 5*X(79)-4*X(1836) = 2*X(1836)-5*X(16152)

X(16154) lies on these lines:
{1,30}, {12,16138}, {21,14803}, {80,3255}, {119,3065}, {758,12648}, {2475,10940}, {2771,12749}, {3256,15228}, {3647,5552}, {3648,10528}, {3652,10942}, {5499,10958}, {5559,14988}, {5902,6925}, {6923,15016}, {10531,16125}, {10803,16115}, {10805,16116}, {10834,16119}, {10878,16123}, {10915,11684}, {10929,16130}, {10930,16131}, {10955,16139}, {10956,16140}, {10965,16142}, {11248,16113}, {11400,16114}, {11509,16117}, {11881,16121}, {11882,16122}, {11914,16129}, {12000,16150}, {12608,14526}, {13906,16148}, {13964,16149}

X(16154) = X(79) of inner-Yff tangents triangle
X(16154) = X(16138) of 1st Johnson-Yff triangle
X(16154) = {X(79), X(5441)}-harmonic conjugate of X(16155)


X(16155) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 2nd FUHRMANN

Barycentrics    a^7-(b+2*c)*(2*b+c)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4+(b^2+c^2)*(b^2+3*b*c+c^2)*a^3+(b^2-c^2)*(b-c)*(2*b^2-b*c+2*c^2)*a^2+2*(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(16155) = X(79)+4*X(12701) = 2*X(12701)+X(16153)

X(16155) lies on these lines:
{1,30}, {11,16139}, {35,12609}, {46,12875}, {72,80}, {191,9614}, {442,5119}, {758,1479}, {2771,12374}, {3434,5697}, {3586,16126}, {3612,11281}, {3647,10527}, {3648,10529}, {3651,14798}, {3652,10943}, {3746,11218}, {5427,11373}, {5428,11376}, {5705,7741}, {5902,6836}, {10532,16125}, {10624,11263}, {10707,10916}, {10804,16115}, {10806,16116}, {10835,16119}, {10879,16123}, {10931,16130}, {10932,16131}, {10941,14450}, {10949,16138}, {10957,16140}, {10959,16141}, {10966,13743}, {11249,16113}, {11401,16114}, {11510,16117}, {11604,12758}, {11883,16121}, {11915,16129}, {12001,16150}, {13907,16148}, {13965,16149}

X(16155) = X(79) of outer-Yff tangents triangle
X(16155) = X(16139) of 2nd Johnson-Yff triangle
X(16155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (79, 5441, 16154), (10543, 16159, 16152), (16152, 16159, 79)


X(16156) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st PARRY TO 2nd FUHRMANN

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

X(16156) lies on these lines:
{351,16157}, {523,9810}, {6003,13250}, {8674,13263}

X(16156) = X(79) of 1st Parry triangle
X(16156) = X(16113) of 2nd Parry triangle


X(16157) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd PARRY TO 2nd FUHRMANN

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

X(16157) lies on these lines:
{351,16156}, {523,9811}, {6003,13251}, {8674,13264}

X(16157) = X(79) of 2nd Parry triangle
X(16157) = X(16113) of 1st Parry triangle


X(16158) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO 2nd FUHRMANN

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

X(16158) lies on these lines:
{21,2787}, {23,385}, {98,105}, {100,110}, {404,14419}, {804,5985}, {1283,13265}, {1621,4010}, {1635,13256}, {2254,6003}, {2775,3651}, {2975,4922}, {3716,8645}, {3871,4730}, {5047,14431}

X(16158) = X(54) of 2nd Sharygin triangle


X(16159) = X(3) OF 2nd FUHRMANN TRIANGLE

Barycentrics    a^7-(2*b^2+b*c+2*c^2)*a^5-(b^3+c^3)*a^4+(b^4+c^4)*a^3+(b^2-c^2)*(b-c)*(2*b^2+b*c+2*c^2)*a^2+(b^2-c^2)^2*b*c*a-(b^2-c^2)^3*(b-c) : :
X(16159) = 2*X(21)-3*X(5886) = 2*X(79)+X(12699) = 3*X(165)-4*X(11277) = X(355)-4*X(16125) = 3*X(381)-X(13465) = 2*X(946)+X(16150) = 3*X(1699)-X(7701) = 3*X(1699)-2*X(16160) = X(3648)-4*X(9955) = 3*X(5426)-4*X(5901) = 3*X(5603)-X(15680) = 5*X(8227)-4*X(10021) = 6*X(11230)-5*X(15674)

X(16159) lies on these lines:
{1,30}, {3,11263}, {4,2771}, {5,191}, {21,5886}, {40,5499}, {55,14526}, {165,11277}, {355,758}, {381,13465}, {442,5812}, {516,16117}, {517,2475}, {946,12600}, {952,16126}, {1385,12877}, {1699,7701}, {1749,7741}, {1770,5172}, {2886,13852}, {3648,3916}, {3652,5805}, {3654,6175}, {5426,5901}, {5428,16113}, {5536,13089}, {5554,10526}, {5603,15680}, {5694,6839}, {5843,15909}, {5880,6701}, {5885,6840}, {6928,8261}, {8227,10021}, {11230,15674}, {11374,16142}

X(16159) = midpoint of X(4) and X(14450)
X(16159) = reflection of X(i) in X(j) for these (i,j): (3, 11263), (40, 5499), (3652, 6841), (3654, 6175)
X(16159) = X(191) of Johnson triangle
X(16159) = X(8254) of 2nd Conway triangle
X(16159) = X(11263) of X3-ABC reflections triangle
X(16159) = X(13423) of 3rd Euler triangle
X(16159) = X(14450) of Euler triangle
X(16159) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (79, 16153, 3649), (79, 16155, 16152), (1699, 7701, 16160), (16152, 16155, 10543)


X(16160) = X(5) OF 2nd FUHRMANN TRIANGLE

Barycentrics    (b^2-4*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(2*b^4+2*c^4-b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b^2+3*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :
X(16160) = 3*X(3)-5*X(15674) = 3*X(4)+X(15680) = 3*X(5)-2*X(442) = 3*X(1699)+X(7701) = 3*X(1699)-X(16159) = 3*X(3656)-X(16126) = 3*X(5886)-X(16132) = 5*X(8227)-X(16143)

As a point on the Euler line, X(16160) has Shinagawa coefficients (R+2*r, 9*R+2*r)

X(16160) lies on these lines:
{2,3}, {11,79}, {12,5441}, {191,12699}, {495,10543}, {496,3649}, {758,3813}, {946,1484}, {1385,12558}, {1699,7701}, {2886,3647}, {3648,11680}, {3652,5536}, {3656,16126}, {3742,9955}, {3816,6701}, {4860,11544}, {5180,11684}, {5427,15446}, {5659,16139}, {5886,16132}, {6147,10122}, {7681,16125}, {7741,16118}, {8227,16143}, {10266,10308}, {10957,16140}, {12611,12615}, {15908,16113}

X(16160) = midpoint of X(191) and X(12699)
X(16160) = anticomplement of X(11277)
X(16160) = complement of X(16117)
X(16160) = X(195) of 3rd Euler triangle
X(16160) = X(5499) of Johnson triangle
X(16160) = X(10021) of X3-ABC reflections triangle
X(16160) = X(12307) of 4th Euler triangle
X(16160) = X(13365) of 2nd Conway triangle
X(16160) = X(13743) of Euler triangle
X(16160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 16117, 11277), (3, 381, 6900), (5, 8703, 8728), (140, 8226, 5), (546, 6831, 5), (1699, 7701, 16159), (2475, 4193, 442), (3850, 6882, 5), (4187, 5066, 5), (6841, 13852, 8226)


X(16161) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 2nd FUHRMANN

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

X(16161) lies on these lines:
{30,12440}, {79,493}, {758,12636}, {2771,12741}, {3647,8222}, {3648,6462}, {3649,11377}, {3652,8220}, {5441,8210}, {6461,16162}, {8188,16118}, {8194,16119}, {8212,16125}, {8214,11684}, {8216,16130}, {8218,16131}, {10875,16123}, {10945,16138}, {10951,16139}, {11394,16114}, {11503,16117}, {11828,16113}, {11840,16115}, {11846,16116}, {11907,16129}, {11930,16140}, {11932,16141}, {11947,16142}, {11949,16150}, {11951,16152}, {11953,16153}, {11955,16154}, {11957,16155}, {13899,16148}, {13956,16149}

X(16161) = X(79) of Lucas homothetic triangle


X(16162) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 2nd FUHRMANN

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

X(16162) lies on these lines:
{30,12441}, {79,494}, {758,12637}, {2771,12742}, {3647,8223}, {3649,11378}, {3652,8221}, {5441,8211}, {6461,16161}, {8189,16118}, {8213,16125}, {10876,16123}, {10946,16138}, {11395,16114}, {11847,16116}, {11931,16140}, {11933,16141}, {11948,16142}, {11950,16150}, {11952,16152}, {11954,16153}, {11956,16154}, {13900,16148}

X(16162) = X(79) of Lucas(-1) homothetic triangle


X(16163) =  MIDPOINT OF X(20) AND X(110)

Barycentrics    (-a^2+b^2+c^2)*(2*a^4-(b^2+c^ 2)*a^2-(b^2-c^2)^2)^2 : :
Barycentrics    (sin 2A)(cos A - 2 cos B cos C)^2 : :
X(16163) = 3*X(2) - 5*X(15051) = 3*X(3) - X(265) = 3*X(3) - 2*X(6699) = 5*X(3) - X(12902) = 11*X(3) - 5*X(15027) = 5*X(3) - 3*X(15061) = 3*X(125) - 2*X(265) = 3*X(125) - 4*X(6699) = X(125)+2*X(12121) = 5*X(125) - 2*X(12902) = 11*X(125) - 10*X(15027) = 5*X(125) - 6*X(15061) = X(265)+3*X(12121) = 5*X(265) - 3*X(12902) = 11*X(265) - 15*X(15027) = 5*X(265) - 9*X(15061) = 2*X(6699)+3*X(12121) = 10*X(6699) - 3*X(12902) = 22*X(6699) - 15*X(15027) = 10*X(6699) - 9*X(15061) = 2*X(7687) - 5*X(15051) = X(10733) - 5*X(15051) = 5*X(12121)+X(12902) = 11*X(12121)+5*X(15027) = 5*X(12121)+3*X(15061) = X(12902) - 3*X(15061)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27182.

Let A'B'C' be as at X(13202). The X(16163) = X(4)-of-A'B'C'. (Randy Hutson, March 14, 2018)

The trilinear polar of X(16163) passes through X(14401). (Randy Hutson, March 14, 2018)

X(16163) lies on these lines:
{2, 7687}, {3, 125}, {4, 5972}, {5, 12295}, {20, 110}, {22, 13289}, {30, 113}, {36, 12896}, {51, 9826}, {52, 14708}, {69, 74}, {114, 7422}, {133, 4240}, {140, 10113}, {154, 11744}, {159, 2935}, {165, 13211}, {184, 4846}, {343, 8703}, {378, 3818}, {381, 12900}, {382, 14643}, {394, 399}, {511, 1986}, {516, 11720}, {548, 12041}, {550, 5562}, {631, 6723}, {632, 15088}, {648, 5667}, {974, 6467}, {1060, 12888}, {1099, 1354}, {1154, 14049}, {1204, 12118}, {1216, 7723}, {1300, 8754}, {1503, 5181}, {1657, 7728}, {1974, 15462}, {1993, 12227}, {2071, 12827}, {2407, 16075}, {2420, 6793}, {2693, 13494}, {2771, 3650}, {2781, 3313}, {2854, 15151}, {2968, 3916}, {2979, 12219}, {3028, 15326}, {3043, 13619}, {3070, 8998}, {3071, 13990}, {3146, 15020}, {3163, 9408}, {3184, 9033}, {3289, 6781}, {3448, 3522}, {3523, 15059}, {3524, 15081}, {3529, 10721}, {3530, 11801}, {3576, 11735}, {3629, 14831}, {3917, 12358}, {3937, 13369}, {4235, 15595}, {4299, 10088}, {4302, 10091}, {5054, 15042}, {5076, 15046}, {5085, 15118}, {5204, 12904}, {5217, 12903}, {5609, 12103}, {5655, 15681}, {5731, 7984}, {5907, 12292}, {6000, 12825}, {6101, 15332}, {6361, 7978}, {6560, 10819}, {6561, 10820}, {7722, 11412}, {8907, 11413}, {9140, 10304}, {9517, 14689}, {9529, 14697}, {9729, 11800}, {9730, 12236}, {10117, 11414}, {10257, 13851}, {10303, 15023}, {10620, 15696}, {10625, 11562}, {10628, 15644}, {10706, 11001}, {10984, 13198}, {11204, 11442}, {11430, 14389}, {11561, 13391}, {11723, 12699}, {11807, 12824}, {12028, 14595}, {12054, 12201}, {12086, 13419}, {12133, 15030}, {12228, 13352}, {12244, 14094}, {12261, 13624}, {12273, 15072}, {12661, 15941}, {12702, 12898}, {13346, 15463}, {13416, 15738}, {14683, 15054}, {14847, 15774}, {15160, 15461}, {15161, 15460}

X(16163) = midpoint of X(i) and X(j) for these {i,j}: {3, 12121}, {20, 110}, {1657, 7728}, {3529, 10721}, {5655, 15681}, {6361, 7978}, {7722, 11412}, {10625, 11562}, {10706, 11001}, {12244, 14094}, {12702, 12898}, {14683, 15054}
X(16163) = reflection of X(i) in X(j) for these (i,j): (4, 5972), (52, 14708), (113, 1511), (12261, 13624), (13202, 113)
X(16163) = complement of X(10733)
X(16163) = anticomplement of X(7687)
X(16163) = X(125)-of-ABC-X3-reflections triangle
X(16163) = X(5972) of anti-Euler triangle
X(16163) = X(12295)-of -Johnson-triangle
X(16163) = X(15030)-of-anti-orthocentroidal-triangle
X(16163) = crossdifference of every pair of points on line X(2433)X(8749)
X(16163) = orthopole of line X(20)X(523)
X(16163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 10733, 7687), (3, 265, 6699), (3, 12902, 15061), (4, 15035, 5972), (113, 1511, 5642), (146, 6053, 15063), (265, 6699, 125), (376, 12383, 74), (616, 617, 1494), (1511, 1539, 10272), (1511, 13392, 11693), (1531, 11064, 1568), (1539, 10272, 113), (5642, 13202, 113), (10733, 15051, 2), (14499, 14500, 1568)


X(16164) =  MIDPOINT OF X(21) AND X(110)

Barycentrics    a*(a^3-(b+c)*a^2-(b^2+b*c+c^2) *a+(b+c)*(b^2+c^2))*(2*a^4-(b^ 2+c^2)*a^2-(b^2-c^2)^2)*(a+c)* (a+b) : :
X(16164) = X(2948) + 3*X(5426) = X(3448) - 5*X(15674) = X(3651) - 3*X(15035) = X(5609) + 2*X(12104) = X(9140) - 3*X(15671) = X(9143) + 3*X(15672) = X(14683) + 7*X(15676) = 5*X(15040) - X(16117)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27182.

X(16164) lies on these lines:
{21, 104}, {30, 113}, {125, 6675}, {442, 5972}, {517, 12826}, {542, 15670}, {758, 11720}, {2948, 5426}, {3028, 5427}, {3448, 15674}, {3651, 15035}, {5428, 5663}, {5609, 12104}, {9140, 15671}, {9143, 15672}, {14683, 15676}, {15040, 16117}

X(16164) = midpoint of X(21) and X(110)
X(16164) = reflection of X(i) in X(j) for these (i,j): (125, 6675), (442, 5972)


X(16165) =  MIDPOINT OF X(22) AND X(110)

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^4-b^4+b^ 2*c^2-c^4)*(2*a^4-(b^2+c^2)*a^ 2-(b^2-c^2)^2) : :
X(16165) = X(378) - 3*X(15035) = X(5609) + 2*X(7555) = X(12082) + 5*X(15034)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27182.

X(16165) lies on these lines:
{3, 15738}, {22, 110}, {23, 6593}, {25, 15462}, {30, 113}, {69, 5648}, {125, 6676}, {182, 12099}, {184, 14984}, {343, 542}, {378, 15035}, {427, 5972}, {974, 2931}, {1370, 15131}, {1503, 12827}, {3549, 15133}, {3796, 5622}, {5422, 11746}, {5562, 5609}, {5663, 7502}, {7387, 16105}, {9140, 15080}, {9934, 12168}, {11598, 13445}, {12041, 14855}, {12082, 15034}, {12310, 13198}, {13394, 15760}

X(16165) = midpoint of X(22) and X(110)
X(16165) = reflection of X(i) in X(j) for these (i,j): (125, 6676), (427, 5972)


X(16166) =  X(5)X(477)∩X(74)X(2070)

Barycentrics    (SB+SC)*(SA-SB)*(SA-SC)*(15* R^2+2*SB-4*SW)*(15*R^2+2*SC-4* SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27185.

X(16166) lies on the circumcircle and these lines:
{5, 477}, {74, 2070}, {842, 13595}, {930, 7471}, {933, 7480}, {1141, 10096}, {1291, 15329}, {1294, 3153}, {2693, 7488}, {7426, 9076}, {14670, 14979}

X(16166) = trilinear pole of the line X(6)X(11559)


X(16167) =  X(20)X(841)∩X(22)X(477)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27185.

X(16167) lies on the circumcircle and these lines:
{20, 841}, {22, 477}, {23, 1300}, {74, 858}, {112, 7471}, {468, 1299}, {842, 7493}, {915, 7469}, {1289, 7480}, {1370, 2693}, {2373, 3260}, {3563, 7426}, {3658, 10100}, {4226, 10098}, {4240, 10423}

X(16167) = trilinear pole of the line X(6)X(7706)


X(16168) =  INFINITY POINT OF X(3)X(476)

Trilinears    (10*cos(2*A)+cos(4*A)+9)*cos( B-C)-4*cos(A)*cos(2*(B-C))+1/ 2*cos(3*(B-C))-12*cos(A)-4* cos(3*A) : :
Barycentrics    S^4+(-3*R^2*(9*R^2+3*SA-4*SW)+ 2*SA^2+SB*SC-SW^2)*S^2+(27*R^ 2*(3*R^2-SW)+SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27192.

X(16168) lies on these lines:
{3, 476}, {4, 14670}, {5, 3258}, {30, 511}, {381, 15111}, {382, 14264}, {399, 14480}, {1511, 7471}, {3447, 12028}, {5609, 14611}, {6070, 10264}, {9179, 14650}, {10095, 12052}, {10620, 14508}, {14851, 14993}, {14895, 15807}

X(16168) = isogonal conjugate of X(16169)


X(16169) =  ISOGONAL CONJUGATE OF X(16168)

Trilinears    sin(A-B)*sin(A-C)*((cos(A-B)- 2*cos(C))^2-(cos(2*C)+1/2)^2)* ((cos(A-C)-2*cos(B))^2-(cos(2* B)+1/2)^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27192.

X(16169) lies on the circumcircle and these lines:
{74, 14809}, {107, 7722}, {110, 14933}, {476, 5663}, {477, 526}

X(16169) = isogonal conjugate of X(16168)
X(16169) = circumcircle-antipode of X(16170)
X(16169) = SR(X(476),X(477))


X(16170) =  X(476)X(526)∩X(477)X(5663)

Trilinears    (SB+SC)*(S^4+(-3*R^2*(9*R^2+3* SB-4*SW)+2*SB^2+SC*SA-SW^2)*S^ 2+(27*R^2*(3*R^2-SW)+SW^2)*SC* SA)*(S^4+(-3*R^2*(9*R^2+3*SC- 4*SW)+2*SC^2+SA*SB-SW^2)*S^2+( 27*R^2*(3*R^2-SW)+SW^2)*SA*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27192.

X(16170) lies on the circumcircle and these lines:
{476, 526}, {477, 5663}, {1294, 12219}

X(16170) = isogonal conjugate of X(16171)
X(16170) = circumcircle-antipode of X(16169)
X(16170) = X(526)-circumcevian-isogonal conjugate of X(526)
X(16170) = X(5663)-circumcevian-isogonal conjugate of X(5663)


X(16171) =  INFINITY POINT OF X(476)X(10412)

Trilinears    sin(B-C)*((cos(B-C)-2*cos(A))^ 2-(cos(2*A)+1/2)^2) : :

Let A1'B1'C1' and A2'B2'C2' be the 1st and 2nd Ehrmann inscribed triangles. Let P1 be the intersection, other than A1', B1', C1', of the Ehrmann conic and the circumcircle of A1'B1'C1'. Let P2 be the intersection, other than A2', B2', C2', of the Ehrmann conic and the circumcircle of A2'B2'C2'. The tangents at P1 and P2 to the respective circumcircles are parallel and meet the line at infinity at X(16171). (Randy Hutson, June 27, 2018)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27192.

X(16171) lies on these lines:
{30, 511}, {351, 9158}, {476, 10412}, {3134, 3258}, {14380, 14989}

X(16171) = isogonal conjugate of X(16170)


X(16172) =  ISOGONAL CONJUGATE OF X(15478)

Barycentrics    SB*SC*(2*R^2-SB)*(2*R^2-SC)* (6*R^2-SA-SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27193.

X(16172) lies on the cubics K025 and K339, and on these lines:
{4, 155}, {30, 13398}, {131, 403}

X(16172) = isogonal conjugate of X(15478)
X(16172) = antigonal conjugate of X(403)
X(16172) = anticomplementary-circle-inverse-of X(12318)
X(16172) = = polar circle-inverse-of X(155)


X(16173) =  (X(1),X(11))-HARMONIC CONJUGATE OF X(80)

Barycentrics    a^4-(b+c)*a^3-(2*b-c)*(b-2*c)* a^2+(b^2-c^2)*(b-c)*a+(b^2-c^ 2)^2 : :
X(16173) = X(1)+2*X(11) = 2*X(1)+X(80) = 5*X(1)-2*X(1317) = X(1)-4*X(1387) = 4*X(1)-X(7972) = 5*X(1)+X(9897) = 5*X(1)+4*X(12019) = 7*X(1)-4*X(12735) = 2*X(5)+X(12737) = 4*X(5)-X(12751) = 4*X(11)-X(80) = 5*X(11)+X(1317) = X(11)+2*X(1387) = 8*X(11)+X(7972) = 10*X(11)-X(9897) = 5*X(11)-2*X(12019) = 7*X(11)+2*X(12735) = 5*X(80)+4*X(1317) = X(80)+8*X(1387) = 2*X(80)+X(7972) = 5*X(80)-2*X(9897) = 5*X(80)-8*X(12019) = 7*X(80)+8*X(12735)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27196.

Let Oa, Ob, Oc be the circles with collinear centers described at X(5531) and Hyacinthos #21433 (Barry Wolk, January 2013). Let A'B'C' be the triangle formed by the radical axes of these circles and the corresponding mixtilinear incircle. A'B'C' is homothetic to the intouch triangle at X(16173). (Randy Hutson, March 14, 2018)

X(16173) lies on these lines:
{1, 5}, {2, 2802}, {3, 14217}, {4, 11715}, {8, 6702}, {10, 1320}, {35, 6940}, {36, 516}, {40, 5442}, {55, 5444}, {79, 104}, {100, 1125}, {106, 3120}, {145, 15863}, {149, 214}, {354, 2771}, {474, 13205}, {484, 15325}, {497, 6951}, {499, 5445}, {517, 3582}, {528, 15015}, {551, 6175}, {942, 11571}, {1001, 3254}, {1145, 1698}, {1156, 5542}, {1210, 11009}, {1319, 3583}, {1385, 4857}, {1388, 9669}, {1420, 10483}, {1478, 9779}, {1479, 5731}, {1482, 12619}, {1537, 1768}, {1699, 2829}, {1702, 13913}, {1703, 13977}, {1706, 3035}, {2646, 13274}, {2800, 5603}, {2801, 11038}, {3036, 3632}, {3057, 6797}, {3065, 3649}, {3086, 5903}, {3244, 12531}, {3245, 3911}, {3303, 12331}, {3304, 12611}, {3336, 12515}, {3340, 12832}, {3485, 5083}, {3576, 5840}, {3584, 5919}, {3585, 12764}, {3622, 6224}, {3628, 13143}, {3646, 7162}, {3679, 5854}, {3825, 4861}, {3874, 12532}, {4297, 10724}, {4309, 13199}, {4316, 5126}, {4317, 12248}, {4330, 13624}, {4870, 5049}, {4973, 5180}, {4996, 5248}, {5131, 5298}, {5259, 13279}, {5425, 11019}, {5433, 11010}, {5550, 9802}, {5904, 10529}, {6595, 12267}, {7280, 12701}, {7284, 11372}, {7343, 13605}, {9964, 10122}, {10039, 10172}, {10074, 12047}, {10246, 11238}, {10265, 10698}, {10399, 12691}, {10404, 16128}, {10589, 12647}, {10595, 12247}, {10768, 11710}, {10769, 11711}, {10770, 11712}, {10771, 11713}, {10772, 11714}, {10773, 11716}, {10774, 11717}, {10775, 11718}, {10776, 11719}, {10777, 11700}, {10778, 11720}, {10779, 11721}, {10780, 11722}, {10912, 12641}, {11012, 16155}, {11256, 12607}, {11570, 14986}, {12767, 13226}, {13463, 13747}

X(16173) = reflection of X(5131) in X(5298)
X(16173) = incircle-inverse-of X(12019)
X(16173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11, 80), (1, 80, 7972), (1, 9897, 1317), (5, 12737, 12751), (11, 1317, 12019), (11, 1387, 1), (149, 3616, 214), (496, 12740, 10073), (946, 5563, 79), (1317, 12019, 9897), (1484, 5901, 6265), (6264, 8227, 119), (6326, 9624, 11729), (9897, 12019, 80), (11373, 11376, 1)


X(16174) =  MIDPOINT OF X(11) AND X(946)

Barycentrics    (b+c)*a^6+(b^2-6*b*c+c^2)*a^5- (b+c)*(4*b^2-9*b*c+4*c^2)*a^4- (2*b-c)*(b-2*c)*(b-c)^2*a^3+( b^2-c^2)*(b-c)*(5*b^2-3*b*c+5* c^2)*a^2+(b^2-c^2)^2*(b^2-3*b* c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(16174) = 3*X(2)+X(14217) = 3*X(11)+X(1537) = 3*X(11)-X(10265) = X(80)+3*X(5603) = X(100)-5*X(8227) = X(104)+3*X(1699) = X(119)-3*X(3817) = X(153)-9*X(9779) = X(214)-3*X(5886) = 3*X(381)+X(12737) = 3*X(946)-X(1537) = 3*X(946)+X(10265) = X(1145)-3*X(10175) = X(1538)+3*X(7743) = 3*X(5886)+X(10738)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27196.

X(16174) lies on these lines:
{1, 6246}, {2, 14217}, {4, 11715}, {5, 2802}, {11, 65}, {80, 5603}, {100, 8227}, {104, 1699}, {119, 3817}, {153, 9779}, {214, 5886}, {381, 12737}, {496, 12005}, {515, 1387}, {516, 6713}, {517, 6702}, {952, 3850}, {1125, 5840}, {1145, 10175}, {1320, 5587}, {1482, 15863}, {1484, 2801}, {1512, 8068}, {2771, 13374}, {3036, 5087}, {3091, 12751}, {3576, 10724}, {3616, 12119}, {3898, 6980}, {5083, 5533}, {5541, 7988}, {6326, 10707}, {6667, 6684}, {6796, 9614}, {6918, 13205}, {7972, 10595}, {7989, 12653}, {7993, 10711}, {10057, 10598}, {10698, 11522}, {11236, 11256}, {11375, 13274}, {11376, 13273}, {12743, 15950}

X(16174) = midpoint of X(i) and X(j) for these {i,j}: {1, 6246}, {4, 11715}, {11, 946}, {214, 10738}, {1482, 15863}, {1484, 12611}
X(16174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 1537, 10265), (946, 10265, 1537), (5533, 12047, 5083), (5886, 10738, 214)


X(16175) =  X(316)X(2393)∩X(598)X(11188)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27198.

X(16175) lies on these lines:
{316, 2393}, {598, 11188}, {671, 2854}, {691, 2936}, {2386, 7799}, {2930, 14246}, {2979, 7850}, {4611, 9157}, {7812, 9971}, {8681, 11054}


X(16176) =  MIDPOINT OF X(11008) AND X(14683)

Barycentrics    (5*a^8-5*(b^2+c^2)*a^6-(3*b^4- 11*b^2*c^2+3*c^4)*a^4+5*(b^4- c^4)*(b^2-c^2)*a^2-2*(b^4-c^4) ^2 : :
X(16176) = 3*X(6)-2*X(67), 5*X(6)-4*X(125), 3*X(6)-4*X(5095), 9*X(6)-8*X(15118), 5*X(67)-6*X(125), 3*X(67)-4*X(15118), 3*X(125)-5*X(5095), 9*X(125)-10*X(15118), 2*X(265)-3*X(5102), 3*X(599)-4*X(6593), 2*X(895)-3*X(15534), 5*X(2930)-6*X(9143), 3*X(5095)-2*X(15118), 3*X(5621)-4*X(8550), 3*X(9143)-5*X(11061)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27198.

X(16176) lies on these lines:
{6, 67}, {23, 524}, {265, 5102}, {382, 542}, {399, 5965}, {599, 6593}, {895, 8877}, {2781, 5925}, {2836, 3901}, {2854, 6144}, {3448, 3629}, {3520, 5621}, {5181, 15533}, {6698, 13169}, {9970, 15069}, {9973, 13417}, {10628, 10938}, {11008, 14683}

X(16176) = midpoint of X(11008) and X(14683)
X(16176) = reflection of X(i) in X(j) for these (i,j): (3448, 3629), (9973, 13417)


X(16177) =  COMPLEMENT OF X(1304)

Barycentrics    SA*(S^2-3*SB*SC)*(2*S^2+SA^2+ 2*SB*SC-SW^2) *(5*S^2-24*R^2*SA+5*SA^2-4*SB* SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27201.

X(16177) lies on the nine-point circle and these lines:
{2, 1304}, {4, 2693}, {30, 133}, {113, 2072}, {114, 5159}, {115, 6587}, {122, 523}, {128, 6760}, {131, 10257}, {132, 858}, {136, 3154}, {1560, 3163}, {1650, 3258}, {2972, 6070}, {3150, 5099}, {3548, 15454}, {5627, 15404}, {12079, 15526}, {13573, 15384}

X(16177) = midpoint of X(i) and X(j) for these {i,j}: {4, 2693}, {13573, 15384}
X(16177) = complement of X(1304)
X(16177) = X(1294) of reflection of Euler triangle in Euler line
X(16177) = reflection of X(122) in Euler line
X(16177) = orthoptic-circle-of-Steiner-inellipse-inverse of X(2697)


X(16178) =  X(4)X(10420)∩X(115)X(6753)

Barycentrics    SB*SC*(6*R^2-SA-SW)*(2*S^2+SA^ 2+2*SB*SC-SW^2)*((14*R^2-3*SW) *S^2-(6*R^2-SW)*SA^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27201.

X(16178) lies on the nine-point circle and these lines:
{4, 10420}, {113, 10151}, {115, 6753}, {131, 403}, {135, 523}

X(16178) = polar circle-inverse of X(10420)
X(16178) = X(1299) of reflection of Euler triangle in Euler line
X(16178) = reflection of X(135) in Euler line


X(16179) =  ORTHOGONAL PROJECTION OF X(13) ON THE EULER LINE

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

As a point on the Euler line, X(16179) has Shinagawa coefficients (-6(3(E + F)F - S2) + 31/2(E - 8F)S, 6((E + F )2 - 3S2) + 3(3)1/2(E - 8F)S ).

See Seiichi Kirikami and César Lozada, Hyacinthos 27204.

X(16179) lies on these lines: {2,3}, {13, 523}


X(16180) =  ORTHOGONAL PROJECTION OF X(14) ON THE EULER LINE

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

As a point on the Euler line, X(16180) has Shinagawa coefficients (-6(3(E + F)F - S2) - 31/2(E - 8F)S, 6((E + F )2 - 3S2) - 3(3)1/2(E - 8F)S ).

See Seiichi Kirikami and César Lozada, Hyacinthos 27204.

X(16180) lies on this line: {2,3}


X(16181) =  ORTHOGONAL PROJECTION OF X(15) ON THE EULER LINE

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

As a point on the Euler line, X(16181) has Shinagawa coefficients (-2(3(E + F)F - S2) + 31/2(E - 8F)S, 2((E + F )2 - 3S2) - 31/2(E - 8F)S ).

See Seiichi Kirikami and César Lozada, Hyacinthos 27204.

X(16181) lies on these lines: {2, 3}, {15, 523}, {2452, 11485}, {2453, 11480}

X(16181) = {X(3), X(1316)}-harmonic conjugate of X(16182)


X(16182) =  ORTHOGONAL PROJECTION OF X(16) ON THE EULER LINE

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

As a point on the Euler line, X(16182) has Shinagawa coefficients (-2(3(E + F)F - S2) + 31/2(E - 8F)S, 2((E + F )2 - 3S2) + 31/2(E - 8F)S ).

See Seiichi Kirikami and César Lozada, Hyacinthos 27204.

X(16182) lies on this line: {2, 3}

X(16182) = {X(3), X(1316)}-harmonic conjugate of X(16181)


X(16183) =  4th HUNG-LOZADA-EULER POINT

Barycentrics    SB*SC*(9*(12*R^2-SA-SW)*S^2-4* SW^3) : :
X(16183) = 9*S^2*(4*R^2-SW)*X(3) - (2*SW^3- 9*S^2*(8*R^2-SW))*X(4)

See Tran Quang Hung and César Lozada, Hyacinthos 27207.

X(16183) lies on these lines: {2, 3}, {5139,6092}


X(16184) =  REFLECTION OF X(3025) IN THE LINE X(1)X(2742)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27208.

X(16184) lies on the incircle and this line: {2348, 3021}

X(16184) = reflection of X(1358) in the line {X(1), X(142)}
X(16184) = reflection of X(3025) in the line {X(1), X(2742)}
X(16184) = reflection of X(3323) in the line {X(1), X(5519)}


X(16185) =  REFLECTION OF X(3025) IN THE LINE X(1)X(2743)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27208.

X(16185) lies on the incircle and these lines: {11, 14112}, {3880, 6018}

X(16185) = reflection of X(11) in the line {X(1), X(13625)}
X(16185) = reflection of X(1357) in the line {X(1), X(474)}
X(16185) = reflection of X(3025) in the line {X(1), X(2743)}
X(16185) = reflection of X(14027) in the line {X(1), X(5516)}


X(16186) =  X(3)X(49)∩X(122)X(125)

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

See Angel Montesdeoca, HG001218.

X(16186) lies on these lines: {3, 49}, {4, 15111}, {30, 14670}, {122, 125}, {186, 7740}, {323, 14355}, {402, 11657}, {511, 15329}, {523, 3134}, {868, 6328}, {1312, 10287}, {1313, 10288}, {1495, 5502}, {1624, 13417}, {3001, 13857}, {3470, 14157}, {5651, 14687}, {6000, 14264}, {7669, 15106}


X(16187) =  X(2)X(98)∩X(3)X(5646)

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

See Angel Montesdeoca, HG0501218.

X(16187) lies on these lines:
{2, 98}, {3, 5646}, {6, 5544}, {25, 14810}, {373, 576}, {394, 5097}, {511, 3066}, {549, 8717}, {567, 5070}, {575, 6090}, {578, 3628}, {1092, 5067}, {1350, 3819}, {1351, 5943}, {1656, 13352}, {1995, 3098}, {3090, 13346}, {3124, 8585}, {3231, 5039}, {3525, 13347}, {3526, 6759}, {3533, 10984}, {4550, 12041}, {5034, 9225}, {5643, 5645}, {5888, 7492}, {7486, 11424}, {7496, 10546}, {8722, 11328}, {10117, 11204}, {10170, 11438}, {10219, 10601}, {10539, 13339}, {11484, 13598}.{3, 49}, {4, 15111}, {30, 14670}, {122, 125}, {186, 7740}, {323, 14355}, {402, 11657}, {511, 15329}, {523, 3134}, {868, 6328}, {1312, 10287}, {1313, 10288}, {1495, 5502}, {1624, 13417}, {3001, 13857}, {3470, 14157}, {5651, 14687}, {6000, 14264}, {7669, 15106}


X(16188) =  COMPLEMENT OF X(842)

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

Let A'B'C' be the orthic triangle. Let La be line X(115)X(125) of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(16188), which is X(3)-of-A"B"C". (Randy Hutson, July 31 2018)

See Angel Montesdeoca, HG120218.

X(16188) lies on the nine-point circle and these lines:
{2, 476}, {4, 691}, {5, 5099}, {11, 6023}, {12, 6027}, {23, 137}, {30, 115}, {113, 512}, {114, 523}, {122, 5159}, {125, 511}, {127, 625}, {136, 468}, {249, 12383}, {403, 5139}, {542, 1550}, {1560, 2501}, {2080, 7574}, {2696, 14659}, {2697, 2715}, {3153, 14712}, {3635, 15519}, {5094, 14687}, {5189, 11792}, {5512, 11799}, {7575, 14693}, {8705, 12494}, {9151, 15820}, {9218, 10733}, {9970, 15899}, {10297, 13449}, {10556, 15398}, {11007, 15819}

X(16188) = midpoint of X(i) and X(j) for these {i,j}: {4, 691}, {1550, 14999}, {2080, 7574}
X(16188) = reflection of X(i) in X(j) for these (i,j): {5099, 5}, {7575, 14693}, {13449, 10297}
X(16188) = complement of X(842)
X(16188) = perspector of circumconic centered at X(2493)
X(16188) = X(2)-Ceva conjugate of X(2493)
X(16188) = orthoptic-circle-of-Steiner-inellipe-inverse of X(476)
X(16188) = nine-point circle antipode of X(5099)
X(16188) = X(99) of reflection of Euler triangle in Euler line
X(16188) = reflection of X(114) in Euler line
X(16188) = Λ(Fermat axis) wrt orthic triangle






leftri  Eulerologic centers 2: X(16189) - X(16220)  rightri

Eulerologic triangles and centers are defined in the preamble just before X10237. Centers X(16189)-X(16220) were contributed by César Eliud Lozada, February 19, 2018.

underbar

X(16189) = EULEROLOGIC CENTER OF THESE TRIANGLES: ABC TO EXCENTERS-REFLECTIONS

Barycentrics    a*(5*a^3-9*(b+c)*a^2-(5*b^2-18*b*c+5*c^2)*a+9*(b^2-c^2)*(b-c)) : :
X(16189) = 9*X(1)-4*X(3) = 7*X(1)-2*X(40) = 8*X(1)-3*X(165) = 13*X(1)-8*X(1385) = X(1)+4*X(1482) = 11*X(1)-6*X(3576) = 23*X(1)-8*X(3579) = 3*X(1)+2*X(7982) = 6*X(1)-X(7991) = 11*X(1)+4*X(8148) = 17*X(1)-12*X(10246) = 7*X(1)-12*X(10247) = 2*X(1)+3*X(11224) = 7*X(1)+8*X(11278) = 4*X(1)+X(11531) = 19*X(1)-4*X(12702) = 31*X(1)-16*X(13624) = 21*X(1)-16*X(15178) = X(1)+9*X(16191) = 17*X(1)-7*X(16192) = X(1)-6*X(16200) = 14*X(3)-9*X(40) = 13*X(3)-18*X(1385) = X(3)+9*X(1482) = 23*X(3)-18*X(3579) = 2*X(3)+3*X(7982) = 8*X(3)-9*X(7987) = 8*X(3)-3*X(7991) = 11*X(3)+9*X(8148) = 19*X(3)-9*X(12702) = 7*X(3)-12*X(15178) = X(40)+14*X(1482) = 3*X(40)+7*X(7982) = 4*X(40)-7*X(7987) = 12*X(40)-7*X(7991) = X(40)-6*X(10247) = X(40)+4*X(11278) = 8*X(40)+7*X(11531) = 19*X(40)-14*X(12702) = 3*X(40)-8*X(15178) = 11*X(165)-16*X(3576) = 3*X(165)-4*X(7987) = 9*X(165)-4*X(7991) = X(165)+4*X(11224) = 3*X(165)+2*X(11531) = X(165)-16*X(16200) = 2*X(1385)+13*X(1482) = 23*X(1385)-13*X(3579) = 16*X(1385)-13*X(7987)

X(16189) lies on these lines:
{1,3}, {5,4677}, {8,7988}, {145,1699}, {355,3857}, {516,3623}, {519,3091}, {546,3656}, {551,9588}, {944,11541}, {946,3633}, {1317,9579}, {1320,5531}, {1389,4900}, {1698,10595}, {3062,3243}, {3085,8275}, {3090,3679}, {3146,3241}, {3244,5691}, {3525,11362}, {3529,5882}, {3544,3632}, {3621,3817}, {3624,12245}, {3628,9624}, {3654,14869}, {3655,12103}, {3951,4861}, {3984,4853}, {4345,6738}, {4668,5844}, {4669,5056}, {4678,10171}, {4745,7486}, {4816,5818}, {4915,5730}, {5223,11682}, {5587,12811}, {5854,15017}, {7993,10698}, {8580,11530}, {9615,10147}, {11519,12635}

X(16189) = reflection of X(i) in X(j) for these (i,j): (1698, 10595), (4668, 8227), (4816, 5818)
X(16189) = X(3091) of excenters-reflections triangle
X(16189) = X(3522) of 6th mixtilinear triangle
X(16189) = X(7987) of 5th mixtilinear triangle
X(16189) = X(15696) of hexyl triangle
X(16189) = X(16195) of Hutson intouch triangle
X(16189) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1482, 11224), (1, 7982, 7991), (1, 11224, 11531), (1, 11280, 2093), (1, 11531, 165), (1, 16191, 1482), (1, 16204, 16206), (1, 16205, 16207), (1482, 10247, 11278), (1482, 16200, 1), (7982, 7991, 11531), (7991, 11224, 7982), (10247, 11278, 40), (16191, 16200, 11224), (16204, 16205, 1482), (16206, 16207, 11224)


X(16190) = EULEROLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO ABC-X3 REFLECTIONS

Barycentrics    SA*(S^2-3*SB*SC)*(3*(12*R^2-SA-2*SW)*S^2-(72*R^2-13*SW)*SB*SC) : :
X(16190) = 4*X(3)-X(1650) = 2*X(3)+X(12113) = 2*X(4)-5*X(15183) = X(20)+2*X(402) = 2*X(40)+X(12626) = 2*X(4297)+X(12438) = 2*X(5188)+X(12794) = 2*X(5894)+X(12791) = 2*X(7689)+X(12418) = 2*X(12041)+X(12790) = X(12369)+2*X(16111) = X(13212)+2*X(16163) = X(13281)+2*X(14689)

X(16190) lies on these lines:
{2,3}, {40,12626}, {165,11900}, {515,16210}, {516,11831}, {517,16211}, {3184,5642}, {4297,12438}, {4299,11912}, {4302,11913}, {5188,12794}, {5204,11906}, {5217,11905}, {5731,11910}, {5894,12791}, {7689,12418}, {9778,16212}, {11909,15338}, {12041,12790}, {12369,16111}, {13212,16163}, {13281,14689}

X(16190) = midpoint of X(9778) and X(16212)
X(16190) = The reciprocal eulerologic center of these triangles is X(11845)
X(16190) = {X(3), X(12113)}-harmonic conjugate of X(1650)


X(16191) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO EXCENTERS-REFLECTIONS

Barycentrics    a*(9*a^3-17*(b+c)*a^2-(9*b^2-34*b*c+9*c^2)*a+17*(b^2-c^2)*(b-c)) : :
X(16191) = 17*X(1)-8*X(3) = 13*X(1)-4*X(40) = 5*X(1)-2*X(165) = 25*X(1)-16*X(1385) = X(1)+8*X(1482) = 7*X(1)-4*X(3576) = 5*X(1)+4*X(7982) = 19*X(1)-10*X(7987) = 11*X(1)-2*X(7991) = 11*X(1)-8*X(10246) = 5*X(1)-8*X(10247) = X(1)+2*X(11224) = 7*X(1)+2*X(11531) = X(1)-10*X(16189) = 16*X(1)-7*X(16192) = X(1)-4*X(16200) = 26*X(3)-17*X(40) = 20*X(3)-17*X(165) = X(3)+17*X(1482) = 14*X(3)-17*X(3576) = 11*X(3)-17*X(10246) = 5*X(3)-17*X(10247) = 35*X(3)-17*X(12702) = 2*X(3)-17*X(16200) = 10*X(40)-13*X(165) = 7*X(40)-13*X(3576) = 5*X(40)+13*X(7982) = 22*X(40)-13*X(7991) = 2*X(40)+13*X(11224) = X(40)-13*X(16200) = 5*X(165)-8*X(1385) = 7*X(165)-10*X(3576) = X(165)+2*X(7982) = 11*X(165)-5*X(7991) = 11*X(165)-20*X(10246) = X(165)-4*X(10247) = X(165)+5*X(11224) = 7*X(165)+5*X(11531) = 7*X(165)-4*X(12702) = X(165)-10*X(16200) = 4*X(1385)+5*X(7982) = 2*X(1385)-5*X(10247) = 14*X(1385)-5*X(12702) = 14*X(1482)+X(3576) = 10*X(1482)-X(7982) = 19*X(1482)-X(8148) = 11*X(1482)+X(10246) = 5*X(1482)+X(10247) = 4*X(1482)-X(11224) = 11*X(1482)-2*X(11278)

X(16191) lies on these lines:
{1,3}, {519,9779}, {3244,9812}, {3632,3817}, {3656,3860}, {3679,10171}, {4668,10175}, {4677,5603}, {5844,7988}

X(16191) = X(3545) of excenters-reflections triangle
X(16191) = X(10245) of Hutson intouch triangle
X(16191) = X(15688) of 6th mixtilinear triangle
X(16191) = X(15689) of hexyl triangle
X(16191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (165, 10247, 1), (165, 11224, 7982), (1482, 16189, 1), (1482, 16200, 11224), (3576, 12702, 165), (7982, 10247, 165), (7982, 12702, 11531), (7982, 16200, 10247), (11224, 16189, 16200), (11224, 16200, 1)


X(16192) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO EXCENTRAL

Barycentrics    a*(7*a^3+(b+c)*a^2-(7*b^2+2*b*c+7*c^2)*a-(b^2-c^2)*(b-c)) : :
X(16192) = X(1)-8*X(3) = 3*X(1)+4*X(40) = X(1)+6*X(165) = 9*X(1)-16*X(1385) = 15*X(1)-8*X(1482) = 5*X(1)-12*X(3576) = 11*X(1)-4*X(7982) = 3*X(1)-10*X(7987) = 5*X(1)+2*X(7991) = 13*X(1)-6*X(11224) = 9*X(1)-2*X(11531) = 17*X(1)-10*X(16189) = 16*X(1)-9*X(16191) = 19*X(1)-12*X(16200) = 6*X(3)+X(40) = 4*X(3)+3*X(165) = 9*X(3)-2*X(1385) = 15*X(3)-X(1482) = 10*X(3)-3*X(3576) = 5*X(3)+2*X(3579) = 12*X(3)-5*X(7987) = 17*X(3)-3*X(10246) = 13*X(3)+X(12702) = 11*X(3)-4*X(13624) = 2*X(40)-9*X(165) = 3*X(40)+4*X(1385) = 5*X(40)+2*X(1482) = 5*X(40)+9*X(3576) = 5*X(40)-12*X(3579) = 11*X(40)+3*X(7982) = 2*X(40)+5*X(7987) = 10*X(40)-3*X(7991) = 6*X(40)+X(11531) = 13*X(40)-6*X(12702) = 5*X(165)+2*X(3576) = 15*X(165)-8*X(3579) = 9*X(165)+5*X(7987) = 15*X(165)-X(7991) = 13*X(165)+X(11224) = 10*X(1385)-3*X(1482) = 5*X(1385)+9*X(3579) = 8*X(1385)-15*X(7987) = 8*X(1385)-X(11531) = 11*X(1385)-18*X(13624) = 25*X(1385)-18*X(15178) = 2*X(1482)-9*X(3576) = X(1482)+6*X(3579) = 22*X(1482)-15*X(7982) = 4*X(1482)+3*X(7991) = 29*X(1482)-15*X(8148)

Let A'B'C' and A"B"C" be as at X(12512). A'B'C' is homothetic to ABC at X(16192) and A"B"C" is homothetic to the excentral triangle at X(16192). (Randy Hutson, March 14, 2018)

X(16192) lies on these lines:
{1,3}, {2,10248}, {4,10172}, {10,3522}, {20,1698}, {30,7989}, {72,10178}, {100,4882}, {140,7988}, {154,9899}, {187,9593}, {200,4652}, {355,8703}, {371,9584}, {372,9582}, {376,5691}, {404,4512}, {411,2951}, {474,11495}, {515,3528}, {516,3523}, {549,8227}, {550,5587}, {551,15705}, {631,1699}, {944,4677}, {946,3524}, {962,15692}, {991,5312}, {1092,9587}, {1125,9589}, {1152,9616}, {1376,5234}, {1483,15714}, {1571,5206}, {1572,15515}, {1657,11231}, {1702,6396}, {1703,6200}, {1742,3216}, {1750,3651}, {2136,11194}, {2948,15055}, {2975,4915}, {3053,9574}, {3097,5188}, {3146,3634}, {3516,7713}, {3529,10175}, {3530,12699}, {3534,9956}, {3583,6865}, {3585,6916}, {3616,5493}, {3632,5731}, {3652,5720}, {3653,15711}, {3655,15759}, {3656,14891}, {3678,11220}, {3679,4297}, {3715,12684}, {3817,10303}, {3916,5223}, {4188,8583}, {4298,5281}, {4300,5313}, {4312,13411}, {4314,5435}, {4324,6987}, {4333,6908}, {4355,13405}, {4640,5438}, {4668,5657}, {4855,12526}, {5044,5918}, {5218,5290}, {5250,13587}, {5265,12575}, {5267,9623}, {5432,9579}, {5433,9580}, {5692,9943}, {5726,7354}, {5886,15712}, {5904,10167}, {6199,9618}, {6361,10165}, {6411,9615}, {6449,9585}, {7262,8951}, {7411,8580}, {8109,12518}, {8589,9619}, {8666,11519}, {9541,13975}, {9575,15815}, {9578,15326}, {9581,15338}, {9590,10323}, {9620,15513}, {9904,15035}, {9955,15720}, {9961,10176}, {10595,15715}, {11276,16159}, {12245,15710}, {12675,15104}, {12767,15015}

X(16192) = complement of X(10248)
X(16192) = X(3090) of excentral triangle
X(16192) = X(3523) of 1st circumperp triangle
X(16192) = X(3526) of 6th mixtilinear triangle
X(16192) = X(3528) of 2nd circumperp triangle
X(16192) = X(3851) of hexyl triangle
X(16192) = X(10244) of intouch triangle
X(16192) = 2nd-Conway-to-excentral similarity image of X(10248)
X(16192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (35, 15803, 1), (40, 1385, 11531), (40, 1482, 7991), (55, 3361, 1), (165, 10980, 7964), (1155, 3601, 3339), (1385, 11531, 1), (1420, 9819, 1), (1482, 3579, 40), (1697, 5204, 13462), (1697, 13462, 1), (2093, 3612, 1), (3339, 3601, 1), (3576, 3579, 7991), (7987, 11531, 1385), (10246, 16189, 1)


X(16193) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-AQUILA TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2+b*c+c^2)*a^3+2*(b^4+c^4-3*b*c*(b^2+c^2))*a^2+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16193) = (2*R^2+4*R*r+r^2)*X(1)-r^2*X(3) = 3*X(354)+X(2646)

X(16193) lies on these lines:
{1,3}, {11,12757}, {226,12675}, {518,4999}, {938,5086}, {946,10391}, {971,12047}, {1210,3742}, {1728,4423}, {1858,15950}, {1864,8227}, {1900,4196}, {3085,3555}, {3086,5728}, {3487,10785}, {3523,7672}, {3873,5703}, {3881,13405}, {4295,10167}, {5219,14872}, {5603,12711}, {5777,11375}, {5806,10572}, {7951,9947}, {9612,12680}, {9614,14100}, {9844,10591}, {9956,10954}, {10122,11281}, {11020,14986}, {12053,12710}

X(16193) = midpoint of X(1) and X(13750)
X(16193) = inverse of X(5535) in the incircle
X(16193) = X(1594) of inverse-in-incircle triangle
X(16193) = X(7542) of intouch triangle
X(16193) = X(10024) of incircle-circles triangle
X(16193) = X(13750) of anti-Aquila triangle
X(16193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 354, 942), (2446, 2447, 5535), (5045, 11018, 1), (5045, 13373, 942), (5045, 16201, 16215), (5045, 16216, 5049), (11018, 16215, 16201), (16201, 16215, 1)


X(16194) = EULEROLOGIC CENTER OF THESE TRIANGLES: 2nd EULER TO ANTI-EXCENTERS-REFLECTIONS

Barycentrics    a^2*((b^2+c^2)*a^6-3*(b^2-c^2)^2*a^4+(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(16194) = X(3)+2*X(13474) = 4*X(4)-X(52) = 3*X(4)-X(3060) = 5*X(4)-2*X(5446) = 7*X(4)-X(5889) = X(4)+5*X(11439) = 17*X(4)-8*X(12002) = 5*X(4)+X(12111) = 2*X(4)+X(12162) = 3*X(52)-4*X(3060) = 5*X(52)-8*X(5446) = 7*X(52)-4*X(5889) = 5*X(52)+4*X(12111) = X(52)+2*X(12162) = X(52)+4*X(15305) = 5*X(3060)-6*X(5446) = 7*X(3060)-3*X(5889) = X(3060)+15*X(11439) = 5*X(3060)+3*X(12111) = 2*X(3060)+3*X(12162) = X(3060)+3*X(15305) = 14*X(5446)-5*X(5889) = 17*X(5446)-20*X(12002) = 2*X(5446)+X(12111) = 4*X(5446)+5*X(12162) = 2*X(5446)+5*X(15305) = 5*X(5889)+7*X(12111) = 2*X(5889)+7*X(12162) = X(5889)+7*X(15305) = 10*X(11439)-X(12162) = 5*X(11439)-X(15305) = 2*X(11455)+X(14855) = 2*X(12111)-5*X(12162) = X(12111)-5*X(15305)

X(16194) lies on these lines:
{2,11455}, {3,13474}, {4,52}, {5,10575}, {22,4550}, {25,11472}, {30,3917}, {51,3845}, {64,7529}, {74,13595}, {110,13596}, {113,427}, {155,11403}, {185,546}, {373,5066}, {376,10170}, {381,1853}, {382,5907}, {389,3843}, {511,3830}, {550,11592}, {568,14269}, {569,1498}, {631,14641}, {632,11017}, {1154,15687}, {1204,13861}, {1216,3146}, {1593,10539}, {1597,3167}, {1657,11793}, {1906,12359}, {1994,14094}, {2883,7403}, {2979,15682}, {3090,12279}, {3091,12290}, {3357,7506}, {3426,5020}, {3518,15062}, {3529,5447}, {3534,3819}, {3543,11459}, {3545,5892}, {3627,5562}, {3796,9818}, {3832,5462}, {3839,5890}, {3850,13491}, {3851,9729}, {3853,5876}, {3854,15024}, {3855,10574}, {3856,15026}, {3857,12006}, {3858,13630}, {3861,6102}, {4549,7500}, {4846,6997}, {5012,12112}, {5055,10219}, {5059,7999}, {5072,11695}, {5073,15644}, {5076,13598}, {5097,12308}, {5198,12163}, {5448,15559}, {5650,8703}, {5878,7528}, {7394,7706}, {7485,8717}, {7527,14157}, {7689,10594}, {7998,11001}, {9306,10564}, {10110,13321}, {10263,12102}, {10282,14130}, {10540,11430}, {11479,13336}, {11562,12292}, {12038,14865}, {12134,13488}, {12294,12295}, {12897,14516}, {13340,15684}, {13567,16003}, {14128,15704}, {14831,14893}, {15082,15693}

X(16194) = midpoint of X(i) and X(j) for these {i,j}: {2, 11455}, {4, 15305}, {2979, 15682}, {3543, 11459}, {13340, 15684}
X(16194) = reflection of X(i) in X(j) for these (i,j): (51, 3845), (185, 5946), (376, 10170), (389, 13570), (3534, 3819), (11562, 12824)
X(16194) = X(15305) of Euler triangle
X(16194) = X(16111) of orthocentroidal triangle
X(16194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12111, 5446), (4, 12162, 52), (5, 11381, 10575), (381, 9730, 14845), (382, 5907, 10625), (3146, 15058, 1216), (3529, 15056, 5447), (3545, 15072, 5892), (3832, 6241, 5462), (3917, 15030, 15060), (3917, 15060, 5891)


X(16195) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO ARA

Barycentrics    a^2*(5*a^8-10*(b^2+c^2)*a^6+4*b^2*c^2*a^4+2*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2-(5*b^4+2*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :
X(16195) = X(3)+4*X(26) = 3*X(3)-8*X(1658) = 3*X(3)+2*X(7387) = 2*X(3)+3*X(9909) = 21*X(3)-16*X(10226) = 3*X(3)+7*X(10244) = X(3)+9*X(10245) = 13*X(3)-8*X(11250) = 9*X(3)-4*X(12084) = 7*X(3)-2*X(12085) = X(3)-16*X(12107) = X(3)-6*X(14070) = 11*X(3)-16*X(15331) = X(4)-6*X(10154) = 6*X(154)-X(12164)

As a point on the Euler line, X(16195) has Shinagawa coefficients (-E-5*F, 3*E+5*F)

X(16195) lies on these lines:
{2,3}, {54,5093}, {154,12164}, {2917,5965}, {3167,10282}, {5562,8780}, {6417,11266}, {6418,11265}, {7689,12315}, {8193,9590}, {9707,12160}, {9919,12893}, {10312,15851}, {12310,13289}, {13754,14530}

X(16195) = X(3091) of Ara triangle
X(16195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 26, 9909), (3, 7517, 1597), (3, 10244, 7387), (22, 3515, 3), (22, 16199, 9909), (24, 7556, 9715), (24, 9715, 3), (26, 1658, 7387), (26, 7387, 10244), (26, 12107, 14070), (26, 14070, 3), (1658, 7387, 3), (2937, 10243, 9909), (6642, 7502, 3), (7387, 10244, 9909), (7387, 14070, 1658)


X(16196) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO MEDIAL

Barycentrics    (-a^2+b^2+c^2)*(2*a^8-3*(b^2+c^2)*a^6-(b^4-10*b^2*c^2+c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4) : :
X(16196) = 3*X(2)+X(11413)

As a point on the Euler line, X(16196) has Shinagawa coefficients (E-3*F, -E+F)

X(16196) lies on these lines:
{2,3}, {185,11064}, {1038,5432}, {1040,5433}, {1062,15325}, {1092,3564}, {1154,15120}, {1216,6699}, {1578,5418}, {1579,5420}, {5085,15812}, {5158,9606}, {5305,14961}, {5504,12421}, {5907,6696}, {6247,9306}, {8263,8549}, {9820,14156}, {10519,14914}, {11431,11482}, {13142,13346}, {13470,15114}

X(16196) = complement of X(235)
X(16196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 6823), (2, 20, 6622), (2, 1593, 5), (2, 11413, 235), (3, 140, 6676), (3, 3526, 3547), (3, 6640, 15760), (3, 7542, 16197), (3, 12605, 548), (4, 5, 13487), (5, 3548, 5159), (5, 12084, 13488), (3147, 11414, 10154), (6640, 15760, 3628), (7542, 16197, 6676), (14784, 14785, 6623)


X(16197) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO MEDIAL

Barycentrics    (-a^2+b^2+c^2)*(2*a^8-3*(b^2+c^2)*a^6-(b^4+14*b^2*c^2+c^4)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4) : :
X(16197) = 3*X(2)+X(11414)

As a point on the Euler line, X(16197) has Shinagawa coefficients (-2*E-3*F, 2*E+F)

X(16197) lies on these lines:
{2,3}, {141,6759}, {216,5305}, {343,10984}, {569,13142}, {1038,15325}, {1092,13394}, {1578,13966}, {1579,8981}, {3527,3618}, {3564,12229}, {3589,10110}, {5446,11574}, {5447,9820}, {7583,11514}, {7584,11513}, {9967,10263}, {10272,13416}, {11515,11543}, {11516,11542}, {14516,15080}, {14530,14826}

X(16197) = complement of X(1595)
X(16197) = X(16198) of Johnson triangle
X(16197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1598, 5), (3, 3547, 5), (3, 3549, 1368), (3, 6676, 140), (3, 15760, 12362), (5, 549, 7393), (5, 550, 14790), (22, 7399, 6756), (140, 13383, 6677), (1368, 3549, 3628), (3530, 10020, 140), (6676, 16196, 7542), (6803, 10565, 3517), (7401, 9909, 7715), (12088, 14788, 428), (12362, 15760, 546)


X(16198) = EULEROLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO EULER

Barycentrics    (SA-4*R^2+5*SW)*SB*SC : :
X(16198) = 3*X(4)+X(1593) = X(4)+3*X(5064) = 5*X(4)-X(12173)

As a point on the Euler line, X(16198) has Shinagawa coefficients (F, 4*E+5*F)

X(16198) lies on these lines:
{2,3}, {125,11566}, {1843,6101}, {3564,3867}, {5090,5844}, {5305,6748}, {5412,13925}, {5413,13993}, {7718,10283}, {10263,12294}, {11393,15172}, {11470,13292}

X(16198) = X(1595) of Euler triangle
X(16198) = X(7525) of anti-Ara triangle
X(16198) = X(16197) of Johnson triangle
X(16198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 427, 6756), (4, 1594, 428), (4, 1597, 3627), (4, 1907, 13488), (4, 3090, 7408), (4, 5064, 1595), (4, 7378, 3), (4, 7507, 1596), (4, 7547, 1906), (4, 13488, 3853), (4, 15559, 3575), (5, 3627, 7387), (381, 6643, 5), (1596, 7507, 3850), (5576, 7553, 6676), (7391, 7566, 7399)


X(16199) = EULEROLOGIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO ARA

Barycentrics    (SB+SC)*((8*R^2+SW)*SB*SC-SW*(S^2+SA^2)) : :

X(16199) lies on the line {2,3}

X(16199) = X(7392) of Ara triangle
X(16199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25, 7484, 7398), (25, 7494, 5020), (26, 10243, 3), (5020, 9909, 7387), (7387, 10154, 5020), (9909, 16195, 22)


X(16200) = EULEROLOGIC CENTER OF THESE TRIANGLES: AQUILA TO EXCENTERS-REFLECTIONS

Trilinears    5 r - 2 R cos A : :
Trilinears    3 cos A + 5 cos B + 5 cos C - 5 : :
Barycentrics    a*(3*a^3-5*(b+c)*a^2-(3*b-c)*(b-3*c)*a+5*(b^2-c^2)*(b-c)) : :
X(16200) = 5*X(1)-2*X(3) = 4*X(1)-X(40) = 3*X(1)-X(165) = 7*X(1)-4*X(1385) = X(1)+2*X(1482) = 13*X(1)-4*X(3579) = 2*X(1)+X(7982) = 11*X(1)-5*X(7987) = 7*X(1)-X(7991) = 7*X(1)+2*X(8148) = 3*X(1)-2*X(10246) = 5*X(1)+4*X(11278) = 5*X(1)+X(11531) = 11*X(1)-2*X(12702) = 17*X(1)-8*X(13624) = 11*X(1)-8*X(15178) = X(1)+5*X(16189) = X(1)+3*X(16191) = 19*X(1)-7*X(16192) = 8*X(3)-5*X(40) = 6*X(3)-5*X(165) = 7*X(3)-10*X(1385) = X(3)+5*X(1482) = 4*X(3)-5*X(3576) = 13*X(3)-10*X(3579) = 4*X(3)+5*X(7982) = 14*X(3)-5*X(7991) = 7*X(3)+5*X(8148) = 3*X(3)-5*X(10246) = X(3)-5*X(10247) = 2*X(3)+5*X(11224) = X(3)+2*X(11278) = 2*X(3)+X(11531) = 11*X(3)-5*X(12702) = 17*X(3)-20*X(13624) = 11*X(3)-20*X(15178) = 2*X(3)+15*X(16191) = 3*X(40)-4*X(165) = 7*X(40)-16*X(1385) = X(40)+8*X(1482) = 13*X(40)-16*X(3579) = X(40)+2*X(7982) = 11*X(40)-20*X(7987) = 7*X(40)-4*X(7991) = 7*X(40)+8*X(8148) = 3*X(40)-8*X(10246)

X(16200) lies on these lines:
{1,3}, {4,3244}, {5,3632}, {8,5056}, {9,14497}, {10,5067}, {20,13607}, {84,1392}, {145,946}, {200,11525}, {355,3633}, {374,2324}, {515,3241}, {516,7967}, {518,5102}, {519,3545}, {547,3679}, {551,5657}, {631,3636}, {944,4301}, {952,1699}, {962,3623}, {990,9519}, {1000,13405}, {1022,3309}, {1125,3533}, {1317,1836}, {1320,3577}, {1389,3680}, {1483,12678}, {1572,5008}, {1698,5901}, {1709,12737}, {2800,3873}, {2801,3243}, {2802,3158}, {2809,15735}, {2818,11189}, {3090,3626}, {3488,4342}, {3525,15808}, {3555,5693}, {3560,5288}, {3616,11362}, {3622,6684}, {3624,5690}, {3646,5330}, {3654,11812}, {3655,15686}, {3740,5289}, {3751,5097}, {3853,5691}, {3884,5436}, {3889,5884}, {3894,14988}, {3899,3929}, {3940,4915}, {3957,7966}, {4312,12119}, {4668,9956}, {4669,10171}, {4677,5790}, {4853,5730}, {4857,10526}, {4861,11682}, {5041,9575}, {5219,12647}, {5270,10525}, {5732,15570}, {5761,9581}, {6265,12653}, {6431,7969}, {6432,7968}, {6437,9583}, {6480,9616}, {6484,9615}, {6765,10912}, {7701,16126}, {7971,10864}, {7972,12831}, {9549,10440}, {9612,10944}, {9614,10950}, {9951,12757}, {9955,12645}, {10164,15719}, {10165,15708}, {10176,15829}, {10283,11539}, {10404,12700}, {10580,14563}, {10914,13374}, {11019,11041}, {11231,15723}, {11827,15172}, {12531,16174}, {12559,12705}, {12629,12635}, {15174,16113}

X(16200) = midpoint of X(1) and X(11224)
X(16200) = reflection of X(i) in X(j) for these (i,j): (1, 10247), (8, 10175), (4669, 10171), (4677, 5790)
X(16200) = X(376) of hexyl triangle
X(16200) = X(381) of excenters-reflections triangle
X(16200) = X(3543) of 2nd circumperp triangle
X(16200) = X(3576) of 5th mixtilinear triangle
X(16200) = X(3830) of excentral triangle
X(16200) = X(8703) of 6th mixtilinear triangle
X(16200) = X(10247) of Aquila triangle
X(16200) = X(11001) of 1st circumperp triangle
X(16200) = X(11224) of anti-Aquila triangle
X(16200) = X(14070) of Hutson intouch triangle
X(16200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1482, 7982), (1, 7982, 40), (1, 11521, 10476), (1, 11531, 3), (3, 1482, 11278), (3, 11278, 11531), (57, 12703, 40), (1385, 8148, 7991), (1697, 12704, 40), (5048, 16205, 7982), (5119, 5535, 40), (10247, 11224, 3576), (11224, 16189, 16191), (11224, 16191, 1482), (11248, 12001, 5563), (11278, 11531, 7982)


X(16201) = EULEROLOGIC CENTER OF THESE TRIANGLES: AQUILA TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2+b*c+c^2)*a^3+2*(b^4+c^4-3*b*c*(b^2+4*b*c+c^2))*a^2+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16201) = X(65)+3*X(10389) = 3*X(354)+X(1697) = 3*X(3475)+X(12711) = X(5082)-5*X(5439)

X(16201) lies on these lines:
{1,3}, {10,5572}, {72,10578}, {495,9947}, {518,12564}, {971,12710}, {2551,10177}, {3475,12711}, {3487,9856}, {3555,11020}, {3812,5853}, {4355,5918}, {5044,13405}, {5082,5439}, {5261,7671}, {5290,14100}, {5542,9943}, {5686,5728}, {6260,10241}, {9948,11035}, {10167,11037}

X(16201) = X(1595) of inverse-in-incircle triangle
X(16201) = X(6823) of incircle-circles triangle
X(16201) = X(16197) of intouch triangle
X(16201) = X(16216) of inner-Yff triangle
X(16201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11018, 5045), (1, 16193, 16215), (354, 3339, 942), (5261, 7671, 9844), (11018, 16215, 16193), (16193, 16215, 5045)


X(16202) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO 2nd CIRCUMPERP

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b+c)^2*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+(b^4+c^4+2*b*c*(2*b^2-3*b*c+2*c^2))*a-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)) : :
X(16202) = X(3)+2*X(3303) = 3*X(3)-2*X(5584) = 3*X(3303)+X(5584) = 7*X(3526)-4*X(9710) = 2*X(5584)+3*X(12000)

X(16202) lies on these lines:
{1,3}, {2,10806}, {4,10587}, {5,10585}, {8,6883}, {20,10597}, {21,7967}, {24,11401}, {30,10532}, {125,12906}, {140,5687}, {145,1006}, {149,6937}, {182,12595}, {355,1001}, {388,7491}, {390,6850}, {405,952}, {411,10595}, {495,6928}, {496,6863}, {497,6842}, {498,10957}, {499,10959}, {549,11240}, {551,6796}, {631,3871}, {912,5250}, {944,1621}, {956,1483}, {993,13607}, {1056,6868}, {1058,6825}, {1125,11499}, {1191,5396}, {1480,4300}, {1656,3816}, {2080,10804}, {2478,10942}, {3035,3526}, {3058,10525}, {3085,6882}, {3149,5901}, {3357,13095}, {3616,6911}, {3622,6905}, {3653,4421}, {3655,4428}, {3913,10916}, {4254,8609}, {4309,5840}, {4423,9956}, {5050,9049}, {5082,6989}, {5248,5882}, {5259,5881}, {5281,6961}, {5284,5818}, {5432,10949}, {5436,7966}, {5603,6985}, {5690,12649}, {5761,10578}, {5790,11108}, {5886,11500}, {6642,10835}, {6713,10087}, {6771,13107}, {6774,13106}, {6838,10596}, {6872,10805}, {6880,10586}, {6907,15172}, {6923,15171}, {6947,10528}, {6954,14986}, {6967,10530}, {6980,9669}, {6986,12245}, {7583,13907}, {7584,13965}, {8715,10165}, {9708,12645}, {10526,15888}, {10610,13122}, {12041,12382}, {12042,12190}, {12359,12431}, {12619,12750}

X(16202) = midpoint of X(6872) and X(10805)
X(16202) = X(12000) of outer-Yff tangents triangle
X(16202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 10680), (1, 35, 10966), (1, 10267, 3), (1, 10902, 11249), (1, 11249, 12001), (1, 14798, 56), (1, 16208, 12704), (3, 3295, 10679), (3, 6767, 1482), (3,10246,16203), (3, 12001, 11249), (55, 1388, 8071), (1388, 8071, 999), (3576, 11248, 3), (6585, 10247, 10680), (11249, 12001, 10680), (12704, 16208, 3579)


X(16203) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO 2nd CIRCUMPERP

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2-4*b*c+c^2)*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4+c^4-2*b*c*(4*b^2-5*b*c+4*c^2))*a-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)) : :
X(16203) = X(3)+2*X(3304) = 2*X(1385)+X(3338) = 7*X(3526)-4*X(9711)

X(16203) lies on these lines:
{1,3}, {2,10805}, {4,10586}, {5,10584}, {20,10596}, {24,11400}, {30,10531}, {104,3560}, {119,1656}, {125,12905}, {140,956}, {145,6940}, {153,6975}, {182,12594}, {377,10943}, {381,6256}, {388,6882}, {404,7967}, {474,952}, {495,6958}, {496,6923}, {498,6713}, {499,10958}, {549,11239}, {551,5450}, {601,1149}, {631,10528}, {944,5253}, {1012,5901}, {1056,1476}, {1058,6948}, {1483,5687}, {1511,13217}, {2080,10803}, {2975,6883}, {3086,6842}, {3357,13094}, {3526,4999}, {3600,6827}, {3622,6906}, {3653,11194}, {3655,11500}, {4190,10806}, {4293,7491}, {5050,9026}, {5261,6978}, {5265,6954}, {5433,10955}, {5434,10526}, {5690,12648}, {5731,5804}, {5761,11037}, {5882,11499}, {5886,6259}, {6642,10834}, {6771,13105}, {6774,13104}, {6850,14986}, {6863,15325}, {6889,10530}, {6890,10597}, {6897,10529}, {6909,10595}, {6971,9654}, {6977,10587}, {7583,13906}, {7584,13964}, {8666,10165}, {9709,12645}, {10610,13121}, {10915,12513}, {11281,13743}, {12041,12381}, {12042,12189}, {12359,12430}, {12619,12749}, {14529,14530}

X(16203) = midpoint of X(4190) and X(10806)
X(16203) = X(7506) of 2nd circumperp triangle
X(16203) = X(12001) of inner-Yff tangents triangle
X(16203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35, 10965), (1, 36, 11509), (1, 10269, 3), (1, 11248, 12000), (1, 14803, 55), (3, 999, 10680), (3, 7373, 1482), (3, 8148, 6244), (3,10246,16202), (3, 10247, 10306), (3, 12000, 11248), (36, 10267, 3), (56, 1385, 3), (56, 3304, 3338), (3576, 5563, 11249), (11248, 12000, 10679), (12703, 16209, 3579)


X(16204) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO EXCENTERS-REFLECTIONS

Barycentrics    a*(5*a^6-14*(b+c)*a^5-(b^2-38*b*c+c^2)*a^4+4*(b+c)*(7*b^2-12*b*c+7*c^2)*a^3-(13*b^4+13*c^4+10*b*c*(2*b-c)*(b-2*c))*a^2-2*(b^2-c^2)*(b-c)*(7*b^2-10*b*c+7*c^2)*a+9*(b^2-c^2)^2*(b-c)^2) : :
X(16204) = (8*R+9*r)*X(1)-4*(R+r)*X(3)

X(16204) lies on these lines:
{1,3}, {200,1389}, {3632,5761}

X(16204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1482, 16189, 16205), (7982, 16200, 11011), (16189, 16206, 1)


X(16205) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO EXCENTERS-REFLECTIONS

Barycentrics    a*(5*a^6-14*(b+c)*a^5-(b^2-54*b*c+c^2)*a^4+4*(b+c)*(7*b^2-20*b*c+7*c^2)*a^3-(13*b^4+13*c^4+6*b*c*(6*b^2-19*b*c+6*c^2))*a^2-2*(b^2-c^2)*(b-c)*(7*b^2-26*b*c+7*c^2)*a+9*(b^2-c^2)^2*(b-c)^2) : :
X(16205) = (8*R-9*r)*X(1)-4*(R-r)*X(3)

X(16205) lies on these lines:
{1,3}, {10698,13227}

X(16205) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1482, 16189, 16204), (2093, 11224, 7982), (3359, 10247, 1), (7982, 16200, 5048), (10269, 10306, 2077), (16189, 16207, 1)


X(16206) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO EXCENTERS-REFLECTIONS

Barycentrics    a*(5*a^6-14*(b+c)*a^5-(b^2-40*b*c+c^2)*a^4+2*(b+c)*(14*b^2-25*b*c+14*c^2)*a^3-(13*b^4+13*c^4+2*b*c*(11*b^2-27*b*c+11*c^2))*a^2-2*(b^2-c^2)*(b-c)*(7*b^2-11*b*c+7*c^2)*a+9*(b^2-c^2)^2*(b-c)^2) : :
X(16206) = (7*R+9*r)*X(1)-4*(R+r)*X(3)

X(16206) lies on the line {1,3}

X(16206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16204, 16189), (11224, 16189, 16207)


X(16207) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO EXCENTERS-REFLECTIONS

Barycentrics    a*(5*a^6-14*(b+c)*a^5-(b^2-52*b*c+c^2)*a^4+2*(b+c)*(14*b^2-39*b*c+14*c^2)*a^3-(13*b^4+13*c^4+2*b*c*(17*b^2-55*b*c+17*c^2))*a^2-2*(b^2-c^2)*(b-c)*(7*b^2-25*b*c+7*c^2)*a+9*(b^2-c^2)^2*(b-c)^2) : :
X(16207) = (7*R-9*r)*X(1)-4*(R-r)*X(3)

X(16207) lies on these lines:
{1,3}, {1519,3633}

X(16207) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16205, 16189), (11224, 16189, 16206)


X(16208) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO EXCENTRAL

Barycentrics    a*(3*a^6-2*(b+c)*a^5-(7*b^2+8*b*c+7*c^2)*a^4+2*(b+c)*(2*b^2+b*c+2*c^2)*a^3+(5*b^4+5*c^4+6*b*c*(b^2-b*c+c^2))*a^2-2*(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16208) = (R-r)*X(1)+4*(R+r)*X(3)

X(16208) lies on these lines:
{1,3}, {10,6992}, {1698,6947}, {1699,6838}, {2478,7989}, {3624,6880}, {4512,5691}, {4640,10085}, {5288,7966}, {5531,12691}, {6684,12116}, {6734,9588}, {6834,7988}, {6987,10039}, {9778,10587}, {10164,10527}, {12514,12528}, {14054,15104}

X(16208) = X(7558) of 6th mixtilinear triangle
X(16208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10268, 165), (1, 15932, 10980), (1, 16192, 11012), (40, 10267, 1), (165, 7987, 16209), (3579, 16202, 12704), (5119, 14798, 1), (12704, 16202, 1)


X(16209) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO EXCENTRAL

Barycentrics    a*(3*a^6-2*(b+c)*a^5-(7*b^2-12*b*c+7*c^2)*a^4+2*(b+c)*(2*b^2-b*c+2*c^2)*a^3+(5*b^4+5*c^4-2*b*c*(7*b^2-b*c+7*c^2))*a^2-2*(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16209) = (R+r)*X(1)+4*(R-r)*X(3)

X(16209) lies on these lines:
{1,3}, {377,7989}, {404,9961}, {474,1709}, {631,12608}, {936,1768}, {1158,6940}, {1376,10085}, {1519,3624}, {1698,6256}, {1699,6890}, {1770,6926}, {2057,5223}, {4188,12520}, {4190,5691}, {4297,5554}, {4333,6827}, {4512,10940}, {4652,5552}, {5438,15071}, {6684,12115}, {6735,9588}, {6833,7988}, {6955,12616}, {6966,12609}, {9778,10586}

X(16209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10270, 165), (1, 16192, 2077), (40, 10269, 1), (46, 14803, 1), (165, 3361, 7991), (165, 7987, 16208), (3579, 16203, 12703), (12703, 16203, 1)


X(16210) = EULEROLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO OUTER-GARCIA

Barycentrics    (b+c)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^8-2*(b^2+c^2)*a^6+(b^2-c^2)*(b-c)*a^5-(5*b^4-12*b^2*c^2+5*c^4)*a^4-2*(b^4-c^4)*(b-c)*a^3+8*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-c^4)*(b^2+c^2)*(b-c)*a-(3*b^4+8*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :
X(16210) = 2*X(1)-5*X(15183) = X(8)+2*X(402) = 2*X(8)+X(12626) = 4*X(10)-X(1650) = 2*X(10)+X(12438) = 2*X(355)+X(12113) = 4*X(402)-X(12626) = 2*X(1145)+X(13268) = X(1650)+2*X(12438) = X(1651)+2*X(3679) = X(3081)+8*X(4745) = 5*X(3617)+X(4240) = 2*X(5690)+X(11251) = 7*X(9780)-4*X(15184) = 2*X(11362)+X(12696)

X(16210) lies on these lines:
{1,15183}, {2,11910}, {8,402}, {10,1650}, {30,5657}, {355,12113}, {515,16190}, {517,11897}, {519,11831}, {958,11848}, {1145,13268}, {1651,3679}, {3081,4745}, {3617,4240}, {5554,11914}, {5690,11251}, {9780,15184}, {10573,11912}, {11362,12696}, {11913,12647}, {12729,15863}

X(16210) = midpoint of X(8) and X(16212)
X(16210) = reflection of X(1651) in X(11852)
X(16210) = X(16212) of Gossard triangle
X(16210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 402, 12626), (10, 12438, 1650)


X(16211) = EULEROLOGIC CENTER OF THESE TRIANGLES: GOSSARD TO 5th MIXTILINEAR

Barycentrics    (S^2-3*SB*SC)*(((132*SA-4*SW)*R^2-26*S^2-33*SA^2+22*SB*SC+SW^2)*a-(4*(SW+3*SA)*R^2-2*S^2-3*SA^2+2*SB*SC-SW^2)*(b+c)) : :
X(16211) = 4*X(1)-X(1650) = 2*X(1)+X(12626) = 2*X(8)-5*X(15183) = X(145)+2*X(402) = 2*X(1317)+X(13268) = 2*X(1482)+X(12113) = 2*X(1483)+X(11251) = X(1650)+2*X(12626) = X(1651)+2*X(3241) = 7*X(3622)-4*X(15184) = 5*X(3623)+X(4240) = 2*X(5882)+X(12696)

X(16211) lies on these lines:
{1,1650}, {8,15183}, {30,7967}, {145,402}, {517,16190}, {519,11831}, {952,11897}, {1317,13268}, {1482,12113}, {1483,11251}, {1651,3241}, {3622,15184}, {3623,4240}, {5882,12696}

X(16211) = reflection of X(1651) in X(16212)
X(16211) = {X(1), X(12626)}-harmonic conjugate of X(1650)


X(16212) = EULEROLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO GOSSARD

Barycentrics    (S^2-3*SB*SC)*((4*(21*SA-2*SW)*R^2-17*S^2-21*SA^2+14*SB*SC+2*SW^2)*a-(4*(-3*SA+2*SW)*R^2+3*S^2+3*SA^2-2*SB*SC-2*SW^2)*(b+c)) : :
X(16212) = 2*X(1)+X(4240) = X(8)-4*X(402) = X(8)+2*X(12626) = X(20)+2*X(12696) = X(145)+2*X(12438) = X(149)+2*X(12729) = 2*X(402)+X(12626) = 4*X(551)-X(11050) = X(944)+2*X(11251) = X(962)+2*X(12113) = 2*X(1650)-5*X(3616) = 2*X(1651)+X(3241) = 11*X(5550)-8*X(15184) = 5*X(5734)+4*X(15774) = X(6224)+2*X(13268)

X(16212) lies on these lines:
{1,4240}, {2,11831}, {8,402}, {20,12696}, {30,5603}, {145,12438}, {149,12729}, {517,11845}, {519,11852}, {551,11050}, {944,11251}, {952,11911}, {962,12113}, {1650,3616}, {1651,3241}, {3485,11905}, {3486,11909}, {3871,11848}, {5550,15184}, {5734,15774}, {6224,13268}, {7718,11832}, {9778,16190}, {9780,15183}

X(16212) = midpoint of X(1651) and X(16211)
X(16212) = reflection of X(i) in X(j) for these (i,j): (2, 11831), (8, 16210), (9778, 16190)
X(16212) = X(16210) of Gossard triangle
X(16212) = {X(402), X(12626)}-harmonic conjugate of X(8)


X(16213) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-SODDY TO INTOUCH

Trilinears    2*(17*cos(A/2)-cos(3*A/2)+12*sin(A/2)+4*sin(3*A/2))*cos((B-C)/2)+2*(cos(A)+2*sin(A)+3)*cos(B-C)+6*cos(A)-cos(2*A)+18*sin(A)+sin(2*A)+19 : :

X(16213) lies on these lines:
{34,1850}, {2362,13456}


X(16214) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-SODDY TO INTOUCH

Trilinears    2*(17*cos(A/2)-cos(3*A/2)-12*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)-2*(cos(A)-2*sin(A)+3)*cos(B-C)-6*cos(A)+cos(2*A)+18*sin(A)+sin(2*A)-19 : :

X(16214) lies on these lines: {34,1849}, {13427,16232}


X(16215) = EULEROLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)*(b^2+b*c+c^2)*a^3+2*(b^4+c^4-3*b*c*(b^2-4*b*c+c^2))*a^2+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16215) = (4*R^2-r^2+4*R*r)*X(1)+r^2*X(3) = 3*X(354)+X(2098) = 3*X(3742)-X(8256)

X(16215) lies on these lines:
{1,3}, {11,9947}, {226,11035}, {388,12128}, {497,6259}, {946,16007}, {971,12053}, {1056,5804}, {1329,11019}, {1699,9850}, {3436,10580}, {3555,14986}, {3742,8256}, {3812,5854}, {3889,5728}, {4342,9943}, {5082,9858}, {5777,11373}, {5806,10106}, {6691,13405}, {8581,11522}, {9785,10167}, {10866,15071}

X(16215) = X(235) of inverse-in-incircle triangle
X(16215) = X(5045) of outer-Yff triangle
X(16215) = X(9947) of 2nd Johnson-Yff triangle
X(16215) = X(11018) of outer-Yff tangents triangle
X(16215) = X(11585) of incircle-circles triangle
X(16215) = X(16196) of intouch triangle
X(16215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5045, 11018), (1, 16193, 16201), (354, 3340, 942), (942, 5049, 7373), (5045, 16201, 16193), (16193, 16201, 11018), (16217, 16218, 11018)


X(16216) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b+c)^3*a^3+2*(b^4+c^4-b*c*(b+3*c)*(3*b+c))*a^2+(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(16216) = (r^2-5*R*r+8*R^2)*X(1)+r*(R-r)*X(3) = 3*X(354)+X(3295) = 3*X(3892)+X(5837)

X(16216) lies on these lines:
{1,3}, {5,5572}, {495,10395}, {912,12564}, {1068,1827}, {3555,10587}, {3824,15733}, {3826,10916}, {3892,5837}, {5542,13369}, {5714,7671}, {5791,15185}, {6147,12710}, {7686,15935}

X(16216) = midpoint of X(6147) and X(12710)
X(16216) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (46, 354, 942), (5045, 11018, 13373), (5049, 16193, 5045), (11018, 16217, 1)


X(16217) = EULEROLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b+c)*(b^2+3*b*c+c^2)*a^6+2*(3*b^4+3*c^4-2*b*c*(b^2+c^2))*a^5+4*b*c*(b+c)*(4*b^2-b*c+4*c^2)*a^4-2*(3*b^6+3*c^6-b*c*(b^2+8*b*c+c^2)*(b^2-3*b*c+c^2))*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4-b*c*(5*b^2+6*b*c+5*c^2))*a^2+2*(b^2-c^2)^4*a-(b^2-c^2)^3*(b-c)^3) : :
X(16217) = 3*X(354)+X(10965)

X(16217) lies on the line {1,3}

X(16217) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16216, 11018), (11018, 16215, 16218)


X(16218) = EULEROLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INVERSE-IN-INCIRCLE

Barycentrics    a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b+c)*(b^2+b*c+c^2)*a^6+2*(3*b^4+3*c^4-4*b*c*(b-c)^2)*a^5+4*b*c*(b+c)*(2*b^2-3*b*c+2*c^2)*a^4-2*(3*b^2-8*b*c+3*c^2)*(b^4+c^4+b*c*(b^2+4*b*c+c^2))*a^3+2*(b^4-c^4)*(b-c)*(b^2-3*b*c+c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*(b-c)^2*a-(b^2-c^2)^3*(b-c)^3) : :
X(16218) = 3*X(354)+X(10966)

X(16218) lies on these lines:
{1,3}, {9947,10958}

X(16218) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 13373, 11018), (11018, 16215, 16217)


X(16219) = EULEROLOGIC CENTER OF THESE TRIANGLES: X3-ABC REFLECTIONS TO KOSNITA

Barycentrics    (SB+SC)*((3*R^2-5*SA-SW)*S^2-(3*R^2*(108*R^2-6*SA-49*SW)+5*SA^2-5*SB*SC+16*SW^2)*SA) : :
X(16219) = 5*X(74)+X(9934) = 2*X(74)+X(13289) = 2*X(9934)-5*X(13289) = 2*X(10282)+X(15054) = X(10606)-3*X(15041) = 4*X(12041)-X(13293)

X(16219) lies on these lines:
{74,186}, {154,10620}, {182,2781}, {381,2777}, {1154,12901}, {3357,12106}, {3532,10293}, {5643,7527}, {5655,10182}, {5663,11202}, {10193,15131}, {10282,15054}, {10628,15055}, {11438,12099}

X(16219) = midpoint of X(154) and X(10620)
X(16219) = reflection of X(5655) in X(10182)


X(16220) = EULEROLOGIC CENTER OF THESE TRIANGLES: X-PARABOLA-TANGENTIAL TO SCHROETER

Barycentrics    (2*a^8-3*(b^2+c^2)*a^6-(b^4-8*b^2*c^2+c^4)*a^4+(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :
X(16220) = X(4)-3*X(5466) = X(4)-4*X(10279) = 2*X(5)-3*X(8371) = 4*X(140)-3*X(1649) = 5*X(631)-2*X(8151) = 5*X(631)-3*X(9168) = 5*X(3091)-8*X(10280) = 3*X(5054)-2*X(10190) = 3*X(5055)-4*X(10189) = 3*X(5466)-4*X(10279) = X(5664)-4*X(15543) = X(7728)-4*X(12064) = 2*X(8151)-3*X(9168) = 3*X(9185)-2*X(11615) = X(12188)+2*X(13187)

X(16220) lies on these lines:
{2,14214}, {3,523}, {4,1499}, {5,8371}, {30,8029}, {39,8704}, {140,1649}, {381,10278}, {512,9730}, {549,11123}, {631,8151}, {669,12106}, {690,16003}, {2780,9979}, {2793,10991}, {3091,10280}, {5054,10190}, {5055,10189}, {5663,13291}, {7728,12064}, {9126,9131}, {9185,11615}, {12188,13187}, {12317,14695}, {14094,14932}

X(16220) = reflection of X(381) in X(10278)


X(16221) =  COMPLEMENT OF X(10420)

Barycentrics    SB*SC*(S^2-3*SA^2)*(2*S^2+SA^ 2+2*SB*SC-SW^2)*(6*R^2-SA-SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27212.

X(16221) lies on the nine-point circle and these lines:
{2, 10420}, {4, 476}, {5, 12052}, {30, 131}, {113, 403}, {114, 468}, {115, 2501}, {122, 3154}, {125, 924}, {128, 186}, {133, 10151}, {136, 523}, {137, 2970}, {230, 1560}, {14120, 14672}

X(16221) = complement of X(10420)
X(16221) = Dou-circles-radical-circle-inverse of X(115)
X(16221) = polar circle-inverse of X(476)
X(16221) = X(1300)-of-reflection-of-Euler-triangle in Euler line
X(16221) = reflection of X(136) in Euler line


X(16222) =  X(5)X(1986)∩X(113)X(389)

Barycentrics    (SB+SC)*((14*R^2-SA-3*SW)*S^2+ (6*R^2*(3*R^2+SA)-2*SW*(7*R^2- SW)-SA^2+SB*SC)*SA) : :
X(16222) = X(3)+2*X(1112), X(3)-4*X(9826), X(4)+2*X(14708), 2*X(5)+X(1986), 4*X(5)-X(7723), X(52)+2*X(5972), X(74)-7*X(15043), X(110)+5*X(3567), X(110)+2*X(12236), X(113)+2*X(389), X(125)-4*X(5462), X(125)+2*X(11557), X(1112)+2*X(9826), 2*X(1986)+X(7723), 5*X(3567)-2*X(12236), 2*X(5462)+X(11557)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16222) lies on thes lines:
{3, 1112}, {4, 14708}, {5, 1986}, {24, 12228}, {52, 5972}, {74, 15043}, {110, 3567}, {113, 389}, {125, 5462}, {143, 1511}, {265, 11746}, {381, 5640}, {399, 7529}, {546, 12292}, {569, 13289}, {973, 11597}, {974, 7728}, {1199, 3047}, {1539, 13630}, {1656, 12358}, {2070, 11416}, {2777, 9730}, {2781, 14561}, {3060, 15035}, {3090, 12219}, {3091, 7722}, {3448, 7528}, {3526, 13416}, {3843, 12133}, {3851, 13148}, {5093, 13321}, {5446, 16163}, {5562, 12900}, {5609, 13358}, {5644, 15041}, {5943, 10628}, {6102, 12825}, {6153, 14049}, {6644, 15463}, {6699, 13417}, {7403, 10264}, {7553, 11566}, {7687, 11562}, {7731, 15024}, {9729, 11807}, {9781, 10733}, {10020, 12606}, {10095, 10113}, {10110, 12295}, {10539, 12227}, {10574, 10721}, {10982, 12302}, {11424, 12901}, {11806, 15063}, {12006, 12041}, {13201, 15028}, {15045, 15055}

X(16222) = midpoint of X(i) and X(j) for these {i,j}: {568, 14643}, {3060, 15035}
X(16222) = X(2072)-of-orthocentroidal-triangle
X(16222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 1986, 7723), (1112, 9826, 3), (5462, 11557, 125), (7731, 15024, 15059), (9729, 11807, 16111), (10095, 11561, 10113)


X(16223) =  X(5)X(113)∩X(110)X(389)

Barycentrics    (SB+SC)*(2*S^2*(5*R^2-SW)+(3* R^2*(12*R^2+SA)-SW*(19*R^2-2* SW))*SA) : :
X(16223) = X(3)+2*X(11557), 2*X(3)+X(13417), X(5)+2*X(11561), 2*X(5)+X(11562), X(20)+2*X(11807), X(52)+2*X(1511), X(74)-4*X(9729), 2*X(113)+X(185), X(113)+2*X(14708), X(125)-4*X(9826), X(185)-4*X(14708), 2*X(974)+X(15063), X(10264)-4*X(12006), 4*X(11557)-X(13417), 4*X(11561)-X(11562)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16223) lies on thes lines:
{2, 10628}, {3, 11557}, {5, 113}, {20, 11807}, {52, 1511}, {74, 9729}, {110, 389}, {146, 10574}, {182, 1205}, {186, 249}, {195, 568}, {265, 5462}, {399, 11806}, {631, 7731}, {1112, 16163}, {1539, 10575}, {1986, 5562}, {2777, 12824}, {3090, 12281}, {3091, 12270}, {3448, 15043}, {3523, 13201}, {3567, 11800}, {5446, 12121}, {5642, 14831}, {5892, 15061}, {5907, 7722}, {5943, 14644}, {6102, 10272}, {6240, 15473}, {6243, 15040}, {7723, 12900}, {9786, 12168}, {10110, 10733}, {10114, 14516}, {10117, 10984}, {10125, 15067}, {11424, 12302}, {11459, 14940}, {11695, 15059}, {11793, 12219}, {11801, 15026}, {12228, 13367}, {12358, 14448}, {13293, 15055}, {13348, 15036}, {13754, 14643}, {14094, 15012}, {15024, 15081}, {15028, 15100}, {15051, 15644}

X(16223) = X(1568)-of-orthocentroidal-triangle
X(16223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11557, 13417), (5, 11561, 11562), (113, 14708, 185), (1986, 5972, 5562), (3567, 12383, 11800), (15024, 15102, 15081)


X(16224) =  X(51)X(2794)∩X(132)X(389)

Barycentrics    (SB+SC)*((4*R^2+3*SW)*S^4-(8*( -SW+3*SA)*R^4-2*(5*SW^2+5*SA^ 2-6*SB*SC)*R^2-(SA+SW)*(SA-3* SW)*SW)*S^2+(4*R^2-SW)*(4*R^2+ SA-2*SW)*SA*SW^2) : :
X(16224) = X(52)+2*X(6720), X(112)+5*X(3567), X(127)-4*X(5462), X(132)+2*X(389), X(1297)-7*X(15043), 2*X(5446)+X(14689), 7*X(9781)-X(10735)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16224) lies on thes lines:
{51, 2794}, {52, 6720}, {112, 3567}, {127, 5462}, {132, 389}, {1297, 15043}, {5446, 14689}, {9781, 10735}


X(16225) =  X(51)X(2794)∩X(112)X(389)

Barycentrics    (SB+SC)*((7*R^2+SW)*S^4-(8*(3* SA-SW)*R^4-(10*SA^2-9*SB*SC+2* SW^2)*R^2+(SA+SW)*SW^2)*S^2+( 4*R^2-SW)^2*SA*SW^2) : :
X(16225) = X(112)+2*X(389), 2*X(132)+X(185), X(1297)-4*X(9729), 5*X(3567)+X(13200), 4*X(5462)-X(10749), X(5562)-4*X(6720), 4*X(10110)-X(10735), 5*X(10574)+X(12384), X(13219)-7*X(15043)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16225) lies on thes lines:
{51, 2794}, {112, 389}, {132, 185}, {1297, 9729}, {3567, 13200}, {5462, 10749}, {5562, 6720}, {10110, 10735}, {10574, 12384}, {13219, 15043}


X(16226) =  X(2)X(389)∩X(30)X(51)

Barycentrics    (SB+SC)*(4*S^2+SA*(SA+12*R^2- 3*SW)) : :
X(16226) = X(2)+2*X(389), 4*X(2)-X(5562), 5*X(2)+X(5889), 11*X(2)-5*X(11444), 5*X(2)-8*X(11695), 7*X(2)-4*X(11793), 2*X(2)+X(14831), 7*X(2)-13*X(15028), X(2)-7*X(15043), 8*X(389)+X(5562), 10*X(389)-X(5889), 5*X(389)+4*X(11695), 7*X(389)+2*X(11793), 4*X(389)-X(14831), 2*X(389)+7*X(15043), 5*X(5562)+4*X(5889), 11*X(5562)-20*X(11444), 7*X(5562)-16*X(11793), X(5562)+2*X(14831), X(5889)+8*X(11695), 2*X(5889)-5*X(14831), 14*X(11695)-5*X(11793), 8*X(11793)+7*X(14831), 4*X(11793)-13*X(15028), X(14831)+14*X(15043)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16226) lies on thes lines:
{2, 389}, {3, 15004}, {4, 15010}, {30, 51}, {52, 549}, {140, 14531}, {143, 8703}, {185, 381}, {186, 575}, {373, 5055}, {376, 3567}, {511, 3524}, {547, 6102}, {568, 3917}, {578, 15078}, {1092, 11432}, {1154, 5650}, {1216, 15694}, {1843, 11179}, {2071, 15019}, {2979, 15708}, {3060, 10304}, {3091, 13382}, {3534, 5446}, {3543, 10110}, {3545, 5890}, {3581, 15038}, {3796, 10245}, {3819, 15709}, {3839, 5640}, {3845, 11381}, {5066, 12162}, {5071, 5907}, {5422, 11438}, {5447, 15701}, {5476, 12294}, {5642, 9826}, {5655, 11806}, {5663, 14845}, {5876, 10109}, {5891, 13363}, {6101, 11812}, {6243, 15693}, {6644, 13366}, {6688, 11459}, {7527, 12834}, {7706, 13851}, {9781, 15682}, {9822, 11180}, {9909, 10984}, {10095, 10575}, {10303, 15606}, {10625, 12100}, {11001, 13598}, {11412, 15702}, {11430, 15053}, {11455, 13570}, {13321, 15688}, {13340, 15706}, {13348, 15698}, {13364, 16194}, {13491, 14893}, {14449, 14891}, {15644, 15692}

X(16226) = midpoint of X(i) and X(j) for these {i,j}: {568, 5054}, {3060, 10304}, {3545, 5890}
X(16226) = reflection of X(i) in X(j) for these (i,j): (3545, 5943), (5891, 15699)
X(16226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 389, 14831), (2, 14831, 5562), (389, 11695, 5889), (568, 5892, 3917), (5890, 5943, 15030), (5946, 9730, 51)


X(16227) =  X(30)X(51)∩X(389)X(468)

Barycentrics    (SB+SC)*((12*R^2+5*SW)*S^2-6* R^2*(12*R^2*SA+6*SB*SC-5*SA^2) +(SA-4*SW)*SA*SW) : :
X(16227) = 2*X(389)+X(468), X(858)-7*X(15043), 2*X(974)+X(1514), 5*X(3567)+X(10295), 4*X(5462)-X(10297), 4*X(9826)-X(11064), 4*X(12006)-X(15122)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27223.

X(16227) lies on thes lines:
{6, 186}, {30, 51}, {389, 468}, {403, 5890}, {597, 15045}, {858, 15043}, {974, 1514}, {1154, 9826}, {2071, 5422}, {3567, 10295}, {5462, 10297}, {5892, 10257}, {7464, 10982}, {7729, 11455}, {11695, 15739}, {12006, 15122}, {13352, 15646}

X(16227) = midpoint of X(403) and X(5890)


X(16228) =  X(4)X(513)∩X(523)X(10151)

Barycentrics    (a^3-(b+c)^2*a+2*(b+c)*b*c)*( b-c)*(a^2-b^2+c^2)*(a^2+b^2-c^ 2) : :
Barycentrics    (tan A)[(c - a) cos B + (a - b) cos C] : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27225.

X(16228) lies on thes lines:
{4, 513}, {523, 10151}, {900, 7649}, {3064, 14321}, {4132, 14618}

X(16228) = center of inverse-in-polar-circle-of-line-X(1)X(3)


X(16229) =  MIDPOINT OF X(4) AND X(14618)

Barycentrics    (SA^2+SB*SC)*(SB-SC)*SB*SC : :
Barycentrics    (b^2 - c^2)(a^4 + 2b^2c^2 - a^2b^2 - a^2c^2)/(b^2 + c^2 - a^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27225.

X(16229) lies on thes lines:
{4, 512}, {523, 10151}, {804, 2489}, {924, 13851}, {1882, 7178}, {2501, 3566}, {2506, 5254}, {4108, 6995}, {4367, 5307}, {5523, 7651}

X(16229) = midpoint of X(4) and X(14618)
X(16229) = crosspoint of X(4) and X(6528)
X(16229) = X(663)-of-orthic-triangle if ABC is acute
X(16229) = center of inverse-in-polar-circle-of-Brocard-axis


X(16230) =  X(4)X(690)∩X(230)X(231)

Barycentrics    (SA^2-SB*SC)*(SB-SC)*SB*SC : :
Barycentrics    sec A cos(A + ω) (b^2 - c^2) : :
Barycentrics    (b^2 - c^2)(b^4 + c^4 - a^2b^2 - a^2c^2)/(b^2 + c^2 - a^2) : :
X(16230) = 3*X(1637) - 2*X(6130)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27225.

The trilinear polar of X(16230) passes through X(868). (Randy Hutson, March 14, 2018)

X(16230) lies on thes lines:
{4, 690}, {5, 6334}, {24, 14270}, {98, 3563}, {107, 110}, {114, 132}, {125, 136}, {230, 231}, {264, 14295}, {526, 1112}, {804, 12131}, {826, 14618}, {850, 6368}, {879, 6531}, {1177, 15328}, {2848, 9409}, {4232, 9185}, {5095, 9003}, {5466, 16080}

X(16230) = polar conjugate of X(2966)
X(16230) = Dao-Moses-Telv-circle-inverse-of X(2501)
X(16230) = polar-circle-inverse-of X(11005)
X(16230) = pole wrt polar circle of trilinear polar of X(2966) (line X(2)X(98))
X(16230) = isoconjugate of X(j) and X(j) for these {i,j}: {48,2966}, {63,2715}
X(16230) = center of inverse-in-polar-circle-of-Fermat-axis


X(16231) =  MIDPOINT OF X(4) AND X(7649)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27225.

X(16231) lies on thes lines:
{4, 2457}, {1598, 4057}, {2969, 5510}, {4786, 6994}

X(16231) = midpoint of X(4) and X(7649)
X(16231) = center of inverse-in-polar-circle-of-Nagel-line


X(16232) =  X(1)X(371)∩X(6)X(19)

Barycentrics    a (a^2 + b^2 - c^2 + 2 a b - 2 S) (a^2 - b^2 + c^2 + 2 a c - 2 S) : :
Barycentrics    a ((a + b - c) (a - b + c) (b + c)-2 a S) : :
Barycentrics    Sin[A] / (1 - Cot[A/2]) : :

X(16232) lies on the circumconic {{A,B,C,X(1),X(2)}}, the cubics K233 and K632, and on these lines:
{1,371}, {2,175}, {4,1123}, {6,19}, {12,13911}, {40,5414}, {46,372}, {56,7968}, {57,6502}, {72,1378}, {81,1806}, {226,13883}, {278,13459}, {354,3297}, {485,12047}, {486,1737}, {517,1335}, {590,11375}, {605,1451}, {942,1124}, {1151,2646}, {1152,1155}, {1159,6417}, {1377,3753}, {1452,5413}, {1587,4295}, {1685,5530}, {1703,2093}, {1722,6203}, {1770,6560}, {1788,3069}, {1805,16049}, {1836,3070}, {1837,3071}, {1875,13438}, {1905,3092}, {2099,7969}, {3057,3298}, {3068,3485}, {3299,5902}, {3300,7951}, {3301,5903}, {3474,6460}, {3486,6459}, {3601,9616}, {3612,6200}, {3911,13971}, {4305,9541}, {4848,13936}, {4870,13846}, {5219,13893}, {5440,9679}, {5886,9661}, {6561,10572}, {6565,10826}, {7288,13959}, {9615,13384}, {9646,11374}, {13411,13912}, {13427,16214}

X(16232) = Ceva conjugate of X(2362)
X(16232) = X(i)-cross conjugate of X(j) for these (i,j): {56, 2362}, {7968, 1}
X(16232) = crosspoint of X(4) and X(1336)
X(16232) = crosssum of X(3) and X(1335)
X(16232) = X(i)-isoconjugate of X(j) for these (i,j): {2, 5414}, {3, 7090}, {8, 2067}, {9, 13388}, {10, 1805}, {63, 7133}, {78, 2362}, {219, 1659}, {1335, 14121}, {2066, 13387}
X(16232) = X(2362)-Hirst inverse of X(14571)
X(16232) = barycentric product X(i)*X(j) for these {i,j}: {1, 13390}, {4, 13389}, {57, 14121}, {92, 6502}, {273, 2066}, {1336, 13388}, {1659, 6212}, {2362, 13386}
X(16232) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 7090}, {25, 7133}, {31, 5414}, {34, 1659}, {56, 13388}, {604, 2067}, {608, 2362}, {1333, 1805}, {1806, 1812}, {2066, 78}, {2067, 3084}, {2362, 13387}, {6502, 63}, {13388, 5391}, {13389, 69}, {13390, 75}, {13460, 13390}, {14121, 312}
X(16232) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1702, 2066), (6, 65, 2362), (19, 34, 2362), (221, 2262, 2362), (607, 1876, 2362), (608, 1829, 2362), (1880, 4185, 2362), (2082, 2263, 2362)


X(16233) =  REFLECTION OF X(5608) IN X(11176)

Barycentrics    (2*S^2-sqrt(3)*(3*R^2-SB-SC)* S-3*(3*SA-2*SW)*R^2-3*SB*SC- SW^2)*(SB-SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27230.

X(16233) lies on these lines:
{396, 11617}, {526,16234}, {619, 1649}, {690, 6771}, {5608, 11176}

X(16233) = reflection of X(5608) in X(11176)


X(16234) =  REFLECTION OF X(5607) IN X(11176)

Barycentrics    (2*S^2+sqrt(3)*(3*R^2-SB-SC)*S-3*(3*SA-2*SW)*R^2-3*SB*SC- SW^2)*(SB-SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27230.

X(16234) lies on these lines:
{395, 11618}, {526,16233, {618, 1649}, {690, 6774}, {5607, 11176}

X(16234) = reflection of X(5607) in X(11176)


X(16235) =  MIDPOINT OF X(16233) AND X(16234)

Barycentrics    (SB-SC)*(2*S^2-3*(3*SA-2*SW)* R^2-3*SB*SC-SW^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27230.

X(16235) lies on these lines:
{2, 2780}, {3, 9148}, {140, 11176}, {351, 5054}, {523, 7623}, {526,16233}, {549, 804}, {620, 2793}, {631, 9147}, {690, 6036}, {3268, 16220}, {9188, 10168}

X(16235) = midpoint of X(i) and X(j) for these {i,j}: {3, 9148}, {3268, 16220}
X(16235) = reflection of X(9188) in X(10168)


X(16236) =  X(1)X(631)∩X(7)X(519)

Barycentrics    (a-2b-2c)(5a-b-c)/(b+c-a) : :
X(16236) = 3 X(1) - 2 X(1000)

See Angel Montesdeoca, HG200218.

X(16236) lies on these lines:
{1, 631}, {7, 519}, {8, 3947}, {46, 7966}, {57, 1317}, {65, 3633}, {80, 1537}, {145, 3339}, {226, 4677}, {517, 14100}, {952, 4312}, {1071, 5903}, {1405, 4752}, {2099, 3679}, {2802, 7672}, {3241, 13462}, {3243, 5854}, {3244, 3361}, {3340, 3632}, {3476, 4031}, {3485, 4668}, {3621, 3671}, {3622, 4675}, {3624, 11011}, {3626, 4323}, {3635, 15519}, {3880, 15185}, {3885, 12432}, {4304, 7991}, {4355, 10944}, {4669, 5226}, {4701, 5261}, {4816, 9578}, {5561, 9897}, {5586, 10106}, {5722, 5763}, {5804, 10573}, {5844, 11529}, {9589, 10950}, {9613, 10052}, {9614, 11280}, {11009, 15079}

X(16236) = reflection of X(i) in X(j) for these (i,j): {1, 11041}, {1000, 14563}, {3632, 11525}, {8275, 1}


X(16237) =  ISOTOMIC CONJUGATE OF X(15421)

Barycentrics    SB*SC*(SA-SB)*(SA-SC)*(2*SA+ SB+SC-6*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27233.

X(16237) lies on these lines:
{2, 216}, {99, 112}, {250, 4226}, {378, 6795}, {476, 1304}, {523, 4230}, {687, 15421}, {11061, 13200}

X(16237) = isotomic conjugate of X(15421)
X(16237) = polar conjugate of X(15328)
X(16237) = trilinear pole of the line {113, 403}
X(16237) = barycentric product X(i)*X(j) for these {i,j}: {99, 403}, {113, 16077}, {264, 15329}, {648, 3580}, {811, 1725}, {892, 12828}, {3003, 6331}, {6528, 13754}
X(16237) = barycentric quotient X(i)/X(j) for these (i,j): (4, 15328), (107, 1300), (110, 5504), (112, 14910), (113, 9033), (186, 15470), (250, 10420), (403, 523), (476, 12028), (648, 2986), (686, 3269), (1304, 10419), (1725, 656), (1986, 526), (2315, 822), (3003, 647), (3580, 525), (4240, 15454), (6334, 15526), (12824, 9517), (12826, 2850), (12828, 690), (13754, 520), (14264, 14380), (15329, 3)
X(16237) = trilinear product X(i)*X(j) for these {i,j}: {92, 15329}, {162, 3580}, {403, 662}, {648, 1725}, {811, 3003}, {823, 13754}, {2315, 6528}
X(16237) = trilinear quotient X(i)/X(j) for these (i,j): (92, 15328), (113, 2631), (162, 14910), (403, 661), (662, 5504), (811, 2986), (823, 1300), (1725, 647), (1986, 2624), (3003, 810), (3580, 656), (6334, 2632), (12828, 2642), (13754, 822), (15329, 48)


X(16238) =  COMPLEMENT OF X(11585)

Barycentrics    (14*R^2-3*SW)*S^2-(2*R^2-SW)* SB*SC : :
X(16238) = 3*X(2) + X(24) = X(4) + 3*X(15078)

As a point on the Euler line, X(16238) has Shinagawa coefficients (E-6*F, E+2*F)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27235.

X(16238) lies on these lines:
{2, 3}, {49, 11245}, {52, 11064}, {125, 12134}, {389, 5972}, {1147, 13292}, {1493, 5181}, {1511, 12370}, {3589, 6153}, {5446, 14156}, {5448, 13568}, {5654, 9786}, {6696, 6699}, {8263, 8548}, {9306, 12359}, {10272, 14708}, {10280, 14341}, {11449, 12022}, {12038, 12241}, {13336, 13394}, {14389, 15024}, {14984, 15120}

X(16238) = complement of X(11585)
X(16238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24, 11585), (3, 468, 13383), (3, 1656, 6816), (5, 549, 7526), (5, 6642, 10127), (140, 10020, 6676), (140, 10096, 548), (140, 13383, 3), (468, 7499, 7493), (468, 11799, 10096), (858, 3518, 7553), (3546, 6353, 7387), (5159, 6756, 13371), (6640, 7506, 427), (6643, 14070, 550), (12106, 13371, 6756)


X(16239) =  COMPLEMENT OF X(3628)

Trilinears    b*c*(6*a^4-11*(b^2+c^2)*a^2+5* (b^2-c^2)^2) : :
Trilinears    11 cos A + 10 cos B cos C : :
Barycentrics    11*S^2-SB*SC : :
X(16239) = 15*X(2)+X(3) = 9*X(2)-X(5) = 3*X(2)+X(140) = 17*X(2)-X(381) = 21*X(2)-X(546) = 5*X(2)-X(547) = 7*X(2)+X(549) = 3*X(2)+5*X(632) = 21*X(2)-5*X(1656) = 9*X(2)+7*X(3526) = 9*X(2)+X(3530) = 15*X(2)-X(3850) = 18*X(2)-X(3856) = 19*X(2)-X(3860) = 13*X(2)+3*X(5054) = 19*X(2)-3*X(5055) = 13*X(2)-X(5066) = 27*X(2)-11*X(5070) = 7*X(2)-X(10109) = 5*X(2)+3*X(11539) = 2*X(2)+X(11540) = 11*X(2)-X(11737) = 5*X(2)+X(11812) = 11*X(2)+X(12100) = 6*X(2)+X(12108) = 12*X(2)-X(12811) = 11*X(2)+3*X(14890) = 13*X(2)+X(14891) = 11*X(2)+5*X(15694) = 11*X(2)-3*X(15699) = 23*X(2)-7*X(15703) = 5*X(2)+11*X(15723) = 17*X(2)+X(15759) = 11*X(3)+5*X(4) = 3*X(3)+5*X(5) = 21*X(3)-5*X(20) = X(3)-5*X(140) = 31*X(3)-15*X(376) = 7*X(3)+5*X(546) = X(3)+3*X(547) = 9*X(3)-5*X(548) = 7*X(3)-15*X(549) = 13*X(3)-5*X(550) = 3*X(3)-5*X(3530) = X(3)-17*X(3533) = 13*X(3)+3*X(3543) = 7*X(3)+9*X(3545) = X(3)+5*X(3628)

As a point on the Euler line, X(16239) has Shinagawa coefficients (11,-1)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27235.

