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This is PART 10: Centers X(18001) - X(20000)

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


X(18001) = X(42)X(8663)∩X(351)X(2054)

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

X(18001) lies on the cubic K978 and these lines: {42, 8663}, {351, 2054}, {514, 1125}, {649, 2308}, {659, 1929}, {661, 1962}, {804, 11599}, {875, 9506}, {4983, 6155}

X(18001) = X(i)-cross conjugate of X(j) for these (i,j): {3121, 9506}, {4455, 512}
X(18001) = crossdifference of every pair of points on line {1931, 6157}
X(18001) = X(3882)-zayin conjugate of X(9508)
X(18001) = X(i)-isoconjugate of X(j) for these (i,j): {99, 1757}, {100, 17731}, {190, 1931}, {423, 1332}, {662, 6542}, {668, 1326}, {799, 17735}, {2786, 4567}, {4584, 6651}, {4589, 8298}, {4600, 9508}, {4601, 5029}
X(18001) = barycentric product X(i)*X(j) for these {i,j}: {512, 6650}, {513, 9278}, {514, 2054}, {649, 11599}, {661, 1929}, {2702, 3120}, {3733, 6543}, {4010, 9506}
X(18001) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 6542}, {649, 17731}, {667, 1931}, {669, 17735}, {798, 1757}, {1919, 1326}, {1929, 799}, {2054, 190}, {2702, 4600}, {3121, 9508}, {3122, 2786}, {4079, 6541}, {4455, 6651}, {6650, 670}, {9278, 668}, {9505, 4639}, {9506, 4589}, {11599, 1978}


X(18002) = X(512)X(2092)∩X(513)X(3666)

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

X(18002) lies on the cubic K978 and these lines: {512, 2092}, {513, 3666}, {667, 2300}, {798, 3725}, {804, 11611}, {893, 2483}, {900, 11609}

X(18002) = crossdifference of every pair of points on line {5209, 5291}
X(18002) = X(i)-isoconjugate of X(j) for these (i,j): {99, 17763}, {100, 5209}, {422, 4561}, {662, 17790}, {799, 5291}, {1978, 5006}, {2787, 4600}, {5061, 7257}
X(18002) = barycentric product X(i)*X(j) for these {i,j}: {667, 11611}, {2703, 3125}, {7180, 11609}
X(18002) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 17790}, {649, 5209}, {669, 5291}, {798, 17763}, {1980, 5006}, {2703, 4601}, {3121, 2787}, {11611, 6386}


X(18003) = X(321)X(690)∩X(523)X(1577)

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

X(18003) lies on the cubic K979 and these lines: {321, 690}, {523, 1577}, {668, 891}, {756, 14430}, {1089, 3762}

X(18003) = crossdifference of every pair of points on line {1333, 1977}
X(18003) = X(58)-isoconjugate of X(2703)
X(18003) = X(523)-Hirst inverse of X(4036)
X(18003) = barycentric product X(i)*X(j) for these {i,j}: {321, 2787}, {523, 17790}, {850, 5291}, {1577, 17763}, {4024, 5209}
X(18003) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 2703}, {2680, 3310}, {2787, 81}, {3700, 11609}, {4036, 11611}, {5040, 1333}, {5061, 4565}, {5209, 4610}, {5291, 110}, {17763, 662}, {17790, 99}


X(18004) = X(10)X(690)∩X(100)X(190)

Barycentrics    (b - c)*(b + c)*(-a^2 - a*b + b^2 - a*c + b*c + c^2) : :
X(18004) = X[4010] - 3 X[4120], X[4088] + 3 X[4120], 3 X[4944] - X[7662], X[4707] - 3 X[14431], X[4922] - 3 X[14432]

X(18004) lies on the cubic K979 and these lines: {10, 690}, {100, 190}, {313, 14295}, {513, 4522}, {523, 661}, {756, 2254}, {826, 4129}, {918, 3837}, {1089, 3762}, {1215, 3716}, {1826, 16230}, {2786, 9508}, {3239, 4874}, {3678, 3887}, {4080, 5466}, {4170, 4808}, {4500, 4802}, {4705, 7265}, {4707, 14431}, {4922, 14432}, {4944, 7662}, {6366, 13272}, {6370, 12078}

X(18004) = midpoint of X(i) and X(j) for these {i,j}: {661, 4122}, {4010, 4088}, {4024, 4824}, {4170, 4808}, {4705, 7265}
X(18004) = reflection of X(i) in X(j) for these {i,j}: {4806, 14321}, {4874, 3239}
X(18004) = X(4562)-Ceva conjugate of X(594)
X(18004) = crosspoint of X(i) and X(j) for these (i,j): {100, 15168}, {321, 4583}, {4444, 4608}
X(18004) = crossdifference of every pair of points on line {58, 1015}
X(18004) = crosssum of X(3733) and X(5009)
X(18004) = X(i)-isoconjugate of X(j) for these (i,j): {81, 2702}, {110, 1929}, {163, 6650}, {4556, 9278}
X(18004) = X(i)-Hirst inverse of X(j) for these (i,j): {523, 4024}, {3943, 4062}
X(18004) = barycentric product X(i)*X(j) for these {i,j}: {10, 2786}, {313, 5029}, {321, 9508}, {423, 4064}, {514, 6541}, {523, 6542}, {850, 17735}, {1577, 1757}, {1931, 4036}, {4024, 17731}
X(18004) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 2702}, {523, 6650}, {661, 1929}, {1326, 4556}, {1757, 662}, {2681, 676}, {2786, 86}, {4024, 11599}, {4079, 2054}, {4705, 9278}, {5029, 58}, {6541, 190}, {6542, 99}, {9508, 81}, {17731, 4610}, {17735, 110}
{X(4088),X(4120)}-harmonic conjugate of X(4010)


X(18005) = X(523)X(3120)∩X(690)X(4049)

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

X(18005) lies on the cubic K979 and these lines: {523, 3120}, {690, 4049}, {900, 903}, {4080, 5466}

X(18005) = barycentric product X(2796)*X(4049)


X(18006) = X(100)X(658)∩X(225)X(16230)

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

X(18006) lies on the cubic K979 and these lines: {100, 658}, {225, 16230}, {226, 690}, {349, 14295}, {523, 656}, {900, 13273}, {1254, 2254}, {7234, 17094}

X(18006) = crossdifference of every pair of points on line {284, 14936}
X(18006) = X(i)-isoconjugate of X(j) for these (i,j): {21, 2701}, {110, 2648}, {2652, 4636}
X(18006) = barycentric product X(i)*X(j) for these {i,j}: {226, 2785}, {349, 5075}, {1577, 1758}
X(18006) = barycentric quotient X(i)/X(j) for these {i,j}: {661, 2648}, {1400, 2701}, {1758, 662}, {2651, 4612}, {2785, 333}, {5060, 4636}, {5075, 284}


X(18007) = X(111)X(9189)∩X(115)X(523)

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

X(18007) = 3 X[9166] - X[9168], 3 X[115] - 2 X[9183], 4 X[10189] - X[15300]

X(18007) lies on the cubic K979 and these lines: {111, 9189}, {115, 523}, {543, 8371}, {671, 690}, {1499, 9880}, {1649, 5461}, {2793, 6094}, {5465, 9214}, {9166, 9168}, {10189, 15300}

X(18007) = midpoint of X(671) and X(5466)
X(18007) = reflection of X(1649) in X(5461)
X(18007) = X(922)-isoconjugate of X(9170)
X(18007) = barycentric product X(i)*X(j) for these {i,j}: {543, 5466}, {671, 8371}
X(18007) = barycentric quotient X(i)/X(j) for these {i,j}: {543, 5468}, {671, 9170}, {2502, 5467}, {8371, 524}, {9171, 187}, {9178, 843}


X(18008) = (name pending)

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

X(18008) lies oon the cubic K979 and the line {523, 14603}


X(18009) = X(523)X(1086)∩X(690)X(4444)

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

X(18009) lies on the cubic K979 and these lines: {523, 1086}, {690, 4444}

X(18009) = X(3573)-isoconjugate of X(12031)
X(18009) = barycentric quotient X(3572)/X(12031)


X(18010) = X(23)X(385)∩X(83)X(690)

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

X(18010) lies on the cubic K979 and these lines: {23, 385}, {83, 690}, {308, 14295}, {804, 11606}, {882, 14970}, {4577, 4630}, {8290, 9479}, {9185, 10130}

X(18010) = cevapoint of X(5113) and X(9479)
X(18010) = crosspoint of X(4577) and X(14970)
X(18010) = crosssum of X(3005) and X(8623)
X(18010) = barycentric product X(i)*X(j) for these {i,j}: {83, 9479}, {308, 5113}, {420, 4580}
X(18010) = barycentric quotient X(i)/X(j) for these {i,j}: {2076, 1634}, {5113, 39}, {7779, 4576}, {9479, 141}


X(18011) = X(10)X(523)∩X(690)X(4080)

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

X(18011) lies on the cubic K979 and these lines: {10, 523}, {690, 4080}, {4555, 4618}


X(18012) = X(523)X(7625)∩X(690)X(5485)

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

X(18012) lies on the cubic K979 and these lines: {523, 7625}, {690, 5485}

X(18012) = barycentric product X(2793)*X(5485)
X(18012) = barycentric quotient X(i)/X(j) for these {i,j}: {2793, 1992}, {9135, 1384}


X(18013) = X(522)X(5745)∩X(523)X(17056)

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

X(18013) lies on the cubic K979 and these lines: {522, 5745}, {523, 17056}, {690, 11608}, {2689, 2701}

X(18013) = X(i)-isoconjugate of X(j) for these (i,j): {109, 2651}, {110, 1758}, {651, 5060}
X(18013) = barycentric product X(i)*X(j) for these {i,j}: {522, 11608}, {1577, 2648}, {2652, 4391}
X(18013) = barycentric quotient X(i)/X(j) for these {i,j}: {650, 2651}, {661, 1758}, {663, 5060}, {2648, 662}, {2652, 651}, {3064, 415}, {11608, 664}


X(18014) = X(10)X(6367)∩X(86)X(4977)

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

X(18014) lies on the cubic K979 and these lines: {10, 6367}, {86, 4977}, {514, 1125}, {523, 1213}, {690, 4049}, {900, 6650}, {1269, 3261}, {1577, 4647}, {1839, 7649}, {2690, 2702}, {3649, 4806}

X(18014) = X(4010)-cross conjugate of X(523)
X(18014) = crossdifference of every pair of points on line {1326, 17735}
X(18014) = X(i)-isoconjugate of X(j) for these (i,j): {100, 1326}, {101, 1931}, {110, 1757}, {163, 6542}, {423, 906}, {662, 17735}, {692, 17731}, {4567, 5029}, {4570, 9508}
X(18014) = trilinear pole of line {3120, 4988}
X(18014) = barycentric product X(i)*X(j) for these {i,j}: {514, 11599}, {523, 6650}, {693, 9278}, {1577, 1929}, {2054, 3261}, {6543, 7192}
X(18014) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 17735}, {513, 1931}, {514, 17731}, {523, 6542}, {649, 1326}, {661, 1757}, {1929, 662}, {2054, 101}, {2702, 4570}, {3120, 2786}, {3122, 5029}, {3125, 9508}, {4010, 6651}, {4024, 6541}, {6543, 3952}, {6650, 99}, {7649, 423}, {9278, 100}, {9505, 4584}, {11599, 190}


X(18015) = X(81)X(6371)∩X(513)X(3666)

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

X(18015) lies on the cubic K979 and these lines: {81, 6371}, {513, 3666}, {523, 1211}, {661, 2292}, {690, 11611}, {1290, 2703}, {1829, 6591}

X(18015) = X(i)-isoconjugate of X(j) for these (i,j): {110, 17763}, {163, 17790}, {190, 5006}, {422, 1331}, {643, 5061}, {662, 5291}, {692, 5209}, {2787, 4570}, {4600, 5040}
X(18015) = crossdifference of every pair of points on line {5006, 5291} X(18015) = barycentric product X(i)*X(j) for these {i,j}: {513, 11611}, {2703, 16732}, {7178, 11609}
X(18015) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 5291}, {514, 5209}, {523, 17790}, {661, 17763}, {667, 5006}, {2703, 4567}, {3121, 5040}, {3125, 2787}, {6591, 422}, {7180, 5061}, {11609, 645}, {11611, 668}


X(18016) = MIDPOINT OF X(3) AND X(15345)

Trilinears    cos(B-C)*((4*cos(A)-2*cos(3*A))*cos(B-C)-4*cos(2*A)-3*cos(4*A)+1/2) : :
Barycentrics    (SB+SC)*(S^2+SB*SC)*((27*R^2+16*SA-6*SW)*S^2-(17*R^2-2*SW)*SA^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 27560.

X(18016) lies on these lines: {3, 54}, {30, 13856}, {140, 6592}, {539, 14143}, {1209, 14072}, {13434, 15770}

X(18016) = midpoint of X(3) and X(15345)
X(18016) = {X(3), X(54)}-harmonic conjugate of X(6150)


X(18017) = REFLECTION OF X(20) IN X(16273)

Barycentrics    (b^2+c^2-a^2) (13 a^14 -22 a^12 (b^2+c^2) +a^10 (-13 b^4+58 b^2 c^2-13 c^4) +20 a^8 (b^2-c^2)^2 (b^2+c^2) +a^6 (b^2-c^2)^2 (35 b^4-22 b^2 c^2+35 c^4) -2 a^4 (b^2-c^2)^2 (19 b^6-3 b^4 c^2-3 b^2 c^4+19 c^6) -a^2 (b^2-c^2)^4 (3 b^4-22 b^2 c^2+3 c^4) +8 (b^2-c^2)^6 (b^2+c^2)) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27570.

X(18017) lies on this line: {2,3}

X(18017) = reflection of X(20) in X(16273)


X(18018) = ISOTOMIC CONJUGATE OF X(22)

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

X(18018) lies on the cubics K141 and K555 and on these lines: {2, 1235}, {4, 13575}, {22, 76}, {25, 339}, {66, 69}, {95, 7485}, {253, 7378}, {264, 5133}, {287, 1993}, {305, 858}, {401, 9983}, {850, 2419}, {2052, 6330}, {3266, 6340}, {7396, 9464}

X(18018) = isogonal conjugate of X(206)
X(18018) = isotomic conjugate of X(22)
X(18018) = polar conjugate of X(8743)
X(18018) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2156, 8878}, {16277, 5905}
X(18018) = X(i)-cross conjugate of X(j) for these (i,j): {6, 76}, {427, 2}, {523, 1289}, {1853, 2052}, {1899, 5392}, {11245, 11140}
X(18018) = X(i)-isoconjugate of X(j) for these (i,j): {1, 206}, {2, 17453}, {6, 2172}, {9, 7251}, {19, 10316}, {22, 31}, {32, 1760}, {37, 17186}, {48, 8743}, {57, 4548}, {63, 17409}, {163, 2485}, {315, 560}, {798, 4611}, {1333, 4456}, {1397, 4123}, {1755, 11610}, {2175, 7210}, {2206, 4463}, {9247, 17907}, {9447, 17076}
X(18018) = X(2)-Hirst inverse of X(16097)
X(18018) = cevapoint of X(i) and X(j) for these (i,j): {2, 7391}, {6, 2353}, {66, 14376}, {339, 523}, {826, 15526}, {13854, 17407}
X(18018) = barycentric product X(i)*X(j) for these {i,j}: {66, 76}, {264, 14376}, {305, 13854}, {561, 2156}, {1289, 3267}, {1502, 2353}, {8024, 16277}
X(18018) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2172}, {2, 22}, {3, 10316}, {4, 8743}, {6, 206}, {10, 4456}, {25, 17409}, {31, 17453}, {55, 4548}, {56, 7251}, {58, 17186}, {66, 6}, {75, 1760}, {76, 315}, {85, 7210}, {98, 11610}, {99, 4611}, {141, 3313}, {264, 17907}, {312, 4123}, {313, 4150}, {321, 4463}, {339, 127}, {523, 2485}, {525, 8673}, {693, 16757}, {1289, 112}, {2156, 31}, {2353, 32}, {6063, 17076}, {9033, 14396}, {13854, 25}, {14376, 3}, {16097, 15013}, {16277, 251}, {16747, 16715}, {17407, 3162}
X(18018) = cevapoint of circumcircle intercepts of de Longchamps line
X(18018) = {X(13854),X(14376)}-harmonic conjugate of X(2)


X(18019) = ISOTOMIC CONJUGATE OF X(23)

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

X(18019) lies on the hyperbola {{A,B,C,X(2),X(69)}} and these lines: {2, 339}, {23, 935}, {67, 69}, {76, 7664}, {95, 7496}, {99, 1799}, {264, 5169}, {287, 323}, {306, 4568}, {325, 328}, {1236, 3266}, {1494, 10989}, {7391, 11605}

X(18019) = isogonal conjugate of X(18374)
X(18019) = isotomic conjugate of X(23)
X(18019) = polar conjugate of X(8744)
X(18019) = X(i)-cross conjugate of X(j) for these (i,j): {524, 76}, {858, 2}, {9140, 94}
X(18019) = X(i)-isoconjugate of X(j) for these (i,j): {19, 10317}, {23, 31}, {32, 16568}, {48, 8744}, {163, 2492}, {316, 560}, {922, 14246}, {923, 6593}, {9447, 17088}
X(18019) = X(23)-vertex conjugate of X(8791)
X(18019) = cevapoint of X(i) and X(j) for these (i,j): {2, 5189}, {6, 5938}, {39, 2393}, {524, 14357}, {3266, 8024}
X(18019) = trilinear pole of line {141, 525} (the line through the symmedian points of the 1st and 2nd Ehrmann inscribed triangles)
X(18019) = barycentric product X(i)*X(j) for these {i,j}: {67, 76}, {305, 8791}, {561, 2157}, {599, 10512}, {850, 17708}, {935, 3267}, {1502, 3455}, {3266, 10415}, {8024, 9076}, {9464, 10511}
X(18019) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 23}, {3, 10317}, {4, 8744}, {67, 6}, {75, 16568}, {76, 316}, {141, 9019}, {523, 2492}, {524, 6593}, {525, 9517}, {599, 10510}, {671, 14246}, {850, 9979}, {935, 112}, {2157, 31}, {3266, 7664}, {3455, 32}, {3580, 12824}, {5466, 10561}, {6063, 17088}, {8791, 25}, {9076, 251}, {10415, 111}, {10511, 1383}, {10512, 598}, {11064, 16165}, {11605, 8743}, {14357, 187}, {17708, 110}


X(18020) = ISOTOMIC CONJUGATE OF X(125)

Barycentrics    (a - b)^2*(a + b)^2*(a - c)^2*(a + c)^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :
Barycentrics    (csc 2A) csc^2(B - C) : :

Line X(99)X(112) (the trilinear polar of X(18020) is the locus of the trilinear pole of the tangent at P to hyperbola {{A,B,C,X(4),P}}, as P moves on the Euler line. (Randy Hutson, June 27, 2018)

X(18020) lies on these lines: {99, 1304}, {107, 10425}, {110, 685}, {112, 9150}, {249, 297}, {250, 325}, {327, 5651}, {415, 4620}, {422, 4601}, {423, 4600}, {450, 3260}, {460, 5203}, {647, 2966}, {648, 892}, {877, 4240}, {1098, 7045}, {2396, 2409}, {4230, 17941}, {4235, 9170}, {5641, 5642}, {6563, 7471}, {7769, 14366}

X(18020) = isogonal conjugate of X(20975)
X(18020) = isotomic conjugate of X(125)
X(18020) = polar conjugate of X(115)
X(18020) = X(5539)-zayin conjugate of X(656)
X(18020) = X(i)-cross conjugate of X(j) for these (i,j): {2, 6331}, {4, 648}, {24, 16813}, {69, 99}, {76, 4577}, {186, 687}, {249, 4590}, {315, 670}, {316, 892}, {317, 6528}, {340, 16077}, {403, 15459}, {419, 685}, {451, 6335}, {511, 2966}, {1092, 4558}, {1330, 190}, {1974, 112}, {2893, 664}, {3043, 14590}, {3144, 653}, {3542, 15352}, {4213, 1897}, {5095, 4235}, {5972, 2}, {6353, 107}, {7058, 4610}, {9306, 110}, {15462, 5649}, {15595, 2396}, {16163, 2407}
X(18020) = pole wrt polar circle of trilinear polar of X(115) (line X(1648)X(8029))
X(18020) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2643}, {6, 3708}, {19, 3269}, {25, 2632}, {31, 125}, {48, 115}, {63, 3124}, {71, 3125}, {72, 3122}, {73, 4516}, {181, 7004}, {184, 1109}, {201, 3271}, {212, 1365}, {213, 4466}, {228, 3120}, {244, 3690}, {255, 8754}, {304, 1084}, {305, 4117}, {306, 3121}, {326, 2971}, {338, 9247}, {339, 560}, {512, 656}, {523, 810}, {525, 798}, {603, 4092}, {647, 661}, {667, 4064}, {669, 14208}, {756, 3937}, {822, 2501}, {872, 1565}, {905, 4079}, {1015, 3949}, {1096, 2972}, {1254, 3270}, {1356, 3718}, {1367, 2212}, {1395, 7068}, {1425, 2310}, {1459, 4705}, {1500, 3942}, {1562, 2155}, {1577, 3049}, {1924, 3267}, {1973, 15526}, {1974, 17879}, {2084, 4580}, {2170, 2197}, {2171, 7117}, {2200, 16732}, {2433, 2631}, {2616, 15451}, {2624, 14582}, {2642, 10097}, {3248, 3695}, {4575, 8029}, {7063, 7182}, {7180, 8611}
X(18020) = cevapoint of X(i) and X(j) for these (i,j): {2, 110}, {4, 648}, {69, 99}, {107, 11547}, {112, 1974}, {249, 250}, {662, 1098}, {1092, 4558}, {1113, 8116}, {1114, 8115}, {2407, 16163}, {2409, 15595}, {3043, 14590}, {3233, 11064}, {4235, 5095}, {4240, 14920}, {5468, 7664}, {6148, 10411}, {15164, 15165}
X(18020) = trilinear pole of line {99, 112}
X(18020) = barycentric product X(i)*X(j) for these {i,j}: {4, 4590}, {27, 4600}, {28, 4601}, {29, 4620}, {76, 250}, {99, 648}, {107, 4563}, {108, 4631}, {110, 6331}, {112, 670}, {162, 799}, {249, 264}, {274, 5379}, {278, 6064}, {281, 7340}, {286, 4567}, {662, 811}, {685, 2396}, {823, 4592}, {877, 2966}, {892, 4235}, {1101, 1969}, {1509, 15742}, {1783, 4623}, {1897, 4610}, {2407, 16077}, {4558, 6528}, {6035, 7473}
X(18020) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3708}, {2, 125}, {3, 3269}, {4, 115}, {19, 2643}, {20, 1562}, {25, 3124}, {27, 3120}, {28, 3125}, {59, 2197}, {60, 7117}, {63, 2632}, {69, 15526}, {76, 339}, {86, 4466}, {92, 1109}, {99, 525}, {107, 2501}, {110, 647}, {112, 512}, {162, 661}, {163, 810}, {186, 2088}, {190, 4064}, {249, 3}, {250, 6}, {264, 338}, {270, 2170}, {275, 8901}, {278, 1365}, {281, 4092}, {286, 16732}, {297, 868}, {304, 17879}, {315, 127}, {323, 16186}, {345, 7068}, {348, 1367}, {393, 8754}, {394, 2972}, {468, 1648}, {476, 14582}, {525, 5489}, {593, 3937}, {643, 8611}, {648, 523}, {662, 656}, {670, 3267}, {685, 2395}, {687, 15328}, {691, 10097}, {757, 3942}, {765, 3949}, {799, 14208}, {811, 1577}, {877, 2799}, {892, 14977}, {933, 2623}, {1016, 3695}, {1101, 48}, {1172, 4516}, {1252, 3690}, {1262, 1425}, {1275, 6356}, {1304, 2433}, {1474, 3122}, {1509, 1565}, {1576, 3049}, {1625, 15451}, {1783, 4705}, {1897, 4024}, {1974, 1084}, {2052, 2970}, {2185, 7004}, {2189, 3271}, {2203, 3121}, {2207, 2971}, {2326, 2310}, {2396, 6333}, {2407, 9033}, {2420, 9409}, {2421, 684}, {2501, 8029}, {2715, 878}, {2966, 879}, {3233, 14401}, {4213, 6627}, {4230, 3569}, {4232, 6791}, {4235, 690}, {4240, 1637}, {4242, 2610}, {4556, 1459}, {4558, 520}, {4563, 3265}, {4564, 201}, {4567, 72}, {4570, 71}, {4573, 17094}, {4575, 822}, {4576, 2525}, {4577, 4580}, {4590, 69}, {4600, 306}, {4610, 4025}, {4611, 8673}, {4612, 521}, {4620, 307}, {4623, 15413}, {4636, 652}, {5094, 8288}, {5379, 37}, {5468, 14417}, {6064, 345}, {6331, 850}, {6335, 4036}, {6353, 6388}, {6528, 14618}, {7012, 2171}, {7054, 3270}, {7058, 2968}, {7115, 181}, {7128, 1254}, {7340, 348}, {7473, 1640}, {8750, 4079}, {10311, 6784}, {10411, 8552}, {11064, 1650}, {11547, 136}, {14129, 137}, {14570, 6368}, {14587, 14533}, {14590, 526}, {14591, 14270}, {14920, 3258}, {15329, 686}, {15388, 2353}, {15395, 11079}, {15460, 15166}, {15461, 15167}, {15742, 594}, {16077, 2394}, {16080, 12079}, {17206, 17216}
X(18020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (647, 9514, 2966), (4240, 5468, 877)


X(18021) = ISOTOMIC CONJUGATE OF X(181)

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

X(18021) lies on these lines: {76, 940}, {86, 310}, {99, 16678}, {171, 5209}, {274, 3666}, {286, 6385}, {305, 16992}, {313, 17731}, {314, 3706}, {333, 3691}, {561, 670}, {799, 14829}, {1909, 2668}, {3596, 4042}, {3996, 7257}, {4631, 7058}

X(18021) = isotomic conjugate of X(181)
X(18021) = X(i)-cross conjugate of X(j) for these (i,j): {3271, 4560}, {3794, 2185}
X(18021) = X(i)-isoconjugate of X(j) for these (i,j): {12, 560}, {31, 181}, {32, 2171}, {42, 1402}, {56, 872}, {57, 7109}, {65, 1918}, {201, 1974}, {213, 1400}, {226, 2205}, {604, 1500}, {669, 4551}, {756, 1397}, {762, 16947}, {765, 1356}, {798, 4559}, {1084, 4564}, {1106, 7064}, {1253, 7143}, {1254, 2175}, {1395, 3690}, {1409, 2333}, {1415, 4079}, {1425, 2212}, {1501, 6358}, {1880, 2200}, {1924, 4552}, {1973, 2197}, {2149, 3124}, {4117, 4998}, {6059, 7138}, {6354, 9447}, {7045, 7063}, {7147, 14827}, {7235, 14598}, {8736, 9247}
X(18021) = cevapoint of X(3271) and X(4560)
X(18021) = barycentric product X(i)*X(j) for these {i,j}: {21, 6385}, {60, 1502}, {76, 261}, {274, 314}, {310, 333}, {312, 873}, {561, 2185}, {670, 4560}, {693, 4631}, {1509, 3596}, {1928, 2150}, {3705, 7307}, {3737, 4602}, {4391, 4623}, {4609, 7252}, {6063, 7058}, {7199, 7257}
X(18021) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 181}, {8, 1500}, {9, 872}, {11, 3124}, {21, 213}, {29, 2333}, {55, 7109}, {60, 32}, {69, 2197}, {75, 2171}, {76, 12}, {81, 1402}, {85, 1254}, {86, 1400}, {99, 4559}, {261, 6}, {264, 8736}, {270, 1973}, {274, 65}, {279, 7143}, {283, 2200}, {284, 1918}, {286, 1880}, {304, 201}, {310, 226}, {312, 756}, {314, 37}, {332, 71}, {333, 42}, {345, 3690}, {346, 7064}, {348, 1425}, {522, 4079}, {552, 1407}, {561, 6358}, {593, 1397}, {645, 4557}, {670, 4552}, {757, 604}, {763, 1408}, {799, 4551}, {873, 57}, {1015, 1356}, {1043, 1334}, {1088, 7147}, {1098, 41}, {1434, 1042}, {1444, 1409}, {1509, 56}, {1812, 228}, {1920, 7211}, {1921, 7235}, {2150, 560}, {2185, 31}, {2189, 1974}, {2194, 2205}, {2326, 2212}, {3271, 1084}, {3596, 594}, {3701, 762}, {3718, 3949}, {3737, 798}, {3786, 3774}, {3794, 16584}, {3926, 7066}, {4087, 4037}, {4391, 4705}, {4560, 512}, {4572, 4605}, {4590, 59}, {4610, 109}, {4612, 692}, {4623, 651}, {4625, 1020}, {4631, 100}, {4858, 2643}, {4976, 8663}, {6061, 14827}, {6063, 6354}, {6064, 1252}, {6385, 1441}, {6514, 4055}, {6628, 1412}, {7017, 7140}, {7054, 2175}, {7058, 55}, {7155, 6378}, {7183, 7138}, {7192, 7180}, {7199, 4017}, {7252, 669}, {7253, 3709}, {7257, 1018}, {7258, 4069}, {7304, 1403}, {7340, 1262}, {8735, 2971}, {14936, 7063}, {17096, 7250}, {17185, 3725}, {17197, 3122}, {17206, 73}, {17880, 3708}


X(18022) = ISOTOMIC CONJUGATE OF X(184)

Barycentrics    b^4*c^4*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :
Barycentrics    csc^2 A csc 2A : :
Barycentrics    sec A csc^3 A : :
Barycentrics    sec A csc(A - ω) : :

X(18022) lies on these lines: {2, 6331}, {25, 8840}, {69, 8795}, {76, 297}, {183, 16089}, {264, 305}, {276, 7763}, {290, 1899}, {308, 13854}, {324, 8024}, {393, 1241}, {458, 3978}, {683, 6524}, {685, 2001}, {1235, 5117}, {1975, 9291}, {3410, 13485}, {6528, 11185}, {9230, 9308}

X(18022) = isogonal conjugate of X(14575)
X(18022) = isotomic conjugate of X(184)
X(18022) = polar conjugate of X(32)
X(18022) = X(i)-cross conjugate of X(j) for these (i,j): {76, 1502}, {311, 76}, {850, 6331}, {1235, 264}, {8754, 14618}, {17864, 75}
X(18022) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14575}, {3, 560}, {6, 9247}, {19, 14585}, {31, 184}, {32, 48}, {63, 1501}, {69, 1917}, {77, 9448}, {163, 3049}, {212, 1397}, {217, 2148}, {222, 9447}, {228, 2206}, {248, 9417}, {255, 1974}, {293, 9418}, {304, 9233}, {577, 1973}, {603, 2175}, {656, 14574}, {669, 4575}, {810, 1576}, {906, 1919}, {922, 14908}, {1176, 1923}, {1331, 1980}, {1333, 2200}, {1395, 6056}, {1437, 1918}, {1755, 14600}, {1790, 2205}, {1924, 4558}, {1933, 17970}, {1964, 10547}, {2179, 14533}, {2196, 14599}, {2203, 4055}, {2207, 4100}, {2212, 7335}, {4592, 9426}, {7099, 14827}, {7193, 14598}
X(18022) = cevapoint of X(i) and X(j) for these (i,j): {2, 11442}, {4, 17907}, {69, 7763}, {76, 264}, {8754, 14618}
X(18022) = trilinear pole of line {2799, 3267}
X(18022) = pole wrt polar circle of trilinear polar of X(32) (line X(669)X(688))
X(18022) = perspector of ABC and orthoanticevian triangle of X(1502)
X(18022) = barycentric product X(i)*X(j) for these {i,j}: {4, 1502}, {19, 1928}, {75, 1969}, {76, 264}, {92, 561}, {276, 311}, {305, 2052}, {308, 1235}, {331, 3596}, {670, 14618}, {850, 6331}, {2501, 4609}, {3267, 6528}, {6063, 7017}, {6386, 17924}
X(18022) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 9247}, {2, 184}, {3, 14585}, {4, 32}, {5, 217}, {6, 14575}, {10, 2200}, {19, 560}, {25, 1501}, {27, 2206}, {33, 9447}, {69, 577}, {75, 48}, {76, 3}, {83, 10547}, {85, 603}, {92, 31}, {95, 14533}, {98, 14600}, {112, 14574}, {158, 1973}, {232, 9418}, {240, 9417}, {242, 14599}, {253, 14642}, {264, 6}, {273, 604}, {274, 1437}, {276, 54}, {278, 1397}, {281, 2175}, {286, 1333}, {290, 248}, {297, 237}, {304, 255}, {305, 394}, {306, 4055}, {308, 1176}, {310, 1790}, {311, 216}, {312, 212}, {313, 71}, {314, 2193}, {315, 10316}, {316, 10317}, {317, 571}, {318, 41}, {321, 228}, {324, 51}, {325, 3289}, {326, 4100}, {331, 56}, {334, 2196}, {339, 3269}, {340, 50}, {341, 1802}, {342, 2199}, {343, 418}, {345, 6056}, {348, 7335}, {349, 73}, {393, 1974}, {419, 14602}, {427, 3051}, {468, 14567}, {492, 8911}, {523, 3049}, {561, 63}, {607, 9448}, {626, 4173}, {648, 1576}, {668, 906}, {670, 4558}, {671, 14908}, {799, 4575}, {811, 163}, {850, 647}, {877, 14966}, {1088, 7099}, {1093, 2207}, {1235, 39}, {1236, 14961}, {1240, 2359}, {1441, 1409}, {1446, 1410}, {1502, 69}, {1577, 810}, {1784, 9406}, {1824, 2205}, {1826, 1918}, {1847, 1106}, {1861, 9454}, {1896, 2204}, {1916, 17970}, {1920, 3955}, {1921, 7193}, {1928, 304}, {1930, 4020}, {1969, 1}, {1973, 1917}, {1974, 9233}, {1978, 1331}, {1990, 9407}, {2052, 25}, {2489, 9426}, {2501, 669}, {2888, 8565}, {2967, 9419}, {2969, 1977}, {2970, 3124}, {2971, 9427}, {2973, 1015}, {2998, 15389}, {3260, 3284}, {3261, 1459}, {3266, 3292}, {3267, 520}, {3596, 219}, {3718, 2289}, {3926, 1092}, {4391, 1946}, {4572, 1813}, {4602, 4592}, {4609, 4563}, {5089, 9455}, {5117, 3117}, {5254, 682}, {5392, 2351}, {6063, 222}, {6143, 9697}, {6331, 110}, {6335, 692}, {6344, 11060}, {6384, 15373}, {6385, 1444}, {6386, 1332}, {6521, 1096}, {6528, 112}, {6530, 2211}, {6531, 14601}, {6591, 1980}, {7017, 55}, {7018, 7116}, {7020, 7118}, {7046, 14827}, {7101, 1253}, {7140, 7109}, {7141, 1500}, {7182, 7125}, {7649, 1919}, {7763, 1147}, {7769, 49}, {8024, 3917}, {8039, 4121}, {8754, 1084}, {8756, 9459}, {8795, 8882}, {8882, 14573}, {9291, 1970}, {13450, 3199}, {14208, 822}, {14249, 3172}, {14603, 12215}, {14615, 15905}, {14618, 512}, {15415, 6368}, {15466, 154}, {16081, 1976}, {16089, 1971}, {16230, 2491}, {17442, 1923}, {17907, 206}, {17980, 8789}, {17984, 1691}


X(18023) = ISOTOMIC CONJUGATE OF X(187)

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

X(18023) lies on these lines: {67, 316}, {76, 338}, {111, 308}, {264, 2970}, {290, 892}, {313, 1978}, {349, 4572}, {691, 2367}, {1502, 4609}, {2453, 5152}, {3978, 17948}, {5466, 14295}, {7771, 11643}, {9211, 9214}

X(18023) = isogonal conjugate of X(14567)
X(18023) = isotomic conjugate of X(187)
X(18023) = X(i)-cross conjugate of X(j) for these (i,j): {625, 2}, {3266, 76}, {14977, 892}
X(18023) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14567}, {6, 922}, {31, 187}, {32, 896}, {163, 351}, {468, 9247}, {524, 560}, {798, 5467}, {1501, 14210}, {1576, 2642}, {1917, 3266}, {1918, 16702}, {1924, 5468}, {1927, 5026}, {1973, 3292}, {2205, 6629}, {4760, 14598}, {5967, 9417}, {7181, 9447}, {9406, 9717}
X(18023) = cevapoint of X(i) and X(j) for these (i,j): {2, 316}, {76, 3266}, {115, 9134}
X(18023) = trilinear pole of line {76, 850}
X(18023) = barycentric product X(i)*X(j) for these {i,j}: {76, 671}, {111, 1502}, {305, 17983}, {561, 897}, {670, 5466}, {850, 892}, {923, 1928}, {4609, 9178}, {6331, 14977}
X(18023) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 922}, {2, 187}, {6, 14567}, {69, 3292}, {75, 896}, {76, 524}, {99, 5467}, {111, 32}, {264, 468}, {274, 16702}, {290, 5967}, {305, 6390}, {310, 6629}, {313, 4062}, {316, 6593}, {325, 9155}, {338, 1648}, {523, 351}, {561, 14210}, {670, 5468}, {671, 6}, {691, 1576}, {693, 14419}, {850, 690}, {892, 110}, {895, 184}, {897, 31}, {923, 560}, {1236, 5181}, {1494, 9717}, {1502, 3266}, {1577, 2642}, {1920, 7267}, {1921, 4760}, {2408, 8644}, {3260, 5642}, {3261, 4750}, {3266, 2482}, {3267, 14417}, {3596, 3712}, {3978, 5026}, {5380, 692}, {5466, 512}, {5547, 2175}, {5968, 237}, {6063, 7181}, {6331, 4235}, {6385, 16741}, {7316, 1397}, {8024, 7813}, {8430, 2491}, {8753, 1974}, {9154, 1976}, {9178, 669}, {9213, 14270}, {9214, 1495}, {10097, 3049}, {10415, 3455}, {14295, 11183}, {14364, 10417}, {14618, 14273}, {14908, 14575}, {14977, 647}, {15398, 14908}, {16092, 5191}, {17948, 2502}, {17983, 25}, {18007, 9171}


X(18024) = ISOTOMIC CONJUGATE OF X(237)

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

X(18024) lies on the hyperbola {{A,B,C,X(2),X(69)}} and these lines: {2, 6331}, {69, 290}, {95, 6394}, {98, 689}, {237, 17984}, {287, 3978}, {305, 4609}, {306, 1978}, {307, 4572}, {1976, 14382}

X(18024) = isogonal conjugate of X(9418)
X(18024) = isotomic conjugate of X(237)
X(18024) = polar conjugate of X(2211)
X(18024) = X(325)-cross conjugate of X(76)
X(18024) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9418}, {6, 9417}, {31, 237}, {32, 1755}, {48, 2211}, {163, 2491}, {232, 9247}, {240, 14575}, {325, 1917}, {511, 560}, {798, 14966}, {1501, 1959}, {1910, 9419}, {1924, 2421}, {1933, 14251}, {1973, 3289}, {2205, 17209}, {2206, 5360}
X(18024) = cevapoint of X(i) and X(j) for these (i,j): {2, 14957}, {76, 325}, {1502, 14603}
X(18024) = trilinear pole of line {76, 525}
X(18024) = barycentric product X(i)*X(j) for these {i,j}: {76, 290}, {98, 1502}, {305, 16081}, {336, 1969}, {561, 1821}, {1910, 1928}, {2395, 4609}
X(18024) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 9417}, {2, 237}, {4, 2211}, {6, 9418}, {69, 3289}, {75, 1755}, {76, 511}, {98, 32}, {99, 14966}, {248, 14575}, {264, 232}, {287, 184}, {290, 6}, {293, 9247}, {310, 17209}, {321, 5360}, {325, 11672}, {336, 48}, {511, 9419}, {523, 2491}, {561, 1959}, {670, 2421}, {850, 3569}, {879, 3049}, {1502, 325}, {1821, 31}, {1910, 560}, {1916, 14251}, {1969, 240}, {1976, 1501}, {2395, 669}, {2422, 9426}, {2715, 14574}, {2966, 1576}, {3266, 9155}, {3267, 684}, {3404, 1923}, {4609, 2396}, {5967, 14567}, {6331, 4230}, {6394, 577}, {6531, 1974}, {14265, 1692}, {14382, 1691}, {14601, 9233}, {14603, 5976}, {14618, 17994}, {15630, 9427}, {16081, 25}, {17974, 14585}


X(18025) = ISOTOMIC CONJUGATE OF X(516)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 + a*c^2 + b*c^2 - 2*c^3)*(a^3 + a*b^2 - 2*b^3 - a^2*c + b^2*c - a*c^2 + c^3) : :
Barycentrics    (csc A)/(a^2 - b^2 cos C - c^2 cos B) : :

X(18025) lies on the Steiner circumellipse and these lines: {7, 4081}, {8, 348}, {69, 144}, {75, 4569}, {85, 318}, {86, 648}, {99, 103}, {280, 6604}, {304, 341}, {319, 6606}, {337, 4562}, {350, 14727}, {658, 2968}, {666, 2338}, {903, 2400}, {911, 4586}, {1121, 3904}, {1222, 6613}, {2424, 3226}, {2966, 17731}, {4555, 15634}

X(18025) = isotomic conjugate of X(516)
X(18025) = X(i)-cross conjugate of X(j) for these (i,j): {516, 2}, {4872, 86}, {9436, 75}, {15634, 2400}
X(18025) = polar conjugate of X(1886)
X(18025) = X(314)-beth conjugate of X(4569)
X(18025) = X(43)-zayin conjugate of X(910)
X(18025) = X(i)-isoconjugate of X(j) for these (i,j): {6, 910}, {31, 516}, {48, 1886}, {55, 1456}, {213, 14953}, {513, 2426}, {667, 2398}, {676, 692}, {810, 4241}, {1333, 17747}, {1438, 9502}
X(18025) = cevapoint of X(i) and X(j) for these (i,j): {1, 7291}, {2, 516}, {8, 3912}, {103, 1815}, {2340, 3730}, {2400, 15634}
X(18025) = trilinear pole of line {2, 2400}
X(18025) = barycentric product X(i)*X(j) for these {i,j}: {76, 103}, {190, 2400}, {264, 1815}, {561, 911}, {677, 3261}, {1016, 15634}, {1978, 2424}, {2338, 6063}, {3263, 9503}
X(18025) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 910}, {2, 516}, {4, 1886}, {10, 17747}, {57, 1456}, {86, 14953}, {101, 2426}, {103, 6}, {190, 2398}, {514, 676}, {518, 9502}, {648, 4241}, {677, 101}, {911, 31}, {1815, 3}, {2338, 55}, {2398, 3234}, {2400, 514}, {2424, 649}, {9503, 105}, {15634, 1086}


X(18026) = ISOTOMIC CONJUGATE OF X(521)

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

X(18026) lies on the Steiner circumellipse and these lines: {4, 150}, {7, 264}, {30, 16090}, {34, 3226}, {65, 290}, {75, 342}, {76, 14257}, {85, 318}, {92, 1121}, {99, 108}, {101, 1981}, {107, 13395}, {190, 653}, {196, 7017}, {226, 1947}, {273, 903}, {278, 3227}, {286, 7282}, {324, 17483}, {329, 15466}, {348, 7952}, {458, 5228}, {527, 1948}, {648, 651}, {664, 1897}, {666, 1783}, {693, 934}, {1305, 13589}, {1415, 2966}, {1441, 1494}, {1785, 9436}, {1861, 10030}, {1880, 3228}, {2052, 5905}, {6180, 9308}

X(18026) = isogonal conjugate of X(1946)
X(18026) = isotomic conjugate of X(521)
X(18026) = polar conjugate of X(650)
X(18026) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 1946}, {43, 652}, {1754, 822}, {2947, 649}
X(18026) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {7045, 6527}, {7128, 20}
X(18026) = X(9394)-complementary conjugate of X(141)
X(18026) = X(811)-Ceva conjugate of X(664)
X(18026) = X(i)-cross conjugate of X(j) for these (i,j): {521, 2}, {651, 4554}, {653, 13149}, {693, 264}, {1897, 6335}, {3868, 4564}, {4391, 85}, {4566, 664}, {5905, 1275}, {12649, 1016}, {14544, 15455}, {17496, 276}, {17896, 75}, {17924, 331}
X(18026) = cevapoint of X(i) and X(j) for these (i,j): {2, 521}, {4, 17924}, {7, 693}, {108, 651}, {226, 522}, {318, 4391}, {342, 17896}, {442, 525}, {513, 3772}, {514, 1210}, {653, 1897}
X(18026) = crosspoint of X(i) and X(j) for these (i,j): {811, 6528}
X(18026) = trilinear pole of line {2, 92}
X(18026) = pole wrt polar circle of trilinear polar of X(650) (line X(926)X(2170))
X(18026) = Brianchon point (perspector) of inscribed parabola with focus X(108)
X(18026) = crossdifference of PU(101)
X(18026) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1946}, {3, 663}, {6, 652}, {21, 810}, {31, 521}, {32, 6332}, {41, 905}, {48, 650}, {55, 1459}, {63, 3063}, {71, 7252}, {77, 8641}, {78, 667}, {101, 7117}, {108, 2638}, {109, 3270}, {184, 522}, {212, 513}, {219, 649}, {222, 657}, {228, 3737}, {283, 512}, {284, 647}, {332, 669}, {333, 3049}, {345, 1919}, {520, 2299}, {577, 3064}, {603, 3900}, {607, 4091}, {656, 2194}, {661, 2193}, {692, 7004}, {798, 1812}, {822, 1172}, {884, 1818}, {906, 2170}, {1015, 4587}, {1021, 1409}, {1331, 3271}, {1333, 8611}, {1334, 7254}, {1364, 8750}, {1436, 10397}, {1437, 4041}, {1790, 3709}, {1802, 3669}, {1803, 10581}, {1807, 8648}, {1808, 4455}, {1813, 14936}, {1980, 3718}, {2159, 14395}, {2175, 4025}, {2188, 6129}, {2196, 4435}, {2200, 4560}, {2212, 4131}, {2289, 6591}, {2318, 3733}, {2327, 7180}, {2342, 8677}, {2489, 6514}, {2605, 8606}, {3248, 4571}, {3287, 7116}, {3937, 3939}, {4105, 7053}, {4130, 7099}, {4391, 9247}, {4516, 4575}, {5075, 17973}, {6056, 7649}, {9447, 15413}, {9456, 14418}, {14331, 14642}, {14432, 14908}
X(18026) = barycentric product X(i)*X(j) for these {i,j}: {4, 4554}, {7, 6335}, {8, 13149}, {19, 4572}, {34, 1978}, {65, 6331}, {75, 653}, {76, 108}, {85, 1897}, {92, 664}, {100, 331}, {107, 1231}, {109, 1969}, {162, 349}, {190, 273}, {225, 799}, {226, 811}, {264, 651}, {278, 668}, {281, 4569}, {286, 4552}, {307, 823}, {318, 658}, {608, 6386}, {646, 1119}, {648, 1441}, {670, 1880}, {934, 7017}, {1214, 6528}, {1783, 6063}, {1826, 4625}, {1847, 3699}, {1874, 4639}, {2052, 6516}, {3261, 7012}, {4623, 8736}, {4624, 5342}, {4626, 7101}, {4998, 17924}, {6517, 6521}, {7282, 15455}
X(18026) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 652}, {2, 521}, {4, 650}, {6, 1946}, {7, 905}, {10, 8611}, {19, 663}, {25, 3063}, {27, 3737}, {28, 7252}, {29, 1021}, {30, 14395}, {33, 657}, {34, 649}, {40, 10397}, {57, 1459}, {59, 906}, {65, 647}, {73, 822}, {75, 6332}, {77, 4091}, {85, 4025}, {92, 522}, {99, 1812}, {100, 219}, {101, 212}, {107, 1172}, {108, 6}, {109, 48}, {110, 2193}, {112, 2194}, {158, 3064}, {162, 284}, {190, 78}, {196, 6129}, {225, 661}, {226, 656}, {242, 4435}, {264, 4391}, {273, 514}, {278, 513}, {281, 3900}, {286, 4560}, {314, 15411}, {331, 693}, {342, 14837}, {348, 4131}, {349, 14208}, {388, 2522}, {461, 4827}, {513, 7117}, {514, 7004}, {519, 14418}, {527, 14414}, {607, 8641}, {643, 2327}, {644, 1260}, {645, 1792}, {646, 1265}, {648, 21}, {650, 3270}, {651, 3}, {652, 2638}, {653, 1}, {655, 1807}, {658, 77}, {662, 283}, {664, 63}, {668, 345}, {765, 4587}, {799, 332}, {811, 333}, {823, 29}, {905, 1364}, {906, 6056}, {927, 1814}, {934, 222}, {1014, 7254}, {1016, 4571}, {1018, 2318}, {1020, 73}, {1025, 1818}, {1118, 6591}, {1119, 3669}, {1214, 520}, {1231, 3265}, {1275, 6516}, {1295, 2431}, {1331, 2289}, {1332, 1259}, {1395, 1919}, {1396, 3733}, {1400, 810}, {1402, 3049}, {1414, 1790}, {1415, 184}, {1426, 7180}, {1441, 525}, {1446, 17094}, {1461, 603}, {1465, 8677}, {1633, 7124}, {1783, 55}, {1784, 14400}, {1813, 255}, {1824, 3709}, {1826, 4041}, {1827, 10581}, {1847, 3676}, {1848, 17420}, {1855, 6608}, {1870, 654}, {1875, 3310}, {1876, 665}, {1877, 1635}, {1880, 512}, {1895, 14331}, {1896, 17926}, {1897, 9}, {1947, 8062}, {1978, 3718}, {1981, 1936}, {2405, 6001}, {2501, 4516}, {2635, 2637}, {2639, 2636}, {2720, 14578}, {3064, 2310}, {3261, 17880}, {3596, 15416}, {3669, 3937}, {3676, 3942}, {3699, 3692}, {3732, 1040}, {3939, 1802}, {3952, 3694}, {4033, 3710}, {4077, 4466}, {4242, 2323}, {4391, 2968}, {4551, 71}, {4552, 72}, {4554, 69}, {4559, 228}, {4561, 3719}, {4564, 1331}, {4565, 1437}, {4566, 1214}, {4569, 348}, {4572, 304}, {4573, 1444}, {4584, 1808}, {4592, 6514}, {4605, 201}, {4617, 7053}, {4620, 4592}, {4625, 17206}, {4626, 7177}, {4998, 1332}, {5089, 926}, {5236, 2254}, {5307, 17418}, {5342, 4765}, {5379, 5546}, {6063, 15413}, {6198, 9404}, {6331, 314}, {6335, 8}, {6358, 4064}, {6516, 394}, {6517, 6507}, {6591, 3271}, {6614, 7099}, {6648, 1791}, {7009, 3287}, {7012, 101}, {7017, 4397}, {7045, 1813}, {7046, 4130}, {7079, 4105}, {7101, 4163}, {7115, 692}, {7128, 109}, {7199, 17219}, {7282, 14838}, {7649, 2170}, {7952, 14298}, {8736, 4705}, {8750, 41}, {8751, 884}, {8756, 4895}, {10404, 2523}, {13136, 1809}, {13138, 268}, {13149, 7}, {14257, 6588}, {14594, 5227}, {15352, 1896}, {15742, 644}, {17896, 16596}, {17905, 11934}, {17906, 3057}, {17923, 3738}, {17924, 11}


X(18027) = ISOTOMIC CONJUGATE OF X(577)

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

X(18027) lies on these lines: {2, 276}, {3, 9291}, {4, 290}, {5, 264}, {68, 317}, {76, 297}, {107, 2367}, {308, 393}, {327, 1235}, {349, 1969}, {685, 2909}, {1968, 10684}, {2207, 3114}, {3767, 16081}, {6331, 7763}

X(18027) = isogonal conjugate of X(14585)
X(18027) = isotomic conjugate of X(577)
X(18027) = X(i)-cross conjugate of X(j) for these (i,j): {324, 264}, {6334, 16077}, {14618, 6528}, {15526, 850}
X(18027) = cevapoint of X(i) and X(j) for these (i,j): {2, 317}, {264, 2052}, {850, 15526}
X(18027) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14585}, {3, 9247}, {25, 4100}, {31, 577}, {32, 255}, {41, 7335}, {48, 184}, {63, 14575}, {217, 2169}, {326, 1501}, {394, 560}, {418, 2148}, {563, 2351}, {604, 6056}, {822, 1576}, {1092, 1973}, {1333, 4055}, {1397, 2289}, {1437, 2200}, {1804, 9447}, {1917, 3926}, {1974, 6507}, {2175, 7125}, {2206, 3990}, {3049, 4575}, {4020, 10547}, {7183, 9448}, {9417, 17974}
X(18027) = trilinear pole of line {850, 6368}
X(18027) = barycentric square of X(264)
X(18027) = {X(9291),X(16089)}-harmonic conjugate of X(3)
X(18027) = barycentric product X(i)*X(j) for these {i,j}: {76, 2052}, {92, 1969}, {158, 561}, {264, 264}, {276, 324}, {304, 6521}, {305, 1093}, {311, 8795}, {331, 7017}, {393, 1502}, {850, 6528}, {1096, 1928}, {3267, 15352}, {6331, 14618}, {15415, 16813}
X(18027) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 577}, {4, 184}, {5, 418}, {6, 14585}, {7, 7335}, {8, 6056}, {10, 4055}, {19, 9247}, {25, 14575}, {53, 217}, {63, 4100}, {69, 1092}, {75, 255}, {76, 394}, {85, 7125}, {92, 48}, {107, 1576}, {158, 31}, {253, 14379}, {264, 3}, {273, 603}, {275, 14533}, {276, 97}, {286, 1437}, {290, 17974}, {297, 3289}, {304, 6507}, {305, 3964}, {311, 5562}, {312, 2289}, {313, 3682}, {317, 1147}, {318, 212}, {321, 3990}, {324, 216}, {331, 222}, {338, 3269}, {339, 2972}, {342, 7114}, {393, 32}, {459, 14642}, {561, 326}, {811, 4575}, {823, 163}, {847, 2351}, {850, 520}, {1093, 25}, {1096, 560}, {1118, 1397}, {1235, 3917}, {1367, 1363}, {1502, 3926}, {1577, 822}, {1585, 8911}, {1748, 563}, {1826, 2200}, {1847, 7099}, {1857, 2175}, {1896, 2194}, {1969, 63}, {2052, 6}, {2207, 1501}, {2501, 3049}, {2973, 3937}, {3261, 4091}, {3596, 1259}, {4572, 6517}, {6059, 9448}, {6063, 1804}, {6331, 4558}, {6335, 906}, {6520, 1973}, {6521, 19}, {6524, 1974}, {6528, 110}, {6530, 237}, {6531, 14600}, {6747, 6752}, {7017, 219}, {7020, 2188}, {7068, 7065}, {7101, 1802}, {7141, 3690}, {8747, 2206}, {8794, 8882}, {8795, 54}, {11547, 571}, {13450, 51}, {14165, 50}, {14249, 154}, {14618, 647}, {15352, 112}, {15466, 15905}, {16081, 248}, {16082, 14578}, {16813, 14586}, {17858, 820}, {17907, 10316}, {17983, 14908}


X(18028) = ISOGONAL CONJUGATE OF X(17956)

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

X(18028) lies on the cubic K221 and this line: {1, 561}

X(18028) = isogonal conjugate of X(17956)
X(18028) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17956}, {2, 17965}
X(18028) = barycentric product X(i)*X(j) for these {i,j}: {662, 18008}, {799, 17995}
X(18028) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17956}, {31, 17965}, {17995, 661}, {18008, 1577}


X(18029) = X(185)X(9033)∩X(5562)X(7723)

Trilinears    cos(A)*(2*(56*cos(A)+19*cos(3* A)+cos(5*A))*cos(B-C)-24*(cos( 2*A)+1)*cos(2*(B-C))+10*cos(A) *cos(3*(B-C))-cos(4*(B-C))-12* cos(4*A)-56*cos(2*A)-45) : :
Barycentrics    SA*(6*S^4+(216*R^4-36*R^2*(SA+ SW)+5*SA^2-2*SB*SC-SW^2)*S^2+( SB+SC)*(648*R^6-9*R^4*(9*SA+ 41*SW)+2*R^2*(9*SA^2-9*SB*SC+ 28*SW^2)-SW^2*(SB+SC))) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27574.

X(18029) lies on these lines: {185, 9033}, {5562, 7723}


X(18030) = (name pending)

Barycentrics    SA*(6*S^4+(144*R^4-8*R^2*(3* SA+4*SW)+5*SA^2-2*SB*SC-SW^2)* S^2-(SB+SC)*(9*R^4*(3*SA+7*SW) -2*R^2*(3*SA^2-3*SB*SC+10*SW^ 2)+SW^2*(SB+SC))) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27574.

X(18030) lies on this line: {185, 10111}


X(18031) = ISOTOMIC CONJUGATE OF X(672)

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

X(18031) lies on the cubic K986 and these lines:
{2, 4554}, {8, 76}, {29, 811}, {75, 4712}, {85, 17451}, {105, 789}, {264, 1863}, {310, 333}, {312, 561}, {349, 6559}, {350, 14942}, {672, 10030}, {927, 1311}, {1438, 4593}, {1921, 3263}, {4572, 4858}

X(18031) = isogonal conjugate of X(9454)
X(18031) = isotomic conjugate of X(672)
X(18031) = polar conjugate of X(2356)
X(18031) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 9454}, {1740, 672}
X(18031) = X(18031) = X(i)-cross conjugate of X(j) for these (i,j): {350, 310}, {3912, 75}
X(18031) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9454}, {2, 9455}, {6, 2223}, {31, 672}, {32, 518}, {41, 1458}, {48, 2356}, {184, 5089}, {213, 3286}, {241, 2175}, {560, 3912}, {604, 2340}, {651, 8638}, {665, 692}, {667, 2284}, {926, 1415}, {1026, 1919}, {1397, 3693}, {1501, 3263}, {1818, 1973}, {1861, 9247}, {1922, 8299}, {2206, 3930}, {2210, 3252}, {2283, 3063}, {2428, 8642}, {3049, 4238}, {4447, 7104}, {9436, 9447}, {14598, 17755}
X(18031) = cevapoint of X(i) and X(j) for these (i,j): {75, 3912}, {76, 1921}
X(18031) = trilinear pole of line {75, 522}
X(18031) = barycentric product X(i)*X(j) for these {i,j}: {75, 2481}, {76, 673}, {105, 561}, {310, 13576}, {666, 3261}, {885, 4572}, {1027, 6386}, {1438, 1502}, {1814, 1969}, {6063, 14942}
X(18031) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2223}, {2, 672}, {4, 2356}, {6, 9454}, {7, 1458}, {8, 2340}, {31, 9455}, {69, 1818}, {75, 518}, {76, 3912}, {85, 241}, {86, 3286}, {92, 5089}, {105, 31}, {190, 2284}, {264, 1861}, {273, 1876}, {294, 41}, {312, 3693}, {313, 3932}, {321, 3930}, {331, 5236}, {335, 3252}, {350, 8299}, {514, 665}, {522, 926}, {561, 3263}, {663, 8638}, {664, 2283}, {666, 101}, {668, 1026}, {673, 6}, {693, 2254}, {811, 4238}, {850, 4088}, {885, 663}, {927, 109}, {1024, 3063}, {1027, 667}, {1111, 3675}, {1269, 4966}, {1416, 1397}, {1438, 32}, {1462, 604}, {1814, 48}, {1909, 4447}, {1921, 17755}, {2195, 2175}, {2481, 1}, {3261, 918}, {3263, 4712}, {3596, 3717}, {3912, 6184}, {4358, 14439}, {4554, 1025}, {4572, 883}, {4858, 17435}, {5377, 1110}, {5936, 14626}, {6063, 9436}, {6185, 1438}, {6559, 220}, {6654, 1914}, {8751, 1973}, {9436, 1362}, {9503, 911}, {10099, 810}, {13576, 42}, {14942, 55}


X(18032) = ISOTOMIC CONJUGATE OF X(1757)

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

X(18032) lies on the cubic K986 and these lines:
{75, 1654}, {76, 4485}, {85, 11375}, {274, 1111}, {286, 2905}, {334, 3948}, {668, 4647}, {767, 2702}, {870, 17962}, {1218, 2054}, {3766, 18014}, {4623, 16709}

X(18032) = isotomic conjugate of X(1757)
X(18032) = X(i)-cross conjugate of X(j) for these (i,j): {350, 75}, {11599, 6650}
X(18032) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17735}, {25, 17976}, {31, 1757}, {32, 6542}, {42, 1326}, {101, 5029}, {110, 17990}, {184, 17927}, {213, 1931}, {423, 2200}, {512, 17943}, {669, 17934}, {692, 9508}, {1576, 18004}, {1911, 8298}, {1918, 17731}, {1922, 6651}, {2206, 6541}
X(18032) = cevapoint of X(i) and X(j) for these (i,j): {1111, 3766}, {3948, 4647}
X(18032) = trilinear pole of line {693, 4359}
X(18032) = barycentric product X(i)*X(j) for these {i,j}: {75, 6650}, {76, 1929}, {274, 11599}, {304, 17982}, {310, 9278}, {561, 17962}, {799, 18014}, {873, 6543}, {1577, 17930}, {1921, 9505}, {1969, 17972}, {2054, 6385}, {4602, 18001}
X(18032) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17735}, {2, 1757}, {63, 17976}, {75, 6542}, {81, 1326}, {86, 1931}, {92, 17927}, {239, 8298}, {274, 17731}, {286, 423}, {321, 6541}, {350, 6651}, {513, 5029}, {514, 9508}, {661, 17990}, {662, 17943}, {693, 2786}, {799, 17934}, {1577, 18004}, {1929, 6}, {2054, 213}, {2702, 692}, {6543, 756}, {6650, 1}, {9278, 42}, {9505, 292}, {9506, 1911}, {11599, 37}, {17930, 662}, {17940, 163}, {17962, 31}, {17972, 48}, {17982, 19}, {18001, 798}, {18014, 661}


X(18033) = ISOTOMIC CONJUGATE OF X(7077)

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

X(18033) lies on the cubics K865 and K986 and these lines:
{7, 871}, {75, 4073}, {76, 85}, {241, 4554}, {310, 7249}, {561, 3212}, {982, 3673}, {1088, 6384}, {1443, 4625}, {1921, 3975}, {1926, 4087}, {4059, 18021}

X(18033) = isotomic conjugate of X(7077)
X(18033) = X(i)-beth conjugate of X(j) for these (i,j): {799, 241}, {3766, 3676}, {4623, 1443}
X(18033) = X(350)-cross conjugate of X(1921)
X(18033) = X(i)-isoconjugate of X(j) for these (i,j): {8, 14598}, {9, 1922}, {31, 7077}, {32, 4876}, {41, 292}, {55, 1911}, {213, 2311}, {291, 2175}, {295, 2212}, {334, 9448}, {335, 9447}, {560, 4518}, {607, 2196}, {813, 3063}, {875, 3939}, {1927, 7081}, {1967, 2330}, {2329, 9468}, {8789, 17787}
X(18033) = X(85)-Hirst inverse of X(6063)
X(18033) = cevapoint of X(350) and X(10030)
X(18033) = X(18033) = barycentric product X(i)*X(j) for these {i,j}: {7, 1921}, {75, 10030}, {76, 1447}, {85, 350}, {239, 6063}, {279, 4087}, {310, 16609}, {561, 1429}, {670, 7212}, {812, 4572}, {1088, 3975}, {1284, 6385}, {1428, 1502}, {1431, 14603}, {1432, 1926}, {3766, 4554}, {3978, 7249}, {7205, 17493}
X(18033) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7077}, {7, 292}, {56, 1922}, {57, 1911}, {75, 4876}, {76, 4518}, {77, 2196}, {85, 291}, {86, 2311}, {238, 41}, {239, 55}, {242, 607}, {348, 295}, {350, 9}, {385, 2330}, {604, 14598}, {659, 3063}, {664, 813}, {740, 1334}, {812, 663}, {874, 644}, {1284, 213}, {1428, 32}, {1429, 31}, {1431, 9468}, {1432, 1967}, {1434, 741}, {1447, 6}, {1874, 2333}, {1914, 2175}, {1921, 8}, {1926, 17787}, {1966, 2329}, {2201, 2212}, {2210, 9447}, {3570, 3939}, {3669, 875}, {3676, 3572}, {3684, 1253}, {3685, 220}, {3716, 657}, {3766, 650}, {3797, 4517}, {3948, 210}, {3975, 200}, {3978, 7081}, {4010, 3709}, {4037, 7064}, {4087, 346}, {4124, 14936}, {4148, 4105}, {4435, 8641}, {4495, 4390}, {4554, 660}, {4572, 4562}, {4625, 4584}, {4839, 8653}, {6063, 335}, {6654, 2195}, {7179, 3862}, {7212, 512}, {7235, 1500}, {7249, 694}, {9436, 3252}, {10030, 1}, {12835, 14599}, {14024, 2332}, {14295, 4140}, {14296, 3287}, {14599, 9448}, {16591, 5360}, {16609, 42}, {17206, 1808}, {17755, 2340}


X(18034) = ISOTOMIC CONJUGATE OF X(9472)

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

X(18034) lies on the cubic K986 and these lines:
{334, 3948}, {335, 350}, {1921, 3263}

X(18034) = isotomic conjugate of X(9472)
X(18034) = X(76)-Ceva conjugate of X(335)
X(18034) = X(i)-isoconjugate of X(j) for these (i,j): {31, 9472}, {2113, 2210}
X(18034) = X(334)-Hirst inverse of X(17789)
X(18034) = barycentric product X(i)*X(j) for these {i,j}: {76, 9470}, {334, 17738}
X(18034) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9472}, {335, 2113}, {2112, 2210}, {8301, 1914}, {9470, 6}, {17738, 238}


X(18035) = ISOTOMIC CONJUGATE OF X(9506)

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

X(18035) lies on the cubic K986 and these lines:
{76, 4485}, {310, 321}, {350, 740}, {874, 1281}, {4505, 4518}

X(18035) = isotomic conjugate of X(9506)
X(18035) = X(76)-Ceva conjugate of X(1921)
X(18035) = X(i)-isoconjugate of X(j) for these (i,j): {31, 9506}, {32, 9505}, {875, 2702}, {1911, 17962}, {1922, 1929}, {6650, 14598}
X(18035) = barycentric product X(i)*X(j) for these {i,j}: {76, 6651}, {561, 8298}, {1921, 6542}
X(18035) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9506}, {75, 9505}, {239, 17962}, {350, 1929}, {740, 2054}, {1757, 1911}, {1921, 6650}, {2786, 3572}, {3570, 2702}, {3948, 9278}, {4010, 18001}, {6542, 292}, {6651, 6}, {8298, 31}, {8843, 7121}, {9508, 875}, {17731, 741}, {17735, 1922}


X(18036) = ISOTOMIC CONJUGATE OF X(17798)

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

X(18036) lies on the cubic K986 and these lines:
{76, 3496}, {314, 7261}

X(18036) = isotomic conjugate of X(17798)
X(18036) = X(350)-cross conjugate of X(76)
X(18036) = X(i)-isoconjugate of X(j) for these (i,j): {31, 17798}, {32, 3509}, {560, 4645}, {1281, 14598}, {1501, 17789}, {2175, 5018}
X(18036) = barycentric product X(i)*X(j) for these {i,j}: {76, 7261}, {561, 3512}, {1502, 8852}
X(18036) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17798}, {75, 3509}, {76, 4645}, {85, 5018}, {313, 4071}, {561, 17789}, {1269, 4987}, {1921, 1281}, {3261, 4458}, {3512, 31}, {7061, 172}, {7261, 6}, {7281, 41}, {8852, 32}


X(18037) = X(76)-CEVA CONJUGATE OF X(350)

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

X(18037) lies on the cubics K356 and K986 and these lines:
{69, 17788}, {71, 190}, {75, 1654}, {76, 3496}, {239, 732}, {257, 335}, {310, 333}, {321, 6653}, {350, 385}, {1086, 15985}, {1921, 3975}, {1966, 3948}, {4645, 17789}, {16696, 16706}

X(18037) = X(3975)-beth conjugate of X(6651)
X(18037) = X(76)-Ceva conjugate of X(350)
X(18037) = X(i)-isoconjugate of X(j) for these (i,j): {292, 8852}, {1911, 3512}, {1922, 7261}, {7061, 9468}
X(18037) = X(i)-Hirst inverse of X(j) for these (i,j): {1921, 3978}, {4645, 17789}
X(18037) = crosspoint of X(3509) and X(8868)
X(18037) = X(18037) = crosssum of X(3512) and X(8875)
X(18037) = barycentric product X(i)*X(j) for these {i,j}: {75, 1281}, {239, 17789}, {350, 4645}, {874, 4458}, {1921, 3509}, {4087, 5018}
X(18037) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 8852}, {239, 3512}, {350, 7261}, {1281, 1}, {1966, 7061}, {3509, 292}, {3685, 7281}, {4458, 876}, {4645, 291}, {17789, 335}, {17798, 1911}
X(18037) = {X(17738),X(17739)}-harmonic conjugate of X(3512)


X(18038) = X(6)-CEVA CONJUGATE OF X(2210)

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

X(18038) lies on the cubic K987 and these lines:
{6, 8852}, {31, 3121}, {42, 1976}, {58, 163}, {239, 16985}, {291, 825}, {1428, 1691}, {1580, 8847}, {1922, 1967}, {2210, 14602}, {4027, 16364}, {8300, 8853}

X(18038) = X(6)-Ceva conjugate of X(2210)
X(18038) = X(i)-isoconjugate of X(j) for these (i,j): {334, 3512}, {335, 7261}, {1916, 7061}
X(18038) = X(1691)-Hirst inverse of X(1914)
X(18038) = crosspoint of X(6) and X(17798)
X(18038) = crosssum of X(2) and X(7261)
X(18038) = barycentric product X(i)*X(j) for these {i,j}: {31, 1281}, {238, 17798}, {1914, 3509}, {2210, 4645}, {14599, 17789}
X(18038) = barycentric quotient X(i)/X(j) for these {i,j}: {1281, 561}, {1933, 7061}, {2210, 7261}, {14599, 3512}, {17798, 334}


X(18039) = X(4)X(526)∩X(5)X(523)

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

At the suggestion of Seiichi Kirikami, the line H through X(5) perpendicular to the Euler line of a triangle ABC is here named the Hatzipolakis axis of ABC. In the plane of a triangle ABDC, let
L = Euler line of ABC;
H = Hatzipolakis axis of ABC;
D = L∩BC, and define E and F cyclically;
LA = L-of-AEF, and define LB and LC cyclically;
MA = H-of-AEF, and define MB and MC cyclically.
The lines MA, MB, MC concur in X(18039).

See Seiichi Kirikami and Angel Montesdeoca, Hyacinthos 27589.

The Hatzipolakis axis H is parallel to orthic axis and the De Londchamps axis; they meet in X(523). An equation for H, in barycentric coordinates, follows:

h(a,b,c) x + h(b,c,a) y + h(c,a,b) = 0, where h(a,b,c) = a2(b2c2 - (a2 - b2 - c2)2 .

H passes through X(i) for these i: 5, 523, 6757, 8151, 10287, 10288, 10412, 14254, 14356, 14566, 14592, 14670, 15475, and X(18114) - X(18122). (Peter Moses, April 27, 2018)

The isogonal conjugate of H is the circumconic with center X(11597), perspector X(50). An equation for this conic follows:

g(a,b,c) y z + g(b,c,a) z x + g(c,a,b) x y = 0, where g(a,b,c) = a4(b2c2 - (a2 - b2 - c2)2.

The conic passes through X(i) for these i: 54, 110, 10411, 14355, 14385, 14591, 17104. (Peter Moses, April 27, 2018)

X(18039) lies on these lines: {3,16171}, {4,526}, {5,523}, {1510,3521}, {5448,9033}


X(18040) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18040) lies on these lines: {1, 18044}, {10, 4022}, {75, 141}, {76, 4043}, {142, 3264}, {308, 18053}, {312, 1230}, {313, 3912}, {314, 17295}, {320, 17787}, {321, 17229}, {341, 495}, {350, 17315}, {561, 18045}, {668, 17277}, {1269, 2321}, {1909, 17289}, {3596, 17234}, {3759, 3780}, {3761, 17286}, {3765, 17279}, {3770, 17280}, {3836, 4710}, {3948, 17243}, {3975, 17263}, {4377, 17231}, {4494, 17298}, {4687, 6376}, {16709, 17303}, {17300, 17790}, {17788, 17791}, {18041, 18055}, {18042, 18047}, {18057, 18059}, {18067, 18069}


X(18041) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18041) lies on these lines: {1, 82}, {2, 17443}, {19, 662}, {48, 16568}, {63, 17471}, {75, 1953}, {192, 17444}, {561, 18051}, {1740, 2643}, {1964, 17472}, {2170, 3759}, {2171, 3758}, {2234, 17891}, {2294, 17394}, {3061, 17289}, {4053, 17363}, {4664, 17452}, {4687, 17451}, {7146, 16706}, {7202, 17364}, {18040, 18055}, {18044, 18061}, {18058, 18060}


X(18042) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18042) lies on these lines: {1, 82}, {38, 1933}, {41, 3759}, {48, 75}, {63, 2148}, {101, 17277}, {190, 572}, {192, 2278}, {239, 2174}, {284, 4360}, {584, 4393}, {604, 3758}, {894, 7113}, {897, 17891}, {922, 1582}, {1429, 16706}, {1580, 1964}, {1918, 11364}, {1953, 16568}, {1959, 17438}, {2173, 17868}, {2267, 17336}, {2268, 4664}, {2302, 11683}, {2304, 17144}, {2329, 17289}, {3204, 17349}, {3573, 8053}, {4268, 17350}, {4287, 17262}, {4687, 9310}, {8772, 17446}, {12195, 18082}, {16788, 17381}, {18040, 18047}, {18053, 18062}, {18064, 18079}


X(18043) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18043) lies on these lines: {1, 18045}, {69, 674}, {75, 7243}, {673, 2319}, {3262, 7788}, {4417, 4766}, {16284, 17294}, {18044, 18056}


X(18044) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18044) lies on these lines: {1, 18040}, {2, 313}, {75, 4494}, {76, 17289}, {190, 1423}, {273, 6335}, {308, 561}, {312, 17285}, {314, 17228}, {321, 17293}, {344, 349}, {350, 17233}, {646, 3644}, {668, 3759}, {1269, 2345}, {1909, 17381}, {3264, 4000}, {3589, 3765}, {3596, 16706}, {3662, 17790}, {3760, 4043}, {3770, 17368}, {3875, 4033}, {3912, 4150}, {3948, 17279}, {3963, 4657}, {3975, 17352}, {4110, 17160}, {4358, 17267}, {4360, 17786}, {4377, 17384}, {4389, 17787}, {6376, 17277}, {6381, 17353}, {18041, 18061}, {18043, 18056}, {18045, 18078}


X(18045) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18045) lies on these lines: {1, 18043}, {75, 3703}, {561, 18040}, {1233, 5249}, {1760, 18046}, {6063, 17234}, {18044, 18078}, {18057, 18066}


X(18046) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18046) lies on these lines: {1, 18040}, {2, 37}, {76, 17381}, {141, 17202}, {313, 17023}, {314, 17307}, {872, 17793}, {1269, 5750}, {1760, 18045}, {3264, 3946}, {3589, 3948}, {3596, 17380}, {3759, 6376}, {3963, 17045}, {4033, 4360}, {7377, 17234}, {8060, 18070}, {16709, 17398}, {17393, 17786}, {18057, 18058}


X(18047) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18047) lies on these lines: {1, 83}, {2, 9259}, {6, 9263}, {8, 8301}, {48, 17786}, {75, 4390}, {99, 813}, {100, 932}, {101, 668}, {172, 17752}, {190, 644}, {344, 3476}, {660, 3903}, {662, 4033}, {718, 8625}, {815, 835}, {894, 7200}, {1237, 14382}, {1909, 2329}, {1914, 10027}, {1958, 4110}, {1975, 4513}, {2275, 17743}, {2295, 6645}, {3759, 9457}, {3807, 4561}, {4128, 4154}, {4164, 4579}, {4551, 8707}, {4554, 4621}, {4919, 17738}, {5773, 14829}, {6224, 17233}, {6376, 9310}, {16720, 17741}, {16916, 16969}, {18040, 18042}


X(18048) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18048) lies on these lines: {2, 6}, {75, 7225}, {100, 17142}, {1078, 16574}, {1760, 18055}, {18040, 18042}


X(18049) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18049) lies on these lines: {1, 82}, {92, 304}, {662, 16545}, {2172, 16568}


X(18050) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18050) lies on these lines: {1, 18091}, {39, 75}, {257, 312}, {321, 1107}, {350, 17489}, {561, 18055}, {984, 3159}, {1237, 17451}, {4613, 16681}, {17144, 17475}


X(18051) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18051) lies on these lines: {1, 75}, {561, 18041}, {1760, 18062}, {4118, 18058}, {18060, 18069}


X(18052) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18052) lies on these lines: {1, 18057}, {2, 16707}, {75, 15523}, {76, 85}, {141, 16739}, {305, 17234}, {561, 18040}, {3720, 17149}, {18059, 18066}, {18065, 18078}


X(18053) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18053) lies on these lines: {1, 18058}, {75, 1581}, {82, 18079}, {308, 18040}, {561, 18041}, {18042, 18062}


X(18054) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18054) lies on these lines: {1, 18066}, {561, 18040}


X(18055) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18055) lies on these lines: {1, 83}, {2, 3721}, {169, 3570}, {312, 1237}, {335, 2275}, {344, 3485}, {561, 18050}, {1078, 1759}, {1760, 18048}, {1909, 3061}, {1953, 17786}, {2140, 4568}, {2243, 7793}, {2896, 4799}, {3096, 17211}, {3662, 16720}, {3670, 7786}, {3702, 17233}, {3876, 17277}, {3930, 17144}, {3954, 17030}, {4110, 17868}, {4561, 17682}, {4595, 14923}, {4950, 7785}, {6376, 17451}, {18040, 18041}


X(18056) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18056) lies on these lines: {1, 561}, {38, 3403}, {63, 1966}, {75, 16750}, {612, 1920}, {614, 1921}, {799, 1707}, {811, 1096}, {982, 4495}, {984, 7244}, {1740, 1926}, {1760, 18058}, {3305, 6376}, {3306, 6384}, {4011, 6381}, {18043, 18044}


X(18057) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18057) lies on these lines: {1, 18052}, {75, 4972}, {257, 312}, {305, 4429}, {319, 350}, {799, 6043}, {17027, 17149}, {18040, 18059}, {18043, 18044}, {18045, 18066}, {18046, 18058}


X(18058) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18058) lies on these lines: {1, 18053}, {38, 75}, {82, 18064}, {560, 18062}, {1760, 18056}, {4118, 18051}, {6376, 17371}, {6384, 17370}, {18041, 18060}, {18046, 18057}


X(18059) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18059) lies on these lines: {1, 561}, {2, 3121}, {42, 1920}, {75, 3873}, {81, 1966}, {310, 740}, {321, 1909}, {334, 4972}, {668, 756}, {799, 846}, {874, 4418}, {1215, 1978}, {1621, 1965}, {1921, 3720}, {2667, 6385}, {3770, 4037}, {3936, 7018}, {4038, 4495}, {4649, 7244}, {4850, 6384}, {18040, 18057}, {18052, 18066}


X(18060) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18060) lies on these lines: {1, 18062}, {75, 1581}, {18041, 18058}, {18051, 18069}


X(18061) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18061) lies on these lines: {1, 83}, {2, 3125}, {75, 4568}, {76, 3061}, {99, 17738}, {257, 3934}, {335, 1015}, {344, 5603}, {668, 2170}, {673, 4561}, {986, 7786}, {1016, 4919}, {1078, 3496}, {2087, 9263}, {2802, 4595}, {3570, 5540}, {3807, 4986}, {3944, 7790}, {4518, 14839}, {10176, 17277}, {18041, 18044}


X(18062) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18062) lies on these lines: {1, 18060}, {75, 2640}, {304, 16563}, {560, 18058}, {662, 799}, {1760, 18051}, {4593, 18063}, {18042, 18053}


X(18063) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(125), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18063) lies on these lines: {662, 811}, {4593, 18062}


X(18064) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18064) lies on these lines: {1, 561}, {31, 799}, {38, 1966}, {82, 18058}, {92, 811}, {310, 5263}, {748, 6376}, {750, 6384}, {874, 17147}, {1920, 3920}, {1921, 7191}, {1926, 1964}, {1965, 17469}, {3891, 6382}, {4495, 17598}, {18042, 18079}


X(18065) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18065) lies on these lines: {1, 18040}, {76, 17286}, {313, 17284}, {344, 6381}, {1826, 3912}, {3264, 4859}, {3596, 17282}, {3662, 4494}, {3760, 17233}, {3761, 17289}, {3763, 4377}, {3875, 17786}, {3963, 17306}, {4033, 17151}, {10447, 17228}, {17274, 17787}, {17298, 17790}, {18052, 18078}, {18067, 18068}


X(18066) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18066) lies on these lines: {1, 18054}, {2, 18037}, {76, 3120}, {149, 350}, {561, 18067}, {3952, 6376}, {18045, 18057}, {18052, 18059}


X(18067) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18067) lies on these lines: {1, 18052}, {76, 2887}, {226, 6381}, {305, 3836}, {350, 4865}, {561, 18066}, {1215, 6376}, {18040, 18069}, {18065, 18068}


X(18068) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18068) lies on these lines: {1, 561}, {9, 7244}, {57, 4495}, {92, 14210}, {799, 16570}, {1707, 1966}, {1920, 5268}, {1921, 5272}, {1926, 16571}, {3403, 17149}, {18065, 18067}


X(18069) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18069) lies on these lines: {1, 18053}, {561, 4118}, {17149, 17445}, {18040, 18067}, {18051, 18060}


X(18070) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18070) lies on these lines: {75, 8061}, {82, 2618}, {308, 3572}, {523, 3963}, {661, 786}, {798, 812}, {1086, 4374}, {2084, 18080}, {4010, 4036}, {4079, 4129}, {4580, 7212}, {8060, 18046}, {18071, 18072}


X(18071) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(513), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18071) lies on these lines: {2, 650}, {75, 16892}, {514, 17789}, {2526, 4397}, {4106, 4391}, {18070, 18072}, {18077, 18081}


X(18072) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(514), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18072) lies on these lines: {522, 4389}, {693, 4036}, {2517, 4828}, {3261, 3596}, {18070, 18071}


X(18073) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18073) lies on these lines: {1, 18040}, {3264, 17067}, {4033, 17160}, {18070, 18071}


X(18074) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(522), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18074) lies on this line: {18070, 18071}


X(18075) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18075) lies on these lines: {1, 561}, {799, 896}, {1920, 5297}, {1921, 7292}, {1926, 2234}, {18070, 18071}


X(18076) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(525), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18076) lies on this line: {18070, 18071}


X(18077) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18077) lies on these lines: {75, 826}, {514, 1921}, {814, 7255}, {2533, 7192}, {4560, 16705}, {18071, 18081}


X(18078) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(391), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18078) lies on these lines: {1, 561}, {1920, 17022}, {1921, 2999}, {1966, 4512}, {4495, 17594}, {18044, 18045}, {18052, 18065}


X(18079) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(626), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18079) lies on these lines: {1, 75}, {82, 18053}, {560, 18058}, {18042, 18064}


X(18080) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18080) lies on these lines: {1, 10566}, {192, 513}, {514, 4079}, {786, 7199}, {2084, 18070}


X(18081) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18081) lies on these lines: {75, 2530}, {76, 514}, {693, 2533}, {812, 1019}, {2084, 18070}, {3766, 4992}, {4398, 4406}, {18071, 18077}


X(18082) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(2), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18082) lies on these lines: {1, 18040}, {2, 16684}, {10, 82}, {12, 1284}, {42, 308}, {86, 334}, {141, 17153}, {190, 256}, {513, 894}, {528, 18101}, {560, 10791}, {594, 2238}, {740, 872}, {827, 2372}, {1010, 1224}, {1176, 15232}, {1826, 2201}, {4366, 18092}, {4972, 18090}, {5046, 17500}, {5846, 17751}, {9903, 16556}, {12195, 18042}, {15523, 17285}, {16890, 18100}, {17045, 17724}, {18004, 18010}, {18083, 18084}


X(18083) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18083) lies on these lines: {2, 18095}, {7363, 18097}, {16889, 18090}, {18082, 18084}


X(18084) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18084) lies on these lines: {1176, 15320}, {5244, 6354}, {18082, 18083}, {18088, 18089}


X(18085) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18085) lies on this line: {18082, 18083}


X(18086) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18086) lies on these lines: {83, 226}, {1334, 3975}, {5046, 17500}, {17192, 17681}


X(18087) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

(18087) lies on these lines: {2, 16684}, {83, 226}, {673, 4599}, {1176, 15320}, {1233, 1475}, {3925, 17672}


X(18088) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18088) lies on these lines: {2, 16683}, {83, 13576}, {3112, 4388}, {3434, 18098}, {5046, 17500}, {18084, 18089}


X(18089) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18089) lies on these lines: {2, 16684}, {81, 6385}, {83, 213}, {3891, 7770}, {4972, 6656}, {18084, 18088}, {18096, 18100}


X(18090) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18090) lies on these lines: {4972, 18082}, {16889, 18083}, {16890, 18104}


X(18091) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18091) lies on these lines: {1, 18050}, {2, 16683}, {10, 82}, {384, 16689}, {904, 7018}, {2176, 18098}, {8299, 18101}, {16918, 18092}, {17685, 18100}


X(18092) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18092) lies on these lines: {2, 3613}, {6, 76}, {183, 251}, {297, 10550}, {1799, 15271}, {4366, 18082}, {10549, 17907}, {16918, 18091}


X(18093) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18093) lies on these lines: {2, 18103}, {31, 83}, {594, 2238}, {16889, 18083}


X(18094) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18094) lies on these lines: {2, 16684}, {308, 16606}, {2295, 3589}, {16819, 17385}, {16889, 16890}, {18100, 18103}


X(18095) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18095) lies on these lines: {2, 18083}, {16889, 16890}


X(18096) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18096) lies on these lines: {2, 3613}, {16889, 18083}, {18089, 18100}


X(18097) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18097) lies on these lines: {12, 1284}, {56, 18102}, {82, 225}, {83, 226}, {308, 349}, {664, 1432}, {3669, 7176}, {5244, 6354}, {6358, 16609}, {7363, 18083}, {7557, 17500}, {18006, 18010}


X(18098) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18098) lies on these lines: {6, 3891}, {37, 82}, {81, 335}, {83, 213}, {100, 733}, {171, 649}, {292, 16717}, {594, 2238}, {756, 3294}, {894, 16707}, {2161, 4628}, {2176, 18091}, {2240, 16587}, {2295, 4972}, {3434, 18088}, {3914, 3997}, {3995, 4366}, {5244, 6354}, {17989, 17997}


X(18099) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(141), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18099) lies on these lines: {1, 83}, {12, 1284}, {1215, 1580}, {1237, 1966}, {4367, 6645}


X(18100) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18100) lies on these lines: {2, 18083}, {16890, 18082}, {17685, 18091}, {18089, 18096}, {18094, 18103}


X(18101) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18101) lies on these lines: {82, 17500}, {83, 13576}, {528, 18082}, {2170, 3907}, {3120, 4107}, {4124, 11988}, {8299, 18091}, {9448, 10798}, {16889, 18102}


X(18102) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18102) lies on these lines: {10, 82}, {56, 18097}, {667, 18107}, {16889, 18101}


X(18103) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(291), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18103) lies on these lines: {2, 18093}, {83, 14621}, {308, 3765}, {4366, 18082}, {18094, 18100}


X(18104) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18104) lies on these lines: {2, 16683}, {3589, 4972}, {16890, 18090}


X(18105) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18105) lies on these lines: {6, 688}, {23, 385}, {82, 876}, {251, 9178}, {512, 1691}, {689, 9150}, {691, 827}, {733, 5970}, {1176, 1510}, {1634, 4577}, {2451, 9009}, {2489, 3804}, {2492, 2514}, {2770, 9076}, {3267, 4108}, {4455, 4705}, {4630, 14560}, {14041, 18107}


X(18106) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18106) lies on this line: {18107, 18108}


X(18107) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(513), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18107) lies on these lines: {667, 18102}, {798, 812}, {3669, 7176}, {4083, 17752}, {14041, 18105}, {18106, 18108}


X(18108) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(514), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18108) lies on these lines: {23, 385}, {171, 649}, {513, 1980}, {661, 830}, {667, 693}, {812, 8635}, {827, 1290}, {1176, 3657}, {1633, 4628}, {2752, 9076}, {3803, 6591}, {4380, 8646}, {4401, 6590}, {18106, 18107}


X(18109) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18109) lies on this line:{2, 16684}, {18106, 18107}


X(18110) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(522), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18110) lies on these lines: {2530, 17686}, {18106, 18107}


X(18111) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(523), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18111) lies on these lines: {83, 1019}, {2533, 4164}, {4367, 6645}, {7178, 10566}, {18106, 18107}


X(18112) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18112) lies on these lines: {12, 1284}, {18106, 18107}


X(18113) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18113) lies on this line:
{3120, 4107}


X(18114) = HATZIPOLAKIS AXIS ∩ BROCARD AXIS

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

The Hatzipolakis axis is defined at X(18039).

X(18114) lies on these lines: {1,7136}, {2,16186}, {3,6}, {4,16237}, {5,523}, {30,15358}, {51,15329}, {54,14587}, {237,11649}, {381,15356}

X(18114) = crossdifference of every pair of points on line {50, 523}
X(18114) = X(i)-line conjugate of X(j) for these (i,j): {3, 50}, {5, 523}
X(18114) = X(i)-isoconjugate of X(j) for these (i,j): {54, 2621}, {110, 2627}
X(18114) = X(i)-Hirst inverse of X(j) for these (i,j): {5, 523}
X(18114) = barycentric product X(i)*X(j) for these {i,j}: {1577, 2626}, {2620, 14213}
X(18114) = barycentric quotient X(i)/X(j) for these {i,j}: {661, 2627}, {1953, 2621}, {2620, 2167}, {2626, 662}
X(18114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 18121, 14356), (3557, 3558, 568), (18121, 18122, 5)


X(18115) = HATZIPOLAKIS AXIS ∩ X(1)X(3)

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

X(18115) lies on these lines: {1,3}, {5,523}, {442,16186}

X(18115) = crossdifference of every pair of points on line {50, 650}


X(18116) = HATZIPOLAKIS AXIS ∩ ANTI-ORTHIC AXIS

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

X(18116) lies on these lines: {5,523}, {44,513}

X(18116) = X(9221)-Ceva conjugate of X(11)
X(18116) = crossdifference of every pair of points on line {1, 50}
X(18116) = X(567)-zayin conjugate of X(4551)
X(18116) = barycentric product X(7951)*X(14838)
X(18116) = barycentric quotient X(7951)/X(15455)


X(18117) = HATZIPOLAKIS AXIS ∩ LEMOINE AXIS

Barycentrics    a^2 (b^2-c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : :
X(18117) = X[3005] + 3 X[15451]

X(18117) lies on these lines: {5,523}, {187,237}

X(18117) = reflection of X(i) in X(j) for these {i,j}: (669, 6140), (14270, 647)
X(18117) = crossdifference of every pair of points on line {2, 50}
X(18117) = crosssum of X(523) and X(14389)
X(18117) = X(662)-isoconjugate of X(7578)
X(18117) = barycentric product X(i*X(j) for these {i,j}: {523, 566}, {647, 7577}
X(18117) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 7578}, {566, 99}, {7577, 6331}


X(18118) = HATZIPOLAKIS AXIS ∩ GERGONNE LINE

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

X(18118) lies on these lines: {5,523}, {241,514}

X(18118) = crosssum of X(2278) and X(9404)
X(18118) = crossdifference of every pair of points on line {50, 55}


X(18119) = HATZIPOLAKIS AXIS ∩ SODDY LINE

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

X(18119) lies on these lines: {1,7}, {5,523}

X(18119) = crossdifference of every pair of points on line {50, 657}


X(18120) = HATZIPOLAKIS AXIS ∩ NAGEL LINE

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

X(18120) lies on these lines: {1,2}, {5,523}

X(18120) = crossdifference of every pair of points on line {50, 649}


X(18121) = HATZIPOLAKIS AXIS ∩ VAN AUBEL LINE

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

X(18121) lies on these lines: {4,6}, {5,523}, {136,11746}, {567,14560}

X(18121) = crossdifference of every pair of points on line {50, 520}
X(18121) = PU(5)-harmonic conjugate of X(14566)


X(18122) = HATZIPOLAKIS AXIS ∩ X(2)X(6)

Barycentrics    a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2-2 a^2 b^4 c^2+2 b^6 c^2-3 a^4 c^4-2 a^2 b^2 c^4-2 b^4 c^4+3 a^2 c^6+2 b^2 c^6-c^8 : :
X(18122) = X[9145] - 3 X[15561]

X(18122) lies on these lines: {2,6}, {5,523}, {53,16237}, {114,2854}, {297,16328}

X(18122) = midpoint of X(6033) and X(9142)
X(18122) = crossdifference of every pair of points on line {50, 512}
X(18122) = X(2)-daleth conjugate of X(3580) X(18122) = X(9160)-complementary conjugate of X(4369)
X(18122) = barycentric product X(i)*X(j) for these {i,j}: {76, 15544}
X(18122) = barycentric quotient X(i)/X(j) for these {i,j}: {15544, 6}
X(18122) = {X(5),X(18114)}-harmonic conjugate of X(18121)


X(18123) = X(6)X(2476)∩X(54)X(6853)

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

See Kadir Altintas and César Lozada, Hyacinthos 27592.

X(18123) lies on the Jerabek hyperbola and these lines: {6, 2476}, {54, 6853}, {65, 2475}, {3448, 3869}, {5080, 15232}, {8044, 17139}


X(18124) = X(6)X(5133)∩X(54)X(1352)

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

See Kadir Altintas and César Lozada, Hyacinthos 27592.

X(18124) lies on the Jerabek hyperbola and these lines: {6, 5133}, {54, 1352}, {66, 7391}, {70, 511}, {74, 1286}, {1176, 1899}, {1177, 3448}, {3431, 3619}, {3564, 15317}, {7544, 14542}


X(18125) = X(6)X(3448)∩X(54)X(542)

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

See Kadir Altintas and César Lozada, Hyacinthos 27592.

X(18125) lies on the Jerabek hyperbola and these lines: {6, 3448}, {54, 542}, {66, 13203}, {67, 5189}, {74, 1287}, {125, 1176}, {1173, 9970}, {1177, 9140}, {1352, 3431}, {2781, 6145}, {2854, 13622}, {2930, 3410}, {3519, 14984}, {3818, 15102}, {5486, 11442}

X(18125) = antigonal conjugate of X(1176)


X(18126) = X(3)X(15317)∩X(20)X(13579)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27594.

X(18126) lies on these lines: {3, 15317}, {20, 13579}, {26, 2165}, {186, 254}

X(18126) = barycentric product X(i)*X(j) for these {i,j}: {155, 13579}, {1993, 15242}, {6515, 15317}
X(18126) = barycentric quotient X(i)/X(j) for these (i,j): (1609, 7505), (15242, 5392), (15317, 6504)
X(18126) = trilinear product X(i)*X(j) for these {i,j}: {47, 15242}, {920, 15317}
X(18126) = trilinear quotient X(i)/X(j) for these (i,j): (920, 7505), (15242, 91), (15317, 921)


X(18127) = X(50)X(2070)∩X(186)X(14731)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27594.

X(18127) lies on these lines: {50, 2070}, {186, 14731}, {6644, 14385}, {12028, 12091}


X(18128) = MIDPOINT OF X(3) AND X(10116)

Barycentrics    (4*S^2-(SB+SC)*(2*R^2+SA+3*SW))*SA : :
X(18128) = 3*X(389)-X(11819), X(5446)-3*X(11245), 3*X(5890)+X(11750), 3*X(5892)-X(12134), 3*X(5946)-X(13419), X(10263)-3*X(11225), X(10575)+3*X(12022), 3*X(12022)-X(12897)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 27606.

X(18128) lies on these lines: {3, 539}, {4, 15019}, {30, 11565}, {49, 14156}, {68, 7400}, {140, 542}, {184, 6640}, {389, 11819}, {511, 17712}, {548, 11264}, {550, 10112}, {569, 11457}, {575, 14864}, {974, 6146}, {1147, 3546}, {1181, 5448}, {1209, 3448}, {1503, 5462}, {1899, 3549}, {3564, 5447}, {5012, 6689}, {5446, 11245}, {5890, 11750}, {5892, 12134}, {5946, 13419}, {5965, 10627}, {6643, 15083}, {6699, 13367}, {7386, 9936}, {10114, 12041}, {10263, 11225}, {10264, 10610}, {10575, 12022}, {11442, 13336}, {12241, 14915}, {12362, 13754}, {13399, 14130}, {13403, 13491}

X(18128) = midpoint of X(i) and X(j) for these {i,j}: {3, 10116}, {548, 11264}, {550, 10112}, {10114, 12041}, {10575, 12897}, {13403, 13491}
X(18128) = {X(10575), X(12022)}-harmonic conjugate of X(12897)


X(18129) = X(32)X(127)∩X(626)X(11574)

Barycentrics    SA*((8*R^2-SW)*S^6-(8*R^4*(2* SA-3*SW)-4*R^2*(2*SA^2-5*SW^2) +SW*(SA^2+SB*SC-3*SW^2))*S^4-( 2*R^2*(3*SA^2-5*SW^2)-(SA^2+2* SB*SC-2*SW^2)*SW)*SW^2*S^2-SB* SC*SW^5) : :

See César Lozada, Hyacinthos 27607.

X(18129) lies on these lines: {32, 127}, {626, 11574}


X(18130) = X(53)X(571)∩X(136)X(216)

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

See César Lozada, Hyacinthos 27607.

X(18130) lies on these lines: {25, 5593}, {53, 571}, {136, 216}, {6751, 8754}

X(18130) = X(5593)-of-anti-Ara triangle
X(18130) = (2nd anti-Conway)-isotomic conjugate of X(235)


X(18131) = X(226)X(14597)∩X(1445)X(15487)

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

See César Lozada, Hyacinthos 27607.

X(18131) lies on these lines: {226, 15497}, {1445, 15487}


X(18132) = 6th HUNG-LOZADA-EULER POINT

Trilinears    16*p^5*(p-2*q)-8*(3*q^2-1)*p^ 4+4*(8*q^2+1)*q*p^3-(40*q^4- 58*q^2+23)*p^2-(10*q^2-3)*q*p+ 6+4*q^2*(3*q^2-4) : : , where p=sin(A/2), q=cos((B-C)/2)
Barycentrics    2*a^9-6*(b+c)*a^8-(3*b^2-22*b* c+3*c^2)*a^7+(b+c)*(19*b^2-36* b*c+19*c^2)*a^6-(9*b^4+9*c^4+ 25*b*c*(b-c)^2)*a^5-(b+c)*(3* b^2-8*b*c+3*c^2)*(5*b^2-8*b*c+ 5*c^2)*a^4+(19*b^4+19*c^4+4*b* c*(4*b^2-5*b*c+4*c^2))*(b-c)^ 2*a^3-(b^2-c^2)*(b-c)*(3*b^4+ 3*c^4+10*b*c*(2*b^2-3*b*c+2*c^ 2))*a^2-(b^2-c^2)^2*(b-c)^2*( 9*b^2-7*b*c+9*c^2)*a+(b^2-c^2) ^3*(b-c)*(5*b^2-4*b*c+5*c^2) : :

As a point on the Euler line, X(18132) has Shinagawa coefficients (7*R*r+8*r^2-E-3*F, 3*R*r-3*F).

See Tran Quang Hung and César Lozada, Hyacinthos 27608.

X(18132) lies on this line: {2,3}


X(18133) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(2), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18133) lies on these lines: {2, 3770}, {6, 18046}, {9, 18044}, {10, 75}, {37, 18040}, {45, 18073}, {69, 2478}, {86, 13741}, {141, 3948}, {192, 4033}, {312, 17228}, {314, 17196}, {319, 350}, {321, 17239}, {668, 4360}, {757, 799}, {908, 17241}, {1909, 17322}, {1964, 17793}, {3644, 4110}, {3661, 4043}, {3701, 3844}, {3720, 17149}, {3731, 18065}, {3760, 17270}, {3765, 4657}, {3963, 4364}, {3975, 16706}, {4358, 17231}, {4664, 17786}, {6646, 17790}, {14994, 15991}, {17195, 17378}, {17234, 18150}, {17258, 17787}, {18072, 18158}


X(18134) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18134) lies on these lines: {1, 977}, {2, 6}, {7, 345}, {8, 3475}, {19, 8897}, {42, 4429}, {43, 3836}, {55, 4645}, {57, 3719}, {63, 320}, {75, 306}, {76, 85}, {92, 914}, {120, 3699}, {142, 3687}, {171, 3771}, {189, 1997}, {190, 5905}, {192, 3782}, {209, 3868}, {223, 7210}, {224, 5174}, {278, 664}, {286, 469}, {305, 18157}, {319, 5271}, {321, 17233}, {329, 344}, {354, 3705}, {377, 1043}, {404, 1792}, {405, 1330}, {442, 10449}, {464, 8822}, {551, 16498}, {643, 1754}, {740, 17889}, {752, 8616}, {846, 4655}, {857, 18135}, {908, 17241}, {986, 3178}, {1001, 4388}, {1086, 3210}, {1125, 16478}, {1352, 7413}, {1434, 3926}, {1621, 6327}, {1730, 3882}, {1738, 4028}, {1836, 3685}, {1848, 18156}, {1999, 3772}, {2550, 3996}, {2886, 4966}, {2893, 7522}, {2999, 17282}, {3006, 3873}, {3011, 3769}, {3060, 3909}, {3175, 17242}, {3187, 17377}, {3219, 17347}, {3262, 17862}, {3305, 17263}, {3416, 3757}, {3662, 3666}, {3695, 6147}, {3712, 11246}, {3729, 4654}, {3750, 4660}, {3752, 3834}, {3758, 5294}, {3821, 17592}, {3823, 4849}, {3824, 5295}, {3840, 17717}, {3944, 4892}, {3957, 5014}, {3966, 16823}, {4001, 17361}, {4054, 17240}, {4259, 5208}, {4384, 4886}, {4385, 13407}, {4389, 17184}, {4398, 17147}, {4415, 17243}, {4641, 17364}, {4684, 4847}, {4972, 17018}, {5256, 16706}, {5287, 17317}, {5814, 16817}, {6505, 17923}, {6626, 16349}, {7081, 17718}, {7262, 17770}, {7377, 10478}, {8728, 9534}, {10371, 16824}, {11679, 17296}, {16787, 17023}, {16826, 16974}, {17011, 17380}, {17312, 17720}, {17336, 17781}, {17715, 17766}


X(18135) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18135) lies on these lines: {1, 6381}, {2, 39}, {6, 17541}, {7, 2899}, {8, 350}, {10, 3760}, {21, 183}, {32, 16920}, {69, 2478}, {75, 3701}, {85, 5226}, {86, 5192}, {99, 4188}, {141, 17550}, {145, 668}, {192, 1921}, {264, 4194}, {279, 4554}, {304, 4358}, {308, 941}, {313, 17321}, {314, 5232}, {315, 5046}, {325, 4193}, {331, 6335}, {344, 349}, {348, 1997}, {384, 16997}, {385, 16916}, {404, 1975}, {406, 1235}, {452, 15589}, {497, 18057}, {857, 18134}, {1007, 6931}, {1078, 4189}, {1125, 3761}, {1278, 10009}, {1909, 3616}, {2176, 4465}, {2295, 4713}, {2475, 11185}, {3161, 10030}, {3214, 3875}, {3263, 3673}, {3314, 17669}, {3403, 17261}, {3596, 3672}, {3617, 17143}, {3618, 18046}, {3619, 18137}, {3691, 17026}, {3718, 17863}, {3734, 5277}, {3785, 6872}, {3933, 4187}, {3975, 5222}, {3992, 7264}, {4044, 17308}, {4396, 4426}, {4410, 4798}, {4419, 17790}, {4648, 18144}, {4721, 17750}, {5047, 16992}, {5051, 5224}, {5154, 7752}, {5275, 17686}, {5276, 7770}, {5305, 17540}, {5712, 18052}, {6337, 6921}, {6390, 13747}, {7750, 11114}, {7751, 17002}, {7767, 11113}, {7771, 17548}, {7776, 17556}, {7793, 17692}, {14907, 15680}, {16915, 16999}, {16918, 16998}, {17777, 18066}


X(18136) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18136) lies on these lines: {2, 3770}, {63, 18044}, {76, 16739}, {81, 18046}, {141, 312}, {321, 17237}, {333, 17681}, {341, 4202}, {940, 18148}, {2185, 18048}, {4033, 17147}, {8033, 18140}, {17027, 17149}, {18139, 18150}


X(18137) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18137) lies on these lines: {2, 37}, {76, 17234}, {141, 3948}, {142, 1269}, {190, 16574}, {239, 16685}, {313, 3912}, {314, 17277}, {319, 3975}, {341, 10449}, {392, 4673}, {518, 3701}, {668, 17295}, {740, 3216}, {984, 4075}, {1230, 18139}, {1909, 17317}, {1921, 18050}, {2300, 3759}, {2321, 3264}, {3230, 17144}, {3250, 18080}, {3596, 4033}, {3619, 18135}, {3685, 5132}, {3696, 3702}, {3760, 17282}, {3765, 4851}, {3770, 17300}, {3840, 4022}, {3963, 17243}, {4417, 14615}, {4676, 5156}, {4828, 18154}, {6376, 17228}, {6385, 18138}, {7283, 16453}, {13476, 17165}, {15668, 16709}, {17232, 18144}, {17240, 17786}, {17284, 18044}, {17307, 18140}, {17788, 18151}


X(18138) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18138) lies on these lines: {2, 16703}, {42, 75}, {76, 85}, {210, 3263}, {305, 4417}, {314, 14828}, {1215, 1930}, {2887, 18057}, {3266, 5741}, {3475, 4441}, {3936, 8024}, {6385, 18137}


X(18139) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18139) lies on these lines: {1, 4202}, {2, 6}, {7, 17776}, {8, 8728}, {37, 17184}, {42, 3836}, {51, 3909}, {63, 17298}, {75, 3969}, {92, 2973}, {100, 16056}, {142, 306}, {190, 17483}, {226, 4358}, {264, 445}, {312, 17241}, {320, 3219}, {321, 1930}, {344, 5905}, {354, 3006}, {386, 17674}, {553, 3977}, {750, 3771}, {1001, 6327}, {1086, 17147}, {1230, 18137}, {1330, 5047}, {1621, 4450}, {1738, 3896}, {1962, 3821}, {1999, 17312}, {2321, 4980}, {2325, 3982}, {2887, 3720}, {3187, 4851}, {3475, 10327}, {3616, 13728}, {3664, 5294}, {3666, 3834}, {3701, 13407}, {3702, 12609}, {3703, 17140}, {3782, 3995}, {3826, 4651}, {3873, 4260}, {3925, 4966}, {3932, 17165}, {3933, 17169}, {4197, 10449}, {4340, 17526}, {4388, 5284}, {4427, 11246}, {4429, 17018}, {4442, 17889}, {4641, 17376}, {4884, 17154}, {5256, 17282}, {5271, 17296}, {8024, 18052}, {9776, 17740}, {16706, 17011}, {16752, 17670}, {17019, 17317}, {17174, 18046}, {18136, 18150} /p>


X(18140) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18140) lies on these lines: {1, 668}, {2, 39}, {10, 350}, {21, 1078}, {32, 16916}, {37, 308}, {69, 5084}, {75, 1089}, {83, 5276}, {85, 4554}, {86, 13741}, {99, 404}, {148, 17565}, {183, 405}, {187, 17692}, {190, 16549}, {192, 10009}, {264, 406}, {286, 5142}, {304, 17284}, {312, 17308}, {313, 17322}, {314, 5224}, {315, 2478}, {316, 5046}, {325, 4187}, {377, 11185}, {384, 5277}, {385, 16918}, {451, 1235}, {452, 3785}, {474, 1975}, {626, 17669}, {646, 4664}, {799, 1509}, {894, 4721}, {940, 11353}, {1125, 1909}, {1574, 17759}, {1577, 18077}, {1961, 1965}, {1969, 6335}, {2238, 17034}, {2295, 4465}, {2896, 17685}, {3096, 17550}, {3230, 17752}, {3247, 17786}, {3264, 17320}, {3293, 4360}, {3403, 3731}, {3570, 4251}, {3596, 17321}, {3624, 3761}, {3679, 17144}, {3734, 16915}, {3765, 17397}, {3770, 17398}, {3831, 4357}, {3875, 6048}, {3912, 4109}, {3931, 4087}, {3933, 17527}, {3952, 17141}, {3972, 16920}, {3975, 17023}, {4188, 7782}, {4189, 7771}, {4193, 7752}, {4358, 17292}, {4364, 17790}, {4441, 9780}, {4482, 9327}, {4687, 18044}, {5129, 15589}, {5209, 6626}, {5254, 17670}, {5275, 7770}, {5284, 18064}, {6179, 17001}, {6337, 17567}, {6872, 14907}, {7750, 11113}, {7751, 16998}, {7773, 17556}, {7787, 16995}, {7789, 17694}, {7792, 17540}, {7793, 16914}, {7802, 11114}, {7815, 17684}, {7816, 17693}, {8033, 18136}, {11108, 16992}, {14296, 14431}, {15031, 17577}, {15668, 18144}, {16917, 17128}, {17000, 17129}, {17048, 17755}, {17234, 17671}, {17277, 18046}, {17283, 18157}, {17307, 18137}, {17451, 18061}


X(18141) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18141) lies on these lines: {2, 6}, {7, 312}, {8, 354}, {57, 345}, {58, 13742}, {63, 344}, {75, 9776}, {142, 11679}, {145, 17597}, {190, 9965}, {200, 4684}, {226, 17298}, {306, 3306}, {320, 329}, {443, 10449}, {487, 16433}, {488, 16432}, {497, 4645}, {553, 3729}, {908, 1997}, {938, 7270}, {980, 16043}, {1043, 6904}, {1265, 3868}, {1330, 5084}, {1376, 4966}, {1445, 3719}, {1999, 4000}, {2094, 17264}, {2550, 10453}, {2999, 3879}, {3210, 3726}, {3218, 17776}, {3305, 4001}, {3416, 3742}, {3474, 3685}, {3475, 7081}, {3616, 3745}, {3666, 17316}, {3687, 5437}, {3695, 5708}, {3703, 4860}, {3752, 4851}, {3772, 3834}, {3785, 11343}, {3819, 10477}, {3873, 10327}, {3883, 10582}, {4035, 6692}, {4188, 5347}, {4340, 13740}, {4357, 17022}, {4358, 5905}, {4387, 11246}, {4415, 7232}, {4416, 7308}, {4514, 10580}, {4656, 17274}, {5272, 5847}, {5287, 17321}, {5308, 17137}, {5324, 16048}, {5337, 14001}, {5744, 17241}, {5918, 9801}, {6542, 17490}, {6776, 16434}, {6862, 11487}, {6890, 12324}, {6958, 11411}, {7172, 11038}, {7195, 17789}, {9534, 17582}, {10452, 18229}, {16602, 17374}, {18135, 18136}


X(18142) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18142) lies on these lines: {2, 16708}, {75, 210}, {76, 2051}, {85, 5226}, {226, 18045}, {908, 1233}, {1699, 18043}, {6385, 18137}, {17234, 18153}


X(18143) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18143) lies on these lines: {2, 3770}, {75, 141}, {76, 17234}, {86, 18046}, {142, 313}, {274, 17307}, {308, 3572}, {310, 18052}, {312, 17241}, {314, 17297}, {321, 17231}, {341, 8728}, {350, 17317}, {662, 18048}, {1269, 3912}, {1909, 16706}, {3761, 17282}, {3765, 17278}, {3823, 4696}, {3844, 4968}, {3948, 17245}, {4359, 17239}, {4410, 17357}, {4553, 17142}, {4648, 18147}, {4751, 6376}, {7321, 17787}, {8024, 16708}, {10436, 18044}, {17143, 17295}, {17144, 17386}


X(18144) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18144) lies on these lines: {2, 3770}, {7, 17790}, {75, 4377}, {76, 141}, {142, 6381}, {192, 18040}, {274, 17327}, {312, 17231}, {313, 3662}, {314, 599}, {321, 17228}, {341, 3823}, {350, 4851}, {513, 7155}, {536, 17786}, {668, 4361}, {894, 18044}, {1086, 3596}, {1269, 3661}, {1278, 4033}, {1909, 4657}, {2228, 17157}, {2278, 18048}, {3729, 18065}, {3739, 6376}, {3760, 17296}, {3761, 17306}, {3765, 16706}, {3844, 4385}, {3948, 17234}, {3963, 4389}, {3975, 17278}, {4043, 17230}, {4110, 4686}, {4112, 18209}, {4358, 17241}, {4410, 17385}, {4445, 17143}, {4494, 4862}, {4648, 18135}, {6383, 6386}, {9464, 16727}, {15668, 18140}, {17144, 17372}, {17232, 18137}, {17276, 17787}, {17300, 18147}, {17313, 18145}, {17379, 18046}, {17788, 18159}


X(18145) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18145) lies on these lines: {2, 39}, {75, 3992}, {99, 13587}, {183, 16370}, {313, 17320}, {314, 17196}, {325, 17533}, {350, 519}, {536, 1921}, {545, 17790}, {551, 1909}, {1078, 17549}, {1577, 4960}, {1975, 16371}, {3679, 3760}, {3734, 16997}, {3972, 17001}, {4193, 7796}, {4358, 4945}, {4554, 17078}, {4677, 17144}, {4740, 10009}, {5046, 7768}, {5154, 7814}, {5277, 17128}, {6179, 16920}, {7751, 16916}, {7760, 17541}, {7780, 17692}, {7788, 17556}, {7794, 17669}, {7811, 11114}, {14210, 18159}, {16857, 16992}, {16915, 17130}, {16998, 17131}, {17313, 18144}, {17342, 18044}, {17378, 18147}


X(18146) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18146) lies on these lines: {2, 39}, {10, 4479}, {75, 3828}, {99, 16371}, {183, 16418}, {350, 3679}, {519, 6376}, {536, 10009}, {551, 6381}, {668, 3241}, {1078, 16370}, {1921, 4664}, {1975, 16417}, {2478, 7768}, {3596, 17320}, {3734, 16999}, {3972, 16997}, {4187, 7796}, {4193, 7814}, {4554, 17079}, {4669, 17144}, {5046, 7860}, {6179, 16916}, {7751, 16918}, {7752, 17533}, {7771, 17549}, {7780, 16914}, {7782, 13587}, {7804, 16995}, {7809, 17556}, {7811, 11113}, {7854, 17685}, {7856, 17540}, {7878, 17541}, {7922, 17669}, {11057, 11114}, {16917, 17130}, {16992, 17542}, {17000, 17131}, {17195, 17378}, {17271, 18147}


X(18147) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18147) lies on these lines: {1, 313}, {2, 37}, {6, 3948}, {29, 264}, {69, 2478}, {76, 86}, {190, 579}, {239, 16520}, {286, 469}, {314, 5224}, {319, 5722}, {320, 18148}, {326, 1226}, {341, 387}, {386, 3596}, {668, 17377}, {1100, 3765}, {1228, 5192}, {1234, 5736}, {1269, 3760}, {1333, 11320}, {1441, 11375}, {1909, 17394}, {3216, 3264}, {3718, 5292}, {3723, 4377}, {3759, 3975}, {3770, 17379}, {3879, 6381}, {3912, 4150}, {3963, 16777}, {4033, 17314}, {4044, 5750}, {4286, 17262}, {4648, 18143}, {4869, 18150}, {5165, 17350}, {5564, 9534}, {14615, 18153}, {17160, 17749}, {17271, 18146}, {17300, 18144}, {17315, 17786}, {17316, 18040}, {17378, 18145}


X(18148) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18148) lies on these lines: {2, 330}, {38, 1237}, {75, 3670}, {76, 16887}, {86, 13741}, {313, 3831}, {320, 18147}, {350, 17137}, {668, 3216}, {940, 18136}, {986, 4485}, {1966, 16574}, {3721, 18050}, {3760, 17274}, {4261, 17786}, {10449, 17144}


X(18149) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18149) lies on these lines: {2, 3121}, {11, 334}, {75, 244}, {312, 335}, {693, 18066}, {874, 1054}, {1920, 3840}, {1921, 4871}, {3226, 17477}, {3766, 4927}, {4728, 18150}, {5718, 6376}, {6382, 17063}, {7033, 17597}


X(18150) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18150) lies on these lines: {2, 16723}, {75, 141}, {312, 4080}, {646, 903}, {3762, 18151}, {4043, 17231}, {4417, 14554}, {4728, 18149}, {4869, 18147}, {6376, 17758}, {16709, 17307}, {17232, 18137}, {17234, 18133}, {17298, 18044}, {17300, 18046}, {17790, 18073}, {18136, 18139}


X(18151) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18151) lies on these lines: {2, 16732}, {9, 75}, {92, 4997}, {239, 17796}, {312, 3969}, {319, 1229}, {322, 17240}, {645, 14616}, {765, 1090}, {1121, 16284}, {1441, 17263}, {2607, 5150}, {3262, 17264}, {3673, 7827}, {3762, 18150}, {3912, 17791}, {4422, 4957}, {6996, 16568}, {17352, 17861}, {17788, 18137}


X(18152) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18152) lies on these lines: {2, 39}, {32, 16955}, {42, 308}, {43, 3760}, {69, 6818}, {75, 756}, {99, 4210}, {183, 1011}, {264, 4207}, {312, 561}, {321, 1921}, {385, 16957}, {668, 17135}, {799, 14829}, {870, 5311}, {1078, 4184}, {1235, 4213}, {1240, 2296}, {1255, 17032}, {1909, 3720}, {1920, 4358}, {1965, 17763}, {1975, 4191}, {3596, 4441}, {3734, 16954}, {3741, 6381}, {3765, 17027}, {3972, 16959}, {4651, 17143}, {4671, 6382}, {5226, 6063}, {6385, 18137}, {16956, 17128}


X(18153) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(220), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18153) lies on these lines: {2, 39}, {75, 4082}, {183, 13615}, {200, 350}, {264, 461}, {1909, 10582}, {1997, 4554}, {3760, 8580}, {4847, 6376}, {6381, 11019}, {14615, 18147}, {17234, 18142}


X(18154) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18154) lies on these lines: {2, 650}, {75, 4024}, {661, 7199}, {850, 4789}, {1019, 1577}, {2517, 7662}, {3261, 6590}, {3700, 4374}, {4057, 4874}, {4359, 17161}, {4811, 7659}, {4815, 4913}, {4828, 18137}, {9534, 14077}


X(18155) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(514), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    b c (a + c) (a + b) (b - c) (-a + b + c) : :

X(18155) lies on these lines: {2, 16751}, {75, 850}, {85, 17094}, {99, 2222}, {312, 3700}, {320, 350}, {333, 1021}, {522, 4087}, {643, 799}, {650, 3975}, {811, 1414}, {812, 18071}, {814, 7255}, {1019, 1577}, {2787, 7234}, {3004, 3766}, {3261, 4025}, {3676, 18033}, {3716, 3737}, {3733, 4874}, {3835, 4481}, {4036, 9508}, {4086, 4913}, {4170, 5216}, {4374, 4897}, {4960, 4978}, {7203, 17218}, {14208, 18160}, {16754, 17496}, {18070, 18106}

X(18155) = isotomic conjugate of X(4551)


X(18156) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(1444), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18156) lies on these lines: {1, 75}, {2, 16605}, {19, 2128}, {34, 664}, {47, 4592}, {69, 960}, {85, 350}, {92, 18056}, {145, 3263}, {158, 811}, {192, 7187}, {239, 16781}, {312, 1909}, {320, 17170}, {330, 3797}, {344, 1212}, {491, 14121}, {492, 7090}, {662, 1973}, {742, 16969}, {1848, 18134}, {1959, 6508}, {1975, 3685}, {3061, 3452}, {3633, 4986}, {3718, 3879}, {3759, 16502}, {3760, 4975}, {3761, 7278}, {3811, 4561}, {3877, 17137}, {3889, 17141}, {3890, 17152}, {6337, 17081}, {6384, 7146}, {12514, 17206}, {16503, 16822}, {16524, 16827}, {17336, 17742}, {17442, 18041}, {18049, 18058}

X(18156) = anticomplement of X(16605)
X(18156) = {[trilinear product of PU(116)],[trilinear product of PU(117)]}-harmonic conjugate of X(75)


X(18157) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(1914), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18157) lies on these lines: {1, 75}, {2, 16703}, {37, 16705}, {38, 17208}, {76, 17234}, {85, 6385}, {99, 2725}, {241, 16728}, {305, 18134}, {310, 312}, {321, 16748}, {514, 1921}, {518, 3263}, {536, 16711}, {537, 17179}, {726, 17205}, {984, 16887}, {1434, 2285}, {2887, 17203}, {3006, 17198}, {3266, 3936}, {3739, 17497}, {3842, 17210}, {4358, 16727}, {4564, 4601}, {4639, 17789}, {4664, 16712}, {4751, 16611}, {5233, 11059}, {8024, 18052}, {8025, 16707}, {13476, 17141}, {16704, 16741}, {16891, 18057}, {17283, 18140}, {17755, 18206}


X(18158) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18158) lies on these lines: {2, 16755}, {75, 4036}, {76, 3261}, {522, 5224}, {1577, 4960}, {4140, 4664}, {4828, 15413}, {18072, 18133}


X(18159) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18159) lies on these lines: {2, 7200}, {12, 85}, {75, 537}, {76, 4485}, {80, 17361}, {150, 320}, {350, 4742}, {514, 18061}, {1909, 5484}, {2481, 3680}, {3570, 9317}, {3753, 7321}, {3762, 18150}, {3766, 4927}, {4919, 6631}, {9318, 18047}, {14210, 18145}, {17788, 18144}


X(18160) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18160) lies on these lines: {2, 15419}, {69, 7253}, {75, 4086}, {86, 8062}, {190, 4529}, {656, 5224}, {693, 4806}, {799, 4590}, {816, 8632}, {918, 3261}, {1577, 4960}, {2517, 4406}, {3762, 4509}, {4036, 4374}, {4171, 17233}, {4625, 15455}, {5641, 14616}, {6002, 18077}, {6003, 18076}, {7799, 14838}, {14208, 18155}


X(18161) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18161) lies on these lines: {1, 159}, {6, 2114}, {7, 1953}, {19, 77}, {48, 1442}, {57, 2164}, {69, 1959}, {86, 423}, {141, 3061}, {189, 6508}, {241, 2262}, {320, 18041}, {394, 1762}, {982, 18168}, {984, 8679}, {986, 16696}, {1423, 8609}, {1565, 16608}, {1633, 4336}, {1726, 2003}, {2170, 4000}, {2171, 4644}, {2294, 3945}, {2323, 16551}, {3056, 17447}, {3879, 9028}, {4051, 4361}, {4419, 17452}, {4475, 7032}, {4648, 17451}, {4675, 17443}, {5845, 17390}, {5902, 18164}, {7087, 17716}, {7119, 7210}, {7201, 17365}, {7204, 8608}, {8680, 10446}, {14963, 17052}, {17170, 17442}, {17197, 17861}, {17276, 17444}, {17592, 18185}, {18194, 18208}


X(18162) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18162) lies on these lines: {1, 159}, {6, 1423}, {7, 48}, {19, 7190}, {37, 16560}, {41, 4000}, {57, 2178}, {101, 142}, {141, 2329}, {144, 2267}, {171, 18209}, {198, 5228}, {222, 8761}, {284, 3663}, {320, 18042}, {527, 572}, {584, 17301}, {604, 4644}, {610, 4328}, {651, 2317}, {662, 7321}, {692, 9440}, {1086, 2174}, {1442, 3942}, {1565, 17043}, {1633, 2293}, {1953, 7269}, {2187, 3475}, {2261, 8545}, {2266, 3598}, {2268, 4419}, {2278, 17276}, {2304, 17753}, {3204, 17278}, {3423, 10934}, {3684, 4361}, {3946, 4251}, {4073, 16799}, {4357, 9028}, {4648, 9310}, {4649, 8679}, {5563, 18164}, {5845, 16503}, {7113, 7175}, {7119, 7247}, {9441, 15624}, {16788, 17306}, {17787, 18048}


X(18163) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18163) lies on these lines: {1, 859}, {2, 3882}, {6, 1764}, {9, 312}, {21, 1697}, {40, 58}, {55, 17194}, {57, 77}, {63, 3187}, {86, 5437}, {165, 3286}, {239, 16722}, {940, 1730}, {1010, 1706}, {1043, 2136}, {1334, 5325}, {1396, 8829}, {1420, 4225}, {2170, 16579}, {2185, 4612}, {2193, 10319}, {2257, 4269}, {2269, 5745}, {2270, 2303}, {2328, 5324}, {2347, 3452}, {3158, 3794}, {3243, 5208}, {3306, 8025}, {3333, 4658}, {3576, 4276}, {3666, 18177}, {3750, 18174}, {3928, 9311}, {3929, 4921}, {4281, 10476}, {4516, 17611}, {4603, 6654}, {5219, 17167}, {5235, 7308}, {5535, 9275}, {5802, 14552}, {6762, 10461}, {8056, 16736}, {9580, 14956}, {14010, 15845}, {14936, 16721}, {17594, 18169}, {17596, 18192}


X(18164) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18164) lies on these lines: {1, 3286}, {6, 16726}, {7, 17197}, {9, 86}, {21, 10390}, {40, 4658}, {57, 77}, {58, 2191}, {63, 8025}, {65, 18177}, {142, 1475}, {165, 18185}, {239, 16710}, {314, 4659}, {333, 5437}, {354, 17194}, {527, 17183}, {579, 3945}, {583, 17392}, {738, 1434}, {757, 2150}, {876, 13610}, {894, 17178}, {940, 5120}, {942, 18176}, {1010, 6762}, {1018, 17390}, {1400, 4667}, {1437, 3338}, {2257, 17189}, {2260, 3664}, {2999, 16700}, {3158, 13588}, {3306, 16704}, {3339, 18178}, {3361, 4267}, {3928, 17185}, {3946, 17474}, {4000, 17205}, {4184, 10389}, {4253, 4648}, {4384, 16709}, {4654, 17167}, {4851, 16549}, {4860, 18191}, {5228, 7053}, {5333, 7308}, {5563, 18162}, {5902, 18161}, {10436, 16738}, {10980, 18165}, {15668, 16552}, {16574, 17379}, {16723, 17313}, {16887, 17306}, {17202, 17274}, {17296, 17754}


X(18165) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18165) lies on these lines: {1, 859}, {2, 4259}, {11, 17167}, {21, 65}, {25, 940}, {27, 243}, {28, 1905}, {29, 1875}, {37, 4269}, {46, 17524}, {51, 5718}, {57, 3286}, {58, 942}, {81, 105}, {86, 1431}, {171, 1283}, {181, 6690}, {210, 5235}, {244, 16700}, {284, 910}, {333, 518}, {511, 17056}, {517, 4653}, {958, 10461}, {960, 11110}, {982, 16696}, {1001, 10473}, {1010, 3812}, {1043, 5836}, {1155, 4184}, {1412, 3660}, {1730, 5132}, {1755, 2294}, {1817, 17603}, {1836, 14956}, {1887, 3559}, {2245, 8731}, {2328, 5173}, {2360, 16193}, {2646, 4225}, {3666, 4215}, {3736, 3752}, {3740, 3786}, {3784, 4675}, {3816, 17182}, {3819, 17245}, {3838, 14009}, {3869, 17588}, {3873, 16704}, {3924, 10457}, {4224, 5135}, {4229, 10178}, {4650, 5902}, {4658, 5045}, {5737, 10477}, {6675, 10974}, {7195, 17169}, {7289, 18166}, {7290, 11021}, {9335, 16753}, {10980, 18164}, {11019, 17197}, {11680, 17173}, {14008, 17605}, {16699, 17451}, {16736, 17063}, {18175, 18181}


X(18166) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (a b + a c + 2 b c) : :

X(18166) lies on these lines: {1, 3286}, {2, 6}, {37, 18206}, {58, 1001}, {110, 9110}, {171, 18185}, {239, 16709}, {274, 4361}, {314, 4363}, {320, 17202}, {354, 18182}, {894, 4043}, {942, 18179}, {1014, 5228}, {1100, 16726}, {2295, 17390}, {2663, 4557}, {2667, 4436}, {2999, 16736}, {3293, 4649}, {3664, 17197}, {3736, 3913}, {3739, 17175}, {3946, 17205}, {4000, 17169}, {4038, 18169}, {4057, 18196}, {4393, 16710}, {4644, 17183}, {4653, 11194}, {4657, 16887}, {4670, 10455}, {4833, 18199}, {4851, 17750}, {4852, 16971}, {5256, 16700}, {7289, 18165}, {16685, 17394}, {16705, 17045}, {16752, 17366}, {16753, 17012}, {17139, 17365}, {17179, 17382}


X(18167) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (b^4 + c^4) : :

X(18167) lies on these lines: {1, 16681}, {81, 18182}, {257, 274}, {626, 3118}, {762, 4469}, {982, 18171}, {1437, 18187}, {2085, 4118}, {3670, 16696}, {3721, 16887}, {3735, 16705}, {3953, 18172}, {3981, 17203}, {18168, 18195}, {18175, 18203}


X(18168) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18168) lies on these lines: {1, 2916}, {6, 18207}, {75, 4475}, {982, 18161}, {3554, 18193}, {3662, 4118}, {4389, 17470}, {4443, 17447}, {7202, 18194}, {7237, 17227}, {7263, 17891}, {16696, 18190}, {18167, 18195}


X(18169) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18169) lies on these lines: {1, 21}, {2, 5145}, {42, 16704}, {43, 333}, {86, 87}, {171, 3286}, {261, 1178}, {572, 5276}, {741, 1961}, {873, 17175}, {940, 16058}, {980, 11328}, {982, 16696}, {986, 18180}, {1019, 8027}, {1054, 8849}, {1107, 1197}, {1185, 16552}, {1201, 17588}, {1613, 5283}, {2185, 5009}, {2267, 2303}, {2309, 3741}, {3120, 17173}, {3216, 5278}, {3550, 4184}, {3666, 18191}, {3720, 4368}, {3737, 4448}, {3742, 16726}, {3750, 18185}, {3944, 17167}, {4038, 18166}, {4425, 17202}, {5235, 16569}, {5255, 17524}, {10459, 11115}, {10980, 18186}, {13323, 13732}, {15569, 18211}, {16700, 17063}, {17594, 18163}, {17600, 18170}, {18182, 18204}, {18190, 18203}


X(18170) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18170) lies on these lines: {1, 6}, {36, 8266}, {75, 7032}, {82, 7122}, {86, 670}, {87, 4363}, {171, 16679}, {183, 614}, {239, 1964}, {308, 1909}, {330, 8264}, {385, 7191}, {571, 10801}, {612, 11174}, {869, 3759}, {872, 17121}, {894, 3248}, {982, 18161}, {1015, 17049}, {1045, 4852}, {1086, 7184}, {1333, 12194}, {1429, 18209}, {1582, 7113}, {1740, 4361}, {2170, 17446}, {2174, 8300}, {2210, 18042}, {2220, 11364}, {2234, 17117}, {2275, 4446}, {2309, 4360}, {2664, 17348}, {3009, 17277}, {3056, 4443}, {3329, 3920}, {3661, 7189}, {3770, 12263}, {3783, 17362}, {5019, 10800}, {5069, 12782}, {5201, 5563}, {5272, 15271}, {7202, 18183}, {7766, 17024}, {9359, 17351}, {10802, 13345}, {16571, 17119}, {16696, 18205}, {17600, 18169}


X(18171) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (a b^3 + b^3 c + a c^3 + b c^3) : :

X(18171) lies on these lines: {1, 3286}, {32, 81}, {86, 5283}, {194, 17178}, {213, 4641}, {274, 330}, {982, 18167}, {1015, 16705}, {1107, 16726}, {2275, 16887}, {3670, 18184}, {4384, 16700}, {16604, 17210}, {16709, 16819}, {16736, 16832}, {16753, 16815}, {17050, 17205}


X(18172) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18172) lies on these lines: {1, 3286}, {81, 172}, {86, 1107}, {239, 16700}, {274, 670}, {330, 17178}, {980, 3053}, {982, 18189}, {1015, 16744}, {2176, 18206}, {3953, 18167}, {3976, 18176}, {4384, 16736}, {16746, 17208}, {16753, 16816}, {16975, 17175}, {18178, 18205}


X(18173) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18173) lies on these lines: {1, 994}, {43, 5208}, {244, 16753}, {667, 18197}, {982, 16696}, {1647, 17174}, {3286, 18201}, {3976, 18180}, {3999, 18191}, {10458, 17591}, {16704, 17449}, {17194, 18193}


X(18174) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18174) lies on these lines: {1, 18191}, {171, 17194}, {354, 18192}, {982, 16696}, {3750, 18163}, {10458, 17592}, {16497, 18206}, {16753, 17063}


X(18175) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18175) lies on these lines: {81, 18210}, {982, 18177}, {18165, 18181}, {18167, 18203}


X(18176) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18176) lies on these lines: {1, 16680}, {81, 16787}, {942, 18164}, {982, 18167}, {986, 16696}, {3061, 16887}, {3501, 16728}, {3673, 17197}, {3976, 18172}, {7146, 18206}, {16708, 17050}, {17046, 17177}, {17169, 17451}, {18192, 18208}


X(18177) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18177) lies on these lines: {65, 18164}, {75, 17197}, {81, 16947}, {982, 18175}, {986, 16696}, {2170, 16713}, {3666, 18163}, {17183, 17452}, {18184, 18186}


X(18178) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18178) lies on these lines: {1, 859}, {4, 5820}, {6, 10441}, {10, 4553}, {12, 17167}, {21, 643}, {40, 3286}, {58, 517}, {65, 81}, {86, 3812}, {284, 501}, {314, 3714}, {333, 960}, {387, 4259}, {511, 1834}, {524, 10381}, {958, 10480}, {986, 16696}, {1010, 5836}, {1043, 3794}, {1071, 2831}, {1108, 4269}, {1319, 4225}, {1329, 17182}, {1385, 4276}, {1682, 4999}, {1697, 17194}, {2170, 16699}, {2194, 3193}, {2262, 2303}, {2551, 17183}, {3339, 18164}, {3579, 4278}, {3698, 14005}, {3736, 4646}, {3786, 4662}, {3869, 16704}, {3890, 17588}, {3893, 4720}, {3959, 16716}, {4256, 5482}, {4271, 13731}, {4642, 17187}, {4653, 9957}, {5119, 17524}, {5123, 14011}, {5292, 5752}, {5324, 17642}, {5562, 5721}, {5706, 11414}, {5885, 15792}, {8679, 13161}, {10461, 12513}, {11115, 14923}, {11681, 17174}, {12701, 14956}, {18172, 18205}


X(18179) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a b^3 + b^4 + a c^3 + c^4) : :

X(18179) lies on these lines: {1, 1631}, {6, 1760}, {141, 3721}, {343, 3782}, {674, 17446}, {942, 18166}, {982, 18161}, {2228, 7237}, {3122, 17470}, {3125, 3739}, {3666, 18202}, {3670, 16696}, {3726, 17390}, {3727, 17045}, {3735, 4657}, {3778, 4118}, {3954, 17239}, {3959, 4361}, {4016, 4357}, {4022, 4475}, {18181, 18190}


X(18180) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18180) lies on these lines: {1, 859}, {2, 5752}, {3, 1243}, {5, 51}, {21, 517}, {25, 5707}, {27, 1071}, {28, 60}, {40, 17194}, {46, 3286}, {58, 65}, {72, 333}, {79, 513}, {86, 5439}, {110, 6583}, {185, 8727}, {284, 2262}, {354, 2360}, {373, 17527}, {389, 6831}, {392, 11110}, {394, 7535}, {404, 5482}, {405, 10441}, {442, 511}, {495, 16980}, {500, 851}, {573, 16455}, {946, 2779}, {956, 10461}, {957, 3622}, {970, 7483}, {986, 18169}, {1010, 3753}, {1043, 10914}, {1155, 4278}, {1210, 17197}, {1216, 6881}, {1325, 5885}, {1385, 4225}, {1532, 10110}, {1714, 4259}, {1715, 7416}, {1730, 16453}, {1764, 16287}, {1790, 10202}, {1817, 9940}, {1824, 14016}, {1872, 3559}, {1884, 13408}, {1953, 2179}, {2194, 5358}, {2392, 11263}, {2476, 3060}, {2646, 4276}, {2771, 12826}, {2979, 4197}, {3057, 4653}, {3125, 16716}, {3193, 4228}, {3555, 5208}, {3567, 6830}, {3579, 4184}, {3670, 16696}, {3697, 3786}, {3791, 3874}, {3819, 17529}, {3868, 16704}, {3877, 17588}, {3917, 8728}, {3931, 10458}, {3976, 18173}, {4183, 5908}, {4187, 5943}, {4193, 5640}, {5044, 5235}, {5084, 17183}, {5141, 11002}, {5173, 5324}, {5249, 11573}, {5398, 13733}, {5446, 6842}, {5462, 6882}, {5499, 13391}, {5663, 16160}, {5706, 13730}, {5709, 8021}, {5751, 14018}, {5810, 6515}, {5889, 6828}, {5890, 6845}, {5907, 8226}, {6688, 17575}, {6829, 11412}, {6841, 13754}, {6941, 9781}, {6943, 15043}, {6963, 15024}, {6990, 11459}, {6991, 11444}, {7419, 10222}, {8679, 13407}, {9955, 14008}, {10473, 16466}, {10883, 12111}, {11113, 15488}, {12699, 14956}, {13374, 17188}, {13587, 14131}, {14011, 17619}, {14964, 16699}, {15329, 18115}, {16608, 17171}


X(18181) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18181) lies on these lines: {81, 1576}, {1634, 16701}, {3675, 16726}, {7202, 18191}, {17197, 17463}, {18165, 18175}, {18179, 18190}


X(18182) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18182) lies on these lines: {81, 18167}, {354, 18166}, {3125, 16707}, {17193, 17456}, {18169, 18204}


X(18183) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18183) lies on these lines: {1, 2916}, {6, 982}, {38, 141}, {69, 4392}, {244, 3589}, {354, 18166}, {518, 3293}, {522, 3663}, {742, 4022}, {984, 3763}, {986, 3242}, {1086, 17446}, {1193, 9021}, {1386, 3953}, {1401, 3313}, {1634, 16696}, {3619, 7226}, {3677, 7289}, {4475, 17445}, {4642, 9053}, {7202, 18170}, {7263, 17872}, {12329, 17595}


X(18184) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18184) lies on these lines: {1, 16680}, {81, 163}, {86, 3732}, {116, 17198}, {1018, 16728}, {1111, 3942}, {2087, 16742}, {2170, 7208}, {3125, 7202}, {3670, 18171}, {3675, 18191}, {3953, 18167}, {4424, 16696}, {5902, 18161}, {16727, 17761}, {18177, 18186}


X(18185) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(142), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18185) lies on these lines: {1, 859}, {3, 4658}, {6, 16058}, {21, 3241}, {55, 81}, {58, 3052}, {86, 1376}, {100, 8025}, {165, 18164}, {171, 18166}, {284, 11051}, {333, 1001}, {518, 17185}, {524, 4199}, {584, 9306}, {846, 4068}, {940, 4191}, {999, 4276}, {1010, 3913}, {1191, 4281}, {1621, 16704}, {1961, 4557}, {2177, 17187}, {3058, 14956}, {3304, 4225}, {3666, 18210}, {3746, 17524}, {3748, 18191}, {3750, 18169}, {3791, 16684}, {4413, 5333}, {4421, 13588}, {4423, 5235}, {4436, 4697}, {4640, 18206}, {4653, 6767}, {5396, 5891}, {10389, 17194}, {11235, 14009}, {11238, 14008}, {13405, 17197}, {16056, 17392}, {16696, 17594}, {17167, 17718}, {17592, 18161}, {17601, 18198}


X(18186) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18186) lies on these lines: {1, 3286}, {9, 16726}, {86, 3731}, {333, 8056}, {1743, 18206}, {3729, 17178}, {4384, 16710}, {4859, 16713}, {4862, 17197}, {10980, 18169}, {16673, 17207}, {16709, 16832}, {18177, 18184}, {18192, 18193}


X(18187) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(162), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18187) lies on these lines: {918, 1086}, {1437, 18167}


X(18188) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(163), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18188) lies on this line: {1, 16873}


X(18189) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(171), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18189) lies on these lines: {1, 16681}, {274, 3959}, {942, 18166}, {982, 18172}, {986, 16696}, {3125, 17175}, {3670, 18171}, {3727, 16705}, {3735, 16887}


X(18190) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(172), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (b^5 + a b^3 c + a b c^3 + c^5) : :

X(18190) lies on these lines: {982, 18175}, {4475, 16738}, {16696, 18168}, {17202, 17470}, {18169, 18203}, {18179, 18181}, {18206, 18207}


X(18191) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18191) lies on these lines: {1, 18174}, {2, 3909}, {11, 124}, {21, 643}, {28, 1408}, {55, 17194}, {58, 65}, {81, 105}, {86, 9432}, {140, 10035}, {210, 333}, {244, 659}, {255, 2218}, {284, 17603}, {314, 4519}, {513, 3120}, {518, 16704}, {654, 2170}, {859, 1319}, {867, 8286}, {940, 5020}, {982, 18192}, {1010, 3698}, {1014, 4637}, {1043, 3893}, {1086, 2969}, {1122, 17189}, {1155, 3286}, {1333, 2262}, {1351, 4383}, {1357, 1358}, {1437, 5358}, {2328, 17642}, {2646, 4267}, {2810, 17724}, {3011, 8679}, {3025, 7336}, {3030, 6174}, {3056, 16713}, {3216, 5482}, {3220, 5137}, {3666, 18169}, {3675, 18184}, {3683, 17185}, {3733, 15635}, {3737, 14115}, {3742, 8025}, {3748, 18185}, {3752, 17187}, {3838, 17173}, {3952, 16729}, {3999, 18173}, {4003, 16696}, {4653, 5919}, {4658, 17609}, {4679, 17183}, {4860, 18164}, {5087, 17174}, {5650, 17337}, {5836, 11115}, {7192, 16727}, {7202, 18181}, {13751, 17104}, {14204, 15343}, {14923, 17539}, {17167, 17605}


X(18192) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18192) lies on these lines: {1, 21}, {43, 16704}, {87, 16738}, {333, 16569}, {354, 18174}, {741, 8706}, {979, 1698}, {982, 18191}, {984, 18211}, {1613, 16552}, {2162, 16975}, {3248, 6682}, {3286, 3550}, {3840, 17178}, {3944, 17197}, {7184, 16713}, {16696, 17591}, {16726, 17063}, {17596, 18163}, {18176, 18208}, {18186, 18193}


X(18193) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(193), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18193) lies on these lines: {1, 3}, {9, 17063}, {38, 3306}, {45, 3848}, {63, 244}, {72, 11512}, {88, 3681}, {200, 1054}, {226, 14494}, {238, 3928}, {240, 1435}, {291, 5223}, {329, 5121}, {511, 7248}, {612, 4392}, {613, 1407}, {614, 1707}, {846, 10582}, {984, 5437}, {1086, 17064}, {1111, 7182}, {1357, 3784}, {1376, 16496}, {1743, 3509}, {3052, 4906}, {3062, 3551}, {3219, 9335}, {3403, 6384}, {3554, 18168}, {3729, 3840}, {3731, 17754}, {3751, 3752}, {3782, 17728}, {3816, 17276}, {3817, 4887}, {3846, 17274}, {3870, 17449}, {3929, 17123}, {3944, 4862}, {4310, 5435}, {4414, 4666}, {4421, 4864}, {4438, 17282}, {4650, 7290}, {4654, 17717}, {5220, 16602}, {5231, 17889}, {6682, 10436}, {7174, 17122}, {7204, 7271}, {7293, 7298}, {11246, 17721}, {17156, 17495}, {17194, 18173}, {18186, 18192}


X(18194) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18194) lies on these lines: {1, 6}, {43, 1964}, {48, 8300}, {75, 87}, {183, 5272}, {239, 1740}, {308, 3761}, {330, 2998}, {385, 614}, {386, 4991}, {513, 3551}, {577, 10801}, {604, 1582}, {612, 3329}, {869, 17121}, {978, 4974}, {979, 4385}, {982, 18207}, {1015, 17065}, {1045, 16834}, {1333, 10789}, {2309, 4393}, {3009, 17349}, {3097, 5069}, {3510, 17026}, {3550, 3941}, {3729, 9359}, {3758, 17445}, {3783, 5839}, {4000, 7184}, {4361, 16571}, {5019, 12194}, {5042, 10800}, {5268, 11174}, {7189, 17363}, {7191, 7766}, {7202, 18168}, {7280, 8266}, {11364, 16946}, {16569, 17348}, {16696, 17591}, {18161, 18208}


X(18195) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(213), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (a b^4 + b^4 c + a c^4 + b c^4) : :

X(18195) lies on these lines: {81, 560}, {244, 16709}, {982, 16696}, {1386, 3953}, {3122, 17202}, {4022, 16738}, {18167, 18168}, {18179, 18181}


X(18196) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(512), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18196) lies on these lines: {1, 16692}, {81, 1919}, {798, 17212}, {1019, 4762}, {3733, 4782}, {4057, 18166}


X(18197) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(513), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a + b) (a + c) (b - c) (a b + a c - b c) : :

X(18197) lies on these lines: {1, 669}, {57, 7180}, {239, 514}, {650, 4481}, {659, 3737}, {667, 18173}, {798, 4369}, {812, 18071}, {1054, 9361}, {1635, 16751}, {2978, 4040}, {3733, 4782}, {3835, 17217}, {4083, 8640}, {4375, 17185}, {4784, 8672}


X(18198) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(519), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18198) lies on these lines: {1, 3286}, {37, 17207}, {44, 16726}, {45, 86}, {81, 89}, {88, 16704}, {330, 4361}, {3733, 4782}, {3834, 16723}, {4286, 17378}, {4395, 16711}, {4480, 17195}, {4887, 17197}, {16709, 16815}, {16710, 16816}, {16714, 17366}, {17067, 17205}, {17601, 18185}


X(18199) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(522), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18199) lies on these lines: {6, 4369}, {81, 6654}, {222, 4077}, {514, 7254}, {661, 940}, {1019, 8712}, {1021, 7203}, {1396, 17926}, {3733, 4782}, {4160, 5711}, {4833, 18166}, {4885, 17218}, {15419, 17069}


X(18200) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(523), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18200) lies on these lines: {1, 16874}, {81, 649}, {86, 3835}, {171, 7234}, {385, 4369}, {661, 9810}, {1019, 1429}, {1412, 3676}, {1580, 4367}, {1919, 4932}, {3733, 4782}, {3776, 4817}, {4107, 16737}


X(18201) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(524), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18201) lies on these lines: {1, 3}, {7, 17717}, {38, 5297}, {44, 1052}, {45, 17754}, {58, 17513}, {63, 17063}, {88, 291}, {89, 985}, {100, 17449}, {190, 4871}, {226, 7608}, {238, 244}, {511, 1357}, {518, 1054}, {527, 5121}, {535, 6788}, {537, 5205}, {614, 4650}, {748, 9335}, {750, 4392}, {752, 5211}, {846, 3742}, {902, 3315}, {984, 3306}, {1086, 17070}, {1647, 5057}, {1707, 5573}, {1757, 16610}, {2243, 16786}, {3286, 18173}, {3733, 4782}, {3752, 4663}, {3756, 17768}, {3911, 17719}, {3928, 5272}, {3938, 9352}, {3944, 17728}, {4031, 17722}, {4310, 17725}, {4414, 16484}, {4495, 18075}, {4649, 4850}, {4661, 9350}, {4716, 17162}, {4722, 17020}, {4887, 9436}, {4902, 7988}, {6384, 7244}, {7064, 15082}, {7226, 17124}, {7293, 7302}, {9324, 9451}, {17767, 17777}


X(18202) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(757), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18202) lies on these lines: {1, 199}, {2, 3125}, {306, 3721}, {982, 18210}, {1211, 4016}, {3666, 18179}, {3670, 18203}, {3782, 14213}, {3959, 5271}, {6155, 17011}


X(18203) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(1333), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

Barycentrics    a (a b^4 + b^5 + a c^4 + c^5) : :

X(18203) lies on these lines: {1, 22}, {982, 18161}, {2887, 4118}, {3670, 18202}, {3741, 4475}, {4137, 17184}, {4425, 17470}, {18167, 18175}, {18169, 18190}


X(18204) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(1914), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18204) lies on these lines: {1, 16876}, {982, 18167}, {3670, 18202}, {16696, 18168}, {18169, 18182}, {18206, 18208}


X(18205) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(291), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18205) lies on these lines: {1, 16683}, {81, 4586}, {1100, 16726}, {3670, 18171}, {16696, 18170}, {18172, 18178}


X(18206) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(350), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18206) lies on these lines: {1, 21}, {2, 2350}, {6, 980}, {7, 16713}, {9, 86}, {37, 18166}, {40, 4229}, {44, 16726}, {57, 85}, {69, 579}, {71, 3879}, {110, 2725}, {141, 583}, {144, 17183}, {193, 573}, {194, 1764}, {200, 13588}, {213, 4641}, {239, 514}, {284, 1444}, {310, 17026}, {314, 3729}, {329, 17182}, {518, 2223}, {524, 2245}, {527, 17139}, {553, 17050}, {599, 5043}, {660, 1757}, {666, 2311}, {672, 3912}, {869, 17187}, {894, 10455}, {940, 5021}, {982, 16476}, {988, 4281}, {1014, 1445}, {1018, 6542}, {1020, 17950}, {1026, 4447}, {1043, 6762}, {1282, 8849}, {1333, 16973}, {1400, 4416}, {1412, 1708}, {1475, 16705}, {1723, 17189}, {1724, 16048}, {1730, 4209}, {1743, 18186}, {1756, 17770}, {1765, 10446}, {1778, 16970}, {1992, 4266}, {2111, 17738}, {2149, 4564}, {2176, 18172}, {2260, 4357}, {2303, 16517}, {2323, 6518}, {3008, 16752}, {3219, 3294}, {3285, 16702}, {3305, 5333}, {3306, 5235}, {3333, 11110}, {3501, 17294}, {3555, 17524}, {3629, 4271}, {3661, 16549}, {3666, 6155}, {3730, 17316}, {3736, 3751}, {3786, 5223}, {3811, 4278}, {3870, 4184}, {3928, 9311}, {4258, 16436}, {4269, 7289}, {4282, 4558}, {4383, 16700}, {4640, 18185}, {4715, 16723}, {4921, 16833}, {5036, 6144}, {5228, 16699}, {5231, 14009}, {5236, 9436}, {5247, 16735}, {5278, 17683}, {5905, 17167}, {6646, 17202}, {6904, 9534}, {7146, 18176}, {7719, 14013}, {9965, 17753}, {10471, 11679}, {13476, 16684}, {16050, 17742}, {16497, 18174}, {16709, 17277}, {16710, 17349}, {17173, 17483}, {17174, 17484}, {17178, 17350}, {17308, 17754}, {17755, 18157}, {18190, 18207}, {18204, 18208}


X(18207) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18207) lies on these lines: {1, 159}, {6, 18168}, {256, 17447}, {320, 4118}, {894, 4475}, {982, 18194}, {1582, 7291}, {2643, 7321}, {3942, 7184}, {6646, 17470}, {7202, 18170}, {7237, 17288}, {18190, 18206}


X(18208) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(385), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18208) lies on these lines: {1, 3}, {6, 18168}, {58, 17517}, {226, 3399}, {238, 17799}, {239, 4475}, {244, 1959}, {304, 6384}, {335, 726}, {518, 3507}, {1111, 18033}, {1580, 3218}, {1920, 1930}, {3061, 16604}, {3673, 3944}, {3674, 3865}, {3726, 4876}, {3752, 17795}, {3840, 17760}, {4118, 16706}, {7237, 17291}, {7291, 8300}, {17302, 17470}, {18161, 18194}, {18176, 18192}, {18204, 18206}


X(18209) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(626), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18209) lies on these lines: {1, 2916}, {6, 7166}, {141, 1580}, {171, 18162}, {320, 7122}, {560, 3662}, {982, 1631}, {1030, 17596}, {1086, 1582}, {1429, 18170}, {2210, 16706}, {3821, 5009}, {4112, 18144}, {7113, 7184}, {8300, 17366}


X(18210) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18210) lies on these lines: {1, 1283}, {2, 16100}, {4, 2828}, {11, 2969}, {56, 11383}, {63, 17972}, {65, 1410}, {77, 17973}, {81, 18175}, {122, 125}, {228, 1214}, {244, 665}, {517, 15626}, {982, 18202}, {1364, 3270}, {1365, 2611}, {1427, 1824}, {1437, 7100}, {1565, 2968}, {1566, 4988}, {2631, 2632}, {2771, 18115}, {3122, 17476}, {3666, 18185}, {3827, 8758}, {7202, 18181}, {8731, 16585}


X(18211) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION IMAGE OF X(646), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

X(18211) lies on these lines: {81, 643}, {244, 659}, {984, 18192}, {15569, 18169}


X(18212) = DAO PERSPECTOR OF X(5)

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

In the plane of a triangle ABC, let P be a point. Let
(BCP) = cirumcircle of BCP, and define (CAP) and (ABP) cyclically;
A' = the point of interection, other than P, or AP and (BCP), and define B' and C' cyclically;
A'' = reflection of A' in the center of (BCP), and define B'' and C'' cyclically.
Then A''B''C'' is perspective to ABC. (Thanh Oai Dao, April 29, 2018)

X(18212) and X(18213) were contributed by César Lozada, April 30, 2018, with the following notes. The perspector of ABC and A''B''C'' is here named the Dao perspector of P, denoted by D(P). The appearance of (i,j) in the following list means that D(X(i)) = X(j): (1, 1), (2, 13608), (3, 24), (4, 68), (5,18212), (6, 14262), (13, 18), (14,17), (15, 62), (16, 61), (20,18213), (36, 35), (59, 11247), (80, 79), (249, 10279), (265, 14111). In general, if P = x : y : z (barycentrics), then

D(P) = (-x^2*SA+SB*x*y+SC*x*z+2*a^2*y*z)*(-SC*c^2*x*y+SB*b^2*x*z-2*a^2*c^2*y^2-SA*a^2*y*z)*(SC*c^2*x*y-SB*b^2*x*z-SA*a^2*y*z-2*a^2*b^2*z^2) : :

X(18212) lies on these lines: {3, 12325}, {5, 10227}, {49, 6150}, {186, 3432}, {1141, 3520}, {15620, 17506}


X(18213) = DAO PERSPECTOR OF X(20)

Barycentrics    SA*(S^2-4*SB*SC)*(3*S^2-16*R^2*SB+3*SB^2-2*SA*SC)*(3*S^2-16*R^2*SC+3*SC^2-2*SA*SB) : :

See X(18212).

