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This is PART 9: Centers X(16001) -

PART 1: Introduction and Centers X(1) - X(1000)
PART 2: Centers X(1001) - X(3000)
PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000)
PART 5: Centers X(7001) - X(10000)
PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000)
PART 8: Centers X(14001) - X(16000)
PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000)
PART 11: Centers X(20001) - X(22000)
PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000)
PART 14: Centers X(26001) - X(28000)
PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000)
PART 17: Centers X(32001) - X(34000)
PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000)
PART 20: Centers X(38001) - X(40000)


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 CÚsar Lozada, Hyacinthos 27108.

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 CÚsar Lozada, Hyacinthos 27108.

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 CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27119.

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


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

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)

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 one 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 CÚsar 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) = {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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27134.

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


X(16027) =  X(5)X(19)&capsX(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(16028) =  X(6)X(63)&capsX(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(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(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(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(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(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(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(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}


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(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 CÚsar 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 CÚsar 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(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(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) : :

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(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) = 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) = 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(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 cubic K019, K289, K501, and on 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 cubics K019, K289, K501, and on 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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 pnt 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) : :
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). (CÚsar 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) : :

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) = 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) : :

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) = 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) : :

X(16082) lies on the circumconic {{A,B,C,X(1),X(2)}} 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) = crosspoint of X(i) and X(j) for these (i,j): {} X(16082) = trilinear pole of line {4, 513}
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(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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar Eliud Lozada, February 13, 2018.

Let ABC be a triangle and A' 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 A" the midpoint of the arc BC of its circumcircle containing A; 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 on the sidelines of ABC. Its A-vertex these 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) = {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) = {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) = {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 : :
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 CÚsar Lozada, Hyacinthos 27182.

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), (12261, 13624) 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) = {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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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(16170)
X(16169) = circumcircle-antipode of X(16171)


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 CÚsar 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(16168)


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) : :

See Antreas Hatzipolakis and CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27196.

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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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 CÚsar 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(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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar Lozada, Hyacinthos 27207.

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

X(16183) = {X(3), X(1316)}-harmonic conjugate of X(16181)


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 CÚsar 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 CÚsar 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) : :

See Angel Montesdeoca, HG120218.

X(16188) lies on 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)






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 CÚsar 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)

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) = {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

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, 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, 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: {34,1850}, {2362,16232}


dfs

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 OX 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 CÚsar 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(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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) : :

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 27225.

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


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

Barycentrics    (SA^2+SB*SC)*(SB-SC)*SB*SC : :

See Antreas Hatzipolakis and CÚsar 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(16230) =  X(4)X(690)∩X(230)X(231)

Barycentrics    (SA^2-SB*SC)*(SB-SC)*SB*SC : :
X(16230) = 3*X(1637)-2*X(6130)

See Antreas Hatzipolakis and CÚsar Lozada, Hyacinthos 27225.

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(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 CÚsar 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(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 CÚsar Lozada, Hyacinthos 27230.

X(16233) lies on these lines:
{396, 11617}, {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 CÚsar Lozada, Hyacinthos 27230.

X(16234) lies on these lines:
{395, 11618}, {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 CÚsar Lozada, Hyacinthos 27230.

X(16235) lies on these lines:
{2, 2780}, {3, 9148}, {140, 11176}, {351, 5054}, {523, 7623}, {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 CÚsar 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 CÚsar 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) : :
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 CÚsar 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): (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)

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 CÚsar Lozada, Hyacinthos 27240.

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) = {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 CÚsar 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 CÚsar 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 CÚsar 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 CÚsar 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) = 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(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) : :    (CÚsar 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).

X(16247) lies on 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 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 CÚsar 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 CÚsar 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 CÚsar 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}



underbar

 

This is the end of 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)
PART 17: Centers X(32001) - X(34000)
PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000)
PART 20: Centers X(38001) - X(40000)