X(16239) lies on these lines:
{2, 3}, {125, 13392}, {143, 3819}, {230, 5041}, {323, 15047}, {373, 10263}, {395, 3412}, {396, 3411}, {485, 6438}, {486, 6437}, {952, 3634}, {1125, 5844}, {1131, 6446}, {1132, 6445}, {1154, 11695}, {1216, 13363}, {1353, 3619}, {1483, 9780}, {3054, 5305}, {3055, 5008}, {3070, 6485}, {3071, 6484}, {3316, 6395}, {3317, 6199}, {3589, 5097}, {3592, 10194}, {3594, 10195}, {3624, 5690}, {3828, 15178}, {3917, 14449}, {4301, 11230}, {5432, 15172}, {5447, 6688}, {5462, 15606}, {5550, 10283}, {5609, 13393}, {5650, 6101}, {5704, 15935}, {5843, 6666}, {5886, 9588}, {5892, 11591}, {5893, 10193}, {5901, 11231}, {5943, 10627}, {5946, 14531}, {6243, 11465}, {6429, 9680}, {6431, 8252}, {6432, 8253}, {6433, 9681}, {6668, 6681}, {6671, 6674}, {6672, 6673}, {6689, 15605}, {6704, 14693}, {7294, 15325}, {7746, 9607}, {7751, 15597}, {7759, 9771}, {7767, 7814}, {7888, 11168}, {8254, 11064}, {8972, 13961}, {9624, 11531}, {9657, 10592}, {9670, 10593}, {9693, 10137}, {9706, 13353}, {9729, 14128}, {10110, 12045}, {10170, 13630}, {10172, 13624}, {10219, 13391}, {10272, 16003}, {10386, 10589}, {10625, 13451}, {11017, 14915}, {11793, 12006}, {13364, 15644}, {13903, 13941}, {14643, 15057}

X(16239) = midpoint of X(i) and X(j) for these {i,j}: {2, 10124}, {125, 13392}, {5447, 10095}, {5609, 13393}, {9729, 14128}, {11793, 12006}
X(16239) = complement of X(3628)
X(16239) = X(3856) of Johnson triangle
X(16239) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3526, 5), (3, 4, 15686), (3, 3543, 550), (4, 12812, 11737), (4, 15699, 12812), (5, 382, 3859), (5, 3627, 3855), (5, 3853, 3850), (382, 3859, 3861), (382, 5070, 7486), (382, 7486, 5), (547, 3853, 5), (1656, 3830, 15022), (2041, 2042, 5054), (3627, 15713, 3523), (3628, 12102, 3090), (3832, 3853, 3861), (12102, 14891, 550)


X(16240) =  X(4)X(74)∩X(132)(468)

Trilinears    (tan A)(sin A)(cos A - 2 cos B cos C)^2 : :
Trilinears    (tan A)(cos A - 2 cos B cos C)[b(cos B - 2 cos C cos A) + c(cos C - 2 cos A cos B)] : :
Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2) ^2)^2*(a^2-b^2+c^2)*(a^2+b^2- c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27240.

From Hyacinthos #21522 (2/11/2013, Antreas Hatzipolakis): Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections of the nine-point circle in the altitudes HHa,HHb,HHc, resp. and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp. The triangles HaHbHc and A'B'C' are perspective at X(16240). (Randy Hutson, March 14, 2018)

X(16240) lies on the orthic inconic, the cubic K496, and these lines:
{4, 74}, {25, 1989}, {51, 6749}, {132, 468}, {184, 6525}, {1495, 1990}, {1637, 9409}, {1842, 2969}, {1859, 3270}, {3081, 3163}, {3517, 13558}, {4232, 9752}, {4240, 5642}, {5095, 9003}, {6618, 15004}, {13857, 15144}

X(16240) = crosspoint of X(4) and X(1990)
X(16240) = X(88)-of-orthic-triangle if ABC is acute
X(16240) = orthic-isogonal conjugate of X(1990)
X(16240) = X(4)-Ceva conjugate of X(1990)
X(16240) = barycentric product X(4)*X(30)*X(30)
X(16240) = bariyentric product X(4)*X(3163)
X(16240) = {X(13202), X(14847)}-harmonic conjugate of X(125)


X(16241) =  X(2)X(14)∩X(3)(13)

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

X(16241) = {X(6),X(5054)}-harmonic conjugate of X(16242)

See Tran Quang Hung, Dao Thanh Oai, and César Lozada, ADGEOM 4392.

X(16241) lies on these lines:
{2, 14}, {3, 13}, {4, 5352}, {5, 5238}, {6, 5054}, {16, 396}, {18, 3526}, {30, 10645}, {61, 140}, {62, 631}, {203, 5432}, {298, 11132}, {299, 618}, {303, 3643}, {381, 11480}, {397, 3530}, {398, 632}, {547, 5321}, {624, 11299}, {629, 633}, {630, 11290}, {1606, 3131}, {3104, 7786}, {3111, 14182}, {3201, 5012}, {3364, 5420}, {3365, 5418}, {3390, 15765}, {3523, 5237}, {3524, 10646}, {3533, 10187}, {3534, 12816}, {3582, 10638}, {3584, 7051}, {3851, 10188}, {4045, 11298}, {5070, 5339}, {5318, 8703}, {5335, 15692}, {5350, 12103}, {5433, 7005}, {5444, 7052}, {5463, 5569}, {5470, 6772}, {5474, 9735}, {5642, 10658}, {6669, 11303}, {6694, 11308}, {6774, 9117}, {6778, 12042}, {7619, 9761}, {7761, 11297}, {9116, 9885}, {10124, 11543}, {10182, 11244}, {11134, 13339}, {11268, 15330}, {11481, 15693}, {11485, 15694}, {11486, 15701}, {11489, 15709}, {11542, 12100}

X(16241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 617, 623), (2, 13083, 5464), (6, 5054, 16242), (396, 549, 16), (397, 3530, 5351), (619, 5981, 5464), (619, 6671, 2)


X(16242) =  X(2)X(13)∩X(3)(14)

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

X(16242) = {X(6),X(5054)}-harmonic conjugate of X(16241)

See Tran Quang Hung, Dao Thanh Oai, and César Lozada, ADGEOM 4392.

X(16242) lies on these lines:
{2, 13}, {3, 14}, {4, 5351}, {5, 5237}, {6, 5054}, {15, 395}, {17, 3526}, {30, 10646}, {61, 631}, {62, 140}, {202, 5432}, {298, 619}, {299, 11133}, {302, 3642}, {381, 11481}, {397, 632}, {398, 3530}, {547, 5318}, {623, 11300}, {629, 11289}, {630, 634}, {1250, 3582}, {1605, 3132}, {2306, 5442}, {3105, 7786}, {3111, 14178}, {3200, 5012}, {3364, 15765}, {3389, 5420}, {3390, 5418}, {3523, 5238}, {3524, 10645}, {3533, 10188}, {3534, 12817}, {3851, 10187}, {4045, 11297}, {5070, 5340}, {5321, 8703}, {5334, 15692}, {5349, 12103}, {5433, 7006}, {5464, 5569}, {5469, 6775}, {5473, 9736}, {5642, 10657}, {6670, 11304}, {6695, 11307}, {6771, 9115}, {7619, 9763}, {7761, 11298}, {10124, 11542}, {10182, 11243}, {11137, 13339}, {11267, 15330}, {11480, 15693}, {11485, 15701}, {11486, 15694}, {11488, 15709}, {11543, 12100}

X(16242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 616, 624), (2, 13084, 5463), , (6, 5054, 16241), (395, 549, 15), (398, 3530, 5352), (618, 5980, 5463), (618, 6672, 2)


X(16243) =  X(74)X(140)∩X(98)(468)

Barycentrics    (SB+3*SC)*(SC+3*SB)*(S^2-3*SA* SB)*(S^2-3*SA*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27242.

X(16243) lies on these lines:
{74, 140}, {98, 468}, {5627, 10096}, {6676, 14919}


X(16244) =  X(140)X(2777)∩X(523)(6140)

Barycentrics    3*S^4+(504*R^4+6*R^2*(2*SA-41* SW)-SB*SC+27*SW^2)*S^2+3*(72* R^4+2*R^2*SW-3*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27242.

X(16244) lies on these lines: {140, 2777}, {523, 6140}


X(16245) =  POLAR CONJUGATE OF X(5404)

Barycentrics    sec A sin(A - ω/2) : :
Barycentrics    (-a^2+b^2-c^2) (a^2+b^2-c^2) (2 a^2+2 Sqrt[a^2 b^2+a^2 c^2+b^2 c^2]) : :
X(16245) = 3 (S + 4 R^2 Sin[w]) X[2] - 2 S (1 + Cos[w]) X[3]

X(16245) lies on these lines:
{2,3}, {275,5403}, {1343,1629}, {1677,2052}, {2546,3796}, {2547,10601}

X(16245) = polar conjugate of X(5404)
X(16245) = orthocentroidal-circle-inverse of X(16246)
X(16245) = barycentric product X(264)*X(1343)
X(16245) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5404}, {1343, 3}, {10312, 1342}


X(16246) =  POLAR CONJUGATE OF X(1676)

Barycentrics    sec A sin(A + ω/2) : :
Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (b^2+c^2+Sqrt[a^2 b^2+a^2 c^2+b^2 c^2]) : :

X(16246) lies on these lines:
{2, 3}, {275, 1677}, {2052, 5403}

X(16246) = polar conjugate of X(1676)
X(16246) = orthocentroidal circle-inverse-of X(16245)
X(16246) = barycentric product X(264)*X(1671)
X(16246) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1676}, {1671, 3}


X(16247) =  PERSPECTOR OF ABC AND 1ST FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    ((3*R^2*(18*R^2+3*SA-4*SW)-2*SA^2+2*SB*SC+2*SW^2)*S^2+sqrt(3)*(-2*(6*R^2-SW)*S^2+R^2*(54*R^2*SA-21*SA^2+12*SB*SC+SW^2)+2*SA^2*SW)*S-(9*R^2-2*SW)*SB*SC*SW)*(SB+SC) : :    (César Lozada, February 25, 2018)

In the plane of a triangle ABC, let F = X(13) or F = X(14), and let

OAB = circumcircle of AFB;
OAC = circumcircle of AFC;
LA = line through F parallel to BC;
AC = the point, other than F, in LA∩OAB;
AB = the point, other than F, in LA∩OAC;
Define BA and CB cyclically;
Define BC and CA cyclically;
Let A' = CBAB∩ACBC, and define B' and C' cyclically.

Theorem (Thanh Oai Dao): The triangle A'B'C' is equilateral and perspective to ABC.

If F = X(13), the the triangle A'B'C' is the 1st Fermat-Dao triangle, and the perspector is X(16247). If F = X(14), the the triangle A'B'C' is the 2nd Fermat-Dao triangle, and the perspector is X(16248). Ffor other Fermat-Dao equilateral triangles, see X(16267), X(16459), X(395), and X(16536).

You can view 1st and 2nd Fermat-Dao equilateral triangles.

X(16247) lies on the cubic K061b and these lines: {14, 5640}, {2378, 14186}


X(16248) =  PERSPECTOR OF ABC AND 2ND FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    ((3*R^2*(18*R^2+3*SA-4*SW)-2*SA^2+2*SB*SC+2*SW^2)*S^2-sqrt(3)*(-2*(6*R^2-SW)*S^2+R^2*(54*R^2*SA-21*SA^2+12*SB*SC+SW^2)+2*SW*SA^2)*S-(9*R^2-2*SW)*SB*SC*SW)*(SB+SC) : :

See X(16247).

X(16248) lies on the cubic K061a and these lines: {13, 5640}, {2379, 14188}


X(16249) =  EULER LINE INTERCEPT OF X(264)X(8739)

Barycentrics    sec A sin(A - ω + π/3) : :

X(16249) lies on these lines: {2,3}, {264,8739}

X(16249) = {X(25),X(458)}-harmonic conjugate of X(16250)


X(16250) =  EULER LINE INTERCEPT OF X(264)X(8740)

Barycentrics    sec A sin(A - ω - π/3) : :

X(16250) lies on these lines: {2,3}, {264,8740}

X(16250) = {X(25),X(458)}-harmonic conjugate of X(16249)


X(16251) =  ISOGONAL CONJUGATE OF X(10606)

Barycentrics    (5*S^2-24*R^2*SB+6*SB^2-4*SC*S A)*(5*S^2-24*R^2*SC+6*SC^2-4*S A*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27251.

X(16251) lies on the cubic K706 and these lines:
{2, 3184}, {20, 11064}, {30, 1249}, {253, 15311}, {3146, 14249}, {3346, 5895}, {12250, 15319}

X(16251) = isogonal conjugate of X(10606)
X(16251) = trilinear pole of the line {6587, 9033}


X(16252) =  COMPLEMENT OF X(6247)

Barycentrics    8*(S^2+SB*SC)*R^2+(SA-3*SW)*S^ 2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27251.

X(16252) lies on these lines:
{2, 1498}, {3, 1661}, {4, 154}, {5, 182}, {6, 3089}, {12, 10535}, {20, 11064}, {23, 2917}, {24, 13568}, {25, 11745}, {30, 5448}, {49, 11799}, {64, 631}, {113, 12605}, {140, 6000}, {141, 3547}, {155, 524}, {156, 15761}, {159, 1598}, {161, 10594}, {184, 235}, {185, 468}, {221, 3086}, {343, 11441}, {376, 5895}, {381, 9833}, {397, 11244}, {398, 11243}, {403, 1614}, {427, 15152}, {428, 3574}, {511, 15585}, {546, 8254}, {548, 2777}, {549, 3357}, {550, 11202}, {578, 1596}, {590, 12970}, {597, 8549}, {1093, 1990}, {1125, 6001}, {1181, 3542}, {1216, 2781}, {1249, 3349}, {1375, 1715}, {1495, 3575}, {1514, 11464}, {1576, 14152}, {1594, 14157}, {1619, 7395}, {1656, 14216}, {1853, 3090}, {1885, 13367}, {1906, 11424}, {1971, 7745}, {2192, 3085}, {2393, 10110}, {3071, 10533}, {3088, 15811}, {3091, 11206}, {3147, 10605}, {3522, 5925}, {3523, 6225}, {3524, 8567}, {3526, 12315}, {3530, 10182}, {3576, 12779}, {3631, 15068}, {3796, 6816}, {3827, 13374}, {3832, 14389}, {3853, 15806}, {5054, 13093}, {5085, 6804}, {5204, 12940}, {5217, 12950}, {5432, 6285}, {5433, 7355}, {5596, 10516}, {5654, 7387}, {5657, 7973}, {5663, 10020}, {5799, 7497}, {5907, 6676}, {5972, 16196}, {6143, 12112}, {6241, 10018}, {6288, 10024}, {6293, 11459}, {6353, 9786}, {6622, 6776}, {6677, 9729}, {6703, 6824}, {6823, 9306}, {6907, 14925}, {7495, 15056}, {7503, 13394}, {7505, 11456}, {7512, 10117}, {7516, 15578}, {7530, 15582}, {7542, 12162}, {7568, 15060}, {7680, 10537}, {8546, 15581}, {9924, 14853}, {9934, 14643}, {10019, 13851}, {10165, 12262}, {10193, 12108}, {10201, 12359}, {10257, 10575}, {10274, 11563}, {10297, 11750}, {10539, 15760}, {11204, 15712}, {11430, 13488}, {11744, 15035}, {11793, 16197}, {13383, 13754}, {14156, 14641}, {14561, 15583}, {15062, 15138}

X(16252) = midpoint of X(i) and X(j) for these {i,j}: {3, 2883}, {5, 6759}, {113, 15647}, {156, 15761}, {159, 5480}, {7680, 10537}
X(16252) = complement of X(6247)
X(16252) = X(11745)-of-Ara-triangle
X(16252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1498, 6247), (3, 5878, 5894), (6, 3089, 15873), (25, 12233, 11745), (184, 235, 12241), (381, 14530, 9833), (403, 1614, 6146), (631, 5656, 64), (1181, 3542, 13567), (1249, 6621, 6523), (2883, 5894, 5878), (2883, 10192, 3), (3523, 6225, 10606), (6225, 10606, 15105), (13568, 15448, 24)


X(16253) =  X(376)X(1515)∩X(800)X(30887)

Barycentrics    (2*S^2-3*SB*SC)*(5*S^2-24*R^2* SA+6*SA^2-4*SB*SC)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27251.

X(16253) lies on these lines:
{2, 3184}, {4, 8567}, {376, 1515}, {800, 3087}, {1249, 15311}, {2777, 10002}, {3146, 14918}, {5667, 6525}, {5894, 6523}


X(16254) =  COMPLEMENT OF X(14528)

Barycentrics    (S^2+2*SB*SC)*(8*R^2-3*SA-3* SW) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27255.

X(16254) lies on these lines:
{2, 14528}, {6, 3090}, {546, 4550}, {631, 15153}, {632, 1511}, {942, 3947}, {1147, 3628}, {3091, 15741}, {3146, 15431}, {3525, 15748}

X(16254) = complement of X(14528)
X(16254) = complementary conjugate of X(631)


X(16255) =  X(13)X(690)∩X(14)X(5917)

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

X(16255) lies on the cubic K061a and these lines:
{13,690}, {14,5917}, {99,299}, {6780,11586}

X(16255) = {X(5917),X(9140)}-harmonic conjugate of X(14)
X(16255) = barycentric quotient X(i)/X(j) for these {i,j}: {395, 531}, {2379, 6151}, {8015, 11549}


X(16256) =  X(13)X(5916)∩X(14)X(690)

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

X(16256) lies on the cubic K061b and these lines:
{13,5916}, {14,690}, {99,298}, {6779,11600}

X(16256) = {X(5915),X(9140)}-harmonic conjugate of X(13)
X(16256) = barycentric quotient X(i)/X(j) for these {i,j}: {396, 530}, {2378, 2981}, {8014, 11537}


X(16257) =  ISOGONAL CONJUGATE OF X(5617)

Barycentrics    (SB+SC)*(2*S^3+sqrt(3)*(S^2* SB+SC*SA*SW))*(2*S^3+sqrt(3)*( SC*S^2+SW*SA*SB)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27257.

X(16257) lies on these lines:
{232, 10633}, {325, 5980}, {463, 6530}, {511, 11127}, {8836, 14356}

X(16257) = isogonal conjugate of X(5617)


X(16258) =  ISOGONAL CONJUGATE OF X(5613)

Barycentrics    (SB+SC)*(-2*S^3+sqrt(3)*(S^2* SB+SC*SA*SW))*(-2*S^3+sqrt(3)* (SC*S^2+SW*SA*SB)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27257.

X(16258) lies on these lines:
{232, 10632}, {325, 5981}, {462, 6530}, {511, 11126}, {8838, 14356}

X(16258) = isogonal conjugate of X(5613)


X(16259) =  CENTER OF 1st FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB + SC)[2 SA^3 (SB + SC) + 11 SA^2 SB SC + 7 SA SB SC (SB + SC) + 7 SB^2 SC^2 + Sqrt[3] S (SA^3 + SA^2 (SB + SC) + 5 SA SB SC + 2 SB SC (SB + SC))] : :

The 1st and 2nd Fermat-Dao equilateral triangles are introduced at X(16247). Centers X(16259)-X(16262) were contributed by Randy Hutson, February 28, 2018..

X(16259) lies on these lines:
{3,3440}, {4,69}, {14,5640}, {15,10546}, {623,7703}, {3642,7998}

X(16259) = reflection of X(16260) in X(6787)
X(16259) = X(616)-of-orthocentroidal-triangle
X(16259) = {X(11188),X(16261)}-harmonic conjugate of X(16260)


X(16260) =  CENTER OF 2nd FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB + SC)[2 SA^3 (SB + SC) + 11 SA^2 SB SC + 7 SA SB SC (SB + SC) + 7 SB^2 SC^2 - Sqrt[3] S (SA^3 + SA^2 (SB + SC) + 5 SA SB SC + 2 SB SC (SB + SC))] : :

The 1st and 2nd Fermat-Dao equilateral triangles are introduced at X(16247). Centers X(16259)-X(16262) were contributed by Randy Hutson, February 28, 2018..

X(16260) lies on these lines:
{3,3441}, {4,69}, {13,5640}, {16,10546}, {624,7703}, {3643,7998}

X(16260) = reflection of X(16259) in X(6787)
X(16260) = X(617)-of-orthocentroidal-triangle
X(16260) = {X(11188),X(16261)}-harmonic conjugate of X(16259)


X(16261) =  X(376) OF ORTHOCENTROIDAL TRIANGLE

Barycentrics    (SB + SC) [2 SA^3 (SB + SC) + 11 SA^2 SB SC + 7 SA SB SC (SB + SC) + 7 SB^2 SC^2] : :
X(16261) = 4 X[546) - X(568) = 2 X(373) - 3 X(3545) = X(2979) + 2 X(3830) = 5 X(3567) - 14 X(3832) = X(3060) - 4 X(3845) = 2 X(185) - 11 X(3855) = 17 X(3854) - 8 X(5462) = 7 X(4) + 2 X(5562) = 10 X(3843) - X(5889) = 4 X(381) - X(5890) = X(3543) + 2 X(5891) = 5 X(5562) - 14 X(5907) = 5 X(4) + 4 X(5907) = 5 X(5890) - 8 X(5946) = 5 X(5640) - 4 X(5946) = 5 X(381) - 2 X(5946) = 10 X(3091) - X(6241) = 2 X(3146) + 7 X(7999) = X(6241) - 4 X(9730) = 5 X(3091) - 2 X(9730) = 16 X(546) - 7 X(9781) = 4 X(568) - 7 X(9781) = X(20) - 4 X(10170) = 14 X(3851) - 5 X(10574) = 11 X(5056) - 2 X(10575) = 4 X(3819) - X(11001) = 3 X(3839) - X(11002) = 7 X(3526) - 16 X(11017) = 4 X(3818) - X(11188) = 7 X(3090) + 2 X(11381) = 16 X(5562) - 7 X(11412) = 8 X(4) + X(11412) = 4 X(5) + 5 X(11439)

The 1st and 2nd Fermat-Dao equilateral triangles are introduced at X(16247). Centers X(16259)-X(16262) were contributed by Randy Hutson, February 28, 2018..

X(16261) lies on these lines:
{2,11455}, {3,5888}, {4,69}, {5,7703}, {6,14094}, {20,10170}, {23,4550}, {30,7998}, {74,1995}, {113,5169}, {146,7533}, {182,12112}, {185,3855}, {373,3545}, {376,5650}, {378,15035}, {381,5640}, {382,15056}, {399,11422}, {512,10033}, {546,568}, {631,13474}, {1154,14269}, {1597,6090}, {1656,12279}, {2979,3830}, {3060,3845}, {3090,11381}, {3091,6241}, {3146,7999}, {3167,13482}, {3426,11284}, {3524,15082}, {3526,11017}, {3543,5891}, {3567,3832}, {3627,11444}, {3819,11001}, {3839,11002}, {3843,5889}, {3850,15043}, {3851,10574}, {3854,5462}, {3917,15682}, {4549,7519}, {5050,11456}, {5056,10575}, {5066,11451}, {5068,11465}, {5072,13491}, {5073,14128}, {5076,11591}, {5093,11443}, {5094,12133}, {5651,7464}, {6644,15055}, {6800,9818}, {7395,8718}, {7496,8717}, {7506,15062}, {7509,15811}, {7527,11464}, {9027,11180}, {9306,13596}, {9775,13860}, {10303,14641}, {10545,15054}, {11440,13861}, {11454,12106}, {11541,13348}, {13352,15052}, {13570,14831}

X(16261) = midpoint of X(5640) and X(15305)
X(16261) = reflection of X(i) in X(j) for these {i,j}: {376, 5650}, {5640, 381}, {5890, 5640}, {15045, 3545}
> X(16261) = X(376)-of-orthocentroidal-triangle
X(16261) = inverse-in-orthocentroidal-circle of X(10706)
X(16261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 16194, 11455), (4, 15030, 11459), (4, 15058, 11412), (5, 11439, 12290), (113, 5169, 7699), (146, 7533, 7706), (381, 15305, 5890), (546, 12111, 9781), (1995, 11472, 74), (3091, 6241, 15024), (3830, 15060, 2979), (3832, 12162, 3567), (5072, 13491, 15028), (11459, 15030, 15058), (16259, 16260, 11188)


X(16262) =  CENTER OF INVERSE SIMILITUDE OF 1st AND 2nd FERMAT-DAO EQUILATERAL TRIANGLES

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

The 1st and 2nd Fermat-Dao equilateral triangles are introduced at X(16247). Centers X(16259)-X(16262) were contributed by Randy Hutson, February 28, 2018..

X(16262) lies on these lines: {4,69}, {3005,5888}

X(16262) = insimilicenter of circumcircles of 1st and 2nd Fermat-Dao equilateral triangles


X(16263) =  X(4)X(1495)∩X(30)X(264)

Barycentrics    SB^2*SC^2*(S^2+3*SA*SB)*(S^2+3 *SA*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27265.

X(16263) lies on these lines:
{3, 14860}, {4, 1495}, {30, 264}, {382, 1105}, {393, 7737}, {847, 6240}, {1093, 3575}, {1217, 3146}, {1300, 1629}, {2052, 6344}, {6526, 7487}, {8884, 12173}, {14249, 15424}

X(16263) = trilinear pole of the line X(2501)X(14398)


X(16264) =  X(4)X(6)∩X(30)X(264)

Barycentrics    SB^2*SC^2 *(S^2+3*SW*SA) : :
X(16264) = 3 X(4) - 2 X(6748)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27265.

X(16264) lies on these lines:
{4, 6}, {30, 264}, {107, 10301}, {160, 378}, {297, 3818}, {427, 1629}, {428, 2052}, {1595, 8884}, {3575, 9873}, {5064, 11547}, {6000, 6752}, {6524, 7408}


X(16265) =  X(4)X(1495)∩X(30)X(14767)

Barycentrics    SB*SC*(S^2+3*SB*SC)*(6*S^2-12* R^2*SA+3*SA^2-5*SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27265.

X(16265) lies on these lines:
{4, 1495}, {30, 14767}, {6000, 6752}


X(16266) =  X(3)X(543)∩X(6)X(140)

Barycentrics    a^2 (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2+8 a^4 b^2 c^2-2 a^2 b^4 c^2-2 b^6 c^2+6 a^4 c^4-2 a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-2 b^2 c^6+c^8) : :
X(16266) = (J^2 - 7) X(3) - 2 (J^2 - 4) X(54)
X(16266) = 3 X(155) - X(1498) = 2 X(156) - 3 X(3167) = 3 X(3167) - X(7387) = 4 X(9820) - 3 X(10201) = 3 X(26) - 4 X(10282) = 3 X(1147) - 2 X(10282) = 5 X(8567) - 6 X(11250) = 2 X(3357) - 3 X(12084) = 5 X(8567) - 3 X(12163) = 3 X(12085) - X(13093) = 3 X(12164) + X(13093) = X(3357) - 3 X(13346) = 6 X(156) - 5 X(14530) = 9 X(3167) - 5 X(14530) = 3 X(7387) - 5 X(14530) = 4 X(10224) - 3 X(14852) = 3 X(5654) - 2 X(15761)

X(16266) lies on the cubic K956 and these lines:
{3,54}, {4,323}, {5,394}, {6,140}, {22,49}, {24,6243}, {25,10263}, {26,206}, {30,155}, {52,1092}, {66,3564}, {68,13371}, {110,7517}, {143,1351}, {156,3167}, {161,9935}, {182,5447}, {184,10625}, {186,2904}, {193,3546}, {382,11441}, {399,5073}, {524,8548}, {550,1181}, {567,7509}, {569,3917}, {576,5462}, {578,1216}, {631,1994}, {632,10601}, {1069,8144}, {1173,11451}, {1199,3523}, {1204,10564}, {1350,7525}, {1368,13292}, {1511,3515}, {1593,5876}, {1598,11387}, {1614,12083}, {1656,15066}, {1657,11456}, {2393,9925}, {2937,9703}, {3060,7506}, {3193,6923}, {3292,7530}, {3357,9938}, {3517,13421}, {3522,15032}, {3526,5422}, {3527,13364}, {3533,15018}, {3548,6515}, {3580,6640}, {3819,13154}, {5020,10095}, {5070,15038}, {5093,13363}, {5097,11695}, {5446,9306}, {5562,7526}, {5654,15761}, {5663,12085}, {5891,11424}, {5944,9715}, {6000,15083}, {6090,7529}, {6146,14791}, {6193,14790}, {6800,9704}, {7393,11426}, {7395,15067}, {7485,13353}, {7512,9545}, {7999,13434}, {8567,11250}, {8909,11265}, {9544,12088}, {9586,9626}, {9591,9621}, {9777,15026}, {9818,11591}, {9820,10201}, {9936,14216}, {9937,14984}, {10224,14852}, {10323,13340}, {11432,12006}, {11444,15033}, {11465,15019}, {11477,12106}, {11479,14128}, {11999,16013}, {12325,13420}, {13336,13366}, {15037,15720}, {15047,15694}

X(16266) = midpoint of X(i) and X(j) for these {i,j}: {6193, 14790}, {9936, 14216}, {12085, 12164}
X(16266) = reflection of X(i) in X(j) for these {i,j}: {26, 1147}, {68, 13371}, {7387, 156}, {12084, 13346}, {12163, 11250}
X(16266) = X(15318)-Ceva conjugate of X(3)
X(16266) = X(12699)-of-tangential-triangle
X(16266) = X(10263)-of-Ara-triangle; see X(5594)
X(16266) = tangential-isogonal conjugate of X(10626)
X(16266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 195, 7592), (3, 1993, 12161), (3, 12160, 6102), (52, 1092, 6644), (54, 2979, 3), (195, 7592, 12161), (569, 3917, 7516), (578, 1216, 7514), (1351, 6642, 143), (1993, 7592, 195), (2937, 9703, 9707), (3167, 7387, 156), (3523, 11004, 1199), (3526, 14627, 5422), (5446, 9306, 13861), (5562, 13352, 7526), (9704, 13564, 6800)


X(16267) =  CENTER OF 3rd FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    a^4-5 a^2 b^2+4 b^4-5 a^2 c^2-8 b^2 c^2+4 c^4-6 Sqrt[3] a^2 S : :
X(16267) = 2 X(13) + X(15) = X(15) - 4 X(396) = X(13) + 2 X(396) = 2 X(623) + X(3180) = 5 X(13) - 2 X(5318) = 5 X(396) + X(5318) = 5 X(15) + 4 X(5318) = X(298) - 4 X(6669) = X(616) - 4 X(6671) = 2 X(6109) + X(6778) = X(14) + 2 X(6783) = X(6770) + 2 X(7684) = X(5318) - 10 X(11542) = X(13) - 4 X(11542) = X(396) + 2 X(11542) = X(15) + 8 X(11542) = X(9901) + 2 X(11707) = X(13103) + 2 X(13350) = 4 X(6771) - X(14538)

Let F = X(13), the 1st Fermat point. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle. Let A' be the centroid of FAbAc, and define B' and C' cyclically. Let A'' be the centroid of AAbAc, and define B'' and C'' cyclically. The triangle A'B'C' is here named the 3rd Fermat-Dao equilateral triangle. (The 4th is defined at X(16268); the triangle A''B''C'' is not generally equilateral.) The triangles A'B'C' and A''B''C'' are perspective, and the perspector is X(16267), which is X(13)-of-A'B'C' and X(13)-of-A''B''C''. Moreover, the triangles ABC and A'B'C' are perspective, with perspector X(17); and ABC and A''B''C'' are perspective, with perspector X(2). (Thanh Oai Dao, March 2, 2018)

Barycentrics, combos, and harmonics conjugacies found by Peter Moses, March 3, 2018.

If you have GeoGebra, you can view X(16267).

You can view 3rd and 4th Fermat-Dao equilateral triangles.

X(16267) lies on these lines:
{2,17}, {4,3412}, {6,5055}, {13,15}, {14,3545}, {16,5054}, {18,547}, {61,381}, {202,3582}, {203,11237}, {230,9112}, {298,6669}, {303,7799}, {376,5352}, {395,15699}, {397,549}, {398,5066}, {531,5470}, {533,5459}, {538,3107}, {546,12817}, {616,6671}, {623,3180}, {1154,11624}, {3389,13846}, {3390,13847}, {3411,3628}, {3524,10646}, {3534,5238}, {3543,12816}, {3584,7006}, {3839,10654}, {5335,10304}, {5344,15683}, {5351,15693}, {5366,15640}, {5859,7751}, {6109,6778}, {6770,7684}, {6771,14538}, {7005,11238}, {9901,11707}, {10188,16239}, {10645,15688}, {11480,15689}, {11481,15707}, {11485,14269}, {11539,16242}, {13103,13350}

X(16267) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5055,16268), (13, 396, 15), (396, 11542, 13), (10653, 11488, 16241), (10653, 16241, 10646)


X(16268) =  CENTER OF 4th FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    a^4-5 a^2 b^2+4 b^4-5 a^2 c^2-8 b^2 c^2+4 c^4+6 Sqrt[3] a^2 S : :
X(16268) = 2 X(14) + X(16) = X(16) - 4 X(395) = X(14) + 2 X(395) = 2 X(624) + X(3181) = 5 X(14) - 2 X(5321) = 5 X(395) + X(5321) = 5 X(16) + 4 X(5321) = X(299) - 4 X(6670) = X(617) - 4 X(6672) = 2 X(6108) + X(6777) = X(13) + 2 X(6782) = X(6773) + 2 X(7685) = X(5321) - 10 X(11543) = X(14) - 4 X(11543) = X(395) + 2 X(11543) = X(16) + 8 X(11543) = X(9900) + 2 X(11708) = X(13102) + 2 X(13349) = 4 X(6774) - X(14539)

Let F = X(14), the 2nd Fermat point. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle. Let A' be the centroid of FAbAc, and define B' and C' cyclically. Let A'' be the centroid of AAbAc, and define B'' and C'' cyclically. The triangle A'B'C' is here named the 4th Fermat-Dao equilateral triangle. (The 3rd is defined at X(16267); the triangle A''B''C'' is not generally equilateral.) The triangles A'B'C' and A''B''C'' are perspective, and the perspector is X(16268), which is X(14)-of-A'B'C' and X(14)-of-A''B''C''. Moreover, the triangles ABC and A'B'C' are perspective, with perspector X(18); and ABC and A''B''C'' are perspective, with perspector X(2). (Thanh Oai Dao, March 2, 2018)

Barycentrics, combos, and harmonics conjugacies found by Peter Moses, March 3, 2018.

X(16268) lies on these lines:
{2,18}, {4,3411}, {6,5055}, {13,3545}, {14,16}, {15,5054}, {17,547}, {62,381}, {202,11237}, {203,3582}, {230,9113}, {299,6670}, {302,7799}, {376,5351}, {396,15699}, {397,5066}, {398,549}, {530,5469}, {532,5460}, {538,3106}, {546,12816}, {617,6672}, {624,3181}, {1154,11626}, {3364,13846}, {3365,13847}, {3412,3628}, {3524,10645}, {3534,5237}, {3543,12817}, {3584,7005}, {3839,10653}, {5334,10304}, {5343,15683}, {5352,15693}, {5365,15640}, {5858,7751}, {6108,6777}, {6773,7685}, {6774,14539}, {7006,11238}, {9900,11708}, {10187,16239}, {10646,15688}, {11480,15707}, {11481,15689}, {11486,14269}, {11539,16241}, {13102,13349}

X(16268) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5055,16267), (14, 395, 16), (395, 11543, 14), (10654, 11489, 16242), (10654, 16242, 10645)


X(16269) =  (Name pending)

Barycentrics    a^2 (a^16 b^4-7 a^14 b^6+21 a^12 b^8-35 a^10 b^10+35 a^8 b^12-21 a^6 b^14+7 a^4 b^16-a^2 b^18-2 a^16 b^2 c^2+6 a^14 b^4 c^2-5 a^12 b^6 c^2+2 a^10 b^8 c^2-5 a^8 b^10 c^2+2 a^6 b^12 c^2+9 a^4 b^14 c^2-10 a^2 b^16 c^2+3 b^18 c^2+a^16 c^4+6 a^14 b^2 c^4-18 a^12 b^4 c^4+15 a^10 b^6 c^4-7 a^8 b^8 c^4+16 a^6 b^10 c^4-40 a^4 b^12 c^4+43 a^2 b^14 c^4-16 b^16 c^4-7 a^14 c^6-5 a^12 b^2 c^6+15 a^10 b^4 c^6-6 a^8 b^6 c^6-5 a^6 b^8 c^6+31 a^4 b^10 c^6-59 a^2 b^12 c^6+36 b^14 c^6+21 a^12 c^8+2 a^10 b^2 c^8-7 a^8 b^4 c^8-5 a^6 b^6 c^8-14 a^4 b^8 c^8+27 a^2 b^10 c^8-48 b^12 c^8-35 a^10 c^10-5 a^8 b^2 c^10+16 a^6 b^4 c^10+31 a^4 b^6 c^10+27 a^2 b^8 c^10+50 b^10 c^10+35 a^8 c^12+2 a^6 b^2 c^12-40 a^4 b^4 c^12-59 a^2 b^6 c^12-48 b^8 c^12-21 a^6 c^14+9 a^4 b^2 c^14+43 a^2 b^4 c^14+36 b^6 c^14+7 a^4 c^16-10 a^2 b^2 c^16-16 b^4 c^16-a^2 c^18+3 b^2 c^18) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27277.

X(16269) lies on this line: {4,52}


X(16270) =  MIDPOINT OF X(74) AND X(1112)

Barycentrics    a^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+2 a^8 b^2 c^2-a^6 b^4 c^2-13 a^4 b^6 c^2+16 a^2 b^8 c^2-5 b^10 c^2-3 a^8 c^4-a^6 b^2 c^4+22 a^4 b^4 c^4-13 a^2 b^6 c^4+11 b^8 c^4+2 a^6 c^6-13 a^4 b^2 c^6-13 a^2 b^4 c^6-14 b^6 c^6+2 a^4 c^8+16 a^2 b^2 c^8+11 b^4 c^8-3 a^2 c^10-5 b^2 c^10+c^12) : :
X(16270) = 3 X[125] + X[185], X[185] - 3 X[974], 3 X[1112] - 5 X[3567], 3 X[74] + 5 X[3567], X[1216] - 3 X[6699], 3 X[9140] + 5 X[10574], 3 X[51] + X[10990], 2 X[10110] - 3 X[11746], X[1216] + 3 X[11806], 3 X[9826] - 4 X[12006], X[10263] + 3 X[12041], X[4] - 3 X[12099], X[10263] - 3 X[12236], 3 X[12133] - X[12290], 3 X[5890] - X[13148], 2 X[1216] - 3 X[13416], 2 X[11806] + X[13416], 3 X[7687] - X[13474], X[12290] - 9 X[14644], X[12133] - 3 X[14644], 3 X[3060] + 5 X[15021], 3 X[10706] - 11 X[15024], 3 X[12824] - 7 X[15043], X[12279] + 7 X[15044], X[14094] - 9 X[15045], 3 X[12824] + X[15054], 7 X[15043] + X[15054], 3 X[12825] - 7 X[15056], X[5889] + 7 X[15057], 7 X[15056] - 15 X[15059], X[12825] - 5 X[15059], X[12358] - 3 X[15061], 9 X[5622] - X[15073], X[12292] - 5 X[15081], 2 X[11017] - 3 X[15088], 2 X[10110] + 3 X[15151], 11 X[15025] - 3 X[15305], 3 X[12099] - 2 X[15465], 3 X[125] - X[15738], 3 X[974] + X[15738], 3 X[9730] + X[16003], 3 X[51] - X[16105], X[10620] + 3 X[16222]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27282.

X(16270) lies on these lines:
{3,895}, {4,10293}, {5,113}, {51,10990}, {74,1112}, {143, 1204}, {265,15740}, {389,2781}, {541,5462}, {542,9729}, {1154, 15122}, {1181,5609}, {1216,6699} ,{1511,13198}, {1986,3541}, {2777,10110}, {2790,14894}, {3060 ,15021}, {3448,6815}, {3548,1235 8}, {5094,5890}, {5159,13754}, { 5621,9786}, {5889,15057}, {5946, 10605}, {7399,12827}, {7687, 13474}, {9140,10574}, {10263, 12041}, {10620,16222}, {10706, 15024}, {10816,10821}, {10984, 16165}, {11284,14094}, {12133, 12290}, {12279,15044}, {12292, 15081}, {12824,15043}, {12825, 15056}, {15025,15305}

X(16270) = midpoint of X(i) in X(j) for these {i,j}: {74,1112}, {125,974}, {1 85,15738}, {6699,11806}, {10264, 14708}, {10990,16105}, {11746, 15151}, {12041,12236}
X(16270) = reflection of X(i) in X(j) for these {i,j}: {{4,15465}, {13416,6699}
X(16270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12099, 15465), (51, 10990, 16105), (125, 185, 15738), (974, 15738, 185), (6699, 15115, 16196), (15043, 15054, 12824)


X(16271) =  CENTER OF THE MORLEY CONIC

Trilinears    sin(B/3 - C/3)*(a*sin(B/3 - C/3) - b*sin(C/3 - A/3) - c*sin(A/3 - B/3) : :
Trilinears    sin(B/3 - C/3)*(sin A)*sin(B/3 - C/3) - (sin B)*sin(C/3 - A/3) - (sin C)*sin(A/3 - B/3) : :

The 1st Morley Triangle, 2nd Morley Triangle, and 3rd Morley Triangle are perspective to ABC, with perspectors X(357), X(1136), and X(1134), respectively.

Theorem 1. The points A, B, C, X(357), X(1136), X(1134) lie on a conic. (Dao Thanh Oai, March 1, 2018)

The conic is here named the Morley conic.

Theorem 2. Let P be a point on the Morley conic. There exists a family of equilateral triangles UVW, all homothetic to the 1st Morley equilateral triangle, with U on PA, V on PB, and W on PC. (Dao Thanh Oai, March 1, 2018)

A trilinear equation for the Morley conic is

sin(B/3 - C/3) y z + sin(C/3 - A/3) z x + sin(A/3 - B/3)x y = 0.

The conic passes through X(i) for i = 14, 357, 1134, 1136, 3602, 3603, 3604, 13593, and has perspector X(5637). The isogonal conjugate of the conic is the tripolar of X(14146), which passes through the perspectors of ABC and the 1st Morley adjunct triangle, the 2nd Morley adjunct triangle, and the 3rd Morley adjunct triangle. (César Lozada, March 7, 2018.)

X(16271) lies on no line X(i)X(j) for 1 <= i < j <= 16272.

X(16271) = X(2)-Ceva conjugate of X(5637)


X(16272) =  X(1)X(30)∩X(230,231)

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

X(16272) lies on these lines:
{1,30}, {186,196}, {225,10151}, {230,231}, {403,1068}, {1360,2078}, {2074,8885}, {3580,6742} X(16272) = midpoint of X(i) and X(j) for these {i,j}: {{1, 11809}, {3580, 6742}
X(16272) = incircle-inverse of X(1836)
X(16272) = crossdifference of every pair of points on line {3, 9404}
X(16272) = orthogonal projection of X(1) on orthic axis


X(16273) =  EULER LINE INTERCEPT OF X(5922)X(14216)

Barycentrics    3*S^4+(256*R^4-80*R^2*SW-5*SB* SC+4*SW^2)*S^2-4*(16*R^2+SW)*( 4*R^2-SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27284.

X(16273) lies on these lines: {2, 3}, {5922, 14216}


X(16274) =  (name pending)

Barycentrics    (6125*R^4-2800*R^2*SW+8*SB*SC+ 320*SW^2)*S^2-5*(2875*R^4- 1320*R^2*SW+152*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27284.

X(16274) lies on this line: {2, 3}


X(16275) =  X(2)X(187)∩X(22)X(7802)

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

X(16275) lies on these lines:
{2, 187}, {22, 7802}, {39, 8878}, {69, 11550}, {76, 7391}, {183, 5064}, {305, 315}, {325, 7667}, {427, 1799}, {1078, 5133}, {1180, 7847}, {1184, 7841}, {1194, 6655}, {1196, 7842}, {1369, 5189}, {1627, 7828}, {2896, 8891}, {3785, 7378}, {3917, 5207}, {5359, 7790}, {7386, 11059}, {7484, 7773}, {7485, 7752}, {7769, 15246}, {7784, 11324}, {7860, 16063}, {15437, 16043}

X(16275) = cevapoint of X(1369) and X(1370)
X(16275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (315, 1370, 305), (427, 1799, 11056), (427, 7750, 1799), (1369, 5189, 8024), (1369, 8024, 7768)


X(16276) =  X(2)X(99)∩X(22)X(76)

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

X(16276) lies on these lines:
on lines {2, 99}, {22, 76}, {23, 8024}, {25, 305}, {51, 12215}, {83, 1180}, {251, 7760}, {315, 7500}, {325, 428}, {350, 5322}, {384, 1194}, {698, 1915}, {1003, 1184}, {1078, 6636}, {1196, 7816}, {1909, 5310}, {3266, 13595}, {3760, 5345}, {3761, 7298}, {3926, 6995}, {3972, 5359}, {3981, 4048}, {4121, 5207}, {4563, 9306}, {5012, 10330}, {5020, 11059}, {6337, 7392}, {6676, 11056}, {6997, 7763}, {7394, 7752}, {7485, 7782}, {7519, 7796}, {7747, 8878}

X(16276) = cevapoint of X(22) and X(8267)
X(16276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22, 76, 1799), (25, 1975, 305), (251, 8267, 7760)


X(16277) =  X(2)X(66)∩X(4)X(251)

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

X(16277) lies on the Kiepert hyperbola, the cubic K959, and on these lines:
{2, 66}, {4, 251}, {22, 76}, {83, 5133}, {96, 1513}, {427, 10547}, {1289, 9076}, {1916, 5986}, {2996, 7500}, {5403, 8880}, {5404, 8881}, {7378, 10548}, {7494, 10130}, {7495, 10159}

X(16277) = isogonal conjugate of X(3313)
X(16277) = X(i)-cross conjugate of X(j) for these (i,j): {25, 251}, {3767, 308}
X(16277) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3313}, {22, 38}, {39, 1760}, {141, 2172}, {206, 1930}, {315, 1964}, {1401, 4123}, {3688, 7210}, {4611, 8061}
X(16277) = X(i)-vertex conjugate of X(j) for these (i,j): {2, 1799}, {96, 3425}
X(16277) = cevapoint of X(i) and X(j) for these (i,j): {25, 13854}, {66, 2353}
X(16277) = barycentric product X(i)*X(j) for these {i,j}: {66, 83}, {308, 2353}, {1289, 4580}, {1799, 13854}, {2156, 3112}
X(16277) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3313}, {66, 141}, {82, 1760}, {83, 315}, {251, 22}, {827, 4611}, {2156, 38}, {2353, 39}, {10547, 10316}, {13854, 427}, {14376, 3933}


X(16278) =  X(4)X(542)∩X(6)X(14832)

Barycentrics    (b - c)^2*(b + c)^2*(3*a^6 - 4*a^4*b^2 + b^6 - 4*a^4*c^2 + 7*a^2*b^2*c^2 - 2*b^4*c^2 - 2*b^2*c^4 + c^6) : :
X(16278) = 3 X[9166] - X[11006], X[10990] - 4 X[11623], 4 X[6723] - 5 X[14061], 2 X[7687] - 3 X[14639], X[11005] - 3 X[14639], X[13188] - 3 X[14643], X[74] - 3 X[14651], X[10620] - 3 X[14849], 5 X[1656] - 3 X[14850], X[13172] - 3 X[15035], 3 X[6034] - 2 X[15118], 3 X[9144] - X[15342], 3 X[671] + X[15342], 3 X[125] - 2 X[15357], 3 X[115] - X[15357], 3 X[125] - 4 X[15359], 3 X[115] - 2 X[15359], 4 X[12900] - 3 X[15561]

X(16278) lies on the cubic K875 and these lines:
{4, 542}, {6, 14832}, {30, 11656}, {74, 14651}, {98, 2777}, {99, 5972}, {110, 148}, {113, 2782}, {115, 125}, {523, 2682}, {525, 14120}, {541, 11632}, {543, 1316}, {1656, 14850}, {2394, 9180}, {2482, 15000}, {2780, 6784}, {2794, 13202}, {3906, 5099}, {5181, 5969}, {5489, 14443}, {6034, 15118}, {6055, 7422}, {6723, 14061}, {7687, 11005}, {7728, 12188}, {9033, 13179}, {9166, 11006}, {9830, 15303}, {9862, 10721}, {10620, 14849}, {10990, 11623}, {12042, 16111}, {12064, 13187}, {12900, 15561}, {13172, 15035}, {13188, 14643}, {15535, 16003}

X(16278) = midpoint of X(i) and X(j) for these {i,j}: {110, 148}, {671, 9144}, {1992, 14833}, {7728, 12188}, {9862, 10721}, {10706, 12243}
X(16278) = reflection of X(i) in X(j) for these {i,j}: {99, 5972}, {125, 115}, {5642, 5465}, {11005, 7687}, {13187, 12064}, {15357, 15359}, {16003, 15535}, {16111, 12042}
X(16278) = polar-circle-inverse of X(648)
X(16278) = X(i)-Hirst inverse of X(j) for these (i,j): {125, 1648}, {1637, 8371}
X(16278) = X(148)-line conjugate of X(110)
X(16278) = barycentric product X(1648)*X(16093)
X(16278) = barycentric quotient X(1648)/X(16103)
X(16278) = orthogonal projection of X(671) on line X(115)X(125)
X(16278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 15357, 15359), (11005, 14639, 7687), (15357, 15359, 125)


X(16279) =  X(2)X(11657)∩X(6)X(30)

Barycentrics    a^12 + a^10*b^2 - a^8*b^4 - 11*a^6*b^6 + 14*a^4*b^8 - 2*a^2*b^10 - 2*b^12 + a^10*c^2 - a^8*b^2*c^2 + 9*a^6*b^4*c^2 - 18*a^4*b^6*c^2 + 2*a^2*b^8*c^2 + 7*b^10*c^2 - a^8*c^4 + 9*a^6*b^2*c^4 + 12*a^4*b^4*c^4 - 10*b^8*c^4 - 11*a^6*c^6 - 18*a^4*b^2*c^6 + 10*b^6*c^6 + 14*a^4*c^8 + 2*a^2*b^2*c^8 - 10*b^4*c^8 - 2*a^2*c^10 + 7*b^2*c^10 - 2*c^12 : :

X(16279) lies on these lines:
{2, 11657}, {6, 30}, {381, 523}, {542, 2452}, {1316, 5476}, {1551, 11163}, {3830, 15358}, {3845, 15356}, {6055, 6103}, {7426, 9753}, {9193, 16188}, {9214, 15928}, {9880, 14832}, {13860, 16092}

X(16279) = reflection of X(1316) in X(5476)


X(16280) =  X(4)X(541)∩X(115)X(125)

Barycentrics    (b - c)^2*(b + c)^2*(-3*a^10 + 8*a^8*b^2 - 7*a^6*b^4 + 3*a^4*b^6 - 2*a^2*b^8 + b^10 + 8*a^8*c^2 - 13*a^6*b^2*c^2 + 6*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 2*b^8*c^2 - 7*a^6*c^4 + 6*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 3*b^6*c^4 + 3*a^4*c^6 - 3*a^2*b^2*c^6 - 3*b^4*c^6 - 2*a^2*c^8 + 2*b^2*c^8 + c^10) : :

X(16280) lies on the cubic K876 and these lines:
{2, 11656}, {4, 541}, {115, 125}, {512, 6070}, {525, 3154}, {542, 1316}, {879, 15453}, {1499, 12079}, {3258, 3906}, {5642, 14981}

X(16280) = X(381)-Hirst inverse of X(523)


X(16281) =  (name pending)

Barycentrics    a^10 - 6*a^8*b^2 - 7*a^6*b^4 + 8*a^4*b^6 + 6*a^2*b^8 - 2*b^10 - 6*a^8*c^2 + 47*a^6*b^2*c^2 - 25*a^4*b^4*c^2 - 31*a^2*b^6*c^2 + 9*b^8*c^2 - 7*a^6*c^4 - 25*a^4*b^2*c^4 + 62*a^2*b^4*c^4 - 7*b^6*c^4 + 8*a^4*c^6 - 31*a^2*b^2*c^6 - 7*b^4*c^6 + 6*a^2*c^8 + 9*b^2*c^8 - 2*c^10 : :

X(16281) lies on this line: {2, 3}


X(16282) =  (name pending)

Barycentrics    (b - c)^2*(b + c)^2*(a^12 - 8*a^10*b^2 + 2*a^8*b^4 + 10*a^6*b^6 - 4*a^4*b^8 - 2*a^2*b^10 + b^12 - 8*a^10*c^2 + 62*a^8*b^2*c^2 - 72*a^6*b^4*c^2 + 12*a^4*b^6*c^2 + 8*a^2*b^8*c^2 - 2*b^10*c^2 + 2*a^8*c^4 - 72*a^6*b^2*c^4 + 105*a^4*b^4*c^4 - 36*a^2*b^6*c^4 + 2*b^8*c^4 + 10*a^6*c^6 + 12*a^4*b^2*c^6 - 36*a^2*b^4*c^6 + 10*b^6*c^6 - 4*a^4*c^8 + 8*a^2*b^2*c^8 + 2*b^4*c^8 - 2*a^2*c^10 - 2*b^2*c^10 + c^12) : :


X(16283) =  CROSSSUM OF X(7) AND X(4000)

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

X(16283) lies on these lines:
{2, 1252}, {9, 983}, {31, 6602}, {32, 220}, {55, 2195}, {213, 1802}, {219, 5019}, {902, 8012}, {1212, 2241}, {2242, 6603}, {2256, 5065}, {4421, 6181}, {9310, 9316}, {14589, 15813}

X(16283) = isogonal of the isotomic conjugate of X(4513)
X(16283) = X(i)-isoconjugate of X(j) for these (i,j): {7, 9311}, {85, 9309}, {6063, 9315}
X(16283) = crosspoint of X(55) and X(7123)
X(16283) = crosssum of X(7) and X(4000)
X(16283) = barycentric product X(i)*X(j) for these {i,j}: {6, 4513}, {9, 9310}, {41, 3729}, {55, 1376}, {200, 9316}, {220, 6180}, {1253, 9312}, {2194, 3967}, {3939, 4449}, {4014, 6065}
X(16283) = barycentric quotient X(i)/X(j) for these {i,j}: {41, 9311}, {1376, 6063}, {2175, 9309}, {4513, 76}, {9310, 85}, {9316, 1088}, {9447, 9315}
X(16283) = X(55),X(5452)}-harmonic conjugate of X(14936)


X(16284) =  ISOTOMIC CONJUGATE OF X(3062)

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

X(16284) lies on these lines:
{2, 4875}, {7, 8}, {76, 4301}, {78, 664}, {190, 728}, {200, 1088}, {264, 5342}, {304, 341}, {312, 10405}, {313, 14615}, {325, 9711}, {348, 7080}, {349, 4101}, {480, 14189}, {519, 3673}, {1111, 3632}, {1330, 6259}, {1376, 7176}, {1447, 12513}, {1698, 7278}, {1930, 4737}, {2481, 3680}, {3061, 3452}, {3177, 3693}, {3263, 10513}, {3436, 4872}, {3633, 7264}, {3879, 6738}, {4352, 4646}, {4384, 5437}, {4513, 10025}, {5088, 5687}, {5806, 10449}, {6736, 9436}, {7179, 12607}

X(16284) = reflection of X(9311) in X(3061)
X(16284) = isotomic conjugate of X(3062)
X(16284) = X(7257)-beth conjugate of X(341)
X(16284) = X(14493)-anticomplementary conjugate of X(5905)
X(16284) = X(312)-Ceva conjugate of X(75)
X(16284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 85, 75), (69, 322, 75), (304, 668, 341)
X(16284) = X(i)-isoconjugate of X(j) for these (i,j): {6, 11051}, {31, 3062}, {32, 10405}
X(16284) = barycentric product X(i)*X(j) for these {i,j}: {75, 144}, {76, 165}, {312, 3160}, {341, 9533}, {561, 3207}, {668, 7658}, {1419, 3596}
X(16284) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 11051}, {2, 3062}, {75, 10405}, {144, 1}, {165, 6}, {1419, 56}, {3160, 57}, {3207, 31}, {7658, 513}, {9533, 269}, {13609, 2310}


X(16285) =  ISOGONAL CONJUGATE OF X(1239)

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

X(17285) lies on these lines:
{2, 6}, {32, 206}, {110, 9101}, {160, 3117}, {1176, 1627}, {1186, 1691}, {1194, 3313}, {1196, 9969}, {1843, 14580}, {3499, 4048}, {5007, 6375}, {7787, 10339}, {9427, 13210}

X(16285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1613, 141)
X(16285) = isogonal conjugate of X(1239)
X(16285) = X(7954)-Ceva conjugate of X(669)
X(16285) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1239}, {661, 6572}
X(16285) = barycentric product X(i)*X(j) for these {i,j}: {6, 1180}, {32, 3096}
X(16285) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1239}, {110, 6572}, {1180, 76}, {3096, 1502}




leftri  Collineation images on the Euler line: X(16286) - X(16302)  rightri

A regular collineation m is defined by its action on four points, P1, P2, P3, P4, no three of which are collinear. Given four such points, let Pk = m(Pk-4) for k = 5,6,7,8. The eight points Pk determine m; conversely, given eight such points, a collineation m is uniquely determined. For every point X, the point m(X) is here called the (P1, P2, P3, P4; P5, P6, P7, P8) collineation image of X. (Regular collineations are discussed in Clark Kimberling, Collineations, Conjugacies and Cubics).

Collineations map lines to lines. Thus, for example the collineation m indicated by (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) maps the Nagel line, X(1)X(2), onto the Euler line, X(2)X(3). Examples of triangle centers m(X) found in this way are X(16286)-X(16302). For more collineation images on the Euler line, see X(16342)-X(16355), X(16367)-X(16384), and X(16393)-X(16458). See also the preamble just before X(16544).

underbar




X(16286) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(8)

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

X(16286) lies on these lines:
{2, 3}, {55, 3216}, {386, 1191}, {500, 3819}, {3646, 10434}, {9708, 10479}

X(16286) = {X(2),X(3)}-harmonic conjugate of X(16414)


X(16287) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(10)

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

X(16287) lies on these lines:
{2, 3}, {35, 238}, {55, 386}, {228, 5044}, {255, 2267}, {500, 3917}, {572, 1437}, {956, 10449}, {958, 10479}, {975, 2352}, {1402, 6051}, {1724, 4267}, {1780, 5135}, {3746, 15621}, {5687, 9534}, {10267, 15623}, {15622, 15931}

X(16287) = {X(2),X(3)}-harmonic conjugate of X(16453)


X(16288) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(42)

Barycentrics    a*(-(a^5*b) - a^4*b^2 + a^3*b^3 + a^2*b^4 - a^5*c - 2*a^4*b*c + 2*a^3*b^2*c + 6*a^2*b^3*c + 3*a*b^4*c - a^4*c^2 + 2*a^3*b*c^2 + 12*a^2*b^2*c^2 + 11*a*b^3*c^2 + 2*b^4*c^2 + a^3*c^3 + 6*a^2*b*c^3 + 11*a*b^2*c^3 + 4*b^3*c^3 + a^2*c^4 + 3*a*b*c^4 + 2*b^2*c^4) : :

X(16288) lies on these lines:
2, 3}, {386, 1001}, {3216, 5259}, {3295, 3996}, {5132, 5248}, {9708, 10449}


X(16289) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(43)

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

X(16289) lies on these lines:
{2, 3}, {55, 9534}, {238, 386}, {958, 10449}, {986, 12567}, {1724, 4281}, {5251, 10479}


X(16290) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(78)

Barycentrics    a*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 + a^5*c + 3*a^4*b*c - 4*a^2*b^3*c - a*b^4*c + b^5*c + a^4*c^2 - 6*a^2*b^2*c^2 - 5*a*b^3*c^2 - a^3*c^3 - 4*a^2*b*c^3 - 5*a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 - a*b*c^4 + b*c^5) : : {1, 209}, {2, 3}, {10, 15624}, {35, 2218}, {37, 579}, {55, 1714}, {386, 1104}, {387, 3295}, {1255, 15934}, {3216, 3601}, {3695, 10449}, {5248, 6679}, {5259, 5285}, {5755, 10441}, {5791, 10479}, {12572, 15669}

X(16290) lies on these lines:
{1, 209}, {2, 3}, {10, 15624}, {35, 2218}, {37, 579}, {55, 1714}, {386, 1104}, {387, 3295}, {1255, 15934}, {3216, 3601}, {3695, 10449}, {5248, 6679}, {5259, 5285}, {5755, 10441}, {5791, 10479}, {12572, 15669}

X(16290) = {X(2),X(3)}-harmonic conjugate of X(16415)


X(16291) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(145)

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

X(16291) lies on these lines:
{2, 3}, {386, 1616}, {3216, 3295}, {5248, 15625}


X(16292) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(239)

Barycentrics    -(a^2*(a^5*b^2 + a^4*b^3 - a^3*b^4 - a^2*b^5 + 2*a^5*b*c + 4*a^4*b^2*c - 4*a^2*b^4*c - 2*a*b^5*c + a^5*c^2 + 4*a^4*b*c^2 - 10*a^2*b^3*c^2 - 10*a*b^4*c^2 - 3*b^5*c^2 + a^4*c^3 - 10*a^2*b^2*c^3 - 16*a*b^3*c^3 - 7*b^4*c^3 - a^3*c^4 - 4*a^2*b*c^4 - 10*a*b^2*c^4 - 7*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5)) : :

X(16292) lies on these lines: on lines {2, 3}, {386, 1500}


X(16293) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(387)

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

X(16293) lies on these lines:
{2, 3}, {55, 936}, {78, 392}, {200, 7160}, {283, 10601}, {938, 956}, {958, 1210}, {965, 4254}, {993, 9843}, {1001, 13411}, {1125, 1617}, {1259, 3305}, {1260, 5044}, {1445, 3916}, {1728, 15823}, {3624, 7742}, {3646, 10902}, {3683, 11509}, {4512, 10310}, {5248, 6700}, {5250, 10306}, {5251, 8071}, {5259, 8069}, {6734, 9708}

X(16293) = {X(2),X(3)}-harmonic conjugate of X(16410)


X(16294) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(498)

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

X(16294) lies on this line: {2, 3}

X(16294) = {X(2),X(3)}-harmonic conjugate of X(16295)


X(16295) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(499)

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

X(16295) lies on these lines: {2, 3}, {386, 8071}, {579, 15817}, {3216, 14793}

X(16295) = {X(2),X(3)}-harmonic conjugate of X(16294)


X(16296) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(519)

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

X(16296) lies on these lines:
{2, 3}, {386, 3303}, {500, 5650}, {3216, 3746}, {5132, 15485}

X(16296) = {X(2),X(3)}-harmonic conjugate of X(16297)


>

X(16297) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(551)

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

X(16297) lies on these lines:
{2, 3}, {106, 386}, {373, 500}, {582, 5651}, {1222, 9534}, {3216, 5563}, {3746, 5132}

X(16297) = {X(2),X(3)}-harmonic conjugate of X(16296)


X(16298) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(612)

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

X(16298) lies on these lines: {2, 3}, {386, 1386}

X(16298) = {X(2),X(3)}-harmonic conjugate of X(16299)


X(16299) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(614)

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

X(16299) lies on these lines: {2, 3}, {386, 518}, {988, 3216}, {3836, 5248}, {4682, 5266}

X(16299) = {X(2),X(3)}-harmonic conjugate of X(16298)


X(16300) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(899)

Barycentrics    a*(-3*a^5*b - 3*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4 - 3*a^5*c - 6*a^4*b*c + 2*a^3*b^2*c + 10*a^2*b^3*c + 5*a*b^4*c - 3*a^4*c^2 + 2*a^3*b*c^2 + 16*a^2*b^2*c^2 + 13*a*b^3*c^2 + 2*b^4*c^2 + 3*a^3*c^3 + 10*a^2*b*c^3 + 13*a*b^2*c^3 + 4*b^3*c^3 + 3*a^2*c^4 + 5*a*b*c^4 + 2*b^2*c^4) : :

X(16300) lies on these lines: {2, 3}, {386, 4749}, {595, 4428}


X(16301) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(976)

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

X(16301) lies on this line: {2, 3}


X(16302) =  (X(1),X(2),X(3),X(6); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(995)

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

X(16302) lies on these lines: {2, 3}, {56, 10408}, {386, 956}, {958, 3216}, {1001, 5264}, {5132, 5687}


X(16303) =  ORTHOGONAL PROJECTION OF X(6) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 - 7*a^4*b^4 + 3*a^2*b^6 + b^8 + a^6*c^2 + 10*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 4*b^6*c^2 - 7*a^4*c^4 - 3*a^2*b^2*c^4 + 6*b^4*c^4 + 3*a^2*c^6 - 4*b^2*c^6 + c^8 : :
X(16303) = 3 X[468] - X[16312], 3 X[468] - 2 X[16321], X[16312] - 6 X[16324], X[16321] - 3 X[16324], 3 X[468] + X[16326], 2 X[16321] + X[16326], 6 X[16324] + X[16326], X[16326] - 6 X[16333], X[468] + 2 X[16333], X[16321] + 3 X[16333], X[16312] + 6 X[16333], 2 X[16312] - 3 X[16334], 4 X[16321] - 3 X[16334], 4 X[16324] - X[16334], 4 X[16333] + X[16334], 2 X[16326] + 3 X[16334]

X(16303) lies on these lines:
{6, 30}, {23, 5304}, {53, 10151}, {186, 1249}, {187, 3163}, {216, 10257}, {230, 231}, {393, 403}, {566, 15122}, {858, 7736}, {1503, 1555}, {2070, 8573}, {2452, 5112}, {3284, 6781}, {5158, 5475}, {5189, 14930}, {5523, 8749}, {6748, 13473}, {7426, 7735}, {7747, 15860}, {8553, 15646}, {8557, 11809}

X(16303) = midpoint of X(i) and X(j) for these {i,j}: {2452, 5112}, {16312, 16326}, {16324, 16333}
reflection of X(i) in X(j) for these {i,j}: {468, 16324}, {16312, 16321}, {16334, 468}
X(16303) = crosssum of X(6) and X(14915)
X(16303) = crossdifference of every pair of points on line {3, 8675}
X(16303) = reflection of X(16334) in Euler line
X(16303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (230, 1990, 3018), (230, 2493, 16317), (230, 3018, 16310), (3003, 3018, 230), (468, 16312, 16321), (468, 16326, 16312), (468, 16327, 16331), (1990, 3003, 16310), (3003, 3018, 230), (16306, 16308, 468), (16312, 16321, 16334), (16313, 16314, 468), (16315, 16329, 468)


X(16304) =  ORTHOGONAL PROJECTION OF X(8) ON ORTHIC AXIS

Barycentrics    2*a^6 - a^4*b^2 + 4*a^3*b^3 - 2*a^2*b^4 - 4*a*b^5 + b^6 - 4*a^3*b^2*c + 4*b^5*c - a^4*c^2 - 4*a^3*b*c^2 + 4*a^2*b^2*c^2 + 4*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 + 4*a*b^2*c^3 - 8*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 4*a*c^5 + 4*b*c^5 + c^6 : :
X(16304) = X[468] - 2 X[16272], 3 X[468] - 4 X[16305], X[16272] - 3 X[16309], 2 X[16305] - 3 X[16309], 3 X[468] - X[16322], 4 X[16305] - X[16322], 6 X[16309] - X[16322], 5 X[16322] - 18 X[16323], 5 X[16272] - 9 X[16323], 10 X[16305] - 9 X[16323], 5 X[468] - 6 X[16323], 5 X[16309] - 3 X[16323], 5 X[16322] - 12 X[16332], 5 X[16272] - 6 X[16332], 5 X[468] - 4 X[16332], 5 X[16305] - 3 X[16332], 5 X[16309] - 2 X[16332], 3 X[16323] - 2 X[16332]

X(16304) lies on these lines:
{8, 30}, {230, 231}

X(16304) = reflection of X(i) in X(j) for these {i,j}: {468, 16309}, {16272, 16305}, {16322, 16272}
X(16304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16322, 16272), (16272, 16305, 468), (16272, 16309, 16305), (16323, 16332, 468)
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X(16305) =  ORTHOGONAL PROJECTION OF X(10) ON ORTHIC AXIS

Barycentrics    2*a^6 - a^4*b^2 + a^3*b^3 - 2*a^2*b^4 - a*b^5 + b^6 - a^3*b^2*c + b^5*c - a^4*c^2 - a^3*b*c^2 + 4*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + a^3*c^3 + a*b^2*c^3 - 2*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - a*c^5 + b*c^5 + c^6 : :
X(16305) = 3 X[468] - X[16272], 3 X[468] + X[16304], X[16304] - 3 X[16309], X[16272] + 3 X[16309], 9 X[468] - X[16322], 3 X[16272] - X[16322], 3 X[16304] + X[16322], 9 X[16309] + X[16322], X[16272] - 9 X[16323], X[468] - 3 X[16323], X[16309] + 3 X[16323], X[16304] + 9 X[16323], 2 X[16322] - 9 X[16332], 2 X[16272] - 3 X[16332], 6 X[16323] - X[16332], 2 X[16309] + X[16332], 2 X[16304] + 3 X[16332]

X(16305) lies on these lines:
{10, 30}, {230, 231}, {976, 10149}, {1495, 6741}, {1842, 10151}, {3580, 14985}

X(16305) = midpoint of X(i) and X(j) for these {i,j}: {468, 16309}, {1495, 6741}, {3580, 14985}, {16272, 16304}
X(16305) = reflection of X(16332) in X(468)
X(16305) = reflection of X(16332) in Euler line
X(16305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16304, 16272), (16272, 16309, 16304), (16309, 16323, 468)


X(16306) =  ORTHOGONAL PROJECTION OF X(32) ON ORTHIC AXIS

Barycentrics    2*a^10 - a^8*b^2 - 2*a^6*b^4 + b^10 - a^8*c^2 + 8*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 2*a^2*b^6*c^2 - 3*b^8*c^2 - 2*a^6*c^4 - 2*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^2*b^2*c^6 + 2*b^4*c^6 - 3*b^2*c^8 + c^10 : :
X(16306) = 3 X[468] - X[16331], 3 X[468] - 2 X[16335]

X(16306) lies on these lines:
{6, 858}, {23, 7735}, {30, 32}, {39, 15122}, {230, 231}, {403, 2207}, {1692, 16188}, {2071, 7738}, {2072, 2548}, {3053, 10295}, {3767, 11799}, {3815, 5158}, {5189, 5304}, {5286, 7464}, {7745, 10297}, {15820, 15860}

X(16306) = reflection of X(16331) in X(16335)
X(16306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (230, 1990, 2493), (230, 16308, 468), (468, 16303, 16308), (468, 16313, 16321), (468, 16327, 16316), (468, 16331, 16335), (3003, 6103, 230)


X(16307) =  ORTHOGONAL PROJECTION OF X(37) ON ORTHIC AXIS

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

X(16307) lies on these lines:
{30, 37}, {45, 7286}, {186, 2178}, {230, 231}, {1781, 3247}, {1841, 10151}


X(16308) =  ORTHOGONAL PROJECTION OF X(39) ON ORTHIC AXIS

Barycentrics    a^2*(2*a^6*b^2 - 2*a^4*b^4 - 2*a^2*b^6 + 2*b^8 + 2*a^6*c^2 + 2*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 + 2*b^4*c^4 - 2*a^2*c^6 - 3*b^2*c^6 + 2*c^8) : :
X(16308) = 3 X[468] - X[16313], 3 X[468] + X[16327], X[16313] - 9 X[16329], X[468] - 3 X[16329], X[16327] + 9 X[16329]

X(16308) lies on these lines:
{6, 23}, {30, 39}, {32, 7575}, {186, 3053}, {216, 3055}, {230, 231}, {566, 858}, {842, 1691}, {2071, 15815}, {2275, 7286}, {2276, 5160}, {2548, 7574}, {3054, 15355}, {5007, 12105}, {5013, 7464}, {5189, 7736}, {5206, 15646}, {5254, 11799}, {5306, 7426}

X(16308) = midpoint of X(16313) and X(16327)
X(16308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (232, 3003, 2493), (468, 16303, 16306), (468, 16306, 230), (468, 16316, 16335), (468, 16327, 16313), (468, 16331, 16321), (2493, 3003, 230), (16314, 16316, 468)
X(16308) = Moses-circle-inverse of X(14537)
X(16308) = complement of the isotomic of X(14388)
X(16308) = X(14388)-complementary conjugate of X(2887)
X(16308) = crosspoint of X(i) and X(j) for these (i,j): {2, 14388}
X(16308) = crossdifference of every pair of points on line {3, 3906} X(16308) = crosssum of X(6) and X(11645)


X(16309) =  ORTHOGONAL PROJECTION OF X(40) ON ORTHIC AXIS

Barycentrics    2*a^6 - a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 - 2*a*b^5 + b^6 - 2*a^3*b^2*c + 2*b^5*c - a^4*c^2 - 2*a^3*b*c^2 + 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6 : :
X(16309) = X[16272] + 2 X[16304], X[16272] - 4 X[16305], X[16304] + 2 X[16305], 5 X[16272] - 2 X[16322], 5 X[468] - X[16322], 10 X[16305] - X[16322], 5 X[16304] + X[16322], 2 X[16322] - 15 X[16323], 2 X[468] - 3 X[16323], X[16272] - 3 X[16323], 4 X[16305] - 3 X[16323], 2 X[16304] + 3 X[16323], 3 X[16322] - 10 X[16332], 3 X[16272] - 4 X[16332], 9 X[16323] - 4 X[16332], 3 X[468] - 2 X[16332], 3 X[16305] - X[16332], 3 X[16304] + 2 X[16332]

X(16309) lies on these lines:
{30, 40}, {230, 231}, {1503, 6741}, {3564, 14985}

X(16309) = midpoint of X(468) and X(16304)
X(16309) = reflection of X(i) in X(j) for these {i,j}: {468, 16305}, {16272, 468}
X(16309) = reflection of X(16272) in Euler
X(16309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16305, 16323), (16272, 16323, 468), (16304, 16305, 16272)


X(16310) =  ORTHOGONAL PROJECTION OF X(50) ON ORTHIC AXIS

Barycentrics    2*a^8 - 3*a^6*b^2 + a^4*b^4 - a^2*b^6 + b^8 - 3*a^6*c^2 + 2*a^4*b^2*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8 : :
Barycentrics    (cot B)(1 + cos 2C + cos 2A) + (cot C)(1 + cos 2A + cos 2B) : :
X(16310) = X[50] + 3 X[1989]

X(16310) is the intersection, other than X(523), of the orthic axis and the polar conic of X(30) with respect to the Neuberg cubic. This conic is a rectangular hyperbola passing through X(1), X(5), X(30), X(523) and the excenters. It has center X(476). (Randy Hutson, March 14, 2018)

X(16310) lies on the cubic K491 and these lines:
{1, 7110}, {2, 14836}, {5, 6}, {24, 254}, {30, 50}, {53, 571}, {111, 16167}, {115, 3284}, {140, 566}, {184, 15508}, {216, 7542}, {230, 231}, {338, 441}, {460, 1576}, {546, 9220}, {577, 7748}, {1249, 7505}, {1658, 8553}, {1781, 8557}, {1879, 6748}, {1995, 7735}, {2003, 7363}, {2079, 7575}, {2323, 6506}, {2450, 9512}, {3087, 7547}, {5063, 5254}, {5158, 7746}, {7503, 7738}, {7506, 8573}, {8818, 13408}, {9605, 14787}, {11079, 12079}, {15341, 15738} X(16310) = midpoint of X(2450) and X(9512)
X(16310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 2165, 9722), (230, 1990, 3003), (230, 2493, 468), (230, 3018, 16303), (231, 3003, 230), (231, 3018, 3003), (485, 486, 14852), (1990, 3003, 16303), (3003, 3018, 1990), (3291, 6103, 230)
X(16310) = complement of the isotomic of X(1300)
X(16310) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 131}, {1300, 2887}
X(16310) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 131}, {3580, 30}
X(16310) = X(63)-isoconjugate of X(1299)
X(16310) = crosspoint of X(i) and X(j) for these (i,j): {2, 1300}, {94, 847}
X(16310) = crossdifference of every pair of points on line {3, 924}
X(16310) = crosssum of X(i) and X(j) for these (i,j): {6, 13754}, {50, 1147}, {520, 2088}, {2245, 3157}
X(16310) = barycentric product X(i)X(j) for these {i,j}: {92, 2314}, {131, 1300}, {847, 12095}
X(16310) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1299}, {2314, 63}, {12095, 9723}
X(16310) = perspector of ABC and orthocevian triangle of X(1300)
X(16310) = perspector of hyperbola {{A,B,C,X(4),X(925)}} (the circumconic centered at X(131))


X(16311) =  ORTHOGONAL PROJECTION OF X(51) ON ORTHIC AXIS

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

X(16311) lies on these lines:
{23, 6795}, {30, 51}, {186, 3168}, {230, 231}, {262, 858}, {403, 2052}

X(16311) = inverse-in-2nd-Lozada-circle of X(51)


X(16312) =  ORTHOGONAL PROJECTION OF X(69) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 + 5*a^4*b^4 - 9*a^2*b^6 + b^8 + a^6*c^2 - 14*a^4*b^2*c^2 + 9*a^2*b^4*c^2 + 8*b^6*c^2 + 5*a^4*c^4 + 9*a^2*b^2*c^4 - 18*b^4*c^4 - 9*a^2*c^6 + 8*b^2*c^6 + c^8 : :
X(16312) = 3 X[468] - 2 X[16303], 3 X[468] - 4 X[16321], 5 X[16303] - 6 X[16324], 5 X[468] - 4 X[16324], 5 X[16321] - 3 X[16324], 12 X[16324] - 5 X[16326], 3 X[468] - X[16326], 4 X[16321] - X[16326], 7 X[16326] - 12 X[16333], 7 X[16303] - 6 X[16333], 7 X[16324] - 5 X[16333], 7 X[468] - 4 X[16333], 7 X[16321] - 3 X[16333], 2 X[16333] - 7 X[16334], X[16326] - 6 X[16334], 2 X[16324] - 5 X[16334], X[16303] - 3 X[16334], 2 X[16321] - 3 X[16334]

X(16312) lies on these lines:
{30, 69}, {230, 231}

X(16312) = reflection of X(i) in X(j) for these {i,j}: {468, 16334}, {16303, 16321}, {16326, 16303}
X(16312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16326, 16303), (16303, 16321, 468), (16303, 16334, 16321), (16313, 16316, 16325), (16313, 16331, 468), (16316, 16325, 468), (16325, 16331, 16316)


X(16313) =  ORTHOGONAL PROJECTION OF X(76) ON ORTHIC AXIS

Barycentrics    2*a^8*b^2 + a^6*b^4 - 2*a^4*b^6 - a^2*b^8 + 2*a^8*c^2 - 4*a^6*b^2*c^2 + a^4*b^4*c^2 + 3*b^8*c^2 + a^6*c^4 + a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 3*b^6*c^4 - 2*a^4*c^6 - 3*b^4*c^6 - a^2*c^8 + 3*b^2*c^8 : :
X(16313) = 3 X[468] - 2 X[16308], 3 X[468] - X[16327], 8 X[16308] - 9 X[16329], 4 X[16327] - 9 X[16329], 4 X[468] - 3 X[16329]

X(16313) lies on these lines:
{30, 76}, {230, 231}, {858, 3314}, {3313, 3631}, {10151, 11397} X(16313) = reflection of X(16327) in X(16308)
X(16313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (230, 16335, 468), (468, 16303, 16314), (468, 16312, 16331), (468, 16327, 16308), (468, 16331, 16316), (16306, 16321, 468), (16312, 16325, 16316), (16325, 16331, 468)


X(16314) =  ORTHOGONAL PROJECTION OF X(83) ON ORTHIC AXIS

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

X(16314) lies on these lines:
{30, 83}, {230, 231}

X(16314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16303, 16313), (468, 16308, 16316), (468, 16327, 16321)


X(16315) =  ORTHOGONAL PROJECTION OF X(98) ON ORTHIC AXIS

Barycentrics    2*a^10 - 3*a^8*b^2 - a^6*b^4 + 2*a^4*b^6 - a^2*b^8 + b^10 - 3*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 - 4*b^8*c^2 - a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 3*b^6*c^4 + 2*a^4*c^6 + 4*a^2*b^2*c^6 + 3*b^4*c^6 - a^2*c^8 - 4*b^2*c^8 + c^10 : :
X(16315) = X[7426] - 3 X[8859], X[691] + 3 X[14568], 4 X[230] - X[16316], 3 X[16316] - 4 X[16320], 3 X[468] - 2 X[16320], 3 X[230] - X[16320]

X(16315) lies on these lines:
{2, 2452}, {30, 98}, {125, 524}, {132, 10151}, {183, 11007}, {230, 231}, {325, 5159}, {385, 858}, {895, 14834}, {1316, 7735}, {3564, 11005}, {5305, 14700}, {5912, 14120}, {7426, 8859}, {9214, 14694}

X(16315) = midpoint of X(385) and X(858)
X(16315) = reflection of X(i) in X(j) for these {i,j}: {325, 5159}, {468, 230}, {16316, 468}
X(16315) = reflection of X(16316) in Euler line
X(16315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16303, 16329)
X(16315) = Dao-Moses-Telv circle inverse of X(10418)
X(16315) = X(523)-Hirst inverse of X(10418)


X(16316) =  ORTHOGONAL PROJECTION OF X(99) ON ORTHIC AXIS

Barycentrics    2*a^10 - 3*a^8*b^2 + 3*a^6*b^4 + 2*a^4*b^6 - 5*a^2*b^8 + b^10 - 3*a^8*c^2 - 3*a^4*b^4*c^2 + 8*a^2*b^6*c^2 + 3*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 - b^6*c^4 + 2*a^4*c^6 + 8*a^2*b^2*c^6 - b^4*c^6 - 5*a^2*c^8 + c^10 : :
X(16316) = 2 X[230] - 3 X[468], X[385] - 3 X[7426], 3 X[23] + X[7779], 3 X[858] - 5 X[7925], 4 X[230] - 3 X[16315], X[16315] - 4 X[16320], X[230] - 3 X[16320]

X(16316) lies on these lines:
{23, 7779}, {30, 99}, {126, 3258}, {230, 231}, {385, 7426}, {524, 1495}, {858, 7925}, {1304, 2374}, {1316, 7736}, {2452, 5304}, {2770, 16103}, {5139, 10151}, {7868, 11007}

X(16316) = reflection of X(i) in X(j) for these {i,j}: {468, 16320}, {16315, 468}
X(16316) = reflection of X(16315) in Euler line
X(16316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16308, 16314), (468, 16312, 16325), (468, 16327, 16306), (468, 16331, 16313), (16308, 16335, 468), (16312, 16325, 16313), (16325, 16331, 16312)
X(16316) = Moses Radical circle inverser of X(3291)
X(16316) = crossdifference of every pair of points on line {3, 6041}


X(16317) =  ORTHOGONAL PROJECTION OF X(111) ON ORTHIC AXIS

Barycentrics    2*a^6 - 5*a^4*b^2 - 6*a^2*b^4 + b^6 - 5*a^4*c^2 + 20*a^2*b^2*c^2 - b^4*c^2 - 6*a^2*c^4 - b^2*c^4 + c^6 : :
X(16317) = 3 X[2] + X[9870]

X(16317) lies on these lines:
{2, 2418}, {5, 9745}, {30, 111}, {230, 231}, {427, 5203}, {524, 5914}, {538, 6719}, {1196, 5355}, {1560, 10151}, {3564, 6792}, {3815, 8585}, {5066, 6032}, {6531, 15144}, {7426, 11580}, {7736, 11284}, {7767, 16055}

X(16317) = midpoint of X(111) and X(5913)
X(16317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (230, 2493, 16303), (230, 10418, 468), (3291, 10418, 230)
X(16317) = complement of the isotomic of X(9084)
X(16317) = X(9084)-complementary conjugate of X(2887)
X(16317) = crosspoint of X(2) and X(9084)
X(16317) = crossdifference of every pair of points on line {3, 8644}
X(16317) = crosssum of X(6) and X(9027)


X(16318) =  ORTHOGONAL PROJECTION OF X(112) ON ORTHIC AXIS

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^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(16318) = X[6793] - X[8779]

X(16318) lies on the cubic K471 and these lines:
{2, 253}, {4, 3172}, {5, 8743}, {6, 66}, {19, 3772}, {20, 8778}, {25, 393}, {30, 112}, {32, 3575}, {53, 428}, {98, 6530}, {115, 10151}, {132, 1503}, {216, 7499}, {230, 231}, {235, 2138}, {251, 10549}, {264, 7792}, {297, 385}, {317, 14614}, {325, 648}, {378, 15048}, {403, 8744}, {441, 9475}, {444, 3209}, {460, 6531}, {577, 7667}, {607, 5230}, {612, 2331}, {614, 7129}, {858, 13573}, {1196, 14091}, {1235, 7819}, {1297, 15312}, {1370, 15905}, {1560, 3163}, {1562, 15311}, {1593, 5286}, {1834, 2332}, {1885, 1968}, {1907, 5319}, {2386, 13166}, {3087, 5064}, {3199, 7755}, {3516, 7738}, {3535, 8974}, {3536, 13950}, {3541, 9605}, {5094, 7736}, {5359, 15809}, {5702, 14930}, {6000, 15341}, {6677, 15355}, {6756, 10312}, {7710, 15258}, {8746, 9722}, {8749, 8791}, {9756, 15274}, {9993, 16264}, {12143, 13357}

X(16318) = midpoint of X(112) and X(5523)
X(16318) = reflection of X(1529) in X(3 X(16318) = X(i)-daleth conjugate of X(j) for these (i,j): {393, 25}, {6330, 16318}, {6531, 460}
X(16318) = X(i)-Ceva conjugate of X(j) for these (i,j): {98, 25}, {6330, 4}, {6530, 460}
X(16318) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1297}, {163, 2419}, {255, 6330}, {394, 8767}, {662, 2435}, {1959, 15407}
X(16318) = X(i)-Hirst inverse of X(j) for these (i,j): {132, 1503}, {393, 6525}
X(16318) = X(i)-vertex conjugate of X(j) for these (i,j): {25, 6587}, {6587, 25}
X(16318) = crosspoint of X(i) and X(j) for these (i,j): {4, 6330}, {393, 6531}
X(16318) = crossdifference of every pair of points on line {3, 2435}
X(16318) = crosssum of X(3) and X(8779)
X(16318) = PU(4)-harmonic conjugate of X(6587)
X(16318) = X(63)-isoconjugate of X(1297)
X(16318) = barycentric product X(i)*X(j) for these {i,j}: {4, 1503}, {92, 2312}, {98, 132}, {158, 8766}, {393, 441}, {523, 2409}, {850, 2445}, {1529, 3424}, {2052, 8779}, {6525, 16096}, {6531, 15595}, {6793, 16080}, {9475, 16081}
X(16318) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1297}, {132, 325}, {393, 6330}, {441, 3926}, {512, 2435}, {523, 2419}, {1096, 8767}, {1503, 69}, {1976, 15407}, {2312, 63}, {2409, 99}, {2445, 110}, {6525, 14944}, {6531, 9476}, {6793, 11064}, {8766, 326}, {8779, 394}, {15595, 6393}
X(16318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (53, 5306, 10311), (53, 10311, 428), (230, 232, 468), (230, 1990, 232), (232, 6103, 230), (393, 7735, 25), (1968, 5254, 1885), (1990, 6103, 468), (2207, 3767, 235), (3162, 13854, 427)


X(16319) =  ORTHOGONAL PROJECTION OF X(113) ON ORTHIC AXIS

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

X(16319) lies on the cubic k884 and these lines:
{2, 6795}, {30, 113}, {186, 5667}, {230, 231}, {403, 1300}, {842, 1302}, {1316, 2549}, {1503, 3154}, {3580, 14611}, {5651, 9996}, {11799, 14934}, {13367, 14895}

X(16319) = midpoint of X(i) and X(j) for these {i,j}: {1495, 3258}, {3580, 14611}, {11799, 14934}
X(16319) = reflection of X(11657) in X(468)
X(16319) = reflection of X(11657) in Euler
X(16319) = {X(1304),X(14165)}-harmonic conjugate of X(403)
X(16319) = Moses-radical-circle-inverse of X(3003)
X(16319) = X(30)-daleth conjugate of X(13202)
X(16319) = X(30)-Hirst inverse of X(113)
X(16319) = crossdifference of every pair of points on line {3, 2433}


X(16320) =  ORTHOGONAL PROJECTION OF X(114) ON ORTHIC AXIS

Barycentrics    2*a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 4*a^2*b^8 + b^10 - 3*a^8*c^2 + 2*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*a^4*c^6 + 7*a^2*b^2*c^6 - 4*a^2*c^8 - b^2*c^8 + c^10 : :
X(16320) = X[5189] - 5 X[7925], 3 X[10256] - 2 X[15122], 3 X[230] - 2 X[16315], 3 X[468] - X[16315], X[230] + 2 X[16316], X[16315] + 3 X[16316]

X(16320) lies on these lines:
{2, 2453}, {23, 325}, {30, 114}, {110, 524}, {141, 9832}, {186, 13200}, {230, 231}, {403, 935}, {842, 1513}, {1316, 3815}, {1503, 11005}, {2452, 5306}, {5189, 7925}, {10011, 16188}, {10256, 15122}

X(16320) = midpoint of X(i) and X(j) for these {i,j}: {23, 325}, {468, 16316}, {842, 1513}
X(16320) = reflection of X(i) in X(j) for these {i,j}: {230, 468}, {16188, 10011}
X(16320) = reflection of X(230) in Euler line
X(16320) = crossdifference of every pair of points on line {3, 8574}


X(16321) =  ORTHOGONAL PROJECTION OF X(141) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 - a^4*b^4 - 3*a^2*b^6 + b^8 + a^6*c^2 - 2*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + 2*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 + 2*b^2*c^6 + c^8 : :
X(16321) = 3 X[468] - X[16303], 3 X[468] + X[16312], 2 X[16303] - 3 X[16324], 2 X[16312] + 3 X[16324], 9 X[16324] - 2 X[16326], 9 X[468] - X[16326], 3 X[16303] - X[16326], 3 X[16312] + X[16326], 4 X[16326] - 9 X[16333], 4 X[16303] - 3 X[16333], 4 X[468] - X[16333], 4 X[16312] + 3 X[16333], X[16312] - 3 X[16334], X[16324] + 2 X[16334], X[16303] + 3 X[16334], X[16333] + 4 X[16334], X[16326] + 9 X[16334]

X(16321) lies on these lines:
{30, 141}, {157, 186}, {183, 7426}, {230, 231}, {858, 7868}, {2453, 5112}

X(16321) = midpoint of X(i) and X(j) for these {i,j}: {468, 16334}, {2453, 5112}, {16303, 16312}
X(16321) = reflection of X(i) in X(j) for these {i,j}: {16324, 468}, {16333, 16324}
X(16321) = reflection of X(16324) in Euler
X(16321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16312, 16303), (468, 16313, 16306), (468, 16325, 230), (468, 16327, 16314), (468, 16331, 16308), (16303, 16334, 16312)


X(16322) =  ORTHOGONAL PROJECTION OF X(145) ON ORTHIC AXIS

Barycentrics    2*a^6 - a^4*b^2 - 8*a^3*b^3 - 2*a^2*b^4 + 8*a*b^5 + b^6 + 8*a^3*b^2*c - 8*b^5*c - a^4*c^2 + 8*a^3*b*c^2 + 4*a^2*b^2*c^2 - 8*a*b^3*c^2 - b^4*c^2 - 8*a^3*c^3 - 8*a*b^2*c^3 + 16*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 + 8*a*c^5 - 8*b*c^5 + c^6 : :
X(16322) = 3 X[468] - 4 X[16272], 3 X[468] - 2 X[16304], 9 X[468] - 8 X[16305], 3 X[16304] - 4 X[16305], 3 X[16272] - 2 X[16305], 10 X[16305] - 9 X[16309], 5 X[16304] - 6 X[16309], 5 X[468] - 4 X[16309], 5 X[16272] - 3 X[16309], 13 X[16304] - 18 X[16323], 13 X[16309] - 15 X[16323], 13 X[468] - 12 X[16323], 13 X[16272] - 9 X[16323], 7 X[16304] - 12 X[16332], 7 X[16309] - 10 X[16332], 7 X[16305] - 9 X[16332], 7 X[468] - 8 X[16332], 7 X[16272] - 6 X[16332]

X(16322) lies on these lines:
{30, 145}, {230, 231}

X(16322) = reflection of X(16304) in X(16272)
X(16322) = {X(16272),X(16304)}-harmonic conjugate of X(468)


X(16323) =  ORTHOGONAL PROJECTION OF X(165) ON ORTHIC AXIS

Barycentrics    6*a^6 - 3*a^4*b^2 + 2*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + 3*b^6 - 2*a^3*b^2*c + 2*b^5*c - 3*a^4*c^2 - 2*a^3*b*c^2 + 12*a^2*b^2*c^2 + 2*a*b^3*c^2 - 3*b^4*c^2 + 2*a^3*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - 6*a^2*c^4 - 3*b^2*c^4 - 2*a*c^5 + 2*b*c^5 + 3*c^6 : :
X(16323) = X[6741] + 2 X[15448], 4 X[468] - X[16272], 5 X[468] + X[16304], 5 X[16272] + 4 X[16304], X[16304] - 10 X[16305], X[468] + 2 X[16305], X[16272] + 8 X[16305], 2 X[16304] - 5 X[16309], 4 X[16305] - X[16309], 2 X[468] + X[16309], X[16272] + 2 X[16309], 13 X[16272] - 4 X[16322], 13 X[468] - X[16322], 13 X[16309] + 2 X[16322], 13 X[16304] + 5 X[16322], 5 X[16272] - 8 X[16332], 5 X[468] - 2 X[16332], 5 X[16305] + X[16332], X[16304] + 2 X[16332], 5 X[16309] + 4 X[16332]

X(16323) lies on these lines:
{30, 165}, {230, 231}, {6741, 15448}

X(16323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16304, 16332), (468, 16305, 16309), (468, 16309, 16272)


X(16324) =  ORTHOGONAL PROJECTION OF X(182) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 - 5*a^4*b^4 + a^2*b^6 + b^8 + a^6*c^2 + 6*a^4*b^2*c^2 - a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - 2*b^2*c^6 + c^8 : :
X(16324) = 5 X[468] - X[16312], 5 X[16303] + X[16312], 2 X[16312] - 5 X[16321], 2 X[16303] + X[16321], 7 X[16303] - X[16326], 7 X[468] + X[16326], 7 X[16321] + 2 X[16326], 7 X[16312] + 5 X[16326], 2 X[16326] - 7 X[16333], 2 X[468] + X[16333], 2 X[16312] + 5 X[16333], 3 X[16312] - 5 X[16334], 3 X[16321] - 2 X[16334], 3 X[468] - X[16334], 3 X[16303] + X[16334], 3 X[16333] + 2 X[16334], 3 X[16326] + 7 X[16334]

X(16324) lies on these lines:
{6, 5112}, {30, 182}, {230, 231}, {858, 11174}, {3589, 11594}, {5159, 15491}, {6795, 11799}

X(16324) = midpoint of X(i) and X(j) for these {i,j}: {6, 5112}, {468, 16303}, {6795, 11799}, {16321, 16333}
X(16324) = reflection of X(i) in X(j) for these {i,j}: {16321, 468}, {16333, 16303}
X(16324) = reflection of X(16321) in Euler line


X(16325) =  ORTHOGONAL PROJECTION OF X(183) ON ORTHIC AXIS

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

X(16325) lies on these lines:
{30, 183}, {230, 231}, {5159, 7868}, {9996, 10297}

X(16325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (230, 16321, 468), (468, 16312, 16316), (468, 16313, 16331), (16312, 16316, 16331), (16313, 16316, 16312)


X(16326) =  ORTHOGONAL PROJECTION OF X(193) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 - 19*a^4*b^4 + 15*a^2*b^6 + b^8 + a^6*c^2 + 34*a^4*b^2*c^2 - 15*a^2*b^4*c^2 - 16*b^6*c^2 - 19*a^4*c^4 - 15*a^2*b^2*c^4 + 30*b^4*c^4 + 15*a^2*c^6 - 16*b^2*c^6 + c^8 : :
X(16326) = 3 X[468] - 4 X[16303], 3 X[468] - 2 X[16312], 9 X[468] - 8 X[16321], 3 X[16312] - 4 X[16321], 3 X[16303] - 2 X[16321], 7 X[16312] - 12 X[16324], 7 X[16321] - 9 X[16324], 7 X[468] - 8 X[16324], 7 X[16303] - 6 X[16324], 5 X[16312] - 12 X[16333], 5 X[16321] - 9 X[16333], 5 X[468] - 8 X[16333], 5 X[16324] - 7 X[16333], 5 X[16303] - 6 X[16333], 10 X[16321] - 9 X[16334], 10 X[16324] - 7 X[16334], 5 X[16312] - 6 X[16334], 5 X[468] - 4 X[16334], 5 X[16303] - 3 X[16334]

X(16326) lies on these lines:
{30, 193}, {230, 231}

X(16326) = reflection of X(i) in X(j) for these {i,j}: {16312, 16303}, {16334, 16333}
X(16326) = {X(16303),X(16312)}-harmonic conjugate of X(468)


X(16327) =  ORTHOGONAL PROJECTION OF X(194) ON ORTHIC AXIS

Barycentrics    2*a^8*b^2 - 5*a^6*b^4 - 2*a^4*b^6 + 5*a^2*b^8 + 2*a^8*c^2 + 8*a^6*b^2*c^2 + a^4*b^4*c^2 - 6*a^2*b^6*c^2 - 3*b^8*c^2 - 5*a^6*c^4 + a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 3*b^6*c^4 - 2*a^4*c^6 - 6*a^2*b^2*c^6 + 3*b^4*c^6 + 5*a^2*c^8 - 3*b^2*c^8 : :
X(16327) = 3 X[468] - 4 X[16308], 3 X[468] - 2 X[16313], 10 X[16308] - 9 X[16329], 5 X[16313] - 9 X[16329], 5 X[468] - 6 X[16329]

X(16327) lies on these lines:
{30, 194}, {230, 231}

X(16327) = reflection of X(16313) in X(16308)
X(16327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16303, 16331, 468), (16306, 16316, 468), (16308, 16313, 468), (16314, 16321, 468)


X(16328) =  ORTHOGONAL PROJECTION OF X(216) ON ORTHIC AXIS

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6*b^2 - 6*a^4*b^4 + 6*a^2*b^6 - 2*b^8 + 2*a^6*c^2 - 2*a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 6*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + 6*a^2*c^6 + b^2*c^6 - 2*c^8) : :
X(16328) = 3 X[468] - X[16330]

X(16328) lies on these lines:
{4, 13531}, {6, 186}, {30, 216}, {53, 403}, {230, 231}, {566, 6749}, {577, 15646}, {3087, 13619}, {5158, 7575}

X(16328) = polar circle inverse of X(13531)
X(16328) = X(63)-isoconjugate of X(13530)
X(16328) = barycentric quotient X(25)/X(13530)
X(16328) = {X(3003),X(11062)}-harmonic conjugate of X(1990)


X(16329) =  ORTHOGONAL PROJECTION OF X(262) ON ORTHIC AXIS

Barycentrics    6*a^8*b^2 - 5*a^6*b^4 - 6*a^4*b^6 + 5*a^2*b^8 + 6*a^8*c^2 + 4*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + b^8*c^2 - 5*a^6*c^4 + 3*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - b^6*c^4 - 6*a^4*c^6 - 8*a^2*b^2*c^6 - b^4*c^6 + 5*a^2*c^8 + b^2*c^8 : :
X(16329) = X[468] + 2 X[16308], 4 X[468] - X[16313], 8 X[16308] + X[16313], 10 X[16308] - X[16327], 5 X[468] + X[16327], 5 X[16313] + 4 X[16327]

X(16329) lies on these lines:
{30, 262}, {51, 8705}, {230, 231}, {10162, 11594}

X(16329) = {X(468),X(16303)}-harmonic conjugate of X(16315)


X(16330) =  ORTHOGONAL PROJECTION OF X(264) ON ORTHIC AXIS

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^8*b^2 - 3*a^6*b^4 + a^2*b^8 + 2*a^8*c^2 - 8*a^6*b^2*c^2 + 5*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 3*b^8*c^2 - 3*a^6*c^4 + 5*a^4*b^2*c^4 - 10*a^2*b^4*c^4 + 3*b^6*c^4 + 4*a^2*b^2*c^6 + 3*b^4*c^6 + a^2*c^8 - 3*b^2*c^8) : :
X(16330) = 3 X[468] - 2 X[16328]

X(16330) lies on these lines:
{30, 264}, {230, 231}, {403, 3186}, {1843, 10151}


X(16331) =  ORTHOGONAL PROJECTION OF X(315) ON ORTHIC AXIS

Barycentrics    2*a^10 - a^8*b^2 + 4*a^6*b^4 - 6*a^2*b^8 + b^10 - a^8*c^2 - 4*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 8*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 2*a^4*b^2*c^4 - 4*a^2*b^4*c^4 - 4*b^6*c^4 + 8*a^2*b^2*c^6 - 4*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 + c^10 : :
X(16331) = 3 X[468] - 2 X[16306], 3 X[468] - 4 X[16335]

X(16331) lies on these lines:
{30, 315}, {230, 231}

X(16331) = reflection of X(16306) in X(16335)
X(16331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16312, 16313), (468, 16313, 16325), (468, 16327, 16303), (16306, 16335, 468), (16308, 16321, 468), (16312, 16316, 16325), (16313, 16316, 468)


X(16332) =  ORTHOGONAL PROJECTION OF X(551) ON ORTHIC AXIS

Barycentrics    2*a^6 - a^4*b^2 - a^3*b^3 - 2*a^2*b^4 + a*b^5 + b^6 + a^3*b^2*c - b^5*c - a^4*c^2 + a^3*b*c^2 + 4*a^2*b^2*c^2 - a*b^3*c^2 - b^4*c^2 - a^3*c^3 - a*b^2*c^3 + 2*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 + a*c^5 - b*c^5 + c^6 : :
X(16332) = 5 X[468] - X[16304], 5 X[16272] + X[16304], 2 X[16304] - 5 X[16305], 2 X[16272] + X[16305], 3 X[16304] - 5 X[16309], 3 X[16305] - 2 X[16309], 3 X[468] - X[16309], 3 X[16272] + X[16309], 7 X[16272] - X[16322], 7 X[468] + X[16322], 7 X[16305] + 2 X[16322], 7 X[16309] + 3 X[16322], 7 X[16304] + 5 X[16322], 5 X[16309] - 9 X[16323], 5 X[16305] - 6 X[16323], 5 X[468] - 3 X[16323], X[16304] - 3 X[16323], 5 X[16272] + 3 X[16323]

X(16332) lies on these lines:
{30, 551}, {230, 231}, {3109, 11809}

X(16332) = midpoint of X(i) and X(j) for these {i,j}: {468, 16272}, {3109, 11809}
X(16332) = reflection of X(16305) in X(468)
X(16332) = reflection of X(16305) in Euler line
X(16332) = {X(468),X(16304)}-harmonic conjugate of X(16323)


X(16333) =  ORTHOGONAL PROJECTION OF X(576) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 - 9*a^4*b^4 + 5*a^2*b^6 + b^8 + a^6*c^2 + 14*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 6*b^6*c^2 - 9*a^4*c^4 - 5*a^2*b^2*c^4 + 10*b^4*c^4 + 5*a^2*c^6 - 6*b^2*c^6 + c^8 : :
X(16333) = X[468] - 3 X[16303], 7 X[468] - 3 X[16312], 7 X[16303] - X[16312], 4 X[16312] - 7 X[16321], 4 X[468] - 3 X[16321], 4 X[16303] - X[16321], 2 X[16312] - 7 X[16324], 2 X[468] - 3 X[16324], 5 X[16303] + X[16326], 5 X[16324] + 2 X[16326], 5 X[468] + 3 X[16326], 5 X[16321] + 4 X[16326], 5 X[16312] + 7 X[16326], 5 X[16312] - 7 X[16334], 5 X[16321] - 4 X[16334], 5 X[468] - 3 X[16334], 5 X[16324] - 2 X[16334], 5 X[16303] - X[16334]

X(16333) lies on these lines:
{30, 576}, {230, 231}, {858, 11163}, {5159, 9771}

X(16333) = midpoint of X(16326) and X(16334)
X(16333) = reflection of X(i) in X(j) for these {i,j}: {16321, 16324}, {16324, 16303}


X(16334) =  ORTHOGONAL PROJECTION OF X(599) ON ORTHIC AXIS

Barycentrics    2*a^8 + a^6*b^2 + a^4*b^4 - 5*a^2*b^6 + b^8 + a^6*c^2 - 6*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + 4*b^6*c^2 + a^4*c^4 + 5*a^2*b^2*c^4 - 10*b^4*c^4 - 5*a^2*c^6 + 4*b^2*c^6 + c^8 : :
X(16334) = X[16303] + 2 X[16312], X[16303] - 4 X[16321], X[16312] + 2 X[16321], 3 X[16303] - 4 X[16324], 3 X[468] - 2 X[16324], 3 X[16321] - X[16324], 3 X[16312] + 2 X[16324], 10 X[16324] - 3 X[16326], 5 X[16303] - 2 X[16326], 5 X[468] - X[16326], 10 X[16321] - X[16326], 5 X[16312] + X[16326], 5 X[16303] - 4 X[16333], 5 X[16324] - 3 X[16333], 5 X[468] - 2 X[16333], 5 X[16321] - X[16333], 5 X[16312] + 2 X[16333]

X(16334) lies on these lines:
{23, 15574}, {30, 599}, {230, 231}, {403, 10002}

X(16334) = midpoint of X(468) and X(16312)
X(16334) = reflection of X(i) in X(j) for these {i,j}: {468, 16321}, {16303, 468}, {16326, 16333}
X(16334) = reflection of X(16303) in Euler
X(16334) = {X(16312),X(16321)}-harmonic conjugate of X(16303)


X(16335) =  ORTHOGONAL PROJECTION OF X(626) ON ORTHIC AXIS

Barycentrics    2*a^10 - a^8*b^2 + a^6*b^4 - 3*a^2*b^8 + b^10 - a^8*c^2 + 2*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 5*a^2*b^6*c^2 + a^6*c^4 - 2*a^4*b^2*c^4 - 4*a^2*b^4*c^4 - b^6*c^4 + 5*a^2*b^2*c^6 - b^4*c^6 - 3*a^2*c^8 + c^10 : :
X(16335) = 3 X[468] - X[16306], 3 X[468] + X[16331]

X(16335) lies on these lines:
{23, 3314}, {30, 626}, {230, 231}, {3631, 13562}

X(16335) = midpoint of X(16306) and X(16331)
X(16335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 16313, 230), (468, 16316, 16308), (468, 16331, 16306)


X(16336) =  NINE-POINT-CIRCLE-INVERSE OF X(1209)

Barycentrics    (S^2+SB*SC)*(2*SA^2-R^2*(5*SA+ 6*R^2-3*SW)-2*SB*SC) : :
X(16336) = 3*X(5) + X(14141) = 3*X(14140) - X(14141)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27292.

X(16336) lies on these lines:
{2, 1157}, {3, 3432}, {5, 51}, {30, 128}, {140, 6150}, {233, 2965}, {539, 1141}, {925, 3153}, {2070, 15561}, {3459, 12325}, {7488, 15848}, {10745, 14980}

X(16336) = midpoint of X(5) and X(14140)
X(16336) = anticomplement of X(10615)
X(16336) = complement of X(1157)
X(16336) = complementary conjugate of X(6592)
X(16336) = nine-points circle-inverse-of X(1209)
X(16336) = {X(2), X(1157)}-harmonic conjugate of X(10615)


X(16337) =  NINE-POINT-CIRCLE-INVERSE OF X(3574)

Barycentrics    (S^2+SB*SC)*(4*S^2-R^2*(24*R^ 2+5*SA-19*SW)+2*SA^2-2*SB*SC- 4*SW^2) : :
X(16337) = 3*X(5)-X(14140) = 5*X(5)-X(14141) = 5*X(14140)-3*X(14141)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27292.

X(16337) lies on these lines:
{3, 10615}, {4, 1157}, {5, 51}, {30, 137}, {107, 14979}, {231, 11077}, {403, 10214}, {2070, 5961}, {3153, 14639}, {3518, 6750}, {3583, 14102}, {10610, 14142}

X(16337) = midpoint of X(i) and X(j) for these {i,j}: {4, 1157}, {2070, 14980}
X(16337) = reflection of X(3) in X(10615)
X(16337) = nine-points-circle-inverse of X(3574)
X(16337) = polar-circle-inverse of X(6801)
X(16337) = X(1157)-of-Euler-triangle
X(16337) = X(10615)-of-X3-ABC-reflections-triangle
X(16337) = crosspoint of X(5) and X(1141)
X(16337) = crosssum of X(54) and X(1154)


X(16338) =  MIDPOINT OF X(80) AND X(2718)

Barycentrics    (b+c)*a^6-(3*b^2+2*b*c+3*c^2)* a^5-(b+c)*(b^2-8*b*c+c^2)*a^4+ (2*b-c)*(b-2*c)*(3*b^2+2*b*c+ 3*c^2)*a^3-(b+c)*(b^4+c^4+b*c* (7*b^2-17*b*c+7*c^2))*a^2-(b^ 2-c^2)^2*(3*b^2-11*b*c+3*c^2)* a+(b^2-c^2)^2*(b+c)*(b^2-3*b* c+c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27305.

X(16338) lies on these lines: {1, 2}, {80, 2718}, {3667, 10265}

X(16338) = midpoint of X(80) and X(2718)


X(16339) =  MIDPOINT OF X(67) AND X(2770)

Barycentrics    (9*R^2-SW)*(27*R^2*(SW+SA)-(9* SA+5*SW)*SW)*S^2-(6*R^2*(SW+3* SA)-SW*(SW+5*SA))*SW^3 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27305.

X(16339) lies on these lines: {2, 6}, {67, 2770}

X(16339) = midpoint of X(67) and X(2770)


X(16340) =  MIDPOINT OF X(265) AND X(477)

Barycentrics    S^4+(135*R^4-54*R^2*SW-3*SB* SC+5*SW^2)*S^2-(81*R^4-18*R^2* SW-SW^2)*SB*SC : :
X(16340) = X(265) + 3 X(14851) = X(476) - 3 X(15061) = X(477) - 3 X(14851) = 3 X(5627) - 5 X(15027) = X(11749) + 2X(12079)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27305.

X(16340) lies on these lines:
{2, 3}, {125, 16168}, {265, 477}, {476, 15061}, {523, 10264}, {3258, 5663}, {5627, 15027}, {7728, 14508}, {11749, 12079}

X(16340) = midpoint of X(i) and X(j) for these {i,j}: {265, 477}, {7728, 14508}
X(16340) = {X(265), X(14851)}-harmonic conjugate of X(477)


X(16341) =  MIDPOINT OF X(671) AND X(843)

Barycentrics    9*(36*R^2-3*SA-7*SW)*S^4+12*( 9*R^2*(3*SA-2*SW)-3*SA^2+3*SB* SC+SW^2)*SW*S^2-(9*SA-7*SW)* SW^4 : :
X(16341) = X(352) - 3 X(8859)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27305.

X(16341) lies on these lines:
{2, 6}, {523, 6094}, {671, 843}, {1499, 11632}

X(16341) = midpoint of X(671) and X(843)


X(16342) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(8)

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

For a discussion of collineation images, see the preamble just before X(16286).

