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This is PART 23: Centers X(44001) - X(46000)

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

X(44001) = VIJAY 11TH PARALLEL TRANSFORM OF X(100)

Barycentrics    (a - b)^2*(a - c)^2*(2*a^5 - 2*a^4*b - 3*a^3*b^2 + 3*a^2*b^3 + 2*a*b^4 - 2*b^5 - 2*a^4*c + 8*a^3*b*c - 3*a^2*b^2*c - 9*a*b^3*c + 6*b^4*c - 3*a^3*c^2 - 3*a^2*b*c^2 + 14*a*b^2*c^2 - 4*b^3*c^2 + 3*a^2*c^3 - 9*a*b*c^3 - 4*b^2*c^3 + 2*a*c^4 + 6*b*c^4 - 2*c^5) : :

X(44001) lies on these lines: {100, 693}, {1252, 31272}

X(44002) = VIJAY 11TH PARALLEL TRANSFORM OF X(101)

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

X(44002) lies on these lines: {101, 3261}, {23990, 31273}

X(44003) = VIJAY 12TH PARALLEL TRANSFORM OF X(3)

Barycentrics    a^10*b^2 - 6*a^6*b^6 + 8*a^4*b^8 - 3*a^2*b^10 + a^10*c^2 - 4*a^8*b^2*c^2 + 7*a^6*b^4*c^2 - 5*a^4*b^6*c^2 + b^10*c^2 + 7*a^6*b^2*c^4 - 6*a^4*b^4*c^4 + 3*a^2*b^6*c^4 - 4*b^8*c^4 - 6*a^6*c^6 - 5*a^4*b^2*c^6 + 3*a^2*b^4*c^6 + 6*b^6*c^6 + 8*a^4*c^8 - 4*b^4*c^8 - 3*a^2*c^10 + b^2*c^10 : :
X(44003) = 3 X[2] - 4 X[2972], 5 X[3091] - 4 X[34334], 5 X[3522] - 4 X[40948]

X(44003) lies on these lines: {2, 1972}, {3, 35311}, {4, 6662}, {20, 5663}, {107, 14919}, {253, 18019}, {1294, 15054}, {2979, 42329}, {3091, 34334}, {3164, 11794}, {3448, 9033}, {3522, 40948}, {5889, 15318}, {6563, 25053}, {7500, 20213}, {14570, 41673}, {23061, 43768}

X(44003) = reflection of X(35360) in X(2972)
X(44003) = anticomplement of X(35360)
X(44003) = anticomplement of the isogonal conjugate of X(23286)
X(44003) = X(44715)-of-anti-Euler-triangle
X(44003) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {54, 7253}, {95, 21300}, {97, 7192}, {656, 2888}, {810, 17035}, {2148, 525}, {2167, 850}, {2168, 14618}, {2169, 523}, {2190, 520}, {2616, 4}, {2623, 5905}, {14533, 4560}, {15412, 21270}, {15958, 6758}, {23286, 8}, {34386, 17217}, {36134, 110}
X(44003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2972, 35360, 2}, {3448, 39352, 44004}, {34186, 39352, 3448}

X(44004) = VIJAY 12TH PARALLEL TRANSFORM OF X(5)

Barycentrics    2*a^12 - 5*a^10*b^2 + 2*a^8*b^4 + 6*a^6*b^6 - 10*a^4*b^8 + 7*a^2*b^10 - 2*b^12 - 5*a^10*c^2 + 10*a^8*b^2*c^2 - 9*a^6*b^4*c^2 + 11*a^4*b^6*c^2 - 10*a^2*b^8*c^2 + 3*b^10*c^2 + 2*a^8*c^4 - 9*a^6*b^2*c^4 - 2*a^4*b^4*c^4 + 3*a^2*b^6*c^4 + 6*b^8*c^4 + 6*a^6*c^6 + 11*a^4*b^2*c^6 + 3*a^2*b^4*c^6 - 14*b^6*c^6 - 10*a^4*c^8 - 10*a^2*b^2*c^8 + 6*b^4*c^8 + 7*a^2*c^10 + 3*b^2*c^10 - 2*c^12 : :
X(44004) = 3 X[2] - 4 X[35442]

X(44004) lies on these lines: {2, 35311}, {20, 10620}, {3448, 9033}, {5965, 43768}, {15319, 15801}

X(44004) = reflection of X(35311) in X(35442)
X(44004) = anticomplement of X(35311)
X(44004) = anticomplement of the isogonal conjugate of X(39180)
X(44004) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {656, 2889}, {1173, 7253}, {31626, 7192}, {39180, 8}, {39181, 21271}, {39183, 21270}, {40410, 21300}
X(44004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3448, 39352, 44003}, {35311, 35442, 2}

X(44005) = VIJAY 12TH PARALLEL TRANSFORM OF X(9)

Barycentrics    a^5*b - 6*a^3*b^3 + 8*a^2*b^4 - 3*a*b^5 + a^5*c - 4*a^4*b*c + 7*a^3*b^2*c - 5*a^2*b^3*c + b^5*c + 7*a^3*b*c^2 - 6*a^2*b^2*c^2 + 3*a*b^3*c^2 - 4*b^4*c^2 - 6*a^3*c^3 - 5*a^2*b*c^3 + 3*a*b^2*c^3 + 6*b^3*c^3 + 8*a^2*c^4 - 4*b^2*c^4 - 3*a*c^5 + b*c^5 : :
X(44005) = 3 X[2] - 4 X[3119]

X(44005) lies on these lines: {2, 3119}, {144, 20095}, {149, 6366}, {658, 41798}, {3448, 14732}, {10405, 18359}

X(44005) = reflection of X(35312) in X(3119)
X(44005) = anticomplement of X(35312)
X(44005) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {650, 2890}, {1174, 693}, {2346, 21302}, {6605, 20295}, {10482, 513}
X(44005) = {X(3119),X(35312)}-harmonic conjugate of X(2)

X(44006) = VIJAY 12TH PARALLEL TRANSFORM OF X(10)

Barycentrics    2*a^3 - a^2*b - a*b^2 - 2*b^3 - a^2*c + 3*b^2*c - a*c^2 + 3*b*c^2 - 2*c^3 : :
X(44006) = 3 X[2] - 4 X[3120], 5 X[4442] - 2 X[4831], 4 X[4442] - X[31301], 4 X[4831] - 5 X[16704], 8 X[4831] - 5 X[31301], 2 X[4938] - 5 X[17491]

X(44006) lies on these lines: {2, 846}, {100, 4080}, {145, 9802}, {148, 690}, {149, 900}, {244, 24429}, {516, 20045}, {726, 21282}, {740, 4938}, {896, 28546}, {902, 28550}, {903, 3315}, {962, 20041}, {1836, 17147}, {3006, 28526}, {3434, 20068}, {3663, 29823}, {3729, 31079}, {3936, 28530}, {3952, 24715}, {3995, 20292}, {4365, 20290}, {4442, 4831}, {4651, 33099}, {4683, 17163}, {4781, 17719}, {4854, 8025}, {5057, 17495}, {5695, 31017}, {5880, 31035}, {5905, 20011}, {9263, 29340}, {17135, 33098}, {17140, 33095}, {17162, 17770}, {17164, 24851}, {17165, 33094}, {17355, 31098}, {17484, 19998}, {17690, 25253}, {17764, 32856}, {17767, 33136}, {17780, 21093}, {20064, 30699}, {20077, 20084}, {20098, 38514}, {24311, 27704}, {24697, 27812}, {24723, 31025}, {26073, 30578}, {27804, 33097}, {28542, 32848}, {29824, 32857}, {31011, 32846}

X(44006) = reflection of X(i) in X(j) for these {i,j}: {4427, 3120}, {16704, 4442}, {31301, 16704}
X(44006) = anticomplement of X(4427)
X(44006) = anticomplement of the isotomic conjugate of X(4608)
X(44006) = X(4608)-Ceva conjugate of X(2)
X(44006) = crossdifference of every pair of points on line {5029, 20976}
X(44006) = barycentric product X(86)*X(24070)
X(44006) = barycentric quotient X(24070)/X(10)
X(44006) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 14779}, {513, 2891}, {649, 41821}, {692, 24074}, {1126, 513}, {1171, 7192}, {1255, 20295}, {1268, 21301}, {3248, 39348}, {4608, 6327}, {6578, 21295}, {8701, 3952}, {28615, 514}, {31010, 21287}, {32014, 17217}, {32018, 21304}, {33635, 4462}, {37212, 668}, {40438, 512}
X(44006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {149, 4440, 17154}, {149, 17154, 20042}, {3120, 4427, 2}, {4683, 17163, 43990}

X(44007) = VIJAY 12TH PARALLEL TRANSFORM OF X(512)

Barycentrics    a^2*(b - c)*(b + c)*(4*a^4*b^4 - a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 4*a^4*c^4 - 7*a^2*b^2*c^4 + 7*b^4*c^4) : :
X(44007) = 7 X[2] - 4 X[23610], 11 X[2] - 8 X[38237], 11 X[23610] - 14 X[38237]

X(44007) lies on these lines: {2, 512}, {69, 9009}, {669, 8617}, {888, 39361}, {1272, 2872}, {9463, 38366}, {9998, 14606}

X(44007) = anticomplement of the isogonal conjugate of X(23342)
X(44007) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {538, 21221}, {662, 538}, {2234, 148}, {3231, 21220}, {5118, 192}, {23342, 8}, {24037, 888}, {24041, 9147}, {30736, 21294}, {30938, 150}, {36133, 3228}

X(44008) = VIJAY 12TH PARALLEL TRANSFORM OF X(513)

Barycentrics    a*(b - c)*(4*a^2*b^2 - a^2*b*c - 7*a*b^2*c + 4*a^2*c^2 - 7*a*b*c^2 + 7*b^2*c^2) : :
X(44008) = 7 X[2] - 4 X[8027], 3 X[2] - 4 X[14434], 5 X[2] - 4 X[14474], 11 X[2] - 8 X[38238], 3 X[8027] - 7 X[14434], 5 X[8027] - 7 X[14474], 11 X[8027] - 14 X[38238], 6 X[8027] - 7 X[43928], 5 X[14434] - 3 X[14474], 11 X[14434] - 6 X[38238], 11 X[14474] - 10 X[38238], 6 X[14474] - 5 X[43928], 12 X[38238] - 11 X[43928]

X(44008) lies on these lines: {2, 513}, {8, 20295}, {891, 39360}, {1655, 31290}, {3240, 38349}, {17794, 21297}, {21219, 26824}

X(44008) = reflection of X(43928) in X(14434)
X(44008) = anticomplement of X(43928)
X(44008) = anticomplement of the isogonal conjugate of X(23343)
X(44008) = anticomplement of the isotomic conjugate of X(41314)
X(44008) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {100, 29824}, {101, 536}, {536, 150}, {765, 891}, {899, 149}, {1293, 3999}, {3230, 4440}, {3994, 3448}, {4009, 33650}, {6381, 21293}, {6632, 41314}, {7045, 30704}, {23343, 8}, {23891, 69}, {34075, 3227}, {41314, 6327}
X(44008) = X(41314)-Ceva conjugate of X(2)
{X(14434),X(43928)}-harmonic conjugate of X(2)

X(44009) = VIJAY 12TH PARALLEL TRANSFORM OF X(514)

Barycentrics    (b - c)*(7*a^2 - 7*a*b + 4*b^2 - 7*a*c - b*c + 4*c^2) : :
X(44009) = 3 X[2] - 4 X[6544], 7 X[2] - 4 X[6545], X[2] - 4 X[6546], 5 X[2] - 8 X[10196], 5 X[2] - 4 X[14475], 11 X[2] - 8 X[21204], X[3621] + 8 X[5592], 8 X[4468] + X[26853], 7 X[4678] - 16 X[32212], 7 X[6544] - 3 X[6545], X[6544] - 3 X[6546], 5 X[6544] - 6 X[10196], 5 X[6544] - 3 X[14475], 11 X[6544] - 6 X[21204], 2 X[6544] - 3 X[31992], X[6545] - 7 X[6546], 6 X[6545] - 7 X[6548], 5 X[6545] - 14 X[10196], 5 X[6545] - 7 X[14475], 11 X[6545] - 14 X[21204], 2 X[6545] - 7 X[31992], 6 X[6546] - X[6548], 5 X[6546] - 2 X[10196], 5 X[6546] - X[14475], 11 X[6546] - 2 X[21204], 5 X[6548] - 12 X[10196], 5 X[6548] - 6 X[14475], 11 X[6548] - 12 X[21204], X[6548] - 3 X[31992], 11 X[10196] - 5 X[21204], 4 X[10196] - 5 X[31992], 11 X[14475] - 10 X[21204], 2 X[14475] - 5 X[31992], 4 X[21204] - 11 X[31992]

X(44009) lies on these lines: {2, 514}, {144, 4468}, {192, 4777}, {513, 17488}, {900, 17487}, {1654, 28209}, {3621, 5592}, {4448, 28151}, {4678, 32212}, {4776, 31056}, {6009, 20533}, {14779, 28179}, {20042, 33920}, {24616, 27484}, {30579, 33888}

X(44009) = reflection of X(i) in X(j) for these {i,j}: {2, 31992}, {6548, 6544}, {14475, 10196}, {31992, 6546}
X(44009) = anticomplement of X(6548)
X(44009) = anticomplement of the isogonal conjugate of X(23344)
X(44009) = anticomplement of the isotomic conjugate of X(17780)
X(44009) = X(17780)-Ceva conjugate of X(2)
X(44009) = crosssum of X(649) and X(1017)
X(44009) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 20042}, {44, 150}, {100, 21282}, {101, 320}, {110, 17145}, {163, 17160}, {519, 21293}, {692, 519}, {902, 149}, {906, 3007}, {1023, 69}, {1110, 900}, {1252, 21297}, {1415, 1266}, {2149, 4453}, {2251, 4440}, {2429, 21296}, {3257, 32032}, {3689, 33650}, {3939, 5176}, {3943, 21294}, {4169, 21287}, {9459, 9263}, {17780, 6327}, {21805, 3448}, {23344, 8}, {23703, 3434}, {23990, 21222}, {24004, 315}, {30731, 21286}, {32665, 903}, {32666, 24841}, {32739, 17495}, {34067, 24715}, {34080, 4887}
X(44009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6544, 6548, 2}, {6548, 31992, 6544}

X(44010) = VIJAY 12TH PARALLEL TRANSFORM OF X(523)

Barycentrics    (b - c)*(b + c)*(7*a^4 - 7*a^2*b^2 + 4*b^4 - 7*a^2*c^2 - b^2*c^2 + 4*c^4) : :
X(44010) = 3 X[2] - 4 X[1649], 7 X[2] - 4 X[8029], 5 X[2] - 4 X[8371], 19 X[2] - 16 X[10189], 5 X[2] - 8 X[10190], 11 X[2] - 8 X[10278], X[2] - 4 X[11123], 7 X[2] + 8 X[34752], X[20] + 8 X[8151], X[148] + 8 X[36955], 7 X[1649] - 3 X[8029], 5 X[1649] - 3 X[8371], 2 X[1649] - 3 X[9168], 19 X[1649] - 12 X[10189], 5 X[1649] - 6 X[10190], 11 X[1649] - 6 X[10278], X[1649] - 3 X[11123], 7 X[1649] + 6 X[34752], 7 X[3523] - 4 X[16220], 7 X[3523] - 16 X[32204], 7 X[5466] - 6 X[8029], 5 X[5466] - 6 X[8371], X[5466] - 3 X[9168], 19 X[5466] - 24 X[10189], 5 X[5466] - 12 X[10190], 11 X[5466] - 12 X[10278], X[5466] - 6 X[11123], 7 X[5466] + 12 X[34752], 8 X[6563] + X[31299], 5 X[8029] - 7 X[8371], 2 X[8029] - 7 X[9168], 19 X[8029] - 28 X[10189], 5 X[8029] - 14 X[10190], 11 X[8029] - 14 X[10278], X[8029] - 7 X[11123], X[8029] + 2 X[34752], 2 X[8371] - 5 X[9168], 19 X[8371] - 20 X[10189], 11 X[8371] - 10 X[10278], X[8371] - 5 X[11123], 7 X[8371] + 10 X[34752], X[8596] - 4 X[14443], 19 X[9168] - 8 X[10189], 5 X[9168] - 4 X[10190], 11 X[9168] - 4 X[10278], 7 X[9168] + 4 X[34752], X[9180] + 4 X[36955], 10 X[10189] - 19 X[10190], 22 X[10189] - 19 X[10278], 4 X[10189] - 19 X[11123], 14 X[10189] + 19 X[34752], 11 X[10190] - 5 X[10278], 2 X[10190] - 5 X[11123], 7 X[10190] + 5 X[34752], 2 X[10278] - 11 X[11123], 7 X[10278] + 11 X[34752], 7 X[11123] + 2 X[34752], X[14683] + 8 X[36739], X[16220] - 4 X[32204]

X(44010) lies on the Steiner-Wallace right hyperbola, the Kiepert circumhyperbola of the anticomplementary triangle, and these lines: {1, 17161}, {2, 523}, {20, 1499}, {147, 2793}, {148, 9180}, {194, 3906}, {512, 33884}, {525, 11148}, {616, 27551}, {617, 27550}, {690, 8591}, {2799, 9123}, {2896, 12073}, {3413, 30508}, {3414, 30509}, {3523, 16220}, {4226, 14611}, {6031, 6194}, {7616, 41298}, {7665, 9185}, {8596, 14443}, {8723, 11003}, {8782, 9147}, {9125, 9979}, {9742, 30474}, {14683, 36739}, {33921, 39356}

X(44010) = anticomplement of X(5466)
X(44010) = reflection of X(i) in X(j) for these {i,j}: {2, 9168}, {148, 9180}, {2408, 18311}, {5466, 1649}, {8371, 10190}, {9168, 11123}, {9485, 9131}, {9979, 9125}
X(44010) = anticomplement of the isogonal conjugate of X(5467)
X(44010) = anticomplement of the isotomic conjugate of X(5468)
X(44010) = X(5468)-Ceva conjugate of X(2)
X(44010) = crosssum of X(512) and X(39689)
X(44010) = crossdifference of every pair of points on line {187, 20977}
X(44010) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {99, 21298}, {110, 17491}, {162, 41724}, {163, 524}, {187, 21221}, {524, 21294}, {662, 316}, {896, 3448}, {922, 148}, {1101, 690}, {1576, 17497}, {4235, 21270}, {4570, 30709}, {4575, 858}, {5467, 8}, {5468, 6327}, {6629, 21293}, {14567, 21220}, {16702, 150}, {23889, 69}, {24039, 315}, {36034, 9140}, {36134, 23061}, {36142, 671}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1649, 5466, 2}, {5466, 9168, 1649}

X(44011) = X(4)X(69)∩X(115)X(512)

Barycentrics    a^2*(b - c)^2*(b + c)^2*(a^8 - a^6*b^2 + a^4*b^4 - a^2*b^6 - a^6*c^2 + 3*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 3*b^6*c^2 + a^4*c^4 - 3*a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + 3*b^2*c^6) : :
X(44011) = 3 X[6787] - X[12833], 3 X[6787] + X[31513], X[13137] - 3 X[14639]

See Antreas Hatzipolakis and Peter Moses, euclid 1836.

X(44011) lies on these lines: {4, 69}, {115, 512}, {3111, 7844}, {5139, 36471}, {13137, 14639}

X(44011) = midpoint of X(12833) and X(31513)
X(44011) = reflection of X(15630) in X(115)
X(44011) = polar circle inverse of X(877)
X(44011) = crosssum of X(99) and X(34473)
X(44011) = crossdifference of every pair of points on line {2421, 3049}
X(44011) = {X(6787),X(31513)}-harmonic conjugate of X(12833)

X(44012) = X(4)X(9)∩X(116)X(514)

Barycentrics    (a - b - c)*(b - c)^2*(a^5 - 4*a^4*b + 4*a^3*b^2 - 2*a^2*b^3 + 3*a*b^4 - 2*b^5 - 4*a^4*c + 5*a^3*b*c - a^2*b^2*c - a*b^3*c + b^4*c + 4*a^3*c^2 - a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 - 2*a^2*c^3 - a*b*c^3 + b^2*c^3 + 3*a*c^4 + b*c^4 - 2*c^5) : :
X(44012) = X[927] - 5 X[31640]

See Antreas Hatzipolakis and Peter Moses, euclid 1836.

X(44012) lies on these lines: {4, 9}, {11, 14330}, {116, 514}, {523, 31648}, {927, 31640}, {40483, 40554}

X(44012) = midpoint of X(1146) and X(1566)
X(44012) = reflection of X(i) in X(j) for these {i,j}: {15634, 116}, {40554, 40483}
X(44012) = polar circle inverse of X(41321)
X(44012) = crosssum of X(101) and X(38692)
X(44012) = crossdifference of every pair of points on line {1459, 2426}

X(44013) = X(4)X(8)∩X(11)X(513)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1836.

X(44013) lies on these lines: {4, 8}, {11, 513}, {2969, 3326}, {10893, 31849}, {13273, 17101}, {15313, 38385}

X(44013) = midpoint of X(i) and X(j) for these {i,j}: {3259, 38389}, {31512, 34151}
X(44013) = reflection of X(i) in X(j) for these {i,j}: {3937, 33646}, {15635, 11}
X(44013) = crosssum of X(100) and X(38693)
X(44013) = crossdifference of every pair of points on line {2427, 22383}

X(44014) = X(1)X(4)∩X(124)X(522)

Barycentrics    (a - b - c)*(b - c)^2*(a^9 + 2*a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - 2*b^9 + 2*a^8*c - 7*a^7*b*c + 7*a^6*b^2*c + 5*a^5*b^3*c - 17*a^4*b^4*c + 11*a^3*b^5*c + 5*a^2*b^6*c - 9*a*b^7*c + 3*b^8*c - 4*a^7*c^2 + 7*a^6*b*c^2 - 10*a^5*b^2*c^2 + 13*a^4*b^3*c^2 + 8*a^3*b^4*c^2 - 23*a^2*b^5*c^2 + 6*a*b^6*c^2 + 3*b^7*c^2 - 4*a^6*c^3 + 5*a^5*b*c^3 + 13*a^4*b^2*c^3 - 30*a^3*b^3*c^3 + 14*a^2*b^4*c^3 + 9*a*b^5*c^3 - 7*b^6*c^3 + 6*a^5*c^4 - 17*a^4*b*c^4 + 8*a^3*b^2*c^4 + 14*a^2*b^3*c^4 - 14*a*b^4*c^4 + 3*b^5*c^4 + 11*a^3*b*c^5 - 23*a^2*b^2*c^5 + 9*a*b^3*c^5 + 3*b^4*c^5 - 4*a^3*c^6 + 5*a^2*b*c^6 + 6*a*b^2*c^6 - 7*b^3*c^6 + 4*a^2*c^7 - 9*a*b*c^7 + 3*b^2*c^7 + a*c^8 + 3*b*c^8 - 2*c^9) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1836.

X(44014) lies on these lines: {1, 4}, {11, 21172}, {124, 522}, {2829, 39762}, {3318, 3326}, {7004, 21189}

X(44014) = midpoint of X(10017) and X(38357)
X(44014) = reflection of X(15633) in X(124)
X(44014) = polar circle inverse of X(23987)
X(44014) = crosssum of X(109) and X(38691)
X(44014) = crossdifference of every pair of points on line {652, 2425}

X(44015) = GIBERT (5,6,1) POINT

Barycentrics    5*a^2*S/Sqrt[3] + a^2*SA + 12*SB*SC : :

X(44015) lies on the cubic K1236 and these lines: {2, 43240}, {3, 42629}, {4, 42779}, {5, 16}, {6, 17505}, {13, 3830}, {14, 3839}, {15, 3146}, {17, 1657}, {20, 42546}, {61, 12102}, {62, 42106}, {376, 16241}, {381, 16961}, {382, 16960}, {396, 35404}, {397, 42692}, {546, 42782}, {550, 42683}, {631, 43467}, {3090, 42954}, {3091, 34755}, {3412, 42085}, {3523, 5366}, {3525, 10646}, {3526, 42891}, {3534, 43199}, {3545, 43777}, {3627, 34754}, {3628, 43106}, {3845, 43418}, {3854, 5344}, {3855, 42935}, {3858, 43775}, {3860, 11543}, {3861, 42695}, {5054, 16966}, {5056, 43485}, {5070, 42689}, {5073, 42691}, {5076, 42630}, {5237, 42114}, {5238, 42109}, {5321, 14893}, {5339, 43030}, {5340, 16809}, {5350, 11542}, {5352, 42113}, {9862, 36962}, {10124, 42123}, {10645, 12103}, {10653, 41106}, {11001, 42795}, {11481, 42962}, {11488, 42434}, {11489, 41974}, {12100, 37832}, {12101, 42520}, {12108, 42146}, {12811, 43241}, {12820, 38335}, {12821, 42969}, {15022, 43300}, {15682, 33607}, {15686, 43483}, {15687, 42781}, {15688, 43544}, {15697, 42952}, {15699, 42686}, {15703, 33416}, {15718, 42528}, {15722, 42625}, {16239, 42685}, {16242, 43249}, {16267, 42119}, {16644, 42429}, {16943, 22901}, {16962, 42130}, {16963, 42095}, {16964, 42815}, {19709, 43545}, {19710, 41121}, {21734, 42092}, {22236, 43010}, {23302, 33923}, {31074, 37776}, {33699, 42777}, {35409, 43542}, {35731, 42177}, {36967, 42817}, {40693, 43466}, {41100, 42129}, {41108, 43368}, {41113, 42480}, {41120, 43006}, {41943, 43401}, {41973, 42970}, {41987, 43111}, {41990, 42953}, {42087, 42916}, {42090, 43403}, {42091, 42488}, {42093, 42991}, {42098, 42158}, {42099, 42156}, {42101, 42923}, {42117, 42992}, {42120, 42915}, {42125, 42897}, {42133, 43292}, {42145, 42598}, {42150, 42806}, {42152, 42909}, {42160, 42982}, {42163, 42922}, {42180, 42243}, {42182, 42242}, {42433, 43029}, {42436, 42818}, {42474, 42996}, {42495, 42994}, {42613, 43782}, {42684, 42960}, {42775, 42937}, {42799, 43475}, {42814, 43031}, {42888, 43311}, {42896, 43033}, {42913, 43644}, {42928, 43193}, {42946, 43499}, {42957, 42979}, {42998, 43424}, {43032, 43235}, {43398, 43556}

X(44015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 42094, 19107}, {13, 42154, 42506}, {16, 42138, 16808}, {16, 42813, 42138}, {396, 35404, 43645}, {5318, 16808, 16965}, {5318, 42138, 16}, {5318, 42693, 5}, {5318, 42813, 16808}, {5335, 43364, 42103}, {10645, 43637, 12103}, {10646, 42142, 42581}, {12816, 19107, 42094}, {16808, 16965, 16967}, {16808, 42905, 42813}, {16964, 42815, 43014}, {16966, 42127, 36968}, {18582, 36969, 42100}, {18582, 42100, 16241}, {19106, 42128, 17}, {34754, 43195, 3627}, {41121, 42941, 42529}, {42105, 42903, 43004}, {42114, 43465, 5237}, {42120, 42921, 42915}, {42125, 42990, 42897}, {42134, 42162, 15}, {42137, 42166, 10645}, {42142, 42161, 10646}, {43328, 43630, 396}

X(44016) = GIBERT (-5,6,1) POINT

Barycentrics    5*a^2*S/Sqrt[3] - a^2*SA - 12*SB*SC : :

X(44016) lies on the cubic K1236 and these lines: {2, 43241}, {3, 42630}, {4, 42780}, {5, 15}, {6, 17505}, {13, 3839}, {14, 3830}, {16, 3146}, {18, 1657}, {20, 42545}, {61, 42103}, {62, 12102}, {376, 16242}, {381, 16960}, {382, 16961}, {395, 35404}, {398, 42693}, {546, 42781}, {550, 42682}, {631, 43468}, {3090, 42955}, {3091, 34754}, {3411, 42086}, {3523, 5365}, {3525, 10645}, {3526, 42890}, {3534, 43200}, {3545, 43778}, {3627, 34755}, {3628, 43105}, {3845, 43419}, {3854, 5343}, {3855, 42934}, {3858, 43776}, {3860, 11542}, {3861, 42694}, {5054, 16967}, {5056, 43486}, {5070, 42688}, {5073, 42690}, {5076, 42629}, {5237, 42108}, {5238, 42111}, {5318, 14893}, {5339, 16808}, {5340, 43031}, {5349, 11543}, {5351, 42112}, {9862, 36961}, {10124, 42122}, {10646, 12103}, {10654, 41106}, {11001, 42796}, {11480, 42963}, {11488, 41973}, {11489, 42433}, {12100, 37835}, {12101, 42521}, {12108, 42143}, {12811, 43240}, {12820, 42968}, {12821, 38335}, {15022, 43301}, {15682, 33606}, {15686, 43484}, {15687, 42782}, {15688, 43545}, {15697, 42953}, {15699, 42687}, {15703, 33417}, {15718, 42529}, {15722, 42626}, {16239, 42684}, {16241, 43248}, {16268, 42120}, {16645, 42430}, {16942, 22855}, {16962, 42098}, {16963, 42131}, {16965, 42816}, {19709, 43544}, {19710, 41122}, {21734, 42089}, {22238, 43011}, {23303, 33923}, {31074, 37775}, {33699, 42778}, {35409, 43543}, {36968, 42818}, {40694, 43465}, {41101, 42132}, {41107, 43369}, {41112, 42481}, {41119, 43007}, {41944, 43402}, {41974, 42971}, {41987, 43110}, {41990, 42952}, {42088, 42917}, {42090, 42489}, {42091, 43404}, {42094, 42990}, {42095, 42157}, {42100, 42153}, {42102, 42922}, {42118, 42993}, {42119, 42914}, {42128, 42896}, {42134, 43293}, {42144, 42599}, {42149, 42908}, {42151, 42805}, {42161, 42983}, {42166, 42923}, {42179, 42245}, {42181, 42244}, {42434, 43028}, {42435, 42817}, {42475, 42997}, {42494, 42995}, {42612, 43781}, {42685, 42961}, {42776, 42936}, {42800, 43476}, {42813, 43030}, {42889, 43310}, {42897, 43032}, {42912, 43649}, {42929, 43194}, {42947, 43500}, {42956, 42978}, {42999, 43425}, {43033, 43234}, {43397, 43557}

X(44016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 42093, 19106}, {14, 42155, 42507}, {15, 42135, 16809}, {15, 42814, 42135}, {395, 35404, 43646}, {5321, 16809, 16964}, {5321, 42135, 15}, {5321, 42692, 5}, {5321, 42814, 16809}, {5334, 43365, 42106}, {10645, 42139, 42580}, {10646, 43636, 12103}, {12817, 19106, 42093}, {16809, 16964, 16966}, {16809, 42904, 42814}, {16965, 42816, 43015}, {16967, 42126, 36967}, {18581, 36970, 42099}, {18581, 42099, 16242}, {19107, 42125, 18}, {34755, 43196, 3627}, {41122, 42940, 42528}, {42104, 42902, 43005}, {42111, 43466, 5238}, {42119, 42920, 42914}, {42128, 42991, 42896}, {42133, 42159, 16}, {42136, 42163, 10646}, {42139, 42160, 10645}, {43329, 43631, 395}

X(44017) = GIBERT (51,8,-5) POINT

Barycentrics    17*Sqrt[3]*a^2*S - 5*a^2*SA + 16*SB*SC : :

X(44017) lies on the cubic K1236 and these lines: {6, 15684}, {13, 11737}, {14, 3839}, {15, 8703}, {16, 15707}, {17, 42805}, {61, 43646}, {62, 1656}, {396, 12108}, {397, 3857}, {398, 3853}, {3412, 5237}, {3529, 10653}, {5340, 43784}, {5352, 43018}, {11481, 42892}, {11488, 15702}, {11542, 42952}, {12820, 43417}, {16242, 41977}, {16267, 33416}, {16960, 42501}, {16963, 43542}, {16966, 42636}, {18582, 43250}, {36967, 42935}, {37641, 42921}, {37835, 41119}, {40693, 42592}, {40694, 42961}, {41107, 42093}, {41108, 42104}, {41992, 42488}, {42087, 43230}, {42091, 42520}, {42094, 42968}, {42118, 43245}, {42131, 43305}, {42147, 42966}, {42153, 42612}, {42154, 42967}, {42155, 43007}, {42163, 43033}, {42499, 43554}, {42502, 42917}, {42506, 42512}, {42510, 43014}, {42516, 42529}, {42543, 42585}, {42581, 43025}, {42628, 43104}, {42634, 42915}, {42778, 42918}, {42796, 42912}, {42893, 42984}, {42923, 42940}, {42993, 43416}

X(44017) = {X(15),X(42800)}-harmonic conjugate of X(41972)

X(44018) = GIBERT (51,-8,5) POINT

Barycentrics    17*Sqrt[3]*a^2*S + 5*a^2*SA - 16*SB*SC : :

X(44018) lies on the cubic K1236 and these lines: {6, 15684}, {13, 3839}, {14, 11737}, {15, 15707}, {16, 8703}, {18, 42806}, {61, 1656}, {62, 43645}, {395, 12108}, {397, 3853}, {398, 3857}, {3411, 5238}, {3529, 10654}, {5339, 43783}, {5351, 43019}, {11480, 42893}, {11489, 15702}, {11543, 42953}, {12821, 43416}, {16241, 41978}, {16268, 33417}, {16961, 42500}, {16962, 43543}, {16967, 42635}, {18581, 43251}, {36968, 42934}, {37640, 42920}, {37832, 41120}, {40693, 42960}, {40694, 42593}, {41107, 42105}, {41108, 42094}, {41992, 42489}, {42088, 43231}, {42090, 42521}, {42093, 42969}, {42117, 43244}, {42130, 43304}, {42148, 42967}, {42154, 43006}, {42155, 42966}, {42156, 42613}, {42166, 43032}, {42498, 43555}, {42503, 42916}, {42507, 42513}, {42511, 43015}, {42517, 42528}, {42544, 42584}, {42580, 43024}, {42627, 43101}, {42633, 42914}, {42777, 42919}, {42795, 42913}, {42892, 42985}, {42922, 42941}, {42992, 43417}

X(44018) = {X(16),X(42799)}-harmonic conjugate of X(41971)

X(44019) = GIBERT (59,30,39) POINT

Barycentrics    59*a^2*S/Sqrt[3] + 39*a^2*SA + 60*SB*SC : :

X(44019) lies on the cubic K1236 and these lines: {14, 11737}, {15, 3146}, {17, 42121}, {3412, 43013}, {5339, 16960}, {5351, 42922}, {10646, 42926}, {11489, 16267}, {11542, 43635}, {15701, 16644}, {16241, 43242}, {16808, 43293}, {33607, 42632}, {35401, 43245}, {41101, 43478}, {42100, 43491}, {42111, 42993}, {42156, 42630}, {42158, 42817}, {42511, 43366}, {42530, 43008}, {42581, 43241}, {42900, 42916}, {42920, 43426}, {42998, 43371}, {43226, 43301}

X(44020) = GIBERT (-59,30,39) POINT

Barycentrics    59*a^2*S/Sqrt[3] - 39*a^2*SA - 60*SB*SC : :

X(44020) lies on the cubic K1236 and these lines: {13, 11737}, {16, 3146}, {18, 42124}, {3411, 43012}, {5340, 16961}, {5352, 42923}, {10645, 42927}, {11488, 16268}, {11543, 43634}, {15701, 16645}, {16242, 43243}, {16809, 43292}, {33606, 42631}, {35401, 43244}, {41100, 43477}, {42099, 43492}, {42114, 42992}, {42153, 42629}, {42157, 42818}, {42510, 43367}, {42531, 43009}, {42580, 43240}, {42901, 42917}, {42921, 43427}, {42999, 43370}, {43227, 43300}

leftri

Points associated with Vijay-Hutson triangles: X(44021)-X(44025)

rightri

This preamble is contributed by Clark Kimberling (July 21, 2021), based on notes from Dasari Naga Vijay Krishna, July 21, 2021 and later from Peter Moses and César Lozada.

In the plane of a triangle ABC, let
Pa = parabola with A as its focus and BC as its directrix, define Pb and Pc cyclically;
A' = apex of equilateral triangle erected externally on the side BC, define B' and C' cyclically;
A'' = apex of equilateral triangle erected internally on the side BC, define B'' and C'' cyclically;
Ab = (triangle BCA')∩Pb, and define Bc and Ca cyclically; see Vijay 3 parabolas and 6 points.png Ac = (triangle BCA') ∩Pc, and define Ba and Cb cyclically;
A'b = (triangle BCA")∩Pb, and define B'c and C'a cyclically;
A'c = (triangle BCA")∩Pc, and define B'a and C'b cyclically;
A1 = CaCb∩BcBa, B1 = CaCb∩AbAc, C1 = AbAc∩BcBa;
A2 = AbCb∩BcAc, B2 = CaBa∩BcAc, C2 = CaBa∩AbCb;
A3 = C'aC'b∩B'cB'a, B3 = C'aC'b∩A'bA'c, C3 = A'bA'c∩B'cB'a;
A4 = A'bC'b∩B'cA'c, B4 = C'aB'a∩B'cA'c, C4 = C'aB'a∩A'bC'b;

Barycentrics for points defined above:

A' = -a^2*sqrt(3) : S+SC*sqrt(3) : S+SB*sqrt(3)
Ab = -c*a^2*sqrt(3) : 4*R*S : c*(S + SB*sqrt(3))
Ac = -b*a^2*sqrt(3) : b*(S + SC*sqrt(3)) : 4*R*S
A1 = - (4096*R^4*S^4 - 256*a*c*R^2*S^4 - 256*a*b*R^2*S^4 + 16*a^2*b*c*S^4 - 256*sqrt(3)*a*b*c^2*R^2*S^3 - 256*sqrt(3)*a*b^2*c*R^2*S^3 + 16*sqrt(3)*a^2*b*c^3*S^3 + 16*sqrt(3)*a^2*b^3*c*S^3 + 192*a*c^5*R^2*S^2 - 192*a*b*c^4*R^2*S^2 - 384*a^3*c^3*R^2*S^2 - 768*a^2*b^2*c^2*R^2*S^2 - 192*a*b^4*c*R^2*S^2 + 192*a^5*c*R^2*S^2 + 192*a*b^5*R^2*S^2 - 384*a^3*b^3*R^2*S^2 + 192*a^5*b*R^2*S^2 + 192*a^2*b^2*c^3*R*S^2 + 192*a^2*b^3*c^2*R*S^2 + 24*a^4*b*c^3*S^2 + 24*a^4*b^3*c*S^2 - 24*a^6*b*c*S^2 - 32*3^(3/2)*a^2*b^2*c^5*R*S + 32*3^(3/2)*a^2*b^3*c^4*R*S + 32*3^(3/2)*a^2*b^4*c^3*R*S + 32*3^(3/2)*a^4*b^2*c^3*R*S - 32*3^(3/2)*a^2*b^5*c^2*R*S + 32*3^(3/2)*a^4*b^3*c^2*R*S - 4*3^(3/2)*a^2*b*c^7*S + 4*3^(3/2)*a^2*b^3*c^5*S + 8*3^(3/2)*a^4*b*c^5*S + 4*3^(3/2)*a^2*b^5*c^3*S - 16*3^(3/2)*a^4*b^3*c^3*S - 4*3^(3/2)*a^6*b*c^3*S - 4*3^(3/2)*a^2*b^7*c*S + 8*3^(3/2)*a^4*b^5*c*S - 4*3^(3/2)*a^6*b^3*c*S - 9*a^2*b*c^9 + 36*a^2*b^3*c^7 + 18*a^4*b*c^7 - 54*a^2*b^5*c^5 - 18*a^4*b^3*c^5 + 36*a^2*b^7*c^3 - 18*a^4*b^5*c^3 - 18*a^8*b*c^3 - 9*a^2*b^9*c + 18*a^4*b^7*c - 18*a^8*b^3*c + 9*a^10*b*c) : 2*b^2*c*(64*R^2*S^2 - 4*a*b*S^2 - 16*sqrt(3)*a*b*c*R*S - 9*a*b*c^4 + 6*a*b^3*c^2 + 6*a^3*b*c^2 + 3*a*b^5 - 6*a^3*b^3 + 3*a^5*b) *(8*sqrt(3)*R*S - 2*sqrt(3)*a*S - 3*a*b^2 - 3*a*c^2 + 3*a^3) : 2*b*c^2*(64*R^2*S^2 - 4*a*c*S^2 - 16*sqrt(3)*a*b*c*R*S + 3*a*c^5 + 6*a*b^2*c^3 - 6*a^3*c^3 - 9*a*b^4*c + 6*a^3*b^2*c + 3*a^5*c)*(8*sqrt(3)*R*S - 2*sqrt(3)*a*S - 3*a*b^2 - 3*a*c^2 + 3*a^3)

A2 = a*(9*b*c^9 - 36*b^3*c^7 - 36*a^2*b*c^7 + 54*b^5*c^5 + 36*a^2*b^3*c^5 + 54*a^4*b*c^5 + 8*3^(3/2)*S*a^2*b*c^5 - 24*S^2*b*c^5 + 192*R^2*S^2*c^4 - 36*b^7*c^3 + 36*a^2*b^5*c^3 + 36*a^4*b^3*c^3 - 16*3^(3/2)*S*a^2*b^3*c^3 + 48*S^2*b^3*c^3 - 36*a^6*b*c^3 - 16*3^(3/2)*S*a^4*b*c^3 - 48*S^2*a^2*b*c^3 + 384*R^2*S^2*a*c^3 - 384*R^2*S^2*b^2*c^2 - 384*R^2*S^2*a*b*c^2 + 9*b^9*c - 36*a^2*b^7*c + 54*a^4*b^5*c + 8*3^(3/2)*S*a^2*b^5*c - 24*S^2*b^5*c - 36*a^6*b^3*c - 16*3^(3/2)*S*a^4*b^3*c - 48*S^2*a^2*b^3*c - 384*R^2*S^2*a*b^2*c + 9*a^8*b*c + 8*3^(3/2)*S*a^6*b*c + 72*S^2*a^4*b*c + 32*sqrt(3)*S^3*a^2*b*c - 768*R^2*S^2*a^2*b*c + 16*S^4*b*c - 384*R^2*S^2*a^3*c - 256*sqrt(3)*R^2*S^3*a*c + 192*R^2*S^2*b^4 + 384*R^2*S^2*a*b^3 - 384*R^2*S^2*a^3*b - 256*sqrt(3)*R^2*S^3*a*b - 192*R^2*S^2*a^4 - 256*sqrt(3)*R^2*S^3*a^2 - 256*R^2*S^4) : 8*R*S*b*(3^(3/2)*c^7 + 2*3^(3/2)*a*c^6 - 3^(5/2)*b^2*c^5 - 3^(3/2)*a^2*c^5 - 6*S*c^5 - 4*3^(3/2)*a*b^2*c^4 - 4*3^(3/2)*a^3*c^4 - 24*S*a*c^4 - 24*R*S*c^4 + 3^(5/2)*b^4*c^3 - 2*3^(3/2)*a^2*b^2*c^3 + 12*S*b^2*c^3 - 3^(3/2)*a^4*c^3 - 12*S*a^2*c^3 - 48*R*S*a*c^3 - 4*sqrt(3)*S^2*c^3 + 2*3^(3/2)*a*b^4*c^2 - 4*3^(3/2)*a^3*b^2*c^2 + 24*S*a*b^2*c^2 + 48*R*S*b^2*c^2 + 48*R*S*a*b*c^2 + 2*3^(3/2)*a^5*c^2 + 24*S*a^3*c^2 + 8*sqrt(3)*S^2*a*c^2 - 3^(3/2)*b^6*c + 3^(5/2)*a^2*b^4*c - 6*S*b^4*c - 3^(5/2)*a^4*b^2*c - 12*S*a^2*b^2*c + 48*R*S*a*b^2*c + 4*sqrt(3)*S^2*b^2*c + 96*R*S*a^2*b*c + 3^(3/2)*a^6*c + 18*S*a^4*c + 48*R*S*a^3*c + 4*3^(3/2)*S^2*a^2*c + 32*sqrt(3)*R*S^2*a*c + 8*S^3*c - 24*R*S*b^4 - 48*R*S*a*b^3 + 48*R*S*a^3*b + 32*sqrt(3)*R*S^2*a*b + 24*R*S*a^4 + 32*sqrt(3)*R*S^2*a^2 + 32*R*S^3) : -8*R*S*c*(3^(3/2)*b*c^6 - 3^(5/2)*b^3*c^4 - 2*3^(3/2)*a*b^2*c^4 - 3^(5/2)*a^2*b*c^4 + 6*S*b*c^4 + 24*R*S*c^4 + 48*R*S*a*c^3 + 3^(5/2)*b^5*c^2 + 4*3^(3/2)*a*b^4*c^2 + 2*3^(3/2)*a^2*b^3*c^2 - 12*S*b^3*c^2 + 4*3^(3/2)*a^3*b^2*c^2 - 24*S*a*b^2*c^2 - 48*R*S*b^2*c^2 + 3^(5/2)*a^4*b*c^2 + 12*S*a^2*b*c^2 - 48*R*S*a*b*c^2 - 4*sqrt(3)*S^2*b*c^2 - 48*R*S*a*b^2*c - 96*R*S*a^2*b*c - 48*R*S*a^3*c - 32*sqrt(3)*R*S^2*a*c - 3^(3/2)*b^7 - 2*3^(3/2)*a*b^6 + 3^(3/2)*a^2*b^5 + 6*S*b^5 + 4*3^(3/2)*a^3*b^4 + 24*S*a*b^4 + 24*R*S*b^4 + 3^(3/2)*a^4*b^3 + 12*S*a^2*b^3 + 48*R*S*a*b^3 + 4*sqrt(3)*S^2*b^3 - 2*3^(3/2)*a^5*b^2 - 24*S*a^3*b^2 - 8*sqrt(3)*S^2*a*b^2 - 3^(3/2)*a^6*b - 18*S*a^4*b - 48*R*S*a^3*b - 4*3^(3/2)*S^2*a^2*b - 32*sqrt(3)*R*S^2*a*b - 8*S^3*b - 24*R*S*a^4 - 32*sqrt(3)*R*S^2*a^2 - 32*R*S^3)

Barycentrics for A″, A'b, A'c, A3, A4 are obtained by replacing sqrt(3) by -sqrt(3) in the barycentrics for A', Ab, Ac, A1, A2, respectively. In this manner, X(44023) is obtained from X(44021) and X(4402) from X(44022).

Related triangles are here named as follows:

A1B1C1 = 1st Vijay-Hutson triangle;
A2B2C2 = 2nd Vijay-Hutson triangle;
A3B3C3 = 3rd Vijay-Hutson triangle;
A4B4C4 = 4th Vijay-Hutson triangle.

Perspectors:

X(44021) = AA1∩BB1∩CC1;
X(44022) = AA2∩BB2∩CC2;
X(44023) = AA3∩BB3∩CC3;
X(44024) = AA4∩BB4∩CC4;
X(44025) = A2A4∩B2B∩C2C4.

X(44021) = PERSPECTOR OF THESE TRIANGLES: ABC AND 1ST VIJAY-HUTSON TRIANGLE

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

Construction: X(44021)

X(44021) lies on these lines: {}

X(44022) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND VIJAY-HUTSON TRIANGLE

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

Constructions: X(44022) and, if you have GeoGebra, X(44022)A

X(44022) lies on these lines: {44024, 44025}

X(44023) = PERSPECTOR OF THESE TRIANGLES: ABC AND 3RD VIJAY-HUTSON TRIANGLE

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

Construction: X(44023)

X(44023) lies on these lines: {}

X(44024) = PERSPECTOR OF THESE TRIANGLES: ABC AND 4TH VIJAY-HUTSON TRIANGLE

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

Construction: X(44024)

X(44024) lies on these lines: {44022, 44025}

X(44025) = PERSPECTOR OF THESE TRIANGLES: 2ND AND 4TH VIJAY-HUTSON TRIANGLE

Barycentrics    -4*a*((b^2+c^2)*a^5-(b+c)*(b^2-3*b*c+c^2)*a^4-(2*b^4+2*c^4-b*c*(3*b^2+2*b*c+3*c^2))*a^3+(b+c)*(2*b^4+2*c^4-b*c*(5*b^2+2*b*c+5*c^2))*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c))*S+a^10-3*(b+c)*a^9+(b^2+15*b*c+c^2)*a^8+6*(b+c)*(b^2-3*b*c+c^2)*a^7-(8*b^4+8*c^4+b*c*(21*b^2+10*b*c+21*c^2))*a^6+4*(b+c)*(7*b^2-6*b*c+7*c^2)*b*c*a^5+(8*b^6+8*c^6-b*c*(b^2-6*b*c+c^2)*(b+c)^2)*a^4-2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+b*c*(7*b^2+4*b*c+7*c^2))*a^3-(b^4+c^4-5*(b^2+c^2)*b*c)*(b^2-c^2)^2*a^2+(b^2-c^2)^3*(b-c)*(3*b^2-2*b*c+3*c^2)*a-(b^2-c^2)^4*(b-c)^2 : :

Construction: X(44025)

X(44025) lies on these lines: {44022, 44024}

X(44026) = X(51)X(31392)∩X(140)X(389)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8)*(a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12 + a^10*c^2 - 4*a^8*b^2*c^2 + a^6*b^4*c^2 + 12*a^4*b^6*c^2 - 16*a^2*b^8*c^2 + 6*b^10*c^2 - 5*a^8*c^4 + a^6*b^2*c^4 + 2*a^4*b^4*c^4 + 11*a^2*b^6*c^4 - 15*b^8*c^4 + 10*a^6*c^6 + 12*a^4*b^2*c^6 + 11*a^2*b^4*c^6 + 20*b^6*c^6 - 10*a^4*c^8 - 16*a^2*b^2*c^8 - 15*b^4*c^8 + 5*a^2*c^10 + 6*b^2*c^10 - c^12) : :
X(44026) = 3 X[51] - X[31392]

See Antreas Hatzipolakis and Peter Moses, euclid 1862.

X(44026) lies on these lines: {51, 31392}, {140, 389}, {143, 32638}, {195, 14367}, {10095, 14051}, {11557, 43966}

X(44026) = reflection of X(14051) in X(10095)

X(44027) = X(30)X(511)∩X(6153)X(34804)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + 3*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(a^16 - 6*a^14*b^2 + 14*a^12*b^4 - 14*a^10*b^6 + 14*a^6*b^10 - 14*a^4*b^12 + 6*a^2*b^14 - b^16 - 6*a^14*c^2 + 25*a^12*b^2*c^2 - 38*a^10*b^4*c^2 + 27*a^8*b^6*c^2 - 18*a^6*b^8*c^2 + 23*a^4*b^10*c^2 - 18*a^2*b^12*c^2 + 5*b^14*c^2 + 14*a^12*c^4 - 38*a^10*b^2*c^4 + 29*a^8*b^4*c^4 - 2*a^6*b^6*c^4 - 15*a^4*b^8*c^4 + 26*a^2*b^10*c^4 - 14*b^12*c^4 - 14*a^10*c^6 + 27*a^8*b^2*c^6 - 2*a^6*b^4*c^6 + 3*a^4*b^6*c^6 - 14*a^2*b^8*c^6 + 27*b^10*c^6 - 18*a^6*b^2*c^8 - 15*a^4*b^4*c^8 - 14*a^2*b^6*c^8 - 34*b^8*c^8 + 14*a^6*c^10 + 23*a^4*b^2*c^10 + 26*a^2*b^4*c^10 + 27*b^6*c^10 - 14*a^4*c^12 - 18*a^2*b^2*c^12 - 14*b^4*c^12 + 6*a^2*c^14 + 5*b^2*c^14 - c^16) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1867.

X(44027) lies on these lines: {30, 511}, {6153, 34804}, {13418, 25043}

X(44028) = X(2)X(3459)∩X(30)X(511)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 5*a^10*b^2 + 10*a^8*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10 - 5*a^10*c^2 + 16*a^8*b^2*c^2 - 17*a^6*b^4*c^2 + 8*a^4*b^6*c^2 - 4*a^2*b^8*c^2 + 2*b^10*c^2 + 10*a^8*c^4 - 17*a^6*b^2*c^4 + a^4*b^4*c^4 + 5*a^2*b^6*c^4 - 8*b^8*c^4 - 10*a^6*c^6 + 8*a^4*b^2*c^6 + 5*a^2*b^4*c^6 + 12*b^6*c^6 + 5*a^4*c^8 - 4*a^2*b^2*c^8 - 8*b^4*c^8 - a^2*c^10 + 2*b^2*c^10) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1869.

X(44028) lies on these lines: {2, 3459}, {30, 511}, {547, 13856}, {549, 18016}, {930, 6150}, {1157, 13512}, {1263, 16336}, {6592, 10615}, {10109, 34768}, {11671, 19552}, {14051, 34804}, {14072, 16337}, {16766, 21230}, {25148, 38899}, {30482, 30483}, {38615, 40631}

X(44028) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {930, 19553, 6150}, {6592, 24385, 10615}, {15345, 25043, 32551}

X(44029) = X(17)X(619)∩X(61)X(618)

Barycentrics    (a^2 - 3*b^2 - 3*c^2 - 2*Sqrt[3]*S)*(7*a^2 - b^2 - c^2 + 2*Sqrt[3]*S) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44029) lies on the Kiepert circumhyperbola of the medial triangle and these lines: {2, 5487}, {3, 22113}, {15, 22916}, {17, 619}, {61, 618}, {114, 16002}, {148, 33413}, {630, 42672}, {3618, 44031}, {6292, 11132}, {8259, 30472}, {11298, 33620}, {11300, 33618}, {11301, 33621}, {11307, 42989}

X(44029) = complement of X(5487)
X(44029) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 34540}, {163, 30215}, {30215, 21253}
X(44029) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 34540}, {99, 30215}
X(44029) = Dao image of X(17)
X(44029) = barycentric quotient X(34540)/X(5487)
X(44029) = {X(35689),X(36782)}-harmonic conjugate of X(627)

X(44030) = X(5)X(302)∩X(17)X(299)

Barycentrics    3*a^4 - 7*a^2*b^2 + 4*b^4 - 7*a^2*c^2 - 14*b^2*c^2 + 4*c^4 - 2*Sqrt[3]*(a^2 + 2*b^2 + 2*c^2)*S : :
X(44030) = 3 X[2] + X[5487]

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44030) lies on these lines: {2, 5487}, {5, 302}, {17, 299}, {99, 33387}, {140, 14144}, {298, 22113}, {383, 16626}, {532, 41121}, {629, 6670}, {3763, 44032}, {5981, 5982}, {6673, 11132}, {7796, 40334}, {7891, 43028}, {7901, 22893}, {11303, 22911}, {11308, 42672}, {11309, 30471}, {16645, 33020}, {33959, 42813}

X(44030) = {X(629),X(22891)}-harmonic conjugate of X(11290)

X(44031) = X(18)X(618)∩X(62)X(619)

Barycentrics    (7*a^2 - b^2 - c^2 - 2*Sqrt[3]*S)*(a^2 - 3*b^2 - 3*c^2 + 2*Sqrt[3]*S) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44031) lies on the Kiepert circumhyperbola of the medial triangle and these lines: {2, 5488}, {3, 22114}, {16, 22871}, {18, 618}, {62, 619}, {114, 16001}, {148, 33412}, {629, 42673}, {3411, 36781}, {3618, 44029}, {6292, 11133}, {8260, 30471}, {11297, 33621}, {11299, 33619}, {11302, 33620}, {11308, 42988}

X(44031) = complement of X(5488)
X(44031) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 34541}, {163, 30216}, {30216, 21253}
X(44031) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 34541}, {99, 30216}
X(44031) = Dao image of X(18)
X(44031) = barycentric quotient X(34541)/X(5488)

X(44032) = X(5)X(303)∩X(18)X(298)

Barycentrics    3*a^4 - 7*a^2*b^2 + 4*b^4 - 7*a^2*c^2 - 14*b^2*c^2 + 4*c^4 + 2*Sqrt[3]*(a^2 + 2*b^2 + 2*c^2)*S : :
X(44032) = 3 X[2] + X[5488]

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44032) lies on these lines: {2, 5488}, {5, 303}, {18, 298}, {99, 33386}, {140, 14145}, {299, 22114}, {533, 41122}, {630, 6669}, {1080, 16627}, {3763, 44030}, {5980, 5983}, {6674, 11133}, {7796, 40335}, {7891, 43029}, {7901, 22847}, {11304, 22866}, {11307, 42673}, {11310, 30472}, {16644, 33020}, {33960, 42814}

X(44032) = {X(630),X(22846)}-harmonic conjugate of X(11289)

X(44033) = X(17)X(623)∩X(61)X(33526)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44033) lies on these lines: {17, 623}, {61, 33526}

X(44033) = barycentric quotient X(2004)/X(18813)

X(44034) = (name pending)

Barycentrics    (a^2 - b^2 - c^2 - 2*Sqrt[3]*S)*(3*a^6*b^2 - 5*a^4*b^4 + a^2*b^6 + b^8 + 3*a^6*c^2 - 18*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 2*b^6*c^2 - 5*a^4*c^4 - 7*a^2*b^2*c^4 - 6*b^4*c^4 + a^2*c^6 + 2*b^2*c^6 + c^8 - 2*Sqrt[3]*(b^2 + c^2)*(a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44034) lies on this line: {61, 302}

X(44035) = X(18)X(624)∩X(62)X(33527)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44035) lies on these lines: {18, 624}, {62, 33527}

X(44035) = barycentric quotient X(2005)/X(18814)

X(44036) = (name pending)

Barycentrics    (a^2 - b^2 - c^2 + 2*Sqrt[3]*S)*(3*a^6*b^2 - 5*a^4*b^4 + a^2*b^6 + b^8 + 3*a^6*c^2 - 18*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 2*b^6*c^2 - 5*a^4*c^4 - 7*a^2*b^2*c^4 - 6*b^4*c^4 + a^2*c^6 + 2*b^2*c^6 + c^8 + 2*Sqrt[3]*(b^2 + c^2)*(a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 + 2*b^2*c^2 - c^4)*S) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1884.

X(44036) lies on this line: {62,303}

X(44037) = X(3)X(6)∩X(2911)X(5405)

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

See Stanley Rabinowitz, Antreas Hatzipolakis and Peter Moses, euclid 1887.

X(44037) lies on these lines: {3,6}, {3911,5405}, {8953,17102}

X(44038) = X(1)X(485)∩X(4)X(9)

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

See Stanley Rabinowitz, Antreas Hatzipolakis and Peter Moses, euclid 1887.

X(44038) lies on these lines: {1, 485}, {4, 9}, {5, 30556}, {80, 30432}, {355, 30557}, {388, 8957}, {908, 13386}, {1336, 5727}, {1378, 5778}, {1478, 6204}, {1698, 32555}, {1737, 6203}, {1867, 12938}, {3300, 7951}, {3577, 13426}, {5219, 5393}, {5292, 18992}, {5691, 32556}, {5747, 13883}, {5822, 13936}, {8953, 35800}, {10590, 30324}, {18391, 30325}

X(44038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 14121, 6212}, {10, 31561, 6213}, {19925, 31594, 31562}

leftri

Points associated with Vijay incentral circles and excentral circles: X(44039)-X(44041)

rightri

This preamble is contributed by Clark Kimberling (July 26, 2021), based on notes from Dasari Naga Vijay Krishna, July 25, 2021.

In the plane of a triangle ABC, let
A'B'C' = excentral triangle
Oa = circle with diameter BC, and define Ob and Oc cyclically;
Ab = BB'∩Oa, and define Bc and Ca cyclically;
Ac = CC'∩Oa, and define Ba and Cb cyclically;
A'b = A'C'∩Oa, and define B'c and C'a cyclically;
A'c = A'B'∩Oa, and define B'a and C'b cyclically;
Oe = circle {{A'b, A'c, B'c, B'a, C'a, C'b}}, here named the Vijay excentral circle;
OIa = circle {{C'a, B'a, Ab, Ac, Bc, Cb}}, here named the Vijay a-incentral circle;
OIb = circle {{A'b, C'b, Ba, Bc, Ca, Ac}}, here named the Vijay b-incentral circle;
OIc = circle {{B'c, A'c, Ca, Cb, Ab, Ba}}, here named the Vijay c-incentral circle;
Pa = polar of A wrt Oe, and define Pb and Pc cyclically;
L'a = polar of A wrt OIa, and define L'b and L'c cyclically.

Barycentric equations for Vijay incentral and excentral circles and their centers:

Vijay excentral circle:
(s - b)(s - c)x^2 + (s -c)(s - a)y^2 + (s - a)(s - b)z^2 + s(ayz + bzx + cxy) = 0, with center X(10) = b + c : c + a : a + b

Vijay a-incentral circle:
(s - b)(s - c)x^2 - (s)(s - b)y^2 - (s)(s - c)z^2 + (s - a)(- ayz + bzx + cxy) = 0, with center b + c : c - a : b - a

Vijay b-incentral circle:
-s(s - a)x^2 +(s - a)(s - c)y^2 - (s)(s - c)z^2 + (s - b)(ayz - bzx + cxy) = 0, with center c - b : c + a : a - b

Vijay c-incentral circle:
-s(s - a)x^2 - (s)(s - b)y^2 + (s - a)(s - b)z^2 + (s - c)(ayz + bzx - cxy) = 0, with center b - c : a - c : a + b

Define 6 points by the following intersections :

A1 = B'cB'a∩C'aC'b, B1 = C'aC'b∩ A'bA'c, C1 = A'bA'c∩B'cB'a;
A2 = BcBa∩CaCb, B2 = CaCb∩ AbAc, C2 = AbAc∩BcBa;
A3 = C'aA'c∩ B'aA'b∩CaAc∩BaAb∩BC = midpoint of BC,
B3 = B'aA'b∩B'cC'b∩ BaAb∩ BcCb∩CA = midpoint of CA,
C3 = B'cC'b∩ C'aA'c∩ BcCb∩ CaAc∩AB = midpoint of AB;
A4 = center of Vijay a-incentral circle, and define B4 and C4 cyclically;
A5 = Pb∩Pc, and define B5 and C5 cyclically;
A6 = L'b∩L'c, and define B6 and C6 cyclically

Barycentrics for points defined above:

Ab = a : c - a : c, Ac = a : b : b - a
A'b = -a : a + c : c, A'c = -a : b : a + b
A1 = -a(b + c) : SC : SB
A2 = 0 : s - c : s - b
A3 = 0 : 1 : 1;
A4 = b + c : c - a : b - a
A5 = 4*(s - a)^2*(s - b)*(s - c) - s^2*a^2 : s^2*a*b - 2*s*c*(s - a)*(s - b) : s^2*a*c - 2*s*b*(s - c)*(s - a)
A6 = ((b + c)*(b + c - 2*a)*(s - b)*(s - c)) : ((s - b)*(a*b*(s - c) + 2*c*(s - a)*(s - b))) : ((s - c)*(a*c*(s - b) + 2*b*(s - a)*(s - c)))

Related triangles are here named:

A1B1C1 = Vijay excentral triangle;
A2B2C2 = intouch triangle;
A3B3C3 = medial triangle ;
A4B4C4 = Vijay abc-incentral triangle; Also, A4B4C4 = Wasat triangle
A5B5C5 = Vijay polar excentral triangle;
A6B6C6 = Vijay polar incentral triangle.

Collinearities:

A1, A2, A4 are collinear;
A1, A3, A' are collinear.

Perspectors :

AA'∩BB'∩CC' = X(1) = incenter = a : b : c
AA3∩BB3∩CC3 = A4A'∩B4B'∩C4C' = X(2) = centroid = 1 : 1 : 1,
AA1∩BB1∩CC1 = X(4) = orthocenter= SB SC : SC SA : SA SB
AA2∩BB2∩CC2 = X(7) = Gergonne point = 1/(s-a) : 1/(s-b) : 1/(s-c)
A1A3A'∩B1B3B'∩C1C3C' = X(9) = mittenpunkt = a(s -a) : b(s-b) : c(s-c)
A3A4∩B3B4∩C3C4 = X(10) = Spieker center = center of Vijay excentral eircle = b + c : c + a : a + b
A2A'∩B2B'∩C2C'= X(57) = a(s-b)(s-c) : b(s-a)(s-c) : c(s-a)(s-b)
A1A2A4∩B1B2B4∩C1C2C4 = X(226) = (b+c)(s-b)(s-c) : (c+a)(s-a)(s-c) : (a+b)(s-a)(s-b)
AA4∩BB4∩CC4 = X(514) = b-c : c-a : a-b
AA5∩BB5∩CC5 = X(2051) = 1/(a^3 - a (b^2 - b c + c^2) - b c(b + c)) : :
A5A'∩B5B'∩C5C' = X(44039)
AA6∩BB6∩CC6 = X(44040)
A5A6∩B5B6∩C5C6; = X(44041)

Constructions:

X(44039)
Vijay a-incentral circle.pdf
Vijay b-incentral circle.pdf
Vijay c-incentral circle.pdf
Vijay exncentral circle.pdf

The Vijay excentral circle is the Spieker radical circle. The Vijay incentral circles are the extraversions of the Spieker radical circle. That is, the Vijay a-incentral circle is the radical circle of the incircle and the B- and C-excircles, and cyclically for the Vijay b- and c-incentral circles. (Randy Hutson, September 30, 2021)

The Vijay excentral circle and the three Vijay incentral circles are Taylor circles of the excentral triangle. (Dao Thanh Oai, October 26, 2021)

A1B1C1 = Vijay excentral triangle is the 2nd extouch triangle of ABC (César Lozada, December 6, 2022)

X(44039) = PERSPECTOR OF EXCENTRAL TRIANGLE AND VIJAY POLAR EXCENTRAL TRIANGLE

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

X(44039) lies on these lines: {1, 2051}, {3, 10}, {4, 30116}, {8, 573}, {65, 29069}, {145, 9535}, {181, 10950}, {386, 944}, {516, 31785}, {517, 2901}, {572, 1220}, {950, 5710}, {952, 970}, {1503, 31799}, {1682, 10944}, {1695, 3632}, {1742, 5691}, {1746, 5260}, {1764, 17751}, {2098, 9554}, {2099, 9553}, {3436, 22020}, {3667, 9961}, {3869, 22022}, {4276, 11491}, {5264, 10572}, {5587, 19858}, {5690, 35203}, {5853, 10443}, {5881, 9548}, {5882, 37698}, {6999, 41232}, {9566, 12645}, {9567, 18526}, {9569, 37727}, {10406, 37734}, {10407, 15950}, {10440, 28236}, {10470, 26115}, {11322, 24996}, {29825, 30389}

X(44039) = midpoint of X(8) and X(10454)
X(44039) = X(23512)-Ceva conjugate of X(1766)
X(44039) = barycentric product X(321)*X(40456)
X(44039) = barycentric quotient X(40456)/X(81)
X(44039) = {X(2536),X(2537)}-harmonic conjugate of X(23361)

X(44040) = PERSPECTOR OF ABC AND VIJAY POLAR INCENTRAL TRIANGLE

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

X(44040) lies on these lines: on lines {1, 28997}, {10, 7069}, {11, 596}, {55, 4075}, {72, 519}, {78, 30568}, {307, 1210}, {497, 24068}, {522, 1329}, {1089, 2310}, {2325, 3694}, {2757, 6789}, {3710, 4723}, {3971, 4314}, {6534, 11238}, {6700, 7515}, {30144, 34587}

X(44040) = X(i)-cross conjugate of X(j) for these (i,j): {3270, 3239}, {21031, 8}
X(44040) = cevapoint of X(2310) and X(3700)
X(44040) = trilinear pole of line {1639, 8611}
X(44040) = X(i)-isoconjugate of X(j) for these (i,j): {56, 404}, {604, 32939}, {7128, 39006}
X(44040) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 32939}, {9, 404}, {3239, 20293}, {3270, 39006}, {3710, 42705}

X(44041) = PERSPECTOR OF VIJAY POLAR INCENTRAL TRIANGLE AND VIJAY POLAR EXCENTRAL TRIANGLE

Barycentrics    (-3*b^2*c^11-2*a*b*c^11+a^2*c^11-3*b^3*c^10+6*a*b^2*c^10+3*a^2*b*c^10-2*a^3*c^10+12*b^4*c^9+6*a*b^3*c^9-6*a^2*b^2*c^9+6*a^3*b*c^9-6*a^4*c^9+12*b^5*c^8-24*a*b^4*c^8+6*a^2*b^3*c^8-2*a^3*b^2*c^8-10*a^4*b*c^8+6*a^5*c^8-18*b^6*c^7-4*a*b^5*c^7+21*a^2*b^4*c^7-32*a^3*b^3*c^7+11*a^4*b^2*c^7-6*a^5*b*c^7+12*a^6*c^7-18*b^7*c^6+36*a*b^6*c^6-25*a^2*b^5*c^6+4*a^3*b^4*c^6-a^4*b^3*c^6-18*a^5*b^2*c^6+12*a^6*b*c^6-6*a^7*c^6+12*b^8*c^5-4*a*b^7*c^5-25*a^2*b^6*c^5+52*a^3*b^5*c^5-26*a^4*b^4*c^5-2*a^5*b^3*c^5+a^6*b^2*c^5+2*a^7*b*c^5-10*a^8*c^5+12*b^9*c^4-24*a*b^8*c^4+21*a^2*b^7*c^4+4*a^3*b^6*c^4-26*a^4*b^5*c^4+8*a^5*b^4*c^4-a^6*b^3*c^4+10*a^7*b^2*c^4-6*a^8*b*c^4+2*a^9*c^4-3*b^10*c^3+6*a*b^9*c^3+6*a^2*b^8*c^3-32*a^3*b^7*c^3-a^4*b^6*c^3-2*a^5*b^5*c^3-a^6*b^4*c^3+28*a^7*b^3*c^3-4*a^8*b^2*c^3+3*a^10*c^3-3*b^11*c^2+6*a*b^10*c^2-6*a^2*b^9*c^2-2*a^3*b^8*c^2+11*a^4*b^7*c^2-18*a^5*b^6*c^2+a^6*b^5*c^2+10*a^7*b^4*c^2-4*a^8*b^3*c^2+4*a^9*b^2*c^2+a^10*b*c^2-2*a*b^11*c+3*a^2*b^10*c+6*a^3*b^9*c-10*a^4*b^8*c-6*a^5*b^7*c+12*a^6*b^6*c+2*a^7*b^5*c-6*a^8*b^4*c+a^10*b^2*c+a^2*b^11-2*a^3*b^10-6*a^4*b^9+6*a^5*b^8+12*a^6*b^7-6*a^7*b^6-10*a^8*b^5+2*a^9*b^4+3*a^10*b^3) : :

X(44041) lies on this line: {2051,44040}

X(44042) = X(11)X(2679)∩X(55)X(805)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44042) lies on the incircle and these lines: {11, 2679}, {12, 33330}, {55, 805}, {56, 2698}, {511, 3027}, {512, 3023}, {5148, 6022}, {5217, 38703}, {5432, 22103}, {13183, 31513}, {20403, 33965}

X(44043) = X(11)X(650)∩X(55)X(927)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44043) lies on the incircle and these lines: {11, 650}, {12, 33331}, {55, 927}, {56, 2724}, {497, 14732}, {514, 3022}, {516, 1362}, {3322, 30331}, {5432, 40554}, {31891, 40629}

X(44043) = reflection of X(3022) in the Soddy line

X(44044) = X(11)X(6129)∩X(55)X(1309)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44044) lies on the incircle and these lines: {11, 6129}, {12, 39535}, {55, 1309}, {56, 2734}, {515, 1361}, {522, 1364}, {1464, 3324}, {3025, 35013}, {3319, 5882}, {5432, 40558}, {6284, 13756}

X(44045) = X(11)X(5519)∩X(55)X(6078)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44045) lies on the incircle and these lines: {11, 5519}, {55, 6078}, {56, 28914}, {518, 3021}, {1358, 3309}, {3689, 5580}

X(44046) = X(11)X(5516)∩X(55)X(6079)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44046) lies on the incircle and these lines: {11, 5516}, {55, 6079}, {350, 5579}, {519, 6018}, {1357, 3667}, {1358, 4106}, {3021, 4009}, {5577, 24840}, {34194, 34587}

X(44046) = reflection of X(3022) in the Nagel line

X(44047) = X(11)X(35580)∩X(55)X(6081)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44047) lies on the incircle these lines: {11, 35580}, {55, 6081}, {521, 3318}, {1359, 6001}, {3319, 12680}

X(44048) = X(11)X(31654)∩X(55)X(6082)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44048) lies on the incircle and these lines: {11, 31654}, {12, 6092}, {55, 6082}, {56, 6093}, {524, 6019}, {1499, 3325}

X(44049) = X(11)X(33504)∩X(55)X(2867)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44049) lies on the incircle these lines: {11, 33504}, {55, 2867}, {525, 6020}, {1503, 3320}

X(44050) = X(11)X(35581)∩X(55)X(16170)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44050) lies on the incircle and these lines: {11, 35581}, {55, 16170}, {56, 16169}, {526, 33965}, {5663, 33964}

X(44051) = X(11)X(35582)∩X(55)X(20404)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44051) lies on the incircle and these lines: {11, 35582}, {55, 20404}, {542, 6023}, {690, 6027}, {804, 33965}, {2782, 33964}

X(44052) = X(11)X(35587)∩X(900)X(3025)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44052) lies on the incircle and these lines: {11, 35587}, {900, 3025}, {952, 13756}, {6018, 12743}

X(44052) = reflection of X(3025) in the X(1)X(5) line

X(44053) = X(11)X(35591)∩X(12)X(33333)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44053) lies on the incircle and these lines: {11, 35591}, {12, 33333}, {56, 15907}, {1154, 7159}, {1510, 3327}

X(44054) = X(526)X(3025)∩X(900)X(33965)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44054) lies on the incircle and these lines: {526, 3025}, {900, 33965}, {952, 33964}, {2771, 31524}, {5663, 13756}, {8674, 31522}

X(44054) = reflection of X(33965) in the X(1)X(5) line

X(44055) = X(55)X(35011)∩X(1361)X(25485)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1907.

X(44055) lies on the incircle and these lines: {55, 35011}, {1361, 25485}, {2800, 3319}, {2818, 13756}, {3025, 8677}, {3326, 3738}

X(44056) = X(5)X(51)∩X(23)X(1493)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 7*a^4*b^4*c^2 - 2*a^2*b^6*c^2 - 2*b^8*c^2 + 2*a^6*c^4 + 7*a^4*b^2*c^4 + a^2*b^4*c^4 + b^6*c^4 + 2*a^4*c^6 - 2*a^2*b^2*c^6 + b^4*c^6 - 3*a^2*c^8 - 2*b^2*c^8 + c^10) : :
X(44056) = 3 X[51] - 2 X[13365], 3 X[51] - X[21230], 3 X[52] + X[43581], 3 X[143] - 2 X[973], X[195] + 3 X[3060], 3 X[195] + X[13423], X[1493] + 2 X[16982], 9 X[3060] - X[13423], 3 X[3060] - X[32196], 5 X[3567] - X[12307], 3 X[5946] - X[7691], X[6153] - 3 X[21849], 3 X[7730] + X[12316], 9 X[11002] - X[12325], 2 X[12242] + X[13421], 3 X[13363] - 2 X[32348], 3 X[13364] - 2 X[13565], X[13423] - 3 X[32196], 3 X[20424] - X[43581]

See Antreas Hatzipolakis and Peter Moses, euclid 1909.

X(44056) lies on these lines: {5, 51}, {23, 1493}, {30, 10115}, {54, 2937}, {156, 195}, {511, 8254}, {568, 43816}, {1112, 2914}, {3519, 7533}, {3567, 12307}, {3581, 12006}, {5446, 32423}, {5946, 7691}, {6102, 15800}, {6153, 21849}, {6288, 18427}, {6689, 10627}, {6746, 12300}, {7512, 10610}, {7545, 13368}, {7730, 12316}, {10628, 11262}, {11002, 12325}, {11802, 16881}, {11803, 25338}, {12242, 13421}, {12291, 26864}, {13363, 32348}, {14449, 22051}, {36853, 38898}

X(44056) = midpoint of X(i) and X(j) for these {i,j}: {52, 20424}, {54, 10263}, {195, 32196}, {6102, 15800}, {13368, 15801}, {14449, 22051}, {36853, 38898}
X(44056) = reflection of X(i) in X(j) for these {i,j}: {1209, 10095}, {10627, 6689}, {11802, 16881}, {21230, 13365}
X(44056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 21230, 13365}, {195, 3060, 32196}

X(44057) = X(4)X(54)∩X(30)X(18402)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1909.

X(44057) lies on the cubic K464 and these lines: {4, 54}, {30, 18402}, {133, 35592}, {137, 403}, {143, 6798}, {186, 16337}, {1263, 32215}, {1986, 14106}, {6368, 18314}, {6801, 37766}, {21268, 32410}, {33643, 37943}

X(44057) = polar circle inverse of X(54)
X(44057) = orthic isogonal conjugate of X(10214)
X(44057) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 10214}, {37766, 11062}
X(44057) = X(2169)-isoconjugate of X(39431)
X(44057) = crossdifference of every pair of points on line {14533, 17434}
X(44057) = barycentric quotient X(53)/X(39431)

X(44058) = X(30)X(5171)∩X(32)X(1316)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1913.

X(44058) lies on these lines: {30, 5171}, {32, 1316}, {182, 523}, {1078, 36163}, {1691, 2453}, {2452, 5034}, {7793, 36181}, {7815, 11007}, {10796, 15539}, {13356, 36157}, {32761, 36822}

X(44059) = ISOGONAL CONJUGATE OF X(8760)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44059) lies on the circumcircle and these lines: {3, 43363}, {4, 20623}, {103, 18446}, {104, 5848}, {105, 5603}, {929, 1633}, {1295, 5759}, {11456, 32726}

X(44059) = reflection of X(i) in X(j) for these {i,j}: {4, 20623}, {43363, 3}
X(44059) = isogonal conjugate of X(8760)
X(44059) = Thomson-isogonal conjugate of X(44670)
X(44059) = Collings transform of X(20623)
X(44059) = Λ(PU(125))
X(44059) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8760}, {513, 2000}, {650, 2002}
X(44059) = trilinear pole of line X6)X(8758)
X(44059) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8760}, {101, 2000}, {109, 2002}

X(44060) = X(3)X(5896)∩X(74)X(1498)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44060) lies on the circumcircle and these lines: {3, 5896}, {74, 1498}, {98, 38253}, {110, 13302}, {1294, 3529}, {1297, 9909}, {1301, 5502}, {1302, 35311}, {2373, 35510}, {2693, 12096}, {2697, 5159}, {7396, 34168}

X(44060) = reflection of X(5896) in X(3)
X(44060) = X(15750)-cross conjugate of X(250)
X(44060) = cevapoint of X(154) and X(647)
X(44060) = Λ(radical axis of circumcircle and midheight circle)
X(44060) = trilinear pole of line X(6)X(1204)
X(44060) = X(i)-isoconjugate of X(j) for these (i,j): {162, 13611}, {525, 18594}, {656, 3146}, {1577, 38292}, {8611, 18624}, {24018, 33630}
X(44060) = barycentric product X(i)*X(j) for these {i,j}: {107, 36609}, {110, 38253}, {112, 35510}, {648, 3532}, {1301, 40170}
X(44060) = barycentric quotient X(i)/X(j) for these {i,j}: {112, 3146}, {647, 13611}, {1301, 14572}, {1576, 38292}, {3532, 525}, {15400, 14638}, {32676, 18594}, {32713, 33630}, {35510, 3267}, {36609, 3265}, {38253, 850}

X(44061) = X(3)X(9076)∩X(5)X(111)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44061) lies on the circumcircle and these lines: {3, 9076}, {5, 111}, {74, 32233}, {98, 35921}, {110, 32312}, {376, 29011}, {691, 14570}, {733, 10796}, {755, 1352}, {930, 11634}, {933, 4235}, {935, 1634}, {1141, 43084}, {1291, 7472}, {2070, 2770}, {2373, 7488}, {2374, 3518}, {3563, 7576}, {9084, 13595}, {10102, 37760}, {12107, 43663}, {12122, 14388}, {12584, 36833}, {37943, 40119}

X(44061) = reflection of X(9076) in X(3)
X(44061) = Thomson isogonal conjugate of X(9019)
X(44061) = Collings transform of X(37454)
X(44061) = X(798)-isoconjugate of X(11056)
X(44061) = cevapoint of X(i) and X(j) for these (i,j): {523, 37454}, {599, 2525}
X(44061) = trilinear pole of line X(6)X(16511)
X(44061) = barycentric quotient X(99)/X(11056)

X(44062) = X(2)X(15240)∩X(74)X(12118)

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 6*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 6*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 2*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) : :
X(44062) = 3 X[2] - 4 X[15240]

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44062) lies on the circumcircle and these lines: {2, 15240}, {74, 12118}, {110, 13223}, {1299, 35471}, {1300, 12084}, {2374, 26284}, {3563, 7391}, {21213, 40120}, {37978, 40118}

X(44062) = anticomplement of X(15241)
X(44062) = reflection of X(15241) in X(15240)
X(44062) = Collings transform of X(6640)
X(44062) = cevapoint of X(523) and X(6640)
X(44062) = circumnormal-isogonal conjugate of X(45780)
X(44062) = circumcircle antipode of X(45780)
X(44062) = {X(15240),X(15241)}-harmonic conjugate of X(2)

X(44063) = X(74)X(1071)∩X(105)X(7965)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44063) lies on the circumcircle and these lines: {3, 43659}, {74, 1071}, {102, 16132}, {105, 7965}, {1295, 30264}, {1297, 30271}, {2752, 14083}, {5951, 11491}

X(44063) = reflection of X(43659) in X(3)
X(44063) = Collings transform of X(33178)
X(44063) = cevapoint of X(513) and X(33178)

X(44064) = X(3)X(39437)∩X(20)X(1299)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44064) lies on the circumcircle and these lines: {3, 39437}, {20, 1299}, {22, 40120}, {74, 11411}, {1300, 11413}, {1301, 30512}, {1370, 3563}, {2374, 26283}, {2383, 12225}, {4226, 39417}, {11634, 30251}, {15329, 30249}, {16386, 32710}, {16387, 40119}, {22239, 40049}, {37929, 40118}

X(44064) = reflection of X(39437) in X(3)
X(44064) = Collings transform of X(3548)
X(44064) = cevapoint of X(523) and X(3548)
X(44064) = trilinear pole of line X(6)X(9820)

X(44065) = X(20)X(759)∩X(915)X(3561)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44065) lies on the circumcircle and these lines: {3, 39439}, {20, 759}, {104, 30267}, {107, 13589}, {915, 3651}, {1301, 4242}, {1304, 36167}, {2071, 12030}, {4220, 15344}, {9061, 26253}, {11413, 39435}, {14987, 30268}, {22239, 37964}

X(44065) = reflection of X(39439) in X(3)
X(44065) = cevapoint of X(3) and X(6003)

X(44066) = X(3)X(43657)∩X(548)X(1141)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1915.

X(44066) lies on the circumcircle and these lines: {3, 43657}, {548, 1141}, {550, 13597}, {2383, 14865}, {5966, 14706}, {14979, 35452}, {23096, 37920}, {33643, 34864}

X(44066) = reflection of X(43657) in X(3)

X(44067) = X(54)X(2575)∩X(110)X(1113)

Barycentrics    a^2*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(-((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)) + a^2*(-a^2 + b^2 + c^2)*(1 - J)) : :

See Tran Viet Hung and Peter Moses, euclid 1936.

X(44067) lies on these lines: {54, 2575}, {110, 1113}, {184, 31954}, {186, 323}, {576, 24650}, {1092, 25407}, {1114, 15463}, {1312, 32423}, {1347, 13415}, {3357, 32616}, {8116, 15462}, {13414, 34507}, {14500, 18400}

X(44067) = reflection of X(44068) in X(11597)
X(44067) = {X(186),X(3043)}-harmonic conjugate of X(44068)
X(44067) = {X(1151),X(22115)}-harmonic conjugate of X(44068)
X(44067) = X(i)-isoconjugate of X(j) for these (i,j): {94, 2578}, {265, 2588}, {1823, 10412}, {1989, 2582}, {2166, 2574}, {2577, 14592}, {2581, 14582}, {2584, 6344}, {2587, 43083}, {36061, 39240}
X(44067) = barycentric product X(i)*X(j) for these {i,j}: {50, 15164}, {186, 8115}, {323, 1113}, {526, 39298}, {2575, 14590}, {2580, 6149}, {8106, 10411}, {14591, 22340}
X(44067) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 2574}, {186, 2592}, {323, 22339}, {1113, 94}, {2575, 14592}, {2576, 2166}, {6149, 2582}, {8106, 10412}, {8115, 328}, {14590, 15165}, {14591, 1114}, {15164, 20573}, {19627, 42668}, {34397, 8105}, {39298, 35139}, {42667, 14582}
X(44067) = {X(110),X(15461)}-harmonic conjugate of X(1113)

X(44068) = X(54)X(2574)∩X(110)X(1114)

Barycentrics    a^2*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(-((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)) + a^2*(-a^2 + b^2 + c^2)*(1 + J)) : :

See Tran Viet Hung and Peter Moses, euclid 1936.

X(44068) lies on these lines: {54, 2574}, {110, 1114}, {184, 31955}, {186, 323}, {576, 24651}, {1092, 25408}, {1113, 15463}, {1313, 32423}, {1346, 13414}, {3357, 32617}, {8115, 15462}, {13415, 34507}, {14499, 18400}

X(44068) = reflection of X(44067) in X(11597)
X(44068) = X(i)-isoconjugate of X(j) for these (i,j): {94, 2579}, {265, 2589}, {1822, 10412}, {1989, 2583}, {2166, 2575}, {2576, 14592}, {2580, 14582}, {2585, 6344}, {2586, 43083}, {36061, 39241}
X(44068) = barycentric product X(i)*X(j) for these {i,j}: {50, 15165}, {186, 8116}, {323, 1114}, {526, 39299}, {2574, 14590}, {2581, 6149}, {8105, 10411}, {14591, 22339}
X(44068) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 2575}, {186, 2593}, {323, 22340}, {1114, 94}, {2574, 14592}, {2577, 2166}, {6149, 2583}, {8105, 10412}, {8116, 328}, {14590, 15164}, {14591, 1113}, {15165, 20573}, {19627, 42667}, {34397, 8106}, {39299, 35139}, {42668, 14582}
X(44068) = {X(110),X(15460)}-harmonic conjugate of X(1114)
X(44068) = {X(186),X(3043)}-harmonic conjugate of X(44067)
X(44068) = {X(1151),X(22115)}-harmonic conjugate of X(44067)

X(44069) = ANTIGONAL IMAGE OF X(1276)

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

X(44069) lies on the cubic K060 and these lines: {5, 6192}, {13, 79}, {14, 39151}, {30, 5673}, {80, 11582}, {265, 36933}, {517, 36932}, {621, 14206}, {5627, 36910}, {7026, 12699}, {11600, 14452}, {34299, 38944}

X(44069) = antigonal image of X(1276)
X(44069) = symgonal image of X(33397)
X(44069) = X(5357)-isoconjugate of X(7060)
X(44069) = cevapoint of X(5673) and X(6192)
X(44069) = barycentric quotient X(i)/X(j) for these {i,j}: {11073, 7060}, {19304, 5357}

X(44070) = ANTIGONAL IMAGE OF X(1277)

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

X(44070 lies on the cubic K060 and these lines: {5, 6191}, {13, 39150}, {14, 79}, {30, 5672}, {80, 11581}, {265, 36932}, {517, 36933}, {622, 14206}, {5627, 36910}, {7043, 12699}, {11601, 14452}, {34299, 38943}

X(44070) = antigonal image of X(1277)
X(44070) = symgonal image of X(33396)
X(44070) = X(5353)-isoconjugate of X(7059)
X(44070) = cevapoint of X(5672) and X(6191)
X(44070) = barycentric quotient X(i)/X(j) for these {i,j}: {11072, 7059}, {19305, 5353}

X(44071) = ANTIGONAL IMAGE OF X(8444)

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

X(44071) lies on the cubic K060 and these lines: {13, 80}, {30, 1276}, {79, 11600}, {101, 1141}, {203, 37710}, {265, 17405}, {355, 11752}, {621, 14213}, {1834, 11072}, {11099, 36941}, {11582, 19658}, {34303, 38944}

X(44071) = antigonal image of X(8444)
X(44071) = X(i)-isoconjugate of X(j) for these (i,j): {5353, 7344}, {6149, 17405}
X(44071) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 17405}, {11072, 7344}

X(44072) = ANTIGONAL IMAGE OF X(8454)

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

X(44072) lies on the cubic K060 and these lines: {14, 80}, {30, 1277}, {79, 11601}, {101, 1141}, {202, 37710}, {265, 17406}, {355, 11789}, {622, 14213}, {1834, 11073}, {11100, 36940}, {11581, 19658}, {34303, 38943}

X(44072) = antigonal image of X(8454)
X(44072) = X(i)-isoconjugate of X(j) for these (i,j): {5357, 7345}, {6149, 17406}
X(44072) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 17406}, {11073, 7345}

X(44073) = CROSSSUM OF X(64) AND X(5923)

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

X(44073) = Q(X(20)), where Q is defined at X(34815).

X(44073) lies on these lines: {3, 41489}, {20, 14615}, {22, 110}, {3164, 3522}, {3515, 34815}, {10607, 11413}, {31388, 37198}

X(44073) = crosssum of X(64) and X(5922)

X(44074) = CROSSSUM OF X(84) AND X(5923)

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

X(44074) = Q(X(40)), where Q is defined at X(34815).

X(44074) lies on these lines: {3, 1167}, {36, 7114}, {40, 329}, {101, 102}, {165, 1745}, {1617, 9786}, {1785, 5119}

X(44074) = crosssum of X(84) and X(5923)

X(44075) = CROSSSUM OF X(40) AND X(5881)

Barycentrics    a^2*(a^7*b + a^6*b^2 - 3*a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 + 3*a^2*b^6 - a*b^7 - b^8 + a^7*c - 5*a^6*b*c + 3*a^5*b^2*c + 9*a^4*b^3*c - 9*a^3*b^4*c - 3*a^2*b^5*c + 5*a*b^6*c - b^7*c + a^6*c^2 + 3*a^5*b*c^2 - 12*a^4*b^2*c^2 + 6*a^3*b^3*c^2 + 9*a^2*b^4*c^2 - 9*a*b^5*c^2 + 2*b^6*c^2 - 3*a^5*c^3 + 9*a^4*b*c^3 + 6*a^3*b^2*c^3 - 18*a^2*b^3*c^3 + 5*a*b^4*c^3 + b^5*c^3 - 3*a^4*c^4 - 9*a^3*b*c^4 + 9*a^2*b^2*c^4 + 5*a*b^3*c^4 - 2*b^4*c^4 + 3*a^3*c^5 - 3*a^2*b*c^5 - 9*a*b^2*c^5 + b^3*c^5 + 3*a^2*c^6 + 5*a*b*c^6 + 2*b^2*c^6 - a*c^7 - b*c^7 - c^8) : :

X(44075) = Q(X(84)), where Q is defined at X(34815).

X(44075) lies on these lines: {3, 947}, {4, 34589}, {36, 7114}, {40, 145}, {56, 102}, {945, 38667}, {946, 21228}, {1753, 10085}, {12704, 36986}, {34040, 38674}

X(44075) = crosssum of X(40) and X(5881)
X(44075) = {X(3),X(34046)}-harmonic conjugate of X(947)

X(44076) = X(3)X(68)∩X(5)X(49)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^8 - 3*a^6*b^2 + a^4*b^4 - a^2*b^6 + b^8 - 3*a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) : :
X(44076) = 2 X[5] - 3 X[12022], 3 X[5] - 4 X[43575], 4 X[143] - 3 X[7576], 3 X[185] - 2 X[43577], 3 X[381] - 2 X[12134], 3 X[381] - 4 X[12241], 4 X[389] - 3 X[38321], 3 X[568] - 2 X[3575], 3 X[568] - 4 X[13292], 3 X[3060] - 2 X[11819], 7 X[3526] - 12 X[12024], 5 X[3567] - 4 X[31830], 3 X[3830] - 2 X[16655], 5 X[3843] - 6 X[16657], 4 X[3853] - 3 X[16658], 5 X[5076] - 4 X[16621], 4 X[5446] - 3 X[7540], 3 X[5890] - X[12278], 3 X[5890] - 4 X[43588], X[6240] - 4 X[11264], 7 X[9781] - 6 X[13490], 3 X[10116] - X[43577], 3 X[11245] - 2 X[31833], 6 X[11245] - 5 X[37481], 8 X[11745] - 9 X[13321], 3 X[12022] - X[14516], 9 X[12022] - 8 X[43575], X[12278] - 4 X[43588], X[12289] + 2 X[32358], 4 X[12362] - 3 X[23039], 2 X[12370] + X[34799], 4 X[13630] - 3 X[38323], 3 X[14516] - 8 X[43575], 8 X[16656] - 9 X[38335], 4 X[16881] - 3 X[38322], 4 X[17712] - 3 X[36987], 2 X[31831] - 3 X[34664], 4 X[31833] - 5 X[37481]-

See Antreas Hatzipolakis and Peter Moses, euclid 1943.

X(44076) lies on these lines: {3, 68}, {4, 1994}, {5, 49}, {20, 18917}, {30, 5889}, {52, 10112}, {125, 12038}, {140, 26913}, {143, 7576}, {155, 18396}, {156, 403}, {184, 9927}, {185, 10111}, {195, 31724}, {235, 10540}, {378, 32140}, {381, 11426}, {382, 1351}, {389, 38321}, {427, 37472}, {511, 11750}, {539, 5562}, {542, 12162}, {546, 11422}, {550, 41724}, {568, 3575}, {569, 37347}, {578, 5576}, {895, 3521}, {1092, 37452}, {1147, 2072}, {1154, 12225}, {1181, 12293}, {1353, 8537}, {1498, 31725}, {1568, 41597}, {1614, 15761}, {1658, 3580}, {1885, 18439}, {1993, 18569}, {2055, 36245}, {2070, 9920}, {2883, 31726}, {2888, 35921}, {2931, 33563}, {3060, 11819}, {3410, 35500}, {3448, 3520}, {3526, 12024}, {3530, 38397}, {3549, 18925}, {3564, 12605}, {3567, 31830}, {3627, 16659}, {3830, 16655}, {3843, 16657}, {3853, 16658}, {5076, 16621}, {5254, 22146}, {5446, 7540}, {5448, 13851}, {5449, 13367}, {5609, 43865}, {5663, 18560}, {5890, 12278}, {5894, 10620}, {6101, 13470}, {6102, 6240}, {6193, 18531}, {6242, 12899}, {6639, 14852}, {6644, 18912}, {6759, 11799}, {6776, 8548}, {7399, 13353}, {7488, 12254}, {7506, 39571}, {7517, 9833}, {7526, 11442}, {7527, 43818}, {7545, 15873}, {7553, 13142}, {7577, 9545}, {8550, 39562}, {9140, 43608}, {9544, 16868}, {9703, 9820}, {9704, 10254}, {9707, 10201}, {9781, 13490}, {10018, 32171}, {10020, 11464}, {10114, 11562}, {10224, 16000}, {10226, 10264}, {10539, 18390}, {10619, 18475}, {10627, 11565}, {10897, 35836}, {10898, 35837}, {10938, 12421}, {11245, 31833}, {11381, 12897}, {11432, 14542}, {11449, 26917}, {11457, 12084}, {11564, 16665}, {11585, 22115}, {11745, 13321}, {12121, 43616}, {12233, 15087}, {12362, 23039}, {12383, 22467}, {12428, 18455}, {12902, 19362}, {13160, 32046}, {13352, 18381}, {13371, 25739}, {13488, 15531}, {13561, 37118}, {13630, 38323}, {13754, 18563}, {15032, 34007}, {15034, 43866}, {15062, 43895}, {15559, 34514}, {15760, 31804}, {15800, 31802}, {16163, 43604}, {16266, 37444}, {16656, 38335}, {16881, 38322}, {17712, 36987}, {17845, 37489}, {17928, 18952}, {18356, 18570}, {18383, 34986}, {18403, 22660}, {18420, 36753}, {18430, 23047}, {18447, 18970}, {18494, 37493}, {18533, 18951}, {18914, 22663}, {20126, 43903}, {20417, 43907}, {23294, 23336}, {23328, 35498}, {23335, 37495}, {26879, 37814}, {30714, 43817}, {31723, 36747}, {31831, 34664}, {32138, 35491}, {34153, 43615}, {36253, 43839}

X(44076) = midpoint of X(i) and X(j) for these {i,j}: {4, 34799}, {5889, 12289}
X(44076) = reflection of X(i) in X(j) for these {i,j}: {3, 6146}, {4, 12370}, {52, 10112}, {185, 10116}, {3575, 13292}, {5889, 32358}, {6101, 13470}, {6102, 11264}, {6240, 6102}, {6242, 12899}, {7553, 13142}, {10627, 11565}, {11381, 12897}, {11562, 10114}, {12134, 12241}, {12162, 13403}, {14516, 5}, {16659, 3627}, {18436, 12605}, {18439, 1885}, {18563, 21659}, {31656, 32410}
X(44076) = crossdifference of every pair of points on line {2081, 6753}
X(44076) = X(34799)-of-Euler-triangle
X(44076) = X(14516)-of-Johnson-triangle
X(44076) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {49, 265, 5}, {68, 12118, 9932}, {68, 19467, 3}, {155, 18396, 18404}, {184, 9927, 10024}, {427, 43595, 37472}, {567, 6288, 5}, {578, 18474, 5576}, {1899, 12118, 3}, {3575, 13292, 568}, {6193, 18945, 18531}, {9703, 10255, 9820}, {10226, 10264, 43607}, {11245, 31833, 37481}, {12022, 14516, 5}, {12134, 12241, 381}, {12383, 43808, 22467}, {13434, 41171, 5}, {13561, 43394, 37118}, {13851, 43844, 5448}, {14852, 19357, 6639}, {15317, 18445, 12161}, {18350, 43821, 5}, {18533, 18951, 37490}, {23236, 43821, 18350}, {25739, 34148, 13371}, {34782, 41587, 2070}

leftri

Points of the form tgX or gtX, where X is on the Euler line: X(44077)-X(44155)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, August 8-11, 2021.

For any triangle center P, let gP and tP denote the isogonal conjugate of P and the isotomic conjugate of P, respectively. Centers X(44077)-X(44127) are points of the form tg(X), and X(44128)-X(44155) of the form gt(X), where X lies on the Euler line. The appearance of (h,i,j,k,m) in the following list means that X(j) is on the Euler line, and (X(h), X(i), X(j), X(k), X(m)) = (gtX, tX, X, gX, tgX):

(6,2,2,6,76)
(184,264,3,4,69)
(25,69,4,3,264)
(51,95,5,54,311)
(154,253,20,64,14615)
(2194,1441,21,65,314)
(206,18018,22,66,315)
(18374,18019,23,67,316)
(44077,20563,24,68,317)
(1974,305,25,69,4)
(44078,20564,26,70,44128)
(1474,306,27,71,44129)
(2203,20336,28,72,286)
(2299,307,29,73,44130)
(1495,1494,30,74,3260)
(13366,40410,140,1173,1232)
(34397,328,186,265,340)
(9418,18024,237,290,511)
(232,287,297,248,44132)
(26864,36889,376,3426,44133)
(1915,9229,384,695,9230)
(1971,1972,401,1987,44137)
(44089,40708,419,36214,17984)
(1843,1799,427,1176,1235)
(40952,40412,442,1175,1234)
(10311,42313,458,43718,44144)
(44102,30786,468,895,44146)
(11402,8797,631,3527,44149)
(2393,2373,858,1177,1236)
(44123,22339,1113,2574,15164)
(44124,22340,1114,2575,15165)

Recall that if L is a line, then gL and tL are conics, and tgL and gtL are lines. Each X on the Euler lines is given by a combo X(2) + k*(X(3), and the locus of gtX is the line

4*(a^2 + b^2 + c^2)*(2 + 3*k)*S^2 X[6] - 3*a^2*b^2*c^2*(3 + J^2)*k*X[25], which, for each k, is a point on the line X(6)X(25).

The locus of tgX is the line

16*S^4*X[6] - a^2*b^2*c^2*(a^2 + b^2 + c^2)*(J^2 - 9 - 12*k)*X[69], which, for each k, is a point on the line X(6)X(69).

X(44077) = ISOGONAL CONJUGATE OF X(20563)

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

X(44077) lies on these lines: {2, 19128}, {3, 12058}, {4, 569}, {6, 25}, {22, 9967}, {23, 8538}, {24, 52}, {49, 3517}, {54, 7487}, {68, 3542}, {97, 37114}, {110, 6353}, {136, 11547}, {156, 13292}, {182, 427}, {186, 2979}, {235, 6146}, {237, 10316}, {297, 10349}, {343, 468}, {378, 14855}, {393, 14593}, {394, 37488}, {403, 18474}, {419, 2001}, {421, 2052}, {428, 5476}, {460, 2909}, {511, 21213}, {567, 18494}, {577, 3135}, {578, 3575}, {973, 10274}, {1092, 3515}, {1112, 16165}, {1204, 6293}, {1209, 7505}, {1501, 2211}, {1593, 10984}, {1614, 3089}, {1899, 41603}, {1976, 13854}, {1994, 6403}, {2175, 14975}, {2207, 39109}, {2351, 3003}, {2917, 21660}, {3131, 10634}, {3132, 10635}, {3133, 39110}, {3155, 10897}, {3156, 10898}, {3162, 40825}, {3518, 11422}, {3541, 13336}, {4232, 9544}, {5063, 23195}, {5064, 19124}, {5094, 43650}, {5200, 13440}, {5622, 32064}, {5651, 37453}, {6524, 32713}, {6620, 40366}, {6623, 14157}, {6676, 19154}, {6756, 32046}, {6995, 11003}, {7494, 19121}, {7499, 19126}, {8717, 35481}, {8780, 41615}, {9677, 35765}, {10151, 18376}, {11383, 20986}, {11424, 12173}, {11470, 21284}, {11550, 37981}, {11819, 34114}, {12140, 38534}, {12228, 15473}, {12294, 22352}, {13198, 31383}, {13352, 18533}, {13567, 41729}, {13851, 15125}, {14580, 42295}, {15139, 26958}, {15750, 43652}, {18381, 20303}, {19137, 37439}, {27369, 34396}, {32078, 37457}, {32379, 39571}, {32734, 39111}, {33581, 40352}, {34945, 39575}, {37460, 43574}, {37480, 37931}, {41609, 42463}

X(44077) = isogonal conjugate of X(20563)
X(44077) = isogonal conjugate of anticomplement of X(40939)
X(44077) = isogonal conjugate of isotomic conjugate of X(24)
X(44077) = isogonal conjugate of polar conjugate of X(8745)
X(44077) = polar conjugate of isotomic conjugate of X(571)
X(44077) = anticomplement of complementary conjugate of X(40939)
X(44077) = trilinear product X(i)*X(j) for these {i,j}: {19, 571}, {24, 31}, {25, 47}, {32, 1748}, {48, 8745}, {163, 6753}, {317, 560}, {393, 563}, {1096, 1147}, {1973, 1993}, {2148, 14576}, {2180, 8882}, {2333, 18605}, {9247, 11547}
X(44077) = X(i)-Ceva conjugate of X(j) for these (i,j): {24, 571}, {393, 32}
X(44077) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20563}, {3, 20571}, {63, 5392}, {68, 75}, {69, 91}, {76, 1820}, {96, 18695}, {304, 2165}, {326, 847}, {561, 2351}, {925, 14208}, {2168, 28706}, {3267, 36145}, {24018, 30450}, {32132, 33808}
X(44077) = crosspoint of X(i) and X(j) for these (i,j): {6, 34438}, {24, 8745}
X(44077) = crosssum of X(i) and X(j) for these (i,j): {2, 37444}, {339, 3265}, {394, 16391}
X(44077) = barycentric product X(i)*X(j) for these {i,j}: {3, 8745}, {4, 571}, {6, 24}, {19, 47}, {25, 1993}, {31, 1748}, {32, 317}, {52, 8882}, {54, 14576}, {68, 36416}, {107, 30451}, {110, 6753}, {112, 924}, {136, 23357}, {158, 563}, {184, 11547}, {371, 5412}, {372, 5413}, {393, 1147}, {512, 41679}, {648, 34952}, {1304, 14397}, {1609, 34756}, {1783, 34948}, {1824, 18605}, {1974, 7763}, {2180, 2190}, {2203, 42700}, {2207, 9723}, {2211, 31635}, {2965, 14111}, {14591, 43088}, {15423, 32734}, {18883, 34397}
X(44077) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20563}, {19, 20571}, {24, 76}, {25, 5392}, {32, 68}, {47, 304}, {52, 28706}, {136, 23962}, {317, 1502}, {560, 1820}, {563, 326}, {571, 69}, {924, 3267}, {1147, 3926}, {1501, 2351}, {1748, 561}, {1973, 91}, {1974, 2165}, {1993, 305}, {2180, 18695}, {2207, 847}, {5412, 34391}, {5413, 34392}, {6753, 850}, {7763, 40050}, {8745, 264}, {8882, 34385}, {11547, 18022}, {14576, 311}, {14585, 16391}, {30451, 3265}, {32713, 30450}, {34397, 37802}, {34948, 15413}, {34952, 525}, {36416, 317}, {36417, 14593}, {41679, 670}
X(44077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {24, 35603, 52}, {25, 34397, 184}, {52, 34116, 1147}, {184, 1495, 1660}, {184, 1974, 25}, {1501, 2211, 17409}, {5412, 5413, 14576}

X(44078) = ISOGONAL CONJUGATE OF X(20564)

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

X(44078) lies on these lines: {2, 43811}, {6, 25}, {26, 34116}, {70, 7505}, {110, 41674}, {156, 32358}, {182, 5133}, {237, 20968}, {343, 19154}, {381, 11572}, {403, 6759}, {418, 9407}, {569, 11818}, {578, 7576}, {1092, 14070}, {1147, 2070}, {1204, 37954}, {1370, 15462}, {1576, 3135}, {1614, 39571}, {1899, 19128}, {1976, 40366}, {2211, 22075}, {3542, 32377}, {5012, 7394}, {6152, 10274}, {7426, 41628}, {7499, 19127}, {7528, 40441}, {9306, 37636}, {9818, 10984}, {10201, 10539}, {10540, 14852}, {11206, 13198}, {11424, 18494}, {13490, 32046}, {15139, 37453}, {19121, 43653}, {21213, 34117}, {22151, 27084}

X(44078) = isogonal conjugate of X(20564)
X(44078) = isogonal conjugate of isotomic conjugate of X(26)
X(44078) = isogonal conjugate of polar conjugate of X(8746)
X(44078) = X(2165)-Ceva conjugate of X(32)
X(44078) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20564}, {70, 75}, {76, 2158}, {1288, 14208}
X(44078) = crosspoint of X(i) and X(j) for these (i,j): {6, 34439}, {26, 8746}
X(44078) = crosssum of X(i) and X(j) for these (i,j): {2, 14790}, {339, 6563}
X(44078) = trilinear product X(i)*X(j) for these {i,j}: {26, 31}, {48, 8746}
X(44078) = barycentric product X(i)*X(j) for these {i,j}: {3, 8746}, {6, 26}, {70, 36418}, {2165, 34116}
X(44078) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 20564}, {26, 76}, {32, 70}, {560, 2158}, {8746, 264}, {34116, 7763}
X(44078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {154, 34397, 184}, {184, 1974, 51}, {18374, 19153, 1974}

X(44079) = POLAR CONJUGATE OF X(40830)

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

Let A'B'C' be the midheight triangle. Let AB, AC be the orthogonal projections of A' on CA, AB, resp. Define BC, BA, CA, CB cyclically. Let A" = BCBA∩CACB, and define B", C" cyclically. Triangle A"B"C" is homothetic to the orthic triangle at X(44079). (Randy Hutson, August 24, 2021)

X(44079) lies on these lines: {2, 12058}, {4, 5943}, {6, 25}, {24, 13346}, {52, 21841}, {181, 2212}, {185, 235}, {186, 36987}, {237, 22401}, {373, 427}, {389, 3089}, {394, 11470}, {403, 15030}, {460, 15575}, {468, 3917}, {511, 6353}, {1112, 5642}, {1196, 2211}, {1395, 3271}, {1585, 12298}, {1586, 12299}, {1593, 17825}, {1596, 9730}, {1597, 5892}, {1598, 5462}, {1853, 11381}, {1899, 41735}, {2356, 23638}, {3060, 4232}, {3088, 11695}, {3167, 21313}, {3168, 43976}, {3517, 5446}, {3542, 5562}, {3819, 38282}, {5200, 6291}, {5640, 6995}, {5650, 37453}, {5889, 16879}, {5891, 37942}, {5907, 6622}, {6000, 6623}, {6403, 21849}, {6524, 41762}, {6525, 6620}, {6677, 37511}, {6688, 8889}, {6754, 16240}, {6756, 11750}, {7378, 11451}, {7398, 9822}, {7487, 10110}, {7507, 27355}, {7714, 11179}, {7715, 10095}, {7718, 23841}, {8754, 14569}, {8780, 34382}, {9967, 10154}, {10151, 32062}, {10539, 21651}, {10565, 11574}, {10601, 19124}, {11363, 16980}, {12133, 17853}, {13598, 27082}, {15462, 41671}, {16194, 37984}, {21746, 40976}, {21971, 31670}, {26958, 34146}, {30443, 34944}, {32263, 41588}, {35603, 43844}, {36417, 42295}, {37643, 41715}, {37894, 40413}

X(44079) = polar conjugate of X(40830)
X(44079) = isogonal conjugate of isotomic conjugate of X(235)
X(44079) = polar conjugate of isotomic conjugate of X(800)
X(44079) = X(i)-Ceva conjugate of X(j) for these (i,j): {235, 800}, {6529, 2489}, {30249, 647}
X(44079) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40830}, {63, 801}, {69, 775}, {304, 41890}, {326, 1105}, {821, 3964}
X(44079) = crosspoint of X(i) and X(j) for these (i,j): {4, 41489}, {6, 43695}, {25, 6524}
X(44079) = crosssum of X(i) and X(j) for these (i,j): {2, 11413}, {3, 37669}, {69, 3964}
X(44079) = barycentric product X(i)*X(j) for these {i,j}: {4, 800}, {6, 235}, {19, 774}, {25, 13567}, {53, 16035}, {185, 393}, {512, 41678}, {820, 6520}, {1096, 6508}, {1624, 2501}, {1824, 18603}, {1973, 17858}, {2207, 41005}, {2883, 41489}, {3199, 19166}, {6509, 6524}, {13854, 41580}, {14091, 43695}, {14569, 19180}, {40144, 41602}
X(44079) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40830}, {25, 801}, {185, 3926}, {235, 76}, {774, 304}, {800, 69}, {820, 1102}, {1624, 4563}, {1973, 775}, {1974, 41890}, {2207, 1105}, {6509, 4176}, {13567, 305}, {16035, 34386}, {17858, 40364}, {41580, 34254}, {41678, 670}
X(44079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 51, 1843}, {25, 19118, 154}, {235, 2883, 22970}, {1843, 15010, 51}, {5943, 9729, 18928}, {6620, 34854, 40325}, {13567, 41580, 185}

X(44080) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(378)

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

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anti-Ara triangle at X(44080). (Randy Hutson, August 24, 2021)

X(44080) lies on these lines: {4, 110}, {6, 25}, {22, 37511}, {23, 6403}, {24, 6800}, {32, 40351}, {49, 1598}, {54, 3089}, {112, 9463}, {156, 6756}, {182, 468}, {186, 15080}, {235, 578}, {378, 4550}, {403, 14389}, {421, 43976}, {427, 3818}, {568, 19456}, {569, 3542}, {576, 1112}, {692, 11383}, {974, 11438}, {1092, 1593}, {1204, 7729}, {1351, 41615}, {1397, 14975}, {1597, 22115}, {1614, 7487}, {1885, 13346}, {1968, 35325}, {1993, 40914}, {1995, 39588}, {2935, 15106}, {3047, 8537}, {3051, 17409}, {3091, 22750}, {3135, 26880}, {3147, 13336}, {3148, 14961}, {3292, 12294}, {3515, 10984}, {3516, 43652}, {3520, 7998}, {3575, 6759}, {4232, 11003}, {4846, 18533}, {5012, 6353}, {5094, 5651}, {5622, 37643}, {5640, 37777}, {6622, 13434}, {6623, 15033}, {6995, 9544}, {7493, 19131}, {8717, 10295}, {9465, 41363}, {9703, 18535}, {9826, 12106}, {10117, 37473}, {10250, 12099}, {10274, 11576}, {10540, 18494}, {11250, 13416}, {11382, 19119}, {11424, 37197}, {11425, 38396}, {11470, 19504}, {11472, 15136}, {11550, 32125}, {12165, 17838}, {12173, 26883}, {12192, 16080}, {12324, 43617}, {12828, 37644}, {14826, 28419}, {15018, 37962}, {15056, 30100}, {15139, 36990}, {15920, 43462}, {18385, 18390}, {19126, 41584}, {19127, 41585}, {19154, 37897}, {21284, 35268}, {21766, 35477}, {21841, 32046}, {32235, 32250}, {35473, 41462}, {37453, 43650}, {37933, 40280}

X(44080) = isogonal conjugate of isotomic conjugate of X(378)
X(44080) = polar conjugate of isotomic conjugate of X(5063)
X(44080) = X(i)-Ceva conjugate of X(j) for these (i,j): {378, 5063}, {18850, 577}
X(44080) = X(i)-isoconjugate of X(j) for these (i,j): {63, 34289}, {75, 4846}, {304, 34288}, {1302, 14208}, {3267, 36149}
X(44080) = crossdifference of every pair of points on line {525, 686}
X(44080) = barycentric product X(i)*X(j) for these {i,j}: {4, 5063}, {6, 378}, {25, 15066}, {112, 8675}, {232, 11653}, {648, 42660}, {1974, 32833}, {2203, 42704}, {4846, 36429}, {5891, 8882}, {8749, 10564}
X(44080) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 34289}, {32, 4846}, {378, 76}, {1974, 34288}, {5063, 69}, {5891, 28706}, {8675, 3267}, {15066, 305}, {32833, 40050}, {42660, 525}
X(44080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 34397, 1974}, {154, 20987, 1495}, {184, 1495, 206}, {184, 1974, 34397}, {1495, 1843, 25}, {4232, 11003, 19128}, {5651, 19124, 5094}, {10539, 13352, 5654}

X(44081) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(379)

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

X(44081) lies on these lines: {4, 24019}, {6, 25}, {32, 56}, {48, 354}, {65, 1973}, {163, 5398}, {284, 940}, {572, 7580}, {579, 5347}, {604, 1427}, {662, 30962}, {851, 2278}, {942, 2172}, {1400, 9447}, {1449, 1763}, {1836, 2201}, {1837, 7119}, {1914, 2352}, {2185, 37419}, {2187, 2266}, {2260, 4275}, {2360, 4251}, {3002, 11334}, {3198, 3745}, {5546, 19245}, {5802, 37388}, {9247, 40955}, {11365, 22131}

X(44081) = isogonal conjugate of isotomic conjugate of X(379)
X(44081) = crosssum of X(2) and X(31015)
X(44081) = crossdifference of every pair of points on line {525, 1734}
X(44081) = barycentric product X(6)*X(379)
X(44081) = barycentric quotient X(379)/X(76)
X(44081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 25, 39690}, {910, 1100, 17441}

X(44082) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(382)

Barycentrics    a^2*(3*a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 4*b^2*c^2 - 2*c^4) : :
X(44082) = 2 X[6] - 5 X[1974], X[6] + 5 X[20987], 4 X[24] - X[1204], 2 X[24] + X[26883], X[1092] + 2 X[7517], X[1204] + 2 X[26883], X[1974] + 2 X[20987], X[10539] + 2 X[37440]

X(44082) lies on these lines: {2, 6030}, {3, 32062}, {4, 11202}, {6, 25}, {22, 3819}, {23, 2979}, {24, 1204}, {26, 5891}, {125, 6353}, {182, 11451}, {185, 3517}, {186, 11204}, {237, 5206}, {343, 37897}, {373, 3796}, {382, 33556}, {394, 20850}, {403, 18376}, {427, 15448}, {428, 10192}, {462, 5349}, {463, 5350}, {468, 11550}, {511, 35264}, {549, 16654}, {569, 13364}, {576, 9544}, {578, 26882}, {1092, 7517}, {1147, 18378}, {1154, 10539}, {1501, 34481}, {1598, 13367}, {1899, 4232}, {1995, 6688}, {2070, 18435}, {3060, 35265}, {3066, 5644}, {3089, 21659}, {3091, 32340}, {3098, 37913}, {3129, 5238}, {3130, 5237}, {3148, 37512}, {3167, 21969}, {3292, 8780}, {3357, 11270}, {3515, 10606}, {3518, 5890}, {3549, 32332}, {3574, 37122}, {3845, 39242}, {3917, 9909}, {5012, 14002}, {5020, 22112}, {5066, 34513}, {5157, 40670}, {5198, 17821}, {5200, 23253}, {5562, 9714}, {5892, 7506}, {5943, 6800}, {5972, 7391}, {6515, 24981}, {6524, 16240}, {6636, 10546}, {6644, 14855}, {7387, 36987}, {7487, 43831}, {7505, 13419}, {7714, 35260}, {9157, 9998}, {9707, 10110}, {9934, 17853}, {10096, 34514}, {10250, 19128}, {10282, 10594}, {10301, 23292}, {11206, 18950}, {11438, 14157}, {11442, 32223}, {11459, 37939}, {12083, 43586}, {12099, 15647}, {12292, 16219}, {12310, 41619}, {13474, 32534}, {13861, 14845}, {14070, 15030}, {15035, 15682}, {15246, 16187}, {15305, 37940}, {15750, 15811}, {16194, 18324}, {16658, 23329}, {19219, 23275}, {20897, 35007}, {21356, 43653}, {21663, 35450}, {22165, 37904}, {23039, 37956}, {23293, 37760}, {23325, 37943}, {26880, 32078}, {27355, 37476}, {33849, 37687}, {36990, 37453}, {37070, 42400}, {37478, 37936}, {37480, 37925}, {38435, 43614}, {40350, 42295}

X(44082) = isogonal conjugate of isotomic conjugate of X(382)
X(44082) = X(16835)-Ceva conjugate of X(6)
X(44082) = X(i)-isoconjugate of X(j) for these (i,j): {75, 11270}, {14208, 33640}
X(44082) = crosspoint of X(6) and X(43719)
X(44082) = crosssum of X(2) and X(3529)
X(44082) = barycentric product X(i)*X(j) for these {i,j}: {6, 382}, {31, 14212}, {11270, 36431}
X(44082) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 11270}, {382, 76}, {14212, 561}
X(44082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6030, 17508}, {24, 26883, 1204}, {25, 154, 51}, {25, 184, 34417}, {25, 1495, 184}, {25, 9777, 31860}, {25, 26864, 17810}, {51, 154, 184}, {51, 1495, 154}, {184, 34417, 15004}, {186, 11455, 11204}, {5020, 22352, 22112}, {6353, 31383, 125}, {8780, 33586, 3292}, {9909, 35259, 3917}, {10282, 10594, 11424}, {11270, 16835, 3357}, {13366, 26864, 184}, {13595, 26881, 182}, {17810, 26864, 13366}, {26882, 34484, 578}

X(44083) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(383)

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

X(44083) lies on these lines: {6, 25}, {15, 3148}, {16, 237}, {62, 20897}, {182, 3129}, {263, 3457}, {418, 11516}, {460, 5321}, {462, 1503}, {463, 5480}, {511, 3130}, {1352, 33529}, {1976, 3458}, {2871, 11081}, {3098, 3132}, {3131, 5092}, {5191, 41407}, {5334, 6620}, {6641, 11515}, {10645, 37457}, {10646, 41275}, {11486, 41266}, {34098, 37776}

X(44083) = isogonal conjugate of isotomic conjugate of X(383)
X(44083) = barycentric product X(6)*X(383)
X(44083) = barycentric quotient X(383)/X(76)

X(44084) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(403)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
X(44084) = 3 X[51] + X[1495], 3 X[403] + X[1986], X[974] - 3 X[16227], X[1514] + 3 X[16227], 3 X[3060] - 4 X[21847], X[3580] + 3 X[12824], 3 X[10151] - X[12133], X[11064] - 3 X[41670], X[11799] + 3 X[16222]

X(44084) lies on these lines: {2, 37511}, {4, 4846}, {6, 25}, {23, 19128}, {24, 5446}, {30, 9826}, {52, 3542}, {107, 421}, {110, 34382}, {113, 403}, {125, 15126}, {136, 6530}, {143, 21841}, {181, 14975}, {185, 26869}, {186, 10564}, {235, 389}, {237, 14961}, {323, 37962}, {373, 5094}, {378, 5892}, {419, 41253}, {420, 33873}, {427, 5943}, {428, 11645}, {460, 512}, {462, 6111}, {463, 6110}, {468, 511}, {685, 13137}, {974, 1514}, {1113, 24651}, {1114, 24650}, {1154, 37942}, {1216, 7505}, {1503, 11746}, {1560, 2679}, {1593, 37475}, {1595, 15026}, {1596, 5946}, {1885, 9729}, {1899, 41580}, {2211, 3124}, {2356, 20962}, {2854, 15471}, {2904, 41597}, {2935, 21663}, {2979, 38282}, {3060, 6353}, {3088, 15024}, {3089, 3567}, {3147, 10625}, {3291, 35325}, {3292, 19504}, {3515, 37497}, {3575, 10110}, {3917, 37453}, {4232, 6403}, {5093, 21313}, {5095, 9027}, {5447, 10018}, {5504, 37951}, {5643, 14865}, {5651, 11470}, {5663, 37984}, {5889, 6622}, {5890, 6623}, {6524, 14593}, {6620, 41370}, {6756, 10095}, {6785, 14165}, {7487, 9781}, {7493, 9967}, {7729, 11381}, {8889, 11451}, {9827, 23411}, {10539, 12235}, {10540, 19456}, {11060, 14581}, {11799, 16222}, {11807, 25564}, {12006, 13488}, {12058, 30771}, {12135, 23841}, {13391, 37935}, {13416, 37911}, {13417, 15131}, {14595, 18384}, {14641, 35490}, {15073, 35260}, {15151, 15311}, {15360, 37943}, {15887, 43831}, {17409, 42295}, {18947, 41724}, {20961, 40976}, {23291, 41715}, {27365, 35264}, {32715, 40388}, {34114, 37440}, {37477, 37933}, {39024, 41363}, {40135, 40352}

X(44084) = midpoint of X(i) and X(j) for these {i,j}: {468, 1112}, {974, 1514}
X(44084) = reflection of X(13416) in X(37911)
X(44084) = polar conjugate of X(40832)
X(44084) = isogonal conjugate of isotomic conjugate of X(403)
X(44084) = polar conjugate of isotomic conjugate of X(3003)
X(44084) = pole wrt polar circle of trilinear polar of X(40832) (line X(69)X(850))
X(44084) = X(i)-Ceva conjugate of X(j) for these (i,j): {403, 3003}, {22239, 647}, {32715, 512}
X(44084) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40832}, {63, 2986}, {69, 36053}, {75, 5504}, {304, 14910}, {326, 1300}, {656, 18878}, {662, 15421}, {687, 24018}, {1577, 43755}, {3265, 36114}, {4592, 15328}, {10420, 14208}, {18879, 20902}, {36062, 39988}
X(44084) = crosspoint of X(i) and X(j) for these (i,j): {4, 8749}, {6, 11744}, {25, 18384}, {250, 32695}
X(44084) = crosssum of X(i) and X(j) for these (i,j): {2, 2071}, {3, 11064}, {125, 41077}
X(44084) = crossdifference of every pair of points on line {394, 525}
X(44084) = barycentric product X(i)*X(j) for these {i,j}: {4, 3003}, {6, 403}, {19, 1725}, {25, 3580}, {107, 686}, {111, 12828}, {113, 8749}, {158, 2315}, {393, 13754}, {512, 16237}, {648, 21731}, {1609, 16172}, {1824, 18609}, {1986, 1989}, {1990, 14264}, {2501, 15329}, {6334, 32713}, {8791, 12824}, {18384, 34834}, {34104, 40388}
X(44084) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40832}, {25, 2986}, {32, 5504}, {112, 18878}, {403, 76}, {512, 15421}, {686, 3265}, {1576, 43755}, {1725, 304}, {1973, 36053}, {1974, 14910}, {1986, 7799}, {2207, 1300}, {2315, 326}, {2489, 15328}, {3003, 69}, {3580, 305}, {8749, 40423}, {11060, 12028}, {12824, 37804}, {12828, 3266}, {13754, 3926}, {14581, 15454}, {15329, 4563}, {16237, 670}, {18384, 40427}, {21731, 525}, {32713, 687}, {40354, 10419}
X(44084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 34397, 1495}, {51, 34417, 9969}, {186, 15472, 10564}, {373, 12294, 5094}, {1514, 16227, 974}, {1974, 34417, 25}, {2211, 3124, 14580}, {3575, 43823, 10110}, {4232, 11002, 6403}, {9777, 40114, 21639}

X(44085) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(404)

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

X(44085) lies on these lines: {2, 5820}, {3, 1408}, {6, 25}, {11, 1746}, {22, 37516}, {31, 692}, {37, 26890}, {42, 2317}, {44, 26885}, {49, 37509}, {54, 65}, {55, 572}, {56, 580}, {57, 36059}, {58, 7428}, {81, 5012}, {110, 6043}, {181, 215}, {182, 940}, {199, 4271}, {209, 2323}, {212, 604}, {226, 5137}, {228, 7113}, {354, 1428}, {386, 1437}, {394, 36741}, {511, 5347}, {569, 5707}, {578, 5706}, {643, 4203}, {1006, 1319}, {1011, 4268}, {1092, 36745}, {1147, 36754}, {1402, 2361}, {1404, 14547}, {1412, 13329}, {1466, 7335}, {1790, 5132}, {1864, 10535}, {1993, 4259}, {2056, 21779}, {2174, 23201}, {2206, 2220}, {2213, 14528}, {2261, 10537}, {2264, 10536}, {2328, 5053}, {2330, 3745}, {2979, 33844}, {3057, 37399}, {3271, 20988}, {3666, 3955}, {3752, 26884}, {3796, 36740}, {3917, 5096}, {4224, 18191}, {4260, 34986}, {4265, 22352}, {4383, 9306}, {4387, 24265}, {4579, 32926}, {4641, 7193}, {4697, 24253}, {5061, 5432}, {5091, 11246}, {5124, 22080}, {5322, 8679}, {5323, 34148}, {5371, 14567}, {5651, 37679}, {5718, 37527}, {5800, 11427}, {10984, 36746}, {11003, 37685}, {13323, 19765}, {13346, 37537}, {14599, 21757}, {18178, 37231}, {19554, 40972}, {20989, 23638}, {22112, 37682}, {23202, 40956}, {23292, 26020}, {34880, 37115}, {37275, 37566}, {37674, 43650}

X(44085) = isogonal conjugate of isotomic conjugate of X(404)
X(44085) = X(i)-Ceva conjugate of X(j) for these (i,j): {3453, 1333}, {7012, 1415}
X(44085) = X(7)-isoconjugate of X(44040)
X(44085) = crosspoint of X(59) and X(163)
X(44085) = crosssum of X(i) and X(j) for these (i,j): {2, 5046}, {11, 1577}
X(44085) = crossdifference of every pair of points on line {525, 3762}
X(44085) = barycentric product X(i)*X(j) for these {i,j}: {6, 404}, {31, 32939}, {1415, 20293}, {2203, 42705}, {7012, 39006}
X(44085) = barycentric quotient X(i)/X(j) for these {i,j}: {41, 44040}, {404, 76}, {32939, 561}, {39006, 17880}
X(44085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 184, 2194}, {81, 5012, 5135}, {212, 604, 2352}, {13366, 40952, 6}

X(44086) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(406)

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

X(44086) lies on these lines: {1, 41609}, {4, 32911}, {6, 25}, {24, 60}, {31, 1195}, {33, 2911}, {42, 2212}, {55, 14975}, {81, 6353}, {213, 607}, {235, 5706}, {406, 5739}, {427, 4383}, {429, 16471}, {451, 32782}, {468, 940}, {608, 2355}, {1061, 41740}, {1172, 4207}, {1185, 2211}, {1191, 11396}, {1193, 22479}, {1203, 7713}, {1386, 41611}, {1395, 2308}, {1593, 36745}, {1598, 37509}, {1783, 7102}, {1829, 16466}, {1885, 37537}, {1892, 34048}, {1993, 35973}, {2204, 2271}, {3194, 37384}, {3515, 36746}, {3517, 36750}, {3542, 5707}, {3681, 6198}, {3690, 7071}, {4232, 37685}, {5089, 16972}, {5094, 37679}, {7378, 14997}, {8889, 37680}, {15750, 37501}, {22131, 26893}, {22132, 26885}, {23122, 26892}, {37453, 37674}, {37633, 38282}

X(44086) = isogonal conjugate of isotomic conjugate of X(406)
X(44086) = polar conjugate of isotomic conjugate of X(36744)
X(44086) = X(406)-Ceva conjugate of X(36744)
X(44086) = crosssum of X(905) and X(26933)
X(44086) = crossdifference of every pair of points on line {525, 4131}
X(44086) = barycentric product X(i)*X(j) for these {i,j}: {4, 36744}, {6, 406}, {9, 1452}, {19, 12514}, {25, 5739}, {1824, 27174}, {2203, 42707}
X(44086) = barycentric quotient X(i)/X(j) for these {i,j}: {406, 76}, {1452, 85}, {5739, 305}, {12514, 304}, {36744, 69}
X(44086) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 19118, 2203}, {31, 40976, 11383}, {607, 3195, 1824}, {2299, 3192, 25}