X(18213) lies on these lines: {3, 1661}, {5896, 6622}





leftri  Zaniah triangles and related centers: X(18214) - X(18261)  rightri

This preamble and centers X(18214)-X(18261) were contributed by César Eliud Lozada, May 01, 2018.

In a triangle ABC, let AmBmCm be the medial triangle, I the incenter and A' the touchpoint of the incircle with the side BC. Then the lines AA', IAm and BmCm are concurrent at a point A1. Denote B1 and C1 cyclically. The triangle A1B1C1 is here named the 1st Zaniah triangle of ABC.

Continuing with the previous construction, let Ja be the A-excenter and A" the touchpoint of the A-excircle with the side BC. Then the lines AA", JaAm and BmCm are concurrent at a point A2. Denote B2 and C2 cyclically. The triangle A2B2C2 is here named the 2nd Zaniah triangle of ABC. [See note (*) below.]

Barycentric coordinates of the first vertex of each triangle are:

  A1 = 2*a : a+b-c : a-b+c

  A2 = 2*a : a-b+c : a+b-c

The vertices of both triangles lie on a conic with center X(18214) and perspector X(18215).

  • Perspective triangles and perspectors (An asterisk * means that the triangles are homothetic; two asterisks means that triangles are inversely similar; two dashes -- means not calculated center)

  • 1st Zaniah triangle:
    (ABC, 7), (Andromeda, 18216), (anti-Aquila, 946), (Aquila, 18217), (Ascella, 942), (Conway, 7), (2nd Conway, 7), (3rd Conway, 18218), (extouch*, 2), (outer-Garcia, 1), (hexyl, 18219), (Honsberger, 7), (Hutson extouch, 7), (Hutson intouch, 18220), (incircle-circles, 942), (intouch, 7), (medial, 1), (5th mixtilinear, 18221), (6th mixtilinear, 18222), (inner-Yff, 18223), (outer-Yff, 18224), (inner-Yff tangents, 18225), (outer-Yff tangents, 18226)

    2nd Zaniah triangle:
    (ABC, 8), (Ascella*, 5745), (Atik*, 18227), (1st circumperp*, 1376), (2nd circumperp*, 958), (inner-Conway*, 2), (Conway*, 5273), (2nd Conway*, 18228), (3rd Conway*, 18229), (3rd Euler*, 2886), (4th Euler*, 1329), (excenters-midpoints, 10), (excenters-reflections*, 15829), (excentral*, 9), (extouch, 8), (2nd extouch*, 9), (Garcia-reflection, 9), (hexyl*, 936), (Honsberger*, 18230), (Hutson extouch, 18231), (inner-Hutson*, --), (Hutson intouch*, 8), (outer-Hutson*, --), (incircle-circles*, 1125), (intouch*, 2), (inverse-in-incircle*, 518), (K798e **, 18232), (K798i, 18233), (medial, 9), (6th mixtilinear*, 8580), (2nd Pamfilos-Zhou*, 18234), (1st Sharygin*, 18235), (tangential-midarc*, 188), (2nd tangential-midarc*, 7028), (Ursa-major*, 18236), (Ursa-minor*, 210), (Yff central*, 236)

  • Orthologic triangles and orthologic centers

  • 1st Zaniah triangle:
    (ABC, 946, 84), (ABC-X3 reflections, 946, 1490), (anti-Aquila, 946, 12114), (anti-Ara, 946, 12136), (5th anti-Brocard, 946, 12196), (anti-Euler, 946, 12246), (anti-Mandart-incircle, 946, 12330), (anticomplementary, 946, 6223), (Aquila, 946, 7992), (Ara, 946, 9910), (Ascella, 942, 9942), (Atik, 942, 9948), (1st Auriga, 946, 12456), (2nd Auriga, 946, 12457), (5th Brocard, 946, 12496), (2nd circumperp tangential, 946, 18237), (1st circumperp, 942, 1158), (2nd circumperp, 942, 6261), (inner-Conway, 942, 12528), (Conway, 942, 9960), (2nd Conway, 942, 9799), (3rd Conway, 942, 12547), (Euler, 946, 6245), (3rd Euler, 942, 12608), (4th Euler, 942, 12616), (excenters-midpoints, 1, 3), (excenters-reflections, 942, 12650), (excentral, 942, 1490), (extouch, 18238, 18239), (2nd extouch, 942, 12664), (Fuhrmann, 1387, 6256), (2nd Fuhrmann, 11544, 16127), (inner-Garcia, 15528, 12666), (outer-Garcia, 946, 12667), (Garcia-reflection, 1, 4), (Gossard, 946, 12668), (inner-Grebe, 946, 6258), (outer-Grebe, 946, 6257), (hexyl, 942, 84), (Honsberger, 942, 12669), (Hutson extouch, 18241, 12671), (inner-Hutson, 942, 12673), (Hutson intouch, 942, 12672), (outer-Hutson, 942, 12674), (incircle-circles, 942, 12675), (intouch, 942, 1071), (inverse-in-incircle, 942, 946), (Johnson, 946, 6259), (inner-Johnson, 946, 12676), (outer-Johnson, 946, 12677), (1st Johnson-Yff, 946, 12678), (2nd Johnson-Yff, 946, 12679), (K798e, 1387, 18242), (K798i, 11544, 18243), (Lucas homothetic, 946, 18245), (Lucas(-1) homothetic, 946, 18246), (Mandart-incircle, 946, 12680), (medial, 946, 6260), (5th mixtilinear, 946, 7971), (6th mixtilinear, 942, 7992), (2nd Pamfilos-Zhou, 942, 12681), (1st Schiffler, 18244, 10308), (2nd Schiffler, 1, 104), (1st Sharygin, 942, 12683), (tangential-midarc, 942, 8095), (2nd tangential-midarc, 942, 8096), (3rd tri-squares-central, 946, 8987), (4th tri-squares-central, 946, 13974), (Ursa-major, 942, 17649), (Ursa-minor, 942, 12688), (X3-ABC reflections, 946, 12684), (Yff central, 942, 12685), (inner-Yff, 946, 1709), (outer-Yff, 946, 10085), (inner-Yff tangents, 946, 12686), (outer-Yff tangents, 946, 12687), (2nd Zaniah, 942, 5777)

    2nd Zaniah triangle:
    (ABC, 10, 1), (ABC-X3 reflections, 10, 40), (Andromeda, 18247, 1), (anti-Aquila, 10, 1), (anti-Ara, 10, 1829), (5th anti-Brocard, 10, 12194), (anti-Euler, 10, 944), (anti-Mandart-incircle, 10, 3), (anticomplementary, 10, 8), (Antlia, 18248, 1), (Aquila, 10, 1), (Ara, 10, 9798), (Ascella, 960, 942), (Atik, 960, 8), (1st Auriga, 10, 55), (2nd Auriga, 10, 55), (Ayme, 5044, 10), (5th Brocard, 10, 9941), (2nd circumperp tangential, 10, 3), (1st circumperp, 960, 40), (2nd circumperp, 960, 1), (inner-Conway, 960, 3869), (Conway, 960, 3868), (2nd Conway, 960, 8), (3rd Conway, 960, 12435), (4th Conway, 18249, 1), (5th Conway, 18250, 1), (Euler, 10, 946), (3rd Euler, 960, 946), (4th Euler, 960, 10), (excenters-midpoints, 8, 1), (excenters-reflections, 960, 1), (excentral, 960, 1), (extouch, 5777, 72), (2nd extouch, 960, 72), (4th extouch, 18251, 65), (5th extouch, 18252, 65), (Fuhrmann, 3036, 8), (2nd Fuhrmann, 18253, 4), (inner-Garcia, 18254, 3869), (outer-Garcia, 10, 8), (Garcia-reflection, 8, 8), (Gossard, 10, 12438), (inner-Grebe, 10, 3641), (outer-Grebe, 10, 3640), (hexyl, 960, 40), (Honsberger, 960, 7672), (Hutson extouch, 18255, 3555), (inner-Hutson, 960, 9805), (Hutson intouch, 960, 3057), (outer-Hutson, 960, 9806), (incentral, 5044, 1), (incircle-circles, 960, 942), (intouch, 960, 65), (inverse-in-incircle, 960, 1), (Johnson, 10, 355), (inner-Johnson, 10, 355), (outer-Johnson, 10, 355), (1st Johnson-Yff, 10, 5252), (2nd Johnson-Yff, 10, 1837), (K798e, 3036, 5690), (K798i, 18253, 5), (Lucas homothetic, 10, 12440), (Lucas(-1) homothetic, 10, 12441), (Malfatti, 18256, 1), (Mandart-excircles, 18257, 3555), (Mandart-incircle, 10, 3057), (medial, 10, 10), (midarc, 18258, 1), (2nd midarc, 18258, 1), (mixtilinear, 3452, 1), (2nd mixtilinear, 3452, 1), (5th mixtilinear, 10, 1), (6th mixtilinear, 960, 7991), (2nd Pamfilos-Zhou, 960, 9808), (1st Schiffler, 18259, 21), (2nd Schiffler, 8, 1320), (1st Sharygin, 960, 2292), (tangential-midarc, 960, 8093), (2nd tangential-midarc, 960, 8094), (3rd tri-squares-central, 10, 8983), (4th tri-squares-central, 10, 13971), (Ursa-major, 960, 10914), (Ursa-minor, 960, 3057), (X3-ABC reflections, 10, 1482), (Yff central, 960, 12445), (inner-Yff, 10, 1), (outer-Yff, 10, 1), (inner-Yff tangents, 10, 1), (outer-Yff tangents, 10, 1), (1st Zaniah, 5777, 942)

  • Parallelogic triangles and parallelogic centers

  • 1st Zaniah triangle:
    (1st Parry, 946, 13254), (2nd Parry, 946, 13255), (2nd Sharygin, 942, 13256)

    2nd Zaniah triangle:
    (Fuhrmann, 3035, 4), (K798e, 3035, 5), (1st Parry, 10, 9810), (2nd Parry, 10, 9811), (2nd Sharygin, 960, 2254)

  • Inverse similar triangles and centers of inverse similitude

  • 2nd Zaniah triangle:
    (K798e, 10)

    (*) Note: the 1st Zaniah triangle is the cevian triangle of X(1) wrt the medial triangle, or equivalently, the extouch triangle of the medial triangle, or the complement of the extouch triangle. The 2nd Zaniah triangle is the cevian triangle of X(9) wrt the medial triangle, or equivalently, the intouch triangle of the medial triangle, or the complement of the intouch triangle. (Randy Hutson, June 27, 2018)
    underbar

    X(18214) = CENTER OF THE CIRCUMCONIC OF THESE TRIANGLES: 1st ZANIAH AND 2nd ZANIAH

    Barycentrics    (b^2+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b-c)*a+(b-c)^2*(b^2+c^2)^2 : :
    X(18214) = 3*X(2)+X(13577)

    This circumconic, as the complement of the Privalov conic, is an ellipse for all ABC.

    X(18214) lies on these lines: {2,1814}, {141,3740}, {1376,17060}, {3035,15260}, {3816,4885}, {3827,15497}, {4682,13405}

    X(18214) = complement of X(5452)
    X(18214) = X(14713) of 2nd Zaniah triangle
    X(18214) = {X(2), X(13577)}-harmonic conjugate of X(5452)


    X(18215) = PERSPECTOR OF THE CIRCUMCONIC OF THESE TRIANGLES: 1st ZANIAH AND 2nd ZANIAH

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

    X(18215) lies on the line {1376,17060}


    X(18216) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND ANDROMEDA

    Barycentrics    a*(a^4-2*(b+c)*a^3+4*(b^2+c^2)*a^2-6*(b^2-c^2)*(b-c)*a+(3*b^2+2*b*c+3*c^2)*(b-c)^2) : :
    X(18216) = (16*R^2-r^2-2*SW)*X(1)+SW*X(6)

    X(18216) lies on these lines: {1,6}, {7,4907}, {57,4319}, {77,7671}, {241,4326}, {269,14100}, {354,4328}, {497,3668}, {938,3755}, {950,7273}, {990,3333}, {1418,2951}, {2263,10384}, {3672,3677}, {4000,5573}, {4356,6744}, {7190,11025}, {7271,15726}, {7322,10578}, {9442,10390}


    X(18217) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND AQUILA

    Barycentrics    3*a^4+11*(b+c)*a^3-(b^2-26*b*c+c^2)*a^2-11*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
    X(18217) = 4*X(946)-3*X(11379)

    X(18217) lies on these lines: {1,3522}, {65,8275}, {142,1698}, {191,3338}, {354,5586}, {942,4355}, {944,14563}, {946,7992}, {1387,1768}, {1699,11544}, {3296,7991}, {3339,9898}, {3624,5273}, {3671,18220}, {3679,11024}, {3894,10855}, {4298,18221}, {4312,5572}, {5045,11034}, {9897,12736}, {10122,11220}, {10543,11518}, {10609,12653}, {10825,11021}, {12409,12917}

    X(18217) = reflection of X(1) in X(5558)
    X(18217) = X(5558) of Aquila triangle
    X(18217) = (inverse-in-incircle)-isotomic conjugate of-X(3945)
    X(18217) = {X(354), X(5586)}-harmonic conjugate of X(9589)


    X(18218) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND 3rd CONWAY

    Barycentrics   a^8+3*(b+c)*a^7-(11*b^2+18*b*c+11*c^2)*a^6-3*(b+c)*(11*b^2-2*b*c+11*c^2)*a^5-(13*b^4+13*c^4+2*(12*b^2+35*b*c+12*c^2)*b*c)*a^4+(b+c)*(25*b^4+25*c^4-2*(22*b^2-3*b*c+22*c^2)*b*c)*a^3+(23*b^4+23*c^4+6*(14*b^2+23*b*c+14*c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(5*b^4+5*c^4+6*(8*b^2+9*b*c+8*c^2)*b*c)*a+4*(b^2-c^2)^2*(b^2+6*b*c+c^2)*b*c : :

    X(18218) lies on these lines: {142,10442}, {942,12547}, {5836,10456}, {10446,11024}, {11281,16124}, {12551,12736}


    X(18219) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND HEXYL

    Barycentrics   a*(a^9-3*(b+c)*a^8+4*b*c*a^7+8*(b^3+c^3)*a^6-2*(3*b^4+3*c^4-2*(b^2-7*b*c+c^2)*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4-2*(2*b-c)*(b-2*c)*b*c)*a^4+4*(2*b^2+3*b*c+2*c^2)*(b-c)^4*a^3+8*(b^2-c^2)^2*(b+c)*b*c*a^2-(b^2-c^2)^2*(3*b^4+3*c^4-2*(6*b^2+7*b*c+6*c^2)*b*c)*a+(b^2-c^2)^3*(b-c)*(b^2-6*b*c+c^2)) : :
    X(18219) = (8*R^2-r^2)*X(4)+2*r*(4*R+r)*X(142)

    X(18219) lies on these lines: {1,9799}, {4,142}, {20,11024}, {40,958}, {63,12651}, {84,942}, {515,12855}, {517,12654}, {936,6847}, {952,12658}, {1490,8727}, {1768,3339}, {3062,9960}, {3295,7966}, {4292,11023}, {5261,7675}, {6001,18241}, {6837,8583}, {7992,12560}, {9624,11281}, {9948,14563}, {10431,12565}, {10582,10884}


    X(18220) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND HUTSON INTOUCH

    Barycentrics    (-a+b+c)*(3*a^3-(b+c)*a^2-7*(b-c)^2*a-3*(b^2-c^2)*(b-c)) : :
    X(18220) = 2*X(1)+X(7319)

    X(18220) lies on these lines: {1,3091}, {2,3057}, {4,1387}, {7,11522}, {10,4345}, {11,145}, {55,17572}, {142,390}, {496,6830}, {497,2475}, {519,5828}, {551,4313}, {938,13464}, {942,5603}, {944,7704}, {946,3600}, {962,5265}, {999,11544}, {1000,1656}, {1058,5901}, {1125,5281}, {1210,5734}, {1319,3146}, {1320,6931}, {1388,5225}, {1420,9812}, {1482,6978}, {1697,5550}, {1699,4308}, {1837,3623}, {1854,3315}, {1898,3485}, {2098,3617}, {3086,5903}, {3241,9581}, {3295,6946}, {3476,3832}, {3486,10129}, {3487,9844}, {3522,12701}, {3621,5048}, {3624,4342}, {3671,18217}, {3742,10866}, {3876,5686}, {3877,18231}, {4293,16118}, {4295,18223}, {4301,5435}, {4321,18222}, {4323,11019}, {4678,17606}, {4853,5328}, {4861,6919}, {5056,10051}, {5068,5252}, {5119,10303}, {5691,6049}, {5703,9624}, {5704,7982}, {5731,9614}, {5886,6964}, {6745,12541}, {7080,17648}, {7486,10039}, {7962,9780}, {7967,9669}, {9779,10106}, {10122,11036}, {10966,16865}, {11375,13867}

    X(18220) = X(3532) of Hutson intouch triangle
    X(18220) = X(15077) of inverse-in-incircle triangle
    X(18220) = (inverse-in-incircle)-isotomic conjugate of-X(4862)


    X(18221) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND 5th MIXTILINEAR

    Barycentrics    3*a^4-8*(b+c)*a^3-2*(b^2+4*b*c+c^2)*a^2+8*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
    X(18221) = 4*X(10390)-3*X(11038)

    X(18221) lies on these lines: {1,3523}, {2,11281}, {4,11544}, {7,5691}, {8,142}, {20,5441}, {40,15933}, {65,390}, {100,3304}, {145,354}, {519,11024}, {758,5129}, {938,946}, {942,944}, {952,3296}, {999,6942}, {1056,12645}, {1058,1159}, {1125,5775}, {1387,6833}, {2136,3241}, {3086,5425}, {3090,16137}, {3091,9803}, {3212,3945}, {3333,7966}, {3336,10304}, {3339,4313}, {3340,10580}, {3475,3617}, {3487,9956}, {3522,5221}, {3528,15174}, {3543,18244}, {3616,5837}, {3649,3832}, {4295,18224}, {4298,18217}, {4308,7990}, {4323,11019}, {4678,15888}, {4778,14812}, {4848,10578}, {5056,15079}, {5059,11246}, {5226,12563}, {5261,6993}, {5665,5809}, {5690,6989}, {5883,17580}, {6744,9785}, {6872,12917}, {7972,12736}, {7991,8236}, {8000,12855}, {8422,11033}, {9949,12560}, {10890,11021}, {11023,11037}

    X(18221) = midpoint of X(65) and X(13867)
    X(18221) = reflection of X(8) in X(11530)
    X(18221) = X(7320) of 5th mixtilinear triangle
    X(18221) = X(15740) of inverse-in-incircle triangle
    X(18221) = X(16936) of Hutson intouch triangle
    X(18221) = (inverse-in-incircle)-isotomic conjugate of-X(4888)
    X(18221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 11518, 11038), (11529, 17706, 938)


    X(18222) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND 6th MIXTILINEAR

    Barycentrics   a*(a^7-5*(b+c)*a^6+(9*b^2+14*b*c+9*c^2)*a^5-(b+c)*(5*b^2+14*b*c+5*c^2)*a^4-(5*b-c)*(b-5*c)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(9*b^2-50*b*c+9*c^2)*a^2-(5*b^4+5*c^4-6*(6*b^2+11*b*c+6*c^2)*b*c)*(b-c)^2*a+(b^2-c^2)*(b-c)^3*(b^2-10*b*c+c^2)) : :
    X(18222) = X(3062)+2*X(10390)

    X(18222) lies on these lines: {4,12855}, {9,5836}, {142,1699}, {516,5129}, {942,7992}, {1750,11218}, {3062,5572}, {4312,11023}, {4321,18220}, {4326,5226}, {4626,7271}, {5223,8001}, {5691,15006}, {9580,12859}, {9949,12560}, {11281,16143}, {12736,12767}


    X(18223) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND INNER-YFF

    Barycentrics   a^7+(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5-(b+c)*(3*b^2-4*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*(3*b^2-5*b*c+3*c^2)*b*c)*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(18223) = 2*r*(R+r)*X(63)+(3*R^2-8*R*r-3*r^2)*X(499) = 4*r*(4*R+r)*X(142)-(3*R^2+10*R*r+3*r^2)*X(498)

    X(18223) lies on these lines: {1,4190}, {4,15528}, {7,10591}, {11,10052}, {56,1387}, {63,499}, {65,10043}, {142,498}, {354,10044}, {553,946}, {942,1478}, {1479,7702}, {1737,11023}, {3086,14450}, {3333,10042}, {3874,10057}, {4295,18220}, {4860,11544}, {5045,11045}, {5836,12647}, {5902,10532}, {8071,11281}, {10039,11024}, {10045,10980}, {10059,12855}, {10090,11517}, {10106,14563}, {10122,16152}, {11019,18224}, {12917,13128}

    X(18223) = X(10940) of outer-Yff triangle
    X(18223) = (inverse-in-incircle)-isotomic conjugate of-X(77)


    X(18224) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND OUTER-YFF

    Barycentrics   a^7+(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5-3*(b+c)*(b^2+c^2)*a^4+(3*b^4+3*c^4-2*(b^2+7*b*c+c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(3*b^2+4*b*c+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
    X(18224) = 4*r*(4*R+r)*X(142)+(5*R^2-14*R*r-3*r^2)*X(499) = (3*r^2+16*R*r+5*R^2)*X(498)-2*r*(7*R+r)*X(3305)

    X(18224) lies on these lines: {1,5905}, {11,10044}, {65,10051}, {142,499}, {354,10052}, {498,3305}, {942,1479}, {946,3982}, {950,14563}, {1387,3304}, {1478,1898}, {1737,11024}, {2099,15171}, {3333,10050}, {3419,5836}, {4295,18221}, {5045,11046}, {10056,10075}, {10073,12736}, {10092,10980}, {10122,16153}, {10531,15528}, {11019,18223}, {12647,15862}, {12917,13129}

    X(18224) = X(10941) of inner-Yff triangle
    X(18224) = (outer-Yff)-isogonal conjugate of-X(8071)
    X(18224) = (inverse-in-incircle)-isotomic conjugate of-X(7190)


    X(18225) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND INNER-YFF TANGENTS

    Barycentrics   a^10-(5*b^2-6*b*c+5*c^2)*a^8+8*(b+c)*b*c*a^7+2*(5*b^4+5*c^4-2*(7*b^2-6*b*c+7*c^2)*b*c)*a^6-16*(b^2-c^2)*(b-c)*b*c*a^5-2*(b^2+c^2)*(5*b^4+5*c^4-2*(10*b^2-9*b*c+10*c^2)*b*c)*a^4+8*(b^2-c^2)*(b-c)^3*b*c*a^3+(b^2-c^2)^2*(5*b^4+5*c^4-2*(10*b^2-17*b*c+10*c^2)*b*c)*a^2-(b^2-c^2)^4*(b-c)^2 : :

    X(18225) lies on these lines: {65,10935}, {142,5552}, {354,10940}, {942,12115}, {946,12686}, {1387,12775}, {5045,11047}, {5261,11023}, {5836,12648}, {7956,11544}, {10122,16154}, {10915,11024}, {10970,10971}, {12736,12749}, {12855,12874}, {12917,13130}

    X(18225) = (inner-Yff tangents)-isogonal conjugate of-X(3303)


    X(18226) = PERSPECTOR OF THESE TRIANGLES: 1st ZANIAH AND OUTER-YFF TANGENTS

    Barycentrics   a^10-(5*b^2+2*b*c+5*c^2)*a^8-8*(b+c)*b*c*a^7+2*(5*b^4+5*c^4+6*(b-c)^2*b*c)*a^6+16*(b+c)^3*b*c*a^5-2*(5*b^6+5*c^6+(8*b^4+8*c^4-25*(b^2+c^2)*b*c)*b*c)*a^4-8*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*b*c*a^3+(b^2-c^2)^2*(5*b^4+5*c^4+2*(2*b^2-7*b*c+2*c^2)*b*c)*a^2-(b^2-c^2)^4*(b-c)^2 : :

    X(18226) lies on these lines: {1,6992}, {65,10936}, {142,10527}, {354,10941}, {942,12116}, {946,12687}, {1387,12776}, {3304,11281}, {3333,11920}, {4863,5836}, {5045,11048}, {10122,16155}, {10916,11024}, {10970,10971}, {12736,12750}, {12855,12875}, {12917,13131}


    X(18227) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND ATIK

    Barycentrics    a*(-a+b+c)*((b+c)*a^3-(b^2+c^2)*a^2-(b+c)*(b^2-6*b*c+c^2)*a+(b^2+4*b*c+c^2)*(b-c)^2) : :
    X(18227) = 3*X(210)+X(497) = X(1376)-3*X(3740) = 3*X(5927)+X(10860)

    X(18227) lies on these lines: {1,18247}, {2,8581}, {8,210}, {9,165}, {10,9842}, {57,8169}, {65,8165}, {188,11858}, {200,10384}, {226,1329}, {236,11860}, {354,5328}, {515,5044}, {516,10241}, {518,3452}, {936,10864}, {958,1420}, {1125,11035}, {1699,5836}, {2886,10863}, {3035,13227}, {3036,9951}, {3742,10569}, {3820,6001}, {3925,5123}, {5204,5302}, {5658,9943}, {5745,10855}, {5777,9948}, {7028,11859}, {9949,18251}, {9950,18252}, {9952,18254}, {9953,18255}, {10862,18229}, {10865,18230}, {10868,18235}, {11519,15829}, {12386,18248}, {12446,18249}, {12449,18257}, {12450,18258}, {12451,18259}, {16120,18253}

    X(18227) = X(13567) of Atik triangle
    X(18227) = X(13567) of 2nd Zaniah triangle
    X(18227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11678, 8581), (8, 10866, 12448), (9, 18236, 3740), (210, 17604, 8), (210, 18228, 960), (3740, 15587, 8580), (5044, 9947, 12447), (5927, 8580, 15587), (8580, 10860, 1376)


    X(18228) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND 2nd CONWAY

    Barycentrics    (-a+b+c)*(a^2+2*(b+c)*a+(b-c)^2) : :
    X(18228) = 12*R*X(2)-(4*R+r)*X(7) = (4*R+r)*X(8)-12*R*X(210) = 8*R*X(10)-r*X(962)

    X(18228) is the homothetic center of the complement of the intouch triangle and the anticomplement of the excentral triangle. (Randy Hutson, June 27, 2018)

    Let A'B'C' be as defined at X(5658). A'B'C' is homothetic to the medial triangle and the orthic-of-intouch triangle at X(226), and to the anticomplementary triangle at X(18228). (Randy Hutson, June 27, 2018)

    Let A"B"C" be the Hutson-extouch triangle. Let La be the tangent to the A-excircle at A", and define Lb and Lc cyclically. Let A* = Lb∩Lc, B* = Lc∩La, C* = La∩Lb. Then A*B*C* is homothetic to ABC and the orthic-of-intouch triangle at X(57), to the anticomplementary triangle at X(144), and to the medial triangle at X(18228). (Randy Hutson, June 27, 2018)

    X(18228) lies on these lines: {1,5129}, {2,7}, {3,5658}, {4,5044}, {8,210}, {10,962}, {11,3715}, {20,936}, {72,938}, {78,452}, {145,3984}, {188,9793}, {200,390}, {220,4383}, {223,3160}, {236,11891}, {281,469}, {333,6557}, {344,4417}, {345,3161}, {346,3687}, {391,11679}, {392,3421}, {405,5703}, {516,8580}, {517,6939}, {518,10580}, {612,4344}, {651,17811}, {857,1211}, {942,17559}, {946,6766}, {950,12536}, {958,3304}, {997,5731}, {1001,10578}, {1058,6764}, {1125,5234}, {1212,5308}, {1260,5766}, {1329,2476}, {1376,5698}, {1532,3820}, {1698,4295}, {1737,5775}, {1848,7079}, {1997,14829}, {2324,5256}, {2345,5743}, {2550,3740}, {2886,9779}, {2999,3672}, {3030,3038}, {3035,9809}, {3036,9802}, {3058,3711}, {3061,17316}, {3083,17805}, {3084,17802}, {3090,5791}, {3241,5289}, {3434,18236}, {3474,4413}, {3487,11108}, {3488,3940}, {3522,5438}, {3523,6700}, {3600,8583}, {3601,11106}, {3617,5837}, {3683,5218}, {3689,10385}, {3697,5082}, {3752,4419}, {3816,5220}, {3869,13601}, {3870,8236}, {3878,15104}, {3883,7172}, {3890,7320}, {3916,17567}, {3927,17527}, {3945,17022}, {3983,12701}, {4000,4415}, {4023,4387}, {4187,5704}, {4208,9612}, {4292,17580}, {4307,5268}, {4310,5272}, {4339,5293}, {4342,4915}, {4358,5739}, {4512,5281}, {4847,5274}, {4855,17576}, {4882,12575}, {5046,5175}, {5056,5705}, {5086,7319}, {5123,5180}, {5195,5199}, {5223,11019}, {5241,7229}, {5250,7080}, {5302,5550}, {5440,11111}, {5603,9708}, {5691,12447}, {5709,6964}, {5714,8728}, {5719,16857}, {5720,6987}, {5741,17776}, {5768,6947}, {5777,6865}, {5784,10430}, {5804,6898}, {5812,6864}, {5817,8727}, {5828,10039}, {5850,10980}, {6147,16853}, {6349,16596}, {6361,9709}, {6734,6919}, {6838,12514}, {6840,10176}, {6910,15823}, {6926,7330}, {7028,9795}, {8582,12526}, {9789,18234}, {9791,18235}, {9797,18247}, {9800,18251}, {9801,18252}, {9803,18254}, {9804,18255}, {9807,18258}, {10446,18229}, {10582,11038}, {11374,16845}, {12391,18248}, {12542,18257}, {12543,18259}, {13411,17558}, {14450,18253}, {16020,17123}, {16602,17276}, {16713,17182}, {16832,17753}, {17170,17284}, {17358,17482}

    X(18228) = polar conjugate of X(11546)
    X(18228) = anticomplement of X(5437)
    X(18228) = X(17810) of 2nd Conway triangle
    X(18228) = X(17810) of 2nd Zaniah triangle
    X(18228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9, 5273), (2, 63, 5435), (2, 144, 57), (2, 329, 7), (2, 908, 5226), (2, 3219, 5744), (2, 3305, 18230), (2, 3452, 5328), (9, 3452, 2), (57, 5316, 2), (226, 7308, 2), (329, 9776, 5905), (908, 3305, 2), (3306, 17781, 9965), (5273, 5328, 2), (5435, 6172, 63)


    X(18229) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND 3rd CONWAY

    Barycentrics    a^3-2*(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a-4*(b+c)*b*c : :
    X(18229) = s^2*X(1)-3*(r^2+s^2)*X(2)

    X(18229) lies on these lines: {1,2}, {9,1764}, {40,2050}, {188,11894}, {210,10473}, {226,17272}, {236,11896}, {312,3731}, {333,1743}, {518,11021}, {940,5783}, {958,10882}, {960,12435}, {966,3452}, {1211,5219}, {1215,5223}, {1329,10887}, {1376,10434}, {1402,4413}, {2297,16713}, {2345,5745}, {2886,10886}, {3035,13244}, {3036,12550}, {3305,5235}, {3666,17151}, {3739,5437}, {3740,10439}, {3772,17306}, {3928,4363}, {4417,17270}, {4859,10468}, {5044,10441}, {5226,5232}, {5234,6996}, {5273,10444}, {5438,10470}, {5777,12547}, {7028,11895}, {10436,14829}, {10446,18228}, {10862,18227}, {10889,18230}, {10891,18234}, {10892,18235}, {11521,15829}, {12126,18247}, {12392,18248}, {12544,18249}, {12545,18250}, {12548,18251}, {12551,18254}, {12552,18255}, {12553,18257}, {12554,18258}, {12557,18259}, {16124,18253}, {17056,17296}, {17617,18236}

    X(18229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8, 12546), (2, 5271, 2999), (2, 11679, 1), (9, 10472, 10456), (1764, 10888, 10442), (2999, 5271, 16833), (4384, 16831, 16827)


    X(18230) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND HONSBERGER

    Barycentrics    3*a^2-4*(b+c)*a+(b-c)^2 : :
    X(18230) = 2*X(1)+3*X(5686) = 6*X(2)-X(7) = 3*X(2)+2*X(9) = 9*X(2)-4*X(142) = 9*X(2)+X(144) = 4*X(2)+X(6172) = 7*X(2)-2*X(6173) = 3*X(2)-8*X(6666) = X(7)+4*X(9) = 3*X(7)-8*X(142) = 3*X(7)+2*X(144) = 2*X(7)+3*X(6172) = 7*X(7)-12*X(6173) = X(7)-16*X(6666) = 3*X(9)+2*X(142) = 6*X(9)-X(144) = 8*X(9)-3*X(6172) = 7*X(9)+3*X(6173) = X(9)+4*X(6666) = 4*X(142)+X(144) = 14*X(142)-9*X(6173) = X(142)-6*X(6666) = 4*X(144)-9*X(6172) = X(329)+4*X(8257) = 4*X(3452)+X(12848) = 4*X(5745)+X(8545) = 7*X(6172)+8*X(6173)

    X(18230) lies on these lines: {1,4924}, {2,7}, {3,5817}, {5,5759}, {6,5308}, {8,344}, {10,390}, {37,5222}, {43,4343}, {44,4648}, {45,4000}, {69,17241}, {72,17552}, {75,3161}, {140,5779}, {149,6594}, {188,7022}, {192,4402}, {193,17244}, {210,5572}, {220,5543}, {238,4344}, {239,4460}, {346,4384}, {391,3912}, {392,4345}, {405,4313}, {427,7717}, {497,15837}, {516,1698}, {518,3616}, {631,971}, {632,5843}, {673,2345}, {857,1213}, {936,7675}, {938,954}, {958,4308}, {960,4323}, {962,6886}, {966,5838}, {984,16020}, {1125,5223}, {1156,3035}, {1212,3160}, {1329,7679}, {1376,7676}, {1621,6600}, {1656,5762}, {1743,3945}, {1992,17317}, {2287,16053}, {2324,7269}, {2325,4461}, {2476,3826}, {2478,2550}, {2886,7678}, {2951,10164}, {3008,3672}, {3036,12730}, {3059,3740}, {3085,15299}, {3086,15298}, {3090,5805}, {3241,3759}, {3243,3622}, {3247,17014}, {3358,6223}, {3475,3715}, {3488,16857}, {3523,5732}, {3600,5234}, {3601,10392}, {3617,5853}, {3619,17256}, {3620,17266}, {3624,5542}, {3646,14986}, {3664,3973}, {3681,15185}, {3683,9778}, {3707,17296}, {3739,7229}, {3757,5423}, {3763,4748}, {3816,6067}, {3925,9812}, {3943,4371}, {3946,16676}, {3950,16833}, {4208,12572}, {4321,5265}, {4326,5281}, {4335,16569}, {4346,4859}, {4370,17118}, {4413,11495}, {4416,4869}, {4419,16814}, {4423,10580}, {4473,4699}, {4488,17336}, {4644,16885}, {4657,6687}, {4675,15492}, {4679,9779}, {4853,7320}, {4877,14953}, {4916,4969}, {5044,5703}, {5084,5766}, {5218,14100}, {5220,5550}, {5232,17284}, {5251,5731}, {5261,12573}, {5440,17561}, {5704,5791}, {5729,6675}, {5735,7486}, {5758,6887}, {5777,12669}, {5784,6910}, {5785,6700}, {5811,6989}, {5825,6857}, {5832,6931}, {5836,7673}, {5839,17243}, {6068,6667}, {6684,11372}, {6840,17057}, {6921,10861}, {7028,8388}, {7288,8581}, {7670,18258}, {8167,17051}, {8237,18234}, {8238,18235}, {8582,18231}, {9846,18247}, {9945,16418}, {10398,13411}, {10865,18227}, {10889,18229}, {11008,17387}, {11024,12514}, {11526,15829}, {12399,18248}, {12536,16859}, {12560,18249}, {12649,17570}, {12706,18251}, {12718,18252}, {12755,18254}, {12846,18255}, {12847,18257}, {12850,18259}, {16133,18253}, {16675,17366}, {16677,17395}, {16815,17339}, {16832,17355}, {17265,17332}, {17267,17330}, {17314,17348}, {17316,17349}, {17321,17352}, {17620,18236}

    X(18230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9, 7), (2, 144, 142), (2, 3219, 9776), (2, 3305, 18228), (2, 5273, 5435), (2, 17260, 5296), (2, 18228, 5226), (7, 9, 6172), (8, 1001, 8236), (9, 142, 144), (9, 6666, 2), (142, 144, 7), (480, 1001, 2346), (1445, 8232, 7), (8545, 8732, 7), (17260, 17338, 2)


    X(18231) = PERSPECTOR OF THESE TRIANGLES: 2nd ZANIAH AND HUTSON EXTOUCH

    Barycentrics    (-a+b+c)*(3*a^3+7*(b+c)*a^2+(b^2+6*b*c+c^2)*a-3*(b^2-c^2)*(b-c)) : :
    X(18231) = X(5556)-7*X(9780)

    X(18231) lies on these lines: {1,5775}, {2,65}, {8,3158}, {9,5128}, {10,20}, {12,144}, {46,4208}, {63,5261}, {100,958}, {191,10590}, {390,6734}, {452,10395}, {962,5705}, {997,10303}, {1212,17756}, {1698,4295}, {1737,5129}, {1770,18250}, {1837,11106}, {2475,2551}, {2550,6895}, {3085,5904}, {3091,12514}, {3146,4640}, {3522,5794}, {3555,11018}, {3600,5744}, {3616,5837}, {3622,4999}, {3634,5328}, {3651,9709}, {3672,5230}, {3679,4305}, {3820,6937}, {3832,5698}, {3877,18220}, {3927,8164}, {4847,12632}, {5082,12732}, {5086,17576}, {5226,12526}, {5231,9785}, {5250,5274}, {5550,15829}, {5657,5791}, {5658,5777}, {5659,6888}, {5660,18254}, {5686,7080}, {5690,6892}, {6850,10742}, {6906,9708}, {8580,12520}, {8582,18230}, {10198,11036}, {10304,17647}, {10527,15558}, {10585,11684}, {12671,14647}

    X(18231) = X(14528) of 2nd Zaniah triangle
    X(18231) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 9780, 8165), (1698, 18249, 18228)


    X(18232) = PERSPECTOR OF THESE TRIANGLES: 2nd ZANIAH AND K798I

    Barycentrics    a*(a^6-3*(b^2+c^2)*a^4+(b+c)*b*c*a^3+(3*b^4+3*c^4+(b^2+c^2)*b*c)*a^2-(b+c)*(b^2+c^2)*b*c*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :
    X(18232) = R*X(8)+(R-2*r)*X(90) = (R^2-R*r-2*r^2)*X(63)+R*(R-3*r)*X(499)

    Let (Oa) be the reflection of the A-excircle in line BC. Define (Ob) and (Oc) cyclically. Then X(18232) is the radical center of circles (Oa), (Ob), (Oc).

    X(18232) lies on these lines: {2,10052}, {3,18254}, {8,90}, {9,2252}, {10,6923}, {63,499}, {912,960}, {956,2098}, {1376,3652}, {1727,5552}, {1776,3811}, {3036,5690}, {3647,5217}, {3814,7702}, {3820,5499}, {5692,15446}, {5745,18233}, {7082,10916}, {7330,17647}, {10395,12572}, {12849,18259}

    X(18232) = complement of X(10052)
    X(18232) = X(15317) of 2nd Zaniah triangle


    X(18233) = PERSPECTOR OF THESE TRIANGLES: 2nd ZANIAH AND K798E

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

    X(18233) lies on these lines: {2,10044}, {5,18253}, {8,7162}, {9,6832}, {10,6928}, {191,6933}, {405,10176}, {498,3305}, {958,6265}, {993,5777}, {1158,6825}, {3091,12514}, {3149,3647}, {5745,18232}, {6987,17647}

    X(18233) = complement of X(10044)


    X(18234) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND 2nd PAMFILOS-ZHOU

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

    X(18234) lies on these lines: {2,8243}, {8,7090}, {9,7595}, {10,7596}, {37,615}, {210,17610}, {236,11996}, {518,11030}, {936,8234}, {958,8225}, {960,9808}, {1125,11042}, {1329,8230}, {1376,8224}, {2886,8228}, {3036,12744}, {3452,12610}, {5273,10885}, {5745,10858}, {7028,8248}, {8237,18230}, {8244,8580}, {8246,18235}, {9789,18228}, {10891,18229}, {11532,15829}, {13090,18258}, {16144,18253}, {17627,18236}

    X(18234) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11687, 8243), (8, 8239, 12638)


    X(18235) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND 1st SHARYGIN

    Barycentrics    a*(a^2+b*c)*(-a+b+c)*((b+c)*a+b^2+c^2) : :
    X(18235) = 3*(2*(r+2*R)*SW+2*r^3-S*s)*X(2)-4*((R+r)*SW-S*s)*X(1284) = 2*(S*s+SW*(4*R-r))*X(9)-(S*s-2*r*(3*r^2+2*SW))*X(43)

    X(18235) lies on these lines: {2,1284}, {8,21}, {9,43}, {10,9840}, {63,1469}, {71,1755}, {100,8852}, {165,3501}, {171,172}, {188,8249}, {198,1376}, {210,17611}, {236,8425}, {518,11031}, {851,4418}, {936,8235}, {960,1193}, {1125,11043}, {1329,5051}, {2886,8229}, {3035,13265}, {3036,12746}, {3061,17592}, {3214,4689}, {3452,4425}, {3740,11203}, {3741,4154}, {3923,4192}, {3980,16056}, {4685,5325}, {5044,9959}, {5289,17599}, {5777,12683}, {7028,8250}, {7081,17787}, {8238,18230}, {8245,8580}, {8246,18234}, {9791,18228}, {10868,18227}, {10892,18229}, {11533,15829}, {12405,18248}, {12567,18249}, {12579,18250}, {12713,18251}, {12725,18252}, {12770,18254}, {12869,18255}, {13071,18257}, {13091,18258}, {13123,18259}, {17628,18236}

    X(18235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11688, 1284), (8, 21, 8240), (8, 8240, 12642), (43, 846, 256), (1376, 8424, 4220)


    X(18236) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ZANIAH AND URSA-MAJOR

    Barycentrics    a*(-a+b+c)*((b+c)*a^3-(b+c)^2*a^2-(b+c)*(b^2-4*b*c+c^2)*a+(b^2+4*b*c+c^2)*(b-c)^2) : :
    X(18236) = (8*R^2-10*R*r-3*r^2)*X(9)+3*r^2*X(165) = (R-2*r)*X(11)+3*(R-r)*X(210)

    X(18236) lies on these lines: {2,3660}, {8,17622}, {9,165}, {10,1532}, {11,210}, {72,1329}, {188,17629}, {236,17631}, {355,2551}, {392,12647}, {517,6973}, {518,17626}, {936,12114}, {958,17614}, {960,3679}, {1125,17624}, {1864,6745}, {2550,10157}, {2886,17618}, {3035,10167}, {3036,17652}, {3434,18228}, {3555,10072}, {3681,5328}, {3753,3838}, {3876,8165}, {3967,4858}, {3983,5837}, {4662,4915}, {4679,10947}, {5173,5748}, {5220,8169}, {5273,17616}, {5435,11678}, {5552,12711}, {5745,17612}, {5777,12666}, {5784,15064}, {5795,10944}, {5811,12676}, {5836,7989}, {5853,17604}, {6692,8581}, {6700,14872}, {7028,17630}, {10177,13405}, {10866,12640}, {11826,12572}, {17617,18229}, {17620,18230}, {17627,18234}, {17628,18235}, {17644,18247}, {17645,18248}, {17646,18249}, {17647,18250}, {17650,18251}, {17651,18252}, {17653,18253}, {17654,18254}, {17655,18255}, {17656,18257}, {17657,18258}, {17659,18259}

    X(18236) = midpoint of X(5435) and X(11678)
    X(18236) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 17615, 17625), (8, 17622, 17648), (11, 210, 17658), (210, 17642, 14740), (1376, 5927, 17668), (1709, 8580, 1376), (3740, 18227, 9)


    X(18237) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 1st ZANIAH

    Barycentrics   a^2*(a^8-2*(b+c)*a^7-2*(b^2-3*b*c+c^2)*a^6+6*(b^2-c^2)*(b-c)*a^5-6*b*c*(b-c)^2*a^4-6*(b^2-c^2)*(b-c)^3*a^3+2*(b^4+c^4-b*c*(b^2+8*b*c+c^2))*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)^5*a-(b^4-c^4)*(b^2-c^2)*(b^2-6*b*c+c^2)) : :
    X(18237) = (4*R^2-r^2)*X(3)-4*R^2*X(960) = (R-r)*X(56)-R*X(84) = (4*R^2-SW)*X(64)-(4*R^2+2*r^2-SW)*X(102)

    X(18237) lies on these lines: {1,1035}, {3,960}, {36,7992}, {55,7971}, {56,84}, {64,102}, {104,10309}, {382,2829}, {474,14647}, {515,12513}, {946,999}, {956,12667}, {958,6260}, {971,11249}, {1001,5450}, {1012,3485}, {1490,3428}, {1498,10571}, {1728,12664}, {1788,3149}, {2800,10306}, {2886,6256}, {6743,11362}, {6913,12608}, {6918,12616}, {8071,15071}, {9708,18242}, {10966,12680}, {11281,13743}

    X(18237) = X(7971) of anti-Mandart-incircle triangle
    X(18237) = X(9937) of 2nd circumperp triangle
    X(18237) = X(12301) of 1st circumperp triangle
    X(18237) = X(12330) of ABC-X3 reflections triangle
    X(18237) = {X(1158), X(6261)}-harmonic conjugate of X(960)


    X(18238) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ZANIAH TO EXTOUCH

    Barycentrics   a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b^3+c^3)*a^6+2*(3*b^2-2*b*c+3*c^2)*(b^2-b*c+c^2)*a^5-2*(b^2-c^2)*(b-c)*b*c*a^4-2*(3*b^4+3*c^4-b*c*(b^2+c^2))*(b-c)^2*a^3+2*(b^3+c^3)*(b^2-c^2)^2*a^2+2*(b^2-c^2)^2*(b^4+c^4-3*b*c*(b^2+c^2))*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
    X(18238) = X(1490)-3*X(10167) = 3*X(3742)-2*X(12608) = 3*X(3742)-4*X(18260) = X(6259)-3*X(10202) = X(9799)+3*X(11220) = 3*X(11220)-X(12671) = 3*X(14647)-X(14872)

    X(18238) lies on these lines: {1,84}, {2,18239}, {4,10305}, {5,142}, {7,10309}, {20,14923}, {63,10310}, {65,2096}, {377,9799}, {474,1490}, {515,5836}, {518,1158}, {960,5450}, {1387,9856}, {1902,3937}, {2800,13600}, {2823,5908}, {2829,4292}, {2949,3916}, {3742,12608}, {3812,6256}, {5123,12616}, {5252,12680}, {5572,12005}, {5658,6983}, {5732,11500}, {5777,6700}, {5787,6923}, {5884,14563}, {6223,6957}, {6259,10202}, {6833,12664}, {6837,9960}, {6966,12528}, {7080,14647}, {8582,18242}, {9948,12855}, {11376,12688}

    X(18238) = midpoint of X(1) and X(17649)
    X(18238) = reflection of X(i) in X(j) for these (i,j): (960, 5450), (5777, 6705)
    X(18238) = complement of X(18239)
    X(18238) = X(4) of 1st Zaniah triangle
    X(18238) = X(9820) of Conway triangle
    X(18238) = X(12420) of 2nd Zaniah triangle
    X(18238) = X(15316) of inverse-in-incircle triangle
    X(18238) = X(17649) of anti-Aquila triangle
    X(18238) = (1st Zaniah)-isogonal conjugate of-X(12608)
    X(18238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9799, 11220, 12671), (12608, 18260, 3742)


    X(18239) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXTOUCH TO 1st ZANIAH

    Barycentrics    a*((b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a-(b^2-c^2)^2*(b+c)) : :
    X(18239) = 3*X(210)-2*X(1158) = 3*X(354)-4*X(12608) = 3*X(5658)-2*X(9942) = X(5691)-4*X(16007) = 3*X(5927)-2*X(6245)

    X(18239) lies on these lines: {1,10092}, {2,18238}, {3,9}, {8,6001}, {10,17649}, {65,6256}, {210,1158}, {354,12608}, {515,3057}, {912,6259}, {946,8581}, {1145,17661}, {1319,1898}, {1858,12678}, {2057,10310}, {2478,9799}, {2829,14110}, {4187,5927}, {5658,6834}, {5691,16007}, {5787,6929}, {5882,9848}, {6838,9960}, {6921,11220}, {6959,13369}, {7971,12629}, {8545,11496}, {9856,12650}, {10167,13747}, {10306,12686}, {10394,12675}, {11500,13528}, {12330,17857}

    X(18239) = reflection of X(i) in X(j) for these (i,j): (65, 6256), (1071, 6260)
    X(18239) = anticomplement of X(18238)
    X(18239) = X(4)-of-extouch-triangle
    X(18239) = X(12420)-of-intouch-triangle
    X(18239) = X(12421)-of-excentral-triangle
    X(18239) = X(17649)-of-outer-Garcia-triangle
    X(18239) = extouch-isogonal conjugate of X(1158)
    X(18239) = X(8)-Ceva conjugate of X(1210)


    X(18240) = CENTER OF THE CIRCUMCONIC OF THESE TRIANGLES: 1st ZANIAH AND INVERSE-IN-INCIRCLE

    Barycentrics    a*((b+c)*a^4-2*(b^2+c^2)*a^3+b*c*(b+c)*a^2+2*(b^2-b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)) : :
    X(18240) = X(11)+3*X(354), 3*X(11)+X(17660), 3*X(354)-X(5083), 9*X(354)-X(17660), X(1145)-5*X(5439), X(1317)-5*X(17609), X(3035)-3*X(3742), 3*X(3892)+X(15863), 3*X(5049)+X(6797), 3*X(5049)-X(12735), 3*X(5083)-X(17660), 3*X(5902)+X(12758), 5*X(8227)-X(12665), X(11570)+3*X(16173)

    X(18240) lies on these lines: {1,88}, {2,14740}, {11,118}, {65,15558}, {80,938}, {104,3333}, {149,10580}, {153,11037}, {165,7673}, {496,12005}, {499,3678}, {516,3660}, {518,6667}, {528,11018}, {676,3738}, {758,5570}, {942,1387}, {946,15528}, {952,5045}, {999,11715}, {1056,12751}, {1058,14217}, {1145,5439}, {1210,3881}, {1317,17609}, {1537,3671}, {1736,17449}, {1768,10980}, {2829,4298}, {3035,3742}, {3086,3874}, {3338,10058}, {3488,12119}, {3812,5854}, {3878,5744}, {3887,14760}, {3892,15863}, {3968,12647}, {5049,6797}, {5531,14151}, {5533,10122}, {5536,7677}, {5660,11038}, {5708,12515}, {5840,13373}, {5880,13271}, {5884,11373}, {5902,12758}, {6147,12611}, {6265,15934}, {6713,12432}, {6744,16193}, {7373,12737}, {8083,8104}, {8227,12665}, {9940,12575}, {10164,17642}, {10404,12764}, {10589,15064}, {10698,11529}, {10707,11020}, {11033,13267}, {11570,14986}

    X(18240) = midpoint of X(i) and X(j) for these {i,j}: {1, 12736}, {11, 5083}, {65, 15558}, {942, 1387}, {946, 15528}
    X(18240) = complement of X(14740)
    X(18240) = incircle-inverse-of X(1054)
    X(18240) = X(113) of incircle-circles triangle
    X(18240) = X(125) of inverse-in-incircle triangle
    X(18240) = X(5972) of intouch triangle
    X(18240) = X(6723) of Ursa-minor triangle
    X(18240) = X(12736) of anti-Aquila triangle
    X(18240) = X(15473) of 2nd circumperp triangle
    X(18240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11, 354, 5083), (354, 17626, 11019), (3742, 12915, 13405), (5049, 6797, 12735)


    X(18241) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ZANIAH TO HUTSON EXTOUCH

    Barycentrics   a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b+c)*(b^2+7*b*c+c^2)*a^6+2*(3*b^4+3*c^4+b*c*(3*b^2-8*b*c+3*c^2))*a^5+10*b*c*(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-2*(3*b^6+3*c^6+(9*b^4+9*c^4-b*c*(19*b^2+18*b*c+19*c^2))*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4-b*c*(7*b^2+24*b*c+7*c^2))*a^2+2*(b^2-c^2)^2*(b^4+c^4+b*c*(5*b^2-8*b*c+5*c^2))*a-(b^4-c^4)*(b^2-c^2)^2*(b-c)) : :
    X(18241) = (10*R+3*r)*X(1)-(2*R+r)*X(5920)

    X(18241) lies on these lines: {1,5920}, {2,12670}, {142,5045}, {354,11023}, {5173,10122}, {5687,12658}, {5785,15185}, {5836,17706}, {6001,18219}, {9804,11024}

    X(18241) = complement of X(12670)


    X(18242) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798I TO 1st ZANIAH

    Barycentrics    (b^2+4*b*c+c^2)*a^5-(b+c)^3*a^4-2*(b^2-c^2)^2*a^3+2*(b^2-c^2)^2*(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(18242) = 3*X(2)+X(12667) = 2*X(10)+X(18243) = X(84)-5*X(1698) = X(1490)+3*X(5587) = 3*X(3679)+X(7971) = 3*X(3829)-2*X(10943) = 3*X(5658)+5*X(5818) = X(6223)+7*X(9780) = X(6223)+3*X(14647) = X(6245)-3*X(10175) = 5*X(8227)-X(12650) = 7*X(9780)-3*X(14647) = 3*X(10711)-X(12762)

    X(18242) lies on these lines: {1,1532}, {2,12114}, {3,119}, {4,12}, {5,515}, {8,6932}, {10,5777}, {11,944}, {20,11681}, {30,6796}, {40,17757}, {56,6834}, {65,1512}, {84,1698}, {100,11826}, {104,5433}, {140,5450}, {153,2975}, {198,15849}, {226,7686}, {227,1785}, {355,2886}, {381,16202}, {388,6848}, {411,5080}, {442,1490}, {495,946}, {496,5882}, {497,10893}, {498,1012}, {517,10915}, {528,10525}, {529,11249}, {908,14110}, {952,3813}, {958,6825}, {971,3826}, {1001,6893}, {1158,3652}, {1210,3660}, {1376,6850}, {1478,3149}, {1519,3057}, {1537,5697}, {1621,13729}, {1768,5445}, {2098,10956}, {2099,10955}, {2551,6908}, {2800,3678}, {3086,6969}, {3303,10531}, {3304,10805}, {3419,17857}, {3428,3436}, {3475,5804}, {3545,7958}, {3560,6690}, {3576,4187}, {3612,5691}, {3614,6830}, {3616,6945}, {3634,6705}, {3679,7971}, {3814,4297}, {3820,6684}, {3829,10943}, {3925,5658}, {4193,5731}, {4197,9799}, {4293,6927}, {4413,6897}, {4423,6898}, {4999,6863}, {5123,9943}, {5204,6880}, {5217,6938}, {5253,6979}, {5432,6906}, {5552,6925}, {5603,15888}, {5693,13257}, {5720,5794}, {5787,6881}, {5790,9710}, {5791,12677}, {6223,9780}, {6245,8728}, {6668,6862}, {6691,6959}, {6734,14872}, {6837,10585}, {6847,10588}, {6905,7354}, {6913,10198}, {6923,11499}, {6929,10267}, {6934,12943}, {6942,15326}, {6947,8273}, {6968,10896}, {6985,10526}, {7682,12915}, {7705,11220}, {7952,10271}, {7956,13464}, {8227,12650}, {8582,18238}, {8727,10592}, {9708,18237}, {9709,12330}, {10039,12672}, {10165,17527}, {10167,17619}, {10523,10572}, {10532,11237}, {10598,10806}, {10711,12762}, {10864,17529}, {10902,11113}, {10954,12047}, {11014,12751}, {12680,17606}

    X(18242) = midpoint of X(i) and X(j) for these {i,j}: {3, 6256}, {4, 11500}, {10, 6260}, {100, 12761}, {355, 6261}, {1158, 6259}, {6985, 10526}
    X(18242) = complement of X(12114)
    X(18242) = X(155)-of-4th-Euler-triangle
    X(18242) = X(5504)-of-K798i-triangle
    X(18242) = X(7680)-of-outer-Johnson-triangle
    X(18242) = X(9932)-of-2nd-Zaniah-triangle
    X(18242) = X(11500)-of-Euler-triangle
    X(18242) = X(12163)-of-3rd-Euler-triangle
    X(18242) = (K798i)-isogonal conjugate of-X(6713)
    X(18242) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1532, 7681), (2, 12667, 12114), (3, 119, 1329), (4, 12, 7680), (4, 3085, 11496), (4, 10590, 10894), (4, 10786, 55), (4, 11491, 6284), (5, 1385, 3816), (8, 6932, 15908), (104, 6949, 5433), (153, 6960, 2975), (355, 6842, 2886), (944, 6941, 11), (6834, 12115, 56)


    X(18243) = ORTHOLOGIC CENTER OF THESE TRIANGLES: K798E TO 1st ZANIAH

    Barycentrics    2*(b+c)*a^6-(b^2-4*b*c+c^2)*a^5-(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+2*(b^2-c^2)^2*a^3+4*(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(18243) = X(10)-3*X(6260) = 2*X(10)-3*X(18242) = 3*X(84)-7*X(3624) = X(145)+3*X(12667) = 5*X(3616)+3*X(6223) = 5*X(3616)-3*X(12114) = 9*X(5658)-X(6361) = 3*X(5658)-X(11500) = X(6361)-3*X(11500) = 2*X(9955)-3*X(12608)

    X(18243) lies on these lines: {3,16127}, {4,3649}, {5,3833}, {10,5777}, {40,13257}, {84,3624}, {145,12667}, {226,12710}, {411,3648}, {515,1483}, {944,9670}, {946,5049}, {971,9955}, {1490,5842}, {1519,12680}, {1532,15071}, {1768,5442}, {2800,4127}, {2829,6259}, {3616,6223}, {3671,16616}, {3816,13369}, {5087,9942}, {5658,6361}, {5768,10893}, {6147,6744}, {6824,16112}, {6985,17768}, {7483,7701}, {7680,12688}, {7956,12005}, {9960,10129}, {11113,16132}, {11246,16116}, {12528,15908}, {12701,12831}

    X(18243) = midpoint of X(3) and X(16127)


    X(18244) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st ZANIAH TO 1st SCHIFFLER

    Barycentrics   a^7+(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5-(b+c)*(3*b^2-b*c+3*c^2)*a^4+(3*b^4+3*c^4-b*c*(3*b^2+5*b*c+3*c^2))*a^3+3*(b^3-c^3)*(b^2-c^2)*a^2-(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :
    X(18244) = 3*X(10266)+X(14450) = 3*X(12913)-X(14450)

    X(18244) lies on these lines: {1,5180}, {2,12682}, {35,13080}, {56,13126}, {79,942}, {142,12444}, {191,3584}, {498,12535}, {499,12849}, {946,16116}, {1387,3649}, {1479,12255}, {1749,14526}, {2475,3919}, {2771,5270}, {3065,12047}, {3543,18221}, {3582,11263}, {3585,12600}, {3746,13995}, {4325,16132}, {4857,16159}, {5441,11011}, {5572,16153}, {5836,16152}, {6869,16143}, {7280,12556}, {7741,12919}, {7951,12947}, {10572,11552}, {11024,12543}, {12660,17700}

    X(18244) = reflection of X(3746) in X(13995)
    X(18244) = complement of X(12682)
    X(18244) = X(13418) of inverse-in-incircle triangle
    X(18244) = (inverse-in-incircle)-isotomic conjugate of-X(7269)
    X(18244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (79, 17637, 3583), (10266, 13129, 1)


    X(18245) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO 1st ZANIAH

    Barycentrics   a*((a^10-(5*b^2-6*b*c+5*c^2)*a^8+4*(b+c)*(b^2+c^2)*a^7+2*(5*b^4+8*b^2*c^2+5*c^4)*a^6-4*(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^5-2*(5*b^4+5*c^4+4*b*c*(b^2+3*b*c+c^2))*(b-c)^2*a^4+4*(b^4-c^4)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3+(b^2+c^2)*(5*b^4+5*c^4-2*b*c*(3*b^2+7*b*c+3*c^2))*(b-c)^2*a^2-4*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b+c)^2*(b^4-c^4)^2)*S+a*(a^11-(b+c)*a^10-(5*b^2-2*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4-4*(b-c)^2*b*c)*a^7-2*(b+c)*(5*b^4+5*c^4-4*(b^2-b*c+c^2)*b*c)*a^6-2*(5*b^6+5*c^6-3*(2*b^4+2*c^4-(3*b^2-8*b*c+3*c^2)*b*c)*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-3*(2*b^4+2*c^4+(b^2+c^2)*b*c)*b*c)*a^4+(b^2+c^2)*(5*b^4+5*c^4+2*(b^2+b*c+c^2)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^4+2*c^4-(17*b^2+12*b*c+17*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6-(2*b^4+2*c^4-b*c*(3*b^2+28*b*c+3*c^2))*b*c)*a+(b^2-c^2)^3*(b-c)*(-4*b^2*c^2+(b^2-c^2)^2))) : :

    X(18245) lies on these lines: {84,493}, {515,12636}, {971,10669}, {1490,11828}, {1709,11951}, {2829,12741}, {6001,12440}, {6223,6462}, {6245,8212}, {6257,8218}, {6258,8216}, {6259,8220}, {6260,8222}, {6461,18246}, {7971,8210}, {7992,8188}, {8194,9910}, {8201,12456}, {8208,12457}, {8214,12667}, {8987,13899}, {10085,11953}, {10875,12496}, {10945,12676}, {10951,12677}, {11377,12114}, {11394,12136}, {11503,12330}, {11840,12196}, {11846,12246}, {11907,12668}, {11930,12678}, {11932,12679}, {11947,12680}, {11949,12684}, {11955,12686}, {11957,12687}, {13956,13974}

    X(18245) = X(84) of Lucas homothetic triangle


    X(18246) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO 1st ZANIAH

    Barycentrics   a*(-(a^10-(5*b^2-6*b*c+5*c^2)*a^8+4*(b+c)*(b^2+c^2)*a^7+2*(5*b^4+8*b^2*c^2+5*c^4)*a^6-4*(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^5-2*(5*b^4+5*c^4+4*b*c*(b^2+3*b*c+c^2))*(b-c)^2*a^4+4*(b^4-c^4)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^3+(b^2+c^2)*(5*b^4+5*c^4-2*b*c*(3*b^2+7*b*c+3*c^2))*(b-c)^2*a^2-4*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-(b+c)^2*(b^4-c^4)^2)*S+a*(a^11-(b+c)*a^10-(5*b^2-2*b*c+5*c^2)*a^9+(b+c)*(5*b^2-2*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4-4*(b-c)^2*b*c)*a^7-2*(b+c)*(5*b^4+5*c^4-4*(b^2-b*c+c^2)*b*c)*a^6-2*(5*b^6+5*c^6-3*(2*b^4+2*c^4-(3*b^2-8*b*c+3*c^2)*b*c)*b*c)*a^5+2*(b+c)*(5*b^6+5*c^6-3*(2*b^4+2*c^4+(b^2+c^2)*b*c)*b*c)*a^4+(b^2+c^2)*(5*b^4+5*c^4+2*(b^2+b*c+c^2)*b*c)*(b-c)^2*a^3-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(2*b^4+2*c^4-(17*b^2+12*b*c+17*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6-(2*b^4+2*c^4-b*c*(3*b^2+28*b*c+3*c^2))*b*c)*a+(b^2-c^2)^3*(b-c)*(-4*b^2*c^2+(b^2-c^2)^2))) : :

    X(18246) lies on these lines: {84,494}, {515,12637}, {971,10673}, {1490,11829}, {1709,11952}, {2829,12742}, {6001,12441}, {6223,6463}, {6245,8213}, {6257,8219}, {6258,8217}, {6259,8221}, {6260,8223}, {6461,18245}, {7971,8211}, {7992,8189}, {8195,9910}, {8202,12456}, {8209,12457}, {8215,12667}, {8987,13900}, {10085,11954}, {10876,12496}, {10946,12676}, {10952,12677}, {11378,12114}, {11395,12136}, {11504,12330}, {11841,12196}, {11847,12246}, {11908,12668}, {11931,12678}, {11933,12679}, {11948,12680}, {11950,12684}, {11956,12686}, {11958,12687}, {13957,13974}

    X(18246) = X(84) of Lucas(-1) homothetic triangle


    X(18247) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO ANDROMEDA

    Barycentrics    a*(-a+b+c)*((b+c)*a^4+6*b*c*a^3-2*(b+c)*(b^2-3*b*c+c^2)*a^2+2*b*c*(b^2+6*b*c+c^2)*a+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)) : :
    X(18247) = 3*X(2)+X(12125) = 3*X(210)+X(17632) = 2*X(936)-3*X(3740) = 3*X(3742)-4*X(9843) = 7*X(3983)-X(9859) = 2*X(4662)+X(9844) = X(9846)-5*X(18230)

    X(18247) lies on these lines: {1,18227}, {2,9850}, {4,5836}, {8,9848}, {9,3913}, {10,971}, {188,9853}, {210,3486}, {354,8165}, {518,938}, {519,960}, {936,958}, {1125,12128}, {1329,3742}, {1376,9841}, {1706,15726}, {2886,9842}, {3333,8169}, {3697,5234}, {3812,5290}, {3820,12675}, {3983,5273}, {4711,5837}, {5044,5882}, {5123,8728}, {5328,17609}, {5745,9858}, {6261,9708}, {6738,9954}, {7028,9854}, {8580,9851}, {9797,18228}, {9846,18230}, {9852,18235}, {12126,18229}, {12127,15829}, {17644,18236}

    X(18247) = midpoint of X(8) and X(9848)
    X(18247) = complement of X(9850)
    X(18247) = {X(2), X(12125)}-harmonic conjugate of X(9850)


    X(18248) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO ANTLIA

    Barycentrics   a*((b+c)*a^8-2*(b^2+5*b*c+c^2)*a^7+2*(b+c)*(b^2+8*b*c+c^2)*a^6-2*(b^4+c^4+b*c*(9*b^2+8*b*c+9*c^2))*a^5+4*b*c*(b+c)*(5*b^2-8*b*c+5*c^2)*a^4+2*(b^6+c^6-(7*b^4+7*c^4-b*c*(11*b^2+10*b*c+11*c^2))*b*c)*a^3-2*(b+c)*(b^6+c^6-(4*b^4+4*c^4-(7*b^2+4*b*c+7*c^2)*b*c)*b*c)*a^2+2*(b^2+c^2)*(b^4+c^4-b*c*(b+c)^2)*(b-c)^2*a-(b^4-c^4)*(b-c)*(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c)) : :
    X(18248) = 3*X(2)+X(12389) = 3*X(210)+X(17633) = X(12399)-5*X(18230)

    X(18248) lies on these lines: {2,12389}, {8,12400}, {9,12396}, {210,17633}, {518,12403}, {936,12398}, {958,12388}, {1125,12401}, {1329,12394}, {1376,12387}, {2886,12393}, {5273,12390}, {5745,12385}, {8580,12404}, {12386,18227}, {12391,18228}, {12392,18229}, {12395,15829}, {12399,18230}, {12405,18235}, {17645,18236}

    X(18248) = midpoint of X(8) and X(12400)
    X(18248) = complement of X(12402)
    X(18248) = {X(2), X(12389)}-harmonic conjugate of X(12402)


    X(18249) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 4th CONWAY

    Barycentrics    (-a+b+c)*(2*a^3+5*(b+c)*a^2+2*(b+c)^2*a-(b^2-c^2)*(b-c)) : :
    X(18249) = 3*X(2)+X(12526) = X(8)+3*X(4512) = 3*X(210)+X(12711) = X(388)+3*X(3929) = 3*X(551)-X(12559) = X(958)-3*X(5325) = 5*X(1698)-X(4295) = 3*X(3679)+X(4294) = 3*X(3740)-X(18251) = 3*X(3828)-2*X(3841) = X(4314)-3*X(4512) = X(4326)+3*X(5686) = 3*X(5325)+X(5837) = 3*X(5657)+X(12705) = 3*X(10164)-X(12520)

    X(18249) lies on these lines: {1,5273}, {2,3339}, {3,9948}, {4,9}, {8,4314}, {21,6737}, {55,6743}, {63,4298}, {72,13405}, {144,5290}, {188,12568}, {191,4292}, {210,12711}, {220,3997}, {236,12570}, {270,2328}, {329,3947}, {387,4356}, {388,3929}, {405,6738}, {518,12564}, {519,958}, {551,12559}, {758,942}, {936,10164}, {946,5791}, {950,3683}, {986,3008}, {997,8726}, {1001,6744}, {1046,3664}, {1212,3931}, {1329,3828}, {1376,12511}, {1698,4295}, {1788,7308}, {2325,3714}, {2886,12558}, {3219,12527}, {3305,8582}, {3452,3634}, {3626,5302}, {3636,5289}, {3647,15823}, {3679,4294}, {3686,3704}, {3740,18251}, {3812,6666}, {3817,5705}, {3876,6745}, {3911,12709}, {3927,5850}, {4047,5750}, {4208,4312}, {4297,9799}, {4326,4882}, {4357,10521}, {4640,12512}, {4656,5230}, {4662,15733}, {4847,5250}, {5044,6001}, {5129,5775}, {5220,12855}, {5249,11684}, {5261,6172}, {5692,13411}, {5744,8583}, {5785,7992}, {6700,10176}, {7028,12569}, {8580,9949}, {9708,11362}, {9858,10178}, {11529,16845}, {12446,18227}, {12544,18229}, {12560,18230}, {12567,18235}, {12607,15481}, {17646,18236}

    X(18249) = midpoint of X(8) and X(4314)
    X(18249) = complement of X(3671)
    X(18249) = X(578) of 2nd Zaniah triangle
    X(18249) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12526, 3671), (8, 4512, 4314), (9, 10, 18250), (10, 5493, 2550), (960, 5745, 1125), (960, 18253, 5745), (4847, 5250, 12575), (5325, 5837, 958), (18228, 18231, 1698)


    X(18250) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 5th CONWAY

    Barycentrics    (-a+b+c)*(2*a^3+3*(b+c)*a^2+2*(b+c)^2*a+(b^2-c^2)*(b-c)) : :
    X(18250) = 3*X(2)+X(12527) = 3*X(210)+X(950) = 3*X(210)-X(6743) = 5*X(1698)-X(4292) = 3*X(3679)+X(10624) = 5*X(3697)+3*X(11113) = 5*X(3876)-X(6737) = 7*X(3983)+X(6284) = X(12573)-5*X(18230)

    X(18250) lies on these lines: {1,5129}, {2,3361}, {4,9}, {8,4082}, {20,8580}, {21,6745}, {56,5316}, {63,8582}, {72,6738}, {144,3339}, {188,12580}, {200,452}, {210,950}, {236,12582}, {329,3671}, {341,3883}, {388,7308}, {390,4882}, {405,13405}, {515,5044}, {518,6744}, {519,960}, {527,3812}, {908,5260}, {936,4297}, {938,5223}, {942,5850}, {946,8158}, {958,999}, {993,6700}, {1056,3646}, {1212,5717}, {1329,3634}, {1376,12512}, {1698,4208}, {1722,3663}, {1770,18231}, {1788,3929}, {1837,3715}, {2325,3704}, {2478,4847}, {2886,12571}, {2899,11679}, {3008,13161}, {3059,9844}, {3244,15829}, {3305,3436}, {3333,17559}, {3624,5328}, {3626,5837}, {3679,10624}, {3686,3714}, {3697,11113}, {3820,6684}, {3828,5325}, {3876,6737}, {3983,6284}, {4301,9623}, {4312,11024}, {4315,8583}, {4342,4853}, {4512,7080}, {4538,4662}, {4640,9711}, {4679,12053}, {4915,9785}, {5084,11019}, {5123,18253}, {5231,6919}, {5250,6736}, {5251,13411}, {5261,12573}, {5716,7322}, {5779,9948}, {5785,9799}, {5791,10175}, {6685,16850}, {7028,12581}, {8169,16408}, {12545,18229}, {12579,18235}, {12607,15254}, {17647,18236}

    X(18250) = midpoint of X(i) and X(j) for these {i,j}: {8, 12575}, {72, 6738}
    X(18250) = complement of X(4298)
    X(18250) = X(389) of 2nd Zaniah triangle
    X(18250) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12527, 4298), (9, 10, 18249), (9, 2551, 10), (10, 5493, 1706), (200, 452, 4314), (210, 950, 6743), (958, 3452, 1125), (1329, 5302, 5745), (1329, 5745, 3634), (1706, 5698, 5493), (5129, 5815, 1), (5273, 8165, 1698)


    X(18251) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 4th EXTOUCH

    Barycentrics    a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^3+c^3)*(b+c)*a^2+(b+c)*(b^4+c^4-2*b*c*(b^2-5*b*c+c^2))*a-(b^2-c^2)^2*(b+c)^2) : :
    X(18251) = 3*X(2)+X(12529) = 3*X(210)-X(12526) = 3*X(210)+X(17634) = 3*X(392)-X(4294) = X(960)+2*X(12446) = 3*X(3740)-2*X(18249) = 3*X(3742)-2*X(12564) = 2*X(5044)+X(17646) = 3*X(10157)-2*X(12617) = X(12706)-5*X(18230)

    X(18251) lies on these lines: {1,5696}, {2,12529}, {8,12709}, {9,5584}, {10,5777}, {20,17668}, {72,2550}, {210,1706}, {236,12716}, {392,4294}, {516,960}, {517,5794}, {518,3671}, {758,3626}, {936,10310}, {942,2886}, {958,971}, {1125,12710}, {1329,10157}, {1376,3579}, {1818,6051}, {1858,3925}, {2551,5927}, {2975,17616}, {3059,11523}, {3600,10861}, {3617,17615}, {3660,10527}, {3740,18249}, {3742,12564}, {3813,12915}, {4292,5857}, {4326,9848}, {4512,5217}, {4640,12511}, {4662,9954}, {4999,11227}, {5087,12558}, {5248,9858}, {5273,9961}, {5436,14100}, {5745,9943}, {5842,17647}, {6762,8581}, {7288,17612}, {9623,14872}, {9800,18228}, {9949,18227}, {11036,15185}, {12527,16120}, {12548,18229}, {12651,15829}, {12706,18230}, {12713,18235}, {17650,18236}

    X(18251) = midpoint of X(i) and X(j) for these {i,j}: {8, 12709}, {72, 4295}, {3059, 12560}
    X(18251) = reflection of X(942) in X(12609)
    X(18251) = complement of X(12711)
    X(18251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12529, 12711), (210, 17634, 12526)


    X(18252) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 5th EXTOUCH

    Barycentrics    a*((b+c)*a^3-(b^2+c^2)*a^2+(b^2-c^2)*(b-c)*a-b^4-c^4-2*b^3*c+2*b^2*c^2-2*b*c^3) : :
    X(18252) = 3*X(2)+X(12530) = 3*X(210)-X(3729) = 3*X(210)+X(17635) = 3*X(354)-5*X(17304) = 3*X(3740)-2*X(17355) = X(12718)-5*X(18230)

    X(18252) lies on these lines: {2,12530}, {8,12721}, {9,1721}, {188,12726}, {210,3729}, {236,12728}, {354,17304}, {516,960}, {517,4660}, {518,3663}, {936,12717}, {942,3821}, {958,990}, {1125,12722}, {1155,16566}, {1329,12618}, {1376,1766}, {1716,16583}, {1742,3061}, {2805,17229}, {2886,12610}, {3740,17355}, {3912,11997}, {3923,5044}, {4523,4655}, {5273,9962}, {5745,9944}, {5784,10444}, {7028,12727}, {7996,8580}, {9004,17345}, {9564,10445}, {9801,18228}, {9950,18227}, {9957,17766}, {12652,15829}, {12718,18230}, {12725,18235}, {17651,18236}

    X(18252) = midpoint of X(i) and X(j) for these {i,j}: {8, 12721}, {4523, 4655}
    X(18252) = reflection of X(i) in X(j) for these (i,j): (942, 3821), (3923, 5044)
    X(18252) = complement of X(12723)
    X(18252) = X(317) of 2nd Zaniah triangle
    X(18252) = (2nd Zaniah)-isotomic conjugate of-X(5777)
    X(18252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12530, 12723), (210, 17635, 3729)


    X(18253) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 2nd FUHRMANN

    Barycentrics    (-a+b+c)*(2*a^3+4*(b+c)*a^2+(b^2+4*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
    X(18253) = X(1)-3*X(15670) = 3*X(2)+X(11684) = X(8)+3*X(21) = 3*X(21)-X(10543) = X(79)+3*X(191) = X(79)-3*X(442) = X(79)-5*X(1698) = X(145)-9*X(15672) = 3*X(191)+5*X(1698) = 3*X(191)-X(3650) = 3*X(210)+X(17637) = 3*X(442)-5*X(1698) = 3*X(442)+X(3650) = 5*X(1698)+X(3650) = X(15174)-3*X(15673)

    X(18253) lies on these lines: {1,15670}, {2,3649}, {5,18233}, {8,21}, {9,46}, {10,30}, {12,3219}, {40,9710}, {44,5530}, {63,10404}, {72,6690}, {140,5694}, {145,15672}, {188,16146}, {210,17637}, {236,16151}, {517,16617}, {518,10122}, {519,15174}, {524,3178}, {758,942}, {846,1834}, {908,6668}, {936,16132}, {946,5771}, {952,12104}, {993,5428}, {1046,17056}, {1158,3652}, {1210,15254}, {1376,3651}, {1749,17757}, {2475,2551}, {2771,3035}, {2795,3039}, {2886,5791}, {2975,5427}, {3036,4691}, {3085,5220}, {3241,15675}, {3244,5837}, {3305,16140}, {3336,17529}, {3452,11263}, {3474,3648}, {3616,15671}, {3617,15677}, {3622,5289}, {3623,15676}, {3633,5426}, {3634,6701}, {3679,5441}, {3683,6734}, {3715,5552}, {3740,5777}, {3813,5250}, {3820,5499}, {3829,16155}, {3831,4422}, {3876,5432}, {3878,10021}, {3927,10198}, {3951,17718}, {4067,5719}, {4197,11246}, {4420,4995}, {4913,6362}, {5123,18250}, {5506,17575}, {5587,16113}, {5692,7483}, {5852,13407}, {6745,14454}, {6857,12635}, {7028,16147}, {8256,9708}, {8580,16143}, {9588,10860}, {9709,16117}, {9948,10164}, {10175,16125}, {12607,17699}, {14450,18228}, {15829,16126}, {16120,18227}, {16124,18229}, {16133,18230}, {16142,17606}, {16144,18234}, {17653,18236}

    X(18253) = midpoint of X(i) and X(j) for these {i,j}: {8, 10543}, {10, 3647}, {1749, 17757}, {3679, 17525}
    X(18253) = complement of X(3649)
    X(18253) = X(54) of 2nd Zaniah triangle
    X(18253) = X(6152) of 4th Euler triangle
    X(18253) = (2nd Zaniah)-isogonal conjugate of-X(5044)
    X(18253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11684, 3649), (8, 21, 10543), (10, 5325, 5302), (79, 191, 3650), (79, 1698, 442), (191, 1698, 79), (442, 3650, 79), (960, 5745, 4999), (1125, 16137, 11281), (3648, 9780, 6175), (5745, 18249, 960), (5791, 12514, 2886), (6675, 16137, 1125)


    X(18254) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO INNER-GARCIA

    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*(3*b^2-2*b*c+3*c^2))*a^2+(b+c)*(b^2+c^2)*(b^2-3*b*c+c^2)*a-(b+c)*(b^2-c^2)*(b^3-c^3)) : :
    X(18254) = 3*X(2)+X(12532) = X(80)+3*X(5692) = X(100)-5*X(3876) = 3*X(210)-X(1145) = 3*X(210)+X(17638) = X(214)-3*X(10176) = 3*X(392)-X(1317) = X(1320)+3*X(3681) = 5*X(1698)-X(11571) = 6*X(3678)+X(9951) = 3*X(3877)+X(12531) = 7*X(4533)-X(13996) = X(5904)+3*X(16173) = X(9951)+3*X(14740) = 3*X(11113)-X(12743)

    Let A'B'C' be the Inner Garcia triangle. Let A" be the orthogonal projection of A on line B'C', and define B", C" cyclically. Triangle A"B"C" is perpsective to the extouch triangle at X(18254). (Randy Hutson, June 27, 2018)

    X(18254) lies on these lines: {2,11570}, {3,18232}, {8,80}, {9,48}, {10,119}, {11,72}, {63,10090}, {78,90}, {100,3876}, {188,12771}, {210,1145}, {236,12774}, {392,1317}, {405,12739}, {517,3036}, {518,1387}, {519,15558}, {758,908}, {912,6713}, {936,1768}, {942,6667}, {952,960}, {956,12740}, {958,6265}, {1125,5083}, {1158,5720}, {1320,3681}, {1329,5694}, {1376,12515}, {1698,11571}, {2478,10073}, {2551,12247}, {2771,3035}, {2829,5777}, {2886,12611}, {3032,4115}, {3086,3874}, {3219,4996}, {3419,12764}, {3436,10057}, {3452,10265}, {3647,17100}, {3754,7951}, {3877,12531}, {3968,17057}, {4533,13996}, {5123,14988}, {5219,5883}, {5250,10087}, {5273,9964}, {5289,12737}, {5660,18231}, {5698,13199}, {5705,15017}, {5745,9946}, {5794,10742}, {5904,10529}, {6127,16586}, {6264,15829}, {7028,12772}, {8580,12767}, {8666,12059}, {9623,13253}, {9803,18228}, {9952,18227}, {11113,12743}, {11813,16174}, {12447,13227}, {12551,18229}, {12755,18230}, {12770,18235}, {17654,18236}

    X(18254) = midpoint of X(i) and X(j) for these {i,j}: {8, 12758}, {11, 72}
    X(18254) = reflection of X(942) in X(6667)
    X(18254) = complement of X(11570)
    X(18254) = X(265)-of-2nd-Zaniah-triangle
    X(18254) = X(10)-of-A"B"C", as described at X(8068)
    X(18254) = 2nd Zaniah-isogonal conjugate of-X(5123)
    X(18254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12532, 11570), (104, 997, 214), (210, 17638, 1145)


    X(18255) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO HUTSON EXTOUCH

    Barycentrics   a*(-a+b+c)*((b+c)*a^7-(b^2+c^2)*a^6-3*(b+c)*(b^2+4*b*c+c^2)*a^5+(3*b^4+2*b^2*c^2+3*c^4)*a^4+(b+c)*(3*b^4+3*c^4+2*b*c*(12*b^2+13*b*c+12*c^2))*a^3-(b^2+c^2)*(3*b^4-38*b^2*c^2+3*c^4)*a^2-(b^2-c^2)*(b-c)*(b^4+c^4+2*b*c*(7*b^2+9*b*c+7*c^2))*a+(b^2-c^2)^4) : :
    X(18255) = 3*X(2)+X(12533) = 3*X(210)+X(17639) = X(12846)-5*X(18230)

    X(18255) lies on these lines: {2,12533}, {8,5920}, {9,3295}, {10,12864}, {210,17639}, {518,12855}, {936,12842}, {958,12521}, {1125,12853}, {1329,12620}, {1376,12516}, {2886,12612}, {3555,11018}, {4882,9898}, {5273,12537}, {5745,12439}, {8001,8580}, {9804,18228}, {9953,18227}, {12552,18229}, {12654,15829}, {12846,18230}, {12869,18235}, {17655,18236}

    X(18255) = midpoint of X(8) and X(5920)
    X(18255) = complement of X(12854)
    X(18255) = (2nd Zaniah)-isogonal conjugate of-X(9710)
    X(18255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12533, 12854), (6765, 7160, 12631)


    X(18256) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO MALFATTI

    Barycentrics    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) = -b*c*(b-c)*(3*a-b-c)*(2*S-(-a+b+c)*(a+b+c))
    G(a,b,c) = a*c*((4*a^2-2*(b-3*c)*a-12*b*c+2*c^2+2*b^2)*S-(a-b+c)*(a+b+c)*((5*b-c)*a-(b+c)*(3*b-c)))
    H(a,b,c) = -(b-c)*S*(a^3-(b^2+8*b*c+c^2-6*S)*a+2*b*c*(b+c))

    X(18256) lies on these lines: {145,188}, {483,7028}, {557,16017}, {2090,14121}


    X(18257) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO MANDART-EXCIRCLES

    Barycentrics    (b^2-4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4+8*b^2*c^2*a^3+2*b*c*(b+c)*(b^2-4*b*c+c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
    X(18257) = 3*X(2)+X(12534) = 3*X(210)+X(17640) = X(12847)-5*X(18230)

    X(18257) lies on these lines: {2,12534}, {8,12876}, {9,12659}, {19,7359}, {210,17640}, {518,12914}, {522,596}, {936,12843}, {958,12522}, {1125,12907}, {1329,12621}, {1376,12517}, {2886,12613}, {5273,12538}, {5745,12442}, {8580,13069}, {12449,18227}, {12542,18228}, {12553,18229}, {12655,15829}, {12847,18230}, {13071,18235}, {17656,18236}

    X(18257) = midpoint of X(8) and X(12876)
    X(18257) = complement of X(12912)
    X(18257) = {X(2), X(12534)}-harmonic conjugate of X(12912)


    X(18258) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO MIDARC

    Trilinears    (-a+b+c)*(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*(3*a^2-4*(b+c)*a+(b-c)^2)
    G(a,b,c) = c*(a-b+c)*(a^2-2*(4*b+c)*a-b^2+c^2)
    H(a,b,c) = (a-b+c)*(a+b-c)*((b+c)*a-b^2-c^2-4*b*c)
    Barycentrics    Cos[A/2] (1 + Sin[B/2] + Sin[C/2]) : : (Peter Moses, May 2 2018)
    X(18258) = 3*X(2)+X(11691) = X(145)-3*X(11234) = 3*X(210)+X(17641) = 5*X(3616)-3*X(11191) = X(7670)-5*X(18230)

    X(18258) lies on these lines: {1,188}, {2,177}, {8,7048}, {9,164}, {10,2090}, {142,178}, {145,11234}, {167,8580}, {210,17641}, {236,13092}, {518,5571}, {936,12844}, {958,12523}, {1125,12908}, {1329,12622}, {1376,12518}, {2886,12614}, {3616,11191}, {5273,12539}, {7670,18230}, {9807,18228}, {11530,12879}, {12450,18227}, {12554,18229}, {12656,15829}, {13090,18234}, {13091,18235}, {17657,18236}

    X(18258) = {8, 8422}, {164, 12694}, {177, 11691}
    X(18258) = reflection of X(12908) in X(1125)
    X(18258) = complement of X(177)
    X(18258) = X(i)-complementary conjugate of X(j) for these (i,j): {260, 10}, {10492, 116}
    X(18258) = X(1)-of-2nd-Zaniah-triangle
    X(18258) = X(12262)-of-Hutson-intouch-triangle
    X(18258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11691, 177), (188, 7028, 1), (188, 12646, 10233)


    X(18259) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO 1st SCHIFFLER

    Barycentrics    a*(a^5+(b+c)*a^4-2*(b^2+c^2)*a^3-(b+c)*(2*b^2+b*c+2*c^2)*a^2+(b^4+c^4-b*c*(2*b^2+3*b*c+2*c^2))*a+(b^3-c^3)*(b^2-c^2))*(-a+b+c) : :
    X(18259) = 3*X(2)+X(12535) = 3*X(210)+X(17643) = X(12850)-5*X(18230)

    X(18259) lies on these lines: {2,10044}, {8,6597}, {9,10266}, {10,191}, {20,7701}, {21,60}, {63,14450}, {210,17643}, {518,12917}, {936,12845}, {958,12524}, {1125,12909}, {1329,12623}, {1376,12519}, {1749,14526}, {2886,12615}, {3218,11263}, {3647,17100}, {3651,3652}, {3715,12342}, {3876,12745}, {4189,12786}, {5273,12540}, {5745,12444}, {6734,11604}, {8580,13101}, {12451,18227}, {12514,15680}, {12543,18228}, {12557,18229}, {12657,15829}, {12849,18232}, {12850,18230}, {13123,18235}, {17659,18236}

    X(18259) = midpoint of X(8) and X(12877)
    X(18259) = complement of X(12913)
    X(18259) = {X(2), X(12535)}-harmonic conjugate of X(12913)


    X(18260) = X(5)-OF-1st ZANIAH TRIANGLE

    Barycentrics   a*((b+c)*a^8-2*(b^2-b*c+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6+2*(3*b^2-2*b*c+3*c^2)*(b^2-b*c+c^2)*a^5+4*(b^3+c^3)*b*c*a^4-2*(b^3-c^3)*(b-c)*(3*b^2-4*b*c+3*c^2)*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4-2*b*c*(b^2+c^2))*a^2+2*(b^4-c^4)*(b^2-c^2)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^3*(b-c)^3) : :
    X(18260) = 3*X(354)+X(1158) = 3*X(3742)-X(12608) = 3*X(3742)+X(18238) = 5*X(5439)-X(6256) = X(6796)-3*X(11227) = 3*X(10202)+X(12114)

    X(18260) lies on these lines: {104,13750}, {354,1158}, {495,12616}, {515,3812}, {942,5450}, {946,3660}, {2800,5045}, {3742,12608}, {5439,6256}, {5570,6906}, {5884,16193}, {5901,6001}, {6705,11018}, {6796,11227}, {10202,12114}

    X(18260) = midpoint of X(i) and X(j) for these {i,j}: {942, 5450}, {6705, 12005}
    X(18260) = X(5) of 1st Zaniah triangle
    X(18260) = {X(3742), X(18238)}-harmonic conjugate of X(12608)


    X(18261) = X(6)-OF-1st ZANIAH TRIANGLE

    Barycentrics    a*(a^4-2*(b^2-b*c+c^2)*a^2+b*c*(b+c)*a+(b^2+3*b*c+c^2)*(b-c)^2) : :
    X(18261) = s*(2*R*s-S)*X(1)+2*R*(SW-s^2)*X(142)

    X(18261) lies on these lines: {1,142}, {527,3554}, {1449,7269}, {4667,7190}, {4861,17306}

    X(18261) = X(6) of 1st Zaniah triangle
    X(18261) = (1st Zaniah)-isogonal conjugate of-X(3742)


    X(18262) = ISOGONAL CONJUGATE OF X(18036)

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

    X(18262) lies on the cubic K988 and theselines: {1, 3492}, {31, 184}, {32, 1917}, {385, 1492}, {692, 17735}, {846, 3955}, {1911, 1933}, {1980, 3063}, {3506, 3510}

    X(18262) = isogonal conjugate of X(18036)
    X(18262) = X(1911)-Ceva conjugate of X(32)
    X(18262) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18036}, {75, 7261}, {76, 3512}, {561, 8852}, {6063, 7281}, {7018, 7061}
    X(18262) = X(31)-Hirst inverse of X(7122)
    X(18262) = barycentric product X(i)*X(j) for these {i,j}: {6, 17798}, {31, 3509}, {32, 4645}, {41, 5018}, {291, 18038}, {560, 17789}, {1281, 1922}, {2206, 4071}, {14598, 18037}
    X(18262) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18036}, {32, 7261}, {560, 3512}, {1501, 8852}, {3509, 561}, {4645, 1502}, {9447, 7281}, {17789, 1928}, {17798, 76}, {18038, 350}


    X(18263) = ISOGONAL CONJUGATE OF X(18035)

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

    X(18263) lies on the cubic K988 and these lines: {741, 1326}, {9505, 17962}

    X(18263) = isogonal conjugate of X(18035)
    X(18263) = X(32)-cross conjugate of X(1922)
    X(18263) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18035}, {75, 6651}, {76, 8298}, {350, 6542}, {874, 2786}, {1757, 1921}, {3948, 17731}, {6382, 8843}
    X(18263) = barycentric product X(i)*X(j) for these {i,j}: {6, 9506}, {31, 9505}, {292, 17962}, {741, 2054}, {1911, 1929}, {1922, 6650}, {2702, 3572}, {14598, 18032}
    X(18263) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18035}, {32, 6651}, {560, 8298}, {1922, 6542}, {9505, 561}, {9506, 76}, {14598, 1757}, {17962, 1921}


    X(18264) = ISOGONAL CONJUGATE OF X(18034)

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

    X(18264) lies on the cubic K988

    X(18264) = isogonal conjugate of X(18034)
    X(18264) = X(32)-cross conjugate of X(1914)
    X(18264) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18034}, {75, 9470}, {334, 8301}, {335, 17738}
    X(18264) = barycentric product X(i)*X(j) for these {i,j}: {6, 9472}, {1914, 2113}
    X(18264) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18034}, {32, 9470}, {2210, 17738}, {9472, 76}, {14599, 8301}


    X(18265) = ISOGONAL CONJUGATE OF X(18033)

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

    X(18265) lies on the cubics K866 and K988 and on these lines: {3, 291}, {21, 4518}, {32, 1922}, {35, 3864}, {55, 2344}, {284, 2311}, {292, 1438}, {604, 1911}, {983, 1582}, {1918, 1927}, {3862, 15624}

    X(18265) = isogonal conjugate of X(18033)
    X(18265) = X(1253)-beth conjugate of X(3939)
    X(18265) = X(1911)-Ceva conjugate of X(1922)
    X(18265) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18033}, {2, 10030}, {7, 350}, {57, 1921}, {75, 1447}, {76, 1429}, {85, 239}, {238, 6063}, {242, 7182}, {269, 4087}, {274, 16609}, {279, 3975}, {310, 1284}, {561, 1428}, {659, 4572}, {664, 3766}, {799, 7212}, {812, 4554}, {873, 7235}, {874, 3676}, {1088, 3685}, {1431, 1926}, {1432, 3978}, {1434, 3948}, {1966, 7249}, {3716, 4569}, {4010, 4625}, {7196, 17493}
    X(18265) = X(2330)-Hirst inverse of X(7077)
    X(18265) = crosssum of X(350) and X(10030)
    X(18265) = X(18264) = barycentric product X(i)*X(j) for these {i,j}: {6, 7077}, {8, 1922}, {9, 1911}, {31, 4876}, {32, 4518}, {33, 2196}, {41, 291}, {42, 2311}, {55, 292}, {295, 607}, {312, 14598}, {334, 9447}, {335, 2175}, {644, 875}, {660, 3063}, {663, 813}, {694, 2330}, {741, 1334}, {1808, 2333}, {1927, 17787}, {1967, 2329}, {2195, 3252}, {3572, 3939}, {4140, 17938}, {7081, 9468}, {7233, 14827}
    barycentric quotient X(i)/X(j) for these {i,j}: {6, 18033}, {31, 10030}, {32, 1447}, {41, 350}, {55, 1921}, {220, 4087}, {292, 6063}, {560, 1429}, {669, 7212}, {813, 4572}, {1253, 3975}, {1501, 1428}, {1911, 85}, {1918, 16609}, {1922, 7}, {1927, 1432}, {2175, 239}, {2196, 7182}, {2205, 1284}, {2311, 310}, {2329, 1926}, {2330, 3978}, {3063, 3766}, {4518, 1502}, {4876, 561}, {7077, 76}, {7081, 14603}, {7109, 7235}, {8789, 1431}, {9447, 238}, {9448, 1914}, {9468, 7249}, {14598, 57}, {14827, 3685}


    X(18266) = ISOGONAL CONJUGATE OF X(18032)

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

    X(18266) lies on the cubic K988 and these lines: {21, 756}, {31, 32}, {36, 386}, {42, 172}, {48, 2209}, {100, 1580}, {101, 3747}, {239, 11364}, {244, 11349}, {560, 15624}, {692, 922}, {872, 1333}, {976, 13723}, {1326, 1757}, {1691, 7077}, {1914, 2109}, {1918, 2174}, {1964, 2220}, {2177, 4262}, {3285, 4557}, {6542, 8298}, {7032, 16946}

    X(18266) = isogonal conjugate of X(18032)
    X(18266) = X(55)-beth conjugate of X(17798)
    X(18266) = X(i)-Ceva conjugate of X(j) for these (i,j): {1326, 17735}, {1911, 31}
    X(18266) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18032}, {2, 6650}, {69, 17982}, {75, 1929}, {76, 17962}, {86, 11599}, {99, 18014}, {264, 17972}, {274, 9278}, {310, 2054}, {350, 9505}, {523, 17930}, {670, 18001}, {850, 17940}, {1509, 6543}, {1921, 9506}, {2702, 3261}
    X(18266) = X(31)-Hirst inverse of X(213)
    X(18266) = crossdifference of every pair of points on line {693, 4359}
    X(18266) = crosssum of X(i) and X(j) for these (i,j): {1111, 3766}, {3948, 4647}
    X(18266) = barycentric product X(i)*X(j) for these {i,j}: {1, 17735}, {6, 1757}, {19, 17976}, {31, 6542}, {37, 1326}, {42, 1931}, {48, 17927}, {100, 5029}, {101, 9508}, {163, 18004}, {213, 17731}, {228, 423}, {292, 8298}, {661, 17943}, {662, 17990}, {692, 2786}, {798, 17934}, {1333, 6541}, {1911, 6651}, {14598, 18035}
    X(18266) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 18032}, {31, 6650}, {32, 1929}, {163, 17930}, {213, 11599}, {560, 17962}, {798, 18014}, {872, 6543}, {1326, 274}, {1757, 76}, {1918, 9278}, {1922, 9505}, {1924, 18001}, {1931, 310}, {1973, 17982}, {2205, 2054}, {5029, 693}, {6542, 561}, {8298, 1921}, {9247, 17972}, {9508, 3261}, {14598, 9506}, {17731, 6385}, {17735, 75}, {17927, 1969}, {17934, 4602}, {17943, 799}, {17976, 304}, {17990, 1577}
    X(18266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 869, 31), (2210, 2223, 31), (2223, 2251, 2210)


    X(18267) = X(32)-CROSS CONJUGATE OF X(1927)

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

    X(18267) lies on the cubic K988 and these lines: {1911, 1914}, {1922, 9454}, {9455, 14598}

    X(18267) = X(32)-cross conjugate of X(1927)
    X(18267) = X(i)-isoconjugate of X(j) for these (i,j): {76, 4366}, {239, 1921}, {310, 4368}, {561, 8300}, {871, 3802}, {874, 3766}, {1447, 4087}, {1978, 4375}, {3027, 18021}, {3685, 18033}, {3975, 10030}, {3978, 17493}
    X(18267) = barycentric product X(i)*X(j) for these {i,j}: {291, 1922}, {292, 1911}, {335, 14598}, {813, 875}
    X(18267) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 4366}, {1501, 8300}, {1911, 1921}, {1922, 350}, {1927, 17493}, {1980, 4375}, {2205, 4368}, {14598, 239}


    X(18268) = ISOGONAL CONJUGATE OF X(3948)

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

    X(18268) lies on the cubics K988 and K997 and on these lines: {6, 662}, {32, 163}, {58, 101}, {81, 4586}, {83, 1509}, {99, 17475}, {110, 1977}, {593, 3051}, {643, 1979}, {692, 1333}, {694, 1169}, {875, 923}, {876, 1910}, {1408, 1415}, {1931, 16514}, {3114, 7307}, {4876, 16785}

    X(18268) = isogonal conjugate of X(3948)
    X(18268) = cevapoint of X(1911) and X(1922)
    X(18268) = trilinear pole of line {31, 669}
    X(18268) = crossdifference of every pair of points on line {4010, 4155}
    X(18268) = crosssum of X(740) and X(3985)
    X(18268) = X(284)-beth conjugate of X(662)
    X(18268) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3948}, {63, 2238}
    X(18268) = X(i)-cross conjugate of X(j) for these (i,j): {237, 56}, {1911, 741}
    X(18268) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3948}, {2, 740}, {7, 3985}, {8, 16609}, {10, 239}, {37, 350}, {42, 1921}, {65, 3975}, {75, 2238}, {76, 3747}, {85, 4433}, {86, 4037}, {88, 4783}, {190, 4010}, {210, 10030}, {226, 3685}, {238, 321}, {242, 306}, {257, 4039}, {304, 862}, {308, 4093}, {312, 1284}, {313, 1914}, {333, 7235}, {335, 4368}, {345, 1874}, {523, 3570}, {659, 4033}, {661, 874}, {799, 4155}, {812, 3952}, {1018, 3766}, {1215, 17493}, {1334, 18033}, {1400, 4087}, {1429, 3701}, {1441, 3684}, {1447, 2321}, {1577, 3573}, {1916, 4154}, {1978, 4455}, {2054, 18035}, {3699, 7212}, {3716, 4552}, {3932, 6654}, {4080, 4432}, {4148, 4566}, {4486, 4613}, {4771, 5936}, {4974, 6539}, {6651, 11599}, {13576, 17755}
    X(18268) = X(58)-Hirst inverse of X(741)
    X(18268) = barycentric product X(i)*X(j) for these {i,j}: {1, 741}, {27, 2196}, {28, 295}, {34, 1808}, {57, 2311}, {58, 291}, {81, 292}, {86, 1911}, {99, 875}, {110, 876}, {163, 4444}, {274, 1922}, {310, 14598}, {334, 2206}, {335, 1333}, {337, 2203}, {649, 4584}, {660, 3733}, {662, 3572}, {667, 4589}, {805, 4367}, {813, 1019}, {1014, 7077}, {1326, 9505}, {1408, 4518}, {1412, 4876}, {1919, 4639}, {1931, 9506}, {1967, 17103}, {2194, 7233}, {4374, 17938}, {8033, 9468}
    X(18268) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3948}, {21, 4087}, {31, 740}, {32, 2238}, {41, 3985}, {58, 350}, {81, 1921}, {110, 874}, {163, 3570}, {213, 4037}, {284, 3975}, {291, 313}, {292, 321}, {560, 3747}, {604, 16609}, {667, 4010}, {669, 4155}, {741, 75}, {813, 4033}, {875, 523}, {876, 850}, {902, 4783}, {1014, 18033}, {1333, 239}, {1395, 1874}, {1397, 1284}, {1402, 7235}, {1408, 1447}, {1412, 10030}, {1576, 3573}, {1808, 3718}, {1911, 10}, {1922, 37}, {1923, 4093}, {1931, 18035}, {1933, 4154}, {1974, 862}, {1980, 4455}, {2175, 4433}, {2194, 3685}, {2196, 306}, {2203, 242}, {2206, 238}, {2210, 4368}, {2311, 312}, {3572, 1577}, {3733, 3766}, {4367, 14295}, {4584, 1978}, {4589, 6386}, {7077, 3701}, {7122, 4039}, {8033, 14603}, {14598, 42}, {16947, 1429}, {17103, 1926}, {17938, 3903}
    X(18268) = {X(741),X(2331)}-harmonic conjugate of X(292)


    X(18269) = X(604)X(1403)∩X(1922)(X(1927)

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

    X(18269) lies on the cubic K988 and these lines: {604, 1403}, {1922, 1927}

    X(18269) = barycentric product X(7121)*X(17792)


    X(18270) = X(31)-CEVA CONJUGATE OF X(1966)

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

    X(18270) lies on the cubic K989 and these lines: {1581, 1965}, {1923, 4602}

    X(18270) = X(31)-Ceva conjugate of X(1966)


    X(18271) = X(1967)-CEVA CONJUGATE OF X(75)

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

    X(18271) lies on the cubic K990 and these lines: {1, 1965}, {38, 1925}, {256, 9230}, {291, 3978}

    X(18271) = X(1967)-Ceva conjugate of X(75)
    X(18271) = X(1)-Hirst inverse of X(1965)
    X(18271) = barycentric product X(1926)*X(8871)
    X(18271) = barycentric quotient X(8871)/X(1967)


    X(18272) = X(1927)-CEVA CONJUGATE OF X(1)

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

    X(18272) lies on the cubic K991 and these lines: {1, 1925}, {31, 1582}, {384, 904}, {385, 1911}, {1964, 1965}, {1967, 2236}, {5999, 8927}

    X(18272) = X(1967)-aleph conjugate of X(17799)
    X(18272) = X(560)-he conjugate of X(1740)
    X(18272) = X(1927)-Ceva conjugate of X(1)
    X(18272) = X(31)-Hirst inverse of X(1582)
    X(18272) = barycentric product X(1966)*X(8871)
    X(18272) = barycentric quotient X(8871)/X(1581)
    {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3510, 7168, 385), (6196, 7346, 384)


    X(18273) = X(1)-CEVA CONJUGATE OF X(1926)

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

    X(18273) lies on the cubic K992 and these lines: {1925, 1934}

    X(18273) = X(1)-Ceva conjugate of X(1926)


    X(18274) = X(32)-CEVA CONJUGATE OF X(1914)

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

    X(18274) lies on the cubics K788 and K993 and on these lines: {31, 1979}, {32, 6196}, {100, 699}, {172, 694}, {238, 8623}, {350, 385}, {662, 1333}, {1580, 8845}, {1691, 2210}

    X(18274) = X(32)-Ceva conjugate of X(1914)
    X(18274) = crosspoint of X(3510) and X(8875)
    X(18274) = crosssum of X(7168) and X(8868)
    X(18274) = X(335)-isoconjugate of X(7168)
    X(18274) = X(1691)-Hirst inverse of X(2210)
    X(18274) = barycentric product X(238)*X(3510)
    X(18274) = barycentric quotient X(i)/X(j) for these {i,j}: {2210, 7168}, {3510, 334}, {18038, 8868}


    X(18275) = X(335)-CEVA CONJUGATE OF X(76)

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

    X(18275) lies on the cubic K994 and these lines: {1, 7346}, {75, 982}, {76, 1928}, {335, 1926}, {561, 3662}, {1978, 17759}, {3261, 3835}

    X(18275) = X(335)-Ceva conjugate of X(76)
    X(18275) = X(32)-isoconjugate of X(7168)
    X(18275) = X(75)-Hirst inverse of X(1920)
    X(18275) = barycentric product X(561)*X(3510)
    X(18275) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 7168}, {3510, 31}, {17789, 8868}


    X(18276) = X(1581)-CEVA CONJUGATE OF X(561)

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

    X(18276) lies on the cubic K995 and these lines: {75, 1925}, {334, 14603}

    X(18275) = X(1581)-Ceva conjugate of X(561)
    X(18275) = X(75)-Hirst inverse of X(1925)
    X(18275) = barycentric quotient X(8871)/X(1927)


    X(18277) = X(2)-CEVA CONJUGATE OF X(1921)

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

    X(18277) lies on the cubic K996 and these lines: {37, 6386}, {75, 3122}, {76, 3061}, {274, 4602}, {334, 1581}, {350, 3978}, {1921, 14603}, {1926, 3948}

    X(18277) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 1921}, {3510, 141}
    X(18277) = X(2)-Ceva conjugate of X(1921)
    X(18277) = X(1922)-isoconjugate of X(7168)
    X(18277) = X(350)-Hirst inverse of X(3978)
    X(18277) = barycentric quotient X(i)/X(j) for these {i,j}: {350, 7168}, {3510, 1911}, {18037, 8868}


    X(18278) = X(1)X(3495)∩X(6)X(75)

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

    X(18278) lies on the cubics K432 and K997 and on these lines: {1, 3495}, {6, 75}, {31, 172}, {171, 213}, {292, 1755}, {825, 14602}, {1403, 3224}, {1429, 2111}, {1580, 1922}, {1911, 3747}, {1967, 16365}, {1979, 2223}, {2231, 17790}

    X(18278) = X(i)-Ceva conjugate of X(j) for these (i,j): {1580, 17798}, {1922, 6}
    X(18278) = crosspoint of X(i) and X(j) for these (i,j): {813, 5383}
    X(18278) = crossdifference of every pair of points on line {788, 3741}
    X(18278) = crosssum of X(812) and X(6377)
    X(18278) = X(i)-isoconjugate of X(j) for these (i,j): {2, 7168}, {7261, 8868}
    X(18278) = X(i)-Hirst inverse of X(j) for these (i,j): {31, 172}
    X(18278) = X(18278) = barycentric product X(i)*X(j) for these {i,j}: {1, 3510}, {3509, 8875}
    X(18278) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 7168}, {3510, 75}


    X(18279) = X(4)X(74)∩X(5)X(523)

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

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27613.

    X(18279) lies on these lines: {3,16177}, {4,74}, {5,523}, {402,7740}, {5502,11911}, {5972,15454}, {6000,11251}, {6070,14264}, {6640,14379}, {10255,14059}


    X(18280) = (name pending)

    Barycentrics    a^2 (a^11+a^10 (b+c) + a^9 (-5 b^2+b c-5 c^2) - a^8 (5 b^3+4 b^2 c+4 b c^2+5 c^3) +a^7 (10 b^4-4 b^3 c+11 b^2 c^2-4 b c^3+10 c^4) + a^6 (10 b^5+6 b^4 c+9 b^3 c^2+9 b^2 c^3+6 b c^4+10 c^5) + a^5 (-10 b^6+6 b^5 c-4 b^4 c^2+9 b^3 c^3-4 b^2 c^4+6 b c^5-10 c^6) - a^4 (10 b^7+4 b^6 c+2 b^5 c^2+5 b^4 c^3+5 b^3 c^4+2 b^2 c^5+4 b c^6+10 c^7) + a^3 (5 b^8-4 b^7 c-5 b^6 c^2-b^5 c^3+2 b^4 c^4-b^3 c^5-5 b^2 c^6-4 b c^7+5 c^8) + a^2 (b-c)^2 (5 b^7+11 b^6 c+10 b^5 c^2+11 b^4 c^3+11 b^3 c^4+10 b^2 c^5+11 b c^6+5 c^7) - a (b^2-c^2)^4 (b^2-b c+c^2) - (b-c)^4 (b+c)^5 (b^2-b c+c^2)) : :
    X(18280) = R (3 R^2-4 s^2)*X(1) + 2 (r R^2+2 R^3+4 r s^2)*X(3)

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27614.

    X(18280) lies on this line: {1,3}


    X(18281) = (name pending)

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

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27615.

    X(18281) lies on these lines: {2,3}, {49,11457}, {68,13561}, {70,16867}, {125,13352}, {156,14216}, {499,8144}, {524,8548}, {541,3357}, {542,1147}, {568,15061}, {576,15118}, {2888,15124}, {5418,11265}, {5420,11266}, {5449,13346}, {5462,5476}, {5562,13857}, {5642,10539}, {5654,5663}, {5878,15138}, {6102,16270}, {6247,9820}, {6662,9214}, {6699,11438}, {9306,14156}, {9630,10072}, {9927,15123}, {10264,15106}, {10282,11645}, {10510,11255}, {11064,15068}, {12038,15126}, {12118,15114}, {12161,15120}, {12325,15137}, {13363,14561}, {15113,17702}


    X(18282) = MIDPOINT OF X(5) AND X(12107)

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

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27612 and Hyacinthos 27612.

    X(18282) lies on these lines: {2, 3}, {17, 11268}, {18, 11267}, {51, 8254}, {52, 11803}, {110, 3519}, {143, 12242}, {5562, 10272}, {5663, 14862}, {5944, 10619}, {5972, 10627}, {6689, 13364}, {6723, 17712}, {7691, 11805}, {8718, 15061}, {8960, 11266}, {10192, 10274}, {13561, 14864}

    X(18282) = midpoint of X(i) and X(j), for these {i, j}: {5,12107}, {26,10224}, {1658,13406}, {10020,13383}, {15331,15761}
    X(18282) = reflection of X(i) in X(j), for these {i, j}: {5,12010}, {5498,10125}, {10125,10020} , {11250,10212}


    X(18283) = X(1)X(4)∩X(3)X(280)

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

    See Telv Cohl and Angel Montesdeoca, Hyacinthos 27620 and HG040518.

    X(18283) lies on these lines: {1,4}, {3,280}, {20,1897}, {24,1604}, {104,963}, {108,3176}, {198,7003}, {318,5731}, {451,10786}, {1071,5908}, {3183,3198}

    X(18283) = reflection of X(i) in X(j) for these (i,j): (4, 7952), (280, 3)


    X(18284) = X(5)X(13289)∩X(107)X(7488)

    Barycentrics    a^2 (a^26-5 a^24 b^2+6 a^22 b^4+10 a^20 b^6-29 a^18 b^8+9 a^16 b^10+36 a^14 b^12-36 a^12 b^14-9 a^10 b^16+29 a^8 b^18-10 a^6 b^20-6 a^4 b^22+5 a^2 b^24-b^26-5 a^24 c^2+24 a^22 b^2 c^2-36 a^20 b^4 c^2-6 a^18 b^6 c^2+76 a^16 b^8 c^2-74 a^14 b^10 c^2-14 a^12 b^12 c^2+82 a^10 b^14 c^2-59 a^8 b^16 c^2-6 a^6 b^18 c^2+34 a^4 b^20 c^2-20 a^2 b^22 c^2+4 b^24 c^2+6 a^22 c^4-36 a^20 b^2 c^4+83 a^18 b^4 c^4-70 a^16 b^6 c^4-49 a^14 b^8 c^4+152 a^12 b^10 c^4-105 a^10 b^12 c^4-18 a^8 b^14 c^4+75 a^6 b^16 c^4-56 a^4 b^18 c^4+22 a^2 b^20 c^4-4 b^22 c^4+10 a^20 c^6-6 a^18 b^2 c^6-70 a^16 b^4 c^6+156 a^14 b^6 c^6-102 a^12 b^8 c^6-76 a^10 b^10 c^6+166 a^8 b^12 c^6-84 a^6 b^14 c^6-4 a^4 b^16 c^6+10 a^2 b^18 c^6-29 a^18 c^8+76 a^16 b^2 c^8-49 a^14 b^4 c^8-102 a^12 b^6 c^8+216 a^10 b^8 c^8-118 a^8 b^10 c^8-57 a^6 b^12 c^8+98 a^4 b^14 c^8-25 a^2 b^16 c^8-10 b^18 c^8+9 a^16 c^10-74 a^14 b^2 c^10+152 a^12 b^4 c^10-76 a^10 b^6 c^10-118 a^8 b^8 c^10+164 a^6 b^10 c^10-66 a^4 b^12 c^10-22 a^2 b^14 c^10+31 b^16 c^10+36 a^14 c^12-14 a^12 b^2 c^12-105 a^10 b^4 c^12+166 a^8 b^6 c^12-57 a^6 b^8 c^12-66 a^4 b^10 c^12+60 a^2 b^12 c^12-20 b^14 c^12-36 a^12 c^14+82 a^10 b^2 c^14-18 a^8 b^4 c^14-84 a^6 b^6 c^14+98 a^4 b^8 c^14-22 a^2 b^10 c^14-20 b^12 c^14-9 a^10 c^16-59 a^8 b^2 c^16+75 a^6 b^4 c^16-4 a^4 b^6 c^16-25 a^2 b^8 c^16+31 b^10 c^16+29 a^8 c^18-6 a^6 b^2 c^18-56 a^4 b^4 c^18+10 a^2 b^6 c^18-10 b^8 c^18-10 a^6 c^20+34 a^4 b^2 c^20+22 a^2 b^4 c^20-6 a^4 c^22-20 a^2 b^2 c^22-4 b^4 c^22+5 a^2 c^24+4 b^2 c^24-c^26) : :

    See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27630.