X(16342) lies on these lines:
{1, 1150}, {2, 3}, {8, 5737}, {10, 10448}, {31, 1125}, {386, 5278}, {940, 1191}, {965, 5296}, {968, 3702}, {980, 16823}, {1724, 10457}, {1764, 5250}, {2268, 5257}, {3685, 10472}, {3811, 4981}, {3877, 10441}, {4357, 5736}, {4652, 10436}, {4653, 10479}, {4850, 16817}, {5235, 9534}, {5550, 15668}, {10470, 13478}


X(16343) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(10)

Barycentrics    a^4 - 2 a^3 b - 5 a^2 b^2 - 2 a b^3 - 2 a^3 c - 8 a^2 b c - 8 a b^2 c - 2 b^3 c - 5 a^2 c^2 - 8 a b c^2 - 4 b^2 c^2 - 2 a c^3 - 2 b c^3: :

X(16343) lies on these lines:
{2, 3}, {6, 10458}, {31, 940}, {42, 958}, {43, 5251}, {55, 5737}, {614, 980}, {748, 2309}, {965, 2268}, {1150, 1621}, {1764, 4512}, {3741, 5248}, {4423, 15668}, {4651, 9708}, {5250, 10441}


X(16344) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(6)) COLLINEATION IMAGE OF X(65)

Barycentrics    a(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 + 4 a^3 b^2 c + 8 a^2 b^3 c + 5 a b^4 c - 2 a^4 c^2 + 4 a^3 b c^2 + 18 a^2 b^2 c^2 + 16 a b^3 c^2 + 4 b^4 c^2 + 2 a^3 c^3 + 8 a^2 b c^3 + 16 a b^2 c^3 + 8 b^3 c^3 + a^2 c^4 + 5 a b c^4 + 4 b^2 c^4 - a c^5) : :

X(16344) lies on these lines: {2, 3}, {997, 6051}, {1737, 5737}


X(16345) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(43)

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

X(16345) lies on these lines:
{2, 3}, {42, 9708}, {238, 940}, {980, 5272}, {1001, 3741}, {1150, 5284}, {3720, 16466}, {8167, 15668}


X(16346) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(78)

Barycentrics    a(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 - 2 a^4 b c + 4 a^3 b^2 c + 10 a^2 b^3 c + 5 a b^4 c - 2 a^4 c^2 + 4 a^3 b c^2 + 18 a^2 b^2 c^2 + 16 a b^3 c^2 + 4 b^4 c^2 + 2 a^3 c^3 + 10 a^2 b c^3 + 16 a b^2 c^3 + 8 b^3 c^3 + a^2 c^4 + 5 a b c^4 + 4 b^2 c^4 - a c^5) : :

X(16346) lies on these lines:
{2, 3}, {37, 78}, {938, 1150}, {940, 1104}, {1001, 1036}, {3616, 5736}, {5737, 6734}


X(16347) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(145)

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

X(16347) lies on these lines:
{2, 3}, {145, 1150}, {392, 5482}, {595, 3616}, {940, 1616}, {3617, 5737}, {3989, 8669}, {4255, 5278}


X(16348) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(200)

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

X(16348) lies on these lines:
{2, 3}, {940, 7290}, {1150, 10580}, {4423, 7083}, {4847, 5737}


X(16349) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(239)

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

X(16349) lies on these lines:
{2, 3}, {940, 2176}, {2271, 5278}, {3661, 5737}, {3662, 15668}


X(16350) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(306)

Barycentrics    a^6 - 4 a^4 b^2 - 6 a^3 b^3 - 5 a^2 b^4 - 2 a b^5 - 6 a^4 b c - 16 a^3 b^2 c - 16 a^2 b^3 c - 8 a b^4 c - 2 b^5 c - 4 a^4 c^2 - 16 a^3 b c^2 - 22 a^2 b^2 c^2 - 14 a b^3 c^2 - 4 b^4 c^2 - 6 a^3 c^3 - 16 a^2 b c^3 - 14 a b^2 c^3 - 4 b^3 c^3 - 5 a^2 c^4 - 8 a b c^4 - 4 b^2 c^4 - 2 a c^5 - 2 b c^5 : :

X(16350) lies on these lines:
{2, 3}, {940, 2214}, {3666, 4361}


X(16351) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(519)

Barycentrics    3 a^4 - 2 a^3 b - 7 a^2 b^2 - 2 a b^3 - 2 a^3 c - 8 a^2 b c - 8 a b^2 c - 2 b^3 c - 7 a^2 c^2 - 8 a b c^2 - 4 b^2 c^2 - 2 a c^3 - 2 b c^3 : :

X(16351) lies on these lines:
{2, 3}, {551, 940}, {1150, 3241}, {3679, 5737}, {15485, 15668}


X(16352) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(612)

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

X(16352) lies on these lines:
{2, 3}, {614, 940}, {1036, 1125}, {1473, 10436}, {2339, 2355}


X(16353) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(614)

Barycentrics    a(a^5 - a b^4 - 4 a^2 b^2 c - 4 a b^3 c - 4 a^2 b c^2 - 10 a b^2 c^2 - 4 b^3 c^2 - 4 a b c^3 - 4 b^2 c^3 - a c^4) : :

X(16353) lies on these lines:
{2, 3}, {55, 3739}, {518, 612}, {614, 4719}, {967, 5268}, {1001, 3914}, {7085, 10436}, {9798, 16828}


X(16354) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(869)

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

X(16354) lies on this line: {2, 3}


X(16355) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(899)

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

X(16355) lies on these lines: {2, 3}, {748, 940}, {4423, 5737}


X(16356) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(976)

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

X(16356) lies on these lines: {2, 3}, {56, 16684}, {958, 5347}, {2218, 4657}


X(16357) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(995)

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

X(16357) lies on these lines: {2, 3}, {940, 956}, {1150, 9708}, {1764, 3753}


X(16358) =  (X(1),X(2),X(3),X(6); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(1026)

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

X(16358) lies on this line: {2, 3}


X(16359) =  (name pending)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (3 a^22 b^2-24 a^20 b^4+84 a^18 b^6-168 a^16 b^8+210 a^14 b^10-168 a^12 b^12+84 a^10 b^14-24 a^8 b^16+3 a^6 b^18+3 a^22 c^2-42 a^20 b^2 c^2+183 a^18 b^4 c^2-389 a^16 b^6 c^2+460 a^14 b^8 c^2-303 a^12 b^10 c^2+82 a^10 b^12 c^2+37 a^8 b^14 c^2-51 a^6 b^16 c^2+25 a^4 b^18 c^2-5 a^2 b^20 c^2-24 a^20 c^4+183 a^18 b^2 c^4-464 a^16 b^4 c^4+515 a^14 b^6 c^4-204 a^12 b^8 c^4-91 a^10 b^10 c^4+115 a^8 b^12 c^4-3 a^6 b^14 c^4-42 a^4 b^16 c^4+12 a^2 b^18 c^4+3 b^20 c^4+84 a^18 c^6-389 a^16 b^2 c^6+515 a^14 b^4 c^6-216 a^12 b^6 c^6+6 a^10 b^8 c^6-68 a^8 b^10 c^6+120 a^6 b^12 c^6-43 a^4 b^14 c^6+15 a^2 b^16 c^6-24 b^18 c^6-168 a^16 c^8+460 a^14 b^2 c^8-204 a^12 b^4 c^8+6 a^10 b^6 c^8+42 a^8 b^8 c^8-69 a^6 b^10 c^8+150 a^4 b^12 c^8-67 a^2 b^14 c^8+84 b^16 c^8+210 a^14 c^10-303 a^12 b^2 c^10-91 a^10 b^4 c^10-68 a^8 b^6 c^10-69 a^6 b^8 c^10-180 a^4 b^10 c^10+45 a^2 b^12 c^10-168 b^14 c^10-168 a^12 c^12+82 a^10 b^2 c^12+115 a^8 b^4 c^12+120 a^6 b^6 c^12+150 a^4 b^8 c^12+45 a^2 b^10 c^12+210 b^12 c^12+84 a^10 c^14+37 a^8 b^2 c^14-3 a^6 b^4 c^14-43 a^4 b^6 c^14-67 a^2 b^8 c^14-168 b^10 c^14-24 a^8 c^16-51 a^6 b^2 c^16-42 a^4 b^4 c^16+15 a^2 b^6 c^16+84 b^8 c^16+3 a^6 c^18+25 a^4 b^2 c^18+12 a^2 b^4 c^18-24 b^6 c^18-5 a^2 b^2 c^20+3 b^4 c^20) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27308.

X(16359) lies on this line:: {5,128}


X(16360) =  (name pending)

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

X(16360) lies on the cubics K960 and K962 and on these lines:
{8844, 8845}, {1281, 16361}, {8424, 16362}, {8852, 16363}

X(16360) = X(291)-isoconjugate of 16362
X(16360) = barycentric quotient X(1914)/16362


X(16361) =  (name pending)

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

X(16361) lies on the cubics K961 and K962 and on these lines:
{2106, 2108}, {1281, 16360}, {8301, 16363}, {8852, 16362}

X(16361) = X(894)-cross conjugate of X(1580)
X(16361) = X(291)-isoconjugate of 16363
X(16361) = barycentric quotient X(1914)/16363


X(16362) =  (name pending)

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

X(16362) lies on the cubic K960 and these lines:
{21, 741}, {171, 8932}, {256, 291}, {1284, 8934}, {1580, 1922}, {1911, 11688}, {2295, 8846}, {3510, 8844}, {8424, 16360}, {8852, 16361}

X(16362) = X(8424)-Ceva conjugate of X(8933)
X(16362) = X(291)-isoconjugate of 16360
X(16362) = barycentric quotient X(1914)/16360
X(16362) = {X(894),X(1967)}-harmonic conjugate of X(291)


X(16363) =  (name pending)

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

X(16363) lies on the cubic K961 and these lines:
{105, 8848}, {256, 291}, {893, 8299}, {904, 3903}, {7104, 8300}, {8936, 9472}, {8301, 16361}, {8852, 16360}

X(16363) = X(8301)-Ceva conjugate of X(8936)
X(16363) = X(291)-isoconjugate of 16361
X(16363) = barycentric quotient X(1914)/16361


X(16364) =  (name pending)

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

X(16364) lies on the cubics K699 and K962 and on these lines: {1967, 7061}, {3508, 8782}

X(16364) = X(256)-cross conjugate of X(239)


X(16365) =  (name pending)

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

X(16365) lies on the cubic K962 and this line: {75,1281}


X(16366) =  (name pending)

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

X(16366) lies on the cubic K960 and these lines:
{846,1334}, {894,8424}, {1281,3978}, {1580,8847}, {4433,8844}, {8845,8932}, {8933,16362}

X(16366) = X(291)-isoconjugate of X(8424)
X(16366) = barycentric quotient X(1914)/X(8424)


X(16367) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(239)

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

For a discussion of collineation images, see the preamble just before X(16286).

X(16367) lies on these lines:
{2, 3}, {35, 4384}, {55, 239}, {81, 5021}, {183, 3948}, {956, 6542}, {958, 3661}, {993, 3912}, {1444, 4648}, {2178, 4687}, {3295, 4393}, {5124, 15668}, {5256, 16478}, {5283, 5337}

X(16367) = {X(2),X(3)}-harmonic conjugate of X(11329)


X(16368) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(306)

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

X(16368) lies on these lines:
{2, 3}, {6, 63}, {9, 3998}, {55, 306}, {608, 1214}, {940, 1333}, {956, 3187}, {958, 5271}, {1001, 2352}, {1259, 3687}, {1621, 4344}, {1724, 2999}, {1751, 5745}, {4254, 5739}, {4261, 4383}, {4657, 5249}, {5273, 5278}, {10477, 16465}, {12514, 16471}


X(16369) =  (X(1),X(2),X(6),X(99); X(1),X(6),X(2),X(99)) COLLINEATION IMAGE OF X(239)

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

X(16369) lies on these lines: {1, 6}, {239, 4037}, {292, 1931}, {2238, 3747}, {2664, 9509}


X(16370) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(519)

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

X(16370) lies on these lines:
{1, 3052}, {2, 3}, {9, 5440}, {10, 5217}, {35, 958}, {36, 1001}, {55, 519}, {56, 551}, {72, 3601}, {78, 15650}, {100, 9708}, {154, 392}, {165, 3753}, {187, 5275}, {191, 12635}, {527, 954}, {528, 10058}, {529, 8069}, {535, 5172}, {599, 4265}, {940, 4257}, {942, 4652}, {960, 3612}, {965, 4877}, {997, 3683}, {999, 1621}, {1125, 1836}, {1376, 5010}, {1384, 5276}, {1385, 5250}, {1388, 3884}, {1470, 5298}, {1478, 6690}, {1479, 3829}, {1482, 3897}, {1709, 7987}, {1724, 4255}, {1727, 5289}, {2646, 5730}, {2886, 4302}, {2932, 6174}, {2975, 3241}, {3053, 5283}, {3058, 10959}, {3207, 3294}, {3218, 15934}, {3219, 3940}, {3255, 15175}, {3303, 8666}, {3306, 5122}, {3419, 4304}, {3421, 5281}, {3488, 5744}, {3582, 14793}, {3584, 11236}, {3616, 5303}, {3624, 3838}, {3653, 10269}, {3654, 11248}, {3655, 10267}, {3656, 11249}, {3697, 5234}, {3746, 12513}, {3813, 4309}, {3822, 12943}, {3869, 4930}, {3877, 10246}, {3913, 4677}, {4004, 5128}, {4256, 4383}, {4669, 8715}, {4720, 5361}, {4855, 5044}, {4860, 4973}, {4995, 10955}, {4996, 10707}, {5023, 5277}, {5259, 7280}, {5260, 9709}, {5325, 11517}, {5414, 9678}, {5426, 5902}, {5436, 5439}, {5450, 9948}, {5711, 10448}, {5719, 5905}, {5985, 13188}, {7354, 10198}, {8071, 10072}, {9668, 11680}, {9881, 13173}, {10031, 12773}, {10527, 15171}, {10529, 15172}, {10902, 12114}, {11012, 11496}, {11240, 15170}, {12326, 13178}

X(16370) = {X(2),X(3)}-harmonic conjugate of X(16371)


X(16371) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(551)

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

X(16371) lies on these lines:
{1, 4004}, {2, 3}, {10, 5204}, {35, 4428}, {36, 956}, {46, 5730}, {55, 551}, {56, 519}, {57, 5440}, {63, 5122}, {72, 3928}, {100, 999}, {165, 392}, {214, 2099}, {484, 5289}, {499, 3829}, {528, 2932}, {553, 1466}, {574, 5275}, {599, 5096}, {936, 3916}, {940, 4256}, {942, 4855}, {954, 6173}, {958, 7280}, {993, 3828}, {997, 1155}, {1001, 5010}, {1125, 5217}, {1145, 3476}, {1260, 2094}, {1329, 4299}, {1420, 10914}, {1470, 5434}, {1478, 3035}, {1479, 6691}, {2096, 13257}, {2975, 9709}, {3216, 4252}, {3218, 3940}, {3295, 5253}, {3304, 8715}, {3336, 12635}, {3361, 3555}, {3419, 3911}, {3434, 15325}, {3576, 3753}, {3582, 11235}, {3584, 14793}, {3601, 5439}, {3612, 3812}, {3653, 10267}, {3654, 11249}, {3655, 10269}, {3656, 11248}, {3814, 12943}, {3816, 4302}, {3825, 12953}, {3871, 7373}, {3872, 5126}, {4257, 4383}, {4304, 6692}, {4317, 12607}, {4511, 4930}, {4650, 5529}, {4652, 5044}, {4669, 8666}, {4677, 12513}, {4881, 10246}, {4900, 13462}, {5013, 5277}, {5024, 5276}, {5082, 5265}, {5131, 5692}, {5283, 15815}, {5298, 10949}, {5303, 9780}, {5902, 15015}, {6284, 10200}, {6502, 9679}, {8071, 10056}, {9655, 11681}, {9943, 16209}, {10031, 12331}, {10199, 11238}, {10586, 15172}, {12017, 15988}, {12641, 15180}

X(16371) = {X(2),X(3)}-harmonic conjugate of X(16370)


X(16372) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(869)

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

X(16372) lies on these lines:
{2, 3}, {35, 2664}, {55, 869}, {1030, 2110}, {1185, 2271}, {2223, 5277}, {5364, 7085}


X(16373) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(899)

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

X(16373) lies on these lines:
{2, 3}, {42, 1191}, {43, 3746}, {55, 748}, {228, 7308}, {373, 573}, {572, 5651}, {991, 5650}, {993, 4871}, {3240, 3295}, {3304, 3720}, {4413, 8053}


X(16374) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(995)

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

X(16374) lies on these lines:
{1, 15621}, {2, 3}, {55, 995}, {956, 14829}, {957, 6244}, {1385, 15623}, {3216, 4267}, {3576, 15626}, {3917, 5396}, {5010, 8053}, {7987, 15622}


X(16375) =  (X(1),X(2),X(6),X(31); X(3),X(2),X(6),X(31)) COLLINEATION IMAGE OF X(1026)

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

X(16375) lies on these lines: {2, 3}, {55, 1026}


X(16376) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(8)

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

X(16376) lies on these lines: {2, 3}, {8, 5091}, {78, 9318}, {3616, 14267}


X(16377) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(10)

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

X(16377) lies on these lines:
{2, 3}, {10, 5091}, {72, 9318}, {3570, 9534}


X(16378) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(42)

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

X(16378) lies on these lines:
{2, 3}, {42, 5091}, {228, 9318}


X(16379) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(43)

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

X(16379) lies on these lines:
{2, 3}, {43, 5091}, {1376, 4713}


X(16380) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(145)

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

X(16380) lies on these lines:
{2, 3}, {145, 5091}, {4855, 9318}


X(16381) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(239)

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

X(16381) lies on these lines:
{2, 3}, {239, 5091}, {6654, 14267}


X(16382) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(386)

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

X(16382) lies on these lines:
{2, 3}, {386, 5091}, {1491, 3216}


X(16383) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(519)

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

X(16383) lies on these lines:
{2, 3}, {519, 5091}, {2795, 15015}, {5440, 9318}


X(16384) =  (X(1),X(2),X(6),X(513); X(3),X(2),X(6),X(513)) COLLINEATION IMAGE OF X(551)

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

X(16384) lies on these lines:
{2, 3}, {551, 5091}


X(16385) =  REFLECTION OF X(5031) IN X(6680)

Barycentrics    a^2 ((a^4 - b^2 c^2) (2 a^4 + b^4 + c^4) + a^2 (b^6 + c^6 - 2 a^2 b^2 c^2)) : :
X(16385) = 3*X(32)+X(2458) = 3 X[1691] - X[2458]

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27309.

X(16385) lies on these lines: {3,6},{699,2715},{5031,6680},{5103,18907}

X(16385) = midpoint of X(32) and X(1691)
X(16385) = reflection of X(5031) in X(6680)


X(16386) =  ANTICOMPLEMENT OF X(10151)

Barycentrics    (14*R^2-3*SW)*S^2-(24*R^2-5* SW)*SB*SC : :
X(16386) = 2*X(20) + X(858) = 3*X(20) + X(3153) = 5*X(20) + X(10296) = 2*X(1514) - 5*X(15051)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27309.

X(16386) lies on these lines:
{2, 3}, {110, 15311}, {343, 11454}, {476, 5897}, {477, 13398}, {523, 14329}, {925, 2693}, {1294, 10420}, {1503, 13445}, {1514, 15051}, {2694, 13397}, {2697, 3565}, {2883, 11449}, {3100, 10149}, {3357, 14516}, {4296, 9627}, {5894, 12111}, {6000, 12825}, {6247, 12278}, {6390, 13219}, {10606, 11442}, {11064, 11744}, {11468, 12359}, {12219, 14677}, {13754, 16111}

X(16386) = midpoint of X(20) and X(2071)
X(16386) = anticomplement of X(10151)
X(16386) = circumcircle-inverse-of X(11413)
X(16386) = 2nd-Droz-Farny circle-inverse-of X(20)
X(16386) = X(403)-of-ABC-X3-reflections-triangle
X(16386) = X(10257)-of-anti-Euler-triangle
X(16386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 376, 22), (20, 7396, 15683), (20, 11413, 12225), (22, 858, 7426), (376, 7386, 3522), (550, 15704, 15332), (1113, 1114, 11413), (3153, 7488, 10096)


X(16387) =  EULER LINE INTERCEPT OF X(325)X(2373)

Barycentrics    (36*R^4-22*R^2*SW+3*SW^2)*S^2+ (6*R^2-SW)*SB*SC*SW : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27309.

X(16387) lies on these lines:
{2, 3}, {325, 2373}, {1297, 16167}, {1503, 12827}, {2770, 13398}, {2781, 11064}, {3580, 5622}, {9019, 11746}, {11416, 15360}

X(16387) = midpoint of X(22) and X(858)
X(16387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 23, 403), (2, 2071, 858), (468, 10257, 2), (3153, 10565, 23)


X(16388) =  X(4)X(990)∩X(10)X(6823)

Trilinears    2*q*p^5-(1-q^2)*(2*p^3*(p+q)+( 2*q^2-5)*p^2-q^2+2) : : , where p = sin(A/2), q = cos(B/2 - C/2)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27310.

X(16388) lies on these lines:
{4, 990}, {10, 6823}, {496, 942}, {515, 1062}, {5706, 13478}


X(16389) =  X(20)X(346)∩X(40)X(64)

Trilinears    2*p^6+p*q*(2*p^4-q^2)+2*(q^2- 3)*p^4+(2*q^2-1)*q*p^3-2*(2*q^ 2-3)*p^2-2+2*q^2 : : , where p = sin(A/2), q = cos(B/2 - C/2)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27310.

X(16389) lies on these lines:
{1, 1037}, {20, 346}, {40, 64}, {46, 3465}, {84, 1726}, {170, 2938}, {220, 610}, {223, 1697}, {226, 11471}, {515, 1753}, {517, 12659}, {1020, 3182}, {1745, 2943}, {1750, 7713}, {2944, 2947}, {5777, 15941}

X(16389) = {X(40), X(1490)}-harmonic conjugate of X(1763)


X(16390) =  (name pending)

Barycentrics    (SB+SC)*(S^4+(2*R^2*(SA+3*SW)+ SB*SC-2*SW^2)*S^2+(16*R^4*(R^ 2-SW)+4*R^2*(SA^2-SB*SC)-SW^2* (SA-SW))*SA) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27310.

X(16390) lies on this line: {5446, 6756}


X(16391) =  X(3)X(68)∩X(76)X(95)

Barycentrics    SA^3* (SB+SC)*(S^2-SB^2)*(S^2-SC^2) : :
Barycentrics    Cos[A]^3 Sec[2 A] : :    (Peter Moses, March 15, 2018)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27310.

X(16391) lies on these lines:
{3, 68}, {24, 5962}, {76, 95}, {378, 847}, {426, 1092}, {454, 9908}, {577, 5562}, {925, 1294}, {1593, 14593}, {2071, 5879}, {2165, 5254}, {4558, 11412}, {5392, 7503}, {13855, 15781}, {14264, 14366}

X(16391) = barycentric product X(i)*X(j) for these {i,j}: {68, 394}, {91, 6507}, {326, 1820}, {2165, 3964}, {2351, 3926}
X(16391) = barycentric quotient X(i)/X(j) for these (i,j): (3, 11547), (68, 2052), (91, 6521), (96, 8794), (184, 8745), (255, 1748), (394, 317), (418, 14576), (577, 24), (925, 15352), (1092, 1993), (1820, 158), (2165, 1093), (2351, 393), (3269, 136), (3964, 7763), (4100, 47)
X(16391) = trilinear product X(i)*X(j) for these {i,j}: {68, 255}, {91, 1092}, {326, 2351}, {394, 1820}, {2165, 6507}, {4100, 5392}
X(16391) = trilinear quotient X(i)/X(j) for these (i,j): (48, 8745), (63, 11547), (68, 158), (91, 1093), (255, 24), (326, 317), (394, 1748), (822, 6753), (1092, 47), (1102, 7763), (1820, 393), (2165, 6520), (2351, 1096), (2632, 136), (4100, 571)
X(16391) = {X(3), X(68)}-harmonic conjugate of X(2351)


X(16392) =  (name pending)

Barycentrics    70*a^10-275*(b^2+c^2)*a^8-80*( b^2-4*b*c+c^2)*(b^2+4*b*c+c^2) *a^6-2*(b^2+c^2)*(109*b^4-418* b^2*c^2+109*c^4)*a^4+2*(329*b^ 8+329*c^8-2*b^2*c^2*(890*b^4- 1131*b^2*c^2+890*c^4))*a^2+(b^ 4-c^4)*(b^2-c^2)*(-155*b^4+ 662*b^2*c^2-155*c^4) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27312.

X(16392) lies on this line: {524, 3091}


X(16393) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(8)

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

For a discussion of collineation images, see the preamble just before X(16286).

X(16393) lies on these lines:
{2, 3}, {8, 4252}, {519, 1468}, {537, 976}, {1086, 3616}, {1150, 4257}, {3241, 5710}


X(16394) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(10)

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

X(16394) lies on these lines:
{1, 536}, {2, 3}, {218, 5782}, {519, 5710}, {535, 5078}, {956, 5263}, {1104, 4688}, {1220, 5687}, {3679, 5247}, {4026, 4302}, {4252, 10479}, {4304, 5750}, {4664, 7283}


X(16395) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(42)

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

X(16395) lies on these lines:
{2, 3}, {42, 4421}, {536, 2352}, {2229, 3053}


X(16396) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(43)

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

X(16396) lies on these lines:
{2, 3}, {43, 16477}, {3741, 11194}


X(16397) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(145)

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

X(16397) lies on these lines:
{1, 4781}, {2, 3}, {3241, 5264}


X(16398) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(200)

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

X(16398) lies on these lines:
{2, 3}, {200, 4641}


X(16399) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(239)

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

X(16399) lies on this line: {2, 3}


X(16400) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(386)

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

X(16400) lies on these lines:
{2, 3}, {4421, 5264}, {5217, 9554}


X(16401) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(519)

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

X(16401) lies on this line: {2,3}


X(16402) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(551)

Barycentrics    9 a^4 - 2 a^3 b - 13 a^2 b^2 - 2 a b^3 - 2 a^3 c - 4 a^2 b c - 4 a b^2 c - 2 b^3 c - 13 a^2 c^2 - 4 a b c^2 - 4 b^2 c^2 - 2 a c^3 - 2 b c^3 : :

X(16402) lies on this line: {2,3}


X(16403) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(612)

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

X(16403) lies on these lines:
{2, 3}, {55, 536}, {612, 4640}


X(16404) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(614)

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

X(16404) lies on this line: {2,3}


X(16405) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(899)

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

X(16405) lies on these lines:
{2, 3}, {6, 2229}, {31, 899}, {42, 3913}, {43, 1203}, {672, 5782}, {750, 2309}, {1403, 4418}, {3240, 5687}, {3741, 8666}


X(16406) =  (X(1),X(2),X(6),X(100); X(3),X(2),X(6),X(100)) COLLINEATION IMAGE OF X(976)

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

X(16406) lies on these lines:
{2, 3}, {4363, 5217}


X(16407) =  (X(1),X(2),X(3),X(6); X(3),X(37),X(2),X(31)) COLLINEATION IMAGE OF X(35)

Barycentrics    a^6 + 3 a^5 b + 8 a^4 b^2 + 11 a^3 b^3 + 7 a^2 b^4 + 2 a b^5 + 3 a^5 c + 17 a^4 b c + 33 a^3 b^2 c + 33 a^2 b^3 c + 16 a b^4 c + 2 b^5 c + 8 a^4 c^2 + 33 a^3 b c^2 + 50 a^2 b^2 c^2 + 34 a b^3 c^2 + 8 b^4 c^2 + 11 a^3 c^3 + 33 a^2 b c^3 + 34 a b^2 c^3 + 12 b^3 c^3 + 7 a^2 c^4 + 16 a b c^4 + 8 b^2 c^4 + 2 a c^5 + 2 b c^5 : :

X(16407) lies on these lines:
{2, 3}, {4362, 15668}


X(16408) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(10)

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

X(16408) lies on these lines:
{1, 3689}, {2, 3}, {8, 7373}, {10, 999}, {35, 4423}, {55, 3624}, {56, 1698}, {57, 3927}, {72, 3306}, {78, 5439}, {84, 10157}, {100, 5550}, {142, 6700}, {165, 3646}, {200, 5045}, {210, 3338}, {355, 8582}, {388, 3820}, {392, 12702}, {496, 2550}, {499, 3925}, {517, 8583}, {551, 3913}, {750, 16466}, {912, 5780}, {936, 942}, {940, 3216}, {946, 6244}, {956, 5253}, {958, 3634}, {975, 3752}, {978, 5711}, {997, 3812}, {1125, 1376}, {1159, 5730}, {1213, 5120}, {1329, 9654}, {1466, 3824}, {1482, 3753}, {1490, 11227}, {1617, 5433}, {1706, 9957}, {1770, 4679}, {2886, 10200}, {3035, 10198}, {3218, 15650}, {3304, 3679}, {3305, 3916}, {3333, 8580}, {3452, 12436}, {3616, 5687}, {3635, 8168}, {3715, 6763}, {3742, 3811}, {3754, 5289}, {3816, 9669}, {3819, 5752}, {3826, 6691}, {3828, 8666}, {3872, 4002}, {3911, 5791}, {4004, 11682}, {4292, 5316}, {4313, 9945}, {4860, 5904}, {5024, 5283}, {5049, 6765}, {5204, 5251}, {5217, 5259}, {5221, 5692}, {5248, 8167}, {5266, 5272}, {5268, 11512}, {5275, 9605}, {5328, 5714}, {5450, 10172}, {5554, 12645}, {5603, 11024}, {5657, 8158}, {5720, 9940}, {5722, 9843}, {5726, 13370}, {5777, 10855}, {5790, 16203}, {5806, 6282}, {5883, 12635}, {5886, 10306}, {5927, 12684}, {6147, 9776}, {7308, 15803}, {8227, 10310}, {8726, 10156}, {9956, 10269}, {10165, 11500}, {10175, 12114}, {11230, 11248}, {11231, 11249}, {12699, 16004}, {15297, 15346}

X(16408) = {X(2),X(3)}-harmonic conjugate of X(11108)


X(16409) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(43)

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

X(16409) lies on these lines:
{2, 3}, {42, 7373}, {43, 999}, {55, 9337}, {991, 6688}, {1376, 3840}, {2328, 16187}, {3720, 6767}, {3741, 9709}, {3848, 15624}


X(16410) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(78)

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

X(16410) lies on these lines:
{1, 6600}, {2, 3}, {10, 1617}, {36, 5234}, {56, 210}, {72, 1445}, {78, 999}, {938, 5687}, {942, 1260}, {956, 4308}, {958, 4311}, {965, 5120}, {1001, 10624}, {1210, 1376}, {1259, 3306}, {1621, 11024}, {1698, 7742}, {1728, 5784}, {2360, 5085}, {2900, 5438}, {3191, 5228}, {3295, 3753}, {3428, 8583}, {3646, 7688}, {3698, 11510}, {3885, 6767}, {4298, 6700}, {4314, 9843}, {4413, 5705}, {5253, 11037}, {5439, 11517}, {6734, 9709}, {8582, 11500}, {10310, 12651}, {12436, 15804}

X(16410) = {X(2),X(3)}-harmonic conjugate of X(16293)
X(16410) = {X(3),X(11108)}-harmonic conjugate of X(13615)


X(16411) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(200)

Barycentrics    a^2 (a^4 - 2 a^3 b + 2 a b^3 - b^4 - 2 a^3 c - 2 a^2 b c + 2 a b^2 c + 2 b^3 c + 2 a b c^2 - 18 b^2 c^2 + 2 a c^3 + 2 b c^3 - c^4) : :

X(16411) lies on these lines:
{1, 12631}, {2, 3}, {9, 10855}, {56, 3983}, {200, 999}, {480, 10980}, {1260, 3306}, {1376, 5853}, {1617, 4413}, {3295, 10582}, {3742, 6600}, {3753, 3895}, {3870, 7373}, {4321, 9954}, {4423, 9580}, {4847, 9709}, {5219, 15804}, {5687, 10580}, {9858, 10396}


X(16412) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(239)

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

X(16412) lies on these lines:
{2, 3}, {56, 4384}, {86, 4254}, {100, 5308}, {198, 10436}, {239, 999}, {391, 1014}, {940, 2271}, {980, 5275}, {1376, 3912}, {1696, 3729}, {2178, 3739}, {2999, 11512}, {3661, 9709}, {4383, 5021}, {4393, 7373}, {5222, 5253}


X(16413) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(306)

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

X(16413) lies on these lines:
{2, 3}, {306, 999}, {1211, 5120}, {2352, 4413}, {2999, 5438}, {3247, 3666}, {3306, 3998}, {5256, 5440}, {5271, 9709}, {5287, 5439}


X(16414) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(386)

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

X(16414) lies on these lines:
{2, 3}, {56, 3216}, {228, 5439}, {386, 999}, {500, 5943}, {582, 9306}, {1617, 1714}, {2183, 11573}, {3295, 5132}, {3874, 4557}, {5495, 13364}, {9709, 10479}

X(16414) = {X(2),X(3)}-harmonic conjugate of X(16286)


X(16415) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(387)

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

X(16415) lies on these lines:
{2, 3}, {44, 579}, {56, 1714}, {57, 3216}, {88, 5708}, {386, 942}, {387, 999}, {1214, 9895}, {1257, 3940}, {1730, 5752}, {3185, 12609}

X(16415) = {X(2),X(3)}-harmonic conjugate of X(16290)


X(16416) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(6)) COLLINEATION IMAGE OF X(57)

Barycentrics    a(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 - 2 a^4 b c + 8 a^3 b^2 c + 18 a^2 b^3 c + 9 a b^4 c - 2 a^4 c^2 + 8 a^3 b c^2 + 34 a^2 b^2 c^2 + 32 a b^3 c^2 + 8 b^4 c^2 + 2 a^3 c^3 + 18 a^2 b c^3 + 32 a b^2 c^3 + 16 b^3 c^3 + a^2 c^4 + 9 a b c^4 + 8 b^2 c^4 - a c^5) : :

X(16416) lies on these lines:
{1, 965}, {2, 3}, {940, 1453}, {1210, 5737}


X(16417) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(519)

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

X(16417) lies on these lines:
{1, 3922}, {2, 3}, {9, 5122}, {10, 11194}, {36, 4413}, {56, 3679}, {57, 3940}, {65, 4930}, {78, 5708}, {100, 6767}, {519, 999}, {551, 3295}, {936, 3927}, {942, 5438}, {958, 3828}, {1125, 4428}, {1159, 4511}, {1329, 9655}, {1466, 4654}, {1470, 11237}, {1617, 5298}, {1698, 5204}, {2550, 15325}, {2932, 10707}, {3158, 5049}, {3241, 5253}, {3306, 5440}, {3361, 9850}, {3488, 9945}, {3576, 10156}, {3579, 8583}, {3582, 8069}, {3584, 8071}, {3624, 5217}, {3646, 16192}, {3655, 11499}, {3656, 10306}, {3753, 10246}, {3816, 9668}, {3820, 4293}, {3829, 6691}, {3929, 5044}, {4423, 5010}, {4669, 12513}, {4677, 5563}, {4745, 8666}, {4855, 5439}, {4870, 11509}, {4973, 5220}, {5024, 5275}, {5126, 9623}, {5266, 11512}, {5277, 9605}, {5719, 9776}, {5722, 6692}, {6174, 10056}, {9669, 10200}, {9856, 10270}, {10199, 11235}, {11374, 12436}, {11711, 12326}, {12258, 13173}, {12688, 16209}, {15815, 16589}

X(16417) = {X(2),X(3)}-harmonic conjugate of X(16418)


X(16418) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(551)

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

X(16418) lies on these lines:
{1, 3683}, {2, 3}, {6, 4653}, {9, 3940}, {10, 4421}, {35, 9709}, {36, 4423}, {55, 3679}, {56, 4355}, {63, 15934}, {329, 5719}, {392, 912}, {517, 4512}, {519, 958}, {527, 551}, {759, 15322}, {942, 3928}, {950, 5791}, {954, 6172}, {956, 1621}, {971, 3576}, {997, 15254}, {1056, 8171}, {1376, 3828}, {1384, 5275}, {1385, 7330}, {1482, 5250}, {1617, 5434}, {1698, 5217}, {1864, 3601}, {2328, 5398}, {2646, 7082}, {2886, 9668}, {2975, 7373}, {3303, 5258}, {3305, 5440}, {3488, 5273}, {3582, 8071}, {3584, 8069}, {3616, 6147}, {3624, 3824}, {3646, 7987}, {3654, 10306}, {3746, 4677}, {3811, 5302}, {3820, 5218}, {3829, 9669}, {3877, 10247}, {3913, 4669}, {3916, 5708}, {3925, 4302}, {4254, 5822}, {4413, 5010}, {4652, 5439}, {4745, 8715}, {5082, 10386}, {5122, 5437}, {5260, 5687}, {5303, 5550}, {5426, 5692}, {5603, 5762}, {5722, 5745}, {6174, 10058}, {8583, 13624}, {9654, 10198}, {10197, 11236}, {10448, 16466}, {10884, 12684}, {11374, 12572}

X(16418) = {X(2),X(3)}-harmonic conjugate of X(16417)
X(16418) = 2nd-extouch-to-2nd-circumperp similarity image of X(2)


X(16419) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(612)

Barycentrics    a^2 (a^4 - b^4 - 10 b^2 c^2 - c^4) : :

X(16419) lies on these lines:
{2, 3}, {6, 3787}, {39, 1611}, {55, 5272}, {56, 5268}, {154, 5092}, {182, 3167}, {184, 12017}, {394, 5050}, {612, 999}, {614, 3295}, {1125, 12410}, {1151, 8855}, {1152, 8854}, {1184, 9605}, {1194, 5024}, {1196, 5013}, {1216, 11432}, {1350, 5544}, {1351, 3917}, {1473, 3305}, {1486, 8167}, {1498, 13347}, {1660, 10249}, {2936, 9167}, {2979, 5644}, {3098, 6688}, {3108, 5359}, {3306, 7085}, {3426, 14855}, {3527, 10625}, {3624, 8193}, {3634, 9798}, {3742, 12329}, {3796, 5651}, {3815, 8573}, {3920, 7373}, {5012, 6090}, {5085, 9306}, {5093, 5422}, {5120, 5275}, {5432, 16541}, {5449, 12309}, {5621, 5972}, {6118, 12979}, {6119, 12978}, {6340, 9723}, {6667, 13222}, {6683, 9917}, {6721, 9861}, {6722, 13175}, {6723, 12310}, {6767, 7191}, {7999, 12160}, {8192, 9780}, {8252, 8281}, {8253, 8280}, {8553, 15820}, {8770, 15815}, {9919, 12900}, {10219, 14810}, {11245, 11898}, {11402, 15066}, {11793, 12164}, {11820, 16194}, {12174, 15056}

X(16419) = {X(2),X(3)}-harmonic conjugate of X(5020)


X(16420) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(869)

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

X(16420) lies on these lines:
{2, 3}, {56, 2664}, {869, 999}


X(16421) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(899)

Barycentrics    a^2 (a^3 b - a b^3 + a^3 c + a^2 b c - a b^2 c - b^3 c - a b c^2 + 12 b^2 c^2 - a c^3 - b c^3) : :

X(16421) lies on these lines:
{2, 3}, {43, 3304}, {573, 15082}, {899, 999}, {1376, 4871}, {3240, 7373}, {13329, 16187}


X(16422) =  (X(1),X(2),X(6),X(101); X(3),X(2),X(6),X(101)) COLLINEATION IMAGE OF X(976)

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

X(16422) lies on these lines:
{2, 3}, {56, 5293}, {283, 5650}, {580, 3819}, {976, 999}, {2360, 5092}, {3295, 4642}


X(16423) =  (X(1),X(2),X(3),X(6); X(3),X(37),X(2),X(31)) COLLINEATION IMAGE OF X(36)

Barycentrics    a^6 - a^5 b - 8 a^4 b^2 - 13 a^3 b^3 - 9 a^2 b^4 - 2 a b^5 - a^5 c - 15 a^4 b c - 35 a^3 b^2 c - 35 a^2 b^3 c - 16 a b^4 c - 2 b^5 c - 8 a^4 c^2 - 35 a^3 b c^2 - 54 a^2 b^2 c^2 - 34 a b^3 c^2 - 8 b^4 c^2 - 13 a^3 c^3 - 35 a^2 b c^3 - 34 a b^2 c^3 - 12 b^3 c^3 - 9 a^2 c^4 - 16 a b c^4 - 8 b^2 c^4 - 2 a c^5 - 2 b c^5 : :

X(16423) lies on these lines:
{2, 3}, {3791, 5737}


X(16424) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(8)

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

X(16424) lies on this line: {2,3}


X(16425) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(10)

Barycentrics    a(a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c + 3 a^4 b c + 6 a^3 b^2 c + 9 a^2 b^3 c + 5 a b^4 c + 6 a^3 b c^2 + 22 a^2 b^2 c^2 + 22 a b^3 c^2 + 6 b^4 c^2 + 9 a^2 b c^3 + 22 a b^2 c^3 + 12 b^3 c^3 - a^2 c^4 + 5 a b c^4 + 6 b^2 c^4 - a c^5) : :

X(16425) lies on these lines: {2, 3}, {940, 3833}


X(16426) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(145)

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

X(16426) lies on this line: {2,3}


X(16427) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(386)

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

X(16427) lies on these lines:
{2, 3}, {575, 1437}, {978, 5363}, {2930, 3216}


X(16428) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(387)

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

X(16428) lies on these lines:
{2, 3}, {4260, 11188}


X(16429) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(519)

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

X(16429) lies on these lines:
{2, 3}, {36, 3739}, {940, 5883}, {2805, 15015}


X(16430) =  (X(1),X(2),X(6),X(110); X(3),X(2),X(6),X(110)) COLLINEATION IMAGE OF X(551)

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

X(16430) lies on these lines:
{2, 3}, {191, 4650}, {758, 940}


X(16431) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(187)

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

X(16431) lies on these lines:
{2, 3}, {81, 5024}, {187, 4383}, {574, 940}, {599, 5124}, {980, 15815}, {1992, 5120}, {2223, 4421}, {3912, 5204}, {5013, 5337}

X(16431) = {X(2),X(3)}-harmonic conjugate of X(16436)


X(16432) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(371)

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

X(16432) lies on these lines:
{2, 3}, {37, 8231}, {55, 5405}, {56, 5393}, {81, 3312}, {198, 8233}, {244, 8337}, {355, 6347}, {371, 4383}, {372, 940}, {487, 14555}, {517, 3083}, {1030, 8252}, {1385, 3084}, {1465, 13388}, {3068, 5120}, {3069, 4254}, {5124, 8253}, {6199, 14997}, {6213, 8965}, {6395, 14996}

X(16432) = {X(2),X(3)}-harmonic conjugate of X(16433)


X(16433) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(372)

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

X(16433) lies on these lines:
{2, 3}, {55, 5393}, {56, 5405}, {81, 3311}, {244, 8336}, {355, 6348}, {371, 940}, {372, 4383}, {488, 14555}, {517, 3084}, {1030, 8253}, {1385, 3083}, {1465, 13389}, {3068, 4254}, {3069, 5120}, {4423, 8225}, {5124, 8252}, {6199, 14996}, {6212, 13388}, {6395, 14997}

X(16433) = {X(2),X(3)}-harmonic conjugate of X(16432)


X(16434) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(511)

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

X(16434) lies on these lines:
{2, 3}, {40, 5272}, {81, 5050}, {98, 8690}, {104, 9104}, {105, 6244}, {165, 15485}, {182, 940}, {197, 3035}, {499, 8193}, {511, 4383}, {517, 614}, {572, 5275}, {612, 1385}, {908, 1473}, {952, 10327}, {956, 4737}, {982, 6211}, {1482, 7191}, {1486, 3816}, {1766, 3290}, {2783, 4387}, {3011, 11249}, {3086, 12410}, {3576, 5268}, {3705, 5100}, {3920, 10246}, {4254, 7736}, {5120, 7735}, {5121, 10310}, {5552, 8192}, {7292, 12702}, {10200, 11365}, {10519, 14555}

X(16434) = {X(2),X(3)}-harmonic conjugate of X(19544)


X(16435) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(573)

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

X(16435) lies on these lines:
{2, 3}, {6, 1764}, {9, 10856}, {40, 2999}, {165, 238}, {198, 3452}, {517, 5256}, {572, 940}, {573, 4383}, {956, 11679}, {958, 10882}, {986, 2944}, {1001, 10434}, {1220, 10465}, {1385, 5287}, {1449, 12555}, {1465, 10319}, {1746, 5737}, {1754, 5085}, {1766, 3666}, {3008, 5584}, {3687, 5687}, {4679, 15494}, {5786, 10479}


X(16436) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(574)

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

X(16436) lies on these lines:
{2, 3}, {81, 1384}, {187, 940}, {574, 4383}, {599, 1030}, {980, 3053}, {1444, 1992}, {2223, 4428}, {3912, 5217}, {5023, 5337}, {5222, 5303}, {5739, 6390}

X(16436) = {X(2),X(3)}-harmonic conjugate of X(16431)


X(16437) =  (X(1),X(2),X(3),X(6); X(3),X(37),X(2),X(31)) COLLINEATION IMAGE OF X(55)

Barycentrics    a^6 + 2 a^5 b + 4 a^4 b^2 + 5 a^3 b^3 + 3 a^2 b^4 + a b^5 + 2 a^5 c + 9 a^4 b c + 16 a^3 b^2 c + 16 a^2 b^3 c + 8 a b^4 c + b^5 c + 4 a^4 c^2 + 16 a^3 b c^2 + 24 a^2 b^2 c^2 + 17 a b^3 c^2 + 4 b^4 c^2 + 5 a^3 c^3 + 16 a^2 b c^3 + 17 a b^2 c^3 + 6 b^3 c^3 + 3 a^2 c^4 + 8 a b c^4 + 4 b^2 c^4 + a c^5 + b c^5 : :

X(16437) lies on these lines:
{2, 3}, {3739, 3745}


X(16438) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(583)

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

X(16438) lies on these lines:
{2, 3}, {198, 5249}, {354, 5256}, {394, 1730}, {583, 4383}, {584, 940}, {2262, 6505}, {2999, 3338}, {3306, 15509}, {5880, 15494}

X(16438) = {X(2),X(3)}-harmonic conjugate of X(16439)


X(16439) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(584)

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

X(16439) lies on these lines:
{2, 3}, {354, 5287}, {583, 940}, {584, 4383}, {1001, 5314}

X(16439) = {X(2),X(3)}-harmonic conjugate of X(16438)


X(16440) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(1151)

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

X(16440) lies on these lines:
{2, 3}, {35, 5405}, {36, 5393}, {40, 3083}, {81, 372}, {176, 13437}, {487, 5739}, {491, 1444}, {515, 6347}, {590, 5124}, {615, 1030}, {940, 1152}, {1151, 4383}, {1621, 8225}, {2178, 6351}, {3084, 3576}, {4254, 7586}, {5120, 7585}, {6221, 14997}, {6348, 6684}, {6398, 14996}, {7133, 13389}


X(16441) =  (X(1),X(2),X(3),X(6); X(6),X(1),X(3),X(2)) COLLINEATION IMAGE OF X(1152)

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

X(16441) lies on these lines:
{2, 3}, {35, 5393}, {36, 5405}, {40, 3084}, {81, 371}, {175, 13459}, {488, 5739}, {492, 1444}, {515, 6348}, {590, 1030}, {615, 5124}, {940, 1151}, {1152, 4383}, {2178, 6352}, {3083, 3576}, {4254, 7585}, {5120, 7586}, {5284, 8225}, {6221, 14996}, {6347, 6684}, {6398, 14997}


X(16442) =  (X(1),X(6),X(513),X(514); X(1),X(6),X(2),X(3)) COLLINEATION IMAGE OF X(512)

Barycentrics    a(a^6 b - 2 a^5 b^2 + 2 a^3 b^4 - a^2 b^5 + a^6 c - 5 a^4 b^2 c + a^3 b^3 c - 5 a b^5 c - 2 a^5 c^2 - 5 a^4 b c^2 - 2 a^3 b^2 c^2 + 11 a^2 b^3 c^2 - 2 a b^4 c^2 - 4 b^5 c^2 + a^3 b c^3 + 11 a^2 b^2 c^3 + 10 a b^3 c^3 + 2 a^3 c^4 - 2 a b^2 c^4 - a^2 c^5 - 5 a b c^5 - 4 b^2 c^5) : :

X(16442) lies on this line: {2, 3}


X(16443) =  (X(1),X(6),X(513),X(514); X(1),X(6),X(2),X(3)) COLLINEATION IMAGE OF X(521)

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

X(16443) lies on this line: {2, 3}


X(16444) =  (X(1),X(6),X(513),X(514); X(1),X(6),X(2),X(3)) COLLINEATION IMAGE OF X(522)

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

X(16444) lies on these lines:
{2, 3}, {1026, 3295}


X(16445) =  (X(1),X(3),X(6),X(2); X(3),X(2),X(31),X(37)) COLLINEATION IMAGE OF X(56)

Barycentrics    a^6 - 4 a^4 b^2 - 7 a^3 b^3 - 5 a^2 b^4 - a b^5 - 7 a^4 b c - 18 a^3 b^2 c - 18 a^2 b^3 c - 8 a b^4 c - b^5 c - 4 a^4 c^2 - 18 a^3 b c^2 - 28 a^2 b^2 c^2 - 17 a b^3 c^2 - 4 b^4 c^2 - 7 a^3 c^3 - 18 a^2 b c^3 - 17 a b^2 c^3 - 6 b^3 c^3 - 5 a^2 c^4 - 8 a b c^4 - 4 b^2 c^4 - a c^5 - b c^5 : :

X(16445) lies on these lines:
{2, 3}, {5153, 10458}


X(16446) =  (X(1),X(6),X(513),X(514); X(1),X(6),X(2),X(3)) COLLINEATION IMAGE OF X(900)

Barycentrics    a^2 (a^5 b - 2 a^4 b^2 + 2 a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - 2 a^4 c^2 - a^3 b c^2 - 10 a^2 b^2 c^2 + 11 a b^3 c^2 - 10 b^4 c^2 + a^2 b c^3 + 11 a b^2 c^3 + 10 b^3 c^3 + 2 a^2 c^4 - 10 b^2 c^4 - a c^5 - b c^5) : :

X(16446) lies on these lines:
{2, 3}, {1026, 3303}


X(16447) =  (X(1),X(3),X(6),X(513); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(35)

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

X(16447) lies on these lines:
{2, 3}, {56, 4564}


X(16448) =  (X(1),X(3),X(6),X(513); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(36)

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

X(16448) lies on these lines:
{2, 3}, {3286, 16500}


X(16449) =  (X(1),X(3),X(6),X(513); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(55)

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

X(16449) lies on this line: {2, 3}


X(16450) =  (X(1),X(3),X(6),X(513); X(2),X(3),X(6),X(1)) COLLINEATION IMAGE OF X(56)

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

X(16450) lies on this line: {2, 3}


X(16451) =  (X(1),X(2),X(3),X(6); X(2),X(1),X(3),X(6)) COLLINEATION IMAGE OF X(35)

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

X(16451) lies on these lines:
{2, 3}, {35, 750}, {36, 386}, {41, 579}, {56, 5132}, {100, 10449}, {172, 4261}, {228, 3868}, {387, 7742}, {500, 3060}, {580, 1790}, {582, 5012}, {986, 3724}, {1193, 5156}, {1724, 4278}, {2352, 5262}, {2975, 9534}, {3216, 7280}, {5303, 15654}


X(16452) =  (X(1),X(2),X(3),X(6); X(2),X(1),X(3),X(6)) COLLINEATION IMAGE OF X(36)

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

X(16452) lies on these lines:
{2, 3}, {31, 35}, {36, 10448}, {60, 2278}, {100, 9534}, {228, 3876}, {283, 572}, {500, 2979}, {579, 2268}, {993, 10479}, {1724, 4276}, {2309, 5156}, {2975, 10449}, {3216, 5010}, {4257, 10457}, {4278, 10458}, {4420, 15624}, {5132, 5217}, {5250, 10434}, {5495, 13340}


X(16453) =  (X(1),X(2),X(3),X(6); X(2),X(1),X(3),X(6)) COLLINEATION IMAGE OF X(55)

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

X(16453) lies on these lines:
{1, 5132}, {2, 3}, {36, 3216}, {46, 3185}, {51, 500}, {56, 181}, {184, 582}, {197, 1714}, {198, 218}, {228, 942}, {387, 1617}, {580, 1437}, {956, 9534}, {1376, 10479}, {1626, 8185}, {1724, 3286}, {2178, 4261}, {2183, 4303}, {2283, 4347}, {2360, 13329}, {3556, 10076}, {3996, 5687}, {4557, 5904}, {5156, 16466}, {5204, 15654}, {5259, 8053}, {11499, 15623}, {11500, 15626}

X(16453) = {X(2),X(3)}-harmonic conjugate of X(16287)


X(16454) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(31)) COLLINEATION IMAGE OF X(55)

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

X(16454) lies on these lines:
{1, 3896}, {2, 3}, {8, 940}, {10, 750}, {41, 5750}, {58, 5278}, {78, 5736}, {81, 9534}, {321, 975}, {612, 4968}, {894, 3876}, {965, 5749}, {1038, 1441}, {1125, 3914}, {1220, 5737}, {2292, 3980}, {3616, 4000}, {3664, 4101}, {3701, 5268}, {3739, 4372}, {4340, 5739}, {4385, 5297}, {5061, 9552}


X(16455) =  (X(1),X(2),X(3),X(6); X(2),X(1),X(3),X(6)) COLLINEATION IMAGE OF X(65)

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

X(16455) lies on these lines:
{2, 3}, {55, 5292}, {65, 6051}, {228, 5791}, {386, 2646}, {1479, 8053}, {1714, 5132}, {2217, 5251}, {3216, 3612}, {4261, 5336}


X(16456) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(6)) COLLINEATION IMAGE OF X(35)

Barycentrics    a^4 + 4 a^3 b + 7 a^2 b^2 + 4 a b^3 + 4 a^3 c + 16 a^2 b c + 16 a b^2 c + 4 b^3 c + 7 a^2 c^2 + 16 a b c^2 + 8 b^2 c^2 + 4 a c^3 + 4 b c^3 : :

X(16456) lies on these lines:
{2, 3}, {10, 4445}, {940, 1698}, {3634, 5737}, {3927, 10436}


X(16457) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(6)) COLLINEATION IMAGE OF X(36)

Barycentrics    a^4 - 4 a^3 b - 9 a^2 b^2 - 4 a b^3 - 4 a^3 c - 16 a^2 b c - 16 a b^2 c - 4 b^3 c - 9 a^2 c^2 - 16 a b c^2 - 8 b^2 c^2 - 4 a c^3 - 4 b c^3 : :

X(16457) lies on these lines:
{2, 3}, {940, 1203}, {1125, 5737}, {1150, 5550}, {1764, 3646}, {5257, 11374}


X(16458) =  (X(1),X(2),X(3),X(6); X(3),X(1),X(2),X(6)) COLLINEATION IMAGE OF X(55)

Barycentrics    a^4 + 2 a^3 b + 3 a^2 b^2 + 2 a b^3 + 2 a^3 c + 8 a^2 b c + 8 a b^2 c + 2 b^3 c + 3 a^2 c^2 + 8 a b c^2 + 4 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(16458) lies on these lines:
{1, 3696}, {2, 3}, {10, 940}, {72, 10436}, {86, 9534}, {218, 965}, {894, 15650}, {966, 4340}, {1150, 9780}, {1698, 5247}, {1714, 6703}, {3753, 10441}, {4292, 5257}, {4687, 7283}, {5287, 5295}


X(16459) =  PERSPECTOR OF ABC AND 5th FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(5*S^2+sqrt(3)*(SB+SW)*S-3*SA*SC)*(5*S^2+sqrt(3)*(SC+SW)*S-3*SA*SB) : :    (César Lozada, March 14, 2018)

Let P be a point in the plane of a triangle ABC. Let Ba and Ca be points on BC such that the triangle PBaCa is equilateral. Define the pairs {Cb, Ab} and {Ab, Bc}cyclically. Let A' be the point other than P that lies on the circles (PAcBc) and (PCbAb), and define B' and C' cyclically. If P = X(13) or P = X(14) then the triangle A'B'C' is equilateral. (Dao Thanh Oai, March 7, 2018). The converse is true; i.e., if A'B'C' is equilateral, then P = X(13) or P = X(14). (Dao Thanh Oai, March 14, 2018)

If P = X(13), then A'B'C' is the 5th Fermat-Dao equilateral triangle. If P = X(14), then A'B'C' is the 6th Fermat-Dao equilateral triangle. See also X(16247) and X(16267).

Barycentrics for the A-vertex of the 5th Fermat-Dao equilateral triangle:

A' = ((7 SA + 4 SW)*S^2 + 5 sqrt(3) S (S^2 + SA^2) + 3 SA^3)/(S + sqrt(3) SA)^2 :
   (sqrt(3)(SA + SB) + 2 S)(SC + SA)/(S + sqrt(3) SB) :
   (sqrt(3)(SC + SA) + 2 S)(SA + SB)/(S + sqrt(3) SC).

Barycentrics for the A-vertex of the 6th Fermat-Dao equilateral triangle:

A' = ((7 SA + 4 SW)*S^2 - 5 sqrt(3) S (S^2 + SA^2) + 3 SA^3)/(- S + sqrt(3) SA)^2 :
   (sqrt(3)(SA + SB) - 2 S)(SC + SA)/(- S + sqrt(3) SB) :
   (sqrt(3)(SC + SA) - 2 S)(SA + SB)/(- S + sqrt(3) SC).

(These barycentrics and centers X(16461)-X(16466) contributed by César Lozada, March 14, 2018)

Click here for an image of the 5th Fermat-Dao equilateral triangle.

You can view 5th and 6th Fermat-Dao equilateral triangles.

X(16459) lies on the cubic K261a and these lines:
{13, 298}, {14, 8014}, {15, 1337}, {2380, 5995}, {2381, 5612}, {8603, 11081}

X(16459) = isogonal conjugate of X(618)
X(16459) = trilinear pole of the line X(6137)X(11081)
X(16459) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(15)


X(16460) =  PERSPECTOR OF ABC AND 6th FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(5*S^2-sqrt(3)*(SB+SW)*S-3*SA*SC)*(5*S^2-sqrt(3)*(SC+SW)*S-3*SA*SB) : :    (César Lozada, March 14, 2018)

See X(16459).

X(16460) lies on the cubic K261b and these lines:
{13, 8015}, {14, 299}, {16, 1338}, {2380, 5616}, {2381, 5994}, {8604, 11086}

X(16460) = isogonal conjugate of X(619)
X(16460) = trilinear pole of the line X(6138)X(11086)
X(16460) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(16)


X(16461) =  CENTER OF 5TH FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(sqrt(3)*(6*S^4+3*(SA^2+2*SB*SC-SW^2)*S^2-(SA^2-SB*SC)^2)-2*S*(3*SA-SW)*(SA^2-SB*SC)) : :

X(16461) lies on these lines: {4, 11581}, {13, 511}, {568, 11555}, {5640, 8014}, {11080, 11624}


X(16462) =  CENTER OF 6TH FERMAT-DAO EQUILATERAL TRIANGLE

Barycentrics    (SB+SC)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(6*S^4+3*(SA^2+2*SB*SC-SW^2)*S^2-(SA^2-SB*SC)^2)+2*S*(3*SA-SW)*(SA^2-SB*SC)) : :

X(16462) lies on these lines: {4, 11582}, {14, 511}, {568, 11556}, {5640, 8015}, {11085, 11626}


X(16463) =  PERSPECTOR OF THESE TRIANGLES: 5TH FERMAT-DAO EQUILATERAL AND OUTER LE VIET AN

Barycentrics    (SB+SC)*(sqrt(3)*SB+S)*(sqrt(3)*SC+S)*(sqrt(3)*(S^2+SB*SC)-(6*R^2+SA-3*SW)*S) : :
Barycentrics    Sin[A]^2 Csc[A + Pi/3]^2 Cos[A - Pi/3] : :    (Peter Moses, March 15, 2018)

X(16463) lies on these lines:
{1495, 3457}, {3129, 11063}, {3130, 8014}, {3458, 11084}, {6104, 6671}, {11083, 11137}


X(16464) =  PERSPECTOR OF THESE TRIANGLES: 6TH FERMAT-DAO EQUILATERAL AND INNER LE VIET AN

Barycentrics    (SB+SC)*(sqrt(3)*SB-S)*(sqrt(3)*SC-S)*(sqrt(3)*(S^2+SB*SC)+(6*R^2+SA-3*SW)*S) : :
Barycentrics    Sin[A]^2 Csc[A - Pi/3]^2 Cos[A + Pi/3] : :    (Peter Moses, March 15, 2018)

X(16464) lies on these lines:
{1495, 3458}, {3130, 11063}, {3457, 11089}, {6105, 6672}, {11088, 11134}


X(16465) =  X(1)X(394)∩X(2)X(955)

Barycentrics    a (a^4 b-2 a^3 b^2+2 a b^4-b^5+a^4 c-2 a^2 b^2 c+b^4 c-2 a^3 c^2-2 a^2 b c^2+2 a c^4+b c^4-c^5) : :
Barycentrics    Sin[A]^2 Csc[A + Pi/3]^2 Cos[A - Pi/3] : :
X(16465) = 3 X[354] - 2 X[2886], X[3434] - 3 X[3873], 3 X[3873] - 2 X[5173], 3 X[210] - 4 X[6690], 3 X[2] - 4 X[11018]

X(16465) lies on these lines:
{1,394}, {2,955}, {3,14054}, {7,3434}, {10,10122}, {20,145}, {21,72}, {27,295}, {42,11031}, {55,63}, {56,224}, {57,1004}, {210,6690}, {329,10394}, {354,2886}, {377,942}, {674,3313}, {758,4304}, {908,1864}, {912,1012}, {936,10399}, {940,2000}, {962,9960}, {971,5905}, {1259,3811}, {1320,9964}, {1331,4641}, {1858,12635}, {2099,11520}, {3059,8255}, {3060,14557}, {3218,7411}, {3428,10884}, {3681,5273}, {3869,4313}, {3874,4292}, {3881,12446}, {3889,11036}, {4197,5439}, {4666,5572}, {4863,5832}, {5045,10529}, {5046,9844}, {5250,12710}, {5284,10177}, {5777,6837}, {5785,10582}, {5842,12671}, {5927,10883}, {7191,14523}, {7680,14872}, {10373,12111}, {10430,12669}, {10477,16368}, {10527,16193}, {10530,13373}, {10569,10861}, {10587,16201}, {12437,15556}, {12536,14923}

X(16465) = reflection of X(i) in X(j) for these {i,j}: {63, 10391}, {3059, 8255}, {3419, 942}, {3428, 12675}, {3434, 5173}, {14872, 7680}
X(16465) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {269, 2894}, {2982, 329}, {15439, 4468}
X(16465) = crosspoint of X(4567) and X(4569)
X(16465) = crosssum of X(3125) and X(8641)
X(16465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5784, 5249), (1071, 3555, 3868), (3434, 3873, 5173), (3868, 11220, 9965), (4430, 9965, 3868)


X(16466) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(8)

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

X(16466) lies on these lines:
{1, 6}, {2, 5711}, {3, 31}, {4, 3195}, {5, 5230}, {8, 13740}, {10, 3966}, {11, 5292}, {28, 608}, {35, 3052}, {36, 4252}, {38, 3927}, {40, 2999}, {42, 3295}, {43, 5255}, {46, 3752}, {47, 8071}, {55, 386}, {56, 58}, {57, 221}, {73, 1617}, {78, 5266}, {81, 3616}, {109, 1466}, {171, 474}, {198, 4264}, {206, 3556}, {244, 5708}, {278, 3194}, {387, 497}, {443, 4307}, {496, 11269}, {580, 3428}, {603, 1450}, {607, 8743}, {609, 3207}, {612, 5044}, {614, 942}, {651, 3600}, {672, 9605}, {748, 11108}, {750, 16408}, {899, 9709}, {936, 5269}, {940, 1125}, {946, 5706}, {957, 961}, {962, 5222}, {975, 3745}, {976, 3940}, {982, 1046}, {988, 1707}, {990, 12688}, {991, 8273}, {999, 1201}, {1012, 3073}, {1040, 12711}, {1042, 1471}, {1072, 5812}, {1096, 1871}, {1126, 2334}, {1149, 7373}, {1376, 3216}, {1399, 1470}, {1400, 13737}, {1407, 3361}, {1420, 2003}, {1448, 1456}, {1451, 1457}, {1479, 1834}, {1480, 4642}, {1621, 16289}, {1697, 7074}, {1714, 2886}, {1722, 3753}, {1914, 2271}, {2098, 15955}, {2178, 4275}, {2275, 5021}, {2276, 14974}, {2298, 5783}, {2594, 11510}, {2650, 15934}, {2901, 4387}, {2915, 5329}, {2951,16936}, {2964, 14793}, {3011, 11374}, {3017, 11238}, {3072, 3149}, {3187, 3702}, {3192, 11398}, {3240, 3871}, {3293, 3913}, {3562, 14986}, {3617, 14997}, {3666, 12514}, {3671, 5228}, {3720, 16345}, {3744, 3811}, {3746, 5312}, {3755, 10624}, {3759, 4673}, {3772, 12047}, {3868, 7191}, {3869, 5262}, {3876, 3920}, {3914, 12699}, {3931, 5250}, {4000, 4295}, {4202, 6327}, {4256, 5217}, {4257, 5204}, {4258, 7031}, {4259, 4749}, {4298, 6180}, {4361, 4647}, {4388, 16062}, {4640, 4719}, {4646, 5119}, {4676, 7283}, {4766, 7866}, {5156, 16453}, {5263, 9534}, {5272, 5439}, {5396, 10267}, {5398, 11249}, {5584, 13329}, {5707, 5886}, {5718, 10198}, {7083, 13730}, {9370, 10106}, {9708, 10459}, {10448, 16418}, {11321, 14621}


X(16467) =  (X(1),X(2),X(6),X(99); X(1),X(6),X(2),X(2)) COLLINEATION IMAGE OF X(8)

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

X(16467) lies on this line: {1,6}


X(16468) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(43)

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

X(16468) lies on these lines:
{1, 6}, {2, 2308}, {31, 43}, {42, 8616}, {57, 1929}, {58, 87}, {75, 4672}, {81, 748}, {86, 3624}, {171, 4383}, {182, 6210}, {239, 3923}, {312, 3791}, {386, 2309}, {519, 3790}, {527, 4989}, {595, 2209}, {597, 4026}, {651, 1471}, {672, 3097}, {673, 4312}, {740, 3759}, {751, 3736}, {752, 4429}, {846, 5256}, {894, 16825}, {896, 4850}, {899, 14997}, {902, 3240}, {940, 9332}, {982, 4641}, {990, 9355}, {997, 5429}, {1125, 4416}, {1193, 5145}, {1281, 10353}, {1423, 1428}, {1445, 5018}, {1699, 1746}, {1707, 2999}, {1740, 2228}, {1742, 13329}, {1918, 3293}, {1961, 3305}, {1999, 4011}, {2163, 4604}, {2307, 10647}, {3244, 4899}, {3271, 4260}, {3286, 7280}, {3361, 7175}, {3501, 12194}, {3629, 4966}, {3632, 4535}, {3666, 7262}, {3679, 5263}, {3752, 4650}, {3764, 4283}, {3873, 4722}, {3993, 4393}, {4038, 4423}, {4253, 5144}, {4366, 16834}, {4384, 14621}, {4417, 6679}, {4655, 16706}, {4716, 5695}, {5010, 5132}, {5264, 6048}, {5332, 8299}, {6419, 8225}, {9340, 9352}


X(16469) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(200)

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

X(16469) lies on these lines:
{1, 6}, {2, 4349}, {10, 4344}, {31, 165}, {56, 1419}, {57, 1456}, {58, 269}, {81, 10582}, {144, 4353}, {386, 2293}, {516, 5222}, {595, 1253}, {614, 2308}, {651, 4321}, {985, 8056}, {990, 3062}, {991, 1193}, {995, 1458}, {1103, 1497}, {1125, 3945}, {1699, 3332}, {1992, 4684}, {2263, 3339}, {3008, 4307}, {3241, 4899}, {3244, 4929}, {3293, 4097}, {3616, 4416}, {3618, 3883}, {3624, 4648}, {3664, 16020}, {3677, 4641}, {3685, 16834}, {3745, 7308}, {3759, 3886}, {3875, 4676}, {3946, 5698}, {4000, 4312}, {4383, 5269}, {4512, 5256}, {4974, 16833}, {5132, 16688}, {5230, 7989}, {5262, 12526}, {5293, 8951}, {5563, 15287}


X(16470) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(306)

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

X(16470) lies on these lines:
{1, 6}, {19, 8743}, {31, 579}, {32, 2277}, {35, 4261}, {36, 1333}, {41, 5037}, {48, 995}, {57, 608}, {58, 1474}, {71, 595}, {81, 4001}, {251, 1400}, {284, 1193}, {380, 2999}, {572, 1064}, {573, 602}, {604, 10571}, {609, 2178}, {672, 4284}, {1125, 2303}, {1172, 1848}, {1420, 2286}, {1423, 2003}, {1761, 3670}, {1765, 3073}, {1838, 5317}, {1914, 2092}, {2078, 2197}, {2268, 5105}, {2275, 5019}, {2280, 4270}, {2281, 3496}, {2298, 4071}, {2359, 5053}, {3694, 3744}, {4329, 5222}, {5257, 5276}, {5279, 7191}, {5332,16946}, {7295, 9591}, {16547, 16583}


X(16471) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(387)

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

X(16471) lies on these lines:
{1, 6}, {3, 1779}, {5, 1714}, {25, 10974}, {31, 3682}, {46, 11347}, {56, 3173}, {58, 394}, {65, 7535}, {155, 5398}, {379, 4295}, {386, 2328}, {387, 2478}, {579, 2360}, {580, 1181}, {595, 3190}, {940, 6675}, {946, 1751}, {1036, 1794}, {1498, 1754}, {1737, 7532}, {2175, 8193}, {2982, 3485}, {3157, 4641}, {3293, 7074}, {3332, 6835}, {4259, 13730}, {5187, 14997}, {5453, 16266}, {5707, 6861}, {5721, 6928}, {7522, 12047}, {12514, 16368}


X(16472) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(498)

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

X(16472) lies on these lines:
{1, 6}, {10, 5422}, {47, 2308}, {51, 8185}, {58, 14793}, {81, 499}, {155, 8227}, {222, 3337}, {323, 5550}, {394, 3624}, {569, 15177}, {920, 5256}, {946, 7592}, {1125, 1993}, {1181, 1699}, {1199, 5603}, {1698, 10601}, {1994, 3616}, {2003, 3338}, {2964, 11507}, {3583, 5706}, {3796, 9591}, {3817, 11441}, {4252, 14792}, {5050, 8193}, {5292, 8070}, {5691, 10982}, {5707, 7741}, {5708, 8614}, {5886, 12161}, {9777, 9798}, {9780, 15018}, {10246, 14627}, {11365, 11402}, {12702, 15037}


X(16473) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(499)

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

X(16473) lies on these lines:
{1, 6}, {8, 1994}, {10, 1993}, {42, 47}, {46, 2003}, {52, 15177}, {55, 2964}, {57, 1079}, {65, 15317}, {79, 8757}, {81, 498}, {154, 9587}, {155, 5587}, {184, 8185}, {195, 5790}, {222, 3336}, {323, 9780}, {355, 12161}, {386, 14793}, {394, 1698}, {515, 7592}, {944, 1199}, {1125, 5422}, {1181, 5691}, {1351, 8193}, {1482, 14627}, {1699, 10982}, {2594, 5398}, {2807, 11424}, {3157, 5902}, {3585, 5706}, {3617, 11004}, {3624, 10601}, {3634, 15066}, {4255, 14792}, {4354, 10394}, {5093, 12410}, {5270, 9370}, {5292, 8068}, {5550, 15018}, {5707, 7951}, {6149, 11507}, {9777, 11365}, {9798, 11402}


X(16474) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(551)

Barycentrics    a^2 (a^2 + 2 a b + b^2 + 2 a c + 5 b c + c^2) : :

X(16474) lies on these lines:
{1, 6}, {3, 2163}, {8, 14996}, {35, 1468}, {36, 42}, {56, 5312}, {58, 902}, {81, 519}, {106, 1126}, {386, 5563}, {517, 2941}, {940, 3679}, {999, 5313}, {1201, 5643}, {1480, 7982}, {1834, 5270}, {1999, 4692}, {2003, 2099}, {2276, 9346}, {2650, 6126}, {3218, 4868}, {3336, 4646}, {3616, 14997}, {3632, 5711}, {3633, 5710}, {3881, 5262}, {3892, 7191}, {3931, 6763}, {3979, 5429}, {4424, 4880}, {4658, 10459}, {5707, 5881}, {7951, 11269}


X(16475) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(612)

Barycentrics    a (-3 a^2 - 2 a b - b^2 - 2 a c - c^2) : :
X(16475) = X(1) + 2 X(6)

X(16475) lies on these lines:
{1, 6}, {2, 5847}, {10, 3618}, {31, 5256}, {35, 16688}, {40, 182}, {42, 3749}, {43, 5269}, {46, 5135}, {57, 985}, {58, 988}, {63, 2308}, {69, 1125}, {77, 1471}, {81, 614}, {141, 3624}, {142, 4989}, {145, 3790}, {165, 5085}, {171, 2999}, {193, 3616}, {214, 10755}, {354, 3167}, {511, 3576}, {515, 14853}, {517, 5050}, {551, 1992}, {575, 7982}, {597, 3679}, {651, 4327}, {740, 16834}, {748, 5287}, {895, 11720}, {940, 5272}, {946, 6776}, {1051, 9052}, {1054, 8297}, {1193, 7032}, {1350, 7987}, {1351, 1385}, {1352, 8227}, {1353, 5901}, {1419, 4334}, {1420, 1469}, {1437, 3338}, {1456, 5228}, {1479, 5800}, {1503, 1699}, {1572, 1692}, {1682, 3056}, {1691, 10789}, {1697, 2330}, {1698, 3416}, {1707, 3666}, {1722, 5711}, {1738, 4307}, {1961, 7308}, {1974, 7713}, {2030, 5184}, {2274, 3248}, {2802, 5150}, {2948, 6593}, {3008, 4349}, {3097, 13331}, {3158, 3795}, {3305, 5311}, {3564, 5886}, {3579, 12017}, {3612, 4259}, {3620, 5550}, {3632, 6329}, {3685, 4393}, {3729, 4672}, {3745, 4383}, {3746, 12329}, {3759, 5263}, {3791, 11679}, {3827, 5902}, {3875, 3923}, {3886, 4991}, {3894, 9021}, {3941, 5132}, {3945, 16020}, {4038, 10582}, {4252, 4719}, {4265, 7280}, {4360, 4676}, {4384, 4974}, {4852, 5695}, {5010, 5096}, {5026, 13174}, {5028, 9619}, {5034, 9620}, {5039, 11364}, {5093, 10246}, {5095, 11735}, {5219, 12588}, {5297, 14997}, {5313, 9024}, {5347, 7298}, {5477, 11725}, {5480, 5691}, {5587, 14561}, {5603, 14912}, {5848, 16173}, {7221, 10394}, {7292, 14996}, {7988, 10516}, {8540, 13384}, {8550, 11522}, {8593, 12258}, {10165, 10519}, {10436, 16825}, {10752, 11709}, {10753, 11710}, {10754, 11711}, {10756, 11712}, {10757, 11713}, {10758, 11714}, {10759, 11715}, {10760, 11716}, {10761, 11717}, {10762, 11718}, {10763, 11719}, {10764, 11700}, {10765, 11721}, {10766, 11722}, {11008, 15808}, {11061, 13605}, {11363, 12167}, {11482, 15178}, {12723, 15076}, {13211, 15118}

X(16475) = {X(1),X(6)}-harmonic conjugate of X(3751)


X(16476) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(869)

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

X(16476) lies on these lines:
{1, 6}, {31, 239}, {32, 8300}, {41, 11364}, {43, 2223}, {56, 2665}, {58, 274}, {171, 4384}, {291, 4253}, {583, 4446}, {595, 3802}, {672, 12782}, {748, 16826}, {750, 16815}, {978, 3510}, {1468, 16823}, {1471, 7176}, {1582, 5019}, {1918, 3759}, {2664, 4383}, {3293, 3795}, {3747, 4393}, {3780, 8299}, {4257, 8297}, {4443, 4749}, {4974, 5156}


X(16477) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(899)

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

X(16477) lies on these lines:
{1, 6}, {31, 3240}, {43, 16396}, {58, 662}, {145, 4439}, {171, 899}, {239, 4672}, {748, 4038}, {750, 14997}, {894, 4974}, {2999, 4650}, {3286, 9343}, {3624, 4708}, {3626, 5263}, {3736,16948}, {3758, 16825}, {3759, 3923}, {4003, 4641}, {4026, 6329}, {4360, 4991}, {4676, 4693}, {4722, 7191}, {5050, 6210}, {5256, 7262}, {14621, 16816}


X(16478) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(976)

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

X(16478) lies on these lines:
{1, 6}, {28, 985}, {31, 986}, {43, 5266}, {56, 5429}, {58, 977}, {1010, 16825}, {1445, 4348}, {1468, 3976}, {1471, 4296}, {1722, 5269}, {1961, 11108}, {2308, 3868}, {3670, 4650}, {3769, 3831}, {3791, 10449}, {3931, 8616}, {4339, 5222}, {4362, 13740}, {4383, 5293}, {4974, 9534}, {5047, 5311}, {5256, 16367}, {5272, 16852}


X(16479) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(1026)

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

X(16479) lies on these lines: {1, 6}, {58, 1019}, {4089, 4989}


X(16480) =  (X(1),X(2),X(6),X(99); X(1),X(6),X(2),X(99)) COLLINEATION IMAGE OF X(10)

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

X(16480) lies on these lines: {1, 6}, {58, 763}


X(16481) =  (X(1),X(2),X(6),X(99); X(1),X(6),X(2),X(99)) COLLINEATION IMAGE OF X(519)

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

X(16481) lies on these lines: {1, 6}, {106, 4622}, {551, 6629}


X(16482) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(513),X(2)) COLLINEATION IMAGE OF X(42)

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

X(16482) lies on these lines:
{1, 6}, {2, 513}, {765, 1621}, {846, 1052}, {2802, 4432}, {3110, 6789}, {3271, 4422}, {3908, 10755}, {4370, 14839}, {4670, 9318}, {4672, 5883}, {4890, 6329}, {5284, 6163}, {9359, 16726}, {11231, 15310}


X(16483) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(8)

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

X(16483) lies on these lines:
{1, 6}, {3, 902}, {8, 13741}, {31, 999}, {32, 8649}, {36, 3052}, {42, 6767}, {55, 995}, {56, 106}, {58, 3304}, {105, 957}, {145, 14997}, {221, 1420}, {222, 1319}, {386, 3303}, {474, 5255}, {496, 5230}, {517, 614}, {519, 4383}, {551, 940}, {739, 14074}, {748, 9708}, {978, 5687}, {997, 3744}, {1055, 1384}, {1125, 5710}, {1193, 2177}, {1334, 9605}, {1407, 13462}, {1457, 1617}, {1468, 7373}, {1482, 3924}, {1572, 3290}, {1714, 3813}, {1722, 10914}, {1870, 3195}, {2093, 5573}, {2163, 3445}, {2275, 14974}, {2742, 9097}, {3011, 5886}, {3187, 4742}, {3207, 7031}, {3216, 3913}, {3550, 16371}, {3616, 5711}, {3622, 14996}, {3746, 4255}, {3749, 5440}, {3752, 5119}, {3753, 5272}, {3877, 7191}, {3890, 5262}, {3938, 3940}, {4315, 6180}, {5007, 9351}, {5603, 7413}, {5706, 13464}, {7074, 7962}, {7280, 8572}, {8616, 16370}, {10246, 11203}, {10459, 11108}


X(16484) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(42)

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

X(16484) lies on these lines:
{1, 6}, {2, 2177}, {21, 8296}, {31, 4038}, {36, 8053}, {42, 5284}, {43, 4423}, {55, 16059}, {75, 4693}, {86, 99}, {105, 8297}, {171, 902}, {210, 3979}, {354, 846}, {740, 16823}, {748, 14997}, {756, 3957}, {894, 4432}, {940, 8616}, {968, 982}, {1054, 4689}, {1125, 1738}, {1149, 2309}, {1201, 3736}, {1319, 7175}, {1458, 8543}, {1721, 3576}, {1961, 3744}, {1962, 7191}, {2163, 10013}, {3286, 5563} ,{3485,16888}, {3550, 4428}, {3622, 4310}, {3631, 4966}, {3683, 4883}, {3722, 5297}, {3739, 4702}, {3746, 5132}, {3748, 3961}, {3915, 5156}, {4257, 5248}, {4366, 16826}, {4512, 4650}, {4716, 16825}, {4732, 16815}, {4859, 15668}, {4907, 13384}, {5268, 10389}, {5919, 13541}, {6048, 16842}, {6396, 8225}, {7274, 13462}, {8167, 16569}


X(16485) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(78)

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

X(16485) lies on these lines:
{1, 6}, {31, 11529}, {34, 106}, {36, 5573}, {40, 902}, {57, 4257}, {58, 11518}, {204, 1870}, {212, 7962}, {223, 1319}, {551, 5712}, {580, 3915}, {581, 1201}, {595, 1451}, {614, 3576}, {948, 4315}, {993, 3677}, {995, 13384}, {1125, 5716}, {1427, 13462}, {1714, 12625}, {2093, 3052}, {2163, 3338}, {2730, 9097}, {3011, 5587}, {3586, 3772}, {3601, 4256}, {3616, 5717}, {3663, 11111}, {3729, 13735}, {3744, 9623}, {4000, 4304}, {4054, 4217}, {4859, 11112}, {4906, 11194}, {5713, 9624}, {7987, 15852}, {8056, 16371}


X(16486) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(145)

Barycentrics    a^2 (a^2 + 2 a b + b^2 + 2 a c - 10 b c + c^2) : :

X(16486) lies on these lines:
{1, 6}, {3, 106}, {35, 8572}, {42, 8162}, {55, 1149}, {56, 902}, {221, 1388}, {595, 7373}, {614, 5919}, {995, 6767}, {999, 3052}, {1017, 9351}, {1201, 2177}, {1319, 1407}, {1385, 1480}, {2241, 3207}, {3241, 4383}, {3295, 4256}, {3304, 3915}, {3622, 5710}, {3623, 14997}, {3636, 5711}, {3756, 5657}, {3880, 5272}, {3895, 16610}, {5048, 7074}, {5573, 9819}, {7987, 15854}, {8168, 16569}, {8616, 11194}, {10912, 13541}


X(16487) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(200)

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

X(16487) lies on these lines:
{1, 6}, {31, 10980}, {36, 15287}, {56, 5575}, {106, 269}, {165, 614}, {551, 3945}, {595, 1471}, {991, 1201}, {995, 2293}, {1125, 4344}, {1253, 9819}, {1319, 1419}, {1420, 1456}, {2163, 2191}, {2177, 2999}, {2263, 3361}, {2736, 9097}, {3011, 7988}, {3052, 5573}, {3332, 11522}, {3550, 8056}, {3616, 4349}, {3619, 3883}, {3744, 8580}, {3870, 14997}, {3915, 7991}, {3924, 11531}, {3928, 4906}, {4512, 7191}, {4666, 14996}, {4684, 11008}


X(16488) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(306)

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

X(16488) lies on these lines:
{1, 6}, {36, 5301}, {106, 112}, {284, 1201}, {551, 2303}, {579, 3915}, {595, 2260}, {608, 1420}, {902, 5285}, {1333, 5563}, {1334, 4284}, {1421, 1880}, {1761, 3953}, {2178, 7031}, {2220, 8610}, {2241, 2277}, {3290, 16547}, {3746, 4261}, {5037, 9310}


X(16489) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(519)

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

X(16489) lies on these lines:
{1, 6}, {36, 106}, {244, 3245}, {995, 2177}, {999, 2163}, {1201, 3746}, {1480, 3576}, {1914, 8649}, {2802, 7292}, {3011, 16173}, {3241, 14997}, {3898, 7191}, {3915, 4257}, {4694, 4880}, {5313, 6767}, {5541, 16610}, {6126, 11720}, {9336, 14974}


X(16490) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(551)

Barycentrics    a^2 (a^2 + 2 a b + b^2 + 2 a c + 11 b c + c^2) : :

X(16490) lies on these lines:
{1, 6}, {36, 2177}, {42, 106}, {55, 2163}, {1480, 16200}, {3241, 14996}, {3746, 4257}, {4256, 5563}, {5312, 7373}, {6126, 7984}


X(16491) =  (X(2),X(1),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(612)

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

X(16491) lies on these lines:
{1, 6}, {40, 5092}, {69, 551}, {106, 1310}, {182, 7982}, {517, 12017}, {519, 3618}, {988, 4257}, {1125, 3619}, {1351, 15178}, {1352, 9624}, {1428, 3340}, {1503, 11522}, {1698, 5846}, {1738, 4344}, {2177, 3749}, {2330, 7962}, {3056, 13384}, {3098, 3576}, {3416, 3624}, {3589, 3679}, {3616, 3620}, {3636, 11008}, {3745, 5272}, {3920, 14997}, {5033, 10800}, {5050, 10222}, {5085, 7991}, {5138, 11518}, {5881, 14561}, {5882, 14853}, {6776, 13464}, {7191, 14996}, {9574, 12055}, {12722, 15076}


X(16492) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(8)

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

X(16492) lies on these lines: {1, 6}, {56, 4057}


X(16493) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(10)

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

X(16493) lies on these lines: {1, 6}, {58, 3733}, {4383, 9458}


X(16494) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(42)

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

X(16494) lies on these lines: {1, 6}, {86, 4833}, {4585, 5284}


X(16495) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(43)

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

X(16495) lies on these lines: {1, 6}, {75, 889}, {748, 4585}, {9362, 9458}


X(16496) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(614)

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

X(16496) lies on these lines:
{1, 6}, {8, 1738}, {10, 3619}, {36, 12329}, {38, 2177}, {40, 3098}, {43, 3677}, {57, 3961}, {63, 902}, {69, 519}, {106, 997}, {141, 3679}, {145, 5847}, {193, 3241}, {200, 982}, {210, 5272}, {354, 5268}, {511, 7982}, {517, 1721}, {537, 3729}, {551, 3618}, {599, 4677}, {612, 3873}, {614, 3681}, {726, 3886}, {756, 4666}, {846, 10389}, {912, 1480}, {936, 3976}, {968, 3957}, {971, 12652}, {975, 3881}, {986, 6765}, {988, 3811}, {1201, 3984}, {1282, 8297}, {1350, 7991}, {1351, 10222}, {1352, 5881}, {1385, 12017}, {1469, 3340}, {1707, 3744}, {2330, 13384}, {2810, 3022}, {3052, 16570}, {3094, 4050}, {3244, 11008}, {3333, 5293}, {3416, 3631}, {3550, 3928}, {3551, 3680}, {3576, 5092}, {3630, 3633}, {3711, 16610}, {3779, 11529}, {3782, 4863}, {3827, 5697}, {3915, 3951}, {3920, 4430}, {3929, 8616}, {3935, 4392}, {3953, 11512}, {3999, 4413}, {4259, 9049}, {4260, 11518}, {4327, 7672}, {4659, 9055}, {4661, 7191}, {5050, 15178}, {5119, 7289}, {5480, 11522}, {5573, 16569}, {5686, 16020}, {5727, 12589}, {5848, 7972}, {5882, 6776}, {8186, 12453}, {8187, 12452}, {8679, 15430}, {9024, 12653}, {9047, 11280}, {9620, 9997}, {9624, 14561}, {9819, 10387}, {10459, 11520}, {10519, 11362}, {11477, 16189}, {12721, 15076}, {13464, 14853}


X(16497) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(869)

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

X(16497) lies on these lines:
{1, 6}, {106, 789}, {239, 2177}, {574, 8334}, {3750, 16834}, {4256, 16825}, {4262, 8297}


X(16498) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(976)

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

X(16498) lies on these lines:
{1, 6}, {106, 833}, {902, 986}, {982, 4257}, {2163, 3953}, {2177, 5262}, {4881, 7191}


X(16499) =  (X(1),X(2),X(6),X(100); X(1),X(6),X(2),X(100)) COLLINEATION IMAGE OF X(995)

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

X(16499) lies on these lines:
{1, 6}, {2, 106}, {8, 4256}, {171, 2163}, {519, 1150}, {846, 13541}, {902, 993}, {988, 3987}, {1573, 8649}, {1739, 9623}, {2802, 4414}, {2975, 4257}, {3241, 4653}, {3244, 10448}, {3445, 16842}, {3632, 5774}, {3872, 4424}, {4745, 9350}, {5767, 5882}, {8666, 10459}


X(16500) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(386)

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

X(16500) lies on these lines:
{1, 6}, {3286, 16448}, {4049, 4778}, {4585, 5047}


X(16501) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(519)

Barycentrics    a^2 (a^3 b + 2 a^2 b^2 + a b^3 + a^3 c - 10 a^2 b c + 2 a b^2 c - 5 b^3 c + 2 a^2 c^2 + 2 a b c^2 + 8 b^2 c^2 + a c^3 - 5 b c^3) : :

X(16501) lies on these lines:
{1, 6}, {105, 739}, {106, 1960}, {840, 9097}


X(16502) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(8)

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

X(16502) lies on these lines:
{1, 6}, {3, 1914}, {5, 9599}, {11, 3767}, {12, 2548}, {30, 9597}, {31, 1475}, {32, 56}, {34, 2207}, {35, 5013}, {36, 3053}, {39, 55}, {41, 1201}, {65, 1572}, {81, 2221}, {115, 9665}, {169, 3290}, {172, 999}, {187, 5204}, {192, 7839}, {194, 4366} ,{198, 16946}, {222, 1424}, {230, 499}, {232, 11398}, {239, 304}, {279, 1462}, {330, 384}, {350, 7754}, {474, 4386}, {495, 9596}, {496, 5305}, {497, 5286}, {498, 3815}, {574, 5217}, {595, 4253}, {604, 608}, {607, 2170}, {609, 5563}, {614, 1184}, {672, 3915}, {738, 1407}, {748, 3691}, {978, 3684}, {982, 3496}, {995, 4251}, {1125, 5275}, {1149, 9310}, {1188, 9449}, {1193, 2271}, {1432, 7104}, {1459, 14825}, {1478, 7745}, {1479, 5254}, {1500, 3303}, {1570, 5148}, {1575, 5687}, {1611, 5272}, {1672, 12050}, {1692, 5194}, {1697, 9593}, {1759, 3953}, {1870, 8743}, {1909, 7770}, {1930, 4361}, {2023, 10053}, {2066, 6422}, {2067, 6424}, {2162, 3502}, {2172, 7113}, {2178, 2220}, {2242, 3304}, {2251, 3207}, {2276, 3295}, {2277, 4254}, {2 356, 3195}, {2440, 3669}, {2549, 6284}, {2646, 9619}, {3023, 10798}, {3052, 5022}, {3057, 9620}, {3058, 7739}, {3085, 7736}, {3086, 7735}, {3238, 12051}, {3509, 3976}, {3601, 9592}, {3616, 5276}, {3674, 5228}, {3912, 4383}, {3975, 11353}, {4294, 7738}, {4309, 9607}, {4354, 9594}, {4423, 16589}, {5010, 15815}, {5023, 7280}, {5024, 10987}, {5120, 12410}, {5301, 8193}, {5304, 14986}, {5306, 10072}, {5309, 11238}, {5359, 7191}, {5414, 6421}, {5475, 10895}, {6423, 6502}, {6645, 7787}, {7296, 7373}, {7297, 16545}, {7354, 7737}, {7741, 13881}, {7747, 9651}, {7748, 12953}, {7753, 9650}, {7765, 9664}, {9300, 10056}, {9456, 14260}, {9598, 15048}, {9654, 15484}, {10069, 12829}, {10311, 11399}, {11508, 13006}, {15668, 16818}

X(16502) = isogonal conjugate of X(30701)
X(16502) = crossdifference of every pair of points on line X(513)X(4468) (the polar of X(1851) wrt polar circle, and the perspectrix of Gemini triangles 35 and 37)
X(16502) = perspector of unary cofactor triangles of Gemini triangles 35 and 37


X(16503) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(42)

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

X(16503) lies on these lines:
{1, 6}, {2, 2280}, {7, 604}, {21, 1475}, {41, 3616}, {48, 5819}, {86, 142}, {87, 4335}, {101, 551}, {171, 1914}, {319, 3686}, {354, 3509}, {390, 2268}, {516, 572}, {672, 1621}, {894, 4366}, {910, 3742}, {942, 3496}, {966, 3775}, {978, 2271}, {1125, 4251}, {1400, 7677}, {1404, 8543}, {1445, 7146}, {1474, 1890}, {2241, 5255}, {2276, 3750}, {2278, 5880}, {2309, 4343}, {2346, 4876}, {2550, 4085}, {3169, 6600}, {3208, 3303}, {3241, 4390}, {3295, 3501}, {3589, 16593}, {3622, 9310}, {3691, 5047}, {3693, 3748}, {3720, 5276}, {3746, 16549}, {3873, 5282}, {3930, 3957}, {4038, 5332}, {4071, 4514}, {4253, 5248}, {4919, 5919}, {5011, 5883}, {5263, 5750}, {5749, 8236}


X(16504) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(614)

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

X(16504) lies on these lines:
{1, 6}, {141, 9458}, {3681, 4585}, {3729, 4777}, {4511, 9039}


X(16505) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(899)

Barycentrics    a (a^3 b + 5 a^2 b^2 - 2 a b^3 + a^3 c - 16 a^2 b c + 5 a b^2 c - 2 b^3 c + 5 a^2 c^2 + 5 a b c^2 + 2 b^2 c^2 - 2 a c^3 - 2 b c^3) : :

X(16505) lies on these lines:
{1, 6}, {88, 659}, {6165, 14637}


X(16506) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(995)

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

X(16506) lies on these lines:
{1, 6}, {190, 1320}, {214, 4557}, {514, 996}, {1150, 3711}, {1376, 15635}, {4436, 5541}


X(16507) =  (X(1),X(2),X(6),X(513); X(1),X(6),X(2),X(513)) COLLINEATION IMAGE OF X(1026)

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

X(16507) lies on these lines: {1, 6}, {105, 9456}, {244, 659}


X(16508) =  X(3)X(543)∩X(6)X(2482)

Barycentrics    7*a^8-20*(b^2+c^2)*a^6+3*(7*b^ 4+3*b^2*c^2+7*c^4)*a^4-5*(b^6+ c^6)*a^2+(b^4-4*b^2*c^2+c^4)*( b^2+c^2)^2 : :
X(16508) = 2 X(9890) + X(14830) = 2 X(11184) - 3 X(15561)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27320.

X(16508) lies on these lines:
{2, 9486}, {3, 543}, {6, 2482}, {30, 9877}, {83, 5503}, {99, 9136}, {194, 9741}, {351, 5652}, {524, 2080}, {597, 15483}, {620, 14535}, {690, 14653}, {3849, 6033}, {5485, 8587}, {5969, 7618}, {6321, 7615}, {7771, 11167}, {8182, 9830}, {8359, 9166}, {8592, 9889}, {11184, 15561}, {14928, 15655}

X(16508) = midpoint of X(i) and X(j) for these {i,j}: {5485, 8591}, {8182, 9890}
X(16508) = reflection of X(6321) in X(7615)


X(16509) =  COMPLEMENT OF X(11165)

Barycentrics   4*a^4-7*(b^2+c^2)*a^2+7*b^4- 22*b^2*c^2+7*c^4 : :
X(16509) = 3*X(2)+X(5485), 5*X(2)-X(9741), 9*X(2)-X(11148), 7*X(5)+2*X(7751), 11*X(5)-2*X(7759), 5*X(5)-2*X(7775), 3*X(5)-2*X(8176), 5*X(5485)+3*X(9741), 3*X(5485)+X(11148), 2*X(5485)+3*X(12040), 7*X(7617)+X(7751), 11*X(7617)-X(7759), 5*X(7617)-X(7775), 3*X(7617)-X(8176), 11*X(7751)+7*X(7759), 5*X(7751)+7*X(7775), 3*X(7751)+7*X(8176), 5*X(7759)-11*X(7775), 3*X(7759)-11*X(8176), 3*X(7775)-5*X(8176), 9*X(9741)-5*X(11148), 3*X(9741)-5*X(11165), 2*X(9741)-5*X(12040), X(11148)-3*X(11165), 2*X(11148)-9*X(12040), 2*X(11165)-3*X(12040)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27320.

X(16509) lies on these lines:
{2, 2418}, {3, 7620}, {5, 524}, {30, 7610}, {115, 11168}, {140, 7618}, {141, 5461}, {538, 9771}, {543, 549}, {547, 11184}, {597, 14762}, {598, 3363}, {599, 8355}, {632, 7619}, {2023, 9466}, {2482, 3054}, {3545, 9740}, {3634, 4364}, {3767, 8367}, {3845, 3849}, {3858, 7780}, {5055, 9770}, {5066, 8667}, {5503, 10302}, {5569, 8703}, {6704, 7817}, {7621, 8728}, {7622, 11539}, {7758, 12812}, {7857, 8369}, {8360, 13881}, {8370, 8859}, {8716, 10124}, {8860, 11164}, {9166, 9478}, {9766, 10109}, {9877, 11632}, {12811, 14023}

X(16509) = midpoint of X(i) and X(j) for these {i,j}: {3, 7620}, {9877, 11632}
X(16509) = reflection of X(5) in X(7617)
X(16509) = complement of X(11165)
X(16509) = {X(2), X(5485)}-harmonic conjugate of X(11165)


X(16510) =  MIDPOINT OF X(5486) AND X(11061)

Barycentrics    (SB+SC)*(9*S^4+(162*R^4-9*R^2*(3*SA+7*SW)+18*SA^2-18*SB*SC+5*SW^2)*S^2-(SA^2-SB*SC)*(9*R^2*(3*SA-SW)-9*SA^2+9*SB*SC+4*SW^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27320.

X(16510) lies on these lines:
{25, 2930}, {110, 9027}, {524, 7575}, {542, 13233}, {575, 15132}, {576, 2854}, {690, 9966}, {895, 1994}, {1995, 15303}, {2393, 9970}, {2781, 13247}, {5486, 11061}, {5505, 12039}, {6593, 8542}, {7496, 13169}, {9517, 12593}

X(16510) = midpoint of X(5486) and X(11061)
X(16510) = reflection of X(5505) in X(12039)


X(16511) =  COMPLEMENT OF X(8542)

Barycentrics    2*(b^2+c^2)*a^6-(b^4+12*b^2*c^ 2+c^4)*a^4-2*(b^2+c^2)*(b^4+c^ 4)*a^2+(b^4-c^4)^2 : :
X(16511) = 3 X(2) + X(5486) = X(5505) - 5 X(15059)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27320.

X(16511) lies on these lines:
{2, 895}, {5, 2393}, {140, 524}, {141, 9027}, {182, 6699}, {523, 4045}, {542, 8546}, {597, 8262}, {620, 9516}, {3549, 8538}, {3589, 12039}, {3818, 8547}, {5012, 11061}, {5044, 9004}, {5449, 8681}, {5505, 15059}, {6689, 9977}, {7395, 15069}, {7820, 9145}, {11179, 16003}, {14789, 15073}

X(16511) = midpoint of X(3818) and X(8547)
X(16511) = complement of X(8542)
X(16511) = {X(2), X(5486)}-harmonic conjugate of X(8542)


X(16512) =  (name pending)

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

X(16512) lies on this line: {19, 27}


X(16513) =  (name pending)

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

X(16513) lies on this line: {19, 27}


X(16514) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(42)

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

X(16514) lies on these lines:
{1, 6}, {63, 1613}, {71, 695}, {101, 8624}, {190, 2235}, {239, 350}, {292, 672}, {610, 3360}, {846, 1197}, {869, 2276}, {1194, 3690}, {1206, 1962}, {1575, 2664}, {1691, 7193}, {1707, 2162}, {1914, 2210}, {2056, 3955}, {2076, 3220}, {2243, 8621}, {3051, 3219}, {3094, 3781}, {3218, 3231}, {3783, 3802}, {3920, 7109}, {4455, 4775}, {5314, 10329}


X(16515) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(43)

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

X(16515) lies on these lines:
{1, 6}, {846, 2162}, {870, 4713}, {2235, 4664}, {2238, 4393}, {2276, 3009}


X(16516) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(78)

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

X(16516) lies on these lines:
{1, 6}, {239, 965}, {292, 2336}, {2287, 4393}


X(16517) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(145)

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

X(16517) lies on these lines:
{1, 6}, {10, 5286}, {39, 936}, {57, 5275}, {63, 5276}, {165, 4386}, {200, 2276}, {239, 391}, {292, 2297}, {573, 990}, {594, 4901}, {612, 672}, {965, 980}, {966, 4000}, {968, 2280}, {975, 4253}, {997, 9592}, {1376, 9574}, {1400, 4327}, {1423, 4328}, {1500, 6765}, {1573, 9620}, {1575, 8580}, {1914, 4512}, {2082, 2292}, {2238, 2999}, {2269, 4319}, {2275, 8583}, {2345, 3717}, {3290, 3677}, {3452, 7736}, {3686, 3755}, {3767, 5705}, {3883, 5839}, {4419, 5819}, {5013, 5438}, {5044, 9605}, {5273, 5304}, {5277, 15803}, {5305, 5791}, {5745, 7735}


X(16518) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(200)

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

X(16518) lies on this line: {1,6}


X(16519) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(306)

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

X(16519) lies on these lines:
{1, 6}, {38, 172}, {239, 1211}, {976, 2276}, {986, 4386}, {1575, 5293}, {1829, 2201}, {1914, 2292}, {2185, 7305}, {2238, 5262}, {2295, 3920}, {3670, 5277}, {3721, 5276}, {4393, 5739}, {4415, 7745}, {4517, 5311}, {5282, 7296}


X(16520) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(386)

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

X(16520) lies on these lines:
{1, 6}, {71, 3009}, {560, 3747}, {2238, 3187}, {4364, 15989}


X(16521) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(519)

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

X(16521) lies on these lines:
{1, 6}, {239, 4364}, {2243, 5276}, {4384, 4708}


X(16522) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(551)

Barycentrics    a (-a^3 - 6 a^2 b - 3 a b^2 - b^3 - 6 a^2 c - 9 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 - c^3) : :

X(16522) lies on these lines:
{1, 6}, {81, 2243}, {239, 4472}


X(16523) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(612)

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

X(16523) lies on these lines:
{1, 6}, {239, 5275}, {4393, 5276}


X(16524) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(614)

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

X(16524) lies on these lines:
{1, 6}, {55, 292}, {999, 8624}, {2162, 2344}, {2280, 3009}


X(16525) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(869)

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

X(16525) lies on these lines:
{1, 6}, {192, 2235}, {560, 1914}, {1613, 3666}, {1964, 2276}, {2162, 4640}, {2238, 3759}, {3231, 4850}


X(16526) =  (X(1),X(2),X(6),X(75); X(6),X(1),X(2),X(75)) COLLINEATION IMAGE OF X(899)

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

X(16526) lies on these lines: {1, 6}, {8626, 10987}


X(16527) =  (X(1),X(2),X(6),X(99); X(1),X(6),X(2),X(99)) COLLINEATION IMAGE OF X(145)

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

X(16527) lies on this line: {1,6}

X(16527) = antipode of X(5400) in excentral-hexyl ellipse
X(16527) = trilinear pole, wrt hexyl triangle, of Brocard axis


X(16528) =  X(3)X(5400)∩X(40)X(550)

Barycentrics    a (a^5 (b+c)+a^3 b c (b+c)+3 b c (b^2-c^2)^2+2 a^4 (b^2-6 b c+c^2)-a (b+c)^3 (b^2-b c+c^2)+a^2 (-2 b^4+9 b^3 c-2 b^2 c^2+9 b c^3-2 c^4)) : :
X(16528) = (r^2+2 r R+s^2) X(40) - 16 r R X(550)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27324 and Francisco Javier Garcia Capitan, Some results about the hexyl triangle.

X(16528) lies on the hexyl-excentral ellipse and these lines:
{3, 5400}, {40, 550}, {376, 573}, {1742, 3576}, {1764, 3534}, {1992, 11737}, {2827, 6326}, {7982, 13541}

X(16528) = reflection of X(5400) in X(3)


X(16529) = X(5)X(14) ∩ X(62)X(619)

Barycentrics    3*(2*a^6-4*(b^2+c^2)*a^4+3*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))+2*(4*a^4-2*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)*S : :
Barycentrics    3*(5*SA-2*SW)*S^2+3*SB*SC*SW-(10*S^2+3*SA^2-6*SB*SC-3*SW^2)*sqrt(3)*S : :
X(16529) = 2*X(13)+X(6780), X(13)+2*X(9117), X(14)-4*X(396), 2*X(15)+X(6778), X(15)+2*X(6783), 2*X(619)+X(3180), X(6778)-4*X(6783), X(6780)-4*X(9117)

Let F be the 1st Fermat point of ABC. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle with the same orientation than ABC. Build Bc, Ba, Ca, Cb cyclically. Then the Euler lines of triangles FBaCa, FCbAb and FAcBc are concurrent at X(16529). Also, the acute angle between each pair of these Euler lines is π/3. (Dao Thanh Oai, March 14, 2018). Coordinates and properties provided by César Eliud Lozada, March 15, 2018.

X(16529) lies on these lines:
{5,14}, {13,5611}, {15,6770}, {62,619}, {147,6109}, {182,16241}, {203,12941}, {531,5470}, {3107,5464}, {5965,15561}, {7005,12951}, {8014,14185}

X(16529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 9117, 6780), (15, 6783, 6778), (61, 5613, 14)


X(16530) = X(5)X(13) ∩ X(61)X(618)

Barycentrics    3*(2*a^6-4*(b^2+c^2)*a^4+3*(b^4+c^4)*a^2-(b^4-c^4)*(b^2-c^2))-2*(4*a^4-2*(b^2+c^2)*a^2+(b^2-c^2)^2)*sqrt(3)*S : :
Barycentrics    3*(5*SA-2*SW)*S^2+3*SB*SC*SW+(10*S^2+3*SA^2-6*SB*SC-3*SW^2)*sqrt(3)*S : :
X(16530) = X(13)-4*X(395), 2*X(618)+X(3181)

Let F be the 2nd Fermat point of ABC. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle with opposite orientation than ABC. Build Bc, Ba, Ca, Cb cyclically. Then the Euler lines of triangles FBaCa, FCbAb and FAcBc are concurrent at X(16530). Also, the acute angle between each pair of these Euler lines is π/3. (Dao Thanh Oai, March 14, 2018). Coordinates and properties provided by César Eliud Lozada, March 15, 2018.

X(16530) lies on these lines:
{5,13}, {14,5615}, {16,6773}, {61,618}, {147,6108}, {182,16242}, {202,12942}, {530,5469}, {3106,5463}, {5965,15561}, {7006,12952}, {8015,14187}

X(16530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 9115, 6779), (16, 6782, 6777), (62, 5617, 13)


X(16531) =  (name pending)

Barycentrics    (29*R^2-7*SW)*S^2-5*(3*R^2-SW) *SB*SC : :
X(16531) = 3 X(2) + 5 X(186) = 9 X(2) - 5 X(2072) = 21 X(2) - 5 X(3153)

As a point on the Euler line, this center has Shinagawa coefficients (E-28*F, 5*E+20*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27332.

X(16531) lies on this line: {2,3}


X(16532) =  (name pending)

Barycentrics    (41*R^2-10*SW)*S^2-3*(5*R^2-2* SW)*SB*SC : :
X(16532) = X(3)+2*X(10096), X(5)+2*X(186), 11*X(5)-2*X(10296), 5*X(5)-8*X(15350), X(20)+2*X(11558), 2*X(11692)-5*X(15026)

As a point on the Euler line, this center has Shinagawa coefficients (E-40*F, 9*E+24*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27332.

X(16532) lies on these lines:
{2, 3}, {1154, 16223}, {5642, 11561}, {5946, 10182}, {11597, 11694}, {11692, 15026}

X(16532) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (468, 15646, 11563), (549, 10154, 550), (11563, 15646, 550)


X(16533) =  (name pending)

Barycentrics    (2*a^6+7*(b+c)*a^5+(5*b^2+6*b* c+5*c^2)*a^4-2*(b+c)*(3*b^2+2* b*c+3*c^2)*a^3-4*(2*b^2+7*b*c+ 2*c^2)*(b-c)^2*a^2-(b^2-c^2)*( b-c)^3*a+(b^2+6*b*c+c^2)*(b^2- c^2)^2)*(3*a+b+c) : :
X(16533) = 3*(-8*R*r+24*R^2-r^2-2*s^2)*X(2) - (-r^2+4*s^2-16*R*r)*X(40)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27333.

X(16533) lies on this line: {2,40}


X(16534) =  MIDPOINT OF X(110) AND X(113)

Barycentrics    2a^10 -8a^8(b^2+c^2) +a^6(11b^4+4b^2c^2+11c^4)- a^4(5b^6+b^4c^2+b^2c^4+5c^6) -a^2(b^2-c^2)^2(b^4-6b^2c^2+c^4) +(b^2-c^2)^4(b^2+c^2) : :
X(16534) = X(4) + 3 X(110)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27334 and Reflexiones de rectas de Euler, HG150318.

X(16534) lies on these lines:
{2, 14094}, {3, 541}, {4, 110}, {5, 542}, {17, 10658}, {18, 10657}, {20, 10706}, {30, 15152}, {52, 10294}, {74, 3523}, {125, 399}, {140, 5663}, {146, 3522}, {155, 5095}, {265, 3851}, {376, 11693}, {382, 15039}, {403, 539}, {468, 13148}, {548, 11694}, {550, 1511}, {631, 15054}, {690, 6132}, {1216, 2781}, {1498, 15131}, {1553, 14934}, {1568, 7574}, {1625, 6103}, {1657, 7728}, {2771, 11281}, {2836, 13374}, {2914, 13431}, {2931, 3517}, {2948, 11522}, {3090, 9140}, {3091, 9143}, {3292, 11799}, {3448, 5056}, {3515, 12168}, {3516, 12901}, {3524, 15021}, {3525, 15057}, {3542, 12828}, {3545, 15029}, {3564, 15471}, {3628, 13393}, {3635, 15519}, {3850, 7687}, {3855, 15044}, {3858, 10113}, {5055, 15027}, {5068, 14644}, {5071, 15025}, {5181, 9970}, {5189, 14157}, {5446, 14984}, {5449, 11441}, {5621, 7393}, {5648, 11477}, {5882, 11720}, {5891, 7495}, {6000, 14156}, {6622, 9936}, {6689, 15738}, {6723, 10264}, {6759, 14791}, {7530, 12584}, {8254, 11017}, {8960, 12376}, {9306, 15132}, {9820, 15115}, {9826, 11806}, {10018, 11562}, {10299, 15055}, {10619, 11597}, {10620, 15720}, {11064, 14915}, {11598, 15105}, {11723, 13464}, {11803, 11808}, {12041, 15712}, {5073, 12121}, {12244, 15051}, {12308, 15061}, {12317, 15059}, {13382, 14708}, {1657, 7728}

X(16534) = midpoint of X(i) and X(j) for these {i,j}: {3, 15063}, {5, 5609}, {110, 113}, {125, 399}, {146, 16111}, {1553, 14934}, {1568, 10540}, {3292, 11799}, {5181, 9970}, {5642, 5655}, {5972, 6053}, {7728, 16163}, {11562, 12825}, {12121, 13202}, {12295, 12383}, {14094, 16003}
X(16534) = reflection of X(i) in X(j) for these (i,j): {125, 12900}, {5972, 10272}, {6699, 5972}, {10264, 6723}, {11806, 9826}, {15115, 9820}


X(16535) =  X(2)X(5627)∩X(5)X(542)

Barycentrics    2 a^16-8 a^14 (b^2+c^2)+14 a^12 (b^2+c^2)^2-(b^2-c^2)^6 (b^4+5 b^2 c^2+c^4)+a^2 (b^2-c^2)^4 (b^6-4 b^4 c^2-4 b^2 c^4+c^6)-a^10 (19 b^6+27 b^4 c^2+27 b^2 c^4+19 c^6)+a^4 (b^2-c^2)^2 (8 b^8+25 b^6 c^2-6 b^4 c^4+25 b^2 c^6+8 c^8)+a^8 (25 b^8-2 b^6 c^2+24 b^4 c^4-2 b^2 c^6+25 c^8)+a^6 (-22 b^10+7 b^8 c^2+5 b^6 c^4+5 b^4 c^6+7 b^2 c^8-22 c^10) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27334 and Reflexiones de rectas de Euler, HG150318

X(16535) lies on these lines: {2, 5627}, {5, 542}, {3635, 15519}, {8675, 10170}


X(16536) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 9th FERMAT-DAO EQUILATERAL

Barycentrics    (36*R^2-SA+2*SW)*S^2+sqrt(3)*(6*(SB+SC)*R^2+4*S^2+SA^2+2*SB*SC+SW^2)*S+3*SB*SC*SW : :

Let F = X(13) be the 1st Fermat point. Let Ba, Bc be the points on BC such that FBaCa is an equilateral triangle. Define Cb, Ab and Ac, Bc cyclically. Let A', B', C' be the midpoints of AbAc, BcBa, CaCb, respectively. Then the triangle A'B'C' is equilateral and is here named the 9th Dao-Fermat equilateral triangle. (Dao Thanh Oai, March 15, 2018)

When F=X(14)=the 2nd Fermat point of ABC, the resulting triangle is also equilateral and is here named the 10th Dao-Fermat equilateral triangle.