X(44087) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(411)

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

X(44087) lies on these lines: {6, 25}, {33, 10537}, {34, 14529}, {54, 36009}, {110, 35996}, {182, 25514}, {198, 6056}, {212, 692}, {223, 36059}, {572, 8021}, {578, 7497}, {580, 859}, {581, 1437}, {1035, 7335}, {1212, 26890}, {1427, 26884}, {1503, 25985}, {1859, 2182}, {1875, 26888}, {1905, 40660}, {2175, 20991}, {2262, 11428}, {2360, 7420}, {3220, 20122}, {3556, 19349}, {4224, 5135}, {4228, 5012}, {5752, 41608}, {7113, 23204}, {9306, 19544}, {14547, 20986}, {18621, 19354}, {19649, 33883}, {23292, 37362}

X(44087) = isogonal conjugate of isotomic conjugate of X(411)
X(44087) = X(24033)-Ceva conjugate of X(1415)
X(44087) = crosssum of X(i) and X(j) for these (i,j): {2, 6895}, {1577, 2968}
X(44087) = barycentric product X(i)*X(j) for these {i,j}: {1, 1630}, {6, 411}, {19, 3561}, {55, 34035}
X(44087) = barycentric quotient X(i)/X(j) for these {i,j}: {411, 76}, {1630, 75}, {3561, 304}, {34035, 6063}
X(44087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 184, 2194}, {10536, 11429, 2182}

X(44088) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(418)

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

X(44088) lies on these lines: {6, 25}, {182, 20792}, {216, 6751}, {217, 27374}, {237, 6752}, {418, 31353}, {436, 8795}, {1147, 23163}, {1176, 17974}, {1576, 14533}, {1614, 13450}, {3518, 9792}, {6759, 8887}, {10282, 19189}, {14575, 14585}, {20775, 34980}

X(44088) = isogonal conjugate of isotomic conjugate of X(418)
X(44088) = isogonal conjugate of polar conjugate of X(217)
X(44088) = X(184)-Ceva conjugate of X(217)
X(44088) = X(i)-isoconjugate of X(j) for these (i,j): {75, 8795}, {92, 276}, {158, 34384}, {264, 40440}, {275, 1969}, {304, 8794}, {561, 8884}, {822, 42369}, {1577, 42405}, {2167, 18027}, {2190, 18022}, {4602, 15422}, {6521, 34386}, {16813, 20948}, {18833, 19174}, {24018, 42401}
X(44088) = crosspoint of X(i) and X(j) for these (i,j): {184, 14585}, {217, 418}
X(44088) = crosssum of X(i) and X(j) for these (i,j): {264, 18027}, {276, 8795}, {525, 41219}
X(44088) = barycentric product X(i)*X(j) for these {i,j}: {3, 217}, {5, 14585}, {6, 418}, {32, 5562}, {51, 577}, {53, 23606}, {110, 42293}, {184, 216}, {255, 2179}, {343, 14575}, {394, 40981}, {1092, 3199}, {1576, 17434}, {1625, 39201}, {2181, 4100}, {3049, 23181}, {6798, 8565}, {10316, 27372}, {13450, 36433}, {14586, 34983}, {15451, 32661}, {22075, 41168}, {23963, 35442}, {23964, 41219}, {27374, 28724}, {28706, 40373}, {40823, 42353}
X(44088) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 8795}, {51, 18027}, {107, 42369}, {184, 276}, {216, 18022}, {217, 264}, {418, 76}, {577, 34384}, {1501, 8884}, {1576, 42405}, {1974, 8794}, {5562, 1502}, {9247, 40440}, {9426, 15422}, {14574, 16813}, {14575, 275}, {14585, 95}, {23606, 34386}, {32713, 42401}, {34983, 15415}, {40373, 8882}, {40981, 2052}, {41219, 36793}, {41331, 19174}, {42293, 850}

X(44089) = ISOGONAL CONJUGATE OF X(40708)

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

X(44089) lies on the cubic K1033 and these lines: {2, 37893}, {4, 3398}, {6, 25}, {24, 3095}, {32, 11325}, {112, 5970}, {186, 35002}, {230, 1576}, {237, 10313}, {250, 325}, {378, 26316}, {385, 419}, {403, 6033}, {427, 7792}, {428, 32085}, {460, 6531}, {577, 20885}, {827, 14568}, {1316, 30737}, {1691, 2679}, {2086, 18902}, {2211, 8789}, {2456, 19128}, {2489, 3804}, {2971, 8744}, {3001, 21284}, {3186, 7766}, {3767, 20968}, {4027, 39927}, {5117, 7875}, {5140, 14581}, {5186, 15014}, {5254, 15257}, {5304, 6620}, {5305, 10547}, {6353, 7774}, {7467, 19121}, {7735, 14575}, {7778, 37453}, {10312, 27369}, {10317, 21177}, {12052, 37980}, {12054, 35476}, {12131, 41204}, {16315, 38861}, {16950, 37891}

X(44089) = isogonal conjugate of X(40708)
X(44089) = polar conjugate of X(18896)
X(44089) = isogonal conjugate of isotomic conjugate of X(419)
X(44089) = polar conjugate of isotomic conjugate of X(1691)
X(44089) = polar conjugate of isogonal conjugate of X(14602)
X(44089) = X(419)-Ceva conjugate of X(1691)
X(44089) = X(14602)-cross conjugate of X(1691)
X(44089) = crosspoint of X(i) and X(j) for these (i,j): {6, 43721}, {250, 32696}, {6531, 32085}
X(44089) = crosssum of X(i) and X(j) for these (i,j): {125, 6333}, {3917, 36212}
X(44089) = crossdifference of every pair of points on line {525, 3933}
X(44089) = pole wrt polar circle of trilinear polar of X(18896) (line X(850)X(2528))
X(44089) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40708}, {3, 1934}, {48, 18896}, {63, 1916}, {69, 1581}, {75, 36214}, {256, 337}, {291, 7019}, {295, 7018}, {304, 694}, {305, 1967}, {334, 7015}, {336, 40810}, {525, 37134}, {561, 17970}, {656, 18829}, {805, 14208}, {1927, 40050}, {3708, 39292}, {3933, 43763}, {7116, 18895}, {9468, 40364}
X(44089) = trilinear product X(i)*X(j) for these {i,j}: {4, 1933}, {19, 1691}, {25, 1580}, {31, 419}, {92, 14602}, {162, 5027}, {172, 2201}, {242, 7122}, {385, 1973}, {560, 17984}, {804, 32676}, {1914, 7119}, {1966, 1974}, {1969, 18902}, {2203, 4039}, {2210, 7009}, {4164, 8750}
X(44089) = barycentric product X(i)*X(j) for these {i,j}: {4, 1691}, {6, 419}, {19, 1580}, {25, 385}, {32, 17984}, {92, 1933}, {112, 804}, {171, 2201}, {172, 242}, {232, 40820}, {238, 7119}, {264, 14602}, {648, 5027}, {1474, 4039}, {1783, 4164}, {1840, 5009}, {1914, 7009}, {1966, 1973}, {1974, 3978}, {1976, 39931}, {2086, 18020}, {2207, 12215}, {2211, 14382}, {2489, 17941}, {3563, 12829}, {4027, 17980}, {4107, 8750}, {5026, 8753}, {6531, 36213}, {8623, 32085}, {8744, 36820}, {11325, 39927}, {18022, 18902}, {20964, 31905}, {22061, 34856}, {24284, 32713}, {32542, 41204}
X(44089) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18896}, {6, 40708}, {19, 1934}, {25, 1916}, {32, 36214}, {112, 18829}, {172, 337}, {250, 39292}, {385, 305}, {419, 76}, {804, 3267}, {1501, 17970}, {1580, 304}, {1691, 69}, {1914, 7019}, {1933, 63}, {1966, 40364}, {1973, 1581}, {1974, 694}, {2086, 125}, {2201, 7018}, {2211, 40810}, {3978, 40050}, {4039, 40071}, {4164, 15413}, {5027, 525}, {7009, 18895}, {7119, 334}, {8623, 3933}, {10311, 8842}, {14599, 7015}, {14601, 15391}, {14602, 3}, {14603, 40360}, {17984, 1502}, {18892, 7116}, {18902, 184}, {32676, 37134}, {32696, 39291}, {36213, 6393}, {36417, 17980}
X(44089) = {X(1974),X(10311)}-harmonic conjugate of X(25)

X(44090) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(420)

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

X(44090) lies on these lines: {2, 37895}, {6, 25}, {39, 11380}, {83, 12143}, {183, 37453}, {186, 9301}, {250, 21284}, {297, 39101}, {385, 468}, {403, 12188}, {419, 39089}, {420, 7779}, {427, 1031}, {685, 36897}, {2207, 33874}, {2211, 32748}, {2967, 19128}, {5064, 37765}, {5094, 11174}, {5201, 37920}, {6240, 13111}, {6353, 7766}, {8743, 11325}, {16318, 39095}, {19556, 41533}, {37197, 39646}, {37912, 41676}

X(44090) = isogonal conjugate of isotomic conjugate of X(420)
X(44090) = polar conjugate of isotomic conjugate of X(2076)
X(44090) = X(i)-Ceva conjugate of X(j) for these (i,j): {420, 2076}, {17980, 25}
X(44090) = X(i)-isoconjugate of X(j) for these (i,j): {63, 11606}, {1799, 17957}, {17949, 34055}
X(44090) = crosssum of X(125) and X(24284)
X(44090) = crossdifference of every pair of points on line {525, 7767}
X(44090) = barycentric product X(i)*X(j) for these {i,j}: {4, 2076}, {6, 420}, {19, 17799}, {25, 7779}, {112, 9479}, {648, 5113}, {1843, 40850}, {3563, 12830}, {8290, 17980}, {17442, 34054}, {17997, 41676}, {18010, 35325}
X(44090) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 11606}, {420, 76}, {1843, 17949}, {2076, 69}, {5113, 525}, {7779, 305}, {9479, 3267}, {17799, 304}, {17980, 9477}, {17997, 4580}

X(44091) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(428)

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

X(44091) lies on these lines: {4, 5092}, {6, 25}, {22, 19137}, {23, 11574}, {24, 3098}, {182, 10594}, {187, 27369}, {235, 20300}, {237, 22052}, {373, 5157}, {419, 32085}, {428, 3589}, {468, 34573}, {511, 3518}, {577, 20897}, {800, 34416}, {1176, 5943}, {1209, 18358}, {1511, 11566}, {1598, 12017}, {1976, 33631}, {1995, 19126}, {3089, 18381}, {3129, 11515}, {3130, 11516}, {3148, 10979}, {3517, 33878}, {3618, 7714}, {3619, 6353}, {3620, 4232}, {3630, 41584}, {3819, 41435}, {3867, 10301}, {5085, 5198}, {5140, 38010}, {5480, 13367}, {5651, 37485}, {5888, 37977}, {6756, 37513}, {6995, 15080}, {7487, 11430}, {7505, 42786}, {7545, 19129}, {7576, 19130}, {7713, 16491}, {9822, 13595}, {9967, 37440}, {10282, 14853}, {10564, 37458}, {10986, 41413}, {11387, 37505}, {11424, 23041}, {12106, 37511}, {12272, 32127}, {13417, 38851}, {13562, 32269}, {13861, 19131}, {14561, 37122}, {17508, 35502}, {18533, 43621}, {18912, 39874}, {19128, 34484}, {20190, 26863}, {20772, 32114}, {20832, 33844}, {28666, 39784}, {28708, 31670}, {32217, 41579}, {33578, 40947}, {35259, 37491}, {40325, 41412}, {40981, 42671}

X(44091) = isogonal conjugate of isotomic conjugate of X(428)
X(44091) = polar conjugate of isotomic conjugate of X(5007)
X(44091) = X(428)-Ceva conjugate of X(5007)
X(44091) = X(i)-isoconjugate of X(j) for these (i,j): {63, 10159}, {75, 41435}, {304, 3108}, {656, 35137}, {4592, 31065}, {7953, 14208}
X(44091) = crosssum of X(69) and X(3933)
X(44091) = barycentric product X(i)*X(j) for these {i,j}: {4, 5007}, {6, 428}, {19, 17469}, {25, 3589}, {28, 21802}, {112, 7927}, {393, 22352}, {607, 7198}, {608, 4030}, {648, 8664}, {1974, 39998}, {2207, 7767}, {2333, 17200}, {2489, 10330}, {11205, 32085}
X(44091) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 10159}, {32, 41435}, {112, 35137}, {428, 76}, {1974, 3108}, {2489, 31065}, {3589, 305}, {5007, 69}, {7927, 3267}, {8664, 525}, {11205, 3933}, {17469, 304}, {21802, 20336}, {22352, 3926}, {39998, 40050}
X(44091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 1974, 1843}, {25, 19118, 7716}, {1974, 8541, 19118}, {7716, 8541, 1843}, {7716, 19118, 8541}, {9969, 18374, 21637}, {10641, 10642, 232}, {13595, 19121, 9822}, {19136, 20987, 6467}

X(44092) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(429)

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

X(44092) lies on these lines: {4, 970}, {6, 25}, {19, 2238}, {24, 13323}, {33, 862}, {81, 7438}, {181, 213}, {209, 21874}, {210, 430}, {235, 5799}, {242, 41234}, {407, 1828}, {427, 5743}, {429, 960}, {468, 6703}, {511, 1812}, {1973, 20970}, {2092, 2354}, {2212, 40984}, {2273, 23638}, {2339, 4199}, {2355, 42067}, {2356, 2653}, {3966, 5130}, {4104, 39579}, {4185, 4383}, {5139, 20623}, {7713, 10974}, {8754, 42072}, {11383, 22080}, {16589, 17442}, {16980, 26377}, {18591, 28266}, {21810, 40966}, {42068, 42071}

X(44092) = polar conjugate of X(40827)
X(44092) = isogonal conjugate of isotomic conjugate of X(429)
X(44092) = polar conjugate of isotomic conjugate of X(2092)
X(44092) = pole wrt polar circle of trilinear polar of X(40827) (line X(850)X(4374))
X(44092) = X(i)-Ceva conjugate of X(j) for these (i,j): {429, 2092}, {1783, 2489}, {40097, 647}, {40976, 3725}
X(44092) = X(i)-isoconjugate of X(j) for these (i,j): {48, 40827}, {63, 14534}, {69, 2363}, {75, 1798}, {86, 1791}, {274, 2359}, {283, 31643}, {304, 1169}, {332, 961}, {662, 15420}, {1220, 1444}, {1240, 1437}, {1790, 30710}, {2298, 17206}, {4581, 4592}, {15419, 36147}
X(44092) = crosspoint of X(i) and X(j) for these (i,j): {4, 1880}, {6, 43703}, {25, 1824}, {1829, 2354}
X(44092) = crosssum of X(i) and X(j) for these (i,j): {2, 16049}, {3, 1812}, {69, 1444}
X(44092) = crossdifference of every pair of points on line {525, 7254}
X(44092) = barycentric product X(i)*X(j) for these {i,j}: {4, 2092}, {6, 429}, {10, 2354}, {19, 2292}, {25, 1211}, {28, 21810}, {34, 21033}, {37, 1829}, {42, 1848}, {92, 3725}, {225, 2269}, {226, 40976}, {278, 40966}, {393, 22076}, {607, 41003}, {608, 3704}, {648, 42661}, {960, 1880}, {1193, 1826}, {1228, 1974}, {1426, 3965}, {1474, 20653}, {1824, 3666}, {1843, 27067}, {1973, 18697}, {2300, 41013}, {2333, 4357}, {4267, 8736}, {7140, 40153}, {8750, 21124}, {20967, 40149}
X(44092) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 40827}, {25, 14534}, {32, 1798}, {213, 1791}, {429, 76}, {444, 8033}, {512, 15420}, {1193, 17206}, {1211, 305}, {1228, 40050}, {1824, 30710}, {1826, 1240}, {1829, 274}, {1848, 310}, {1880, 31643}, {1918, 2359}, {1973, 2363}, {1974, 1169}, {2092, 69}, {2269, 332}, {2292, 304}, {2300, 1444}, {2333, 1220}, {2354, 86}, {2489, 4581}, {3725, 63}, {6371, 15419}, {18697, 40364}, {20653, 40071}, {20967, 1812}, {21033, 3718}, {21810, 20336}, {22076, 3926}, {40966, 345}, {40976, 333}, {42661, 525}

X(44093) = ISOGONAL CONJUGATE OF X(40414)

Barycentrics    a^2*(b + c)*(a^2 - b^2 - c^2)*(2*a^3 + a^2*b + b^3 + a^2*c - b^2*c - b*c^2 + c^3) : :
Trilinears    a ((tan B)/(c + a) + (tan C)/(a + b)) : :

X(44093) lies on these lines: {3, 2327}, {6, 25}, {9, 37225}, {41, 2092}, {48, 18591}, {71, 228}, {101, 2983}, {125, 1213}, {185, 573}, {198, 2245}, {213, 1042}, {219, 22076}, {284, 3145}, {391, 6776}, {407, 1901}, {408, 22344}, {440, 18650}, {572, 13367}, {579, 13738}, {610, 851}, {855, 40979}, {966, 1899}, {1017, 3269}, {1204, 37499}, {1426, 30456}, {1473, 22440}, {1713, 13724}, {1765, 13734}, {1834, 1842}, {2197, 2200}, {2261, 3330}, {2269, 3270}, {2287, 4220}, {3142, 40942}, {3292, 22133}, {3937, 14597}, {5279, 10381}, {5746, 37384}, {5755, 7420}, {5776, 37194}, {5802, 28076}, {5929, 7291}, {9119, 41609}, {11064, 18648}, {13851, 32431}, {18210, 18675}, {18592, 26934}, {21663, 37508}, {21810, 21811}, {22073, 22356}, {22088, 23222}, {23526, 40733}

X(44093) = isogonal conjugate of X(40414)
X(44093) = isogonal conjugate of isotomic conjugate of X(440)
X(44093) = isotomic conjugate of polar conjugate of X(40984)
X(44093) = isogonal conjugate of polar conjugate of X(1834)
X(44093) = X(i)-Ceva conjugate of X(j) for these (i,j): {101, 647}, {1834, 40984}, {2264, 40977}
X(44093) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40414}, {2, 40431}, {27, 1257}, {81, 40445}, {286, 2983}, {951, 31623}
X(44093) = crosspoint of X(i) and X(j) for these (i,j): {6, 71}, {440, 1834}
X(44093) = crosssum of X(i) and X(j) for these (i,j): {2, 27}, {4, 2322}, {333, 18134}
X(44093) = crossdifference of every pair of points on line {447, 525}
X(44093) = trilinear product X(i)*X(j) for these {i,j}: {6, 18673}, {31, 440}, {48, 1834}, {71, 1104}, {73, 2264}, {213, 18650}, {810, 14543}, {950, 1409}, {1333, 21671}, {1842, 3990}, {2200, 17863}
X(44093) = barycentric product X(i)*X(j) for these {i,j}: {1, 18673}, {3, 1834}, {6, 440}, {42, 18650}, {58, 21671}, {63, 40977}, {69, 40984}, {71, 40940}, {72, 1104}, {73, 950}, {228, 17863}, {647, 14543}, {1214, 2264}, {1842, 3682}, {4574, 29162}
X(44093) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40414}, {31, 40431}, {42, 40445}, {228, 1257}, {440, 76}, {1104, 286}, {1834, 264}, {2200, 2983}, {2264, 31623}, {14543, 6331}, {18650, 310}, {18673, 75}, {21671, 313}, {40977, 92}, {40984, 4}
X(44093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1400, 1409, 1425}

X(44094) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(443)

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

X(44094) lies on these lines: {1, 7085}, {3, 81}, {6, 25}, {21, 19783}, {22, 37492}, {31, 1475}, {37, 26867}, {42, 604}, {55, 1100}, {56, 228}, {57, 17441}, {65, 1398}, {86, 16353}, {199, 4254}, {212, 2260}, {218, 26885}, {333, 16352}, {386, 37257}, {387, 4185}, {394, 4260}, {427, 5800}, {474, 41014}, {940, 7484}, {967, 5156}, {999, 1260}, {1002, 1617}, {1011, 5120}, {1014, 37262}, {1203, 11365}, {1249, 37386}, {1410, 1466}, {1449, 5285}, {1593, 5706}, {1824, 2285}, {1834, 4214}, {2082, 2355}, {2187, 10460}, {2192, 42447}, {2256, 3690}, {2308, 7083}, {2328, 4253}, {3745, 12329}, {3796, 5138}, {3945, 37261}, {4224, 37666}, {4231, 14912}, {4383, 11284}, {4649, 5329}, {5020, 32911}, {5042, 40984}, {5208, 37248}, {5256, 37581}, {5278, 19309}, {5347, 36740}, {5707, 7395}, {5746, 11323}, {5802, 37385}, {6642, 37509}, {7387, 36750}, {7485, 14996}, {9709, 33078}, {10037, 16473}, {10046, 16472}, {11350, 37502}, {11414, 36742}, {11435, 19354}, {16408, 33172}, {16419, 37633}, {17379, 37090}, {17778, 37099}, {19310, 37652}, {20009, 39696}, {20018, 37091}, {20835, 37507}, {22080, 37500}, {24597, 25514}, {36746, 37198}

X(44094) = isogonal conjugate of isotomic conjugate of X(443)
X(44094) = crosspoint of X(6) and X(2213)
X(44094) = crosssum of X(2) and X(452)
X(44094) = crossdifference of every pair of points on line {449, 525}
X(44094) = barycentric product X(6)*X(443)
X(44094) = barycentric quotient X(443)/X(76)
X(44094) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 154, 5320}, {6, 37538, 25}, {22, 37685, 37492}, {940, 36741, 7484}

X(44095) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(445)

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

X(44095) lies on these lines: {4, 583}, {6, 25}, {19, 1990}, {24, 584}, {186, 17454}, {216, 16453}, {230, 30677}, {393, 14018}, {403, 8818}, {468, 17718}, {607, 2178}, {1030, 2332}, {1172, 2245}, {1841, 2260}, {1865, 15762}, {2174, 2594}, {2911, 11399}, {3284, 20840}, {4275, 8743}, {5153, 39575}, {16777, 41320}

X(44095) = isogonal conjugate of isotomic conjugate of X(445)
X(44095) = polar conjugate of isotomic conjugate of X(500)
X(44095) = X(i)-Ceva conjugate of X(j) for these (i,j): {19, 1841}, {445, 500}
X(44095) = X(i)-isoconjugate of X(j) for these (i,j): {1794, 30690}, {7100, 40435}
X(44095) = barycentric product X(i)*X(j) for these {i,j}: {1, 1844}, {4, 500}, {6, 445}, {19, 16585}, {34, 31938}, {35, 1838}, {942, 6198}, {1442, 1859}, {1841, 3219}, {1865, 40214}, {7282, 14547}
X(44095) = barycentric quotient X(i)/X(j) for these {i,j}: {445, 76}, {500, 69}, {1838, 20565}, {1841, 30690}, {1844, 75}, {6198, 40422}, {14975, 943}, {16585, 304}, {31938, 3718}, {40956, 7100}
X(44095) = {X(186),X(41502)}-harmonic conjugate of X(17454)

X(44096) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(450)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - a^2*b^2 + a^2*b*c - b^3*c - a^2*c^2 + 2*b^2*c^2 - b*c^3)*(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(44096) lies on these lines: {6, 25}, {30, 250}, {112, 23200}, {237, 32713}, {450, 40888}, {1093, 14152}, {1249, 14575}, {1316, 37778}, {1576, 1990}, {2489, 42658}, {2790, 41204}, {2967, 22151}, {3284, 34854}, {4558, 15143}, {6524, 23606}, {6644, 30258}, {10788, 18533}, {15262, 20975}, {15274, 37196}, {17907, 37893}, {34396, 40138}, {37458, 42873}

X(44096) = isogonal conjugate of isotomic conjugate of X(450)
X(44096) = isogonal conjugate of polar conjugate of X(41368)
X(44096) = X(i)-isoconjugate of X(j) for these (i,j): {75, 1942}, {2713, 14208}, {7108, 40843}
X(44096) = crosspoint of X(450) and X(41368)
X(44096) = crossdifference of every pair of points on line {525, 41005}
X(44096) = barycentric product X(i)*X(j) for these {i,j}: {3, 41368}, {6, 450}, {25, 40888}, {48, 41497}, {112, 2797}, {243, 1950}, {1935, 2202}, {1936, 7120}, {1940, 1951}, {1981, 21761}, {23582, 35236}
X(44096) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1942}, {450, 76}, {2797, 3267}, {35236, 15526}, {40888, 305}, {41368, 264}, {41497, 1969}

X(44097) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(451)

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

X(44097) lies on these lines: {4, 37509}, {6, 25}, {24, 36750}, {42, 14975}, {81, 468}, {407, 3194}, {427, 32911}, {430, 1172}, {451, 2895}, {940, 37453}, {1203, 1829}, {1593, 36754}, {1783, 7140}, {1824, 21353}, {1994, 35973}, {2204, 20970}, {2211, 21753}, {2308, 40976}, {3515, 36742}, {3516, 36745}, {4383, 5094}, {5706, 37197}, {6353, 37685}, {8889, 14997}, {11396, 16466}, {11398, 16472}, {11399, 16473}, {14996, 38282}, {15750, 36746}, {21779, 35325}, {22122, 26893}, {22123, 26885}

X(44097) = isogonal conjugate of isotomic conjugate of X(451)
X(44097) = polar conjugate of isotomic conjugate of X(1030)
X(44097) = X(i)-Ceva conjugate of X(j) for these (i,j): {451, 1030}, {1824, 25}
X(44097) = X(i)-isoconjugate of X(j) for these (i,j): {63, 1029}, {69, 267}, {304, 3444}, {306, 40143}, {502, 1444}, {17206, 21353}
X(44097) = crosspoint of X(250) and X(1783)
X(44097) = crosssum of X(125) and X(905)
X(44097) = barycentric product X(i)*X(j) for these {i,j}: {4, 1030}, {6, 451}, {19, 191}, {25, 2895}, {28, 21873}, {37, 2906}, {281, 8614}, {393, 22136}, {501, 1826}, {607, 41808}, {648, 42653}, {1474, 21081}, {1783, 31947}, {1824, 40592}, {1973, 20932}, {2203, 42710}, {8750, 21192}
X(44097) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 1029}, {191, 304}, {451, 76}, {501, 17206}, {1030, 69}, {1973, 267}, {1974, 3444}, {2203, 40143}, {2333, 502}, {2895, 305}, {2906, 274}, {8614, 348}, {20932, 40364}, {21081, 40071}, {21873, 20336}, {22136, 3926}, {31947, 15413}, {42653, 525}

X(44098) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(452)

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

X(44098) lies on these lines: {1, 40970}, {6, 25}, {31, 198}, {33, 2264}, {41, 55}, {58, 13737}, {65, 5338}, {81, 35259}, {199, 37500}, {218, 5285}, {221, 1395}, {228, 3052}, {284, 13615}, {461, 5802}, {940, 4223}, {1011, 37504}, {1184, 5371}, {1724, 37320}, {1754, 37412}, {1834, 28076}, {2175, 7074}, {2256, 26885}, {2328, 4254}, {2352, 3207}, {3053, 21779}, {3145, 4255}, {3745, 40131}, {3796, 32911}, {4220, 4383}, {4252, 13738}, {4260, 9909}, {5020, 5138}, {5135, 17825}, {7412, 11425}, {9306, 37492}, {17811, 36740}, {19309, 19727}, {19732, 37149}, {19764, 37284}, {35264, 37685}, {35273, 37553}, {37367, 37646}, {37516, 37672}, {37519, 40956}

X(44098) = isogonal conjugate of isotomic conjugate of X(452)
X(44098) = X(i)-isoconjugate of X(j) for these (i,j): {75, 2213}, {85, 2336}
X(44098) = crosspoint of X(380) and X(1453)
X(44098) = crosssum of X(2) and X(37435)
X(44098) = crossdifference of every pair of points on line {525, 3676}
X(44098) = barycentric product X(i)*X(j) for these {i,j}: {1, 380}, {6, 452}, {9, 1453}
X(44098) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 2213}, {380, 75}, {452, 76}, {1453, 85}, {2175, 2336}
X(44098) = {X(25),X(5320)}-harmonic conjugate of X(6)

X(44099) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(460)

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

X(44099) lies on these lines: {4, 3972}, {6, 25}, {24, 9737}, {32, 40325}, {112, 5140}, {230, 460}, {250, 37777}, {317, 1007}, {468, 14052}, {669, 2489}, {1196, 14575}, {2032, 15630}, {2971, 14581}, {3080, 40981}, {3148, 10314}, {3517, 10983}, {7714, 37765}, {7735, 41762}, {8754, 16318}, {9407, 40350}, {11470, 40801}, {12294, 35387}, {27369, 33874}, {32696, 34854}, {40121, 43291}

X(44099) = isogonal conjugate of isotomic conjugate of X(460)
X(44099) = polar conjugate of isotomic conjugate of X(1692)
X(44099) = X(i)-Ceva conjugate of X(j) for these (i,j): {460, 1692}, {32696, 2489}
X(44099) = X(i)-isoconjugate of X(j) for these (i,j): {63, 8781}, {69, 8773}, {75, 43705}, {304, 2987}, {305, 36051}, {326, 35142}, {561, 42065}, {3265, 36105}, {10425, 14208}, {32654, 40364}
X(44099) = crosspoint of X(i) and X(j) for these (i,j): {4, 39645}, {18020, 20031}
X(44099) = crosssum of X(69) and X(6393)
X(44099) = crossdifference of every pair of points on line {525, 3926}
X(44099) = barycentric product X(i)*X(j) for these {i,j}: {4, 1692}, {6, 460}, {19, 8772}, {25, 230}, {648, 42663}, {1733, 1973}, {2207, 3564}, {2211, 14265}, {2489, 4226}, {5477, 8753}, {12829, 17980}, {14581, 36875}, {39072, 39645}
X(44099) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 8781}, {32, 43705}, {230, 305}, {460, 76}, {1501, 42065}, {1692, 69}, {1733, 40364}, {1973, 8773}, {1974, 2987}, {2207, 35142}, {8772, 304}, {14581, 36891}, {36417, 3563}, {42663, 525}
X(44099) = {X(25),X(10311)}-harmonic conjugate of X(1843)

X(44100) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(461)

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

X(44100) lies on these lines: {4, 37681}, {6, 25}, {33, 3715}, {235, 3332}, {391, 461}, {427, 37650}, {468, 4648}, {607, 1253}, {991, 3515}, {1172, 28044}, {1279, 11396}, {1398, 1471}, {1452, 1456}, {1593, 13329}, {1829, 7290}, {2293, 40976}, {3945, 6353}, {5094, 17337}, {5222, 7717}, {7713, 16469}, {11383, 21002}, {11406, 14975}, {14004, 17349}, {17245, 37453}

X(44100) = isogonal conjugate of isotomic conjugate of X(461)
X(44100) = polar conjugate of isotomic conjugate of X(4258)
X(44100) = X(461)-Ceva conjugate of X(4258)
X(44100) = X(i)-isoconjugate of X(j) for these (i,j): {77, 5936}, {222, 40023}, {348, 25430}, {905, 4624}, {2334, 7182}, {4614, 17094}, {4866, 7056}, {5545, 14208}
X(44100) = barycentric product X(i)*X(j) for these {i,j}: {4, 4258}, {6, 461}, {9, 5338}, {19, 4512}, {25, 391}, {33, 1449}, {41, 5342}, {108, 4827}, {112, 4843}, {607, 3616}, {648, 8653}, {1172, 37593}, {1334, 31903}, {1474, 4061}, {1973, 4673}, {2203, 42712}, {2212, 19804}, {2299, 5257}, {2332, 3671}, {3361, 7079}, {4765, 8750}, {4832, 36797}, {7071, 21454}, {14625, 37908}
X(44100) = barycentric quotient X(i)/X(j) for these {i,j}: {33, 40023}, {391, 305}, {461, 76}, {607, 5936}, {1449, 7182}, {2212, 25430}, {4061, 40071}, {4258, 69}, {4512, 304}, {4673, 40364}, {4827, 35518}, {4832, 17094}, {4843, 3267}, {5338, 85}, {5342, 20567}, {8653, 525}, {8750, 4624}, {37593, 1231}
X(44100) = {X(607),X(2212)}-harmonic conjugate of X(7071)

X(44101) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(464)

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

X(44101) lies on these lines: {3, 48}, {6, 25}, {19, 5706}, {56, 1409}, {185, 37499}, {281, 5767}, {572, 19357}, {573, 1181}, {579, 2360}, {608, 26888}, {610, 1754}, {965, 3740}, {966, 6776}, {1213, 1899}, {1400, 19349}, {1473, 14597}, {1826, 5786}, {2260, 16466}, {2261, 5776}, {2269, 19354}, {3167, 22133}, {3190, 38868}, {4269, 37250}, {5746, 37383}, {5816, 6146}, {10605, 37508}, {18396, 32431}, {18675, 26934}

X(44101) = isogonal conjugate of isotomic conjugate of X(464)
X(44101) = isogonal conjugate of polar conjugate of X(387)
X(44101) = crosspoint of X(387) and X(464)
X(44101) = crosssum of X(2) and X(6994)
X(44101) = crossdifference of every pair of points on line {525, 7649}
X(44101) = barycentric product X(i)*X(j) for these {i,j}: {3, 387}, {6, 464}
X(44101) = barycentric quotient X(i)/X(j) for these {i,j}: {387, 264}, {464, 76}
X(44101) = {X(6),X(154)}-harmonic conjugate of X(1474)

X(44102) = ISOGONAL CONJUGATE OF X(30786)

Barycentrics    a^2*(2*a^2 - b^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) : :
Trilinears    (tan A) (sin A - 3 cos A tan ω) : :
X(44102) = 2 X[6] + X[1495], 5 X[6] + X[12367], 3 X[6] + X[19596], 2 X[110] + X[32127], X[186] - 3 X[19128], 2 X[468] + X[5095], X[468] + 2 X[15471], X[858] - 4 X[32300], X[1205] - 4 X[35371], 5 X[1495] - 2 X[12367], 3 X[1495] - 2 X[19596], X[3292] - 4 X[6593], X[5095] - 4 X[15471], 2 X[5972] + X[32220], X[8262] + 2 X[41595], X[12167] + 5 X[34397], X[12367] - 5 X[18374], 3 X[12367] - 5 X[19596], 2 X[12367] + 5 X[21639], 2 X[15126] + X[32264], 2 X[15303] + X[32225], 3 X[18374] - X[19596], 2 X[18374] + X[21639], 2 X[19596] + 3 X[21639], X[32250] - 4 X[37984], X[32260] - 4 X[35370]

The trilinear polar of X(44102) passes through X(351).

Let A'B'C' be the medial triangle. Let OA be the circumcircle of AB'C'. Let A" be the perspector of OA. Let LA be the polar of A" wrt OA. Define LB, LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. The lines AA*, BB*, CC* concur in X(44102). (Randy Hutson, August 24, 2021)

X(44102) lies on these lines: {3, 11470}, {4, 575}, {6, 25}, {22, 11511}, {23, 11416}, {24, 576}, {26, 8538}, {39, 9515}, {49, 32284}, {69, 38282}, {110, 8681}, {112, 843}, {182, 378}, {185, 34117}, {186, 249}, {187, 23200}, {193, 19122}, {216, 37457}, {235, 8550}, {237, 3284}, {340, 420}, {403, 542}, {419, 648}, {427, 597}, {460, 1990}, {462, 23713}, {463, 23712}, {468, 524}, {512, 1692}, {538, 37912}, {577, 41275}, {599, 37453}, {800, 14575}, {858, 32300}, {895, 37777}, {1112, 9019}, {1177, 1205}, {1249, 41762}, {1353, 19155}, {1503, 10151}, {1576, 3003}, {1594, 25555}, {1597, 5050}, {1691, 9217}, {1899, 41719}, {1968, 10568}, {1976, 8749}, {1986, 25556}, {1992, 6353}, {1995, 9813}, {2070, 18449}, {2080, 23164}, {2207, 39238}, {2356, 20958}, {2450, 23583}, {2781, 21663}, {3098, 35472}, {3148, 5158}, {3199, 34154}, {3455, 39840}, {3515, 11477}, {3516, 10541}, {3517, 11482}, {3518, 8537}, {3520, 20190}, {3542, 43844}, {3564, 37942}, {3618, 8889}, {3629, 41584}, {4232, 5032}, {4235, 14608}, {4663, 11363}, {5007, 27369}, {5038, 11380}, {5085, 11410}, {5092, 35473}, {5097, 6403}, {5140, 8744}, {5182, 15014}, {5562, 35603}, {5621, 34146}, {5622, 6000}, {5967, 37778}, {5972, 32220}, {6620, 40138}, {6623, 6776}, {6636, 11574}, {7485, 19126}, {7487, 37505}, {7502, 9967}, {7505, 34507}, {7514, 19131}, {8548, 10539}, {8549, 26883}, {8584, 41585}, {8743, 40325}, {9306, 41614}, {9407, 20975}, {9544, 15531}, {9729, 43815}, {10018, 40107}, {10168, 37118}, {10282, 15073}, {10295, 19924}, {10317, 21419}, {10423, 41511}, {10510, 21284}, {10540, 39562}, {10594, 22234}, {11206, 18919}, {11255, 37440}, {11550, 23327}, {12007, 39871}, {12220, 37913}, {13198, 41744}, {14273, 33919}, {14580, 32740}, {14848, 18494}, {14912, 19123}, {15118, 32239}, {15126, 15128}, {15139, 32251}, {15387, 32729}, {16868, 18553}, {18386, 38072}, {18533, 20423}, {18800, 37855}, {18860, 22087}, {19127, 22352}, {20959, 40976}, {21213, 21969}, {23061, 37977}, {25406, 40196}, {32113, 32226}, {32250, 37984}, {32260, 35370}, {32713, 34854}, {34774, 41602}, {36696, 41363}, {39561, 39588}

X(44102) = midpoint of X(i) and X(j) for these {i,j}: {6, 18374}, {23, 11416}, {110, 37784}, {1495, 21639}, {2070, 18449}, {10540, 39562}
X(44102) = reflection of X(i) in X(j) for these {i,j}: {1495, 18374}, {21639, 6}, {32127, 37784}
X(44102) = isogonal conjugate of X(30786)
X(44102) = polar conjugate of X(18023)
X(44102) = isogonal conjugate of complement of X(7665)
X(44102) = isogonal conjugate of isotomic conjugate of X(468)
X(44102) = polar conjugate of isotomic conjugate of X(187)
X(44102) = polar conjugate of isogonal conjugate of X(14567)
X(44102) = X(i)-Ceva conjugate of X(j) for these (i,j): {468, 187}, {8744, 14580}, {8753, 25}, {10423, 647}, {32709, 8644}
X(44102) = X(i)-cross conjugate of X(j) for these (i,j): {14567, 187}, {21905, 110}, {21906, 14273}, {41911, 4}
X(44102) = crosspoint of X(i) and X(j) for these (i,j): {4, 8791}, {6, 1177}, {25, 8753}
X(44102) = crosssum of X(i) and X(j) for these (i,j): {2, 858}, {3, 22151}, {69, 6390}, {125, 14417}
X(44102) = crossdifference of every pair of points on line {69, 525}
X(44102) = pole wrt polar circle of trilinear polar of X(18023) (line X(76)X(850))
X(44102) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30786}, {48, 18023}, {63, 671}, {69, 897}, {75, 895}, {76, 36060}, {111, 304}, {305, 923}, {326, 17983}, {336, 5968}, {525, 36085}, {561, 14908}, {656, 892}, {662, 14977}, {691, 14208}, {799, 10097}, {3267, 36142}, {3718, 7316}, {3926, 36128}, {4025, 5380}, {4563, 23894}, {4592, 5466}, {5547, 7182}, {14210, 15398}, {20884, 41511}, {31125, 34055}, {32740, 40364}
X(44102) = barycentric product X(i)*X(j) for these {i,j}: {4, 187}, {6, 468}, {19, 896}, {25, 524}, {28, 21839}, {92, 922}, {110, 14273}, {111, 5095}, {112, 690}, {162, 2642}, {184, 37778}, {232, 5967}, {250, 1648}, {264, 14567}, {351, 648}, {393, 3292}, {419, 18872}, {512, 4235}, {607, 7181}, {608, 3712}, {1177, 1560}, {1474, 4062}, {1783, 14419}, {1824, 16702}, {1973, 14210}, {1974, 3266}, {1990, 9717}, {2052, 23200}, {2203, 42713}, {2207, 6390}, {2333, 6629}, {2482, 8753}, {2489, 5468}, {2501, 5467}, {3053, 5203}, {3563, 5477}, {4750, 8750}, {5026, 17980}, {5140, 34161}, {5642, 8749}, {6531, 9155}, {6593, 8791}, {8744, 14357}, {8882, 41586}, {12828, 14910}, {14248, 32459}, {14417, 32713}, {14432, 32674}, {14581, 36890}, {14776, 42760}, {15471, 21448}, {17983, 39689}, {18020, 21906}, {22105, 35325}, {32740, 34336}, {34397, 43084}, {35282, 43717}, {36128, 42081}, {40347, 41616}
X(44102) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 18023}, {6, 30786}, {25, 671}, {32, 895}, {112, 892}, {187, 69}, {351, 525}, {468, 76}, {512, 14977}, {524, 305}, {560, 36060}, {669, 10097}, {690, 3267}, {896, 304}, {922, 63}, {1501, 14908}, {1560, 1236}, {1648, 339}, {1843, 31125}, {1973, 897}, {1974, 111}, {2207, 17983}, {2211, 5968}, {2489, 5466}, {2642, 14208}, {3266, 40050}, {3292, 3926}, {4062, 40071}, {4235, 670}, {5095, 3266}, {5467, 4563}, {6593, 37804}, {8541, 42008}, {9155, 6393}, {14210, 40364}, {14273, 850}, {14419, 15413}, {14567, 3}, {14581, 9214}, {15471, 11059}, {18872, 40708}, {21839, 20336}, {21906, 125}, {23200, 394}, {32676, 36085}, {32740, 15398}, {36417, 8753}, {37778, 18022}, {39689, 6390}, {40354, 9139}, {41586, 28706}, {41616, 37803}, {41911, 625}, {42671, 36894}
X(44102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 25, 8541}, {6, 154, 10602}, {6, 184, 40673}, {6, 206, 6467}, {6, 1974, 1843}, {6, 9973, 39125}, {6, 19118, 1974}, {6, 19132, 19459}, {6, 19136, 51}, {6, 19153, 184}, {6, 41593, 21637}, {25, 8541, 1843}, {460, 1990, 8754}, {468, 12828, 32225}, {468, 15471, 5095}, {1177, 34470, 1205}, {1974, 8541, 25}, {6593, 41612, 5642}, {6593, 41616, 5095}, {8739, 8740, 232}, {9407, 20975, 42671}, {12828, 15303, 5095}, {40135, 42671, 20975}

X(44103) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(469)

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

X(44103) lies on these lines: {2, 17171}, {4, 9}, {6, 25}, {24, 572}, {27, 17277}, {28, 579}, {32, 3437}, {33, 2269}, {34, 1400}, {37, 1829}, {48, 581}, {216, 28348}, {219, 5752}, {284, 14017}, {378, 37508}, {391, 6995}, {427, 1213}, {428, 17330}, {444, 992}, {468, 17398}, {469, 5224}, {577, 37259}, {583, 17523}, {607, 4277}, {608, 28615}, {672, 5338}, {941, 1039}, {1100, 11363}, {1108, 40964}, {1172, 4222}, {1409, 19366}, {1452, 2285}, {1453, 2260}, {1593, 37499}, {1696, 2336}, {1730, 37388}, {1765, 37395}, {1779, 2253}, {1828, 1841}, {1865, 1894}, {1875, 1880}, {1901, 37376}, {1973, 4270}, {2178, 22479}, {2189, 3453}, {2193, 11334}, {2204, 2220}, {2245, 4185}, {2278, 20832}, {2287, 7466}, {2332, 4254}, {4186, 4271}, {4214, 5036}, {4253, 17562}, {4288, 20846}, {5090, 17275}, {5227, 5739}, {5738, 7289}, {5742, 37362}, {5755, 7497}, {5839, 7718}, {6748, 37226}, {6985, 15945}, {7017, 29395}, {7490, 37650}, {7649, 20979}, {7714, 37654}, {9306, 22133}, {11396, 16777}, {12135, 17362}, {13738, 18591}, {14597, 26892}, {21767, 42448}, {26671, 37279}, {34265, 37390}, {37245, 37500}, {43739, 43742}

X(44103) = isogonal conjugate of isotomic conjugate of X(469)
X(44103) = polar conjugate of isotomic conjugate of X(386)
X(44103) = X(469)-Ceva conjugate of X(386)
X(44103) = X(i)-isoconjugate of X(j) for these (i,j): {63, 43531}, {69, 2214}, {835, 905}, {1332, 43927}, {1459, 37218}
X(44103) = crosssum of X(2) and X(7560)
X(44103) = crossdifference of every pair of points on line {525, 1459}
X(44103) = barycentric product X(i)*X(j) for these {i,j}: {4, 386}, {6, 469}, {19, 28606}, {25, 5224}, {34, 3876}, {112, 23879}, {607, 33949}, {648, 42664}, {834, 1897}, {1783, 14349}, {1973, 33935}, {2203, 42714}
X(44103) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 43531}, {386, 69}, {469, 76}, {834, 4025}, {1783, 37218}, {1973, 2214}, {3876, 3718}, {5224, 305}, {8637, 1459}, {8750, 835}, {14349, 15413}, {23879, 3267}, {28606, 304}, {33935, 40364}, {42664, 525}
X(44103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 25, 1474}, {9, 7713, 19}, {2333, 2354, 19}

X(44104) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(474)

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

X(44104) lies on these lines: {1, 26890}, {6, 25}, {31, 5042}, {41, 20230}, {42, 1397}, {54, 7501}, {55, 4268}, {81, 182}, {199, 4266}, {212, 40956}, {228, 604}, {569, 39271}, {578, 37305}, {940, 43650}, {1011, 5053}, {1092, 36754}, {1147, 37509}, {1408, 4255}, {1412, 4191}, {1743, 26885}, {1790, 37502}, {1993, 4260}, {2175, 2308}, {2206, 16946}, {2256, 26867}, {2260, 6056}, {2999, 26884}, {3796, 37492}, {3917, 36741}, {3920, 43146}, {3955, 5256}, {4383, 5651}, {5012, 5138}, {5347, 37516}, {5706, 11424}, {9306, 32911}, {10984, 36742}, {11429, 40971}, {13323, 19767}, {16187, 37687}, {19354, 42447}, {22080, 36743}, {22112, 37674}, {22352, 36740}, {24265, 32915}, {36745, 43652}

X(44104) = isogonal conjugate of isotomic conjugate of X(474)
X(44104) = crosssum of X(2) and X(2478)
X(44104) = crossdifference of every pair of points on line {525, 4462}
X(44104) = barycentric product X(6)*X(474)
X(44104) = barycentric quotient X(474)/X(76)
X(44104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 184, 5320}, {6, 37538, 51}, {5012, 37685, 5138}

X(44105) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(475)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 - c^3) : :
Trilinears    sin A (sin A + sin B + sin C - tan A) : :
Trilinears    (tan A) (s cos A - R sin A) : :
Trilinears    a tan A - 2 s sin A : :

X(44105) lies on these lines: {4, 81}, {6, 25}, {24, 36754}, {31, 2356}, {42, 1395}, {56, 22348}, {222, 1876}, {394, 10477}, {427, 940}, {468, 4383}, {607, 2355}, {608, 1824}, {1396, 4196}, {1398, 34046}, {1468, 22479}, {1593, 36746}, {1598, 36750}, {1870, 3873}, {1892, 37543}, {2212, 2308}, {3157, 14054}, {3194, 28076}, {3195, 20231}, {3515, 36745}, {3516, 37501}, {3517, 37509}, {3575, 5706}, {4231, 39588}, {4663, 41611}, {5090, 5711}, {5094, 37674}, {5347, 21213}, {5422, 35973}, {5710, 12135}, {6353, 32911}, {6995, 37685}, {7378, 14996}, {8889, 37633}, {11363, 16466}, {22131, 26885}, {22132, 26893}, {23122, 26884}, {37453, 37679}, {37680, 38282}

X(44105) = isogonal conjugate of isotomic conjugate of X(475)
X(44105) = polar conjugate of isotomic conjugate of X(36743)
X(44105) = X(475)-Ceva conjugate of X(36743)
X(44105) = crosssum of X(2) and X(27505)
X(44105) = crossdifference of every pair of points on line {525, 20296}
X(44105) = barycentric product X(i)*X(j) for these {i,j}: {4, 36743}, {6, 475}, {2203, 42715}
X(44105) = barycentric quotient X(i)/X(j) for these {i,j}: {475, 76}, {36743, 69}
X(44105) = {X(42),X(1395)}-harmonic conjugate of X(11383)

X(44106) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(546)

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

X(44106) lies on these lines: {2, 14488}, {3, 27355}, {4, 11270}, {6, 25}, {22, 373}, {23, 5643}, {107, 42400}, {110, 21849}, {125, 428}, {143, 43844}, {185, 10594}, {237, 37512}, {389, 14157}, {462, 5350}, {463, 5349}, {511, 13595}, {575, 26881}, {576, 35264}, {1204, 5198}, {1216, 18369}, {1598, 10605}, {1899, 7714}, {1995, 3917}, {3051, 40350}, {3060, 3292}, {3066, 9909}, {3129, 5237}, {3130, 5238}, {3148, 5206}, {3517, 11424}, {3518, 10110}, {3574, 21841}, {3819, 15107}, {3845, 32110}, {4224, 37687}, {5012, 32237}, {5020, 5650}, {5076, 43604}, {5092, 11451}, {5097, 9544}, {5133, 32223}, {5200, 23263}, {5446, 13621}, {5462, 18378}, {5562, 13861}, {5651, 33586}, {5892, 5899}, {6636, 6688}, {6756, 11572}, {6995, 11550}, {7398, 43653}, {7496, 10219}, {7502, 14845}, {7527, 13570}, {7545, 13754}, {7576, 13851}, {9306, 21969}, {9781, 10282}, {10117, 34468}, {10170, 21308}, {10193, 35484}, {10301, 13567}, {10601, 20850}, {11002, 34986}, {11160, 14826}, {11438, 32062}, {11695, 12088}, {11745, 43831}, {12002, 37495}, {12041, 12101}, {12087, 17704}, {13363, 37947}, {13364, 37513}, {13399, 16654}, {13474, 16835}, {15873, 21659}, {16240, 24862}, {16836, 37925}, {19219, 23269}, {21243, 32225}, {26276, 33798}, {26882, 37505}, {34481, 40130}, {37649, 37897}, {37945, 43584}

X(44106) = isogonal conjugate of isotomic conjugate of X(546)
X(44106) = crosspoint of X(6) and X(16835)
X(44106) = crosssum of X(2) and X(550)
X(44106) = barycentric product X(6)*X(546)
X(44106) = barycentric quotient X(546)/X(76)
X(44106) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {23, 5943, 22352}, {25, 51, 1495}, {25, 11402, 41424}, {25, 17810, 184}, {25, 34417, 51}, {51, 154, 34566}, {51, 184, 34565}, {51, 1495, 13366}, {184, 17810, 51}, {184, 34417, 17810}, {184, 34565, 13366}, {1495, 34565, 184}, {1843, 19136, 21639}, {3066, 9909, 43650}, {3518, 10110, 13367}, {6030, 20190, 22352}, {6636, 10545, 6688}, {10601, 20850, 35268}, {11451, 37913, 5092}, {13364, 37936, 37513}, {34484, 38848, 389}

X(44107) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(547)

Barycentrics    a^2*(2*a^4 - 7*a^2*b^2 + 5*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 5*c^4) : :
X(44107) = 3 X[7496] - 7 X[15018], X[7496] - 7 X[15019], 9 X[7496] - 7 X[41462], X[15018] - 3 X[15019], 3 X[15018] - X[41462], 9 X[15019] - X[41462]

X(44107) lies on these lines: {4, 14491}, {6, 25}, {23, 15516}, {52, 33533}, {74, 389}, {110, 22330}, {143, 37513}, {323, 5943}, {373, 576}, {511, 7496}, {575, 11002}, {578, 35479}, {1351, 5650}, {1570, 13410}, {1994, 10545}, {1995, 15520}, {3060, 5092}, {3066, 11482}, {3098, 5422}, {3292, 5097}, {3527, 3531}, {3567, 11430}, {3581, 15038}, {3630, 37439}, {3819, 12834}, {3917, 37517}, {4550, 14831}, {5007, 5191}, {5093, 5651}, {5446, 15037}, {5476, 37644}, {5643, 15082}, {5892, 37496}, {5946, 10564}, {6800, 22234}, {10095, 43844}, {10110, 15032}, {10546, 34986}, {11381, 11432}, {11438, 35477}, {11464, 37505}, {11477, 22112}, {12112, 14483}, {13202, 16657}, {13321, 14805}, {13367, 22233}, {13474, 43596}, {13857, 37648}, {14389, 32225}, {14810, 16981}, {14848, 37638}, {15107, 21849}, {16226, 37470}, {16655, 34564}, {18583, 41586}, {33878, 43650}

X(44107) = isogonal conjugate of isotomic conjugate of X(547)
X(44107) = crosspoint of X(i) and X(j) for these (i,j): {4, 30537}, {6, 14483}
X(44107) = crosssum of X(i) and X(j) for these (i,j): {2, 549}, {3, 15018}
X(44107) = barycentric product X(6)*X(547)
X(44107) = barycentric quotient X(547)/X(76)
X(44107) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 51, 1495}, {6, 1495, 13366}, {6, 31860, 11402}, {51, 15004, 34565}, {51, 34565, 13366}, {184, 34417, 41424}, {184, 34566, 13366}, {1495, 34565, 6}, {5097, 5640, 3292}, {9777, 15004, 51}, {21849, 34545, 22352}

X(44108) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(548)

Barycentrics    a^2*(6*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(44108) = 4 X[6] - 7 X[21637], 5 X[1614] + X[3520], 2 X[1614] + X[13367], 2 X[3520] - 5 X[13367]

X(44108) lies on these lines: {6, 25}, {54, 26863}, {110, 3819}, {156, 5891}, {185, 9707}, {186, 43596}, {373, 35264}, {1147, 36987}, {1154, 43844}, {1511, 17853}, {1614, 3520}, {1994, 32237}, {2979, 3292}, {3167, 35268}, {3796, 5650}, {3917, 6800}, {5012, 6688}, {5890, 10282}, {5943, 35265}, {6455, 10132}, {6456, 10133}, {6676, 24981}, {6759, 32062}, {7409, 31383}, {7426, 11225}, {8780, 43650}, {9306, 40916}, {9705, 15644}, {9706, 13598}, {10619, 16252}, {11003, 11451}, {11204, 11456}, {11381, 19357}, {11424, 14530}, {11430, 11455}, {11464, 21663}, {14845, 32046}, {15602, 37457}, {15647, 32226}, {18435, 18475}, {18555, 36966}, {18950, 35260}, {26881, 34986}, {31652, 42671}, {32078, 41212}

X(44108) = isogonal conjugate of isotomic conjugate of X(548)
X(44108) = crosssum of X(2) and X(3627)
X(44108) = barycentric product X(6)*X(548)
X(44108) = barycentric quotient X(548)/X(76)
X(44108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {51, 154, 1495}, {51, 11402, 34566}, {154, 184, 51}, {184, 1495, 13366}, {184, 34417, 17809}, {1495, 34565, 25}, {11402, 34566, 13366}

X(44109) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(549)

Barycentrics    a^2*(4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4) : :
Trilinears    a^2 (4 cos A + cos(B - C)) : :
Trilinears    a^2 (2 sec A + 5 sec B sec C) : :
X(44109) = X[7492] - 5 X[11003], 3 X[7492] + 5 X[11004], X[7492] + 5 X[11422], 3 X[7492] - 5 X[15080], 3 X[11003] + X[11004], 3 X[11003] - X[15080], X[11004] - 3 X[11422], 3 X[11422] + X[15080]

X(44109) lies on these lines: {2, 43150}, {5, 24981}, {6, 25}, {22, 37517}, {23, 5097}, {39, 5191}, {49, 15037}, {52, 32136}, {54, 74}, {110, 373}, {125, 8550}, {182, 3292}, {187, 34396}, {217, 9408}, {237, 5008}, {323, 3917}, {389, 11423}, {394, 12017}, {399, 567}, {468, 12007}, {511, 7492}, {542, 14389}, {569, 15068}, {576, 6800}, {578, 11381}, {1181, 13093}, {1199, 10282}, {1351, 35268}, {1353, 13394}, {1493, 10625}, {1511, 9730}, {1614, 26863}, {1692, 40130}, {1970, 9412}, {1976, 39389}, {1993, 3098}, {1994, 15107}, {1995, 39561}, {2030, 9463}, {2317, 23202}, {2930, 12039}, {3060, 7712}, {3131, 34754}, {3132, 34755}, {3167, 43650}, {3431, 5890}, {3531, 10982}, {3574, 31804}, {3581, 14831}, {3631, 7499}, {3796, 33878}, {4550, 18445}, {5050, 5651}, {5052, 14567}, {5462, 9704}, {5477, 30516}, {5562, 32046}, {5640, 15516}, {5642, 37648}, {5892, 9703}, {5943, 9544}, {5965, 7495}, {5972, 33749}, {6090, 22112}, {6221, 10133}, {6398, 10132}, {7394, 42785}, {7494, 11008}, {7592, 11438}, {7687, 12022}, {7998, 9716}, {9545, 9729}, {9706, 43584}, {9781, 13472}, {10111, 12900}, {10539, 27355}, {10545, 34545}, {10619, 12233}, {10984, 37483}, {11002, 11663}, {11126, 36755}, {11127, 36756}, {11179, 13857}, {11403, 14490}, {11424, 19347}, {11426, 26883}, {11427, 11550}, {11449, 15012}, {11935, 40280}, {12038, 43845}, {12112, 15033}, {12161, 14531}, {12242, 34224}, {13196, 30749}, {13353, 41597}, {13434, 15052}, {13630, 41673}, {13754, 14805}, {14683, 18553}, {14810, 23061}, {14912, 37643}, {15019, 35265}, {17847, 34468}, {18358, 37649}, {20080, 43653}, {20583, 37904}, {21849, 26881}, {22052, 23606}, {26316, 36212}, {32225, 37644}, {35497, 43612}, {36987, 37496}

X(44109) = midpoint of X(i) and X(j) for these {i,j}: {11003, 11422}, {11004, 15080}
X(44109) = isogonal conjugate of isotomic conjugate of X(549)
X(44109) = isogonal conjugate of polar conjugate of X(6749)
X(44109) = X(75)-isoconjugate of X(14483)
X(44109) = crosspoint of X(i) and X(j) for these (i,j): {6, 3431}, {549, 6749}
X(44109) = crosssum of X(i) and X(j) for these (i,j): {2, 381}, {3, 11004}
X(44109) = crossdifference of every pair of points on line {525, 14391}
X(44109) = barycentric product X(i)*X(j) for these {i,j}: {3, 6749}, {6, 549}
X(44109) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 14483}, {549, 76}, {6749, 264}
X(44109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 184, 1495}, {6, 1495, 51}, {6, 26864, 34417}, {6, 31860, 9777}, {25, 34565, 51}, {49, 15037, 43586}, {54, 15032, 11430}, {110, 575, 373}, {154, 34566, 51}, {182, 3292, 5650}, {184, 11402, 13366}, {184, 13366, 51}, {184, 15004, 154}, {184, 34417, 26864}, {323, 5012, 5092}, {323, 5092, 3917}, {1353, 13394, 41586}, {1495, 13366, 6}, {3520, 15032, 43596}, {5012, 34986, 3917}, {5092, 34986, 323}, {11003, 11004, 15080}, {11402, 17809, 184}, {11422, 15080, 11004}, {11430, 15032, 185}, {15087, 18475, 14831}, {22330, 32237, 11002}, {26864, 34417, 1495}

X(44110) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(550)

Barycentrics    a^2*(4*a^4 - 3*a^2*b^2 - b^4 - 3*a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(44110) = 3 X[9544] + X[37913], 3 X[11464] - X[35473], 3 X[26881] - X[37913]

X(44110) lies on these lines: {2, 18553}, {3, 33556}, {4, 36809}, {5, 11565}, {6, 25}, {22, 3292}, {23, 21969}, {26, 14531}, {49, 5899}, {52, 37936}, {110, 3917}, {125, 10192}, {156, 5562}, {182, 35264}, {185, 186}, {235, 10619}, {237, 35007}, {343, 24981}, {373, 5012}, {378, 6759}, {389, 26882}, {394, 35268}, {511, 9544}, {549, 13399}, {569, 27355}, {575, 12834}, {1092, 35243}, {1147, 12083}, {1196, 14567}, {1204, 17821}, {1368, 5642}, {1370, 13857}, {1498, 11410}, {1501, 40130}, {1511, 14855}, {1597, 14530}, {1899, 35260}, {2070, 14831}, {2937, 41597}, {2979, 7712}, {3060, 32237}, {3131, 5238}, {3132, 5237}, {3155, 6453}, {3156, 6454}, {3431, 11455}, {3520, 16835}, {3787, 8627}, {3796, 5651}, {3819, 15080}, {5181, 22165}, {5206, 41275}, {5446, 9704}, {5650, 6800}, {5943, 11003}, {5944, 12162}, {6000, 11464}, {6143, 14864}, {6146, 37942}, {6241, 11270}, {6449, 10132}, {6450, 10133}, {6623, 19467}, {6677, 35266}, {6688, 10546}, {7378, 31383}, {7494, 21356}, {7499, 20582}, {7514, 10539}, {8024, 35356}, {8703, 10990}, {8877, 32729}, {8889, 11206}, {8908, 26886}, {9545, 13598}, {9705, 12088}, {9833, 11572}, {10151, 16252}, {10154, 41586}, {10540, 15030}, {10565, 11160}, {10575, 32171}, {11202, 11456}, {11245, 15448}, {11403, 14528}, {11422, 21849}, {11424, 18535}, {11430, 13596}, {11645, 31074}, {12038, 18859}, {12106, 16226}, {13417, 15647}, {13491, 37968}, {13620, 43605}, {14581, 14585}, {14862, 18560}, {15712, 23060}, {16003, 34477}, {20773, 21649}, {22115, 36987}, {31255, 43273}, {32379, 43581}, {32737, 40352}, {34148, 37945}, {34484, 37505}, {34782, 43831}, {35259, 43650}, {37457, 37512}, {37904, 41149}

X(44110) = midpoint of X(9544) and X(26881)
X(44110) = isogonal conjugate of isotomic conjugate of X(550)
X(44110) = X(33640)-Ceva conjugate of X(647)
X(44110) = X(75)-isoconjugate of X(16835)
X(44110) = crosspoint of X(6) and X(11270)
X(44110) = crosssum of X(2) and X(382)
X(44110) = crossdifference of every pair of points on line {525, 31072}
X(44110) = barycentric product X(6)*X(550)
X(44110) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 16835}, {550, 76}
X(44110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {23, 34986, 21969}, {25, 184, 13366}, {25, 13366, 51}, {25, 17809, 15004}, {26, 43844, 14531}, {154, 184, 1495}, {154, 26864, 184}, {184, 1495, 51}, {184, 15004, 17809}, {184, 34417, 11402}, {184, 41424, 34566}, {1495, 13366, 25}, {1614, 10282, 185}, {3796, 8780, 5651}, {6759, 9707, 13367}, {6759, 13367, 11381}, {6800, 9306, 22352}, {9306, 22352, 5650}, {10540, 18475, 15030}, {11202, 11456, 21663}, {11402, 34417, 34565}, {11430, 14157, 32062}, {14530, 19357, 26883}, {15004, 17809, 13366}, {34417, 34565, 51}

X(44111) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(632)

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

X(44111) lies on these lines: {2, 11668}, {6, 25}, {22, 15520}, {54, 34567}, {185, 1199}, {373, 34545}, {389, 21844}, {567, 14831}, {569, 14531}, {575, 1994}, {576, 22352}, {578, 21663}, {1993, 5650}, {3060, 22330}, {3284, 26907}, {3292, 5422}, {3567, 13472}, {3796, 11482}, {3819, 11004}, {5012, 5097}, {5041, 34396}, {5462, 9703}, {5943, 11422}, {6427, 10133}, {6428, 10132}, {7499, 32455}, {7592, 11381}, {10110, 11423}, {10605, 11426}, {10625, 36153}, {11003, 21849}, {11225, 14389}, {11550, 14912}, {13367, 35479}, {15030, 15087}, {15032, 32062}, {22112, 37672}, {27355, 43844}

X(44111) = isogonal conjugate of isotomic conjugate of X(632)
X(44111) = crosspoint of X(6) and X(13472)
X(44111) = crosssum of X(2) and X(1656)
X(44111) = barycentric product X(i)*X(j) for these {i,j}: {6, 632}, {233, 39667}
X(44111) = barycentric quotient X(i)/X(j) for these {i,j}: {632, 76}, {39667, 31617}
X(44111) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 184, 34565}, {6, 11402, 15004}, {6, 13366, 51}, {184, 15004, 17810}, {184, 17810, 1495}, {184, 34565, 51}, {575, 1994, 3917}, {1199, 37505, 185}, {1495, 13366, 11402}, {1495, 15004, 51}, {5012, 5097, 21969}, {11402, 15004, 1495}, {11402, 17810, 184}, {13366, 34565, 184}, {13366, 34566, 6}, {34545, 34986, 373}

X(44112) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(851)

Barycentrics    a^3*(b + c)*(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(44112) lies on these lines: {6, 25}, {86, 7413}, {667, 788}, {692, 2245}, {851, 1936}, {862, 3330}, {1253, 22369}, {1397, 40984}, {1402, 1918}, {1834, 14529}, {3948, 4579}, {3955, 4199}, {4204, 26890}, {18610, 23440}, {18611, 23526}, {18619, 23420}

X(44112) = isogonal conjugate of isotomic conjugate of X(851)
X(44112) = X(26884)-Ceva conjugate of X(42669)
X(44112) = X(i)-isoconjugate of X(j) for these (i,j): {2, 35145}, {75, 37142}, {76, 2249}, {314, 1937}, {333, 1952}, {1945, 28660}, {4391, 41206}, {6332, 41207}, {31623, 40843}
X(44112) = crosssum of X(2) and X(14956)
X(44112) = crossdifference of every pair of points on line {75, 525}
X(44112) = barycentric product X(i)*X(j) for these {i,j}: {1, 42669}, {6, 851}, {31, 8680}, {37, 26884}, {65, 1951}, {71, 1430}, {73, 2202}, {112, 9391}, {213, 5088}, {243, 1409}, {647, 23353}, {669, 15418}, {810, 1981}, {1400, 1936}, {1402, 1944}
X(44112) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 35145}, {32, 37142}, {560, 2249}, {851, 76}, {1402, 1952}, {1936, 28660}, {1944, 40072}, {1951, 314}, {5088, 6385}, {8680, 561}, {9391, 3267}, {15418, 4609}, {23353, 6331}, {26884, 274}, {42669, 75}

X(44113) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(860)

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

X(44113) lies on these lines: {1, 429}, {4, 5396}, {6, 25}, {19, 4272}, {24, 5398}, {33, 430}, {34, 407}, {42, 1824}, {58, 20832}, {73, 1426}, {81, 4231}, {199, 2193}, {225, 2594}, {228, 40590}, {235, 5721}, {386, 4185}, {427, 5718}, {431, 1834}, {442, 37697}, {468, 35466}, {500, 7414}, {580, 20837}, {581, 37194}, {607, 20970}, {608, 2092}, {661, 663}, {851, 1465}, {860, 1870}, {1068, 5399}, {1193, 40985}, {1395, 20966}, {1427, 22348}, {1464, 1835}, {1718, 37982}, {1897, 17987}, {2238, 5089}, {2245, 2361}, {2355, 40976}, {3002, 37908}, {3136, 37695}, {3195, 40984}, {4194, 19783}, {4204, 40937}, {5090, 5725}, {5724, 12135}, {6353, 24597}, {10974, 11398}, {22134, 26893}, {31187, 37453}, {37368, 37732}

X(44113) = isogonal conjugate of isotomic conjugate of X(860)
X(44113) = polar conjugate of isotomic conjugate of X(2245)
X(44113) = X(i)-Ceva conjugate of X(j) for these (i,j): {860, 2245}, {1299, 2178}, {8749, 607}, {14776, 512}, {36067, 647}
X(44113) = X(i)-isoconjugate of X(j) for these (i,j): {3, 14616}, {7, 1793}, {63, 24624}, {69, 759}, {77, 6740}, {80, 1444}, {86, 1807}, {283, 18815}, {304, 34079}, {328, 17104}, {332, 1411}, {339, 9274}, {348, 2341}, {525, 37140}, {1437, 20566}, {1790, 18359}, {1812, 2006}, {2161, 17206}, {3267, 32671}, {4467, 36061}, {7254, 36804}, {9273, 20902}, {14208, 36069}, {18160, 32662}, {23189, 35174}, {23226, 35139}
X(44113) = crossdifference of every pair of points on line {63, 525}
X(44113) = barycentric product X(i)*X(j) for these {i,j}: {4, 2245}, {6, 860}, {9, 1835}, {19, 758}, {25, 3936}, {28, 4053}, {33, 18593}, {36, 1826}, {37, 1870}, {42, 17923}, {92, 3724}, {112, 6370}, {162, 2610}, {186, 8818}, {225, 2323}, {281, 1464}, {320, 2333}, {607, 41804}, {648, 42666}, {661, 4242}, {1400, 5081}, {1824, 3218}, {1832, 7127}, {1845, 2250}, {1880, 4511}, {1897, 21828}, {1973, 35550}, {1983, 24006}, {2171, 17515}, {2361, 40149}, {4707, 8750}, {6739, 8749}, {7113, 41013}, {36125, 40988}
X(44113) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 14616}, {25, 24624}, {36, 17206}, {41, 1793}, {186, 34016}, {213, 1807}, {607, 6740}, {758, 304}, {860, 76}, {1464, 348}, {1824, 18359}, {1826, 20566}, {1835, 85}, {1870, 274}, {1880, 18815}, {1973, 759}, {1974, 34079}, {1983, 4592}, {2212, 2341}, {2245, 69}, {2323, 332}, {2333, 80}, {2361, 1812}, {2610, 14208}, {3724, 63}, {3936, 305}, {4053, 20336}, {4242, 799}, {5081, 28660}, {6370, 3267}, {7113, 1444}, {8818, 328}, {8882, 39277}, {17923, 310}, {18593, 7182}, {21828, 4025}, {32676, 37140}, {34397, 40214}, {35550, 40364}, {42666, 525}

X(44114) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(868)

Barycentrics    a^2*(b - c)^2*(b + c)^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4) : :
Trilinears    a^2 (sin 2B sin(A - C) csc(A - B) - sin 2C sin(A - B) csc(A - C)) : :
X(44114) = X[2421] - 3 X[5968]

X(44114) lies on these lines: {4, 9513}, {6, 25}, {74, 8753}, {125, 136}, {182, 37930}, {235, 43278}, {287, 36898}, {325, 40810}, {351, 865}, {511, 2421}, {512, 2088}, {525, 3143}, {577, 42847}, {868, 35088}, {878, 15630}, {1562, 5139}, {1625, 39846}, {1648, 8029}, {2433, 9178}, {2493, 2871}, {2679, 38974}, {2971, 3269}, {3292, 33928}, {3455, 11060}, {3563, 17974}, {6786, 36790}, {7417, 34211}, {13210, 16175}, {14995, 32225}, {16186, 34291}, {21046, 22212}, {23350, 32112}, {34980, 38356}, {39374, 40083}, {39691, 41221}, {41670, 42742}

X(44114) = polar conjugate of X(41174)
X(44114) = isogonal conjugate of isotomic conjugate of X(868)
X(44114) = polar conjugate of isotomic conjugate of X(41172)
X(44114) = tripolar centroid of X(14998)
X(44114) = pole wrt polar circle of trilinear polar of X(41174) (line X(110)X(685))
X(44114) = X(i)-Ceva conjugate of X(j) for these (i,j): {232, 2491}, {511, 3569}, {868, 41172}, {882, 22260}, {1976, 512}, {3124, 2679}, {3563, 647}, {34854, 17994}, {39644, 669}, {40810, 41167}, {43717, 2489}
X(44114) = X(i)-isoconjugate of X(j) for these (i,j): {48, 41174}, {98, 24041}, {99, 36084}, {110, 36036}, {162, 17932}, {163, 43187}, {249, 1821}, {250, 336}, {290, 1101}, {293, 18020}, {662, 2966}, {685, 4592}, {799, 2715}, {811, 43754}, {1910, 4590}, {1976, 24037}, {4563, 36104}, {4575, 22456}, {6394, 24000}, {17974, 23999}, {18024, 23995}
X(44114) = crosspoint of X(i) and X(j) for these (i,j): {4, 2395}, {6, 35364}, {232, 16230}, {511, 3569}, {512, 1976}, {850, 1916}, {882, 40810}, {17994, 34854}
X(44114) = crosssum of X(i) and X(j) for these (i,j): {2, 4226}, {3, 2421}, {98, 2966}, {99, 325}, {287, 43754}, {1576, 1691}, {2715, 14601}, {6394, 17932}, {17941, 40820}
X(44114) = crossdifference of every pair of points on line {99, 249}
X(44114) = barycentric product X(i)*X(j) for these {i,j}: {4, 41172}, {6, 868}, {115, 511}, {125, 232}, {237, 338}, {240, 3708}, {297, 20975}, {325, 3124}, {339, 2211}, {512, 2799}, {523, 3569}, {525, 17994}, {647, 16230}, {684, 2501}, {690, 8430}, {850, 2491}, {1109, 1755}, {1637, 32112}, {1640, 23350}, {1648, 5968}, {1916, 2679}, {1959, 2643}, {1976, 35088}, {2028, 14501}, {2029, 14502}, {2088, 14356}, {2395, 41167}, {2396, 22260}, {2421, 8029}, {2489, 6333}, {2970, 3289}, {2971, 6393}, {3121, 42703}, {3269, 6530}, {3563, 41181}, {4092, 43034}, {5360, 16732}, {6041, 34765}, {6328, 34349}, {8754, 36212}, {9417, 23994}, {9418, 23962}, {14618, 39469}, {14966, 23105}, {15526, 34854}, {15630, 32458}, {17209, 21043}, {36471, 39644}
X(44114) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 41174}, {115, 290}, {232, 18020}, {237, 249}, {325, 34537}, {338, 18024}, {511, 4590}, {512, 2966}, {523, 43187}, {647, 17932}, {661, 36036}, {669, 2715}, {684, 4563}, {798, 36084}, {868, 76}, {882, 39291}, {1084, 1976}, {1755, 24041}, {1959, 24037}, {2086, 40820}, {2211, 250}, {2421, 31614}, {2422, 41173}, {2489, 685}, {2491, 110}, {2501, 22456}, {2643, 1821}, {2679, 385}, {2799, 670}, {2971, 6531}, {3049, 43754}, {3124, 98}, {3269, 6394}, {3569, 99}, {3708, 336}, {5360, 4567}, {6041, 34761}, {8029, 43665}, {8430, 892}, {8754, 16081}, {9417, 1101}, {9418, 23357}, {9427, 14601}, {15630, 41932}, {16230, 6331}, {17994, 648}, {20975, 287}, {21906, 5967}, {22260, 2395}, {23099, 2422}, {23216, 14600}, {23350, 6035}, {34854, 23582}, {39469, 4558}, {40810, 39292}, {41167, 2396}, {41172, 69}, {43034, 7340}
X(44114) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3124, 20975, 6784}

X(44115) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(964)

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

X(44115) lies on these lines: {1, 37317}, {6, 25}, {31, 37}, {32, 55}, {42, 560}, {58, 405}, {65, 1395}, {81, 37325}, {171, 29674}, {172, 2352}, {199, 4261}, {209, 2273}, {354, 1104}, {386, 2915}, {580, 37320}, {757, 30962}, {1011, 1333}, {1098, 37314}, {1724, 4205}, {1834, 37398}, {2206, 2278}, {2288, 10537}, {2328, 4264}, {2363, 4195}, {3666, 13723}, {3695, 5264}, {4204, 5115}, {4239, 32911}, {4252, 37246}, {5156, 37327}, {5269, 17742}, {5280, 5285}, {5398, 37527}, {6043, 37652}, {9306, 40153}, {10457, 13733}, {16466, 27802}, {16974, 20985}, {17126, 17776}, {17526, 18141}, {17698, 37522}, {19309, 19728}, {35612, 36011}, {37315, 37646}

X(44115) = isogonal conjugate of isotomic conjugate of X(964)
X(44115) = crosssum of X(2) and X(17676)
X(44115) = crossdifference of every pair of points on line {525, 3004}
X(44115) = barycentric product X(6)*X(964)
X(44115) = barycentric quotient X(964)/X(76)

X(44116) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1003)

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

X(44116) lies on these lines: {2, 5033}, {6, 25}, {23, 5028}, {32, 110}, {39, 6800}, {182, 14567}, {187, 15066}, {251, 9544}, {287, 10603}, {574, 15080}, {576, 20976}, {1184, 8780}, {1196, 35264}, {1383, 2987}, {1501, 3231}, {1691, 5651}, {1692, 1995}, {1799, 3620}, {2001, 10546}, {2030, 8585}, {2374, 40819}, {2502, 3506}, {3053, 6090}, {3094, 35268}, {3098, 8627}, {3291, 35259}, {3292, 5017}, {3552, 4563}, {5034, 11003}, {5206, 7998}, {5207, 30747}, {5475, 14389}, {5477, 37644}, {5640, 39764}, {7737, 37645}, {7781, 10330}, {7798, 35356}, {8588, 41462}, {9465, 35265}, {13410, 39561}, {14002, 39024}, {14609, 32729}, {15513, 21766}, {20080, 40405}, {22112, 39560}, {26881, 34945}

X(44116) = isogonal conjugate of isotomic conjugate of X(1003)
X(44116) = crosssum of X(2) and X(33017)
X(44116) = crossdifference of every pair of points on line {525, 9148}
X(44116) = barycentric product X(6)*X(1003)
X(44116) = barycentric quotient X(1003)/X(76)
X(44116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1501, 3231, 41412}, {9306, 41412, 3231}, {35259, 40825, 3291}

X(44117) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1006)

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

X(44117) lies on these lines: {6, 25}, {31, 21741}, {41, 9247}, {54, 7412}, {110, 4223}, {199, 5755}, {228, 2148}, {389, 20837}, {405, 41608}, {578, 37194}, {1243, 36009}, {1426, 19365}, {1437, 5398}, {1465, 26889}, {1824, 11428}, {1827, 10535}, {2193, 23606}, {2355, 10536}, {3145, 5396}, {4220, 5012}, {5135, 5718}, {6759, 37387}, {11003, 35988}, {11383, 19350}, {24597, 37367}, {26866, 34042}, {26885, 40937}, {33718, 34396}

X(44117) = isogonal conjugate of isotomic conjugate of X(1006)
X(44117) = X(75)-isoconjugate of X(1243)
X(44117) = crosssum of X(2) and X(6839)
X(44117) = crossdifference of every pair of points on line {525, 36038}
X(44117) = barycentric product X(i)*X(j) for these {i,j}: {1, 2302}, {6, 1006}
X(44117) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1243}, {1006, 76}, {2302, 75}
X(44117) = {X(184),X(5320)}-harmonic conjugate of X(25)

X(44118) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1008)

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

X(44118) lies on these lines: {2, 31}, {6, 25}, {22, 3736}, {32, 16372}, {39, 33714}, {58, 19310}, {82, 18147}, {386, 37576}, {612, 1918}, {968, 2339}, {2274, 5322}, {2328, 4279}, {4261, 23868}, {5132, 37577}, {10457, 37254}, {16466, 36025}, {16478, 20985}, {26885, 40728}, {28660, 33737}

X(44118) = isogonal conjugate of isotomic conjugate of X(1008)
X(44118) = crossdifference of every pair of points on line {525, 3250}
X(44118) = barycentric product X(6)*X(1008)
X(44118) = barycentric quotient X(1008)/X(76)

X(44119) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1010)

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

X(44119) lies on these lines: {1, 21}, {6, 25}, {27, 3914}, {42, 284}, {55, 1333}, {86, 33124}, {110, 28476}, {112, 38883}, {171, 306}, {172, 228}, {199, 2092}, {213, 26885}, {314, 14012}, {333, 6043}, {386, 11337}, {387, 4198}, {593, 4184}, {612, 2303}, {941, 4264}, {1011, 5019}, {1043, 2363}, {1096, 41364}, {1193, 2360}, {1396, 2263}, {1397, 39780}, {1402, 4215}, {1408, 34046}, {1412, 1458}, {1430, 1848}, {1437, 23122}, {1460, 2286}, {1724, 37314}, {1754, 37419}, {1781, 40973}, {1790, 3736}, {1951, 20967}, {2177, 33628}, {2285, 4206}, {2305, 22080}, {2308, 16470}, {2332, 7169}, {2355, 41015}, {3751, 40571}, {4320, 5323}, {4418, 18697}, {4749, 20988}, {6051, 37322}, {7093, 13588}, {16488, 21747}, {17016, 17521}, {17587, 32929}, {17594, 27174}, {20986, 23381}, {22389, 40956}, {24632, 26034}, {24941, 30851}, {26266, 27041}, {30941, 33171}, {30965, 32783}

X(44119) = isogonal conjugate of isotomic conjugate of X(1010)
X(44119) = X(i)-Ceva conjugate of X(j) for these (i,j): {907, 3733}, {5331, 284}
X(44119) = X(1460)-cross conjugate of X(4206)
X(44119) = X(i)-isoconjugate of X(j) for these (i,j): {65, 30479}, {75, 1245}, {76, 2281}, {226, 2339}, {307, 1039}, {313, 1472}, {321, 2221}, {523, 1310}, {525, 36099}, {661, 37215}, {1036, 1441}, {14208, 32691}, {21750, 40831}, {31993, 34260}
X(44119) = crosspoint of X(2303) and X(5323)
X(44119) = crosssum of X(i) and X(j) for these (i,j): {2, 26117}, {10, 4656}, {1214, 12709}
X(44119) = crossdifference of every pair of points on line {525, 661}
X(44119) = barycentric product X(i)*X(j) for these {i,j}: {1, 2303}, {6, 1010}, {9, 5323}, {21, 2285}, {27, 7085}, {28, 5227}, {29, 2286}, {58, 2345}, {63, 4206}, {81, 612}, {99, 2484}, {110, 6590}, {112, 23874}, {162, 2522}, {163, 2517}, {284, 388}, {333, 1460}, {662, 8678}, {799, 8646}, {1038, 1172}, {1098, 8898}, {1333, 4385}, {1412, 3974}, {1790, 7102}, {2203, 19799}, {2287, 4320}, {2327, 7103}, {2328, 7365}, {5331, 34261}, {7252, 14594}, {19459, 40411}
X(44119) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1245}, {110, 37215}, {163, 1310}, {284, 30479}, {388, 349}, {560, 2281}, {612, 321}, {1010, 76}, {1038, 1231}, {1184, 3914}, {1460, 226}, {2194, 2339}, {2204, 1039}, {2206, 2221}, {2285, 1441}, {2286, 307}, {2303, 75}, {2345, 313}, {2484, 523}, {2517, 20948}, {2522, 14208}, {3974, 30713}, {4206, 92}, {4320, 1446}, {4385, 27801}, {5227, 20336}, {5323, 85}, {6590, 850}, {7085, 306}, {8646, 661}, {8678, 1577}, {19459, 18589}, {23874, 3267}, {32676, 36099}
X(44119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {42, 2206, 284}, {58, 2328, 31}, {2194, 37538, 1474}

X(44120) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1011)

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

X(44120) lies on these lines: {6, 25}, {32, 560}, {48, 33718}, {86, 9306}, {110, 17379}, {182, 6998}, {584, 692}, {1092, 37474}, {1098, 13323}, {1185, 9455}, {1437, 37507}, {4269, 33714}, {5012, 17349}, {5145, 17104}, {5651, 15668}, {7193, 13723}, {9544, 37677}, {17259, 43650}, {20986, 20992}, {26885, 37316}, {33745, 41248}, {37317, 42463}

X(44120) = isogonal conjugate of isotomic conjugate of X(1011)
X(44120) = X(i)-isoconjugate of X(j) for these (i,j): {75, 1246}, {76, 2282}, {28624, 40495}
X(44120) = crossdifference of every pair of points on line {525, 3261}
X(44120) = barycentric product X(i)*X(j) for these {i,j}: {1, 2304}, {6, 1011}, {32, 10449}, {41, 37523}, {2175, 27339}, {28623, 32739}
X(44120) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 1246}, {560, 2282}, {1011, 76}, {2304, 75}, {10449, 1502}, {27339, 41283}, {37523, 20567}
X(44120) = {X(6),X(25)}-harmonic conjugate of X(40954)

X(44121) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1012)

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

X(44121) lies on these lines: {4, 14925}, {6, 25}, {19, 10535}, {28, 6759}, {31, 2208}, {56, 3937}, {110, 37254}, {182, 33849}, {578, 4222}, {1092, 13730}, {1147, 20831}, {1437, 37260}, {1498, 37245}, {1503, 37432}, {1614, 17562}, {2192, 11406}, {2324, 26885}, {2360, 28348}, {3270, 11383}, {4185, 26883}, {4186, 11424}, {4224, 9306}, {5651, 25934}, {10282, 14017}, {10539, 39271}, {10984, 37034}, {11206, 37394}, {13323, 28376}, {13346, 28029}, {16252, 37376}, {20986, 20991}, {33810, 37397}, {37366, 43650}

X(44121) = isogonal conjugate of isotomic conjugate of X(1012)
X(44121) = crosssum of X(i) and X(j) for these (i,j): {2, 6925}, {8, 6350}
X(44121) = barycentric product X(6)*X(1012)
X(44121) = barycentric quotient X(1012)/X(76)
X(44121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 2208, 23204}, {154, 2194, 184}

X(44122) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1080)

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

X(44122) lies on these lines: {6, 25}, {15, 237}, {16, 3148}, {61, 20897}, {182, 3130}, {263, 3458}, {418, 11515}, {460, 5318}, {462, 5480}, {463, 1503}, {511, 3129}, {1352, 33530}, {1976, 3457}, {2871, 11086}, {3098, 3131}, {3132, 5092}, {5191, 41406}, {5335, 6620}, {6641, 11516}, {10645, 41275}, {10646, 37457}, {11485, 41266}, {34098, 37775}

X(44122) = isogonal conjugate of isotomic conjugate of X(1080)
X(44122) = barycentric product X(6)*X(1080)
X(44122) = barycentric quotient X(1080)/X(76)

X(44123) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1113)

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

X(44123) lies on the cubic K1067 and these lines: {6, 25}, {23, 24650}, {110, 1113}, {182, 1345}, {186, 31954}, {468, 13414}, {685, 39241}, {692, 2576}, {1114, 19128}, {1177, 2575}, {1312, 1503}, {1492, 2580}, {1576, 42668}, {1976, 8106}, {2105, 15460}, {4577, 15164}, {14500, 36201}, {15167, 42671}, {32717, 39298}, {40352, 42667}

X(44123) = reflection of X(44124) in X(206)
X(44123) = isogonal conjugate of X(22339)
X(44123) = isogonal conjugate of anticomplement of X(8105)
X(44123) = isogonal conjugate of isotomic conjugate of X(1113)
X(44123) = X(i)-Ceva conjugate of X(j) for these (i,j): {15461, 6}, {41941, 32}
X(44123) = X(i)-cross conjugate of X(j) for these (i,j): {32, 41941}, {42668, 25}
X(44123) = X(i)-isoconjugate of X(j) for these (i,j): {1, 22339}, {2, 2582}, {63, 2592}, {69, 2588}, {75, 2574}, {76, 2578}, {264, 2584}, {304, 8105}, {525, 2581}, {561, 42668}, {656, 15165}, {850, 1823}, {1114, 14208}, {1577, 8116}, {2577, 3267}, {2587, 3265}, {4592, 39240}, {20902, 39299}
X(44123) = cevapoint of X(184) and X(42668)
X(44123) = crosssum of X(2) and X(14807)
X(44123) = trilinear pole of line {32, 42667}
X(44123) = trilinear product X(i)*X(j) for these {i,j}: {6, 2576}, {25, 1822}, {31, 1113}, {32, 2580}, {112, 2579}, {163, 8106}, {184, 2586}, {560, 15164}, {1576, 2589}, {1973, 8115}
X(44123) = barycentric product X(i)*X(j) for these {i,j}: {1, 2576}, {6, 1113}, {19, 1822}, {25, 8115}, {31, 2580}, {32, 15164}, {48, 2586}, {110, 8106}, {112, 2575}, {162, 2579}, {163, 2589}, {512, 39298}, {648, 42667}, {1576, 2593}, {1989, 44067}, {2574, 41941}, {2583, 32676}, {2585, 24019}, {8105, 15461}, {23357, 39241}
X(44123) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 22339}, {25, 2592}, {31, 2582}, {32, 2574}, {112, 15165}, {560, 2578}, {1113, 76}, {1501, 42668}, {1576, 8116}, {1822, 304}, {1973, 2588}, {1974, 8105}, {2489, 39240}, {2575, 3267}, {2576, 75}, {2579, 14208}, {2580, 561}, {2586, 1969}, {2589, 20948}, {8106, 850}, {8115, 305}, {9247, 2584}, {15164, 1502}, {32676, 2581}, {39241, 23962}, {39298, 670}, {41941, 15164}, {42667, 525}, {44067, 7799}
X(44123) = {X(6),X(18374)}-harmonic conjugate of X(44124)
X(44123) = {X(25),X(34397)}-harmonic conjugate of X(44124)

X(44124) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1114)

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

X(44124) lies on the cubic K1067 and these lines: {6, 25}, {23, 24651}, {110, 1114}, {182, 1344}, {186, 31955}, {468, 13415}, {685, 39240}, {692, 2577}, {1113, 19128}, {1177, 2574}, {1313, 1503}, {1492, 2581}, {1576, 42667}, {1976, 8105}, {2104, 15461}, {4577, 15165}, {14499, 36201}, {15166, 42671}, {32717, 39299}, {40352, 42668}

X(44124) = reflection of X(44123) in X(206)
X(44124) = isogonal conjugate of X(22340)
X(44124) = isogonal conjugate of anticomplement of X(8106)
X(44124) = isogonal conjugate of isotomic conjugate of X(1114)
X(44124) = X(i)-Ceva conjugate of X(j) for these (i,j): {15460, 6}, {41942, 32}
X(44124) = X(i)-cross conjugate of X(j) for these (i,j): {32, 41942}, {42667, 25}
X(44124) = X(i)-isoconjugate of X(j) for these (i,j): {1, 22340}, {2, 2583}, {63, 2593}, {69, 2589}, {75, 2575}, {76, 2579}, {264, 2585}, {304, 8106}, {525, 2580}, {561, 42667}, {656, 15164}, {850, 1822}, {1113, 14208}, {1577, 8115}, {2576, 3267}, {2586, 3265}, {4592, 39241}, {20902, 39298}
X(44124) = cevapoint of X(184) and X(42667)
X(44124) = crosssum of X(2) and X(14808)
X(44124) = trilinear pole of line {32, 42668}
X(44124) = trilinear product X(i)*X(j) for these {i,j}: {6, 2577}, {25, 1823}, {31, 1114}, {32, 2581}, {112, 2578}, {163, 8105}, {184, 2587}, {560, 15165}, {1576, 2588}, {1973, 8116}
X(44124) = barycentric product X(i)*X(j) for these {i,j}: {1, 2577}, {6, 1114}, {19, 1823}, {25, 8116}, {31, 2581}, {32, 15165}, {48, 2587}, {110, 8105}, {112, 2574}, {162, 2578}, {163, 2588}, {512, 39299}, {648, 42668}, {1576, 2592}, {1989, 44068}, {2575, 41942}, {2582, 32676}, {2584, 24019}, {8106, 15460}, {23357, 39240}
X(44124) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 22340}, {25, 2593}, {31, 2583}, {32, 2575}, {112, 15164}, {560, 2579}, {1114, 76}, {1501, 42667}, {1576, 8115}, {1823, 304}, {1973, 2589}, {1974, 8106}, {2489, 39241}, {2574, 3267}, {2577, 75}, {2578, 14208}, {2581, 561}, {2587, 1969}, {2588, 20948}, {8105, 850}, {8116, 305}, {9247, 2585}, {15165, 1502}, {32676, 2580}, {39240, 23962}, {39299, 670}, {41942, 15165}, {42668, 525}, {44068, 7799}
X(44124) = {X(6),X(18374)}-harmonic conjugate of X(44123)
X(44124) = {X(25),X(34397)}-harmonic conjugate of X(44123)

X(44125) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1312)

Barycentrics    a^2*(a^2*(-a^2 + b^2 + c^2)*(-1 + J) + (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(1 + J)) : :

X(44125) lies on the orthic inconic and these lines: {4, 41519}, {6, 25}, {125, 1312}, {237, 15166}, {373, 1345}, {511, 1113}, {524, 20405}, {895, 15461}, {1344, 12294}, {1976, 41941}, {2574, 5095}, {3978, 15164}, {6000, 31955}, {6784, 8106}, {8115, 8681}, {15167, 20975}

X(44125) = midpoint of X(1113) and X(24650)
X(44125) = reflection of X(44126) in X(6)
X(44125) = isogonal conjugate of isotomic conjugate of X(1312)
X(44125) = polar conjugate of isotomic conjugate of X(15167)
X(44125) = orthic isogonal conjugate of X(8106)
X(44125) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 8106}, {25, 42667}, {1113, 647}, {1312, 15167}
X(44125) = X(i)-isoconjugate of X(j) for these (i,j): {75, 15460}, {304, 41942}, {1313, 24041}, {1823, 15165}, {2581, 8116}, {2582, 39299}
X(44125) = crosspoint of X(i) and X(j) for these (i,j): {4, 8106}, {6, 2575}
X(44125) = crosssum of X(i) and X(j) for these (i,j): {2, 1114}, {3, 8116}
X(44125) = crossdifference of every pair of points on line {525, 8116}
X(44125) = barycentric product X(i)*X(j) for these {i,j}: {4, 15167}, {6, 1312}, {115, 15461}, {125, 41941}, {2575, 8106}, {2579, 2589}, {2593, 42667}, {8749, 14500}
X(44125) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 15460}, {1312, 76}, {1974, 41942}, {3124, 1313}, {8106, 15165}, {15167, 69}, {15461, 4590}, {41941, 18020}, {42667, 8116}
X(44125) = {X(25),X(44084)}-harmonic conjugate of X(44126)
X(44125) = {X(51),X(1495)}-harmonic conjugate of X(44126)

X(44126) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1313)

Barycentrics    a^2*((a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(1 - J) - a^2*(-a^2 + b^2 + c^2)*(1 + J)) : :

X(44126) lies on the orthic inconic and these lines: {4, 41518}, {6, 25}, {125, 1313}, {237, 15167}, {373, 1344}, {511, 1114}, {524, 20406}, {895, 15460}, {1345, 12294}, {1976, 41942}, {2575, 5095}, {3978, 15165}, {6000, 31954}, {6784, 8105}, {8116, 8681}, {15166, 20975}

X(44126) = midpoint of X(1114) and X(24651)
X(44126) = reflection of X(44125) in X(6)
X(44126) = isogonal conjugate of isotomic conjugate of X(1313)
X(44126) = polar conjugate of isotomic conjugate of X(15166)
X(44126) = orthic isogonal conjugate of X(8105)
X(44126) = X(i)-Ceva conjugate of X(j) for these (i,j): {4, 8105}, {25, 42668}, {1114, 647}, {1313, 15166}
X(44126) = X(i)-isoconjugate of X(j) for these (i,j): {75, 15461}, {304, 41941}, {1312, 24041}, {1822, 15164}, {2580, 8115}, {2583, 39298}
X(44126) = crosspoint of X(i) and X(j) for these (i,j): {4, 8105}, {6, 2574}
X(44126) = crosssum of X(i) and X(j) for these (i,j): {2, 1113}, {3, 8115}
X(44126) = crossdifference of every pair of points on line {525, 8115}
X(44126) = barycentric product X(i)*X(j) for these {i,j}: {4, 15166}, {6, 1313}, {115, 15460}, {125, 41942}, {2574, 8105}, {2578, 2588}, {2592, 42668}, {8749, 14499}
X(44126) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 15461}, {1313, 76}, {1974, 41941}, {3124, 1312}, {8105, 15164}, {15166, 69}, {15460, 4590}, {41942, 18020}, {42668, 8115}
X(44126) = {X(25),X(44084)}-harmonic conjugate of X(44125)
X(44126) = {X(51),X(1495)}-harmonic conjugate of X(44125)

X(44127) = ISOGONAL CONJUGATE OF ISOTOMIC CONJUGATE OF X(1316)

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

X(44127) lies on these lines: {4, 23582}, {6, 25}, {32, 512}, {110, 34383}, {115, 14574}, {182, 7418}, {263, 32716}, {511, 37930}, {525, 36156}, {1501, 2086}, {1976, 6784}, {2421, 9306}, {2679, 2715}, {2682, 7737}, {3053, 10568}, {3231, 23200}, {3506, 9149}, {3511, 19576}, {3575, 43278}, {3767, 40373}, {4558, 6786}, {5167, 10317}, {5467, 5651}, {6785, 14355}, {6793, 8754}, {7735, 41932}, {9218, 36182}, {10312, 40951}, {11645, 39750}, {14898, 41412}, {17932, 30226}, {22146, 39846}

X(44127) = isogonal conjugate of isotomic conjugate of X(1316)
X(44127) = X(i)-isoconjugate of X(j) for these (i,j): {75, 9513}, {1934, 40077}
X(44127) = crosssum of X(2) and X(36163)
X(44127) = crossdifference of every pair of points on line {325, 525}
X(44127) = barycentric product X(i)*X(j) for these {i,j}: {6, 1316}, {512, 40866}, {1691, 38947}, {2715, 31953}, {3569, 43113}
X(44127) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 9513}, {1316, 76}, {14602, 40077}, {38947, 18896}, {40866, 670}, {43113, 43187}
X(44127) = {X(25),X(2445)}-harmonic conjugate of X(1974)

X(44128) = ISOTOMIC CONJUGATE OF X(70)

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

X(44128) lies on these lines: {2, 571}, {3, 39113}, {4, 69}, {6, 41237}, {20, 1273}, {22, 160}, {66, 14957}, {86, 37156}, {95, 7558}, {141, 41231}, {183, 5133}, {290, 18124}, {297, 8745}, {343, 6748}, {394, 467}, {1007, 7494}, {1176, 20022}, {1225, 31723}, {1238, 1975}, {1369, 15589}, {2373, 13398}, {2393, 41757}, {3547, 32816}, {3785, 7404}, {3926, 31305}, {3933, 7553}, {3964, 7387}, {5392, 13579}, {5596, 25046}, {6527, 13219}, {7403, 7767}, {7500, 37668}, {7503, 7750}, {7512, 7763}, {7773, 13160}, {7788, 34603}, {12220, 41761}, {12225, 20477}, {12605, 41005}, {14165, 37669}, {14790, 28706}, {14907, 35921}, {15760, 41008}, {17907, 22151}, {20563, 37444}, {27377, 41614}, {28419, 41253}, {28712, 37448}, {35296, 42406}

X(44128) = isotomic conjugate of X(70)
X(44128) = anticomplement of X(571)
X(44128) = anticomplement of isogonal conjugate of X(5392)
X(44128) = isotomic conjugate of anticomplement of X(34116)
X(44128) = isotomic conjugate of isogonal conjugate of X(26)
X(44128) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {68, 6360}, {75, 40697}, {91, 2}, {92, 6193}, {96, 17479}, {847, 5905}, {925, 4560}, {1820, 3164}, {2165, 192}, {5392, 8}, {14593, 21216}, {20563, 4329}, {20571, 69}, {30450, 7253}, {34385, 21271}, {36145, 31296}
X(44128) = X(34116)-cross conjugate of X(2)
X(44128) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2158}, {31, 70}, {560, 20564}, {810, 1288}
X(44128) = crosssum of X(20975) and X(34952)
X(44128) = X(19)-of-dual-of-orthic-triangle if ABC is acute
X(44128) = barycentric product X(i)*X(j) for these {i,j}: {26, 76}, {305, 8746}, {1502, 44078}
X(44128) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2158}, {2, 70}, {26, 6}, {76, 20564}, {648, 1288}, {8746, 25}, {34116, 571}, {36418, 44078}, {44078, 32}
X(44128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 311}, {315, 317, 69}, {340, 14615, 69}, {637, 638, 11412}, {32002, 38434, 69}

X(44129) = ISOTOMIC CONJUGATE OF X(71)

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

X(44129) lies on these lines: {4, 69}, {27, 310}, {29, 811}, {58, 21207}, {86, 8747}, {92, 304}, {99, 917}, {274, 278}, {324, 26541}, {333, 349}, {337, 16747}, {447, 33954}, {469, 18152}, {648, 43093}, {799, 37203}, {823, 37214}, {1043, 35517}, {1172, 30940}, {1396, 8033}, {1838, 10471}, {2064, 27801}, {2700, 22456}, {2973, 17982}, {6331, 6336}, {6528, 18025}, {6999, 26166}, {7379, 30737}, {16749, 23989}, {17911, 17913}, {18134, 40011}, {36659, 41009}, {40071, 40445}

X(44129) = isogonal conjugate of X(2200)
X(44129) = isotomic conjugate of X(71)
X(44129) = polar conjugate of X(42)
X(44129) = isotomic conjugate of anticomplement of X(34830)
X(44129) = isotomic conjugate of complement of X(17220)
X(44129) = isotomic conjugate of isogonal conjugate of X(27)
X(44129) = anticomplement of isotomic conjugate of polar conjugate of X(40954)
X(44129) = anticomplement of crosspoint of X(2) and X(71)
X(44129) = anticomplement of crosssum of X(6) and X(27)
X(44129) = anticomplement of X(2)-Ceva conjugate of X(34830)
X(44129) = pole wrt polar circle of trilinear polar of X(42) (line X(512)X(798))
X(44129) = perspector of ABC and orthoanticevian triangle of X(310)
X(44129) = polar conjugate of isotomic conjugate of X(310)
X(44129) = polar conjugate of isogonal conjugate of X(86)
X(44129) = X(i)-cross conjugate of X(j) for these (i,j): {85, 274}, {86, 310}, {92, 286}, {1848, 273}, {5249, 75}, {17167, 86}, {17171, 27}, {18161, 81}, {18650, 15467}, {19786, 1240}, {21137, 514}, {34830, 2}
X(44129) = cevapoint of X(i) and X(j) for these (i,j): {2, 17220}, {4, 17911}, {6, 23339}, {27, 86}, {75, 18134}, {85, 331}, {92, 264}, {286, 31623}, {514, 21207}
X(44129) = crosssum of X(3) and X(23176)
X(44129) = trilinear pole of line {3261, 4025}
X(44129) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2200}, {3, 213}, {6, 228}, {10, 9247}, {19, 4055}, {25, 3990}, {31, 71}, {32, 72}, {37, 184}, {41, 73}, {42, 48}, {55, 1409}, {63, 1918}, {69, 2205}, {100, 3049}, {101, 810}, {181, 2193}, {212, 1400}, {219, 1402}, {220, 1410}, {248, 5360}, {255, 2333}, {295, 41333}, {306, 560}, {307, 9447}, {321, 14575}, {512, 906}, {577, 1824}, {603, 1334}, {604, 2318}, {607, 22341}, {647, 692}, {656, 32739}, {661, 32656}, {667, 4574}, {669, 1332}, {798, 1331}, {822, 8750}, {872, 1790}, {904, 22061}, {1042, 1802}, {1176, 21814}, {1214, 2175}, {1231, 9448}, {1258, 23212}, {1333, 3690}, {1397, 3694}, {1437, 1500}, {1439, 14827}, {1444, 7109}, {1501, 20336}, {1783, 39201}, {1794, 40978}, {1880, 6056}, {1917, 40071}, {1924, 4561}, {1946, 4559}, {1973, 3682}, {1974, 3998}, {1976, 42702}, {2176, 22381}, {2187, 41087}, {2194, 2197}, {2196, 3747}, {2204, 7066}, {2206, 3949}, {2212, 40152}, {2281, 7085}, {2332, 7138}, {2359, 3725}, {3063, 23067}, {3198, 14642}, {3709, 36059}, {3954, 10547}, {3955, 40729}, {4041, 32660}, {4079, 4575}, {4557, 22383}, {4601, 23216}, {4705, 32661}, {7084, 23620}, {7116, 20964}, {7123, 22363}, {8606, 21741}, {14533, 21807}, {14573, 42698}, {14585, 41013}, {14908, 21839}, {15389, 21877}, {17743, 22364}, {18098, 20775}, {18210, 23990}, {20683, 32658}, {20760, 21759}, {21046, 23995}, {21805, 32659}, {21874, 40319}, {22080, 28615}, {27801, 40373}, {34055, 41267}, {36057, 39258}
X(44129) = barycentric product X(i)*X(j) for these {i,j}: {4, 310}, {19, 6385}, {27, 76}, {28, 561}, {29, 6063}, {34, 40072}, {58, 18022}, {75, 286}, {81, 1969}, {85, 31623}, {86, 264}, {92, 274}, {162, 40495}, {225, 18021}, {273, 314}, {276, 17167}, {278, 28660}, {305, 8747}, {308, 17171}, {331, 333}, {514, 6331}, {648, 3261}, {670, 7649}, {689, 21108}, {693, 811}, {799, 17924}, {823, 15413}, {873, 41013}, {1172, 20567}, {1396, 28659}, {1434, 7017}, {1474, 1502}, {1790, 18027}, {1848, 40827}, {1896, 7182}, {1928, 2203}, {1978, 17925}, {2052, 17206}, {2299, 41283}, {2973, 4600}, {3112, 16747}, {4025, 6528}, {4602, 6591}, {4610, 14618}, {4623, 24006}, {5317, 40364}, {6335, 7199}, {6628, 7141}, {15149, 18031}, {15352, 30805}, {18020, 21207}, {18026, 18155}, {18827, 40717}, {18895, 31905}, {36419, 40071}
X(44129) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 228}, {2, 71}, {3, 4055}, {4, 42}, {6, 2200}, {7, 73}, {8, 2318}, {10, 3690}, {19, 213}, {21, 212}, {25, 1918}, {27, 6}, {28, 31}, {29, 55}, {34, 1402}, {57, 1409}, {58, 184}, {63, 3990}, {69, 3682}, {75, 72}, {76, 306}, {77, 22341}, {81, 48}, {85, 1214}, {86, 3}, {87, 22381}, {92, 37}, {99, 1331}, {107, 8750}, {110, 32656}, {112, 32739}, {158, 1824}, {162, 692}, {189, 41087}, {190, 4574}, {225, 181}, {226, 2197}, {240, 5360}, {242, 3747}, {261, 283}, {264, 10}, {269, 1410}, {270, 2194}, {273, 65}, {274, 63}, {278, 1400}, {281, 1334}, {283, 6056}, {285, 2188}, {286, 1}, {304, 3998}, {307, 7066}, {310, 69}, {312, 3694}, {313, 3695}, {314, 78}, {318, 210}, {321, 3949}, {324, 21011}, {331, 226}, {332, 1259}, {333, 219}, {338, 21046}, {342, 227}, {348, 40152}, {349, 26942}, {393, 2333}, {423, 17735}, {427, 21035}, {513, 810}, {514, 647}, {561, 20336}, {614, 22363}, {645, 4587}, {648, 101}, {649, 3049}, {653, 4559}, {662, 906}, {664, 23067}, {670, 4561}, {693, 656}, {757, 1437}, {799, 1332}, {811, 100}, {823, 1783}, {850, 4064}, {873, 1444}, {894, 22061}, {905, 822}, {1010, 7085}, {1014, 603}, {1019, 22383}, {1043, 1260}, {1088, 1439}, {1111, 18210}, {1119, 1042}, {1125, 22080}, {1172, 41}, {1210, 3611}, {1235, 15523}, {1269, 41014}, {1333, 9247}, {1396, 604}, {1414, 36059}, {1430, 44112}, {1434, 222}, {1439, 7138}, {1441, 201}, {1442, 22342}, {1444, 255}, {1446, 37755}, {1459, 39201}, {1474, 32}, {1502, 40071}, {1509, 1790}, {1790, 577}, {1812, 2289}, {1824, 872}, {1826, 1500}, {1829, 3725}, {1838, 40952}, {1839, 20970}, {1841, 40978}, {1842, 40984}, {1843, 41267}, {1847, 1427}, {1848, 2092}, {1851, 40934}, {1855, 21795}, {1860, 40954}, {1861, 20683}, {1870, 3724}, {1895, 3198}, {1896, 33}, {1897, 4557}, {1920, 4019}, {1959, 42702}, {1969, 321}, {1973, 2205}, {2052, 1826}, {2185, 2193}, {2201, 41333}, {2203, 560}, {2204, 9447}, {2206, 14575}, {2287, 1802}, {2299, 2175}, {2309, 23212}, {2322, 220}, {2332, 14827}, {2333, 7109}, {2501, 4079}, {2669, 20796}, {2905, 18755}, {2969, 3122}, {2970, 21043}, {2973, 3120}, {3064, 3709}, {3120, 20975}, {3194, 2187}, {3261, 525}, {3596, 3710}, {3615, 8606}, {3662, 20727}, {3668, 1425}, {3673, 17441}, {3720, 22369}, {3737, 1946}, {3794, 20753}, {3944, 22169}, {4000, 23620}, {4025, 520}, {4091, 32320}, {4146, 7591}, {4183, 1253}, {4233, 21059}, {4241, 2426}, {4248, 3052}, {4357, 22076}, {4359, 3958}, {4391, 8611}, {4466, 3269}, {4556, 32661}, {4560, 652}, {4565, 32660}, {4573, 1813}, {4610, 4558}, {4623, 4592}, {4625, 6516}, {5089, 39258}, {5125, 209}, {5209, 17977}, {5249, 18591}, {5317, 1973}, {5342, 37593}, {5379, 1110}, {6063, 307}, {6331, 190}, {6335, 1018}, {6385, 304}, {6528, 1897}, {6591, 798}, {6625, 15377}, {6626, 22139}, {6629, 3292}, {7009, 20964}, {7017, 2321}, {7032, 22364}, {7058, 2327}, {7101, 4515}, {7141, 6535}, {7191, 23203}, {7192, 1459}, {7199, 905}, {7257, 4571}, {7282, 2594}, {7292, 23230}, {7649, 512}, {8025, 22054}, {8743, 21034}, {8747, 25}, {8748, 607}, {8822, 7078}, {11125, 9409}, {13149, 1020}, {14006, 2330}, {14012, 26924}, {14534, 2359}, {14616, 1807}, {14618, 4024}, {15149, 672}, {15413, 24018}, {15419, 4091}, {15466, 8804}, {15467, 28786}, {16082, 2250}, {16099, 43693}, {16696, 4020}, {16704, 22356}, {16705, 22097}, {16709, 3916}, {16727, 3942}, {16732, 3708}, {16738, 22065}, {16747, 38}, {16749, 26934}, {16750, 7289}, {16887, 3917}, {16891, 20819}, {17103, 3955}, {17139, 22350}, {17167, 216}, {17168, 22052}, {17169, 22053}, {17170, 22057}, {17171, 39}, {17172, 14961}, {17173, 22058}, {17174, 22059}, {17175, 22060}, {17176, 22062}, {17177, 22064}, {17178, 22066}, {17179, 22067}, {17180, 22068}, {17181, 22069}, {17182, 22071}, {17183, 22072}, {17184, 22073}, {17185, 22074}, {17186, 22075}, {17187, 20775}, {17192, 22077}, {17193, 22078}, {17194, 22079}, {17195, 22082}, {17196, 22083}, {17197, 7117}, {17198, 22084}, {17199, 22085}, {17200, 22352}, {17202, 22447}, {17204, 22087}, {17205, 3937}, {17206, 394}, {17208, 22412}, {17209, 3289}, {17211, 22420}, {17212, 22093}, {17215, 22089}, {17216, 2972}, {17217, 22090}, {17218, 22091}, {17219, 1364}, {17220, 40591}, {17442, 21814}, {17515, 2361}, {17555, 22276}, {17731, 17976}, {17770, 20754}, {17863, 18673}, {17907, 4456}, {17911, 40586}, {17921, 20979}, {17923, 2245}, {17924, 661}, {17925, 649}, {17926, 657}, {17982, 2054}, {17984, 4039}, {18020, 4570}, {18021, 332}, {18022, 313}, {18026, 4551}, {18155, 521}, {18157, 25083}, {18169, 22389}, {18206, 20752}, {18604, 4100}, {18605, 563}, {18609, 2315}, {18645, 22055}, {18646, 22056}, {18648, 22401}, {18653, 3284}, {18792, 20777}, {18827, 295}, {19804, 4047}, {20567, 1231}, {20883, 3954}, {21102, 15451}, {21108, 3005}, {21109, 42665}, {21172, 42658}, {21178, 8673}, {21191, 2524}, {21205, 9517}, {21207, 125}, {23788, 8677}, {23989, 4466}, {23999, 5379}, {24006, 4705}, {24046, 43218}, {26818, 22088}, {26860, 22357}, {26871, 836}, {26959, 22409}, {28660, 345}, {29767, 22126}, {30939, 5440}, {30940, 20769}, {30941, 1818}, {30966, 3781}, {31008, 22370}, {31623, 9}, {31900, 2308}, {31905, 1914}, {31906, 21764}, {31908, 5332}, {31909, 2276}, {31912, 21793}, {31917, 2275}, {31926, 2280}, {32010, 7015}, {32014, 1796}, {33295, 7193}, {33296, 20760}, {33947, 3784}, {33955, 7293}, {36419, 1474}, {36421, 2332}, {36797, 3939}, {37128, 2196}, {37168, 902}, {38462, 21805}, {39915, 23079}, {40011, 40161}, {40072, 3718}, {40149, 2171}, {40395, 2259}, {40411, 7123}, {40412, 1794}, {40414, 2983}, {40432, 7116}, {40495, 14208}, {40684, 21012}, {40717, 740}, {40836, 2357}, {40940, 44093}, {40975, 21753}, {41013, 756}, {41083, 198}, {41364, 205}, {41629, 20818}, {42394, 21022}, {42396, 4628}, {43925, 1919}

X(44130) = ISOTOMIC CONJUGATE OF X(73)

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

X(44130) lies on these lines: {4, 69}, {27, 1240}, {29, 332}, {75, 158}, {86, 811}, {99, 7436}, {273, 310}, {274, 40836}, {281, 345}, {313, 1043}, {318, 3718}, {333, 1948}, {799, 43764}, {1441, 14009}, {1896, 30479}, {2708, 22456}, {6528, 34393}, {7019, 40717}, {7141, 17927}, {7498, 30022}, {8822, 35516}, {18697, 40703}, {30737, 37443}

X(44130) = isotomic conjugate of X(73)
X(44130) = polar conjugate of X(1400)
X(44130) = anticomplement of isogonal conjugate of polar conjugate of X(3142)
X(44130) = anticomplement of crosspoint of X(2) and X(73)
X(44130) = anticomplement of crosssum of X(6) and X(29)
X(44130) = anticomplement of X(2)-Ceva conjugate of X(34831)
X(44130) = pole wrt polar circle of trilinear polar of X(1400) (line X(512)X(810))
X(44130) = isotomic conjugate of anticomplement of X(34831)
X(44130) = isotomic conjugate of isogonal conjugate of X(29)
X(44130) = polar conjugate of isotomic conjugate of X(28660)
X(44130) = polar conjugate of isogonal conjugate of X(333)
X(44130) = X(i)-cross conjugate of X(j) for these (i,j): {75, 314}, {318, 31623}, {333, 28660}, {6734, 312}, {18155, 811}, {24430, 21}, {34831, 2}, {37774, 18031}
X(44130) = cevapoint of X(i) and X(j) for these (i,j): {29, 333}, {75, 264}, {92, 17555}, {318, 7017}
X(44130) = trilinear pole of line {3064, 6332}
X(44130) = X(i)-isoconjugate of X(j) for these (i,j): {3, 1402}, {6, 1409}, {25, 22341}, {31, 73}, {32, 1214}, {34, 4055}, {42, 603}, {48, 1400}, {55, 1410}, {56, 228}, {57, 2200}, {65, 184}, {71, 604}, {72, 1397}, {77, 1918}, {108, 39201}, {109, 810}, {181, 1437}, {201, 2206}, {212, 1042}, {213, 222}, {226, 9247}, {296, 44112}, {307, 560}, {348, 2205}, {512, 36059}, {577, 1880}, {608, 3990}, {647, 1415}, {651, 3049}, {661, 32660}, {667, 23067}, {669, 6516}, {798, 1813}, {822, 32674}, {906, 7180}, {1037, 22363}, {1106, 2318}, {1231, 1501}, {1333, 2197}, {1334, 7099}, {1395, 3682}, {1403, 22381}, {1408, 3690}, {1425, 2194}, {1426, 6056}, {1439, 2175}, {1441, 14575}, {1824, 7335}, {1949, 42669}, {1973, 40152}, {2199, 41087}, {2203, 7066}, {2281, 2286}, {2299, 7138}, {2333, 7125}, {2357, 7114}, {3949, 16947}, {4017, 32656}, {4559, 22383}, {6186, 22342}, {14585, 40149}, {14642, 30456}, {20336, 41280}
X(44130) = barycentric product X(i)*X(j) for these {i,j}: {4, 28660}, {19, 40072}, {21, 1969}, {27, 3596}, {28, 28659}, {29, 76}, {33, 6385}, {75, 31623}, {86, 7017}, {92, 314}, {264, 333}, {270, 27801}, {274, 318}, {281, 310}, {283, 18027}, {284, 18022}, {286, 312}, {304, 1896}, {305, 8748}, {331, 1043}, {332, 2052}, {522, 6331}, {561, 1172}, {648, 35519}, {670, 3064}, {811, 4391}, {823, 35518}, {1474, 40363}, {1502, 2299}, {1826, 18021}, {1928, 2204}, {2322, 6063}, {2332, 41283}, {3261, 36797}, {4183, 20567}, {4572, 17926}, {4602, 18344}, {4620, 21666}, {4631, 24006}, {5931, 15466}, {6332, 6528}, {6335, 18155}, {7257, 17924}, {14024, 18895}, {36800, 40717}
X(44130) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1409}, {2, 73}, {4, 1400}, {8, 71}, {9, 228}, {10, 2197}, {19, 1402}, {21, 48}, {27, 56}, {28, 604}, {29, 6}, {33, 213}, {55, 2200}, {57, 1410}, {63, 22341}, {69, 40152}, {75, 1214}, {76, 307}, {78, 3990}, {81, 603}, {85, 1439}, {86, 222}, {92, 65}, {99, 1813}, {107, 32674}, {110, 32660}, {158, 1880}, {162, 1415}, {190, 23067}, {219, 4055}, {226, 1425}, {243, 42669}, {261, 1790}, {264, 226}, {270, 1333}, {273, 1427}, {274, 77}, {278, 1042}, {280, 41087}, {281, 42}, {283, 577}, {284, 184}, {286, 57}, {306, 7066}, {310, 348}, {312, 72}, {313, 26942}, {314, 63}, {318, 37}, {321, 201}, {331, 3668}, {332, 394}, {333, 3}, {341, 3694}, {345, 3682}, {346, 2318}, {349, 6356}, {415, 17966}, {497, 23620}, {521, 822}, {522, 647}, {556, 7591}, {561, 1231}, {607, 1918}, {643, 906}, {645, 1331}, {648, 109}, {650, 810}, {652, 39201}, {662, 36059}, {663, 3049}, {799, 6516}, {811, 651}, {823, 108}, {950, 44093}, {1010, 2286}, {1014, 7099}, {1021, 1946}, {1039, 2281}, {1043, 219}, {1098, 2193}, {1172, 31}, {1214, 7138}, {1396, 1106}, {1434, 7053}, {1441, 37755}, {1444, 7125}, {1474, 1397}, {1790, 7335}, {1792, 2289}, {1812, 255}, {1817, 7114}, {1826, 181}, {1857, 2333}, {1859, 40978}, {1895, 30456}, {1896, 19}, {1897, 4559}, {1948, 851}, {1969, 1441}, {2052, 225}, {2082, 22363}, {2185, 1437}, {2189, 2206}, {2194, 9247}, {2202, 44112}, {2204, 560}, {2212, 2205}, {2287, 212}, {2299, 32}, {2319, 22381}, {2321, 3690}, {2322, 55}, {2326, 2194}, {2327, 6056}, {2332, 2175}, {2907, 2305}, {3064, 512}, {3194, 2199}, {3219, 22342}, {3261, 17094}, {3559, 2178}, {3596, 306}, {3686, 22080}, {3687, 22076}, {3691, 22369}, {3699, 4574}, {3701, 3949}, {3702, 3958}, {3705, 20727}, {3718, 3998}, {3737, 22383}, {4183, 41}, {4391, 656}, {4397, 8611}, {4560, 1459}, {4563, 6517}, {4612, 4575}, {4631, 4592}, {4636, 32661}, {4673, 4047}, {4858, 18210}, {5081, 2245}, {5249, 39791}, {5317, 1395}, {5379, 2149}, {5546, 32656}, {5931, 1073}, {6198, 21741}, {6331, 664}, {6332, 520}, {6335, 4551}, {6385, 7182}, {6514, 1092}, {6528, 653}, {6708, 2658}, {6734, 18591}, {7003, 2357}, {7017, 10}, {7020, 1903}, {7046, 1334}, {7058, 283}, {7081, 22061}, {7101, 210}, {7253, 652}, {7256, 4587}, {7257, 1332}, {7258, 4571}, {7452, 2425}, {7649, 7180}, {8735, 3122}, {8747, 608}, {8748, 25}, {8822, 7011}, {11107, 2174}, {11393, 21744}, {14006, 172}, {14024, 1914}, {14331, 42658}, {14361, 8803}, {14400, 9409}, {15146, 1951}, {15149, 1458}, {15352, 36127}, {15466, 5930}, {16713, 22053}, {17167, 30493}, {17171, 1401}, {17182, 23154}, {17185, 22345}, {17197, 3937}, {17206, 1804}, {17515, 7113}, {17555, 40590}, {17923, 1464}, {17924, 4017}, {17925, 43924}, {17926, 663}, {18021, 17206}, {18022, 349}, {18026, 1020}, {18155, 905}, {18163, 22344}, {18344, 798}, {20665, 22364}, {21044, 20975}, {21300, 22443}, {21666, 21044}, {23661, 18675}, {25128, 2524}, {27398, 7078}, {27509, 22057}, {27527, 22090}, {27958, 3955}, {28659, 20336}, {28660, 69}, {30713, 3695}, {31623, 1}, {31631, 3157}, {31905, 1428}, {31909, 1469}, {31917, 7248}, {31926, 1471}, {33950, 23203}, {34387, 4466}, {35145, 296}, {35196, 14533}, {35518, 24018}, {35519, 525}, {36421, 2299}, {36797, 101}, {36800, 295}, {37142, 1949}, {37168, 1404}, {37908, 9454}, {40011, 28786}, {40072, 304}, {40149, 1254}, {40363, 40071}, {40411, 1037}, {40414, 951}, {40571, 3215}, {40717, 16609}, {40882, 17975}, {40979, 23204}, {40987, 21750}, {41013, 2171}, {41083, 221}

X(44131) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(235)

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

X(44131) lies on these lines: {4, 69}, {6, 37778}, {24, 20477}, {25, 30737}, {53, 338}, {74, 8795}, {95, 3520}, {185, 19166}, {235, 41005}, {273, 34387}, {297, 26156}, {318, 34388}, {324, 3580}, {339, 1596}, {393, 41760}, {427, 11197}, {458, 26206}, {459, 2052}, {800, 41678}, {1249, 21447}, {1593, 26166}, {1885, 41008}, {2322, 26592}, {2373, 20626}, {3089, 6527}, {3542, 40680}, {9308, 40318}, {15066, 40684}, {27377, 40316}

X(44131) = polar conjugate of X(41890)
X(44131) = isotomic conjugate of isogonal conjugate of X(235)
X(44131) = polar conjugate of isogonal conjugate of X(13567)
X(44131) = X(264)-Ceva conjugate of X(41005)
X(44131) = X(i)-isoconjugate of X(j) for these (i,j): {48, 41890}, {184, 775}, {801, 9247}, {821, 23606}
X(44131) = cevapoint of X(235) and X(13567)
X(44131) = crosssum of X(184) and X(23606)
X(44131) = barycentric product X(i)*X(j) for these {i,j}: {76, 235}, {92, 17858}, {185, 18027}, {264, 13567}, {324, 19166}, {774, 1969}, {800, 18022}, {850, 41678}, {1502, 44079}, {2052, 41005}
X(44131) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 41890}, {92, 775}, {185, 577}, {235, 6}, {264, 801}, {774, 48}, {800, 184}, {820, 4100}, {1624, 32661}, {2052, 1105}, {2883, 15905}, {6508, 255}, {6509, 1092}, {6521, 821}, {13567, 3}, {14091, 1660}, {16035, 14533}, {17773, 22089}, {17858, 63}, {18022, 40830}, {18603, 1437}, {19166, 97}, {19180, 19210}, {36424, 44079}, {41005, 394}, {41580, 10316}, {41602, 23115}, {41603, 14961}, {41678, 110}, {44079, 32}
X(44131) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 264, 32000}, {264, 311, 1235}, {264, 317, 3260}, {21447, 40814, 1249}

X(44132) = ISOTOMIC CONJUGATE X(248)

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

X(44132) lies on these lines: {4, 69}, {276, 308}, {290, 41074}, {297, 6393}, {305, 2052}, {324, 8024}, {325, 2967}, {339, 15980}, {384, 26166}, {393, 3926}, {436, 37894}, {648, 41363}, {687, 40832}, {1297, 5999}, {1502, 18027}, {1629, 16276}, {1972, 18024}, {1975, 33971}, {1990, 35549}, {2211, 39931}, {2799, 3267}, {2987, 16081}, {3266, 6331}, {4121, 6747}, {4176, 6820}, {4590, 15014}, {5025, 40822}, {7763, 17907}, {7769, 34349}, {7799, 37765}, {10002, 32816}, {11547, 34254}, {12215, 41204}, {14165, 37804}, {16264, 32819}, {30506, 33798}, {31636, 34137}, {32830, 43981}, {39998, 40684}, {43710, 43714}

X(44132) = isogonal conjugate of X(14600)
X(44132) = isotomic conjugate of X(248)
X(44132) = polar conjugate of X(1976)
X(44132) = isotomic conjugate of isogonal conjugate of X(297)
X(44132) = polar conjugate of isogonal conjugate of X(325)
X(44132) = X(41174)-Ceva conjugate of X(6331)
X(44132) = X(i)-cross conjugate of X(j) for these (i,j): {2799, 877}, {35088, 850}
X(44132) = cevapoint of X(297) and X(325)
X(44132) = crosspoint of X(6331) and X(41174)
X(44132) = trilinear pole of line {6333, 16230}
X(44132) = crossdifference of every pair of points on line {3049, 14575}
X(44132) = pole wrt polar circle of trilinear polar of X(1976) (line X(32)X(512), or PU(39))
X(44132) = perspector of circumconic through the polar conjugates of PU(39)
X(44132) = trilinear product X(i)*X(j) for these {i,j}: {75, 297}, {76, 240}, {92, 325}, {158, 6393}, {232, 561}, {264, 1959}, {304, 6530}, {511, 1969}, {811, 2799}, {823, 6333}, {877, 1577}, {1235, 3405}, {1928, 2211}
X(44132) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14600}, {31, 248}, {32, 293}, {48, 1976}, {63, 14601}, {98, 9247}, {163, 878}, {184, 1910}, {287, 560}, {336, 1501}, {798, 43754}, {810, 2715}, {822, 32696}, {1821, 14575}, {1924, 17932}, {1933, 15391}, {1973, 17974}, {2422, 4575}, {3049, 36084}, {3404, 10547}, {14585, 36120}, {36104, 39201}
X(44132) = barycentric product X(i)*X(j) for these {i,j}: {75, 40703}, {76, 297}, {232, 1502}, {240, 561}, {264, 325}, {286, 42703}, {305, 6530}, {511, 18022}, {670, 16230}, {850, 877}, {1235, 20022}, {1959, 1969}, {2052, 6393}, {2211, 40362}, {2396, 14618}, {2799, 6331}, {2967, 18024}, {4609, 17994}, {6333, 6528}, {16081, 32458}, {18027, 36212}, {18896, 39931}, {34384, 39569}, {34854, 40050}, {35088, 41174}
X(44132) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 248}, {4, 1976}, {6, 14600}, {25, 14601}, {69, 17974}, {75, 293}, {76, 287}, {92, 1910}, {99, 43754}, {107, 32696}, {132, 42671}, {232, 32}, {237, 14575}, {240, 31}, {264, 98}, {297, 6}, {305, 6394}, {325, 3}, {340, 14355}, {511, 184}, {523, 878}, {561, 336}, {648, 2715}, {670, 17932}, {684, 39201}, {811, 36084}, {823, 36104}, {850, 879}, {868, 20975}, {877, 110}, {1235, 20021}, {1755, 9247}, {1916, 15391}, {1959, 48}, {1969, 1821}, {2052, 6531}, {2211, 1501}, {2396, 4558}, {2421, 32661}, {2450, 40947}, {2501, 2422}, {2799, 647}, {2967, 237}, {2973, 43920}, {3260, 35912}, {3289, 14585}, {3569, 3049}, {4230, 1576}, {5968, 14908}, {6331, 2966}, {6333, 520}, {6393, 394}, {6528, 685}, {6530, 25}, {7017, 15628}, {8754, 15630}, {8840, 43722}, {9155, 23200}, {9418, 40373}, {14618, 2395}, {15352, 20031}, {15595, 8779}, {16081, 41932}, {16089, 32545}, {16230, 512}, {17875, 8766}, {17907, 11610}, {17984, 40820}, {17994, 669}, {18022, 290}, {18027, 16081}, {20022, 1176}, {20883, 3404}, {22456, 41173}, {30737, 34156}, {32458, 36212}, {33752, 42659}, {34765, 35909}, {34854, 1974}, {35088, 41172}, {35140, 15407}, {35142, 2065}, {35908, 40352}, {35910, 18877}, {36212, 577}, {36426, 232}, {36790, 3289}, {39569, 51}, {39931, 1691}, {40703, 1}, {40810, 17970}, {40887, 1968}, {41167, 39469}, {42703, 72}, {42717, 906}, {42751, 23220}
X(44132) = {X(324),X(8024)}-harmonic conjugate of X(18022)
X(44132) = {P,U}-harmonic conjugate of X(76), where P, U are the polar conjugates of X(5638) and X(5639)

X(44133) = ISOTOMIC CONJUGATE X(3426)

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

X(44133) lies on these lines: {4, 69}, {95, 16035}, {99, 43660}, {183, 11284}, {290, 35179}, {305, 670}, {322, 3264}, {325, 11059}, {338, 15533}, {394, 648}, {524, 40814}, {1273, 7871}, {3262, 33933}, {3266, 37668}, {6148, 11057}, {6527, 30698}, {8681, 37190}, {9307, 20819}, {9464, 10513}, {15066, 34289}, {15589, 26235}, {16284, 35516}, {32831, 40680}, {34386, 43752}, {37671, 40022}, {40341, 41760}, {40996, 41009}

X(44133) = isotomic conjugate of X(3426)
X(44133) = isotomic conjugate of isogonal conjugate of X(376)
X(44133) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3426}, {560, 36889}, {810, 9064}
X(44133) = cevapoint of X(15066) and X(21312)
X(44133) = barycentric product X(i)*X(j) for these {i,j}: {76, 376}, {305, 40138}, {670, 9209}, {1502, 26864}, {6331, 9007}, {32833, 39263}
X(44133) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3426}, {76, 36889}, {376, 6}, {648, 9064}, {9007, 647}, {9209, 512}, {26864, 32}, {36427, 26864}, {39263, 34288}, {40138, 25}
X(44133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 315, 340}, {69, 3260, 76}, {69, 14615, 264}, {76, 3260, 264}, {76, 14615, 3260}

X(44134) = ISOTOMIC CONJUGATE X(4846)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) : :
X(44134) = 3 X[458] - 2 X[6749]

X(44134) lies on these lines: {2, 648}, {4, 69}, {5, 40996}, {25, 37671}, {53, 3631}, {95, 253}, {99, 35485}, {140, 41005}, {141, 1990}, {183, 468}, {193, 36794}, {231, 41770}, {232, 16990}, {273, 319}, {290, 5486}, {297, 599}, {298, 471}, {299, 470}, {318, 320}, {325, 5094}, {376, 36889}, {378, 32833}, {393, 3620}, {427, 7788}, {458, 524}, {550, 20477}, {670, 18022}, {1007, 37803}, {1078, 35486}, {1093, 11487}, {1119, 42696}, {1249, 3619}, {1272, 14558}, {1273, 37119}, {1585, 32809}, {1586, 32808}, {1656, 40995}, {1897, 4389}, {1899, 2992}, {2322, 17234}, {3087, 20080}, {3096, 41361}, {3516, 3964}, {3522, 6527}, {3535, 32811}, {3536, 32810}, {3541, 7796}, {3618, 5702}, {3630, 6748}, {3642, 6111}, {3643, 6110}, {3785, 37460}, {4232, 15589}, {5081, 17360}, {5117, 38294}, {6148, 35481}, {7046, 42697}, {7282, 17361}, {7408, 40002}, {7750, 37196}, {7758, 37337}, {7763, 37118}, {7767, 37458}, {7811, 18533}, {7868, 16318}, {7879, 27376}, {7896, 27371}, {8795, 42021}, {9723, 35477}, {10295, 14907}, {11008, 40065}, {11109, 17378}, {11547, 37636}, {13577, 18816}, {15454, 40423}, {16063, 30737}, {16263, 41465}, {16264, 33878}, {17271, 17555}, {17297, 37448}, {17346, 26003}, {18384, 34405}, {21356, 37765}, {24206, 41371}, {27377, 40341}, {31635, 41359}, {31886, 41374}, {32817, 35483}, {35142, 35179}, {35510, 40410}, {37645, 43530}

X(44134) = isotomic conjugate of X(4846)
X(44134) = anticomplement of X(5158)
X(44134) = polar conjugate of X(34288)
X(44134) = anticomplement of isogonal conjugate of X(43530)
X(44134) = isotomic conjugate of anticomplement of X(4550)
X(44134) = isotomic conjugate of isogonal conjugate of X(378)
X(44134) = polar conjugate of isotomic conjugate of X(32833)
X(44134) = polar conjugate of isogonal conjugate of X(15066)
X(44134) = pole wrt polar circle of trilinear polar of X(34288) (line X(512)X(1637))
X(44134) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {92, 18387}, {3431, 6360}, {16263, 5905}, {22455, 18668}, {43530, 8}
X(44134) = X(i)-cross conjugate of X(j) for these (i,j): {4550, 2}, {5891, 15066}, {15066, 32833}
X(44134) = X(i)-isoconjugate of X(j) for these (i,j): {31, 4846}, {48, 34288}, {647, 36149}, {656, 32738}, {810, 1302}, {2631, 32681}, {9247, 34289}, {9409, 36083}
X(44134) = cevapoint of X(i) and X(j) for these (i,j): {376, 37645}, {378, 15066}
X(44134) = crossdifference of every pair of points on line {3049, 9409}
X(44134) = barycentric product X(i)*X(j) for these {i,j}: {4, 32833}, {76, 378}, {264, 15066}, {276, 5891}, {286, 42704}, {648, 30474}, {1502, 44080}, {5063, 18022}, {6331, 8675}
X(44134) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4846}, {4, 34288}, {112, 32738}, {162, 36149}, {264, 34289}, {378, 6}, {648, 1302}, {1304, 32681}, {4550, 5158}, {5063, 184}, {5891, 216}, {8675, 647}, {10564, 3284}, {11653, 248}, {15066, 3}, {30474, 525}, {32833, 69}, {36429, 44080}, {42660, 3049}, {42704, 72}, {44080, 32}
X(44134) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 340}, {4, 340, 317}, {69, 264, 317}, {69, 3260, 315}, {69, 32000, 264}, {141, 1990, 11331}, {141, 9308, 17907}, {264, 340, 4}, {1235, 3260, 264}, {1990, 11331, 17907}, {2992, 2993, 1899}, {9308, 11331, 1990}, {32001, 32002, 317}

X(44135) = ISOTOMIC CONJUGATE X(3431)

Barycentrics    b^2*c^2*(-a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 4*b^2*c^2 + 2*c^4) : :
Barycentrics    A-power of anti-orthocentroidal circle : :
Barycentrics    A-power of polar circle of A-altimedial triangle : :
Barycentrics    A'-power of circumcircle : : , where A'B'C' = orthocentroidal triangle
Barycentrics    A'-power of nine-point circle : : , where A'B'C' = orthocentroidal triangle
Barycentrics    A'-power of polar circle : : , where A'B'C' = orthocentroidal triangle
Barycentrics    A'-power of polar circle : : , where A'B'C' = reflection triangle

The anti-orthocentroidal circle is here defined as the circumcircle of the anti-orthocentroidal triangle, with center X(399). (Randy Hutson, August 24, 2021)

X(44135) lies on the cubic K504 and these lines: {2, 94}, {4, 69}, {23, 183}, {99, 13530}, {290, 5967}, {308, 14387}, {313, 27558}, {325, 5169}, {327, 18023}, {458, 22151}, {1007, 3266}, {1078, 7556}, {1273, 7752}, {1975, 7527}, {3262, 3701}, {3549, 32838}, {3618, 41760}, {3702, 30596}, {5392, 11427}, {7488, 20477}, {7493, 26235}, {7519, 15589}, {7526, 9723}, {7540, 7767}, {7552, 32832}, {7565, 7773}, {7578, 11004}, {8754, 21243}, {8797, 20563}, {9220, 18375}, {10024, 41005}, {11059, 34803}, {15574, 26284}, {18300, 18387}, {18563, 41008}, {18581, 43086}, {18582, 43085}, {20023, 40074}, {32805, 34391}, {32806, 34392}, {32829, 40697}, {34289, 37643}, {34387, 42697}, {34388, 42696}, {37669, 40684}

X(44135) = isotomic conjugate of X(3431)
X(44135) = anticomplement of X(566)
X(44135) = anticomplement of isogonal conjugate of X(7578)
X(44135) = isotomic conjugate of complement of X(18387)
X(44135) = isotomic conjugate of isogonal conjugate of X(381)
X(44135) = polar conjugate of isogonal conjugate of X(37638)
X(44135) = X(7578)-anticomplementary conjugate of X(8)
X(44135) = cevapoint of X(i) and X(j) for these (i,j): {2, 18387}, {381, 37638}, {10298, 11004}
X(44135) = crosssum of X(18117) and X(20975)
X(44135) = crossdifference of every pair of points on line {3049, 14270}
X(44135) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3431}, {9247, 43530}
X(44135) = trilinear product X(i)*X(j) for these {i,j}: {2, 80067}, {75, 381}, {92, 37638}, {264, 18477}, {561, 34417}, {1494, 18486}, {1928, 34416}, {1969, 5158}, {4993, 14213}, {18487, 33805}
X(44135) = trilinear quotient X(i)/X(j) for these (i,j): (75, 3431), (381, 31), (1969, 43530), (4993, 2148), (5158, 9247), (18477, 184), (18486, 1495), (18487, 9406), (34416, 1917), (34417, 560), (37638, 48)
X(44135) = barycentric product X(i)*X(j) for these {i,j}: {76, 381}, {264, 37638}, {311, 4993}, {1502, 34417}, {1969, 18477}, {3581, 20573}, {5158, 18022}, {18023, 32225}, {18486, 33805}, {34416, 40362}
X(44135) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3431}, {94, 18316}, {264, 43530}, {381, 6}, {1531, 3284}, {2052, 16263}, {3581, 50}, {4550, 5063}, {4993, 54}, {5158, 184}, {16080, 22455}, {18477, 48}, {18484, 18487}, {18486, 2173}, {18487, 1495}, {21970, 3053}, {32225, 187}, {34416, 1501}, {34417, 32}, {36430, 34417}, {37638, 3}
X(44135) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 264, 3260}, {76, 3260, 69}, {264, 311, 69}, {300, 301, 2}, {311, 3260, 76}, {1232, 14615, 69}, {9220, 18375, 34827}

X(44136) = ISOTOMIC CONJUGATE X(11270)

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

X(44136) lies on these lines: {4, 69}, {99, 20480}, {183, 13595}, {193, 338}, {253, 328}, {324, 37669}, {325, 13481}, {339, 32827}, {393, 28408}, {1007, 30737}, {1272, 32837}, {1992, 41760}, {2071, 20477}, {2072, 41005}, {3548, 32839}, {3964, 18354}, {6527, 32835}, {9308, 22151}, {9723, 12084}, {14570, 40896}, {20563, 36889}, {28419, 43981}, {32810, 34391}, {32811, 34392}

X(44136) = isotomic conjugate of X(11270)
X(44136) = isotomic conjugate of isogonal conjugate of X(382)
X(44136) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {12092, 4560}, {16867, 6360}
X(44136) = X(i)-isoconjugate of X(j) for these (i,j): {31, 11270}, {810, 33640}
X(44136) = barycentric product X(i)*X(j) for these {i,j}: {75, 14212}, {76, 382}, {1502, 44082}
X(44136) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11270}, {382, 6}, {648, 33640}, {14212, 1}, {36431, 44082}, {44082, 32}
X(44136) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {264, 3260, 69}, {264, 14615, 311}, {311, 3260, 14615}, {311, 14615, 69}

X(44137) = ISOTOMIC CONJUGATE X(1987)

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

X(44137) lies on the cubic K779 and these lines: {2, 40815}, {4, 69}, {6, 40822}, {147, 30737}, {276, 34850}, {287, 3978}, {290, 3564}, {323, 23962}, {325, 14941}, {394, 18022}, {670, 6393}, {2456, 14382}, {2896, 26166}, {3267, 7799}, {5661, 7769}, {6331, 11064}, {6374, 6389}, {8783, 32458}, {9308, 18027}, {16089, 41204}, {32428, 39682}, {36901, 41724}

X(44137) = isotomic conjugate of X(1987)
X(44137) = isotomic conjugate of isogonal conjugate of X(401)
X(44137) = isotomic conjugate of polar conjugate of X(16089)
X(44137) = X(325)-Ceva conjugate of X(3978)
X(44137) = X(401)-cross conjugate of X(16089)
X(44137) = crossdifference of every pair of points on line {3049, 40981}
X(44137) = X(i)-isoconjugate of X(j) for these (i,j): {31, 1987}, {32, 1956}, {560, 1972}, {1298, 2179}, {1967, 32542}, {1973, 14941}, {3402, 39683}
X(44137) = barycentric product X(i)*X(j) for these {i,j}: {69, 16089}, {76, 401}, {305, 41204}, {561, 1955}, {670, 6130}, {1502, 1971}, {20023, 39682}, {32428, 34384}
X(44137) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1987}, {69, 14941}, {75, 1956}, {76, 1972}, {95, 1298}, {183, 39683}, {325, 40804}, {385, 32542}, {401, 6}, {1955, 31}, {1971, 32}, {2313, 2179}, {6130, 512}, {16089, 4}, {32428, 51}, {32545, 1976}, {39682, 263}, {41204, 25}, {42405, 41210}
X(44137) = {X(8920),X(14615)}-harmonic conjugate of X(315)

X(44138) = ISOTOMIC CONJUGATE X(5504)

Barycentrics    b^2*c^2*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6) : :
Barycentrics    (csc 2A) (1 + cos 2B + cos 2C) : :

The trilinear polar of X(44138) passes through X(6334).