    X(18284) lies on these lines: {5,13289}, {107,7488}


    X(18285) = MIDPOINT OF X(399) AND X(1138)

    Barycentrics    4 a^16-17 a^14 b^2+19 a^12 b^4+19 a^10 b^6-65 a^8 b^8+61 a^6 b^10-23 a^4 b^12+a^2 b^14+b^16-17 a^14 c^2+58 a^12 b^2 c^2-81 a^10 b^4 c^2+76 a^8 b^6 c^2-59 a^6 b^8 c^2+18 a^4 b^10 c^2+13 a^2 b^12 c^2-8 b^14 c^2+19 a^12 c^4-81 a^10 b^2 c^4+54 a^8 b^4 c^4-11 a^6 b^6 c^4+36 a^4 b^8 c^4-45 a^2 b^10 c^4+28 b^12 c^4+19 a^10 c^6+76 a^8 b^2 c^6-11 a^6 b^4 c^6-62 a^4 b^6 c^6+31 a^2 b^8 c^6-56 b^10 c^6-65 a^8 c^8-59 a^6 b^2 c^8+36 a^4 b^4 c^8+31 a^2 b^6 c^8+70 b^8 c^8+61 a^6 c^10+18 a^4 b^2 c^10-45 a^2 b^4 c^10-56 b^6 c^10-23 a^4 c^12+13 a^2 b^2 c^12+28 b^4 c^12+a^2 c^14-8 b^2 c^14+c^16 : :
    X(18285) = 3 X[5] - 2 X[5627] = 2 X[399] + X[11749] = X[5] + 2 X[14480] = X[5627] + 3 X[14480]

    See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27632.

    X(18285) lies on these lines: {5,1117}, {30,146}

    X(18285) = midpoint of X(399) and X(1138)
    X(18285) = reflection of X(i) in X(j) for these {i,j}: {11749, 1138}, {14993,10272}
    X(18285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3470, 3471, 5)
    X(18285) = X(484)-aleph conjugate of X(1048)
    X(18285) = X(i)-Ceva conjugate of X(j) for these (i,j): {10272, 5}, {14993, 30}
    X(18285) = barycentric quotient X(i)/X(j) for these {i,j}: {5671, 13582}, {11063, 8487}


    X(18286) = (name pending)

    Barycentrics    (a^10-5*(b^2+c^2)*a^8-6*(b^ 2+3*b*c+c^2)*(b^2-3*b*c+c^2)* a^6+2*(b^2+c^2)*(3*b^4-13*b^2* c^2+3*c^4)*a^4+(5*b^8+5*c^8-2* (29*b^4-49*b^2*c^2+29*c^4)*b^ 2*c^2)*a^2-(b^2+c^2)*(b^8+c^8- 10*(b^4-b^2*c^2+c^4)*b^2*c^2)) *a^2 : :
    X(18286) = (6*R^2-SW)*(32*S^2*R^2-SW*( SW^2+2*S^2))*X(25) - SW^3*R^2*X(193)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27638.

    X(18286) lies on this line: {25, 193}


    X(18287) = ANTICOMPLEMENT OF X(6340)

    Barycentrics    5*a^6-5*(b^2+c^2)*a^4-(9*b^4- 14*b^2*c^2+9*c^4)*a^2+(b^2+c^ 2)^3 : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27638.

    X(18287) lies on these lines: {2, 1975}, {4, 8892}, {25, 193}, {385, 17037}, {439, 1611}, {3164, 10565}, {6353, 6392}, {6462, 8854}, {6995, 8878}, {7398, 17035}

    X(18287) = anticomplement of X(6340)
    X(18287) = {X(6337), X(8770)}-harmonic conjugate of X(2)


    X(18288) = X(2)X(107)∩X(25)X(64)

    Barycentrics    tan(A)*(2*(6*cos(A)-7*cos(3*A) +cos(5*A))*cos(B-C)-4*(3*cos( 2*A)+cos(4*A)+4)*cos(2*(B-C))+ 8*cos(4*A)-cos(6*A)+21*cos(2*A)+4) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27638.

    X(18288) lies on these lines: {2, 107}, {4, 17830}, {25, 64}, {232, 1033}, {3183, 6353}


    X(18289) = X(2)X(371)∩X(25)X(485)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27638.

    X(18289) lies on these lines: {2, 371}, {4, 8280}, {6, 6676}, {22, 6560}, {25, 485}, {372, 7494}, {427, 6561}, {590, 5020}, {858, 9681}, {1151, 1368}, {1587, 10565}, {3068, 5413}, {3070, 9909}, {4232, 8960}, {5013, 8964}, {5159, 6425}, {5420, 7499}, {6200, 7386}, {6409, 10691}, {6423, 8944}, {6453, 16051}, {6459, 8889}, {6564, 6995}, {6677, 8981}, {7392, 10576}, {7396, 9541}, {7583, 10154}

    X(18289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1588, 8281), (6, 6676, 18290), (3068, 6353, 8854)


    X(18290) = X(2)X(372)∩X(25)X(486)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27638.

    X(18290) lies on these lines: {2, 372}, {4, 8281}, {6, 6676}, {22, 6561}, {25, 486}, {371, 7494}, {427, 6560}, {1152, 1368}, {1588, 10565}, {3053, 8964}, {3069, 5412}, {3071, 9909}, {5159, 6426}, {5418, 7499}, {6410, 10691}, {6424, 8940}, {6454, 16051}, {6460, 8889}, {6677, 13966}, {7392, 10577}, {7584, 10154}

    X(18290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1587, 8280), (6, 6676, 18289), (3069, 6353, 8855)


    X(18291) = X(1)X(164)∩X(168)X(3973)

    Trilinears    Cos[A/2] (Csc[B/2] - Csc[C/2]) - Cos[B/2] (Csc[C/2] - Csc[A/2]) - Cos[C/2] (Csc[A/2] - Csc[B/2]) - 2 Cot[B/2] (1 - Csc[C/2] Sin[A/2]) + 2 Cot[C/2] (1 - Csc[B/2] Sin[A/2]) : :
    Barycentrics    Cos[A/2] Sin[A/2] (1 + Sin[B/2] + Sin[C/2] - 3 Sin[A/2]) : :    (Peter Moses, May 11, 2018)
    Barycentrics    a(1 + Sin[B/2] + Sin[C/2] - 3 Sin[A/2]) : :

    See Randy Hutson, ADGEOM 4554.

    X(18291) lies on these lines: {1,164}, {168,3973}, {361,1743}, {8688,12518}


    X(18292)  = EULER LINE INTERCEPT OF X(1)X(359)

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

    X(18292) lies on these lines: {1, 359}, {2, 3}


    X(18293)  = EULER LINE INTERCEPT OF X(1)X(360)

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

    X(18293) lies on these lines: {1, 360}, {2, 3}


    X(18294)  = COMPLEMENT OF X(1115)

    Barycentrics    A + π : B + π : C + π
    X(18294) = 3 X[2] + X[360]

    X(18294) lies on this line: {2, 360}

    X(18294) = midpoint of X(360) and X(1115)
    X(18294) = complement X(1115)
    X(18294) = X(i)-complementary conjugate of X(j) for these (i,j): {7021, 10}, {7041, 141}
    X(18294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 360, 1115)


    X(18295)  = (name pending)

    Barycentrics    2A + π : 2B + π : 2C + π
    X(18295) = 3 X[2] + 2 X[360] = 9 X[2] - 4 X[1115] = 3 X[360] + 2 X[1115]

    X(18295) lies on this line: {2, 360}


    X(18296) = X(65)X(5556)∩X(3091)X(14528)

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

    See Angel Montesdeoca, Hyacinthos 27653.

    X(18296) lies on these lines:
    {65,5556}, {3091,14528}, {3146,3532}, {3426,12102}, {3431,3544}, {11270,11541}, {13851,15740}


    X(18297) = ISOTOMIC CONJUGATE OF X(366)

    Barycentrics    a-1/2 : b-1/2 : c-1/2

    X(18297) lies on the cubics K744, K766, and the line {510, 1759}

    X(18297) = isogonal conjugate of X(18753)
    X(18297) = isotomic conjugate of X(366)
    X(18297) = X(4179)-cross conjugate of X(366)
    X(18297) = X(i)-isoconjugate of X(j) for these (i,j): {6, 365}, {31, 366}, {56, 4166}, {604, 4182}, {1333, 4179}
    X(18297) = cevapoint of X(i) and X(j) for these (i,j): {1, 510}, {366, 4182}
    X(18297) = barycentric product X(i)*X(j) for these {i,j}: {75, 366}, {76, 365}, {85, 4182}, {274, 4179}, {508, 556}, {4166, 6063}
    X(18297) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 365}, {2, 366}, {8, 4182}, {9, 4166}, {10, 4179}, {174, 509}, {365, 6}, {366, 1}, {508, 174}, {509, 266}, {4146, 508}, {4166, 55}, {4179, 37}, {4181, 4180}, {4182, 9}


    X(18298) = ISOTOMIC CONJUGATE OF X(1045)

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

    X(18298) lies on the cubic K744 and these lines: {1, 2668}, {10, 1920}, {37, 1655}, {65, 7196}

    X(18298) = isogonal conjugate of X(18755)
    X(18298) = isotomic conjugate of X(1045)
    X(18298) = X(i)-cross conjugate of X(j) for these (i,j): {257, 85}, {310, 75}
    X(18298) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1045}, {32, 1655}, {110, 9402}
    X(18298) = trilinear pole of line {661, 4374}
    X(18298) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1045}, {75, 1655}, {661, 9402}


    X(18299) = ISOTOMIC CONJUGATE OF X(17792)

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

    X(18299) lies on the the Feuerbach hyperbola, the cubic K744, and these lines: {8, 3978}, {9, 1966}, {76, 7155}, {256, 10030}, {1909, 4876}, {3551, 3673}

    X(18299) = isogonal conjugate of X(18758)
    X(18299) = isotomic conjugate of X(17792)
    X(18299) = X(i)-cross conjugate of X(j) for these (i,j): {257, 274}, {18033, 2481}
    X(18299) = X(i)-isoconjugate of X(j) for these (i,j): {31, 17792}, {32, 17760}, {192, 18269}, {1911, 8844}
    X(18299) = cevapoint of X(i) and X(j) for these (i,j): {2, 3056}, {693, 4459}
    X(18299) = trilinear pole of line {650, 14296}
    X(18299) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 17792}, {75, 17760}, {239, 8844}, {7121, 18269}




    leftri  Triangles and centers related to the Ehrmann pivots (PU(5)): X(18300) - X(18587)  rightri

    This preamble and centers X(18300)-X(18587) were contributed by Randy Hutson, May 2, 2018.

    Let TC1=A1B1C1 be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of 2π/3, so that TC1 circumscribes ABC. Triangle TC1 is here named the 1st Ehrmann circumscribing triangle. Let TC2=A2B2C2 be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of -2π/3, so that TC2 circumscribes ABC. Triangle TC2 is here named the 2nd Ehrmann circumscribing triangle. TC1 and TC2 are inscribed in the Johnson circle (centered at X(4)).

    Let TI1=A'1B'1C'1 be the triangle obtained by rotating ABC about the 1st Ehrmann pivot, P(5), by an angle of -2π/3, so that TI1 is inscribed in ABC. Triangle TI1 is here named the 1st Ehrmann inscribed triangle. Let TI2=A'2B'2C'2 be the triangle obtained by rotating ABC about the 2nd Ehrmann pivot, U(5), by an angle of 2π/3, so that TI2 is inscribed in ABC. Triangle TI2 is here named the 2nd Ehrmann inscribed triangle. TI1 and TI2 are inscribed in a common conic, here named the Ehrmann conic. The center of the Ehrmann conic is X(14993).


    The orthocenter of TC1 is the circumcenter of TI1, and is here introduced (and at Bicentric Pairs) as P(173). P(173) is therefore the intersection of the Euler lines of TC1 and TI1. The orthocenter of TC2 is the circumcenter of TI2, and is here introduced as U(173). U(173) is therefore the intersection of the Euler lines of TC2 and TI2. PU(173) lie on the Hatzipolakis axis (line PU(5) or X(5)X(523)). The midpoint of PU(173) is X(5). P(173), X(3) and X(4) are the vertices of an equilateral triangle with center P(5). U(173), X(3) and X(4) are the vertices of an equilateral triangle with center U(5).


    The vertex-triangle of TC1 and TC2 is here named the Ehrmann vertex-triangle. The Ehrmann vertex-triangle is the Kosnita triangle of the Johnson triangle (or equivalently, the reflection of the Kosnita triangle in X(5)), and the tangential triangle of the Ehrmann mid-triangle (defined below). The vertex-triangle of TI1 and TI2 is ABC.

    Let VAVBVC be the Ehrmann vertex-triangle. Then VA is the isogonal conjugate of A'1 wrt TC1, and the isogonal conjugate of A'2 wrt TC2, and cyclically for VB and VC.

    The A-vertex of the Ehrmann vertex-triangle has barycentric coordinates:
    VA = a^6 - a^4(b^2 + c^2) - a^2(b^4 + c^4) + (b^2 - c^2)^2(b^2 + c^2)   :   -(a^2 + b^2 - c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2]   :   -(a^2 - b^2 + c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2]

    The Ehrmann vertex-triangle is orthologic to ABC with orthology center X(4).

    The Ehrmann vertex-triangle is Eulerologic to ABC at X(7577).

    The Ehrmann vertex-triangle is perspective to the reflections-of-P triangle for all P.

    The appearance of (T,i) in the following list means that the Ehrmann vertex-triangle is perspective to triangle T at X(i). An asterisk indicates the triangles are homothetic.

    (ABC, 4), (anticomplementary, 18387), (anti-Ara, 18385), (anti-Ascella*, 18386), (1st anti-Conway*, 18388), (2nd anti-Conway*, 18390), (anti-Euler, 4), (3rd anti-Euler*, 18392), (4th anti-Euler*, 18394), (anti-excenters-incenter reflections*, 4), (2nd anti-extouch*, 18396), (anti-Hutson intouch*, 382), (anti-incircle-circles*, 3843), (anti-inverse-in-incircle*, 4), (6th anti-mixtilinear*, 18531), (anti-tangential midarc*, 3585), (circumorthic*, 4), (dual of orthic*, 3153), (2nd Ehrmann*, 542), (X(2)-Ehrmann*, 7574), (X(3)-Ehrmann*, 18400), (X(4)-Ehrmann*, 18403), (Ehrmann mid, 3818), (Ehrmann side*, 18403), (Euler, 4), (2nd Euler*, 18404), (1st excosine*, 18405), (extangents*, 18406), (2nd extouch, 4), (3rd extouch, 4), (intangents*, 3583), (Johnson, 6288), (1st Kenmotu diagonals*, 6564), (2nd Kenmotu diagonals*, 6565), (Kosnita*, 5), (Lucas antipodal tangents*, 18414), (Lucas(-1) antipodal tangents*, 18415), (midheight, 4), (mid-triangle of 1st and 2nd Kenmotu diagonals*, 115), (mid-triangle of inner and outer tri-equilateral*, 18424), (orthic*, 4), (orthoanticevian of X(3)*, 18418), (orthocentroidal, 4), (orthocevian of X(3)*, 18416), (1st orthosymmedial, 4), (reflection, 4), (reflections-of-X(1), 18426), (reflections-of-X(2), 18427), (reflections-of-X(5), 18428), (reflections-of-X(6), 18429), (submedial*, 18420), (tangential*, 381), (inner tri-equilateral*, 16808), (outer tri-equilateral*, 16809), (Trinh*, 30)


    The side-triangle of TI1 and TI2 is here named the Ehrmann side-triangle. The Ehrmann side-triangle is the circumorthic triangle of the Johnson triangle (or equivalently, the reflection of the circumorthic triangle in X(5)), and also the 2nd isogonal triangle of X(4) (see X(36)). It is inscribed (along with TC1 and TC2) in the Johnson circle. The side-triangle of TC1 and TC2 is ABC.

    The A-vertex of the Ehrmann side-triangle has barycentric coordinates:
    SA = a^2[a^4 + a^2(b^2 + c^2) - 2(b^2 - c^2)^2]   :   -(a^2 - b^2 + c^2)[(a^2 + b^2 - c^2)^2 - a^2b^2]   :   -(a^2 + b^2 - c^2)[(a^2 - b^2 + c^2)^2 - a^2c^2]

    The Ehrmann side-triangle is orthologic to ABC with orthology center X(3).

    The Ehrmann side-triangle is Eulerologic to ABC at X(3). ABC is Eulerologic to the Ehrmann side-triangle at X(18403).

    The appearance of (T,i) in the following list means that the Ehrmann side-triangle is perspective to triangle T at X(i). An asterisk indicates the triangles are homothetic.

    (AAOA, 18441), (ABC, 265), (ABC-X3 reflections, 18442), (anti-Ascella*, 9818), (1st anti-Conway*, 567), (2nd anti-Conway*, 568), (3rd anti-Euler*, 11459), (4th anti-Euler*, 12111), (anti-excenters-incenter reflections*, 382), (2nd anti-extouch*, 18445), (anti-Hutson intouch*, 3), (anti-incircle-circles*, 3), (anti-inverse-in-incircle*, 18531), (6th anti-mixtilinear*, 3), (anti-tangential midarc*, 18447), (Ara, 3), (Ascella, 3), (circumorthic*, 5), (1st circumperp, 3), (2nd circumperp, 3), (dual of orthic*, 30), (2nd Ehrmann*, 18449), (X(3)-Ehrmann*, 10540), (X(4)-Ehrmann*, 18403), (Ehrmann vertex*, 18403), (2nd Euler*, 3), (1st excosine*, 18451), (extangents*, 18453), (Fuhrmann, 3), (intangents*, 18455), (Johnson, 3), (1st Kenmotu diagonals*, 18457), (2nd Kenmotu diagonals*, 18459), (Kosnita*, 3), (Lucas antipodal tangents*, 18462), (Lucas(-1) antipodal tangents*, 18462), (medial, 3), (mid-triangle of 1st and 2nd Kenmotu diagonals*, 10317), (mid-triangle of inner and outer tri-equilateral*, 18472), (orthic, 381), (orthoanticevian of X(3)*, 18466), (orthocevian of X(3)*, 18464), (submedial*, 5055), (tangential*, 3), (inner tri-equilateral*, 18468), (outer tri-equilateral*, 18470), (Trinh*, 3), (X3-ABC reflections, 12429)


    The mid-triangle of TC1 and TC2 is here named the Ehrmann mid-triangle. The Ehrmann mid-triangle is the Johnson triangle of the Euler triangle (or equivalently, the reflection of the Euler triangle in X(546)), and the intouch triangle of the Ehrmann vertex-triangle if ABC is acute. The mid-triangle of TI1 and TI2 is the medial triangle.

    The A-vertex of the Ehrmann mid-triangle has barycentric coordinates:
    MA = 2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2   :   -2a^4 + b^4 - 2c^4 + a^2b^2 + 4a^2c^2 + b^2c^2   :   -2a^4 - 2b^4 + c^4 + 4a^2b^2 + a^2c^2 + b^2c^2

    ABC is Eulerologic to the Ehrmann mid-triangle at X(4).

    The appearance of (T,i) in the following list means that the Ehrmann mid-triangle is perspective to triangle T at X(i). An asterisk indicates the triangles are homothetic.

    (ABC*, 381), (ABC-X3 reflections*, 382), (anticomplementary*, 4), (anti-Ara*, 4), (anti-Aquila*, 9955), (5th anti-Brocard*, 18502), (anti-Euler*, 3091), (anti-excenters-incenter reflections, 18488), (anti-inverse-in-incircle, 18489), (anti-Mandart-incircle*, 18491), (3rd antipedal of X(1)*, 18529), (3rd antipedal of X(3)*, 18535), (3rd antipedal of X(4)*, 18537), (Ara*, 9818), (Aquila*, 18492), (1st Auriga*, 18495), (2nd Auriga*, 18497), (5th Brocard*, 18500), (circumorthic, 18504), (Danneels-Bevan*, 18505), (Ehrmann vertex, 3818), (Euler*, 546), (3rd Euler, 5), (4th Euler, 5), (outer Garcia*, 12699), (Gossard*, 18507), (inner Grebe*, 18509), (outer Grebe*, 18511), (2nd isogonal of X(1)*, 18513), (Johnson*, 4), (inner Johnson*, 18516), (outer Johnson*, 18517), (1st Johnson-Yff*, 1479), (2nd Johnson-Yff*, 1478), (Lucas homothetic*, 18520), (Lucas(-1) homothetic*, 18522), (Mandart-incircle*, 3583), (medial*, 30), (5th mixtilinear*, 18525), (orthic, 113), (orthocevian of X(2)*, 5), (3rd pedal of X(1)*, 18527), (3rd pedal of X(3)*, 18531), (3rd pedal of X(4)*, 1596), (X3-ABC reflections*, 3843), (submedial, 5), (inner Yff*, 10895), (outer Yff*, 10896), (inner Yff tangents*, 18542), (outer Yff tangents*, 18544)


    The cross-triangle of TI1 and TI2 is here named the Ehrmann cross-triangle. The Ehrmann cross-triangle is degenerate, its vertices lying on line X(3)X(523) (the trilinear polar of X(2986)). It is perspective to the Johnson triangle at X(10745). The vertices of the cross triangle of TC1 and TC2 are the infinite points of lines BC, CA, AB.

    The A-vertex of the Ehrmann cross-triangle has barycentric coordinates:
    XA = (b^2 - c^2)(2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)   :   -(a^2 - b^2 + c^2)[(a^2 + b^2 - c^2)^2 - a^2b^2]   :   (a^2 + b^2 - c^2)[(a^2 - b^2 + c^2)^2 - a^2c^2]



    underbar



    X(18300) = X(4)X(94)∩X(328)X(3153)

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

    Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. X(18300) is the radical center of the circumcircles of AA1A2, BB1B2, CC1C2.

    X(18300) lies on these lines: {4,94}, {328,3153}, {381,7578}, {476,10296}, {1989,3839} et al


    X(18301) = ANTICOMPLEMENT OF X(94)

    Barycentrics    sin A csc 3A - sin B csc 3B - sin C csc 3C : :
    Barycentrics    a^10(b^2 + c^2) + a^2(b^2 - c^2)^2(b^2 + c^2)^3 - a^8(4b^4 + 3b^2c^2 + 4c^4) + 2a^6(3b^6 + b^4c^2 + b^2c^4 + 3c^6) - a^4(4b^8 - b^4c^4 + 4c^8) - b^2c^2(b^2 - c^2)^4 : :

    Let A'1B'1C'1 and A'2B'2C'2 be the 1st and 2nd Ehrmann inscribed triangles. X(18301) is the radical center of circumcircles of AA'1A'2, BB'1B'2, CC'1C'2.

    X(18301) lies on these lines: {2,94}, {20,5663}, {147,5189}, {148,13582}, {194,11004}, {6194,7492} et al

    X(18301) = anticomplement of X(94)


    X(18302) = X(4)X(8)∩X(6265)X(18342)

    Barycentrics    [a^2(b^2 - bc + c^2) - (b^2 - c^2)^2]*[a^6 - a^4(2b^2 + bc + 2c^2) + 3a^3bc(b + c) + a^2(b^4 - 4b^2c^2 + c^4) - 3abc(b - c)^2(b + c) + bc(b^2 - c^2)^2] : :

    X(18302) is the radical center of the incircles of ABC and the 1st and 2nd Ehrmann circumscribing triangles.

    X(18302) lies on these lines: {4,8}, {6265,18342}


    X(18303) = X(3)X(10)∩X(2800)X(18341)

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

    X(18303) is the radical center of the incircles of ABC and the 1st and 2nd Ehrmann inscribed triangles.

    X(18303) lies on these lines: {3,10}, {2800,18341}


    X(18304) = X(4)X(69)∩X(110)X(1316)

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

    X(18304) is the radical center of the Brocard circles of ABC and the 1st and 2nd Ehrmann circumscribing triangles.

    X(18304) lies on these lines: {4,69}, {39,6794}, {110,1316}, {194,5654}, {262,2394}, {868,7697}, {4549,6194} et al


    X(18305) = X(3)X(2916)∩X(690)X(6033)

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

    X(18305) is the radical center of the Brocard circles of ABC and the 1st and 2nd Ehrmann inscribed triangles.

    X(18305) lies on these lines: {3,2916}, {690,6033}, {2420,11646}


    X(18306) = X(4)X(141)∩X(1499)X(5664)

    Barycentrics    a^14 + 3a^12(b^2 + c^2) - a^10(3b^4 + b^2c^2 + 3c^4) - 5a^8(3b^6 + b^4c^2 + b^2c^4 + 3c^6) + a^6(19b^8 + 10b^6c^2 + 14b^4c^4 + 10b^2c^6 + 19c^8) - a^4(b^2 + c^2)(3b^8 + 12b^6c^2 - 14b^4c^4 + 12b^2c^6 + 3c^8) - a^2(b^2 - c^2)^2(b^8 - 5b^6c^2 + 16b^4c^4 - 5b^2c^6 + c^8) - (b^2 - c^2)^4(b^2 + c^2)^3 : :

    X(18306) is the radical center of the orthosymmedial circles of ABC and the 1st and 2nd Ehrmann circumscribing triangles.

    X(18306) lies on these lines: {4,141}, {1499,5664}, {2088,2548} et al


    X(18307) = X(3)X(66)∩X(1316)X(6792)

    Barycentrics    3a^12(b^2 + c^2) - a^10(5b^4 + 27b^2c^2 + 5c^4) + a^8(b^2 + c^2)(2b^4 + 35b^2c^2 + 2c^4) + a^6(2b^8 - 36b^6c^2 - 17b^4c^4 - 36b^2c^6 + 2c^8) - a^4(b^2 + c^2)(b^8 - 21b^6c^2 + 13b^4c^4 - 21b^2c^6 + c^8) - a^2(b^2 + c^2)^2(b^8 + 2b^6c^2 - 2b^4c^4 + 2b^2c^6 + c^8) - b^2c^2(b^10 - b^8c^2 - b^2c^8 + c^10) : :

    X(18307) is the radical center of the orthosymmedial circles of ABC and the 1st and 2nd Ehrmann inscribed triangles.

    X(18307) lies on these lines: {3,66}, {1316,6792}, {2088,2453} et al


    X(18308) = X(5)X(1116)∩X(113)X(114)

    Barycentrics    (b^2 - c^2)[2a^8 + a^6(b^2 + c^2) - a^4(9b^4 - 8b^2c^2 + 9c^4) + a^2(b^2 + c^2)(7b^4 - 11b^2c^2 + 7c^4) - (b^2 - c^2)^2(b^4 + 4b^2c^2 + c^4)] : :
    X(18308) = 3*X(5)-X(15543), 3*X(381)-X(15475), 3*X(1116)-2*X(15543)

    X(18308) is the intersection, other than X(4), of the Lester circles of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18308) lies on these lines: {5, 1116}, {113, 114}, {381, 15475}, {512, 14566}, {523, 546}, {826, 2394}


    X(18309) = X(2)X(8644)∩X(5)X(9175)

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

    X(18309) is the intersection, other than X(4), of circles {{X(2),X(3),X(6),X(111),X(691)}} of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18309) lies on these lines: {2,8644}, {5,9175}, {381,9178}, {512,3818}, {523,3845}, {4108,10033} et al


    X(18310) = X(2)X(523)∩X(141)X(525)

    Barycentrics    (b^2 - c^2)[a^6 + a^4(b^2 + c^2) - a^2(b^4 + 5b^2c^2 + c^4) - (b^2 + c^2)(b^4 - 4b^2c^2 + c^4)] : :

    X(18310) is the intersection, other than X(3), of circles {{X(2),X(4),X(6)}} of the 1st and 2nd Ehrmann inscribed triangles.

    X(18310) lies on these lines: {2,523}, {141,525}, {599,1640}, {2492,6719} et al

    X(18310) = complement of X(18311)


    X(18311) = ANTICOMPLEMENT OF X(18310)

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

    X(18311) lies on these lines: {2,523}, {6,525}, {126,1560}, {233,13162}, {597,1640}, {648,5649}, {690,15303}, {804,14443}, {826,9171}, {1196,2485} et al

    X(18311) = anticomplement of X(18310)
    X(18311) = radical center of polar circle and circles O(13,14) and O(15,16) (the latter being the Schoute circle)
    X(18311) = barycentric product X(10)*X(514)*X(316)*X(524)
    X(18311) = barycentric product X(i)*X(j) for these {i,j}: {316,690}, {523,7664}, {524,9979}


    X(18312) = ISOTOMIC CONJUGATE OF X(5649)

    Trilinears    (csc 2A)[sin 3B csc 2B tan C cos(C + ω) - sin 3C csc 2C tan B cos(B + ω)] : :
    Barycentrics    (b^2 - c^2)(2a^6 - b^6 - c^6 - 2a^4b^2 - 2a^4c^2 + a^2b^4 + a^2c^4 + b^4c^2 + b^2c^4)/a^2 : :

    X(18312) lies on these lines: {2,647}, {5,523}, {76,2394}, {115,127}, {141,525}, {264,8430}, {512,3818}, {804,12042}, {879,1352}, {2485,7746} et al

    X(18312) = isotomic conjugate of X(5649)
    X(18312) = intersection, other than X(4), of circles {{X(3),X(4),X(6),X(112),X(842)}} of the 1st and 2nd Ehrmann circumscribing triangles
    X(18312) = intersection, other than X(3), of circles {{X(3),X(4),X(6),X(112),X(842)}} of the 1st and 2nd Ehrmann inscribed triangles


    X(18313) = X(512)X(3818)∩X(523)X(546)

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

    X(18313) is the intersection, other than X(4), of circles {{X(3),X(5),X(6),X(115)}} of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18313) lies on these lines: {512,3818}, {523,546}, {647,15820}, {804,6140}, {924,18553}, {3566,18358} et al


    X(18314) = POLAR CONJUGATE OF X(933)

    Barycentrics    sin(2B - 2C) : :
    Barycentrics    (b^2 - c^2)(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2)/a^2 : :

    X(18314) is the intersection, other than X(3), of circles {{X(4),X(15),X(16),X(186),X(3484),X(15412)}} of the 1st and 2nd Ehrmann inscribed triangles.

    The trilinear polar of X(18314) passes through X(137).

    X(18314) lies on the circumconic centered at X(12077), and these lines: {2,2413}, {5,15451}, {83,2422}, {128,16188}, {233,13162}, {297,525}, {512,13449}, {523,2072}, {826,1209}, {1510,11583}, {1577,6358} et al

    X(18314) = pole wrt polar circle of trilinear polar of X(933) (line X(6)X(24))
    X(18314) = polar conjugate of X(933)
    X(18314) = isogonal conjugate of X(14586)
    X(18314) = isotomic conjugate of X(18315)
    X(18314) = complement of X(15412)
    X(18314) = X(i)-isoconjugate of X(j) for these {i,j}: {1,14586}, {31,18315}, {48,933}, {54,163}
    X(18314) = barycentric product X(5)*X(10)*X(76)*X(514)
    X(18314) = barycentric product X(i)*X(j) for these {i,j}: {5,850}, {76,12077}, {311,523}


    X(18315) = TRILINEAR POLE OF LINE X(3)X(54)

    Barycentrics    csc(2B - 2C) : :
    Barycentrics    a^2/[(b^2 - c^2)(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2)] : :

    Line X(3)X(54) is the isogonal conjugate of hyperbola {{A,B,C,X(4),X(5)}}.

    X(18315) lies on the MacBeath circumconic and these lines: {54,575}, {95,141}, {97,6509}, {110,933}, {249,14570}, {275,2986}, {648,16813}, {1141,12383} et al

    X(18315) = isogonal conjugate of X(12077)
    X(18315) = isotomic conjugate of X(18314)
    X(18315) = trilinear pole of line X(3)X(54)
    X(18315) = X(i)-isoconjugate of X(j) for these {i,j}: {1,12077}, {19,6368}, {31,18314}, {51,1577}, {92,15451}
    X(18315) = barycentric product X(6)*X(86)*X(95)*X(190)
    X(18315) = barycentric product X(i)*X(j) for these {i,j}: {54,99}, {95,110}


    X(18316) = ISOGONAL CONJUGATE OF X(3581)

    Barycentrics    1/{[2b^4 + 2c^4 - a^4 - a^2b^2 - a^2c^2 - 4b^2c^2][(b^2 + c^2 - a^2)^2 - b^2c^2]} : :

    Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. Let A' be the cevapoint of A1 and A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X(18316).

    X(18316) lies on the Kiepert hyperbola and these lines: {2,265}, {4,1989}, {30,94}, {76,328}, {186,5627}, {275,1141}, {381,7578}, {526,2394}, {598,14356}, {1513,10511}, {3839,18576} et al

    X(18316) = barycentric quotient X(1989)/X(381)


    X(18317) = PERSPECTOR OF EHRMANN CONIC

    Barycentrics    1/[a^8 - 4a^6(b^2 + c^2) + a^4(6b^4 - 5b^2c^2 + 6c^4) - 4a^2(b^4 - c^4)(b^2 - c^2) + (b^4 + 7b^2c^2 + c^4)(b^2 - c^2)^2] : :

    Let A'1B'1C'1 and A'2B'2C'2 be the 1st and 2nd Ehrmann inscribed triangles. Let A' be the intersection of the tangents to the Ehrmann conic at A'1 and A'2. Define B', C' cyclically. The lines AA', BB', CC' concur in X(18317).

    X(18317) lies on these lines: {3,3163}, {30,14919}, {97,8703}, {394,399}, {1073,3830}, {1272,3926} et al

    X(18317) = isogonal conjugate of X(12112)
    X(18317) = perspector of Ehrmann conic


    X(18318) = INVERSE-IN-EHRMANN-CONIC OF X(3)

    Barycentrics    (a^2 - b^2 - c^2)[a^18 - 2a^16(b^2 + c^2) - 6a^14(b^4 - 3b^2c^2 + c^4) + a^12(b^2 + c^2)(23b^4 - 48b^2c^2 + 23c^4) - a^10(26b^8 + 29b^6c^2 - 111b^4c^4 + 29b^2c^6 + 26c^8) + 3a^8(b^2 - c^2)^2(b^2 + c^2)(3b^4 + 28b^2c^2 + 3c^4) + 2a^6(b^2 - c^2)^2(b^8 - 12b^6c^2 - 57b^4c^4 - 12b^2c^6 + c^8) + a^4(b^2 - c^2)^2(b^2 + c^2)(b^8 - 22b^6c^2 + 66b^4c^4 - 22b^2c^6 + c^8) - 3a^2(b^2 - c^2)^4(b^8 - b^6c^2 - 7b^4c^4 - b^2c^6 + c^8) + (b^2 - c^2)^6(b^2 + c^2)(b^4 + 4b^2c^2 + c^4)]/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :

    X(18318) lies on these lines: {3,14993}, {265,6334}, {476,12113}, {1650,5627}


    X(18319) = X(30)X(74)∩X(140)X(477)

    Barycentrics    3a^14(b^2 + c^2) - a^12(13b^4 - 2b^2c^2 + 13c^4) + 3a^10(b^2 + c^2)(7b^4 - 8b^2c^2 + 7c^4) - 3a^8(5b^8 + 8b^6c^2 - 18b^4c^4 + 8b^2c^6 + 5c^8) + a^6(b^2 + c^2)(5b^8 + 18b^6c^2 - 43b^4c^4 + 18b^2c^6 + 5c^8) - 3a^4(b^2 - c^2)^2(b^8 - 2b^6c^2 + 11b^4c^4 - 2b^2c^6 + c^8) + 3a^2(b^2 - c^2)^6(b^2 + c^2) - (b^2 - c^2)^6(b^4 + 4b^2c^2 + c^4) : :
    X(18319) = 2 X(140) - X(477)

    Let A'1B'1C'1 and A'2B'2C'2 be the 1st and 2nd Ehrmann inscribed triangles. Let P1 be the intersection, other than A'1, B'1, C'1, of the Ehrmann conic and the circumcircle of A'1B'1C'1. Let P2 be the intersection, other than A'2, B'2, C'2, of the Ehrmann conic and the circumcircle of A'2B'2C'2. X(18319) is the midpoint of P1 and P2.

    Let LA, LB, LC be the lines through A, B, C, resp. parallel to the Euler line. Let MA, MB, MC be the reflections of BC, CA, AB in LA, LB, LC, resp. Let A' = MB∩MC, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. X(18319) = X(5)-of-A'B'C'. (See Hyacinthos #16741/16782, Sep 2008.)

    X(18319) lies on these lines: {5,3258}, {30,74}, {140,477}, {381,14731}, {1117,1291}, {12091,15760}, {14851,15059}

    X(18319) = reflection of X(477) in X(140)
    X(18319) = X(14731)-of-Ehrmann-mid-triangle


    X(18320) = X(3)X(6)∩X(647)X(2436)

    Barycentrics    a^2[a^12 - 2a^10(b^2 + c^2) - 2a^8(b^4 - 5b^2c^2 + c^4) + 2a^6(4b^6 - 5b^4c^2 - 5b^2c^4 + 4c^6) - a^4(7b^8 + b^6c^2 - 17b^4c^4 + b^2c^6 + 7c^8) + 2a^2(b^2 - c^2)^2(b^2 + c^2)^3 + b^2c^2(b^2 - c^2)^4] : :

    Let P1 and P2 be as at X(18319). X(18320) is the crossdifference of P1 and P2.

    X(18320) lies on these lines: {3,6}, {647,2436}, {6128,10413}


    X(18321) = INTERSECTION OF BROCARD AXES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18321) lies on these lines: {3,512}, {5,6787}, {382,511}, {3111,3526}, {13449,18322}

    X(18321) = reflection of X(18322) in X(13449)
    X(18321) = reflection of X(3) in line X(4)X(69)


    X(18322) = INTERSECTION OF LEMOINE AXES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18322) lies on these lines: {3,6}, {265,290}, {385,1154}, {13449,18321}

    X(18322) = reflection of X(18321) in X(13449)
    X(18322) = reflection of X(3) in line X(512)X(13449)


    X(18323) = INTERSECTION OF ORTHIC AXES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    [4a^8 - a^6(b^2 + c^2) - a^4(7b^4 - 12b^2c^2 + 7c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + 3(b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18323) = 3 X(2) - 2 X(3) - 4 X(4) + X(23) = 3 X(2) + 2 X(3) - 8 X(5) + X(23) = X(3) - 2 X(10297)

    X(18323) lies on these lines: {2,3}, {476,18576}, {1503,7728}, {3018,3284}, {8705,18438}, {11557,13202}, {13376,15012}, {14537,15860}

    X(18323) = reflection of X(3) in X(10297)
    X(18323) = reflection of X(3) in line X(523)X(6334)
    X(18323) = inverse-in-circumcircle of X(18324)
    X(18323) = {X(1113),X(1114)}-harmonic conjugate of X(18324)
    X(18323) = X(10297)-of-X3-ABC-reflections-triangle


    X(18324) = INVERSE-IN-CIRCUMCIRCLE OF X(18323)

    Barycentrics    a^2[3a^8 - 6a^6(b^2 + c^2) + 8a^4b^2c^2 + a^2(6b^6 - 4b^4c^2 - 4b^2c^4 + 6c^6) - (b^2 - c^2)^2(3b^4 + 4b^2c^2 + 3c^4)] : :
    X(18324) = 2 X(3) + X(26) = X(3) + X(14070) = 2 X(5) - X(18568)

    X(18324) lies on these lines: {2,3}, {143,11425}, {154,5663}, {156,12163}, {159,13680}, {394,1511}, {539,9932}, {541,13289}, {542,15577}, {569,16226}, {1151,11266}, {1152,11265} et al

    X(18324) = midpoint of X(3) and X(14070)
    X(18324) = reflection of X(18568) in X(5)
    X(18324) = harmonic center of Kosnita and tangential circles
    X(18324) = inverse-in-circumcircle of X(18323)
    X(18324) = {X(1113),X(1114)}-harmonic conjugate of X(18323)
    X(18324) = X(18568)-of-Johnson-triangle
    X(18324) = X(3576)-of-Kosnita-triangle if ABC is acute
    X(18324) = X(10246)-of-tangential-triangle if ABC is acute
    X(18324) = orthic-to-Kosnita similarity image of X(381)
    X(18324) = orthic-to-tangential similarity image of X(3845)
    X(18324) = Ehrmann-mid-to-ABC similarity image of X(18566)
    X(18324) = X(26)-Gibert-Moses centroid; see the preamble just before X(21153)


    X(18325) = INTERSECTION OF DE LONGCHAMPS LINES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    a^10 - a^8(b^2 + c^2) - a^6(2b^4 - 11b^2c^2 + 2c^4) + a^4(b^2 + c^2)(2b^4 - 7b^2c^2 + 2c^4) + a^2(b^2 - c^2)^2(b^4 - 6b^2c^2 + c^4) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18325) = 3*R^2*X(3) - (12*R^2-2*SW)*X(4) = X(3) - 2 X(11799)

    X(18325) lies on these lines: {2,3}, {265,10293}, {1478,5160}, {1479,7286}, {1503,12902}, {2777,3581}, {3521,5446}, {3580,10620}, {5270,9628}, {18358,18551} et al

    X(18325) = reflection of X(3) in X(11799)
    X(18325) = reflection of X(3) in line X(523)X(11799)
    X(18325) = X(11799)-of-X3-ABC-reflections-triangle


    X(18326) = INTERSECTION OF NAGEL LINES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18326) lies on these lines: {3,3667}, {30,6790}, {381,6788}, {519,3830} et al

    X(18326) = reflection of X(3) in line X(355)X(381)


    X(18327) = INTERSECTION OF GERGONNE LINES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18327) lies on these lines: {3,142}, {381,8074}, {5179,7687}


    X(18328) = INTERSECTION OF SODDY LINES OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

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

    X(18328) lies on the Fuhrmann circle and these lines: {8,144}, {103,1146}, {118,664}, {382,18329}, {1308,18491} et al

    X(18328) = reflection of X(4) in line X(355)X(382)
    X(18328) = X(99)-of-Fuhrmann-triangle
    X(18328) = intersection of Soddy lines of anticevian triangles of PU(4)


    X(18329) = INTERSECTION OF SODDY LINES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18329) lies on these lines: {3,514}, {382,18328}, {516,4701}, {5252,18340} et al

    X(18329) = reflection of X(3) in line X(355)X(382)


    X(18330) = INTERSECTION OF ANTIORTHIC AXES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Trilinears    a^7(b^2 + bc + c^2) - 2a^6bc(b + c) - a^5(3b^4 - b^2c^2 + 3c^4) + a^4bc(b + c)(3b^2 - 4bc + 3c^2) + a^3(3b^6 - 3b^5c + 2b^3c^3 - 3bc^5 + 3c^6) + 2a^2b^2c^2(b - c)^2(b + c) - a(b^2 - c^2)^2(b^4 - 2b^3c + 4b^2c^2 - 2bc^3 + c^4) - bc(b - c)^4(b + c)^3 : :

    X(18330) lies on these lines: {1,3}, {568,5722}, {5787,18439}, {8679,10742} et al


    X(18331) = INTERSECTION OF FERMAT AXES OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    a^14 - 2a^12(b^2 + c^2) + a^10(4b^4 + 4c^4 - 2b^2c^2) - a^8(11b^6 + 11c^6 - 9b^4c^2 - 9b^2c^4) + a^6(15b^8 + 15c^8 - 10b^6c^2 - 10b^2c^6 - 9b^4c^4) - a^4(b^2 - c^2)^2(b^2 + c^2)(10b^4 + 10c^4 + 3b^2c^2) + a^2(b^2 - c^2)^2(4b^8 + 4c^8 + 4b^6c^2 + 4b^2c^6 - b^4c^4) - (b^2 - c^2)^4(b^6 + c^6 + 2b^4c^2 + 2b^2c^4) : :
    X(18331) = X(4) + 2 X(98) - 4 X(125) = X(4) + 2 X(110) - 4 X(114) = X(4) - 2 X(11005)

    X(18331) lies on these lines: {2,18332}, {4,690}, {69,74}, {114,15342}, {115,15081}, {147,5663}, {246,2782}, {1511,14850}, {3545,9144}, {3569,6792}, {9140,12243}

    X(18331) = reflection of X(4) in X(11005)
    X(18331) = reflection of X(4) in line X(2)X(98)
    X(18331) = anticomplement of X(18332)
    X(18331) = intersection, other than X(4), of X(2)-, X(15)-, and X(16)-Fuhrmann circles (aka Hagge circles)
    X(18331) = X(11005)-of-anti-Euler-triangle
    X(18331) = circummedial-to-X(2)-Fuhrmann similarity image of X(842)
    X(18331) = intersection of Fermat axes of anticevian triangles of PU(4)


    X(18332) = INTERSECTION OF FERMAT AXES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

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

    X(18332) lies on the Brocard circle, the Brocard (seventh) cubic (K023), and these lines: {2,18331}, {3,690}, {6,13}, {30,9144}, {74,12042}, {98,5663}, {110,1316}, {146,9862}, {3023,10091}, {3448,14651}, {5642,8724}, {5972,15561}

    X(18332) = reflection of X(3) in line X(2)X(98)
    X(18332) = isogonal conjugate of X(18333)
    X(18332) = complement of X(18331)
    X(18332) = inverse-in-circumcircle of X(14270)
    X(18332) = inverse-in-orthocentroidal-circle of X(14356)
    X(18332) = crossdifference of every pair of points on line X(526)X(2493)
    X(18332) = X(842)-of-1st-Brocard-triangle
    X(18332) = intersection, other than X(6), of Brocard circle and Fermat axis
    X(18332) = intersection, other than X(3), of circles O(3,13) and O(3,14)
    X(18332) = similitude center of X(13)- and X(14)-Brocard triangles
    X(18332) = {X(1113),X(1114)}-harmonic conjugate of X(14270)
    X(18332) = intersection of Fermat axes of antipedal triangles of PU(1)
    X(18332) = {P15,P16}-harmonic conjugate of X(6), where P15 and P16 are the orthogonal projections of X(15) and X(16) on the Fermat axis


    X(18333) = ANTITOMIC CONJUGATE OF X(842)

    Barycentrics    a^2/[a^14 - 3a^12(b^2 + c^2) + a^10(5b^4 + 5c^4 + 4b^2c^2) - a^8(7b^6 + 7c^6 + b^4c^2 + b^2c^4) + a^6(6b^8 + 6c^8 - 3b^4c^4) - a^4(b^2 + c^2)(2b^8 + 2c^8 - 3b^4c^4) + a^2b^2c^2(b^2 - c^2)^2(3b^4 + 3c^4 - 2b^2c^2) - b^2c^2(b^2 - c^2)^4(b^2 + c^2)] : :
    X(18333) = X(842) - 2 X(18334)

    X(18333) lies on hyperbola {{A,B,C,X(2),X(15),X(16)}} and these lines: {323,7468}, {842,18334}, {7799,14221}

    X(18333) = reflection of X(842) in X(18334)
    X(18333) = isogonal conjugate of X(18332)
    X(18333) = antitomic conjugate of X(842)
    X(18333) = antipode of X(842) in hyperbola {{A,B,C,X(2),X(15),X(16)}}
    X(18333) = trilinear pole of line X(526)X(2493)


    X(18334) = CENTER OF HYPERBOLA {{A,B,C,X(2),X(15),X(16)}}

    Trilinears    f(A,B,C)[- a f(A,B,C) + b f(B,C,A) + c f(C,A,B)] : :, where f(A,B,C) = (1 + 2 cos 2A) sin(B - C)
    Barycentrics    csc B csc(C - A)/(1 + 2 cos 2B) + csc C csc(A - B)/(1 + 2 cos 2C) : :
    Barycentrics    1/(sin C csc 3C - sin A csc 3A) + 1/(sin A csc 3A - sin B csc 3B) : :
    Barycentrics    a^4(b^2 - c^2)^2[(a^2 - b^2 - c^2)^2 - b^2c^2]^2 : :

    Hyperbola {{A,B,C,X(2),X(15),X(16)}} is the isogonal conjugate of the Fermat axis and the isotomic conjugate of line X(2)X(94), and intersects the circumcircle, other than at A, B, and C, at X(842). The perspector of hyperbola {{A,B,C,X(2),X(15),X(16)}} is X(526).

    Let A'B'C' be the orthic triangle. Let La be the Fermat axis of triangle AB'C', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(18334). (Randy Hutson, July 31 2018)

    X(18334) lies on the Steiner inellipse and these lines: {32,14385}, {115,647}, {187,1511}, {842,18333}, {2088,16186} et al

    X(18334) = perspector of circumparabola centered at X(526)
    X(18334) = X(2)-Ceva conjugate of X(526)
    X(18334) = barycentric square of X(526)
    X(18334) = crossdifference of every pair of points on line X(476)X(10412) (the tangent to circumcircle at X(476))


    X(18335) = INTERSECTION OF NAPOLEON LINES OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    a^14 - 4a^12(b^2 + c^2) + 2a^10(5b^4 + b^2c^2 + 5c^4) - a^8(19b^6 - 7b^4c^2 - 7b^2c^4 + 19c^6) + a^6(23b^8 - 18b^6c^2 - b^4c^4 - 18b^2c^6 + 23c^8) - a^4(b^2 - c^2)^2(16b^6 + 7b^4c^2 + 7b^2c^4 + 16c^6) + a^2(b^2 - c^2)^2(6b^8 - 4b^6c^2 + 3b^4c^4 - 4b^2c^6 + 6c^8) - (b^2 - c^2)^4(b^6 + c^6) : :

    X(18335) lies on these lines: {54,69}, {3091,18336}

    X(18335) = reflection of X(4) in line X(193)X(576)
    X(18335) = intersection of Napoleon axes of anticevian triangles of PU(4)


    X(18336) = INTERSECTION OF NAPOLEON LINES OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    a^14 - 5a^12(b^2 + c^2) + 19a^10(b^4 + c^4) - a^8(39b^6 - 5b^4c^2 - 5b^2c^4 + 39c^6) + a^6(38b^8 - 8b^6c^2 - 3b^4c^4 - 8b^2c^6 + 38c^8) - a^4(b^2 + c^2)(16b^8 - 12b^6c^2 + b^4c^4 - 12b^2c^6 + 16c^8) + a^2(b^2 - c^2)^2(2b^8 + 19b^6c^2 - 14b^4c^4 + 19b^2c^6 + 2c^8) - 3b^2c^2(b^2 - c^2)^4(b^2 + c^2) : :

    X(18336) lies on these lines: {3091,18335}, {5076,5965}

    X(18336) = reflection of X(3) in line X(193)X(576)


    X(18337) = INTERSECTION OF VAN AUBEL LINES OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    a^14 - a^12(b^2 + c^2) + a^10(b^4 - b^2c^2 + c^4) - 7a^8(b^2 - c^2)^2(b^2 + c^2) + a^6(b^2 - c^2)^2(11b^4 + 16b^2c^2 + 11c^4) - a^4(b^2 - c^2)^2(b^2 + c^2)(7b^4 + 6b^2c^2 + 7c^4) + a^2(b^2 - c^2)^2(3b^8 + 5b^6c^2 + 5b^2c^6 + 3c^8) - (b^2 - c^2)^4(b^2 + c^2)^3 : :

    X(18337) lies on these lines: {2,18338}, {4,525}, {5,6794}, {20,64}, {110,14981}, {2421,11441} et al

    X(18337) = reflection of X(4) in line X(3)X(66)
    X(18337) = anticomplement of X(18338)
    X(18337) = X(12384)-of-1st-Brocard-triangle
    X(18337) = intersection, other than X(4), of X(2)- and X(3)-Fuhrmann circles (aka -Hagge circles)
    X(18337) = X(112)-of-X(3)-Fuhrmann-triangle
    X(18337) = circummedial-to-X(2)-Fuhrmann similarity image of X(1297)
    X(18337) = intersection of van Aubel lines of anticevian triangles of PU(4)


    X(18338) = COMPLEMENT OF X(18337)

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

    X(18338) lies on the Brocard circle and these lines: {2,18337}, {3,525}, {4,6}, {76,15407}, {98,3269}, {125,11623}, {154,2409}, {184,1316}, {648,10762}, {868,1899}, {935,11464}, {1562,2794} et al

    X(18338) = complement of X(18337)
    X(18338) = intersection, other than X(6), of van Aubel line and Brocard circle
    X(18338) = orthogonal projection of X(3) on van Aubel line
    X(18338) = X(1297)-of-1st-Brocard-triangle
    X(18338) = intersection of van Aubel lines of antipedal triangles of PU(1)


    X(18339) = INTERSECTION OF LINES X(1)X(4) OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    a^10 - a^9(b + c) + a^8bc + a^7(b - c)^2(b + c) - a^6(b - c)^2(5b^2 + 7bc + 5c^2) + 3a^5(b - c)^2(b + c)^3 + a^4(b - c)^2(5b^4 + b^3c - 4b^2c^2 + bc^3 + 5c^4) - a^3(b - c)^2(b + c)^3(5b^2 - 6bc + 5c^2) + a^2bc(b^2 - c^2)^2(5b^2 - 6bc + 5c^2) + 2a(b - c)^4(b + c)^3(b^2 + c^2) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18339) = 2 X(355) - X(18340)

    X(18339) lies on the Fuhrmann circle and these lines: {4,522}, {8,20}, {102,2968}, {109,952}, {117,1897}, {355,18340}, {1785,10826} et al

    X(18339) = reflection of X(18340) in X(355)
    X(18339) = reflection of X(4) in line X(3)X(10)
    X(18339) = X(925)-of-Fuhrmann-triangle
    X(18339) = X(109)-of-X(3)-Fuhrmann-triangle
    X(18339) = antipode in Fuhrmann circle of X(18340)
    X(18339) = intersection, other than X(4), of Fuhrman circle and X(3)-Fuhrmann circle


    X(18340) = INVERSE-IN-POLAR-CIRCLE OF X(1877)

    Barycentrics    (b + c - a)[a^6 - a^5(b + c) - a^4(b - 2c)(2b - c) + a^3(b - c)^2(b + c) - 3a^2bc(b - c)^2 + 2abc(b - c)^2(b + c) + (b - c)^4(b + c)^2] : :
    X(18340) = 2 X(355) - X(18339)

    X(18340) lies on the Fuhrmann circle, circle {{X(1),X(3),X(355)}}, and these lines: {1,4}, {3,2222}, {8,522}, {11,106}, {55,13744}, {80,1772}, {109,2829}, {153,4551}, {355,18339}, {535,1936}, {952,10703}, {1324,7428}, {1388,3319}, {1647,9581}, {1837,6788}, {1854,18525}, {2098,3326}, {5252,18328}, {5587,15737}, {5727,18343} et al

    X(18340) = reflection of X(18339) in X(355)
    X(18340) = reflection of X(4) in line X(355)X(522)
    X(18340) = X(1300)-of-Fuhrmann-triangle
    X(18340) = antipode in Fuhrmann circle of X(18339)
    X(18340) = intersection of lines X(1)X(522) of 1st and 2nd Ehrmann circumscribing triangles


    X(18341) = INTERSECTION OF LINES X(1)X(5) OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    a^10 - 2a^9(b + c) + a^8(b^2 + 4bc + c^2) + a^7(b + c)(3b^2 - 8bc + 3c^2) - a^6(8b^4 - 4b^3c - 9b^2c^2 - 4bc^3 + 8c^4) + a^5(b - c)^2(b + c)(3b^2 + 13bc + 3c^2) + a^4(8b^6 - 19b^5c + 22b^3c^3 - 19bc^5 + 8c^6) - a^3(b - c)^2(b + c)(7b^4 + 4b^3c - 10b^2c^2 + 4bc^3 + 7c^4) - a^2(b^2 - c^2)^2(b^4 - 10b^3c + 15b^2c^2 - 10bc^3 + c^4) + 3a(b - c)^4(b + c)^3(b^2 - bc + c^2) - (b^2 - c^2)^4(b^2 - bc + c^2) : :
    X(18341) = 2 X(5) - X(18342)

    X(18341) lies on the Fuhrmann circle, the circle O(3,4), the circle {{X(1),X(3),X(355)}}, and these lines: {3,8}, {4,900}, {5,18342}, {80,1772}, {108,12832}, {119,15343}, {1769,6788}, {2800,18303} et al

    X(18341) = reflection of X(4) in line X(1)X(5)
    X(18341) = reflection of X(18342) in X(5)
    X(18341) = X(476)-of-Fuhrmann-triangle
    X(18341) = X(18342)-of-Johnson-triangle
    X(18341) = intersection of lines X(1)X(5) of anticevian triangles of PU(4)


    X(18342) = INTERSECTION OF LINES X(1)X(5) OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    a^10 - 2a^9(b + c) + 4a^8bc + a^7(b + c)(5b^2 - 8bc + 5c^2) - a^6(6b^4 + 4b^3c - 9b^2c^2 + 4bc^3 + 6c^4) - a^5(b + c)(3b^4 - 17b^3c + 24b^2c^2 - 17bc^3 + 3c^4) + a^4(8b^6 - 5b^5c - 13b^4c^2 + 18b^3c^3 - 13b^2c^4 - 5bc^5 + 8c^6) - a^3(b - c)^2(b + c)(b^4 + 12b^3c - 2b^2c^2 + 12bc^3 + c^4) - a^2(b^2 - c^2)^2(3b^4 - 6b^3c + 2b^2c^2 - 6bc^3 + 3c^4) + a(b - c)^4(b + c)^3(b^2 + 3bc + c^2) - bc(b^2 - c^2)^4 : :
    X(18342) = 2 X(5) - X(18341)

    X(18342) lies on these lines: {3,900}, {4,145}, {5,18341}, {6265,18302}

    X(18342) = reflection of X(18341) in X(5)
    X(18342) = reflection of X(3) in line X(1)X(5)
    X(18342) = X(18341)-of-Johnson-triangle


    X(18343) = INTERSECTION OF LINES X(1)X(6) OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

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

    X(18343) lies on the Fuhrmann circle and these lines: {2,1083}, {4,885}, {7,8}, {10,9318}, {105,4904}, {120,644}, {150,13576}, {279,3323}, {840,929}, {1352,7613}, {2711,2864}, {2784,9317}, {5727,18340} et al

    X(18343) = reflection of X(4) in line X(355)X(518)
    X(18343) = X(112)-of-Fuhrmann-triangle
    X(18343) = inverse-in-polar-circle of X(18344)
    X(18343) = intersection, other than X(4), of Fuhrmann circle and X(2)-Fuhrmann circle (aka Hagge circle)
    X(18343) = circummedial-to-X(2)-Fuhrmann similarity image of X(105)


    X(18344) = POLAR CONJUGATE OF X(4554)

    Trilinears    (tan A)(cos B - cos C) : :
    Trilinears    (b - c)(b + c - a)/(b^2 + c^2 - a^2) : :

    The trilinear polar of X(18344) passes through X(4516). (Randy Hutson, October 15, 2018)

    X(18344) lies on these lines: {4,885}, {25,667}, {33,4162}, {34,2424}, {107,2714}, {108,7128}, {112,9090}, {460,512}, {468,10006}, {513,1835}, {521,1948}, {650,1946}, {657,4041}, {661,663}, {905,8760}, {1146,11988}, {2432,7008}, {3064,3700} et al

    X(18344) = isogonal conjugate of X(6516)
    X(18344) = pole wrt polar circle of trilinear polar of X(4554) (line X(7)X(8))
    X(18344) = polar conjugate of X(4554)
    X(18344) = inverse-in-polar-circle of X(18343)
    X(18344) = crossdifference of every pair of points on line X(63)X(77)
    X(18344) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 6516}, {48, 4554}, {63, 651}
    X(18344) = trilinear product X(i)*X(j) for these {i,j}: {9,6591}, {19,650}, {33,513}
    X(18344) = barycentric product X(4)*X(650)
    X(18344) = intersection of perspectrices of [ABC and Gemini triangle 37] and [ABC and Gemini triangle 38]


    X(18345) = INTERSECTION OF LINES X(1)X(6) OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Trilinears    a^8(b^2 - bc + c^2) - a^7(b + c)(3b^2 - 4bc + 3c^2) + a^6(b^4 + b^2c^2 + c^4) + a^5(b + c)(b^2 + bc + c^2)(5b^2 - 9bc + 5c^2) - 5 a^4(b^6 + c^6) - a^3(b + c)[(b^2 - c^2)^2 - b^2c^2](b^2 + c^2) + a^2(3b^8 - 4b^5c^3 + 2b^4c^4 - 4b^3c^5 + 3c^8) - a(b^9 + b^8c - 8b^6c^3 + 6b^5c^4 + 6b^4c^5 - 8b^3c^6 + bc^8 + c^9) + bc(b - c)^4(b + c)^2(b^2 + c^2) : :

    X(18345) lies on these lines: {3,667}, {518,18440}

    X(18345) = reflection of X(3) in line X(355)X(518)


    X(18346) = INTERSECTION OF LINES X(2)X(6) OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    a^14 - 14a^12(b^2 + c^2) + a^10(21b^4 + 80b^2c^2 + 21c^4) - 3a^8(b^2 + c^2)(4b^4 + 31b^2c^2 + 4c^4) + a^6(b^8 + 111b^6c^2 + 41b^4c^4 + 111b^2c^6 + c^8) + a^4(b^2 + c^2)(2b^8 - 66b^6c^2 + 37b^4c^4 - 66b^2c^6 + 2 c^8) + a^2(b^12 + 22b^10c^2 - 7b^8c^4 + 24b^6c^6 - 7b^4c^8 + 22b^2c^10 + c^12) - 2b^2c^2(b^10 - b^8c^2 - b^2c^8 + c^10) : :

    X(18346) lies on these lines: {3,669}, {381,6792}, {524,3830}, {3534,14916}, {9169,15703} et al

    X(18346) = reflection of X(3) in line X(381)X(524)


    X(18347) = INTERSECTION OF LINES X(5)X(6) OF 1st AND 2nd EHRMANN CIRCUMSCRIBING TRIANGLES

    Barycentrics    (b^2 + c^2 - a^2)[a^12 - 2a^10(b^2 + c^2) + 5a^8(b^4 - b^2c^2 + c^4) - a^6(10b^6 - 9b^4c^2 - 9b^2c^4 + 10c^6) + 3a^4(b^2 - c^2)^2(3b^4 + b^2c^2 + 3c^4) - a^2(b^2 - c^2)^2(b^2 + c^2)(4b^4 - 5b^2c^2 + 4c^4) + (b^2 - c^2)^4(b^4 + c^4)] : :
    X(18347) = 2 X(5) - X(18348)

    X(18347) lies on these lines: {3,69}, {4,3566}, {5,18348}, {5622,15357} et al

    X(18347) = reflection of X(18348) in X(5)
    X(18347) = reflection of X(4) in line X(5)X(6)
    X(18347) = X(18348)-of-Johnson-triangle


    X(18348) = INTERSECTION OF LINES X(5)X(6) OF 1st AND 2nd EHRMANN INSCRIBED TRIANGLES

    Barycentrics    a^14 - 4a^12(b^2 + c^2) + a^10(11b^4 + 3b^2c^2 + 11c^4) - a^8(20b^6 - b^4c^2 - b^2c^4 + 20c^6) + a^6(19b^8 - 3b^6c^2 - 4b^4c^4 - 3b^2c^6 + 19c^8) - a^4(b^2 + c^2)(8b^8 - 5b^6c^2 - 2b^4c^4 - 5b^2c^6 + 8c^8) + a^2(b^2 - c^2)^2(b^8 + 10b^6c^2 - 6b^4c^4 + 10b^2c^6 + c^8) - 2b^2c^2(b^2 - c^2)^4(b^2 + c^2) : :
    X(18348) = 2 X(5) - X(18347)

    X(18348) lies on these lines: {3,3566}, {4,193}, {5,18347}, {1975,10425}, {4226,8780} et al

    X(18348) = reflection of X(3) in line X(5)X(6)
    X(18348) = reflection of X(18347) in X(5)
    X(18348) = X(18347)-of-Johnson-triangle


    X(18349) = CEVAPOINT OF PU(173)

    Barycentrics    1/[a^8 - 3a^6(b^2 + c^2) + 3a^4(b^4 + b^2c^2 + c^4) - a^2(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) + 2b^2c^2(b^2 - c^2)^2] : :

    X(18349) lies on these lines: {20,1154}, {94,18351}, {1249,11062}, {1273,14615}, {3238,7661}

    X(18349) = isogonal conjugate of X(18350)


    X(18350) = CROSSSUM OF PU(173)

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

    X(18350) lies on these lines: {2,156}, {3,64}, {5,49}, {17,11137}, {18,11134}, {23,6101}, {24,3581}, {25,6243}, {30,13489}, {51,195}, {52,12316}, {115,9603}, {140,1614}, {143,13595}, {155,568}, {182,5070}, {184,1656}, {185,399}, {186,5876}, {215,7741}, {323,10263}, {373,15047}, {381,1147}, {382,1092}, {394,7517}, {436,14978}, {499,9652}, {550,14157}, {569,5055}, {578,3851}, {1154,3518}, {1199,15026}, {1216,1495}, {1352,6639}, {1437,7489}, {1493,13364}, {1506,9604}, {1511,3520}, {3167,3527}, {3521,5655}, {3526,5651}, {6642,18445}, {7488,11591}, {10255,18474}, {12038,12302}, {12121,18560} et al

    X(18350) = isogonal conjugate of X(18349)


    X(18351) = CROSSPOINT OF PU(173)

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

    X(18351) lies on these lines: {5,93}, {94,18349}, {382,6243}

    X(18351) = barycentric product X(18353)*X(18354)


    X(18352) = TRILINEAR PRODUCT OF PU(173)

    Trilinears    1 - 2 cos(2B - 2C) : :
    Trilinears    cos(B - C + π/3) cos(B - C - π/3) : :
    Trilinears    cos2(B - C) - 3 sin2(B - C) : :
    Trilinears    b^2c^2[a^4(b^4 - b^2c^2 + c^4) - 2a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4] : :

    X(18352) lies on these lines: {1,564}, {63,2962}, {91,6149}, {92,2964}, {2169,9377} et al


    X(18353) = BARYCENTRIC PRODUCT OF PU(173)

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

    X(18353) lies on these lines: {3,2963}, {4,2965}, {5,13351}, {6,13}, {50,2165}, {566,13160}, {7387,8553} et al

    X(18353) = barycentric quotient X(18351)/X(18354)


    X(18354) = X(20)X(64)∩X(95)X(99)

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

    X(18354) lies on these lines: {20,64}, {30,1238}, {95,99}, {264,14865}, {317,18559}, {1225,7488}, {3260,13489} et al

    X(18354) = barycentric quotient X(18351)/X(18353)


    X(18355) = VERTEX CONJUGATE OF PU(173)

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

    X(18355) lies on these lines: {}

    X(18355) = isogonal connjugate of X(18356)


    X(18356) = ISOGONAL CONJUGATE OF X(18355)

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

    X(18356) lies on these lines: {3,2888}, {4,16880}, {5,5422}, {30,64}, {140,1899}, {155,10224}, {156,542}, {265,12111}, {343,7525}, {550,11457}, {578,11264}, {973,6102}, {1147,13561}, {1154,18381}, {5655,18504}, {10226,12118}, {11411,18569}, {11818,16881}, {13754,18377}, {13861,18440}, {18379,18566} et al

    X(18356) = isogonal conjugate of X(18355)


    X(18357) = MIDPOINT OF X(5) AND X(355)

    Barycentrics    2a^4 - 2a^3(b + c) + a^2(b^2 + 4bc + c^2) + 2a(b - c)^2(b + c) - 3(b^2 - c^2)^2 : :
    X(18357) = X(1) - 3 X(5) = X(5) + X(355) = 2 X(5) - X(5901)

    X(18357) is the centroid of the incenters of the 1st and 2nd Ehrmann circumscribing triangles and the 1st and 2nd Ehrmann inscribed triangles.

    X(18357) lies on these lines: {1,5}, {2,18525}, {3,5260}, {4,5690}, {8,381}, {10,30}, {21,18524}, {40,3627}, {45,5816}, {65,11544}, {100,13743}, {140,515}, {145,3545}, {153,6901}, {165,15704}, {382,5657}, {403,12135}, {405,18518}, {474,18519}, {484,3652}, {516,3853}, {517,546}, {518,18358}, {519,5066}, {547,1125}, {548,6684}, {549,1698}, {550,5691}, {551,10109}, {632,3576}, {912,9947}, {943,6913}, {944,1656}, {946,3625}, {958,18491}, {962,3843}, {1159,5714}, {1385,3628}, {1482,3091}, {1699,3858}, {1706,18540}, {3090,10246}, {3584,10543}, {3586,10386}, {3616,5055}, {3622,5071}, {3624,3655}, {3654,15687}, {3679,3845}, {3839,4678}, {3851,5603}, {4188,18515}, {5056,7967}, {5554,18542}, {5599,18497}, {5600,18495}, {5694,15064}, {5704,6918}, {5708,6826}, {6147,9654}, {7354,18395}, {7969,18538}, {16210,18507} et al

    X(18357) = midpoint of X(i) and X(j) for these {i,j}: {5,355}, {10,18480}
    X(18357) = reflection of X(5901) in X(5)
    X(18357) = X(140)-of-Fuhrmann-triangle
    X(18357) = X(5901)-of-Johnson-triangle


    X(18358) = MIDPOINT OF X(5) AND X(1352)

    Barycentrics    2a^6 - a^4(b^2 + c^2) + 2a^2(b^4 + 4b^2c^2 + c^4) - 3(b^2 - c^2)^2(b^2 + c^2) : :
    X(18358) = 3 X(5) - X(6) = X(5) + X(1352)

    X(18358) is the centroid of the symmedian points of the 1st and 2nd Ehrmann circumscribing triangles and the 1st and 2nd Ehrmann inscribed triangles.

    X(18358) lies on these lines: {2,8780}, {3,14927}, {4,3620}, {5,6}, {30,141}, {69,381}, {114,3055}, {140,1503}, {156,182}, {159,7514}, {184,11548}, {193,3545}, {323,5133}, {382,10519}, {427,15066}, {428,15107}, {468,10546}, {495,12589}, {496,12588}, {511,546}, {518,18357}, {524,5066}, {542,547}, {549,3763}, {575,12812}, {576,12811}, {597,10109}, {599,3845}, {611,10592}, {613,10593}, {632,5085}, {1154,9969}, {1176,10540}, {1495,6676}, {3054,5033}, {3090,5050}, {3544,11482}, {3566,18313}, {3580,10545}, {3618,5055}, {3630,3850}, {3851,11008}, {3857,11477}, {5056,14912}, {5072,5093}, {5181,10113}, {7687,14913}, {10297,11188}, {10691,11550}, {15435,18420}, {18325,18551} et al

    X(18358) = midpoint of X(5) and X(1352)


    X(18359) = POLAR CONJUGATE OF X(1870)

    Barycentrics    1/(1 - 2 cos A) : :
    Barycentrics    bc/(a^2 - b^2 - c^2 + bc) : :

    X(18359) is the trilinear pole of the line through the incenters of the 1st and 2nd Ehrmann inscribed triangles (line X(10)X(522)).

    Let A38B38C38 be Gemini triangle 38. Let LA be the line through A38 parallel to BC, and define LB and LC cyclically. Let A'38 = LBnLC, and define B'38 and C'38' cyclically. Triangle A'38B'38C'38 is homothetic to ABC at X(18359). (Randy Hutson, January 15, 2019)

    X(18359) lies on these lines: {2,2006}, {8,80}, {29,1807}, {92,324}, {94,226}, {189,5905}, {190,321}, {312,3969}, {329,2994}, {476,15168}, {655,3218}, {759,835}, {1220,1411} et al

    X(18359) = isogonal conjugate of X(7113)
    X(18359) = isotomic conjugate of X(3218)
    X(18359) = anticomplement of X(16586)
    X(18359) = polar conjugate of X(1870)
    X(18359) = trilinear pole of line X(10)X(522)
    X(18359) = X(i)-isoconjugate of X(j) for these {i,j}: {1,7113}, {31,3218}, {48,1870}


    X(18360) = X(46)X(582)∩X(58)X(65)

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

    X(18360) is the crosssum of the incenters of the 1st and 2nd Ehrmann inscribed triangles.

    X(18360) lies on these lines: {6,8775}, {31,5221}, {35,1464}, {42,8614}, {46,582}, {55,1066}, {56,1149}, {58,65}, {73,14882}, {171,3649}, {221,4255}, {603,2099}, {1042,5172} et al


    X(18361) = ISOGONAL CONJUGATE OF X(10546)

    Barycentrics    1/(2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 5b^2c^2) : :

    X(18361) is the cevapoint of the centroids of the 1st and 2nd Ehrmann inscribed triangles.

    X(18361) lies on these lines: {468,5306}, {523,11648}, {524,3098}, {3266,7788}

    X(18361) = isogonal conjugate of X(10546)


    X(18362) = X(2)X(99)∩X(32)X(381)

    Barycentrics    a^4 - 2a^2(b^2 + c^2) + 4(b^2 - c^2)^2 : :
    Barycentrics    3(2 SA (SB + SC) + SB^2 + SC^2) - 10 S^2 : :
    X(18362) = X(5206)-4*X(7746)

    X(18362) is the barycentric product of the centroids of the 1st and 2nd Ehrmann inscribed triangles.

    X(18362) lies on these lines: {2, 99}, {5, 5309}, {30, 5206}, {32, 381}, {39, 5055}, {187, 3830}, {230, 3845}, {376, 7749}, {547, 5254}, {549, 7748}, {591, 13850}, {625, 7788}, {631, 12815}, {1506, 5071}, {1989, 5158}, {1991, 13932}, {3053, 14269}, {3054, 12100}, {3090, 7765}, {3091, 7755}, {3406, 14458}, {3524, 7756}, {3534, 8588}, {3545, 3767}, {3815, 10109}, {3839, 7747}, {3851, 5007}, {4995, 9664}, {5013, 15703}, {5023, 15684}, {5025, 7865}, {5028, 6034}, {5033, 11645}, {5054, 15515}, {5056, 9698}, {5066, 5306}, {5068, 5319}, {5107, 15533}, {5210, 15685}, {5298, 9651}, {5305, 11737}, {5569, 8353}, {6781, 15682}, {7703, 11647}, {7751, 7809}, {7775, 7837}, {7799, 7862}, {7808, 7884}, {7810, 16041}, {7811, 7825}, {7815, 7924}, {7880, 7887}, {7902, 16921}, {8354, 15597}, {8589, 15701}, {9466, 11318}, {9606, 12812}, {10979, 18353}, {11057, 14041}, {11121, 11132}, {11122, 11133}, {14075, 15484}, {15513, 15681}, {15544, 18435}

    X(18362) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 115, 11648), (2, 11648, 574), (3545, 3767, 7753), (5066, 5306, 5475), (5071, 7739, 1506), (6034, 11178, 5028)


    X(18363) = X(546)X(11559)∩X(3153)X(14861)

    Barycentrics    1/[3a^8 - 6a^6(b^2 + c^2) + 11a^4b^2c^2 + 3a^2(b^2 + c^2)(2b^4 - 3b^2c^2 + 2c^4) - (b^2 - c^2)^2(3b^4 + 8b^2c^2 + 3c^4)] : :

    X(18363) is the cevapoint of the nine-point centers of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18363) lies on the Jerabek hyperbola and these lines: {546,11559}, {3153,14861}, {3532,7577}, {7283,14471}

    X(18363) = isogonal conjugate of X(18364)


    X(18364) = X(2)X(3)∩X(74)X(10610)

    Barycentrics    a^2[3a^8 - 6a^6(b^2 + c^2) + 11a^4b^2c^2 + 3a^2(b^2 + c^2)(2b^4 - 3b^2c^2 + 2c^4) - (b^2 - c^2)^2(3b^4 + 8b^2c^2 + 3c^4)] : :

    X(18364) is the crosssum of the nine-point centers of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18364) lies on these lines: {2,3}, {74,10610}, {110,11559}, {195,11430}, {399,13367}, {1204,14805}, {1209,12121} et al

    X(18364) = isogonal conjugate of X(18363)


    X(18365) = X(3)X(6)∩X(2493)X(10313)

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

    X(18365) is the crossdifference of the nine-point centers of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18365) lies on these lines: {3,6}, {2493,10313}, {14579,14910}, {16310,18572}

    X(18365) = isogonal conjugate of X(18366)
    X(18365) = crossdifference of every pair of points on line X(523)X(546)


    X(18366) = TRILINEAR POLE OF LINE X(523)X(546)

    Barycentrics    1/[3a^6 - 5a^4(b^2 + c^2) + a^2(b^4 + 5b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)] : :

    Line X(523)X(546) is the line through the nine-point centers of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18366) lies on the Kiepert hyperbola and these lines: {4,10264}, {98,16166}, {801,15108}, {3580,13582} et al

    X(18366) = isogonal conjugate of X(18365)
    X(18366) = polar conjugate of X(13619)
    X(18366) = trilinear pole of line X(523)X(546)
    X(18366) = X(i)-isoconjugate of X(j) for these {i,j}: {1,18365}, {48,13619}


    X(18367) = X(6)X(17505)∩X(53)X(112)

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

    X(18367) is the barycentric product of the nine-point centers of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18367) lies on these lines: {6,17505}, {53,112}, {577,18562}


    X(18368) = X(3)X(2889)∩X(4)X(12291)

    Barycentrics    1/[a^8 - 2a^6(b^2 + c^2) + a^4b^2c^2 + a^2(b^2 + c^2)(2b^4 - 7b^2c^2 + 2c^4) - (b^2 - c^2)^2(b^4 - 4b^2c^2 + c^4)] : :
    X(18368) = 3*X(1173)-4*X(12242)

    X(18368) is the cevapoint of the nine-point centers of the 1st and 2nd Ehrmann inscribed triangles.

    X(18368) lies on the Jerabek hyperbola and these lines: {3, 2889}, {4, 12291}, {54, 13420}, {64, 12254}, {74, 10619}, {140, 10821}, {265, 14128}, {1154, 14861}, {1173, 12242}, {1994, 18282}, {3519, 15108}, {3521, 12226}, {13582, 18370}, {15002, 15047}

    X(18368) = isogonal conjugate of X(18369)


    X(18369) = X(2)X(3)∩X(51)X(195)

    Barycentrics    a^2[a^8 - 2a^6(b^2 + c^2) + a^4b^2c^2 + a^2(b^2 + c^2)(2b^4 - 7b^2c^2 + 2c^4) - (b^2 - c^2)^2(b^4 - 4b^2c^2 + c^4)] : :

    X(18369) is the crosssum of the nine-point centers of the 1st and 2nd Ehrmann inscribed triangles.

    X(18369) lies on these lines: {2,3}, {49,15038}, {51,195}, {54,13364}, {107,14978}, {110,1173}, {143,15801}, {155,13321}, {156,5640}, {389,399}, {568,15083}, {669,10280}, {1495,13353}, {5609,13358}, {10282,14845} et al

    X(18369) = isogonal conjugate of X(18368)


    X(18370) = X(4)X(93)∩X(140)X(930)

    Barycentrics    [a^8 - 2a^6(b^2 + c^2) + a^4b^2c^2 + a^2(b^2 + c^2)(2b^4 - 7b^2c^2 + 2c^4) - (b^2 - c^2)^2(b^4 - 4b^2c^2 + c^4)]/(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2 - b^2c^2) : :

    X(18370) is the crosspoint of the nine-point centers of the 1st and 2nd Ehrmann inscribed triangles.

    Let P17 be the X(17)-Ceva conjugate of X(18). Let P18 be the X(18)-Ceva conjugate of X(17). Then X(18370) = X(17)P17 ∩ X(18)P18.

    X(18370) lies on these lines: {4,93}, {140,930}, {252,550}, {13582,18368} et al


    X(18371) = X(2)X(6)∩X(50)X(237)

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

    X(18371) is the crossdifference of the symmedian points of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18371) lies on these lines: {2,6}, {50,237}, {186,2076}, {526,3049}, {566,14096}, {694,14910}, {1691,15462} et al

    X(18371) = isogonal conjugate of X(18372)
    X(18371) = crossdifference of every pair of points on line X(512)X(3818)


    X(18372) = TRILINEAR POLE OF LINE X(512)X(3818)

    Barycentrics    b^2c^2/[a^6(b^2 + c^2) - a^4(b^2 + c^2)^2 - a^2(b^2 + c^2)(b^4 + c^4) + b^8 + 4b^4c^4 + c^8] : :

    Line X(512)X(3818) is the line through the symmedian points of the 1st and 2nd Ehrmann circumscribing triangles.

    X(18372) lies on these lines: {25,12188}, {385,14910}, {694,3580}, {850,14998}

    X(18372) = isogonal conjugate of X(18371)
    X(18372) = trilinear pole of line X(512)X(3818)


    X(18373) = X(32)X(381)∩X(112)X(251)

    Barycentrics    a^2[a^8 - a^6(b^2 + c^2) - a^4(b^4 - 7b^2c^2 + c^4) + a^2(b^6 + c^6) - 2b^2c^2(b^2 - c^2)^2] : :

    X(18373) is the crosssum of the symmedian points of the 1st and 2nd Ehrmann inscribed triangles.

    X(18373) lies on these lines: {32,381}, {112,251}, {1384,5020} et al


    X(18374) = X(6)X(25)∩X(50)X(237)

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

    X(18374) is the crossdifference of the symmedian points of the 1st and 2nd Ehrmann inscribed triangles.

    X(18374) lies on these lines: {6,25}, {23,6593}, {30,15462}, {49,576}, {50,237}, {67,468}, {110,524}, {182,381}, {186,2781}, {215,8540}, {338,419}, {403,1503}, {511,2070}, {512,1691}, {542,10540}, {566,3148}, {567,5476}, {575,7545}, {597,5012}, {599,9306}, {1084,14602}, {15356,18487} et al

    X(18374) = isogonal conjugate of X(18019)
    X(18374) = crossdifference of every pair of points on line X(141)X(525)


    X(18375) = X(3)X(67)∩X(50)X(69)

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

    X(18375) is the barycentric product of the symmedian points of the 1st and 2nd Ehrmann inscribed triangles.

    X(18375) lies on these lines: {3,67}, {50,69}, {141,566}, {338,5025}, {7574,7818}, {10510,14003}


    X(18376) = X(2) OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    3a^10 - 4a^8(b^2 + c^2) - a^6(3b^4 - 8b^2c^2 + 3c^4) + 3a^4(b^2 - c^2)^2(b^2 + c^2) + 2 a^2(b^2 - c^2)^2(2b^4 - b^2c^2 + 2c^4) - 3(b^2 - c^2)^4(b^2 + c^2) : :
    X(18376) = 2 X(5) - X(11202)

    X(18376) lies on these lines: {4,51}, {5,11202}, {6,18429}, {64,5076}, {265,18434}, {1154,9927}, {2393,3818}, {3534,10193}, {5663,18566}, {13754,18568}, {18403,18430} et al

    X(18376) = reflection of X(11202) in X(5)
    X(18376) = X(2)-of-Ehrmann-vertex-triangle
    X(18376) = X(154)-of-Ehrmann-mid-triangle
    X(18376) = X(11202)-of-Johnson-triangle


    X(18377) = X(3) OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    SA^2 (SB + SC) - SA (SB^2 + SC^2 + 2 S^2) - SB SC (SB + SC + 52 R^2 - 16 SW) : :
    X(18377) = 3 X(2) - X(3) + X(4) - X(26) = X(3) + 2 X(4) - X(26) = X(4) + 2 X(5) - X(26) = X(4) - 2 X(18567) = X(4) + X(18569) = 2 X(5) - X(1658) = 2 X(546) - X(15761) = X(18566)- 2 X(18568)

    X(18377) lies on these lines: {2,3}, {68,18434}, {143,18390}, {155,18405}, {1154,9927}, {3564,18382}, {5876,18474}, {6564,11265}, {13754,18356}, {18430,18436} et al

    X(18377) = midpoint of X(4) and X(18569)
    X(18377) = reflection of X(i) in X(j) for these (i,j): (4,18567), (1658,5), (15761,546), (18566,18568)
    X(18377) = anticomplement of X(15331)
    X(18377) = Ehrmann-side-to-Ehrmann-vertex similarity image of X(4)
    X(18377) = orthic-to-Ehrmann-side similarity image of X(13406)
    X(18377) = X(3)-of-Ehrmann-vertex-triangle
    X(18377) = X(26)-of-Ehrmann-mid-triangle
    X(18377) = X(1658)-of-Johnson-triangle
    X(18377) = X(18569)-of-Euler-triangle
    X(18377) = X(18567)-of-anti-Euler-triangle
    X(18377) = {X(381),X(382)}-harmonic conjugate of X(18378)


    X(18378) = {X(381),X(382)}-HARMONIC CONJUGATE OF X(18377)

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

    X(18378) is the intersection of the Euler lines of ABC and the reflection triangle.

    X(18378) lies on these lines: {2,3}, {49,1495}, {52,10540}, {64,9919}, {110,10263}, {143,1614}, {154,9920}, {155,5898}, {156,195}, {159,5093}, {161,14530}, {184,14627}, {399,5889}, {511,18350}, {567,10110}, {568,6759}, {999,9658}, {1482,8185}, {6243,10539}, {6767,10037}, {7373,10046}, {9591,9956}, {9625,18480}, {9626,9955}, {10117,18381} et al

    X(18378) = {X(381),X(382)}-harmonic conjugate of X(18377)


    X(18379) = X(5) OF EHRMANN VERTEX-TRIANGLE

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

    X(18379) lies on these lines: {3,18392}, {4,94}, {5,13367}, {26,18405}, {30,5449}, {68,18568}, {1154,9927}, {5876,18403}, {10095,18390}, {14852,18434}, {18356,18566} et al

    X(18379) = X(5)-of-Ehrmann-vertex-triangle
    X(18379) = X(156)-of-Ehrmann-mid-triangle
    X(18379) = {X(18392),X(18394)}-harmonic conjugate of X(3)


    X(18380) = X(6) OF EHRMANN VERTEX-TRIANGLE

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

    X(18380) lies on these lines: {4,6}, {157,381}, {2871,3818}, {6751,13851}, {18403,18437} et al

    X(18380) = X(6)-of-Ehrmann-vertex-triangle
    X(18380) = X(157)-of-Ehrmann-mid-triangle


    X(18381) = X(8) OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    a^10 - 2a^8(b^2 + c^2) + a^6(b^4 + c^4) - a^4(b^2 - c^2)^2(b^2 + c^2) + 2a^2(b^8 - b^6c^2 - b^2c^6 + c^8) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18381) = 2 X(3) - X(4) - 2 X(5) - X(64) = 3 X(4) - X(5878) = X(4) + X(14216) = 2 X(5) - X(6759) = 2 X(546) - X(2883)

    X(18381) lies on these lines: {2,9833}, {4,51}, {5,182}, {17,11243}, {18,11244}, {24,125}, {26,5449}, {30,3357}, {64,265}, {154,1656}, {155,542}, {156,10224}, {221,9654}, {381,1498}, {546,2883}, {576,13292}, {1147,13371}, {1154,18356}, {3521,18434}, {3575,11438}, {5446,18382}, {13491,18428}, {14915,18431}, {18403,18439} et al

    X(18381) = midpoint of X(4) and X(14216)
    X(18381) = reflection of X(i) in X(j) for these (i,j): (2883,546), (6759,5)
    X(18381) = complement of X(9833)
    X(18381) = anticomplement of X(10282)
    X(18381) = X(8)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18381) = X(1498)-of-Ehrmann-mid-triangle
    X(18381) = X(3811)-of-orthic-triangle if ABC is acute
    X(18381) = X(6759)-of-Johnson-triangle
    X(18381) = X(14216)-of-Euler-triangle


    X(18382) = X(9) OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    a^12 - a^8(3b^4 + 2b^2c^2 + 3c^4) + 2a^6b^2c^2(b^2 + c^2) + a^4(b^2 - c^2)^2(3b^4 + 2b^2c^2 + 3c^4) + 2a^2b^2c^2(b^2 - c^2)^2(b^2 + c^2) - (b^2 - c^2)^4(b^2 + c^2)^2 : :
    X(18382) = X(3) + 2 X(4) - X(159) = 3 X(3) - 2 X(4) - X(6) - X(64) - X(159) = 2 X(5) - X(15577) = X(20) - 2 X(15578)

    X(18382) lies on these lines: {2,18427}, {4,6}, {5,15577}, {20,15578}, {66,265}, {69,3153}, {141,18531}, {154,7394}, {159,381}, {161,5133}, {182,18428}, {206,11818}, {511,9927}, {542,12596}, {546,15581}, {1352,6288}, {1843,13851}, {2393,3818}, {3564,18377}, {3589,18420}, {3827,18480}, {5446,18381}, {6403,18394}, {9969,18390}, {12220,18392}, {15585,18537} et al

    X(18382) = reflection of X(i) in X(j) for these (i,j): (20,15578), (15577,5)
    X(18382) = X(9)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18382) = X(159)-of-Ehrmann-mid-triangle
    X(18382) = X(15577)-of-Johnson-triangle


    X(18383) = X(10) OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    2 a^10 - 3a^8(b^2 + c^2) - a^6(b^4 - 4b^2c^2 + c^4) + a^4(b^2 - c^2)^2(b^2 + c^2) + 3a^2(b^2 - c^2)^2(b^4 + c^4) - 2(b^2 - c^2)^4(b^2 + c^2) : :
    X(18383) = X(3) - 4 X(4) + X(64) = 2 X(5) - X(10282) = X(382) + X(3357)

    X(18383) lies on these lines: {3,18405}, {4,51}, {5,5944}, {20,18392}, {30,5449}, {52,265}, {64,3830}, {125,6240}, {154,3851}, {184,7547}, {235,7687}, {381,569}, {382,1853}, {427,13403}, {511,9927}, {546,575}, {548,10193}, {567,10274}, {578,7507}, {924,10412}, {1181,18386}, {5663,18567}, {5876,18572}, {5907,18404}, {6146,18388}, {7355,18513}, {11695,18420}, {11793,18531}, {12162,18403}, {13754,18356} et al

    X(18383) = midpoint of X(382) and X(3357)
    X(18383) = reflection of X(10282) in X(5)
    X(18383) = X(10)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18383) = X(6759)-of-Ehrmann-mid-triangle
    X(18383) = X(10282)-of-Johnson-triangle


    X(18384) = BARYCENTRIC PRODUCT OF VERTICES OF EHRMANN VERTEX-TRIANGLE

    Barycentrics    1/{(a^2 - b^2 - c^2)[(a^2 - b^2 - c^2)^2 - b^2c^2]} : :

    X(18384) lies on these lines: {4,94}, {25,1989}, {51,11138}, {403,14993}, {460,8753}, {468,476}, {1990,14560} et al

    X(18384) = pole wrt polar circle of trilinear polar of X(7799) (line X(526)X(3268))
    X(18384) = polar conjugate of X(7799)
    X(18384) = isogonal conjugate of isotomic conjugate of X(6344)
    X(18384) = barycentric product X(4)*X(79)*X(80)
    X(18384) = barycentric product X(i)*X(j) for these {i,j}: {4,1989}, {6,6344}
    X(18384) = barycentric quotient X(4)/X(7799)


    X(18385) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND ANTI-ARA

    Barycentrics    [a^18 - 4a^16(b^2 + c^2) + a^14(5b^4 + 16b^2c^2 + 5c^4) - a^12(b^2 + c^2)(b^4 + 20b^2c^2 + c^4) - a^10(b^8 - 4b^6c^2 - 34b^4c^4 - 4b^2c^6 + c^8) - a^8(b^2 + c^2)(b^4 - 8b^2c^2 + c^4)(b^4 - 4b^2c^2 + c^4) - a^6(b^12 + 7b^8c^4 - 16b^6c^6 + 7b^4c^8 + c^12) + a^4(b^2 - c^2)^2(5b^10 - 5b^8c^2 + 4b^6c^4 + 4b^4c^6 - 5b^2c^8 + 5c^10) - 4a^2(b^2 - c^2)^4(b^2 + c^2)^2(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^6(b^2 + c^2)^3]/(b^2 + c^2 - a^2) : :

    X(18385) lies on these lines: {25,265}, {235,9927}, {1843,13851}, {3575,11438}, {6623,18387}

    X(18385) = X(18532)-of-anti-Ara-triangle


    X(18386) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ANTI-ASCELLA

    Barycentrics    [3a^6 - 2a^4(b^2 + c^2) - a^2(5b^4 - 2b^2c^2 + 5c^4) + 4(b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :

    X(18386) lies on these lines: {2,3} {184,18405}, {1181,18383}, {1498,11572}, {1853,13399}, {1986,13321}, {3172,18429}, {9777,18390} et al

    X(18386) = {X(381),X(382)}-harmonic conjugate of X(10254)


    X(18387) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND ANTICOMPLEMENTARY

    Barycentrics    2 SA(S^4 + 9 SB^2 SC^2) + 3 S^2 (SB + SC)(SA^2 - SA (SB - SC) + SB SC) : :
    Barycentrics    4a^10 - 8a^8(b^2 + c^2) + a^6(4b^4 + 15b^2c^2 + 4c^4) - 2a^4(b^2 + c^2)(2b^4 + b^2c^2 + 2c^4) + a^2(b^2 - c^2)^2(8b^4 + 3b^2c^2 + 8c^4) - 4(b^2 - c^2)^4(b^2 + c^2) : :

    X(18387) lies on these lines: {2,265}, {4,14449}, {5,11935}, {20,11454}, {146,11442}, {193,3818}, {3091,5654}, {3543,18474}, {6623,18385} et al

    X(18387) = anticomplement of X(3431)
    X(18387) = perspector of Johnson circle wrt Ehrmann vertex-triangle
    X(18387) = X(5561)-of-Ehrmann-vertex-triangle if ABC is acute


    X(18388) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 1ST ANTI-CONWAY

    Barycentrics    2a^8(b^2 + c^2) - 5a^6(b^4 + c^4) + 3a^4(b^2 - c^2)^2(b^2 + c^2) + a^2(b^2 - c^2)^2(b^4 + c^4) - (b^2 - c^2)^4 (b^2 + c^2) : :
    X(18388) = X(4) + X(184)

    X(18388) lies on these lines: {2,1568}, {4,54}, {5,389}, {6,13}, {20,6030}, {30,11430}, {51,403}, {125,5890}, {140,13568}, {154,18494}, {182,18531}, {185,1594}, {186,10182}, {567,18403}, {1181,7507}, {6146,18383}, {17809,18405}, {18439,18488} et al

    X(18388) = midpoint of X(4) and X(184)
    X(18388) = {X(6),X(381)}-harmonic conjugate of X(18390)
    X(18388) = X(993)-of-orthic-triangle if ABC is acute
    X(18388) = X(18389)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18388) = X(18389)-of-1st-anti-Conway-triangle if ABC is acute
    X(18388) = X(1478)-of-2nd-anti-Conway-triangle if ABC is acute
    X(18388) = X(18474)-of-Ehrmann-mid-triangle
    X(18388) = X(184)-of-Euler-triangle
    X(18388) = X(11430)-of-orthocentroidal-triangle
    X(18388) = Ehrmann-side-to-orthic similarity image of X(18474)


    X(18389) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 1ST ANTI-CONWAY

    Trilinears    a^5(b + c) - a^4(b^2 + c^2) - a^3(2b^3 + b^2c + bc^2 + 2c^3) + a^2(b - c)^2(2b^2 + 3bc + 2c^2) + a(b^5 - b^3c^2 - b^2c^3 + c^5) - (b^2 - c^2)^2(b^2 - bc + c^2) : :
    X(18389) = X(1478)-3*X(5902), 2*X(3822)-3*X(5883)

    The homothetic center of these triangles is X(18388).

    X(18389) lies on these lines: {1, 21}, {2, 18397}, {3, 15556}, {5, 226}, {7, 80}, {10, 343}, {20, 5903}, {36, 18444}, {41, 1729}, {42, 1735}, {46, 10884}, {56, 12005}, {57, 6905}, {65, 515}, {72, 5745}, {84, 1389}, {158, 1844}, {224, 17700}, {377, 3754}, {484, 7411}, {496, 6583}, {498, 3678}, {517, 4304}, {518, 8255}, {519, 16465}, {527, 5728}, {938, 5046}, {946, 1858}, {950, 7491}, {960, 16193}, {999, 5083}, {1012, 2099}, {1159, 18519}, {1454, 6796}, {1470, 15528}, {1708, 18443}, {1737, 3822}, {1741, 3553}, {1765, 2171}, {1777, 4332}, {1788, 15016}, {1870, 2003}, {1898, 18483}, {2093, 5732}, {2096, 11041}, {3085, 5904}, {3086, 5443}, {3419, 5832}, {3485, 5693}, {3487, 6852}, {3586, 5735}, {3679, 5833}, {3753, 5784}, {3911, 10202}, {3919, 17616}, {4197, 18395}, {4295, 9799}, {4306, 7138}, {4311, 12675}, {4312, 12669}, {4313, 5697}, {4973, 14793}, {5119, 7675}, {5173, 6001}, {5273, 5692}, {5434, 17660}, {5570, 11019}, {5691, 9960}, {5709, 10393}, {6284, 17637}, {6830, 12691}, {6911, 9946}, {6920, 11518}, {7098, 10902}, {7489, 15934}, {7508, 17010}, {7548, 9612}, {7951, 15064}, {8545, 10398}, {10247, 15558}, {10624, 12711}, {10883, 18393}, {11571, 13243}, {11888, 18399}, {11889, 18409}, {11890, 18408}, {13601, 18238}, {13739, 17104}

    X(18389) = reflection of X(72) in X(5745)
    X(18389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 920, 5248), (5902, 18391, 12736), (5902, 18412, 18391)


    X(18390) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd ANTI-CONWAY

    Barycentrics    a^10 - 2a^8(b^2 + c^2) + a^6(b^4 + 4b^2c^2 + c^4) - a^4(b^2 - c^2)^2(b^2 + c^2) + 2a^2(b^2 - c^2)^2(b^4 - b^2c^2 + c^4) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18390) = X(4) + X(1899) = 2 X(5) - X(9306)

    X(18390) lies on these lines: {2,11430}, {3,2929}, {4,51}, {5,578}, {6,13}, {25,18396}, {30,11438}, {52,18404}, {68,5907}, {125,378}, {143,18377}, {155,10112}, {182,15760}, {184,403}, {235,6146}, {382,9786}, {394,16072}, {468,11202}, {511,18531}, {539,15068}, {546,12233}, {567,10254}, {568,18403}, {569,10024}, {576,10297}, {800,18437}, {1173,16000}, {1204,18560}, {1350,18536}, {1352,8681}, {1593,15121}, {1620,15696}, {2888,15056}, {3448,15305}, {3531,15321}, {3581,18564}, {5446,18569}, {5449,7526}, {5640,16223}, {5943,18420}, {6653,15644}, {6756,15873}, {7507,10982}, {7577,14644}, {9777,18386}, {9969,18382}, {10095,18379}, {10201,18475}, {10625,18555}, {10733,15053}, {11435,18406}, {11442,15030}, {11444,15108}, {13364,15465}, {13561,15807}, {15019,15044}, {15078,16163}, {17810,18405} et al

    X(18390) = midpoint of X(4) and X(1899)
    X(18390) = reflection of X(9306) in X(5)
    X(18390) = {X(6),X(381)}-harmonic conjugate of X(18388)
    X(18390) = orthic-to-2nd-Euler similarity image of X(9306)
    X(18390) = X(997)-of-orthic-triangle if ABC is acute
    X(18390) = X(18391)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18390) = X(18391)-of-2nd-anti-Conway-triangle if ABC is acute
    X(18390) = X(1899)-of-Euler-triangle
    X(18390) = X(9306)-of-Johnson-triangle
    X(18390) = X(11438)-of-orthocentroidal-triangle


    X(18391) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd ANTI-CONWAY

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

    The homothetic center of these triangles is X(18390).

    X(18391) lies on these lines: {1,2}, {3,1788}, {4,65}, {5,3485}, {6,281}, {7,80}, {11,2099}, {12,3487}, {19,5802}, {20,46}, {21,11507}, {30,3474}, {35,4313}, {36,5435}, {40,950}, {55,1006}, {56,944}, {57,515}, {58,1771}, {72,2551}, {79,7319}, {81,11103}, {100,8069}, {104,1470}, {1512,18446}, {2771,18516}, {3871,11508}, {5708,18525}, {6147,9654} et al

    X(18391) = anticomplement of X(997)
    X(18391) = polar conjugate of isotomic conjugate of X(6350)


    X(18392) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 3rd ANTI-EULER

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

    X(18392) lies on these lines: {3,18379}, {4,52}, {5,11449}, {20,18383}, {22,18405}, {110,381}, {10539,18504}, {12220,18382}, {18436,18567} et al

    X(18392) = X(18393)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18392) = X(18393)-of-3rd-anti-Euler-triangle if ABC is acute
    X(18392) = {X(3),X(18379)}-harmonic conjugate of X(18394)


    X(18393) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 3rd ANTI-EULER

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

    The homothetic center of these triangles is X(18392).

    X(18393) lies on these lines: {1,4}, {2,484}, {5,5903}, {7,3065}, {8,11280}, {10,3899}, {11,11571}, {12,5697}, {13,7052}, {30,15950}, {35,11375}, {36,1836}, {40,6863}, {46,6862}, {56,79}, {57,1727}, {65,7741}, {80,381}, {165,6954} et al

    X(18393) = {X(5),X(5903)}-harmonic conjugate of X(18395)


    X(18394) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 4th ANTI-EULER

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

    X(18394) lies on these lines: {3,18379}, {4,51}, {5,10546}, {24,14644}, {64,10721}, {186,11704}, {381,1614}, {6403,18382}, {10540,18504}, {12111,18403}, {15058,18474}, {18436,18572} et al

    X(18394) = X(18395)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18394) = X(18395)-of-4th-anti-Euler-triangle if ABC is acute
    X(18394) = {X(3),X(18379)}-harmonic conjugate of X(18392)


    X(18395) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 4th ANTI-EULER

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

    The homothetic center of these triangles is X(18394).

    X(18395) lies on these lines: {1,2}, {3,80}, {4,484}, {5,5903}, {11,5690}, {12,5902}, {18,7052}, {35,1837}, {36,355}, {40,3583}, {46,3585}, {56,5790}, {57,5270}, {65,5694}, {72,5123}, {79,10895}, {90,3359}, {119,5693}, {140,10950}, {5204,5442}, {7354,18357}, {7701,18516} et al

    X(18395) = {X(5),X(5903)}-harmonic conjugate of X(18393)


    X(18396) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd ANTI-EXTOUCH

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

    X(18396) lies on these lines: {3,125}, {4,6}, {20,3580}, {24,12289}, {25,18390}, {30,1899}, {51,18494}, {64,11457}, {68,4549}, {113,12419}, {184,381}, {185,382}, {378,1853}, {394,18531}, {974,10733}, {1568,3167}, {5562,11898}, {6467,15030}, {9818,18474}, {18403,18445} et al

    X(18396) = X(18397)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18396) = X(18397)-of-2nd-anti-extouch-triangle if ABC is acute


    X(18397) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd ANTI-EXTOUCH

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

    The homothetic center of these triangles is X(18396).

    X(18397) lies on these lines: {1,6}, {2,18389}, {4,80}, {35,920}, {36,1708}, {40,1858}, {43,1735}, {46,1490}, {56,6326}, {57,912}, {65,5587}, {191,11507}, {201,581}, {226,1737}, {329,758}, {452,3878} et al


    X(18398) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ANTI-INCIRCLE-CIRCLES

    Trilinears    a^2(b + c) + 3abc - (b - c)^2(b + c) : :

    The homothetic center of these triangles is X(3843).

    X(18398) lies on these lines: {1,3}, {2,3678}, {4,5557}, {7,79}, {8,3881}, {10,3873}, {11,6147}, {37,5043}, {63,5259}, {72,3624}, {80,388}, {81,5358}, {104,15173}, {145,3754}, {1210,3947} et al


    X(18399) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ANTI-TANGENTIAL MIDARC

    Trilinears    (b - c) (a^5 - 5 a^4 (b + c) + a^3 (4 b^2 + 9 b c + 4 c^2) + a^2 (b + c) (4 b^2 - 9 b c + 4 c^2) - a (5 b^4 + 3 b^3 c - 10 b^2 c^2 + 3 b c^3 + 5 c^4) + (b - c)^2 (b + c) (b^2 + 5 b c + c^2)) - 2 (b - c) (a^2 - b^2 - c^2 + b c) (a^2 (b + c) - a b c - (b + c) (b^2 + c^2)) Sin[A/2] - 2 (c^6 - c^5 a + c^4 (a - b) (3 a + 2 b) + c^3 a (a^2 - 6 a b + 4 b^2) + c^2 (-4 a^4 + a^3 b + 3 a^2 b^2 + b^4) + a b c (a^2 - b^2) (a + 3 b) + a b (a^2 - b^2)^2) Sin[B/2] + 2 (b^6 - b^5 a - b^4 (c - a) (2 c + 3 a) + b^3 a (4 c^2 - 6 c a + a^2) + b^2 (c^4 + 3 c^2 a^2 + c a^3 - 4 a^4) - a b c (c^2 - a^2) (3 c + a) + c a (c^2 - a^2)^2) Sin[C/2] : : (Randy Hutson, June 27, 2018)

    The anti-tangential midarc triangle is the reflection of the intangents triangle in X(1).

    The homothetic center of these triangles is X(3585).

    X(18399) lies on these lines: {1,164}, {46,8081}, {65,8099}, {80,177}, {188,5692}, {484,8075}, {517,10503}, {758,11690}, {942,10506}, {2089,5902} et al


    X(18400) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND X(3)-EHRMANN

    Trilinears    b[sec A sec(C - A) - sec B sec(B - C)] - c[sec C sec(B - C) - sec A sec(A - B)] : :
    Barycentrics    2a^10 - 4a^8(b^2 + c^2) + a^6(b^4 + c^4 + 4b^2c^2) + a^4(b^2 - c^2)^2(b^2 + c^2) + a^2(b^2 - c^2)^2(b^4 + c^4) - (b^2 - c^2)^4(b^2 + c^2) : :

    The X(3)-Ehrmann triangle is the medial triangle of the tangential triangle, and also the cross-triangle of the following pairs of triangles: {1st and 2nd Kenmotu diagonals}, {inner and outer tri-equilateral}, {1st and 2nd anti-Conway}. It is also the anti-inner-Conway triangle and the reflection of the Kosnita triangle in X(10282). See X(25) for a generalization.

    X(18400) lies on these (parallel) lines: {2,10182}, {3,161}, {4,54}, {5,5944}, {13,11243}, {14,11244}, {20,2888}, {25,18390}, {26,9927}, {30,511}, {51,7576}, {52,10112}, {64,1657}, {66,3098}, {74,10421}, {107,6761}, {110,1568}, {113,10540}, {115,1971}, {125,186}, {140,13470}, {143,11262}, {154,381}, {195,382}, {265,2070} et al

    X(18400) = isogonal conjugate of X(18401)
    X(18400) = X(758)-of-orthic-triangle if ABC is acute


    X(18401) = ISOGONAL CONJUGATE OF X(18400)

    Trilinears    1/{b[sec A sec(C - A) - sec B sec(B - C)] - c[sec C sec(B - C) - sec A sec(A - B)]} : :
    Barycentrics    a^2/[2a^10 - 4a^8(b^2 + c^2) + a^6(b^4 + c^4 + 4b^2c^2) + a^4(b^2 - c^2)^2(b^2 + c^2) + a^2(b^2 - c^2)^2(b^4 + c^4) - (b^2 - c^2)^4(b^2 + c^2)] : :
    X(18401) = 2 X(3) - X(933)

    X(18401) lies on the circumcircle and these lines: {2,18402}, {3,933}, {5,107}, {110,5562}, {112,216}, {476,3153}, {925,12225}, {1289,7576}, {1291,2071}, {1301,3518}, {1304,2070}, {3520,6799} et al

    X(18401) = circumcircle antipode of X(933)
    X(18401) = reflection of X(933) in X(3)
    X(18401) = anticomplement of X(18402)
    X(18401) = Λ(X(4), X(54))
    X(18401) = the point of intersection, other than A, B, and C, of the circumcircle and conic {{A,B,C,X(3),X(5)}}
    X(18401) = Collings transform of X(2972)


    X(18402) = INVERSE-IN-POLAR-CIRCLE OF X(1141)

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

    X(18402) lies on the nine-point circle and these lines: {2,18401}, {4,137}, {11,1944}, {115,6748}, {122,140}, {125,389}, {127,14767}, {130,6752}, {136,3575}, {139,571}, {186,3258}, {216,8439}, {403,10214}, {1540,18400} et al

    X(18402) = complement of X(18401)
    X(18402) = inverse-in-polar-circle of X(1141)
    X(18402) = Λ(X(4), X(54)), wrt orthic triangle


    X(18403) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND EHRMANN SIDE

    Barycentrics    [(a^4 - b^4 - c^4 + 2b^2c^2)^2 - a^4b^2c^2](b^2 + c^2 - a^2) : :
    X(18403) = X(3) + X(4) - X(186) = X(3) - 2 X(2072) = 2 X(4) + 2 X(5) - X(23) = 2 X(4) - X(23) + X(186) = X(4) + X(3153) = 2 X(5) - X(186) = X(23) - 2 X(11563) = 2 X(403) - X(2070) = 2 X(546) - X(11563)
    X(18403) = a^2b^2c^2 X(3) + (a^2 - b^2 - c^2)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2) X(4)

    X(18403) is the perspector of ABC and the reflection of the circumorthic tangential triangle in line X(403)X(523) (which is the perspectrix of ABC and the circumorthic tangential triangle).

    X(18403) lies on these lines: {2,3}, {49,5448}, {113,10540}, {115,10317}, {156,12289}, {265,1531}, {389,13376}, {541,13399}, {542,18449}, {567,18388}, {568,18390}, {577,9220}, {1060,18513}, {1989,3284}, {3581,7687}, {3583,9629}, {3585,18447}, {5446,15800}, {5876,18379}, {5964,9927}, {6000,7728}, {6564,18457}, {6565,18459}, {10721,13445}, {11440,13561}, {12022,15087}, {12111,18394}, {12162,18383}, {12358,13391}, {14156,16163}, {14915,16223}, {16808,18468}, {16809,18470}, {18376,18430}, {18380,18437}, {18381,18439}, {18382,18440}, {18396,18445}, {18406,18453}, {18414,18462}, {18415,18463}, {18416,18464}, {18418,18466}, {18424,18472}

    X(18403) = midpoint of X(4) and X(3153)
    X(18403) = reflection of X(i) in X(j) for these (i,j): (3,2072), (23,11563), (186,5), (2070,403), (11563,546)
    X(18403) = complement of X(13619)
    X(18403) = anticomplement of X(15646)
    X(18403) = inverse-in-Johnson-circle of X(3)
    X(18403) = inverse-in-circumcircle of X(1658)
    X(18403) = inverse-in-nine-point-circle of X(10024)
    X(18403) = inverse-in-polar-circle of X(6240)
    X(18403) = inverse-in-2nd-Droz-Farny-circle of X(15761)
    X(18403) = pole wrt Johnson circle of line X(3)X(523) (the line of the Ehrmann cross-triangle)
    X(18403) = homothetic center of Ehrmann vertex-triangle and X(4)-Ehrmann triangle
    X(18403) = homothetic center of Ehrmann side-triangle and X(4)-Ehrmann triangle
    X(18403) = X(36)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18403) = X(36)-of-Ehrmann-side-triangle if ABC is acute
    X(18403) = X(36)-of-X(4)-Ehrmann-triangle if ABC is acute
    X(18403) = X(3153)-of-Euler-triangle
    X(18403) = X(2072)-of-X3-ABC-reflections-triangle
    X(18403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,382,25), (1113,1114,1658), (1312,1313,10024), (10750,10751,3)
    X(18403) = barycentric product X(69)*X(3583)*X(3585)
    X(18403) = Eulerologic center of these triangles: ABC to Ehrmann side


    X(18404) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd EULER

    Barycentrics    [a^8 - 2a^4(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18404) = 2a^2b^2c^2 X(3) + (a^2 - b^2 - c^2)(a^2 + b^2 - c^2)(a^2 - b^2 + c^2) X(4) = 2 X(5) - X(24)

    X(18404) lies on these lines: {2,3}, {52,18390}, {113,6759}, {125,7689}, {127,7825}, {485,18457}, {486,18459}, {1062,3583}, {1147,1568}, {1352,6288}, {3521,4846}, {5878,7728}, {5907,18383}, {7748,14961}, {14216,18439}, {17814,18405} et al

    X(18404) = reflection of X(24) in X(5)
    X(18404) = inverse-in-Johnson-circle of X(2072)
    X(18404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (1,j,k): (381,382,1598), (10750,10751,2072)
    X(18404) = X(46)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18404) = X(46)-of-2nd-Euler-triangle if ABC is acute
    X(18404) = X(24)-of-Johnson-triangle
    X(18404) = X(7517)-of-Ehrmann-mid-triangle


    X(18405) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 1st EXCOSINE

    Barycentrics    5a^10 - 8a^8(b^2 + c^2) - 2a^6(b^4 - 6b^2c^2 + c^4) + 4a^4(b^2 - c^2)^2(b^2 + c^2) + a^2(b^2 - c^2)^2(5b^4 - 2b^2c^2 + 5c^4) - 4(b^2 - c^2)^4(b^2 + c^2) : :
    X(18405) = X(2) + X(4) - X(154) = X(154) - 2 X(381)

    X(18405) lies on these lines: {3,18383}, {4,6}, {5,17821}, {22,18392}, {24,14644}, {25,13851}, {26,18379}, {30,1853}, {64,265}, {154,381}, {155,18377}, {161,9818}, {184,18386}, {221,3585}, {389,10938}, {394,3153}, {542,17813}, {546,9833}, {567,3843}, {568,3830}, {1192,6240}, {1619,18535}, {3197,18406}, {5907,18438}, {6145,9927}, {6288,17846}, {6564,17819}, {6565,17820}, {12293,18569}, {15068,18572}, {16808,17826}, {16809,17827}, {17809,18388}, {17810,18390}, {17811,18531}, {17814,18404}, {17825,18420}, {18403,18451} et al

    X(18405) = reflection of X(154) in X(381)
    X(18405) = orthic-to-Ehrmann-vertex similarity image of X(12022)
    X(18405) = X(5692)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18405) = X(5692)-of-1st-excosine-triangle if ABC is acute


    X(18406) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND EXTANGENTS

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

    X(18406) lies on these lines: {1,18517}, {4,9}, {55,381}, {65,79}, {908,4420}, {1478,4654}, {1479,5219}, {2093,18513}, {3197,18405}, {5270,18525}, {9816,18420}, {11435,18390}, {18403,18453} et al

    X(18406) = X(18408)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18406) = X(18408)-of-extangents-triangle if ABC is acute
    X(18406) = {X(381),X(18407)}-harmonic conjugate of X(3583)


    X(18407) = X(4)X(8)∩X(55)X(381)

    Barycentrics    a^7-a^6 b-a^5 b^2+a^4 b^3-a^3 b^4+a^2 b^5+a b^6-b^7-a^6 c+b^6 c-a^5 c^2-a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+a^4 c^3-a^2 b^2 c^3-3 b^4 c^3-a^3 c^4-a b^2 c^4-3 b^3 c^4+a^2 c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
    X(18407) = X[55] - 3 X[381], 3 X[4] + X[3434], 3 X[5] - 2 X[6690], 5 X[3843] - X[10679], X[5119] - 5 X[18492], 3 X[381] + X[18499]

    X(18407) lies on these lines: {4,8}, {5,5248}, {30,993}, {55,381}, {56,18544}, {149,3656}, {377,13624}, {382,3428}, {528,3845}, {546,7680}, {674,3818}, {1385,3824}, {1479,9955}, {1699,6326}, {1836,2771}, {2099,3585}, {2475,3897}, {3304,18543}, {3715,5790}, {3843,10679}, {3868,16159}, {3873,11604}, {3874,16125}, {4294,6866}, {4316,18515}, {4857,18493}, {5119,18492}, {5172,7741}, {5225,6849}, {5270,18526}, {5840,8727}, {5886,6839}, {6253,6842}, {6284,6841}, {6681,6924}, {6826,11230}, {6827,11231}, {6928,9956}, {7548,11491}, {8069,10896}, {9656,18545}, {9897,10742}, {10537,18400}, {10895,18518}, {11018,18527}, {11366,18496}, {11367,18498}, {12116,15178}, {12943,18519}, {15733,18482}

    X(18407) = midpoint of X(i) and X(j) for these {i,j}: {55, 18499}, {382, 3428}, {2099, 18525}, {3419, 12699}, {12943, 18519}
    X(18407) = reflection of X(i) in X(j) for these {i,j}: {7680, 546}
    X(18407) = X(55)-of-Ehrmann-mid-triangle
    X(18407) = X(174)-of-Ehrmann-vertex-triangle

    X(18407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18517, 18480), (355, 12699, 3869), (381, 18499, 55), (381, 18524, 7951), (3583, 18406, 381)


    X(18408) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND EXTANGENTS

    Trilinears    (a - b - c) (a - b + c) (a + b - c) (a + b + c) (b - c) (a^2 - b^2 - c^2 + b c) + 2 (a - b - c) (b - c) (a^2 - b^2 - c^2 + b c) (a^2 (b + c) + a b c - (b - c)^2 (b + c)) Sin[A/2] - 2 (a - b + c) (c^6 - 3 c^5 a + c^4 (a - b) (a + 2 b) + c^3 a (3 a^2 - 2 a b + 4 b^2) + c^2 (-2 a^4 + a^3 b - a^2 b^2 - 2 a b^3 + b^4) - a b c (a - b)^2 (a + b) + a b (a^2 - b^2)^2) Sin[B/2] + 2 (a + b - c) (b^6 - 3 b^5 a - b^4 (c - a) (2 c + a) + b^3 a (4 c^2 - 2 c a + 3 a^2) + b^2 (c^4 - 2 c^3 a - c^2 a^2 + c a^3 - 2 a^4) - a b c (c - a)^2 (c + a) + c a (c^2 - a^2)^2) Sin[C/2] : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(18406).

    X(18408) lies on these lines: {1, 168}, {36, 18454}, {46, 7590}, {65, 12491}, {80, 177}, {174, 5902}, {236, 5692}, {484, 7589}, {517, 10502}, {758, 8126}, {942, 10501}, {2093, 8423}, {3337, 7588}, {5425, 18456}, {5691, 12685}, {5697, 11924}, {5883, 8125}, {5903, 8351}, {7593, 18397}, {8092, 18398}, {8129, 15016}, {8379, 18393}, {8382, 18395}, {11571, 13267}, {11890, 18389}, {11891, 18391}

    X(18408) = {X(174),X(5902)}-harmonic conjugate of X(18409)


    X(18409) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND INTANGENTS

    Trilinears    a (a - b - c) (a - b + c) (a + b - c) (a + b + c) (b - c) (a^2 - b^2 - c^2 + b c) - 2 a (a - b - c) (b - c) (a^2 - b^2 - c^2 + b c) (a^2 (b + c) + a b c - (b - c)^2 (b + c)) Sin[A/2] + 2 (a - b + c) (a^5 (b^2 + b c + c^2) - a^4 b c (b + 2 c) - a^3 (2 b^4 - b c^3 + 2 c^4) + a^2 b c (2 b^3 - b c^2 + 2 c^3) + a (b - c)^2 (b + c) (b^3 - b c^2 + c^3) - b^2 c (b^2 - c^2)^2) Sin[B/2] - 2 (a + b - c) (a^5 (b^2 + b c + c^2) - a^4 b c (2 b + c) - a^3 (2 b^4 - b^3 c + 2 c^4) + a^2 b c (2 b^3 - b^2 c + 2 c^3) + a (b - c)^2 (b + c) (b^3 - b^2 c + c^3) - b c^2 (b^2 - c^2)^2) Sin[C/2] : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(3583).

    X(18409) lies on these lines: {1,164}, {46,8082}, {65,8100}, {80,12772}, {174,5902}, {484,8076}, {517,10501}, {758,8125}, {942,10502}, {2093,8090} et al

    X(18409) = {X(174),X(5902)}-harmonic conjugate of X(18408)


    X(18410) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 1st KENMOTU DIAGONALS

    Trilinears    (a - b - c)(a - b + c)(a + b - c)(a + b + c)[a^2(b + c) + abc - (b - c)^2(b + c)] + [a^4(b + c) - a^3(2b^2 + bc + 2c^2) + a(b - c)^2(2b^2 + 3bc + 2c^2) - b^5 + b^4c + bc^4 - c^5]*2S : :

    The homothetic center of these triangles is X(6564).

    X(18410) lies on these lines: {1,372}, {7,80}, {36,18458}

    X(18410) = {X(7),X(18413)}-harmonic conjugate of X(18411)
    X(18410) = {X(5902),X(18412)}-harmonic conjugate of X(18411)


    X(18411) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd KENMOTU DIAGONALS

    Trilinears    (a - b - c)(a - b + c)(a + b - c)(a + b + c)[a^2(b + c) + abc - (b - c)^2(b + c)] - [a^4(b + c) - a^3(2b^2 + bc + 2c^2) + a(b - c)^2(2b^2 + 3bc + 2c^2) - b^5 + b^4c + bc^4 - c^5]*2S : :

    The homothetic center of these triangles is X(6565).

    X(18411) lies on these lines: {1,371}, {7,80}, {36,18460}

    X(18411) = {X(7),X(18413)}-harmonic conjugate of X(18410)
    X(18411) = {X(5902),X(18412)}-harmonic conjugate of X(18413)


    X(18412) = X(1)X(6)∩X(7)X(80)

    Trilinears    a^4(b + c) - a^3(2b^2 + bc + 2c^2) + a(b - c)^2(2b^2 + 3bc + 2c^2) - b^5 + b^4c + bc^4 - c^5 : :
    X(18412) = X(1) - 2 X(5728)

    X(18412) lies on these lines: {1,6}, {4,15909}, {7,80}, {20,12432}, {35,7675}, {36,1445}, {46,5732}, {55,15104}, {57,11502}, {65,971}, {142,1737}, {144,758}, {165,10391}, {200,16465}, {210,11018}, {354,5219}, {390,5697}, {484,11495}, {516,5903}, {517,14100}, {908,3873}, {912,11529}, {938,3874}, {942,5290}, {946,10392}, {999,6326}, {1002,11028}, {1210,3947}, {3062,3577}, {3826,18395}, {5708,18491} et al

    X(18412) = reflection of X(1) in X(5728)
    X(18412) = {X(18410),X(18411)}-harmonic conjugate of X(5902)


    X(18413) = ENDO-HOMOTHETIC CENTER OF EHRMANN VERTEX-TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

    Trilinears    a^6(b + c) - 2a^5(b^2 + bc + c^2) + a^4(b + c)(b^2 + bc + c^2) - 2a^3b^2c^2 - a^2(b - c)^2(b + c)^3 + 2a(b^6 - b^5c - bc^5 + c^6) - (b - c)^4(b + c)(b^2 + bc + c^2) : :
    X(18413) = X(1) - 2 X(11028)

    The homothetic center of these triangles is X(115).

    X(18413) lies on these lines: {1,41}, {7,80}, {36,11714}, {46,103}, {65,2808}, {72,3041}, {116,1737}, {118,12047}, {152,4295}, {354,5723}, {517,3022}, {518,1146}, {758,10025}, {926,10015}, {942,1362}, {1721,2093}, {3033,5530} et al

    X(18413) = reflection of X(1) in X(11028)
    X(18413) = {X(18410),X(18411)}-harmonic conjugate of X(7)


    X(18414) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND LUCAS ANTIPODAL TANGENTS

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

    X(18414) lies on these lines: {381, 8939}, {382, 13021}, {3818, 12590}, {9723, 18415}, {18403, 18462}


    X(18415) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND LUCAS(-1) ANTIPODAL TANGENTS

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

    X(18415) lies on these lines: {381, 8943}, {382, 13022}, {3818, 12591}, {9723, 18414}, {18403, 18463}


    X(18416) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ORTHOCEVIAN OF X(3)

    Barycentrics    a^22 - 7a^20(b^2 + c^2) + a^18(21b^4 + 37b^2c^2 + 21c^4) - 2a^16(b^2 + c^2)(17b^4 + 19b^2c^2 + 17c^4) + a^14(29b^8 + 57b^6c^2 + 82b^4c^4 + 57b^2c^6 + 29c^8) - a^12(b^2 + c^2)(7b^8 + 5b^6c^2 + 24b^4c^4 + 5b^2c^6 + 7c^8) - a^10(b^2 - c^2)^2(7b^8 + 5b^6c^2 + 18b^4c^4 + 5b^2c^6 + 7c^8) - a^8(b^2 - c^2)^2(b^2 + c^2)(b^4 - b^2c^2 + c^4)(b^4 + 14b^2c^2 + c^4) + a^6(b^2 - c^2)^4(14b^8 + 23b^6c^2 + 32b^4c^4 + 23b^2c^6 + 14c^8) - 7a^4(b^2 - c^2)^6(b^2 + c^2)(2b^4 + b^2c^2 + 2c^4) + 2a^2(b^2 - c^2)^6(3b^8 - b^6c^2 - 3b^4c^4 - b^2c^6 + 3c^8) - (b^2 - c^2)^8(b^6 + c^6) : :

    X(18416) lies on these lines: {4,54}, {265,8612}, {18403,18464}

    X(18416) = X(18417)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18416) = X(18417)-of-orthocevian-triangle-of-X(3) if ABC is acute


    X(18417) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ORTHOCEVIAN OF X(3)

    Trilinears    (a^2 (b^2 + b c + c^2) - a b c (b + c) - (b + c)^2 (b^2 - b c + c^2))/(b + c) : : : :

    The homothetic center of these triangles is X(18416).

    X(18417) lies on these lines: {1,21}, {5,1211}, {80,313}, {86,5902}, {333,5692}, {484,13588}, {524,10477}, {859,5289}, {1010,5903} et al


    X(18418) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ORTHOANTICEVIAN OF X(3)

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

    X(18418) lies on these lines: {5,3357}, {6,13}, {185,3091}, {1568,2063}, {3858,12370}, {18403,18466} et al

    X(18418) = X(18419)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18418) = X(18419)-of-orthoanticevian-triangle-of-X(3) if ABC is acute


    X(18419) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND ORTHOANTICEVIAN OF X(3)

    Trilinears    [2a^3(b + c) - a^2(2b^2 + bc + 2c^2) - 2a(b - c)^2(b + c) + (b + c)^2(2b^2 - 3bc + 2c^2)]/(b + c - a) : :

    The homothetic center of these triangles is X(18418).

    X(18419) lies on these lines: {7,80}, {57,4511}, {65,145}, {758,5435}, {938,5884}, {942,5603}, {999,10698}, {1476,1482} et al


    X(18420) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND SUBMEDIAL

    Barycentrics    a^10 - a^8(b^2 + c^2) - 2a^6(b^2 - c^2)^2 + 2a^4(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) + a^2(b^4 - c^4)^2 - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18420) = 2*R^2*X(3) - (2*R^2 - SW)*X(4) = X(4) - 2 X(11818) = 2 X(5) - X(9818)

    X(18420) lies on these lines: {2,3}, {66,3818}, {68,389}, {69,1154}, {571,7737}, {1352,7706}, {1568,5651}, {1899,9730}, {3589,18382}, {5422,12022}, {5462,9815}, {5943,18390}, {6193,12161}, {9816,18406}, {9895,18517}, {10574,11457}, {11487,11591}, {11695,18383}, {12163,13568}, {12324,13491}, {14826,15068}, {17825,18405} et al

    X(18420) = reflection of X(i) in X(j) fir these (i,j): (4,11818), (9818,5)
    X(18420) = anticomplement of X(7514)
    X(18420) = inverse-in-orthocentroidal-circle of X(18531)
    X(18420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,18531), (3,4,14790)
    X(18420) = X(18421)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18420) = X(18421)-of-submedial-triangle if ABC is acute
    X(18420) = X(1597)-of-Ehrmann-mid-triangle
    X(18420) = X(9818)-of-Johnson-triangle
    X(18420) = X(11818)-of-anti-Euler-triangle


    X(18421) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND SUBMEDIAL

    Trilinears    (5b + 5c - a)/(b + c - a) : :

    The homothetic center of these triangles is X(18420).

    X(18421) lies on these lines: {1,3}, {7,519}, {8,3671}, {10,5226}, {72,4866}, {145,4298}, {226,3679}, {3062,3577} et al

    X(18421) = trilinear product X(57)*X(16676)
    X(18421) = trilinear quotient X(16676)/X(9)


    X(18422) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND INNER TRI-EQUILATERAL

    Trilinears    Sqrt[3] (a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 (b + c) + a b c - (b - c)^2 (b + c)) + 2 S (a^4 (b + c) - a^3 (2 b^2 + b c + 2 c^2) + a (b - c)^2 (2 b^2 + 3 b c + 2 c^2) - b^5 + b^4 c + b c^4 - c^5) : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(16808).

    X(18422) lies on these lines: {1,16}, {7,80}, {36,18469}

    X(18422) = {X(7),X(18425)}-harmonic conjugate of X(18423)


    X(18423) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN VERTEX AND OUTER TRI-EQUILATERAL

    Trilinears    Sqrt[3] (a - b - c) (a + b - c) (a - b + c) (a + b + c) (a^2 (b + c) + a b c - (b - c)^2 (b + c)) - 2 S (a^4 (b + c) - a^3 (2 b^2 + b c + 2 c^2) + a (b - c)^2 (2 b^2 + 3 b c + 2 c^2) - b^5 + b^4 c + b c^4 - c^5) : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(16809).

    X(18423) lies on these lines: {1,15}, {7,80}, {36,18471}

    X(18423) = {X(7),X(18425)}-harmonic conjugate of X(18422)


    X(18424) = HOMOTHETIC CENTER OF EHRMANN VERTEX-TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

    Barycentrics    a^4 + a^2(b^2 + c^2) - 3(b^2 - c^2)^2 : :
    X(18424) = X(6564) + X(6565)

    X(18424) lies on these lines: {2,8589}, {3,11742}, {4,187}, {5,574}, {6,13}, {32,546}, {39,3091}, {69,7615}, {111,5169}, {230,3845}, {316,9939}, {325,18546}, {382,5210}, {577,10297}, {625,7801}, {631,11614}, {671,7777}, {858,8585}, {1003,6722}, {1078,14062}, {1352,5107}, {1384,3843}, {1506,3851}, {2548,3855}, {2549,3545}, {18403,18472} et al

    X(18424) = midpoint of X(6564) and X(6565)
    X(18424) = X(18425)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18424) = X(8588)-of-orthocentroidal-triangle


    X(18425) = ENDO-HOMOTHETIC CENTER OF EHRMANN VERTEX-TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

    Trilinears    2a^6(b + c) - a^5(2b^2 + bc + 2c^2) - a^4(b + c)(2b^2 - 3bc + 2c^2) - 2a^3bc(b^2 + bc + c^2) + 2a^2(b - c)^2(b + c)(b^2 + c^2) + a(b - c)^2(b^2 + c^2)(2b^2 + 3bc + 2c^2) - (b - c)^4(b + c)(2b^2 + 3bc + 2c^2) : :

    The homothetic center of these triangles is X(18424).

    X(18425) lies on these lines: {1,1055}, {7,80}, {36,18473}, {4336,5119} et al

    X(18425) = {X(18422),X(18423)}-harmonic conjugate of X(7)


    X(18426) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND REFLECTIONS-OF-X(1)

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

    X(18426) lies on these lines: {46,1749}, {109,3585}, {15430,18514} et al


    X(18427) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND REFLECTIONS-OF-X(2)

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

    X(18427) lies on these lines: {2,18382}, {110,7533}, {265,5946}, {381,5898}, {3060,3410}, {5169,12310}, {5189,7703} et al


    X(18428) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND REFLECTIONS-OF-X(5)

    Barycentrics    a^16 - 2a^14(b^2 + c^2) - a^12(2b^4 - 3b^2c^2 + 2c^4) + a^10(b^2 + c^2)(6b^4 - 7b^2c^2 + 6c^4) - a^8(3b^6c^2 - 8b^4c^4 + 3b^2c^6) - a^6(6b^10 - 11b^8c^2 + b^6c^4 + b^4c^6 - 11b^2c^8 + 6c^10) + 2a^4(b^12 - 4b^10c^2 + 6b^6c^6 - 4b^2c^10 + c^12) + 2a^2(b^2 - c^2)^4(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) - (b^2 - c^2)^6(b^2 + c^2)^2 : :

    X(18428) lies on these lines: {4,14860}, {52,6288}, {110,3574}, {143,9927}, {182,18382}, {265,389}, {3818,5876}, {13491,18381} et al


    X(18429) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND REFLECTIONS-OF-X(6)

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

    X(18429) lies on these lines: {6,18376}, {381,3053}, {1989,3845}, {3172,18386} et al


    X(18430) = HOMOTHETIC CENTER EHRMANN VERTEX-TRIANGLE AND CROSS-TRIANGLE OF EHRMANN VERTEX- AND EHRMANN SIDE-TRIANGLES

    Barycentrics    2a^10 - 3a^8(b^2 + c^2) - a^6(b^4 - 5b^2c^2 + c^4) + a^4(b^6 - 2b^4c^2 - 2b^2c^4 + c^6) + 3a^2(b^2 - c^2)^2(b^4 + c^4) - 2(b^2 - c^2)^4(b^2 + c^2) : :
    X(18430) = 2 X(5) - X(11464)

    X(18430) lies on these lines: {3,18383}, {4,94}, {5,10546}, {6,3843}, {30,11454}, {49,7547}, {184,381}, {382,3581}, {1352,6288}, {3521,11457}, {6243,9927}, {11442,18568}, {13449,18321}, {18376,18403}, {18377,18436}, {18440,18449} et al

    X(18430) = reflection of X(11464) in X(5)
    X(18430) = X(7951)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18430) = X(11464)-of-Johnson-triangle


    X(18431) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN VERTEX AND AOA

    Barycentrics    a^16 - 2a^12(5b^4 - 8b^2c^2 + 5c^4) + 2a^10(b^2 + c^2)(8b^4 - 15b^2c^2 + 8c^4) - 8a^8b^2c^2(7b^6 - 15b^2c^2 + 7c^6) - 8a^6(b^2 - c^2)^2(b^2 + c^2)(2b^4 - 9b^2c^2 + 2c^4) + 2a^4(b^2 - c^2)^2(5b^8 - 4b^6c^2 - 38b^4c^4 - 4b^2c^6 + 5c^8) - 10a^2b^2c^2(b^2 - c^2)^4(b^2 + c^2) - (b^2 - c^2)^6(b^2 + c^2)^2 : :

    X(18431) lies on these lines: {4,10293}, {5,64}, {30,8548}, {541,3357}, {5894,12084}, {6000,18553}, {6288,12324}, {10606,15122}, {14915,18381}, {15311,15579}


    X(18432) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN VERTEX AND AAOA

    Barycentrics    a^16 - 3a^14(b^2 + c^2) + a^12(2b^2 + c^2)(b^2 + 2c^2) + a^10(b^6 - 2b^4c^2 - 2b^2c^4 + c^6) + 5a^8b^4c^4 - a^6(b^10 - 3b^8c^2 + b^6c^4 + b^4c^6 - 3b^2c^8 + c^10) - a^4(2b^12 + b^10c^2 - 6b^6c^6 + b^2c^10 + 2c^12) + 3a^2(b^2 - c^2)^4(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) - (b^2 - c^2)^6(b^2 + c^2)^2 : :

    X(18432) lies on these lines: {265,576}, {569,6145}, {1147,15133}, {2888,18569}, {2904,7507}, {6243,9927}, {7574,11649}


    X(18433) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN VERTEX AND 1st HYACINTH

    Barycentrics    a^22 - 4a^20(b^2 + c^2) + a^18(4b^4 + 21b^2c^2 + 4c^4) + a^16(b^2 + c^2)(3b^4 - 35b^2c^2 + 3c^4) - a^14(6b^8 + 5b^6c^2 - 78b^4c^4 + 5b^2c^6 + 6c^8) + a^12b^2c^2(b^2 + c^2)(47b^4 - 92b^2c^2 + 47c^4) - a^10b^2c^2(11b^8 + 88b^6c^2 - 142b^4c^4 + 88b^2c^6 + 11c^8) + a^8(b^2 + c^2)(6b^12 - 47b^10c^2 + 174b^8c^4 - 258b^6c^6 + 174b^4c^8 - 47b^2c^10 + 6c^12) - a^6(b^2 - c^2)^2(3b^12 - 11b^10c^2 - 19b^8c^4 + 70b^6c^6 - 19b^4c^8 - 11b^2c^10 + 3c^12) - a^4(b^2 - c^2)^4(b^2 + c^2)(4b^8 - 13b^6c^2 + 28b^4c^4 - 13b^2c^6 + 4c^8) + 2a^2(b^2 - c^2)^6(b^2 + c^2)^2(2b^4 - 3b^2c^2 + 2c^4) - (b^2 - c^2)^8(b^2 + c^2)^3 : :

    X(18433) lies on these lines: {54,15760}, {265,1885}, {5449,7526}, {12370,13198}


    X(18434) = ISOGONAL CONJUGATE OF X(10298)

    Barycentrics    1/[2a^8 - 4a^6(b^2 + c^2 + 5a^4b^2c^2) + a^2(4b^6 - 2b^4c^2 - 2b^2c^4 + 4c^6) - (b^2 - c^2)^2(2b^4 + 3b^2c^2 + 2c^4)] : :

    Let VAVBVC be the Ehrmann vertex-triangle. Let A' be the center of conic {{A,B,C,VB,VC}} and define B', C' cyclically. The lines AA', BB', CC' concur in X(18434).