The centers of the 9th and 10th Fermat-Dao equilateral triangles are X(11624) and X(11626), respectively and their sides are congruent. Both triangles are perspective with perspector X(6) to these triangles: ABC, anti-Conway, 2nd anti-Conway, 2nd Brocard, circumsymmedial, 2nd Ehrmann, inner-Grebe, outer-Grebe, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd orthosymmedial, symmedial, tangential, inner tri-equilateral and outer tri-equilateral. Also, the 9th triangle is homothetic to the outer-Le Viet An triangle with center X(11142) and perspective to the medial and orthic triangles with perspectors X(16534) and X(16536), respectively. Finally, the 10th triangle is homothetic to the inner-Le Viet An triangle with center X(11141) and perspective to the medial and orthic triangles with perspectors X(16535) and X(16537), respectively.

This introduction and centers X(16534) to X(16537) were submitted by César Lozada, March 18, 2018.

You can view 9th and 10th equilateral triangles.

X(16534) lies on these lines:
{2,11080}, {115,619}, {303,14922}, {396,12097}, {623,10413}


X(16537) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 10th FERMAT-DAO EQUILATERAL

Barycentrics    (36*R^2-SA+2*SW)*S^2-sqrt(3)*(6*(SB+SC)*R^2+4*S^2+SA^2+2*SB*SC+SW^2)*S+3*SB*SC*SW : :

X(16535) lies on these lines:
{2,11085}, {115,618}, {302,14921}, {395,12098}, {624,10413}


X(16538) = PERSPECTOR OF THESE TRIANGLES: ORTHIC AND 9th FERMAT-DAO EQUILATERAL

Barycentrics    SB*SC*(SB+SC)*((18*R^2-4*SA-7*SW)*S^2-3*sqrt(3)*(SA^2+S^2+2*R^2*SA)*S-3*SW*SA^2) : :

X(16536) lies on these lines:
{4,11581}, {462,1843}, {2501,6748}


X(16539) =  X(1056)X(1061)∩X(1058)X(1063)

Trilinears    f(B,C)*f(C,B) : : , where f(B,C) = 8+3*cos(B)+cos(3*B)+cos(B-2*C) +2*cos(B+2*C)+cos(3*B+2*C)
Barycentrics    (a-b+c) *(a+b-c)*(a^6-2*c*a^5-(b^2-3* c^2)*a^4-4*c^3*a^3-(b^4-6*b^2* c^2-3*c^4)*a^2+2*(b^4-c^4)*c* a+(b^4-c^4)*(b^2-c^2)) *(a^6-2*b*a^5+(3*b^2-c^2)*a^4- 4*b^3*a^3+(3*b^4+6*b^2*c^2-c^ 4)*a^2-2*(b^4-c^4)*b*a+(b^4-c^ 4)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27340.

X(16539) lies on the Feuerbach hyperbola and these lines:
{1, 10996}, {388, 1039}, {497, 1041}, {1056, 1061}, {1058, 1063}

X(16539) = isogonal conjugate of X(16541)


X(16540) =  (name pending)

Trilinears    f(B,C)*f(C,B) : : , where f(B,C) = 8+3*cos(B)+cos(3*B)+cos(B-2*C) +2*cos(B+2*C)+cos(3*B+2*C)
Barycentrics    SA*((4*R^2*S^2+SW*SB^2)^2-c^2* a^2*SW^2*SB^2)*((4*R^2*S^2+SW* SC^2)^2-a^2*b^2*SW^2*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27340.

X(16540) lies on the Jerabek hyperbola and no lines X(i)X(j) fgor 1 <= i < j <= 16539.

X(16540) = isogonal conjugate of X(16542)


X(16541) =  ISOGONAL CONJUGATE OF X(16539)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27340.

X(16541) lies on these lines:
{1, 3}, {11, 5020}, {12, 11479}, {22, 390}, {24, 1058}, {25, 497}, {26, 15172}, {378, 1056}, {388, 1593}, {495, 9818}, {496, 6642}, {611, 11429}, {613, 11436}, {950, 9798}, {1057, 3422}, {1059, 7163}, {1478, 1597}, {1479, 1598}, {1486, 10829}, {1619, 2192}, {1914, 8573}, {1995, 5274}, {3058, 9909}, {3085, 7395}, {3486, 8192}, {3517, 10046}, {3600, 11413}, {4186, 10834}, {4254, 4548}, {4294, 11414}, {4319, 5322}, {5198, 5225}, {5218, 7484}, {5229, 11403}, {5281, 7485}, {5432, 16419}, {7387, 15171}, {7529, 9669}, {7741, 11484}, {9658, 9670}, {9911, 10624}, {9912, 15558}, {10589, 11284}, {11365, 12053}, {12309, 12428}, {12979, 13082}, {14070, 15170}

X(16541) = isogonal conjugate of X(16539)
X(16541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 56, 1040), (55, 10832, 3), (1479, 10037, 1598), (3303, 9672, 10831), (9672, 10831, 3)


X(16542) =  ISOGONAL CONJUGATE OF X(16540)

Barycentrics    SB*SC*(SB+SC)*(16*R^4*S^2+SW*( 8*R^2-SW)*SA^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27340.

X(16542) lies on this line: {2,3}

X(16542) = isogonal conjugate of X(16540)


X(16543) =  X(1352)X(1598)∩X(1597)X(5878)

Barycentrics    SB*SC*(SB+SC)*((8*R^4-4*R^2* SW-SW^2)*SA^2+4*R^2*S^2*SA+8*( 2*R^2-SW)*R^2*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27341.

X(16543) lies on these lines: {25, 11487}, {1352, 1598}, {1597, 5878}, {5887, 12410}




leftri  Collineation images: X(16544) - X(16614)  rightri

This section follows the discussion of collineations just before X(16286). If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A,B,C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A,B,C,P).

The collineation images at X(16544)-X(16576) result from A'B'C' = excentral triangle, P = X(2), and U = X(1). We write the image of X as m(X); let m-1 denote the inverse collineation. Then centers X(16544)-X(16576) are examples of m(X), and X(16577)-X(16614) are examples of m-1(X). Other examples are given by the following list, in which the appearance of (i,j) means that m(X(i)) = X(j):

(1,9), (2,1), (3,19), 4,610), (5,48), (6,63), (7,165), (8,1743), (9,57), (10,6), (11,101), (12,572)

A collineation maps lines to lines. The appearance of {h,i} -> {j,k} in the next list means that m(X(h)X(i)) = X(j)X(k):

{1,2} -> {1,6}
{1,3} -> {4,9}
{1,4} -> {3,9}
{1,6} -> {2,7}
{2,3} -> {1,19}
{2,6} -> {1,21}
{2,7} -> {1,3}
{2,37} -> {1,2}
{3,6} -> {19,27}
{30,511} -> {44,513}
{37,226} -> {2,3)

In the following list, the appearance of (i,j) means that m-1(X(i)) = X(j):

(1,2), (2,37), (3,226), (4,1214), (5,16577), (6,10), (7,1212), (8,3752), (9,1), (10,3666), (11,16578), (12,16579)

If X = x : y : z (barycentrics), then m(X) = a(y + z - x) : b(z + x - y) : c(x + y - z)    and    m-1(X) = a(cy + bz) : b(az + cx) : c(bx + ay).

More generally, let P = p : q : r and U = u : v : w, and write M(X) for the image of X under the collineation (A,B,C,P; A',B',C',U), where A'B'C' = excentral triangle. Then

M(X) = a(-aqr(cv + bw)x + brp(aw + cu)y + cpq(bu + av)z : :

M-1(X) = u(bp + aq)(cp + ar)(cy + bz) : :

underbar




X(16544) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16544) lies on these lines: {1, 2156}, {19, 31}, {63, 2172}


X(16545) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16545) lies on these lines:
{1, 19}, {47, 2312}, {169, 1698}, {304, 16568}, {920, 16562}, {1722, 5540}, {1760, 1930}, {2128, 16563}, {4873, 16548}, {5089, 8185}, {5268, 15487}, {7297, 16502}


X(16546) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16546) lies on these lines:
{1, 19}, {1749, 1755}, {2312, 6149}, {3943, 5525}, {5280, 5356}, {5299, 7300}, {5341, 16785}, {5540, 16611}, {7297, 16784}, {13146, 16550}, {14210, 16563}


X(16547) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16547) lies on these lines:
{1, 584}, {3, 16553}, {4, 9}, {5, 7110}, {6, 1718}, {37, 3746}, {44, 5341}, {57, 15474}, {82, 4628}, {101, 1953}, {142, 7291}, {198, 10267}, {572, 2173}, {583, 2160}, {910, 2919}, {1100, 7300}, {1400, 1731}, {1419, 2002}, {1449, 2082}, {1719, 7262}, {1729, 15656}, {1730, 1762}, {1752, 2257}, {1760, 4384}, {1761, 16552}, {1763, 9816}, {2171, 2246}, {2262, 2323}, {2273, 3959}, {2285, 16670}, {2294, 4251}, {2355, 5285}, {2911, 5903}, {3219, 4967}, {3247, 3920}, {3290, 16488}, {3686, 5279}, {3924, 5037}, {4034, 5227}, {5124, 15586}, {5356, 16669}, {6173, 7289}, {16470, 16583}, {16566, 16568}

X(16547) = {X(9),X(19)}-harmonic conjugate of X(16548)
X(16547) = X(2965)-of-excentral-triangle


X(16548) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16548) lies on these lines:
{1, 2278}, {4, 9}, {6, 5903}, {27, 6358}, {35, 37}, {36, 8609}, {44, 3245}, {46, 8557}, {63, 4659}, {75, 16566}, {101, 2173}, {171, 1719}, {190, 16568}, {198, 11248}, {226, 3101}, {284, 2171}, {329, 9536}, {484, 2161}, {517, 2182}, {522, 649}, {527, 7291}, {572, 1953}, {672, 1731}, {910, 5537}, {1055, 3100}, {1100, 5356}, {1172, 1825}, {1266, 3218}, {1404, 1449}, {1710, 1761}, {1760, 3729}, {1817, 16577}, {1824, 5285}, {2082, 16670}, {2149, 7012}, {2170, 5053}, {2252, 5535}, {2321, 5279}, {2640, 2664}, {2939, 3191}, {2941, 4640}, {3508, 16564}, {3943, 5525}, {4007, 5227}, {4552, 14953}, {4858, 6996}, {4873, 16545}, {5130, 10251}, {5297, 16676}, {5528, 16550}, {5715, 8251}, {5812, 8141}, {7300, 16669}, {9573, 9612}, {11349, 16578}

X(16548) = {X(9),X(19)}-harmonic conjugate of X(16547)
X(16548) = inverse-in-circumconic-centered-at-X(9) of X(10)
X(16548) = X(50)-of-excentral-triangle


X(16549) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16549) lies on these lines:
{1, 39}, {2, 2140}, {3, 16788}, {6, 3293}, {8, 4253}, {9, 46}, {10, 672}, {36, 2329}, {37, 3670}, {43, 2229}, {55, 16783}, {57, 7198}, {71, 5750}, {85, 1025}, {100, 4251}, {101, 404}, {171, 5280}, {213, 1575}, {218, 1376}, {220, 474}, {354, 3991}, {484, 3496}, {518, 4006}, {519, 1475}, {573, 5749}, {579, 2345}, {583, 594}, {644, 5253}, {728, 3333}, {942, 3693}, {946, 8568}, {956, 5022}, {999, 4513}, {1019, 7255}, {1023, 9310}, {1125, 1334}, {1193, 3997}, {1212, 3753}, {1400, 3831}, {1449, 5109}, {1574, 2238}, {1706, 16572}, {1730, 4456}, {1739, 16583}, {1743, 4274}, {2260, 2321}, {2280, 8715}, {2975, 5030}, {3061, 5903}, {3169, 16667}, {3207, 16371}, {3230, 16604}, {3336, 3509}, {3337, 5525}, {3550, 7031}, {3555, 4515}, {3633, 4050}, {3729, 3760}, {3746, 16503}, {3758, 3882}, {3812, 16601}, {3874, 3930}, {3959, 4674}, {3980, 5364}, {4095, 4692}, {4390, 8666}, {4752, 9327}, {5046, 5134}, {5255, 5299}, {5710, 9605}, {5883, 14439}, {6168, 104 81}, {11115, 14964}

X(16549) = bicentric sum of PU(146)
X(16549) = PU(146)-harmonic conjugate of X(1019)


X(16550) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(105), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16550) lies on these lines:
{1, 41}, {9, 141}, {19, 3174}, {37, 16686}, {40, 728}, {191, 3730}, {200, 15487}, {942, 5280}, {1045, 1781}, {1766, 2951}, {2285, 7271}, {3732, 6559}, {3912, 7291}, {5011, 5525}, {5528, 16548}, {13146, 16546}


X(16551) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16551) lies on these lines:
{1, 2175}, {9, 141}, {40, 3358}, {57, 1723}, {63, 321}, {169, 1445}, {362, 4146}, {573, 7291}, {1253, 2809}, {1731, 4000}, {1754, 9944}, {1759, 1760}, {5223, 6211}


X(16552) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16552) lies on these lines:
{1, 6}, {2, 2350}, {8, 1018}, {10, 672}, {20, 391}, {21, 4251}, {35, 3684}, {38, 16600}, {39, 2238}, {41, 993}, {57, 4059}, {58, 5276}, {63, 169}, {71, 3686}, {101, 2975}, {147, 7385}, {163, 1098}, {191, 2795}, {194, 16722}, {200, 1011}, {210, 2223}, {239, 3219}, {274, 16574}, {333, 1746}, {404, 5030}, {443, 579}, {474, 5022}, {517, 4875}, {519, 1334}, {572, 2287}, {583, 1213}, {596, 5282}, {894, 16819}, {910, 3916}, {965, 5120}, {980, 4383}, {1078, 3570}, {1125, 1475}, {1150, 16729}, {1400, 4298}, {1500, 3780}, {1573, 2295}, {1654, 2896}, {1714, 5286}, {1739, 16605}, {1761, 16547}, {1778, 4264}, {1985, 5231}, {2082, 12514}, {2141, 3681}, {2170, 3878}, {2183, 3707}, {2245, 11112}, {2260, 5257}, {2262, 4047}, {2267, 2304}, {2269, 4314}, {2270, 15656}, {2276, 3293}, {2280, 5248}, {3006, 4153}, {3208, 3632}, {3218, 16815}, {3290, 3953}, {3305, 16831}, {3333, 16844}, {3501, 3679}, {3509, 6763}, {3670, 16583}, {3693, 4006}, {3929, 16833}, {3997 , 10459}, {4050, 4677}, {4051, 5697}, {4185, 7719}, {4189, 4262}, {4258, 16370}, {4266, 11111}, {4357, 16818}, {4520, 9957}, {5021, 5275}, {5136, 7079}, {5179, 6734}, {5279, 16817}, {5296, 11037}, {5699, 6191}, {5700, 6192}, {5739, 14021}, {5750, 16828}, {5816, 6835}, {8580, 11358}, {8666, 9310}, {16560, 16563}

X(16552) = X(217)-of-excentral-triangle


X(16553) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(79), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16553) lies on these lines:
{1, 2160}, {3, 16547}, {9, 2173}, {19, 186}, {30, 7110}, {35, 37}, {57, 77}, {101, 5951}, {219, 9904}, {484, 2174}, {573, 16554}, {583, 5131}, {584, 3336}, {1375, 6707}, {1743, 2245}, {5124, 5540}, {5358, 16611}, {6155, 16884}, {8818, 16118}


X(16554) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(80), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16554) lies on these lines:
{1, 2161}, {6, 1718}, {9, 48}, {36, 44}, {57, 88}, {110, 2341}, {374, 10202}, {517, 2182}, {573, 16553}, {583, 1743}, {597, 4795}, {610, 5755}, {1023, 16561}, {1404, 1731}, {1449, 2170}, {1635, 3738}, {2173, 5535}, {2245, 5131}, {2850, 14395}, {3681, 4579}, {5425, 16666}, {5587, 12022}, {13384, 16676}


X(16555) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16555) lies on these lines:
{1, 251}, {6, 982}, {43, 1759}, {573, 5282}, {1781, 10469}, {3496, 3954}


X(16556) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16556) lies on these lines:
{1, 82}, {3, 984}, {6, 982}, {38, 1582}, {48, 16559}, {63, 1740}, {75, 2640}, {610, 16567}, {846, 8053}, {1757, 3216}, {1918, 3099}, {2175, 9941}, {2916, 3961}, {3496, 3499}, {3511, 3512}, {4438, 5224}, {7351, 8925}


X(16557) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16557) lies on these lines:
{1, 2162}, {9, 1575}, {57, 4554}, {63, 194}, {165, 3508}, {726, 2319}, {846, 3229}, {1742, 8844}, {1766, 3509}, {4598, 8026}


X(16558) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16558) lies on these lines:
{1, 89}, {9, 484}, {40, 1657}, {46, 3646}, {63, 4677}, {165, 6326}, {200, 13146}, {214, 3899}, {2136, 11010}, {3895, 6763}, {4867, 5010}, {5223, 5528}, {5289, 7280}, {5506, 10129}


X(16559) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(98), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16559) lies on these lines:
{1, 163}, {3, 3061}, {6, 2114}, {19, 1581}, {48, 16556}, {326, 610}, {910, 9509}, {1959, 2312}, {2930, 4053}, {3216, 5540}, {3497, 3499}, {3509, 3511}, {6211, 8925}


X(16560) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16560) lies on these lines:
{1, 692}, {2, 1762}, {6, 2114}, {9, 141}, {19, 273}, {40, 528}, {57, 1020}, {63, 190}, {77, 2261}, {101, 16578}, {169, 8257}, {238, 3827}, {241, 2182}, {335, 2196}, {513, 2957}, {518, 6211}, {545, 3928}, {610, 911}, {651, 2265}, {900, 1768}, {909, 1813}, {1053, 4083}, {1054, 2640}, {1210, 1782}, {1282, 4557}, {1429, 8609}, {1442, 2317}, {1633, 2310}, {1710, 3336}, {1731, 3008}, {1736, 3220}, {1738, 2385}, {1757, 3792}, {1761, 16566}, {2173, 11349}, {2183, 7291}, {2809, 3939}, {2939, 16453}, {3218, 4440}, {3219, 4473}, {3708, 16599}, {3912, 9028}, {3929, 4370}, {4432, 12514}, {4437, 5227}, {4516, 5091}, {4552, 5773}, {5540, 6084}, {6762, 9041}, {6996, 8680}, {16552, 16563}, {16562, 16565}


X(16561) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(106), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16561) lies on these lines:
{1, 9456}, {6, 4919}, {9, 80}, {37, 9259}, {44, 3880}, {45, 846}, {57, 545}, {63, 190}, {515, 2325}, {644, 2265}, {903, 3306}, {1023, 16554}, {1086, 5437}, {1761, 16562}, {2316, 2802}, {3465, 5440}, {4422, 7308}, {4553, 9355}, {4752, 12034}


X(16562) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16562) lies on these lines:
{1, 163}, {19, 2166}, {63, 2157}, {101, 2948}, {191, 3730}, {610, 1820}, {920, 16545}, {1707, 2156}, {1725, 2312}, {1749, 1755}, {1761, 16561}, {2629, 2631}, {2640, 2642}, {16560, 16565}


X(16563) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(111), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16563) lies on these lines:
{1, 662}, {2, 5540}, {63, 2157}, {2128, 16545}, {5525, 8591}, {14210, 16546}, {16552, 16560}


X(16564) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(237), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16564) lies on these lines:
{1, 19}, {798, 812}, {1733, 1755}, {1747, 2083}, {1760, 3403}, {3508, 16548}


X(16565) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(163), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16565) lies on these lines:
{1, 1576}, {163, 16599}, {6089, 12078}, {16560, 16562}


X(16566) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16566) lies on these lines:
{1, 7122}, {2, 1781}, {9, 1760}, {19, 4384}, {37, 5337}, {40, 5015}, {63, 321}, {75, 16548}, {83, 3405}, {191, 3923}, {484, 4660}, {516, 6734}, {572, 1959}, {726, 6763}, {760, 2330}, {983, 9941}, {990, 4652}, {1719, 3980}, {1761, 16560}, {3101, 3687}, {3218, 3663}, {3336, 3821}, {3739, 5341}, {3912, 5279}, {4416, 7291}, {4463, 5314}, {4640, 12723}, {4670, 5356}, {6996, 11683}, {16547, 16568}


X(16567) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(182), WHERE A'B'C' = EXCENTRAL TRIANGLE

Trilinears    (cot A + cot B + cot C) tan A + (tan A + tan B + tan C) cot A : :
Barycentrics    a (a^6 - a^4 b^2 + a^2 b^4 - b^6 - a^4 c^2 + 2 a^2 b^2 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 - c^6) : :

X(16567) lies on these lines:
{1, 163}, {9, 3846}, {19, 27}, {31, 1820}, {38, 48}, {610, 16556}, {920, 2179}, {1747, 3402}, {2183, 5282}

X(16567) = {X(19),X(63)}-harmonic conjugate of X(1755)


X(16568) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16568) lies on these lines:
{1, 922}, {2, 16581}, {19, 27}, {82, 1933}, {86, 1781}, {190, 16548}, {239, 7297}, {240, 1101}, {304, 16545}, {319, 5279}, {320, 7291}, {345, 9536}, {662, 1959}, {798, 812}, {894, 5341}, {896, 897}, {1580, 2244}, {1582, 4118}, {1733, 1749}, {1747, 4008}, {2234, 2640}, {14210, 16546}, {16547, 16566}

X(16568) = isogonal conjugate of X(2157)
X(16568) = trilinear product X(2)*X(23)


X(16569) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (a b + a c - 3 b c) : :

X(16569) lies on these lines:
{1, 2}, {9, 1575}, {35, 16058}, {36, 16059}, {44, 4650}, {55, 15485}, {56, 16409}, {57, 1463}, {63, 1054}, {100, 748}, {165, 2108}, {171, 4383}, {210, 982}, {220, 8169}, {238, 1376}, {244, 3681}, {291, 5223}, {391, 7184}, {474, 5247}, {518, 16602}, {668, 6384}, {672, 3973}, {750, 9342}, {756, 4850}, {846, 3305}, {872, 4751}, {896, 9352}, {984, 3740}, {986, 5044}, {1011, 5010}, {1155, 7262}, {1278, 4135}, {1478, 6821}, {1479, 6822}, {1574, 3501}, {1707, 2239}, {1724, 11358}, {1738, 3452}, {1739, 5692}, {1742, 10164}, {1743, 2238}, {2177, 5284}, {2276, 3731}, {2308, 14997}, {2887, 5233}, {3061, 16605}, {3175, 4706}, {3210, 3971}, {3339, 8951}, {3583, 6818}, {3585, 6817}, {3684, 16779}, {3742, 4849}, {3750, 4423}, {3751, 5437}, {3769, 4974}, {3836, 4417}, {3846, 4429}, {3877, 4695}, {3899, 4674}, {3975, 9902}, {3989, 9330}, {3993, 4734}, {4050, 16969}, {4191, 7280}, {4281, 14007}, {4334, 5435}, {4659, 4713}, {4661, 9335}, {5255, 9709}, {5563, 16421}, {5573, 16496}, {816 7, 16484}, {8168, 16486}, {15803, 16056}

X(16569) = SS(a->bc) of X(1743)


X(16570) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16570) lies on these lines:
{1, 21}, {9, 4650}, {57, 7262}, {165, 1757}, {171, 3929}, {238, 3928}, {240, 8765}, {326, 922}, {672, 3973}, {1473, 7280}, {1755, 16571}, {1955, 2184}, {3052, 16496}, {3218, 5272}, {3219, 5268}, {3550, 5223}, {3751, 4640}, {5010, 7085}, {10980, 15485}


X(16571) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16571) lies on these lines:
{1, 75}, {7, 3783}, {8, 7184}, {9, 1575}, {43, 894}, {46, 1757}, {63, 2227}, {87, 239}, {193, 4489}, {978, 3923}, {1278, 3009}, {1580, 1958}, {1581, 8769}, {1706, 3751}, {1707, 2236}, {1743, 2235}, {1755, 16570}, {1760, 2640}, {2309, 4699}, {2664, 3729}, {3596, 9902}, {4436, 16690}


X(16572) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16572) lies on these lines:
{1, 6}, {20, 5838}, {34, 1783}, {40, 672}, {41, 3576}, {46, 5540}, {56, 2348}, {57, 169}, {63, 5222}, {84, 294}, {101, 1420}, {223, 241}, {279, 1445}, {346, 6764}, {519, 728}, {579, 2270}, {610, 5120}, {651, 4350}, {906, 7031}, {910, 5022}, {1018, 2136}, {1190, 10857}, {1210, 6554}, {1323, 2124}, {1475, 3333}, {1697, 3730}, {1706, 16549}, {1707, 9441}, {1729, 5723}, {1732, 2183}, {1759, 3928}, {1788, 8074}, {2170, 7982}, {2264, 5584}, {2297, 5783}, {2999, 9605}, {3161, 9797}, {3305, 5308}, {3601, 4251}, {3677, 16600}, {3691, 5717}, {3693, 6765}, {4292, 5819}, {4513, 12629}, {4641, 5228}, {4875, 9623}, {5011, 5128}, {5179, 9581}, {5712, 7308}


X(16573) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(648), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16573) lies on these lines:
{2,811},{3,9509},{115,123},{122,5517},{125,7117},{656,3269},{856,2238},{993,9254},{1015,15526},{2170,2972},{2968,6377},{3139,8735},{8287,16595},{16758,17216}

X(16573) = complement X(811)
X(16573) = complement of the isogonal conjugate of X(810)
X(16573) = complement of the isotomic conjugate of X(656)
X(16573) = X(i)-complementary conjugate of X(j) for these (i,j): {3, 512}, {25, 520}, {31, 8062}, {32, 525}, {48, 4369}, {71, 3835}, {73, 17072}, {184, 523}, {217, 18314}, {222, 17066}, {228, 513}, {351, 5181}, {512, 5}, {520, 1368}, {525, 626}, {560, 16612}, {647, 141}, {656, 2887}, {667, 942}, {669, 6}, {798, 226}, {810, 10}, {822, 18589}, {878, 511}, {1084, 6388}, {1402, 521}, {1409, 4885}, {1410, 3900}, {1459, 3741}, {1501, 2485}, {1576, 5972}, {1799, 688}, {1924, 16583}, {1946, 960}, {1974, 6587}, {1976, 6130}, {2200, 514}, {2205, 2509}, {2351, 924}, {2353, 8673}, {2489, 13567}, {2491, 15595}, {2623, 14767}, {3049, 2}, {3063, 6708}, {3269, 127}, {3455, 9517}, {3504, 3221}, {7180, 16608}, {8789, 2507}, {8858, 9429}, {9247, 14838}, {9407, 14401}, {9409, 113}, {9426, 1196}, {10097, 625}, {10547, 826}, {14270, 1511}, {14567, 18311}, {14573, 16040}, {14575, 647}, {14600, 2799}, {14908, 690}, {15451, 1209}, {17094, 17047}, {17970, 804}, {18105, 5943}
X(16573) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 8062}, {330, 525}, {2994, 520}, {7219, 512}, {7361, 523}
X(16573) = X(1981)-isoconjugate of X(2713)
X(16573) = crosspoint of X(2) and X(656)
X(16573) = crosssum of X(6) and X(162)
X(16573) = barycentric product X(i)*X(j) for these {i,j}: {647, 17899}, {656, 8062}, {7076, 17216}, {7283, 18210}
X(16573) = barycentric quotient X(i)/X(j) for these {i,j}: {7364, 4620}, {8062, 811}, {17899, 6331}


X(16574) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16574) lies on these lines:
{1, 1918}, {2, 7}, {3, 10461}, {6, 980}, {40, 3886}, {46, 10479}, {69, 573}, {71, 3912}, {72, 16453}, {78, 16451}, {81, 4264}, {141, 2245}, {193, 4266}, {274, 16552}, {314, 1764}, {333, 1730}, {344, 3730}, {386, 988}, {404, 3786}, {518, 5132}, {524, 4271}, {572, 1444}, {583, 3589}, {599, 5036}, {965, 16412}, {1158, 12717}, {1756, 4655}, {1757, 3216}, {1759, 1760}, {1761, 16560}, {1765, 6996}, {2183, 4416}, {2239, 3778}, {2269, 3879}, {2287, 11349}, {2305, 5337}, {2893, 6999}, {3294, 4687}, {3618, 4253}, {3688, 4447}, {3916, 16287}, {3927, 16414}, {4652, 16452}, {5138, 13723}, {10381, 13731}


X(16575) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(645), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16575) lies on these lines:
{1, 643}, {9, 16592}, {56, 1046}, {57, 16591}, {846, 982}, {1054, 2640}, {1155, 2959}, {1768, 5539}, {2629, 9359}


X(16576) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16576) lies on these lines:
{1, 1120}, {9, 6377}, {43, 3158}, {659, 1054}, {978, 1420}, {979, 11512}, {1403, 4383}, {1979, 9360}, {3216, 15015}, {8054, 9458}, {9299, 9361}


X(16577) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (a + b - c) (a - b + c) (b + c) (a^2 - b^2 - b c - c^2) : :
Barycentrics    cos(A - B) + cos(A - C) : :

The trilinear polar of X(16577) meets the line at infinity at X(526). (Randy Hutson, June 27, 2018)

X(16577) lies on these lines:
{1, 201}, {2, 2006}, {12, 502}, {15, 1082}, {16, 559}, {35, 186}, {37, 226}, {38, 5083}, {48, 1726}, {55, 11028}, {57, 1255}, {65, 3743}, {101, 1762}, {109, 846}, {222, 4559}, {223, 3731}, {241, 553}, {249, 2185}, {323, 1442}, {498, 1068}, {756, 4551}, {842, 2222}, {942, 15443}, {968, 8270}, {1215, 16598}, {1399, 3647}, {1418, 4114}, {1421, 5284}, {1458, 3989}, {1577, 15412}, {1730, 1953}, {1736, 14547}, {1758, 1961}, {1817, 16548}, {2078, 3920}, {2321, 3998}, {2594, 3678}, {2823, 7416}, {3074, 8555}, {3666, 3911}, {3931, 4848}, {4605, 16599}, {5173, 15569}, {8758, 13405}, {16587, 16591}, {16600, 16609}, {16603, 16607}

X(16577) = complement of X(14213)
X(16577) = isotomic conjugate of polar conjugate of X(1825)
X(16577) = polar conjugate of isogonal conjugate of X(22342)
X(16577) = {X(16578),X(16579)}-harmonic conjugate of X(2)


X(16578) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = EXCENTRAL TRIANGLE

Trilinears    b(a + b - c)(a - b)^2 + c(c + a - b)(c - a)^2 : :
Barycentrics    2 - cos(A - B) - cos(A - C) : :
Barycentrics    a (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - a c^3 - b c^3 + c^4) : :

X(16578) lies on these lines:
{1, 3939}, {2, 2006}, {3, 2823}, {9, 77}, {37, 142}, {44, 6510}, {88, 4606}, {101, 16560}, {214, 1807}, {220, 15730}, {241, 527}, {662, 2341}, {676, 2804}, {692, 11712}, {918, 3960}, {1214, 3452}, {1736, 1818}, {2809, 4557}, {2835, 15507}, {3008, 8609}, {3666, 6692}, {3743, 3812}, {4069, 4712}, {4755, 6706}, {5723, 6666}, {6745, 8758}, {11349, 16548}

X(16578) = complement of X(4858)
X(16578) = polar conjugate of isogonal conjugate of X(22346)
X(16578) = {X(2),X(16577)}-harmonic conjugate of X(16579)


X(16579) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = EXCENTRAL TRIANGLE

Trilinears    b(a + b)^2/(a + b - c) + c(c + a)^2/(c + a - b) : :
Barycentrics    2 + cos(A - B) + cos(A - C) : :
Barycentrics    cos^2(A/2 - B/2) + cos^2(A/2 - C/2) : :
Barycentrics    a (a - b - c) (a^3 b + a^2 b^2 - a b^3 - b^4 + a^3 c + 2 a^2 b c + 2 a b^2 c + b^3 c + a^2 c^2 + 2 a b c^2 - a c^3 + b c^3 - c^4) : :

X(16579) lies on these lines:
{2, 2006}, {9, 5256}, {37, 3452}, {124, 4425}, {142, 1214}, {960, 3743}, {1062, 5248}, {1212, 5325}, {1764, 1953}, {3218, 16585}, {3666, 3946}, {3931, 5837}

X(16579) = complement of X(6358)
X(16579) = polar conjugate of isogonal conjugate of X(22347)
X(16579) = {X(2),X(16577)}-harmonic conjugate of X(16578)


X(16580) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16580) lies on these lines:
{2, 1760}, {11, 1827}, {37, 226}, {56, 1892}, {307, 2245}, {946, 1279}, {1001, 7742}, {1086, 12610}, {1400, 4466}, {1486, 1836}, {1826, 16732}, {1848, 3772}, {3610, 4035}, {4026, 12609}, {4657, 5249}, {7382, 15474}

X(16580) = polar conjugate of isogonal conjugate of X(22348)


X(16581) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16581) lies on these lines:
{2, 16568}, {37, 226}, {812, 14838}, {857, 16732}, {1375, 15586}, {2245, 4466}, {4657, 5333}, {4892, 16597}, {8287, 16609}

X(16581) = polar conjugate of isogonal conjugate of X(22349)


X(16582) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16582) lies on these lines: {37, 16607}, {206, 942}, {226, 16600}

X(16582) = polar conjugate of isogonal conjugate of X(22362)


X(16583) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16583) lies on these lines:
{1, 2271}, {2, 304}, {3, 16968}, {6, 169}, {9, 986}, {10, 37}, {19, 2207}, {31, 2355}, {32, 910}, {38, 3691}, {39, 1212}, {40, 14974}, {41, 3924}, {57, 5021}, {58, 16716}, {65, 213}, {72, 2238}, {198, 5336}, {210, 3954}, {220, 9620}, {238, 3496}, {244, 1475}, {257, 16827}, {392, 3727}, {517, 2176}, {595, 5011}, {614, 1184}, {762, 3983}, {800, 1108}, {960, 3735}, {978, 3061}, {1018, 3987}, {1107, 16825}, {1191, 1572}, {1201, 2170}, {1214, 16584}, {1254, 1400}, {1279, 2241}, {1333, 5358}, {1334, 4642}, {1418, 10521}, {1724, 1759}, {1739, 16549}, {1841, 7713}, {2178, 9798}, {2245, 4456}, {2262, 2300}, {2276, 16601}, {2295, 3753}, {2887, 4109}, {3057, 3230}, {3214, 3930}, {3293, 3970}, {3294, 4424}, {3509, 5247}, {3555, 3726}, {3666, 4384}, {3670, 16552}, {3673, 4000}, {3754, 3997}, {3767, 3772}, {4386, 5266}, {5089, 5230}, {5179, 5254}, {5262, 5276}, {5299, 5540}, {7719, 8557}, {8287, 16607}, {9957, 16969}, {16470, 16547}


X(16584) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16584) lies on these lines:
{1, 7075}, {2, 561}, {6, 9017}, {10, 718}, {31, 1501}, {32, 2352}, {37, 714}, {38, 8620}, {39, 712}, {42, 3121}, {141, 720}, {171, 292}, {213, 1402}, {226, 16591}, {238, 9285}, {321, 2229}, {354, 1015}, {626, 724}, {696, 8265}, {707, 789}, {722, 2887}, {910, 9287}, {982, 2275}, {1196, 3290}, {1197, 1964}, {1214, 16583}, {2205, 3724}, {2225, 3051}, {3116, 7032}, {3752, 6377}, {3846, 9284}, {6586, 10196}, {8624, 16678}

X(16584) = isogonal conjugate of isotomic conjugate of X(3721)
X(16584) = complement of X(561)
X(16584) = center of inellipse that is the trilinear square of the Lemoine axis


X(16585) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(79), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (a^2 - b^2 - b c - c^2) (a^2 b - b^3 + a^2 c + 2 a b c + b^2 c + b c^2 - c^3) : :
Barycentrics    (1 + 2 cos A) (1 + cos B + cos C) : :

X(16585) lies on these lines:
{1, 21}, {2, 7110}, {9, 6505}, {72, 5453}, {223, 3305}, {323, 1442}, {1214, 6357}, {1511, 13151}, {1762, 1790}, {2071, 3101}, {3218, 16579}, {3650, 8143}, {4733, 6741}, {5745, 16586}

X(16585) = isotomic conjugate of polar conjugate of X(1844)
X(16585) = complement of polar conjugate of X(6198)


X(16586) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(80), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (a^2 - b^2 + b c - c^2) (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3) : :
Barycentrics    (1 - 2 cos A)(1 - cos B - cos C) : :

X(16586) lies on these lines:
{1, 88}, {2, 2006}, {57, 1813}, {63, 223}, {283, 3466}, {323, 1443}, {908, 1465}, {1086, 3666}, {1212, 5723}, {2968, 6734}, {3160, 5744}, {3310, 10015}, {3904, 3960}, {4706, 8758}, {4996, 11700}, {5745, 16585}, {8609, 16610}


X(16587) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16587) lies on these lines:
{2, 3112}, {10, 718}, {37, 744}, {38, 8041}, {39, 3703}, {171, 172}, {306, 3774}, {594, 1575}, {661, 756}, {984, 9285}, {1214, 16603}, {1215, 16592}, {1500, 3666}, {1920, 1926}, {2239, 7109}, {16577, 16591}

X(16587) = polar conjugate of isogonal conjugate of X(22367)


X(16588) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = EXCENTRAL TRIANGLE

Trilinears    sin B sec^2(C/2) + sin C sec^2(B/2) : :
Barycentrics    a^2 (a - b - c) (a b^2 - b^3 + b^2 c + a c^2 + b c^2 - c^3) : :
Barycentrics    a^2 (a - b - c) (b^2 (a - b + c) + c^2 (a + b - c)) : :

X(16588) lies on these lines:
{1, 9445}, {2, 4554}, {6, 13404}, {9, 43}, {31, 15263}, {37, 800}, {39, 1212}, {42, 8012}, {51, 2225}, {55, 2195}, {181, 5364}, {220, 1500}, {650, 5432}, {756, 3119}, {984, 3041}, {1125, 9367}, {1146, 1573}, {1196, 3290}, {1200, 2293}, {1376, 6181}, {1621, 5375}, {1743, 15838}, {1939, 6684}, {2177, 6602}, {2275, 5573}, {3731, 5574}, {3816, 5701}, {4646, 15853}, {5283, 6554}

X(16588) = complement of X(6063)
X(16588) = crosssum of X(6) and X(7)
X(16588) = crosspoint of X(2) and X(55)
X(16588) = crossdifference of every pair of points on the polar of X(7) wrt the circumcircle
X(16588) = polar conjugate of isogonal conjugate of X(22368)


X(16589) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (b + c) (a b + a c + 2 b c) : :

X(16589) lies on these lines:
{1, 1573}, {2, 39}, {6, 4658}, {9, 1046}, {10, 37}, {21, 187}, {32, 405}, {45, 2245}, {83, 16918}, {99, 16917}, {115, 120}, {172, 5251}, {230, 6675}, {232, 451}, {321, 7230}, {350, 16819}, {377, 7748}, {385, 16912}, {386, 16846}, {406, 3199}, {429, 5089}, {443, 2549}, {452, 7737}, {474, 574}, {573, 15488}, {612, 4204}, {661, 764}, {756, 3954}, {762, 3930}, {899, 9280}, {958, 2242}, {966, 4263}, {975, 16968}, {1001, 2241}, {1015, 1107}, {1084, 4755}, {1211, 3912}, {1212, 3452}, {1384, 16866}, {1506, 4187}, {1570, 15988}, {1574, 1698}, {1575, 3634}, {1914, 5259}, {1962, 6155}, {2240, 5297}, {2275, 3624}, {2292, 3125}, {2295, 3294}, {2305, 4877}, {2478, 5475}, {2548, 5084}, {2642, 4526}, {2667, 4111}, {3053, 16418}, {3230, 10459}, {3496, 5184}, {3501, 3731}, {3646, 9575}, {3691, 3720}, {3728, 4890}, {3734, 11321}, {3735, 9560}, {3972, 16914}, {4037, 4647}, {4099, 4714}, {4129, 14991}, {4188, 8589}, {4189, 15513}, {4193, 7603}, {4199, 5268}, {4272, 16777}, {4386, 5248}, {4423, 16502}, {4687, 6376}, {4698, 6381}, {4791, 6586}, {5007, 5047}, {5008, 16859}, {5013, 16408}, {5024, 16863}, {5179, 14873}, {5206, 16370}, {5254, 8728}, {5260, 5291}, {6707, 16696}, {7483, 7749}, {7735, 16845}, {7747, 11113}, {7756, 11112}, {7772, 16842}, {7804, 16916}, {7816, 16915}, {8583, 9619}, {9605, 16853}, {15515, 16371}, {15815, 16417}, {16592, 16594}

X(16589) = complement of X(274)
X(16589) = polar conjugate of isogonal conjugate of X(22369)
X(16589) = trilinear pole, wrt medial triangle, of antiorthic axis


X(16590) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16590) lies on these lines:
{2, 44}, {9, 484}, {10, 4370}, {37, 519}, {45, 3679}, {391, 3723}, {513, 6544}, {545, 4688}, {551, 3707}, {751, 9330}, {903, 16815}, {966, 3161}, {1100, 5296}, {1213, 15492}, {1405, 4870}, {2325, 4745}, {3196, 5251}, {3943, 4669}, {4034, 16677}, {4677, 4727}, {4945, 5235}, {5257, 16669}, {5513, 15614}, {11231, 12034}

X(16590) = complement of isotomic conjugate of X(3679)
X(16590) = polar conjugate of isogonal conjugate of X(22372)


X(16591) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(98), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16591) lies on these lines:
{2, 1821}, {11, 114}, {37, 8287}, {57, 16575}, {226, 16584}, {238, 1284}, {511, 7062}, {828, 16595}, {1086, 11672}, {1214, 2887}, {1581, 7146}, {4357, 15595}, {16577, 16587}

X(16591) = complement of X(1821)


X(16592) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16592) lies on these lines:
{2, 799}, {9, 16575}, {11, 115}, {37, 16597}, {39, 5718}, {187, 15447}, {226, 16584}, {244, 661}, {512, 7063}, {942, 2653}, {1084, 1086}, {1215, 16587}, {1365, 7180}, {1574, 4023}, {1575, 10026}, {1577, 7208}, {2092, 10427}, {2229, 3936}, {2645, 9508}, {2670, 3664}, {2887, 16606}, {3120, 3121}, {5241, 6537}, {6627, 8287}, {9280, 15569}, {16589, 16594}


X(16593) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(105), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16593) lies on these lines:
{2, 11}, {7, 190}, {9, 141}, {37, 142}, {144, 4473}, {516, 3836}, {518, 3717}, {527, 4370}, {537, 4078}, {545, 6173}, {900, 3126}, {1018, 4904}, {1213, 6666}, {1279, 3008}, {3039, 3732}, {3243, 4929}, {3452, 13609}, {3589, 16503}, {4728, 6009}, {5580, 16184}

X(16593) = polar conjugate of isogonal conjugate of X(20749)


X(16594) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(106), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16594) lies on these lines:
{2, 45}, {10, 11}, {120, 3259}, {145, 3699}, {519, 4152}, {528, 9458}, {678, 1644}, {908, 3834}, {1211, 8287}, {1266, 16610}, {1332, 1997}, {1639, 3762}, {2885, 14923}, {3021, 16185}, {3264, 3943}, {3665, 5219}, {3716, 10427}, {3756, 3952}, {3837, 14434}, {3912, 13466}, {4009, 5121}, {4432, 6174}, {4728, 6009}, {4767, 9041}, {4854, 6686}, {5328, 6554}, {5718, 6376}, {6789, 10609}, {16589, 16592}

X(16594) = isotomic conjugate of polar conjugate of X(5151)
X(16594) = complement of X(88)


X(16595) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(107), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16595) lies on these lines:
{2, 823}, {11, 122}, {520, 7065}, {828, 16591}, {1214, 16599}, {2972, 7004}


X(16596) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(108), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16596) lies on these lines:
{2, 196}, {3, 1633}, {7, 268}, {9, 1020}, {11, 123}, {122, 124}, {127, 5190}, {253, 7003}, {278, 1073}, {329, 7011}, {441, 1944}, {1146, 8287}, {1214, 3452}, {1565, 3942}, {2969, 2972}, {5328, 6350}, {6506, 16732}


X(16597) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(111), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16597) lies on these lines:
{2, 897}, {10, 8287}, {11, 126}, {37, 16592}, {524, 7067}, {960, 2836}, {1086, 1125}, {1649, 4010}, {4422, 4934}, {4442, 4956}, {4892, 16581}


X(16598) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16598) lies on these lines:
{1, 643}, {2, 1109}, {10, 6739}, {37, 16592}, {100, 2611}, {523, 3035}, {740, 12080}, {758, 1319}, {1054, 2643}, {1214, 2887}, {1215, 16577}, {3636, 3743}, {3647, 16164}, {4458, 6370}


X(16599) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(125), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16599) lies on these lines:
{37, 8287}, {163, 16565}, {1214, 16595}, {2294, 7269}, {3708, 16560}, {4458, 6370}, {4605, 16577}, {5236, 8680}


X(16600) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16600) lies on these lines:
{1, 41}, {2, 1930}, {6, 3874}, {10, 37}, {31, 1759}, {38, 16552}, {42, 3970}, {58, 3509}, {65, 3997}, {213, 758}, {226, 16582}, {241, 10521}, {519, 4168}, {595, 3496}, {672, 3670}, {762, 3956}, {910, 5266}, {982, 4253}, {986, 3730}, {993, 16968}, {995, 3061}, {1018, 4642}, {1125, 3290}, {1334, 4424}, {1400, 4456}, {1475, 3953}, {1722, 3731}, {1724, 5282}, {1781, 2298}, {2176, 3735}, {2238, 3678}, {2292, 3294}, {2295, 3125}, {2303, 5358}, {2887, 4153}, {3008, 3666}, {3159, 3985}, {3214, 4006}, {3230, 3727}, {3293, 3930}, {3454, 4109}, {3677, 16572}, {3726, 3881}, {3764, 14620}, {3898, 16969}, {3924, 16788}, {4805, 4950}, {5011, 5255}, {5179, 13161}, {5262, 5280}, {5269, 5338}, {5283, 16825}, {5299, 7191}, {7859, 16706}, {9237, 14838}, {12514, 16970}, {16577, 16609}

X(16600) = complement of X(1930)
X(16600) = polar conjugate of isogonal conjugate of X(23203)


X(16601) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(142), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16601) lies on these lines:
{1, 6}, {2, 277}, {4, 5089}, {8, 3991}, {10, 3693}, {12, 5179}, {35, 910}, {39, 3290}, {55, 169}, {65, 3730}, {101, 2646}, {115, 120}, {198, 13730}, {226, 241}, {279, 8232}, {294, 943}, {304, 344}, {329, 5308}, {354, 4253}, {374, 4266}, {500, 5777}, {517, 1334}, {519, 4875}, {573, 7957}, {612, 13615}, {614, 9605}, {672, 942}, {728, 9623}, {756, 2340}, {846, 9441}, {936, 16851}, {948, 1214}, {966, 3694}, {968, 1260}, {1005, 5297}, {1018, 5836}, {1055, 13624}, {1111, 6706}, {1146, 10039}, {1174, 3748}, {1385, 9310}, {1475, 5045}, {1696, 8273}, {1708, 5228}, {1759, 4640}, {1766, 5584}, {2082, 3295}, {2170, 9957}, {2240, 4199}, {2276, 16583}, {2348, 4251}, {3008, 3666}, {3011, 5305}, {3085, 6554}, {3207, 3612}, {3208, 10914}, {3338, 5022}, {3501, 3753}, {3509, 3916}, {3679, 4515}, {3691, 3930}, {3746, 5540}, {3812, 16549}, {3878, 4520}, {3965, 6743}, {4006, 4662}, {4294, 5819}, {4350, 8545}, {4653, 16699}, {5044, 16850}, {5257, 13728}, {5266, 5276}, {5268, 7580}, {5296, 13725}, {5749, 13742}, {5817, 11200}, {7322, 10382}, {8012, 11018}, {8568, 9843}, {10482, 15837}, {11113, 14537}, {14482, 16020}

X(16601) = complement of X(20880)
X(16601) = {X(1),X(9)}-harmonic conjugate of X(218)
X(16601) = X(217)-of-2nd-extouch-triangle


X(16602) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16602) lies on these lines:
{1, 3848}, {2, 37}, {6, 5437}, {9, 8056}, {36, 16427}, {43, 3742}, {44, 57}, {63, 15492}, {88, 3219}, {200, 4864}, {210, 244}, {227, 5433}, {333, 16726}, {354, 899}, {375, 1401}, {392, 1739}, {474, 1104}, {518, 16569}, {614, 4413}, {631, 15852}, {650, 6545}, {748, 1155}, {756, 4003}, {940, 16666}, {958, 11512}, {975, 16863}, {978, 3812}, {982, 3740}, {991, 10156}, {1054, 4640}, {1086, 3452}, {1100, 2999}, {1107, 16832}, {1125, 4646}, {1201, 3698}, {1279, 1376}, {1418, 5435}, {1427, 3911}, {1574, 4515}, {1616, 1706}, {1738, 3816}, {1834, 9843}, {2886, 5121}, {3008, 6692}, {3216, 5439}, {3240, 4883}, {3242, 5573}, {3246, 3550}, {3306, 4383}, {3445, 4853}, {3681, 3999}, {3689, 9350}, {3696, 3840}, {3697, 3953}, {3705, 3823}, {3744, 7292}, {3756, 4847}, {3834, 4417}, {3928, 16885}, {3961, 4906}, {3976, 4662}, {4049, 14838}, {4689, 5284}, {4695, 5919}, {5235, 16700}, {5400, 10167}, {7191, 9342}, {7308, 16814}, {9367, 15853}, {9371, 10589}

X(16602) = complement of X(18743)


X(16603) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(182), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16603) lies on these lines:
{1, 7380}, {2, 1429}, {10, 12}, {37, 8287}, {57, 7247}, {306, 4095}, {307, 4032}, {321, 4136}, {349, 6358}, {857, 1334}, {984, 5117}, {1214, 16587}, {1423, 5224}, {1441, 16888}, {1469, 3775}, {3314, 3661}, {3982, 10521}, {4384, 5219}, {11375, 16825}, {16577, 16607}


X(16604) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16604) lies on these lines:
{1, 1575}, {2, 330}, {6, 978}, {10, 1015}, {37, 39}, {44, 4253}, {56, 4426}, {87, 14823}, {115, 3825}, {172, 5253}, {230, 6691}, {244, 3721}, {274, 4602}, {292, 16823}, {377, 9599}, {386, 1100}, {404, 1914}, {474, 4386}, {513, 3249}, {519, 1574}, {551, 1500}, {574, 5248}, {899, 3780}, {992, 2260}, {1001, 5013}, {1086, 4920}, {1201, 2295}, {1212, 3039}, {1376, 16781}, {1475, 2238}, {1506, 3822}, {1573, 3634}, {2276, 3616}, {2329, 9259}, {2478, 9597}, {3008, 6692}, {3055, 6668}, {3230, 16549}, {3501, 16969}, {3624, 5283}, {3679, 9336}, {3739, 6374}, {3752, 6703}, {3767, 10200}, {3816, 5254}, {3953, 3954}, {5272, 16968}, {5277, 5299}, {5291, 5563}, {5437, 9575}, {6681, 7749}, {16716, 16736}, {16726,16887}

X(16604) = complement of X(6376)
X(16604) = midpoint of centers of 1st and 2nd bicentrics of the circumcircle


X(16605) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16605) lies on these lines:
{6, 1722}, {8, 3290}, {10, 37}, {44, 169}, {65, 2238}, {72, 3125}, {210, 3721}, {213, 3753}, {333, 16716}, {354, 3780}, {762, 3921}, {910, 4426}, {960, 3959}, {980, 1107}, {992, 2262}, {1104, 4386}, {1212, 1575}, {1334, 4695}, {1376, 16968}, {1427, 16606}, {1706, 16970}, {1738, 5254}, {1739, 16552}, {2176, 5836}, {2245, 2333}, {2275, 4875}, {2295, 3698}, {3008, 6692}, {3061, 16569}, {3230, 10914}, {3294, 3987}, {3697, 3954}, {3735, 5044}, {3880, 16969}, {3918, 3997}, {5272, 16781}, {9259, 11260}

X(16605) = complement of X(18156)


X(16606) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    a (b + c) (a b - a c - b c) (a b - a c + b c) : :

X(16606) lies on these lines:
{1, 7275}, {2, 330}, {6, 43}, {25, 2053}, {37, 714}, {39, 6685}, {42, 2229}, {75, 2998}, {111, 932}, {291, 694}, {292, 7081}, {310, 16742}, {321, 3121}, {333, 4598}, {650, 3572}, {893, 894}, {899, 2350}, {941, 2276}, {958, 8770}, {1015, 3840}, {1218, 3739}, {1400, 2238}, {1427, 16605}, {1908, 4697}, {1914, 4203}, {2248, 5247}, {2887, 16592}, {3931, 9281}

X(16606) = complement of X(17149)


X(16607) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(206), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16607) lies on these lines:
{2, 2172}, {10, 4523}, {37, 16582}, {116, 1210}, {141, 11573}, {857, 4456}, {8287, 16583}, {16577, 16603}

X(16607) = complement of X(2172)


X(16608) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(212), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16608) is the center of the inconic that is the polar conjugate of the isogonal conjugate of the incircle. This inconic is an ellipse if ABC is acute, and a hyperbola if ABC is obtuse. The Brianchon point (perspector) of the inconic is X(331). (Randy Hutson, November 30, 2018)

X(16608) lies on these lines:
{2, 219}, {5, 916}, {7, 281}, {10, 141}, {48, 1375}, {116, 117}, {200, 8271}, {226, 6708}, {278, 5932}, {343, 5249}, {496, 14056}, {946, 6247}, {1001, 5803}, {1439, 5236}, {1861, 5728}, {1953, 4466}, {3085, 4648}, {3694, 3912}, {3811, 4851}, {3879, 6510}, {3946, 6738}, {4000, 4904}, {4682, 13405}, {4869, 7080}, {5707, 10198}, {6260, 15873}, {6355, 7003}, {6907, 15669}, {7289, 7719}, {7686, 12610}

X(16608) = complement of X(219)


X(16609) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(511), WHERE A'B'C' = EXCENTRAL TRIANGLE

Barycentrics    (a + b - c) (a - b + c) (b + c) (a^2 - b c) : :
Barycentrics    cos(B + ω) + cos(C + ω) : :

X(16609) lies on these lines:
{1, 6998}, {2, 257}, {7, 1654}, {9, 11683}, {10, 12}, {56, 16825}, {57, 85}, {75, 1423}, {169, 1708}, {223, 1047}, {238, 242}, {239, 385}, {241, 514}, {278, 3144}, {306, 4136}, {307, 16888}, {321, 4095}, {448, 5060}, {553, 4059}, {673, 3512}, {740, 1284}, {948, 1788}, {1214, 16583}, {1400, 1441}, {1427, 16605}, {1428, 4974}, {1733, 1756}, {1831, 1848}, {1835, 5236}, {1921, 3975}, {2245, 8680}, {3739, 15985}, {3772, 3959}, {3912, 5977}, {3948, 3985}, {3982, 4955}, {4008, 6210}, {8287, 16581}, {9317, 11349}, {16577, 16600}

X(16609) = isogonal conjugate of X(2311)
X(16609) = complement of X(1959)
X(16609) = isotomic conjugate of polar conjugate of X(1874)


X(16610) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16610) lies on these lines:
{1, 3689}, {2, 37}, {6, 3306}, {11, 1738}, {38, 3740}, {42, 3742}, {43, 354}, {44, 88}, {55, 5272}, {56, 1722}, {57, 1122}, {63, 16885}, {65, 978}, {81, 16668}, {100, 1279}, {142, 5718}, {200, 5573}, {210, 982}, {227, 7288}, {238, 1054}, {239, 6631}, {241, 514}, {244, 518}, {333, 16700}, {386, 5439}, {404, 1104}, {513, 8661}, {517, 1739}, {614, 1376}, {726, 4009}, {740, 4706}, {748, 4640}, {750, 1386}, {902, 3246}, {908, 1086}, {940, 1449}, {942, 3216}, {968, 8167}, {971, 5400}, {975, 16862}, {980, 16832}, {984, 4003}, {995, 3753}, {1001, 4689}, {1107, 16815}, {1125, 4868}, {1149, 3880}, {1193, 3812}, {1201, 5836}, {1215, 6686}, {1266, 16594}, {1427, 5435}, {1456, 9364}, {2275, 4875}, {2352, 16059}, {3006, 3823}, {3011, 3035}, {3120, 5087}, {3293, 5045}, {3315, 3935}, {3452, 3782}, {3523, 15852}, {3616, 4646}, {3624, 3931}, {3660, 4551}, {3662, 5233}, {3663, 5316}, {3670, 5044}, {3677, 8580}, {3706, 3840}, {3711, 16496}, {3720, 3848}, {3741, 4732}, {3751, 4860}, {3816, 3914}, {3834, 3936}, {3873, 4849}, {3895, 16486}, {3896, 4891}, {3912, 6547}, {3920, 9342}, {3938, 4906}, {3976, 6048}, {3977, 4422}, {3987, 9957}, {4054, 7263}, {4070, 5750}, {4357, 5241}, {4395, 8610}, {4414, 15254}, {4684, 5212}, {4859, 5219}, {5218, 16020}, {5235, 16696}, {5256, 16884}, {5541, 16489}, {8609, 16586}, {14996, 16666}, {14997, 16669}, {16704, 16726}

X(16610) = complement of X(4358)
X(16610) = isotomic conjugate of polar conjugate of X(1878)


X(16611) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16611) lies on these lines:
{2, 14210}, {6, 5883}, {10, 37}, {169, 1046}, {213, 3754}, {238, 5011}, {241, 514}, {519, 3290}, {540, 4987}, {672, 1739}, {758, 2238}, {762, 4540}, {897, 10630}, {1018, 4695}, {1054, 5030}, {1334, 3987}, {1449, 2303}, {1738, 5179}, {1783, 1835}, {2295, 3918}, {2802, 3230}, {3214, 3970}, {3294, 4642}, {3670, 3691}, {3678, 3721}, {3735, 10176}, {3753, 3997}, {3780, 3881}, {3878, 3959}, {3954, 4015}, {4384, 4850}, {5358, 16553}, {5540, 16546}, {6629, 16756}, {7292, 11580}, {8691, 9136}


X(16612) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(525), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16612) lies on these lines:
{1, 8611}, {2, 14208}, {241, 514}, {522, 2522}, {579, 822}, {647, 4458}, {656, 1021}, {661, 3737}, {1751, 4444}, {2509, 3239}, {4025, 16757}, {6003, 7252}


X(16613) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(645), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16613) lies on these lines:
{2, 7257}, {11, 115}, {512, 1356}, {1084, 1146}, {1145, 2092}, {2170, 3124}, {2642, 2643}, {2802, 5213}, {2968, 6377}, {3013, 9456}, {3057, 9560}, {3756, 16614}, {6388, 7358}


X(16614) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

X(16614) lies on these lines: {115, 3141}, {244, 665}, {3756, 16613}


X(16615) =  ISOGONAL CONJUGATE OF X(13624)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27348.

X(16615) lies on the Feuerbach hyperbola and these lines:
{7, 9655}, {8, 381}, {12, 1000}, {21, 3579}, {65, 10308}, {104, 7686}, {515, 5557}, {944, 5558}, {946, 5559}, {1159, 5556}, {1320, 12738}, {1329, 6873}, {2320, 5253}, {3296, 5434}, {3899, 4866}, {5260, 12702}, {5603, 7320}, {10698, 13143}, {10742, 11604}

X(16615) = isogonal conjugate of X(13624)


X(16616) =  MIDPOINT OF X(4) AND X(7686)

Barycentrics    a*((b+c)*a^5-(b^2+8*b*c+c^2)* a^4-2*(b^2-c^2)*(b-c)*a^3+2*( b^2-c^2)^2*a^2+(b^2-c^2)*(b-c) ^3*a-(b^2-8*b*c+c^2)*(b^2-c^2) ^2) : :
X(16616) = 3*X(4)+X(65), 5*X(4)-X(12688), 3*X(40)-7*X(4002), X(65)-3*X(7686), 5*X(65)+3*X(12688), 3*X(381)-X(960), 3*X(946)-X(9957), X(1657)-3*X(10178), 9*X(1699)-X(5697), 5*X(3091)-X(14110), 5*X(3697)-9*X(5587), 5*X(3698)-X(6361), X(5045)-3*X(5806), 2*X(5045)-3*X(13374), 5*X(7686)+X(12688)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27348.

X(16616) lies on these lines:
{4, 65}, {30, 3812}, {40, 4002}, {355, 12731}, {381, 960}, {382, 9943}, {495, 946}, {515, 5045}, {516, 3918}, {517, 546}, {518, 3818}, {1389, 14496}, {1657, 10178}, {1699, 5697}, {2778, 7687}, {3057, 10590}, {3091, 14110}, {3149, 3612}, {3361, 12114}, {3586, 12710}, {3627, 15726}, {3697, 5587}, {3698, 6361}, {3822, 9955}, {3839, 3869}, {3843, 5887}, {3848, 13624}, {3861, 14988}, {4067, 5777}, {4537, 15064}, {5691, 11034}, {5794, 6849}, {5805, 6256}, {5818, 7957}, {5836, 12699}

X(16616) = midpoint of X(i) and X(j) for these {i,j}: {4, 7686}, {382, 9943}, {5691, 12675}, {5836, 12699}
X(16616) = X(7686)-of-Euler-triangle


X(16617) =  MIDPOINT OF X(946) AND X(3647)

Barycentrics    a^4*(-a+b+c)*(5*b^2-4*b*c+5*c^ 2-2*a^2)+4*(b^4+b^2*c^2+c^4)*a ^3-2*(b^2-c^2)*(b-c)*(2*b^2+b* c+2*c^2)*a^2-(b^2-c^2)^2*(b^2+ 4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :
X(16617) = X(3)-3*X(15670), X(4)+3*X(21), X(4)-3*X(6841), X(20)-9*X(15672), X(79)-5*X(8227), 3*X(191)+5*X(11522), 3*X(1699)+X(16113), X(3649)-3*X(5886), X(3652)+3*X(5886), 3*X(3817)-X(16125), X(5441)+3*X(5587), 11*X(5550)+X(10308), 3*X(5603)+X(11684), 9*X(7988)-X(16118)

As a point on the Euler line, this center has Shinagawa coefficients (3*(R+r), 3*R-r).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27352.

X(16617) lies on these lines:
{2, 3}, {79, 8227}, {90, 11375}, {191, 11522}, {354, 5887}, {355, 10543}, {758, 13464}, {912, 10122}, {946, 3647}, {952, 15174}, {1125, 13369}, {1385, 12617}, {1699, 16113}, {2771, 11281}, {3338, 3649}, {3624, 7171}, {3650, 11415}, {3817, 16125}, {4999, 9955}, {5441, 5587}, {5482, 10170}, {5550, 10308}, {5603, 11684}, {6701, 6713}, {7988, 16118}, {11230, 12608}, {11376, 16140}, {11544, 12047}

X(16617) = midpoint of X(i) and X(j) for these {i,j}: {355, 10543}, {946, 3647}
X(16617) = X(12606)-of-3rd-Euler-triangle
X(16617) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3560, 6824, 5), (3652, 5886, 3649), (6824, 6930, 6855), (6828, 7491, 546), (6832, 6974, 3), (6842, 6852, 3628), (6846, 6892, 6911), (6846, 6911, 5), (6852, 6912, 6842), (6857, 6985, 549), (6862, 6913, 5), (6884, 6906, 6881), (6888, 6920, 6882)


X(16618) =  EULER LINE INTERCEPT OF X(1216)X(2781)

Barycentrics    2*a^10-5*(b^2+c^2)*a^8+2*(b^4- 4*b^2*c^2+c^4)*a^6+4*(b^4+b^2* c^2+c^4)*(b^2+c^2)*a^4-4*(b^2- c^2)^2*(b^4+c^4)*a^2+(b^4-c^4) *(b^2-c^2)^3 : :
X(16618) = 3*X(2)+X(12082), X(4)+3*X(22), X(4)-3*X(15760), X(13352)-3*X(13394)

As a point on the Euler line, this center has Shinagawa coefficients (-3*E-6*F, 5*E+2*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27352.

X(16618) lies on these lines:
{2, 3}, {1216, 2781}, {1350, 5654}, {3003, 5305}, {5181, 5609}, {5319, 14836}, {5446, 9019}, {5486, 12161}, {6696, 14641}, {8550, 13292}, {9820, 15644}, {12022, 15080}, {13352, 13394}

X(16618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 468, 140), (3, 13383, 16238), (5, 550, 14791), (140, 468, 16238), (140, 13383, 468), (548, 10020, 16196), (3089, 7393, 5), (3547, 7387, 5), (7403, 7558, 3628), (7494, 9818, 549), (7512, 12605, 548), (7525, 15761, 12362), (7553, 13160, 546), (12088, 13160, 7553)


X(16619) =  REFLECTION OF X(1511) IN X(15448)

Barycentrics    2*a^10-5*(b^2+c^2)*a^8+2*(b^4- 4*b^2*c^2+c^4)*a^6+2*(b^2+c^2) *(2*b^4-b^2*c^2+2*c^4)*a^4-2*( b^2-c^2)^2*(2*b^4-3*b^2*c^2+2* c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(16619) = X(3)-3*X(7426), X(4)+3*X(23), 7*X(4)-3*X(10296), X(4)-3*X(11799), 3*X(1533)+X(10990), X(14094)+3*X(15360), 5*X(15027)-9*X(15362)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27352.

X(16619) lies on these lines:
{2, 3}, {511, 16534}, {523, 11615}, {524, 5609}, {1511, 15448}, {1533, 10990}, {1990, 16104}, {5305, 16303}, {5446, 11649}, {5648, 9970}, {7755, 16306}, {8705, 15074}, {10263, 16252}, {12002, 12242}, {14094, 15360}, {15027, 15362}

X(16619) = reflection of X(1511) in X(15448)
X(16619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 26, 550), (4, 10024, 3850), (7517, 15761, 11819)


X(16620) =  (name pending)

Barycentrics    ((16*R^2+4*SW-3*SB)*SB-3*S^2)* ((16*R^2+4*SW-3*SC)*SC-3*S^2) : :
X(16620) = 2*R^2*(7*R^2-8*SW)*X(10575)-( 27*R^2+2*SW)*(2*R^2-SW)*X( 13623)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16620) lies on the Jerabek hyperbola and this line: {10575, 13623}

X(16620) = isogonal conjugate of X(16661)


X(16621) =  (name pending)

Barycentrics    (SB+SC)*S^2-2*(4*R^2+SW)*SB*SC : :
X(16621) = 3*X(4)-X(6146), 7*X(4)-3*X(12022), 8*X(4)-3*X(12024), X(185)-3*X(428), 3*X(428)-2*X(11745), 3*X(3543)+X(14516), 7*X(6146)-9*X(12022), 8*X(6146)-9*X(12024), 2*X(6146)-3*X(12241), X(6240)+3*X(11455), 3*X(7576)+X(12290), 5*X(11439)-X(12225), 8*X(12022)-7*X(12024), 6*X(12022)-7*X(12241), 3*X(12024)-4*X(12241)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16621) lies on these lines:
{4, 6}, {24, 6696}, {25, 6247}, {30, 1216}, {64, 7487}, {66, 9914}, {140, 15448}, {141, 11414}, {154, 3088}, {184, 1907}, {185, 428}, {235, 11550}, {382, 12134}, {427, 15152}, {542, 13142}, {550, 4550}, {1596, 15153}, {1597, 9833}, {1598, 13567}, {1853, 3089}, {1899, 5198}, {3529, 11821}, {3541, 10192}, {3543, 14516}, {3564, 13598}, {3575, 11381}, {3589, 10984}, {3627, 15083}, {3818, 6823}, {6000, 6756}, {6240, 11455}, {6995, 9786}, {7400, 10516}, {7530, 12359}, {7553, 12162}, {7576, 12290}, {7715, 11438}, {8718, 14788}, {10151, 11572}, {11206, 11425}, {11439, 12225}, {12370, 15687}, {12605, 16194}, {13380, 14458}, {13490, 13491}, {14157, 15559}

X(16621) = midpoint of X(i) and X(j) for these {i,j}: {382, 12134}, {3575, 11381}, {7553, 12162}
X(16621) = reflection of X(185) in X(11745)
X(16621) = {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (4, 1181, 5480), (4, 1498, 12233), (185, 428, 11745), (1598, 14216, 13567), (1899, 5198, 15873), (6995, 12324, 9786)


X(16622) =  (name pending)

Barycentrics    (3*SA-16*R^2-SW)*(8*R^2+3*SA- SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16622) lies on these lines:
{5895, 9777}, {7715, 12006}, {9729, 11817}


X(16623) =  (name pending)

Barycentrics    (3*SA-16*R^2-SW)*(8*R^2+3*SA- SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16623) lies on the Jerabek hyperbola and these lines:
{54, 6696}, {3519, 15644}, {7391, 15749}, {12162, 14861}, {14483, 15559}

X(16623) = isogonal conjugate of X(12087)


X(16624) =  X(4)X(11225)∩X(2914)X(5972)

Trilinears    (3*(b^2+c^2)*a^6-(9*b^4+4*b^2* c^2+9*c^4)*a^4+(b^2+c^2)*(9*b^ 4-14*b^2*c^2+9*c^4)*a^2-3*(b^ 4+c^4)*(b^2-c^2)^2)*a : :
Barycentrics    (5*SA-3*SW+8*R^2)*(5*SA+SW-16* R^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16624) lies on these lines: {4, 11225}, {2914, 5972}, {11442, 15751}


X(16625) =  (name pending)

Barycentrics    (SB+SC)*(3*SA^2-8*R^2*SA-3*SB* SC) : :
X(16625) = 3*X(2)+X(14531), X(3)+3*X(52), X(3)-3*X(389), X(3)-9*X(568), 5*X(3)+3*X(6243), 2*X(3)-3*X(9729), 5*X(3)-9*X(9730), 7*X(3)-3*X(10625), 17*X(3)-9*X(13340), 4*X(3)-3*X(13348), 5*X(3)-3*X(15644), X(52)+3*X(568), 5*X(52)-X(6243), 2*X(52)+X(9729), 5*X(52)+3*X(9730), 7*X(52)+X(10625), 17*X(52)+3*X(13340), 4*X(52)+X(13348), 3*X(52)+2*X(15012), 5*X(52)+X(15644), X(389)-3*X(568), 5*X(389)+X(6243), 5*X(389)-3*X(9730), 7*X(389)-X(10625), 17*X(389)-3*X(13340), 4*X(389)-X(13348), 3*X(389)-2*X(15012), 5*X(389)-X(15644), 15*X(568)+X(6243), 6*X(568)-X(9729), 5*X(568)-X(9730), 21*X(568)-X(10625), 17*X(568)-X(13340), 12*X(568)-X(13348), 9*X(568)-2*X(15012), 15*X(568)-X(15644)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27353.

X(16625) lies on these lines:
{2, 14531}, {3, 6}, {4, 14831}, {5, 16254}, {30, 11565}, {51, 3091}, {69, 9815}, {140, 15606}, {143, 546}, {185, 3060}, {373, 11444}, {399, 13402}, {524, 9825}, {542, 6756}, {548, 13421}, {631, 16226}, {632, 1216}, {1154, 3628}, {1493, 7575}, {1986, 11800}, {1994, 13367}, {3090, 3567}, {3292, 15801}, {3525, 3819}, {3529, 5890}, {3544, 11459}, {3564, 11745}, {3574, 3580}, {3575, 10112}, {3627, 5446}, {3818, 11411}, {3853, 12002}, {3857, 5876}, {3917, 10303}, {5072, 13321}, {5076, 13474}, {5079, 5891}, {5447, 12006}, {5476, 7404}, {5609, 11557}, {5640, 15022}, {5650, 15028}, {5663, 12102}, {5892, 6101}, {6146, 11225}, {6622, 15010}, {7488, 13366}, {7503, 15004}, {7542, 12242}, {7553, 11645}, {7556, 11423}, {7999, 15082}, {9306, 12160}, {9781, 15030}, {10095, 12811}, {10116, 11819}, {10170, 15026}, {10263, 15704}, {10282, 12161}, {10297, 13376}, {10628, 12236}, {11002, 12111}, {11004, 11449}, {11591, 12812}, {12045, 15067}, {12103, 13630}, {12219, 15025}, {13142, 13568}, {13417, 15054}, {13861, 15083}, {15034, 16223}

X(16625) = midpoint of X(i) and X(j) for these {i,j}: {185, 13598}, {548, 13421}, {1986, 11800}, {3575, 10112}, {10116, 11819}, {13142, 13568}
X(16625) = reflection of X(i) in X(j) for these (i,j): (3, 15012), (399, 13402), (3853, 12002), (5447, 12006)
X(16625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 389, 15012), (3, 15012, 9729), (51, 5889, 5907), (52, 568, 389), (52, 9730, 6243), (185, 3060, 13598), (389, 15644, 9730), (1216, 5946, 11695), (1351, 9786, 13346), (3567, 5562, 5943), (5462, 11793, 6688), (6243, 9730, 15644)


X(16626) = PERSPECTOR OF THESE TRIANGLES: 11th FERMAT-DAO AND JOHNSON

Barycentrics    2*S*(S^2+2*SB*SC)+sqrt(3)*(S^2*SA+SB*SC*SW) : :
X(16626) = 5*X(1656)-4*X(6673) = 3*X(5886)-2*X(11739)

Let F = X(13) be the 1st Fermat point of ABC. Let Ab and Ac be the points where the circle FBC cuts again AC and AB, respectively. Define Bc, Ba and Ca, Cb cyclically. Denote A', B', C' the circumcenters of ABcCb, BCaAc and CBaAb, respectively. Then the triangle A'B'C' is equilateral and is here named the 11th Dao-Fermat equilateral triangle. (Dao Thanh Oai, March 19, 2018)

When F=X(14), i.e., the 2nd Fermat point of ABC, the resulting triangle is also equilateral and is here named the 12th Dao-Fermat equilateral triangle.

Both 11th and 12th Fermat-Dao equilateral triangles have center X(381), which also is their center of inverse similitude.

The appearance of (T,n) in the following list means that T and the 11th Dao-equilateral triangle are perspective with perspector X(n); in this list and the next, an asterisk * means that the triangles are homothetic, and ** means that they are inversely similar.

(ABC, 4), (anti-Euler, 4), (anti-excenters-reflections, 4), (anti-inverse-in-incircle, 4), (circumorthic, 4), (Euler, 4), (2nd extouch, 4), (3rd extouch, 4), (3er Fermat-Dao*, 61), (4th Fermat-Dao**, 16630), (7th Fermat-Dao*, 10654), (8th Fermat-Dao**, 16634), (12th Fermat-Dao**, 4), (Johnson, 16626), (midheight, 4), (outer-Napoleon*, 5), (orthic, 4), (orthocentroidal, 4), (1st orthosymmedial, 4), (reflection, 4), (X(3)-ABC reflections, 16628)

The appearance of (T,n) in the following list means that T and the 12th Dao-equilateral triangle are perspective with perspector X(n):

(ABC, 4), (anti-Euler, 4), (anti-excenters-reflections, 4), (anti-inverse-in-incircle, 4), (circumorthic, 4), (Euler, 4), (2nd extouch, 4), (3rd extouch, 4), (3rd Fermat-Dao**, 16631), (4th Fermat-Dao*, 62), (7th Fermat-Dao**, 16635), (8th Fermat-Dao*, 10653), (11th Fermat-Dao**, 4), (Johnson, 16627), (midheight, 4), (inner-Napoleon*, 5), (orthic, 4), (orthocentroidal, 4), (1st orthosymmedial, 4), (reflection, 4), (X(3)-ABC reflections, 16629)

This introduction and centers X(16626) to X(16645) were contributed by César Lozada, March 23, 2018.

X(16626) lies on these lines:
{3,623}, {4,616}, {5,14}, {13,5872}, {114,16002}, {193,576}, {621,7752}, {622,15031}, {633,7809}, {1656,6673}, {2782,11602}, {5487,14492}, {5886,11739}, {6287,7685}, {11555,16459}

X(16626) = midpoint of X(4) and X(627)
X(16626) = reflection of X(3) in X(629)


X(16627) = PERSPECTOR OF THESE TRIANGLES: 12th FERMAT-DAO AND JOHNSON

Barycentrics    -2*S*(S^2+2*SB*SC)+sqrt(3)*(S^2*SA+SB*SC*SW) : :
X(16627) = 5*X(1656)-4*X(6674) = 3*X(5886)-2*X(11740)

X(16627) lies on these lines:
{3,624}, {4,617}, {5,13}, {14,5873}, {114,16001}, {193,576}, {621,15031}, {622,7752}, {634,7809}, {1656,6674}, {2782,11603}, {5886,11740}, {6287,7684}, {11556,16460}

X(16627) = midpoint of X(4) and X(628)
X(16627) = reflection of X(3) in X(630)


X(16628) = PERSPECTOR OF THESE TRIANGLES: 11th FERMAT-DAO AND X3-ABC REFLECTIONS

Barycentrics    S*(S^2+7*SB*SC)+sqrt(3)*((2*SA-SW)*S^2-SB*SC*SW) : :
X(16628) = 4*X(630)-5*X(1656) = 7*X(3526)-8*X(6674) = 3*X(10246)-4*X(11740)

X(16628) lies on these lines:
{3,14}, {4,3181}, {5,303}, {62,14537}, {382,5615}, {630,1656}, {1351,3818}, {1975,11133}, {3526,6674}, {7747,11486}, {10246,11740}, {11485,13881}

X(16628) = reflection of X(3) in X(18)


X(16629) = PERSPECTOR OF THESE TRIANGLES: 12th FERMAT-DAO AND X3-ABC REFLECTIONS

Barycentrics    -S*(S^2+7*SB*SC)+sqrt(3)*((2*SA-SW)*S^2-SB*SC*SW) : :
X(16629) = 4*X(629)-5*X(1656) = 7*X(3526)-8*X(6673) = 3*X(10246)-4*X(11739)

X(16629) lies on these lines:
{3,13}, {4,3180}, {5,302}, {61,14537}, {382,5611}, {629,1656}, {1351,3818}, {1975,11132}, {3526,6673}, {7747,11485}, {10246,11739}, {11486,13881}

X(16629) = reflection of X(3) in X(17)


X(16630) = PERSPECTOR OF THESE TRIANGLES: 11th FERMAT-DAO AND 4th FERMAT-DAO

Barycentrics    S*(8*S^2-3*SW^2+3*SA^2+8*SB*SC)+sqrt(3)*(S^2*SA-SB*SC*SW) : :

X(16630) lies on these lines:
{5,3107}, {13,5872}, {15,13881}, {62,381}, {115,3104}, {511,16631}, {636,5025}, {7617,11305}, {11304,13084}


X(16631) = PERSPECTOR OF THESE TRIANGLES: 12th FERMAT-DAO AND 3rd FERMAT-DAO

Barycentrics    -S*(8*S^2-3*SW^2+3*SA^2+8*SB*SC)+sqrt(3)*(S^2*SA-SB*SC*SW) : :

X(16631) lies on these lines:
{5,3106}, {14,5873}, {16,13881}, {61,381}, {62,16629}, {115,3105}, {511,16630}, {635,5025}, {7617,11306}, {11303,13083}


X(16632) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 12th FERMAT-DAO AND 13th FERMAT-DAO

Barycentrics    3*S^4-3*(R^2*(SW+3*SA)-SA^2-SB*SC)*S^2-sqrt(3)*((9*R^2-5*SW)*SB*SC-(9*R^2-SA)*S^2)*S-SB*SC*SW^2 : :

The 13th and 14th Fermat-Dao equilateral triangles are defined at X(16636).

X(16632) lies on these lines:
{621,11002}, {1316,11586}


X(16633) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 11th FERMAT-DAO AND 14th FERMAT-DAO

Barycentrics    3*S^4-3*(R^2*(SW+3*SA)-SA^2-SB*SC)*S^2+sqrt(3)*((9*R^2-5*SW)*SB*SC-(9*R^2-SA)*S^2)*S-SB*SC*SW^2 : :

The 13th and 14th Fermat-Dao equilateral triangles are defined at X(16636).

X(16633) lies on these lines:
{622,11002}, {1316,15743}


X(16634) = PERSPECTOR OF THESE TRIANGLES: 11th FERMAT-DAO AND 8th FERMAT-DAO

Barycentrics    sqrt(3)*S*(3*S^2+SA^2+5*SB*SC-SW^2)+(S^2-3*SB*SC)*SW : :

X(16634) lies on these lines:
{2,11154}, {4,6108}, {13,1992}, {14,6776}, {30,5210}, {115,16632}, {381,395}, {3643,16041}, {6115,9741}, {7615,9763}


X(16635) = PERSPECTOR OF THESE TRIANGLES: 12th FERMAT-DAO AND 7th FERMAT-DAO

Barycentrics    -sqrt(3)*S*(3*S^2+SA^2+5*SB*SC-SW^2)+(S^2-3*SB*SC)*SW : :

X(16635) lies on these lines:
{2,11153}, {13,6776}, {14,1992}, {115,16633}, {381,396}, {3642,16041}, {7615,9761}


X(16636) = PERSPECTOR OF THESE TRIANGLES: 13th FERMAT-DAO AND 1st FERMAT-DAO

Barycentrics    (-2*(2*SA^2-2*SB*SC-3*R^2*(SW+3*SA))*S+sqrt(3)*(SA^2*(9*R^2-2*SW)+(21*R^2-2*SW)*S^2))*(SB+SC) : :

Let F = X(13) be the 1st Fermat point of ABC. Let Ab and Ac be the points where the circle FBC cuts again AC and AB, respectively. Define Bc, Ba and Ca, Cb cyclically. Let A', B', C' be centroids of ABcCb, BCaAc, CAbBa respectively. Then the triangle A'B'C' is equilateral and will be named here the 13th Fermat-Dao equilateral triangle. (Dao Thanh Oai, March 19, 2018)

When F=X(14), i.e., the 2nd Fermat point of ABC, the resulting triangle is also equilateral and will be named here the 14th Dao-Fermat equilateral triangle.

The perspector of the 13th and 14th Fermat-Dao triangles is X(6) and both are perspective with the same perspector to these triangles: ABC, anti-Conway, 2nd anti-Conway, 2nd Brocard, circumsymmedial, 2nd Ehrmann, inner-Grebe, outer-Grebe, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd orthosymmedial, symmedial, tangential, inner tri-equilateral, outer tri-equilateral.

The 13th Fermat-Dao is also perspective to these triangles: 1st Fermat-Dao* at X(16636), 5th Fermat-Dao* at X(8014), outer-Le Viet An* at X(3130), 9th Fermat-Dao* at X(6) and 10th Fermat-Dao** at X(6).

The 14th Fermat-Dao is also perspective to these triangles: 2nd Fermat-Dao* at X(16637), 6th Fermat-Dao* at X(8015), inner-Le Viet An* at X(3129), 9th Fermat-Dao** at X(6) and 10th Fermat-Dao* at X(6).

Note: an asterisk * means that the triangles are homothetic, and ** means that they are inversely similar.

(This introduction and centers X(16636) and X(16637) were contributed by César Lozada, March 23, 2018.)

X(16636) lies on these lines:
{6,2981}, {14,5640}, {543,16637}


X(16637) = PERSPECTOR OF THESE TRIANGLES: 14th FERMAT-DAO AND 2nd FERMAT-DAO

Barycentrics    (2*(2*SA^2-2*SB*SC-3*R^2*(SW+3*SA))*S+sqrt(3)*(SA^2*(9*R^2-2*SW)+(21*R^2-2*SW)*S^2))*(SB+SC) : :

X(16637) lies on these lines:
{6,6151}, {13,5640}, {543,16636}


X(16638) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO AND 5th FERMAT-DAO

Barycentrics
(3*(-SW+9*R^2)*SW*SA^2+(9*S^2+3*R^2*(18*SA+23*SW)+5*SA^2+4*SB*SC-3*SW^2)*S^2+sqrt(3)*S*((63*R^2+4*SA)*S^2+3*R^2*(17*SA^2-2*SB*SC+2*SW^2)-4*SA^2*SW))*(SB+SC)*(S+sqrt(3)*SB)*(S+sqrt(3)*SC) : :

X(16638) lies on these lines:
{16,16459}, {16259,16461}, {16463,16642}


X(16639) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO AND 6th FERMAT-DAO

Barycentrics
(3*(-SW+9*R^2)*SW*SA^2+(9*S^2+3*R^2*(18*SA+23*SW)+5*SA^2+4*SB*SC-3*SW^2)*S^2-sqrt(3)*S*((63*R^2+4*SA)*S^2+3*R^2*(17*SA^2-2*SB*SC+2*SW^2)-4*SA^2*SW))*(SB+SC)*(-S+sqrt(3)*SB)*(-S+sqrt(3)*SC) : :

X(16639) lies on these lines:
{15,16460}, {16260,16462}, {16464,16643}


X(16640) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO AND 9th FERMAT-DAO

Barycentrics    (2*S*(3*R^2*(9*R^2+3*SA+2*SW)-2*SA^2+2*SB*SC)+(3*R^2*(SW^2+10*S^2+3*SA^2)-2*SW*(S^2+SA^2))*sqrt(3))*(SB+SC) : :

X(16640) lies on these lines:
{6,2981}, {11080,16247}, {11624,16259}


X(16641) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO AND 10th FERMAT-DAO

Barycentrics    (-2*S*(3*R^2*(9*R^2+3*SA+2*SW)-2*SA^2+2*SB*SC)+(3*R^2*(SW^2+10*S^2+3*SA^2)-2*SW*(S^2+SA^2))*sqrt(3))*(SB+SC) : :

X(16641) lies on these lines:
{6,6151}, {11626,16260}


X(16642) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st FERMAT-DAO AND OUTER-LE VIET AN

Barycentrics    (SB+SC)*(sqrt(3)*S*((21*R^2+3*SA-4*SW)*S^2+9*R^2*(SA^2-2*SB*SC)-SA*SW*(-3*SW+4*SA))-3*SB*SC*SW^2+(6*R^2*(SW+6*SA)-2*SA^2-7*SB*SC)*S^2+2*S^4) : :

X(16642) lies on these lines:
{3130,16636}, {14170,16259}, {16463,16638}


X(16643) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd FERMAT-DAO AND INNER-LE VIET AN

Barycentrics    (SB+SC)*(-sqrt(3)*S*((21*R^2+3*SA-4*SW)*S^2+9*R^2*(SA^2-2*SB*SC)-SA*SW*(-3*SW+4*SA))-3*SB*SC*SW^2+(6*R^2*(SW+6*SA)-2*SA^2-7*SB*SC)*S^2+2*S^4) : :

X(16643) lies on these lines:
{3129,16637}, {16464,16639}


X(16644) = HOMOTHETIC CENTER OF THESE TRIANGLES: 7th FERMAT-DAO AND OUTER-NAPOLEON

Barycentrics    4*S+(SB+SC)*sqrt(3) : :

X(16644) lies on these lines:
{2,6}, {3,13}, {5,5339}, {14,5055}, {15,381}, {16,5054}, {18,3412}, {30,11480}, {61,1656}, {62,3526}, {115,5464}, {182,11134}, {376,5318}, {382,5238}, {397,631}, {398,3090}, {549,11481}, {619,6775}, {622,10616}, {624,11297}, {633,8259}, {1152,15765}, {1657,5352}, {1853,11243}, {3364,13951}, {3365,8976}, {3366,10577}, {3367,10576}, {3524,5335}, {3528,5344}, {3529,5350}, {3534,10645}, {3544,5343}, {3545,5321}, {3642,6300}, {3643,6671}, {3855,5349}, {5050,6774}, {5064,10641}, {5071,5334}, {5094,8740}, {5237,15720}, {5459,6772}, {5463,5472}, {5613,6109}, {5617,15069}, {6412,15764}, {6694,11312}, {6780,13102}, {7051,11237}, {7998,11624}, {9112,9115}, {9306,11137}, {10638,11238}, {10646,15693}, {11486,15694}, {11543,15699}, {12816,15685}

X(16644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 16645), (2, 396, 6), (2, 3180, 302), (2, 9763, 599), (2, 11488, 396), (302, 3180, 5858), (549, 16633, 11481), (590, 615, 11488), (3054, 15533, 16645), (3180, 5858, 6144), (3642, 6669, 11305), (6300, 6304, 11305), (8252, 13847, 16645), (8253, 13846, 16645), (8860, 9763, 15534)


X(16645) = HOMOTHETIC CENTER OF THESE TRIANGLES: 8th FERMAT-DAO AND INNER-NAPOLEON

Barycentrics    -4*S+(SB+SC)*sqrt(3) : :

X(16645) lies on these lines:
{2,6}, {3,14}, {5,5340}, {13,5055}, {15,5054}, {16,381}, {17,3411}, {30,11481}, {61,3526}, {62,1656}, {115,5463}, {182,11137}, {376,5321}, {382,5237}, {397,3090}, {398,631}, {549,11480}, {618,6772}, {621,10617}, {623,11298}, {634,8260}, {1151,15765}, {1250,11238}, {1657,5351}, {1853,11244}, {3389,13951}, {3390,8976}, {3391,10577}, {3392,10576}, {3524,5334}, {3528,5343}, {3529,5349}, {3534,10646}, {3544,5344}, {3545,5318}, {3642,6672}, {3643,6301}, {3855,5350}, {5050,6771}, {5064,10642}, {5071,5335}, {5094,8739}, {5238,15720}, {5460,6775}, {5464,5471}, {5613,15069}, {6411,15764}, {6694,11309}, {6695,11311}, {7998,11626}, {9113,9117}, {9306,11134}, {10645,15693}, {11485,15694}, {11542,15699}, {12817,15685}

X(16645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 16644), (2, 395, 6), (2, 3181, 303), (2, 9761, 599), (2, 11489, 395), (303, 3181, 5859), (549, 16632, 11480), (590, 615, 11489), (3054, 15533, 16644), (3181, 5859, 6144), (3642, 6672, 11302), (6301, 6305, 11306), (8252, 13847, 16644), (8253, 13846, 16644), (8860, 9761, 15534)


X(16646) = PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT AND ANTI-ARTZT

Barycentrics    sqrt(3)*(6*S^2-(3*SA-SW)*(3*SA-2*SW))*SW-(6*SA-7*SW)*(3*SA+SW)*S : :

In a triangle ABC, build equilateral triangles BCA', CAB', ABC' inwards ABC. The triangle A'B'C' is named the inner-Fermat triangle of ABC. Similarly, if BCA", CAB", ABC" are equilateral triangles built outwards ABC, the triangle A"B"C" is the outer-Fermat triangle of ABC.

Both triangles are perspective with perspector X(3) to the following triangles: (anti-Hutson intouch, anti-incircle-circles, 6th anti-mixtilinear, Ara, Ascella, 1st Brocard, 1st circumperp, 2nd circumperp, 1st Ehrmann, 2nd Euler, Fuhrmann, 2nd Fuhrmann, Johnson, Kosnita, McCay, medial, inner-Napoleon, outer-Napoleon, 1st Neuberg, 2nd Neuberg, tangential, Trinh, inner-Vecten and outer-Vecten. The perspector of the inner- and outer-Fermat triangles is also X(3).

The inner-Fermat triangle is also perspective to the following triangles with given perspectors: (ABC, X(14)), (anti-Artzt, X(16646)), (1st anti-Brocard, X(16648)), (anti-McCay, X(16650)), (anticomplementary, X(616)), (Artzt, X(16652)), (excentral, X(1277)), (reflection, X(15)). The outer-Fermat triangle is also perspective to the following triangles with given perspectors: (ABC, X(13)), (anti-Artzt, X(16647)), (1st anti-Brocard, X(16649)), (anti-McCay, X(16651)), (anticomplementary, X(617)), (Artzt, X(16653)), (excentral, X(1276)), (reflection, X(16)).
(This introduction and centers X(16646)-X(16653) were contributed by César Lozada, March 23, 2018)

X(16646) lies on these lines:
{4045,16647}, {5858,7813}, {6775,11154}


X(16647) = PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT AND ANTI-ARTZT

Barycentrics    sqrt(3)*(6*S^2-(3*SA-SW)*(3*SA-2*SW))*SW+(6*SA-7*SW)*(3*SA+SW)*S : :

X(16647) lies on these lines:
{4045,16646}, {5859,7813}, {6772,11153}


X(16648) = PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT AND 1st ANTI-BROCARD

Barycentrics    sqrt(3)*(SW^2*SA*(5*SA-4*SW)+(SA^2-4*SB*SC+SW^2)*S^2+5*S^4)*S*SW+S^2*(-SW^2*(SA^2-6*SB*SC-SW^2)-SW*(SA+3*SW)*S^2+5*S^4)-3*SB*SC*SW^4 : :

X(16648) lies on these lines:
{114,8292}, {1916,3104}, {5152,6033}, {5978,5989}, {5982,16626}, {8295,9749}


X(16649) = PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT AND 1st ANTI-BROCARD

Barycentrics    -sqrt(3)*(SW^2*SA*(5*SA-4*SW)+(SA^2-4*SB*SC+SW^2)*S^2+5*S^4)*S*SW+S^2*(-SW^2*(SA^2-6*SB*SC-SW^2)-SW*(SA+3*SW)*S^2+5*S^4)-3*SB*SC*SW^4 : :

X(16649) lies on these lines:
{3,5982}, {114,8291}, {1916,3105}, {5152,6033}


X(16650) = PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT AND ANTI-MCCAY

Barycentrics    S*(135*S^4-3*(9*SA^2+72*SB*SC+13*SW^2)*S^2-(3*SA^2-48*SB*SC-2*SW^2)*SW^2)+3*sqrt(3)*(3*S^2*(9*SA-8*SW)+SW*(3*SA^2+42*SB*SC+2*SW^2))*S^2-sqrt(3)*(9*SB*SC+SW^2)*SW^3 : :

X(16650) lies on these lines:
{2482,10809}, {3104,11152}, {8594,10488}


X(16651) = PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT AND ANTI-MCCAY

Barycentrics    -S*(135*S^4-3*(9*SA^2+72*SB*SC+13*SW^2)*S^2-(3*SA^2-48*SB*SC-2*SW^2)*SW^2)+3*sqrt(3)*(3*S^2*(9*SA-8*SW)+SW*(3*SA^2+42*SB*SC+2*SW^2))*S^2-sqrt(3)*(9*SB*SC+SW^2)*SW^3 : :

X(16651) lies on these lines:
{2482,10808}, {3105,11152}, {8595,10488}


X(16652) = PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT AND ARTZT

Barycentrics    3*sqrt(3)*(S^2-SB*SC)*S*SW+9*S^4+(-SA^2+SB*SC+SW^2)*S^2-2*SB*SC*SW^2 : :

X(16652) lies on these lines:
{3,10242}, {15,9754}, {17,262}, {627,9742}, {629,7710}, {5858,5965}, {5982,9772}, {9749,9756}, {9752,14138}


X(16653) = PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT AND ARTZT

Barycentrics    -3*sqrt(3)*(S^2-SB*SC)*S*SW+9*S^4+(-SA^2+SB*SC+SW^2)*S^2-2*SB*SC*SW^2 : :

X(16653) lies on these lines:
{3,10242}, {16,9754}, {18,262}, {628,9742}, {630,7710}, {5859,5965}, {9750,9756}, {9752,14139}


X(16654) =  X(4)X(6)∩X(30)X(3917)

Barycentrics    (SB+SC)*S^2-2*(6*R^2+SW)*SB*SC : :
X(16654) = 4*X(4)-X(6146), 3*X(4)-X(12022), 7*X(4)-2*X(12024), 5*X(4)-2*X(12241), X(1885)+2*X(13419), X(3575)+2*X(13474), 2*X(3627)+X(12134), 3*X(6146)-4*X(12022), 7*X(6146)-8*X(12024), 5*X(6146)-8*X(12241), X(6241)-4*X(11745), 2*X(6756)+X(11381), 7*X(12022)-6*X(12024), 5*X(12022)-6*X(12241), 5*X(12024)-7*X(12241)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16654) lies on these lines:
{4, 6}, {30, 3917}, {141, 12082}, {428, 6000}, {1204, 7715}, {1539, 3853}, {1596, 11550}, {1885, 13419}, {1907, 6759}, {2777, 3575}, {3518, 6696}, {3627, 12134}, {5198, 14216}, {5965, 13598}, {6241, 11745}, {6247, 10594}, {6756, 11381}, {6995, 10605}, {7576, 11455}, {9833, 11403}, {10127, 14855}, {10301, 11438}, {11457, 15873}, {12102, 12370}, {12290, 13568}, {15559, 16252}

X(16654) = midpoint of X(7576) and X(11455)
X(16654) = {X(4), X(11456)}-harmonic conjugate of X(5480)


X(16655) =  X(4)X(6)∩X(30)X(5562)

Barycentrics    (SB+SC)*S^2-2*(2*R^2+SW)*SB*SC : :
X(16655) = 5*X(4)-3*X(12022), 11*X(4)-6*X(12024), 3*X(4)-2*X(12241), 2*X(389)-3*X(428), 3*X(5890)-4*X(11745), 5*X(6146)-6*X(12022), 11*X(6146)-12*X(12024), 3*X(6146)-4*X(12241), X(6146)-4*X(16621), 11*X(12022)-10*X(12024), 9*X(12022)-10*X(12241), 3*X(12022)-10*X(16621), 9*X(12024)-11*X(12241), 3*X(12024)-11*X(16621), X(12241)-3*X(16621)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16655) lies on these lines:
{4, 6}, {5, 10984}, {24, 6247}, {25, 14216}, {30, 5562}, {140, 1495}, {141, 10323}, {154, 3541}, {184, 1595}, {185, 6756}, {186, 6696}, {343, 7387}, {382, 9936}, {389, 428}, {542, 13598}, {546, 6346}, {578, 1907}, {1209, 16618}, {1352, 11414}, {1593, 9833}, {1594, 14157}, {1598, 1899}, {1614, 15559}, {1657, 3426}, {1853, 3542}, {1885, 11577}, {3088, 11206}, {3146, 14516}, {3410, 12087}, {3574, 16198}, {3575, 6000}, {3796, 7404}, {3818, 7399}, {3853, 12370}, {5663, 11819}, {5878, 12173}, {5890, 11745}, {6240, 12290}, {6241, 7576}, {7383, 10516}, {7391, 11441}, {7487, 10605}, {7517, 12359}, {7553, 13754}, {7577, 15152}, {7667, 11793}, {9820, 10540}, {10018, 15448}, {10110, 11245}, {10539, 11064}, {10594, 11457}, {11750, 16194}, {12225, 15305}, {12362, 15030}, {13202, 13431}, {13417, 14449}, {13490, 13630}, {13599, 14458}

X(16655) = midpoint of X(i) and X(j) for these {i,j}: {3146, 14516}, {6240, 12290}
X(16655) = reflection of X(i) in X(j) for these (i,j): (4, 16621), (1885, 13474), (3575, 13419), (6241, 13568)
X(16655) = X(14216)-of-antiAra-triangle
X(16655) = X(16621)-of-antiEuler-triangle
X(16655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6776, 10982), (4, 7592, 5480), (4, 11456, 12233), (1594, 14157, 16252), (6241, 7576, 13568), (7487, 12324, 10605), (10594, 11457, 13567)


X(16656) =  X(4)X(6)∩X(30)X(5447)

Barycentrics    (SB+SC)*S^2-2*(8*R^2+SW)*SB*SC : :
X(16656) = 5*X(4)-X(6146), 11*X(4)-3*X(12022), 13*X(4)-3*X(12024), 3*X(4)-X(12241), 3*X(428)+X(11381), 3*X(428)-X(13568), 11*X(6146)-15*X(12022), 13*X(6146)-15*X(12024), 3*X(6146)-5*X(12241), X(6146)+5*X(16621), 13*X(12022)-11*X(12024), 9*X(12022)-11*X(12241), 3*X(12022)+11*X(16621), 9*X(12024)-13*X(12241), 3*X(12024)+13*X(16621), X(12241)+3*X(16621)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16656) lies on these lines:
{4, 6}, {5, 13347}, {25, 6696}, {30, 5447}, {64, 6995}, {428, 11381}, {524, 13598}, {1192, 7714}, {1204, 10301}, {1595, 16252}, {1598, 6247}, {1906, 11550}, {3088, 10192}, {3357, 7715}, {3426, 15105}, {3830, 12134}, {5059, 14490}, {5198, 13567}, {5894, 7487}, {6000, 11745}, {6225, 7408}, {6756, 13474}, {7553, 16194}, {10127, 14641}, {13419, 13488}, {14216, 15873}

X(16656) = midpoint of X(i) and X(j) for these {i,j}: {4, 16621}, {6756, 13474}, {13419, 13488}
X(16656) = X(6696)-of-antiAra-triangle
X(16656) = X(16621)-of-Euler-triangle
X(16656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1498, 5480), (4, 15811, 2883), (428, 11381, 13568)


X(16657) =  X(4)X(6)∩X(30)X(51)

Barycentrics    (SB+SC)*S^2+2*(6*R^2-SW)*SB*SC : :
X(16657) = 2*X(4)+X(6146), 3*X(4)+2*X(12024), X(4)+2*X(12241), 5*X(4)-2*X(16621), 4*X(546)-X(12134), 2*X(546)+X(12370), 3*X(6146)-4*X(12024), X(6146)-4*X(12241), 5*X(6146)+4*X(16621), 3*X(12022)-2*X(12024), 5*X(12022)+2*X(16621), X(12024)-3*X(12241), 5*X(12024)+3*X(16621), X(12134)+2*X(12370), 5*X(12241)+X(16621)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16657) lies on these lines:
{4, 6}, {5, 1092}, {24, 15873}, {30, 51}, {54, 16252}, {113, 137}, {140, 10564}, {143, 15807}, {184, 1596}, {185, 13488}, {195, 12364}, {235, 578}, {343, 9818}, {378, 13567}, {381, 3167}, {382, 3527}, {389, 974}, {403, 15033}, {468, 10182}, {524, 11459}, {567, 11799}, {973, 5446}, {1173, 11744}, {1594, 14644}, {1597, 1899}, {1906, 6759}, {2935, 6696}, {3542, 11425}, {3564, 15030}, {3567, 13568}, {3575, 10110}, {3580, 7527}, {3832, 14516}, {5198, 9833}, {5305, 6793}, {5462, 12897}, {5504, 9820}, {5562, 13142}, {5890, 7729}, {5907, 5965}, {6000, 11245}, {6240, 9781}, {6623, 11427}, {7403, 9927}, {7528, 12293}, {7529, 12118}, {9826, 10095}, {10127, 14845}, {10605, 11433}, {11403, 14216}, {11464, 15448}, {11572, 16198}, {12162, 13292}, {15053, 16386}

X(16657) = midpoint of X(4) and X(12022)
X(16657) = X(974)-of-orthocentroidal-triangle
X(16657) = X(12022)-of-Euler-triangle
X(16657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 7592, 2883), (4, 12241, 6146), (4, 14912, 5656), (5, 13352, 11064), (546, 12370, 12134), (5656, 14912, 1181), (6240, 9781, 11745), (10110, 13403, 3575), (14644, 15472, 15131)


X(16658) =  X(4)X(6)∩X(30)X(2979)

Barycentrics    (SB+SC)*S^2-2*(3*R^2+SW)*SB*SC : :
X(16658) = 5*X(4)-2*X(6146), 9*X(4)-4*X(12024), 7*X(4)-4*X(12241), X(4)-4*X(16621), 2*X(382)+X(14516), 4*X(6146)-5*X(12022), 9*X(6146)-10*X(12024), 7*X(6146)-10*X(12241), X(6146)-10*X(16621), 9*X(12022)-8*X(12024), 7*X(12022)-8*X(12241), X(12022)-8*X(16621), 7*X(12024)-9*X(12241), X(12024)-9*X(16621), X(12241)-7*X(16621)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16658) lies on these lines:
{4, 6}, {5, 13339}, {20, 11472}, {30, 2979}, {54, 1907}, {140, 10546}, {235, 14644}, {382, 14516}, {403, 11550}, {427, 14157}, {428, 5890}, {549, 6030}, {1352, 12082}, {1495, 10182}, {1595, 1614}, {1598, 11457}, {2777, 6240}, {3088, 9707}, {3146, 12134}, {3518, 6247}, {3575, 12290}, {3580, 7530}, {3627, 7728}, {5076, 12370}, {5189, 15052}, {5663, 7540}, {6000, 7576}, {6241, 6756}, {7399, 8718}, {7553, 12111}, {10594, 14216}, {11439, 12605}, {12289, 13488}, {13371, 14643}

X(16658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 15032, 5480), (11381, 13419, 6240)


X(16659) =  X(4)X(6)∩X(30)X(11412)

Barycentrics    (SB+SC)*S^2-2*(R^2+SW)*SB*SC : :
X(16659) = 3*X(4)-2*X(6146), 4*X(4)-3*X(12022), 17*X(4)-12*X(12024), 5*X(4)-4*X(12241), 3*X(4)-4*X(16621), 2*X(185)-3*X(7576), 8*X(6146)-9*X(12022), 17*X(6146)-18*X(12024), 5*X(6146)-6*X(12241), 3*X(7576)-4*X(13419), 17*X(12022)-16*X(12024), 15*X(12022)-16*X(12241), 9*X(12022)-16*X(16621), 15*X(12024)-17*X(12241), 9*X(12024)-17*X(16621), 3*X(12241)-5*X(16621)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27355.

X(16659) lies on these lines:
{4, 6}, {5, 14157}, {20, 12134}, {23, 12359}, {24, 14216}, {25, 11457}, {30, 11412}, {54, 1595}, {125, 14864}, {140, 15080}, {155, 7391}, {184, 15559}, {185, 7576}, {186, 6247}, {343, 12088}, {378, 9833}, {382, 12160}, {427, 1614}, {428, 3567}, {858, 10539}, {1352, 10323}, {1495, 10018}, {1594, 6759}, {1853, 7505}, {1885, 11455}, {1899, 10594}, {1907, 15033}, {2888, 12087}, {2918, 6636}, {3357, 10295}, {3541, 9707}, {3575, 6241}, {3580, 7517}, {3818, 10984}, {3830, 12370}, {5012, 7403}, {5094, 14530}, {5889, 7553}, {5890, 6756}, {5894, 13619}, {5907, 11645}, {6000, 6240}, {6102, 7540}, {6143, 10192}, {6823, 8718}, {7387, 11442}, {7500, 11411}, {7577, 16252}, {7667, 7999}, {9781, 11245}, {10540, 13371}, {11232, 12002}, {11381, 12300}, {11441, 14790}, {12162, 12225}, {12173, 12315}, {12244, 15105}, {12254, 13596}, {12362, 15058}, {12605, 15305}

X(16659) = reflection of X(i) in X(j) for these (i,j): (20, 12134), (185, 13419), (5889, 7553)
X(16659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1199, 5480), (4, 12112, 2883), (185, 13419, 7576), (3541, 11206, 9707), (3818, 10984, 14788), (6146, 16621, 4), (6759, 11550, 1594), (11455, 12289, 1885)


X(16660) =  5th HUNG-LOZADA-EULER POINT

Barycentrics    2*(17*R^2-20*SW)*S^4+(R^4*( 347*R^2-410*SW)+2*(79*SW^2+29* SB*SC)*R^2-4*(7*SA^2-7*SA*SW+ 5*SW^2)*SW)*S^2+(R^4*(239*R^2- 242*SW)+2*SW^2*(35*R^2-2*SW))* SB*SC : :

See Tran Quang Hung and César Lozada, Hyacinthos 27359.

X(16660) lies on this line: {2,3}


X(16661) =  ISOGONAL CONJUGATE OF X(16620)

Barycentrics    a^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2-9 a^4 b^2 c^2+8 a^2 b^4 c^2+3 b^6 c^2+8 a^2 b^2 c^4-4 b^4 c^4+2 a^2 c^6+3 b^2 c^6-c^8) : :
X(16661) = 5 X[3091] - 6 X[14788] = 9 X[2] + (J^2 - 17) X[3]

X(16661) lies on these lines: {2,3}, {110,13348}, {323,15644}, {1199,13391}, {1216,8718}, {1994,10984}, {3098,12111}, {3622,9911}, {5265,10833}, {5447,14157}, {5640,13347}, {6030,13367}, {6225,15577}, {6241,8717}, {7999,15052}, {9729,15107}, {11381,14810}, {11591,12112}, {13346,15080}

X(16661) = isogonal conjugate of X(16620)
X(16661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11414, 12087), (3, 20, 12086), (3, 3091, 7496), (3, 3529, 7527), (3, 3627, 7550), (3, 7530, 3525), (3, 7545, 14869), (3, 10594, 10303), (3, 12082, 3091), (3, 12083, 13861), (3, 12086, 14118), (3, 12103, 7464), (3, 13861, 631), (3, 15704, 14865), (20, 6636, 14118), (20, 7400, 7391), (20, 10323, 6636), (548, 13564, 186), (550, 7512, 2071), (3523, 7387, 13595), (3534, 7525, 3520), (6636, 12086, 3), (10303, 10594, 16042), (12086, 16042, 3088)


X(16662) =  X(2)X(15891)∩X(7)X(13389)

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

X(16662) lies on the cubic K965 and these lines:
{2, 15891}, {7, 13389}, {57, 279}, {175, 9778}, {176, 10580}, {482, 1699}, {557, 5431}, {558, 7371}, {3160, 13388}

X(16662) = X(6365)-zayin conjugate of X(657)
X(16662) = X(55)-isoconjugate of X(15891)
X(16662) = barycentric product X(7)*X(175)
X(16662) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 15891}, {175, 8}


X(16663) =  X(2)X(15892)∩X(7)X(1659)

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

X(16663) lies on the cubic K965 and these lines:
{2, 15892}, {7, 1659}, {57, 279}, {175, 10580}, {176, 9778}, {481, 1699}, {557, 7371}, {3160, 13389}

X(16663) = X(6364)-zayin conjugate of X(657)
X(16663) = X(55)-isoconjugate of X(15892)
X(16663) = barycentric product X(7)*X(176)
X(16663) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 15892}, {176, 8}, {7371, 5451}


X(16664) =  X(2)X(15495)∩X(174)X(223)

Barycentrics    Sin[A/2]/(-Sin[A/2]+Sin[B/2]+Sin[C/2]) : :

X(16664) lies on the cubic K965 and these lines:
{2, 15495}, {174, 223}, {505, 2089}

X(16664) = X(57)-cross conjugate of X(174)
X(16664) = X(i)-isoconjugate of X(j) for these (i,j): {55, 15495}, {164, 259}
X(16664) = barycentric product X(505)*X(4146)
X(16664) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 15495}, {174, 16017}, {266, 164}, {505, 188}


X(16665) =  X(49)X(74)∩X(54)X(15646)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27366.