X(44138) lies on these lines: {4, 69}, {23, 30716}, {26, 20477}, {94, 16080}, {95, 14118}, {186, 328}, {290, 685}, {297, 338}, {300, 470}, {301, 471}, {325, 37981}, {339, 11799}, {421, 18020}, {648, 37778}, {850, 6368}, {1272, 37943}, {3003, 16237}, {5392, 11547}, {7527, 26166}, {8795, 43689}, {15761, 41005}, {16077, 40423}, {16089, 41203}, {17907, 41760}

X(44138) = isotomic conjugate of X(5504)
X(44138) = polar conjugate of X(14910)
X(44138) = isotomic conjugate of isogonal conjugate of X(403)
X(44138) = polar conjugate of isogonal conjugate of X(3580)
X(44138) = X(38534)-anticomplementary conjugate of X(6360)
X(44138) = X(16077)-Ceva conjugate of X(850)
X(44138) = X(16221)-cross conjugate of X(14618)
X(44138) = cevapoint of X(i) and X(j) for these (i,j): {403, 3580}, {35235, 41079}
X(44138) = crosspoint of X(264) and X(18817)
X(44138) = crossdifference of every pair of points on line {3049, 14585}
X(44138) = pole wrt polar circle of trilinear polar of X(14910) (line X(184)X(512))
X(44138) = X(i)-isoconjugate of X(j) for these (i,j): {31, 5504}, {48, 14910}, {184, 36053}, {798, 43755}, {810, 10420}, {822, 32708}, {2986, 9247}, {36114, 39201}
X(44138) = barycentric product X(i)*X(j) for these {i,j}: {76, 403}, {264, 3580}, {850, 16237}, {1502, 44084}, {1725, 1969}, {1986, 20573}, {3003, 18022}, {6334, 6528}, {12828, 18023}, {13754, 18027}, {18817, 34834}
X(44138) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 5504}, {4, 14910}, {92, 36053}, {94, 12028}, {99, 43755}, {107, 32708}, {113, 3284}, {264, 2986}, {403, 6}, {648, 10420}, {686, 39201}, {823, 36114}, {850, 15421}, {1725, 48}, {1986, 50}, {2052, 1300}, {3003, 184}, {3580, 3}, {6331, 18878}, {6334, 520}, {6515, 15478}, {6528, 687}, {11557, 9380}, {12824, 10317}, {12827, 14961}, {12828, 187}, {13754, 577}, {14165, 38936}, {14264, 18877}, {14618, 15328}, {14920, 39371}, {15329, 32661}, {16080, 10419}, {16221, 2088}, {16237, 110}, {18020, 18879}, {18022, 40832}, {18609, 1437}, {18817, 40427}, {21731, 3049}, {23290, 35361}, {34834, 22115}, {39985, 32663}, {41512, 32662}, {41665, 14889}, {44084, 32}
X(44138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {264, 340, 3260}, {15164, 15165, 317}

X(44139) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(404)

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

X(44139) lies on these lines: {2, 34283}, {4, 69}, {6, 18144}, {7, 3596}, {9, 20917}, {75, 537}, {77, 4554}, {85, 3718}, {86, 13741}, {141, 3770}, {142, 3975}, {183, 16434}, {190, 18040}, {261, 799}, {274, 4283}, {312, 17296}, {313, 320}, {319, 1269}, {321, 17287}, {341, 5290}, {350, 3879}, {527, 17787}, {572, 18048}, {645, 37659}, {646, 3729}, {894, 4503}, {1078, 1444}, {1107, 25538}, {1230, 32863}, {1240, 20245}, {1264, 6063}, {1654, 20913}, {1746, 14829}, {1909, 4357}, {3264, 7321}, {3662, 3765}, {3664, 6381}, {3688, 17794}, {3758, 18044}, {3875, 24524}, {3945, 18135}, {3948, 17300}, {3963, 6646}, {4001, 19810}, {4043, 17295}, {4110, 4659}, {4358, 17312}, {4360, 39995}, {4363, 30473}, {4377, 17345}, {4410, 17239}, {4441, 32099}, {4648, 30830}, {4690, 20174}, {4710, 32857}, {4869, 28809}, {4967, 25280}, {5232, 34284}, {6173, 30090}, {6376, 10436}, {6385, 33769}, {10401, 31643}, {11353, 28014}, {14548, 18153}, {14828, 18052}, {16709, 17210}, {17117, 25298}, {17178, 31026}, {17277, 18143}, {17279, 30866}, {17283, 18150}, {17288, 20891}, {17297, 18137}, {17298, 20923}, {17310, 22016}, {17361, 30596}, {17365, 17790}, {17375, 31060}, {17378, 18145}, {17392, 25660}, {17483, 28654}, {18152, 30941}, {18697, 20955}, {19768, 37492}, {20336, 20924}, {21226, 26149}, {21299, 35892}, {21352, 36856}, {24437, 24688}, {25278, 32087}, {27184, 30710}, {29447, 41681}, {31008, 37632}, {32911, 40013}, {34020, 37678}

X(44139) = isotomic conjugate of isogonal conjugate of X(404)
X(44139) = cevapoint of X(i) and X(j) for these (i,j): {69, 18133}, {32911, 35998}
X(44139) = X(1397)-isoconjugate of X(44040)
X(44139) = barycentric product X(i)*X(j) for these {i,j}: {75, 32939}, {76, 404}, {286, 42705}, {1502, 44085}, {4554, 20293}, {4623, 21721}
X(44139) = barycentric quotient X(i)/X(j) for these {i,j}: {312, 44040}, {404, 6}, {6516, 40518}, {20293, 650}, {21721, 4705}, {32939, 1}, {42705, 72}, {44085, 32}
X(44139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 76, 314}, {69, 21287, 7768}, {86, 18133, 18140}, {319, 1269, 17143}, {1232, 1234, 76}, {3729, 17786, 646}, {3761, 17272, 75}

X(44140) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(405)

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

X(44140) lies on these lines: {2, 37}, {4, 69}, {7, 349}, {8, 313}, {9, 4044}, {28, 19838}, {86, 964}, {141, 33736}, {183, 4220}, {193, 3770}, {273, 1231}, {274, 37037}, {310, 30962}, {319, 5016}, {322, 4673}, {386, 3875}, {387, 4385}, {579, 3729}, {941, 27042}, {966, 3948}, {1089, 1714}, {1228, 2478}, {1230, 5739}, {1240, 1246}, {1265, 3596}, {1284, 5695}, {1441, 3485}, {1444, 1975}, {1654, 31060}, {1766, 16574}, {1909, 5716}, {1930, 24204}, {1992, 34283}, {2303, 19281}, {3216, 17151}, {3262, 5761}, {3264, 32087}, {3620, 18144}, {3701, 38057}, {3760, 4357}, {3761, 3879}, {3765, 5839}, {3945, 30939}, {3963, 17314}, {4066, 20083}, {4360, 19767}, {4373, 40010}, {4377, 17299}, {4431, 4494}, {4648, 20913}, {4869, 18143}, {4967, 32104}, {5019, 24271}, {5051, 5224}, {5165, 17351}, {5232, 18133}, {5317, 9308}, {5747, 20236}, {6381, 17270}, {7253, 20948}, {7283, 13726}, {8062, 17893}, {8822, 14829}, {10436, 20888}, {16752, 25504}, {17144, 20018}, {17220, 17751}, {17277, 28809}, {17861, 18697}, {17907, 18685}, {18040, 29616}, {18044, 29611}, {18065, 29594}, {19766, 39731}, {20477, 36029}, {26045, 27269}, {26243, 36744}, {30882, 36022}

X(44140) = isotomic conjugate of isogonal conjugate of X(405)
X(44140) = anticomplement of X(4261)
X(44140) = X(839)-anticomplementary conjugate of X(20295)
X(44140) = X(5295)-cross conjugate of X(5271)
X(44140) = cevapoint of X(7520) and X(37685)
X(44140) = crossdifference of every pair of points on line {667, 3049}
X(44140) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2215}, {604, 2335}, {649, 36080}, {810, 36077}
X(44140) = barycentric product X(i)*X(j) for these {i,j}: {75, 5271}, {76, 405}, {274, 5295}, {286, 42706}, {304, 39585}, {668, 23882}, {1451, 28659}, {1502, 5320}, {3596, 37543}, {3952, 15417}, {14549, 18152}
X(44140) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2215}, {8, 2335}, {100, 36080}, {405, 6}, {648, 36077}, {1451, 604}, {1882, 1880}, {5271, 1}, {5295, 37}, {5320, 32}, {14549, 2350}, {15417, 7192}, {23882, 513}, {37543, 56}, {39585, 19}, {42706, 72}
X(44140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 69, 21287}, {75, 312, 20336}, {75, 350, 17321}, {75, 18147, 2}, {76, 264, 1234}, {76, 314, 69}, {312, 19792, 2}, {321, 17863, 75}, {3596, 17143, 42696}, {3760, 10447, 4357}

X(44141) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(426)

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

X(44141) lies on these lines: {4, 69}, {68, 339}, {394, 441}, {525, 15075}, {1092, 3926}, {1181, 41005}, {1568, 32816}, {1899, 41009}, {3289, 7758}, {3964, 16391}, {6389, 39643}, {6390, 35602}, {6527, 34781}, {7800, 22416}, {8779, 28696}, {10984, 40680}, {12164, 40995}, {14216, 30737}, {15595, 41361}, {32817, 34403}, {32818, 37669}

X(44141) = isotomic conjugate of isogonal conjugate of X(426)
X(44141) = isotomic conjugate of polar conjugate of X(6389)
X(44141) = X(69)-Ceva conjugate of X(1899)
X(44141) = X(426)-cross conjugate of X(6389)
X(44141) = crosspoint of X(69) and X(4176)
X(44141) = barycentric product X(i)*X(j) for these {i,j}: {69, 6389}, {76, 426}, {305, 39643}, {394, 41009}, {1102, 17871}, {1632, 4143}, {1899, 3926}, {3767, 4176}, {3964, 41760}
X(44141) = barycentric quotient X(i)/X(j) for these {i,j}: {426, 6}, {1632, 6529}, {1899, 393}, {2083, 1096}, {3767, 6524}, {3926, 34405}, {4176, 42407}, {6389, 4}, {6751, 3199}, {17871, 6520}, {39643, 25}, {40947, 2207}, {41009, 2052}, {41760, 1093}, {41762, 36434}

X(44142) = ISOTOMIC CONJUGATE OF X(41435)

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

X(44142) lies on these lines: {2, 40674}, {4, 69}, {5, 30737}, {25, 26235}, {30, 26166}, {83, 8744}, {94, 41253}, {99, 14865}, {183, 10594}, {186, 43459}, {232, 6683}, {276, 35474}, {290, 1173}, {297, 14129}, {305, 32823}, {324, 458}, {325, 15559}, {338, 7745}, {339, 546}, {427, 3266}, {428, 7767}, {472, 41000}, {473, 41001}, {1078, 3518}, {1870, 25303}, {1883, 16747}, {1907, 3933}, {1975, 35502}, {2052, 18841}, {3088, 32831}, {3091, 41009}, {3199, 31239}, {3541, 32829}, {3542, 32838}, {3785, 37122}, {4911, 34387}, {5015, 34388}, {5064, 7776}, {5395, 41370}, {5523, 14770}, {7141, 40717}, {7378, 9464}, {7509, 20477}, {7576, 7750}, {7714, 40022}, {7762, 41628}, {7773, 28706}, {7780, 10985}, {7782, 35475}, {7793, 10986}, {7802, 18559}, {7841, 26214}, {7859, 37765}, {8370, 26164}, {8753, 18023}, {8889, 11059}, {11361, 26179}, {14387, 18027}, {15355, 28407}, {27376, 37778}, {32450, 41676}, {37337, 39575}, {41361, 43981}

X(44142) = isotomic conjugate of X(41435)
X(44142) = polar conjugate of X(3108)
X(44142) = isotomic conjugate of isogonal conjugate of X(428)
X(44142) = polar conjugate of isotomic conjugate of X(39998)
X(44142) = polar conjugate of isogonal conjugate of X(3589)
X(44142) = pole wrt polar circle of trilinear polar of X(3108) (line X(523)X(2076))
X(44142) = X(3589)-cross conjugate of X(39998)
X(44142) = cevapoint of X(428) and X(3589)
X(44142) = crosssum of X(184) and X(20775)
X(44142) = X(i)-isoconjugate of X(j) for these (i,j): {31, 41435}, {48, 3108}, {810, 7953}, {9247, 10159}
X(44142) = barycentric product X(i)*X(j) for these {i,j}: {4, 39998}, {76, 428}, {264, 3589}, {331, 4030}, {1502, 44091}, {1969, 17469}, {2052, 7767}, {5007, 18022}, {6331, 7927}, {7017, 7198}, {10330, 14618}, {16707, 41013}, {18027, 22352}, {18062, 24006}, {32085, 42554}
X(44142) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 41435}, {4, 3108}, {264, 10159}, {428, 6}, {648, 7953}, {3589, 3}, {4030, 219}, {5007, 184}, {6292, 3917}, {6331, 35137}, {7198, 222}, {7767, 394}, {7927, 647}, {8664, 3049}, {10330, 4558}, {11205, 20775}, {14618, 31065}, {16707, 1444}, {17200, 1790}, {17457, 4020}, {17469, 48}, {18062, 4592}, {21802, 228}, {22352, 577}, {28666, 8041}, {39998, 69}, {42554, 3933}, {44091, 32}
X(44142) = {X(4),X(264)}-harmonic conjugate of X(1235)

X(44143) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(430)

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

X(44143) lies on these lines: {4, 69}, {25, 26266}, {318, 860}, {324, 17555}, {338, 1834}, {339, 21665}, {387, 41760}, {407, 38462}, {429, 3695}, {430, 1230}, {451, 31623}, {942, 34387}, {1577, 23104}, {1826, 21070}, {1893, 5295}, {6998, 30737}, {8747, 37778}, {11109, 40684}, {16589, 17911}, {17905, 27040}, {36672, 41009}

X(44143) = isotomic conjugate of isogonal conjugate of X(430)
X(44143) = polar conjugate of X(1171)
X(44143) = polar conjugate of isotomic conjugate of X(1230)
X(44143) = polar conjugate of isogonal conjugate of X(1213)
X(44143) = pole wrt polar circle of trilinear polar of X(1171) (line X(512)X(1326))
X(44143) = X(1213)-cross conjugate of X(1230)
X(44143) = X(i)-isoconjugate of X(j) for these (i,j): {48, 1171}, {184, 40438}, {810, 6578}, {1126, 1437}, {1333, 1796}, {1790, 28615}, {4629, 22383}, {9247, 32014}
X(44143) = cevapoint of X(430) and X(1213)
X(44143) = barycentric product X(i)*X(j) for these {i,j}: {4, 1230}, {76, 430}, {92, 4647}, {264, 1213}, {313, 1839}, {331, 4046}, {1269, 1826}, {1962, 1969}, {2052, 41014}, {2355, 27801}, {3649, 7017}, {3702, 40149}, {4359, 41013}, {4427, 14618}, {6331, 6367}, {6335, 30591}, {7141, 8025}, {18022, 20970}, {18027, 22080}, {28654, 31900}
X(44143) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1171}, {10, 1796}, {92, 40438}, {264, 32014}, {430, 6}, {648, 6578}, {1100, 1437}, {1125, 1790}, {1213, 3}, {1230, 69}, {1269, 17206}, {1824, 28615}, {1826, 1126}, {1839, 58}, {1897, 4629}, {1962, 48}, {2355, 1333}, {3649, 222}, {3683, 2193}, {3686, 283}, {3702, 1812}, {3916, 18604}, {3958, 255}, {4046, 219}, {4115, 1331}, {4359, 1444}, {4427, 4558}, {4647, 63}, {4976, 23189}, {4977, 7254}, {4983, 22383}, {4988, 1459}, {4990, 23090}, {4992, 23092}, {6335, 4596}, {6367, 647}, {7141, 6539}, {8013, 71}, {8040, 22054}, {8663, 3049}, {14618, 4608}, {20970, 184}, {21816, 228}, {22080, 577}, {30591, 905}, {31900, 593}, {35327, 32661}, {35342, 4575}, {41013, 1255}, {41014, 394}

X(44144) = ISOTOMIC CONJUGATE OF X(43718)

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

X(44144) lies on these lines: {2, 6331}, {4, 69}, {25, 42394}, {183, 33971}, {276, 1502}, {290, 6776}, {308, 393}, {324, 39998}, {327, 40330}, {339, 37348}, {458, 20023}, {670, 10008}, {1629, 1799}, {2052, 37187}, {2211, 9308}, {3087, 33769}, {6528, 10002}, {7750, 16264}, {7763, 37125}, {7791, 26166}, {8024, 40684}, {11056, 14165}, {12215, 37124}, {16089, 34229}, {17907, 32832}, {18911, 36901}, {28706, 40073}, {30737, 37182}, {32834, 43981}

X(44144) = isotomic conjugate of X(43718)
X(44144) = polar conjugate of X(263)
X(44144) = isotomic conjugate of isogonal conjugate of X(458)
X(44144) = polar conjugate of isotomic conjugate of X(20023)
X(44144) = polar conjugate of isogonal conjugate of X(183)
X(44144) = X(183)-cross conjugate of X(20023)
X(44144) = cevapoint of X(183) and X(458)
X(44144) = X(i)-isoconjugate of X(j) for these (i,j): {3, 3402}, {31, 43718}, {48, 263}, {184, 2186}, {262, 9247}, {560, 42313}, {810, 26714}, {4020, 42288}, {36132, 39469}
X(44144) = barycentric product X(i)*X(j) for these {i,j}: {4, 20023}, {76, 458}, {92, 3403}, {182, 18022}, {183, 264}, {286, 42711}, {305, 33971}, {1502, 10311}, {6331, 23878}, {8842, 17984}, {34384, 39530}
X(44144) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43718}, {4, 263}, {19, 3402}, {76, 42313}, {92, 2186}, {182, 184}, {183, 3}, {264, 262}, {276, 42300}, {458, 6}, {648, 26714}, {685, 32716}, {3288, 3049}, {3403, 63}, {8842, 36214}, {10311, 32}, {14096, 20775}, {14994, 3917}, {16089, 39682}, {18022, 327}, {20023, 69}, {22456, 6037}, {23878, 647}, {32085, 42288}, {33971, 25}, {34396, 14575}, {39530, 51}, {42711, 72}
X(44144) = {X(264),X(17984)}-harmonic conjugate of X(4)

X(44145) = ISOTOMIC CONJUGATE OF X(43705)

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

X(44145) lies on these lines: {4, 69}, {5, 23635}, {24, 157}, {25, 324}, {68, 41757}, {93, 32085}, {94, 35265}, {98, 39644}, {107, 37777}, {110, 421}, {186, 30716}, {237, 32428}, {338, 1503}, {393, 847}, {403, 523}, {419, 685}, {427, 40684}, {458, 39588}, {460, 3564}, {468, 2970}, {935, 1300}, {1093, 6622}, {1513, 30737}, {1594, 3001}, {1692, 14265}, {2052, 6353}, {2974, 10011}, {3089, 43981}, {3563, 22456}, {5201, 10594}, {6240, 16264}, {6344, 17983}, {6531, 41363}, {6756, 14978}, {6776, 41760}, {7505, 17907}, {8754, 39569}, {10608, 30549}, {14767, 37121}, {14912, 40814}, {15466, 38282}, {16868, 18114}, {34334, 37984}, {35360, 44084}, {37114, 42329}, {37446, 40822}, {37765, 37943}

X(44145) = isogonal conjugate of X(42065)
X(44145) = isotomic conjugate of X(43705)
X(44145) = polar conjugate of X(2987)
X(44145) = polar-circle-inverse of X(31848)
X(44145) = pole wrt polar circle of trilinear polar of X(2987) (line X(3)X(512))
X(44145) = isotomic conjugate of isogonal conjugate of X(460)
X(44145) = polar conjugate of isogonal conjugate of X(230)
X(44145) = X(22456)-Ceva conjugate of X(14618)
X(44145) = X(114)-cross conjugate of X(4)
X(44145) = cevapoint of X(230) and X(460)
X(44145) = crosspoint of X(264) and X(16081)
X(44145) = crosssum of X(184) and X(3289)
X(44145) = crossdifference of every pair of points on line {577, 3049}
X(44145) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42065}, {3, 36051}, {31, 43705}, {48, 2987}, {63, 32654}, {184, 8773}, {255, 3563}, {293, 34157}, {810, 10425}, {822, 32697}, {4575, 35364}, {8781, 9247}, {36105, 39201}
X(44145) = barycentric product X(i)*X(j) for these {i,j}: {76, 460}, {92, 1733}, {114, 16081}, {230, 264}, {297, 14265}, {1502, 44099}, {1692, 18022}, {1969, 8772}, {2052, 3564}, {4226, 14618}, {10011, 42298}
X(44145) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 43705}, {4, 2987}, {6, 42065}, {19, 36051}, {25, 32654}, {92, 8773}, {107, 32697}, {114, 36212}, {230, 3}, {232, 34157}, {264, 8781}, {393, 3563}, {460, 6}, {648, 10425}, {823, 36105}, {1692, 184}, {1733, 63}, {2052, 35142}, {2501, 35364}, {3564, 394}, {4226, 4558}, {5477, 3292}, {6531, 2065}, {8772, 48}, {14265, 287}, {16081, 40428}, {36875, 14919}, {42663, 3049}, {44099, 32}
X(44145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3186, 6403}, {264, 43976, 4}, {419, 41204, 19128}, {847, 3542, 13450}, {1352, 41762, 4}, {1843, 39530, 4}, {6248, 40325, 4}

X(44146) = ISOTOMIC CONJUGATE OF X(895)

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

X(44146) lies on these lines: {2, 14961}, {4, 69}, {5, 26166}, {20, 41009}, {23, 935}, {24, 1975}, {25, 8024}, {27, 1230}, {28, 1228}, {30, 339}, {33, 3761}, {34, 3760}, {74, 290}, {95, 7550}, {99, 186}, {112, 385}, {148, 40889}, {183, 378}, {187, 4235}, {193, 41370}, {194, 39575}, {232, 538}, {235, 3933}, {242, 38457}, {274, 451}, {276, 4994}, {287, 13509}, {297, 525}, {305, 6353}, {308, 32581}, {310, 4213}, {323, 41253}, {325, 403}, {350, 1870}, {384, 10312}, {406, 34284}, {419, 18020}, {420, 3978}, {427, 39998}, {429, 16747}, {458, 15066}, {460, 14052}, {468, 3266}, {470, 41000}, {471, 41001}, {475, 18135}, {524, 37778}, {648, 8744}, {732, 2211}, {892, 8753}, {1078, 3520}, {1172, 3770}, {1861, 6381}, {1885, 7767}, {1909, 6198}, {1968, 7751}, {2052, 5485}, {2967, 32515}, {2973, 40717}, {2996, 43678}, {3088, 32834}, {3089, 32830}, {3144, 28660}, {3147, 6337}, {3263, 37989}, {3541, 32828}, {3542, 3926}, {3734, 10311}, {3934, 37125}, {3948, 15149}, {4212, 18152}, {4232, 9464}, {5025, 26179}, {5081, 34387}, {5094, 26235}, {5641, 6528}, {5971, 37962}, {6240, 32819}, {6344, 35139}, {6392, 41361}, {6527, 34621}, {6622, 32818}, {6623, 37668}, {6656, 26156}, {6664, 27376}, {7282, 34388}, {7500, 18018}, {7505, 7763}, {7750, 18560}, {7752, 16868}, {7754, 8743}, {7762, 40316}, {7769, 14940}, {7770, 14965}, {7771, 35473}, {7773, 35488}, {7776, 37197}, {7782, 21844}, {7799, 37943}, {8267, 40938}, {8370, 27377}, {8889, 40022}, {9466, 33843}, {10313, 15013}, {10317, 40856}, {11054, 37765}, {12082, 20477}, {12215, 19128}, {12243, 16089}, {14711, 33842}, {14712, 40890}, {14907, 35481}, {17555, 26541}, {18533, 32815}, {20880, 25987}, {23115, 26226}, {26592, 37448}, {28809, 37382}, {31276, 37337}, {32225, 36890}, {32269, 36793}, {32832, 37119}, {34664, 41008}, {35940, 37667}, {37118, 37688}, {37643, 40814}, {37912, 44089}, {38664, 41377}, {40894, 42811}, {40895, 42812}, {41760, 43448}

X(44146) = reflection of X(i) in X(j) for these {i,j}: {30737, 339}, {41676, 232}
X(44146) = isogonal conjugate of X(14908)
X(44146) = isotomic conjugate of X(895)
X(44146) = anticomplement of X(14961)
X(44146) = polar conjugate of X(111)
X(44146) = isogonal conjugate of anticomplement of X(34517)
X(44146) = isogonal conjugate of complement of X(34518)
X(44146) = isotomic conjugate of anticomplement of X(5181)
X(44146) = isotomic conjugate of isogonal conjugate of X(468)
X(44146) = isotomic conjugate of polar conjugate of X(37778)
X(44146) = polar conjugate of isotomic conjugate of X(3266)
X(44146) = polar conjugate of isogonal conjugate of X(524)
X(44146) = pole wrt polar circle of trilinear polar of X(111) (line X(6)X(512))
X(44146) = {P,U}-harmonic conjugate of X(264), where P, U are the polar conjugates of X(5638) and X(5639)
X(44146) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {92, 2892}, {158, 34163}, {1177, 6360}, {2373, 4329}, {10423, 4560}, {36095, 523}, {37220, 1370}
X(44146) = X(264)-Ceva conjugate of X(34336)
X(44146) = X(i)-cross conjugate of X(j) for these (i,j): {468, 37778}, {524, 3266}, {690, 4235}, {5181, 2}, {5642, 43084}, {34336, 264}, {41586, 524}
X(44146) = cevapoint of X(i) and X(j) for these (i,j): {23, 37784}, {468, 524}, {2393, 3291}
X(44146) = crosssum of X(i) and X(j) for these (i,j): {184, 23200}, {20975, 42665}
X(44146) = trilinear pole of line {126, 1560}
X(44146) = crossdifference of every pair of points on line {184, 3049}
X(44146) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14908}, {3, 923}, {6, 36060}, {31, 895}, {48, 111}, {63, 32740}, {163, 10097}, {184, 897}, {212, 7316}, {255, 8753}, {304, 19626}, {560, 30786}, {577, 36128}, {603, 5547}, {647, 36142}, {656, 32729}, {671, 9247}, {691, 810}, {922, 15398}, {3049, 36085}, {4575, 9178}, {6091, 38252}, {23894, 32661}, {34055, 41272}
X(44146) = barycentric product X(i)*X(j) for these {i,j}: {4, 3266}, {69, 37778}, {76, 468}, {92, 14210}, {187, 18022}, {264, 524}, {276, 41586}, {286, 42713}, {331, 3712}, {340, 43084}, {648, 35522}, {670, 14273}, {671, 34336}, {690, 6331}, {850, 4235}, {896, 1969}, {1502, 44102}, {2052, 6390}, {3292, 18027}, {5095, 18023}, {5468, 14618}, {6528, 14417}, {7017, 7181}, {10604, 24855}, {12828, 40832}, {16741, 41013}, {17924, 42721}, {17983, 36792}, {24006, 24039}
X(44146) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36060}, {2, 895}, {4, 111}, {6, 14908}, {19, 923}, {25, 32740}, {76, 30786}, {92, 897}, {112, 32729}, {126, 8681}, {158, 36128}, {162, 36142}, {187, 184}, {193, 6091}, {264, 671}, {278, 7316}, {281, 5547}, {297, 5968}, {351, 3049}, {393, 8753}, {468, 6}, {523, 10097}, {524, 3}, {648, 691}, {671, 15398}, {690, 647}, {811, 36085}, {850, 14977}, {896, 48}, {922, 9247}, {1235, 31125}, {1560, 2393}, {1648, 20975}, {1843, 41272}, {1974, 19626}, {2052, 17983}, {2373, 41511}, {2374, 15387}, {2393, 34158}, {2482, 3292}, {2501, 9178}, {2642, 810}, {3266, 69}, {3292, 577}, {3712, 219}, {3793, 3796}, {4062, 71}, {4235, 110}, {4750, 1459}, {4760, 7193}, {5094, 42007}, {5095, 187}, {5181, 14961}, {5203, 8770}, {5467, 32661}, {5468, 4558}, {5642, 3284}, {5967, 248}, {6331, 892}, {6335, 5380}, {6390, 394}, {6593, 10317}, {6629, 1790}, {7181, 222}, {7267, 3955}, {7664, 22151}, {7813, 3917}, {8753, 41936}, {9155, 3289}, {9717, 18877}, {11053, 22143}, {12828, 3003}, {14210, 63}, {14273, 512}, {14417, 520}, {14419, 22383}, {14432, 652}, {14559, 32662}, {14567, 14575}, {14618, 5466}, {15471, 1384}, {16080, 9139}, {16081, 9154}, {16230, 8430}, {16702, 1437}, {16741, 1444}, {17925, 43926}, {17983, 10630}, {18022, 18023}, {18311, 9517}, {18872, 17970}, {19577, 8869}, {21839, 228}, {23200, 14585}, {23287, 30491}, {23889, 4575}, {24006, 23894}, {24039, 4592}, {24855, 10602}, {30247, 35188}, {30737, 36894}, {31013, 1796}, {31068, 41435}, {32225, 5158}, {32459, 3167}, {34163, 19330}, {34336, 524}, {35282, 8779}, {35522, 525}, {36306, 9206}, {36309, 9207}, {36792, 6390}, {36890, 14919}, {37765, 14246}, {37778, 4}, {37784, 39169}, {37855, 1995}, {39689, 23200}, {41586, 216}, {41616, 41336}, {41676, 36827}, {42713, 72}, {42721, 1332}, {42760, 8677}, {43084, 265}, {44102, 32}
X(44146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 76, 1235}, {76, 316, 1236}, {76, 11185, 311}, {316, 1236, 3260}, {317, 11185, 4}, {385, 15014, 112}, {15164, 15165, 4}

X(44147) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(474)

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

X(44147) lies on these lines: {2, 3770}, {4, 69}, {7, 313}, {8, 1269}, {75, 3617}, {85, 20336}, {86, 5192}, {183, 1444}, {319, 4441}, {320, 30596}, {341, 4208}, {344, 20917}, {346, 18040}, {668, 42696}, {894, 41316}, {966, 20913}, {1909, 17321}, {1975, 37328}, {3264, 31995}, {3596, 42697}, {3618, 34283}, {3672, 39995}, {3718, 20925}, {3729, 29716}, {3760, 3879}, {3761, 4357}, {3765, 4000}, {3945, 18147}, {3948, 4648}, {3963, 4419}, {4033, 4461}, {4043, 29616}, {4044, 17296}, {4202, 5224}, {4371, 25298}, {4377, 17276}, {4410, 17303}, {4869, 18137}, {5564, 25278}, {5749, 18044}, {6381, 10436}, {6385, 20023}, {9965, 19807}, {14552, 19792}, {17014, 29764}, {17234, 28809}, {17270, 20888}, {17300, 31060}, {17355, 18065}, {18142, 37668}, {18152, 30962}, {18749, 28808}, {19825, 40603}, {24599, 29756}, {29484, 37681}

X(44147) = isotomic conjugate of isogonal conjugate of X(474)
X(44147) = anticomplement of X(5069)
X(44147) = barycentric product X(i)*X(j) for these {i,j}: {76, 474}, {1502, 44104}
X(44147) = barycentric quotient X(i)/X(j) for these {i,j}: {474, 6}, {44104, 32}
X(44147) = {X(3770),X(18144)}-harmonic conjugate of X(2)

X(44148) = ISOTOMIC CONJUGATE OF X(14483)

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

X(44148) lies on these lines: {4, 69}, {183, 3266}, {290, 42367}, {298, 41001}, {299, 41000}, {325, 26235}, {338, 22165}, {343, 36789}, {1273, 3933}, {3917, 25051}, {6148, 8024}, {7788, 39998}, {9464, 15589}, {11059, 34229}, {17271, 26541}, {17297, 26592}, {17360, 34387}, {17361, 34388}, {21356, 40814}, {28975, 30939}, {32814, 34392}, {32840, 40697}

X(44148) = isotomic conjugate of X(14483)
X(44148) = isotomic conjugate of isogonal conjugate of X(549)
X(44148) = X(31)-isoconjugate of X(14483)
X(44148) = crosssum of X(32) and X(34416)
X(44148) = barycentric product X(i)*X(j) for these {i,j}: {76, 549}, {305, 6749}, {1502, 44109}
X(44148) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14483}, {549, 6}, {6749, 25}, {44109, 32}
X(44148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 76, 3260}, {69, 1232, 311}, {76, 3260, 311}, {1232, 3260, 76}

X(44149) = ISOTOMIC CONJUGATE OF X(3527)

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

X(44149) lies on these lines: {4, 69}, {75, 6735}, {95, 183}, {141, 40814}, {290, 34817}, {322, 1269}, {325, 37439}, {339, 18536}, {343, 15466}, {394, 36794}, {599, 41760}, {1078, 9723}, {1230, 37655}, {1494, 40032}, {1975, 37198}, {4869, 26592}, {5232, 26541}, {7667, 37671}, {7752, 34939}, {7796, 39113}, {8024, 15589}, {11059, 37688}, {11257, 22062}, {15394, 42333}, {15574, 16276}, {20775, 22712}, {21296, 34388}, {30596, 35516}, {32085, 37491}, {32099, 34387}, {32808, 34392}, {32809, 34391}, {32830, 40680}, {32833, 40697}, {37668, 39998}, {41008, 41009}

X(44149) = isotomic conjugate of X(3527)
X(44149) = polar conjugate of X(34818)
X(44149) = isotomic conjugate of isogonal conjugate of X(631)
X(44149) = cevapoint of X(5422) and X(11414)
X(44149) = crosssum of X(32) and X(33578)
X(44149) = pole wrt polar circle of trilinear polar of X(34818) (line X(512)X(30442))
X(44149) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3527}, {48, 34818}, {560, 8797}, {8796, 9247}
X(44149) = barycentric product X(i)*X(j) for these {i,j}: {76, 631}, {305, 3087}, {1502, 11402}, {18022, 36748}
X(44149) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3527}, {4, 34818}, {76, 8797}, {264, 8796}, {631, 6}, {3087, 25}, {6755, 3199}, {11402, 32}, {26907, 217}, {36748, 184}
X(44149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 76, 264}, {69, 311, 14615}, {69, 1232, 76}, {76, 14615, 311}, {183, 3964, 95}, {311, 14615, 264}

X(44150) = ISOTOMIC CONJUGATE OF X(37142)

Barycentrics    b*c*(b + c)*(-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(44150) lies on these lines: {2, 3002}, {4, 69}, {5, 18738}, {10, 307}, {65, 17867}, {72, 349}, {85, 442}, {140, 29477}, {169, 17739}, {274, 25446}, {333, 36019}, {348, 37154}, {379, 1150}, {440, 18750}, {514, 661}, {517, 35517}, {664, 16090}, {668, 16086}, {758, 21207}, {851, 5088}, {860, 23674}, {948, 43059}, {1230, 34255}, {1231, 17864}, {1834, 3673}, {1944, 1948}, {3142, 17181}, {3177, 27021}, {3212, 10974}, {3661, 26605}, {3732, 39690}, {3761, 11679}, {3933, 21596}, {4223, 37670}, {5051, 26563}, {5763, 16284}, {6376, 18749}, {6554, 16589}, {6734, 17866}, {7112, 14505}, {15988, 17499}, {16091, 18026}, {16552, 25002}, {16749, 24883}, {16822, 27966}, {18641, 40702}, {20347, 34387}, {20888, 30101}, {20943, 38298}, {27022, 30694}, {27049, 30695}, {27250, 27267}, {29962, 29981}, {30011, 34847}, {30031, 30034}, {30809, 30828}, {30879, 44081}, {31042, 31060}

X(44150) = anticomplement of X(3002)
X(44150) = isotomic conjugate of X(37142)
X(44150) = isotomic conjugate of isogonal conjugate of X(851)
X(44150) = crosssum of X(31) and X(44112)
X(44150) = crossdifference of every pair of points on line {31, 3049}
X(44150) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2249}, {31, 37142}, {32, 35145}, {284, 1945}, {296, 2299}, {1172, 1949}, {1937, 2194}, {2204, 40843}, {2713, 21761}, {3063, 41206}
X(44150) = barycentric product X(i)*X(j) for these {i,j}: {75, 8680}, {76, 851}, {243, 1231}, {307, 1948}, {321, 5088}, {349, 1936}, {523, 15418}, {561, 42669}, {1430, 40071}, {1441, 1944}, {1446, 7360}, {1502, 44112}, {1981, 14208}, {3267, 23353}, {6331, 9391}, {26884, 27801}
X(44150) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2249}, {2, 37142}, {65, 1945}, {73, 1949}, {75, 35145}, {226, 1937}, {243, 1172}, {307, 40843}, {664, 41206}, {851, 6}, {1214, 296}, {1430, 1474}, {1441, 1952}, {1936, 284}, {1944, 21}, {1948, 29}, {1951, 2194}, {1981, 162}, {2202, 2299}, {5088, 81}, {6518, 283}, {7360, 2287}, {8680, 1}, {9391, 647}, {15146, 2326}, {15418, 99}, {18026, 41207}, {23353, 112}, {26884, 1333}, {35075, 851}, {39035, 2651}, {39036, 2659}, {42669, 31}, {44112, 32}
X(44150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {72, 349, 21403}, {1231, 41013, 17864}

X(44151) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(863)

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

X(44151) lies on these lines: {4, 69}, {7, 40954}, {44, 513}, {73, 1284}, {651, 44112}, {1716, 1745}, {1742, 22369}, {1985, 30945}, {3136, 26892}, {3286, 13738}, {3888, 3948}, {3962, 20718}, {4644, 40952}, {5360, 8680}, {5698, 22076}, {10974, 24695}, {13724, 28350}, {21246, 22412}, {23212, 32462}, {26027, 26041}, {36287, 41333}

X(44151) = isotomic conjugate of isogonal conjugate of X(863)
X(44151) = crossdifference of every pair of points on line {1, 3049}
X(44151) = barycentric product X(76)*X(863)
X(44151) = barycentric quotient X(863)/X(6)

X(44152) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(1003)

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

X(44152) lies on these lines: {2, 670}, {4, 69}, {6, 30736}, {193, 1502}, {308, 3620}, {524, 20023}, {1992, 3978}, {3117, 6379}, {3266, 7774}, {3618, 6374}, {4470, 40087}, {4577, 9544}, {4644, 18891}, {6331, 40138}, {7736, 11059}, {7779, 9464}, {14033, 16084}, {16989, 35524}, {16990, 26235}, {20080, 33769}, {20105, 40907}, {35906, 43187}, {36794, 40405}

X(44152) = isotomic conjugate of isogonal conjugate of X(1003)
X(44152) = crossdifference of every pair of points on line {887, 3049}
X(44152) = barycentric product X(i)*X(j) for these {i,j}: {76, 1003}, {1502, 44116}
X(44152) = barycentric quotient X(i)/X(j) for these {i,j}: {1003, 6}, {44116, 32}

X(44153) = ISOTOMIC CONJUGATE OF X(1244)

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

X(44153) lies on these lines: {4, 69}, {6, 3948}, {10, 24162}, {75, 141}, {193, 31060}, {305, 30962}, {313, 518}, {350, 1965}, {668, 30893}, {942, 40071}, {3263, 18139}, {3596, 4260}, {3618, 30830}, {3765, 16973}, {3912, 20336}, {8024, 30941}, {13728, 39731}, {16085, 32939}, {18052, 37664}, {29841, 30963}, {29983, 41240}

X(44153) = isotomic conjugate of X(1244)
X(44153) = isotomic conjugate of isogonal conjugate of X(1009)
X(44153) = X(31)-isoconjugate of X(1244)
X(44153) = barycentric product X(76)*X(1009)
X(44153) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1244}, {1009, 6}

X(44154) = ISOTOMIC CONJUGATE OF X(1245)

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

X(44154) lies on these lines: {4, 69}, {38, 75}, {86, 313}, {274, 3596}, {332, 1065}, {1230, 17778}, {1269, 17273}, {1920, 20336}, {1978, 37842}, {3761, 10455}, {3765, 27644}, {3770, 28369}, {3948, 26110}, {3963, 40773}, {4377, 16696}, {4476, 21080}, {5224, 34265}, {17321, 31008}, {18147, 18152}, {20891, 30965}, {28604, 28654}, {30596, 30939}

X(44154) = isotomic conjugate of X(1245)
X(44154) = isotomic conjugate of isogonal conjugate of X(1010)
X(44154) = crossdifference of every pair of points on line {1924, 3049}
X(44154) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2281}, {31, 1245}, {42, 1472}, {213, 2221}, {669, 1310}, {810, 32691}, {1036, 1402}, {1924, 37215}, {3049, 36099}
X(44154) = barycentric product X(i)*X(j) for these {i,j}: {76, 1010}, {274, 4385}, {286, 19799}, {310, 2345}, {388, 28660}, {561, 2303}, {612, 6385}, {670, 6590}, {799, 2517}, {1502, 44119}, {2285, 40072}, {2484, 4609}, {4206, 40364}, {4602, 8678}, {5323, 28659}, {6331, 23874}
X(44154) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2281}, {2, 1245}, {81, 1472}, {86, 2221}, {314, 2339}, {333, 1036}, {388, 1400}, {612, 213}, {648, 32691}, {670, 37215}, {799, 1310}, {811, 36099}, {1010, 6}, {1038, 1409}, {2285, 1402}, {2303, 31}, {2345, 42}, {2484, 669}, {2517, 661}, {2522, 810}, {3610, 3690}, {3974, 1334}, {4206, 1973}, {4385, 37}, {5227, 228}, {5286, 40934}, {5323, 604}, {6590, 512}, {7085, 2200}, {7102, 2333}, {7365, 1042}, {7386, 23620}, {8646, 1924}, {8678, 798}, {14594, 4559}, {19799, 72}, {23874, 647}, {28660, 30479}, {31623, 1039}, {44119, 32}
X(44154) = {X(86),X(313)}-harmonic conjugate of X(28660)

X(44155) = ISOTOMIC CONJUGATE OF X(9513)

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

X(44155) is the intersection of the 1st and 2nd isobarycs of the Brocard axis (lines P(11)P(38) and U(11)U(38)). (Randy Hutson, August 24, 2021)

X(44155) lies on these lines: {2, 647}, {4, 69}, {99, 37991}, {110, 23962}, {141, 34359}, {182, 14382}, {183, 7418}, {290, 542}, {325, 36183}, {327, 5641}, {338, 11646}, {339, 6033}, {880, 1502}, {3448, 36901}, {3788, 28407}, {5972, 6331}, {6375, 7746}, {6390, 30736}, {6776, 14265}, {7761, 35923}, {9306, 18020}, {10479, 35044}, {10684, 14966}, {14356, 35139}, {15595, 16081}, {18024, 36213}, {24206, 39058}, {26276, 34098}, {30737, 43460}, {31636, 41255}

X(44155) = midpoint of X(i) and X(j) for these {i,j}: {69, 30226}, {15164, 15165}
X(44155) = reflection of X(34359) in X(141)
X(44155) = isotomic conjugate of X(9513)
X(44155) = anticomplement of X(5661)
X(44155) = antitomic conjugate of X(40866)
X(44155) = isotomic conjugate of isogonal conjugate of X(1316)
X(44155) = crossdifference of every pair of points on line {237, 3049}
X(44155) = X(i)-isoconjugate of X(j) for these (i,j): {31, 9513}, {1967, 40077}, {36084, 43112}
X(44155) = barycentric product X(i)*X(j) for these {i,j}: {76, 1316}, {850, 40866}, {1502, 44127}, {3978, 38947}, {31953, 43187}
X(44155) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9513}, {385, 40077}, {1316, 6}, {3569, 43112}, {31953, 3569}, {38947, 694}, {40866, 110}, {43113, 2715}, {44127, 32}

X(44156) = X(230)X(577)∩X(1092)X(3564)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1971.

X(44156) lies on these lines: {230, 577}, {1092, 3564}, {2790, 40082}, {3546, 19210}

X(44156) = cevapoint of X(i) and X(j) for these (i,j): {3, 30771}, {125, 22089}

X(44157) = X(549)X(1147)∩X(571)X(6749)

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

See Antreas Hatzipolakis and Peter Moses, euclid 1972.

X(44157) lies on these lines: {549, 1147}, {571, 6749}, {5961, 7525}, {14533, 26937}

X(44158) = X(2)X(12163)∩X(3)X(68)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^8 - a^6*b^2 - 5*a^4*b^4 + 5*a^2*b^6 - b^8 - a^6*c^2 + 6*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 4*b^6*c^2 - 5*a^4*c^4 - 5*a^2*b^2*c^4 - 6*b^4*c^4 + 5*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(44158) =3 X[2] + X[12163], 3 X[3] + X[68], 5 X[3] - X[12118], 7 X[3] + X[12429], X[20] + 3 X[14852], 5 X[68] + 3 X[12118], X[68] - 3 X[12359], 7 X[68] - 3 X[12429], 3 X[140] - 2 X[43839], X[155] - 5 X[631], X[156] - 3 X[34477], 3 X[376] + X[12293], 3 X[549] - X[1147], X[2883] - 3 X[10201], 3 X[3167] - 11 X[15720], X[3521] + 3 X[43689], 7 X[3523] + X[11411], 9 X[3524] - X[6193], 7 X[3526] - 3 X[5654], 9 X[5054] - X[12164], X[5504] - 5 X[38728], 3 X[6699] - X[15115], X[6759] - 3 X[34351], X[9820] - 4 X[20191], 3 X[9820] - 4 X[43839], 3 X[10192] - X[32139], X[12084] - 3 X[23328], 4 X[12108] - X[41597], X[12118] + 5 X[12359], 7 X[12118] + 5 X[12429], 7 X[12359] - X[12429], 3 X[14070] + X[14216], X[14790] - 5 X[40686], 7 X[14869] - X[15083], 3 X[15061] - X[23306], 3 X[18324] + X[32140], 3 X[18324] - X[34782], X[18569] - 3 X[23332], 3 X[20191] - X[43839], 3 X[23329] - X[23335]

See Antreas Hatzipolakis and Peter Moses, euclid 1972.

X(44158) lies on these lines: {2, 12163}, {3, 68}, {5, 4550}, {20, 14852}, {26, 6247}, {30, 5449}, {54, 15136}, {125, 12605}, {140, 9729}, {141, 19908}, {155, 631}, {156, 34477}, {185, 7542}, {186, 12134}, {376, 12293}, {378, 41587}, {382, 32269}, {403, 11440}, {428, 18488}, {468, 12162}, {511, 25563}, {539, 10213}, {546, 20193}, {548, 13470}, {549, 1147}, {550, 9927}, {858, 43608}, {912, 3678}, {1154, 20376}, {1192, 18420}, {1204, 15760}, {1216, 6699}, {1368, 20302}, {1503, 1658}, {2070, 16655}, {2777, 15114}, {2883, 10201}, {2888, 37941}, {2931, 7512}, {3089, 11472}, {3147, 18451}, {3167, 15720}, {3448, 38448}, {3520, 3580}, {3521, 10024}, {3523, 11411}, {3524, 6193}, {3526, 5654}, {3530, 3564}, {3538, 12318}, {3541, 37489}, {3547, 15740}, {3549, 10605}, {3575, 32110}, {3579, 12259}, {3589, 12006}, {3628, 5448}, {5010, 12428}, {5054, 12164}, {5204, 10055}, {5217, 10071}, {5432, 7352}, {5433, 6238}, {5504, 34483}, {5562, 10257}, {5663, 10020}, {5889, 37118}, {5893, 13406}, {5907, 16238}, {6000, 13383}, {6101, 15122}, {6102, 23292}, {6221, 19061}, {6240, 23293}, {6398, 19062}, {6643, 22661}, {6676, 40647}, {6759, 34351}, {6804, 32620}, {7280, 18970}, {7526, 13567}, {10018, 12111}, {10192, 32139}, {10264, 12893}, {10298, 34224}, {11064, 18436}, {11381, 37971}, {11425, 18951}, {11430, 13292}, {11442, 32534}, {11454, 18560}, {11457, 38444}, {12084, 23328}, {12108, 41597}, {12161, 18580}, {12225, 23294}, {12235, 15644}, {12241, 18570}, {12421, 38727}, {13474, 32223}, {13909, 42216}, {13970, 42215}, {14070, 14216}, {14118, 26879}, {14130, 16657}, {14516, 21844}, {14788, 15053}, {14790, 40686}, {14806, 41523}, {14869, 15083}, {15061, 23306}, {15123, 43577}, {15311, 15761}, {15873, 31861}, {16618, 20417}, {16621, 37440}, {16654, 18378}, {16836, 32348}, {18324, 32140}, {18475, 18914}, {18562, 21400}, {18569, 23332}, {18916, 37506}, {18917, 19357}, {19128, 43894}, {21243, 31833}, {23329, 23335}, {31829, 43589}, {32358, 43394}, {33547, 40685}, {34664, 43817}, {37481, 37649},

X(44158) = complement of X(22660)
X(44158) = midpoint of X(i) and X(j) for these {i,j}: {3, 12359}, {5, 7689}, {26, 6247}, {550, 9927}, {3579, 12259}, {10264, 12893}, {12163, 22660}, {12235, 15644}, {15761, 32138}, {32140, 34782}
X(44158) = reflection of X(i) in X(j) for these {i,j}: {140, 20191}, {5448, 3628}, {5893, 13406}, {9820, 140}, {12038, 3530}, {16252, 10020}, {33547, 40685}
X(44158) = crosspoint of X(69) and X(42410)
X(44158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12163, 22660}, {1216, 6699, 16196}, {18324, 32140, 34782}

leftri

Isogonal conjugates and isotomic conjugates: X(44159)-X(44173)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, August 10-12, 2021.

For any triangle center P, let gP and tP denote the isogonal conjugate of P and the isotomic conjugate of P, respectively. Centers X(44159)-X(44173) are members of chains of points

X, gX, tgX, gtgX, tgtgX, gtgtgX, ...

Following is a list of such chains, in which 0 signifies a point that is not in ETC:

(41288, 0, 41286, 0, 41281, 44159, 41280, 40363, 1397, 3596, 56, 8, 7, 55, 6063, 2175, 41283, 9448, 41287, 0, 41289, 0, 41290)

(9233, 40362, 1501, 1502, 32, 76, 6, 2, 2, 6, 76, 32, 1502, 1501, 40362, 9233, 40359)

(18903, 0, 14604, 18901, 8789, 14603, 9468, 3978, 694, 385, 1916, 1691, 18896, 14602, 44160, 18902)

(1917, 1928, 560, 561, 31, 75, 1, 1, 75, 31, 561, 560, 1928, 1917)

(8023, 44163, 44164, 44165, 8265, 38830, 20859, 40416, 626, 38826, 44166, 44167, 8039)

(40372, 0, 20968, 40421, 206, 18018, 22, 66, 315, 2353, 40073, 40146)

(23963, 23962, 23357, 338, 249, 115, 4590, 3124, 34537, 1084, 44168, 9427)

(18897, 44169, 1922, 1921, 292, 239, 335, 1914, 18895, 14599, 44170, 18894)

(18893, 44171, 14598, 18891, 1911, 350, 291, 238, 334, 2210, 44172, 18892)

(14574, 44173, 1576, 850, 110, 523, 99, 512, 670, 669, 4609, 9426)

(1691, 1916, 385, 694, 3978, 9468, 14603, 8789, 18901, 14604, 0, 18903)

X(44159) = ISOGONAL CONJUGATE OF X(41281)

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

X(44159) lies on these lines: {561, 4920}, {1228, 40367}, {1928, 35523}, {4136, 28659}, {33788, 35527}, {40360, 41287}

X(44159) = isogonal conjugate of X(41281)
X(44159) = isotomic conjugate of X(41280)
X(44159) = isotomic conjugate of the isogonal conjugate of X(40363)
X(44159) = X(i)-isoconjugate of X(j) for these (i,j): {1, 41281}, {31, 41280}, {34, 40373}, {56, 1917}, {57, 9233}, {75, 41286}, {560, 1397}, {561, 41288}, {604, 1501}, {1106, 9448}, {1356, 23995}, {1395, 14575}, {1428, 18893}, {2205, 16947}
X(44159) = barycentric product X(i)*X(j) for these {i,j}: {8, 40362}, {55, 40359}, {76, 40363}, {220, 41289}, {281, 40360}, {312, 1928}, {346, 41287}, {561, 28659}, {1502, 3596}, {7017, 40050}, {14827, 41290}, {27801, 40072}
X(44159) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 41280}, {6, 41281}, {8, 1501}, {9, 1917}, {32, 41286}, {55, 9233}, {76, 1397}, {219, 40373}, {310, 16947}, {312, 560}, {338, 1356}, {341, 9447}, {345, 14575}, {346, 9448}, {561, 604}, {645, 14574}, {1264, 14585}, {1501, 41288}, {1502, 56}, {1928, 57}, {1969, 1395}, {3596, 32}, {3685, 18894}, {3700, 9426}, {3701, 2205}, {3703, 41331}, {3718, 9247}, {3975, 18892}, {4086, 1924}, {4087, 14599}, {4092, 9427}, {4136, 8022}, {4391, 1980}, {4518, 18897}, {4609, 4565}, {4876, 18893}, {6064, 23963}, {6385, 1408}, {6386, 1415}, {7017, 1974}, {18022, 608}, {18027, 7337}, {20567, 1106}, {27801, 1402}, {28659, 31}, {28660, 2206}, {30713, 1918}, {34387, 1977}, {35519, 1919}, {40050, 222}, {40072, 1333}, {40073, 7251}, {40359, 6063}, {40360, 348}, {40362, 7}, {40363, 6}, {40364, 603}, {40367, 1403}, {41283, 1407}, {41287, 279}

X(44160) = ISOGONAL CONJUGATE OF X(18902)

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

X(44160) lies on these lines: {76, 694}, {305, 8842}, {308, 9468}, {338, 1502}, {384, 14604}, {1916, 4609}, {2970, 18022}, {8039, 23962}, {14603, 18024}, {16081, 39292}, {18829, 40074}, {20023, 20027}, {35549, 42822}, {36214, 40073}

X(44160) = isogonal conjugate of X(18902)
X(44160) = isotomic conjugate of X(14602)
X(44160) = isotomic conjugate of the isogonal conjugate of X(18896)
X(44160) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18902}, {31, 14602}, {32, 1933}, {171, 18894}, {172, 18892}, {385, 1917}, {560, 1691}, {1501, 1580}, {1966, 9233}, {2086, 23995}, {7122, 14599}, {9247, 44089}
X(44160) = trilinear product X(i)*X(j) for these {i,j}: {76, 1934}, {561, 1916}, {694, 1928}, {1502, 1581}
X(44160) = barycentric product X(i)*X(j) for these {i,j}: {76, 18896}, {561, 1934}, {694, 40362}, {1502, 1916}, {1581, 1928}, {9468, 40359}, {17980, 40360}, {18022, 40708}, {18901, 41517}, {23962, 39292}
X(44160) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14602}, {6, 18902}, {75, 1933}, {76, 1691}, {256, 18892}, {257, 14599}, {264, 44089}, {334, 7122}, {338, 2086}, {561, 1580}, {694, 1501}, {805, 14574}, {850, 5027}, {882, 9426}, {893, 18894}, {1502, 385}, {1581, 560}, {1916, 32}, {1928, 1966}, {1934, 31}, {1967, 1917}, {3493, 19556}, {4609, 17941}, {5207, 19575}, {7018, 2210}, {8024, 8623}, {8842, 34396}, {9468, 9233}, {14603, 4027}, {17970, 40373}, {18022, 419}, {18024, 40820}, {18829, 1576}, {18895, 172}, {18896, 6}, {36214, 14575}, {36897, 14601}, {39292, 23357}, {40050, 12215}, {40359, 14603}, {40362, 3978}, {40495, 4164}, {40708, 184}, {40810, 9418}, {41209, 4630}, {41517, 8789}

X(44161) = ISOGONAL CONJUGATE OF X(40373)

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

X(44161) lies on these lines: {76, 22416}, {183, 41488}, {1235, 5117}, {1502, 18027}, {6331, 39575}, {14603, 44144}, {18024, 41009}, {22456, 38907}, {23128, 43187}, {40016, 43678}, {40359, 40360}

X(44161) = isogonal conjugate of X(40373)
X(44161) = isotomic conjugate of X(14575)
X(44161) = polar conjugate of X(1501)
X(44161) = isotomic conjugate of the isogonal conjugate of X(18022)
X(44161) = polar conjugate of the isotomic conjugate of X(40362)
X(44161) = polar conjugate of the isogonal conjugate of X(1502)
X(44161) = X(1502)-cross conjugate of X(40362)
X(44161) = cevapoint of X(1502) and X(18022)
X(44161) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40373}, {3, 1917}, {31, 14575}, {32, 9247}, {48, 1501}, {63, 9233}, {78, 41281}, {184, 560}, {212, 41280}, {603, 9448}, {810, 14574}, {1101, 23216}, {1923, 10547}, {1924, 32661}, {1973, 14585}, {1980, 32656}, {2196, 18894}, {3718, 41286}, {4100, 36417}, {4575, 9426}, {7193, 18893}, {9417, 14600}
X(44161) = barycentric product X(i)*X(j) for these {i,j}: {4, 40362}, {25, 40359}, {76, 18022}, {92, 1928}, {264, 1502}, {281, 41287}, {305, 18027}, {331, 40363}, {393, 40360}, {561, 1969}, {607, 41289}, {826, 42395}, {1235, 40016}, {2052, 40050}, {4609, 14618}, {7017, 41283}, {18024, 44132}
X(44161) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 14575}, {4, 1501}, {6, 40373}, {19, 1917}, {25, 9233}, {69, 14585}, {75, 9247}, {76, 184}, {92, 560}, {115, 23216}, {242, 18894}, {264, 32}, {275, 14573}, {278, 41280}, {281, 9448}, {290, 14600}, {297, 9418}, {305, 577}, {308, 10547}, {311, 217}, {313, 2200}, {315, 22075}, {318, 9447}, {324, 40981}, {331, 1397}, {340, 19627}, {343, 44088}, {419, 18902}, {427, 41331}, {561, 48}, {608, 41281}, {626, 23209}, {648, 14574}, {670, 32661}, {850, 3049}, {1093, 36417}, {1235, 3051}, {1502, 3}, {1928, 63}, {1969, 31}, {1978, 32656}, {2052, 1974}, {2501, 9426}, {2967, 36425}, {2970, 1084}, {2973, 1977}, {3266, 23200}, {3267, 39201}, {3926, 23606}, {3964, 36433}, {4572, 32660}, {4602, 4575}, {4609, 4558}, {5117, 18899}, {6331, 1576}, {6385, 1437}, {6386, 906}, {7017, 2175}, {7141, 7109}, {8024, 20775}, {8039, 20819}, {8743, 40372}, {8754, 9427}, {14618, 669}, {15415, 15451}, {16081, 14601}, {17907, 20968}, {17924, 1980}, {17980, 14604}, {17983, 19626}, {17984, 14602}, {18020, 23963}, {18022, 6}, {18023, 14908}, {18024, 248}, {18027, 25}, {18817, 11060}, {18896, 17970}, {18901, 12215}, {20234, 22364}, {20567, 603}, {20883, 1923}, {20948, 810}, {23962, 20975}, {23989, 22096}, {24006, 1924}, {27801, 228}, {28659, 212}, {28706, 418}, {34384, 14533}, {36793, 34980}, {40016, 1176}, {40050, 394}, {40071, 4055}, {40072, 2193}, {40073, 10316}, {40074, 10317}, {40162, 15389}, {40359, 305}, {40360, 3926}, {40362, 69}, {40363, 219}, {40364, 255}, {40367, 20760}, {40495, 22383}, {40703, 9417}, {40717, 14599}, {41013, 2205}, {41283, 222}, {41287, 348}, {41530, 14642}, {42395, 4577}, {43678, 40146}, {44129, 2206}, {44132, 237}, {44144, 34396}, {44146, 14567}

X(44162) = ISOGONAL CONJUGATE OF X(40050)

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

X(44162) lies on these lines: {3, 19118}, {4, 3398}, {6, 15257}, {24, 2967}, {25, 251}, {32, 682}, {39, 9515}, {76, 37912}, {110, 19597}, {112, 699}, {186, 9821}, {237, 10316}, {264, 3115}, {468, 7767}, {827, 6179}, {1501, 40372}, {1576, 3053}, {1843, 5007}, {2205, 9448}, {2207, 2971}, {2353, 20975}, {2868, 10423}, {3080, 36417}, {3199, 44099}, {5008, 44091}, {5095, 7890}, {6287, 16868}, {7487, 41371}, {9418, 14585}, {10317, 20960}, {11470, 30270}, {12167, 43136}, {14001, 37893}, {14581, 40325}, {15270, 18374}, {20993, 40947}, {23216, 40351}, {39575, 44090}, {42671, 44079}

X(44162) = isogonal conjugate of X(40050)
X(44162) = isotomic conjugate of X(40360)
X(44162) = polar conjugate of X(40362)
X(44162) = isogonal conjugate of the isotomic conjugate of X(1974)
X(44162) = isogonal conjugate of the polar conjugate of X(36417)
X(44162) = polar conjugate of the isotomic conjugate of X(1501)
X(44162) = polar conjugate of the isogonal conjugate of X(9233)
X(44162) = X(1974)-Ceva conjugate of X(1501)
X(44162) = X(i)-cross conjugate of X(j) for these (i,j): {9233, 1501}, {23216, 9426}
X(44162) = cevapoint of X(9426) and X(23216)
X(44162) = crosspoint of X(1974) and X(36417)
X(44162) = crosssum of X(i) and X(j) for these (i,j): {69, 34254}, {125, 3267}
X(44162) = crossdifference of every pair of points on line {2525, 3267}
X(44162) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40050}, {2, 40364}, {3, 1928}, {31, 40360}, {48, 40362}, {63, 1502}, {69, 561}, {75, 305}, {76, 304}, {77, 40363}, {78, 41283}, {212, 41287}, {274, 40071}, {306, 6385}, {307, 40072}, {310, 20336}, {326, 18022}, {337, 18891}, {339, 24037}, {345, 20567}, {348, 28659}, {525, 4602}, {656, 4609}, {670, 14208}, {799, 3267}, {1102, 18027}, {1231, 28660}, {1926, 40708}, {1969, 3926}, {1978, 15413}, {2525, 37204}, {3596, 7182}, {3718, 6063}, {3933, 18833}, {4025, 6386}, {4561, 40495}, {4563, 20948}, {4572, 35518}, {9247, 40359}, {17206, 27801}, {18695, 34384}, {18837, 43714}, {20902, 34537}, {33778, 36952}
X(44162) = barycentric product X(i)*X(j) for these {i,j}: {3, 36417}, {4, 1501}, {6, 1974}, {19, 560}, {25, 32}, {28, 2205}, {30, 40351}, {31, 1973}, {34, 9447}, {41, 1395}, {53, 14573}, {92, 1917}, {112, 669}, {162, 1924}, {184, 2207}, {213, 2203}, {232, 14601}, {242, 18897}, {249, 42068}, {250, 1084}, {251, 27369}, {264, 9233}, {278, 9448}, {281, 41280}, {393, 14575}, {419, 8789}, {468, 19626}, {604, 2212}, {607, 1397}, {608, 2175}, {648, 9426}, {798, 32676}, {878, 34859}, {1096, 9247}, {1398, 14827}, {1402, 2204}, {1474, 1918}, {1495, 40354}, {1576, 2489}, {1783, 1980}, {1919, 8750}, {1976, 2211}, {2052, 40373}, {2201, 14598}, {2206, 2333}, {2353, 17409}, {2491, 32696}, {2501, 14574}, {2971, 23357}, {2998, 41293}, {3049, 32713}, {3172, 33581}, {6524, 14585}, {6531, 9418}, {6620, 40823}, {7017, 41281}, {8743, 40146}, {8749, 9407}, {8751, 9455}, {8752, 9459}, {8753, 14567}, {8754, 23963}, {8882, 40981}, {9427, 18020}, {9468, 44089}, {9494, 42396}, {11060, 34397}, {13854, 20968}, {14398, 32715}, {14581, 40352}, {14600, 34854}, {14602, 17980}, {14604, 17984}, {18384, 19627}, {20975, 41937}, {23216, 23582}, {23606, 36434}, {23975, 34980}, {23990, 42067}, {32085, 41331}, {32654, 44099}, {32740, 44102}, {40372, 43678}
X(44162) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40360}, {4, 40362}, {6, 40050}, {19, 1928}, {25, 1502}, {31, 40364}, {32, 305}, {112, 4609}, {264, 40359}, {278, 41287}, {331, 41289}, {419, 18901}, {560, 304}, {607, 40363}, {608, 41283}, {669, 3267}, {1084, 339}, {1395, 20567}, {1501, 69}, {1917, 63}, {1918, 40071}, {1924, 14208}, {1973, 561}, {1974, 76}, {1980, 15413}, {2203, 6385}, {2204, 40072}, {2205, 20336}, {2207, 18022}, {2212, 28659}, {2971, 23962}, {4117, 20902}, {8023, 20819}, {8789, 40708}, {9233, 3}, {9418, 6393}, {9426, 525}, {9427, 125}, {9447, 3718}, {9448, 345}, {9494, 2525}, {14573, 34386}, {14574, 4563}, {14575, 3926}, {14585, 4176}, {14604, 36214}, {17409, 40073}, {18897, 337}, {18902, 12215}, {18903, 17970}, {19626, 30786}, {20968, 34254}, {23216, 15526}, {27369, 8024}, {32676, 4602}, {36417, 264}, {40351, 1494}, {40372, 20806}, {40373, 394}, {40981, 28706}, {41280, 348}, {41281, 222}, {41293, 194}, {41331, 3933}, {42068, 338}, {44089, 14603}
X(44162) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 1974, 27369}, {32, 20968, 14575}, {10547, 30435, 34396}, {11380, 44089, 4}

X(44163) = ISOGONAL CONJUGATE OF X(8023)

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

X(44163) lies on these lines: {6, 35530}, {39, 14603}, {141, 18901}, {2353, 38842}, {6664, 40016}

X(44163) = isogonal conjugate of X(8023)
X(44163) = isotomic conjugate of X(44164)
X(44163) = X(76)-cross conjugate of X(38830)
X(44163) = X(i)-isoconjugate of X(j) for these (i,j): {1, 8023}, {560, 8265}, {1501, 2085}, {1917, 20859}, {1973, 23209}, {4118, 9233}
X(44163) = cevapoint of X(i) and X(j) for these (i,j): {76, 40362}, {38830, 38842}
X(44163) = barycentric product X(i)*X(j) for these {i,j}: {1502, 38830}, {1928, 38847}, {38826, 40359}, {38842, 40421}, {40362, 40416}
X(44163) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 8023}, {69, 23209}, {76, 8265}, {305, 4173}, {561, 2085}, {1502, 20859}, {1928, 4118}, {6385, 16717}, {8024, 3118}, {38826, 9233}, {38830, 32}, {38842, 206}, {38847, 560}, {40050, 20819}, {40360, 4121}, {40362, 626}, {40416, 1501}, {41287, 7217}

X(44164) = ISOTOMIC CONJUGATE OF ISOGONAL CONJUGATE OF X(8023)

Barycentrics    a^6*(b^4 + c^4) : :

X(44164) lies on these lines: {6, 76}, {32, 8789}, {39, 32748}, {217, 1692}, {575, 6310}, {711, 827}, {1084, 27374}, {1207, 3329}, {1501, 20968}, {1613, 6179}, {1691, 15926}, {2086, 7755}, {2387, 14820}, {3051, 3229}, {3118, 4173}, {3224, 3972}, {3231, 7780}, {5012, 14885}, {8023, 23209}, {8619, 21751}

X(44164) = isogonal conjugate of X(44165)
X(44164) = isotomic conjugate of X(44163)
X(44164) = isotomic conjugate of the isogonal conjugate of X(8023)
X(44164) = isogonal conjugate of the isotomic conjugate of X(8265)
X(44164) = polar conjugate of the isotomic conjugate of X(23209)
X(44164) = X(i)-Ceva conjugate of X(j) for these (i,j): {6, 20859}, {827, 9494}, {8265, 23209}
X(44164) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44165}, {75, 38830}, {76, 38847}, {561, 40416}, {1928, 38826}, {1930, 3115}
X(44164) = crosspoint of X(6) and X(1501)
X(44164) = crosssum of X(2) and X(1502)
X(44164) = crossdifference of every pair of points on line {688, 14295}
X(44164) = trilinear product X(i)*X(j) for these {i,j}: {19, 23209}, {31, 8265}, {32, 2085}, {75, 8023}, {560, 20859}, {626, 1917}, {1501, 4118}, {1918, 16717}, {1973, 4173}, {9233, 20627}
X(44164) = barycentric product X(i)*X(j) for these {i,j}: {4, 23209}, {6, 8265}, {25, 4173}, {31, 2085}, {32, 20859}, {76, 8023}, {213, 16717}, {251, 3118}, {560, 4118}, {626, 1501}, {1917, 20627}, {1974, 20819}, {2205, 18167}, {4178, 41280}, {7217, 9448}, {16890, 41331}
X(44164) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 38830}, {206, 38842}, {560, 38847}, {626, 40362}, {1501, 40416}, {2085, 561}, {3118, 8024}, {4118, 1928}, {4121, 40360}, {4173, 305}, {7217, 41287}, {8023, 6}, {8265, 76}, {9233, 38826}, {16717, 6385}, {20819, 40050}, {20859, 1502}, {23209, 69}
X(44164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1186, 42548}, {6, 3499, 7760}, {6, 33786, 76}, {3051, 9490, 5007}, {4173, 8265, 3118}

X(44165) = ISOTOMIC CONJUGATE OF X(8265)

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

X(44165) lies on these lines: {2, 40359}, {32, 710}, {39, 14603}, {626, 14946}, {3051, 3978}, {4609, 7909}, {7832, 18901}, {9230, 38821}, {17984, 27369}, {40146, 40421}

X(44165) = isogonal conjugate of X(44164)
X(44165) = isotomic conjugate of X(8265)
X(44165) = isotomic conjugate of the complement of X(1502)
X(44165) = isotomic conjugate of the isogonal conjugate of X(38830)
X(44165) = X(i)-cross conjugate of X(j) for these (i,j): {2, 40416}, {826, 42371}, {31296, 670}
X(44165) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44164}, {19, 23209}, {31, 8265}, {32, 2085}, {75, 8023}, {560, 20859}, {626, 1917}, {1501, 4118}, {1918, 16717}, {1973, 4173}, {9233, 20627}
X(44165) = cevapoint of X(2) and X(1502)
X(44165) = trilinear pole of line {688, 14295}
X(44165) = trilinear product X(i)*X(j) for these {i,j}: {75, 38830}, {76, 38847}, {561, 40416}, {1928, 38826}, {1930, 3115}
X(44165) = barycentric product X(i)*X(j) for these {i,j}: {76, 38830}, {561, 38847}, {1502, 40416}, {3115, 8024}, {18018, 38842}, {38826, 40362}
X(44165) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8265}, {3, 23209}, {32, 8023}, {69, 4173}, {75, 2085}, {76, 20859}, {141, 3118}, {274, 16717}, {305, 20819}, {561, 4118}, {1502, 626}, {1928, 20627}, {3115, 251}, {6385, 18167}, {33515, 4630}, {38826, 1501}, {38830, 6}, {38842, 22}, {38847, 31}, {40016, 16890}, {40050, 4121}, {40359, 8039}, {40363, 4178}, {40416, 32}, {41283, 7217}

X(44166) = ISOTOMIC CONJUGATE OF X(38826)

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

X(44166) lies on these lines: {6, 76}, {53, 44132}, {69, 2871}, {141, 35542}, {305, 7778}, {311, 325}, {313, 35551}, {321, 35547}, {338, 1502}, {736, 41331}, {1989, 32833}, {3721, 35537}, {4159, 9233}, {5976, 8266}, {6664, 34294}, {7763, 13351}, {7792, 39998}, {8039, 16893}, {9230, 40074}, {16890, 20859}, {16894, 20627}, {17500, 33798}

X(44166) = isogonal conjugate of X(44167)
X(44166) = isotomic conjugate of X(38826)
X(44166) = isotomic conjugate of the isogonal conjugate of X(626)
X(44166) = isogonal conjugate of the isotomic conjugate of X(8039)
X(44166) = polar conjugate of the isogonal conjugate of X(4121)
X(44166) = X(76)-Ceva conjugate of X(20859)
X(44166) = X(16893)-cross conjugate of X(626)
X(44166) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44167}, {31, 38826}, {560, 40416}, {1501, 38847}, {1917, 38830}
X(44166) = cevapoint of X(626) and X(4121)
X(44166) = crosspoint of X(76) and X(40362)
X(44166) = crosssum of X(32) and X(9233)
X(44166) = trilinear product X(i)*X(j) for these {i,j}: {2, 20627}, {31, 8039}, {75, 626}, {76, 4118}, {85, 4178}, {92, 4121}, {274, 16894}, {312, 7217}, {313, 18167}, {321, 16891}, {561, 20859}, {668, 21110}, {1502, 2085}, {1928, 8265}, {1930, 16890}, {1969, 20819}, {3112, 16893}
X(44166) = barycentric product X(i)*X(j) for these {i,j}: {6, 8039}, {75, 20627}, {76, 626}, {264, 4121}, {308, 16893}, {310, 16894}, {313, 16891}, {561, 4118}, {1502, 20859}, {1928, 2085}, {1978, 21110}, {3596, 7217}, {4178, 6063}, {8024, 16890}, {8265, 40362}, {18022, 20819}, {18167, 27801}, {35530, 40847}
X(44166) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 38826}, {76, 40416}, {561, 38847}, {626, 6}, {689, 33515}, {1502, 38830}, {2085, 560}, {3118, 41331}, {4118, 31}, {4121, 3}, {4173, 14575}, {4178, 55}, {7217, 56}, {8039, 76}, {8264, 38838}, {8265, 1501}, {16890, 251}, {16891, 58}, {16893, 39}, {16894, 42}, {18167, 1333}, {20627, 1}, {20819, 184}, {20859, 32}, {21110, 649}, {23209, 40373}, {35530, 16985}, {40016, 3115}, {40847, 711}
X(44166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 40073, 35549}, {76, 40035, 33769}, {76, 40073, 6}

X(44167) = ISOTOMIC CONJUGATE OF X(8039)

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

X(44167) lies on these lines: {2, 9233}, {32, 40366}, {39, 5012}, {3051, 18902}, {4630, 8023}, {33515, 43094}

X(44167) = isogonal conjugate of X(44166)
X(44167) = isotomic conjugate of X(8039)
X(44167) = isogonal conjugate of the isotomic conjugate of X(38826)
X(44167) = X(32)-cross conjugate of X(40416)
X(44167) = X(i)-isoconjugate of X(j) for these (i,j): {1, 44166}, {2, 20627}, {31, 8039}, {75, 626}, {76, 4118}, {85, 4178}, {92, 4121}, {274, 16894}, {312, 7217}, {313, 18167}, {321, 16891}, {561, 20859}, {668, 21110}, {1502, 2085}, {1928, 8265}, {1930, 16890}, {1969, 20819}, {3112, 16893}
X(44167) = cevapoint of X(32) and X(9233)
X(44167) = crosssum of X(626) and X(4121)
X(44167) = trilinear product X(i)*X(j) for these {i,j}: {31, 38826}, {560, 40416}, {1501, 38847}, {1917, 38830}
X(44167) = barycentric product X(i)*X(j) for these {i,j}: {6, 38826}, {32, 40416}, {560, 38847}, {688, 33515}, {1501, 38830}, {3115, 41331}
X(44167) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8039}, {31, 20627}, {32, 626}, {184, 4121}, {560, 4118}, {1397, 7217}, {1501, 20859}, {1917, 2085}, {1918, 16894}, {1919, 21110}, {2175, 4178}, {2206, 16891}, {3051, 16893}, {9233, 8265}, {14575, 20819}, {33515, 42371}, {38826, 76}, {38830, 40362}, {38838, 19562}, {38847, 1928}, {40373, 4173}, {40416, 1502}

X(44168) = ISOTOMIC CONJUGATE OF X(1084)

Barycentrics    (a - b)^2*b^4*(a + b)^2*(a - c)^2*c^4*(a + c)^2 : :
X(44168) = 2 X[670] + X[886], 3 X[3228] - 4 X[31646], 15 X[31639] - 16 X[40507]

X(44168) lies on these lines: {99, 9491}, {512, 670}, {538, 14603}, {799, 21763}, {1502, 14609}, {3228, 31646}, {3231, 3978}, {6374, 30229}, {14382, 40050}, {16084, 17984}, {18829, 42371}, {31639, 40507}

X(44168) = midpoint of X(670) and X(9428)
X(44168) = reflection of X(886) in X(9428)
X(44168) = isogonal conjugate of X(9427)
X(44168) = isotomic conjugate of X(1084)
X(44168) = polar conjugate of X(42068)
X(44168) = isotomic conjugate of the anticomplement of X(36950)
X(44168) = isotomic conjugate of the complement of X(670)
X(44168) = isotomic conjugate of the isogonal conjugate of X(34537)
X(44168) = barycentric square of X(670)
X(44168) = X(i)-cross conjugate of X(j) for these (i,j): {2, 670}, {76, 42371}, {194, 99}, {538, 886}, {1502, 4609}, {1655, 668}, {6392, 6528}, {8267, 4577}, {10010, 9063}, {19570, 35139}, {31088, 42367}, {34022, 799}, {36950, 2}, {40858, 18829}
X(44168) = cevapoint of X(i) and X(j) for these (i,j): {2, 670}, {99, 7760}, {668, 32026}, {1502, 4609}, {1613, 1634}
X(44168) = trilinear pole of line {670, 888} (the tangent to the Steiner circumellipse at X(670))
X(44168) = X(i)-isoconjugate of X(j) for these (i,j): {1, 9427}, {6, 4117}, {19, 23216}, {31, 1084}, {41, 1356}, {48, 42068}, {115, 1917}, {163, 23099}, {512, 1924}, {560, 3124}, {604, 7063}, {661, 9426}, {662, 23610}, {669, 798}, {872, 1977}, {1109, 9233}, {1501, 2643}, {1918, 3121}, {1927, 2086}, {1980, 4079}, {2151, 41993}, {2152, 41994}, {2205, 3122}, {2971, 9247}, {3248, 7109}, {9417, 15630}, {33918, 36133}
X(44168) = barycentric product X(i)*X(j) for these {i,j}: {76, 34537}, {99, 4609}, {249, 40362}, {250, 40360}, {561, 24037}, {670, 670}, {799, 4602}, {1502, 4590}, {1928, 24041}, {4576, 42371}, {4601, 6385}, {4623, 6386}, {6064, 41283}, {7340, 40363}, {14603, 39292}, {18020, 40050}, {23357, 40359}
X(44168) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4117}, {2, 1084}, {3, 23216}, {4, 42068}, {6, 9427}, {7, 1356}, {8, 7063}, {13, 41993}, {14, 41994}, {76, 3124}, {99, 669}, {110, 9426}, {249, 1501}, {264, 2971}, {274, 3121}, {290, 15630}, {305, 20975}, {310, 3122}, {512, 23610}, {523, 23099}, {538, 1645}, {561, 2643}, {662, 1924}, {670, 512}, {689, 18105}, {799, 798}, {850, 22260}, {873, 3248}, {880, 5027}, {888, 33918}, {1016, 7109}, {1101, 1917}, {1502, 115}, {1509, 1977}, {1634, 9494}, {1920, 21823}, {1928, 1109}, {1978, 4079}, {2396, 2491}, {3266, 21906}, {3978, 2086}, {4176, 34980}, {4563, 3049}, {4567, 2205}, {4576, 688}, {4590, 32}, {4600, 1918}, {4601, 213}, {4602, 661}, {4609, 523}, {4610, 1919}, {4623, 667}, {4631, 3063}, {6064, 2175}, {6331, 2489}, {6385, 3125}, {6386, 4705}, {7035, 872}, {7304, 21762}, {7340, 1397}, {7760, 38996}, {8033, 21755}, {9428, 38237}, {16084, 865}, {18020, 1974}, {18021, 3271}, {18022, 8754}, {18829, 881}, {20023, 6784}, {23342, 887}, {23357, 9233}, {23582, 36417}, {24037, 31}, {24041, 560}, {27801, 21833}, {31008, 21835}, {31614, 1576}, {31625, 1500}, {31639, 31646}, {34537, 6}, {35073, 39010}, {35540, 41178}, {39292, 9468}, {40016, 34294}, {40050, 125}, {40072, 4516}, {40359, 23962}, {40360, 339}, {40362, 338}, {40363, 4092}, {40364, 3708}, {41283, 1365}, {43187, 2422}

X(44169) = ISOTOMIC CONJUGATE OF X(1922)

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

X(44169) lies on these lines: {75, 4493}, {76, 1928}, {313, 561}, {321, 40016}, {350, 1926}, {768, 3261}, {1909, 1925}, {3263, 6386}, {3978, 41250}, {4602, 20924}, {6385, 16703}, {7242, 33788}, {18833, 33941}, {18837, 33930}, {18891, 35544}, {28659, 40367}, {33778, 33939}

X(44169) = isogonal conjugate of X(18897)
X(44169) = isotomic conjugate of X(1922)
X(44169) = isotomic conjugate of the isogonal conjugate of X(1921)
X(44169) = X(20630)-cross conjugate of X(75)
X(44169) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18897}, {2, 18893}, {6, 14598}, {31, 1922}, {32, 1911}, {171, 8789}, {172, 1927}, {291, 1501}, {292, 560}, {334, 9233}, {335, 1917}, {604, 18265}, {741, 2205}, {813, 1980}, {875, 32739}, {1909, 14604}, {1914, 18267}, {1918, 18268}, {1919, 34067}, {1974, 2196}, {4584, 9426}, {4876, 41280}, {7122, 9468}, {18263, 18266}, {18894, 30663}
X(44169) = cevapoint of X(75) and X(20644)
X(44169) = trilinear 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(44169) = barycentric product X(i)*X(j) for these {i,j}: {75, 18891}, {76, 1921}, {238, 1928}, {239, 1502}, {242, 40050}, {257, 14603}, {305, 40717}, {310, 35544}, {350, 561}, {874, 40495}, {893, 18901}, {1447, 40363}, {1914, 40362}, {1926, 7018}, {3261, 27853}, {3596, 18033}, {3685, 41283}, {3766, 6386}, {3948, 6385}, {3975, 20567}, {4010, 4609}, {4087, 6063}, {10030, 28659}, {14599, 40359}, {27801, 30940}, {39914, 40367}
X(44169) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14598}, {2, 1922}, {6, 18897}, {8, 18265}, {31, 18893}, {75, 1911}, {76, 292}, {238, 560}, {239, 32}, {242, 1974}, {256, 1927}, {257, 9468}, {274, 18268}, {291, 18267}, {304, 2196}, {305, 295}, {310, 741}, {350, 31}, {561, 291}, {659, 1980}, {668, 34067}, {693, 875}, {740, 1918}, {812, 1919}, {874, 692}, {893, 8789}, {1281, 18262}, {1428, 41280}, {1447, 1397}, {1502, 335}, {1914, 1501}, {1921, 6}, {1926, 171}, {1928, 334}, {1966, 7122}, {1978, 813}, {2210, 1917}, {2238, 2205}, {3261, 3572}, {3263, 40730}, {3570, 32739}, {3596, 7077}, {3684, 9447}, {3685, 2175}, {3766, 667}, {3783, 18900}, {3797, 40728}, {3948, 213}, {3975, 41}, {3978, 172}, {4010, 669}, {4037, 7109}, {4087, 55}, {4366, 14599}, {4432, 9459}, {4455, 9426}, {4594, 17938}, {4602, 4584}, {4609, 4589}, {4760, 14567}, {6385, 37128}, {6386, 660}, {6650, 18263}, {7018, 1967}, {7019, 17970}, {7104, 14604}, {7193, 14575}, {8299, 9455}, {8300, 18892}, {10030, 604}, {14295, 7234}, {14599, 9233}, {14603, 894}, {17493, 7104}, {17755, 9454}, {18033, 56}, {18035, 17735}, {18036, 30648}, {18037, 19554}, {18277, 18278}, {18891, 1}, {18901, 1920}, {18904, 21751}, {20769, 9247}, {21832, 1924}, {24459, 3049}, {27853, 101}, {27918, 1977}, {28659, 4876}, {28660, 2311}, {30665, 8630}, {30940, 1333}, {33295, 2206}, {35538, 40155}, {35544, 42}, {39044, 2210}, {40050, 337}, {40362, 18895}, {40363, 4518}, {40367, 40848}, {40495, 876}, {40717, 25}, {41283, 7233}

X(44170) = ISOTOMIC CONJUGATE OF X(14599)

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

X(44170) lies on these lines: {292, 1920}, {334, 561}, {335, 6385}, {18895, 27801}, {40367, 41283}

X(44170) = isogonal conjugate of X(18894)
X(44170) = isotomic conjugate of X(14599)
X(44170) = isotomic conjugate of the isogonal conjugate of X(18895)
X(44170) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18894}, {6, 18892}, {31, 14599}, {32, 2210}, {238, 1501}, {239, 1917}, {256, 18902}, {350, 9233}, {560, 1914}, {904, 14602}, {1428, 9447}, {1429, 9448}, {1933, 7104}, {2201, 14575}, {2205, 5009}, {2206, 41333}, {3684, 41280}, {3975, 41281}, {4366, 18893}, {8300, 18897}, {14574, 21832}
X(44170) = trilinear product X(i)*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(44170) = barycentric product X(i)*X(j) for these {i,j}: {76, 18895}, {291, 1928}, {292, 40362}, {334, 561}, {335, 1502}, {337, 18022}, {1920, 18896}, {1922, 40359}, {4518, 41283}, {4583, 40495}, {4609, 35352}, {4639, 20948}, {7077, 41287}, {7233, 40363}, {18265, 41289}, {27801, 40017}, {30870, 41072}
X(44170) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18892}, {2, 14599}, {6, 18894}, {75, 2210}, {76, 1914}, {172, 18902}, {291, 560}, {292, 1501}, {295, 14575}, {305, 7193}, {310, 5009}, {313, 3747}, {321, 41333}, {334, 31}, {335, 32}, {337, 184}, {561, 238}, {850, 4455}, {876, 1980}, {894, 14602}, {1502, 239}, {1909, 1933}, {1911, 1917}, {1916, 7104}, {1920, 1691}, {1922, 9233}, {1928, 350}, {1934, 904}, {1969, 2201}, {3261, 8632}, {3864, 18900}, {4444, 1919}, {4518, 2175}, {4562, 32739}, {4583, 692}, {4589, 1576}, {4639, 163}, {4876, 9447}, {6063, 1428}, {6386, 3573}, {7077, 9448}, {7233, 1397}, {17789, 18038}, {18022, 242}, {18033, 12835}, {18275, 18274}, {18827, 2206}, {18891, 8300}, {18895, 6}, {18896, 893}, {19567, 30634}, {20567, 1429}, {20908, 38367}, {20948, 21832}, {22116, 9455}, {27801, 2238}, {28659, 3684}, {30663, 14598}, {30870, 30665}, {35352, 669}, {35538, 20663}, {40017, 1333}, {40098, 1922}, {40217, 9454}, {40362, 1921}, {40363, 3685}, {40364, 20769}, {40495, 659}, {41072, 34069}, {41283, 1447}, {41287, 18033}, {43534, 1918}

X(44171) = ISOTOMIC CONJUGATE OF X(14598)

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

X(44171) lies on these lines: {561, 2887}, {1502, 1928}, {1921, 14603}, {4609, 40075}, {7034, 33165}, {35559, 40495}

X(44171) = isogonal conjugate of X(18893)
X(44171) = isotomic conjugate of X(14598)
X(44171) = isotomic conjugate of the isogonal conjugate of X(18891)
X(44171) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18893}, {6, 18897}, {31, 14598}, {32, 1922}, {172, 8789}, {291, 1917}, {292, 1501}, {335, 9233}, {560, 1911}, {894, 14604}, {1397, 18265}, {1920, 18903}, {1927, 7122}, {1980, 34067}, {2205, 18268}, {2210, 18267}, {4518, 41281}, {7077, 41280}
X(44171) = trilinear product X(i)*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(44171) = barycentric product X(i)*X(j) for these {i,j}: {76, 18891}, {238, 40362}, {239, 1928}, {256, 18901}, {350, 1502}, {561, 1921}, {2201, 40360}, {2210, 40359}, {3684, 41287}, {3975, 41283}, {4087, 20567}, {6385, 35544}, {7018, 14603}, {10030, 40363}, {18033, 28659}, {27853, 40495}, {40364, 40717}
X(44171) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18897}, {2, 14598}, {6, 18893}, {75, 1922}, {76, 1911}, {238, 1501}, {239, 560}, {256, 8789}, {257, 1927}, {305, 2196}, {310, 18268}, {312, 18265}, {335, 18267}, {350, 32}, {561, 292}, {740, 2205}, {812, 1980}, {874, 32739}, {904, 14604}, {1429, 41280}, {1502, 291}, {1914, 1917}, {1921, 31}, {1926, 172}, {1928, 335}, {1978, 34067}, {2210, 9233}, {3261, 875}, {3684, 9448}, {3685, 9447}, {3766, 1919}, {3797, 18900}, {3948, 1918}, {3975, 2175}, {3978, 7122}, {4010, 1924}, {4087, 41}, {4366, 18892}, {4486, 8630}, {4609, 4584}, {6385, 741}, {6386, 813}, {7018, 9468}, {7260, 17938}, {8300, 18894}, {10030, 1397}, {14603, 171}, {17755, 9455}, {18032, 18263}, {18033, 604}, {18035, 18266}, {18037, 18262}, {18891, 6}, {18901, 1909}, {18904, 8022}, {20769, 14575}, {21832, 9426}, {27853, 692}, {28659, 7077}, {30870, 30671}, {30940, 2206}, {35544, 213}, {39044, 14599}, {40072, 2311}, {40362, 334}, {40363, 4876}, {40364, 295}, {40367, 41531}, {40495, 3572}, {40717, 1973}

X(44172) = ISOTOMIC CONJUGATE OF X(2210)

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

X(44172) lies on these lines: {76, 335}, {291, 310}, {305, 4518}, {313, 334}, {561, 8024}, {1909, 1911}, {1926, 18034}, {3761, 18787}, {4583, 40075}, {17755, 25758}, {18037, 37133}, {20567, 34388}

X(44172) = isogonal conjugate of X(18892)
X(44172) = isotomic conjugate of X(2210)
X(44172) = isotomic conjugate of the isogonal conjugate of X(334)
X(44172) = X(20629)-cross conjugate of X(75)
X(44172) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18892}, {2, 18894}, {6, 14599}, {31, 2210}, {32, 1914}, {238, 560}, {239, 1501}, {242, 14575}, {257, 18902}, {350, 1917}, {893, 14602}, {904, 1933}, {1333, 41333}, {1428, 2175}, {1429, 9447}, {1447, 9448}, {1576, 4455}, {1691, 7104}, {1918, 5009}, {1921, 9233}, {1974, 7193}, {1980, 3573}, {2201, 9247}, {2206, 3747}, {3685, 41280}, {4010, 14574}, {4087, 41281}, {4366, 18897}, {4760, 19626}, {8300, 14598}, {8632, 32739}, {12835, 18265}, {18893, 39044}, {40373, 40717}
X(44172) = cevapoint of X(i) and X(j) for these (i,j): {75, 20643}, {16892, 21140}
X(44172) = trilinear pole of line {23100, 23596}
X(44172) = trilinear 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(44172) = barycentric product X(i)*X(j) for these {i,j}: {75, 18895}, {76, 334}, {291, 1502}, {292, 1928}, {313, 40017}, {335, 561}, {337, 1969}, {850, 4639}, {1909, 18896}, {1911, 40362}, {1920, 1934}, {3261, 4583}, {4444, 6386}, {4518, 20567}, {4562, 40495}, {4589, 20948}, {4602, 35352}, {4876, 41283}, {6385, 43534}, {7233, 28659}, {14598, 40359}, {18275, 30633}, {18827, 27801}, {18891, 40098}, {30870, 37207}
X(44172) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14599}, {2, 2210}, {6, 18892}, {10, 41333}, {31, 18894}, {75, 1914}, {76, 238}, {85, 1428}, {171, 14602}, {264, 2201}, {274, 5009}, {291, 32}, {292, 560}, {295, 9247}, {304, 7193}, {305, 20769}, {313, 2238}, {321, 3747}, {334, 6}, {335, 31}, {337, 48}, {349, 1284}, {561, 239}, {660, 32739}, {693, 8632}, {850, 21832}, {876, 1919}, {894, 1933}, {1502, 350}, {1577, 4455}, {1581, 7104}, {1909, 1691}, {1911, 1501}, {1916, 904}, {1920, 1580}, {1921, 8300}, {1922, 1917}, {1928, 1921}, {1934, 893}, {1969, 242}, {1978, 3573}, {2196, 14575}, {3252, 9455}, {3261, 659}, {3572, 1980}, {3596, 3684}, {3837, 38367}, {3862, 18900}, {3864, 40728}, {4444, 667}, {4518, 41}, {4562, 692}, {4583, 101}, {4584, 1576}, {4589, 163}, {4639, 110}, {4645, 18038}, {4876, 2175}, {5378, 23990}, {6063, 1429}, {6383, 34252}, {6385, 33295}, {6386, 3570}, {7077, 9447}, {7122, 18902}, {7233, 604}, {10030, 12835}, {14598, 9233}, {15413, 22384}, {17789, 19561}, {18034, 2112}, {18275, 19580}, {18827, 1333}, {18891, 4366}, {18895, 1}, {18896, 256}, {19565, 30634}, {19567, 18274}, {20567, 1447}, {20643, 39029}, {20948, 4010}, {21207, 39786}, {22116, 9454}, {23596, 788}, {23989, 27846}, {27801, 740}, {28659, 3685}, {30663, 1922}, {30669, 7122}, {30671, 8630}, {30713, 4433}, {30870, 4486}, {33931, 16514}, {35352, 798}, {35519, 4435}, {35538, 17475}, {36800, 2194}, {36806, 4636}, {37128, 2206}, {37207, 34069}, {40017, 58}, {40075, 27950}, {40093, 2220}, {40094, 4251}, {40095, 33882}, {40098, 1911}, {40217, 2223}, {40362, 18891}, {40363, 3975}, {40495, 812}, {40708, 7116}, {40834, 38813}, {40848, 2209}, {41072, 825}, {41283, 10030}, {43534, 213}

X(44173) = ISOTOMIC CONJUGATE OF X(1576)

Barycentrics    b^4*(b - c)*c^4*(b + c) : :
Barycentrics    (directed distance from A to Brocard axis)*(directed distance from A to Lemoine axis) : : (Randy Hutson, November 30, 2021)

X(44173) lies on these lines: {76, 525}, {99, 22089}, {305, 30474}, {512, 14295}, {523, 14603}, {670, 14221}, {689, 1287}, {778, 9494}, {804, 30492}, {826, 850}, {1078, 39201}, {2531, 23301}, {2799, 3267}, {7771, 39228}, {8029, 40362}, {17415, 42291}, {23285, 30870}

X(44173) = reflection of X(i) in X(j) for these {i,j}: {2531, 23301}, {17415, 42291}
X(44173) = isogonal conjugate of X(14574)
X(44173) = isotomic conjugate of X(1576)
X(44173) = isotomic conjugate of the isogonal conjugate of X(850)
X(44173) = polar conjugate of the isogonal conjugate of X(3267)
X(44173) = X(2980)-anticomplementary conjugate of X(21220)
X(44173) = X(i)-Ceva conjugate of X(j) for these (i,j): {1502, 23962}, {4609, 1502}, {40362, 338}
X(44173) = X(i)-cross conjugate of X(j) for these (i,j): {338, 40362}, {339, 76}, {868, 18896}, {8029, 338}, {21430, 75}, {23285, 850}, {23962, 1502}
X(44173) = cevapoint of X(i) and X(j) for these (i,j): {75, 21603}, {338, 8029}, {523, 33294}, {850, 3267}
X(44173) = crosspoint of X(i) and X(j) for these (i,j): {308, 6331}, {670, 34384}, {1502, 4609}
X(44173) = crosssum of X(i) and X(j) for these (i,j): {669, 40981}, {1501, 9426}, {3049, 3051}
X(44173) = trilinear pole of line {338, 23962}
X(44173) = pole wrt polar circle of line X(32)X(682)
X(44173) = crossdifference of every pair of points on line {1501, 9233}
X(44173) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14574}, {31, 1576}, {32, 163}, {99, 1917}, {110, 560}, {112, 9247}, {162, 14575}, {184, 32676}, {249, 1924}, {512, 23995}, {643, 41280}, {661, 23963}, {662, 1501}, {669, 1101}, {692, 2206}, {798, 23357}, {799, 9233}, {811, 40373}, {822, 41937}, {827, 1923}, {922, 32729}, {1333, 32739}, {1414, 9448}, {1933, 17938}, {1964, 4630}, {1973, 32661}, {1974, 4575}, {1980, 4570}, {2179, 14586}, {2203, 32656}, {2204, 32660}, {2205, 4556}, {2617, 14573}, {2715, 9417}, {3051, 34072}, {4565, 9447}, {4584, 18894}, {4599, 41331}, {7257, 41281}, {9406, 32640}, {9407, 36034}, {9418, 36084}, {9426, 24041}, {14567, 36142}, {14585, 24019}, {14601, 23997}, {18902, 37134}, {19626, 23889}, {19627, 32678}, {36134, 40981}
X(44173) = trilinear product X(i)*X(j) for these {i,j}: {75, 850}, {76, 1577}, {92, 3267}, {115, 4602}, {264, 14208}, {310, 4036}, {313, 693}, {321, 3261}, {338, 799}, {339, 811}, {349, 4391}, {512, 1928}, {523, 561}, {525, 1969}, {661, 1502}, {670, 1109}, {871, 4122}, {1934, 14295}, {2643, 4609}, {3120, 6386}, {3596, 4077}, {3801, 7034}, {4024, 6385}, {4086, 6063}, {4143, 6521}
X(44173) = barycentric product X(i)*X(j) for these {i,j}: {75, 20948}, {76, 850}, {95, 15415}, {99, 23962}, {115, 4609}, {264, 3267}, {305, 14618}, {308, 23285}, {313, 3261}, {321, 40495}, {338, 670}, {339, 6331}, {349, 35519}, {512, 40362}, {523, 1502}, {525, 18022}, {561, 1577}, {661, 1928}, {669, 40359}, {693, 27801}, {799, 23994}, {826, 40016}, {882, 18901}, {1109, 4602}, {1969, 14208}, {1978, 21207}, {2489, 40360}, {2501, 40050}, {2799, 18024}, {3265, 18027}, {3268, 20573}, {3700, 41283}, {3709, 41287}, {4036, 6385}, {4077, 28659}, {4086, 20567}, {6386, 16732}, {6528, 36793}, {7178, 40363}, {14295, 18896}, {18023, 35522}, {18314, 34384}, {23105, 34537}, {23989, 27808}, {24006, 40364}, {33294, 40421}, {39691, 42371}
X(44173) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 1576}, {6, 14574}, {10, 32739}, {69, 32661}, {75, 163}, {76, 110}, {83, 4630}, {92, 32676}, {94, 14560}, {95, 14586}, {99, 23357}, {107, 41937}, {110, 23963}, {115, 669}, {125, 3049}, {264, 112}, {276, 933}, {290, 2715}, {300, 5995}, {301, 5994}, {304, 4575}, {305, 4558}, {306, 32656}, {307, 32660}, {308, 827}, {310, 4556}, {311, 1625}, {313, 101}, {321, 692}, {325, 14966}, {327, 26714}, {328, 32662}, {338, 512}, {339, 647}, {340, 14591}, {349, 109}, {512, 1501}, {514, 2206}, {520, 14585}, {523, 32}, {525, 184}, {526, 19627}, {561, 662}, {647, 14575}, {656, 9247}, {661, 560}, {662, 23995}, {669, 9233}, {670, 249}, {671, 32729}, {690, 14567}, {693, 1333}, {798, 1917}, {799, 1101}, {804, 14602}, {826, 3051}, {850, 6}, {868, 2491}, {879, 14600}, {881, 14604}, {882, 8789}, {1109, 798}, {1230, 35327}, {1231, 36059}, {1232, 35324}, {1235, 35325}, {1441, 1415}, {1494, 32640}, {1502, 99}, {1577, 31}, {1637, 9407}, {1916, 17938}, {1928, 799}, {1969, 162}, {1978, 4570}, {2052, 32713}, {2394, 40352}, {2395, 14601}, {2485, 20968}, {2501, 1974}, {2525, 20775}, {2592, 44124}, {2593, 44123}, {2618, 2179}, {2623, 14573}, {2643, 1924}, {2799, 237}, {2970, 2489}, {2973, 43925}, {3005, 41331}, {3049, 40373}, {3112, 34072}, {3120, 1919}, {3124, 9426}, {3125, 1980}, {3260, 2420}, {3261, 58}, {3265, 577}, {3266, 5467}, {3267, 3}, {3268, 50}, {3569, 9418}, {3596, 5546}, {3676, 16947}, {3700, 2175}, {3709, 9448}, {3952, 23990}, {4010, 14599}, {4024, 1918}, {4033, 1110}, {4036, 213}, {4041, 9447}, {4064, 2200}, {4077, 604}, {4080, 32719}, {4086, 41}, {4088, 9454}, {4120, 9459}, {4122, 40728}, {4143, 1092}, {4391, 2194}, {4455, 18894}, {4566, 23979}, {4580, 10547}, {4602, 24041}, {4609, 4590}, {4705, 2205}, {5027, 18902}, {5392, 32734}, {5466, 32740}, {6063, 4565}, {6330, 32649}, {6331, 250}, {6333, 3289}, {6368, 217}, {6386, 4567}, {6528, 23964}, {6530, 34859}, {6563, 571}, {7178, 1397}, {7180, 41280}, {7199, 849}, {8024, 1634}, {8029, 1084}, {8061, 1923}, {8673, 22075}, {9148, 33875}, {9178, 19626}, {9464, 9145}, {9979, 18374}, {10412, 11060}, {11140, 32737}, {12077, 40981}, {14208, 48}, {14295, 1691}, {14316, 19558}, {14417, 23200}, {14603, 17941}, {14616, 32671}, {14618, 25}, {14638, 14379}, {14977, 14908}, {15413, 1437}, {15414, 19210}, {15415, 5}, {15449, 2531}, {15526, 39201}, {16080, 32715}, {16081, 32696}, {16230, 2211}, {16732, 667}, {17096, 7342}, {17434, 44088}, {17879, 822}, {17924, 2203}, {18022, 648}, {18023, 691}, {18024, 2966}, {18027, 107}, {18155, 2150}, {18160, 17104}, {18312, 5191}, {18314, 51}, {18808, 40354}, {18833, 4599}, {18896, 805}, {18901, 880}, {20336, 906}, {20567, 1414}, {20571, 36145}, {20573, 476}, {20902, 810}, {20948, 1}, {21178, 17186}, {21207, 649}, {21832, 18892}, {22260, 9427}, {23105, 3124}, {23107, 2972}, {23285, 39}, {23290, 3199}, {23616, 34980}, {23870, 34394}, {23871, 34395}, {23878, 34396}, {23881, 23208}, {23962, 523}, {23978, 21789}, {23989, 3733}, {23994, 661}, {24002, 1408}, {24006, 1973}, {24290, 9455}, {26235, 35357}, {27801, 100}, {27808, 1252}, {28654, 4557}, {28659, 643}, {28660, 4636}, {28706, 23181}, {30474, 5063}, {30713, 3939}, {30730, 6066}, {30735, 40825}, {30736, 5118}, {30870, 40773}, {33294, 206}, {33805, 36034}, {34087, 32717}, {34289, 32738}, {34384, 18315}, {34385, 32692}, {34386, 15958}, {34387, 7252}, {34388, 4559}, {34389, 16806}, {34390, 16807}, {34391, 39383}, {34392, 39384}, {34767, 18877}, {35352, 1922}, {35442, 42293}, {35518, 2193}, {35519, 284}, {35522, 187}, {35524, 41337}, {35550, 1983}, {36035, 9406}, {36793, 520}, {36901, 3050}, {39691, 688}, {40016, 4577}, {40050, 4563}, {40071, 1331}, {40072, 4612}, {40073, 4611}, {40359, 4609}, {40362, 670}, {40363, 645}, {40364, 4592}, {40495, 81}, {40822, 35278}, {40826, 11636}, {40828, 931}, {40832, 10420}, {41000, 35329}, {41001, 35330}, {41079, 1495}, {41167, 9419}, {41283, 4573}, {41298, 2965}, {42331, 1970}, {42761, 23220}, {43665, 1976}, {44132, 4230}

X(44174) = ISOGONAL CONJUGATE OF X(136)

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

X(44174) lies on the cubic K568, the curve Q120, and these lines: {68, 39170}, {249, 14587}, {250, 403}, {523, 925}, {924, 4558}, {3003, 23357}, {3564, 13557}, {14264, 14366}

X(44174) = isogonal conjugate of X(136)
X(44174) = isogonal conjugate of the anticomplement of X(34844)
X(44174) = isogonal conjugate of the complement of X(925)
X(44174) = anticomplement of complementary conjugate of X(34844)
X(44174) = X(i)-cross conjugate of X(j) for these (i,j): {3, 925}, {6, 4558}, {155, 110}, {184, 32692}, {3564, 43754}, {10132, 39384}, {10133, 39383}, {10661, 38414}, {10662, 38413}, {13557, 10420}, {23128, 4563}
X(44174) = X(i)-isoconjugate of X(j) for these (i,j): {1, 136}, {24, 1109}, {25, 17881}, {47, 2970}, {91, 34338}, {115, 1748}, {135, 921}, {317, 2643}, {924, 24006}, {1577, 6753}, {3708, 11547}, {6754, 20571}, {8745, 20902}, {23994, 44077}
X(44174) = cevapoint of X(i) and X(j) for these (i,j): {6, 32734}, {110, 5889}, {1576, 1609}
X(44174) = trilinear pole of line {686, 23181}
X(44174) = barycentric product X(i)*X(j) for these {i,j}: {68, 249}, {925, 4558}, {1820, 24041}, {2351, 4590}, {4563, 32734}, {4592, 36145}, {16391, 23582}, {20563, 23357}
X(44174) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 136}, {63, 17881}, {68, 338}, {249, 317}, {250, 11547}, {571, 34338}, {925, 14618}, {1101, 1748}, {1576, 6753}, {1609, 135}, {1820, 1109}, {2165, 2970}, {2351, 115}, {4558, 6563}, {16391, 15526}, {20563, 23962}, {23357, 24}, {23963, 44077}, {32661, 924}, {32662, 43088}, {32734, 2501}, {36145, 24006}

X(44175) = ISOGONAL CONJUGATE OF X(157)

Barycentrics    (a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 - b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(44175) = 3 X[2] - 4 X[14725]

X(44174) lies on these lines: {2, 11610}, {22, 160}, {69, 41765}, {110, 34405}, {297, 1993}, {315, 2979}, {317, 3060}, {850, 1899}, {4463, 42703}, {9544, 13485}, {11442, 18022}

X(44175) = reflection of X(22391) in X(14725)
X(44175) = isogonal conjugate of X(157)
X(44175) = isotomic conjugate of X(11442)
X(44175) = anticomplement of X(22391)
X(44175) = isogonal conjugate of the anticomplement of X(23333)
X(44175) = isogonal conjugate of the complement of X(41761)
X(44175) = isotomic conjugate of the anticomplement of X(184)
X(44175) = isotomic conjugate of the isogonal conjugate of X(1485)
X(44175) = X(184)-cross conjugate of X(2)
X(44175) = X(i)-isoconjugate of X(j) for these (i,j): {1, 157}, {6, 21374}, {19, 23128}, {31, 11442}, {32, 21593}, {75, 2909}, {92, 22391}
X(44175) = cevapoint of X(i) and X(j) for these (i,j): {127, 520}, {524, 34517}, {35088, 39469}
X(44175) = crosssum of X(2909) and X(22391)
X(44175) = trilinear pole of line {2485, 2799}
X(44175) = barycentric product X(76)*X(1485)
X(44175) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21374}, {2, 11442}, {3, 23128}, {6, 157}, {32, 2909}, {75, 21593}, {184, 22391}, {1485, 6}, {41765, 41760}
X(44175) = {X(14725),X(22391)}-harmonic conjugate of X(2)

X(44176) = ISOGONAL CONJUGATE OF X(160)

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

X(44176) lies on these lines: {2, 40588}, {22, 157}, {25, 36901}, {251, 41760}, {264, 5012}, {290, 3060}, {315, 2387}, {324, 458}, {338, 1501}, {850, 1853}, {1297, 18018}, {2979, 34384}, {4463, 42711}, {11550, 23962}

X(44176) = isogonal conjugate of X(160)
X(44176) = isotomic conjugate of X(2979)
X(44176) = anticomplement of X(40588)
X(44176) = polar conjugate of X(39575)
X(44176) = isogonal conjugate of the anticomplement of X(34845)
X(44176) = isotomic conjugate of the anticomplement of X(51)
X(44176) = isotomic conjugate of the isogonal conjugate of X(2980)
X(44176) = X(i)-cross conjugate of X(j) for these (i,j): {51, 2}, {5007, 308}, {27366, 2980}, {34396, 42354}, {42671, 16081}
X(44176) = X(i)-isoconjugate of X(j) for these (i,j): {1, 160}, {31, 2979}, {48, 39575}, {75, 3202}, {560, 7796}, {2148, 41480}, {2167, 40588}
X(44176) = cevapoint of X(i) and X(j) for these (i,j): {127, 6368}, {338, 512}
X(44176) = crosssum of X(23208) and X(34452)
X(44176) = trilinear pole of line {2485, 18314}
X(44176) = barycentric product X(i)*X(j) for these {i,j}: {76, 2980}, {308, 27366}
X(44176) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2979}, {4, 39575}, {5, 41480}, {6, 160}, {32, 3202}, {51, 40588}, {76, 7796}, {1676, 1670}, {1677, 1671}, {2980, 6}, {3199, 15897}, {16732, 18188}, {27366, 39}

X(44177) = ISOGONAL CONJUGATE OF X(161)

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

X(44177) lies on these lines: {69, 7488}, {253, 7487}, {264, 7544}, {317, 20564}, {328, 13219}, {11090, 11417}, {11091, 11418}, {15526, 36416}, {18019, 32001}, {20563, 37444}

X(44177) = isogonal conjugate of X(161)
X(44177) = isotomic conjugate of X(37444)
X(44177) = isotomic conjugate of the anticomplement of X(24)
X(44177) = isotomic conjugate of the complement of X(31304)
X(44177) = isotomic conjugate of the isogonal conjugate of X(34438)
X(44177) = X(i)-cross conjugate of X(j) for these (i,j): {24, 2}, {70, 13579}
X(44177) = X(i)-isoconjugate of X(j) for these (i,j): {1, 161}, {6, 18595}, {31, 37444}, {41, 18628}
X(44177) = cevapoint of X(i) and X(j) for these (i,j): {2, 31304}, {924, 15526}
X(44177) = trilinear pole of line {525, 16040}
X(44177) = barycentric product X(76)*X(34438)
X(44177) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 18595}, {2, 37444}, {6, 161}, {7, 18628}, {1993, 8907}, {34438, 6}

X(44178) = ISOGONAL CONJUGATE OF X(169)

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

X(44178) lies on these lines: {1, 36057}, {3, 518}, {9, 34847}, {19, 14377}, {27, 21370}, {57, 5236}, {58, 30733}, {63, 3730}, {85, 169}, {101, 7131}, {103, 26706}, {116, 7079}, {218, 222}, {295, 22116}, {514, 2082}, {1445, 4253}, {1790, 18206}, {2725, 35185}, {4251, 39273}, {8257, 36949}, {17206, 18157}, {17758, 40131}, {26934, 42467}, {32462, 39344}

X(44178) = isogonal conjugate of X(169)
X(44178) = isotomic conjugate of X(20927)
X(44178) = isogonal conjugate of the anticomplement of X(34847)
X(44178) = isogonal conjugate of the complement of X(17170)
X(44178) = X(41)-cross conjugate of X(1)
X(44178) = X(i)-isoconjugate of X(j) for these (i,j): {1, 169}, {2, 1486}, {3, 17905}, {4, 22131}, {6, 3434}, {7, 5452}, {9, 34036}, {25, 28420}, {31, 20927}, {37, 4228}, {55, 37800}, {58, 21073}, {81, 21867}, {101, 21185}, {218, 14268}, {650, 40576}, {651, 11934}, {692, 26546}, {2298, 41581}, {3052, 27826}, {18098, 41582}
X(44178) = cevapoint of X(i) and X(j) for these (i,j): {6, 1473}, {650, 17463}, {657, 7004}, {798, 17476}, {3433, 40141}
X(44178) = crosssum of X(1486) and X(5452)
X(44178) = trilinear pole of line {1459, 2254}
X(44178) = barycentric product X(i)*X(j) for these {i,j}: {1, 13577}, {75, 3433}, {85, 40141}, {100, 26721}, {4025, 26706}, {7123, 41788}
X(44178) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3434}, {2, 20927}, {6, 169}, {19, 17905}, {31, 1486}, {37, 21073}, {41, 5452}, {42, 21867}, {48, 22131}, {56, 34036}, {57, 37800}, {58, 4228}, {63, 28420}, {109, 40576}, {513, 21185}, {514, 26546}, {663, 11934}, {1193, 41581}, {2191, 14268}, {3433, 1}, {8056, 27826}, {13577, 75}, {17187, 41582}, {26706, 1897}, {26721, 693}, {35185, 36086}, {40141, 9}

X(44179) = ISOTOMIC CONJUGATE OF X(91)

Barycentrics    a*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
Barycentrics    cos A cot 2A : :
Trilinears    cot A cot 2A : :

X(44179) lies on these lines: {1, 75}, {19, 662}, {48, 1760}, {63, 2148}, {69, 1442}, {77, 320}, {78, 319}, {92, 31631}, {273, 664}, {306, 19795}, {318, 20570}, {344, 26668}, {560, 17467}, {610, 16568}, {914, 28793}, {997, 5224}, {1332, 27396}, {1444, 3869}, {1953, 1958}, {1969, 20571}, {1993, 42700}, {2324, 17336}, {3083, 32791}, {3084, 32792}, {3553, 3758}, {3554, 3759}, {3666, 26625}, {3811, 17377}, {3872, 5564}, {3879, 22836}, {3912, 28738}, {4000, 26639}, {4357, 30144}, {4861, 42696}, {5552, 7318}, {5738, 34772}, {6350, 28935}, {6505, 17923}, {6510, 28965}, {7190, 7321}, {7269, 42697}, {16678, 18614}, {16876, 18611}, {17073, 37796}, {17134, 17139}, {17136, 17220}, {17181, 21276}, {17221, 20245}, {17263, 25930}, {17322, 19861}, {17776, 37645}, {17791, 25716}, {18049, 18596}, {18151, 27384}, {18161, 20769}, {19860, 28653}, {20883, 21593}, {21277, 37700}, {25538, 30140}, {26637, 28606}