    X(18434) lies on the Jerabek hyperbola and these lines: {3,18383}, {6,13851}, {54,7547}, {64,11572}, {68,18377}, {69,3153}, {74,1853}, {265,18376}, {3521,18381}, {14852,18379} et al

    X(18434) = X(17057)-of-Ehrmann-vertex-triangle is ABC is acute


    X(18435) = X(2)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    a^2[a^6(b^2 + c^2) - a^4(3b^4 - b^2c^2 + 3c^4) + 3a^2(b^6 + c^6) - (b^2 - c^2)^2(b^4 + 4b^2c^2 + c^4)] : :
    X(18435) = X(2) - 2 X(15060) = X(3) + 2 X(4) - 2 X(51) = X(3) - 2 X(5891) = X(381) - 2 X(15030) = 2 X(381) - X(568) = 2 X(5) - X(5890)

    X(18435) lies on these lines: {2,5655}, {3,64}, {4,93}, {5,5890}, {25,3581}, {30,2979}, {51,381}, {110,18570}, {140,6241}, {389,3851}, {547,15045}, {548,7999}, {550,11444}, {631,13491}, {1971,18472}, {2393,18438}, {11190,18453}, {11241,18457}, {11242,18459}, {18376,18403} et al

    X(18435) = reflection of X(i) in X(j) for these (i,j): (2,15060), (3,5891), (381,15030), (568,381), (5890,5)
    X(18435) = X(2)-of-Ehrmann-side-triangle
    X(18435) = X(5890)-of-Johnson-triangle
    X(18435) = X(5891)-of-X3-ABC-reflections-triangle


    X(18436) = X(4)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    a^2[a^4(b^2 + c^2) - a^2(2b^4 - b^2c^2 + 2c^4) + (b^2 - c^2)^2(b^2 + c^2)] : :
    X(18436) = X(3) + 2 X(4) - 2 X(52) = X(3) - 2 X(5562) = X(4) - 2 X(5876) = 2 X(4) - X(6243) = 2 X(5) - X(5889) = X(20) - 2 X(6101) = X(52) - 2 X(5907) = X(382) - 2 X(12162)

    X(18436) lies on these lines: {2,6102}, {3,49}, {4,93}, {5,568}, {20,5663}, {24,3581}, {26,10540}, {30,11412}, {52,381}, {54,14805}, {68,265}, {69,3521}, {110,1658}, {140,5890}, {143,3091}, {156,7488}, {195,578}, {323,3520}, {382,511}, {389,1656}, {399,2917}, {539,18564}, {542,11750}, {546,3060}, {547,15024}, {548,15072}, {549,7999}, {550,2979}, {1657,5925}, {1986,7505}, {1993,7526}, {2070,10539}, {2072,12359}, {2781,5878}, {2807,12702}, {3090,5946}, {3534,10575}, {3545,10095}, {3548,12358}, {3564,12605}, {3627,15305}, {3819,13382}, {5076,13598}, {5964,9927}, {6237,18453}, {7352,18447}, {7517,17834}, {10665,18457}, {10666,18459}, {11442,18569}, {11449,15331}, {11465,15699}, {11592,15692}, {12118,18442}, {12270,15332}, {12279,15704}, {14585,18472}, {15033,15801}, {15133,18441}, {15606,15696}, {18377,18430}, {18392,18567}, {18394,18572} et al

    X(18436) = reflection of X(i) in X(j) for these (i,j): (3,5562), (4,5876), (20,6101), (52,5907), (382,12162), (5889,5), (6243,4)
    X(18436) = anticomplement of X(6102)
    X(18436) = Cundy-Parry Phi transform of X(49)
    X(18436) = Cundy-Parry Psi transform of X(93)
    X(18436) = X(4)-of-Ehrmann-side-triangle
    X(18436) = X(5889)-of-Johnson-triangle
    X(18436) = X(5562)-of-X3-ABC-reflections-triangle
    X(18436) = X(5876)-of-anti-Euler-triangle
    X(18436) = orthic-to-circumorthic similarity image of X(5876)


    X(18437) = X(6)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    [a^10 + a^8(b^2 + c^2) - a^6(4b^4 - 2b^2c^2 + 4c^4) + 2a^4(b^2 - c^2)^2(b^2 + c^2) - a^2(b^2 - c^2)^2(b^4 + c^4) + (b^2 - c^2)^4(b^2 + c^2)]/(b^2 + c^2 - a^2) : :

    X(18437) lies on these lines: {3,66}, {4,3164}, {30,317}, {53,381}, {147,14941}, {216,3818}, {418,11442}, {542,577}, {800,18390}, {1899,6638}, {2871,18438}, {17849,18451}, {18380,18403} et al

    X(18437) = X(6)-of-Ehrmann-side-triangle
    X(18437) = perspector of Johnson circle wrt Ehrmann side-triangle


    X(18438) = X(7)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    a^2(b^2 + c^2 - a^2)[a^6(b^2 + c^2) - a^4(b^4 + b^2c^2 + c^4) - a^2(b^6 + c^6) + (b^2 - c^2)^2(b^4 + c^4)] : :
    Barycentrics    SA(SB + SC)[SA^3 (SB + SC) + 2 SA^2 (SB^2 + 4 SB SC + SC^2) + SA (SB + SC) (SB^2 + 3 SB SC + SC^2) + SB SC (SB - SC)^2] : :
    X(18438) = (3*R^2-SW)*X(3) - (4*R^2-SW)*X(6) = X(3) - 4 X(6) + 2 X(52) = X(3) - 2 X(9967) = X(382) - 2 X(12294) = 2 X(5) - X(6403)

    X(18438) lies on these lines: {3,6}, {5,6403}, {30,12220}, {67,18441}, {69,265}, {141,2072}, {159,10540}, {193,1154}, {206,11597}, {376,1986}, {381,1843}, {382,12294}, {542,18564}, {1112,7493}, {1352,6288}, {1353,5889}, {1469,18447}, {1503,18439}, {2393,18435}, {2871,18437}, {3056,18455}, {3091,11576}, {3564,12605}, {3779,18453}, {5562,11898}, {5907,18405}, {6776,15074}, {7502,12228}, {8705,18323}, {9924,18451}, {10297,11188}, {11645,18561}, {12590,18462}, {12591,18463} et al

    X(18438) = reflection of X(i) in X(j) for these (i,j): (3,9967), (382,12294), (6403,5)
    X(18438) = X(7)-of-Ehrmann-side-triangle if ABC is acute
    X(18438) = X(6403)-of-Johnson-triangle
    X(18438) = X(9967)-of-X3-ABC-reflections-triangle


    X(18439) = X(8)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    a^2[a^6(b^2 + c^2) - 3a^4(b^4 - b^2c^2 + c^4) + a^2(3b^6 - 2b^4c^2 - 2b^2c^4 + 3c^6) - (b^2 - c^2)^2(b^4 + 4b^2c^2 + c^4)] : :
    X(18439) = X(3)+6 X(4)-6 X(51) = 11*X(3)-12*X(3819), 5*X(3)-6*X(5891), 3*X(3)-4*X(5907), 3*X(3)-2*X(10575), 7*X(3)-8*X(11793), 7*X(3)-6*X(14855), 2*X(3)-3*X(18435), 10*X(3819)-11*X(5891), 9*X(3819)-11*X(5907), 18*X(3819)-11*X(10575), 6*X(3819)-11*X(12162), 14*X(3819)-11*X(14855), 8*X(3819)-11*X(18435), 9*X(5891)-10*X(5907), 9*X(5891)-5*X(10575), 21*X(5891)-20*X(11793), 3*X(5891)-5*X(12162), 7*X(5891)-5*X(14855), 4*X(5891)-5*X(18435), 7*X(5907)-6*X(11793), 2*X(5907)-3*X(12162), 14*X(5907)-9*X(14855), 8*X(5907)-9*X(18435), 7*X(10575)-12*X(11793), X(10575)-3*X(12162), 7*X(10575)-9*X(14855), 4*X(10575)-9*X(18435), 4*X(11793)-7*X(12162), 4*X(11793)-3*X(14855), 7*X(12162)-3*X(14855), 4*X(12162)-3*X(18435), 4*X(14855)-7*X(18435)

    X(18439) lies on these lines: {2, 13491}, {3, 64}, {4, 94}, {5, 6241}, {20, 5876}, {30, 11412}, {49, 378}, {52, 3830}, {70, 3521}, {110, 11250}, {113, 7729}, {140, 15058}, {156, 3520}, {184, 14130}, {185, 381}, {376, 11591}, {382, 6243}, {389, 3843}, {399, 1147}, {511, 5073}, {546, 5890}, {548, 11444}, {549, 15056}, {550, 11459}, {567, 1181}, {631, 15060}, {1154, 3146}, {1204, 10620}, {1216, 3534}, {1503, 18438}, {1593, 18445}, {1614, 15062}, {1656, 15030}, {1657, 5562}, {1658, 11440}, {2070, 7689}, {2072, 6247}, {2777, 18565}, {2807, 18525}, {2883, 10024}, {2979, 15704}, {3060, 3853}, {3091, 13630}, {3426, 12164}, {3522, 15067}, {3523, 14128}, {3529, 6101}, {3543, 10263}, {3545, 12006}, {3549, 5656}, {3567, 3845}, {3581, 7517}, {3627, 5889}, {3832, 5946}, {3839, 10095}, {3850, 15043}, {3851, 9730}, {3855, 15026}, {3858, 5640}, {3861, 9781}, {3917, 14641}, {4550, 10984}, {5055, 9729}, {5066, 15024}, {5068, 13363}, {5071, 11017}, {5076, 5446}, {5079, 5892}, {5448, 15063}, {5655, 18281}, {5787, 18330}, {5878, 6288}, {6254, 18453}, {6285, 18455}, {6293, 18474}, {6640, 14643}, {6699, 17856}, {7355, 18447}, {7395, 13339}, {7488, 12112}, {7506, 10605}, {7525, 8718}, {7526, 11456}, {7540, 16621}, {7999, 8703}, {8549, 18449}, {9704, 11430}, {9818, 12174}, {10110, 14269}, {10170, 15720}, {10224, 12270}, {10226, 11449}, {10625, 17800}, {11413, 15068}, {11424, 15087}, {11441, 12084}, {11451, 12811}, {11454, 15331}, {11468, 15646}, {11750, 18564}, {11799, 12359}, {11819, 16658}, {12106, 15054}, {12121, 12825}, {12134, 15311}, {12308, 13352}, {12324, 18531}, {13561, 16868}, {14216, 18404}, {14845, 15012}, {15061, 17854}, {15462, 15579}, {15463, 16867}, {15644, 15681}, {18381, 18403}, {18388, 18488}, {18400, 18562}

    X(18439) = reflection of X(i) in X(j) for these (i,j): (3, 12162), (20, 5876), (52, 13474), (382, 11381), (1657, 5562), (3529, 6101), (12121, 12825)
    X(18439) = anticomplement of X(13491)
    X(18439) = X(8)-of-Ehrmann-side-triangle if ABC is acute
    X(18439) = X(20)-of-OAOBOC, as described at X(7666)
    X(18439) = X(6241)-of-Johnson-triangle
    X(18439) = X(12162)-of-X3-ABC-reflections-triangle
    X(18439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 12162, 18435), (3, 18451, 18350), (52, 13474, 3830), (64, 18451, 3), (389, 16194, 3843), (1539, 18379, 4), (1614, 15062, 18570), (3357, 10539, 3), (5890, 11439, 546), (5907, 10575, 3), (6241, 15305, 5), (10575, 12162, 5907), (11459, 12279, 550), (11793, 14855, 3), (15058, 15072, 140)


    X(18440) = X(9)-OF-EHRMANN-SIDE-TRIANGLE

    Barycentrics    3a^6 - 2a^4(b^2 + c^2) + a^2(b^2 + c^2)^2 - 2(b^2 - c^2)^2(b^2 + c^2) : :
    X(18440) = X(3) + 2 X(4) - 2 X(6) = X(3) - 2 X(1352) = 2 X(4) - X(1351) = X(4) + X(5921) = 2 X(5) - X(6776) = X(6) - 2 X(3818) = 2 X(67) - X(10620) = X(382) + X(11898)

    X(18440) lies on these lines: {2,8780}, {3,66}, {4,193}, {5,3618}, {6,13}, {22,3410}, {25,3580}, {30,69}, {67,10620}, {68,1598}, {110,5094}, {114,9756}, {147,7777}, {182,1656}, {206,10540}, {317,16264}, {343,9909}, {376,3620}, {378,12168}, {382,511}, {394,11550}, {427,3167}, {428,6515}, {518,18345}, {524,3830}, {546,1353}, {550,10519}, {674,18499}, {1350,1657}, {1386,18493}, {1843,13754}, {2393,18435}, {3091,7920}, {3094,18503}, {3242,18526}, {3751,18480}, {5039,18502}, {5076,5965}, {5891,11574}, {6467,15030}, {12212,18501}, {12329,18524}, {12583,18508}, {12586,18519}, {12587,18518}, {12594,18545}, {12595,18543}, {13562,18531}, {13861,18356}, {15141,18441}, {18382,18403}, {18430,18449} et al

    X(18440) = midpoint of X(i) and X(j) for these {i,j}: {4,5921}, {382,11898}
    X(18440) = reflection of X(i) in X(j) for these (i,j): (3,1352), (6,3818), (1351,4), (6776,5), (10620,67)
    X(18440) = X(9)-of-Ehrmann-side-triangle if ABC is acute
    X(18440) = X(6776)-of-Johnson-triangle
    X(18440) = X(5921)-of-Euler-triangle
    X(18440) = X(1352)-of-X3-ABC-reflections-triangle


    X(18441) = PERSPECTOR OF THESE TRIANGLES: EHRMANN SIDE AND AAOA

    Barycentrics    (b^2 + c^2 - a^2)[a^26 - 4a^24(b^2 + c^2) + 3a^22(b^4 + 5b^2c^2 + c^4) + a^20(b^2 + c^2)(7b^4 - 19b^2c^2 + 7c^4) - 2a^18(5b^8 + 11b^6c^2 - 10b^4c^4 + 11b^2c^6 + 5c^8) - a^16(3b^10 - 37b^8c^2 - 37b^2c^8 + 3c^10) + 2a^14(3b^12 + 7b^10c^2 - 29b^8c^4 + 26b^6c^6 - 29b^4c^8 + 7b^2c^10 + 3c^12) + 2a^12(b^2 + c^2)(3b^12 - 31b^10c^2 + 62b^8c^4 - 70b^6c^6 + 62b^4c^8 - 31b^2c^10 + 3c^12) - a^10(3b^16 - 4b^14c^2 - 56b^12c^4 + 128b^10c^6 - 146b^8c^8 + 128b^6c^10 - 56b^4c^12 - 4b^2c^14 + 3c^16) - 2a^8(b^2 - c^2)^2(b^2 + c^2)(5b^12 - 26b^10c^2 + 48b^8c^4 - 56b^6c^6 + 48b^4c^8 - 26b^2c^10 + 5c^12) + a^6(b^2 - c^2)^4(7b^12 - 9b^10c^2 - 31b^8c^4 + 6b^6c^6 - 31b^4c^8 - 9b^2c^10 + 7c^12) + a^4(b^2 - c^2)^6(b^2 + c^2)(3b^8 - 5b^6c^2 + 18b^4c^4 - 5b^2c^6 + 3c^8) - 2a^2(b^2 - c^2)^8(b^2 + c^2)^2(2b^4 - b^2c^2 + 2c^4) + (b^2 - c^2)^10(b^2 + c^2)^3] : :

    X(18441) lies on these lines: {67,18438}, {265,15136}, {15133,18436}, {15141,18440}


    X(18442) = PERSPECTOR OF THESE TRIANGLES: EHRMANN SIDE AND ABC-X3 REFLECTIONS

    Barycentrics    (b^2 + c^2 - a^2)[3a^8 - a^6(b^2 + c^2) - a^4(6b^4 - 7b^2c^2 + 6c^4) + 3a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4] : :
    X(18442) = 2 X(3) - X(3521)

    X(18442) lies on these lines: {3,1568}, {20,5876}, {30,6288}, {110,550}, {265,12359}, {548,6030}, {1350,1657}, {6776,15074}, {9927,18561}, {12118,18436} et al

    X(18442) = reflection of X(3521) in X(3)
    X(18442) = X(1389)-of-Ehrmann-side-triangle
    X(18442) = X(3521)-of-ABC-X3-reflections-triangle


    X(18443) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ANTI-ASCELLA

    Trilinears    a^6 - 2a^5(b + c) - a^4(b + c)^2 + 4a^3(b^3 + c^3) - a^2(b^4 - 4b^3c - 2b^2c^2 - 4bc^3 + c^4) - 2a(b - c)(b^4 - c^4) + (b - c)^4(b + c)^2 : :

    The homothetic center of these triangles is X(9818).

    X(18443) lies on these lines: {1,3}, {2,5720}, {4,5249}, {5,1490}, {7,6987}, {9,912}, {10,5534}, {28,1790}, {30,5732}, {34,4303}, {63,1006}, {77,1119}, {78,631}, {84,3560}, {104,9946}, {990,18506}, {997,5745}, {1001,3358}, {1012,7171} et al

    X(18443) = X(11818)-of-excentral-triangle


    X(18444) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 1st ANTI-CONWAY

    Trilinears    a^6 - 2a^5(b + c) - a^4(b^2 + bc + c^2) + 4a^3(b^3 + c^3) - a^2(b^4 - 2b^3c - 2b^2c^2 - 2bc^3 + c^4) - 2a(b^5 - b^4c - bc^4 + c^5) + (b^2 - c^2)^2(b^2 - bc + c^2) : :

    The homothetic center of these triangles is X(567).

    X(18444) lies on these lines: {1,7}, {2,5720}, {3,3218}, {8,224}, {10,5531}, {21,104}, {35,11570}, {36,18389}, {40,11520}, {63,3576}, {72,6986}, {78,3523}, {84,2320}, {200,5775}, {226,6840}, {329,6992}, {355,4197}, {377,944}, {404,9940}, {405,5779}, {411,942}, {412,6198}, {515,5249}, {517,3957}, {572,5279}, {581,5262}, {631,5770}, {758,15931}, {912,1006}, {936,10303}, {938,6838}, {946,16132}, {971,6912}, {997,5273}, {999,11020}, {1001,12669}, {1012,10246}, {2094,6282}, {2801,5251}, {2894, 4861}, {2975,12675}, {3487,6836}, {3616,6261}, {5248,15071}, {5260,14872}, {5761,6899}, {5787,6828}, {5884,10902}, {5905,6987}, {7580,15934}, {11012,12005}, {11888,18448}, {11889,18456}, {11890,18454} et al


    X(18445) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd ANTI-EXTOUCH

    Barycentrics    a^2*(a^2-b^2-c^2)*(a^6-3*(b^2+c^2)*a^4+(3*b^4-4*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)) : :
    X(18445) = 3*X(3)-4*X(18475), 3*X(184)-2*X(18475), 3*X(381)-4*X(18388), 3*X(381)-2*X(18474), 3*X(5093)-2*X(8541), 3*X(6800)-2*X(7502), 2*X(12827)-3*X(14643)

    X(18445) lies on these lines: {2, 15032}, {3, 49}, {4, 1994}, {5, 5422}, {6, 13}, {20, 16266}, {22, 1154}, {24, 156}, {25, 568}, {26, 1614}, {30, 1993}, {52, 161}, {54, 7526}, {64, 16867}, {68, 10024}, {110, 5890}, {143, 10594}, {146, 2914}, {154, 2070}, {182, 5891}, {186, 9544}, {195, 382}, {235, 13292}, {323, 376}, {378, 5663}, {389, 7506}, {511, 12083}, {549, 15066}, {567, 9818}, {569, 5907}, {578, 12162}, {1069, 18447}, {1199, 3091}, {1351, 2393}, {1353, 1596}, {1495, 14831}, {1511, 15078}, {1593, 18439}, {1656, 15038}, {1658, 9707}, {1899, 2072}, {1995, 5609}, {2192, 9642}, {2875, 10679}, {2937, 17834}, {3047, 7722}, {3060, 7530}, {3095, 14917}, {3157, 18455}, {3448, 7577}, {3520, 9545}, {3521, 15316}, {3532, 16665}, {3543, 11004}, {3545, 15052}, {3549, 11411}, {3564, 15760}, {3567, 13861}, {3580, 10201}, {3581, 14070}, {3843, 10982}, {4846, 12364}, {5012, 7514}, {5054, 17811}, {5055, 10601}, {5070, 15805}, {5071, 15018}, {5076, 15811}, {5079, 15047}, {5093, 8541}, {5448, 10116}, {5651, 5892}, {5876, 7503}, {5986, 13860}, {6000, 13352}, {6101, 10323}, {6146, 18404}, {6241, 12084}, {6243, 7387}, {6639, 12359}, {6640, 9820}, {6642, 18350}, {6776, 18531}, {6800, 7502}, {7395, 13353}, {7464, 9716}, {7484, 13339}, {7485, 15067}, {7509, 11591}, {7516, 11444}, {7529, 11432}, {7545, 13321}, {7723, 13198}, {9306, 9730}, {9705, 11449}, {9706, 11440}, {9714, 14530}, {9955, 16472}, {10254, 14852}, {10575, 13346}, {10602, 18449}, {10606, 10620}, {11413, 13491}, {11422, 14094}, {11423, 13434}, {11425, 14130}, {11457, 13371}, {11464, 18324}, {11472, 12308}, {11477, 11649}, {11597, 12168}, {11793, 13336}, {12082, 13391}, {12085, 12174}, {12134, 12233}, {12165, 12412}, {12228, 12825}, {12291, 15801}, {12827, 14643}, {13366, 15030}, {13630, 17928}, {14480, 15111}, {14528, 18364}, {14912, 18537}, {16473, 18480}, {18386, 18430}, {18396, 18403}

    X(18445) = reflection of X(i) in X(j) for these (i,j): (3,184)
    X(18445) = X(18446)-of-Ehrmann-side-triangle
    X(18445) = X(18446)-of-2nd-anti-extouch-triangle
    X(18445) = X(184)-of-X3-ABC-reflections-triangle
    X(18445) = X(11442)-of-Johnson-triangle
    X(18445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 12164, 18436), (6, 18451, 381), (113, 10111, 265), (113, 18390, 381), (155, 1181, 3), (156, 6102, 24), (185, 1147, 3), (381, 399, 18451), (381, 15087, 6), (399, 15087, 381), (1204, 12038, 3), (1216, 10984, 3), (7592, 11441, 5), (7689, 13367, 3), (18388, 18474, 381)


    X(18446) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd ANTI-EXTOUCH

    Trilinears    sec B + sec C - 1 : :
    Trilinears    (b^2 + c^2 - a^2)[a^4 - 2 a^3(b + c) + 2a(b - c)^2(b + c) - (b^2 - c^2)^2] : :
    X(18446) = (r + 2R)*X(1) - 3R*X(2) + 2R*X(3) = 2 X(3) - X(63) = X(4) - 2 X(226) = X(20) + X(5905)

    The homothetic center of these triangles is X(18445).

    X(18446) lies on these lines: {1,4}, {2,5720}, {3,63}, {8,5534}, {9,48}, {10,6889}, {20,5758}, {21,7330}, {35,1158}, {36,1708}, {37,5776}, {40,758}, {46,5884}, {55,6001}, {56,12675}, {57,6905}, {65,11500}, {77,1060}, {84,943}, {100,3359}, {142,6854}, {198,9119}, {200,5657}, {207,1148}, {218,7124}, {284,4227}, {329,4511}, {355,442}, {376,527}, {392,13615}, {405,1385}, {411,3868}, {452,5811}, {474,9940}, {498,12616}, {499,10395}, {500,13442}, {517,3870}, {518,3428}, {519,2900}, {549,13226}, {1389,5665}, {1512,18391}, {1538,18527}, {3295,12672}, {3358,12669}, {5881,6937}, {5927,6913}, {6843,18528}, {6912,18540}, {6983,9843}, {7593,18454}, {8079,18448}, {8080,9837}, {12848,18450} et al

    X(18446) = midpoint of X(20) and X(5905)
    X(18446) = reflection of X(i) in X(j) for these (i,j): (4,226), (63,3)
    X(18446) = X(184)-of-hexyl-triangle
    X(18446) = X(226)-of-anti-Euler-triangle
    X(18446) = X(18474)-of-excentral-triangle
    X(18446) = Ehrmann-mid-to-ABC similarity image of X(18388)
    X(18446) = hexyl-isogonal conjugate of X(6210)
    X(18446) = Cundy-Parry Phi transform of X(78)
    X(18446) = Cundy-Parry Psi transform of X(34)
    X(18446) = trilinear product X(63)*X(8557)


    X(18447) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ANTI-TANGENTIAL MIDARC

    Trilinears    (b^2 + c^2 - a^2)(a^4 - b^4 - c^4 + a^2bc + 2b^2c^2) : :

    X(18447) lies on these lines: {1,3}, {5,1870}, {11,10024}, {12,2072}, {20,8144}, {33,382}, {34,381}, {73,265}, {172,10317}, {221,18451}, {227,18524}, {603,18477}, {1442,5453}, {1469,18438}, {3585,18403}, {7352,18436}, {7355,18439} et al

    X(18447) = X(18448)-of-Ehrmann-side-triangle if ABC is acute
    X(18447) = X(18448)-of-anti-tangential-midarc-triangle if ABC is acute
    X(18447) = trilinear product X(63)*X(5341)


    X(18448) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ANTI-TANGENTIAL MIDARC

    Trilinears    (a - b - c) (a - b + c) (a + b - c) (a + b + c) (b - c) + 2 (b - c) (a^4 - 2 a^3 (b + c) + 4 a^2 b c + 2 a (b^3 + c^3) - (b + c)^2 (b^2 + c^2)) Sin[A/2] + 2 (a - b + c) (a^4 - a^3 (b + c) - a^2 b (b - 2 c) + a (b^3 + b^2 c - 3 b c^2 + c^3) + c^2 (b^2 - c^2)) Sin[B/2] - 2 (a + b - c) (a^4 - a^3 (b + c) + a^2 c (2 b - c) + a (b^3 - 3 b^2 c + b c^2 + c^3) - b^2 (b^2 - c^2)) Sin[C/2] : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(18447).

    X(18448) lies on these lines: {1,167}, {3,8093}, {40,11534}, {104,12771}, {188,997}, {355,8087}, {517,8075}, {999,11032}, {1319,10503}, {8079,18446}, {11888,18444} et al


    X(18449) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd EHRMANN

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

    X(18449) lies on these lines: {3,6}, {4,11255}, {5,8537}, {30,11416}, {265,895}, {1503,7728}, {1986,7464}, {8539,18453}, {8549,18439}, {17813,18451}, {18430,18440} et al

    X(18449) = X(18450)-of-Ehrmann-side-triangle if ABC is acute
    X(18449) = X(18450)-of-2nd-Ehrmann-triangle if ABC is acute


    X(18450) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd EHRMANN

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

    The homothetic center of these triangles is X(18449).

    X(18450) lies on these lines: {1,7}, {9,1055}, {36,2801}, {56,10394}, {100,518}, {104,971}, {517,14151}, {527,4511}, {663,6006}, {997,6172}, {999,7671}, {1319,15726}, {12848,18446} et al


    X(18451) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 1st EXCOSINE

    Barycentrics    a^2[a^8 - 4a^6(b^2 + c^2) + 6a^4(b^4 + c^4) - 4a^2(b^6 + c^6) + (b^2 - c^2)^2(b^4 + 6b^2c^2 + c^4)] : :
    X(18451) = X(3) + X(4) - X(1899) = X(3) - X(25) - X(394) = X(3) - 2 X(9306) = 2 X(5) - X(1899)

    X(18451) lies on these lines: {2,11456}, {3,64}, {4,155}, {5,1181}, {6,13}, {22,11459}, {24,12111}, {25,13754}, {26,5876}, {30,394}, {33,3157}, {34,1069}, {49,11425}, {52,1598}, {68,235}, {70,13160}, {74,15078}, {110,378}, {156,7526}, {184,9818}, {185,6642}, {221,18447}, {323,3543}, {376,12112}, {382,13419}, {389,7529}, {403,11442}, {427,5654}, {511,18534}, {541,15106}, {546,10982}, {567,17809}, {568,17810}, {569,11479}, {1092,11381}, {1351,8681}, {1503,18531}, {2003,18540}, {2192,18455}, {3197,18453}, {4550,18475}, {5055,17825}, {6288,17824}, {6776,18537}, {7517,17834}, {7728,17847}, {9924,18438}, {12429,17836}, {17813,18449}, {17819,18457}, {17826,18468}, {17827,18470}, {17845,18563}, {17849,18437}, {18403,18405} et al

    X(18451) = reflection of X(i) in X(j) for these (i,j): (3,9306), (1899,5)
    X(18451) = X(18452)-of-Ehrmann-side-triangle
    X(18451) = X(18452)-of-1st-excosine-triangle
    X(18451) = X(1899)-of-Johnson-triangle


    X(18452) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 1st EXCOSINE

    Trilinears    a^6 + 2a^5(b + c) - a^4(b - 5c)(5b - c) - 4a^3(b + c)(b^2 + 5bc + c^2) + a^2(7b^4 - 16b^3c + 50b^2c^2 - 16bc^3 + 7c^4) + 2a(b - c)^2(b + c)(b^2 + 12bc + c^2) - (b^2 - c^2)^2(3b + c)(b + 3c) : :

    The homothetic center of these triangles is X(18451).

    X(18452) lies on these lines: {1,1864}, {20,519}, {145,7995}, {200,2975}, {952,2093}, {1320,2801}, {10980,18391} et al


    X(18453) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND EXTANGENTS

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

    X(18453) lies on these lines: {1,3}, {4,8141}, {5,6197}, {19,381}, {30,3101}, {71,265}, {382,11471}, {567,11428}, {568,11435}, {2550,18531}, {3197,18451}, {3198,18524}, {3779,18438}, {5415,18457}, {5416,18459}, {6237,18436}, {6253,18563}, {6254,18439}, {8539,18449}, {10636,18468}, {10637,18470}, {10988,18472}, {11190,18435}, {18403,18406} et al

    X(18453) = X(18454)-of-Ehrmann-side-triangle
    X(18453) = X(18454)-of-extangents-triangle


    X(18454) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND EXTANGENTS

    Trilinears    a^3 + b^3 + c^3 - a^2b - ab^2 - a^2c - ac^2 - b^2c - bc^2 + 2bc(b + c - a)*sin(A/2) : :

    The homothetic center of these triangles is X(18453).

    X(18454) lies on these lines: {1,167}, {3,12445}, {40,11535}, {104,7707}, {173,3576}, {236,997}, {258,11529}, {355,8382}, {517,7589}, {758,1130}, {942,7588}, {999,8083}, {1319,10502}, {7593,18446}, {11890,18444} et al

    X(18454) = {X(1),X(174)}-harmonic conjugate of X(18456)


    X(18455) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND INTANGENTS

    Trilinears    a^6 - a^4(b^2 + bc + c^2) - a^2(b^4 - b^3c - 2b^2c^2 - bc^3 + c^4) + b^6 - b^4c^2 - b^2c^4 + c^6 : :

    X(18455) lies on these lines: {1,3}, {4,8144}, {21,2906}, {212,18477}, {339,350}, {1398,12085}, {2192,18451}, {3056,18438}, {6285,18439}, {10987,18472} et al

    X(18455) = {X(9627),X(9630)}-harmonic conjugate of X(1)
    X(18455) = X(18456)-of-Ehrmann-side-triangle
    X(18455) = X(18456)-of-intangents-triangle


    X(18456) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND INTANGENTS

    Trilinears    a^3 + b^3 + c^3 - a^2b - ab^2 - a^2c - ac^2 - b^2c - bc^2 + 2bc(a + b + c)*sin(A/2) : :

    The homothetic center of these triangles is X(18455).

    X(18456) lies on these lines: {1,167}, {3,8094}, {40,11899}, {104,12772}, {173,11529}, {214,10231}, {258,3576}, {355,8088}, {517,8076}, {942,7587}, {997,7028}, {999,11033}, {1319,10501}, {8080,9837}, {11889,18444} et al

    X(18456) = {X(1),X(174)}-harmonic conjugate of X(18454)


    X(18457) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 1st KENMOTU DIAGONALS

    Trilinears    (cos A)[2 - cos 2A + 2 cos 2B + 2 cos 2C + 2 cos(2B - 2C) + cos(2A - 2B) + cos(2A - 2C) + cos 4A + cos(2B + 2C) - cos(2A + 2B) - cos(2A + 2C) + 2 sin 2A + sin 2B + sin 2C + sin(2A - 2B) + sin(2A - 2C) + sin(4A + 4B) + sin(4A + 4C)] : :

    X(18457) lies on these lines: {3,6}, {4,11265}, {5,10880}, {30,11417}, {49,10666}, {265,6413}, {381,5412}, {382,11473}, {485,18404}, {486,6639}, {590,2072}, {1658,10881}, {2070,5413}, {3092,7517}, {5055,10961}, {5415,18453}, {5418,6640}, {6564,18403}, {6565,10254}, {7488,11266}, {7502,11418}, {7728,13287}, {10255,10576}, {10533,10540}, {10665,18436}, {10962,11597}, {11241,18435}, {11447,11459}, {11448,11464}, {11462,12111}, {11474,14130}, {12424,12429}, {17819,18451} et al

    X(18457) = {X(3),X(6)}-harmonic conjugate of X(18459)
    X(18457) = X(18458)-of-Ehrmann-side-triangle
    X(18457) = X(18458)-of-1st-Kenmotu-diagonals-triangle


    X(18458) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 1st KENMOTU DIAGONALS

    Trilinears    a^7 - a^6(b + c) - a^5(3b^2 + 5bc + 3c^2) + a^4(3b^3 - 2b^2c - 2bc^2 + 3c^3) + a^3(3b^4 + 6b^3c + 2b^2c^2 + 6bc^3 + 3c^4) - a^2(3b^5 - 3b^4c - 4b^3c^2 - 4b^2c^3 - 3bc^4 + 3c^5) - a(b^2 - c^2)^2(b^2 + bc + c^2) + (b - c)^2(b + c)^3(b^2 - bc + c^2) - 2*S*[2a^4(b + c) - a^3(2b^2 + bc + 2c^2) - a^2(2b^3 + 3b^2c + 3bc^2 + 2c^3) + a(2b^4 - 3b^3c - 2b^2c^2 - 3bc^3 + 2c^4) + bc(b - c)^2(b + c)] : :

    The homothetic center of these triangles is X(18457).

    X(18458) lies on these lines: {1,7}, {36,18410}, {104,7133}, {3576,6204} et al


    X(18459) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd KENMOTU DIAGONALS

    Trilinears    (cos A)[2 - cos 2A + 2 cos 2B + 2 cos 2C + 2 cos(2B - 2C) + cos(2A - 2B) + cos(2A - 2C) + cos 4A + cos(2B + 2C) - cos(2A + 2B) - cos(2A + 2C) - 2 sin 2A - sin 2B - sin 2C - sin(2A - 2B) - sin(2A - 2C) - sin(4A + 4B) - sin(4A + 4C)] : :

    X(18459) lies on these lines: {3,6}, {4,11266}, {5,10881}, {30,11418}, {49,10665}, {265,6414}, {381,5413}, {382,11474}, {485,6639}, {486,18404}, {615,2072}, {1658,10880}, {2070,5412}, {3093,7517}, {5055,10963}, {5416,18453}, {5420,6640}, {6564,10254}, {6565,18403}, {7488,11265}, {7502,11417}, {7728,13288}, {10255,10577}, {10534,10540}, {10666,18436}, {10960,11597}, {11242,18435}, {11447,11464}, {11448,11459}, {11463,12111}, {11473,14130}, {12425,12429} et al

    X(18459) = {X(3),X(6)}-harmonic conjugate of X(18457)
    X(18459) = X(18460)-of-Ehrmann-side-triangle
    X(18459) = X(18460)-of-2nd-Kenmotu-diagonals-triangle


    X(18460) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND 2nd KENMOTU DIAGONALS

    Trilinears    a^7 - a^6(b + c) - a^5(3b^2 + 5bc + 3c^2) + a^4(3b^3 - 2b^2c - 2bc^2 + 3c^3) + a^3(3b^4 + 6b^3c + 2b^2c^2 + 6bc^3 + 3c^4) - a^2(3b^5 - 3b^4c - 4b^3c^2 - 4b^2c^3 - 3bc^4 + 3c^5) - a(b^2 - c^2)^2(b^2 + bc + c^2) + (b - c)^2(b + c)^3(b^2 - bc + c^2) + 2*S*[2a^4(b + c) - a^3(2b^2 + bc + 2c^2) - a^2(2b^3 + 3b^2c + 3bc^2 + 2c^3) + a(2b^4 - 3b^3c - 2b^2c^2 - 3bc^3 + 2c^4) + bc(b - c)^2(b + c)] : :

    The homothetic center of these triangles is X(18459).

    X(18460) lies on these lines: {1,7}, {36,18411}, {3576,6203} et al


    X(18461) = ENDO-HOMOTHETIC CENTER OF EHRMANN SIDE-TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

    Trilinears    a^7 - 3a^6(b + c) + a^5(3b^2 + 5bc + 3c^2) - a^4(b^3 + c^3) - a^3(b^4 + 2b^3c - 2b^2c^2 + 2bc^3 + c^4) + 3a^2(b - c)^2(b + c)(b^2 + c^2) - a(b - c)^2(b^2 + bc + c^2)(3b^2 - 2bc + 3c^2) + (b - c)^4(b + c)(b^2 + bc + c^2) : :

    The homothetic center of these triangles is X(10317).

    X(18461) lies on these lines: {1,7}, {104,294}, {515,9317}, {517,840}, {1055,3576} et al


    X(18462) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND LUCAS ANTIPODAL TANGENTS

    Barycentrics    (pending)

    X(18462) lies on these lines: {3,485}, {9723,18463}, {12590,18438}, {18403,18414}


    X(18463) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND LUCAS(-1) ANTIPODAL TANGENTS

    Barycentrics    (pending)

    X(18463) lies on these lines: {3,486}, {9723,18462}, {12591,18438}, {18403,18415}


    X(18464) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ORTHOCEVIAN OF X(3)

    Barycentrics    a^2[a^18(b^2 + c^2) - a^16(7b^4 + 9b^2c^2 + 7c^4) + 3a^14(b^2 + c^2)(7b^4 + 3b^2c^2 + 7c^4) - a^12(35b^8 + 49b^6c^2 + 47b^4c^4 + 49b^2c^6 + 35c^8) + a^10(b^2 + c^2)(35b^8 + 6b^6c^2 + 27b^4c^4 + 6b^2c^6 + 35c^8) - a^8(21b^12 + 16b^10c^2 + 11b^8c^4 + 11b^4c^8 + 16b^2c^10 + 21c^12) + a^6(b^8 - c^8)(b^2 - c^2)(7b^4 + 11b^2c^2 + 7c^4) - a^4(b^2 - c^2)^4(b^8 + 9b^6c^2 + 11b^4c^4 + 9b^2c^6 + c^8) + a^2b^2c^2(b^2 - c^2)^4(b^2 + c^2)(4b^4 - 7b^2c^2 + 4c^4) - b^2c^2(b^2 - c^2)^6(b^4 + c^4)] : :

    X(18464) lies on these lines: {3,8612}, {5,49}, {18403,18416}

    X(18464) = X(18465)-of-Ehrmann-side-triangle
    X(18464) = X(18465)-of-orthocevian-triangle-of-X(3)


    X(18465) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ORTHOCEVIAN OF X(3)

    Trilinears    (b - c)[a^3(b + c) - a^2(b^2 + bc + c^2) - a(b - c)^2(b + c) + b^4 + b^3c + bc^3 + c^4] : :

    The homothetic center of these triangles is X(18464).

    X(18465) lies on these lines: {1,75}, {2,5396}, {21,104}, {81,4511}, {214,4276}, {333,997}, {355,14011}, {517,13588}, {859,3794}, {956,3786}, {999,5208}, {1420,10461} et al


    X(18466) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ORTHOANTICEVIAN OF X(3)

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

    X(18466) lies on these lines: {3,6}, {265,3548}, {1147,10938}, {18403,18418} et al

    X(18466) = X(18467)-of-Ehrmann-side-triangle
    X(18466) = X(18467)-of-orthoanticevian-triangle-of-X(3)


    X(18467) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND ORTHOANTICEVIAN OF X(3)

    Trilinears    [2a^4 - 4a^3(b + c) + 3a^2bc + 2a(b + c)(2b^2 - 3bc + 2c^2) - (b + c)^2(2b^2 - 3bc + 2c^2)]/(b + c - a) : :

    The homothetic center of these triangles is X(18466).

    X(18467) lies on these lines: {1,7}, {56,4996}, {997,5775}, {1319,3877} et al


    X(18468) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND INNER TRI-EQUILATERAL

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

    X(18468) lies on these lines: {3, 6}, {5, 10632}, {13, 18564}, {30, 11420}, {381, 10641}, {382, 11475}, {465, 3580}, {466, 14389}, {1658, 10633}, {2070, 10642}, {3166, 10640}, {3549, 5334}, {5055, 10643}, {5318, 18563}, {5321, 10024}, {6639, 18581}, {7051, 18447}, {7502, 11421}, {7542, 11543}, {9818, 11408}, {10254, 16809}, {10255, 16966}, {10636, 18453}, {10638, 18455}, {10661, 18436}, {11409, 14070}, {11452, 11459}, {11453, 11464}, {11466, 12111}, {11476, 14130}, {11488, 18531}, {11542, 12605}, {16808, 18403}, {18404, 18582}

    X(18468) = Brocard-circle-inverse-of X(18470)
    X(18468) = X(18469)-of-Ehrmann-side-triangle
    X(18468) = X(18469)-of-inner-tri-equilateral-triangle
    X(18468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 18470), (15, 10634, 3), (18438, 18472, 18470)


    X(18469) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND INNER TRI-EQUILATERAL

    Trilinears    Sqrt[3] (a - b - c) (a - b + c) (a + b - c) (a + b + c) (a^3 - a^2 (b + c) - a (b^2 + c^2) + (b - c)^2 (b + c)) + 2 S (a^5 - 3 a^4 (b + c) + 2 a^3 (b^2 + c^2) + 2 a^2 (b^3 + c^3) - a (b - c)^2 (3 b^2 + 2 b c + 3 c^2) + b^5 - b^4 c - b c^4 + c^5) : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(18468).

    X(18469) lies on these lines: {1,7}, {36,18422}, {1653,3576}


    X(18470) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND OUTER TRI-EQUILATERAL

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

    X(18470) lies on these lines: {3, 6}, {5, 10633}, {14, 18564}, {30, 11421}, {49, 10661}, {381, 10642}, {382, 11476}, {465, 14389}, {466, 3580}, {1250, 18455}, {1658, 10632}, {2070, 10641}, {3549, 5335}, {5055, 10644}, {5318, 10024}, {5321, 18563}, {6639, 18582}, {7502, 11420}, {7542, 11542}, {9818, 11409}, {10254, 16808}, {10255, 16967}, {10637, 18453}, {11408, 14070}, {11452, 11464}, {11453, 11459}, {11467, 12111}, {11475, 14130}, {11489, 18531}, {11543, 12605}, {16809, 18403}, {18404, 18581}

    X(18470) = Brocard circle-inverse-of X(18468)
    X(18470) = X(18471)-of-Ehrmann-side-triangle
    X(18470) = X(18471)-of-outer-tri-equilateral-triangle
    X(18470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6, 18468), (16, 10635, 3), (18438, 18472, 18468)


    X(18471) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN SIDE AND OUTER TRI-EQUILATERAL

    Trilinears    Sqrt[3] (a - b - c) (a - b + c) (a + b - c) (a + b + c) (a^3 - a^2 (b + c) - a (b^2 + c^2) + (b - c)^2 (b + c)) - 2 S (a^5 - 3 a^4 (b + c) + 2 a^3 (b^2 + c^2) + 2 a^2 (b^3 + c^3) - a (b - c)^2 (3 b^2 + 2 b c + 3 c^2) + b^5 - b^4 c - b c^4 + c^5) : : (Randy Hutson, June 27, 2018)

    The homothetic center of these triangles is X(18470).

    X(18471) lies on these lines: {1,7}, {36,18423}, {104,1251}, {1652,3576}


    X(18472) = HOMOTHETIC CENTER OF EHRMANN SIDE-TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

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

    X(18472) lies on these lines: {3,6}, {5,10986}, {112,7502}, {115,18564}, {550,5523}, {1968,2937}, {1971,18435}, {2072,3054}, {10987,18455}, {10988,18453}, {14585,18436}, {18403,18424} et al

    X(18472) = X(18473)-of-Ehrmann-side-triangle if ABC is acute


    X(18473) = ENDO-HOMOTHETIC CENTER OF EHRMANN SIDE-TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

    Trilinears    2a^7 - 4a^6(b + c) + 5a^5bc + a^4(b + c)(2b^2 - bc + 2c^2) + 2a^3(b^2 + c^2)(b^2 - bc + c^2) - 2a^2bc(b - c)^2(b + c) - a(b - c)^2(4b^4 + 3b^3c + 6b^2c^2 + 3bc^3 + 4c^4) + (b - c)^4(b + c)(2b^2 + 3bc + 2c^2) : :

    The homothetic center of these triangles is X(18472).

    X(18473) lies on these lines: {1,7}, {36,18425}, {5902,11714}


    X(18474) = HOMOTHETIC CENTER OF EHRMANN SIDE-TRIANGLE AND CROSS-TRIANGLE OF EHRMANN VERTEX- AND EHRMANN SIDE-TRIANGLES

    Barycentrics    a^10 - 2a^8(b^2 + c^2) + a^6(b^2 + c^2)^2 - a^4(b^2 + c^2)(b^4 + c^4) + 2a^2(b^8 - b^6c^2 - b^2c^6 + c^8) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18474) = (2R^2 - SW)*X(4) + R^2*X(52) = X(4) + X(11442) = 2 X(5) - X(184)

    X(18474) lies on these lines: {2,11464}, {3,161}, {4,52}, {5,156}, {6,13}, {30,343}, {51,11818}, {128,6069}, {578,5576}, {973,6102}, {1092,13371}, {1352,2393}, {1531,18568}, {1593,9937}, {1899,9730}, {3543,18387}, {5562,11572}, {5876,18377}, {5907,18383}, {6293,18439}, {7574,11649}, {9818,18396}, {10255,18350}, {15058,18394}, {18376,18403} et al

    X(18474) = midpoint of X(4) and X(11442)
    X(18474) = reflection of X(184) in X(5)
    X(18474) = anticomplement of X(18475)
    X(18474) = X(18446)-of-orthic-triangle if ABC is acute
    X(18474) = X(993)-of-Ehrmann-side-triangle if ABC is acute
    X(18474) = X(184)-of-Johnson-triangle
    X(18474) = X(11442)-of-Euler-triangle
    X(18474) = X(18445)-of-Ehrmann-mid-triangle
    X(18474) = Ehrmann-side-to-orthic similarity image of X(18445)
    X(18474) = orthic-to-Ehrmann-side similarity image of X(18388)


    X(18475) = COMPLEMENT OF X(18474)

    Barycentrics    a^2(b^2 + c^2 - a^2)[2a^6 - 3a^4(b^2 + c^2) - 2a^2b^2c^2 + (b^2 - c^2)^2(b^2 + c^2)] : :
    X(18475) = X(3) + X(184) = X(22) + X(13352)

    Let OAOBOC be the Kosnita triangle. Let A' be the trilinear pole, wrt OAOBOC, of line BC, and define B', C' cyclically. The lines OAA', OBB', OCC' concur in X(18475).

    X(18475) lies on these lines: {2,11464}, {3,49}, {5,5944}, {6,14070}, {22,13352}, {23,15033}, {24,569}, {26,578}, {30,11430}, {51,567}, {52,54}, {110,5891}, {140,13561}, {141,542}, {143,12107}, {154,9818}, {156,5907}, {161,6642}, {1531,18564}, {4550,18451}, {10201,18390} et al

    X(18475) = midpoint of X(i) and X(j) for these {i,j}: {3,184}, {22,13352}
    X(18475) = complement of X(18474)
    X(18475) = {X(24),X(569)}-harmonic conjugate of X(5462)
    X(18475) = X(226)-of-Kosnita-triangle if ABC is acute


    X(18476) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN SIDE AND 1st HYACINTH

    Barycentrics    a^2[a^12(b^2 + c^2) - a^10(4b^4 + 13b^2c^2 + 4c^4) + a^8(b^2 + c^2)(5b^4 + 27b^2c^2 + 5c^4) - a^6(34b^6c^2 + 33b^4c^4 + 34b^2c^6) - 5a^4(b^2 + c^2)(b^4 - 5b^2c^2 + c^4)(b^4 - b^2c^2 + c^4) + a^2(b^2 - c^2)^2(4b^8 - 9b^6c^2 - 9b^2c^6 + 4c^8) - (b^2 - c^2)^4(b^6 - 2b^4c^2 - 2b^2c^4 + c^6)] : :

    X(18476) lies on these lines: {110,1173}, {1598,2904}, {1657,5889}, {9937,15047}, {11935,15024}


    X(18477) = X(1)X(21)∩X(75)X(811)

    Trilinears    tan B tan C + 3 : :
    Trilinears    (b^2 + c^2 - a^2)(a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2) : :

    Let SASBSC be the Ehrmann side-triangle. Let A' be the trilinear product SB*SC, and define B', C' cyclically. The lines AA', BB', CC' concur in X(18477).

    X(18477) lies on these lines: {1,21}, {34,18540}, {40,9643}, {48,3708}, {75,811}, {158,1087}, {212,18455}, {603,18447} et al

    X(18477) = barycentric product X(75)*X(5158)
    X(18477) = barycentric quotient X(5158)/X(1)


    X(18478) = TRILINEAR PRODUCT OF VERTICES OF EHRMANN SIDE-TRIANGLE

    Barycentrics    (b^2 + c^2 - a^2)^2(a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2)/[(b^2 + c^2 - a^2)^2 - b^2c^2] : :

    X(18478) lies on these lines: {3,125}, {328,1494}, {476,10296}, {6344,14860} et al

    X(18478) = barycentric product X(69)^2*X(381)*X(1989)


    X(18479) = BARYCENTRIC PRODUCT OF VERTICES OF EHRMANN SIDE-TRIANGLE

    Barycentrics    a^2(b^2 + c^2 - a^2)^2(a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2)/[(b^2 + c^2 - a^2)^2 - b^2c^2] : :

    X(18479) lies on these lines: {30,74}, {184,5158}

    X(18479) = barycentric product X(6)*X(69)^2*X(381)*X(1989)


    X(18480) = X(1) OF EHRMANN MID-TRIANGLE

    Trilinears    (r/R) - 2 cos(B - C) + cos A : :
    Trilinears    (r/R) + 3 cos A - 4 sin B sin C : :
    Trilinears    2 cos A + cos B + cos C - 2 cos(B - C) - 1 : :
    Trilinears    4 cos A + cos B + cos C - 4 sin B sin C - 1 : :
    Barycentrics    2a^4 - a^3(b + c) + 2a^2bc + a(b - c)^2(b + c) - 2(b^2 - c^2)^2 : :
    X(18480) = 2 X(1) - 3 X(2) - 3 X(4) = X(1) - 3 X(381) = X(1) - 2 X(9955) = X(1) + X(18525) = X(3) + X(5691) = X(3) - 2 X(9956) = 3 X(4) + X(8) = X(4) + X(355) = 2 X(5) - X(1385) = X(8) + X(12699) = 2 X(12) - X(3579) = X(40) + X(382) = X(140) - 2 X(4297) = 2 X(546) - X(946)

    X(18480) lies on these lines: {1,381}, {2,13624}, {3,1698}, {4,8}, {5,515}, {10,30}, {12,6841}, {20,5818}, {35,13743}, {40,382}, {55,10827}, {56,10826}, {57,9655}, {65,79}, {140,4297}, {518,3818}, {519,3845}, {1482,1699}, {3751,18440}, {3827,18382}, {7713,18494}, {9941,18500}, {11852,18508}, {12438,18507}, {12440,18520}, {12441,18522} et al

    X(18480) = midpoint of X(i) and X(j) for these {i,j}: {1,18525}, {3,5691}, {4,355}, {8,12699}, {40,382}
    X(18480) = reflection of X(i) in X(j) for these (i,j): (1,9955), (3,9956), (10,18357), (140,4297), (946,546), (1385,5), (3579,12)
    X(18480) = complement of X(18481)
    X(18480) = anticomplement of X(13624)
    X(18480) = inverse-in-Johnson-circle of X(5080)
    X(18480) = X(1)-of-Ehrmann-mid-triangle
    X(18480) = X(355)-of-Euler-triangle
    X(18480) = X(1385)-of-Johnson-triangle
    X(18480) = X(177)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18480) = X(3579)-of-outer-Garcia-triangle
    X(18480) = X(9955)-of-Aquila-triangle
    X(18480) = X(9956)-of-X3-ABC-reflections-triangle
    X(18480) = X(18525)-of-anti-Aquila-triangle
    X(18480) = X(1511)-of-Fuhrmann-triangle
    X(18480) = orthic-to-Ehrmann-side similarity image of X(9955)
    X(18480) = orthic-to-2nd-Euler similarity image of X(1385)
    X(18480) = excentral-to-Fuhrmann similarity image of X(12515)
    X(18480) = endo-homothetic center of these triangles: Ehrmann mid and anti-Aquila; The homothetic center is X(9955).
    X(18480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,9955), (4,8,12699), (10750,10751,5080), (18495,18497,55), (18516,18517,4)


    X(18481) = ANTICOMPLEMENT OF X(18480)

    Trilinears    1 - 2 cos A - cos B - cos C + 2 cos B cos C
    Barycentrics    3a^4 - a^3(b + c) - 2a^2(b^2 - bc + c^2) + a(b - c)^2(b + c) - (b^2 - c^2)^2 : :
    X(18481) = X(1) - 3 X(2) + 3 X(3) = 2 X(1) + 3 X(2) - 3 X(4) = X(1) + X(3) - X(4) = X(1) - 6 X(3) + 2 X(8) = 4 X(1) - 3 X(4) + X(8) = X(1) + 2 X(20) - X(40) = 2 X(1) - 3 X(3655) = 4 X(1) - 3 X(3656) = 2 X(1) - X(12699) = 3 X(3) - 2 X(10) = 2 X(3) - X(355) = X(3) - 2 X(4297) = X(4) - X(8) + 2 X(20) = X(4) - 2 X(1385) = 2 X(5) - X(5691) = X(8) - 2 X(3579) = 2 X(10) - X(18525) = 2 X(20) - 2 X(40) + X(355) = X(20) + X(944) = X(145) + X(6361) = 2 X(376) - X(3654)

    X(18481) lies on these lines: {1,30}, {2,13624}, {3,10}, {4,1385}, {5,3576}, {8,376}, {12,3612}, {20,145}, {140,5587}, {381,1125}, {549,1698} et al

    X(18481) = midpoint of X(i) and X(j) for these {i,j}: {20,944}, {145,6361}
    X(18481) = reflection of X(i) in X(j) for these (i,j): (3,4297), (4,1385), (8,3579), (355,3), (3654,376), (5691,5), (12699,1), (18525,10)
    X(18481) = X(5876)-of-excentral-triangle
    X(18481) = X(12902)-of-Fuhrmann-triangle
    X(18481) = X(1385)-of-anti-Euler-triangle
    X(18481) = X(355)-of-ABC-X3-reflections-triangle
    X(18481) = X(4297)-of-X3-ABC-reflections-triangle
    X(18481) = X(5691)-of-Johnson-triangle


    X(18482) = X(9) OF EHRMANN MID-TRIANGLE

    Barycentrics    2a^6 - 3a^5(b + c) - 2a^4bc - 2a^2bc(b - c)^2 + 3a(b - c)^2(b + c)^3 - 2(b - c)^4(b + c)^2 : :
    X(18482) = 3 X(4) + X(7) = X(4) + X(5805)

    X(18482) lies on these lines: {4,7}, {5,516}, {9,381}, {30,142}, {46,11372}, {55,1538}, {80,15909}, {144,3839}, {518,3818}, {3243,18525} et al

    X(18482) = midpoint of X(4) and X(5805)
    X(18482) = X(9)-of-Ehrmann-mid-triangle
    X(18482) = X(5805)-of-Euler-triangle


    X(18483) = X(10) OF EHRMANN MID-TRIANGLE

    Trilinears    (r/R) + 6 cos B cos C : :
    Barycentrics    2a^4 + a^3(b + c) + a^2(b - c)^2 - a(b - c)^2(b + c) - 3(b^2 - c^2)^2 : :
    X(18483) = X(1) + 3 X(4) = X(4) + X(946) = 2 X(5) - X(6684) = X(1125) - 2 X(9955)

    X(18483) lies on these lines: {1,4}, {3,3817}, {5,516}, {8,3839}, {10,381}, {11,1354}, {12,10624}, {20,5550}, {30,1125}, {40,3091}, {45,10445}, {57,10591}, {79,553}, {84,5556}, {124,133}, {140,10171}, {142,6851}, {165,3090}, {355,3625}, {376,3624}, {382,4297}, {495,12575}, {496,4298}, {517,546}, {519,3845} et al

    X(18483) = midpoint of X(4) and X(946)
    X(18483) = reflection of X(i) in X(j) for these (i,j): (1125,9955), (6684,5)
    X(18483) = QA-P33 (Centroid of the Orthocenter Quadrangle) of quadrangle ABCX(1)
    X(18483) = X(10)-of-Ehrmann-mid-triangle
    X(18483) = X(6684)-of-Johnson-triangle
    X(18483) = X(946)-of-Euler-triangle


    X(18484) = TRILINEAR PRODUCT OF VERTICES OF EHRMANN MID-TRIANGLE

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

    X(18484) lies on this line: {1495,14254}

    X(18484) = barycentric product X(30)*X(76)*X(381)^2


    X(18485) = BARYCENTRIC PRODUCT OF VERTICES OF EHRMANN MID-TRIANGLE

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

    X(18485) lies on these lines: {1990,3081}, {9407,14583}

    X(18485) = barycentric product X(30)*X(381)^2


    X(18486) = X(75)X(158)∩X(1099)X(1784)

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

    Let A'B'C' be the Ehrmann mid-triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(18486).

    X(18486) lies on these lines: {75,158}, {1099,1784}, {1725,2166} et al

    X(18486) = barycentric product X(30)*X(75)*X(381)
    X(18486) = barycentric product X(381)*X(14206)


    X(18487) = X(2)X(216)∩X(30)X(1990)

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

    Let A'B'C' be the Ehrmann mid-triangle. Let A" be the barycentric product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(18487).

    X(18487) lies on these lines: {2,216}, {4,15860}, {6,3830}, {30,1990}, {53,3845}, {115,16303}, {187,3018}, {233,10109}, {381,5158}, {577,3534}, {1249,15682}, {6748,12101}, {6749,15687}, {10979,15693}, {15356,18374}, {15685,15905} et al

    X(18487) = barycentric product X(30)*X(381)


    X(18488) = PERSPECTOR OF THESE TRIANGLES: EHRMANN MID AND ANTI-EXCENTERS-INCENTER REFLECTIONS

    Barycentrics    a^6(b^4 - 6b^2c^2 + c^4) - a^4(3b^6 - b^4c^2 - b^2c^4 + 3c^6) + a^2(b^2 - c^2)^2(3b^4 + 8b^2c^2 + 3c^4) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18488) = X(4) + X(15062)

    X(18488) lies on these lines: {2,8718}, {3,2916}, {4,5449}, {5,10575}, {30,1209}, {52,1595}, {64,381}, {113,1594}, {125,546}, {378,8907}, {389,16003}, {1593,9937}, {7577,11439}, {18388,18439} et al

    X(18488) = midpoint of X(4) and X(15062)
    X(18488) = X(3521)-of-Ehrmann-mid-triangle
    X(18488) = X(1389)-of-anti-excenters-incenter-reflections-triangle
    X(18488) = X(15062)-of-Euler-triangle


    X(18489) = PERSPECTOR OF THESE TRIANGLES: EHRMANN MID AND ANTI-INVERSE-IN-INCIRCLE

    Barycentrics    a^10 - 5a^8(b^2 + c^2) + 2a^6(5b^4 - 6b^2c^2 + 5c^4) - 10a^4(b^2 + c^2)^3 + a^2(b^2 - c^2)^2(5b^4 + 54b^2c^2 + 5c^4) - (b^2 - c^2)^4(b^2 + c^2) : :

    X(18489) lies on these lines: {4,5447}, {5,5544}, {8,6849}, {69,381}, {113,3545}, {376,5888}, {3544,18504}, {3818,18537} et al

    X(18489) = perspector of Johnson circle wrt Ehrmann mid-triangle
    X(18489) = X(3531)-of-Ehrmann-mid-triangle
    X(18489) = X(18490)-of-anti-inverse-in-incircle-triangle
    X(18489) = X(5544)-of-Johnson-triangle


    X(18490) = ISOGONAL CONJUGATE OF X(6767)

    Trilinears    1/(4 + cos A) : :

    The trilinear polar of X(18490) passes through X(650).

    X(18490) lies on the Feuerbach hyperbola and these lines: {1,3528}, {7,5049}, {8,4002}, {9,551}, {79,1058}, {80,1056}, {84,10595}, {354,1000}, {388,5560} et al

    X(18490) = isogonal conjugate of X(6767)


    X(18491) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND ANTI-MANDART-INCIRCLE

    Trilinears    a^6 - a^5(b + c) - 2a^4(b^2 + bc + c^2) + 2a^3(b + c)(b^2 + c^2) + a^2(b^4 - 2b^3c - 2b^2c^2 - 2bc^3 + c^4) - a(b - c)^2(b + c)^3 + 4bc(b^2 - c^2)^2 : :
    X(18491) = X(57) + X(18528)

    X(18491) lies on these lines: {1,18518}, {3,1698}, {4,100}, {5,1001}, {30,1376}, {55,381}, {56,80}, {57,18528}, {113,12334}, {165,18529}, {355,956}, {382,10310}, {515,6692}, {958,18357}, {1308,18328}, {1466,9655}, {1479,11501}, {3585,11509}, {7741,11510}, {8068,10896}, {11504,18522}, {11848,18507} et al

    X(18491) = midpoint of X(57) and X(18528)
    X(18491) = X(9668)-of-Ehrmann-mid-triangle
    X(18491) = X(9668)-of-anti-Mandart-incircle-triangle
    X(18491) = {X(381),X(18499)}-harmonic conjugate of X(3583)


    X(18492) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND AQUILA

    Trilinears    (r/R) - 4 cos(B - C) + 2 cos A : :
    Trilinears    4 cos A + cos B + cos C - 4 cos(B - C) - 1 : :
    Barycentrics    3a^4 - a^3(b + c) + a^2(b + c)^2 + a(b - c)^2(b + c) - 4(b^2 - c^2)^2 : :
    X(18492) = X(1) - 6 X(381) = X(1) - 2 X(18493) = X(4) + X(5818)

    X(18492) lies on these lines: {1,381}, {3,7989}, {4,9}, {5,3576}, {12,3586}, {20,10175}, {30,1698}, {46,1749}, {57,3585}, {84,6839}, {113,12407}, {145,946}, {165,382}, {354,9656}, {355,546}, {376,3634}, {515,3091}, {517,3843}, {944,3636}, {950,10590}, {962,4678}, {1385,3851}, {1478,3333}, {1479,9578}, {1656,7987}, {1697,3583}, {1837,3649}, {2478,3646}, {3090,4297}, {3361,9655}, {3488,3947}, {3529,10164}, {3543,9780}, {3601,5441}, {3653,11737}, {3654,14893}, {3679,3845}, {3715,12702}, {3814,5438}, {3822,5436}, {3828,15682}, {3854,5882}, {3857,5901}, {3861,5690}, {4004,6001}, {4314,8164}, {4677,8148}, {4746,12245}, {5010,13743}, {5055,13624}, {5090,10151}, {5219,6866}, {5252,9614}, {5290,5722}, {8188,18520}, {8189,18522}, {11852,18507} et al

    X(18492) = midpoint of X(4) and X(5818)
    X(18492) = reflection of X(1) in X(18493)
    X(18492) = X(18493)-of-Ehrmann-mid-triangle
    X(18492) = X(18493)-of-Aquila-triangle
    X(18492) = {X(381),X(18525)}-harmonic conjugate of X(9955)
    X(18492) = X(5818)-of-Euler-triangle


    X(18493) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND AQUILA

    Trilinears    (2r/R) + 2 cos(B - C) - cos A : :
    Trilinears    cos A + 2 cos B + 2 cos C + 2 cos(B - C) - 2 : :
    Barycentrics    a^4 - 2a^3(b + c) - a^2(3b^2 - 4bc + 3c^2) + 2a(b - c)^2(b + c) + 2(b^2 - c^2)^2 : :
    X(18493) = 2 X(1) + 3 X(381) = X(1) + X(18492)

    The homothetic center of these triangles is X(18492).

    X(18493) lies on these lines: {1,381}, {2,12702}, {3,142}, {4,3622}, {5,8}, {10,3656}, {30,3616}, {40,3526}, {55,5443}, {56,79}, {140,962}, {145,3545}, {149,6900}, {165,15720}, {226,7373}, {265,11723}, {355,3244}, {382,1385}, {388,1387}, {399,12261}, {403,11396}, {496,3485}, {515,3843}, {517,1656}, {546,944}, {547,9780}, {549,5550}, {550,9812}, {551,3830}, {952,3091}, {958,11813}, {982,5492}, {999,10404}, {1058,5719}, {1386,18440}, {1388,3585}, {1657,3576}, {3241,5066}, {3295,11375}, {3487,5809}, {3560,5057}, {7687,12898}, {7704,18549}, {11720,12902}, {11831,18508}, {12164,12259} et al

    X(18493) = midpoint of X(1) and X(18492)
    X(18493) = {X(1),X(381)}-harmonic conjugate of X(18525)


    X(18494) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND ARA

    Barycentrics    [3a^6 - 4a^4(b^2 + c^2) - a^2(b^2 + c^2)^2 + 2(b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :

    The homothetic center of these triangles is X(9818).

    X(18494) lies on these lines: {2,3}, {6,18400}, {34,9655}, {51,18396}, {52,12429}, {53,7737}, {66,3426}, {154,18388}, {1829,18525}, {1843,13754}, {7713,18480}, {11383,18524}, {17810,18390}


    X(18495) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 1st AURIGA

    Barycentrics    a^2(a - b - c)^2(a + b - c)(a - b + c)(a + b + c) + [2a^4 - a^3(b + c) + 2a^2bc + a(b - c)^2(b + c) - 2(b^2 - c^2)^2]*4S(rR + 4R^2)^1/2 : :

    X(18495) lies on these lines: {4,5601}, {5,9834}, {30,5599}, {55,10827}, {113,12466}, {355,8204}, {381,5597}, {382,11822}, {5598,18525}, {5600,18357}, {8201,18520}, {8202,18522), {11865,18516} et al

    X(18495) = {X(55),X(18480)}-harmonic conjugate of X(18497)
    X(18495) = X(18496)-of-Ehrmann-mid-triangle
    X(18495) = X(18496)-of-1st-Auriga-triangle


    X(18496) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 1st AURIGA

    Barycentrics    -2*(-a+b+c)*(a-b+c)*(a+b-c)*a*D+3*a^7-3*(b+c)*a^6-4*(b^2+c^2)*a^5+2*(b+c)*(2*b^2-b*c+2*c^2)*a^4-(b^2+c^2)^2*a^3+(b^4-c^4)*(b-c)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-2*(b^2-c^2)^3*(b-c) : :, where D = 4*S*sqrt(R*(4*R+r)) : :

    The homothetic center of these triangles is X(18495).

    X(18496) lies on these lines: {1,18498}, {381,5597}

    X(18496) = {X(1),X(18499)}-harmonic conjugate of X(18498)


    X(18497) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 2nd AURIGA

    Barycentrics    a^2(a - b - c)^2(a + b - c)(a - b + c)(a + b + c) - [2a^4 - a^3(b + c) + 2a^2bc + a(b - c)^2(b + c) - 2(b^2 - c^2)^2]*4S(rR + 4R^2)^1/2 : :

    X(18497) lies on these lines: {4,5602}, {5,9835}, {30,5600}, {55,10827}, {113,12467}, {355,8197}, {381,5598}, {382,11823}, {5597,18525}, {5599,18357}, {8208,18520}, {8209,18522}, {11866,18516} et al

    X(18497) = {X(55),X(18480)}-harmonic conjugate of X(18495)
    X(18497) = X(18498)-of-Ehrmann-mid-triangle
    X(18497) = X(18498)-of-2nd-Auriga-triangle


    X(18498) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 2nd AURIGA

    Barycentrics    2*(-a+b+c)*(a-b+c)*(a+b-c)*a*D+3*a^7-3*(b+c)*a^6-4*(b^2+c^2)*a^5+2*(b+c)*(2*b^2-b*c+2*c^2)*a^4-(b^2+c^2)^2*a^3+(b^4-c^4)*(b-c)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-2*(b^2-c^2)^3*(b-c) : :, where D = 4*S*sqrt(R*(4*R+r)) : :

    The homothetic center of these triangles is X(18497).

    X(18498) lies on these lines: {1,18496}, {381,5598}

    X(18498) = {X(1),X(18499)}-harmonic conjugate of X(18496)


    X(18499) = {X(18496),X(18498)}-HARMONIC CONJUGATE OF X(1)

    Barycentrics    3a^7 - 3a^6(b + c) - 4a^5(b^2 + c^2) + 2a^4(b + c)(2b^2 - bc + 2c^2) - a^3(b^2 + c^2)^2 + a^2(b - c)^2(b + c)(b^2 + c^2) + 2a(b^2 - c^2)^2(b^2 + c^2) - 2(b - c)^4(b + c)^3 : :
    X(18499) = 3*X(3)-4*X(2886), 2*X(55)-3*X(381), 5*X(3843)-4*X(7680), 9*X(5055)-8*X(6690)

    X(18499) lies on these lines: {1, 18496}, {3, 2886}, {4, 3871}, {9, 5790}, {30, 956}, {55, 381}, {56, 18543}, {79, 2099}, {382, 517}, {528, 3830}, {674, 18440}, {952, 5905}, {1656, 5259}, {1657, 3428}, {1824, 18494}, {3149, 11928}, {3419, 12702}, {3534, 18515}, {3585, 18545}, {3814, 11499}, {3820, 6928}, {3843, 7680}, {4294, 6841}, {5055, 6690}, {5119, 12953}, {5173, 18541}, {5249, 10246}, {5535, 11661}, {6253, 10525}, {6284, 18517}, {6917, 16202}, {6934, 10943}, {8069, 9669}, {9614, 18493}, {9670, 9955}, {11018, 18530}, {12116, 16203}, {12635, 12699}

    X(18499) = reflection of X(i) in X(j) for these (i,j): (1657, 3428), (5119, 18480)
    X(18499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 18518, 18542), (3583, 18491, 381), (18496, 18498, 1)


    X(18500) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th BROCARD

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

    X(18500) lies on these lines: {3,7914}, {4,2896}, {5,7846}, {30,3096}, {32,381}, {113,12501}, {382,3098}, {1479,10873}, {3091,9862}, {3545,10583}, {3583,10877}, {3830,7842}, {3851,9756}, {7899,10000}, {7918,14458}, {7923,10345}, {9941,18480}, {9997,18525}, {10047,10896}, {10871,18516}, {10872,18517}, {10875,18520}, {10876,18522}, {10878,18542}, {10879,18544}, {11885,18507} et al

    X(18500) = X(18501)-of-Ehrmann-mid-triangle
    X(18500) = X(18501)-of-5th-Brocard-triangle


    X(18501) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th BROCARD

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

    The homothetic center of these triangles is X(18500).

    X(18501) lies on these lines: {3,83}, {4,11842}, {5,7793}, {30,7787}, {32,381}, {98,3843}, {182,1657}, {382,3398}, {399,12201}, {9766,18548}, {11490,18524}, {11839,18508}, {12194,18525}, {12212,18440} et al


    X(18502) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th ANTI-BROCARD

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

    X(18502) lies on these lines: {3,6683}, {4,3398}, {5,316}, {30,83}, {32,381}, {98,546}, {113,12201}, {182,382}, {384,14881}, {538,18548}, {1478,10798}, {1656,5171}, {2782,7839}, {3583,10799}, {3843,11842}, {3849,8150}, {5039,18440}, {5476,7748}, {7770,9821}, {10113,13193}, {10792,18509}, {10793,18511}, {10794,18516}, {10800,18525}, {10802,10896}, {11839,18507}, {11840,18520}, {11841,18522}

    X(18502) = X(18503)-of-Ehrmann-mid-triangle
    X(18502) = X(18503)-of-5th-anti-Brocard-triangle


    X(18503) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th ANTI-BROCARD

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

    The homothetic center of these triangles is X(18502).

    X(18503) lies on these lines: {3,3096}, {4,9301}, {5,9862}, {30,2896}, {32,381}, {382,6248}, {399,12501}, {3094,18440}, {9941,18525}, {11494,18524}, {11885,18508} et al


    X(18504) = PERSPECTOR OF THESE TRIANGLES: EHRMANN MID AND CIRCUMORTHIC

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

    X(18504) lies on these lines: {3,1539}, {4,11449}, {5,6241}, {54,156}, {110,12293}, {113,5449}, {235,6403}, {3544,18489}, {5655,18356}, {7577,11439}, {10539,18392}, {10540,18394} et al


    X(18505) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND DANNEELS-BEVAN

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

    X(18505) lies on these lines: {10,30}, {381,2999}, {2270,18540}

    X(18505) = X(18506)-of-Ehrmann-mid-triangle
    X(18505) = X(18506)-of-Danneels-Bevan-triangle


    X(18506) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND DANNEELS-BEVAN

    Trilinears    a^6 + 2a^5(b + c) - a^4(b^2 + 6bc + c^2) - 4a^3(b^3 + b^2c + bc^2 + c^3) - a^2(b^4 + 6b^2c^2 + c^4) + 2a(b - c)^2(b + c)^3 + (b^2 - c^2)^2(b^2 + 6bc + c^2) : :

    The homothetic center of these triangles is X(18505).

    X(18506) lies on these lines: {1,30}, {4,5256}, {6,18540}, {37,3587}, {42,18528}, {84,5707}, {376,5287}, {381,2999}, {517,7174}, {940,7171}, {942,3182}, {990,18443}, {1103,9654} et al


    X(18507) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND GOSSARD

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

    X(18507) lies on these lines: {2,3}, {113,12790}, {1478,11906}, {1479,11905}, {1539,12369}, {3818,12583}, {9955,11831}, {10113,13212}, {10895,11912}, {10896,11913}, {11839,18502}, {11848,18491}, {11852,18492}, {11863,18495}, {11864,18497}, {11885,18500}, {11900,12699}, {11901,18509}, {11902,18511}, {11903,18516}, {11904,18517}, {11907,18520}, {11908,18522}, {11910,18525}, {11914,18542}, {11915,18544}, {12438,18480}, {12611,12729}, {12794,14881}, {16210,18357}

    X(18507) = X(18508)-of-Ehrmann-mid-triangle
    X(18507) = X(18508)-of-Gossard-triangle
    X(18507) = X(12113)-of-Johnson-triangle


    X(18508) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND GOSSARD

    Barycentrics    (2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)[3a^10 - 3a^8(b^2 + c^2) - a^6(7b^4 - 17b^2c^2 + 7c^4) + 9a^4(b^2 - c^2)^2(b^2 + c^2) - 11a^2b^2c^2(b^2 - c^2)^2 - 2(b^2 - c^2)^4(b^2 + c^2)](b^2 + c^2 - a^2) : :

    The homothetic center of these triangles is X(18507).

    X(18508) lies on these lines: {2,3}, {399,12790}, {999,11906}, {2777,5502}, {3295,11905}, {8148,12626}, {9033,12121}, {9654,11912}, {11831,18493}, {11839,18501}, {11848,18524}, {11852,18480}, {11885,18503}, {11900,12702}, {12164,12418}, {12438,18525}, {12583,18440}, {12902,13212}, {15526,16111}


    X(18509) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND INNER GREBE

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

    X(18509) lies on these lines: {3,10261}, {4,1161}, {5,5871}, {6,13}, {30,5591}, {1478,10925}, {1479,10923}, {3583,10927}, {5605,18525}, {8216,18520}, {8217,18522}, {9955,11370}, {10048,10896}, {10792,18502}, {10919,18516}, {10929,18542}, {10931,18544}, {11901,18507}

    X(18509) = {X(381),X(3818)}-harmonic conjugate of X(18511)
    X(18509) = X(18510)-of-Ehrmann-mid-triangle
    X(18509) = X(18510)-of-inner-Grebe-triangle


    X(18510) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND INNER GREBE

    Barycentrics    a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2 - 8a^2 S : :

    The homothetic center of these triangles is X(18509).

    X(18510) lies on these lines: {2,6199}, {3,1588}, {4,6418}, {5,6417}, {6,13}, {30,6395}, {371,3526}, {372,1657}, {376,6446}, {382,3071}, {485,5072}, {486,590}, {546,1132}, {547,8972}, {549,6445}, {550,6408}, {615,5054}, {631,6407}, {632,3317}, {1131,3858}, {1151,15720}, {1152,15696} et al

    X(18510) = {X(6),X(381)}-harmonic conjugate of X(18512)
    X(18510) = X(6395)-of-orthocentroidal-triangle


    X(18511) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND OUTER GREBE

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

    X(18511) lies on these lines: {3,10262}, {4,1160}, {5,5870}, {6,13}, {30,5590}, {1478,10926}, {1479,10924}, {3583,10928}, {8218,18520}, {8219,18522}, {9955,11371}, {10049,10896}, {10793,18502}, {10920,18516}, {10930,18542}, {10932,18544}, {11902,18507}

    X(18511) = {X(381),X(3818)}-harmonic conjugate of X(18509)
    X(18511) = X(18512)-of-Ehrmann-mid-triangle
    X(18511) = X(18512)-of-outer-Grebe-triangle


    X(18512) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND OUTER GREBE

    Barycentrics    a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2 + 8a^2 S : :

    The homothetic center of these triangles is X(18511).

    X(18512) lies on these lines: {2,6395}, {3,1587}, {4,6417}, {5,6418}, {6,13}, {30,6199}, {371,1657}, {372,3526}, {376,6445}, {382,3070}, {485,615}, {486,5072}, {546,1131}, {549,6446}, {550,6407}, {590,5054}, {631,6408}, {632,3316}, {1132,3858}, {1151,15696}, {1152,8960} et al

    X(18512) = {X(6),X(381)}-harmonic conjugate of X(18510)
    X(18512) = X(6199)-of-orthocentroidal-triangle


    X(18513) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 2nd ISOGONAL OF X(1)

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

    X(18513) lies on these lines: {1,4}, {5,7280}, {8,4525}, {11,3845}, {30,5010}, {35,382}, {36,381}, {46,1749}, {55,3830}, {56,3843}, {57,13273}, {79,1837}, {80,1836}, {498,3146}, {1060,18403}, {1709,12761}, {2093,18406}, {3295,9656}, {3336,9579}, {3337,9581}, {3543,3584}, {3746,5076}, {3822,11114}, {3853,6284}, {4214,8185}, {4309,5261}, {4317,10591}, {5073,5217}, {5141,5267}, {5268,7391}, {5272,7394}, {5326,8703}, {5441,11374}, {5563,9655}, {6645,14044}, {7355,18383}, {9897,10742}, {10037,11403}, {10827,11010}, {11009,18525}, {12102,15171} et al

    X(18513) = X(18515)-of-Ehrmann-mid-triangle
    X(18513) = X(18515)-2nd-isogonal-triangle-of-X(1)


    X(18514) = {X(1),X(4)}-HARMONIC CONJUGATE OF X(18513)

    Barycentrics    2a^4 - a^2bc - 2(b^2 - c^2)^2 : :

    X(18514) lies on these lines: {1,4}, {2,4324}, {5,5010}, {8,4537}, {11,3627}, {12,3845}, {30,5433}, {35,381}, {36,382}, {46,12764}, {65,18550}, {15430,18426} et al

    X(18514) = {X(1),X(4)}-harmonic conjugate of X(18513)


    X(18515) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 2nd ISOGONAL OF X(1)

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

    The homothetic center of these triangles is X(18513).

    X(18515) lies on these lines: {2,10742}, {3,10}, {30,11680}, {35,18526}, {36,381}, {55,7972}, {56,79}, {104,1621}, {952,6950}, {999,11551}, {1385,15071}, {4188,18357} et al


    X(18516) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND INNER JOHNSON

    Barycentrics    a^7 - a^6(b + c) - a^5(b^2 - 6bc + c^2) + a^4(b + c)(b^2 - 4bc + c^2) - a^3(b^4 - 6b^2c^2 + c^4) + a^2(b - c)^2(b + c)(b^2 + 4bc + c^2) + a(b^2 - c^2)^2(b^2 - 6bc + c^2) - (b - c)^4(b + c)^3 : :
    X(18516) = R*X(1) - R*X(3) - (2R - r)*X(4) = X(1) - 2 X(3) - 4 X(4) + X(57) = 2 X(5) - X(10269) = 2 X(546) - X(7956)

    X(18516) lies on these lines: {4,8}, {5,6256}, {10,16004}, {11,381}, {12,18542}, {30,1376}, {57,3585}, {104,6945}, {113,12889}, {119,1012}, {153,5603}, {382,6244}, {388,9955}, {498,13743}, {546,7956}, {1385,6893}, {1479,10944}, {1709,3359}, {2093,18406}, {2096,6839}, {2551,3579}, {2771,18391}, {3091,10785}, {3545,10584}, {3560,6690}, {3583,7962}, {3586,18528}, {3843,11928}, {4302,18524}, {5084,13624}, {5450,6681}, {5731,6965}, {5886,6957}, {6282,18529}, {6284,18518}, {6841,10523}, {6850,9956}, {6916,11231}, {6939,11230}, {7701,18395}, {10629,12915}, {10794,18502}, {10871,18500}, {10896,10948}, {10919,18509}, {10920,18511}, {10942,11496}, {10945,18520}, {10946,18522}, {10949,18544}, {11865,18495}, {11866,18497}, {11903,18507}, {12619,14647}, {15888,18545} et al

    X(18516) = reflection of X(i) in X(j) for these (i,j): (7956,546), (10269,5)
    X(18516) = X(999)-of-Ehrmann-mid-triangle
    X(18516) = X(999)-of-inner-Johnson-triangle
    X(18516) = X(10269)-of-Johnson-triangle
    X(18516) = {X(4),X(18480)}-harmonic conjugate of X(18517)


    X(18517) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND OUTER JOHNSON

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

    X(18517) lies on these lines: {1,18406}, {3,3925}, {4,8}, {5,1001}, {11,18544}, {12,381}, {30,958}, {55,6841}, {113,12890}, {443,13624}, {497,6849}, {498,18524}, {515,6917}, {1385,6826}, {1478,3649}, {1621,6990}, {1697,3583}, {2550,3579}, {2771,4295}, {3091,10786}, {3340,3585}, {3545,10585}, {3616,6900}, {3818,9052}, {3832,10599}, {3843,10738}, {4302,13743}, {4317,12773}, {4846,15232}, {5603,6894}, {5657,6895}, {5731,6901}, {5768,5885}, {5880,13369}, {5886,6835}, {6691,6911}, {6796,6862}, {6827,9956}, {6861,10902}, {6864,11230}, {6865,11231}, {6903,9780}, {9895,18420}, {10523,10896}, {10872,18500}, {10895,10954}, {10951,18520}, {10952,18522}, {10955,18542}, {11904,18507} et al

    X(18517) = X(3295)-of-Ehrmann-mid-triangle
    X(18517) = X(3295)-of-outer-Johnson-triangle
    X(18517) = {X(4),X(18480)}-harmonic conjugate of X(18516)


    X(18518) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 1st JOHNSON-YFF

    Trilinears    a^6 - a^5(b + c) - 2a^4(b + c)^2 + 2a^3(b + c)(b^2 + bc + c^2) + a^2(b^4 - 6b^2c^2 + c^4) - a(b - c)^2(b + c)(b^2 + 4bc + c^2) + 4bc(b^2 - c^2)^2 : :

    The homothetic center of these triangles is X(1479).

    X(18518) lies on these lines: {1,18491}, {3,10}, {4,3871}, {5,10585}, {8,6985}, {11,18543}, {12,381}, {30,3436}, {40,18528}, {55,10827}, {72,3426}, {405,18357}, {6284,18516}, {12587,18440} et al

    X(18518) = inner-Johnson-to-outer-Johnson similarity image of X(3)
    X(18518) = {X(3),X(18525)}-harmonic conjugate of X(18519)


    X(18519) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 2nd JOHNSON-YFF

    Trilinears    a^6 - a^5(b + c) - 2a^4(b^2 - 4bc + c^2) + 2a^3(b^3 - 2b^2c - 2bc^2 + c^3) + a^2(b^4 - 4b^3c + 10b^2c^2 - 4bc^3 + c^4) - a(b - c)^2(b^3 - 3b^2c - 3bc^2 + c^3) - 4bc(b^2 - c^2)^2 : :

    The homothetic center of these triangles is X(1478).

    X(18519) lies on these lines: {1,1898}, {3,10}, {4,10529}, {5,10584}, {11,381}, {12,18545}, {30,956}, {36,18491}, {56,10826}, {63,10914}, {80,1470}, {104,6911}, {153,6830}, {226,7373}, {382,5841}, {388,6841}, {399,12889}, {474,18357}, {5927,6913}, {12586,18440} et al

    X(18519) = outer-Johnson-to-inner-Johnson similarity image of X(3)
    X(18519) = {X(3),X(18525)}-harmonic conjugate of X(18518)


    X(18520) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND LUCAS HOMOTHETIC

    Barycentrics    6a^2b^2c^2(a - b - c)(a + b - c)(a - b + c)(a + b + c)[2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2] - 2S^3*[11a^8 + a^6(b^2 + c^2) - a^4(9b^4 + 10b^2c^2 + 9c^4) + a^2(7b^6 + 5b^4c^2 + 5b^2c^4 + 7c^6) - 2(b^2 - c^2)^2(5b^4 + 2b^2c^2 + 5c^4)] + 3S^2*[a^10 - 16a^8(b^2 + c^2) + 4a^6(3b^4 - 2b^2c^2 + 3c^4) - 2a^4(b^6 - 5b^4c^2 - 5b^2c^4 + c^6) + a^2(3b^8 + 4b^6c^2 - 30b^4c^4 + 4b^2c^6 + 3c^8) + 2(b^2 - c^2)^2(b^2 + c^2)^3] - S*[a^12 - 9a^10(b^2 + c^2) + 12a^8(2b^4 + 7b^2c^2 + 2c^4) - 2a^6(13b^6 + 9b^4c^2 + 9b^2c^4 + 13c^6) + 3a^4(b^2 - c^2)^2(3b^4 - 2b^2c^2 + 3c^4) + 3a^2(b^2 - c^2)^2(b^6 - 13b^4c^2 - 13b^2c^4 + c^6) - 2(b^2 - c^2)^6] : :

    X(18520) lies on these lines: {4,6462}, {5,9838}, {30,8222}, {113,12894}, {381,493}, {546,8212}, {1478,11932}, {1479,11930}, {3091,11846}, {3583,11947}, {3818,12590}, {3843,11949}, {6461,18522}, {8188,18492}, {8194,9818}, {8201,18495}, {8208,18497}, {8210,18525}, {8214,12699}, {8216,18509}, {8218,18511}, {9955,11377}, {10113,13215}, {10875,18500}, {10895,11951}, {10896,11953}, {10945,18516}, {10951,18517}, {11503,18491}, {11840,18502}, {11907,18507}, {11955,18542}, {11957,18544}, {12440,18480}

    X(18520) = X(18521)-of-Ehrmann-mid-triangle
    X(18520) = X(18521)-of-Lucas-homothetic-triangle


    X(18521) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND LUCAS HOMOTHETIC

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

    The homothetic center of these triangles is X(18520).

    X(18521) lies on these lines: {3,5490}, {381,493}, {6464,18523}


    X(18522) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND LUCAS(-1) HOMOTHETIC

    Barycentrics    6a^2b^2c^2(a - b - c)(a + b - c)(a - b + c)(a + b + c)[2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2] + 2S^3*[11a^8 + a^6(b^2 + c^2) - a^4(9b^4 + 10b^2c^2 + 9c^4) + a^2(7b^6 + 5b^4c^2 + 5b^2c^4 + 7c^6) - 2(b^2 - c^2)^2(5b^4 + 2b^2c^2 + 5c^4)] + 3S^2*[a^10 - 16a^8(b^2 + c^2) + 4a^6(3b^4 - 2b^2c^2 + 3c^4) - 2a^4(b^6 - 5b^4c^2 - 5b^2c^4 + c^6) + a^2(3b^8 + 4b^6c^2 - 30b^4c^4 + 4b^2c^6 + 3c^8) + 2(b^2 - c^2)^2(b^2 + c^2)^3] + S*[a^12 - 9a^10(b^2 + c^2) + 12a^8(2b^4 + 7b^2c^2 + 2c^4) - 2a^6(13b^6 + 9b^4c^2 + 9b^2c^4 + 13c^6) + 3a^4(b^2 - c^2)^2(3b^4 - 2b^2c^2 + 3c^4) + 3a^2(b^2 - c^2)^2(b^6 - 13b^4c^2 - 13b^2c^4 + c^6) - 2(b^2 - c^2)^6] : :

    X(18522) lies on these lines: {4,6463}, {5,9839}, {30,8223}, {113,12895}, {381,494}, {546,8213}, {1478,11933}, {1479,11931}, {3091,11847}, {3583,11948}, {3818,12591}, {3843,11950}, {6461,18520}, {8189,18492}, {8195,9818}, {8202,18495}, {8209,18497}, {8211,18525}, {8215,12699}, {8217,18509}, {8219,18511}, {9955,11378}, {10113,13216}, {10876,18500}, {10895,11952}, {10896,11954}, {10946,18516}, {10952,18517}, {11504,18491}, {11841,18502}, {11908,18507}, {11956,18542}, {11958,18544}, {12441,18480}

    X(18522) = X(18523)-of-Ehrmann-mid-triangle
    X(18522) = X(18523)-of-Lucas(-1)-homothetic-triangle


    X(18523) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND LUCAS(-1) HOMOTHETIC

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

    The homothetic center of these triangles is X(18522).

    X(18523) lies on these lines: {3,5491}, {381,494}, {6464,18521}


    X(18524) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND MANDART-INCIRCLE

    Trilinears    a^6 - a^5(b + c) - a^4(2b^2 + bc + 2c^2) + 2a^3(b + c)(b^2 + c^2) + a^2(b^4 - b^3c - 2b^2c^2 - bc^3 + c^4) - a(b - c)^2(b + c)^3 + 2bc(b^2 - c^2)^2 : :

    The homothetic center of these triangles is X(3583).

    X(18524) lies on these lines: {3,10}, {4,11849}, {5,1621}, {21,18357}, {30,100}, {35,13743}, {36,9897}, {40,5694}, {55,381}, {56,18526}, {80,5172}, {119,5842}, {140,9342}, {165,18528}, {191,210}, {227,18447}, {382,11248}, {399,12334}, {411,5690}, {474,3897}, {484,2771}, {498,18517}, {517,3689}, {547,5284}, {692,10540}, {758,12738}, {859,6740}, {944,6924}, {952,6905}, {999,11502}, {1001,5055}, {1482,3149}, {1656,4423}, {3198,18453}, {3295,11375}, {3869,3940}, {4302,18516}, {4413,5054}, {5531,5535}, {9654,11507}, {9655,11509}, {11231,15931}, {11249,12645}, {11383,18494}, {11490,18501}, {11494,18503}, {11510,18543}, {11848,18508}, {12164,12328}, {12329,18440} et al


    X(18525) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th MIXTILINEAR

    Trilinears    (2r/R) - 2 cos(B - C) + cos A : :
    Trilinears    3 cos A + 2 cos B + 2 cos C - 2 cos(B - C) - 2 : :
    Barycentrics    3a^4 - 2a^3(b + c) - a^2(b^2 - 4bc + c^2) + 2a(b - c)^2(b + c) - 2(b^2 - c^2)^2 : :
    X(18525) = 4 X(1) - 3 X(2) - 3 X(4) = 2 X(1) - X(3) - 2 X(4) = 2 X(1) - X(4) - 2 X(5) = 2 X(1) - 3 X(4) - X(8) = 2 X(1) - 3 X(381) = X(1) - 2 X(18480) = 2 X(546) - X(1483)

    X(18525) lies on these lines: {1,381}, {3,10}, {4,145}, {5,944}, {8,30}, {20,4678}, {40,1657}, {55,5441}, {56,80}, {65,9655}, {104,6924}, {113,12898}, {140,5731}, {382,517}, {399,12368}, {495,3486}, {496,3476}, {516,4701}, {518,18345}, {519,3830}, {546,1483}, {547,5550}, {942,5727}, {950,6767}, {956,5086}, {999,1837}, {1012,11015}, {1056,6849}, {1071,4004}, {1125,3655}, {1385,1656}, {1387,10591}, {1478,3649}, {1479,10944}, {1697,18540}, {1829,18494}, {1854,18340}, {2099,3585}, {2807,18439}, {3241,3845}, {3242,3818}, {3243,18482}, {3244,3656}, {5204,5442}, {5270,18406}, {5597,18497}, {5598,18495}, {5604,18511}, {5605,18509}, {5708,18391}, {8210,18520}, {8211,18522}, {9941,18503}, {9997,18500}, {10800,18502}, {11009,18513}, {11910,18507}, {12194,18501}, {12410,18534}, {12438,18508} et al

    X(18525) = reflection of X(i) in X(j) for these (i,j): (1,18480), (1483,546)
    X(18525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,18493), (9955,18492,381), (18518,18519,3), (18527,18529,381), (18542,18544,381)
    X(18525) = X(18526)-of-Ehrmann-mid-triangle
    X(18525) = X(18526)-of-5th-mixtilinear-triangle
    X(18525) = anti-Aquila-to-ABC similarity image of X(18480)


    X(18526) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 5th MIXTILINEAR

    Trilinears    (4r/R) - 2 cos(B - C) + cos A : :
    Trilinears    5 cos A + 4 cos B + 4 cos C - 2 cos(B - C) - 4 : :
    Barycentrics    5a^4 - 4a^3(b + c) - a^2(3b^2 - 8bc + 3c^2) + 4a(b - c)^2(b + c) - 2(b^2 - c^2)^2 : :
    X(18526) = 8 X(1) - 3 X(2) - 3 X(4) = 4 X(1) - 3 X(381)

    The homothetic center of these triangles is X(18525).

    X(18526) lies on these lines: {1,381}, {3,8}, {4,1483}, {5,3622}, {10,3655}, {20,5844}, {30,145}, {35,18515}, {40,15696}, {56,18524}, {79,2099}, {80,1388}, {119,3847}, {355,1125}, {376,3621}, {382,515}, {399,12898}, {517,1657}, {519,3534}, {3242,18440}, {3616,5055}, {7984,12902}, {11928,12737} et al

    X(18526) = {X(18543),X(18545)}-harmonic conjugate of X(381)


    X(18527) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd PEDAL OF X(1)

    Barycentrics    2a^4 - a^3(b + c) - 6a^2bc + a(b - c)^2(b + c) - 2(b^2 - c^2)^2 : :
    X(18527) = X(1) - X(57) - 4 X(497) = X(497) + X(5722) = X(999) + X(3586)

    X(18527) lies on these lines: {1,381}, {4,5045}, {8,7317}, {30,11019}, {65,14861}, {355,1058}, {382,3333}, {497,517}, {942,1479}, {999,3586}, {1478,5049}, {1538,18446} et al

    X(18527) = midpoint of X(i) and X(j) for these {i,j}: {497,5722}, {999,3586}
    X(18527) = {X(381),X(18525)}-harmonic conjugate of X(18529)
    X(18527) = X(18528)-of-Ehrmann-mid-triangle
    X(18527) = X(18528)-of-3rd-pedal-triangle-of-X(1)


    X(18528) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd PEDAL OF X(1)

    Trilinears    a^6 - 2a^5(b + c) - a^4(b^2 + 6bc + c^2) + 4a^3(b + c)(b^2 + c^2) - a^2(b^4 + 6b^2c^2 + c^4) - 2a(b - c)^2(b + c)^3 + (b^2 - c^2)^2(b^2 + 6bc + c^2) : :
    X(18528) = X(57) - 2 X(18491)

    The homothetic center of these triangles is X(18527).

    X(18528) lies on these lines: {1,381}, {3,5234}, {4,3870}, {5,10582}, {30,200}, {40,18518}, {42,18506}, {55,18540}, {57,18491}, {84,11499}, {165,18524}, {210,3587}, {3586,18516}, {6843,18446} et al

    X(18528) = reflection of X(57) in X(18491)


    X(18529) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd ANTIPEDAL OF X(1)

    Trilinears    a^6 - 2a^5(b + c) - a^4(b^2 + 10bc + c^2) + 4a^3(b + c)(b^2 + c^2) - a^2(b + c)^4 - 2a(b - c)^2(b + c)^3 + (b^2 - c^2)^2(b^2 + 14bc + c^2) : :

    X(18529) lies on these lines: {1,381}, {4,200}, {30,8580}, {40,3715}, {165,18491}, {1490,6843}, {1750,3925}, {6282,18516} et al

    X(18529) = {X(381),X(18525)}-harmonic conjugate of X(18527)
    X(18529) = X(18530)-of-Ehrmann-mid-triangle
    X(18529) = X(18530)-of-3rd-antipedal-triangle-of-X(1)


    X(18530) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd ANTIPEDAL OF X(1)

    Barycentrics    3a^4 - 2a^3(b + c) - a^2(b^2 + 12bc + c^2) + 2a(b - c)^2(b + c) - 2(b^2 - c^2)^2 : :

    The homothetic center of these triangles is X(18529).

    X(18530) lies on these lines: {1,381}, {3,4314}, {5,10578}, {30,10580}, {80,8162}, {382,5045} et al


    X(18531) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd PEDAL OF X(3)

    Barycentrics    [(a^4 - b^4 - c^4 + 2b^2c^2)^2 - 4a^4b^2c^2](b^2 + c^2 - a^2) : :
    X(18531) = 6 R^2 X(2) - 6 R^2 X(3) - (6 R^2 - SW) X(4) = X(3) - 2 X(1368) = X(4) + X(1370) = 2 X(5) - X(25)

    X(18531) lies on these lines: {2, 3}, {68, 5562}, {69, 265}, {113, 206}, {115, 577}, {125, 4549}, {127, 131}, {141, 18382}, {155, 6146}, {156, 13470}, {182, 18388}, {184, 1568}, {216, 5475}, {343, 14852}, {388, 18447}, {394, 18396}, {486, 10898}, {497, 18455}, {511, 18390}, {542, 11511}, {567, 11427}, {568, 11433}, {626, 14376}, {1038, 3585}, {1040, 3583}, {1060, 1478}, {1062, 1479}, {1092, 12118}, {1154, 6515}, {1216, 9927}, {1352, 2393}, {1503, 18451}, {1531, 4846}, {1660, 9833}, {1899, 13754}, {1992, 18449}, {1993, 12022}, {2549, 14961}, {2550, 18453}, {2790, 6033}, {2834, 10743}, {2888, 12606}, {2971, 13556}, {2974, 10748}, {3068, 18457}, {3069, 18459}, {3098, 7687}, {3284, 5309}, {3448, 7723}, {3519, 15077}, {3521, 15740}, {3564, 10602}, {3767, 10316}, {3818, 11574}, {3819, 18376}, {5158, 7753}, {5512, 15560}, {5878, 10575}, {5892, 7706}, {5907, 18381}, {5972, 11202}, {6288, 11487}, {6564, 11513}, {6565, 11514}, {6776, 18445}, {7728, 13203}, {7735, 10317}, {7748, 15075}, {7818, 15526}, {7998, 18392}, {7999, 18394}, {9306, 18400}, {9969, 15812}, {10113, 13416}, {10116, 15083}, {10319, 18406}, {10540, 11206}, {10634, 18582}, {10635, 18581}, {11411, 18436}, {11442, 11459}, {11457, 12111}, {11488, 18468}, {11489, 18470}, {11515, 16808}, {11516, 16809}, {11550, 15030}, {11793, 18383}, {12091, 14731}, {12134, 17814}, {12160, 13292}, {12162, 14216}, {12318, 12429}, {12324, 18439}, {12370, 16266}, {13346, 13403}, {13562, 18440}, {14649, 14689}, {14841, 14843}, {14912, 15087}, {15133, 15738}, {17811, 18405}

    X(18531) = midpoint of X(4) and X(1370)
    X(18531) = reflection of X(i) in X(j) for these (i,j): (3,1368), (25,5)
    X(18531) = isogonal conjugate of X(18532)
    X(18531) = complement of X(18533)
    X(18531) = anticomplement of X(6644)
    X(18531) = inverse-in-Johnson-circle of X(10297)
    X(18531) = inverse-in-orthocentroidal-circle of X(18420)
    X(18531) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(16387)
    X(18531) = homothetic center of Ehrmann vertex-triangle and 6th anti-mixtilinear
    X(18531) = homothetic center of Ehrmann side-triangle and anti-inverse-in-incircle triangle
    X(18531) = X(18534)-of-Ehrmann-mid-triangle
    X(18531) = X(18534)-of-3rd-pedal-triangle-of-X(3)
    X(18531) = X(2093)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18531) = X(999)-of-Ehrmann-side-triangle if ABC is acute
    X(18531) = X(25)-of-Johnson-triangle
    X(18531) = X(1370)-of-Euler-triangle
    X(18531) = X(1368)-of-X3-ABC-reflections-triangle
    X(18531) = X(6282)-of-orthic-triangle if ABC is acute
    X(18531) = orthic-to-Ehrmann-vertex similarity image of X(18533)
    X(18531) = barycentric product X(69)*X(1478)*X(1479)
    X(18531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18563, 20), (4, 3545, 7394), (4, 18537, 381), (5, 7502, 10201), (381, 382, 18535), (381, 12083, 11799), (381, 18534, 1596), (382, 7529, 6756), (1597, 1598, 16542), (3091, 3547, 10024), (6826, 6851, 7490), (7502, 10201, 7493), (7526, 13371, 3541), (10297, 15760, 381), (10750, 10751, 10297), (14807, 14808, 7464)


    X(18532) = ISOGONAL CONJUGATE OF X(18531)

    Barycentrics    a^2/{[(a^4 - b^4 - c^4 + 2b^2c^2)^2 - 4a^4b^2c^2](b^2 + c^2 - a^2)} : :

    X(18532) lies on the Jerabek hyperbola and these lines: {24,68}, {25,265}, {65,1061}, {69,186}, {73,3422}, {378,4846} et al

    X(18532) = isogonal conjugate of X(18531)
    X(18532) = anti-Ara-to-ABC similarity image of X(18385)


    X(18533) = ANTICOMPLEMENT OF X(18531)

    Barycentrics    [3a^6 - 5a^4(b^2 + c^2) + a^2(b^2 + c^2)^2 + (b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :
    X(18533) = 12 R^2 X(2) + 2 SW X(3) - (6 R^2 - SW) X(4) = 2 X(3) - X(1370) = X(4) - 2 X(25) = X(20) + X(7500)
    X(18533) = X(3)-3*X(11178), 5*X(4)+3*X(69), X(4)+3*X(1352), X(4)-3*X(3818), 5*X(5)-3*X(597), 3*X(5)-X(8550), 3*X(6)-7*X(3851), X(69)-5*X(1352), X(69)+5*X(3818), 4*X(140)-3*X(5092), X(140)-3*X(18358), 5*X(575)-6*X(597), 3*X(575)-2*X(8550), 9*X(597)-5*X(8550), X(5092)-4*X(18358), 3*X(11188)+5*X(15058)

    X(18533) lies on these lines: {2,3}, {33,4302}, {34,4299}, {35,11392}, {36,11393}, {52,12118}, {54,14542}, {64,16655}, {66,74}, {99,317}, {112,393}, {132,14649}, {185,9833}, {193,1986}, {232,7737}, {254,8883}, {264,14907}, {477,10423}, {570,3087}

    X(18533) = midpoint of X(i) and X(j) for these {i,j}: {20,7500}, {183,18440)
    X(18533) = reflection of X(i) in X(j) for these (i,j): (4,25), (1370,3)
    X(18533) = anticomplement of X(18531)
    X(18533) = inverse-in-polar-circle of X(10297)
    X(18533) = {X(3),X(4)}-harmonic conjugate of X(3541)
    X(18533) = X(25)-of-anti-Euler-triangle
    X(18533) = X(57)-of-circumorthic-triangle if ABC is acute
    X(18533) = X(1370)-of-ABC-X3-reflections-triangle
    X(18533) = X(2093)-of-orthic-triangle if ABC is acute
    X(18533) = orthic-to-circumorthic similarity image of X(25)
    X(18533) = Ehrmann-vertex-to-orthic similarity image of X(18531)
    X(18533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 15069, 576), (10516, 18440, 182)


    X(18534) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd PEDAL OF X(3)

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

    The homothetic center of these triangles is X(18531).

    X(18534) lies on these lines: {2,3}, {34,9645}, {49,1660}, {52,1498}, {115,1609}, {154,13352}, {265,9919}, {511,18451}, {12410,18525} et al

    X(18534) = {X(381),X(1657)}-harmonic conjugate of X(18536)


    X(18535) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd ANTIPEDAL OF X(3)

    Trilinears    cos A - 4 sec A : :
    Trilinears    (sec A)(4 - cos2A) : :
    Barycentrics    a^2[(a^2 - b^2 - c^2)^2 - 16b^2c^2]/(b^2 + c^2 - a^2) : :

    X(18535) lies on these lines: {2,3}, {6,3531}, {33,6767}, {34,7373}, {115,8573}, {182,13570}, {389,12315}, {399,2971}, {1112,12308}, {1159,1905}, {1181,3527}, {1351,8681}, {1619,18405}, {3066,11820}, {3092,6417}, {3093,6418}, {3199,9605} et al

    X(18535) = {X(381),X(382)}-harmonic conjugate of X(18531)
    X(18535) = X(18536)-of-Ehrmann-mid-triangle
    X(18535) = X(18536)-of-3rd-antipedal-triangle-of-X(3)
    X(18535) = endo-homothetic center of these triangles: Ehrmann mid and 3rd antipedal of X(4); The homothetic center is X(18537).
    X(18535) = trilinear product X(4)*X(999)*X(3295)


    X(18536) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd ANTIPEDAL OF X(3)

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

    The homothetic center of these triangles is X(18535).

    X(18536) lies on these lines: {2,3}, {216,15484}, {1038,9655}, {1216,11850}, {1350,18390}, {5891,11574} et al

    X(18536) = {X(381),X(1657)}-harmonic conjugate of X(18534)


    X(18537) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND 3rd ANTIPEDAL OF X(4)

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

    X(18537) lies on these lines: {2,3}, {113,3618}, {146,9826}, {800,5475}, {1352,8681}, {1899,15030}, {3818,18489}, {6337,15505}, {6776,18451}, {15585,18382} et al

    X(18537) = {X(3),X(381)}-harmonic conjugate of X(1596)
    X(18537) = X(18535)-of-Ehrmann-mid-triangle
    X(18537) = X(18535)-of-3rd-antipedal-triangle-of-X(4)


    X(18538) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND X(2)-QUADSQUARES

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

    X(18538) lies on these lines: {2,6398}, {4,3590}, {5,6}, {20,3316}, {30,590}, {140,3070}, {371,546}, {372,3628}, {381,3068}, {631,1131}, {632,1152}, {1151,3627}, {3069,5055}, {3071,3850}, {3090,3312}, {3091,3311}, {7969,18357} et al

    X(18538) = X(18539)-of-Ehrmann-mid-triangle
    X(18538) = X(18539)-of-X(2)-quadsquares-triangle


    X(18539) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND X(2)-QUADSQUARES

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

    The homothetic center of these triangles is X(18538).

    X(18539) lies on these lines: {4,193}, {30,492}, {230,6565}, {381,3068}, {3830,5860} et al


    X(18540) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND TANGENTIAL-OF-EXCENTRAL

    Trilinears    a^6 - 3a^4(b - c)^2 + a^2(3b^4 + 2b^2c^2 + 3c^4) - (b^2 - c^2)^2(b^2 + 6bc + c^2) : :

    X(18540) lies on these lines: {1,1898}, {2,7171}, {3,1750}, {4,63}, {5,84}, {6,18506}, {9,30}, {34,18477}, {40,382}, {46,1749}, {55,18528}, {57,381}, {165,18491}, {515,6930}, {517,5223}, {1697,18525}, {1706,18357}, {1709,3359}, {2003,18451}, {2270,18505}, {6912,18446} et al

    X(18540) = X(18541)-of-Ehrmann-mid-triangle
    X(18540) = X(18541)-of-tangential-of-excentral-triangle


    X(18541) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND TANGENTIAL-OF-EXCENTRAL

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

    The homothetic center of these triangles is X(18540).

    X(18541) lies on these lines: {1,1657}, {3,226}, {4,5708}, {5,5435}, {7,30}, {20,6147}, {46,9654}, {55,15228}, {56,79}, {57,381}, {65,9655}, {140,5714}, {376,5719}, {377,3927}, {382,942}, {388,12702} et al


    X(18542) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND INNER YFF TANGENTS

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

    X(18542) lies on these lines: {1,381}, {3,119}, {4,3871}, {5,10584}, {8,10711}, {12,18516}, {30,5552}, {113,12905}, {382,11248}, {546,10531}, {944,11729}, {1470,9655}, {1478,10958}, {1479,10956}, {1656,10269}, {1657,2077}, {2098,12611}, {3091,10805}, {3545,10586}, {3583,10965}, {3585,11509}, {3818,12594}, {3832,10596}, {3843,12000}, {3845,11239}, {5055,10200}, {5554,18357}, {5587,15071}, {5687,11698}, {5777,5790}, {6735,12702}, {6841,10590}, {6882,12667}, {6929,16202}, {6968,10943}, {7681,12001}, {9818,10834}, {10113,13217}, {10878,18500}, {10929,18509}, {10930,18511}, {10955,18517}, {11231,16209}, {11914,18507}, {11955,18520}, {11956,18522}, {12645,12751}

    X(18542) = {X(381),X(18525)}-harmonic conjugate of X(18544)
    X(18542) = X(18543)-of-Ehrmann-mid-triangle
    X(18542) = X(18543)-of-inner-Yff-tangents-triangle
    X(18542) = endo-homothetic center of these triangles: Ehrmann mid and outer Yff; The homothetic center is X(10896).


    X(18543) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND INNER YFF TANGENTS

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

    The homothetic center of these triangles is X(18542).

    X(18543) lies on these lines: {1,381}, {3,3434}, {4,12001}, {5,10587}, {11,18518}, {30,10529}, {56,18499}, {382,2829}, {6831,12000}, {11510,18524}, {12595,18440} et al

    X(18543) = {X(381),X(18526)}-harmonic conjugate of X(18545)


    X(18544) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND OUTER YFF TANGENTS

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

    X(18544) lies on these lines: {1,381}, {3,2886}, {4,10529}, {5,10585}, {11,18517}, {30,10527}, {56,18407}, {113,12906}, {382,11249}, {497,6841}, {546,10532}, {1478,10959}, {1479,10957}, {1656,4423}, {1657,11012}, {3091,10806}, {3526,10902}, {3545,10587}, {3583,10966}, {3818,12595}, {3832,10597}, {3843,10742}, {3845,11240}, {3847,5055}, {5044,5790}, {6734,12702}, {6917,16203}, {7548,7967}, {7680,12000}, {7741,11510}, {8148,13463}, {9655,12773}, {9818,10835}, {10113,13218}, {10879,18500}, {10916,11235}, {10931,18509}, {10932,18511}, {10949,18516}, {11231,16208}, {11915,18507}, {11957,18520}, {11958,18522}, {12611,12750}

    X(18544) = {X(381),X(18525)}-harmonic conjugate of X(18542)
    X(18544) = X(18545)-of-Ehrmann-mid-triangle
    X(18544) = X(18545)-of-outer-Yff-tangents-triangle
    X(18544) = endo-homothetic center of these triangles: Ehrmann mid and inner Yff; The homothetic center is X(10895).


    X(18545) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN MID AND OUTER YFF TANGENTS

    Barycentrics    a^7 - a^6(b + c) + 12a^5bc - 10a^4bc(b + c) - a^3(3b^4 + 4b^3c - 18b^2c^2 + 4bc^3 + 3c^4) + 3a^2(b - c)^2(b + c)(b^2 + 4bc + c^2) + 2a(b^2 - c^2)^2(b^2 - 4bc + c^2) - 2(b - c)^4(b + c)^3 : :

    The homothetic center of these triangles is X(18544).

    X(18545) lies on these lines: {1,381}, {3,3436}, {4,12000}, {5,10586}, {12,18519}, {30,10528}, {119,1656}, {153,3560}, {355,12609}, {382,5842}, {399,12905}, {1478,10955}, {1482,12608}, {1532,12001}, {1657,11248}, {9655,11509}, {12594,18440}, {15888,18516} et al

    X(18545) = {X(381),X(18526)}-harmonic conjugate of X(18543)


    X(18546) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND 1st BROCARD

    Barycentrics    a^4 - 2b^4 - 2c^4 + 6b^2c^2 : :
    X(18546) = X(2) + X(4) - X(7775) = 2 X(381) - X(7775)

    X(18546) lies on these lines: {2,99}, {4,754}, {5,7781}, {30,5171}, {32,11361}, {76,7818}, {194,15031}, {316,17131}, {325,18424}, {381,538}, {382,7780}, {524,3818}, {546,7759}, {591,6565}, {625,7908}, {626,16041}, {1007,14148} et al

    X(18546) = reflection of X(7775) in X(381)
    X(18546) = X(9766)-of-Ehrmann-mid-triangle
    X(18546) = X(9766)-of-1st-Brocard-triangle


    X(18547) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND 6th BROCARD

    Barycentrics    4a^12 - 2a^10(b^2 + c^2) - 4a^8b^2c^2 + 5a^6b^2c^2(b^2 + c^2) + a^4(2b^8 + 4b^6c^2 - 27b^4c^4 + 4b^2c^6 + 2c^8) + 21a^2b^2c^2(b^2 - c^2)^2(b^2 + c^2) - (b^2 - c^2)^2(4b^8 + 2b^6c^2 - 3b^4c^4 + 2b^2c^6 + 4c^8) : :

    X(18547) lies on these lines: {194,3818}, {3845,7812}, {9166,14880}


    X(18548) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND 6th ANTI-BROCARD

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

    X(18548) lies on these lines: {2,32}, {538,18502}, {576,732}, {3095,7781}, {3398,7843}, {3849,12054}, {5034,7872}, {7697,7751}, {7759,10796}, {7764,12110}, {8178,14881}, {9766,18501}


    X(18549) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND INNER GARCIA

    Barycentrics    4a^7 - 6a^6(b + c) - a^5(2b^2 - 13bc + 2c^2) + 2a^4(b + c)(4b^2 - 7bc + 4c^2) - a^3(8b^4 + b^3c - 12b^2c^2 + bc^3 + 8c^4) + 2a^2(b - c)^2(b + c)(b^2 + 5bc + c^2) + 6a(b - c)^4(b + c)^2 - 4(b - c)^4(b + c)^3 : :

    X(18549) lies on these lines: {3,15079}, {8,10738}, {517,5560}, {1482,1699}, {5076,16150}, {6246,12645}, {6913,11517}, {6980,10543}, {7704,18493}, {11928,12737}


    X(18550) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND ORTHOCENTROIDAL

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

    X(18550) lies on these lines: {3,1531}, {4,15003}, {5,11270}, {6,3830}, {30,3431}, {54,382}, {64,3843}, {65,18514}, {67,3818}, {74,381} et al


    X(18551) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: EHRMANN MID AND ANTI-ORTHOCENTROIDAL

    Barycentrics    a^2[a^8 + 2a^6(b^2 + c^2) - a^4(12b^4 - 25b^2c^2 + 12c^4) + a^2(14b^6 + b^4c^2 + b^2c^4 + 14c^6) - (b^2 - c^2)^2(5b^4 + 38b^2c^2 + 5c^4)] : :

    X(18551) lies on these lines: {3,5888}, {6,12308}, {146,381}, {399,11702}, {599,3818}, {3426,5055}, {18325,18358} et al


    X(18552) = CENTER OF THE PERSPECONIC OF THESE TRIANGLES: ABC AND EHRMANN MID

    Barycentrics    (a^4 - 2b^4 - 2c^4 + a^2b^2 + a^2c^2 + 4b^2c^2)[2a^8 + 7a^6(b^2 + c^2) - a^4(15b^4 + 4b^2c^2 + 15c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^2(5b^4 + 8b^2c^2 + 5c^4)] : :

    X(18552) lies on these lines: {30,3589}, {264,3545}, {381,5158}, et al


    X(18553) = CENTER OF THE PERSPECONIC OF THESE TRIANGLES: EHRMANN MID AND JOHNSON

    Barycentrics    2a^6 - a^4(b^2 + c^2) + a^2(b^4 + 4b^2c^2 + c^4) - 2(b^2 - c^2)^2(b^2 + c^2) : :
    X(18553) = 12 X(2) - 5 X(3) - 3 X(6) = 3 X(3) + 4 X(4) - 3 X(6) = 2 X(5) - X(575) = 2 X(141) - X(14810) = X(576) + X(15069)

    X(18553) lies on these lines: {3, 11178}, {4, 69}, {5, 542}, {6, 3851}, {39, 6287}, {51, 3410}, {67, 3521}, {98, 16984}, {114, 5939}, {140, 1503}, {141, 550}, {182, 1656}, {343, 10301}, {373, 3448}, {381, 576}, {382, 599}, {524, 546}, {924, 18313}, {1350, 5073}, {1351, 18555}, {1657, 3098}, {1899, 6688}, {1992, 3855}, {2030, 7746}, {2393, 18383}, {3090, 11179}, {3091, 5032}, {3292, 5169}, {3519, 11808}, {3529, 7936}, {3564, 3850}, {3619, 10299}, {3628, 10168}, {3763, 15720}, {3819, 11550}, {3843, 11477}, {3854, 14853}, {3858, 5480}, {3917, 5189}, {5012, 7570}, {5038, 7603}, {5056, 6776}, {5059, 10519}, {5068, 5921}, {5070, 10541}, {5094, 9306}, {5891, 7574}, {5943, 11442}, {6000, 18431}, {6054, 7608}, {6102, 16776}, {6249, 7762}, {7401, 15012}, {7514, 15581}, {7527, 12584}, {7528, 16625}, {7547, 8541}, {7607, 8590}, {7747, 15993}, {8675, 18039}, {8681, 9927}, {8786, 10486}, {9019, 11591}, {9140, 16042}, {9225, 15820}, {9730, 14982}, {9751, 16988}, {9822, 13382}, {9971, 18436}, {9976, 14644}, {9977, 15052}, {10991, 13335}, {11257, 11261}, {11572, 15056}, {11649, 15060}, {11793, 14791}, {13108, 18500}, {13851, 15073}, {14216, 17704}, {14269, 15533}, {14866, 15098}, {16194, 18325}

    X(18553) = midpoint of X(576) and X(15069)
    X(18553) = reflection of X(1) in X(j) for these (i,j): (575,5), (14810,141)
    X(18553) = X(575)-of-Johnson-triangle


    X(18554) = X(30)X(16240)∩X(264)X(3545)

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

    Let MAMBMC be the Ehrmann mid-triangle. Let A' be the center of conic {{A,B,C,MB,MC}} and define B', C' cyclically. The lines AA', BB', CC' concur in X(18554).

    X(18554) lies on these lines: {30,16240}, {264,3545}

    X(18554) = barycentric quotient X(30)*X(381)/X(376)
    X(18554) = barycentric quotient X(18487)/X(376)


    X(18555) = X(4)X(52)∩X(113)X(13431)

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

    Let MAMBMC be the Ehrmann mid-triangle and JAJBJC the Johnson triangle. Let A' be the center of conic {{JA,JB,JC,MB,MC}} and define B', C' cyclically. The lines AA', BB', CC' concur in X(4). The lines JAA', JBB', JCC', concur in X(18555).

    X(18555) lies on these lines: {4,52}, {113,13431}, {265,13598}, {382,14864}, {1216,2889}, {10619,12370}, {10625,18390}, {13421,18572} et al


    X(18556) = X(2) OF EHRMANN CROSS-TRIANGLE

    Barycentrics    (b^2 - c^2)[4a^8 - 7a^6(b^2 + c^2) + a^4(3b^2 + c^2)(b^2 + 3c^2) - a^2(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) + b^8 - b^6c^2 - b^2c^6 + c^8] : :
    X(18556) = 2 X(3) - X(5664) = X(20) + X(2394)

    X(18556) lies on these lines: {3,523}, {20,2394}, {122,127}, {512,5891}, {1640,2420}, {1649,11007}, {3268,9517}, {7493,9209}

    X(18556) = midpoint of X(20) and X(2394)
    X(18556) = reflection of X(5664) in X(3)
    X(18556) = X(2)-of-Ehrmann-cross-triangle
    X(18556) = X(14566)-of-anti-Euler-triangle
    X(18556) = X(5664)-of-ABC-X3-reflections-triangle


    X(18557) = TRILINEAR PRODUCT OF VERTICES OF EHRMANN CROSS-TRIANGLE

    Barycentrics    b^2c^2(b^2 - c^2)(b^2 + c^2 - a^2)^2[2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2]/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :

    X(18557) lies on these lines: {265,8673}, {343,525}, {648,14618}, {1568,9033} et al

    X(18557) = barycentric product X(30)*X(69)^2*X(76)*X(523)*X(1989)
    X(18557) = barycentric product X(i)*X(j) for these {i,j}: {76,18558}, {328,9033}, {3265,14254}, {11064,14592}
    X(18557) = barycentric quotient X(18558)/X(6)


    X(18558) = BARYCENTRIC PRODUCT OF VERTICES OF EHRMANN CROSS-TRIANGLE

    Barycentrics    (b^2 - c^2)(b^2 + c^2 - a^2)^2[2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2]/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :

    X(18558) lies on these lines: {112,476}, {216,647}, {850,5664}, {1989,6587} et al

    X(18558) = barycentric product X(30)*X(69)^2*X(523)*X(1989)
    X(18558) = barycentric product X(i)*X(j) for these {i,j}: {6,18557}, {265,9033}, {3265,14583}, {11064,14582}
    X(18558) = barycentric quotient X(i)/X(j) for these (i,j): (9033,340), (14583,107), (18557,76)


    X(18559) = EHRMANN-VERTEX-TO-ORTHIC SIMILARITY IMAGE OF X(2)

    Barycentrics    [2a^6 - 3a^4(b^2 + c^2) + a^2b^2c^2 + (b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :
    X(18559) = 2 X(5) - X(18564)

    X(18559) lies on these lines: {2,3}, {13,10632}, {14,10633}, {52,12278}, {74,11550}, {115,10986}, {232,14537}, {264,11057}, {317,18354}, {389,12289}, {539,5889}, {541,12140}, {542,6403}, {1235,7811} et al

    X(18559) = reflection of X(18564) in X(5)
    X(18559) = inverse-in-polar-circle of X(18572)
    X(18559) = X(18564)-of-Johnson-triangle


    X(18560) = EHRMANN-VERTEX-TO-ORTHIC SIMILARITY IMAGE OF X(382)

    Barycentrics    [2a^6 - 3a^4(b^2 + c^2) + 6a^2b^2c^2 + (b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :

    X(18560) lies on these lines: {2,3}, {33,10483}, {49,7728}, {54,10721}, {64,11457}, {74,5894}, {112,5254}, {113,12038}, {185,1986}, {232,7756}, {541,10116}, {567,3521}, {1105,1300}, {1204,18390}, {12121,18350} et al

    X(18560) = Kosnita-to-orthic similarity image of X(20)
    X(18560) = harmonic center of polar and second Droz-Farny circles
    X(18560) = inverse-in-2nd-Droz-Farny-circle of X(186)
    X(18560) = X(5697)-of-orthic-triangle if ABC is acute
    X(18560) = {X(3),X(4)}-harmonic conjugate of X(403)


    X(18561) = EHRMANN-VERTEX-TO-EHRMANN-SIDE SIMILARITY IMAGE OF X(2)

    Barycentrics    (b^2 + c^2 - a^2)[4a^8 - 2a^6(b^2 + c^2) - a^4(6b^4 - 9b^2c^2 + 6c^4) + 2a^2(b^2 - c^2)^2(b^2 + c^2) + 2(b^2 - c^2)^4] : :
    X(18561) = X(3) - 2 X(18564) = X(18562) + X(18564) = 2 X(18563) - X(18564)

    X(18561) lies on these lines: {2,3}, {339,11057}, {9927,18442}, {10317,11648}, {11645,18438}, {12111,12291}, {12902,15085}

    X(18561) = midpoint of X(18562) and X(18564)
    X(18561) = reflection of X(i) in X(j) for these (i,j): (3,18564), (18564,18563)
    X(18561) = {X(18562),X(18563)}-harmonic conjugate of X(3)
    X(18561) = Ehrmann-mid-to-ABC similarity image of X(18559)
    X(18561) = X(18564)-of-X3-ABC-reflections-triangle
    X(18561) = X(18559)-of-Ehrmann-side-triangle


    X(18562) = EHRMANN-VERTEX-TO-EHRMANN-SIDE SIMILARITY IMAGE OF X(3)

    Barycentrics    [2a^8 - a^6(b^2 + c^2) - a^4(3b^4 - 5b^2c^2 + 3c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18562) = (7*R^2-2*SW)*X(3) - 2*(4*R^2-SW)*X(4) = X(3) - 2 X(18563) = 2 X(3) - X(18565) = 2 X(18561) - X(18564)

    X(18562) lies on these lines: {2,3}, {68,11559}, {265,7689}, {339,7802}, {568,13403}, {1092,12121}, {18439,18400} et al

    X(18562) = reflection of X(i) in X(j) for these (i,j): (3,18563), (1209,18561), (18565,3)
    X(18562) = X(5903)-of-Ehrmann-side-triangle if ABC is acute
    X(18562) = X(18563)-of-X3-ABC-reflections-triangle
    X(18562) = {X(3),X(4)}-harmonic conjugate of X(10254)


    X(18563) = EHRMANN-VERTEX-TO-EHRMANN-SIDE SIMILARITY IMAGE OF X(5)

    Barycentrics    [2a^8 - a^6(b^2 + c^2) - a^4(3b^4 - 4b^2c^2 + 3c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18563) = (3*R^2-SW)*X(3) - (4*R^2-SW)*X(4) = X(3) - 2 X(12605) = X(3) + X(18562) = 2 X(5) - X(6240) = X(18561) + X(18564)

    X(18563) lies on these lines: {2,3}, {52,13403}, {113,10282}, {127,7842}, {131,14103}, {265,12359}, {339,7750}, {567,12233}, {568,12241}, {1060,10483}, {1503,18438}, {6253,18453}, {17845,18451} et al

    X(18563) = midpoint of X(i) and X(j) for these {i,j}: {3,18562}, {18561,18564}
    X(18563) = reflection of X(i) in X(j) for these (i,j): (3,12605), (6240,5)
    X(18563) = X(65)-of-Ehrmann-side-triangle if ABC is acute
    X(18563) = X(6240)-of-Johnson-triangle
    X(18563) = X(12605)-of-X3-ABC-reflections-triangle
    X(18563) = {X(3),X(4)}-harmonic conjugate of X(10024)


    X(18564) = EHRMANN-VERTEX-TO-EHRMANN-SIDE SIMILARITY IMAGE OF X(381)

    Barycentrics    [2a^8 - a^6(b^2 + c^2) - 3a^4(b^4 - b^2c^2 + c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18564) = X(3) + X(18561) = 2 X(5) - X(18559) = 2 X(18561) - X(18562) = X(18561) - 2 X(18563)

    X(18564) lies on these lines: {2,3}, {13,18468}, {14,18470}, {115,18472}, {339,7811}, {539,18436}, {542,18438}, {577,1989}, {1531,18475}, {3581,18390}, {11750,18439} et al

    X(18564) = midpoint of X(3) and X(18561)
    X(18564) = reflection of X(i) in X(j) for these (i,j): (1207,18561), (18559,5), (18561,18563)
    X(18564) = X(5902)-of-Ehrmann-side-triangle if ABC is acute
    X(18564) = X(18559)-of-Johnson-triangle
    X(18564) = {X(3),X(18562)}-harmonic conjugate of X(18565)


    X(18565) = EHRMANN-VERTEX-TO-EHRMANN-SIDE SIMILARITY IMAGE OF X(382)

    Barycentrics    [2a^8 - a^6(b^2 + c^2) - a^4(3b^4 - 7b^2c^2 + 3c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^4](b^2 + c^2 - a^2) : :
    X(18565) = (9*R^2-2*SW)*X(3) - 2*(4*R^2-SW)*X(4) = 2 X(3) - X(18562)

    X(18565) lies on these lines: {2,3}, {184,3521}, {265,1204}, {1147,12121}, {2777,18439}, {4846,15002} et al

    X(18565) = reflection of X(18562) in X(3)
    X(18565) = X(5697)-of-Ehrmann-side-triangle if ABC is acute
    X(18565) = {X(3),X(4)}-harmonic conjugate of X(10255)
    X(18565) = {X(3),X(18562)}-harmonic conjugate of X(18564)


    X(18566) = EHRMANN-SIDE-TO-EHRMANN-VERTEX SIMILARITY IMAGE OF X(2)

    Barycentrics    [2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2]*[a^6 - a^2(3b^4 - 5b^2c^2 + 3c^4) + 2(b^2 - c^2)^2(b^2 + c^2)] : :
    X(18566) = X(4) + X(18568) = X(18377) - 2 X(18568) = 2 X(18567) - X(18568)

    X(18566) lies on these lines: {2,3}, {1539,11550}, {5663,18376}, {18356,18379}

    X(18566) = midpoint of X(4) and X(18568)
    X(18566) = reflection of X(i) in X(j) for these (i,j): (18377,18568), (18568,18567)
    X(18566) = orthic-to-Ehrmann-vertex similarity image of X(3845)
    X(18566) = X(18568)-of-Euler-triangle
    X(18566) = X(10246)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18566) = X(18324)-of-Ehrmann-mid-triangle


    X(18567) = EHRMANN-SIDE-TO-EHRMANN-VERTEX SIMILARITY IMAGE OF X(5)

    Barycentrics    a^2[2a^8 - 4a^6(b^2 + c^2) + 6a^4b^2c^2 + a^2(4b^6 - 3b^4c^2 - 3b^2c^4 + 4c^6) - (b^2 - c^2)^2(2b^4 + 3b^2c^2 + 2c^4)] : :
    X(18567) = X(4) + X(18377) = 2 X(5) - X(15331) = X(18566) + X(18568)

    X(18567) lies on these lines: {2,3}, {52,10113}, {1531,5876}, {1539,11381}, {3818,12061}, {5663,18383}, {18392,18436} et al

    X(18567) = midpoint of X(i) and X(j) for these {i,j}: {4,18377}, {18566,18568}
    X(18567) = reflection of X(15331) in X(5)
    X(18567) = X(1385)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18567) = X(1658)-of-Ehrmann-mid-triangle
    X(18567) = X(15331)-of-Johnson-triangle
    X(18567) = circumorthic-to-orthic similarity image of X(18377)


    X(18568) = EHRMANN-SIDE-TO-EHRMANN-VERTEX SIMILARITY IMAGE OF X(381)

    Barycentrics    3a^10 - 3a^8(b^2 + c^2) - a^6(6b^4 - 8b^2c^2 + 6c^4) + 2a^4(b^2 + c^2)(3b^4 - 5b^2c^2 + 3c^4) + a^2(b^2 - c^2)^2(3b^4 - 4b^2c^2 + 3c^4) - 3(b^2 - c^2)^4(b^2 + c^2) : :
    X(18568) = 3(J^2 - 1) X(2) - 2(J^2 - 2) X(3) + 2(J^2 - 2) X(4) = 3(J^2 - 3) X(2) - 2(J^2 - 4) X(3) + 2(J^2 - 1) X(4) = 3 X(2) - X(3) + 2 X(4) - X(26) = X(4) - 2 X(18566) = 2 X(5) - X(18324) = 2 X(381) - X(10201) = X(18377) + X(18566) = X(18566) - 2 X(18567)

    X(18568) lies on these lines: {2,3}, {68,18379}, {542,12596}, {1531,18474}, {11442,18430}, {13754,18376}, {13851,14831}

    X(18568) = midpoint of X(18377) and X(18566)
    X(18568) = reflection of X(i) in X(j) for these (i,j): (4,18566), (10201,381), (18324,5), (18566,18567)
    X(18568) = X(3576)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18568) = X(14070)-of-Ehrmann-mid-triangle
    X(18568) = X(18324)-of-Johnson-triangle
    X(18568) = X(18566)-of-anti-Euler-triangle
    X(18568) = inverse-in-Johnson-circle of X(10295)
    X(18568) = {X(10750),X(10751)}-harmonic conjugate of X(10295)


    X(18569) = EHRMANN-SIDE-TO-EHRMANN-VERTEX SIMILARITY IMAGE OF X(382)

    Barycentrics    a^10 - a^8(b^2 + c^2) - 2a^6(b^4 + c^4) + 2a^4(b^6 + c^6) + a^2(b^2 - c^2)^2(b^4 + c^4) - (b^2 - c^2)^4(b^2 + c^2) : :
    X(18569) = 3(J^2 - 3) X(2) - 2(J^2 - 4) X(3) = 3 X(2) - X(3) - X(26) = 2 X(3) + (J^2 - 3) X(4) = X(3) - 2 X(13371) = R^2*X(3) + (3*R^2-SW)*X(4) = X(4) + X(14790) = X(4) - 2 X(18377) = X(20) - 2 X(11250) = 2 X(5) - X(26) = X(382) + X(12085)

    X(18569) lies on these lines: {2,3}, {50,2165}, {68,1154}, {70,265}, {156,5654}, {184,11750}, {485,11265}, {486,11266}, {511,9927}, {542,15083}, {568,15800}, {569,3574}, {1147,18400}, {1352,9973}, {1479,8144}, {1531,11381}, {1568,10539}, {1853,12163}, {1899,6102}, {2888,18432}, {5446,18390}, {5448,6759}, {5562,11572}, {6146,12161}, {6193,12319}, {7706,9729}, {11411,18356}, {12293,18405} et al

    X(18569) = midpoint of X(i) and X(j) for these {i,j}: {4,14790}, {382,12085}
    X(18569) = reflection of X(i) in X(j) for these (i,j): (3,13371), (4,18377), (20,11250), (26,5)
    X(18569) = anticomplement of X(1658)
    X(18569) = X(40)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18569) = X(11249)-of-Ehrmann-side-triangle if ABC is acute
    X(18569) = X(7387)-of-Ehrmann-mid-triangle
    X(18569) = X(26)-of-Johnson-triangle
    X(18569) = X(14790)-of-Euler-triangle
    X(18569) = X(18377)-of-anti-Euler-triangle
    X(18569) = X(18377)-of-circumorthic-triangle if ABC is acute
    X(18569) = X(13371)-of-X3-ABC-reflections-triangle
    X(18569) = inverse-in-Johnson-circle of X(403)
    X(18569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5,11818), (5,26,10201), (10750,10751,403)


    X(18570) = MIDPOINT OF X(3) AND X(378)

    Barycentrics    a^2[a^8 - 2a^6(b^2 + c^2) + 4a^4b^2c^2 + a^2(b^2 + c^2)(2b^4 - 3b^2c^2 + 2c^4) - (b^2 - c^2)^2(b^4 + 3b^2c^2 + c^4)] : :
    X(18570) = 3*R^2*X(2) - (9*R^2-2*SW)*X(3) = 3 X(3) - X(22) = X(3) + X(378) = 2 X(3) - X(7502)

    Let A'1B'1C'1, A'2B'2C'2 be the 1st and 2nd Ehrmann inscribed triangles, and VAVBVC the Ehrmann vertex-triangle. X(18570) is the radical center of the circumcircles of A'1B'1C'1, A'2B'2C'2, and VAVBVC.