X(16665) lies on these lines:
{49, 74}, {54, 15646}, {265, 12038}, {3431, 13630}, {3521, 13367}, {10610, 13418}


X(16666) =  (name pending)

Trilinears    4a + b + c : :
Barycentrics    a (4a + b + c) : :

X(16666) lies on these lines:

X(16666) = {X(1),X(6)}-harmonic conjugate of X(44)


X(16667) =  (name pending)

Trilinears    5a + b + c : :
Barycentrics    a (5a + b + c) : :

X(16667) lies on these lines:


X(16668) =  (name pending)

Trilinears    6a + b + c : :
Barycentrics    a (6a + b + c) : :

X(16668) lies on these lines:


X(16669) =  (name pending)

Trilinears    -4a + b + c : :
Barycentrics    a (-4a + b + c) : :

X(16669) lies on these lines:


X(16670) =  (name pending)

Trilinears    -5a + b + c : :
Barycentrics    a (-5a + b + c) : :

X(16670) lies on these lines:


X(16671) =  (name pending)

Trilinears    -6a + b + c : :
Barycentrics    a (-6a + b + c) : :

X(16671) lies on these lines:


X(16672) =  (name pending)

Trilinears    a + 4b + 4c : :
Barycentrics    a (a + 4b + 4c) : :

X(16672) lies on these lines:


X(16673) =  (name pending)

Trilinears    a + 5b + 5c : :
Barycentrics    a (a + 5b + 5c) : :

X(16673) lies on these lines:


X(16674) =  (name pending)

Trilinears    a + 6b + 6c : :
Barycentrics    a (a + 6b + 6c) : :

X(16674) lies on these lines:


X(16675) =  (name pending)

Trilinears    -a + 4b + 4c : :
Barycentrics    a (- a + 4b + 4c) : :

X(16675) lies on these lines:
{1,6}, {2,4398}, {190,15668}, {344,3763}, {346,1213}, {594,5296}, {756,3711}, {966,3943}, {968,3689}, {1962,3715}, {2321,4691}, {2325,3986}, {3196,4254}, {3204,4289}, {3686,4098}, {3729,4698}, {3828,5257}, {3842,5695}, {3950,4060}, {3989,4423}, {4029,4701}, {4034,4727}, {4361,4664}, {4363,4687}, {4370,5749}, {4384,4681}, {4755,10436}, {4851,15533}, {6172,7277}, {6351,8253}, {6352,8252}, {11063,15817}


X(16676) =  (name pending)

Trilinears    -a + 5b + 5c : :
Barycentrics    a(- a + 5b + 5c) : :

X(16676) is the center of the conic described in Hyacinthos #20547 (Francisco Javier García Capitán, 12/19/2011) when P = X(1). This is the same conic described in ADGEOM #833 (César Lozada, 11/12/2013) when P = X(1). (Randy Hutson, June 27, 2018)

X(16676) lies on these lines:
{1,6}, {2,1266}, {8,4029}, {10,4873}, {63,89}, {88,14439}, {142,4346}, {145,3707}, {201,5665}, {346,5257}, {374,9957}, {527,5308}, {672,9334}, {966,3626}, {968,3158}, {1156,9502}, {1443,8545}, {1696,5217}, {1731,10389}, {2238,9333}, {2321,3617}, {2345,3634}, {3161,5550}, {3241,4700}, {3621,3686}, {3625,4034}, {3672,6666}, {3677,3989}, {3679,3943}, {3729,4687}, {3875,4704}, {3929,5287}, {4277,9331}, {4363,4755}, {4384,4664}, {4419,4887}, {4648,4896}, {4667,6172}, {4677,4727}, {5297,16548}, {5714,8804}, {13384,16554}


X(16677) =  (name pending)

Trilinears    -a + 6b + 6c : :
Barycentrics    a (- a + 6b + 6c) : :

X(16677) lies on these lines:
{1,6}, {1030,1696}, {1334,5036}, {2321,4745}, {3619,4364}, {3729,4755}, {3943,5296}, {4029,4746}, {4034,16590}, {4098,4545}, {4287,9310}, {4361,4704}





leftri  Collineation images: X(16678) - X(16759)  rightri

This section follows the discussion of collineations just before X(16286), and especially just before X(16544). If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A,B,C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A,B,C,P).

The collineation images at X(16678)-X(16695) result from A'B'C' = tangential triangle, P = X(2), and U = X(1). We write the image of X as m(X); let m-1 denote the inverse collineation. Then centers X(16678)-X(16695) are examples of m(X), and X(16696)-X(16759) are examples of m-1(X). Other examples are given by the following list, in which the appearance of (i,j) means that m(X(i)) = X(j):

(1,8053), (2,1), (7,3185), (8,3941), (37,1621), (58,1631), (69,2352), (81,3), (85,48), (86,55), (274,6), (333,56), (348,19), (1107,86), (2275,75),, (3121,99), (4560,513)

A collineation maps lines to lines. The appearance of {h,i} -> {j,k} in the next list means that m(X(h)X(i)) = X(j)X(k):

{1,3286} -> {1,2}
{2,6} -> {1,3}
{1,859} -> {2,7}
{2,274} -> {1,6}
{81,274} -> {3,6}
{81,16696} -> {2,3}

underbar




X(16678) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16678) lies on these lines:
{1, 3}, {6, 2205}, {8, 16451}, {10, 16453}, {11, 4192}, {12, 13731}, {22, 1602}, {25, 5307}, {31, 2274}, {38, 3724}, {42, 5132}, {63, 3185}, {86, 1621}, {100, 3996}, {197, 5271}, {199, 8301}, {228, 518}, {333, 1610}, {411, 15622}, {515, 7420}, {516, 7416}, {595, 4278}, {669, 4897}, {851, 2886}, {859, 993}, {956, 4042}, {958, 13738}, {970, 2594}, {1001, 1011}, {1125, 16287}, {1284, 3782}, {1376, 4191}, {1444, 16872}, {1468, 4267}, {1698, 16414}, {2178, 5275}, {2361, 3955}, {3187, 11340}, {3616, 16452}, {3624, 16286}, {3634, 16297}, {3681, 4557}, {3870, 15624}, {3925, 16056}, {4245, 5251}, {4413, 16059}, {4423, 16058}, {5124, 10329}, {5263, 13588}, {5267, 7428}, {5434, 14636}, {6796, 15623}, {7354, 9840}, {7411, 14942}, {8167, 16373}, {8624, 16584}, {10198, 16455}, {16681, 16708}

X(16678) = X(274)-Ceva conjugate of X(6) X(16678) = crosspoint of X(59) and X(99) X(16678) = crossdifference of every pair of points on line {650, 784}
X(16678) = crosssum of X(11) and X(512)
X(16678) = tangential-isogonal conjugate of X(1030)
X(16678) = X(8714)-zayin conjugate of X(513)
X(16678) = barycentric product X(1)*X(16574)
X(16678) = barycentric quotient X(16574)/X(75)
X(16678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (35, 3750, 55), (55, 56, 940), (1621, 4184, 8053), (1626, 1631, 22)


X(16679) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16679) lies on these lines:
{1, 3286}, {6, 292}, {31, 16685}, {55, 4497}, {56, 1631}, {58, 16683}, {81, 16687}, {82, 876}, {86, 16684}, {583, 3688}, {674, 2260}, {757, 1634}, {999, 1486}, {1100, 2223}, {1376, 3791}, {1449, 15624}, {2178, 4471}, {3589, 4447}, {4360, 4436}, {5269, 15621}, {16691, 16872}


X(16680) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16680) lies on these lines:
{110, 3565}, {643, 3573}, {644, 4557}, {664, 1633}, {692, 2498}, {1486, 2099}, {2283, 8638}, {3877, 8053}


X(16681) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16681) lies on these lines:
{1, 16689}, {3, 142}, {31, 2275}, {674, 2200}, {2085, 8625}, {2271, 4471}, {2936, 16873}, {3721, 8628}, {4497, 5021}, {9259, 16365}, {16678, 16708}


X(16682) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16682) lies on these lines:
{1, 2916}, {22, 1602}, {1030, 4386}, {8053, 16876}, {16685, 16877}, {16686, 16690}


X(16683) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16683) lies on these lines:
{2, 16889}, {3, 142}, {21, 16693}, {31, 172}, {58, 16679}, {859, 16689}, {1509, 1634}, {2975, 16691}, {4251, 4557}


X(16684) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16684) lies on these lines:
{1, 6}, {2, 16687}, {10, 16301}, {22, 1602}, {55, 4361}, {56, 16356}, {75, 4436}, {86, 16679}, {261, 1634}, {594, 8299}, {1030, 8301}, {1582, 3285}, {1621, 4068}, {2223, 3739}, {3746, 4716}, {3941, 10436}, {4286, 4446}, {4384, 15624}, {4399, 4433}, {5132, 16825}, {5271, 16439}, {6224, 12746}, {8424, 15588}


X(16685) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16685) lies on these lines:
{1, 6}, {31, 16679}, {48, 3285}, {71, 1201}, {101, 2220}, {154, 2352}, {583, 1015}, {584, 2241}, {594, 992}, {595, 1333}, {604, 4559}, {995, 4261}, {1149, 2260}, {1319, 1409}, {1400, 8610}, {1500, 5153}, {1613, 8620}, {1740, 4436}, {1914, 2174}, {1918, 3009}, {1964, 3747}, {1979, 5163}, {2209, 4557}, {2215, 3445}, {2242, 4275}, {2305, 9259}, {2605, 3050}, {3187, 4358}, {3204, 16946}, {3730, 5069}, {3878, 4016}, {4290, 9351}, {5124, 10329}, {16682, 16877}


X(16686) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    a^2 (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(16686) lies on these lines:
{1, 2836}, {6, 692}, {23, 5078}, {25, 3052}, {31, 3122}, {35, 5506}, {36, 3246}, {37, 16550}, {45, 55}, {100, 4422}, {105, 1086}, {238, 5096}, {513, 3446}, {1001, 4265}, {1030, 2110}, {1083, 4553}, {1191, 13730}, {1279, 3220}, {1332, 9024}, {1621, 4364}, {1631, 5124}, {1769, 4491}, {1960, 9259}, {1979, 5040}, {2054, 3444}, {2174, 2293}, {2183, 8647}, {3683, 5310}, {4252, 11365}, {4436, 8301}, {4455, 7669}, {5341, 12723}, {5358, 15171}, {8424, 15588}, {10117, 14667}, {12329, 16885}, {16682, 16690}


X(16687) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16687) lies on these lines:
{1, 3}, {2, 16684}, {38, 3286}, {63, 3941}, {81, 16679}, {228, 1386}, {593, 1634}, {1009, 3703}, {1613, 8620}, {3891, 11322}, {5256, 15624}


X(16688) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16688) lies on these lines:
{1, 3286}, {3, 7290}, {9, 2223}, {31, 284}, {35, 16475}, {36, 1486}, {55, 1449}, {56, 12560}, {579, 2293}, {902, 7032}, {1011, 5269}, {1014, 1621}, {1037, 2078}, {1253, 5053}, {1631, 7280}, {1743, 15624}, {2300, 3052}, {2352, 4512}, {3973, 4557}, {5132, 16469}, {8616, 16693}, {10934, 11012}


X(16689) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16689) lies on these lines:
{1, 16681}, {3, 8301}, {21, 5263}, {56, 1631}, {859, 16683}, {904, 1964}, {3727, 8628}, {4225, 16693}


X(16690) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16690) lies on these lines:
{1, 6}, {2, 1918}, {21, 2274}, {31, 86}, {71, 4446}, {75, 3747}, {171, 15668}, {748, 2209}, {978, 5132}, {992, 8299}, {1125, 5156}, {1582, 5301}, {1740, 8053}, {2664, 15624}, {3056, 15994}, {3736, 5248}, {3915, 5263}, {4038, 10013}, {4436, 16571}, {7175, 16878}, {16682, 16686}, {16875, 16877}


X(16691) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(210), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16691) lies on these lines:
{1, 16693}, {595, 741}, {2110, 3780}, {2975, 16683}, {3915, 3941}, {5563, 8618}, {16679, 16872}


X(16692) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16692) lies on these lines: {31, 1919}, {669, 2106}, {4057, 8053}


X(16693) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(518), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16693) lies on these lines:
{1, 16691}, {3, 10186}, {21, 16683}, {31, 48}, {36, 8618}, {81, 16679}, {86, 1621}, {292, 1438}, {669, 2106}, {1486, 16064}, {1631, 6636}, {2110, 2238}, {2280, 15624}, {4225, 16689}, {7113, 9455}, {8616, 16688}


X(16694) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16694) lies on these lines:
{1, 3286}, {31, 3285}, {36, 14190}, {44, 2223}, {669, 2106}, {1631, 5204}, {5096, 5217}, {15624, 16670}


X(16695) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16695) lies on these lines:
{3, 6002}, {6, 1924}, {36, 238}, {99, 8709}, {514, 8637}, {659, 4560}, {669, 4367}, {814, 7255}, {1021, 8662}, {3777, 16754}, {4083, 8640}, {4378, 4960}, {4401, 8689}, {4775, 5216}, {4784, 8639}, {7252, 8632}, {9259, 9265}


X(16696) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(2), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16696) lies on these lines:
{1, 3286}, {2, 3770}, {6, 980}, {21, 1279}, {37, 86}, {38, 1964}, {39, 141}, {58, 1386}, {69, 4261}, {75, 16738}, {81, 593}, {193, 4277}, {241, 1014}, {256, 513}, {274, 1107}, {314, 536}, {333, 3752}, {354, 10458}, {518, 3736}, {524, 2092}, {988, 4267}, {1214, 1412}, {1418, 1434}, {1790, 4280}, {1959, 4016}, {2234, 3728}, {2275, 4657}, {2276, 4851}, {2277, 4643}, {2309, 4022}, {3121, 16733}, {3629, 4263}, {3743, 5625}, {3744, 4184}, {3759, 4850}, {3931, 4658}, {4000, 16699}, {4278, 5266}, {4281, 4719}, {4363, 10455}, {5124, 5337}, {5235, 16610}, {5283, 15668}, {6707, 16589}, {16707, 16717}, {16711, 16724}, {16712, 16723}, {16714, 16752}, {16715, 16747}, {16725, 16732}

,p> X(16696) = complement X(3770)
X(16696) = {X(81),X(1444)}-harmonic conjugate of X(1333)
X(16696) = X(4623)-Ceva conjugate of X(513)
X(16696) = X(i)-isoconjugate of X(j) for these (i,j): {10, 251}, {37, 82}, {42, 83}, {213, 3112}, {308, 1918}, {523, 4628}, {733, 4039}, {827, 4024}, {1176, 1826}, {1799, 2333}, {4079, 4577}, {4456, 16277}, {4557, 10566}, {4580, 8750}, {4599, 4705}
X(16696) = cevapoint of X(38) and X(39)
X(16696) = crosspoint of X(81) and X(274)
X(16696) = crossdifference of every pair of points on line {4455, 4705}
X(16696) = crosssum of X(i) and X(j) for these (i,j): {6, 5277}, {37, 213}
X(16696) = barycentric product X(i)*X(j) for these {i,j}: {21, 3665}, {28, 3933}, {38, 86}, {39, 274}, {58, 1930}, {81, 141}, {99, 2530}, {286, 3917}, {310, 1964}, {314, 1401}, {427, 1444}, {513, 4576}, {693, 1634}, {757, 15523}, {1014, 3703}, {1019, 4568}, {1235, 1437}, {1333, 8024}, {1509, 3954}, {3005, 4623}, {3051, 6385}, {4553, 7192}, {4610, 8061}
X(16696) = barycentric quotient X(i)/X(j) for these {i,j}: {38, 10}, {39, 37}, {58, 82}, {81, 83}, {86, 3112}, {141, 321}, {163, 4628}, {274, 308}, {826, 4036}, {905, 4580}, {1019, 10566}, {1333, 251}, {1401, 65}, {1437, 1176}, {1444, 1799}, {1634, 100}, {1843, 1824}, {1923, 1918}, {1930, 313}, {1964, 42}, {2084, 4079}, {2236, 4039}, {2530, 523}, {3005, 4705}, {3051, 213}, {3313, 4463}, {3665, 1441}, {3688, 210}, {3703, 3701}, {3917, 72}, {3954, 594}, {4020, 71}, {4553, 3952}, {4556, 4599}, {4568, 4033}, {4576, 668}, {4610, 4593}, {4623, 689}, {8041, 3954}, {8061, 4024}, {15523, 1089}


X(16697) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16697) lies on these lines:
{2, 16743}, {81, 593}, {216, 343}, {332, 3998}, {333, 16701}, {1214, 1790}, {3772, 16700}, {16703, 16725}, {16716, 16717}


X(16698) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16698) lies on these lines:
{2, 16701}, {81, 593}, {570, 1238}, {16707, 16734}, {16740, 16747}


X(16699) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16699) lies on these lines:
{21, 294}, {37, 2287}, {169, 859}, {216, 5742}, {239, 257}, {241, 16054}, {274, 16759}, {284, 1731}, {391, 4277}, {442, 3002}, {607, 958}, {910, 4225}, {1043, 3693}, {1108, 2303}, {1213, 3003}, {2082, 4267}, {2262, 4269}, {2271, 5283}, {2275, 16700}, {2322, 14571}, {4000, 16696}, {4515, 4720}, {4653, 16601}, {4999, 7117}


X(16700) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16700) lies on these lines:
{2, 3770}, {6, 16438}, {37, 5333}, {58, 14158}, {81, 88}, {86, 3666}, {274, 1920}, {333, 16610}, {354, 3736}, {614, 3286}, {1014, 1169}, {1279, 4184}, {1412, 1465}, {2275, 16699}, {3742, 10458}, {3744, 13588}, {3772, 16697}, {3999, 5208}, {4850, 8025}, {5235, 16602}, {1211, 16887}


X(16701) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16701) lies on these lines:
{2, 16698}, {37, 86}, {81, 2990}, {333, 16697}, {1444, 4565}, {2323, 6518}


X(16702) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16702) lies on these lines:
{6, 16436}, {28, 16745}, {36, 238}, {44, 662}, {58, 4719}, {81, 593}, {99, 536}, {110, 2721}, {187, 524}, {241, 4565}, {261, 3739}, {518, 1326}, {674, 3110}, {849, 3916}, {896, 922}, {1414, 6610}, {1634, 2223}, {1963, 3723}, {3941, 8616}, {4383, 11350}, {4653, 15569}, {4708, 6626}, {4760, 16733}, {5127, 6510}, {9004, 9145}


X(16703) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16703) lies on these lines:
{2, 16739}, {38, 1930}, {75, 3873}, {81, 239}, {86, 7191}, {141, 6665}, {304, 16705}, {310, 321}, {314, 16750}, {693, 7018}, {1211, 3266}, {1231, 1434}, {1962, 14210}, {2887, 16891}, {3121, 16746}, {3263, 4981}, {5208, 10471}, {16697, 16725}


X(16704) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16704) lies on these lines:
{2, 6}, {4, 5769}, {8, 58}, {20, 5767}, {21, 145}, {27, 4373}, {38, 3791}, {44, 4358}, {63, 3187}, {89, 274}, {99, 2384}, {100, 3286}, {110, 2726}, {149, 2651}, {162, 14954}, {171, 4651}, {239, 514}, {244, 4974}, {283, 12649}, {284, 5744}, {314, 4671}, {321, 4641}, {346, 1778}, {519, 902}, {593, 7058}, {740, 896}, {1010, 3617}, {1016, 1252}, {1043, 3621}, {1171, 6539}, {1215, 4722}, {1333, 5839}, {1404, 3911}, {1408, 1788}, {1412, 5435}, {1757, 3952}, {1931, 6630}, {1999, 3219}, {2094, 4402}, {2193, 6350}, {2226, 4615}, {2308, 3741}, {2975, 4267}, {3006, 5847}, {3193, 10529}, {3240, 3736}, {3241, 4653}, {3257, 4080}, {3285, 4969}, {3564, 8229}, {3616, 4658}, {3622, 11110}, {3666, 16718}, {3681, 3769}, {3745, 4981}, {3759, 4850}, {3794, 4661}, {3896, 4640}, {4234, 4720}, {4257, 16397}, {4430, 5208}, {4434, 4753}, {4442, 4831}, {4452, 8822}, {4589, 4607}, {8033, 16748}, {8047, 15149}, {10449, 11319}, {12245, 15952}, {16610, 16726}, {16707, 16739}

X(16704) = isotomic conjugate of X(4080)


X(16705) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16705) lies on these lines:
{1, 16887}, {2, 39}, {7, 21}, {37, 16720}, {81, 2221}, {99, 5992}, {304, 16703}, {314, 3672}, {325, 5051}, {333, 5222}, {593, 763}, {764, 7192}, {964, 1975}, {1107, 16742}, {1125, 4368}, {1193, 4357}, {1621, 16876}, {2275, 4657}, {3663, 10455}, {3666, 16739}, {3743, 14210}, {3933, 13728}, {4080, 4632}, {4441, 10471}, {5276, 16060}, {5333, 16050}, {6155, 8682}, {11110, 16020}, {16709, 16714}, {16726, 16733}


X(16706) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    a^2 + b^2 - b c + c^2 : :

X(16706) lies on these lines:
{1, 3836}, {2, 37}, {6, 320}, {7, 3618}, {8, 3619}, {9, 4389}, {10, 10159}, {19, 3306}, {44, 6646}, {57, 1760}, {69, 3759}, {82, 171}, {83, 4911}, {86, 142}, {141, 239}, {190, 3663}, {238, 3821}, {274, 16818}, {348, 8732}, {499, 4008}, {583, 3218}, {594, 4395}, {894, 1086}, {1100, 3834}, {1104, 4201}, {1125, 1738}, {1213, 16521}, {1233, 16750}, {1386, 4645}, {1447, 7792}, {2550, 3616}, {2999, 4417}, {3008, 4357}, {3096, 5015}, {3248, 7184}, {3620, 5839}, {3631, 4969}, {3661, 3763}, {3670, 4283}, {3673, 7803}, {3705, 7868}, {3717, 4353}, {3729, 4398}, {3826, 16830}, {3912, 3946}, {3936, 4272}, {4026, 16823}, {4202, 5262}, {4384, 5224}, {4393, 4851}, {4514, 4972}, {4655, 16468}, {4715, 16671}, {4852, 6542}, {4859, 10436}, {5701, 10030}, {6329, 7238}, {6687, 16814}, {6996, 12610}, {7061, 9478}, {7179, 11174}, {7859, 16600}, {9791, 15254}, {13728, 16817}, {16709, 16752}


X(16707) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16707) lies on these lines:
{81, 239}, {86, 3263}, {319, 4651}, {2203, 16747}, {3266, 6703}, {3589, 11205}, {8033, 16727}, {16696, 16717}, {16698, 16734}, {16704, 16739}


X(16708) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16708) lies on these lines:
{2, 16727}, {57, 85}, {75, 3873}, {86, 2191}, {142, 1233}, {310, 312}, {314, 10390}, {873, 2185}, {1418, 16713}, {1444, 4228}, {10471, 11021}, {16678, 16681}


X(16709) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    b c (a + b) (a + c) (2 a + b + c) : :

X(16709) lies on these lines:
{1, 75}, {2, 3770}, {81, 3759}, {85, 1014}, {286, 4248}, {312, 5333}, {319, 4651}, {333, 3306}, {341, 14007}, {350, 16748}, {668, 1268}, {757, 873}, {1100, 4359}, {1125, 1269}, {1279, 11115}, {1444, 4228}, {3286, 16823}, {3739, 16726}, {3948, 6707}, {3963, 4472}, {4043, 16826}, {4974, 6533}, {5224, 16887}, {16705, 16714}, {16706, 16752}, {16727, 16733}


X(16710) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16710) lies on these lines:
{2, 3770}, {75, 16726}, {86, 192}, {274, 330}, {314, 4740}, {3210, 8025}


X(16711) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16711) lies on these lines:
{2, 39}, {75, 16714}, {99, 9097}, {1149, 1266}, {1975, 11346}, {2226, 4615}, {3669, 4560}, {3679, 16887}, {16696, 16724}


X(16712) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16712) lies on these lines:
{2, 39}, {86, 99}, {325, 16052}, {376, 10446}, {519, 16887}, {552, 553}, {995, 4389}, {1975, 11354}, {3945, 14907}, {4201, 7768}, {4229, 4301}, {7760, 16060}, {7788, 11359}, {7796, 16062}, {16696, 16723}


X(16713) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16713) lies on these lines:
{2, 6}, {21, 390}, {58, 4307}, {75, 16728}, {77, 4384}, {142, 1475}, {261, 4612}, {274, 279}, {283, 10527}, {314, 346}, {1014, 8732}, {1212, 1229}, {1418, 16708}, {1444, 14953}, {2269, 5745}, {2293, 4847}, {2322, 15149}, {2550, 3286}, {3739, 4875}, {3786, 5686}, {4000, 16696}, {4296, 16824}, {5749, 10455}, {8165, 14011}, {11110, 14986}, {16723, 16759}


X(16714) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(210), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16714) lies on these lines:
{75, 16711}, {86, 3445}, {274, 3596}, {1014, 1325}, {1444, 11102}, {16696, 16752}, {16705, 16709}


X(16715) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16715) lies on these lines: {28, 60}, {16696, 16747}


X(16716) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16716) lies on these lines:
{21, 3290}, {28, 1104}, {37, 1010}, {39, 8728}, {58, 16583}, {274, 1107}, {333, 16605}, {443, 4261}, {1196, 1368}, {2275, 16699}, {3286, 16968}, {3752, 16054}, {3772, 15149}, {5277, 16429}, {5283, 16458}, {16604, 16736}, {16697, 16717}, {16732, 16747}


X(16717) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    a^3 (a + b) (a + c) (b^4 + c^4) : :

X(16717) lies on these lines:
{81, 893}, {16696, 16707}, {16697, 16716}, {16721, 16743}


X(16718) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(79), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16718) lies on these lines: {3666, 16704}, {8025, 16743}


X(16719) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(80), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16719) lies on these lines: {2, 16698}, {3666, 4760}


X(16720) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16720) lies on these lines:
{3, 4376}, {6, 4372}, {10, 514}, {35, 4760}, {37, 16705}, {39, 1930}, {56, 4363}, {75, 2275}, {141, 3665}, {172, 894}, {274, 292}, {304, 2276}, {325, 16886}, {348, 2345}, {536, 3702}, {538, 1089}, {712, 3670}, {742, 1193}, {1107, 3263}, {1111, 3934}, {1215, 4447}, {1500, 14210}, {1909, 7187}, {3293, 8682}, {3758, 7296}, {3876, 4643}, {3954, 4568}, {4056, 7761}, {4400, 7081}, {4670, 5764}, {4680, 7759}, {7181, 7227}


X(16721) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16721) lies on these lines: {333, 2669}, {2275, 16699}, {16717, 16743}


X(16722) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16722) lies on these lines:
{2, 16744}, {192, 2176}, {194, 16552}, {274, 330}, {333, 16816}, {4850, 16729}, {6376, 16742}


X(16723) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16723) lies on these lines:
{2, 16726}, {274, 16724}, {536, 16728}, {662, 4921}, {3739, 7200}, {16696, 16712}, {16713, 16759}


X(16724) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16724) lies on these lines:
{2, 4277}, {274, 16723}, {333, 4715}, {4688, 16728}, {4945, 5235}, {16696, 16711}


X(16725) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(98), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16725) lies on these lines:
{81, 893}, {1444, 16730}, {16696, 16732}, {16697, 16703}, {16738, 16743}


X(16726) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16726) lies on these lines:
{1, 4436}, {2, 16723}, {37, 86}, {75, 16710}, {81, 88}, {244, 659}, {274, 670}, {291, 4553}, {314, 4686}, {333, 16602}, {513, 3122}, {527, 8610}, {812, 1015}, {1014, 1333}, {1279, 3286}, {1412, 1427}, {1964, 13476}, {2275, 4675}, {3124, 9810}, {3125, 7202}, {3666, 4760}, {3739, 16709}, {3945, 4261}, {4646, 4658}, {4648, 5069}, {4858, 7208}, {6627, 8287}, {7200, 16732}, {9359, 16482}, {16604, 16887}, {16610, 16704}, {16705, 16733}


X(16727) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16727) lies on these lines:
{2, 16708}, {81, 1462}, {85, 4850}, {88, 274}, {244, 1111}, {310, 321}, {658, 1434}, {693, 3120}, {2481, 3315}, {3121, 7200}, {4359, 16739}, {4403, 6377}, {4609, 6385}, {8033, 16707}, {16709, 16733}


X(16728) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(105), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16728) lies on these lines:
{2, 16708}, {37, 86}, {63, 3185}, {75, 16713}, {81, 643}, {274, 1212}, {518, 2223}, {536, 16723}, {662, 911}, {665, 918}, {1333, 2991}, {2205, 4641}, {3002, 6390}, {3121, 3666}, {3618, 5069}, {3739, 16732}, {4437, 6184}, {4688, 16724}


X(16729) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(106), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16729) lies on these lines:
{2, 16723}, {44, 4358}, {81, 645}, {88, 274}, {190, 321}, {678, 4738}, {1107, 3121}, {1150, 16552}, {1635, 3762}, {4359, 16732}, {4850, 16722}


X(16730) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(107), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16730) lies on these lines: {1444, 16725}, {16732, 16758}


X(16731) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(108), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16731) lies on these lines:
{333, 16697}, {662, 911}, {1812, 4558}, {3998, 6514}, {16732, 16759}


X(16732) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16732) lies on these lines:
{4, 2831}, {11, 2969}, {37, 1441}, {75, 1654}, {76, 11611}, {85, 4675}, {92, 3772}, {115, 127}, {226, 4605}, {274, 4623}, {286, 1333}, {321, 4033}, {513, 4459}, {514, 7202}, {523, 2486}, {536, 3262}, {744, 4039}, {857, 16581}, {918, 1086}, {1109, 2632}, {1826, 16580}, {2245, 8680}, {2795, 4436}, {3121, 14296}, {3739, 16728}, {3782, 14213}, {4112, 4381}, {4359, 16729}, {4415, 6358}, {4466, 8287}, {4647, 4783}, {6155, 7264}, {6506, 16596}, {7200, 16726}, {14953, 15586}, {16696, 16725}, {16716, 16747}, {16730, 16758}, {16731, 16759}


X(16733) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(111), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16733) lies on these lines:
{81, 7306}, {274, 4623}, {3121, 16696}, {4760, 16702}, {16705, 16726}, {16709, 16727}


X(16734) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16734) lies on these lines:
{3121, 16696}, {16697, 16703}, {16698, 16707}


X(16735) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16735) lies on these lines:
{28, 60}, {274, 1107}, {980, 11321}, {1010, 16830}, {1194, 6656}


X(16736) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16736) lies on these lines:
{2, 3770}, {81, 16610}, {86, 3752}, {333, 16602}, {1279, 13588}, {1427, 6359}, {3286, 5272}, {3666, 5333}, {3736, 3742}, {3772, 16744}, {4267, 11512}, {5743, 16887}, {16604, 16716}


X(16737) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(798), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16737) lies on these lines:
{86, 663}, {274, 514}, {310, 4379}, {804, 1966}, {1019, 4817}, {1909, 2533}, {3669, 4560}, {3978, 4369}


X(16738) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16738) lies on these lines:
{2, 6}, {8, 3736}, {75, 16696}, {192, 314}, {194, 10471}, {274, 330}, {894, 10455}, {956, 1010}, {2140, 3662}, {2309, 3741}, {2975, 3286}, {3739, 16709}, {5145, 10479}, {10453, 10458}, {16725, 16743}


X(16739) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16739) lies on these lines:
{2, 16703}, {38, 75}, {57, 85}, {86, 614}, {305, 5224}, {757, 873}, {982, 16887}, {1218, 2350}, {3666, 16705}, {3673, 10471}, {4359, 16727}, {6545, 7199}, {8025, 16741}, {10180, 14210}, {16704, 16707}


X(16740) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(182), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16740) lies on these lines:
{81, 239}, {1010, 4861}, {16696, 16725}, {16698, 16747}


X(16741) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16741) lies on these lines:
{81, 239}, {86, 7292}, {99, 2721}, {320, 350}, {321, 8033}, {518, 4576}, {524, 3266}, {799, 4358}, {896, 6629}, {4760, 16702}, {8025, 16739}


X(16742) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16742) lies on these lines:
{37, 4568}, {274, 4602}, {310, 16606}, {812, 1015}, {1107, 16705}, {3121, 7200}, {6376, 16722}, {16696, 16712}


X(16743) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(226), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16743) lies on these lines:
{2, 16697}, {81, 3554}, {333, 4850}, {1014, 2002}, {1444, 1760}, {1630, 1790}, {2287, 16517}, {3739, 16759}, {4000, 16696}, {8025, 16718}, {16717, 16721}, {16725, 16738}


X(16744) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16744) lies on these lines:
{2, 16722}, {274, 4602}, {1015, 16887}, {2275, 4657}, {3772, 16736}


X(16745) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16745) lies on these lines: {28, 16702}, {274, 1107}, {3772, 16736}


X(16746) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16746) lies on these lines: {2, 16722}, {310, 16606}, {3121, 16703}, {16696, 16707}


X(16747) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(206), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16747) lies on these lines:
{28, 242}, {29, 16749}, {76, 5142}, {92, 15149}, {427, 1235}, {1010, 1441}, {2203, 16707}, {16696, 16715}, {16698, 16740}, {16716, 16732}


X(16748) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    b c (a + b) (a + c) (a b + a c + 2 b c) : :

X(16748) lies on these lines:
{2, 39}, {75, 3873}, {86, 4441}, {99, 9110}, {350, 16709}, {873, 8025}, {1909, 4651}, {4359, 16727}, {4699, 6385}, {7192, 8027}, {8033, 16704}, {10453, 10471}


X(16749) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(219), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16749) lies on these lines:
{2, 39}, {21, 3673}, {29, 16747}, {58, 1111}, {81, 85}, {273, 1014}, {658, 1434}, {1447, 4225}, {4653, 7264}, {6734, 16887}


X(16750) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(220), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16750) lies on these lines:
{2, 39}, {27, 1088}, {81, 1462}, {85, 5256}, {86, 2191}, {314, 16703}, {614, 3673}, {1233, 16706}, {2481, 7191}, {4847, 16887}


X(16751) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(514), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16751) lies on these lines:
{81, 1021}, {647, 4467}, {649, 4481}, {650, 7192}, {661, 1019}, {693, 905}, {1491, 3733}, {2254, 3737}, {4565, 7115}, {4824, 8043}


X(16752) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(518), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16752) lies on these lines:
{2, 39}, {21, 16020}, {28, 1851}, {81, 277}, {86, 4000}, {105, 16876}, {278, 14013}, {693, 905}, {948, 1434}, {1010, 3616}, {1193, 5249}, {1975, 11342}, {2275, 16699}, {3772, 16736}, {3786, 4310}, {4236, 14267}, {4384, 16887}, {6185, 14953}, {7292, 14956}, {16696, 16714}, {16706, 16709}


X(16753) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16753) lies on these lines:
{2, 3770}, {81, 2999}, {86, 4850}, {693, 905}, {3286, 7292}, {3752, 8025}, {5333, 10436}, {16610, 16704}


X(16754) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(522), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16754) lies on these lines:
{693, 905}, {1019, 3960}, {3669, 7192}, {3737, 4017}, {3777, 16695}, {6129, 7253}


X(16755) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16755) lies on these lines:
{86, 522}, {274, 3261}, {2605, 3268}, {3669, 4560}, {4782, 7192}, {7799, 14838}


X(16756) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16756) lies on these lines:
{99, 3290}, {126, 3291}, {274, 1107}, {517, 2106}, {693, 905}, {1010, 3685}, {6629, 16611}


X(16757) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(525), WHERE A'B'C' = TANGENTIAL TRIANGLE

Barycentrics    a (b - c) (a^4 - b^4 - c^4) : :

X(16757) lies on these lines:
{2, 6588}, {81, 4131}, {513, 1980}, {693, 905}, {2522, 4467}, {4025, 16612}, {6590, 14838}


X(16758) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(648), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16758) lies on this line: {16730, 16732}


X(16759) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(651), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16759) lies on these lines:
{274, 16699}, {3739, 16743}, {4858, 11998}, {16713, 16723}, {16731, 16732}


X(16760) =  MIDPOINT OF X(3) AND X(5099)

Barycentrics    2 a^14 - 6 a^12 (b^2+c^2) + 4 a^10 (2 b^4+3 b^2 c^2+2 c^4) - a^8 (5 b^6+11 b^4 c^2+11 b^2 c^4+5 c^6) + a^6 (-4 b^8+10 b^6 c^2+6 b^4 c^4+10 b^2 c^6-4 c^8) + a^4 (10 b^10-15 b^8 c^2+3 b^6 c^4+3 b^4 c^6-15 b^2 c^8+10 c^10) - 2 a^2 (b^2-c^2)^2 (3 b^8-b^6 c^2+4 b^4 c^4-b^2 c^6+3 c^8) + (b^2-c^2)^4 (b^6+c^6) : :
X(16760) = 3 X(2) + X(842)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27373.

X(16760) lies on these lines:
{2,476}, {3,5099}, {30,620}, {69,10425}, {186,316}, {468,511}, {512,6699}, {523,6036}, {631,691}, {858,13372}, {5432,6023}, {5433,6027}, {10295,13449}

X(16760) = midpoint of X(i) and X(j) for these {i,j}: {3,5099}, {842,16188}, {10295,13449}


X(16761) =  MIDPOINT OF X(12342) AND X(12524)

Barycentrics    a^2 (a^8 - 2 a^7 (b+c) - 2 a^6 (b^2+c^2) + a^5 (6 b^3+4 b^2 c+4 b c^2+6 c^3) + a^4 b c (2 b^2+5 b c+2 c^2) - 2 a^3 (3 b^5+b^4 c+3 b^3 c^2+3 b^2 c^3+b c^4+3 c^5) + a^2 (2 b^6-4 b^5 c-5 b^4 c^2-5 b^2 c^4-4 b c^5+2 c^6) + 2 a (b^7+b^5 c^2-b^4 c^3-b^3 c^4+b^2 c^5+c^7) - (b^2-c^2)^2 (b^4-2 b^3 c-2 b c^3+c^4)) : :
X(16761) = R(R-2r) X(11) + (2 r^2+5 r R-R^2) X(21)

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27376.

X(16761) lies on these lines:
{3,12342}, {11,21}, {35,72}, {3467,6597}, {3876,12745}, {5010,12660}, {5172,12913}, {6905,12615}

X(16761) = midpoint of X(12342) and X(12524)


X(16762) =  X(3)X(54)∩X(381)X(1263)

Barycentrics    (SB+SC)*(2*S^2*(5*R^2-2*SW)+(( 3*R^2-10*SA)*R^2+4*SA^2-4*SB* SC)*SA) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16762) lies on these lines:
{3, 54}, {381, 1263}, {1656, 3459}, {1994, 15770}, {3843, 15307}, {11271, 14143}

X(16762) = {X(195), X(15345)}-harmonic conjugate of X(3)


X(16763) =  X(1)X(3)∩X(381)X(3065)

Barycentrics    a*(a^6-(3*b^2-2*b*c+3*c^2)*a^ 4+(b^2+b*c+c^2)*(3*b^2-7*b*c+ 3*c^2)*a^2-(b^2-c^2)^2*(b-c)^ 2) : :
X(16763) = (3*R^2+8*R*r+4*r^2)*X(1)-8*r^ 2*X(3)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16763) lies on these lines:
{1, 3}, {2, 1749}, {81, 15767}, {381, 3065}, {499, 9782}, {1656, 3467}, {1768, 6830}, {2964, 9340}, {3218, 3584}

X(16762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5221, 5903), (999, 5221, 5902), (3336, 3337, 5221), (13750, 14792, 1)


X(16764) = X(3)X(1263)∩X(381)X(1157)

Barycentrics   
X(16764) = 20*S^4-(9*R^4-20*(SB+SC)*R^2- 8*SA^2+4*SB*SC+8*SW^2)*S^2-3* R^4*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16764) lies on these lines:
: {3, 1263}, {6, 17}, {381, 1157}, {3526, 15345}, {5072, 15307}


X(16765) =  X(3)X(3460)∩X(381)X(3483)

Barycentrics    a*(2*(2*R^2*(4*R^2-3*SW)+S^2+ SW^2)*b*c+(11*R^2+2*SA+4*SW)* S^2+R^2*(16*R^4-8*(3*SA+SW)*R^ 2+14*SA^2-24*SB*SC+SW^2)+2*SA* SW*(2*SA-3*SW)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16765) lies on these lines:
: {3, 3460}, {381, 3483}, {1656, 3461}


X(16766) =  X(5)X(128)∩X(140)X(3459)

Barycentrics    (4*S^2-11*R^4-12*R^2*SA+8*R^2* SW-8*SB*SC)*(S^2+SB*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16766)lies on these lines:
: {5, 128}, {30, 195}, {140, 3459}, {550, 1157}, {3845, 15307}, {10285, 11671}

X(16766) = {X(1263), X(15345)}-harmonic conjugate of X(5)


X(16767) =  X(5)X(3065)∩X(140)X(3467)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16767) lies on these lines:
: {1, 104}, {3, 1749}, {5, 3065}, {30, 3336}, {140, 3467}, {411, 5131}, {484, 550}, {496, 3337}, {1727, 7508}, {3218, 4330}, {3871, 6763}, {5844, 11010}, {7483, 13089}


X(16768) =  X(5)X(252)∩X(140)X(195)

Barycentrics    20*S^4-(9*R^4-20*(SB+SC)*R^2- 8*SA^2+12*SB*SC+8*SW^2)*S^2-( 17*R^2-8*SW)*R^2*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16768) lies on these lines:
: {5, 252}, {30, 3459}, {140, 195}, {549, 15345}, {550, 1263}


X(16769) =  X(5)X(3483)∩X(30)X(3460)

Barycentrics    a*(a^18-6*(b^2+c^2)*a^16+(15* b^4+15*c^4+4*(b^2+6*b*c+c^2)* b*c)*a^14-(21*b^6+21*c^6+(20* b^4+20*c^4+(37*b^2+24*b*c+37* c^2)*b*c)*b*c)*a^12+(21*b^8+ 21*c^8+(40*b^6+40*c^6+(28*b^4+ 28*c^4+(44*b^2+31*b*c+44*c^2)* b*c)*b*c)*b*c)*a^10-(21*b^10+ 21*c^10+(40*b^8+40*c^8+(6*b^6+ 6*c^6+(24*b^4+24*c^4+(9*b^2+ 16*b*c+9*c^2)*b*c)*b*c)*b*c)* b*c)*a^8+(b^2-c^2)^2*(21*b^8+ 21*c^8+2*(10*b^6+10*c^6+(11*b^ 4+11*c^4+2*(7*b^2+6*b*c+7*c^2) *b*c)*b*c)*b*c)*a^6-(b^2-c^2)^ 2*(15*b^10+15*c^10+(b^2-b*c+c^ 2)*(4*b^6+4*c^6-(b^4+c^4+(13* b^2+18*b*c+13*c^2)*b*c)*b*c)* b*c)*a^4+(b^2-c^2)^4*(6*b^8+6* c^8-(4*b^2+3*b*c+4*c^2)*b^3*c^ 3)*a^2-(b^6+c^6)*(b^2-c^2)^6) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16769) lies on these lines:
: {5, 3483}, {30, 3460}, {140, 3461}, {550, 7165}


X(16770) =  ANTICOMLEMENT OF X(11131)

Barycentrics    (sqrt(3)*(SB+SC)*SA-(SB-3*SC)* S)*(sqrt(3)*(SB+SC)*SA+(3*SB- SC)*S)*(SA^2-2*sqrt(3)*S*SA+3* S^2) : :
Barycentrics    tan(A - π/6) : tan(B - π/6) : tan(C - π/6)   (Peter Moses March 29, 2018)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16770) lies on the cubic K67 and these lines:
{2, 13}, {4, 94}, {62, 8836}, {69, 300}, {125, 16001}, {303, 11142}, {397, 14389}, {472, 1994}, {621, 11581}, {628, 8929}, {2981, 6772}, {3180, 11080}, {3580, 5318}, {11304, 15018}

X(16770) = anticomplement of X(11131)


X(16771) =  ANTICOMLEMENT OF X(11130)

Barycentrics    (sqrt(3)*(SB+SC)*SA+(SB-3*SC)* S)*(sqrt(3)*(SB+SC)*SA-(3*SB- SC)*S)*(SA^2+2*sqrt(3)*S*SA+3* S^2) : :
Barycentrics    tan(A + π/6) : tan(B + π/6) : tan(C + π/6)   (Peter Moses March 29, 2018)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16771) lies on the cubic K67 and these lines:
{2, 14}, {4, 94}, {61, 8838}, {69, 301}, {125, 16002}, {302, 11141}, {398, 14389}, {473, 1994}, {622, 11582}, {627, 8930}, {3181, 11085}, {3580, 5321}, {6151, 6775}, {11303, 15018}

X(16771)= anticomplement of X(11130)


X(16772) =  X(5)X(15)∩X(17)X(30)

Barycentrics    4*S^2+sqrt(3)*(SB+SC)*S-2*SB* SC : :
X(16772) = 4*X(17)-X(5350), 11*X(17)-3*X(12816), 4*X(5238)+X(5350), 11*X(5238)+3*X(12816), 11*X(5350)-12*X(12816)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16772) lies on these lines:
{2, 398}, {3, 396}, {4, 16644}, {5, 15}, {6, 631}, {13, 550}, {14, 3628}, {16, 3412}, {17, 30}, {18, 632}, {20, 5318}, {61, 140}, {62, 549}, {141, 628}, {303, 7750}, {376, 5340}, {511, 8259}, {524, 627}, {548, 10645}, {590, 2041}, {615, 2042}, {619, 6694}, {635, 6671}, {636, 7810}, {1656, 16632}, {1906, 11475}, {1907, 10641}, {2307, 5432}, {3090, 5339}, {3091, 5349}, {3364, 13966}, {3525, 16645}, {3526, 11485}, {3528, 5335}, {3544, 5365}, {3589, 11308}, {5067, 5334}, {5071, 5343}, {5237, 15712}, {5351, 12100}, {5366, 11001}, {5472, 15513}, {6247, 11243}, {7005, 15325}, {7051, 15888}, {10124, 16268}, {10632, 15559}, {11481, 15717}, {11543, 16239}, {14869, 16242}

X(16772) = midpoint of X(17) and X(5238)
X(16772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 396, 397), (6, 631, 16773), (61, 140, 395), (61, 16241, 140), (628, 11307, 141), (11480, 11488, 5318)


X(16773) =  X(5)X(16)∩X(18)X(30)

Barycentrics    4*S^2-sqrt(3)*(SB+SC)*S-2*SB* SC : :
X(16773) = 4*X(18)-X(5349), 11*X(18)-3*X(12817), 4*X(5237)+X(5349), 11*X(5237)+3*X(12817), 11*X(5349)-12*X(12817)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16773) lies on these lines:
{2, 397}, {3, 395}, {4, 16645}, {5, 16}, {6, 631}, {13, 3628}, {14, 550}, {15, 3411}, {17, 632}, {18, 30}, {20, 5321}, {61, 549}, {62, 140}, {141, 627}, {302, 7750}, {376, 5339}, {511, 8260}, {524, 628}, {548, 10646}, {590, 2042}, {615, 2041}, {618, 6695}, {635, 7810}, {636, 6672}, {1656, 16633}, {1906, 11476}, {1907, 10642}, {3090, 5340}, {3091, 5350}, {3389, 13966}, {3390, 8981}, {3525, 16644}, {3526, 11486}, {3528, 5334}, {3544, 5366}, {3589, 11307}, {5067, 5335}, {5071, 5344}, {5238, 15712}, {5352, 12100}, {5365, 11001}, {5433, 7127}, {5471, 15513}, {7006, 15325}, {8703, 16268}, {10124, 16267}, {10633, 15559}, {11480, 15717}, {11542, 16239}, {14869, 16241}

X(16773) = midpoint of X(18) and X(5237)
X(16773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 395, 398), (6, 631, 16772), (62, 140, 396), (62, 16242, 140), (627, 11308, 141), (11481, 11489, 5321)


X(16774) =  ISOGONAL CONJUGATE OF X(9909)

Barycentrics    (S^2-3*SA*SC+SB^2)*(S^2-3*SA* SB+SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16775) lies on the Jerabek hyperbola and these lines:
{3, 3620}, {6, 8889}, {511, 15077}, {895, 11008}, {1176, 3619}, {1352, 15740}, {1503, 3532}, {6776, 14528}

X(16774) = isogonal conjugate of X(9909)
X(16774) = trilinear pole of the line {647, 2506}


X(16775) =  X(511)X(5894)∩X(2854)X(10990)

Barycentrics    X(16775) = (48*R^2-5*SA-9*SW)*S^2-4*(24* R^2-5*SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27377.

X(16775) lies on these lines:
: {511, 5894}, {2854, 10990}, {14915, 15105}


X(16776) =  X(5)X(141)∩X(6)X(110)

Barycentrics    a^2 (a^4 b^2-b^6+a^4 c^2+2 a^2 b^2 c^2+2 b^4 c^2+2 b^2 c^4-c^6) : :
X(16776) = 3 X[373] + X[1843], 3 X[373] - 2 X[3589], X[1843] + 2 X[3589], X[6] - 3 X[5640], X[141] - 4 X[9822], 2 X[9822] + X[9969], X[141] + 2 X[9969], X[576] - 4 X[10095], X[69] + 3 X[11002], 3 X[5640] + X[11188], 5 X[11188] - X[12272], 5 X[6] + X[12272], 15 X[5640] + X[12272], 2 X[575] - 5 X[15026], 3 X[6] - X[15531], 9 X[5640] - X[15531], 3 X[11188] + X[15531], 3 X[12272] + 5 X[15531]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27369.

Let A"B"C" be as at X(5640). Then X(16776) = X(3)-of-A"B"C".

X(16776) lies on these lines:
{2,9019}, {5,141}, {6,110}, {24,5085}, {51,524}, {66,3521}, {67,7533}, {69,7693}, {159,5050}, {160,11171}, {182,12106}, {373,468}, {381,2781}, {542,5946}, {568,1352}, {569,15582}, {575,15026}, {576,10095}, {597,2393}, {599,3060}

X(16776) = midpoint of X(i) and X(j) for these {i,j}: {2, 9971}, {6, 11188}, {568, 1352}, {599, 3060}
X(16776) = reflection of X(i) in X(j) for these {i,j}: {{182, 13363}, {597, 5943}, {5476,13364}
X(16776) = crossdifference of every pair of points on line {690, 3050}.
X(16776) = X(5)-of-reflection-triangle-of-X(2)
X(16776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5640, 11188, 6), (9822, 9969, 141).


X(16777) =  (name pending)

Trilinears    a + 2b + 2c : :
Barycentrics: a(a + 2b + 2c) : :
X(16777) = 4 s^2 X[1]-SW X[6]

X(16777) lies on these lines:
{1,6}, {2,594}, {8,1213}, {10,4060}, {19,3207}, {33,1841}, {42,3711}, {53,7952}, {55,199}, {56,2171}, {69,4364}, {75,15668}, {86,192}, {101,584}, {142,4021}, {144,7277}, {145,966}, {169,7300}, {172,5301}, {190,4704}, {198,1953}, {214,5110}, {239,4687}, {241,7190}, {281,1990}, {292,10013}, {344,3589}, {345,6703}, {346,3622}, {391,3623}, {519,5257}, {536,10436}, {551,3950}, {559,1653}, {572,10246}, {573,1482}, {579,15934}, {583,3730}, {590,6352}, {595,4275}, {599,4357}, {604,1388}, {612,3290}, {615,6351}, {869,2667}, {894,4664}, {910,10389}, {940,3218}, {941,1320}, {952,5816}, {968,3052}, {975,4255}, {988,15815}, {995,5153}, {1015,5069}, {1068,1865}, {1082,1652}, {1086,3672}, {1125,2321}

X(16777) = isogonal conjugate of X(25417)
X(16777) = complement of isotomic conjugate of X(3296)
X(16777) = X(i)-Ceva conjugate of X(j) for these (i,j): {1698, 3715}, {2214, 6}, {4606, 513}, {4654, 5221}, {4756, 4834}, {5248, 1011}, {5333, 1698}, {15322, 661}
X(16777) = X(4834)-cross conjugate of X(4756)
X(16777) = crosspoint of X(i) and X(j) for these (i,j): {2, 3296}, {835, 1016}, {1698, 4654}, {4658, 5333}
X(16777) = crossdifference of every pair of points on line {513, 4401}
X(16777) = crosssum of X(i) and X(j) for these (i,j): {1, 1449}, {6, 3295}, {834, 1015}
X(16777) = complement of isotomic conjugate of X(3296)
X(16777) = X(514)-isoconjugate of X(8652)
X(16777) = X(1698)-Hirst inverse of X(4716)
X(16777) = trilinear pole of line X(4813)X(4834)
X(16777) = X(9)-beth conjugate of X(3731)
X(16777) = X(4813)-zayin conjugate of X(513)
X(16777) = X(3296)-complementary conjugate of X(2887)
X(16777) = barycentric product X(i)*X(j) for these {i,j}: {1, 1698}, {4, 3927}, {7, 3715}, {8, 5221}, {9, 4654}, {10, 4658}, {37, 5333}, {57, 4007}, {58, 4066}, {80, 4880}, {88, 4727}, {100, 4802}, {101, 4823}, {190, 4813}, {226, 4877}, {291, 4716}, {513, 4756}, {651, 4820}, {660, 4810}, {662, 4838}, {668, 4834}, {799, 4826}, {897, 4938}, {943, 3824}, {1018, 4960}, {3257, 4958}, {3952, 4840}, {4866, 5586}, {4898, 8056}, {4942, 9309}
X(16777) = barycentric quotient X(i)/X(j) for these {i,j}: {692, 8652}, {1698, 75}, {3715, 8}, {3927, 69}, {4007, 312}, {4066, 313}, {4654, 85}, {4658, 86}, {4716, 350}, {4727, 4358}, {4756, 668}, {4802, 693}, {4810, 3766}, {4813, 514}, {4820, 4391}, {4823, 3261}, {4826, 661}, {4834, 513}, {4838, 1577}, {4840, 7192}, {4877, 333}, {4880, 320}, {4938, 14210}, {4949, 4462}, {4958, 3762}, {4960, 7199}, {5221, 7}, {5333, 274}
X(16777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9, 1100), (1, 37, 6), (1, 3247, 37), (1, 3731, 1449), (1, 6051, 1191), (1, 8965, 3298), (2, 4360, 4361), (6, 37, 45), (9, 1100, 6), (37, 44, 3731), (37, 1100, 9), (37, 3723, 1), (37, 4864, 16517), (44, 1449, 6), (55, 2178, 1030), (86, 192, 4363), (551, 3950, 5750), (968, 3745, 3052), (1108, 3553, 6), (1124, 1335, 16473), (1449, 3731, 44), (1962, 5311, 55), (3247, 3723, 6), (3672, 4648, 1086), (4357, 4851, 599), (5239, 5240, 958)


X(16778) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16778) lies on these lines:
{1, 3}, {31, 1790}, {58, 2258}, {63, 3724}, {228, 3751}, {968, 4184}, {1626, 5345}, {1631, 7298}, {1707, 3185}, {1721, 7416}, {1722, 16453}, {1836, 15447}, {4231, 5307}, {4278, 12514}, {8616, 16688}


X(16779) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(43)

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

X(16779) lies on these lines:
{1, 6}, {43, 2280}, {87, 572}, {604, 4334}, {672, 8616}, {673, 4859}, {978, 4251}, {1429, 6180}, {1914, 3550}, {2241, 3501}, {2344, 3551}, {3662, 3664}, {3684, 16569}, {3729, 4366}, {4307, 4660}, {5838, 16020}, {7175, 7271}


X(16780) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(78)

Barycentrics    a(-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) : :

X(16780) lies on these lines:
{1, 6}, {32, 57}, {34, 1438}, {35, 9574}, {39, 3601}, {40, 1914}, {41, 614}, {46, 7031}, {55, 9593}, {172, 3333}, {204, 2356}, {223, 1429}, {604, 4320}, {609, 3338}, {938, 5304}, {950, 5286}, {1015, 1420}, {1024, 8578}, {1210, 7735}, {1462, 4350}, {1500, 10389}, {1572, 3340}, {1697, 2241}, {1722, 3684}, {1876, 3172}, {2082, 3924}, {2251, 5573}, {2270, 16946}, {2271, 2999}, {2275, 3576}, {2285, 4332}, {2548, 5219}, {2646, 9592}, {3053, 15803}, {3501, 3749}, {3586, 5254}, {3752, 4258}, {3767, 9581}, {4304, 7738}, {5007, 11518}, {5277, 5437}, {5305, 5722}, {5332, 11529}, {7736, 13411}, {7737, 9579}, {7745, 9612}, {8227, 9599}, {9619, 13384}


X(16781) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(145)

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

X(16781) lies on these lines:
{1, 6}, {3, 1015}, {11, 13881}, {12, 9599}, {32, 999}, {35, 15815}, {36, 5023}, {39, 3295}, {41, 1149}, {55, 2275}, {56, 1914}, {115, 9669}, {172, 3304}, {230, 3086}, {304, 4361}, {330, 1975}, {381, 9665}, {382, 9651}, {388, 7745}, {390, 7738}, {495, 2548}, {496, 3767}, {497, 5254}, {595, 5021}, {614, 1611}, {942, 1572}, {995, 2271}, {1058, 5286}, {1184, 7191}, {1201, 2280}, {1376, 16604}, {1398, 1968}, {1407, 1429}, {1438, 3207}, {1462, 3160}, {1475, 3915}, {1500, 6767}, {1573, 11108}, {1575, 3913}, {1691, 5194}, {1759, 4694}, {1870, 2207}, {1973, 7113}, {2082, 3290}, {2170, 7124}, {2242, 7373}, {2276, 3303}, {2549, 15171}, {3058, 9598}, {3085, 3815}, {3094, 10387}, {3496, 3976}, {3616, 5275}, {3622, 5276}, {3780, 4383}, {4253, 14974}, {5111, 5148}, {5124, 8193}, {5204, 5210}, {5217, 10987}, {5272, 16605}, {5475, 9654}, {5563, 7031}, {6284, 9597}, {7735, 14986}, {7739, 15170}, {7747, 9655}, {7748, 9668}, {9263, 16916}, {9596, 15888}, {9620, 9957}, {9650, 154 84}, {15048, 15172}


X(16782) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(239)

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

X(16782) lies on these lines:
{1, 6}, {239, 3263}, {292, 1438}, {672, 3747}, {869, 2280}, {1015, 8624}, {1429, 2111}, {1968, 1973}, {2112, 2210}, {2238, 3912}, {2664, 3684}, {5275, 16831}, {5276, 16826}


X(16783) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(386)

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

X(16783) lies on these lines:
{1, 6}, {2, 4251}, {10, 2280}, {20, 572}, {21, 4253}, {41, 1125}, {55, 16549}, {79, 2344}, {101, 3616}, {171, 7031}, {284, 443}, {379, 2140}, {404, 4262}, {474, 4258}, {475, 2332}, {551, 9310}, {573, 6986}, {604, 4298}, {672, 5248}, {910, 5439}, {942, 1759}, {993, 1475}, {1018, 3295}, {1429, 4654}, {1438, 13576}, {1509, 12150}, {1621, 3730}, {1698, 3684}, {1839, 1973}, {1855, 5136}, {1914, 5264}, {2241, 2295}, {2268, 4314}, {2271, 3216}, {2278, 11112}, {2303, 5037}, {3217, 3986}, {3244, 4390}, {3496, 5902}, {3501, 3746}, {3579, 6205}, {3734, 4754}, {3748, 3991}, {3870, 4006}, {3874, 5282}, {3912, 5278}, {3915, 3997}, {4071, 4894}, {4189, 5030}, {4254, 16410}, {4513, 6767}, {5022, 16370}, {5053, 11111}, {5816, 6886}, {6996, 10478}, {14210, 16822}


X(16784) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(519)

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

X(16784) lies on these lines:
{1, 6}, {32, 5563}, {35, 574}, {36, 187}, {39, 3746}, {55, 5024}, {56, 1384}, {81, 6629}, {101, 1149}, {106, 919}, {172, 5008}, {230, 3582}, {239, 14210}, {244, 5011}, {551, 5276}, {595, 1475}, {609, 999}, {739, 1308}, {902, 5030}, {995, 2280}, {1019, 1429}, {1201, 4251}, {1323, 1462}, {1415, 5193}, {1428, 2030}, {1572, 5902}, {1759, 3976}, {1870, 8744}, {2003, 7349}, {2210, 5168}, {2242, 5332}, {2251, 9259}, {3058, 15048}, {3290, 5540}, {3303, 9605}, {3496, 3953}, {3509, 4694}, {3584, 3815}, {3915, 4253}, {4309, 7738}, {4857, 5254}, {5088, 7208}, {5148, 8540}, {5204, 15655}, {5210, 7280}, {5270, 7745}, {5354, 7191}, {6767, 9331}, {7292, 11580}, {7296, 14075}, {7297, 16546}, {7343, 14901}, {7735, 10072}, {7736, 10056}, {7951, 9599}, {11237, 15484}


X(16785) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(551)

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

X(16785) lies on these lines:
{1, 6}, {32, 3746}, {35, 172}, {36, 574}, {39, 5563}, {42, 101}, {55, 609}, {56, 5024}, {58, 1334}, {81, 644}, {171, 1018}, {230, 3584}, {386, 9310}, {519, 5276}, {759, 8693}, {894, 14210}, {1055, 4256}, {1201, 9327}, {1285, 10385}, {1415, 3256}, {1468, 3730}, {1914, 5008}, {2003, 4559}, {2030, 2330}, {2177, 4262}, {2241, 7296}, {2298, 3950}, {2303, 2321}, {2667, 8625}, {3208, 5264}, {3295, 7031}, {3304, 9605}, {3338, 9593}, {3509, 4424}, {3582, 3815}, {3679, 5275}, {3920, 5354}, {4006, 5293}, {4099, 7283}, {4317, 7738}, {4513, 5711}, {4857, 7745}, {5010, 5210}, {5217, 15655}, {5254, 5270}, {5297, 11580}, {5305, 15888}, {5332, 14075}, {5341, 16546}, {5434, 15048}, {5902, 9620}, {6126, 14901}, {6198, 8744}, {7373, 9336}, {7735, 10056}, {7736, 10072}, {7741, 9596}, {9598, 10483}, {11238, 15484}


X(16786) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(899)

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

X(16786) lies on these lines:
{1, 6}, {171, 5332}, {320, 4667}, {651, 1404}, {899, 3684}, {1438, 5053}, {2246, 7292}, {2280, 3240}, {3509, 3999}, {3912, 4700}, {4070, 5211}, {4690, 6687}


X(16787) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(976)

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

X(16787) lies on these lines:
{1, 6}, {32, 982}, {41, 7191}, {172, 3976}, {335, 7787}, {604, 4296}, {609, 3953}, {977, 1438}, {986, 1914}, {1870, 1973}, {2280, 5262}, {3501, 3744}, {3662, 7893}, {3670, 7031}, {3721, 5332}, {3726, 7296}, {3749, 9593}


X(16788) =  (X(1),X(2),X(6),X(101); X(1),X(6),X(2),X(101)) COLLINEATION IMAGE OF X(995)

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

X(16788) lies on these lines:
{1, 6}, {2, 101}, {3, 16549}, {8, 4251}, {10, 41}, {21, 3730}, {31, 3997}, {32, 2295}, {35, 3501}, {48, 5750}, {55, 1018}, {65, 1759}, {80, 2344}, {100, 4262}, {171, 609}, {284, 2345}, {474, 3207}, {519, 2280}, {572, 5731}, {584, 594}, {604, 4315}, {644, 1621}, {672, 993}, {758, 5282}, {825, 13194}, {894, 5088}, {910, 3753}, {996, 1438}, {1125, 9310}, {1150, 3912}, {1155, 6205}, {1213, 3204}, {1215, 4112}, {1220, 10454}, {1334, 5248}, {1429, 5219}, {1475, 8666}, {1783, 3192}, {1826, 1973}, {1930, 16822}, {2172, 2908}, {2251, 4386}, {2271, 3293}, {2278, 10609}, {2301, 2550}, {2975, 4253}, {3208, 3746}, {3295, 4513}, {3496, 5903}, {3509, 5902}, {3622, 9327}, {3679, 3684}, {3750, 9331}, {3811, 4006}, {3924, 16600}, {4071, 4680}, {4153, 5016}, {4258, 5687}, {4424, 9620}, {4876, 15175}, {5134, 11114}, {5255, 7031}, {8568, 10165}


X(16789) =  MIDPOINT OF X(22) AND X(69)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27381.

X(16789) lies on these lines:
{5,9971}, {6,6676}, {22,69}, {30,599}, {67,550}, {141,427}, {155,16618}, {184,524}, {343,2393}, {378,10519}, {511,15760}, {1632,7750}, {1634,3933}, {2353,7767}, {2781,2883}, {2918,2930}, {3564,7502}, {3620,7391}, {5188,15526}, {9833,15069}, {10169,11416}

X(16789) = midpoint of X(22) and X(69)
X(16789) = reflection of X(i) in X(j) for these {i,j}: {6, 6676}, {427, 141}
X(16789) = barycentric product X(141)*X(7493)
X(16789) = barycentric quotient X(7493)/X(83)


X(16790) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(8)

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

X(16790) lies on these lines:
{1, 6}, {31, 1626}, {595, 4259}, {674, 3915}, {995, 5135}, {1469, 7246}, {3006, 4383}, {5300, 5710}


X(16791) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(10)

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

X(16791) lies on these lines:
{1, 6}, {31, 4259}, {58, 1576}, {595, 674}, {1193, 5135}


X(16792) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(42)

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

X(16792) lies on these lines:
{1, 6}, {11, 2330}, {86, 110}, {182, 5886}, {674, 1621}, {1125, 5135}, {1428, 15950}, {2175, 4657}, {3219, 9020}, {4259, 5248}, {8539, 9054}, {10755, 15988}


X(16793) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(43)

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

X(16793) lies on these lines:
{1, 6}, {674, 8616}, {978, 5135}


X(16794) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(145)

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

X(16794) lies on these lines:
{1, 6}, {3052, 16064}, {4202, 5710}


X(16795) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(239)

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

X(16795) lies on these lines:
{1, 6}, {239, 9022}, {292, 7113}, {513, 3050}, {674, 3747}, {700, 4154}, {742, 15994}, {1409, 1463}, {1914, 2225}, {2223, 3285}, {2238, 3006}, {2245, 8624}, {5846, 15990}


X(16796) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(519)

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

X(16796) lies on these lines:
{1, 6}, {1201, 5135}, {3834, 5711}, {3915, 4259}


X(16797) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(551)

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

X(16797) lies on this line: {1, 6}


X(16798) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(612)

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

X(16798) lies on these lines: {1, 6}, {3006, 3618}


X(16799) =  (X(1),X(2),X(6),X(692); X(1),X(6),X(2),X(692)) COLLINEATION IMAGE OF X(614)

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

X(16799) lies on these lines:
{1, 6}, {63, 674}, {69, 3006}, {2267, 4712}, {3811, 5135}, {4847, 9028}, {6734, 12587}


X(16800) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(306)

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

X(16800) lies on these lines: {1, 6}, {86, 2887}, {4446, 5301}


X(16801) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(519)

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

X(16801) lies on these lines:
{1, 6}, {86, 752}, {1914, 8297}, {3099, 3290}, {8649, 11364}


X(16802) = PERSPECTOR OF THESE TRIANGLES: 1st ISODYNAMIC-DAO AND ANTI-EXCENTERS-REFLECTIONS

Barycentrics    (SB+SC)*(3*(30*R^2-5*SA-7*SW)*S^2-sqrt(3)*(2*SW^2-7*SB*SC+21*SA^2+18*S^2-2*(39*SA+4*SW)*R^2)*S+6*(SA^2-SB*SC)*(12*R^2-2*SA-SW)) : :

Let P1 = X(15) be the 1st isodynamic point of ABC. Circles (A, P1), (B, P1) and (C, P1) intersect by pairs at P1, A', B', C'. Then the triangle A'B'C' is equilateral.

Let P2 = X(16) be the 2nd isodynamic point of ABC. Circles (A, P2), (B, P2) and (C, P2) intersect by pairs at P2, A", B", C". Then the triangle A"B"C" is equilateral. (Both conclusions by Dao Thanh Oai, March 25, 2018)

Triangles A'B'C' and A"B"C" are here named the 1st isodynamic-Dao equilateral triangle and the 2nd isodynamic-Dao equilateral triangle, respectively.

Barycentric coordinates of A' and A" are:

 A' = -(SB+SC)*(sqrt(3)*SA+S) : sqrt(3)*(S^2+SA*SC)+(SA+3*SC)*S : sqrt(3)*(S^2+SA*SB)+(SA+3*SB)*S

 A" = -(SB+SC)*(sqrt(3)*SA-S) : sqrt(3)*(S^2+SA*SC)-(SA+3*SC)*S : sqrt(3)*(S^2+SA*SB)-(SA+3*SB)*S

The squared sides of A'B'C' and A"B"C" are a'2 = 4*S^2/(SW+sqrt(3)*S) and a"2 = 4*S^2/(SW-sqrt(3)*S),respectively, and their centers are X(13) and X(14), respectively.

The appearance of (T,n) in the following list means that A'B'C' and triangle T are perspective with perspector X(n):
(ABC, 13), (anti-excenters-reflections, 16802), (anti-inverse-in-incircle, 16804), (anti-orthocentroidal, 10657), (circumorthic, 16806), (outer-Fermat, 13), (3rd Fermat-Dao*, 11542), (7th Fermat-Dao*, 16267), (11th Fermat-Dao*, 16808), (outer-Napoleon*, 16), (orthic, 2902), (inner tri-equilateral, 15)

The appearance of (T,n) in the following list means that A"B"C" and triangle T are perspective with perspector X(n):
(ABC, 14), (anti-excenters-reflections, 16803), (anti-inverse-in-incircle, 16805), (anti-orthocentroidal, 10658), (circumorthic, 16807), (inner-Fermat, 14), (4th Fermat-Dao*, 11543), (8th Fermat-Dao*, 16268), (12th Fermat-Dao*, 16809), (inner-Napoleon*, 15), (orthic, 2903), (outer tri-equilateral, 16)

In both lists, an asterisk * means that the triangles are homothetic.

(This introduction and centers X(16802) to X(16809) were contributed by César Lozada, March 28, 2018)

X(16802) lies on the line {11475,16806}


X(16803) = PERSPECTOR OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO AND ANTI-EXCENTERS-REFLECTIONS

Barycentrics    (SB+SC)*(3*(30*R^2-5*SA-7*SW)*S^2+sqrt(3)*(2*SW^2-7*SB*SC+21*SA^2+18*S^2-2*(39*SA+4*SW)*R^2)*S+6*(SA^2-SB*SC)*(12*R^2-2*SA-SW)) : :

X(16803) lies on the line {11476,16807}


X(16804) = PERSPECTOR OF THESE TRIANGLES: 1st ISODYNAMIC-DAO AND ANTI-INVERSE-IN-INCIRCLE

Barycentrics    -sqrt(3)*(-SW^2*SB*SC+(-2*(SB+SC)*R^2+SB*SC-SW^2)*S^2-S^4)+S*(-(SB+SC)*(6*R^2*SA+2*SA^2-2*SB*SC-SW^2)+(8*R^2+5*SA+SW)*S^2) : :

X(16804) lies on the line {11488,16806}


X(16805) = PERSPECTOR OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO AND ANTI-INVERSE-IN-INCIRCLE

Barycentrics    -sqrt(3)*(-SW^2*SB*SC+(-2*(SB+SC)*R^2+SB*SC-SW^2)*S^2-S^4)-S*(-(SB+SC)*(6*R^2*SA+2*SA^2-2*SB*SC-SW^2)+(8*R^2+5*SA+SW)*S^2) : :

X(16805) lies on the line {11489,16807}


X(16806) = PERSPECTOR OF THESE TRIANGLES: 1st ISODYNAMIC-DAO AND CIRCUMORTHIC

Barycentrics    (SC+sqrt(3)*S)*(SB+sqrt(3)*S)*(SA-SC)*(SA-SB)*(SB+SC) : :

Let La be the reflection of the line X(4)X(15) in line BC, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(16808). (Randy Hutson, June 27, 2018)

Let P be a point on line X(2)X(17) other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(16806). (Randy Hutson, June 27, 2018)

X(16806) lies on the circumcircle and these lines:
{6,2380}, {13,1141}, {17,98}, {62,5966}, {74,3166}, {115,11087}, {930,14187}, {1297,14540}, {1300,8741}, {1625,5994}, {2378,8603}, {2383,2902}, {6777,11600}, {11475,16802}, {11488,16804}

X(16806) = trilinear pole of the line {6, 3132}
X(16806) = X(61)-isoconjugate of X(1577)
X(16806) = Λ(trilinear polar of X(473))
X(16806) = Ψ(X(i), X(j)) for these (i,j): (2,17), (4,15), (76,17)
X(16806) = barycentric product X(17)*X(110)
X(16806) = barycentric quotient X(17)/X(850)


X(16807) = PERSPECTOR OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO AND CIRCUMORTHIC

Barycentrics    (SC-sqrt(3)*S)*(SB-sqrt(3)*S)*(SA-SC)*(SA-SB)*(SB+SC) : :

Let La be the reflection of the line X(4)X(16) in line BC, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(16807). (Randy Hutson, June 27, 2018)

Let P be a point on line X(2)X(18) other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC∩B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(16807). (Randy Hutson, June 27, 2018)

X(16807) lies on the circumcircle and these lines:
{6,2381}, {14,1141}, {18,98}, {61,5966}, {74,3165}, {115,11082}, {930,14185}, {1297,14541}, {1300,8742}, {1576,6140}, {1625,5995}, {2379,8604}, {2383,2903}, {11476,16803}, {11489,16805}

X(16807) = trilinear pole of the line {6, 3131}
X(16807) = X(62)-isoconjugate of X(1577)
X(16807) = Λ(trilinear polar of X(472))
X(16807) = Ψ(X(i), X(j)) for these (i,j): (2,18), (4,16), (76,18)
X(16807) = barycentric product X(18)*X(110)
X(16807) = barycentric quotient X(18)/X(850)


X(16808) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ISODYNAMIC-DAO AND 11th FERMAT-DAO

Barycentrics    (SB+SC)*S+sqrt(3)*(S^2+3*SB*SC) : :
X(16808) = 2*X(10645)-3*X(16241)

X(16808) lies on these lines:
{2,10646}, {4,15}, {5,16}, {6,13}, {18,3851}, {30,10645}, {61,546}, {62,3091}, {140,5350}, {148,6299}, {262,11603}, {303,5464}, {382,11480}, {383,5478}, {395,5066}, {396,3845}, {397,3850}, {398,3858}, {403,10642}, {622,623}, {624,11303}, {1250,7951}, {1594,11476}, {1656,11481}, {3090,5237}, {3105,16627}, {3146,5352}, {3153,11420}, {3201,15033}, {3412,3843}, {3545,10653}, {3574,10678}, {3583,10638}, {3585,7051}, {3627,5238}, {3628,5351}, {3643,7934}, {3830,16644}, {3832,5334}, {3839,10654}, {5055,16242}, {5056,5366}, {5068,5344}, {5448,10662}, {5479,6783}, {5615,16001}, {6288,10677}, {6669,11299}, {7777,9760}, {9116,9762}, {9927,10661}, {10024,10635}, {11268,13406}, {11516,15760}, {12820,15681}

X(16808) = inverse of X(6777) in the orthocentroidal circle
X(16808) = homothetic center of Ehrmann vertex-triangle and inner tri-equilateral triangle
X(16808) = X(10645)-of-orthcentroidal-triangle
X(16808) = X(18422)-of-Ehrmann-vertex-triangle if ABC is acute
X(16808) = X(18422)-of-inner-tri-equilateral-triangle if ABC is acute
X(16808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5318, 16), (6, 381, 16809), (3851, 5340, 18)


X(16809) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ISODYNAMIC-DAO AND 12th FERMAT-DAO

Barycentrics    -(SB+SC)*S+sqrt(3)*(S^2+3*SB*SC) : :
X(16809) = 2*X(10646)-3*X(16242)

X(16809) lies on these lines:
{2,10645}, {4,16}, {5,15}, {6,13}, {17,3851}, {30,10646}, {61,3091}, {62,546}, {140,5349}, {148,6298}, {262,11602}, {302,5463}, {382,11481}, {395,3845}, {396,5066}, {397,3858}, {398,3850}, {403,10641}, {621,624}, {623,11304}, {1080,5479}, {1250,3583}, {1594,11475}, {1656,11480}, {3090,5238}, {3104,16626}, {3146,5351}, {3153,11421}, {3200,15033}, {3411,3843}, {3545,10654}, {3574,10677}, {3627,5237}, {3628,5352}, {3642,7934}, {3830,16645}, {3832,5335}, {3839,10653}, {5055,16241}, {5056,5365}, {5068,5343}, {5448,10661}, {5478,6782}, {5611,16002}, {6288,10678}, {6670,11300}, {7051,7741}, {7777,9762}, {7951,10638}, {9927,10662}, {10024,10634}, {11267,13406}, {11515,15760}, {12821,15681}

X(16809) = inverse of X(6778) in the orthocentroidal circle
X(16809) = homothetic center of Ehrmann vertex-triangle and outer tri-equilateral triangle
X(16809) = X(10646)-of-orthcentroidal-triangle
X(16809) = X(18423)-of-Ehrmann-vertex-triangle if ABC is acute
X(16809) = X(18423)-of-inner-tri-equilateral-triangle if ABC is acute
X(16809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 5321, 15), (6, 381, 16808), (3851, 5339, 17)


X(16810) =  (name pending)

Barycentrics    2 a^22-10 a^20 b^2+17 a^18 b^4-8 a^16 b^6-a^14 b^8-21 a^12 b^10+49 a^10 b^12-37 a^8 b^14+3 a^6 b^16+11 a^4 b^18-6 a^2 b^20+b^22-10 a^20 c^2+38 a^18 b^2 c^2-46 a^16 b^4 c^2+7 a^14 b^6 c^2+32 a^12 b^8 c^2-47 a^10 b^10 c^2+42 a^8 b^12 c^2+a^6 b^14 c^2-42 a^4 b^16 c^2+33 a^2 b^18 c^2-8 b^20 c^2+17 a^18 c^4-46 a^16 b^2 c^4+42 a^14 b^4 c^4-11 a^12 b^6 c^4-15 a^10 b^8 c^4+16 a^8 b^10 c^4-16 a^6 b^12 c^4+61 a^4 b^14 c^4-76 a^2 b^16 c^4+28 b^18 c^4-8 a^16 c^6+7 a^14 b^2 c^6-11 a^12 b^4 c^6+26 a^10 b^6 c^6-21 a^8 b^8 c^6+7 a^6 b^10 c^6-41 a^4 b^12 c^6+96 a^2 b^14 c^6-55 b^16 c^6-a^14 c^8+32 a^12 b^2 c^8-15 a^10 b^4 c^8-21 a^8 b^6 c^8+10 a^6 b^8 c^8+11 a^4 b^10 c^8-78 a^2 b^12 c^8+62 b^14 c^8-21 a^12 c^10-47 a^10 b^2 c^10+16 a^8 b^4 c^10+7 a^6 b^6 c^10+11 a^4 b^8 c^10+62 a^2 b^10 c^10-28 b^12 c^10+49 a^10 c^12+42 a^8 b^2 c^12-16 a^6 b^4 c^12-41 a^4 b^6 c^12-78 a^2 b^8 c^12-28 b^10 c^12-37 a^8 c^14+a^6 b^2 c^14+61 a^4 b^4 c^14+96 a^2 b^6 c^14+62 b^8 c^14+3 a^6 c^16-42 a^4 b^2 c^16-76 a^2 b^4 c^16-55 b^6 c^16+11 a^4 c^18+33 a^2 b^2 c^18+28 b^4 c^18-6 a^2 c^20-8 b^2 c^20+c^22 : :

See Antreas Hatzipolakis, Peter Moses, and César Lozada, Hyacinthos 27387 and Hyacinthos 27388.

X(16810) lies on this line: {4,54}

X(16810) = polar-circle-inverse-of X(3462)


X(16811) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(645), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16811) lies on this line: {3121, 7200}


X(16812) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16812) lies on no line X(i)X(j) for 0 < i < j <= 16900.


X(16813) =  TRILINEAR POLE OF X(4)X(54)

Trilinears    1/[sec B sec(A - B) - sec C sec(A - C)] : :
Barycentrics    (tan A)/(cos^2 2B - cos^2 2C) : :
Barycentrics    (tan A)/(b^2 cos^3 B - c^2 cos^3 C) : :
Barycentrics    (sec A)/[b sec(A - B) - c sec(A - C)] : :
Barycentrics    SB^2*SC^2*(SA-SB)*(SA-SC)*(S^ 2+SA*SB)*(S^2+SA*SC) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27388.

X(16813) lies on these lines
{95, 6330}, {107, 933}, {112, 15352}, {248, 8795}, {275, 6749}, {906, 6335}, {2052, 11077}, {4558, 6331}, {8794, 14910}, {8884, 14908}

X(16813) = trilinear pole of the line {4, 54}
X(16813) = polar conjugate of X(6368)
X(16813) = perspector of ABC and tangential triangle, wrt anticevian triangle of X(4), of bianticevian conic of X(4) and X(6)
X(16813) = barycentric product X(i)*X(j) for these {i,j}: {54, 6528}, {95, 107}, {97, 15352}, {99, 8884}, {110, 8795}, {112, 276}, {264, 933}, {275, 648}, {811, 2190}, {823, 2167}
X(16813) = barycentric quotient X(i)/X(j) for these (i,j): (4, 6368), (25, 15451), (54, 520), (95, 3265), (107, 5), (110, 5562), (112, 216), (158, 2618), (275, 525), (276, 3267), (393, 12077), (648, 343), (823, 14213), (933, 3), (1301, 8798)
X(16813) = trilinear product X(i)*X(j) for these {i,j}: {54, 823}, {92, 933}, {107, 2167}, {162, 275}, {163, 8795}, {648, 2190}, {662, 8884}, {811, 8882}
X(16813) = trilinear quotient X(i)/X(j) for these (i,j): (19, 15451), (54, 822), (92, 6368), (107, 1953), (158, 12077), (162, 216), (163, 418), (275, 656), (276, 14208), (662, 5562), (811, 343), (823, 5), (933, 48)


X(16814) =  (name pending)

Trilinears    -2 a + 3b + 3c : :
Barycentrics    a (-2a + 3b + 3c) : :
X(16814) = 3 X[1] - 5 X[16484]

X(16814) lies on these lines:
{1,6}, {2,4912}, {71,5183}, {86,4755}, {115,121}, {144,4675}, {190,3739}, {210,2177}, {239,4681}, {344,3620}, {346,4678}, {584,3217}, {594,2325}, {674,7064}, {756,902}, {846,3740}, {894,4698}, {966,3161}, {968,3715}, {1086,6666}, {1334,4271}, {1418,8545}, {2268,3204}, {2310,15837}, {2321,4669}, {3052,7322}, {3175,5278}, {3305,3752}, {3630,4416}, {3631,3912}, {3644,16816}, {3686,3943}, {3707,3950}, {3729,4688}, {3759,4704}, {3834,6646}, {3977,5241}, {3986,15828}, {4256,5044}, {4268,9310}, {4357,4422}, {4361,4718}, {4384,4686}, {4473,4708}, {4641,14996}, {4648,6172}, {4664,4852}, {4670,4687}, {4682,7262}, {4739,16815}, {5008,5266}, {6351,8972}, {6352,13941}, {6687,16706}, {7308,16602}

X(16814) = complement of X(7321)
X(16814) = X(5559)-complementary conjugate of X(2887)
X(16814) = crosspoint of X(2) and X(5559)
X(16814) = crosssum of X(i) and X(j) for these (i,j): 1, 16669}, {6, 5563}
X(16814) = barycentric product X(i)*X(j) for these {i,j}: 1, 3626}, {8, 11011}
X(16814) = barycentric quotient X(i)/X(j) for these {i,j}: 3626, 75}, {11011, 7}
X(16814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9, 16885), (1, 16668, 1100), (1, 16669, 16668), (1, 16675, 37), (1, 16885, 16669), (6, 9, 15492), (6, 37, 3723), (6, 45, 3731), (6, 3723, 1100), (6, 3731, 37), (6, 15492, 44), (6, 16674, 1), (6, 16675, 16674), (6, 16677, 3247), (9, 37, 44), (9, 45, 37), (9, 3247, 3973), (9, 3731, 6), (9, 16675, 16669), (9, 16676, 1743), (37, 44, 1100), (37, 15492, 6), (37, 16666, 16777), (37, 16669, 1), (37, 16885, 16668), (44, 1100, 16671), (44, 3723, 6), (44, 16668, 16669), (45, 16885, 16675), (968, 3715, 4849), (984, 15254, 1279), (1743, 16676, 16777), (1743, 16777, 16666), (3247, 3731, 16677), (3247, 3973, 6), (3247, 16677, 37), (3731, 3973, 3247), (3731, 15492, 3723), (16669, 16885, 44), (16670, 16673, 16884), (16674, 16885, 6), (16675, 16885, 1), (16676, 16777, 37)


X(16815) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(44)

Barycentrics    a^2 - 2 a b - 2 a c - 3 b c : :

X(16815) lies on these lines:
{1, 2}, {6, 4751}, {9, 4699}, {44, 894}, {45, 75}, {86, 16666}, {88, 274}, {142, 1654}, {190, 4688}, {192, 16676}, {673, 6651}, {750, 16476}, {903, 16590}, {966, 3662}, {1107, 16610}, {1213, 16521}, {1278, 3731}, {3218, 16552}, {3246, 5263}, {3579, 6996}, {3644, 16675}, {3729, 4772}, {3759, 15668}, {3911, 7176}, {4360, 4698}, {4361, 4687}, {4416, 4896}, {4508, 4763}, {4732, 16484}, {4739, 16814}, {4741, 6173}, {4850, 5283}, {4887, 6646}, {4967, 6666}, {5204, 11329}, {5217, 16367}, {5247, 16911}, {10436, 16670}


X(16816) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(45)

Barycentrics    2 a^2 - a b - a c - 3 b c : :

X(16816) lies on these lines:
{1, 2}, {6, 4699}, {9, 1278}, {44, 75}, {45, 192}, {88, 330}, {89, 274}, {190, 4740}, {213, 14997}, {333, 16722}, {344, 4371}, {391, 4346}, {673, 5220}, {894, 4772}, {966, 16521}, {1043, 16047}, {1086, 4741}, {1100, 4751}, {1107, 4850}, {1266, 3707}, {1654, 4000}, {3210, 5278}, {3246, 3696}, {3644, 16814}, {3662, 3686}, {3729, 4821}, {3739, 3759}, {3758, 4688}, {3797, 15254}, {3875, 4704}, {3943, 4405}, {4360, 16672}, {4389, 4395}, {4416, 4887}, {4687, 4852}, {4709, 15485}, {4726, 15492}, {4739, 16669}, {4747, 5032}, {5247, 16913}, {5708, 16054}, {6996, 12702}, {14621, 16477}


X(16817) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(72)

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

X(16817) lies on these lines:
{1, 2}, {21, 4359}, {28, 242}, {36, 6533}, {75, 405}, {312, 11108}, {321, 5047}, {333, 942}, {345, 16845}, {391, 11036}, {894, 1724}, {1010, 1104}, {1213, 16519}, {1330, 5249}, {1398, 6359}, {1453, 10436}, {2345, 13742}, {2891, 4684}, {3487, 14555}, {3666, 11110}, {3685, 4647}, {3702, 5284}, {3746, 4714}, {3797, 16912}, {3868, 5278}, {3925, 5015}, {4000, 13725}, {4195, 4699}, {4197, 5016}, {4388, 12609}, {4688, 13735}, {4751, 16458}, {4850, 16342}, {4968, 5260}, {4980, 16861}, {5100, 9710}, {5233, 11374}, {5279, 16552}, {5439, 14829}, {7270, 8728}, {13728, 16706}


X(16818) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(5280)

Barycentrics    a^3 b + a b^3 + 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 : :

X(16818) lies on these lines:
{1, 2}, {35, 16061}, {36, 16060}, {86, 5299}, {213, 3589}, {274, 16706}, {304, 4751}, {673, 1010}, {1475, 16887}, {1930, 3739}, {2223, 8362}, {4357, 16552}, {4657, 5283}, {7889, 8624}, {15668, 16502}


X(16819) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(213)

Barycentrics    a^2 b^2 + a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + b^2 c^2 : :

X(16819) lies on these lines:
{1, 2}, {36, 16917}, {75, 5283}, {194, 4699}, {274, 1107}, {350, 16589}, {384, 5251}, {894, 16552}, {958, 11321}, {993, 16915}, {1213, 16514}, {1573, 1909}, {1724, 14621}, {2223, 16061}, {3774, 4687}, {3797, 4647}, {3925, 6656}, {4366, 5259}, {4413, 11285}, {5258, 6645}


X(16820) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(2087)

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

X(16820) lies on these lines:
{1, 2}, {213, 4585}, {274, 4601}, {894, 1023}, {1083, 4676}, {4089, 6646}


X(16821) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(392)

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

X(16821) lies on these lines:
{1, 2}, {36, 4714}, {75, 956}, {274, 3262}, {312, 9708}, {333, 517}, {405, 4673}, {958, 5695}, {1621, 3902}, {2099, 4042}, {3685, 5251}, {3702, 5260}, {3753, 14829}, {3877, 5278}, {4023, 15950}, {4647, 5258}, {4672, 5247}, {4742, 5284}, {5233, 5886}, {5603, 14555}


X(16822) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(3061)

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

X(16822) lies on these lines:
{1, 2}, {37, 3905}, {75, 2329}, {274, 7132}, {333, 7146}, {384, 3923}, {986, 16060}, {1580, 3980}, {1930, 16788}, {1959, 5278}, {2295, 4372}, {3120, 16910}, {3674, 4416}, {4011, 16916}, {4418, 16919}, {4766, 5016}, {6651, 16914}, {14210, 16783}


X(16823) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(518)

Barycentrics    a^3 - a^2 b - a^2 c - 3 a b c - b^2 c - b c^2 : :

X(16823) lies on these lines:
{1, 2}, {21, 99}, {25, 5342}, {35, 6533}, {56, 85}, {75, 1001}, {86, 1386}, {142, 3883}, {183, 16284}, {190, 15254}, {238, 894}, {292, 16604}, {312, 4423}, {319, 4966}, {321, 5284}, {325, 11281}, {333, 354}, {344, 3790}, {391, 11038}, {404, 2223}, {515, 7385}, {675, 8691}, {740, 16484}, {946, 7379}, {958, 16048}, {968, 3210}, {980, 16342}, {1107, 3290}, {1212, 6559}, {1279, 3739}, {1385, 6998}, {1441, 7677}, {1468, 16476}, {1621, 4359}, {2975, 4223}, {3263, 3702}, {3286, 16709}, {3294, 4115}, {3315, 5235}, {3475, 14555}, {3485, 6604}, {3579, 13635}, {3663, 9791}, {3686, 4684}, {3717, 6666}, {3742, 14829}, {3748, 3996}, {3791, 4038}, {3873, 5278}, {3923, 15485}, {3925, 4514}, {3980, 8616}, {4026, 16706}, {4360, 15569}, {4363, 4676}, {4385, 11108}, {4388, 5249}, {4416, 5542}, {4649, 4974}, {4968, 5047}, {5015, 8728}, {5088, 5144}, {5259, 7283}, {5260, 9369}, {5296, 16517}, {5731, 7390}, {5749, 16970}, {5886, 7380}, {7290, 10436}, {7987, 9746}, { 9105, 15322}, {13624, 13634}


X(16824) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(960)

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

X(16824) lies on these lines:
{1, 2}, {35, 4714}, {65, 333}, {75, 958}, {92, 4185}, {190, 5302}, {274, 961}, {321, 5260}, {405, 3685}, {429, 5174}, {894, 5247}, {1001, 4673}, {1043, 3696}, {1104, 5263}, {1330, 12609}, {1738, 4201}, {1880, 2322}, {2345, 16968}, {2975, 4359}, {3485, 14555}, {3671, 4416}, {3699, 3983}, {3702, 5047}, {3729, 5234}, {3812, 14829}, {3869, 5278}, {3886, 5436}, {3925, 7270}, {3931, 11110}, {4296, 16713}, {4385, 9708}, {4647, 5251}, {4968, 9369}, {5233, 11375}, {5563, 6533}, {6559, 15853}


X(16825) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(984)

Barycentrics    a^3 - 2 a b c - b^2 c - b c^2 : :

X(16825) lies on these lines:
{1, 2}, {3, 8301}, {6, 4974}, {9, 726}, {31, 3980}, {38, 5278}, {56, 16609}, {58, 274}, {75, 238}, {142, 5847}, {169, 3509}, {244, 1150}, {321, 748}, {330, 1929}, {333, 982}, {344, 6541}, {391, 4310}, {536, 15254}, {537, 5220}, {596, 5282}, {740, 1001}, {752, 5880}, {756, 3891}, {846, 3210}, {885, 8714}, {894, 16468}, {940, 3791}, {968, 4970}, {993, 4124}, {1010, 16478}, {1086, 4655}, {1107, 16583}, {1215, 4383}, {1279, 3696}, {1386, 3739}, {1441, 1471}, {1449, 4991}, {1738, 3883}, {1966, 10009}, {2796, 5698}, {2887, 3966}, {3159, 3294}, {3305, 3971}, {3361, 7176}, {3416, 3836}, {3685, 15485}, {3747, 4647}, {3758, 16477}, {3759, 4649}, {3772, 3846}, {3782, 4703}, {3821, 4000}, {3826, 5846}, {3875, 3993}, {3886, 4709}, {3925, 4865}, {4078, 6666}, {4095, 16969}, {4256, 16497}, {4283, 4446}, {4353, 16517}, {4363, 4672}, {4368, 4441}, {4413, 4434}, {4426, 8624}, {4432, 5695}, {4659, 4759}, {4689, 4706}, {4716, 16484}, {4852, 15569}, {4989, 5750}, {5132, 16684}, { 5283, 16600}, {5372, 9335}, {5625, 16884}, {5737, 6682}, {10436, 16475}, {11375, 16603}, {12263, 16514}


X(16826) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1100)

Barycentrics    a^2 + 2 a b + 2 a c + b c : :

X(16826) lies on these lines:
{1, 2}, {6, 4687}, {27, 11363}, {37, 86}, {44, 4755}, {45, 3758}, {55, 11329}, {56, 16367}, {69, 4748}, {72, 16053}, {75, 15668}, {81, 213}, {171, 3747}, {192, 3247}, {193, 5296}, {226, 6625}, {261, 1963}, {274, 321}, {319, 1213}, {320, 4364}, {515, 7384}, {594, 6707}, {671, 4597}, {748, 16476}, {940, 2176}, {946, 6999}, {1001, 14621}, {1100, 4698}, {1385, 6996}, {1447, 7146}, {1509, 1931}, {1621, 2223}, {1654, 3879}, {1824, 14013}, {1909, 3948}, {2325, 4758}, {3219, 3294}, {3250, 4817}, {3295, 16412}, {3589, 4437}, {3662, 4648}, {3664, 6646}, {3723, 3739}, {3729, 4704}, {3759, 16884}, {3842, 4649}, {3875, 4699}, {3943, 4472}, {3945, 16970}, {3986, 4416}, {4043, 16709}, {4361, 4751}, {4363, 4664}, {4366, 16484}, {4378, 4776}, {4389, 4675}, {4422, 16521}, {4508, 4928}, {4702, 5263}, {4851, 5224}, {5074, 5088}, {5247, 16912}, {5266, 16060}, {5275, 16524}, {5276, 16782}, {5283, 11342}, {5295, 14007}, {5337, 8624}, {5712, 16968}, {5731, 7406}, {5886, 7377}, {6198, 15149}

X(16826) = isogonal conjugate of X(25426)
X(16826) = isotomic conjugate of X(27483)
X(16826) = anticomplement of X(24603)
X(16826) = perspector of Gemini triangle 3 and cross-triangle of ABC and Gemini triangle 3
X(16826) = trilinear pole of perspectrix of ABC and Gemini triangle 4


X(16827) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1107)

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

X(16827) lies on these lines:
{1, 2}, {9, 194}, {31, 16915}, {72, 335}, {75, 2176}, {171, 16917}, {213, 274}, {238, 384}, {257, 16583}, {748, 16916}, {992, 16514}, {1400, 7176}, {1909, 2238}, {3570, 4400}, {4361, 16969}, {5247, 6645}, {6651, 7283}, {11321, 14621}


X(16828) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1203)

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

X(16828) lies on these lines:
{1, 2}, {35, 11110}, {37, 4099}, {55, 16844}, {71, 3294}, {213, 1213}, {274, 313}, {405, 8053}, {406, 1869}, {475, 1826}, {958, 16458}, {993, 16454}, {1010, 5251}, {1220, 14007}, {3826, 13728}, {3841, 5051}, {3874, 4981}, {3925, 4205}, {3931, 4714}, {4078, 7206}, {4295, 5296}, {4302, 13736}, {4698, 6051}, {5259, 5263}, {5260, 14005}, {5750, 16552}, {9548, 10478}, {9708, 16456}, {9709, 16457}, {9798, 16353}


X(16829) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(3230)

Barycentrics    a^2 b^2 - a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + b^2 c^2 : :

X(16829) lies on these lines:
{1, 2}, {75, 716}, {194, 4740}, {274, 670}, {350, 1573}, {384, 5258}, {536, 1107}, {1213, 16526}, {3934, 13466}, {4366, 5251}, {4664, 5283}, {5288, 6645}, {5563, 16917}, {8666, 16915}, {11321, 12513}


X(16830) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1386)

Barycentrics    a^3 + a^2 b + 2 a b^2 + a^2 c + 3 a b c + b^2 c + 2 a c^2 + b c^2 : :

X(16830) lies on these lines:
{1, 2}, {21, 2223}, {29, 5089}, {37, 3685}, {45, 4676}, {65, 1447}, {81, 4981}, {83, 5047}, {86, 518}, {105, 8708}, {213, 5276}, {228, 1621}, {238, 3842}, {274, 1390}, {292, 1107}, {333, 3745}, {348, 388}, {355, 7380}, {515, 7379}, {516, 8245}, {517, 6998}, {870, 6376}, {894, 984}, {946, 7385}, {956, 16849}, {962, 7390}, {964, 5283}, {980, 16454}, {1001, 4687}, {1010, 16735}, {1150, 9347}, {1213, 5846}, {1279, 4698}, {1281, 2292}, {1654, 5847}, {2049, 4385}, {2176, 5275}, {2295, 16514}, {2345, 3790}, {3242, 15668}, {3247, 3886}, {3416, 5224}, {3496, 5250}, {3579, 13634}, {3696, 4360}, {3717, 5750}, {3747, 5255}, {3750, 10180}, {3758, 5220}, {3769, 5737}, {3826, 16706}, {3883, 5257}, {3915, 12194}, {3989, 4418}, {4205, 5015}, {4339, 13736}, {4344, 5296}, {4349, 4416}, {4357, 4645}, {4389, 5880}, {4429, 4657}, {4664, 5695}, {4682, 14829}, {4716, 4732}, {4733, 5564}, {4969, 16522}, {5266, 11110}, {5277, 8624}, {5687, 16852}, { 5716, 16968}, {5749, 16517}, {5988, 14949}, {6707, 9053}, {7174, 10436}, {7410, 12245}, {7991, 9746}, {13624, 13635}


X(16831) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1449)

Barycentrics    a^2 + 3 a b + 3 a c + 2 b c : :

X(16831) lies on these lines:
{1, 2}, {6, 4698}, {9, 86}, {33, 15149}, {35, 11329}, {36, 16367}, {37, 980}, {45, 4670}, {55, 16412}, {63, 3294}, {69, 5257}, {75, 3247}, {190, 16676}, {192, 16673}, {213, 940}, {226, 348}, {274, 312}, {292, 4465}, {344, 5750}, {536, 16672}, {599, 4708}, {750, 3747}, {894, 3731}, {966, 3879}, {988, 16061}, {1001, 2223}, {1213, 4851}, {1621, 11349}, {1699, 6999}, {1992, 3707}, {3305, 16552}, {3306, 6205}, {3576, 6996}, {3601, 16054}, {3618, 6666}, {3664, 3986}, {3723, 4361}, {3739, 3875}, {3751, 3842}, {3761, 3948}, {3886, 15569}, {3945, 4416}, {4297, 7406}, {4312, 9791}, {4357, 4648}, {4360, 4751}, {4364, 4675}, {4371, 4464}, {4389, 6173}, {4659, 4664}, {4681, 16674}, {4690, 16522}, {4747, 6172}, {4888, 6646}, {5226, 7176}, {5249, 14021}, {5275, 16782}, {5295, 16456}, {5691, 7384}, {7377, 8227}, {11350, 11365}, {14621, 15485}


X(16832) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(1743)

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

X(16832) lies on these lines:
{1, 2}, {9, 3739}, {36, 16412}, {45, 4659}, {57, 4059}, {75, 3731}, {86, 16667}, {142, 966}, {165, 6996}, {274, 8056}, {344, 4967}, {391, 3664}, {536, 16676}, {573, 10442}, {597, 4798}, {673, 4432}, {894, 3973}, {980, 16610}, {993, 11349}, {1107, 16602}, {1213, 16517}, {1449, 15668}, {1453, 16458}, {1743, 3758}, {2223, 4413}, {2345, 6666}, {3247, 4361}, {3294, 7308}, {3306, 5235}, {3663, 5296}, {3672, 3986}, {3686, 4648}, {3707, 4644}, {3729, 4699}, {3752, 5283}, {3761, 3975}, {3875, 4687}, {4000, 5257}, {4021, 4402}, {4034, 4851}, {4357, 4859}, {4416, 4888}, {4643, 6173}, {4665, 4873}, {4670, 16670}, {4686, 16675}, {4718, 16677}, {4902, 6646}, {5010, 16367}, {5088, 5199}, {5219, 5241}, {5251, 11343}, {5437, 5737}, {5745, 6554}, {6684, 7397}, {7280, 11329}, {7377, 7989}, {7402, 10175}, {8185, 11350}, {15803, 16054}


X(16833) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(3731)

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

X(16833) lies on these lines:
{1, 2}, {6, 4688}, {9, 536}, {44, 4659}, {57, 7223}, {63, 5540}, {69, 4859}, {75, 1743}, {141, 4034}, {142, 5839}, {193, 4888}, {312, 4986}, {391, 3663}, {524, 6173}, {527, 5819}, {537, 673}, {553, 7195}, {966, 3946}, {1449, 3739}, {1453, 16394}, {2321, 4371}, {3175, 3294}, {3227, 8056}, {3247, 4755}, {3618, 4967}, {3677, 4042}, {3686, 4000}, {3696, 7290}, {3707, 4419}, {3729, 3973}, {3731, 3875}, {3759, 10436}, {3760, 3975}, {3928, 5792}, {3929, 16552}, {3945, 4856}, {4007, 4399}, {4021, 5296}, {4360, 16673}, {4363, 16670}, {4383, 14535}, {4395, 4643}, {4405, 4422}, {4416, 4862}, {4644, 4700}, {4654, 5244}, {4675, 4969}, {4686, 16885}, {4795, 8584}, {4974, 16469}, {5258, 11343}, {5563, 16412}, {5695, 15601}, {6996, 7991}, {7397, 11362}, {7406, 9589}, {8666, 11349}


X(16834) =  (X(1),X(2),X(6),X(75); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(3247)

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

X(16834) lies on these lines:
{1, 2}, {6, 536}, {9, 3759}, {57, 664}, {58, 16046}, {69, 3946}, {75, 1449}, {165, 4734}, {190, 16670}, {192, 1743}, {193, 3663}, {213, 3175}, {346, 4460}, {379, 11520}, {527, 1992}, {537, 3751}, {545, 8584}, {597, 4971}, {599, 4725}, {666, 6654}, {673, 3243}, {740, 16475}, {894, 4740}, {1100, 4361}, {1266, 4644}, {1386, 3886}, {1453, 13735}, {1699, 2784}, {1707, 4970}, {2094, 5088}, {2223, 4421}, {2279, 3226}, {2321, 3618}, {2809, 3873}, {3230, 4383}, {3304, 16412}, {3672, 4416}, {3685, 16469}, {3739, 16884}, {3746, 16367}, {3750, 16497}, {3758, 4659}, {3760, 3765}, {3879, 4000}, {3905, 9575}, {3923, 4991}, {3928, 9311}, {3945, 4402}, {4034, 5224}, {4052, 5395}, {4301, 7406}, {4357, 5839}, {4363, 16666}, {4366, 16468}, {4371, 4967}, {4395, 4675}, {4405, 4472}, {4431, 5749}, {4643, 4969}, {4654, 7247}, {4667, 4982}, {4681, 16885}, {4686, 16668}, {4715, 15534}, {4718, 16671}, {4755, 16777}, {5228, 9312}, {5264, 16399}, {5563, 11329}, {5881, 7377}, {6996, 7982}, {7 384, 11522}, {11194, 16436}, {11343, 12513}, {11518, 16054}


X(16835) =  ISOGONAL CONJUGATE OF X(550)

Barycentrics    (SB+SC)*(3*S^2-5*SA*SC)*(3*S^ 2-5*SA*SB) : :
X(16835) = 10*X(3)-9*X(6030), 4*X(3)-3*X(8718), 2*X(3)-3*X(15062), 3*X(54)-4*X(14865), 4*X(546)-3*X(3521), 6*X(6030)-5*X(8718), 3*X(6030)-5*X(15062)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27397.

X(16835) lies on the cubic K664 and these lines:
{3, 6030}, {4, 13399}, {5, 14861}, {6, 6241}, {24, 3532}, {54, 6000}, {64, 10594}, {66, 12250}, {67, 6240}, {68, 3146}, {73, 3746}, {74, 3518}, {185, 1173}, {235, 10293}, {265, 3627}, {378, 14528}, {389, 14483}, {546, 3521}, {1204, 13452}, {1593, 11423}, {1614, 3431}, {3090, 15740}, {3091, 4846}, {3357, 11270}, {3426, 5198}, {3527, 5890}, {3531, 9781}, {3628, 13623}, {5073, 14841}, {5349, 11138}, {5350, 11139}, {5504, 12086}, {5900, 12244}, {6145, 15311}, {6247, 11744}, {6413, 6453}, {6414, 6454}, {7576, 15105}, {9786, 14490}, {10706, 13371}, {11438, 11738}, {11457, 14457}, {11550, 16000}, {13472, 15033}, {15704, 16659}, {16621, 16623}

X(16835) = isogonal conjugate of X(550)


X(16836) =  COMPLEMENT OF X(15030)

Barycentrics    (SA*(12*R^2-SW)+S^2)*(SB+SC) : :
X(16836) = 3*X(2)+X(15072), 5*X(2)-X(15305), 5*X(3)+X(52), 2*X(3)+X(389), 3*X(3)+X(568), 11*X(3)+X(6243), X(3)+2*X(9729), 7*X(3)-X(10625), 5*X(3)-X(13340), 5*X(3)-2*X(13348), 5*X(3)+4*X(15012), 4*X(3)-X(15644), 7*X(3)+2*X(16625), 2*X(52)-5*X(389), 3*X(52)-5*X(568), 11*X(52)-5*X(6243), X(52)-10*X(9729), X(52)-5*X(9730), 7*X(52)+5*X(10625), X(52)+2*X(13348), X(52)-4*X(15012), 4*X(52)+5*X(15644), 7*X(52)-10*X(16625), 3*X(389)-2*X(568), 11*X(389)-2*X(6243), X(389)-4*X(9729), 7*X(389)+2*X(10625), 5*X(389)+2*X(13340), 5*X(389)+4*X(13348), 5*X(389)-8*X(15012), 2*X(389)+X(15644), 7*X(389)-4*X(16625), 11*X(568)-3*X(6243), X(568)-6*X(9729), X(568)-3*X(9730), 7*X(568)+3*X(10625), 5*X(568)+3*X(13340), 5*X(568)+6*X(13348), 5*X(568)-12*X(15012), 4*X(568)+3*X(15644), 7*X(568)-6*X(16625), 5*X(15030)-3*X(15305), 5*X(15072)+3*X(15305)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27397.

X(16836) lies on these lines:
{2, 5656}, {3, 6}, {4, 373}, {5, 13474}, {20, 5640}, {30, 5892}, {51, 376}, {74, 7550}, {140, 5663}, {185, 631}, {381, 6688}, {1181, 6090}, {2854, 16270}, {2979, 14831}, {3292, 15032}, {3796, 11202}, {3917, 5890}, {5446, 12006}, {5447, 6102}, {5462, 12002}, {5562, 7998}, {5889, 15606}, {6800, 10282}, {8550, 9027}, {8681, 11179}, {9026, 12675}, {9826, 11807}, {11002, 15043}, {11381, 16261}, {13198, 13367}

X(16836) = midpoint of X(i) and X(j) for these {i,j}: {3, 9730}, {51, 376}, {185, 11459}, {381, 14855}, {2979, 14831}, {3917, 5890}
X(16836) = reflection of X(i) in X(j) for these (i,j): (381, 6688), (3819, 549)
X(16836) = complement of X(15030)
X(16836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 15072, 15030), (3, 52, 13348), (3, 182, 11430), (3, 389, 15644), (3, 567, 10564), (3, 9729, 389), (52, 15012, 389), (140, 10170, 15082), (185, 631, 11793), (185, 5650, 11459), (376, 15045, 51), (631, 11459, 5650), (5650, 11459, 11793), (5907, 15082, 10170), (9729, 13348, 15012), (13348, 15012, 52)


X(16837) =  X(53)X(566)∩X(252)X(571)

Barycentrics    (S^2-SB*(SB+2*R^2-2*SW))*(S^2- SC*(SC+2*R^2-2*SW)) : :

See Tran Quang Hung, César Lozada, and Peter Moses, Hyacinthos 27390 and Hyacinthos 27396.

X(16837) lies on the conic {{A,B,C,X(4),X(5)}} these lines:
{5, 3060}, {53, 566}, {184, 11816}, {252, 571}, {311, 7752}, {578, 1141}, {1614, 2980}, {5133, 8800}, {7577, 13450}, {12289, 15619}



X(16838) =  (name pending)

Trilinears    (b+c)*a^11+(9*b^2+4*b*c+9*c^2) *a^10-(b+c)*(13*b^2-71*b*c+13* c^2)*a^9-(37*b^4+37*c^4+8*b*c* (2*b^2-33*b*c+2*c^2))*a^8+(b+c )*(42*b^4+42*c^4-b*c*(202*b^2- 355*b*c+202*c^2))*a^7+(58*b^6+ 58*c^6+(14*b^4+14*c^4-b*c*(211 *b^2-152*b*c+211*c^2))*b*c)*a^ 6-(b+c)*(58*b^6+58*c^6-(180*b^ 4+180*c^4-b*c*(384*b^2-443*b* c+384*c^2))*b*c)*a^5-2*(21*b^ 8+21*c^8-(7*b^6+7*c^6-(120*b^ 4+120*c^4+b*c*(45*b^2-331*b*c+ 45*c^2))*b*c)*b*c)*a^4+(b^2-c^ 2)*(b-c)*(37*b^6+37*c^6+(36*b^ 4+36*c^4-b*c*(102*b^2+437*b*c+ 102*c^2))*b*c)*a^3+(b^2-c^2)^ 2*(13*b^6+13*c^6-2*(13*b^4+13* c^4-b*c*(73*b^2-34*b*c+73*c^2) )*b*c)*a^2-(b^2-c^2)^2*(b+c)*( 9*b^6+9*c^6+(11*b^4+11*c^4-b* c*(161*b^2-66*b*c+161*c^2))*b* c)*a-(b^2-c^2)^4*(b^2-14*b*c+ c^2)*(b^2+4*b*c+c^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27401.

X(16838) lies on this line: {4, 9782}


X(16839) =  PERSPECTOR OF THESE TRIANGLES: ABC AND 1st MORLEY-ADJUNCT-MIDPOINT

Barycentrics    sin 2A csc(2A/3 - π/3) : :
Barycentrics    Sin[A] (2 Sin[A]+Sec[C/3] Sin[B]+Sec[B/3] Sin[C]) : :

Let A'B'C' be the 1st Morley. Let Ba and Ca be points on BC such that A'BaCa is an equilateral triangle having the same orientation as ABC; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 1st Morley-adjunct-midpoint triangle, is perspective to ABC, with perspector X(16839). See X(3604). (Thanh Oai Dao, barycentrics by Peter Moses, March 30, 2018)

If you have GeoGebra, you can view X(16839).

X(16839) lies on these lines: {3,1135}, {3273,3603}

X(16839) = isogonal conjugate of X(16871)
X(16839) = barycentric product X(5456)*X(7309)


X(16840) =  PERSPECTOR OF THESE TRIANGLES: ABC AND 2nd MORLEY-ADJUNCT-MIDPOINT

Barycentrics    sin 2A csc(2A/3 + π/3) : :
Barycentrics    Sin[A] (2 Sin[A]+Csc[C/3-Pi/6] Sin[B]+Csc[B/3-Pi/6] Sin[C]) : :

Let A'B'C' be the 2nd Morley triangle. Let Ba and Ca be points on BC such that A'BaCa is an equilateral triangle having the same orientation as ABC; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'', here named the 2nd Morley-adnjunct-midpoint triangle, is perspective to ABC with perspector X(16840). See X(3602). (Thanh Oai Dao, barycentrics by Peter Moses, March 30, 2018)

If you have GeoGebra, you can view X(16840).

X(16840) lies on these lines: {3,358}, {3274,3602}

X(16840) = X(19)-isoconjugate of X(7309)
X(16840) = barycentric quotient X(3)/X(7309)


X(16841) =  PERSPECTOR OF THESE TRIANGLES: ABC AND 3rd MORLEY-ADJUNCT-MIDPOINT

Barycentrics    sin 2A csc(2A/3) : :
Barycentrics    Sin[A] (2 Sin[A]-Csc[C/3+Pi/6] Sin[B]-Csc[B/3+Pi/6] Sin[C]) : :

Let A'B'C' be the 3rd Morley triangle. Let Ba and Ca be points on BC such that A'BaCa is an equilateral triangle having the same orientation as ABC; define Cb and Ac cyclically, and define Ab and Bc cyclically. Let A'' be the midpoint of Ab and Ac, and define B'' and C'' cyclically. Then A''B''C'' here named the 3rd Morley-adjunct-midpoint triangle, is perspective to ABC, with perspector X(16841). See X(3603). (Thanh Oai Dao, barycentrics by Peter Moses, March 30, 2018)

If you have GeoGebra, you can view X(16841).