X(44179) = isotomic conjugate of X(91)
X(44179) = isotomic conjugate of the isogonal conjugate of X(47)
X(44179) = isotomic conjugate of the polar conjugate of X(1748)
X(44179) = X(7130)-anticomplementary conjugate of X(2475)
X(44179) = X(i)-Ceva conjugate of X(j) for these (i,j): {1969, 63}, {40440, 75}
X(44179) = X(i)-cross conjugate of X(j) for these (i,j): {47, 1748}, {42700, 7763}
X(44179) = X(i)-isoconjugate of X(j) for these (i,j): {3, 14593}, {4, 2351}, {5, 41271}, {6, 2165}, {19, 1820}, {25, 68}, {31, 91}, {32, 5392}, {51, 96}, {184, 847}, {485, 8576}, {486, 8577}, {512, 925}, {523, 32734}, {560, 20571}, {661, 36145}, {1799, 27367}, {1953, 2168}, {1974, 20563}, {3049, 30450}, {3426, 40348}, {6413, 41516}, {6414, 41515}, {6524, 16391}, {11060, 37802}, {12077, 32692}, {34385, 40981}, {34428, 39111}, {34853, 39109}
X(44179) = crosspoint of X(811) and X(24041)
X(44179) = crosssum of X(810) and X(2643)
X(44179) = barycentric product X(i)*X(j) for these {i,j}: {1, 7763}, {24, 304}, {47, 76}, {63, 317}, {69, 1748}, {75, 1993}, {86, 42700}, {92, 9723}, {249, 17881}, {313, 18605}, {326, 11547}, {561, 571}, {563, 18022}, {662, 6563}, {799, 924}, {1147, 1969}, {1959, 31635}, {1978, 34948}, {2167, 39113}, {2180, 34384}, {4602, 34952}, {14208, 41679}, {40364, 44077}
X(44179) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2165}, {2, 91}, {3, 1820}, {19, 14593}, {24, 19}, {47, 6}, {48, 2351}, {52, 1953}, {54, 2168}, {63, 68}, {75, 5392}, {76, 20571}, {92, 847}, {110, 36145}, {163, 32734}, {304, 20563}, {317, 92}, {563, 184}, {571, 31}, {662, 925}, {811, 30450}, {924, 661}, {1147, 48}, {1599, 3378}, {1600, 3377}, {1748, 4}, {1993, 1}, {2148, 41271}, {2167, 96}, {2180, 51}, {3133, 2180}, {6507, 16391}, {6563, 1577}, {7763, 75}, {8745, 1096}, {8907, 18595}, {9723, 63}, {11547, 158}, {12095, 2314}, {14576, 2181}, {17881, 338}, {18605, 58}, {18883, 2166}, {30451, 810}, {31635, 1821}, {33808, 39116}, {34948, 649}, {34952, 798}, {36134, 32692}, {39113, 14213}, {41679, 162}, {41770, 17871}, {42700, 10}, {44077, 1973}
X(44179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 326, 75}, {48, 1959, 1760}, {75, 18156, 33808}, {610, 18713, 16568}, {662, 18041, 19}, {1442, 4511, 69}, {14210, 17859, 18695}, {17859, 18695, 75}

X(44180) = ISOTOMIC CONJUGATE OF X(93)

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

X(44180) lies on these lines: {2, 9609}, {3, 69}, {6, 35296}, {22, 1007}, {25, 34803}, {26, 32829}, {75, 4996}, {95, 99}, {141, 15109}, {160, 33801}, {183, 15246}, {186, 317}, {193, 8553}, {216, 4558}, {253, 35493}, {264, 3520}, {302, 11141}, {303, 11142}, {325, 6636}, {340, 17506}, {343, 34433}, {577, 22151}, {1078, 1232}, {1176, 43705}, {1273, 15620}, {1583, 32812}, {1584, 32813}, {1599, 32805}, {1600, 32806}, {1609, 1992}, {1975, 37126}, {1994, 2965}, {2071, 20477}, {3432, 7488}, {3518, 7769}, {3629, 11063}, {4357, 14792}, {5224, 37293}, {5562, 20574}, {6148, 22468}, {6467, 14060}, {6503, 37068}, {7393, 36948}, {7485, 34229}, {7496, 14360}, {7506, 32839}, {7512, 7763}, {7514, 32815}, {7516, 32828}, {7550, 11185}, {7752, 12088}, {7771, 44149}, {8266, 39099}, {8797, 9818}, {9734, 11188}, {10323, 32006}, {10607, 36751}, {12083, 32827}, {13154, 32867}, {13595, 37647}, {14118, 44135}, {14558, 14615}, {14793, 17321}, {15031, 40410}, {17500, 35919}, {20572, 20573}, {20806, 36748}, {21844, 32001}, {22052, 36212}, {22085, 37893}, {23333, 36163}, {27377, 41758}, {28408, 37188}, {32000, 35473}, {32835, 38435}, {43459, 44148}

X(44180) = isotomic conjugate of X(93)
X(44180) = isotomic conjugate of the anticomplement of X(34833)
X(44180) = isotomic conjugate of the isogonal conjugate of X(49)
X(44180) = isotomic conjugate of the polar conjugate of X(1994)
X(44180) = isogonal conjugate of the polar conjugate of X(7769)
X(44180) = X(i)-Ceva conjugate of X(j) for these (i,j): {7769, 1994}, {20573, 323}
X(44180) = X(i)-cross conjugate of X(j) for these (i,j): {49, 1994}, {34833, 2}
X(44180) = X(i)-isoconjugate of X(j) for these (i,j): {19, 2963}, {25, 2962}, {31, 93}, {252, 2181}, {560, 20572}, {798, 38342}, {1096, 3519}, {1973, 11140}, {2501, 36148}, {24006, 32737}
X(44180) = crosssum of X(512) and X(41221)
X(44180) = barycentric product X(i)*X(j) for these {i,j}: {3, 7769}, {49, 76}, {69, 1994}, {143, 34386}, {304, 2964}, {305, 2965}, {394, 32002}, {1510, 4563}, {3518, 3926}, {4558, 41298}, {6331, 37084}, {25044, 28706}
X(44180) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 93}, {3, 2963}, {49, 6}, {61, 8742}, {62, 8741}, {63, 2962}, {69, 11140}, {76, 20572}, {97, 252}, {99, 38342}, {143, 53}, {323, 562}, {343, 25043}, {394, 3519}, {1493, 6748}, {1510, 2501}, {1993, 14111}, {1994, 4}, {2964, 19}, {2965, 25}, {3518, 393}, {4558, 930}, {4575, 36148}, {7769, 264}, {14129, 13450}, {14577, 14569}, {20577, 23290}, {25044, 8882}, {30529, 6344}, {31626, 1487}, {32002, 2052}, {32661, 32737}, {37084, 647}, {41298, 14618}
X(44180) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 9723, 69}, {95, 99, 311}, {99, 311, 18354}, {4996, 7279, 75}, {10607, 36751, 41614}

X(44181) = ISOTOMIC CONJUGATE OF X(122)

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

X(44181) lies on these lines: {2, 23590}, {107, 3265}, {253, 42308}, {441, 23582}, {1301, 22456}, {6528, 39464}, {6529, 14341}, {15384, 18020}, {16096, 32230}, {34407, 39297}

X(44181) = isotomic conjugate of X(122)
X(44181) = polar conjugate of X(1562)
X(44181) = isotomic conjugate of the anticomplement of X(6716)
X(44181) = isotomic conjugate of the complement of X(107)
X(44181) = isotomic conjugate of the isogonal conjugate of X(15384)
X(44181) = X(i)-cross conjugate of X(j) for these (i,j): {20, 648}, {69, 6528}, {2060, 36841}, {2897, 18026}, {6527, 99}, {6716, 2}, {20477, 18831}, {32230, 23582}, {34170, 15459}
X(44181) = X(i)-isoconjugate of X(j) for these (i,j): {31, 122}, {48, 1562}, {154, 2632}, {204, 2972}, {610, 3269}, {656, 42658}, {798, 20580}, {810, 8057}, {822, 6587}, {1249, 37754}, {1895, 34980}, {2155, 39020}, {2643, 35602}, {3708, 15905}, {14249, 42080}, {17898, 39201}
X(44181) = cevapoint of X(i) and X(j) for these (i,j): {2, 107}, {20, 648}, {459, 1301}, {1968, 32713}
X(44181) = trilinear pole of line {648, 2404}
X(44181) = barycentric product X(i)*X(j) for these {i,j}: {76, 15384}, {253, 23582}, {459, 18020}, {1301, 6331}, {2184, 23999}, {4590, 6526}, {15394, 34538}, {23964, 41530}, {32230, 34403}
X(44181) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 122}, {4, 1562}, {20, 39020}, {64, 3269}, {99, 20580}, {107, 6587}, {112, 42658}, {249, 35602}, {250, 15905}, {253, 15526}, {459, 125}, {648, 8057}, {823, 17898}, {1073, 2972}, {1301, 647}, {2184, 2632}, {4240, 14345}, {6526, 115}, {6616, 13613}, {13157, 35442}, {14379, 35071}, {14572, 13611}, {14642, 34980}, {15384, 6}, {18020, 37669}, {19614, 37754}, {23582, 20}, {23590, 6525}, {23964, 154}, {23999, 18750}, {24000, 610}, {32230, 1249}, {34538, 14249}, {38956, 39008}, {41489, 20975}, {41530, 36793}

X(44182) = ISOTOMIC CONJUGATE OF X(126)

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

X(44182) lies on these lines: {2, 34161}, {111, 3266}, {468, 691}, {523, 15398}, {524, 9225}, {892, 3291}, {4590, 10418}, {5967, 40132}, {8585, 40826}, {9172, 18823}

X(44182) = isotomic conjugate of X(126)
X(44182) = isotomic conjugate of the anticomplement of X(6719)
X(44182) = isotomic conjugate of the complement of X(111)
X(44182) = isotomic conjugate of the isogonal conjugate of X(15387)
X(44182) = X(i)-cross conjugate of X(j) for these (i,j): {69, 671}, {512, 892}, {6388, 5466}, {6719, 2}, {34161, 41909}
X(44182) = X(i)-isoconjugate of X(j) for these (i,j): {6, 17466}, {31, 126}, {662, 21905}, {896, 3291}, {2642, 11634}, {14263, 42081}
X(44182) = cevapoint of X(i) and X(j) for these (i,j): {2, 111}, {115, 14977}, {34161, 41909}
X(44182) = trilinear pole of line {690, 895}
X(44182) = barycentric product X(i)*X(j) for these {i,j}: {76, 15387}, {671, 41909}, {2374, 30786}, {9154, 36892}
X(44182) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17466}, {2, 126}, {111, 3291}, {512, 21905}, {691, 11634}, {895, 8681}, {2374, 468}, {5466, 9134}, {8753, 5140}, {9154, 36874}, {10630, 14263}, {15387, 6}, {34161, 2482}, {41909, 524}

X(44183) = ISOTOMIC CONJUGATE OF X(127)

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

X(44183) lies on these lines: {2, 41937}, {112, 3267}, {249, 15388}, {250, 858}, {691, 1289}, {2966, 41679}, {4143, 15639}, {5649, 16237}, {15013, 23582}, {17907, 32230}, {39295, 43678}

X(44183) = isogonal conjugate of X(38356)
X(44183) = isotomic conjugate of X(127)
X(44183) = isotomic conjugate of the anticomplement of X(6720)
X(44183) = isotomic conjugate of the complement of X(112)
X(44183) = isotomic conjugate of the isogonal conjugate of X(15388)
X(44183) = X(i)-cross conjugate of X(j) for these (i,j): {69, 648}, {1370, 99}, {2979, 18831}, {6720, 2}, {10316, 110}, {11442, 6528}, {12220, 4577}, {13854, 1289}, {28696, 4563}, {28766, 4554}, {30737, 2966}, {41363, 685}
X(44183) = X(i)-isoconjugate of X(j) for these (i,j): {1, 38356}, {22, 3708}, {31, 127}, {42, 18187}, {125, 2172}, {206, 20902}, {339, 17453}, {656, 2485}, {661, 8673}, {810, 33294}, {1109, 10316}, {1760, 20975}, {2632, 8743}, {2643, 20806}, {4456, 18210}, {17409, 17879}, {22075, 23994}
X(44183) = cevapoint of X(i) and X(j) for these (i,j): {2, 112}, {3, 35325}, {110, 10316}, {441, 15639}, {1289, 13854}, {1560, 4235}, {4558, 28419}, {14966, 15595}
X(44183) = trilinear pole of line {110, 1289}
X(44183) = trilinear product X(i)*X(j) for these {i,j}: {75, 15388}, {662, 1289}
X(44183) = barycentric product X(i)*X(j) for these {i,j}: {66, 18020}, {76, 15388}, {99, 1289}, {249, 43678}, {250, 18018}, {4590, 13854}, {14376, 23582}
X(44183) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 127}, {6, 38356}, {66, 125}, {81, 18187}, {110, 8673}, {112, 2485}, {249, 20806}, {250, 22}, {648, 33294}, {1289, 523}, {2156, 3708}, {2353, 20975}, {2420, 14396}, {4590, 34254}, {5379, 4463}, {13854, 115}, {14376, 15526}, {15388, 6}, {18018, 339}, {18020, 315}, {23357, 10316}, {23582, 17907}, {23963, 22075}, {23964, 8743}, {41168, 35442}, {41676, 23881}, {41937, 17409}, {43678, 338}

X(44184) = ISOTOMIC CONJUGATE OF X(150)

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

X(44184) lies on these lines: {2, 39026}, {149, 7253}, {150, 3261}, {513, 33650}, {517, 16086}, {518, 1875}, {521, 25048}, {859, 34179}, {1457, 1818}, {2183, 3006}, {4388, 36278}, {5730, 6790}

X(44184) = isogonal conjugate of X(20999)
X(44184) = isotomic conjugate of X(150)
X(44184) = anticomplement of X(39026)
X(44184) = cyclocevian conjugate of X(190)
X(44184) = isotomic conjugate of the anticomplement of X(101)
X(44184) = isotomic conjugate of the complement of X(20096)
X(44184) = isotomic conjugate of the isogonal conjugate of X(34179)
X(44184) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {100, 14888}, {34179, 17494}
X(44184) = X(101)-cross conjugate of X(2)
X(44184) = X(i)-isoconjugate of X(j) for these (i,j): {1, 20999}, {6, 16560}, {19, 22145}, {31, 150}, {32, 20940}, {58, 22321}, {100, 8578}, {244, 14887}, {292, 27943}, {513, 39026}, {692, 21202}, {1333, 21091}
X(44184) = cevapoint of X(i) and X(j) for these (i,j): {2, 20096}, {8, 17777}, {101, 40150}, {519, 3259}
X(44184) = trilinear pole of line {3310, 13006}
X(44184) = barycentric product X(i)*X(j) for these {i,j}: {76, 34179}, {3261, 40150}
X(44184) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16560}, {2, 150}, {3, 22145}, {6, 20999}, {10, 21091}, {37, 22321}, {75, 20940}, {101, 39026}, {238, 27943}, {514, 21202}, {649, 8578}, {1252, 14887}, {34179, 6}, {40150, 101}

X(44185) = ISOTOMIC CONJUGATE OF X(160)

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

X(44185) lies on these lines: {32, 23962}, {76, 41480}, {315, 2387}, {1078, 18022}, {1502, 7917}, {3001, 40073}, {4150, 4174}, {7752, 18024}, {17907, 32832}, {35140, 40421}

X(44185) = isogonal conjugate of X(3202)
X(44185) = isotomic conjugate of X(160)
X(44185) = isotomic conjugate of the anticomplement of X(34845)
X(44185) = X(i)-cross conjugate of X(j) for these (i,j): {5, 76}, {3589, 40016}, {23300, 2052}, {23333, 5392}, {34845, 2}
X(44185) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3202}, {31, 160}, {560, 2979}, {1917, 7796}, {2148, 40588}, {2169, 15897}, {9247, 39575}
X(44185) = cevapoint of X(i) and X(j) for these (i,j): {76, 7814}, {523, 23962}, {1676, 1677}
X(44185) = trilinear pole of line {15415, 33294}
X(44185) = barycentric product X(i)*X(j) for these {i,j}: {1502, 2980}, {27366, 40016}
X(44185) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 160}, {5, 40588}, {6, 3202}, {53, 15897}, {76, 2979}, {264, 39575}, {311, 41480}, {1502, 7796}, {1676, 41379}, {1677, 41378}, {2980, 32}, {27366, 3051}

X(44186) = ISOTOMIC CONJUGATE OF X(165)

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

X(44186) lies on these lines: {75, 31627}, {312, 10405}, {314, 3062}, {1088, 24026}, {1699, 18025}, {3596, 35517}, {18738, 34258}, {18743, 19605}, {20935, 42034}, {20942, 33677}, {25507, 31623}, {33672, 40422}

X(44186) = isotomic conjugate of X(165)
X(44186) = isotomic conjugate of the anticomplement of X(3817)
X(44186) = isotomic conjugate of the complement of X(9812)
X(44186) = isotomic conjugate of the isogonal conjugate of X(3062)
X(44186) = X(i)-cross conjugate of X(j) for these (i,j): {85, 75}, {3817, 2}, {20905, 76}, {32023, 6384}
X(44186) = X(i)-isoconjugate of X(j) for these (i,j): {6, 3207}, {25, 22117}, {31, 165}, {32, 144}, {41, 1419}, {560, 16284}, {1253, 17106}, {1333, 21872}, {1475, 33634}, {2175, 3160}, {2206, 21060}, {7658, 32739}, {9447, 31627}, {9533, 14827}, {13609, 23979}
X(44186) = cevapoint of X(i) and X(j) for these (i,j): {2, 9812}, {693, 24026}
X(44186) = trilinear pole of line {4391, 20907}
X(44186) = barycentric product X(i)*X(j) for these {i,j}: {75, 10405}, {76, 3062}, {312, 36620}, {561, 11051}, {6063, 19605}
X(44186) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3207}, {2, 165}, {7, 1419}, {10, 21872}, {63, 22117}, {75, 144}, {76, 16284}, {85, 3160}, {279, 17106}, {321, 21060}, {693, 7658}, {1088, 9533}, {2346, 33634}, {3062, 6}, {6063, 31627}, {10405, 1}, {11051, 31}, {17862, 41561}, {19605, 55}, {20905, 43182}, {24026, 13609}, {36620, 57}, {42872, 221}

X(44187) = ISOTOMIC CONJUGATE OF X(172)

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

X(44187) lies on these lines: {75, 256}, {76, 20236}, {239, 7104}, {257, 1921}, {304, 1432}, {313, 1934}, {314, 4594}, {561, 35524}, {789, 17797}, {1228, 28659}, {1240, 1441}, {1916, 18895}, {1930, 3865}, {1969, 44132}, {3261, 21110}, {3596, 6382}, {4451, 20895}, {6383, 33930}, {6385, 16732}, {7019, 40717}, {9230, 17788}, {18891, 21442}, {20892, 27447}

X(44187) = isotomic conjugate of X(172)
X(44187) = isotomic conjugate of the isogonal conjugate of X(257)
X(44187) = polar conjugate of the isogonal conjugate of X(7019)
X(44187) = X(i)-cross conjugate of X(j) for these (i,j): {18891, 18895}, {21442, 75}
X(44187) = X(i)-isoconjugate of X(j) for these (i,j): {6, 7122}, {31, 172}, {32, 171}, {163, 7234}, {184, 7119}, {291, 14602}, {292, 1933}, {334, 18902}, {385, 14598}, {560, 894}, {604, 2330}, {692, 20981}, {1101, 21823}, {1333, 20964}, {1397, 2329}, {1501, 1909}, {1580, 1922}, {1691, 1911}, {1917, 1920}, {1919, 4579}, {1927, 27982}, {1966, 18897}, {1973, 3955}, {1980, 18047}, {2175, 7175}, {2196, 44089}, {2203, 22061}, {2205, 17103}, {2206, 2295}, {3978, 18893}, {4367, 32739}, {4570, 21755}, {7009, 9247}, {7176, 9447}, {7196, 9448}, {14599, 18787}, {17787, 41280}, {18262, 41534}, {18892, 30669}, {18900, 40745}, {21725, 23357}
X(44187) = cevapoint of X(i) and X(j) for these (i,j): {75, 17788}, {257, 7019}, {312, 33938}, {3004, 21138}, {3261, 16732}
X(44187) = trilinear pole of line {29017, 35519}
X(44187) = barycentric product X(i)*X(j) for these {i,j}: {75, 7018}, {76, 257}, {239, 18896}, {256, 561}, {264, 7019}, {313, 32010}, {350, 1934}, {694, 44169}, {850, 4594}, {893, 1502}, {904, 1928}, {1431, 40363}, {1432, 28659}, {1577, 7260}, {1581, 18891}, {1914, 44160}, {1916, 1921}, {1920, 40099}, {1967, 44171}, {3261, 27805}, {3596, 7249}, {3865, 7034}, {3903, 40495}, {4451, 6063}, {4603, 20948}, {6382, 27447}, {7015, 18022}, {7104, 40362}, {17493, 18895}, {18786, 44172}, {20234, 40835}, {27801, 40432}, {30670, 30870}, {40708, 40717}
X(44187) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7122}, {2, 172}, {8, 2330}, {10, 20964}, {69, 3955}, {75, 171}, {76, 894}, {85, 7175}, {92, 7119}, {115, 21823}, {238, 1933}, {239, 1691}, {242, 44089}, {256, 31}, {257, 6}, {264, 7009}, {306, 22061}, {310, 17103}, {312, 2329}, {313, 1215}, {321, 2295}, {334, 18787}, {349, 4032}, {350, 1580}, {514, 20981}, {523, 7234}, {561, 1909}, {668, 4579}, {693, 4367}, {694, 1922}, {850, 2533}, {893, 32}, {904, 560}, {1089, 21803}, {1109, 21725}, {1178, 2206}, {1228, 27697}, {1269, 4697}, {1431, 1397}, {1432, 604}, {1502, 1920}, {1581, 1911}, {1914, 14602}, {1916, 292}, {1920, 6645}, {1921, 385}, {1927, 18893}, {1934, 291}, {1967, 14598}, {1978, 18047}, {3120, 4128}, {3125, 21755}, {3261, 4369}, {3263, 4447}, {3264, 4434}, {3266, 7267}, {3596, 7081}, {3766, 4164}, {3865, 7032}, {3903, 692}, {3954, 21752}, {3978, 27982}, {4010, 5027}, {4025, 22093}, {4391, 3287}, {4397, 4477}, {4451, 55}, {4486, 30654}, {4496, 2241}, {4572, 6649}, {4594, 110}, {4603, 163}, {6063, 7176}, {6382, 17752}, {6385, 8033}, {7015, 184}, {7018, 1}, {7019, 3}, {7081, 10799}, {7104, 1501}, {7116, 9247}, {7199, 18200}, {7249, 56}, {7260, 662}, {7303, 849}, {8024, 16720}, {9468, 18897}, {14599, 18902}, {15523, 40936}, {16732, 16592}, {17493, 1914}, {17788, 40597}, {18036, 7061}, {18210, 22373}, {18786, 2210}, {18835, 17797}, {18891, 1966}, {18895, 30669}, {18896, 335}, {20234, 18905}, {20567, 7196}, {20906, 24533}, {20911, 28369}, {23989, 7200}, {27447, 2162}, {27801, 3963}, {27805, 101}, {28654, 21021}, {28659, 17787}, {28660, 27958}, {30643, 18786}, {30670, 34069}, {30713, 4095}, {30966, 40731}, {32010, 58}, {33930, 7184}, {33931, 40790}, {34387, 4459}, {34388, 7211}, {35519, 3907}, {35544, 4039}, {37137, 1415}, {40071, 4019}, {40098, 30657}, {40099, 893}, {40106, 40096}, {40432, 1333}, {40495, 4374}, {40708, 295}, {40717, 419}, {40729, 2205}, {40738, 40746}, {40845, 41534}, {40849, 18278}, {40873, 19554}, {41283, 7205}, {41532, 18262}, {43263, 7031}, {44130, 14006}, {44160, 18895}, {44169, 3978}, {44171, 1926}

X(44188) = ISOTOMIC CONJUGATE OF X(191)

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)*(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(44188) lies on these lines: {75, 267}, {92, 17394}, {313, 502}, {319, 321}, {561, 20450}, {757, 1109}, {1441, 17095}, {18040, 20648}, {18133, 20939}, {18151, 25660}

X(44188) = isotomic conjugate of X(191)
X(44188) = isotomic conjugate of the anticomplement of X(11263)
X(44188) = isotomic conjugate of the complement of X(14450)
X(44188) = isotomic conjugate of the isogonal conjugate of X(267)
X(44188) = X(i)-cross conjugate of X(j) for these (i,j): {86, 75}, {502, 1029}, {11263, 2}, {30690, 85}
X(44188) = X(i)-isoconjugate of X(j) for these (i,j): {3, 44097}, {6, 1030}, {25, 22136}, {31, 191}, {32, 2895}, {42, 501}, {55, 8614}, {110, 42653}, {184, 451}, {213, 40592}, {228, 2906}, {560, 20932}, {692, 31947}, {1333, 21873}, {2175, 41808}, {2206, 21081}, {21192, 32739}
X(44188) = cevapoint of X(i) and X(j) for these (i,j): {2, 14450}, {514, 1109}, {693, 17886}, {39149, 41910}
X(44188) = trilinear pole of line {1577, 4467}
X(44188) = barycentric product X(i)*X(j) for these {i,j}: {75, 1029}, {76, 267}, {274, 502}, {310, 21353}, {313, 40143}, {561, 3444}, {30602, 33939}
X(44188) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1030}, {2, 191}, {10, 21873}, {19, 44097}, {27, 2906}, {57, 8614}, {63, 22136}, {75, 2895}, {76, 20932}, {81, 501}, {85, 41808}, {86, 40592}, {92, 451}, {267, 6}, {313, 42710}, {321, 21081}, {502, 37}, {514, 31947}, {661, 42653}, {693, 21192}, {1029, 1}, {3444, 31}, {21353, 42}, {30602, 2160}, {39149, 2245}, {40143, 58}, {41493, 2294}, {41910, 34586}

X(44189) = ISOTOMIC CONJUGATE OF X(196)

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

X(44189) lies on these lines: {2, 23982}, {69, 189}, {271, 1265}, {280, 285}, {304, 7055}, {314, 7003}, {326, 345}, {346, 394}, {348, 34403}, {1422, 30701}, {1433, 20745}, {6225, 7219}, {7101, 34413}

X(44189) = isogonal conjugate of X(3209)
X(44189) = isotomic conjugate of X(196)
X(44189) = isotomic conjugate of the isogonal conjugate of X(268)
X(44189) = isotomic conjugate of the polar conjugate of X(280)
X(44189) = X(i)-cross conjugate of X(j) for these (i,j): {8, 69}, {63, 345}, {268, 280}
X(44189) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3209}, {4, 2199}, {6, 208}, {19, 221}, {25, 223}, {31, 196}, {32, 342}, {33, 6611}, {34, 198}, {40, 608}, {56, 2331}, {57, 3195}, {227, 1474}, {278, 2187}, {329, 1395}, {347, 1973}, {393, 7114}, {560, 40701}, {604, 7952}, {1096, 7011}, {1398, 2324}, {1400, 3194}, {1402, 41083}, {1407, 40971}, {1435, 7074}, {1880, 2360}, {1974, 40702}, {2149, 38362}, {2207, 7013}, {2212, 14256}, {3213, 41088}, {6087, 32667}, {6129, 32674}, {7151, 40212}
X(44189) = cevapoint of X(i) and X(j) for these (i,j): {63, 41081}, {6332, 23983}
X(44189) = barycentric product X(i)*X(j) for these {i,j}: {63, 34404}, {69, 280}, {75, 271}, {76, 268}, {78, 309}, {84, 3718}, {189, 345}, {282, 304}, {285, 20336}, {305, 2192}, {312, 41081}, {326, 7020}, {332, 39130}, {346, 34400}, {561, 2188}, {1264, 40836}, {1265, 1440}, {1433, 3596}, {3926, 7003}, {7118, 40364}, {13138, 35518}, {15416, 37141}, {28660, 41087}
X(44189) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 208}, {2, 196}, {3, 221}, {6, 3209}, {8, 7952}, {9, 2331}, {11, 38362}, {21, 3194}, {48, 2199}, {55, 3195}, {63, 223}, {69, 347}, {72, 227}, {75, 342}, {76, 40701}, {78, 40}, {84, 34}, {189, 278}, {200, 40971}, {212, 2187}, {219, 198}, {222, 6611}, {255, 7114}, {268, 6}, {271, 1}, {280, 4}, {282, 19}, {283, 2360}, {285, 28}, {304, 40702}, {309, 273}, {326, 7013}, {332, 8822}, {333, 41083}, {345, 329}, {348, 14256}, {394, 7011}, {521, 6129}, {1259, 7078}, {1260, 7074}, {1265, 7080}, {1413, 1398}, {1422, 1435}, {1433, 56}, {1436, 608}, {1440, 1119}, {1565, 38374}, {1809, 15501}, {1812, 1817}, {1903, 1880}, {2188, 31}, {2192, 25}, {2208, 1395}, {2968, 38357}, {3341, 207}, {3692, 2324}, {3694, 21871}, {3710, 21075}, {3718, 322}, {6081, 36067}, {6332, 14837}, {6355, 6046}, {7003, 393}, {7008, 1096}, {7020, 158}, {7118, 1973}, {7151, 7337}, {7154, 2207}, {7358, 3318}, {7367, 607}, {13138, 108}, {23983, 16596}, {34162, 42451}, {34400, 279}, {34404, 92}, {35518, 17896}, {36049, 32674}, {37141, 32714}, {39130, 225}, {39471, 6087}, {40836, 1118}, {41081, 57}, {41087, 1400}

X(44190) = ISOTOMIC CONJUGATE OF X(198)

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

X(44190) lies on these lines: {76, 7182}, {189, 28660}, {274, 40836}, {279, 23978}, {282, 30022}, {304, 309}, {305, 28659}, {348, 7017}, {1240, 1440}, {1969, 6063}

X(44190) = isotomic conjugate of X(198)
X(44190) = polar conjugate of X(3195)
X(44190) = isotomic conjugate of the anticomplement of X(21239)
X(44190) = isotomic conjugate of the complement of X(21279)
X(44190) = isotomic conjugate of the isogonal conjugate of X(189)
X(44190) = X(i)-cross conjugate of X(j) for these (i,j): {85, 76}, {264, 6063}, {1226, 561}, {21239, 2}
X(44190) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2187}, {31, 198}, {32, 40}, {41, 221}, {48, 3195}, {55, 2199}, {184, 2331}, {212, 3209}, {213, 2360}, {223, 2175}, {322, 1501}, {329, 560}, {347, 9447}, {604, 7074}, {607, 7114}, {1106, 7368}, {1253, 6611}, {1397, 2324}, {1817, 1918}, {1973, 7078}, {2200, 3194}, {2205, 8822}, {2206, 21871}, {2212, 7011}, {6129, 32739}, {7952, 9247}, {9448, 40702}
X(44190) = cevapoint of X(i) and X(j) for these (i,j): {2, 21279}, {75, 20921}, {309, 34404}, {693, 23978}
X(44190) = trilinear pole of line {15413, 17896}
X(44190) = barycentric product X(i)*X(j) for these {i,j}: {75, 309}, {76, 189}, {84, 561}, {85, 34404}, {280, 6063}, {282, 20567}, {305, 40836}, {310, 39130}, {1413, 40363}, {1422, 28659}, {1433, 18022}, {1436, 1502}, {1440, 3596}, {1903, 6385}, {1928, 2208}, {1969, 41081}, {2192, 41283}, {7017, 34400}, {7020, 7182}, {7129, 40364}, {7151, 40050}, {8808, 28660}, {13138, 40495}, {41084, 41530}
X(44190) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2187}, {2, 198}, {4, 3195}, {7, 221}, {8, 7074}, {57, 2199}, {69, 7078}, {75, 40}, {76, 329}, {77, 7114}, {84, 31}, {85, 223}, {86, 2360}, {92, 2331}, {189, 6}, {253, 41088}, {264, 7952}, {271, 212}, {273, 208}, {274, 1817}, {278, 3209}, {279, 6611}, {280, 55}, {282, 41}, {285, 2194}, {286, 3194}, {309, 1}, {310, 8822}, {312, 2324}, {313, 21075}, {318, 40971}, {321, 21871}, {322, 1103}, {331, 196}, {332, 1819}, {346, 7368}, {348, 7011}, {561, 322}, {693, 6129}, {1226, 6260}, {1256, 2208}, {1413, 1397}, {1422, 604}, {1433, 184}, {1436, 32}, {1440, 56}, {1441, 227}, {1903, 213}, {2192, 2175}, {2208, 560}, {2357, 1918}, {2973, 38362}, {3261, 14837}, {3596, 7080}, {4391, 14298}, {6063, 347}, {6332, 10397}, {6355, 1425}, {7003, 607}, {7008, 2212}, {7020, 33}, {7118, 9447}, {7129, 1973}, {7151, 1974}, {7182, 7013}, {7367, 14827}, {8808, 1400}, {13138, 692}, {13156, 1475}, {18816, 15501}, {20567, 40702}, {23978, 5514}, {28660, 27398}, {34387, 38357}, {34400, 222}, {34404, 9}, {34413, 42019}, {35519, 8058}, {36049, 32739}, {37141, 1415}, {39130, 42}, {40495, 17896}, {40702, 40212}, {40836, 25}, {41081, 48}, {41084, 154}, {41087, 2200}, {42549, 20228}, {44129, 41083}

X(44191) = (name pending)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4)*(a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12 - 4*a^10*c^2 + 8*a^8*b^2*c^2 - 4*a^6*b^4*c^2 + 4*a^4*b^6*c^2 - 8*a^2*b^8*c^2 + 4*b^10*c^2 + 5*a^8*c^4 - 4*a^6*b^2*c^4 - a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 7*b^8*c^4 + 4*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 8*b^6*c^6 - 5*a^4*c^8 - 8*a^2*b^2*c^8 - 7*b^4*c^8 + 4*a^2*c^10 + 4*b^2*c^10 - c^12)*(2*a^16 - 11*a^14*b^2 + 23*a^12*b^4 - 19*a^10*b^6 - 5*a^8*b^8 + 23*a^6*b^10 - 19*a^4*b^12 + 7*a^2*b^14 - b^16 - 11*a^14*c^2 + 42*a^12*b^2*c^2 - 57*a^10*b^4*c^2 + 38*a^8*b^6*c^2 - 31*a^6*b^8*c^2 + 36*a^4*b^10*c^2 - 21*a^2*b^12*c^2 + 4*b^14*c^2 + 23*a^12*c^4 - 57*a^10*b^2*c^4 + 36*a^8*b^4*c^4 - a^6*b^6*c^4 - 18*a^4*b^8*c^4 + 21*a^2*b^10*c^4 - 4*b^12*c^4 - 19*a^10*c^6 + 38*a^8*b^2*c^6 - a^6*b^4*c^6 + 2*a^4*b^6*c^6 - 7*a^2*b^8*c^6 - 4*b^10*c^6 - 5*a^8*c^8 - 31*a^6*b^2*c^8 - 18*a^4*b^4*c^8 - 7*a^2*b^6*c^8 + 10*b^8*c^8 + 23*a^6*c^10 + 36*a^4*b^2*c^10 + 21*a^2*b^4*c^10 - 4*b^6*c^10 - 19*a^4*c^12 - 21*a^2*b^2*c^12 - 4*b^4*c^12 + 7*a^2*c^14 + 4*b^2*c^14 - c^16) : :

See Antreas Hatzipolakis and Peter Moses, euclid 1999.

X(44191) lies on this line: {125, 1493}

leftri

Points associated with Vijay orthic polar triangle of circumcircle: X(44192)-X(44200)

rightri

This preamble is contributed by Dasari Naga Vijay Krishna, July 26, 2021.

In the plane of a triangle ABC, O = circumcircle of triangle ABC. let
Oa = circle with diameter BC, and define Ob and Oc cyclically;
La = The perpendicular from X(3) of triangle ABC to side BC, define Lb and Lc cyclically;
A1, A2 = La ∩Oa such that A2 is nearer to A than A1, and define B1, B2, C1, C2 cyclically;
Ta, T'a = polar of A1, A2 wrt Oa, define Tb, T'b, Tc, T'c cyclically;
Pa, P'a = polar of A1, A2 wrt O, define Pb, P'b, Pc and P'c cyclically;
A3 = Tb ∩ Tc, B3 = Tc ∩ Ta, C3 = Ta ∩ Tb;
A4 = T'b ∩ T'c, B4= T'c ∩ T'a, C4 = T'a ∩ T'b;
A5 = Pb ∩ Pc, B5= Pc ∩ Pa, C5 = Pa ∩ Pb;
A6 = P'b ∩ P'c, B6= P'c ∩ P'a, C6 = P'a ∩ P'b;

Barycentrics:

A1 = -a^2 : S + SC : S + SB
A3 = -(2S + b^2 + c^2) : b^2 : c^2
A5 = -(2*S*(S + SB)*(S + SC) + b^2*(S - SB)*(S + SC) + c^2*(S - SC)*(S + SB)) : b^2*(S - SB)*(S + SC) : c^2*(S - SC)*(S + SB)

Barycentrics for A2, A4, A6 are obtained by replacing S by -S in the barycentrics for A1, A3, A5 respectively.

Related triangles are here named as follows:

A1B1C1 = 1st Vijay orthic polar triangle of circumcircle;
A2B2C2 = 2nd Vijay orthic polar trriangle of circumcircle;
A3B3C3 = 3rd Vijay orthic polar trriangle of circumcircle;
A4B4C4 = 4th Vijay orthic polar trriangle of circumcircle;
A5B5C5 = 5th Vijay orthic polar trriangle of circumcircle;
A6B6C6 = 6th Vijay orthic polar trriangle of circumcircle.

The first four of those triangles have been introduced previously:
A1B1C1 is the outer Vecten triangle.
A2B2C2 is the inner Vecten triangle.
A3B3C3 is the 1st anti-Kenmotu-centers triangle.
A4B4C4 is the 2nd anti-Kenmotu-centers triangle.
(Randy Hutson, January 11, 2022)

Collinearities:

A3, A4, A are collinear.
X(485), X(486), X(6) are collinear.
X(6), X(1991), X(591) are collinear.
X(44192), X(44193), X(44200) are collinear.
X(6), X(44193), X(44197), X(44199), are collinear.
X(6), X(44192), X(44196), X(44198), are collinear.

Perspectors :

AA1 ∩ BB1∩ CC1 = X(485);
AA2 ∩ BB2∩ CC2 = X(486);
AA3 ∩ BB3∩ CC3 = AA4 ∩ BB4∩ CC4 = A3A4 ∩ B3B4∩ C3C4 = X(6);
A1A4 ∩ B1B4∩ C1C4 = X(1991);
A2A3 ∩ B2B3 ∩ C2C3 = X(591);
X(44192) = AA5∩ BB5∩ CC5;
X(44193) = AA6∩ BB6∩ CC6;
X(44194) = A1A5∩ B1B5∩ C1C5;
X(44195) = A2A6∩ B2B6∩ C2C6;
X(44196) = A3A5∩ B3B5∩ C3C5;
X(44197) = A3A6∩ B3B6∩ C3C6;
X(44198) = A4A5∩ B4B5∩ C4C5;
X(44199) = A4A6∩ B4B6∩ C4C6;
X(44200) = A5A6∩ B5B6∩ C5C6:

X(44192) = PERSPECTOR OF THESE TRIANGLES: ABC AND 5TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44192) lies on these lines: {2, 44194}, {3, 485}, {6, 3156}, {22, 13638}, {24, 13440}, {25, 53}, {141, 1584}, {183, 34391}, {237, 45429}, {371, 5417}, {372, 1147}, {491, 1600}, {571, 5412}, {1321, 1593}, {1586, 41770}, {2164, 34125}, {2178, 34121}, {3003, 5413}, {3053, 8946}, {3155, 8553}, {8573, 44599}, {8770, 45596}, {8943, 8944}, {10533, 12968}, {14533, 15846}, {32420, 39383}, {35302, 44393}, {40947, 45428}

X(44192) = perspector of ABC and cross-triangle of ABC and 1st Kenmotu diagonals triangle
X(44192) = {X(25),X(1609)}-harmonic conjugate of X(44193)

X(44193) = PERSPECTOR OF THESE TRIANGLES: ABC AND 6TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44193) lies on these lines: {2, 44195}, {3, 486}, {6, 3155}, {22, 13758}, {24, 13429}, {25, 53}, {141, 1583}, {183, 34392}, {237, 45428}, {371, 1147}, {372, 5419}, {492, 1599}, {571, 5413}, {1151, 26922}, {1322, 1593}, {1585, 41770}, {2164, 34121}, {2178, 34125}, {3003, 5412}, {3053, 8948}, {3156, 8553}, {8573, 44598}, {8770, 45595}, {8908, 9675}, {8939, 8940}, {10534, 12963}, {14533, 15847}, {32422, 39384}, {35302, 44400}, {40947, 45429}

X(44193) = perspector of ABC and cross-triangle of ABC and 2nd Kenmotu diagonals triangle
X(44193) = {X(25),X(1609)}-harmonic conjugate of X(44192)

X(44194) = PERSPECTOR OF THESE TRIANGLES: 1ST AND 5TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

Barycentrics    (S+SB)*(S+SC)*(c^10+b^2*c^8-4*a^2*c^8+6*S*c^8-2*b^4*c^6+4*a^4*c^6-8*S*a^2*c^6+16*S^2*c^6-2*b^6*c^4+8*a^2*b^4*c^4-12*S*b^4*c^4-8*a^4*b^2*c^4+8*S*a^2*b^2*c^4+2*a^6*c^4+8*S*a^4*c^4-8*S^2*a^2*c^4+32*S^3*c^4+b^8*c^2-8*a^4*b^4*c^2+8*S*a^2*b^4*c^2+12*a^6*b^2*c^2+8*S*a^4*b^2*c^2-16*S^2*a^2*b^2*c^2+32*S^3*b^2*c^2-5*a^8*c^2-16*S*a^6*c^2-24*S^2*a^4*c^2+48*S^4*c^2+b^10-4*a^2*b^8+6*S*b^8+4*a^4*b^6-8*S*a^2*b^6+16*S^2*b^6+2*a^6*b^4+8*S*a^4*b^4-8*S^2*a^2*b^4+32*S^3*b^4-5*a^8*b^2-16*S*a^6*b^2-24*S^2*a^4*b^2+48*S^4*b^2+2*a^10+10*S*a^8+16*S^2*a^6+16*S^3*a^4+32*S^4*a^2+32*S^5) : :

X(44194) lies on these lines: {2, 44192}, {6290, 44198}

X(44195) = PERSPECTOR OF THESE TRIANGLES: 2ND AND 6TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

Barycentrics    (S-SB)*(S-SC)*(c^10+b^2*c^8-4*a^2*c^8-6*S*c^8-2*b^4*c^6+4*a^4*c^6+8*S*a^2*c^6+16*S^2*c^6-2*b^6*c^4+8*a^2*b^4*c^4+12*S*b^4*c^4-8*a^4*b^2*c^4-8*S*a^2*b^2*c^4+2*a^6*c^4-8*S*a^4*c^4-8*S^2*a^2*c^4-32*S^3*c^4+b^8*c^2-8*a^4*b^4*c^2-8*S*a^2*b^4*c^2+12*a^6*b^2*c^2-8*S*a^4*b^2*c^2-16*S^2*a^2*b^2*c^2-32*S^3*b^2*c^2-5*a^8*c^2+16*S*a^6*c^2-24*S^2*a^4*c^2+48*S^4*c^2+b^10-4*a^2*b^8-6*S*b^8+4*a^4*b^6+8*S*a^2*b^6+16*S^2*b^6+2*a^6*b^4-8*S*a^4*b^4-8*S^2*a^2*b^4-32*S^3*b^4-5*a^8*b^2+16*S*a^6*b^2-24*S^2*a^4*b^2+48*S^4*b^2+2*a^10-10*S*a^8+16*S^2*a^6-16*S^3*a^4+32*S^4*a^2-32*S^5) : :

X(44195) lies on these lines: {2, 44193}, {6289, 44197}

X(44196) = PERSPECTOR OF THESE TRIANGLES: 3RD AND 5TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44196) lies on these lines: {3, 639}, {6, 3156}, {22, 160}, {25, 8939}, {51, 19358}, {155, 9733}, {157, 492}, {195, 45488}, {1498, 12305}, {1584, 31521}, {1597, 19454}, {1598, 18980}, {1995, 19406}, {3517, 19440}, {7716, 12590}, {8904, 33582}, {8943, 44197}, {10494, 38034}, {10594, 19424}, {11793, 17668}, {15047, 45411}, {15805, 43119}, {18414, 18494}, {19216, 45416}, {40947, 44392}

X(44196) = crosspoint of circumcircle intercepts of outer Vecten circle
X(44196) = {X(22),X(160)}-harmonic conjugate of X(44199)

X(44197) = PERSPECTOR OF THESE TRIANGLES: 3RD AND 6TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44197) lies on these lines: {2, 44198}, {3, 591}, {6, 3155}, {157, 3069}, {570,18194}, {615, 40947}, {1583, 32621}, {3148, 32788}, {3156, 7669}, {6289, 44195}, {6410, 32071}, {8943, 44196}, {15624, 45416}, {33029, 38997}, {33582, 45428}

X(44198) = PERSPECTOR OF THESE TRIANGLES: 4TH AND 5TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44198) lies on these lines: {2, 44197}, {3, 1991}, {6, 3156}, {157, 3068}, {590, 40947}, {1584, 32621}, {3148, 32787}, {3155, 7669}, {6290, 44194}, {6409, 32070}, {8939, 44199}, {8969, 33582}, {15624, 45417}

X(44199) = PERSPECTOR OF THESE TRIANGLES: 4TH AND 6TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44199) lies on these lines: {3, 640}, {6, 3155}, {22, 160}, {25, 8943}, {51, 19359}, {155, 9732}, {157, 491}, {195, 45489}, {1498, 12306}, {1583, 31521}, {1598, 18981}, {1995, 19407}, {7716, 12591}, {8903, 33582}, {8939, 44198}, {10594, 19425}, {15047, 45410}, {15805, 43118}, {18415, 18494}, {19215, 45417}, {26922, 30427}, {40947, 44394}

X(44199) = crosspoint of circumcircle intercepts of inner Vecten circle
X(44199) = {X(22),X(160)}-harmonic conjugate of X(44196)

X(44200) = PERSPECTOR OF THESE TRIANGLES: 5TH AND 6TH VIJAY ORTHIC POLAR TRIANGLE OF CIRCUMCIRCLE

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

X(44200) lies on these lines: {3, 69}, {22, 33974}, {25, 53}, {50, 34777}, {98, 20477}, {159, 7669}, {184, 36751}, {186, 16312}, {216, 19125}, {237, 33582}, {511, 10608}, {570, 11402}, {571, 12167}, {577, 10602}, {1151, 19358}, {1152, 19359}, {1593, 35717}, {1599, 19422}, {1600, 19423}, {1843, 3053}, {1899, 34828}, {2974, 20563}, {3003, 19118}, {3148, 8573}, {3155, 13889}, {3156, 13943}, {3186, 21445}, {3515, 15653}, {4558, 6391}, {5023, 9924}, {6409, 19430}, {6410, 19431}, {6467, 36748}, {7484, 7778}, {8266, 8667}, {8681, 10607}, {8939, 44198}, {8943, 44196}, {9715, 15512}, {9777, 13345}, {9909, 27364}, {11063, 20987}, {11284, 33980}, {13558, 33801}, {14060, 20806}, {14615, 34473}, {15655, 34106}, {15905, 20975}, {21312, 38749}, {23200, 38292}, {33580, 40981}, {37183, 37491}, {41489, 44096}

X(44200) = X(34208)-Ceva conjugate of X(6)
X(44200) = crosssum of X(4) and X(32001)
X(44200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 19588, 9723}, {3, 39653, 40321}, {3, 40947, 19459}, {157, 1609, 25}, {7669, 8553, 159}, {15512, 31381, 9715}, {19446, 19447, 9723}

X(44201) = X(3)X(68)∩X(140)X(389)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^8 - a^6*b^2 - 5*a^4*b^4 + 5*a^2*b^6 - b^8 - a^6*c^2 - 2*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 4*b^6*c^2 - 5*a^4*c^4 - 5*a^2*b^2*c^4 - 6*b^4*c^4 + 5*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(44201) = 5 X[631] - X[1993], 3 X[10519] + X[41614], 3 X[13394] - X[18445]

See Antreas Hatzipolakis and Peter Moses, euclid 2005.

X(44201) lies these lines: {2, 37489}, {3, 68}, {5, 11745}, {30, 21243}, {140, 389}, {141, 6644}, {182, 524}, {185, 34002}, {186, 37636}, {381, 4549}, {427, 37478}, {468, 5891}, {548, 30522}, {550, 6696}, {568, 37649}, {631, 1199}, {1209, 3575}, {1352, 14070}, {1503, 7502}, {1511, 3631}, {1594, 7691}, {1596, 4550}, {2937, 16655}, {2979, 37118}, {3098, 23300}, {3410, 7488}, {3522, 43607}, {3523, 18916}, {3530, 32165}, {3541, 37486}, {3547, 12163}, {3549, 22660}, {3564, 18475}, {3580, 35921}, {3581, 37347}, {3589, 5946}, {3796, 18917}, {3917, 10257}, {5447, 16196}, {5449, 12362}, {5562, 7542}, {5663, 25337}, {5876, 16252}, {5890, 7495}, {5899, 16654}, {5907, 13383}, {6000, 16618}, {6102, 7568}, {6515, 37506}, {6676, 13754}, {6677, 10170}, {6699, 11574}, {6823, 7689}, {7493, 18451}, {7499, 9730}, {7503, 41587}, {7514, 13567}, {7789, 36952}, {9306, 34351}, {10018, 11444}, {10020, 11591}, {10127, 24206}, {10192, 15068}, {10282, 31831}, {10300, 20397}, {10519, 41614}, {10610, 32358}, {10627, 23336}, {11064, 23039}, {11202, 34507}, {11245, 37513}, {11793, 16238}, {12605, 13851}, {13348, 25563}, {13394, 18445}, {14791, 23332}, {14852, 18918}, {15030, 37971}, {15361, 20582}, {15585, 34776}, {15720, 42021}, {16197, 40647}, {16621, 17714}, {16658, 37913}, {16659, 38435}, {18531, 35254}, {18537, 32620}, {18951, 37476}, {21230, 23358}, {26879, 37126}, {31834, 34577}, {34200, 38726}, {37935, 43586}

X(44201) = midpoint of X(i) and X(j) for these {i,j}: {3, 343}, {427, 37478}
X(44201) = reflection of X(23292) in X(140)
X(44201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {389, 32348, 140}, {5447, 20191, 16196}, {5562, 7542, 9820}

leftri

Centers of circles through X(24007) and X(24008): X(44202)-X(44205)

rightri

This preamble is contributed by Peter Moses, August 13, 2021.

In addition to the Dao-Moses-Telv circle, {13,14,5000,5001,6104,6105,6106,6107,6108,6109,6110,6111,24007,24008}, the following four circles also pass thorugh X(24007) and X(24008):

{{2,98,112,5913,10295,24007,24008}}, with center X(44202)
{{4,107,111,671,5523,7426,9979,20410,24007,24008,41125}}, with center X(44203)
{{51,115,132,24007,24008}, with center X(44204)}
{{125,187,1560,6055,9730,10162,24007,24008}}, with center X(44205)

X(44202) = MIDPOINT OF X(376) AND X(9979)

Barycentrics    (b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 + c^4) : :
X(44202) = X[3268] - 3 X[3524]

X(44202) lies on these lines: {3, 2799}, {30, 1637}, {376, 9979}, {541, 14697}, {549, 14417}, {690, 6055}, {1499, 9189}, {2793, 14666}, {3268, 3524}, {3534, 9529}, {5664, 9033}, {6644, 42659}, {9003, 11179}, {9730, 39469}, {11621, 32473}, {15469, 35912}

X(44202) = midpoint of X(376) and X(9979)
X(44202) = reflection of X(14417) in X(549)

X(44203) = MIDPOINT OF X(4) AND X(9979)

Barycentrics    (a^4-b^4+4*b^2*c^2-c^4)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2) : :
X(44203) = X[3268] - 3 X[3545], 2 X[9126] - 3 X[9189]

X(44203) lies on these lines: {3, 9529}, {4, 9979}, {5, 14417}, {30, 1637}, {113, 133}, {381, 2799}, {525, 16229}, {690, 9880}, {2780, 9134}, {2793, 6094}, {3268, 3545}, {5512, 14672}, {7530, 42659}, {9003, 20423}, {9126, 9189}, {12083, 25644}, {14697, 17702}, {30209, 37855}

X(44203) = midpoint of X(4) and X(9979)
X(44203) = reflection of X(14417) in X(5)
X(44203) = X(i)-isoconjugate of X(j) for these (i,j): {5486, 36034}, {18877, 37217}, {30247, 35200}
X(44203) = crossdifference of every pair of points on line {9717, 18877}
X(44203) = X(9979)-of-Euler-triangle
X(44203) = X(14417)-of-Johnson-triangle
X(44203) = barycentric product X(i)*X(j) for these {i,j}: {1637, 11185}, {1784, 14209}, {1995, 41079}
X(44203) = barycentric quotient X(i)/X(j) for these {i,j}: {1637, 5486}, {1784, 37217}, {1990, 30247}, {19136, 32640}, {30209, 14919}

X(44204) = MIDPOINT OF X(381) AND X(9979)

Barycentrics    (b^2 - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^2*b^2 - b^4 + a^2*c^2 + 3*b^2*c^2 - c^4) : :
X(44204) = X[3268] - 3 X[5055]

X(44204) lies on these lines: {5, 2799}, {30, 1637}, {381, 9979}, {547, 14417}, {2780, 10278}, {3268, 5055}, {8029, 19912}, {8703, 9529}, {10189, 16235}, {24978, 39491}

X(44204) = midpoint of X(i) and X(j) for these {i,j}: {381, 9979}, {8029, 19912}, {24978, 39491}
X(44204) = reflection of X(i) in X(j) for these {i,j}: {14417, 547}, {16235, 10189}
X(44204) = barycentric product X(5640)*X(41079)
X(44204) = barycentric quotient X(33885)/X(1304)

X(44205) = MIDPOINT OF X(9979) AND X(44206)

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

X(44205) lies on this line: {2799, 12042}

X(44205) = midpoint of X(9979) and X(44205)

X(44206) = REFLECTION OF X(9979) IN X(44205)

Barycentrics    (b^2 - c^2)*(5*a^10 + a^8*b^2 - 13*a^6*b^4 + a^4*b^6 + 8*a^2*b^8 - 2*b^10 + a^8*c^2 - 13*a^6*b^2*c^2 + 25*a^4*b^4*c^2 - 17*a^2*b^6*c^2 + 4*b^8*c^2 - 13*a^6*c^4 + 25*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - 2*b^6*c^4 + a^4*c^6 - 17*a^2*b^2*c^6 - 2*b^4*c^6 + 8*a^2*c^8 + 4*b^2*c^8 - 2*c^10) : :
X(44206) = 3 X[3] - 2 X[18311], 3 X[381] - 4 X[18310]

X(44206) lies on these lines: {3, 18311}, {30, 14977}, {381, 18310}, {523, 3534}, {525, 33878}, {2799, 14830}, {9979, 44205}

X(44206) = reflection of X(9979) in X(44205)

X(44207) = (name pending)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2016.

X(44207) lies on the Jerabek circumhyperbola and these lines: { }

X(44207) = isogonal conjugate of X(44208)

X(44208) = 61ST HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, euclid 2016.

X(44208) lies on these lines: {2,3}, {2351,11442}

X(44208) = isogonal conjugate of X(44207)

X(44209) = 62ND HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^16 - 9*a^14*b^2 + 17*a^12*b^4 - 19*a^10*b^6 + 15*a^8*b^8 - 7*a^6*b^10 - a^4*b^12 + 3*a^2*b^14 - b^16 - 9*a^14*c^2 + 22*a^12*b^2*c^2 - 19*a^10*b^4*c^2 + 10*a^8*b^6*c^2 - 7*a^6*b^8*c^2 + 10*a^4*b^10*c^2 - 13*a^2*b^12*c^2 + 6*b^14*c^2 + 17*a^12*c^4 - 19*a^10*b^2*c^4 - 2*a^8*b^4*c^4 + 6*a^6*b^6*c^4 - 7*a^4*b^8*c^4 + 21*a^2*b^10*c^4 - 16*b^12*c^4 - 19*a^10*c^6 + 10*a^8*b^2*c^6 + 6*a^6*b^4*c^6 - 4*a^4*b^6*c^6 - 11*a^2*b^8*c^6 + 26*b^10*c^6 + 15*a^8*c^8 - 7*a^6*b^2*c^8 - 7*a^4*b^4*c^8 - 11*a^2*b^6*c^8 - 30*b^8*c^8 - 7*a^6*c^10 + 10*a^4*b^2*c^10 + 21*a^2*b^4*c^10 + 26*b^6*c^10 - a^4*c^12 - 13*a^2*b^2*c^12 - 16*b^4*c^12 + 3*a^2*c^14 + 6*b^2*c^14 - c^16 : :

See Antreas Hatzipolakis and Peter Moses, euclid 2016.

X(44209) lies on this line: {2,3}

leftri

Midpoints on the Euler line: X(44210)-X(44290)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, August 16, 2021.

If U and V are points on the Euler line, then their midpoint, given by the combo U + V, also lies on the Euler line.

X(44210) = MIDPOINT OF X(2) AND X(22)

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

X(44210) lies on these lines: {2, 3}, {51, 597}, {69, 26864}, {74, 32227}, {113, 35254}, {141, 1495}, {154, 599}, {160, 15652}, {182, 32269}, {184, 524}, {343, 542}, {511, 13394}, {612, 4995}, {614, 5298}, {1112, 9967}, {1177, 34319}, {1194, 3003}, {1353, 11003}, {1494, 1799}, {1503, 35268}, {1879, 3054}, {1899, 43273}, {1992, 11402}, {2781, 3917}, {2916, 23300}, {2979, 40112}, {3058, 5310}, {3098, 11064}, {3163, 40938}, {3564, 6800}, {3580, 15080}, {3589, 34417}, {3629, 44109}, {3763, 41424}, {3796, 11179}, {3815, 35345}, {3933, 26233}, {5012, 15360}, {5063, 9300}, {5090, 34712}, {5092, 32223}, {5181, 22165}, {5191, 7664}, {5322, 5434}, {5370, 7354}, {5476, 37649}, {5486, 15534}, {5640, 38110}, {5650, 21167}, {5651, 15448}, {5943, 10168}, {5972, 14810}, {6030, 9140}, {6031, 14929}, {6090, 10519}, {6284, 7302}, {6723, 33751}, {6781, 15820}, {7749, 40350}, {7750, 37804}, {7788, 34254}, {7799, 33651}, {7810, 42671}, {8192, 34667}, {8193, 34656}, {8263, 40114}, {8550, 41586}, {8584, 13366}, {8588, 24855}, {9157, 10718}, {9306, 35266}, {9591, 34657}, {10601, 38064}, {11056, 32819}, {11180, 11206}, {11396, 34730}, {11513, 13937}, {11514, 13884}, {11574, 44084}, {11645, 21243}, {12017, 21970}, {13567, 22352}, {14389, 15107}, {14569, 37765}, {14826, 21356}, {15880, 43618}, {16316, 40879}, {16317, 21843}, {16331, 19221}, {19121, 34397}, {19126, 41584}, {20423, 33586}, {20582, 44082}, {24206, 32237}, {25406, 26869}, {26881, 37636}, {32218, 37283}, {33878, 37645}, {37775, 42121}, {37776, 42124}, {41587, 43573}

X(44210) = midpoint of X(2) and X(22)
X(44210) = complement of X(31133)

X(44211) = MIDPOINT OF X(2) AND X(24)

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

X(44211) lies on these lines: {2, 3}, {343, 43586}, {539, 33563}, {542, 20771}, {800, 3163}, {1353, 9703}, {5642, 14831}, {5651, 44201}, {5892, 13394}, {5943, 10182}, {6148, 39113}, {8780, 18917}, {9730, 10192}, {11179, 23041}, {11449, 43595}, {11597, 19138}, {16194, 23328}, {18475, 37648}, {21850, 38794}, {22115, 41588}, {23324, 23515}, {32062, 38727}, {34782, 43817}, {41628, 43572}

X(44211) = midpoint of X(2) and X(24)
X(44211) = complement of X(31180)

X(44212) = MIDPOINT OF X(2) AND X(25)

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

X(44212) lies on these lines: {2, 3}, {51, 5642}, {69, 21970}, {110, 1353}, {125, 39884}, {141, 32223}, {154, 11179}, {182, 15448}, {184, 35266}, {343, 32225}, {373, 13394}, {524, 8263}, {542, 13567}, {597, 2393}, {1084, 1196}, {1494, 40413}, {1495, 37648}, {1992, 3167}, {2790, 6055}, {3054, 36412}, {3060, 40112}, {3066, 18583}, {3564, 35259}, {3580, 10546}, {4995, 5268}, {5012, 40114}, {5050, 35260}, {5254, 40350}, {5272, 5298}, {5309, 34481}, {5476, 23292}, {5480, 5972}, {5651, 32269}, {6090, 34380}, {6593, 8584}, {6688, 10168}, {7713, 34643}, {7718, 34713}, {8185, 34634}, {8262, 22165}, {8770, 16310}, {8780, 11433}, {8854, 32787}, {8855, 32788}, {9300, 33871}, {10272, 32227}, {10418, 18907}, {10545, 14389}, {11064, 21850}, {11245, 35264}, {11427, 14848}, {11439, 43903}, {11746, 41714}, {12294, 13416}, {13846, 18289}, {13847, 18290}, {13857, 44106}, {14908, 19661}, {14929, 26276}, {15060, 15361}, {15082, 21167}, {17810, 20423}, {17825, 38064}, {18358, 37638}, {18440, 37643}, {23332, 36201}, {31670, 31860}, {31804, 43573}, {32819, 37803}, {33552, 33962}, {33651, 37671}, {34224, 43836}, {34319, 38851}, {37488, 40917}, {37649, 38079}

X(44212) = midpoint of X(2) and X(25)
X(44212) = complement of X(31152)

X(44213) = MIDPOINT OF X(2) AND X(26)

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

X(44213) lies on these lines:s {2, 3}, {49, 41628}, {395, 11267}, {396, 11268}, {524, 19154}, {539, 10282}, {542, 20773}, {1154, 10192}, {1989, 42459}, {4995, 8144}, {5298, 32047}, {5944, 41587}, {5946, 13394}, {9707, 32358}, {9971, 38110}, {11265, 32788}, {11266, 32787}, {15448, 44201}, {15806, 31802}, {18475, 32223}, {19153, 34380}, {36987, 38793}

X(44213) = midpoint of X(2) and X(26)
X(44213) = complement of X(31181)

X(44214) = MIDPOINT OF X(2) AND X(186)

Barycentrics    4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 14*a^6*b^2*c^2 - 10*a^4*b^4*c^2 + 8*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 10*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 8*a^4*c^6 + 8*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44214) lies on these lines: {2, 3}, {182, 32113}, {187, 16760}, {511, 16222}, {524, 15462}, {541, 21663}, {568, 16227}, {1154, 15361}, {1353, 11935}, {1495, 6699}, {1503, 15061}, {1511, 3580}, {1514, 20127}, {1531, 12900}, {3003, 3163}, {3564, 32609}, {3581, 11064}, {4995, 10149}, {5181, 11693}, {5642, 13754}, {5972, 32110}, {6795, 16321}, {9730, 10182}, {10168, 11649}, {10193, 16194}, {10540, 20126}, {10564, 32223}, {11657, 14934}, {11694, 40111}, {12028, 14993}, {12041, 32111}, {12893, 32123}, {13289, 32125}, {13367, 43573}, {13394, 40280}, {13857, 14156}, {14693, 38613}, {14805, 37648}, {14915, 38727}, {15034, 41724}, {15360, 43574}, {15448, 38728}, {18350, 44158}, {26879, 32171}, {32269, 37477}, {38227, 38704}, {40352, 40630}

X(44214) = midpoint of X(2) and X(186)
X(44214) = X(44452)-of-anti-Euler-triangle

X(44215) = MIDPOINT OF X(2) AND X(237)

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

X(44215) lies on these lines: {2, 3}, {230, 5106}, {373, 40108}, {512, 11176}, {524, 36213}, {620, 32223}, {1495, 12042}, {1625, 3231}, {3117, 5306}, {4995, 40790}, {5968, 16329}, {9155, 32515}, {10168, 34236}, {11673, 40112}, {16237, 16330}, {22329, 35146}, {35278, 38225}

X(44215) = midpoint of X(2) and X(237)

X(44216) = MIDPOINT OF X(2) AND X(297)

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

X(44216) lies on these lines: {2, 3}, {317, 20204}, {524, 3163}, {525, 1637}, {1494, 6330}, {1503, 41145}, {1990, 40996}, {5309, 13567}, {5641, 22329}, {7615, 34360}, {7753, 23292}, {8859, 39359}, {15526, 18487}, {16076, 16080}, {16321, 38393}, {34573, 36412}

X(44216) = midpoint of X(2) and X(297)
X(44216) = complement of X(40884)
X(44216) = anticomplement of X(44346)

X(44217) = MIDPOINT OF X(2) AND X(377)

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

X(44217) lies on these lines: {1, 21949}, {2, 3}, {10, 553}, {56, 3841}, {72, 4654}, {78, 3824}, {142, 3419}, {274, 7788}, {392, 31162}, {518, 599}, {540, 19723}, {551, 31140}, {940, 3017}, {956, 3925}, {958, 41859}, {997, 4870}, {999, 33108}, {1125, 11238}, {1376, 3584}, {1478, 3826}, {1698, 3916}, {2094, 38058}, {2886, 10072}, {3219, 18541}, {3336, 3928}, {3421, 40333}, {3434, 15170}, {3582, 25524}, {3583, 8167}, {3586, 20195}, {3634, 10895}, {3653, 24541}, {3654, 24987}, {3697, 5290}, {3822, 4413}, {3828, 31141}, {3940, 31019}, {4002, 9578}, {4292, 5325}, {4995, 10198}, {5131, 19876}, {5250, 28198}, {5253, 31493}, {5260, 9655}, {5275, 5309}, {5284, 9668}, {5298, 26363}, {5362, 42974}, {5367, 42975}, {5440, 25525}, {5550, 9669}, {5587, 10167}, {5687, 10056}, {5730, 12609}, {6739, 15668}, {6767, 33110}, {7610, 7621}, {7811, 16992}, {7989, 9841}, {8583, 38021}, {9654, 9780}, {9709, 26060}, {9782, 11024}, {10390, 38024}, {10707, 32558}, {10711, 34122}, {10712, 34124}, {10894, 31423}, {10896, 19862}, {11231, 21165}, {11235, 17614}, {11648, 16589}, {12699, 24564}, {15934, 27186}, {16118, 41872}, {16483, 33109}, {16763, 17057}, {17757, 26040}, {19860, 28204}, {25973, 34697}, {31473, 35823}, {32833, 37664}, {34648, 38204}

X(44217) = midpoint of X(2) and X(377)
X(44217) = complement of X(31156)

X(44218) = MIDPOINT OF X(2) AND X(378)

Barycentrics    4*a^10 - 9*a^8*b^2 + 2*a^6*b^4 + 8*a^4*b^6 - 6*a^2*b^8 + b^10 - 9*a^8*c^2 + 20*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 4*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 + 20*a^2*b^4*c^4 + 2*b^6*c^4 + 8*a^4*c^6 - 4*a^2*b^2*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44218) lies on these lines: {2, 3}, {39, 3163}, {49, 43894}, {74, 14389}, {141, 10564}, {373, 38727}, {511, 30539}, {524, 13352}, {542, 11430}, {567, 5622}, {569, 6696}, {575, 20417}, {597, 2781}, {599, 37497}, {1503, 39242}, {3581, 21850}, {3589, 37470}, {4550, 11064}, {5476, 11438}, {5480, 32110}, {5642, 15030}, {5943, 10193}, {6699, 37648}, {8550, 12506}, {8589, 36412}, {9140, 12022}, {10168, 16836}, {10192, 16194}, {10249, 11179}, {10625, 40929}, {11424, 44158}, {11459, 40112}, {13394, 14915}, {13434, 43607}, {13482, 41628}, {13630, 43896}, {16165, 43586}, {16789, 37480}, {18488, 34782}, {20423, 37489}, {37477, 41721}, {37487, 38072}, {38110, 40280}, {43608, 43836}

X(44218) = midpoint of X(2) and X(378)

X(44219) = MIDPOINT OF X(2) AND X(383)

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

X(44219) lies on these lines: {2, 3}, {13, 3815}, {14, 230}, {16, 41043}, {98, 42063}, {114, 530}, {302, 18358}, {303, 21850}, {325, 11128}, {395, 542}, {396, 5476}, {397, 32907}, {524, 5613}, {531, 7685}, {597, 9749}, {1352, 9761}, {1499, 9195}, {3054, 16809}, {3055, 16808}, {3564, 37785}, {3818, 23303}, {5321, 6109}, {5460, 6055}, {5480, 33475}, {5617, 33474}, {5979, 23234}, {6774, 11645}, {7610, 22491}, {7735, 42975}, {7736, 42974}, {8550, 36383}, {9113, 43274}, {9762, 9771}, {9763, 20423}, {11184, 22492}, {11489, 18440}, {11537, 14356}, {14848, 37640}, {15597, 20428}, {16644, 38072}, {19130, 23302}, {20112, 22576}, {21159, 41025}, {22796, 43101}, {22797, 42943}, {23698, 41047}, {37689, 42816}, {37835, 41042}

X(44219) = midpoint of X(2) and X(383)

X(44220) = MIDPOINT OF X(3) AND X(28)

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

X(44220) lies on these lines: {2, 3}, {284, 5453}, {942, 1511}, {2828, 38602}, {2838, 38603}, {3601, 37729}, {5267, 6708}, {7280, 37695}, {7740, 18115}, {9895, 11259}, {15803, 37697}, {15943, 18481}, {30282, 37696}, {38606, 38612}, {40111, 41608}

X(44220) = midpoint of X(3) and X(28)

X(44221) = MIDPOINT OF X(3) AND X(237)

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

X(44221) lies on these lines: {2, 3}, {39, 5946}, {50, 15462}, {187, 1511}, {249, 2080}, {327, 7771}, {511, 34990}, {512, 5926}, {1154, 36212}, {1634, 3564}, {2387, 13335}, {2482, 15361}, {3003, 14984}, {3580, 36829}, {5504, 41336}, {5892, 13334}, {7789, 36952}, {9826, 14961}, {10317, 12228}, {23217, 32269}, {32219, 37470}

X(44221) = midpoint of X(3) and X(237)

X(44222) = MIDPOINT OF X(3) AND X(377)

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

X(44222) lies on these lines: {2, 3}, {10, 13369}, {46, 495}, {119, 16209}, {496, 3612}, {518, 5690}, {960, 41540}, {1385, 3813}, {1698, 7171}, {2886, 13624}, {3576, 10943}, {3579, 25466}, {3653, 37722}, {3654, 15888}, {3820, 5302}, {3826, 18480}, {3925, 18481}, {5122, 15844}, {5249, 37585}, {5433, 14803}, {5794, 37705}, {7280, 26481}, {7680, 31663}, {7987, 26470}, {8273, 37820}, {9710, 28204}, {9956, 31805}, {10198, 35238}, {10942, 21031}, {11231, 18242}, {11246, 16139}, {11698, 26066}, {12609, 22791}, {12616, 38042}, {12625, 18443}, {15325, 22768}, {17647, 31419}, {26475, 37616}, {30389, 37726}

X(44222) = midpoint of X(3) and X(377)
X(44222) = complement of X(37234)

X(44223) = MIDPOINT OF X(3) AND X(383)

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

X(44223) lies on these lines: {2, 3}, {14, 38230}, {16, 5613}, {182, 33390}, {187, 6114}, {299, 5615}, {303, 20425}, {395, 11135}, {396, 3106}, {511, 619}, {617, 20426}, {618, 24206}, {620, 624}, {636, 7880}, {1353, 42634}, {1503, 33389}, {2080, 5978}, {2782, 6108}, {3564, 42913}, {3643, 9736}, {5237, 37825}, {5321, 20253}, {5476, 13083}, {5617, 16242}, {5872, 42149}, {5873, 22238}, {5979, 15561}, {6109, 14693}, {6672, 6774}, {6771, 33479}, {11178, 13084}, {11543, 19781}, {13103, 14145}, {13349, 41023}, {16626, 42489}, {16627, 42158}, {16964, 36959}, {16966, 23005}, {18581, 21843}, {18583, 42912}, {20252, 37832}, {20428, 30560}, {20429, 36968}, {22509, 22998}, {33416, 36765}

X(44223) = midpoint of X(3) and X(383)
X(44223) = outer-Napoleon-circle-inverse of X(44462)

X(44224) = MIDPOINT OF X(3) AND X(384)

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

X(44224) lies on these lines: {2, 3}, {32, 32521}, {76, 26316}, {83, 35002}, {99, 12054}, {141, 32151}, {182, 698}, {325, 34885}, {511, 32134}, {736, 7780}, {1503, 35422}, {1511, 37890}, {2023, 37512}, {3095, 7878}, {3098, 42534}, {3398, 7760}, {3734, 14880}, {3934, 12042}, {3972, 9821}, {5031, 7830}, {5092, 7816}, {5116, 43456}, {5149, 7789}, {5162, 7745}, {5206, 42535}, {6033, 7832}, {6292, 38749}, {7804, 14881}, {7822, 9996}, {7847, 38730}, {7861, 22515}, {7944, 10722}, {8290, 35464}, {10357, 14712}, {10359, 32447}, {10583, 43453}, {10796, 30270}, {11272, 18860}, {11842, 12251}, {12176, 40238}, {12188, 17128}, {13334, 33813}, {18358, 24273}, {18769, 35701}, {21167, 43152}

X(44224) = midpoint of X(3) and X(384)
X(44224) = complement of X(37243)

X(44225) = MIDPOINT OF X(4) AND X(29)

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

X(44225) lies on these lines: {2, 3}, {34, 7100}, {92, 22791}, {113, 20620}, {124, 133}, {158, 39542}, {240, 5492}, {243, 37737}, {278, 18493}, {281, 12702}, {355, 39531}, {517, 39574}, {946, 39529}, {1159, 3176}, {1547, 13474}, {1784, 3649}, {1838, 9955}, {1844, 2771}, {1859, 31937}, {1895, 6147}, {5174, 18357}, {5708, 40836}, {7070, 18492}, {10572, 42387}, {12047, 42385}, {12699, 39585}, {21666, 34334}

X(44225) = midpoint of X(4) and X(29)
X(44225) = X(29)-of-Euler-triangle

X(44226) = MIDPOINT OF X(4) AND X(235)

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

X(44226) lies on these lines: {2, 3}, {113, 22970}, {185, 1514}, {389, 5893}, {1829, 40273}, {1902, 18357}, {1968, 43291}, {2883, 18390}, {3564, 11470}, {6225, 26944}, {6746, 13451}, {6748, 39590}, {7687, 13474}, {8537, 39871}, {8548, 39884}, {9820, 12897}, {10110, 43392}, {10575, 44079}, {10641, 42136}, {10642, 42137}, {11245, 43602}, {11363, 28186}, {11473, 18538}, {11474, 18762}, {11475, 42146}, {11476, 42143}, {11572, 16654}, {11801, 12133}, {12294, 18358}, {12295, 20771}, {13093, 23291}, {13142, 22660}, {13346, 18418}, {13403, 16252}, {13419, 15125}, {13567, 22802}, {13568, 43589}, {13630, 44084}, {13851, 16655}, {14864, 15153}, {15114, 15473}, {15120, 30522}, {16621, 18383}, {16657, 43831}, {16658, 18394}, {18918, 34780}, {18945, 32063}, {20427, 26958}, {22115, 22750}, {26879, 43806}, {35764, 42283}, {35765, 42284}

X(44226) = midpoint of X(4) and X(235)
X(44226) = X(235)-of-Euler-triangle

X(44227) = MIDPOINT OF X(4) AND X(237)

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

X(44227) lies on these lines: {2, 3}, {113, 2679}, {133, 38974}, {217, 5305}, {512, 6130}, {1503, 7668}, {2387, 18388}, {2790, 3003}, {3331, 43291}, {3767, 32445}, {6530, 15352}, {13881, 38297}, {15048, 43718}, {16330, 35908}, {23039, 32521}, {30258, 43976}

X(44227) = midpoint of X(2) and X(237)
X(44227) = X(237)-of-Euler-triangle

X(44228) = MIDPOINT OF X(4) AND X(297)

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

X(44228) lies on these lines: {2, 3}, {53, 3818}, {113, 38970}, {133, 36471}, {264, 18358}, {265, 5523}, {317, 21850}, {393, 18440}, {399, 8744}, {525, 16229}, {542, 1990}, {648, 3564}, {1249, 39899}, {1351, 10002}, {1353, 41371}, {1503, 39569}, {2207, 18451}, {2966, 39663}, {5254, 18390}, {5476, 6749}, {6531, 43291}, {6748, 19130}, {7745, 18388}, {8743, 18445}, {11442, 14569}, {13142, 43995}, {13754, 34854}, {15262, 39562}, {16077, 35142}, {16240, 32225}, {16326, 36875}, {18437, 42459}, {18474, 27376}, {31670, 42854}, {33971, 39884}, {34334, 44146}

X(44228) = midpoint of X(4) and X(297)
X(44228) = X(297)-of-Euler-triangle

X(44229) = MIDPOINT OF X(4) AND X(377)

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

X(44229) lies on these lines: {1, 18406}, {2, 3}, {6, 13408}, {8, 20886}, {10, 37584}, {12, 18491}, {57, 3585}, {68, 1243}, {80, 18962}, {119, 10894}, {142, 31673}, {225, 37696}, {278, 18447}, {355, 518}, {388, 15934}, {495, 18518}, {496, 18544}, {497, 18493}, {515, 30143}, {579, 32431}, {942, 1478}, {946, 12437}, {952, 10532}, {960, 12699}, {1056, 18526}, {1060, 1838}, {1068, 37729}, {1466, 13273}, {1479, 9955}, {1483, 10597}, {1699, 10525}, {1728, 9579}, {1836, 31937}, {2550, 12702}, {3085, 18524}, {3193, 15068}, {3434, 5730}, {3436, 18357}, {3583, 3601}, {3587, 41869}, {3600, 12773}, {3818, 4260}, {3925, 35239}, {4292, 10395}, {4293, 26321}, {4295, 40266}, {4857, 38021}, {5082, 8148}, {5138, 19130}, {5229, 5708}, {5251, 35250}, {5270, 11518}, {5290, 18528}, {5302, 5791}, {5396, 5713}, {5492, 24248}, {5587, 5709}, {5654, 41608}, {5691, 18443}, {5706, 18451}, {5707, 5721}, {5715, 5720}, {5755, 5816}, {5777, 37826}, {5778, 5798}, {5787, 6256}, {5800, 18440}, {5886, 24299}, {5891, 10441}, {5901, 12116}, {6245, 37612}, {6246, 9946}, {6253, 10267}, {6326, 33594}, {7354, 18761}, {7680, 11499}, {7688, 41859}, {7741, 37583}, {7951, 10953}, {7956, 11928}, {7965, 11826}, {9654, 10629}, {9945, 22938}, {10283, 10806}, {10531, 38034}, {10805, 28224}, {11459, 41723}, {12667, 38107}, {13151, 18481}, {13226, 22799}, {15171, 18499}, {15803, 18513}, {15888, 34746}, {18343, 18345}, {18483, 21616}, {18514, 30282}, {18519, 18990}, {19925, 37532}, {22753, 26470}, {22793, 31793}, {22798, 41697}, {26333, 33596}, {28204, 37724}, {37534, 41698}, {37623, 38140}, {37697, 40950}

X(44229) = midpoint of X(4) and X(377)
X(44229) = X(377)-of-Euler-triangle

X(44230) = MIDPOINT OF X(4) AND X(384)

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

X(44230) lies on these lines: {2, 3}, {32, 3818}, {39, 19130}, {76, 9993}, {83, 43460}, {98, 20576}, {113, 35971}, {114, 6249}, {115, 13357}, {141, 9821}, {325, 14881}, {511, 7794}, {542, 5007}, {698, 3095}, {736, 6248}, {1352, 14023}, {1503, 3398}, {2080, 6287}, {3098, 7822}, {3589, 12054}, {3933, 18906}, {3972, 9873}, {5092, 7889}, {5103, 7789}, {5188, 24206}, {5207, 7767}, {5305, 12188}, {5475, 13356}, {5476, 7772}, {6033, 7745}, {6054, 7858}, {6393, 40252}, {7750, 9996}, {7758, 20423}, {7761, 10356}, {7792, 14880}, {7795, 31670}, {7799, 14492}, {7810, 25561}, {7826, 43150}, {7854, 11178}, {7855, 37517}, {7860, 34733}, {8721, 14561}, {9478, 12042}, {9863, 10788}, {9971, 18375}, {11842, 39884}, {12176, 12206}, {13754, 27374}, {17128, 43453}, {18440, 30435}, {18501, 18907}, {22681, 39663}, {32447, 38136}, {37479, 38317}, {39899, 43136}

X(44230) = midpoint of X(4) and X(384)
X(44230) = midpoint of X(37332) and X(37333)
X(44230) = complement of X(7470)
X(44230) = X(384)-of-Euler-triangle

X(44231) = MIDPOINT OF X(4) AND X(401)

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

X(44231) lies on these lines: {2, 3}, {6, 18437}, {32, 18390}, {39, 18388}, {113, 11672}, {131, 2679}, {157, 18382}, {187, 7687}, {216, 19130}, {248, 265}, {316, 6394}, {399, 13509}, {511, 15526}, {525, 684}, {542, 3284}, {577, 3818}, {1503, 1576}, {1568, 36212}, {2055, 12134}, {3564, 22143}, {5158, 5476}, {5480, 30258}, {5523, 9475}, {6389, 31670}, {6530, 32428}, {10110, 10600}, {10316, 18474}, {10745, 35002}, {12022, 34396}, {12241, 36245}, {15905, 18440}, {18358, 41008}, {18400, 42671}, {18445, 22120}, {18451, 23115}, {20208, 33878}, {21850, 41005}, {38292, 39899}

X(44231) = midpoint of X(4) and X(401)
X(44231) = X(401)-of-Euler-triangle

X(44232) = MIDPOINT OF X(5) AND X(24)

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

X(44232) lies on these lines: {2, 3}, {110, 32358}, {143, 9820}, {154, 18952}, {156, 13567}, {973, 22051}, {1216, 32223}, {1495, 43817}, {3564, 33563}, {3580, 18350}, {5446, 5972}, {6101, 32269}, {6696, 32137}, {6699, 13474}, {8254, 18583}, {10095, 23292}, {10110, 43839}, {10192, 32046}, {10263, 11064}, {10540, 26879}, {11255, 41585}, {11572, 23515}, {12241, 32171}, {13292, 34116}, {13451, 20193}, {13598, 14156}, {13630, 16252}, {14128, 44201}, {16654, 34128}, {16657, 43394}, {20771, 32423}, {25738, 35264}, {26958, 32140}, {35265, 43808}

X(44232) = midpoint of X(5) and X(24)

X(44233) = MIDPOINT OF X(5) AND X(25)

Barycentrics    2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 - 8*a^4*b^4*c^2 + 16*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 8*a^4*b^2*c^4 - 24*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 16*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44233) lies on these lines: {2, 3}, {1147, 15873}, {1568, 44106}, {1660, 32046}, {2386, 20576}, {2393, 13364}, {3564, 19136}, {5448, 11745}, {5462, 16252}, {5654, 17810}, {5891, 32269}, {8263, 34380}, {9820, 10110}, {9969, 10272}, {10539, 13292}, {10540, 11245}, {14845, 37649}, {14862, 15012}, {15053, 32111}, {15068, 41588}, {15448, 18475}, {16655, 43817}, {16658, 26913}, {20304, 36201}, {20772, 32423}, {31831, 41587}, {32223, 44201}

X(44233) = midpoint of X(5) and X(25)
X(44233) = complement of complement of X(18534)

X(44234) = 1ST HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 + 6*a^6*b^2*c^2 - 5*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 5*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 7*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(44899) = X[1495]+2*X[40685], 3*X[5943]-X[11692], X[10264]+X[10540], X[13445]-5*X[38728], X[14157]+3*X[15061], 5*X[38794]-X[43574]

See Antreas Hatzipolakis and Francisco Javier García Capitán, Euclid 2479 .

X(44234) lies on these lines: {2, 3}, {49, 32165}, {143, 43839}, {389, 15806}, {523, 31667}, {1154, 5972}, {1495, 40685}, {3580, 11597}, {3589, 11649}, {5462, 8254}, {5943, 11692}, {5944, 43817}, {6689, 13365}, {10110, 20193}, {10264, 10540}, {10272, 11561}, {11801, 30522}, {11803, 16625}, {13367, 43575}, {13391, 14156}, {13445, 38728}, {14157, 15061}, {16243, 40630}, {16881, 22051}, {17433, 31945}, {18400, 20304}, {22104, 34837}, {23292, 32411}, {25563, 32137}, {32218, 40670}, {38794, 43574}

X(44234) = midpoint of X(5) and X(186)
X(44234) = complement of X(37938)

X(44235) = MIDPOINT OF X(5) AND X(235)

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

X(44235) lies on these lines: {2, 3}, {113, 6102}, {143, 5448}, {156, 18390}, {184, 43575}, {576, 11803}, {578, 15806}, {1147, 10272}, {1568, 10263}, {1614, 43821}, {3292, 18555}, {3521, 15053}, {3567, 18504}, {3574, 13368}, {5946, 43831}, {5972, 12897}, {7687, 18379}, {7730, 20424}, {9544, 36966}, {10095, 18388}, {10112, 16534}, {10113, 20771}, {10264, 18439}, {10539, 32423}, {11430, 15807}, {11439, 11704}, {11805, 32339}, {13403, 32171}, {13445, 43866}, {13451, 15873}, {13491, 43817}, {13630, 32050}, {13754, 15887}, {14643, 34148}, {14862, 18128}, {15002, 15038}, {15030, 34826}, {18356, 18451}, {18445, 32165}, {20299, 20304}, {21659, 43865}, {22251, 38942}, {22660, 33563}

X(44235) = midpoint of X(5) and X(235)

X(44236) = MIDPOINT OF X(5) AND X(378)

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

X(442) lies on these lines: {2, 3}, {143, 44158}, {578, 43588}, {2781, 18583}, {3815, 19220}, {5462, 25563}, {5663, 23292}, {5943, 6699}, {5944, 16655}, {6247, 32046}, {6696, 13630}, {7706, 11204}, {10110, 20191}, {10193, 19130}, {10264, 11245}, {11064, 15060}, {11425, 32140}, {11550, 39242}, {12134, 43394}, {12233, 32138}, {12241, 13561}, {12834, 15057}, {13366, 16003}, {13367, 18488}, {13391, 44201}, {13482, 41724}, {13568, 32210}, {15739, 22051}, {16252, 32137}, {18356, 43595}, {18952, 40686}, {32358, 37472}, {37477, 37636}, {37481, 43607}

X(44236) = midpoint of X(5) and X(378)

X(44237) = MIDPOINT OF X(5) AND X(384)

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

X(44237) lies on these lines: {2, 3}, {620, 42788}, {698, 18583}, {736, 20576}, {2782, 7829}, {3314, 18501}, {3564, 42421}, {3589, 32516}, {3972, 32151}, {7759, 10796}, {7820, 14881}, {7832, 18502}, {7888, 10358}, {10583, 12188}, {32515, 35437}

X(44237) = midpoint of X(5) and X(384)

X(44238) = MIDPOINT OF X(20) AND X(21)

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

X(44238) lies on these lines: {2, 3}, {7, 16137}, {11, 17009}, {46, 5441}, {63, 16139}, {65, 4304}, {74, 43354}, {79, 3612}, {84, 191}, {224, 16006}, {515, 21677}, {516, 35016}, {517, 39772}, {758, 1071}, {950, 41547}, {1259, 35250}, {2077, 33961}, {2245, 40979}, {2646, 3649}, {2771, 3650}, {2777, 16164}, {2795, 38738}, {2829, 35204}, {3184, 9528}, {3576, 11281}, {3647, 15823}, {3655, 11520}, {3868, 34773}, {4313, 6361}, {4324, 41853}, {5249, 13624}, {5427, 6284}, {5731, 34195}, {5732, 16113}, {5787, 21165}, {5794, 18253}, {7987, 26725}, {10391, 41549}, {10884, 33858}, {11491, 31799}, {11544, 37606}, {11604, 38693}, {11826, 12511}, {12512, 24466}, {12664, 40661}, {16155, 37618}, {16465, 37585}, {26066, 35242}, {28628, 41869}, {33594, 34123}, {35252, 43740}, {37623, 41574}

X(44238) = midpoint of X(20) and X(21)

X(44239) = MIDPOINT OF X(20) AND X(22)

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

X(44239) lies on these lines: {2, 3}, {54, 31802}, {66, 10606}, {74, 41464}, {99, 18401}, {112, 42459}, {343, 18400}, {477, 11635}, {827, 1294}, {1141, 3565}, {1287, 2693}, {1295, 26712}, {1297, 44061}, {1350, 36989}, {1503, 16789}, {1568, 10192}, {2777, 16165}, {2781, 3313}, {4549, 18451}, {5562, 34782}, {5889, 31804}, {5890, 12220}, {5891, 35254}, {5966, 20187}, {6243, 43595}, {7691, 14516}, {8550, 14831}, {9019, 19161}, {9538, 10386}, {10313, 15048}, {10605, 26926}, {10619, 14531}, {10984, 13568}, {11064, 11202}, {11245, 37489}, {11381, 35240}, {11417, 42216}, {11418, 42215}, {11420, 42118}, {11421, 42117}, {11425, 14542}, {11605, 38717}, {11750, 12359}, {12022, 41588}, {12118, 37486}, {12160, 18925}, {12226, 36966}, {13394, 18388}, {14907, 20477}, {15033, 19121}, {17712, 43604}, {17834, 19467}, {18390, 32269}, {18474, 44201}, {18907, 22240}, {19127, 29181}, {24466, 36984}, {26864, 41465}, {30264, 36986}, {41468, 41469}

X(44239) = midpoint of X(20) and X(22)

X(44240) = MIDPOINT OF X(20) AND X(24)

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

X(44240) lies on these lines: {2, 3}, {49, 38723}, {74, 14516}, {113, 43898}, {131, 33553}, {185, 16163}, {1060, 15338}, {1062, 15326}, {1941, 5667}, {2777, 20771}, {3053, 15075}, {3357, 12134}, {4846, 19357}, {5663, 43896}, {5890, 43595}, {5894, 12162}, {6696, 18474}, {6781, 22401}, {7723, 14677}, {7747, 40349}, {8263, 38885}, {9934, 22955}, {10539, 15311}, {10575, 34782}, {10605, 12118}, {12022, 43601}, {12038, 38726}, {12041, 43903}, {12121, 43616}, {12293, 26937}, {12359, 21663}, {12363, 43581}, {13142, 37490}, {13352, 13568}, {13445, 16659}, {14855, 40928}, {15034, 43599}, {15055, 43607}, {17702, 33563}, {18350, 20127}, {18451, 20427}, {23328, 43907}, {31802, 37495}, {35240, 35254}, {37511, 40929}

X(44240) = midpoint of X(20) and X(24)

X(44241) = MIDPOINT OF X(20) AND X(25)

Barycentrics    (a^2 - b^2 - c^2)*(4*a^8 - 3*a^6*b^2 - 5*a^4*b^4 + 3*a^2*b^6 + b^8 - 3*a^6*c^2 + 18*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 4*b^6*c^2 - 5*a^4*c^4 - 3*a^2*b^2*c^4 + 6*b^4*c^4 + 3*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(44241) lies on these lines: {2, 3}, {99, 22468}, {112, 41890}, {141, 11598}, {343, 21663}, {974, 6467}, {1038, 15338}, {1040, 15326}, {1060, 4319}, {1062, 4320}, {1181, 34966}, {1209, 43907}, {1285, 38292}, {1352, 10606}, {1353, 5890}, {1503, 8263}, {1578, 42260}, {1579, 42261}, {2777, 20772}, {2790, 3184}, {2883, 22966}, {3564, 10605}, {3815, 40349}, {5305, 15075}, {5651, 20725}, {5656, 8780}, {5891, 13416}, {5894, 5907}, {5925, 32602}, {9306, 15311}, {9722, 34866}, {9786, 13142}, {9967, 36987}, {10386, 18447}, {10602, 14914}, {11438, 41588}, {11454, 37636}, {12118, 12421}, {12162, 30443}, {12358, 14677}, {12359, 43604}, {12429, 18913}, {13346, 13568}, {13562, 34778}, {13630, 43595}, {14855, 34750}, {14907, 41008}, {14961, 18907}, {15644, 31807}, {15740, 27082}, {17814, 20427}, {18475, 38726}, {18850, 36876}, {19136, 29181}, {19467, 22663}, {21243, 23328}, {22581, 22833}, {22660, 43577}, {31730, 37613}, {31804, 40647}, {32817, 40995}, {32833, 40996}, {39120, 39812}

X(44241) = midpoint of X(20) and X(25)
X(44241) = complement of X(44438)
X(44241) = anticomplement of X(44920)

X(44242) = MIDPOINT OF X(20) AND X(26)

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

X(44242) lies on these lines: {2, 3}, {50, 42459}, {74, 41482}, {511, 32903}, {1192, 18952}, {1503, 32138}, {2777, 20773}, {4351, 9630}, {5663, 16775}, {5891, 35240}, {5944, 34798}, {6247, 32210}, {7689, 34785}, {8144, 15338}, {9934, 15132}, {10625, 11562}, {11265, 42259}, {11266, 42258}, {11267, 42088}, {11268, 42087}, {11597, 38723}, {11660, 36966}, {11750, 21663}, {12359, 30522}, {13561, 41362}, {13568, 32046}, {15326, 32047}, {15807, 15873}, {16111, 21650}, {17845, 32140}, {18350, 18442}, {18383, 20191}, {19154, 29181}, {19467, 43588}, {21659, 32110}, {22660, 32171}, {25487, 38726}, {32142, 35254}, {32330, 32423}

X(44242) = midpoint of X(20) and X(26)
X(44242) = 1st-Droz-Farny-circle-inverse of X(44905)

X(44243) = MIDPOINT OF X(20) AND X(27)

Barycentrics    (a^2 - b^2 - c^2)*(4*a^8 - 2*a^7*b - 3*a^6*b^2 + 4*a^5*b^3 - 5*a^4*b^4 - 2*a^3*b^5 + 3*a^2*b^6 + b^8 - 2*a^7*c - 2*a^6*b*c + 4*a^5*b^2*c + 4*a^4*b^3*c - 2*a^3*b^4*c - 2*a^2*b^5*c - 3*a^6*c^2 + 4*a^5*b*c^2 + 18*a^4*b^2*c^2 + 4*a^3*b^3*c^2 - 3*a^2*b^4*c^2 - 4*b^6*c^2 + 4*a^5*c^3 + 4*a^4*b*c^3 + 4*a^3*b^2*c^3 + 4*a^2*b^3*c^3 - 5*a^4*c^4 - 2*a^3*b*c^4 - 3*a^2*b^2*c^4 + 6*b^4*c^4 - 2*a^3*c^5 - 2*a^2*b*c^5 + 3*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(44243) lies on these lines: {2, 3}, {63, 15941}, {84, 1762}, {1060, 7675}, {1214, 4304}, {1565, 16163}, {2328, 15311}, {2822, 3184}, {4653, 18643}, {6356, 24929}, {8680, 30271}, {10543, 41393}, {15326, 23207}

X(44243) = midpoint of X(20) and X(27)

X(44244) = MIDPOINT OF X(20) AND X(29)

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

X(44244) lies on these lines: {2, 3}, {77, 1062}, {390, 38290}, {1214, 31730}, {2360, 15311}, {2816, 3184}, {2968, 3916}, {3345, 5732}, {4294, 7011}, {10386, 38284}, {15326, 40946}, {15338, 22341}, {16553, 23058}, {17073, 41869}, {17102, 18593}

X(44244) = midpoint of X(20) and X(29)

X(44245) = MIDPOINT OF X(20) AND X(140)

Barycentrics    14*a^4 - 11*a^2*b^2 - 3*b^4 - 11*a^2*c^2 + 6*b^2*c^2 - 3*c^4 : :

X(44245) lies on these lines: {2, 3}, {61, 42123}, {62, 42122}, {156, 8717}, {371, 43321}, {372, 43320}, {395, 42996}, {396, 42997}, {397, 42529}, {398, 42528}, {952, 12512}, {1125, 28182}, {1385, 28216}, {1483, 9778}, {1503, 32903}, {1587, 6519}, {1588, 6522}, {2777, 13392}, {3098, 32138}, {3411, 42792}, {3412, 42791}, {3579, 28224}, {3592, 42261}, {3594, 42260}, {3746, 15326}, {3819, 32137}, {4297, 5844}, {4324, 15325}, {5237, 42087}, {5238, 42088}, {5351, 11543}, {5352, 11542}, {5355, 35007}, {5563, 15172}, {5609, 16111}, {5663, 13348}, {5901, 31666}, {6053, 38726}, {6101, 14855}, {6102, 36987}, {6390, 7917}, {6409, 13925}, {6410, 13993}, {6425, 42216}, {6426, 42215}, {6427, 9541}, {6429, 42643}, {6430, 42644}, {6445, 43787}, {6446, 43788}, {6447, 19117}, {6448, 19116}, {6453, 42259}, {6454, 42258}, {6455, 43407}, {6456, 43408}, {6496, 23249}, {6497, 23259}, {6684, 28190}, {6781, 31652}, {7991, 34773}, {8960, 43209}, {10095, 17704}, {10222, 28212}, {10541, 21850}, {10575, 31834}, {10645, 42165}, {10646, 42164}, {10721, 15023}, {11495, 32153}, {11591, 14641}, {11801, 38729}, {12046, 13570}, {12121, 15021}, {13353, 43576}, {13391, 16625}, {13624, 28178}, {14094, 14677}, {14915, 32142}, {15012, 16881}, {15034, 20127}, {15044, 38728}, {15048, 22331}, {15054, 34153}, {15178, 28174}, {15513, 43291}, {16192, 38042}, {16808, 42590}, {16809, 42591}, {16962, 43207}, {16963, 43208}, {16964, 42497}, {16965, 42496}, {17502, 40273}, {17508, 42785}, {17712, 30522}, {17852, 43323}, {18510, 43884}, {18512, 43883}, {18907, 22332}, {20125, 22250}, {20190, 29181}, {22236, 42091}, {22238, 42090}, {22251, 38790}, {22676, 32448}, {22791, 30389}, {23235, 38731}, {23302, 42889}, {23303, 42888}, {28186, 31663}, {32423, 37853}, {35255, 42267}, {35256, 42266}, {36836, 42118}, {36843, 42117}, {36967, 43233}, {36968, 43232}, {38664, 38742}, {38668, 38766}, {38669, 38754}, {38674, 38778}, {38688, 38798}, {42099, 42163}, {42100, 42166}, {42108, 43102}, {42109, 43103}, {42115, 43307}, {42116, 43306}, {42121, 42160}, {42124, 42161}, {42136, 42599}, {42137, 42598}, {42144, 42159}, {42145, 42162}, {42150, 42625}, {42151, 42626}, {42157, 42913}, {42158, 42912}, {42429, 42936}, {42430, 42937}, {42431, 43372}, {42432, 43373}, {42433, 42924}, {42434, 42925}, {42488, 43401}, {42489, 43402}, {42641, 43568}, {42642, 43569}, {42795, 42939}, {42796, 42938}, {42896, 42966}, {42897, 42967}, {42916, 43465}, {42917, 43466}, {42922, 43777}, {42923, 43778}, {42944, 43417}, {42945, 43416}, {42970, 43005}, {42971, 43004}, {42990, 43109}, {42991, 43108}, {43110, 43645}, {43111, 43646}, {43292, 43643}, {43293, 43638}, {43304, 43635}, {43305, 43634}, {43463, 43648}, {43464, 43647}, {43483, 43546}, {43484, 43547}

X(44245) = midpoint of X(20) and X(140)
X(44245) = anticomplement of X(3856)

X(44246) = MIDPOINT OF X(20) AND X(186)

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

X(44246) lies on these lines: {2, 3}, {97, 19651}, {127, 32456}, {1204, 44076}, {1272, 40996}, {1288, 2693}, {1503, 38788}, {1531, 14156}, {3184, 12091}, {4351, 15326}, {4354, 10149}, {5447, 35240}, {5894, 18439}, {6000, 7723}, {6699, 13851}, {6781, 14961}, {9730, 32411}, {9934, 10540}, {10620, 25740}, {11204, 18474}, {11468, 12278}, {11649, 37511}, {12041, 30522}, {13367, 43577}, {13376, 17704}, {13557, 14934}, {13568, 37472}, {13754, 16163}, {14516, 32138}, {14677, 22584}, {15055, 25739}, {15075, 16306}, {15905, 16303}, {17702, 21663}, {18400, 37853}, {18430, 23332}, {18914, 43807}, {20299, 24572}, {21659, 43604}, {22115, 38723}, {32110, 32282}, {32710, 44062}, {35254, 35257}

X(44246) = midpoint of X(20) and X(186)
X(44246) = anticomplement of X(23323)
X(44246) = circumcircle-inverse of X(12084)

X(44247) = MIDPOINT OF X(20) AND X(235)

Barycentrics    (a^2 - b^2 - c^2)*(6*a^8 - 5*a^6*b^2 - 7*a^4*b^4 + 5*a^2*b^6 + b^8 - 5*a^6*c^2 + 22*a^4*b^2*c^2 - 5*a^2*b^4*c^2 - 4*b^6*c^2 - 7*a^4*c^4 - 5*a^2*b^2*c^4 + 6*b^4*c^4 + 5*a^2*c^6 - 4*b^2*c^6 + c^8) : :

X(44247) lies on these lines: {2, 3}, {1192, 41588}, {1204, 3564}, {1352, 8567}, {3532, 15069}, {5893, 5972}, {5894, 9306}, {5907, 13416}, {6146, 16163}, {6225, 8780}, {6776, 27082}, {7745, 40349}, {9307, 30262}, {9820, 43577}, {11245, 43601}, {11438, 13142}, {11442, 43903}, {12235, 16270}, {12429, 18931}, {12512, 37613}, {15074, 37511}, {15647, 22966}, {16111, 20771}, {19126, 23326}, {22581, 32223}, {31802, 37497}, {31831, 32138}, {32820, 40996}, {38726, 40647}

X(44247) = midpoint of X(20) and X(235)

X(44248) = MIDPOINT OF X(20) AND X(297)

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

X(442) lies on these lines: {2, 3}, {32, 41369}, {74, 2867}, {122, 15448}, {187, 3184}, {525, 42658}, {542, 40996}, {1294, 32646}, {1503, 15526}, {1990, 9530}, {3098, 41008}, {3793, 40948}, {6390, 16163}, {6776, 34815}, {10991, 34109}, {15311, 42671}, {15312, 41204}, {39874, 40995}

X(44248) = midpoint of X(20) and X(297)

X(44249) = MIDPOINT OF X(20) AND X(378)

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

X(44249) lies on these lines: {2, 3}, {49, 18442}, {113, 10192}, {216, 6781}, {343, 17702}, {394, 4549}, {516, 24301}, {569, 13568}, {577, 19220}, {1060, 15326}, {1062, 15338}, {2781, 9967}, {3098, 16789}, {3284, 14836}, {3581, 41588}, {3796, 4846}, {3917, 12358}, {5562, 35240}, {5889, 43595}, {5894, 10575}, {6146, 7689}, {6247, 11750}, {7706, 37649}, {7723, 34153}, {8717, 20725}, {9019, 37511}, {10317, 15048}, {10610, 34798}, {10634, 42088}, {10635, 42087}, {10897, 42259}, {10898, 42258}, {11003, 34796}, {11440, 34224}, {11472, 31383}, {11597, 20127}, {11793, 32903}, {12134, 34785}, {12162, 34782}, {12163, 19467}, {12359, 21659}, {13367, 22660}, {13403, 41587}, {13470, 32210}, {13491, 31807}, {13567, 32110}, {14675, 23240}, {14983, 38699}, {15062, 16659}, {15063, 44110}, {16165, 38726}, {18438, 35257}, {18457, 42216}, {18459, 42215}, {18468, 42118}, {18470, 42117}, {19129, 21850}, {19131, 29181}, {22165, 32275}, {22815, 36966}, {23292, 39242}, {26881, 32111}, {31802, 37472}, {31804, 34783}, {32423, 41730}

X(44249) = midpoint of X(20) and X(378)

X(44250) = MIDPOINT OF X(20) AND X(383)

Barycentrics    Sqrt[3]*(4*a^8 - 15*a^6*b^2 + 13*a^4*b^4 - a^2*b^6 - b^8 - 15*a^6*c^2 + 10*a^4*b^2*c^2 + 5*a^2*b^4*c^2 + 4*b^6*c^2 + 13*a^4*c^4 + 5*a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 4*b^2*c^6 - c^8) + 2*(a^2 + b^2 + c^2)*(8*a^4 - 7*a^2*b^2 - b^4 - 7*a^2*c^2 + 2*b^2*c^2 - c^4)*S : :

X(44250) lies on these lines: {2, 3}, {14, 21159}, {16, 5474}, {62, 22843}, {182, 42942}, {299, 36995}, {395, 13349}, {396, 9735}, {511, 25179}, {617, 3564}, {1296, 34376}, {1350, 42625}, {5085, 42626}, {5321, 6774}, {5479, 6672}, {5865, 42151}, {5979, 21166}, {6036, 31710}, {6108, 23698}, {11543, 13102}, {14538, 42528}, {14539, 16529}, {14541, 42433}, {20425, 42118}, {20426, 42913}, {21156, 23005}, {21157, 36970}, {21401, 42791}, {32553, 38749}, {33517, 36756}

X(44250) = midpoint of X(20) and X(383)

X(44251) = MIDPOINT OF X(20) AND X(384)

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

X(44251) lies on these lines: {2, 3}, {141, 9873}, {698, 11257}, {736, 5188}, {1350, 36998}, {2794, 7873}, {3098, 6393}, {5305, 43453}, {5976, 38749}, {7738, 40825}, {7759, 30270}, {7767, 9862}, {7787, 21850}, {7789, 43460}, {9983, 32521}, {12110, 29181}, {14712, 35456}, {14810, 32152}, {15048, 38905}, {16111, 38650}, {16163, 38641}, {20065, 33878}, {24466, 38646}, {32134, 40238}, {32476, 32516}, {32830, 39874}, {33706, 35700}, {38642, 38738}, {38643, 38761}, {38644, 38773}, {38651, 38805}

X(44251) = midpoint of X(20) and X(384)

X(44252) = MIDPOINT OF X(20) AND X(401)

Barycentrics    4*a^12 - 9*a^10*b^2 + 7*a^8*b^4 - 6*a^6*b^6 + 6*a^4*b^8 - a^2*b^10 - b^12 - 9*a^10*c^2 + 16*a^8*b^2*c^2 - 2*a^6*b^4*c^2 - 4*a^4*b^6*c^2 - 5*a^2*b^8*c^2 + 4*b^10*c^2 + 7*a^8*c^4 - 2*a^6*b^2*c^4 - 4*a^4*b^4*c^4 + 6*a^2*b^6*c^4 - 7*b^8*c^4 - 6*a^6*c^6 - 4*a^4*b^2*c^6 + 6*a^2*b^4*c^6 + 8*b^6*c^6 + 6*a^4*c^8 - 5*a^2*b^2*c^8 - 7*b^4*c^8 - a^2*c^10 + 4*b^2*c^10 - c^12 : :

X(44252) lies on these lines: {2, 3}, {511, 34980}, {1294, 2715}, {1503, 1632}, {1975, 44141}, {3284, 9530}, {3564, 39352}, {6527, 39874}, {6530, 23583}, {10605, 36998}, {16163, 43952}, {36988, 42329}

X(44252) = midpoint of X(20) and X(401)

X(44253) = MIDPOINT OF X(21) AND X(28)

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

X(44253) lies on these lines: {2, 3}, {110, 24475}, {229, 24470}, {270, 18455}, {1780, 41697}, {2328, 16139}, {2360, 33858}, {16164, 18180}, {17188, 33592}, {39772, 41608}

X(44253) = midpoint of X(21) and X(28)

X(44254) = MIDPOINT OF X(21) AND X(140)

Barycentrics    6*a^7 - 6*a^6*b - 13*a^5*b^2 + 13*a^4*b^3 + 8*a^3*b^4 - 8*a^2*b^5 - a*b^6 + b^7 - 6*a^6*c + 2*a^5*b*c + a^4*b^2*c + 3*a^3*b^3*c + 6*a^2*b^4*c - 5*a*b^5*c - b^6*c - 13*a^5*c^2 + a^4*b*c^2 + 12*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 - 3*b^5*c^2 + 13*a^4*c^3 + 3*a^3*b*c^3 + 2*a^2*b^2*c^3 + 10*a*b^3*c^3 + 3*b^4*c^3 + 8*a^3*c^4 + 6*a^2*b*c^4 + a*b^2*c^4 + 3*b^3*c^4 - 8*a^2*c^5 - 5*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7 : :

X(44254) lies on these lines: {2, 3}, {191, 38028}, {2771, 13392}, {5426, 5690}, {5844, 35016}, {5901, 22937}, {10165, 22936}, {10543, 11545}, {21677, 34352}, {37737, 41542}

X(44254) = midpoint of X(21) and X(140)

X(44255) = MIDPOINT OF X(21) AND X(376)

Barycentrics    8*a^7 - 8*a^6*b - 15*a^5*b^2 + 15*a^4*b^3 + 6*a^3*b^4 - 6*a^2*b^5 + a*b^6 - b^7 - 8*a^6*c - 2*a^5*b*c - a^4*b^2*c + 4*a^3*b^3*c + 8*a^2*b^4*c - 2*a*b^5*c + b^6*c - 15*a^5*c^2 - a^4*b*c^2 + 16*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 15*a^4*c^3 + 4*a^3*b*c^3 - 2*a^2*b^2*c^3 + 4*a*b^3*c^3 - 3*b^4*c^3 + 6*a^3*c^4 + 8*a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 - 6*a^2*c^5 - 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7 : :

X(44255) lies on these lines: {2, 3}, {541, 16164}, {553, 13151}, {3058, 5427}, {3189, 34718}, {3579, 10543}, {3649, 13624}, {3650, 4511}, {3653, 11281}, {3655, 16139}, {4677, 10268}, {5441, 35242}, {7987, 16113}, {10122, 37585}, {11237, 35250}, {12702, 15174}, {17502, 38033}, {18253, 18481}, {21677, 28204}, {28194, 35016}, {35459, 41549}

X(44255) = midpoint of X(21) and X(376)

X(44256) = MIDPOINT OF X(21) AND X(377)

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

X(44256) lies on these lines: {2, 3}, {7, 11684}, {10, 10122}, {12, 5302}, {63, 10404}, {224, 33857}, {518, 8261}, {960, 3649}, {1259, 10198}, {1837, 3826}, {3647, 4292}, {3683, 31936}, {3841, 4304}, {3925, 10543}, {4313, 33108}, {5427, 24953}, {5441, 41859}, {5730, 16137}, {5745, 41547}, {5784, 17637}, {6598, 37723}, {6701, 21616}, {8583, 26725}, {9710, 19860}, {11031, 21674}, {11064, 25526}, {11281, 19861}, {15174, 31419}, {15823, 41542}, {24541, 37722}, {24982, 34501}, {26531, 27483}, {31446, 37719}, {40661, 41571}

X(44256) = midpoint of X(21) and X(377)

X(44257) = MIDPOINT OF X(21) AND X(381)

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

X(44257) lies on these lines: {2, 3}, {191, 38021}, {499, 11544}, {1125, 22798}, {2771, 3742}, {3582, 3649}, {3584, 10543}, {3647, 9955}, {3652, 8227}, {3656, 3899}, {5441, 5560}, {6701, 26202}, {10056, 15174}, {10072, 16137}, {10171, 34126}, {10199, 33668}, {11281, 38022}, {11684, 18493}, {16139, 31162}, {16140, 23708}, {16141, 37692}, {18253, 22791}, {19919, 33592}, {25055, 33858}, {28204, 35016}, {38114, 38140}

X(44257) = midpoint of X(21) and X(381)

X(44258) = MIDPOINT OF X(21) AND X(382)

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

X(44258) lies on these lines: {2, 3}, {79, 5561}, {758, 22793}, {1478, 15174}, {1479, 15935}, {1539, 2771}, {1699, 33858}, {2795, 22505}, {3583, 3649}, {3585, 5719}, {3678, 18480}, {5427, 10483}, {5441, 11374}, {5806, 16125}, {9581, 16118}, {11281, 38034}, {12679, 16159}, {16139, 41869}, {18483, 33592}, {19919, 22798}, {21677, 28174}, {28160, 35016}

X(44258) = midpoint of X(21) and X(382)

X(44259) = MIDPOINT OF X(22) AND X(24)

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

X(44259) lies on these lines: {2, 3}, {156, 8907}, {2781, 20771}, {2931, 10605}, {7723, 10117}, {9699, 14961}, {9932, 34783}, {12358, 40914}, {16165, 18445}, {25738, 33563}, {32048, 44076}

X(44259) = midpoint of X(22) and X(24)
X(44259) = harmonic center of circumcircle and tangential circle
X(44259) = center of circle {{X(22),X(24),PU(4)}}
X(44259) = tangential-circle-inverse of X(11810)
X(44259) = X(8069)-of-tangential-triangle if ABC is acute

X(44260) = MIDPOINT OF X(22) AND X(25)

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

X(44260) lies on these lines: {2, 3}, {182, 11746}, {184, 14984}, {1352, 10117}, {1383, 22121}, {1495, 37511}, {1660, 15582}, {2393, 19127}, {2781, 9306}, {2931, 4846}, {3060, 18449}, {5651, 13416}, {6776, 12310}, {9019, 11511}, {15080, 19129}, {15107, 18438}, {19126, 41579}, {19131, 35268}, {19220, 34481}, {21243, 36201}, {21849, 37827}, {26881, 40114}, {31804, 32048}

X(44260) = midpoint of X(22) and X(25)

X(44261) = MIDPOINT OF X(22) AND X(376)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 4*a^6*b^2*c^2 + 4*a^4*b^4*c^2 + 4*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 + 4*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 + 4*a^2*b^2*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44261) lies on these lines: {2, 3}, {32, 14836}, {52, 8584}, {524, 37478}, {541, 16165}, {542, 16789}, {597, 37513}, {1495, 35254}, {3933, 6148}, {9019, 9730}, {11179, 37489}, {11430, 19924}, {11464, 40112}, {12022, 15360}, {13352, 19127}, {14805, 21850}, {15534, 17834}, {18317, 28724}, {18472, 42459}, {20423, 37506}, {22165, 34782}, {29181, 39242}, {30714, 31729}, {37853, 40291}

X(44261) = midpoint of X(22) and X(376)

X(44262) = MIDPOINT OF X(22) AND X(381)

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

X(44262) lies on these lines: {2, 3}, {113, 33533}, {155, 15533}, {265, 15080}, {542, 19127}, {568, 15360}, {599, 15068}, {2781, 15067}, {3767, 18573}, {5305, 14836}, {5476, 9019}, {5609, 31744}, {5655, 11459}, {6795, 34209}, {6800, 32423}, {7998, 14643}, {9730, 32225}, {10264, 37638}, {10272, 15066}, {11438, 15361}, {11645, 34514}, {12161, 15534}, {13339, 15362}, {14852, 43273}, {14880, 39170}, {15072, 20126}, {16534, 40107}, {17702, 34513}, {21230, 32139}, {21357, 32063}

X(44262) = midpoint of X(22) and X(381)

X(44263) = MIDPOINT OF X(22) AND X(382)

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

X(44263) lies on these lines: {2, 3}, {49, 12278}, {146, 18550}, {156, 43831}, {184, 30522}, {265, 5890}, {1204, 13561}, {1511, 19479}, {1531, 5891}, {1539, 2781}, {3521, 6241}, {3583, 37729}, {4846, 18434}, {5654, 40111}, {5663, 18474}, {5944, 34785}, {5946, 7706}, {6000, 34514}, {6102, 9927}, {6288, 12111}, {7689, 34798}, {7699, 12121}, {7703, 20127}, {7728, 12281}, {9220, 38872}, {9703, 12383}, {9730, 13851}, {9826, 13364}, {10264, 10605}, {10574, 18394}, {10575, 11572}, {10606, 14677}, {10610, 32365}, {10733, 12228}, {10749, 14983}, {10984, 13470}, {11472, 32316}, {12041, 19506}, {12042, 39847}, {12161, 12293}, {12233, 12370}, {12902, 15087}, {13403, 34114}, {13491, 18381}, {13630, 18379}, {14356, 18380}, {14644, 15053}, {15026, 43865}, {15043, 43821}, {15111, 20957}, {15800, 32338}, {17702, 18388}, {18356, 34783}, {18383, 40647}, {18430, 25739}, {18435, 41171}, {18445, 32423}, {18513, 37697}, {18514, 37696}, {19127, 29012}, {20299, 43577}, {20424, 36747}, {21400, 43816}, {21659, 32046}, {22802, 22804}, {32767, 43604}, {33813, 39818}, {36753, 43575}

X(44263) = midpoint of X(22) and X(382)
X(44263) = reflection of X(18570) in X(5)
X(44263) = X(18570)-of-Johnson-triangle
X(44263) = orthocentroidal-circle-inverse of X(18403)
X(44263) = {X(2),X(4)}-harmonic conjugate of X(18403)
X(44263) = {X(3),X(4)}-harmonic conjugate of X(18377)

X(44264) = MIDPOINT OF X(23) AND X(140)

Barycentrics    6*a^10 - 13*a^8*b^2 + 2*a^6*b^4 + 12*a^4*b^6 - 8*a^2*b^8 + b^10 - 13*a^8*c^2 + 6*a^6*b^2*c^2 - 7*a^4*b^4*c^2 + 17*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 7*a^4*b^2*c^4 - 18*a^2*b^4*c^4 + 2*b^6*c^4 + 12*a^4*c^6 + 17*a^2*b^2*c^6 + 2*b^4*c^6 - 8*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44264) lies on these lines: {2, 3}, {143, 40632}, {511, 13392}, {3564, 32218}, {7286, 38458}, {10610, 20193}, {13353, 43579}, {14979, 33505}, {22251, 37496}, {29012, 40685}, {32217, 34380}, {32223, 32423}

X(44264) = midpoint of X(23) and X(140)
X(44264) = X(468)-of-Napoleon-Feuerbach-triangle

X(44265) = MIDPOINT OF X(23) AND X(376)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 22*a^6*b^2*c^2 - 14*a^4*b^4*c^2 + 4*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 14*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 + 4*a^2*b^2*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44265) lies on these lines: {2, 3}, {187, 3018}, {511, 15303}, {524, 3581}, {541, 1495}, {542, 32110}, {568, 8584}, {597, 14805}, {1384, 16303}, {1503, 20126}, {1511, 40112}, {2080, 14934}, {2777, 32267}, {3580, 15361}, {5476, 39242}, {5655, 35266}, {7728, 15448}, {8547, 11179}, {8705, 40280}, {9730, 11649}, {10564, 19924}, {12121, 32269}, {14653, 16320}, {15462, 32217}, {15534, 37489}, {16111, 32237}, {16279, 16324}, {16760, 31173}, {17702, 32225}, {18316, 34209}, {18317, 18876}, {21158, 34314}, {21159, 34313}, {22151, 37496}, {32111, 32124}, {32417, 42671}

X(44265) = midpoint of X(23) and X(376)
X(44265) = circumcircle-inverse of X(31861)
X(44265) = nine-point-circle-inverse of X(7579)
X(44265) = orthocentroidal-circle-inverse of X(39484)
X(44265) = {X(2),X(4)}-harmonic conjugate of X(39484)

X(44266) = MIDPOINT OF X(23) AND X(381)

Barycentrics    2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 6*a^8*c^2 - 8*a^6*b^2*c^2 + a^4*b^4*c^2 + 19*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 + a^4*b^2*c^4 - 26*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 19*a^2*b^2*c^6 + 4*b^4*c^6 - 6*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

X(44266) lies on these lines: {2, 3}, {511, 25566}, {523, 11622}, {524, 19140}, {541, 15361}, {542, 32217}, {2682, 32219}, {3581, 10706}, {3582, 7286}, {3584, 5160}, {5309, 16308}, {5476, 8705}, {5655, 15360}, {5663, 32225}, {9140, 15362}, {10113, 32237}, {10272, 40112}, {15068, 15533}, {15448, 34153}, {16252, 41149}, {17702, 32267}

X(44266) = midpoint of X(23) and X(381)
X(44266) = {X(13626),X(13627)}-harmonic conjugate of X(5)

X(44267) = MIDPOINT OF X(23) AND X(382)

Barycentrics    2*a^10 - 2*a^8*b^2 - 4*a^6*b^4 + 4*a^4*b^6 + 2*a^2*b^8 - 2*b^10 - 2*a^8*c^2 + 16*a^6*b^2*c^2 - 7*a^4*b^4*c^2 - 13*a^2*b^6*c^2 + 6*b^8*c^2 - 4*a^6*c^4 - 7*a^4*b^2*c^4 + 22*a^2*b^4*c^4 - 4*b^6*c^4 + 4*a^4*c^6 - 13*a^2*b^2*c^6 - 4*b^4*c^6 + 2*a^2*c^8 + 6*b^2*c^8 - 2*c^10 : :

X(44267) lies on these lines: {2, 3}, {323, 38789}, {511, 1539}, {1154, 16105}, {1503, 9976}, {1531, 41673}, {1533, 12295}, {3521, 16881}, {3564, 16176}, {3581, 10721}, {3583, 7286}, {3585, 5160}, {5270, 10149}, {5446, 43392}, {7748, 16308}, {10113, 14915}, {12112, 12902}, {13382, 43585}, {13851, 16270}, {15118, 29012}, {15125, 15647}, {16534, 40111}, {32110, 34584}, {32111, 32423}, {32220, 32255}

X(44267) = midpoint of X(23) and X(382)

X(44268) = MIDPOINT OF X(24) AND X(376)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 36*a^6*b^2*c^2 - 24*a^4*b^4*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 24*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44268) lies on these lines: {2, 3}, {541, 20771}, {5085, 10169}, {8584, 19161}, {12134, 43903}, {13567, 16163}, {16270, 27365}, {22660, 43898}, {23332, 38727}, {31804, 43601}

X(44268) = midpoint of X(24) and X(376)

X(44269) = MIDPOINT OF X(24) AND X(378)

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

X(44269) lies on these lines: {2, 3}, {74, 18532}, {230, 21397}, {232, 19220}, {974, 38534}, {1204, 43896}, {1619, 12112}, {1853, 19457}, {1993, 15463}, {2904, 6102}, {3043, 3167}, {3060, 15472}, {8537, 13482}, {9938, 14516}, {11550, 32607}, {12111, 19908}, {12140, 12901}, {12290, 32321}, {13171, 35450}, {15473, 25564}, {20771, 35264}

X(44269) = midpoint of X(24) and X(378)
X(44269) = X(45179)-of-anti-Euler-triangle

X(44270) = MIDPOINT OF X(24) AND X(381)

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

X(44270) lies on these lines: {2, 3}, {1993, 10272}, {3060, 14643}, {9707, 43575}, {10264, 26958}, {11455, 15061}, {11550, 20304}, {14576, 18487}, {15534, 19139}, {26882, 43821}, {32423, 35264}, {34785, 43865}, {34796, 38789}

X(44270) = midpoint of X(24) and X(381)

X(44271) = MIDPOINT OF X(24) AND X(382)

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

X(44271) lies on these lines: {2, 3}, {64, 10264}, {155, 22972}, {265, 12290}, {569, 15807}, {1533, 11750}, {1539, 10263}, {2883, 12370}, {3521, 3567}, {5889, 7728}, {6102, 22802}, {6288, 11439}, {8548, 36990}, {10113, 18381}, {11550, 18379}, {13203, 21970}, {13423, 15800}, {13474, 34514}, {13491, 18390}, {15072, 43821}, {15811, 19458}, {18356, 18439}, {18383, 31978}, {18474, 32137}, {18550, 43891}, {22968, 43577}, {26883, 30522}, {32111, 44076}

X(44271) = midpoint of X(24) and X(382)

X(44272) = MIDPOINT OF X(25) AND X(186)

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

X(44272) lies on these lines: {2, 3}, {184, 16227}, {974, 1495}, {1511, 44084}, {2493, 14581}, {5140, 38611}, {5926, 41357}, {11062, 40135}, {12584, 15471}, {13754, 20772}, {14984, 44102}

X(44272) = midpoint of X(25) and X(186)

X(44273) = MIDPOINT OF X(25) AND X(376)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 40*a^6*b^2*c^2 - 32*a^4*b^4*c^2 + 4*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 32*a^4*b^2*c^4 + 4*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 + 4*a^2*b^2*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44273) lies on these lines: {2, 3}, {389, 8584}, {524, 11438}, {541, 20772}, {542, 8263}, {597, 11430}, {599, 37487}, {1192, 15533}, {2393, 16836}, {3053, 34288}, {9730, 14984}, {9786, 15534}, {10564, 21850}, {11179, 37475}, {11180, 18931}, {12022, 43804}, {14389, 15051}, {14836, 14910}, {16163, 37648}, {19136, 37480}, {20423, 37497}, {23328, 36201}, {38110, 39242}, {43601, 43897}

X(44273) = midpoint of X(25) and X(376)

X(44274) = MIDPOINT OF X(25) AND X(378)

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

X(44274) lies on these lines: {2, 3}, {32, 39176}, {1511, 44080}, {1843, 10564}, {1974, 37470}, {2393, 11430}, {2781, 11438}, {2790, 3455}, {6403, 37477}, {8541, 13352}, {9019, 37480}, {9730, 44102}, {10986, 40115}, {11464, 40114}, {12168, 37645}, {12294, 32110}, {15655, 41758}, {16836, 19127}, {19124, 39242}, {19128, 40280}

X(44274) = midpoint of X(25) and X(378)

X(44275) = MIDPOINT OF X(25) AND X(381)

Barycentrics    a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 - 4*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 14*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 - 22*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 14*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44275) lies on these lines: {2, 3}, {113, 34417}, {155, 15534}, {399, 37644}, {524, 15068}, {541, 11438}, {542, 19136}, {567, 40114}, {568, 5655}, {576, 16534}, {1533, 37470}, {1989, 3767}, {2393, 5476}, {3199, 18487}, {4550, 32223}, {5609, 34319}, {5642, 13352}, {5648, 5654}, {6759, 43573}, {8584, 12161}, {9220, 43620}, {10264, 37643}, {10272, 37645}, {10602, 14848}, {11004, 20125}, {11179, 18374}, {11459, 15360}, {12228, 44080}, {15030, 32225}, {15533, 17814}, {16657, 35266}, {18952, 26883}, {23325, 36201}, {26937, 32137}, {31860, 40909}

X(44275) = midpoint of X(25) and X(381)

X(44276) = MIDPOINT OF X(25) AND X(382)

Barycentrics    a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 12*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 10*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 4*a^4*b^2*c^4 + 18*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 10*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44276) lies on these lines: {2, 3}, {1353, 41735}, {1478, 9629}, {1498, 12370}, {1539, 9973}, {2790, 22515}, {2883, 12161}, {4846, 5946}, {5446, 22802}, {5878, 6102}, {6225, 18951}, {6759, 12897}, {9927, 13474}, {10575, 18952}, {11381, 32140}, {11744, 12236}, {12174, 43588}, {12290, 25738}, {12293, 15811}, {13419, 22661}, {13491, 39571}, {13598, 31815}, {14677, 18933}, {14915, 18390}, {18382, 36201}, {18445, 32111}, {18474, 32062}, {19136, 29012}, {21850, 34777}, {22660, 34966}, {30522, 31383}

X(44276) = midpoint of X(25) and X(382)

X(44277) = MIDPOINT OF X(26) AND X(140)

Barycentrics    6*a^10 - 13*a^8*b^2 + 2*a^6*b^4 + 12*a^4*b^6 - 8*a^2*b^8 + b^10 - 13*a^8*c^2 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 12*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 4*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 + 12*a^4*c^6 + 12*a^2*b^2*c^6 + 2*b^4*c^6 - 8*a^2*c^8 - 3*b^2*c^8 + c^10 : :

X(44277) lies on these lines: {2, 3}, {1147, 19154}, {2883, 16775}, {3564, 10282}, {5907, 15448}, {5944, 13292}, {10619, 32225}, {12164, 35260}, {12241, 32223}, {12359, 34776}, {13142, 13367}, {16621, 32237}, {19357, 41588}

X(44277) = midpoint of X(26) and X(140)

X(44278) = MIDPOINT OF X(26) AND X(381)

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

X(44278) lies on these lines: {2, 3}, {13, 11268}, {14, 11267}, {154, 32423}, {156, 539}, {265, 26881}, {394, 10272}, {542, 19154}, {2979, 14643}, {3582, 32047}, {3584, 8144}, {6759, 18356}, {8254, 10982}, {11265, 35823}, {11266, 35822}, {15806, 36747}, {19347, 32165}, {26883, 34826}, {35237, 40685}

X(44278) = midpoint of X(26) and X(381)

X(44279) = MIDPOINT OF X(26) AND X(382)