    X(18570) lies on these lines: {2,3}, {49,12111}, {54,11440}, {74,5012}, {99,14558}, {110,18435}, {143,11424}, {154,11472}, {156,12162}, {182,2781}, {184,5663}, {399,3431} et al

    X(18570) = midpoint of X(3) and X(378)
    X(18570) = reflection of X(7502) in X(3)
    X(18570) = inverse-in-orthocentroidal-circle of X(10254)
    X(18570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,10254), (3,4,1658), (3,1995,18571)
    X(18570) = X(55)-of-Trinh-triangle if ABC is acute
    X(18570) = harmonic center of nine-point circle and Trinh circle
    X(18570) = Trinh-isogonal conjugate of X(3098)
    X(18570) = center of circle that is the circumperp conjugate of the de Longchamps line
    X(18570) = circumperp conjugate of X(7574)


    X(18571) = INVERSE-IN-CIRCUMCIRCLE OF X(3830)

    Barycentrics    a^2[4a^8 - 8a^6(b^2 + c^2) + 14a^4b^2c^2 + a^2(8b^6 - 9b^4c^2 - 9b^2c^4 + 8c^6) - (b^2 - c^2)^2(4b^4 + 5b^2c^2 + 4c^4)] : :
    X(18571) = 3 X(3) + X(23)

    Let A'1B'1C'1, A'2B'2C'2 be the 1st and 2nd Ehrmann inscribed triangles, and MAMBMC the Ehrmann mid-triangle. X(18571) is the radical center of the circumcircles of A'1B'1C'1, A'2B'2C'2, and MAMBMC.

    X(18571) lies on these lines: {2,3}, {182,15826}, {187,16308}, {323,15040}, {1511,3292} et al

    X(18571) = complement of X(18572)
    X(18571) = inverse-in-circumcircle of X(3830)
    X(18571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1995,18570), (1113,1114,3830)


    X(18572) = INVERSE-IN-JOHNSON-CIRCLE OF X(2)

    Barycentrics    2a^10 - 2a^8(b^2 + c^2) - 4a^6(b^4 - b^2c^2 + c^4) + a^4(b^2 + c^2)(4b^4 - 5b^2c^2 + 4c^4) + a^2(b^2 - c^2)^2(2b^4 - 3b^2c^2 + 2c^4) - 2(b^2 - c^2)^4(b^2 + c^2) : :
    X(18572) = X(4) + 3 X(3153) = X(4) + X(7574) = 2 X(5) - X(7575)

    X(18572) lies on these lines: {2,3}, {265,11564}, {323,12902}, {511,10113}, {542,15826}, {1154,13851}, {5876,18383}, {13421,18555}, {15068,18405}, {18394,18436} et al

    X(18572) = midpoint of X(4) and X(7574)
    X(18572) = reflection of X(7575) in X(5)
    X(18572) = anticomplement of X(18571)
    X(18572) = inverse-in-Johnson-circle of X(2)
    X(18572) = inverse-in-polar-circle of X(18559)
    X(18572) = {X(10750),X(10751)}-harmonic conjugate of X(2)
    X(18572) = X(1155)-of-Ehrmann-vertex-triangle if ABC is acute
    X(18572) = X(23)-of-Ehrmann-mid-triangle
    X(18572) = X(7575)-of-Johnson-triangle
    X(18572) = X(7574)-of-Euler-triangle


    X(18573) = X(3)X(6)∩X(393)X(14940)

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

    Let SASBSC be the Ehrmann side-triangle and MAMBMC the Ehrmann mid-triangle. X(18573) is the radical center of the circumcircles of ASAMA, BSBMB, and CSCMC.

    X(18573) lies on these lines: {2,16308}, {3,6}, {4,16328}, {140,16303}, {393,14940}, {1989,7746}, {3018,7749}, {6128,7756}, {7495,16306}


    X(18574) = X(4)X(18575)∩X(5)X(11456)

    Barycentrics    a^16 - 4a^14(b^2 + c^2) + a^12(8b^4 + 11b^2c^2 + 8c^4) - 2a^10(b^2 + c^2)(7b^4 + 8b^2c^2 + 7c^4) + a^8(20b^8 + 44b^6c^2 - 6b^4c^4 + 44b^2c^6 + 20c^8) + 2 a^2 (b - c)^4 (b + c)^4 (b^2 + c^2) (b^4 + 10 b^2 c^2 + c^4) - 4a^6(b^2 - c^2)^2(b^2 + c^2)(4b^4 + 5b^2c^2 + 4c^4) + a^4 (b^2 - c^2)^2 (4 b^8 - 25 b^6 c^2 - 25 b^2 c^6 + 4 c^8) - (b^2 - c^2)^6(b^4 + 4b^2c^2 + c^4) : :

    Let VAVBVC, SASBSC, MAMBMC be the Ehrmann vertex-triangle, Ehrmann side-triangle, and Ehrmann mid-triangle, resp. X(18574) is the radical center of the circumcircles of VASAMA, VBSBMB, and VCSCMC.

    X(18574) lies on these lines: {4,18575}, {5,11456}


    X(18575) = PERSPECTOR OF ORTHOCENTROIDAL CIRCLE

    Barycentrics    1/(2a^4 - 2a^2b^2 - 2a^2c^2 - b^2c^2) : :

    X(18575) lies on hyperbola {{A,B,C,X(4),X(381)}} and these lines: {2,11594}, {4,18574}, {67,5475}, {232,566}, {250,458}, {262,338}, {325,5169}, {378,8719}, {381,511} et al

    X(18575) = isogonal conjugate of X(11003)
    X(18575) = isotomic conjugate of X(7771)
    X(18575) = trilinear pole of line X(3569)X(3906)
    X(18575) = perspector of orthocentroidal circle
    X(18575) = pole of Lemoine axis wrt orthocentroidal circle


    X(18576) = X(4)X(94)∩X(476)X(18323)

    Barycentrics    [3a^8 - 6a^6(b^2 + c^2) + 8a^4b^2c^2 + a^2(6b^6 - 4b^4c^2 - 4b^2c^4 + 6c^6) - (b^2 - c^2)^2(3b^4 + 4b^2c^2 + 3c^4)]/[(a^2 - b^2 - c^2)^2 - b^2c^2] : :

    Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles, resp., and VAVBVC be the Ehrmann vertex-triangle. X(18576) is the radical center of the circumcircles of A1A2VA, B1B2VB, and C1C2VC.

    X(18576) lies on these lines: {4,94}, {476,18323}, {1989,3845}, {3839,18316}, {5961,7526}


    X(18577) = X(4)X(477)∩X(113)X(10540)

    Barycentrics    a^20(b^2 + c^2) - 5a^18(b^4 + c^4) + 2a^16(4b^6 + b^4c^2 + b^2c^4 + 4c^6) - 4a^14(7b^6c^2 - 9b^4c^4 + 7b^2c^6) - a^12(b^2 + c^2)(14b^8 - 58b^6c^2 + 83b^4c^4 - 58b^2c^6 + 14c^8) + a^10(14b^12 + 5b^10c^2 - 97b^8c^4 + 154b^6c^6 - 97b^4c^8 + 5b^2c^10 + 14c^12) - 11a^8b^2c^2(b^2 - c^2)^2(b^2 + c^2)(5b^4 - 7b^2c^2 + 5c^4) - a^6(b^2 - c^2)^2(8b^12 - 22b^10c^2 - 39b^8c^4 + 80b^6c^6 - 39b^4c^8 - 22b^2c^10 + 8c^12) + a^4(b^2 - c^2)^6(b^2 + c^2)(5b^4 + 18b^2c^2 + 5c^4) - a^2(b^2 - c^2)^6(b^8 + 5b^6c^2 + 12b^4c^4 + 5b^2c^6 + c^8) - b^2c^2(b^2 - c^2)^8(b^2 + c^2) : :

    Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles, resp., and XAXBXC be the Ehrmann cross-triangle. X(18577) is the radical center of the circumcircles of A1A2XA, B1B2XB, and C1C2XC.

    X(18577) lies on these lines: {4,477}, {113,10540}, {1553,3153}


    X(18578) = PERSPECTOR OF HYPERBOLA {{A,B,C,X(110),PU(5)}}

    Trilinears    csc(C - A - π/3) csc(B - A - π/3) - csc(A - B - π/3) csc(A - C - π/3) : :

    X(18578) lies on these lines: {3,6}, {94,3580}, {1986,11062} et al

    X(18578) = crossdifference of the isogonal conjugates of PU(5)


    X(18579) = EULER-PONCELET POINT OF QUADRANGLE PU(4)PU(5)

    Barycentrics    10a^10 - 21a^8(b^2 + c^2) + 2a^6(b^4 + 16b^2c^2 + c^4) + 2a^4(b^2 + c^2)(10b^4 - 21b^2c^2 + 10c^4) - 2a^2(b^2 - c^2)^2(6b^4 + 5b^2c^2 + 6c^4) + (b^2 - c^2)^4(b^2 + c^2) : :

    X(18579) lies on these lines: {2,3}, {523,9126}, {524,1511}, {3292,11693}, {3564,5648}, {5892,11649} et al

    X(18579) = QA-P2 (Euler-Poncelet Point) of quadrangle PU(4)PU(5)
    X(18579) = anticenter of cyclic quadrilateral PU(4)PU(5)


    X(18580) = MIDPOINT OF X(3) AND X(5094)

    Barycentrics    a^10 - 7a^8(b^2 + c^2) + 2a^6(b^4 + 6b^2c^2 + c^4) + 2a^4(b^2 + c^2)(3b^4 - 5b^2c^2 + 3c^4) - a^2(b^2 - c^2)^2(5b^4 + 8b^2c^2 + 5c^4) + (b^2 - c^2)^4(b^2 + c^2) : :
    X(18580) = X(3) + X(5094)

    X(18580) lies on these lines: {2,3}, {67,11179}, {182,6699}, {184,15132}, {1352,1511} et al

    X(18580) = midpoint of X(3) and X(5094)
    X(18580) = center of conic {{X(3),X(5094),PU(4),PU(5)}}


    X(18581) = X(2)X(14)∩X(5)X(6)

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

    X(18581) is the PU(5)PU(45) trapezoid point - the point in which the extended legs P(5)U(45) and U(5)P(45) of the trapezoid PU(5)PU(45) meet.

    X(18581) lies on these lines: {2,14}, {3,5321}, {4,16}, {5,6}, {13,3545}, {17,5056}, {20,10646}, {30,11481}, {61,3090}, {62,3091}, {69,624}, {115,5617}, {140,5339}, {141,11306}, {381,395}, {393,6116}, {396,5055}, {397,3851}, {398,1656}, {498,10638}, {499,7051}, {631,10645}, {1250,1479}, {3850,5340}, {3589,11305}, {5471,5613}, {6783,9113} et al

    X(18581) = {X(5),X(6)}-harmonic conjugate of X(18582)
    X(18581) = {X(18583),X(18584)}-harmonic conjugate of X(18582)


    X(18582) = X(2)X(13)∩X(5)X(6)

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

    X(18582) is the diagonal crosspoint of trapezoid PU(5)PU(45) - the point in which the diagonals P(5)P(45) and U(5)U(45) of the trapezoid PU(5)PU(45) intersect.

    X(18582) lies on these lines: {2,13}, {3,5318}, {4,15}, {5,6}, {14,3545}, {18,5056}, {20,10645}, {30,11480}, {61,3091}, {62,3090}, {69,623}, {115,5613}, {140,5340}, {141,11305}, {381,396}, {393,6117}, {395,5055}, {397,1656}, {398,3851}, {498,1250}, {631,10646}, {1478,7051}, {1479,10638}, {3850,5339}, {3589,11306}, {5472,5617}, {6782,9112} et al

    X(18582) = {X(5),X(6)}-harmonic conjugate of X(18581)
    X(18582) = {X(18583),X(18584)}-harmonic conjugate of X(18581)


    X(18583) = MIDPOINT OF X(5) AND X(6)

    Barycentrics    2a^6 - 5a^4(b^2 + c^2) + 2a^2(b^4 - 4b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2) : :
    X(18583) = 3 X(2) - X(3) + 2 X(6) = X(3) + X(4) + 2 X(6) = X(5) + X(6) = X(140) - 2 X(3589) = X(141) + X(576) = X(141) - 2 X(3628) = X(182) + X(5480) = X(548) - 2 X(5092)

    X(18583) lies on these lines: {2,1351}, {3,3618}, {4,5050}, {5,6}, {20,12017}, {30,182}, {69,1656}, {114,9300}, {140,143}, {141,576}, {524,547}, {546,575}, {548,5092}, {549,1350}, {550,5085}, {3091,7920} et al

    X(18583) = midpoint of X(i) and X(j) for these {i,j}: {5,6}, {141,576}, {182,5480}
    X(18583) = reflection of X(i) in X(j) for these (i,j): (140,3589), (141,3628), (548,5092)
    X(18583) = complement of complement of X(1351)
    X(18583) = centroid of PU(5)PU(45)
    X(18583) = {X(14136),X(14137)}-harmonic conjugate of X(9300)
    X(18583) = {X(18581),X(18582)}-harmonic conjugate of X(18584)


    X(18584) = {X(18581),X(18582)}-HARMONIC CONJUGATE OF X(18583)

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

    X(18584) lies on these lines: {2, 5210}, {4, 3055}, {5, 6}, {32, 5079}, {39, 5072}, {140, 5585}, {182, 14162}, {187, 1656}, {230, 5071}, {381, 574}, {382, 8589}, {546, 15815}, {547, 7737}, {599, 8176}, {625, 3763}, {1383, 7570}, {1384, 5055}, {1506, 3851}, {2079, 5020}, {2549, 5066}, {3053, 3054}, {3091, 5013}, {3526, 8588}, {3545, 3815}, {3628, 5023}, {5008, 15484}, {5056, 7745}, {5068, 5254}, {5070, 7747}, {5076, 15515}, {5206, 11614}, {6671, 11306}, {6672, 11305}, {6781, 15694}, {7530, 15109}, {7617, 15534}, {7739, 14892}, {7773, 7929}, {7784, 16921}, {7844, 10485}, {8375, 10576}, {8376, 10577}, {11184, 11185}, {11737, 15048}, {15491, 16041}

    X(18584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (382, 8589, 11742), (1506, 18424, 5024), (3851, 5024, 18424), (18581, 18582, 18583)


    X(18585) = 31/2X(2) - X(3)

    Barycentrics    Sqrt(3) a^2 (- a^2 + b^2 + c^2) - 4 S^2 : :
    X(18585) = sqrt(3) X(2) - X(3) = 2 X(5) - X(15765)

    The Euler line intersects the circle O(PU(5)) (the circle with segment PU(5) as diameter) in two points, X(15765) and X(18585). Of the two, X(18585) is the farthest from X(3).

    Let A'B'C' be the triangle obtained by rotaing ABC about the 1st Ehrmann pivot, P(5), by π/2. Let A"B"C" be the triangle obtained by rotaing ABC about the 2nd Ehrmann pivot, U(5), by -π/2. Then A', B', C', A", B", C" lie on a common ellipse, with center X(18585). Also, the lines A'A", B'B", C'C" concur in X(18585) and X(18585) is the similitude center of A'B'C' and A"B"C".

    X(18585) lies on these lines: {2,3}, {13,3071}, {14,3070}, {15,590}, {16,615}, {371,396}, {372,395}, {485,10654}, {486,10653}, {618,641}, {619,642}, {629,13701}, {630,13821}, {639,3643}, {640,3642}, {3068,11485}, {3069,11486}, {5318,6565}, {5321,6564}, {5334,13665}, {5335,13785}, {5478,6251}, {5479,6250}, {6221,11488}, {6303,6774}, {6306,6771}, {6398,11489}, {8252,11481}, {8253,11480}

    X(18585) = reflection of X(15765) in X(5)
    X(18585) = X(15765)-of-Johnson-triangle
    X(18585) = X(18586)-of-Ehrmann-mid-triangle
    X(18585) = {X(i)X(j)}-harmonic conjugate of X(15765) for these {i,j}: {2,3}, {4,381}, {20,5055}, {140,549}, {376,1656}, {382,3545}, {546,3845}, {547,550}, {548,15699}, {631,5054}, {632,12100} et al
    X(18585) = {X(2043),X(2044)}-harmonic conjugate of X(18586)


    X(18586) = REFLECTION OF X(2044) IN X(5)

    Barycentrics    a^4 + a^2(b^2 + c^2) - 2(b^2 - c^2)^2 + Sqrt(3) a^2(b^2 + c^2 - a^2) : :
    X(18586) = 3 X(2) - 3 X(3) - sqrt(3) X(4) = X(3) - 2 X(15765) = 2 X(4) - X(18587) = 2 X(5) - X(2044)

    X(18586) is the intersection nearest to X(2) of the Euler line and the circle centered at X(4) and passing through PU(5).

    X(18586) lies on these lines: {2,3}, {13,372}, {14,371}, {15,6565}, {16,6564}, {395,485}, {396,486}, {590,18581}, {615,18582}, {3068,11543}, {3069,11542} et al

    X(18586) = reflection of X(i) in X(j) for these (i,j): (3,15765), (2044,5), (18587,4)
    X(18586) = X(1) of X(2)PU(5)
    X(18586) = X(2044)-of-Johnson-triangle
    X(18586) = X(15765)-of-X3-ABC-reflections-triangle
    X(18586) = Ehrmann-mid-to-ABC similarity image of X(18585)
    X(18586) = {X(i),X(j)}-harmonic conjugate of X(18587) for these {i,j}: {2,5}, {3,381}, {20,3845}, {140,3545}, {376,546}, {382,3830}, {547,3090}, {549,3091}, {550,3839}, {2043,18585}, {2044,15765}
    X(18586) = {X(2043),X(2044)}-harmonic conjugate of X(18585)


    X(18587) = REFLECTION OF X(2043) IN X(5)

    Barycentrics    a^4 + a^2(b^2 + c^2) - 2(b^2 - c^2)^2 - Sqrt(3) a^2(b^2 + c^2 - a^2) : :
    X(18587) = 3 X(2) - 3 X(3) + sqrt(3) X(4) = X(3) - 2 X(18585) = 2 X(4) - X(18586) = 2 X(5) - X(2043)

    X(18587) is the intersection farthest from X(2) of the Euler line and the circle centered at X(4) and passing through PU(5).

    X(18587) lies on these lines: {2,3}, {13,371}, {14,372}, {15,6564}, {16,6565}, {395,486}, {396,485}, {590,18582}, {615,18581}, {3068,11542}, {3069,11543}, {5318,6561}, {5321,6560} et al

    X(18587) = reflection of X(i) in X(j) for these (i,j): (3,18585), (2043,5), (18586,4)
    X(18587) = X(2043)-of-Johnson-triangle
    X(18587) = X(18585)-of-X3-ABC-reflections-triangle
    X(18587) = Ehrmann-mid-to-ABC similarity image of X(15765)
    X(18587) = {X(i),X(j)}-harmonic conjugate of X(18586) for these {i,j}: {2,5}, {3,381}, {20,3845}, {140,3545}, {376,546}, {382,3830}, {547,3090}, {549,3091}, {550,3839}, {2043,18585}, {2044,15765}
    X(18587) = X(2043),X(2044)}-harmonic conjugate of X(15765)






    leftri  Collineations inverse-images: X(18588) - X(18752)  rightri

    This preamble and centers X(18588)-X(18752) were contributed by César Eliud Lozada, May 4, 2018.
    \

    Suppose that m is a collination. If P = m(Q), then P is the m collineation-image of Q, as in the preambles just before X(16286) and X(16504), and Q is here named the m inverse colllineation-image of P. Explicitly, if

    P = (A',B',C',U; A'',B'',C''V) collineation image of Q, then Q = (A'',B'',C'',V; A',B',c",U) collination inverse-image of P.

    Let Q be the (T1, X1; T2, X2)-collineation image of P, where T1 and T2 are two central triangles, and X1 and X2 two centers. The following table shows the general expressions of the first trilinear coordinate of the image Q when P = u : v : w (trilinears) is given, and the first trilinear coordinate of the inverse-image P, when Q = u : v : w (trilinears) is given.

    Collineation image of u:v:w inverse-image of u:v:w
    (ABC, X(2); excentral, X(1)) -u*a+v*b+w*c b*c*(v+w)
    (excentral, X(1); ABC, X(2)) b*c*(v+w) -u*a+v*b+w*c
    (ABC, X(2); tangential, X(1)) (u*(b+c)*a^2-v*(a+c)*b^2-w*(a+b)*c^2)*a b*c*(w*b+v*c)*(a+c)*(a+b)
    (tangential, X(1); ABC, X(2)) b*c*(w*b+v*c)*(a+c)*(a+b) a*(u*(b+c)*a^2-v*(a+c)*b^2-w*(a+b)*c^2)
    (ABC, X(2); medial, X(1)) b*c*((a-b+c)*b*v+c*(a+b-c)*w) b*c*(u*a-v*b-w*c)*(a+b-c)*(a-b+c)
    (medial, X(1); ABC, X(2)) b*c*(u*a-v*b-w*c)*(a+b-c)*(a-b+c) b*c*((a-b+c)*b*v+c*(a+b-c)*w)
    (ABC, X(2); anticomplementary, X(1)) b*c*(-(b+c)*a*u+b*(a+c)*v+(a+b)*c*w) b*c*(v*b+w*c)*(a+c)*(a+b)
    (anticomplementary, X(1); ABC, X(2)) b*c*(v*b+w*c)*(a+c)*(a+b) b*c*(-(b+c)*a*u+b*(a+c)*v+(a+b)*c*w)
    (ABC, X(2); incentral, X(1)) v*b+w*c b*c*(u-v-w)
    (incentral, X(1); ABC, X(2)) b*c*(u-v-w) v*b+w*c
    (ABC, X(2); cevian-of-X(75), X(1)) b^2*c^2*(b*(a^2-b^2+c^2)*v+c*(a^2+b^2-c^2)*w) b*c*(u*a^2-v*b^2-w*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)
    (cevian-of-X(75), X(1); ABC, X(2)) b*c*(u*a^2-v*b^2-w*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) b^2*c^2*(b*(a^2-b^2+c^2)*v+c*(a^2+b^2-c^2)*w)
    (ABC, X(2); anticevian-of-X(75), X(1)) b^2*c^2*(-(b^2+c^2)*u*a+(a^2+c^2)*v*b+(a^2+b^2)*w*c) b*c*(v*b^2+w*c^2)*(a^2+c^2)*(a^2+b^2)
    (anticevian-of-X(75), X(1); ABC, X(2)) b*c*(v*b^2+w*c^2)*(a^2+c^2)*(a^2+b^2) b^2*c^2*(-(b^2+c^2)*u*a+(a^2+c^2)*v*b+(a^2+b^2)*w*c)
    (ABC, X(1); anticevian-of-X(75), X(2)) b^2*c^2*((-b-c)*u+(a+c)*v+(a+b)*w) (v*b^2+w*c^2)*(a+c)*(a+b)
    (anticevian-of-X(75), X(2); ABC, X(1)) (v*b^2+w*c^2)*(a+c)*(a+b) b^2*c^2*((-b-c)*u+(a+c)*v+(a+b)*w)

    underbar

    X(18588) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    (b+c)*(a^4-2*b*c*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2)*(a+b-c)*(a-b+c) : :
    X(18588) = 3*X(2)+X(18664)

    X(18588) lies on these lines: {2,1748}, {3,1770}, {5,1859}, {37,226}, {222,18651}, {343,18638}, {498,8251}, {1040,1699}, {1062,1479}, {2193,17167}, {5219,10319}, {5249,17073}, {12609,18641}, {18604,18647}

    X(18588) = complement of X(1748)
    X(18588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18664, 1748), (226, 18589, 1214), (226, 18590, 16580)


    X(18589) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    (b+c)*(a^2+(b-c)^2)*(-a^2+b^2+c^2) : :
    X(18589) = 3*X(2)+X(4329)

    X(18589) lies on these lines: {1,5800}, {2,19}, {3,142}, {10,4523}, {21,17171}, {37,226}, {40,18634}, {48,18650}, {63,18651}, {65,18635}, {71,307}, {141,960}, {219,9028}, {306,3610}, {344,908}, {347,5236}, {379,1839}, {464,5249}, {497,614}, {517,16608}, {828,16591}, {857,1441}, {1038,2263}, {1062,3946}, {1279,12053}, {1368,2886}, {1385,17043}, {1444,18648}, {1890,4223}, {2193,17197}, {2385,6389}, {2822,10741}, {2876,11574}, {2883,9943}, {3452,17279}, {3579,18644}, {3812,5799}, {4026,18641}, {4075,4078}, {4851,12635}, {4859,9614}, {5316,17357}, {6684,8251}, {6706,6823}, {7289,17170}, {7536,17384}, {8062,8768}, {10691,17382}, {16593,16596}, {18671,18733}

    X(18589) = complementary conjugate of X(226)
    X(18589) = complement of X(19)
    X(18589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4329, 19), (37, 16580, 226), (37, 16581, 16580), (71, 4466, 307), (857, 1441, 1826), (1214, 18588, 226), (16577, 18590, 226)


    X(18590) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    (b+c)*(a+b-c)*(a-b+c)*(a^6-(b+c)^2*a^4-(b^4+c^4-2*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b^2-c^2)) : :
    X(18590) = 3*X(2)+X(18665)

    X(18590) lies on these lines: {2,18665}, {37,226}, {16607,16609}, {18605,18649}

    X(18590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (226, 18589, 16577), (16580, 18588, 226)


    X(18591) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    a^2*((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(b+c)*(-a^2+b^2+c^2) : :
    X(18591) = 3*X(2)+X(18666)

    X(18591) lies on these lines: {2,286}, {3,6}, {9,1745}, {19,851}, {37,226}, {45,15831}, {53,6907}, {71,73}, {198,5452}, {199,2299}, {212,8606}, {232,4220}, {233,6882}, {241,18635}, {393,6908}, {408,2183}, {442,1838}, {464,5712}, {478,1035}, {1108,1834}, {1172,3651}, {1212,1213}, {1474,3145}, {1826,3142}, {2260,14547}, {2276,10319}, {2335,3487}, {2345,6350}, {3087,6987}, {4303,14597}, {5179,14873}, {5929,18161}, {6184,15526}, {7515,17398}, {8963,16440}, {18210,18674}, {18606,18650}

    X(18591) = complement of X(286)
    X(18591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18666, 286), (71, 73, 3990), (440, 1214, 18592), (579, 581, 6)


    X(18592) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    a*((b+c)*a^4+(b^2+c^2)*a^3-(b^2-c^2)*(b-c)*a^2-(b^2-c^2)^2*a-2*(b^2-c^2)*(b-c)*b*c)*(b+c)*(-a^2+b^2+c^2) : :
    X(18592) = 3*X(2)+X(18667)

    X(18592) lies on these lines: {2,216}, {3,4653}, {33,851}, {37,226}, {81,3284}, {223,3330}, {225,3142}, {408,2654}, {441,6703}, {442,1785}, {464,4648}, {577,940}, {856,5437}, {857,17080}, {1210,1834}, {1211,15526}, {1465,1865}, {1856,3136}, {2092,18642}, {3666,18635}, {3772,17053}, {4383,5158}, {11064,16697}, {16588,16589}, {18608,18652}

    X(18592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (440, 1214, 18591), (440, 18643, 17056)


    X(18593) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = EXCENTRAL TRIANGLE

    Barycentrics    a*(b+c)*(a+b-c)*(a-b+c)*(a^2-b^2+b*c-c^2) : :
    Barycentrics    (1 - 2 cos A) (cos B + cos C) : :
    X(18593) = 3*X(2)+X(18668)

    X(18593) lies on these lines: {1,3651}, {2,7110}, {36,186}, {37,226}, {56,2915}, {57,77}, {65,4868}, {73,15556}, {109,1758}, {223,1708}, {227,4848}, {241,514}, {278,18679}, {323,1443}, {500,10122}, {516,8758}, {553,3664}, {580,3468}, {758,1464}, {851,18210}, {908,16578}, {942,5453}, {1418,4031}, {1421,7677}, {1458,5083}, {1735,12016}, {1736,2635}, {1787,5053}, {1795,3466}, {1873,7741}, {2078,4318}, {2594,12432}, {3649,3743}, {3998,4035}, {4292,13408}, {4892,16598}, {5249,16579}, {5427,7286}, {5745,18607}, {7799,17078}, {8143,11544}, {14165,17923}

    X(18593) = midpoint of X(851) and X(18210)
    X(18593) = isogonal conjugate of X(2341)
    X(18593) = complement of X(14206)
    X(18593) = trilinear pole of the line {526, 3028}
    X(18593) = isotomic conjugate of polar conjugate of X(1835)
    X(18593) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18668, 14206), (36, 4351, 11700), (226, 1214, 16577), (241, 1465, 3911), (1214, 1427, 226), (4850, 17092, 57), (17080, 17092, 4850)


    X(18594) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18594) lies on these lines: {1,19}, {6,3339}, {9,165}, {35,198}, {36,1436}, {46,1743}, {57,2264}, {200,5279}, {219,7991}, {269,7291}, {281,5691}, {579,2272}, {662,18713}, {1108,13462}, {1580,8769}, {1604,8069}, {1707,2312}, {1723,15803}, {1741,1768}, {1755,16570}, {1944,10442}, {1959,8771}, {2093,3197}, {2183,3973}, {2256,9819}, {2257,3361}, {2262,5902}, {2287,12526}, {2324,16548}, {3207,3247}, {3496,15479}, {3553,5341}, {3554,7297}, {4312,5746}, {4882,5227}, {5223,5781}, {5750,5819}, {5776,7992}, {6197,9121}, {7271,7289}, {8809,11347}, {9119,15071}, {14543,18655}

    X(18594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 610, 1), (19, 2173, 610), (2182, 2270, 1743)


    X(18595) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18595) lies on these lines: {1,19}, {63,17865}, {662,18716}, {1748,17858}, {1760,18695}, {1763,5219}, {1820,2158}, {14543,18658}

    X(18595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 18596, 610), (19, 18597, 16545)


    X(18596) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18596) lies on the cubic K605 and these lines: {1,19}, {2,169}, {6,18732}, {20,346}, {37,13730}, {63,1930}, {77,18727}, {194,5088}, {255,2312}, {304,1760}, {662,18717}, {920,1755}, {921,1910}, {1448,2285}, {1490,16550}, {1759,1764}, {2082,5299}, {2128,14210}, {4020,16567}, {5089,9798}, {7297,16781}, {14543,18659}, {14953,17147}, {16551,16552}, {16568,18156}

    X(18596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 610, 2172), (1, 16545, 19), (1, 16546, 16545), (48, 17442, 1), (48, 18597, 19), (610, 18595, 19), (1755, 2083, 920), (1973, 18671, 1), (2172, 18669, 1), (2173, 18671, 1973)


    X(18597) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18597) lies on these lines: {1,19}, {662,18718}, {1755,1820}, {2083,2180}

    X(18597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 18596, 48), (16545, 18595, 19)


    X(18598) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18598) lies on these lines: {1,19}, {9,440}, {40,1503}, {71,191}, {223,2285}, {306,2897}, {573,2949}, {662,18720}, {1045,1719}, {1490,1766}, {2264,16470}, {2831,6326}, {2941,2951}, {3198,5285}, {4329,14543}, {7291,18650}, {11683,18697}

    X(18598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1474, 18674, 1), (2173, 18674, 1474), (2939, 18599, 610)


    X(18599) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = EXCENTRAL TRIANGLE

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

    X(18599) lies on these lines: {1,19}, {6,3468}, {9,1745}, {37,3465}, {191,2956}, {219,1761}, {223,1708}, {347,14543}, {662,18721}, {846,1709}, {978,1047}, {1046,1744}, {1108,15854}, {1214,1762}, {1490,3731}, {1940,5930}, {3682,5279}, {8915,9121}

    X(18599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (610, 18598, 2939), (1172, 18675, 1), (2173, 18675, 1172)


    X(18600) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(9), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18600) lies on these lines: {1,17136}, {2,39}, {7,10571}, {8,16887}, {41,18723}, {81,279}, {85,4850}, {86,3445}, {99,9109}, {192,18157}, {304,17147}, {314,4452}, {347,1014}, {348,16697}, {1201,3663}, {1444,17521}, {1509,4610}, {1975,11319}, {2170,18176}, {2275,16742}, {3210,16703}, {3241,17179}, {3598,4225}, {3616,17175}, {3666,16708}, {3736,4310}, {3933,4202}, {3945,4190}, {3946,17474}, {4000,16696}, {4059,4719}, {4267,7195}, {4346,17139}, {4653,17201}, {4868,7278}, {5088,5262}, {5222,18206}, {7176,17016}, {7225,18724}, {7774,16910}, {7891,17003}, {7906,16906}, {9780,17210}, {10453,17208}, {16347,16992}, {16704,17206}, {16709,17321}, {16710,17302}, {16726,17301}, {16737,18081}, {16739,17490}, {16919,17379}, {16931,17000}, {17141,17154}

    X(18600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17205, 17169), (274, 16705, 2), (274, 16712, 16705), (274, 16750, 16749), (4000, 16696, 16713), (16705, 16711, 274), (16711, 16712, 2)


    X(18601) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(10), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18601) lies on these lines: {2,3770}, {57,77}, {239,18171}, {244,18169}, {274,561}, {321,16720}, {348,16697}, {982,17187}, {1086,17173}, {2275,16717}, {2999,18186}, {3210,17178}, {3286,7191}, {3666,4760}, {3736,3873}, {3752,16704}, {3782,17174}, {4359,16738}, {4393,18172}, {5249,17205}, {5333,16831}, {16714,16743}, {16725,16727}, {16735,16752}, {17011,18166}, {17012,18198}, {17863,18608}

    X(18601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 16700, 16753), (3666, 16726, 8025), (5256, 18164, 81), (16696, 16700, 2)


    X(18602) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18602) lies on these lines: {2,16698}, {86,16697}, {333,3752}, {8025,18609}, {16577,18645}, {16579,18646}, {16714,16743}

    X(18602) = {X(2), X(16698)}-harmonic conjugate of X(16701)


    X(18603) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18603) lies on these lines: {1,8021}, {2,16699}, {27,1427}, {58,17102}, {81,593}, {241,1817}, {284,1214}, {440,3002}, {774,820}, {800,6509}, {859,1763}, {1040,3286}, {2194,8758}, {4000,16700}, {4215,17441}, {16745,16746}, {18175,18180}

    X(18603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (81, 16697, 16696), (81, 18609, 16697)


    X(18604) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18604) lies on these lines: {2,18746}, {58,8071}, {81,593}, {222,1790}, {255,820}, {283,1433}, {394,577}, {967,5019}, {1412,4282}, {1455,16049}, {1800,7078}, {1812,4558}, {1817,4565}, {3286,10832}, {18588,18647}

    X(18604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (81, 1444, 16697), (81, 18605, 1333)


    X(18605) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18605) lies on these lines: {2,18748}, {58,14793}, {81,593}, {571,1993}, {1396,4565}, {1790,4282}, {18590,18649}

    X(18605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (81, 1444, 16698), (1333, 18604, 81)


    X(18606) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18606) lies on these lines: {2,18749}, {37,5736}, {48,1214}, {63,3990}, {75,3164}, {77,14597}, {81,593}, {216,307}, {255,3916}, {348,2275}, {1804,5228}, {2197,9028}, {3739,16699}, {3827,16872}, {7193,18734}, {18210,18733}, {18591,18650}

    X(18606) = {X(16697), X(18607)}-harmonic conjugate of X(18608)


    X(18607) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18607) lies on these lines: {2,85}, {3,6511}, {27,5088}, {37,5905}, {57,1804}, {63,77}, {69,3998}, {75,6360}, {81,593}, {169,16438}, {283,1789}, {306,3933}, {307,343}, {440,1565}, {464,17170}, {469,17181}, {500,14054}, {517,7416}, {527,16577}, {536,17479}, {553,16579}, {942,8021}, {960,1464}, {967,980}, {991,16465}, {1445,10601}, {1455,2975}, {1763,11350}, {1817,7291}, {2000,7580}, {2895,3965}, {3151,4872}, {3219,6610}, {3739,18668}, {3782,8609}, {3827,16678}, {3945,9965}, {4359,16713}, {4640,8758}, {4652,17102}, {5249,16585}, {5744,17080}, {5745,18593}, {6708,14206}, {9776,17092}, {10319,18733}, {11064,18652}, {12649,15852}

    X(18607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (63, 77, 394), (63, 6505, 219), (440, 1565, 18651), (18606, 18608, 16697)


    X(18608) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18608) lies on these lines: {2,18751}, {81,593}, {348,4352}, {603,17102}, {940,1804}, {1107,16721}, {1214,7125}, {16700,16749}, {17863,18601}, {18592,18652}

    X(18608) = {X(16697), X(18607)}-harmonic conjugate of X(18606)


    X(18609) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18609) lies on these lines: {2,16718}, {58,1789}, {81,593}, {110,8758}, {693,905}, {1736,2617}, {3003,3580}, {8025,18602}, {16701,16704}, {16734,16741}

    X(18609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (81, 16697, 16698), (16697, 18603, 81)


    X(18610) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(3), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18610) lies on these lines: {1,159}, {3,3739}, {21,5263}, {22,1602}, {25,1841}, {75,16876}, {157,14017}, {885,4057}, {1576,2189}, {1633,17220}, {16685,16686}

    X(18610) = X(1841) of Ara triangle
    X(18610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18611, 18612), (1, 18615, 18611), (1, 18616, 18619), (16682, 16684, 16678), (18611, 18615, 18622), (18611, 18618, 18619), (18612, 18620, 18619), (18612, 18622, 18611), (18616, 18618, 18620)


    X(18611) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(4), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18611) lies on these lines: {1,159}, {154,2352}, {206,2175}, {595,3941}, {1503,15976}, {1619,16678}, {1630,2328}, {1633,17134}, {3692,4557}, {5250,8053}, {8618,15270}, {16679,18613}

    X(18611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18615, 18610), (3556, 18621, 159), (18610, 18612, 1), (18610, 18619, 18618), (18610, 18622, 18615), (18612, 18622, 18610)


    X(18612) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(5), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18612) lies on these lines: {1,159}, {1633,17221}, {3877,8053}

    X(18612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18611, 18610), (16680, 18614, 8053), (18610, 18611, 18622), (18610, 18619, 18620)


    X(18613) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(9), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18613) lies on these lines: {1,859}, {2,15621}, {3,551}, {22,3007}, {55,750}, {56,1149}, {86,1621}, {228,3748}, {404,15625}, {519,4245}, {528,16056}, {851,3058}, {855,5434}, {946,15622}, {1001,16058}, {1279,1402}, {1376,3840}, {1486,1617}, {1497,14529}, {1699,15626}, {2318,9049}, {3286,8616}, {3303,13738}, {3746,16453}, {3750,5132}, {3870,4557}, {3941,16878}, {4362,15571}, {4421,16059}, {5563,7428}, {7373,15654}, {8227,15623}, {8715,16414}, {10389,15624}, {13724,15888}, {16679,18611}

    X(18613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1486, 1617, 1626), (1621, 16678, 8053)


    X(18614) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18614) lies on these lines: {1,18177}, {3877,8053}, {16679,16691}

    X(18614) = {X(8053), X(18612)}-harmonic conjugate of X(16680)


    X(18615) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18615) lies on these lines: {1,159}, {31,1474}, {154,14597}, {326,16876}, {610,3185}, {1633,18655}, {1661,2352}, {16690,16778}

    X(18615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18610, 18611, 1), (18610, 18622, 18611)


    X(18616) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18616) lies on these lines: {1,159}, {25,16583}, {304,16876}, {1633,18656}, {3696,8193}

    X(18616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18617, 18619), (18610, 18619, 1), (18610, 18620, 18618)


    X(18617) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18617) lies on these lines: {1,159}, {23,17497}, {1019,16874}, {1633,18657}, {14210,16876}

    X(18617) = {X(18616), X(18619)}-harmonic conjugate of X(1)


    X(18618) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18618) lies on these lines: {1,159}, {1633,18658}

    X(18618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18610, 18619, 18611), (18610, 18620, 18616)


    X(18619) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18619) lies on these lines: {1,159}, {1043,1610}, {1633,18659}, {2393,4749}, {16876,18156}

    X(18619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18616, 18610), (1, 18617, 18616), (18611, 18618, 18610), (18612, 18620, 18610)


    X(18620) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18620) lies on the line {1,159}

    X(18620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18610, 18619, 18612), (18616, 18618, 18610)


    X(18621) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = TANGENTIAL TRIANGLE

    Barycentrics    a^2*(a^6-2*(b+c)*a^5+(b-c)^2*a^4+2*(b+c)*b*c*a^3-(b-c)^2*(b^2+c^2)*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
    X(18621) = 3*X(154)-X(3197)

    X(18621) lies on these lines: {1,159}, {3,12335}, {6,2212}, {25,2262}, {48,55}, {64,8273}, {197,7070}, {198,9502}, {206,219}, {221,1458}, {347,1633}, {478,8750}, {1001,1503}, {1108,7083}, {1319,10934}, {1376,10192}, {1495,11383}, {1498,8053}, {1617,3433}, {1621,11206}, {1631,3428}, {1661,3185}, {1853,4423}, {2328,7169}, {3126,15313}, {5584,7973}, {6759,10267}, {10282,11248}, {10310,17821}, {11500,16252}, {12328,13383}

    X(18621) = X(2262) of Ara triangle
    X(18621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154, 2192, 10537), (159, 18611, 3556)


    X(18622) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = TANGENTIAL TRIANGLE

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

    X(18622) lies on these lines: {1,159}, {669,2106}, {1633,18661}

    X(18622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18610, 18611, 18612), (18611, 18615, 18610)


    X(18623) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(4), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18623) lies on these lines: {1,9799}, {2,77}, {6,7365}, {7,27}, {8,1943}, {20,1394}, {34,938}, {56,5324}, {57,279}, {63,347}, {73,5703}, {109,9778}, {221,962}, {226,1419}, {307,14552}, {329,394}, {333,348}, {345,664}, {387,1448}, {388,3745}, {411,1035}, {497,1456}, {934,7011}, {940,948}, {1079,10321}, {1118,7335}, {1214,3160}, {1407,4000}, {1433,6223}, {1435,7289}, {1439,7490}, {1455,5731}, {1465,5435}, {1804,1817}, {1870,5768}, {1895,6616}, {2264,7197}, {3100,10430}, {3157,5758}, {3616,10571}, {3772,6610}, {4573,7055}, {4644,6354}, {5287,8232}, {5423,14594}, {5658,15252}, {5744,17080}, {5811,8757}, {5815,9370}, {6358,7229}, {6611,15509}, {7053,11347}, {9776,17074}, {17081,17082}

    X(18623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 18624, 278), (222, 278, 7), (222, 6357, 278), (223, 1422, 77), (278, 6357, 18624), (278, 18629, 18628), (1394, 5930, 20), (3160, 5273, 1214), (16662, 16663, 14256), (18631, 18633, 18632)


    X(18624) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18624) lies on these lines: {2,3160}, {7,27}, {189,17923}, {223,1442}, {347,5273}, {934,11347}, {1068,6223}, {1118,7338}, {1422,1443}, {1435,7291}, {1456,9812}, {4313,5930}, {5222,7365}, {5748,6505}, {7055,17088}, {7490,14256}

    X(18624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (278, 6357, 18623), (278, 18623, 7)


    X(18625) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18625) lies on these lines: {2,7110}, {7,27}, {85,17087}, {226,1029}, {323,5905}, {651,6354}, {857,948}, {1068,13408}, {1443,5249}, {1478,3153}, {3193,14450}, {3487,5453}, {3616,17584}, {3772,7200}, {4854,16133}, {5435,7365}


    X(18626) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18626) lies on these lines: {7,27}, {4000,7297}

    X(18626) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 18627, 18629), (278, 18629, 7), (278, 18630, 18628)


    X(18627) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18627) lies on the line {7,27}

    X(18627) = {X(18626), X(18629)}-harmonic conjugate of X(7)


    X(18628) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18628) lies on the line {7,27}

    X(18628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (278, 18629, 18623), (278, 18630, 18626)


    X(18629) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18629) lies on these lines: {7,27}, {85,16706}, {4000,7291}

    X(18629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 18626, 278), (7, 18627, 18626), (222, 18630, 278), (18623, 18628, 278)


    X(18630) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18630) lies on the line {7,27}

    X(18630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (278, 18629, 222), (18626, 18628, 278)


    X(18631) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18631) lies on these lines: {7,27}, {73,1442}, {307,1943}, {2260,7291}, {2897,14544}, {3160,5932}, {3212,7105}

    X(18631) = {X(18623), X(18632)}-harmonic conjugate of X(18633)


    X(18632) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18632) lies on these lines: {7,27}, {347,18697}, {4296,5930}, {7291,10521}, {7365,14256}

    X(18632) = {X(18631), X(18633)}-harmonic conjugate of X(18623)


    X(18633) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18633) lies on these lines: {7,27}, {938,15763}, {1375,5435}, {5226,5932}, {5273,6349}, {5603,9799}

    X(18633) = {X(18623), X(18632)}-harmonic conjugate of X(18631)


    X(18634) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18634) lies on these lines: {1,16608}, {2,7}, {19,4466}, {29,9579}, {40,18589}, {78,17296}, {141,936}, {223,13567}, {269,282}, {273,4858}, {278,459}, {281,3668}, {610,1375}, {857,18655}, {938,3946}, {965,17272}, {1119,8894}, {1210,1861}, {1439,9119}, {1449,5738}, {1565,18725}, {1736,4859}, {3086,3341}, {3739,5705}, {4292,7498}, {4648,13411}, {4851,17044}, {5125,9581}, {5704,17067}, {5742,16832}, {6706,17327}, {7515,15803}, {7532,9612}, {9843,12618}, {17064,17065}

    X(18634) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 307, 9), (16608, 17073, 1), (16608, 18644, 17073)


    X(18635) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18635) lies on these lines: {1,16608}, {2,6}, {3,5803}, {5,5751}, {7,857}, {37,307}, {57,440}, {65,18589}, {78,4851}, {142,442}, {226,1439}, {241,18591}, {273,1865}, {284,1375}, {429,1876}, {936,17296}, {938,1834}, {948,5932}, {1086,17863}, {1445,2245}, {1565,18726}, {2092,17058}, {2262,5929}, {2294,4466}, {2893,16054}, {3142,15844}, {3330,6180}, {3454,9843}, {3666,18592}, {3739,6734}, {3946,4904}, {4205,5808}, {4340,7498}, {4361,12649}, {5051,5807}, {5244,16580}, {17171,18165}

    X(18635) = complement of X(2287)
    X(18635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 69, 965), (2, 5738, 6), (7, 857, 1901), (142, 17052, 442)


    X(18636) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18636) lies on these lines: {1,16608}, {56,14376}, {57,7198}, {72,141}, {220,3763}, {427,16607}, {857,18656}, {1375,2172}, {1565,18727}, {1902,12610}, {3912,3998}, {4466,17442}, {4904,17054}, {11517,17060}, {13567,14557}

    X(18636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18637, 18639), (16608, 18639, 1), (16608, 18640, 18638)


    X(18637) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18637) lies on these lines: {1,16608}, {141,5692}, {857,18657}, {1565,18728}, {1734,17069}, {4466,18669}

    X(18637) = {X(18636), X(18639)}-harmonic conjugate of X(1)


    X(18638) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18638) lies on these lines: {1,16608}, {343,18588}, {857,18658}, {1565,18729}, {4466,18670}

    X(18638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16608, 18639, 17073), (16608, 18640, 18636)


    X(18639) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18639) lies on these lines: {1,16608}, {2,607}, {3,8299}, {141,960}, {304,6393}, {857,18659}, {1214,3912}, {1368,3741}, {1375,1973}, {1565,18730}, {3061,16596}, {3742,18214}, {4466,18671}, {6349,17316}

    X(18639) = complement of X(607)
    X(18639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18636, 16608), (1, 18637, 18636), (17043, 18640, 16608), (17073, 18638, 16608)


    X(18640) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18640) lies on these lines: {1,16608}, {1565,18731}

    X(18640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16608, 18639, 17043), (18636, 18638, 16608)


    X(18641) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = MEDIAL TRIANGLE

    Barycentrics    (2*a^4+(b+c)*a^3-(b-c)^2*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*(b+c)*(-a^2+b^2+c^2) : :
    Barycentrics    (tan B)/(cos C + cos A) + (tan C)/(cos A + cos B) : :
    X(18641) = 3*X(2)+X(3152)

    X(18641) lies on these lines: {1,16608}, {2,3}, {8,6349}, {9,3182}, {10,227}, {72,307}, {78,1060}, {117,122}, {223,936}, {283,11064}, {388,7011}, {392,10373}, {581,13567}, {960,16596}, {1210,1834}, {1212,1213}, {1503,2360}, {1565,18732}, {1838,6708}, {1901,4292}, {2968,6734}, {3160,5932}, {3454,6700}, {3695,3998}, {4026,18589}, {4340,15905}, {4466,18673}, {5044,10380}, {5250,15941}, {5296,15831}, {5777,10379}, {6350,9780}, {8885,18687}, {12609,18588}, {13411,17056}

    X(18641) = isotomic conjugate of polar conjugate of X(1901)
    X(18641) = complement of X(29)
    X(18641) = inverse of X(7532) in the orthocentroidal circle
    X(18641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 7515), (2, 4, 7532), (2, 20, 7498), (2, 3152, 29), (2, 5125, 5), (2, 7572, 140), (2, 13725, 16416), (1375, 13442, 28), (17073, 18642, 18643)


    X(18642) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = MEDIAL TRIANGLE

    Barycentrics    ((b+c)*a^4+2*b*c*a^3-(b^4-c^4)*(b-c))*(b+c)*(-a^2+b^2+c^2) : :
    X(18642) = 3*X(2)+X(2897)

    X(18642) lies on these lines: {1,16608}, {2,1172}, {3,66}, {9,440}, {10,4523}, {119,127}, {281,857}, {306,307}, {441,2193}, {442,1861}, {1038,17296}, {1040,17306}, {1060,4851}, {1062,4657}, {1368,10472}, {1375,1474}, {1565,18733}, {1848,6708}, {2092,18592}, {3454,6260}, {4254,13567}, {4466,18674}, {6184,15526}, {6882,14767}, {14961,16696}

    X(18642) = complementary conjugate of X(6708)
    X(18642) = complement of X(1172)
    X(18642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2897, 1172), (18641, 18643, 17073)


    X(18643) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = MEDIAL TRIANGLE

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

    X(18643) lies on these lines: {1,16608}, {2,253}, {3,3332}, {37,226}, {86,441}, {127,16052}, {142,17102}, {216,17245}, {347,857}, {442,7952}, {577,17392}, {1172,1375}, {1213,15526}, {1565,18734}, {2968,3739}, {3945,15905}, {4466,18675}, {5158,17337}, {6389,15668}, {14376,17698}

    X(18643) = complement of X(2322)
    X(18643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17056, 18592, 440), (17073, 18642, 18641)


    X(18644) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = MEDIAL TRIANGLE

    Barycentrics    2*a^5-(b^2+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2-(b^2-c^2)^2*a+(b^4-c^4)*(3*b-3*c) : :
    X(18644) = 3*X(857)+X(18661) = 3*X(1375)-X(2173) = X(2173)+3*X(4466)

    X(18644) lies on these lines: {1,16608}, {2,7359}, {142,3647}, {522,676}, {857,18661}, {1375,2173}, {1565,18735}, {3579,18589}, {10200,17290}

    X(18644) = midpoint of X(1375) and X(4466)
    X(18644) = complement of X(7359)
    X(18644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16608, 17073, 17043), (17073, 18634, 16608)


    X(18645) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(11), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18645) lies on these lines: {2,17168}, {86,142}, {575,13747}, {1790,17182}, {16577,18602}, {16578,16701}

    X(18645) = {X(2), X(17168)}-harmonic conjugate of X(18646)


    X(18646) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18646) lies on these lines: {2,17168}, {21,551}, {86,226}, {1010,5882}, {2185,17197}, {3218,8025}, {16577,16701}, {16579,18602}, {17183,17483}

    X(18646) = {X(2), X(17168)}-harmonic conjugate of X(18645)


    X(18647) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18647) lies on these lines: {27,86}, {18588,18604}

    X(18647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17167, 18648, 1790), (17167, 18649, 17171)


    X(18648) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18648) lies on these lines: {27,86}, {1368,6467}, {1444,18589}, {16738,17177}, {17170,17175}

    X(18648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (86, 17171, 17167), (86, 17172, 17171), (1790, 18647, 17167), (17168, 18649, 17167)


    X(18649) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18649) lies on these lines: {27,86}, {18590,18605}

    X(18649) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17167, 18648, 17168), (17171, 18647, 17167)


    X(18650) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18650) lies on these lines: {1,7}, {2,610}, {3,307}, {9,14021}, {21,3220}, {27,86}, {48,18589}, {57,5738}, {63,69}, {85,1891}, {142,379}, {159,1626}, {224,326}, {226,2268}, {320,8822}, {377,10436}, {412,7282}, {511,14053}, {515,1441}, {534,1953}, {553,15936}, {674,3313}, {950,17863}, {1266,11015}, {1439,10167}, {1565,16163}, {2269,15982}, {2278,16580}, {2772,12825}, {2893,6734}, {3868,3879}, {3911,5740}, {3912,5279}, {3942,18674}, {5232,5273}, {5285,7411}, {6467,17441}, {7291,18598}, {10452,10461}, {10572,17861}, {17647,18698}, {17880,18697}, {18591,18606}

    X(18650) = X(570) of Conway triangle
    X(18650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 20, 18655), (7, 3188, 3668), (7, 4313, 3672), (1790, 18651, 18652), (3664, 4292, 7), (3668, 4297, 17134), (3942, 18674, 18733)


    X(18651) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18651) lies on these lines: {1,1370}, {2,169}, {27,86}, {63,18589}, {77,226}, {85,469}, {222,18588}, {304,305}, {348,464}, {440,1565}, {908,17241}, {940,16580}, {946,4666}, {1125,4228}, {1368,17441}, {3007,18662}, {3151,5088}, {3687,17880}, {3720,4303}, {5236,7210}, {5311,13407}, {7573,17095}, {17183,17184}

    X(18651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (440, 1565, 18607), (18650, 18652, 1790)


    X(18652) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18652) lies on these lines: {2,77}, {27,86}, {63,348}, {222,17073}, {226,7125}, {306,326}, {307,394}, {1125,4303}, {1445,11427}, {1812,4001}, {3616,10430}, {4359,17880}, {6357,6708}, {9776,17023}, {11064,18607}, {18592,18608}

    X(18652) = {X(1790), X(18651)}-harmonic conjugate of X(18650)


    X(18653) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

    Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a+c)*(a+b) : :
    X(18653) = 3*X(1325)-X(6740)

    X(18653) lies on these lines: {2,17190}, {27,86}, {30,113}, {60,4292}, {81,553}, {99,3977}, {110,516}, {229,950}, {239,514}, {501,12047}, {515,1325}, {662,908}, {1326,3011}, {1770,17104}, {1817,5235}, {2173,14206}, {2617,2635}, {4001,7058}, {4225,5267}, {4297,11101}, {11350,17259}, {13407,15792}, {16049,17647}, {17173,18646}, {17174,18645}, {17199,17204}, {18593,18609}

    X(18653) = midpoint of X(110) and X(5196)
    X(18653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (27, 1790, 17167), (1790, 17167, 17168)


    X(18654) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18654) lies on these lines: {1,17183}, {75,17136}, {2975,4360}, {3869,17393}, {4460,5744}, {5176,17322}, {14543,18042}, {14953,17868}

    X(18654) = {X(75), X(17221)}-harmonic conjugate of X(17136)


    X(18655) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18655) lies on these lines: {1,7}, {2,8804}, {4,307}, {9,379}, {19,27}, {21,10436}, {40,1441}, {46,17861}, {57,17863}, {84,2997}, {142,14021}, {150,2822}, {273,412}, {326,17139}, {377,4357}, {464,5249}, {484,17885}, {553,15956}, {610,14953}, {857,18634}, {950,5738}, {1445,6996}, {1486,16678}, {1633,18615}, {1741,4858}, {1958,2327}, {3187,4452}, {3601,5736}, {3729,5279}, {3868,3875}, {4360,11520}, {5128,17895}, {5740,9581}, {6356,10400}, {9436,10431}, {10447,10461}, {12514,18698}, {14543,18594}, {17156,17157}, {17274,17579}

    X(18655) = anticomplement of X(8804)
    X(18655) = X(571) of Conway triangle
    X(18655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 20, 18650), (7, 3188, 269), (7, 4313, 3945), (75, 8822, 63), (3663, 4292, 7), (5088, 10446, 77), (17134, 17220, 1), (17220, 18661, 17134)


    X(18656) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18656) lies on these lines: {1,7}, {2,4456}, {4,349}, {75,16747}, {150,2825}, {304,17139}, {517,1231}, {857,18636}, {1369,17135}, {1633,18616}, {2172,14953}, {7391,17492}, {8680,17442}, {14543,16545}

    X(18656) = anticomplement of X(4456)
    X(18656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18657, 18659), (17220, 18659, 1), (17220, 18660, 18658)


    X(18657) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18657) lies on these lines: {1,7}, {857,18637}, {1633,18617}, {8680,18669}, {14210,17139}, {14543,16546}, {14712,17482}

    X(18657) = {X(18656), X(18659)}-harmonic conjugate of X(1)


    X(18658) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18658) lies on these lines: {1,7}, {857,18638}, {1633,18618}, {8680,18670}, {14543,18595}

    X(18658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17220, 18659, 17134), (17220, 18660, 18656)


    X(18659) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18659) lies on these lines: {1,7}, {2,2333}, {857,18639}, {1370,17135}, {1633,18619}, {1973,14953}, {5307,7406}, {8680,18671}, {14543,18596}, {17139,18156}

    X(18659) = anticomplement of X(2333)
    X(18659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18656, 17220), (1, 18657, 18656), (17134, 18658, 17220), (17221, 18660, 17220)


    X(18660) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

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

    X(18660) lies on these lines: {}


    X(18661) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICOMPLEMENTARY TRIANGLE

    Barycentrics    2*a^5+3*(b+c)*a^4-(b^2+c^2)*a^3-3*(b^3+c^3)*a^2-(b^2-c^2)^2*a-3*(b^2-c^2)*(b-c)*b*c : :
    X(18661) = 3*X(857)-4*X(18644) = 4*X(2173)-3*X(14543) = 2*X(2173)-3*X(14953)

    X(18661) lies on these lines: {1,7}, {523,4467}, {857,18644}, {1155,17895}, {1441,3579}, {1633,18622}, {2173,8680}, {3647,18698}, {5307,8756}, {8822,11684}, {10543,15936}, {17136,17139}

    X(18661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17134, 17220, 17221), (17134, 18655, 17220)


    X(18662) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    (b+c)*a^5-(b+c)*(2*b^2-b*c+2*c^2)*a^3+(b-c)^2*b*c*a^2+(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*b*c : :
    X(18662) = 3*X(2)-4*X(16579)

    X(18662) lies on these lines: {2,2006}, {7,6360}, {63,3187}, {92,18677}, {192,329}, {908,3995}, {914,17184}, {1214,17862}, {1790,17221}, {2167,4560}, {2185,14570}, {2975,17512}, {3007,18651}, {3210,5744}, {6758,17140}, {17483,18668}

    X(18662) = anticomplement of X(6358)
    X(18662) = polar conjugate of isogonal conjugate of X(20803)
    X(18662) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4858, 16577, 2), (6358, 16579, 2)


    X(18663) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    (b+c)*a^5+3*b*c*a^4-2*(b^2-c^2)*(b-c)*a^3-2*(b^2+c^2)*b*c*a^2+(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*b*c : :
    X(18663) = 3*X(2)-4*X(1427)

    X(18663) lies on these lines: {2,85}, {20,17441}, {192,3151}, {193,3210}, {1763,5088}, {3218,17490}, {3732,11347}, {7500,17480}

    X(18663) = anticomplement of X(18750)
    X(18663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1427, 18750, 2), (5905, 6360, 192), (5905, 18668, 6360)


    X(18664) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics   a^9+(b+c)*a^8-2*(b^2+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6+2*b^2*c^2*a^5+2*(b+c)*(b^2+c^2)*b*c*a^4+2*(b^4-c^4)*(b^2-c^2)*a^3+2*(b^2-c^2)*(b-c)*(b^4+c^4)*a^2-(b^2-c^2)^2*(b^4+c^4)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
    X(18664) = 3*X(2)-4*X(18588)

    X(18664) lies on these lines: {2,1748}, {7,7125}, {192,3151}, {347,17483}, {3152,4295}, {7538,11415}

    X(18664) = anticomplement of X(1748)
    X(18664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1748, 18588, 2), (4329, 5905, 6360), (5905, 18665, 17481)


    X(18665) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    a^9+(b+c)*a^8-2*(b^2+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6+2*(b^3+c^3)*b*c*a^4+2*(b^6+c^6)*a^3+2*(b^6-c^6)*(b-c)*a^2-(b^2-c^2)^2*(b^4+c^4)*a-(b^4-c^4)*(b^2-c^2)^2*(b-c) : :
    X(18665) = 3*X(2)-4*X(18590)

    X(18665) lies on these lines: {2,18590}, {192,3151}

    X(18665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4329, 5905, 17479), (17481, 18664, 5905)


    X(18666) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    (b^2+3*b*c+c^2)*a^6+3*(b+c)*b*c*a^5-(2*b^4+2*c^4+(2*b^2-b*c+2*c^2)*b*c)*a^4-2*(b+c)*(b^2+c^2)*b*c*a^3+(b^2-c^2)*(b-c)*(b^3+c^3)*a^2-(b^2-c^2)^2*(b+c)*b*c*a-(b^2-c^2)^2*b^2*c^2 : :
    X(18666) = 3*X(2)-4*X(18591)

    X(18666) lies on these lines: {2,286}, {20,185}, {22,16998}, {192,3151}, {401,15988}, {1654,3152}, {3101,17759}, {4269,7560}, {6840,17035}, {7538,17379}, {7580,9308}

    X(18666) = anticomplement of X(286)
    X(18666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (286, 18591, 2), (3151, 6360, 18667)


    X(18667) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    (b^2+b*c+c^2)*a^7+(b+c)*(b^2+c^2)*a^6-(2*b^4+2*c^4+(b^2-b*c+c^2)*b*c)*a^5-(b+c)*(2*b^4-b^2*c^2+2*c^4)*a^4+(b^2-c^2)*(b-c)*(b^3+c^3)*a^3+(b^4-c^4)*(b^2-c^2)*(b+c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*b*c*a-(b^2-c^2)^2*(b+c)*b^2*c^2 : :
    X(18667) = 3*X(2)-4*X(18592)

    X(18667) lies on these lines: {2,216}, {20,5208}, {81,401}, {192,3151}, {1214,1947}, {2897,17950}, {3152,3210}

    X(18667) = {X(3151), X(6360)}-harmonic conjugate of X(18666)


    X(18668) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    (b+c)*a^5+2*b*c*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3-(b^2+c^2)*b*c*a^2+(b^3+c^3)*(b-c)^2*a-(b^2-c^2)^2*b*c : :
    X(18668) = 3*X(2)-4*X(18593)

    X(18668) lies on these lines: {2,7110}, {192,3151}, {239,514}, {3739,18607}, {4552,17484}, {6758,17491}, {14956,18210}, {17147,17364}, {17483,18662}

    X(18668) = reflection of X(14956) in X(18210)
    X(18668) = anticomplement of X(14206)
    X(18668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5905, 6360, 17479), (6360, 18663, 5905), (14206, 18593, 2)


    X(18669) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = INCENTRAL TRIANGLE

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

    X(18669) lies on these lines: {1,19}, {163,2312}, {304,18717}, {514,661}, {896,2157}, {1725,1755}, {1763,5287}, {1930,17865}, {2170,16784}, {2171,16785}, {3002,18210}, {3942,18728}, {4466,18637}, {5497,17464}, {6149,16562}, {8680,18657}, {9406,17468}, {17170,18727}, {17466,17472}

    X(18669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16545, 1973), (1, 18596, 2172), (1755, 3708, 1725), (2576, 2577, 2172), (14210, 18715, 1959), (17442, 18671, 1)


    X(18670) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = INCENTRAL TRIANGLE

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

    X(18670) lies on these lines: {1,19}, {255,1820}, {774,2180}, {1959,18716}, {2314,4100}, {3942,18729}, {4466,18638}, {6508,14206}, {8680,18658}

    X(18670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1953, 18671, 48), (1953, 18672, 17442)


    X(18671) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = INCENTRAL TRIANGLE

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

    X(18671) lies on these lines: {1,19}, {38,17473}, {255,2083}, {326,2128}, {336,9239}, {774,1755}, {775,1910}, {820,17462}, {1107,17447}, {1496,16567}, {1959,6508}, {2170,7124}, {2171,2286}, {3061,6554}, {3708,4020}, {3720,17441}, {3942,17170}, {4466,18639}, {7146,7365}, {8680,18659}, {16716,17872}, {18589,18733}

    X(18671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17442, 1953), (1, 18596, 1973), (1, 18669, 17442), (48, 18670, 1953), (1973, 18596, 2173), (17170, 18730, 3942), (17438, 18672, 1953), (18156, 18717, 1959)


    X(18672) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = INCENTRAL TRIANGLE

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

    X(18672) lies on these lines: {1755,17473}, {1959,18718}


    X(18673) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    a*(2*a^3+(b+c)*a^2+(b^2-c^2)*(b-c))*(b+c)*(-a^2+b^2+c^2) : :
    X(18673) = 2*X(684)-3*X(14414)

    X(18673) lies on these lines: {1,19}, {20,8680}, {21,1762}, {35,1782}, {40,758}, {42,65}, {55,976}, {63,1792}, {71,72}, {78,10319}, {100,1257}, {201,228}, {226,1869}, {678,2632}, {684,14414}, {740,3189}, {774,3185}, {950,1842}, {960,8299}, {1043,11683}, {1104,2264}, {1191,7124}, {1490,11471}, {1715,5884}, {1763,10393}, {1829,14547}, {1859,2654}, {1888,2635}, {1959,18719}, {2265,3074}, {2357,10901}, {3057,17832}, {3251,4139}, {3611,7066}, {3827,4300}, {3869,6508}, {3942,4303}, {4101,8896}, {4267,11031}, {4466,18641}, {5494,10902}, {5584,12329}, {6986,16560}, {7289,10884}, {7971,17831}, {17164,17784}

    X(18673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2939, 28), (28, 2939, 2173), (48, 18674, 18675), (4303, 18732, 3942)


    X(18674) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = INCENTRAL TRIANGLE

    Barycentrics    a*((b+c)*a^3+(b^2+c^2)*a^2+(b^2-c^2)*(b-c)*a+(b^2-c^2)^2)*(b+c)*(-a^2+b^2+c^2) : :
    X(18674) = 2*X(1486)-3*X(1962)

    X(18674) lies on these lines: {1,19}, {37,17441}, {73,2171}, {192,3151}, {306,3610}, {516,4065}, {740,11677}, {1486,1962}, {1959,18720}, {2099,2286}, {2292,3827}, {3057,4016}, {3159,3950}, {3942,18650}, {3951,3958}, {4431,18697}, {4466,18642}, {18210,18591}

    X(18674) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18598, 1474), (1474, 18598, 2173), (18650, 18733, 3942), (18673, 18675, 48)


    X(18675) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = INCENTRAL TRIANGLE

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

    X(18675) lies on these lines: {1,19}, {37,73}, {71,1214}, {201,3990}, {219,3157}, {221,2256}, {347,8680}, {500,971}, {1060,2289}, {1108,1201}, {1826,5930}, {1841,2654}, {1959,18721}, {1962,2293}, {2285,7114}, {3682,3949}, {3743,12705}, {3942,18734}, {4466,18643}, {7004,14597}, {8803,10901}

    X(18675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18599, 1172), (48, 18674, 18673), (1172, 18599, 2173)


    X(18676) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(5), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18676) lies on these lines: {2,17906}, {4,2181}, {92,4850}, {281,17902}, {653,17074}, {1148,2658}, {17892,17915}, {17912,17914}

    X(18676) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (281, 17903, 17902), (17902, 17903, 18688), (17902, 18683, 18684), (17906, 18677, 2)


    X(18677) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18677) lies on these lines: {2,17906}, {19,18163}, {92,18662}, {7952,10056}

    X(18677) = {X(2), X(18676)}-harmonic conjugate of X(17906)


    X(18678) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18678) lies on these lines: {2,280}, {4,204}, {33,2999}, {108,11347}, {196,8755}, {223,7129}, {225,7490}, {226,1249}, {278,393}, {281,17902}, {469,5222}

    X(18678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (393, 3772, 278), (17902, 17903, 281), (17902, 18688, 17903)


    X(18679) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18679) lies on these lines: {2,216}, {4,580}, {27,1865}, {29,1834}, {33,43}, {81,445}, {92,3772}, {199,14192}, {225,1247}, {278,18593}, {281,17902}, {297,333}, {440,8748}, {442,8747}, {451,498}, {469,4383}, {648,17778}, {653,6354}, {1211,2322}, {1249,5712}, {1990,17056}, {4641,7282}, {6530,7413}, {9308,18134}, {17911,17919}


    X(18680) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18680) lies on these lines: {281,17902}, {386,6198}

    X(18680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (281, 18681, 18683), (17902, 18683, 281), (17902, 18684, 18682)


    X(18681) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18681) lies on these lines: {281,17902}, {17908,17925}

    X(18681) = {X(18680), X(18683)}-harmonic conjugate of X(281)


    X(18682) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18682) lies on the line {281,17902}

    X(18682) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17902, 18683, 17903), (17902, 18684, 18680)


    X(18683) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18683) lies on the line {281,17902}

    X(18683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (281, 18680, 17902), (281, 18681, 18680), (17903, 18682, 17902), (18676, 18684, 17902)


    X(18684) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

    Barycentrics    (a^2-b^2+c^2)*(a^2+b^2-c^2)*(a^11-2*(b^2+c^2)*a^9+(b^3+c^3)*a^8-2*(b^3+c^3)*(b^2+c^2)*a^6+2*(b^6+c^6)*a^5+2*(b+c)*b^3*c^3*a^4-(b^2-c^2)^2*(b^4+c^4)*a^3+2*(b^3+c^3)*(b-c)^2*(b^2+b*c+c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^2*(b^3-c^3)) : :

    X(18684) lies on the line {281,17902}

    X(18684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17902, 18683, 18676), (18680, 18682, 17902)


    X(18685) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18685) lies on these lines: {4,6}, {37,451}, {281,17902}, {2322,5051}, {2345,7952}, {4220,16318}

    X(18685) = {X(17903), X(18686)}-harmonic conjugate of X(18687)


    X(18686) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18686) lies on these lines: {2,1235}, {27,5523}, {42,4213}, {112,3151}, {281,17902}, {440,16318}, {469,8743}

    X(18686) = {X(18685), X(18687)}-harmonic conjugate of X(17903)


    X(18687) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18687) lies on these lines: {2,253}, {81,15262}, {281,17902}, {451,3085}, {1075,6853}, {3183,6908}, {4383,8743}, {6964,8888}, {8885,18641}

    X(18687) = {X(17903), X(18686)}-harmonic conjugate of X(18685)


    X(18688) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18688) lies on these lines: {281,17902}, {693,905}

    X(18688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (17902, 17903, 18676), (17903, 18678, 17902)


    X(18689) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(11), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18689) lies on these lines: {1,17877}, {8,4566}, {75,77}, {339,6739}, {1441,17859}, {4453,4986}, {4858,9317}

    X(18689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 664, 17880), (1441, 17859, 18690)


    X(18690) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18690) lies on these lines: {7,5906}, {75,78}, {86,17880}, {284,4858}, {1441,17859}, {3687,3936}, {4511,18697}

    X(18690) = {X(1441), X(17859)}-harmonic conjugate of X(18689)


    X(18691) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18691) lies on these lines: {1,75}, {269,17880}, {1441,17860}, {4081,6046}, {8769,17901}, {17861,17869}, {17890,17891}

    X(18691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 17858, 1), (17858, 18699, 75)


    X(18692) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18692) lies on these lines: {1,75}, {1441,12609}, {3664,17880}, {17860,17874}, {17863,17877}


    X(18693) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18693) lies on these lines: {1,75}, {774,17879}, {1748,16545}

    X(18693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18694, 1930), (1930, 17858, 1), (17858, 18696, 18695)


    X(18694) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18694) lies on these lines: {1,75}, {1725,17879}, {14208,17901}

    X(18694) = {X(1930), X(18693)}-harmonic conjugate of X(1)


    X(18695) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18695) lies on these lines: {1,75}, {63,1820}, {69,17880}, {92,18713}, {1760,18595}, {1953,14213}, {1959,17865}

    X(18695) = isotomic conjugate of X(2190)
    X(18695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 304, 326), (1930, 17858, 75), (17858, 18696, 18693)


    X(18696) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18696) lies on the line {1,75}

    X(18696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1930, 17858, 17859), (18693, 18695, 17858)


    X(18697) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18697) lies on these lines: {1,75}, {7,17164}, {8,2893}, {69,758}, {141,4016}, {226,306}, {239,16470}, {312,17286}, {313,1089}, {322,4385}, {347,18632}, {1111,1269}, {1231,3668}, {1240,1978}, {1848,3687}, {2292,4357}, {2294,3912}, {2650,3879}, {3263,4967}, {3596,6382}, {3702,17863}, {3718,17270}, {3729,5227}, {3743,17321}, {3946,4359}, {3952,4538}, {3958,4416}, {4021,4065}, {4085,4714}, {4137,7237}, {4431,18674}, {4511,18690}, {4980,17133}, {4986,5564}, {5285,17797}, {10319,11679}, {11683,18598}, {17880,18650}

    X(18697) = isotomic conjugate of X(2363)
    X(18697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 304, 10436), (75, 17762, 314), (4647, 18698, 75)


    X(18698) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18698) lies on these lines: {1,75}, {2,17861}, {7,758}, {9,8680}, {10,307}, {63,1744}, {85,17272}, {142,2294}, {191,8822}, {219,4363}, {273,5705}, {313,3992}, {321,3950}, {322,3679}, {519,15936}, {523,3126}, {527,3958}, {993,17134}, {1086,4016}, {1089,4078}, {1108,4688}, {1111,4357}, {1125,17863}, {1213,16732}, {1214,6358}, {1698,17885}, {1723,4384}, {1754,3980}, {1781,11683}, {1962,4021}, {2256,17118}, {2292,3663}, {2328,4418}, {2650,3664}, {3262,4967}, {3634,17895}, {3672,3743}, {3718,3761}, {3739,4858}, {3878,17220}, {3879,7278}, {4466,17052}, {4847,17874}, {5235,14206}, {5496,7269}, {5936,6757}, {7264,17321}, {12514,18655}, {17647,18650}

    X(18698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 304, 10447), (75, 18697, 4647), (11683, 16054, 1781)


    X(18699) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = CEVIAN TRIANGLE OF X(75)

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

    X(18699) lies on these lines: {1,75}, {1443,17880}, {4397,14208}

    X(18699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 17858, 17859), (75, 18691, 17858)


    X(18700) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18700) lies on these lines: {2,18085}, {4557,18099}, {18082,18095}

    X(18700) = {X(2), X(18085)}-harmonic conjugate of X(18701)


    X(18701) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18701) lies on these lines: {2,18085}, {3112,6654}, {18084,18095}

    X(18701) = {X(2), X(18085)}-harmonic conjugate of X(18700)


    X(18702) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18702) lies on these lines: {2,18086}, {13853,18097}, {17500,18087}, {18082,18083}

    X(18702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18083, 18084, 18082), (18083, 18712, 18084)


    X(18703) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18703) lies on these lines: {2,3613}, {83,1751}, {308,18134}, {857,18086}, {18082,18083}, {18087,18095}


    X(18704) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18704) lies on the line {18082,18083}

    X(18704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18082, 18705, 18707), (18083, 18707, 18082), (18083, 18708, 18706)


    X(18705) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18705) lies on the line {18082,18083}

    X(18705) = {X(18704), X(18707)}-harmonic conjugate of X(18082)


    X(18706) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18706) lies on the line {18082,18083}

    X(18706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18083, 18707, 18084), (18083, 18708, 18704)


    X(18707) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18707) lies on these lines: {18082,18083}, {18091,18095}

    X(18707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18082, 18704, 18083), (18082, 18705, 18704), (18084, 18706, 18083), (18085, 18708, 18083)


    X(18708) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18708) lies on the line {18082,18083}

    X(18708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18083, 18707, 18085), (18704, 18706, 18083)


    X(18709) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18709) lies on these lines: {4,83}, {18082,18083}

    X(18709) = {X(18084), X(18710)}-harmonic conjugate of X(18711)


    X(18710) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18710) lies on these lines: {2,15270}, {83,469}, {18082,18083}

    X(18710) = {X(18709), X(18711)}-harmonic conjugate of X(18084)


    X(18711) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18711) lies on these lines: {2,66}, {18082,18083}

    X(18711) = {X(18084), X(18710)}-harmonic conjugate of X(18709)


    X(18712) = (A,B,C,X(2); A',B',C',X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18712) lies on these lines: {18082,18083}, {18106,18107}

    X(18712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18083, 18084, 18085), (18084, 18702, 18083)


    X(18713) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18713) lies on these lines: {1,82}, {19,326}, {63,1953}, {92,18695}, {320,18725}, {610,16568}, {662,18594}, {1707,17472}, {2234,8769}, {18040,18043}, {18068,18069}

    X(18713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 1959, 326), (1760, 18041, 1), (18041, 18722, 1760)


    X(18714) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18714) lies on these lines: {1,82}, {86,1959}, {190,2171}, {239,17443}, {284,16568}, {320,18726}, {662,1781}, {1045,2643}, {1654,4053}, {1953,4360}, {2667,17472}, {3061,17381}, {3970,17315}, {4137,10458}, {7146,17234}, {17160,17868}, {17233,17762}, {17277,17451}, {17319,17444}, {17378,18161}, {18040,18050}, {18043,18059}, {18044,18055}

    X(18714) = {X(1959), X(2294)}-harmonic conjugate of X(86)


    X(18715) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18715) lies on these lines: {1,82}, {320,18728}, {514,661}, {662,16546}, {1930,16747}, {2644,17799}, {17019,17456}

    X(18715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1959, 18669, 14210), (18049, 18717, 1)


    X(18716) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18716) lies on these lines: {1,82}, {320,18729}, {662,18595}, {1959,18670}

    X(18716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18041, 18717, 1760), (18041, 18718, 18049)


    X(18717) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18717) lies on these lines: {1,82}, {75,17442}, {304,18669}, {320,18730}, {662,18596}, {1959,6508}, {18052,18055}

    X(18717) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18049, 18041), (1, 18715, 18049), (1760, 18716, 18041), (1959, 18671, 18156), (18042, 18718, 18041)


    X(18718) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18718) lies on these lines: {1,82}, {320,18731}, {662,18597}, {1959,18672}

    X(18718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18041, 18717, 18042), (18049, 18716, 18041)


    X(18719) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18719) lies on these lines: {1,82}, {20,2831}, {28,16568}, {72,319}, {75,16747}, {320,18732}, {662,2939}, {1046,6043}, {1098,1762}, {1959,18673}, {3868,18178}, {3869,4673}, {4463,7270}, {5692,18747}

    X(18719) = reflection of X(3868) in X(18178)
    X(18719) = {X(1760), X(18720)}-harmonic conjugate of X(18721)


    X(18720) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18720) lies on these lines: {1,82}, {75,17171}, {320,18733}, {662,18598}, {1474,16568}, {1959,18674}

    X(18720) = {X(18719), X(18721)}-harmonic conjugate of X(1760)


    X(18721) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18721) lies on these lines: {1,82}, {75,15149}, {320,18734}, {662,18599}, {1172,16568}, {1761,16599}, {1959,18675}

    X(18721) = {X(1760), X(18720)}-harmonic conjugate of X(18719)


    X(18722) = (A',B',C',X(1); A,B,C,X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18722) lies on these lines: {1,82}, {320,18735}, {662,1959}, {896,2644}, {897,1581}, {1580,17472}, {2349,14206}, {2643,17799}, {18070,18071}

    X(18722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1760, 18041, 18042), (1760, 18713, 18041)


    X(18723) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18723) lies on these lines: {1,16680}, {6,16742}, {41,18600}, {101,17205}, {999,18164}, {1429,18206}, {2329,16887}, {7225,16713}, {7289,18177}, {9259,16726}, {9267,9299}, {9310,17169}, {9317,16727}, {16696,18162}

    X(18723) = {X(16696), X(18162)}-harmonic conjugate of X(18724)


    X(18724) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18724) lies on these lines: {1,18177}, {41,16713}, {56,18164}, {81,1400}, {86,142}, {940,1730}, {2268,17183}, {4267,18166}, {7083,17194}, {7225,18600}, {7289,18176}, {16696,18162}

    X(18724) = {X(16696), X(18162)}-harmonic conjugate of X(18723)


    X(18725) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18725) lies on these lines: {1,159}, {19,269}, {57,1422}, {77,610}, {142,2391}, {189,2184}, {222,3197}, {241,2270}, {320,18713}, {513,3062}, {1086,5575}, {1108,2097}, {1419,2182}, {1467,1829}, {1565,18634}, {1743,16560}, {1953,4328}, {2385,4312}, {2809,3174}, {3554,7202}, {4445,7323}, {4851,5845}, {5223,8679}, {6245,13156}, {8680,10442}, {16696,18163}, {18164,18180}, {18193,18194}

    X(18725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19, 3942, 269), (77, 7291, 610), (7289, 18161, 1), (18161, 18735, 7289)


    X(18726) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18726) lies on these lines: {1,159}, {6,16551}, {9,7146}, {37,4503}, {57,1804}, {81,2150}, {142,17451}, {284,7291}, {320,18714}, {527,2171}, {942,18164}, {1086,17443}, {1100,7202}, {1266,17868}, {1565,18635}, {1953,3663}, {1959,4357}, {2170,3946}, {2294,3664}, {2309,4475}, {3061,17306}, {3665,16608}, {3666,18163}, {3670,16696}, {3970,4851}, {4006,4445}, {4053,17344}, {4389,18041}, {17197,17863}, {17246,17444}, {18166,18184}, {18169,18190}

    X(18726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2294, 3942, 3664), (16696, 18179, 3670)


    X(18727) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18727) lies on these lines: {1,159}, {77,18596}, {81,17186}, {1565,18636}, {2003,5280}, {2172,7291}, {3942,17442}, {17170,18669}

    X(18727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18728, 18730), (18161, 18730, 1), (18161, 18731, 18729)


    X(18728) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18728) lies on these lines: {1,159}, {320,18715}, {1565,18637}, {3942,18669}, {7202,16784}, {18200,18208}

    X(18728) = {X(18727), X(18730)}-harmonic conjugate of X(1)


    X(18729) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18729) lies on these lines: {1,159}, {320,18716}, {1565,18638}, {3942,18670}

    X(18729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18161, 18730, 7289), (18161, 18731, 18727)


    X(18730) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18730) lies on these lines: {1,159}, {7,17442}, {222,3497}, {320,18717}, {1565,18639}, {1973,7291}, {3942,17170}, {7202,16781}, {18169,18176}

    X(18730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18727, 18161), (1, 18728, 18727), (3942, 18671, 17170), (7289, 18729, 18161), (18162, 18731, 18161)


    X(18731) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18731) lies on these lines: {1,159}, {320,18718}, {1565,18640}

    X(18731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18161, 18730, 18162), (18727, 18729, 18161)


    X(18732) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18732) lies on these lines: {1,159}, {3,6511}, {6,18596}, {20,145}, {28,60}, {65,222}, {69,72}, {279,1439}, {320,18719}, {511,14054}, {518,3313}, {912,5562}, {960,2836}, {971,1902}, {1062,1473}, {1364,10544}, {1385,8907}, {1565,18641}, {1828,5722}, {2771,12825}, {2835,10624}, {3074,16560}, {3670,18163}, {3937,13369}, {3942,4303}, {5045,17024}, {5249,9895}, {5728,7717}, {5904,9004}

    X(18732) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3942, 18673, 4303), (7289, 18733, 18734)


    X(18733) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18733) lies on these lines: {1,159}, {320,18720}, {1474,7291}, {1565,18642}, {3942,18650}, {10319,18607}, {16696,18175}, {18178,18179}, {18210,18606}, {18589,18671}

    X(18733) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3942, 18674, 18650), (18732, 18734, 7289)


    X(18734) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18734) lies on these lines: {1,159}, {63,77}, {81,2189}, {241,579}, {284,3666}, {320,18721}, {1172,7291}, {1437,16696}, {1565,18643}, {2175,8758}, {3942,18675}, {4357,15595}, {5089,16608}, {7193,18606}, {18179,18187}

    X(18734) = {X(7289), X(18733)}-harmonic conjugate of X(18732)


    X(18735) = (A,B,C,X(1); A',B',C',X(2)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

    Barycentrics    a*(a^4+(b^2-3*b*c+c^2)*a^2-(2*b^2+b*c+2*c^2)*(b-c)^2) : :
    X(18735) = 2*X(44)-3*X(16560)

    X(18735) lies on these lines: {1,159}, {44,16560}, {320,18722}, {1429,7202}, {1443,2173}, {1565,18644}, {2391,17067}, {3733,4782}

    X(18735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1443, 7291, 2173), (2173, 3942, 1443), (7289, 18161, 18162), (7289, 18725, 18161)


    X(18736) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(4), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18736) lies on these lines: {2,16697}, {76,7182}, {333,18738}, {908,17241}, {1577,1764}, {4417,18740}, {17234,18741}, {18135,18136}

    X(18736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18133, 18742, 18134), (18134, 18737, 18133), (18134, 18747, 18746), (18134, 18752, 18742), (18737, 18752, 18134), (18749, 18751, 18750)


    X(18737) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(5), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18737) lies on these lines: {2,16698}, {908,17241}

    X(18737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18133, 18736, 18134), (18134, 18736, 18752), (18134, 18747, 18748), (18740, 18741, 2)


    X(18738) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(7), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18738) lies on these lines: {2,16699}, {69,2478}, {75,225}, {85,2476}, {183,16048}, {333,18736}, {469,18750}, {811,1098}, {1577,16552}, {3061,3452}, {5224,14615}, {5701,17263}, {14829,18148}, {17346,18740}

    X(18738) = {X(349), X(6734)}-harmonic conjugate of X(75)


    X(18739) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(8), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18739) lies on these lines: {2,3770}, {57,18044}, {226,17234}, {312,17184}, {321,17235}, {940,18046}, {3210,4033}, {3666,18040}, {4359,4377}, {14829,18148}, {17595,18073}, {18134,18150}, {18141,18147}

    X(18739) = {X(2), X(18136)}-harmonic conjugate of X(18133)


    X(18740) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(11), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18740) lies on these lines: {2,16698}, {190,653}, {645,4585}, {1577,3882}, {3936,18752}, {4417,18736}, {17234,18133}, {17346,18738}

    X(18740) = {X(2), X(18737)}-harmonic conjugate of X(18741)


    X(18741) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(12), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18741) lies on these lines: {2,16698}, {4417,18133}, {14829,18136}, {17234,18736}, {18139,18752}

    X(18741) = {X(2), X(18737)}-harmonic conjugate of X(18740)


    X(18742) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(20), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18742) lies on these lines: {2,16699}, {76,1088}, {908,17241}, {4869,18136}, {18141,18147}

    X(18742) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18134, 18736, 18133), (18134, 18752, 18736)


    X(18743) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(21), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

    Barycentrics    (3*a-b-c)*b*c : :

    X(18743) lies on the cubic K972 and these lines: {1,341}, {2,37}, {7,8051}, {8,3740}, {9,14829}, {10,4673}, {55,5205}, {57,190}, {63,17336}, {69,18228}, {85,5226}, {86,2297}, {92,4997}, {100,9083}, {145,4487}, {171,4011}, {210,10453}, {226,17234}, {238,3769}, {239,16969}, {304,17284}, {306,5233}, {314,18229}, {319,14555}, {320,329}, {322,5328}, {333,3305}, {354,4009}, {388,2899}, {474,7283}, {668,14759}, {726,17063}, {740,16569}, {748,17763}, {908,17241}, {936,1043}, {938,1265}, {940,3758}, {975,13740}, {982,3971}, {984,3840}, {1001,7081}, {1088,4554}, {1089,3624}, {1125,4125}, {1211,17228}, {1376,3685}, {1698,4714}, {1836,17777}, {1909,5308}, {1999,3759}, {2051,18061}, {2325,6692}, {2403,4462}, {2478,7270}, {2901,17749}, {2999,4360}, {3008,17158}, {3061,3452}, {3158,4939}, {3161,5435}, {3241,4723}, {3304,9369}, {3550,4432}, {3616,3701}, {3622,4696}, {3661,5743}, {3662,4415}, {3679,4975}, {3687,5316}, {3695,17527}, {3699,3870}, {3702,9780}, {3705,3816}, {3717,11019}, {3718,17353}, {3729,5437}, {3742,3967}, {3756,4884}, {3757,4423}, {3836,3944}, {3873,3952}, {3886,8580}, {3891,7292}, {3923,17122}, {3948,17056}, {3975,17316}, {3985,17754}, {3993,6686}, {3994,17155}, {4033,16594}, {4095,4384}, {4110,17242}, {4248,4855}, {4362,17123}, {4387,4413}, {4388,4679}, {4389,4656}, {4418,17124}, {4427,9352}, {4434,8616}, {4514,10327}, {4968,5550}, {4981,9330}, {5044,10449}, {5219,18044}, {5256,17393}, {5263,5268}, {5287,17394}, {5423,10580}, {5712,17317}, {5718,18040}, {5722,16086}, {5737,17260}, {5739,17360}, {6063,18153}, {6703,17368}, {7308,11679}, {7321,9776}, {8167,16823}, {16593,18045}, {16817,16842}, {16832,17143}, {17149,18149}, {17243,17786}, {17266,17789}, {17267,17788}, {17308,17762}, {17387,17778}, {18136,18139}

    X(18743) = isotomic conjugate of X(8056)
    X(18743) = anticomplement of X(16602)
    X(18743) = complement of X(17490)
    X(18743) = trilinear pole of the line {3667, 4404}
    X(18743) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3992, 4737), (2, 192, 3752), (2, 312, 75), (2, 3210, 16610), (2, 3995, 4850), (2, 4358, 312), (2, 4671, 4359), (2, 17490, 16602), (344, 1997, 2), (3175, 3210, 3644), (3175, 16610, 3210), (3992, 4737, 341), (4043, 4751, 75), (4687, 18137, 75), (4849, 4891, 145)


    X(18744) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(22), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18744) lies on these lines: {2,1333}, {75,4150}, {857,18147}, {908,17241}, {5224,17335}, {13741,17283}, {16062,17370}

    X(18744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18133, 18745, 18747), (18134, 18747, 18133), (18134, 18748, 18746)


    X(18745) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(23), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18745) lies on these lines: {2,16702}, {908,17241}, {4129,4481}

    X(18745) = {X(18744), X(18747)}-harmonic conjugate of X(18133)


    X(18746) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(24), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18746) lies on these lines: {2,18604}, {908,17241}

    X(18746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18134, 18747, 18736), (18134, 18748, 18744)


    X(18747) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(25), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18747) lies on these lines: {2,1444}, {9,1760}, {69,857}, {75,1826}, {76,5179}, {86,5747}, {333,469}, {908,17241}, {3718,4150}, {5692,18719}, {5816,7377}, {17234,17671}, {18142,18143}

    X(18747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18133, 18744, 18134), (18133, 18745, 18744), (18736, 18746, 18134), (18737, 18748, 18134)


    X(18748) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(26), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18748) lies on these lines: {2,18605}, {908,17241}

    X(18748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18134, 18747, 18737), (18744, 18746, 18134)


    X(18749) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(27), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18749) lies on these lines: {2,18606}, {75,225}, {319,16090}, {320,18147}, {326,4554}, {908,17241}, {1760,16560}, {7013,18026}

    X(18749) = isotomic conjugate of X(3362)
    X(18749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (264, 307, 75), (18736, 18750, 18751)


    X(18750) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(28), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

    Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*b*c : :
    Barycentrics    cos A - cos B cos C : :
    Barycentrics    sec A - sec B sec C : :

    X(18750) lies on the cubic K605 and these lines: {2,85}, {8,7957}, {19,27}, {20,3198}, {69,189}, {144,321}, {158,255}, {190,3719}, {204,1097}, {219,1943}, {222,1944}, {304,2184}, {306,16284}, {322,345}, {341,1370}, {469,18738}, {662,6507}, {896,17871}, {908,17241}, {1441,5273}, {1707,4008}, {1733,16570}, {1746,16551}, {1763,3732}, {1959,6508}, {1999,10025}, {2000,14004}, {2064,3718}, {2975,4228}, {3187,17158}, {3869,4673}, {3928,4858}, {3929,6358}, {3952,11678}, {4385,12527}, {5088,11347}, {5287,14828}, {5342,6734}, {7112,7182}, {7360,7580}, {8804,14615}, {9535,14557}, {9965,17862}

    X(18750) = isogonal conjugate of X(2155)
    X(18750) = isotomic conjugate of X(2184)
    X(18750) = anticomplement of X(1427)
    X(18750) = complement of X(18663)
    X(18750) = X(6)-isoconjugate of X(64)
    X(18750) = trilinear product X(2)*X(20)
    X(18750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 18663, 1427), (63, 92, 75), (63, 14206, 92), (63, 14212, 14213), (189, 329, 69), (6554, 7365, 2), (14206, 14213, 14212), (14212, 14213, 92), (18749, 18751, 18736)


    X(18751) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(29), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18751) lies on these lines: {2,18608}, {63,6335}, {75,7017}, {312,343}, {908,17241}, {4391,17080}

    X(18751) = {X(18736), X(18750)}-harmonic conjugate of X(18749)


    X(18752) = (A',B',C',X(2); A,B,C,X(1)) COLLINEATION INVERSE-IMAGE OF X(30), WHERE A'B'C' = ANTICEVIAN TRIANGLE OF X(75)

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

    X(18752) lies on these lines: {2,16718}, {908,17241}, {1019,1577}, {3936,18740}, {18139,18741}

    X(18752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18134, 18736, 18737), (18736, 18742, 18134)


    X(18753) = CROSSSUM OF X(1) AND X(510)

    Barycentrics    a^(5/2) : :

    X(18753) lies on the cubics K771 and K1006, and on these lines: {1, 510}, {365, 4166}

    X(18753) = isogonal conjugate of X(18297)
    X(18753) = X(i)-isoconjugate of X(j) for these (i,j): {2, 366}, {7, 4182}, {75, 365}, {85, 4166}, {86, 4179}, {188, 508}, {509, 556}
    X(18753) = crosssum of X(i) and X(j) for these (i,j): {1, 510}, {366, 4182}
    X(18753) = barycentric product X(i)*X(j) for these {i,j}: {1, 365}, {6, 366}, {56, 4182}, {57, 4166}, {58, 4179}, {259, 509}
    X(18753) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 366}, {32, 365}, {41, 4182}, {213, 4179}, {365, 75}, {366, 76}, {2175, 4166}, {4166, 312}, {4179, 313}, {4182, 3596}


    X(18754) = X(1)X(257)∩X(41)X(43)

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

    X(18754) lies on the cubic K1006 and these lines: {1, 257}, {32, 6196}, {41, 43}, {56, 2665}, {58, 87}, {171, 213}, {172, 3510}, {609, 18272}, {1045, 8424}, {4116, 6179}

    X(18754) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 3496}, {365, 17596}, {366, 16566}
    X(18754) = X(894)-he conjugate of X(2664)
    X(18754) = X(171)-Hirst inverse of X(18278)
    X(18754) = barycentric product X(239)*X(16362)
    X(18754) = barycentric quotient X(16362)/X(335)
    X(18754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (385, 904, 1)


    X(18755) = X(1)X(1929)∩X(3)X(6)

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

    X(18755) lies on the cubics K1003 and K1006, and on and these lines: {1, 1929}, {3, 6}, {21, 2238}, {35, 213}, {37, 1247}, {41, 2276}, {42, 172}, {43, 4426}, {55, 869}, {65, 17966}, {86, 16917}, {99, 17499}, {100, 2295}, {101, 1500}, {141, 16060}, {197, 3207}, {230, 1834}, {232, 2332}, {332, 15985}, {524, 17206}, {609, 5312}, {940, 11329}, {988, 16973}, {995, 2241}, {1045, 8424}, {1078, 17034}, {1100, 4719}, {1107, 3684}, {1126, 9341}, {1193, 1914}, {1213, 11110}, {1330, 10026}, {1415, 2594}, {1654, 6626}, {1655, 3570}, {1792, 15984}, {1968, 3192}, {2110, 16683}, {2174, 2200}, {2177, 5168}, {2251, 5280}, {2275, 2280}, {2329, 3507}, {2975, 3780}, {3230, 3746}, {3293, 5291}, {3295, 16969}, {3589, 16061}, {3601, 16968}, {3666, 16519}, {3727, 4511}, {3915, 10987}, {4383, 16367}, {4559, 14882}, {4653, 16589}, {5254, 13727}, {5306, 13634}, {5313, 7031}, {5563, 16971}, {6996, 7745}, {8649, 9327}, {9300, 13635}, {16054, 17056}, {16503, 16604}, {17103, 17693}, {17337, 17687}

    X(18755) = isogonal conjugate of X(6625)
    X(18755) = X(i)-Ceva conjugate of X(j) for these (i,j): {42, 6}, {172, 2176}
    X(18755) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6625}, {2, 13610}, {75, 2248}, {286, 15377}
    X(18755) = X(213)-Hirst inverse of X(17735)
    X(18755) = cevapoint of X(8298) and X(8845)
    X(18755) = crosspoint of X(i) and X(j) for these (i,j): {101, 249}, {1654, 4213}
    X(18755) = crossdifference of every pair of points on line {523, 2487}
    X(18755) = crosssum of X(i) and X(j) for these (i,j): {115, 514}, {513, 16592}, {523, 6627}
    X(18755) = tangential-isogonal conjugate of X(16681)
    X(18755) = polar conjugate of isotomic conjugate of X(22139)
    X(18755) = barycentric product X(i)*X(j) for these {i,j}: {1, 846}, {3, 4213}, {6, 1654}, {31, 17762}, {35, 14844}, {42, 6626}, {55, 17084}, {71, 2905}, {249, 6627}, {2664, 8937}
    X(18755) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6625}, {31, 13610}, {32, 2248}, {846, 75}, {1654, 76}, {2200, 15377}, {4213, 264}, {6626, 310}, {6627, 338}, {17084, 6063}, {17762, 561}
    X(18755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2271, 6), (6, 1030, 2305), (6, 5023, 4252), (6, 15815, 5022), (32, 386, 6), (35, 213, 17735), (39, 4251, 6), (284, 2092, 6), (386, 4262, 32), (572, 4263, 6), (584, 4261, 6), (1333, 4272, 6), (1654, 17689, 6626), (2220, 5153, 6), (2245, 4273, 6), (2278, 4277, 6), (4251, 4256, 39), (4255, 4258, 6), (4266, 5114, 6), (4270, 5019, 6), (4272, 17454, 1333), (4285, 5035, 6), (5105, 16946, 6)


    X(18756) = X(1)X(21)∩X(6)X(9403)

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

    X(18756) lies on the cubic K1006 and these lines: {1, 21}, {6, 9403}, {41, 904}, {42, 16372}, {172, 1918}, {237, 2187}, {748, 17030}, {5255, 8622}

    X(18756) = isogonal conjugate of X(18298)
    X(18756) = X(i)-Ceva conjugate of X(j) for these (i,j): {172, 41}, {1918, 31}
    X(18756) = crossdifference of every pair of points on line {661, 4374}
    X(18756) = X(55)-beth conjugate of X(16372)
    X(18756) = barycentric product X(i)*X(j) for these {i,j}: {6, 1045}, {31, 1655}, {662, 9402}
    X(18756) = barycentric quotient X(i)/X(j) for these {i,j}: {1045, 76}, {1655, 561}, {9402, 1577}
    {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (595, 1923, 31)


    X(18757) = X(1)X(1326)∩X(42)X(172)

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

    X(18757) lies on the cubic K1006 and these lines: {1, 1326}, {42, 172}, {213, 7122}, {1967, 18268}, {2107, 2112}

    X(18757) = isogonal conjugate of X(17762)
    X(18757) = X(i)-cross conjugate of X(j) for these (i,j): {904, 7121}, {1333, 31}
    X(18757) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17762}, {2, 1654}, {8, 17084}, {10, 6626}, {69, 4213}, {75, 846}, {306, 2905}, {319, 14844}, {4590, 6627}
    X(18757) = barycentric product X(i)*X(j) for these {i,j}: {1, 2248}, {6, 13610}, {28, 15377}, {31, 6625}
    X(18757) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17762}, {31, 1654}, {32, 846}, {604, 17084}, {1333, 6626}, {1973, 4213}, {2203, 2905}, {2248, 75}, {6625, 561}, {13610, 76}


    X(18758) = X(1)X(3)∩X(32)X(2209)

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

    X(18758) lies on the cubic K1006 and these lines: {1, 3}, {32, 2209}, {39, 2309}, {42, 237}, {43, 11328}, {100, 17752}, {172, 18265}, {213, 904}, {1197, 3117}, {1755, 4531}, {1923, 8022}, {2200, 9417}, {3720, 14096}, {8844, 17760}, {16604, 16683}

    X(18758) = isogonal conjugate of X(18299)
    X(18758) = X(i)-Ceva conjugate of X(j) for these (i,j): {172, 213}, {18265, 2223}
    X(18758) = crossdifference of every pair of points on line {650, 14296}
    X(18758) = crosssum of X(i) and X(j) for these (i,j): {2, 3056}, {693, 4459}
    X(18758) = X(643)-beth conjugate of X(17752)
    X(18758) = barycentric product X(i)*X(j) for these {i,j}: {6, 17792}, {31, 17760}, {292, 8844}, {6376, 18269}
    X(18758) = barycentric quotient X(i)/X(j) for these {i,j}: {8844, 1921}, {17760, 561}, {17792, 76}, {18269, 87}


    X(18759) = X(1)X(3506)∩X(56)X(58)

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

    X(18759) lies on the cubic K1006 and these lines: {1, 3506}, {32, 904}, {41, 18038}, {56, 58}, {101, 10799}, {172, 18262}, {213, 7122}, {1492, 6645}, {1691, 2175}, {2174, 4531}

    X(18759) = isogonal conjugate of X(18760)
    X(18759) = X(172)-Ceva conjugate of X(32)
    X(18759) = X(334)-isoconjugate of X(16366)
    X(18759) = barycentric product X(i)*X(j) for these {i,j}: {6, 8424}, {31, 17739}
    X(18759) = barycentric quotient X(i)/X(j) for these {i,j}: {8424, 76}, {14599, 16366}, {17739, 561}
    X(18759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (904, 1933, 32)


    X(18760) = ISOTOMIC CONJUGATE OF X(8424)

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

    X(18760) lies on the cubic K744 and these lines: {1909, 3509}, {2321, 17760}

    X(18760) = isogonal conjugate of X(18759)
    X(18760) = isotomic conjugate of X(8424)
    X(18760) = X(257)-cross conjugate of X(76)
    X(18760) = X(i)-isoconjugate of X(j) for these (i,j): {31, 8424}, {32, 17739}
    X(18760) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8424}, {75, 17739}, {16366, 1914}


    X(18761) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-MID AND 2nd CIRCUMPERP-TANGENTIAL

    Barycentrics    a*(a^6-(b+c)*a^5-2*(b^2-3*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c)*a^2-(b^2-c^2)*(b-c)^3*a-4*(b^2-c^2)^2*b*c) : :
    X(18761) = 3*X(381)-X(9655), 3*X(3560)-2*X(5248), 4*X(5248)-3*X(10267), 3*X(7330)-X(12526)

    Centers X(18761)-X(18770) were contributed by César Lozada, May 10, 2018.

    X(18761) lies on these lines: {1,1898}, {3,1698}, {4,2975}, {5,6256}, {30,958}, {35,18518}, {36,18492}, {55,5441}, {56,381}, {80,11509}, {104,3091}, {119,6833}, {191,12702}, {355,1012}, {376,5260}, {382,3428}, {405,18481}, {515,3560}, {517,3927}, {912,12559}, {944,6912}, {952,11496}, {956,12699}, {993,6985}, {999,9612}, {1376,18357}, {1385,1490}, {1470,10826}, {1478,6841}, {1482,5693}, {1657,5584}, {1699,10680}, {1706,3579}, {2829,6917}, {3303,18526}, {3304,12611}, {3545,5253}, {3556,18400}, {3583,10966}, {3587,5234}, {3601,18528}, {3845,11194}, {4293,6849}, {4297,6883}, {4857,18543}, {5080,6845}, {5204,18515}, {5217,18524}, {5229,6866}, {5450,6911}, {5694,5779}, {5731,6920}, {5790,10310}, {5818,6909}, {5881,10679}, {5887,11682}, {5903,7701}, {6668,6862}, {6713,6944}, {6824,12667}, {6837,12115}, {6900,12248}, {6906,11499}, {6914,11500}, {6957,10785}, {6974,10786}, {7951,18542}, {7987,18529}, {8068,10742}, {8227,16203}, {8666,18483}, {8727,10526}, {10058,11501}, {10085,10202}, {10198,16617}, {10483,18406}, {10864,18443}, {11108,13624}, {11492,18497}, {11493,18495}, {11495,15704}, {11522,12001}

    X(18761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18480, 18491), (5, 12114, 10269), (355, 1012, 11248), (12773, 18493, 3304), (13743, 18525, 55)


    X(18762) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-MID AND 4th TRI-SQUARES-CENTRAL

    Barycentrics    -4*S*a^2+3*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
    X(18762) = 3*S*X(5)-SW*X(6)

    X(18762) lies on these lines: {2,6221}, {4,3591}, {5,6}, {20,3317}, {140,3071}, {323,15233}, {371,3628}, {372,546}, {381,3069}, {382,6446}, {547,590}, {549,6411}, {550,5420}, {631,1132}, {632,1151}, {640,3631}, {1124,10592}, {1152,3627}, {1328,8703}, {1335,10593}, {1478,13955}, {1479,13954}, {1587,3851}, {1588,1656}, {1592,15066}, {3068,5055}, {3070,3850}, {3090,3311}, {3091,3312}, {3146,6450}, {3299,3614}, {3301,7173}, {3367,10646}, {3392,10645}, {3525,6449}, {3526,6445}, {3529,6456}, {3544,6428}, {3545,7586}, {3583,13958}, {3594,3857}, {3619,11314}, {3845,6438}, {3853,6481}, {5054,9541}, {5056,7582}, {5066,6564}, {5068,7581}, {5070,9540}, {5071,7585}, {5072,6418}, {5076,6408}, {5079,6417}, {5418,6437}, {6409,14869}, {6410,15704}, {6419,12812}, {6420,12811}, {6427,13886}, {6433,11539}, {6435,10109}, {6441,13846}, {6454,12102}, {6455,10303}, {6477,12101}, {6480,16239}, {6497,17538}, {6677,8281}, {7968,18357}, {8253,15699}, {9818,13943}, {9955,13936}, {10194,15712}, {10297,18459}, {10895,13962}, {10896,13963}, {11316,12322}, {11488,18586}, {12699,13947}, {13938,18502}, {13940,18491}, {13942,18492}, {13946,18500}, {13952,18516}, {13959,18525}, {13964,18542}, {13965,18544}, {13971,18480}, {15018,15234}

    X(18762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 13941, 6398), (4, 13951, 13966), (5, 6, 18538), (5, 486, 7584), (5, 5874, 6290), (5, 7584, 7583), (6, 18538, 7583), (546, 13993, 372), (1588, 1656, 8981), (3071, 10577, 140), (6398, 13941, 13966), (6398, 13951, 13941), (6561, 8252, 549), (7584, 18538, 6), (11542, 11543, 7584)


    X(18763) = PERSPECTOR OF THESE TRIANGLES: EHRMANN-SIDE AND INNER GARCIA

    Trilinears         2*sin(3*A/2)*cos((B-C)/2)+(-4*cos(A)-2*cos(2*A))*cos(B-C)+(-4*sin(A/2)+2*sin(3*A/2))*cos(3*(B-C)/2)+cos(2*A)+cos(3*A)+2*cos(A)-1 : :
    Barycentrics   a*(a^9-(2*b^2+b*c+2*c^2)*a^7+(b+c)*(b^2+c^2)*a^6+3*b^2*c^2*a^5-(b+c)*(3*b^4+3*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*a^4+(2*b^6+2*c^6+(3*b^4+3*c^4-(3*b^2-2*b*c+3*c^2)*b*c)*b*c)*a^3+(b^2-c^2)*(b-c)*(3*b^4+2*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*(b^4+c^4+2*(b^2+c^2)*b*c)*a-(b^2-c^2)^2*(b-c)^2*(b^3+c^3)) : :

    X(18763) lies on these lines: {40,6127}, {3869,3940}


    X(18764) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND 1st ISODYNAMIC-DAO

    Barycentrics    sqrt(3)*(S^2+R^2*SA-4*R^2*SW-SB*SC+SW^2)*S^2-S*((SA-SW)*(3*R^2*(19*SA-SW)-20*SA^2+20*SB*SC+SW^2)+(64*R^2-23*SA-SW)*S^2)-7*sqrt(3)*(3*R^2-SW)*SB*SC*SW : :

    X(18764) lies on the line {16806,16808}


    X(18765) = PERSPECTOR OF THESE TRIANGLES: EHRMANN VERTEX AND 2nd ISODYNAMIC-DAO

    Barycentrics    sqrt(3)*(S^2+R^2*SA-4*R^2*SW-SB*SC+SW^2)*S^2+S*((SA-SW)*(3*R^2*(19*SA-SW)-20*SA^2+20*SB*SC+SW^2)+(64*R^2-23*SA-SW)*S^2)-7*sqrt(3)*(3*R^2-SW)*SB*SC*SW : :

    X(18765) lies on the line {16807,16809}


    X(18766) = CENTER OF THE BISECTING CIRCLE OF MIXTILINEAR CIRCLES

    Trilinears    2*p^5*(p+2*q)-7*(2*q^2-1)*p^4-2*q*p^3+(15*q^2+1)*p^2+q*(q+2*p)-8 : : , where p=sin(A/2), q=cos((B-C)/2)
    Barycentrics    a^2*(a^6-(11*b^2-4*b*c+11*c^2)*a^4+16*(b+c)*(b^2+c^2)*a^3+(3*b^4+3*c^4-2*b*c*(28*b^2-25*b*c+28*c^2))*a^2-16*(b^2-c^2)*(b-c)^3*a+(7*b^4+7*c^4+2*b*c*(b^2-13*b*c+c^2))*(b-c)^2) : :

    X(18766) lies on these lines: {672,6244}, {991,999}


    X(18767) = CENTER OF THE BISECTING CIRCLE OF 2nd MIXTILINEAR CIRCLES

    Trilinears    2*p^7*(p+q)-(18*q^2-7)*p^6+7*(2*q^2-3)*q*p^5+(25*q^2-9)*p^4-(27*q^2-25)*q*p^3-3*(q^2+6)*p^2+(11*q^2-12)*q*p+16-10*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

    X(18767) lies on the line {220,6244}


    X(18768) = CENTER OF THE BISECTING CIRCLE OF NEUBERG CIRCLES

    Barycentrics    (b^2+c^2)*a^6-3*b^2*c^2*a^4+(b^2+c^2)^3*a^2-b^2*c^2*(3*b^4-2*b^2*c^2+3*c^4) : :
    X(18768) = 4*X(3934)-3*X(13085)

    X(18768) lies on these lines: {2,39}, {98,7781}, {543,9873}, {698,1352}, {736,12251}, {1569,6337}, {7694,7758}, {7747,14645}

    X(18768) = {X(76), X(194)}-harmonic conjugate of X(3767)


    X(18769) = CENTER OF THE BISECTING CIRCLE OF 2nd NEUBERG CIRCLES

    Barycentrics    3*a^8-(b^2+c^2)*a^6-5*b^2*c^2*a^4+(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^4+c^4+(b^2-c^2)*b*c)*(b^4+c^4-(b^2-c^2)*b*c) : :
    X(18769) = 5*X(83)-3*X(9765), 2*X(2896)-3*X(13086), 4*X(8150)-3*X(13086)

    X(18769) lies on these lines: {2,32}, {5480,13111}, {11257,12252}

    X(18769) = {X(2896), X(8150)}-harmonic conjugate of X(13086)


    X(18770) = CENTER OF THE BISECTING CIRCLE OF YIU CIRCLES

    Trilinears    6*(cos(2*A)+cos(4*A)+1)*cos(B-C)-2*(cos(A)+5*cos(3*A)-cos(5*A)+2*cos(7*A))*cos(2*(B-C))+(6*cos(2*A)+2*cos(6*A)+1)*cos(3*(B-C))+2*cos(3*A)*cos(4*(B-C))-cos(9*A)-7*cos(A)-8*cos(5*A)+cos(7*A) : :

    X(18770) lies on the line {1157,8154}

    X(18770) = reflection of X(1157) in X(8154)


    X(18771) = ISOGONAL CONJUGATE OF X(3035)

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

    Suppose that P = p : q : r (barycentyrics) is a point in the plane of a triangle ABC, and let A'B'C' be the cevian triangle of P. Let A'' be the point, other than P, of the interserction of the circumcircles of PBC' and PCB', and define B'' and C'' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is the point

    f(P) = a2(p + q)(p + r) : b2(q + r)(q + p) : c2(r + p)(r + q).

    The appearance of (i,j) in the following list means that f(X(i)) = X(j): (1,58), (2,6), (3,54), (4,4), (5,1173), (6,251), (7,57), (8,1), (9,1174), (10,1126), (11,18771), (12,18772), (13,16459), (14,16460), (20,3), (21,1175), (22,1176), (23,1177), (30,74), (38,1178), (40,947), (52,1179), (54,1166), (55,3449), (56,3450), (57,3451), (58,3453), (63,284), (65,961), (66,18018), (67,10415), (68,847), (69,2), (72,943), (74,10419), (75,81), (76,83), (78,1167), (80,1168), (81,1169), (85,1170), (86,1171), (92,1172), (95,288), (98,2065), (99,249), (100,59), (101,15378), (102,15379), (103,15380), (104,15381), (105,15382), (106,15383), (107,15384), (108,15385), (109,15386), (110,250), (111,15387), (112,15388), etc.

    Let "conj" abbreviate "conjugate"; then
         f(P) = complement of isotomic conj of cyclocevian conj of isotomic conj of P
         f(P) = complement of anticomplementary conj of P
         f(P) = isogonal conj of midpoint of X(2) and (reflection of P in X(2))
         f(P) = isogonal conj of midpoint of X(3) and (reflection of P in X(5))
         f(P) = isogonal conj of midpoint of X(4) and (reflection of P in X(3))

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27654.

    X(18771) lies on these lines: {6, 1618}, {7, 7336}, {59, 3271}, {518, 5048}, {672, 5537}, {1458, 5193}, {5091, 9309}

    X(18771) = isogonal conjugate of X(3035)
    X(18771) = trilinear pole of the line {665, 9259}


    X(18772) = ISOGONAL CONJUGATE OF X(4999)

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

    See X(18771) and Antreas Hatzipolakis and César Lozada, Hyacinthos 27654.

    X(18772) lies on these lines: {8, 6058}, {60, 181}, {960, 5260}, {2269, 3746}, {4267, 5172}

    X(18772) = isogonal conjugate of X(4999)


    X(18773) = 1st MOSES-SALMON POINT

    Barycentrics    a^2 (b^2 - c^2) (f(a,b,c) + (a^4 - b^2 c^2) Sqrt[f(a,b,c)]) : : , where f(a,b,c) = (b^2 c^2 + c^2 a^2 + a^2 b^2)^2 - 3 a^2 b^2 c^2 (a^2 + b^2 + c^2)

    See Bernard Gibert, Q142.

    If you have GeoGebra, you can view X(18773)&X(18774) .

    X(18773) lies on the Feuerbach hyperbola of the tangential triangle (i.e., the Stammler hypoerbola), the Salmon quartic Q142, the cubics K035 and K150, and on this line: {187,237}

    X(18773) = reflection of X(18774) in X(5027)
    X(18773) = X(99)-Ceva conjugate of X(18774)
    X(18773) = X(9427)-cross conjugate of X(18774)
    X(18773) = X(4602)-isoconjugate of X(18774)
    X(18773) = X(3231)-Hirst inverse of X(18774)
    X(18773) = X(6)-vertex conjugate of X(18774)
    X(18773) = barycentric quotient X(9426)/X(18774)
    {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (187, 5106, 18774), (669, 887, 18774)


    X(18774) = 2nd MOSES-SALMON POINT

    Barycentrics   a^2 (b^2 - c^2) (f(a,b,c) - (a^4 - b^2 c^2) Sqrt[f(a,b,c)]) : : , where f(a,b,c) = (b^2 c^2 + c^2 a^2 + a^2 b^2)^2 - 3 a^2 b^2 c^2 (a^2 + b^2 + c^2)

    See X(18774) and Bernard Gibert, Q142.

    X(18774) lies on the Feuerbach hyperbola of the tangential triangle, the curve Q142, the cubics K035 and K150, and on this line: {187,237}

    X(18774) = reflection of X(18773) in X(5027)
    X(18774) = X(99)-Ceva conjugate of X(18773)
    X(18774) = X(9427)-cross conjugate of X(18773)
    X(18774) = X(4602)-isoconjugate of X(18773)
    X(18774) = X(3231)-Hirst inverse of X(18773)
    X(18774) = X(6)-vertex conjugate of X(18773)
    X(18774) = barycentric quotient X(9426)/X(18773)
    X(18774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (187, 5106, 18773), (669, 887, 18773)


    X(18775) = X(2)X(6082)∩X(6)X(2482)

    Barycentrics    (4 a^4-a^2 b^2+b^4-a^2 c^2-4 b^2 c^2+c^4)/ ((a^2+b^2+c^2)^2-9 b^2c^2) : :

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27657.

    X(18775) lies on these lines: {2,6082}, {6,2482}, {597,1499}


    X(18776) = X(3)X(11659)∩X(6)X(13)

    Barycentrics    Sqrt[3] (-3 a^10 (b^2+c^2)-9 a^6 b^2 c^2 (b^2+c^2)+(b^2-c^2)^4 (b^4-b^2 c^2+c^4)+2 a^8 (2 b^4+5 b^2 c^2+2 c^4)-a^2 (b^2-c^2)^2 (3 b^6+2 b^4 c^2+2 b^2 c^4+3 c^6)+a^4 (b^8+5 b^6 c^2-6 b^4 c^4+5 b^2 c^6+c^8))-Sqrt[-a^4-(b^2-c^2)^2+2 a^2 (b^2+c^2)] (4 a^10-5 a^8 (b^2+c^2)+a^6 (b^4+4 b^2 c^2+c^4)+a^4 (-5 b^6+6 b^4 c^2+6 b^2 c^4-5 c^6)+(b^2-c^2)^2 (b^6+c^6)+a^2 (4 b^8-8 b^6 c^2+6 b^4 c^4-8 b^2 c^6+4 c^8)) : :

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27657.

    X(18776) lies on these lines: {3,11659}, {6,13}, {15,15743}, {396,523}, {5917,6775}, {5994,6772}, {9214,11085}, {11485,15442}


    X(18777) = X(3)X(11658)∩X(6)X(13)

    Barycentrics    Sqrt[3] (-3 a^10 (b^2+c^2)-9 a^6 b^2 c^2 (b^2+c^2)+(b^2-c^2)^4 (b^4-b^2 c^2+c^4)+2 a^8 (2 b^4+5 b^2 c^2+2 c^4)-a^2 (b^2-c^2)^2 (3 b^6+2 b^4 c^2+2 b^2 c^4+3 c^6)+a^4 (b^8+5 b^6 c^2-6 b^4 c^4+5 b^2 c^6+c^8))+Sqrt[-a^4-(b^2-c^2)^2+2 a^2 (b^2+c^2)] (4 a^10-5 a^8 (b^2+c^2)+a^6 (b^4+4 b^2 c^2+c^4)+a^4 (-5 b^6+6 b^4 c^2+6 b^2 c^4-5 c^6)+(b^2-c^2)^2 (b^6+c^6)+a^2 (4 b^8-8 b^6 c^2+6 b^4 c^4-8 b^2 c^6+4 c^8)) : :

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27657.

    X(18777) lies on these lines: {3,11658}, {6,13}, {16,11586}, {395,523}, {5916,6772}, {5995,6775}, {9214,11080}, {11486,15441}


    X(18778) = X(6)X(15526)∩X(525)X(5894)

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

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27657.

    X(18778) lies on these lines: {6,15526}, {525,5894}


    X(18779) = X(6)X(281)∩X(522)X(1158)

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

    See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 27657.

    X(18779) lies on these lines: {6,281}, {522,1158}


    X(18780) = X(30)X(113)∩X(74)X(1989)

    Barycentrics    ((6*R^2-SA)*S^2+3*(3*R^2-SW)*( 36*R^4+3*(SA-4*SW)*R^2-SA^2+ SB*SC+SW^2))*(S^2-3*SB*SC) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27662.

    X(18780) lies on these lines: {30, 113}, {74, 1989}, {1986, 1990}


    X(18781) = X(30)X(146)∩X(74)X(1989)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27662.