X(16841) lies on these lines: {3,1137}, {3275,3602}

X(16841) = X(19)-isoconjugate of X(5456)
X(16841) = barycentric quotient X(3)/X(5459)


X(16842) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(44)

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

X(16842) lies on these lines:
{1, 3697}, {2, 3}, {9, 5439}, {10, 3303}, {45, 3670}, {46, 15254}, {55, 3634}, {56, 3947}, {72, 7308}, {373, 5752}, {392, 3646}, {748, 5711}, {942, 3305}, {954, 1210}, {956, 1125}, {958, 3624}, {999, 5260}, {1001, 1698}, {1376, 5259}, {1470, 7294}, {1479, 3826}, {1617, 10588}, {1621, 9709}, {1697, 4002}, {1722, 6051}, {3219, 5708}, {3295, 5284}, {3338, 3848}, {3445, 16499}, {3555, 10582}, {3616, 9708}, {3617, 6767}, {3625, 8162}, {3715, 3874}, {3753, 7991}, {3817, 5584}, {3833, 5221}, {3841, 10896}, {3876, 15934}, {3916, 5437}, {3921, 6765}, {3984, 5044}, {4006, 16777}, {4413, 5248}, {4679, 12609}, {5007, 5275}, {5223, 10390}, {5436, 5440}, {5506, 5902}, {5927, 8726}, {6048, 16484}, {7772, 16589}, {9711, 10056}, {9844, 10383}, {10157, 10884}, {10914, 11530}, {11482, 15988}, {12045, 15489}, {13323, 16187}, {13887, 13947}, {13893, 13940}


X(16843) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(72)

Barycentrics    a (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c - 8 a^3 b^2 c - 14 a^2 b^3 c - 9 a b^4 c - 2 b^5 c - 8 a^3 b c^2 - 26 a^2 b^2 c^2 - 26 a b^3 c^2 - 8 b^4 c^2 - 14 a^2 b c^3 - 26 a b^2 c^3 - 12 b^3 c^3 - a^2 c^4 - 9 a b c^4 - 8 b^2 c^4 - a c^5 - 2 b c^5) : :

X(16843) lies on these lines:
{2, 3}, {6, 975}, {1724, 7308}, {5778, 13323}


X(16844) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(213)

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

X(16844) lies on these lines:
{1, 4042}, {2, 3}, {6, 1125}, {55, 16828}, {58, 15668}, {975, 4698}, {1213, 2271}, {1724, 3624}, {3159, 16675}, {3290, 5283}, {3333, 16552}, {3487, 5296}, {3616, 5278}, {3646, 10476}, {4281, 5737}, {5044, 10477}, {5788, 6176}, {10165, 15486}


X(16845) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(220)

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

X(16845) lies on these lines:
{1, 2318}, {2, 3}, {9, 1125}, {10, 3158}, {58, 4648}, {72, 3616}, {144, 6147}, {226, 3361}, {329, 5550}, {345, 16817}, {388, 5251}, {392, 10595}, {480, 12260}, {497, 5259}, {551, 11523}, {936, 6666}, {938, 5791}, {942, 5273}, {943, 1001}, {946, 5759}, {950, 1698}, {954, 14986}, {958, 1056}, {966, 4251}, {1213, 4258}, {1490, 5817}, {1621, 5082}, {1708, 3485}, {1724, 5712}, {2550, 5248}, {2551, 8164}, {3086, 4423}, {3295, 12632}, {3419, 9780}, {3421, 5260}, {3486, 10395}, {3579, 11024}, {3618, 10477}, {3683, 4295}, {3876, 14054}, {3916, 9776}, {3925, 4294}, {3927, 11036}, {4428, 9710}, {4512, 6361}, {4999, 8167}, {5022, 5746}, {5044, 5703}, {5265, 8232}, {5281, 9709}, {5284, 10527}, {5439, 5744}, {5657, 6769}, {5698, 12609}, {5714, 12572}, {5758, 5886}, {5812, 11230}, {5837, 11041}, {7308, 10396}, {7735, 16589}, {9623, 12640}, {12053, 12864}, {14482, 16020}


X(16846) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(238)

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

X(16846) lies on these lines:
{2, 3}, {386, 16589}, {3846, 8167}


X(16847) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(392)

Barycentrics    a (a^6 + a^5 b - a^2 b^4 - a b^5 + a^5 c + 4 a^4 b c - 8 a^3 b^2 c - 18 a^2 b^3 c - 9 a b^4 c - 2 b^5 c - 8 a^3 b c^2 - 26 a^2 b^2 c^2 - 26 a b^3 c^2 - 8 b^4 c^2 - 18 a^2 b c^3 - 26 a b^2 c^3 - 12 b^3 c^3 - a^2 c^4 - 9 a b c^4 - 8 b^2 c^4 - a c^5 - 2 b c^5) : :

X(16847) lies on these lines:


X(16848) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(405)

Barycentrics    a (a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + a^4 b c - 4 a^3 b^2 c - 8 a^2 b^3 c - 5 a b^4 c - b^5 c + a^4 c^2 - 4 a^3 b c^2 - 14 a^2 b^2 c^2 - 13 a b^3 c^2 - 4 b^4 c^2 - a^3 c^3 - 8 a^2 b c^3 - 13 a b^2 c^3 - 6 b^3 c^3 - a^2 c^4 - 5 a b c^4 - 4 b^2 c^4 - b c^5) : :

X(16848) lies on these lines:
{2, 3}, {37, 386}, {387, 9708}, {2218, 5259}, {3216, 7308}, {3695, 9534}, {5722, 10479}, {10449, 12433}


X(16849) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(518)

Barycentrics    a (a^5 - a b^4 - 2 a^3 b c - 6 a^2 b^2 c - 6 a b^3 c - 2 b^4 c - 6 a^2 b c^2 - 14 a b^2 c^2 - 6 b^3 c^2 - 6 a b c^3 - 6 b^2 c^3 - a c^4 - 2 b c^4) : :

X(16849) lies on these lines:
{2, 3}, {58, 5275}, {218, 5044}, {956, 16830}, {965, 5138}, {1001, 3739}, {5247, 5268}


X(16850) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(1001)

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

X(16850) lies on these lines:
{2, 3}, {9, 4260}, {141, 1001}, {386, 2238}, {992, 3736}, {1213, 5132}, {1245, 6051}, {3789, 3811}, {4388, 5284}, {5044, 16601}, {5248, 8299}


X(16851) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(1279)

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

X(16851) lies on these lines:
{2, 3}, {936, 16601}, {954, 4357}, {4255, 5283}


X(16852) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(1386)

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

X(16852) lies on these lines:
{2, 3}, {386, 5275}, {965, 4260}, {975, 3290}, {5272, 16478}, {5687, 16830}


X(16853) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(1743)

Barycentrics    a (a^3 - a b^2 - 8 a b c - 8 b^2 c - a c^2 - 8 b c^2) : :

X(16853) lies on these lines:
{1, 3711}, {2, 3}, {9, 5708}, {10, 6767}, {84, 10156}, {392, 8148}, {517, 3646}, {942, 7308}, {954, 5704}, {956, 5550}, {960, 1159}, {970, 10219}, {999, 3624}, {1001, 3634}, {1125, 7373}, {1698, 3295}, {3305, 3927}, {3697, 4666}, {3828, 3913}, {3925, 9669}, {4413, 5259}, {4668, 8162}, {5044, 11523}, {5284, 5687}, {5316, 11374}, {5534, 8583}, {5584, 7988}, {5752, 6688}, {5779, 9940}, {5791, 6666}, {5886, 8158}, {7989, 8273}, {8726, 10157}, {9605, 16589}, {10172, 11500}, {10306, 11231}, {11227, 12684}, {11495, 12571}


X(16854) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16669)

Barycentrics    a (a^3 - a b^2 - 10 a b c - 10 b^2 c - a c^2 - 10 b c^2) : :

X(16854) lies on these lines:
{2, 3}, {392, 11531}, {956, 3624}, {1437, 16187}, {1698, 3913}, {3303, 3828}, {3555, 4866}, {3634, 4423}, {3646, 3753}, {3697, 10582}, {4383, 4658}, {4691, 8162}, {5044, 11520}, {5284, 9709}, {5439, 7308}, {5550, 9708}, {5584, 10171}


X(16855) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16670)

Barycentrics    a (a^3 - a b^2 - 12 a b c - 12 b^2 c - a c^2 - 12 b c^2) : :

X(16855) lies on these lines:
{2, 3}, {970, 12045}, {1698, 3303}, {3295, 3634}, {3304, 3624}, {3305, 5708}, {3646, 7991}, {3746, 4423}, {3826, 9669}, {3927, 7308}, {3951, 5439}, {3984, 15934}, {5044, 11518}, {5221, 5506}, {5550, 7373}, {6767, 9780}, {8583, 15178}, {9957, 11530}


X(16856) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16671)

Barycentrics    a (a^3 - a b^2 - 14 a b c - 14 b^2 c - a c^2 - 14 b c^2) : :

X(16856) lies on these lines:
{2, 3}, {10, 8162}, {3624, 12513}, {4423, 8715}, {5687, 8167}


X(16857) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(3731)

Barycentrics    a (3 a^3 - 3 a b^2 - 8 a b c - 8 b^2 c - 3 a c^2 - 8 b c^2) : :

X(16857) lies on these lines:
{1, 3715}, {2, 3}, {9, 15934}, {10, 4428}, {392, 5644}, {495, 8171}, {519, 1001}, {551, 958}, {942, 3929}, {954, 15933}, {956, 5284}, {960, 4930}, {993, 8167}, {999, 4423}, {1125, 11194}, {1159, 15254}, {1385, 3646}, {1617, 11237}, {3241, 5260}, {3295, 3679}, {3303, 4677}, {3305, 3940}, {3576, 10157}, {3656, 8158}, {3828, 4421}, {3913, 4745}, {3925, 9668}, {3928, 5708}, {5044, 5436}, {5045, 5234}, {8726, 12684}


X(16858) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16674)

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

X(16858) lies on these lines:
{1, 4127}, {2, 3}, {8, 4428}, {35, 3828}, {104, 3653}, {519, 1621}, {551, 2975}, {958, 3241}, {993, 5284}, {1001, 6172}, {3616, 11194}, {3624, 5303}, {3679, 3871}, {3746, 4669}, {3868, 3929}, {3877, 16200}, {4511, 15254}, {4720, 5278}, {4980, 7283}, {5008, 5276}, {5010, 9342}, {5250, 11531}, {5426, 10176}


X(16859) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16675)

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

X(16859) lies on these lines:
{1, 4134}, {2, 3}, {8, 5259}, {9, 11520}, {145, 1001}, {392, 5645}, {958, 3622}, {970, 11451}, {976, 9330}, {993, 5550}, {1392, 7161}, {1621, 3617}, {1724, 4658}, {2975, 4423}, {3295, 4678}, {3305, 5436}, {3616, 5251}, {3621, 9708}, {3869, 15254}, {3870, 4866}, {3873, 5302}, {3877, 11278}, {3924, 11533}, {4666, 5234}, {5008, 16589}, {5041, 5283}, {5102, 15988}, {5217, 9342}, {5248, 9780}, {5253, 8167}, {10248, 12511}, {10459, 15485}


X(16860) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16676)

Barycentrics    a (5 a^3 - 5 a b^2 - 12 a b c - 12 b^2 c - 5 a c^2 - 12 b c^2) : :

X(16860) lies on these lines:
{2, 3}, {958, 3636}, {999, 15808}, {1001, 3244}, {3295, 3626}, {3303, 3632}, {3304, 5251}, {3927, 11518}, {3940, 5436}, {3951, 15934}, {4423, 5563}, {5260, 6767}, {5284, 7373}


X(16861) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16677)

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

X(16861) lies on these lines:
{1, 3988}, {2, 3}, {100, 3828}, {519, 5259}, {551, 5251}, {1001, 3241}, {1621, 3679}, {3746, 4745}, {3871, 4428}, {3876, 5436}, {3877, 11224}, {3889, 5234}, {4421, 9780}, {4423, 11194}, {4980, 16817}, {5434, 7677}, {6666, 9963}


X(16862) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16666)

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

X(16862) lies on these lines:
{1, 3848}, {2, 3}, {10, 3304}, {35, 8167}, {56, 3634}, {57, 15650}, {72, 5437}, {392, 7991}, {499, 3826}, {936, 5439}, {942, 3984}, {956, 1698}, {970, 15082}, {975, 16610}, {999, 9780}, {1125, 3303}, {1376, 3624}, {2098, 3918}, {2932, 6667}, {3295, 5550}, {3306, 3951}, {3333, 3697}, {3337, 5220}, {3338, 3740}, {3419, 9843}, {3555, 8580}, {3616, 9342}, {3617, 7373}, {3678, 4860}, {3711, 3881}, {3753, 7982}, {3812, 5730}, {3876, 5708}, {3916, 7308}, {3921, 6762}, {3925, 10200}, {4004, 15829}, {5221, 10176}, {5234, 8169}, {5253, 9708}, {5275, 7772}, {5316, 12436}, {5650, 5752}, {9710, 10072}, {10156, 10884}


X(16863) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16667)

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

X(16863) lies on these lines:
{1, 4731}, {2, 3}, {10, 7373}, {56, 5726}, {936, 15934}, {956, 5828}, {975, 16602}, {999, 1698}, {1125, 3913}, {1159, 3812}, {1482, 8583}, {1490, 10156}, {3295, 3624}, {3306, 3927}, {3333, 4866}, {3337, 3715}, {3579, 3646}, {3634, 8666}, {3753, 8148}, {3811, 3848}, {3826, 10200}, {3828, 12513}, {3833, 12635}, {3940, 5439}, {3968, 10912}, {5024, 16589}, {5044, 5437}, {5045, 8580}, {5550, 5687}, {5780, 10202}, {5790, 8582}, {5791, 6692}, {6244, 8227}, {10157, 12684}, {10172, 12114}, {10306, 11230}, {12433, 12536}


X(16864) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16668)

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

X(16864) lies on these lines:
{2, 3}, {956, 3634}, {1698, 12513}, {3304, 3828}, {3306, 15650}, {3624, 5687}, {3753, 11531}, {4413, 8715}, {4533, 10980}, {5041, 5275}, {5439, 11523}, {5550, 9709}, {8583, 16200}


X(16865) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16672)

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

X(16865) lies on these lines:
{1, 2308}, {2, 3}, {8, 3746}, {9, 3984}, {35, 9780}, {36, 5550}, {55, 3617}, {56, 5284}, {58, 14996}, {63, 5436}, {110, 13323}, {144, 1001}, {145, 958}, {238, 10448}, {373, 15489}, {386, 14997}, {392, 3897}, {846, 3924}, {940, 16948}, {956, 3623}, {970, 5640}, {988, 7292}, {993, 3616}, {1043, 5278}, {1201, 5145}, {1385, 12528}, {1420, 8545}, {1724, 4653}, {2320, 3467}, {2646, 10394}, {2650, 7262}, {3241, 5258}, {3295, 3621}, {3305, 3601}, {3619, 4265}, {3624, 5267}, {3634, 5010}, {3647, 5902}, {3681, 5302}, {3683, 3869}, {3826, 15338}, {3870, 5234}, {3871, 4678}, {3877, 10222}, {3929, 11520}, {4423, 5253}, {4512, 7991}, {4855, 7308}, {4881, 8583}, {4995, 9711}, {5007, 5283}, {5080, 10198}, {5172, 10588}, {5204, 8167}, {5250, 7982}, {5273, 12649}, {5293, 9330}, {5361, 10449}, {5450, 6223}, {5752, 11002}, {6690, 11681}, {8616, 10459}, {11477, 15988}


X(16866) =  (X(1),X(6),X(2),X(75); X(3),X(2),X(6),X(1)) COLLINEATION IMAGE OF X(16673)

Barycentrics    a (5 a^3 - 5 a b^2 - 8 a b c - 8 b^2 c - 5 a c^2 - 8 b c^2) : :

X(16866) lies on these lines:
{2, 3}, {958, 3244}, {993, 15808}, {999, 5259}, {1001, 3636}, {1159, 12514}, {1384, 16589}, {1385, 5779}, {3295, 3632}, {3626, 3913}, {3646, 13624}, {3927, 11520}, {4512, 12702}, {5250, 8148}, {5267, 8167}, {5273, 12433}, {5436, 15934}, {6259, 10165}


X(16867) =  X(4)X(49)∩X(54)X(15053)

Barycentrics    (S^2-SB*SC)*(SB-5*R^2+SW)*(SC+ SW-5*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27406.

X(16867) lies on the Jerabek hyperbola and these lines:
{4, 49}, {54, 15053}, {68, 6640}, {74, 11250}, {182, 13622}, {184, 3521}, {265, 1147}, {631, 13418}, {1092, 3519}, {1173, 12106}, {10224, 16000}, {10937, 13198}, {10984, 13623}, {11744, 13352}

X(16867) = isogonal conjugate of X(16868)


X(16868) =  ISOGONAL CONJUGATE OF X(16867)

Barycentrics    ((b^2+c^2)*a^4-(2*b^4-b^2*c^2+ 2*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(16868) = X(4) + 2*X(10018) = X(4) - 4*X(10019)

As a point on the Euler line, X(16868) center has Shinagawa coefficients (-4*F, E-4*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27406.

X(16868) lies on these lines:
{2, 3}, {93, 1093}, {110, 9927}, {112, 7746}, {113, 5449}, {125, 6241}, {156, 265}, {264, 15424}, {281, 15833}, {847, 6344}, {1112, 11591}, {1209, 15056}, {1249, 9722}, {1514, 6696}, {1568, 11412}, {1614, 13198}, {1625, 8571}, {1870, 7741}, {1986, 5876}, {2777, 11468}, {2888, 2904}, {3043, 14643}, {3527, 13418}, {3574, 6242}, {5448, 5889}, {5475, 10312}, {5476, 8537}, {5895, 12244}, {6198, 7951}, {6530, 13481}, {6564, 10881}, {6565, 10880}, {6746, 13364}, {7687, 11464}, {8743, 13881}, {9220, 11062}, {9545, 12370}, {9707, 12254}, {10282, 12289}, {10632, 16809}, {10633, 16808}, {11178, 11470}, {11441, 14852}, {11456, 15081}, {12112, 14216}, {12131, 15092}, {12133, 15088}, {12292, 13491}, {12293, 12383}, {12300, 13565}

X(16868) = isogonal conjugate of X(16867)
X(16868) = orthocentroidal-circle-inverse-of X(3520)
X(16868) = polar-circle-inverse-of X(15646)
X(16868) = X(10255)-of-Johnson-triangle
X(16868) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 3520), (4, 5, 7577), (4, 3541, 13596), (4, 3542, 3518), (4, 6143, 378), (4, 7505, 186), (4, 13619, 382), (4, 14940, 3), (5, 10024, 2), (5, 11563, 10224), (5, 15761, 2072), (468, 546, 6240), (1596, 15559, 4), (2072, 15761, 20), (10224, 11563, 382), (11799, 13371, 3146)


X(16869) =  X(1)X(4)∩X(124)X(519)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27409.

X(16869) lies on these lines:
{1, 4}, {124, 519}, {496, 4906}, {522, 8062}, {1324, 1486}, {1394, 16127}, {1854, 12616}, {6001, 15252}, {6735, 15633}

X(16869)-= incircle-inverse-of X(1479)
X(16869) = polar-circle-inverse-of X(1068)


X(16870) =  X(1)X(4)∩X(10)X(1854)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27409.

X(16870) lies on the cubic K806 and these lines:
{1, 4}, {10, 1854}, {55, 4656}, {118, 10017}, {142, 9817}, {208, 16389}, {329, 7070}, {522, 650}, {527, 1936}, {908, 3100}, {971, 15252}, {972, 2222}, {1040, 3452}, {1394, 6223}, {1736, 3911}, {2310, 3011}, {3021, 3326}, {3660, 11028}, {3710, 5423}, {3731, 5218}, {6700, 7515}, {7037, 8806}, {8055, 10538}, {10306, 12912}

X(16870) = incircle-inverse-of X(497)
X(16870) = polar circle-inverse-of X(278)
X(16870) = {X(1490), X(7952)}-harmonic conjugate of X(5930)


X(16871) =  ISOGONAL CONJUGATE OF X(16839)

Barycentrics    sin(2A/3 - π/3) : :

X(16871) = isogonal conjugate of X(16839)


X(16872) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16872) lies on these lines:
{21, 5263}, {31, 48}, {160, 1631}, {197, 1011}, {198, 2110}, {1444, 16678}, {2148, 14573}, {2183, 2309}, {2268, 15624}, {2933, 16452}, {16679, 16691}


X(16873) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(163), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16873) lies on these lines: {157, 1602}, {2936, 16681}


X(16874) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(523), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16874) lies on these lines:
{31, 649}, {55, 7234}, {512, 5006}, {513, 5078}, {667, 4790}, {669, 2106}, {1019, 16877}, {3566, 4367}, {4979, 8635}, {7659, 8641}


X(16875) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(7793), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16875) lies on these lines: {1, 2916}, {16690, 16877}


X(16876) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(1914), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16876) lies on these lines:
{2, 3}, {86, 1486}, {99, 2862}, {105, 16752}, {814, 7255}, {1444, 1602}, {1473, 5208}, {1621, 16705}, {3736, 7295}, {3747, 16877}, {8053, 16682}, {16678, 16681}


X(16877) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(385), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16877) lies on these lines:
{1, 3}, {1019, 16874}, {1580, 3286}, {1959, 3724}, {3747, 16876}, {16682, 16685}, {16690, 16875}


X(16878) =  (A,B,C,X(2),A',B',C',X(1)) COLLINEATION IMAGE OF X(391), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

X(16878) lies on these lines:
{1, 3}, {31, 1412}, {108, 5307}, {669, 7203}, {1014, 1621}, {1284, 4654}, {1400, 4270}, {1468, 2258}, {3286, 4512}, {3886, 13588}, {5248, 5323}, {7175, 16690}


X(16879) =  X(4)X(14831)∩X(6)X(1204)

Barycentrics    (SB+SC)*(2*S^2*(SA+SW-6*R^2)+S A*(S^2+SB*SC))*(S^2-2*SA^2)*SB *SC : :
X(16879) = X(15077) - 4*X(16625)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27413.

X(16879) lies on these lines:
{4, 14831}, {6, 1204}, {155, 3517}, {3515, 14531}, {3574, 13567}, {13148, 13202}


X(16880) =  X(6)X(3520)∩X(24)X(2914)

Barycentrics    (SB+SC)*((-6*R^2+2*SA+SW)*S^2+ SA*SB*SC)*(3*S^2-5*SA^2)*SB*SC : :
X(16880) = 5*X(3567) - 3*X(11704)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27413.

X(16880) lies on these lines:
{6, 3520}, {24, 2914}, {52, 12278}, {113, 5889}, {155, 3518}, {186, 2904}, {378, 11999}, {568, 13561}, {1986, 10263}, {3567, 3574}, {3575, 15110}, {6242, 13431}, {7722, 12290}


X(16881) =  MIDPOINT OF X(52) AND X(140)

Barycentrics    (S^2+10*R^2*SA-3*SA^2+4*SB*SC) *(SB+SC) : :
X(16881) = X(4)-9*X(13321), X(4)-3*X(13451), X(5)+3*X(568), X(5)-5*X(3567), 5*X(5)-9*X(5640), 3*X(5)+X(5889), 7*X(5)-3*X(11459), 11*X(5)-7*X(15056), 3*X(568)+5*X(3567), 5*X(568)+3*X(5640), 9*X(568)-X(5889), 7*X(568)+X(11459), 25*X(3567)-9*X(5640), 15*X(3567)+X(5889), 21*X(5640)-5*X(11459), 7*X(5889)+9*X(11459), 3*X(13321)-X(13451)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27413.

X(16881) lies on these lines:
{3, 14449}, {4, 13321}, {5, 568}, {6, 1658}, {26, 11432}, {30, 143}, {51, 546}, {52, 140}, {54, 7575}, {185, 3853}, {186, 14627}, {382, 11002}, {468, 15806}, {511, 3530}, {547, 5562}, {548, 9730}, {549, 6243}, {550, 3060}, {578, 15331}, {632, 11412}, {973, 12236}, {1112, 13488}, {1154, 3628}, {1199, 2070}, {1216, 13363}, {1986, 11801}, {2979, 14869}, {3518, 15087}, {3581, 13434}, {3627, 5890}, {3845, 9781}, {3850, 10095}, {3858, 12111}, {3859, 15030}, {3861, 5663}, {5066, 5876}, {5097, 12038}, {5447, 11812}, {5449, 13413}, {5891, 12812}, {5892, 10627}, {5907, 12811}, {5943, 11591}, {5944, 13366}, {6000, 12102}, {6241, 15687}, {6746, 6756}, {7512, 15037}, {7526, 9777}, {7542, 8254}, {7550, 12307}, {9729, 13391}, {9786, 11250}, {10109, 14128}, {10124, 11695}, {10224, 13567}, {10226, 11438}, {10264, 15559}, {10272, 16222}, {10574, 15704}, {10625, 12100}, {11225, 11264}, {11245, 11819}, {11381, 12101}, {11444, 15699}, {11539, 15028}, {11557, 13358}, {11561, 11800}, {12106, 12161}, {12233, 13406}, {13348, 14891}, {13421, 15644}, {14531, 15067}, {15003, 16194}, {15045, 15712}

X(16881) = midpoint of X(i) and X(j) for these {i,j}: {3, 14449}, {52, 140}, {185, 3853}, {548, 10263}, {1986, 11801}, {5066, 14831}, {11557, 13358}, {11561, 11800}, {13421, 15644}
X(16881) = reflection of X(i) in X(j) for these (i,j): (1216, 16239), (3850, 10095), (3861, 10110), (5907, 12811)
X(16881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 6102, 546), (52, 5946, 140), (143, 13630, 5446), (389, 5446, 13630), (568, 3567, 5), (1216, 13363, 16239), (5562, 15026, 547), (5892, 10627, 12108), (5907, 13364, 12811), (6243, 15043, 549), (9730, 10263, 548)


X(16882) =  ISOGONAL CONJUGATE OF X(8444)

Barycentrics    a*(-2*sqrt(3)*(a^9-2*(b+c)*a^8 -(b^2+c^2)*a^7+(b+c)*(5*b^2-2* b*c+5*c^2)*a^6-3*(b^4+3*b^2*c^ 2+c^4)*a^5-(b+c)*(b^2+b*c+c^2) *(3*b^2-4*b*c+3*c^2)*a^4+(b^2+ c^2)*(5*b^2+6*b*c+5*c^2)*(b-c) ^2*a^3-(b^4-c^4)*(b^2+c^2)*(b- c)*a^2-(b^2-c^2)^2*(2*b^4+2*c^ 4-b*c*(4*b^2+3*b*c+4*c^2))*a+( b^4-c^4)*(b^2-c^2)*(b^3+c^3))* S+a^11+2*(b+c)*a^10-8*(b^2+c^2 )*a^9-(b+c)*(b^2+6*b*c+c^2)*a^ 8+(16*b^4+16*c^4-b*c*(4*b^2-21 *b*c+4*c^2))*a^7-(b+c)*(10*b^4 +10*c^4-b*c*(19*b^2-12*b*c+19* c^2))*a^6-(10*b^6+10*c^6-(12*b ^4+12*c^4+b*c*(3*b^2+8*b*c+3*c ^2))*b*c)*a^5+(b^2-c^2)*(b-c)* (16*b^4+16*c^4+b*c*(11*b^2+18* b*c+11*c^2))*a^4-(b^6+c^6+2*( 7*b^4+7*c^4+b*c*(20*b^2+21*b* c+20*c^2))*b*c)*(b-c)^2*a^3-( b^2-c^2)^2*(b+c)*(8*b^4+8*c^4- 3*b*c*(3*b^2-4*b*c+3*c^2))*a^ 2+(b^2-c^2)^2*(2*b^6+2*c^6+(4* b^4+4*c^4+b*c*(b^2-20*b*c+c^2) )*b*c)*a+(b^3+c^3)*(b^2-c^2)^ 4) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27419.

X(16882) lies on the cubic K001 (the Neuberg cubic) and these lines:
{1, 8174}, {3, 1276}, {13, 3465}, {16, 3483}, {1263, 5673}, {1277, 1337}, {3482,16883}, {3484, 7059}, {5624, 7329}, {5667, 7326}, {8173, 8480}

X(16882) = isogonal conjugate of X(8444)


X(16883) =  ISOGONAL CONJUGATE OF X(8454)

Barycentrics    a*(2*sqrt(3)*(a^9-2*(b+c)*a^8- (b^2+c^2)*a^7+(b+c)*(5*b^2-2*b *c+5*c^2)*a^6-3*(b^4+3*b^2*c^2 +c^4)*a^5-(b+c)*(b^2+b*c+c^2)* (3*b^2-4*b*c+3*c^2)*a^4+(b^2+c ^2)*(5*b^2+6*b*c+5*c^2)*(b-c)^ 2*a^3-(b^4-c^4)*(b^2+c^2)*(b-c )*a^2-(b^2-c^2)^2*(2*b^4+2*c^4 -b*c*(4*b^2+3*b*c+4*c^2))*a+(b ^4-c^4)*(b^2-c^2)*(b^3+c^3))*S +a^11+2*(b+c)*a^10-8*(b^2+c^2) *a^9-(b+c)*(b^2+6*b*c+c^2)*a^8 +(16*b^4+16*c^4-b*c*(4*b^2-21* b*c+4*c^2))*a^7-(b+c)*(10*b^4+ 10*c^4-b*c*(19*b^2-12*b*c+19*c ^2))*a^6-(10*b^6+10*c^6-(12*b^ 4+12*c^4+b*c*(3*b^2+8*b*c+3*c^ 2))*b*c)*a^5+(b^2-c^2)*(b-c)*( 16*b^4+16*c^4+b*c*(11*b^2+18*b *c+11*c^2))*a^4-(b^6+c^6+2*(7* b^4+7*c^4+b*c*(20*b^2+21*b*c+ 20*c^2))*b*c)*(b-c)^2*a^3-(b^ 2-c^2)^2*(b+c)*(8*b^4+8*c^4-3* b*c*(3*b^2-4*b*c+3*c^2))*a^2+( b^2-c^2)^2*(2*b^6+2*c^6+(4*b^ 4+4*c^4+b*c*(b^2-20*b*c+c^2))* b*c)*a+(b^3+c^3)*(b^2-c^2)^4) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27419.

X(16883) lies on the cubic K001 (the Neuberg cubic) and these lines:
{1, 8175}, {3, 1277}, {14, 3465}, {15, 3483}, {1263, 5672}, {1276, 1338}, {3065, 5669}, {3482,16882}, {3484, 7060}, {5623, 7329}, {5667, 7325}, {8172, 8480}

X(16883) = isogonal conjugate of X(8454)


X(16884) =  (name pending)

Trilinears    3 a + 2 b + 2 c : :
Barycentrics    a (3 a + 2 b + 2 c) : :

X(16884) lies on these lines:
{1, 6}, {2, 4445}, {10, 4545}, {48, 4289}, {55, 4497}, {56, 1030}, {71, 5043}, {86, 4361}, {142, 4909}, {145, 594}, {193, 4364}, {239, 4751}, {284, 15934}, {344, 597}, {551, 3686}, {572, 1482}, {573, 10246}, {595, 5115}, {599, 3879}, {604, 2099}, {894, 3644}, {940, 4850}, {966, 3622}, {988, 5023}, {995, 4272}, {1015, 4261}, {1086, 3945}, {1213, 3616}, {1278, 4360}, {1333, 2241}, {1388, 1400}, {1392, 2298}, {1442, 5228}, {1474, 11396}, {1500, 5069}, {1696, 3196}, {1766, 10222}, {1901, 3488}, {2098, 2268}, {2174, 2280}, {2178, 3304}, {2220, 2242}, {2262, 3207}, {2269, 5036}, {2285, 11011}, {2321, 3635}, {2345, 3241}, {2667, 7032}, {3244, 4058}, {3624, 4034}, {3636, 4856}, {3713, 4861}, {3739, 16834}, {3759, 16826}, {3763, 4851}, {3875, 4670}, {3943, 5749}, {3946, 4675}, {3986, 4700}, {4021, 4667}, {4038, 4386}, {4254, 7373}, {4258, 5045}, {4328, 6610}, {4383, 17019}, {4416, 15534}, {4419, 7277}, {4643, 6144}, {4648, 17014}, {4653, 9346}, {4739, 4852}, {4798, 4967}, {486 8, 5110}, {5013, 5266}, {5120, 6767}, {5256, 16610}, {5275, 7191}, {5276, 17024}, {5393, 8252}, {5405, 8253}, {5625, 16825}, {5747, 12433}, {5816, 5901}, {5949, 11680}, {6155, 16553}, {6180, 7269}, {6748, 7952}, {8071, 8553}, {10445, 13607}


X(16885) =  (name pending)

Trilinears    -3 a + 2 b + 2 c : :
Barycentrics    a (-3 a + 2 b + 2 c) : :

X(16885) lies on these lines:
{1, 6}, {2, 7232}, {19, 1878}, {31, 3715}, {63, 16610}, {69, 4422}, {100, 8696}, {101, 4268}, {144, 1086}, {169, 5341}, {190, 1278}, {198, 2265}, {210, 3052}, {239, 3644}, {281, 6748}, {344, 524}, {346, 4370}, {391, 594}, {572, 3204}, {573, 12034}, {599, 4416}, {894, 4751}, {896, 4413}, {902, 3711}, {1213, 5749}, {1376, 7262}, {1400, 5043}, {1707, 3740}, {1766, 7297}, {2174, 2267}, {2183, 5036}, {2261, 3207}, {2268, 4289}, {2287, 3285}, {2305, 5783}, {2321, 4701}, {2325, 4072}, {3161, 3943}, {3219, 4383}, {3305, 4641}, {3618, 4364}, {3686, 4058}, {3707, 4691}, {3729, 4726}, {3730, 4271}, {3752, 3929}, {3758, 15668}, {3763, 4643}, {3879, 15534}, {3923, 4732}, {3928, 16602}, {3950, 4700}, {4000, 6172}, {4029, 4856}, {4252, 5044}, {4273, 4877}, {4363, 4699}, {4373, 4409}, {4384, 4739}, {4512, 4849}, {4648, 7277}, {4675, 6666}, {4681, 16834}, {4686, 16833}, {4709, 5695}, {4851, 6144}, {5779, 13329}, {7074, 7082}, {7075, 14470}, {9355, 11495}, {9441, 16112}, {10645, 11790 }, {10646, 11791}, {12329, 16686}

X(16885) = {X(6),X(9)}-harmonic conjugate of X(45)


X(16886) =  (name pending)

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

X(16886) lies on these lines:
{2, 4372}, {10, 213}, {12, 594}, {32, 4680}, {37, 5051}, {115, 1089}, {172, 7270}, {315, 4376}, {319, 4400}, {325, 16720}, {345, 9598}, {626, 1930}, {1107, 3006}, {1111, 7794}, {1914, 5015}, {2241, 4894}, {2275, 3705}, {2345, 9596}, {2887, 3721}, {3061, 7239}, {3136, 15523}, {3454, 3954}, {3695, 4037}, {3703, 5254}, {3959, 4165}, {4056, 7818}, {4386, 5300}, {4426, 5016}, {7206, 7230}


X(16887) =  (name pending)

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

X(16887) lies on these lines:
{1, 16705}, {2, 2350}, {7, 10455}, {10, 274}, {38, 1930}, {39, 141}, {58, 86}, {69, 386}, {75, 596}, {81, 5299}, {150, 5484}, {257, 514}, {307, 1014}, {310, 3741}, {314, 3663}, {325, 3454}, {333, 3008}, {348, 4306}, {519, 16712}, {982, 16739}, {1010, 1434}, {1015, 16744}, {1211, 16700}, {1401, 3665}, {1444, 4278}, {1475, 16818}, {2140, 3662}, {2292, 14210}, {3679, 16711}, {3831, 6381}, {3954, 4568}, {4251, 16060}, {4352, 10449}, {4384, 16752}, {4847, 16750}, {4920, 12263}, {5021, 15668}, {5030, 16061}, {5224, 16709}, {5743, 16736}, {6734, 16749}, {13725, 14548}, {16604, 16726}


X(16888) =  (name pending)

Barycentrics    (b^3 + c^3)/(b + c - a) : :

X(16888) lies on these lines:
{7, 604}, {37, 226}, {57, 1760}, {65, 4085}, {307, 16609}, {1441, 16603}, {3061, 3662}, {3485, 16484}, {6063, 6374}


X(16889) =  (name pending)

Barycentrics    (b^3 + c^3)/(b^2 + c^2) : :

X(16889) lies on these lines:
{2, 16683}, {12, 1284}, {2295, 4972}, {3777, 7187}


X(16890) =  (name pending)

Barycentrics    (b^4 + c^4)/(b^2 + c^2): :

X(16890) lies on these lines:
{2, 3613}, {76, 6664}, {83, 316}, {141, 308}, {230, 1799}, {251, 7792}, {264, 10549}, {315, 16285}, {458, 10550}, {523, 9229}, {626, 710}, {6375, 7853}


X(16891) =  (name pending)

Barycentrics    (b^4 + c^4)/(b + c): :

X(16891) lies on these lines:
{86, 2206}, {310, 3120}, {626, 3118}, {2887, 16703}


X(16892) =  (name pending)

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

X(16892) lies on these lines:
{239, 514}, {321, 693}, {522, 4382}, {523, 2254}, {650, 3752}, {661, 918}, {690, 8663}, {768, 3261}, {812, 4467}, {826, 2474}, {1491, 4088}, {2522, 3669}, {3676, 4379}, {3700, 4728}, {3835, 4120}, {3837, 4122}, {4367, 8635}, {4369, 4453}, {4378, 8646}, {4380, 4984}, {4468, 4893}, {4686, 4777}, {4885, 14475}, {4897, 4979}, {4927, 4931}, {4976, 6084}, {10914, 14077}

X(16892) = isogonal conjugate of X(4628)
X(16892) = barycentric product X(10)*X(1086)
X(16892) = barycentric product X(141)*X(514)


X(16893) =  (name pending)

Barycentrics    (b^4 + c^4) (b^2 + c^2) : :

X(16893) lies on these lines:
{2, 6}, {51, 7821}, {626, 3118}, {1369, 2076}, {5207, 10328}, {7801, 11550}


X(16894) =  (name pending)

Barycentrics    (b + c)(b^4 + c^4) : :

X(16894) lies on these lines:
{2, 4412}, {10, 82}, {313, 1934}, {560, 4769}, {626, 4118}, {3613, 15523}


X(16895) =  (name pending)

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

X(16895) lies on these lines:
{2, 3}, {76, 5355}, {83, 3314}, {115, 7943}, {141, 7787}, {183, 10583}, {194, 3589}, {316, 7914}, {1506, 7930}, {1613, 10339}, {2548, 7931}, {2896, 3763}, {3096, 7804}, {3329, 7795}, {3618, 7839}, {3734, 7859}, {3815, 7945}, {3934, 7806}, {3972, 6292}, {5475, 7944}, {6680, 17004}, {6683, 7835}, {6704, 7786}, {7736, 7947}, {7737, 7928}, {7745, 7938}, {7747, 7937}, {7752, 7915}, {7753, 7922}, {7777, 7808}, {7785, 7868}, {7794, 7837}, {7812, 7849}, {7836, 11174}, {7854, 10159}, {7856, 9466}, {7858, 7869}, {7870, 9698}, {7923, 11185}


X(16896) =  (name pending)

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

X(16896) lies on these lines:
{2, 3}, {83, 7845}, {385, 7889}, {3231, 10339}, {3329, 7796}, {3589, 7839}, {3619, 7893}, {3763, 7787}, {5007, 10159}, {5319, 7875}, {6704, 7832}, {7765, 7859}, {7804, 7928}, {7808, 7814}, {7836, 9606}, {7868, 7941}, {7885, 7914}, {7915, 7925}, {7930, 17005}, {7947, 11174}, {9983, 16984}, {10351, 15069}


X(16897) =  (name pending)

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

X(16897) lies on these lines:
{2, 3}, {83, 7873}, {141, 7839}, {194, 3763}, {316, 6704}, {385, 5368}, {538, 10159}, {2896, 3589}, {3096, 3329}, {3618, 7893}, {3934, 7923}, {6683, 7925}, {7745, 9990}, {7783, 7822}, {7786, 7888}, {7800, 7875}, {7808, 7885}, {7815, 7943}, {7831, 7889}, {7840, 7849}, {7865, 7878}, {7867, 17005}, {7868, 7947}, {7930, 15482}, {7932, 15271}, {7938, 7941}, {7942, 17006}


X(16898) =  (name pending)

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

X(16898) lies on these lines:
{2, 3}, {69, 7787}, {76, 5319}, {83, 7774}, {194, 3618}, {315, 7804}, {574, 6704}, {1007, 7945}, {1975, 3589}, {2548, 7814}, {2549, 7859}, {2896, 3619}, {3096, 7737}, {3329, 3926}, {3620, 7893}, {3734, 7765}, {3763, 7750}, {3767, 7846}, {3934, 17008}, {3972, 7800}, {5031, 14712}, {5286, 7875}, {5475, 7915}, {6292, 14907}, {6392, 7920}, {7735, 9983}, {7736, 7836}, {7738, 12055}, {7745, 7868}, {7747, 7914}, {7753, 7869}, {7758, 7878}, {7763, 7808}, {7789, 9606}, {7811, 10159}, {7834, 11185}, {9463, 10339}, {10334, 10359}, {10351, 14912}, {10352, 14981}, {12150, 14023}


X(16899) =  (name pending)

Barycentrics    2 a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 + 4 a^3 c + 12 a^2 b c + 12 a b^2 c + 4 b^3 c + 6 a^2 c^2 + 12 a b c^2 + 7 b^2 c^2 + 4 a c^3 + 4 b c^3 + c^4 : :

X(16899) lies on this line: {2, 3}


X(16900) =  (name pending)

Barycentrics    -4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4 - 4 a^3 c - 12 a^2 b c - 12 a b^2 c - 4 b^3 c - 6 a^2 c^2 - 12 a b c^2 - 5 b^2 c^2 - 4 a c^3 - 4 b c^3 - c^4 : :

X(16900) lies on this line: {2, 3}


X(16901) =  (name pending)

Barycentrics    3 a^4 + 8 a^3 b + 12 a^2 b^2 + 8 a b^3 + 2 b^4 + 8 a^3 c + 24 a^2 b c + 24 a b^2 c + 8 b^3 c + 12 a^2 c^2 + 24 a b c^2 + 13 b^2 c^2 + 8 a c^3 + 8 b c^3 + 2 c^4 : :

X(16901) lies on this line: {2, 3}


X(16902) =  (name pending)

Barycentrics    -a^4 - 8 a^3 b - 12 a^2 b^2 - 8 a b^3 - 2 b^4 - 8 a^3 c - 24 a^2 b c - 24 a b^2 c - 8 b^3 c - 12 a^2 c^2 - 24 a b c^2 - 11 b^2 c^2 - 8 a c^3 - 8 b c^3 - 2 c^4 : :

X(16902) lies on this line: {2, 3}


X(16903) =  (name pending)

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

X(16903) lies on this line: {2, 3}


X(16904) =  (name pending)

Barycentrics    a^4 - 4 a^3 b - 6 a^2 b^2 - 4 a b^3 - b^4 - 4 a^3 c - 12 a^2 b c - 12 a b^2 c - 4 b^3 c - 6 a^2 c^2 - 12 a b c^2 - 4 b^2 c^2 - 4 a c^3 - 4 b c^3 - c^4 : :

X(16904) lies on this line: {2, 3}


X(16905) =  (name pending)

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

X(16905) lies on these lines:
{2, 3}, {32, 16991}, {76, 17003}, {6679, 17033}


X(16906) =  (name pending)

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

X(16906) lies on these lines:
{2, 3}, {32, 17003}, {76, 16991}, {2887, 17033}


X(16907) =  (name pending)

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

X(16907) lies on this line: {2, 3}


X(16908) =  (name pending)

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

X(16908) lies on this line: {2, 3}


X(16909) =  (name pending)

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

X(16909) lies on these lines: {2, 3}, {32, 17007}


X(16910) =  (name pending)

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

X(16910) lies on these lines:
{2, 3}, {76, 17007}, {3120, 16822}, {6327, 17033}


X(16911) =  (name pending)

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

X(16911) lies on these lines:
{2, 3}, {32, 16994}, {76, 16993}, {274, 7839}, {4426, 4751}, {5247, 16815}, {5258, 6645}


X(16912) =  (name pending)

Barycentrics    a^4 - 2 a^2 b^2 - 4 a^2 b c - 4 a b^2 c - 2 a^2 c^2 - 4 a b c^2 - b^2 c^2 : :

X(16912) lies on these lines:
{2, 3}, {32, 16993}, {76, 16994}, {385, 16589}, {3797, 16817}, {4366, 5259}, {4426, 4687}, {5247, 16826}, {5251, 6645}, {5283, 7839}


X(16913) =  (name pending)

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

X(16913) lies on these lines:
{2, 3}, {32, 16996}, {76, 16995}, {274, 7787}, {4426, 4699}, {5247, 16816}, {6645, 12513}


X(16914) =  (name pending)

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

X(16914) lies on these lines:
{2, 3}, {32, 16995}, {76, 16996}, {192, 4426}, {958, 4366}, {1001, 6645}, {1655, 7766}, {1975, 17000}, {3053, 16999}, {3972, 16589}, {4294, 6653}, {4393, 5247}, {5283, 7787}, {6651, 16822}


X(16915) =  (name pending)

Barycentrics    a^4 + a^2 b c + a b^2 c + a b c^2 + b^2 c^2 : :

X(16915) lies on these lines:
{2, 3}, {6, 2669}, {8, 6645}, {31, 16827}, {32, 274}, {36, 17030}, {41, 894}, {75, 172}, {76, 5277}, {99, 5283}, {171, 17033}, {194, 5276}, {239, 1468}, {335, 976}, {993, 16819}, {1193, 14621}, {1580, 3980}, {1655, 1975}, {1909, 4386}, {1931, 5278}, {3053, 16992}, {3616, 4366}, {7816, 16589}, {8666, 16829}


X(16916) =  (name pending)

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

X(16916) lies on these lines:
{2, 3}, {6, 1655}, {8, 4366}, {83, 5283}, {32, 16997}, {76, 16998}, {238, 17033}, {274, 3734}, {350, 4426}, {668, 2241}, {748, 16827}, {1724, 17034}, {1914, 6376}, {3616, 6645}, {3972, 5277}, {4011, 16822}, {5247, 17027}, {5251, 17030}, {5276, 7787}, {7804, 16589}, {9263, 16781}


X(16917) =  (name pending)

Barycentrics    a^4 + 2 a^2 b c + 2 a b^2 c + 2 a b c^2 + b^2 c^2 : :

X(16917) lies on these lines:
{2, 3}, {10, 6645}, {32, 17000}, {36, 16819}, {76, 16999}, {99, 16589}, {171, 16827}, {194, 5275}, {274, 385}, {335, 5293}, {750, 17033}, {978, 14621}, {1125, 4366}, {1213, 6626}, {3570, 4754}, {5276, 7839}, {5283, 7783}, {5563, 16829}, {7754, 16995}, {7793, 16992}


X(16918) =  (name pending)

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

X(16918) lies on these lines:
{2, 3}, {10, 4366}, {32, 16999}, {76, 17000}, {83, 16589}, {748, 17033}, {1125, 6645}, {1655, 7839}, {3329, 5283}, {5275, 7787}, {6653, 15171}


X(16919) =  (name pending)

Barycentrics    2 a^4 + a^2 b c + a b^2 c + a b c^2 + 2 b^2 c^2 : :

X(16919) lies on these lines:
{2, 3}, {32, 17002}, {76, 17001}, {145, 6645}, {172, 4441}, {274, 3972}, {1975, 5276}, {3622, 4366}, {3734, 5277}, {4418, 16822}, {5283, 7816}


X(16920) =  (name pending)

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

X(16920) lies on these lines:
{2, 3}, {32, 17001}, {76, 17002}, {145, 4366}, {1655, 7787}, {3622, 6645}, {4426, 4441}, {5283, 7804}, {6381, 7031}


X(16921) =  (name pending)

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

X(16921) lies on these lines:
{2, 3}, {32, 17004}, {69, 7941}, {76, 1506}, {83, 7746}, {115, 7786}, {141, 7912}, {148, 5013}, {183, 7785}, {194, 3815}, {230, 7787}, {316, 7815}, {385, 2548}, {498, 4366}, {499, 6645}, {625, 3096}, {1007, 7947}, {1078, 5475}, {1184, 10339}, {2896, 7773}, {3314, 3934}, {3329, 3767}, {3589, 7932}, {3734, 7769}, {3763, 5103}, {3814, 17030}, {3972, 7749}, {6179, 7753}, {6292, 7934}, {6683, 7790}, {6704, 7943}, {6722, 7889}, {7617, 7827}, {7736, 7839}, {7745, 7793}, {7747, 7771}, {7748, 15031}, {7751, 7837}, {7755, 7878}, {7757, 9698}, {7763, 17005}, {7767, 7900}, {7768, 7775}, {7772, 14568}, {7780, 7812}, {7783, 11185}, {7794, 7814}, {7795, 7925}, {7796, 9466}, {7797, 11174}, {7800, 7885}, {7804, 7857}, {7808, 7828}, {7809, 7854}, {7810, 7860}, {7811, 7843}, {7820, 7940}, {7822, 7899}, {7825, 7831}, {7826, 7926}, {7832, 7862}, {7834, 10352}, {7844, 7859}, {7846, 7886}, {7847, 15482}, {7883, 8176}, {9939, 11168}, {11163, 13571}


X(16922) =  (name pending)

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

X(16922) lies on these lines:
{2, 3}, {32, 17006}, {76, 17005}, {183, 7941}, {385, 1506}, {625, 7928}, {1078, 7603}, {2548, 17004}, {3329, 7746}, {3815, 7839}, {3934, 7909}, {5007, 8859}, {6683, 7923}, {6722, 7859}, {7752, 7854}, {7758, 7777}, {7769, 7863}, {7786, 7902}, {7808, 16984}, {7815, 7885}, {7829, 12815}, {7852, 10352}, {7862, 7931}, {7912, 15271}, {7921, 17008}, {9698, 14568}


X(16923) =  (name pending)

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

X(16923) lies on these lines:
{2, 3}, {32, 17005}, {76, 17006}, {83, 10631}, {183, 7947}, {230, 7839}, {385, 7749}, {543, 12815}, {1007, 7893}, {1078, 7821}, {1153, 7883}, {3329, 7857}, {3589, 15514}, {6722, 7847}, {7619, 7827}, {7746, 7783}, {7760, 8859}, {7763, 17004}, {7771, 7862}, {7780, 7840}, {7786, 16984}, {7793, 7941}, {7815, 7931}, {7868, 10351}, {7874, 10352}, {7886, 7923}, {7899, 7928}, {7906, 17008}, {7942, 15482}, {7945, 15271}, {8587, 10159}


X(16924) =  (name pending)

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

X(16924) lies on these lines:
{2, 3}, {32, 17008}, {39, 11185}, {69, 7785}, {76, 2548}, {83, 3767}, {115, 7803}, {141, 7773}, {148, 2023}, {183, 7745}, {193, 7921}, {194, 7736}, {315, 3934}, {316, 7800}, {350, 9596}, {625, 7822}, {1007, 7836}, {1078, 7737}, {1506, 3734}, {1909, 9599}, {1975, 3815}, {2549, 7786}, {3085, 4366}, {3086, 6645}, {3329, 5286}, {3589, 7851}, {3618, 5038}, {3619, 5031}, {3620, 7939}, {3785, 7823}, {3788, 7603}, {3926, 7777}, {4027, 10359}, {5254, 11174}, {5319, 7878}, {5939, 7932}, {5943, 6310}, {6248, 9744}, {6292, 7825}, {6392, 7839}, {6683, 7748}, {6704, 7913}, {7615, 7827}, {7735, 7787}, {7746, 7804}, {7747, 7815}, {7750, 15271}, {7751, 7753}, {7752, 7795}, {7756, 15482}, {7759, 9466}, {7762, 15484}, {7775, 7794}, {7781, 9698}, {7790, 15031}, {7792, 13881}, {7812, 14023}, {7820, 7862}, {7844, 7889}, {7846, 14061}, {7891, 17005}, {7893, 15589}, {9753, 10350}


X(16925) =  (name pending)

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

X(16925) lies on these lines:
{2, 3}, {32, 620}, {69, 1691}, {76, 2021}, {99, 3767}, {183, 7789}, {187, 315}, {193, 6393}, {194, 5976}, {230, 1975}, {316, 7940}, {325, 3053}, {385, 3926}, {491, 12968}, {492, 12963}, {574, 6680}, {626, 5206}, {637, 2460}, {638, 2459}, {754, 7888}, {1007, 7785}, {1078, 7795}, {1384, 7762}, {1992, 13571}, {2080, 10349}, {2482, 7755}, {2548, 3972}, {2549, 7782}, {2896, 7945}, {3085, 6645}, {3086, 4366}, {3094, 3618}, {3314, 3785}, {3734, 7749}, {4045, 15515}, {5013, 7792}, {5023, 7750}, {5171, 10350}, {5210, 7784}, {5286, 7783}, {5304, 7839}, {5319, 7757}, {6179, 7758}, {6390, 7754}, {6781, 7825}, {7617, 12815}, {7618, 7827}, {7736, 7787}, {7737, 7752}, {7738, 7797}, {7739, 7856}, {7746, 7816}, {7747, 7862}, {7748, 7886}, {7751, 7863}, {7756, 7844}, {7761, 7874}, {7767, 7881}, {7768, 7870}, {7771, 7800}, {7775, 9167}, {7779, 10351}, {7780, 7801}, {7796, 14023}, {7802, 7899}, {7810, 7869}, {7811, 7909}, {7815, 7820}, {7823, 7925}, {7826, 7908}, {7 830, 7867}, {7831, 7930}, {7847, 7942}, {7852, 8589}, {7854, 7880}, {7864, 16984}, {7883, 8182}, {7889, 15482}, {7893, 7947}, {7904, 7931}, {7912, 14712}, {9737, 9753}, {9744, 13335}


X(16926) =  (name pending)

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

X(16926) lies on this line: {2,3}


X(16927) =  (name pending)

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

X(16927) lies on these lines: {2, 3}, {16705, 17000}


X(16928) =  (name pending)

Barycentrics    a^4 + 2 a^3 b + 4 a^2 b^2 + 2 a b^3 + 2 a^3 c + 8 a^2 b c + 8 a b^2 c + 2 b^3 c + 4 a^2 c^2 + 8 a b c^2 + 5 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(16928) lies on this line: {2,3}


X(16929) =  (name pending)

Barycentrics    a^4 - 2 a^3 b - 4 a^2 b^2 - 2 a b^3 - 2 a^3 c - 8 a^2 b c - 8 a b^2 c - 2 b^3 c - 4 a^2 c^2 - 8 a b c^2 - 3 b^2 c^2 - 2 a c^3 - 2 b c^3 : :

X(16929) lies on this line: {2,3}


X(16930) =  (name pending)

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

X(16930) lies on this line: {2,3}


X(16931) =  (name pending)

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

X(16931) lies on this line: {2,3}


X(16932) =  (name pending)

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

X(16932) lies on these lines:
{2, 3}, {6, 6664}, {76, 251}, {83, 1180}, {1194, 4159}, {1369, 7823}, {1627, 3972}, {3108, 7757}, {3734, 8024}, {6645, 17024}


X(16933) =  X(113)X(10151)∩X(114)X(468)

Barycentrics    SB*SC*((90*R^2+7*SA-14*SW)*S^ 2-36*R^4*(3*SA-5*SW)+6*R^2*(5* SA^2-11*SB*SC-15*SW^2)+(8*SA- 9*SW)*(SA^2-SB*SC)+11*SW^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27420.

X(16933) lies on these lines:
{4, 14611}, {113, 10151}, {114, 468}, {520, 1112}, {5186, 14052}


X(16934) =  X(115)X(3284)∩X(125)X(10257)

Barycentrics    SA*((18*R^2-5*SW)*(SA+SB)*(SB+ SC)+2*(3*SW+SB-12*R^2)*S^2)*(( 18*R^2-5*SW)*(SA+SC)*(SB+SC)+ 2*(3*SW+SC-12*R^2)*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27420.

X(16934) lies on these lines:
{3, 12079}, {30, 2970}, {115, 3284}, {125, 10257}, {520, 12358}, {2072, 8901}


X(16935) =  X(20)X(64)∩X(631)X(6688)

Barycentrics    2*(8*R^2+SW)*(S^2-2*SB*SC)+3*S ^2*SA : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27423.

X(16935) lies on these lines:
{20, 64}, {631, 6688}, {1907, 3619}


X(16936) =  X(6)X(20)∩X(64)X(159)

Barycentrics    (4*SA*(8*R^2-SA)-3*S^2)*(SB+SC ) : :
X(16936) = 3*X(20)+X(15741), 3*X(376)-X(11821), 5*X(3522)-X(11469), 3*X(15740)-X(15741)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27423.

X(16936) lies on the Stammler hyperbola and these lines:
{1, 1407}, {3, 13474}, {6, 20}, {22, 1620}, {30, 9815}, {64, 159}, {155, 550}, {170, 1740}, {195, 3534}, {220, 610}, {221, 5918}, {376, 1498}, {399, 15696}, {599, 12324}, {1191, 12565}, {1192, 11414}, {1657, 15047}, {2916, 15062}, {2917, 10323}, {2930, 15054}, {2931, 12041}, {2935, 15035}, {2951, 16466}, {3216, 7580}, {3522, 11469}, {3537, 16621}, {5059, 10601}, {5895, 7667}, {6409, 8939}, {6410, 8943}, {7689, 9937}, {8053, 8273}, {8717, 12038}, {10982, 11001}, {10996, 15435}, {11793, 11820}, {15691, 16266}


X(16937) =  (name pending)

Barycentrics    44*S^4-(64*R^2*(5*SA-4*SW)+16* SA^2+59*SB*SC-32*SW^2)*S^2-24* SB*SC*SW^2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27423.

X(16937) lies on this line: {511, 3528}


X(16938) =  COMPLEMENT OF X(8600)

Barycentrics    (b^2-c^2)^2*(7*a^4-7*(b^2+c^2) *a^2+2*(b^2-2*c^2)*(2*b^2-c^2) )*(4*a^6-14*(b^2+c^2)*a^4+7*( 2*b^4+b^2*c^2+2*c^4)*a^2-(b^2+ c^2)*(4*b^4-7*b^2*c^2+4*c^4)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27424.

X(16938) lies on the nine-point circle and these lines: {2, 8600}, {114, 15597}

X(16938) = complement of X(8600)


X(16939) =  (name pending)

Barycentrics    (4*(2*a^2-b^2-c^2)^2-9*b^2*c^ 2)*(8*a^6-33*(b^2+c^2)*a^4+9*( b^4+14*b^2*c^2+c^4)*a^2-4*(b^ 2+c^2)^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27424.

X(16939) lies on this line: {8584, 12040}


X(16940) = PERSPECTOR OF THESE TRIANGLES: 1st LEMOINE-DAO AND ANTI-EULER

Barycentrics    sqrt(3)*((SB+SC)*S^2+SW*SB*SC)-S*(S^2+2*SW*(SB+SC)) : :

Like in the construction of the 2nd Lemoine circle of a triangle ABC, let Ab, Ac be the points where the antiparallel to BC through X(6) cuts the sides AC and AB, respectively. Define Bc, Ba, Ca, Cb cyclically. Build equilateral triangles BcCbA', CaAcB' and AbBaC' all with the same orientation than ABC and build equilateral triangles BcCbA", CaAcB" and AbBaC", all with opposite orientation than ABC. Then A'B'C' and A"B"C" are both equilateral triangles. (Dao Thanh Oai, March 25, 2018)

A'B'C' and A"B"C" will be named here the 1st Lemoine-Dao and the 2nd Lemoine-Dao equilateral triangles, respectively. Their respective centers are X(10654) and X(10653). The 1st vertices of each have barycentric coordinates:

  A' = -(sqrt(3)*SA+2*S)*(SB+SC)/SA : sqrt(3)*SC-S : sqrt(3)*SB-S

  A" = -(sqrt(3)*SA-2*S)*(SB+SC)/SA : sqrt(3)*SC+S : sqrt(3)*SB+S

and their sidelengths are a'=(S/SW)*sqrt(|SW+sqrt(3)*S|) and a"=(S/SW)*sqrt(|SW-sqrt(3)*S|)

List of triangles perspective to A'B'C' with index of perspector: (ABC, 14), (anti-Euler, 16940), (4th Brocard, 15), (Euler, 16942), (inner-Fermat, 14), (3rd Fermat-Dao*, 16808), (7th Fermat-Dao*, 381), (11th Fermat-Dao*, 5321), (1st isodynamic-Dao*, 61), (2nd isodynamic-Dao**, 14), (2nd Lemoine-Dao**, 6), (outer-Napoleon*, 15)

List of triangles perspective to A"B"C" with index of perspector: (ABC, 13), (anti-Euler, 16941), (4th Brocard, 16), (Euler, 16943), (outer-Fermat, 13), (4th Fermat-Dao*, 16809), (8th Fermat-Dao*, 381), (12th Fermat-Dao*, 5318), (1st isodynamic-Dao**, 13), (2nd isodynamic-Dao*, 62), (1st Lemoine-Dao**, 6), (inner-Napoleon*, 16)

Note: In both lists, * means that triangles are homothetic and ** means that triangles are inversely similar.

(This introduction and centers X(16940) to X(16943) were contributed by César Lozada, April 1, 2018)

X(16940) lies on these lines:
{6,383}, {15,14137}, {16,376}, {62,12252}, {298,11543}, {616,11295}, {2549,5334}, {3104,12251}, {5335,7753}, {9113,9862}, {9744,9993}, {11302,11485}


X(16941) = PERSPECTOR OF THESE TRIANGLES: 2nd LEMOINE-DAO AND ANTI-EULER

Barycentrics    sqrt(3)*((SB+SC)*S^2+SW*SB*SC)+S*(S^2+2*SW*(SB+SC)) : :

X(16941) lies on these lines:
{6,1080}, {13,12243}, {15,376}, {16,14136}, {61,12252}, {299,11542}, {617,11296}, {2549,5335}, {3105,12251}, {5334,7753}, {9112,9862}, {9744,9993}, {11301,11486}


X(16942) = PERSPECTOR OF THESE TRIANGLES: 1st LEMOINE-DAO AND EULER

Barycentrics    -S*(2*S^2+SA^2+8*SB*SC-SW^2)+sqrt(3)*((SB+SC)*S^2-2*SB*SC*SW) : :

X(16942) lies on these lines:
{6,5478}, {14,9880}, {381,396}, {618,11295}, {1503,16943}, {3104,6248}


X(16943) = PERSPECTOR OF THESE TRIANGLES: 2nd LEMOINE-DAO AND EULER

Barycentrics    S*(2*S^2+SA^2+8*SB*SC-SW^2)+sqrt(3)*((SB+SC)*S^2-2*SB*SC*SW) : :

X(16943) lies on these lines:
{6,5479}, {13,9880}, {381,395}, {619,11296}, {1503,16942}, {3105,6248}


X(16944) =  X(56)X(106)∩X(57)X(1168)

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

X(16944) lies on the cubic K967 and these lines:
{1, 901}, {3, 3446}, {32, 9456}, {56, 106}, {57, 1168}, {58, 3733}, {1320, 5903}, {1481, 2316}, {1983, 7113}, {2163, 2226}, {3336, 4674}, {4591, 9275}, {6549, 10521}

X(16944) = X(i)-Ceva conjugate of X(j) for these (i,j): {2226, 9456}, {10428, 106}
X(16944) = crossdifference of every pair of points on line {1639, 3943}
X(16944) = crosssum of X(3992) and X(4738)
X(16944) = X(16548)-zayin conjugate of X(44)
X(16944) = X(i)-isoconjugate of X(j) for these (i,j): {8, 14584}, {9, 14628}, {80, 519}, {655, 1639}, {759, 3992}, {1168, 4738}, {1411, 4723}, {2006, 2325}, {2161, 4358}, {2222, 4768}, {3264, 6187}
X(16944) = barycentric product X(i)*X(j) for these {i,j}: {36, 88}, {106, 3218}, {214, 2226}, {320, 9456}, {901, 3960}, {903, 7113}, {1443, 2316}, {1797, 1870}, {1983, 6548}, {10428, 16586}
X(16944) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 4358}, {56, 14628}, {604, 14584}, {654, 4768}, {1417, 2006}, {2245, 3992}, {2323, 4723}, {2361, 2325}, {3218, 3264}, {3724, 3943}, {7113, 519}, {8648, 1639}, {9456, 80}
X(16944) = {X(56),X(14260)}-harmonic conjugate of X(106)


X(16945) =  X(56)X(1149)∩X(57)X(1476)

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

X(16945) lies on the cubic K967 and these lines:
{1, 1293}, {56, 1149}, {57, 1476}, {603, 1417}, {961, 1722}, {1193, 15375}, {2137, 3158}, {3600, 6556}, {5563, 14261}

X(16945) = cevapoint of X(1357) and X(6363)
X(16945) = crosssum of X(i) and X(j) for these (i,j): {8, 6552}, {3161, 6555}
X(16945) = X(58)-beth conjugate of X(3915)
X(16945) = X(31)-cross conjugate of X(604)
X(16945) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3161}, {7, 6555}, {8, 145}, {75, 3158}, {85, 4936}, {190, 4521}, {312, 1743}, {314, 4849}, {318, 4855}, {333, 3950}, {341, 1420}, {346, 5435}, {643, 4404}, {644, 4462}, {645, 14321}, {646, 4394}, {664, 4546}, {668, 4162}, {765, 4939}, {1016, 4534}, {1043, 4848}, {1222, 12640}, {1320, 4487}, {3052, 3596}, {3667, 3699}, {3710, 4248}, {3756, 4076}, {4102, 4856}, {4373, 15519}, {4582, 14425}, {4729, 7257}, {4899, 14942}, {4953, 4998}, {6049, 6556}, {8706, 14284}
X(16945) = barycentric product X(i)*X(j) for these {i,j}: {56, 8056}, {57, 3445}, {604, 4373}, {1106, 6557}, {1293, 3669}, {1357, 5382}, {1407, 3680}, {1408, 4052}, {1743, 16079}, {6556, 7366}
X(16945) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3161}, {32, 3158}, {41, 6555}, {604, 145}, {667, 4521}, {1015, 4939}, {1106, 5435}, {1293, 646}, {1397, 1743}, {1402, 3950}, {1404, 4487}, {1919, 4162}, {2175, 4936}, {3063, 4546}, {3248, 4534}, {3445, 312}, {7180, 4404}, {8056, 3596}


X(16946) =  X(3)X(6)∩X(9)X(983)

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

X(16946) lies on these lines:
{3, 6}, {9, 983}, {37, 2241}, {41, 2300}, {44, 3694}, {48, 2251}, {172, 1449}, {198, 16502}, {251, 941}, {380, 10315}, {609, 16667}, {966, 13742}, {985, 2663}, {1015, 2178}, {1100, 2242}, {1172, 10311}, {1572, 3553}, {1743, 7031}, {2082, 5336}, {2175, 2209}, {2205, 2280}, {2269, 2273}, {2270, 16780}, {2277, 5299}, {3052, 6600}, {3204, 16685}, {3217, 3915}, {3618, 5337}, {3686, 4426}, {3734, 3770}, {4386, 5750}, {5291, 5839}, {5332, 16470}

X(16946) = crosspoint of X(i) and X(j) for these (i,j): {1252, 1415}, {4186, 4383}
X(16946) = crossdifference of every pair of points on line {523, 3777}
X(16946) = isogonal conjugate of the isotomic conjugate of X(4383)
X(16946) = X(284)-beth conjugate of X(4268)
X(16946) = X(1408)-Ceva conjugate of X(31)
X(16946) = X(1577)-isoconjugate of X(8690)
X(16946) = crosssum of X(1086) and X(4391)
X(16946) = barycentric product X(i)*X(j) for these {i,j}: {1, 3915}, {3, 4186}, {6, 4383}, {31, 3875}, {56, 3913}, {57, 3217}, {58, 3214}, {101, 4498}, {110, 4139}, {692, 4106}, {1333, 3175}
X(16946) = barycentric quotient X(i)/X(j) for these {i,j}: {1576, 8690}, {3214, 313}, {3217, 312}, {3875, 561}, {3913, 3596}, {3915, 75}, {4139, 850}, {4186, 264}, {4383, 76}, {4498, 3261}
X(16946) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 32, 5019), (6, 284, 5114), (6, 1030, 5069), (6, 1333, 5042), (6, 2220, 32), (6, 2305, 4253), (6, 3053, 5120), (6, 3285, 4268), (6, 4254, 2092), (6, 4261, 7772), (32, 5042, 1333), (1030, 5069, 574), (1333, 5042, 5019), (2209, 2210, 2175), (4263, 5007, 6), (4266, 5037, 6)


X(16947) =  X(58)X(1169)∩X(284)X(1404)

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

X(16947) lies on these lines:
{58, 1169}, {284, 1404}, {593, 1412}, {603, 604}, {1397, 2206}

X(16947) = isogonal conjugate of X(30713)
X(16947) = X(1408)-Ceva conjugate of X(2206)
X(16947) = X(1397)-cross conjugate of X(1408)
X(16947) = crosssum of X(2321) and X(3710)
X(16947) = {X(1333),X(1408)}-harmonic conjugate of X(604)
X(16947) = isogonal conjugate of the isotomic conjugate of X(1412)
X(16947) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3701}, {8, 321}, {9, 313}, {10, 312}, {37, 3596}, {72, 7017}, {75, 2321}, {76, 210}, {85, 4082}, {92, 3710}, {190, 4086}, {200, 349}, {226, 341}, {264, 3694}, {274, 6057}, {306, 318}, {307, 7101}, {314, 594}, {333, 1089}, {334, 3985}, {346, 1441}, {522, 4033}, {523, 646}, {561, 1334}, {644, 850}, {645, 4036}, {668, 3700}, {1043, 6358}, {1231, 7046}, {1446, 5423}, {1577, 3699}, {1812, 7141}, {1826, 3718}, {1969, 2318}, {1978, 4041}, {3261, 4069}, {3702, 6539}, {3709, 6386}, {3948, 4518}, {3952, 4391}, {3963, 4451}, {3992, 4997}, {4024, 7257}, {4076, 16732}, {4077, 6558}, {4080, 4723}, {4092, 4601}, {4095, 7018}, {4102, 4647}, {4136, 7033}, {4171, 4572}, {4397, 4552}, {4515, 6063}, {4571, 14618}, {6385, 7064}
X(16947) = barycentric product X(i)*X(j) for these {i,j}: {1, 1408}, {6, 1412}, {7, 2206}, {10, 7342}, {21, 1106}, {28, 603}, {31, 1014}, {32, 1434}, {34, 1437}, {42, 7341}, {48, 1396}, {56, 58}, {57, 1333}, {60, 1042}, {65, 849}, {77, 2203}, {81, 604}, {86, 1397}, {109, 3733}, {163, 3669}, {222, 1474}, {269, 2194}, {270, 1410}, {283, 1398}, {284, 1407}, {552, 1918}, {593, 1400}, {608, 1790}, {649, 4565}, {667, 1414}, {692, 7203}, {741, 1428}, {757, 1402}, {1019, 1415}, {1172, 7099}, {1357, 4570}, {1395, 1444}, {1413, 2360}, {1416, 3286}, {1427, 2150}, {1435, 2193}, {1461, 7252}, {1472, 5323}, {1576, 3676}, {1919, 4573}, {1977, 4620}, {1980, 4625}, {2149, 16726}, {2204, 7177}, {2287, 7366}, {2299, 7053}, {2328, 7023}, {3063, 4637}, {4556, 7180}, {4636, 7250}, {5317, 7125}, {7335, 8747}
X(16947) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3701}, {32, 2321}, {56, 313}, {58, 3596}, {163, 646}, {184, 3710}, {560, 210}, {604, 321}, {667, 4086}, {849, 314}, {1014, 561}, {1106, 1441}, {1333, 312}, {1396, 1969}, {1397, 10}, {1402, 1089}, {1407, 349}, {1408, 75}, {1412, 76}, {1414, 6386}, {1415, 4033}, {1434, 1502}, {1437, 3718}, {1474, 7017}, {1501, 1334}, {1576, 3699}, {1918, 6057}, {1919, 3700}, {1980, 4041}, {2175, 4082}, {2194, 341}, {2203, 318}, {2204, 7101}, {2206, 8}, {4565, 1978}, {5009, 4087}, {7099, 1231}, {7251, 4150}, {7341, 310}, {7342, 86}, {7366, 1446}, {9247, 3694}, {9447, 4515}, {14574, 3939}, {14575, 2318}, {14599, 3985}


X(16948) =  X(1)X(21)∩X(2)X(4252)

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

X(16948) lies on these lines:
{1, 21}, {2, 4252}, {6, 4189}, {8, 4234}, {20, 5721}, {28, 88}, {29, 5704}, {44, 1333}, {45, 2303}, {56, 7419}, {60, 13624}, {100, 3214}, {110, 1408}, {145, 3052}, {171, 5260}, {229, 4228}, {238, 5253}, {239, 16046}, {284, 16670}, {333, 3617}, {404, 1724}, {580, 6909}, {593, 1098}, {759, 8697}, {899, 13588}, {940, 16865}, {1010, 5235}, {1043, 3621}, {1104, 3218}, {1125, 4683}, {1150, 4195}, {1155, 5324}, {1193, 5303}, {1203, 5267}, {1325, 5358}, {1396, 13739}, {1412, 2137}, {1437, 5126}, {1453, 4652}, {1722, 9352}, {1743, 4855}, {1792, 3935}, {1834, 15680}, {2053, 2106}, {2330, 4663}, {2363, 5297}, {3017, 15678}, {3216, 13587}, {3240, 4184}, {3286, 4225}, {3579, 4221}, {3625, 4720}, {3634, 14005}, {3736, 16477}, {3916, 5262}, {3924, 4650}, {4188, 4383}, {4248, 5435}, {4346, 8822}, {4575, 14868}, {4877, 16676}, {5221, 11101}, {5225, 14956}, {5292, 11114}, {5333, 5550}, {5398, 6906}, {5427, 8614}, {7173, 14008}, {9534, 16393}, {11319, 14829}, {12702, 15952}

X(16948) = cevapoint of X(1743) and X(3052)
X(16948) = crosspoint of X(1414) and X(4567)
X(16948) = crosssum of X(3125) and X(4041)
X(16948) = X(4636)-beth conjugate of X(1408)
X(16948) = X(4394)-zayin conjugate of X(661)
X(16948) = X(1014)-Ceva conjugate of X(81)
X(16948) = X(1420)-cross conjugate of X(4248)
X(16948) = X(i)-isoconjugate of X(j) for these (i,j): {6, 4052}, {10, 3445}, {37, 8056}, {42, 4373}, {65, 3680}, {523, 1293}, {1042, 6556}, {1400, 6557}, {2429, 4049}, {3125, 5382}
X(16948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 896, 11684), (21, 58, 81), (1333, 1778, 2287), (1453, 4652, 4850), (1724, 4257, 404), (4228, 5323, 229)
X(16948) = barycentric product X(i)*X(j) for these {i,j}: {21, 5435}, {27, 4855}, {63, 4248}, {81, 145}, {86, 1743}, {99, 4394}, {110, 4462}, {274, 3052}, {333, 1420}, {662, 3667}, {757, 3950}, {799, 8643}, {1014, 3161}, {1414, 4521}, {1434, 3158}, {1509, 4849}, {2185, 4848}, {3756, 4567}, {4162, 4573}, {4404, 4556}, {4504, 4603}, {4546, 4637}, {4610, 4729}, {4622, 14425}
X(16948) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4052}, {21, 6557}, {58, 8056}, {81, 4373}, {145, 321}, {163, 1293}, {284, 3680}, {1333, 3445}, {1420, 226}, {1743, 10}, {2287, 6556}, {3052, 37}, {3158, 2321}, {3161, 3701}, {3667, 1577}, {3756, 16732}, {3950, 1089}, {4162, 3700}, {4248, 92}, {4394, 523}, {4462, 850}, {4521, 4086}, {4570, 5382}, {4729, 4024}, {4848, 6358}, {4849, 594}, {4855, 306}, {4856, 4647}, {4881, 3936}, {4898, 4066}, {4936, 4082}, {5435, 1441}, {8643, 661}, {14321, 4036}


X(16949) =  (name pending)

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

X(16949) lies on these lines:
{2, 3}, {32, 8024}, {76, 1627}, {99, 1180}, {187, 8891}, {251, 305}, {1194, 7816}, {1369, 3314}, {1501, 4074}, {1613, 10328}, {1975, 5359}, {3051, 4048}, {3108, 7878}, {4366, 17024}, {7832, 16275}, {7836, 8878}, {9465, 16276}


X(16950) =  (name pending)

Barycentrics    a^6 + a^4 b^2 + a^4 c^2 + 3 a^2 b^2 c^2 + b^4 c^2 + b^2 c^4 : :

X(16950) lies on these lines:
{2, 3}, {39, 16276}, {83, 1194}, {251, 308}, {305, 3734}, {1180, 3329}, {1196, 7804}, {1799, 3934}, {3589, 10329}, {3920, 4366}, {5359, 7787}, {6645, 7191}, {7745, 8878}, {7747, 16275}, {7839, 8267}, {10328, 12215}


X(16951) =  (name pending)

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

X(16951) lies on these lines:
{2, 3}, {32, 305}, {99, 1194}, {187, 1799}, {194, 5359}, {251, 3266}, {325, 8878}, {385, 1627}, {626, 16275}, {1007, 15437}, {1078, 8891}, {1180, 7783}, {1184, 1975}, {1196, 7816}, {1369, 7939}, {1613, 4048}, {1691, 4074}, {3051, 12215}, {3108, 3228}, {3231, 10328}, {3920, 6645}, {4366, 7191}, {5026, 14153}, {5182, 13366}, {5354, 8267}, {10191, 12055}


X(16952) =  (name pending)

Barycentrics    2 a^6 + 2 a^4 b^2 + 2 a^4 c^2 + 3 a^2 b^2 c^2 + 2 b^4 c^2 + 2 b^2 c^4 : :

X(16952) lies on these lines:
{2, 3}, {251, 3734}, {1180, 7804}, {1369, 7737}, {3108, 7781}, {7787, 8267}


X(16953) =  (name pending)

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

X(16953) lies on these lines:
{2, 3}, {251, 9464}, {1180, 7816}, {1369, 7795}, {1627, 3734}, {3972, 8024}, {9463, 10328}


X(16954) =  (name pending)

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

X(16954) lies on these lines: {2, 3}, {32, 310}, {6645, 17018}, {16748, 16998}


X(16955) =  (name pending)

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

X(16955) lies on these lines: {2, 3}, {310, 3734}, {4366, 17018}


X(16956) =  (name pending)

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

X(16956) lies on these lines: {2, 3}, {42, 6645}, {310, 385}, {3720, 4366}


X(16957) =  (name pending)

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

X(16957) lies on these lines: {2, 3}, {42, 4366}, {3720, 6645}


X(16958) =  (name pending)

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

X(16958) lies on these lines: {2, 3}, {310, 3972}


X(16959) =  (name pending)

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

X(16959) lies on this line: {2,3}



X(16960) = HOMOTHETIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO AND 3rd FERMAT-DAO

Barycentrics    9*S^2+5*sqrt(3)*(SB+SC)*S+3*SB*SC : :
X(16960) = 3*X(13)+2*X(15) = X(13)+4*X(396) = 9*X(13)-4*X(5318) = 3*X(13)-8*X(11542) = X(13)-6*X(16267) = 2*X(13)+3*X(16962) = X(15)-6*X(396) = 3*X(15)+2*X(5318) = X(15)+4*X(11542) = X(15)+9*X(16267) = 4*X(15)-9*X(16962) = 9*X(396)+X(5318) = 3*X(396)+2*X(11542) = 2*X(396)+3*X(16267) = 8*X(396)-3*X(16962) = X(5318)-6*X(11542) = 4*X(11542)-9*X(16267) = 4*X(16267)+X(16962)

Let AoBoCo be the outer-Fermat triangle of ABC. Denote p'a the polar of A with respect to the circle with center Ao and radius a=BC. Define similarly p'b and p'c. Then the triangle A'B'C' bounded by these polars is equilateral.

Let AiBiCi be the inner-Fermat triangle of ABC. Denote p"a the polar of A with respect to the circle with center Ai and radius a=BC. Define similarly p"b and p"c. Then the triangle A"B"C" bounded by these polars is equilateral. (Both conclusions by Dao Thanh Oai, March 25, 2018)

The triangles A'B'C' and A"B"C" are here named the 15th Fermat-Dao equilateral triangle and the 16th Fermat-Dao equilateral triangle, respectively. Their respective centers are X(13) and X(14), with sidelengths a'=4*S/sqrt(SW+sqrt(3)*S) and a"=4*S/sqrt(SW-sqrt(3)*S). The 1st vertices of each have barycentric coordinates:

  A' = (sqrt(3)*(3*S^2+SB*SC)+5*(SB+SC)*S)/(SA-sqrt(3)*S) : sqrt(3)*SC+S : sqrt(3)*SB+S

  A" = (sqrt(3)*(3*S^2+SB*SC)-5*(SB+SC)*S)/(SA+sqrt(3)*S) : sqrt(3)*SC-S : sqrt(3)*SB-S

List of triangles perspective to A'B'C' with index of perspector: (ABC, 13), (outer-Fermat, 13), (3rd Fermat-Dao*, 16960), (7th Fermat-Dao*, 16962), (11th Fermat-Dao*, 16809), (1st isodynamic-Dao*, 13), (2nd isodynamic-Dao**, 6777), (1st Lemoine-Dao*, 16964), (2nd Lemoine-Dao**, 13), (outer-Napoleon*, 16966)

List of triangles perspective to A"B"C" with index of perspector: (ABC, 14), (inner-Fermat, 14), (4th Fermat-Dao*, 16961), (8th Fermat-Dao*, 16963), (12th Fermat-Dao*, 16808), (1st isodynamic-Dao**, 6778), (2nd isodynamic-Dao*, 14), (1st Lemoine-Dao**, 14), (2nd Lemoine-Dao*, 16965), (inner-Napoleon*, 16967)

Note: In both lists, * means that triangles are homothetic and ** means that triangles are inversely similar.

(This introduction and centers X(16960) to X(16967) were contributed by César Lozada, April 2, 2018)

X(16960) lies on these lines:
{6,17}, {13,15}, {16,631}, {61,3091}, {62,632}, {115,16529}, {230,16530}, {397,10646}, {621,5459}, {3180,6669}, {3412,3843}, {3522,5335}, {3830,12820}, {3858,5321}, {5463,6671}, {5470,9117}, {6777,6783}, {10653,15692}, {10654,12817}, {11480,15696}, {11481,15693}, {11486,15694}, {11543,12812}

X(16960) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 17, 16966), (6, 1656, 16961), (6, 16966, 18), (3412, 16808, 11485), (11485, 16808, 16964)


X(16961) = HOMOTHETIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO AND 4th FERMAT-DAO

Barycentrics    9*S^2-5*sqrt(3)*(SB+SC)*S+3*SB*SC : :
X(16961) = 3*X(14)+2*X(16) = X(14)+4*X(395) = 9*X(14)-4*X(5321) = 3*X(14)-8*X(11543) = X(14)-6*X(16268) = 2*X(14)+3*X(16963) = X(16)-6*X(395) = 3*X(16)+2*X(5321) = X(16)+4*X(11543) = X(16)+9*X(16268) = 4*X(16)-9*X(16963) = 9*X(395)+X(5321) = 3*X(395)+2*X(11543) = 2*X(395)+3*X(16268) = 8*X(395)-3*X(16963) = X(5321)-6*X(11543) = 4*X(11543)-9*X(16268) = 4*X(16268)+X(16963)

X(16961) lies on these lines:
{6,17}, {14,16}, {15,631}, {61,632}, {62,3091}, {115,16530}, {230,16529}, {398,10645}, {622,5460}, {3181,6670}, {3411,3843}, {3522,5334}, {3830,12821}, {3858,5318}, {5464,6672}, {5469,9115}, {6778,6782}, {10653,12816}, {10654,15692}, {11480,15693}, {11481,15696}, {11485,15694}, {11542,12812}

X(16961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 18, 16967), (6, 1656, 16960), (6, 16967, 17), (3411, 16809, 11486), (11486, 16809, 16965)


X(16962) = HOMOTHETIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO AND 7th FERMAT-DAO

Barycentrics    7*S^2+3*sqrt(3)*(SB+SC)*S-3*SB*SC : :
X(16962) = X(13)+2*X(15) = X(13)-4*X(396) = 7*X(13)-4*X(5318) = 5*X(13)-8*X(11542) = 2*X(13)-5*X(16960) = X(15)+2*X(396) = 7*X(15)+2*X(5318) = 5*X(15)+4*X(11542) = 4*X(15)+5*X(16960) = 7*X(396)-X(5318) = 5*X(396)-2*X(11542) = 8*X(396)-5*X(16960) = 5*X(5318)-14*X(11542) = 2*X(5318)-7*X(16267) = 4*X(11542)-5*X(16267) = 4*X(16267)-5*X(16960)

X(16962) lies on these lines:
{2,18}, {3,3412}, {6,5054}, {13,15}, {14,5055}, {16,3524}, {17,381}, {62,549}, {115,6780}, {140,3411}, {202,5298}, {203,10056}, {298,6671}, {303,7809}, {376,5238}, {382,12816}, {395,11539}, {397,5352}, {398,547}, {542,16529}, {618,3180}, {621,6669}, {754,9763}, {2307,3584}, {3106,13083}, {3534,16965}, {3545,10654}, {3628,10188}, {3642,7884}, {3860,5349}, {4995,7006}, {5237,12100}, {5340,15681}, {5351,15692}, {5362,15671}, {5464,11298}, {5473,13350}, {5611,6771}, {5858,7764}, {5859,11301}, {7005,10072}, {7975,11707}, {10304,10645}, {11480,15688}, {11481,15706}, {11486,15707}, {11624,13391}, {11812,16773}, {12155,13586}, {14269,16808}

X(16962) = reflection of X(13) in X(16267)
X(16962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5054, 16963), (6, 16241, 16242), (381, 16964, 12817), (5054, 16963, 16242), (16241, 16963, 5054)


X(16963) = HOMOTHETIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO AND 8th FERMAT-DAO

Barycentrics    7*S^2-3*sqrt(3)*(SB+SC)*S-3*SB*SC : :
X(16963) = X(13)+2*X(9115) = X(14)+2*X(16) = X(14)-4*X(395) = 7*X(14)-4*X(5321) = 5*X(14)-8*X(11543) = 2*X(14)-5*X(16961) = X(16)+2*X(395) = 7*X(16)+2*X(5321) = 5*X(16)+4*X(11543) = 4*X(16)+5*X(16961) = 7*X(395)-X(5321) = 5*X(395)-2*X(11543) = 8*X(395)-5*X(16961) = 5*X(5321)-14*X(11543) = 2*X(5321)-7*X(16268) = 4*X(11543)-5*X(16268) = 4*X(16268)-5*X(16961)

X(16963) lies on these lines:
{2,17}, {3,3411}, {6,5054}, {13,5055}, {14,16}, {15,3524}, {18,381}, {61,549}, {140,3412}, {187,6780}, {202,10056}, {203,5298}, {299,6672}, {302,7809}, {376,5237}, {382,12817}, {396,11539}, {397,547}, {398,5351}, {542,16530}, {619,3181}, {622,6670}, {754,9761}, {3107,13084}, {3534,16964}, {3545,10653}, {3582,7127}, {3628,10187}, {3643,7884}, {3860,5350}, {4995,7005}, {5238,12100}, {5339,15681}, {5352,15692}, {5367,15671}, {5463,11297}, {5474,13349}, {5615,6774}, {5858,11302}, {5859,7764}, {7006,10072}, {7974,11708}, {10304,10646}, {11480,15706}, {11481,15688}, {11485,15707}, {11626,13391}, {11812,16772}, {12154,13586}, {14269,16809}

X(16963) = reflection of X(14) in X(16268)
X(16963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5054, 16962), (6, 16242, 16241), (381, 16965, 12816), (5054, 16962, 16241), (16242, 16962, 5054)


X(16964) = HOMOTHETIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO AND 1st LEMOINE-DAO

Barycentrics    S^2+sqrt(3)*(SB+SC)*S-5*SB*SC : :
X(16964) = 4*X(6695)-3*X(11300)

X(16964) lies on these lines:
{2,5238}, {3,14}, {4,13}, {5,15}, {6,382}, {16,20}, {17,381}, {30,62}, {140,5352}, {156,3201}, {202,7354}, {203,1479}, {299,7860}, {302,7782}, {376,5351}, {395,550}, {396,546}, {397,3627}, {548,10646}, {621,635}, {623,11307}, {624,628}, {627,5463}, {631,10645}, {633,3643}, {1250,4330}, {1478,7005}, {1605,3443}, {1614,3206}, {1656,16241}, {2041,3390}, {2042,3389}, {2307,3583}, {2782,3104}, {3070,3364}, {3071,3365}, {3091,5365}, {3106,11257}, {3107,14881}, {3146,10653}, {3412,3843}, {3526,11480}, {3528,11489}, {3534,16963}, {3642,11290}, {3830,5340}, {3845,16267}, {3851,16644}, {3853,5318}, {3855,11488}, {3861,11542}, {5056,10188}, {5350,15687}, {5464,11306}, {5469,16630}, {5471,7756}, {6107,8837}, {6240,8739}, {6284,7006}, {6695,11300}, {7841,12154}, {8918,15442}, {8929,15441}, {10657,15063}, {11475,15559}, {11481,15696}

X(16964) = reflection of X(16965) in X(7765)
X(16964) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18, 16242), (4, 61, 13), (4, 10654, 61), (6, 382, 16965), (15, 5321, 16809), (15, 16809, 16966), (395, 550, 5237), (396, 5349, 546), (548, 16773, 10646), (3366, 3367, 16809), (11485, 16808, 16960), (12817, 16962, 381)


X(16965) = HOMOTHETIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO AND 2nd LEMOINE-DAO

Barycentrics    S^2-sqrt(3)*(SB+SC)*S-5*SB*SC : :
X(16965) = 4*X(6694)-3*X(11299)

X(16965) lies on these lines:
{2,5237}, {3,13}, {4,14}, {5,16}, {6,382}, {15,20}, {18,381}, {30,61}, {140,5351}, {156,3200}, {202,1479}, {203,7354}, {298,7860}, {303,7782}, {376,5352}, {395,546}, {396,550}, {398,3627}, {548,10645}, {622,636}, {623,627}, {624,11308}, {628,5464}, {631,10646}, {634,3642}, {1478,7006}, {1606,3442}, {1614,3205}, {1656,16242}, {2041,3364}, {2042,3365}, {2307,10483}, {2782,3105}, {3070,3389}, {3071,3390}, {3091,5366}, {3106,14881}, {3107,11257}, {3146,10654}, {3411,3843}, {3526,11481}, {3528,11488}, {3534,16962}, {3585,7127}, {3643,11289}, {3830,5339}, {3845,16268}, {3851,16645}, {3853,5321}, {3855,11489}, {3861,11543}, {4325,7051}, {4330,10638}, {5056,10187}, {5349,15687}, {5463,11305}, {5470,16631}, {5472,7756}, {5474,16529}, {6106,8839}, {6240,8740}, {6284,7005}, {6694,11299}, {7841,12155}, {8919,15441}, {8930,15442}, {10658,15063}, {11476,15559}, {11480,15696}

X(16965) = reflection of X(16964) in X(7765)
X(16965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 17, 16241), (4, 62, 14), (4, 10653, 62), (6, 382, 16964), (16, 5318, 16808), (16, 16808, 16967), (395, 5350, 546), (396, 550, 5238), (548, 16772, 10645), (3391, 3392, 16808), (11486, 16809, 16961), (12816, 16963, 381)


X(16966) = HOMOTHETIC CENTER OF THESE TRIANGLES: 15th FERMAT-DAO AND OUTER-NAPOLEON

Barycentrics    9*S^2+sqrt(3)*(SB+SC)*S+3*SB*SC : :

X(16966) lies on these lines:
{2,13}, {3,16808}, {4,10188}, {5,15}, {6,17}, {14,5055}, {61,3090}, {62,3628}, {125,10657}, {140,5318}, {303,623}, {381,11480}, {395,15699}, {396,547}, {403,11475}, {470,6116}, {546,5352}, {632,5237}, {636,7899}, {1594,10641}, {2072,10634}, {3091,5238}, {3411,5067}, {3525,5351}, {3526,11481}, {3549,11516}, {5056,5334}, {5070,11486}, {5071,10654}, {5350,15712}, {5362,7504}, {5449,10661}, {5471,16529}, {5613,6777}, {6639,10635}, {6670,6783}, {6671,11304}, {6673,11307}, {6704,11312}, {6774,6778}, {7051,7951}, {7505,10642}, {7577,10632}, {7741,10638}, {9112,16530}, {10224,11267}, {10633,14940}, {10658,14643}, {11515,11585}, {12816,15693}, {12820,15688}, {15703,16645}

X(16966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 15, 16809), (6, 17, 16960), (6, 1656, 16967), (6, 16967, 18), (15, 16809, 16964), (17, 1656, 18), (17, 16967, 6), (18, 16960, 6), (140, 5318, 10646), (10576, 10577, 17)


X(16967) = HOMOTHETIC CENTER OF THESE TRIANGLES: 16th FERMAT-DAO AND INNER-NAPOLEON

Barycentrics    9*S^2-sqrt(3)*(SB+SC)*S+3*SB*SC : :

X(16967) lies on these lines:
{2,14}, {3,16809}, {4,10187}, {5,16}, {6,17}, {13,5055}, {61,3628}, {62,3090}, {125,10658}, {140,5321}, {302,624}, {381,11481}, {395,547}, {396,15699}, {403,11476}, {471,6117}, {546,5351}, {632,5238}, {635,7899}, {1250,7741}, {1594,10642}, {2072,10635}, {3091,5237}, {3412,5067}, {3525,5352}, {3526,11480}, {3549,11515}, {5056,5335}, {5070,11485}, {5071,10653}, {5349,15712}, {5367,7504}, {5449,10662}, {5472,16530}, {5617,6778}, {6639,10634}, {6669,6782}, {6672,11303}, {6674,11308}, {6704,11311}, {6771,6777}, {7505,10641}, {7577,10633}, {9113,16529}, {10224,11268}, {10632,14940}, {10657,14643}, {11516,11585}, {12817,15693}, {12821,15688}, {15703,16644}

X(16967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 16, 16808), (6, 18, 16961), (6, 1656, 16966), (6, 16966, 17), (16, 16808, 16965), (17, 16961, 6), (18, 1656, 17), (18, 16966, 6), (140, 5321, 10645), (10576, 10577, 18)


X(16968) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(78)

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

X(16968) lies on these lines:
{1, 6}, {3, 16583}, {19, 1968}, {32, 169}, {34, 292}, {40, 3959}, {46, 3125}, {56, 3290}, {63, 3721}, {78, 2238}, {197, 2223}, {212, 3747}, {239, 345}, {517, 14974}, {595, 1572}, {614, 1194}, {672, 3924}, {869, 14547}, {910, 3053}, {942, 5021}, {948, 7176}, {975, 16589}, {976, 3691}, {993, 16600}, {995, 9619}, {1376, 16605}, {1420, 9259}, {1427, 8770}, {1575, 1722}, {1914, 2082}, {2170, 3915}, {2345, 16824}, {3286, 16716}, {3727, 5250}, {3730, 9620}, {3735, 12514}, {3744, 4875}, {3752, 5013}, {3767, 5179}, {3772, 5254}, {3780, 3870}, {3914, 9598}, {4000, 7738}, {4136, 4438}, {5272, 16604}, {5540, 7031}, {5712, 16826}, {5716, 16830}, {6554, 7735}


X(16969) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(145)

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

X(16969) lies on these lines:
{1, 6}, {3, 9259}, {55, 3009}, {101, 2241}, {145, 2238}, {172, 3915}, {190, 330}, {292, 3445}, {595, 2242}, {995, 1500}, {999, 14974}, {1015, 3730}, {1149, 1334}, {1201, 2276}, {1388, 4559}, {1486, 2076}, {1575, 3208}, {1613, 1621}, {1909, 4713}, {1914, 9310}, {2162, 8616}, {2223, 5023}, {2271, 6767}, {2295, 3616}, {3052, 3747}, {3053, 8624}, {3057, 3290}, {3125, 5697}, {3241, 3780}, {3360, 8053}, {3501, 16604}, {3636, 3997}, {3721, 3877}, {3726, 3869}, {3727, 3890}, {3735, 3884}, {3863, 10387}, {3880, 16605}, {3898, 16600}, {4050, 16569}, {4095, 16825}, {4361, 16827}, {4383, 4393}, {5021, 7373}, {5134, 9651}, {6376, 10027}, {8572, 15815}, {9957, 16583}

X(16969) = polar conjugate of isotomic conjugate of X(22149)


X(16970) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(200)

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

X(16970) lies on these lines:
{1, 6}, {19, 2195}, {40, 14974}, {57, 3290}, {169, 595}, {198, 2223}, {200, 2238}, {239, 346}, {269, 292}, {614, 672}, {869, 2293}, {910, 3052}, {966, 3883}, {995, 9592}, {1253, 3747}, {1400, 2263}, {1418, 5575}, {1445, 1462}, {1471, 2285}, {1706, 16605}, {1707, 3509}, {1716, 2664}, {1722, 3501}, {1766, 13329}, {1914, 14827}, {2082, 3915}, {2093, 3125}, {2178, 3220}, {2191, 2260}, {2207, 7719}, {2276, 2999}, {2345, 4384}, {3195, 5089}, {3333, 5021}, {3684, 3749}, {3693, 4383}, {3721, 12526}, {3730, 9593}, {3752, 9574}, {3945, 16826}, {3959, 7991}, {3986, 4349}, {4344, 5296}, {4357, 4648}, {5120, 15287}, {5269, 5275}, {5749, 16823}, {9259, 13462}, {10315, 16283}, {12514, 16600}


X(16971) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(551)

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

X(16971) lies on these lines:
{1, 6}, {31, 9346}, {42, 1015}, {81, 99}, {86, 16829}, {354, 3125}, {551, 2238}, {574, 2177}, {940, 16834}, {1125, 3780}, {1468, 2241}, {1475, 1500}, {1573, 3720}, {1979, 9332}, {2242, 2251}, {2271, 3304}, {2295, 3244}, {3290, 5049}, {3293, 16604}, {3303, 5021}, {3721, 3881}, {3726, 3892}, {3727, 3874}, {3735, 3873}, {4257, 10987}, {4285, 8610}, {4393, 14996}, {5008, 8624}, {5312, 9336}


X(16972) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(612)

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

X(16972) lies on these lines:
{1, 6}, {19, 1974}, {69, 4748}, {141, 16831}, {182, 1766}, {239, 2345}, {292, 2277}, {346, 4393}, {583, 2279}, {597, 4971}, {612, 2238}, {672, 17017}, {742, 10436}, {940, 3290}, {966, 16830}, {1213, 3416}, {1428, 2285}, {1575, 2999}, {1691, 2959}, {2271, 5266}, {2276, 5256}, {2308, 5282}, {3589, 4384}, {3745, 5275}, {3931, 14974}, {3997, 9620}, {4386, 5269}, {4719, 5013}, {4989, 5750}, {5019, 8624}, {5257, 5847}, {5711, 16583}


X(16973) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(614)

Barycentrics    a (-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) : :

X(16973) lies on these lines:
{1, 6}, {38, 2280}, {39, 3811}, {57, 4386}, {63, 1914}, {69, 239}, {78, 2275}, {141, 4384}, {193, 3672}, {200, 1575}, {354, 5275}, {511, 990}, {519, 7739}, {524, 16834}, {599, 16833}, {614, 2238}, {672, 3938}, {742, 3875}, {758, 1572}, {869, 4878}, {908, 9599}, {936, 16604}, {966, 16823}, {976, 1475}, {982, 3684}, {997, 1015}, {1024, 9000}, {1469, 4327}, {1571, 8715}, {2082, 3721}, {2223, 12329}, {2241, 12514}, {2276, 3870}, {2876, 3056}, {3094, 3169}, {3158, 9574}, {3189, 7738}, {3338, 5277}, {3589, 16831}, {3618, 16826}, {3619, 16815}, {3620, 16816}, {3686, 16825}, {3729, 9055}, {3747, 3958}, {3755, 4660}, {3763, 16832}, {3767, 10916}, {3873, 5276}, {4353, 4856}, {5021, 5266}, {6763, 7031}, {6765, 9593}, {8624, 16946}


X(16974) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(976)

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

X(16974) lies on these lines:
{1, 6}, {2, 4372}, {31, 3721}, {32, 16600}, {292, 977}, {595, 3735}, {614, 16604}, {975, 16846}, {976, 2238}, {1125, 4153}, {1180, 2275}, {1333, 11102}, {1468, 3726}, {2276, 5262}, {2295, 3924}, {3125, 5264}, {3616, 4144}, {3666, 16367}, {3727, 3915}, {3780, 3938}, {3881, 9346}, {3959, 5255}, {4136, 6679}, {4204, 5311}, {4386, 5266}, {7876, 16706}


X(16975) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(75)) COLLINEATION IMAGE OF X(995)

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

X(16975) lies on these lines:
{1, 6}, {2, 668}, {8, 39}, {10, 2275}, {21, 2241}, {32, 2975}, {36, 4386}, {38, 2170}, {55, 11998}, {56, 5277}, {63, 1572}, {75, 716}, {78, 9619}, {81, 9346}, {100, 574}, {115, 11680}, {145, 1500}, {149, 9664}, {172, 8666}, {194, 1278}, {200, 9592}, {239, 980}, {274, 330}, {292, 996}, {386, 3780}, {513, 4492}, {519, 2276}, {538, 4441}, {594, 5069}, {982, 3125}, {986, 4051}, {993, 1914}, {995, 2238}, {999, 5275}, {1201, 3691}, {1475, 10459}, {1506, 11681}, {1574, 3617}, {1575, 3679}, {1698, 16604}, {1909, 17030}, {2092, 5839}, {2242, 5276}, {2277, 3686}, {2295, 4253}, {2298, 5042}, {2475, 9651}, {2548, 3436}, {2549, 3434}, {3421, 7736}, {3616, 16589}, {3624, 9336}, {3666, 16834}, {3670, 3959}, {3728, 7032}, {3752, 16833}, {3767, 10527}, {3872, 9620}, {4277, 4969}, {4361, 16696}, {4384, 16610}, {4393, 16704}, {4751, 16819}, {4853, 9593}, {4875, 16583}, {5013, 5687}, {5021, 5710}, {5046, 9665}, {5080, 5475}, {5082, 7738}, {5206, 5303}, {10468, 15 985}, {12647, 13006}, {16611, 16825}


X(16976) =  COMPLEMENT OF X(10151)

Barycentrics    ((SB+SC)*(2*SA+24*R^2-5*SW)-2* S^2)*SA : :
X(16976) = 3*X(2)+X(16386), 3*X(3)+X(2072), 2*X(3)+X(5159), 5*X(3)+X(10297)

As a point on the Euler line, this center has Shinagawa coefficients (E-7*F, -E+5*F).

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27430.

X(16976) lies on these lines:
{2, 3}, {1040, 10149}, {5972, 15311}, {8263, 10249}, {13416, 13754}

X(16976) = complement of X(10151)
X(16976) = orthoptic-circle-of-Steiner-inellipse-inverse-of X(7396)
X(16976) = polar circle-inverse-of X(6622)
X(16976) = inverse-in-complement-of-polar-circle of X(4)
X(16976) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 16386, 10151), (3, 549, 6676), (3, 631, 6823), (3, 3548, 550), (3, 11585, 548), (3, 15720, 3547), (3, 16196, 12362), (140, 548, 13406), (3523, 7484, 549), (7386, 15692, 3), (7494, 7503, 15760), (11250, 16238, 13488)


X(16977) =  EULER LINE INTERCEPT OF X(216)X(16306)

Barycentrics    SA*((SB+SC)*(2*SA^2-R^2*(6*SA- 5*SW)-2*SB*SC-SW^2)+2*(3*R^2- SA)*S^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27430.

X(16977) lies on these lines:
{2, 3}, {216, 16306}, {3564, 13198}, {11064, 13416}

X(16977) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7495, 6676), (3, 10257, 15122), (468, 10257, 5159), (5159, 6676, 468), (7495, 7499, 7568)


X(16978) =  REFLECTION OF X(6070) IN X(11800)

Barycentrics    (S^4-(198*R^4+2*R^2*(3*SA-46* SW)-SA^2+11*SW^2)*S^2-((18*R^ 4+SW^2)*(3*SA-8*SW)-6*R^2*SW*( 4*SA-11*SW))*SA)*(SB+SC) : :
X(26978) = 3*X(2)-4*X(12052), X(476)-3*X(3060), 2*X(3233)-3*X(12824)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16978) lies on these lines:
{2, 12052}, {23, 2967}, {30, 1986}, {476, 3060}, {511, 3258}, {1112, 7471}, {1553, 11807}, {3233, 12824}, {6070, 11800}, {10263, 16168}, {14611, 14984}

X(16978) = reflection of X(i) in X(j) for these (i,j): (1553, 11807), (6070, 11800)


X(16979) =  X(114)X(325)∩X(805)X(3060)

Barycentrics    (SB+SC)*(SA^2-SB*SC)*(13*S^4+( 3*SA^2-SW*(24*R^2+5*SW))*S^2-( SA^2-2*SW^2)*SW^2) : :
X(26979) = X(805)-3*X(3060)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16979) lies on these lines: {114, 325}, {805, 3060}


X(16980) =  X(1)X(51)∩X(8)X(511)

Barycentrics    a^2*((b^2+c^2)*a^3+(b+c)*(b^2+ c^2)*a^2-(b^2+c^2)^2*a-(b^2-c^ 2)^2*(b+c)) : :
X(26980) = 2*X(1)-3*X(51), 4*X(10)-3*X(3917), X(145)-3*X(3060), 9*X(373)-8*X(1125), 2*X(1216)-3*X(5790), 10*X(1698)-9*X(5650), 3*X(2979)-5*X(3617), 5*X(3567)-3*X(7967), 5*X(3616)-6*X(5943), 7*X(3622)-9*X(5640), 5*X(3623)-9*X(11002), 2*X(3655)-3*X(16226), 3*X(3753)-2*X(11573), 6*X(3819)-7*X(9780), 3*X(3873)-4*X(12109)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16980) lies on these lines:
{1, 51}, {6, 8192}, {8, 511}, {10, 3917}, {46, 3937}, {52, 952}, {65, 8679}, {143, 1483}, {145, 3060}, {181, 1468}, {184, 9798}, {185, 515}, {355, 5562}, {373, 1125}, {389, 944}, {518, 1829}, {859, 5399}, {956, 5752}, {962, 13598}, {970, 2975}, {976, 10544}, {1216, 5790}, {1482, 5446}, {1495, 8185}, {1698, 5650}, {1824, 14872}, {2807, 5691}, {2810, 3868}, {2979, 3617}, {3242, 9969}, {3271, 3915}, {3436, 10441}, {3555, 14557}, {3567, 7967}, {3616, 5943}, {3622, 5640}, {3623, 11002}, {3655, 16226}, {3751, 6467}, {3753, 11573}, {3819, 9780}, {3873, 12109}, {3884, 15049}, {4185, 9370}, {5080, 15488}, {5090, 12294}, {5462, 10246}, {5550, 6688}, {5603, 10110}, {5657, 15644}, {5690, 10625}, {5731, 9729}, {5818, 11793}, {5836, 9037}, {5844, 10263}, {5881, 14531}, {6243, 12645}, {7978, 11807}, {7979, 11808}, {7984, 11800}, {9781, 10595}, {10095, 10283}, {11557, 12898}, {13366, 16473}

X(16980) = midpoint of X(6243) and X(12645)
X(16980) = reflection of X(i) in X(j) for these (i,j):
X(16980) = anticomplement of X(1) wrt orthic triangle


X(16981) =  REFLECTION OF X(2) IN X(11002)

Barycentrics    a^2*(4*(b^2+c^2)*a^2-4*b^4+3* b^2*c^2-4*c^4) : :
X(26981) = 5*X(2)-8*X(51), 7*X(2)-8*X(373), 7*X(2)-4*X(2979), X(2)-4*X(3060), 19*X(2)-16*X(3819), 11*X(2)-8*X(3917), 3*X(2)-4*X(5640), 9*X(2)-8*X(5650), 13*X(2)-16*X(5943), 5*X(2)-4*X(7998), 17*X(2)-20*X(11451), 17*X(2)-16*X(15082), 7*X(51)-5*X(373), 14*X(51)-5*X(2979), 2*X(51)-5*X(3060), 19*X(51)-10*X(3819), 11*X(51)-5*X(3917), 6*X(51)-5*X(5640), 9*X(51)-5*X(5650), 13*X(51)-10*X(5943), 29*X(51)-20*X(6688), 4*X(51)-5*X(11002), 31*X(51)-20*X(12045), 17*X(51)-10*X(15082), 2*X(373)-7*X(3060), 19*X(373)-14*X(3819), 11*X(373)-7*X(3917), 6*X(373)-7*X(5640), 9*X(373)-7*X(5650), 13*X(373)-14*X(5943), 10*X(373)-7*X(7998), 4*X(373)-7*X(11002), 17*X(373)-14*X(15082), X(2979)-7*X(3060), 11*X(2979)-14*X(3917), 3*X(2979)-7*X(5640), 9*X(2979)-14*X(5650), 5*X(2979)-7*X(7998), 2*X(2979)-7*X(11002), 19*X(3060)-4*X(3819), 11*X(3060)-2*X(3917), 3*X(3060)-X(5640), 9*X(3060)-2*X(5650), 13*X(3060)-4*X(5943), 5*X(3060)-X(7998), 17*X(3060)-5*X(11451)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16981) lies on these lines:
{2, 51}, {4, 14449}, {6, 7492}, {20, 568}, {22, 5093}, {23, 1351}, {52, 3146}, {69, 7533}, {143, 3523}, {193, 2854}, {323, 11477}, {576, 11003}, {1112, 4232}, {1154, 3839}, {1350, 15018}, {1383, 5107}, {1495, 9716}, {1992, 8705}, {1994, 5102}, {3091, 6243}, {3098, 15019}, {3522, 9730}, {3524, 13321}, {3528, 16881}, {3543, 5663}, {3567, 15717}, {3620, 16776}, {3832, 5446}, {3854, 5562}, {5012, 15520}, {5032, 9019}, {5050, 6636}, {5056, 13421}, {5059, 13382}, {5068, 11412}, {5071, 13451}, {5097, 15080}, {5889, 13474}, {5890, 15683}, {5946, 15692}, {5987, 10754}, {6090, 13595}, {6101, 7486}, {6997, 15108}, {7712, 11422}, {7766, 13207}, {9777, 15246}, {9781, 10170}, {9971, 11160}, {10303, 13363}, {10304, 13391}, {11432, 16661}, {11935, 12105}, {15045, 15705}

X(16981) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (23, 1351, 11004), (193, 7519, 14683), (576, 15107, 11003), (5102, 6800, 1994)


X(16982) =  REFLECTION OF X(3861) IN X(12002)

Barycentrics    (SB+SC)*(3*S^2-SA*(2*R^2+3*SA) +6*SB*SC) : :
X(26982) = X(3)-3*X(143), X(3)-9*X(3060), 7*X(3)-15*X(3567), 5*X(3)-9*X(5946), X(3)+3*X(10263), 2*X(3)-3*X(12006), X(143)-3*X(3060), 7*X(143)-5*X(3567), 5*X(143)-3*X(5946), 11*X(143)-9*X(13321), 13*X(143)-7*X(15043), 19*X(143)-9*X(15045), 21*X(3060)-5*X(3567), 5*X(3060)-X(5946), 3*X(3060)+X(10263), 6*X(3060)-X(12006), 11*X(3060)-3*X(13321), 19*X(3060)-3*X(15045), 5*X(3567)+7*X(10263), 10*X(3567)-7*X(12006), 3*X(5946)+5*X(10263), 6*X(5946)-5*X(12006), 11*X(5946)-15*X(13321), 19*X(5946)-15*X(15045), 2*X(10263)+X(12006), 11*X(10263)+9*X(13321), 13*X(10263)+7*X(15043), 11*X(12006)-18*X(13321), 13*X(12006)-14*X(15043), 19*X(12006)-18*X(15045), 19*X(13321)-11*X(15045)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16982) lies on these lines:
{3, 143}, {5, 13421}, {23, 1493}, {30, 11565}, {51, 632}, {52, 3627}, {156, 1351}, {389, 12103}, {511, 3628}, {546, 1154}, {568, 3529}, {576, 11536}, {1112, 3043}, {1216, 12812}, {3090, 6101}, {3091, 6243}, {3146, 6102}, {3525, 11002}, {3845, 14531}, {3857, 5562}, {3861, 12002}, {5072, 11412}, {5076, 5889}, {5079, 9781}, {5462, 12108}, {5609, 14668}, {5876, 16261}, {6746, 14865}, {7553, 11264}, {10110, 12811}, {10625, 11592}, {11477, 13861}, {12086, 12236}, {12102, 13754}, {13630, 14855}, {14627, 15107}, {15012, 16881}

X(16982) = midpoint of X(i) and X(j) for these {i,j}: {5, 13421}, {7553, 11264}
X(16982) = reflection of X(i) in X(j) for these (i,j): (3861, 12002), (10625, 11592)
X(16982) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3060, 10263, 143), (10625, 13363, 11592)


X(16983) =  MIDPOINT OF X(7737) AND X(9973)

Barycentrics    ((b^2+c^2)*a^6+(b^4+4*b^2*c^2+ c^4)*a^4-(b^6+c^6)*a^2-(b^4+3* b^2*c^2+c^4)*(b^2-c^2)^2)*a^2 :
X(26983) = X(2549)-3*X(9971), 2*X(4045)-3*X(16776), 3*X(11286)-X(12220)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27433.