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

X(44279) lies on these lines: {2, 3}, {156, 17702}, {265, 6241}, {974, 10113}, {1514, 12134}, {1531, 10625}, {1539, 5876}, {1614, 10733}, {2777, 5449}, {3357, 13561}, {3521, 5890}, {3583, 32047}, {3585, 8144}, {4846, 18952}, {5663, 9927}, {5878, 32140}, {5895, 14852}, {6288, 15305}, {6759, 30522}, {7706, 10095}, {7728, 7731}, {10540, 12278}, {10574, 43821}, {10575, 13851}, {10721, 11440}, {11265, 35821}, {11266, 35820}, {11267, 19107}, {11268, 19106}, {11381, 34514}, {11449, 12121}, {11468, 20127}, {11598, 19506}, {11704, 15061}, {12279, 18394}, {12290, 18392}, {12293, 17824}, {12295, 21659}, {12897, 18388}, {12902, 34799}, {13403, 32046}, {13630, 18390}, {14643, 18504}, {14831, 18555}, {14915, 18383}, {18379, 18381}, {18418, 22816}, {19154, 29012}, {19479, 20773}, {26881, 40242}

X(44279) = midpoint of X(26) and X(382)
X(44279) = reflection of X(18377) in X(4)
X(44279) = reflection of X(18569) in X(18567)
X(44279) = complement of X(34350)
X(44279) = anticomplement of X(10226)
X(44279) = {X(4),X(20)}-harmonic conjugate of X(18403)

X(44280) = MIDPOINT OF X(186) AND X(376)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 34*a^6*b^2*c^2 - 20*a^4*b^4*c^2 - 2*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 20*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 - 2*a^2*b^2*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44280) lies on these lines: {2, 3}, {542, 21663}, {1350, 32220}, {1495, 37853}, {1503, 15055}, {3060, 16227}, {3580, 16163}, {5023, 16306}, {6091, 16092}, {8584, 37473}, {11064, 15051}, {11693, 39083}, {15136, 43572}, {15311, 35266}, {15448, 20725}, {16111, 32111}, {16226, 32411}, {32110, 38726}, {38699, 38701}, {38702, 38718}

X(44280) = midpoint of X186) and X(376)

X(44281) = MIDPOINT OF X(186) AND X(378)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 6*a^8*c^2 + 16*a^6*b^2*c^2 - 11*a^4*b^4*c^2 - 2*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 11*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 5*b^6*c^4 + 4*a^4*c^6 - 2*a^2*b^2*c^6 - 5*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 + 2*c^10) : :

X(44281) lies on these lines: {2, 3}, {74, 34397}, {1112, 32110}, {1495, 12133}, {1503, 32607}, {1514, 13289}, {1968, 16308}, {2781, 21663}, {3003, 39176}, {3581, 15472}, {8705, 19124}, {11204, 44077}, {11438, 16227}, {11807, 25564}, {12294, 32217}, {15055, 19128}, {16013, 16659}, {22455, 40352}, {32113, 39871}

X(44281) = midpoint of X(186) and X(378)

X(44282) = MIDPOINT OF X(186) AND X(381)

Barycentrics    2*a^10 - 6*a^8*b^2 + 4*a^6*b^4 + 4*a^4*b^6 - 6*a^2*b^8 + 2*b^10 - 6*a^8*c^2 + 4*a^6*b^2*c^2 - 5*a^4*b^4*c^2 + 13*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 5*a^4*b^2*c^4 - 14*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 13*a^2*b^2*c^6 + 4*b^4*c^6 - 6*a^2*c^8 - 6*b^2*c^8 + 2*c^10 : :

X(44282) lies on these lines: {2, 3}, {524, 34155}, {1154, 32225}, {1495, 20304}, {1514, 14677}, {3580, 10272}, {3584, 10149}, {5099, 14693}, {5642, 40111}, {7746, 16308}, {8705, 38317}, {9140, 10540}, {11062, 18487}, {11649, 16776}, {12900, 32223}, {13391, 13857}, {14915, 34128}, {14993, 18883}, {15311, 16219}, {15362, 22115}, {15826, 25555}, {16310, 18285}, {16319, 34209}, {18583, 32113}, {24206, 32217}, {35265, 38724}, {38795, 41586}

X(44282) = midpoint of X(186) and X(381)

X(44283) = MIDPOINT OF X(186) AND X(382)

Barycentrics    2*a^10 - 2*a^8*b^2 - 4*a^6*b^4 + 4*a^4*b^6 + 2*a^2*b^8 - 2*b^10 - 2*a^8*c^2 + 12*a^6*b^2*c^2 - 5*a^4*b^4*c^2 - 11*a^2*b^6*c^2 + 6*b^8*c^2 - 4*a^6*c^4 - 5*a^4*b^2*c^4 + 18*a^2*b^4*c^4 - 4*b^6*c^4 + 4*a^4*c^6 - 11*a^2*b^2*c^6 - 4*b^4*c^6 + 2*a^2*c^8 + 6*b^2*c^8 - 2*c^10 : :

X(44283) lies on these lines: {2, 3}, {113, 40111}, {539, 38791}, {974, 13851}, {1531, 13391}, {1539, 11807}, {3585, 10149}, {5893, 12370}, {6000, 10113}, {10110, 22833}, {10264, 15311}, {10540, 10733}, {10688, 23956}, {11381, 18379}, {11455, 18430}, {12295, 30522}, {13321, 18550}, {21663, 34584}, {40647, 43865}

X(44283) = midpoint of X(186) and X(382)
X(44283) = center of the Vu pedal-centroidal circle of X(23)

X(44284) = MIDPOINT OF X(376) AND X(377)

Barycentrics    a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 - a^6*c - 16*a^5*b*c + a^4*b^2*c + 14*a^3*b^3*c + a^2*b^4*c + 2*a*b^5*c - b^6*c - 3*a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 + 14*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 3*a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7 : :

X(44284) lies on these lines: {2, 3}, {518, 3654}, {553, 37584}, {3579, 10056}, {3584, 35242}, {3587, 4654}, {3655, 11260}, {3656, 10179}, {4677, 30503}, {4745, 43151}, {4995, 35238}, {5434, 35239}, {10072, 13624}, {10178, 26446}, {10525, 35202}, {16192, 26487}, {37496, 37685}

X(44284) = midpoint of X(376) and X(377)

X(44285) = MIDPOINT OF X(376) AND X(378)

Barycentrics    8*a^10 - 15*a^8*b^2 - 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - b^10 - 15*a^8*c^2 + 28*a^6*b^2*c^2 - 8*a^4*b^4*c^2 - 8*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 8*a^4*b^2*c^4 + 28*a^2*b^4*c^4 - 2*b^6*c^4 + 16*a^4*c^6 - 8*a^2*b^2*c^6 - 2*b^4*c^6 - 6*a^2*c^8 + 3*b^2*c^8 - c^10 : :

X(44285) lies on these lines: {2, 3}, {141, 16163}, {541, 18475}, {2777, 13394}, {2781, 15303}, {5063, 14836}, {5092, 37853}, {8584, 14831}, {9019, 36987}, {10564, 35254}, {10605, 11179}, {10606, 43273}, {11202, 35266}, {11440, 31804}, {15472, 19129}

X(44285) = midpoint of X(376) and X(378)

X(44286) = MIDPOINT OF X(377) AND X(382)

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

X(44286) lies on these lines: {2, 3}, {518, 39884}, {3874, 31673}, {5557, 37723}, {5761, 10248}, {5806, 26201}, {10404, 12433}, {10572, 15935}, {17098, 37730}, {18482, 25557}

X(44286) = midpoint of X(377) and X(382)

X(44287) = MIDPOINT OF X(377) AND X(381)

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

X(44287) lies on these lines: {2, 3}, {6, 10264}, {156, 18488}, {182, 20113}, {265, 7703}, {541, 18388}, {1511, 3818}, {2781, 5476}, {3574, 32138}, {5640, 15061}, {5655, 15305}, {5890, 20126}, {5891, 13857}, {6699, 19130}, {7699, 7728}, {7706, 12041}, {8548, 15534}, {9140, 15033}, {9143, 9703}, {10168, 19127}, {10546, 38794}, {11424, 13561}, {11430, 34514}, {11645, 18475}, {11935, 14683}, {12163, 20424}, {14643, 16261}, {18356, 37472}, {20299, 43573}, {21230, 37498}, {29012, 34513}, {31840, 41731}

X(44287) = midpoint of X(377) and X(381)

X(44288) = MIDPOINT OF X(378) AND X(382)

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

X(44288) lies on these lines: {2, 3}, {52, 11572}, {68, 31815}, {195, 34799}, {265, 3060}, {568, 25739}, {569, 13470}, {1154, 18474}, {1353, 11216}, {1531, 16194}, {1539, 19479}, {1853, 10264}, {1993, 32423}, {2781, 10113}, {3167, 12319}, {3357, 34798}, {3521, 12279}, {3574, 11750}, {3581, 23293}, {3818, 9019}, {5446, 18383}, {5448, 13419}, {5663, 11550}, {5889, 15800}, {6101, 22804}, {6102, 11262}, {6288, 11412}, {7699, 26881}, {7728, 11455}, {9707, 15806}, {9781, 43821}, {9927, 10263}, {10272, 35264}, {10620, 34796}, {11605, 12918}, {12161, 17824}, {12278, 37495}, {12289, 37472}, {12295, 32235}, {12370, 41362}, {13203, 35450}, {13352, 30522}, {13754, 34514}, {16789, 18358}, {16881, 18912}, {17505, 17711}, {18382, 21850}, {18396, 39522}, {19127, 19130}, {19139, 36990}, {20299, 32393}, {22505, 22823}, {22515, 39847}, {23039, 41171}, {31802, 32358}, {32346, 36966}, {32533, 38433}, {34785, 43394}

X(44288) = midpoint of X(378) and X(382)
X(44288) = Johnson-circle-inverse of X(23)

X(44289) = MIDPOINT OF X(381) AND X(383)

Barycentrics    3*a^10 - 12*a^6*b^4 + 6*a^4*b^6 + 9*a^2*b^8 - 6*b^10 - 12*a^6*b^2*c^2 - 30*a^4*b^4*c^2 + 24*a^2*b^6*c^2 + 18*b^8*c^2 - 12*a^6*c^4 - 30*a^4*b^2*c^4 - 66*a^2*b^4*c^4 - 12*b^6*c^4 + 6*a^4*c^6 + 24*a^2*b^2*c^6 - 12*b^4*c^6 + 9*a^2*c^8 + 18*b^2*c^8 - 6*c^10 + 2*Sqrt[3]*(a^8 + 7*a^4*b^4 - 9*a^2*b^6 + b^8 + 10*a^4*b^2*c^2 + 10*a^2*b^4*c^2 - 9*b^6*c^2 + 7*a^4*c^4 + 10*a^2*b^2*c^4 + 16*b^4*c^4 - 9*a^2*c^6 - 9*b^2*c^6 + c^8)*S : :

X(44289) lies on these lines: {2, 3}, {530, 22566}, {542, 7685}, {623, 25561}, {624, 19924}, {2548, 41112}, {3767, 41113}, {5459, 19130}, {5460, 11645}, {5873, 15534}, {6298, 8176}, {14881, 33483}, {16268, 36382}, {22496, 36362}, {22568, 22575}, {22796, 33477}, {22803, 33484}, {25164, 41037}, {43291, 43417}

X(44289) = midpoint of X(381) and X(383)

X(44290) = MIDPOINT OF X(28) AND X(29)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^7 - 2*a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + 2*a*b^6 - b^7 - 2*a^5*b*c + 3*a^4*b^2*c - 2*a^2*b^4*c + 2*a*b^5*c - b^6*c - 2*a^5*c^2 + 3*a^4*b*c^2 + 4*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - 2*a*b^4*c^2 + b^5*c^2 - a^4*c^3 - 4*a^2*b^2*c^3 - 4*a*b^3*c^3 + b^4*c^3 - 2*a^3*c^4 - 2*a^2*b*c^4 - 2*a*b^2*c^4 + b^3*c^4 + 2*a^2*c^5 + 2*a*b*c^5 + b^2*c^5 + 2*a*c^6 - b*c^6 - c^7) : :
X(44290) = X(28) + X(29)

X(44290) lies on these lines: {2, 3}, {162, 34753}, {14192, 15171}

X(44290) = midpoint of X(28) and X(29)

X(44291) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^7 - a^6*b - 3*a^5*b^2 - 3*a^3*b^4 + 3*a^2*b^5 + 3*a*b^6 - 2*b^7 - a^6*c - 3*a^5*b*c + 6*a^4*b^2*c - 3*a^2*b^4*c + 3*a*b^5*c - 2*b^6*c - 3*a^5*c^2 + 6*a^4*b*c^2 + 6*a^3*b^2*c^2 - 8*a^2*b^3*c^2 - 3*a*b^4*c^2 + 2*b^5*c^2 - 8*a^2*b^2*c^3 - 6*a*b^3*c^3 + 2*b^4*c^3 - 3*a^3*c^4 - 3*a^2*b*c^4 - 3*a*b^2*c^4 + 2*b^3*c^4 + 3*a^2*c^5 + 3*a*b*c^5 + 2*b^2*c^5 + 3*a*c^6 - 2*b*c^6 - 2*c^7) : :
X(44291) = X(28) + 2 X(29)

X(44291) lies on this line: {2, 3}

X(44291) = midpoint of X(29) and X(44293)
X(44291) = reflection of X(i) in X(j) for these (i, j): (28, 44293), (44293, 44290)
X(44291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (29, 44290, 28), (4248, 15763, 28)

X(44292) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^7 - 3*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 + a^2*b^5 + a*b^6 - 2*b^7 - 3*a^6*c - a^5*b*c + 6*a^4*b^2*c - a^2*b^4*c + a*b^5*c - 2*b^6*c - a^5*c^2 + 6*a^4*b*c^2 + 2*a^3*b^2*c^2 - 8*a^2*b^3*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 8*a^2*b^2*c^3 - 2*a*b^3*c^3 + 2*b^4*c^3 - a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 + 2*b^3*c^4 + a^2*c^5 + a*b*c^5 + 2*b^2*c^5 + a*c^6 - 2*b*c^6 - 2*c^7) : :
X(44292) = X(28) - 2 X(29)

X(44292) lies on this line: {2, 3}

X(44293) = X(2)X(3)∩X(3058)X(14192)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^6 - 2*a^5*b - a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + 4*a*b^5 - b^6 - 2*a^5*c + a^4*b*c + 3*a^3*b^2*c - a^2*b^3*c - a*b^4*c - a^4*c^2 + 3*a^3*b*c^2 - 3*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - a^2*b*c^3 - 3*a*b^2*c^3 - a^2*c^4 - a*b*c^4 + b^2*c^4 + 4*a*c^5 - c^6) : :
X(44293) = 2 X(28) + X(29)

X(44293) lies on these lines: {2, 3}, {3058, 14192}

X(44294) = X(2)X(3)∩X(270)X(18653)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^7 + 3*a^6*b - a^5*b^2 - 5*a^4*b^3 - a^3*b^4 + a^2*b^5 + a*b^6 + b^7 + 3*a^6*c - a^5*b*c - 3*a^4*b^2*c - a^2*b^4*c + a*b^5*c + b^6*c - a^5*c^2 - 3*a^4*b*c^2 + 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 - 5*a^4*c^3 + 4*a^2*b^2*c^3 - 2*a*b^3*c^3 - b^4*c^3 - a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + a^2*c^5 + a*b*c^5 - b^2*c^5 + a*c^6 + b*c^6 + c^7) : :
X(44294) = 2 X(28) - X(29)

X(44294) lies on these lines: {2, 3}, {270, 18653}, {7354, 14192}

X(44295) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^9 + a^7*b^2 - a^6*b^3 - 4*a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 - b^9 + 2*a^7*b*c + a^6*b^2*c - 4*a^5*b^3*c - a^4*b^4*c + a^2*b^6*c + 2*a*b^7*c - b^8*c + a^7*c^2 + a^6*b*c^2 - a^5*b^2*c^2 + 2*a^4*b^3*c^2 - a^3*b^4*c^2 - 3*a^2*b^5*c^2 + a*b^6*c^2 - a^6*c^3 - 4*a^5*b*c^3 + 2*a^4*b^2*c^3 - 7*a^2*b^4*c^3 - 2*a*b^5*c^3 - 4*a^5*c^4 - a^4*b*c^4 - a^3*b^2*c^4 - 7*a^2*b^3*c^4 - 4*a*b^4*c^4 + 2*b^5*c^4 + a^4*c^5 - 3*a^2*b^2*c^5 - 2*a*b^3*c^5 + 2*b^4*c^5 + a^3*c^6 + a^2*b*c^6 + a*b^2*c^6 + a^2*c^7 + 2*a*b*c^7 + a*c^8 - b*c^8 - c^9) : :
X(44295) = (a*b + a*c + b*c)X(28) + (a^2 + b^2 + c^2)X(29)

X(44295) lies on this line: {2, 3}

X(44296) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^9 - 2*a^8*b - a^7*b^2 + a^6*b^3 + 3*a^4*b^5 - a^3*b^6 - a^2*b^7 + a*b^8 - b^9 - 2*a^8*c - 4*a^7*b*c + 3*a^6*b^2*c + 2*a^5*b^3*c + 5*a^4*b^4*c + 2*a^3*b^5*c - 5*a^2*b^6*c - b^8*c - a^7*c^2 + 3*a^6*b*c^2 + a^5*b^2*c^2 - 2*a^4*b^3*c^2 + a^3*b^4*c^2 - a^2*b^5*c^2 - a*b^6*c^2 + a^6*c^3 + 2*a^5*b*c^3 - 2*a^4*b^2*c^3 - 4*a^3*b^3*c^3 - a^2*b^4*c^3 + 5*a^4*b*c^4 + a^3*b^2*c^4 - a^2*b^3*c^4 + 2*b^5*c^4 + 3*a^4*c^5 + 2*a^3*b*c^5 - a^2*b^2*c^5 + 2*b^4*c^5 - a^3*c^6 - 5*a^2*b*c^6 - a*b^2*c^6 - a^2*c^7 + a*c^8 - b*c^8 - c^9) : :
X(44296) = (a*b + a*c + b*c)X(28) - (a^2 + b^2 + c^2)X(29)

X(44296) lies on this line: {2, 3}

X(44297) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^9 + 2*a^8*b - a^7*b^2 - 2*a^6*b^3 - a^5*b^4 - 2*a^4*b^5 + a^3*b^6 + 2*a^2*b^7 + 2*a^8*c - 2*a^7*b*c - 2*a^6*b^2*c + 2*a^5*b^3*c - a^4*b^4*c + 2*a^2*b^6*c - b^8*c - a^7*c^2 - 2*a^6*b*c^2 + 5*a^5*b^2*c^2 + 3*a^4*b^3*c^2 - 5*a^3*b^4*c^2 + a*b^6*c^2 - b^7*c^2 - 2*a^6*c^3 + 2*a^5*b*c^3 + 3*a^4*b^2*c^3 - 8*a^3*b^3*c^3 - 8*a^2*b^4*c^3 + b^6*c^3 - a^5*c^4 - a^4*b*c^4 - 5*a^3*b^2*c^4 - 8*a^2*b^3*c^4 - 2*a*b^4*c^4 + b^5*c^4 - 2*a^4*c^5 + b^4*c^5 + a^3*c^6 + 2*a^2*b*c^6 + a*b^2*c^6 + b^3*c^6 + 2*a^2*c^7 - b^2*c^7 - b*c^8) : :
X(44297) = (a^2 + b^2 + c^2)X(28) + (a*b + a*c + b*c)X(29)

X(44297) lies on this line: {2, 3}

X(44298) = (pending)

Barycentrics    (a + b)*(a + c)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^8 - a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - a^4*b^4 + a^3*b^5 - 2*a^2*b^6 + 2*a*b^7 - a^7*c + 2*a^6*b*c - 2*a^4*b^3*c + 2*a^3*b^4*c - a^2*b^5*c - a*b^6*c + b^7*c + 2*a^6*c^2 - 5*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 2*a^5*c^3 - 2*a^4*b*c^3 + 4*a^2*b^3*c^3 - a*b^4*c^3 - b^5*c^3 - a^4*c^4 + 2*a^3*b*c^4 + 3*a^2*b^2*c^4 - a*b^3*c^4 + a^3*c^5 - a^2*b*c^5 - b^3*c^5 - 2*a^2*c^6 - a*b*c^6 + 2*a*c^7 + b*c^7) : :
X(44297) = (a^2 + b^2 + c^2)X(28) - (a*b + a*c + b*c)X(29)

X(44298) lies on this line: {2, 3}

X(44299) = X(2)X(51)∩X(110)X(7484)

Barycentrics    a^2*(a^2*b^2 - b^4 + a^2*c^2 - 7*b^2*c^2 - c^4) : :
X(44299) = 9 X[2] - 2 X[51], 13 X[2] - 6 X[373], 6 X[2] + X[2979], 8 X[2] - X[3060], 3 X[2] + 4 X[3819], 5 X[2] + 2 X[3917], 10 X[2] - 3 X[5640], X[2] + 6 X[5650], 11 X[2] - 4 X[5943], 15 X[2] - 8 X[6688], 4 X[2] + 3 X[7998], 23 X[2] - 16 X[10219], 17 X[2] - 3 X[11002], 12 X[2] - 5 X[11451], 31 X[2] - 24 X[12045], 5 X[2] - 12 X[15082], 31 X[2] - 3 X[16981], 25 X[2] - 4 X[21849], 23 X[2] - 2 X[21969], 2 X[2] - 9 X[33879], 11 X[2] + 3 X[33884], 16 X[3] + 5 X[11439], 6 X[3] + X[11455], 17 X[3] + 4 X[32137], 13 X[51] - 27 X[373], 4 X[51] + 3 X[2979], 16 X[51] - 9 X[3060], X[51] + 6 X[3819], 5 X[51] + 9 X[3917], 20 X[51] - 27 X[5640], X[51] + 27 X[5650], 11 X[51] - 18 X[5943], 5 X[51] - 12 X[6688], 8 X[51] + 27 X[7998], 23 X[51] - 72 X[10219], 34 X[51] - 27 X[11002], 8 X[51] - 15 X[11451], 31 X[51] - 108 X[12045]

See Francisco Javier García Capitán and Peter Moses, euclid 2027.

X(44299) lies on these lines: {2, 51}, {3, 11439}, {9, 26910}, {25, 41462}, {54, 13154}, {57, 26911}, {110, 7484}, {140, 5890}, {141, 26913}, {154, 5888}, {394, 5646}, {549, 15072}, {568, 10124}, {573, 16057}, {631, 5891}, {632, 11412}, {1154, 3526}, {1180, 21001}, {1194, 8617}, {1216, 3533}, {1368, 7703}, {1401, 9330}, {1613, 15302}, {1994, 22112}, {2393, 3619}, {3091, 36987}, {3522, 32062}, {3523, 6000}, {3524, 10170}, {3525, 5889}, {3530, 15058}, {3567, 32142}, {3620, 40673}, {3688, 9335}, {3763, 12220}, {3784, 35595}, {3843, 11592}, {5012, 10541}, {5020, 21766}, {5054, 11459}, {5067, 5447}, {5068, 13348}, {5070, 13364}, {5092, 44108}, {5644, 15019}, {5651, 15246}, {5663, 15701}, {6030, 35259}, {6101, 11465}, {6241, 15720}, {6617, 39243}, {6636, 10546}, {7486, 15644}, {7495, 41715}, {7496, 9306}, {7509, 11449}, {8703, 16261}, {9730, 15709}, {9971, 34573}, {10303, 10574}, {10601, 23061}, {11004, 34566}, {11064, 33523}, {11188, 20582}, {11202, 37126}, {11205, 36650}, {11402, 15066}, {11422, 43650}, {11539, 23039}, {12279, 15717}, {12290, 15712}, {13340, 15699}, {13363, 15723}, {13391, 15703}, {13595, 16187}, {13754, 15702}, {14449, 41992}, {14915, 15698}, {15024, 16239}, {15030, 15692}, {15045, 15067}, {15060, 15693}, {15100, 38794}, {15531, 21356}, {15713, 40280}, {15721, 16836}, {15801, 15805}, {16194, 19708}, {16658, 43934}, {17534, 37482}, {17582, 41723}, {18950, 41724}, {20859, 39576}, {23293, 30739}, {24206, 31101}, {31255, 40920}, {31274, 39836}, {33540, 35502}

X(44299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2979, 11451}, {2, 3819, 2979}, {2, 3917, 5640}, {2, 7998, 3060}, {2, 33884, 5943}, {631, 5891, 20791}, {1216, 3533, 15028}, {2979, 3819, 7998}, {2979, 11451, 3060}, {3524, 10170, 15305}, {3526, 7999, 15043}, {3819, 6688, 3917}, {3819, 15082, 6688}, {3917, 15082, 2}, {5640, 6688, 11451}, {5640, 33879, 15082}, {5650, 33879, 7998}, {5651, 15246, 26881}, {5891, 20791, 12111}, {7998, 11451, 2979}, {10303, 11793, 10574}, {15067, 15694, 15045}, {17811, 40916, 5012}

X(44300) = X(5)X(113)∩X(20)X(22112)

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

See Francisco Javier García Capitán and Peter Moses, euclid 2033.

X(44300) lies on these lines: {5, 113}, {20, 22112}, {542, 5643}, {631, 34417}, {632, 20192}, {858, 6688}, {1506, 6791}, {2548, 22111}, {3066, 3526}, {3124, 9698}, {3628, 32225}, {3843, 5544}, {5181, 40670}, {5640, 40107}, {5642, 16042}, {5943, 37636}, {5972, 7605}, {7693, 29317}, {10168, 14002}, {11451, 38397}, {15018, 24981}, {16239, 32269}, {16240, 37124}, {25565, 32271}

X(44300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 13363, 16003}, {16042, 25555, 5642}

leftri

Perspectors involving 1st and 2nd Savin triangles: X(44301)-X(44308)

rightri

This preamble is based on notes from Andrey Savin, August 19, 2021 and Peter Moses, August 20, 2021. Let A' = a : s - a : s - a = 2a : -a + b + c : -a + b + c (barycentrics) , and define B' and C' cyclically. The triangle A'B'C' is introduced here as the 1st Savin triangle.
Let A" = a : s + a : s + a = 2a : 3a + b + c : 3a + b + c (barycentrics), and define B" and C" cyclically. The triangle A"B"C" is introduced here as the 2nd Savin triangle.

1st Savin triangle
2nd Savin triangle

Let T1 = 1st Savin triangle.
T1 is perspective to the following triangles, with perspector X(2): ABC, medial, anticomplementary, circum-medial, Gemini 1,2, 9-14, 20-24, 27, 28, 31-61, 65-70, 72-111.
T2 is perspective to the following triangles with perspector X(2): intouch, intangents, hexyl, infinite altitude, 6th mixtilinear, Hutson intouch, 3rd Conway, Garcia reflection, Gemini 8, Bevan-antipodal (see X(34488)).
The appearance of (T,k) in the following list means that T1 is perspective to T and the perspector is X(k):
(Andromeda, 3677)
(Jenkins, 5530)
(2nd outer Soddy, 31582)
(2nd inner Soddy, 31583)
(anticevian of X(8051), 44301)
(pedal of X(24851), 44302)

Let T2 = 2nd Savin triangle.
T2 is perspective to the following triangles, with perspector X(1): ABC, medial, anticomplementary, circum-medial, Gemini 1,2, 9-14, 20-24, 27, 28, 31-61, 65-70, 72-111.
The appearance of (T,k) in the following list means that T2 is perspective to T and the perspector is X(k):
(excentral, 7308)
(Gemini 7, 25430)
(Soddy, 44303) (2nd Sharygin, 44304)
(Aquila, 44305)
(2nd extouch, 44306)
(Gemini 15, 44307)
(9th Vijay-Paasche-Hutson, 44308)

X(44301) = PERSPECTOR OF THESE TRIANGLES: 1ST SAVIN AND ANTICEVIAN OF X(8051)

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

X(44301) lies on the cubic K365 and these lines: {56, 15519}, {57, 145}, {1420, 24150}, {3340, 35577}, {6049, 37743}

Construction: X(44301)

X(44301) = X(i)-cross conjugate of X(j) for these (i,j): {1, 145}, {3161, 5435}, {24150, 8051}
X(44301) = cevapoint of X(1) and X(2137)
X(44301) = X(i)-isoconjugate of X(j) for these (i,j): {6, 24151}, {41, 27828}, {1616, 3680}, {2136, 3445}, {6552, 16945}, {8055, 38266}
X(44301) = barycentric product X(i)*X(j) for these {i,j}: {7, 24150}, {145, 8051}, {2137, 18743}, {5435, 6553}
X(44301) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 24151}, {7, 27828}, {145, 8055}, {1420, 23511}, {1743, 2136}, {2137, 8056}, {3161, 6552}, {5435, 4452}, {6553, 6557}, {8051, 4373}, {24150, 8}, {39126, 33780}
X(44301) = {X(2137),X(6553)}-harmonic conjugate of X(8051)

X(44302) = PERSPECTOR OF THESE TRIANGLES: 1ST SAVIN AND PEDAL OF X(24851)

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

X(44302) lies on these lines: {1, 256}, {6, 1959}, {37, 1332}, {81, 593}, {257, 27958}, {319, 31089}, {894, 33946}, {1442, 37596}, {1790, 18202}, {1958, 3959}, {2277, 44179}, {2650, 23928}, {3570, 41779}, {3672, 20090}, {3924, 28365}, {3936, 4851}, {7083, 23193}, {7113, 18179}, {8424, 8772}, {11683, 27970}, {15903, 27691}, {17231, 30831}, {17778, 21287}, {25898, 27697}, {26639, 28358}, {37549, 40765}

X(44302) = crossdifference of every pair of points on line {3287, 4705}
X(44302) = barycentric product X(81)*X(27688)
X(44302) = barycentric quotient X(27688)/X(321)

X(44303) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND SODDY

Barycentrics    (a + b - c)*(a - b + c)*(3*a^4 + 12*a^3*b - 2*a^2*b^2 - 12*a*b^3 - b^4 + 12*a^3*c + 24*a^2*b*c - 36*a*b^2*c - 2*a^2*c^2 - 36*a*b*c^2 + 2*b^2*c^2 - 12*a*c^3 - c^4) : :

X44303 is also the perspector of the 2nd Savin triangle and the Soddy triangle.

X(44303) lies on these lines: {7, 25430}, {279, 5226}, {3666, 5435}, {5273, 16578}, {18230, 18623}

X(44304) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND 2ND SHARYGIN

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

Construction: X(44304)

X(44304) lies on these lines: {2, 11}, {37, 244}, {42, 17477}, {291, 3677}, {344, 30947}, {350, 21404}, {354, 4712}, {659, 4728}, {899, 1279}, {1282, 7308}, {1284, 3911}, {2108, 25502}, {2238, 21341}, {3008, 4433}, {3675, 24036}, {3932, 29824}, {3937, 24494}, {4011, 36538}, {4124, 21232}, {4370, 24405}, {4422, 21320}, {4436, 40480}, {4447, 17266}, {4465, 9318}, {4869, 36635}, {5251, 29637}, {7484, 20871}, {8300, 17123}, {8849, 25507}, {9024, 16494}, {9345, 24512}, {9451, 10582}, {16373, 20999}, {16468, 37676}, {17279, 30942}, {17357, 30970}, {17754, 35341}, {17777, 30997}, {21010, 29627}, {21580, 30963}, {23407, 29629}, {24330, 31005}, {29677, 37319}

X(44305) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND AQUILA

Barycentrics    3*a^3 + 20*a^2*b + 27*a*b^2 + 6*b^3 + 20*a^2*c + 82*a*b*c + 26*b^2*c + 27*a*c^2 + 26*b*c^2 + 6*c^3 : :

X(44305) lies on these lines: {1698, 37548}, {5316, 12047}

X(44306) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND 2ND EXTOUCH

Barycentrics    a*(a^5 + 4*a^4*b + 2*a^3*b^2 - 4*a^2*b^3 - 3*a*b^4 + 4*a^4*c + 6*a^3*b*c - 14*a^2*b^2*c - 18*a*b^3*c - 2*b^4*c + 2*a^3*c^2 - 14*a^2*b*c^2 - 30*a*b^2*c^2 - 14*b^3*c^2 - 4*a^2*c^3 - 18*a*b*c^3 - 14*b^2*c^3 - 3*a*c^4 - 2*b*c^4) : :

X(44306) lies on these lines: {2, 1901}, {3, 1750}, {9, 940}, {329, 15668}, {386, 405}, {965, 3305}, {1211, 37169}, {1213, 37185}, {3683, 9816}, {4422, 5737}, {4877, 11347}, {6913, 21363}, {9612, 16456}, {12572, 16458}, {16053, 19701}, {17123, 19758}, {18230, 19645}

X(44306) = {X(3306),X(37323)}-harmonic conjugate of X(965)

X(44307) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND GEMINI 15

Barycentrics    a*(a*b + b^2 + a*c + 4*b*c + c^2) : :
Barycentrics    c a (2 b + c + a) + a b (2 c + a + b) : :
X(44307) = 3 X[2] + X[3995], 2 X[756] + X[4883]

X(44307) lies on these lines: {1, 210}, {2, 37}, {5, 37528}, {6, 3305}, {9, 940}, {10, 3706}, {31, 4682}, {35, 20988}, {38, 3742}, {42, 3740}, {43, 37593}, {44, 81}, {45, 63}, {55, 5020}, {57, 3731}, {86, 27064}, {142, 3782}, {171, 3683}, {223, 43064}, {226, 241}, {227, 10588}, {238, 1961}, {239, 34064}, {244, 3848}, {278, 17916}, {306, 5743}, {329, 4648}, {333, 17260}, {354, 984}, {392, 30116}, {405, 975}, {469, 1841}, {517, 21363}, {518, 756}, {519, 4113}, {551, 4090}, {612, 1001}, {614, 8167}, {650, 4500}, {748, 1386}, {750, 4640}, {846, 1155}, {899, 1962}, {905, 4944}, {908, 16585}, {910, 37675}, {936, 16416}, {960, 22275}, {966, 34255}, {968, 1376}, {980, 25066}, {982, 25502}, {991, 5927}, {1089, 25512}, {1100, 17019}, {1104, 5047}, {1107, 16826}, {1125, 1215}, {1203, 5506}, {1211, 3912}, {1212, 5308}, {1214, 5219}, {1255, 3723}, {1279, 3920}, {1407, 8545}, {1427, 5226}, {1465, 16577}, {1468, 5302}, {1500, 24603}, {1573, 29574}, {1621, 5297}, {1698, 3931}, {1724, 37594}, {1736, 11018}, {1738, 4854}, {1754, 31658}, {1757, 4038}, {1848, 5089}, {1920, 18140}, {1999, 17277}, {2256, 17825}, {2292, 3812}, {2295, 4520}, {2352, 16058}, {2895, 17374}, {2999, 3247}, {3008, 27800}, {3091, 15852}, {3187, 17348}, {3218, 33761}, {3219, 16814}, {3242, 4666}, {3246, 17469}, {3306, 16675}, {3452, 5718}, {3616, 27538}, {3624, 37592}, {3634, 3743}, {3661, 41015}, {3663, 40688}, {3681, 9330}, {3687, 5241}, {3689, 3750}, {3696, 26037}, {3698, 37598}, {3700, 25084}, {3702, 19874}, {3703, 4078}, {3714, 31339}, {3715, 3751}, {3741, 3842}, {3748, 3961}, {3765, 24656}, {3816, 29639}, {3823, 4972}, {3826, 3914}, {3828, 4868}, {3834, 17184}, {3836, 4425}, {3844, 29687}, {3846, 29653}, {3925, 24210}, {3938, 42819}, {3965, 11679}, {3967, 32771}, {3971, 24325}, {3974, 39581}, {3985, 5750}, {3991, 16832}, {4001, 17332}, {4003, 17063}, {4023, 4028}, {4042, 39594}, {4413, 17594}, {4414, 17124}, {4415, 5249}, {4419, 9776}, {4422, 5294}, {4503, 29418}, {4512, 37540}, {4518, 25531}, {4646, 9780}, {4651, 28581}, {4653, 5440}, {4670, 26223}, {4675, 5905}, {4679, 26098}, {4702, 32945}, {4706, 4970}, {4719, 27627}, {4849, 17018}, {4851, 5739}, {4864, 29817}, {4871, 6682}, {4875, 14555}, {4886, 6542}, {4891, 17135}, {4903, 5550}, {4914, 32847}, {4981, 29824}, {5069, 28640}, {5087, 33105}, {5129, 5716}, {5135, 26885}, {5256, 16777}, {5259, 5266}, {5262, 17536}, {5271, 17259}, {5272, 17599}, {5283, 16831}, {5293, 37080}, {5316, 37663}, {5333, 16696}, {5393, 9646}, {5405, 8965}, {5432, 9371}, {5437, 16676}, {5492, 40296}, {5710, 31435}, {5745, 37634}, {5847, 41002}, {6354, 21617}, {6536, 32781}, {6666, 40940}, {6685, 10180}, {6687, 29833}, {6706, 26100}, {7069, 10391}, {7174, 10582}, {7226, 21342}, {7262, 37604}, {7365, 8232}, {8055, 25082}, {8580, 37553}, {8609, 37662}, {8758, 29640}, {9047, 20961}, {9345, 15481}, {9347, 17127}, {9791, 33068}, {10436, 30568}, {14996, 15492}, {15082, 40649}, {15485, 17716}, {16569, 17592}, {16578, 25080}, {16583, 17308}, {16600, 25086}, {16604, 29612}, {16669, 37685}, {16673, 23511}, {16699, 27398}, {16700, 25507}, {16823, 32926}, {16830, 32942}, {16975, 29597}, {17012, 37687}, {17013, 39260}, {17017, 17125}, {17074, 29007}, {17120, 42028}, {17140, 28582}, {17231, 32782}, {17234, 27184}, {17237, 33172}, {17239, 41809}, {17244, 18134}, {17256, 37653}, {17257, 18141}, {17258, 26840}, {17261, 32939}, {17267, 39982}, {17287, 41816}, {17300, 33066}, {17317, 17778}, {17335, 37652}, {17337, 26723}, {17338, 29841}, {17344, 32863}, {17365, 17781}, {17376, 32859}, {17381, 41850}, {17448, 29570}, {17449, 42039}, {17450, 42041}, {17529, 23537}, {17603, 24430}, {17605, 33111}, {17721, 26105}, {18139, 26580}, {18607, 31053}, {19684, 28639}, {19717, 41241}, {19730, 21371}, {20106, 25081}, {20138, 20166}, {20195, 23681}, {20487, 24239}, {20691, 29576}, {20716, 32772}, {20917, 27269}, {21611, 27193}, {21838, 31336}, {21870, 42042}, {21949, 33134}, {21965, 25007}, {23632, 31052}, {24331, 32920}, {24471, 28387}, {24575, 25420}, {24697, 33085}, {25068, 25089}, {25069, 43063}, {25072, 35466}, {25098, 31250}, {25109, 27044}, {25614, 29593}, {25760, 29854}, {25960, 29643}, {25961, 32776}, {26627, 32933}, {27186, 33151}, {27253, 30854}, {27255, 30830}, {28626, 39956}, {29610, 41817}, {29624, 40133}, {29654, 31289}, {29837, 33118}, {29845, 33115}, {29851, 32775}, {30567, 37660}, {30615, 36479}, {30710, 32009}, {30821, 39957}, {30823, 41878}, {31445, 37522}, {32017, 37870}, {37597, 41867}, {39703, 39738}

X(44307) = midpoint of X(i) and X(j) for these {i,j}: {756, 3720}, {3995, 4359}
X(44307) = reflection of X(4883) in X(3720)
X(44307) = complement of X(4359)
X(44307) = complement of the isogonal conjugate of X(28615)
X(44307) = complement of the isotomic conjugate of X(1255)
X(44307) = isotomic conjugate of the polar conjugate of X(1900)
X(44307) = X(i)-complementary conjugate of X(j) for these (i,j): {1126, 141}, {1171, 3741}, {1255, 2887}, {1268, 626}, {1796, 1368}, {1919, 35076}, {2206, 41820}, {4596, 42327}, {4629, 512}, {4632, 23301}, {6540, 21262}, {8701, 3835}, {28615, 10}, {32018, 21235}, {32635, 21244}, {32739, 4988}, {33635, 1329}, {37212, 21260}, {40438, 21240}
X(44307) = X(i)-Ceva conjugate of X(j) for these (i,j): {4967, 4662}, {6540, 513}
X(44307) = crosspoint of X(2) and X(1255)
X(44307) = crosssum of X(6) and X(1100)
X(44307) = crossdifference of every pair of points on line {667, 4790}
X(44307) = barycentric product X(i)*X(j) for these {i,j}: {1, 4967}, {7, 4662}, {69, 1900}, {40438, 42437}
X(44307) = barycentric quotient X(i)/X(j) for these {i,j}: {1900, 4}, {4662, 8}, {4967, 75}, {27597, 6533}, {42437, 4647}
X(44307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 7308, 4383}, {2, 37, 3666}, {2, 192, 19804}, {2, 312, 31993}, {2, 321, 3739}, {2, 344, 32777}, {2, 3666, 16610}, {2, 3995, 4359}, {2, 4704, 17490}, {2, 4850, 16602}, {2, 17147, 24589}, {2, 17280, 19808}, {2, 18743, 30818}, {2, 19785, 17278}, {2, 28606, 3752}, {2, 31035, 321}, {2, 32774, 17356}, {2, 33155, 26724}, {2, 41839, 75}, {6, 5287, 37595}, {9, 940, 4641}, {9, 17022, 940}, {10, 6051, 37548}, {10, 27784, 6051}, {37, 3752, 28606}, {38, 3742, 3999}, {38, 30950, 3742}, {45, 37674, 63}, {63, 37674, 37520}, {75, 41839, 3175}, {81, 27065, 44}, {142, 4656, 3782}, {192, 19804, 42051}, {238, 1961, 3745}, {321, 31035, 35652}, {405, 975, 37539}, {612, 1001, 3744}, {748, 5311, 1386}, {846, 17122, 1155}, {968, 1376, 4689}, {984, 26102, 354}, {1255, 17011, 3723}, {1255, 37680, 17011}, {1698, 27785, 3931}, {2999, 3247, 20182}, {3305, 5287, 6}, {3739, 35652, 321}, {3740, 15569, 42}, {3752, 28606, 3666}, {3920, 5284, 1279}, {3961, 16484, 3748}, {3971, 25501, 24325}, {4415, 17245, 5249}, {4422, 6703, 5294}, {4682, 15254, 31}, {4687, 18743, 2}, {4688, 22034, 28605}, {5283, 16831, 37596}, {5308, 18228, 5712}, {5743, 17243, 306}, {7174, 10582, 17597}, {7308, 25430, 1}, {9330, 29814, 3681}, {10180, 24003, 6685}, {16601, 29571, 241}, {16777, 37679, 5256}, {17019, 32911, 1100}, {17019, 35595, 32911}, {17021, 27065, 81}, {17341, 19812, 2}, {17720, 25067, 16610}, {25089, 25092, 25068}, {25507, 40773, 16700}, {26037, 32915, 3696}

X(44308) = PERSPECTOR OF THESE TRIANGLES: 2ND SAVIN AND 9TH VIJAY-PAASCHE-HUTSON

Barycentrics    5*a^5*b + a^4*b^2 - 10*a^3*b^3 - 2*a^2*b^4 + 5*a*b^5 + b^6 + 5*a^5*c + 14*a^4*b*c - 10*a^3*b^2*c - 20*a^2*b^3*c + 5*a*b^4*c + 6*b^5*c + a^4*c^2 - 10*a^3*b*c^2 - 52*a^2*b^2*c^2 - 10*a*b^3*c^2 - b^4*c^2 - 10*a^3*c^3 - 20*a^2*b*c^3 - 10*a*b^2*c^3 - 12*b^3*c^3 - 2*a^2*c^4 + 5*a*b*c^4 - b^2*c^4 + 5*a*c^5 + 6*b*c^5 + c^6 + 2*(a + b + c)*(a^3 + a^2*b - 3*a*b^2 + b^3 + a^2*c - 18*a*b*c - b^2*c - 3*a*c^2 - b*c^2 + c^3)*S : :

X(44308) lies on these lines: {2, 30416}, {1123, 5405}

X(44309) = EULER LINE INTERCEPT OF X(1503)X(15461)

Barycentrics   (8*OH*((-a^2+b^2+c^2)^2-b^2*c^2)*S*a^3*b*c+(b^2+c^2)*a^10-3*(b^4+4*b^2*c^2+c^4)*a^8+2*(b^2+c^2)*(b^4+7*b^2*c^2+c^4)*a^6+2*(b^4+3*b^2*c^2+c^4)*(b^4-4*b^2*c^2+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*(3*b^4+4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    (2*(3*R^2-SB-SC)*OH*R-6*(3*R^2+SA)*R^2+S^2+SA^2-SB*SC+SW^2)*SB*SC : :
X(44309) = X(858)+2*X(20408)

See Antreas Hatzipolakis and César Lozada, euclid 2048.

X(44309) lies on these lines: {2, 3}, {1503, 15461}, {6000, 14500}

X(44309) = midpoint of X(i) and X(j) for these {i, j}: {2071, 14808}, {15154, 18403}
X(44309) = reflection of X(i) in X(j) for these (i, j): {403, 1312}, {1114, 10257}, {10736, 13473}, {10750, 23323}, {15646, 31681}, {44214, 13626}, {44246, 35231}
X(44309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): {1113, 10736, 20}, {1312, 20408, 1113}

X(44310) = EULER LINE INTERCEPT OF X(1503)X(15460)

Barycentrics    (-8*OH*((-a^2+b^2+c^2)^2-b^2*c^2)*S*a^3*b*c+(b^2+c^2)*a^10-3*(b^4+4*b^2*c^2+c^4)*a^8+2*(b^2+c^2)*(b^4+7*b^2*c^2+c^4)*a^6+2*(b^4+3*b^2*c^2+c^4)*(b^4-4*b^2*c^2+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*(3*b^4+4*b^2*c^2+3*c^4)*a^2+(b^4-c^4)^2*(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    (-2*(3*R^2-SB-SC)*OH*R-6*(3*R^2+SA)*R^2+S^2+SA^2-SB*SC+SW^2)*SB*SC : :
X(44310) = X(858)+2*X(20409)

See Antreas Hatzipolakis and César Lozada, euclid 2048.

X(44310) lies on these lines: {2, 3}, {1503, 15460}, {6000, 14499}

X(44310) = midpoint of X(i) and X(j) for these {i, j}: {2071, 14807}, {15155, 18403}
X(44310) = reflection of X(i) in X(j) for these (i, j): {403, 1313}, {1113, 10257}, {10737, 13473}, {10751, 23323}, {15646, 31682}, {44214, 13627}, {44246, 35232}
X(44310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): {1114, 10737, 20}, {1313, 20409, 1114}

leftri

Points associated with the Moses-Soddy triangle: X(44301)-X(44305)

rightri

This preamble is based on notes from Peter Moses, August 21, 2021. Let A' = pole of Soddy line in the Soddy A-circle, and define B' and C' cyclically, so that A' = b - c : a - c : b - a
B' = c - b : c - a : b - a
C' = c - b : a - c : a - b

The triangle A'B'C' is here named the Moses-Soddy triangle. This triangle is also the complement of the Yff contact triangle. The vertices A', B', C' lie on the cubic K927.

The appearance of (T,n) in the following list means that A'B'C' is perspective to T, and the perspector is X(k): (ABC, 514), (medial, 1086), (orthic, 116), (incentral, 17761), (intouch, 11), (extouch, 4904), (McBeath, 44311), (symmedial, 44312), (Steiner, 1125), (3rd Euler, 11), (2nd Hatzipolakis, 44313), (Yff contact,2), (Garcia reflection, 3667), (Gemini 7, 7658), (Gemini 8, 3667), (anti-Ursa-minor, 44316), (Lemoine, 44317), (Ursa-major, 44318), (Ursa-minor, 44319), (24th Vijay-Paasche-Hutson, 44320)

The Moses-Soddy triangle is also perspective to the Wasat triangle.

X(2)-of-A'B'C' = X(21204)
X(3)-of-A'B'C' = X(44314)
X(4)-of-A'B'C' = X(1)
X(5)-of-A'B'C' = X(44315)

X(44311) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND MACBEATH

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

X(44311) lies on these lines: {8, 596}, {10, 1772}, {11, 522}, {116, 136}, {200, 17155}, {244, 17888}, {318, 24046}, {514, 3937}, {537, 14740}, {726, 6745}, {900, 38390}, {982, 17860}, {1086, 2968}, {1111, 24031}, {1125, 34834}, {1357, 23772}, {1421, 24410}, {1845, 4292}, {2611, 4151}, {3159, 27385}, {3190, 3210}, {3663, 18690}, {3667, 38389}, {3670, 23661}, {3705, 31075}, {3953, 23528}, {4025, 23989}, {4075, 27529}, {4193, 44040}, {4467, 40619}, {4765, 14936}, {4847, 20882}, {4853, 9457}, {4858, 7004}, {4957, 28623}, {4976, 38347}, {5515, 40626}, {5552, 24068}, {6734, 24176}, {7649, 21666}, {10538, 30117}, {17205, 17880}, {18210, 24237}, {21196, 38987}, {21208, 24186}, {21318, 40687}, {22084, 23820}, {22148, 36205}, {24177, 24218}, {24431, 41797}, {27009, 27486}, {34467, 39210}

X(44311) = polar conjugate of the isogonal conjugate of X(39006)
X(44311) = X(i)-complementary conjugate of X(j) for these (i,j): {54, 513}, {95, 21260}, {513, 1209}, {667, 233}, {1333, 18314}, {2148, 514}, {2167, 3835}, {2169, 20315}, {2190, 20316}, {2203, 17434}, {2616, 3454}, {2623, 1211}, {6591, 34836}, {15412, 21245}, {18210, 20625}, {23224, 10600}, {23286, 21530}, {34948, 34835}, {39177, 21246}
X(44311) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 514}, {40010, 4391}, {40012, 3239}
X(44311) = X(i)-isoconjugate of X(j) for these (i,j): {1783, 40518}, {24027, 44040}
X(44311) = crosspoint of X(75) and X(4560)
X(44311) = crosssum of X(31) and X(4559)
X(44311) = barycentric product X(i)*X(j) for these {i,j}: {11, 32939}, {264, 39006}, {404, 4858}, {514, 20293}, {2170, 44139}
X(44311) = barycentric quotient X(i)/X(j) for these {i,j}: {404, 4564}, {1146, 44040}, {1459, 40518}, {20293, 190}, {32939, 4998}, {39006, 3}, {44085, 2149}

X(44312) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND SYMMEDIAL

Barycentrics    (b - c)^2*(-(a^3*b) + a^2*b^2 - a^3*c + a^2*b*c + a^2*c^2 + b^2*c^2) : :
X(44312) = 3 X[2] + X[25049]

X(44312) lies on these lines: {2, 25049}, {11, 3835}, {75, 7239}, {116, 125}, {141, 36951}, {226, 34253}, {244, 38995}, {514, 20974}, {649, 27009}, {661, 40619}, {675, 32739}, {812, 38347}, {1084, 1086}, {1125, 38998}, {2140, 5773}, {3120, 4107}, {3124, 21208}, {3741, 39080}, {3937, 4932}, {4904, 38992}, {6593, 34830}, {20295, 26846}, {20999, 24279}, {21196, 38987}, {23638, 24787}, {23803, 24198}, {24220, 29658}, {24237, 38991}, {29654, 36213}, {40601, 40940}

X(44312) = midpoint of X(20974) and X(23989)
X(44312) = complement of X(46148)
X(44312) = complement of the isogonal conjugate of X(10566)
X(44312) = X(i)-complementary conjugate of X(j) for these (i,j): {28, 23285}, {81, 3005}, {82, 514}, {83, 513}, {251, 650}, {308, 21260}, {513, 6292}, {514, 21249}, {649, 16587}, {693, 21248}, {3112, 3835}, {3121, 35971}, {3125, 15449}, {3669, 17055}, {4580, 21530}, {4628, 24036}, {10566, 10}, {18070, 3454}, {18082, 4129}, {18098, 661}, {18101, 26932}, {18105, 16589}, {18107, 34832}, {18108, 2}, {18113, 40617}, {18833, 21262}, {34055, 20315}, {39179, 1125}, {39276, 9508}
X(44312) = X(6)-Ceva conjugate of X(514)
X(44312) = crosspoint of X(6) and X(21791)
X(44312) = barycentric product X(i)*X(j) for these {i,j}: {514, 21225}, {3261, 21791}, {7199, 21901}
X(44312) = barycentric quotient X(i)/X(j) for these {i,j}: {21225, 190}, {21791, 101}, {21901, 1018}, {23093, 1331}

X(44313) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND 2ND HATZIPOLAKIS

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

X(44313) lies on these lines: {10, 2840}, {11, 1357}, {116, 5521}, {117, 2829}, {124, 513}, {514, 2968}, {867, 3937}, {946, 15528}, {1364, 6003}, {3667, 38357}, {3825, 12014}, {4292, 7683}, {5190, 39006}, {7192, 40618}, {8226, 40687}, {10167, 40677}, {11028, 21621}

X(44313) = X(1119)-Ceva conjugate of X(514)

X(44314) = X(3)-OF-MOSES-SODDY TRIANGLE

Barycentrics    (b - c)*(a^3 - 2*a^2*b + 2*a*b^2 + b^3 - 2*a^2*c + a*b*c - 2*b^2*c + 2*a*c^2 - 2*b*c^2 + c^3) : :
X(44314) = X[1] - 3 X[21204], X[8] + 3 X[6545], X[145] - 9 X[6548], 5 X[1698] - 3 X[10196], 5 X[3616] - 9 X[14475], 3 X[4049] - X[21201], 9 X[6544] - 13 X[19877], 3 X[6546] - 7 X[9780]

X(44314) lies on these lines: {1, 21204}, {2, 5592}, {4, 2457}, {8, 6545}, {10, 514}, {145, 6548}, {513, 3812}, {676, 28521}, {942, 37998}, {1027, 1722}, {1698, 10196}, {2785, 3837}, {2789, 3960}, {3309, 5806}, {3616, 14475}, {3676, 10106}, {3835, 12047}, {4406, 33944}, {4778, 6133}, {6544, 19877}, {6546, 9780}, {7192, 17589}, {7658, 11512}, {8713, 40551}, {13464, 28292}, {19950, 21211}, {19951, 21143}, {21188, 28470}, {21198, 28225}, {21212, 29066}, {25380, 29240}

X(44314) = reflection of X(1) in X(44315)
X(44314) = reflection of X(32212) in X(10)
X(44314) = complement of X(5592)
X(44314) = crossdifference of every pair of points on line {1914, 22356}
X(44314) = Moses-Soddy-isogonal conjugate of X(1)

X(44315) = X(5)-OF-MOSES-SODDY TRIANGLE

Barycentrics    (b - c)*(3*a^3 - 2*a^2*b - 2*a*b^2 - b^3 - 2*a^2*c + 3*a*b*c + 2*b^2*c - 2*a*c^2 + 2*b*c^2 - c^3) : :
X(44315) = X[1] + 3 X[21204], X[8] - 9 X[14475], 5 X[3616] - X[5592], 5 X[3616] + 3 X[6545], 7 X[3622] + 9 X[6548], 7 X[3624] - 3 X[10196], 11 X[5550] - 3 X[6546], X[5592] + 3 X[6545]

X(44315) lies on these lines: {1, 21204}, {2, 32212}, {8, 14475}, {514, 1125}, {522, 19947}, {946, 3667}, {3616, 5592}, {3622, 6548}, {3624, 10196}, {3835, 13407}, {4449, 5293}, {4786, 26839}, {5045, 37998}, {5550, 6546}, {6789, 36205}, {7658, 24174}, {11726, 38019}, {19884, 28161}, {19949, 21196}, {23345, 28225}, {29350, 33815}

X(44315) = midpoint of X(1) and X(44314)
X(44315) = midpoint of X(11726) and X(38019)
X(44315) = complement of X(32212)
X(44315) = {X(3616),X(6545)}-harmonic conjugate of X(5592)

X(44316) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND ANTI-URSA-MINOR

Barycentrics    ((b^2+c^2)*a^2+(b+c)*(b^2+c^2)*a-(b+c)^2*b*c)*(b-c) : :
X(44316) = X(3777)+2*X(21714) = 2*X(4472)+X(24698)

Centers X(44316)-X(44320) were contributed by César Lozada, August 21, 2021.

X(44316) lies on these lines: {2, 4057}, {5, 3667}, {11, 42312}, {12, 43924}, {86, 21304}, {140, 39225}, {141, 21191}, {325, 523}, {427, 7649}, {514, 40086}, {594, 17458}, {656, 14288}, {834, 17072}, {900, 21189}, {1213, 20979}, {1368, 20315}, {1595, 16231}, {1919, 17398}, {2530, 4036}, {3733, 21301}, {3777, 21714}, {3814, 3836}, {4472, 24698}, {4665, 23886}, {4802, 23815}, {4926, 23809}, {4977, 21051}, {6004, 24959}, {9002, 20316}, {11681, 23345}, {17303, 21389}, {21053, 21123}, {21187, 21193}, {21261, 42327}, {30795, 31003}, {31096, 31097}

X(44316) = midpoint of X(i) and X(j) for these {i, j}: {656, 14288}, {2530, 4036}, {3733, 21301}
X(44316) = reflection of X(i) in X(j) for these (i, j): (5, 39508), (31946, 21260), (39225, 140)
X(44316) = complement of X(4057)
X(44316) = complementary conjugate of X(8054)
X(44316) = barycentric product X(514)*X(24068)
X(44316) = barycentric quotient X(514)/X(39693)
X(44316) = trilinear product X(513)*X(24068)
X(44316) = trilinear quotient X(693)/X(39693)
X(44316) = pole of the Nagel line wrt the nine-point circle
X(44316) = pole of the trilinear polar of X(20295) with respect to circumhyperbola dual of Yff parabola
X(44316) = crossdifference of every pair of points on line {X(32), X(16685)}
X(44316) = X(i)-complementary conjugate of-X(j) for these (i, j): (1, 8054), (100, 4075), (596, 11)
X(44316) = X(692)-isoconjugate-of-X(39693)
X(44316) = X(514)-reciprocal conjugate of-X(39693)

X(44317) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND LEMOINE

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

X(44317) lies on these lines: {115, 116}, {6789, 30790}, {38988, 44312}

X(44317) = pole of the trilinear polar of X(598) with respect to circumhyperbola dual of Yff parabola
X(44317) = X(598)-Ceva conjugate of-X(514)
X(44317) = X(1333)-complementary conjugate of-X(17436)

X(44318) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND URSA-MAJOR

Barycentrics    a*((b+c)*a^6-(3*b^2+4*b*c+3*c^2)*a^5+(b+c)*(2*b^2+9*b*c+2*c^2)*a^4+2*(b^4+c^4-2*(3*b^2+4*b*c+3*c^2)*b*c)*a^3-(b+c)*(3*b^4+3*c^4-2*(3*b^2+4*b*c+3*c^2)*b*c)*a^2+(b^6+c^6+(7*b^2-24*b*c+7*c^2)*b^2*c^2)*a+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*b*c)*(b-c) : :

X(44318) lies on these lines: {2254, 13252}, {3676, 17625}, {3835, 17618}, {3900, 23813}

X(44319) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND URSA-MINOR

Barycentrics    a*((b+c)*a^3-2*(b^2+b*c+c^2)*a^2+(b^3+c^3)*a+(b-c)^2*b*c)*(b-c) : :
X(44319) = 3*X(210)-2*X(4468) = 3*X(354)-4*X(3676) = 3*X(1638)-2*X(2488) = 3*X(30724)-2*X(39541)

X(44319) lies on these lines: {210, 4468}, {354, 3676}, {512, 20507}, {926, 21104}, {1638, 2488}, {1836, 8049}, {2223, 4905}, {3057, 28292}, {3309, 4077}, {3474, 26853}, {3835, 17605}, {6003, 40467}, {6006, 31391}, {15636, 43921}, {17660, 37998}, {18071, 21302}, {23806, 42325}, {30724, 39541}

X(44319) = crossdifference of every pair of points on line {X(1), X(14746)}
X(44319) = crosspoint of X(651) and X(42311)
X(44319) = crosssum of X(i) and X(j) for these (i, j): {55, 4040}, {354, 4905}
X(44319) = barycentric product X(513)*X(28742)
X(44319) = trilinear product X(649)*X(28742)

X(44320) = PERSPECTOR OF THESE TRIANGLES: MOSES-SODDY AND 24TH VIJAY-PAASCHE-HUTSON

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

X(44320) lies on these lines: {514, 22107}, {812, 1015}

X(44320) = pole of the trilinear polar of X(1123) with respect to circumhyperbola dual of Yff parabola
X(44320) = X(1123)-Ceva conjugate of-X(514)

X(44321) = X(51)X(15059)∩X(125)X(3819)

Barycentrics    ((b^2+c^2)*a^8+7*(b^2+c^2)*a^4*b^2*c^2-2*(b^4+3*b^2*c^2+c^4)*a^6+(2*b^8+2*c^8+(7*b^4-24*b^2*c^2+7*c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+10*b^2*c^2+c^4))*a^2 : :
Barycentrics    (SB+SC)*((33*R^2-10*SA-8*SW)*S^2+(3*(12*SA+5*SW)*R^2-10*SA^2+10*SB*SC-2*SW^2)*SA) : :
X(44321) = X(51)-5*X(15059), X(3448)+7*X(44299), X(5447)+2*X(20396), X(5642)-3*X(15082), 3*X(5650)+X(9140), X(5891)+3*X(15061), X(5892)-3*X(34128), X(5907)+5*X(38729), 4*X(6723)-X(41671), X(9143)-9*X(33879), 5*X(11451)-X(13417), X(11793)+2*X(20397), X(11807)-3*X(14845), 3*X(14644)+X(36987), X(14855)-5*X(38728), 3*X(15055)+X(32062), X(15738)+2*X(17704), 3*X(20791)+X(21650)

See Antreas Hatzipolakis and César Lozada, euclid 2058.

X(44321) lies on these lines: {51, 15059}, {125, 3819}, {511, 12099}, {1154, 40685}, {2393, 6698}, {2777, 10127}, {2781, 6688}, {2979, 11800}, {3448, 44299}, {5447, 20396}, {5642, 15082}, {5650, 9140}, {5663, 10124}, {5891, 15061}, {5892, 10628}, {5907, 38729}, {6000, 6699}, {9143, 33879}, {10219, 41670}, {11451, 13417}, {11793, 20397}, {11807, 14845}, {14644, 36987}, {14855, 38728}, {15055, 32062}, {15738, 17704}, {17855, 18435}, {20791, 21650}

X(44321) = midpoint of X(i) and X(j) for these {i, j}: {125, 3819}, {2979, 11800}, {17855, 18435}
X(44321) = reflection of X(i) in X(j) for these (i, j): (6688, 6723), (41670, 10219), (41671, 6688)

X(44322) = X(49)X(41586)∩X(54)X(37644)

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

See Antreas Hatzipolakis and César Lozada, euclid 2058.

X(44322) lies on these lines: {49, 41586}, {54, 37644}, {206, 5965}, {382, 6288}, {539, 2917}, {1154, 15761}, {1209, 1216}, {2888, 31304}, {3574, 10254}, {5449, 34751}, {6689, 10601}, {7689, 18400}, {7691, 35481}, {8254, 8262}, {13419, 15605}, {18580, 32348}

X(44322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1209, 41599, 6153), (41594, 41598, 1209)

X(44323) = X(6)X(6636)∩X(51)X(6329)

Barycentrics    a^2*(2*(b^2+c^2)*a^4-10*b^2*c^2*a^2-(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)) : :
X(44323) = 5*X(141)-7*X(44299), 5*X(575)-2*X(16881), 5*X(597)-3*X(5640), 3*X(597)-X(9971), 5*X(3589)-4*X(6688), 7*X(3589)-4*X(9822), 3*X(3589)-2*X(40670), 5*X(3589)-2*X(41579), X(3631)+2*X(32366), 3*X(3917)+5*X(40673), 9*X(5640)-5*X(9971), 2*X(6329)+X(17710), X(6467)+2*X(34573), 7*X(6688)-5*X(9822), 6*X(6688)-5*X(40670), 5*X(8550)+X(18436), 6*X(9822)-7*X(40670), 10*X(9822)-7*X(41579), 6*X(15082)-5*X(20582), 5*X(40670)-3*X(41579)

See Antreas Hatzipolakis and César Lozada, euclid 2058.

X(44323) lies on these lines: {6, 6636}, {51, 6329}, {141, 26913}, {182, 15646}, {511, 20583}, {524, 3917}, {575, 16881}, {597, 5640}, {1154, 12007}, {2393, 3589}, {2781, 40647}, {2854, 15082}, {2979, 3629}, {3631, 3819}, {6467, 34573}, {6656, 16175}, {8550, 18436}, {9019, 21849}, {9973, 11451}, {10541, 40929}, {11574, 32455}, {15531, 22165}, {37283, 41614}

X(44323) = midpoint of X(i) and X(j) for these {i, j}: {51, 17710}, {2979, 3629}, {3819, 32366}, {15531, 22165}
X(44323) = reflection of X(i) in X(j) for these (i, j): (51, 6329), (3631, 3819), (41579, 6688)

X(44324) = X(3)X(9544)∩X(5)X(2979)

Barycentrics    a^2*(2*(b^2+c^2)*a^6-2*(3*b^4+5*b^2*c^2+3*c^4)*a^4+3*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :
X(44324) = 7*X(2)-3*X(13321), 2*X(3)+X(31834), X(5)-7*X(7999), 5*X(5)+X(37484), X(2979)+7*X(7999), 5*X(2979)-X(37484), 3*X(3917)+X(5891), 7*X(3917)+X(15030), 5*X(3917)+X(15060), 11*X(3917)+X(16194), 15*X(3917)+X(32062), 9*X(3917)-X(36987), 7*X(5891)-3*X(15030), 5*X(5891)-3*X(15060), X(5891)-3*X(15067), 11*X(5891)-3*X(16194), 5*X(5891)-X(32062), 3*X(5891)+X(36987), 5*X(15030)-7*X(15060), X(15030)-7*X(15067), 11*X(15030)-7*X(16194), 15*X(15030)-7*X(32062), 9*X(15030)+7*X(36987)

See Antreas Hatzipolakis and César Lozada, euclid 2058.

X(44324) lies on these lines: {2, 13321}, {3, 9544}, {5, 2979}, {30, 3917}, {51, 3628}, {52, 16239}, {140, 389}, {141, 39504}, {143, 6688}, {154, 7525}, {381, 33884}, {511, 547}, {546, 10627}, {548, 5447}, {549, 5890}, {550, 11444}, {568, 11539}, {632, 11412}, {2781, 10272}, {3060, 15699}, {3530, 5562}, {3845, 13340}, {3850, 10625}, {3853, 14128}, {3859, 13598}, {5066, 10170}, {5446, 12812}, {5650, 5946}, {5663, 34200}, {5876, 14855}, {5889, 14869}, {5907, 12103}, {6102, 12108}, {6243, 11451}, {6592, 34828}, {7496, 15087}, {7502, 15066}, {7516, 11402}, {7555, 9306}, {8703, 11459}, {9730, 11812}, {9820, 34004}, {10201, 10519}, {10226, 43652}, {10263, 14845}, {11002, 15703}, {11455, 15704}, {11465, 41992}, {11592, 40647}, {11801, 41673}, {12100, 13754}, {12106, 17811}, {12162, 44245}, {14915, 15691}, {15045, 15713}, {15305, 15686}, {15712, 18436}, {16261, 35404}, {16836, 41983}, {18445, 21766}, {21230, 37452}, {21357, 37636}, {32137, 40247}, {34006, 35265}

X(44324) = midpoint of X(i) and X(j) for these {i, j}: {5, 2979}, {51, 6101}, {549, 23039}, {550, 18435}, {1216, 3819}, {3845, 13340}, {3917, 15067}, {5876, 14855}, {6688, 15606}, {8703, 11459}, {11455, 15704}, {15305, 15686}
X(44324) = reflection of X(i) in X(j) for these (i, j): (51, 3628), (140, 3819), (143, 6688), (3819, 32142), (5066, 10170), (5946, 10124), (9730, 11812), (13451, 547), (14449, 51), (14855, 33923)
X(44324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (632, 11412, 16881), (1216, 32142, 140), (3628, 6101, 14449), (5447, 11591, 548), (5650, 5946, 10124), (5891, 32062, 15060), (7998, 23039, 549), (10627, 11793, 546), (14128, 15644, 3853)

X(44325) = X(3)X(54)∩X(51)X(8254)

Barycentrics    a^2*((b^2+c^2)*a^12-2*(b^2+2*c^2)*(2*b^2+c^2)*a^10+(b^2+c^2)*(5*b^4+16*b^2*c^2+5*c^4)*a^8-3*(5*b^4+4*b^2*c^2+5*c^4)*b^2*c^2*a^6-(b^2+c^2)*(5*b^8+5*c^8-(9*b^4-17*b^2*c^2+9*c^4)*b^2*c^2)*a^4+(b^6-c^6)*(b^2-c^2)*(4*b^4+b^2*c^2+4*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3*(b^4+b^2*c^2+c^4)) : :
Barycentrics    (SB+SC)*((11*R^2+2*SA-2*SW)*S^2+(6*R^4-(11*SA+10*SW)*R^2+2*SA^2-2*SB*SC+4*SW^2)*SA) : :
X(44325) = 2*X(140)+X(21660), 2*X(143)+X(12226), 2*X(1209)+X(15532), 2*X(1493)+X(6101), X(1493)+2*X(12363), 5*X(1656)+X(12291), X(3519)-4*X(32142), 7*X(3526)-X(12280), X(5876)+2*X(10619), X(6101)-4*X(12363), X(6102)+2*X(12606), 2*X(6152)-5*X(15026), X(6242)-4*X(12006), 4*X(6689)-X(13368), 4*X(8254)-X(32196), X(10263)-4*X(12242), X(10625)+2*X(11803), 2*X(10627)+X(15801), 2*X(13630)+X(22815), X(21230)+2*X(40632)

See Antreas Hatzipolakis and César Lozada, euclid 2058.

X(44325) lies on these lines: {3, 54}, {51, 8254}, {140, 21660}, {141, 21357}, {143, 12226}, {539, 15067}, {1209, 15532}, {1656, 12291}, {2781, 11702}, {3519, 15108}, {3526, 12280}, {3819, 21230}, {5876, 10619}, {5891, 32423}, {5946, 15330}, {6152, 15026}, {6153, 6688}, {6242, 12006}, {6689, 13368}, {10263, 12242}, {10625, 11803}, {11451, 13365}, {12234, 36153}, {12254, 18435}, {13364, 37943}, {14855, 43581}, {15089, 35921}, {18376, 22804}, {18475, 38898}, {23048, 38317}, {37649, 44234}

X(44325) = midpoint of X(i) and X(j) for these {i, j}: {195, 2979}, {3819, 40632}, {12254, 18435}, {14855, 43581}
X(44325) = reflection of X(i) in X(j) for these (i, j): (51, 8254), (6153, 6688), (21230, 3819), (32196, 51)
X(44325) = {X(1493), X(12363)}-harmonic conjugate of X(6101)

X(44326) = ISOTOMIC CONJUGATE OF X(6587)

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

See Antreas Hatzipolakis and César Lozada, euclid 2060.

X(44326) lies on these lines: {2, 20313}, {99, 1301}, {107, 3265}, {110, 35571}, {122, 35140}, {253, 30769}, {325, 16096}, {459, 8781}, {1073, 41530}, {4561, 7256}, {4563, 34211}, {7763, 34403}, {8858, 33581}, {34410, 40995}

X(44326) = isotomic conjugate of X(6587)
X(44326) = isogonal conjugate of PK-transform of X(20)
X(44326) = isogonal conjugate of PK-transform of X(64)
X(44326) = polar conjugate of X(44705)
X(44326) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(107)}} and {{A, B, C, X(99), X(4554)}}
X(44326) = trilinear pole of the line {20, 64}
X(44326) = barycentric product X(i)*X(j) for these {i, j}: {64, 670}, {99, 253}, {110, 41530}, {305, 1301}, {459, 4563}, {648, 34403}
X(44326) = barycentric quotient X(i)/X(j) for these (i, j): (3, 42658), (8, 14308), (64, 512), (75, 17898), (86, 21172), (99, 20)
X(44326) = trilinear product X(i)*X(j) for these {i, j}: {64, 799}, {99, 2184}, {162, 34403}, {163, 41530}, {253, 662}, {304, 1301}
X(44326) = trilinear quotient X(i)/X(j) for these (i, j): (63, 42658), (64, 798), (76, 17898), (99, 610), (162, 3172), (190, 3198)

X(44327) = ISOTOMIC CONJUGATE OF X(14837)

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

See Antreas Hatzipolakis and César Lozada, euclid 2060.

X(44327) lies on these lines: {2, 23982}, {84, 36798}, {189, 1997}, {190, 2406}, {282, 309}, {336, 41087}, {645, 4592}, {646, 4561}, {653, 6332}, {662, 7452}, {664, 6335}, {1310, 40117}, {1332, 3699}, {1422, 36805}, {1436, 36799}, {1440, 28753}, {1461, 3239}, {1903, 36800}, {4626, 15413}, {6081, 9056}, {7020, 8777}, {8059, 8707}, {9376, 19582}, {16596, 34393}, {18743, 34404}

X(44327) = isotomic conjugate of X(14837)
X(44327) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1461)}} and {{A, B, C, X(2), X(653)}}
X(44327) = trilinear pole of the line {8, 20} (the line through X(8) perpendicular to the trilinear polar of X(8))
X(44327) = barycentric product X(i)*X(j) for these {i, j}: {75, 13138}, {76, 36049}, {84, 668}, {99, 39130}, {100, 309}, {101, 44190}
X(44327) = barycentric quotient X(i)/X(j) for these (i, j): (1, 6129), (9, 14298), (75, 17896), (84, 513), (99, 8822), (100, 40)
X(44327) = trilinear product X(i)*X(j) for these {i, j}: {2, 13138}, {8, 37141}, {69, 40117}, {75, 36049}, {76, 32652}, {84, 190}
X(44327) = trilinear quotient X(i)/X(j) for these (i, j): (2, 6129), (8, 14298), (76, 17896), (78, 10397), (84, 649), (99, 1817)

leftri

Steiner-ellipse-inverses of points on the Euler line: X(44328)-X(44349)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, August 23, 2021.

Suppose that P is a point on the Euler line. Then P is given by the combo X(2) + t*x(3) for some t, and the Steiner-circumellipse-inverse of P is given by the following combo:

(3*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6)*k - 16*S^4)*X[2] + 16*(1 + k)*S^4*X[3]

The appearance of (i,j) in the following list means that X(j) = Steiner-circumellipse-inverse of X(i):

(2,30), (3,401), (4,297), (5,40853), (20,441), (21,448), (22,15013), (23,40856), (24,44328), (25,15014), (26,44329), (27,447), (28,44330), (29,44331), (237,10684), (376,40884), (381,40885), (384,6660), (427,40889), (449,452), (458,35474), (468,40890), (472,11093), (473,11094), (858,35923), (2479, 2479), (2480,2480), (3543,44216), (4235,7473), (6655,21536), (8613,15781), (1113,44332), (1114,44333), (14953,37045), (37174,44228), (37188,44252)

Continuing with a point P on the Euler line, with combo X(2) + t*x(3) for some t, the Steiner-inellipse-inverse of P is given by the following combo:

(3*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - a^2*b^2*c^4 - 2*b^4*c^4 + a^2*c^6 + b^2*c^6)*k - 16*S^4)*X[2] + 16*(1 + k)*S^4*X[3]

The appearance of (i,j) in the following list means that X(j) = Steiner-inellipse-inverse of X(i):

(2,30), (3,441), (4,44334), (5,297), (20,44335), (21,44336), (22,44337), (23,44338), (24,44339), (25,44340), (26,44341), (27,44342), (28,44343), (29,44344), (140,401), (237,44345), (376,44346), (381,44216), (384,44347), (447,6678), (448,6675), (449,11108), (468,40856), (547,40885), (549,40884), (1113,44348), (1114,44349), (1375,37045), (2454,2454), (2455, 2455), (3628,40853), (5159,35923), (6656,21536), (6660,7819), (6676,15013), (6677,15014), (37911,40890)

X(44328) = STEINER-CIRCUMELLIPSE-INVERSE OF X(24)

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

X(44328) lies on these lines: {2, 3}, {287, 34146}, {4558, 37778}, {20563, 36416}, {36212, 41253}, {37784, 41678}, {41679, 44138}

X(44328) = anticomplement of anticomplement of X(44339)
X(44328) = {X(2479),X(2480)}-harmonic conjugate of X(24)

X(44329) = STEINER-CIRCUMELLIPSE-INVERSE OF X(26)

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

X(44329) lies on these lines: {2, 3}, {20564, 36418}

X(44329) = anticomplement of anticomplement of X(44341)
X(44329) = {X(2479),X(2480)}-harmonic conjugate of X(26)

X(44330) = STEINER-CIRCUMELLIPSE-INVERSE OF X(28)

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

X(44330) lies on these lines: {2, 3}, {162, 239}, {525, 17498}, {536, 648}, {894, 2326}, {3227, 16077}, {4567, 23582}, {20336, 36420}, {25257, 41676}

X(44330) = anticomplement of anticomplement of X(44343)
X(44330) = antitomic conjugate of X(16085)
X(44330) = {X(2479),X(2480)}-harmonic conjugate of X(28)

X(44331) = STEINER-CIRCUMELLIPSE-INVERSE OF X(29)

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

X(44331) lies on these lines: {2, 3}, {307, 36421}, {525, 17926}, {527, 648}, {823, 1948}, {1121, 16077}, {2326, 7282}, {3496, 31623}, {3912, 36797}, {8748, 8822}, {40843, 41207}

X(44331) = anticomplement of anticomplement of X(44344)
X(44331) = {X(2479),X(2480)}-harmonic conjugate of X(29)

X(44332) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1113)

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

X(44332) lies on these lines: {2, 3}, {112, 2592}, {287, 2575}, {525, 8115}, {2966, 15164}, {22339, 41941}, {23582, 39299}

X(44332) = relfection of X(44333) in X(40856)
X(44332) = anticomplement of anticomplement of X(44348)
X(44332) = {X(2),X(401)}-harmonic conjugate of X(44333)
X(44332) = {X(2479),X(2480)}-harmonic conjugate of X(1113)

X(44333) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1114)

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

X(44333) lies on these lines: {2, 3}, {112, 2593}, {287, 2574}, {525, 8116}, {2966, 15165}, {22340, 41942}, {23582, 39298}

X(44333) = reflection of X(44332) in X(40856)
X(44333) = anticomplement of anticomplement of X(44349)
X(44333) = {X(2),X(401)}-harmonic conjugate of X(44332)
X(44333) = {X(2479),X(2480)}-harmonic conjugate of X(1114)

X(44334) = STEINER-INELLIPSE-INVERSE OF X(4)

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

X(44334) lies on these lines: {2, 3}, {6, 20204}, {98, 9476}, {230, 23976}, {253, 33630}, {393, 20208}, {524, 23583}, {525, 3239}, {648, 40996}, {1249, 40995}, {1503, 40542}, {1990, 15526}, {3564, 15595}, {3767, 20207}, {3912, 15252}, {5305, 13567}, {6330, 6530}, {6389, 42459}, {7776, 37669}, {10002, 34815}, {14572, 38253}, {14725, 23292}, {14743, 17282}, {14767, 34573}, {15466, 41009}, {16318, 43717}, {17907, 41005}, {23967, 41139}, {32001, 38292}, {34360, 43620}

X(44334) = complement of X(441)
X(44334) = anticomplement of X(44335)
X(44334) = {X(2454),X(2455)}-harmonic conjugate of X(4)

X(44335) = STEINER-INELLIPSE-INVERSE OF X(20)

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

X(44335) lies on these lines: {2, 3}, {524, 40484}, {525, 7658}, {3788, 20203}, {6389, 20204}, {11064, 40512}, {18440, 42287}

X(44335) = complement of X(44334)
X(44335) = {X(2454),X(2455)}-harmonic conjugate of X(20)

X(44336) = STEINER-INELLIPSE-INVERSE OF X(21)

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

X(44336) lies on these lines: {2, 3}, {525, 14838}

X(44336) = {X(2454),X(2455)}-harmonic conjugate of X(21)

X(44337) = STEINER-INELLIPSE-INVERSE OF X(22)

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

X(44337) lies on these lines: {2, 3}, {525, 7630}, {754, 40484}, {3564, 41255}

X(44337) = {X(2454),X(2455)}-harmonic conjugate of X(22)

X(44338) = STEINER-INELLIPSE-INVERSE OF X(23)

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

X(44338) lies on these lines: {2, 3}, {525, 3589}, {15595, 32423}

X(44338) = midpoint of X(44348) and X(44349)
X(44338) = complement of complement of X(40856)
X(44338) = {X(2454),X(2455)}-harmonic conjugate of X(23)

X(44339) = STEINER-INELLIPSE-INVERSE OF X(24)

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

X(44339) lies on these lines: {2, 3}, {525, 16040}

X(44339) = complement of complement of X(44328)
X(44339) = {X(2454),X(2455)}-harmonic conjugate of X(24)

X(44340) = STEINER-INELLIPSE-INVERSE OF X(25)

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

X(44340) lies on these lines: {2, 3}, {230, 6720}, {339, 16318}, {525, 2485}, {538, 23583}, {1575, 40561}, {2207, 14376}, {14581, 15526}, {15048, 20204}

X(44340) = {X(2454),X(2455)}-harmonic conjugate of X(25)

X(44341) = STEINER-INELLIPSE-INVERSE OF X(26)

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

X(44341) lies on these lines: {2, 3}

X(44341) = complement of complement of X(44329)
X(44341) = {X(2454),X(2455)}-harmonic conjugate of X(26)

X(44342) = STEINER-INELLIPSE-INVERSE OF X(27)

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

X(44342) lies on these lines: {2, 3}, {519, 23583}, {525, 8062}, {30117, 35122}

X(44342) = complement of isotomic conjugate of cevapoint of X(2) and X(447)
X(44342) = complement of complement of X(447)
X(44342) = {X(2454),X(2455)}-harmonic conjugate of X(27)

X(44343) = STEINER-INELLIPSE-INVERSE OF X(28)

Barycentrics    2*a^11 + 2*a^10*b - a^9*b^2 - a^8*b^3 - 2*a^7*b^4 - 2*a^6*b^5 + a*b^10 + b^11 + 2*a^10*c + 2*a^9*b*c - a^8*b^2*c - a^7*b^3*c - 2*a^6*b^4*c - a^5*b^5*c - 3*a^3*b^7*c + 3*a*b^9*c + b^10*c - a^9*c^2 - a^8*b*c^2 + 4*a^7*b^2*c^2 + 4*a^6*b^3*c^2 - 4*a^3*b^6*c^2 - 4*a^2*b^7*c^2 + a*b^8*c^2 + b^9*c^2 - a^8*c^3 - a^7*b*c^3 + 4*a^6*b^2*c^3 + 2*a^5*b^3*c^3 + 3*a^3*b^5*c^3 - 4*a^2*b^6*c^3 - 4*a*b^7*c^3 + b^8*c^3 - 2*a^7*c^4 - 2*a^6*b*c^4 + 8*a^3*b^4*c^4 + 8*a^2*b^5*c^4 - 2*a*b^6*c^4 - 2*b^7*c^4 - 2*a^6*c^5 - a^5*b*c^5 + 3*a^3*b^3*c^5 + 8*a^2*b^4*c^5 + 2*a*b^5*c^5 - 2*b^6*c^5 - 4*a^3*b^2*c^6 - 4*a^2*b^3*c^6 - 2*a*b^4*c^6 - 2*b^5*c^6 - 3*a^3*b*c^7 - 4*a^2*b^2*c^7 - 4*a*b^3*c^7 - 2*b^4*c^7 + a*b^2*c^8 + b^3*c^8 + 3*a*b*c^9 + b^2*c^9 + a*c^10 + b*c^10 + c^11 : :

X(44343) lies on these lines: {2, 3}, {525, 16612}, {536, 23583}, {3008, 40532}

X(44343) = complement of complement of X(44330)
X(44343) = {X(2454),X(2455)}-harmonic conjugate of X(28)

X(44344) = STEINER-INELLIPSE-INVERSE OF X(29)

Barycentrics    2*a^11 - 3*a^9*b^2 - a^7*b^4 - a^6*b^5 + a^5*b^6 + 3*a^4*b^7 + 3*a^3*b^8 - 3*a^2*b^9 - 2*a*b^10 + b^11 - 2*a^9*b*c - 2*a^8*b^2*c + a^7*b^3*c + a^5*b^5*c + 4*a^4*b^6*c + 3*a^3*b^7*c - 3*a*b^9*c - 2*b^10*c - 3*a^9*c^2 - 2*a^8*b*c^2 + 6*a^7*b^2*c^2 + 3*a^6*b^3*c^2 - a^5*b^4*c^2 - 2*a^4*b^5*c^2 - 4*a^3*b^6*c^2 + 3*a^2*b^7*c^2 + 2*a*b^8*c^2 - 2*b^9*c^2 + a^7*b*c^3 + 3*a^6*b^2*c^3 - 2*a^5*b^3*c^3 - 5*a^4*b^4*c^3 - 3*a^3*b^5*c^3 - 3*a^2*b^6*c^3 + 4*a*b^7*c^3 + 5*b^8*c^3 - a^7*c^4 - a^5*b^2*c^4 - 5*a^4*b^3*c^4 + 2*a^3*b^4*c^4 + 3*a^2*b^5*c^4 + 2*b^7*c^4 - a^6*c^5 + a^5*b*c^5 - 2*a^4*b^2*c^5 - 3*a^3*b^3*c^5 + 3*a^2*b^4*c^5 - 2*a*b^5*c^5 - 4*b^6*c^5 + a^5*c^6 + 4*a^4*b*c^6 - 4*a^3*b^2*c^6 - 3*a^2*b^3*c^6 - 4*b^5*c^6 + 3*a^4*c^7 + 3*a^3*b*c^7 + 3*a^2*b^2*c^7 + 4*a*b^3*c^7 + 2*b^4*c^7 + 3*a^3*c^8 + 2*a*b^2*c^8 + 5*b^3*c^8 - 3*a^2*c^9 - 3*a*b*c^9 - 2*b^2*c^9 - 2*a*c^10 - 2*b*c^10 + c^11 : :

X(44344) lies on this line {2, 3}, {527, 23583}

X(44344) = complement of complement of X(44331)
X(44344) = {X(2454),X(2455)}-harmonic conjugate of X(29)

X(44345) = STEINER-INELLIPSE-INVERSE OF X(237)

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

X(44345) lies on these lines: {2, 3}, {39, 525}, {83, 2966}, {141, 14966}, {339, 9475}, {1506, 35088}, {1576, 41255}, {2421, 3933}, {3589, 5661}, {5013, 34360}

X(44345) = {X(2454),X(2455)}-harmonic conjugate of X(237)

X(44346) = STEINER-INELLIPSE-INVERSE OF X(376)

Barycentrics    10*a^8 - 13*a^6*b^2 + 3*a^4*b^4 - 7*a^2*b^6 + 7*b^8 - 13*a^6*c^2 + 10*a^4*b^2*c^2 + 7*a^2*b^4*c^2 - 4*b^6*c^2 + 3*a^4*c^4 + 7*a^2*b^2*c^4 - 6*b^4*c^4 - 7*a^2*c^6 - 4*b^2*c^6 + 7*c^8 : :

X(44346) lies on these lines: {2, 3}, {525, 14345}, {2966, 41133}, {3564, 41145}, {11180, 42287}, {22110, 23967}

X(44346) = complement of X(44216)
X(44346) = {X(2454),X(2455)}-harmonic conjugate of X(376)

X(44347) = STEINER-INELLIPSE-INVERSE OF X(384)

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

X(44347) lies on these lines: {2, 3}, {325, 19576}, {525, 5113}, {3506, 3564}, {9019, 40559}, {41273, 43291}

X(44347) = complement of X(21536)
X(44347) = {X(2454),X(2455)}-harmonic conjugate of X(384)

X(44348) = STEINER-INELLIPSE-INVERSE OF X(1113)

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

X(44348) lies on this line: {2, 3}

X(44348) = reflection of X(44349) in X(44338)
X(44348) = complement of complement of X(44332)
X(44348) = {X(2),X(441)}-harmonic conjugate of X(44349)
X(44348) = {X(2454),X(2455)}-harmonic conjugate of X(1113)

X(44349) = STEINER-INELLIPSE-INVERSE OF X(1114)

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

X(44349) lies on this line: {2, 3}

X(44349) = reflection of X(44348) in X(44338)
X(44349) = complement of complement of X(44333)
X(44349) = {X(2),X(441)}-harmonic conjugate of X(44348)
X(44349) = {X(2454),X(2455)}-harmonic conjugate of X(1114)

leftri

Steiner-ellipse-inverses of points on the line X(1)X(3): X(44350)-X(44360)

rightri

This preamble is contributed by Peter Moses, August 24, 2021.

Let f(a,b,c,x,y,z) = b c (b - c) (b + c - a) (x2 - y z). The inverse of the line X(1)X(3) in the Steiner circumellipse is the ellipse given by

f(a,b,c,x,y,z) + f(b,c,a,y,z x) + f(c,a,b,z,x,y) = 0.

Let g(a,b,c,x,y,z) = (b - c) (a3 - a b2 - a c2 - 2 a b c + 2 b2 c + 2 b c2) (x2 - y z). The inverse of the line X(1)X(3) in the Steiner inellipse is the ellipse given by

g(a,b,c,x,y,z) + g(b,c,a,y,z x) + g(c,a,b,z,x,y) = 0.

The appearance of (j,.k) in the following list means that X(j) lies on the line X(1)X(3) and the X(k) = Steiner-circumellipse-inverse of X(j):

(1,239), (3,401), (55,40861), (57,40862), (65, 44350), (241,44351), (517,2), (940,44352), (982,44353), (1214,44354), (5662,17496)

The appearance of (j,.k) in the following list means that X(j) lies on the line X(1)X(3) and the X(k) = Steiner-inellipse-inverse of X(j):

(1,3008), (3,441), (55,44355), (57,44356), (241,44357), (942, 44358), (982,44359), (1214,44360), (517,2), (5662,905)

X(44350) = STEINER-CIRCUMELLIPSE-INVERSE OF X(65)

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

X(44350) lies on these lines: {2, 65}, {73, 1999}, {239, 1457}, {241, 350}, {348, 4352}, {693, 3669}, {948, 1909}, {1284, 26113}, {3911, 25510}, {11349, 40861}, {15149, 17985}, {17139, 17950}, {17321, 40784}, {26048, 40663}, {26801, 41245}

X(44351) = STEINER-CIRCUMELLIPSE-INVERSE OF X(241)

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

X(44351) lies on these lines: {2, 85}, {145, 3900}, {651, 40861}, {664, 39350}, {1025, 3501}, {6604, 35167}, {7056, 21218}, {24410, 40862}

X(44352) = STEINER-CIRCUMELLIPSE-INVERSE OF X(940)

Barycentrics    a^5 + a^4*b + a^4*c + 4*a^3*b*c - 2*a*b^3*c - a*b^2*c^2 - b^3*c^2 - 2*a*b*c^3 - b^2*c^3 : :
X(44352) = 3 X[13586] - 4 X[16702]

X(44352) lies on these lines: {2, 6}, {314, 24271}, {523, 17496}, {1227, 4366}, {1943, 7175}, {5019, 34282}, {5327, 20077}, {13586, 16702}

X(44352) = reflection of X(385) in X(19623)
X(44352) = crossdifference of every pair of points on line {512, 4263}
X(44352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 940}

X(44353) = STEINER-CIRCUMELLIPSE-INVERSE OF X(982)

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

X(44353) lies on these lines: {2, 38}, {7, 9263}, {190, 2275}, {239, 2810}, {330, 4440}, {668, 3662}, {812, 17496}, {894, 1015}, {1086, 1909}, {1469, 32029}, {5299, 24815}, {9055, 40875}, {10027, 14839}, {17291, 27076}, {17368, 27195}

X(44354) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1214)

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

X(44354) lies on these lines: {2, 92}, {7, 3164}, {226, 18667}, {401, 651}, {416, 2659}, {521, 17496}, {1943, 40152}, {6180, 20477}, {6516, 40888}, {7361, 20078}, {8680, 40843}, {10538, 40862}, {17483, 43988}, {17950, 35145}

X(44354) = anticomplement of X(1948)
X(44354) = anticomplement of the isogonal conjugate of X(1949)
X(44354) = anticomplement of the isotomic conjugate of X(40843)
X(44354) = isotomic conjugate of the anticomplement of X(39036)
X(44354) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {296, 69}, {1937, 21270}, {1945, 4}, {1949, 8}, {1952, 11442}, {40843, 6327}, {41206, 21300}
X(44354) = X(i)-Ceva conjugate of X(j) for these (i,j): {8680, 17950}, {40843, 2}
X(44354) = X(39036)-cross conjugate of X(2)
X(44354) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2656}, {1172, 2660}
X(44354) = barycentric product X(i)*X(j) for these {i,j}: {75, 2655}, {307, 2659}, {416, 1441}, {39036, 40843}
X(44354) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2656}, {73, 2660}, {416, 21}, {2655, 1}, {2659, 29}, {39036, 1948}

X(44355) = STEINER-INELLIPSE-INVERSE OF X(55)

Barycentrics    2*a^6 - 4*a^5*b + 3*a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 + b^6 - 4*a^5*c + 4*a^4*b*c + 2*a*b^4*c - 2*b^5*c + 3*a^4*c^2 - 4*a^2*b^2*c^2 + 3*b^4*c^2 - 2*a^3*c^3 - 4*b^3*c^3 + 2*a^2*c^4 + 2*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6 : :
X(44355) = 3 X[2] + X[40861]

X(44355) lies on these lines: {2, 11}, {664, 17352}, {905, 918}, {1086, 20269}, {1146, 17337}, {3589, 17044}, {4422, 7789}, {14827, 18214}, {28850, 31289}

X(44355) = crossdifference of every pair of points on line {665, 1486}

X(44356) = STEINER-INELLIPSE-INVERSE OF X(57)

Barycentrics    2*a^5 - a^4*b - 2*a^3*b^2 + b^5 - a^4*c + 4*a^3*b*c - 4*a*b^3*c + b^4*c - 2*a^3*c^2 + 8*a*b^2*c^2 - 2*b^3*c^2 - 4*a*b*c^3 - 2*b^2*c^3 + b*c^4 + c^5 : :
X(44356) = 3 X[2] + X[40862], X[10025] + 3 X[40892]

X(44356) lies on these lines: {2, 7}, {269, 281}, {519, 6510}, {522, 905}, {536, 17044}, {651, 26001}, {1111, 37805}, {1125, 25375}, {1146, 6610}, {1407, 20205}, {1420, 24565}, {1422, 7003}, {2325, 16578}, {3008, 36949}, {3663, 17073}, {3664, 16608}, {3752, 20201}, {3942, 8756}, {4021, 17043}, {4472, 6706}, {4670, 21258}, {4858, 43035}, {4887, 18644}, {6180, 20262}, {6907, 25365}, {7190, 24553}, {8074, 34371}, {10106, 24537}, {15149, 17197}, {17067, 24781}, {17862, 18652}, {17917, 23681}, {20206, 24213}, {23986, 35094}, {26011, 34050}

X(44356) = midpoint of X(i) and X(j) for these {i,j}: {1146, 6610}, {1944, 9436}, {40862, 40880}
X(44356) = complement of X(40880)
X(44356) = X(9372)-complementary conjugate of X(141)
X(44356) = crossdifference of every pair of points on line {198, 663}
X(44356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 40862, 40880}

X(44357) = STEINER-INELLIPSE-INVERSE OF X(241)

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

X(44357) lies on these lines: {1, 905}, {2, 85}, {142, 5701}, {220, 1025}, {5662, 29571}, {6184, 17044}, {21258, 35094}, {25242, 42719}

X(44357) = crossdifference of every pair of points on line {910, 8641}

X(44358) = STEINER-INELLIPSE-INVERSE OF X(942)

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

X(44358) lies on these lines: {2, 72}, {36, 448}, {241, 16090}, {350, 25083}, {441, 15325}, {693, 905}, {17095, 19786}, {20935, 31997}, {26971, 27334}

X(44359) = STEINER-INELLIPSE-INVERSE OF X(982)

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

X(44359) lies on these lines: {2, 38}, {142, 1015}, {668, 17282}, {812, 905}, {1086, 4920}, {2810, 3008}, {5750, 40479}, {10436, 27195}, {27191, 31997}

X(44359) = crossdifference of every pair of points on line {8632, 34247}

X(44360) = STEINER-INELLIPSE-INVERSE OF X(1214)

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

X(44360) lies on these lines: {2, 92}, {142, 216}, {226, 6509}, {394, 7364}, {441, 36949}, {521, 656}, {527, 35072}, {6617, 34048}, {16608, 41005}, {25365, 42353}

X(44360) = complement of X(1948)
X(44360) = complement of the isogonal conjugate of X(1949)
X(44360) = complement of the isotomic conjugate of X(40843)
X(44360) = X(i)-complementary conjugate of X(j) for these (i,j): {296, 141}, {1937, 20305}, {1945, 5}, {1949, 10}, {1952, 21243}, {2249, 34831}, {40843, 2887}, {41206, 21259}
X(44360) = X(6)-isoconjugate of X(8764)
X(44360) = crosspoint of X(2) and X(40843)
X(44360) = crosssum of X(6) and X(2202)
X(44360) = crossdifference of every pair of points on line {19, 1946}
X(44360) = barycentric product X(75)*X(8763)
X(44360) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 8764}, {8763, 1}

X(44361) = STEINER-CIRCUMELLIPSE-INVERSE OF X(302)

Barycentrics    a^2*b^2 - b^4 + a^2*c^2 - c^4 + 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(44361) = 4 X[230] - 3 X[37785], 3 X[298] - 4 X[325], 3 X[299] - 2 X[325], 2 X[385] - 3 X[37786], 4 X[6671] - 3 X[16530], 2 X[6781] - 3 X[8594], 3 X[8595] - 4 X[32456]

X(44361) lies on these lines: {2, 6}, {61, 22737}, {76, 22701}, {99, 532}, {316, 533}, {340, 23712}, {511, 22509}, {621, 20425}, {892, 11117}, {5965, 5983}, {6671, 16530}, {6781, 8594}, {8595, 32456}, {11132, 16529}, {18813, 34389}

X(44361) = midpoint of X(3180) and X(40899)
X(44361) = reflection of X(i) in X(j) for these {i,j}: {298, 299}, {3181, 396}
X(44361) = isotomic conjugate of X(11602)
X(44361) = isotomic conjugate of the isogonal conjugate of X(39554)
X(44361) = X(31)-isoconjugate of X(11602)
X(44361) = barycentric product X(76)*X(39554)
X(44361) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11602}, {39554, 6}
X(44361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 302}, {33459, 37647, 302}, {39365, 39366, 40900}

X(44362) = STEINER-CIRCUMELLIPSE-INVERSE OF X(303)

Barycentrics    a^2*b^2 - b^4 + a^2*c^2 - c^4 - 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(44362) = 4 X[230] - 3 X[37786], 3 X[298] - 2 X[325], 3 X[299] - 4 X[325], 2 X[385] - 3 X[37785], 4 X[6672] - 3 X[16529], 2 X[6781] - 3 X[8595], 3 X[8594] - 4 X[32456]

X(44362) lies on these lines: {2, 6}, {62, 22736}, {76, 22702}, {99, 533}, {316, 532}, {340, 23713}, {511, 22507}, {622, 20426}, {892, 11118}, {5965, 5982}, {6672, 16529}, {6781, 8595}, {8594, 32456}, {11133, 16530}, {18814, 34390}

X(44362) = midpoint of X(3181) and X(40898)
X(44362) = reflection of X(i) in X(j) for these {i,j}: {299, 298}, {3180, 395}
X(44362) = isotomic conjugate of X(11603)
X(44362) = isotomic conjugate of the isogonal conjugate of X(39555)
X(44362) = X(31)-isoconjugate of X(11603)
X(44362) = barycentric product X(76)*X(39555)
X(44362) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 11603}, {39555, 6}
X(44362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 303}, {33458, 37647, 303}, {39365, 39366, 40901}

X(44363) = STEINER-CIRCUMELLIPSE-INVERSE OF X(343)

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

X(44363) lies on these lines: {2, 6}, {147, 2393}, {311, 17035}, {340, 36212}, {523, 3153}, {1273, 4558}, {3164, 44128}, {3260, 39352}, {7577, 36207}, {9019, 40236}, {14570, 40853}, {18420, 32515}, {22087, 35922}, {40074, 44144}, {40897, 44180}

X(44363) = reflection of X(40888) in X(325)
X(44363) = isotomic conjugate of the polar conjugate of X(41203)
X(44363) = isotomic conjugate of antigonal conjugate of X(275)
X(44363) = barycentric product X(i)*X(j) for these {i,j}: {69, 41203}, {670, 42651}
X(44363) = barycentric quotient X(i)/X(j) for these {i,j}: {41203, 4}, {42651, 512}
X(44363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 343}

X(44364) = STEINER-CIRCUMELLIPSE-INVERSE OF X(491)

Barycentrics    a^2*b^2 - b^4 + a^2*c^2 - c^4 - 2*(2*a^2 - b^2 - c^2)*S : :
X(44364) lies on these lines: {2, 6}, {99, 32419}, {316, 32421}, {317, 41516}, {372, 6229}, {487, 21445}, {489, 12256}, {511, 6231}, {637, 8982}, {3564, 9867}, {5491, 13939}, {7771, 41490}, {8036, 18820}, {8598, 13797}, {12323, 26331}, {22623, 42009}, {24243, 38294}, {32515, 33434}

X(44364) = isotomic conjugate of the isogonal conjugate of X(2460)
X(44364) = isotomic conjugate of antigonal conjugate of X(486)
X(44364) = crossdifference of every pair of points on line {512, 5058}
X(44364) = barycentric product X(76)*X(2460)
X(44364) = barycentric quotient X(2460)/X(6)
X(44364) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 385, 44365}, {69, 3069, 491}, {1007, 5861, 491}, {6189, 6190, 491}

X(44365) = STEINER-CIRCUMELLIPSE-INVERSE OF X(492)

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

X(44365) lies on these lines: {2, 6}, {99, 32421}, {316, 32419}, {317, 41515}, {371, 6228}, {488, 21445}, {490, 12257}, {511, 6230}, {638, 26441}, {3564, 9868}, {5490, 13886}, {7771, 41491}, {8035, 18819}, {8598, 13677}, {8960, 42009}, {12322, 26330}, {22594, 42060}, {24244, 38294}, {32515, 33435}

X(44365) = isotomic conjugate of the isogonal conjugate of X(2459)
X(44365) = isotomic conjugate of antigonal conjugate of X(485)
X(44365) = crossdifference of every pair of points on line {512, 5062}
X(44365) = barycentric product X(76)*X(2459)
X(44365) = barycentric quotient X(2459)/X(6)
X(44365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 385, 44364}, {69, 3068, 492}, {1007, 5860, 492}, {6189, 6190, 492}

X(44366) = STEINER-CIRCUMELLIPSE-INVERSE OF X(591)

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

X(44366) lies on these lines: {2, 6}, {187, 13640}, {1585, 38294}, {5107, 13773}, {9855, 33343}, {11054, 13676}, {13586, 32421}, {14041, 32419}, {26289, 33017}, {33273, 41491}

X(44366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3068, 5861, 1992}, {6189, 6190, 591}

X(44367) = STEINER-CIRCUMELLIPSE-INVERSE OF X(597)

Barycentrics    5*a^4 + a^2*b^2 - b^4 + a^2*c^2 - 5*b^2*c^2 - c^4 : :
X(44367) = 7 X[2] - 8 X[230], 5 X[2] - 4 X[325], 11 X[2] - 10 X[7925], 5 X[2] - 6 X[8859], X[2] - 8 X[15480], 9 X[2] - 8 X[22110], 3 X[2] - 4 X[22329], 13 X[2] - 12 X[41133], 4 X[2] - 3 X[41136], 23 X[2] - 24 X[41139], 10 X[230] - 7 X[325], 4 X[230] - 7 X[385], 16 X[230] - 7 X[7779], 12 X[230] - 7 X[7840], 44 X[230] - 35 X[7925], 20 X[230] - 21 X[8859], X[230] - 7 X[15480], 9 X[230] - 7 X[22110], 6 X[230] - 7 X[22329], 26 X[230] - 21 X[41133], 32 X[230] - 21 X[41136], 23 X[230] - 21 X[41139], 2 X[316] - 3 X[41135], 2 X[325] - 5 X[385], 8 X[325] - 5 X[7779], 6 X[325] - 5 X[7840], 22 X[325] - 25 X[7925], 2 X[325] - 3 X[8859], X[325] - 10 X[15480], 9 X[325] - 10 X[22110], 3 X[325] - 5 X[22329], 13 X[325] - 15 X[41133], 16 X[325] - 15 X[41136], 23 X[325] - 30 X[41139], 4 X[385] - X[7779], 3 X[385] - X[7840], 11 X[385] - 5 X[7925], 5 X[385] - 3 X[8859], X[385] - 4 X[15480], 9 X[385] - 4 X[22110], 3 X[385] - 2 X[22329], 13 X[385] - 6 X[41133], 8 X[385] - 3 X[41136], 23 X[385] - 12 X[41139], 2 X[671] - 3 X[19570], 3 X[5032] - 2 X[39099], 4 X[5461] - 3 X[7809], 3 X[7779] - 4 X[7840], 11 X[7779] - 20 X[7925], 5 X[7779] - 12 X[8859], X[7779] - 16 X[15480], 9 X[7779] - 16 X[22110], 3 X[7779] - 8 X[22329], 13 X[7779] - 24 X[41133], 2 X[7779] - 3 X[41136], 23 X[7779] - 48 X[41139], 2 X[7813] - 3 X[41134], 11 X[7840] - 15 X[7925], 5 X[7840] - 9 X[8859], X[7840] - 12 X[15480], 3 X[7840] - 4 X[22110], 13 X[7840] - 18 X[41133], 8 X[7840] - 9 X[41136], 23 X[7840] - 36 X[41139], 25 X[7925] - 33 X[8859], 5 X[7925] - 44 X[15480], 45 X[7925] - 44 X[22110], 15 X[7925] - 22 X[22329], 65 X[7925] - 66 X[41133], 40 X[7925] - 33 X[41136], 115 X[7925] - 132 X[41139], 2 X[8591] - 3 X[33265], 3 X[8859] - 20 X[15480], 27 X[8859] - 20 X[22110], 9 X[8859] - 10 X[22329], 13 X[8859] - 10 X[41133], 8 X[8859] - 5 X[41136], 23 X[8859] - 20 X[41139], 4 X[11054] - X[40246], 3 X[14568] - 2 X[31173], 9 X[15480] - X[22110], 6 X[15480] - X[22329], 26 X[15480] - 3 X[41133], 32 X[15480] - 3 X[41136], 23 X[15480] - 3 X[41139], 5 X[15692] - 6 X[21445], 2 X[22110] - 3 X[22329], 26 X[22110] - 27 X[41133], 32 X[22110] - 27 X[41136], 23 X[22110] - 27 X[41139], 13 X[22329] - 9 X[41133], 16 X[22329] - 9 X[41136], 23 X[22329] - 18 X[41139], 3 X[26613] - 2 X[39785], 16 X[41133] - 13 X[41136], 23 X[41133] - 26 X[41139], 23 X[41136] - 32 X[41139]

X(44367) lies on these lines: {2, 6}, {30, 5984}, {76, 34604}, {114, 10487}, {148, 3849}, {316, 41135}, {376, 32515}, {511, 11177}, {519, 1281}, {523, 37901}, {538, 8591}, {542, 40236}, {543, 14712}, {671, 754}, {736, 19686}, {2482, 4027}, {2896, 7805}, {3552, 32824}, {3793, 8598}, {4577, 18823}, {5189, 25051}, {5211, 26280}, {5368, 32027}, {5461, 7809}, {5965, 6054}, {5976, 8787}, {5992, 28558}, {6034, 9866}, {6179, 7801}, {6194, 11179}, {6392, 33192}, {6653, 37857}, {6655, 9939}, {6995, 38294}, {7751, 7812}, {7754, 7833}, {7758, 33259}, {7760, 7810}, {7762, 33013}, {7768, 7817}, {7775, 7877}, {7780, 13571}, {7793, 34511}, {7797, 7826}, {7811, 41748}, {7813, 41134}, {7823, 34505}, {7839, 8359}, {7841, 7893}, {7855, 7870}, {7890, 34506}, {7900, 33006}, {7926, 8176}, {7929, 33190}, {7939, 8360}, {8370, 17129}, {8587, 35005}, {9164, 31068}, {9855, 20094}, {10162, 11056}, {10353, 12151}, {14568, 31173}, {14907, 32480}, {15527, 35087}, {15692, 21445}, {20065, 32826}, {20081, 32822}, {22253, 35955}, {26613, 39785}, {32833, 37809}

X(44367) = reflection of X(i) in X(j) for these {i,j}: {2, 385}, {148, 11054}, {7779, 2}, {7840, 22329}, {8598, 3793}, {8782, 22564}, {20094, 9855}, {40246, 148}
X(44367) = anticomplement of X(7840)
X(44367) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(9169)
X(44367) = anticomplement of the isotomic conjugate of X(43535)
X(44367) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {32694, 7192}, {43535, 6327}
X(44367) = X(43535)-Ceva conjugate of X(2)
X(44367) = crosssum of X(3124) and X(9208)
X(44367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7779, 41136}, {193, 9740, 2}, {325, 8859, 2}, {385, 7840, 22329}, {5032, 15589, 2}, {6189, 6190, 597}, {7610, 7777, 2}, {7837, 8667, 2}, {7840, 22329, 2}, {7875, 21358, 2}, {9770, 17008, 2}, {9771, 17006, 2}, {11184, 17004, 2}, {15597, 17005, 2}, {16989, 21356, 2}, {39107, 39108, 3589}, {39365, 39366, 1992}

X(44368) = STEINER-CIRCUMELLIPSE-INVERSE OF X(615)

Barycentrics    a^4 - b^2*c^2 - 2*(2*a^2 - b^2 - c^2)*S : :
X(44368) = 4 X[230] - 3 X[44374], 2 X[325] - 3 X[44366]

X(44368) lies on these lines: {2, 6}, {148, 32419}, {511, 33431}, {5965, 6231}, {8591, 13797}, {14712, 32421}, {35944, 43133}

X(44368) = anticomplement of X(44364)
X(44368) = reflection of X(7779) in X(44365)
X(44368) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {325, 44365, 1271}, {6189, 6190, 615}

X(44369) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1007)

Barycentrics    4*a^6 - 9*a^4*b^2 + 8*a^2*b^4 - 3*b^6 - 9*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + 8*a^2*c^4 + b^2*c^4 - 3*c^6 : :
X(44369) = 3 X[69] - X[39099], 3 X[325] - 2 X[39099], 4 X[599] - 3 X[41133], 2 X[1351] - 3 X[39663], 3 X[1570] - 4 X[6722], 2 X[5107] - 3 X[33228], 2 X[5477] - 3 X[35297], 4 X[15993] - 3 X[22329]

X(44369) lies on these lines: {2, 6}, {99, 3564}, {340, 892}, {511, 39809}, {1351, 39663}, {1570, 6722}, {5107, 33228}, {5477, 35297}, {5965, 6393}, {6394, 40996}, {9146, 41724}, {10008, 21445}, {10553, 35266}, {14929, 32515}, {15069, 32819}, {32001, 34208}, {40074, 44149}

X(44369) = midpoint of X(385) and X(20080)
X(44369) = reflection of X(i) in X(j) for these {i,j}: {193, 230}, {325, 69}, {41146, 22165}
X(44369) = crossdifference of every pair of points on line {512, 39764}
X(44369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {298, 299, 22110}, {325, 37688, 41133}, {6189, 6190, 1007}

X(44370) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1211)

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

X(44370) lies on these lines: {2, 6}, {148, 536}, {239, 26081}, {740, 20558}, {894, 21076}, {984, 36223}, {986, 1330}, {3770, 17788}, {3912, 26147}, {4053, 6542}, {4363, 6625}, {4440, 20349}, {20654, 28604}, {35511, 42713}, {39356, 39360}

X(44370) = reflection of X(40882) in X(10026)
X(44370) = anticomplement of X(19623)
X(44370) = anticomplement of the isotomic conjugate of X(11611)
X(44370) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2703, 7192}, {11609, 20245}, {11611, 6327}, {17929, 17159}, {17939, 17161}, {17946, 17135}, {17954, 75}, {17961, 1}, {17971, 17134}, {17981, 17220}, {18002, 4440}, {18015, 150}, {35147, 17217}
X(44370) = X(11611)-Ceva conjugate of X(2)
X(44370) = crosspoint of X(4590) and X(35147)
X(44370) = crosssum of X(3124) and X(5040)
X(44370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 1211}, {39365, 39366, 2895}

X(44371) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1613)

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

X(44371) lies on these lines: {2, 6}, {99, 6379}, {237, 25054}, {670, 33875}, {706, 12215}, {1916, 2393}, {4590, 39927}, {8264, 20794}

X(44371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 1613}

X(44372) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1641)

Barycentrics    5*a^8 - 10*a^6*b^2 + 5*a^2*b^6 - b^8 - 10*a^6*c^2 + 30*a^4*b^2*c^2 - 15*a^2*b^4*c^2 - b^6*c^2 - 15*a^2*b^2*c^4 + 9*b^4*c^4 + 5*a^2*c^6 - b^2*c^6 - c^8 : :
X(44372) = X[671] - 3 X[31998], 2 X[2482] - 3 X[4590], 3 X[4590] - X[18823], 8 X[9164] - 3 X[35511], 3 X[9166] - 4 X[40553], 2 X[14588] + X[31372], 2 X[23992] - 3 X[41134], 3 X[31998] - 2 X[35087]

X(44372) lies on these lines: {2, 6}, {148, 17948}, {523, 8591}, {543, 892}, {671, 31998}, {2482, 4590}, {9164, 35511}, {9166, 40553}, {9180, 33915}, {14588, 31372}, {23992, 41134}

X(44372) = reflection of X(i) in X(j) for these {i,j}: {2, 9182}, {148, 17948}, {671, 35087}, {18823, 2482}
X(44372) = barycentric product X(670)*X(39527)
X(44372) = barycentric quotient X(39527)/X(512)
X(44372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {671, 31998, 35087}, {4590, 18823, 2482}, {6189, 6190, 1641}, {39107, 39108, 11053}

X(44373) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1648)

Barycentrics    a^8 - 2*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - b^8 - 2*a^6*c^2 + 10*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 5*a^2*b^2*c^4 + b^4*c^4 + 3*a^2*c^6 + b^2*c^6 - c^8 : :
X(44373) = X[148] + 3 X[35511], 4 X[620] - 3 X[4590], 3 X[9166] - 2 X[35087], 5 X[14061] - 6 X[23991], 5 X[14061] - 3 X[31998], 5 X[14061] - 4 X[40553], 2 X[17948] - 3 X[41135], 3 X[23991] - 2 X[40553], 3 X[31998] - 4 X[40553], 3 X[35511] - X[39356]

X(44373) lies on these lines: {2, 6}, {99, 23992}, {115, 892}, {148, 523}, {543, 18823}, {620, 4590}, {888, 31513}, {9166, 35087}, {9170, 41176}, {9180, 33921}, {14061, 23991}, {17948, 41135}

X(44373) = midpoint of X(148) and X(39356)
X(44373) = reflection of X(i) in X(j) for these {i,j}: {99, 23992}, {892, 115}, {31998, 23991}
X(44373) = anticomplement of X(9182)
X(44373) = anticomplement of the isotomic conjugate of X(9180)
X(44373) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {843, 7192}, {9180, 6327}, {18823, 17217}
X(44373) = X(9180)-Ceva conjugate of X(2)
X(44373) = crosspoint of X(4590) and X(18823)
X(44373) = crosssum of X(2502) and X(3124)
X(44373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {148, 35511, 39356}, {325, 5912, 2}, {325, 22329, 24855}, {6189, 6190, 1648}, {14061, 31998, 40553}, {23991, 40553, 14061}

X(44374) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1991)

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

X(44374) lies on these lines: {2, 6}, {187, 13760}, {1586, 38294}, {5107, 13653}, {9855, 33342}, {11054, 13796}, {13586, 32419}, {14041, 32421}, {26288, 33017}, {33273, 41490}

X(44374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3069, 5860, 1992}, {6189, 6190, 1991}

X(44375) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1993)

Barycentrics    a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + a^4*b^2*c^2 - b^6*c^2 + a^4*c^4 + 2*b^4*c^4 - b^2*c^6 : :
X(44375) = X[385] + 2 X[40879], 3 X[21445] + X[38294]

X(44375) lies on these lines: {2, 6}, {50, 338}, {95, 570}, {98, 2393}, {112, 37778}, {157, 3186}, {186, 523}, {231, 340}, {237, 9512}, {264, 571}, {297, 16310}, {317, 2165}, {389, 19179}, {419, 1576}, {577, 41760}, {648, 3003}, {687, 2966}, {1316, 23200}, {1609, 9308}, {1879, 32002}, {1989, 40885}, {3018, 37765}, {3164, 8553}, {5063, 40814}, {5999, 9019}, {6644, 36207}, {7514, 32515}, {7793, 19221}, {9722, 27377}, {9755, 32621}, {11676, 39231}, {14570, 35296}, {18533, 33971}, {41679, 44138}

X(44375) = midpoint of X(385) and X(40888)
X(44375) = reflection of X(i) in X(j) for these {i,j}: {11676, 39231}, {40888, 40879}
X(44375) = isotomic conjugate of the polar conjugate of X(421)
X(44375) = crossdifference of every pair of points on line {216, 512}
X(44375) = barycentric product X(69)*X(421)
X(44375) = barycentric quotient X(421)/X(4)
X(44375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {50, 338, 401}, {6189, 6190, 1993}, {39022, 39023, 23292}, {41679, 44138, 44328}

X(44376) = STEINER-CIRCUMELLIPSE-INVERSE OF X(1994)

Barycentrics    a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 - a^4*b^2*c^2 + a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - b^2*c^6 : :
X(44376) = 2 X[385] + X[40879], 3 X[385] + X[40888], 3 X[40879] - 2 X[40888]

X(44376) lies on these lines: {2, 6}, {98, 9019}, {311, 2965}, {523, 2070}, {648, 11062}, {2782, 39231}, {2966, 11077}, {5201, 9512}, {11063, 14570}

X(44376) = crossdifference of every pair of points on line {512, 570}
X(44376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 1994}, {39022, 39023, 37649}

X(44377) = STEINER-INELLIPSE-INVERSE OF X(69)

Barycentrics    2*a^4 - 3*a^2*b^2 + 3*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 3*c^4 : :
X(44377) = 3 X[2] + X[325], 9 X[2] - X[385], 15 X[2] + X[7779], 7 X[2] + X[7840], 3 X[2] + 5 X[7925], 11 X[2] - 3 X[8859], 15 X[2] - X[15480], 5 X[2] - X[22329], X[2] + 3 X[41133], 13 X[2] + 3 X[41136], 5 X[2] - 3 X[41139], X[3] - 3 X[10256], X[98] - 5 X[40336], X[99] + 3 X[33228], X[115] - 5 X[31275], X[187] - 5 X[31274], 3 X[230] - X[385], 5 X[230] + X[7779], 7 X[230] + 3 X[7840], X[230] + 5 X[7925], 11 X[230] - 9 X[8859], 5 X[230] - X[15480], X[230] + 3 X[22110], 5 X[230] - 3 X[22329], X[230] + 9 X[41133], 13 X[230] + 9 X[41136], 5 X[230] - 9 X[41139], X[316] + 3 X[35297], 3 X[325] + X[385], 5 X[325] - X[7779], 7 X[325] - 3 X[7840], X[325] - 5 X[7925], 11 X[325] + 9 X[8859], 5 X[325] + X[15480], X[325] - 3 X[22110], 5 X[325] + 3 X[22329], X[325] - 9 X[41133], 13 X[325] - 9 X[41136], 5 X[325] + 9 X[41139], 5 X[385] + 3 X[7779], 7 X[385] + 9 X[7840], X[385] + 15 X[7925], 11 X[385] - 27 X[8859], 5 X[385] - 3 X[15480], X[385] + 9 X[22110], 5 X[385] - 9 X[22329], X[385] + 27 X[41133], 13 X[385] + 27 X[41136], 5 X[385] - 27 X[41139], 3 X[620] - X[32456], 3 X[625] + X[32456], 2 X[625] + X[32459], 7 X[3090] - 3 X[39663], 11 X[3525] - 3 X[21445], 7 X[3619] + X[39099], 5 X[3763] - X[15993], X[5026] + 3 X[5031], X[5461] - 3 X[10150], 3 X[5461] + X[14148], 3 X[5461] - X[32457], X[6390] + 5 X[31275], 3 X[6393] + X[10754], X[6781] - 9 X[9167], X[6781] - 3 X[27088], X[6781] + 3 X[31173], 7 X[7779] - 15 X[7840], X[7779] - 25 X[7925], 11 X[7779] + 45 X[8859], X[7779] - 15 X[22110], X[7779] + 3 X[22329], X[7779] - 45 X[41133], 13 X[7779] - 45 X[41136], X[7779] + 9 X[41139], 3 X[7799] + 5 X[14061], 3 X[7840] - 35 X[7925], 11 X[7840] + 21 X[8859], 15 X[7840] + 7 X[15480], X[7840] - 7 X[22110], 5 X[7840] + 7 X[22329], X[7840] - 21 X[41133], 13 X[7840] - 21 X[41136], 5 X[7840] + 21 X[41139], 55 X[7925] + 9 X[8859], 25 X[7925] + X[15480], 5 X[7925] - 3 X[22110], 25 X[7925] + 3 X[22329], 5 X[7925] - 9 X[41133], 65 X[7925] - 9 X[41136], 25 X[7925] + 9 X[41139], X[8352] + 3 X[41134], 45 X[8859] - 11 X[15480], 3 X[8859] + 11 X[22110], 15 X[8859] - 11 X[22329], X[8859] + 11 X[41133], 13 X[8859] + 11 X[41136], 5 X[8859] - 11 X[41139], 3 X[9167] - X[27088], 3 X[9167] + X[31173], 9 X[10150] + X[14148], 9 X[10150] - X[32457], X[13449] + 3 X[38748], 3 X[14971] + X[39785], X[15480] + 15 X[22110], X[15480] - 3 X[22329], X[15480] + 45 X[41133], 13 X[15480] + 45 X[41136], X[15480] - 9 X[41139], 3 X[15561] + X[15980], X[18860] + 3 X[36519], 3 X[21358] + X[41146], 5 X[22110] + X[22329], X[22110] - 3 X[41133], 13 X[22110] - 3 X[41136], 5 X[22110] + 3 X[41139], X[22329] + 15 X[41133], 13 X[22329] + 15 X[41136], X[22329] - 3 X[41139], 9 X[23234] - X[43460], 2 X[32456] - 3 X[32459], 13 X[41133] - X[41136], 5 X[41133] + X[41139], 5 X[41136] + 13 X[41139]

X(44377) lies on these lines: {2, 6}, {3, 7694}, {5, 3734}, {30, 620}, {39, 8361}, {53, 42406}, {76, 33249}, {98, 13196}, {99, 33228}, {114, 1503}, {115, 6390}, {126, 3258}, {127, 10257}, {140, 626}, {187, 31274}, {232, 34990}, {287, 40428}, {315, 33233}, {316, 35297}, {441, 35067}, {468, 14052}, {511, 6721}, {523, 4885}, {538, 6722}, {543, 8355}, {546, 7816}, {547, 7880}, {548, 7842}, {549, 7761}, {550, 7825}, {574, 33184}, {631, 7784}, {632, 7815}, {698, 2023}, {736, 6683}, {858, 16320}, {1368, 23333}, {1447, 7238}, {1506, 7819}, {1513, 5103}, {1656, 7795}, {1975, 32961}, {2482, 37350}, {2548, 32954}, {2549, 11318}, {2896, 16923}, {2996, 39143}, {3035, 20541}, {3053, 32816}, {3090, 39663}, {3266, 40511}, {3525, 21445}, {3526, 7800}, {3529, 39142}, {3530, 7830}, {3564, 6036}, {3628, 3934}, {3705, 4399}, {3767, 22253}, {3785, 32977}, {3793, 7845}, {3849, 22247}, {3926, 13881}, {3933, 7746}, {3972, 7745}, {4045, 8360}, {4048, 13860}, {4478, 7081}, {4643, 36407}, {5013, 14064}, {5023, 32006}, {5024, 33240}, {5210, 33216}, {5254, 7763}, {5286, 32955}, {5305, 7764}, {5461, 10150}, {5475, 8369}, {5480, 37071}, {5866, 34866}, {5988, 28530}, {6337, 32972}, {6389, 30771}, {6393, 8781}, {6656, 7769}, {6667, 20530}, {6676, 14725}, {6781, 9167}, {7179, 7228}, {7499, 15822}, {7603, 7820}, {7737, 11288}, {7738, 32835}, {7749, 7767}, {7750, 7907}, {7762, 7814}, {7773, 16925}, {7775, 18907}, {7782, 33229}, {7785, 33245}, {7786, 8363}, {7799, 14061}, {7803, 9606}, {7804, 8368}, {7808, 33185}, {7832, 32992}, {7834, 31406}, {7835, 8370}, {7836, 32967}, {7844, 15048}, {7848, 34506}, {7849, 16239}, {7851, 9607}, {7852, 9698}, {7853, 8359}, {7863, 39565}, {7865, 11539}, {7866, 31401}, {7867, 8362}, {7881, 32832}, {7885, 33259}, {7891, 32819}, {7898, 33274}, {7934, 8356}, {7938, 33015}, {7945, 16921}, {8352, 41134}, {8354, 8589}, {8357, 37512}, {8716, 43448}, {8787, 25486}, {9306, 19156}, {9752, 11477}, {9767, 13785}, {9768, 13665}, {10155, 18840}, {11286, 31415}, {11812, 40344}, {13449, 38748}, {14069, 31404}, {14971, 39785}, {15483, 15561}, {15815, 32974}, {15819, 15850}, {16043, 32839}, {17045, 24239}, {18584, 32983}, {18860, 36519}, {22998, 38412}, {23234, 43460}, {30103, 31460}, {30739, 30747}, {30749, 37454}, {31400, 32951}, {32815, 32984}, {32817, 34505}, {32818, 32958}, {32823, 32959}, {32827, 32985}, {32828, 32976}, {32871, 33202}, {35387, 37451}, {37450, 43461}

X(44377) = midpoint of X(i) and X(j) for these {i,j}: {2, 22110}, {115, 6390}, {230, 325}, {441, 35088}, {620, 625}, {858, 16320}, {2482, 37350}, {3793, 7845}, {7779, 15480}, {14148, 32457}, {27088, 31173}, {44382, 44383}, {44384, 44385}, {44390, 44391} to midpoints
X(44377) = reflection of X(i) in X(j) for these {i,j}: {10011, 6721}, {32459, 620}, {43291, 6722}
X(44377) = complement of X(230)
X(44377) = orthoptic-circle-of-Steiner-inellipe-inverse of X(38940)
X(44377) = complement of the isogonal conjugate of X(2987)
X(44377) = complement of the isotomic conjugate of X(8781)
X(44377) = isotomic conjugate of the isogonal conjugate of X(1570)
X(44377) = X(i)-complementary conjugate of X(j) for these (i,j): {48, 35067}, {63, 31842}, {661, 36472}, {2065, 16609}, {2987, 10}, {3563, 226}, {8773, 141}, {8781, 2887}, {10425, 4369}, {32654, 37}, {32697, 8062}, {34157, 16591}, {35142, 20305}, {35364, 8287}, {36051, 2}, {36105, 30476}, {42065, 1214}, {43705, 18589}
X(44377) = X(4226)-Ceva conjugate of X(2799)
X(44377) = crosspoint of X(2) and X(8781)
X(44377) = crosssum of X(6) and X(1692)
X(44377) = crossdifference of every pair of points on line {512, 3053}
X(44377) = barycentric product X(76)*X(1570)
X(44377) = barycentric quotient X(1570)/X(6)
X(44377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 37637}, {2, 183, 3054}, {2, 325, 230}, {2, 599, 15597}, {2, 1007, 6}, {2, 3314, 37688}, {2, 3815, 3589}, {2, 7777, 7792}, {2, 7778, 141}, {2, 7897, 17004}, {2, 7925, 325}, {2, 11184, 597}, {2, 22329, 41139}, {2, 30760, 1211}, {2, 30761, 1213}, {2, 31489, 15491}, {2, 34803, 31489}, {2, 37647, 3055}, {2, 37690, 7778}, {2, 41133, 22110}, {5, 3788, 7789}, {69, 37637, 13468}, {230, 15480, 22329}, {230, 22110, 325}, {325, 7925, 22110}, {325, 22329, 7779}, {325, 41133, 7925}, {599, 34229, 15598}, {1506, 7874, 7819}, {3788, 7862, 5}, {3926, 32969, 13881}, {5306, 7774, 32455}, {5461, 14148, 32457}, {6189, 6190, 20080}, {7735, 9766, 3629}, {7746, 7888, 3933}, {7749, 7821, 7767}, {7752, 7807, 7745}, {7752, 7940, 7807}, {7763, 7887, 5254}, {7764, 7886, 5305}, {7769, 7899, 6656}, {7777, 7792, 9300}, {7779, 22329, 15480}, {7792, 9300, 6329}, {7814, 7857, 7762}, {7867, 31455, 8362}, {7891, 32966, 32819}, {7897, 17004, 37671}, {7907, 7912, 7750}, {8667, 37668, 3630}, {9167, 31173, 27088}, {9771, 15491, 31489}, {14064, 32829, 5013}, {15480, 41139, 230}, {15597, 15598, 34229}, {31406, 33186, 7834}, {31489, 34803, 9771}, {32006, 32989, 5023}, {32816, 32970, 3053}, {32835, 33199, 7738}, {39022, 39023, 69}, {39107, 39108, 11160}