    X(18781) lies on these lines: {30, 146}, {74, 1989}, {94, 10264}


    X(18782) = X(4)X(7)∩X(55)X(381)

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

    X(18782) lies on these lines: {4,7}, {46,12764}, {55,381}, {1479,17718}, {3579,6928}, {5122,10431}, {7742,10896}, {12611,13274}

    X(18782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3583, 18406, 9668)


    X(18783) = ISOGONAL CONJUGATE OF X(17738

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

    X(18783) lies on the cubics K771 and K1025 and on these lines: {350, 3509}, {672, 1282}, {846, 8934}, {1914, 18262}, {2223, 16514}, {8868, 17739}, {17731, 18206}

    X(18783) = isogonal conjugate of X(17738)
    X(18783) = X(1911)-cross conjugate of X(1)
    X(18783) = crosspoint of X(2109) and X(9500)
    X(18783) = X(i)-he conjugate of X(j) for these (i,j): {238, 1757}, {291, 2108}
    X(18783) = X(2113)-Hirst inverse of X(9472)
    X(18783) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17738}, {2, 8301}, {75, 2112}, {239, 9470}, {1914, 18034}
    X(18783) = barycentric product X(i)*X(j) for these {i,j}: {1, 2113}, {291, 9472}, {334, 18264}
    X(18783) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17738}, {31, 8301}, {32, 2112}, {291, 18034}, {1911, 9470}, {2113, 75}, {9472, 350}, {18264, 238}


    X(18784) = ISOGONAL CONJUGATE OF X(17739

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

    X(18784) lies on the cubics K1006 and K1025 and on these lines: {9, 6626}, {172, 18262}, {846, 1334}, {1282, 8937}, {1909, 3509}, {8868, 17738}, {18755, 18758}

    X(18784) = isogonal conjugate of X(17739)
    X(18784) = X(904)-cross conjugate of X(1)
    X(18784) = X(7061)-he conjugate of X(9)
    X(18784) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17739}, {2, 8424}, {76, 18759}
    X(18784) = barycentric product X(i)*X(j) for these {i,j}: {31, 18760}, {291, 16366}
    X(18784) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 17739}, {31, 8424}, {560, 18759}, {16366, 350}, {18760, 561}


    X(18785) = ISOGONAL CONJUGATE OF X(18206

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

    X(18785) lies on the cubic K1025 and these lines: {1, 41}, {6, 13476}, {9, 75}, {10, 1018}, {19, 2195}, {37, 4068}, {57, 2279}, {65, 213}, {82, 4628}, {158, 7719}, {225, 2333}, {292, 659}, {596, 5282}, {666, 2311}, {672, 3008}, {759, 919}, {897, 16548}, {910, 2223}, {969, 1814}, {1020, 1400}, {1024, 2161}, {1026, 4876}, {1416, 1451}, {1462, 2285}, {1581, 2664}, {1697, 14942}, {1730, 5364}, {1766, 8769}, {1781, 13610}, {2214, 16972}, {2217, 16968}, {3124, 6044}, {3290, 16782}, {3730, 14267}, {3731, 17038}, {4674, 16611}, {9278, 16369}, {17739, 18298}

    X(18785) = isogonal conjugate of X(18206)
    X(18785) = isotomic conjugate of X(18157)
    X(18785) = X(i)-Ceva conjugate of X(j) for these (i,j): {666, 1027}, {673, 13576}, {919, 1024}
    X(18785) = X(3747)-cross conjugate of X(1)
    X(18785) = cevapoint of X(37) and X(2238)
    X(18785) = crosspoint of X(105) and X(673)
    X(18785) = trilinear pole of line {42, 661}
    X(18785) = crossdifference of every pair of points on line {2254, 8299}
    X(18785) = crosssum of X(518) and X(672)
    X(18785) = X(9)-beth conjugate of X(4557)
    X(18785) = X(105)-daleth conjugate of X(1)
    X(18785) = X(i)-he conjugate of X(j) for these (i,j): {2481, 672}, {7233, 3509}
    X(18785) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 18206}, {2, 672}, {661, 2254}, {3252, 1757}, {17451, 17799}, {17719, 1025}
    X(18785) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18206}, {2, 3286}, {3, 15149}, {21, 241}, {27, 1818}, {31, 18157}, {58, 3912}, {81, 518}, {86, 672}, {99, 665}, {105, 16728}, {110, 918}, {274, 2223}, {283, 5236}, {284, 9436}, {310, 9454}, {333, 1458}, {593, 3932}, {662, 2254}, {741, 17755}, {757, 3930}, {883, 7252}, {905, 4238}, {926, 4573}, {1014, 3693}, {1019, 1026}, {1025, 3737}, {1171, 4966}, {1333, 3263}, {1412, 3717}, {1434, 2340}, {1444, 5089}, {1790, 1861}, {1812, 1876}, {2283, 4560}, {2284, 7192}, {2356, 17206}, {3675, 4567}, {4088, 4556}, {6385, 9455}
    X(18785) = barycentric product X(i)*X(j) for these {i,j}: {1, 13576}, {10, 105}, {37, 673}, {42, 2481}, {65, 14942}, {213, 18031}, {226, 294}, {306, 8751}, {321, 1438}, {661, 666}, {885, 4551}, {919, 1577}, {927, 4041}, {1024, 4552}, {1027, 3952}, {1416, 3701}, {1427, 6559}, {1441, 2195}, {1462, 2321}, {1814, 1826}, {1897, 10099}, {3120, 5377}, {3709, 4528}, {9503, 17747}
    X(18785) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18157}, {6, 18206}, {10, 3263}, {19, 15149}, {31, 3286}, {37, 3912}, {42, 518}, {65, 9436}, {105, 86}, {210, 3717}, {213, 672}, {228, 1818}, {294, 333}, {512, 2254}, {661, 918}, {666, 799}, {672, 16728}, {673, 274}, {756, 3932}, {798, 665}, {884, 3737}, {885, 18155}, {919, 662}, {927, 4625}, {1024, 4560}, {1027, 7192}, {1334, 3693}, {1400, 241}, {1402, 1458}, {1416, 1014}, {1438, 81}, {1462, 1434}, {1500, 3930}, {1814, 17206}, {1824, 1861}, {1880, 5236}, {1918, 2223}, {1962, 4966}, {2195, 21}, {2205, 9454}, {2238, 17755}, {2333, 5089}, {2481, 310}, {3122, 3675}, {3747, 8299}, {3930, 4437}, {4551, 883}, {4557, 1026}, {4559, 1025}, {4705, 4088}, {4729, 4925}, {4849, 4899}, {5377, 4600}, {8750, 4238}, {8751, 27}, {10099, 4025}, {13576, 75}, {14942, 314}, {18031, 6385}
    X(18785) = {X(105),X(294)}-harmonic conjugate of X(1438)


    X(18786) = X(1)X(257)∩X(9)X(43)

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

    X(18786) lies on the cubics K1025 and K1026 and on these lines: {1, 257}, {9, 43}, {57, 87}, {200, 4451}, {261, 1178}, {659, 3737}, {694, 3509}, {695, 3496}, {978, 3865}, {1282, 16363}, {1581, 2664}, {1697, 7220}, {1740, 3863}, {1757, 1967}, {1759, 2233}, {2236, 17596}, {2665, 18206}, {2999, 4835}, {3570, 3961}, {3684, 16514}, {3685, 3783}, {4076, 5524}, {5272, 7249}

    X(18786) = X(694)-Ceva conjugate of X(1)
    X(18786) = X(4366)-cross conjugate of X(1)
    X(18786) = cevapoint of X(2238) and X(4093)
    X(18786) = crossdifference of every pair of points on line {2295, 4367}
    X(18786) = crosssum of X(385) and X(6645)
    X(18786) = X(i)-he conjugate of X(j) for these (i,j): {2, 43}, {6, 9}, {893, 17596}, {3224, 46}
    X(18786) = X(i)-zayin conjugate of X(j) for these (i,j): {511, 43}, {665, 3287}, {2086, 5539}, {3229, 9}, {5113, 3737}
    X(18786) = trilinear product of vertices of 1st Sharygin triangle
    X(18786) = X(i)-isoconjugate of X(j) for these (i,j): {171, 291}, {172, 335}, {292, 894}, {295, 7009}, {334, 7122}, {660, 4367}, {694, 6645}, {741, 1215}, {813, 4369}, {876, 4579}, {1909, 1911}, {1920, 1922}, {1926, 18267}, {2311, 4032}, {2330, 7233}, {3572, 18047}, {3963, 18268}, {4589, 7234}, {4876, 7175}, {7077, 7176}, {7205, 18265}
    X(18786) = X(256)-Hirst inverse of X(893)
    X(18786) = X(i)-he conjugate of X(j) for these (i,j): {2, 43}, {6, 9}, {893, 17596}, {3224, 46}
    X(18786) = X(i)-zayin conjugate of X(j) for these (i,j): {511, 43}, {665, 3287}, {2086, 5539}, {3229, 9}, {5113, 3737}
    X(18786) = barycentric product X(i)*X(j) for these {i,j}: {1, 17493}, {238, 257}, {239, 256}, {350, 893}, {812, 3903}, {904, 1921}, {1178, 3948}, {1429, 4451}, {1431, 3975}, {1432, 3685}, {1581, 4366}, {1914, 7018}, {1916, 8300}, {2201, 7019}, {3684, 7249}, {4010, 4603}, {4037, 7303}, {4455, 7260}
    X(18786) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 894}, {239, 1909}, {256, 335}, {257, 334}, {350, 1920}, {659, 4369}, {740, 3963}, {812, 4374}, {862, 1840}, {893, 291}, {904, 292}, {1284, 4032}, {1428, 7175}, {1429, 7176}, {1432, 7233}, {1447, 7196}, {1580, 6645}, {1914, 171}, {2201, 7009}, {2210, 172}, {2238, 1215}, {3573, 18047}, {3684, 7081}, {3685, 17787}, {3747, 2295}, {3903, 4562}, {3948, 1237}, {4093, 16587}, {4366, 1966}, {4375, 14296}, {4433, 4095}, {4435, 3907}, {4594, 4639}, {4603, 4589}, {4810, 4842}, {7104, 1911}, {7116, 295}, {8300, 385}, {8632, 4367}, {8789, 18267}, {10030, 7205}, {14599, 7122}, {17493, 75}
    X(18786) = {X(257),X(904)}-harmonic conjugate of X(1)


    X(18787) = X(1)X(335)∩X(9)X(87)

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

    X(18787) lies on the cubic K1025 and K1026 and on these lines: {1, 335}, {9, 87}, {43, 57}, {269, 7233}, {334, 10436}, {660, 1757}, {694, 3509}, {741, 1961}, {813, 6015}, {846, 16362}, {894, 7184}, {1018, 5539}, {1045, 13610}, {1215, 8033}, {1740, 3862}, {4128, 4154}, {4518, 5268}, {5018, 9416}, {5378, 7312}

    X(18787) = X(i)-cross conjugate of X(j) for these (i,j): {385, 1}, {2236, 17103}, {4447, 171}
    X(18787) = cevapoint of X(385) and X(6645)
    X(18787) = crosssum of X(2238) and X(4093)
    X(18787) = trilinear pole of line {2295, 4367}
    X(18787) = X(i)-aleph conjugate of X(j) for these (i,j): {291, 9}, {292, 43}, {335, 1759}, {660, 1018}, {4876, 10860}, {7077, 170}
    X(18787) = X(i)-he conjugate of X(j) for these (i,j): {291, 2664}, {292, 1757}, {2113, 43}
    X(18787) = X(i)-zayin conjugate of X(j) for these (i,j): {39, 2238}, {512, 659}, {3252, 2664}
    X(18787) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17493}, {238, 256}, {239, 893}, {242, 7015}, {257, 1914}, {350, 904}, {659, 3903}, {694, 4366}, {740, 1178}, {1428, 4451}, {1431, 3685}, {1432, 3684}, {1581, 8300}, {1921, 7104}, {2210, 7018}, {4455, 4594}
    X(18787) = barycentric product X(i)*X(j) for these {i,j}: {171, 335}, {172, 334}, {291, 894}, {292, 1909}, {337, 7119}, {660, 4369}, {741, 3963}, {813, 4374}, {876, 18047}, {1237, 18268}, {1581, 6645}, {1911, 1920}, {2329, 7233}, {2533, 4584}, {4367, 4562}, {4444, 4579}, {4518, 7175}, {4639, 7234}, {4876, 7176}, {5378, 7200}, {7077, 7196}, {14603, 18267}
    X(18787) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17493}, {171, 239}, {172, 238}, {291, 257}, {292, 256}, {335, 7018}, {813, 3903}, {894, 350}, {1215, 3948}, {1580, 4366}, {1691, 8300}, {1909, 1921}, {1911, 893}, {1922, 904}, {2196, 7015}, {2295, 740}, {2329, 3685}, {2330, 3684}, {3287, 3716}, {3805, 4486}, {4164, 4375}, {4367, 812}, {4369, 3766}, {4447, 17755}, {4477, 4148}, {4579, 3570}, {4584, 4594}, {4589, 7260}, {4876, 4451}, {6645, 1966}, {7081, 3975}, {7119, 242}, {7122, 1914}, {7175, 1447}, {7176, 10030}, {7196, 18033}, {14598, 7104}, {17787, 4087}, {18047, 874}, {18267, 9468}, {18268, 1178}
    X(18787) = {X(335),X(1911)}-harmonic conjugate of X(1)


    X(18788) = X(1)X(3)∩X(9)X(17792)

    Barycentrics    a*(a^4 - a^3*b + 2*a^2*b^2 - a*b^3 - b^4 - a^3*c + a^2*b*c - a*b^2*c + b^3*c + 2*a^2*c^2 - a*b*c^2 - a*c^3 + b*c^3 - c^4) : :
    X(18788) = 3 X[165] - 2 X[9441]

    X(18788) lies on the cubic K1025 and these lines: {1, 3}, {9, 17792}, {10, 7379}, {37, 8245}, {43, 2082}, {100, 1959}, {511, 1757}, {516, 3685}, {674, 16560}, {846, 1334}, {984, 1350}, {1282, 2340}, {1376, 3061}, {1580, 9323}, {1581, 2664}, {1698, 7380}, {1699, 2887}, {1709, 17742}, {1742, 1766}, {1961, 4220}, {2329, 4640}, {2784, 6542}, {2938, 13610}, {2944, 4300}, {2947, 18596}, {2951, 7996}, {3309, 4790}, {3430, 5293}, {3509, 4447}, {3674, 13405}, {3769, 3905}, {4039, 9860}, {4219, 17442}, {5691, 7270}, {7081, 17739}, {9004, 18735}, {9746, 16831}, {9778, 17316}, {10164, 17023}, {10860, 16557}, {12329, 18161}

    X(18788) = reflection of X(1757) in X(6211)
    X(18788) = X(i)-Ceva conjugate of X(j) for these (i,j): {3509, 1757}, {4447, 2664}, {4876, 1}
    X(18788) = crosspoint of X(660) and X(7045)
    X(18788) = crosssum of X(659) and X(2310)
    X(18788) = X(i)-he conjugate of X(j) for these (i,j): {8, 165}, {9, 40}, {2319, 9}, {17787, 8931}
    X(18788) = X(i)-zayin conjugate of X(j) for these (i,j): {3693, 165}, {7077, 1757}
    X(18788) = X(9)-Hirst inverse of X(17792)
    X(18788) = barycentric product X(335)*X(8932)
    X(18788) = barycentric quotient X(8932)/X(239)
    X(18788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 7146, 1), (2340, 7291, 1282), (3507, 17799, 1757)


    X(18789) = (name pending)

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

    X(18789) lies on the cubic K1025 and this line: {57, 846}


    X(18790) = X(1)X(3506)∩X(9)X(8245)

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

    X(18790) lies on the cubic K1025) and these lines: {1, 3506}, {9, 8245}, {57, 1929}, {846, 8847}, {1282, 8853}, {1757, 17798}, {1768, 5150}, {1781, 13610}, {9499, 18206}

    X(18790) = X(3509)-Ceva conjugate of X(1)
    X(18790) = barycentric product X(i)*X(j) for these {i,j}: {239, 8933}, {894, 8936}
    X(18790) = barycentric quotient X(i)/X(j) for these {i,j}: {8933, 335}, {8936, 257}


    X(18791) = X(9)X(86)∩X(1002)X(1282)

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

    X(18791) lies on the cubic K1025 and these lines: {9, 86}, {1002, 1282}, {2938, 13610}

    X(18791) = barycentric quotient X(9279)/X(4804)


    X(18792) = X(58)-HE CONJUGATE OF X(6)

    Barycentrics    a*(a + b)*(a + c)*(a*b^2 - b^2*c + a*c^2 - b*c^2) : :
    X(18792) = X[1] + 2 X[2234]

    X(18792) lies on the cubic K1026 and these lines: {1, 75}, {2, 5145}, {6, 474}, {8, 17178}, {10, 16738}, {21, 3551}, {36, 238}, {38, 18601}, {42, 8025}, {43, 81}, {58, 87}, {244, 16753}, {284, 16779}, {333, 16569}, {386, 17379}, {404, 4279}, {518, 16726}, {596, 17142}, {660, 1757}, {662, 5009}, {726, 3009}, {741, 3510}, {899, 16704}, {982, 16700}, {984, 16696}, {1014, 4334}, {1125, 2309}, {1201, 11115}, {1738, 17197}, {2106, 2108}, {2229, 3231}, {3061, 16716}, {3097, 4476}, {3120, 17174}, {3248, 4974}, {3293, 4649}, {3550, 13588}, {3576, 10892}, {3725, 4697}, {3751, 18164}, {3821, 17202}, {3944, 17182}, {4184, 8616}, {4383, 16409}, {4551, 5061}, {5223, 18186}, {5253, 10457}, {5272, 17194}, {5333, 10458}, {7032, 16825}, {10459, 17589}, {16610, 18191}, {16736, 17063}, {16748, 17176}, {17167, 17889}, {17185, 17596}, {17349, 17749}

    X(18792) = cevapoint of X(1575) and X(3009)
    X(18792) = crossdifference of every pair of points on line {37, 798}
    X(18792) = crosssum of X(i) and X(j) for these (i,j): {42, 2238}, {659, 8054}
    X(18792) = X(i)-Ceva conjugate of X(j) for these (i,j): {741, 1}, {4589, 1019}
    X(18792) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 13174}, {4584, 3570}
    X(18792) = X(86)-daleth conjugate of X(1)
    X(18792) = X(i)-he conjugate of X(j) for these (i,j): {58, 6}, {81, 3216}, {9506, 2664}, {12031, 3737}
    X(18792) = X(i)-zayin conjugate of X(j) for these (i,j): {292, 2238}, {1755, 1743}, {3286, 3216}, {3747, 43}, {8632, 661}
    X(18792) = X(17793)-cross conjugate of X(1)
    X(18792) = X(i)-isoconjugate of X(j) for these (i,j): {10, 727}, {42, 3226}, {65, 8851}, {512, 8709}
    X(18792) = X(i)-vertex conjugate of X(j) for these (i,j): {1, 16695}, {16695, 1}
    X(18792) = barycentric product X(i)*X(j) for these {i,j}: {81, 726}, {86, 1575}, {274, 3009}, {333, 1463}, {662, 3837}, {799, 6373}
    X(18792) = barycentric quotient X(i)/X(j) for these {i,j}: {81, 3226}, {284, 8851}, {662, 8709}, {726, 321}, {1333, 727}, {1463, 226}, {1575, 10}, {3009, 37}, {3837, 1577}, {6373, 661}, {8850, 16609}, {17475, 740}, {17793, 3948}
    X(18792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 17187, 18169), (86, 3736, 1), (87, 978, 16468), (2667, 5625, 1), (16569, 18192, 333), (16736, 18165, 17063)


    X(18793) = X(740)-CROSS CONJUGATE OF X(1)

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

    X(18793) lies on the cubic K1026 and these lines:L
    {1, 668}, {2, 8050}, {31, 43}, {42, 3952}, {213, 1018}, {291, 659}, {741, 3510}, {923, 5380}, {1042, 4566}, {1402, 4551}, {1743, 3402}, {1757, 1967}, {1783, 1973}, {3112, 18093}, {3216, 16476}, {16569, 16576}, {18754, 18757}

    X(18793) = cevapoint of X(i) and X(j) for these (i,j): {42, 2238}, {659, 8054}
    X(18793) = crosssum of X(1575) and X(3009)
    X(18793) = X(3226)-daleth conjugate of X(1)
    X(18793) = X(i)-he conjugate of X(j) for these (i,j): {727, 1575}, {2109, 43}
    X(18793) = X(6)-zayin conjugate of X(1575)
    X(18793) = X(i)-cross conjugate of X(j) for these (i,j): {740, 1}, {4455, 1018}
    X(18793) = X(i)-isoconjugate of X(j) for these (i,j): {21, 1463}, {58, 726}, {81, 1575}, {86, 3009}, {99, 6373}, {110, 3837}, {741, 17793}
    X(18793) = trilinear pole of line {37, 798}
    X(18793) = barycentric product X(i)*X(j) for these {i,j}: {37, 3226}, {226, 8851}, {321, 727}, {661, 8709}
    X(18793) = barycentric quotient X(i)/X(j) for these {i,j}: {37, 726}, {42, 1575}, {213, 3009}, {661, 3837}, {727, 81}, {798, 6373}, {1400, 1463}, {2238, 17793}, {3226, 274}, {3747, 17475}, {8709, 799}, {8851, 333}


    X(18794) = X(1)X(335)∩X(31)X(34)

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

    X(18794) lies on the cubics K771 and K1026 and on these lines: {1, 335}, {6, 1045}, {31, 43}, {32, 6196}, {87, 572}, {171, 1979}, {238, 2110}, {1914, 3510}, {2209, 4579}, {3972, 4116}

    X(18794) = X(i)-Ceva conjugate of X(j) for these (i,j): {1914, 1}, {3510, 18754}
    X(18794) = X(1581)-isoconjugate of X(16361)
    X(18794) = X(i)-aleph conjugate of X(j) for these (i,j): {1, 3509}, {6, 2664}, {365, 1757}, {18753, 3510}
    X(18794) = X(i)-he conjugate of X(j) for these (i,j): {6, 2108}, {238, 43}, {2210, 1740}, {9472, 2664}
    X(18794) = X(3747)-zayin conjugate of X(2108)
    X(18794) = X(31)-Hirst inverse of X(8300)
    X(18794) = barycentric product X(894)*X(16363)
    X(18794) = barycentric quotient X(i)/X(j) for these {i,j}: {1691, 16361}, {16363, 257}
    X(18794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 6196, 18754), (1911, 4366, 1), (7168, 18274, 18754)


    X(18795) = X(335)-CROSS CONJUGATE OF X(1)

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

    X(18795) lies on the cubics K766 and K1026, and on these lines: {350, 18275}, {726, 3783}, {1575, 2664}, {1914, 3510}, {2106, 2108}, {2111, 2665}, {8875, 18754}

    X(18795) = X(335)-cross conjugate of X(1)
    X(18795) = X(171)-isoconjugate of X(16363) X(18795) = X(i)-he conjugate of X(j) for these (i,j): {239, 2664}, {2145, 43}
    X(18795) = barycentric product X(1916)*X(16361)
    X(18795) = barycentric quotient X(i)/X(j) for these {i,j}: {893, 16363}, {16361, 385}


    X(18796) = X(76)-CEVA CONJUGATE OF X(32)

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

    X(18796) lies on the cubic K1027 and these lines: {1, 7096}, {5, 182}, {32, 14820}, {49, 3095}, {110, 7796}, {184, 3456}, {211, 14575}, {1092, 8922}, {3492, 3734}, {3506, 7751}, {7869, 9306}, {9544, 13571}, {14574, 15270}

    X(18796) = X(76)-Ceva conjugate of X(32)
    X(18796) = barycentric product X(2206)X(4174)


    X(18797) = X(32)-CROSS CONJUGATE OF X(76)

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

    X(18797) lies on the cubic K1027 and on this line: {2979, 7768}

    X(18797) = X(32)-cross conjugate of X(76)
    X(18797) = barycentric quotient X(313)/X(4174)


    X(18798) = X(1)X(1670)∩X(76)X(1423)

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

    X(18798) lies on the cubic K1027 and these lines: {1, 1670}, {76, 1423}, {1676, 1759}

    X(18798) = isogonal conjugate of X(18799)
    X(18798) = X(67)-isoconjugate of X(1672)
    X(18798) = barycentric product X(561)*X(1673)
    X(18798) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 1672}, {1673, 31}
    X(18798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 3403, 18799), (1423, 3503, 18799), (3501, 3508, 18799)


    X(18799) = X(1)X(1671)∩X(76)X(1423)

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

    X(18799) lies on the cubic K1027 and these lines: {1, 1671}, {76, 1423}, {1677, 1759}

    X(18799) = isogonal conjugate of X(18798)
    X(18799) = X(76)-isoconjugate of X(1673)
    X(18799) = barycentric product X(561)*X(1672)
    X(18799) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 1673}, {1672, 31}
    X(18799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 3403, 18798), (1423, 3503, 18798), (3501, 3508, 18798)


    X(18800) = REFLECTION OF X(115) IN X(597)

    Barycentrics    (3*SA-SW)*(3*S^2+(3*SA-2*SW)*S W) : :
    X(18800) = X(2)-3*X(5182), 2*X(6)+X(14928), 2*X(141)-3*X(9167), 2*X(576)+X(10992), X(2482)+2*X(8787), 4*X(3589)-3*X(14971), 5*X(3618)-3*X(9166), 2*X(5026)+X(5477), 3*X(5032)+X(8591), 3*X(5032)-X(10754), 3*X(5050)-X(11632), 3*X(5182)+X(8593), 9*X(5182)-X(11161), 2*X(8584)+X(15300), 3*X(8593)+X(11161)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18800) lies on these lines: {2, 98}, {6, 543}, {99, 1285}, {115, 597}, {126, 1641}, {141, 9167}, {187, 524}, {376, 10753}, {511, 8598}, {575, 8370}, {576, 10992}, {599, 620}, {690, 15303}, {754, 12151}, {1384, 16508}, {1569, 5052}, {1995, 9966}, {2434, 15304}, {2502, 9172}, {2682, 8352}, {2793, 9135}, {2794, 5077}, {2796, 4991}, {3589, 14971}, {3618, 9166}, {4235, 5095}, {5017, 14645}, {5032, 8591}, {5050, 11632}, {5461, 10488}, {5468, 12036}, {5476, 9880}, {6321, 14848}, {7735, 9877}, {8182, 15483}, {8359, 10991}, {8369, 8550}, {9169, 10418}, {10554, 14916}, {17952, 17979}

    X(18800) = midpoint of X(i) and X(j) for these {i,j}: {2, 8593}, {99, 1992}, {376, 10753}
    X(18800) = reflection of X(i) in X(j) for these (i,j): (115, 597), (599, 620)
    X(18800) = complement of X(11161)
    X(18800) = orthoptic-circle-of-Steiner-inellipse-inverse-of X(9759)
    X(18800) = barycentric quotient X(524)/X(5503)
    X(18800) = X(115)-of-anti-Artzt-triangle
    X(18800) = X(597)-of-anti-McCay-triangle
    X(18800) = X(8370)-of-6th-anti-Brocard-triangle
    X(18800) = X(8598)-of-1st-Brocard-triangle
    X(18800) = center of circle {{X(i), X(j), X(k)}} for these {i, j, k}: {67, 10488, 11646}, {99, 1992, 15342}, {5026, 6593, 8787}
    X(18800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5032, 8591, 10754), (5182, 8593, 2), (13642, 13761, 2)


    X(18801) = REFLECTION OF X(11) IN X(8255)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18801) lies on these lines: {7, 528}, {11, 7671}, {55, 5851}, {100, 12848}, {119, 15607}, {3660, 10427}, {12831, 15726}

    X(18801) = reflection of X(11) in X(8255)
    X(18801) = {X(12730), X(14151)}-harmonic conjugate of X(3241)


    X(18802) = REFLECTION OF X(11) IN X(8256)

    Barycentrics    (4*a^4-5*(b+c)*a^3-(3*b^2-16*b *c+3*c^2)*a^2+(b+c)*(5*b^2-12* b*c+5*c^2)*a-(b^2-c^2)^2)*((b+ c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18802) lies on these lines: {8, 2829}, {11, 8256}, {56, 100}, {57, 12641}, {119, 517}, {1210, 2802}, {2098, 3035}, {3036, 3434}, {3880, 12832}, {4855, 12735}, {5083, 12640}, {12531, 17784}

    X(18802) = reflection of X(i) in X(j) for these (i,j): (11, 8256), (2098, 3035)
    X(18802) = {X(1145), X(1537)}-harmonic conjugate of X(6735)


    X(18803) = MIDPOINT OF X(3180) IN X(10409)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18803) lies on these lines: {115, 6783}, {396, 15609}, {3180, 10409}, {5612, 11601}, {11600, 16267}

    X(18803) = midpoint of X(3180) and X(10409)


    X(18804) = MIDPOINT OF X(3180) IN X(10409)

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

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18804) lies on these lines: {115, 6782}, {395, 15610}, {3181, 10410}, {5616, 11600}, {11601, 16268}

    X(18804) = midpoint of X(3181) and X(10410)


    X(18805) = MIDPOINT OF X(31) AND X(75)

    Barycentrics    (b+c)*a^4+2*b*c*a^3+(b^3+c^3)* b*c : :
    X(18805) = 5*X(4699)-X(6327)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18805) lies on these lines: {31, 75}, {37, 6679}, {752, 4688}, {2210, 4836}, {2887, 3739}, {4412, 5019}, {4699, 6327}

    X(18805) = midpoint of X(31) and X(75)
    X(18805) = reflection of X(i) in X(j) for these (i,j): (37, 6679), (2887, 3739)


    X(18806) = MIDPOINT OF X(32) AND X(76)

    Barycentrics    (b^2+c^2)*a^6+2*b^2*c^2*a^4+(b ^4+c^4)*b^2*c^2 : :
    X(18806) = 5*X(4699)-X(6327)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27670.

    X(18806) lies on the cubic K459 and these lines: {5, 141}, {6, 8149}, {32, 76}, {39, 620}, {194, 5319}, {262, 7862}, {315, 16044}, {538, 5306}, {698, 5305}, {732, 7805}, {754, 8370}, {760, 12263}, {1078, 5162}, {1916, 7828}, {2023, 7886}, {2458, 8150}, {2782, 7816}, {2794, 6248}, {3094, 7834}, {3095, 3788}, {3398, 5149}, {5028, 7808}, {5052, 7838}, {5188, 7830}, {5969, 7817}, {6194, 7800}, {6309, 7798}, {7759, 13330}, {7760, 9865}, {7761, 9821}, {7795, 9753}, {7797, 8782}, {7920, 10335}

    X(18806) = midpoint of X(32) and X(76)
    X(18806) = reflection of X(39) in X(6680)
    X(18806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76, 6179, 9983), (5403, 5404, 5031)


    X(18807) = X(5)X(128)∩X(1154)X(14071)

    Barycentrics    (S^2+SB*SC)*(5*S^4-(27*R^4+3* R^2*(3*SA-5*SW)+6*SB*SC+4*SW^ 2)*S^2-63*R^8+3*(SA+31*SW)*R^ 6-(6*SA^2-8*SB*SC+49*SW^2)*R^ 4-(SA+SW)*(4*SA-11*SW)*SW*R^2+ (2*SB*SC-SW^2)*SW^2) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27674.

    X(18807) lies on these lines: {5, 128}, {1154, 14071}


    X(18808) = X(4)X(523)∩X(74)X(1300)

    Barycentrics    (S^2+SB*SC)*(5*S^4-(27*R^4+3* R^2*(3*SA-5*SW)+6*SB*SC+4*SW^ 2)*S^2-63*R^8+3*(SA+31*SW)*R^ 6-(6*SA^2-8*SB*SC+49*SW^2)*R^ 4-(SA+SW)*(4*SA-11*SW)*SW*R^2+ (2*SB*SC-SW^2)*SW^2) : :

    See Angel Montesdeoca, HG160518.

    X(18808) lies on these lines: {4,523}, {74,1300}, {93,18039}, {264,850}, {297,14977}, {393,2433}, {476,1304}, {512,16263}, {520,12111}, {525,16253}, {685,879}, {847,11704}, {877,892}, {924,12290}, {1105,6368}, {1826,4024}, {2395,6531}, {5466,16080}, {6344,10412}, {9033,10152}


    X(18809) = MIDPOINT OF X(4) AND X(1304)

    Barycentrics    2 a^20 (b^2+c^2) -8 a^18 (b^4+c^4) +a^16 (5 b^6+7 b^4 c^2+7 b^2 c^4+5 c^6) +a^14 (22 b^8-66 b^6 c^2+72 b^4 c^4-66 b^2 c^6+22 c^8) -a^12 (42 b^10-87 b^8 c^2+43 b^6 c^4+43 b^4 c^6-87 b^2 c^8+42 c^10) +2 a^10 (b^2-c^2)^2 (7 b^8+41 b^6 c^2-59 b^4 c^4+41 b^2 c^6+7 c^8) +a^8 (b^2-c^2)^2 (28 b^10-119 b^8 c^2+79 b^6 c^4+79 b^4 c^6-119 b^2 c^8+28 c^10) -2 a^6 (b^2-c^2)^4 (15 b^8+11 b^6 c^2-46 b^4 c^4+11 b^2 c^6+15 c^8) +a^4 (b^2-c^2)^4 (8 b^10+43 b^8 c^2-41 b^6 c^4-41 b^4 c^6+43 b^2 c^8+8 c^10) +2 a^2 (b^2-c^2)^6 (b^8-5 b^6 c^2-15 b^4 c^4-5 b^2 c^6+c^8) -(b^2-c^2)^8 (b^6+4 b^4 c^2+4 b^2 c^4+c^6) : :

    See Angel Montesdeoca, HG160518.

    X(18809) lies on these lines: {2,2693}, {4,477}, {5,16177}, {30,122}, {113,520}, {115,1990}, {125,403}, {127,11799}, {133,523}, {136,10151}, {235,16221}, {1552,2777}, {1553,7480}, {1560,9209}, {1596,5099}, {5520,15763}, {6761,11792}

    X(18809) = reflection of X(16177) in X(5)
    X(18809) = reflection of X(133) in Euler line
    X(18809) = complement of X(2693)
    X(18809) = inverse-in-polar-circle of X(477)
    X(18809) = X(107) of reflection of Euler triangle in Euler line





    leftri  PERSPECTORS OF INCONICS: X(18810)-X(88131)  rightri

    Let P be a point in the plane of a triangle ABC. Let
    A' = reflection of P in BC, and define B' and C' cyclically;
    Q = circumcenter of A'B'C' = isogonal conjugate of P;
    A'' = QA'∩BC, and define B'' and C'' cyclically.
    The triangle A''B''C'' is perspective to ABC. Let D(P) denote the perspector. The points P and Q are the foci of the inconic tangent to BC, CA, AB at A'', B'', C'', respectively. (Based on notes from Thanh Oai Dao, May 17, 2018)

    If P = p : q : r (barycentrics), then D(P) = q r/(b2r2+ c2q2 + (b2 - c2 - a2) q r) : : , and

    D(P) = isotomic conjugate of the anticomplement of (midpoint of P and isogonal conjugate of P)
    D(P) = cevapoint of P and the orthocorrespondent of P
    D(P) = cross conjugate of (midpoint of P and isogonal conjugate of P) and G
    If P is on the infinity line, then D(P) = isotomic conjugate of P
    If P on the circumcircle or infinity line, then D(P) lies on the Steiner circumellipse.
    (Peter Moses, May 21, 2018)

    The appearance of (i,j) in the following list means that D(X(i)) = X(j):
    (67,10512), (69,10604), (74,1494), (98,290), (99,670), (100,668), (101,190), (103,18025), (105,2481} {106,903), (107,6528), (108,18026), (109,664), (110,99), (111,671), (112,648), (511,290), (512,670), (513,668), (514,190), (516,18025), (518,2481), (519,903), (520,6528), (521,18026), (522,664), (523,99), (524,671), (525,648), (527,1121), (532,11117), (533,11118), (536,3227), (538,3228), (542,5641), (690,892), (691,892), (698,3225), (699,3225), (726,3226), (727,3226), (729,3228), (732,14970), (733,14970), (739,3227), (758,14616), (759,14616), (812,4562), (813,4562), (824,4586), (825,4586), (826,4577), (827,4577), (842,5641), (888,886), (891,889), (898,889), (900,4555), (901,4555), (918,666), (919,666), (934,4569), (1113,15164), (1114,15165), (1304,16077), (1379,6190), (1380,6189), (2291,1121), (2380,11117), (2381,11118), (2574,15164), (2575,15165), (2715,2966), (2799,2966), (3413,6190), (3414,6189), (3900,4569), (3910,6648), (4588,4597), (4777,4597), (4977,6540), (6362,6606), (6550,6635), (6551,6635), (8687,6648), (8701,6540), (9033,16077), (9136,9487), (9150,886)

    underbar

    X(18810) = PERSPECTOR OF THE INCONIC WITH FOCI X(7) AND X(55)

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

    X(18810) lies on these lines: {85, 527}, {1088, 1323}

    X(18810) = isotomic conjugate of the anticomplement X(8255)
    X(18810) = X(8255)-cross conjugate of X(2)
    X(18810) = cevapoint of X(7) and X(1996)
    X(18810) = Brianchon point (perspector) of inconic centered at X(8255)
    X(18810) = isotomic conjugate of isogonal conjugate of vertex conjugate of X(7) and X(55)
    X(18810) = barycentric quotient X(i)/X(j) for these {i,j}: {85, 5231}, {279, 4860}, {513, 17425}, {1088, 6173}, {1996, 15346}
    X(18810) = X(i)-isoconjugate of X(j) for these (i,j): {101, 17425}, {1253, 4860}, {2175, 5231}, {6173, 14827}


    X(18811) = PERSPECTOR OF THE INCONIC WITH FOCI X(8) AND X(56)

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

    X(18811) lies on this line: {312, 3911}

    X(18811) = isotomic conjugate of X(2098)
    X(18811) = cevapoint of X(8) and X(1997)
    X(18811) = X(8256)-cross conjugate of X(2)
    X(18811) = X(i)-cross conjugate of X(j) for these (i,j): {4462, 4554}, {8256, 2}
    X(18811) = Brianchon point (perspector) of inconic centered at X(8256)
    X(18811) = isotomic conjugate of isogonal conjugate of vertex conjugate of X(8) and X(56)
    X(18811) = X(i)-isoconjugate of X(j) for these (i,j): {31, 2098}, {101, 17424}, {2175, 4862}
    X(18811) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2098}, {85, 4862}, {513, 17424}, {1997, 15347}


    X(18812) = PERSPECTOR OF THE INCONIC WITH FOCI X(10) AND X(58)

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

    X(18812) lies on these lines: {6, 6543}, {10, 261}, {12, 86}, {313, 17731}, {314, 1089}, {333, 594}

    X(18812) = isotomic conjugate of the anticomplement X(8258)
    X(18812) = isotomic conjugate of the complement X(1046)
    X(18812) = X(8258)-cross conjugate of X(2)
    X(18812) = X(i)-isoconjugate of X(j) for these (i,j): {81, 9560}, {662, 17411}
    X(18812) = cevapoint of X(i) and X(j) for these (i,j): {2, 1046}, {10, 1999}
    X(18812) = trilinear pole of line {4024, 4560}
    X(18812) = Brianchon point (perspector) of inconic centered at X(8258)
    X(18812) = isotomic conjugate of isogonal conjugate of vertex conjugate of X(10) and X(58)
    X(18812) = barycentric quotient X(i)/X(j) for these {i,j}: {42, 9560}, {512, 17411}, {1999, 15349}


    X(18813) = PERSPECTOR OF THE INCONIC WITH FOCI X(17) AND X(61)

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

    X(18813) lies on this line: {17, 622}

    X(18813) = isotomic conjugate of the anticomplement X(8259)
    X(18813) = X(8259)-cross conjugate of X(2)
    X(18813) = cevapoint of X(61) and X(2004)


    X(18814) = PERSPECTOR OF THE INCONIC WITH FOCI X(18) AND X(62)

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

    X(18814) lies on this line: {18, 621}

    X(18814) = isotomic conjugate of the anticomplement X(8260)
    X(18814) = X(8260)-cross conjugate of X(2)
    X(18814) = cevapoint of X(62) and X(2005)


    X(18815) = PERSPECTOR OF THE INCONIC WITH FOCI X(36) AND X(80)

    Barycentrics    b*(-a + b - c)*(a + b - c)*c*(a^2 - a*b + b^2 - c^2)*(-a^2 + b^2 + a*c - c^2) : :
    Barycentrics    1/(Cos[A] + Cos[2 A]) : :

    X(18815) lies on the conic {{A,B,C,X(2),X(7)}} and on these lines: {2, 2006}, {7, 80}, {27, 653}, {75, 311}, {77, 17885}, {85, 14584}, {86, 664}, {272, 759}, {273, 2973}, {310, 4572}, {347, 7318}, {651, 16732}, {655, 673}, {675, 1447}, {1325, 14194}, {1443, 17895}, {1807, 7269}, {1944, 2989}, {2166, 3668}, {3262, 4511}, {4957, 5723}, {5936, 15065}, {6650, 17950}

    X(18815) = isogonal conjugate of X(2361)
    X(18815) = isotomic conjugate of X(4511)
    X(18815) = cevapoint of X(i) and X(j) for these (i,j): {11, 10015}, {36, 2003}, {65, 1465}, {80, 2006}, {912, 1214}, {3911, 14584}, {4858, 14304}
    X(18815) = X(i)-cross conjugate of X(j) for these (i,j): {80, 18359}, {515, 92}, {1737, 2}, {3911, 85}
    X(18815) = X(75)-beth conjugate of X(664)
    X(18815) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 2361}, {4040, 654}
    X(18815) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2361}, {6, 2323}, {9, 7113}, {21, 3724}, {31, 4511}, {36, 55}, {37, 4282}, {41, 3218}, {50, 7110}, {80, 215}, {100, 8648}, {101, 654}, {184, 5081}, {186, 8606}, {212, 1870}, {228, 17515}, {284, 2245}, {320, 2175}, {650, 1983}, {692, 3738}, {758, 2194}, {1253, 1443}, {1464, 2328}, {1946, 4242}, {2150, 4053}, {2316, 17455}, {3063, 4585}, {3689, 16944}, {4089, 6066}, {4996, 6187}, {6149, 7073}, {14827, 17078}
    X(18815) = trilinear pole of line {226, 514}
    X(18815) = barycentric product X(i)*X(j) for these {i,j}: {7, 18359}, {75, 2006}, {76, 1411}, {80, 85}, {94, 1442}, {226, 14616}, {331, 1807}, {349, 759}, {655, 693}, {903, 14628}, {1434, 15065}, {1446, 6740}, {2161, 6063}, {2166, 17095}, {2222, 3261}
    X(18815) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2323}, {2, 4511}, {6, 2361}, {7, 3218}, {12, 4053}, {27, 17515}, {56, 7113}, {57, 36}, {58, 4282}, {65, 2245}, {80, 9}, {85, 320}, {92, 5081}, {109, 1983}, {226, 758}, {273, 17923}, {278, 1870}, {279, 1443}, {513, 654}, {514, 3738}, {553, 4973}, {649, 8648}, {653, 4242}, {655, 100}, {664, 4585}, {693, 3904}, {759, 284}, {1088, 17078}, {1168, 2316}, {1319, 17455}, {1399, 50}, {1400, 3724}, {1411, 6}, {1427, 1464}, {1441, 3936}, {1442, 323}, {1793, 2327}, {1807, 219}, {1989, 7073}, {2003, 6149}, {2006, 1}, {2161, 55}, {2166, 7110}, {2222, 101}, {2341, 2328}, {3218, 4996}, {3668, 18593}, {3676, 3960}, {3911, 214}, {4077, 4707}, {4654, 4880}, {5219, 4867}, {5435, 4881}, {6187, 41}, {6740, 2287}, {7113, 215}, {7146, 3792}, {14584, 44}, {14616, 333}, {14628, 519}, {15065, 2321}, {18359, 8}


    X(18816) = PERSPECTOR OF THE INCONIC WITH FOCI X(104) AND X(517)

    Barycentrics    b*c*(a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(-a^3 + a*b^2 + a^2*c - 2*a*b*c + b^2*c + a*c^2 - c^3) : :
    Barycentrics    (csc A)/(cos B + cos C - 1) : :

    X(18816) lies on the Steiner circumellipse and on these lines: {7, 264}, {63, 190}, {69, 150}, {75, 77}, {76, 14266}, {81, 648}, {99, 104}, {286, 6528}, {317, 8048}, {666, 1814}, {889, 15635}, {909, 4586}, {2401, 3227}, {2966, 14578}, {3262, 4555}, {4124, 9432}, {4569, 6063}, {4597, 17143}, {10538, 14198}

    X(18816) = isotomic conjugate of X(517)
    X(18816) = cevapoint of X(i) and X(j) for these (i,j): {2, 517}, {8, 4358}, {75, 320}, {2401, 15635}
    X(18816) = X(i)-cross conjugate of X(j) for these (i,j): {104, 16082}, {517, 2}, {2804, 6335}, {3218, 274}, {3904, 4554}, {15635, 2401}
    X(18816) = X(314)-beth conjugate of X(664)
    X(18816) = X(43)-zayin conjugate of X(2183)
    X(18816) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2183}, {31, 517}, {32, 908}, {41, 1465}, {42, 859}, {48, 14571}, {55, 1457}, {101, 3310}, {184, 1785}, {212, 1875}, {560, 3262}, {649, 2427}, {692, 1769}, {810, 4246}, {902, 14260}, {1361, 2342}, {1397, 6735}, {1911, 15507}, {1918, 17139}, {1919, 2397}, {2206, 17757}, {8677, 8750}
    X(18816) = trilinear pole of line {2, 905}
    X(18816) = polar conjugate of X(14571)
    X(18816) = barycentric product X(i)*X(j) for these {i,j}: {69, 16082}, {76, 104}, {310, 2250}, {331, 1809}, {561, 909}, {668, 2401}, {693, 13136}, {1309, 15413}, {1795, 1969}, {2423, 6386}, {14578, 18022}
    X(18816) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2183}, {2, 517}, {4, 14571}, {7, 1465}, {57, 1457}, {75, 908}, {76, 3262}, {81, 859}, {88, 14260}, {92, 1785}, {100, 2427}, {104, 6}, {239, 15507}, {274, 17139}, {278, 1875}, {312, 6735}, {320, 16586}, {321, 17757}, {513, 3310}, {514, 1769}, {648, 4246}, {668, 2397}, {693, 10015}, {905, 8677}, {909, 31}, {1309, 1783}, {1465, 1361}, {1795, 48}, {1809, 219}, {2250, 42}, {2342, 41}, {2397, 15632}, {2401, 513}, {2423, 667}, {2720, 1415}, {4358, 1145}, {4391, 2804}, {10428, 9456}, {13136, 100}, {14266, 8609}, {14578, 184}, {15501, 198}, {15635, 1015}, {16082, 4}, {17862, 1532}, {17923, 1845}


    X(18817) = PERSPECTOR OF THE INCONIC WITH FOCI X(186) AND X(265)

    Barycentrics   b^4*c^4*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2) : :, Csc[A] Csc[3 A] Sec[A] : :

    X(18817) lies on these lines: {94, 2052}, {264, 328}, {265, 6528}, {476, 16089}, {1989, 16081}, {14254, 18027}

    X(18817) = isotomic conjugate of the isogonal conjugate of X(6344)
    X(18817) = cevapoint of X(i) and X(j) for these (i,j): {5, 3580}, {94, 6344}, {186, 1994}
    X(18817) = X(i)-isoconjugate of X(j) for these (i,j): {48, 50}, {184, 6149}, {323, 9247}, {822, 14591}, {2290, 14533}, {4575, 14270}
    X(18817) = trilinear pole of line {324, 14592}
    X(18817) = polar conjugate of X(50)
    X(18817) = barycentric product X(i)*X(j) for these {i,j}: {76, 6344}, {94, 264}, {265, 18027}, {328, 2052}, {1502, 18384}, {1969, 2166}, {1989, 18022}, {6331, 10412}, {6528, 14592}
    X(18817) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 50}, {92, 6149}, {94, 3}, {107, 14591}, {264, 323}, {265, 577}, {324, 1154}, {328, 394}, {338, 16186}, {850, 8552}, {1141, 14533}, {1989, 184}, {2052, 186}, {2166, 48}, {2501, 14270}, {2970, 2088}, {6331, 10411}, {6344, 6}, {6528, 14590}, {10412, 647}, {11060, 14575}, {13450, 11062}, {14165, 3043}, {14254, 3284}, {14356, 3289}, {14592, 520}, {14618, 526}, {14859, 11077}, {15475, 3049}, {16080, 14385}, {16081, 14355}, {18022, 7799}, {18027, 340}, {18384, 32}


    X(18818) = PERSPECTOR OF THE INCONIC WITH FOCI X(187) AND X(671)

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

    X(18818) lies on the conic {{A,B,C,X(2),X(56)) and these lines: {2, 18023}, {6, 598}, {25, 17983}, {111, 8859}, {263, 9214}, {524, 8785}, {694, 17948}, {1976, 9154}, {4590, 11162}, {6094, 14568}

    X(18818) = isotomic conjugate of the complement X(11054)
    X(18818) = X(i)-cross conjugate of X(j) for these (i,j): {2408, 892}, {6088, 670}, {9465, 10630}, {9485, 99}
    X(18818) = cevapoint of X(i) and X(j) for these (i,j): {2, 11054}, {187, 11422}
    X(18818) = X(i)-isoconjugate of X(j) for these (i,j): {574, 896}, {599, 922}, {2642, 9145}
    X(18818) = trilinear pole of line {512, 598}
    X(18818) = barycentric product X(i)*X(j) for these {i,j}: {598, 671}, {892, 8599}, {1383, 18023}, {10512, 14246}
    X(18818) = barycentric quotient X(i)/X(j) for these {i,j}: {111, 574}, {598, 524}, {671, 599}, {691, 9145}, {892, 9146}, {1383, 187}, {5380, 3908}, {5466, 3906}, {8599, 690}, {8753, 8541}, {9178, 17414}, {9214, 13857}, {10511, 14357}, {11636, 5467}, {14246, 10510}, {17983, 5094}, {18023, 9464}


    X(18819) = PERSPECTOR OF THE INCONIC WITH FOCI X(371) AND X(485)

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

    X(18819) lies on this line: {485, 490}

    X(18819) = cevapoint of X(371) and X(8577)
    X(18819) = trilinear pole of line {9131, 13320}
    X(18819) = barycentric quotient X(8577)/X(1504)


    X(18820) = PERSPECTOR OF THE INCONIC WITH FOCI X(372) AND X(486)

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

    X(18820) lies on this line: {486, 489}

    X(18820) = cevapoint of X(372) and X(8576)
    X(18820) = trilinear pole of line {9131, 13319}
    X(18820) = barycentric quotient X(8576)/X(1505)


    X(18821) = PERSPECTOR OF THE INCONIC WITH FOCI X(528) AND X(840)

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

    X(18821) lies on the Steiner circumellipse and these lines: {2, 666}, {8, 4555}, {99, 840}, {190, 320}, {514, 1121}, {519, 664}, {522, 903}, {648, 15149}, {668, 3263}, {693, 2481}, {4076, 6635}, {4562, 17294}, {4597, 10031}, {4998, 6174}, {7199, 14616}

    X(18821) = reflection of X(666) in X(2)
    X(18821) = isotomic conjugate of X(528)
    X(18821) = X(43)-zayin conjugate of X(2246)
    X(18821) = X(528)-cross conjugate of X(2)
    X(18821) = cevapoint of X(i) and X(j) for these (i,j): {2, 528}
    X(18821) = trilinear pole of line {2, 918}
    X(18821) = barycentric product X(76)*X(840)
    X(18821) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2246}, {2, 528}, {7, 5723}, {88, 14190}, {513, 1643}, {518, 1642}, {840, 6}, {926, 14411}, {5723, 3322}, {14191, 44}
    X(18821) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2246}, {31, 528}, {41, 5723}, {101, 1643}, {902, 14190}, {1438, 1642}


    X(18822) = PERSPECTOR OF THE INCONIC WITH FOCI X(537) AND X(2382)

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

    X(18822) lies on the Steiner circumellipse and these lines: {1, 4555}, {2, 4562}, {44, 190}, {75, 889}, {99, 2382}, {350, 519}, {513, 903}, {514, 3227}, {664, 1319}, {666, 6654}, {765, 6635}, {1877, 18026}

    X(18822) = reflection of X(4562) in X(2)
    X(18822) = isotomic conjugate of X(537)
    X(18822) = X(537)-cross conjugate of X(2) X(18822) = cevapoint of X(2) and X(537)
    X(18822) = X(314)-beth conjugate of X(889)
    X(18822) = trilinear pole of line {2, 812}
    X(18822) = barycentric product X(76)*X(2382)
    X(18822) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 537}, {2382, 6}


    X(18823) = PERSPECTOR OF THE INCONIC WITH FOCI X(543) AND X(843)

    Barycentrics    (a^4 - 4*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 2*c^4)*(a^4 + 2*a^2*b^2 - 2*b^4 - 4*a^2*c^2 + 2*b^2*c^2 + c^4) : :
    X(18823) = 4 X[2482] - 3 X[4590], 4 X[9164] - 3 X[9182], 16 X[9165] - 15 X[14061], 3 X[9166] - 2 X[17948]

    X(18823) lies on the Steiner circumellipse and these lines: {2, 892}, {30, 16103}, {76, 14255}, {99, 524}, {190, 4062}, {468, 648}, {523, 671}, {670, 3266}, {1992, 2966}, {2482, 4590}, {5461, 14728}, {9164, 9182}, {9165, 14061}, {9166, 17948}, {13479, 16092}

    X(18823) = reflection of X(892) in X(2)
    X(18823) = isogonal conjugate of X(2502)
    X(18823) = isotomic conjugate of X(543)
    X(18823) = cevapoint of X(2) and X(543)
    X(18823) = X(9170)-Ceva conjugate of X(9180)
    X(18823) = X(i)-cross conjugate of X(j) for these (i,j): {543, 2}, {9168, 99}, {9180, 9170}
    X(18823) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2502}, {31, 543}, {163, 8371}, {187, 17955}, {661, 9181}, {662, 9171}, {798, 9182}, {896, 17964}, {922, 17948}, {923, 1641}
    X(18823) = trilinear pole of line {2, 690}
    X(18823) = barycentric product X(i)*X(j) for these {i,j}: {76, 843}, {99, 9180}, {523, 9170}
    X(18823) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 543}, {6, 2502}, {99, 9182}, {110, 9181}, {111, 17964}, {512, 9171}, {523, 8371}, {524, 1641}, {671, 17948}, {843, 6}, {897, 17955}, {5466, 18007}, {9170, 99}, {9178, 17993}, {9180, 523}


    X(18824) = PERSPECTOR OF THE INCONIC WITH FOCI X(696) AND X(697)

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

    X(18824) lies on the Steiner circumellipse and these lines: {31, 668}, {32, 190}, {58, 670}, {99, 697}, {664, 1397}, {1106, 4569}, {1395, 18026}, {1922, 4562}

    X(18824) = isogonal conjugate of X(8619)
    X(18824) = isotomic conjugate of X(696)
    X(18824) = X(696)-cross conjugate of X(2)
    X(18824) = cevapoint of X(2) and X(696)
    X(18824) = trilinear pole of line {2, 1919}
    X(18824) = barycentric product X(76)*X(697)
    X(18824) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 696}, {6, 8619}, {697, 6}


    X(18825) = PERSPECTOR OF THE INCONIC WITH FOCI X(712) AND X(713)

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

    X(18825) lies on the Steiner circumellipse and these lines: {6, 668}, {31, 190}, {81, 670}, {99, 713}, {385, 17961}, {604, 664}, {608, 18026}, {648, 2203}, {739, 889}, {1407, 4569}, {1911, 4562}, {2162, 4363}, {4555, 9456}, {5317, 6528}

    X(18825) = isogonal conjugate of X(8620)
    X(18825) = isotomic conjugate of X(712)
    X(18825) = cevapoint of X(i) and X(j) for these (i,j): {2, 712}
    X(18825) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 8620}, {43, 2228}
    X(18825) = X(712)-cross conjugate of X(2)
    X(18825) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8620}, {6, 2228}, {31, 712}, {190, 9297}
    X(18825) = trilinear pole of line {2, 667}
    X(18825) = barycentric product X(i)*X(j) for these {i,j}: {76, 713}
    X(18825) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2228}, {2, 712}, {6, 8620}, {667, 9297}, {713, 6}


    X(18826) = PERSPECTOR OF THE INCONIC WITH FOCI X(714) AND X(715)

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

    X(18826) lies on the Steiner circumellipse and these lines: {1, 670}, {31, 99}, {42, 668}, {190, 213}, {648, 1973}, {664, 1402}, {892, 923}, {1042, 4569}, {1096, 6528}

    X(18826) = isotomic conjugate of X(714)
    X(18826) = X(43)-zayin conjugate of X(2229)
    X(18826) = X(i)-cross conjugate of X(j) for these (i,j): {714, 2}, {3768, 799}
    X(18826) = cevapoint of X(2) and X(714)
    X(18826) = trilinear pole of line {2, 798}
    X(18826) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2229}, {31, 714}
    X(18826) = barycentric product X(76)*X(715)
    X(18826) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2229}, {2, 714}, {715, 6}


    X(18827) = PERSPECTOR OF THE INCONIC WITH FOCI X(740) AND X(741)

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

    X(18827) lies on the Steiner circumellipse, the cubics K768 and K769, and on these lines: on lines {1, 99}, {10, 274}, {19, 648}, {37, 86}, {38, 873}, {65, 664}, {75, 670}, {81, 4586}, {82, 757}, {158, 6528}, {225, 18026}, {239, 9278}, {244, 799}, {518, 2669}, {596, 17143}, {666, 2311}, {671, 4444}, {813, 2368}, {870, 982}, {876, 18009}, {889, 4639}, {892, 897}, {1910, 2966}, {1916, 6625}, {2106, 16514}, {2481, 4458}, {2588, 15165}, {2589, 15164}, {2663, 18787}, {2668, 3666}, {3228, 3572}, {3668, 4569}, {4360, 13476}, {4555, 4589}, {4562, 6542}, {4576, 17154}, {6540, 17175}, {6626, 16823}, {9505, 17770}, {10436, 17038}

    X(18827) = isogonal conjugate of X(3747)
    X(18827) = isotomic conjugate of X(740)
    X(18827) = cevapoint of X(i) and X(j) for these (i,j): {1, 18206}, {2, 740}, {86, 17731}, {244, 812}, {291, 335}, {4155, 16592}
    X(18827) = X(i)-cross conjugate of X(j) for these (i,j): {740, 2}, {812, 799}, {876, 4562}, {1959, 85}, {2227, 87}, {2254, 662}, {4645, 14534}
    X(18827) = X(314)-beth conjugate of X(670)
    X(18827) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 3747}, {43, 2238}
    X(18827) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3747}, {3, 862}, {6, 2238}, {10, 2210}, {31, 740}, {32, 3948}, {37, 1914}, {41, 16609}, {42, 238}, {55, 1284}, {56, 4433}, {71, 2201}, {82, 4093}, {100, 4455}, {110, 4155}, {210, 1428}, {212, 1874}, {213, 239}, {228, 242}, {321, 14599}, {350, 1918}, {512, 3573}, {604, 3985}, {659, 4557}, {669, 874}, {692, 4010}, {741, 4094}, {756, 5009}, {798, 3570}, {904, 4039}, {1018, 8632}, {1333, 4037}, {1334, 1429}, {1400, 3684}, {1402, 3685}, {1824, 7193}, {1911, 4368}, {1921, 2205}, {1967, 4154}, {2054, 8298}, {2194, 7235}, {3903, 5027}, {4435, 4559}
    X(18827) = X(741)-Hirst inverse of X(17103)
    X(18827) = trilinear pole of line {2, 661}
    X(18827) = barycentric product X(i)*X(j) for these {i,j}: {27, 337}, {76, 741}, {81, 334}, {86, 335}, {99, 4444}, {274, 291}, {292, 310}, {331, 1808}, {333, 7233}, {513, 4639}, {514, 4589}, {561, 18268}, {660, 7199}, {670, 3572}, {693, 4584}, {799, 876}, {875, 4602}, {1019, 4583}, {1434, 4518}, {1581, 8033}, {1911, 6385}, {1916, 17103}, {2311, 6063}, {4562, 7192}, {5378, 16727}
    X(18827) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2238}, {2, 740}, {6, 3747}, {7, 16609}, {8, 3985}, {9, 4433}, {10, 4037}, {19, 862}, {21, 3684}, {27, 242}, {28, 2201}, {39, 4093}, {57, 1284}, {58, 1914}, {75, 3948}, {81, 238}, {86, 239}, {99, 3570}, {226, 7235}, {239, 4368}, {274, 350}, {278, 1874}, {291, 37}, {292, 42}, {295, 71}, {310, 1921}, {314, 3975}, {333, 3685}, {334, 321}, {335, 10}, {337, 306}, {385, 4154}, {514, 4010}, {593, 5009}, {649, 4455}, {660, 1018}, {661, 4155}, {662, 3573}, {741, 6}, {799, 874}, {813, 4557}, {875, 798}, {876, 661}, {894, 4039}, {1014, 1429}, {1019, 659}, {1333, 2210}, {1412, 1428}, {1434, 1447}, {1790, 7193}, {1808, 219}, {1911, 213}, {1922, 1918}, {1931, 8298}, {2196, 228}, {2206, 14599}, {2238, 4094}, {2311, 55}, {3572, 512}, {3616, 4771}, {3676, 7212}, {3733, 8632}, {3736, 16514}, {3737, 4435}, {4128, 2086}, {4358, 4783}, {4369, 804}, {4444, 523}, {4518, 2321}, {4560, 3716}, {4562, 3952}, {4583, 4033}, {4584, 100}, {4589, 190}, {4639, 668}, {4649, 16369}, {4778, 4839}, {4876, 210}, {4960, 4810}, {5235, 4693}, {5257, 4829}, {5333, 4716}, {6629, 4760}, {7077, 1334}, {7192, 812}, {7199, 3766}, {7233, 226}, {7253, 4148}, {8025, 4974}, {8033, 1966}, {9505, 9278}, {9506, 2054}, {14598, 2205}, {16609, 3027}, {16704, 4432}, {16737, 14296}, {17103, 385}, {17197, 4124}, {17212, 4107}, {17731, 6651}, {18200, 4164}, {18206, 8299}, {18268, 31}, {18787, 2295}, {18792, 17475}


    X(18828) = PERSPECTOR OF THE INCONIC WITH FOCI X(782) AND X(783)

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

    X(18828) lies on the Steiner circumellipse and these lines: {99, 783}, {688, 4577}, {754, 14946}

    X(18828) = isotomic conjugate of X(782)
    X(18828) = X(782)-cross conjugate of X(2)
    X(18828) = cevapoint of X(i) and X(j) for these (i,j): {2, 782}, {882, 14970}
    X(18828) = X(i)-isoconjugate of X(j) for these (i,j): {31, 782}, {2084, 16985}
    X(18828) = trilinear pole of line {2, 14946}
    X(18828) = barycentric product X(76)*X(783)
    X(18828) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 782}, {783, 6}, {4577, 16985}, {14946, 688}


    X(18829) = PERSPECTOR OF THE INCONIC WITH FOCI X(804) AND X(805)

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

    X(18829) lies on the Steiner circumellipse and these lines: {76, 14251}, {99, 512}, {190, 4079}, {290, 325}, {351, 9150}, {385, 3225}, {523, 670}, {524, 694}, {538, 671}, {648, 2489}, {664, 4589}, {668, 4705}, {881, 886}, {882, 2396}, {892, 9178}, {1634, 4577}, {1934, 14616}, {2421, 2422}, {3226, 17731}, {3493, 3933}, {4584, 4586}, {5641, 7788}, {7779, 14970}, {9467, 12215}, {9487, 11163}

    X(18829) = reflection of X(i) in X(j) for these {i,j}: {385, 3229}, {3978, 325}
    X(18829) = isogonal conjugate of X(5027)
    X(18829) = isotomic conjugate of X(804)
    X(18829) = cevapoint of X(i) and X(j) for these (i,j): {2, 804}, {69, 684}, {325, 523}, {512, 3229}, {694, 881}, {732, 3005}, {812, 6682}, {1634, 2421}, {3569, 14251}
    X(18829) = X(i)-cross conjugate of X(j) for these (i,j): {804, 2}, {876, 1581}, {881, 694}, {882, 14970}, {2395, 8781}, {2396, 99}, {3569, 76}, {5207, 18020}, {7779, 4590}
    X(18829) = crossdifference of every pair of points on line {2086, 2679}
    X(18829) = crosssum of X(2491) and X(9429)
    X(18829) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5027}, {31, 804}, {42, 4164}, {171, 4455}, {213, 4107}, {238, 7234}, {385, 798}, {419, 810}, {512, 1580}, {523, 1933}, {560, 14295}, {661, 1691}, {662, 2086}, {667, 4039}, {669, 1966}, {875, 4154}, {880, 4117}, {923, 11183}, {1577, 14602}, {1918, 14296}, {1924, 3978}, {1926, 9426}, {2210, 2533}, {2236, 18105}, {2295, 8632}, {3573, 4128}, {3747, 4367}, {4010, 7122}
    X(18829) = X(99)-Hirst inverse of X(805)
    X(18829) = X(17938)-vertex conjugate of X(17941)
    X(18829) = trilinear pole of line {2, 694}
    X(18829) = barycentric product X(i)*X(j) for these {i,j}: {76, 805}, {99, 1916}, {256, 4639}, {257, 4589}, {291, 7260}, {334, 4603}, {335, 4594}, {662, 1934}, {670, 694}, {799, 1581}, {1502, 17938}, {1967, 4602}, {4576, 14970}, {4584, 7018}, {4609, 9468}
    X(18829) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 804}, {6, 5027}, {76, 14295}, {81, 4164}, {86, 4107}, {99, 385}, {110, 1691}, {163, 1933}, {190, 4039}, {257, 4010}, {274, 14296}, {292, 7234}, {335, 2533}, {512, 2086}, {524, 11183}, {648, 419}, {660, 2295}, {662, 1580}, {670, 3978}, {694, 512}, {733, 18105}, {799, 1966}, {805, 6}, {876, 16592}, {881, 1084}, {882, 3124}, {893, 4455}, {1178, 8632}, {1576, 14602}, {1581, 661}, {1634, 8623}, {1916, 523}, {1927, 1924}, {1934, 1577}, {1967, 798}, {2396, 5976}, {3569, 2679}, {3570, 4154}, {3572, 4128}, {3903, 2238}, {4074, 782}, {4226, 12829}, {4518, 4140}, {4562, 1215}, {4563, 12215}, {4576, 732}, {4583, 3963}, {4584, 171}, {4589, 894}, {4590, 17941}, {4594, 239}, {4602, 1926}, {4603, 238}, {4609, 14603}, {4639, 1909}, {4835, 4839}, {5468, 5026}, {6331, 17984}, {7249, 7212}, {7260, 350}, {8789, 9426}, {9468, 669}, {11654, 9009}, {14251, 2491}, {15391, 878}, {17938, 32}, {17941, 4027}, {17970, 3049}, {17980, 2489}


    X(18830) = PERSPECTOR OF THE INCONIC WITH FOCI X(932) AND X(4083)

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

    X(18830) lies on the Steiner circumellipse and these lines: {75, 87}, {76, 4014}, {99, 932}, {190, 4598}, {330, 3227}, {350, 4947}, {513, 6386}, {646, 4562}, {664, 874}, {666, 5383}, {668, 3888}, {894, 3225}, {903, 4479}, {1221, 7168}, {1930, 3494}, {2162, 4363}, {2481, 6383}, {3228, 16606}

    X(18830) = isogonal conjugate of X(8640)
    X(18830) = isotomic conjugate of X(4083)
    X(18830) = X(4598)-Ceva conjugate of X(668)
    X(18830) = X(i)-cross conjugate of X(j) for these (i,j): {513, 87}, {649, 274}, {1978, 668}, {4083, 2}, {7155, 5383}, {10453, 1016}, {17144, 7035}
    X(18830) = cevapoint of X(i) and X(j) for these (i,j): {2, 4083}, {75, 513}, {321, 2533}, {514, 3840}
    X(18830) = X(4598)-aleph conjugate of X(15966)
    X(18830) = X(314)-beth conjugate of X(3226)
    X(18830) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8640}, {31, 4083}, {32, 3835}, {42, 16695}, {43, 667}, {101, 6377}, {192, 1919}, {213, 18197}, {513, 2209}, {649, 2176}, {663, 1403}, {692, 3123}, {1397, 4147}, {1423, 3063}, {1918, 17217}, {1977, 4595}, {1980, 6376}, {2200, 17921}, {3051, 18107}, {9456, 14408}
    X(18830) = trilinear pole of line {2, 330}
    X(18830) = barycentric product X(i)*X(j) for these {i,j}: {75, 4598}, {76, 932}, {87, 1978}, {100, 6383}, {190, 6384}, {330, 668}, {670, 16606}, {693, 5383}, {2162, 6386}, {2319, 4572}, {3699, 7209}, {4554, 7155}
    X(18830) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4083}, {6, 8640}, {75, 3835}, {81, 16695}, {86, 18197}, {87, 649}, {100, 2176}, {101, 2209}, {190, 43}, {274, 17217}, {286, 17921}, {312, 4147}, {330, 513}, {513, 6377}, {514, 3123}, {519, 14408}, {536, 14426}, {651, 1403}, {664, 1423}, {668, 192}, {932, 6}, {1978, 6376}, {2053, 3063}, {2162, 667}, {2319, 663}, {3112, 18107}, {3699, 3208}, {4033, 3971}, {4359, 4992}, {4554, 3212}, {4598, 1}, {4623, 7304}, {5383, 100}, {6383, 693}, {6384, 514}, {6386, 6382}, {7035, 4595}, {7121, 1919}, {7148, 4079}, {7155, 650}, {7192, 16742}, {7209, 3676}, {7275, 4502}, {16606, 512}


    X(18831) = PERSPECTOR OF THE INCONIC WITH FOCI X(933) AND X(6368)

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

    X(18831) lies on the Steiner circumellipse and these lines: {49, 9291}, {54, 276}, {95, 549}, {99, 933}, {110, 6528}, {275, 671}, {648, 16813}, {877, 4577}, {1147, 18027}, {2966, 14586}, {3228, 8882}, {5504, 8795}

    X(18831) = isogonal conjugate of X(15451)
    X(18831) = isotomic conjugate of X(6368)
    X(18831) = X(i)-cross conjugate of X(j) for these (i,j): {110, 18315}, {264, 18020}, {933, 16813}, {1993, 4590}, {5012, 250}, {6368, 2}, {15412, 276}
    X(18831) = cevapoint of X(i) and X(j) for these (i,j): {2, 6368}, {54, 15412}, {110, 648}, {140, 525}, {933, 18315}, {9033, 14920}
    X(18831) = X(i)-vertex conjugate of X(j) for these (i,j): {4, 14586}, {1576, 6528}
    X(18831) = trilinear pole of line {2, 95}
    X(18831) = center of bianticevian conic of X(2) and X(3)
    X(18831) = polar conjugate of X(12077)
    X(18831) = orthic-to-ABC barycentric image of X(130)
    X(18831) = barycentric product X(i)*X(j) for these {i,j}: {54, 6331}, {69, 16813}, {76, 933}, {95, 648}, {97, 6528}, {99, 275}, {110, 276}, {264, 18315}, {670, 8882}, {799, 2190}, {811, 2167}, {4558, 8795}, {4563, 8884}, {14586, 18022}, {15412, 18020}, {15958, 18027}
    X(18831) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6368}, {3, 17434}, {4, 12077}, {6, 15451}, {30, 14391}, {54, 647}, {92, 2618}, {95, 525}, {97, 520}, {99, 343}, {107, 53}, {110, 216}, {112, 51}, {162, 1953}, {186, 2081}, {250, 1625}, {264, 18314}, {275, 523}, {276, 850}, {648, 5}, {799, 18695}, {811, 14213}, {933, 6}, {1141, 14582}, {1576, 217}, {2148, 810}, {2167, 656}, {2169, 822}, {2190, 661}, {2407, 1568}, {2616, 3708}, {4558, 5562}, {6331, 311}, {6528, 324}, {6529, 14569}, {8795, 14618}, {8882, 512}, {8884, 2501}, {14586, 184}, {14590, 1154}, {15352, 13450}, {15412, 125}, {15422, 8754}, {15958, 577}, {16813, 4}, {17515, 2600}, {18020, 14570}, {18022, 15415}, {18315, 3}
    X(18831) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15451}, {5, 810}, {19, 17434}, {31, 6368}, {48, 12077}, {51, 656}, {53, 822}, {184, 2618}, {216, 661}, {217, 1577}, {343, 798}, {520, 2181}, {525, 2179}, {647, 1953}, {669, 18695}, {1625, 3708}, {2159, 14391}, {2290, 14582}, {3049, 14213}, {4079, 16697}, {9247, 18314}


    X(18832) = ISOTOMIC CONJUGATE OF X(1740)

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

    X(18832) lies on the cubics K995 and K1028 and on these lines: {1, 1965}, {10, 6382}, {19, 1966}, {37, 2998}, {75, 1925}, {304, 1581}, {759, 3222}, {2186, 18156}, {3403, 8769}

    X(18832) = isotomic conjugate of X(1740)
    X(18832) = X(3223)-Ceva conjugate of X(75)
    X(18832) = X(i)-cross conjugate of X(j) for these (i,j): {561, 75}, {4117, 18070}
    X(18832) = X(i)-isoconjugate of X(j) for these (i,j): {3, 11325}, {6, 1613}, {31, 1740}, {32, 194}, {41, 1424}, {99, 9491}, {110, 3221}, {112, 2524}, {184, 3186}, {560, 17149}, {604, 7075}, {1501, 6374}, {2175, 17082}
    X(18832) = cevapoint of X(693) and X(3123)
    X(18832) = trilinear pole of line {661, 17893}
    X(18832) = barycentric product X(i)*X(j) for these {i,j}: {75, 2998}, {76, 3223}, {561, 3224}, {1577, 3222}, {1969, 3504}
    X(18832) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1613}, {2, 1740}, {7, 1424}, {8, 7075}, {19, 11325}, {75, 194}, {76, 17149}, {85, 17082}, {92, 3186}, {561, 6374}, {656, 2524}, {661, 3221}, {798, 9491}, {2998, 1}, {3222, 662}, {3223, 6}, {3224, 31}, {3504, 48}, {15389, 9247}


    X(18833) = ISOGONAL CONJUGATE OF X(1923)

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

    X(18833) lies on the cubics K865 and K1028 and on these lines: {1, 561}, {19, 1969}, {37, 308}, {65, 18033}, {75, 1928}, {82, 1966}, {304, 2186}, {310, 596}, {689, 759}, {1581, 1925}, {1910, 4593}, {1923, 1965}, {6385, 13476}

    X(18833) = isogonal conjugate of X(1923)
    X(18833) = isotomic conjugate of X(1964)
    X(18833) = cevapoint of X(75) and X(561)
    X(18833) = X(i)-zayin conjugate of X(j) for these (i,j): {1, 1923}, {18272, 2236}
    X(18833) = X(i)-cross conjugate of X(j) for these (i,j): {75, 3112}, {1237, 76}, {1577, 4602}, {16889, 83}, {17446, 92}
    X(18833) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1923}, {6, 3051}, {31, 1964}, {32, 39}, {38, 560}, {99, 9494}, {110, 688}, {141, 1501}, {163, 2084}, {184, 1843}, {427, 14575}, {669, 1634}, {732, 8789}, {826, 14574}, {827, 2531}, {1397, 3688}, {1401, 2175}, {1576, 3005}, {1917, 1930}, {1918, 17187}, {1927, 2236}, {1973, 4020}, {1974, 3917}, {1980, 4553}, {2205, 16696}, {3404, 9417}, {3665, 9448}, {4576, 9426}, {8024, 9233}, {8623, 9468}, {9247, 17442}
    X(18833) = X(1)-Hirst inverse of X(18028)
    X(18833) = trilinear pole of line {661, 786}
    X(18833) = barycentric product X(i)*X(j) for these {i,j}: {75, 308}, {76, 3112}, {82, 1502}, {83, 561}, {251, 1928}, {670, 18070}, {689, 1577}, {850, 4593}, {1799, 1969}, {1926, 14970}, {3405, 18024}, {6385, 18082}, {6386, 10566}
    X(18833) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3051}, {2, 1964}, {6, 1923}, {69, 4020}, {75, 39}, {76, 38}, {82, 32}, {83, 31}, {85, 1401}, {92, 1843}, {251, 560}, {264, 17442}, {274, 17187}, {290, 3404}, {304, 3917}, {308, 1}, {310, 16696}, {312, 3688}, {313, 3954}, {523, 2084}, {561, 141}, {661, 688}, {689, 662}, {733, 1927}, {798, 9494}, {799, 1634}, {850, 8061}, {1176, 9247}, {1237, 16587}, {1502, 1930}, {1577, 3005}, {1799, 48}, {1925, 4074}, {1926, 732}, {1928, 8024}, {1930, 8041}, {1966, 8623}, {1969, 427}, {1978, 4553}, {3112, 6}, {3261, 2530}, {3403, 14096}, {3405, 237}, {3948, 4093}, {3978, 2236}, {4118, 3118}, {4577, 163}, {4580, 810}, {4593, 110}, {4599, 1576}, {4602, 4576}, {6385, 16887}, {6386, 4568}, {8061, 2531}, {10566, 667}, {14970, 1967}, {16889, 16584}, {16890, 2085}, {17500, 2179}, {18041, 211}, {18042, 3203}, {18070, 512}, {18082, 213}, {18091, 1197}, {18097, 1402}, {18098, 1918}, {18105, 1924}, {18107, 8640}, {18108, 1919}, {18156, 3787}
    X(18833) = {X(1925),X(1930)}-harmonic conjugate of X(4602)


    X(18834) = ISOTOMIC CONJUGATE OF X(16556)

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

    X(18834) lies on the cubic K1028 and these lines: {1031, 17788}, {1930, 1965}

    X(18834) = isotomic conjugate of X(16556)
    X(18834) = X(3112)-cross conjugate of X(75)
    X(18834) = X(i)-isoconjugate of X(j) for these (i,j): {6, 10329}, {31, 16556}, {32, 2896}, {39, 14885}, {2175, 17083}
    X(18834) = barycentric product X(i)*X(j) for these {i,j}: {75, 1031}, {561, 14370}
    X(18834) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 10329}, {2, 16556}, {75, 2896}, {82, 14885}, {85, 17083}, {1031, 1}, {14370, 31}


    X(18835) = X(1)X(76)∩ X(264)X(1969)

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

    X(18835) lies on the cubic K1028 and on these lines: {1, 76}, {264, 1969}, {305, 3705}, {313, 3178}, {1848, 2064}, {1928, 18036}, {1930, 7018}, {4109, 17788}

    X(18835) = isotomic conjugate of the isogonal conjugate of X(4388)
    X(18835) = X(i)-isoconjugate of X(j) for these (i,j): {32, 3497}, {560, 7224}
    X(18835) = barycentric product X(i)*X(j) for these {i,j}: {75, 17788}, {76, 4388}, {310, 4109}, {561, 3496}, {1978, 4142}, {3596, 17086}
    X(18835) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 3497}, {76, 7224}, {3496, 31}, {4109, 42}, {4142, 649}, {4388, 6}, {17086, 56}, {17788, 1}, {17797, 172}


    X(18836) = X(1)X(18036)∩ X(76)X(3497)

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

    X(18836) lies on the cubic K1028 and these lines: {1, 18036}, {76, 3497}, {304, 17786}, {4385, 7182}

    X(18836) = isotomic conjugate of the isogonal conjugate of X(7224)
    X(18836) = X(1909)-cross conjugate of X(76)
    X(18836) = X(i)-isoconjugate of X(j) for these (i,j): {32, 3496}, {560, 4388}, {1501, 17788}, {9447, 17086}
    X(18836) = barycentric product X(i)*X(j) for these {i,j}: {76, 7224}, {561, 3497}
    X(18836) = barycentric quotient X(i)/X(j) for these {i,j}: {75, 3496}, {76, 4388}, {313, 4109}, {561, 17788}, {1920, 17797}, {3261, 4142}, {3497, 31}, {6063, 17086}, {7224, 6}


    X(18837) = X(1)X(4602)∩ X(75)X(1925)

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

    X(18837) lies on the cubic K1028 and on these lines: {1, 4602}, {75, 1925}, {76, 3662}, {304, 1926}, {305, 3705}, {561, 1928}, {6376, 6386}

    X(18837) = {X(1928),X(1930)}-harmonic conjugate of X(561)
    X(18837) = isotomic conjugate of the isogonal conjugate of X(17149)
    X(18837) = X(75)-Ceva conjugate of X(561)
    X(18837) = crosspoint of X(75) and X(17149)
    X(18837) = X(i)-isoconjugate of X(j) for these (i,j): {25, 15389}, {32, 3224}, {560, 3223}, {1501, 2998}, {1974, 3504}, {3222, 9426}
    X(18837) = X(304)-Hirst inverse of X(1926)
    X(18837) = barycentric product X(i)*X(j) for these {i,j}: {75, 6374}, {76, 17149}, {194, 561}, {1502, 1740}, {1613, 1928}
    X(18837) = barycentric quotient X(i)/X(j) for these {i,j}: {63, 15389}, {75, 3224}, {76, 3223}, {194, 31}, {304, 3504}, {561, 2998}, {1424, 1397}, {1613, 560}, {1740, 32}, {3186, 1973}, {3221, 1924}, {4602, 3222}, {6374, 1}, {7075, 2175}, {17082, 604}, {17149, 6}


    X(18838) = INCIRCLE-INVERSE OF X(56)

    Barycentrics    a*((b+c)*a^3-(b^2+c^2)*a^2-(b^ 2-c^2)*(b-c)*a+(b^2-c^2)^2)*( a+b-c)*(a-b+c) : :
    X(18838) = X(36)+3*X(5902), X(65)+2*X(3660), 3*X(354)-X(5048), 3*X(3582)+X(11571), 4*X(3812)-X(17615), X(3814)-3*X(5883)

    See Mihajlon, Angel Montesdeoca, and César Lozada, Hyacinthos 27693 and Hyacinthos 27694.

    X(18838) lies on these lines: {1, 3}, {2, 18419}, {4, 5553}, {7, 5080}, {11, 1519}, {12, 3812}, {34, 1406}, {119, 912}, {221, 17054}, {226, 3814}, {244, 1457}, {388, 5176}, {499, 5887}, {513, 1835}, {515, 12736}, {518, 6735}, {519, 5083}, {535, 553}, {603, 3924}, {758, 3911}, {915, 1870}, {960, 5433}, {971, 13273}, {1042, 1393}, {1104, 1399}, {1118, 5146}, {1210, 1858}, {1317, 3880}, {1357, 3319}, {1359, 3025}, {1361, 14027}, {1404, 2173}, {1408, 18180}, {1411, 1455}, {1426, 14593}, {1428, 3827}, {1464, 1465}, {1739, 4551}, {1788, 3868}, {1790, 18178}, {1898, 9581}, {2170, 2272}, {2252, 8609}, {3290, 4559}, {3582, 11571}, {3649, 5087}, {3671, 11813}, {3742, 15950}, {3753, 5252}, {3754, 10106}, {3869, 7288}, {3873, 12648}, {3874, 4848}, {3919, 4315}, {3922, 9850}, {4295, 10531}, {5225, 9961}, {5439, 10200}, {5440, 12739}, {5777, 17606}, {5836, 10944}, {6173, 8581}, {6284, 9943}, {7335, 18732}, {7354, 7686}, {7672, 11239}, {10122, 17706}, {10572, 13369}, {10896, 12688}, {10940, 12649}, {10950, 12675}, {11219, 17638}, {11376, 12672}, {11502, 18446}, {11729, 14988}, {12115, 18391}, {12679, 17649}, {12740, 17654}, {12751, 17660}, {15904, 16869}

    X(18838) = midpoint of X(i) and X(j), for these {i, j}: {65,1319}, {1737,11570}
    X(18838) = reflection of X(i) in X(j), for these (i,j): (1319,3660), (5123,3812), (5570,942), (17615,5123), (18839,5570)
    X(18838) = incircle-inverse-of X(56)
    X(18838) = X(403)-of-intouch-triangle
    X(18838) = X(2071)-of-inverse-in-incircle-triangle
    X(18838) = X(10257)-of-Ursa-minor-triangle
    X(18838) = X(13528)-of-Mandart-incircle-triangle
    X(18838) = X(16386)-of-Hutson-intouch-triangle
    X(18838) = X(17615)-of-1st-Johnson-Yff-triangle
    X(18838) = inverse-in-{circumcircle, incircle}-inverter of X(1617)
    X(18838) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 46, 11248), (1, 57, 1470), (1, 2093, 12703), (1, 5193, 1319), (36, 5535, 1155), (36, 11529, 5048), (56, 5221, 1454), (57, 5902, 65), (65, 354, 2099), (65, 11011, 13601), (65, 13751, 11011), (1155, 1319, 5172), (2446, 2447, 56), (5045, 13601, 11011), (5172, 11509, 2077), (11011, 13751, 5045)


    X(18839) = INCIRCLE-INVERSE OF X(55)

    Barycentrics    a* (-a+b+c)*((b+c)*a^3-(b^2+c^2)* a^2-(b^2-c^2)*(b-c)*a+(b-c)^4) : :
    X(18839) = 3*X(354)-X(1155), 3*X(354)-2*X(3660), 9*X(354)-4*X(11575), 2*X(942)-3*X(5570), 3*X(3873)+X(5057), 2*X(4662)-3*X(5123)

    See César Lozada, Hyacinthos 27694.

    X(18839) lies on these lines: {1, 3}, {11, 518}, {12, 13374}, {72, 11376}, {105, 2990}, {210, 5231}, {497, 3873}, {513, 11934}, {516, 5083}, {535, 950}, {672, 8609}, {840, 3100}, {971, 13274}, {1279, 2361}, {1362, 3326}, {1364, 3322}, {1512, 10956}, {1797, 14190}, {1830, 1876}, {1836, 10947}, {1837, 3555}, {1858, 3874}, {1864, 11238}, {1898, 9614}, {2323, 2348}, {2342, 15635}, {3021, 3025}, {3058, 10391}, {3193, 5324}, {3254, 15733}, {3328, 5580}, {3486, 3889}, {3583, 12750}, {3681, 10589}, {3742, 5432}, {3814, 10916}, {3827, 10535}, {3848, 5326}, {3868, 11240}, {3870, 11502}, {3876, 10527}, {3911, 18240}, {4018, 17622}, {4430, 5274}, {4511, 13279}, {4662, 5123}, {5176, 12649}, {6284, 12675}, {7004, 17449}, {7686, 10944}, {7965, 10949}, {8540, 9004}, {9657, 9850}, {10529, 18220}, {10624, 12005}, {10896, 14872}, {11813, 14054}, {11997, 13476}, {12680, 12953}, {17626, 17728}

    X(18839) = reflection of X(3911) in X(18240)
    X(18839) = incircle-inverse-of X(55)
    X(18839) = X(23)-of-inverse-in-incircle-triangle
    X(18839) = X(468)-of-Ursa-minor-triangle
    X(18839) = X(858)-of-intouch-triangle
    X(18839) = X(1155)-of-Mandart-incircle-triangle
    X(18839) = X(7574)-of-incircle-circles triangle
    X(18839) = X(10295)-of-Hutson-intouch-triangle
    X(18839) = X(17615)-of-2nd-Johnson-Yff triangle
    X(18839) = reflection of X(18838) in X(5570)
    X(18839) = inverse-in-{circumcircle, incircle}-inverter of X(3)
    X(18839) = anticomplement, wrt intouch triangle, of X(3660)
    X(18839) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5536, 2078), (1, 5709, 11510), (57, 5537, 1155), (65, 354, 4860), (354, 3057, 17603), (354, 3748, 11018), (354, 17642, 55), (2078, 5536, 1155), (2098, 4860, 55), (2446, 2447, 55), (3057, 17603, 55), (5173, 12915, 354), (5597, 5598, 8069), (6583, 9957, 13750), (17603, 17642, 3057)






    leftri  Dao-bipedal-perspectors: X(18840) - X(18855)  rightri

    This preamble and centers X(18840)-X(18855) were contributed by César Eliud Lozada, May 26, 2018.

    Let ABC be a triangle, P, Q two isogonal conjugate points and PaPbPc, QaQbQc their respective pedal triangles. Let t be a real number, P’a the point on PPa such that PP’a/PPa=t and Q'a the point on QQa such that QQ'a/QQa=t; define P’b, P’c, Q’b, Q’c cyclically. Denote A’=PQ’a ∩ QP’a and similarly B' and C'. Then A’B’C’ and ABC are perspective and, for given P, Q, the locus of the perspectors is a rectangular hyperbola. (Dao Thanh Oai, May 22, 2018)

    Suppose P=U:V:W (trilinears). Vertex A' has trilinear coordinates:

      A' = -(t-2)*u*v*w : (u^2+v^2+t*cos(C)*u*v)*w : (u^2+w^2+t*cos(B)*u*w)*v

    The perspector A’B’C’, ABC is:

      Z(P, t) = V*W*(U^2+t*cos(C)*U*V+V^2) *(U^2+t*cos(B)*U*W+W^2) : : (trilinears)

    When P is fixed and t varies, the perspector Z(P, t) lies on the rectangular hyperbola Κ(P) with trilinear equation:

      Κ(P) = ∑ ( ((U^2+V^2)*W*cos(B)-(U^2+W^2)*V*cos(C))*v*w )= 0

    Κ(P) has center:

      O*(P) = b*c*((U^2+W^2)*V*a*b-(U^2+V^2)*W*a*c+(b^2-c^2)*U*(V^2+W^2))*((U^2+V^2)*W*cos(B)-(U^2+W^2)*V*cos(C)) : :

    and perspector:

      P*(P) = (U^2+V^2)*W*cos(B)-(U^2+W^2)*V*cos(C) : :

    Κ(P) is the isogonal conjugate of the line:

       ∑(((U^2+V^2)*W*cos(B)-(U^2+W^2)*V*cos(C)) u ) = 0

    The perspector Z(P, t) is here named here the Dao-bipedal-perspector-of-(P, t) and the conic Κ(P) is here named here the Dao-bipedal-conic-of-P.

    The appearance of (i, t, j) in the following list means that the Dao-bipedal-perspector-of-(X(i), t)=X(j):
    (2, -3, 2996), (2, -5/2, 5485), (2, -2, 76), (2, -3/2, 18840), (2, -1, 2), (2, -1/2, 18841), (2, 0, 83), (2, 1/2, 18842), (2, 1, 5395), (2, 3/2, 18843), (2, 2, 598), (2, 5/2, 18844), (2, 3, 18845), (3, -3, 18846), (3, -5/2, 18847), (3, -2, 18848), (3, -3/2, 18849), (3, -1, 18850), (3, -1/2, 18851), (3, 0, 1105), (3, 1/2, 18852), (3, 1, 1217), (3, 3/2, 18853), (3, 2, 264), (3, 5/2, 18854), (3, 3, 18855)

    The appearance of (i, j) in the following list means that the O*(X(i)) = X(j): (1, 11), (2, 115), (3, 136)

    underbar


    X(18840) = DAO-BIPEDAL-PERSPECTOR OF (X(2), -3/2)

    Trilinears    1/(2 sin A - cos A tan ω) : :
    Trilinears    1/(cos A - 2 sin A cot ω) : :
    Trilinears    1/(a - R cos A tan ω) : :
    Barycentrics    1/(cot B + cot C + cot ω) : :
    Barycentrics    1/(3a^2 + b^2 + c^2) : :
    Barycentrics    (a^2+3*b^2+c^2)*(a^2+b^2+3*c^2) : :

    X(18840) lies on the Kiepert hyperbola and these lines: {2,3933}, {3,3424}, {4,141}, {5,14484}, {6,18841}, {10,4000}, {69,83}, {76,3619}, {98,620}, {183,14069}, {226,7195}, {262,3090}, {297,8796}, {315,598}, {321,3673}, {376,7800}, {485,5591}, {486,5590}, {599,18842}, {626,3545}, {671,3096}, {1131,7389}, {1132,7388}, {1235,2052}, {1285,7767}, {1916,14064}, {2051,7402}, {2896,14033}, {2996,6656}, {3406,14912}, {3407,7793}, {3524,7789}, {3525,7612}, {3529,3734}, {3533,3788}, {3618,7894}, {3620,5395}, {3763,5286}, {3785,14039}, {3855,7849}, {4869,13740}, {5067,7778}, {5082,13576}, {5232,17681}, {5254,5485}, {5503,7867}, {6292,7738}, {6625,17232}, {7397,13478}, {7608,7869}, {7735,7822}, {7736,7794}, {7745,18843}, {7783,16043}, {7791,11606}, {7801,11167}, {7803,10159}, {7819,15589}, {7865,15682}, {7878,11008}, {7879,18845}, {7880,15709}, {7895,9770}, {7938,16041}, {8889,8891}, {14534,18141}

    X(18840) = isotomic conjugate of X(3618)
    X(18840) = polar conjugate of X(6995)
    X(18840) = trilinear pole of the line {523, 2525}


    X(18841) = DAO-BIPEDAL-PERSPECTOR OF (X(2), -1/2)

    Trilinears    1/(2 sin A + cos A tan ω) : :
    Trilinears    1/(cos A + 2 sin A cot ω) : :
    Trilinears    1/(a + R cos A tan ω) : :
    Barycentrics    1/(a^2 + 3b^2 + 3c^2) : :
    Barycentrics    (3*a^2+b^2+3*c^2)*(3*a^2+3*b^2+c^2) : :

    X(18841) lies on the Kiepert hyperbola and these lines: {2,7762}, {3,14484}, {4,3589}, {5,3424}, {6,18840}, {10,7290}, {69,10159}, {76,3618}, {98,3090}, {262,631}, {321,5222}, {376,14492}, {458,8796}, {459,8743}, {485,7376}, {486,7375}, {598,7859}, {671,7803}, {1131,7388}, {1132,7389}, {1285,8362}, {1916,14001}, {2051,7397}, {2996,7770}, {3091,14535}, {3407,14064}, {3525,14494}, {3529,7804}, {3533,6680}, {3545,7834}, {3619,7878}, {3855,12252}, {5067,7612}, {5286,5485}, {5395,6656}, {5503,9167}, {7402,13478}, {7736,7888}, {7745,18842}, {7746,11167}, {7846,8781}, {7915,9770}, {10155,15491}, {10357,15709}, {10358,14485}, {11174,14069}, {11606,16924}

    X(18841) = isogonal conjugate of X(9605)
    X(18841) = isotomic conjugate of X(3619)
    X(18841) = polar conjugate of X(7378)
    X(18841) = trilinear pole of the line {523, 3804}


    X(18842) = DAO-BIPEDAL-PERSPECTOR OF (X(2), 1/2)

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

    X(18842) lies on the Kiepert hyperbola and these lines: {2,1285}, {4,597}, {6,5485}, {10,15601}, {30,14484}, {69,10302}, {76,1992}, {98,3545}, {182,14485}, {262,376}, {381,3424}, {543,14482}, {598,3618}, {599,18840}, {631,7608}, {671,18800}, {1916,8591}, {2394,18311}, {2444,5466}, {2482,5503}, {2996,8370}, {3317,13783}, {3407,16041}, {3524,14494}, {3590,7388}, {3591,7389}, {3855,7817}, {4080,17014}, {5071,7612}, {5395,7841}, {5490,13757}, {5491,13637}, {7375,10194}, {7376,10195}, {7735,11167}, {7745,18841}, {7770,11160}, {7804,9770}, {7812,10159}, {8787,14912}, {10155,15702}, {11163,14039}, {11669,15709}, {14492,15682}

    X(18842) = reflection of X(2) in X(14535)
    X(18842) = isogonal conjugate of X(5024)
    X(18842) = trilinear pole of the line {523, 8644} (the orthic axis of the Thomson triangle)


    X(18843) = DAO-BIPEDAL-PERSPECTOR OF (X(2), 3/2)

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

    X(18843) lies on the Kiepert hyperbola and these lines: {4,6329}, {76,11008}, {98,3855}, {262,3529}, {382,14484}, {546,3424}, {631,11669}, {3528,14494}, {3544,7612}, {7608,10299}, {7745,18840}


    X(18844) = DAO-BIPEDAL-PERSPECTOR OF (X(2), 5/2)

    Barycentrics    (9*a^2-5*b^2+9*c^2)*(9*a^2+9*b^2-5*c^2) : :

    X(18844) lies on the Kiepert hyperbola and these lines: {3090,11668}, {3424,3843}, {3627,14484}, {5485,7745}, {14494,17538}


    X(18845) = DAO-BIPEDAL-PERSPECTOR OF (X(2), 3)

    Barycentrics    (5*a^2-3*b^2+5*c^2)*(5*a^2+5*b^2-3*c^2) : :
    Trilinears    1/(3 sin(A - ω) - 5 sin(A + ω)) : :
    Trilinears    1/(4 cos A + sin A cot ω) : :
    Trilinears    1/(sin A + 4 cos A tan ω) : :
    Trilinears    1/(a + 8R cos A tan ω) : :

    X(18845) lies on the Kiepert hyperbola and these lines: {2,5023}, {3,10155}, {20,14494}, {98,3832}, {262,3146}, {315,10302}, {439,5475}, {1916,14068}, {2996,7745}, {3091,7612}, {3522,7608}, {5068,7607}, {5485,7754}, {7879,18840}, {11669,15717}, {14484,17578}

    X(18845) = isogonal conjugate of X(15815)


    Barycentrics   

    X(18846) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -3)

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

    X(18846) lies on these lines: {30,6526}, {382,393}, {1093,3146}, {1885,18855}, {3543,8884}, {8801,13488}, {16263,17578}

    X(18846) = trilinear pole of the line {2501, 14345}


    X(18847) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -5/2)

    Barycentrics   -(a^2-b^2+c^2)*(9*a^6-(13*b^2+9*c^2)*a^4-(b^4-26*b^2*c^2+9*c^4)*a^2+(5*b^2+9*c^2)*(b^2-c^2)^2)*(a^2+b^2-c^2)*(9*a^6-(9*b^2+13*c^2)*a^4-(9*b^4-26*b^2*c^2+c^4)*a^2+(9*b^2+5*c^2)*(b^2-c^2)^2) : :

    X(18847) lies on these lines: {393,15682}, {1885,18854}, {3529,6526}, {9007,18808}


    X(18848) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -2)

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

    X(18848) lies on th3e hyperbola {{A,B,C,X(4),X(93)}} these lines: {4,18418}, {20,6526}, {30,1093}, {264,1885}, {382,8884}, {393,3146}, {520,12111}, {648,5895}, {847,18560}, {1593,14860}, {3627,16263}, {10152,11441}

    X(18848) = isogonal conjugate of X(1204)
    X(18848) = cevapoint of X(4) and X(20)
    X(18848) = trilinear pole of the line {1636, 2501}


    X(18849) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -3/2)

    Barycentrics    (a^2-b^2+c^2)*(7*a^6-(11*b^2+7*c^2)*a^4+(b^4+22*b^2*c^2-7*c^4)*a^2+(3*b^2+7*c^2)*(b^2-c^2)^2)*(a^2+b^2-c^2)*(7*a^6-(7*b^2+11*c^2)*a^4-(7*b^4-22*b^2*c^2-c^4)*a^2+(7*b^2+3*c^2)*(b^2-c^2)^2) : :

    X(18849) lies on these lines: {376,6526}, {1093,3529}, {1885,18853}, {8884,15682}


    X(18850) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -1)

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

    X(18850) lies on these lines: {3,6526}, {4,11064}, {20,1093}, {30,393}, {254,18560}, {525,16253}, {1217,1885}, {1593,18855}, {1597,8801}, {1826,15942}, {3088,14860}, {3146,8884}, {3543,16263}

    X(18850) = isogonal conjugate of X(10605)
    X(18850) = trilinear pole of the line {2501, 9033}


    X(18851) = DAO-BIPEDAL-PERSPECTOR OF (X(3), -1/2)

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

    X(18851) lies on these lines: {376,1093}, {393,3529}, {631,6526}, {1593,18854}, {1885,18852}


    X(18852) = DAO-BIPEDAL-PERSPECTOR OF (X(3), 1/2)

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

    X(18852) lies on these lines: {4,6090}, {376,393}, {631,1093}, {1593,18853}, {1885,18851}, {3088,18854}, {3090,6526}, {3147,15424}, {3529,8884}, {15682,16263}

    X(18852) = trilinear pole of the line {2501, 9007}


    X(18853) = DAO-BIPEDAL-PERSPECTOR OF (X(3), 3/2)

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

    X(18853) lies on these lines: {376,8884}, {393,631}, {427,18854}, {847,8889}, {1093,3090}, {1593,18852}, {1885,18849}, {3545,6526}


    X(18854) = DAO-BIPEDAL-PERSPECTOR OF (X(3), 5/2)

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

    X(18854) lies on these lines: {254,8889}, {393,3090}, {427,18853}, {631,8884}, {1093,3545}, {1179,6353}, {1593,18851}, {1885,18847}, {3088,18852}, {3529,16263}, {3855,6526}


    X(18855) = DAO-BIPEDAL-PERSPECTOR OF (X(3), 3)

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

    X(18855) lies on these lines: {2,8884}, {4,343}, {5,393}, {20,16263}, {68,3087}, {254,1594}, {324,1093}, {381,6526}, {427,1217}, {1105,3088}, {1179,3542}, {1300,3541}, {1593,18850}, {1885,18846}, {6525,7528}

    X(18855) = polar conjugate of X(11427)
    X(18855) = trilinear pole of the line {2501, 6368}


    X(18856) = X(1)X(3)∩X(912)X(3035)

    Trilinears    8*q*p^5-4*(4*q^2-3)*p^4+4*(2* q^2-3)*q*p^3-2*p^2+4*q*p-1 : : , where p=sin(A/2), q=cos(B/2 - C/2)
    Barycentrics    a*((b+c)*a^8-2*(b^2-b*c+c^2)* a^7-2*(b^3+c^3)*a^6+2*(3*b^4+ 3*c^4-2*b*c*(3*b^2-2*b*c+3*c^ 2))*a^5+8*b^2*c^2*(b+c)*a^4-2* (3*b^6+3*c^6-(9*b^4+9*c^4-b*c* (7*b^2+2*b*c+7*c^2))*b*c)*a^3+ 2*(b^2-c^2)^2*(b+c)*(b^2-3*b* c+c^2)*a^2+2*(b^2-c^2)^2*(b-c) ^4*a-(b^2-c^2)^3*(b-c)^3) : :
    X(18856) = X(5570) - 3*X(10202)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27696.

    X(18856) lies on these lines: {1, 3}, {912, 3035}, {6001, 6713}, {13369, 18242}


    X(18857) = CIRCUMCIRCLE-INVERSE OF X(10680)

    Trilinears    4*p^3*(4*p-5*q)+2*(2*q^2-1)*p^ 2+4*q*p-1 : : , where p=sin(A/2), q=cos(B/2 - C/2)
    Barycentrics    a*(4*a^6-5*(b+c)*a^5-(7*b^2- 16*b*c+7*c^2)*a^4+2*(b+c)*(5* b^2-8*b*c+5*c^2)*a^3+2*(b^4+c^ 4-b*c*(7*b^2-10*b*c+7*c^2))*a^ 2-(b^2-c^2)*(b-c)*(5*b^2-6*b* c+5*c^2)*a+(b^2-c^2)^2*(b-c)^ 2) : :
    X(18857) = 3*X(3)-X(13528), X(36)+3*X(3576), 5*X(36)-X(5535), X(104)+3*X(4881), 5*X(631)-X(5176), 3*X(1319)+X(13528), 3*X(1385)+X(10225), X(2077)-5*X(7987), 15*X(3576)+X(5535), 3*X(3582)+X(12119), X(3814)-3*X(10165), X(5048)-3*X(10246), X(5126)+2*X(13624), 3*X(5193)+X(6282)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27696.

    X(18857) lies on these lines: {1, 3}, {104, 4881}, {140, 5123}, {214, 912}, {355, 6921}, {513, 17099}, {515, 6681}, {631, 5176}, {3582, 12119}, {3814, 10165}, {5080, 6947}, {5731, 6880}, {5886, 6938}, {6834, 18481}, {6929, 11230}, {6959, 18480}, {9956, 13747}

    X(18857) = midpoint of X(3) and X(1319)
    X(18857) = circumcircle-inverse-of X(10680)
    X(18857) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 10246, 5119), (3, 16203, 10966), (1381, 1382, 10680), (3576, 10269, 1385)


    X(18858) = X(98)X(4467)∩X(805)X(2966)

    Barycentrics    1/((b^2 - c^2) (-a^4 + b^2 c^2) (b^4 + c^4 - a^2 (b^2 + c^2))^2 : :

    See Angel Montesdeoca, HG260518.

    X(18858) lies on these lines: {98,446}, {805,2966}, {1297,1916}, {2698,12176}, {6037,17938}




    leftri  Circumperp conjugates: X(18859) - X(18864)  rightri

    This preamble and centers X(18859)-X(18864) were contributed by César Eliud Lozada, May 28, 2018.

    Let ABC be a triangle and P a point. The perpendicular bisectors of BC, CA, AB intersect the circumcircle at (A1, A2), (B1, B2), (C1, C2) ,respectively. Then the circumcircles of PA1A2, PB1B2, PC1C2 are coaxial. (Antreas Hatzipolakis, May 26, 2018, Anopolis 7568)

    Let Q be the point of intersection (other than P) of the three indicated circles. Then:

    For P other than X(3), the mapping P -> Q is a conjugacy. The point Q is here named the Q=circumperp conjugate of P (as A1, A2, B1, B2, C1, C2 are the vertices of the circumperp triangles). For P= u:v:w (trilinears), coordinates of Q(P) are:

      Q(P) = (-a*(-b*c*SC*SB*u^2+2*w*v*SA^3)-SA*a*b*c*(b^2*v^2+c^2*w^2)+b*((SA+SB)*S^2-2*SA*SB^2)*w*u+c*((SA+SC)*S^2-2*SA*SC^2)*u*v)*a : :

    Some properties:

    The appearance of (i,j) in the following list means that the circumperp conjugate of X(i) is X(j), for X(i) not on the circumcircle of ABC:
    (1,2077), (2,7464), (4,2071), (5,18859), (6,18860), (8,18861), (11,18862), (13,18863), (14,18864), (15,14539), (16,14538), (20,186), (22,10295), (23,376), (24,16386), (36,40), (40,36), (56,13528), (64,12096), (122,2935), (131,12302), (186,20), (187,1350), (265,13496), (376,23), (378,858), (399,12041), (403,11413), (484,11012), (548,5899), (550,2070), (858,378), (944,17100), (1155,3428), (1157,7691), (1319,10310), (1324,4297), (1325,3651), (1326,3430), (1350,187), (1498,11589), (1511,10620), (1657,15646), (2070,550), (2071,4), (2072,12084), (2076,5188), (2077,1), (2078,6282), (2080,3098), (2456,9737), (2459,11825), (2460,11824), (2482,16010), (2935,122), (3098,2080), (3153,3520), (3184,10117), (3357,6760), (3428,1155), (3430,1326), (3520,3153), (3534,7575), (3576,5537), (3651,1325), (4297,1324), (4996,12247), (5104,8722), (5126,6244), (5144,11495), (5152,11257), (5172,14110), (5188,2076), (5196,7430), (5473,6104), (5474,6105), (5536,7688), (5537,3576), (5538,10902), (5609,15041), (5621,14981), (5866,6776), (5899,548), (5961,12121), (6150,12307), (6244,5126), (6282,2078), (6566,12305), (6567,12306), (6760,3357), (6770,14368), (6773,14369), (6776,5866), (7418,7472), (7421,7424), (7422,7468), (7424,7421), (7425,7475), (7429,7477), (7430,5196), (7440,7479), (7464,2), (7468,7422), (7472,7418), (7475,7425), (7477,7429), (7479,7440), (7488,13619), (7574,18570), (7575,3534), (7688,5536), (7689,13557), (7691,1157), (8722,5104), (9301,14810), (9737,2456), (10117,3184), (10257,12085), (10260,12119), (10295,22), (10310,1319), (10564,14687), (10620,1511), (10745,13293), (10902,5538), (11012,484), (11250,18403), (11257,5152), (11413,403), (11495,5144), (11589,1498), (11643,12117), (11676,15915), (11713,12332), (11824,2460), (11825,2459), (12041,399), (12042,13188), (12084,2072), (12085,10257), (12095,12163), (12096,64), (12105,15689), (12117,11643), (12119,10260), (12121,5961), (12163,12095), (12247,4996), (12302,131), (12305,6566), (12306,6567), (12307,6150), (12332,11713), (12383,14652), (12584,14830), (13188,12042), (13293,10745), (13496,265), (13528,56), (13557,7689), (13558,16163), (13619,7488), (14094,15055), (14110,5172), (14538,16), (14539,15), (14652,12383), (14687,10564), (14703,16111), (14810,9301), (14830,12584), (14981,5621), (15035,15054), (15041,5609), (15054,15035), (15055,14094), (15646,1657), (15681,18571), (15689,12105), (15915,11676), (16010,2482), (16111,14703), (16163,13558), (16386,24), (17100,944), (18403,11250), (18570,7574), (18571,15681), (18859,5), (18860,6), (18861,8), (18862,11), (18863,13), (18864,14)

    underbar

    X(18859) = CIRCUMPERP CONJUGATE OF X(5)

    Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+9*b^2 *c^2*a^4+(b^2+c^2)*(2*b^4-7*b^ 2*c^2+2*c^4)*a^2-(b^4+4*b^2*c^ 2+c^4)*(b^2-c^2)^2) : :
    X(18859) = 9*X(2)-8*X(15350), 5*X(3)-2*X(23), 3*X(3)-2*X(186), 3*X(3)-X(5899), X(3)+2*X(7464), 7*X(3)-4*X(7575), 17*X(3)-8*X(12105), 5*X(3)-4*X(15646), 11*X(3)-8*X(18571), 3*X(5)-2*X(11558), X(399)-4*X(10564), 4*X(14156)-3*X(14643)

    As a point on the Euler line, X(18859) has Shinagawa coefficients (3*E-8*F, -7*E+8*F).

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18859) is the crosspoint of the intersections of the circumcircle and Trinh circle. (Randy Hutson, June 27, 2018)

    X(18859) lies on these lines: {2, 3}, {36, 9629}, {39, 18373}, {49, 10575}, {56, 9641}, {74, 1154}, {156, 12279}, {185, 195}, {399, 2935}, {477, 930}, {511, 5621}, {539, 13399}, {933, 2693}, {999, 10149}, {1092, 18439}, {1204, 6243}, {1350, 11649}, {1511, 14157}, {1568, 7728}, {2696, 9076}, {3098, 9973}, {3357, 18436}, {5663, 13445}, {5944, 8718}, {6101, 11440}, {6128, 11063}, {9703, 11456}, {9729, 15047}, {10540, 14915}, {10620, 12302}, {10625, 12307}, {11381, 18350}, {11430, 14855}, {11591, 15062}, {12006, 12834}, {12041, 13358}, {12099, 15107}, {12121, 13293}, {12242, 14861}, {13352, 15087}, {13367, 14641}, {13630, 14627}, {13863, 18401}, {14156, 14643}, {14581, 14961}, {15033, 15037}, {15089, 17855}, {18863, 18864}

    X(18859) = reflection of X(382) in X(18403)
    X(18859) = anticomplement of X(11563)
    X(18859) = circumcircle-inverse of X(550)
    X(18859) = Stammler circle-inverse of X(1657)
    X(18859) = Stammler circles radical circle-inverse of X(3)
    X(18859) = X(2070)-of ABC-X3-reflections-triangle
    X(18859) = X(2071)-of-X3-ABC-reflections-triangle
    X(18859) = anti-Hutson intouch-isogonal conjugate of X(10620)
    X(18859) = pole wrt circumcircle of line X(523)X(550)
    X(18859) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1593, 1656), (3, 3830, 6644), (3, 3843, 17928), (3, 5073, 24), (3, 5899, 186), (3, 9818, 5054), (20, 18569, 18565), (186, 5899, 2070), (1113, 1114, 550), (3830, 6644, 7545), (7488, 10226, 3), (7502, 18569, 25), (12087, 17506, 12107), (15122, 18325, 1656), (15154, 15155, 1657), (18565, 18569, 382)


    X(18860) = CIRCUMPERP CONJUGATE OF X(6)

    Trilinears    sin(A + ω) e^2 sec^2 ω (sin^2 A + sin^2 B + sin^2 C) - (sin A - 3 cos A tan ω)[sin A sin(A - ω) + sin B sin(B - ω) + sin C sin(C - ω)] : :
    Barycentrics    a^2*(2*a^6-3*(b^2+c^2)*a^4+4*( b^4+b^2*c^2+c^4)*a^2-(b^2+c^2) *(3*b^4-2*b^2*c^2+3*c^4)) : :
    X(18860) = 3*X(3)-X(2080), 5*X(3)-X(9301), X(147)-3*X(7799), 3*X(165)-X(5184), 4*X(182)-3*X(1692), 3*X(187)-2*X(2080), 5*X(187)-2*X(9301), 4*X(549)-3*X(5215), 3*X(549)-2*X(14693), X(691)-3*X(2071), 2*X(1351)-3*X(1570), X(1351)-3*X(2456), 5*X(3522)-X(14712), 5*X(5071)-6*X(10150), 9*X(5215)-8*X(14693)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18860) lies on these lines: {3, 6}, {4, 625}, {5, 7820}, {20, 316}, {30, 114}, {55, 5194}, {56, 5148}, {74, 2709}, {98, 538}, {99, 5999}, {103, 2705}, {110, 2710}, {122, 441}, {140, 7852}, {147, 7799}, {165, 5184}, {262, 7804}, {325, 2794}, {376, 3849}, {512, 684}, {549, 5215}, {620, 1513}, {631, 7834}, {691, 1297}, {842, 1296}, {1007, 7694}, {1092, 2909}, {1293, 2700}, {1352, 7801}, {1495, 9155}, {1503, 6390}, {1593, 5140}, {2393, 9145}, {2936, 6000}, {2967, 14581}, {3148, 5651}, {3292, 5191}, {3455, 13754}, {3522, 14712}, {3523, 7803}, {3564, 7813}, {3734, 13860}, {3926, 5921}, {5031, 7789}, {5071, 10150}, {5149, 11676}, {5207, 6337}, {5480, 8369}, {6194, 7771}, {7575, 11642}, {7780, 12251}, {7796, 9863}, {7798, 9755}, {7835, 13862}, {8149, 11257}, {8719, 14532}, {8724, 11645}, {9027, 9142}, {9890, 13172}, {10011, 10256}, {10352, 13586}, {10723, 14041}, {11799, 16760}, {13241, 14388}, {15122, 16188}

    X(18860) = midpoint of X(i) and X(j) for these {i,j}: {20, 316}, {99, 5999}, {842,7464}
    X(18860) = reflection of X(i) in X(j) for these (i,j): (4, 625), (187,3), (11799, 16760)
    X(18860) = circumcircle-inverse of X(1350)
    X(18860) = Moses-circle-inverse of X(5028)
    X(18860) = X(187)-of ABC-X3-reflections-triangle
    X(18860) = X(625)-of-anti-Euler-triangle
    X(18860) = X(5148)-of-2nd-circumperp-tangential-triangle
    X(18860) = X(5194)-of-anti-Mandart-incircle-triangle
    X(18860) = (ABC-X3 reflections)-isogonal conjugate of X(12117)
    X(18860) = X(187)-of-circumcevian-triangle-of-X(511)
    X(18860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1350, 8722), (3, 3095, 13335), (3, 5024, 5085), (3, 5171, 15513), (3, 9734, 8589), (3, 9737, 39), (3, 11171, 5092), (15, 16, 2030), (1350, 8722, 5188), (1379, 1380, 1350), (3095, 13335, 5007), (3098, 9734, 3), (5097, 13335, 11842), (6566, 6567, 1692), (10631, 15513, 187), (12305, 12306, 11477)


    X(18861) = CIRCUMPERP CONJUGATE OF X(8)

    Trilinears    (12*sin(A/2)-8*sin(3*A/2)+2*si n(5*A/2))*cos((B-C)/2)+(2*cos( A)-cos(2*A)-3/2)*cos(B-C)-4* cos(2*A)+1/2*cos(3*A)+9*cos(A) -5 : :
    Barycentrics    a^2*(a^8-2*(b+c)*a^7-2*(b^2-4* b*c+c^2)*a^6+2*(b+c)*(3*b^2-5* b*c+3*c^2)*a^5-(14*b^2-19*b*c+ 14*c^2)*b*c*a^4-2*(b^3+c^3)*(3 *b^2-7*b*c+3*c^2)*a^3+2*(b^4+c ^4+(4*b^2-b*c+4*c^2)*b*c)*(b-c )^2*a^2+2*(b^2-c^2)*(b-c)*(b^4 +c^4-3*(b^2-b*c+c^2)*b*c)*a-(b ^2-c^2)^2*(b^2-b*c+c^2)^2) : : : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18861) lies on these lines: {3, 8}, {4, 10090}, {11, 6906}, {21, 6713}, {35, 11715}, {36, 2800}, {56, 10698}, {80, 5450}, {119, 404}, {149, 10785}, {153, 4188}, {497, 5533}, {953, 2841}, {1320, 11248}, {1385, 17654}, {1470, 5603}, {1768, 6261}, {2077, 2802}, {2829, 6905}, {3035, 6940}, {3149, 10728}, {3885, 12737}, {5253, 11729}, {5840, 6909}, {5884, 14800}, {6667, 6920}, {6830, 13273}, {6924, 10742}, {6941, 12761}, {6942, 12248}, {6952, 8068}, {6977, 10629}, {7967, 10087}, {10265, 10572}, {10711, 16371}, {12116, 13199}, {14803, 15528}, {15015, 18446}

    X(18861) = circumcircle-inverse of X(944)
    X(18861) = X(17100)-of-ABC-X3-reflections-triangle
    X(18861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 12332, 10698), (100, 104, 944)


    X(18862) = CIRCUMPERP CONJUGATE OF X(11)

    Trilinears    8*p^7*(p-q)+2*(4*q^2-9)*p^6-4* (2*q^2-5)*q*p^5-5*(2*q^2-3)*p^ 4+8*(q^2-2)*q*p^3+4*(q^2-1)*p^ 2-(3*q^2-5)*q*p-1/2 : : , where p=sin(A/2), q=cos(B/2 - C/2) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18862) lies on these lines: {3, 11}, {40, 12302}, {378, 915}, {1295, 11413}, {1593, 5521}, {3428, 11714}

    X(18862) = X(14667)-of ABC-X3-reflections-triangle


    X(18863) = CIRCUMPERP CONJUGATE OF X(13)

    Barycentrics    (SB+SC)*((3*R^2+2*SA)*S^2+sqrt (3)*(3*S^2+3*SA^2-2*SB*SC-15* R^2*SA)*S-6*R^2*SA*SW) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18863) lies on these lines: {3, 13}, {7691, 14541}, {12041, 14539}, {18859, 18864}

    X(18863) = circumcircle-inverse of X(5473)
    X(18863) = X(6104)-of-ABC-X3-reflections-triangle


    X(18864) = CIRCUMPERP CONJUGATE OF X(14)

    Barycentrics    (SB+SC)*((3*R^2+2*SA)*S^2-sqrt (3)*(3*S^2+3*SA^2-2*SB*SC-15* R^2*SA)*S-6*R^2*SA*SW) : :

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27704.

    X(18864) lies on these lines: {3, 14}, {7691, 14540}, {12041, 14538}, {18859, 18863}

    X(18864) = circumcircle-inverse of X(5474)
    X(18864) = X(6105)-of-ABC-X3-reflections-triangle


    X(18865) = X(4)X(5535)∩X(5)X(580)

    Barycentrics    a^10-(b+c)*a^9-2*(b^2+b*c+c^2)*a^8+(b+c)*(3*b^2-2*b*c+3*c^2)*a^7+(b^4+c^4+(4*b^2+7*b*c+4*c^2)*b*c)*a^6-3*(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)*a^5-(b^3+c^3)*(b+c)*(b^2+3*b*c+c^2)*a^4+(b+c)*(b^6+c^6-2*(b^4+b^2*c^2+c^4)*b*c)*a^3+2*(b^2-c^2)^2*(b+c)*(b^3+c^3)*a^2+(b^4-c^4)*(b^2-c^2)*(b+c)*b*c*a-(b^4-c^4)*(b^2-c^2)^3 : :
    X(18865) = 2*X(502)-3*X(5587)

    See Antreas Hatzipolakis and César Lozada, Hyacinthos 27705.

    X(18865) lies on the Fuhrmann circle, the cubic K800, and these lines:
    {4, 5535}, {5, 580}, {355, 13514}, {502, 5587}

    X(18865) = Fuhrmann-circle-antipode of X(13514)
    X(18865) = X(6798)-of-excentral-triangle


    X(18866) = (name pending)

    Barycentrics    a^16-2 a^14 (b^2+c^2)-3 a^12 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^4 (b^2+c^2)^2 (2 b^4-3 b^2 c^2+2 c^4)+2 a^10 (4 b^6-5 b^4 c^2-5 b^2 c^4+4 c^6)-2 a^6 (b^2-c^2)^2 (5 b^6-2 b^4 c^2-2 b^2 c^4+5 c^6)+a^8 (b^8-15 b^6 c^2+29 b^4 c^4-15 b^2 c^6+c^8)+a^4 (b^2-c^2)^2 (3 b^8+2 b^6 c^2-15 b^4 c^4+2 b^2 c^6+3 c^8)+4 a^2 (b^2-c^2)^2 (b^10-b^8 c^2+2 b^6 c^4+2 b^4 c^6-b^2 c^8+c^10) : :

    See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4608.

    X(18866) lies on this line: {2,3}


    X(18867) = REFLECTION OF X(2) IN X(13448)

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

    See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4608.

    X(18867) lies on these lines: {2,3}, {94,9140}, {2088,14846}

    X(18867) = reflection of X(2) in X(13448)


    X(18868) = (name pending)

    Barycentrics    (b^2-c^2)^2 (-a^2+b^2+c^2) (a^8+a^4 b^2 c^2-2 a^6 (b^2+c^2)-(b^2-c^2)^2 (b^4+c^4)+a^2 (b^2+c^2) (2 b^4-3 b^2 c^2+2 c^4)) (a^10-4 a^8 (b^2+c^2)+2 (b^2-c^2)^4 (b^2+c^2)+a^6 (4 b^4+b^2 c^2+4 c^4)-a^2 (b^2-c^2)^2 (5 b^4+4 b^2 c^2+5 c^4)+a^4 (2 b^6+3 b^4 c^2+3 b^2 c^4+2 c^6))+(a^2-b^2) (a^2-c^2) (a^4-(b^2-c^2)^2) (a^16-4 a^14 (b^2+c^2)-(b^2-c^2)^6 (b^2+c^2)^2-4 a^10 (b^2+c^2) (b^4+b^2 c^2+c^4)+2 a^6 (b^2-c^2)^2 (b^2+c^2) (2 b^4+b^2 c^2+2 c^4)+2 a^2 (b^2-c^2)^4 (b^2+c^2) (2 b^4+b^2 c^2+2 c^4)+2 a^12 (3 b^4+5 b^2 c^2+3 c^4)+a^8 (2 b^6 c^2+5 b^4 c^4+2 b^2 c^6)-a^4 (b^2-c^2)^2 (6 b^8+4 b^6 c^2+5 b^4 c^4+4 b^2 c^6+6 c^8)) : :

    See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4608.

    X(18868) lies on this line: {2,3}


    X(18869) = (name pending)

    Barycentrics    (b^2-c^2)^2 (-a^2+b^2+c^2) (-a^8-7 a^4 b^2 c^2+2 a^6 (b^2+c^2)+(b^2-c^2)^2 (b^4+3 b^2 c^2+c^4)-2 a^2 (b^6-2 b^4 c^2-2 b^2 c^4+c^6)) (-5 a^10+8 a^8 (b^2+c^2)+2 (b^2-c^2)^4 (b^2+c^2)+4 a^6 (b^4-8 b^2 c^2+c^4)+a^2 (b^2-c^2)^2 (b^4+14 b^2 c^2+c^4)-2 a^4 (5 b^6-9 b^4 c^2-9 b^2 c^4+5 c^6))-(a^2-b^2) (a^2-c^2) (a^4-(b^2-c^2)^2) (a^16-34 a^8 b^2 c^2 (b^2-c^2)^2-a^14 (b^2+c^2)+11 a^10 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^6 (b^2+c^2)^2+a^12 (-6 b^4+13 b^2 c^2-6 c^4)+a^2 (b^2-c^2)^4 (b^6-9 b^4 c^2-9 b^2 c^4+c^6)-a^6 (b^2-c^2)^2 (11 b^6-27 b^4 c^2-27 b^2 c^4+11 c^6)+a^4 (b^2-c^2)^2 (6 b^8+5 b^6 c^2-38 b^4 c^4+5 b^2 c^6+6 c^8)) : :

    See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4608.

    X(18869) lies on this line: {2,3}


    X(18870) = (name pending)

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

    See Tran Quang Hung and Angel Montesdeoca, ADGEOM 4608.

    X(18870) lies on these lines: {2,3}, {125,9530}, {523,1853}, {1899,2452}, {2972,10714}, {3258,11550}


    X(18871) = (name pending)

    Barycentrics    (a^3-a^2 b-a b^2+b^3-a^2 c+3 a b c-b^2 c-a c^2-b c^2+c^3) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)+a b c (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) J: : , where J = |OH|/R   (Peter Moses, May 30, 2018)
    X(18871) = (3 (2 r - R - R* J)) X[2] - (6 r - 3 R - R*J)) X[3]

    See Tran Quang Hung and Francisco Javier García Capitán, ADGEOM 4585.

    X(18871) lies on this line: {2,3}


    X(18872) = X(6)X(694)∩X(39)X(512)

    Barycentrics    a^2 (2 a^2 - b^2 - c^2) / (a^4 - b^2 c^2) : :

    See Tran Quang Hung and Randy Hutson, ADGEOM 4613.

    X(18872) lies on these lines: {3,9217}, {6,694}, {32,249}, {39,512}, {543,598}, {574,805}, {733,12074}, {2086,3229}


    X(18873) = X(262)X(6036)∩X(523)X(3629)

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

    See Tran Quang Hung and Randy Hutson, ADGEOM 4613.

    X(18873) lies on these lines: {262,6036}, {523,3629}, {576,2065}


    X(18874) = MIDPOINT OF X(5) AND X(10095)

    Barycentrics    a^2 (a^2-b^2-3 b c-c^2) (a^2-b^2+3 b c-c^2) (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : :
    X(18874) = 5 X[5] + 3 X[51], 7 X[5] + X[52], 3 X[52] - 7 X[143], 9 X[51] - 5 X[143], 3 X[5] + X[143], 9 X[373] - X[550], X[185] + 7 X[3857], X[389] + 3 X[5066], 3 X[568] + 13 X[5068], 3 X[547] + X[5446], 9 X[5] - X[5562], 3 X[143] + X[5562], 9 X[52] + 7 X[5562], 11 X[5] - 3 X[5891], 11 X[51] + 5 X[5891], 11 X[143] + 9 X[5891], X[546] + 3 X[5943], 15 X[381] + X[6241], X[52] - 7 X[10095], 3 X[51] - 5 X[10095]

    See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27715.

    X(18874) lies on these lines: {4,7693}, {5,51}, {30,11695}, { 140,13598}, {156,17809}, {185, 3857}, {373,550}, {381,6241}, { 382,11451}, {389,5066}, {546, 5943}, {547,5446}, {568,5068}

    X(18874) = midpoint of X(i) and X(j) for these {i,j}: {5,10095}, {143,14128}, {546, 12006}, {1216,16982}, {3628, 10110}, {3850,5462}, {3861,9729}
    X(18874) = reflection of X(i) in X(j) for these {i,j}: {3628,12046}, {11017,12811}, { 11592,16239}
    X(18874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 51, 11591), (5, 143, 14128), (5, 13364, 10095), (381, 15026, 13630), (546, 5943, 12006), (10095, 14128, 143)


    X(18875) = MIDPOINT OF X(11583) AND X(16336)

    Barycentrics    a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10+a^10 c^2-4 a^8 b^2 c^2+2 a^6 b^4 c^2+9 a^4 b^6 c^2-13 a^2 b^8 c^2+5 b^10 c^2-4 a^8 c^4+2 a^6 b^2 c^4+2 a^4 b^4 c^4+3 a^2 b^6 c^4-9 b^8 c^4+6 a^6 c^6+9 a^4 b^2 c^6+3 a^2 b^4 c^6+8 b^6 c^6-4 a^4 c^8-13 a^2 b^2 c^8-9 b^4 c^8+a^2 c^10+5 b^2 c^10

    See Antreas Hatzipolakis and Peter Moses, Hyacinthos 27718.