X(16983) lies on these lines:
{30, 1843}, {2549, 9971}, {3734, 9019}, {4045, 16776}, {5140, 15687}, {6240, 11397}, {7737, 9973}, {9969, 15048}, {10796, 15074}, {11286, 12220}, {14913, 14929}

X(16983) = midpoint of X(7737) and X(9973)


X(16984) =  X(2)X(6)∩X(3)X(7923)

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

X(16984) lies on these lines:
{2, 6}, {3, 7923}, {5, 9862}, {32, 7860}, {76, 14043}, {83, 7603}, {99, 7817}, {115, 384}, {148, 8369}, {187, 7919}, {315, 14065}, {316, 14046}, {574, 7884}, {620, 7827}, {625, 12150}, {626, 14047}, {1078, 7852}, {1285, 7823}, {1384, 7898}, {2896, 8363}, {3053, 7933}, {3090, 9863}, {3526, 12251}, {3552, 7851}, {3767, 7892}, {3788, 7839}, {3972, 7844}, {5007, 7899}, {5008, 7809}, {5025, 7737}, {5206, 7918}, {5286, 7891}, {5305, 7836}, {5309, 7835}, {5319, 7906}, {5346, 7796}, {5355, 7799}, {5368, 7905}, {5999, 9993}, {6179, 7867}, {7746, 7846}, {7747, 14045}, {7749, 7859}, {7751, 7930}, {7754, 7945}, {7755, 7832}, {7760, 7874}, {7763, 7920}, {7769, 7829}, {7771, 7913}, {7772, 7940}, {7780, 7944}, {7782, 7902}, {7783, 7797}, {7785, 8361}, {7786, 16923}, {7787, 7887}, {7790, 13586}, {7793, 7866}, {7795, 14067}, {7798, 7870}, {7803, 7907}, {7804, 14061}, {7805, 7909}, {7808, 16922}, {7815, 7943}, {7820, 14568}, {7824, 7834}, {7862, 7878}, {7864, 16925}, {7888, 7894}, {9983, 16896}, {10104, 10345}, {10722, 13335}, {11185, 14036}


X(16985) =  X(2)X(32)∩X(184)X(194)

Barycentrics    a^8 - b^4 c^4 : :

X(16985) lies on these lines:
{2, 32}, {6, 9229}, {184, 194}, {206, 8264}, {384, 1915}, {385, 419}, {702, 4577}, {826, 5027}, {894, 1580}, {1502, 9233}, {1916, 6660}, {1974, 2998}, {2966, 15391}, {3407, 11338}, {3504, 11325}, {3978, 4027}, {5117, 7823}, {7839, 14153}, {9218, 9514}


X(X16986) =  X(2)X(6)∩X(4)X(7928)

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

X(16986) lies on these lines:
{2, 6}, {4, 7928}, {5, 6194}, {32, 16895}, {76, 4045}, {83, 7854}, {115, 7937}, {140, 7945}, {148, 11287}, {160, 15246}, {187, 14036}, {194, 8362}, {316, 7865}, {327, 14387}, {384, 7800}, {620, 8289}, {626, 16921}, {1078, 3407}, {1447, 17292}, {1506, 7922}, {2548, 7939}, {2896, 7770}, {3096, 3934}, {3424, 15717}, {3526, 9755}, {3661, 4119}, {3734, 7831}, {3767, 7948}, {3785, 16898}, {3972, 7810}, {5054, 11177}, {5085, 5984}, {5475, 7883}, {6179, 7889}, {6683, 7796}, {6704, 7826}, {7081, 17291}, {7486, 9748}, {7745, 7929}, {7746, 7944}, {7747, 7936}, {7749, 7930}, {7751, 7859}, {7752, 7849}, {7753, 7850}, {7755, 7943}, {7761, 11361}, {7767, 7787}, {7768, 7808}, {7769, 7869}, {7771, 7820}, {7780, 7846}, {7783, 16043}, {7784, 16044}, {7785, 7879}, {7786, 7794}, {7790, 9466}, {7791, 17128}, {7793, 7819}, {7795, 7824}, {7799, 15482}, {7802, 14034}, {7803, 16897}, {7804, 7811}, {7812, 7848}, {7815, 7832}, {7827, 17131}, {7828, 7914}, {7836, 11285}, {7842, 14066}, {7847, 17130}, {7857, 7915}, {7858, 7896}, {7871, 9698}, {7885, 16924}, {7898, 8370}, {7911, 14062}, {7913, 14568}, {7924, 11185}, {7932, 8364}, {8782, 9478}, {11286, 14712}


X16987) =  X(2)X(6)∩X(5)X(12252)

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

X(16987) lies on these lines:
{2, 6}, {5, 12252}, {32, 16897}, {76, 16896}, {83, 7843}, {384, 4045}, {1506, 14047}, {3407, 7808}, {3526, 6194}, {5070, 9755}, {6704, 7828}, {7770, 7923}, {7783, 7819}, {7785, 8364}, {7786, 14043}, {7787, 7928}, {7803, 16895}, {7804, 7924}, {7805, 10159}, {7822, 7839}, {7824, 7846}, {7829, 17129}, {7864, 16898}, {7876, 14907}, {7878, 7914}, {7913, 14041}, {7915, 7947}, {7918, 14042}, {7941, 7944}, {7942, 16922}, {8289, 14061}, {8362, 10583}, {11177, 15699}


X(16988) =  X(2)X(6)∩X(5)X(10357)

Barycentrics    -a^4 - 4 a^2 b^2 - 2 b^4 - 4 a^2 c^2 - 5 b^2 c^2 - 2 c^4 : :

X(16988) lies on these lines:
{2, 6}, {5, 10357}, {32, 16896}, {39, 10159}, {76, 16897}, {83, 7848}, {384, 6292}, {1656, 6194}, {2549, 7876}, {3096, 5475}, {3407, 7815}, {3934, 7919}, {5355, 7859}, {6683, 7947}, {6704, 7768}, {7770, 7898}, {7783, 8362}, {7786, 7908}, {7800, 16895}, {7808, 7926}, {7822, 7824}, {7823, 16045}, {7849, 7941}, {7867, 16922}, {7901, 7914}, {7904, 16898}, {7923, 10335}, {7930, 16923}, {7935, 14042}, {7937, 14041}, {8290, 11646}, {11177, 11539}


X(16989) =  X(2)X(6)∩X(4)X(3398)

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

X(16989) lies on these lines:
{2, 6}, {4, 3398}, {5, 9755}, {20, 7864}, {23, 16324}, {32, 4045}, {39, 16925}, {76, 5319}, {83, 3767}, {98, 14561}, {99, 7739}, {147, 14912}, {148, 8289}, {182, 9753}, {194, 10336}, {315, 5007}, {316, 7884}, {384, 5286}, {458, 16318}, {575, 9744}, {631, 3095}, {754, 7913}, {1003, 15048}, {1285, 14712}, {1384, 8356}, {1447, 17367}, {1513, 5050}, {1975, 14037}, {2548, 7828}, {2549, 3972}, {2980, 7394}, {3096, 14023}, {3424, 3832}, {3545, 6033}, {3552, 7738}, {3705, 17121}, {3734, 5355}, {3785, 7876}, {3788, 5041}, {3926, 7839}, {3934, 5346}, {3974, 17280}, {5008, 7761}, {5034, 10352}, {5254, 14035}, {5305, 7770}, {5309, 7804}, {5368, 7751}, {5475, 7817}, {5939, 6034}, {5976, 13331}, {5999, 14853}, {6179, 7800}, {6337, 10335}, {6392, 17128}, {6680, 7763}, {6776, 13862}, {7081, 17368}, {7179, 17120}, {7737, 7790}, {7745, 7851}, {7747, 7902}, {7753, 7844}, {7754, 7819}, {7755, 7808}, {7758, 7832}, {7759, 7852}, {7760, 7795}, {7762, 7866}, {7768, 7943}, {7776, 8363}, {7785, 7932}, {7793, 16043}, {7798, 7820}, {7805, 7822}, {7807, 9605}, {7812, 7919}, {7818, 14075}, {7823, 7923}, {7826, 7914}, {7836, 14069}, {7838, 7867}, {7855, 7915}, {7858, 7942}, {7869, 7890}, {7877, 7944}, {7879, 8364}, {7893, 7948}, {7901, 7921}, {7905, 7930}, {7906, 14043}, {7941, 14065}, {7945, 13571}, {7947, 14067}, {10311, 17907}, {16895, 17129}


X(16990) =  X(2)X(6)∩X(4)X(2896)

Barycentrics    a^4 - 2 a^2 b^2 - b^4 - 2 a^2 c^2 - 4 b^2 c^2 - c^4 : :

X(16990) lies on these lines:
{2, 6}, {4, 2896}, {5, 7879}, {20, 7904}, {23, 16321}, {32, 16898}, {76, 2549}, {83, 14023}, {115, 7865}, {140, 7881}, {157, 6636}, {194, 16043}, {315, 3934}, {384, 3785}, {574, 14148}, {631, 7836}, {858, 16325}, {1078, 7795}, {1384, 6661}, {1447, 3661}, {1506, 7896}, {2548, 7768}, {3053, 14037}, {3090, 7912}, {3091, 7885}, {3096, 3767}, {3407, 7793}, {3424, 3522}, {3523, 7891}, {3524, 8289}, {3662, 7081}, {3705, 17287}, {3734, 6781}, {3926, 7824}, {3933, 11285}, {4045, 17131}, {5056, 9748}, {5286, 7876}, {5319, 7859}, {5355, 6292}, {5477, 10352}, {5976, 11646}, {5984, 8290}, {5999, 10519}, {6392, 7864}, {6683, 7855}, {7179, 17288}, {7737, 7811}, {7738, 10335}, {7746, 7849}, {7749, 7869}, {7750, 14035}, {7754, 8362}, {7755, 7914}, {7758, 7786}, {7761, 9466}, {7763, 7794}, {7767, 7770}, {7780, 7822}, {7784, 14063}, {7787, 16045}, {7808, 7826}, {7813, 15482}, {7830, 17130}, {7846, 10159}, {7916, 9698}, {7920, 16897}, {7929, 16044}, {7932, 10336}, {7937, 14568}, {7938, 14064}, {7939, 16921}, {8182, 10302}, {9744, 15819}, {9832, 16316}, {11331, 16318}, {14033, 14712}


X(16991) =  X(2)X(6)∩X(32)X(16905)

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

X(16991) lies on these lines:
{2, 6}, {32, 16905}, {76, 16906}, {192, 4972}, {194, 4202}, {1655, 16062}, {4429, 17759}, {5192, 7785}, {7823, 11319}, {16908, 17129}, {16910, 17128}


X(16992) =  X(2)X(6)∩X(3)X(274)

Barycentrics    a^4 - a^2 b^2 - 2 a^2 b c - 2 a b^2 c - a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 : :

X(16992) lies on these lines:
{2, 6}, {3, 274}, {21, 1975}, {22, 1602}, {25, 286}, {32, 11321}, {55, 75}, {56, 85}, {76, 405}, {99, 16370}, {171, 10436}, {261, 8033}, {310, 16343}, {315, 442}, {316, 17532}, {319, 4042}, {350, 1001}, {377, 7750}, {443, 3785}, {474, 1078}, {480, 7081}, {668, 9708}, {958, 1909}, {1444, 4224}, {1621, 4441}, {2476, 7773}, {2663, 5268}, {3053, 16915}, {3295, 17143}, {3303, 17144}, {3684, 4384}, {3739, 4386}, {3750, 3875}, {3760, 5259}, {3761, 5251}, {3926, 6857}, {3933, 6675}, {4252, 17103}, {4363, 17735}, {5013, 17684}, {5023, 17693}, {5283, 7754}, {5985, 5989}, {6063, 16090}, {7413, 10446}, {7483, 7763}, {7751, 16589}, {7767, 8728}, {7771, 16371}, {7793, 16917}, {7800, 17670}, {7851, 17550}, {7904, 17565}, {11112, 14907}, {11113, 11185}, {13881, 17669}, {14017, 16747}, {16342, 16705}, {16351, 16712}, {16502, 17030}, {16503, 17026}, {16912, 17129}, {16914, 17128}


X(16993) =  X(2)X(6)∩X(32)X(16912)

Barycentrics    a^4 + 2 a^2 b^2 + 4 a^2 b c + 4 a b^2 c + 2 a^2 c^2 + 4 a b c^2 + b^2 c^2 : :

X(16993) lies on these lines:
{2, 6}, {32, 16912}, {76, 16911}, {171, 17260}, {384, 16589}, {443, 7864}, {3684, 16826}, {4038, 17121}, {4386, 4687}, {5283, 7783}, {7787, 11108}, {7797, 8728}, {11321, 17128}


X(16994) =  X(2)X(6)∩X(32)X(16911)

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

X(16994) lies on these lines:
{2, 6}, {32, 16911}, {55, 4699}, {76, 16912}, {274, 7783}, {405, 17128}, {442, 7885}, {443, 7904}, {2896, 8728}, {3684, 16815}, {3750, 17117}, {4042, 17373}, {4386, 4751}, {6675, 7836}, {6857, 7891}, {7898, 17528}, {16589, 17129}


X(16995) =  X(2)X(6)∩X(32)X(16914)

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

X(16995) lies on these lines:
{2, 6}, {32, 16914}, {55, 4704}, {76, 16913}, {171, 17350}, {192, 4386}, {194, 5277}, {474, 7839}, {1655, 3552}, {3550, 17261}, {3684, 4393}, {4187, 7921}, {5283, 7793}, {5286, 17565}, {6179, 16589}, {7754, 16917}, {7900, 17669}, {7906, 17694}, {7920, 17670}, {7929, 17550}, {11321, 17129}


X(16996) =  X(2)X(6)∩X(32)X(16913)

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

X(16996) lies on these lines:
{2, 6}, {32, 16913}, {55, 1278}, {76, 16914}, {274, 7793}, {405, 17129}, {442, 7893}, {2476, 7900}, {3550, 17116}, {3684, 16816}, {3785, 17565}, {4352, 17689}, {4386, 4699}, {5985, 8782}, {7483, 7906}


X(16997) =  X(2)X(6)∩X(3)X(1655)

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

X(16997) lies on these lines:
{2, 6}, {3, 1655}, {21, 7793}, {32, 16916}, {75, 4396}, {76, 5277}, {100, 192}, {148, 17579}, {172, 6376}, {194, 404}, {274, 7751}, {315, 17669}, {330, 5253}, {350, 4386}, {474, 7754}, {668, 2242}, {750, 894}, {985, 17793}, {999, 9263}, {1078, 5283}, {1376, 17759}, {1447, 7225}, {1975, 17693}, {2177, 17319}, {2896, 17550}, {3053, 17692}, {3684, 17027}, {3933, 17694}, {4187, 7762}, {4188, 7783}, {4193, 7785}, {5046, 7823}, {5305, 17670}, {6180, 6649}, {6392, 6904}, {7780, 16589}, {7787, 17541}, {9310, 17752}, {11114, 14712}, {16917, 17129}, {16919, 17128}


X(16998) =  X(2)X(6)∩X(21)X(194)

Barycentrics    a^4 - a^2 b c - a b^2 c - a b c^2 - b^2 c^2 : :

X(16998) lies on these lines:
{2, 6}, {3, 8849}, {21, 194}, {31, 894}, {32, 274}, {39, 17684}, {41, 16827}, {55, 17759}, {75, 1914}, {76, 16916}, {105, 330}, {148, 11114}, {192, 1621}, {239, 2280}, {251, 1218}, {257, 3924}, {335, 5282}, {404, 7793}, {405, 1655}, {442, 7762}, {452, 6392}, {604, 1447}, {612, 2663}, {956, 9263}, {1468, 16476}, {1724, 17499}, {1909, 4426}, {1916, 5985}, {1975, 17692}, {2241, 17143}, {2475, 7823}, {2476, 7785}, {3053, 17693}, {3729, 8616}, {3767, 17669}, {4189, 7783}, {4366, 4441}, {4390, 10027}, {4400, 6376}, {5277, 6179}, {5283, 7760}, {5299, 17030}, {6652, 9318}, {7767, 17670}, {7787, 17686}, {7797, 17550}, {7805, 16589}, {8267, 16684}, {8300, 16825}, {14712, 17579}, {16503, 17027}, {16748, 16954}, {16779, 17026}, {16783, 17034}, {16786, 17028}, {16918, 17129}, {16920, 17128}


X(16999) =  X(2)X(6)∩X(32)X(16918)

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

X(16999) lies on these lines:
{2, 6}, {32, 16918}, {76, 16917}, {148, 11112}, {192, 1376}, {194, 474}, {274, 17129}, {384, 5277}, {404, 1655}, {405, 7793}, {894, 17122}, {1078, 16589}, {2478, 7823}, {3053, 16914}, {4187, 7785}, {4366, 4386}, {5254, 17565}, {5283, 7824}, {6376, 6645}, {6392, 17580}, {7750, 17685}, {7754, 16408}, {7762, 17527}, {7797, 17670}, {7836, 17694}, {7885, 17669}, {7928, 17550}, {10583, 17540}, {11113, 14712}, {16915, 17128}


X(17000) =  X(2)X(6)∩X(9)X(335)

Barycentrics    a^4 - 2 a^2 b c - 2 a b^2 c - 2 a b c^2 - b^2 c^2 : :

X(17000) lies on these lines:
{2, 6}, {9, 335}, {21, 7783}, {32, 16917}, {75, 4366}, {76, 16918}, {148, 11113}, {192, 1001}, {194, 405}, {238, 894}, {239, 16503}, {274, 384}, {330, 958}, {377, 7823}, {427, 2905}, {442, 7785}, {474, 7793}, {1447, 7175}, {1621, 17759}, {1655, 5047}, {1975, 16914}, {2896, 17670}, {3729, 15485}, {4384, 16779}, {4426, 6645}, {4649, 16830}, {5129, 6392}, {5254, 17685}, {5283, 7839}, {5299, 16819}, {7750, 17565}, {7754, 11108}, {7760, 16589}, {7762, 8728}, {7787, 11321}, {7923, 17550}, {10436, 14621}, {11112, 14712}, {16020, 17257}, {16484, 17319}, {16705, 16927}, {16784, 16829}, {16786, 16815}, {16916, 17128}


X(17001) =  X(2)X(6)∩X(32)X(16920)

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

X(17001) lies on these lines:
{2, 6}, {32, 16920}, {76, 16919}, {192, 15624}, {194, 4188}, {404, 7754}, {609, 6381}, {1655, 4189}, {3793, 11113}, {4190, 6392}, {4193, 7762}, {4386, 4396}, {5154, 7785}, {5277, 7751}, {5283, 7780}, {7767, 17550}, {7893, 17669}, {16915, 17129}


X(17002) =  X(2)X(6)∩X(21)X(7754)

Barycentrics    2 a^4 - a^2 b c - a b^2 c - a b c^2 - 2 b^2 c^2 : :

X(17002) lies on these lines:
{2, 6}, {21, 7754}, {32, 16919}, {76, 16920}, {194, 4189}, {274, 6179}, {536, 10987}, {894, 17126}, {902, 3729}, {1655, 16865}, {1914, 4441}, {2476, 7762}, {3006, 17363}, {3011, 4416}, {3793, 11112}, {3924, 17739}, {4188, 7793}, {4400, 4426}, {5141, 7785}, {5283, 7805}, {5305, 17550}, {6392, 6872}, {7783, 17548}, {7839, 17684}, {16916, 17129}


X(17003) =  X(2)X(6)∩X(32)X(16906)

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

X(17003) lies on these lines:
{2, 6}, {32, 16906}, {76, 16905}, {4202, 7793}, {16907, 17129}, {16909, 17128}


X(17004) =  X(2)X(6)∩X(3)X(148)

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

X(17004) lies on these lines:
{2, 6}, {3, 148}, {5, 7793}, {32, 16921}, {76, 620}, {115, 7771}, {140, 194}, {187, 11361}, {192, 5432}, {330, 5433}, {381, 14712}, {547, 3793}, {574, 14568}, {625, 7811}, {631, 7783}, {671, 5569}, {1078, 5025}, {1352, 9754}, {1506, 6179}, {1656, 7785}, {1916, 6036}, {2548, 16922}, {2896, 7887}, {2996, 15717}, {3053, 16044}, {3096, 7886}, {3291, 10163}, {3526, 7754}, {3628, 7762}, {3767, 7824}, {3785, 7885}, {3934, 7857}, {5023, 6658}, {6033, 9863}, {6292, 7942}, {6392, 10303}, {6655, 13881}, {6680, 16895}, {6683, 7856}, {6722, 7810}, {7571, 8878}, {7603, 7812}, {7615, 9855}, {7622, 11054}, {7697, 14693}, {7751, 7769}, {7752, 7780}, {7755, 7786}, {7763, 16923}, {7767, 7912}, {7768, 7862}, {7794, 7940}, {7797, 11285}, {7799, 17131}, {7800, 7901}, {7802, 14062}, {7814, 7826}, {7815, 7828}, {7822, 14067}, {7827, 15482}, {7831, 7844}, {7835, 9466}, {7854, 7899}, {7860, 12815}, {7894, 9698}, {7923, 16043}, {7928, 14064}, {7932, 8362}, {7938, 8361}, {7941, 14023}, {8182, 8597}, {8587, 8592}, {8598, 16509}, {10335, 15819}, {11185, 13586}, {14041, 14907}, {14066, 15031}, {16925, 17128}


X(17005) =  X(2)X(6)∩X(5)X(148)

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

X(17005) lies on these lines:
{2, 6}, {5, 148}, {32, 16923}, {39, 14061}, {76, 16922}, {98, 10486}, {99, 7603}, {140, 7785}, {194, 1656}, {384, 620}, {549, 14712}, {574, 14041}, {625, 7924}, {631, 7823}, {632, 7762}, {1078, 7845}, {1916, 7608}, {2548, 7907}, {3526, 7793}, {3793, 10124}, {3934, 7947}, {4045, 14046}, {5070, 7754}, {5475, 13586}, {6683, 7899}, {6722, 7827}, {7622, 9855}, {7746, 7839}, {7749, 7858}, {7752, 7761}, {7756, 14044}, {7763, 16921}, {7764, 17129}, {7771, 7775}, {7782, 14042}, {7786, 7862}, {7808, 7940}, {7814, 7815}, {7828, 9698}, {7847, 14045}, {7859, 14047}, {7867, 16897}, {7887, 7923}, {7891, 16924}, {7912, 7928}, {7930, 16896}, {7934, 15482}, {8176, 8597}, {8366, 14535}, {8591, 12040}, {8592, 8786}


X(17006) =  X(2)X(6)∩X(5)X(14712)

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

X(17006) lies on these lines:
{2, 6}, {5, 14712}, {32, 16922}, {76, 16923}, {126, 11056}, {140, 7783}, {148, 549}, {182, 7607}, {194, 3526}, {384, 7749}, {511, 10486}, {625, 1078}, {671, 1153}, {1656, 7793}, {1916, 15819}, {3090, 7823}, {3628, 3793}, {5097, 7608}, {5206, 14042}, {5569, 8597}, {6292, 14047}, {6722, 7831}, {7495, 7665}, {7617, 8588}, {7619, 11054}, {7746, 7790}, {7769, 7813}, {7771, 14041}, {7780, 7941}, {7815, 7901}, {7830, 12815}, {7862, 7939}, {7886, 7948}, {7887, 7928}, {7907, 17128}, {7923, 11285}, {7924, 14061}, {7942, 16897}, {8587, 8593}, {8591, 16509}, {9754, 13862}, {9830, 10487}, {11317, 15655}, {15031, 15513}


X(17007) =  X(2)X(6)∩X(32)X(16909)

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

X(17007) lies on these lines:
{2, 6}, {32, 16909}, {76, 16910}, {1655, 17676}, {3263, 4643}, {3290, 4690}, {3920, 17248}, {4202, 7754}, {5192, 7762}, {7191, 17363}, {10327, 17257}, {16906, 17129}


X(17008) =  X(2)X(6)∩X(4)X(2080)

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

X(17008) lies on these lines:
{2, 6}, {4, 2080}, {5, 3793}, {23, 157}, {32, 16924}, {76, 2021}, {114, 9754}, {115, 14907}, {140, 7754}, {148, 376}, {160, 9149}, {187, 11185}, {192, 5218}, {194, 631}, {315, 625}, {330, 7288}, {468, 9308}, {543, 8588}, {620, 17131}, {671, 8182}, {1078, 3767}, {1384, 8370}, {1655, 6910}, {1656, 7762}, {1916, 6194}, {2548, 6179}, {2549, 7771}, {2896, 14064}, {2996, 3522}, {3053, 14035}, {3090, 7785}, {3091, 7823}, {3098, 6055}, {3164, 7493}, {3523, 6392}, {3533, 13571}, {3545, 9996}, {3785, 5025}, {3926, 7907}, {3934, 16898}, {5286, 7824}, {5305, 11285}, {5319, 7786}, {5346, 6683}, {5355, 15482}, {5485, 8587}, {5569, 8589}, {5984, 7710}, {6103, 17907}, {6722, 7818}, {7426, 16325}, {7492, 14652}, {7618, 11054}, {7620, 9855}, {7749, 7751}, {7750, 13881}, {7752, 14023}, {7755, 7803}, {7758, 7769}, {7767, 7887}, {7795, 7857}, {7797, 16043}, {7800, 7828}, {7810, 7844}, {7811, 14061}, {7826, 7862}, {7854, 7886}, {7879, 8361}, {7898, 16041}, {7906, 16923}, {7921, 16922}, {8598, 15655}, {8722, 11623}, {9752, 13862}, {9832, 16315}, {10583, 16045}, {11317, 16509}


X(17009) =  X(3)X(10265)∩X(21)X(10165)

Barycentrics    a (-2a^9 +4a^8(b+c) +2 a^7(2b^2-5b c+2c^2) -3a^6(4b^3-b^2 c-b c^2+4c^3) + 2a^5b c(7b^2-8b c+7c^2) + a^4(12b^5-17b^4 c+10b^3 c^2+10b^2c^3-17b c^4+12c^5) + a^3(-4b^6+2b^5 c+9b^4 c^2-22b^3 c^3+9b^2c^4+2b c^5-4c^6) - a^2(b-c)^2(4b^5-b^4c+6b^3c^2+6b^2c^3-b c^4+4c^5) + a(b^2-c^2)^2(2b^4-6b^3c+7b^2c^2-6b c^3+2c^4) + b c(b-c)^4(b+c)^3) : :
X(17009) = (2r^2+r R-2R^2) X(214) + 2R(r+R) X(960)

See Dao Thanh Oai and Angel Montesdeoca, HG030418.

X(17009) lies on these lines:
{3,10265}, {21,10165}, {30,6713}, {104,15931}, {214,960}, {758,15528}, {1155,5427}, {1768,6875}, {2829,6675}, {3256,14563}, {3651,10090}, {5426,6950}, {10543,12832}, {14526,14800}


X(17010) =  X(1)X(89)∩X(36)X(516)

Barycentrics    a(2a^6 - 2a^5(b+c) - 4a^4(b^2-b c+c^2) + a^3(4b^3-b^2 c-b c^2+4c^3) + a^2(b-c)^2(2b^2+b c+2c^2) - a(b-c)^2(2b^3+b^2c+b c^2+2c^3) - b c(b^2-c^2)^2) : :
X(17010) = R X(1)+r X(104)

See Dao Thanh Oai and Angel Montesdeoca, HG030418.

X(17010) lies on these lines:
{1,89}, {3,950}, {21,908}, {36,516}, {56,13464}, {57,6950}, {104,2078}, {226,6914}, {499,4190}, {514,1946}, {515,5172}, {519,10087}, {527,954}, {993,8069}, {1155,5427}, {1319,2800}, {1478,6974}, {1728,4855}, {1776,6326}, {3086,3522}, {3601,6875}, {4292,6906}, {4311,5450}, {4973,5570}, {5193,12775}, {5840,15325}, {6942,9581}, {7288,15866}, {7675,10398}, {8666,11508}, {8758,11700}, {10624,11012}, {11019,14793}, {12047,14804}

X(17010) = midpoint of X(36) and X(10058)


X(17011) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(37)

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

X(17011) lies on these lines:
{1, 2}, {6, 3219}, {27, 1870}, {31, 17592}, {34, 6994}, {38, 4649}, {56, 11340}, {57, 1442}, {63, 1449}, {81, 593}, {86, 4359}, {100, 3745}, {223, 7190}, {226, 7269}, {238, 1962}, {244, 4038}, {312, 17393}, {321, 4360}, {350, 1230}, {380, 9536}, {388, 7382}, {464, 1062}, {469, 6198}, {497, 7381}, {553, 1443}, {559, 7051}, {846, 2308}, {894, 17147}, {940, 4850}, {942, 1817}, {968, 16475}, {999, 11350}, {1051, 1757}, {1171, 1931}, {1185, 16525}, {1203, 3743}, {1211, 17045}, {1255, 3723}, {1376, 9347}, {1386, 1621}, {1453, 16865}, {1482, 16435}, {1796, 5030}, {2177, 17716}, {2323, 16579}, {2475, 5717}, {2667, 3797}, {2895, 4357}, {3099, 17449}, {3100, 3151}, {3210, 17379}, {3247, 3305}, {3315, 4883}, {3618, 17776}, {3663, 17483}, {3670, 4658}, {3672, 5905}, {3739, 5333}, {3750, 17469}, {3751, 7226}, {3759, 5278}, {3782, 17395}, {3868, 16368}, {3873, 17599}, {3889, 11343}, {3896, 5263}, {3946, 5249}, {3969, 17289}, {3995, 17319}, {4021, 17484}, {4270, 17248}, {4272, 4886}, {4285, 17256}, {4296, 7560}, {4383, 16777}, {4418, 4970}, {4641, 16666}, {4661, 7174}, {4719, 5253}, {4854, 5057}, {4974, 10180}, {4980, 17160}, {5045, 11349}, {5284, 15569}, {5739, 17321}, {6505, 9776}, {6651, 8054}, {8025, 17495}, {8540, 17611}, {8555, 14953}, {9345, 17063}, {9539, 10382}, {11347, 15934}, {11680, 17723}, {17126, 17594}, {17184, 17302}

X(17011) = {X(1),X(2)}-harmonic conjugate of X(17019)


X(17012) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(44)

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

X(17012) lies on these lines:
{1, 2}, {6, 2243}, {9, 14997}, {31, 17601}, {44, 3219}, {45, 4383}, {57, 89}, {63, 16670}, {81, 88}, {100, 1386}, {149, 3755}, {244, 4649}, {320, 4285}, {321, 17160}, {678, 17469}, {748, 17592}, {756, 17600}, {894, 17495}, {896, 16477}, {908, 3946}, {1100, 16610}, {1126, 3953}, {1150, 3759}, {1155, 5135}, {1442, 3911}, {1449, 3306}, {1453, 4189}, {1621, 3246}, {1962, 17123}, {2238, 16521}, {2308, 17596}, {2975, 4719}, {3305, 16676}, {3618, 17740}, {3662, 4270}, {3663, 17484}, {3677, 4430}, {3681, 17599}, {3751, 4392}, {3875, 4671}, {3936, 4272}, {4003, 4663}, {4346, 5905}, {4358, 4360}, {4413, 9347}, {4414, 16468}, {4682, 9342}, {4784, 8034}, {4868, 5315}, {4887, 17483}, {5204, 11340}, {5219, 7269}, {5225, 7381}, {5229, 7382}, {5233, 17380}, {5235, 17348}, {5241, 17045}, {5249, 17067}, {5708, 11347}, {5718, 17366}, {12702, 16435}, {16475, 17126}, {16704, 17121}, {16972, 17756}, {17127, 17594}


X(17013) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(45)

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

X(17013) lies on these lines:
{1, 2}, {37, 14997}, {81, 89}, {88, 940}, {678, 17716}, {1100, 4850}, {1449, 3218}, {1817, 5708}, {3219, 16670}, {3666, 5332}, {3672, 17484}, {3936, 17380}, {4038, 9335}, {4346, 17483}, {4358, 17393}, {4360, 4671}, {4383, 16672}, {4392, 4649}, {4430, 17599}, {7226, 17600}, {8025, 17490}, {8148, 16435}, {17126, 17601}, {17127, 17592}, {17379, 17495}


X(17014) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(220)

Barycentrics    5 a^2 + 2 a b + b^2 + 2 a c - 2 b c + c^2 : :

X(17014) lies on these lines:
{1, 2}, {6, 144}, {7, 1419}, {57, 3160}, {69, 17305}, {81, 279}, {193, 4741}, {226, 5543}, {241, 4850}, {321, 17158}, {344, 17393}, {346, 3618}, {379, 11036}, {390, 1386}, {391, 3759}, {597, 17318}, {673, 11038}, {894, 4452}, {948, 7247}, {952, 7402}, {966, 17045}, {999, 11349}, {1100, 3945}, {1429, 2280}, {1442, 8732}, {1453, 11106}, {1482, 7397}, {1743, 4021}, {1992, 4389}, {2238, 16518}, {2321, 4460}, {2345, 4852}, {3219, 16572}, {3589, 17269}, {3619, 17377}, {3620, 17383}, {3629, 17323}, {3663, 16667}, {3755, 4344}, {3758, 4454}, {3875, 4461}, {4310, 4649}, {4346, 4644}, {4356, 16469}, {4361, 4472}, {4371, 17303}, {4373, 14621}, {4402, 10436}, {4464, 17286}, {4470, 17119}, {4648, 16884}, {4657, 4690}, {4719, 5265}, {4856, 17272}, {4869, 16706}, {4910, 17229}, {4916, 17231}, {4969, 17325}, {6172, 16670}, {6329, 17262}, {7229, 17151}, {7269, 8232}, {7490, 11396}, {7960, 9965}, {8158, 16435}, {9441, 11200}, {11008, 17273}, {16668, 17276}, {17121, 17257}


X(17015) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(392)

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

X(17015) lies on these lines:
{1, 2}, {6, 3877}, {21, 6043}, {34, 4323}, {36, 4868}, {65, 17074}, {81, 517}, {404, 4646}, {529, 4854}, {758, 16474}, {902, 5429}, {964, 4673}, {989, 1392}, {999, 4850}, {1060, 11041}, {1100, 1320}, {1203, 3884}, {1220, 3702}, {1386, 5919}, {1449, 7962}, {1468, 4650}, {1824, 1870}, {2099, 4318}, {2334, 12635}, {2975, 3931}, {3218, 4424}, {3315, 5049}, {3340, 4296}, {3743, 5258}, {3745, 3880}, {3885, 5710}, {3890, 16466}, {3892, 16490}, {3895, 5269}, {3898, 5315}, {3902, 5263}, {3946, 9317}, {4256, 4881}, {4689, 17549}, {4695, 17122}, {5119, 17126}, {5130, 6198}, {5260, 6051}, {5711, 14923}, {5725, 11680}, {14571, 17519}


X(17016) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(960)

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

X(17016) lies on these lines:
{1, 2}, {6, 3727}, {21, 3931}, {35, 4868}, {37, 5260}, {40, 17126}, {56, 4850}, {58, 4424}, {65, 81}, {100, 4646}, {171, 4642}, {321, 1220}, {429, 6198}, {651, 12709}, {894, 17164}, {941, 5336}, {968, 16865}, {977, 2334}, {986, 1468}, {998, 11415}, {1039, 6995}, {1043, 3896}, {1096, 7518}, {1104, 1621}, {1172, 1880}, {1191, 3890}, {1203, 3878}, {1319, 4719}, {1386, 2330}, {1402, 4225}, {1404, 1449}, {1411, 5331}, {1441, 16749}, {1453, 5250}, {1717, 10572}, {1721, 5059}, {1757, 11533}, {1834, 5086}, {1854, 10394}, {1870, 4185}, {2292, 3219}, {2475, 3914}, {2476, 5725}, {2650, 4649}, {2975, 3666}, {3100, 3486}, {3315, 17609}, {3434, 5716}, {3685, 11319}, {3698, 4682}, {3702, 13740}, {3743, 5251}, {3745, 5836}, {3752, 5253}, {3780, 16519}, {3871, 5266}, {3874, 16474}, {3877, 16466}, {3884, 5315}, {3915, 16478}, {3946, 10106}, {3962, 4663}, {4189, 17594}, {4640, 16948}, {4641, 11684}, {4658, 17191}, {4972, 7270}, {5047, 6051}, {5396, 5797}, {5484, 17302}, {5710, 14923}, {7438, 11363}, {10448, 17592}, {11681, 17720}, {12513, 17599}, {12607, 17602}, {15888, 17061}, {16600, 16785}


X(17017) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(984)

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

X(17017) lies on these lines:
{1, 2}, {6, 38}, {9, 3989}, {21, 16478}, {31, 1386}, {37, 748}, {48, 354}, {55, 5096}, {56, 199}, {63, 2308}, {81, 982}, {100, 17716}, {171, 4850}, {223, 4327}, {238, 17600}, {244, 940}, {310, 870}, {497, 4336}, {559, 10648}, {597, 4884}, {613, 11031}, {672, 16972}, {750, 3745}, {756, 4383}, {846, 17127}, {894, 17155}, {902, 16491}, {968, 7290}, {977, 5331}, {1001, 1962}, {1040, 2293}, {1051, 4430}, {1082, 10647}, {1104, 10448}, {1150, 3791}, {1214, 1471}, {1215, 3891}, {1449, 3509}, {1468, 1472}, {1621, 17592}, {1757, 7226}, {1836, 17301}, {2177, 3744}, {2241, 6155}, {2292, 16466}, {2886, 17726}, {3210, 4418}, {3218, 17591}, {3219, 16468}, {3333, 8555}, {3589, 3703}, {3706, 4852}, {3742, 9345}, {3772, 17723}, {3821, 6327}, {3873, 4649}, {3875, 4365}, {3914, 3946}, {3915, 3931}, {3923, 17147}, {3925, 17366}, {3930, 4423}, {3966, 4657}, {3980, 17495}, {3995, 4011}, {4085, 5014}, {4387, 17318}, {4388, 17302}, {4389, 4683}, {4642, 5710}, {4682, 16610}, {4854, 17395}, {4865, 4972}, {4883, 4906}, {4974, 5278}, {5132, 16687}, {5718, 17061}, {6536, 17321}, {7032, 17187}, {7073, 11238}, {7194, 9277}, {7221, 10382}, {9347, 17122}, {11680, 17722}, {16884, 17450}, {17126, 17596}


X(17018) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(1001)

Barycentrics    a (2 a b + 2 a c + b c) : :

X(17018) lies on these lines:
{1, 2}, {6, 1621}, {7, 8049}, {31, 3750}, {37, 3681}, {38, 4430}, {55, 81}, {56, 4210}, {75, 3896}, {86, 3996}, {89, 17601}, {100, 940}, {101, 9094}, {144, 4343}, {171, 2177}, {192, 714}, {210, 9330}, {291, 3722}, {350, 17393}, {354, 4850}, {390, 14547}, {495, 3136}, {500, 6361}, {518, 7226}, {581, 962}, {607, 4233}, {672, 1449}, {748, 14997}, {749, 1100}, {750, 4038}, {956, 16343}, {958, 2334}, {968, 3219}, {984, 1962}, {991, 9778}, {993, 16474}, {999, 4191}, {1011, 3295}, {1056, 6817}, {1058, 6818}, {1066, 11036}, {1126, 1724}, {1197, 9463}, {1214, 7672}, {1215, 4671}, {1386, 3748}, {1442, 8270}, {1468, 4189}, {1482, 4192}, {1870, 4196}, {2238, 16777}, {2239, 17600}, {2308, 8616}, {2318, 5686}, {2335, 2346}, {2356, 6995}, {2594, 3485}, {2663, 17350}, {3191, 5815}, {3210, 17140}, {3218, 17594}, {3434, 5712}, {3475, 13576}, {3486, 14956}, {3487, 5399}, {3666, 3873}, {3683, 4663}, {3689, 4682}, {3736, 8025}, {3743, 5904}, {3752, 4883}, {3755, 5249}, {3868, 3931}, {3871, 5711}, {3876, 6051}, {3891, 4360}, {3945, 17784}, {3997, 9331}, {4042, 5235}, {4207, 6198}, {4255, 5253}, {4270, 5296}, {4281, 17588}, {4323, 10571}, {4336, 9539}, {4366, 16955}, {4383, 5284}, {4551, 5226}, {4658, 5264}, {4719, 17609}, {4722, 7262}, {4734, 17495}, {4868, 5902}, {4970, 17155}, {5173, 17080}, {5247, 16865}, {5396, 5603}, {5453, 12702}, {5707, 11491}, {5710, 11322}, {5718, 11680}, {6327, 17778}, {6645, 16954}, {6767, 16058}, {7373, 16059}, {9345, 17122}, {10458, 16704}, {10950, 14008}, {15934, 16056}, {16884, 17756}, {17063, 17450}, {17379, 17759}, {17449, 17591}, {17469, 17715}


X(17019) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(1100)

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

X(17019) lies on these lines:
{1, 2}, {27, 6198}, {33, 6994}, {37, 81}, {38, 4038}, {55, 9347}, {57, 7269}, {63, 3247}, {86, 321}, {89, 3928}, {100, 4682}, {111, 1310}, {171, 1962}, {226, 1029}, {244, 17600}, {312, 17394}, {335, 741}, {388, 7381}, {464, 1060}, {469, 1870}, {497, 7382}, {593, 1963}, {750, 17592}, {756, 4649}, {894, 3995}, {940, 3218}, {967, 14974}, {968, 17126}, {982, 9345}, {1211, 17390}, {1215, 5625}, {1230, 1909}, {1386, 5284}, {1443, 4654}, {1449, 3305}, {1453, 16859}, {1621, 3745}, {1824, 14014}, {2248, 3721}, {2294, 3101}, {2605, 4789}, {2895, 3879}, {3100, 7560}, {3151, 4296}, {3175, 4670}, {3295, 11350}, {3578, 17256}, {3664, 17483}, {3666, 3723}, {3782, 17392}, {3945, 5905}, {3969, 17315}, {3993, 4418}, {4359, 4360}, {4383, 16884}, {4430, 7174}, {4656, 17484}, {4667, 17781}, {4687, 5278}, {5046, 5717}, {5295, 14005}, {7308, 14997}, {7675, 9539}, {10246, 16435}, {16484, 17469}, {17147, 17319}, {17184, 17300}, {17450, 17598}

X(17019) = {X(1),X(2)}-harmonic conjugate of X(17011)


X(17020) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(16669)

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

X(17020) lies on these lines:
{1, 2}, {63, 14997}, {81, 16610}, {1054, 2308}, {1453, 4188}, {1817, 5122}, {3218, 3752}, {3219, 4383}, {3666, 16814}, {3677, 4661}, {3745, 9342}, {4719, 5260}, {5437, 14996}, {5741, 16706}, {9350, 17716}, {17125, 17592}


X(17021) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(16666)

Barycentrics    a (a^2 + 2 a b + b^2 + 2 a c + 5 b c + c^2) : :

X(17021) lies on these lines:
{1, 2}, {9, 14996}, {37, 2666}, {44, 81}, {45, 940}, {63, 89}, {86, 4358}, {88, 1255}, {100, 15569}, {226, 1443}, {678, 3750}, {750, 17601}, {756, 4038}, {984, 9345}, {1001, 9347}, {1150, 4687}, {1442, 5219}, {1449, 14997}, {1453, 17570}, {1621, 4682}, {1962, 17122}, {2238, 16522}, {3246, 3745}, {3247, 3306}, {3305, 16670}, {3664, 17484}, {3723, 16610}, {3751, 9330}, {3911, 7269}, {3936, 17317}, {4359, 17160}, {4656, 4896}, {4661, 7322}, {4671, 10436}, {4698, 5235}, {4850, 16777}, {5217, 11340}, {5225, 7382}, {5229, 7381}, {5241, 17390}, {5483, 17056}, {9539, 10383}, {16672, 17595}, {16704, 17260}, {17124, 17592}, {17319, 17495}


X(17022) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(16667)

Barycentrics    a (a^2 + 2 a b + b^2 + 2 a c + 6 b c + c^2) : :

X(17022) lies on these lines:
{1, 2}, {6, 7308}, {7, 4656}, {9, 940}, {33, 7490}, {35, 11350}, {37, 57}, {38, 10980}, {40, 6051}, {45, 3929}, {63, 3731}, {77, 5226}, {81, 1743}, {86, 2297}, {165, 750}, {171, 4512}, {223, 5219}, {226, 269}, {312, 10436}, {329, 3664}, {354, 7174}, {380, 9816}, {440, 1038}, {518, 7322}, {553, 4419}, {573, 12555}, {748, 16469}, {756, 5223}, {990, 10857}, {991, 1750}, {1001, 4682}, {1040, 7536}, {1076, 9612}, {1211, 17296}, {1255, 4850}, {1376, 15569}, {1386, 8167}, {1412, 2267}, {1449, 4383}, {1453, 11108}, {1468, 5234}, {1817, 4653}, {1962, 17124}, {2292, 3339}, {2324, 5745}, {2334, 3983}, {3175, 4659}, {3243, 4883}, {3247, 3666}, {3306, 16673}, {3338, 16439}, {3452, 5712}, {3576, 16435}, {3583, 7382}, {3585, 7381}, {3601, 11347}, {3646, 16466}, {3663, 9776}, {3677, 3742}, {3723, 16602}, {3745, 4423}, {3749, 16484}, {3751, 4038}, {3752, 16777}, {3772, 17245}, {3782, 6173}, {3879, 14555}, {3928, 16676}, {3973, 14996}, {4340, 12572}, {4359, 17151}, {4415, 4654}, {4417, 17317}, {4687, 14829}, {4698, 5737}, {4851, 5743}, {4888, 5905}, {5010, 11340}, {5084, 5717}, {5284, 9347}, {5435, 7190}, {5573, 17599}, {6703, 17279}, {6831, 9121}, {7203, 14321}, {7522, 9817}, {7987, 10448}, {8545, 17074}, {16475, 17123}, {16487, 17469}, {17122, 17594}, {17319, 17490}


X(17023) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(5280)

Barycentrics    2 a^2 + a b + b^2 + a c + c^2 : :

X(17023) lies on these lines:
{1, 2}, {6, 4357}, {7, 4747}, {9, 3618}, {37, 3589}, {39, 712}, {44, 597}, {56, 11343}, {57, 348}, {63, 4253}, {69, 1449}, {75, 3946}, {81, 5299}, {83, 226}, {86, 142}, {141, 1100}, {171, 12194}, {190, 17320}, {192, 4021}, {193, 16667}, {319, 17307}, {320, 4667}, {344, 3247}, {350, 4044}, {379, 2140}, {441, 17102}, {458, 1785}, {515, 7377}, {524, 16666}, {527, 3758}, {536, 17369}, {572, 12610}, {594, 4852}, {692, 16792}, {894, 3663}, {908, 16788}, {910, 5834}, {940, 16502}, {944, 7402}, {946, 6996}, {980, 2275}, {988, 14001}, {1086, 4670}, {1213, 17348}, {1215, 12263}, {1266, 4363}, {1375, 5439}, {1386, 3883}, {1453, 13725}, {1475, 16705}, {1580, 4425}, {1621, 5314}, {1654, 17121}, {1699, 7406}, {1743, 17257}, {1826, 17918}, {1838, 11341}, {1848, 1973}, {1914, 5337}, {1930, 4359}, {2260, 16574}, {2321, 4360}, {2325, 4664}, {2329, 3452}, {2345, 3875}, {3061, 5745}, {3305, 17742}, {3306, 5011}, {3619, 17296}, {3629, 16668}, {3662, 3664}, {3668, 17086}, {3672, 3729}, {3685, 4356}, {3686, 3759}, {3707, 17256}, {3723, 17243}, {3739, 17366}, {3752, 6703}, {3755, 5263}, {3763, 4851}, {3765, 6381}, {3797, 3993}, {3817, 7384}, {3821, 14621}, {3834, 17392}, {3911, 7146}, {3943, 17359}, {3945, 17298}, {3950, 17280}, {3986, 17260}, {4000, 10436}, {4029, 17264}, {4297, 6999}, {4298, 17691}, {4349, 4645}, {4361, 4967}, {4395, 4472}, {4407, 4753}, {4419, 4480}, {4452, 7229}, {4464, 17293}, {4644, 17274}, {4648, 17282}, {4653, 16050}, {4675, 17290}, {4681, 17340}, {4686, 7227}, {4687, 6666}, {4690, 4969}, {4698, 17337}, {4700, 17250}, {4704, 17339}, {4708, 17330}, {4755, 6687}, {4758, 17067}, {4796, 17235}, {4856, 17238}, {4899, 7174}, {4909, 17232}, {4982, 17360}, {5204, 16436}, {5217, 16431}, {5228, 9436}, {5248, 16367}, {5253, 11349}, {5257, 17277}, {5266, 8362}, {5603, 7397}, {5712, 16780}, {5717, 16062}, {5839, 17270}, {5988, 10352}, {6329, 16669}, {6646, 17120}, {7277, 17345}, {7770, 13161}, {8025, 17192}, {9776, 17170}, {15668, 17278}, {16609, 17048}, {16752, 17175}, {16777, 17279}, {17227, 17378}, {17228, 17377}, {17229, 17388}, {17231, 17390}, {17233, 17371}, {17234, 17370}, {17236, 17364}, {17239, 17362}, {17242, 17358}, {17245, 17356}, {17246, 17351}, {17247, 17350}, {17248, 17349}, {17249, 17347}, {17275, 17327}, {17276, 17323}, {17281, 17318}, {17283, 17317}, {17285, 17315}, {17286, 17314}, {17291, 17300}

X(17023) = complement of X(3661)
X(17023) = isotomic conjugate of polar conjugate of X(1890)


X(17024) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(3242)

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

X(17024) lies on these lines:
{1, 2}, {6, 4430}, {22, 999}, {23, 3304}, {25, 7373}, {31, 4392}, {33, 7409}, {34, 7408}, {38, 7262}, {55, 15246}, {56, 6636}, {81, 17597}, {105, 8652}, {238, 7226}, {244, 17716}, {251, 2242}, {330, 8267}, {354, 7712}, {388, 7394}, {496, 5133}, {497, 7391}, {613, 1994}, {750, 9335}, {902, 17591}, {940, 3315}, {982, 17126}, {988, 17548}, {1015, 1180}, {1056, 6997}, {1058, 1370}, {1386, 3873}, {1421, 5226}, {1442, 3598}, {1621, 17599}, {1627, 2241}, {1870, 6995}, {3218, 3677}, {3219, 7290}, {3242, 4661}, {3263, 17393}, {3295, 7485}, {3303, 7496}, {3410, 12589}, {3553, 14930}, {3600, 7500}, {3681, 14997}, {3742, 9347}, {3744, 4850}, {3745, 4906}, {3891, 4671}, {3989, 15485}, {4188, 5266}, {4220, 10246}, {4224, 15934}, {4310, 17483}, {4366, 16949}, {4741, 4749}, {5014, 16706}, {5276, 16884}, {5310, 5563}, {5359, 16781}, {6198, 7378}, {6645, 16932}, {6767, 7484}, {7667, 15172}, {9330, 17123}, {9595, 15437}, {9630, 16063}, {10149, 10989}, {10247, 16434}, {11680, 17061}


X(17025) =  (X(1),X(2),X(6),X(31); X(1),X(6),X(2),X(31)) COLLINEATION IMAGE OF X(5220)

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

X(17025) lies on these lines:
{1, 2}, {6, 4392}, {31, 17593}, {81, 4860}, {88, 8301}, {89, 985}, {244, 8297}, {748, 17600}, {940, 9335}, {984, 14997}, {1155, 1386}, {1255, 8167}, {2308, 17591}, {3218, 16475}, {3666, 17127}, {3791, 5372}, {3999, 16666}, {4189, 16478}, {4430, 17598}, {5057, 17301}, {5145, 8054}, {5220, 7226}, {5282, 16670}, {5361, 6682}, {9347, 16610}, {17366, 17726}


X(17026) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(9)

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

X(17026) lies on these lines:
{1, 2}, {9, 350}, {44, 4713}, {57, 6063}, {63, 17738}, {75, 17754}, {183, 3684}, {190, 4479}, {312, 17755}, {672, 3729}, {673, 2319}, {940, 1197}, {1008, 1453}, {1468, 17686}, {1575, 4361}, {2082, 17739}, {2276, 3875}, {3208, 17144}, {3501, 17143}, {3760, 16552}, {3795, 4716}, {3886, 8299}, {4119, 17233}, {4366, 8616}, {4766, 11680}, {5223, 17794}, {5247, 7770}, {16503, 16992}, {16525, 17348}, {16779, 16998}, {17151, 17759}


X(17027) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(37)

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

X(17027) lies on these lines:
{1, 2}, {6, 350}, {31, 4366}, {81, 310}, {192, 672}, {194, 1475}, {320, 4799}, {330, 17474}, {335, 3873}, {384, 1468}, {385, 2280}, {894, 4441}, {1575, 4852}, {2238, 3759}, {2276, 4360}, {2295, 17144}, {2350, 17147}, {3684, 16997}, {3734, 9346}, {3751, 17794}, {3758, 4479}, {3760, 17499}, {3780, 6376}, {3875, 17754}, {4368, 16468}, {5228, 7196}, {5247, 16916}, {16503, 16998}, {17143, 17750}


X(17028) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(44)

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

X(17028) lies on these lines:
{1, 2}, {45, 350}, {2276, 17160}, {4465, 17335}, {16786, 16998}, {17117, 17756}


X(17029) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(45)

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

X(17029) lies on these lines:
{1, 2}, {44, 350}, {799, 16704}, {17160, 17759}


X(17030) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(213)

Barycentrics    a^2 b^2 + a b^2 c + a^2 c^2 + a b c^2 + b^2 c^2 : :

X(17030) lies on these lines:
{1, 2}, {35, 17684}, {36, 16915}, {39, 75}, {56, 11321}, {58, 14621}, {76, 1107}, {83, 4426}, {274, 2275}, {350, 5283}, {384, 993}, {673, 16060}, {894, 4253}, {958, 7770}, {984, 12263}, {1078, 4386}, {1213, 16525}, {1376, 11285}, {1500, 17144}, {1573, 3934}, {1574, 6683}, {1575, 7786}, {1655, 3760}, {1909, 16975}, {2140, 3662}, {2276, 17143}, {2550, 16043}, {2886, 6656}, {2975, 17686}, {3552, 5267}, {3739, 6374}, {3814, 16921}, {3925, 17670}, {4366, 5248}, {4999, 7807}, {5251, 16916}, {5260, 17541}, {5263, 16061}, {5299, 16998}, {5433, 17694}, {6645, 8666}, {7280, 17693}, {7741, 17669}, {11680, 17550}, {16502, 16992}


X(17031) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(984)

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

X(17031) lies on these lines:
{1, 2}, {58, 310}, {76, 16476}, {213, 12263}, {238, 350}, {385, 8300}, {596, 2350}, {672, 726}, {740, 8299}, {1008, 16478}, {1757, 17794}, {1929, 8937}, {2108, 17759}, {2238, 4974}, {3923, 4441}, {4479, 4676}, {4762, 4782}, {13576, 17766}


X(17032) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(1100)

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

X(17032) lies on these lines:
{1, 2}, {86, 2276}, {335, 2296}, {350, 16777}, {672, 17379}, {1621, 14621}, {2238, 4687}, {4441, 17319}, {4713, 16672}, {6645, 10448}, {10436, 17759}, {16525, 17394}


X(17033) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(1107)

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

X(17033) lies on these lines:
{1, 2}, {6, 1909}, {9, 1655}, {31, 384}, {41, 385}, {75, 2295}, {76, 213}, {171, 16915}, {192, 1334}, {194, 672}, {218, 7754}, {238, 16916}, {274, 17750}, {330, 1475}, {335, 3868}, {350, 2176}, {607, 9308}, {730, 16476}, {748, 16918}, {750, 16917}, {992, 16525}, {1400, 2998}, {1468, 6645}, {2238, 6376}, {2251, 6179}, {2887, 16906}, {3208, 3875}, {3501, 17759}, {3662, 17137}, {3691, 17349}, {3758, 4754}, {3759, 3780}, {3761, 17499}, {3769, 4447}, {3905, 4876}, {3915, 4366}, {3975, 4383}, {4645, 17680}, {4766, 5025}, {4805, 7768}, {5711, 11321}, {6327, 16910}, {6679, 16905}, {7770, 16466}, {9902, 16468}, {14621, 17686}, {16919, 17126}, {16920, 17127}


X(17034) =  (X(1),X(2),X(6),X(76); X(1),X(6),X(2),X(76)) COLLINEATION IMAGE OF X(5283)

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

X(17034) lies on these lines:
{1, 2}, {6, 76}, {58, 384}, {75, 17750}, {81, 17686}, {183, 2271}, {192, 3730}, {194, 4253}, {213, 350}, {335, 3874}, {385, 4251}, {595, 4366}, {668, 3780}, {940, 11321}, {1003, 4252}, {1043, 16061}, {1046, 17738}, {1500, 4360}, {1655, 16552}, {1724, 16916}, {1834, 6656}, {1975, 5021}, {2295, 17143}, {2901, 3797}, {3496, 16574}, {3501, 3875}, {3552, 4257}, {3759, 6376}, {3948, 17541}, {4074, 7304}, {4255, 11285}, {4256, 7824}, {4262, 7793}, {4283, 12782}, {5030, 7783}, {5156, 12194}, {6381, 17121}, {6996, 15488}, {14829, 16060}, {16549, 17759}, {16589, 17277}, {16783, 16998}, {17300, 17758}


X(17035) =  ANTICOMPLEMENT OF X(95)

Barycentrics    S^2-4*(2*SA-SW)*R^2+2*SA^2-4* SB*SC-SW^2 : :
X(17035) = 3*X(2)-4*X(233), 9*X(2)-8*X(6709), 5*X(4)-4*X(15800), 3*X(95)-4*X(6709), 3*X(233)-2*X(6709)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17035) lies on these lines:
{2, 95}, {4, 3164}, {193, 576}, {194, 7544}, {340, 14767}, {381, 9308}, {385, 5133}, {1995, 7665}, {3180, 11144}, {3181, 11143}, {3832, 10002}, {6997, 7774}, {7527, 14712}

X(17035) = anticomplement of X(95)
X(17035) = anticomplementary conjugate of X(2979)
X(17035) = anticomplementary isotomic conjugate of X(3)
X(17035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (95, 233, 2), (4993, 14918, 2)


X(17036) =  ANTICOMPLEMENT OF X(4998)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17036) lies on these on lines: {2, 1252}, {149, 14732}, {150, 3835}, {239, 908}, {4025, 4440}

,

X(17036) = anticomplement of X(4998)
X(17036) = anticomplementary conjugate of X(3888)


X(17037) =  ANTICOMPLEMENT OF X(253)

Barycentrics    7*a^8-4*(b^2+c^2)*a^6-14*(b^2- c^2)^2*a^4+12*(b^4-c^4)*(b^2- c^2)*a^2-(b^4+14*b^2*c^2+c^4)* (b^2-c^2)^2 : :
Barycentrics    7*S^2-16*(2*SA-SW)*R^2+8*SA^2- 4*SB*SC-4*SW^2 : : (barys)
X(17037) = 3*X(2)-4*X(1249), 5*X(3522)-8*X(15258), 7*X(3832)-8*X(10002), 13*X(5068)-16*X(15274), 16*X(15576)-11*X(15717)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17037) lies on these lines:
{2, 253}, {20, 15312}, {193, 1503}, {648, 6527}, {2883, 5922}, {3164, 3522}, {3187, 4452}, {3832, 10002}, {5068, 15274}, {7665, 15594}, {7774, 8892}, {9530, 15683}, {12324, 15238}, {15576, 15717}

X(17037) = reflection of X(12324) in X(15238)
X(17037) = anticomplement of X(253)
X(17037) = {X(253), X(1249)}-harmonic conjugate of X(2)


X(17038) =  X(10)X(192)∩X(37)X(43)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17038) lies on these lines:
{1, 3728}, {10, 192}, {19, 846}, {37, 43}, {63, 13610}, {65, 984}, {82, 8616}, {87, 1107}, {876, 8672}, {941, 4489}, {1221, 6376}, {1278, 3989}, {3668, 7179}, {3993, 9534}

X(17038) = trilinear pole of the line {661, 4083}


X(17039) =  X(3)X(9792)∩X(5)X(3164)

Barycentrics    (SB+SC)*SA*(3*S^2-(4*R^2-SW)*( 2*SC-SW)-2*SA*SB)*(3*S^2-(4*R^ 2-SW)*(2*SB-SW)-2*SA*SC)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17039) lies on these lines: {3, 9792}, {5, 3164}, {216, 5943}, {264, 10184}


X(17040) =  ISOGONAL CONJUGATE OF X(5020)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17040) lies on the Jerabek hyperbola and these lines:
{2, 6391}, {3, 193}, {4, 6467}, {6, 6353}, {64, 6776}, {68, 6804}, {69, 3819}, {71, 4028}, {248, 5065}, {511, 15740}, {895, 3618}, {1176, 1992}, {1352, 15077}, {1596, 3531}, {1899, 16774}, {2854, 15435}, {3089, 3527}, {3532, 8550}, {3800, 10097}, {4846, 12220}, {5032, 10154}, {5644, 6677}, {6413, 8944}, {6414, 8940}, {7392, 12272}, {7714, 9924}, {7734, 11160}, {7738, 9307}, {10565, 11402}, {14542, 15073}, {15316, 15805}

X(17040) = isogonal conjugate of X(5020)
X(17040) = trilinear pole of the line {647, 3566}


X(17041) =  (name pending)

Barycentrics    (S^2+SB*SC)*(7*S^4-(SA-SB-5* SC)*SA*S^2+(SB+SC)*(SW-2*SB)* SA^2)*(7*S^4-(SA-SC-5*SB)*SA* S^2+(SB+SC)*(SW-2*SC)*SA^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17041) lies on this line: {233, 12012}


X(17042) =  ISOGONAL CONJUGATE OF X(7787)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27439.

X(17042) lies on these lines:
{22, 14370}, {39, 1613}, {141, 194}, {1843, 3094}, {2353, 10329}, {7770, 14970}

X(17042) = isogonal conjugate of X(7787)
X(17042) = trilinear pole of the line {3005, 3221}




leftri  Collineation images: X(17043) - X(17097)  rightri

If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A,B,C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A,B,C,P).

The collineation images at X(17043)-X(17097) result from A'B'C' = medial triangle, P = X(2), and U = X(1). We write the image of X as m(X); let m-1 denote the inverse collineation. Then centers X(17043)-X(17073) are examples of m(X), and X(17074)-X(17097) are examples of m-1(X). Other examples are given by the following list, in which the appearance of (i,j) means that m(X(i)) = X(j):

(1,142), (2,1), (3,16608), (4, 17073), (5, 17043), (6,2886), (7,2), (8,4000), (9,11019), (10,3946), (37,3742), (56,141), (57,10), (85,37), (651,11)

A collineation maps lines to lines. The appearance of {h,i} -> {j,k} in the next list means that m(X(h)X(i)) = X(j)X(k):

{1,2} -> {1,142}
{1,3} -> {10,141}
{2,3} -> {1,16608}
{2,7} -> {1,2}
{2,85} -> {1,6}
{2,222} -> {1,5}
{7,8} -> {2,37}
{7,27} -> {2,3}

A regular collineation m is a permutation of the set S of triangle centers. That is, m maps S onto S, and if U and V are in S and m(U) = m(V), then U = V. As an illustration, the appearance of (i,j,k) in the following list means that

X(17102) = (A,B,C,X(i); A',B',C',X(j)) collineation image of X(k), where A'B'C' = medial triangle.

In particular, for every i and j, such a k is uniquely determined. (List contributed by Peter Moses, April 6, 2018)

underbar




X(17043) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17043) lies on these lines:
{1, 16608}, {142, 214}, {465, 559}, {466, 1082}, {857, 17221}, {1375, 1953}, {3589, 16578}, {4357, 6510}, {4466, 17438}


X(17044) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17044) lies on these lines:
{1, 4904}, {2, 664}, {11, 9317}, {30, 5074}, {101, 1565}, {116, 952}, {141, 997}, {142, 214}, {220, 348}, {514, 6710}, {516, 11728}, {524, 6510}, {529, 6647}, {597, 8257}, {620, 2785}, {672, 7181}, {905, 13006}, {918, 3960}, {1001, 10186}, {1086, 9259}, {1125, 6706}, {1358, 9318}, {2140, 5901}, {3035, 6366}, {3160, 6554}, {3207, 17170}, {3234, 14116}, {3616, 14942}, {3665, 9310}, {4437, 4561}, {5022, 17081}, {5088, 17747}, {6167, 7079}, {6349, 17811}, {6505, 13567}, {6603, 9436}, {10025, 17078}, {16608, 17390}


X(17045) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    2 a^2 + 2 a b + b^2 + 2 a c + c^2 : :

X(17045) lies on these lines:
{1, 141}, {2, 594}, {6, 4364}, {7, 17323}, {8, 17327}, {9, 597}, {10, 4399}, {37, 3589}, {44, 6329}, {45, 3618}, {69, 16884}, {75, 4472}, {86, 1086}, {142, 214}, {145, 4445}, {192, 17369}, {193, 17253}, {198, 5834}, {239, 1213}, {319, 17326}, {320, 17324}, {344, 16672}, {519, 4478}, {523, 8060}, {524, 1100}, {536, 4021}, {545, 894}, {740, 1125}, {966, 17014}, {1015, 16696}, {1062, 17073}, {1211, 17011}, {1385, 12610}, {1449, 3629}, {1621, 5347}, {1654, 4969}, {1698, 4405}, {2321, 17385}, {2345, 17318}, {3008, 4698}, {3244, 17372}, {3247, 17279}, {3616, 4000}, {3619, 17311}, {3622, 4648}, {3630, 17272}, {3631, 3879}, {3632, 4910}, {3635, 4889}, {3636, 3834}, {3661, 17388}, {3662, 17392}, {3663, 4670}, {3664, 7238}, {3665, 7225}, {3666, 6703}, {3672, 4363}, {3686, 4708}, {3723, 3912}, {3758, 17247}, {3759, 17248}, {3763, 17316}, {3875, 4665}, {3943, 17289}, {3945, 7232}, {3950, 17359}, {3963, 18046}, {4068, 8299}, {4371, 9780}, {4389, 17365}, {4393, 5224}, {4402, 5550}, {4416, 16666}, {4452, 4470}, {4644, 17255}, {4664, 17340}, {4667, 17345}, {4675, 17304}, {4681, 17355}, {4687, 17337}, {4704, 17354}, {4716, 4733}, {4755, 6666}, {5222, 17259}, {5241, 17012}, {5256, 5743}, {5257, 17348}, {5308, 17265}, {5749, 17262}, {5839, 17251}, {5845, 16503}, {6542, 17307}, {6646, 7277}, {7263, 10436}, {8584, 16667}, {10022, 17118}, {10186, 11495}, {12433, 17052}, {14767, 15252}, {16706, 16826}, {16831, 17278}, {16834, 17275}, {17120, 17258}, {17121, 17256}, {17227, 17391}, {17228, 17389}, {17234, 17383}, {17236, 17378}, {17238, 17377}, {17242, 17371}, {17244, 17370}, {17249, 17364}, {17250, 17363}, {17291, 17317}, {17292, 17315}, {17299, 17308}, {17300, 17305}, {17724, 18082}


X(17046) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17046) lies on these lines:
{1, 17062}, {2, 41}, {9, 17671}, {10, 116}, {141, 1329}, {142, 442}, {150, 2329}, {626, 766}, {1368, 3741}, {1759, 4056}, {1930, 4136}, {2389, 2886}, {3061, 17181}, {3496, 4872}, {3509, 4911}, {3662, 5025}, {3721, 4920}, {3822, 17758}, {3878, 5074}, {3912, 16603}, {4372, 4950}, {4766, 17137}, {5086, 9317}, {7272, 17736}, {8287, 17237}, {9437, 17072}


X(17047) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    -a b^4 + b^5 - b^4 c - a c^4 - b c^4 + c^5 : :

X(17047) lies on these lines:
{1, 17055}, {2, 2175}, {141, 2876}, {760, 16580}, {2886, 16608}, {17058, 17065}


X(17048) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17048) lies on these lines:
{142, 442}, {1215, 3934}, {1737, 17062}, {2140, 5883}, {3509, 17681}, {3673, 17754}, {3739, 3831}, {3742, 6706}, {3754, 17761}, {3812, 17050}, {5253, 9317}, {5563, 6647}, {7264, 16549}, {16609, 17023}, {16825, 16852}


X(17049) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17049) lies on these lines:
{1, 2092}, {2, 3688}, {10, 9052}, {37, 14839}, {39, 4446}, {65, 3883}, {75, 6007}, {142, 17792}, {181, 3757}, {284, 8301}, {354, 3879}, {518, 3686}, {524, 13476}, {674, 3739}, {894, 3271}, {942, 5847}, {1215, 5943}, {2245, 16684}, {2886, 16608}, {3056, 10436}, {3122, 17445}, {3125, 17446}, {3664, 9025}, {3753, 14523}, {3754, 17766}, {3779, 4384}, {3873, 17363}, {3948, 17142}, {4014, 7321}, {4260, 16825}, {4360, 4890}, {4516, 17868}, {4553, 17245}, {5572, 5836}, {5933, 10473}, {7064, 17260}, {10822, 16817}, {12194, 16800}, {17055, 17058}


X(17050) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17050) lies on these lines:
{1, 142}, {2, 1334}, {9, 17753}, {10, 2140}, {141, 9049}, {213, 3008}, {226, 4384}, {239, 2890}, {274, 1432}, {442, 4904}, {519, 17758}, {527, 16552}, {908, 16815}, {960, 3739}, {993, 14377}, {1086, 1107}, {1125, 8299}, {1429, 16054}, {2176, 17278}, {2329, 17682}, {2388, 3741}, {2389, 2886}, {3294, 6666}, {3452, 16832}, {3663, 5283}, {3812, 17048}, {3912, 17143}, {4059, 4875}, {4357, 16819}, {4858, 17866}, {5267, 17729}, {5750, 16818}, {5836, 6706}, {9310, 17683}, {9534, 11523}, {12609, 16825}, {17144, 17234}, {17169, 17474}, {17175, 17197}


X(17051) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    2 a^2 b - 3 a b^2 + b^3 + 2 a^2 c + 8 a b c - b^2 c - 3 a c^2 - b c^2 + c^3 : :

X(17051) lies on these lines:
{1, 1145}, {2, 3711}, {11, 10129}, {142, 2886}, {214, 15935}, {354, 908}, {495, 6702}, {518, 5316}, {528, 3306}, {551, 14563}, {982, 17246}, {1001, 5744}, {1329, 5045}, {1376, 10580}, {1647, 5718}, {3086, 11281}, {3315, 17602}, {3705, 17241}, {3813, 5439}, {3820, 3892}, {3829, 5249}, {3847, 13407}, {3848, 4847}, {3881, 17527}, {4428, 5435}, {4666, 6690}, {4679, 5852}, {4860, 17768}, {5087, 5542}, {5719, 10199}, {6667, 17718}, {6745, 15570}, {7681, 13373}, {9345, 17726}, {9711, 9843}, {9776, 11235}, {12607, 17609}, {12611, 15528}, {15842, 17626}, {17392, 17722}


X(17052) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17052) lies on these lines:
{2, 272}, {5, 141}, {9, 857}, {10, 4523}, {69, 5747}, {116, 3739}, {117, 127}, {142, 442}, {440, 5745}, {527, 1901}, {1211, 3452}, {1213, 6666}, {1834, 3946}, {2321, 16603}, {2886, 16608}, {3936, 5219}, {3940, 4445}, {4648, 6856}, {4657, 5722}, {4851, 11374}, {5044, 5074}, {5051, 9581}, {5142, 10479}, {5719, 17390}, {5794, 17073}, {6510, 17056}, {11108, 17327}, {12433, 17045}


X(17053) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17053) is the center of the inellipse that is the isogonal conjugate of the isotomic conjugate of the incircle. The Brianchon point (perspector) of the inellipse is X(56). (Randy Hutson, November 30, 2018)

X(17053) lies on these lines:
{1, 2092}, {2, 1240}, {6, 101}, {9, 1050}, {31, 1495}, {32, 2178}, {37, 39}, {45, 5069}, {56, 478}, {71, 3230}, {172, 16470}, {187, 5301}, {198, 16502}, {213, 2260}, {216, 7561}, {244, 2171}, {292, 7194}, {579, 2176}, {594, 1574}, {595, 2305}, {614, 5336}, {800, 1108}, {941, 3622}, {966, 16975}, {980, 17321}, {1100, 4263}, {1107, 5257}, {1149, 2269}, {1196, 3290}, {1201, 1400}, {1213, 1573}, {1449, 15839}, {1500, 4261}, {1575, 2321}, {1740, 6007}, {1841, 3199}, {1914, 16488}, {1953, 3125}, {1964, 3122}, {2182, 9434}, {2197, 3924}, {2245, 16685}, {2273, 9310}, {2276, 3247}, {2298, 5253}, {2886, 17055}, {3009, 3688}, {3271, 7032}, {3686, 17448}, {3752, 3946}, {3772, 17073}, {3782, 16700}, {3948, 17148}, {4254, 16781}, {4277, 16884}, {4364, 16696}, {4446, 14839}, {4516, 17872}, {4749, 16679}, {4850, 17396}, {6739, 16613}, {9336, 16667}, {11364, 16800}, {13006, 16614}, {16608, 17058}, {16726, 17365}

X(17053) = isogonal conjugate of X(2985)
X(17053) = complement of X(3596)


X(17054) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(345), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17054) lies on these lines:
{1, 474}, {2, 1257}, {4, 1086}, {6, 169}, {8, 17597}, {10, 3242}, {11, 1854}, {21, 17595}, {31, 5221}, {34, 1407}, {37, 9593}, {40, 1279}, {45, 11108}, {46, 3052}, {56, 244}, {57, 1104}, {58, 5708}, {65, 614}, {78, 16610}, {88, 4188}, {145, 3315}, {220, 3290}, {386, 15934}, {387, 17366}, {405, 3670}, {517, 1616}, {518, 1722}, {580, 2095}, {599, 5814}, {910, 16780}, {936, 16602}, {938, 1834}, {940, 5262}, {956, 3953}, {958, 982}, {960, 5272}, {976, 4413}, {978, 12635}, {986, 1001}, {990, 5806}, {1043, 17490}, {1149, 2098}, {1201, 2099}, {1210, 3772}, {1330, 7232}, {1385, 8572}, {1427, 1467}, {1428, 14529}, {1435, 8899}, {1468, 4860}, {1480, 13145}, {1500, 4261}, {1772, 11508}, {2292, 4423}, {2478, 3782}, {2551, 4310}, {2999, 11518}, {3057, 16486}, {3086, 3756}, {3120, 10896}, {3125, 16502}, {3271, 17114}, {3295, 15287}, {3303, 4642}, {3339, 7290}, {3695, 17267}, {3755, 6744}, {3868, 4383}, {3869, 7292}, {3927, 16885}, {3940, 17749}, {3959, 16781}, {3976, 12513}, {4205, 17325}, {4259, 12109}, {4273, 4658}, {4286, 16290}, {4361, 10449}, {4363, 13740}, {4392, 5260}, {4415, 5084}, {4419, 5129}, {4662, 16496}, {4675, 5717}, {4864, 6765}, {5022, 16968}, {5253, 9335}, {5295, 17119}, {5552, 17724}, {5710, 7191}, {5711, 5883}, {5716, 9776}, {5737, 16817}, {5902, 16466}, {5903, 16483}, {7373, 15955}, {7986, 9955}, {8726, 15852}, {12572, 17276}, {15803, 16485}, {16062, 17290}, {16605, 16973}


X(17055) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17055) lies on these lines:
{1, 17047}, {1125, 1279}, {2886, 17053}, {17049, 17058}


X(17056) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17056) lies on these lines:
{1, 442}, {2, 6}, {3, 5713}, {5, 581}, {10, 4035}, {11, 3136}, {12, 73}, {30, 4653}, {34, 429}, {37, 226}, {42, 3925}, {43, 3826}, {45, 329}, {55, 851}, {57, 2245}, {58, 6675}, {63, 17365}, {115, 15903}, {140, 580}, {142, 2092}, {171, 6690}, {212, 750}, {223, 5219}, {278, 1865}, {284, 6678}, {306, 594}, {312, 17243}, {321, 3943}, {345, 4363}, {386, 8728}, {407, 2646}, {430, 17605}, {443, 4255}, {451, 3194}, {468, 2299}, {498, 7078}, {500, 6841}, {523, 18013}, {528, 3750}, {551, 16052}, {612, 17718}, {614, 17723}, {661, 6545}, {664, 17947}, {846, 17768}, {857, 948}, {860, 17917}, {908, 16585}, {942, 10974}, {946, 15852}, {960, 10381}, {968, 1836}, {975, 1060}, {991, 8727}, {1001, 4199}, {1030, 1817}, {1086, 3666}, {1104, 1125}, {1146, 6708}, {1212, 3452}, {1215, 3932}, {1230, 4358}, {1330, 11110}, {1375, 5277}, {1451, 5433}, {1453, 3624}, {1503, 7413}, {1638, 2610}, {1730, 4271}, {1841, 1848}, {1961, 17719}, {1962, 3120}, {1999, 17390}, {2051, 17758}, {2292, 3649}, {2887, 4026}, {2999, 4272}, {3011, 3745}, {3035, 17122}, {3052, 4307}, {3101, 5341}, {3175, 4054}, {3178, 3704}, {3210, 7263}, {3216, 17529}, {3242, 3475}, {3286, 8731}, {3616, 5051}, {3664, 5745}, {3687, 3739}, {3696, 4028}, {3712, 4418}, {3721, 5244}, {3741, 4966}, {3742, 3756}, {3743, 11263}, {3757, 5846}, {3782, 17246}, {3812, 5530}, {3816, 17717}, {3836, 6685}, {3838, 15569}, {3848, 5121}, {3920, 17724}, {3928, 4888}, {3931, 12609}, {3948, 17244}, {3967, 4078}, {4046, 4062}, {4052, 4098}, {4184, 15447}, {4204, 4423}, {4224, 4265}, {4252, 4340}, {4414, 11246}, {4425, 4892}, {4641, 7277}, {4644, 5273}, {4654, 17276}, {4666, 17721}, {4851, 11679}, {4918, 17164}, {5132, 16056}, {5256, 17366}, {5271, 17362}, {5287, 5949}, {5311, 17602}, {5347, 7465}, {5396, 6881}, {5483, 17021}, {5706, 6889}, {5711, 10198}, {5721, 6829}, {5880, 17594}, {5905, 17334}, {6051, 12047}, {6510, 17052}, {6692, 17058}, {7191, 17726}, {7354, 10448}, {7567, 12241}, {8286, 13405}, {10459, 15888}, {15674, 16948}, {16592, 16593}, {16831, 16968}, {17340, 17776}, {17592, 17889}

X(17056) = complement of X(333)


X(17057) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = MEDIAL TRIANGLE

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

Let A'B'C' be the excentral triangle. X(17057) is the radical center of the orthocentroidal circles of triangles A'BC, B'CA, C'AB. (Randy Hutson, June 27, 2018)

X(17057) lies on these lines:
{2, 80}, {3, 1698}, {9, 484}, {10, 908}, {12, 5904}, {100, 15175}, {119, 3925}, {142, 1737}, {144, 10590}, {191, 10895}, {377, 5445}, {1125, 15079}, {1145, 2886}, {1478, 5744}, {1512, 5316}, {2099, 3679}, {2475, 3647}, {2550, 6594}, {3419, 3584}, {3577, 7988}, {3624, 5727}, {3634, 7705}, {3822, 5902}, {3826, 10427}, {3828, 17579}, {3878, 5141}, {3899, 17605}, {5258, 5705}, {5259, 10826}, {5288, 9578}, {5432, 9945}, {5443, 6933}, {5660, 12247}, {5790, 6326}, {5818, 6853}, {6260, 6937}, {6763, 9654}, {6856, 10573}, {6980, 12611}, {9709, 14882}, {10039, 12640}, {10051, 16173}, {10472, 17239}, {15298, 15348}

X(17057) = complement of X(2320)
X(17057) = trilinear product X(45)*X(5902)


X(17058) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17058) lies on these lines:
{2, 645}, {115, 116}, {125, 244}, {661, 6791}, {1015, 15526}, {2092, 17060}, {2643, 4934}, {3125, 4466}, {3756, 8286}, {4904, 16613}, {6388, 16592}, {6537, 17237}, {6692, 17056}, {16608, 17053}, {17047, 17065}, {17049, 17055}


X(17059) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17059) lies on these lines:
{2, 3939}, {5, 2810}, {11, 124}, {116, 926}, {142, 17060}, {244, 13256}, {522, 1086}, {946, 2835}, {2316, 10589}, {3120, 4939}, {3756, 8286}, {3834, 15733}, {3836, 5853}, {3888, 11680}, {4422, 5856}, {5510, 6085}, {6600, 17265}


X(17060) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(105), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17060) lies on these lines:
{1, 4904}, {2, 294}, {3, 8299}, {9, 141}, {10, 116}, {142, 17059}, {241, 3693}, {442, 6706}, {2092, 17058}, {3126, 4925}, {3732, 6554}, {3834, 10427}, {4851, 8271}, {6600, 16608}, {7123, 13577}


X(17061) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17061) lies on these lines:
{1, 442}, {2, 1390}, {11, 5133}, {12, 5262}, {31, 3782}, {42, 17724}, {43, 17366}, {55, 7465}, {141, 4362}, {142, 4682}, {171, 1086}, {226, 1386}, {238, 4415}, {388, 7557}, {496, 17111}, {524, 3791}, {528, 3744}, {596, 6693}, {612, 3826}, {614, 3816}, {726, 6679}, {846, 17246}, {851, 16687}, {1104, 13161}, {1215, 3589}, {1376, 4000}, {1621, 4854}, {1707, 17276}, {1722, 9711}, {1961, 17245}, {1999, 4966}, {2887, 5846}, {3008, 3740}, {3011, 3666}, {3035, 3752}, {3120, 17469}, {3662, 3769}, {3663, 4640}, {3712, 17147}, {3745, 5249}, {3757, 4026}, {3817, 15251}, {3829, 17721}, {3920, 3925}, {3936, 17150}, {3946, 13405}, {3967, 17353}, {3971, 4422}, {3980, 7263}, {4028, 4852}, {4030, 4972}, {4104, 17348}, {4199, 16684}, {4353, 5745}, {4438, 4884}, {4641, 5852}, {4656, 15254}, {4719, 13411}, {4850, 5432}, {4906, 11019}, {5254, 16974}, {5256, 17718}, {5268, 17278}, {5269, 5880}, {5305, 16600}, {5718, 17017}, {5743, 16825}, {7081, 16706}, {7262, 17334}, {8167, 16020}, {11246, 17126}, {11269, 17597}, {11680, 17024}, {15888, 17016}, {16608, 17068}, {17301, 17594}, {17395, 17592}, {17716, 17889}


X(17062) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17062) lies on these lines:
{1, 17046}, {2, 1429}, {10, 141}, {75, 4136}, {114, 116}, {320, 17739}, {442, 4904}, {1737, 17048}, {1759, 7272}, {1834, 3946}, {2140, 3822}, {3061, 7179}, {3496, 4911}, {3509, 7247}, {3663, 5254}, {3735, 4920}, {3884, 5074}, {3912, 4095}, {4384, 4417}, {4544, 16822}, {4660, 7784}, {4766, 17152}, {5249, 16609}, {5437, 17308}, {6647, 10106}


X(17063) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17063) lies on these lines:
{1, 474}, {2, 38}, {10, 3976}, {11, 17889}, {31, 7292}, {36, 1626}, {37, 3848}, {43, 354}, {55, 1054}, {57, 238}, {63, 17123}, {75, 3840}, {88, 1621}, {100, 17715}, {142, 17065}, {171, 614}, {210, 3999}, {226, 5121}, {240, 17917}, {277, 9445}, {312, 4871}, {499, 1725}, {518, 16569}, {612, 17598}, {740, 17490}, {748, 3218}, {750, 7191}, {846, 4423}, {899, 3873}, {902, 9352}, {910, 16779}, {942, 978}, {968, 17593}, {976, 17531}, {986, 1125}, {995, 5883}, {1001, 17596}, {1046, 5708}, {1086, 3816}, {1155, 8616}, {1254, 5265}, {1279, 3550}, {1393, 7288}, {1401, 5943}, {1447, 7204}, {1647, 11680}, {1698, 3953}, {1722, 3333}, {1724, 3337}, {1738, 11019}, {1742, 11227}, {1921, 6384}, {1961, 17599}, {2292, 5550}, {2886, 3756}, {2999, 4649}, {3061, 16604}, {3074, 17437}, {3214, 3889}, {3219, 17125}, {3241, 4695}, {3290, 17754}, {3315, 3938}, {3338, 5247}, {3555, 6048}, {3622, 4642}, {3624, 3670}, {3662, 3846}, {3677, 5268}, {3679, 4694}, {3681, 17449}, {3705, 3836}, {3720, 4850}, {3750, 4666}, {3751, 10980}, {3920, 17124}, {3924, 5253}, {3935, 9350}, {3961, 4413}, {4022, 4751}, {4038, 5256}, {4358, 17155}, {4383, 4860}, {4414, 5284}, {4446, 17245}, {4640, 15485}, {4722, 14997}, {4859, 17064}, {4865, 5211}, {5249, 17717}, {5274, 7613}, {5277, 16787}, {5287, 17600}, {5293, 16408}, {5435, 16020}, {6533, 10479}, {7004, 10589}, {7868, 17282}, {8580, 16496}, {9345, 17011}, {9843, 13161}, {10582, 16484}, {14829, 16825}, {15601, 16570}, {16753, 17187}, {17018, 17450}


X(17064) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17064) lies on these lines:
{1, 442}, {2, 968}, {5, 1722}, {6, 3838}, {9, 3944}, {10, 1265}, {11, 1040}, {43, 5219}, {57, 1711}, {63, 3120}, {69, 4138}, {200, 17719}, {226, 3751}, {238, 1699}, {499, 11512}, {614, 11680}, {851, 16778}, {975, 3841}, {978, 8227}, {982, 5231}, {986, 5705}, {1074, 3086}, {1279, 11235}, {1698, 4424}, {1707, 1836}, {1714, 12047}, {1721, 8727}, {2006, 8270}, {2887, 11679}, {2999, 17717}, {3008, 3817}, {3011, 3434}, {3624, 16458}, {3689, 17783}, {3729, 4438}, {3771, 3886}, {3792, 10439}, {3816, 17278}, {3840, 17282}, {3846, 4384}, {3925, 5268}, {3936, 17156}, {4312, 4650}, {4383, 17605}, {4847, 16496}, {4859, 17063}, {4863, 17724}, {5121, 10589}, {5247, 9612}, {5249, 11269}, {5274, 16020}, {5292, 12609}, {5435, 7613}, {5530, 6856}, {6682, 17304}, {7741, 17111}, {8616, 9580}, {10472, 17306}, {13881, 16605}, {16570, 17768}


X(17065) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17065) lies on these lines:
{1, 2092}, {2, 3778}, {9, 291}, {10, 17786}, {32, 16800}, {37, 4446}, {38, 17248}, {57, 1716}, {75, 3122}, {86, 3764}, {87, 3271}, {142, 17063}, {238, 579}, {244, 3662}, {256, 10436}, {978, 4260}, {982, 4357}, {984, 5257}, {985, 16470}, {1210, 1738}, {1266, 4941}, {2178, 11364}, {2228, 17234}, {2245, 16690}, {2260, 16476}, {2664, 3779}, {3094, 16604}, {3739, 4443}, {3948, 17157}, {4022, 5224}, {4270, 4649}, {4484, 17259}, {4516, 17891}, {4698, 4735}, {6007, 16571}, {7241, 17262}, {8056, 9445}, {17047, 17058}


X(17066) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17066) lies on these lines:
{2, 3709}, {522, 676}, {523, 3739}, {665, 3261}, {926, 17072}, {2605, 15668}, {3063, 17215}, {3287, 10436}, {4369, 8672}, {4384, 17218}, {4411, 6586}, {4751, 7199}, {4832, 17217}


X(17067) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    2 a^2 - a b + 3 b^2 - a c - 6 b c + 3 c^2 : :

X(17067) lies on these lines:
{1, 142}, {2, 1266}, {6, 4896}, {7, 16670}, {9, 4346}, {10, 17290}, {44, 527}, {45, 3663}, {57, 1731}, {88, 655}, {141, 3626}, {320, 4700}, {515, 15898}, {516, 3246}, {519, 3834}, {522, 676}, {545, 6687}, {673, 16786}, {726, 3634}, {903, 4480}, {1125, 17382}, {1155, 7336}, {2321, 17282}, {3244, 17313}, {3264, 18073}, {3579, 12442}, {3621, 4402}, {3625, 4361}, {3662, 3686}, {3664, 16666}, {3685, 5550}, {3707, 17274}, {3717, 9780}, {3772, 6692}, {3826, 4353}, {3912, 17133}, {3950, 17265}, {3986, 17323}, {4021, 17245}, {4060, 17117}, {4357, 16815}, {4371, 4816}, {4405, 4701}, {4422, 17132}, {4431, 17283}, {4464, 17312}, {4545, 17287}, {4667, 5222}, {4758, 17023}, {4856, 17376}, {4858, 17895}, {4967, 17291}, {4982, 17378}, {5249, 17012}, {5257, 17304}, {5745, 17595}, {5750, 16706}, {7263, 17355}, {7290, 7613}, {15668, 15808}, {16672, 17301}, {16752, 17197}


X(17068) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17068) lies on these lines:
{6708, 16582}, {16608, 17061}


X(17069) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(523), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17069) lies on these lines:
{2, 3700}, {57, 1021}, {63, 9404}, {513, 3798}, {514, 4394}, {518, 4524}, {522, 676}, {523, 2487}, {525, 14838}, {647, 3666}, {649, 3004}, {650, 918}, {661, 4750}, {662, 14999}, {664, 9358}, {693, 1638}, {850, 4359}, {900, 3835}, {905, 3910}, {1376, 4477}, {1635, 16892}, {2490, 4763}, {2516, 11068}, {2786, 14321}, {3667, 4940}, {3676, 4762}, {3776, 6084}, {4142, 6362}, {4380, 4773}, {4382, 4927}, {4453, 17494}, {4560, 7178}, {4782, 4932}, {4786, 4790}, {4789, 17161}, {4802, 7653}, {4841, 7192}, {4850, 16751}


X(17070) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17070) lies on these lines:
{1, 442}, {2, 3712}, {11, 858}, {226, 4663}, {522, 676}, {524, 4892}, {528, 3011}, {614, 3829}, {896, 3120}, {897, 16092}, {1738, 3035}, {3008, 5087}, {3826, 17720}, {3914, 4689}, {3925, 5297}, {3936, 17162}, {4831, 17491}, {5204, 16049}, {5524, 17719}, {5550, 14005}, {6667, 16610}, {10593, 17111}, {17366, 17717}


X(17071) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17071) lies on these lines:
{244, 665}, {1357, 17477}


X(17072) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17072) lies on these lines:
{2, 663}, {8, 4449}, {10, 514}, {141, 9029}, {240, 522}, {512, 625}, {513, 3823}, {525, 4522}, {693, 4041}, {814, 9508}, {905, 3907}, {926, 17066}, {1698, 4040}, {2254, 4391}, {2785, 6332}, {3309, 3716}, {3634, 4794}, {3667, 14431}, {3676, 4163}, {3762, 4905}, {3810, 10015}, {3831, 4874}, {3837, 4083}, {3900, 4885}, {4017, 4397}, {4129, 6005}, {4151, 4823}, {4369, 8678}, {4379, 17166}, {4406, 5224}, {4462, 14430}, {4474, 17496}, {4521, 8713}, {4546, 4847}, {4724, 9780}, {4728, 4729}, {4761, 14349}, {4763, 6050}, {4776, 4822}, {4785, 4834}, {4791, 8714}, {4925, 6362}, {8062, 15313}, {9437, 17046}


X(17073) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    (a^2 - b^2 - c^2) (a^3 - b^3 + b^2 c + b c^2 - c^3) : :
Barycentrics    2 + sec B + sec C : :

X(17073) is the center of the inconic that is the isotomic conjugate of the polar conjugate of the incircle. The Brianchon point (perspector) of the inconic is X(348). (Randy Hutson, November 30, 2018)

X(17073) lies on these lines:
{1, 16608}, {2, 92}, {3, 142}, {9, 1020}, {19, 1375}, {48, 4466}, {63, 11064}, {69, 6510}, {86, 2193}, {140, 14743}, {141, 997}, {216, 17278}, {219, 307}, {226, 7011}, {343, 6505}, {441, 10436}, {577, 4675}, {857, 17134}, {859, 17171}, {1038, 8583}, {1040, 10582}, {1062, 17045}, {1158, 16252}, {1565, 7289}, {1836, 17188}, {1838, 7532}, {1848, 11347}, {2178, 16580}, {2968, 5231}, {3086, 4000}, {3419, 6739}, {3589, 8257}, {3624, 7515}, {3664, 15905}, {3739, 6389}, {3772, 17053}, {3946, 11019}, {3986, 15831}, {4361, 10916}, {5437, 7536}, {5794, 17052}, {6247, 15836}, {6691, 7561}, {15526, 17275}, {16578, 17279}

X(17073) = complement of X(281)
X(17073) = isotomic conjugate of polar conjugate of X(1836)


X(17074) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17074) lies on these lines:
{1, 1106}, {2, 222}, {3, 3562}, {6, 5435}, {7, 940}, {21, 603}, {31, 7677}, {33, 11220}, {34, 17519}, {42, 9364}, {56, 4216}, {57, 77}, {63, 2324}, {65, 17015}, {73, 404}, {89, 2982}, {97, 1214}, {109, 1621}, {171, 1458}, {172, 241}, {221, 3616}, {223, 3306}, {255, 6986}, {273, 1396}, {333, 17077}, {347, 7560}, {354, 4318}, {394, 5744}, {411, 3075}, {631, 3157}, {873, 4573}, {942, 4296}, {1038, 3868}, {1262, 5662}, {1401, 5061}, {1406, 3485}, {1419, 5437}, {1427, 1443}, {1442, 3666}, {1456, 3742}, {1462, 2162}, {1617, 17126}, {1745, 6915}, {1870, 10202}, {1935, 5047}, {1936, 7411}, {1943, 4359}, {2003, 3911}, {2185, 4565}, {3090, 8757}, {3100, 10167}, {3160, 5228}, {3173, 15066}, {3523, 7078}, {3600, 5711}, {3660, 7191}, {3720, 8543}, {3784, 4220}, {3873, 8270}, {3920, 17625}, {3938, 14151}, {4308, 5710}, {4322, 5255}, {4617, 7056}, {4682, 8581}, {5221, 15832}, {5226, 6180}, {5253, 10571}, {5265, 16466}, {5273, 17811}, {5433, 8614}, {5444, 6126}, {6198, 13369}, {8545, 17022}, {9345, 16133}, {9363, 10459}, {9370, 9780}, {14594, 17165}


X(17075) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17075) lies on these lines:
{2, 7110}, {7, 604}, {77, 17298}, {85, 17400}, {241, 17356}, {307, 3686}, {346, 347}, {348, 17077}, {651, 17347}, {664, 17295}, {1150, 6357}, {1441, 17303}, {1442, 17391}, {1443, 3662}, {1766, 3007}, {4149, 6327}, {6180, 17255}, {16706, 17078}


X(17076) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17076) lies on these lines:
{2, 7112}, {7, 1397}, {274, 278}, {305, 4554}, {315, 4123}, {1804, 7337}, {2064, 3926}, {4573, 7055}, {7009, 17181}, {17085, 17091}


X(17077) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17077) lies on these lines:
{2, 7}, {8, 1818}, {10, 1458}, {48, 5773}, {56, 16454}, {75, 4552}, {77, 4384}, {85, 4751}, {222, 5278}, {239, 1442}, {241, 1441}, {261, 4565}, {269, 16832}, {273, 15149}, {333, 17074}, {348, 17075}, {604, 16738}, {651, 17277}, {653, 17913}, {1214, 4359}, {1284, 5433}, {1443, 16815}, {1450, 3616}, {1466, 16342}, {1698, 4334}, {2283, 16684}, {2284, 17234}, {3674, 16818}, {4296, 16817}, {4298, 16828}, {4318, 16823}, {4343, 11019}, {4981, 17625}, {5228, 15668}, {5263, 7677}, {5701, 10030}, {6180, 17259}, {7176, 16819}, {7190, 16831}, {7269, 16826}, {7676, 14942}, {14543, 16551}, {16577, 17147}


X(17078) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17078) lies on these lines:
{2, 85}, {7, 1319}, {30, 1565}, {36, 4089}, {56, 7185}, {77, 17378}, {269, 17274}, {307, 17271}, {320, 1443}, {376, 17170}, {381, 17181}, {519, 664}, {527, 1275}, {528, 14189}, {551, 10481}, {552, 553}, {738, 3928}, {752, 5018}, {1111, 3582}, {1358, 1447}, {1418, 17086}, {2094, 7056}, {3160, 3241}, {3188, 17579}, {3656, 17753}, {3665, 5434}, {3669, 4560}, {3673, 10072}, {3679, 9312}, {4059, 4870}, {4715, 6610}, {6180, 17333}, {6516, 13587}, {7179, 7223}, {7195, 17081}, {7270, 7788}, {10025, 17044}, {14548, 15933}, {16706, 17075}


X(17079) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17079) lies on these lines:
{2, 85}, {7, 528}, {30, 17170}, {347, 17320}, {376, 5088}, {381, 1565}, {388, 7185}, {519, 6604}, {551, 1323}, {553, 16834}, {1111, 10072}, {1434, 16711}, {1478, 4089}, {1788, 17090}, {3543, 4872}, {3545, 17181}, {3665, 11237}, {3668, 17274}, {3674, 4654}, {3679, 9436}, {3945, 15956}, {4000, 7200}, {4403, 5309}, {4795, 6610}, {5298, 17081}, {6516, 16371}, {7176, 7195}


X(17080) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17080) lies on these lines:
{1, 411}, {2, 92}, {3, 1398}, {7, 941}, {8, 227}, {20, 17102}, {21, 34}, {31, 1758}, {35, 4347}, {37, 5226}, {42, 7672}, {55, 4318}, {56, 5262}, {57, 77}, {63, 223}, {73, 3868}, {85, 16705}, {100, 8270}, {201, 3876}, {222, 1993}, {225, 2476}, {241, 2275}, {255, 3468}, {307, 3687}, {312, 4552}, {404, 1038}, {497, 8758}, {561, 4554}, {603, 15777}, {614, 7677}, {664, 14829}, {940, 1442}, {968, 8543}, {970, 1425}, {980, 7176}, {982, 1458}, {986, 1042}, {988, 4320}, {1040, 7411}, {1060, 6905}, {1062, 3651}, {1068, 6825}, {1076, 6943}, {1150, 1943}, {1319, 4906}, {1393, 1816}, {1394, 4652}, {1407, 1443}, {1419, 3928}, {1445, 2999}, {1456, 4640}, {1457, 3877}, {1617, 7191}, {1682, 7143}, {1745, 12528}, {1785, 6932}, {1804, 6611}, {1838, 6828}, {1877, 11114}, {2263, 17594}, {2647, 10448}, {3100, 7580}, {3562, 5709}, {3670, 4306}, {3681, 4551}, {3869, 10571}, {3976, 4322}, {4313, 15852}, {4334, 17591}, {4351, 14793}, {4392, 17625}, {5018, 9316}, {5173, 17018}, {5219, 16577}, {5312, 12432}, {5718, 6354}, {5930, 6734}, {6198, 6985}, {6838, 7952}, {7004, 11220}, {7011, 11350}, {7291, 15509}, {9371, 9778}, {11020, 14547}, {13388, 16440}, {13389, 16441}


X(17081) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17081) lies on these lines:
{2, 7176}, {7, 21}, {36, 17170}, {77, 1193}, {85, 7288}, {241, 2275}, {278, 14013}, {279, 1447}, {388, 17095}, {664, 1788}, {927, 14267}, {934, 1398}, {1038, 1442}, {1319, 6604}, {1397, 7055}, {1420, 9436}, {1445, 1475}, {2646, 14548}, {2898, 3188}, {3086, 5088}, {3361, 3674}, {3598, 7185}, {3600, 7179}, {3911, 9312}, {4293, 17181}, {5022, 17044}, {5261, 7268}, {5298, 17079}, {5433, 7223}, {7175, 17257}, {7195, 17078}, {7229, 16720}, {7279, 10831}, {8732, 14189}, {9778, 17798}, {12649, 17136}


X(17082) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17082) lies on these lines:
{7, 310}, {57, 239}, {77, 614}, {194, 1424}, {278, 17085}, {348, 17083}, {664, 1403}, {1397, 4573}, {1469, 7196}, {4320, 7093}, {7192, 17127}, {7248, 10030}


X(17083) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17083) lies on these lines:
{7, 1397}, {57, 7185}, {77, 612}, {348, 17082}, {6063, 17085}


X(17084) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17084) lies on these lines:
{1, 147}, {2, 257}, {7, 21}, {8, 325}, {12, 664}, {35, 5195}, {57, 17397}, {65, 17095}, {73, 1442}, {85, 11375}, {226, 6625}, {857, 948}, {934, 15168}, {1111, 5443}, {1125, 1447}, {1319, 7247}, {1385, 4911}, {2475, 17136}, {2646, 4872}, {3487, 15972}, {3673, 5886}, {4059, 4870}, {5088, 12047}, {5219, 9312}, {5435, 6703}, {6542, 16603}, {7282, 11363}


X(17085) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17085) lies on these lines:
{7, 1365}, {278, 17082}, {5219, 9312}, {6063, 17083}, {17076, 17091}


X(17086) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17086) lies on these lines:
{2, 92}, {6, 17950}, {7, 604}, {77, 3662}, {85, 4657}, {141, 664}, {142, 14189}, {175, 11293}, {176, 11294}, {239, 307}, {241, 16706}, {269, 17304}, {348, 2275}, {651, 6646}, {948, 17321}, {1418, 17078}, {1419, 17274}, {1435, 2339}, {1442, 17300}, {1445, 17367}, {3668, 17023}, {3739, 17095}, {3821, 5018}, {3946, 9436}, {4201, 4296}, {4388, 17797}, {4389, 6180}, {4552, 17280}, {4572, 9230}, {5088, 12610}, {5228, 17380}, {5723, 17277}, {6610, 17235}, {6999, 17134}, {7190, 17396}, {8545, 17247}, {9312, 17306}


X(17087) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17087) lies on these lines:
{2, 7112}, {348, 349}, {4554, 8024}, {5061, 7217}


X(17088) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17088) lies on these lines:
{274, 278}, {1365, 7214}, {3266, 4554}, {4573, 6357}, {7112, 17923}


X(17089) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17089) lies on these lines:
{7, 528}, {85, 11375}, {150, 4089}, {279, 291}, {348, 17090}, {1111, 16173}, {1323, 1447}, {1565, 12019}, {3160, 3445}, {4440, 4919}, {5435, 5723}, {5726, 7179}, {7185, 9312}


X(17090) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    (a + b - c) (a - b + c) (a b + a c - 3 b c) : :

X(17090) lies on these lines:
{2, 17091}, {7, 8}, {10, 7185}, {57, 16816}, {226, 17230}, {241, 16602}, {348, 17089}, {664, 1388}, {1111, 5697}, {1266, 12640}, {1278, 4050}, {1420, 1447}, {1788, 17079}, {3160, 16020}, {3361, 7176}, {3673, 9957}, {3674, 3947}, {5226, 7146}, {5435, 16609}, {16888, 17238}


X(17091) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17091) lies on these lines:
{2, 17090}, {7, 310}, {3212, 7196}, {3741, 7185}, {4362, 7176}, {17076, 17085}


X(17092) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(210), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17092) lies on these lines:
{1, 7673}, {6, 1443}, {7, 37}, {56, 4318}, {57, 77}, {85, 4751}, {100, 8271}, {219, 3218}, {269, 651}, {277, 279}, {1427, 5435}, {1441, 4699}, {1442, 5228}, {1458, 7672}, {1471, 5018}, {1742, 7671}, {1818, 3868}, {2263, 7677}, {2293, 11025}, {3339, 4868}, {3644, 4552}, {3743, 5586}, {6180, 16885}, {6610, 16671}, {7176, 16975}, {7271, 8545}, {7289, 11349}, {16706, 17075}


X(17093) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(220), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17093) lies on these lines:
{2, 85}, {7, 55}, {11, 2898}, {57, 7056}, {69, 200}, {77, 14547}, {278, 13149}, {479, 658}, {497, 14189}, {934, 9061}, {1323, 11019}, {1407, 2991}, {1434, 1817}, {1997, 4554}, {3160, 10580}, {3188, 10431}, {3668, 17321}, {3870, 4350}, {4847, 9312}, {5218, 9446}, {7580, 17170}, {10481, 13405}


X(17094) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(525), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17094) lies on these lines:
{7, 4467}, {57, 1021}, {226, 3700}, {241, 514}, {521, 4025}, {523, 4077}, {525, 8611}, {647, 1214}, {656, 8057}, {850, 1441}, {934, 2766}, {1708, 9404}, {2804, 17896}, {3265, 14208}, {4897, 6003}, {7234, 18006}


X(17095) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(1100), WHERE A'B'C' = MEDIAL TRIANGLE

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

X(17095) lies on these lines:
{2, 85}, {3, 4872}, {5, 5088}, {7, 5550}, {9, 7183}, {10, 664}, {12, 7176}, {21, 6516}, {36, 4911}, {56, 7179}, {65, 17084}, {75, 7318}, {77, 5224}, {86, 283}, {140, 1565}, {150, 1385}, {226, 1434}, {274, 349}, {304, 7763}, {312, 3926}, {319, 1273}, {325, 7270}, {331, 17923}, {388, 17081}, {499, 3673}, {631, 17170}, {651, 17256}, {658, 2349}, {934, 5260}, {1025, 3294}, {1038, 7210}, {1125, 9436}, {1323, 3634}, {1358, 7294}, {1414, 1935}, {1445, 17381}, {1447, 3665}, {1698, 9312}, {2185, 7364}, {2329, 4564}, {3160, 9780}, {3188, 4197}, {3305, 7177}, {3579, 5195}, {3582, 7264}, {3584, 7278}, {3616, 5543}, {3674, 3911}, {3739, 17086}, {3826, 14189}, {4056, 7280}, {4352, 17720}, {4670, 14564}, {4708, 6610}, {4870, 4955}, {5086, 17136}, {5228, 17397}, {5298, 7198}, {5552, 16284}, {5701, 10030}, {5703, 14548}, {5723, 16815}, {5886, 17753}, {6180, 17248}, {6356, 6359}, {6390, 7283}, {7282, 11107}, {7799, 16577}, {10529, 17158}, {13411, 14828}

X(17095) = isotomic conjugate of X(7110)


X(17096) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    (a + b) (a + c) (b - c) (a + b - c) (a - b + c) : :

X(17096) lies on these lines:
{2, 14207}, {7, 4017}, {1014, 3733}, {1443, 1447}, {3669, 4560}, {3945, 6003}, {4552, 4625}, {4565, 4616}, {4573, 4615}, {7253, 17218}


X(17097) =  X(8)X(12)∩X(21)X(65)

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

See Dao Thanh Oai and Angel Montesdeoca, HG030418.

X(17097) lies on these lines:
{1,411}, {4,15911}, {7,3486}, {8,12}, {9,1405}, {21,65}, {35,5424}, {56,2320}, {57,5303}, {72,12867}, {79,3671}, {80,7548}, {84,5884}, {90,6912}, {100,6596}, {104,942}, {145,6601}, {226,5086}, {314,1441}, {354,1476}, {517,943}, {651,2647}, {758,15910}, {938,3427}, {987,4332}, {1000,1482}, {1156,1858}, {1159,3560}, {1172,1880}, {1320,11011}, {1937,2654}, {2098,7320}, {2298,4318}, {2346,3057}, {3062,10394}, {3065,11571}, {3243,10865}, {3254,10106}, {3296,3600}, {3577,6261}, {3649,11604}, {3678,4866}, {3680,3870}, {4298,5557}, {4308,5558}, {4853,11526}, {4900,6765}, {5045,15179}, {5559,11009}, {5665,10393}, {5730,6856}, {5902,15446}, {5903,15175}, {6909,13750}, {7091,11518}, {7160,7982}, {12709,16133}


X(17098) =  X(1)X(6985)∩X(21)X(46)

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

See Dao Thanh Oai and Angel Montesdeoca, HG030418.

X(17098) lies on these lines:
{1,6985}, {7,10572}, {8,6871}, {9,5903}, {21,46}, {40,15175}, {57,15446}, {65,90}, {79,11529}, {80,3340}, {84,5902}, {104,3338}, {517,7162}, {942,7284}, {943,5119}, {1000,3485}, {1159,1898}, {1389,6261}, {1392,3957}, {1478,11520}, {2346,13375}, {3254,9613}, {3255,4312}, {3296,3486}, {3427,5804}, {3601,5424}, {3612,6876}, {3671,5555}, {3680,11009}, {4338,7491}, {5425,5665}, {5531,13143}, {5557,11518}, {5558,15933}, {5559,6842}, {5697,7160}, {5726,11280}, {5734,6838}, {6598,7700}, {6866,10826}, {7962,13606}


X(17099) =  (name pending)

Barycentrics    a (4 a^6-7 a^5 b-7 a^4 b^2+14 a^3 b^3+2 a^2 b^4-7 a b^5+b^6-7 a^5 c+26 a^4 b c-11 a^3 b^2 c-25 a^2 b^3 c+18 a b^4 c-b^5 c-7 a^4 c^2-11 a^3 b c^2+34 a^2 b^2 c^2-9 a b^3 c^2-b^4 c^2+14 a^3 c^3-25 a^2 b c^3-9 a b^2 c^3+2 b^3 c^3+2 a^2 c^4+18 a b c^4-b^2 c^4-7 a c^5-b c^5+c^6) : :

See Antreas Hatzipolakis and Peter Moses, < Hyacinthos 27443.

X(17099) lies on this line: {1,88}


X(17100) =  CIRCUMCIRCLE-INVERSE OF X(8)

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

See Antreas Hatzipolakis and Peter Moses, < Hyacinthos 27443.

X(17100) lies on these lines:
{2,10058}, {3,8}, {11,404}, {21,3035}, {35,214}, {36,2802}, {56,1320}, {78,1768}, {119,6906}, {145,10074}, {149,3086}, {153,5552}, {224,9964}, {528,5172}, {900,405 7}, {901,2841}, {997,5010}, {1158 ,4855}, {1259,13243}, {1317,3871 }, {1387,5253}, {1470,3241}, { 1811,3435}, {2077,2800}, {2475,8 068}, {2771,5440}, {2829,5080}, { 3149,10724}, {3436,12248}, {3869 ,12515}, {4190,10321}, {4193, 12764}, {4421,10031}, {4861, 11715}, {5086,12619}, {5096, 9024}, {5204,8668}, {5450,12751} ,{5541,7280}, {5840,6905}, { 6713,6940}, {6796,12119}, {6924, 10738}, {6942,13199}, {7972,8715}, {9802,13279}, {9809,10309 }, {10087,14793}, {10698,11248}, {10707,16371}, {10742,11681}, {11015,12743}, {12737,14923}

X(17100) = reflection of X(149) in X(5533)
X(17100) = isogonal conjugate of X(17101)
X(17100) = circumcircle-inverse of X(8)
X(17100) = crosssum of X(1329) and X(5854)
X(17100) = incircle-of-anticomplmementary-triange-inverse of X(5687)
X(17100) = X(643)-beth conjugate of X(1145)
X(17100) = X(8)-vertex conjugate of X(900)
X(17100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 100, 4996), (3, 2932, 100), (56, 13205, 1320), (78, 1768, 12532), (100, 104, 8), (100, 2975, 1145), (100, 12531, 5687), (149, 4188, 10090), (5687, 12773, 12531)


X(17101) =  ISOGONAL CONJUGATE OF X(17100)

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

See Antreas Hatzipolakis and Peter Moses, < Hyacinthos 27443.