X(44378) = STEINER-INELLIPSE-INVERSE OF X(81)

Barycentrics    2*a^5 + 2*a^4*b - a^3*b^2 - a^2*b^3 + a*b^4 + b^5 + 2*a^4*c + 2*a^3*b*c - a^2*b^2*c - a*b^3*c + b^4*c - a^3*c^2 - a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - a^2*c^3 - a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5 : :
X(44378) = 3 X[2] + X[19623]

X(44378) lies on these lines: {2, 6}, {115, 16702}, {523, 8043}, {536, 620}, {1931, 24957}, {3008, 40539}, {4665, 24384}, {8609, 34990}, {24636, 37756}, {36953, 42713}

X(44378) = midpoint of X(115) and X(16702)
X(44378) = crosssum of X(6) and X(5164)
X(44378) = crossdifference of every pair of points on line {512, 1030}
X(44378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 20086}, {39022, 39023, 81}

X(44379) = STEINER-INELLIPSE-INVERSE OF X(86)

Barycentrics    2*a^4 + 2*a^3*b - a^2*b^2 - a*b^3 + b^4 + 2*a^3*c + 2*a^2*b*c - a*b^2*c - b^3*c - a^2*c^2 - a*b*c^2 - 2*b^2*c^2 - a*c^3 - b*c^3 + c^4 : :
X(44379) = X[2] + X[17731], 9 X[2] - X[20536], 3 X[10026] - X[20536], 3 X[17731] + X[20536], 5 X[29590] - X[35148]

X(44379) lies on these lines: {2, 6}, {8, 24384}, {115, 6629}, {239, 24636}, {519, 620}, {523, 2487}, {540, 625}, {1834, 35916}, {1931, 23947}, {4062, 36953}, {4399, 24374}, {4434, 4478}, {4987, 25383}, {5209, 25472}, {6626, 23905}, {6682, 17045}, {6690, 17390}, {8608, 34990}, {11599, 17768}, {13174, 28530}, {17103, 23897}, {17290, 36223}, {17366, 35960}, {20530, 40546}, {24345, 34824}, {24366, 24883}, {29590, 35148}

X(44379) = midpoint of X(i) and X(j) for these {i,j}: {115, 6629}, {239, 35080}, {10026, 17731}
X(44379) = complement of X(10026)
X(44379) = X(i)-complementary conjugate of X(j) for these (i,j): {1333, 35114}, {28482, 1211}, {35162, 21245}
X(44379) = crosssum of X(6) and X(20666)
X(44379) = crossdifference of every pair of points on line {512, 18755}
X(44379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 17731, 10026}, {6189, 6190, 20090}, {39022, 39023, 86}

X(44380) = STEINER-INELLIPSE-INVERSE OF X(183)

Barycentrics    2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 + b^6 - 3*a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 + 2*a^2*c^4 + b^2*c^4 + c^6 : :
X(44380) = 3 X[2] + X[39099], X[2] + 3 X[41137], X[69] - 5 X[7925], X[148] + 3 X[12215], X[316] + 3 X[5182], X[385] - 5 X[3618], X[599] - 3 X[41133], X[1352] + 3 X[22525], 3 X[1692] + X[7845], 3 X[2456] + X[6033], X[5104] - 3 X[35297], 3 X[7799] + X[10754], X[11646] + 3 X[12151], X[11646] - 3 X[33228], X[15993] + 9 X[41137], X[15993] + 3 X[41146], X[39099] - 9 X[41137], X[39099] - 3 X[41146], 3 X[41137] - X[41146]

X(44380) lies on these lines: {2, 6}, {30, 5026}, {126, 13857}, {148, 12215}, {182, 7761}, {316, 5182}, {511, 620}, {523, 24284}, {542, 625}, {575, 626}, {576, 3788}, {732, 2025}, {754, 2030}, {1352, 22525}, {1503, 2456}, {1692, 7845}, {1990, 39931}, {2882, 35060}, {3001, 38987}, {3095, 7789}, {3564, 5031}, {3734, 5476}, {3934, 25555}, {4048, 5480}, {4364, 36405}, {5034, 7913}, {5038, 6656}, {5104, 35297}, {5111, 6393}, {5112, 32217}, {5969, 6390}, {6593, 34827}, {7745, 10349}, {7750, 39560}, {7799, 10754}, {7807, 13330}, {7830, 20190}, {7835, 22486}, {7862, 34507}, {8705, 16320}, {9132, 25328}, {9830, 37350}, {10256, 22677}, {11173, 11288}, {11245, 40379}, {11646, 12151}, {11676, 29181}, {15118, 40553}, {15483, 32459}, {18583, 24256}, {18800, 31173}, {19924, 32456}, {31636, 34369}, {40074, 41760}

X(44380) = midpoint of X(i) and X(j) for these {i,j}: {2, 41146}, {6, 325}, {5103, 13196}, {5111, 6393}, {12151, 33228}, {12177, 15980}, {15993, 39099}, {18800, 31173}
X(44380) = reflection of X(230) in X(3589)
X(44380) = complement of X(15993)
X(44380) = crosssum of X(6) and X(2021)
X(44380) = crossdifference of every pair of points on line {512, 5017}
X(44380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 39099, 15993}, {2, 41137, 41146}, {15483, 37461, 32459}, {15993, 41146, 39099}, {39022, 39023, 183}

X(44381) = STEINER-INELLIPSE-INVERSE OF X(193)

Barycentrics    6*a^4 - 5*a^2*b^2 + 5*b^4 - 5*a^2*c^2 - 6*b^2*c^2 + 5*c^4 : :
X(44381) = 3 X[2] + X[230], 9 X[2] - X[325], 15 X[2] + X[385], 33 X[2] - X[7779], 17 X[2] - X[7840], 21 X[2] - 5 X[7925], 13 X[2] + 3 X[8859], 27 X[2] + X[15480], 5 X[2] - X[22110], 7 X[2] + X[22329], 11 X[2] - 3 X[41133], 35 X[2] - 3 X[41136], X[2] + 3 X[41139], 3 X[230] + X[325], 5 X[230] - X[385], 11 X[230] + X[7779], 17 X[230] + 3 X[7840], 7 X[230] + 5 X[7925], 13 X[230] - 9 X[8859], 9 X[230] - X[15480], 5 X[230] + 3 X[22110], 7 X[230] - 3 X[22329], 11 X[230] + 9 X[41133], 35 X[230] + 9 X[41136], X[230] - 9 X[41139], 5 X[325] + 3 X[385], 11 X[325] - 3 X[7779], 17 X[325] - 9 X[7840], 7 X[325] - 15 X[7925], 13 X[325] + 27 X[8859], 3 X[325] + X[15480], 5 X[325] - 9 X[22110], 7 X[325] + 9 X[22329], 11 X[325] - 27 X[41133], 35 X[325] - 27 X[41136], X[325] + 27 X[41139], 11 X[385] + 5 X[7779], 17 X[385] + 15 X[7840], 7 X[385] + 25 X[7925], 13 X[385] - 45 X[8859], 9 X[385] - 5 X[15480], X[385] + 3 X[22110], 7 X[385] - 15 X[22329], 11 X[385] + 45 X[41133], 7 X[385] + 9 X[41136], X[385] - 45 X[41139], 5 X[620] - X[15301], 3 X[620] + X[32457], 5 X[631] + 3 X[39663], 7 X[3526] - 3 X[10256], 13 X[5067] + 3 X[21445], 9 X[5215] - X[6781], 3 X[5215] + X[37350], 3 X[5461] + X[32456], X[6390] - 5 X[31274], X[6781] + 3 X[37350], 17 X[7779] - 33 X[7840], 7 X[7779] - 55 X[7925], 13 X[7779] + 99 X[8859], 9 X[7779] + 11 X[15480], 5 X[7779] - 33 X[22110], 7 X[7779] + 33 X[22329], X[7779] - 9 X[41133], 35 X[7779] - 99 X[41136], X[7779] + 99 X[41139], 21 X[7840] - 85 X[7925], 13 X[7840] + 51 X[8859], 27 X[7840] + 17 X[15480], 5 X[7840] - 17 X[22110], 7 X[7840] + 17 X[22329], 11 X[7840] - 51 X[41133], 35 X[7840] - 51 X[41136], X[7840] + 51 X[41139], 65 X[7925] + 63 X[8859], 45 X[7925] + 7 X[15480], 25 X[7925] - 21 X[22110], 5 X[7925] + 3 X[22329], 55 X[7925] - 63 X[41133], 25 X[7925] - 9 X[41136], 5 X[7925] + 63 X[41139], 81 X[8859] - 13 X[15480], 15 X[8859] + 13 X[22110], 21 X[8859] - 13 X[22329], 11 X[8859] + 13 X[41133], 35 X[8859] + 13 X[41136], X[8859] - 13 X[41139], 5 X[14061] + 3 X[35297], 3 X[14971] + X[27088], 3 X[15301] + 5 X[32457], X[15301] + 5 X[43291], 5 X[15480] + 27 X[22110], 7 X[15480] - 27 X[22329], 11 X[15480] + 81 X[41133], 35 X[15480] + 81 X[41136], X[15480] - 81 X[41139], 7 X[22110] + 5 X[22329], 11 X[22110] - 15 X[41133], 7 X[22110] - 3 X[41136], X[22110] + 15 X[41139], 11 X[22329] + 21 X[41133], 5 X[22329] + 3 X[41136], X[22329] - 21 X[41139], X[32457] - 3 X[43291], 3 X[34127] + X[37459], 3 X[38227] + 5 X[40336], 35 X[41133] - 11 X[41136], X[41133] + 11 X[41139], X[41136] + 35 X[41139]

X(44381) lies on these lines: {2, 6}, {30, 6722}, {115, 32459}, {140, 4045}, {439, 39143}, {468, 40511}, {523, 14341}, {547, 7804}, {549, 7844}, {620, 15301}, {631, 39663}, {632, 7834}, {1503, 6036}, {3053, 32827}, {3526, 10256}, {3530, 7861}, {3564, 6721}, {3628, 6680}, {5013, 32977}, {5023, 32972}, {5066, 32414}, {5067, 21445}, {5210, 16041}, {5215, 6781}, {5254, 33233}, {5461, 32456}, {5585, 33272}, {6390, 31274}, {6683, 16239}, {6719, 22104}, {7612, 15069}, {7745, 7857}, {7746, 7789}, {7749, 7853}, {7784, 32955}, {7815, 33186}, {7851, 33000}, {7872, 15712}, {7887, 14907}, {7914, 33212}, {11288, 43620}, {11318, 21843}, {11539, 15482}, {13881, 32815}, {14061, 35297}, {14971, 27088}, {32817, 32959}, {32838, 33222}, {34127, 37459}, {34369, 40428}, {38227, 40336}, {40477, 40486}

X(44381) = midpoint of X(i) and X(j) for these {i,j}: {115, 32459}, {620, 43291}, {6036, 10011}
X(44381) = crosssum of X(6) and X(1570)
X(44381) = crossdifference of every pair of points on line {512, 5023}
X(44381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7792, 3055}, {2, 7806, 37647}, {2, 15597, 20582}, {2, 23053, 21358}, {2, 37637, 141}, {230, 22110, 385}, {385, 7925, 41136}, {7778, 13468, 3631}, {7806, 37647, 9300}, {7857, 33249, 7745}, {39022, 39023, 193}

X(44382) = STEINER-INELLIPSE-INVERSE OF X(298)

Barycentrics    Sqrt[3]*(b^2 + c^2) - 4*S : :
X(44382) = 3 X[2] + X[299], 9 X[2] - X[3181], 5 X[2] - X[37785], 15 X[2] + X[40899], X[14] - 5 X[40335], X[16] + 3 X[21360], X[298] - 5 X[7925], 3 X[299] + X[3181], 5 X[299] + 3 X[37785], 5 X[299] - X[40899], 3 X[395] - X[3181], 5 X[395] - 3 X[37785], 5 X[395] + X[40899], 5 X[1656] - X[20426], 5 X[3181] - 9 X[37785], 5 X[3181] + 3 X[40899], 3 X[5464] + X[36970], X[9115] - 5 X[31274], 3 X[14971] - X[22574], X[22998] - 5 X[36770], 3 X[31694] - X[36970], 3 X[35303] - X[36968], 3 X[37785] + X[40899]


X(44382) lies on these lines: {2, 6}, {5, 3642}, {13, 7789}, {14, 40335}, {15, 37351}, {16, 21360}, {30, 619}, {140, 618}, {383, 29181}, {397, 11308}, {398, 628}, {466, 34828}, {530, 620}, {531, 625}, {532, 6672}, {533, 6670}, {547, 623}, {549, 3643}, {617, 5321}, {622, 42943}, {626, 42912}, {629, 16239}, {633, 42599}, {634, 16773}, {635, 3628}, {641, 34551}, {642, 34552}, {1503, 5613}, {1656, 20426}, {3564, 6774}, {3788, 34509}, {3934, 6669}, {4399, 40713}, {4478, 40714}, {5031, 6109}, {5103, 5978}, {5237, 33386}, {5318, 11300}, {5464, 31694}, {6292, 22892}, {6673, 42590}, {6694, 7849}, {7761, 13083}, {7799, 40706}, {7862, 34508}, {7865, 42124}, {7880, 11542}, {8355, 33476}, {9115, 31274}, {10410, 33507}, {10653, 11302}, {10654, 11306}, {11085, 40710}, {11092, 11120}, {11121, 44032}, {11129, 22847}, {11132, 22893}, {11289, 42598}, {11290, 16772}, {11298, 18582}, {11301, 42092}, {11304, 42942}, {11305, 42911}, {11310, 42149}, {11312, 42152}, {14539, 41017}, {14905, 33228}, {14971, 22574}, {15685, 33618}, {16241, 37340}, {18586, 23311}, {18587, 23312}, {18840, 43554}, {20429, 44250}, {22511, 36763}, {22573, 39785}, {22845, 42993}, {22998, 36770}, {32459, 35303}, {32515, 33479}, {33560, 42146}, {33615, 33617}, {34827, 41888}, {37170, 42098}, {37171, 42154}, {37173, 42155}, {37177, 43238}, {37178, 42156}, {37352, 37832}, {38745, 41022}, {41887, 43962}

X(44382) = midpoint of X(i) and X(j) for these {i,j}: {299, 395}, {325, 396}, {617, 5321}, {619, 624}, {622, 42943}, {5464, 31694}, {10410, 33507}, {11132, 22893}, {14539, 41017}, {20429, 44250}, {22573, 39785}, {22797, 36756}, {41887, 43962}
X(44382) = reflection of X(i) in X(j) for these {i,j}: {11543, 6670}, {42913, 6672}, {43417, 33561}, {44383, 44377}
X(44382) = complement of X(395)
X(44382) = circumcircle-of-outer-Napoleon-triangle inverse of X(38940)
X(44382) = complement of the isogonal conjugate of X(6151)
X(44382) = complement of the isotomic conjugate of X(40706)
X(44382) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 22848}, {163, 35444}, {661, 15610}, {2154, 16537}, {6151, 10}, {10410, 4369}, {19301, 40580}, {38427, 20305}, {40706, 2887}
X(44382) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 22848}, {35315, 23871}
X(44382) = crosspoint of X(2) and X(40706)
X(44382) = crosssum of X(32) and X(11134)
X(44382) = crossdifference of every pair of points on line {512, 19780}
X(44382) = barycentric product X(i)*X(j) for these {i,j}: {16022, 34390}, {22848, 40706}
X(44382) = barycentric quotient X(i)/X(j) for these {i,j}: {16022, 62}, {22848, 395}
X(44382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 16645}, {2, 141, 44383}, {2, 298, 23303}, {2, 299, 395}, {2, 303, 396}, {2, 396, 3589}, {2, 599, 33474}, {2, 9763, 597}, {2, 34541, 298}, {69, 16645, 33459}, {299, 37785, 40899}, {630, 636, 140}, {3180, 43229, 32455}, {5859, 37641, 3629}, {6189, 6190, 40898}, {11092, 11120, 34326}, {15685, 33620, 33618}, {23303, 34541, 3631}, {39022, 39023, 298}

X(44383) = STEINER-INELLIPSE-INVERSE OF X(299)

Barycentrics    Sqrt[3]*(b^2 + c^2) + 4*S : :
X(44383) = 3 X[2] + X[298], 9 X[2] - X[3180], 5 X[2] - X[37786], 15 X[2] + X[40898], X[13] - 5 X[40334], X[15] + 3 X[21359], X[15] - 5 X[36770], 3 X[298] + X[3180], 5 X[298] + 3 X[37786], 5 X[298] - X[40898], X[299] - 5 X[7925], 3 X[396] - X[3180], 5 X[396] - 3 X[37786], 5 X[396] + X[40898], 5 X[1656] - X[20425], 5 X[3180] - 9 X[37786], 5 X[3180] + 3 X[40898], 3 X[5463] + X[36969], X[9117] - 5 X[31274], X[14538] + 3 X[36765], 3 X[14971] - X[22573], 3 X[21359] + 5 X[36770], 3 X[31693] - X[36969], 3 X[35304] - X[36967], 3 X[36765] - X[41016], 3 X[37786] + X[40898]

X(44383) lies on these lines: {2, 6}, {5, 3643}, {13, 40334}, {14, 7789}, {15, 21359}, {16, 37352}, {30, 618}, {140, 619}, {397, 627}, {398, 11307}, {465, 34828}, {530, 625}, {531, 620}, {532, 6669}, {533, 6671}, {547, 624}, {549, 3642}, {616, 5318}, {621, 42942}, {626, 42913}, {630, 16239}, {633, 16772}, {634, 42598}, {636, 3628}, {641, 34552}, {642, 34551}, {1080, 29181}, {1503, 5617}, {1656, 20425}, {3564, 6771}, {3788, 34508}, {3934, 6670}, {4399, 40714}, {4478, 40713}, {5031, 6108}, {5103, 5979}, {5238, 33387}, {5321, 11299}, {5463, 31693}, {6292, 22848}, {6674, 42591}, {6695, 7849}, {7761, 13084}, {7799, 40707}, {7862, 34509}, {7865, 42121}, {7880, 11543}, {8355, 33477}, {9117, 31274}, {10409, 33506}, {10653, 11305}, {10654, 11301}, {11078, 11119}, {11080, 40709}, {11122, 44030}, {11128, 22893}, {11133, 22847}, {11289, 16773}, {11290, 42599}, {11297, 18581}, {11302, 42089}, {11303, 42943}, {11306, 42910}, {11309, 42152}, {11311, 42149}, {14538, 36765}, {14904, 33228}, {14971, 22573}, {15685, 33619}, {16242, 37341}, {18586, 23312}, {18587, 23311}, {18840, 43555}, {22510, 42489}, {22574, 39785}, {22844, 42992}, {32459, 35304}, {32515, 33478}, {33561, 42143}, {33614, 33616}, {34827, 41887}, {37170, 42155}, {37171, 42095}, {37172, 42154}, {37177, 42153}, {37178, 43239}, {37351, 37835}, {38745, 41023}, {41888, 43961}

X(44383) = midpoint of X(i) and X(j) for these {i,j}: {298, 396}, {325, 395}, {616, 5318}, {618, 623}, {621, 42942}, {5463, 31693}, {10409, 33506}, {11133, 22847}, {14538, 41016}, {22574, 39785}, {22796, 36755}, {41888, 43961}
X(44383) = reflection of X(i) in X(j) for these {i,j}: {11542, 6669}, {42912, 6671}, {43416, 33560}, {44382, 44377}
X(44383) = complement of X(396)
X(44383) = circumcircle of inner Napoleon triangle inverse of X(38940)
X(44383) = complement of the isogonal conjugate of X(2981)
X(44383) = complement of the isotomic conjugate of X(40707)
X(44383) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 22892}, {163, 35443}, {661, 15609}, {2153, 16536}, {2981, 10}, {10409, 4369}, {19300, 40581}, {38428, 20305}, {40707, 2887}
X(44383) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 22892}, {35314, 23870}
X(44383) = crosspoint of X(2) and X(40707)
X(44383) = crosssum of X(32) and X(11137)
X(44383) = crossdifference of every pair of points on line {512, 19781}
X(44383) = barycentric product X(i)*X(j) for these {i,j}: {16021, 34389}, {22892, 40707}
X(44383) = barycentric quotient X(i)/X(j) for these {i,j}: {16021, 61}, {22892, 396}
X(44383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 16644}, {2, 141, 44382}, {2, 298, 396}, {2, 299, 23302}, {2, 302, 395}, {2, 395, 3589}, {2, 599, 33475}, {2, 9761, 597}, {2, 34540, 299}, {69, 16644, 33458}, {298, 37786, 40898}, {629, 635, 140}, {3181, 43228, 32455}, {5858, 37640, 3629}, {6189, 6190, 40899}, {11078, 11119, 34325}, {14538, 36765, 41016}, {15685, 33621, 33619}, {21359, 36770, 15}, {23302, 34540, 3631}, {39022, 39023, 299}

X(44384) = STEINER-INELLIPSE-INVERSE OF X(302)

Barycentrics    2*a^4 - 3*a^2*b^2 + 3*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 3*c^4 - 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(44384) = 3 X[299] + X[385], X[385] - 3 X[396], X[7779] + 3 X[37786], 3 X[21360] + X[22997]

X(44384) lies on these lines: {2, 6}, {30, 32552}, {530, 32456}, {532, 620}, {533, 625}, {1503, 22509}, {3564, 25560}, {3734, 34509}, {5026, 6783}, {6779, 32459}, {8598, 33376}, {13196, 22508}, {21360, 22997}, {32515, 33462}

X(44384) = midpoint of X(299) and X(396)
X(44384) = reflection of X(44385) in X(44377)
X(44384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 385, 44385}, {6189, 6190, 40900}, {39022, 39023, 302}

X(44385) = STEINER-INELLIPSE-INVERSE OF X(303)

Barycentrics    2*a^4 - 3*a^2*b^2 + 3*b^4 - 3*a^2*c^2 - 2*b^2*c^2 + 3*c^4 + 2*Sqrt[3]*(2*a^2 - b^2 - c^2)*S : :
X(44385) = 3 X[298] + X[385], X[385] - 3 X[395], X[7779] + 3 X[37785], 3 X[21359] + X[22998]

X(44385) lies on these lines: {2, 6}, {30, 32553}, {531, 32456}, {532, 625}, {533, 620}, {1503, 22507}, {3564, 25559}, {3734, 34508}, {5026, 6782}, {6780, 32459}, {8598, 33377}, {13196, 22506}, {21359, 22998}, {32515, 33463}

X(44385) = midpoint of X(298) and X(395)
X(44385) = reflection of X(44384) in X(44377)
X(44385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 385, 44384}, {6189, 6190, 40901}, {39022, 39023, 303}

X(44386) = STEINER-INELLIPSE-INVERSE OF X(323)

Barycentrics    2*a^8 - 5*a^6*b^2 + 5*a^4*b^4 - 3*a^2*b^6 + b^8 - 5*a^6*c^2 + 4*a^4*b^2*c^2 - 4*b^6*c^2 + 5*a^4*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + c^8 : :
X(44386) = 3 X[2] + X[40879], 7 X[3526] + X[36207], X[9142] - 5 X[38739], X[9145] + 3 X[38224]

X(44386) lies on these lines: {2, 6}, {140, 523}, {2854, 6036}, {3018, 34990}, {3526, 36207}, {3934, 36597}, {4558, 34989}, {6748, 14590}, {7495, 16320}, {7499, 16316}, {7789, 36953}, {7847, 40429}, {9142, 38739}, {9145, 38224}, {11594, 37459}, {19221, 33233}, {37283, 37451}

X(44386) = midpoint of X(18122) and X(40879)
X(44386) = complement of X(18122)
X(44386) = X(39448)-complementary conjugate of X(4369)
X(44386) = crosssum of X(6) and X(15544)
X(44386) = crossdifference of every pair of points on line {512, 11063}
X(44386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 24975, 3589}, {2, 40879, 18122}, {39022, 39023, 323}

X(44387) = STEINER-INELLIPSE-INVERSE OF X(333)

Barycentrics    2*a^5 - 3*a^3*b^2 + 2*a*b^4 + b^5 - 2*a^3*b*c - 2*a^2*b^2*c + a*b^3*c + 2*b^4*c - 3*a^3*c^2 - 2*a^2*b*c^2 - b^3*c^2 + a*b*c^3 - b^2*c^3 + 2*a*c^4 + 2*b*c^4 + c^5 : :
X(44387) = 3 X[2] + X[40882]

X(44387) lies on these lines: {2, 6}, {523, 8045}, {527, 620}, {662, 20337}, {1944, 35086}, {3834, 40539}, {4363, 24384}, {8607, 34990}, {23947, 24957}

X(44387) = midpoint of X(i) and X(j) for these {i,j}: {1944, 35086}, {10026, 19623}
X(44387) = crossdifference of every pair of points on line {512, 2305}
X(44387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 333}

X(44388) = STEINER-INELLIPSE-INVERSE OF X(343)

Barycentrics    a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - a^2*b^4*c^2 + 2*b^6*c^2 - 3*a^4*c^4 - a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 2*b^2*c^6 - c^8 : :
X(44388) = X[325] + 2 X[18122], 5 X[7925] - X[40888]

X(44388) lies on these lines: {2, 6}, {114, 2393}, {297, 11062}, {317, 42406}, {523, 2072}, {566, 41237}, {868, 22087}, {1513, 9019}, {2450, 3001}, {8553, 44128}, {16320, 37980}, {18531, 42353}, {37347, 41169}, {37459, 39231}, {40074, 40822}

X(44388) = reflection of X(39231) in X(37459)
X(44388) = crossdifference of every pair of points on line {512, 571}
X(44388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {34827, 34990, 297}, {39022, 39023, 343}

X(44389) = STEINER-INELLIPSE-INVERSE OF X(394)

Barycentrics    2*a^8 - 5*a^6*b^2 + 5*a^4*b^4 - 3*a^2*b^6 + b^8 - 5*a^6*c^2 + 6*a^4*b^2*c^2 - a^2*b^4*c^2 - 4*b^6*c^2 + 5*a^4*c^4 - a^2*b^2*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + c^8 : :
X(44389) = 3 X[2] + X[40888]

X(44389) lies on these lines: {2, 6}, {523, 7663}, {1990, 34990}, {2165, 3964}, {6036, 8681}, {9308, 42406}, {16310, 36212}, {16320, 16387}, {36953, 37778}

X(44389) = crossdifference of every pair of points on line {512, 1609}
X(44389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 394}

X(44390) = STEINER-INELLIPSE-INVERSE OF X(491)

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

X(44390) lies on these lines: {2, 6}, {620, 32419}, {625, 32421}, {639, 43121}, {641, 9738}, {1503, 6231}, {7751, 13880}, {7761, 41490}, {13748, 35944}, {14230, 23311}

X(44390) = relection of X(44391) in X(44377)
X(44390) = crossdifference of every pair of points on line {512, 12963}
X(44390) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 230, 44391}, {492, 13758, 591}, {39022, 39023, 491}

X(44391) = STEINER-INELLIPSE-INVERSE OF X(492)

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

X(44391) lies on these lines: {2, 6}, {620, 32421}, {625, 32419}, {640, 43120}, {642, 9739}, {1503, 6230}, {7751, 13921}, {7761, 41491}, {13749, 35945}, {14233, 23312}

X(44391) = relection of X(44390) in X(44377)
X(44391) = crossdifference of every pair of points on line {512, 12968}
X(44391) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 230, 44390}, {491, 13638, 1991}, {39022, 39023, 492}

X(44392) = STEINER-INELLIPSE-INVERSE OF X(590)

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

X(44392) lies on these lines: {2, 6}, {115, 32421}, {187, 13989}, {316, 19108}, {523, 17431}, {574, 41490}, {637, 12968}, {639, 5062}, {641, 1504}, {1503, 33430}, {3070, 6289}, {3071, 9733}, {3102, 7584}, {3564, 6230}, {6423, 13644}, {6560, 13771}, {7746, 13880}, {7799, 19109}, {12256, 13748}, {12305, 35945}, {12962, 39387}, {13677, 33342}, {13934, 13961}, {13935, 21445}, {13988, 18362}, {22594, 35684}, {22616, 42276}, {26288, 43448}, {32494, 39679}, {39663, 42262}, {40947, 44196}

X(44392) = relection of X(44394) in X(230)
X(44392) = crosssum of X(6) and X(2459)
X(44392) = crossdifference of every pair of points on line {371, 512}
X(44392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 325, 44394}, {6, 13847, 13758}, {1991, 37637, 590}, {3069, 5860, 7735}, {39022, 39023, 590}, {39107, 39108, 13637}

X(44393) = STEINER-INELLIPSE-INVERSE OF X(591)

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

X(44393) lies on these lines: {2, 6}, {3, 11157}, {3070, 35306}, {3933, 13921}, {12158, 13785}, {13665, 13669}, {32419, 32435}, {33184, 41491}, {35302, 44192}

X(44393) = reflection of X(44400) in X(2)
X(44393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 591}, {39107, 39108, 5860}

X(44394) = STEINER-INELLIPSE-INVERSE OF X(615)

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

X(44394) lies on these lines: {2, 6}, {115, 32419}, {187, 8997}, {316, 19109}, {523, 17432}, {574, 41491}, {638, 12963}, {640, 5058}, {642, 1505}, {1503, 33431}, {3070, 9732}, {3071, 6290}, {3103, 7583}, {3564, 6231}, {6424, 13763}, {6561, 13650}, {7746, 13921}, {7799, 19108}, {9540, 21445}, {12257, 13749}, {12306, 35944}, {12969, 39388}, {13797, 33343}, {13848, 18362}, {13882, 13903}, {22623, 35685}, {22645, 42275}, {26289, 43448}, {31454, 32497}, {39663, 42265}, {40947, 44199}

X(44394) = relection of X(44392) in X(230)
X(44394) = crosssum of X(6) and X(2460)
X(44394) = crossdifference of every pair of points on line {372, 512}
X(44394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 325, 44392}, {6, 13846, 13638}, {591, 37637, 615}, {3068, 5861, 7735}, {39022, 39023, 615}, {39107, 39108, 13757}

X(44395) = STEINER-INELLIPSE-INVERSE OF X(1007)

Barycentrics    2*a^6 - 9*a^4*b^2 + 8*a^2*b^4 - 5*b^6 - 9*a^4*c^2 + 8*a^2*b^2*c^2 + b^4*c^2 + 8*a^2*c^4 + b^2*c^4 - 5*c^6 : :
X(44395) = X[325] - 5 X[3620], 3 X[599] + X[15993], X[1992] - 3 X[41139], 3 X[21356] - X[22110], 9 X[21356] - X[39099], 3 X[22110] - X[39099]

X(44395) lies on these lines: {2, 6}, {542, 32459}, {620, 3564}, {1503, 38749}, {7789, 34507}, {14645, 43291}, {15069, 39647}, {41585, 41762}

X(44395) = midpoint of X(69) and X(230)
X(44395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15271, 21356, 141}, {39022, 39023, 1007}

X(44396) = STEINER-INELLIPSE-INVERSE OF X(1211)

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

X(44396) lies on these lines: {2, 6}, {75, 5949}, {115, 536}, {338, 3262}, {442, 28611}, {523, 1577}, {620, 16702}, {740, 20546}, {1030, 21287}, {1086, 20337}, {2092, 21245}, {3454, 25374}, {3695, 14873}, {3712, 33329}, {3729, 8818}, {3834, 17058}, {3842, 24348}, {3912, 8287}, {3943, 23947}, {3948, 18151}, {4665, 23897}, {4708, 6537}, {5164, 41179}, {5695, 37049}, {13466, 23992}, {15526, 35083}, {17318, 23903}, {17790, 35147}, {20654, 27697}, {20975, 35552}, {21076, 27691}, {23980, 35088}, {23991, 25357}, {24345, 33159}, {28654, 42710}, {30860, 41310}

X(44396) = midpoint of X(17790) and X(35147)
X(44396) = reflection of X(16702) in X(620)
X(44396) = complement of X(19623)
X(44396) = complement of the isotomic conjugate of X(11611)
X(44396) = isotomic conjugate of the isogonal conjugate of X(5164)
X(44396) = isotomic conjugate of the polar conjugate of X(424)
X(44396) = isotomic conjugate of antigonal conjugate of X(1029)
X(44396) = X(i)-complementary conjugate of X(j) for these (i,j): {798, 35079}, {2703, 4369}, {4079, 41179}, {11609, 21246}, {11611, 2887}, {17939, 21196}, {17946, 3741}, {17954, 3739}, {17961, 1125}, {17981, 34830}, {18002, 1086}, {18015, 116}, {35147, 42327}
X(44396) = X(i)-Ceva conjugate of X(j) for these (i,j): {35147, 523}, {43189, 525}
X(44396) = X(5164)-cross conjugate of X(424)
X(44396) = crosspoint of X(2) and X(11611)
X(44396) = crosssum of X(6) and X(5006)
X(44396) = crossdifference of every pair of points on line {512, 1333}
X(44396) = barycentric product X(i)*X(j) for these {i,j}: {69, 424}, {76, 5164}
X(44396) = barycentric quotient X(i)/X(j) for these {i,j}: {424, 4}, {5164, 6}
X(44396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6189, 6190, 2895}, {27556, 27688, 6}, {39022, 39023, 1211}, {39107, 39108, 31143}

X(44397) = STEINER-INELLIPSE-INVERSE OF X(1641)

Barycentrics    8*a^8 - 16*a^6*b^2 + 9*a^4*b^4 - a^2*b^6 + 2*b^8 - 16*a^6*c^2 + 30*a^4*b^2*c^2 - 15*a^2*b^4*c^2 - 7*b^6*c^2 + 9*a^4*c^4 - 15*a^2*b^2*c^4 + 18*b^4*c^4 - a^2*c^6 - 7*b^2*c^6 + 2*c^8 : :
X(44397) = X[671] + 3 X[4590], 2 X[671] - 3 X[31644], X[892] + 3 X[41134], 2 X[4590] + X[31644], X[8591] - 3 X[14588], 2 X[9164] + X[35087], 4 X[9165] - 3 X[23991], 3 X[9167] - X[23992], X[18823] + 3 X[31998], X[18823] - 6 X[36953], X[31998] + 2 X[36953]

X(44397) lies on these lines: {2, 6}, {99, 17948}, {523, 2482}, {543, 40553}, {671, 4590}, {892, 41134}, {6786, 9044}, {8591, 14588}, {8598, 34205}, {9165, 23991}, {9167, 23992}, {10717, 16092}, {14120, 31173}, {18823, 31998}

X(44397) = midpoint of X(i) and X(j) for these {i,j}: {2, 9182}, {99, 17948}, {2482, 35087}
X(44397) = reflection of X(2482) in X(9164)
X(44397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 1641}, {39107, 39108, 5468}

X(44398) = STEINER-INELLIPSE-INVERSE OF X(1648)

Barycentrics    (b - c)^2*(b + c)^2*(5*a^4 - 5*a^2*b^2 + 2*b^4 - 5*a^2*c^2 + b^2*c^2 + 2*c^4) : :
X(44398) = X[115] - 3 X[23991], 4 X[115] - 3 X[31644], X[892] - 5 X[14061], 2 X[9164] - 3 X[9167], 4 X[9165] - 3 X[14971], 4 X[9165] - X[35087], 3 X[9166] - X[17948], 3 X[9166] + X[18823], 3 X[14971] - X[35087], 3 X[23991] + X[23992], 4 X[23991] - X[31644], 4 X[23992] + 3 X[31644], X[35511] + 4 X[40511]

X(44398) lies on these lines: {2, 6}, {115, 523}, {125, 9193}, {526, 14113}, {542, 5915}, {620, 40486}, {892, 14061}, {2679, 9009}, {2872, 44011}, {4590, 36953}, {5914, 10418}, {5996, 8288}, {6722, 40553}, {6784, 9044}, {6791, 9209}, {9007, 41181}, {9012, 15630}, {9164, 9167}, {9165, 14971}, {9166, 17948}, {16320, 36168}, {17416, 35078}, {28209, 41180}, {35088, 35133}, {35511, 40511}, {39689, 41102}

X(44398) = midpoint of X(i) and X(j) for these {i,j}: {115, 23992}, {17948, 18823}
X(44398) = reflection of X(i) in X(j) for these {i,j}: {620, 40486}, {4590, 36953}, {40553, 6722}
X(44398) = complement of X(9182)
X(44398) = complement of the isotomic conjugate of X(9180)
X(44398) = X(i)-complementary conjugate of X(j) for these (i,j): {798, 35087}, {843, 4369}, {9180, 2887}, {18823, 42327}
X(44398) = X(i)-Ceva conjugate of X(j) for these (i,j): {9166, 8371}, {18823, 523}, {41134, 9168}
X(44398) = crosspoint of X(i) and X(j) for these (i,j): {2, 9180}, {523, 9164}, {9168, 41134}
X(44398) = crosssum of X(6) and X(9181)
X(44398) = crossdifference of every pair of points on line {512, 5467}
X(44398) = barycentric product X(i)*X(j) for these {i,j}: {115, 41134}, {523, 9168}
X(44398) = barycentric quotient X(i)/X(j) for these {i,j}: {9168, 99}, {41134, 4590}, {41177, 1641}
X(44398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5912, 230}, {230, 22110, 24855}, {9166, 18823, 17948}, {23991, 23992, 115}, {39022, 39023, 1648}

X(44399) = STEINER-INELLIPSE-INVERSE OF X(1654)

Barycentrics    2*a^4 - 2*a^3*b - 3*a^2*b^2 + a*b^3 + 3*b^4 - 2*a^3*c - 2*a^2*b*c + a*b^2*c + b^3*c - 3*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + a*c^3 + b*c^3 + 3*c^4 : :
X(44399) = 3 X[2] + X[10026], 9 X[2] - X[17731], 15 X[2] + X[20536], X[6629] - 5 X[31274], 3 X[10026] + X[17731], 5 X[10026] - X[20536], 5 X[17266] - X[35080], 5 X[17731] + 3 X[20536]

X(44399) lies on these lines: {2, 6}, {519, 6722}, {523, 25666}, {4062, 40511}, {6629, 31274}, {11599, 28530}, {17266, 35080}, {20337, 24617}, {24348, 31285}

X(44399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 1654}

X(44400) = STEINER-INELLIPSE-INVERSE OF X(1991)

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

X(44400) lies on these lines: {2, 6}, {3, 11158}, {3071, 35305}, {3933, 13880}, {12159, 13665}, {13785, 13789}, {32421, 32432}, {33184, 41490}, {35302, 44193}

X(44400) = reflection of X(44393) in X(2)
X(44400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39022, 39023, 1991}, {39107, 39108, 5861}

X(44401) = STEINER-INELLIPSE-INVERSE OF X(1992)

Barycentrics    10*a^4 - 7*a^2*b^2 + 7*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 7*c^4 : :
X(44401) = 5 X[2] - X[325], 7 X[2] + X[385], 17 X[2] - X[7779], 9 X[2] - X[7840], 13 X[2] - 5 X[7925], 5 X[2] + 3 X[8859], 13 X[2] + X[15480], 3 X[2] + X[22329], 7 X[2] - 3 X[41133], 19 X[2] - 3 X[41136], X[2] - 3 X[41139], X[115] + 3 X[5215], X[187] + 3 X[14971], 5 X[230] + X[325], 7 X[230] - X[385], 17 X[230] + X[7779], 9 X[230] + X[7840], 13 X[230] + 5 X[7925], 5 X[230] - 3 X[8859], 13 X[230] - X[15480], 3 X[230] + X[22110], 3 X[230] - X[22329], 7 X[230] + 3 X[41133], 19 X[230] + 3 X[41136], X[230] + 3 X[41139], 7 X[325] + 5 X[385], 17 X[325] - 5 X[7779], 9 X[325] - 5 X[7840], 13 X[325] - 25 X[7925], X[325] + 3 X[8859], 13 X[325] + 5 X[15480], 3 X[325] - 5 X[22110], 3 X[325] + 5 X[22329], 7 X[325] - 15 X[41133], 19 X[325] - 15 X[41136], X[325] - 15 X[41139], X[376] + 3 X[39663], 17 X[385] + 7 X[7779], 9 X[385] + 7 X[7840], 13 X[385] + 35 X[7925], 5 X[385] - 21 X[8859], 13 X[385] - 7 X[15480], 3 X[385] + 7 X[22110], 3 X[385] - 7 X[22329], X[385] + 3 X[41133], 19 X[385] + 21 X[41136], X[385] + 21 X[41139], X[671] + 3 X[35297], X[3793] + 5 X[31275], 5 X[5071] + 3 X[21445], 3 X[5215] - X[27088], X[6390] - 3 X[9167], 9 X[7779] - 17 X[7840], 13 X[7779] - 85 X[7925], 5 X[7779] + 51 X[8859], 13 X[7779] + 17 X[15480], 3 X[7779] - 17 X[22110], 3 X[7779] + 17 X[22329], 7 X[7779] - 51 X[41133], 19 X[7779] - 51 X[41136], X[7779] - 51 X[41139], 13 X[7840] - 45 X[7925], 5 X[7840] + 27 X[8859], 13 X[7840] + 9 X[15480], X[7840] - 3 X[22110], X[7840] + 3 X[22329], 7 X[7840] - 27 X[41133], 19 X[7840] - 27 X[41136], X[7840] - 27 X[41139], 25 X[7925] + 39 X[8859], 5 X[7925] + X[15480], 15 X[7925] - 13 X[22110], 15 X[7925] + 13 X[22329], 35 X[7925] - 39 X[41133], 95 X[7925] - 39 X[41136], 5 X[7925] - 39 X[41139], X[8352] - 5 X[14061], X[8352] + 3 X[26613], X[8598] + 3 X[9166], 39 X[8859] - 5 X[15480], 9 X[8859] + 5 X[22110], 9 X[8859] - 5 X[22329], 7 X[8859] + 5 X[41133], 19 X[8859] + 5 X[41136], X[8859] + 5 X[41139], 3 X[10256] - 5 X[15694], 5 X[14061] + 3 X[26613], 3 X[14971] - X[37350], 3 X[15480] + 13 X[22110], 3 X[15480] - 13 X[22329], 7 X[15480] + 39 X[41133], 19 X[15480] + 39 X[41136], X[15480] + 39 X[41139], 7 X[22110] - 9 X[41133], 19 X[22110] - 9 X[41136], X[22110] - 9 X[41139], 7 X[22329] + 9 X[41133], 19 X[22329] + 9 X[41136], X[22329] + 9 X[41139], 5 X[31274] - X[39785], X[32459] + 2 X[43291], X[37461] + 3 X[38224], 19 X[41133] - 7 X[41136], X[41133] - 7 X[41139], X[41136] - 19 X[41139]

X(44401) lies on these lines: {2, 6}, {30, 5461}, {98, 10153}, {114, 8787}, {115, 5215}, {140, 7817}, {187, 14971}, {376, 39663}, {381, 37809}, {468, 8754}, {538, 22247}, {542, 10011}, {543, 32459}, {549, 9734}, {671, 35297}, {858, 34989}, {1153, 4045}, {1503, 6055}, {2030, 19662}, {2549, 11157}, {3053, 32984}, {3291, 34990}, {3734, 16509}, {3793, 31275}, {3849, 6722}, {3934, 8365}, {5071, 21445}, {5077, 21843}, {5159, 9165}, {5475, 19661}, {5569, 7844}, {6390, 9167}, {6680, 8367}, {7612, 11180}, {7622, 15048}, {7746, 8369}, {7749, 8359}, {7810, 8361}, {7812, 33249}, {7829, 16239}, {7857, 8370}, {7886, 8360}, {7902, 14869}, {8176, 18907}, {8352, 14061}, {8366, 32832}, {8598, 9166}, {9607, 33000}, {10124, 32515}, {10162, 37454}, {10256, 15694}, {11159, 20112}, {13665, 13681}, {13785, 13801}, {13881, 32826}, {14726, 40478}, {15525, 35087}, {16092, 16320}, {19130, 32414}, {22331, 32988}, {23583, 37911}, {23967, 44216}, {29181, 38227}, {31274, 39785}, {32456, 36523}, {32457, 36521}, {32822, 34505}, {32824, 32970}, {37461, 38224}

X(44401) = midpoint of X(i) and X(j) for these {i,j}: {2, 230}, {115, 27088}, {187, 37350}, {2030, 19662}, {16092, 16320}, {22110, 22329}, {23967, 44216}, {32456, 36523}, {32457, 36521}
X(44401) = reflection of X(8355) in X(6722)
X(44401) = complement of X(22110)
X(44401) = crosssum of X(6) and X(5107)
X(44401) = crossdifference of every pair of points on line {512, 5210}
X(44401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 9771}, {2, 385, 41133}, {2, 5032, 34803}, {2, 7610, 141}, {2, 7735, 11184}, {2, 8859, 325}, {2, 8860, 11168}, {2, 9740, 37690}, {2, 11168, 20582}, {2, 22329, 22110}, {2, 23055, 599}, {2, 34229, 21358}, {2, 37637, 15597}, {2, 37689, 9770}, {115, 5215, 27088}, {187, 14971, 37350}, {230, 22110, 22329}, {230, 41139, 2}, {599, 23055, 13468}, {3055, 7806, 6329}, {5306, 11163, 20583}, {7735, 11184, 8584}, {7886, 34506, 8360}, {11159, 43620, 20112}, {13663, 13783, 1992}, {14061, 26613, 8352}, {39022, 39023, 1992}, {39107, 39108, 193}

X(44402) = X(2)X(94)∩X(69)X(5621)

Barycentrics    a^14 - 5*a^12*b^2 + 9*a^10*b^4 - 5*a^8*b^6 - 5*a^6*b^8 + 9*a^4*b^10 - 5*a^2*b^12 + b^14 - 5*a^12*c^2 + 15*a^10*b^2*c^2 - 23*a^8*b^4*c^2 + 30*a^6*b^6*c^2 - 27*a^4*b^8*c^2 + 11*a^2*b^10*c^2 - b^12*c^2 + 9*a^10*c^4 - 23*a^8*b^2*c^4 - 2*a^6*b^4*c^4 + 10*a^4*b^6*c^4 + 9*a^2*b^8*c^4 - 3*b^10*c^4 - 5*a^8*c^6 + 30*a^6*b^2*c^6 + 10*a^4*b^4*c^6 - 30*a^2*b^6*c^6 + 3*b^8*c^6 - 5*a^6*c^8 - 27*a^4*b^2*c^8 + 9*a^2*b^4*c^8 + 3*b^6*c^8 + 9*a^4*c^10 + 11*a^2*b^2*c^10 - 3*b^4*c^10 - 5*a^2*c^12 - b^2*c^12 + c^14 : :

See Antreas Hatzipolakis and Peter Moses, euclid 2094.

X(44402) lies on these lines: {2, 94}, {69, 5621}, {99, 40680}, {253, 35520}, {1494, 3926}, {6148, 10304}

X(44402) = isotomic conjugate of the isogonal conjugate of X(17838)
X(44402) = barycentric product X(76)*X(17838)
X(44402) = barycentric quotient X(17838)/X(6)

X(44403) = X(65)X(74)∩X(165)X(2836)

Barycentrics    a*(a^10*b - 2*a^9*b^2 - a^8*b^3 + 4*a^7*b^4 - 2*a^6*b^5 + 2*a^4*b^7 - 4*a^3*b^8 + a^2*b^9 + 2*a*b^10 - b^11 + a^10*c + 2*a^9*b*c - 3*a^7*b^3*c - 5*a^6*b^4*c - 7*a^5*b^5*c + 9*a^4*b^6*c + 15*a^3*b^7*c - 8*a^2*b^8*c - 7*a*b^9*c + 3*b^10*c - 2*a^9*c^2 + 7*a^6*b^3*c^2 - 2*a^5*b^4*c^2 + a^4*b^5*c^2 - 4*a^3*b^6*c^2 - 7*a^2*b^7*c^2 + 8*a*b^8*c^2 - b^9*c^2 - a^8*c^3 - 3*a^7*b*c^3 + 7*a^6*b^2*c^3 + 18*a^5*b^3*c^3 - 12*a^4*b^4*c^3 - 15*a^3*b^5*c^3 + 11*a^2*b^6*c^3 - 5*b^8*c^3 + 4*a^7*c^4 - 5*a^6*b*c^4 - 2*a^5*b^2*c^4 - 12*a^4*b^3*c^4 + 16*a^3*b^4*c^4 + 3*a^2*b^5*c^4 - 10*a*b^6*c^4 + 6*b^7*c^4 - 2*a^6*c^5 - 7*a^5*b*c^5 + a^4*b^2*c^5 - 15*a^3*b^3*c^5 + 3*a^2*b^4*c^5 + 14*a*b^5*c^5 - 2*b^6*c^5 + 9*a^4*b*c^6 - 4*a^3*b^2*c^6 + 11*a^2*b^3*c^6 - 10*a*b^4*c^6 - 2*b^5*c^6 + 2*a^4*c^7 + 15*a^3*b*c^7 - 7*a^2*b^2*c^7 + 6*b^4*c^7 - 4*a^3*c^8 - 8*a^2*b*c^8 + 8*a*b^2*c^8 - 5*b^3*c^8 + a^2*c^9 - 7*a*b*c^9 - b^2*c^9 + 2*a*c^10 + 3*b*c^10 - c^11) : :
X(44403) = (OI - 5*R^2)*X[65] + (3*OI - 11*R^2)*X[74]

See Antreas Hatzipolakis, Francisco Javier García Capitán and Peter Moses, euclid 2098 and euclid 2099 .

X(44403) lies on these lines: {65, 74}, {165, 2836}, {2771, 20417}, {2779, 15151}, {12041, 31793}

X(44404) = (name pending)

Barycentrics    a*(3*a^11*b - 3*a^10*b^2 - 9*a^9*b^3 + 9*a^8*b^4 + 6*a^7*b^5 - 6*a^6*b^6 + 6*a^5*b^7 - 6*a^4*b^8 - 9*a^3*b^9 + 9*a^2*b^10 + 3*a*b^11 - 3*b^12 + 3*a^11*c + 4*a^9*b^2*c - 26*a^7*b^4*c - 4*a^6*b^5*c + 24*a^5*b^6*c + 12*a^4*b^7*c - a^3*b^8*c - 12*a^2*b^9*c - 4*a*b^10*c + 4*b^11*c - 3*a^10*c^2 + 4*a^9*b*c^2 - 8*a^8*b^2*c^2 + 20*a^7*b^3*c^2 + 6*a^6*b^4*c^2 - 48*a^5*b^5*c^2 + 24*a^4*b^6*c^2 + 20*a^3*b^7*c^2 - 19*a^2*b^8*c^2 + 4*a*b^9*c^2 - 9*a^9*c^3 + 20*a^7*b^2*c^3 + 24*a^6*b^3*c^3 + 26*a^5*b^4*c^3 - 4*a^4*b^5*c^3 - 36*a^3*b^6*c^3 - 8*a^2*b^7*c^3 - a*b^8*c^3 - 12*b^9*c^3 + 9*a^8*c^4 - 26*a^7*b*c^4 + 6*a^6*b^2*c^4 + 26*a^5*b^3*c^4 - 52*a^4*b^4*c^4 + 26*a^3*b^5*c^4 + 10*a^2*b^6*c^4 - 26*a*b^7*c^4 + 27*b^8*c^4 + 6*a^7*c^5 - 4*a^6*b*c^5 - 48*a^5*b^2*c^5 - 4*a^4*b^3*c^5 + 26*a^3*b^4*c^5 + 40*a^2*b^5*c^5 + 24*a*b^6*c^5 + 8*b^7*c^5 - 6*a^6*c^6 + 24*a^5*b*c^6 + 24*a^4*b^2*c^6 - 36*a^3*b^3*c^6 + 10*a^2*b^4*c^6 + 24*a*b^5*c^6 - 48*b^6*c^6 + 6*a^5*c^7 + 12*a^4*b*c^7 + 20*a^3*b^2*c^7 - 8*a^2*b^3*c^7 - 26*a*b^4*c^7 + 8*b^5*c^7 - 6*a^4*c^8 - a^3*b*c^8 - 19*a^2*b^2*c^8 - a*b^3*c^8 + 27*b^4*c^8 - 9*a^3*c^9 - 12*a^2*b*c^9 + 4*a*b^2*c^9 - 12*b^3*c^9 + 9*a^2*c^10 - 4*a*b*c^10 + 3*a*c^11 + 4*b*c^11 - 3*c^12) : :

See Antreas Hatzipolakis and Peter Moses, euclid 2100.

X(44404) lies on these lines: { }

X(44405) = X(95)X(3549)∩X(454)X(3964)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2106.

X(44405) lies on these lines: {95, 3549}, {454, 3964}, {924, 10539}, {11585, 44156}, {13754, 16391}

X(44405) = X(158)-isoconjugate of X(3548)
X(44405) = barycentric quotient X(577)/X(3548)

X(44406) = X(5)X(13558)∩X(185)X(567)

Barycentrics    a^2 (a^20-5 a^18 b^2+9 a^16 b^4-6 a^14 b^6+6 a^6 b^14-9 a^4 b^16+5 a^2 b^18-b^20-5 a^18 c^2+20 a^16 b^2 c^2-26 a^14 b^4 c^2+6 a^12 b^6 c^2+9 a^10 b^8 c^2+5 a^8 b^10 c^2-20 a^6 b^12 c^2+16 a^4 b^14 c^2-6 a^2 b^16 c^2+b^18 c^2+9 a^16 c^4-26 a^14 b^2 c^4+21 a^12 b^4 c^4-14 a^8 b^8 c^4+24 a^6 b^10 c^4-9 a^4 b^12 c^4-14 a^2 b^14 c^4+9 b^16 c^4-6 a^14 c^6+6 a^12 b^2 c^6+2 a^8 b^6 c^6-10 a^6 b^8 c^6+4 a^4 b^10 c^6+28 a^2 b^12 c^6-24 b^14 c^6+9 a^10 b^2 c^8-14 a^8 b^4 c^8-10 a^6 b^6 c^8-4 a^4 b^8 c^8-13 a^2 b^10 c^8+24 b^12 c^8+5 a^8 b^2 c^10+24 a^6 b^4 c^10+4 a^4 b^6 c^10-13 a^2 b^8 c^10-18 b^10 c^10-20 a^6 b^2 c^12-9 a^4 b^4 c^12+28 a^2 b^6 c^12+24 b^8 c^12+6 a^6 c^14+16 a^4 b^2 c^14-14 a^2 b^4 c^14-24 b^6 c^14-9 a^4 c^16-6 a^2 b^2 c^16+9 b^4 c^16+5 a^2 c^18+b^2 c^18-c^20) : :

See Francisco Javier García Capitán, euclid 2114

X(44406) lies on these lines: {5,13558}, {185,567}, {3520,5667}, {7503,14687}

X(44407) = X(3)X(2918)∩X(4)X(569)

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

See Francisco Javier García Capitán, euclid 2114

X(44407) lies on these lines: {3,2918}, {4,569}, {5,13419}, {20,3410}, {22,18474}, {26,5449}, {30,511}, {51,7540}, {52,10116}, {113,3153}, {125,2070}, {182,11818}, {186,6699}, {265,5899}, {381,3796}, {382,1181}, {389,11819}, {427,18475}, {546,13470}, {858,14156}, {1112,11692}, {1147,9833}, {1209,7512}, {1216,12134}, {1495,2072}, {1568,7574}, {1658,20191}, {1853,14070}, {3146,12289}, {3529,12278}, {3627,12233}, {3818,7514}, {5157,18420}, {5446,6146}, {5448,6759}, {5462,6756}, {5576,6689}, {5654,11206}, {5943,13490}, {6240,10575}, {6288,13564}, {6688,23410}, {7387,9927}, {7391,13352}, {7502,21243}, {7530,18390}, {7544,13336}, {7556,23293}, {7575,20397}, {7576,9730}, {7687,11563}, {7689,14216}, {7706,18494}, {8717,14927}, {9306,14791}, {9909,14852}, {9934,19479}, {10024,11572}, {10096,20304}, {10112,10263}, {10201,23325}, {10282,13371}, {10313,15340}, {10625,14516}, {11202,18281}, {11264,14449}, {11381,18563}, {11799,13851}, {12038,23335}, {12046,23409}, {12102,15807}, {12107,13561}, {12162,12225}, {12370,13598}, {13445,13619}, {14118,18488}, {15072,18559}, {15761,18383}, {18324,23329}, {18396,18534}

X(44408) = CIRCUMCIRCLE-POLE OF SODDY LINE

Barycentrics    a^2*(b - c)*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + a^2*b*c + a*b^2*c - b^3*c - a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - b*c^3) : :
X(44408) = 3 X[4057] - 4 X[39200], 3 X[39199] - 2 X[39200]

X(44408) lies on these lines: {3, 514}, {6, 22090}, {36, 238}, {55, 4449}, {56, 663}, {58, 22154}, {523, 2071}, {650, 22091}, {664, 14723}, {958, 17072}, {1011, 4379}, {1376, 4147}, {1486, 14878}, {1491, 23361}, {1597, 39532}, {1946, 3669}, {2975, 21302}, {3433, 26721}, {3960, 22160}, {4083, 9441}, {4091, 8676}, {4191, 4893}, {4367, 8638}, {4724, 5204}, {4778, 39226}, {5267, 23789}, {6003, 23187}, {6545, 16064}, {8071, 21185}, {8641, 43932}, {9029, 22769}, {20834, 21204}, {20835, 21183}, {21118, 37564}, {21791, 33863}, {23394, 23867}, {23864, 43067}, {28209, 39478}, {28225, 39225}

X(44408) = reflection of X(i) in X(j) for these {i,j}: {3, 39476}, {3733, 23224}, {4057, 39199}
X(44408) = isogonal conjugate of isotomic conjugate X(46402)
X(44408) = isogonal conjugate of the anticomplement of X(14714)
X(44408) = X(658)-Ceva conjugate of X(6)
X(44408) = crosspoint of X(i) and X(j) for these (i,j): {58, 934}, {109, 3449}, {15380, 24016}
X(44408) = crosssum of X(i) and X(j) for these (i,j): {10, 3900}, {512, 1834}, {513, 11019}, {514, 21258}, {521, 34822}, {522, 2886}, {8058, 20307}
X(44408) = crossdifference of every pair of points on line {37, 800}
X(44408) = barycentric product X(i)*X(j) for these {i,j}: {513, 37659}, {658, 14714}, {905, 4219}
X(44408) = barycentric quotient X(i)/X(j) for these {i,j}: {4219, 6335}, {14714, 3239}, {37659, 668}

X(44409) = INCIRCLE-POLE OF EULER LINE

Barycentrics    (b - c)*(-2*a^4 + a^3*b + a^2*b^2 - a*b^3 + b^4 + a^3*c - a*b^2*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4) : :
X(44409) = 3 X[354] - 2 X[34954], X[656] - 3 X[11125], 2 X[656] - 3 X[41800]

X(44409) lies on these lines: {1, 523}, {57, 5583}, {354, 34954}, {425, 2501}, {447, 525}, {513, 5570}, {521, 7649}, {522, 905}, {656, 11125}, {676, 21189}, {900, 5533}, {1021, 6587}, {2815, 40467}, {2826, 43924}, {3667, 3676}, {3700, 16612}, {3738, 21179}, {3800, 4581}, {3900, 21186}, {4017, 34958}, {4885, 28590}, {6003, 7178}, {6362, 21173}, {6615, 11031}, {7655, 21188}, {8672, 39541}, {21180, 35057}, {21182, 43042}

X(44409) = reflection of X(i) in X(j) for these {i,j}: {1, 39540}, {905, 21172}, {4017, 34958}, {7655, 21188}, {10015, 7649}, {21189, 676}, {41800, 11125}
X(44409) = incircle-inverse of X(3109)
X(44409) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {28, 34548}, {112, 8055}, {162, 42020}, {8056, 13219}, {34080, 3151}, {38266, 39352}, {38828, 2897}
X(44409) = X(i)-complementary conjugate of X(j) for these (i,j): {56, 42425}, {604, 40622}, {6011, 1329}
X(44409) = crosspoint of X(7) and X(648)
X(44409) = crosssum of X(55) and X(647)
X(44409) = crossdifference of every pair of points on line {198, 2245}
X(44409) = barycentric product X(21452)*X(43728)

X(44410) = INCIRCLE-POLE OF BROCARD AXIS

Barycentrics    a^2*(b - c)*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - a*b^2*c - a^2*c^2 - a*b*c^2 - a*c^3 + c^4) : :
X(44410) = 3 X[354] - 2 X[34958]

X(44410) lies on these lines: {1, 512}, {65, 28473}, {354, 34958}, {513, 5570}, {520, 3737}, {654, 22160}, {663, 928}, {905, 8676}, {924, 2605}, {926, 1734}, {942, 7178}, {1210, 4129}, {1401, 38370}, {1737, 21051}, {2488, 4040}, {2774, 14838}, {3022, 7266}, {3309, 4897}, {3738, 4142}, {4017, 34954}, {4807, 31397}, {7215, 11918}, {7252, 8673}, {10122, 42325}, {17899, 21300}, {21259, 24353}, {22093, 42662}, {23599, 39790}, {24472, 40459}

X(44410) = reflection of X(i) in X(j) for these {i,j}: {1, 39541}, {4017, 34954}, {4040, 2488}, {7178, 942}
X(44410) = incircle-inverse of X(3110)
X(44410) = crosspoint of X(7) and X(110)
X(44410) = crosssum of X(55) and X(523)
X(44410) = crossdifference of every pair of points on line {2238, 2911}
X(44410) = barycentric product X(i)*X(j) for these {i,j}: {4556, 21947}, {4570, 23762}
X(44410) = barycentric quotient X(23762)/X(21207)

X(44411) = NINE-POINT-CIRCLE-POLE OF ANTI-ORTHIC AXIS

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

X(44411) lies on these lines: {2, 16678}, {5, 10}, {11, 42}, {12, 73}, {35, 37357}, {55, 1985}, {71, 1213}, {100, 14008}, {235, 1869}, {313, 325}, {427, 1826}, {442, 16828}, {523, 21318}, {692, 17188}, {908, 22275}, {1220, 14011}, {1746, 20986}, {2476, 19874}, {2486, 3914}, {3035, 37365}, {3120, 21936}, {3214, 7173}, {3293, 7741}, {3613, 15523}, {3816, 37355}, {3829, 4685}, {4193, 26115}, {4417, 11681}, {4651, 5233}, {4847, 22271}, {4884, 21080}, {5432, 37354}, {5433, 28268}, {5711, 19754}, {6690, 37370}, {7354, 28377}, {7951, 33111}, {8804, 23305}, {10592, 15666}, {10886, 30827}, {14321, 23301}, {14973, 15281}, {15974, 37619}, {17530, 19870}, {20486, 21025}, {20531, 21098}, {20718, 22000}, {21020, 21801}, {21028, 21920}, {21045, 21911}, {21051, 34964}, {21892, 21949}, {22301, 31330}, {30980, 32943}, {30981, 32918}, {31084, 37353}

X(44411) = reflection of X(5) in X(39505)
X(44411) = complement of X(16678)
X(44411) = nine-point-circle-inverse of X(38472)
X(44411) = medial isogonal conjugate of X(40600)
X(44411) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 40600}, {40516, 1213}
X(44411) = X(4559)-Ceva conjugate of X(523)
X(44411) = barycentric product X(i)*X(j) for these {i,j}: {1, 40564}, {10, 24220}, {14618, 23161}
X(44411) = barycentric quotient X(i)/X(j) for these {i,j}: {23161, 4558}, {24220, 86}, {40564, 75}
X(44411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 946, 22300}, {10, 39583, 5}, {1329, 2886, 5743}, {2051, 41797, 38472}, {2540, 2541, 2051}

X(44412) = NINE-POINT-CIRCLE-POLE OF GERGONNE LINE

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

X(44412) lies on these lines: {2, 1631}, {5, 516}, {11, 4336}, {141, 9018}, {427, 1826}, {523, 41007}, {673, 7678}, {674, 20305}, {857, 8053}, {858, 17134}, {1268, 33108}, {1486, 30808}, {1836, 2160}, {1953, 21045}, {2886, 27798}, {4364, 21536}, {6284, 27555}, {8068, 34845}, {8287, 21746}, {17243, 20531}, {17792, 21237}, {20470, 37050}, {21239, 40646}, {21252, 24220}, {21275, 30882}, {22277, 26012}

X(44412) = reflection of X(5) in X(39507)
X(44412) = complement of X(1631)
X(44412) = complement of the isogonal conjugate of X(7357)
X(44412) = medial isogonal conjugate of X(32664)
X(44412) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 32664}, {7087, 37}, {7096, 2}, {7213, 1}, {7357, 10}, {40145, 39}
X(44412) = crosssum of X(6) and X(40370)

X(44413) = 2ND-LEMOINE-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 12*a^4*b^2*c^2 - 8*b^6*c^2 + 6*a^4*c^4 + 14*b^4*c^4 - 4*a^2*c^6 - 8*b^2*c^6 + c^8) : :
X(44413) = 7 X[6] - X[33534], 4 X[6] - X[35237], 2 X[1351] + X[11472], 4 X[9818] - 3 X[32620], X[11477] + 2 X[31861], 4 X[21850] - X[40909], 4 X[33534] - 7 X[35237], X[33534] - 14 X[39522], X[35237] - 8 X[39522]

X(44413) lies on these lines: {2, 37483}, {3, 51}, {4, 155}, {5, 17811}, {6, 30}, {20, 34545}, {22, 15033}, {25, 13352}, {26, 11425}, {52, 1593}, {64, 6102}, {68, 1595}, {143, 9786}, {154, 7530}, {182, 35243}, {184, 18534}, {185, 37493}, {195, 5076}, {323, 3839}, {376, 5422}, {378, 3060}, {381, 394}, {382, 1181}, {389, 12085}, {399, 38335}, {427, 14852}, {511, 9813}, {539, 18440}, {541, 39562}, {546, 16266}, {549, 17825}, {550, 37514}, {567, 3796}, {568, 10605}, {569, 11414}, {576, 6000}, {578, 7387}, {1092, 7529}, {1112, 12302}, {1147, 1598}, {1154, 9972}, {1192, 11250}, {1199, 33703}, {1204, 19360}, {1216, 11479}, {1350, 7514}, {1351, 1597}, {1498, 3627}, {1511, 31860}, {1539, 17838}, {1596, 5654}, {1620, 10226}, {1657, 36753}, {1994, 3543}, {1995, 43574}, {2071, 11002}, {2323, 18540}, {2931, 15472}, {2935, 12236}, {3066, 37477}, {3088, 12359}, {3089, 9820}, {3146, 7592}, {3167, 18535}, {3357, 16625}, {3426, 38263}, {3431, 37939}, {3517, 12002}, {3534, 15038}, {3541, 41587}, {3545, 15066}, {3567, 11413}, {3830, 18445}, {3845, 15068}, {3853, 15811}, {5012, 12082}, {5054, 21970}, {5055, 13857}, {5064, 18474}, {5073, 14627}, {5093, 14915}, {5102, 5663}, {5198, 10539}, {5480, 18420}, {5504, 20772}, {5707, 37406}, {5889, 35502}, {5943, 37480}, {5944, 14528}, {5946, 37475}, {6247, 18951}, {6642, 10110}, {6644, 17810}, {6756, 12118}, {6800, 37925}, {6985, 36742}, {7391, 12022}, {7393, 15644}, {7395, 10625}, {7503, 37486}, {7506, 35602}, {7517, 19357}, {7526, 10263}, {7553, 19467}, {8909, 35764}, {9645, 19365}, {9714, 13367}, {9730, 9777}, {9781, 17928}, {9833, 43595}, {9909, 18475}, {10113, 17847}, {10116, 34780}, {10304, 15018}, {10323, 13434}, {10594, 34148}, {11003, 37945}, {11284, 14845}, {11403, 12160}, {11410, 32110}, {11426, 39568}, {11430, 14070}, {11432, 40647}, {11438, 21849}, {11439, 15801}, {11464, 13482}, {11591, 33537}, {11807, 12412}, {11819, 17845}, {12241, 14790}, {12295, 19504}, {13202, 19456}, {13292, 14216}, {13321, 18859}, {13336, 37198}, {13488, 19458}, {15019, 20791}, {15032, 15682}, {15037, 15681}, {15045, 43576}, {16473, 41869}, {16657, 18531}, {17702, 18494}, {17821, 37440}, {18281, 26958}, {18396, 31723}, {18400, 19149}, {18405, 44288}, {20424, 44271}, {21971, 34417}, {22115, 35259}, {22505, 39820}, {22515, 39849}, {23060, 32046}, {23335, 39571}, {25739, 31133}, {32276, 34802}, {36750, 37411}, {37784, 39588}, {39806, 39841}, {39809, 39839}, {39810, 39838}, {39812, 39835}

X(44413) = midpoint of X(1351) and X(1597)
X(44413) = reflection of X(i) in X(j) for these {i,j}: {6, 39522}, {1350, 7514}, {11472, 1597}, {18420, 5480}, {35243, 182}
X(44413) = 2nd-Lemoine-circle-inverse of X(16303)
X(44413) = X(18852)-Ceva conjugate of X(3)
X(44413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3527, 5462}, {4, 1993, 18451}, {4, 36747, 155}, {22, 15033, 37506}, {52, 1593, 12163}, {143, 12084, 9786}, {378, 3060, 37489}, {382, 36749, 1181}, {546, 16266, 17814}, {567, 12083, 3796}, {578, 13598, 7387}, {1595, 13142, 68}, {1993, 18451, 155}, {1994, 3543, 11456}, {3627, 12161, 1498}, {3853, 32139, 15811}, {7506, 37495, 35602}, {7517, 37472, 19357}, {7526, 10263, 17834}, {9777, 21312, 9730}, {10110, 13346, 6642}, {11403, 12160, 12162}, {15316, 22660, 155}, {17810, 37497, 6644}, {18451, 36747, 1993}, {36987, 43650, 3}

X(44414) = 2ND-LEMOINE-CIRCLE-POLE OF ANTI-ORTHIC AXIS

Barycentrics    a^2*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 4*a^2*b^2*c + 3*b^4*c - 2*a^3*c^2 - 4*a^2*b*c^2 + 2*a*b^2*c^2 - 4*b^3*c^2 - 2*a^2*c^3 - 4*b^2*c^3 + a*c^4 + 3*b*c^4 + c^5) : :
X(44414) = 4 X[6] - X[1480], X[1480] - 8 X[39523], 2 X[3751] + X[7986]

X(44414) lies on these lines: {1, 6883}, {3, 42}, {5, 33137}, {6, 517}, {10, 5707}, {31, 10679}, {40, 36742}, {43, 6911}, {55, 5398}, {56, 5399}, {58, 11248}, {65, 921}, {81, 5657}, {212, 1497}, {218, 6913}, {219, 9708}, {222, 36279}, {355, 5706}, {386, 11249}, {392, 10601}, {394, 3753}, {495, 37543}, {500, 5584}, {580, 10267}, {581, 1126}, {582, 2334}, {595, 37622}, {601, 35448}, {602, 16202}, {607, 7497}, {613, 5844}, {912, 3751}, {940, 26446}, {999, 1450}, {1006, 17018}, {1060, 41539}, {1069, 10573}, {1159, 23071}, {1191, 10222}, {1193, 10680}, {1201, 12001}, {1203, 7982}, {1385, 36745}, {1482, 16466}, {1616, 33179}, {1656, 24892}, {1834, 10526}, {1870, 7672}, {2003, 2093}, {2323, 9623}, {3193, 5554}, {3240, 6905}, {3293, 11499}, {3428, 5396}, {3526, 29678}, {3560, 5247}, {3576, 16474}, {3579, 36746}, {3877, 5422}, {3914, 37826}, {3915, 12000}, {3931, 26921}, {4252, 26285}, {4255, 26286}, {4295, 8757}, {4383, 5886}, {4663, 6001}, {5173, 37697}, {5312, 11012}, {5315, 16200}, {5603, 32911}, {5690, 5711}, {5697, 16472}, {5790, 21696}, {5903, 16473}, {6583, 17054}, {6829, 33139}, {6830, 33142}, {6882, 11269}, {6985, 37699}, {7074, 24929}, {10247, 16483}, {10982, 12672}, {11230, 37679}, {11231, 37674}, {11929, 21935}, {12702, 36750}, {18481, 37537}, {25413, 36749}, {28466, 42042}, {31663, 37501}, {31786, 37514}, {31788, 37498}, {34046, 37582}, {34048, 39542}, {35238, 37469}, {36747, 37562}

X(44414) = reflection of X(6) in X(39523)
X(44414) = crossdifference of every pair of points on line {9001, 13401}
X(44414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1482, 37509, 16466}, {3293, 37530, 11499}, {5247, 37529, 3560}, {35774, 35775, 21853}

X(44415) = 2ND-LEMOINE-CIRCLE-POLE OF DE LONGCHAMPS AXIS

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

X(44415) lies on these lines: {3, 20859}, {4, 34945}, {6, 30}, {323, 33251}, {394, 33184}, {1003, 5422}, {1180, 15033}, {1184, 37483}, {1194, 13352}, {1501, 12083}, {1915, 7530}, {1993, 7841}, {1994, 33017}, {3094, 7514}, {3981, 6644}, {5017, 13391}, {5028, 13754}, {5254, 23128}, {6800, 37902}, {8360, 17811}, {8368, 17825}, {8369, 10601}, {9465, 43574}, {11004, 33278}, {15018, 33255}, {15045, 39024}, {15066, 33219}, {33007, 34545}, {34481, 43586}, {35243, 40825}

X(44415) = reflection of X(6) in X(39524)

X(44416) = SPIEKER-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    2*a^3 - a*b^2 + b^3 + b^2*c - a*c^2 + b*c^2 + c^3 : :
X(44416) = 3 X[2] + X[32933], 3 X[31] + X[32854], X[31] + 3 X[33161], 3 X[3703] - X[32854], X[3703] - 3 X[33161], X[3744] - 3 X[35263], 5 X[31237] - X[33098], X[32854] - 9 X[33161], X[32912] + 3 X[33156], X[32929] + 3 X[33114]

X(44416) lies on these lines: {1, 4884}, {2, 45}, {6, 345}, {8, 3052}, {9, 5743}, {10, 30}, {11, 32930}, {31, 3703}, {37, 6703}, {38, 29686}, {42, 3712}, {44, 3687}, {55, 33163}, {57, 17279}, {58, 3695}, {63, 141}, {71, 15985}, {81, 17390}, {100, 33166}, {171, 3932}, {226, 17351}, {238, 33167}, {306, 524}, {312, 17340}, {321, 35466}, {333, 594}, {344, 37674}, {346, 37642}, {527, 20106}, {528, 29673}, {536, 40940}, {537, 29656}, {553, 3834}, {597, 5256}, {726, 6679}, {846, 4026}, {894, 17056}, {896, 15523}, {902, 4030}, {940, 17243}, {958, 5835}, {1018, 18163}, {1125, 39544}, {1211, 3219}, {1213, 19808}, {1215, 6690}, {1329, 30448}, {1376, 24320}, {1621, 33170}, {1707, 3416}, {1757, 33160}, {1762, 24335}, {1812, 17796}, {1817, 30906}, {1834, 7283}, {1836, 29857}, {1999, 3943}, {2194, 17977}, {2308, 32848}, {2325, 35652}, {2345, 5273}, {2886, 3923}, {2887, 17768}, {3058, 33120}, {3159, 6693}, {3187, 4971}, {3210, 17366}, {3218, 33157}, {3550, 33165}, {3589, 3666}, {3631, 4001}, {3685, 33121}, {3704, 5247}, {3705, 4676}, {3710, 37539}, {3729, 3772}, {3744, 9053}, {3752, 17353}, {3769, 3790}, {3771, 32935}, {3816, 4011}, {3826, 3980}, {3914, 28530}, {3925, 4418}, {3928, 17284}, {3929, 4643}, {3938, 9041}, {3969, 16704}, {3994, 29683}, {4028, 4663}, {4046, 32864}, {4062, 4722}, {4078, 4682}, {4104, 15481}, {4141, 29819}, {4234, 16086}, {4358, 37634}, {4376, 5845}, {4383, 17740}, {4387, 11269}, {4395, 26723}, {4414, 26061}, {4417, 17350}, {4427, 4972}, {4428, 36479}, {4432, 29655}, {4445, 14552}, {4450, 31079}, {4650, 29674}, {4664, 29841}, {4665, 5271}, {4672, 29671}, {4697, 29653}, {4854, 29631}, {4918, 17016}, {4957, 20919}, {4966, 32913}, {5057, 29872}, {5222, 42049}, {5235, 24146}, {5241, 27065}, {5249, 7228}, {5432, 32931}, {5433, 25591}, {5695, 33137}, {5718, 26223}, {5745, 17355}, {5852, 33064}, {5905, 30811}, {6057, 17763}, {6284, 36568}, {6542, 41629}, {6682, 24295}, {6691, 25079}, {7227, 31993}, {7232, 9965}, {7262, 32778}, {7263, 24789}, {7277, 17778}, {7789, 25083}, {8369, 30108}, {8616, 33169}, {9021, 40959}, {9776, 17265}, {10327, 37540}, {11246, 25957}, {11679, 17281}, {12572, 39559}, {13742, 17054}, {14555, 16885}, {14829, 17280}, {16468, 32855}, {16579, 24036}, {17022, 41313}, {17045, 28606}, {17126, 32862}, {17127, 33089}, {17140, 24542}, {17165, 17724}, {17233, 37683}, {17246, 19786}, {17247, 19812}, {17248, 19827}, {17267, 18141}, {17269, 34255}, {17276, 25527}, {17289, 38000}, {17314, 37666}, {17334, 27184}, {17337, 19804}, {17339, 18743}, {17352, 17490}, {17356, 24177}, {17362, 21793}, {17365, 18134}, {17484, 30831}, {17526, 37549}, {17594, 38047}, {17596, 33159}, {17602, 32925}, {17775, 30834}, {19732, 19822}, {20017, 28337}, {20292, 29873}, {21000, 35261}, {24358, 36483}, {24443, 25992}, {24851, 36499}, {24943, 36263}, {25453, 32934}, {25557, 29642}, {26685, 37679}, {27064, 32851}, {27539, 34524}, {28333, 32859}, {29632, 32940}, {29672, 42055}, {29846, 32938}, {29850, 32845}, {29856, 33154}, {29858, 33103}, {29861, 33095}, {29862, 33097}, {29865, 32856}, {29867, 33145}, {31237, 33098}, {32911, 33168}, {32912, 33156}, {32929, 33114}, {32932, 33118}, {33117, 34612}, {37759, 41806}

X(44416) = midpoint of X(i) and X(j) for these {i,j}: {31, 3703}, {306, 4641}, {3782, 32933}, {19542, 21375}
X(44416) = reflection of X(i) in X(j) for these {i,j}: {17061, 6679}, {39544, 1125}
X(44416) = complement of X(3782)
X(44416) = Spieker-circle-inverse of X(16305)
X(44416) = complement of the isotomic conjugate of X(2985)
X(44416) = isotomic conjugate of the polar conjugate of X(12135)
X(44416) = X(i)-complementary conjugate of X(j) for these (i,j): {2985, 2887}, {3450, 142}
X(44416) = crosspoint of X(2) and X(2985)
X(44416) = crosssum of X(6) and X(17053)
X(44416) = barycentric product X(69)*X(12135)
X(44416) = barycentric quotient X(12135)/X(4)
X(44416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 190, 4415}, {2, 32933, 3782}, {2, 32939, 1086}, {2, 40688, 40480}, {31, 33161, 3703}, {63, 32777, 141}, {171, 33164, 3932}, {345, 26065, 6}, {846, 32780, 4026}, {894, 33116, 17056}, {902, 33162, 4030}, {940, 17776, 17243}, {1211, 3219, 17332}, {1999, 42033, 3943}, {2325, 39595, 35652}, {2345, 5273, 5737}, {3219, 32779, 1211}, {3666, 5294, 3589}, {3923, 4438, 2886}, {3977, 5294, 3666}, {4418, 33115, 3925}, {17340, 37646, 312}, {26223, 33113, 5718}, {26723, 42051, 4395}, {27064, 32851, 37662}, {29631, 32936, 4854}, {32913, 33158, 4966}, {32930, 33119, 11}

X(44417) = SPIEKER-CIRCLE-POLE OF ANTI-ORTHIC AXIS

Barycentrics    a^2*b + a*b^2 + a^2*c + 2*b^2*c + a*c^2 + 2*b*c^2 : :
X(44417) = 3 X[2] + X[321], 9 X[2] - X[17147], X[38] - 5 X[31241], X[42] - 5 X[31264], 3 X[321] + X[17147], 5 X[1698] - X[4424], 3 X[3666] - X[17147], X[3706] + 5 X[31264], 3 X[3740] - X[22325], 3 X[3971] + X[36862], X[11611] - 5 X[14061]

X(44417) lies on these lines: {1, 3714}, {2, 37}, {5, 10}, {6, 11679}, {8, 4849}, {9, 1764}, {38, 28582}, {42, 3706}, {43, 3696}, {44, 333}, {45, 30568}, {55, 29828}, {57, 4363}, {63, 17351}, {72, 10479}, {141, 226}, {190, 38000}, {210, 31330}, {220, 27411}, {241, 16720}, {274, 16736}, {306, 5718}, {329, 4643}, {354, 30942}, {386, 5295}, {518, 1215}, {553, 7228}, {594, 3687}, {712, 3934}, {726, 6682}, {740, 6685}, {756, 4009}, {824, 4885}, {894, 14829}, {899, 3725}, {908, 1211}, {910, 26244}, {940, 4670}, {958, 37415}, {964, 37539}, {965, 27413}, {966, 18228}, {968, 4387}, {975, 2049}, {982, 29827}, {984, 3967}, {1089, 19863}, {1100, 1999}, {1104, 13740}, {1107, 21902}, {1125, 17061}, {1150, 4641}, {1155, 4418}, {1279, 3757}, {1376, 3185}, {1386, 4362}, {1465, 6358}, {1698, 4424}, {1766, 2050}, {1836, 26034}, {1861, 37362}, {2176, 4384}, {2352, 16405}, {2783, 3035}, {2887, 3838}, {2999, 4361}, {3008, 6704}, {3120, 32781}, {3212, 5226}, {3219, 41242}, {3305, 19732}, {3416, 26098}, {3589, 9022}, {3619, 26132}, {3634, 17070}, {3661, 4417}, {3683, 32917}, {3689, 32945}, {3700, 30864}, {3702, 26115}, {3703, 29639}, {3704, 5530}, {3723, 34064}, {3742, 3840}, {3744, 24552}, {3745, 17763}, {3748, 32943}, {3750, 4702}, {3758, 37683}, {3763, 25527}, {3773, 29671}, {3782, 4054}, {3812, 3831}, {3823, 3925}, {3834, 5249}, {3848, 4871}, {3911, 7227}, {3912, 17056}, {3923, 4640}, {3944, 32784}, {3952, 4981}, {3971, 36862}, {3982, 7238}, {3989, 3994}, {3999, 17140}, {4001, 4715}, {4003, 17155}, {4011, 15254}, {4026, 24210}, {4031, 7231}, {4035, 29594}, {4113, 21805}, {4357, 4415}, {4364, 4656}, {4383, 5271}, {4426, 41236}, {4429, 21949}, {4519, 32915}, {4644, 37655}, {4654, 7232}, {4663, 32853}, {4675, 18141}, {4682, 29649}, {4689, 32929}, {4690, 5739}, {4708, 25345}, {4717, 4868}, {4851, 5712}, {4852, 5256}, {4883, 29824}, {4906, 29668}, {4914, 32844}, {4970, 28484}, {4972, 26251}, {5057, 33083}, {5219, 17293}, {5235, 27065}, {5241, 28633}, {5263, 7081}, {5287, 19701}, {5294, 5830}, {5296, 8055}, {5333, 17021}, {5435, 7229}, {5437, 25590}, {5461, 27076}, {5695, 17594}, {5745, 17355}, {5749, 37642}, {5750, 6703}, {5905, 17345}, {5955, 26364}, {6048, 31327}, {6533, 19847}, {6535, 29688}, {6557, 9780}, {6679, 24295}, {6706, 17284}, {7222, 21454}, {7263, 24177}, {7283, 19270}, {7308, 17259}, {8727, 12618}, {9478, 26582}, {10129, 25958}, {10436, 30567}, {11354, 37817}, {11611, 14061}, {11680, 29667}, {13741, 16817}, {13881, 17308}, {14534, 19623}, {14555, 17275}, {15523, 33105}, {15569, 43223}, {15668, 17022}, {16669, 37652}, {16700, 30599}, {16826, 25130}, {17019, 37869}, {17120, 41629}, {17122, 24342}, {17231, 18134}, {17237, 27184}, {17251, 31142}, {17265, 41867}, {17272, 28609}, {17285, 41878}, {17290, 23681}, {17316, 24656}, {17317, 26109}, {17344, 33066}, {17369, 37646}, {17374, 17778}, {17381, 29841}, {17592, 29825}, {17599, 29826}, {17605, 25760}, {17717, 32778}, {17718, 33171}, {17719, 32783}, {17722, 32866}, {17723, 33088}, {17889, 33174}, {18142, 26563}, {19684, 37595}, {19742, 41241}, {20292, 33086}, {20317, 28468}, {20436, 26035}, {20593, 22230}, {21071, 29571}, {21241, 28595}, {21242, 29673}, {21342, 24349}, {21611, 27346}, {21857, 34258}, {24003, 27798}, {24178, 25914}, {24627, 32939}, {24725, 33080}, {24892, 26061}, {24943, 33127}, {25107, 39143}, {25109, 29610}, {25383, 30566}, {25502, 40328}, {25591, 25917}, {26105, 39581}, {27483, 32011}, {28107, 30617}, {28570, 41011}, {29633, 33135}, {29637, 33130}, {29640, 33158}, {29650, 32921}, {29651, 42819}, {29652, 32920}, {29657, 33092}, {29659, 33141}, {29663, 33128}, {29664, 32862}, {29670, 32941}, {29674, 33111}, {29676, 33169}, {29678, 33156}, {29679, 33108}, {29680, 33089}, {29690, 33162}, {29857, 31245}, {30811, 31266}, {30823, 30831}, {30824, 30852}, {31019, 33172}, {31053, 32782}, {32780, 33140}, {32914, 32944}, {33074, 33104}, {33075, 33107}, {33076, 33106}, {33078, 33112}, {33079, 33109}, {33082, 33096}, {33085, 33097}, {33137, 38047}, {33138, 33159}, {37594, 43531}

X(44417) = midpoint of X(i) and X(j) for these {i,j}: {42, 3706}, {321, 3666}, {1215, 3741}, {4717, 4868}
X(44417) = complement of X(3666)
X(44417) = Spieker-circle-inverse of X(38472)
X(44417) = complement of the isogonal conjugate of X(2298)
X(44417) = complement of the isotomic conjugate of X(30710)
X(44417) = X(i)-complementary conjugate of X(j) for these (i,j): {649, 15611}, {961, 142}, {1169, 1125}, {1220, 141}, {1240, 626}, {1791, 18589}, {1919, 39015}, {2298, 10}, {2359, 3}, {2363, 3739}, {4581, 116}, {6648, 17072}, {8687, 522}, {8707, 3835}, {14534, 3741}, {14624, 3454}, {30710, 2887}, {31643, 17046}, {32736, 514}, {36098, 4885}, {36147, 513}, {40454, 12610}
X(44417) = X(10455)-Ceva conjugate of X(10459)
X(44417) = crosspoint of X(2) and X(30710)
X(44417) = crosssum of X(6) and X(2300)
X(44417) = barycentric product X(i)*X(j) for these {i,j}: {10, 10455}, {75, 10459}, {313, 10457}, {3596, 10475}
X(44417) = barycentric quotient X(i)/X(j) for these {i,j}: {10455, 86}, {10457, 58}, {10459, 1}, {10475, 56}
X(44417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 75, 3752}, {2, 312, 37}, {2, 321, 3666}, {2, 4359, 16610}, {2, 4671, 28606}, {2, 17280, 33116}, {2, 19786, 17384}, {2, 19804, 16602}, {2, 24789, 17356}, {2, 26591, 25091}, {2, 28605, 4850}, {2, 31025, 4359}, {2, 31993, 3739}, {2, 37759, 19786}, {9, 18229, 5737}, {10, 3452, 5743}, {37, 312, 35652}, {192, 42034, 22034}, {333, 27064, 44}, {594, 37662, 3687}, {960, 5836, 34434}, {1150, 26223, 4641}, {2887, 25385, 3838}, {3175, 28606, 4681}, {3210, 42029, 4686}, {3452, 6708, 34852}, {3702, 26115, 37548}, {3757, 32942, 1279}, {3838, 3844, 2887}, {3840, 24325, 3742}, {3923, 32916, 4640}, {4362, 25496, 1386}, {4383, 5271, 17348}, {4418, 32918, 1155}, {4519, 37593, 32915}, {4671, 28606, 3175}, {4688, 16602, 19804}, {4850, 28605, 42051}, {5044, 39564, 10}, {5287, 19701, 28639}, {5712, 34255, 4851}, {5750, 39595, 6703}, {6535, 29688, 32848}, {9780, 19582, 31359}, {10436, 30567, 37674}, {16610, 31025, 4739}, {17763, 32772, 3745}, {24552, 26227, 3744}, {25591, 31339, 25917}, {28605, 42051, 4726}, {30818, 31993, 2}, {30942, 32771, 354}, {31330, 32931, 210}, {32917, 32930, 3683}, {33066, 37653, 17344}

X(44418) = SPIEKER-CIRCLE-POLE OF LEMOINE AXIS

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

X(44418) lies on these lines: {2, 32017}, {10, 511}, {39, 3596}, {75, 1574}, {192, 30819}, {313, 538}, {2092, 17790}, {2234, 21022}, {2277, 4494}, {3264, 6683}, {3664, 25102}, {3739, 9055}, {3948, 26764}, {3963, 27102}, {4021, 20530}, {4033, 26979}, {4357, 27076}, {4967, 29991}, {6007, 21238}, {8891, 31130}, {17049, 24327}, {17787, 21796}, {27042, 29388}, {27274, 28653}, {28350, 29381}, {28369, 29511}

X(44418) = {X(3596),X(26042)}-harmonic conjugate of X(39)

X(44419) = SPIEKER-CIRCLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    2*a^3 - 2*a^2*b - a*b^2 - b^3 - 2*a^2*c - b^2*c - a*c^2 - b*c^2 - c^3 : :
X(44419) = 3 X[2] + X[4450]

X(44419) lies on these lines: {2, 3052}, {8, 4884}, {10, 30}, {11, 32918}, {20, 5793}, {31, 3589}, {38, 4030}, {40, 5835}, {42, 524}, {55, 141}, {100, 1211}, {171, 4026}, {200, 4643}, {210, 17332}, {306, 4689}, {321, 28530}, {511, 22325}, {528, 3741}, {594, 32932}, {612, 4364}, {752, 6685}, {846, 3932}, {899, 41002}, {902, 32781}, {958, 37331}, {966, 42316}, {968, 17243}, {1086, 3757}, {1213, 17735}, {1215, 17768}, {1376, 4192}, {1503, 37619}, {1621, 33086}, {1707, 38047}, {1836, 29828}, {2177, 3631}, {2308, 6329}, {2345, 9778}, {2550, 5737}, {2886, 4660}, {2887, 6690}, {3035, 3846}, {3058, 30942}, {3158, 17272}, {3256, 26942}, {3416, 17594}, {3434, 37660}, {3474, 4363}, {3475, 7232}, {3550, 32784}, {3578, 19998}, {3666, 5846}, {3683, 4422}, {3703, 4414}, {3712, 15523}, {3740, 29349}, {3745, 17045}, {3750, 4966}, {3752, 3883}, {3763, 21000}, {3782, 26227}, {3821, 17061}, {3925, 32917}, {3974, 17262}, {3996, 37653}, {4046, 4478}, {4061, 4690}, {4388, 37662}, {4389, 37671}, {4395, 32914}, {4399, 32860}, {4415, 7081}, {4416, 4849}, {4418, 7227}, {4419, 7172}, {4425, 4434}, {4472, 4797}, {4512, 17279}, {4514, 24627}, {4553, 40966}, {4645, 17056}, {4650, 29659}, {4655, 29670}, {4657, 5269}, {4722, 4831}, {4851, 37553}, {4854, 17763}, {4972, 35466}, {4995, 29846}, {5264, 13728}, {5432, 25760}, {5718, 6327}, {6007, 14973}, {6057, 32936}, {6154, 32945}, {6682, 17766}, {7228, 11246}, {7238, 33067}, {7789, 37586}, {8013, 21047}, {8616, 33174}, {16052, 17734}, {17147, 28472}, {17184, 17724}, {17246, 32926}, {17259, 26040}, {17305, 29838}, {17334, 32937}, {17390, 37593}, {17398, 21793}, {17593, 32866}, {17596, 33076}, {17601, 32778}, {17602, 32776}, {17747, 26244}, {17792, 22276}, {18235, 23845}, {18252, 40635}, {20582, 24943}, {25557, 29651}, {29678, 31134}, {29822, 37631}, {31330, 34612}, {32773, 37646}, {32777, 35258}, {32850, 38000}, {33069, 37703}, {33771, 41014}

X(44419) = midpoint of X(38) and X(4030)
X(44419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {55, 26034, 141}, {100, 33083, 1211}, {171, 4026, 6703}, {846, 33079, 3932}, {3750, 33085, 4966}, {3757, 33068, 1086}, {4414, 33074, 3703}, {4660, 32916, 2886}, {7081, 24723, 4415}, {11246, 32771, 7228}, {26227, 32950, 3782}, {32917, 32948, 3925}, {32918, 32947, 11}

X(44420) = SPIEKER-CIRCLE-POLE OF EULER LINE

Barycentrics    a^2*(2*a^6*b^2 - a^4*b^4 - 2*a^2*b^6 + b^8 + 2*a^6*c^2 - 4*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 + 8*b^4*c^4 - 2*a^2*c^6 - 4*b^2*c^6 + c^8) : :
X(44420) = 2 X[2493] + X[9149]

X(44420) lies on these lines: {2, 2782}, {6, 110}, {23, 2080}, {98, 6795}, {311, 7495}, {351, 523}, {373, 36213}, {381, 11187}, {543, 9177}, {574, 13233}, {1634, 12093}, {1976, 6800}, {2021, 3291}, {2770, 14480}, {3164, 7493}, {3186, 4232}, {5663, 7418}, {7998, 36790}, {9176, 10418}, {9486, 11580}, {11002, 34098}, {11159, 37811}, {11634, 33962}, {14611, 36168}, {14934, 36166}, {15329, 40283}

X(44420) = midpoint of X(5968) and X(9149)
X(44420) = reflection of X(i) in X(j) for these {i,j}: {351, 39526}, {5968, 2493}
X(44420) = Parry-circle-inverse of X(7426)
X(44420) = crossdifference of every pair of points on line {574, 690}
X(44420) = barycentric product X(598)*X(17430)
X(44420) = barycentric quotient X(17430)/X(599)
X(44420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 111, 1995}, {110, 11422, 39689}, {111, 9465, 3124}, {3124, 39689, 13410}, {37775, 37776, 2502}

X(44421) = BEVAN-CIRCLE-POLE OF EULER LINE

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

X(44421) lies on these lines: {1, 20793}, {2, 7}, {6, 37555}, {40, 511}, {46, 1757}, {69, 3501}, {71, 4644}, {75, 21384}, {193, 3169}, {194, 3875}, {219, 7175}, {239, 41834}, {291, 1716}, {314, 3729}, {322, 35102}, {347, 34497}, {583, 17276}, {1002, 4343}, {1014, 9310}, {1044, 10822}, {1334, 3945}, {1409, 1419}, {1429, 5120}, {1475, 3672}, {1697, 2293}, {1743, 20367}, {2171, 24635}, {2260, 4419}, {2264, 4641}, {2323, 23125}, {2663, 17594}, {3000, 3214}, {3208, 3879}, {3217, 11349}, {3247, 18164}, {3663, 4253}, {3664, 3730}, {3786, 3951}, {3886, 6762}, {4650, 20368}, {6210, 24695}, {7201, 40937}, {10456, 21061}, {12526, 35628}, {15803, 20805}, {16549, 17272}, {16552, 25590}, {16572, 20605}, {17207, 29597}, {17364, 22370}, {18161, 21853}, {18162, 36743}, {20358, 36635}, {21746, 24708}, {21866, 34371}, {24328, 37500}, {24342, 41229}, {27544, 36854}, {27623, 39970}

X(44421) = reflection of X(i) in X(j) for these {i,j}: {1, 37507}, {1423, 579}
X(44421) = isogonal conjugate of the isotomic conjugate of X(40493)
X(44421) = X(39741)-Ceva conjugate of X(1)
X(44421) = barycentric product X(i)*X(j) for these {i,j}: {1, 36854}, {6, 40493}, {57, 27544}
X(44421) = barycentric quotient X(i)/X(j) for these {i,j}: {27544, 312}, {36854, 75}, {40493, 76}
X(44421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {7, 672, 27626}, {9, 3928, 16574}, {63, 894, 9}, {329, 27334, 21246}, {3218, 17350, 21371}, {17350, 21371, 9}

X(44422) = GALLATLY-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    a^6*b^2 + 4*a^4*b^4 - 5*a^2*b^6 + a^6*c^2 + 12*a^4*b^2*c^2 + 3*a^2*b^4*c^2 - 2*b^6*c^2 + 4*a^4*c^4 + 3*a^2*b^2*c^4 + 4*b^4*c^4 - 5*a^2*c^6 - 2*b^2*c^6 : :
X(44422) = X[2] - 3 X[262], 7 X[2] - 3 X[6194], 4 X[2] - 3 X[15819], 5 X[2] - 3 X[22712], X[39] + 2 X[14881], 7 X[39] - 4 X[32516], X[76] - 3 X[3545], X[194] + 3 X[3839], 7 X[262] - X[6194], 4 X[262] - X[15819], 5 X[262] - X[22712], 9 X[262] - X[33706], 5 X[381] - X[13108], 7 X[381] + X[32520], 2 X[3095] + X[6248], 5 X[3095] + X[13108], 7 X[3095] - X[32520], 3 X[3524] - 5 X[7786], X[3534] - 3 X[11171], X[3534] + 3 X[22728], 2 X[3818] + X[41622], X[3830] + 3 X[32447], 2 X[3845] - 3 X[22682], 4 X[3860] - 3 X[22681], 2 X[3934] - 3 X[5055], 3 X[5054] - 4 X[6683], 3 X[5054] - X[9821], 4 X[5066] - X[14711], 5 X[5071] - X[12251], X[5188] - 4 X[11272], 4 X[6194] - 7 X[15819], 5 X[6194] - 7 X[22712], 9 X[6194] - 7 X[33706], 5 X[6248] - 2 X[13108], 7 X[6248] + 2 X[32520], 4 X[6683] - X[9821], 3 X[7697] - 5 X[19709], 3 X[7709] + X[15682], 5 X[7921] + X[9873], 2 X[8703] - 3 X[21163], X[9764] - 3 X[9770], X[11055] + 5 X[41099], 2 X[12100] - 3 X[40108], 7 X[13108] + 5 X[32520], 3 X[13331] - X[43273], 3 X[14269] + 2 X[32450], 3 X[14853] - X[22486], 7 X[14881] + 2 X[32516], X[15683] - 5 X[32522], 6 X[15699] - 5 X[31239], 3 X[15699] - X[32521], 5 X[15819] - 4 X[22712], 9 X[15819] - 4 X[33706], 5 X[19708] - 3 X[22676], X[22564] - 3 X[38227], 9 X[22712] - 5 X[33706], 5 X[31239] - 2 X[32521]

X(44422) lies on these lines: {2, 51}, {4, 7757}, {5, 7794}, {30, 39}, {76, 3545}, {98, 5097}, {114, 5480}, {183, 37517}, {194, 3839}, {325, 19130}, {376, 13334}, {381, 538}, {524, 35439}, {542, 38383}, {549, 5188}, {575, 5999}, {576, 13860}, {597, 13354}, {754, 35436}, {1003, 9737}, {1351, 8667}, {1570, 42535}, {1916, 6054}, {2023, 5052}, {2782, 3845}, {2967, 39530}, {3098, 11174}, {3102, 35822}, {3103, 35823}, {3104, 16267}, {3105, 16268}, {3106, 41107}, {3107, 41108}, {3329, 5092}, {3524, 7786}, {3534, 11171}, {3543, 11257}, {3656, 14839}, {3815, 21850}, {3818, 7774}, {3830, 32447}, {3860, 22681}, {3934, 5055}, {5008, 12042}, {5041, 14880}, {5054, 6683}, {5066, 14711}, {5071, 12251}, {5093, 9756}, {7697, 19709}, {7709, 15682}, {7736, 31670}, {7764, 44230}, {7776, 10356}, {7777, 9993}, {7779, 43150}, {7788, 11178}, {7804, 35002}, {7816, 18502}, {7833, 34733}, {7840, 25561}, {7843, 37243}, {7845, 9996}, {7921, 9873}, {7976, 34627}, {8029, 8704}, {8556, 11477}, {8703, 21163}, {8716, 10983}, {8859, 32414}, {9151, 15820}, {9734, 39656}, {9755, 15520}, {9764, 9770}, {10011, 15850}, {10358, 11286}, {10711, 32454}, {10796, 18860}, {11055, 41099}, {11163, 44114}, {11180, 32451}, {11237, 12836}, {11238, 12837}, {12100, 40108}, {12110, 13586}, {12150, 13335}, {12782, 31162}, {12816, 32466}, {12817, 32465}, {13331, 43273}, {14160, 14639}, {14269, 32450}, {15683, 32522}, {15687, 32448}, {15699, 31239}, {18907, 38749}, {19708, 22676}, {22521, 34473}, {25555, 37450}, {35437, 41750}, {39095, 41413}

X(44422) = midpoint of X(i) and X(j) for these {i,j}: {4, 7757}, {381, 3095}, {1916, 6054}, {3543, 11257}, {7833, 34733}, {7976, 34627}, {10711, 32454}, {11171, 22728}, {11180, 32451}, {12782, 31162}, {15687, 32448}, {36384, 36385}
X(44422) = reflection of X(i) in X(j) for these {i,j}: {376, 13334}, {549, 11272}, {5188, 549}, {6055, 2023}, {6248, 381}, {9466, 5}, {13354, 597}, {14994, 11178}
X(44422) = complement of X(33706)
X(44422) = Gallatly-circle-inverse of X(16308)
X(44422) = X(7757)-of-Euler-triangle
X(44422) = X(9466)-of-Johnson-triangle

X(44423) = GALLATLY-CIRCLE-POLE OF LEMOINE AXIS

Barycentrics    a^2*(a^6*b^2 - 2*a^4*b^4 + b^8 + a^6*c^2 - 4*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 7*a^2*b^2*c^4 - 2*b^4*c^4 + b^2*c^6 + c^8) : :
X(44423) = X[6] + 3 X[32447], 3 X[39] + X[35439], X[69] - 3 X[11261], X[76] - 3 X[38317], X[182] - 3 X[13331], X[194] + 3 X[14561], 3 X[262] - X[3818], X[3095] + 3 X[13331], X[3098] - 3 X[11171], 4 X[3589] - 3 X[32149], 3 X[7697] + X[41747], 3 X[7709] + X[31670], X[9821] - 3 X[17508], X[13330] - 3 X[15520]

X(44423) lies on these lines: {3, 6}, {4, 32429}, {5, 32449}, {69, 11261}, {76, 38317}, {147, 262}, {194, 14561}, {302, 33478}, {303, 33479}, {698, 18583}, {732, 7764}, {2001, 34986}, {2782, 19130}, {3399, 7760}, {3589, 32149}, {5476, 7757}, {5480, 32448}, {7697, 41747}, {7709, 31670}, {7777, 41622}, {7796, 14994}, {7806, 15819}, {10007, 40107}, {11163, 25561}, {12110, 32476}, {13571, 32451}, {14881, 29012}, {24256, 25555}, {29181, 32516}, {30499, 40810}, {34236, 36212}

X(44423) = midpoint of X(i) and X(j) for these {i,j}: {4, 32429}, {5, 32449}, {182, 3095}, {576, 3094}, {5476, 7757}, {5480, 32448}, {32451, 34507}, {32452, 35431}
X(44423) = reflection of X(i) in X(j) for these {i,j}: {5052, 22330}, {14810, 13334}, {24206, 11272}, {24256, 25555}, {40107, 10007}
X(44423) = Gallatly-circle-inverse of X(187)}
X(44423) = X(5) of X(6)PU(1)
X(44423) = X(32429)-of-Euler-triangle
X(44423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 35436, 13334}, {39, 35437, 3}, {1689, 1690, 2076}, {2026, 2027, 187}, {3095, 13331, 182}, {35426, 43157, 1691}

X(44424) = STEVANOVIC-CIRCLE-POLE OF NAGEL LINE

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

X(44424) lies on these lines: {3, 25068}, {4, 5089}, {5, 25086}, {9, 165}, {20, 25066}, {37, 1699}, {72, 1695}, {210, 573}, {218, 1490}, {226, 1465}, {515, 43065}, {516, 3693}, {517, 3930}, {650, 3667}, {672, 971}, {962, 3991}, {1212, 5691}, {1334, 9856}, {3146, 25082}, {3691, 9947}, {3730, 12688}, {3965, 10443}, {4192, 10157}, {4253, 12680}, {4515, 7991}, {5022, 10085}, {5269, 10382}, {5658, 17756}, {5806, 21808}, {6999, 25083}, {7308, 16435}, {9943, 16549}, {10167, 17754}, {12679, 17732}, {19541, 40131}, {24036, 28164}, {31793, 33299}

X(44425) = STEVANOVIC-CIRCLE-POLE OF LINE X(1)X(6)

Barycentrics    a*(a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c + 3*a^3*b*c - 2*a^2*b^2*c - a*b^3*c + 2*b^4*c - 2*a^2*b*c^2 + 4*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - a*b*c^3 - 2*b^2*c^3 - a*c^4 + 2*b*c^4) : :
X(44425) = 3 X[36] - 2 X[104], X[104] - 3 X[6905], 4 X[119] - 3 X[31160], 3 X[165] - 2 X[17613], 3 X[484] - X[12767], 2 X[908] - 3 X[5660], 4 X[1538] - 3 X[1699], 3 X[2077] - 4 X[33814], 4 X[3911] - 3 X[11219], 4 X[5087] - 5 X[15017], X[5537] + 2 X[36002], 4 X[6882] - 5 X[31263], 5 X[7987] - 6 X[35271], 2 X[9952] - 3 X[40663], X[12331] - 3 X[18524]

X(44425) lies on these lines: {1, 227}, {2, 15931}, {3, 1698}, {4, 35}, {5, 5259}, {9, 165}, {10, 411}, {11, 2078}, {12, 20420}, {20, 8165}, {21, 19925}, {30, 119}, {36, 80}, {40, 5692}, {43, 1754}, {46, 1490}, {55, 1538}, {56, 5727}, {57, 11502}, {100, 516}, {109, 2635}, {125, 851}, {153, 535}, {191, 5777}, {200, 15104}, {238, 5400}, {355, 5258}, {381, 32613}, {382, 26285}, {404, 4297}, {405, 7989}, {474, 7987}, {484, 6001}, {497, 8166}, {499, 6927}, {517, 3689}, {518, 5531}, {546, 38109}, {581, 37559}, {631, 35202}, {650, 1734}, {750, 991}, {899, 13329}, {912, 4880}, {919, 29352}, {944, 5563}, {946, 3746}, {956, 37712}, {958, 37714}, {962, 8715}, {971, 1155}, {1001, 7988}, {1004, 5732}, {1006, 10175}, {1012, 5010}, {1013, 39531}, {1071, 3336}, {1125, 6915}, {1158, 37572}, {1203, 3072}, {1283, 8229}, {1325, 1793}, {1385, 37251}, {1479, 6848}, {1532, 3583}, {1621, 3817}, {1657, 37001}, {1697, 11501}, {1735, 3465}, {1736, 1758}, {1745, 1771}, {1770, 6260}, {1826, 4219}, {1836, 3256}, {1837, 37583}, {1936, 4551}, {1995, 39475}, {2003, 5348}, {2095, 3894}, {2346, 11218}, {2646, 40262}, {2752, 6011}, {2800, 3245}, {2801, 3218}, {2829, 4316}, {3035, 37374}, {3058, 7956}, {3090, 25542}, {3091, 5248}, {3100, 24025}, {3216, 37570}, {3219, 15064}, {3295, 11522}, {3337, 12675}, {3428, 3679}, {3474, 5658}, {3560, 18492}, {3576, 6911}, {3579, 12688}, {3584, 7680}, {3585, 18242}, {3586, 8069}, {3624, 6918}, {3632, 22770}, {3634, 6986}, {3651, 6684}, {3658, 18653}, {3667, 13266}, {3683, 10157}, {3814, 6840}, {3822, 6839}, {3825, 6979}, {3871, 4301}, {3913, 11531}, {4192, 29315}, {4299, 12667}, {4311, 13370}, {4312, 37541}, {4421, 42843}, {4428, 30308}, {4434, 28850}, {4857, 7681}, {4995, 7965}, {5087, 15017}, {5089, 5268}, {5284, 10171}, {5288, 5881}, {5312, 5706}, {5424, 14496}, {5432, 8727}, {5440, 5538}, {5445, 12616}, {5450, 6942}, {5534, 12704}, {5584, 9588}, {5659, 25006}, {5687, 7991}, {5709, 5904}, {5787, 24914}, {5805, 17718}, {5806, 37080}, {5816, 37400}, {5818, 6876}, {5883, 18444}, {5886, 34486}, {5902, 18446}, {5903, 6261}, {6244, 31142}, {6256, 6934}, {6690, 8226}, {6763, 14872}, {6830, 14799}, {6834, 7741}, {6835, 10198}, {6836, 26364}, {6841, 31659}, {6863, 18517}, {6882, 31263}, {6889, 41859}, {6895, 27529}, {6906, 31673}, {6909, 28164}, {6924, 18481}, {6941, 14795}, {6946, 10165}, {6960, 25639}, {6962, 26363}, {6980, 18407}, {6988, 19854}, {7081, 20236}, {7280, 12114}, {7411, 10164}, {7489, 38140}, {7503, 39582}, {7688, 26446}, {7967, 37602}, {8071, 9613}, {8227, 10267}, {8273, 16408}, {9318, 29069}, {9352, 11220}, {9578, 26357}, {9579, 11509}, {9581, 37579}, {9589, 10306}, {9612, 11507}, {9614, 11508}, {9624, 16202}, {9778, 31018}, {9841, 16209}, {9856, 37568}, {9897, 22775}, {9943, 16143}, {9952, 40663}, {9955, 37621}, {10085, 15803}, {10310, 37411}, {10434, 19544}, {10679, 31162}, {10708, 34927}, {10786, 26332}, {10826, 36152}, {10857, 37270}, {10966, 37709}, {11009, 40257}, {11010, 12672}, {11248, 41869}, {11372, 35445}, {11849, 22793}, {12116, 37720}, {12512, 33557}, {12650, 37618}, {12680, 37582}, {12699, 32141}, {13257, 17768}, {13743, 33862}, {14204, 34535}, {15622, 16453}, {15624, 27471}, {15626, 20470}, {16117, 22936}, {16132, 34339}, {16139, 31835}, {16192, 37426}, {16208, 31435}, {16371, 34628}, {16489, 32486}, {16858, 38076}, {17549, 34648}, {17763, 29016}, {18450, 37789}, {18525, 26286}, {18839, 37736}, {20330, 37703}, {21031, 31799}, {22765, 28204}, {22935, 35459}, {23961, 28208}, {24982, 35979}, {25485, 41553}, {25524, 30389}, {25882, 33305}, {26086, 33697}, {26333, 37000}, {26487, 44229}, {28146, 35000}, {28172, 34474}, {28234, 38665}, {29046, 40109}, {31159, 37820}, {31231, 37578}, {31434, 40292}, {34462, 38472}, {35986, 36991}, {36475, 37619}, {37240, 38052}, {37530, 37699}, {37625, 37700}, {41561, 41572}

X(44425) = midpoint of X(i) and X(j) for these {i,j}: {100, 36002}, {5531, 5536}
X(44425) = reflection of X(i) in X(j) for these {i,j}: {36, 6905}, {80, 1512}, {1768, 1155}, {3583, 1532}, {4867, 6326}, {4880, 5535}, {5057, 21635}, {5537, 100}, {5538, 5440}, {6840, 3814}, {34462, 38472}, {35459, 22935}, {37374, 3035}
X(44425) = reflection of X(34464) in the anti-orthic axis
X(44425) = Stevanovic-circle-inverse of X(5526)
X(44425) = crosspoint of X(21453) and X(34234)
X(44425) = crosssum of X(2183) and X(2293)
X(44425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 5587, 5251}, {3, 18491, 5587}, {3, 20989, 9590}, {4, 6796, 35}, {5, 10902, 5259}, {40, 5720, 5692}, {46, 1490, 15071}, {55, 19541, 1699}, {165, 1750, 1709}, {165, 7580, 41853}, {165, 41860, 10860}, {200, 41338, 15104}, {355, 11012, 5258}, {946, 11491, 3746}, {1376, 7580, 165}, {1745, 1771, 34043}, {3072, 37732, 1203}, {3149, 11500, 1}, {5584, 9709, 9588}, {5709, 17857, 5904}, {5881, 11249, 5288}, {6256, 6934, 10483}, {6924, 18481, 37561}, {6985, 11499, 40}, {7686, 33597, 1}, {10090, 21578, 36}, {10786, 26332, 37719}, {11249, 18518, 5881}, {14872, 37623, 6763}, {18242, 37468, 3585}, {37625, 37700, 41696}

X(44426) = POLAR-CIRCLE-POLE OF LINE X(1)X(3)

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

X(44426) lies on the ABC-inscribed parabola having focus X(108) and these lines: {4, 513}, {11, 21666}, {19, 21390}, {107, 2766}, {108, 1309}, {240, 522}, {273, 2400}, {281, 28132}, {318, 7705}, {403, 523}, {451, 33528}, {469, 4776}, {514, 16231}, {521, 1948}, {650, 17926}, {693, 17094}, {811, 17931}, {885, 1863}, {1172, 3063}, {2399, 7020}, {2804, 4397}, {3064, 3239}, {3667, 39532}, {3810, 21108}, {3900, 4036}, {4823, 23595}, {6335, 15742}, {6591, 17737}, {6941, 42769}, {8672, 16229}, {8760, 14294}, {15633, 24026}, {17925, 23880}, {18026, 35157}, {23614, 40165}, {23615, 40149}, {36059, 36113}, {43737, 43742}

X(44426) = reflection of X(4) in X(16228)
X(44426) = isogonal conjugate of X(36059)
X(44426) = isotomic conjugate of X(6516)
X(44426) = polar-circle-inverse of X(31849)
X(44426) = polar conjugate of X(651)
X(44426) = isotomic conjugate of the anticomplement of X(6506)
X(44426) = isotomic conjugate of the isogonal conjugate of X(18344)
X(44426) = polar conjugate of the isotomic conjugate of X(4391)
X(44426) = polar conjugate of the isogonal conjugate of X(650)
X(44426) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {90, 34188}, {7040, 33650}, {36082, 20}
X(44426) = X(i)-Ceva conjugate of X(j) for these (i,j): {273, 4858}, {1897, 41013}, {2052, 1146}, {6335, 281}, {7017, 8735}, {7020, 24026}, {18026, 92}, {40165, 34591}
X(44426) = X(i)-cross conjugate of X(j) for these (i,j): {11, 4}, {523, 522}, {650, 4391}, {1146, 2052}, {2968, 43742}, {3064, 17924}, {3318, 7149}, {4081, 7003}, {4516, 1172}, {5514, 459}, {6506, 2}, {8735, 7017}, {14312, 43728}, {21044, 40149}, {35014, 36121}, {42069, 281}, {42455, 21666}
X(44426) = cevapoint of X(i) and X(j) for these (i,j): {11, 42455}, {523, 24006}, {650, 18344}, {21044, 23615}
X(44426) = crosspoint of X(i) and X(j) for these (i,j): {29, 1897}, {92, 18026}, {264, 6335}
X(44426) = crosssum of X(i) and X(j) for these (i,j): {3, 23187}, {48, 1946}, {73, 1459}, {184, 22383}, {212, 36054}, {520, 22076}, {652, 40945}, {7335, 23224}, {32656, 32660}
X(44426) = trilinear pole of line {1146, 8735}
X(44426) = crossdifference of every pair of points on line {48, 577}
X(44426) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36059}, {2, 32660}, {3, 109}, {6, 1813}, {7, 32656}, {25, 6517}, {31, 6516}, {48, 651}, {56, 1331}, {57, 906}, {58, 23067}, {59, 1459}, {63, 1415}, {65, 4575}, {71, 4565}, {73, 110}, {77, 692}, {100, 603}, {101, 222}, {108, 255}, {112, 40152}, {162, 22341}, {163, 1214}, {184, 664}, {212, 934}, {219, 1461}, {226, 32661}, {228, 1414}, {307, 1576}, {348, 32739}, {394, 32674}, {521, 24027}, {577, 653}, {604, 1332}, {643, 1410}, {644, 7099}, {652, 1262}, {662, 1409}, {905, 2149}, {1020, 2193}, {1025, 32658}, {1092, 36127}, {1106, 4571}, {1260, 6614}, {1397, 4561}, {1400, 4558}, {1402, 4592}, {1407, 4587}, {1412, 4574}, {1425, 4636}, {1437, 4551}, {1783, 7125}, {1790, 4559}, {1795, 23981}, {1796, 36075}, {1802, 4617}, {1803, 35326}, {1804, 8750}, {1818, 32735}, {1897, 7335}, {1946, 7045}, {2197, 4556}, {2200, 4573}, {2283, 36057}, {2289, 32714}, {2701, 17975}, {2720, 22350}, {3049, 4620}, {3157, 36082}, {3215, 13397}, {3451, 23113}, {3784, 8685}, {3939, 7053}, {3955, 29055}, {4091, 7115}, {4303, 15439}, {4554, 9247}, {4564, 22383}, {4572, 14575}, {4619, 7117}, {6056, 36118}, {6332, 23979}, {6510, 36141}, {7011, 36049}, {7012, 23224}, {7013, 32652}, {7078, 8059}, {7114, 13138}, {7128, 36054}, {7352, 33600}, {8687, 22097}, {13486, 22342}, {14578, 24029}, {20752, 36146}, {20818, 38828}, {22128, 32675}, {22345, 36098}, {23071, 34921}, {23207, 36048}, {23225, 39293}, {23703, 36058}
X(44426) = trilinear product X(i)*X(j) for these {i,j}: {2, 3064}, {4, 522}, {8, 7649}, {9, 17924}, {11, 1897}, {19, 4391}, {21, 24006}, {27, 3700}, {28, 4086}, {29, 523}, {33, 693}, {34, 4397}, {75, 18344}, {92, 650}, {108, 24026}, {158, 521}, {190, 8735}, {225, 7253}, {226, 17926}, {264, 663}, {273, 3900}, {278, 3239}, {281, 514}, {284, 14618}, {286, 4041}, {312, 6591}, {318, 513}, {331, 657}, {333, 2501}, {393, 6332}, {525, 8748}, {607, 3261}, {649, 7017}, {652, 2052}, {653, 1146}, {656, 1896}, {661, 31623}, {850, 2299}, {885, 1861}, {1172, 1577}, {1783, 4858}, {1824, 18155}, {1826, 4560}, {1857, 4025}, {1969, 3063}, {2170, 6335}, {2204, 20948}, {2310, 18026}, {2399, 8755}, {2489, 28660}, {6129, 7020}, {7003, 14837}, {7008, 17896}, {8750, 34387}, {23978, 32674}
X(44426) = barycentric product X(i)*X(j) for these {i,j}: {4, 4391}, {8, 17924}, {11, 6335}, {19, 35519}, {21, 14618}, {27, 4086}, {29, 1577}, {33, 3261}, {75, 3064}, {76, 18344}, {92, 522}, {108, 23978}, {158, 6332}, {264, 650}, {273, 3239}, {278, 4397}, {281, 693}, {286, 3700}, {312, 7649}, {314, 2501}, {318, 514}, {331, 3900}, {333, 24006}, {393, 35518}, {513, 7017}, {521, 2052}, {523, 31623}, {525, 1896}, {607, 40495}, {644, 2973}, {646, 2969}, {651, 21666}, {653, 24026}, {661, 44130}, {663, 1969}, {668, 8735}, {811, 21044}, {850, 1172}, {1118, 15416}, {1146, 18026}, {1441, 17926}, {1783, 34387}, {1826, 18155}, {1847, 4163}, {1857, 15413}, {1897, 4858}, {1946, 18027}, {2204, 44173}, {2299, 20948}, {2322, 4077}, {2489, 40072}, {2804, 16082}, {2970, 4612}, {3063, 18022}, {3596, 6591}, {3676, 7101}, {3701, 17925}, {4041, 44129}, {4081, 13149}, {4516, 6331}, {4554, 42069}, {4560, 41013}, {4631, 8754}, {4768, 6336}, {7003, 17896}, {7020, 14837}, {7046, 24002}, {7128, 23104}, {7253, 40149}, {8748, 14208}, {15633, 24035}, {15742, 40166}, {16732, 36797}, {36795, 39534}
X(44426) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1813}, {2, 6516}, {4, 651}, {6, 36059}, {8, 1332}, {9, 1331}, {11, 905}, {19, 109}, {21, 4558}, {25, 1415}, {27, 1414}, {28, 4565}, {29, 662}, {31, 32660}, {33, 101}, {34, 1461}, {37, 23067}, {41, 32656}, {55, 906}, {63, 6517}, {92, 664}, {108, 1262}, {158, 653}, {200, 4587}, {210, 4574}, {225, 1020}, {264, 4554}, {270, 4556}, {273, 658}, {278, 934}, {281, 100}, {284, 4575}, {286, 4573}, {312, 4561}, {314, 4563}, {318, 190}, {331, 4569}, {333, 4592}, {346, 4571}, {393, 108}, {512, 1409}, {513, 222}, {514, 77}, {521, 394}, {522, 63}, {523, 1214}, {607, 692}, {647, 22341}, {649, 603}, {650, 3}, {652, 255}, {653, 7045}, {656, 40152}, {657, 212}, {661, 73}, {663, 48}, {693, 348}, {811, 4620}, {850, 1231}, {884, 32658}, {885, 1814}, {905, 1804}, {926, 20752}, {1021, 283}, {1024, 36057}, {1096, 32674}, {1118, 32714}, {1119, 4617}, {1146, 521}, {1172, 110}, {1435, 6614}, {1459, 7125}, {1577, 307}, {1639, 5440}, {1783, 59}, {1785, 24029}, {1824, 4559}, {1826, 4551}, {1827, 35326}, {1847, 4626}, {1855, 35338}, {1857, 1783}, {1861, 1025}, {1896, 648}, {1897, 4564}, {1946, 577}, {1969, 4572}, {2052, 18026}, {2170, 1459}, {2194, 32661}, {2204, 1576}, {2212, 32739}, {2299, 163}, {2310, 652}, {2322, 643}, {2326, 4636}, {2355, 36075}, {2432, 36055}, {2489, 1402}, {2501, 65}, {2969, 3669}, {2973, 24002}, {3057, 23113}, {3063, 184}, {3064, 1}, {3239, 78}, {3261, 7182}, {3270, 36054}, {3271, 22383}, {3287, 3955}, {3309, 23144}, {3669, 7053}, {3676, 7177}, {3700, 72}, {3709, 228}, {3716, 20769}, {3737, 1790}, {3738, 22128}, {3900, 219}, {4024, 201}, {4025, 7183}, {4036, 26942}, {4041, 71}, {4086, 306}, {4105, 1802}, {4130, 1260}, {4147, 22370}, {4162, 20818}, {4163, 3692}, {4171, 2318}, {4183, 5546}, {4391, 69}, {4397, 345}, {4435, 7193}, {4516, 647}, {4521, 4855}, {4560, 1444}, {4705, 2197}, {4765, 4652}, {4768, 3977}, {4820, 3927}, {4843, 4047}, {4858, 4025}, {4895, 22356}, {4944, 3940}, {4976, 3916}, {4985, 4001}, {5081, 4585}, {5089, 2283}, {5190, 23800}, {5236, 41353}, {6129, 7011}, {6332, 326}, {6335, 4998}, {6366, 6510}, {6520, 36127}, {6590, 1038}, {6591, 56}, {7003, 13138}, {7004, 4091}, {7008, 36049}, {7012, 4619}, {7017, 668}, {7046, 644}, {7079, 3939}, {7101, 3699}, {7117, 23224}, {7129, 8059}, {7140, 21859}, {7154, 32652}, {7178, 1439}, {7180, 1410}, {7252, 1437}, {7253, 1812}, {7649, 57}, {8062, 7364}, {8611, 3682}, {8678, 2286}, {8735, 513}, {8748, 162}, {8750, 2149}, {8751, 32735}, {8756, 23703}, {10581, 22079}, {11124, 22055}, {11193, 22144}, {11934, 22131}, {11998, 23187}, {14298, 7078}, {14571, 23981}, {14618, 1441}, {14775, 2982}, {14837, 7013}, {14936, 1946}, {15313, 3173}, {15413, 7055}, {15416, 1264}, {15742, 31615}, {16228, 34048}, {16732, 17094}, {17115, 7124}, {17420, 22097}, {17880, 30805}, {17905, 40576}, {17924, 7}, {17925, 1014}, {17926, 21}, {18026, 1275}, {18155, 17206}, {18191, 7254}, {18344, 6}, {21044, 656}, {21127, 22053}, {21132, 3942}, {21666, 4391}, {21789, 2193}, {22383, 7335}, {23189, 18604}, {23615, 34591}, {23710, 23890}, {23838, 1797}, {23978, 35518}, {24002, 7056}, {24006, 226}, {24026, 6332}, {26932, 4131}, {31623, 99}, {32674, 24027}, {32714, 7339}, {33525, 23207}, {33573, 14414}, {34387, 15413}, {35518, 3926}, {35519, 304}, {36054, 1092}, {36123, 37136}, {36124, 36146}, {36127, 7128}, {36797, 4567}, {38347, 22160}, {39534, 1465}, {40149, 4566}, {40166, 1565}, {40213, 17219}, {40573, 36048}, {40836, 37141}, {41013, 4552}, {42069, 650}, {42455, 26932}, {42462, 7004}, {43923, 1407}, {43924, 7099}, {43925, 1408}, {43933, 34051}, {44129, 4625}, {44130, 799}
X(44426) = {X(7649),X(24006)}-harmonic conjugate of X(17924)

X(44427) = POLAR-CIRCLE-POLE OF FERMAT LINE

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2)*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2) : :
Barycentrics    sec A sin(B - C) (1 + 2 cos 2A) : :
X(44427) = 4 X[2501] - 3 X[14618], 3 X[3268] - 4 X[8552], 3 X[3268] - 2 X[41078], X[5489] - 3 X[42731], 3 X[5664] - 2 X[8552], 3 X[5664] - X[41078], 2 X[6130] - 3 X[42731], 3 X[9979] - 4 X[24978], 3 X[9979] - 2 X[41079]

X(44427) lies on these lines: {2, 6334}, {4, 690}, {74, 1300}, {99, 112}, {186, 14270}, {246, 2970}, {297, 525}, {338, 3269}, {420, 9208}, {468, 9185}, {523, 9409}, {526, 1986}, {671, 35142}, {685, 935}, {1235, 14295}, {1637, 2394}, {1826, 22037}, {2489, 7624}, {2491, 39575}, {2986, 43756}, {3268, 5664}, {5094, 9191}, {5191, 9131}, {5489, 6130}, {5667, 9033}, {6110, 6782}, {6111, 6783}, {9003, 32234}, {9479, 17994}, {14696, 41203}, {16868, 39509}, {21844, 39477}, {33919, 38294}, {43673, 43717}