    X(18875) lies on this line: {5,51}

    X(18875) = midpoint of X(11583) and X(16336)


    X(18876) = ISOGONAL CONJUGATE OF X(5523)

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

    X(18876) lies on the cubics K039, K043, K113, K938 and these lines: {2, 112}, {3, 1177}, {32, 14376}, {97, 14586}, {132, 18420}, {186, 1297}, {248, 6334}, {276, 16813}, {524, 10317}, {577, 2482}, {858, 11605}, {906, 3998}, {1073, 1384}, {1214, 1415}, {2072, 10749}, {2080, 6760}, {2794, 18531}, {2799, 14910}, {2966, 11610}, {3153, 10735}, {3926, 4558}, {5866, 10766}, {9517, 14908}, {10422, 15899}, {10745, 14830}, {13310, 14489}, {13754, 17974}

    X(18876) = isogonal conjugate of X(5523)
    X(18876) = X(2373)-Ceva conjugate of X(1177)
    X(18876) = X(i)-cross conjugate of X(j) for these (i,j): {187, 3}, {14417, 4558}
    X(18876) = X(2)-line conjugate of X(1560)
    X(18876) = X(1177)-vertex conjugate of X(9517)
    X(18876) = cevapoint of X(i) and X(j) for these (i,j): {3, 10317}, {6, 15139}, {577, 3292}
    X(18876) = trilinear pole of line {184, 520}
    X(18876) = circumcircle-inverse of X(1177)
    X(18876) = crosssum of X(i) and X(j) for these (i,j): {6, 8428}, {2393, 14580}
    X(18876) = barycentric product X(i)*X(j) for these {i,j}: {3, 2373}, {69, 1177}, {3265, 10423}, {6390, 10422}
    X(18876) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 858}, {6, 5523}, {32, 14580}, {48, 18669}, {69, 1236}, {184, 2393}, {187, 1560}, {577, 14961}, {1177, 4}, {1204, 15126}, {1790, 17172}, {2373, 264}, {3292, 5181}, {10422, 17983}, {10423, 107}, {13754, 12827}
    X(18876) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5523}, {4, 18669}, {19, 858}, {75, 14580}, {92, 2393}, {158, 14961}, {897, 1560}, {1236, 1973}, {1824, 17172}


    X(18877) = BARYCENTRIC PRODUCT X(3)*X(74)

    Barycentrics    a^4*(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) : :

    Let P and U be the touchpoints of the tangents to the polar circle from X(74). Then X(18877) is the crossdifference of P and U. (Randy Hutson, October 15, 2018)

    X(18877) lies on the cubic K381 and these lines: {6, 74}, {53, 10152}, {125, 6128}, {184, 1576}, {248, 14380}, {275, 6749}, {287, 524}, {394, 4558}, {526, 686}, {541, 3163}, {571, 14642}, {906, 3990}, {1304, 1971}, {1409, 1415}, {1562, 3018}, {1636, 2430}, {1989, 18320}, {1990, 15311}, {2871, 10602}, {3284, 11079}, {5467, 17974}, {14385, 14585}, {14533, 14586}

    X(18877) = X(50)-cross conjugate of X(14533)
    X(18877) = cevapoint of X(686) and X(3269)
    X(18877) = crosspoint of X(i) and X(j) for these (i,j): {74, 14919}, {1294, 2986}
    X(18877) = crossdifference of every pair of points on line {113, 133}
    X(18877) = crosssum of X(i) and X(j) for these (i,j): {6, 14703}, {30, 1990}, {186, 15262}, {1650, 14391}, {3003, 6000}
    X(18877) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1784}, {4, 14206}, {19, 3260}, {30, 92}, {75, 1990}, {158, 11064}, {264, 2173}, {273, 7359}, {318, 6357}, {561, 14581}, {811, 1637}, {823, 9033}, {1099, 16080}, {1495, 1969}, {1577, 4240}, {2166, 14920}, {2631, 6528}, {6335, 11125}, {9406, 18022}, {14400, 18026}
    X(18877) = X(1726)-zayin conjugate of X(2173)
    X(18877) = barycentric product X(i)*X(j) for these {i,j}: {3, 74}, {6, 14919}, {48, 2349}, {63, 2159}, {110, 14380}, {184, 1494}, {265, 14385}, {323, 11079}, {394, 8749}, {520, 1304}, {577, 16080}, {895, 9717}, {1073, 15291}, {2433, 4558}, {3292, 9139}, {5504, 14264}, {6000, 15404}, {10152, 14379}, {10419, 13754}, {15395, 16186}
    X(18877) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3260}, {31, 1784}, {32, 1990}, {48, 14206}, {50, 14920}, {74, 264}, {184, 30}, {418, 1568}, {577, 11064}, {1304, 6528}, {1494, 18022}, {1501, 14581}, {1576, 4240}, {2159, 92}, {2349, 1969}, {2433, 14618}, {3049, 1637}, {5627, 18817}, {8749, 2052}, {9247, 2173}, {9407, 16240}, {11079, 94}, {14380, 850}, {14385, 340}, {14575, 1495}, {14585, 3284}, {14908, 9214}, {14919, 76}, {15291, 15466}, {16080, 18027}
    X(18877) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (74, 15291, 8749), (8749, 15291, 6)


    X(18878) = X(99)X(6563)∩X(249)X(648)

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

    X(18878) lies on the Steiner circumellipse, the cubic K256, and on these lines: {99, 6563}, {249, 648}, {290, 1236}, {315, 5641}, {316, 1300}, {671, 2986}, {892, 15328}, {1494, 7799}, {2966, 15421}, {3228, 14910}, {4590, 15470}, {6528, 18020}

    X(18878) = isotomic conjugate of crosspoint of X(4) and X(476)
    X(18878) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1784}, {4, 14206}, {19, 3260}, {30, 92}, {75, 1990}, {158, 11064}, {264, 2173}, {273, 7359}, {318, 6357}, {561, 14581}, {811, 1637}, {823, 9033}, {1099, 16080}, {1495, 1969}, {1577, 4240}, {2166, 14920}, {2631, 6528}, {6335, 11125}, {9406, 18022}, {14400, 18026}
    X(18878) = X(i)-cross conjugate of X(j) for these (i,j): {323, 4590}, {3260, 18020}, {10420, 687}, {15328, 2986}
    X(18878) = cevapoint of X(i) and X(j) for these (i,j): {69, 3268}, {99, 10411}, {110, 2407}, {323, 15470}, {523, 11064}, {525, 10257}, {2986, 15328}, {5504, 15421}
    X(18878) = X(i)-isoconjugate of X(j) for these (i,j): {19, 686}, {403, 810}, {512, 1725}, {661, 3003}, {798, 3580}, {1973, 6334}, {2315, 2501}, {2643, 15329}, {4079, 18609}
    X(18878) = trilinear pole of line {2, 2986}
    X(18878) = barycentric product X(i)*X(j) for these {i,j}: {69, 687}, {76, 10420}, {99, 2986}, {670, 14910}, {1300, 4563}, {4590, 15328}, {5504, 6331}, {15421, 18020}
    X(18878) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 686}, {69, 6334}, {99, 3580}, {110, 3003}, {249, 15329}, {648, 403}, {662, 1725}, {687, 4}, {1300, 2501}, {2407, 113}, {2986, 523}, {4235, 12828}, {4558, 13754}, {4575, 2315}, {5504, 647}, {10419, 2433}, {10420, 6}, {12028, 14582}, {14590, 1986}, {14910, 512}, {15328, 115}, {15421, 125}, {15454, 1637}, {15470, 2088}, {18020, 16237}


    X(18879) = X(24)X(250)∩X(249)X(1993)

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

    X(18879) lies on these lines: {24, 250}, {249, 1993}, {687, 2407}, {691, 10420}, {1300, 10723}, {2966, 15421}, {4590, 7763}, {4611, 5649}

    X(18879) = X(i)-cross conjugate of X(j) for these (i,j): {186, 99}, {3003, 110}, {14910, 10420}
    X(18879) = cevapoint of X(i) and X(j) for these (i,j): {110, 3003}, {323, 4558}, {10420, 14910}
    X(18879) = X(i)-isoconjugate of X(j) for these (i,j): {115, 1725}, {403, 3708}, {1109, 3003}, {2315, 2970}, {2643, 3580}
    X(18879) = trilinear pole of line {110, 924}
    X(18879) = barycentric product X(i)*X(j) for these {i,j}: {99, 10420}, {249, 2986}, {687, 4558}, {4590, 14910}, {5504, 18020}
    X(18879) = barycentric quotient X(i)/X(j) for these {i,j}: {186, 16221}, {249, 3580}, {250, 403}, {687, 14618}, {1101, 1725}, {1300, 2970}, {2986, 338}, {4558, 6334}, {5504, 125}, {10419, 12079}, {10420, 523}, {14910, 115}


    X(18880) = (name pending)

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

    X(18880) lies on the cubic K375


    X(18881) = (name pending)

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


    X(18882) = X(20)X(49)∩X(23)X(206)

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

    X(18882) lies on the cubic K428 and these lines: {20, 49}, {23, 206}, {69, 11003}, {110, 3818}, {184, 323}, {378, 399}, {1147, 11459}, {2916, 6800}, {3047, 17847}, {7391, 9544}, {9704, 10323}, {11597, 12383}, {12219, 18445}

    X(18882) = barycentric product X(6)X(14558)
    X(18882) = barycentric quotient X(14558)/X(76)


    X(18883) = X(2)X(94)∩X(5)X(49)

    Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
    Barycentrics    Sin[A] Cos[2 A] Csc[3 A] : :

    X(18883) lies on the cubic K491 and these lines: {2, 94}, {5, 49}, {24, 136}, {254, 6344}, {381, 18576}, {403, 12028}, {468, 476}, {626, 11060}, {1995, 3425}, {2006, 7110}, {3091, 18300}, {3545, 18316}, {3580, 16310}, {6132, 10412}

    X(18883) = cevapoint of X(5) and X(16310)
    X(18883) = crossdifference of every pair of points on line {2081, 14270}
    X(18883) = trilinear pole of line {52, 924}
    X(18883) = X(18127)-complementary conjugate of X(18589)
    X(18883) = X(18817)-Ceva conjugate of X(265)
    X(18883) = polar conjugate of X(5962)
    X(18883) = X(i)-isoconjugate of X(j) for these (i,j): {48, 5962}, {50, 91}, {96, 2290}, {186, 1820}, {925, 2624}, {1154, 2168}, {2165, 6149}
    X(18883) = {X(8836),X(8838)}-harmonic conjugate of X(265)
    X(18883) = barycentric product X(i)*X(j) for these {i,j}: {24, 328}, {94, 1993}, {264, 5961}, {265, 317}, {476, 6563}, {1147, 18817}, {1989, 7763}, {6344, 9723}
    X(18883) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5962}, {24, 186}, {47, 6149}, {52, 1154}, {94, 5392}, {265, 68}, {317, 340}, {467, 14918}, {476, 925}, {571, 50}, {924, 526}, {1141, 96}, {1989, 2165}, {1993, 323}, {2166, 91}, {2180, 2290}, {5961, 3}, {6344, 847}, {6563, 3268}, {7763, 7799}, {11547, 14165}, {14111, 562}, {14576, 11062}, {18384, 14593}


    X(18884) = (name pending)

    Barycentrics    Sin[A/2]/(a*(-a + b + c)*(a*(-a + b + c) + 2*(2*b*c*Sin[A/2] + a*c*Sin[B/2] + a*b*Sin[C/2]))) : :

    X(18884) lies on the conic {{A, B, C, X(2), X(7)}}, the cubic K745, and on this line: {7, 10500}


    X(18885) = X(37)X(2089)∩X(173)X(13443)

    Barycentrics    Sin[A] Tan[A/2] (Cos[B/2]+Cos[C/2])^2 : :

    Let (Oa), (Ob), (Oc) be the circles centered at A, B, C, resp., and externally tangent to the incircle. Let A' be the insimilicenter of (Ob) and (Oc), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(18885). (Randy Hutson, June 27, 2018)

    X(18885) lies on the cubics K745, K746, K972, and these lines: {37,2089}, {173,13443}, {178,10489}, {1400,12809}, {1418,7371}, {7707,10502}

    X(18885) = X(555)-Ceva conjugate of X(7)
    X(18885) = X(i)-cross conjugate of X(j) for these (i,j): {173, 7057}, {2089, 7}
    X(18885) = cevapoint of X(i) and X(j) for these (i,j): {234, 14596}
    X(18885) = X(i)-isoconjugate of X(j) for these (i,j): {41, 7048}, {55, 258}, {289, 6726}
    X(18885) = barycentric product X(i)*X(j) for these {i,j}: {7, 7057}, {85, 173}, {236, 555}, {2089, 4146}, {7022, 7048}
    X(18885) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 7048}, {57, 258}, {173, 9}, {174, 7028}, {234, 2090}, {236, 6731}, {2089, 188}, {7022, 7057}, {7057, 8}, {7370, 289}, {7371, 1488}, {10490, 15997}, {13444, 3659}, {14596, 16015}


    X(18886) = X(1)X(10489)∩X(2)X(4146)

    Barycentrics    Tan[A/2] (Sec[A/2]+Tan[A/2]) : :

    X(18886) lies on these lines: {1, 10489}, {2, 4146}, {7, 174}, {177, 11192}, {279, 555}, {2089, 7022}, {5435, 7371}, {8113, 8387}, {10499, 13092}

    X(18886) = X(10489)-Ceva conjugate of X(10502)
    X(18886) = crosspoint of X(i) and X(j) for these (i,j): {10490, 14596}
    X(18886) = X(i)-isoconjugate of X(j) for these (i,j): {1252, 10491}, {4564, 10501}
    X(18886) = barycentric product X(i)*X(j) for these {i,j}: {1, 10489}, {7, 10502}, {177, 177}, {178, 10490}, {234, 7707}, {14596, 16016}
    X(18886) = barycentric quotient X(i)/X(j) for these {i,j}: {244, 10491}, {3271, 10501}, {10489, 75}, {10502, 8}


    X(18887) = X(37)X(259)∩X(234)X(14596)

    Barycentrics    Sin[A] (Cos[B/2]+Cos[C/2])(Sin[B/2]+Sin[C/2])

    X(18887) lies on the cubic K746 and these lines: {37, 259}, {234, 14596}, {7707, 10500}, {8965, 10232}

    X(18887) = crosspoint of X(i) and X(j) for these (i,j): {7707, 16016}, {15997, 16015}
    X(18887) = barycentric product X(i)*X(j) for these {i,j}: {178, 15997}, {2090, 7707}, {7048, 10502}, {16015, 16016}
    X(18887) = barycentric quotient X(10502)/X(7057)


    X(18888) = X(2)X(266)∩X(9)X(173)

    Barycentrics    (Cos[B/2] + Cos[C/2])*Sec[A/2]*Sin[A]^2 : :

    X(18888) lies on these lines: {6, 266}, {9, 173}, {55, 259}, {57, 7371}, {284, 6727}, {2291, 13444}, {7707, 10500}

    X(18888) = X(i)-Ceva conjugate of X(j) for these (i,j): {57, 10490}, {173, 7707}, {177, 16012}
    X(18888) = X(i)-isoconjugate of X(j) for these (i,j): {2, 260}, {100, 10492}
    X(18888) = crosspoint of X(i) and X(j) for these (i,j): {57, 266}, {177, 14596}, {8372, 16015}
    X(18888) = trilinear pole of line {663, 6729}
    X(18888) = crosssum of X(i) and X(j) for these (i,j): {9, 188}, {258, 7028}
    X(18888) = barycentric product X(i)*X(j) for these {i,j}: {1, 177}, {7, 16012}, {9, 14596}, {57, 16016}, {173, 16015}, {174, 7707}, {178, 266}, {188, 10490}, {234, 259}, {260, 10489}, {522, 13444}, {2089, 15997}, {7057, 16011}, {8372, 13443}
    X(18888) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 260}, {177, 75}, {649, 10492}, {7707, 556}, {10490, 4146}, {13444, 664}, {14596, 85}, {16011, 7048}, {16012, 8}, {16016, 312}
    X(18888) = {X(173),X(8078)}-harmonic conjugate of X(9)


    X(18889) = X(6)X(109)∩X(41)X(692)

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

    X(18889) lies on these lines: {6, 109}, {41, 692}, {44, 294}, {220, 3939}, {607, 8750}, {643, 2287}, {911, 7113}

    X(18889) = crosspoint of X(2291) and X(4845)
    X(18889) = crossdifference of every pair of points on line {6366, 10427}
    X(18889) = crosssum of X(527) and X(1323)
    X(18889) = trilinear pole of line {41, 8641}
    X(18889) = X(169)-zayin conjugate of X(1155)
    X(18889) = X(i)-isoconjugate of X(j) for these (i,j): {2, 1323}, {7, 527}, {75, 6610}, {85, 1155}, {273, 6510}, {279, 6745}, {658, 6366}, {664, 1638}, {1055, 6063}, {1088, 6603}, {1121, 3321}, {4554, 14413}, {6647, 7249}, {13149, 14414}
    X(18889) = barycentric product X(i)*X(j) for these {i,j}: {1, 4845}, {9, 2291}, {41, 1121}, {55, 1156}, {3900, 14733}, {5526, 15734}, {10426, 15733}
    X(18889) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 1323}, {32, 6610}, {41, 527}, {1156, 6063}, {1253, 6745}, {2175, 1155}, {2291, 85}, {3063, 1638}, {4845, 75}, {8641, 6366}, {9447, 1055}, {14733, 4569}, {14827, 6603}


    X(18890) = X(154)X(160)∩X(216)X(631)

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

    X(18890) lies on the cubic K307 and these lines: {154, 160}, {216, 631}, {577, 14371}

    X(18890) = X(51)-cross conjugate of X(3)
    X(18890) = crosspoint of X(1073) and X(13855)
    X(18890) = crosssum of X(1075) and X(1249)
    X(18890) = X(i)-isoconjugate of X(j) for these (i,j): {92, 6759}, {2167, 14363}
    X(18890) = barycentric product X(i)*X(j) for these {i,j}: {3, 15318}, {5, 14371}
    X(18890) = barycentric quotient X(i)/X(j) for these {i,j}: {51, 14363}, {184, 6759}, {14371, 95}, {15318, 264}


    X(18891) = ISOGONAL CONJUGATE OF X(14598)

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

    X(18891) lies on the cubic K1024 and these lines: {10, 18833}, {75, 700}, {76, 321}, {239, 3978}, {305, 3705}, {308, 17289}, {310, 3741}, {318, 18022}, {320, 670}, {693, 784}, {894, 9230}, {1269, 6385}, {1281, 8783}, {1921, 3797}, {1926, 4087}, {1978, 3912}, {3264, 6386}, {3662, 6374}, {4044, 18152}, {6063, 7185}, {6376, 7034}

    X(18891) = isogonal conjugate of X(14598)
    X(18891) = isotomic conjugate of X(1911)
    X(18891) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14598}, {6, 1922}, {31, 1911}, {32, 292}, {56, 18265}, {171, 1927}, {172, 9468}, {213, 18268}, {238, 18267}, {291, 560}, {295, 1974}, {334, 1917}, {335, 1501}, {660, 1980}, {692, 875}, {741, 1918}, {813, 1919}, {894, 8789}, {1397, 7077}, {1920, 14604}, {1924, 4584}, {1967, 7122}, {1973, 2196}, {4589, 9426}, {7233, 9448}, {7234, 17938}, {17735, 18263}
    X(18891) = X(76)-Hirst inverse of X(561)
    X(18891) = cevapoint of X(1921) and X(4087)
    X(18891) = crossdifference of every pair of points on line {1980, 2205}
    X(18891) = X(i)-isoconjugate of X(j) for these (i,j): {92, 6759}, {2167, 14363}
    X(18891) = barycentric product X(i)*X(j) for these {i,j}: {75, 1921}, {76, 350}, {85, 4087}, {238, 1502}, {239, 561}, {256, 14603}, {257, 1926}, {310, 3948}, {312, 18033}, {740, 6385}, {812, 6386}, {871, 3797}, {874, 3261}, {1914, 1928}, {1978, 3766}, {3596, 10030}, {3975, 6063}, {3978, 7018}, {4010, 4602}, {7260, 14295}, {18032, 18035}, {18036, 18037}
    X(18891) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1922}, {2, 1911}, {6, 14598}, {9, 18265}, {69, 2196}, {75, 292}, {76, 291}, {86, 18268}, {238, 32}, {239, 31}, {242, 1973}, {256, 9468}, {257, 1967}, {274, 741}, {292, 18267}, {304, 295}, {312, 7077}, {314, 2311}, {350, 6}, {385, 7122}, {514, 875}, {561, 335}, {659, 1919}, {668, 813}, {670, 4584}, {693, 3572}, {740, 213}, {874, 101}, {893, 1927}, {904, 8789}, {1429, 1397}, {1447, 604}, {1502, 334}, {1914, 560}, {1920, 18787}, {1921, 1}, {1926, 894}, {1929, 18263}, {1966, 172}, {1978, 660}, {2201, 1974}, {2210, 1501}, {2238, 1918}, {3261, 876}, {3263, 3252}, {3570, 692}, {3596, 4876}, {3684, 2175}, {3685, 41}, {3716, 3063}, {3747, 2205}, {3766, 649}, {3797, 869}, {3948, 42}, {3975, 55}, {3978, 171}, {4010, 798}, {4037, 872}, {4087, 9}, {4148, 8641}, {4366, 2210}, {4432, 2251}, {4455, 1924}, {4486, 788}, {4495, 2242}, {4602, 4589}, {4603, 17938}, {4609, 4639}, {4760, 922}, {6385, 18827}, {6386, 4562}, {6651, 18266}, {7018, 694}, {7193, 9247}, {7260, 805}, {8299, 9454}, {8300, 14599}, {8632, 1980}, {10030, 56}, {14024, 2204}, {14433, 890}, {14599, 1917}, {14603, 1909}, {16609, 1402}, {17493, 904}, {17755, 2223}, {17984, 7119}, {18032, 9506}, {18033, 57}, {18035, 1757}, {18037, 17798}, {18277, 3510}, {18786, 7104}
    X(18891) = {X(76),X(6382)}-harmonic conjugate of X(3661)


    X(18892) = X(32)X(1917)∩X(172)X(1932)

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

    X(18892) lies on these lines: {32, 1917}, {172, 1932}, {560, 1501}, {1914, 1933}, {4161, 18759}, {6066, 9459}

    X(18892) = X(i)-isoconjugate of X(j) for these (i,j): {75, 334}, {76, 335}, {264, 337}, {291, 561}, {292, 1502}, {295, 18022}, {313, 18827}, {693, 4583}, {850, 4589}, {871, 3864}, {876, 6386}, {1577, 4639}, {1909, 1934}, {1911, 1928}, {1916, 1920}, {1978, 4444}, {3261, 4562}, {3596, 7233}, {4518, 6063}
    X(18892) = X(560)-Hirst inverse of X(1501)
    X(18892) = barycentric product X(i)*X(j) for these {i,j}: {1, 14599}, {6, 2210}, {31, 1914}, {32, 238}, {41, 1428}, {163, 4455}, {184, 2201}, {213, 5009}, {239, 560}, {242, 9247}, {256, 14602}, {350, 1501}, {692, 8632}, {893, 1933}, {904, 1691}, {1333, 3747}, {1397, 3684}, {1429, 2175}, {1447, 9447}, {1580, 7104}, {1917, 1921}, {1919, 3573}, {1922, 8300}, {1973, 7193}, {1980, 3570}, {2112, 18264}, {2206, 2238}, {4366, 14598}, {4433, 16947}, {6652, 18267}, {6654, 9455}, {8852, 18038}, {9448, 10030}
    X(18892) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 334}, {238, 1502}, {239, 1928}, {560, 335}, {1501, 291}, {1576, 4639}, {1914, 561}, {1917, 292}, {1933, 1920}, {1980, 4444}, {2201, 18022}, {2210, 76}, {5009, 6385}, {7104, 1934}, {9233, 1911}, {9247, 337}, {9447, 4518}, {9448, 4876}, {14574, 4584}, {14599, 75}, {14602, 1909}


    X(18893) = X(291)X(7122)∩X(560)X(1922)

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

    X(18893) lies on these lines: {291, 7122}, {560, 1922}, {1911, 2206}

    X(18893) = X(i)-isoconjugate of X(j) for these (i,j): {76, 1921}, {238, 1928}, {239, 1502}, {257, 14603}, {350, 561}, {1926, 7018}, {3596, 18033}, {3766, 6386}, {3948, 6385}, {4010, 4609}, {4087, 6063}
    X(18893) = barycentric product X(i)*X(j) for these {i,j}: {6, 14598}, {31, 1922}, {32, 1911}, {171, 8789}, {172, 1927}, {291, 1501}, {292, 560}, {334, 9233}, {335, 1917}, {604, 18265}, {741, 2205}, {813, 1980}, {1909, 14604}, {1914, 18267}, {1918, 18268}, {1974, 2196}, {4584, 9426}, {7122, 9468}, {18263, 18266}
    X(18893) = barycentric quotient X(i)/X(j) for these {i,j}: {292, 1928}, {560, 1921}, {1501, 350}, {1911, 1502}, {1917, 239}, {1922, 561}, {7122, 14603}, {8789, 7018}, {9233, 238}, {9447, 4087}, {9448, 3975}, {14598, 76}, {14604, 256}


    X(18894) = X(560)X(9233)∩X(1501)X(1917)

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

    X(18894) lies on these lines: {560, 9233}, {1501, 1917}, {2210, 14602}

    X(18894) = X(1501)-Hirst inverse of X(1917)
    X(18894) = X(i)-isoconjugate of X(j) for these (i,j): {76, 334}, {291, 1502}, {292, 1928}, {335, 561}, {337, 1969}, {850, 4639}, {1920, 1934}, {3261, 4583}, {4444, 6386}
    X(18894) = barycentric product X(i)*X(j) for these {i,j}: {6, 14599}, {31, 2210}, {32, 1914}, {238, 560}, {239, 1501}, {242, 14575}, {350, 1917}, {893, 14602}, {904, 1933}, {1428, 2175}, {1429, 9447}, {1447, 9448}, {1576, 4455}, {1691, 7104}, {1918, 5009}, {1921, 9233}, {1974, 7193}, {1980, 3573}, {2201, 9247}, {2206, 3747}, {4010, 14574}, {8300, 14598}, {12835, 18265}
    X(18894) = barycentric quotient X(i)/X(j) for these {i,j}: {238, 1928}, {560, 334}, {1501, 335}, {1914, 1502}, {1917, 291}, {2210, 561}, {9233, 292}, {9448, 4518}, {14574, 4589}, {14575, 337}, {14599, 76}, {14602, 1920}


    X(18895) = ISOGONAL CONJUGATE OF X(14599)

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

    X(18895) lies on these lines: {75, 291}, {76, 334}, {274, 292}, {295, 4589}, {304, 4876}, {310, 321}, {337, 16747}, {561, 8024}, {789, 1281}, {894, 1922}, {1921, 3263}, {3978, 17789}, {4459, 18830}, {4639, 18359}, {6063, 6358}, {6386, 16732}, {10471, 18827}

    X(18895) = isogonal conjugate of X(14599)
    X(18895) = isotomic conjugate of X(1914)
    X(18895) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14599}, {6, 2210}, {31, 1914}, {32, 238}, {41, 1428}, {163, 4455}, {184, 2201}, {213, 5009}, {239, 560}, {242, 9247}, {256, 14602}, {350, 1501}, {692, 8632}, {893, 1933}, {904, 1691}, {1333, 3747}, {1397, 3684}, {1429, 2175}, {1447, 9447}, {1580, 7104}, {1917, 1921}, {1919, 3573}, {1922, 8300}, {1973, 7193}, {1980, 3570}, {2112, 18264}, {2206, 2238}, {4366, 14598}, {4433, 16947}, {6652, 18267}, {6654, 9455}, {8852, 18038}, {9448, 10030}
    X(18895) = X(i)-Hirst inverse of X(j) for these (i,j): {76, 334}, {335, 1920}, {1921, 18034}
    X(18895) = cevapoint of X(i) and X(j) for these (i,j): {75, 17789}, {321, 3263}, {335, 337}
    X(18895) = trilinear pole of line {313, 3261}
    X(18895) = barycentric product X(i)*X(j) for these {i,j}: {75, 334}, {76, 335}, {264, 337}, {291, 561}, {292, 1502}, {295, 18022}, {313, 18827}, {693, 4583}, {850, 4589}, {871, 3864}, {876, 6386}, {1577, 4639}, {1909, 1934}, {1911, 1928}, {1916, 1920}, {1978, 4444}, {3261, 4562}, {3596, 7233}, {4518, 6063}
    X(18895) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2210}, {2, 1914}, {6, 14599}, {7, 1428}, {10, 3747}, {69, 7193}, {75, 238}, {76, 239}, {85, 1429}, {86, 5009}, {92, 2201}, {171, 1933}, {172, 14602}, {264, 242}, {291, 31}, {292, 32}, {295, 184}, {312, 3684}, {313, 740}, {321, 2238}, {334, 1}, {335, 6}, {337, 3}, {349, 16609}, {350, 8300}, {514, 8632}, {523, 4455}, {561, 350}, {660, 692}, {668, 3573}, {693, 659}, {694, 7104}, {741, 2206}, {850, 4010}, {875, 1980}, {894, 1691}, {1237, 4039}, {1269, 4974}, {1441, 1284}, {1447, 12835}, {1502, 1921}, {1581, 904}, {1909, 1580}, {1911, 560}, {1916, 893}, {1920, 385}, {1921, 4366}, {1922, 1501}, {1934, 256}, {1978, 3570}, {2113, 18264}, {2196, 9247}, {2533, 5027}, {3252, 9454}, {3261, 812}, {3262, 15507}, {3263, 8299}, {3264, 4432}, {3266, 4760}, {3509, 18038}, {3572, 1919}, {3596, 3685}, {3661, 16514}, {3701, 4433}, {3864, 869}, {4036, 4155}, {4374, 4164}, {4391, 4435}, {4444, 649}, {4518, 55}, {4562, 101}, {4583, 100}, {4584, 163}, {4589, 110}, {4639, 662}, {4876, 41}, {5378, 1110}, {6063, 1447}, {6386, 874}, {7018, 18786}, {7077, 2175}, {7233, 56}, {7245, 2242}, {14598, 1917}, {15523, 4093}, {16720, 8623}, {18031, 6654}, {18034, 8301}, {18265, 9448}, {18760, 16366}, {18787, 7122}, {18827, 58}


    X(18896) = ISOGONAL CONJUGATE OF X(14602)

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

    Let A'B'C' and A"B"C" be the 1st Brocard and 1st anti-Brocard triangles, resp. Let A* be the barycentric product A'*A", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(18896). (Randy Hutson, June 27, 2018)

    X(18896) lies on the cubic K1023 and these lines: {2, 3114}, {5, 6234}, {76, 115}, {83, 3115}, {141, 308}, {264, 5117}, {290, 325}, {313, 1934}, {315, 17970}, {316, 805}, {334, 1581}, {338, 1502}, {733, 6572}, {880, 11646}, {882, 14295}, {2086, 3225}, {5031, 14603}, {5207, 9467}, {7752, 14251}, {7937, 11654}

    X(18896) = isogonal conjugate of X(14602)
    X(18896) = isotomic conjugate of X(1691)
    X(18896) = X(5031)-cross conjugate of X(2)
    X(18896) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14602}, {6, 1933}, {31, 1691}, {32, 1580}, {163, 5027}, {171, 14599}, {172, 2210}, {385, 560}, {419, 9247}, {1101, 2086}, {1501, 1966}, {1914, 7122}, {1917, 3978}, {1924, 17941}, {1926, 9233}, {1927, 4027}, {9236, 16985}
    X(18896) = X(i)-Hirst inverse of X(j) for these (i,j): {76, 1916}, {694, 9230}
    X(18896) = cevapoint of X(i) and X(j) for these (i,j): {2, 5207}, {141, 325}
    X(18896) = trilinear pole of line {850, 2528}
    X(18896) = {X(334),X(7018)}-harmonic conjugate of X(1581)
    X(18896) = barycentric product X(i)*X(j) for these {i,j}: {75, 1934}, {76, 1916}, {327, 8842}, {334, 7018}, {561, 1581}, {694, 1502}, {850, 18829}, {882, 4609}, {1928, 1967}, {8024, 14970}
    X(18896) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1933}, {2, 1691}, {6, 14602}, {75, 1580}, {76, 385}, {115, 2086}, {141, 8623}, {256, 2210}, {257, 1914}, {264, 419}, {291, 7122}, {305, 12215}, {313, 4039}, {334, 171}, {335, 172}, {337, 3955}, {523, 5027}, {561, 1966}, {670, 17941}, {693, 4164}, {694, 32}, {805, 1576}, {850, 804}, {868, 2679}, {881, 9426}, {882, 669}, {893, 14599}, {1502, 3978}, {1581, 31}, {1916, 6}, {1927, 1917}, {1928, 1926}, {1930, 2236}, {1934, 1}, {1967, 560}, {3261, 4107}, {3266, 5026}, {3978, 4027}, {4518, 2330}, {4583, 4579}, {4609, 880}, {7018, 238}, {7019, 7193}, {7249, 1428}, {8024, 732}, {8789, 9233}, {8842, 182}, {9230, 16985}, {9468, 1501}, {14251, 9418}, {14970, 251}, {15391, 14600}, {17938, 14574}, {17970, 14575}, {17980, 1974}, {18022, 17984}, {18024, 14382}, {18276, 18270}, {18829, 110}


    X(18897) = X(32)X(1911)∩X(172)X(335)

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

    X(18897) lies on these lines: {32, 1911}, {172, 335}, {292, 1333}, {560, 14598}, {609, 18787}, {697, 813}, {741, 769}, {1492, 18274}, {1922, 2205}, {1927, 14604}

    X(18897) = X(i)-isoconjugate of X(j) for these (i,j): {75, 1921}, {76, 350}, {85, 4087}, {238, 1502}, {239, 561}, {256, 14603}, {257, 1926}, {310, 3948}, {312, 18033}, {740, 6385}, {812, 6386}, {871, 3797}, {874, 3261}, {1914, 1928}, {1978, 3766}, {3596, 10030}, {3975, 6063}, {3978, 7018}, {4010, 4602}, {7260, 14295}, {18032, 18035}, {18036, 18037}
    X(18897) = X(i)-Hirst inverse of X(j) for these (i,j): {560, 14598}
    X(18897) = barycentric product X(i)*X(j) for these {i,j}: {1, 14598}, {6, 1922}, {31, 1911}, {32, 292}, {56, 18265}, {171, 1927}, {172, 9468}, {213, 18268}, {238, 18267}, {291, 560}, {295, 1974}, {334, 1917}, {335, 1501}, {660, 1980}, {692, 875}, {741, 1918}, {813, 1919}, {894, 8789}, {1397, 7077}, {1920, 14604}, {1924, 4584}, {1967, 7122}, {1973, 2196}, {4589, 9426}, {7233, 9448}, {7234, 17938}, {17735, 18263}
    X(18897) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1921}, {172, 14603}, {291, 1928}, {292, 1502}, {560, 350}, {1397, 18033}, {1501, 239}, {1911, 561}, {1917, 238}, {1922, 76}, {1927, 7018}, {1980, 3766}, {2175, 4087}, {2205, 3948}, {7122, 1926}, {8789, 257}, {9233, 1914}, {9426, 4010}, {9447, 3975}, {9448, 3685}, {14598, 75}, {14604, 893}, {18265, 3596}, {18267, 334}, {18268, 6385}


    X(18898) = ISOGONAL CONJUGATE OF X(3314)

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

    X(18898) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1013, and on these lines: {2, 1501}, {6, 6660}, {32, 694}, {37, 983}, {251, 3981}, {263, 12212}, {308, 3114}, {2987, 13330}, {3108, 5012}, {5038, 11175}

    X(18898) = isogonal conjugate of X(3314)
    X(18898) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3314}, {63, 5117}, {75, 3094}, {76, 3116}, {561, 3117}, {824, 3888}, {982, 3661}, {984, 3662}, {1581, 9865}, {3061, 7179}, {3705, 7146}, {3776, 3799}, {3777, 3807}, {3786, 16888}, {3794, 16603}, {4602, 17415}
    X(18898) = barycentric product X(i)*X(j) for these {i,j}: {6, 3407}, {31, 3113}, {32, 3114}, {251, 14617}, {983, 985}, {1976, 8840}, {2344, 7132}, {9063, 9426}
    X(18898) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3314}, {25, 5117}, {32, 3094}, {560, 3116}, {1501, 3117}, {1691, 9865}, {3113, 561}, {3114, 1502}, {3407, 76}, {9426, 17415}, {14617, 8024}


    X(18899) = X(2)X(6)∩X(32)X(8789)

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

    X(18899) lies on the cubic K1016 and these lines: {2, 6}, {32, 8789}, {39, 1186}, {110, 707}, {194, 3499}, {1207, 7786}, {1501, 9233}, {3116, 7032}

    X(18899) = crosssum of X(1502) and X(10010)
    X(18899) = crossdifference of every pair of points on line {512, 14295}
    X(18899) = X(i)-isoconjugate of X(j) for these (i,j): {75, 3114}, {76, 3113}, {561, 3407}, {661, 9063}, {871, 17743}, {7034, 14621}, {14617, 18833}
    X(18899) = {X(6),X(1613)}-harmonic conjugate of X(385)
    X(18899) = barycentric product X(i)*X(j) for these {i,j}: {6, 3117}, {31, 3116}, {32, 3094}, {99, 9006}, {110, 17415}, {869, 7032}, {1501, 3314}, {3888, 8630}, {5117, 14575}, {8789, 9865}
    X(18899) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 3114}, {110, 9063}, {560, 3113}, {869, 7034}, {1501, 3407}, {3094, 1502}, {3116, 561}, {3117, 76}, {7032, 871}, {9006, 523}, {9418, 8840}, {17415, 850}


    X(18900) = ISOGONAL CONJUGATE OF X(871)

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

    X(18900) lies on these lines: {6, 31}, {32, 1922}, {100, 701}, {101, 717}, {314, 983}, {560, 1501}, {595, 3802}, {872, 1185}, {985, 4279}, {2175, 7104}, {2176, 18756}, {2236, 4376}, {3187, 3797}, {3510, 3550}, {4386, 8622}, {7032, 16693}

    X(18900) = isogonal conjugate of X(871)
    X(18900) = X(2175)-beth conjugate of X(31)
    X(18900) = X(i)-isoconjugate of X(j) for these (i,j): {1, 871}, {75, 870}, {76, 14621}, {561, 985}, {693, 789}, {1086, 5388}, {1978, 4817}, {3114, 3662}, {3261, 4586}
    X(18900) = barycentric product X(i)*X(j) for these {i,j}: {6, 869}, {31, 2276}, {32, 984}, {41, 1469}, {58, 3774}, {101, 788}, {190, 8630}, {213, 3736}, {560, 3661}, {604, 4517}, {692, 3250}, {901, 14436}, {983, 3117}, {1911, 16514}, {1919, 3799}, {1922, 3783}, {1973, 3781}, {1980, 3807}, {2175, 7146}, {2210, 3862}, {3797, 14598}, {3864, 14599}, {7179, 9447}, {7204, 14827}
    X(18900) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 871}, {32, 870}, {560, 14621}, {788, 3261}, {869, 76}, {984, 1502}, {1110, 5388}, {1501, 985}, {1980, 4817}, {2276, 561}, {3661, 1928}, {3736, 6385}, {3774, 313}, {8630, 514}, {9448, 2344}
    X(18900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31, 2209, 1914), (2205, 9455, 560)


    X(18901) = ISOGONAL CONJUGATE OF X(14604)

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

    X(18901) lies on these lines: {76, 14820}, {325, 4609}, {1502, 3314}, {9865, 14603}, {10010, 16986}

    X(18901) = isogonal conjugate of X(14604)
    X(18901) = isotomic conjugate of X(8789)
    X(18901) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14604}, {31, 8789}, {32, 1927}, {560, 9468}, {694, 1917}, {1501, 1967}, {1581, 9233}, {1924, 17938}, {7104, 14598}
    X(18901) = barycentric product X(i)*X(j) for these {i,j}: {76, 14603}, {561, 1926}, {1502, 3978}, {1928, 1966}, {4609, 14295}
    X(18901) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8789}, {6, 14604}, {75, 1927}, {76, 9468}, {305, 17970}, {385, 1501}, {561, 1967}, {670, 17938}, {804, 9426}, {850, 881}, {880, 1576}, {1502, 694}, {1580, 1917}, {1691, 9233}, {1909, 14598}, {1920, 1922}, {1921, 7104}, {1926, 31}, {1928, 1581}, {1966, 560}, {3978, 32}, {4609, 805}, {5976, 9418}, {12215, 14575}, {14295, 669}, {14296, 1980}, {14382, 14601}, {14603, 6}, {17941, 14574}, {17984, 1974}, {18022, 17980}


    X(18902) = X(32)X(14820)∩X(50)X(3289)

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

    X(18902) lies on these lines: {32, 14820}, {50, 3289}, {1501, 9233}

    X(18902) = X(i)-Hirst inverse of X(j) for these (i,j): {1501, 9233}
    X(18902) = X(i)-isoconjugate of X(j) for these (i,j): {76, 1934}, {561, 1916}, {694, 1928}, {1502, 1581}
    X(18902) = barycentric product X(i)*X(j) for these {i,j}: {6, 14602}, {31, 1933}, {32, 1691}, {172, 14599}, {385, 1501}, {419, 14575}, {560, 1580}, {804, 14574}, {1576, 5027}, {1917, 1966}, {2210, 7122}, {3978, 9233}, {4027, 8789}, {9426, 17941}
    X(18902) = barycentric quotient X(i)/X(j) for these {i,j}: {560, 1934}, {1501, 1916}, {1580, 1928}, {1691, 1502}, {1917, 1581}, {1933, 561}, {9233, 694}, {14574, 18829}, {14602, 76}


    X(18903) = X(1501)X(9468)∩X(8789)X(14574)

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

    X(18903) lies on these lines: {1501, 9468}, {8789, 14574}, {9233, 14604}

    X(18903) = X(i)-Hirst inverse of X(j) for these (i,j): {9233, 14604}
    X(18903) = X(i)-isoconjugate of X(j) for these (i,j): {561, 14603}, {1502, 1926}, {1928, 3978}
    X(18903) = barycentric product X(i)*X(j) for these {i,j}: {6, 14604}, {32, 8789}, {560, 1927}, {694, 9233}, {881, 14574}, {1501, 9468}, {1917, 1967}, {9426, 17938}
    X(18903) = barycentric quotient X(i)/X(j) for these {i,j}: {1501, 14603}, {1917, 1926}, {1927, 1928}, {8789, 1502}, {9233, 3978}, {14604, 76}


    X(18904) = MIDPOINT OF X(1966) AND X(17493)

    Barycentrics    a*(b + c)*(a^2 - b*c)*(b^2 - b*c + c^2) : :
    X(18904) = 3 X[2] + X[17493]

    X(18904) lies on the cubic K1035 and these lines: {2, 893}, {10, 37}, {39, 3821}, {142, 16604}, {226, 16606}, {238, 1691}, {292, 4645}, {722, 2887}, {732, 1107}, {804, 3709}, {812, 14838}, {874, 17279}, {1001, 18755}, {1100, 15989}, {1279, 9423}, {1284, 2238}, {1575, 1738}, {1716, 16968}, {1921, 14603}, {2229, 3120}, {2240, 3724}, {2275, 3662}, {3006, 8620}, {3061, 3094}, {3121, 3936}, {3721, 3778}, {3777, 3808}, {4657, 17030}, {4892, 16592}, {16591, 16609}, {16886, 16889}

    X(18904) = midpoint of X(1966) and X(17493)
    X(18904) = complement X(1966)
    X(18904) = X(i)-complementary conjugate of X(j) for these (i,j): {32, 5976}, {512, 2679}, {694, 141}, {733, 3934}, {805, 512}, {881, 115}, {882, 125}, {904, 17793}, {1581, 2887}, {1916, 626}, {1927, 37}, {1967, 10}, {7104, 17755}, {8789, 39}, {9468, 2}, {14251, 114}, {14604, 8265}, {14946, 6656}, {17938, 523}, {17970, 3}, {17980, 5}
    X(18904) = X(13576)-Ceva conjugate of X(4531)
    X(18904) = crosspoint of X(i) and X(j) for these (i,j): {2, 1581}, {238, 1921}
    X(18904) = crossdifference of every pair of points on line {983, 3733}
    X(18904) = crosssum of X(i) and X(j) for these (i,j): {6, 1580}, {291, 1922}
    X(18904) = X(i)-isoconjugate of X(j) for these (i,j): {741, 17743}, {813, 7255}, {1019, 8684}, {7033, 18268}
    X(18904) = X(3721)-Hirst inverse of X(3778)
    X(18904) = barycentric product X(i)*X(j) for these {i,j}: {238, 2887}, {239, 3721}, {350, 3778}, {740, 982}, {812, 7239}, {1284, 3705}, {1429, 4136}, {1921, 16584}, {2238, 3662}, {2275, 3948}, {3061, 16609}, {3573, 3801}, {3684, 16888}, {3794, 7235}, {3808, 3952}, {3865, 4039}, {3888, 4010}, {4433, 7185}, {4531, 18033}
    X(18904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 17493, 1966), (1921, 18277, 14603)


    X(18905) = MIDPOINT OF X(1965) AND X(17485)

    Barycentrics    a*(b + c)*(a^2 + b*c)*(b^2 - b*c + c^2) : :
    X(18905) = 3 X[2] + X[17485]

    X(18905) lies on the cubic K1036 and these lines: {2, 292}, {10, 16606}, {37, 226}, {171, 1691}, {722, 2887}, {893, 4645}, {982, 3094}, {1215, 16587}, {1575, 3687}, {1920, 14603}, {2229, 15523}, {2275, 3705}, {3121, 4972}, {3663, 17459}, {4847, 17448}, {7187, 9865}, {8620, 17184}

    X(18905) = midpoint of X(1965) and X(17485)
    X(18905) = complement X[1965]
    X(18905) = X(i)-complementary conjugate of X(j) for these (i,j): {695, 141}, {9229, 626}, {9236, 37}, {9285, 2887}, {9288, 10}, {14946, 325}
    X(18905) = crosspoint of X(i) and X(j) for these (i,j): {2, 9285}, {171, 1920}, {7184, 7187}
    X(18905) = crosssum of X(i) and X(j) for these (i,j): {6, 1582}, {256, 7104}
    X(18905) = X(1178)-isoconjugate of X(17743)
    X(18905) = barycentric product X(i)*X(j) for these {i,j}: {10, 7184}, {37, 7187}, {171, 2887}, {894, 3721}, {982, 1215}, {1237, 7032}, {1909, 3778}, {1920, 16584}, {2275, 3963}, {2295, 3662}, {2329, 16888}, {2533, 3888}, {3061, 4032}, {3794, 7211}, {3801, 4579}, {4136, 7175}, {4369, 7239}, {4531, 7205}, {7237, 17103}
    X(18905) = barycentric quotient X(i)/X(j) for these {i,j}: {1215, 7033}, {1237, 7034}, {2295, 17743}, {2887, 7018}, {3721, 257}, {3778, 256}, {3888, 4594}, {4367, 7255}, {7032, 1178}, {7184, 86}, {7187, 274}, {7188, 17103}, {8033, 7307}, {16584, 893}
    X(18905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 17485, 1965), (16587, 16592, 1215)


    X(18906) = REFLECTION OF X(69) IN X(76)

    Barycentrics    a^4*b^2 - a^2*b^4 + a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 : :
    X(18906) = 4 X[39] - 5 X[3618], 7 X[3619] - 8 X[3934], 3 X[1992] - 4 X[5052], 2 X[1569] - 3 X[5182], 2 X[1350] - 3 X[6194], 4 X[182] - 3 X[7709], 9 X[2] - 8 X[10007], 3 X[3094] - 4 X[10007], 2 X[3095] - 3 X[14853], 3 X[69] - 4 X[14994], 3 X[76] - 2 X[14994]

    X(18906) lies on the cubics K677, K708, K757, K1014, K1037 and these lines: {2, 694}, {4, 69}, {5, 6393}, {6, 194}, {9, 16822}, {39, 3618}, {51, 305}, {99, 182}, {141, 5025}, {183, 1350}, {184, 16276}, {193, 732}, {262, 1007}, {325, 5480}, {350, 1469}, {373, 11059}, {385, 5017}, {518, 17144}, {524, 11361}, {538, 1992}, {597, 14036}, {599, 14041}, {670, 10010}, {726, 3751}, {736, 7737}, {1078, 3098}, {1431, 1966}, {1569, 5182}, {1691, 3552}, {1692, 7816}, {1909, 3056}, {1965, 7033}, {2021, 13085}, {2076, 7793}, {2548, 8149}, {2782, 6776}, {3060, 8024}, {3095, 3926}, {3102, 11292}, {3103, 11291}, {3266, 5640}, {3329, 10335}, {3407, 7766}, {3564, 13108}, {3589, 7892}, {3619, 3934}, {3620, 14063}, {3629, 14034}, {3630, 14066}, {3631, 14062}, {3734, 5028}, {3763, 7901}, {3767, 18806}, {3785, 9821}, {4176, 7392}, {4563, 5651}, {5034, 7781}, {5039, 7760}, {5092, 7782}, {5103, 7912}, {5476, 7799}, {5490, 13877}, {5491, 13930}, {5847, 9902}, {6329, 14038}, {6376, 17792}, {6384, 17082}, {6390, 18583}, {6664, 16285}, {7615, 9466}, {7697, 10008}, {7757, 9741}, {7763, 14561}, {7771, 14810}, {7774, 9865}, {7786, 14069}, {8584, 14030}, {9464, 11002}, {10330, 11003}, {11257, 13354}, {13331, 14037}

    X(18906) = reflection of X(i) in X(j) for these {i,j}: {69, 76}, {193, 13330}, {194, 6}, {11257, 13354}
    X(18906) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3113, 69}, {3114, 6327}, {3407, 8}
    X(18906) = X(i)-Ceva conjugate of X(j) for these (i,j): {458, 1007}, {3114, 2}
    X(18906) = X(183)-Hirst inverse of X(5999)
    X(18906) = cevapoint of X(194) and X(6194)
    X(18906) = crossdifference of every pair of points on line {3049, 3221}
    X(18906) = anticomplement of X(3094)
    X(18906) = X(194)-of-1st-Brocard-triangle
    X(18906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 69, 5207), (6, 1975, 12215), (2076, 8177, 7793), (3212, 7155, 1966), (3981, 4074, 2)
    X(18906) = barycentric product X(i)*X(j) for these {i,j}: {76, 11328}, {3978, 6234}
    X(18906) = barycentric quotient X(i)/X(j) for these {i,j}: {6234, 694}, {11328, 6}


    X(18907) = X(5)X(32)∩X(6)X(30)

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

    See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 27722.

    X(18907) lies on these lines: {2,1285}, {3,7736}, {4,3172}, {5,32}, {6,30}, {11,609}, {12,7031}, {20,9605}, {39,550}, {51,15510}, {53,14581}, {69,11286}, {83,7750}, {98,14485}, {99,12156}, {112,251}, {115,3845}, {140,2548}, {141,754}, {172,496}, {183,3793}, {187,549}, {193,14033}, {235,10312}, {315,7819}, {316,7792}, {325,3972}, {376,5024}, {381,7735}, {382,5286}, {384,3933}, {385,8370}, {393,18494}, {428,5359}, {468,9745}, {495,1914}, {524,3734}, {538,3629}, {543,8584}, {546,3767}, {548,5013}, {574,8703}, {597,2030}, {598,3363}, {632,1506}, {952,1572}, {966,11354}, {1003,6390}, {1007,11288}, {1316,6792}, {1353,2782}, {1383,7426}, {1500,10386}, {1503,5039}, {1513,10788}, {1555,8779}, {1595,1968}, {1596,10311}, {1597,3087}, {1609,7514}, {1611,10128}, {1625,3051}, {1657,7738}, {1901,5037}, {1992,11159}, {1995,16317}, {2207,6756}, {2386,9969}, {2393,16983}, {2420,11007}, {2794,5480}, {2896,16988}, {3054,7603}, {3055,11539}, {3058,16785}, {3199,7715}, {3314,6661}, {3329,8356}, {3524,15655}, {3530,5023}, {3541,8778}, {3552,7921}, {3575,8743}, {3589,7761}, {3618,11287}, {3627,5007}, {3820,4386}, {3850,13881}, {3853,5319}, {3858,7755}, {3934,15598}, {5041,7756}, {5103,16385}, {5206,15712}, {5210,12100}, {5276,11113}, {5277,17527}, {5280,6284}, {5299,7354}, {5309,15687}, {5434,16784}, {5471,14137}, {5472,14136}, {5585,14891}, {6423,7584}, {6424,7583}, {6656,7787}, {6658,7839}, {6680,7843}, {6772,9112}, {6775,9113}, {6823,10316}, {7575,9699}, {7576,8744}, {7754,14035}, {7759,7789}, {7766,11361}, {7767,7770}, {7772,15704}, {7773,8361}, {7776,14001}, {7778,8368}, {7784,8364}, {7785,7807}, {7802,7878}, {7803,8357}, {7816,7838}, {7820,7845}, {7829,7842}, {7841,16989}, {7846,7860}, {7873,7889}, {7879,16898}, {7881,14037}, {7885,8363}, {7892,7900}, {7897,14036}, {7929,16895}, {8359,11174}, {8367,15271}, {8573,9818}, {8588,17504}, {8981,12963}, {9465,10301}, {9575,18481}, {9599,15325}, {9698,15513}, {9939,16986}, {10109,18584}, {10317,15760}, {10547,11380}, {11001,14482}, {11163,12040}, {11185,14614}, {11297,11488}, {11298,11489}, {11646,12212}, {11648,14075}, {11842,15980}, {12006,15575}, {12968,13966}, {13357,14881}, {13785,18539}, {15480,17131}, {15603,15700}, {15809,17409}, {16306,18572}

    X(18907) = midpoint of X(i) and X(j) for these {i,j}: {6,7737}, {1992,11159}
    X(18907) = reflection of X(i) in X(j) for these {i,j}: {5,10796}, {141,7804}, {7761,3589}, {14929,141}, {15048,6}
    X(18907) = X(6)-of-1st-orthosymmedial-triangle
    X(18907) = X(6)-of-pedal-triangle-of-X(6)
    X(18907) = pedal isogonal conjugate of X(6)


    X(18908) = X(3)X(3697)∩X(4)X(8)

    Barycentrics    a (-a^4 (b-c)^2+a^5 (b+c)-(b^2-c^2)^2 (b^2+4 b c+c^2)-2 a^3 (b^3+2 b^2 c+2 b c^2+c^3)+a (b-c)^2 (b^3+5 b^2 c+5 b c^2+c^3)+2 a^2 (b^4+b^3 c+4 b^2 c^2+b c^3+c^4)) : :
    X(18908) = (r - 2 R)*X(3) - (r + 4 R)(X(8)

    See Kadir Altintas and Angel Montesdeoca, ADGEOM 4650.

    X(18908) lies on these lines: {3,3697}, {4,8}, {5,3555}, {10,1071}, {12,354}, {40,4662}, {84,165}, {200,1012}, {210,515}, {392,952}, {405,5534}, {495,5728}, {518,5587}, {519,15064}, {912,3753}, {936,958}, {942,5261}, {944,5044}, {956,5720}, {960,5881}, {971,5657}, {997,18236}, {1385,5260}, {1532,4847}, {1698,12675}, {1737,17625}, {2836,13214}, {3036,12665}, {3086,17624}, {3090,5045}, {3295,9844}, {3617,12528}, {3621,13600}, {3678,14110}, {3679,6001}, {3698,5884}, {3870,6913}, {3889,5056}, {3892,10171}, {3916,11499}, {3921,10167}, {3935,6912}, {3956,10164}, {3983,6684}, {4015,4297}, {4533,18525}, {4882,12705}, {5049,11374}, {5173,10590}, {5251,5531}, {5290,5902}, {5302,10902}, {5439,9956}, {5603,10157}, {5687,7330}, {5693,5836}, {5726,18412}, {5770,17612}, {5780,10246}, {5791,10786}, {5886,11240}, {5904,7686}, {5919,10950}, {6927,12125}, {6965,18527}, {7580,18528}, {7989,13374}, {8164,11018}, {8168,12703}, {8728,10202}, {9004,12587}, {9708,18446}, {9780,9940}, {10039,12711}, {10176,18250}, {10573,12709}, {10588,16193}, {11362,12688}, {12599,12692}, {12629,12635}

    X(18908) = reflection of X(i) in X(j) for these {i,j}: {354,10175}, {3576,3740}, {3753,5790}, {3892,10171}, {5603,10157}, {10164,3956}





    leftri  anti-triangles: X(18909) - X(19212)  rightri

    This preamble and centers X(18909)-X(19212) were contributed by César Eliud Lozada, June 4, 2018.

    Let T=AtBtCt be a triangle perspective and orthologic to ABC. Suppose Pt is the perspector (ABC, T) and Ot is the orthologic center ABC to T, both expressed with respect to T. The anti-triangle-of-T is ABC and, if T is taken as the reference triangle, then its anti-triangle T'=A'tB'tC't is given by:

      A't = PtA ∩ (perpendicular to BC through Ot)

    and cyclically for B't and C't.

    Several triangles T satisfying the above conditions are given in the following table, together with the (ABC, T) perspector, the ABC-to-T orthologic center and its anti-triangle:

    Triangle T Perspector (ABC,T)
    (w/r to T)
    Orthologic center ABC to T
    (w/r to T)
    Anti-triangle
    A' barycentric coordinates
    Notes
    Atik X(69) X(18909) anti-Atik triange:
    A' = S^2*(4*R^2-SA)/SA^2 : SB : SC
    Only for ABC acute
    2nd circumperp-tangential X(56) X(7354) 2nd anti-circumperp-tangential triangle:
    A' = (b+c)^2/(-a+b+c) : b^2/(a-b+c) : c^2/(a+b-c)
    For any ABC
    inner Grebe X(6) X(1588) anti-inner-Grebe triangle:
    A' = a^2-S : b^2 : c^2
    For any ABC
    outer Grebe X(6) X(1587) anti-outer-Grebe triangle:
    A' = a^2+S : b^2 : c^2
    For any ABC
    Honsberger X(6) X(182) anti-Honsberger triangle:
    A' = -a^4/(b^2+c^2) : b^2 : c^2
    Only for ABC acute
    1st orthosymmedial X(1297) X(19158) 1st anti-orthosymmedial triangle:
    A' =
    (S^4+3*SA^2*S^2+2*(SA^2-SB*SC-SW^2)*SA^2)*a^2/(2*SA+SB+SC) :
    ((SA-SB)*S^2-2*(SA*SC-SB^2)*SA)*b^2 :
    ((SA-SC)*S^2-2*(SA*SB-SC^2)*SA)*c^2
    Only for ABC acute
    1st Sharygin X(8795) X(8884) 1st anti-Sharygin triangle:
    A' =
    -a^2*SB*SC/(S^2+SB*SC) :
    SC^2*(SA+SB)/(S^2+SA*SC) :
    SB^2*(SA+SC)/(S^2+SA*SB)
    Only for ABC acute
    3rd tri-squares X(1328) X(486) 3rd anti-tri-squares triangle:
    A' =
    2*S*(2*S^2-(18*R^2+SA-4*SW)*S-3*SB*SC)/((3*SA-S)*(SA-S)) :
    3*SC-S :
    3*SB-S
    Only for ABC acute
    4th tri-squares X(1327) X(485) 4th anti-tri-squares triangle:
    A' =
    -2*S*(2*S^2+(18*R^2+SA-4*SW)*S-3*SB*SC)/((3*SA+S)*(SA+S)) :
    3*SC+S :
    3*SB+S
    Only for ABC acute

    Centers X(18909)-X(19212) are perspectors, homothetic centers, orthologic centers and parallelogic centers of these anti-triangles and other triangles. For a complete list, see X(18909) Anti-triangles.pdf.

    underbar

    X(18909) = X(3)X(69) ∩ X(4)X(51)

    Barycentrics    (a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*(a^4-2*(b+c)^2*a^2+(b^2-c^2)^2)*(a^2-b^2-c^2) : :
    X(18909) = X(4)-3*X(18950)

    X(18909) lies on these lines: {1,18915}, {2,1181}, {3,69}, {4,51}, {5,18920}, {6,3088}, {15,18929}, {16,18930}, {20,6146}, {24,11206}, {30,18945}, {40,18921}, {64,12241}, {68,15740}, {125,3090}, {140,11487}, {155,3546}, {182,19119}, {184,631}, {217,7736}, {235,5656}, {287,14001}, {343,7400}, {371,18923}, {372,18924}, {376,1204}, {511,18935}, {569,5622}, {576,18919}, {578,14912}, {942,10360}, {974,3448}, {1056,1425}, {1058,3270}, {1154,18946}, {1217,9308}, {1352,6803}, {1368,12164}, {1370,5889}, {1498,3089}, {1503,7487}, {1593,11245}, {1595,11432}, {1596,12315}, {1614,3147}, {1853,12233}, {1885,12250}, {1907,9777}, {1986,13203}, {2777,18947}, {2883,6623}, {3043,13198}, {3146,18396}, {3167,16196}, {3269,7738}, {3521,15749}, {3524,13367}, {3538,3917}, {3541,7592}, {3542,11456}, {3547,12359}, {3548,18445}, {3618,7404}, {3818,9815}, {4846,9927}, {5072,18489}, {5562,7386}, {5663,18933}, {5739,6908}, {5907,6804}, {6102,14790}, {6353,6759}, {6467,15644}, {6643,13754}, {6696,8550}, {6815,10574}, {6816,12111}, {6825,14555}, {6891,18141}, {6995,16655}, {6997,15043}, {7401,9730}, {7494,10984}, {7689,18128}, {8884,19166}, {9825,18440}, {9833,11438}, {10116,12118}, {10619,12325}, {10625,15073}, {10938,13491}, {11424,13399}, {12085,13292}, {12162,18537}, {12319,18948}, {13093,13488}, {13630,18420}, {15134,18569}, {15811,15873}, {16621,17810}, {17702,18932}, {18296,18550}, {18926,18980}, {18927,18981}, {18937,18944}, {18938,18943}

    X(18909) = reflection of X(15811) in X(15873)
    X(18909) = anticomplement of X(17814)
    X(18909) = excentral-to-ABC functional image of X(4882)
    X(18909) = X(4882)-of-orthic-triangle if ABC is acute
    X(18909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 11411, 69), (3, 18913, 18931), (3, 18914, 6776), (3, 18917, 11411), (4, 1075, 6524), (4, 6241, 6225), (4, 14361, 1093), (4, 18916, 11433), (487, 488, 3964), (5878, 18390, 4), (6241, 18912, 4), (6776, 18913, 3), (11433, 12324, 4), (18913, 18914, 18925), (18915, 18922, 1), (18925, 18931, 3)


    X(18910) = PERSPECTOR OF THESE TRIANGLES: ANTI-ATIK AND ANTI-EULER

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

    X(18910) lies on these lines: {3,18934}, {4,14457}, {20,6146}, {24,159}, {125,11487}, {376,18936}, {631,5181}, {974,15740}, {1181,3089}, {1899,5562}, {3313,15073}, {3448,12825}, {7387,18914}, {8907,18925}, {12250,18560}, {12324,18396}, {12605,18917}, {16063,18946}, {16163,18932}

    X(18910) = reflection of X(4) in X(14457)


    X(18911) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 3rd ANTI-EULER

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

    X(18911) lies on these lines: {2,98}, {3,3580}, {4,4846}, {5,11456}, {6,858}, {20,11438}, {22,13567}, {51,7391}, {54,3548}, {66,3618}, {68,631}, {69,3266}, {70,13353}, {141,8546}, {193,11511}, {343,7485}, {373,3818}, {376,15360}, {427,5422}, {468,6800}, {511,16063}, {567,18281}, {568,14791}, {632,18356}, {850,879}, {852,18437}, {1353,1368}, {1370,3060}, {1503,1995}, {1648,9832}, {1853,5133}, {2088,2549}, {2888,10303}, {2979,6515}, {3091,5643}, {3398,14003}, {3541,13434}, {3564,15066}, {3620,5888}, {4197,5810}, {4550,16003}, {5050,5094}, {5085,7495}, {5169,14561}, {5189,11002}, {5297,12588}, {5449,7558}, {5654,15032}, {5889,6643}, {5890,18531}, {5892,18474}, {5943,7394}, {6030,10565}, {6146,17928}, {6804,15056}, {6816,12111}, {6997,11451}, {7292,12589}, {7378,12834}, {7401,15028}, {7493,15080}, {7509,12359}, {7528,15024}, {7529,16659}, {7544,18381}, {7592,11585}, {7728,12824}, {8550,11064}, {9707,16238}, {9786,12225}, {10113,12099}, {10539,18128}, {11284,18440}, {11411,11444}, {11412,18951}, {11413,12241}, {11422,14912}, {11439,12324}, {11440,18913}, {11441,18914}, {11443,18919}, {11445,18921}, {11446,18922}, {11447,18923}, {11448,18924}, {11449,18925}, {11452,18929}, {11453,18930}, {11454,18931}, {11459,18917}, {12270,18933}, {12271,18934}, {12272,18935}, {12273,18932}, {12274,18937}, {12275,18938}, {12276,18941}, {12277,18942}, {12278,18945}, {12280,18946}, {12828,16111}, {13015,18943}, {13016,18944}, {13201,18947}, {13630,18404}, {14788,15805}, {14805,15061}, {14853,15019}, {15045,18420}, {15053,18533}, {15305,18537}, {18124,19151}, {18392,18918}, {19119,19122}, {19166,19167}

    X(18911) = anticomplement of X(5651)
    X(18911) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1899, 11442), (2, 3448, 1352), (3, 18952, 18912), (125, 182, 2), (1352, 1899, 3448), (1352, 3448, 11442), (1368, 11245, 1993), (1370, 11433, 3060), (1853, 10601, 5133), (5050, 5094, 14389), (5169, 15018, 14561), (5449, 13336, 7558), (6515, 7386, 2979), (6643, 18916, 5889), (7386, 18950, 6515)


    X(18912) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 4th ANTI-EULER

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

    X(18912) lies on these lines: {2,54}, {3,3580}, {4,51}, {5,5422}, {6,70}, {20,17712}, {24,161}, {66,1173}, {69,7999}, {74,14457}, {110,10111}, {113,3091}, {125,578}, {182,7558}, {184,7505}, {193,8538}, {206,1614}, {235,11456}, {265,14708}, {317,19166}, {343,7509}, {378,6696}, {403,1181}, {468,9707}, {568,18569}, {575,1352}, {1092,10112}, {1192,10295}, {1199,7577}, {1204,13403}, {1503,10594}, {1598,16659}, {1656,11402}, {1853,10982}, {1993,11585}, {1995,12134}, {2072,12161}, {2548,16837}, {3087,4994}, {3089,14157}, {3147,11464}, {3410,5056}, {3518,9833}, {3527,5064}, {3541,15033}, {3549,5012}, {3855,18418}, {5094,11426}, {5198,16658}, {5446,7391}, {5462,7544}, {5640,7528}, {5810,6829}, {5889,18531}, {5925,10605}, {6102,18404}, {6240,9786}, {6243,14791}, {6247,16657}, {6515,6643}, {6642,14516}, {6800,13383}, {6815,15045}, {6816,11411}, {7401,15024}, {7503,12359}, {7507,11432}, {7547,12233}, {7552,11179}, {7731,18947}, {9730,9927}, {9815,16223}, {10116,10539}, {10255,15087}, {10601,14788}, {10619,11202}, {10625,16063}, {11458,18919}, {11460,18921}, {11461,18922}, {11462,18923}, {11463,18924}, {11465,18928}, {11466,18929}, {11467,18930}, {11468,18931}, {12099,15026}, {12111,18917}, {12164,16072}, {12227,15081}, {12278,15053}, {12281,18933}, {12282,18934}, {12283,18935}, {12284,18932}, {12285,18937}, {12286,18938}, {12287,18941}, {12288,18942}, {12289,18533}, {12291,18946}, {13017,18943}, {13018,18944}, {13160,14852}, {14542,16000}, {14627,15135}, {14855,18555}, {15032,16868}, {15043,18420}, {15058,18537}, {15121,18281}, {15873,16655}, {18948,19161}, {19119,19123}

    X(18912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18952, 18911), (4, 1899, 11457), (4, 11433, 3567), (4, 12324, 11455), (4, 18909, 6241), (4, 18916, 5890), (4, 18918, 18394), (4, 18950, 18916), (5, 11245, 7592), (51, 18381, 4), (185, 18390, 4), (569, 5449, 2), (6146, 13567, 24), (8538, 12585, 193), (10110, 11550, 4)


    X(18913) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND ANTI-HUTSON INTOUCH

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

    X(18913) lies on these lines: {1,10360}, {2,185}, {3,69}, {4,64}, {6,6696}, {20,1204}, {24,1619}, {25,12324}, {55,18915}, {56,18922}, {74,14457}, {125,146}, {155,2063}, {184,3523}, {193,13346}, {235,6225}, {343,10996}, {376,6146}, {378,18916}, {382,18918}, {389,3088}, {468,12174}, {578,5095}, {631,1181}, {974,7722}, {1151,18923}, {1152,18924}, {1192,1503}, {1350,18935}, {1498,6353}, {1593,11433}, {1596,13093}, {2883,6622}, {2935,18947}, {3089,6000}, {3147,11456}, {3269,5286}, {3270,14986}, {3346,9307}, {3515,11206}, {3516,11245}, {3529,18396}, {3541,5890}, {3542,5656}, {3546,13754}, {4846,5449}, {5085,19119}, {5584,18921}, {5878,6623}, {6392,9289}, {6467,13348}, {6515,11413}, {6643,12163}, {6823,15740}, {6995,13399}, {7386,11821}, {7404,9730}, {7487,11438}, {7714,16621}, {8889,12233}, {9545,13198}, {10606,12241}, {10620,18933}, {11425,14912}, {11440,18911}, {11457,18533}, {11477,18919}, {11479,18928}, {11480,18929}, {11481,18930}, {12084,18951}, {12164,16196}, {12301,18934}, {12302,18932}, {12303,18937}, {12304,18938}, {12305,18941}, {12306,18942}, {12307,18946}, {13021,18926}, {13022,18927}, {13367,15717}, {13851,17578}, {14561,15012}, {14826,17928}, {15073,15644}, {15128,16270}, {19166,19172}

    X(18913) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18909, 6776), (3, 18914, 18925), (3, 18917, 6193), (20, 1899, 18945), (64, 13567, 4), (389, 3088, 14853), (1204, 1899, 20), (1853, 13568, 4), (3542, 6241, 5656), (6247, 9786, 4), (11438, 14216, 7487), (18909, 18925, 18914), (18909, 18931, 3), (18914, 18925, 6776)


    X(18914) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND ANTI-INCIRCLE-CIRCLES

    Barycentrics    ((4*R^2+SW)*(SB+SC)-2*S^2)*SA : :
    X(18914) = X(4)-3*X(11245) = 3*X(51)-X(16655) = 3*X(389)-2*X(11745) = 3*X(389)-X(13419) = 3*X(428)-5*X(3567) = 3*X(428)-X(16659) = 3*X(568)-X(7553) = 2*X(1216)-3*X(10691) = X(1885)-3*X(12022) = 5*X(3567)-X(16659) = X(3575)-3*X(5890) = X(6241)+3*X(12022) = 3*X(6756)-4*X(11745) = 3*X(6756)-2*X(13419) = 3*X(7667)-X(11412)

    X(18914) lies on these lines: {3,69}, {4,3527}, {5,1181}, {6,1595}, {25,18916}, {30,52}, {49,10257}, {51,16655}, {68,6823}, {125,3628}, {140,184}, {155,1368}, {156,16238}, {182,13562}, {235,11456}, {287,7819}, {343,10984}, {382,18945}, {389,1503}, {427,7592}, {428,3567}, {468,1614}, {524,15644}, {542,9729}, {548,1204}, {550,10605}, {568,7553}, {578,6247}, {999,18915}, {1147,16196}, {1199,15559}, {1216,10691}, {1351,18935}, {1370,12160}, {1498,1596}, {1594,15032}, {1597,12324}, {1598,11433}, {1885,6241}, {2777,12024}, {2883,18390}, {3088,11426}, {3089,18950}, {3167,3546}, {3270,15172}, {3295,18922}, {3311,18923}, {3312,18924}, {3448,13160}, {3517,11206}, {3530,13367}, {3541,11402}, {3575,5890}, {3627,18396}, {3843,18918}, {3861,13851}, {4846,12293}, {5050,7404}, {5159,9820}, {5480,9968}, {5622,13353}, {5708,10360}, {5848,12675}, {5921,6803}, {5965,13348}, {6000,12241}, {6353,14530}, {6467,10625}, {6515,11414}, {6643,12164}, {6676,12359}, {6677,10539}, {6696,11430}, {6759,13567}, {7387,18910}, {7399,11442}, {7401,18440}, {7403,18583}, {7405,18358}, {7667,11412}, {9730,9825}, {9781,16658}, {9786,9833}, {9919,18947}, {10110,16621}, {10114,17855}, {10127,12006}, {10264,13198}, {10295,12254}, {10306,18921}, {10574,14516}, {10937,12278}, {11225,13598}, {11232,14641}, {11381,16657}, {11441,18911}, {11482,18919}, {11484,18928}, {11485,18929}, {11486,18930}, {11487,16419}, {11550,16198}, {11585,18445}, {12038,16976}, {12118,12421}, {12233,18381}, {12308,18933}, {12309,18934}, {12310,18932}, {12311,18937}, {12312,18938}, {12313,18941}, {12314,18942}, {12316,18946}, {12362,13754}, {13023,18943}, {13024,18944}, {13382,13568}, {13403,15311}, {14786,19125}, {19166,19173}

    X(18914) = midpoint of X(i) and X(j) for these {i,j}: {1885, 6241}, {10114, 17855}, {12118, 12421}
    X(18914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 14216, 1595), (343, 10984, 16197), (389, 13419, 11745), (1181, 1899, 5), (3088, 14912, 11426), (3567, 16659, 428), (6241, 12022, 1885), (6247, 8550, 578), (6776, 18909, 3), (7592, 11457, 427), (9730, 12134, 9825), (11456, 18912, 235), (11745, 13419, 6756), (18909, 18925, 18913), (18913, 18925, 3)


    X(18915) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND ANTI-TANGENTIAL-MIDARC

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

    X(18915) lies on these lines: {1,18909}, {4,65}, {33,12324}, {34,11433}, {35,18931}, {36,18925}, {55,18913}, {56,6776}, {69,73}, {125,10588}, {184,7288}, {185,497}, {221,13567}, {388,1425}, {966,1409}, {999,18914}, {1060,11411}, {1181,3086}, {1398,11245}, {1428,19119}, {1469,18935}, {1870,18916}, {2067,18923}, {3157,3546}, {3585,18918}, {4293,6146}, {4294,10605}, {4296,6515}, {6502,18924}, {6643,7352}, {7051,18929}, {7353,18942}, {7354,18945}, {7356,18946}, {7362,18941}, {18447,18917}, {19166,19175}

    X(18915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18909, 18922), (1425, 1899, 388)


    X(18916) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND CIRCUMORTHIC

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

    X(18916) lies on these lines: {2,155}, {3,6515}, {4,51}, {5,18917}, {6,3541}, {20,12022}, {24,159}, {25,18914}, {52,1370}, {54,69}, {64,16657}, {68,6815}, {74,18947}, {110,18932}, {140,11402}, {184,3147}, {186,18925}, {343,7383}, {378,18913}, {427,11432}, {568,14790}, {1181,3542}, {1199,11427}, {1353,16196}, {1368,12160}, {1595,9777}, {1596,12174}, {1614,6353}, {1870,18915}, {1906,12315}, {1907,3527}, {1993,3546}, {3089,11456}, {3090,18928}, {3292,3525}, {3448,7544}, {3518,11206}, {3520,18931}, {3538,11574}, {3547,3580}, {3548,12161}, {3855,12317}, {5071,11704}, {5422,7404}, {5462,6997}, {5739,6889}, {5889,6643}, {5894,10605}, {5946,7528}, {6102,18531}, {6146,9786}, {6197,18921}, {6198,18922}, {6239,18941}, {6240,18945}, {6242,18946}, {6247,10982}, {6400,18942}, {6403,18935}, {6640,15087}, {6803,15045}, {6804,11459}, {6816,13754}, {6853,14555}, {6897,9534}, {6995,16659}, {7386,11412}, {7392,15024}, {7401,11442}, {7505,15032}, {7699,18920}, {7722,18933}, {8537,18919}, {9815,16226}, {9935,12254}, {10632,18929}, {10633,18930}, {10880,18923}, {10881,18924}, {11225,13346}, {11431,14853}, {12111,18537}, {12509,18937}, {12510,18938}, {13035,18943}, {13036,18944}, {13568,18396}, {14787,15047}, {15135,18281}, {16655,17810}, {19119,19128}

    X(18916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18951, 6515), (4, 14361, 13450), (4, 18950, 18912), (51, 14216, 4), (68, 9730, 6815), (389, 1899, 4), (631, 14912, 54), (1181, 13567, 3542), (3567, 11457, 4), (5889, 18911, 6643), (5890, 18912, 4), (6102, 18952, 18531), (6146, 9786, 18533), (11433, 18909, 4), (11442, 15043, 7401)


    X(18917) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-SIDE

    Barycentrics    SA*(3*(SB+SC)*R^2-S^2) : :
    X(18917) = 4*X(140)-3*X(6090)

    X(18917) lies on these lines: {2,15032}, {3,69}, {4,94}, {5,18916}, {30,6515}, {49,631}, {52,14216}, {68,185}, {74,10111}, {125,5654}, {140,6090}, {155,3548}, {156,3147}, {381,11433}, {382,12324}, {389,3818}, {542,11438}, {567,14912}, {1092,9936}, {1147,6699}, {1154,1370}, {1181,3549}, {1204,12118}, {1352,9730}, {1593,13292}, {1899,13754}, {2070,11206}, {3167,10257}, {3357,10112}, {3526,11487}, {3528,12325}, {3541,12161}, {3580,11456}, {3589,14786}, {3618,14787}, {3629,6247}, {3851,18489}, {5055,18928}, {5656,11799}, {5876,6816}, {5889,11457}, {5890,11442}, {5900,16867}, {5946,6997}, {6146,12163}, {6353,10540}, {6643,18436}, {6815,13630}, {7403,11432}, {7592,14389}, {7689,10116}, {9786,12134}, {10264,15106}, {11427,15087}, {11459,18911}, {11472,16657}, {11550,14831}, {11579,13352}, {11585,12164}, {12111,18912}, {12301,12421}, {12429,18934}, {12605,18910}, {13567,18451}, {15077,18550}, {15108,15692}, {18403,18918}, {18435,18537}, {18437,18953}, {18438,18935}, {18447,18915}, {18449,18919}, {18453,18921}, {18455,18922}, {18457,18923}, {18459,18924}, {18462,18926}, {18463,18927}, {18468,18929}, {18470,18930}, {18563,18945}, {19119,19129}, {19166,19176}

    X(18917) = anticomplement of X(15068)
    X(18917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1181, 12359, 3549), (5876, 18952, 6816), (5889, 11457, 14790), (5890, 11442, 18420), (6193, 18913, 3), (11411, 18909, 3), (12317, 18932, 265), (14787, 15037, 3618)


    X(18918) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-VERTEX

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

    X(18918) lies on these lines: {2,18396}, {3,15749}, {4,51}, {5,8780}, {30,18931}, {68,15077}, {69,265}, {125,376}, {184,3545}, {287,16041}, {381,6776}, {382,18913}, {403,11206}, {542,18919}, {1181,3832}, {1503,6623}, {1853,15153}, {3091,6146}, {3147,12289}, {3153,6515}, {3521,18296}, {3543,10605}, {3546,12293}, {3583,18922}, {3585,18915}, {3818,18935}, {3843,18914}, {5067,13367}, {5447,6643}, {5656,10151}, {5891,15073}, {5946,10938}, {6288,18946}, {6353,18400}, {6564,18923}, {6565,18924}, {6622,9833}, {7378,16657}, {10201,11801}, {10519,18536}, {10602,11180}, {11245,18386}, {11411,18404}, {11427,12022}, {12024,17809}, {12118,14156}, {13363,18420}, {13567,18405}, {14449,18569}, {14826,16072}, {14912,18388}, {16808,18929}, {16809,18930}, {18377,18951}, {18379,18952}, {18380,18953}, {18392,18911}, {18403,18917}, {18406,18921}, {18414,18926}, {18415,18927}, {18474,18537}, {19119,19130}, {19166,19177}

    X(18918) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6761, 6524), (4, 11457, 6225), (5, 18945, 18925), (1899, 13851, 4), (18394, 18912, 4)


    X(18919) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd EHRMANN

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

    X(18919) lies on these lines: {2,8263}, {4,6}, {69,125}, {193,858}, {373,3618}, {376,5622}, {511,18931}, {542,18918}, {575,9815}, {576,18909}, {631,15073}, {1899,1992}, {2393,6353}, {6515,11416}, {6643,8548}, {7714,19136}, {8537,18916}, {8538,11411}, {8539,18921}, {8540,18922}, {8541,11433}, {9813,18928}, {9924,15448}, {9926,18934}, {9974,18941}, {9975,18942}, {9976,18933}, {9977,18946}, {11216,18950}, {11245,11405}, {11255,18951}, {11443,18911}, {11458,18912}, {11470,12324}, {11477,18913}, {11482,18914}, {11585,11898}, {12596,18932}, {12597,18937}, {12598,18938}, {13037,18943}, {13038,18944}, {13248,18947}, {13567,17813}, {18449,18917}, {19166,19178}

    X(18919) = {X(6), X(18935)}-harmonic conjugate of X(19119)


    X(18920) = PERSPECTOR OF THESE TRIANGLES: ANTI-ATIK AND EULER

    Barycentrics    (4*(SB+SC)*R^2*(4*SA-8*R^2+3*SW)-(SW+16*R^2)*S^2)*SA : :

    X(18920) lies on these lines: {5,18909}, {125,6622}, {1352,6804}, {1598,18931}, {3089,6247}, {3574,11433}, {5447,6643}, {7699,18916}, {10113,14790}


    X(18921) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EXTANGENTS

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

    X(18921) lies on these lines: {4,65}, {19,11433}, {40,18909}, {55,6776}, {63,69}, {1409,5712}, {1899,2550}, {3101,6515}, {3197,13567}, {3779,18935}, {5415,18923}, {5416,18924}, {5584,18913}, {6197,18916}, {6252,18941}, {6253,18945}, {6255,18946}, {6353,10536}, {6404,18942}, {7688,18931}, {7724,18933}, {8141,18951}, {8251,11411}, {8539,18919}, {9816,18928}, {10119,18947}, {10306,18914}, {10636,18929}, {10637,18930}, {10902,18925}, {11190,18950}, {11245,11406}, {11428,14912}, {11445,18911}, {11460,18912}, {11471,12324}, {12417,18934}, {12661,18932}, {12662,18937}, {12663,18938}, {13041,18943}, {13042,18944}, {18406,18918}, {18453,18917}, {19119,19133}, {19166,19181}

    X(18921) = {X(1899), X(3611)}-harmonic conjugate of X(2550)


    X(18922) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND INTANGENTS

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

    X(18922) lies on these lines: {1,18909}, {4,6285}, {33,11433}, {35,18925}, {36,18931}, {55,6776}, {56,18913}, {69,1040}, {125,10589}, {184,5218}, {185,388}, {354,10360}, {497,1899}, {1062,11411}, {1069,3546}, {1181,3085}, {1250,18930}, {2066,18923}, {2192,13567}, {2330,19119}, {3056,18935}, {3100,6515}, {3147,9638}, {3295,18914}, {3583,18918}, {4293,10605}, {4294,6146}, {5414,18924}, {6198,18916}, {6238,6643}, {6283,18941}, {6284,18945}, {6286,18946}, {6353,10535}, {6405,18942}, {7071,11245}, {7727,18933}, {8144,18951}, {8540,18919}, {9817,18928}, {9931,18934}, {10118,18947}, {10638,18929}, {11189,18950}, {11429,14912}, {11446,18911}, {11461,18912}, {12888,18932}, {12910,18937}, {12911,18938}, {13043,18943}, {13044,18944}, {18455,18917}, {19166,19182}

    X(18922) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 18909, 18915), (1899, 3270, 497)


    X(18923) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 1st KENMOTU DIAGONALS

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

    X(18923) lies on these lines: {4,6}, {69,1589}, {184,3069}, {185,6459}, {216,12256}, {371,18909}, {372,18925}, {577,12257}, {1151,18913}, {1899,3068}, {2066,18922}, {2067,18915}, {3156,18997}, {3311,18914}, {3546,8909}, {5410,11245}, {5412,11433}, {5413,11206}, {5415,18921}, {6353,10533}, {6458,11514}, {6515,11417}, {6564,18918}, {6643,10665}, {9541,10605}, {10116,19062}, {10880,18916}, {10897,11411}, {10961,18928}, {11241,18950}, {11265,18951}, {11447,18911}, {11462,18912}, {11473,12324}, {12375,18933}, {12424,18934}, {12891,18932}, {12960,18937}, {12961,18938}, {12962,18941}, {12963,18942}, {12965,18946}, {13045,18943}, {13046,18944}, {13287,18947}, {13567,17819}, {18457,18917}, {19166,19183}

    X(18923) = {X(6), X(6776)}-harmonic conjugate of X(18924)


    X(18924) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd KENMOTU DIAGONALS

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

    X(18924) lies on these lines: {4,6}, {69,1590}, {184,3068}, {185,6460}, {216,12257}, {371,18925}, {372,18909}, {577,12256}, {1152,18913}, {1589,8911}, {1899,3069}, {3155,18998}, {3312,18914}, {5411,11245}, {5412,11206}, {5413,11433}, {5414,18922}, {5416,18921}, {6353,10534}, {6396,18931}, {6457,11513}, {6502,18915}, {6515,11418}, {6565,18918}, {6643,10666}, {10116,19061}, {10881,18916}, {10898,11411}, {10963,18928}, {11242,18950}, {11266,18951}, {11448,18911}, {11463,18912}, {11474,12324}, {12376,18933}, {12425,18934}, {12892,18932}, {12966,18937}, {12967,18938}, {12968,18941}, {12969,18942}, {12971,18946}, {13047,18943}, {13048,18944}, {13288,18947}, {13567,17820}, {18459,18917}, {19166,19184}

    X(18924) = {X(6), X(6776)}-harmonic conjugate of X(18923)


    X(18925) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND KOSNITA

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

    X(18925) lies on these lines: {2,6146}, {3,69}, {4,54}, {5,8780}, {6,7487}, {15,18930}, {16,18929}, {20,1181}, {24,11433}, {35,18922}, {36,18915}, {49,18531}, {52,1992}, {68,18475}, {70,3431}, {110,6816}, {125,3525}, {154,3089}, {182,6803}, {185,376}, {186,18916}, {193,17834}, {287,16043}, {371,18924}, {372,18923}, {378,12324}, {389,6403}, {511,19119}, {567,7528}, {569,3618}, {575,9815}, {631,1899}, {1092,7386}, {1093,6618}, {1147,6643}, {1204,3528}, {1503,3088}, {1596,14530}, {1658,18951}, {1853,14528}, {1885,5656}, {2055,14575}, {3091,14389}, {3135,16035}, {3147,11464}, {3167,12362}, {3515,11245}, {3522,10605}, {3527,7715}, {3542,9707}, {3575,11402}, {3796,7400}, {3855,13851}, {4549,15083}, {5012,6815}, {5050,9825}, {5622,13336}, {5876,10938}, {6225,11456}, {6353,10282}, {6515,7488}, {6622,18390}, {6623,16252}, {6642,18928}, {6676,12429}, {6756,11426}, {6804,9306}, {6995,10982}, {6997,13434}, {7395,14826}, {7404,12134}, {7514,11487}, {7592,18533}, {7714,10110}, {7735,14585}, {8550,9786}, {8797,19176}, {8889,18381}, {8907,18910}, {9143,15738}, {9704,18404}, {9729,11179}, {9909,13142}, {9932,18934}, {9969,11387}, {10539,18537}, {10602,11432}, {10902,18921}, {10984,10996}, {11202,18950}, {11430,14216}, {11449,18911}, {11577,12226}, {12118,12318}, {12174,12250}, {12233,17809}, {12363,18436}, {12383,13198}, {12893,18932}, {12972,18937}, {12973,18938}, {12974,18941}, {12975,18942}, {13049,18943}, {13050,18944}, {13203,15463}, {13289,18947}, {13292,14070}, {13352,14927}, {13567,17821}, {14852,15077}, {15316,15740}, {19166,19185}

    X(18925) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6193, 69), (3, 6776, 18909), (3, 18909, 18931), (3, 18914, 18913), (4, 54, 11427), (5, 18945, 18918), (154, 12241, 3089), (569, 7401, 3618), (578, 9833, 4), (1899, 13367, 631), (6756, 11426, 14853), (6776, 18913, 18914), (9707, 12022, 3542), (11464, 18912, 3147), (18913, 18914, 18909)


    X(18926) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND LUCAS ANTIPODAL TANGENTIAL

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

    X(18926) lies on these lines: {3,18938}, {3070,11846}, {6643,18939}, {6776,8939}, {9723,18927}, {12257,13889}, {12590,18935}, {13021,18913}, {18414,18918}, {18462,18917}, {18909,18980}, {19119,19134}, {19166,19186}


    X(18927) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND LUCAS(-1) ANTIPODAL TANGENTIAL

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

    X(18927) lies on these lines: {3,18937}, {3071,11847}, {6643,18940}, {6776,8943}, {9723,18926}, {12256,13943}, {12591,18935}, {13022,18913}, {18415,18918}, {18463,18917}, {18909,18981}, {19119,19135}, {19166,19187}


    X(18928) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND SUBMEDIAL

    Barycentrics    a^6-3*(b^2+c^2)*a^4+(3*b^4-14*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :

    X(18928) lies on these lines: {2,6}, {4,5943}, {5,18909}, {20,17810}, {51,7386}, {83,459}, {182,6353}, {264,14361}, {329,6604}, {373,1899}, {389,6804}, {631,13346}, {637,3539}, {638,3540}, {641,18938}, {642,18937}, {1352,6688}, {1368,14853}, {1370,5640}, {1503,7398}, {1656,11411}, {1660,5012}, {1853,2883}, {1995,11206}, {2052,6819}, {3066,6995}, {3090,18916}, {3547,15805}, {3628,11487}, {3796,4232}, {3981,7738}, {4176,11059}, {5014,5554}, {5020,6776}, {5050,6677}, {5055,18917}, {5085,10565}, {5462,6643}, {5480,7396}, {5644,18583}, {6225,10574}, {6642,18925}, {6723,18947}, {6803,11695}, {6805,12322}, {6806,12323}, {6815,15028}, {6816,15043}, {6997,11451}, {7401,18474}, {8889,14561}, {9306,14912}, {9730,18537}, {9813,18919}, {9816,18921}, {9817,18922}, {9820,18934}, {9822,18935}, {9823,18941}, {9824,18942}, {9825,18945}, {10128,18440}, {10154,12017}, {10519,16419}, {10643,18929}, {10644,18930}, {10961,18923}, {10963,18924}, {11225,16187}, {11245,11284}, {11431,12160}, {11465,18912}, {11479,18913}, {11484,18914}, {12900,18932}, {13053,18943}, {13054,18944}, {13363,18420}, {14557,17170}, {14790,15026}, {19119,19137}, {19166,19188}

    X(18928) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 193, 17811), (2, 343, 3619), (2, 5422, 11427), (2, 10601, 3618), (2, 11433, 69), (4, 9729, 15740), (373, 1899, 7392), (3628, 18951, 11487), (11245, 11284, 14826), (11451, 18911, 6997), (13567, 17825, 2), (14555, 18141, 4417)


    X(18929) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND INNER TRI-EQUILATERAL

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

    X(18929) lies on these lines: {4,6}, {15,18909}, {16,18925}, {69,11515}, {184,11489}, {1899,11488}, {6515,11420}, {6643,10661}, {7051,18915}, {10632,18916}, {10634,11411}, {10636,18921}, {10638,18922}, {10641,11433}, {10642,11206}, {10643,18928}, {10645,18931}, {10657,18933}, {10659,18934}, {10663,18932}, {10667,18941}, {10671,18942}, {10677,18946}, {10681,18947}, {11243,18950}, {11245,11408}, {11267,18951}, {11452,18911}, {11466,18912}, {11475,12324}, {11480,18913}, {11485,18914}, {12980,18937}, {12982,18938}, {13057,18943}, {13058,18944}, {13567,17826}, {16808,18918}, {18468,18917}, {19166,19190}

    X(18929) = {X(6), X(6776)}-harmonic conjugate of X(18930)


    X(18930) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND OUTER TRI-EQUILATERAL

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

    X(18930) lies on these lines: {4,6}, {15,18925}, {16,18909}, {69,11516}, {184,11488}, {1250,18922}, {1899,11489}, {6515,11421}, {6643,10662}, {10633,18916}, {10635,11411}, {10637,18921}, {10641,11206}, {10642,11433}, {10644,18928}, {10646,18931}, {10658,18933}, {10660,18934}, {10664,18932}, {10668,18941}, {10672,18942}, {10678,18946}, {10682,18947}, {11244,18950}, {11245,11409}, {11268,18951}, {11453,18911}, {11467,18912}, {11476,12324}, {11481,18913}, {11486,18914}, {12981,18937}, {12983,18938}, {13059,18943}, {13060,18944}, {13567,17827}, {16809,18918}, {18470,18917}, {19166,19191}

    X(18930) = {X(6), X(6776)}-harmonic conjugate of X(18929)


    X(18931) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND TRINH

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

    X(18931) lies on these lines: {2,10605}, {3,69}, {4,74}, {20,3580}, {24,12324}, {30,18918}, {35,18915}, {36,18922}, {64,3089}, {184,3524}, {185,631}, {186,11206}, {235,12250}, {376,1899}, {378,11433}, {468,5656}, {511,18919}, {550,18945}, {568,16270}, {974,12219}, {1181,3523}, {1192,6247}, {1498,15448}, {1598,18920}, {1992,5622}, {2071,6515}, {3088,5480}, {3098,18935}, {3147,6241}, {3167,16976}, {3269,7735}, {3520,18916}, {3522,6146}, {3542,6225}, {3546,12163}, {3547,15740}, {3618,9730}, {4549,6643}, {5092,19119}, {5654,6699}, {5878,6622}, {5890,11427}, {6000,6353}, {6396,18924}, {6623,15311}, {6816,11440}, {6997,15053}, {7493,15072}, {7688,18921}, {7690,18941}, {7691,18946}, {7692,18942}, {7729,15151}, {8567,12241}, {9938,18934}, {10299,13367}, {10564,11008}, {10606,13567}, {10645,18929}, {10646,18930}, {11204,18950}, {11245,11410}, {11250,18951}, {11425,12007}, {11430,14912}, {11454,18911}, {11468,18912}, {12293,15077}, {12901,18932}, {12984,18937}, {12985,18938}, {13061,18943}, {13062,18944}, {13293,18947}, {13851,15682}, {19166,19192}

    X(18931) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18909, 18925), (3, 18913, 18909), (1192, 6247, 7487), (6696, 9786, 3088)


    X(18932) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO AAOA

    Barycentrics    SA*(3*(SB+SC)*R^2*(6*R^2-SB-SC)-(13*R^2-2*SW)*S^2) : :
    X(18932) = X(12317)+3*X(18947) = X(12317)+6*X(18951)

    The reciprocal orthologic center of these triangles is X(15136)

    X(18932) lies on these lines: {4,94}, {52,13203}, {68,11806}, {69,5504}, {74,6515}, {110,18916}, {113,11433}, {125,11411}, {193,10264}, {1899,12319}, {2931,6776}, {3047,3147}, {5900,15317}, {6723,11487}, {10111,12383}, {10663,18929}, {10664,18930}, {11245,12168}, {11800,14216}, {12228,14912}, {12273,18911}, {12284,18912}, {12295,12324}, {12302,18913}, {12310,18914}, {12325,15089}, {12596,18919}, {12661,18921}, {12825,18537}, {12888,18922}, {12891,18923}, {12892,18924}, {12893,18925}, {12900,18928}, {12901,18931}, {13567,17838}, {13754,18933}, {14984,18935}, {16163,18910}, {17702,18909}, {19119,19138}, {19166,19193}

    X(18932) = orthologic center of the anti-Atik triangle to these triangles: AOA, 1st Hyacinth
    X(18932) = {X(265), X(18917)}-harmonic conjugate of X(12317)


    X(18933) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO ANTI-ORTHOCENTROIDAL

    Barycentrics    (2*(SB+SC)*(6*R^2-SW)*(3*R^2-SA-SW)+(13*R^2-2*SW)*S^2)*SA : :

    The reciprocal orthologic center of these triangles is X(3581)

    X(18933) lies on these lines: {4,74}, {5,12412}, {69,265}, {110,6816}, {113,3618}, {146,974}, {235,13171}, {399,6776}, {542,18935}, {1370,10733}, {1596,9919}, {1899,12317}, {1986,11433}, {3089,10117}, {3153,3580}, {3448,12825}, {3546,12302}, {5621,6623}, {5663,18909}, {5878,17855}, {5972,6804}, {6146,14683}, {6225,17854}, {6353,13289}, {6515,12219}, {6643,17702}, {6723,6803}, {6815,15059}, {7386,16163}, {7722,18916}, {7723,11411}, {7724,18921}, {7727,18922}, {7728,16270}, {9976,18919}, {10113,14790}, {10620,18913}, {10628,18947}, {10657,18929}, {10658,18930}, {11245,12165}, {11744,15151}, {11801,18569}, {12227,14912}, {12241,17847}, {12270,18911}, {12281,18912}, {12292,12324}, {12308,18914}, {12310,12362}, {12375,18923}, {12376,18924}, {13567,17835}, {13754,18932}, {15066,18396}, {18945,18946}, {19119,19140}, {19166,19195}

    X(18933) = orthologic center of these triangles: anti-Atik to orthocentroidal
    X(18933) = {X(265), X(18531)}-harmonic conjugate of X(12319)


    X(18934) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO ARIES

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

    The reciprocal orthologic center of these triangles is X(7387)

    X(18934) lies on these lines: {3,18910}, {4,52}, {69,3546}, {155,11433}, {1147,14912}, {1899,12318}, {3564,6642}, {5449,11487}, {6776,9937}, {9820,18928}, {9926,18919}, {9931,18922}, {9932,18925}, {9938,18931}, {10659,18929}, {10660,18930}, {11245,12166}, {12118,18128}, {12271,18911}, {12282,18912}, {12293,12324}, {12301,18913}, {12309,18914}, {12417,18921}, {12424,18923}, {12425,18924}, {12429,18917}, {13567,17836}, {19119,19141}, {19166,19196}

    X(18934) = orthologic center of these triangles: anti-Atik to 2nd Hyacinth
    X(18934) = {X(12359), X(15316)}-harmonic conjugate of X(3546)


    X(18935) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO 1st EHRMANN

    Barycentrics    (-a^2+b^2+c^2)*(a^6+3*(b^2+c^2)*a^4+3*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)) : :
    X(18935) = 5*X(3618)-4*X(19137)

    The reciprocal orthologic center of these triangles is X(576)

    X(18935) lies on these lines: {4,6}, {66,17040}, {69,305}, {125,3619}, {159,6353}, {182,6803}, {184,3618}, {193,1370}, {511,18909}, {542,18933}, {800,8721}, {1350,18913}, {1351,18914}, {1352,6804}, {1353,14790}, {1469,18915}, {1843,11433}, {1974,11206}, {2393,18950}, {2871,18953}, {3056,18922}, {3098,18931}, {3313,15073}, {3538,10519}, {3564,6643}, {3589,15435}, {3779,18921}, {3818,18918}, {5050,7401}, {5095,13203}, {5157,5622}, {5486,16774}, {5839,11677}, {5921,6816}, {5965,18946}, {6403,18916}, {6515,12220}, {6997,19125}, {9822,18928}, {9924,13567}, {9967,11411}, {10996,19126}, {11245,12167}, {11513,12257}, {11514,12256}, {12272,18911}, {12283,18912}, {12294,12324}, {12590,18926}, {12591,18927}, {14984,18932}, {18438,18917}, {18440,18537}, {19166,19197}

    X(18935) = reflection of X(69) in X(15812)
    X(18935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3867, 14853), (6, 6776, 19119), (1181, 18396, 16654), (1899, 6467, 69), (6776, 14853, 1181), (18919, 19119, 6)


    X(18936) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO HATZIPOLAKIS-MOSES

    Barycentrics    (R^2*(SB+SC)*(16*R^2-3*SW)*(8*R^2-SB-SC)-(76*R^4-25*R^2*SW+2*SW^2)*S^2)*SA : :

    The reciprocal orthologic center of these triangles is X(9729)

    X(18936) lies on these lines: {376,18910}, {2929,6776}, {13567,17837}, {19119,19142}, {19166,19198}


    X(18937) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS ANTIPODAL

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18937) lies on these lines: {3,18927}, {4,12237}, {69,486}, {487,1584}, {642,18928}, {1598,3564}, {1899,12320}, {6515,12221}, {6776,12978}, {11245,12169}, {11411,12601}, {12229,14912}, {12274,18911}, {12285,18912}, {12296,12324}, {12303,18913}, {12311,18914}, {12509,18916}, {12597,18919}, {12662,18921}, {12910,18922}, {12960,18923}, {12966,18924}, {12972,18925}, {12980,18929}, {12981,18930}, {12984,18931}, {13567,17839}, {18909,18944}, {19119,19143}, {19166,19199}


    X(18938) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS(-1) ANTIPODAL

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18938) lies on these lines: {3,18926}, {4,12238}, {69,485}, {488,1583}, {641,18928}, {1598,3564}, {1899,12321}, {6515,12222}, {6776,12979}, {11245,12170}, {11411,12602}, {12230,14912}, {12275,18911}, {12286,18912}, {12297,12324}, {12304,18913}, {12312,18914}, {12510,18916}, {12598,18919}, {12663,18921}, {12911,18922}, {12961,18923}, {12967,18924}, {12973,18925}, {12982,18929}, {12983,18930}, {12985,18931}, {13567,17842}, {18909,18943}, {19119,19144}, {19166,19200}


    X(18939) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO ANTI-ATIK

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

    The reciprocal orthologic center of these triangles is X(6643)

    X(18939) lies on these lines: {155,371}, {3564,12426}, {5408,11198}, {6643,18926}, {9723,18940}, {9927,18414}, {12163,13021}, {13754,18980}, {18436,18462}, {19134,19139}, {19186,19194}


    X(18940) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO ANTI-ATIK

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

    The reciprocal orthologic center of these triangles is X(6643)

    X(18940) lies on these lines: {155,372}, {3564,12427}, {6643,18927}, {9723,18939}, {9927,18415}, {12163,13022}, {13754,18981}, {18436,18463}, {19135,19139}, {19187,19194}


    X(18941) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS CENTRAL

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18941) lies on these lines: {4,12239}, {69,12360}, {511,18909}, {1151,6776}, {1899,12322}, {6239,18916}, {6252,18921}, {6283,18922}, {6291,11433}, {6515,12223}, {7362,18915}, {7690,18931}, {9823,18928}, {9974,18919}, {10667,18929}, {10668,18930}, {11245,12171}, {11411,12603}, {12231,14912}, {12276,18911}, {12287,18912}, {12298,12324}, {12305,18913}, {12313,18914}, {12962,18923}, {12968,18924}, {12974,18925}, {13567,17840}, {19119,19145}, {19166,19201}

    X(18941) = {X(18909), X(18935)}-harmonic conjugate of X(18942)


    X(18942) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS(-1) CENTRAL

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18942) lies on these lines: {4,12240}, {69,12361}, {511,18909}, {1152,6776}, {1899,12323}, {6400,18916}, {6404,18921}, {6405,18922}, {6406,11433}, {6515,12224}, {7353,18915}, {7692,18931}, {9824,18928}, {9975,18919}, {10671,18929}, {10672,18930}, {11245,12172}, {11411,12604}, {12232,14912}, {12277,18911}, {12288,18912}, {12299,12324}, {12306,18913}, {12314,18914}, {12963,18923}, {12969,18924}, {12975,18925}, {13567,17843}, {19119,19146}, {19166,19202}


    X(18943) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS REFLECTION

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

    The reciprocal orthologic center of these triangles is X(10670)

    X(18943) lies on these lines: {4,13013}, {69,13027}, {1899,13025}, {6515,13009}, {6776,13055}, {11245,13007}, {11411,13039}, {11433,13051}, {12324,13019}, {13011,14912}, {13015,18911}, {13017,18912}, {13021,18913}, {13023,18914}, {13035,18916}, {13037,18919}, {13041,18921}, {13043,18922}, {13045,18923}, {13047,18924}, {13049,18925}, {13053,18928}, {13057,18929}, {13059,18930}, {13061,18931}, {13567,17841}, {18909,18938}, {19119,19147}, {19166,19203}


    X(18944) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO LUCAS(-1) REFLECTION

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

    The reciprocal orthologic center of these triangles is X(10674)

    X(18944) lies on these lines: {4,13014}, {69,13028}, {1899,13026}, {6515,13010}, {6776,13056}, {11245,13008}, {11411,13040}, {11433,13052}, {12324,13020}, {13012,14912}, {13016,18911}, {13018,18912}, {13022,18913}, {13024,18914}, {13036,18916}, {13038,18919}, {13042,18921}, {13044,18922}, {13046,18923}, {13048,18924}, {13050,18925}, {13054,18928}, {13058,18929}, {13060,18930}, {13062,18931}, {13567,17844}, {18909,18937}, {19119,19148}, {19166,19204}


    X(18945) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO MACBEATH

    Barycentrics    (-a^2+b^2+c^2)*(5*a^8-6*(b^2+c^2)*a^6-2*(b^4-c^4)*(b^2-c^2)*a^2+3*(b^2-c^2)^4) : :
    X(18945) = 3*X(4)-2*X(15811) = 3*X(7714)-4*X(15873) = 2*X(9786)-3*X(18950) = 3*X(10519)-4*X(15812)

    The reciprocal orthologic center of these triangles is X(4)

    X(18945) lies on these lines: {2,11449}, {3,15077}, {4,6}, {5,8780}, {20,1204}, {30,18909}, {69,11821}, {125,3523}, {154,6622}, {184,3091}, {185,3060}, {193,10112}, {235,11206}, {265,3549}, {382,18914}, {550,18931}, {1885,12324}, {3088,18381}, {3089,9833}, {3525,15748}, {3529,10605}, {3541,15126}, {3546,12118}, {3547,9927}, {3575,11433}, {3832,13851}, {4292,10360}, {4846,18128}, {5056,10619}, {5562,8681}, {5622,10984}, {5907,5921}, {6102,10938}, {6240,18916}, {6253,18921}, {6284,18922}, {6288,14786}, {6515,12225}, {6623,6759}, {6816,14516}, {7354,18915}, {7378,11424}, {7396,13346}, {7404,18474}, {7487,18400}, {7505,12254}, {7507,11427}, {7714,15873}, {8889,11425}, {9786,18950}, {9825,18928}, {10519,15812}, {10602,12164}, {11245,12173}, {11411,12605}, {12134,18537}, {12250,18560}, {12278,18911}, {12289,18533}, {12370,14790}, {13403,14216}, {13567,17845}, {18563,18917}, {18933,18946}, {19166,19205}

    X(18945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 6146, 6776), (4, 14912, 12233), (20, 1899, 18913), (69, 12362, 11821), (6146, 18396, 4), (6816, 14516, 14826), (9833, 18390, 3089), (12024, 18405, 14912), (12233, 18405, 4), (12289, 18912, 18533), (12362, 12429, 69), (18918, 18925, 5)


    X(18946) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO REFLECTION

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

    The reciprocal orthologic center of these triangles is X(6243)

    X(18946) lies on these lines: {4,54}, {69,3519}, {70,18368}, {195,6776}, {323,6146}, {539,6643}, {1154,18909}, {1209,3619}, {1370,15801}, {1493,14790}, {1899,12325}, {2888,11577}, {2914,13203}, {5965,18935}, {6152,11433}, {6242,18916}, {6255,18921}, {6286,18922}, {6288,18918}, {6403,13433}, {6515,12226}, {7356,18915}, {7691,18931}, {9977,18919}, {10677,18929}, {10678,18930}, {11245,12175}, {11411,12606}, {12234,14912}, {12280,18911}, {12291,18912}, {12300,12324}, {12307,18913}, {12316,18914}, {12319,15089}, {12965,18923}, {12971,18924}, {13567,17846}, {16063,18910}, {18933,18945}, {19119,19150}, {19166,19207}


    X(18947) = PARALLELOGIC CENTER OF THESE TRIANGLES: ANTI-ATIK TO AAOA

    Barycentrics    SB*SC*(2*SA^2-6*R^2*SA+R^2*SW) : :
    X(18947) = X(12317)-4*X(18932) = X(12317)-8*X(18951)

    The reciprocal parallelogic center of these triangles is X(15139)

    X(18947) lies on these lines: {4,94}, {25,14683}, {52,12319}, {68,11557}, {69,5095}, {74,18916}, {110,6353}, {113,11411}, {125,8889}, {235,12165}, {399,3089}, {542,7714}, {631,15463}, {1177,19119}, {1596,12308}, {1899,13203}, {2777,18909}, {2781,18950}, {2904,14940}, {2914,7505}, {2935,18913}, {3043,3147}, {3088,10264}, {6143,15018}, {6723,18928}, {6776,10117}, {6804,12358}, {6816,12219}, {7401,16222}, {7723,18537}, {7731,18912}, {9833,10114}, {9919,18914}, {10118,18922}, {10119,18921}, {10628,18933}, {10681,18929}, {10682,18930}, {11245,13171}, {11487,12900}, {11807,14216}, {12241,17835}, {12324,13202}, {12412,13292}, {13198,14912}, {13201,18911}, {13248,18919}, {13287,18923}, {13288,18924}, {13289,18925}, {13293,18931}, {13567,17847}, {19166,19208}

    X(18947) = parallelogic center of these triangles: anti-Atik to AOA
    X(18947) = parallelogic center of these triangles: anti-Atik to 1st Hyacinth
    X(18947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1112, 3448, 4), (1899, 13417, 13203)


    X(18948) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: ANTI-ATIK AND AAOA

    Barycentrics    SA*((3*R^2-SW)*(5*R^2-SW)*S^2+(SB+SC)*(16*R^6-3*R^4*(4*SA+9*SW)+R^2*SW*(10*SW+7*SA)-SW^2*(SW+SA))) : :

    X(18948) lies on these lines: {10619,16063}, {12319,18909}, {18912,19161}


    X(18949) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: ANTI-ATIK AND 1st HYACINTH

    Barycentrics    (160*R^4+2*R^2*(8*SA-39*SW)+SA*SW+6*SW^2)*R^2*S^2-2*(SW^3-(R^2-SW)*(16*R^2-11*SW)*R^2)*SB*SC : :

    X(18949) lies on the line {1147,6804}


    X(18950) = X(2) OF ANTI-ATIK TRIANGLE

    Barycentrics    3*a^6-5*(b^2+c^2)*a^4+5*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2) : :
    X(18950) = X(4)+2*X(18909) = 7*X(3090)-4*X(17814) = X(6643)+2*X(18951) = X(6643)-4*X(18952) = 2*X(9786)+X(18945) = X(18951)+2*X(18952)

    X(18950) lies on these lines: {2,3167}, {4,51}, {5,5644}, {6,8889}, {54,3525}, {68,5892}, {69,3819}, {125,11427}, {154,6353}, {193,1368}, {343,631}, {376,12022}, {436,459}, {1154,6643}, {1181,6622}, {1351,7396}, {1352,6688}, {1503,7714}, {1853,14853}, {1992,11225}, {1993,16051}, {2393,18935}, {2781,18947}, {2979,6515}, {3089,18914}, {3090,7592}, {3448,6997}, {3546,13292}, {3580,7494}, {3620,16419}, {5020,5921}, {5891,6804}, {7378,9777}, {7392,11442}, {7398,18440}, {9786,18945}, {10606,12241}, {11189,18922}, {11190,18921}, {11202,18925}, {11204,18931}, {11216,18919}, {11241,18923}, {11242,18924}, {11243,18929}, {11244,18930}, {15012,15077}, {18435,18537}, {19119,19153}, {19166,19209}

    X(18950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11245, 14912), (1899, 11433, 4), (6515, 18911, 7386), (6776, 13567, 6353), (18912, 18916, 4), (18951, 18952, 6643)


    X(18951) = X(3) OF ANTI-ATIK TRIANGLE

    Barycentrics    (3*R^2-SA)*S^2-R^2*SB*SC : :
    X(18951) = X(6643)-3*X(18950) = X(12317)-7*X(18932) = X(12317)+7*X(18947) = 3*X(18950)-2*X(18952)

    X(18951) lies on these lines: {2,1199}, {3,6515}, {4,94}, {5,11411}, {6,12359}, {20,12370}, {26,6776}, {30,18909}, {49,3147}, {52,1899}, {68,389}, {69,140}, {141,15805}, {155,13567}, {156,6353}, {193,3546}, {195,6640}, {578,11225}, {631,13353}, {1147,5181}, {1154,6643}, {1352,5462}, {1370,6243}, {1493,3525}, {1658,18925}, {1992,18281}, {1993,3548}, {2895,6989}, {3060,11457}, {3167,16238}, {3542,18445}, {3549,3580}, {3564,6642}, {3567,7528}, {3627,12324}, {3628,11487}, {4846,13382}, {5422,14786}, {5446,14216}, {5876,18537}, {5889,18531}, {5946,7401}, {6101,7386}, {6639,15087}, {6803,12006}, {6804,11591}, {6816,18436}, {7387,18910}, {7392,15026}, {7403,9777}, {7542,11402}, {8141,18921}, {8144,18922}, {8889,13561}, {9306,9936}, {9833,10116}, {10071,11436}, {10112,11438}, {10821,11004}, {11250,18931}, {11255,18919}, {11265,18923}, {11266,18924}, {11267,18929}, {11268,18930}, {11412,18911}, {11585,12160}, {11818,16881}, {12084,18913}, {12085,13142}, {12163,12241}, {12233,14852}, {12235,19161}, {12293,13568}, {14791,15134}, {14912,19129}, {16625,18381}, {18377,18918}, {19119,19154}, {19166,19210}

    X(18951) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (52, 1899, 14790), (68, 389, 18420), (193, 3546, 16266), (3567, 11442, 7528), (3580, 7592, 3549), (5889, 18912, 18531), (6515, 18916, 3), (6643, 18950, 18952), (10112, 11438, 12118), (11411, 11433, 5), (11487, 18928, 3628), (16881, 18356, 11818)


    X(18952) = X(5) OF ANTI-ATIK TRIANGLE

    Barycentrics    (5*R^2-SA)*S^2+(R^2-SW)*SB*SC : :
    X(18952) = X(6643)+3*X(18950) = 3*X(18950)-X(18951)

    X(18952) lies on these lines: {2,49}, {3,3580}, {4,3521}, {5,1181}, {6,13371}, {26,13567}, {30,9786}, {52,14791}, {54,6640}, {66,18583}, {68,140}, {70,14389}, {125,569}, {143,11433}, {156,6776}, {182,5449}, {343,7516}, {381,11457}, {389,18569}, {546,14216}, {578,18281}, {1154,6643}, {1352,3628}, {1368,13292}, {1370,10263}, {1503,13861}, {1656,11442}, {2072,7592}, {2888,3525}, {3090,3448}, {3147,5944}, {3410,5067}, {3548,15121}, {5012,6639}, {5422,5576}, {5462,11818}, {5663,18909}, {5876,6816}, {5890,18404}, {6101,6515}, {6102,18531}, {6146,6644}, {6759,18128}, {6804,14128}, {7386,10627}, {7393,12166}, {7401,13363}, {7514,12359}, {7528,15026}, {7706,15012}, {8550,9820}, {9140,14787}, {9306,10116}, {9729,9927}, {9833,12106}, {9968,19130}, {11003,14940}, {11245,11585}, {11411,11591}, {11511,12585}, {11572,16226}, {12006,18420}, {12084,12241}, {12289,15053}, {12319,13358}, {15114,15128}, {18379,18918}, {19119,19155}, {19166,19211}

    X(18952) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1368, 13292, 16266), (3628, 18356, 1352), (5462, 18381, 11818), (6643, 18950, 18951), (6816, 18917, 5876), (11245, 11585, 12161), (11433, 14790, 143), (15012, 18383, 7706), (18531, 18916, 6102), (18911, 18912, 3)


    X(18953) = X(6) OF ANTI-ATIK TRIANGLE

    Barycentrics    SA*(SA*(4*R^2-SA)+S^2)*(SA^2-SW^2+2*S^2) : :

    X(18953) lies on these lines: {53,11433}, {69,10607}, {157,6776}, {1503,7487}, {1899,6751}, {2871,18935}, {13567,17849}, {18380,18918}, {18437,18917}, {19119,19156}, {19166,19212}


    X(18954) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND ARA

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

    X(18954) lies on these lines: {1,7387}, {3,12}, {4,10832}, {11,1598}, {22,388}, {23,3600}, {24,4293}, {25,34}, {26,18990}, {36,6642}, {55,11414}, {57,8185}, {65,9798}, {159,1469}, {161,221}, {197,10834}, {390,12087}, {496,7530}, {499,7529}, {529,9712}, {999,7517}, {1056,12088}, {1317,13222}, {1319,11365}, {1399,1460}, {1454,1473}, {1470,15654}, {1479,18534}, {1486,11510}, {1593,9672}, {1909,15574}, {1935,5329}, {1995,7288}, {2099,8192}, {3023,9861}, {3027,13175}, {3028,12310}, {3057,9911}, {3085,10323}, {3086,10594}, {3167,9652}, {3295,4354}, {3304,9673}, {3320,11641}, {3324,14673}, {3327,15960}, {4185,10829}, {4214,13273}, {4294,12082}, {4317,9714}, {5020,5433}, {5198,10896}, {5229,7503}, {5252,8193}, {5261,6636}, {5265,13595}, {5281,16661}, {5347,9370}, {5434,9909}, {5594,18960}, {5595,18959}, {5899,7373}, {6020,12413}, {6284,16541}, {6285,9914}, {7071,9628}, {7352,9937}, {7393,7951}, {7395,10895}, {7485,10588}, {7509,10590}, {7516,10592}, {8190,18955}, {8191,18956}, {8194,18963}, {8195,18964}, {9613,15177}, {9657,9659}, {9667,14530}, {9876,18969}, {9908,18970}, {9910,12688}, {9912,18976}, {9915,18975}, {9916,18974}, {9917,18982}, {9918,18983}, {9920,18984}, {9921,18989}, {9922,18988}, {10483,12085}, {10790,12835}, {10828,18957}, {10830,18962}, {10835,18967}, {10944,12410}, {11853,18958}, {12168,12373}, {12411,18979}, {12412,18968}, {12414,18985}, {12903,13171}, {13680,18986}, {13800,18987}, {13861,15325}, {13889,18965}, {13943,18966}, {16119,18977}, {18995,19005}, {18996,19006}

    X(18954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7387, 10833), (22, 388, 10831), (34, 5322, 56), (56, 9658, 25), (999, 7517, 10046), (9672, 12943, 1593)


    X(18955) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 1st AURIGA

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

    X(18955) lies on these lines: {1,3}, {4,11871}, {8,11870}, {11,8196}, {12,5599}, {34,11384}, {80,18497}, {388,5601}, {1317,13228}, {1469,12452}, {1478,8200}, {3023,12179}, {3027,13176}, {3028,13208}, {3320,13229}, {3585,18495}, {4293,11843}, {5252,8197}, {5434,11207}, {6020,12478}, {6285,12468}, {8190,18954}, {8198,18959}, {8199,18960}, {8201,18963}, {8202,18964}, {8207,10573}, {10944,12454}, {11837,12835}, {11861,18957}, {11863,18958}, {11865,18961}, {11867,18962}, {11872,18391}, {12345,18969}, {12415,18970}, {12456,12688}, {12460,18976}, {12464,18979}, {12466,18968}, {12470,18975}, {12472,18974}, {12474,18982}, {12476,18983}, {12480,18984}, {12482,18985}, {12484,18989}, {12486,18988}, {13682,18986}, {13802,18987}, {13890,18965}, {13944,18966}, {16121,18977}, {18995,19007}, {18996,19008}

    X(18955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11252, 11873), (55, 65, 18956), (388, 5601, 11869), (999, 11875, 11879)


    X(18956) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 2nd AURIGA

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

    X(18956) lies on these lines: {1,3}, {4,11872}, {8,11869}, {11,8203}, {12,5600}, {34,11385}, {80,18495}, {388,5602}, {1317,13230}, {1469,12453}, {1478,8207}, {3023,12180}, {3027,13177}, {3320,13231}, {3585,18497}, {4293,11844}, {5252,8204}, {5434,11208}, {6020,12479}, {6285,12469}, {8191,18954}, {8200,10573}, {8205,18959}, {8206,18960}, {8208,18963}, {8209,18964}, {10944,12455}, {11838,12835}, {11862,18957}, {11864,18958}, {11866,18961}, {11868,18962}, {11871,18391}, {12346,18969}, {12416,18970}, {12457,12688}, {12461,18976}, {12465,18979}, {12467,18968}, {12471,18975}, {12473,18974}, {12475,18982}, {12477,18983}, {12481,18984}, {12483,18985}, {12485,18989}, {12487,18988}, {13683,18986}, {13803,18987}, {13891,18965}, {13945,18966}, {16122,18977}, {18995,19009}, {18996,19010}

    X(18956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11253, 11874), (55, 65, 18955), (388, 5602, 11870), (999, 11876, 11880)


    X(18957) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 5th BROCARD

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

    X(18957) lies on these lines: {1,9821}, {3,10038}, {4,10874}, {11,9993}, {12,3096}, {32,56}, {34,11386}, {55,3098}, {57,3099}, {65,9941}, {388,2896}, {999,9301}, {1317,13235}, {1319,11368}, {1401,1403}, {1469,2276}, {1478,9996}, {2099,9997}, {3023,4293}, {3027,8782}, {3028,13210}, {3057,12497}, {3085,10357}, {3320,13236}, {3585,18500}, {5252,9857}, {5433,7846}, {5434,7811}, {6020,12503}, {6285,12502}, {7288,10583}, {7333,9998}, {7354,9873}, {7865,11237}, {9655,18503}, {9878,18969}, {9923,18970}, {9981,18975}, {9982,18974}, {9983,18982}, {9985,18984}, {9986,18989}, {9987,18988}, {9994,18959}, {9995,18960}, {10345,10797}, {10356,10895}, {10828,18954}, {10871,18961}, {10872,18962}, {10875,18963}, {10876,18964}, {10878,11494}, {10879,18967}, {10944,12495}, {11861,18955}, {11862,18956}, {11885,18958}, {12496,12688}, {12498,18976}, {12500,18979}, {12501,18968}, {12504,18985}, {13685,18986}, {13805,18987}, {13892,18965}, {13946,18966}, {16123,18977}, {18995,19011}, {18996,19012}

    X(18957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9821, 10877), (388, 2896, 10873), (999, 9301, 10047)


    X(18958) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND GOSSARD

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

    X(18958) lies on these lines: {1,11251}, {3,11912}, {4,11906}, {11,11897}, {12,1650}, {30,55}, {34,11832}, {56,402}, {57,11852}, {65,12438}, {388,4240}, {999,11911}, {1317,13268}, {1319,11831}, {1469,12583}, {1651,5434}, {2099,11910}, {3023,12181}, {3027,13179}, {3028,13212}, {3057,12696}, {3476,16212}, {3585,18507}, {4293,11845}, {5252,11900}, {5433,15183}, {6020,12796}, {6285,12791}, {7354,12113}, {9655,18508}, {9657,15774}, {10944,12626}, {11509,11848}, {11839,12835}, {11853,18954}, {11863,18955}, {11864,18956}, {11885,18957}, {11901,18959}, {11902,18960}, {11903,18961}, {11904,18962}, {11907,18963}, {11908,18964}, {11915,18967}, {12347,18969}, {12418,18970}, {12668,12688}, {12729,18976}, {12789,18979}, {12790,18968}, {12792,18975}, {12793,18974}, {12794,18982}, {12795,18983}, {12797,18984}, {12798,18985}, {12799,18989}, {12800,18988}, {13689,18986}, {13809,18987}, {13894,18965}, {13948,18966}, {15326,16190}, {16129,18977}, {18995,19017}, {18996,19018}

    X(18958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11251, 11909), (999, 11911, 11913)


    X(18959) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND INNER-GREBE

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

    X(18959) lies on these lines: {1,1161}, {3,10040}, {4,10925}, {6,41}, {11,6202}, {12,5591}, {34,11388}, {55,11824}, {57,5589}, {65,3641}, {388,1271}, {999,10048}, {1317,13269}, {1319,11370}, {1478,6215}, {2099,5605}, {3023,6227}, {3027,6319}, {3028,7732}, {3057,12697}, {3085,10517}, {3303,7353}, {3320,13282}, {3585,18509}, {4293,10783}, {5252,5689}, {5434,5861}, {5595,18954}, {5871,7354}, {5875,18990}, {6020,12805}, {6258,12688}, {6263,18976}, {6267,6285}, {6270,18974}, {6271,18975}, {6273,18982}, {6275,18983}, {6277,18984}, {6279,18988}, {6281,9657}, {8198,18955}, {8205,18956}, {8216,18963}, {8217,18964}, {8396,19038}, {8974,18965}, {9882,18969}, {9929,18970}, {9994,18957}, {10513,10924}, {10514,10895}, {10792,12835}, {10919,18961}, {10921,18962}, {10929,11497}, {10931,18967}, {10944,12627}, {11901,18958}, {12801,18979}, {12803,18968}, {12807,18985}, {13690,18986}, {13810,18987}, {13949,18966}, {16130,18977}

    X(18959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1161, 10927), (56, 1469, 18960), (388, 1271, 10923), (999, 11916, 10048)


    X(18960) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND OUTER-GREBE

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

    X(18960) lies on these lines: {1,1160}, {3,10041}, {4,10926}, {6,41}, {11,6201}, {12,5590}, {34,11389}, {55,11825}, {57,5588}, {65,3640}, {388,1270}, {999,10049}, {1317,13270}, {1319,11371}, {1478,6214}, {2099,5604}, {3023,6226}, {3027,6320}, {3028,7733}, {3057,12698}, {3085,10518}, {3303,7362}, {3320,13283}, {3585,18511}, {4293,10784}, {5252,5688}, {5434,5860}, {5594,18954}, {5870,7354}, {5874,18990}, {6020,12806}, {6257,12688}, {6262,18976}, {6266,6285}, {6268,18974}, {6269,18975}, {6272,18982}, {6274,18983}, {6276,18984}, {6278,9657}, {6280,18989}, {8199,18955}, {8206,18956}, {8218,18963}, {8219,18964}, {8416,19037}, {9883,18969}, {9930,18970}, {9995,18957}, {10513,10923}, {10515,10895}, {10793,12835}, {10920,18961}, {10922,18962}, {10930,11498}, {10932,18967}, {10944,12628}, {11902,18958}, {12802,18979}, {12804,18968}, {12808,18985}, {13691,18986}, {13811,18987}, {13950,18966}, {16131,18977}

    X(18960) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1160, 10928), (56, 1469, 18959), (388, 1270, 10924), (999, 11917, 10049)


    X(18961) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND INNER-JOHNSON

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

    X(18961) lies on these lines: {1,6923}, {3,10320}, {4,11}, {5,1470}, {12,377}, {20,5172}, {28,9658}, {34,11390}, {36,6928}, {46,10526}, {55,6850}, {57,3585}, {65,68}, {145,388}, {226,17647}, {382,1617}, {497,1388}, {499,6929}, {999,10948}, {1317,13271}, {1319,1479}, {1399,5230}, {1415,3767}, {1420,3583}, {1454,4292}, {1466,6826}, {1469,12586}, {1709,9579}, {1788,5080}, {1836,12672}, {1887,5101}, {1898,6259}, {2478,5433}, {3023,12182}, {3027,13180}, {3028,13213}, {3057,12700}, {3085,6951}, {3320,13294}, {3340,5270}, {3361,18513}, {3614,6854}, {3913,10956}, {3916,17619}, {4185,10829}, {4190,10524}, {5046,7288}, {5204,6827}, {5217,6916}, {5221,5229}, {5252,10914}, {5432,6897}, {5434,10949}, {5697,16153}, {5840,11508}, {5903,10057}, {6020,12925}, {6284,6925}, {6285,12920}, {6836,15326}, {6842,8071}, {6863,14793}, {6901,10590}, {6948,10321}, {6957,7173}, {6958,8068}, {6959,10090}, {7491,7742}, {9581,10085}, {9655,18519}, {10043,16159}, {10404,17625}, {10589,13729}, {10742,12832}, {10794,12835}, {10871,18957}, {10919,18959}, {10920,18960}, {10943,18990}, {10945,18963}, {10946,18964}, {10950,12115}, {10966,15908}, {11375,17614}, {11502,18242}, {11604,18467}, {11865,18955}, {11866,18956}, {11903,18958}, {12348,18969}, {12422,18970}, {12676,12688}, {12701,17622}, {12737,18976}, {12857,18979}, {12889,18968}, {12921,18975}, {12922,18974}, {12923,18982}, {12924,18983}, {12926,18984}, {12927,18985}, {12928,18989}, {12929,18988}, {13462,18514}, {13693,18986}, {13813,18987}, {13895,18965}, {13952,18966}, {16138,18977}, {18995,19023}, {18996,19024}

    X(18961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10525, 10947), (11, 7354, 12114), (56, 13273, 4), (65, 1478, 18962), (377, 10522, 1376), (388, 3434, 10944), (999, 11928, 10948), (3086, 10598, 11), (3585, 10826, 18516), (6850, 10629, 55), (10893, 12761, 4)


    X(18962) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND OUTER-JOHNSON

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

    X(18962) lies on these lines: {1,6928}, {2,12}, {3,10954}, {4,1389}, {10,1454}, {11,10532}, {20,14882}, {34,11391}, {55,6868}, {57,5270}, {65,68}, {72,5252}, {153,5229}, {999,6971}, {1056,1388}, {1259,11501}, {1317,13272}, {1319,11374}, {1466,6885}, {1469,12587}, {1470,6924}, {1479,3656}, {1482,10947}, {1836,6256}, {1875,5130}, {2478,15950}, {3023,12183}, {3027,13181}, {3028,13214}, {3057,5812}, {3085,5172}, {3086,10599}, {3256,10483}, {3320,13295}, {3340,3585}, {3485,5080}, {4293,6942}, {5173,18480}, {5221,6901}, {5708,12832}, {5903,6923}, {6020,12935}, {6253,12667}, {6285,12930}, {6867,10895}, {6874,10590}, {6934,7354}, {6936,11510}, {9613,17857}, {9655,18518}, {10404,18838}, {10522,10944}, {10795,12835}, {10830,18954}, {10872,18957}, {10921,18959}, {10922,18960}, {10948,12001}, {10951,18963}, {10952,18964}, {10957,12513}, {11867,18955}, {11868,18956}, {11904,18958}, {12349,18969}, {12423,18970}, {12677,12688}, {12738,18976}, {12858,18979}, {12890,18968}, {12931,18975}, {12932,18974}, {12933,18982}, {12934,18983}, {12936,18984}, {12937,18985}, {12938,18989}, {12939,18988}, {13694,18986}, {13814,18987}, {13896,18965}, {13953,18966}, {16139,18977}, {18995,19025}, {18996,19026}

    X(18962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10526, 10953), (65, 1478, 18961), (388, 3436, 12), (999, 11929, 10523), (1478, 10573, 6917), (7354, 10955, 11500), (15843, 15844, 12)


    X(18963) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND LUCAS HOMOTHETIC

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

    X(18963) lies on these lines: {1,10669}, {3,11951}, {4,11932}, {11,8212}, {12,8222}, {34,11394}, {55,11828}, {56,493}, {57,8188}, {65,12440}, {388,6462}, {999,11949}, {1317,13275}, {1319,11377}, {1469,12590}, {1478,8220}, {2099,8210}, {3023,12186}, {3027,13184}, {3028,13215}, {3320,13298}, {3585,18520}, {4293,11846}, {5252,8214}, {5434,12152}, {6020,12996}, {6285,12986}, {6339,11931}, {6461,18964}, {7354,9838}, {8194,18954}, {8201,18955}, {8208,18956}, {8216,18959}, {8218,18960}, {10875,18957}, {10944,12636}, {10945,18961}, {10951,18962}, {10981,11948}, {11503,11509}, {11840,12835}, {11907,18958}, {11957,18967}, {12352,18969}, {12426,18970}, {12688,18245}, {12741,18976}, {12861,18979}, {12894,18968}, {12988,18975}, {12990,18974}, {12992,18982}, {12994,18983}, {12998,18984}, {13000,18985}, {13002,18989}, {13004,18988}, {13697,18986}, {13817,18987}, {13899,18965}, {13956,18966}, {16161,18977}, {18995,19031}, {18996,19032}

    X(18963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10669, 11947), (388, 6462, 11930)


    X(18964) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND LUCAS(-1) HOMOTHETIC

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

    X(18964) lies on these lines: {1,10673}, {3,11952}, {4,11933}, {11,8213}, {12,8223}, {34,11395}, {55,11829}, {56,494}, {57,8189}, {65,12441}, {388,6463}, {999,11950}, {1317,13276}, {1319,11378}, {1469,12591}, {1478,8221}, {2099,8211}, {3023,12187}, {3027,13185}, {3028,13216}, {3320,13299}, {3585,18522}, {4293,11847}, {5252,8215}, {5434,12153}, {6020,12997}, {6285,12987}, {6339,11930}, {6461,18963}, {7354,9839}, {8195,18954}, {8202,18955}, {8209,18956}, {8217,18959}, {8219,18960}, {10876,18957}, {10944,12637}, {10946,18961}, {10952,18962}, {10981,11947}, {11504,11509}, {11841,12835}, {11908,18958}, {11958,18967}, {12353,18969}, {12427,18970}, {12688,18246}, {12742,18976}, {12862,18979}, {12895,18968}, {12989,18975}, {12991,18974}, {12993,18982}, {12995,18983}, {12999,18984}, {13001,18985}, {13003,18989}, {13005,18988}, {13698,18986}, {13818,18987}, {13900,18965}, {13957,18966}, {16162,18977}, {18995,19033}, {18996,19034}

    X(18964) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10673, 11948), (388, 6463, 11931)


    X(18965) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 3rd TRI-SQUARES-CENTRAL

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

    X(18965) lies on these lines: {1,8981}, {2,13954}, {3,13904}, {4,13898}, {6,5433}, {11,371}, {12,590}, {34,13884}, {36,7583}, {55,9540}, {56,3068}, {57,13888}, {65,8983}, {140,3301}, {388,8972}, {485,7354}, {499,3311}, {615,7294}, {631,19037}, {999,13903}, {1151,6284}, {1317,13922}, {1319,13883}, {1335,5418}, {1388,19066}, {1469,13910}, {1478,8976}, {1479,6221}, {1587,5204}, {1702,11376}, {1837,9583}, {2099,13902}, {3023,8980}, {3027,8997}, {3028,8998}, {3057,13912}, {3058,9648}, {3070,15326}, {3071,7173}, {3086,19038}, {3299,15325}, {3316,10590}, {3320,13923}, {3526,13963}, {3585,18538}, {3614,10576}, {4293,13886}, {4299,13665}, {4302,6449}, {5160,9631}, {5252,13893}, {5298,6502}, {5434,13846}, {6020,13918}, {6285,8991}, {6417,13962}, {6453,9660}, {6459,10896}, {6564,9647}, {7288,7585}, {7582,13955}, {8974,18959}, {8987,12688}, {8988,18976}, {8992,18982}, {8993,18983}, {8995,18984}, {9541,12953}, {9616,12701}, {9646,15888}, {10944,13911}, {11509,13887}, {12835,13885}, {13848,18987}, {13879,18988}, {13889,18954}, {13890,18955}, {13891,18956}, {13892,18957}, {13894,18958}, {13895,18961}, {13896,18962}, {13899,18963}, {13900,18964}, {13907,18967}, {13908,18969}, {13909,18970}, {13914,18979}, {13915,18968}, {13916,18975}, {13917,18974}, {13919,18985}, {13920,18986}, {13921,18989}, {13925,18990}, {15950,16232}, {16148,18977}

    X(18965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8981, 13901), (2, 18996, 19027), (3, 13904, 19030), (6, 5433, 18966), (56, 3068, 19028), (140, 3301, 13958), (371, 9661, 11), (388, 8972, 13897), (499, 3311, 19029), (590, 2067, 12), (999, 13903, 13905), (1335, 5418, 5432), (6284, 9663, 1151), (6453, 9660, 9662), (7288, 7585, 18995)


    X(18966) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 4th TRI-SQUARES-CENTRAL

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

    X(18966) lies on these lines: {1,13958}, {2,13897}, {3,13962}, {4,13955}, {6,5433}, {11,372}, {12,615}, {34,13937}, {36,7584}, {55,13935}, {56,3069}, {57,13942}, {65,13971}, {140,3299}, {388,13941}, {486,7354}, {499,3312}, {590,7294}, {631,19038}, {999,13961}, {1124,5420}, {1152,6284}, {1317,13991}, {1319,13936}, {1388,19065}, {1469,13972}, {1478,13951}, {1479,6398}, {1588,5204}, {1703,11376}, {2067,5298}, {2099,13959}, {2362,15950}, {3023,13967}, {3027,13989}, {3028,13990}, {3057,13975}, {3070,7173}, {3071,15326}, {3086,19037}, {3301,15325}, {3317,10590}, {3320,13992}, {3526,13905}, {3585,18762}, {3592,9663}, {3614,10577}, {4293,13939}, {4299,13785}, {4302,6450}, {5252,13947}, {5326,9646}, {5434,13847}, {6020,13985}, {6285,13980}, {6396,15338}, {6418,13904}, {6420,9661}, {6460,10896}, {7288,7586}, {7581,13898}, {10944,13973}, {11509,13940}, {12688,13974}, {12835,13938}, {13849,18987}, {13880,18988}, {13933,18989}, {13943,18954}, {13944,18955}, {13945,18956}, {13946,18957}, {13948,18958}, {13949,18959}, {13950,18960}, {13952,18961}, {13953,18962}, {13956,18963}, {13957,18964}, {13965,18967}, {13968,18969}, {13970,18970}, {13976,18976}, {13978,18979}, {13979,18968}, {13981,18975}, {13982,18974}, {13983,18982}, {13984,18983}, {13986,18984}, {13987,18985}, {13988,18986}, {13993,18990}, {16149,18977}

    X(18966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 13966, 13958), (2, 18995, 19028), (3, 13962, 19029), (6, 5433, 18965), (56, 3069, 19027), (140, 3299, 13901), (388, 13941, 13954), (499, 3312, 19030), (615, 6502, 12), (999, 13961, 13963), (1124, 5420, 5432), (7288, 7586, 18996)


    X(18967) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND OUTER-YFF TANGENTS

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

    X(18967) lies on these lines: {1,3}, {4,10959}, {11,10532}, {12,6933}, {34,11401}, {145,11501}, {149,3600}, {388,6871}, {956,11375}, {958,15950}, {1149,1451}, {1191,15306}, {1317,13279}, {1398,1887}, {1469,12595}, {1478,10943}, {1696,2323}, {2178,17438}, {3023,12190}, {3027,13190}, {3028,13218}, {3086,6879}, {3320,13314}, {3585,18544}, {3649,10941}, {3890,7098}, {4293,10806}, {4298,7702}, {5219,5288}, {5252,10916}, {5434,10949}, {6020,13119}, {6285,13095}, {6834,10955}, {7288,10587}, {7354,12116}, {9655,18543}, {9657,13273}, {10530,10956}, {10804,12835}, {10835,18954}, {10879,18957}, {10931,18959}, {10932,18960}, {10944,12649}, {11237,17530}, {11915,18958}, {11957,18963}, {11958,18964}, {12357,18969}, {12431,18970}, {12687,12688}, {12750,18976}, {12875,18979}, {12906,18968}, {13106,18975}, {13107,18974}, {13110,18982}, {13113,18983}, {13122,18984}, {13131,18985}, {13133,18989}, {13135,18988}, {13717,18986}, {13840,18987}, {13907,18965}, {13965,18966}, {16155,18977}, {18393,18761}, {18995,19049}, {18996,19050}

    X(18967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 36, 16202), (1, 56, 11510), (1, 10680, 10966), (1, 11249, 55), (1, 12704, 3057), (56, 2099, 11509), (56, 3303, 5172), (56, 14882, 5204), (65, 999, 56), (388, 10529, 10957), (999, 12001, 1), (1470, 5563, 56), (3340, 5563, 1470)


    X(18968) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO AAOA

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

    The reciprocal orthologic center of these triangles is X(7574)

    X(18968) lies on these lines: {1,12888}, {3,12903}, {4,10091}, {11,10113}, {12,1511}, {20,10065}, {30,3024}, {34,12140}, {35,16163}, {36,125}, {46,13211}, {55,12121}, {56,265}, {57,12407}, {74,4299}, {79,1365}, {80,1354}, {110,1478}, {113,3585}, {382,12374}, {388,10088}, {498,15035}, {499,14644}, {542,1469}, {999,12902}, {1319,12261}, {1479,10733}, {2099,12898}, {2771,18976}, {2777,6285}, {2948,9613}, {3028,3327}, {3448,4293}, {3583,12295}, {4311,13605}, {4316,16111}, {4325,16003}, {5204,15061}, {5252,12778}, {5663,7354}, {5972,7951}, {6699,7280}, {7288,15081}, {7343,13202}, {7687,7741}, {7728,12943}, {9651,14901}, {10074,10778}, {10089,11005}, {10895,14643}, {11392,15472}, {11509,12334}, {11720,12047}, {11723,18393}, {11801,15325}, {12041,15326}, {12201,12835}, {12412,18954}, {12466,18955}, {12467,18956}, {12501,18957}, {12790,18958}, {12803,18959}, {12804,18960}, {12889,18961}, {12890,18962}, {12894,18963}, {12895,18964}, {12906,18967}, {13182,18332}, {13915,18965}, {13979,18966}, {18995,19051}, {18996,19052}

    X(18968) = reflection of X(3028) in X(18990)
    X(18968) = orthologic center of these triangles: 2nd anti-circumperp-tangential to AOA
    X(18968) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 1st Hyacinth
    X(18968) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 12383, 10088), (999, 12902, 12904), (3448, 4293, 10081)


    X(18969) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ANTI-MCCAY

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

    The reciprocal orthologic center of these triangles is X(9855)

    X(18969) lies on these lines: {1,12354}, {2,13182}, {3,10054}, {4,12351}, {11,9880}, {12,2482}, {30,3023}, {34,12132}, {55,12117}, {56,671}, {57,9875}, {99,11237}, {115,5298}, {381,10089}, {388,8591}, {530,18975}, {531,18974}, {542,7354}, {543,3027}, {999,10070}, {1319,12258}, {1469,9830}, {1478,8724}, {2099,9884}, {2782,18971}, {2796,10106}, {3534,10053}, {3543,12185}, {3600,8596}, {4293,12243}, {4299,14830}, {4870,11711}, {5182,10797}, {5252,9881}, {5433,5461}, {5969,18982}, {6034,9597}, {6321,10072}, {9114,12941}, {9116,12942}, {9876,18954}, {9878,18957}, {9882,18959}, {9883,18960}, {10056,15452}, {10992,15888}, {11006,12903}, {11152,12837}, {11509,12326}, {12191,12835}, {12345,18955}, {12346,18956}, {12347,18958}, {12348,18961}, {12349,18962}, {12352,18963}, {12353,18964}, {12357,18967}, {13908,18965}, {13968,18966}, {18995,19057}, {18996,19058}

    X(18969) = orthologic center of these triangles: McCay to 2nd anti-circumperp-tangential
    X(18969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 8591, 12350), (999, 12355, 10070)


    X(18970) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ARIES

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

    The reciprocal orthologic center of these triangles is X(9833)

    X(18970) lies on these lines: {1,9931}, {3,10055}, {4,1069}, {11,9927}, {12,1147}, {30,6238}, {34,12134}, {36,12359}, {55,12118}, {56,68}, {57,9896}, {155,1478}, {388,3157}, {499,14852}, {539,5434}, {999,10071}, {1060,6146}, {1319,12259}, {1469,3564}, {1479,12293}, {1870,14516}, {2099,9933}, {3167,9654}, {4293,11411}, {4299,12163}, {5252,9928}, {5432,12038}, {5433,5449}, {5504,12903}, {5654,10895}, {7354,13754}, {7689,15326}, {7951,9820}, {9655,12164}, {9657,9936}, {9908,18954}, {9923,18957}, {9929,18959}, {9930,18960}, {11509,12328}, {12193,12835}, {12415,18955}, {12416,18956}, {12418,18958}, {12422,18961}, {12423,18962}, {12426,18963}, {12427,18964}, {12431,18967}, {13909,18965}, {13970,18966}, {18995,19061}, {18996,19062}

    X(18970) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 2nd Hyacinth
    X(18970) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 6193, 3157), (999, 12429, 10071)


    X(18971) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 1st BROCARD-REFLECTED

    Barycentrics    3*(b^2+c^2)*a^6-(b^4+c^4-2*b*c*(3*b^2+b*c+3*c^2))*a^4-(2*b^4+2*c^4+b*c*(4*b^2+7*b*c+4*c^2))*(b-c)^2*a^2-(b^2-c^2)^2*b^2*c^2 : :
    X(18971) = 2*X(7354)+X(13077) = X(18982)-4*X(18990)

    The reciprocal orthologic center of these triangles is X(3)

    X(18971) lies on these lines: {1,13078}, {12,15819}, {56,262}, {76,9657}, {388,6194}, {511,5434}, {1469,3023}, {2782,18969}, {3095,4317}, {3600,12836}, {4293,7709}, {5188,15888}, {5563,14881}, {7354,13077}, {9655,10079}, {9772,12184}, {11246,14839}, {18982,18990}, {18995,19063}, {18996,19064}


    X(18972) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO INNER-FERMAT

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18972) lies on these lines: {1,13075}, {12,630}, {18,56}, {388,628}, {533,5434}, {999,16628}, {1319,11740}, {1469,5965}, {1478,16627}, {5433,6674}, {11603,13182}, {12942,14145}, {18974,18990}, {18995,19069}, {18996,19072}


    X(18973) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO OUTER-FERMAT

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18973) lies on these lines: {1,13076}, {12,629}, {17,56}, {388,627}, {532,5434}, {1319,11739}, {1469,5965}, {1478,16626}, {5433,6673}, {11602,13182}, {12941,14144}, {18975,18990}, {18995,19071}, {18996,19070}


    X(18974) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 3rd FERMAT-DAO

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

    The reciprocal orthologic center of these triangles is X(13)

    X(18974) lies on these lines: {1,13076}, {3,10062}, {4,12952}, {11,5478}, {12,618}, {13,56}, {14,13182}, {34,12142}, {36,6771}, {55,5473}, {57,9901}, {99,12941}, {388,616}, {530,5434}, {531,18969}, {542,1469}, {999,10078}, {1081,1365}, {1319,11705}, {1478,5617}, {2099,7975}, {4293,6770}, {5252,12781}, {5298,5459}, {5433,6669}, {5463,11237}, {5472,7051}, {5563,16001}, {5613,10089}, {6268,18960}, {6270,18959}, {6321,10077}, {9116,12350}, {9916,18954}, {9982,18957}, {11509,12337}, {12205,12835}, {12472,18955}, {12473,18956}, {12793,18958}, {12922,18961}, {12932,18962}, {12990,18963}, {12991,18964}, {13107,18967}, {13917,18965}, {13982,18966}, {18972,18990}, {18995,19073}, {18996,19074}

    X(18974) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 7th Fermat-Dao
    X(18974) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 11th Fermat-Dao
    X(18974) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 15th Fermat-Dao
    X(18974) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 1st isodynamic-Dao
    X(18974) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 1st Lemoine-Dao
    X(18974) = orthologic center of these triangles: outer-Napoleon to 2nd anti-circumperp-tangential
    X(18974) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 616, 12942), (999, 13103, 10078)


    X(18975) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 4th FERMAT-DAO

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

    The reciprocal orthologic center of these triangles is X(14)

    X(18975) lies on these lines: {1,13075}, {3,10061}, {4,12951}, {11,5479}, {12,619}, {13,13182}, {14,56}, {34,12141}, {36,6774}, {55,5474}, {57,9900}, {99,12942}, {115,7051}, {388,617}, {530,18969}, {531,5434}, {542,1469}, {554,1365}, {999,10077}, {1319,11706}, {1478,5613}, {2099,7974}, {4293,6773}, {5252,12780}, {5298,5460}, {5433,6670}, {5464,11237}, {5563,16002}, {5617,10089}, {6269,18960}, {6271,18959}, {6321,10078}, {9114,12350}, {9915,18954}, {9981,18957}, {11509,12336}, {12204,12835}, {12470,18955}, {12471,18956}, {12792,18958}, {12921,18961}, {12931,18962}, {12988,18963}, {12989,18964}, {13106,18967}, {13916,18965}, {13981,18966}, {18973,18990}, {18995,19075}, {18996,19076}

    X(18975) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 8th Fermat-Dao
    X(18975) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 12th Fermat-Dao
    X(18975) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 16th Fermat-Dao
    X(18975) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 2nd isodynamic-Dao
    X(18975) = orthologic center of these triangles: 2nd anti-circumperp-tangential to 2nd Lemoine-Dao
    X(18975) = orthologic center of these triangles: inner-Napoleon to 2nd anti-circumperp-tangential
    X(18975) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 617, 12941), (999, 13102, 10077)


    X(18976) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO FUHRMANN

    Barycentrics    (2*a^5-3*(b+c)*a^4+6*b*c*a^3+(2*b-c)*(b-2*c)*(b+c)*a^2-2*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c))*(a+b-c)*(a-b+c) : :
    X(18976) = 2*X(5083)-3*X(5434)

    The reciprocal orthologic center of these triangles is X(3)

    X(18976) lies on these lines: {1,10738}, {3,10057}, {4,12740}, {11,515}, {12,214}, {30,12758}, {34,12137}, {36,12619}, {55,12119}, {56,80}, {57,9897}, {65,952}, {100,5252}, {149,3476}, {244,14584}, {355,10090}, {388,6224}, {528,8581}, {999,10073}, {1145,17647}, {1317,1365}, {1385,8068}, {1387,10572}, {1388,9669}, {1470,12751}, {1836,10698}, {2098,14217}, {2099,7972}, {2771,18968}, {2800,7354}, {2802,10944}, {2829,12688}, {2932,11501}, {3057,5840}, {3585,12611}, {4293,12247}, {4299,12515}, {4311,10265}, {4973,15863}, {5083,5434}, {5433,6702}, {5691,12764}, {6262,18960}, {6263,18959}, {6284,15558}, {6326,9613}, {6797,18838}, {8988,18965}, {9578,15015}, {9579,13253}, {9912,18954}, {10058,18481}, {10609,10956}, {10724,12701}, {10950,12005}, {11509,12331}, {11729,17605}, {12198,12835}, {12460,18955}, {12461,18956}, {12498,18957}, {12729,18958}, {12737,18961}, {12738,18962}, {12741,18963}, {12742,18964}, {12750,18967}, {13976,18966}, {18995,19077}, {18996,19078}

    X(18976) = reflection of X(i) in X(j) for these (i,j): (1145, 17647), (1317, 10106), (6284, 15558), (10950, 12736)
    X(18976) = orthologic center of these triangles: 2nd anti-circumperp-tangential to K798e
    X(18976) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 6224, 12739), (999, 12747, 10073), (6326, 9613, 12763)


    X(18977) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 2nd FUHRMANN

    Barycentrics    (2*a^5-(b+c)*a^4-2*(b^2+b*c+c^2)*a^3-b*c*(b+c)*a^2+(b^2-c^2)^2*(b+c))*(a+b-c)*(a-b+c) : :
    X(18977) = 3*X(3649)-4*X(4298) = 3*X(10543)-2*X(10624)

    The reciprocal orthologic center of these triangles is X(3)