X(17101) lies on the cubic K684 and these lines:
{11,10428}, {56,3259}, {513,127 64}, {517,10742}, {901,1329}, {95 3,2829}, {2098,3326}

X(17101) = reflection of X(i) in X(j) for these {i,j}: {56, 3259}, {901, 1329}
X(17101) = isogonal conjugate of X(17100)
X(17101) = cevapoint of X(1329) and X(5854)
X(17101) = symgonal image of X(1329)


X(17102) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(92), WHERE A'B'C' = MEDIAL TRIANGLE

Barycentrics    a (a^2-b^2-c^2) (a^3 b+a^2 b^2-a b^3-b^4+a^3 c-2 a^2 b c+a b^2 c+a^2 c^2+a b c^2+2 b^2 c^2-a c^3-c^4) : :
X(17102) = SA SB SC X[1] + 8 sa^2 sb^2 sc^2 X[3]

Also, X(17102) = (A,B,C,X(1); A',B',C',X(2)) collineation image of X(281), where A'B'C' = medial triangle

X(17102) lies on these lines:
{1,3}, {2,280}, {4,1465}, {5,1785}, {6,1741}, {10,2968}, {21,162}, {33,3149}, {34,1012}, {37,216}, {44,5158}, {63,7078}, {73,1071}, {77,1433}, {84,223}, {123,4187}, {140,15252}, {158,6708}, {221,1158}, {222,1181}, {225,6831}, {227,515}, {255,3916}, {278,6847}, {283,16697}, {404,15500}, {405,1712}, {411,3100}, {441,17023}, {442,1074}, {577,1100}, {581,10391}, {774,1193}, {855,1828}, {856,5439}, {859,1905}, {938,4850}, {971,1745}, {1064,12711}, {1068,6833}, {1125,6509}, {1210,1834}, {1249,6857}

X(17102) = complement X(318)
X(17102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1735, 65), (1, 8069, 5266), (73, 7004, 1071), (13388, 13389, 7011)
X(17102) = X(i)-complementary conjugate of X(j) for these (i,j): {3, 1329}, {25, 15849}, {48, 3452}, {56, 5}, {73, 3454}, {77, 2887}, {184, 9}, {222, 141}, {348, 626}, {603, 10}, {604, 226}, {608, 13567}, {667, 6506}, {1106, 1210}, {1333, 6708}, {1397, 6}, {1407, 16608}, {1408, 942}, {1409, 1211}, {1410, 442}, {1437, 960}, {1459, 124}, {1804, 1368}, {1813, 3835}, {1946, 5514}, {2203, 9119}, {2720, 8677}, {7053, 2886}, {7056, 17047}, {7099, 142}, {7114, 6260}, {7177, 17046}, {7335, 3}, {9247, 1212}, {14575, 16588}
X(17102) = X(1897)-Ceva conjugate of X(521)
X(17102) = X(4)-isoconjugate of X(947)
X(17102) = crosspoint of X(2) and X(77)
X(17102) = crosssum of X(i) and X(j) for these (i,j): {1, 1771}, {6, 33}, {19, 3195}
X(17102) = barycentric product X(i)*X(j) for these {i,j}: {63, 946}, {69, 2262}, {1856, 7183}
X(17102) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 947}, {946, 92}, {2262, 4}
X(17102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1735, 65), (1, 8069, 5266), (73, 7004, 1071), (13388, 13389, 7011)


X(17103) =  X(1)X(99)∩X(7)X(21)

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

X(17103) lies on the cubic K967 and these lines:
{1, 99}, {2, 1931}, {6, 2669}, {7, 21}, {10, 6629}, {31, 873}, {41, 662}, {55, 2668}, {57, 14534}, {58, 274}, {75, 757}, {76, 5209}, {81, 330}, {83, 1019}, {85, 603}, {87, 1178}, {171, 1909}, {172, 894}, {192, 1963}, {261, 10436}, {310, 14012}, {385, 4754}, {750, 799}, {880, 1966}, {940, 1975}, {1580, 4697}, {1621, 16691}, {2106, 2176}, {2238, 16917}, {2275, 14621}, {2295, 6645}, {3570, 5277}, {3618, 4209}, {3926, 4340}, {3972, 16783}, {4252, 16992}, {4418, 6628}, {4573, 9312}, {4622, 16944}, {4670, 16702}, {5021, 11321}, {5224, 16454}, {5712, 6337}, {7196, 14006}, {16583, 16756}

X(17103) = {X(99),X(1509)}-harmonic conjugate of X(1)
X(17103) = X(i)-Ceva conjugate of X(j) for these (i,j): {99, 4367}, {14534, 86}
X(17103) = X(i)-cross conjugate of X(j) for these (i,j): {894, 8033}, {4154, 385}, {4374, 6649}, {4697, 894}, {7207, 4367}
X(17103) = X(i)-isoconjugate of X(j) for these (i,j): {10, 904}, {37, 893}, {42, 256}, {210, 1431}, {213, 257}, {321, 7104}, {512, 3903}, {694, 2238}, {740, 1967}, {756, 1178}, {805, 4155}, {874, 881}, {882, 3573}, {1334, 1432}, {1402, 4451}, {1581, 3747}, {1824, 7015}, {1826, 7116}, {1918, 7018}, {3948, 9468}, {4079, 4603}
X(17103) = cevapoint of X(i) and X(j) for these (i,j): {171, 894}, {385, 4154}
X(17103) = trilinear pole of line {385, 4369}
X(17103) = crossdifference of every pair of points on line {3709, 4093}
X(17103) = barycentric product X(i)*X(j) for these {i,j}: {1, 8033}, {21, 7196}, {58, 1920}, {81, 1909}, {86, 894}, {99, 4369}, {100, 16737}, {171, 274}, {172, 310}, {261, 4032}, {284, 7205}, {314, 7175}, {333, 7176}, {348, 14006}, {552, 4095}, {593, 1237}, {662, 4374}, {741, 3978}, {757, 3963}, {799, 4367}, {873, 2295}, {880, 3572}, {1215, 1509}, {1434, 7081}, {2533, 4610}, {3287, 4625}, {3907, 4573}, {4107, 4589}, {4164, 4639}, {4459, 4620}, {4477, 4635}, {4529, 4616}, {4560, 6649}, {4579, 7199}, {4584, 14296}, {4600, 7200}, {4615, 4922}, {6385, 7122}
X(17103) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 893}, {81, 256}, {86, 257}, {171, 37}, {172, 42}, {274, 7018}, {333, 4451}, {385, 740}, {593, 1178}, {662, 3903}, {741, 694}, {763, 7303}, {894, 10}, {1014, 1432}, {1215, 594}, {1333, 904}, {1412, 1431}, {1434, 7249}, {1437, 7116}, {1580, 2238}, {1691, 3747}, {1790, 7015}, {1840, 7140}, {1909, 321}, {1920, 313}, {1966, 3948}, {2206, 7104}, {2295, 756}, {2329, 210}, {2330, 1334}, {2533, 4024}, {3287, 4041}, {3572, 882}, {3907, 3700}, {3955, 71}, {3963, 1089}, {4019, 3695}, {4027, 4154}, {4032, 12}, {4039, 4037}, {4095, 6057}, {4107, 4010}, {4128, 3124}, {4367, 661}, {4369, 523}, {4374, 1577}, {4400, 4365}, {4434, 3943}, {4447, 3930}, {4477, 4171}, {4504, 14321}, {4579, 1018}, {4610, 4594}, {4623, 7260}, {4697, 1213}, {4774, 4931}, {4922, 4120}, {6645, 1215}, {6649, 4552}, {7009, 1826}, {7081, 2321}, {7119, 1824}, {7122, 213}, {7175, 65}, {7176, 226}, {7184, 3721}, {7187, 2887}, {7196, 1441}, {7200, 3120}, {7205, 349}, {7207, 16592}, {7234, 4079}, {7267, 4062}, {8033, 75}, {8623, 4093}, {14006, 281}, {16592, 2643}, {16720, 15523}, {16737, 693}


X(17104) =  X(1)X(60)∩X(3)X(501)

Trilinears    csc(B - C)[sin(A - C) - sin(A - B)] : :
Barycentrics    a^3*(a + b)*(a + c)*(a^2 - b^2 - b*c - c^2) : :

X(17104) lies on the cubic K967 and these lines:
{1, 60}, {3, 501}, {6, 3444}, {21, 15446}, {31, 849}, {41, 163}, {48, 2150}, {49, 5396}, {54, 2616}, {55, 15792}, {56, 58}, {57, 1175}, {80, 13746}, {81, 5358}, {184, 386}, {229, 5902}, {283, 3422}, {284, 1069}, {581, 1147}, {643, 8715}, {991, 1092}, {993, 1098}, {1048, 2612}, {1101, 9274}, {1178, 3453}, {1193, 15618}, {1325, 5903}, {1326, 2175}, {1780, 1790}, {2176, 5006}, {2185, 5248}, {2477, 2594}, {2606, 6757}, {2608, 9273}, {2651, 5197}, {3109, 10950}, {3216, 5012}, {3615, 7741}, {3647, 10411}, {3736, 7087}, {3754, 11116}, {4658, 15934}, {5312, 9587}, {5400, 13434}, {14355, 14838}

X(17104) = isogonal conjugate of X(6757)
X(17104) = crosspoint of X(110) and X(1101)
X(17104) = crosssum of X(523) and X(1109)
X(17104) = crossdifference of every pair of points on line {2610, 3700}
X(17104) = X(i)-beth conjugate of X(j) for these (i,j): {21, 11101}, {4636, 1}, {6740, 13746}
X(17104) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 6757}, {2624, 2610}
X(17104) = X(i)-Ceva conjugate of X(j) for these (i,j): {110, 2605}, {1171, 1333}, {1175, 58}, {9273, 1983}
X(17104) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6757}, {2, 8818}, {10, 79}, {12, 3615}, {94, 2245}, {226, 7110}, {265, 860}, {313, 6186}, {321, 2160}, {476, 6370}, {523, 6742}, {661, 15455}, {758, 2166}, {1441, 7073}, {1989, 3936}, {4036, 13486}, {5627, 6739}
X(17104) = trilinear pole of line {50, 2624}
X(17104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 60, 9275), (60, 110, 1), (501, 5127, 3), (1437, 2194, 58), (1780, 1790, 4278)
X(17104) = barycentric product X(i)*X(j) for these {i,j}: {21, 2003}, {35, 81}, {50, 14616}, {58, 3219}, {60, 16577}, {86, 2174}, {110, 14838}, {163, 4467}, {222, 11107}, {249, 2611}, {284, 1442}, {319, 1333}, {323, 759}, {333, 1399}, {593, 3678}, {662, 2605}, {692, 16755}, {849, 3969}, {1101, 8287}, {1171, 3647}, {1175, 16585}, {1412, 4420}, {1414, 9404}, {1790, 6198}, {2185, 2594}, {2193, 7282}, {4570, 7202}
X(17104) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6757}, {31, 8818}, {35, 321}, {50, 758}, {110, 15455}, {163, 6742}, {759, 94}, {1333, 79}, {1399, 226}, {1442, 349}, {2003, 1441}, {2150, 3615}, {2174, 10}, {2194, 7110}, {2206, 2160}, {2477, 16577}, {2594, 6358}, {2605, 1577}, {2611, 338}, {2624, 6370}, {3219, 313}, {3647, 1230}, {6149, 3936}, {9404, 4086}, {11107, 7017}, {14270, 2610}, {14591, 4242}, {14838, 850}, {14975, 1826}, {16585, 1234}


X(17105) =  X(1)X(727)∩X(56)X(87)

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

X(17105) lies on the cubic K967 and these lines:
{1, 727}, {56, 87}, {172, 983}, {1580, 9316}, {2162, 8616}

X(17105) = {X(932),X(7121)}-harmonic conjugate of X(1)
X(17105) = X(43)-isoconjugate of X(3551)
X(17105) = barycentric product X(330)*X(3550)
X(17105) = barycentric quotient X(i)/X(j) for these {i,j}: {2162, 3551}, {3550, 192}


X(17106) =  X(1)X(103)∩X(7)X(11522)

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

X(17106) lies on the cubic K967 and these lines:
{1, 103}, {7, 11522}, {41, 1461}, {56, 269}, {57, 7955}, {77, 7987}, {165, 3160}, {279, 3361}, {603, 6614}, {658, 9312}, {910, 2124}, {1025, 4936}, {1323, 14256}, {1419, 3207}, {1439, 3532}, {2999, 6611}, {4350, 13462}, {4512, 7056}, {7183, 12526}

X(17106) = X(i)-Ceva conjugate of X(j) for these (i,j): {57, 269}, {9533, 1419}
X(17106) = crosspoint of X(57) and X(1419)
X(17106) = X(i)-isoconjugate of X(j) for these (i,j): {200, 3062}, {220, 10405}, {346, 11051}
X(17106) = X(269)-Hirst inverse of X(1456)
X(17106) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 738, 269), (57, 8830, 11051), (934, 7177, 1)
X(17106) = barycentric product X(i)*X(j) for these {i,j}: {1, 9533}, {7, 1419}, {57, 3160}, {144, 269}, {165, 279}, {934, 7658}, {1088, 3207}, {1407, 16284}
X(17106) = barycentric quotient X(i)/X(j) for these {i,j}: {144, 341}, {165, 346}, {269, 10405}, {1106, 11051}, {1407, 3062}, {1419, 8}, {3160, 312}, {3207, 200}, {7658, 4397}, {9533, 75}


X(17107) =  X(1)X(1292)∩X(7)X(7131)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - 2*a*b + b^2 - 2*b*c + c^2)*(a^2 + b^2 - 2*a*c - 2*b*c + c^2) : : X(17107) lies on the cubic K967 and these lines:
{1, 1292}, {7, 7131}, {56, 1279}, {57, 169}, {603, 1416}, {1358, 2082}, {3338, 14268}, {6601, 7091}

X(17107) = X(1292)-Ceva conjugate of X(3669)
X(17107) = X(i)-cross conjugate of X(j) for these (i,j): {31, 269}, {1475, 56}, {16502, 34}
X(17107) = X(i)-isoconjugate of X(j) for these (i,j): {2, 6600}, {8, 218}, {9, 3870}, {21, 3991}, {55, 344}, {78, 7719}, {200, 1445}, {220, 6604}, {333, 4878}, {346, 1617}, {644, 3309}, {646, 8642}, {728, 4350}, {3694, 4233}, {3939, 4468}, {4904, 6065}, {6605, 15185}
X(17107) = barycentric product X(i)*X(j) for these {i,j}: {7, 2191}, {57, 277}, {269, 6601}, {1292, 3676}
X(17107) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 6600}, {56, 3870}, {57, 344}, {269, 6604}, {277, 312}, {604, 218}, {608, 7719}, {1106, 1617}, {1292, 3699}, {1400, 3991}, {1402, 4878}, {1407, 1445}, {2191, 8}, {3669, 4468}, {6601, 341}, {7023, 4350}


X(17108) =  X(1)X(831)∩X(2)X(1930)

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

X(17108) lies on the cubic K967 and these lines:
{1, 831}, {2, 1930}, {56, 1631}

X(17108) = X(i)-isoconjugate of X(j) for these (i,j): {1220, 5280}, {2298, 3920}, {2483, 8707}
X(17108) = barycentric product X(831)*X(3004)
X(17108) = barycentric quotient X(i)/X(j) for these {i,j}: {831, 8707}, {1193, 3920}, {2300, 5280}, {6371, 830}


X(17109) =  X(1)X(88)∩X(31)X(16944)

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

X(17109) lies on the cubic K967 and these lines:
{1, 88}, {31, 16944}, {41, 9456}, {56, 4057}, {603, 1417}, {901, 3915}, {1201, 14260}, {4737, 9458}

X(17109) = X(106)-Ceva conjugate of X(1149)
X(17109) = crosssum of X(519) and X(4738)
X(17109) = crossdifference of every pair of points on line {1635, 2325}
X(17109) = X(901)-beth conjugate of X(1420)
X(17109) = X(i)-isoconjugate of X(j) for these (i,j): {519, 1120}, {900, 6079}, {4723, 8686}
X(17109) = crossdifference of every pair of points on line {1635, 2325}
X(17109) = barycentric product X(i)*X(j) for these {i,j}: {88, 1149}, {106, 16610}, {1266, 9456}, {1797, 1878}, {3257, 6085}
X(17109) = barycentric quotient X(i)/X(j) for these {i,j}: {1149, 4358}, {6085, 3762}, {8660, 1635}, {9456, 1120}, {16610, 3264}


X(17110) =  X(1)X(6013)∩X(56)X(7225)

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

X(17110) lies on the cubic K967 and these lines:
{1, 6013}, {56, 7225}, {81, 4384}

X(17110) = X(i)-isoconjugate of X(j) for these (i,j): {941, 17018}, {2258, 4687}
X(17110) = barycentric product X(10013)*X(10436)
X(17110) = barycentric quotient X(i)/X(j) for these {i,j}: {940, 4687}, {1468, 17018}


X(17111) =  X(5)X(10)∩X(11)X(33)

Barycentrics    (b^2+c^2)*a^4-2*b*c*(b-c)^2*a^ 2+2*(b^2-c^2)*(b-c)*b*c*a-(b^ 4-c^4)*(b^2-c^2) : :
X(17111) = (r^2-s^2+6*R*r)*X(5) + 2*R^2*X(10)

See César Lozada, Hyacinthos 27450.

X(17111) lies on these lines:
{2, 1486}, {5, 10}, {11, 33}, {56, 1883}, {429, 10896}, {496, 17061}, {1368, 3816}, {5133, 11680}, {5142, 10591}, {7399, 15908}, {7741, 17064}, {10593, 17070}, {13567, 14717}X(17110) = X(i)-isoconjugate of X(j) for these (i,j): {941, 17018}, {2258, 4687}


X(17112) =  CIRCUMCIRCLE-INVERSE OF X(1604)

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

See César Lozada, Hyacinthos 27450.

X(17112) lies on these lines:
{3, 9}, {517, 5514}, {946, 15849}, {3239, 3900}

X(17112) = circumcircle-inverse-of X(1604)


X(17113) =  MIDPOINT OF X(7) AND X(15913)

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

See César Lozada, Hyacinthos 27450.

X(17113) lies on these lines:
{7, 1699}, {142, 10004}, {144, 658}, {279, 1418}, {346, 4569}, {934, 11495}

X(17113) = midpoint of X(7) and X(15913)
X(17113) = medial-isotomic conjugate of X(279)


X(17114) =  X(7)X(76)∩X(56)X(106)

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

See César Lozada, Hyacinthos 27450.

X(17114) lies on these lines:
{1, 7248}, {4, 4014}, {7, 76}, {10, 1463}, {56, 106}, {57, 978}, {65, 519}, {181, 5221}, {774, 3675}, {942, 15310}, {1122, 10521}, {1319, 15999}, {1397, 1406}, {1403, 4306}, {1407, 7143}, {1469, 3339}, {3271, 17054}, {3562, 5091}, {3784, 10544}, {3924, 3937}

X(17114) = X(1093)-of-intouch-triangle
X(17114) = intouch-isogonal conjugate of X(12053)
X(17114) = intouch-isotomic conjugate of X(4907)


X(17115) =  ISOGONAL CONJUGATE OF X(8269)

Barycentrics    a*(b-c)*(a^2+(b-c)^2)*(-a+b+c) ^2 : :
X(17115) = X(650)+3*X(11193) = 3*X(11193)-X(11934)

See César Lozada, Hyacinthos 27450.

X(17115) lies on these lines:
{1, 9373}, {11, 11927}, {55, 650}, {512, 6587}, {513, 676}, {521, 3716}, {657, 6608}, {661, 663}, {693, 885}, {926, 14298}, {1385, 8760}, {1538, 3309}, {1946, 6050}, {2820, 7658}, {3035, 10006}, {3239, 3900}, {3816, 4885}, {4394, 6139}, {9957, 14077}, {15280, 15845}

X(17115) = isogonal conjugate of X(8269)
X(17115) = medial-isotomic conjugate of X(14936)


X(17116) =  X(1)X(1278)∩X(6)X(75)

Barycentrics    a^2 + 3 b c : :

X(17116) lies on these lines:
{1, 1278}, {2, 2415}, {6, 75}, {7, 3620}, {8, 17364}, {9, 4699}, {10, 6646}, {44, 4739}, {45, 4751}, {69, 7222}, {81, 4980}, {86, 536}, {141, 7321}, {142, 17266}, {144, 17331}, {145, 4349}, {190, 3739}, {192, 3247}, {274, 3230}, {319, 3630}, {320, 594}, {346, 17244}, {524, 5564}, {527, 1654}, {545, 1213}, {726, 16830}, {869, 16571}, {902, 3757}, {903, 17235}, {966, 17333}, {1086, 7227}, {1100, 4726}, {1266, 5750}, {1268, 4708}, {1269, 17790}, {1281, 14931}, {1730, 3219}, {1743, 16816}, {1930, 16785}, {1992, 4371}, {1999, 10447}, {2321, 17300}, {2345, 3619}, {3264, 3770}, {3403, 17752}, {3550, 16996}, {3617, 5223}, {3644, 16777}, {3664, 4431}, {3672, 17397}, {3679, 17343}, {3685, 16484}, {3834, 17285}, {3875, 4740}, {3923, 15485}, {3943, 17317}, {3945, 17389}, {3973, 4384}, {3980, 7081}, {4000, 17368}, {4007, 17373}, {4058, 4896}, {4195, 16485}, {4357, 4440}, {4360, 4670}, {4376, 10987}, {4389, 17303}, {4393, 4821}, {4398, 4657}, {4399, 7277}, {4419, 17248}, {4445, 17361}, {4454, 17257}, {4461, 17316}, {4470, 17321}, {4472, 17246}, {4473, 6666}, {4644, 11008}, {4647, 16474}, {4648, 17242}, {4664, 15668}, {4675, 17233}, {4676, 8692}, {4687, 16677}, {4688, 15492}, {4704, 16831}, {4733, 5852}, {4741, 17270}, {4764, 17318}, {4862, 17236}, {4888, 17294}, {5224, 17254}, {5749, 17367}, {6173, 17232}, {7061, 8289}, {7232, 17228}, {7263, 16706}, {7613, 9780}, {7996, 9812}, {8296, 8626}, {10022, 17320}, {16971, 17143}, {17227, 17293}, {17229, 17297}, {17230, 17298}, {17234, 17268}, {17238, 17274}, {17239, 17273}, {17240, 17313}, {17241, 17269}, {17245, 17264}, {17249, 17327}, {17250, 17255}, {17251, 17329}, {17256, 17334}, {17259, 17336}, {17263, 17340}, {17265, 17342}, {17271, 17345}, {17275, 17347}, {17278, 17354}, {17282, 17358}, {17283, 17359}, {17290, 17371}, {17295, 17376}, {17299, 17378}, {17301, 17381}, {17305, 17385}, {17309, 17387}, {17314, 17391}, {17315, 17392}, {17323, 17400}

X(17116) = {X(6),X(75)}-harmonic conjugate of X(17117)


X(17117) =  X(1)X(4699)∩X(6)X(75)

Barycentrics    a^2 - 3 b c : :

X(17117) lies on these lines: {1, 4699}, {2, 2321}, {6, 75}, {7, 17363}, {8, 1738}, {9, 1278}, {10, 17302}, {37, 16815}, {44, 4726}, {45, 3644}, {69, 4371}, {86, 4688}, {141, 5564}, {142, 6542}, {145, 17391}, {190, 4686}, {192, 3731}, {274, 16971}, {319, 1086}, {320, 3630}, {346, 17338}, {391, 17333}, {519, 17300}, {524, 7321}, {536, 16814}, {594, 4395}, {740, 16484}, {903, 17345}, {966, 17247}, {1100, 4739}, {1213, 17320}, {1266, 3686}, {1281, 8289}, {1654, 3663}, {1930, 16784}, {1992, 7222}, {1999, 4359}, {2177, 3757}, {2345, 17367}, {3008, 4431}, {3187, 14996}, {3210, 5271}, {3230, 16827}, {3243, 3621}, {3617, 7174}, {3619, 3661}, {3632, 17298}, {3672, 17248}, {3679, 17238}, {3685, 15485}, {3723, 3739}, {3729, 3973}, {3750, 16994}, {3782, 4886}, {3834, 17295}, {3943, 17263}, {4007, 17230}, {4034, 17274}, {4060, 17067}, {4389, 17252}, {4393, 4772}, {4398, 4643}, {4416, 4440}, {4419, 17331}, {4445, 17227}, {4452, 17257}, {4647, 5315}, {4648, 17389}, {4659, 4821}, {4664, 16677}, {4665, 17289}, {4673, 16486}, {4675, 17377}, {4687, 16674}, {4690, 17273}, {4741, 4862}, {4751, 16777}, {4764, 17262}, {4859, 17232}, {4971, 17245}, {5222, 17368}, {5224, 17301}, {5695, 8692}, {5839, 11008}, {6173, 17375}, {7032, 16571}, {7061, 14931}, {7191, 17163}, {7232, 17360}, {11679, 17490}, {15668, 17393}, {16499, 16821}, {16738, 16829}, {16834, 17379}, {17028, 17756}, {17228, 17290}, {17229, 17283}, {17233, 17266}, {17234, 17299}, {17235, 17271}, {17236, 17270}, {17239, 17305}, {17240, 17265}, {17241, 17309}, {17244, 17314}, {17246, 17256}, {17249, 17251}, {17250, 17323}, {17255, 17328}, {17258, 17330}, {17264, 17337}, {17269, 17341}, {17276, 17346}, {17281, 17352}, {17285, 17356}, {17293, 17370}, {17297, 17372}, {17303, 17380}, {17307, 17382}, {17308, 17383}, {17313, 17386}, {17317, 17388}, {17322, 17395}, {17327, 17399}

X(17117) = {X(6),X(75)}-harmonic conjugate of X(17116)


X(17118) =  X(1)X(4686)∩X(6)X(75)

Barycentrics    a^2 + 4 b c : :

X(17118) lies on these lines:
{1, 4686}, {2, 4398}, {6, 75}, {7, 594}, {8, 7222}, {9, 4688}, {10, 17253}, {37, 4659}, {45, 3729}, {63, 5341}, {69, 4478}, {86, 1278}, {142, 17267}, {144, 17330}, {190, 4699}, {192, 15668}, {193, 4399}, {274, 16969}, {319, 15533}, {320, 4445}, {346, 17245}, {527, 17275}, {536, 10436}, {545, 17257}, {903, 17236}, {966, 4454}, {1086, 2345}, {1100, 17151}, {1213, 4419}, {1266, 4657}, {1267, 8252}, {1407, 6358}, {1447, 8556}, {2321, 4675}, {3052, 4418}, {3247, 4718}, {3618, 4395}, {3620, 7238}, {3644, 16826}, {3661, 7232}, {3662, 17293}, {3663, 17303}, {3664, 17299}, {3672, 4470}, {3679, 17344}, {3778, 4492}, {3834, 17286}, {3875, 4670}, {3943, 4461}, {3945, 17388}, {3980, 4434}, {4000, 7229}, {4007, 4888}, {4053, 7201}, {4360, 4740}, {4371, 4969}, {4384, 4739}, {4389, 17327}, {4409, 4748}, {4431, 4851}, {4440, 5224}, {4452, 17395}, {4459, 10387}, {4472, 17321}, {4643, 4967}, {4644, 6144}, {4681, 16674}, {4698, 16677}, {4751, 17261}, {4764, 17319}, {4772, 17277}, {4821, 17160}, {4859, 17357}, {4862, 17237}, {5257, 17132}, {5391, 8253}, {5564, 17364}, {5749, 17366}, {5750, 17301}, {5839, 7277}, {6173, 17231}, {6646, 17251}, {7200, 9457}, {10022, 17045}, {16669, 16833}, {16814, 16832}, {16815, 17336}, {17229, 17298}, {17233, 17313}, {17234, 17269}, {17235, 17308}, {17239, 17274}, {17265, 17280}, {17270, 17345}, {17278, 17355}, {17282, 17359}, {17289, 17290}, {17294, 17376}, {17300, 17309}, {17304, 17385}, {17314, 17392}

X(17118) = {X(6),X(75)}-harmonic conjugate of X(17119)


X(17119) =  X(1)X(4688)∩X(6)X(75)

Barycentrics    a^2 - 4 b c : :

X(17119) lies on these lines:
{1, 4688}, {2, 3943}, {6, 75}, {7, 4371}, {8, 599}, {9, 4686}, {10, 17301}, {37, 16832}, {44, 4659}, {45, 536}, {63, 7297}, {69, 4399}, {86, 4772}, {142, 17299}, {145, 17392}, {190, 4740}, {192, 16675}, {220, 4858}, {319, 7232}, {320, 15533}, {346, 17337}, {391, 17334}, {519, 4675}, {524, 4405}, {594, 3763}, {903, 4741}, {966, 4452}, {1001, 4693}, {1191, 4647}, {1213, 3672}, {1266, 4643}, {1267, 8253}, {1278, 17262}, {1654, 4398}, {1930, 16781}, {2321, 17267}, {2345, 4402}, {3008, 17281}, {3210, 5737}, {3242, 3696}, {3618, 7227}, {3620, 4478}, {3632, 6173}, {3644, 17260}, {3661, 17290}, {3662, 4445}, {3663, 17253}, {3679, 17237}, {3686, 17276}, {3707, 17132}, {3729, 4726}, {3731, 4718}, {3739, 3875}, {3764, 4492}, {3834, 17294}, {3946, 17303}, {4007, 4859}, {4034, 4862}, {4042, 17155}, {4051, 7202}, {4360, 4699}, {4365, 4423}, {4389, 17251}, {4419, 17330}, {4431, 17279}, {4432, 5695}, {4440, 17346}, {4441, 4465}, {4461, 17340}, {4470, 17014}, {4644, 4969}, {4648, 17388}, {4657, 4967}, {4664, 16815}, {4670, 16834}, {4681, 16677}, {4690, 17274}, {4698, 16674}, {4739, 4852}, {4751, 17319}, {4764, 17261}, {4821, 17349}, {4971, 17316}, {5222, 17369}, {5224, 17323}, {5295, 17054}, {5391, 8252}, {5839, 6144}, {6542, 17313}, {7081, 8556}, {7222, 7277}, {7321, 17363}, {16706, 17293}, {16969, 17143}, {17229, 17282}, {17233, 17265}, {17234, 17309}, {17235, 17270}, {17239, 17304}, {17245, 17314}, {17286, 17356}, {17298, 17372}, {17302, 17327}, {17308, 17382}

X(17119) = {X(6),X(75)}-harmonic conjugate of X(17118)


X(17120) =  X(1)X(4704)∩X(6)X(75)

Barycentrics    3 a^2 + b c : :

X(17120) lies on these lines:
{1, 4704}, {2, 1743}, {6, 75}, {7, 17367}, {9, 16826}, {44, 86}, {45, 17394}, {69, 17292}, {81, 4358}, {87, 869}, {144, 17247}, {190, 1100}, {192, 1449}, {193, 3661}, {319, 3629}, {320, 3589}, {344, 17391}, {346, 17389}, {524, 17287}, {527, 17302}, {536, 16668}, {597, 16706}, {599, 17371}, {645, 1963}, {749, 4484}, {1051, 4970}, {1086, 6329}, {1278, 16834}, {1447, 3329}, {1654, 5750}, {1757, 16830}, {1992, 2345}, {1999, 4671}, {2298, 2991}, {2308, 3757}, {2309, 2663}, {3618, 3662}, {3685, 4649}, {3729, 4393}, {3739, 16671}, {3763, 17361}, {3879, 17280}, {3945, 17244}, {3946, 4440}, {3973, 16831}, {3997, 10027}, {4021, 4480}, {4360, 4718}, {4419, 17396}, {4422, 17317}, {4431, 4856}, {4445, 15534}, {4464, 4982}, {4643, 17326}, {4648, 17338}, {4657, 17254}, {4663, 5263}, {4664, 16884}, {4667, 17266}, {4670, 16669}, {4675, 17352}, {4687, 16885}, {4700, 4967}, {4715, 17273}, {4741, 17306}, {4772, 16833}, {4795, 17278}, {4851, 17268}, {4969, 5564}, {5280, 14210}, {5294, 17778}, {5356, 16568}, {6144, 17293}, {6381, 17499}, {6542, 17355}, {6646, 17023}, {7081, 7766}, {7179, 16989}, {7191, 17146}, {7232, 17370}, {7321, 17366}, {8584, 17362}, {10436, 16670}, {10455, 16704}, {15668, 17335}, {16468, 16823}, {16777, 17336}, {17045, 17258}, {17253, 17400}, {17255, 17399}, {17256, 17398}, {17257, 17397}, {17262, 17393}, {17263, 17392}, {17264, 17390}, {17267, 17387}, {17269, 17386}, {17271, 17385}, {17274, 17383}, {17276, 17380}, {17279, 17312}, {17281, 17377}, {17283, 17376}, {17284, 17375}, {17285, 17374}, {17286, 17373}, {17295, 17359}, {17296, 17358}, {17297, 17357}, {17303, 17346}, {17305, 17345}, {17307, 17344}, {17308, 17343}, {17311, 17342}, {17313, 17341}, {17315, 17340}, {17316, 17339}, {17320, 17334}, {17321, 17333}, {17322, 17332}, {17325, 17329}, {17327, 17328}

X(17120) = {X(6),X(75)}-harmonic conjugate of X(17121)


X(17121) =  X(1)X(4991)∩X(6)X(75)

Barycentrics    3 a^2 - b c : :

X(17121) lies on these lines:
{1, 4991}, {2, 1449}, {6, 75}, {8, 16475}, {9, 4393}, {43, 7032}, {44, 4360}, {45, 17393}, {69, 17291}, {86, 16666}, {145, 3717}, {190, 4718}, {192, 1743}, {193, 3662}, {319, 3589}, {320, 3629}, {344, 17389}, {391, 17248}, {519, 17280}, {524, 16706}, {536, 16671}, {594, 6329}, {597, 17289}, {599, 17370}, {740, 16477}, {966, 17397}, {1100, 4698}, {1447, 7766}, {1654, 17023}, {1707, 4734}, {1992, 4000}, {1999, 2300}, {2209, 16476}, {2325, 4464}, {3008, 17300}, {3187, 4671}, {3329, 3791}, {3618, 3661}, {3672, 17333}, {3685, 16468}, {3705, 16989}, {3739, 16668}, {3763, 17360}, {3780, 17752}, {3875, 4788}, {3912, 4856}, {3946, 6646}, {3950, 4473}, {4038, 16993}, {4357, 4700}, {4384, 16667}, {4395, 7277}, {4416, 17254}, {4422, 17315}, {4445, 17371}, {4643, 17324}, {4649, 4974}, {4657, 17252}, {4664, 16885}, {4672, 4716}, {4684, 4989}, {4687, 16884}, {4690, 17307}, {4699, 16833}, {4725, 17295}, {4741, 17304}, {4851, 17266}, {4982, 6666}, {5299, 14210}, {5564, 17369}, {6144, 17290}, {6381, 17034}, {6542, 17268}, {7122, 8300}, {7191, 17145}, {7232, 15534}, {7300, 16568}, {8584, 17365}, {10436, 16816}, {16704, 17012}, {16777, 17335}, {17014, 17257}, {17045, 17256}, {17160, 17351}, {17251, 17400}, {17253, 17399}, {17258, 17395}, {17259, 17394}, {17263, 17390}, {17264, 17388}, {17265, 17387}, {17267, 17386}, {17271, 17384}, {17272, 17383}, {17273, 17382}, {17275, 17381}, {17278, 17378}, {17279, 17310}, {17282, 17375}, {17283, 17374}, {17284, 17373}, {17285, 17372}, {17294, 17358}, {17297, 17356}, {17299, 17354}, {17301, 17347}, {17305, 17344}, {17306, 17343}, {17309, 17342}, {17311, 17341}, {17314, 17339}, {17316, 17338}, {17317, 17337}, {17318, 17336}, {17320, 17332}, {17321, 17331}, {17322, 17330}, {17323, 17329}, {17325, 17328}

X(17121) = {X(6),X(75)}-harmonic conjugate of X(17120)


X(17122) =  X(1)X(474)∩X(2)X(32)

Barycentrics    a^3 + 3 a b c : :

X(17122) lies on these lines:
{1, 474}, {2, 31}, {6, 16569}, {9, 4650}, {10, 14829}, {35, 16453}, {36, 16374}, {37, 17596}, {38, 5297}, {42, 4038}, {43, 940}, {51, 7186}, {55, 16059}, {57, 984}, {58, 3634}, {81, 899}, {86, 6685}, {100, 3720}, {140, 3072}, {181, 3792}, {183, 10436}, {226, 9364}, {244, 3920}, {312, 3980}, {354, 3961}, {498, 3075}, {601, 3090}, {602, 3525}, {603, 10588}, {612, 982}, {614, 17716}, {756, 3218}, {846, 1155}, {894, 16999}, {902, 5284}, {942, 5293}, {968, 17601}, {975, 986}, {978, 5711}, {1001, 3550}, {1010, 3831}, {1046, 5044}, {1054, 1961}, {1064, 6946}, {1106, 5261}, {1125, 5255}, {1193, 17531}, {1215, 5205}, {1279, 3848}, {1386, 16602}, {1395, 8889}, {1460, 16419}, {1468, 9780}, {1656, 3073}, {1698, 5247}, {1707, 7308}, {1757, 3740}, {1920, 4495}, {1936, 5432}, {1962, 17021}, {1999, 4716}, {2251, 5277}, {2361, 5326}, {3035, 17056}, {3052, 8167}, {3074, 6149}, {3240, 9350}, {3271, 6688}, {3305, 7262}, {3624, 5264}, {3689, 3979}, {3745, 16610}, {3749, 10582}, {3751, 8580}, {3757, 4434}, {3769, 16825}, {3771, 17234}, {3840, 5263}, {3915, 5550}, {3918, 15955}, {3944, 5880}, {3957, 17450}, {4188, 10448}, {4300, 6915}, {4358, 4418}, {4359, 17763}, {4383, 16477}, {4386, 16503}, {4414, 9352}, {4423, 8616}, {4666, 17715}, {4695, 17015}, {4850, 5311}, {5020, 7295}, {5226, 9316}, {5249, 17719}, {5251, 16357}, {5253, 10459}, {5259, 16302}, {5269, 5272}, {5275, 17754}, {5287, 17592}, {5329, 7484}, {6682, 16830}, {6690, 17245}, {7292, 17469}, {7301, 11284}, {9345, 17018}, {9347, 17017}, {9355, 10157}, {9363, 9578}, {9441, 10164}, {10457, 17589}, {10980, 16496}, {12436, 13161}, {16466, 16862}, {16476, 16832}, {16604, 17795}, {17022, 17594}, {17720, 17889}


X(17123) =  X(1)X(210)∩X(2)X(31)

Barycentrics    a^3 - 3 a b c : :

X(17123) lies on these lines:
{1, 210}, {2, 31}, {6, 4038}, {9, 982}, {10, 4514}, {35, 16286}, {36, 4245}, {37, 17600}, {38, 7292}, {42, 5284}, {43, 1001}, {44, 3742}, {51, 3792}, {55, 15485}, {57, 7262}, {63, 17063}, {75, 4011}, {81, 16477}, {140, 3073}, {181, 6688}, {200, 17715}, {212, 10589}, {244, 3219}, {312, 16825}, {333, 3840}, {354, 1757}, {405, 978}, {499, 3074}, {595, 3634}, {601, 3525}, {602, 3090}, {614, 984}, {756, 7191}, {846, 3752}, {899, 1621}, {940, 9332}, {975, 16478}, {1046, 5439}, {1054, 4640}, {1125, 5247}, {1193, 5047}, {1201, 5260}, {1215, 16823}, {1253, 5274}, {1279, 3740}, {1376, 8616}, {1386, 1961}, {1399, 7294}, {1468, 5550}, {1471, 5226}, {1656, 3072}, {1698, 5255}, {1699, 9441}, {1707, 5437}, {1724, 3624}, {1734, 11193}, {1754, 7988}, {1921, 7244}, {1935, 5433}, {1957, 17917}, {1962, 17012}, {1999, 4974}, {2212, 8889}, {2238, 16503}, {2886, 17337}, {2999, 17592}, {3052, 8692}, {3075, 6149}, {3216, 5259}, {3271, 3819}, {3295, 6048}, {3306, 4650}, {3329, 17260}, {3452, 17719}, {3550, 4413}, {3662, 4703}, {3683, 16610}, {3705, 17338}, {3715, 17597}, {3720, 4649}, {3741, 17277}, {3747, 16815}, {3749, 8580}, {3751, 10582}, {3771, 5233}, {3817, 13329}, {3915, 9780}, {3944, 4679}, {3973, 10980}, {3979, 4849}, {3980, 4676}, {4384, 11353}, {4425, 16706}, {4512, 17601}, {4871, 14829}, {5020, 5329}, {5121, 5745}, {5248, 17749}, {5268, 7290}, {5297, 17469}, {5363, 11284}, {5400, 15931}, {5711, 16853}, {6210, 16434}, {6666, 17722}, {7076, 17923}, {7083, 16419}, {7295, 7484}, {9330, 17024}, {9355, 10167}, {9440, 11019}, {10448, 16859}, {16466, 16842}, {16475, 17022}, {16476, 16831}, {16690, 17259}, {16827, 16918}, {17278, 17889}


X(17124) =  X(1)X(3833)∩X(2)X(31)

Barycentrics    a^3 + 4 a b c : :

X(17124) lies on these lines:
{1, 3833}, {2, 31}, {12, 1106}, {38, 3306}, {42, 4413}, {43, 9342}, {57, 756}, {58, 17551}, {81, 16569}, {88, 17591}, {100, 17782}, {181, 5650}, {244, 612}, {404, 10448}, {498, 1496}, {601, 1656}, {602, 3526}, {605, 8252}, {606, 8253}, {678, 10389}, {846, 9352}, {896, 3305}, {899, 940}, {902, 4423}, {936, 2650}, {976, 5439}, {978, 17535}, {982, 5297}, {1193, 16408}, {1253, 5432}, {1376, 2177}, {1395, 5094}, {1468, 1698}, {1961, 4850}, {1962, 17022}, {2209, 15668}, {3072, 3525}, {3073, 5067}, {3240, 4038}, {3550, 5284}, {3624, 3915}, {3722, 4666}, {3742, 3938}, {3744, 3848}, {3745, 16602}, {3752, 5311}, {3831, 16454}, {3870, 17450}, {3920, 17063}, {3980, 4358}, {3989, 17595}, {4300, 6918}, {4682, 16610}, {4937, 4942}, {5219, 9316}, {5226, 9364}, {5230, 17582}, {5255, 5550}, {5272, 17469}, {5326, 5348}, {5640, 7186}, {5711, 16862}, {7292, 17716}, {9335, 17598}, {10457, 14005}, {16466, 16863}, {17021, 17592}


X(17125) =  X(1)X(4015)∩X(2)X(31)

Barycentrics    a^3 - 4 a b c : :

X(17125) lies on these lines:
{1, 4015}, {2, 31}, {6, 9345}, {9, 244}, {11, 1253}, {38, 3305}, {42, 4423}, {43, 5284}, {55, 9350}, {100, 15485}, {210, 4864}, {499, 1496}, {601, 3526}, {602, 1656}, {605, 8253}, {606, 8252}, {614, 756}, {667, 14434}, {896, 3306}, {899, 1001}, {902, 4413}, {958, 8688}, {978, 5047}, {984, 7292}, {1106, 5433}, {1149, 9708}, {1150, 4871}, {1193, 11108}, {1468, 3624}, {1471, 5219}, {1621, 16569}, {1698, 3915}, {1754, 10171}, {1962, 2999}, {2209, 17259}, {2212, 5094}, {3011, 5316}, {3072, 5067}, {3073, 3525}, {3120, 4679}, {3219, 17063}, {3240, 16484}, {3271, 5650}, {3550, 9342}, {3683, 16602}, {3720, 3789}, {3740, 3938}, {3747, 16832}, {3751, 17450}, {3792, 5640}, {3840, 5278}, {3848, 4641}, {3999, 15481}, {4003, 16814}, {4011, 4359}, {4358, 16825}, {4414, 15254}, {4649, 14997}, {4860, 16885}, {5220, 17449}, {5230, 17559}, {5247, 5550}, {5259, 17749}, {5268, 17469}, {5711, 16854}, {7076, 17917}, {7294, 7299}, {7988, 13329}, {9441, 9779}, {10457, 17557}, {14996, 16477}, {16466, 16853}, {17020, 17592}


X(17126) =  X(1)X(89)∩X(2)X(31)

Barycentrics    2 a^3 + a b c : :
Barycentrics    2 a^2 sin A + S : :

X(17126) lies on these lines:
{1, 89}, {2, 31}, {6, 100}, {7, 109}, {8, 58}, {9, 5297}, {20, 601}, {21, 5711}, {22, 1460}, {23, 7295}, {37, 9347}, {38, 4650}, {40, 17016}, {42, 3097}, {43, 2308}, {46, 5262}, {47, 3085}, {55, 81}, {57, 4318}, {63, 3920}, {82, 4000}, {110, 2175}, {145, 1468}, {149, 11269}, {162, 281}, {165, 5256}, {181, 3060}, {192, 4427}, {212, 5281}, {321, 3769}, {388, 1399}, {390, 1936}, {404, 16466}, {497, 5348}, {498, 2964}, {595, 3616}, {602, 3523}, {603, 3600}, {605, 7586}, {606, 7585}, {608, 7466}, {609, 3809}, {612, 1707}, {614, 9335}, {643, 757}, {756, 7262}, {846, 5311}, {894, 17002}, {896, 984}, {899, 14997}, {901, 5091}, {938, 1771}, {940, 1621}, {958, 16948}, {968, 17019}, {976, 1046}, {978, 17572}, {982, 17024}, {1100, 4689}, {1150, 5263}, {1155, 1386}, {1191, 5253}, {1193, 4188}, {1395, 6995}, {1397, 5012}, {1471, 9364}, {1497, 3075}, {1613, 7109}, {1617, 17074}, {1724, 9780}, {1754, 9778}, {1918, 17379}, {1935, 5261}, {1995, 7083}, {2162, 3051}, {2176, 3231}, {2177, 4649}, {2212, 4232}, {2242, 3573}, {2345, 4275}, {2361, 5218}, {2475, 5230}, {2975, 4252}, {3073, 3091}, {3210, 17150}, {3271, 5640}, {3306, 7290}, {3315, 4860}, {3617, 5247}, {3622, 3915}, {3681, 4641}, {3683, 4682}, {3689, 4663}, {3720, 8616}, {3741, 5372}, {3744, 3873}, {3745, 4640}, {3749, 3957}, {3751, 3935}, {3752, 9352}, {3868, 5266}, {3923, 4671}, {3938, 4430}, {3952, 17350}, {3961, 4661}, {4042, 4921}, {4195, 17751}, {4264, 5749}, {4265, 5078}, {4339, 12649}, {4344, 5744}, {4347, 15932}, {4358, 4676}, {4362, 4418}, {4393, 4781}, {4434, 4672}, {4512, 5287}, {5057, 17720}, {5115, 5839}, {5119, 17015}, {5329, 6636}, {5363, 7492}, {5398, 5657}, {5774, 16394}, {6149, 10056}, {6353, 14975}, {7224, 7357}, {7299, 10588}, {7301, 14002}, {9345, 16484}, {9441, 11200}, {10457, 17539}, {11246, 17061}, {16475, 17012}, {16476, 16816}, {16919, 17033}, {17011, 17594}, {17013, 17601}, {17017, 17596}, {17365, 17724}, {17602, 17768} X(17126) = isogonal conjugate of X(4492)
X(17126) = isotomic conjugate of X(30635)
X(17126) = anticomplement of X(25760)
X(17126) = {X(2),X(31)}-harmonic conjugate of X(17127)


X(17127) =  X(1)X(2308)∩X(2)X(31)

Barycentrics    2 a^3 - a b c : :
Barycentrics    2 a^2 sin A - S : :

X(17127) lies on these lines:
{1, 2308}, {2, 31}, {4, 14975}, {6, 1621}, {8, 595}, {9, 3920}, {20, 602}, {21, 16466}, {22, 7083}, {23, 5329}, {38, 7262}, {42, 8616}, {43, 902}, {44, 3681}, {47, 3086}, {55, 3240}, {56, 7419}, {57, 7292}, {58, 3616}, {63, 3677}, {81, 1001}, {82, 2345}, {100, 3052}, {109, 5435}, {110, 1397}, {145, 3915}, {162, 278}, {181, 5640}, {190, 3891}, {192, 17150}, {212, 390}, {222, 7677}, {244, 4650}, {255, 14986}, {321, 4676}, {354, 3246}, {388, 7299}, {497, 2361}, {499, 2964}, {580, 962}, {582, 6361}, {601, 3523}, {603, 5265}, {605, 7585}, {606, 7586}, {608, 4233}, {612, 9330}, {614, 1707}, {651, 1617}, {756, 17716}, {765, 5387}, {846, 17017}, {896, 982}, {899, 3550}, {940, 5284}, {959, 11101}, {968, 16475}, {978, 4188}, {984, 17469}, {993, 5315}, {995, 17187}, {1104, 3869}, {1191, 2975}, {1193, 4189}, {1203, 5248}, {1279, 3873}, {1386, 3683}, {1395, 4232}, {1399, 7288}, {1453, 5250}, {1460, 1995}, {1468, 3622}, {1497, 3074}, {1612, 16471}, {1613, 1977}, {1708, 4318}, {1743, 3870}, {1754, 9812}, {1757, 3938}, {1771, 5704}, {1780, 11415}, {1918, 4651}, {1935, 3600}, {1936, 5274}, {2162, 3231}, {2175, 5012}, {2176, 3051}, {2212, 6995}, {2292, 16478}, {2310, 9539}, {3060, 3271}, {3072, 3091}, {3187, 3685}, {3210, 4427}, {3305, 5269}, {3617, 5255}, {3666, 17025}, {3720, 14996}, {3741, 5361}, {3745, 15254}, {3747, 4393}, {3748, 4663}, {3749, 3935}, {3750, 16477}, {3751, 3957}, {3759, 3896}, {3769, 4358}, {3772, 5057}, {3791, 4432}, {3876, 5266}, {3883, 5294}, {3971, 4759}, {4011, 17763}, {4252, 5253}, {4264, 5296}, {4362, 4671}, {4418, 16825}, {4423, 8692}, {4429, 4450}, {4512, 5256}, {4640, 4850}, {5046, 5230}, {5047, 5711}, {5260, 5710}, {5262, 12514}, {5263, 5278}, {5264, 9780}, {5347, 16686}, {5348, 10589}, {5363, 14002}, {5398, 5603}, {6149, 10072}, {6636, 7295}, {7192, 17082}, {7261, 7357}, {7301, 7492}, {9352, 16610}, {9778, 13329}, {10389, 16670}, {10453, 16704}, {16690, 17379}, {16827, 16919}, {16920, 17033}, {17012, 17594}, {17013, 17592}, {17165, 17350}, {17697, 17751}, {17735, 17756}

X(17127) = isogonal conjugate of X(7241)
X(17127) = isotomic conjugate of X(30636)
X(17127) = anticomplement of X(25757)
X(17127) = {X(2),X(31)}-harmonic conjugate of X(17126)


X(17128) =  X(2)X(1975)∩X(32)X(76)

Barycentrics    a^4 + 3 b^2 c^2 : :

X(17128) lies on these lines:
{2, 1975}, {4, 3314}, {5, 7836}, {20, 7904}, {30, 2896}, {32, 76}, {69, 7823}, {83, 538}, {99, 3934}, {115, 7832}, {140, 13188}, {141, 6655}, {148, 6656}, {183, 3552}, {194, 3329}, {257, 17738}, {274, 16918}, {310, 16957}, {315, 11361}, {316, 7794}, {325, 16044}, {350, 6645}, {381, 7881}, {382, 7879}, {405, 16994}, {543, 6292}, {598, 7949}, {599, 7929}, {620, 16923}, {625, 7909}, {626, 14041}, {671, 7861}, {1003, 7793}, {1078, 7816}, {1220, 17302}, {1235, 15014}, {1350, 3146}, {1506, 7799}, {1655, 17686}, {1909, 4366}, {2548, 7906}, {2549, 7876}, {2782, 8290}, {3096, 7748}, {3729, 17743}, {3767, 7892}, {3926, 7777}, {3933, 7785}, {4045, 16897}, {4074, 9225}, {5025, 7795}, {5038, 12215}, {5275, 16913}, {5286, 7875}, {5305, 6661}, {5309, 7846}, {5475, 7796}, {5999, 6248}, {6392, 16989}, {6658, 7750}, {6680, 14568}, {7735, 14037}, {7737, 7893}, {7745, 7779}, {7746, 7835}, {7747, 7768}, {7752, 7801}, {7753, 7905}, {7754, 7787}, {7756, 7831}, {7757, 7808}, {7758, 7921}, {7760, 7804}, {7763, 16921}, {7765, 7859}, {7767, 14712}, {7769, 7863}, {7773, 7897}, {7775, 7871}, {7781, 7786}, {7782, 7815}, {7788, 7900}, {7790, 7822}, {7791, 16986}, {7797, 7819}, {7798, 7878}, {7800, 7833}, {7802, 7854}, {7803, 16895}, {7805, 12150}, {7806, 14001}, {7809, 7895}, {7810, 9855}, {7812, 7855}, {7813, 7858}, {7814, 7908}, {7820, 7828}, {7825, 7922}, {7827, 7889}, {7841, 7938}, {7842, 7883}, {7843, 7917}, {7844, 7930}, {7849, 7911}, {7860, 7896}, {7862, 7870}, {7865, 7910}, {7867, 14046}, {7868, 7933}, {7869, 7934}, {7872, 7937}, {7874, 14061}, {7880, 7899}, {7882, 14537}, {7887, 7945}, {7902, 7943}, {7907, 17006}, {7914, 7918}, {7915, 7919}, {7916, 7926}, {8024, 16950}, {8359, 8591}, {8369, 8859}, {8891, 16276}, {9698, 14148}, {9939, 11159}, {10328, 14567}, {11321, 16993}, {14023, 14032}, {16589, 16911}, {16909, 17003}, {16910, 16991}, {16914, 16992}, {16915, 16999}, {16916, 17000}, {16919, 16997}, {16920, 16998}, {16925, 17004}


X(17129) =  X(2)X(3933)∩X(32)X(76)

Barycentrics    a^4 - 3 b^2 c^2 : :

X(17129) lies on these lines:
{2, 3933}, {4, 7893}, {5, 7779}, {32, 76}, {69, 5025}, {83, 7805}, {99, 7780}, {115, 7768}, {141, 7797}, {148, 7750}, {183, 194}, {193, 7921}, {230, 7836}, {274, 16999}, {315, 14041}, {316, 7826}, {350, 4400}, {381, 7900}, {405, 16996}, {524, 7785}, {538, 1078}, {599, 7851}, {625, 7917}, {626, 14046}, {671, 7842}, {732, 5038}, {1278, 5687}, {1285, 14031}, {1351, 3091}, {1506, 7905}, {1909, 4396}, {1975, 5023}, {2548, 7837}, {2549, 7904}, {2896, 5254}, {3096, 5309}, {3314, 3767}, {3329, 3934}, {3630, 15514}, {3760, 4366}, {3761, 6645}, {3770, 5035}, {3785, 7833}, {3815, 13571}, {3926, 7907}, {5275, 16911}, {5286, 7876}, {5304, 16898}, {5306, 10583}, {5319, 7875}, {5346, 7846}, {5355, 7859}, {5475, 7877}, {5999, 12251}, {6292, 7827}, {6392, 7791}, {6655, 7767}, {7470, 8782}, {7735, 7892}, {7746, 7796}, {7748, 7811}, {7749, 7799}, {7752, 7840}, {7755, 7832}, {7757, 7815}, {7758, 7777}, {7762, 16044}, {7763, 16923}, {7764, 17005}, {7765, 7831}, {7766, 7770}, {7769, 7813}, {7771, 7781}, {7773, 7946}, {7774, 16921}, {7775, 7949}, {7786, 7798}, {7787, 14614}, {7788, 7912}, {7790, 7854}, {7794, 7828}, {7795, 7806}, {7800, 7864}, {7801, 7857}, {7803, 16897}, {7808, 7894}, {7809, 7882}, {7810, 7847}, {7814, 7916}, {7816, 14711}, {7817, 7944}, {7821, 14061}, {7822, 7856}, {7823, 11185}, {7825, 7850}, {7829, 16987}, {7841, 7929}, {7843, 15031}, {7844, 7922}, {7848, 7911}, {7849, 7919}, {7858, 7890}, {7861, 7883}, {7862, 7871}, {7865, 7918}, {7868, 7932}, {7869, 7942}, {7872, 7936}, {7879, 7933}, {7884, 7914}, {7886, 7909}, {7887, 7897}, {7895, 7899}, {7896, 7934}, {7902, 7937}, {7908, 7940}, {7910, 11648}, {8177, 12215}, {8597, 9939}, {9607, 15598}, {9865, 12054}, {11321, 16995}, {11676, 13108}, {16589, 16994}, {16895, 16989}, {16906, 17007}, {16907, 17003}, {16908, 16991}, {16912, 16992}, {16915, 17001}, {16916, 17002}, {16917, 16997}, {16918, 16998}


X(17130) =  X(2)X(7765)∩X(32)X(76)

Barycentrics    a^4 + 4 b^2 c^2 : :

X(17130) lies on these lines:
{2, 7765}, {3, 8556}, {4, 7794}, {5, 7801}, {20, 7810}, {30, 7854}, {32, 76}, {69, 7747}, {83, 7798}, {99, 7815}, {115, 7795}, {141, 7748}, {148, 3096}, {182, 13108}, {183, 5206}, {194, 7808}, {315, 14068}, {316, 7896}, {381, 7821}, {382, 599}, {538, 7770}, {543, 7791}, {574, 1975}, {625, 7881}, {626, 11185}, {631, 2482}, {671, 7933}, {754, 14035}, {1003, 7780}, {1078, 8588}, {1506, 3926}, {2241, 3761}, {2242, 3760}, {2548, 7813}, {2549, 6292}, {3314, 7825}, {3530, 11168}, {3767, 7820}, {3785, 6781}, {3933, 5475}, {4048, 5033}, {5007, 11286}, {5025, 7869}, {5254, 7822}, {5286, 7889}, {5309, 7819}, {5355, 6392}, {6655, 7865}, {6656, 11648}, {6658, 7811}, {7617, 7870}, {7697, 9737}, {7737, 7826}, {7739, 16045}, {7745, 7855}, {7746, 7789}, {7752, 7908}, {7753, 7758}, {7754, 7804}, {7755, 14001}, {7756, 7800}, {7759, 8370}, {7764, 16924}, {7768, 11361}, {7773, 7895}, {7775, 7796}, {7783, 15482}, {7785, 7916}, {7787, 14075}, {7788, 7843}, {7790, 7914}, {7799, 16921}, {7827, 16895}, {7828, 14067}, {7829, 16898}, {7830, 16990}, {7832, 7844}, {7836, 7862}, {7841, 7849}, {7842, 7879}, {7847, 16986}, {7851, 7915}, {7860, 14042}, {7861, 7868}, {7874, 13881}, {7880, 7887}, {7892, 14568}, {7912, 15031}, {7922, 14041}, {7945, 14061}, {8367, 9606}, {9902, 10800}, {14023, 14033}


X(17131) =  X(2)X(5355)∩X(32)X(76)

Barycentrics    a^4 - 4 b^2 c^2 : :

X(17131) lies on these lines:
{2, 5355}, {4, 7826}, {5, 7855}, {6, 9466}, {32, 76}, {39, 15271}, {69, 115}, {99, 8588}, {141, 5309}, {148, 7811}, {183, 538}, {187, 8667}, {193, 7753}, {194, 7815}, {230, 7801}, {339, 577}, {381, 7845}, {524, 3363}, {543, 14907}, {576, 7697}, {599, 7853}, {620, 17008}, {625, 7788}, {671, 7898}, {732, 5034}, {754, 11185}, {1078, 7781}, {1506, 7758}, {1569, 15483}, {1975, 5206}, {2241, 3760}, {2242, 3761}, {2548, 7890}, {2549, 7810}, {2782, 8722}, {2896, 7872}, {3096, 7902}, {3098, 12188}, {3314, 7844}, {3767, 7794}, {3770, 5042}, {3785, 7756}, {3926, 7749}, {3933, 7746}, {3934, 7754}, {4045, 16990}, {5008, 11286}, {5024, 8556}, {5025, 7896}, {5026, 5033}, {5028, 14994}, {5171, 13108}, {5254, 7854}, {5286, 6292}, {5305, 7822}, {5319, 7889}, {5346, 7819}, {5971, 8585}, {6144, 15484}, {6390, 13468}, {6392, 7765}, {7603, 9766}, {7615, 11160}, {7617, 7840}, {7735, 7820}, {7747, 14023}, {7748, 7767}, {7752, 7916}, {7755, 7795}, {7757, 15482}, {7760, 7808}, {7761, 11648}, {7766, 14075}, {7768, 7825}, {7770, 7805}, {7773, 7882}, {7775, 7779}, {7790, 7865}, {7796, 7862}, {7797, 7914}, {7799, 17004}, {7804, 14614}, {7817, 7868}, {7821, 13881}, {7827, 16986}, {7828, 7869}, {7831, 11054}, {7838, 16924}, {7841, 7848}, {7849, 7851}, {7850, 14041}, {7861, 7879}, {7877, 16044}, {7881, 7886}, {7887, 7895}, {7897, 14061}, {7900, 15031}, {7905, 16921}, {8182, 15300}, {8359, 15598}, {8586, 15533}, {8589, 8716}

X(17131) = barycentric product X(4361)*X(4363)


X(17132) =  INFINITY LINE INTERCEPT OF X(1)X(4454)

Barycentrics    2 a^2 - 3 a b - b^2 - 3 a c + 6 b c - c^2 : :

X(17132) lies on these lines:
{1,4454}, {2,2415}, {7,3950}, {10,4419}, {30,511}, {190,1266}, {192,3664}, {239,4480}, {346,4862}, {551,3923}, {553,3175}, {597,3946}, {599,2321}, {894,4021}, {903,4582}, {1086,2325}, {1125,4363}, {1278,4416}, {1743,4452}, {1766,3928}, {1992,3875}, {3120,4141}, {3159,12436}, {3244,4644}, {3247,7222}, {3626,4643}, {3634,4364}, {3635,4667}, {3636,4670}, {3644,3879}, {3679,10005}, {3686,4686}, {3821,3828}, {3905,11164}, {3912,4440}, {3943,4409}, {3973,4402}, {4029,4675}, {4058,4461}, {4098,4648}, {4422,17067}, {4431,6646}, {4660,4669}, {4665,4691}, {4690,4746}, {4713,6686}, {4852,8584}, {4869,4902}, {5032,16834}, {5503,11599}, {6172,16833}, {6666,7263}, {7618,8720}, {7961,11019}, {8182,8669}, {10022,12040}

X(17132) = isogonal conjugate of X(17222)
X(17132) = crossdifference of every pair of points on line {6, 8643}
X(17132) = barycentric product X(903)*(12035)
X(17132) = barycentric quotient X(12035)/X(519)
X(17132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (190, 1266, 3008), (3161, 6557, 15223), (3912, 4440, 4887), (4419, 4659, 10), (4452, 4488, 1743), (4779, 15590, 1)


X(17133) =  INFINITY LINE INTERCEPT OF X(1)X(4470)

Barycentrics    2 a^2 + 3 a b - b^2 + 3 a c - 6 b c - c^2 : :

X(17133) lies on these lines:
{1,4470}, {2,2321}, {30,511}, {145,4659}, {190,4700}, {192,3686}, {239,2325}, {551,4133}, {594,4021}, {597,4852}, {599,3663}, {894,4464}, {1086,4727}, {1125,4527}, {1266,6542}, {1278,3879}, {1449,4460}, {1654,4545}, {1992,3729}, {3008,3943}, {3011,4933}, {3178,7621}, {3241,3886}, {3244,4363}, {3625,4643}, {3626,4364}, {3632,4419}, {3633,4644}, {3634,4535}, {3635,4670}, {3636,4472}, {3644,4416}, {3664,4686}, {3672,4007}, {3679,3755}, {3731,4371}, {3758,4982}, {3773,3828}, {3912,17067}, {3950,4361}, {4029,4384}, {4058,4657}, {4060,4357}, {4085,4745}, {4360,4431}, {4399,4681}, {4648,4898}, {4668,4748}, {4669,4780}, {4690,4701}, {4691,4708}, {4856,8584}, {4873,5222}, {4888,4916}, {4923,7174}, {6762,11148}

X(17133) = isogonal conjugate of X(17223)
X(17133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (145, 4659, 4667), (2321, 3875, 3946), (3950, 4361, 6666), (4360, 4431, 5750), (4460, 4461, 1449)




leftri  Collineation images: X(17134) - X(17219)  rightri

If A'B'C' is a central triangle other than ABC and P and U are triangle centers, then (A,B,C,P; A',B',C',U) is a regular collineation, as is its inverse, given by (A',B',C',U; A,B,C,P).

The collineation images at X(17134)-X(17219) result from A'B'C' = medial triangle, P = X(2), and U = X(1). We write the image of X as m(X); let m-1 denote the inverse collineation. Then centers X(17134)-X(17166) are examples of m(X), and X(17167)-X(17219) are examples of m-1(X). Other examples are given by the following list, in which the appearance of (i,j) means that m(X(i)) = X(j):

(1,75), (2,1), (3,17220), (4,17134), (5,17221), (6,17135), (7,63), (8,3875), (9, 3873), (10,4360), (21,7), (58,69), (81,8),

A collineation maps lines to lines. The appearance of {h,i} -> {j,k} in the next list means that m(X(h)X(i)) = X(j)X(k):

{1,2} -> {1,75}
{1,21} -> {8,75}
{2,3} -> {1,7}
{2,7} -> {1,21}
{239,514} -> {2,3}
{513,649} -> {30,513}, the infinity line

underbar




X(17134) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17134) lies on these lines:
{1, 7}, {2, 1826}, {3, 1441}, {19, 14953}, {22, 1602}, {36, 17861}, {48, 8680}, {56, 17863}, {75, 1444}, {92, 1817}, {100, 322}, {104, 1305}, {273, 411}, {286, 4225}, {307, 515}, {326, 17136}, {610, 14543}, {675, 13397}, {857, 17073}, {1447, 16778}, {1766, 4552}, {1837, 5740}, {2268, 4032}, {2646, 5736}, {2897, 6224}, {3101, 3164}, {3152, 5484}, {3262, 9723}, {3486, 5738}, {3869, 8822}, {3873, 4360}, {4566, 7013}, {4897, 6563}, {5204, 17895}, {5271, 5744}, {6527, 17135}, {6999, 17086}, {7280, 17885}, {9018, 12220}

X(17134) = anticomplement of X(1826)


X(17135) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^2 b - a b^2 + a^2 c - b^2 c - a c^2 - b c^2 : :

X(17135) lies on these lines:
{1, 2}, {7, 7243}, {11, 5741}, {31, 16704}, {37, 4981}, {38, 740}, {55, 1150}, {63, 3886}, {69, 674}, {72, 3702}, {75, 3873}, {81, 5263}, {100, 3996}, {110, 7058}, {141, 4972}, {149, 2895}, {150, 2813}, {192, 7226}, {210, 4358}, {310, 2388}, {312, 3681}, {314, 17153}, {319, 350}, {321, 518}, {333, 1621}, {354, 3696}, {390, 14552}, {392, 4742}, {497, 5739}, {516, 4001}, {517, 3902}, {528, 4450}, {672, 2321}, {726, 4365}, {946, 4101}, {952, 4192}, {956, 1011}, {982, 17495}, {984, 3995}, {1001, 4042}, {1043, 2975}, {1278, 17157}, {1330, 2891}, {1441, 5173}, {1468, 11115}, {1740, 17178}, {1836, 17491}, {1943, 4318}, {1957, 14954}, {1985, 5730}, {2238, 16685}, {2239, 17765}, {2276, 17299}, {2350, 16549}, {2352, 17740}, {2886, 3936}, {2997, 6601}, {3056, 15983}, {3058, 3578}, {3210, 4392}, {3219, 3685}, {3242, 3891}, {3295, 16343}, {3416, 4863}, {3421, 6818}, {3555, 4968}, {3666, 3896}, {3683, 4702}, {3703, 3969}, {3714, 4696}, {3739, 4883}, {3740, 4113}, {3782, 4442}, {3789, 3966}, {3791, 17469}, {3816, 4023}, {3868, 17164}, {3869, 4673}, {3874, 4647}, {3914, 17184}, {3919, 4793}, {3925, 4966}, {3967, 4519}, {3989, 3993}, {4007, 17754}, {4191, 5687}, {4361, 17597}, {4387, 5220}, {4417, 11680}, {4479, 17360}, {4661, 4671}, {4672, 4722}, {4684, 5249}, {4709, 17449}, {4714, 5883}, {4716, 17598}, {4720, 13588}, {4732, 17450}, {4975, 10176}, {5016, 10371}, {5082, 6817}, {5178, 7270}, {5247, 11319}, {5284, 17277}, {5435, 16878}, {5904, 11330}, {6527, 17134}, {6767, 16345}, {11322, 12513}, {16684, 17233}, {16690, 17349}, {17144, 17149}, {17148, 17759}, {17154, 17155}

X(17135) = complement of X(20011)
X(17135) = isotomic conjugate of X(8049)
X(17135) = anticomplement of X(42)


X(17136) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17136) lies on these lines:
{1, 17169}, {2, 9317}, {7, 6596}, {75, 17221}, {99, 110}, {100, 658}, {145, 9451}, {150, 6224}, {214, 1111}, {224, 3188}, {326, 17134}, {662, 14543}, {668, 4597}, {883, 4579}, {1310, 13397}, {1447, 4881}, {1565, 10609}, {1813, 2406}, {1959, 14953}, {2475, 17084}, {3120, 15903}, {3212, 4188}, {3930, 6647}, {3952, 4561}, {4511, 5088}, {4596, 4608}, {4614, 6742}, {4620, 17931}, {4855, 9312}, {5086, 17095}, {6606, 6613}, {12649, 17081}

X(17136) = anticomplement of X(21044)


X(17137) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^3 b - a b^3 + a^3 c - b^3 c - a c^3 - b c^3 : :

X(17137) lies on these lines:
{1, 16705}, {2, 213}, {7, 8}, {72, 3263}, {76, 17751}, {141, 2295}, {145, 4352}, {150, 1330}, {304, 3869}, {310, 2388}, {314, 17220}, {315, 766}, {524, 3780}, {742, 3721}, {758, 1930}, {1042, 9436}, {1228, 10381}, {1334, 3912}, {1655, 6646}, {2896, 6542}, {2975, 17206}, {3208, 17296}, {3662, 17033}, {3663, 10452}, {3691, 4416}, {3878, 14210}, {3959, 17497}, {4441, 10449}, {4754, 17365}, {4766, 17046}, {15983, 17364}, {17288, 17752}

X(17137) = complement of X(20109)
X(17137) = anticomplement of X(213)


X(17138) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^4 b - a b^4 + a^4 c - b^4 c - a c^4 - b c^4 : :

X(17138) lies on these lines:
{1, 2896}, {2, 1918}, {69, 674}, {71, 3006}, {75, 17153}, {744, 4118}, {1654, 4388}, {3779, 5014}, {3781, 5300}, {4440, 17157}


X(17139) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17139) lies on these lines:
{2, 2245}, {4, 69}, {7, 21}, {27, 1812}, {57, 17182}, {63, 17167}, {75, 3869}, {81, 4644}, {99, 953}, {110, 2861}, {144, 16713}, {226, 17185}, {274, 957}, {319, 5086}, {320, 350}, {325, 8229}, {329, 333}, {332, 945}, {497, 5208}, {517, 3262}, {527, 17197}, {662, 14953}, {859, 15507}, {894, 17202}, {903, 16711}, {908, 2183}, {946, 10461}, {962, 1043}, {1005, 14828}, {1010, 4295}, {1086, 16752}, {1275, 5379}, {1633, 16876}, {1756, 4357}, {1780, 17189}, {1788, 14011}, {1792, 5758}, {1944, 15149}, {1959, 8680}, {2287, 5819}, {2476, 5224}, {2550, 3786}, {3218, 17174}, {3219, 17173}, {3257, 4080}, {3474, 13588}, {3770, 15983}, {3879, 10572}, {3945, 6872}, {4622, 10428}, {4862, 16714}, {4887, 17205}, {5080, 14616}, {5232, 6871}, {6261, 10444}, {6646, 16738}, {7321, 16709}, {8025, 17483}, {8044, 18123}, {10129, 17250}, {10393, 10889}, {10436, 12514}, {11114, 17378}, {14543, 16568}, {16696, 17276}, {16887, 17274}, {17142, 17153}, {17179, 17195}, {17271, 17577}, {17929, 17971}

X(17139) = anticomplement of X(2245)
X(17139) = isotomic conjugate of isogonal conjugate of X(859)


X(17140) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^2 b - a b^2 + a^2 c + 2 a b c + b^2 c - a c^2 + b c^2 : :

X(17140) lies on these lines:
{1, 596}, {2, 38}, {7, 6327}, {8, 2891}, {42, 17495}, {55, 4781}, {75, 3873}, {81, 17150}, {86, 17142}, {190, 5284}, {274, 17141}, {306, 5542}, {321, 354}, {330, 17486}, {518, 4113}, {536, 4883}, {726, 3720}, {873, 4576}, {875, 3112}, {894, 7191}, {940, 3891}, {942, 4968}, {1086, 4972}, {1125, 3989}, {1441, 17625}, {1621, 4427}, {1930, 17169}, {1964, 16710}, {2975, 11101}, {3006, 5249}, {3210, 17018}, {3218, 3757}, {3219, 16823}, {3240, 17490}, {3475, 17740}, {3678, 6533}, {3699, 9342}, {3701, 5439}, {3702, 5045}, {3706, 4980}, {3729, 4666}, {3739, 4981}, {3741, 17449}, {3742, 4358}, {3812, 4696}, {3833, 3992}, {3848, 4009}, {3881, 4647}, {3938, 3980}, {3968, 4738}, {3969, 4966}, {4054, 11019}, {4363, 17597}, {4388, 17483}, {4450, 11246}, {4514, 7321}, {4692, 5883}, {4697, 17469}, {4722, 4974}, {4742, 5049}, {5014, 5880}, {5016, 10404}, {5083, 6358}, {9776, 10327}, {17193, 17205}

X(17140) = anticomplement of X(756)


X(17141) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17141) lies on these lines:
{1, 17200}, {2, 3954}, {6, 17489}, {7, 8}, {38, 16705}, {76, 17165}, {194, 712}, {274, 17140}, {304, 3873}, {758, 17152}, {942, 3263}, {1468, 3905}, {1475, 17760}, {1509, 4576}, {1930, 3874}, {2295, 9055}, {3754, 4986}, {3780, 17497}, {3881, 14210}, {4561, 5253}, {17143, 17164}


X(17142) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17142) lies on these lines:
{1, 87}, {2, 4446}, {6, 3891}, {38, 16738}, {69, 674}, {86, 17140}, {100, 18048}, {190, 16684}, {714, 17445}, {1654, 17794}, {1740, 17155}, {3778, 12263}, {3948, 17049}, {3952, 17277}, {3963, 14839}, {4427, 8053}, {5263, 17164}, {17139, 17153}, {17154, 17178}


X(17143) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17143) lies on these lines:
{1, 75}, {2, 1500}, {8, 76}, {10, 350}, {37, 16819}, {39, 17759}, {69, 2891}, {83, 213}, {85, 3340}, {99, 2975}, {100, 1078}, {183, 5687}, {190, 16552}, {192, 5283}, {194, 1278}, {310, 2388}, {312, 728}, {313, 5564}, {315, 3434}, {319, 1269}, {330, 3227}, {384, 5291}, {519, 1909}, {536, 1107}, {870, 1126}, {956, 1975}, {980, 3210}, {1089, 4986}, {1265, 3596}, {1573, 1655}, {1920, 3706}, {1921, 3696}, {2176, 4361}, {2241, 16998}, {2242, 16915}, {2276, 17030}, {2295, 17034}, {2896, 6653}, {3230, 16827}, {3262, 17866}, {3263, 3702}, {3294, 4043}, {3295, 16992}, {3436, 11185}, {3501, 17026}, {3626, 6381}, {3632, 3761}, {3679, 3760}, {3770, 17362}, {3780, 17499}, {3783, 12263}, {3785, 17784}, {3802, 16825}, {3912, 17050}, {3923, 16476}, {3975, 4044}, {4007, 17786}, {4087, 5295}, {4352, 4452}, {4358, 16815}, {4359, 16826}, {4410, 4725}, {4431, 17787}, {4665, 17790}, {4671, 16816}, {4686, 17448}, {4730, 14296}, {6063, 6604}, {7752, 11680}, {7763, 10527}, {7786, 17756}, {7828, 17737}, {16818, 17289}, {16828, 17322}, {16969, 17119}, {16971, 17116}, {17027, 17750}, {17141, 17164}

X(17143) = anticomplement of X(1500)


X(17144) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17144) lies on these lines:
{1, 75}, {8, 350}, {76, 519}, {85, 2099}, {99, 8666}, {145, 1909}, {183, 3913}, {192, 1107}, {194, 536}, {213, 3759}, {239, 312}, {320, 17753}, {321, 4393}, {325, 3813}, {330, 1278}, {528, 7750}, {668, 3632}, {870, 2334}, {979, 3226}, {1078, 8715}, {1125, 3795}, {1269, 17377}, {1500, 17030}, {1965, 17156}, {1975, 12513}, {2275, 17759}, {2295, 17027}, {2304, 18042}, {2481, 3680}, {3208, 17026}, {3294, 17335}, {3303, 16992}, {3625, 6381}, {3633, 3761}, {3644, 16975}, {3774, 4687}, {3873, 16741}, {3930, 18055}, {4050, 4595}, {4095, 4384}, {4358, 16816}, {4361, 16827}, {4366, 4426}, {4664, 5283}, {4676, 16476}, {4709, 10009}, {5564, 9534}, {6384, 10453}, {7773, 11235}, {12632, 15589}, {16552, 17336}, {16818, 17371}, {17050, 17234}, {17135, 17149}, {17299, 17786}, {17475, 18050}

X(17144) = anticomplement of X(20691)


X(17145) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17145) lies on these lines:
{1, 16704}, {2, 17450}, {8, 3881}, {75, 3873}, {145, 986}, {149, 17491}, {354, 4651}, {512, 7192}, {518, 3952}, {519, 4674}, {740, 17154}, {3006, 4684}, {3218, 4781}, {3555, 17751}, {3874, 17164}, {4430, 4671}, {4698, 4883}, {4704, 7226}, {7191, 17121}


X(17146) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17146) lies on these lines:
{1, 4427}, {2, 4090}, {38, 10180}, {75, 3873}, {354, 3967}, {537, 17450}, {3006, 5542}, {3306, 17780}, {3889, 17164}, {4022, 7226}, {4430, 4651}, {4681, 4883}, {4738, 5883}, {7191, 17120}


X(17147) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^2 b + a b^2 + a^2 c - b^2 c + a c^2 - b c^2 : :

X(17147) lies on these lines:
{1, 596}, {2, 37}, {8, 4424}, {10, 3989}, {20, 145}, {31, 4427}, {38, 740}, {42, 726}, {43, 3952}, {55, 3891}, {57, 4552}, {63, 3187}, {69, 9022}, {81, 4360}, {99, 593}, {141, 3969}, {147, 149}, {148, 1029}, {194, 712}, {239, 3219}, {306, 3663}, {333, 17160}, {335, 8049}, {518, 3896}, {519, 4001}, {824, 17161}, {874, 18064}, {894, 17011}, {896, 3791}, {899, 3971}, {940, 17318}, {984, 4651}, {986, 17751}, {990, 3870}, {1211, 17246}, {1230, 2092}, {1266, 5249}, {1726, 5773}, {1764, 1999}, {1897, 4219}, {2051, 4080}, {2350, 17027}, {2886, 4442}, {2895, 6646}, {2896, 6542}, {2901, 3670}, {3006, 3914}, {3051, 17475}, {3159, 3216}, {3240, 4734}, {3550, 4781}, {3578, 17362}, {3623, 17480}, {3685, 7191}, {3695, 4202}, {3696, 4981}, {3703, 4972}, {3712, 17061}, {3720, 3993}, {3729, 5256}, {3740, 4706}, {3741, 4365}, {3782, 3936}, {3821, 15523}, {3873, 17154}, {3923, 17017}, {3931, 4968}, {3946, 5294}, {3980, 5311}, {3999, 4891}, {4361, 5278}, {4362, 4414}, {4383, 17262}, {4392, 10453}, {4415, 5741}, {4419, 5739}, {4436, 16687}, {4440, 17483}, {4450, 5846}, {4641, 4852}, {4646, 4696}, {4692, 4868}, {4693, 17598}, {4886, 17258}, {5262, 7283}, {5271, 17151}, {5695, 17599}, {6758, 13174}, {8591, 9263}, {16577, 17077}, {16710, 17394}, {17019, 17319}, {17596, 17763}

X(17147) = anticomplement of X(321)


X(17148) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17148) lies on these lines:
{1, 87}, {2, 313}, {39, 3963}, {75, 16696}, {81, 4360}, {190, 16685}, {193, 8679}, {239, 16574}, {385, 16678}, {604, 4552}, {696, 8264}, {714, 1964}, {730, 3778}, {1150, 4361}, {1654, 5484}, {2277, 3765}, {2975, 11683}, {3101, 3164}, {3948, 17053}, {3995, 16777}, {4440, 17220}, {5271, 17490}, {6224, 9263}, {8267, 16684}, {10436, 16710}, {17135, 17759}


X(17149) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17149) lies on these lines:
{1, 1965}, {2, 330}, {31, 799}, {38, 75}, {43, 668}, {63, 1966}, {69, 350}, {76, 3741}, {92, 1934}, {304, 9239}, {388, 7196}, {726, 6382}, {811, 1957}, {982, 1921}, {984, 1920}, {1469, 17794}, {1959, 6508}, {1978, 8026}, {2276, 17786}, {3403, 18068}, {3720, 17394}, {3840, 6381}, {4110, 17759}, {4386, 16956}, {8620, 17486}, {17027, 18057}, {17135, 17144}, {17445, 18069}

X(17149) = complement of X(21223)
X(17149) = anticomplement of X(16606)


X(17150) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17150) lies on these lines:
{1, 2}, {6, 3891}, {31, 4427}, {38, 3791}, {75, 16707}, {81, 17140}, {100, 16687}, {171, 17495}, {192, 17127}, {238, 3995}, {321, 1386}, {537, 4722}, {696, 8264}, {726, 2308}, {740, 17469}, {756, 4974}, {902, 4970}, {1150, 17599}, {1621, 4068}, {3210, 17126}, {3681, 3759}, {3744, 3896}, {3745, 4359}, {3769, 4850}, {3782, 17491}, {3873, 14544}, {3936, 17061}, {4001, 4353}, {4576, 7304}, {4972, 5846}, {5263, 17163}, {5596, 17220}, {5741, 17602}, {5847, 17184}, {7109, 17475}, {8266, 16678}, {8267, 17489}, {11319, 16478}, {14012, 17164}


X(17151) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17151) lies on these lines:
{1, 75}, {2, 3950}, {6, 4659}, {7, 519}, {8, 3663}, {9, 536}, {10, 3672}, {37, 16832}, {43, 4441}, {45, 4718}, {63, 4921}, {69, 1266}, {85, 7274}, {141, 4007}, {142, 17133}, {144, 17132}, {145, 3664}, {165, 4362}, {190, 3973}, {192, 3731}, {200, 17861}, {239, 1278}, {269, 12629}, {319, 4398}, {320, 4902}, {321, 2999}, {322, 1111}, {346, 3008}, {350, 16569}, {517, 10442}, {527, 5839}, {594, 17301}, {614, 4365}, {726, 5223}, {894, 4740}, {903, 17361}, {936, 17863}, {1086, 17296}, {1100, 17118}, {1122, 3893}, {1441, 4328}, {1449, 4363}, {1698, 4078}, {2321, 4000}, {2324, 4858}, {2345, 3946}, {3175, 7308}, {3210, 11679}, {3244, 3945}, {3247, 3739}, {3262, 17885}, {3263, 5272}, {3340, 12546}, {3361, 17733}, {3596, 3760}, {3623, 4909}, {3625, 4346}, {3626, 5232}, {3633, 3879}, {3634, 5936}, {3644, 17277}, {3661, 17304}, {3662, 17294}, {3673, 4882}, {3677, 3706}, {3679, 4357}, {3686, 4371}, {3696, 7174}, {3718, 4986}, {3751, 4716}, {3811, 7190}, {3834, 17309}, {3912, 4859}, {3923, 16469}, {3943, 17278}, {4033, 18065}, {4034, 4399}, {4312, 5847}, {4359, 17022}, {4389, 4668}, {4393, 4821}, {4395, 4873}, {4405, 17332}, {4436, 16688}, {4440, 17363}, {4445, 17235}, {4461, 5222}, {4657, 4665}, {4667, 7222}, {4675, 17388}, {4681, 16676}, {4688, 16777}, {4690, 17255}, {4699, 16831}, {4704, 16815}, {4727, 17311}, {4739, 15668}, {4772, 16826}, {4788, 16816}, {4851, 4971}, {4898, 17316}, {5271, 17147}, {5695, 7290}, {6542, 17298}, {6764, 10481}, {7229, 17014}, {7232, 17372}, {7271, 9312}, {7321, 17377}, {7991, 10444}, {8056, 17490}, {9819, 10889}, {10446, 11531}, {16670, 17351}, {16706, 17286}, {17026, 17759}, {17155, 17156}, {17229, 17290}, {17233, 17282}, {17239, 17323}, {17246, 17275}, {17269, 17356}, {17276, 17362}, {17281, 17366}, {17293, 17382}, {17302, 17308}, {17303, 17395}


X(17152) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17152) lies on these lines:
{1, 16705}, {2, 1258}, {8, 76}, {69, 145}, {75, 3869}, {304, 3877}, {350, 17751}, {732, 6646}, {742, 3727}, {758, 17141}, {960, 3263}, {1930, 3878}, {3112, 4388}, {3678, 4986}, {3735, 17489}, {3884, 14210}, {3997, 16818}, {4021, 10452}, {4357, 10459}, {4450, 7750}, {4766, 17062}, {5834, 14555}, {8049, 16748}, {9318, 17739}, {17135, 17144}


X(17153) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17153) lies on these lines:
{1, 17202}, {7, 6327}, {38, 256}, {69, 9016}, {75, 17138}, {141, 18082}, {314, 17135}, {320, 13476}, {1330, 3874}, {1469, 5016}, {17139, 17142}


X(17154) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17154) lies on these lines:
{1, 4427}, {2, 38}, {8, 596}, {11, 4080}, {75, 16727}, {88, 3699}, {145, 2802}, {149, 900}, {190, 3315}, {192, 13476}, {354, 3995}, {518, 4706}, {726, 17449}, {740, 17145}, {891, 9263}, {1054, 17780}, {2835, 9965}, {3210, 4430}, {3617, 4738}, {3623, 17460}, {3722, 4781}, {3873, 17147}, {3999, 4358}, {4360, 14757}, {4552, 5083}, {4661, 17490}, {5211, 17484}, {5744, 15590}, {17135, 17155}, {17142, 17178}


X(17155) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a b^2 - a b c - b^2 c + a c^2 - b c^2 : :

X(17155) lies on these lines:
{1, 596}, {2, 726}, {10, 7226}, {38, 75}, {42, 3210}, {43, 17165}, {57, 17763}, {171, 3891}, {190, 748}, {192, 3720}, {244, 312}, {321, 982}, {354, 536}, {519, 3894}, {522, 6545}, {537, 3681}, {614, 3729}, {649, 2319}, {740, 3873}, {894, 17017}, {899, 17490}, {984, 4359}, {986, 4968}, {1086, 3703}, {1215, 4850}, {1266, 3914}, {1269, 4446}, {1278, 4365}, {1740, 17142}, {1978, 6384}, {3006, 17889}, {3120, 3705}, {3159, 3624}, {3175, 3742}, {3218, 4362}, {3219, 16825}, {3644, 17450}, {3662, 15523}, {3677, 4659}, {3699, 9350}, {3702, 3976}, {3706, 4686}, {3741, 4392}, {3757, 4414}, {3759, 4722}, {3840, 4671}, {3917, 14839}, {3920, 3980}, {3923, 7191}, {3925, 4884}, {3952, 16569}, {3966, 4683}, {3967, 16610}, {4009, 16602}, {4011, 7292}, {4042, 17119}, {4135, 4871}, {4188, 8669}, {4189, 8720}, {4358, 17063}, {4363, 17599}, {4388, 4440}, {4423, 17262}, {4427, 8616}, {4434, 9352}, {4661, 4685}, {4695, 4737}, {4706, 4849}, {4970, 17018}, {5695, 17597}, {5846, 11246}, {6536, 17247}, {10436, 17176}, {17135, 17154}, {17151, 17156}


X(17156) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17156) lies on these lines:
{1, 2}, {6, 3706}, {31, 3886}, {37, 4042}, {38, 3875}, {44, 4387}, {63, 740}, {69, 3914}, {75, 969}, {100, 16778}, {321, 3751}, {333, 968}, {354, 4361}, {430, 5130}, {497, 5839}, {518, 1824}, {524, 1836}, {940, 3696}, {982, 4716}, {1150, 3896}, {1707, 16704}, {1834, 10371}, {1943, 2263}, {1965, 17144}, {3175, 5220}, {3434, 5847}, {3703, 17299}, {3729, 4365}, {3769, 3996}, {3891, 16496}, {3925, 4851}, {3936, 17064}, {3966, 17362}, {3980, 4709}, {4388, 17363}, {4423, 4891}, {4427, 16570}, {4641, 5695}, {4643, 4854}, {4693, 7262}, {4819, 5432}, {4852, 17599}, {4863, 5846}, {4865, 17772}, {4884, 4971}, {9022, 16799}, {17151, 17155}


X(17157) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17157) lies on these lines:
{1, 87}, {38, 75}, {76, 3778}, {190, 16690}, {313, 4446}, {596, 17038}, {698, 3056}, {740, 3868}, {984, 4968}, {1269, 4443}, {1278, 17135}, {3739, 4003}, {3764, 3770}, {3948, 17065}, {3963, 12782}, {4440, 17138}, {11320, 16800}


X(17158) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17158) lies on these lines:
{1, 75}, {2, 4515}, {7, 9797}, {69, 6764}, {85, 145}, {190, 16572}, {192, 1212}, {218, 3759}, {220, 239}, {241, 3210}, {279, 4452}, {312, 5222}, {321, 17014}, {341, 350}, {519, 3673}, {664, 4350}, {1088, 6604}, {1111, 3633}, {1266, 10481}, {1441, 4460}, {1447, 3913}, {1999, 5228}, {3212, 3880}, {3434, 7247}, {3555, 17753}, {3598, 12632}, {3632, 7264}, {3760, 4737}, {3769, 9441}, {3813, 7179}, {4664, 16601}, {6766, 10444}, {10529, 17095}


X(17159) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17159) lies on these lines:
{1, 17212}, {2, 4079}, {75, 513}, {512, 4374}, {523, 4467}, {834, 4406}, {4132, 7199}, {4155, 17166}


X(17160) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17160) lies on these lines:
{1, 75}, {2, 3943}, {6, 1278}, {7, 10944}, {8, 4389}, {9, 3644}, {10, 17320}, {37, 16815}, {43, 4479}, {44, 190}, {45, 192}, {69, 3621}, {88, 17495}, {142, 17315}, {145, 17378}, {313, 3214}, {319, 3625}, {320, 519}, {321, 17012}, {333, 17147}, {344, 4402}, {346, 17352}, {350, 899}, {523, 4467}, {524, 4440}, {545, 4969}, {594, 17302}, {664, 1443}, {726, 4716}, {889, 3226}, {894, 4686}, {1086, 4971}, {1100, 4726}, {1111, 17791}, {1268, 3634}, {1654, 4399}, {1992, 4454}, {2276, 17028}, {2321, 16706}, {2345, 17380}, {3008, 17264}, {3210, 14829}, {3240, 4441}, {3246, 3685}, {3247, 4751}, {3262, 3935}, {3263, 7292}, {3617, 3672}, {3618, 4461}, {3626, 4357}, {3632, 17274}, {3661, 17301}, {3662, 17295}, {3664, 4464}, {3679, 17250}, {3686, 17258}, {3723, 4739}, {3729, 3759}, {3739, 17319}, {3758, 4659}, {3834, 4727}, {3879, 4896}, {3891, 3996}, {3912, 17067}, {3932, 9780}, {3946, 4431}, {3950, 17263}, {4000, 17233}, {4007, 17228}, {4033, 18073}, {4034, 17328}, {4110, 18044}, {4359, 17021}, {4362, 17601}, {4363, 4393}, {4371, 17257}, {4384, 4664}, {4405, 17330}, {4419, 17346}, {4434, 9324}, {4436, 16694}, {4445, 17236}, {4480, 4700}, {4675, 17389}, {4681, 17260}, {4688, 16826}, {4690, 17254}, {4699, 16777}, {4704, 17259}, {4706, 5205}, {4718, 17261}, {4772, 15668}, {4788, 17262}, {4816, 17272}, {4821, 17118}, {4859, 17241}, {4862, 17361}, {4873, 17342}, {4908, 6687}, {4980, 17011}, {5222, 17354}, {5839, 17347}, {6173, 17387}, {6646, 17362}, {7232, 17373}, {7263, 17300}, {8148, 10446}, {16833, 17335}, {17029, 17759}, {17121, 17351}, {17227, 17294}, {17229, 17291}, {17230, 17290}, {17232, 17309}, {17234, 17314}, {17235, 17287}, {17238, 17323}, {17239, 17324}, {17240, 17282}, {17242, 17278}, {17247, 17275}, {17249, 17270}, {17255, 17343}, {17268, 17356}, {17276, 17363}, {17280, 17366}, {17281, 17367}, {17286, 17370}, {17288, 17372}, {17292, 17382}, {17293, 17383}, {17298, 17386}, {17303, 17396}, {17308, 17399}


X(17161) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(523), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17161) lies on these lines:
{2, 4024}, {75, 693}, {150, 14731}, {514, 14779}, {523, 4467}, {824, 17147}, {2400, 8049}, {2786, 4988}, {3868, 14077}, {4369, 4838}, {4776, 4820}, {4789, 17069}, {6367, 17166}


X(17162) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17162) lies on these lines:
{1, 2}, {44, 4037}, {75, 16741}, {81, 17163}, {321, 4663}, {523, 4467}, {524, 4442}, {740, 896}, {3578, 4854}, {3896, 4689}, {3936, 17070}, {3994, 4753}, {4716, 17495}, {4734, 5372}, {4892, 4938}


X(17163) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(1100), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17163) lies on these lines:
{1, 17589}, {2, 740}, {8, 79}, {10, 3995}, {42, 4709}, {75, 3873}, {81, 17162}, {210, 321}, {333, 4427}, {518, 4980}, {536, 4981}, {594, 4972}, {714, 4740}, {756, 4732}, {1089, 3956}, {1211, 4442}, {1278, 3728}, {1698, 4065}, {2292, 3617}, {2650, 3621}, {2667, 4699}, {3578, 17768}, {3701, 3921}, {3706, 3742}, {3714, 4731}, {3741, 17495}, {3743, 9780}, {3747, 16816}, {3753, 5295}, {3925, 3969}, {3936, 4046}, {3968, 4714}, {4054, 4061}, {4068, 5284}, {4418, 16704}, {4425, 8013}, {4512, 5271}, {4671, 4903}, {4696, 4711}, {4733, 4854}, {4739, 4883}, {4847, 6758}, {4886, 5057}, {5263, 17150}, {5278, 5695}, {5429, 11115}, {7191, 17117}, {7192, 9279}, {10327, 11221}

X(17163) = anticomplement of X(1962)
X(17163) = centroid of Gemini triangle 18


X(17164) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(3666), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17164) lies on these lines:
{1, 596}, {2, 986}, {8, 79}, {10, 3120}, {21, 4427}, {35, 4781}, {65, 321}, {72, 4651}, {75, 3869}, {145, 740}, {190, 5260}, {306, 3671}, {333, 11684}, {346, 2294}, {349, 4566}, {391, 3958}, {517, 4968}, {523, 1222}, {726, 10459}, {846, 17588}, {894, 17016}, {942, 3702}, {960, 4359}, {1046, 16704}, {1089, 3754}, {1193, 17495}, {1215, 4642}, {1441, 12709}, {1834, 4442}, {1836, 5016}, {1962, 3622}, {2345, 4016}, {3057, 4459}, {3219, 16824}, {3340, 4659}, {3485, 17740}, {3555, 3902}, {3616, 3743}, {3649, 3704}, {3678, 4714}, {3696, 3962}, {3698, 3967}, {3701, 3753}, {3712, 11281}, {3724, 4188}, {3725, 17490}, {3782, 5835}, {3812, 4358}, {3868, 17135}, {3873, 4673}, {3874, 17145}, {3886, 11520}, {3889, 17146}, {3891, 5710}, {3918, 3992}, {3919, 4066}, {3923, 3924}, {4018, 5295}, {4225, 11688}, {4414, 16347}, {4440, 5484}, {4454, 8680}, {4696, 5836}, {4742, 5045}, {4861, 6758}, {4918, 17056}, {5263, 17142}, {5271, 12526}, {5423, 11221}, {11533, 17589}, {12567, 16865}, {12648, 17874}, {14012, 17150}, {14210, 17169}, {17141, 17143}, {17497, 17499}

X(17164) = anticomplement of X(2292)


X(17165) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(16696), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a^2 b - a b^2 + a^2 c + b^2 c - a c^2 + b c^2 : :

X(17165) lies on these lines:
{1, 3159}, {2, 38}, {6, 3891}, {7, 10327}, {8, 79}, {10, 17184}, {42, 726}, {43, 17155}, {55, 4427}, {65, 4696}, {69, 9020}, {72, 4968}, {75, 3681}, {76, 17141}, {190, 1621}, {192, 714}, {210, 4359}, {226, 3006}, {312, 3873}, {321, 518}, {354, 3967}, {519, 4365}, {536, 3896}, {596, 3216}, {883, 6063}, {894, 3920}, {899, 4090}, {942, 3701}, {976, 11115}, {1089, 3874}, {1376, 17780}, {1836, 5014}, {3187, 3751}, {3210, 3240}, {3218, 7081}, {3219, 3757}, {3475, 17776}, {3555, 3702}, {3685, 3957}, {3696, 4980}, {3703, 3936}, {3717, 5249}, {3720, 3971}, {3729, 3870}, {3742, 4009}, {3744, 17351}, {3753, 4723}, {3782, 4972}, {3791, 4722}, {3805, 7033}, {3826, 4126}, {3840, 17449}, {3868, 4385}, {3883, 17781}, {3892, 4975}, {3923, 3938}, {3944, 4080}, {3961, 4418}, {3977, 13405}, {3992, 5883}, {4001, 5850}, {4030, 4450}, {4054, 4847}, {4082, 5542}, {4362, 16704}, {4387, 4942}, {4388, 17484}, {4430, 4671}, {4454, 17784}, {4463, 17220}, {4514, 5057}, {4576, 8033}, {4579, 5012}, {4645, 17483}, {4654, 4901}, {4672, 17469}, {4756, 5284}, {4767, 9342}, {4884, 5718}, {4966, 6057}, {5220, 5278}, {5223, 5271}, {5311, 8025}, {5423, 9776}, {6758, 17479}, {7172, 9965}, {8620, 16606}, {11680, 17774}, {14594, 17074}, {17127, 17350}, {17489, 17499}

X(17165) = complement of X(20068)
X(17165) = anticomplement of X(38)
X(17165) = polar conjugate of isogonal conjugate of X(22164)


X(17166) =  (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17166) lies on these lines:
{1, 514}, {2, 4705}, {8, 2533}, {75, 16737}, {512, 7192}, {513, 4801}, {523, 1325}, {667, 17494}, {693, 8678}, {784, 4378}, {830, 4978}, {1019, 4151}, {1577, 4160}, {3616, 4824}, {4041, 4369}, {4088, 8045}, {4155, 17159}, {4170, 15309}, {4379, 17072}, {4391, 7662}, {4490, 4874}, {4804, 6002}, {4843, 4897}, {4988, 5029}, {6367, 17161}, {7253, 8672}, {9422, 17794}


X(17167) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17167) lies on these lines:
{2, 573}, {5, 51}, {21, 946}, {27, 86}, {58, 12047}, {63, 17139}, {81, 226}, {142, 1817}, {306, 314}, {329, 16713}, {333, 908}, {394, 7522}, {511, 3136}, {516, 4184}, {859, 5886}, {1086, 16700}, {1125, 4225}, {1699, 14956}, {1770, 4278}, {1836, 3286}, {1953, 14213}, {2140, 14953}, {2328, 7474}, {3120, 17187}, {3452, 5235}, {3674, 16749}, {3736, 3914}, {3782, 16696}, {3794, 14009}, {3817, 14008}, {4228, 17188}, {4267, 11375}, {4658, 13407}, {5805, 8021}, {5816, 6515}, {6678, 11064}, {8680, 16585}, {10461, 10527}, {11350, 15668}, {12609, 16049}, {12699, 17524}, {16887, 16891}, {17186, 17200}


X(17168) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17168) lies on these lines:
{2, 17191}, {27, 86}, {81, 3911}, {140, 1493}, {16577, 16698}, {17176, 17199}, {17200, 17209}


X(17169) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17169) lies on these lines:
{1, 17136}, {2, 2350}, {6, 17683}, {7, 21}, {8, 274}, {58, 16020}, {60, 1509}, {69, 16709}, {75, 3889}, {81, 277}, {142, 1475}, {314, 5558}, {329, 5333}, {354, 16708}, {377, 14548}, {404, 14828}, {938, 16749}, {962, 4229}, {1010, 11037}, {1193, 3664}, {1790, 8025}, {1930, 17140}, {2275, 4675}, {3110, 14267}, {3241, 16711}, {3522, 10446}, {3622, 17753}, {3673, 16727}, {3742, 4059}, {3754, 7278}, {3945, 6904}, {4209, 17379}, {4444, 6625}, {4648, 16696}, {5249, 17177}, {5712, 16700}, {5815, 14007}, {6173, 17197}, {7177, 17219}, {9263, 16722}, {10453, 10471}, {10481, 17194}, {10578, 13588}, {10580, 16750}, {14210, 17164}, {16710, 17300}, {17050, 17474}, {17181, 17198}


X(17170) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17170) lies on these lines:
{1, 7}, {2, 169}, {3, 348}, {4, 85}, {8, 150}, {28, 86}, {30, 17079}, {36, 17081}, {40, 9436}, {55, 3665}, {69, 72}, {75, 5082}, {144, 17742}, {220, 5845}, {277, 673}, {329, 3912}, {345, 3933}, {376, 17078}, {388, 4911}, {497, 3673}, {515, 9312}, {517, 6604}, {631, 17095}, {664, 944}, {738, 9841}, {940, 5244}, {942, 14548}, {948, 6996}, {1056, 7247}, {1086, 16781}, {1111, 1479}, {1122, 9848}, {1358, 6284}, {1429, 7125}, {1439, 15740}, {1440, 6359}, {1446, 6836}, {1447, 3086}, {1478, 4056}, {1790, 8025}, {1836, 4059}, {1870, 7210}, {2899, 6381}, {3085, 7179}, {3207, 17044}, {3294, 17257}, {3304, 7198}, {3421, 16284}, {3487, 14828}, {3598, 14986}, {3732, 6554}, {3784, 7055}, {3970, 3995}, {4000, 16502}, {4368, 17216}, {4403, 7748}, {5204, 7181}, {5222, 5299}, {5819, 17682}, {5933, 12432}, {6172, 17744}, {7056, 10167}, {7187, 7791}, {7200, 9597}, {7223, 7354}, {7272, 7278}, {7386, 17441}, {7580, 17093}, {9776, 17023}, {10521, 11019}, {13730, 17321}, {16887, 17185}, {17300, 17481}


X(17171) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17171) lies on these lines:
{4, 991}, {19, 10436}, {25, 15668}, {27, 86}, {28, 142}, {141, 427}, {226, 1396}, {286, 334}, {307, 4269}, {468, 6707}, {469, 17234}, {516, 7431}, {859, 17073}, {1172, 17189}, {1333, 16580}, {1829, 3739}, {1930, 16747}, {3665, 16696}, {4227, 12610}, {4361, 11396}, {4851, 5090}, {4869, 7378}, {5064, 17313}, {5094, 17327}, {12135, 17390}


X(17172) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17172) lies on these lines:
{27, 86}, {514, 1921}, {858, 2393}, {3007, 14570}, {4466, 17209}, {4648, 7381}, {16581, 16702}


X(17173) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17173) lies on these lines:
{2, 573}, {21, 12699}, {81, 3772}, {226, 16704}, {946, 17588}, {2476, 3060}, {3219, 17139}, {5249, 8025}, {5905, 16713}, {16738, 17184}, {17187, 17889}, {17203, 17204}


X(17174) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17174) lies on these lines:
{2, 573}, {21, 5886}, {81, 17720}, {226, 8025}, {908, 16704}, {946, 11115}, {1086, 16753}, {3218, 17139}, {3286, 5057}, {3615, 11102}, {3794, 14008}, {3944, 17187}, {4193, 5640}, {4657, 5333}, {5249, 17190}, {17176, 17203}


X(17175) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    (a + b) (a + c) (a b + a c + 2 b c) : :

X(17175) lies on these lines:
{1, 75}, {2, 2350}, {81, 4384}, {142, 16818}, {194, 16710}, {213, 4670}, {239, 8025}, {310, 3760}, {333, 16832}, {551, 16711}, {894, 3294}, {980, 16700}, {1019, 3249}, {1107, 16726}, {1125, 4368}, {1434, 3361}, {3673, 16708}, {3720, 16748}, {3945, 9534}, {3995, 16826}, {4040, 16737}, {4754, 16589}, {4866, 14007}, {5249, 17203}, {5283, 15668}, {5333, 16831}, {10582, 16750}, {11115, 17201}, {11321, 16783}, {16601, 16728}, {16704, 16815}, {16738, 16819}, {16752, 17023}, {16823, 17200}, {17050, 17197}, {17183, 17753}, {17398, 17540}


X(17176) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17176) lies on these lines:
{86, 310}, {274, 3009}, {1215, 16707}, {3120, 17193}, {3995, 16826}, {10436, 17155}, {16887, 16891}, {17168, 17199}, {17174, 17203}


X(17177) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17177) lies on these lines:
{2, 17198}, {86, 2194}, {2886, 16708}, {3662, 16891}, {3705, 16703}, {5249, 17169}, {11680, 16727}, {16887, 17181}, {17205, 17889}


X(17178) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17178) lies on these lines:
{2, 6}, {75, 16710}, {145, 3736}, {192, 16696}, {274, 4772}, {314, 1278}, {1740, 17135}, {3286, 17539}, {3662, 17197}, {4699, 16709}, {6646, 17183}, {10453, 17187}, {10455, 17207}, {16700, 17490}, {16722, 17448}, {16723, 17342}, {16887, 17202}, {17142, 17154}, {17195, 17333}


X(17179) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17179) lies on these lines:
{1, 16712}, {2, 2350}, {86, 2163}, {274, 3679}, {519, 16711}, {551, 16705}, {1019, 4785}, {1434, 4234}, {3662, 17207}, {3760, 6384}, {3834, 16723}, {3879, 16714}, {16696, 17313}, {16709, 17271}, {17139, 17195}


X(17180) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17180) lies on these lines:
{1, 16711}, {2, 2350}, {86, 99}, {274, 519}, {16709, 17378}


X(17181) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17181) lies on these lines:
{1, 147}, {2, 169}, {3, 4872}, {4, 348}, {5, 85}, {7, 90}, {11, 3665}, {36, 4056}, {40, 5195}, {56, 4911}, {77, 1745}, {86, 1437}, {241, 7377}, {279, 3091}, {304, 325}, {307, 10446}, {312, 3933}, {326, 2893}, {355, 664}, {381, 17078}, {499, 1447}, {911, 17682}, {946, 9436}, {982, 4920}, {999, 7247}, {1060, 7210}, {1088, 8226}, {1111, 7185}, {1210, 3674}, {1212, 17671}, {1358, 7173}, {1446, 6828}, {1478, 7176}, {1737, 3212}, {1804, 7282}, {1930, 3705}, {3061, 17046}, {3177, 5179}, {3188, 6839}, {3487, 14548}, {3545, 17079}, {3662, 17211}, {3730, 5074}, {3797, 7906}, {3817, 10481}, {3926, 7283}, {4059, 17605}, {4293, 17081}, {5025, 7187}, {5563, 7272}, {5587, 9312}, {5603, 6604}, {6063, 17866}, {6516, 6906}, {7009, 17076}, {7096, 17368}, {7181, 7354}, {7183, 7330}, {7195, 10589}, {7223, 10895}, {7270, 7776}, {11374, 14828}, {16284, 17757}, {16887, 17177}, {17169, 17198}, {17210, 17248}


X(17182) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17182) lies on these lines:
{2, 573}, {10, 14011}, {21, 36}, {57, 17139}, {81, 908}, {86, 226}, {314, 3687}, {321, 4568}, {333, 645}, {516, 13588}, {946, 1010}, {1043, 12053}, {1086, 16736}, {1519, 4221}, {2328, 5988}, {3086, 10461}, {3782, 16700}, {3786, 4847}, {3794, 3840}, {3817, 14009}, {4187, 5943}, {4415, 16696}, {5208, 11019}, {6176, 11113}, {16887, 17177}


X(17183) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17183) lies on these lines:
{2, 573}, {7, 21}, {8, 314}, {9, 16713}, {69, 2478}, {75, 3877}, {78, 10889}, {81, 329}, {332, 3478}, {333, 6557}, {452, 3945}, {962, 1010}, {1043, 9785}, {1201, 3663}, {2287, 5838}, {2328, 17189}, {2347, 3452}, {3664, 3720}, {3794, 9309}, {3897, 17394}, {3948, 15983}, {4193, 5224}, {4360, 5330}, {4419, 16696}, {4440, 16710}, {4862, 17205}, {5208, 10580}, {5232, 6919}, {5250, 10436}, {5333, 9776}, {5736, 11344}, {5905, 8025}, {6646, 17178}, {8583, 10442}, {9778, 13588}, {9779, 14009}, {9780, 14011}, {10461, 14986}, {11024, 14007}, {16726, 17276}, {16738, 17257}, {17175, 17753}, {17195, 17219}


X(17184) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

Barycentrics    a b^2 + b^3 + a c^2 + c^3 : :

X(17184) lies on these lines:
{1, 6327}, {2, 7}, {10, 17165}, {31, 4655}, {38, 2887}, {42, 3821}, {69, 3187}, {72, 4202}, {81, 320}, {86, 593}, {141, 321}, {238, 4683}, {239, 2895}, {244, 3846}, {306, 3663}, {312, 17227}, {333, 17273}, {343, 1231}, {518, 4972}, {536, 3969}, {594, 4980}, {726, 15523}, {748, 4703}, {752, 17469}, {756, 3836}, {896, 6679}, {940, 7232}, {942, 5051}, {1086, 1211}, {1125, 6536}, {1150, 3772}, {1255, 17317}, {1330, 5262}, {1396, 17923}, {1457, 4296}, {1738, 4651}, {1999, 10452}, {2308, 17770}, {3120, 3741}, {3175, 17231}, {3242, 5014}, {3416, 3891}, {3454, 3670}, {3578, 17344}, {3616, 4340}, {3664, 8025}, {3666, 3936}, {3681, 4429}, {3687, 17495}, {3705, 4392}, {3706, 4442}, {3720, 4425}, {3744, 4450}, {3752, 5741}, {3771, 4414}, {3868, 16062}, {3912, 3995}, {3914, 17135}, {3920, 4645}, {3925, 4981}, {3938, 4660}, {4000, 5739}, {4001, 16704}, {4062, 4970}, {4292, 11115}, {4358, 4415}, {4383, 17290}, {4388, 7191}, {4417, 4850}, {4419, 17776}, {4641, 17345}, {4643, 5278}, {4722, 17771}, {4854, 4966}, {4892, 6682}, {5044, 17674}, {5253, 5323}, {5256, 17304}, {5271, 17272}, {5287, 17298}, {5333, 17322}, {5847, 17150}, {6147, 13728}, {6703, 7238}, {13161, 17751}, {16738, 17173}, {16887, 16891}, {17011, 17302}, {17019, 17300}, {17198, 17203}


X(17185) =  (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

X(17185) lies on these lines:
{1, 21}, {2, 573}, {9, 312}, {27, 10444}, {29, 7713}, {40, 1010}, {43, 4476}, {57, 86}, {165, 13588}, {200, 3786}, {226, 17139}, {284, 1812}, {392, 859}, {394, 16368}, {405, 10441}, {452, 10449}, {511, 4199}, {940, 16574}, {958, 10480}, {960, 4267}, {997, 4276}, {1001, 10473}, {1043, 1697}, {1211, 3882}, {1402, 3286}, {1412, 1444}, {1698, 14011}, {1699, 14009}, {1756, 4425}, {1817, 10856}, {1848, 2354}, {1999, 3219}, {2150, 2185}, {2245, 6703}, {2269, 3687}, {2287, 15479}, {2300, 3666}, {2478, 10479}, {3218, 8025}, {3305, 5235}, {3306, 5333}, {3512, 4603}, {3674, 16705}, {3736, 17594}, {3741, 4368}, {3895, 4720}, {3929, 4664}, {4001, 10452}, {4184, 10434}, {4189, 10470}, {4225, 10882}, {4229, 10860}, {4266, 14555}, {4271, 5743}, {4384, 10471}, {4414, 17187}, {4666, 11021}, {5271, 10447}, {5273, 16713}, {5745, 17197}, {6872, 10454}, {9306, 13723}, {9965, 10462}, {10463, 17576}, {10476, 11110}, {10477, 13615}, {10884, 12547}, {10886, 14008}, {11343, 17811}, {16700, 17595}, {16887, 17170}, {17190, 17191}