X(44427) = reflection of X(i) in X(j) for these {i,j}: {4, 16230}, {2394, 1637}, {3268, 5664}, {5489, 6130}, {41078, 8552}, {41079, 24978}
X(44427) = isogonal conjugate of X(32662)
X(44427) = anticomplement of X(6334)
X(44427) = polar conjugate of X(476)
X(44427) = anticomplement of the isogonal conjugate of X(32708)
X(44427) = anticomplement of the isotomic conjugate of X(687)
X(44427) = polar conjugate of the isotomic conjugate of X(3268)
X(44427) = polar conjugate of the isogonal conjugate of X(526)
X(44427) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {687, 6327}, {1300, 21294}, {10420, 4329}, {32708, 8}, {36053, 13219}, {36114, 69}
X(44427) = isotomic conjugate of trilinear pole of line X(3)X(125)
X(44427) = intersection of trilinear polars of X(470) and X(471)
X(44427) = radical center of circumcircle and X(15)- and X(16)-Fuhrmann circles (aka -Hagge circles)
X(44427) = X(i)-Ceva conjugate of X(j) for these (i,j): {340, 35235}, {648, 14920}, {687, 2}, {9381, 338}, {14590, 14918}, {16077, 4}, {18831, 3043}, {18878, 38936}, {43752, 3134}
X(44427) = X(i)-cross conjugate of X(j) for these (i,j): {526, 3268}, {2081, 526}, {2088, 186}, {35235, 340}
X(44427) = X(i)-isoconjugate of X(j) for these (i,j): {1, 32662}, {3, 32678}, {6, 36061}, {48, 476}, {63, 14560}, {163, 265}, {184, 32680}, {577, 36129}, {810, 39295}, {1101, 14582}, {1789, 32675}, {1989, 4575}, {2153, 38413}, {2154, 38414}, {2166, 32661}, {2437, 36062}, {2617, 11077}, {2631, 15395}, {4592, 11060}, {5961, 36145}, {9247, 35139}, {14559, 36060}, {14592, 23995}, {35200, 41392}
X(44427) = crosspoint of X(i) and X(j) for these (i,j): {76, 18878}, {648, 16080}, {2052, 15459}, {2394, 15412}
X(44427) = crosssum of X(i) and X(j) for these (i,j): {32, 21731}, {577, 1636}, {647, 3284}, {1625, 2420}
X(44427) = trilinear pole of line {3258, 16186}
X(44427) = crossdifference of every pair of points on line {184, 5158}
X(44427) = barycentric product X(i)*X(j) for these {i,j}: {4, 3268}, {92, 32679}, {99, 35235}, {186, 850}, {264, 526}, {275, 41078}, {276, 2081}, {323, 14618}, {338, 14590}, {340, 523}, {470, 23871}, {471, 23870}, {525, 14165}, {562, 41298}, {860, 4467}, {1969, 2624}, {2052, 8552}, {2088, 6331}, {2394, 14920}, {2501, 7799}, {2970, 10411}, {3258, 16077}, {4242, 17886}, {5664, 16080}, {5962, 6563}, {6148, 18808}, {6528, 16186}, {7265, 17923}, {9213, 44146}, {14270, 18022}, {14591, 23962}, {14918, 15412}, {15470, 44138}, {16221, 18878}, {17924, 42701}, {34397, 44173}
X(44427) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 36061}, {4, 476}, {6, 32662}, {15, 38413}, {16, 38414}, {19, 32678}, {25, 14560}, {50, 32661}, {92, 32680}, {115, 14582}, {125, 43083}, {136, 43088}, {158, 36129}, {186, 110}, {264, 35139}, {323, 4558}, {338, 14592}, {340, 99}, {403, 41512}, {468, 14559}, {470, 23896}, {471, 23895}, {523, 265}, {526, 3}, {562, 930}, {648, 39295}, {850, 328}, {860, 6742}, {924, 5961}, {1154, 23181}, {1304, 15395}, {1825, 2222}, {1835, 26700}, {1870, 13486}, {1986, 15329}, {1990, 41392}, {2081, 216}, {2088, 647}, {2433, 11079}, {2436, 32663}, {2489, 11060}, {2501, 1989}, {2623, 11077}, {2624, 48}, {2970, 10412}, {3258, 9033}, {3268, 69}, {3738, 1789}, {5664, 11064}, {5962, 925}, {6103, 23968}, {6137, 36297}, {6138, 36296}, {6143, 43965}, {6149, 4575}, {7799, 4563}, {8552, 394}, {8739, 5994}, {8740, 5995}, {8754, 15475}, {9213, 895}, {11062, 1625}, {14165, 648}, {14222, 1300}, {14270, 184}, {14355, 43754}, {14590, 249}, {14591, 23357}, {14618, 94}, {14918, 14570}, {14920, 2407}, {15328, 12028}, {15470, 5504}, {15475, 14595}, {16080, 39290}, {16186, 520}, {16230, 14356}, {18808, 5627}, {20188, 31676}, {23283, 10217}, {23284, 10218}, {23870, 40710}, {23871, 40709}, {24006, 2166}, {32679, 63}, {32710, 35189}, {34397, 1576}, {35057, 1793}, {35235, 523}, {36130, 36047}, {36423, 14591}, {38936, 10420}, {39176, 2420}, {41078, 343}, {42701, 1332}
X(44427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2592, 2593, 14618}, {5489, 42731, 6130}, {5664, 41078, 8552}, {8552, 41078, 3268}, {24978, 41079, 9979}

X(44428) = POLAR-CIRCLE-POLE OF LINE X(1)X(5)

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

X(44428) lies on these lines: {4, 900}, {24, 39200}, {100, 108}, {104, 915}, {186, 523}, {240, 522}, {406, 26144}, {427, 31131}, {475, 26078}, {521, 43923}, {676, 14312}, {1119, 43042}, {1172, 4435}, {1845, 3738}, {3064, 4765}, {3144, 28284}, {3904, 6369}, {3907, 21108}, {3910, 17925}, {4194, 27545}, {4213, 4800}, {4560, 14024}, {4858, 7004}, {4926, 16228}, {4962, 16231}, {4976, 17926}, {6089, 16230}, {6353, 26275}, {7577, 39493}, {10015, 39471}, {17555, 25020}, {21192, 23595}, {28114, 37055}, {30091, 31916}, {36121, 36123}

X(44428) = reflection of X(i) in X(j) for these {i,j}: {4, 39534}, {14304, 21180}, {14312, 676}
X(44428) = polar conjugate of X(655)
X(44428) = polar conjugate of the isotomic conjugate of X(3904)
X(44428) = polar conjugate of the isogonal conjugate of X(654)
X(44428) = isotomic conjugate of trilinear pole of line X(63)X(343)
X(44428) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {913, 37781}, {915, 33650}, {32698, 329}, {36052, 34188}, {36106, 3436}
X(44428) = X(i)-Ceva conjugate of X(j) for these (i,j): {1309, 4}, {4242, 860}, {16080, 1146}
X(44428) = X(654)-cross conjugate of X(3904)
X(44428) = crosspoint of X(i) and X(j) for these (i,j): {4242, 17515}, {16082, 18026}
X(44428) = crossdifference of every pair of points on line {48, 216}
X(44428) = X(i)-isoconjugate of X(j) for these (i,j): {3, 2222}, {48, 655}, {63, 32675}, {80, 36059}, {109, 1807}, {184, 35174}, {201, 36069}, {476, 22342}, {759, 23067}, {906, 2006}, {1331, 1411}, {1813, 2161}, {2197, 37140}, {2594, 36061}, {6187, 6516}, {16577, 32662}, {18359, 32660}, {18815, 32656}, {26942, 32671}
X(44428) = barycentric product X(i)*X(j) for these {i,j}: {4, 3904}, {29, 4707}, {92, 3738}, {264, 654}, {275, 6369}, {281, 4453}, {318, 3960}, {320, 3064}, {514, 5081}, {522, 17923}, {860, 4560}, {1577, 17515}, {1870, 4391}, {1969, 8648}, {2600, 40440}, {4242, 4858}, {4511, 17924}, {7649, 32851}, {17926, 41804}, {18344, 20924}, {21828, 44130}
X(44428) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 655}, {19, 2222}, {25, 32675}, {36, 1813}, {92, 35174}, {270, 37140}, {318, 36804}, {650, 1807}, {654, 3}, {860, 4552}, {1021, 1793}, {1835, 1020}, {1845, 24029}, {1870, 651}, {2189, 36069}, {2245, 23067}, {2323, 1331}, {2361, 906}, {2610, 201}, {2624, 22342}, {3064, 80}, {3218, 6516}, {3738, 63}, {3904, 69}, {3960, 77}, {4242, 4564}, {4282, 4575}, {4453, 348}, {4511, 1332}, {4707, 307}, {5081, 190}, {6369, 343}, {6370, 26942}, {6591, 1411}, {7113, 36059}, {7649, 2006}, {8648, 48}, {8882, 36078}, {17515, 662}, {17923, 664}, {17924, 18815}, {17926, 6740}, {18344, 2161}, {21758, 603}, {21828, 73}, {22128, 6517}, {22379, 7125}, {32851, 4561}, {42666, 2197}, {44113, 4559}

X(44429) = ORTHOPTIC-CIRCLE-OF-STEINER-INELLIPSE-POLE OF LINE X(1)X(3)

Barycentrics    (b - c)*(2*a*b^2 + a*b*c - b^2*c + 2*a*c^2 - b*c^2) : :
X(44429) = X[649] - 4 X[25380], 2 X[659] - 5 X[31209], X[661] + 2 X[24720], X[693] + 2 X[1491], X[693] - 4 X[3837], 2 X[905] + X[21301], X[1491] + 2 X[3837], X[2254] + 2 X[3835], X[2526] + 2 X[4885], 2 X[2530] + X[4391], X[2530] + 2 X[21260], 2 X[3126] + X[30804], 2 X[3716] - 5 X[30835], 2 X[3776] + X[4088], 2 X[3777] + X[4462], X[3777] + 2 X[21051], X[4024] + 2 X[4818], 2 X[4129] + X[4905], X[4378] - 4 X[19947], X[4380] - 4 X[9508], X[4380] + 2 X[24719], X[4382] + 2 X[4913], X[4391] - 4 X[21260], X[4462] - 4 X[21051], 2 X[4522] + X[16892], 2 X[4705] + X[4801], X[4705] + 2 X[23815], X[4724] - 4 X[25666], X[4776] + 2 X[36848], X[4801] - 4 X[23815], 2 X[4874] - 5 X[30795], 2 X[4940] + X[7659], 2 X[7662] - 5 X[26985], 2 X[9508] + X[24719], 4 X[10006] - X[13266], 2 X[24718] + X[27469]

X(44429) lies on these lines: {2, 513}, {105, 29348}, {325, 523}, {514, 14430}, {522, 4728}, {649, 25380}, {659, 31209}, {661, 4521}, {663, 28521}, {905, 21301}, {2254, 3667}, {2526, 4885}, {2530, 4391}, {2787, 31149}, {2976, 4806}, {3063, 33854}, {3126, 30804}, {3716, 30835}, {3733, 26249}, {3776, 4088}, {3777, 4462}, {4010, 4926}, {4024, 4818}, {4036, 31096}, {4129, 4905}, {4369, 30764}, {4378, 16830}, {4380, 9508}, {4382, 4913}, {4444, 28851}, {4486, 30519}, {4522, 16892}, {4705, 4801}, {4724, 25666}, {4775, 16823}, {4784, 26277}, {4824, 28199}, {4874, 30795}, {4940, 7659}, {4977, 30792}, {5276, 20980}, {6363, 14426}, {7179, 24002}, {7378, 16228}, {7662, 26985}, {10006, 13266}, {20949, 30758}, {21146, 28195}, {24533, 28399}, {24718, 27469}, {25299, 28374}, {26275, 28217}, {28468, 30574}, {28481, 41800}, {29362, 31150}, {36238, 42722}

X(44429) = midpoint of X(2530) and X(14431)
X(44429) = reflection of X(i) in X(j) for these {i,j}: {4391, 14431}, {14431, 21260}
X(44429) = orthoptic-circle-of-Steiner-inellipe-inverse of X(34583)
X(44429) = isotomic conjugate of X(9067)
X(44429) = isotomic conjugate of the isogonal conjugate of X(9010)
X(44429) = X(31)-isoconjugate of X(9067)
X(44429) = crossdifference of every pair of points on line {32, 3230}
X(44429) = barycentric product X(i)*X(j) for these {i,j}: {76, 9010}, {693, 17756}
X(44429) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9067}, {9010, 6}, {17756, 100}
X(44429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1491, 3837, 693}, {2530, 21260, 4391}, {3777, 21051, 4462}, {4705, 23815, 4801}, {9508, 24719, 4380}

X(44430) = ORTHOPTIC-CIRCLE-OF-STEINER-INELLIPSE-POLE OF ANTI-ORTHIC AXIS

Barycentrics    2*a^3*b^3 - 2*a*b^5 - 3*a^4*b*c + 4*a^3*b^2*c - b^5*c + 4*a^3*b*c^2 + 2*a*b^3*c^2 + 2*a^3*c^3 + 2*a*b^2*c^3 + 2*b^3*c^3 - 2*a*c^5 - b*c^5 : :
X(44430) = X[75] + 2 X[31395], 4 X[3842] - X[6210], 5 X[4687] - 2 X[31394]

X(44430) lies on these lines: {1, 21554}, {2, 392}, {3, 16830}, {4, 5089}, {10, 262}, {40, 6998}, {75, 31395}, {165, 13634}, {355, 7379}, {381, 29365}, {495, 1565}, {612, 7413}, {944, 39587}, {1447, 36279}, {1482, 16823}, {2783, 4664}, {3576, 13635}, {3842, 6210}, {3920, 5767}, {3940, 5774}, {4385, 15973}, {4648, 39898}, {4687, 31394}, {5587, 24808}, {5711, 9755}, {5790, 29331}, {5818, 7407}, {5988, 37716}, {6361, 7390}, {7385, 12699}, {8158, 19319}, {10306, 19309}, {10595, 16020}, {10679, 26241}, {11248, 19310}, {11249, 19314}, {12245, 39581}, {14853, 38057}, {18788, 36531}, {19313, 22770}, {28915, 36722}, {29641, 37360}, {31359, 31785}

X(44430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {10, 39605, 7380}, {40, 39586, 6998}, {7407, 39570, 5818}

X(44431) = ORTHOPTIC-CIRCLE-OF-STEINER-INELLIPSE-POLE OF GERGONNE LINE

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

X(44431) lies on these lines: {2, 165}, {4, 5089}, {7, 43751}, {8, 7985}, {10, 43951}, {40, 7407}, {120, 42356}, {226, 3424}, {262, 13576}, {376, 28897}, {381, 10712}, {390, 7179}, {497, 3666}, {515, 11200}, {612, 1750}, {946, 16020}, {962, 7379}, {1503, 3475}, {1541, 11372}, {2784, 3241}, {3146, 16830}, {3598, 4312}, {3755, 37665}, {3886, 37668}, {4339, 13442}, {4349, 41825}, {4423, 19649}, {5483, 29815}, {5658, 29207}, {5691, 39587}, {5698, 17747}, {5731, 10186}, {6194, 30946}, {6361, 7380}, {7172, 21060}, {7390, 39605}, {7967, 28901}, {9756, 37540}, {9791, 40236}, {18788, 29611}, {19925, 39570}, {24239, 31326}, {25080, 37456}, {28862, 34632}, {28881, 31162}

X(44431) = reflection of X(5731) in X(10186)
X(44431) = anticomplement of X(9746)
X(44431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {962, 7379, 39581}, {39605, 41869, 7390}

X(44432) = ORTHOPTIC-CIRCLE-OF-STEINER-INELLIPSE-POLE OF SODDY LINE

Barycentrics    (b - c)*(-3*a^2 + 4*a*b + b^2 + 4*a*c - 4*b*c + c^2) : :
X(44432) = 2 X[2487] + X[4940], X[3004] + 5 X[31250], X[3239] + 2 X[21212], X[3676] + 2 X[25666], X[3776] + 2 X[4521], X[3798] + 2 X[3835], X[3798] - 4 X[7658], X[3835] + 2 X[7658], X[4025] + 5 X[30835], X[4120] - 5 X[30835], X[4893] + 3 X[14475], 3 X[4893] + X[21116], X[11068] - 4 X[31287], 9 X[14475] - X[21116], 3 X[14475] - X[21183], X[21116] - 3 X[21183]

X(44432) lies on these lines: {2, 514}, {25, 39476}, {427, 39532}, {522, 4928}, {523, 4885}, {650, 4927}, {663, 5272}, {1638, 28846}, {2254, 3667}, {2487, 4940}, {2826, 10006}, {3004, 31250}, {3239, 21212}, {3676, 25666}, {3705, 4546}, {3742, 9029}, {3776, 4521}, {3960, 40134}, {4025, 4120}, {4106, 4773}, {4107, 28296}, {4369, 4778}, {4449, 5268}, {4468, 21115}, {4777, 30792}, {4786, 31147}, {4926, 17069}, {6006, 26275}, {9209, 21188}, {11068, 14425}, {14837, 28468}, {21196, 30764}, {25084, 43051}, {28478, 41800}

X(44432) = midpoint of X(i) and X(j) for these {i,j}: {650, 4927}, {4025, 4120}, {4106, 4773}, {4468, 21115}, {4786, 31147}, {4893, 21183}
X(44432) = reflection of X(i) in X(j) for these {i,j}: {11068, 14425}, {14425, 31287}
X(44432) = orthoptic-circle-of-Steiner-inellipe-inverse of X(38941)
X(44432) = X(i)-complementary conjugate of X(j) for these (i,j): {3478, 26932}, {9088, 226}
X(44432) = crossdifference of every pair of points on line {902, 3053}
X(44432) = barycentric product X(514)*X(20073)
X(44432) = barycentric quotient X(20073)/X(190)
X(44432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3835, 7658, 3798}, {4893, 14475, 21183}

X(44433) = ORTHOPTIC-CIRCLE-OF-STEINER-INELLIPSE-POLE OF LINE X(1)X(5)

Barycentrics    (b - c)*(3*a^3 - a^2*b + a*b^2 - b^3 - a^2*c - a*b*c + a*c^2 - c^3) : :
X(44433) = 5 X[2] - 4 X[30792], 4 X[1960] - X[3904], X[2254] - 4 X[13246], 5 X[26275] - 2 X[30792], 4 X[26275] - X[31131], 8 X[30792] - 5 X[31131]

X(44433) lies on these lines: {1, 23888}, {2, 900}, {22, 39200}, {23, 385}, {100, 2397}, {104, 105}, {513, 4453}, {522, 1635}, {663, 28468}, {665, 26242}, {676, 4927}, {690, 1281}, {1960, 3904}, {2254, 3667}, {2496, 4926}, {3598, 43042}, {3716, 4120}, {3766, 26234}, {3810, 8643}, {4435, 5276}, {4448, 30565}, {4458, 4778}, {4467, 26277}, {4724, 28851}, {6636, 39478}, {6995, 39534}, {10015, 28294}, {24128, 39567}, {28521, 30574}

X(44433) = reflection of X(i) in X(j) for these {i,j}: {2, 26275}, {3904, 30580}, {4120, 3716}, {4453, 4809}, {4927, 676}, {21115, 4458}, {25020, 28396}, {30565, 4448}, {30580, 1960}, {31131, 2}
X(44433) = anticomplement of the isotomic conjugate of X(9089)
X(44433) = X(9089)-anticomplementary conjugate of X(6327)
X(44433) = X(9089)-Ceva conjugate of X(2)
X(44433) = crossdifference of every pair of points on line {39, 8649}

X(44434) = ORTHOPTIC-CIRCLE-OF-STEINER-CIRCUMLLIPSE-POLE OF LEMOINE AXIS

Barycentrics    3*a^6*b^2 + 2*a^4*b^4 - 5*a^2*b^6 + 3*a^6*c^2 + 11*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - 5*a^2*c^6 - b^2*c^6 : :
X(44434) = 3 X[2] - 4 X[262], 9 X[2] - 8 X[15819], 5 X[2] - 4 X[22712], 7 X[2] - 4 X[33706], 5 X[4] - 2 X[13108], 4 X[4] - X[20081], X[20] - 4 X[3095], 8 X[39] - 5 X[3522], 4 X[76] - 7 X[3832], 2 X[194] + X[3146], 3 X[262] - 2 X[15819], 5 X[262] - 3 X[22712], 7 X[262] - 3 X[33706], 4 X[1916] - X[5984], 7 X[3090] - 4 X[32521], 5 X[3091] - 4 X[7697], 5 X[3091] - 2 X[12251], 5 X[3091] - 8 X[14881], 5 X[3522] - 4 X[22676], 7 X[3523] - 4 X[9821], 7 X[3523] - 8 X[40108], X[3529] - 4 X[32448], 5 X[3617] - 4 X[22697], 7 X[3622] - 8 X[22475], 5 X[3623] - 4 X[22713], 2 X[3627] + X[32520], 7 X[3832] - 8 X[22682], 9 X[3839] - 8 X[22681], 16 X[3934] - 19 X[15022], X[5059] - 4 X[11257], 13 X[5068] - 10 X[31276], 8 X[5188] - 11 X[15717], 3 X[6194] - 4 X[15819], 5 X[6194] - 6 X[22712], 7 X[6194] - 6 X[33706], 4 X[7757] - X[15683], 4 X[7843] - X[18768], X[8782] - 4 X[38383], 13 X[10303] - 16 X[11272], 3 X[10304] - 4 X[11171], 3 X[10519] - 4 X[11261], X[12251] - 4 X[14881], 4 X[12782] - X[20070], 8 X[13108] - 5 X[20081], X[13108] - 5 X[22728], 16 X[13334] - 13 X[21734], 3 X[14853] - 2 X[31958], X[14927] - 4 X[32449], 10 X[15819] - 9 X[22712], 14 X[15819] - 9 X[33706], 5 X[17538] - 8 X[32516], 5 X[17578] + X[20105], X[20081] - 8 X[22728], 7 X[22712] - 5 X[33706]

X(44434) lies on these lines: {2, 51}, {3, 22521}, {4, 7779}, {8, 7985}, {20, 3095}, {23, 22655}, {30, 32519}, {39, 3522}, {76, 3832}, {98, 37517}, {147, 31670}, {193, 1916}, {194, 3146}, {376, 32447}, {385, 9756}, {390, 12837}, {538, 23334}, {726, 9812}, {1160, 10839}, {1161, 10840}, {1350, 3329}, {1351, 5999}, {1352, 9866}, {1503, 7837}, {1569, 43618}, {1587, 35838}, {1588, 35839}, {2023, 37689}, {2782, 3543}, {3090, 32521}, {3091, 7697}, {3094, 37665}, {3097, 9778}, {3098, 9751}, {3104, 42998}, {3105, 42999}, {3314, 5480}, {3523, 9821}, {3529, 32448}, {3552, 39101}, {3600, 12836}, {3617, 22697}, {3620, 14484}, {3622, 22475}, {3623, 22713}, {3627, 32520}, {3839, 22681}, {3934, 15022}, {5059, 11257}, {5068, 31276}, {5171, 33022}, {5188, 15717}, {5261, 22705}, {5274, 22706}, {5304, 13330}, {5905, 29840}, {5987, 10752}, {6776, 33693}, {6995, 22480}, {7694, 7785}, {7710, 7774}, {7757, 15683}, {7770, 40268}, {7783, 8719}, {7787, 30270}, {7843, 18768}, {7897, 13862}, {7929, 37336}, {8586, 42535}, {8721, 13571}, {8782, 9742}, {9737, 33014}, {9917, 14118}, {10303, 11272}, {10304, 11171}, {10334, 12110}, {10753, 14931}, {10788, 35002}, {10841, 12974}, {10842, 12975}, {12206, 37479}, {12782, 20070}, {13334, 21734}, {14927, 32449}, {14986, 22730}, {17538, 32516}, {17578, 20105}, {18906, 37668}, {21445, 39089}, {22664, 36849}, {22679, 32965}, {23253, 35866}, {23263, 35867}, {23698, 32469}, {29181, 41624}, {32465, 42085}, {32466, 42086}, {32470, 43408}, {32471, 43407}, {32966, 39663}, {33878, 37455}

X(44434) = reflection of X(i) in X(j) for these {i,j}: {4, 22728}, {8, 22650}, {20, 7709}, {76, 22682}, {376, 32447}, {6194, 262}, {7697, 14881}, {7709, 3095}, {8782, 9772}, {9772, 38383}, {9778, 3097}, {9821, 40108}, {12251, 7697}, {22676, 39}
X(44434) = anticomplement of X(6194)
X(44434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {262, 6194, 2}, {1351, 5999, 7766}, {12251, 14881, 3091}, {22684, 22686, 31958}

X(44435) = ORTHOPTIC-CIRCLE-OF-STEINER-ICIRCUMELLIPSE-POLE OF SODDY LINE

Barycentrics    (b - c)*(a*b + b^2 + a*c - b*c + c^2) : :
X(44435) = X[649] - 4 X[21212], X[661] + 2 X[3776], X[693] + 2 X[3004], 4 X[3239] - 7 X[27138], 4 X[3676] - X[7192], 4 X[3798] - X[26853], 2 X[3835] + X[16892], 4 X[3835] - X[25259], 2 X[4025] + X[20295], 2 X[4106] + X[4467], X[4380] - 4 X[17069], X[4380] + 2 X[23729], X[4382] + 2 X[21196], 3 X[4728] - X[4931], X[4804] + 2 X[4818], 2 X[4932] + X[23731], 3 X[6545] - X[21116], 3 X[6548] - 2 X[21183], 2 X[6590] - 5 X[26985], 8 X[7658] - 5 X[27013], 4 X[11068] - 7 X[27115], 4 X[14425] - 5 X[31209], 2 X[16892] + X[25259], 2 X[17069] + X[23729]

X(44435) lies on these lines: {2, 514}, {325, 523}, {513, 4453}, {522, 21297}, {614, 4040}, {649, 21212}, {661, 3776}, {663, 7191}, {664, 14513}, {675, 953}, {812, 27486}, {824, 4728}, {918, 4776}, {1443, 1447}, {1635, 28882}, {2786, 31147}, {2826, 30804}, {3239, 27138}, {3667, 4025}, {3777, 8034}, {3798, 26853}, {3835, 4120}, {3873, 9029}, {3920, 4449}, {4024, 31094}, {4106, 4467}, {4147, 29679}, {4367, 26249}, {4380, 4773}, {4382, 21196}, {4406, 26234}, {4750, 4785}, {4777, 31131}, {4789, 28894}, {4804, 4818}, {4897, 39386}, {4928, 28863}, {4932, 23731}, {6084, 31150}, {6590, 26985}, {6636, 39476}, {7409, 39532}, {7658, 27013}, {9810, 28840}, {11068, 27115}, {14425, 31209}, {17072, 29667}, {20952, 29427}, {21118, 29680}, {21130, 23888}, {23345, 26240}, {23813, 28205}, {26230, 30580}, {26248, 28195}, {26275, 28209}, {28179, 30792}, {28374, 43051}, {28859, 31148}, {29110, 31149}, {29144, 36848}, {30520, 30565}

X(44435) = midpoint of X(i) and X(j) for these {i,j}: {661, 21115}, {3004, 4927}, {4120, 16892}, {4773, 23729}
X(44435) = reflection of X(i) in X(j) for these {i,j}: {693, 4927}, {4120, 3835}, {4379, 21204}, {4380, 4773}, {4773, 17069}, {21115, 3776}, {25259, 4120}
X(44435) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(38941)
X(44435) = isotomic conjugate of X(9059)
X(44435) = isotomic conjugate of the isogonal conjugate of X(9002)
X(44435) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {3478, 37781}, {9088, 5905}
X(44435) = X(20569)-Ceva conjugate of X(1086)
X(44435) = X(i)-isoconjugate of X(j) for these (i,j): {31, 9059}, {44, 32686}, {101, 40401}, {692, 996}, {902, 36091}
X(44435) = crossdifference of every pair of points on line {32, 902}
X(44435) = barycentric product X(i)*X(j) for these {i,j}: {76, 9002}, {513, 33934}, {514, 4389}, {523, 16712}, {693, 4850}, {903, 23888}, {995, 3261}, {3267, 4247}, {3676, 5233}, {3877, 24002}, {4424, 7199}, {7192, 26580}, {21130, 39704}
X(44435) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 9059}, {88, 36091}, {106, 32686}, {513, 40401}, {514, 996}, {995, 101}, {3877, 644}, {4247, 112}, {4266, 3939}, {4389, 190}, {4424, 1018}, {4850, 100}, {5233, 3699}, {9002, 6}, {16712, 99}, {17461, 4752}, {21130, 3679}, {23206, 906}, {23888, 519}, {26580, 3952}, {33934, 668}
X(44435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3835, 16892, 25259}, {17069, 23729, 4380}

X(44436) = MOSES-RADICAL-CIRCLE-POLE OF VAN AUBEL LINE

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 + a^6*c^2 + 4*a^4*b^2*c^2 - 3*a^2*b^4*c^2 - 2*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 + 6*b^4*c^4 + 3*a^2*c^6 - 2*b^2*c^6 - c^8) : :
Barycentrics    (csc A) (sec B cos C (sin 2A - sin 2B) + cos B sec C (sin 2A - sin 2C)) : :

X(44436) lies on these lines: {2, 216}, {3, 1495}, {20, 16253}, {51, 38283}, {110, 34147}, {122, 858}, {126, 38974}, {131, 2072}, {268, 22129}, {323, 3284}, {373, 30258}, {394, 1073}, {401, 39062}, {417, 5907}, {418, 3819}, {426, 9306}, {441, 525}, {511, 852}, {577, 15066}, {800, 37643}, {1214, 6357}, {1304, 2071}, {1350, 33924}, {1531, 10745}, {1624, 34146}, {1971, 17811}, {3218, 35072}, {3580, 15526}, {3917, 6638}, {5421, 23292}, {5562, 14059}, {5943, 13409}, {6000, 40948}, {6760, 8431}, {7386, 43460}, {7485, 26880}, {8798, 12111}, {10257, 16319}, {10601, 15851}, {11413, 14379}, {11427, 13341}, {11589, 13445}, {11695, 42441}, {16419, 26898}, {18890, 37669}, {20208, 37638}, {26895, 33879}, {30739, 42353}, {35071, 40884}, {37648, 41005}

X(44436) = midpoint of X(852) and X(2972)
X(44436) = complement of X(46106)
X(44436) = complement of polar conjugate of X(74)
X(44436) = Moses-radical-circle-inverse of X(15341)
X(44436) = complement of the isogonal conjugate of X(18877)
X(44436) = complement of the isotomic conjugate of X(14919)
X(44436) = isotomic conjugate of the polar conjugate of X(6000)
X(44436) = X(i)-complementary conjugate of X(j) for these (i,j): {48, 113}, {74, 20305}, {822, 16177}, {2159, 5}, {2349, 21243}, {9247, 3163}, {14380, 21253}, {14919, 2887}, {15291, 20308}, {18877, 10}, {32640, 8062}, {35200, 141}, {36034, 30476}, {36131, 520}, {40352, 226}, {40354, 24005}
X(44436) = X(i)-Ceva conjugate of X(j) for these (i,j): {3260, 13754}, {16077, 520}
X(44436) = X(i)-isoconjugate of X(j) for these (i,j): {19, 1294}, {647, 36043}, {656, 32646}, {2430, 36126}, {24019, 43701}
X(44436) = cevapoint of X(3284) and X(12096)
X(44436) = crosspoint of X(i) and X(j) for these (i,j): {2, 14919}, {276, 43767}
X(44436) = crosssum of X(i) and X(j) for these (i,j): {6, 1990}, {1562, 1637}
X(44436) = crossdifference of every pair of points on line {25, 9209}
X(44436) = barycentric product X(i)*X(j) for these {i,j}: {69, 6000}, {511, 36893}, {1494, 40948}, {1559, 15394}, {2442, 4143}, {3260, 39174}, {6148, 39376}
X(44436) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1294}, {112, 32646}, {162, 36043}, {520, 43701}, {1301, 39464}, {1559, 14249}, {2404, 15352}, {2442, 6529}, {6000, 4}, {32320, 2430}, {36893, 290}, {39174, 74}, {39376, 5627}, {40948, 30}
X(44436) = {X(1073),X(6617)}-harmonic conjugate of X(394)

X(44437) = MOSES-RADICAL-CIRCLE-POLE OF LINE X(2)X(6)

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

X(44437) lies on these lines: {3, 1495}, {4, 39}, {30, 36212}, {32, 11456}, {187, 12112}, {237, 6000}, {566, 36990}, {647, 1499}, {800, 39874}, {842, 37946}, {1181, 8779}, {1199, 34571}, {1498, 1971}, {1503, 3003}, {1513, 3291}, {1514, 14961}, {1561, 11672}, {3088, 14165}, {3926, 34621}, {5007, 15032}, {5188, 11459}, {5562, 9821}, {10605, 41266}, {10984, 19558}, {11438, 20897}, {12082, 30270}, {12290, 37114}, {14157, 42671}, {15072, 37465}, {15305, 37184}, {15811, 15815}, {16194, 35934}, {16261, 21163}, {16836, 37338}, {32062, 32444}

X(44437) = crossdifference of every pair of points on line {9209, 11284}

X(44438) = 2ND-DROZ-FARNY-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^6 - 4*a^4*b^2 - a^2*b^4 + 2*b^6 - 4*a^4*c^2 + 10*a^2*b^2*c^2 - 2*b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + 2*c^6) : :
X(44438) = 2 X[3] - 3 X[16072], 4 X[3] - 5 X[31255], 3 X[4] - 2 X[1596], 3 X[4] - X[18533], 4 X[4] - X[37196], 5 X[4] - 2 X[37458], 3 X[25] - 4 X[1596], 3 X[25] - 2 X[18533], 5 X[25] - 4 X[37458], 2 X[376] - 3 X[32216], 3 X[381] - 2 X[6644], 4 X[403] - 3 X[37917], 8 X[1596] - 3 X[37196], 5 X[1596] - 3 X[37458], 5 X[3091] - 4 X[6677], 3 X[3545] - 2 X[44273], 3 X[3830] - X[18534], 3 X[3830] - 2 X[44276], 3 X[3839] - 2 X[44212], 3 X[3845] - 2 X[44233], X[5073] + 2 X[14791], 5 X[5076] - 2 X[7530], X[7500] - 5 X[17578], 2 X[10605] - 3 X[26869], 3 X[14269] - 2 X[44275], 6 X[16072] - 5 X[31255], 4 X[18390] - 3 X[26869], 4 X[18531] - 3 X[31152], 4 X[18533] - 3 X[37196], 5 X[18533] - 6 X[37458], 2 X[21312] - 3 X[31152], 5 X[37196] - 8 X[37458]

X(44438) lies on these lines: {2, 3}, {6, 1562}, {33, 12943}, {34, 9627}, {64, 13399}, {125, 10606}, {146, 12165}, {148, 9308}, {185, 5895}, {1105, 18848}, {1112, 5890}, {1181, 13403}, {1204, 5925}, {1217, 18846}, {1398, 1479}, {1478, 7071}, {1498, 21659}, {1503, 10602}, {1514, 26864}, {1531, 44080}, {1539, 15472}, {1552, 9717}, {1568, 37497}, {1660, 17845}, {1824, 15942}, {1829, 41869}, {1853, 13851}, {1862, 10728}, {1870, 9668}, {1876, 3586}, {1899, 15311}, {1902, 5691}, {1986, 38790}, {2207, 7748}, {2386, 36997}, {2393, 12294}, {2777, 10605}, {2790, 5186}, {2834, 10729}, {2883, 19467}, {3092, 35820}, {3093, 35821}, {3172, 5254}, {3521, 36753}, {3767, 8778}, {5090, 31673}, {5185, 10727}, {5410, 6561}, {5411, 6560}, {5412, 42263}, {5413, 42264}, {5878, 6146}, {5894, 26937}, {5972, 18418}, {6000, 18396}, {6198, 9655}, {6225, 18945}, {6247, 15153}, {6748, 15433}, {6749, 34288}, {7687, 23329}, {7728, 18445}, {8739, 42155}, {8740, 42154}, {9541, 13884}, {9703, 15463}, {9777, 16657}, {9968, 11470}, {10311, 41336}, {10483, 11399}, {10632, 42130}, {10633, 42131}, {10641, 42096}, {10642, 42097}, {10723, 12131}, {10724, 12138}, {10733, 12133}, {10734, 10735}, {11396, 12699}, {11405, 20423}, {11408, 42085}, {11409, 42086}, {11425, 43831}, {11441, 34966}, {11457, 13093}, {11472, 12295}, {11473, 23251}, {11474, 23261}, {11475, 42094}, {11476, 42093}, {11550, 18405}, {11648, 14581}, {12111, 12282}, {12162, 12293}, {12167, 31670}, {12290, 34780}, {12292, 12902}, {12315, 34224}, {12897, 36747}, {13219, 40995}, {13474, 34786}, {13622, 14490}, {14356, 34233}, {14379, 33553}, {14989, 15111}, {15043, 43823}, {15048, 41370}, {15123, 15127}, {15152, 34782}, {15928, 39809}, {16318, 43448}, {17702, 18451}, {17846, 32340}, {18550, 38534}, {18847, 18852}, {18849, 18851}, {19124, 19136}, {20427, 34469}, {21663, 26958}, {22951, 22970}, {23698, 40801}, {29181, 41585}, {32063, 32111}, {33584, 38956}, {43273, 44102}

X(44438) = midpoint of X(1370) and X(3146)
X(44438) = reflection of X(i) in X(j) for these {i,j}: {20, 1368}, {25, 4}, {10605, 18390}, {17845, 1660}, {18533, 1596}, {18534, 44276}, {21312, 18531}, {37196, 25}
X(44438) = anticomplement of X(44241)
X(44438) = orthocentroidal-circle-inverse of X(10151)
X(44438) = 2nd-Droz-Farny-circle-inverse of X(468)
X(44438) = crosspoint of X(4) and X(18850)
X(44438) = crosssum of X(3) and X(10605)
X(44438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 10151}, {2, 16386, 3}, {3, 4, 37197}, {3, 25, 37917}, {3, 403, 37453}, {3, 3830, 31726}, {3, 16072, 31255}, {4, 20, 235}, {4, 376, 6623}, {4, 378, 381}, {4, 382, 12173}, {4, 427, 18386}, {4, 1593, 7507}, {4, 1594, 3843}, {4, 1597, 5064}, {4, 1885, 1593}, {4, 3088, 23047}, {4, 3146, 3575}, {4, 3520, 35488}, {4, 3529, 3089}, {4, 3541, 546}, {4, 3542, 44226}, {4, 3575, 5198}, {4, 6240, 1598}, {4, 6869, 431}, {4, 7487, 1906}, {4, 7576, 18535}, {4, 8889, 3839}, {4, 13488, 11403}, {4, 14865, 7547}, {4, 18533, 1596}, {4, 18560, 3}, {4, 30100, 7566}, {4, 33703, 7487}, {4, 34797, 10594}, {4, 35480, 18494}, {4, 35481, 403}, {4, 35485, 37984}, {4, 35490, 382}, {5, 16976, 2}, {5, 34350, 3}, {20, 235, 3515}, {20, 6353, 37931}, {25, 32216, 468}, {235, 37931, 6353}, {376, 6623, 468}, {378, 381, 5094}, {381, 35495, 1656}, {382, 18494, 35480}, {403, 18560, 35481}, {403, 35481, 3}, {427, 13473, 4}, {427, 18386, 7507}, {550, 3542, 15750}, {550, 44226, 3542}, {1344, 1345, 1593}, {1593, 18386, 427}, {1595, 3853, 4}, {1596, 18533, 25}, {1597, 3830, 4}, {1598, 5073, 6240}, {1885, 13473, 427}, {3517, 17800, 35471}, {3520, 35488, 1656}, {3627, 13488, 4}, {3830, 18534, 44276}, {5878, 6146, 12174}, {5899, 18561, 1657}, {6353, 37931, 3515}, {7505, 35491, 3}, {7577, 18533, 6644}, {8703, 37942, 35486}, {10605, 18390, 26869}, {12362, 37201, 37198}, {13403, 22802, 1181}, {16868, 35477, 3526}, {18325, 18564, 12083}, {18494, 35480, 12173}, {18531, 21312, 31152}, {18563, 31725, 7387}, {30771, 34622, 2071}, {31861, 44263, 381}, {37197, 37453, 403}

X(44439) = 2ND-DROZ-FARNY-CIRCLE-POLE OF LEMOINE AXIS

Barycentrics    a^2*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10 + a^8*c^2 - 3*a^6*b^2*c^2 + 4*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + b^8*c^2 - 2*a^6*c^4 + 4*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 3*a^2*b^2*c^6 + 2*a^2*c^8 + b^2*c^8 - c^10) : :
X(44439) = 5 X[6] - 4 X[389], 3 X[6] - 2 X[19161], 7 X[6] - 4 X[21851], 2 X[52] - 3 X[5102], 10 X[182] - 9 X[40280], 3 X[381] - 2 X[41714], 6 X[389] - 5 X[19161], 7 X[389] - 5 X[21851], 8 X[389] - 5 X[37473], 3 X[568] - 4 X[5097], 5 X[1352] - 6 X[15060], 5 X[3091] - 4 X[41579], 4 X[3631] - 5 X[11444], 5 X[3843] - 4 X[43129], 3 X[5085] - 2 X[37511], 4 X[5480] - 3 X[9971], 3 X[5890] - 4 X[12007], 8 X[6329] - 7 X[15043], 2 X[6403] - 3 X[9971], 4 X[11574] - 3 X[31884], 5 X[12283] + 3 X[12290], X[12283] - 3 X[15073], X[12290] + 5 X[15073], 5 X[12294] - 3 X[32062], 5 X[18438] - X[37484], 7 X[19161] - 6 X[21851], 4 X[19161] - 3 X[37473], 8 X[21851] - 7 X[37473], 6 X[32062] - 5 X[36990], 5 X[37481] - 6 X[39561]

X(44439) lies on these lines: {2, 11746}, {3, 6}, {4, 9973}, {20, 17710}, {51, 37453}, {69, 22466}, {184, 10117}, {185, 32366}, {206, 11470}, {381, 41714}, {403, 5480}, {427, 34751}, {524, 41716}, {599, 16072}, {974, 35485}, {1154, 4549}, {1352, 10113}, {1503, 12283}, {1593, 34777}, {1843, 37197}, {2393, 12294}, {2781, 6776}, {2854, 5921}, {2929, 43652}, {2979, 13567}, {3056, 9627}, {3060, 23292}, {3091, 41579}, {3589, 40929}, {3629, 5889}, {3631, 11444}, {3843, 43129}, {3917, 26958}, {5486, 11744}, {5562, 40341}, {5622, 15578}, {5890, 12007}, {5965, 13403}, {6101, 39571}, {6293, 19467}, {6329, 15043}, {6467, 30443}, {7505, 14853}, {8550, 35491}, {9306, 12310}, {10201, 20423}, {10628, 10938}, {10752, 32245}, {11412, 12241}, {11649, 31726}, {12220, 29181}, {12236, 18580}, {12902, 18435}, {13310, 19220}, {13367, 41593}, {14157, 15580}, {14913, 15752}, {15069, 34382}, {15072, 20725}, {15074, 34350}, {15140, 15463}, {15577, 18374}, {15761, 21850}, {16386, 32220}, {18390, 23039}, {19128, 35228}, {19136, 37917}, {19151, 38534}, {20987, 34787}, {21243, 32263}, {21660, 26883}, {26913, 32282}, {32246, 40330}, {33884, 37643}

X(44439) = reflection of X(i) in X(j) for these {i,j}: {20, 17710}, {185, 32366}, {1350, 9967}, {5889, 3629}, {6243, 37517}, {6403, 5480}, {9973, 4}, {36990, 12294}, {37473, 6}, {40341, 5562}, {40929, 3589}
X(44439) = 2nd-Droz-Farny-circle-inverse of X(187)
X(44439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1970, 5111, 6}, {5102, 11425, 6}, {5480, 6403, 9971}

X(44440) = 2ND-DROZ-FARNY-CIRCLE-POLE OF DE LONGCHAMPS LINE

Barycentrics    a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 10*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 6*a^4*b^2*c^4 + 10*a^2*b^4*c^4 - 2*b^6*c^4 + 2*a^4*c^6 - 6*a^2*b^2*c^6 - 2*b^4*c^6 + a^2*c^8 + 3*b^2*c^8 - c^10 : :
X(44440) = 3 X[2] - 4 X[15760], 5 X[2] - 4 X[44218], 3 X[3] - 4 X[25337], 3 X[4] - 2 X[31723], 3 X[4] - 4 X[44263], 5 X[4] - 4 X[44288], 3 X[20] - 4 X[44239], 3 X[22] - 2 X[44239], 3 X[376] - 4 X[7502], 5 X[378] - 6 X[44218], 4 X[381] - 3 X[31105], 4 X[427] - 5 X[3091], 5 X[631] - 4 X[18570], 5 X[1656] - 4 X[44236], 3 X[2071] - 4 X[16387], X[3146] + 2 X[12082], 5 X[3522] - 8 X[16618], 5 X[3522] - 4 X[44249], 7 X[3523] - 8 X[6676], 3 X[3524] - 4 X[44262], 17 X[3544] - 16 X[13413], 9 X[3545] - 8 X[39504], 3 X[3839] - 2 X[31133], 11 X[5056] - 10 X[31236], 5 X[5071] - 4 X[44287], 3 X[7391] - 4 X[31723], 3 X[7391] - 8 X[44263], 5 X[7391] - 8 X[44288], 8 X[7555] - 5 X[17538], 3 X[10304] - 4 X[44210], 5 X[15692] - 4 X[44285], 5 X[15760] - 3 X[44218], 4 X[19127] - 3 X[25406], X[20062] + 2 X[35480], 8 X[25337] - 3 X[35481], 5 X[31723] - 6 X[44288], 5 X[44263] - 3 X[44288]

X(44440) lies on these lines: {2, 3}, {68, 6241}, {69, 146}, {148, 3164}, {280, 10522}, {343, 15311}, {388, 9538}, {569, 12897}, {1352, 15305}, {1478, 3100}, {1479, 4296}, {1498, 14516}, {1503, 41614}, {1514, 15066}, {1531, 36987}, {1568, 37480}, {1614, 12118}, {1899, 15072}, {2063, 5893}, {2549, 22240}, {2777, 12827}, {2883, 11441}, {2888, 6225}, {3014, 41761}, {3098, 13202}, {3448, 17854}, {3521, 6243}, {3580, 10605}, {3767, 19220}, {3818, 32062}, {4549, 10721}, {4846, 5890}, {5270, 9643}, {5422, 16657}, {5562, 22802}, {5622, 10733}, {5654, 43574}, {5878, 12111}, {5921, 41735}, {6000, 11442}, {6193, 43605}, {6560, 11418}, {6561, 11417}, {6776, 37784}, {7712, 12319}, {7728, 23039}, {7735, 41336}, {7737, 10313}, {8717, 12295}, {8718, 12289}, {9833, 12278}, {9927, 10575}, {10249, 18382}, {10574, 39571}, {10606, 37638}, {10984, 13403}, {11185, 30737}, {11202, 16163}, {11416, 20423}, {11420, 42085}, {11421, 42086}, {11440, 20427}, {11455, 41171}, {12058, 15030}, {12174, 12429}, {12220, 31670}, {12279, 14216}, {12293, 34224}, {12384, 14983}, {13203, 41398}, {13346, 43831}, {13445, 23293}, {13491, 25738}, {14826, 15052}, {14915, 18474}, {15075, 39575}, {15438, 32605}, {15466, 34170}, {15740, 22466}, {16165, 35260}, {18390, 18911}, {18451, 32111}, {18912, 40647}, {19127, 25406}, {22555, 22951}, {34944, 40196}, {36983, 43695}, {41464, 43621}, {41465, 41466}

X(44440) = midpoint of X(i) and X(j) for these {i,j}: {3146, 20062}, {12082, 35480}
X(44440) = reflection of X(i) in X(j) for these {i,j}: {20, 22}, {378, 15760}, {3146, 35480}, {7391, 4}, {12384, 14983}, {20062, 12082}, {31723, 44263}, {35481, 3}, {44249, 16618}
X(44440) = anticomplement of X(378)
X(44440) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(37980)
X(44440) = 2nd-Droz-Farny-circle-inverse of X(858)
X(44440) = circumcircle-of-anticomplementary-triangle-inverse of X(10296)
X(44440) = de Longchamps circle inverse of X(7464)
X(44440) = anticomplement of the isogonal conjugate of X(4846)
X(44440) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1302, 7253}, {4846, 8}, {32738, 17498}, {34288, 5905}, {34289, 21270}, {36083, 9033}, {36149, 525}
X(44440) = crosssum of X(3269) and X(42660)
X(44440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20, 2071}, {2, 3543, 37077}, {3, 403, 2}, {3, 15761, 7505}, {4, 20, 37444}, {4, 376, 18531}, {4, 1370, 3153}, {4, 3529, 14790}, {4, 6815, 3091}, {4, 6997, 3839}, {4, 7401, 3832}, {4, 10996, 6816}, {4, 18420, 7394}, {4, 35513, 1370}, {4, 37201, 20}, {5, 31725, 4}, {20, 3153, 1370}, {20, 3839, 7396}, {20, 10298, 376}, {235, 31829, 17928}, {376, 7493, 10298}, {376, 18531, 16063}, {378, 15760, 2}, {381, 18325, 44276}, {381, 21312, 858}, {381, 44276, 4}, {382, 11414, 12225}, {468, 44241, 15078}, {550, 10020, 3}, {550, 44279, 18404}, {1370, 3153, 37444}, {1370, 35513, 20}, {1370, 37201, 35513}, {1885, 6823, 7503}, {3146, 34007, 4}, {3534, 18403, 14791}, {3830, 11818, 4}, {5002, 5003, 16063}, {6240, 7387, 31304}, {6816, 10996, 3523}, {7394, 18420, 7544}, {7500, 34621, 37945}, {7576, 18534, 7519}, {9927, 10575, 11457}, {10024, 12084, 37119}, {10254, 18859, 18281}, {10323, 35490, 12605}, {11414, 12225, 20}, {11744, 12825, 146}, {13383, 44240, 32534}, {14784, 14785, 12084}, {14807, 14808, 10296}, {15760, 31723, 18537}, {15760, 44239, 16387}, {16386, 31726, 3153}, {18404, 44279, 4}, {31723, 44239, 1370}, {31723, 44263, 4}, {35732, 42282, 12086}

X(44441) = 1ST-DROZ-FARNY-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 12*a^6*b^2*c^2 - 6*a^4*b^4*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 + 6*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(44441) = 9 X[2] - 8 X[34330], 7 X[2] - 8 X[34331], 5 X[2] - X[34621], 7 X[2] - 4 X[44278], 2 X[3] + X[14790], X[3] + 2 X[23335], X[4] + 2 X[12084], X[4] - 4 X[13371], 2 X[5] + X[12085], X[20] - 4 X[11250], X[20] + 2 X[18569], 2 X[26] - 5 X[631], X[26] - 4 X[23336], 2 X[26] + X[34938], X[64] + 2 X[22660], X[68] + 2 X[13346], X[68] - 4 X[20299], 4 X[140] - X[7387], X[155] + 2 X[6247], 4 X[156] - X[34781], X[376] + 2 X[31181], 5 X[631] - 8 X[23336], 5 X[631] - 4 X[34477], 5 X[631] - X[34608], 5 X[631] + X[34938], 2 X[1147] + X[14216], X[1498] - 4 X[9820], 5 X[1656] - 8 X[32144], 4 X[1658] - 7 X[3523], 4 X[1658] - X[31305], X[2935] + 2 X[23306], 7 X[3090] - 4 X[15761], 7 X[3090] - 10 X[31283], 5 X[3091] - 8 X[10224], X[3146] - 4 X[18377], X[3146] + 2 X[34350], 5 X[3522] - 8 X[10226], 7 X[3523] - X[31305], 3 X[3524] - 2 X[18324], 11 X[3525] - 8 X[10020], 7 X[3526] - 4 X[13383], 7 X[3526] - X[39568], 7 X[3528] - 4 X[44242], 7 X[3832] - 4 X[44279], 3 X[5054] - X[9909], 3 X[5054] - 2 X[34351], 11 X[5056] - 8 X[13406], 11 X[5056] - 16 X[34199], 5 X[5071] - 9 X[30775], 4 X[5448] - X[5878], 16 X[5498] - 13 X[10303], 4 X[5498] - X[17714], X[6193] + 2 X[32140], 4 X[6696] - X[12163], X[9833] - 4 X[12038], X[10201] - 3 X[18281], 3 X[10201] - 4 X[34330], 7 X[10201] - 12 X[34331], 10 X[10201] - 3 X[34621], 7 X[10201] - 6 X[44278], 3 X[10245] - 7 X[15701], 13 X[10303] - 4 X[17714], 2 X[11250] + X[18569], X[11411] + 2 X[16266], X[12084] + 2 X[13371], X[12118] + 2 X[18381], X[12302] + 2 X[23315], X[12324] + 2 X[32139], 2 X[12359] + X[37498], 2 X[12359] - 5 X[40686], X[12383] - 4 X[25487], X[13346] + 2 X[20299], 4 X[13383] - X[39568], X[14790] - 4 X[23335], 7 X[14869] - 4 X[44277], 8 X[15330] - 11 X[15721], 8 X[15331] - 11 X[15717], 5 X[15693] - X[34726], 7 X[15702] - 4 X[44213], 11 X[15720] - 5 X[16195], 2 X[15761] - 5 X[31283], 5 X[17578] - 8 X[18567], X[17834] - 4 X[44158], 9 X[18281] - 4 X[34330], 7 X[18281] - 4 X[34331], 10 X[18281] - X[34621], 7 X[18281] - 2 X[44278], 2 X[18377] + X[34350], 8 X[23336] - X[34608], 8 X[23336] + X[34938], 7 X[34330] - 9 X[34331], 40 X[34330] - 9 X[34621], 14 X[34330] - 9 X[44278], 40 X[34331] - 7 X[34621], 4 X[34477] - X[34608], 4 X[34477] + X[34938], 7 X[34621] - 20 X[44278], X[37498] + 5 X[40686], X[39812] + 2 X[39845], 2 X[39816] + X[39841]

X(44441) lies on these lines: {2, 3}, {52, 26937}, {64, 22660}, {66, 542}, {68, 13346}, {155, 6247}, {156, 34781}, {193, 10264}, {343, 37483}, {511, 23048}, {571, 6128}, {1147, 14216}, {1236, 32833}, {1272, 32837}, {1350, 44201}, {1498, 9820}, {1511, 28408}, {1853, 37497}, {1899, 13352}, {1993, 18917}, {2931, 18382}, {2935, 23306}, {3085, 32047}, {3086, 8144}, {3654, 34643}, {4846, 18388}, {5157, 38064}, {5433, 9645}, {5448, 5878}, {5654, 6000}, {5892, 14561}, {5946, 14853}, {6102, 18913}, {6193, 32140}, {6696, 12163}, {6699, 31670}, {9540, 11265}, {9833, 12038}, {10564, 18474}, {11064, 18451}, {11202, 29012}, {11266, 13935}, {11411, 16266}, {11433, 39522}, {11442, 43574}, {11457, 34148}, {11645, 36989}, {12042, 41770}, {12099, 15061}, {12118, 18381}, {12134, 35602}, {12161, 18909}, {12302, 23315}, {12324, 32139}, {12359, 37498}, {12383, 25487}, {13292, 26944}, {14852, 23332}, {15033, 18911}, {15068, 37669}, {15136, 34118}, {16658, 35264}, {17822, 17836}, {17834, 44158}, {18440, 28419}, {18445, 37645}, {18916, 36749}, {18951, 36747}, {19161, 20423}, {21243, 37480}, {25738, 37495}, {39812, 39845}, {39816, 39841}

X(44441) = midpoint of X(i) and X(j) for these {i,j}: {3, 34609}, {1853, 37497}, {3534, 34725}, {3654, 34643}, {3830, 34622}, {34608, 34938}
X(44441) = reflection of X(i) in X(j) for these {i,j}: {2, 18281}, {26, 34477}, {3543, 18568}, {7387, 10154}, {9909, 34351}, {10154, 140}, {13406, 34199}, {14070, 549}, {14790, 34609}, {14852, 23332}, {33591, 11812}, {34477, 23336}, {34608, 26}, {34609, 23335}, {44278, 34331}
X(44441) = anticomplement of X(10201)
X(44441) = 1st-Droz-Farny-circle-inverse of X(468)
X(44441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 427, 18420}, {3, 5576, 6815}, {3, 23335, 14790}, {3, 31723, 18533}, {5, 44276, 6623}, {20, 37119, 3549}, {26, 23336, 631}, {378, 858, 18531}, {382, 6640, 3542}, {549, 13490, 6644}, {631, 34938, 26}, {1595, 16196, 6642}, {1597, 30771, 5}, {2071, 31074, 4}, {3088, 3546, 5}, {3523, 31305, 1658}, {3526, 39568, 13383}, {5054, 9909, 34351}, {5054, 14787, 2}, {5094, 21312, 15760}, {7500, 35486, 2070}, {11250, 18569, 20}, {12084, 13371, 4}, {13346, 20299, 68}, {13490, 15122, 549}, {15078, 31133, 7576}, {15559, 17928, 7528}, {15761, 31283, 3090}, {15765, 18585, 6642}, {18377, 34350, 3146}, {18533, 31099, 31723}, {18586, 18587, 235}, {34551, 34552, 16197}, {37498, 40686, 12359}

X(44442) = CIRCUMCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF ORTHIC AXIS

Barycentrics    3*a^6 + 3*a^4*b^2 - 3*a^2*b^4 - 3*b^6 + 3*a^4*c^2 - 2*a^2*b^2*c^2 + 3*b^4*c^2 - 3*a^2*c^4 + 3*b^2*c^4 - 3*c^6 : :
X(44442) = 5 X[2] - 4 X[10154], 8 X[2] - 9 X[30775], X[4] - 4 X[14790], 13 X[4] - 16 X[18377], 15 X[4] - 16 X[18566], 29 X[4] - 32 X[18567], 7 X[4] - 8 X[18568], 5 X[4] - 8 X[18569], X[4] + 2 X[34938], 19 X[4] - 16 X[44279], 8 X[26] - 11 X[3525], 4 X[140] - 3 X[10245], 5 X[631] - 4 X[14070], 5 X[631] - 8 X[23335], 5 X[631] - 2 X[31305], 10 X[632] - 7 X[10244], 7 X[3090] - 4 X[7387], 5 X[3091] - 2 X[39568], 9 X[3524] - 8 X[18324], X[3529] - 4 X[12085], 13 X[5067] - 16 X[13371], 5 X[5071] - 8 X[31181], 5 X[9909] - 6 X[10154], 16 X[9909] - 27 X[30775], 4 X[9909] - 3 X[34608], X[9909] - 3 X[34609], 32 X[10154] - 45 X[30775], 8 X[10154] - 5 X[34608], 2 X[10154] - 5 X[34609], 13 X[10303] - 10 X[16195], 8 X[12084] - 5 X[17538], 13 X[14790] - 4 X[18377], 15 X[14790] - 4 X[18566], 29 X[14790] - 8 X[18567], 7 X[14790] - 2 X[18568], 5 X[14790] - 2 X[18569], 2 X[14790] + X[34938], 19 X[14790] - 4 X[44279], 32 X[15330] - 35 X[15702], 4 X[15330] - 5 X[18281], 9 X[15330] - 10 X[34478], 5 X[15694] - 4 X[33591], 7 X[15702] - 8 X[18281], 63 X[15702] - 64 X[34478], 9 X[15709] - 8 X[34351], 9 X[18281] - 8 X[34478], 15 X[18377] - 13 X[18566], 29 X[18377] - 26 X[18567], 14 X[18377] - 13 X[18568], 10 X[18377] - 13 X[18569], 8 X[18377] + 13 X[34938], 19 X[18377] - 13 X[44279], 29 X[18566] - 30 X[18567], 14 X[18566] - 15 X[18568], 2 X[18566] - 3 X[18569], 8 X[18566] + 15 X[34938], 19 X[18566] - 15 X[44279], 28 X[18567] - 29 X[18568], 20 X[18567] - 29 X[18569], 16 X[18567] + 29 X[34938], 38 X[18567] - 29 X[44279], 5 X[18568] - 7 X[18569], 4 X[18568] + 7 X[34938], 19 X[18568] - 14 X[44279], 4 X[18569] + 5 X[34938], 19 X[18569] - 10 X[44279], 4 X[23335] - X[31305], 9 X[30775] - 4 X[34608], 9 X[30775] - 16 X[34609], X[34608] - 4 X[34609], 19 X[34938] + 8 X[44279]

X(44442) is the centroid of the 3rd antipedal triangle of X(4), which is also the polar triangle of the anticomplementary circle, and the anticomplement of the Ara triangle. (Randy Hutson, September 30, 2021)

X(44442) lies on these lines: {2, 3}, {39, 15437}, {69, 11550}, {184, 14927}, {305, 32006}, {388, 4348}, {497, 7221}, {511, 32064}, {524, 34944}, {534, 11677}, {541, 12319}, {542, 13203}, {543, 39842}, {551, 34712}, {612, 5229}, {614, 5225}, {1184, 43448}, {1503, 37672}, {1853, 29181}, {1899, 21969}, {1992, 18935}, {1993, 39874}, {3060, 18950}, {3241, 34643}, {3434, 4980}, {3679, 34730}, {3828, 34642}, {3829, 34702}, {4176, 5207}, {5254, 40179}, {5485, 40178}, {6403, 12058}, {7753, 34722}, {8144, 17024}, {8280, 42260}, {8281, 42261}, {8854, 22644}, {8855, 22615}, {9766, 41761}, {10625, 33523}, {11206, 29012}, {11433, 21849}, {11645, 41736}, {12320, 32419}, {12321, 32421}, {14826, 36990}, {14831, 18909}, {15435, 21358}, {18289, 42266}, {18290, 42267}, {20049, 34729}, {21243, 33522}, {23291, 33586}, {29323, 35260}, {29815, 32047}, {30737, 32001}, {31145, 34713}, {31383, 37669}, {37643, 43621}, {37665, 42459}

X(44442) = reflection of X(i) in X(j) for these {i,j}: {2, 34609}, {3241, 34643}, {3543, 34725}, {14070, 23335}, {15683, 34622}, {20049, 34729}, {31145, 34713}, {31305, 14070}, {34608, 2}, {34621, 381}, {34642, 3828}, {34658, 10691}, {34702, 3829}, {34712, 551}, {34722, 7753}, {34726, 549}, {34730, 3679}
X(44442) = anticomplement of X(9909)
X(44442) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(37941)
X(44442) = circumcircle-of-anticomplementary-triangle-inverse of X(468)
X(44442) = anticomplement of the isogonal conjugate of X(16774)
X(44442) = X(16774)-anticomplementary conjugate of X(8)
X(44442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3543, 428}, {2, 34603, 7714}, {4, 1370, 7386}, {4, 3537, 18420}, {4, 3538, 7401}, {4, 7386, 7392}, {20, 427, 7494}, {22, 31099, 8889}, {25, 7396, 16051}, {376, 15682, 18559}, {381, 10691, 2}, {382, 1368, 6995}, {428, 31152, 2}, {858, 7500, 6353}, {1368, 6995, 40132}, {1370, 6997, 16063}, {1370, 7391, 4}, {2043, 2044, 7400}, {3146, 7396, 25}, {3529, 8889, 22}, {3627, 5020, 7408}, {5064, 7667, 2}, {5159, 21974, 2}, {5189, 7391, 1370}, {6353, 33703, 7500}, {7714, 15682, 34603}, {14790, 34938, 4}, {14807, 14808, 468}, {20062, 31074, 7493}, {23335, 31305, 631}, {34621, 34658, 34608}

X(44443) = CIRCUMCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF LEMOINE AXIS

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

X(44443) lies on these lines: {2, 22062}, {4, 69}, {6, 37190}, {20, 20775}, {30, 20794}, {95, 5171}, {193, 14957}, {216, 7386}, {382, 22152}, {393, 3289}, {420, 28408}, {458, 37491}, {1370, 3164}, {3619, 17500}, {5080, 25311}, {5999, 40947}, {6643, 30258}, {6997, 16990}, {7391, 7779}, {7392, 14767}, {7762, 26926}, {7900, 32747}, {9306, 32085}, {9723, 11676}, {9917, 37337}, {11008, 25051}, {32428, 34938}, {32854, 36855}, {33971, 37498}, {37124, 37488}

X(44443) = circumcircle-of-anticomplementary-triangle-inverse of X(5167)
X(44443) = polar conjugate of the isotomic conjugate of X(28441)
X(44443) = barycentric product X(4)*X(28441)
X(44443) = barycentric quotient X(28441)/X(69)
X(44443) = {X(315),X(9230)}-harmonic conjugate of X(69)

X(44444) = CIRCUMCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF NAGEL LINE

Barycentrics    (b - c)*(-a^4 - a^3*b - a^2*b^2 - a*b^3 - a^3*c + a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - a*c^3 + b*c^3) : :
X(44444) = 5 X[631] - 4 X[39225], 7 X[3090] - 8 X[39508], X[4419] - 4 X[24698]

X(44444) lies on these lines: {2, 4057}, {4, 2457}, {69, 21304}, {388, 43924}, {497, 42312}, {513, 2517}, {523, 2528}, {631, 39225}, {659, 25299}, {834, 21302}, {966, 20979}, {1370, 20294}, {1459, 28470}, {2345, 21389}, {2789, 21103}, {3090, 39508}, {3261, 17159}, {3733, 31291}, {3837, 26097}, {4106, 17896}, {4132, 20950}, {4419, 24698}, {4491, 31946}, {4648, 21191}, {7253, 14288}, {7386, 20315}, {9002, 20293}, {10566, 26853}, {17217, 21303}, {17314, 17458}, {20060, 23345}, {21102, 28487}

X(44444) = reflection of X(i) in X(j) for these {i,j}: {4491, 31946}, {7253, 14288}, {31291, 3733}
X(44444) = anticomplement of X(4057)
X(44444) = circumcircle-of-anticomplementary-triangle-inverse of X(6788)
X(44444) = anticomplement of the isogonal conjugate of X(8050)
X(44444) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {100, 24068}, {190, 18133}, {596, 149}, {8050, 8}, {34594, 1}, {37205, 75}, {39798, 4440}, {39949, 17154}, {40013, 150}, {40085, 21221}, {40148, 9263}, {40519, 192}
X(44444) = crossdifference of every pair of points on line {2300, 5007}

X(44445) = CIRCUMCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF LINE X(2)X(6)

Barycentrics    (b^2 - c^2)*(-a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2) : :
X(44445) = 3 X[2] - 4 X[23301], 9 X[2] - 10 X[31279], 5 X[631] - 4 X[5926], 2 X[647] - 3 X[5996], 4 X[647] - 3 X[9147], X[669] - 3 X[31176], 3 X[669] - 5 X[31279], 8 X[2501] - 9 X[5466], 7 X[3090] - 8 X[39511], 2 X[3804] - 3 X[4108], 3 X[4108] - 4 X[30476], X[8664] - 3 X[9148], 2 X[8664] - 5 X[31072], 6 X[9148] - 5 X[31072], 3 X[9979] - 4 X[12075], 2 X[23301] - 3 X[31176], 6 X[23301] - 5 X[31279], 4 X[23301] - X[31299], 9 X[31176] - 5 X[31279], 6 X[31176] - X[31299], 10 X[31279] - 3 X[31299]


X(44445) lies on these lines: {2, 669}, {4, 1499}, {69, 9009}, {316, 512}, {523, 2528}, {631, 5926}, {647, 5996}, {688, 14295}, {804, 3005}, {924, 30735}, {1370, 6563}, {3090, 39511}, {3221, 9493}, {3288, 30217}, {3566, 33294}, {3569, 13307}, {3804, 4108}, {3907, 27469}, {4010, 27712}, {4132, 24719}, {4151, 4382}, {4367, 26822}, {4455, 27045}, {4774, 31290}, {6655, 14824}, {6776, 30451}, {7234, 26983}, {7392, 14341}, {7533, 8371}, {7785, 23099}, {8639, 31291}, {8640, 30968}, {8664, 9148}, {9168, 16063}, {9491, 26823}, {9979, 12075}, {10278, 37349}, {10359, 39518}, {11550, 21646}, {12077, 32473}, {17072, 27673}, {18020, 32729}, {18197, 31330}, {20953, 22322}, {20983, 25298}, {24721, 40471}, {28470, 30094}, {34944, 43673}

X(44445) = reflection of X(i) in X(j) for these {i,j}: {2, 31176}, {669, 23301}, {3804, 30476}, {9147, 5996}, {31291, 8639}, {31296, 3005}, {31299, 669}
X(44445) = complement of X(31299)
X(44445) = anticomplement of X(669)
X(44445) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(32531)
X(44445) = circumcircle-of-anticomplementary-triangle-inverse of X(6792)
X(44445) = anticomplement of the isogonal conjugate of X(670)
X(44445) = anticomplement of the isotomic conjugate of X(4609)
X(44445) = isotomic conjugate of the anticomplement of X(38996)
X(44445) = isotomic conjugate of the isogonal conjugate of X(21006)
X(44445) = polar conjugate of the isogonal conjugate of X(22159)
X(44445) = anticomplementary isogonal conjugate of X(25054)
X(44445) = X(4609)-Ceva conjugate of X(2)
X(44445) = X(i)-cross conjugate of X(j) for these (i,j): {8711, 21006}, {38996, 2}
X(44445) = X(i)-isoconjugate of X(j) for these (i,j): {163, 6664}, {1964, 6573}
X(44445) = cevapoint of X(21006) and X(22159)
X(44445) = crosspoint of X(i) and X(j) for these (i,j): {83, 670}, {99, 40416}, {40425, 42371}
X(44445) = crosssum of X(i) and X(j) for these (i,j): {39, 669}, {512, 20859}, {667, 23632}, {9494, 11205}
X(44445) = crossdifference of every pair of points on line {3051, 3229}
X(44445) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 25054}, {2, 21220}, {75, 148}, {76, 21221}, {81, 21224}, {86, 9263}, {99, 192}, {110, 17486}, {163, 8264}, {190, 1655}, {274, 4440}, {304, 39352}, {310, 149}, {314, 39351}, {561, 3448}, {643, 21218}, {645, 3177}, {648, 21216}, {662, 194}, {668, 1654}, {670, 8}, {689, 17165}, {789, 40721}, {799, 2}, {811, 193}, {823, 6392}, {873, 17154}, {874, 39367}, {892, 17497}, {1502, 21294}, {1930, 39346}, {1978, 2895}, {3112, 25047}, {3222, 21223}, {3736, 39347}, {4554, 17778}, {4561, 18666}, {4563, 6360}, {4567, 21225}, {4572, 2475}, {4573, 3210}, {4576, 21217}, {4577, 17489}, {4584, 19565}, {4589, 17759}, {4590, 4560}, {4592, 3164}, {4593, 6}, {4594, 21226}, {4599, 8267}, {4600, 17494}, {4601, 514}, {4602, 69}, {4609, 6327}, {4610, 17147}, {4615, 17495}, {4616, 17480}, {4620, 17496}, {4623, 1}, {4625, 145}, {4631, 63}, {4632, 3995}, {4633, 41839}, {4634, 519}, {4635, 4452}, {4639, 6542}, {6331, 5905}, {6385, 150}, {6386, 1330}, {7035, 31290}, {7257, 144}, {7258, 30695}, {7260, 6646}, {14210, 39356}, {16709, 39348}, {18020, 17498}, {18155, 17036}, {18157, 39353}, {18829, 17493}, {18833, 25051}, {23999, 33294}, {24037, 523}, {24039, 8591}, {24041, 31296}, {28660, 37781}, {30939, 39349}, {30940, 39362}, {31614, 6758}, {32680, 19570}, {34537, 7192}, {35137, 28598}, {36036, 385}, {36806, 1959}, {36860, 21219}, {37134, 40858}, {37204, 76}, {40072, 33650}, {40364, 13219}, {42371, 21278}, {44168, 17217}
X(44445) = barycentric product X(i)*X(j) for these {i,j}: {1, 20953}, {76, 21006}, {264, 22159}, {274, 22322}, {308, 8711}, {523, 7760}, {661, 18064}, {850, 1627}, {1577, 33760}, {3005, 41297}, {4609, 38996}
X(44445) = barycentric quotient X(i)/X(j) for these {i,j}: {83, 6573}, {523, 6664}, {1627, 110}, {7760, 99}, {8711, 39}, {18064, 799}, {20953, 75}, {21006, 6}, {22159, 3}, {22322, 37}, {33760, 662}, {38996, 669}, {41297, 689}
X(44445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31299, 669}, {669, 23301, 2}, {669, 31176, 23301}, {3804, 30476, 4108}, {24533, 28401, 27677}, {24533, 31003, 2}, {25299, 26148, 2}

X(44446) = INCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF ORTHIC AXIS

Barycentrics    5*a^3 - a^2*b - 3*a*b^2 - b^3 - a^2*c + 3*b^2*c - 3*a*c^2 + 3*b*c^2 - c^3 : :
X(44446) = 2 X[33144] - 3 X[35261]

X(44446) lies on these lines: {8, 30}, {46, 2899}, {63, 12717}, {144, 32932}, {190, 3474}, {200, 4480}, {344, 11246}, {345, 17768}, {545, 3052}, {612, 20073}, {1707, 28526}, {1836, 30741}, {2796, 33137}, {3616, 26729}, {3683, 42697}, {3685, 9965}, {3757, 4454}, {4418, 17257}, {4427, 5905}, {4438, 28546}, {4488, 9778}, {5435, 17777}, {5698, 32939}, {9369, 20070}, {9780, 32950}, {10453, 28610}, {17316, 32936}, {17336, 26040}, {17767, 33144}, {20017, 31301}, {20078, 32929}, {24248, 25453}, {24695, 32934}, {25734, 27549}, {26132, 29865}, {29579, 33067}, {29864, 33100}

X(44446) = reflection of X(30699) in X(1707)
X(44446) = incircle-of-anticomplementary-triangle-inverse of X(16304)
X(44446) = {X(4488),X(9778)}-harmonic conjugate of X(32937)

X(44447) = INCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    3*a^3 - a^2*b - a*b^2 - b^3 - a^2*c + b^2*c - a*c^2 + b*c^2 - c^3 : :
X(44447) = 9 X[2] - 8 X[3838], 3 X[2] - 4 X[4640], 3 X[63] - 2 X[4847], 3 X[210] - 2 X[41871], 2 X[226] - 3 X[35258], 4 X[946] - 5 X[6974], 3 X[1836] - 4 X[3838], 3 X[3434] - 4 X[4847], 5 X[3616] - 6 X[16370], 5 X[3616] - 4 X[39542], 2 X[3838] - 3 X[4640], 3 X[3873] - 4 X[10391], 3 X[5603] - 4 X[6914], 3 X[5657] - 2 X[6923], 2 X[7580] - 3 X[9778], 4 X[8727] - 3 X[9812], 7 X[9780] - 6 X[17532], 4 X[13405] - 3 X[31164], 3 X[16370] - 2 X[39542], 2 X[22791] - 3 X[28444]

X(44447) lies on these lines: {2, 1155}, {3, 11415}, {4, 1748}, {7, 1617}, {8, 30}, {20, 3869}, {21, 4295}, {22, 1633}, {31, 19785}, {40, 3436}, {42, 24695}, {46, 2478}, {55, 5905}, {63, 516}, {65, 6872}, {69, 20291}, {78, 31730}, {79, 10198}, {100, 329}, {144, 3059}, {145, 28646}, {149, 24477}, {162, 17903}, {165, 908}, {190, 10327}, {192, 20069}, {193, 3896}, {200, 17781}, {210, 41871}, {226, 35258}, {283, 1777}, {321, 24280}, {345, 4427}, {346, 33078}, {376, 4511}, {377, 1770}, {390, 3873}, {497, 3218}, {513, 43991}, {517, 6938}, {518, 20075}, {527, 3870}, {529, 12648}, {535, 12647}, {550, 5730}, {553, 4666}, {752, 32934}, {758, 4302}, {896, 33094}, {902, 33098}, {912, 37000}, {944, 14988}, {946, 4652}, {956, 28174}, {960, 4190}, {962, 1012}, {1001, 11246}, {1145, 38756}, {1158, 6836}, {1376, 31018}, {1479, 1727}, {1707, 3914}, {1759, 17732}, {1760, 11677}, {1788, 5046}, {1889, 5278}, {2094, 10580}, {2096, 5731}, {2194, 26830}, {2345, 33083}, {2550, 3219}, {2796, 4362}, {3000, 25941}, {3052, 3782}, {3057, 20076}, {3146, 5086}, {3152, 12940}, {3185, 35980}, {3189, 20066}, {3241, 34740}, {3306, 40998}, {3419, 28146}, {3475, 17483}, {3476, 20067}, {3485, 4189}, {3486, 15680}, {3487, 14450}, {3534, 10609}, {3550, 33099}, {3556, 16049}, {3579, 5552}, {3600, 3890}, {3616, 5303}, {3647, 19854}, {3652, 18517}, {3717, 25734}, {3744, 17276}, {3827, 20243}, {3868, 4294}, {3872, 28194}, {3874, 4309}, {3877, 4293}, {3878, 4299}, {3884, 4317}, {3886, 4001}, {3899, 4316}, {3901, 4330}, {3911, 10584}, {3916, 10527}, {3920, 4419}, {3923, 26034}, {3928, 9580}, {3929, 25006}, {3935, 34607}, {3937, 35645}, {3957, 10385}, {3962, 20013}, {3996, 17347}, {4000, 17127}, {4184, 17139}, {4191, 15507}, {4292, 5250}, {4297, 11682}, {4307, 28606}, {4312, 4512}, {4313, 34195}, {4314, 11520}, {4329, 8822}, {4333, 17647}, {4338, 12609}, {4388, 17740}, {4414, 26098}, {4415, 37540}, {4418, 19822}, {4463, 12530}, {4644, 17018}, {4645, 17776}, {4650, 11269}, {4655, 33171}, {4660, 33163}, {4661, 20095}, {4756, 5423}, {4855, 12512}, {4865, 28494}, {4972, 26065}, {4973, 10072}, {5080, 5657}, {5128, 24982}, {5180, 5603}, {5187, 24914}, {5218, 31053}, {5220, 34612}, {5230, 24851}, {5273, 33108}, {5284, 9776}, {5289, 15326}, {5314, 24309}, {5327, 27174}, {5493, 12527}, {5535, 26333}, {5554, 37567}, {5692, 15228}, {5697, 36977}, {5739, 32932}, {5744, 8727}, {5794, 31295}, {5852, 41711}, {5887, 6934}, {6224, 38753}, {6284, 12649}, {6734, 41869}, {6840, 14647}, {6871, 26066}, {6907, 11681}, {6910, 12047}, {6921, 21616}, {6936, 34339}, {6962, 12608}, {7262, 24715}, {7406, 24633}, {7613, 26724}, {8616, 32857}, {9579, 24987}, {9612, 10585}, {9780, 17532}, {9791, 37327}, {10052, 14798}, {10056, 16152}, {10164, 30852}, {10200, 37524}, {10404, 10587}, {10528, 37568}, {10529, 12701}, {10531, 37532}, {10586, 32636}, {11113, 36279}, {11114, 18391}, {11206, 17134}, {11220, 43161}, {11362, 40264}, {12116, 24467}, {12532, 13199}, {12635, 15338}, {13405, 31164}, {14206, 17860}, {14923, 20070}, {15310, 26893}, {15803, 41012}, {16558, 18513}, {16865, 28629}, {17126, 33100}, {17147, 20064}, {17150, 42058}, {17156, 28580}, {17484, 25568}, {17594, 41011}, {17601, 33096}, {17724, 21000}, {17747, 26258}, {17753, 23407}, {17764, 32853}, {17767, 32920}, {19535, 37737}, {19819, 32914}, {20011, 31301}, {20012, 20072}, {22060, 31394}, {22791, 28444}, {24692, 29642}, {25527, 35263}, {26040, 27065}, {26105, 27003}, {26132, 35261}, {26364, 37572}, {26842, 38053}, {27385, 35242}, {28508, 32946}, {28609, 35445}, {28610, 30332}, {30295, 35977}, {30652, 33155}, {30653, 33150}, {30985, 35270}, {31631, 35997}, {32842, 42049}, {32940, 36479}, {33134, 37642}, {34610, 38460}, {34647, 37600}, {35915, 40214}, {36277, 40940}, {41539, 41563}

X(44447) = midpoint of X(20075) and X(20078)
X(44447) = reflection of X(i) in X(j) for these {i,j}: {962, 1012}, {1836, 4640}, {3434, 63}, {5905, 55}, {6925, 40}, {10431, 1709}, {14450, 37286}, {33088, 32934}
X(44447) = anticomplement of X(1836)
X(44447) = anticomplement of the isogonal conjugate of X(37741)
X(44447) = anticomplement of the isotomic conjugate of X(34409)
X(44447) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {34409, 6327}, {37741, 8}
X(44447) = X(34409)-Ceva conjugate of X(2)
X(44447) = crosssum of X(512) and X(22094)
X(44447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {31, 24248, 19785}, {144, 17784, 3681}, {329, 9778, 100}, {390, 9965, 3873}, {896, 33094, 33137}, {902, 33098, 33144}, {1155, 24703, 2}, {1707, 3914, 24597}, {1770, 12514, 377}, {1836, 4640, 2}, {3052, 3782, 26228}, {3474, 5698, 2}, {3683, 5880, 2}, {3916, 12699, 10527}, {3928, 9580, 26015}, {4312, 4512, 5249}, {4427, 6327, 345}, {4450, 32933, 8}, {4650, 33095, 11269}, {4676, 33068, 2}, {5744, 9812, 11680}, {16370, 39542, 3616}, {17127, 33102, 4000}, {28610, 30332, 36845}, {30332, 36845, 34611}

X(44448) = INCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF SODDY LINE

Barycentrics    (b - c)*(-a + b + c)*(a^2 - 2*a*b + b^2 - 2*a*c + c^2) : :
X(44448) = X[4959] - 3 X[14432]

X(44448) lies on these lines: {8, 514}, {10, 21185}, {75, 30181}, {78, 663}, {100, 2730}, {200, 4040}, {522, 3717}, {850, 4151}, {1734, 4025}, {2832, 28591}, {3239, 28058}, {3309, 4468}, {3667, 4498}, {3669, 4925}, {3872, 4449}, {3900, 6332}, {4420, 4794}, {4437, 30188}, {4528, 6362}, {4729, 28478}, {4847, 21183}, {4959, 14432}, {6065, 32094}, {6734, 17072}, {8058, 20294}, {17880, 24010}, {21124, 28161}

X(44448) = reflection of X(i) in X(j) for these {i,j}: {8, 4546}, {3669, 4925}, {4025, 1734}, {4391, 4163}, {21185, 10}
X(44448) = X(41790)-anticomplementary conjugate of X(150)
X(44448) = X(i)-Ceva conjugate of X(j) for these (i,j): {3261, 3239}, {6558, 8}
X(44448) = X(4904)-cross conjugate of X(8)
X(44448) = X(i)-isoconjugate of X(j) for these (i,j): {56, 1292}, {101, 17107}, {109, 2191}, {241, 32644}, {277, 1415}, {604, 37206}, {692, 40154}, {1458, 36041}, {1462, 2428}
X(44448) = crosspoint of X(i) and X(j) for these (i,j): {75, 3699}, {190, 8817}
X(44448) = crosssum of X(i) and X(j) for these (i,j): {31, 43924}, {649, 7083}
X(44448) = barycentric product X(i)*X(j) for these {i,j}: {8, 4468}, {218, 35519}, {312, 3309}, {318, 24562}, {341, 43049}, {344, 522}, {346, 31605}, {645, 21945}, {668, 38375}, {1445, 4397}, {2402, 3717}, {3239, 6604}, {3261, 6600}, {3699, 4904}, {3870, 4391}, {3900, 21609}, {3991, 18155}, {4076, 23760}, {4086, 41610}, {4163, 17093}, {6558, 40615}, {7719, 35518}, {8642, 28659}
X(44448) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 37206}, {9, 1292}, {218, 109}, {294, 36041}, {344, 664}, {513, 17107}, {514, 40154}, {522, 277}, {650, 2191}, {1445, 934}, {1617, 1461}, {2195, 32644}, {2340, 2428}, {2440, 1416}, {3239, 6601}, {3309, 57}, {3717, 2414}, {3870, 651}, {3991, 4551}, {4350, 4617}, {4468, 7}, {4878, 4559}, {4904, 3676}, {6600, 101}, {6604, 658}, {7719, 108}, {8642, 604}, {17093, 4626}, {21059, 1415}, {21609, 4569}, {21945, 7178}, {23760, 1358}, {24562, 77}, {31605, 279}, {31638, 927}, {38375, 513}, {41539, 1020}, {41610, 1414}, {43049, 269}

X(44449) = INCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF LINE X(2)X(6)

Barycentrics    (b - c)*(-a^2 - 2*a*b + b^2 - 2*a*c + b*c + c^2) : :
X(44449) 9 X[2] - 8 X[2487], 3 X[2] - 4 X[14321], 2 X[649] - 3 X[30565], 3 X[661] - 2 X[21196], 6 X[1638] - 7 X[27138], 6 X[1639] - 5 X[27013], 4 X[2487] - 3 X[4897], 2 X[2487] - 3 X[14321], 5 X[3616] - 4 X[39545], 4 X[3700] - 3 X[4789], 2 X[3776] - 3 X[31147], 4 X[3798] - 5 X[31209], 4 X[3835] - 3 X[4453], 2 X[4025] - 3 X[4776], 3 X[4120] - 2 X[4369], 3 X[4467] - 4 X[21196], 4 X[4521] - 3 X[4786], 3 X[4750] - 4 X[25666], 3 X[4789] - 2 X[7192], X[14779] - 5 X[31290], 3 X[20295] - 2 X[23729], 2 X[21104] - 3 X[21297]

X(44449) lies on these lines: {2, 2487}, {8, 1499}, {226, 31603}, {513, 4122}, {514, 4838}, {523, 4963}, {649, 28867}, {661, 2786}, {693, 28846}, {824, 4813}, {900, 17494}, {918, 20295}, {1638, 27138}, {1639, 27013}, {3616, 39545}, {3667, 4380}, {3700, 4789}, {3709, 16751}, {3776, 31147}, {3798, 31209}, {3835, 4453}, {3904, 29148}, {4024, 28840}, {4025, 4776}, {4079, 27469}, {4106, 4949}, {4120, 4369}, {4382, 28851}, {4462, 28478}, {4498, 28493}, {4500, 28886}, {4521, 4786}, {4750, 25666}, {4784, 18004}, {4822, 29037}, {4841, 17161}, {4958, 28855}, {4983, 29090}, {5287, 7203}, {7180, 28606}, {7265, 15309}, {21104, 21297}, {23731, 28863}, {25020, 25902}, {26853, 28217}

X(44449) = reflection of X(i) in X(j) for these {i,j}: {4106, 4949}, {4380, 4468}, {4467, 661}, {4784, 18004}, {4897, 14321}, {7192, 3700}, {17161, 4841}
X(44449) = anticomplement of X(4897)
X(44449) = X(8599)-Ceva conjugate of X(4789)
X(44449) = barycentric product X(514)*X(17315)
X(44449) = barycentric quotient X(17315)/X(190)
X(44449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3700, 7192, 4789}, {4897, 14321, 2}

X(44450) = INCIRCLE-OF-ANTICOMPLEMENTARY-TRIANGLE-POLE OF HATZIPOLAKIS AXIS

Barycentrics    a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 13*a^6*b^2*c^2 - 8*a^4*b^4*c^2 + a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 8*a^4*b^2*c^4 + 4*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 + a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(44450) = 17 X[2] - 8 X[44266], 11 X[2] - 8 X[44282], X[4] + 2 X[18859], X[4] - 4 X[37938], 2 X[5] + X[35452], X[20] + 8 X[858], X[20] - 4 X[2071], X[20] + 2 X[3153], 5 X[20] + 4 X[10296], 5 X[20] - 8 X[16386], X[23] - 4 X[10257], 4 X[23] - 13 X[10303], 4 X[140] - X[5899], X[146] - 4 X[1568], X[146] + 2 X[13445], 4 X[186] - 7 X[3523], 2 X[186] + X[5189], X[186] - 4 X[15122], 8 X[403] - 11 X[5056], 2 X[403] - 5 X[30745], 2 X[403] + X[37944], 4 X[468] - X[37945], 5 X[631] - 2 X[2070], 10 X[631] - X[20063], 25 X[631] - 16 X[22249], 2 X[858] + X[2071], 4 X[858] - X[3153], 10 X[858] - X[10296], 5 X[858] + X[16386], 2 X[1568] + X[13445], 5 X[1656] - 2 X[43893], 4 X[2070] - X[20063], 5 X[2070] - 8 X[22249], 2 X[2071] + X[3153], 5 X[2071] + X[10296], 5 X[2071] - 2 X[16386], 8 X[2072] - 5 X[3091], 2 X[2072] + X[7464], 7 X[3090] - 4 X[11563], 7 X[3090] + 2 X[35001], 5 X[3091] + 4 X[7464], X[3146] - 4 X[18403], X[3146] + 8 X[37950], 5 X[3153] - 2 X[10296], 5 X[3153] + 4 X[16386], X[3448] + 2 X[43574], 5 X[3522] + 4 X[7574], 5 X[3522] - 2 X[13619], 5 X[3522] - 8 X[34152], 7 X[3523] + 2 X[5189], 7 X[3523] - 16 X[15122], 3 X[3524] - 2 X[37955], 11 X[3525] - 2 X[37924], 11 X[3525] - 8 X[44234], 7 X[3526] - 4 X[10096], 7 X[3526] - X[37949], 17 X[3533] - 8 X[25338], 7 X[3832] - 4 X[31726], 7 X[3851] - 4 X[11558], 17 X[3854] - 8 X[44267], 3 X[5054] - 2 X[16532], 3 X[5054] - X[37956], 11 X[5056] - 20 X[30745], 11 X[5056] + 4 X[37944], X[5059] + 8 X[18572], 13 X[5068] - 4 X[18325], X[5189] + 8 X[15122], 17 X[7486] - 8 X[11799], 2 X[7574] + X[13619], X[7574] + 2 X[34152], 4 X[10096] - X[37949], 16 X[10257] - 13 X[10303], X[10296] + 2 X[16386], 3 X[10304] - 4 X[37948], 2 X[10564] + X[25739], 4 X[10989] + 5 X[15692], 2 X[11563] + X[35001], X[13619] - 4 X[34152], 4 X[14156] - X[14157], X[14683] - 4 X[22115], 14 X[14869] - 5 X[37923], 8 X[15646] - 11 X[15717], 5 X[15692] - 4 X[37941], 9 X[15708] - 4 X[37940], 11 X[15721] - 2 X[37901], 11 X[15721] - 4 X[37939], 11 X[15721] - 8 X[44214], 8 X[16531] - 5 X[37953], 8 X[16976] - 5 X[37952], X[18403] + 2 X[37950], X[18859] + 2 X[37938], 5 X[20063] - 32 X[22249], 5 X[30745] + X[37944], 5 X[37760] - 2 X[37925], X[37900] - 4 X[37935], X[37901] - 4 X[44214], X[37924] - 4 X[44234], 17 X[37943] - 12 X[44266], 11 X[37943] - 12 X[44282], 11 X[44266] - 17 X[44282]

X(44450) lies on these lines: {2, 3}, {146, 1568}, {539, 3448}, {1157, 43768}, {2697, 7953}, {2888, 20299}, {2979, 23329}, {3100, 3582}, {3584, 4296}, {5900, 10264}, {7691, 25563}, {7799, 30737}, {7809, 13219}, {8718, 43839}, {9143, 43572}, {9538, 10072}, {9833, 38942}, {10149, 14986}, {10168, 19121}, {10420, 13597}, {10519, 11649}, {10564, 25739}, {10625, 43608}, {10627, 12226}, {11420, 16241}, {11421, 16242}, {12058, 16226}, {12219, 20126}, {13391, 15061}, {13482, 43573}, {13857, 15100}, {14072, 33855}, {14156, 14157}, {14683, 22115}, {23293, 37480}, {37472, 43838}, {37495, 43808}

X(44450) = midpoint of X(10989) and X(37941)
X(44450) = reflection of X(i) in X(j) for these {i,j}: {9143, 43572}, {35489, 3}, {37901, 37939}, {37922, 549}, {37939, 44214}, {37956, 16532}
X(44450) = anticomplement of X(37943)
X(44450) = orthoptic-circle-of-Steiner-inellipe-inverse of X(10691)
X(44450) = de Longchamps circle inverse of X(5)
X(44450) = anticomplement of the isogonal conjugate of X(43704)
X(44450) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1291, 7253}, {2169, 13512}, {13582, 21270}, {14579, 5905}, {43704, 8}
X(44450) = crosssum of X(3269) and X(6140)
X(44450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 31181, 18559}, {376, 18281, 2}, {858, 2071, 3153}, {1370, 10298, 20}, {1568, 13445, 146}, {2071, 3153, 20}, {2071, 10296, 16386}, {3526, 37949, 10096}, {5002, 5003, 20063}, {5004, 5005, 37920}, {5054, 37956, 16532}, {5189, 15122, 3523}, {7574, 34152, 13619}, {13619, 34152, 3522}, {14787, 15702, 2}, {18859, 37938, 4}, {28447, 28448, 34006}, {30745, 37944, 403}

X(44451) = NINE-POINT-CIRCLE-OF-MEDIAL-TRIANGLE-POLE OF LINE X(2)X(6)

Barycentrics    (b^2 - c^2)*(2*a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2) : :
X(44451) = 3 X[2] + X[669], 5 X[2] - X[31176], 9 X[2] - 5 X[31279], 15 X[2] + X[31299], 3 X[351] + X[850], X[647] - 3 X[11176], 5 X[669] + 3 X[31176], 3 X[669] + 5 X[31279], 5 X[669] - X[31299], X[3005] + 3 X[4108], 3 X[5996] + X[8664], X[6563] - 3 X[10190], 3 X[8644] + 5 X[31277], 3 X[9147] + 5 X[31072], X[9148] + 3 X[15724], 3 X[10189] - 4 X[14341], 5 X[23301] - 3 X[31176], 3 X[23301] - 5 X[31279], 5 X[23301] + X[31299], 9 X[31176] - 25 X[31279], 3 X[31176] + X[31299], 25 X[31279] + 3 X[31299], 3 X[32193] - X[33294]

X(44451) lies on these lines: {2, 669}, {5, 5926}, {140, 1499}, {230, 231}, {351, 850}, {420, 39201}, {423, 23864}, {512, 31286}, {525, 5113}, {659, 26114}, {804, 8651}, {1576, 18020}, {3005, 4108}, {3221, 24675}, {3265, 9479}, {3566, 24284}, {3589, 9009}, {3628, 39511}, {3837, 25511}, {4367, 27527}, {5972, 22103}, {5996, 8664}, {6563, 10190}, {6677, 10189}, {7234, 21051}, {7857, 23099}, {7907, 14824}, {8062, 9508}, {8644, 31277}, {8655, 30968}, {9147, 31072}, {9148, 15724}, {10104, 39518}, {18282, 32204}, {20979, 24674}, {21841, 39533}, {23655, 25636}, {24666, 24755}, {32193, 33294}

X(44451) = midpoint of X(i) and X(j) for these {i,j}: {5, 5926}, {669, 23301}, {8651, 30476}
X(44451) = reflection of X(39511) in X(3628)
X(44451) = complement of X(23301)
X(44451) = complement of the isotomic conjugate of X(3222)
X(44451) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 6374}, {2998, 21253}, {3222, 2887}, {3223, 125}, {3224, 8287}, {3504, 34846}, {15389, 16573}, {34248, 115}
X(44451) = X(662)-isoconjugate of X(30496)
X(44451) = crosspoint of X(2) and X(3222)
X(44451) = crosssum of X(6) and X(3221)
X(44451) = crossdifference of every pair of points on line {3, 3229}
X(44451) = radical center of polar circles of {ABC, 1st Brocard triangle, 1st anti-Brocard triangle}
X(44451) = barycentric product X(523)*X(3552)
X(44451) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 30496}, {3552, 99}
X(44451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 669, 23301}, {2, 24533, 25126}, {669, 31176, 31299}

X(44452) = NINE-POINT-CIRCLE-OF-MEDIAL-TRIANGLE-POLE OF HATZIPOLAKIS AXIS

Barycentrics    2*a^10 - 5*a^8*b^2 + 2*a^6*b^4 + 4*a^4*b^6 - 4*a^2*b^8 + b^10 - 5*a^8*c^2 + 8*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 - 4*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 6*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(44452) = 3 X[2] + X[186], 9 X[2] - X[3153], 3 X[2] + 2 X[16531], 2 X[2] + X[18579], 5 X[2] + X[44265], 3 X[3] - X[16386], 3 X[3] + X[31726], X[4] + 3 X[37941], X[5] - 4 X[37911], X[23] + 11 X[3525], 2 X[140] + X[468], 3 X[140] + X[10096], 4 X[140] - X[15122], 8 X[140] + X[16619], 5 X[140] + X[25338], 6 X[140] + X[37971], 3 X[186] + X[3153], 2 X[186] - 3 X[18579], X[186] - 3 X[44214], 5 X[186] - 3 X[44265], 3 X[403] + X[16386], 3 X[403] - X[31726], 3 X[468] - 2 X[10096], 2 X[468] + X[15122], 4 X[468] - X[16619], 5 X[468] - 2 X[25338], 3 X[468] - X[37971], 3 X[549] + X[11563], 3 X[549] - 2 X[16976], 3 X[549] - X[34152], 3 X[549] + 2 X[37942], X[550] + 2 X[37984], 5 X[631] - X[2071], 5 X[631] + X[11799], 5 X[631] + 3 X[37943], 5 X[632] - 2 X[5159], 5 X[632] + X[7575], 5 X[632] + 3 X[16532], 5 X[632] + 2 X[37935], 5 X[632] - X[37938], X[858] - 7 X[3526], X[1370] + 3 X[37951], 5 X[1656] + X[10295], 5 X[1656] - X[18403], 5 X[1656] + 3 X[37955], X[2070] + 7 X[3526], X[2071] + 3 X[37943], 3 X[2072] - X[3153], X[2072] + 2 X[16531], 2 X[2072] + 3 X[18579], X[2072] + 3 X[44214], 5 X[2072] + 3 X[44265], 7 X[3090] + X[13619], 7 X[3090] - X[18323], 7 X[3090] + 5 X[37952], X[3153] + 6 X[16531], 2 X[3153] + 9 X[18579], X[3153] + 9 X[44214], 5 X[3153] + 9 X[44265], 7 X[3523] - 3 X[37948], 17 X[3533] - 5 X[30745], 17 X[3533] + 3 X[37940], X[3580] + 5 X[38794], 4 X[3628] - X[10297], 2 X[3628] + X[18571], 4 X[3628] + X[37931], 9 X[5054] - X[18859], 13 X[5067] - X[10296], 13 X[5067] + 3 X[35489], 2 X[5159] + X[7575], 2 X[5159] + 3 X[16532], X[5189] + 3 X[37939], X[5899] - 3 X[7426], X[5899] + 15 X[15694], 3 X[6688] - X[13376], X[7426] + 5 X[15694], X[7464] - 13 X[10303], X[7574] + 3 X[37922], X[7575] - 3 X[16532], 2 X[10096] + 3 X[10257], 4 X[10096] + 3 X[15122], 8 X[10096] - 3 X[16619], 5 X[10096] - 3 X[25338], X[10096] - 3 X[44234], X[10151] - 4 X[15350], X[10151] + 2 X[37968], 4 X[10257] + X[16619], 5 X[10257] + 2 X[25338], 3 X[10257] + X[37971], X[10257] + 2 X[44234], X[10295] - 3 X[37955], X[10296] + 3 X[35489], X[10297] + 2 X[18571], X[10540] + 3 X[15061], 9 X[11539] + X[37936], 8 X[11540] + X[37904], X[11558] + 3 X[12100], X[11563] + 2 X[16976], X[11563] - 3 X[44282], X[11799] - 3 X[37943], X[13399] - 5 X[38729], X[13619] - 5 X[37952], X[13851] - 3 X[23515], 7 X[14869] - X[37950], 7 X[14869] + X[43893], 5 X[15059] - X[25739], 2 X[15122] + X[16619], 5 X[15122] + 4 X[25338], 3 X[15122] + 2 X[37971], X[15122] + 4 X[44234], 2 X[15350] + X[37968], 2 X[15646] + X[23323], X[15646] + 4 X[37911], 21 X[15702] - X[37944], 3 X[15709] + X[37907], 27 X[15709] + X[37945], 5 X[15712] + X[44267], 5 X[15713] + X[44266], 11 X[15720] + X[18325], 2 X[16239] + X[22249], 4 X[16531] - 3 X[18579], 2 X[16531] - 3 X[44214], 10 X[16531] - 3 X[44265], 3 X[16532] - 2 X[37935], 3 X[16532] + X[37938], 5 X[16619] - 8 X[25338], 3 X[16619] - 4 X[37971], X[16619] - 8 X[44234], 2 X[16976] + 3 X[44282], X[18323] + 5 X[37952], X[18403] + 3 X[37955], X[18572] + 2 X[37934], 5 X[18579] - 2 X[44265], X[22115] - 5 X[38794], X[23323] - 8 X[37911], 6 X[25338] - 5 X[37971], X[25338] - 5 X[44234], 5 X[30745] + 3 X[37940], 5 X[31255] + 3 X[37917], X[32111] + 5 X[38728], X[34152] + 2 X[37942], X[34152] + 3 X[44282], X[34170] + 3 X[38719], 5 X[37760] - X[37925], X[37899] - 4 X[44264], X[37900] - 3 X[37956], 9 X[37907] - X[37945], 2 X[37935] + X[37938], 3 X[37941] - X[44246], 2 X[37942] - 3 X[44282], X[37971] - 6 X[44234], X[41724] + 3 X[43572], 5 X[44214] - X[44265]

X(44452) lies on these lines: {2, 3}, {49, 26879}, {113, 21663}, {389, 43839}, {511, 14156}, {1147, 32358}, {1154, 9826}, {1503, 34128}, {1568, 32110}, {3564, 15462}, {3580, 12228}, {5432, 10149}, {5462, 32411}, {5876, 44158}, {5907, 20191}, {5946, 23292}, {5972, 13754}, {6000, 6699}, {6102, 9820}, {6146, 32171}, {6688, 13376}, {6689, 11695}, {6716, 31379}, {6720, 16760}, {6723, 18400}, {7740, 39170}, {7749, 16306}, {10182, 18475}, {10272, 12825}, {10540, 15061}, {11449, 26917}, {11464, 26913}, {11589, 18809}, {11704, 12278}, {11744, 14677}, {12028, 34209}, {12038, 12370}, {12041, 15311}, {12095, 16221}, {12134, 13561}, {12241, 43394}, {13348, 13446}, {13363, 37649}, {13367, 43817}, {13391, 32269}, {13399, 38729}, {13491, 16252}, {13851, 23515}, {15059, 25739}, {15067, 44201}, {18439, 43607}, {18952, 19357}, {19129, 26156}, {20304, 30522}, {20376, 22804}, {21243, 43586}, {22104, 34840}, {22151, 34380}, {23332, 34514}, {26937, 32139}, {31945, 39005}, {32111, 38728}, {32415, 32767}, {34170, 38719}, {41724, 43572}

X(44452) = midpoint of X(i) and X(j) for these {i,j}: {2, 44214}, {3, 403}, {4, 44246}, {5, 15646}, {113, 21663}, {140, 44234}, {186, 2072}, {381, 44280}, {468, 10257}, {549, 44282}, {550, 44283}, {858, 2070}, {1368, 44272}, {1568, 32110}, {2071, 11799}, {3580, 22115}, {5159, 37935}, {7575, 37938}, {10295, 18403}, {10297, 37931}, {11563, 34152}, {11589, 18809}, {12095, 16221}, {13348, 13446}, {13619, 18323}, {15760, 44281}, {16386, 31726}, {16976, 37942}, {37950, 43893}
X(44452) = reflection of X(i) in X(j) for these {i,j}: {186, 16531}, {468, 44234}, {7575, 37935}, {10257, 140}, {11563, 37942}, {13473, 546}, {15122, 10257}, {18579, 44214}, {23323, 5}, {32411, 5462}, {34152, 16976}, {37931, 18571}, {37938, 5159}, {37947, 37897}, {37971, 10096}, {44283, 37984}
X(44452) = complement of X(2072)
X(44452) = circumcircle-inverse of X(7517)
X(44452) = nine-point-circle-inverse of X(13371)
X(44452) = polar-circle-inverse of X(7505)
X(44452) = orthoptic-circle-of-Steiner-inellipe-inverse of X(7391)
X(44452) = 2nd-Droz-Farny-circle-inverse of X(382)
X(44452) = ninepoint-circle-of-medial-triangle-inverse of X(5)
X(44452) = complement of the isogonal conjugate of X(38534)
X(44452) = X(38534)-complementary conjugate of X(10)
X(44452) = crossdifference of every pair of points on line {647, 8553}
X(44452) = radical trace of circumcircle and 1st Droz-Farny circle
X(44452) = X(44214)-of-Euler-triangle
X(44452) = X(23323)-of-Johnson-triangle
X(44452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 186, 2072}, {2, 6644, 5}, {2, 35486, 18531}, {2, 37347, 3628}, {3, 381, 35481}, {3, 7505, 15761}, {3, 10018, 10020}, {3, 31726, 16386}, {3, 37197, 34350}, {3, 37453, 10201}, {4, 37941, 44246}, {5, 549, 18570}, {24, 6640, 13371}, {24, 13371, 11819}, {24, 30744, 31723}, {49, 26879, 43588}, {140, 468, 15122}, {140, 546, 5498}, {140, 6676, 549}, {140, 7542, 7568}, {140, 10020, 3}, {140, 10125, 7542}, {140, 13383, 16196}, {140, 16238, 5}, {140, 34004, 12108}, {140, 34577, 3530}, {140, 44232, 23336}, {186, 7514, 34152}, {186, 16531, 18579}, {186, 44214, 16531}, {381, 37118, 44236}, {403, 16386, 31726}, {427, 12106, 13490}, {427, 44211, 12106}, {468, 7499, 16387}, {468, 15122, 16619}, {468, 37971, 10096}, {549, 11563, 34152}, {549, 34152, 16976}, {631, 37943, 2071}, {632, 7575, 5159}, {632, 16532, 37938}, {1113, 1114, 7517}, {1312, 1313, 13371}, {1368, 34351, 7502}, {1656, 37955, 18403}, {2071, 37943, 11799}, {2072, 44214, 186}, {3090, 37952, 18323}, {3147, 3548, 26}, {3530, 34577, 34002}, {3628, 18571, 10297}, {3628, 31833, 5}, {5054, 9818, 18580}, {6640, 31723, 30744}, {7499, 16387, 16977}, {7568, 15122, 16977}, {7575, 16532, 37935}, {7577, 38321, 546}, {10226, 44235, 1885}, {11449, 26917, 44076}, {11563, 44282, 37942}, {13154, 37943, 37938}, {13383, 16196, 550}, {14070, 30771, 14791}, {14940, 22467, 10024}, {15350, 37968, 10151}, {16532, 37938, 7575}, {18324, 18531, 550}, {18403, 37955, 10295}, {18404, 32534, 44242}, {18531, 35486, 18324}, {23336, 44232, 4}, {30744, 31723, 13371}, {31664, 31665, 15122}, {31681, 31682, 3853}, {34152, 44282, 11563}, {34350, 37197, 3627}, {35231, 35232, 37814}, {42807, 42808, 35477}

X(44453) = 2ND-BROCARD-CIRCLE-POLE OF LEMOINE AXIS

Barycentrics    a^2*(a^2*b^2 - 2*b^4 + a^2*c^2 - b^2*c^2 - 2*c^4) : :
Trilinears    sin A - 2 sin(A + 2ω) : :
X(44453) = 3 X[6] - 4 X[39], 5 X[6] - 4 X[5052], 3 X[6] - 2 X[13330], 5 X[6] - 6 X[13331], 2 X[39] - 3 X[3094], 5 X[39] - 3 X[5052], 10 X[39] - 9 X[13331], 3 X[69] - X[20081], 2 X[76] - 3 X[599], 4 X[140] - 3 X[31958], 6 X[141] - 5 X[31276], 2 X[576] - 3 X[32447], 3 X[1350] - 2 X[9821], 5 X[1656] - 6 X[11261], 5 X[3094] - 2 X[5052], 3 X[3094] - X[13330], 5 X[3094] - 3 X[13331], 3 X[3097] - 2 X[4663], 5 X[3763] - 4 X[24256], 8 X[3934] - 9 X[21358], 6 X[5052] - 5 X[13330], 2 X[5052] - 3 X[13331], 3 X[7709] - 2 X[8550], 5 X[7786] - 3 X[22486], 7 X[10541] - 8 X[13334], 3 X[11160] + X[20105], 3 X[11179] - 4 X[32516], 5 X[13330] - 9 X[13331], 3 X[18906] - 5 X[31276]

X(44453) lies on these lines: {2, 20977}, {3, 6}, {69, 698}, {76, 338}, {115, 40107}, {140, 8179}, {141, 5025}, {183, 1916}, {193, 32449}, {194, 524}, {230, 6194}, {262, 7608}, {320, 33890}, {353, 9716}, {538, 5077}, {542, 7756}, {597, 33274}, {726, 17299}, {732, 33234}, {1078, 10754}, {1196, 36650}, {1569, 38741}, {1613, 2979}, {1656, 11261}, {1992, 14897}, {1993, 10329}, {2023, 22712}, {2162, 7186}, {2176, 3792}, {2393, 10568}, {2493, 40805}, {2549, 32515}, {2781, 32445}, {2782, 15069}, {2810, 38522}, {2854, 38523}, {2882, 4173}, {3056, 23633}, {3060, 8041}, {3097, 4663}, {3124, 7998}, {3231, 33884}, {3269, 15073}, {3291, 3917}, {3297, 12841}, {3298, 12840}, {3314, 8782}, {3399, 11170}, {3589, 7907}, {3618, 33259}, {3629, 33275}, {3763, 7887}, {3764, 23473}, {3767, 32521}, {3864, 5220}, {3906, 10097}, {3934, 7617}, {4643, 17760}, {5026, 7782}, {5108, 16055}, {5254, 12251}, {5475, 22695}, {5476, 31455}, {5480, 37446}, {5976, 7778}, {6034, 7746}, {6144, 32451}, {7492, 14567}, {7709, 8550}, {7747, 19924}, {7748, 11646}, {7757, 15534}, {7767, 31981}, {7776, 8149}, {7779, 10335}, {7783, 39099}, {7786, 22486}, {7788, 9865}, {7830, 14645}, {7866, 18806}, {7873, 37004}, {8288, 38397}, {8705, 36182}, {9009, 14824}, {9024, 38521}, {10007, 33233}, {10488, 11152}, {10766, 14585}, {11160, 20105}, {11179, 32516}, {11257, 32469}, {12837, 31477}, {13192, 16042}, {13881, 22677}, {15066, 20998}, {15080, 20976}, {15915, 16308}, {17042, 30489}, {20423, 31401}, {25235, 32465}, {25236, 32466}, {31670, 40279}, {32429, 39899}, {39024, 41462}

X(44453) = reflection of X(i) in X(j) for these {i,j}: {6, 3094}, {182, 43147}, {193, 32449}, {6144, 32451}, {10488, 11152}, {11477, 3095}, {13108, 34507}, {13330, 39}, {15534, 7757}, {18906, 141}, {33683, 32480}, {39899, 32429}
X(44453) = Brocard-circle-inverse of X(39560)
X(44453) = Ehrmann-circle-inverse of X(5111)
X(44453) = 2nd-Brocard-circle-inverse of X(187)
X(44453) = isogonal conjugate of the isotomic conjugate of X(7897)
X(44453) = crosssum of X(187) and X(8787)
X(44453) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(11842)
X(44453) = barycentric product X(6)*X(7897)
X(44453) = barycentric quotient X(7897)/X(76)
X(44453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 39560}, {6, 1350, 2076}, {39, 9821, 3053}, {39, 13330, 6}, {182, 5111, 6}, {371, 372, 11842}, {574, 576, 5038}, {574, 8586, 6}, {576, 5038, 6}, {1350, 10542, 3053}, {1670, 1671, 187}, {1689, 1690, 575}, {1691, 5028, 6}, {2023, 22712, 37637}, {2028, 8589, 1341}, {2029, 8589, 1340}, {2979, 20859, 1613}, {3053, 10542, 6}, {3060, 15302, 13410}, {3094, 8586, 32447}, {3094, 13330, 39}, {3098, 5028, 1691}, {3102, 3103, 11171}, {3104, 3105, 3}, {3917, 3981, 21001}, {5013, 11477, 6}, {5038, 8586, 576}, {5052, 13331, 6}, {5107, 37512, 575}, {5116, 15514, 6}, {7748, 34507, 11646}, {8041, 13410, 15302}, {12055, 37517, 6}, {13325, 13326, 39}, {22695, 22696, 22728}, {39229, 39230, 32}

X(44454) = STAMMLER-CIRCLE-POLE OF ORTHIC AXIS

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 - 24*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 16*b^6*c^2 + 10*a^2*b^2*c^4 - 30*b^4*c^4 + 2*a^2*c^6 + 16*b^2*c^6 - c^8) : :
X(44454) = 3 X[3] - 4 X[25], 7 X[3] - 8 X[6644], 5 X[3] - 8 X[7530], 13 X[3] - 16 X[12106], 5 X[3] - 4 X[21312], 7 X[25] - 6 X[6644], 5 X[25] - 6 X[7530], 13 X[25] - 12 X[12106], 2 X[25] - 3 X[18534], 5 X[25] - 3 X[21312], 3 X[381] - 2 X[1370], 8 X[1368] - 9 X[5055], 8 X[1596] - 7 X[3851], 7 X[3526] - 8 X[44233], 5 X[3843] - 4 X[18531], 5 X[6644] - 7 X[7530], 13 X[6644] - 14 X[12106], 4 X[6644] - 7 X[18534], 10 X[6644] - 7 X[21312], 16 X[6677] - 15 X[15694], 13 X[7530] - 10 X[12106], 4 X[7530] - 5 X[18534], 8 X[12106] - 13 X[18534], 20 X[12106] - 13 X[21312], 5 X[15693] - 6 X[26255], 7 X[15701] - 8 X[44212], 21 X[15703] - 20 X[31255], 7 X[15703] - 8 X[44275], 5 X[18534] - 2 X[21312], 5 X[19709] - 4 X[31152], 5 X[31255] - 6 X[44275]

X(44454) lies on these lines: {2, 3}, {51, 35237}, {159, 43621}, {1350, 16194}, {1384, 40237}, {1660, 9703}, {3531, 5644}, {3581, 35450}, {5621, 12295}, {6243, 12315}, {8193, 33697}, {8717, 10601}, {8780, 37477}, {10263, 12174}, {10625, 15811}, {12308, 14984}, {12902, 36201}, {13474, 37486}, {13598, 37493}, {14530, 37495}, {14855, 17810}, {14915, 33586}, {18435, 33878}, {18550, 43725}, {19005, 42226}, {19006, 42225}, {31670, 32621}, {35264, 43576}

X(44454) = reflection of X(i) in X(j) for these {i,j}: {3, 18534}, {1657, 18533}, {21312, 7530}, {35001, 37980}
X(44454) = Stammler-circle-inverse of X(468)
X(44454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 33524, 7393}, {20, 7529, 3}, {25, 1370, 16419}, {378, 37945, 7387}, {381, 3534, 43957}, {382, 1657, 1885}, {382, 12083, 1597}, {382, 39568, 3}, {1597, 12083, 3}, {1597, 39568, 12083}, {1598, 1657, 3}, {3146, 37945, 378}, {3534, 5020, 3}, {3543, 12082, 9818}, {3845, 33532, 7484}, {9714, 12085, 3}, {15154, 15155, 468}, {15682, 37946, 22}, {15684, 37949, 3}, {17810, 33534, 14855}

X(44455) = STAMMLER-CIRCLE-POLE OF ANTI-ORTHIC AXIS

Barycentrics    a^2*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c + 8*a^2*b^2*c - 7*b^4*c - 2*a^3*c^2 + 8*a^2*b*c^2 - 14*a*b^2*c^2 + 8*b^3*c^2 + 2*a^2*c^3 + 8*b^2*c^3 + a*c^4 - 7*b*c^4 - c^5) : :
X(44455) = 3 X[3] - 4 X[55], 5 X[3] - 4 X[3428], 7 X[3] - 8 X[32613], 5 X[55] - 3 X[3428], 2 X[55] - 3 X[10679], 7 X[55] - 6 X[32613], 3 X[381] - 2 X[3434], 2 X[956] - 3 X[28444], 3 X[1482] - 2 X[25415], 8 X[2886] - 9 X[5055], 2 X[3428] - 5 X[10679], 7 X[3428] - 10 X[32613], 3 X[3830] - 2 X[18499], 5 X[3843] - 4 X[37820], 7 X[3851] - 8 X[7680], 16 X[6690] - 15 X[15694], 3 X[10247] - 4 X[37533], 7 X[10679] - 4 X[32613], 9 X[14269] - 8 X[18407], 21 X[15703] - 20 X[31245], 5 X[19709] - 4 X[31140], X[25415] - 3 X[37569]

X(44455) lies on these lines: {1, 3}, {8, 37234}, {30, 20075}, {42, 1480}, {355, 8168}, {381, 3434}, {390, 28459}, {519, 18519}, {528, 3830}, {956, 28444}, {1001, 3654}, {1012, 5844}, {1056, 28458}, {1376, 3656}, {1483, 37022}, {1537, 12331}, {1657, 37000}, {1824, 18535}, {2771, 41711}, {2886, 5055}, {3560, 12245}, {3621, 21669}, {3623, 37403}, {3632, 18761}, {3843, 37820}, {3851, 7680}, {3871, 6985}, {3913, 12699}, {3938, 7986}, {4301, 11499}, {5073, 5842}, {5080, 34629}, {5082, 6841}, {5274, 6882}, {5284, 5657}, {5434, 35249}, {5687, 22791}, {5905, 13278}, {6690, 15694}, {6765, 40263}, {6842, 8164}, {6890, 32214}, {6925, 32213}, {7580, 28212}, {8167, 26446}, {9708, 34718}, {9709, 18493}, {9812, 38665}, {10525, 11929}, {10528, 37406}, {10587, 44222}, {12607, 18542}, {12773, 25416}, {14269, 18407}, {15681, 34698}, {15703, 31245}, {16853, 25011}, {17784, 28452}, {18491, 31162}, {18524, 34647}, {18525, 32049}, {18543, 37356}, {19709, 31140}, {22758, 28234}, {25439, 28194}

X(44455) = reflection of X(i) in X(j) for these {i,j}: {3, 10679}, {1482, 37569}, {1657, 37000}, {6925, 32213}, {12702, 5119}
X(44455) = Stammler-circle-inverse of X(1155)
X(44455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40, 40296}, {1, 35448, 3}, {3, 1482, 12001}, {35, 35252, 3}, {40, 16202, 3}, {40, 37622, 16202}, {55, 25415, 999}, {56, 35251, 3}, {999, 10306, 35000}, {999, 35000, 3}, {1482, 10306, 3}, {1482, 12702, 65}, {1482, 35000, 999}, {3295, 12702, 3}, {3913, 12699, 18518}, {5537, 16200, 10269}, {6244, 10246, 3}, {7982, 11248, 10680}, {10222, 10310, 16203}, {10306, 22770, 40245}, {10310, 16203, 3}, {10680, 11248, 3}, {11248, 16207, 16203}, {11849, 22770, 3}, {35459, 37541, 3}, {37533, 37584, 11018}

X(44456) = STAMMLER-CIRCLE-POLE OF LEMOINE AXIS

Barycentrics    a^2*(a^4 - 8*a^2*b^2 + 7*b^4 - 8*a^2*c^2 - 2*b^2*c^2 + 7*c^4) : :
Trilinears    2 a - 3 R cot ω cos A : :
Trilinears    4 sin A - 3 cot ω cos A : :
X(44456) = 3 X[3] - 4 X[6], 7 X[3] - 8 X[182], 13 X[3] - 16 X[575], 5 X[3] - 8 X[576], 5 X[3] - 4 X[1350], 9 X[3] - 8 X[3098], 5 X[3] - 6 X[5050], 11 X[3] - 12 X[5085], 15 X[3] - 16 X[5092], 2 X[3] - 3 X[5093], 11 X[3] - 16 X[5097], 7 X[3] - 12 X[5102], 25 X[3] - 28 X[10541], X[3] - 4 X[11477], 7 X[3] - 10 X[11482], 9 X[3] - 10 X[12017], 17 X[3] - 16 X[14810], 25 X[3] - 32 X[15516], 17 X[3] - 24 X[15520], 23 X[3] - 24 X[17508], 29 X[3] - 32 X[20190], 31 X[3] - 40 X[22234], 23 X[3] - 32 X[22330], 13 X[3] - 12 X[31884], 3 X[3] - 2 X[33878], 3 X[3] - 8 X[37517], 19 X[3] - 24 X[39561], 3 X[4] - X[20080], 6 X[5] - 5 X[3620], 7 X[6] - 6 X[182], 13 X[6] - 12 X[575], 5 X[6] - 6 X[576], 5 X[6] - 3 X[1350], 2 X[6] - 3 X[1351], 3 X[6] - 2 X[3098], 10 X[6] - 9 X[5050], 11 X[6] - 9 X[5085], 5 X[6] - 4 X[5092], 8 X[6] - 9 X[5093], 11 X[6] - 12 X[5097], 7 X[6] - 9 X[5102], 25 X[6] - 21 X[10541], X[6] - 3 X[11477], 14 X[6] - 15 X[11482], 6 X[6] - 5 X[12017], 17 X[6] - 12 X[14810], 25 X[6] - 24 X[15516], 17 X[6] - 18 X[15520], 23 X[6] - 18 X[17508], 29 X[6] - 24 X[20190], 31 X[6] - 30 X[22234], 23 X[6] - 24 X[22330], 13 X[6] - 9 X[31884], 19 X[6] - 18 X[39561], 3 X[52] - 2 X[21851], 2 X[69] - 3 X[381], 3 X[69] - 4 X[18358], 2 X[74] - 3 X[39562], 8 X[141] - 9 X[5055], 2 X[141] - 3 X[20423], 13 X[182] - 14 X[575], 5 X[182] - 7 X[576], 10 X[182] - 7 X[1350], 4 X[182] - 7 X[1351], 9 X[182] - 7 X[3098], 20 X[182] - 21 X[5050], 22 X[182] - 21 X[5085], 15 X[182] - 14 X[5092], 16 X[182] - 21 X[5093], 11 X[182] - 14 X[5097], 2 X[182] - 3 X[5102], 50 X[182] - 49 X[10541], 2 X[182] - 7 X[11477], 4 X[182] - 5 X[11482], 36 X[182] - 35 X[12017], 17 X[182] - 14 X[14810], 25 X[182] - 28 X[15516], 17 X[182] - 21 X[15520], 23 X[182] - 21 X[17508], 29 X[182] - 28 X[20190], 31 X[182] - 35 X[22234], 23 X[182] - 28 X[22330], 26 X[182] - 21 X[31884], 12 X[182] - 7 X[33878], 3 X[182] - 7 X[37517], 19 X[182] - 21 X[39561], 3 X[193] - X[39874], 9 X[381] - 8 X[18358], 3 X[381] - 4 X[21850], 3 X[382] + 2 X[11008], 2 X[550] - 3 X[14912], 3 X[568] - 2 X[37511], 10 X[575] - 13 X[576], 20 X[575] - 13 X[1350], 8 X[575] - 13 X[1351], 18 X[575] - 13 X[3098], 40 X[575] - 39 X[5050], 44 X[575] - 39 X[5085], 15 X[575] - 13 X[5092], 32 X[575] - 39 X[5093], 11 X[575] - 13 X[5097], 28 X[575] - 39 X[5102], 100 X[575] - 91 X[10541], 4 X[575] - 13 X[11477], 56 X[575] - 65 X[11482], 72 X[575] - 65 X[12017], 17 X[575] - 13 X[14810], 25 X[575] - 26 X[15516], 34 X[575] - 39 X[15520], 46 X[575] - 39 X[17508], 29 X[575] - 26 X[20190], 62 X[575] - 65 X[22234], 23 X[575] - 26 X[22330], 4 X[575] - 3 X[31884], 24 X[575] - 13 X[33878], 6 X[575] - 13 X[37517], 38 X[575] - 39 X[39561], 4 X[576] - 5 X[1351], 9 X[576] - 5 X[3098], 4 X[576] - 3 X[5050], 22 X[576] - 15 X[5085], 3 X[576] - 2 X[5092], 16 X[576] - 15 X[5093], 11 X[576] - 10 X[5097], 14 X[576] - 15 X[5102], 10 X[576] - 7 X[10541], 2 X[576] - 5 X[11477], 28 X[576] - 25 X[11482], 36 X[576] - 25 X[12017], 17 X[576] - 10 X[14810], 5 X[576] - 4 X[15516], 17 X[576] - 15 X[15520], 23 X[576] - 15 X[17508], 29 X[576] - 20 X[20190], 31 X[576] - 25 X[22234], 23 X[576] - 20 X[22330], 26 X[576] - 15 X[31884], 12 X[576] - 5 X[33878], 3 X[576] - 5 X[37517], 19 X[576] - 15 X[39561], 8 X[597] - 7 X[15701], 3 X[599] - 4 X[19130], 4 X[599] - 5 X[19709], 2 X[1350] - 5 X[1351], 9 X[1350] - 10 X[3098], 2 X[1350] - 3 X[5050], 11 X[1350] - 15 X[5085], 3 X[1350] - 4 X[5092], 8 X[1350] - 15 X[5093], 11 X[1350] - 20 X[5097], 7 X[1350] - 15 X[5102], 5 X[1350] - 7 X[10541], X[1350] - 5 X[11477], 14 X[1350] - 25 X[11482], 18 X[1350] - 25 X[12017], 17 X[1350] - 20 X[14810], 5 X[1350] - 8 X[15516], 17 X[1350] - 30 X[15520], 23 X[1350] - 30 X[17508], 29 X[1350] - 40 X[20190], 31 X[1350] - 50 X[22234], 23 X[1350] - 40 X[22330], 13 X[1350] - 15 X[31884], 6 X[1350] - 5 X[33878], 3 X[1350] - 10 X[37517], 19 X[1350] - 30 X[39561], 9 X[1351] - 4 X[3098], 5 X[1351] - 3 X[5050], 11 X[1351] - 6 X[5085], 15 X[1351] - 8 X[5092], 4 X[1351] - 3 X[5093], 11 X[1351] - 8 X[5097], 7 X[1351] - 6 X[5102], 25 X[1351] - 14 X[10541], 7 X[1351] - 5 X[11482], 9 X[1351] - 5 X[12017], 17 X[1351] - 8 X[14810], 25 X[1351] - 16 X[15516], 17 X[1351] - 12 X[15520]

X(44456) lies on these lines: {3, 6}, {4, 11898}, {5, 3620}, {20, 1353}, {22, 11004}, {25, 323}, {30, 193}, {49, 10244}, {69, 381}, {74, 38263}, {141, 5055}, {159, 5899}, {206, 9703}, {382, 3564}, {383, 40901}, {394, 21969}, {399, 10752}, {517, 7996}, {518, 8148}, {524, 3830}, {542, 6144}, {550, 14912}, {597, 15701}, {599, 19130}, {895, 10620}, {1080, 40900}, {1154, 1597}, {1216, 3527}, {1352, 3630}, {1469, 7373}, {1482, 16496}, {1495, 3167}, {1503, 5073}, {1511, 19118}, {1598, 6403}, {1656, 3619}, {1657, 6776}, {1843, 18535}, {1974, 22115}, {1992, 3534}, {1993, 9909}, {1995, 16981}, {2104, 15155}, {2105, 15154}, {2781, 34777}, {2854, 12308}, {2979, 5644}, {2987, 5191}, {3056, 6767}, {3060, 5020}, {3426, 6391}, {3515, 38942}, {3517, 13421}, {3526, 10519}, {3528, 33748}, {3589, 14848}, {3618, 5054}, {3627, 5921}, {3629, 15681}, {3631, 3851}, {3751, 12702}, {3763, 5476}, {3818, 14269}, {3845, 11160}, {5032, 8703}, {5070, 14561}, {5072, 38136}, {5076, 39884}, {5095, 12121}, {5198, 15052}, {5422, 41462}, {5477, 38730}, {5544, 5650}, {5655, 32114}, {5847, 18525}, {5965, 36990}, {6033, 14645}, {6090, 10546}, {6329, 15707}, {6467, 35237}, {6515, 34609}, {6642, 14449}, {7484, 15018}, {7777, 40248}, {7788, 9993}, {8541, 35501}, {8584, 15695}, {8780, 37672}, {9306, 31860}, {9655, 39897}, {9668, 39873}, {9714, 19139}, {9924, 32063}, {10113, 32244}, {10246, 16491}, {10249, 39125}, {10601, 44107}, {10753, 13188}, {10754, 12188}, {10755, 12773}, {10756, 38574}, {10757, 38579}, {10758, 38572}, {10759, 12331}, {10760, 38589}, {10761, 38590}, {10762, 38591}, {10763, 38592}, {10764, 38573}, {10765, 38593}, {10766, 13115}, {10985, 40801}, {11002, 11284}, {11064, 21970}, {11174, 33706}, {11179, 15689}, {11180, 38335}, {11402, 15080}, {11412, 11479}, {11414, 15032}, {11456, 12160}, {11464, 16195}, {11470, 44091}, {11649, 37949}, {12083, 19459}, {12161, 41464}, {12164, 34382}, {12174, 12283}, {12220, 13391}, {12294, 18436}, {12295, 32272}, {12316, 39879}, {12601, 42283}, {12602, 42284}, {12699, 34379}, {12902, 34775}, {13093, 34788}, {13634, 37677}, {14531, 14914}, {14996, 19544}, {14997, 16434}, {15019, 21766}, {15533, 43150}, {15534, 15685}, {15696, 25406}, {15718, 38064}, {15720, 38110}, {15723, 38079}, {15988, 16418}, {17800, 29181}, {17811, 21849}, {17813, 35450}, {19140, 20987}, {19154, 34148}, {19277, 25898}, {20425, 42128}, {20426, 42125}, {24206, 42785}, {25321, 34153}, {28146, 39878}, {30771, 37643}, {31133, 37779}, {31152, 37644}, {32001, 44228}, {33533, 39522}, {33851, 38638}, {34207, 43704}, {37925, 41450}, {39893, 42263}, {39894, 42264}, {40107, 42786}, {40995, 44231}