    X(18977) lies on these lines: {1,16142}, {3,16152}, {4,16141}, {11,1354}, {12,3647}, {21,11375}, {30,65}, {34,16114}, {55,16113}, {56,79}, {57,16118}, {191,9578}, {382,5221}, {388,3648}, {758,10944}, {999,16150}, {1319,3636}, {1361,3327}, {1454,10826}, {1478,3652}, {1621,10404}, {1788,2475}, {2099,5441}, {2771,18968}, {3065,13273}, {3485,15677}, {3884,5434}, {4293,16116}, {4308,14450}, {4870,17525}, {5183,16004}, {5252,11684}, {5427,11263}, {5428,14526}, {5433,6701}, {5901,11544}, {8581,17768}, {10543,10624}, {11373,16159}, {11509,16117}, {12835,16115}, {16119,18954}, {16121,18955}, {16122,18956}, {16123,18957}, {16129,18958}, {16130,18959}, {16131,18960}, {16138,18961}, {16139,18962}, {16148,18965}, {16149,18966}, {16155,18967}, {16161,18963}, {16162,18964}, {16617,17605}, {18995,19079}, {18996,19080}

    X(18977) = orthologic center of these triangles: 2nd anti-circumperp-tangential to K798i
    X(18977) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 3648, 16140), (999, 16150, 16153)


    X(18978) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO HATZIPOLAKIS-MOSES

    Barycentrics
    2*a^16-2*(3*b^2-b*c+3*c^2)*a^14+(3*b^4+3*c^4-4*b*c*(b^2-7*b*c+c^2))*a^12+2*(3*b^6+3*c^6-(b^4+c^4+b*c*(16*b^2-11*b*c+16*c^2))*b*c)*a^10-(5*b^8+5*c^8-2*(4*b^6+4*c^6-(5*b^4+5*c^4+b*c*(9*b^2-37*b*c+9*c^2))*b*c)*b*c)*a^8-2*(b^8+c^8+(3*b^6+3*c^6-(9*b^4+9*c^4+2*b*c*(6*b^2-b*c+6*c^2))*b*c)*b*c)*(b-c)^2*a^6+(b^2-c^2)^2*(b-c)^2*(b^6+c^6-(2*b^4+2*c^4+b*c*(b^2-14*b*c+c^2))*b*c)*a^4+2*(b^4-c^4)*(b^2-c^2)^3*(b-c)^2*(b^2+3*b*c+c^2)*a^2-(b^2+c^2)^2*(b^2-c^2)^6 : :

    The reciprocal orthologic center of these triangles is X(12241)

    X(18978) lies on these lines: {18995,19083}, {18996,19084}


    X(18979) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO HUTSON EXTOUCH

    Barycentrics
    a*(a+b-c)*(a-b+c)*((b+c)*a^6-2*(b^2+c^2)*a^5-(b^2-c^2)*(b-c)*a^4+4*(b^4+6*b^2*c^2+c^4)*a^3-(b+c)*(b^4+c^4+2*b*c*(2*b^2+27*b*c+2*c^2))*a^2-2*(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^2*(b+c)^3) : :

    The reciprocal orthologic center of these triangles is X(40)

    X(18979) lies on these lines: {1,12863}, {3,10059}, {4,12860}, {11,12599}, {12,12864}, {34,12139}, {55,12120}, {56,7160}, {57,9898}, {65,1056}, {388,9874}, {999,10075}, {1319,12260}, {1478,12856}, {1864,12853}, {2099,7971}, {3340,5696}, {3649,12854}, {4293,12249}, {5173,9578}, {5252,12777}, {11509,12333}, {12200,12835}, {12411,18954}, {12464,18955}, {12465,18956}, {12500,18957}, {12789,18958}, {12801,18959}, {12802,18960}, {12857,18961}, {12858,18962}, {12861,18963}, {12862,18964}, {12875,18967}, {13914,18965}, {13978,18966}, {18995,19085}, {18996,19086}

    X(18979) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 9874, 12859), (999, 12872, 10075)


    X(18980) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

    The reciprocal orthologic center of these triangles is X(1)

    X(18980) lies on these lines: {3,485}, {4,18414}, {182,19134}, {511,10669}, {7395,9723}, {8884,19186}, {13754,18939}, {18909,18926}


    X(18981) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

    The reciprocal orthologic center of these triangles is X(1)

    X(18981) lies on these lines: {3,486}, {4,18415}, {182,19135}, {511,10673}, {7395,9723}, {8884,19187}, {13754,18940}, {18909,18927}


    X(18982) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 1st NEUBERG

    Barycentrics    ((b^2+c^2)*a^4+b^2*c^2*(b+c)^2)*(a+b-c)*(a-b+c) : :
    X(18982) = 3*X(18971)-4*X(18990)

    The reciprocal orthologic center of these triangles is X(3)

    X(18982) lies on these lines: {1,2782}, {3,10063}, {4,12836}, {6,10797}, {11,6248}, {12,39}, {34,12143}, {55,11257}, {56,76}, {57,9902}, {65,730}, {194,388}, {262,10895}, {384,12835}, {498,11171}, {499,7697}, {511,7354}, {538,5434}, {726,10106}, {732,1469}, {999,10079}, {1319,12263}, {1478,3095}, {1916,13182}, {2099,7976}, {3085,7709}, {3094,9597}, {3097,9578}, {3202,9652}, {3585,14881}, {3934,5433}, {4293,12251}, {4299,9821}, {5188,15326}, {5252,12782}, {5298,9466}, {5432,13334}, {5969,18969}, {6272,18960}, {6273,18959}, {7757,11237}, {7951,11272}, {8992,18965}, {9917,18954}, {9983,18957}, {10944,14839}, {11152,12350}, {11509,12338}, {12474,18955}, {12475,18956}, {12794,18958}, {12923,18961}, {12933,18962}, {12992,18963}, {12993,18964}, {13110,18967}, {13983,18966}, {18971,18990}, {18995,19089}, {18996,19090}

    X(18982) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194, 388, 12837), (999, 13108, 10079)


    X(18983) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 2nd NEUBERG

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

    The reciprocal orthologic center of these triangles is X(3)

    X(18983) lies on these lines: {1,13078}, {3,10064}, {4,12954}, {11,6249}, {12,6292}, {34,12144}, {55,12122}, {56,83}, {57,9903}, {388,2896}, {732,1469}, {754,5434}, {999,10080}, {1319,12264}, {1463,10106}, {1478,6287}, {2099,7977}, {3023,18990}, {4293,12252}, {4299,8725}, {5204,9751}, {5252,12783}, {5433,6704}, {6274,18960}, {6275,18959}, {7333,7334}, {8993,18965}, {9918,18954}, {11509,12339}, {11606,13182}, {12206,12835}, {12476,18955}, {12477,18956}, {12795,18958}, {12924,18961}, {12934,18962}, {12994,18963}, {12995,18964}, {13113,18967}, {13984,18966}, {18995,19091}, {18996,19092}

    X(18983) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 2896, 12944), (999, 13111, 10080)


    X(18984) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO REFLECTION

    Barycentrics    a^2*((b+c)^2*a^6-(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))*a^4+(3*b^4+3*c^4+b*c*(4*b^2+5*b*c+4*c^2))*(b-c)^2*a^2-(b^2-c^2)^2*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))) : :
    X(18984) = 3*X(1)-X(6286) = X(6286)+3*X(7356) = 2*X(6286)-3*X(13079) = 2*X(7356)+X(13079)

    The reciprocal orthologic center of these triangles is X(4)

    X(18984) lies on these lines: {1,1154}, {3,10066}, {4,12956}, {11,3574}, {12,942}, {34,11576}, {36,10610}, {54,56}, {55,7691}, {57,9905}, {195,999}, {388,2888}, {517,17705}, {539,5434}, {1056,12325}, {1060,12363}, {1319,12266}, {1425,10619}, {1478,6288}, {1493,5563}, {1870,6152}, {2099,7979}, {3028,3327}, {3295,12307}, {3304,15801}, {4293,12254}, {5252,12785}, {5433,6689}, {6198,12300}, {6255,11529}, {6276,18960}, {6277,18959}, {7354,18400}, {7373,12316}, {7951,13565}, {8254,15325}, {8995,18965}, {9920,18954}, {9985,18957}, {11509,12341}, {12208,12835}, {12480,18955}, {12481,18956}, {12606,18447}, {12797,18958}, {12926,18961}, {12936,18962}, {12998,18963}, {12999,18964}, {13122,18967}, {13986,18966}, {18995,19095}, {18996,19096}

    X(18984) = midpoint of X(1) and X(7356)
    X(18984) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (195, 999, 10082), (388, 2888, 12946)


    X(18985) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 1st SCHIFFLER

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

    The reciprocal orthologic center of these triangles is X(79)

    X(18985) lies on these lines: {1,13080}, {3,13128}, {4,12957}, {11,79}, {12,13089}, {34,12146}, {55,12556}, {56,10266}, {57,12409}, {65,17643}, {388,12849}, {999,13126}, {1319,12267}, {1478,12919}, {2099,13100}, {4293,12255}, {5252,12786}, {5563,18244}, {6595,13273}, {11509,12342}, {12209,12835}, {12414,18954}, {12482,18955}, {12483,18956}, {12504,18957}, {12798,18958}, {12807,18959}, {12808,18960}, {12927,18961}, {12937,18962}, {13000,18963}, {13001,18964}, {13131,18967}, {13919,18965}, {13987,18966}, {18995,19097}, {18996,19098}

    X(18985) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 12849, 12947), (999, 13126, 13129)


    X(18986) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 1st TRI-SQUARES-CENTRAL

    Barycentrics    (10*a^4-(5*b^2-12*b*c+5*c^2)*a^2-5*(b^2-c^2)^2)*S+3*(b^2+b*c+c^2)*a^2*(a-b+c)*(a+b-c) : :
    X(18986) = 2*X(7362)-5*X(18988)

    The reciprocal orthologic center of these triangles is X(13665)

    X(18986) lies on these lines: {1,13699}, {3,13714}, {4,13696}, {11,13687}, {12,13701}, {30,7362}, {55,13666}, {56,1327}, {57,13679}, {388,13678}, {999,13713}, {1319,13667}, {1469,18987}, {1478,13692}, {2099,13702}, {4293,13674}, {5252,13688}, {11237,13712}, {11509,13675}, {12835,13672}, {13680,18954}, {13682,18955}, {13683,18956}, {13685,18957}, {13689,18958}, {13690,18959}, {13691,18960}, {13693,18961}, {13694,18962}, {13697,18963}, {13698,18964}, {13717,18967}, {13920,18965}, {13988,18966}, {18995,19099}

    X(18986) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 13678, 13695), (999, 13713, 13715)


    X(18987) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 2nd TRI-SQUARES-CENTRAL

    Barycentrics    -(10*a^4-(5*b^2-12*b*c+5*c^2)*a^2-5*(b^2-c^2)^2)*S+3*(b^2+b*c+c^2)*a^2*(a-b+c)*(a+b-c) : :
    X(18987) = 2*X(7353)-5*X(18989)

    The reciprocal orthologic center of these triangles is X(13785)

    X(18987) lies on these lines: {1,13819}, {3,13837}, {4,13816}, {11,13807}, {12,13821}, {30,7353}, {34,13788}, {55,13786}, {56,1328}, {57,13799}, {388,13798}, {999,13836}, {1319,13787}, {1469,18986}, {1478,13812}, {2099,13822}, {4293,13794}, {5252,13808}, {11237,13835}, {11509,13795}, {12835,13792}, {13800,18954}, {13802,18955}, {13803,18956}, {13805,18957}, {13809,18958}, {13810,18959}, {13811,18960}, {13813,18961}, {13814,18962}, {13817,18963}, {13818,18964}, {13840,18967}, {13848,18965}, {13849,18966}, {18995,19101}, {18996,19100}

    X(18987) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 13798, 13815), (999, 13836, 13838)


    X(18988) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 3rd TRI-SQUARES

    Barycentrics    (2*a^4-(b^2-4*b*c+c^2)*a^2-(b^2-c^2)^2)*S+(b^2+b*c+c^2)*a^2*(a-b+c)*(a+b-c) : :
    X(18988) = 2*X(7362)+3*X(18986)

    The reciprocal orthologic center of these triangles is X(485)

    X(18988) lies on these lines: {1,12911}, {3,10068}, {4,12959}, {11,6250}, {12,641}, {30,7362}, {34,12148}, {55,12124}, {56,485}, {57,9907}, {388,488}, {497,12297}, {999,10084}, {1056,12510}, {1319,12269}, {1469,3564}, {1478,6289}, {2099,7981}, {3600,12222}, {4293,12257}, {5252,12788}, {5433,6118}, {6278,9657}, {6279,18959}, {6337,12948}, {9922,18954}, {9987,18957}, {11509,12344}, {12211,12835}, {12304,16541}, {12486,18955}, {12487,18956}, {12800,18958}, {12929,18961}, {12939,18962}, {13004,18963}, {13005,18964}, {13135,18967}, {13879,18965}, {13880,18966}, {18995,19102}, {18996,19103}

    X(18988) = orthologic center of these triangles: 2nd anti-circumperp-tangential to outer-Vecten
    X(18988) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 488, 12949), (999, 12602, 10084), (1469, 18990, 18989)


    X(18989) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 4th TRI-SQUARES

    Barycentrics    -(2*a^4-(b^2-4*b*c+c^2)*a^2-(b^2-c^2)^2)*S+(b^2+b*c+c^2)*a^2*(a-b+c)*(a+b-c) : :
    X(18989) = 2*X(7353)+3*X(18987)

    The reciprocal orthologic center of these triangles is X(486)

    X(18989) lies on these lines: {1,12910}, {3,10067}, {4,12958}, {11,6251}, {12,642}, {30,7353}, {34,12147}, {55,12123}, {56,486}, {57,9906}, {388,487}, {497,12296}, {999,10083}, {1056,12509}, {1319,12268}, {1469,3564}, {1478,6290}, {2099,7980}, {3600,12221}, {4293,12256}, {5252,12787}, {5433,6119}, {6280,18960}, {6281,9657}, {6337,12949}, {9921,18954}, {9986,18957}, {11509,12343}, {12210,12835}, {12303,16541}, {12484,18955}, {12485,18956}, {12799,18958}, {12928,18961}, {12938,18962}, {13002,18963}, {13003,18964}, {13133,18967}, {13921,18965}, {13933,18966}, {18995,19104}, {18996,19105}

    X(18989) = orthologic center of these triangles: 2nd anti-circumperp-tangential to inner-Vecten
    X(18989) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388, 487, 12948), (999, 12601, 10083), (1469, 18990, 18988)


    X(18990) = X(5) OF 2nd ANTI-CIRCUMPERP-TANGENTIAL TRIANGLE

    Barycentrics    2*a^4-(b^2-4*b*c+c^2)*a^2-(b^2-c^2)^2 : :
    X(18990) = 5*X(1)-3*X(3058) = X(1)-3*X(5434) = 3*X(1)-X(6284) = 3*X(1)+X(10483) = 4*X(1)-3*X(15170) = 3*X(1)-2*X(15172) = X(3058)-5*X(5434) = 9*X(3058)-5*X(6284) = 3*X(3058)+5*X(7354) = 9*X(3058)+5*X(10483) = 4*X(3058)-5*X(15170) = 6*X(3058)-5*X(15171) = 9*X(3058)-10*X(15172) = 9*X(5434)-X(6284) = 3*X(5434)+X(7354) = 9*X(5434)+X(10483) = 4*X(5434)-X(15170) = 6*X(5434)-X(15171) = 9*X(5434)-2*X(15172) = X(6284)+3*X(7354) = 4*X(6284)-9*X(15170) = 2*X(6284)-3*X(15171) = 3*X(7354)-X(10483) = 4*X(7354)+3*X(15170) = 2*X(7354)+X(15171) = 3*X(7354)+2*X(15172) = 4*X(10483)+9*X(15170) = 2*X(10483)+3*X(15171) = X(10483)+2*X(15172) = 3*X(15170)-2*X(15171) = 9*X(15170)-8*X(15172) = 3*X(15171)-4*X(15172)

    X(18990) lies on these lines: {1,30}, {2,9654}, {3,388}, {4,496}, {5,56}, {7,944}, {8,2094}, {10,529}, {11,546}, {12,36}, {20,1056}, {26,18954}, {33,13488}, {34,6756}, {35,548}, {46,5252}, {55,550}, {57,355}, {65,952}, {80,3337}, {104,6831}, {145,17579}, {150,1434}, {153,6915}, {172,5305}, {202,398}, {203,397}, {221,9833}, {226,1385}, {229,3109}, {278,7511}, {330,7762}, {354,10572}, {377,956}, {381,3086}, {382,497}, {390,3529}, {404,17757}, {428,7191}, {442,2975}, {443,9708}, {474,3436}, {498,549}, {515,942}, {516,9957}, {517,4292}, {518,17647}, {519,4757}, {528,3244}, {535,1125}, {547,3614}, {612,10691}, {631,5261}, {908,17614}, {946,1387}, {950,5045}, {958,8728}, {971,12573}, {988,5725}, {993,6675}, {1010,5484}, {1012,10532}, {1015,7745}, {1058,3146}, {1060,4320}, {1111,7198}, {1155,10039}, {1159,18526}, {1210,12019}, {1317,11009}, {1319,5901}, {1376,17563}, {1388,10283}, {1420,5886}, {1425,6146}, {1428,18583}, {1466,11499}, {1469,3564}, {1470,6924}, {1479,3304}, {1482,3476}, {1483,2099}, {1484,10074}, {1565,4911}, {1595,11392}, {1596,11399}, {1617,3560}, {1656,7288}, {1657,4294}, {1699,11373}, {1737,18357}, {1770,3057}, {1788,5790}, {1837,3338}, {1870,3575}, {1885,6198}, {1909,7767}, {2066,9647}, {2067,7583}, {2192,5878}, {2242,5254}, {2551,16408}, {2646,5719}, {2886,8666}, {3023,18983}, {3028,3327}, {3090,5265}, {3149,12115}, {3297,6561}, {3298,6560}, {3303,4302}, {3333,5691}, {3339,5881}, {3361,5587}, {3421,6904}, {3474,12702}, {3475,4305}, {3485,7491}, {3486,15934}, {3487,5731}, {3488,11037}, {3523,8164}, {3526,10588}, {3528,5281}, {3530,5432}, {3576,5290}, {3582,5066}, {3583,3853}, {3584,12100}, {3612,17718}, {3616,5714}, {3622,11114}, {3628,5433}, {3654,5128}, {3665,7272}, {3670,5724}, {3671,5842}, {3746,4316}, {3814,6691}, {3815,9650}, {3822,4999}, {3830,5225}, {3843,10591}, {3845,10072}, {3850,7741}, {3851,10589}, {3861,18513}, {3878,17768}, {3911,9956}, {3920,7667}, {3925,5258}, {3947,10165}, {3957,11015}, {4067,5852}, {4084,5855}, {4187,5080}, {4190,5687}, {4308,5603}, {4312,7982}, {4321,5805}, {4355,6253}, {5044,12527}, {5049,12577}, {5083,6583}, {5088,7247}, {5122,6684}, {5134,9327}, {5217,8703}, {5221,10573}, {5260,17529}, {5267,6690}, {5268,7734}, {5272,10128}, {5322,6676}, {5427,10021}, {5435,5818}, {5450,7680}, {5552,16371}, {5693,5843}, {5708,18391}, {5730,5905}, {5762,8581}, {5795,12436}, {5840,12735}, {5844,5903}, {5874,18960}, {5875,18959}, {5902,10950}, {6502,7584}, {6644,10037}, {6645,6656}, {6907,11249}, {6920,7677}, {6922,10269}, {6923,10680}, {6928,16203}, {6934,10805}, {6938,10597}, {6948,10306}, {6958,11929}, {7051,11542}, {7284,17700}, {7502,9659}, {7526,10832}, {7530,10046}, {7737,16781}, {7743,18483}, {8227,13462}, {8727,12114}, {8757,16466}, {9578,15803}, {9597,15048}, {9661,18538}, {9672,18570}, {10065,14677}, {10069,13182}, {10081,10264}, {10089,12184}, {10090,11698}, {10091,12373}, {10385,15681}, {10527,17532}, {10826,17728}, {10943,18961}, {10954,14793}, {11236,17564}, {11238,15687}, {11681,13747}, {12085,16541}, {12102,18514}, {12605,18447}, {12616,13226}, {12945,13312}, {13117,13296}, {13411,13624}, {13925,18965}, {13993,18966}, {18971,18982}, {18972,18974}, {18973,18975}, {18995,19116}, {18996,19117}

    X(18990) = midpoint of X(i) and X(j) for these {i,j}: {1, 7354}, {1770, 3057}, {3028, 18968}
    X(18990) = reflection of X(i) in X(j) for these (i,j): (950, 5045), (5795, 12436)
    X(18990) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6147, 16137), (1, 6284, 15172), (1, 9579, 12699), (1, 10404, 6147), (1, 10483, 6284), (1, 15171, 15170), (2, 9654, 10592), (3, 388, 495), (4, 999, 496), (4, 3600, 999), (4, 14986, 9669), (388, 4293, 3), (3600, 9655, 496), (5434, 7354, 1), (6284, 7354, 10483), (6284, 15172, 15171)


    X(18991) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND ANTI-AQUILA

    Trilinears    r + 2 R sin A : :
    Barycentrics    a*((a+b+c)*a+S) : :
    X(18991) = 3*X(19054)-X(19066)

    X(18991) lies on these lines: {1,6}, {2,8983}, {3,1703}, {8,7585}, {10,3068}, {11,19077}, {30,19080}, {40,371}, {55,19010}, {57,2067}, {58,606}, {65,6415}, {81,3084}, {165,1151}, {171,8941}, {176,5222}, {214,19112}, {355,7583}, {372,3576}, {481,4644}, {482,4000}, {485,5587}, {486,8227}, {515,1587}, {516,6459}, {517,1702}, {519,19054}, {551,13959}, {590,1698}, {595,605}, {615,3624}, {631,13975}, {730,19090}, {936,1377}, {944,7581}, {946,1588}, {952,19078}, {999,19013}, {1086,1373}, {1125,3069}, {1132,9779}, {1152,7987}, {1319,18995}, {1371,17366}, {1372,7277}, {1374,17365}, {1378,9623}, {1385,3312}, {1420,6502}, {1468,6203}, {1482,6417}, {1504,9620}, {1505,9619}, {1572,5058}, {1685,9549}, {1686,9548}, {1697,2066}, {1699,3071}, {1737,13904}, {1829,5410}, {1837,19030}, {2646,19037}, {2800,19082}, {2802,19113}, {2999,13389}, {3057,19038}, {3070,5691}, {3295,18999}, {3340,16232}, {3579,6221}, {3592,7991}, {3601,5414}, {3616,7586}, {3649,19079}, {3679,13911}, {4297,6460}, {5252,19028}, {5393,7090}, {5411,11363}, {5412,7713}, {5550,13941}, {5603,7582}, {5657,13912}, {5818,13886}, {5886,7584}, {5901,19116}, {6001,19068}, {6175,16148}, {6398,13624}, {6409,16192}, {6418,10246}, {6419,7982}, {6421,9592}, {6422,9593}, {6428,15178}, {6429,9584}, {6431,11531}, {6450,17502}, {6500,10247}, {6564,18492}, {6684,9540}, {8396,12697}, {8972,9780}, {8976,9956}, {8987,14647}, {9798,19006}, {9941,19012}, {9955,13785}, {10039,13905}, {10165,13935}, {11230,13951}, {11364,18993}, {11365,19005}, {11366,19007}, {11367,19009}, {11368,19011}, {11373,19023}, {11374,19025}, {11375,19027}, {11376,19029}, {11377,19031}, {11378,19033}, {11705,19073}, {11706,19075}, {11709,19059}, {11710,19055}, {11711,19108}, {11715,19081}, {11720,19110}, {11722,19114}, {11739,19071}, {11740,19069}, {11831,19017}, {12114,19067}, {12194,18994}, {12258,19057}, {12259,19061}, {12260,19085}, {12261,19051}, {12262,19087}, {12263,19089}, {12264,19091}, {12265,19093}, {12266,19095}, {12267,19097}, {12268,19104}, {12269,19102}, {12438,19018}, {12440,19032}, {12441,19034}, {13665,18480}, {13667,19099}, {13787,19101}, {13895,17619}, {13898,17606}, {18493,18510}, {18512,18525}

    X(18991) = midpoint of X(1) and X(5589)
    X(18991) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 18992), (1, 3751, 3640), (1, 16475, 11370), (1, 19003, 7968), (1, 19004, 6), (6, 7968, 19003), (6, 7969, 1), (6, 19048, 3301), (6, 19050, 3299), (1203, 3554, 18992), (1453, 2257, 18992), (3641, 11371, 1), (7968, 19003, 18992), (7969, 19004, 18992), (9575, 16475, 18992)


    X(18992) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ANTI-AQUILA

    Trilinears    r - 2 R sin A : :
    Barycentrics    a*((a+b+c)*a-S) : :
    X(18992) = 3*X(19053)-X(19065)

    X(18992) lies on these lines: {1,6}, {2,13883}, {3,1702}, {8,7586}, {10,3069}, {11,19078}, {30,19079}, {40,372}, {55,19007}, {57,6502}, {58,605}, {65,6416}, {81,3083}, {165,1152}, {171,8945}, {175,5222}, {214,19113}, {355,7584}, {371,3576}, {481,4000}, {482,4644}, {485,8227}, {486,5587}, {515,1588}, {516,6460}, {517,1703}, {519,19053}, {551,13902}, {590,3624}, {595,606}, {615,1698}, {631,13912}, {730,19089}, {936,1378}, {944,7582}, {946,1587}, {952,19077}, {999,19014}, {1086,1374}, {1125,3068}, {1131,9779}, {1151,7987}, {1319,18996}, {1371,7277}, {1372,17366}, {1373,17365}, {1377,9623}, {1385,3311}, {1420,2067}, {1482,6418}, {1505,9620}, {1685,9548}, {1686,9549}, {1697,5414}, {1699,3070}, {1737,13962}, {1829,5411}, {1837,19029}, {2066,3601}, {2362,3340}, {2646,19038}, {2800,19081}, {2802,19112}, {2999,13388}, {3057,19037}, {3071,5691}, {3295,19000}, {3579,6398}, {3594,7991}, {3616,7585}, {3649,19080}, {3679,13973}, {4297,6459}, {5252,19027}, {5405,14121}, {5410,11363}, {5413,7713}, {5550,8972}, {5603,7581}, {5657,13975}, {5818,13939}, {5886,7583}, {5901,19117}, {6001,19067}, {6175,16149}, {6221,13624}, {6395,12702}, {6410,16192}, {6417,10246}, {6420,7982}, {6421,9593}, {6422,9592}, {6427,15178}, {6432,11531}, {6437,9585}, {6449,17502}, {6501,10247}, {6565,18492}, {6684,13935}, {8416,12698}, {8976,11230}, {9540,10165}, {9780,13941}, {9798,19005}, {9941,19011}, {9955,13665}, {9956,13951}, {10039,13963}, {11364,18994}, {11365,19006}, {11366,19008}, {11367,19010}, {11368,19012}, {11373,19024}, {11374,19026}, {11375,19028}, {11376,19030}, {11377,19032}, {11378,19034}, {11705,19074}, {11706,19076}, {11709,19060}, {11710,19056}, {11711,19109}, {11715,19082}, {11720,19111}, {11722,19115}, {11739,19070}, {11740,19072}, {11831,19018}, {12114,19068}, {12194,18993}, {12258,19058}, {12259,19062}, {12260,19086}, {12261,19052}, {12262,19088}, {12263,19090}, {12264,19092}, {12265,19094}, {12266,19096}, {12267,19098}, {12268,19105}, {12269,19103}, {12438,19017}, {12440,19031}, {12441,19033}, {13785,18480}, {13787,19100}, {13952,17619}, {13955,17606}, {13974,14647}, {18493,18512}, {18510,18525}

    X(18992) = midpoint of X(1) and X(5588)
    X(18992) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 18991), (1, 3751, 3641), (1, 16475, 11371), (1, 19003, 6), (1, 19004, 7969), (6, 7968, 1), (6, 7969, 19004), (6, 19047, 3299), (6, 19049, 3301), (1203, 3554, 18991), (1453, 2257, 18991), (3640, 11370, 1), (7968, 19003, 18991), (7969, 19004, 18991), (9575, 16475, 18991)


    X(18993) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 5th ANTI-BROCARD

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

    X(18993) lies on these lines: {2,13938}, {3,6}, {83,3069}, {98,1587}, {183,8992}, {384,19089}, {385,19090}, {486,10358}, {590,7815}, {615,7808}, {641,3815}, {1078,3068}, {1194,8577}, {1588,12110}, {1703,12197}, {3299,10801}, {3301,10802}, {4027,19108}, {5411,11380}, {7582,10788}, {7583,10104}, {7584,10796}, {7585,7793}, {7586,7787}, {7770,13983}, {7968,10800}, {10352,13989}, {10789,19003}, {10790,19005}, {10791,13936}, {10794,19023}, {10795,19025}, {10797,19027}, {10798,19029}, {10799,19037}, {10803,19047}, {11364,18991}, {11490,18999}, {11837,19007}, {11838,19009}, {11839,19017}, {11840,19031}, {11841,19033}, {12150,19053}, {12176,19055}, {12191,19057}, {12192,19059}, {12193,19061}, {12194,18992}, {12195,19065}, {12196,19067}, {12198,19077}, {12199,19081}, {12200,19085}, {12201,19051}, {12202,19087}, {12204,19075}, {12205,19073}, {12206,19091}, {12207,19093}, {12208,19095}, {12209,19097}, {12210,19104}, {12211,19102}, {12835,18995}, {13193,19110}, {13194,19112}, {13195,19114}, {13672,19099}, {13785,18502}, {13792,19101}, {16115,19079}, {18501,18510}

    X(18993) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 32, 18994), (32, 5039, 10792), (39, 372, 9995), (182, 13356, 18994), (577, 1207, 18994), (1342, 1343, 6423), (7585, 7793, 13885)


    X(18994) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 5th ANTI-BROCARD

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

    X(18994) lies on these lines: {2,13885}, {3,6}, {83,3068}, {98,1588}, {183,13983}, {384,19090}, {385,19089}, {485,10358}, {590,7808}, {615,7815}, {642,3815}, {1078,3069}, {1194,8576}, {1587,12110}, {1702,12197}, {3299,10802}, {3301,10801}, {4027,19109}, {5410,11380}, {7581,10788}, {7583,10796}, {7584,10104}, {7585,7787}, {7586,7793}, {7770,8992}, {7969,10800}, {8997,10352}, {10789,19004}, {10790,19006}, {10791,13883}, {10794,19024}, {10795,19026}, {10797,19028}, {10798,19030}, {10799,19038}, {10803,19048}, {10804,19050}, {11364,18992}, {11490,19000}, {11837,19008}, {11838,19010}, {11839,19018}, {11840,19032}, {11841,19034}, {12150,19054}, {12176,19056}, {12191,19058}, {12192,19060}, {12193,19062}, {12194,18991}, {12195,19066}, {12196,19068}, {12198,19078}, {12199,19082}, {12200,19086}, {12201,19052}, {12202,19088}, {12204,19076}, {12205,19074}, {12206,19092}, {12207,19094}, {12208,19096}, {12209,19098}, {12210,19105}, {12211,19103}, {12835,18996}, {13193,19111}, {13194,19113}, {13195,19115}, {13665,18502}, {13792,19100}, {16115,19080}, {18501,18512}

    X(18994) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 32, 18993), (32, 5039, 10793), (39, 371, 9994), (182, 13356, 18993), (577, 1207, 18993), (1342, 1343, 6424), (7586, 7793, 13938)


    X(18995) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

    X(18995) lies on these lines: {1,3312}, {2,13897}, {3,3299}, {4,19029}, {5,13955}, {6,41}, {11,1587}, {12,3069}, {34,5411}, {35,6398}, {36,3311}, {55,372}, {57,19003}, {65,6416}, {140,13905}, {371,5204}, {388,7586}, {486,10895}, {495,13963}, {498,13966}, {499,7583}, {605,1399}, {631,13901}, {999,3301}, {1152,2066}, {1155,1702}, {1317,19112}, {1319,18991}, {1335,3304}, {1378,4413}, {1388,7969}, {1420,19004}, {1478,7584}, {1588,7354}, {1703,3057}, {2099,2362}, {2361,3076}, {3023,19055}, {3027,19108}, {3028,19110}, {3068,5433}, {3070,10896}, {3071,12943}, {3085,13958}, {3086,7581}, {3295,6395}, {3297,3303}, {3298,6432}, {3320,19114}, {3585,13785}, {4293,7582}, {4641,13389}, {5010,6450}, {5219,13942}, {5221,16232}, {5252,13936}, {5298,19054}, {5420,9646}, {5432,13935}, {5434,19053}, {5563,6428}, {6020,19093}, {6221,7280}, {6284,6460}, {6285,19087}, {6459,15326}, {6471,8162}, {6560,12953}, {7288,7585}, {7741,13665}, {7951,13951}, {8416,10928}, {9655,18510}, {10588,13941}, {10590,13939}, {10592,13993}, {10881,11398}, {10944,19065}, {11375,13971}, {11510,19000}, {12688,19067}, {12835,18993}, {13904,15325}, {13959,15950}, {18954,19005}, {18955,19007}, {18956,19009}, {18957,19011}, {18958,19017}, {18961,19023}, {18962,19025}, {18963,19031}, {18964,19033}, {18967,19049}, {18968,19051}, {18969,19057}, {18970,19061}, {18971,19063}, {18972,19069}, {18973,19071}, {18974,19073}, {18975,19075}, {18976,19077}, {18977,19079}, {18978,19083}, {18979,19085}, {18982,19089}, {18983,19091}, {18984,19095}, {18985,19097}, {18986,19099}, {18987,19101}, {18988,19102}, {18989,19104}, {18990,19116}

    X(18995) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3312, 19037), (2, 19028, 13897), (3, 3299, 19038), (5, 13962, 13955), (6, 56, 18996), (6, 6502, 56), (12, 3069, 13954), (372, 1124, 55), (388, 7586, 19027), (499, 7583, 13898), (999, 6418, 3301), (1152, 2066, 5217), (2362, 7968, 2099), (3086, 7581, 19030), (18966, 19028, 2)


    X(18996) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

    X(18996) lies on these lines: {1,3311}, {2,13954}, {3,3301}, {4,19030}, {5,13898}, {6,41}, {11,1588}, {12,3068}, {34,5410}, {35,6221}, {36,3312}, {55,371}, {57,19004}, {65,6415}, {140,13963}, {372,5204}, {388,7585}, {485,10895}, {486,9661}, {495,13905}, {498,8981}, {499,7584}, {606,1399}, {631,13958}, {999,3299}, {1124,3304}, {1151,5217}, {1155,1703}, {1317,19113}, {1319,18992}, {1377,4413}, {1388,7968}, {1420,19003}, {1478,7583}, {1587,7354}, {1702,3057}, {2066,3298}, {2099,7969}, {2361,3077}, {2362,5221}, {2646,9583}, {3023,19056}, {3027,19109}, {3028,19111}, {3069,5433}, {3070,12943}, {3071,10896}, {3085,13901}, {3086,7582}, {3297,6431}, {3320,19115}, {3585,13665}, {4293,7581}, {4641,13388}, {4995,9648}, {5010,6449}, {5219,13888}, {5252,13883}, {5298,19053}, {5432,9540}, {5434,19054}, {5563,6427}, {6020,19094}, {6284,6459}, {6285,19088}, {6398,7280}, {6460,15326}, {6470,8162}, {6560,9647}, {6561,12953}, {7288,7586}, {7741,13785}, {7951,8976}, {8396,10927}, {8972,10588}, {8983,11375}, {9541,15338}, {9655,18512}, {10590,13886}, {10592,13925}, {10880,11398}, {10944,19066}, {11509,19000}, {11510,18999}, {12688,19068}, {12835,18994}, {13902,15950}, {13962,15325}, {18954,19006}, {18955,19008}, {18956,19010}, {18957,19012}, {18958,19018}, {18961,19024}, {18962,19026}, {18963,19032}, {18964,19034}, {18967,19050}, {18968,19052}, {18969,19058}, {18970,19062}, {18971,19064}, {18972,19072}, {18973,19070}, {18974,19074}, {18975,19076}, {18976,19078}, {18977,19080}, {18978,19084}, {18979,19086}, {18982,19090}, {18983,19092}, {18984,19096}, {18985,19098}, {18987,19100}, {18988,19103}, {18989,19105}, {18990,19117}

    X(18996) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3311, 19038), (2, 19027, 13954), (3, 3301, 19037), (5, 13904, 13898), (6, 56, 18995), (6, 2067, 56), (12, 3068, 13897), (371, 1335, 55), (388, 7585, 19028), (499, 7584, 13955), (999, 6417, 3299), (1151, 5414, 5217), (2066, 3298, 3303), (3298, 3592, 2066), (18965, 19027, 2)


    X(18997) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd ANTI-EXTOUCH

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

    X(18997) lies on these lines: {3,19061}, {25,19039}, {1181,3312}, {1588,1593}, {1899,18998}, {3156,18923}, {19005,19041}


    X(18998) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd ANTI-EXTOUCH

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

    X(18998) lies on these lines: {3,19062}, {25,19040}, {1181,3311}, {1587,1593}, {1899,18997}, {3155,18924}, {19006,19042}


    X(18999) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND ANTI-MANDART-INCIRCLE

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

    X(18999) lies on these lines: {1,19014}, {2,13940}, {3,1702}, {6,31}, {35,19003}, {56,7968}, {100,7586}, {197,19005}, {372,10310}, {405,13883}, {474,13971}, {590,4423}, {615,4413}, {958,19066}, {1001,3068}, {1151,8273}, {1376,3069}, {1466,6502}, {1486,19006}, {1587,11496}, {1588,11500}, {1621,7585}, {1703,10306}, {3295,18991}, {3299,11507}, {3301,11508}, {3303,7969}, {3311,10267}, {3312,11248}, {3746,19004}, {3913,19065}, {4421,19053}, {4428,19054}, {5284,8972}, {5411,11383}, {5687,13936}, {6418,11849}, {7582,11491}, {7584,11499}, {9709,13947}, {11108,13893}, {11490,18993}, {11492,19007}, {11493,19009}, {11494,19011}, {11501,19027}, {11502,19029}, {11503,19031}, {11504,19033}, {11510,18996}, {11848,19017}, {12178,19055}, {12326,19057}, {12327,19059}, {12328,19061}, {12330,19067}, {12331,19077}, {12332,19081}, {12333,19085}, {12335,19087}, {12336,19075}, {12337,19073}, {12338,19089}, {12339,19091}, {12340,19093}, {12341,19095}, {12342,19097}, {12343,19104}, {12344,19102}, {13173,19108}, {13204,19110}, {13205,19112}, {13206,19114}, {13675,19099}, {13785,18491}, {13795,19101}, {16117,19079}, {17819,18621}, {18510,18524}

    X(18999) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18992, 19013), (6, 55, 19000), (6, 3052, 606), (1621, 7585, 13887)


    X(19000) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ANTI-MANDART-INCIRCLE

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

    X(19000) lies on these lines: {1,19013}, {2,13887}, {3,1703}, {6,31}, {35,19004}, {56,2362}, {100,7585}, {197,19006}, {371,10310}, {405,13936}, {474,8983}, {590,4413}, {615,4423}, {958,19065}, {1001,3069}, {1108,7133}, {1152,8273}, {1376,3068}, {1466,2067}, {1486,19005}, {1587,11500}, {1588,11496}, {1621,7586}, {1702,10306}, {3295,18992}, {3299,11508}, {3301,11507}, {3303,7968}, {3311,11248}, {3312,10267}, {3746,19003}, {3913,19066}, {4421,19054}, {4428,19053}, {5284,13941}, {5410,11383}, {5687,13883}, {6244,9616}, {6417,11849}, {7581,11491}, {7583,11499}, {9709,13893}, {11108,13947}, {11490,18994}, {11492,19008}, {11493,19010}, {11494,19012}, {11501,19028}, {11502,19030}, {11503,19032}, {11504,19034}, {11509,18996}, {11510,18995}, {11848,19018}, {12178,19056}, {12326,19058}, {12327,19060}, {12328,19062}, {12330,19068}, {12331,19078}, {12332,19082}, {12333,19086}, {12334,19052}, {12335,19088}, {12336,19076}, {12337,19074}, {12338,19090}, {12339,19092}, {12340,19094}, {12341,19096}, {12342,19098}, {12343,19105}, {12344,19103}, {13173,19109}, {13204,19111}, {13205,19113}, {13206,19115}, {13665,18491}, {13795,19100}, {16117,19080}, {17820,18621}, {18512,18524}

    X(19000) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18991, 19014), (6, 55, 18999), (6, 3052, 605), (1621, 7586, 13940)


    X(19001) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND ANTI-ORTHOCENTROIDAL

    Barycentrics    (18*(4*R^2-SA-SW)*S^2+S*(7*S^2-36*R^2*(9*R^2+SA-2*SW)+9*SA^2-6*SB*SC)+18*(R^2*(SW+3*SA)-SA^2+SB*SC)*SA)*(SB+SC) : :

    X(19001) lies on these lines: {74,1588}, {125,19002}, {399,3312}, {2914,19095}, {3581,19051}, {7720,10814}, {12112,19059}, {13288,17812}


    X(19002) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ANTI-ORTHOCENTROIDAL

    Barycentrics    (18*(4*R^2-SA-SW)*S^2-S*(7*S^2-36*R^2*(9*R^2+SA-2*SW)+9*SA^2-6*SB*SC)+18*(R^2*(SW+3*SA)-SA^2+SB*SC)*SA)*(SB+SC) : :

    X(19002) lies on these lines: {74,1587}, {125,19001}, {399,3311}, {2914,19096}, {3581,19052}, {7721,10815}, {12112,19060}, {13287,17812}


    X(19003) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND AQUILA

    Trilinears    r - 4 R sin A : :
    Barycentrics    a*(2*(a+b+c)*a-S) : :

    X(19003) lies on these lines: {1,6}, {2,13942}, {10,7586}, {35,18999}, {36,19013}, {40,3312}, {57,18995}, {165,372}, {355,19116}, {371,7987}, {481,5222}, {485,7988}, {486,7989}, {515,7582}, {517,6418}, {615,13893}, {946,7581}, {1125,7585}, {1131,12571}, {1152,9616}, {1373,4644}, {1374,4000}, {1378,8580}, {1385,6417}, {1420,18996}, {1482,6501}, {1504,9592}, {1505,9593}, {1587,1699}, {1588,5691}, {1697,19037}, {1698,3069}, {1703,6420}, {1768,19081}, {2067,13462}, {2362,18421}, {2948,19110}, {3068,3624}, {3099,19011}, {3311,3576}, {3317,10172}, {3339,16232}, {3361,6502}, {3579,6395}, {3592,9615}, {3601,19038}, {3634,13941}, {3679,13936}, {3746,19000}, {5219,19028}, {5411,7713}, {5541,19112}, {5563,19014}, {5587,7584}, {5886,19117}, {6396,9582}, {6409,9584}, {6419,9583}, {6428,7982}, {6447,9617}, {6455,9618}, {6500,10246}, {7583,8227}, {7992,19067}, {8185,19005}, {8187,19009}, {8188,19031}, {8189,19033}, {8983,13959}, {9578,19027}, {9581,19029}, {9588,13975}, {9860,19055}, {9875,19057}, {9896,19061}, {9897,19077}, {9898,19085}, {9899,19087}, {9900,19075}, {9901,19073}, {9902,19089}, {9903,19091}, {9904,19059}, {9905,19095}, {9906,19104}, {9907,19102}, {9955,18512}, {10175,13939}, {10789,18993}, {10826,19023}, {10827,19025}, {11231,13961}, {11852,19017}, {12407,19051}, {12408,19093}, {12409,19097}, {13174,19108}, {13221,19114}, {13679,19099}, {13785,18492}, {13799,19101}, {13911,13947}, {13912,13935}, {15015,19113}, {16118,19079}, {18480,18510}

    X(19003) = reflection of X(1) in X(11370)
    X(19003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19004), (6, 7968, 18991), (6, 18992, 1), (372, 1702, 165), (1125, 7585, 13888), (1152, 9616, 16192), (1386, 3640, 1), (3068, 13971, 3624), (3069, 13883, 1698), (7968, 18991, 1), (13936, 19066, 3679), (13959, 19054, 8983), (18991, 18992, 7968), (19053, 19066, 13936)


    X(19004) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND AQUILA

    Trilinears    r + 4 R sin A : :
    Barycentrics    a*(2*(a+b+c)*a+S) : :

    X(19004) lies on these lines: {1,6}, {2,13888}, {10,7585}, {35,19000}, {36,19014}, {40,3311}, {57,18996}, {165,371}, {355,19117}, {372,7987}, {482,5222}, {485,7989}, {486,7988}, {515,7581}, {517,6417}, {590,13947}, {946,7582}, {1125,7586}, {1132,12571}, {1151,16192}, {1152,9615}, {1373,4000}, {1374,4644}, {1377,8580}, {1385,6418}, {1420,18995}, {1482,6500}, {1504,9593}, {1505,9592}, {1587,5691}, {1588,1699}, {1697,19038}, {1698,3068}, {1702,6419}, {1768,19082}, {2067,3361}, {2362,3339}, {2948,19111}, {3069,3624}, {3099,19012}, {3312,3576}, {3316,10172}, {3592,9616}, {3601,19037}, {3634,8972}, {3679,13883}, {3746,18999}, {5219,19027}, {5410,7713}, {5541,19113}, {5563,19013}, {5587,7583}, {5886,19116}, {6395,13624}, {6425,9584}, {6427,7982}, {6447,9618}, {6455,9617}, {6501,10246}, {6502,13462}, {7584,8227}, {7992,19068}, {8185,19006}, {8187,19010}, {8188,19032}, {8189,19034}, {9540,13975}, {9578,19028}, {9581,19030}, {9588,13912}, {9860,19056}, {9875,19058}, {9896,19062}, {9897,19078}, {9898,19086}, {9899,19088}, {9900,19076}, {9901,19074}, {9902,19090}, {9903,19092}, {9904,19060}, {9905,19096}, {9906,19105}, {9907,19103}, {9955,18510}, {10175,13886}, {10789,18994}, {10826,19024}, {10827,19026}, {11231,13903}, {11852,19018}, {12407,19052}, {12408,19094}, {12409,19098}, {13174,19109}, {13221,19115}, {13665,18492}, {13799,19100}, {13893,13973}, {13902,13971}, {15015,19112}, {16118,19080}, {16232,18421}, {18480,18512}

    X(19004) = reflection of X(1) in X(11371)
    X(19004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19003), (6, 7969, 18992), (6, 18991, 1), (371, 1703, 165), (372, 9583, 7987), (1125, 7586, 13942), (1386, 3641, 1), (3068, 13936, 1698), (3069, 8983, 3624), (7969, 18992, 1), (13883, 19065, 3679), (13902, 19053, 13971), (18991, 18992, 7969), (19054, 19065, 13883)


    X(19005) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND ARA

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

    X(19005) lies on these lines: {2,13943}, {3,1588}, {6,25}, {22,7586}, {24,7582}, {26,19116}, {197,18999}, {371,8277}, {372,11414}, {486,7395}, {492,1583}, {590,11284}, {605,1460}, {606,7083}, {615,7484}, {1486,19000}, {1587,1598}, {1593,3071}, {1703,9911}, {1995,7585}, {3068,5020}, {3070,5198}, {3156,8573}, {3299,10037}, {3301,10046}, {3311,6642}, {3312,7387}, {3524,9695}, {6395,12083}, {6414,6421}, {6417,7506}, {6418,7517}, {6419,8276}, {6424,8911}, {6501,18378}, {7393,13951}, {7485,13941}, {7509,13939}, {7516,13993}, {7529,7583}, {7581,10594}, {7968,8192}, {8185,19003}, {8190,19007}, {8191,19009}, {8193,13936}, {8194,19031}, {8195,19033}, {8564,8943}, {9798,18992}, {9861,19055}, {9876,19057}, {9908,19061}, {9909,19053}, {9910,19067}, {9912,19077}, {9913,19081}, {9914,19087}, {9915,19075}, {9916,19073}, {9917,19089}, {9918,19091}, {9919,19059}, {9920,19095}, {9921,19104}, {9922,19102}, {10666,12166}, {10790,18993}, {10828,19011}, {10829,19023}, {10830,19025}, {10831,19027}, {10832,19029}, {10833,19037}, {10834,19047}, {10835,19049}, {11365,18991}, {11427,15200}, {11433,15199}, {11641,19114}, {11853,19017}, {12310,19110}, {12410,19065}, {12411,19085}, {12412,19051}, {12413,19093}, {12414,19097}, {13175,19108}, {13222,19112}, {13567,15211}, {13680,19099}, {13800,19101}, {13861,19117}, {18954,18995}, {18997,19041}

    X(19005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25, 19006), (1995, 7585, 13889)


    X(19006) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ARA

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

    X(19006) lies on these lines: {2,13889}, {3,1587}, {6,25}, {22,7585}, {24,7581}, {26,19117}, {197,19000}, {371,11414}, {372,8276}, {485,7395}, {491,1584}, {590,7484}, {605,7083}, {606,1460}, {615,11284}, {1486,18999}, {1588,1598}, {1593,3070}, {1702,9911}, {1995,7586}, {3069,5020}, {3071,5198}, {3155,8573}, {3299,10046}, {3301,10037}, {3311,7387}, {3312,6642}, {3522,9694}, {6413,6422}, {6417,7517}, {6418,7506}, {6420,8277}, {6500,18378}, {7393,8976}, {7485,8972}, {7509,13886}, {7516,13925}, {7529,7584}, {7582,10594}, {7969,8192}, {8185,19004}, {8190,19008}, {8191,19010}, {8193,13883}, {8194,19032}, {8195,19034}, {8563,8939}, {9798,18991}, {9861,19056}, {9876,19058}, {9908,19062}, {9909,19054}, {9910,19068}, {9912,19078}, {9913,19082}, {9914,19088}, {9915,19076}, {9916,19074}, {9917,19090}, {9918,19092}, {9919,19060}, {9920,19096}, {9921,19105}, {9922,19103}, {10665,12166}, {10790,18994}, {10828,19012}, {10829,19024}, {10830,19026}, {10831,19028}, {10832,19030}, {10833,19038}, {10834,19048}, {10835,19050}, {11365,18992}, {11427,15199}, {11433,15200}, {11641,19115}, {11853,19018}, {12410,19066}, {12411,19086}, {12412,19052}, {12413,19094}, {12414,19098}, {13175,19109}, {13222,19113}, {13567,15212}, {13800,19100}, {13861,19116}, {16119,19080}, {18954,18996}, {18998,19042}

    X(19006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 25, 19005), (1995, 7586, 13943)


    X(19007) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 1st AURIGA

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

    X(19007) lies on these lines: {1,19010}, {2,13944}, {6,5597}, {55,18992}, {372,11822}, {1587,8196}, {1703,12458}, {3299,11877}, {3301,11879}, {3312,11252}, {3616,13891}, {5411,11384}, {5598,7968}, {5600,19066}, {5601,7586}, {6418,11875}, {7582,11843}, {7584,8200}, {7585,13890}, {8190,19005}, {8197,13936}, {8201,19031}, {8202,19033}, {11207,19053}, {11366,18991}, {11492,18999}, {11493,19013}, {11837,18993}, {11861,19011}, {11863,19017}, {11865,19023}, {11867,19025}, {11869,19027}, {11871,19029}, {11873,19037}, {11881,19047}, {11883,19049}, {12179,19055}, {12345,19057}, {12365,19059}, {12415,19061}, {12454,19065}, {12456,19067}, {12460,19077}, {12462,19081}, {12464,19085}, {12466,19051}, {12468,19087}, {12470,19075}, {12472,19073}, {12474,19089}, {12476,19091}, {12478,19093}, {12480,19095}, {12482,19097}, {12484,19104}, {12486,19102}, {13176,19108}, {13208,19110}, {13228,19112}, {13229,19114}, {13682,19099}, {13785,18495}, {13802,19101}, {16121,19079}, {18955,18995}

    X(19007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5597, 19008), (55, 18992, 19009)


    X(19008) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 1st AURIGA

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

    X(19008) lies on these lines: {1,19009}, {2,13890}, {6,5597}, {55,18991}, {371,11822}, {1588,8196}, {1702,12458}, {3068,5599}, {3299,11879}, {3301,11877}, {3311,11252}, {3616,13945}, {5410,11384}, {5598,7969}, {5600,19065}, {5601,7585}, {6417,11875}, {7581,11843}, {7583,8200}, {7586,13944}, {8190,19006}, {8197,13883}, {8201,19032}, {8202,19034}, {11366,18992}, {11492,19000}, {11493,19014}, {11837,18994}, {11861,19012}, {11863,19018}, {11865,19024}, {11867,19026}, {11869,19028}, {11871,19030}, {11873,19038}, {11881,19048}, {11883,19050}, {12179,19056}, {12345,19058}, {12365,19060}, {12415,19062}, {12454,19066}, {12456,19068}, {12460,19078}, {12462,19082}, {12464,19086}, {12466,19052}, {12468,19088}, {12470,19076}, {12472,19074}, {12474,19090}, {12476,19092}, {12478,19094}, {12480,19096}, {12482,19098}, {12484,19105}, {12486,19103}, {13176,19109}, {13208,19111}, {13228,19113}, {13665,18495}, {13802,19100}, {16121,19080}, {18955,18996}

    X(19008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5597, 19007), (55, 18991, 19010)


    X(19009) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd AURIGA

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

    X(19009) lies on these lines: {1,19008}, {2,13945}, {6,5598}, {55,18992}, {372,11823}, {1587,8203}, {1703,12459}, {3069,5600}, {3299,11878}, {3301,11880}, {3312,11253}, {3616,13890}, {5411,11385}, {5597,7968}, {5599,19066}, {5602,7586}, {6418,11876}, {7582,11844}, {7584,8207}, {7585,13891}, {8187,19003}, {8191,19005}, {8204,13936}, {8208,19031}, {8209,19033}, {11208,19053}, {11367,18991}, {11492,19013}, {11493,18999}, {11838,18993}, {11862,19011}, {11864,19017}, {11866,19023}, {11868,19025}, {11870,19027}, {11872,19029}, {11874,19037}, {11882,19047}, {11884,19049}, {12180,19055}, {12346,19057}, {12366,19059}, {12455,19065}, {12457,19067}, {12461,19077}, {12463,19081}, {12465,19085}, {12467,19051}, {12469,19087}, {12471,19075}, {12473,19073}, {12475,19089}, {12477,19091}, {12479,19093}, {12481,19095}, {12483,19097}, {12485,19104}, {12487,19102}, {13177,19108}, {13209,19110}, {13230,19112}, {13231,19114}, {13683,19099}, {13785,18497}, {13803,19101}, {16122,19079}, {18956,18995}

    X(19009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5598, 19010), (55, 18992, 19007)


    X(19010) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd AURIGA

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

    X(19010) lies on these lines: {1,19007}, {2,13891}, {6,5598}, {55,18991}, {371,11823}, {1588,8203}, {1702,12459}, {3068,5600}, {3299,11880}, {3301,11878}, {3311,11253}, {3616,13944}, {5410,11385}, {5597,7969}, {5599,19065}, {5602,7585}, {6417,11876}, {7581,11844}, {7583,8207}, {7586,13945}, {8187,19004}, {8191,19006}, {8204,13883}, {8208,19032}, {8209,19034}, {11208,19054}, {11367,18992}, {11492,19014}, {11493,19000}, {11838,18994}, {11862,19012}, {11864,19018}, {11866,19024}, {11868,19026}, {11870,19028}, {11872,19030}, {11874,19038}, {11882,19048}, {11884,19050}, {12180,19056}, {12346,19058}, {12366,19060}, {12416,19062}, {12455,19066}, {12457,19068}, {12461,19078}, {12463,19082}, {12465,19086}, {12467,19052}, {12469,19088}, {12471,19076}, {12473,19074}, {12475,19090}, {12477,19092}, {12479,19094}, {12481,19096}, {12483,19098}, {12485,19105}, {12487,19103}, {13177,19109}, {13209,19111}, {13230,19113}, {13231,19115}, {13665,18497}, {13803,19100}, {16122,19080}, {18956,18996}

    X(19010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5598, 19009), (55, 18991, 19008)


    X(19011) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 5th BROCARD

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

    X(19011) lies on these lines: {2,13946}, {3,6}, {486,10356}, {615,7914}, {1587,9993}, {1588,9873}, {1703,12497}, {2896,7586}, {3068,7846}, {3069,3096}, {3099,19003}, {3299,10038}, {3301,10047}, {5411,11386}, {7582,9862}, {7584,9996}, {7585,10583}, {7811,19053}, {7968,9997}, {8782,19108}, {9857,13936}, {9878,19057}, {9923,19061}, {9941,18992}, {9981,19075}, {9982,19073}, {9983,19089}, {9984,19059}, {9985,19095}, {9986,19104}, {9987,19102}, {10828,19005}, {10871,19023}, {10872,19025}, {10873,19027}, {10874,19029}, {10875,19031}, {10876,19033}, {10877,19037}, {10878,19047}, {10879,19049}, {11368,18991}, {11494,18999}, {11861,19007}, {11862,19009}, {11885,19017}, {12495,19065}, {12496,19067}, {12498,19077}, {12499,19081}, {12500,19085}, {12501,19051}, {12502,19087}, {12503,19093}, {12504,19097}, {13210,19110}, {13235,19112}, {13236,19114}, {13685,19099}, {13785,18500}, {13805,19101}, {16123,19079}, {18503,18510}, {18957,18995}

    X(19011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 32, 19012), (372, 5007, 10793), (7585, 10583, 13892)


    X(19012) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 5th BROCARD

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

    X(19012) lies on these lines: {2,13892}, {3,6}, {485,10356}, {590,7914}, {1587,9873}, {1588,9993}, {1702,12497}, {2896,7585}, {3068,3096}, {3069,7846}, {3099,19004}, {3299,10047}, {3301,10038}, {5410,11386}, {7581,9862}, {7583,9996}, {7586,10583}, {7811,19054}, {7969,9997}, {8782,19109}, {9857,13883}, {9878,19058}, {9923,19062}, {9941,18991}, {9981,19076}, {9982,19074}, {9983,19090}, {9984,19060}, {9985,19096}, {9986,19105}, {9987,19103}, {10828,19006}, {10871,19024}, {10872,19026}, {10873,19028}, {10874,19030}, {10875,19032}, {10876,19034}, {10877,19038}, {10878,19048}, {10879,19050}, {11368,18992}, {11494,19000}, {11861,19008}, {11862,19010}, {11885,19018}, {12495,19066}, {12496,19068}, {12498,19078}, {12499,19082}, {12500,19086}, {12501,19052}, {12502,19088}, {12503,19094}, {12504,19098}, {13210,19111}, {13235,19113}, {13236,19115}, {13665,18500}, {13805,19100}, {16123,19080}, {18503,18512}, {18957,18996}

    X(19012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 32, 19011), (371, 5007, 10792), (7586, 10583, 13946)


    X(19013) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd CIRCUMPERP TANGENTIAL

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

    X(19013) lies on these lines: {1,19000}, {3,1702}, {6,41}, {21,13940}, {36,19003}, {55,7968}, {104,7582}, {372,3428}, {405,13971}, {474,13883}, {605,4252}, {606,1191}, {956,13936}, {958,3069}, {999,18991}, {1001,13959}, {1152,5416}, {1376,19066}, {1466,16232}, {1588,12114}, {2975,7586}, {3297,5415}, {3304,7969}, {3311,10269}, {3312,11249}, {3556,17820}, {3616,13887}, {4413,13911}, {5251,13942}, {5253,7585}, {5260,13941}, {5563,19004}, {9708,13947}, {10966,19037}, {11194,19053}, {11492,19009}, {11493,19007}, {12513,19065}, {12773,19077}, {13743,19079}, {13785,18761}, {13893,16408}, {18237,19067}, {19038,19047}, {19093,19159}, {19114,19162}

    X(19013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18992, 18999), (6, 56, 19014)


    X(19014) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd CIRCUMPERP TANGENTIAL

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

    X(19014) lies on these lines: {1,18999}, {3,1703}, {6,41}, {21,13887}, {36,19004}, {55,7969}, {104,7581}, {371,3428}, {405,8983}, {474,13936}, {605,1191}, {606,4252}, {956,13883}, {958,3068}, {999,18992}, {1001,13902}, {1151,5415}, {1376,19065}, {1466,2362}, {1587,12114}, {2975,7585}, {3298,5416}, {3304,7968}, {3311,11249}, {3312,10269}, {3556,17819}, {3616,13940}, {4413,13973}, {5251,13888}, {5253,7586}, {5260,8972}, {5563,19003}, {9708,13893}, {10966,19038}, {11194,19054}, {11492,19010}, {11493,19008}, {12513,19066}, {12773,19078}, {13665,18761}, {13743,19080}, {13947,16408}, {18237,19068}, {19037,19048}, {19094,19159}, {19115,19162}

    X(19014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18991, 19000), (6, 56, 19013)


    X(19015) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND 1st EXCOSINE

    Barycentrics    (8*(3*R^2-SA-SW)*S^2+(3*S^2-16*R^2*(8*R^2+SA-2*SW)+4*SA^2-2*SB*SC)*S+8*(2*R^2*(SA+SW)-SA^2+SB*SC)*SA)*(SB+SC) : :

    X(19015) lies on these lines: {64,1588}, {1498,3312}, {1853,19016}, {6218,17810}, {17812,19059}, {17824,19095}, {17834,19061}, {17835,19051}


    X(19016) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 1st EXCOSINE

    Barycentrics    (8*(3*R^2-SA-SW)*S^2-(3*S^2-16*R^2*(8*R^2+SA-2*SW)+4*SA^2-2*SB*SC)*S+8*(2*R^2*(SA+SW)-SA^2+SB*SC)*SA)*(SB+SC) : :

    X(19016) lies on these lines: {64,1587}, {1498,3311}, {1853,19015}, {6217,17810}, {17812,19060}, {17824,19096}, {17834,19062}, {17835,19052}


    X(19017) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND GOSSARD

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

    X(19017) lies on these lines: {2,13948}, {6,402}, {30,372}, {615,15184}, {1587,11897}, {1588,12113}, {1650,3069}, {1651,19053}, {1703,12696}, {3068,15183}, {3299,11912}, {3301,11913}, {3312,11251}, {4240,7586}, {5411,11832}, {6418,11911}, {6459,16190}, {7582,11845}, {7585,13894}, {7968,11910}, {11049,13847}, {11831,18991}, {11839,18993}, {11848,18999}, {11852,19003}, {11853,19005}, {11863,19007}, {11864,19009}, {11885,19011}, {11900,13936}, {11903,19023}, {11904,19025}, {11905,19027}, {11906,19029}, {11907,19031}, {11908,19033}, {11909,19037}, {11914,19047}, {11915,19049}, {12181,19055}, {12347,19057}, {12369,19059}, {12418,19061}, {12438,18992}, {12626,19065}, {12668,19067}, {12729,19077}, {12752,19081}, {12789,19085}, {12790,19051}, {12791,19087}, {12792,19075}, {12793,19073}, {12794,19089}, {12795,19091}, {12796,19093}, {12797,19095}, {12798,19097}, {12799,19104}, {12800,19102}, {13179,19108}, {13212,19110}, {13268,19112}, {13281,19114}, {13689,19099}, {13785,18507}, {13809,19101}, {16129,19079}, {16210,19066}, {18508,18510}, {18958,18995}

    X(19017) = {X(6), X(402)}-harmonic conjugate of X(19018)


    X(19018) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND GOSSARD

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

    X(19018) lies on these lines: {2,13894}, {6,402}, {30,371}, {590,15184}, {1587,12113}, {1588,11897}, {1650,3068}, {1651,19054}, {1702,12696}, {3069,15183}, {3299,11913}, {3301,11912}, {3311,11251}, {4240,7585}, {5410,11832}, {6417,11911}, {6460,16190}, {7581,11845}, {7586,13948}, {7969,11910}, {11049,13846}, {11831,18992}, {11839,18994}, {11848,19000}, {11852,19004}, {11853,19006}, {11863,19008}, {11864,19010}, {11885,19012}, {11900,13883}, {11903,19024}, {11904,19026}, {11905,19028}, {11907,19032}, {11908,19034}, {11909,19038}, {11914,19048}, {11915,19050}, {12181,19056}, {12347,19058}, {12369,19060}, {12418,19062}, {12438,18991}, {12626,19066}, {12668,19068}, {12729,19078}, {12752,19082}, {12789,19086}, {12790,19052}, {12791,19088}, {12792,19076}, {12793,19074}, {12794,19090}, {12795,19092}, {12796,19094}, {12797,19096}, {12798,19098}, {12799,19105}, {12800,19103}, {13179,19109}, {13212,19111}, {13268,19113}, {13281,19115}, {13665,18507}, {13809,19100}, {16129,19080}, {16210,19065}, {18508,18512}, {18958,18996}

    X(19018) = {X(6), X(402)}-harmonic conjugate of X(19017)


    X(19019) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND HATZIPOLAKIS-MOSES

    Barycentrics    (2*(26*R^2+SA-5*SW)*S^2+(3*S^2+4*R^2*(112*R^2-SA-42*SW)+SA^2-2*SB*SC+16*SW^2)*S-2*(4*R^2*(2*SA-3*SW)-SA^2+SB*SC+2*SW^2)*SA)*(SB+SC) : :

    X(19019) lies on these lines: {3312,13630}, {12241,19083}


    X(19020) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND HATZIPOLAKIS-MOSES

    Barycentrics    (2*(26*R^2+SA-5*SW)*S^2-(3*S^2+4*R^2*(112*R^2-SA-42*SW)+SA^2-2*SB*SC+16*SW^2)*S-2*(4*R^2*(2*SA-3*SW)-SA^2+SB*SC+2*SW^2)*SA)*(SB+SC) : :

    X(19020) lies on these lines: {3311,13630}, {12241,19084}


    X(19021) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd HYACINTH

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

    X(19021) lies on these lines: {185,3312}, {372,17818}, {1588,1885}, {1899,6416}, {5411,19039}, {6146,19061}, {6415,13198}, {6467,10133}, {15851,19022}


    X(19022) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd HYACINTH

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

    X(19022) lies on these lines: {185,3311}, {371,17818}, {1587,1885}, {1899,6415}, {5410,19040}, {6146,19062}, {6416,13198}, {6467,10132}, {15851,19021}


    X(19023) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND INNER-JOHNSON

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

    X(19023) lies on these lines: {2,13952}, {6,11}, {12,19047}, {355,7584}, {372,11826}, {1376,3069}, {1587,10893}, {1588,12114}, {1703,12700}, {3299,10523}, {3301,10948}, {3312,10525}, {3434,7586}, {5411,11390}, {6418,11928}, {7581,10598}, {7582,10785}, {7585,10584}, {7968,10944}, {10794,18993}, {10826,19003}, {10829,19005}, {10871,19011}, {10912,19065}, {10914,13936}, {10943,19116}, {10945,19031}, {10946,19033}, {10947,19037}, {10949,19049}, {11235,19053}, {11373,18991}, {11865,19007}, {11866,19009}, {11903,19017}, {12182,19055}, {12348,19057}, {12371,19059}, {12422,19061}, {12676,19067}, {12737,19077}, {12761,19081}, {12857,19085}, {12889,19051}, {12920,19087}, {12921,19075}, {12922,19073}, {12923,19089}, {12924,19091}, {12925,19093}, {12926,19095}, {12927,19097}, {12928,19104}, {12929,19102}, {13180,19108}, {13213,19110}, {13271,19112}, {13294,19114}, {13693,19099}, {13785,18516}, {13813,19101}, {13883,17619}, {13971,17614}, {16138,19079}, {18510,18519}, {18961,18995}

    X(19023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11, 19024), (7584, 18992, 19025), (7585, 10584, 13895)


    X(19024) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND INNER-JOHNSON

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

    X(19024) lies on these lines: {2,13895}, {6,11}, {12,19048}, {355,7583}, {371,11826}, {1376,3068}, {1587,12114}, {1588,10893}, {1702,12700}, {3299,10948}, {3301,10523}, {3311,10525}, {3434,7585}, {5410,11390}, {6417,11928}, {7581,10785}, {7582,10598}, {7586,10584}, {7969,10944}, {8983,17614}, {10794,18994}, {10826,19004}, {10829,19006}, {10871,19012}, {10912,19066}, {10914,13883}, {10943,19117}, {10945,19032}, {10946,19034}, {10947,19038}, {10949,19050}, {11235,19054}, {11373,18992}, {11865,19008}, {11866,19010}, {11903,19018}, {12182,19056}, {12348,19058}, {12371,19060}, {12422,19062}, {12676,19068}, {12737,19078}, {12761,19082}, {12857,19086}, {12889,19052}, {12920,19088}, {12921,19076}, {12922,19074}, {12923,19090}, {12924,19092}, {12925,19094}, {12926,19096}, {12927,19098}, {12928,19105}, {12929,19103}, {13180,19109}, {13213,19111}, {13271,19113}, {13294,19115}, {13665,18516}, {13813,19100}, {13936,17619}, {16138,19080}, {18512,18519}, {18961,18996}

    X(19024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11, 19023), (7583, 18991, 19026), (7586, 10584, 13952)


    X(19025) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND OUTER-JOHNSON

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

    X(19025) lies on these lines: {2,13953}, {6,12}, {11,19049}, {72,13936}, {355,7584}, {372,11827}, {958,3069}, {1001,13965}, {1587,10894}, {1588,11500}, {1703,5812}, {3071,5416}, {3299,10954}, {3301,10523}, {3312,10526}, {3436,7586}, {5411,11391}, {5791,13947}, {6418,11929}, {7581,10599}, {7582,10786}, {7585,10585}, {7968,10950}, {10795,18993}, {10827,19003}, {10830,19005}, {10872,19011}, {10942,19116}, {10951,19031}, {10952,19033}, {10953,19037}, {10955,19047}, {11236,19053}, {11374,18991}, {11867,19007}, {11868,19009}, {11904,19017}, {12183,19055}, {12349,19057}, {12372,19059}, {12423,19061}, {12635,19065}, {12677,19067}, {12738,19077}, {12762,19081}, {12858,19085}, {12890,19051}, {12930,19087}, {12931,19075}, {12932,19073}, {12933,19089}, {12934,19091}, {12935,19093}, {12936,19095}, {12937,19097}, {12938,19104}, {12939,19102}, {13181,19108}, {13214,19110}, {13272,19112}, {13295,19114}, {13694,19099}, {13785,18517}, {13814,19101}, {15888,19050}, {16139,19079}, {18510,18518}, {18962,18995}

    X(19025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12, 19026), (7584, 18992, 19023), (7585, 10585, 13896)


    X(19026) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND OUTER-JOHNSON

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

    X(19026) lies on these lines: {2,13896}, {6,12}, {11,19050}, {72,13883}, {355,7583}, {371,11827}, {958,3068}, {1001,13907}, {1587,11500}, {1588,10894}, {1702,5812}, {3070,5415}, {3299,10523}, {3301,10954}, {3311,10526}, {3436,7585}, {5410,11391}, {5791,13893}, {6417,11929}, {7581,10786}, {7582,10599}, {7586,10585}, {7969,10950}, {10795,18994}, {10827,19004}, {10830,19006}, {10872,19012}, {10942,19117}, {10951,19032}, {10952,19034}, {10953,19038}, {10955,19048}, {11236,19054}, {11374,18992}, {11867,19008}, {11868,19010}, {11904,19018}, {12183,19056}, {12349,19058}, {12372,19060}, {12423,19062}, {12635,19066}, {12677,19068}, {12738,19078}, {12762,19082}, {12858,19086}, {12890,19052}, {12930,19088}, {12931,19076}, {12932,19074}, {12933,19090}, {12934,19092}, {12935,19094}, {12936,19096}, {12937,19098}, {12938,19105}, {12939,19103}, {13181,19109}, {13214,19111}, {13272,19113}, {13295,19115}, {13665,18517}, {13814,19100}, {15888,19049}, {16139,19080}, {18512,18518}, {18962,18996}

    X(19026) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12, 19025), (7583, 18991, 19024), (7586, 10585, 13953)


    X(19027) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 1st JOHNSON-YFF

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

    X(19027) lies on these lines: {1,7584}, {2,13954}, {3,13958}, {4,19037}, {5,3301}, {6,12}, {11,486}, {36,13966}, {55,1588}, {56,3069}, {65,13936}, {371,5432}, {372,7354}, {388,7586}, {485,3614}, {495,3299}, {498,3311}, {499,13951}, {606,7299}, {615,2067}, {999,13962}, {1124,15888}, {1132,5225}, {1152,15326}, {1155,13975}, {1319,13971}, {1377,3925}, {1388,13959}, {1420,13942}, {1428,13972}, {1478,3312}, {1479,13785}, {1587,10895}, {1656,13904}, {1686,10406}, {1703,1836}, {2099,19065}, {2362,3649}, {3071,5414}, {3077,5348}, {3085,7582}, {3086,13939}, {3090,13898}, {3157,19061}, {3295,18510}, {4299,6398}, {5204,13935}, {5217,6459}, {5219,19004}, {5252,18992}, {5298,13847}, {5326,5418}, {5411,11392}, {5434,6502}, {6174,9679}, {6203,10911}, {6395,9655}, {6396,9647}, {6409,9649}, {6417,13905}, {6418,9654}, {6419,9646}, {6460,12943}, {6561,15338}, {7288,13941}, {7294,8252}, {7581,10590}, {7583,7951}, {7585,10588}, {7741,18762}, {7968,10944}, {7969,15950}, {9578,19003}, {9661,10577}, {10088,19051}, {10592,19117}, {10797,18993}, {10831,19005}, {10873,19011}, {10956,19047}, {10957,19049}, {11237,19053}, {11375,18991}, {11501,18999}, {11510,13940}, {11869,19007}, {11870,19009}, {11905,19017}, {11930,19031}, {11931,19033}, {12184,19055}, {12350,19057}, {12373,19059}, {12678,19067}, {12739,19077}, {12763,19081}, {12837,19089}, {12859,19085}, {12903,19110}, {12940,19087}, {12941,19075}, {12942,19073}, {12944,19091}, {12945,19093}, {12946,19095}, {12947,19097}, {12948,19104}, {12949,19102}, {13182,19108}, {13273,19112}, {13296,19114}, {13695,19099}, {13815,19101}, {13973,16232}, {13993,15325}, {16140,19079}

    X(19027) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7584, 19029), (2, 18996, 18965), (3, 13963, 13958), (5, 3301, 19030), (6, 12, 19028), (56, 3069, 18966), (388, 7586, 18995), (486, 1335, 11), (495, 19116, 3299), (498, 3311, 13901), (615, 2067, 5433), (3071, 5414, 6284), (3085, 7582, 19038), (3086, 13939, 13955), (13954, 18996, 2)


    X(19028) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 1st JOHNSON-YFF

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

    X(19028) lies on these lines: {1,7583}, {2,13897}, {3,13901}, {4,19038}, {5,3299}, {6,12}, {11,485}, {36,8981}, {55,1587}, {56,3068}, {65,13883}, {371,7354}, {372,5432}, {388,7585}, {486,3614}, {495,3301}, {498,3312}, {499,8976}, {590,5433}, {605,7299}, {999,13904}, {1131,5225}, {1151,15326}, {1155,13912}, {1319,8983}, {1335,15888}, {1378,3925}, {1388,13902}, {1428,13910}, {1478,3311}, {1479,13665}, {1588,10895}, {1656,13962}, {1685,10406}, {1702,1836}, {2066,3070}, {2067,5434}, {2099,19066}, {2362,13911}, {3076,5348}, {3085,7581}, {3086,13886}, {3090,13955}, {3157,19062}, {3295,18512}, {3649,16232}, {4299,6221}, {5204,9540}, {5217,6460}, {5219,19003}, {5252,18991}, {5298,13846}, {5326,5420}, {5410,11392}, {6204,10910}, {6417,9654}, {6418,13963}, {6425,9649}, {6459,12943}, {6560,15338}, {7288,8972}, {7294,8253}, {7582,10590}, {7584,7951}, {7586,10588}, {7741,18538}, {7968,15950}, {7969,10944}, {8960,9661}, {9578,19004}, {10088,19052}, {10592,19116}, {10797,18994}, {10831,19006}, {10873,19012}, {10956,19048}, {10957,19050}, {11237,19054}, {11375,18992}, {11501,19000}, {11510,13887}, {11869,19008}, {11870,19010}, {11905,19018}, {11930,19032}, {11931,19034}, {12184,19056}, {12350,19058}, {12373,19060}, {12678,19068}, {12739,19078}, {12763,19082}, {12837,19090}, {12859,19086}, {12903,19111}, {12940,19088}, {12941,19076}, {12942,19074}, {12944,19092}, {12945,19094}, {12946,19096}, {12947,19098}, {12948,19105}, {12949,19103}, {13182,19109}, {13273,19113}, {13296,19115}, {13815,19100}, {13925,15325}, {16140,19080}

    X(19028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7583, 19030), (2, 18995, 18966), (3, 13905, 13901), (5, 3299, 19029), (6, 12, 19027), (56, 3068, 18965), (372, 9646, 5432), (388, 7585, 18996), (485, 1124, 11), (495, 19117, 3301), (498, 3312, 13958), (590, 6502, 5433), (2066, 3070, 6284), (3085, 7581, 19037), (13897, 18995, 2)


    X(19029) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 2nd JOHNSON-YFF

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

    X(19029) lies on these lines: {1,7584}, {2,13901}, {3,13962}, {4,18995}, {5,3299}, {6,11}, {12,486}, {35,13966}, {55,3069}, {56,1588}, {371,5433}, {372,6284}, {485,7173}, {496,3301}, {497,7586}, {498,13951}, {499,3311}, {605,5348}, {615,2066}, {999,18510}, {1069,19061}, {1132,5229}, {1152,15338}, {1478,13785}, {1479,3312}, {1587,10896}, {1656,13905}, {1703,12701}, {1837,18992}, {2098,19065}, {2330,13972}, {2646,13971}, {3057,13936}, {3058,5414}, {3071,6502}, {3076,7299}, {3085,13939}, {3086,7582}, {3090,13897}, {3295,13963}, {3297,15888}, {3601,13942}, {4302,6398}, {4995,13847}, {5204,6459}, {5217,13935}, {5218,13941}, {5326,8252}, {5411,11393}, {5418,7294}, {6395,9668}, {6396,9660}, {6409,9662}, {6417,13904}, {6418,9669}, {6419,9661}, {6460,12953}, {6561,15326}, {7581,10591}, {7583,7741}, {7585,10589}, {7951,18762}, {7968,10950}, {9581,19003}, {9646,10577}, {10091,19051}, {10593,19117}, {10798,18993}, {10832,19005}, {10874,19011}, {10958,19047}, {10959,19049}, {11238,19053}, {11376,18991}, {11502,18999}, {11871,19007}, {11872,19009}, {11906,19017}, {11932,19031}, {11933,19033}, {12185,19055}, {12351,19057}, {12374,19059}, {12679,19067}, {12740,19077}, {12764,19081}, {12836,19089}, {12904,19110}, {12950,19087}, {12951,19075}, {12952,19073}, {12954,19091}, {12955,19093}, {12956,19095}, {12957,19097}, {12958,19104}, {12959,19102}, {13183,19108}, {13274,19112}, {13297,19114}, {13696,19099}, {13816,19101}, {13883,17606}, {13989,15452}, {16141,19079}

    X(19029) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7584, 19027), (2, 19038, 13901), (3, 13962, 18966), (5, 3299, 19028), (6, 11, 19030), (55, 3069, 13958), (486, 1124, 12), (496, 19116, 3301), (497, 7586, 19037), (499, 3311, 18965), (615, 2066, 5432), (3071, 6502, 7354), (3085, 13939, 13954), (3086, 7582, 18996), (13955, 19038, 2)


    X(19030) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd JOHNSON-YFF

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

    X(19030) lies on these lines: {1,7583}, {2,13898}, {3,13904}, {4,18996}, {5,3301}, {6,11}, {12,485}, {35,8981}, {55,3068}, {56,1587}, {371,6284}, {372,5433}, {486,7173}, {496,3299}, {497,7585}, {498,8976}, {499,3312}, {590,5414}, {606,5348}, {999,18512}, {1069,19062}, {1131,5229}, {1151,15338}, {1478,13665}, {1479,3311}, {1588,10896}, {1656,13963}, {1702,12701}, {1837,18991}, {2066,3058}, {2067,3070}, {2098,19066}, {2330,13910}, {2646,8983}, {3057,13883}, {3077,7299}, {3085,13886}, {3086,7581}, {3090,13954}, {3295,13905}, {3298,15888}, {3601,13888}, {4302,6221}, {4995,13846}, {5204,6460}, {5217,9540}, {5218,8972}, {5326,8253}, {5410,11393}, {5420,7294}, {6417,9669}, {6418,13962}, {6425,9662}, {6459,12953}, {6560,15326}, {7582,10591}, {7584,7741}, {7586,10589}, {7951,18538}, {7969,10950}, {8960,9646}, {8997,15452}, {9581,19004}, {10091,19052}, {10593,19116}, {10798,18994}, {10832,19006}, {10874,19012}, {10958,19048}, {10959,19050}, {11238,19054}, {11376,18992}, {11502,19000}, {11871,19008}, {11872,19010}, {11906,19018}, {11932,19032}, {11933,19034}, {12185,19056}, {12351,19058}, {12374,19060}, {12679,19068}, {12740,19078}, {12764,19082}, {12836,19090}, {12860,19086}, {12904,19111}, {12950,19088}, {12951,19076}, {12952,19074}, {12954,19092}, {12955,19094}, {12956,19096}, {12957,19098}, {12958,19105}, {12959,19103}, {13183,19109}, {13274,19113}, {13297,19115}, {13816,19100}, {13936,17606}, {16141,19080}

    X(19030) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7583, 19028), (2, 19037, 13958), (3, 13904, 18965), (5, 3301, 19027), (6, 11, 19029), (55, 3068, 13901), (372, 9661, 5433), (485, 1335, 12), (496, 19117, 3299), (497, 7585, 19038), (499, 3312, 18966), (590, 5414, 5432), (2067, 3070, 7354), (3085, 13886, 13897), (13898, 19037, 2)


    X(19031) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND LUCAS HOMOTHETIC

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

    X(19031) lies on these lines: {2,13956}, {6,493}, {39,8408}, {372,11828}, {800,19034}, {1587,8212}, {1588,9838}, {3069,8222}, {3299,11951}, {3301,11953}, {3312,10669}, {5411,11394}, {6418,11949}, {6461,19033}, {6462,7586}, {7582,11846}, {7584,8220}, {7585,13899}, {7968,8210}, {8188,19003}, {8194,19005}, {8201,19007}, {8208,19009}, {8214,13936}, {10875,19011}, {10945,19023}, {10951,19025}, {11377,18991}, {11503,18999}, {11840,18993}, {11907,19017}, {11930,19027}, {11932,19029}, {11947,19037}, {11955,19047}, {11957,19049}, {12152,19053}, {12186,19055}, {12352,19057}, {12377,19059}, {12426,19061}, {12440,18992}, {12636,19065}, {12741,19077}, {12765,19081}, {12861,19085}, {12894,19051}, {12986,19087}, {12988,19075}, {12990,19073}, {12992,19089}, {12994,19091}, {12996,19093}, {12998,19095}, {13000,19097}, {13002,19104}, {13004,19102}, {13184,19108}, {13215,19110}, {13275,19112}, {13298,19114}, {13697,19099}, {13785,18520}, {13817,19101}, {16161,19079}, {18245,19067}, {18963,18995}

    X(19031) = {X(6), X(493)}-harmonic conjugate of X(19032)


    X(19032) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND LUCAS HOMOTHETIC

    Barycentrics    a^2*((1/2*b^2+1/2*c^2)*S+b^2*c^2)*(a^2+S) : :

    X(19032) lies on these lines: {2,13899}, {6,493}, {32,8408}, {194,6462}, {371,11828}, {1587,9838}, {1588,8212}, {3068,8222}, {3301,11951}, {3311,10669}, {5410,11394}, {6417,11949}, {6461,19034}, {7581,11846}, {7583,8220}, {7586,13956}, {7969,8210}, {8188,19004}, {8194,19006}, {8201,19008}, {8208,19010}, {8214,13883}, {8393,17475}, {10875,19012}, {10945,19024}, {10951,19026}, {11377,18992}, {11503,19000}, {11840,18994}, {11907,19018}, {11930,19028}, {11932,19030}, {11947,19038}, {11955,19048}, {11957,19050}, {12152,19054}, {12186,19056}, {12352,19058}, {12377,19060}, {12426,19062}, {12440,18991}, {12636,19066}, {12741,19078}, {12765,19082}, {12861,19086}, {12894,19052}, {12986,19088}, {12988,19076}, {12990,19074}, {12994,19092}, {12996,19094}, {12998,19096}, {13000,19098}, {13002,19105}, {13004,19103}, {13184,19109}, {13215,19111}, {13275,19113}, {13298,19115}, {13341,19033}, {13665,18520}, {13817,19100}, {16161,19080}, {18245,19068}, {18963,18996}

    X(19032) = {X(6), X(493)}-harmonic conjugate of X(19031)


    X(19033) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND LUCAS(-1) HOMOTHETIC

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

    X(19033) lies on these lines: {2,13957}, {6,494}, {32,8420}, {194,6463}, {372,11829}, {1587,8213}, {1588,9839}, {3069,8223}, {3299,11952}, {3301,11954}, {3312,10673}, {5411,11395}, {6418,11950}, {6461,19031}, {7582,11847}, {7584,8221}, {7585,13900}, {7968,8211}, {8189,19003}, {8195,19005}, {8202,19007}, {8209,19009}, {8215,13936}, {8394,17475}, {10876,19011}, {10946,19023}, {10952,19025}, {11378,18991}, {11504,18999}, {11841,18993}, {11908,19017}, {11931,19027}, {11933,19029}, {11948,19037}, {11956,19047}, {11958,19049}, {12153,19053}, {12187,19055}, {12353,19057}, {12378,19059}, {12427,19061}, {12441,18992}, {12637,19065}, {12742,19077}, {12766,19081}, {12862,19085}, {12895,19051}, {12987,19087}, {12989,19075}, {12991,19073}, {12995,19091}, {12997,19093}, {12999,19095}, {13001,19097}, {13003,19104}, {13005,19102}, {13185,19108}, {13216,19110}, {13276,19112}, {13299,19114}, {13341,19032}, {13698,19099}, {13785,18522}, {13818,19101}, {16162,19079}, {18246,19067}, {18964,18995}

    X(19033) = {X(6), X(494)}-harmonic conjugate of X(19034)


    X(19034) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND LUCAS(-1) HOMOTHETIC

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

    X(19034) lies on these lines: {2,13900}, {6,494}, {39,8420}, {371,11829}, {800,19031}, {1587,9839}, {1588,8213}, {3068,8223}, {3299,11954}, {3301,11952}, {3311,10673}, {5410,11395}, {6417,11950}, {6461,19032}, {6463,7585}, {7581,11847}, {7583,8221}, {7586,13957}, {7969,8211}, {8189,19004}, {8195,19006}, {8202,19008}, {8209,19010}, {8215,13883}, {10876,19012}, {10946,19024}, {10952,19026}, {11378,18992}, {11504,19000}, {11841,18994}, {11908,19018}, {11931,19028}, {11933,19030}, {11948,19038}, {11956,19048}, {11958,19050}, {12153,19054}, {12187,19056}, {12353,19058}, {12378,19060}, {12427,19062}, {12441,18991}, {12637,19066}, {12742,19078}, {12766,19082}, {12862,19086}, {12895,19052}, {12987,19088}, {12989,19076}, {12991,19074}, {12993,19090}, {12995,19092}, {12997,19094}, {12999,19096}, {13001,19098}, {13003,19105}, {13005,19103}, {13185,19109}, {13216,19111}, {13276,19113}, {13299,19115}, {13665,18522}, {13818,19100}, {16162,19080}, {18246,19068}, {18964,18996}

    X(19034) = {X(6), X(494)}-harmonic conjugate of X(19033)


    X(19035) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND MANDART-EXCIRCLES

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

    X(19035) lies on the line {65,6416}


    X(19036) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND MANDART-EXCIRCLES

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

    X(19036) lies on the line {65,6415}


    X(19037) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND MANDART-INCIRCLE

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

    X(19037) lies on these lines: {1,3312}, {2,13898}, {3,3301}, {4,19027}, {5,13954}, {6,31}, {11,3069}, {12,1587}, {33,5411}, {35,3311}, {36,6398}, {56,372}, {65,1703}, {140,13904}, {371,5217}, {486,10896}, {496,13962}, {497,7586}, {498,7583}, {499,13966}, {631,18965}, {999,6395}, {1100,7133}, {1124,3303}, {1152,2067}, {1317,19081}, {1399,3077}, {1479,7584}, {1588,6284}, {1697,19003}, {1837,13936}, {2098,7968}, {2646,18991}, {3023,19108}, {3027,19055}, {3028,19059}, {3057,18992}, {3058,19053}, {3068,5432}, {3070,10895}, {3071,12953}, {3085,7581}, {3086,18966}, {3295,3299}, {3297,6432}, {3298,3304}, {3320,19093}, {3583,13785}, {3601,19004}, {3746,6428}, {4294,7582}, {4995,19054}, {5010,6221}, {5218,7585}, {5420,9661}, {5433,13935}, {6020,19114}, {6450,7280}, {6459,15338}, {6460,7354}, {6560,12943}, {7355,19087}, {7741,13951}, {7951,13665}, {8416,18960}, {9645,11266}, {9668,18510}, {10535,17820}, {10589,13941}, {10591,13939}, {10593,13993}, {10799,18993}, {10833,19005}, {10877,19011}, {10881,11399}, {10947,19023}, {10950,19065}, {10953,19025}, {10965,19047}, {10966,19013}, {11376,13971}, {11873,19007}, {11874,19009}, {11909,19017}, {11947,19031}, {11948,19033}, {12354,19057}, {12428,19061}, {12680,19067}, {12743,19077}, {12863,19085}, {12896,19051}, {13075,19075}, {13076,19073}, {13077,19089}, {13078,19091}, {13079,19095}, {13080,19097}, {13081,19104}, {13082,19102}, {13699,19099}, {13819,19101}, {13905,19117}, {13947,17606}, {15171,19116}, {15452,19109}, {16142,19079}, {19014,19048}

    X(19037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3312, 18995), (2, 19030, 13898), (3, 3301, 18996), (5, 13963, 13954), (6, 55, 19038), (6, 5414, 55), (11, 3069, 13955), (372, 1335, 56), (497, 7586, 19029), (498, 7583, 13897), (1152, 2067, 5204), (3085, 7581, 19028), (3295, 6418, 3299), (3298, 3594, 6502), (3298, 6502, 3304), (13958, 19030, 2)


    X(19038) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND MANDART-INCIRCLE

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

    X(19038) lies on these lines: {1,3311}, {2,13901}, {3,3299}, {4,19028}, {5,13897}, {6,31}, {11,3068}, {12,1588}, {33,5410}, {35,3312}, {36,6221}, {44,7133}, {56,371}, {65,1702}, {140,13962}, {372,5217}, {485,10896}, {486,9646}, {496,13904}, {497,7585}, {498,7584}, {499,8981}, {631,18966}, {1151,5204}, {1155,9616}, {1317,19082}, {1319,9583}, {1335,3303}, {1399,3076}, {1479,7583}, {1587,6284}, {1697,19004}, {1837,13883}, {2067,3297}, {2098,7969}, {2646,18992}, {3023,19109}, {3027,19056}, {3028,19060}, {3057,18991}, {3058,19054}, {3069,5432}, {3070,12953}, {3071,10895}, {3085,7582}, {3086,18965}, {3295,3301}, {3298,6431}, {3320,19094}, {3583,13665}, {3601,19003}, {3746,6427}, {4294,7581}, {4995,19053}, {5010,6398}, {5218,7586}, {5298,9663}, {5433,9540}, {6020,19115}, {6449,7280}, {6459,7354}, {6460,15338}, {6560,9660}, {6561,12943}, {7355,19088}, {7741,8976}, {7951,13785}, {8396,18959}, {8972,10589}, {8983,11376}, {9541,15326}, {9645,11265}, {9668,18512}, {10535,17819}, {10591,13886}, {10593,13925}, {10799,18994}, {10833,19006}, {10877,19012}, {10880,11399}, {10947,19024}, {10950,19066}, {10953,19026}, {10965,19048}, {10966,19014}, {11873,19008}, {11874,19010}, {11909,19018}, {11947,19032}, {11948,19034}, {12354,19058}, {12428,19062}, {12680,19068}, {12743,19078}, {12863,19086}, {12896,19052}, {13075,19076}, {13076,19074}, {13077,19090}, {13078,19092}, {13079,19096}, {13080,19098}, {13081,19105}, {13082,19103}, {13819,19100}, {13893,17606}, {13963,19116}, {15171,19117}, {15452,19108}, {16142,19080}, {19013,19047}

    X(19038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3311, 18996), (2, 19029, 13955), (3, 3299, 18995), (5, 13905, 13897), (6, 55, 19037), (6, 2066, 55), (11, 3068, 13898), (371, 1124, 56), (497, 7585, 19030), (498, 7584, 13954), (1151, 6502, 5204), (2067, 3297, 3304), (3085, 7582, 19027), (3297, 3592, 2067), (13901, 19029, 2)


    X(19039) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND MIDHEIGHT

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

    X(19039) lies on these lines: {4,19087}, {5,1587}, {6,3536}, {25,18997}, {51,19041}, {125,19042}, {389,1588}, {3092,17822}, {5411,19021}, {5413,5871}, {6218,17810}, {7582,11431}, {7687,19059}, {10783,17820}, {12235,19061}, {12236,19051}, {12242,19095}, {13567,19040}


    X(19040) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND MIDHEIGHT

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

    X(19040) lies on these lines: {4,19088}, {5,1588}, {6,3535}, {25,18998}, {51,19042}, {125,19041}, {389,1587}, {3091,8969}, {3093,17822}, {5410,19022}, {5412,5870}, {6217,17810}, {7581,11431}, {7687,19060}, {10784,17819}, {12235,19062}, {12236,19052}, {12242,19096}, {13567,19039}


    X(19041) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND ORTHIC

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

    X(19041) lies on these lines: {4,1132}, {6,3128}, {25,1163}, {51,19039}, {52,19061}, {125,19040}, {185,1588}, {193,1586}, {427,5304}, {1271,3536}, {1587,3574}, {1843,5200}, {1986,19051}, {3069,10133}, {3071,5895}, {3087,3127}, {13202,19059}, {13431,13440}, {18997,19005}


    X(19042) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ORTHIC

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

    X(19042) lies on these lines: {4,1131}, {6,3127}, {25,1162}, {51,19040}, {52,19062}, {125,19039}, {185,1587}, {193,1585}, {427,5304}, {1270,3535}, {1588,3574}, {1986,19052}, {3068,5200}, {3070,5895}, {3087,3128}, {13202,19060}, {13429,13431}, {18998,19006}


    X(19043) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND ORTHOCENTROIDAL

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

    X(19043) lies on these lines: {4,19059}, {125,19044}, {381,486}, {568,19051}, {1587,7699}, {1588,5890}, {7581,19095}, {7720,10814}


    X(19044) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND ORTHOCENTROIDAL

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

    X(19044) lies on these lines: {4,19060}, {125,19043}, {381,485}, {1587,5890}, {1588,7699}, {7582,19096}, {7721,10815}


    X(19045) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GREBE AND REFLECTION

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

    X(19045) lies on these lines: {4,19095}, {6,10261}, {184,6220}, {382,3071}, {1588,6241}


    X(19046) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-GREBE AND REFLECTION

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

    X(19046) lies on these lines: {4,19096}, {6,10262}, {184,6219}, {382,3070}, {1587,6241}


    X(19047) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND INNER-YFF TANGENTS

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

    X(19047) lies on these lines: {1,6}, {2,13964}, {12,19023}, {119,486}, {371,10269}, {372,11248}, {590,10200}, {615,1378}, {1152,2077}, {1470,6502}, {1587,10531}, {1588,12115}, {1703,12703}, {3069,5552}, {3071,6256}, {3311,16203}, {3312,10679}, {5411,11400}, {5554,19066}, {6418,12000}, {6735,13973}, {7581,10596}, {7582,10805}, {7584,10942}, {7585,10586}, {7586,10528}, {9616,16209}, {10803,18993}, {10834,19005}, {10878,19011}, {10915,13936}, {10955,19025}, {10956,19027}, {10958,19029}, {10965,19037}, {11239,19053}, {11881,19007}, {11882,19009}, {11914,19017}, {11955,19031}, {11956,19033}, {12189,19055}, {12356,19057}, {12381,19059}, {12430,19061}, {12648,19065}, {12686,19067}, {12749,19077}, {12775,19081}, {12874,19085}, {12905,19051}, {13094,19087}, {13104,19075}, {13105,19073}, {13109,19089}, {13112,19091}, {13118,19093}, {13121,19095}, {13130,19097}, {13132,19104}, {13134,19102}, {13189,19108}, {13217,19110}, {13278,19112}, {13313,19114}, {13716,19099}, {13785,18542}, {13839,19101}, {16154,19079}, {16232,18838}, {18510,18545}, {19013,19038}

    X(19047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19048), (6, 1124, 19050), (6, 3297, 7969), (6, 7968, 19049), (3299, 18992, 6), (3640, 10048, 12595), (7585, 10586, 13906)


    X(19048) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND INNER-YFF TANGENTS

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

    X(19048) lies on these lines: {1,6}, {2,13906}, {12,19024}, {119,485}, {371,11248}, {372,10269}, {590,1377}, {615,10200}, {1151,2077}, {1470,2067}, {1587,12115}, {1588,10531}, {1702,12703}, {2362,18838}, {3068,5552}, {3070,6256}, {3311,10679}, {3312,16203}, {5410,11400}, {5554,19065}, {6417,12000}, {6735,13911}, {7581,10805}, {7582,10596}, {7583,10942}, {7585,10528}, {7586,10586}, {10803,18994}, {10834,19006}, {10878,19012}, {10915,13883}, {10955,19026}, {10956,19028}, {10958,19030}, {10965,19038}, {11239,19054}, {11509,18996}, {11881,19008}, {11882,19010}, {11914,19018}, {11955,19032}, {11956,19034}, {12189,19056}, {12356,19058}, {12381,19060}, {12430,19062}, {12648,19066}, {12686,19068}, {12749,19078}, {12775,19082}, {12874,19086}, {12905,19052}, {13094,19088}, {13104,19076}, {13105,19074}, {13109,19090}, {13112,19092}, {13118,19094}, {13121,19096}, {13130,19098}, {13132,19105}, {13134,19103}, {13189,19109}, {13217,19111}, {13278,19113}, {13313,19115}, {13665,18542}, {13839,19100}, {16154,19080}, {18512,18545}, {19014,19037}

    X(19048) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19047), (6, 1335, 19049), (6, 3298, 7968), (6, 7969, 19050), (3301, 18991, 6), (3641, 10049, 12595), (7586, 10586, 13964)


    X(19049) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND OUTER-YFF TANGENTS

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

    X(19049) lies on these lines: {1,6}, {2,13965}, {11,19025}, {371,10267}, {372,11249}, {590,10198}, {615,1377}, {1151,10902}, {1152,11012}, {1587,10532}, {1588,12116}, {1703,12704}, {2067,5416}, {3311,16202}, {3312,10680}, {5411,11401}, {6283,19133}, {6418,12001}, {6734,13973}, {7581,10597}, {7582,10806}, {7584,10943}, {9616,16208}, {10835,19005}, {10879,19011}, {10949,19023}, {10957,19027}, {10959,19029}, {10966,19013}, {11510,18996}, {11883,19007}, {11884,19009}, {11915,19017}, {11957,19031}, {11958,19033}, {12190,19055}, {12357,19057}, {12382,19059}, {12431,19061}, {12649,19065}, {12687,19067}, {12750,19077}, {12776,19081}, {12875,19085}, {12906,19051}, {13095,19087}, {13106,19075}, {13107,19073}, {13110,19089}, {13119,19093}, {13122,19095}, {13131,19097}, {13133,19104}, {13135,19102}, {13190,19108}, {13218,19110}, {13279,19112}, {13314,19114}, {13717,19099}, {13785,18544}, {13840,19101}, {15888,19026}, {16155,19079}, {18510,18543}, {18967,18995}

    X(19049) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19050), (6, 1335, 19048), (6, 3298, 7969), (6, 7968, 19047), (3301, 18992, 6), (3640, 10040, 12594)


    X(19050) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND OUTER-YFF TANGENTS

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

    X(19050) lies on these lines: {1,6}, {2,13907}, {11,19026}, {371,11249}, {372,10267}, {590,1378}, {615,10198}, {1151,11012}, {1152,10902}, {1587,12116}, {1588,10532}, {1702,12704}, {3068,10527}, {3311,10680}, {3312,16202}, {5410,11401}, {5415,6502}, {6405,19133}, {6417,12001}, {6734,13911}, {7581,10806}, {7582,10597}, {7583,10943}, {7585,10529}, {7586,10587}, {10804,18994}, {10835,19006}, {10879,19012}, {10916,13883}, {10949,19024}, {10957,19028}, {10959,19030}, {10966,19014}, {11240,19054}, {11510,18995}, {11883,19008}, {11884,19010}, {11915,19018}, {11957,19032}, {11958,19034}, {12190,19056}, {12357,19058}, {12382,19060}, {12431,19062}, {12649,19066}, {12687,19068}, {12750,19078}, {12776,19082}, {12906,19052}, {13095,19088}, {13106,19076}, {13107,19074}, {13110,19090}, {13113,19092}, {13119,19094}, {13122,19096}, {13131,19098}, {13133,19105}, {13135,19103}, {13190,19109}, {13218,19111}, {13279,19113}, {13314,19115}, {13665,18544}, {13840,19100}, {15888,19025}, {16155,19080}, {18512,18543}, {18967,18996}

    X(19050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6, 19049), (6, 1124, 19047), (6, 3297, 7968), (6, 7969, 19048), (3299, 18991, 6), (3641, 10041, 12594), (7586, 10587, 13965)


    X(19051) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO AAOA

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

    The reciprocal orthologic center of these triangles is X(7574)

    X(19051) lies on these lines: {2,13979}, {3,13969}, {5,19111}, {6,13}, {30,19059}, {110,7584}, {125,3311}, {371,15061}, {372,12121}, {486,14643}, {568,19043}, {615,10819}, {1511,3069}, {1587,10113}, {1588,5663}, {1656,8998}, {1986,19041}, {2777,19087}, {3071,7728}, {3299,12903}, {3301,12904}, {3312,17702}, {3448,7582}, {3581,19001}, {5411,12140}, {5972,13951}, {6221,6699}, {6398,16163}, {6418,12902}, {6419,15027}, {6459,12041}, {7583,14644}, {7585,13915}, {7586,12383}, {7968,12898}, {8981,15059}, {10088,19027}, {10091,19029}, {10264,19060}, {11801,19117}, {11804,19096}, {12201,18993}, {12236,19039}, {12261,18991}, {12407,19003}, {12412,19005}, {12466,19007}, {12467,19009}, {12501,19011}, {12778,13936}, {12790,19017}, {12889,19023}, {12890,19025}, {12894,19031}, {12895,19033}, {12896,19037}, {12905,19047}, {12906,19049}, {13961,15040}, {13966,15035}, {13972,15462}, {15535,19056}, {17835,19015}, {18968,18995}, {19095,19110}

    X(19051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 265, 19052), (7585, 15081, 13915)


    X(19052) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO AAOA

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

    The reciprocal orthologic center of these triangles is X(7574)

    X(19052) lies on these lines: {2,13915}, {3,8994}, {6,13}, {110,7583}, {125,3312}, {371,12121}, {372,15061}, {485,14643}, {568,19044}, {590,10820}, {1511,3068}, {1587,5663}, {1588,10113}, {1656,13990}, {1986,19042}, {2771,19078}, {2777,19088}, {3070,7728}, {3299,12904}, {3301,12903}, {3311,17702}, {3448,7581}, {3581,19002}, {5410,12140}, {5972,8976}, {6221,16163}, {6395,13969}, {6398,6699}, {6417,12902}, {6420,15027}, {6460,12041}, {7584,14644}, {7585,12383}, {7586,13979}, {7969,12898}, {8981,15035}, {10088,19028}, {10091,19030}, {10264,19059}, {11801,19116}, {11804,19095}, {12201,18994}, {12236,19040}, {12261,18992}, {12334,19000}, {12407,19004}, {12412,19006}, {12466,19008}, {12467,19010}, {12501,19012}, {12778,13883}, {12790,19018}, {12889,19024}, {12890,19026}, {12894,19032}, {12895,19034}, {12896,19038}, {12905,19048}, {12906,19050}, {13903,15040}, {13910,15462}, {13966,15059}, {15535,19055}, {17835,19016}, {18968,18996}

    X(19052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 265, 19051), (7586, 15081, 13979)


    X(19053) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO ANTI-ARTZT

    Barycentrics    3*a^2-S : :
    X(19053) = X(1588)+2*X(3312) = 2*X(1588)+X(6460) = X(1588)-4*X(19116) = 4*X(3312)-X(6460) = X(3312)+2*X(19116) = X(6460)+8*X(19116) = 2*X(18992)+X(19065)

    The reciprocal orthologic center of these triangles is X(2)

    X(19053) lies on these lines: {2,6}, {4,1328}, {5,6428}, {20,3594}, {30,1588}, {115,19058}, {140,6427}, {371,3524}, {372,376}, {381,1587}, {428,5411}, {485,5071}, {486,3545}, {511,19063}, {519,18992}, {528,19112}, {530,19073}, {531,19075}, {532,19071}, {533,19069}, {538,19089}, {539,19061}, {541,19059}, {542,19055}, {543,19057}, {547,13951}, {548,6448}, {549,3311}, {551,13959}, {631,6419}, {754,19091}, {1151,15692}, {1152,10304}, {1267,17120}, {1327,6436}, {1505,7739}, {1589,5158}, {1590,3284}, {1651,19017}, {2482,19109}, {3058,19037}, {3070,3839}, {3071,3543}, {3241,7968}, {3299,10056}, {3301,10072}, {3317,10576}, {3522,6426}, {3523,3592}, {3528,6454}, {3534,6395}, {3536,5702}, {3582,13962}, {3584,13963}, {3679,13936}, {3828,13947}, {3830,18510}, {3845,13785}, {4421,18999}, {4428,19000}, {4995,19038}, {5007,11291}, {5054,6417}, {5055,6501}, {5066,13665}, {5067,8960}, {5298,18996}, {5391,17121}, {5405,16670}, {5413,7714}, {5414,10385}, {5418,15709}, {5420,15702}, {5434,18995}, {5459,19074}, {5460,19076}, {5642,19111}, {6055,19056}, {6174,19113}, {6221,12100}, {6398,8703}, {6409,15705}, {6423,13798}, {6425,15717}, {6431,15708}, {6442,15640}, {6445,15716}, {6447,15712}, {6449,17504}, {6451,15711}, {6452,15759}, {6453,10299}, {6455,14891}, {6471,15683}, {6497,15714}, {6499,13925}, {6500,8981}, {6560,15682}, {6561,11001}, {7000,8550}, {7738,12969}, {7772,11292}, {7811,19011}, {8976,13993}, {8983,13942}, {9166,13968}, {9466,19090}, {9530,19093}, {9909,19005}, {10577,13886}, {11194,19013}, {11206,11242}, {11207,19007}, {11208,19009}, {11235,19023}, {11236,19025}, {11237,19027}, {11238,19029}, {11239,19047}, {11240,19049}, {11485,15764}, {12150,18993}, {12152,19031}, {12153,19033}, {13701,13769}, {13712,14482}, {13782,13832}, {13833,13843}, {13902,13971}, {13908,14061}, {18512,18762}

    X(19053) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 19054), (2, 1992, 5861), (2, 7585, 13846), (2, 13759, 591), (2, 19054, 3068), (6, 615, 7585), (6, 3069, 3068), (6, 7586, 3069), (395, 396, 8252), (591, 597, 2), (615, 13846, 2), (1588, 3312, 6460), (3069, 19054, 2), (3312, 19116, 1588), (5032, 5306, 19054), (5304, 8584, 19054)


    X(19054) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO ANTI-ARTZT

    Barycentrics    3*a^2+S : :
    X(19054) = X(1587)+2*X(3311) = 2*X(1587)+X(6459) = X(1587)-4*X(19117) = 4*X(3311)-X(6459) = X(3311)+2*X(19117) = X(6459)+8*X(19117) = 2*X(18991)+X(19066)

    The reciprocal orthologic center of these triangles is X(2)

    X(19054) lies on these lines: {2,6}, {4,1327}, {5,6427}, {20,3592}, {30,1587}, {115,19057}, {140,6428}, {371,376}, {372,3524}, {381,1588}, {428,5410}, {485,3545}, {486,5071}, {511,19064}, {519,18991}, {528,19113}, {530,19074}, {531,19076}, {532,19070}, {533,19072}, {538,19090}, {539,19062}, {541,19060}, {542,19056}, {543,19058}, {547,8976}, {548,6447}, {549,3312}, {551,13902}, {631,6420}, {754,19092}, {1151,10304}, {1152,15692}, {1267,17121}, {1328,6435}, {1504,7739}, {1589,3284}, {1590,5158}, {1651,19018}, {2066,10385}, {2482,19108}, {3058,19038}, {3070,3543}, {3071,3839}, {3090,8960}, {3241,7969}, {3299,10072}, {3301,10056}, {3316,10577}, {3522,6425}, {3523,3594}, {3528,6453}, {3535,5702}, {3582,13904}, {3584,13905}, {3679,13883}, {3828,13893}, {3830,18512}, {3845,13665}, {4421,19000}, {4428,18999}, {4995,19037}, {5007,11292}, {5054,6418}, {5055,6500}, {5066,13785}, {5298,18995}, {5391,17120}, {5393,16670}, {5412,7714}, {5418,15702}, {5420,15709}, {5434,18996}, {5459,19073}, {5460,19075}, {5642,19110}, {6055,19055}, {6174,19112}, {6221,8703}, {6395,15693}, {6396,15698}, {6398,12100}, {6410,15705}, {6424,13678}, {6426,15717}, {6432,15708}, {6441,15640}, {6446,15716}, {6448,15712}, {6450,17504}, {6451,15759}, {6452,15711}, {6454,9680}, {6456,14891}, {6470,15683}, {6496,15714}, {6498,13993}, {6501,13903}, {6560,11001}, {6561,15682}, {7374,8550}, {7738,12962}, {7772,11291}, {7811,19012}, {8963,13341}, {8983,13959}, {9166,13908}, {9466,19089}, {9530,19094}, {9681,17538}, {9909,19006}, {10576,13939}, {11194,19014}, {11206,11241}, {11207,19008}, {11208,19010}, {11235,19024}, {11236,19026}, {11237,19028}, {11238,19030}, {11239,19048}, {11240,19050}, {11486,15764}, {12150,18994}, {12152,19032}, {12153,19034}, {13662,13831}, {13720,13769}, {13821,13833}, {13835,14482}, {13888,13971}, {13925,13951}, {13968,14061}, {18510,18538}

    X(19054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6, 19053), (2, 1992, 5860), (2, 7586, 13847), (2, 13639, 1991), (2, 19053, 3069), (6, 590, 7586), (6, 3068, 3069), (6, 7585, 3068), (395, 396, 8253), (590, 13847, 2), (597, 1991, 2), (1587, 3311, 6459), (3068, 19053, 2), (3311, 19117, 1587), (5032, 5306, 19053), (5304, 8584, 19053)


    X(19055) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 1st ANTI-BROCARD

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

    The reciprocal orthologic center of these triangles is X(5999)

    X(19055) lies on these lines: {2,13967}, {3,19109}, {6,98}, {30,19057}, {99,372}, {114,3069}, {115,1587}, {147,7586}, {485,14061}, {486,6569}, {491,8781}, {542,19053}, {620,13935}, {631,8997}, {690,19059}, {1327,6568}, {1588,2794}, {2782,3312}, {2783,19112}, {2787,19081}, {2799,19093}, {3023,18995}, {3027,19037}, {3068,6036}, {3070,14639}, {3071,10722}, {3299,10053}, {3301,10069}, {3311,12042}, {3545,13968}, {5182,19146}, {5411,12131}, {6033,7584}, {6055,19054}, {6230,13773}, {6231,13760}, {6320,8416}, {6395,13188}, {6418,12188}, {6560,10723}, {7581,14651}, {7582,9862}, {7585,8980}, {7968,7970}, {9767,13926}, {9860,19003}, {9861,19005}, {9864,13936}, {11632,19058}, {11710,18991}, {11724,13959}, {12176,18993}, {12178,18999}, {12179,19007}, {12180,19009}, {12181,19017}, {12182,19023}, {12183,19025}, {12184,19027}, {12185,19029}, {12186,19031}, {12187,19033}, {12189,19047}, {12190,19049}, {13640,13790}, {13966,15561}, {15535,19052}, {19091,19116}

    X(19055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 98, 19056), (98, 10753, 6227)


    X(19056) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 1st ANTI-BROCARD

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

    The reciprocal orthologic center of these triangles is X(5999)

    X(19056) lies on these lines: {2,8980}, {3,19108}, {6,98}, {30,19058}, {99,371}, {114,3068}, {115,1588}, {147,7585}, {485,6568}, {486,14061}, {492,8781}, {542,19054}, {620,9540}, {631,13989}, {690,19060}, {1328,6569}, {1587,2794}, {2782,3311}, {2783,19113}, {2787,19082}, {2799,19094}, {3023,18996}, {3027,19038}, {3069,6036}, {3070,10722}, {3071,14639}, {3299,10069}, {3301,10053}, {3312,12042}, {3545,13908}, {5182,19145}, {5410,12131}, {6033,7583}, {6055,19053}, {6230,13640}, {6231,13653}, {6319,8396}, {6417,12188}, {6561,10723}, {7581,9862}, {7582,14651}, {7586,13967}, {7969,7970}, {8981,15561}, {9583,11711}, {9768,13873}, {9860,19004}, {9861,19006}, {9864,13883}, {11632,19057}, {11710,18992}, {11724,13902}, {12176,18994}, {12178,19000}, {12179,19008}, {12180,19010}, {12181,19018}, {12182,19024}, {12183,19026}, {12184,19028}, {12185,19030}, {12186,19032}, {12187,19034}, {12189,19048}, {12190,19050}, {13670,13760}, {15535,19051}, {19092,19117}

    X(19056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 98, 19055), (98, 10753, 6226)


    X(19057) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO ANTI-MCCAY

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

    The reciprocal orthologic center of these triangles is X(9855)

    X(19057) lies on these lines: {2,8997}, {6,598}, {30,19055}, {115,19054}, {372,12117}, {530,19075}, {531,19073}, {542,1588}, {543,19053}, {1587,9880}, {2482,3069}, {2782,19063}, {3068,5461}, {3299,10054}, {3301,10070}, {3524,13967}, {5411,12132}, {5465,19111}, {5969,19089}, {6418,12355}, {7582,12243}, {7584,8724}, {7585,13908}, {7586,8591}, {7968,9884}, {9875,19003}, {9876,19005}, {9878,19011}, {9881,13936}, {9892,13761}, {11632,19056}, {12158,13927}, {12191,18993}, {12258,18991}, {12326,18999}, {12345,19007}, {12346,19009}, {12347,19017}, {12348,19023}, {12349,19025}, {12350,19027}, {12351,19029}, {12352,19031}, {12353,19033}, {12354,19037}, {12356,19047}, {12357,19049}, {13642,13796}, {13846,14061}, {18969,18995}

    X(19057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 671, 19058), (671, 8593, 9882)


    X(19058) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO ANTI-MCCAY

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

    The reciprocal orthologic center of these triangles is X(9855)

    X(19058) lies on these lines: {2,13908}, {6,598}, {30,19056}, {115,19053}, {371,12117}, {530,19076}, {531,19074}, {542,1587}, {543,19054}, {1588,9880}, {2482,3068}, {2782,19064}, {3069,5461}, {3299,10070}, {3301,10054}, {3524,8980}, {5410,12132}, {5465,19110}, {5969,19090}, {6417,12355}, {7581,12243}, {7583,8724}, {7585,8591}, {7586,13968}, {7969,9884}, {9875,19004}, {9876,19006}, {9878,19012}, {9881,13883}, {9894,13642}, {11632,19055}, {12159,13874}, {12191,18994}, {12258,18992}, {12326,19000}, {12345,19008}, {12346,19010}, {12347,19018}, {12348,19024}, {12349,19026}, {12350,19028}, {12351,19030}, {12352,19032}, {12353,19034}, {12354,19038}, {12356,19048}, {12357,19050}, {13676,13761}, {13847,14061}, {18969,18996}

    X(19058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 671, 19057), (671, 8593, 9883)


    X(19059) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO ANTI-ORTHOCENTROIDAL

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

    The reciprocal orthologic center of these triangles is X(12112)

    X(19059) lies on these lines: {2,13969}, {3,19111}, {4,19043}, {6,74}, {30,19051}, {110,372}, {113,3069}, {125,1587}, {146,7586}, {371,10817}, {381,13979}, {399,6395}, {485,15059}, {541,19053}, {542,19108}, {631,8998}, {690,19055}, {1152,15035}, {1511,6398}, {1539,13785}, {1588,2777}, {2771,19112}, {3028,19037}, {3068,6699}, {3070,14644}, {3071,10721}, {3299,10065}, {3301,10081}, {3311,12041}, {3312,5663}, {3594,14094}, {5411,12133}, {5972,13935}, {6396,10818}, {6410,15036}, {6417,15041}, {6418,10620}, {6419,15021}, {6420,15054}, {6426,15034}, {6454,15020}, {6459,16111}, {6460,17702}, {6560,10733}, {7582,12244}, {7583,15061}, {7584,7728}, {7585,8994}, {7687,19039}, {7733,8416}, {7968,7978}, {8674,19081}, {9517,19093}, {9904,19003}, {9919,19005}, {9984,19011}, {10264,19052}, {10628,19095}, {11709,18991}, {11723,13959}, {12112,19001}, {12192,18993}, {12327,18999}, {12365,19007}, {12366,19009}, {12368,13936}, {12369,19017}, {12371,19023}, {12372,19025}, {12373,19027}, {12374,19029}, {12377,19031}, {12378,19033}, {12381,19047}, {12382,19049}, {13202,19041}, {13915,18512}, {13966,14643}, {17812,19015}

    X(19059) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 74, 19060), (74, 10752, 7725), (372, 12375, 10820)


    X(19060) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO ANTI-ORTHOCENTROIDAL

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

    The reciprocal orthologic center of these triangles is X(12112)

    X(19060) lies on these lines: {2,8994}, {3,19110}, {4,19044}, {6,74}, {110,371}, {113,3068}, {125,1588}, {146,7585}, {372,10818}, {381,13915}, {486,15059}, {541,19054}, {542,19109}, {631,13990}, {690,19056}, {1151,15035}, {1511,6221}, {1539,13665}, {1587,2777}, {2771,19113}, {3028,19038}, {3069,6699}, {3070,10721}, {3071,14644}, {3299,10081}, {3301,10065}, {3311,5663}, {3312,12041}, {3592,14094}, {5410,12133}, {5972,9540}, {6409,15036}, {6417,10620}, {6418,15041}, {6419,15054}, {6420,15021}, {6425,15034}, {6453,15020}, {6459,17702}, {6460,16111}, {6561,10733}, {7581,12244}, {7583,7728}, {7584,15061}, {7586,13969}, {7687,19040}, {7732,8396}, {7969,7978}, {8674,19082}, {8981,14643}, {9517,19094}, {9541,16163}, {9583,11720}, {9904,19004}, {9919,19006}, {9984,19012}, {10264,19051}, {10628,19096}, {11709,18992}, {11723,13902}, {12112,19002}, {12192,18994}, {12327,19000}, {12365,19008}, {12366,19010}, {12368,13883}, {12369,19018}, {12371,19024}, {12372,19026}, {12373,19028}, {12374,19030}, {12377,19032}, {12378,19034}, {12381,19048}, {12382,19050}, {13202,19042}, {13979,18510}, {17812,19016}

    X(19060) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 74, 19059), (74, 10752, 7726), (371, 12376, 10819)


    X(19061) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO ARIES

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

    The reciprocal orthologic center of these triangles is X(9833)

    X(19061) lies on these lines: {2,13970}, {3,18997}, {5,6}, {30,19087}, {52,19041}, {372,12118}, {539,19053}, {615,8909}, {1069,19029}, {1147,3069}, {1587,9927}, {1588,13754}, {3068,5449}, {3157,19027}, {3299,10055}, {3301,10071}, {3311,12359}, {3536,6515}, {5054,8912}, {5411,12134}, {6146,19021}, {6418,12429}, {6458,10898}, {6459,7689}, {6460,17702}, {7505,11447}, {7582,11411}, {7585,13909}, {7968,9933}, {9820,13951}, {9833,11266}, {9896,19003}, {9908,19005}, {9923,19011}, {9928,13936}, {10116,18924}, {12038,13935}, {12164,18510}, {12193,18993}, {12235,19039}, {12259,18991}, {12328,18999}, {12415,19007}, {12418,19017}, {12422,19023}, {12423,19025}, {12426,19031}, {12427,19033}, {12428,19037}, {12430,19047}, {12431,19049}, {13383,17819}, {17834,19015}, {18970,18995}

    X(19061) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 68, 19062), (486, 10665, 5654)


    X(19062) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO ARIES

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

    The reciprocal orthologic center of these triangles is X(9833)

    X(19062) lies on these lines: {2,13909}, {3,18998}, {5,6}, {30,19088}, {52,19042}, {371,12118}, {539,19054}, {1069,19030}, {1147,3068}, {1587,13754}, {1588,9927}, {3069,5449}, {3157,19028}, {3299,10071}, {3301,10055}, {3312,12359}, {3535,6515}, {5410,12134}, {6146,19022}, {6417,12429}, {6457,10897}, {6459,17702}, {6460,7689}, {7505,11448}, {7581,11411}, {7586,13970}, {7969,9933}, {8976,9820}, {9540,12038}, {9833,11265}, {9896,19004}, {9908,19006}, {9923,19012}, {9928,13883}, {10116,18923}, {12164,18512}, {12193,18994}, {12235,19040}, {12259,18992}, {12328,19000}, {12415,19008}, {12416,19010}, {12418,19018}, {12422,19024}, {12423,19026}, {12426,19032}, {12427,19034}, {12428,19038}, {12430,19048}, {12431,19050}, {13383,17820}, {17834,19016}, {18970,18996}

    X(19062) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 68, 19061), (485, 10666, 5654)


    X(19063) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 1st BROCARD-REFLECTED

    Barycentrics    ((2*b^2+c^2)*a^2+(b^2-c^2)*c^2)*((b^2+2*c^2)*a^2-(b^2-c^2)*b^2)-12*S*a^2*((b^2+c^2)*a^2+b^2*c^2) : :
    X(19063) = X(19089)-4*X(19116)

    The reciprocal orthologic center of these triangles is X(3)

    X(19063) lies on these lines: {6,98}, {511,19053}, {2782,19057}, {3069,15819}, {3312,19091}, {6194,7586}, {6419,7786}, {6427,11272}, {6428,14881}, {7582,7709}, {7584,7697}, {8992,13939}, {18971,18995}, {19089,19116}

    X(19063) = {X(6), X(262)}-harmonic conjugate of X(19064)


    X(19064) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 1st BROCARD-REFLECTED

    Barycentrics    ((2*b^2+c^2)*a^2+(b^2-c^2)*c^2)*((b^2+2*c^2)*a^2-(b^2-c^2)*b^2)+12*S*a^2*((b^2+c^2)*a^2+b^2*c^2) : :
    X(19064) = X(19090)-4*X(19117)

    The reciprocal orthologic center of these triangles is X(3)

    X(19064) lies on these lines: {6,98}, {511,19054}, {2782,19058}, {3068,15819}, {3311,19092}, {6194,7585}, {6420,7786}, {6427,14881}, {6428,11272}, {7581,7709}, {7583,7697}, {13886,13983}, {18971,18996}, {19090,19117}

    X(19064) = {X(6), X(262)}-harmonic conjugate of X(19063)


    X(19065) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO EXCENTERS-MIDPOINTS

    Barycentrics    -(-a+b+c)*S+a^2*(a+b+c) : :
    X(19065) = 2*X(18992)-3*X(19053)

    The reciprocal orthologic center of these triangles is X(10)

    X(19065) lies on these lines: {1,1123}, {2,7969}, {6,8}, {10,3068}, {40,6459}, {145,7586}, {355,1587}, {371,5657}, {372,944}, {388,2362}, {485,5818}, {486,5603}, {515,1703}, {517,1588}, {519,18992}, {590,9780}, {605,5255}, {606,5247}, {615,3616}, {758,19079}, {938,3298}, {952,3312}, {958,19000}, {962,3071}, {1125,13947}, {1145,19113}, {1152,5731}, {1335,18391}, {1376,19014}, {1385,13935}, {1388,18966}, {1482,7584}, {1698,8983}, {1702,11362}, {1788,2067}, {1999,13458}, {2098,19029}, {2099,19027}, {2802,19077}, {3189,5416}, {3299,12647}, {3303,13940}, {3311,5690}, {3476,6502}, {3486,5414}, {3576,13975}, {3579,9541}, {3617,7585}, {3622,13941}, {3679,13883}, {3913,18999}, {5411,12135}, {5413,7718}, {5550,8252}, {5554,19048}, {5599,19010}, {5600,19008}, {5790,7583}, {5844,19116}, {5901,13951}, {6361,6561}, {6395,18526}, {6418,12645}, {6684,9583}, {7582,12245}, {8148,18510}, {9615,10164}, {10246,13966}, {10283,13993}, {10595,13939}, {10912,19023}, {10944,18995}, {10950,19037}, {12195,18993}, {12410,19005}, {12454,19007}, {12455,19009}, {12495,19011}, {12513,19013}, {12626,19017}, {12635,19025}, {12636,19031}, {12637,19033}, {12648,19047}, {12649,19049}, {13665,18357}, {13904,18395}, {13954,15950}, {13976,16173}, {14839,19089}, {15863,19078}, {16210,19018}, {18493,18762}

    X(19065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3069, 13959), (1, 7090, 6351), (1, 13936, 3069), (2, 7969, 13902), (6, 8, 19066), (10, 18991, 3068), (145, 7586, 7968), (3617, 7585, 13911), (3679, 19004, 13883), (7969, 13973, 2), (13883, 19004, 19054)


    X(19066) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO EXCENTERS-MIDPOINTS

    Barycentrics    (-a+b+c)*S+a^2*(a+b+c) : :
    X(19066) = 2*X(18991)-3*X(19054)

    The reciprocal orthologic center of these triangles is X(10)

    X(19066) lies on these lines: {1,1336}, {2,7968}, {6,8}, {10,3069}, {40,6460}, {145,7585}, {355,1588}, {371,944}, {372,5657}, {388,16232}, {485,5603}, {486,5818}, {515,1702}, {517,1587}, {519,18991}, {590,3616}, {605,5247}, {606,5255}, {615,9780}, {758,19080}, {938,3297}, {952,3311}, {958,18999}, {962,3070}, {1124,18391}, {1125,13893}, {1145,19112}, {1151,5731}, {1376,19013}, {1385,9540}, {1388,18965}, {1482,7583}, {1698,13971}, {1703,11362}, {1788,6502}, {1999,13425}, {2066,3486}, {2067,3476}, {2098,19030}, {2099,19028}, {2802,19078}, {3189,5415}, {3299,10573}, {3301,12647}, {3303,13887}, {3312,5690}, {3576,13912}, {3617,7586}, {3622,8972}, {3679,13936}, {3913,19000}, {4297,9616}, {5410,12135}, {5412,7718}, {5550,8253}, {5554,19047}, {5599,19009}, {5600,19007}, {5790,7584}, {5844,19117}, {5882,9583}, {5901,8976}, {6361,6560}, {6417,12645}, {7581,12245}, {8148,18512}, {8981,10246}, {9541,18481}, {10283,13925}, {10595,13886}, {10912,19024}, {10944,18996}, {10950,19038}, {12195,18994}, {12410,19006}, {12454,19008}, {12455,19010}, {12495,19012}, {12513,19014}, {12626,19018}, {12635,19026}, {12636,19032}, {12637,19034}, {12648,19048}, {12649,19050}, {13785,18357}, {13897,15950}, {13962,18395}, {14839,19090}, {15863,19077}, {16210,19017}, {18493,18538}

    X(19066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3068, 13902), (1, 13883, 3068), (1, 14121, 6352), (2, 7968, 13959), (6, 8, 19065), (10, 18992, 3069), (145, 7585, 7969), (3617, 7586, 13973), (3679, 19003, 13936), (7968, 13911, 2), (13936, 19003, 19053)


    X(19067) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO EXTOUCH

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

    The reciprocal orthologic center of these triangles is X(40)

    X(19067) lies on these lines: {2,13974}, {6,84}, {372,1490}, {515,1703}, {971,3312}, {1124,12705}, {1158,1702}, {1587,6245}, {1709,3299}, {2829,19077}, {3068,6705}, {3069,6260}, {3301,10085}, {5411,12136}, {5450,9583}, {6001,18992}, {6259,7584}, {6418,12684}, {7582,12246}, {7585,8987}, {7968,7971}, {7992,19003}, {9910,19005}, {12114,18991}, {12196,18993}, {12330,18999}, {12456,19007}, {12457,19009}, {12496,19011}, {12608,13964}, {12667,13936}, {12668,19017}, {12676,19023}, {12677,19025}, {12678,19027}, {12679,19029}, {12680,19037}, {12686,19047}, {12687,19049}, {12688,18995}, {13883,14647}, {13947,18242}, {18237,19013}, {18245,19031}, {18246,19033}

    X(19067) = {X(6), X(84)}-harmonic conjugate of X(19068)


    X(19068) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO EXTOUCH

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

    The reciprocal orthologic center of these triangles is X(40)

    X(19068) lies on these lines: {2,8987}, {6,84}, {371,1490}, {515,1702}, {971,3311}, {1158,1703}, {1335,12705}, {1588,6245}, {1709,3301}, {2829,19078}, {3068,6260}, {3069,6705}, {3299,10085}, {5410,12136}, {6001,18991}, {6259,7583}, {6261,9583}, {6417,12684}, {6796,9582}, {7581,12246}, {7586,13974}, {7969,7971}, {7992,19004}, {9616,11500}, {9910,19006}, {12114,18992}, {12196,18994}, {12330,19000}, {12456,19008}, {12457,19010}, {12496,19012}, {12608,13906}, {12667,13883}, {12668,19018}, {12676,19024}, {12677,19026}, {12678,19028}, {12679,19030}, {12680,19038}, {12686,19048}, {12687,19050}, {12688,18996}, {13893,18242}, {13936,14647}, {18237,19014}, {18245,19032}, {18246,19034}

    X(19068) = {X(6), X(84)}-harmonic conjugate of X(19067)


    X(19069) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO INNER-FERMAT

    Barycentrics    4*S^2-(10+sqrt(3))*(SB+SC)*S-(SB+SC)*(SA-2*sqrt(3)*SW) : :

    The reciprocal orthologic center of these triangles is X(3)

    X(19069) lies on these lines: {6,17}, {533,19053}, {628,7586}, {630,3069}, {3068,6674}, {3312,19075}, {6418,16628}, {7584,16627}, {10612,19076}, {11740,18991}, {18972,18995}, {19073,19116}

    X(19069) = {X(6), X(18)}-harmonic conjugate of X(19072)


    X(19070) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO INNER-FERMAT

    Barycentrics    4*S^2+(10+sqrt(3))*(SB+SC)*S-(SB+SC)*(SA-2*sqrt(3)*SW) : :

    The reciprocal orthologic center of these triangles is X(3)

    X(19070) lies on these lines: {6,17}, {532,19054}, {627,7585}, {629,3068}, {3069,6673}, {3311,19074}, {6417,16629}, {7583,16626}, {10611,19073}, {11739,18992}, {18973,18996}, {19076,19117}

    X(19070) = {X(6), X(17)}-harmonic conjugate of X(19071)


    X(19071) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO OUTER-FERMAT

    Barycentrics    4*S^2-(10-sqrt(3))*(SB+SC)*S-(SB+SC)*(SA+2*sqrt(3)*SW) : :

    The reciprocal orthologic center of these triangles is X(3)

    X(19071) lies on these lines: {6,17}, {532,19053}, {627,7586}, {629,3069}, {3068,6673}, {3312,19073}, {6418,16629}, {7584,16626}, {10611,19074}, {11739,18991}, {18973,18995}, {19075,19116}

    X(19071) = {X(6), X(17)}-harmonic conjugate of X(19070)


    X(19072) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO OUTER-FERMAT

    Barycentrics    4*S^2+(10-sqrt(3))*(SB+SC)*S-(SB+SC)*(SA+2*sqrt(3)*SW) : :

    The reciprocal orthologic center of these triangles is X(3)

    X(19072) lies on these lines: {6,17}, {533,19054}, {628,7585}, {630,3068}, {3069,6674}, {3311,19076}, {6417,16628}, {7583,16627}, {10612,19075}, {11740,18992}, {18972,18996}, {19074,19117}

    X(19072) = {X(6), X(18)}-harmonic conjugate of X(19069)


    X(19073) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE TO 3rd FERMAT-DAO

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

    The reciprocal orthologic center of these triangles is X(3)

    X(19073) lies on these lines: {2,13982}, {6,13}, {372,5473}, {530,19053}, {531,19057}, {616,7586}, {618,3069}, {619,19109}, {1587,5478}, {3068,6669}, {3299,10062}, {3301,10078}, {3311,6771}, {3312,19071}, {5411,12142}, {5459,19054}, {5617,7584}, {6302,13765}, {6418,13103}, {6428,16001}, {6770,7582}, {7585,13917}, {7968,7975}, {9901,19003}, {9916,19005}, {9982,19011}, {10611,19070}, {11705,18991}, {12205,18993}, {12337,18999}, {12472,19007}, {12473,19009}, {12781,13936}, {12793,19017}, {12922,19023}, {12932,19025}, {12942,19027}, {12952,19029}, {12990,19031}, {12991,19033}, {13076,19037}, {13105,19047}, {13107,19049}, {13646,13825}, {13916,14061}, {18974,18995}, {19069,19116}

    X(19073) = {X(6), X(13)}-harmonic conjugate of X(19074)


    X(19074) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE TO 3rd FERMAT-DAO

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

    The reciprocal orthologic center of these triangles is X(3)

    X(19074) lies on these lines: {2,13917}, {6,13}, {371,5473}, {530,19054}, {531,19058}, {616,7585}, {618,3068}, {619,19108}, {1588,5478}, {3069,6669}, {3299,10078}, {3301,10062}, {3311,19070}, {3312,6771}, {5410,12142}, {5459,19053}, {5617,7583}, {6306,13646}, {6417,13103}, {6427,16001}, {6770,7581}, {7586,13982}, {7969,7975}, {9901,19004}, {9916,19006}, {9982,19012}, {10611,19071}, {11705,18992}, {12205,18994}, {12337,19000}, {12472,19008}, {12473,19010}, {12781,13883}, {12793,19018}, {12922,19024}, {12932,19026}, {12942,19028}, {12952,19030}, {12990,19032}, {12991,19034}, {13076,19038}, {13105,19048}, {13107,19050}, {13705,13765}, {13981,14061}, {18974,18996}, {19072,19117}

    X(19074) = {X(6), X(13)}-harmonic co