X(44456) = reflection of X(i) in X(j) for these {i,j}: {3, 1351}, {6, 37517}, {20, 1353}, {69, 21850}, {399, 10752}, {1350, 576}, {1351, 11477}, {1657, 6776}, {3534, 1992}, {5921, 3627}, {6403, 10263}, {9821, 5052}, {9924, 34779}, {10620, 895}, {11160, 3845}, {11898, 4}, {12121, 5095}, {12188, 10754}, {12331, 10759}, {12702, 3751}, {12773, 10755}, {13115, 10766}, {13188, 10753}, {15154, 2105}, {15155, 2104}, {18436, 12294}, {18440, 31670}, {32244, 10113}, {32272, 12295}, {33878, 6}, {35002, 5107}, {35450, 17813}, {35456, 5111}, {35458, 15514}, {37484, 9967}, {38572, 10758}, {38573, 10764}, {38574, 10756}, {38579, 10757}, {38589, 10760}, {38590, 10761}, {38591, 10762}, {38592, 10763}, {38593, 10765}, {38730, 5477}, {39899, 193}, {40341, 3818}
X(44456) = Schoute-circle-inverse of X(5023)
X(44456) = Stammler-circle-inverse of X(187)
X(44456) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(22331)
X(44456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1351, 5093}, {6, 1350, 5092}, {6, 3098, 12017}, {6, 5092, 5050}, {6, 11477, 37517}, {6, 33878, 3}, {6, 37517, 1351}, {15, 16, 5023}, {69, 21850, 381}, {182, 5102, 11482}, {371, 372, 22331}, {576, 1350, 5050}, {576, 5092, 6}, {1350, 5050, 3}, {1351, 5050, 576}, {1351, 11482, 5102}, {1351, 33878, 6}, {1384, 35002, 3}, {1993, 15107, 26864}, {2979, 9777, 16419}, {3098, 12017, 3}, {3167, 33586, 20850}, {3311, 3312, 5007}, {5017, 35458, 3}, {6199, 6395, 21309}, {6200, 6396, 5585}, {6221, 6398, 187}, {9605, 9821, 3}, {9732, 12314, 3}, {9733, 12313, 3}, {9777, 16419, 5644}, {9924, 34779, 32063}, {10519, 18583, 3526}, {10541, 15516, 5050}, {10625, 11432, 3}, {11426, 37486, 3}, {11485, 11486, 32}, {12017, 33878, 3098}, {15107, 26864, 9909}, {18440, 31670, 3830}, {22809, 22810, 382}, {35456, 40825, 3}, {37484, 37493, 3}, {38136, 40330, 5072}, {38596, 38597, 187}, {42115, 42116, 8588}

X(44457) = STAMMLER-CIRCLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 - 14*a^4*b^2*c^2 + 8*a^2*b^4*c^2 + 8*b^6*c^2 + 8*a^2*b^2*c^4 - 14*b^4*c^4 + 2*a^2*c^6 + 8*b^2*c^6 - c^8) : :
X(44457) = 3 X[3] - 4 X[22], 5 X[3] - 4 X[378], 7 X[3] - 8 X[7502], 13 X[3] - 16 X[7555], X[3] - 4 X[12082], 9 X[3] - 8 X[18570], 5 X[22] - 3 X[378], 7 X[22] - 6 X[7502], 13 X[22] - 12 X[7555], X[22] - 3 X[12082], 2 X[22] - 3 X[12083], 3 X[22] - 2 X[18570], 7 X[378] - 10 X[7502], 13 X[378] - 20 X[7555], X[378] - 5 X[12082], 2 X[378] - 5 X[12083], 9 X[378] - 10 X[18570], 3 X[381] - 2 X[7391], 8 X[427] - 9 X[5055], 7 X[3526] - 8 X[25337], 3 X[3534] - 2 X[35481], 5 X[3843] - 4 X[31723], 7 X[3851] - 8 X[15760], 9 X[5054] - 8 X[44236], 16 X[6676] - 15 X[15694], 13 X[7502] - 14 X[7555], 2 X[7502] - 7 X[12082], 4 X[7502] - 7 X[12083], 9 X[7502] - 7 X[18570], 4 X[7555] - 13 X[12082], 8 X[7555] - 13 X[12083], 18 X[7555] - 13 X[18570], 9 X[12082] - 2 X[18570], 9 X[12083] - 4 X[18570], 9 X[14269] - 8 X[44288], 3 X[15684] - 4 X[35480], 9 X[15689] - 8 X[44249], 7 X[15701] - 8 X[44210], 21 X[15703] - 20 X[31236], 7 X[15703] - 8 X[44262], 9 X[15707] - 8 X[44218], 5 X[19709] - 4 X[31133], 5 X[31236] - 6 X[44262]

X(44457) lies on these lines: {2, 3}, {51, 8717}, {154, 37477}, {161, 2777}, {1350, 18435}, {1384, 19220}, {1498, 37484}, {1619, 9919}, {2781, 12308}, {2936, 38730}, {3098, 16194}, {3167, 37496}, {3581, 33534}, {4316, 9673}, {4324, 9658}, {5210, 40237}, {6000, 37494}, {6445, 13889}, {6446, 13943}, {8718, 12161}, {9609, 43618}, {9683, 42272}, {10540, 37483}, {11456, 13391}, {12367, 33878}, {12902, 37488}, {13340, 18451}, {13598, 36753}, {15177, 28168}, {17810, 40280}, {18439, 37486}, {26881, 43576}, {28208, 37546}, {33586, 35237}, {33697, 37557}, {33974, 34106}

X(44457) = reflection of X(i) in X(j) for these {i,j}: {3, 12083}, {12083, 12082}
X(44457) = Stammler-circle-inverse of X(858)
X(44457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 15684, 1597}, {3, 18535, 5055}, {3, 19709, 16419}, {3, 37949, 18534}, {4, 16661, 7516}, {20, 7517, 3}, {20, 37925, 6644}, {22, 7391, 7484}, {25, 3534, 3}, {376, 37945, 7530}, {381, 35243, 3}, {382, 1657, 12225}, {382, 11414, 3}, {382, 34864, 11403}, {550, 7506, 3}, {1593, 13564, 3}, {1656, 37198, 3}, {1657, 2070, 21312}, {1657, 7387, 3}, {2070, 21312, 3}, {2937, 12085, 3}, {3529, 12087, 26}, {3534, 37924, 25}, {5059, 12088, 12084}, {5899, 15681, 3}, {6636, 15682, 31861}, {6642, 15696, 3}, {6644, 37925, 7517}, {7387, 21312, 2070}, {7464, 37913, 18324}, {7492, 15640, 13596}, {13730, 16117, 3}, {14070, 18859, 3}, {15154, 15155, 858}, {18534, 39568, 37949}

X(44458) = STAMMLER-CIRCLES-RADICAL-CIRCLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    2*a^10 - 3*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 - b^10 - 3*a^8*c^2 + 18*a^6*b^2*c^2 - 12*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 12*a^4*b^2*c^4 + 12*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 6*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :
X(44458) = 5 X[2] - 3 X[37077], 4 X[3] - X[18560], 5 X[4] - 8 X[9825], X[4] - 4 X[31829], 2 X[20] + X[6240], 3 X[376] - 2 X[7667], 4 X[548] - X[18563], 4 X[550] - X[12225], 5 X[631] - 2 X[1885], 7 X[3090] - 4 X[13488], X[3146] - 4 X[31833], 5 X[3522] - 2 X[12605], 9 X[3524] - 8 X[7734], 3 X[3524] - 2 X[34664], 7 X[3528] - 4 X[12362], X[3529] + 2 X[3575], 3 X[3839] - 4 X[10127], X[5059] + 2 X[7553], 15 X[5071] - 16 X[13361], X[5073] - 4 X[31830], 4 X[6756] - X[33703], 4 X[7540] - 5 X[7576], 6 X[7540] - 5 X[34603], 8 X[7540] - 5 X[34613], 2 X[7540] - 3 X[38320], 3 X[7540] - 5 X[38321], 7 X[7540] - 10 X[38322], 2 X[7540] - 5 X[38323], 3 X[7576] - 2 X[34603], 5 X[7576] - 6 X[38320], 3 X[7576] - 4 X[38321], 7 X[7576] - 8 X[38322], 4 X[7734] - 3 X[34664], 2 X[9825] - 5 X[31829], 8 X[10128] - 7 X[41106], 2 X[10575] + X[14516], X[10625] + 2 X[43577], 4 X[10691] - 5 X[19708], 2 X[11819] + X[17800], 2 X[12134] + X[12279], 3 X[14269] - 4 X[23410], 3 X[15045] - 2 X[16657], 3 X[15688] - X[18564], 5 X[15695] - X[18561], 5 X[15696] + X[18565], 7 X[15698] - 6 X[43957], 5 X[17538] + X[34797], 4 X[34603] - 3 X[34613], 5 X[34603] - 9 X[38320], 7 X[34603] - 12 X[38322], X[34603] - 3 X[38323], 5 X[34613] - 12 X[38320], 3 X[34613] - 8 X[38321], 7 X[34613] - 16 X[38322], X[34613] - 4 X[38323], 9 X[38320] - 10 X[38321], 21 X[38320] - 20 X[38322], 3 X[38320] - 5 X[38323], 7 X[38321] - 6 X[38322], 2 X[38321] - 3 X[38323], 4 X[38322] - 7 X[38323]

X(44458) lies on these lines: {2, 3}, {74, 343}, {541, 12825}, {542, 17854}, {599, 34778}, {1993, 4846}, {2777, 3917}, {5306, 41336}, {5434, 9627}, {8718, 34782}, {9306, 32111}, {10192, 15035}, {10575, 14516}, {10625, 43577}, {10706, 11744}, {11057, 14615}, {11454, 44201}, {11459, 15311}, {11645, 14913}, {12134, 12279}, {12827, 16111}, {14855, 17702}, {15045, 16657}, {22466, 43836}, {22802, 43652}, {28208, 34668}, {41587, 43601}

X(44458) = midpoint of X(11001) and X(18559)
X(44458) = reflection of X(i) in X(j) for these {i,j}: {382, 13490}, {7576, 38323}, {15682, 428}, {34603, 38321}, {34613, 7576}
X(44458) = 2nd-Droz-Farny-circle-inverse of X(7464)
X(44458) = Stammler-circles-radical-circle-inverse of X(858)
X(44458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20, 35491}, {3, 34350, 34005}, {20, 6636, 44249}, {20, 16661, 550}, {376, 7552, 37948}, {403, 35491, 378}, {550, 15760, 2071}, {2043, 2044, 11413}, {2071, 15760, 37118}, {3534, 34609, 21312}, {3547, 30552, 35477}, {6240, 35491, 18560}, {7512, 16661, 10323}, {7540, 38320, 7576}, {7552, 37948, 549}, {8703, 34477, 3}, {10154, 44241, 44268}, {10154, 44268, 186}, {18533, 35513, 12082}, {34603, 38321, 7576}, {34603, 38323, 38321}

X(44459) = CIRCUMCIRCLE-OF-INNER-NAPOLEON-TRIANGLE-POLE OF ORTHIC AXIS

Barycentrics    Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^2 + b^2 + c^2) - 4*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(44459) = 5 X[631] - 6 X[11297], 7 X[3090] - 6 X[11298], 3 X[3545] - 2 X[11296], 4 X[3642] - 5 X[40330], 4 X[3643] - 3 X[10519], 2 X[10653] - 3 X[14853], X[39874] - 4 X[42117]

X(44459) lies on these lines: {2, 3}, {13, 36994}, {14, 41458}, {15, 36962}, {396, 41039}, {531, 11180}, {538, 36323}, {621, 32815}, {633, 32836}, {1503, 42154}, {2549, 5334}, {2794, 6770}, {3642, 40330}, {3643, 10519}, {5335, 7737}, {5478, 33388}, {5479, 18581}, {5480, 42155}, {5617, 23698}, {5868, 42164}, {5869, 42147}, {6108, 9752}, {6775, 43404}, {6776, 10654}, {7739, 42999}, {9735, 22797}, {10614, 36995}, {10653, 14853}, {14981, 35696}, {16242, 41037}, {22491, 41044}, {22513, 42134}, {23004, 43452}, {23235, 35691}, {32515, 40898}, {39874, 42117}, {41021, 41108}, {41022, 42085}, {41025, 42529}, {41038, 42940}, {42133, 43619}

X(44459) = reflection of X(i) in X(j) for these {i,j}: {376, 11295}, {6770, 6772}, {6773, 22512}, {6776, 10654}, {42155, 5480}, {44463, 4}
X(44459) = circumcircle-of-inner-Napoleon-triangle-inverse of X(468)
X(44459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 376, 383}, {2043, 2044, 37173}, {2549, 22512, 5334}, {3830, 41016, 4}

X(44460) = CIRCUMCIRCLE-OF-INNER-NAPOLEON-TRIANGLE-POLE OF LEMOINE AXIS

Barycentrics    (a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) - 2*Sqrt[3]*a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S : :
X(44460) = 3 X[2] - 4 X[33478], 2 X[2] + X[36323], 4 X[2] - X[36345], 5 X[2] - 2 X[36364], X[2] + 2 X[36384], X[4] + 2 X[3104], X[6194] - 4 X[33463], 4 X[22715] + 3 X[36323], 8 X[22715] - 3 X[36345], 5 X[22715] - 3 X[36364], X[22715] + 3 X[36384], 8 X[33478] + 3 X[36323], 16 X[33478] - 3 X[36345], 10 X[33478] - 3 X[36364], 2 X[33478] + 3 X[36384], 2 X[36323] + X[36345], 5 X[36323] + 4 X[36364], X[36323] - 4 X[36384], 5 X[36345] - 8 X[36364], X[36345] + 8 X[36384], X[36364] + 5 X[36384]

X(44460) lies on these lines: {2, 51}, {4, 3104}, {15, 33389}, {16, 10788}, {39, 42999}, {61, 33388}, {76, 43953}, {194, 22114}, {298, 20426}, {618, 36780}, {621, 37242}, {627, 3095}, {3094, 16940}, {3105, 42149}, {3106, 6773}, {3107, 37641}, {5334, 22707}, {5335, 31701}, {6782, 32465}, {7684, 22696}, {7737, 22708}, {9993, 41071}, {14538, 35925}, {14651, 43454}, {14881, 33413}, {18581, 43539}, {18582, 22701}, {22688, 40693}, {22691, 37640}, {32447, 37785}, {35917, 39656}, {41070, 43461}

X(44460) = midpoint of X(3104) and X(22693)
X(44460) = reflection of X(i) in X(j) for these {i,j}: {4, 22693}, {6194, 22714}, {7709, 3106}, {22686, 33479}, {22714, 33463}, {22715, 33478}, {36780, 618}, {44464, 262}
X(44460) = anticomplement of X(22715)
X(44460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36323, 36345}, {2, 36384, 36323}, {11261, 22714, 2}, {22715, 33478, 2}

X(44461) = CIRCUMCIRCLE-OF-INNER-NAPOLEON-TRIANGLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    Sqrt[3]*a^2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) - 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(44461) = 2 X[5] - 3 X[11298], 4 X[140] - 3 X[11297], 2 X[3098] + X[42086], 3 X[5085] - X[42154], 4 X[5092] - X[42085], 5 X[12017] - 2 X[42117], X[33878] + 2 X[42118]

X(44461) lies on these lines: {2, 3}, {13, 14539}, {15, 2549}, {16, 7737}, {61, 7739}, {182, 6775}, {397, 5865}, {511, 10653}, {531, 11179}, {538, 36364}, {616, 36995}, {617, 2782}, {621, 6773}, {627, 22531}, {1350, 42155}, {1352, 3642}, {2794, 5617}, {3098, 22513}, {3180, 32515}, {3734, 5613}, {5085, 42154}, {5092, 42085}, {5238, 22843}, {5334, 13102}, {5335, 20425}, {5344, 16629}, {5611, 37640}, {5864, 42148}, {5873, 32836}, {5978, 9744}, {5980, 14907}, {5981, 11185}, {6771, 6772}, {6774, 18581}, {7694, 9750}, {9988, 36998}, {10645, 21156}, {10646, 43618}, {10992, 35696}, {11092, 18911}, {11485, 15048}, {11486, 18907}, {12017, 42117}, {14538, 36968}, {14540, 42158}, {14541, 16965}, {16002, 41113}, {16242, 21159}, {18582, 36756}, {20416, 41120}, {20426, 37641}, {21157, 37835}, {21158, 42529}, {21163, 31702}, {21467, 40280}, {22687, 41022}, {22891, 31703}, {33878, 42118}, {36755, 42091}, {36759, 43455}

X(44461) = midpoint of X(1350) and X(42155)
X(44461) = reflection of X(i) in X(j) for these {i,j}: {1352, 3642}, {6772, 6771}, {10654, 182}, {11295, 549}, {22512, 6774}, {44465, 3}
X(44461) = circumcircle-of-inner-Napoleon-triangle-inverse of X(858)
X(44461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 37178, 5}, {11303, 35917, 2}, {15765, 18585, 11306}

X(44462) = CIRCUMCIRCLE-OF-INNER-NAPOLEON-TRIANGLE-POLE OF HATZIPOLAKIS AXIS

Barycentrics    Sqrt[3]*a^2*(a^4 - b^4 + b^2*c^2 - c^4) - 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(44462) = 3 X[2] - 4 X[32460], X[5189] - 4 X[37975], 4 X[32461] - 5 X[37760]

X(44462) lies on these lines: {2, 3}, {14, 14169}, {62, 8015}, {299, 2453}, {476, 16770}, {523, 3180}, {531, 9143}, {621, 35315}, {691, 34376}, {3448, 11092}, {5474, 11130}, {5479, 8836}, {5612, 8173}, {5617, 14187}, {6773, 16771}, {6774, 41473}, {6800, 42154}, {7712, 42085}, {10653, 11002}, {10654, 11003}, {11549, 37641}, {14170, 36967}, {16529, 22738}, {19106, 30468}, {21466, 30466}, {22492, 34316}, {30485, 36993}, {36210, 40694}, {36962, 40710}, {40855, 41022}

X(44462) = reflection of X(i) in X(j) for these {i,j}: {3448, 11092}, {5189, 36186}, {36185, 32460}, {36186, 37975}, {44466, 23}
X(44462) = anticomplement of X(36185)
X(44462) = circumcircle-inverse of X(11145)
X(44462) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(383)
X(44462) = circumcircle-ofinner-Napoleon-triangle-inverse of X(5)
X(44462) = circumcircle-of-outer-Napoleon-triangle-inverse of X(44223)
X(44462) = psi-transform of X(6774)
X(44462) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36181, 44466}, {1113, 1114, 11145}, {1316, 36186, 2}, {32460, 36185, 2}

X(44463) = CIRCUMCIRCLE-OF-OUTER-NAPOLEON-TRIANGLE-POLE OF ORTHIC AXIS

Barycentrics    Sqrt[3]*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2+c^2)+4*(2*a^4-a^2*b^2-b^4-a^2*c^2+2*b^2*c^2-c^4)*S : :
X(44463) = 5 X[631] - 6 X[11298], 7 X[3090] - 6 X[11297], 3 X[3545] - 2 X[11295], 4 X[3642] - 3 X[10519], 4 X[3643] - 5 X[40330], 2 X[10654] - 3 X[14853], X[39874] - 4 X[42118]

X(44463) lies on these lines: {2, 3}, {13, 36761}, {14, 36992}, {16, 36961}, {395, 41038}, {530, 11180}, {538, 36322}, {622, 32815}, {634, 32836}, {1503, 42155}, {2549, 5335}, {2794, 6773}, {3642, 10519}, {3643, 40330}, {5334, 7737}, {5478, 18582}, {5479, 33389}, {5480, 42154}, {5613, 23698}, {5868, 42148}, {5869, 42165}, {6109, 9752}, {6772, 43403}, {6776, 10653}, {7739, 42998}, {9736, 22796}, {10613, 36993}, {10654, 14853}, {14981, 35692}, {16241, 41036}, {22492, 36775}, {22512, 42133}, {23005, 43451}, {23235, 35695}, {32515, 40899}, {39874, 42118}, {41020, 41107}, {41023, 42086}, {41024, 42528}, {41039, 42941}, {42134, 43619}

X(44463) = reflection of X(i) in X(j) for these {i,j}: {376, 11296}, {6770, 22513}, {6773, 6775}, {6776, 10653}, {42154, 5480}, {44459, 4}
X(44463) = circumcircle-of-outer-Napoleon-triangle-inverse of X(468)
X(44463) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 376, 1080}, {2043, 2044, 37172}, {2549, 22513, 5335}, {3830, 41017, 4}

X(44464) = CIRCUMCIRCLE-OF-OUTER-NAPOLEON-TRIANGLE-POLE OF LEMOINE AXIS

Barycentrics    (a^2*b^2 - b^4 + 2*a^2*c^2 + b^2*c^2)*(2*a^2*b^2 + a^2*c^2 + b^2*c^2 - c^4) + 2*Sqrt[3]*a^2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S : :
X(44464) = 3 X[2] - 4 X[33479], 2 X[2] + X[36322], 4 X[2] - X[36347], 5 X[2] - 2 X[36365], X[2] + 2 X[36385], X[4] + 2 X[3105], X[6194] - 4 X[33462], 4 X[22714] + 3 X[36322], 8 X[22714] - 3 X[36347], 5 X[22714] - 3 X[36365], X[22714] + 3 X[36385], 8 X[33479] + 3 X[36322], 16 X[33479] - 3 X[36347], 10 X[33479] - 3 X[36365], 2 X[33479] + 3 X[36385], 2 X[36322] + X[36347], 5 X[36322] + 4 X[36365], X[36322] - 4 X[36385], 5 X[36347] - 8 X[36365], X[36347] + 8 X[36385], X[36365] + 5 X[36385]

X(44464) lies on these lines: {2, 51}, {4, 3105}, {15, 10788}, {16, 33388}, {39, 42998}, {62, 33389}, {76, 43954}, {194, 22113}, {299, 20425}, {622, 37242}, {628, 3095}, {3094, 16941}, {3104, 42152}, {3106, 37640}, {3107, 6770}, {3643, 36780}, {5334, 31702}, {5335, 22708}, {6783, 32466}, {7685, 22695}, {7737, 22707}, {9993, 41070}, {14539, 35925}, {14651, 43455}, {14881, 33412}, {18581, 22702}, {18582, 43538}, {22690, 40694}, {22692, 37641}, {32447, 37786}, {35918, 39656}, {41071, 43461}

X(44464) = midpoint of X(3105) and X(22694)
X(44464) = reflection of X(i) in X(j) for these {i,j}: {4, 22694}, {6194, 22715}, {7709, 3107}, {22684, 33478}, {22714, 33479}, {22715, 33462}, {44460, 262}
X(44464) = anticomplement of X(22714)
X(44464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36322, 36347}, {2, 36385, 36322}, {11261, 22715, 2}, {22714, 33479, 2}

X(44465) = CIRCUMCIRCLE-OF-OUTER-NAPOLEON-TRIANGLE-POLE OF DE LONGCHAMPS AXIS

Barycentrics    Sqrt[3]*a^2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2) + 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(44465) = 2 X[5] - 3 X[11297], 4 X[140] - 3 X[11298], 2 X[3098] + X[42085], 3 X[5085] - X[42155], 4 X[5092] - X[42086], 5 X[12017] - 2 X[42118], X[33878] + 2 X[42117]

X(44465) lies on these lines: {2, 3}, {14, 14538}, {15, 7737}, {16, 2549}, {62, 7739}, {182, 6772}, {398, 5864}, {511, 10654}, {530, 11179}, {538, 36365}, {616, 2782}, {617, 36993}, {622, 6770}, {628, 22532}, {1350, 42154}, {1352, 3643}, {2794, 5613}, {3098, 22512}, {3181, 32515}, {3734, 5617}, {5085, 42155}, {5092, 42086}, {5237, 22890}, {5334, 20426}, {5335, 13103}, {5343, 16628}, {5615, 37641}, {5865, 42147}, {5872, 32836}, {5979, 9744}, {5980, 11185}, {5981, 14907}, {6771, 18582}, {6774, 6775}, {7694, 9749}, {9989, 36998}, {10645, 43618}, {10646, 21157}, {10992, 35692}, {11078, 18911}, {11485, 18907}, {11486, 15048}, {12017, 42118}, {14539, 36967}, {14540, 16964}, {14541, 42157}, {16001, 41112}, {16241, 21158}, {18581, 36755}, {20415, 41119}, {20425, 37640}, {21156, 37832}, {21159, 42528}, {21163, 31701}, {21466, 40280}, {22689, 41023}, {22846, 31704}, {33878, 42117}, {36756, 42090}, {36760, 43454}

X(44465) = midpoint of X(1350) and X(42154)
X(44465) = reflection of X(i) in X(j) for these {i,j}: {1352, 3643}, {6775, 6774}, {10653, 182}, {11296, 549}, {22513, 6771, {44461, 3}
X(44465) = circumcircle-of-outer-Napoleon-triangle-inverse of X(858)
X(44465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3534, 44250}, {4, 37177, 5}, {11304, 35918, 2}, {15765, 18585, 11305}

X(44466) = CIRCUMCIRCLE-OF-OUTER-NAPOLEON-TRIANGLE-POLE OF HATZIPOLAKIS AXIS

Barycentrics    Sqrt[3]*a^2*(a^4 - b^4 + b^2*c^2 - c^4) + 2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*S : :
X(44466) = 3 X[2] - 4 X[32461], X[5189] - 4 X[37974], 4 X[32460] - 5 X[37760]

X(44466) lies on these lines: {2, 3}, {13, 14170}, {61, 8014}, {298, 2453}, {476, 16771}, {523, 3181}, {530, 9143}, {622, 35314}, {691, 34374}, {3448, 11078}, {5473, 11131}, {5478, 8838}, {5613, 14185}, {5616, 8172}, {6770, 16770}, {6771, 41472}, {6800, 42155}, {7712, 42086}, {10653, 11003}, {10654, 11002}, {11537, 37640}, {14169, 36968}, {16530, 22739}, {19107, 30465}, {21467, 30469}, {22491, 34315}, {30486, 36995}, {36211, 40693}, {36961, 40709}, {40854, 41023}

X(44466) = anticomplement of X(36186)
X(44466) = reflection of X(i) in X(j) for these {i,j}: {3448, 11078}, {5189, 36185}, {36185, 37974}, {36186, 32461}, {44462, 23}
X(44466) = circumcircle-inverse of X(11146)
X(44466) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(1080)
X(44466) = circumcircle-of-anticomplementary-triangle-inverse of X(10210)
X(44466) = circumcircle-of-outer-Napoleon-triangle-inverse of X(5)
X(44466) = psi-transform of X(6771)
X(44466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36181, 44462}, {1113, 1114, 11146}, {1316, 36185, 2}, {14807, 14808, 10210}, {32461, 36186, 2}

X(44467) = MOSES-PARRY-CIRCLE-POLE OF EULER LINE

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

X(44467) is the intersection of the diagonals of cyclic quadrilateral X(111)X(112)X(115)X(1560). (Randy Hutson, September 30, 2021)

X(44467) lies on these lines: {2, 339}, {6, 5505}, {25, 111}, {115, 427}, {186, 11580}, {187, 37969}, {230, 231}, {250, 15398}, {1112, 3124}, {1611, 2079}, {3162, 8770}, {5094, 8426}, {5523, 5913}, {6791, 12828}, {8744, 37962}, {8749, 9717}, {10301, 33842}, {10311, 15364}, {10317, 37980}, {10766, 15106}, {12824, 36828}, {13595, 36415}, {13854, 40347}, {19504, 39024}, {28662, 41618}, {32740, 41616}, {34809, 40126}, {39575, 39576}

X(44467) = Moses-Parry-circle-inverse of X(468)
X(44467) = polar conjugate of the isotomic conjugate of X(2854)
X(44467) = X(40119)-Ceva conjugate of X(25)
X(44467) = PU(4)-harmonic conjugate of X(2492)
X(44467) = radical center of {circumcircle, nine-point circle, Moses-Parry circle}
X(44467) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2770}, {69, 36150}, {304, 32741}, {34055, 36824}
X(44467) = crossdifference of every pair of points on line {3, 14417}
X(44467) = barycentric product X(i)*X(j) for these {i,j}: {4, 2854}, {232, 37858}, {523, 7482}, {9177, 17983}, {31655, 40119}
X(44467) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2770}, {1843, 36824}, {1973, 36150}, {1974, 32741}, {2854, 69}, {5140, 34171}, {7482, 99}, {9177, 6390}
X(44467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 41676, 34336}, {111, 112, 25}, {115, 1560, 427}, {2079, 8428, 21213}, {3018, 6103, 16318}, {3124, 35325, 1112}, {3291, 14580, 468}, {8105, 8106, 468}, {8426, 8427, 5094}, {16317, 16318, 468}

X(44468) = MOSES-PARRY-CIRCLE-POLE OF FERMAT LINE

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

X(44468) lies on these lines: {5, 39}, {6, 5663}, {30, 3003}, {111, 5024}, {112, 1593}, {187, 37950}, {230, 15122}, {232, 37984}, {246, 39024}, {543, 34990}, {566, 2549}, {690, 2492}, {1597, 8749}, {1989, 7739}, {2079, 15815}, {3053, 12084}, {3269, 16270}, {3291, 5159}, {3767, 18281}, {5094, 8426}, {5305, 6103}, {5309, 44287}, {5523, 37981}, {6593, 13233}, {6749, 18907}, {6772, 40695}, {6775, 40696}, {7514, 14910}, {8552, 14566}, {10766, 32251}, {13596, 18373}, {39832, 40121}

X(44468) = complement of isotomic conjugate of X(9139)
X(44468) = X(i)-complementary conjugate of X(j) for these (i,j): {923, 113}, {2159, 126}, {9139, 2887}, {36119, 34517}, {40352, 16597}
X(44468) = crosspoint of X(2) and X(9139)
X(44468) = crosssum of X(6) and X(5642)
X(44468) = crossdifference of every pair of points on line {2930, 9003}
X(44468) = {X(39),X(115)}-harmonic conjugate of X(2493)

leftri

Lozada-Lemoine circles: X(44469)-X(44514)

rightri

This preamble and centers X(44469)-X(44514), based on a construction by Anton Zakharov in Mathoverflow, were contributed by César Eliud Lozada, August 28, 2021.

In the three following constructions, let K be the symmedian point X(6)-of-ABC and T' = A'B'C' a triangle.

  1. Let Ab = B'C' ∩ AC and Ac = B'C' ∩ AB, and define Bc, Ca, Ba, Cb cyclically.

    1. The circle {{K, Ab, Ac}} cuts again AC and AB in A'b and A'c, respectively, and points B'c, C'a, B'a, C'b are defined similarly. For some triangles T' these last six points lie on a circle here named Lozada-Lemoine-circles-1A of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (anti-Conway, 44469), (2nd anti-Conway, 44470), (anti-inner-Grebe, 44471), (anti-outer-Grebe, 44472), (anti-Honsberger, 6), (inner-Grebe, 44473), (outer-Grebe, 44474), (1st Kenmotu diagonals, 44475), (2nd Kenmotu diagonals, 44476), (inner tri-equilateral, 44477), (outer tri-equilateral, 44478) (1)

      (1): All listed triangles are perspective to ABC with perspector X(6), however, this is not a sufficient condition leading to the described circle.

    2. The circle {{K, Cb, Bc}} cuts again AC and AB in A'b and A'c, respectively, and points B'c, C'a, B'a, C'b are defined similarly. For some triangles T' these last six points lie on a circle here named Lozada-Lemoine-circles-1B of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (anti-Conway, 44479), (2nd anti-Conway, 44480), (anti-inner-Grebe, 44481), (anti-outer-Grebe, 44482), (anticomplementary, 576), (Bevan antipodal, 576), (2nd Ehrmann, 575), (excentral, 576), (inner-Grebe, 44483), (outer-Grebe, 44484), (1st Kenmotu diagonals, 44485), (2nd Kenmotu diagonals, 44486), (Largest-circumscribed-equilateral, 576), (Moses-Soddy, 576), (Pelletier, 576), (Schroeter, 576), (Soddy, 576), (tangential, 576), (inner tri-equilateral, 44487), (outer tri-equilateral, 44488), (X-parabola-tangential, 576), (Yiu tangents, 576)

  2. Now, let Ab, Ac be the orthogonal projections of A' in AC and AB, respectively, and define Bc, Ca, Ba, Cb cyclically.

    1. For certain triangles T', points A'b, A'c, B'a, B'c, C'a, C'b defined similarly as in (1a), lie on a circle, here named Lozada-Lemoine-circles-2A of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (ABC-X3 reflections, 576), (midheight, 44489), (orthic, 44470), (orthocentroidal, 44490), (reflection, 44491)

    2. Also, points A'b, A'c, B'a, B'c, C'a, C'b defined similarly as in (1b), lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-2B of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (midheight, 44492), (orthic, 44480), (orthocentroidal, 44493), (reflection, 44494)

  3. Finally, let Ab, Ac be the points at which the parallel line through A' to BC cuts AC and AB, respectively, and define Bc, Ca, Ba, Cb cyclically.

    1. Points built as in (1a) lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-3A of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (2nd anti-extouch, 44479), (1st Brocard-reflected, 5097)*, (1st excosine, 44495), (extouch, 576), (inner-Fermat, 44511), (outer-Fermat, 44512), (2nd Hatzipolakis, 576), (incentral, 576), (intouch, 576), (Lemoine, 576), (Macbeath, 576), (McCay, 44496), (medial, 576), (inner-Napoleon, 44497), (outer-Napoleon, 44498), (1st Neuberg, 44499), (2nd Neuberg, 44500), (orthic, 576), (Steiner, 576), (symmedial, 576), (inner-Vecten, 44501), (outer-Vecten, 44502), (Yff contact, 576)

      *This circle has squared radius (25*S^2+SW^2)*R^2/(16*SW^2).

    2. In the same way, points built as in (1b) lie on a circle for some triangles T'. These circles are named here Lozada-Lemoine-circles-3B of T'.

      The appearance of (T', n) in the following list means that for triangle T' the corresponding circle has center X(n):

      (1st anti-circumperp, 576), (2nd anti-extouch, 44469), (1st Brocard-reflected**, 5050), (1st Brocard, 6), (inner-Conway, 576), (1st excosine, 44503), (inner-Fermat, 44513), (outer-Fermat, 44514), (Gemini-017, 576), (Gemini-018, 576), (Gemini-029, 576), (Gemini-030, 576), (McCay, 44504), (inner-Napoleon, 44505), (outer-Napoleon, 44506), (1st Neuberg, 44507), (2nd Neuberg, 44508), (1st orthosymmedial, 6), (inner-Vecten, 44509), (outer-Vecten, 44510)

      **This circle has squared radius (4*S^2+SW^2)*R^2/(8*SW^2).

Notes:

X(44469) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF ANTI-CONWAY TRIANGLE

Barycentrics    a^2*(a^10-3*(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-(3*b^8+2*b^4*c^4+3*c^8)*a^2+(b^4-c^4)*(b^2-c^2)^3) : :
X(44469) = 3*X(6)+X(37498) = 3*X(6)-X(44492) = X(68)-3*X(23327) = 3*X(5050)-X(37488) = X(11477)+3*X(37497) = 3*X(19139)-X(32139)

The squared-radius of this circle is ρ2 = (R^4*SW^2+(3*R^2-SW)^2*S^2)*R^2/((4*R^2-SW)^2*SW^2)

X(44469) lies on these lines: {3, 6}, {4, 22151}, {24, 15462}, {26, 9019}, {30, 34117}, {67, 15317}, {68, 23327}, {69, 37119}, {154, 37972}, {155, 542}, {156, 15581}, {184, 26283}, {382, 15140}, {394, 5094}, {399, 19379}, {524, 8548}, {613, 9630}, {858, 1899}, {895, 11458}, {1092, 8541}, {1147, 2393}, {1352, 1594}, {1498, 11645}, {1503, 18569}, {1992, 18916}, {1994, 16063}, {1995, 43811}, {2781, 12084}, {2854, 9925}, {3167, 15139}, {3564, 13371}, {3567, 14951}, {3618, 7558}, {3818, 7507}, {5422, 7495}, {5446, 19136}, {5476, 10982}, {5504, 13248}, {5622, 5889}, {6240, 31670}, {6593, 7530}, {6776, 37444}, {7387, 19153}, {7506, 9971}, {7517, 18374}, {7542, 16789}, {7568, 38110}, {7574, 18445}, {7592, 11179}, {8537, 43574}, {8550, 12161}, {8681, 15316}, {8705, 15582}, {9306, 43130}, {9932, 11255}, {9968, 14915}, {9976, 15106}, {9977, 32348}, {10168, 15805}, {10249, 12163}, {10250, 19458}, {11216, 15136}, {11416, 15073}, {11649, 34787}, {12085, 41725}, {12167, 41714}, {12293, 23049}, {13160, 14561}, {13564, 19151}, {14912, 41256}, {15135, 31152}, {15137, 15533}, {15577, 32171}, {16977, 41588}, {17814, 18553}, {18580, 44201}, {19140, 38791}, {19149, 22802}, {19150, 32341}, {19360, 32251}, {20423, 38323}, {21284, 33586}, {21970, 41674}, {23336, 34380}, {28695, 41255}, {31724, 36990}, {34155, 37196}, {34938, 41719}, {37118, 41614}

X(44469) = midpoint of X(i) and X(j) for these {i, j}: {155, 8549}, {576, 13346}, {5504, 13248}, {8548, 16266}, {37498, 44492}
X(44469) = reflection of X(i) in X(j) for these (i, j): (15581, 156), (34118, 13371), (44470, 575)
X(44469) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(44480)}} and {{A, B, C, X(67), X(8553)}}
X(44469) = Brocard circle-inverse of-X(44480)
X(44469) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44480), (3, 1351, 37473), (6, 36747, 576), (6, 36752, 44494), (6, 37483, 44493), (6, 37498, 44492), (6, 44503, 575), (182, 576, 389), (182, 44494, 36752), (578, 11511, 44479), (11416, 34148, 15073), (20806, 39588, 1352)

X(44470) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF 2nd ANTI-CONWAY TRIANGLE

Barycentrics    a^2*(a^10-2*(b^2+c^2)*a^8+2*b^2*c^2*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(44470) = 3*X(6)+X(17834) = X(155)-3*X(19153) = 2*X(156)-3*X(206) = X(156)-3*X(19154) = 3*X(182)-X(13346) = 3*X(5050)+X(37491) = 7*X(10541)-3*X(37497) = X(17834)-3*X(37488)

The squared-radius of this circle is ρ2 = R^2*((6*R^2-SW)^2*S^2+(2*R^2-SW)^2*SW^2)/(4*(4*R^2-SW)^2*SW^2)

X(44470) lies on these lines: {3, 6}, {5, 19136}, {20, 5622}, {24, 41614}, {25, 43130}, {26, 2393}, {54, 1992}, {68, 542}, {69, 3147}, {141, 16238}, {155, 19153}, {156, 206}, {184, 6515}, {193, 9545}, {235, 3818}, {343, 468}, {524, 1147}, {631, 43815}, {895, 7556}, {973, 9977}, {1092, 5095}, {1176, 14912}, {1209, 11178}, {1352, 1974}, {1503, 9927}, {1658, 9926}, {1660, 10154}, {2781, 7689}, {2854, 15582}, {2904, 20806}, {2937, 39562}, {3518, 11188}, {3574, 6816}, {4558, 37114}, {5181, 34116}, {5449, 34118}, {5476, 34664}, {5486, 40441}, {5562, 35603}, {5651, 43811}, {5663, 9968}, {5965, 10274}, {6776, 19121}, {7387, 8549}, {7488, 15073}, {7502, 15074}, {7506, 29959}, {8542, 12106}, {8550, 13292}, {8681, 9937}, {9027, 9925}, {9715, 10602}, {9827, 16776}, {9970, 14448}, {10192, 41619}, {10249, 12085}, {10539, 15069}, {10984, 11179}, {11411, 41719}, {11412, 22151}, {11424, 20423}, {11464, 41617}, {11649, 34788}, {11750, 32273}, {11799, 18474}, {13754, 34117}, {14070, 34787}, {14790, 23327}, {14791, 15118}, {14852, 41613}, {15132, 19138}, {15577, 34382}, {18475, 32284}, {19124, 31670}, {19137, 24206}, {19139, 41593}, {21659, 37201}, {25711, 41612}, {29012, 34786}, {30739, 37649}, {32263, 41674}, {32269, 44080}, {32275, 38851}, {34155, 41616}, {34751, 37928}, {39805, 41672}

X(44470) = midpoint of X(i) and X(j) for these {i, j}: {3, 44492}, {6, 37488}, {26, 8548}, {7387, 8549}
X(44470) = reflection of X(i) in X(j) for these (i, j): (206, 19154), (19139, 41593), (34118, 5449), (44469, 575)
X(44470) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(44479)}} and {{A, B, C, X(68), X(14961)}}
X(44470) = Brocard circle-inverse of-X(44479)
X(44470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44479), (6, 52, 576), (6, 19131, 182), (182, 576, 578), (182, 13347, 20190), (182, 44489, 575), (343, 44077, 9306), (569, 17834, 13346), (575, 44490, 44489), (575, 44491, 182), (5050, 5157, 182), (7488, 37784, 15073), (15069, 18374, 10539), (44490, 44491, 575)

X(44471) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF ANTI-INNER-GREBE TRIANGLE

Barycentrics    a^2*((a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44471) = 3*X(6)-X(1161) = 3*X(182)-2*X(9738) = 3*X(182)-4*X(44476) = 2*X(1161)-3*X(42858) = 3*X(3098)-4*X(43141) = 2*X(9733)-3*X(42859) = 5*X(22234)-4*X(44475)

The squared-radius of this circle is ρ2 = (4*SW^4+(9*S^2+5*SW^2)*S^2-(4*(3*S^2+SW^2))*S*SW)*R^2/(4*S^2*SW^2)

X(44471) lies on these lines: {3, 6}, {524, 5875}, {542, 5871}, {5591, 40107}, {6215, 34507}, {10514, 11178}, {15069, 26336}, {18509, 18553}, {29012, 39887}

X(44471) = midpoint of X(1160) and X(11477)
X(44471) = reflection of X(i) in X(j) for these (i, j): (9732, 575), (9738, 44476), (42858, 6), (44472, 576)
X(44471) = Brocard circle-inverse of-X(44483)
X(44471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44483), (6, 11477, 11916), (6, 26341, 575), (182, 576, 44473), (576, 3098, 44485), (576, 44486, 44474), (1351, 44502, 576), (9738, 44476, 182)

X(44472) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF ANTI-OUTER-GREBE TRIANGLE

Barycentrics    a^2*(-(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44472) = 3*X(6)-X(1160) = 3*X(182)-2*X(9739) = 3*X(182)-4*X(44475) = 2*X(1160)-3*X(42859) = 3*X(3098)-4*X(43144) = 2*X(9732)-3*X(42858) = 5*X(22234)-4*X(44476)

The squared-radius of this circle is ρ2 = (4*SW^4+(9*S^2+5*SW^2)*S^2+(4*(3*S^2+SW^2))*S*SW)*R^2/(4*S^2*SW^2)

X(44472) lies on these lines: {3, 6}, {524, 5874}, {542, 5870}, {5590, 40107}, {6214, 34507}, {10515, 11178}, {15069, 26346}, {18511, 18553}, {29012, 39888}

X(44472) = midpoint of X(1161) and X(11477)
X(44472) = reflection of X(i) in X(j) for these (i, j): (9733, 575), (9739, 44475), (42859, 6), (44471, 576)
X(44472) = Brocard circle-inverse of-X(44484)
X(44472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44484), (6, 11477, 11917), (6, 26348, 575), (182, 576, 44474), (576, 3098, 44486), (576, 44485, 44473), (1351, 44501, 576), (9739, 44475, 182)

X(44473) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF INNER-GREBE TRIANGLE

Barycentrics    a^2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4+4*(a^2+b^2+c^2)*S) : :
X(44473) = 3*X(6)-X(3312) = 2*X(3312)-3*X(42832) = 5*X(11482)-X(11917)

The squared-radius of this circle is ρ2 = R^2*S^2*(9*S^2-24*S*SW+17*SW^2)/(4*SW^2*(2*SW-S)^2)

X(44473) lies on these lines: {3, 6}, {141, 13993}, {486, 11178}, {524, 19116}, {542, 1588}, {597, 642}, {3068, 25555}, {3069, 40107}, {5476, 7583}, {6460, 19924}, {6680, 35684}, {7581, 20423}, {7584, 34507}, {9540, 10168}, {13785, 18553}, {13886, 14561}, {13921, 38317}, {15069, 18510}, {19109, 32135}, {19111, 25556}, {19130, 31412}, {20301, 32253}, {29012, 39875}, {29317, 43407}

X(44473) = reflection of X(i) in X(j) for these (i, j): (42832, 6), (43118, 575)
X(44473) = Brocard circle-inverse of-X(44481)
X(44473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44481), (6, 576, 44474), (6, 3311, 575), (6, 6417, 44482), (6, 9975, 44502), (6, 11477, 6418), (6, 44501, 576), (182, 576, 44471), (575, 9738, 182), (576, 39561, 44486), (576, 44485, 44472), (3102, 43125, 7690), (6417, 44482, 42833), (6500, 11482, 6), (9975, 44502, 576), (39561, 44499, 44474), (44501, 44502, 9975)

X(44474) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF OUTER-GREBE TRIANGLE

Barycentrics    a^2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4-4*(a^2+b^2+c^2)*S) : :
X(44474) = 3*X(6)-X(3311) = 2*X(3311)-3*X(42833) = 5*X(11482)-X(11916)

The squared-radius of this circle is ρ2 = R^2*S^2*(9*S^2+24*S*SW+17*SW^2)/(4*SW^2*(2*SW+S)^2)

X(44474) lies on these lines: {3, 6}, {141, 13925}, {485, 11178}, {524, 19117}, {542, 1587}, {597, 641}, {3068, 40107}, {3069, 25555}, {5476, 7584}, {6459, 19924}, {6680, 35685}, {7582, 20423}, {7583, 34507}, {10168, 13935}, {13665, 18553}, {13880, 38317}, {13939, 14561}, {15069, 18512}, {19108, 32135}, {19110, 25556}, {19130, 42561}, {20301, 32252}, {29012, 39876}, {29317, 43408}

X(44474) = reflection of X(i) in X(j) for these (i, j): (42833, 6), (43119, 575)
X(44474) = Brocard circle-inverse of-X(44482)
X(44474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44482), (6, 576, 44473), (6, 3312, 575), (6, 6418, 44481), (6, 9974, 44501), (6, 11477, 6417), (6, 44502, 576), (182, 576, 44472), (575, 9739, 182), (576, 39561, 44485), (576, 44486, 44471), (3103, 43124, 7692), (6418, 44481, 42832), (6501, 11482, 6), (9974, 44501, 576), (39561, 44499, 44473), (44501, 44502, 9974)

X(44475) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF 1st KENMOTU-DIAGONALS TRIANGLE

Barycentrics    a^2*(-2*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44475) = 3*X(6)+X(9732) = 3*X(182)-X(9739) = 3*X(182)+X(44472) = X(1160)-9*X(5050) = 5*X(22234)-X(44471) = 3*X(39561)+X(42858)

The squared-radius of this circle is ρ2 = ((3*S^2+SW^2)^2-4*S^2*SW^2+2*S*SW*(3*S^2-SW^2))*R^2/(16*S^2*SW^2)

X(44475) lies on these lines: {3, 6}, {491, 34507}, {641, 40107}, {6251, 19130}, {6290, 18553}, {11645, 13749}, {14561, 32489}, {22525, 22727}, {22594, 35930}

X(44475) = midpoint of X(i) and X(j) for these {i, j}: {576, 9738}, {9739, 44472}
X(44475) = reflection of X(i) in X(j) for these (i, j): (43141, 20190), (44476, 575)
X(44475) = Brocard circle-inverse of-X(44486)
X(44475) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44486), (6, 3103, 44502), (6, 44509, 575), (182, 576, 372), (182, 44472, 9739), (575, 5092, 44484), (575, 5097, 44481), (44482, 44483, 575)

X(44476) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF 2nd KENMOTU-DIAGONALS TRIANGLE

Barycentrics    a^2*(2*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44476) = 3*X(6)+X(9733) = 3*X(182)-X(9738) = 3*X(182)+X(44471) = X(1161)-9*X(5050) = 5*X(22234)-X(44472) = 3*X(39561)+X(42859)

The squared-radius of this circle is ρ2 = ((3*S^2+SW^2)^2-4*S^2*SW^2-2*S*SW*(3*S^2-SW^2))*R^2/(16*S^2*SW^2)

X(44476) lies on these lines: {3, 6}, {492, 34507}, {642, 40107}, {6250, 19130}, {6289, 18553}, {11645, 13748}, {14561, 32488}, {22525, 22726}, {22623, 35930}

X(44476) = midpoint of X(i) and X(j) for these {i, j}: {576, 9739}, {9738, 44471}
X(44476) = reflection of X(i) in X(j) for these (i, j): (43144, 20190), (44475, 575)
X(44476) = Brocard circle-inverse of-X(44485)
X(44476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 3102, 44501), (6, 44510, 575), (182, 576, 371), (182, 44471, 9738), (575, 5092, 44483), (575, 5097, 44482), (44481, 44484, 575)

X(44477) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF INNER TRI-EQUILATERAL TRIANGLE

Barycentrics    a^2*(-2*sqrt(3)*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44477) = 3*X(182)-X(9736)

The squared-radius of this circle is ρ2 = R^2*(S^2*(27*S^2+4*SW^2)+2*sqrt(3)*SW*S*(3*S^2-SW^2)+SW^4)/(48*SW^2*S^2)

X(44477) lies on these lines: {3, 6}, {303, 34507}, {542, 33476}, {5476, 25164}, {5965, 22866}, {11645, 41039}, {12151, 25157}, {22525, 22715}, {22689, 32135}

X(44477) = midpoint of X(576) and X(9735)
X(44477) = reflection of X(44478) in X(575)
X(44477) = Brocard circle-inverse of-X(44488)
X(44477) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44488), (6, 5611, 576), (6, 44505, 575), (182, 576, 16)

X(44478) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF OUTER TRI-EQUILATERAL TRIANGLE

Barycentrics    a^2*(2*sqrt(3)*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44478) = 3*X(182)-X(9735)

The squared-radius of this circle is ρ2 = R^2*(S^2*(27*S^2+4*SW^2)-2*sqrt(3)*SW*S*(3*S^2-SW^2)+SW^4)/(48*SW^2*S^2)

X(44478) lies on these lines: {3, 6}, {302, 34507}, {542, 33477}, {5476, 25154}, {5965, 22911}, {11645, 41038}, {12151, 25167}, {22525, 22714}, {22687, 32135}

X(44478) = midpoint of X(576) and X(9736)
X(44478) = reflection of X(44477) in X(575)
X(44478) = Brocard circle-inverse of-X(44487)
X(44478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44487), (6, 5615, 576), (6, 44506, 575), (182, 576, 15)

X(44479) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF ANTI-CONWAY TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^8+6*(b^2+c^2)*b^2*c^2*a^4-2*(b^4+3*b^2*c^2+c^4)*a^6+2*(b^4+c^4+(b^2-c^2)*b*c)*(b^4+c^4-(b^2-c^2)*b*c)*a^2+(b^8-c^8)*(-b^2+c^2)) : :
X(44479) = 3*X(2)+X(15073) = 3*X(6)-X(52) = X(52)+3*X(9967) = 3*X(69)-7*X(7999) = 3*X(182)-2*X(9729) = X(185)-3*X(11179) = 3*X(1351)+X(37484) = 3*X(3313)-X(37484) = 3*X(5050)+X(18438) = 3*X(5050)-X(19161) = 9*X(5050)-5*X(37481) = 3*X(5085)-X(37511) = X(6243)-5*X(11482) = 3*X(9730)-X(37473) = 3*X(11574)-X(15644) = 2*X(15074)+X(43130) = 2*X(16625)-5*X(22234) = 3*X(16836)-4*X(20190) = 3*X(18438)+5*X(37481) = 3*X(19161)-5*X(37481)

The squared-radius of this circle is: ρ2 = ((6*R^2-SW)^2*S^2+(2*R^2-SW)^2*SW^2)/(16*SW^2*R^2)

X(44479) lies on these lines: {2, 15073}, {3, 6}, {5, 2393}, {26, 19136}, {51, 7493}, {54, 22151}, {68, 5486}, {69, 7999}, {140, 12235}, {141, 5449}, {155, 32621}, {185, 11179}, {186, 43815}, {235, 19130}, {343, 3819}, {468, 5943}, {524, 1216}, {542, 5907}, {597, 5462}, {895, 7550}, {973, 14763}, {1177, 34155}, {1205, 9970}, {1209, 5181}, {1352, 6467}, {1353, 22829}, {1656, 29959}, {1843, 3542}, {1885, 29012}, {1992, 11412}, {2386, 37242}, {2781, 40647}, {2854, 10170}, {3090, 11188}, {3147, 3618}, {3564, 11264}, {3589, 6153}, {3818, 18383}, {3917, 6515}, {5446, 9019}, {5476, 10110}, {5480, 17710}, {5562, 40673}, {5622, 14118}, {5891, 15069}, {6329, 32191}, {6642, 34787}, {6688, 34751}, {6723, 15082}, {6776, 12111}, {7395, 10602}, {7509, 41614}, {7514, 8548}, {8537, 43651}, {8547, 15581}, {8549, 9818}, {8550, 13754}, {9027, 15067}, {9813, 34788}, {9822, 38317}, {9969, 10095}, {9972, 21660}, {9977, 32368}, {10575, 43273}, {10984, 11470}, {11180, 15056}, {11416, 13434}, {11444, 15531}, {11645, 11750}, {12061, 16776}, {12220, 14853}, {12272, 40330}, {13367, 15462}, {13394, 44084}, {13491, 34146}, {14912, 41716}, {14913, 20303}, {16789, 41587}, {18474, 18553}, {21849, 44210}, {22352, 37929}, {31670, 37201}, {34114, 41593}, {34864, 39562}, {37126, 37784}

X(44479) = midpoint of X(i) and X(j) for these {i, j}: {5, 15074}, {6, 9967}, {1205, 9970}, {1216, 32284}, {1351, 3313}, {1352, 6467}, {5480, 17710}, {9972, 21660}, {10625, 11477}, {18438, 19161}
X(44479) = reflection of X(i) in X(j) for these (i, j): (389, 575), (576, 44495), (1353, 22829), (9969, 18583), (14913, 24206), (32191, 6329), (34507, 11793), (41714, 9822), (43130, 5)
X(44479) = complement of the complement of X(15073)
X(44479) = Brocard circle-inverse of-X(44470)
X(44479) = X(69)-of-X(5)-Brocard triangle
X(44479) = center of Lozada-Lemoine-circle-3A of 2nd anti-extouch triangle
X(44479) = X(43130)-of-Johnson-triangle
X(44479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44470), (3, 44503, 182), (6, 569, 575), (6, 11477, 37493), (52, 37476, 9729), (182, 576, 44480), (569, 8538, 6), (578, 11511, 44469), (5050, 18438, 19161), (38317, 41714, 9822)

X(44480) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF 2nd ANTI-CONWAY TRIANGLE

Barycentrics    a^2*(a^10-3*(b^2+c^2)*a^8+2*(b^4+c^4)*a^6+2*(b^2+c^2)*(b^4+c^4)*a^4-(3*b^8+3*c^8-2*(2*b^4+3*b^2*c^2+2*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3) : :
X(44480) = 3*X(6)-X(36747)

The squared-radius of this circle is: ρ2 = (R^4*SW^2+(3*R^2-SW)^2*S^2)*R^2/((2*R^2-SW)^2*SW^2)

X(44480) lies on these lines: {3, 6}, {5, 34117}, {26, 19127}, {51, 26283}, {67, 1656}, {69, 7558}, {141, 9820}, {155, 34507}, {159, 41714}, {394, 15135}, {524, 12161}, {542, 1181}, {597, 18281}, {611, 9630}, {631, 22151}, {858, 5422}, {1176, 6403}, {1199, 1992}, {1352, 11441}, {1594, 14561}, {1614, 11188}, {1993, 7495}, {2781, 7526}, {3618, 37119}, {3796, 11649}, {3818, 19149}, {5012, 15073}, {5020, 15139}, {5094, 10601}, {5462, 19136}, {5480, 18569}, {5622, 10574}, {5944, 15577}, {5965, 32341}, {6240, 39588}, {6593, 15132}, {6642, 19153}, {6723, 15106}, {6759, 9813}, {6776, 34799}, {7401, 41719}, {7506, 18374}, {7507, 19130}, {7517, 9971}, {7592, 41614}, {7770, 41255}, {8541, 10984}, {8548, 8550}, {8549, 9977}, {8681, 19458}, {10539, 29959}, {10766, 26216}, {11061, 14789}, {11178, 17814}, {11179, 19467}, {11180, 43605}, {12061, 35707}, {12173, 29012}, {12225, 31670}, {13371, 18583}, {13861, 16776}, {14853, 37444}, {14984, 32046}, {15045, 43815}, {15069, 18445}, {15141, 15805}, {15317, 19151}, {15462, 17928}, {16063, 34545}, {16789, 34002}, {17810, 37972}, {18451, 18553}, {25406, 35471}, {32233, 39562}, {32276, 41731}

X(44480) = midpoint of X(i) and X(j) for these {i, j}: {1351, 37485}, {11477, 37486}
X(44480) = reflection of X(i) in X(j) for these (i, j): (3, 44491), (578, 575)
X(44480) = Brocard circle-inverse of-X(44469)
X(44480) = center of Lozada-Lemoine-Circle-2b of orthic triangle
X(44480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44469), (3, 11482, 10510), (6, 11477, 36749), (6, 36752, 575), (6, 37514, 44503), (6, 44492, 576), (182, 576, 44479), (575, 9729, 182), (576, 44493, 44492), (576, 44494, 6), (6759, 9813, 43130), (37514, 44503, 182), (44493, 44494, 576)

X(44481) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF ANTI-INNER-GREBE TRIANGLE

Barycentrics    (2*a^4+b^4-4*b^2*c^2+c^4-3*(b^2+c^2)*a^2+(4*a^2+4*c^2+4*b^2)*S)*a^2 : :
X(44481) = 3*X(6)+X(372) = 5*X(6)-X(35841) = 3*X(6)-X(44501) = 5*X(372)+3*X(35841) = 3*X(597)-X(640) = 5*X(22234)-X(44485) = 3*X(35841)-5*X(44501)

The squared-radius of this circle is: ρ2 = (9*S^2-24*SW*S+17*SW^2)*R^2*S^2/(16*(SW-S)^2*SW^2)

X(44481) lies on these lines: {3, 6}, {486, 18553}, {542, 7584}, {597, 640}, {599, 13961}, {1352, 13939}, {1587, 5476}, {3069, 34507}, {3071, 11645}, {3589, 13925}, {3818, 39875}, {7582, 11179}, {7583, 25555}, {7798, 22594}, {8550, 19116}, {8981, 10168}, {8992, 32149}, {9976, 19110}, {9977, 19095}, {11178, 13951}, {13966, 40107}, {13972, 24206}, {14561, 26469}, {19130, 22596}, {25561, 42262}, {29323, 42271}, {32252, 41731}, {43150, 43431}

X(44481) = midpoint of X(i) and X(j) for these {i, j}: {32, 44502}, {372, 44501}, {576, 43121}
X(44481) = reflection of X(44483) in X(575)
X(44481) = Brocard circle-inverse of-X(44473)
X(44481) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44473), (6, 372, 44501), (6, 575, 44482), (6, 1505, 44499), (6, 3312, 576), (6, 3594, 9975), (6, 5050, 42833), (6, 5062, 44500), (6, 6418, 44474), (6, 6420, 44502), (6, 6432, 9974), (182, 576, 9732), (575, 5097, 44475), (575, 44476, 44484), (6421, 43125, 43141), (39561, 44510, 575), (42832, 44474, 6418)

X(44482) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF ANTI-OUTER-GREBE TRIANGLE

Barycentrics    a^2*(2*a^4+b^4-4*b^2*c^2+c^4-3*(b^2+c^2)*a^2-(4*a^2+4*c^2+4*b^2)*S) : :
X(44482) = 3*X(6)+X(371) = 5*X(6)-X(35840) = 3*X(6)-X(44502) = 5*X(371)+3*X(35840) = 3*X(597)-X(639) = 5*X(22234)-X(44486) = 3*X(35840)-5*X(44502)

The squared-radius of this circle is: ρ2 = (9*S^2+24*SW*S+17*SW^2)*R^2*S^2/(16*(SW+S)^2*SW^2)

X(44482) lies on these lines: {3, 6}, {485, 18553}, {542, 7583}, {597, 639}, {599, 13903}, {1352, 13886}, {1588, 5476}, {3068, 34507}, {3070, 11645}, {3589, 13993}, {3818, 31412}, {7581, 11179}, {7584, 25555}, {7798, 22623}, {8550, 19117}, {8976, 11178}, {8981, 40107}, {9976, 19111}, {9977, 19096}, {10168, 13966}, {13910, 24206}, {13983, 32149}, {14561, 26468}, {19130, 22625}, {25561, 42265}, {29323, 42272}, {32253, 41731}, {43150, 43430}

X(44482) = midpoint of X(i) and X(j) for these {i, j}: {32, 44501}, {371, 44502}, {576, 43120}
X(44482) = reflection of X(44484) in X(575)
X(44482) = Brocard circle-inverse of-X(44474)
X(44482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44474), (6, 371, 44502), (6, 575, 44481), (6, 3311, 576), (6, 3592, 9974), (6, 5050, 42832), (6, 5058, 44500), (6, 6417, 44473), (6, 6419, 44501), (6, 6431, 9975), (6, 44501, 22330), (182, 576, 9733), (575, 5097, 44476), (575, 44475, 44483), (6422, 43124, 43144), (39561, 44509, 575), (42833, 44473, 6417)

X(44483) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF INNER-GREBE TRIANGLE

Barycentrics    (-(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2))*a^2 : :
X(44483) = 3*X(6)+X(11824) = 3*X(182)-X(43121) = 3*X(182)+X(44485) = 3*X(5085)+X(35841)

The squared-radius of this circle is: ρ2 = (9*S^2+24*SW*S+17*SW^2)*R^2*S^2/(16*(SW+S)^2*SW^2)

X(44483) lies on these lines: {3, 6}, {542, 6215}, {640, 5875}, {3818, 39887}, {5476, 6202}, {5591, 34507}, {6277, 9977}, {7732, 9976}, {10514, 18553}, {10783, 11179}, {32280, 41731}

X(44483) = midpoint of X(i) and X(j) for these {i, j}: {3, 44501}, {640, 8550}, {43121, 44485}
X(44483) = reflection of X(44481) in X(575)
X(44483) = Brocard circle-inverse of-X(44471)
X(44483) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44471), (6, 1161, 576), (182, 575, 44484), (182, 576, 43118), (182, 44485, 43121), (182, 44509, 575), (575, 5092, 44476), (575, 44475, 44482), (5092, 44507, 44484), (11824, 26341, 43121)

X(44484) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF OUTER-GREBE TRIANGLE

Barycentrics    a^2*((2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44484) = 3*X(6)+X(11825) = 3*X(182)-X(43120) = 3*X(182)+X(44486) = 3*X(5085)+X(35840)

The squared-radius of this circle is: ρ2 = ((9*S^2+5*SW^2)*S^2+4*(3*S^2+SW^2)*SW*S+4*SW^4)*R^2/(16*(SW+S)^2*SW^2)

X(44484) lies on these lines: {3, 6}, {542, 6214}, {639, 5874}, {3818, 39888}, {5476, 6201}, {5590, 34507}, {6276, 9977}, {7733, 9976}, {10515, 18553}, {10784, 11179}, {32281, 41731}

X(44484) = midpoint of X(i) and X(j) for these {i, j}: {3, 44502}, {639, 8550}, {43120, 44486}
X(44484) = reflection of X(44482) in X(575)
X(44484) = Brocard circle-inverse of-X(44472)
X(44484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44472), (6, 1160, 576), (182, 575, 44483), (182, 576, 43119), (182, 43121, 20190), (182, 44486, 43120), (182, 44510, 575), (575, 5092, 44475), (575, 44476, 44481), (5092, 44507, 44483), (11825, 26348, 43120)

X(44485) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF 1st KENMOTU DIAGONALS TRIANGLE

Barycentrics    a^2*(-2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44485) = 3*X(182)-2*X(43121) = 3*X(182)-4*X(44483) = X(8982)-3*X(11179) = X(11824)+3*X(35841) = 5*X(22234)-4*X(44481)

The squared-radius of this circle is: ρ2 = ((3*S^2+SW^2)^2-4*S^2*SW^2-2*S*SW*(3*S^2-SW^2))*R^2/(4*(SW-S)^2*SW^2)

X(44485) lies on these lines: {3, 6}, {492, 40107}, {640, 34507}, {1352, 32488}, {6036, 13758}, {6118, 38317}, {6289, 11178}, {8982, 11179}

X(44485) = reflection of X(i) in X(j) for these (i, j): (372, 575), (576, 44501), (34507, 640), (43121, 44483)
X(44485) = Brocard circle-inverse of-X(44476)
X(44485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44476), (6, 43119, 44510), (182, 576, 44486), (575, 43120, 182), (576, 3098, 44471), (576, 39561, 44474), (35424, 44499, 44486), (43119, 44510, 182), (43121, 44483, 182), (44472, 44473, 576)

X(44486) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF 2nd KENMOTU DIAGONALS TRIANGLE

Barycentrics    a^2*(2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44486) = 3*X(182)-2*X(43120) = 3*X(182)-4*X(44484) = 3*X(11179)-X(26441) = X(11825)+3*X(35840) = 5*X(22234)-4*X(44482)

The squared-radius of this circle is: ρ2 = ((3*S^2+SW^2)^2-4*S^2*SW^2+2*S*SW*(3*S^2-SW^2))*R^2/(4*(SW+S)^2*SW^2)

X(44486) lies on these lines: {3, 6}, {491, 40107}, {639, 34507}, {1352, 32489}, {6036, 13638}, {6119, 38317}, {6290, 11178}, {11179, 26441}

X(44486) = reflection of X(i) in X(j) for these (i, j): (371, 575), (576, 44502), (34507, 639), (43120, 44484)
X(44486) = Brocard circle-inverse of-X(44475)
X(44486) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44475), (3, 44510, 182), (6, 43118, 44509), (182, 576, 44485), (575, 43121, 182), (576, 3098, 44472), (576, 39561, 44473), (35424, 44499, 44485), (43118, 44509, 182), (43120, 44484, 182), (44471, 44474, 576)

X(44487) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF INNER TRI-EQUILATERAL TRIANGLE

Barycentrics    a^2*(-2*sqrt(3)*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44487) = 3*X(6)-X(5615) = 3*X(182)-2*X(13349) = 3*X(5476)-2*X(7685) = 4*X(8590)-3*X(39554) = 3*X(11179)-X(36995) = 4*X(15516)-3*X(36758) = 4*X(20190)-3*X(21159)

The squared-radius of this circle is: ρ2 = (S^2*(27*S^2+4*SW^2)+SW^4-2*S*sqrt(3)*SW*(3*S^2-SW^2))*R^2/(4*(SW-S*sqrt(3))^2*SW^2)

X(44487) lies on these lines: {3, 6}, {302, 40107}, {383, 25559}, {542, 20429}, {624, 34507}, {5459, 5476}, {5979, 25560}, {5980, 22684}, {6774, 37785}, {11179, 36995}, {11645, 36994}

X(44487) = reflection of X(i) in X(j) for these (i, j): (16, 575), (576, 44497), (34507, 624)
X(44487) = Brocard circle-inverse of-X(44478)
X(44487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44478), (3, 44505, 182), (182, 576, 44488), (575, 13350, 182)

X(44488) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF OUTER TRI-EQUILATERAL TRIANGLE

Barycentrics    a^2*(2*sqrt(3)*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44488) = 3*X(6)-X(5611) = 3*X(182)-2*X(13350) = 3*X(5476)-2*X(7684) = 4*X(8590)-3*X(39555) = 3*X(11179)-X(36993) = 4*X(15516)-3*X(36757) = 4*X(20190)-3*X(21158)

The squared-radius of this circle is: ρ2 = (S^2*(27*S^2+4*SW^2)+SW^4+2*sqrt(3)*SW*S*(3*S^2-SW^2))*R^2/(4*(SW+sqrt(3)*S)^2*SW^2)

X(44488) lies on these lines: {3, 6}, {303, 40107}, {542, 20428}, {623, 34507}, {1080, 25560}, {5460, 5476}, {5978, 25559}, {5981, 22686}, {6771, 37786}, {11179, 36993}, {11645, 36992}

X(44488) = reflection of X(i) in X(j) for these (i, j): (15, 575), (576, 44498), (34507, 623)
X(44488) = Brocard circle-inverse of-X(44477)
X(44488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44477), (3, 44506, 182), (182, 576, 44487), (575, 13349, 182)

X(44489) = CENTER OF LOZADA-LEMOINE-CIRCLE-2A OF MIDHEIGHT TRIANGLE

Barycentrics    a^2*(a^10-2*(b^2+c^2)*a^8+10*b^2*c^2*a^6+2*(b^2+c^2)*(b^4-9*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(b^4-8*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(44489) = 3*X(6)+X(9786)

X(44489) lies on these lines: {3, 6}, {4, 43812}, {24, 40673}, {184, 4232}, {206, 12007}, {542, 39571}, {1092, 1992}, {1614, 1974}, {3564, 19137}, {3567, 8541}, {5032, 34148}, {5462, 8548}, {5476, 12233}, {5480, 22802}, {5622, 10990}, {5890, 11470}, {6642, 8681}, {6644, 32284}, {6759, 8550}, {6776, 26883}, {7592, 44102}, {8549, 10110}, {9306, 11225}, {9925, 43586}, {9976, 11746}, {10169, 15311}, {10282, 32621}, {11179, 34621}, {11255, 16881}, {11431, 15462}, {11579, 38791}, {13567, 34507}, {15033, 35483}, {15043, 37784}, {15577, 22829}, {16187, 37643}, {16270, 41618}, {18583, 22660}, {19121, 33748}, {19124, 32601}, {20423, 43810}, {32046, 33591}

X(44489) = midpoint of X(8550) and X(15873)
X(44489) = reflection of X(44503) in X(575)
X(44489) = Brocard circle-inverse of-X(44495)
X(44489) = The squared-radius of this circle is: ρ2 = ((12*R^2-SW)^2*S^2+(4*R^2-SW)^2*SW^2)*R^2/(4*((8*R^2-SW)^2*SW^2))
X(44489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44495), (6, 389, 576), (182, 576, 13346), (575, 44470, 182), (575, 44490, 44470), (1992, 43815, 1092), (5050, 19126, 182), (5462, 8548, 9813), (8550, 19136, 6759)

X(44490) = CENTER OF LOZADA-LEMOINE-CIRCLE-2A OF ORTHOCENTROIDAL TRIANGLE

Barycentrics    a^2*(a^10-2*(b^2+c^2)*a^8+6*b^2*c^2*a^6+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^4-(b^8+c^8-2*(4*b^4-5*b^2*c^2+4*c^4)*b^2*c^2)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(44490) = 3*X(6)+X(37489) = 3*X(182)-X(37480) = X(11511)-3*X(39561)

The squared-radius of this circle is: ρ2 = ((9*R^2-SW)^2*S^2+(3*R^2-SW)^2*SW^2)*R^2/(4*(6*R^2-SW)^2*SW^2)

X(44490) lies on these lines: {3, 6}, {4, 11579}, {20, 43810}, {26, 8547}, {110, 37644}, {184, 7426}, {542, 19136}, {1147, 19138}, {1986, 11470}, {1992, 15462}, {2777, 18431}, {2854, 12106}, {3580, 5651}, {5462, 12039}, {5622, 20423}, {6759, 41613}, {8550, 16619}, {8584, 18579}, {9027, 43586}, {9971, 39562}, {9976, 34417}, {11225, 44077}, {11459, 32599}, {11645, 18396}, {12007, 19154}, {14389, 22112}, {14852, 18553}, {15534, 22115}, {16187, 37638}, {19137, 43150}, {20301, 23325}, {21852, 39125}, {25328, 34514}, {33851, 37814}, {41618, 44274}

X(44490) = midpoint of X(576) and X(11438)
X(44490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 568, 576), (182, 576, 13352), (567, 37489, 37480), (575, 44470, 44491), (44470, 44489, 575)

X(44491) = CENTER OF LOZADA-LEMOINE-CIRCLE-2A OF REFLECTION TRIANGLE

Barycentrics    a^2*(a^10-2*(b^2+c^2)*a^8-2*b^2*c^2*a^6+2*(b^2+c^2)*(b^4+c^4)*a^4-(b^4+c^4+2*(b^2+c^2)*b*c)*(b^4+c^4-2*(b^2+c^2)*b*c)*a^2-2*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :
X(44491) = 3*X(6)+X(37486) = 3*X(182)-X(578) = X(578)+3*X(19126) = 3*X(5050)+X(37485) = 7*X(10541)-3*X(37506)

The squared-radius of this circle is: ρ2 = ((3*R^2-SW)^2*S^2+(R^2-SW)^2*SW^2)*R^2/(4*(2*R^2-SW)^2*SW^2)

X(44491) lies on these lines: {2, 43811}, {3, 6}, {5, 19127}, {49, 599}, {66, 32344}, {70, 1176}, {141, 7568}, {184, 7495}, {206, 24206}, {524, 32046}, {631, 11061}, {858, 43650}, {1147, 40107}, {1594, 1974}, {1656, 18374}, {2888, 6776}, {2918, 9813}, {2937, 9971}, {3519, 19151}, {3589, 13371}, {3818, 13160}, {4550, 9968}, {5012, 43653}, {5622, 19467}, {5943, 26283}, {6152, 8541}, {6240, 19124}, {6639, 38851}, {6759, 18553}, {7393, 19153}, {7514, 34117}, {7525, 9019}, {8550, 32358}, {10168, 18281}, {10539, 11178}, {10984, 34224}, {11179, 43812}, {12106, 32154}, {12173, 29323}, {13434, 20423}, {14561, 19121}, {14791, 19136}, {15133, 20301}, {16776, 37440}, {18569, 19130}, {19128, 37119}, {19150, 41590}, {19468, 32251}, {38064, 43815}

X(44491) = midpoint of X(i) and X(j) for these {i, j}: {3, 44480}, {182, 19126}
X(44491) = intersection, other than A,B,C, of conics {{A, B, C, X(39), X(70)}} and {{A, B, C, X(67), X(13351)}}
X(44491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 37473, 3098), (6, 6243, 576), (182, 576, 569), (182, 37515, 20190), (182, 44470, 575), (575, 44470, 44490), (5085, 19129, 182), (5157, 19131, 182), (13353, 37486, 578), (17704, 20190, 5092)

X(44492) = CENTER OF LOZADA-LEMOINE-CIRCLE-2B OF MIDHEIGHT TRIANGLE

Barycentrics    (a^10-3*(b^2+c^2)*a^8+2*(b^2+c^2)^2*a^6+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^4-(3*b^8+3*c^8-2*(6*b^4-b^2*c^2+6*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3)*a^2 : :
X(44492) = 3*X(6)-X(37498) = 3*X(6)-2*X(44469) = 2*X(1147)-3*X(19153) = 3*X(19149)-2*X(32139)

The squared-radius of this circle is: ρ2 = (4*R^4*SW^2+(6*R^2-SW)^2*S^2)*R^2*S^4/(SA^2*SB^2*SC^2*SW^2)

X(44492) lies on these lines: {3, 6}, {4, 41614}, {20, 37784}, {22, 15073}, {26, 1177}, {30, 8548}, {69, 3542}, {154, 41615}, {155, 524}, {156, 9925}, {159, 32048}, {235, 1352}, {394, 468}, {542, 1498}, {597, 15805}, {895, 12082}, {1092, 44102}, {1147, 19153}, {1503, 12293}, {1598, 43130}, {1657, 39562}, {1885, 31670}, {1992, 7592}, {1993, 7493}, {2063, 38282}, {2393, 7387}, {2781, 12163}, {2854, 15581}, {3147, 20806}, {3564, 19149}, {5486, 12161}, {5562, 11470}, {5622, 11413}, {5965, 17824}, {6193, 41719}, {6391, 39879}, {6642, 19136}, {6759, 8681}, {6776, 37201}, {6816, 14853}, {7529, 29959}, {7716, 41714}, {8263, 21841}, {9813, 10110}, {9972, 11576}, {9976, 33534}, {10154, 34966}, {10249, 12084}, {10519, 26206}, {10594, 11188}, {10601, 30739}, {10602, 11414}, {10982, 20423}, {10984, 40673}, {11255, 13391}, {11456, 41617}, {11649, 37928}, {11799, 15069}, {13383, 19139}, {14852, 34118}, {15462, 35602}, {16266, 34351}, {17811, 40107}, {17814, 34507}, {17825, 25555}, {18569, 23049}, {19154, 23041}, {19924, 34622}, {21850, 31815}, {22151, 35486}, {23327, 23335}, {31725, 36990}, {32138, 34778}, {32284, 32621}

X(44492) = midpoint of X(i) and X(j) for these {i, j}: {1351, 37491}, {6391, 39879}, {11477, 17834}
X(44492) = reflection of X(i) in X(j) for these (i, j): (3, 44470), (155, 34117), (8549, 8548), (9925, 156), (13346, 575), (34787, 26), (37498, 44469)
X(44492) = Brocard circle-inverse of-X(44503)
X(44492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44503), (6, 11477, 36747), (6, 37498, 44469), (6, 37514, 575), (182, 576, 44495), (575, 17704, 182), (576, 44480, 6), (576, 44493, 44480), (1351, 6243, 11477), (11482, 36753, 6)

X(44493) = CENTER OF LOZADA-LEMOINE-CIRCLE-2B OF ORTHOCENTROIDAL TRIANGLE

Barycentrics    (a^10-3*(b^2+c^2)*a^8+2*(b^4+b^2*c^2+c^4)*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^4-(3*b^8+3*c^8-2*(4*b^4+b^2*c^2+4*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3)*a^2 : :
X(44493) = 3*X(182)-2*X(11430) = 2*X(5092)-3*X(19131) = X(11456)+3*X(41614) = 4*X(20190)-3*X(39242)

The squared-radius of this circle is: ρ2 = (9*R^4*SW^2+(9*R^2-2*SW)^2*S^2)*R^2/(4*(3*R^2-SW)^2*SW^2)

X(44493) lies on these lines: {3, 6}, {22, 11649}, {69, 7552}, {399, 15069}, {524, 44262}, {542, 11456}, {1352, 15052}, {1658, 40929}, {3619, 14940}, {5972, 15066}, {8548, 9976}, {8549, 35237}, {9970, 11459}, {10601, 32216}, {11179, 37784}, {11216, 35243}, {11579, 15072}, {11663, 12088}, {12061, 17714}, {12106, 32217}, {13198, 15073}, {13406, 18358}, {14984, 19127}, {15032, 41617}, {15068, 16534}, {15087, 15534}, {15106, 37638}, {18396, 32251}, {29317, 39588}, {39562, 43273}

X(44493) = midpoint of X(11477) and X(37494)
X(44493) = reflection of X(13352) in X(575)
X(44493) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 37483, 44469), (575, 16836, 182), (576, 44480, 44494), (15068, 34117, 19140), (19140, 34507, 15068), (44480, 44492, 576)

X(44494) = CENTER OF LOZADA-LEMOINE-CIRCLE-2B OF REFLECTION TRIANGLE

Barycentrics    (a^10-3*(b^2+c^2)*a^8+2*(b^4-b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(b^4+3*b^2*c^2+c^4)*a^4-(b^4-3*c^4)*(3*b^4-c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3)*a^2 : :
X(44494) = 3*X(6)-X(36749) = 5*X(22234)-2*X(37505)

The squared-radius of this circle is: ρ2 = (R^4*SW^2+(3*R^2-2*SW)^2*S^2)*R^2/(4*(R^2-SW)^2*SW^2)

X(44494) lies on these lines: {3, 6}, {51, 26284}, {143, 19127}, {155, 11178}, {156, 16776}, {195, 599}, {539, 8548}, {542, 7592}, {597, 12359}, {1199, 14789}, {1899, 5169}, {1993, 40107}, {3818, 32139}, {5422, 21243}, {5449, 19139}, {5476, 18381}, {6689, 19150}, {7566, 19130}, {9972, 15531}, {10168, 32348}, {11188, 11423}, {12161, 12585}, {14561, 18912}, {14763, 19360}, {15024, 43811}, {15043, 15462}, {15069, 15087}, {15074, 36153}, {15118, 34155}, {18383, 19149}, {18445, 18553}, {19140, 36253}, {20301, 25335}, {20303, 26926}, {23300, 33332}, {32251, 41731}

X(44494) = reflection of X(569) in X(575)
X(44494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 11477, 14627), (6, 36752, 44469), (6, 36753, 575), (6, 44480, 576), (389, 575, 182), (576, 44480, 44493), (36752, 44469, 182)

X(44495) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF 1st EXCOSINE TRIANGLE

Barycentrics    a^2*((b^2+c^2)*a^8+18*(b^2+c^2)*b^2*c^2*a^4-2*(b^4+5*b^2*c^2+c^4)*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2*a^2+(b^8-c^8)*(-b^2+c^2)) : :
X(44495) = 3*X(6)-X(389) = 5*X(6)-X(19161) = 7*X(6)-X(21851) = 9*X(6)-X(37473) = X(52)-5*X(11482) = 3*X(182)-2*X(17704) = 5*X(389)-3*X(19161) = 7*X(389)-3*X(21851) = 3*X(389)-X(37473) = 3*X(1351)+X(10625) = X(3313)+3*X(5102) = 9*X(5093)-X(6243) = 3*X(5093)+X(9967) = X(6243)+3*X(9967) = X(10625)-3*X(11574) = 2*X(15012)-5*X(22234) = X(16625)-4*X(22330) = 7*X(19161)-5*X(21851) = 9*X(19161)-5*X(37473) = 9*X(21851)-7*X(37473)

The squared-radius of this circle is: ρ2 = ((12*R^2-SW)^2*S^2+(4*R^2-SW)^2*SW^2)/(64*SW^2*R^2)

X(44495) lies on these lines: {3, 6}, {4, 40673}, {5, 8681}, {51, 4232}, {54, 44102}, {143, 33591}, {193, 11444}, {524, 11793}, {542, 12241}, {597, 11695}, {1199, 1205}, {1353, 5876}, {1503, 22829}, {1843, 9781}, {1974, 9707}, {1992, 5562}, {2393, 10110}, {2781, 13382}, {2854, 15465}, {3091, 15531}, {3819, 11433}, {5032, 5889}, {5446, 15074}, {5462, 14984}, {5476, 43130}, {5480, 32366}, {5890, 35483}, {5943, 11427}, {6000, 8550}, {6241, 12294}, {6467, 14853}, {6688, 23292}, {6759, 32621}, {6776, 11381}, {7592, 11470}, {9820, 9822}, {9969, 12061}, {9977, 22830}, {10282, 19136}, {10602, 10982}, {11487, 34507}, {11645, 13403}, {11649, 37897}, {12007, 34146}, {12242, 32246}, {13434, 37784}, {13567, 40107}, {13568, 19924}, {13570, 18388}, {13598, 20423}, {14561, 14913}, {14865, 43812}, {15082, 37643}, {18390, 18553}, {19596, 43829}, {32142, 34380}, {34565, 37977}, {34854, 42873}

X(44495) = midpoint of X(i) and X(j) for these {i, j}: {5, 32284}, {576, 44479}, {1351, 11574}, {5446, 15074}, {5480, 32366}, {11477, 15644}
X(44495) = reflection of X(i) in X(j) for these (i, j): (9729, 575), (9822, 18583)
X(44495) = Brocard circle-inverse of-X(44489)
X(44495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44489), (6, 578, 575), (6, 11477, 11432), (182, 576, 44492), (389, 11425, 17704)

X(44496) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF MCCAY TRIANGLE

Barycentrics    a^2*(4*a^4-7*(b^2+c^2)*a^2+7*b^4-4*b^2*c^2+7*c^4) : :
X(44496) = 3*X(6)-X(187) = X(6)-3*X(1570) = 7*X(6)-3*X(1691) = 5*X(6)-3*X(1692) = 11*X(6)-3*X(2076) = 5*X(6)-X(5104) = X(6)+3*X(5111) = 3*X(6)+X(8586) = 5*X(6)+3*X(15514) = 17*X(6)-9*X(35006) = 5*X(6)-2*X(38010) = X(187)-9*X(1570) = 7*X(187)-9*X(1691) = 5*X(187)-9*X(1692) = 2*X(187)-3*X(2030) = 11*X(187)-9*X(2076) = 5*X(187)-3*X(5104) = X(187)+3*X(5107) = X(187)+9*X(5111) = 5*X(187)+9*X(15514) = 5*X(187)-6*X(38010)

The squared-radius of this circle is: ρ2 = (27*S^2*(3*S^2-SW^2)+4*SW^4)*R^2*S^2/(4*(3*S^2-SW^2)^2*SW^2)

X(44496) lies on these lines: {2, 42011}, {3, 6}, {30, 41672}, {111, 3292}, {316, 1992}, {352, 39024}, {353, 44109}, {394, 21448}, {524, 625}, {597, 7619}, {599, 31275}, {842, 8779}, {895, 10630}, {1383, 3060}, {1495, 13192}, {2386, 39848}, {3054, 40107}, {3055, 25555}, {3630, 5031}, {3849, 8584}, {5032, 14712}, {5095, 5523}, {5140, 8541}, {5148, 8540}, {5194, 16785}, {5476, 31415}, {5477, 11645}, {5650, 7708}, {5969, 15301}, {6791, 40112}, {6792, 13857}, {7813, 41146}, {8681, 9132}, {8787, 32479}, {9486, 36212}, {10765, 13509}, {10766, 21639}, {11580, 23061}, {13449, 43448}, {15066, 22111}, {15534, 31173}, {18424, 18553}, {20976, 32237}, {20977, 39689}, {34507, 43620}, {34565, 39389}

X(44496) = midpoint of X(i) and X(j) for these {i, j}: {6, 5107}, {187, 8586}, {1570, 5111}, {1692, 15514}, {11477, 18860}, {15534, 31173}, {44497, 44498}
X(44496) = reflection of X(i) in X(j) for these (i, j): (2030, 6), (5104, 38010)
X(44496) = isogonal conjugate of X(10153)
X(44496) = barycentric product X(6)*X(41133)
X(44496) = trilinear product X(31)*X(41133)
X(44496) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(41133)}} and {{A, B, C, X(249), X(15655)}}
X(44496) = circumcircle-inverse of-X(15655)
X(44496) = 2nd Lemoine circle (cosine circle)-inverse of-X(5085)
X(44496) = Moses circle-inverse of-X(8589)
X(44496) = crossdifference of every pair of points on line {X(523), X(15534)}
X(44496) = crosssum of X(2) and X(8859)
X(44496) = X(923)-complementary conjugate of-X(15850)
X(44496) = X(6)-Daleth conjugate of-X(8589)
X(44496) = X(6)-Hirst inverse of-X(15655)
X(44496) = X(512)-vertex conjugate of-X(15655)
X(44496) = radical trace of Brocard circle and Ehrmann circle
X(44496) = radical trace of Brocard circle and circle described in Hyacinthos #20367 (11/15/2011, Quang Tuan Bui)
X(44496) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 574, 575), (6, 5104, 1692), (6, 5111, 5107), (6, 8586, 187), (6, 11477, 1384), (6, 13330, 5008), (6, 15514, 5104), (6, 37517, 41413), (187, 5107, 8586), (576, 44499, 44500), (1379, 1380, 15655), (1570, 5107, 6), (1666, 1667, 5085), (1692, 5104, 38010), (1692, 38010, 2030), (2028, 2029, 8589), (5038, 10631, 8590), (8589, 10485, 20190), (9974, 9975, 11477), (15520, 42852, 6), (21309, 35002, 187), (22242, 22243, 5024), (44501, 44502, 575)

X(44497) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF INNER-NAPOLEON TRIANGLE

Barycentrics    a^2*(2*sqrt(3)*(a^2+b^2+c^2)*S+3*a^4-9*(b^2+c^2)*a^2+6*b^4-6*b^2*c^2+6*c^4) : :
X(44497) = 3*X(6)-X(16) = 5*X(6)-3*X(36758) = 5*X(16)-9*X(36758) = 3*X(597)-2*X(6672) = 3*X(599)-5*X(40335) = X(622)+3*X(1992) = X(5104)-3*X(36757) = X(5615)-5*X(11482) = 2*X(21402)-5*X(22234)

The squared-radius of this circle is: ρ2 = (27*S^2-12*sqrt(3)*S*SW+7*SW^2)*R^2*S^2/(4*(SW-sqrt(3)*S)^2*SW^2)

X(44497) lies on these lines: {3, 6}, {30, 41621}, {51, 37776}, {193, 42983}, {524, 624}, {530, 8584}, {542, 5321}, {597, 6672}, {599, 40335}, {622, 1992}, {625, 34508}, {1352, 42139}, {1353, 42923}, {1503, 42136}, {3180, 16940}, {3292, 37775}, {3589, 43103}, {3618, 43463}, {3818, 42103}, {5334, 20429}, {5335, 20423}, {5353, 8540}, {5357, 19369}, {5476, 18582}, {5480, 42138}, {5965, 31706}, {6108, 43228}, {6776, 43466}, {7685, 11542}, {8550, 42117}, {8681, 10662}, {10632, 44102}, {10678, 32302}, {11178, 42095}, {11179, 42119}, {11645, 19107}, {11646, 22856}, {11649, 36980}, {14848, 42817}, {15069, 42125}, {15073, 21648}, {15534, 42975}, {16809, 18553}, {18581, 34507}, {18583, 42627}, {19130, 42110}, {19924, 42088}, {22496, 31173}, {23302, 25555}, {23303, 40107}, {25561, 42918}, {29012, 42108}, {29181, 42584}, {31670, 42141}, {33749, 42147}, {36994, 42126}, {41037, 42128}, {42130, 43273}

X(44497) = midpoint of X(i) and X(j) for these {i, j}: {15, 8586}, {576, 44487}, {11477, 14539}
X(44497) = reflection of X(i) in X(j) for these (i, j): (13349, 575), (44498, 44496)
X(44497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 15, 575), (6, 576, 44498), (6, 11477, 11486)

X(44498) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF OUTER-NAPOLEON TRIANGLE

Barycentrics    a^2*(-2*sqrt(3)*(a^2+b^2+c^2)*S+3*a^4-9*(b^2+c^2)*a^2+6*b^4-6*b^2*c^2+6*c^4) : :
X(44498) = 3*X(6)-X(15) = 5*X(6)-3*X(36757) = 5*X(15)-9*X(36757) = 3*X(597)-2*X(6671) = 3*X(599)-5*X(40334) = X(621)+3*X(1992) = X(5104)-3*X(36758) = X(5611)-5*X(11482) = 2*X(21401)-5*X(22234)

The squared-radius of this circle is: ρ2 = (27*S^2+12*sqrt(3)*SW*S+7*SW^2)*R^2*S^2/(4*(SW+sqrt(3)*S)^2*SW^2)

X(44498) lies on these lines: {3, 6}, {30, 41620}, {51, 37775}, {193, 42982}, {524, 623}, {531, 8584}, {542, 5318}, {597, 6671}, {599, 40334}, {621, 1992}, {625, 34509}, {1352, 42142}, {1353, 42922}, {1503, 42137}, {3181, 16941}, {3292, 37776}, {3589, 43102}, {3618, 43464}, {3818, 42106}, {5334, 20423}, {5335, 20428}, {5353, 19369}, {5357, 8540}, {5476, 18581}, {5480, 42135}, {5965, 31705}, {6109, 43229}, {6776, 43465}, {7684, 11543}, {8550, 42118}, {8681, 10661}, {10633, 44102}, {10677, 32301}, {11178, 42098}, {11179, 42120}, {11645, 19106}, {11646, 22900}, {11649, 36978}, {14848, 42818}, {15069, 42128}, {15073, 21647}, {15534, 42974}, {16808, 18553}, {18582, 34507}, {18583, 42628}, {19130, 42107}, {19924, 42087}, {22495, 31173}, {23302, 40107}, {23303, 25555}, {25561, 42919}, {29012, 42109}, {29181, 42585}, {31670, 42140}, {33749, 42148}, {36992, 42127}, {41036, 42125}, {42131, 43273}

X(44498) = midpoint of X(i) and X(j) for these {i, j}: {16, 8586}, {576, 44488}, {11477, 14538}
X(44498) = reflection of X(i) in X(j) for these (i, j): (13350, 575), (44497, 44496)
X(44498) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 16, 575), (6, 576, 44497), (6, 11477, 11485)

X(44499) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF 1st NEUBERG TRIANGLE

Barycentrics    a^2*(2*a^4-(b^2+c^2)*a^2+3*b^4+3*c^4) : :
X(44499) = 3*X(6)-X(32) = 5*X(6)-X(5017) = 4*X(6)-X(41413) = 5*X(32)-3*X(5017) = X(32)+3*X(5028) = 4*X(32)-3*X(41413) = X(315)+3*X(1992) = 3*X(597)-2*X(6680) = 3*X(599)-5*X(7867) = X(5017)+5*X(5028) = 4*X(5017)-5*X(41413) = 4*X(5028)+X(41413) = 3*X(5050)-X(35387) = 3*X(5093)-X(35389) = 3*X(15520)-X(35431) = X(35424)-3*X(39561)

The squared-radius of this circle is: ρ2 = R^2*(9*S^4-11*S^2*SW^2+4*SW^4)*S^2/(4*((SW^2+S^2)^2-4*S^2*SW^2)*SW^2)

X(44499) lies on these lines: {3, 6}, {51, 34945}, {69, 7828}, {115, 18553}, {141, 7886}, {185, 10766}, {193, 7797}, {230, 40107}, {251, 21969}, {315, 1992}, {323, 40130}, {373, 39024}, {384, 10754}, {394, 40126}, {524, 626}, {542, 5254}, {597, 6680}, {599, 7867}, {754, 8584}, {760, 4663}, {1180, 13366}, {1194, 34986}, {1196, 9225}, {2548, 5476}, {2794, 8550}, {3292, 9465}, {3329, 36849}, {3618, 7769}, {3767, 34507}, {3815, 6721}, {3819, 42295}, {3852, 39125}, {3981, 40350}, {5032, 20065}, {5182, 7783}, {5280, 19369}, {5299, 8540}, {5354, 23061}, {5477, 7765}, {5969, 7816}, {6034, 25561}, {6467, 34137}, {7738, 11179}, {7748, 11645}, {7755, 15993}, {7760, 39099}, {7779, 10336}, {7787, 22486}, {7792, 32458}, {7818, 15534}, {7839, 10350}, {8574, 8675}, {8681, 23128}, {8743, 11470}, {8779, 15073}, {9607, 33749}, {9753, 37665}, {11178, 13881}, {12294, 41363}, {13754, 39524}, {14561, 31404}, {14567, 20859}, {14700, 32127}, {14994, 17129}, {15819, 39095}, {16984, 36859}, {32237, 44116}, {39575, 44102}

X(44499) = midpoint of X(i) and X(j) for these {i, j}: {6, 5028}, {1351, 13355}, {2458, 15514}, {5162, 8586}, {7818, 15534}, {11477, 30270}, {13330, 32452}, {44501, 44502}
X(44499) = reflection of X(i) in X(j) for these (i, j): (13335, 575), (39750, 15516)
X(44499) = crosssum of X(2) and X(7806)
X(44499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 39, 575), (6, 576, 44500), (6, 1351, 5039), (6, 1504, 44482), (6, 1505, 44481), (6, 1570, 5097), (6, 3094, 1692), (6, 5034, 15516), (6, 5111, 5052), (6, 8586, 5008), (6, 10542, 3), (6, 11477, 30435), (6, 13330, 5007), (6, 15514, 12212), (1351, 40268, 11477), (1692, 3094, 5092), (1692, 37512, 39560), (3094, 39560, 37512), (5007, 5107, 13330), (7772, 43183, 13357), (9974, 9975, 5102), (37512, 39560, 5092), (44473, 44474, 39561), (44485, 44486, 35424), (44496, 44500, 576)

X(44500) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF 2nd NEUBERG TRIANGLE

Barycentrics    a^2*(5*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4) : :
X(44500) = 3*X(6)-X(39) = 5*X(6)-X(3094) = 3*X(6)+X(13330) = 7*X(6)-3*X(13331) = 5*X(39)-3*X(3094) = X(39)+3*X(5052) = 7*X(39)-9*X(13331) = X(76)+3*X(1992) = 3*X(193)+5*X(31276) = 3*X(1351)+X(9821) = X(3094)+5*X(5052) = 3*X(3094)+5*X(13330) = 7*X(3094)-15*X(13331) = X(3095)-5*X(11482) = 3*X(5052)-X(13330) = 7*X(5052)+3*X(13331) = 3*X(5093)-X(35439) = X(9821)-3*X(13354) = 7*X(13330)+9*X(13331) = 3*X(14994)-5*X(31276)

The squared-radius of this circle is: ρ2 = (9*S^2+4*SW^2)*R^2*S^2/(4*(S^2+SW^2)*SW^2)

X(44500) lies on these lines: {3, 6}, {51, 9465}, {69, 7858}, {76, 1992}, {83, 39099}, {193, 14994}, {194, 5032}, {230, 25555}, {251, 13366}, {262, 5304}, {263, 14252}, {373, 9463}, {524, 3934}, {538, 8584}, {542, 7745}, {597, 6683}, {599, 7903}, {732, 32455}, {1180, 21969}, {1194, 20977}, {1506, 15993}, {1613, 6688}, {2023, 35021}, {2393, 27375}, {2548, 34507}, {2782, 41672}, {3051, 3291}, {3068, 22723}, {3069, 22722}, {3202, 19136}, {3629, 24256}, {3767, 5476}, {3815, 40107}, {3819, 20965}, {3917, 15302}, {4663, 14839}, {5280, 8540}, {5286, 20423}, {5299, 19369}, {5305, 11623}, {5354, 15019}, {5359, 15004}, {5475, 18553}, {5640, 40130}, {5643, 11580}, {5969, 20583}, {6194, 14930}, {6329, 10007}, {7736, 15819}, {7747, 11645}, {7839, 10754}, {8541, 8743}, {8550, 18907}, {9466, 15534}, {10219, 21001}, {10312, 44102}, {11646, 39590}, {11649, 16308}, {14928, 19687}, {15069, 15484}, {18906, 41622}, {19063, 26456}, {19064, 26463}, {20081, 32451}, {21760, 39543}, {22712, 37665}, {39024, 44107}, {40332, 40341}

X(44500) = midpoint of X(i) and X(j) for these {i, j}: {6, 5052}, {39, 13330}, {193, 14994}, {1351, 13354}, {3629, 24256}, {5028, 35432}, {5188, 11477}, {9466, 15534}, {18906, 41622}
X(44500) = reflection of X(i) in X(j) for these (i, j): (10007, 6329), (13334, 575)
X(44500) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(8722)}} and {{A, B, C, X(32), X(18842)}}
X(44500) = 2nd Brocard circle-inverse of-X(5024)
X(44500) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 32, 575), (6, 576, 44499), (6, 1692, 15516), (6, 5017, 5034), (6, 5058, 44482), (6, 5062, 44481), (6, 11173, 42852), (6, 11477, 9605), (6, 12212, 1692), (6, 13330, 39), (6, 40825, 39561), (32, 575, 2030), (39, 5052, 13330), (187, 5038, 20190), (371, 372, 8722), (576, 44499, 44496), (1670, 1671, 5024), (1689, 1690, 11477), (3051, 13410, 3291), (3291, 13410, 5943), (5017, 5034, 5092), (11482, 43136, 6), (35007, 39560, 38010), (44501, 44502, 5097)

X(44501) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF INNER-VECTEN TRIANGLE

Barycentrics    a^2*(2*(a^2+b^2+c^2)*(2*a^4-5*(b^2+c^2)*a^2+3*b^4-4*b^2*c^2+3*c^4)*S+4*S^2*(5*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)) : :
X(44501) = 3*X(6)-X(372) = 3*X(6)-2*X(44481) = X(372)+3*X(35841) = X(638)+3*X(1992) = 9*X(5032)-X(43133) = 3*X(35841)+2*X(44481)

The squared-radius of this circle is: ρ2 = (9*S^2-12*S*SW+5*SW^2)*R^2*S^2/(4*(S^2-2*S*SW+SW^2)*SW^2)

X(44501) lies on these lines: {3, 6}, {69, 13939}, {141, 6119}, {485, 5476}, {486, 34507}, {524, 640}, {542, 3071}, {590, 25555}, {597, 8981}, {599, 13951}, {615, 40107}, {638, 1992}, {641, 14645}, {1132, 11180}, {1352, 42561}, {1587, 20423}, {2393, 30427}, {3299, 19369}, {3301, 8540}, {3317, 21356}, {3788, 35684}, {3818, 42268}, {5032, 43133}, {6459, 11179}, {6565, 18553}, {8550, 42215}, {8584, 32421}, {8681, 10666}, {8996, 32621}, {10880, 44102}, {11178, 42262}, {11314, 42060}, {11645, 35821}, {12971, 32292}, {13032, 13038}, {13785, 15069}, {13925, 18583}, {15073, 21641}, {19130, 42273}, {19924, 42259}, {24206, 42583}, {29012, 42271}, {31670, 39875}

X(44501) = midpoint of X(i) and X(j) for these {i, j}: {6, 35841}, {576, 44485}, {11477, 11824}
X(44501) = reflection of X(i) in X(j) for these (i, j): (3, 44483), (32, 44482), (372, 44481), (43121, 575), (44502, 44499)
X(44501) = Brocard-circle-inverse of X(44657)
X(44501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44657), (6, 371, 575), (6, 372, 44481), (6, 576, 44502), (6, 3102, 44476), (6, 6419, 44482), (6, 9974, 44474), (6, 9975, 576), (6, 11477, 3312), (6, 13330, 5062), (6, 19145, 39561), (371, 1505, 43121), (575, 43144, 182), (575, 44496, 44502), (576, 44472, 1351), (576, 44473, 6), (576, 44474, 9974), (3102, 5058, 43120), (3364, 3389, 5058), (5097, 44500, 44502), (6417, 11482, 6), (9974, 44474, 44502), (9975, 44473, 44502), (15520, 42833, 6), (22330, 44482, 6)

X(44502) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF OUTER-VECTEN TRIANGLE

Barycentrics    a^2*(a^4+2*b^4-2*b^2*c^2+2*c^4-3*(b^2+c^2)*a^2-(2*a^2+2*c^2+2*b^2)*S) : :
X(44502) = 3*X(6)-X(371) = 3*X(6)-2*X(44482) = X(371)+3*X(35840) = X(637)+3*X(1992) = 9*X(5032)-X(43134) = 3*X(35840)+2*X(44482)

The squared-radius of this circle is ρ2 = (9*S^2+12*SW*S+5*SW^2)*R^2*S^2/(4*(S^2+2*SW*S+SW^2)*SW^2)

X(44502) lies on these lines: {3, 6}, {69, 13886}, {141, 6118}, {485, 34507}, {486, 5476}, {524, 639}, {542, 3070}, {590, 40107}, {597, 13966}, {599, 8976}, {615, 25555}, {637, 1992}, {642, 14645}, {1131, 11180}, {1352, 31412}, {1588, 20423}, {2393, 30428}, {3299, 8540}, {3301, 19369}, {3316, 21356}, {3788, 35685}, {3818, 42269}, {5032, 43134}, {6460, 11179}, {6564, 18553}, {8550, 42216}, {8584, 32419}, {8681, 10665}, {10881, 44102}, {11178, 42265}, {11313, 42009}, {11645, 35820}, {12965, 32291}, {13030, 13037}, {13665, 15069}, {13993, 18583}, {15073, 21640}, {19130, 42270}, {19924, 42258}, {24206, 42582}, {29012, 42272}, {31670, 39876}

X(44502) = midpoint of X(i) and X(j) for these {i, j}: {6, 35840}, {576, 44486}, {11477, 11825}
X(44502) = reflection of X(i) in X(j) for these (i, j): (3, 44484), (32, 44481), (371, 44482), (43120, 575), (44501, 44499)
X(44502) = Brocard-circle-inverse of X(44656)
X(44502) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44656), (6, 371, 44482), (6, 372, 575), (6, 576, 44501), (6, 3103, 44475), (6, 6420, 44481), (6, 9974, 576), (6, 9975, 44473), (6, 11477, 3311), (6, 13330, 5058), (6, 19146, 39561), (372, 1504, 43120), (575, 43141, 182), (575, 44496, 44501), (576, 44471, 1351), (576, 44473, 9975), (576, 44474, 6), (3103, 5062, 43121), (3365, 3390, 5062), (5097, 44500, 44501), (6418, 11482, 6), (9974, 44474, 44501), (9975, 44473, 44501), (15520, 42832, 6)

X(44503) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF 1st EXCOSINE TRIANGLE

Barycentrics    a^2*(a^10-3*(b^2+c^2)*a^8+2*(b^4+6*b^2*c^2+c^4)*a^6+2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^4-(3*b^8+3*c^8-2*(2*b^4-9*b^2*c^2+2*c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)^3) : :
X(44503) = 3*X(597)-X(15873)

The squared-radius of this circle is: ρ2 = (4*R^4*SW^2+(6*R^2-SW)^2*S^2)*R^2/((8*R^2-SW)^2*SW^2)

X(44503) lies on these lines: {3, 6}, {5, 8549}, {24, 43815}, {69, 26879}, {140, 5486}, {155, 8550}, {235, 14561}, {394, 11245}, {468, 10601}, {524, 18951}, {542, 17814}, {597, 15873}, {631, 41614}, {1092, 40673}, {1147, 32621}, {1181, 11179}, {2393, 6642}, {3523, 37784}, {3526, 5181}, {3542, 3618}, {3564, 18952}, {5020, 43130}, {5422, 7493}, {5622, 7503}, {5892, 11216}, {5944, 23041}, {6644, 15074}, {6723, 9976}, {6776, 6816}, {6803, 18919}, {7387, 19136}, {7393, 19458}, {7526, 10249}, {7592, 22151}, {8537, 15045}, {8547, 15582}, {9813, 11695}, {9826, 13248}, {10168, 19361}, {10519, 40318}, {10605, 43814}, {10984, 44102}, {11255, 12006}, {11416, 15043}, {11459, 43812}, {11645, 15811}, {11649, 37933}, {11793, 17836}, {14853, 37201}, {14912, 20806}, {15069, 25738}, {15073, 17928}, {15121, 31255}, {15311, 34117}, {15462, 19357}, {16238, 38110}, {17809, 41615}, {17811, 34507}, {26221, 39141}, {35259, 43811}

X(44503) = reflection of X(44489) in X(575)
X(44503) = Brocard circle-inverse of-X(44492)
X(44503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44492), (6, 37498, 576), (6, 37514, 44480), (182, 576, 9729), (182, 44479, 3), (182, 44480, 37514), (575, 44469, 6), (6644, 15074, 34787)

X(44504) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF MCCAY TRIANGLE

Barycentrics    a^2*(7*a^8-20*(b^2+c^2)*a^6+2*(13*b^4+4*b^2*c^2+13*c^4)*a^4-4*(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^2+3*b^8-2*(10*b^4-13*b^2*c^2+10*c^4)*b^2*c^2+3*c^8) : :

The squared-radius of this circle is: ρ2 = (9*S^4*(9*S^2-SW^2)-SW^4*(5*S^2-SW^2))*R^2/(4*SW^2*(6*S^2-SW^2)^2)

X(44504) lies on these lines: {3, 6}, {183, 22525}, {5921, 32963}, {6036, 11184}, {6776, 33006}, {8352, 11179}, {8753, 21460}, {21501, 37527}, {32998, 40330}

X(44504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (575, 44507, 44508), (44505, 44506, 6), (44509, 44510, 3)

X(44505) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF INNER-NAPOLEON TRIANGLE

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

The squared-radius of this circle is: ρ2 = (SW^4+(27*S^2+4*SW^2)*S^2-2*sqrt(3)*(3*S^2+SW^2)*S*SW)*R^2/(4*(SW-2*sqrt(3)*S)^2*SW^2)

X(44505) lies on these lines: {3, 6}, {6774, 9763}

X(44505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 44504, 44506), (182, 575, 44506), (182, 576, 13349), (182, 44487, 3), (575, 44477, 6)

X(44506) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF OUTER-NAPOLEON TRIANGLE

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

The squared-radius of this circle is: ρ2 = (SW^4+(27*S^2+4*SW^2)*S^2+2*sqrt(3)*(3*S^2+SW^2)*S*SW)*R^2/(4*(SW+2*sqrt(3)*S)^2*SW^2)

X(44506) lies on these lines: {3, 6}, {6771, 9761}

X(44506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 44504, 44505), (182, 575, 44505), (182, 576, 13350), (182, 44488, 3), (575, 44478, 6)

X(44507) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF 1ST NEUBERG TRIANGLE

Barycentrics    a^2*(3*a^8-6*(b^2+c^2)*a^6+8*(b^4+c^4)*a^4-2*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2+b^8-2*(4*b^4-3*b^2*c^2+4*c^4)*b^2*c^2+c^8) : :
X(44507) = 3*X(5050)-X(40825)

The squared-radius of this circle is: ρ2 = (S^4*(9*S^2-5*SW^2)-SW^4*(S^2-SW^2))*R^2/(4*(2*S^2-SW^2)^2*SW^2)

X(44507) lies on these lines: {3, 6}, {325, 9755}, {542, 11318}, {1352, 7887}, {3618, 37446}, {5025, 6776}, {5182, 11257}, {5622, 38523}, {5967, 41238}, {6036, 7778}, {7841, 11179}, {9752, 16989}, {10349, 12177}, {14912, 32823}, {18583, 40279}

X(44507) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (182, 575, 44508), (182, 576, 13335), (2025, 39764, 6), (44483, 44484, 5092), (44504, 44508, 575), (44509, 44510, 6)

X(44508) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF 2nd NEUBERG TRIANGLE

Barycentrics    a^2*(a^8-8*(b^2+c^2)*a^6+2*(5*b^4+4*b^2*c^2+5*c^4)*a^4-4*(b^2+c^2)*(b^4-5*b^2*c^2+c^4)*a^2+b^8-2*(2*b^4-7*b^2*c^2+2*c^4)*b^2*c^2+c^8) : :
X(44508) = 9*X(5050)-X(10983)

The squared-radius of this circle is: ρ2 = (S^4*(9*S^2+7*SW^2)+SW^4*(3*S^2+SW^2))*R^2/(4*(2*S^2+SW^2)^2*SW^2)

X(44508) lies on these lines: {3, 6}, {631, 39099}, {1352, 32992}, {1975, 31958}, {5921, 33261}, {6776, 16924}, {7770, 12177}, {7786, 10753}, {7851, 14561}, {7864, 14853}, {8177, 12007}, {8370, 11179}, {10982, 14133}, {13111, 14848}, {15271, 34507}, {22525, 39093}

X(44508) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5050, 5038), (182, 575, 44507), (182, 576, 13334), (182, 35424, 5085), (575, 44507, 44504), (35431, 39561, 6), (44509, 44510, 5050)

X(44509) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF INNER-VECTEN TRIANGLE

Barycentrics    a^2*(-2*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44509) = 3*X(6)+X(12306) = 3*X(182)-X(12975) = 9*X(5050)-X(12314) = 3*X(5050)-X(19146) = X(12314)-3*X(19146)

The squared-radius of this circle is: ρ2 = (S^2*(9*S^2+2*SW^2)-2*S*SW*(3*S^2+SW^2)+SW^4)*R^2/(4*(SW-2*S)^2*SW^2)

X(44509) lies on these lines: {3, 6}, {524, 13087}, {542, 6290}, {1503, 22596}, {2782, 22594}, {6776, 26469}, {8550, 23312}, {12299, 44102}, {13749, 22820}

X(44509) = midpoint of X(i) and X(j) for these {i, j}: {3, 9975}, {576, 7692}, {8550, 23312}
X(44509) = Brocard-circle-inverse of X(44655)
X(44509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44655), (3, 44504, 44510), (6, 9732, 576), (6, 43118, 44486), (6, 44507, 44510), (182, 575, 44510), (182, 576, 43121), (182, 12974, 20190), (182, 44486, 43118), (575, 44475, 6), (575, 44482, 39561), (575, 44483, 182), (5050, 44508, 44510), (12306, 43118, 12975)

X(44510) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF OUTER-VECTEN TRIANGLE

Barycentrics    a^2*(2*(2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4)*S+(a^2+b^2+c^2)*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)) : :
X(44510) = 3*X(6)+X(12305) = 3*X(182)-X(12974) = 9*X(5050)-X(12313) = 3*X(5050)-X(19145) = X(12313)-3*X(19145)

The squared-radius of this circle is: ρ2 = (S^2*(9*S^2+2*SW^2)+2*S*SW*(3*S^2+SW^2)+SW^4)*R^2/(4*(SW+2*S)^2*SW^2)

X(44510) lies on these lines: {3, 6}, {524, 13088}, {542, 6289}, {1503, 22625}, {2782, 22623}, {6776, 26468}, {8550, 23311}, {12298, 44102}, {13748, 22819}

X(44510) = midpoint of X(i) and X(j) for these {i, j}: {3, 9974}, {576, 7690}, {8550, 23311}
X(44510) = Brocard-circle-inverse of X(44654)
X(44510) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44654), (3, 44504, 44509), (6, 9733, 576), (6, 43119, 44485), (6, 44507, 44509), (182, 575, 44509), (182, 576, 43120), (182, 12975, 20190), (182, 44485, 43119), (182, 44486, 3), (575, 44476, 6), (575, 44481, 39561), (575, 44484, 182), (5050, 44508, 44509), (9974, 19145, 12962), (12305, 43119, 12974)

X(44511) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF INNER-FERMAT TRIANGLE

Barycentrics    a^2*(2*sqrt(3)*(a^2+b^2+c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : :
X(44511) = 3*X(6)-X(62) = 3*X(597)-2*X(6695) = X(634)+3*X(1992)

The squared-radius of this circle is: ρ2 = (9*S^2-12*sqrt(3)*S*SW+13*SW^2)*R^2*S^2/(4*(sqrt(3)*SW-S)^2*SW^2)

X(44511) lies on these lines: {3, 6}, {14, 18553}, {395, 40107}, {396, 14137}, {398, 542}, {524, 636}, {532, 8584}, {597, 6695}, {599, 42989}, {630, 3589}, {634, 1992}, {2393, 30390}, {3818, 42159}, {5254, 16002}, {5476, 40693}, {7745, 16001}, {7797, 22114}, {7817, 34508}, {8550, 41023}, {10168, 16772}, {11178, 42153}, {11303, 22579}, {11645, 16964}, {15069, 42975}, {19130, 42166}, {19924, 42148}, {20423, 42998}, {24206, 42599}, {29012, 42164}, {34507, 40694}, {34573, 42591}, {37825, 42999}, {43228, 44219}

X(44511) = midpoint of X(11477) and X(14541)
X(44511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 61, 575), (6, 576, 44512), (6, 3364, 44482), (6, 3365, 44481), (6, 36757, 15516), (6, 44497, 44498), (575, 44499, 44512), (576, 44512, 44498), (22330, 44500, 44512), (44497, 44512, 576), (44501, 44502, 44497)

X(44512) = CENTER OF LOZADA-LEMOINE-CIRCLE-3A OF OUTER-FERMAT TRIANGLE

Barycentrics    a^2*(-2*sqrt(3)*(a^2+b^2+c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : :
X(44512) = 3*X(6)-X(61) = 3*X(597)-2*X(6694) = X(633)+3*X(1992)

The squared-radius of this circle is: ρ2 = (9*S^2-12*sqrt(3)*S*SW+13*SW^2)*R^2*S^2/(4*(sqrt(3)*SW-S)^2*SW^2)

X(44512) lies on these lines: {3, 6}, {13, 18553}, {395, 14136}, {396, 40107}, {397, 542}, {524, 635}, {533, 8584}, {597, 6694}, {599, 42988}, {629, 3589}, {633, 1992}, {2393, 30391}, {3818, 42162}, {5254, 16001}, {5476, 40694}, {7745, 16002}, {7797, 22113}, {7817, 34509}, {8550, 41022}, {10168, 16773}, {11178, 42156}, {11304, 22580}, {11645, 16965}, {15069, 42974}, {19130, 42163}, {19924, 42147}, {20423, 42999}, {24206, 42598}, {29012, 42165}, {34507, 40693}, {34573, 42590}, {37824, 42998}

X(44512) = midpoint of X(11477) and X(14540)
X(44512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 62, 575), (6, 576, 44511), (6, 3389, 44482), (6, 3390, 44481), (6, 36758, 15516), (6, 44498, 44497), (575, 44499, 44511), (576, 44511, 44497), (22330, 44500, 44511), (44498, 44511, 576), (44501, 44502, 44498)

X(44513) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF INNER-FERMAT TRIANGLE

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

The squared-radius of this circle is: ρ2 = ((9*S^2+4*SW^2)*S^2-2*sqrt(3)*(3*S^2+SW^2)*SW*S+3*SW^4)*R^2/(4*(sqrt(3)*SW-2*S)^2*SW^2)

X(44513) lies on these lines: {3, 6}, {542, 11306}, {5873, 8550}, {6774, 34507}, {11179, 11304}, {25555, 41041}

X(44513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 44507, 44514), (6, 5865, 576), (182, 575, 44514), (182, 44505, 44506), (575, 44514, 44506), (10541, 44504, 44514), (44505, 44514, 575), (44509, 44510, 44505)

X(44514) = CENTER OF LOZADA-LEMOINE-CIRCLE-3B OF OUTER-FERMAT TRIANGLE

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

The squared-radius of this circle is: ρ2 = ((9*S^2+4*SW^2)*S^2+2*sqrt(3)*(3*S^2+SW^2)*SW*S+3*SW^4)*R^2/(4*(sqrt(3)*SW+2*S)^2*SW^2)

X(44514) lies on these lines: {3, 6}, {542, 11305}, {5872, 8550}, {6771, 34507}, {11179, 11303}, {25555, 41040}

X(44514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 44507, 44513), (6, 5864, 576), (182, 575, 44513), (182, 44506, 44505), (575, 44513, 44505), (10541, 44504, 44513), (44506, 44513, 575), (44509, 44510, 44506)

X(44515) = X(3)X(161)∩X(26)X(195)

Barycentrics    (a^14-3*(b^2+c^2)*a^12+(b^2+c^2)^2*a^10+(b^2+c^2)*(5*b^4+2*b^2*c^2+5*c^4)*a^8-(5*b^8+5*c^8+3*(2*b^4+b^2*c^2+2*c^4)*b^2*c^2)*a^6-(b^2+c^2)*(b^8+c^8-(2*b^2-c^2)*(b^2-2*c^2)*b^2*c^2)*a^4+(3*b^8+3*c^8+2*(b^4+b^2*c^2+c^4)*b^2*c^2)*(b^2-c^2)^2*a^2-(b^8-c^8)*(b^2-c^2)^3)*a^2 : :
Barycentrics    (SB+SC)*(2*(3*R^2-SA-SW)*S^2+(11*R^4+2*(2*SA-7*SW)*R^2-2*(SA^2-SB*SC-2*SW^2))*SA) : :

See Antreas Hatzipolakis and César Lozada, euclid 2116.

X(44515) lies on these lines: {3, 161}, {22, 12307}, {23, 20424}, {26, 195}, {54, 143}, {156, 32338}, {399, 41726}, {567, 11808}, {1147, 15137}, {1154, 1614}, {1658, 12254}, {2888, 7502}, {2918, 3519}, {3518, 8254}, {3574, 18378}, {5663, 7691}, {5899, 15800}, {6242, 37932}, {6639, 32346}, {6746, 15037}, {7488, 32423}, {7512, 21230}, {7517, 32333}, {7555, 25714}, {7730, 32046}, {10282, 11597}, {10610, 13363}, {11804, 18282}, {12107, 36966}, {12325, 38435}, {12380, 13368}, {12899, 41596}, {15647, 22815}, {22051, 37936}, {32349, 40276}, {37922, 43573}

X(44515) = reflection of X(i) in X(j) for these (i, j): (54, 5944), (15800, 43831)
X(44515) = {X(10610), X(13365)}-harmonic conjugate of X(43651)

X(44516) = X(2)X(1614)∩X(3)X(1568)

Barycentrics    (-a^2+b^2+c^2)*(2*a^8-4*(b^2+c^2)*a^6+(b^4-4*b^2*c^2+c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4) : :
X(44516) = 3*X(2)+X(1614), 5*X(631)-X(11440), 3*X(7552)+X(34148)

See Antreas Hatzipolakis and César Lozada, euclid 2116.

X(44516) lies on these lines: {2, 1614}, {3, 1568}, {4, 39242}, {5, 5944}, {49, 539}, {110, 1209}, {113, 14118}, {125, 18128}, {140, 5663}, {143, 12242}, {156, 21243}, {184, 5449}, {185, 20191}, {195, 41586}, {265, 10619}, {343, 41597}, {389, 10020}, {468, 5462}, {542, 34826}, {549, 43604}, {569, 7505}, {578, 10201}, {631, 11440}, {858, 17712}, {1147, 3549}, {1154, 15806}, {1216, 6676}, {1495, 5576}, {1594, 44407}, {1656, 35259}, {1658, 18388}, {2070, 3574}, {2888, 9705}, {2918, 37972}, {3357, 18580}, {3515, 7706}, {3523, 4846}, {3589, 43130}, {5012, 14940}, {5446, 13383}, {5447, 11064}, {5892, 16238}, {5943, 44232}, {6146, 36253}, {6640, 10984}, {7392, 15436}, {7399, 43586}, {7542, 13754}, {7550, 38795}, {7552, 34148}, {7568, 11793}, {7569, 35264}, {7577, 11750}, {7728, 18364}, {8254, 10095}, {9707, 18474}, {9729, 44452}, {9730, 10018}, {9927, 19357}, {10024, 13367}, {10125, 13630}, {10182, 37814}, {10254, 21659}, {10575, 37118}, {10615, 30482}, {11003, 26917}, {11225, 32136}, {11430, 12897}, {11536, 44322}, {11585, 13394}, {11591, 32348}, {11702, 34149}, {11800, 40632}, {12006, 44234}, {12010, 43575}, {12038, 15760}, {12233, 34351}, {12900, 37513}, {13289, 32364}, {13403, 13406}, {13419, 39504}, {13431, 37779}, {13434, 37943}, {13474, 44236}, {13491, 25563}, {13909, 19356}, {13970, 19355}, {14157, 18488}, {14643, 34864}, {14677, 43585}, {15074, 16776}, {15644, 25337}, {20299, 40276}, {22352, 37452}, {32046, 43573}, {32068, 36153}, {32142, 34004}, {36752, 37453}, {37760, 38848}

X(44516) = midpoint of X(i) and X(j) for these {i, j}: {3, 43831}, {5, 5944}, {10024, 13367}, {15806, 34577}
X(44516) = reflection of X(6746) in X(5462)
X(44516) = complement of the complement of X(1614)
X(44516) = barycentric product X(343)*X(36842)

X(44517) = X(1)X(6)∩X(3)X(115)

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

X(44517) lies on these lines: {1, 6}, {3, 115}, {21, 230}, {32, 7489}, {187, 13743}, {404, 3054}, {1006, 5254}, {1012, 5023}, {1030, 9840}, {1213, 25906}, {3053, 3560}, {3055, 17536}, {3815, 5047}, {5013, 6883}, {5124, 13732}, {5306, 16858}, {5428, 43291}, {6920, 7745}, {7735, 16865}, {7736, 16859}, {8553, 13730}, {9300, 16861}, {11108, 31489}, {11342, 19720}, {17544, 37665}

X(44517) = crossdifference of every pair of points on line {513, 6132}

X(44518) = X(3)X(115)∩X(4)X(6)

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(44518) = 2 X[3788] - 3 X[11318], X[3926] - 3 X[16041], 3 X[7776] - 2 X[7916], 3 X[7825] - X[7916], 4 X[7886] - 3 X[11288]

X(44518) lies on these lines: {2, 15815}, {3, 115}, {4, 6}, {5, 2549}, {11, 9597}, {12, 9598}, {20, 230}, {30, 3053}, {32, 382}, {39, 381}, {69, 2996}, {76, 338}, {99, 7887}, {112, 35490}, {140, 43620}, {141, 32974}, {148, 1975}, {154, 460}, {172, 12943}, {183, 6655}, {187, 1657}, {194, 7773}, {232, 37197}, {290, 30496}, {297, 26958}, {315, 33229}, {316, 6144}, {325, 14063}, {384, 7851}, {385, 33019}, {485, 18939}, {486, 18940}, {524, 6392}, {538, 7776}, {543, 3788}, {546, 2548}, {548, 21843}, {550, 5210}, {574, 1656}, {577, 8990}, {578, 9604}, {625, 7781}, {940, 23903}, {999, 9651}, {1003, 7828}, {1007, 32980}, {1015, 9669}, {1078, 33234}, {1150, 23942}, {1184, 7391}, {1194, 5064}, {1196, 34609}, {1368, 8770}, {1370, 1611}, {1384, 5073}, {1479, 16781}, {1500, 9654}, {1504, 13665}, {1505, 13785}, {1506, 3851}, {1569, 38743}, {1570, 39899}, {1571, 9956}, {1572, 22793}, {1597, 27371}, {1609, 39568}, {1613, 14957}, {1698, 31443}, {1853, 3981}, {1879, 11479}, {1914, 12953}, {1968, 44438}, {1971, 17845}, {2023, 11257}, {2088, 15538}, {2165, 12362}, {2241, 9668}, {2242, 9655}, {2275, 10896}, {2276, 10895}, {2393, 40325}, {2475, 5275}, {2886, 31490}, {3054, 3523}, {3055, 5056}, {3091, 3815}, {3094, 6248}, {3146, 7735}, {3291, 31152}, {3295, 9664}, {3329, 33018}, {3436, 21956}, {3522, 5585}, {3526, 37512}, {3534, 5206}, {3543, 5306}, {3545, 31400}, {3583, 16502}, {3589, 32971}, {3614, 31497}, {3618, 32979}, {3627, 5305}, {3734, 7861}, {3763, 6656}, {3785, 33238}, {3830, 5309}, {3832, 7736}, {3839, 9300}, {3843, 5475}, {3845, 7739}, {3850, 31406}, {3853, 5319}, {3855, 9606}, {3856, 31417}, {3926, 16041}, {3934, 7872}, {5007, 5076}, {5022, 36654}, {5028, 18440}, {5038, 10358}, {5054, 15515}, {5055, 31455}, {5059, 37689}, {5072, 7603}, {5077, 7830}, {5079, 31652}, {5111, 13449}, {5116, 7770}, {5124, 37415}, {5134, 14974}, {5280, 18513}, {5283, 17532}, {5299, 18514}, {5304, 17578}, {5346, 21309}, {5355, 43136}, {5395, 41895}, {5461, 34504}, {5737, 23897}, {6034, 9880}, {6103, 8778}, {6337, 32972}, {6409, 21737}, {6421, 6565}, {6422, 6564}, {6423, 35820}, {6424, 35821}, {6531, 18848}, {6658, 7806}, {6772, 11305}, {6775, 11306}, {6781, 17800}, {6823, 9722}, {6871, 37661}, {7388, 8252}, {7389, 8253}, {7395, 18353}, {7396, 40326}, {7406, 37646}, {7503, 9609}, {7509, 15109}, {7571, 38862}, {7610, 7833}, {7615, 8359}, {7620, 21358}, {7750, 8667}, {7751, 7842}, {7752, 31859}, {7753, 14269}, {7759, 22253}, {7762, 8352}, {7763, 8716}, {7772, 15484}, {7774, 32996}, {7775, 32450}, {7777, 32993}, {7782, 14061}, {7783, 32966}, {7785, 14062}, {7786, 15031}, {7787, 14042}, {7788, 7885}, {7789, 14064}, {7791, 15271}, {7792, 14035}, {7793, 33256}, {7795, 33184}, {7797, 11361}, {7798, 7843}, {7800, 8357}, {7802, 14568}, {7803, 8370}, {7804, 7902}, {7810, 40727}, {7816, 7844}, {7817, 11159}, {7823, 14614}, {7827, 11317}, {7832, 33219}, {7834, 11286}, {7835, 33218}, {7839, 14044}, {7840, 20105}, {7847, 11285}, {7852, 33237}, {7853, 17130}, {7857, 33235}, {7864, 11174}, {7867, 33241}, {7868, 7933}, {7873, 17131}, {7874, 33240}, {7881, 7934}, {7886, 11288}, {7891, 20094}, {7893, 19570}, {7898, 17129}, {7903, 31173}, {7912, 14045}, {7919, 33217}, {7920, 14066}, {7924, 31276}, {7935, 9466}, {7942, 33220}, {7945, 14046}, {7947, 33289}, {7951, 31448}, {7989, 9574}, {8356, 32832}, {8553, 11414}, {8556, 32828}, {8589, 15720}, {8719, 39663}, {8754, 10602}, {9289, 35142}, {9465, 31133}, {9540, 9601}, {9593, 18492}, {9600, 10576}, {9603, 10539}, {9608, 10594}, {9619, 9955}, {9620, 18480}, {9699, 18378}, {9745, 31857}, {9924, 41762}, {10097, 23105}, {10311, 12173}, {10312, 35480}, {10329, 41231}, {10583, 14034}, {10722, 12829}, {11063, 12082}, {11184, 33006}, {11231, 31422}, {11235, 17448}, {11236, 20691}, {11289, 43029}, {11290, 43028}, {11303, 16644}, {11304, 16645}, {11331, 41254}, {11585, 15075}, {11646, 15069}, {12188, 43183}, {12203, 39560}, {12293, 23128}, {12601, 35841}, {12602, 35840}, {12902, 14901}, {12963, 42263}, {12968, 42264}, {13108, 32452}, {13161, 16777}, {13468, 33272}, {13567, 37174}, {13711, 42260}, {13834, 42261}, {13850, 41491}, {13932, 41490}, {14001, 32826}, {14068, 16989}, {14537, 38335}, {14880, 22515}, {14907, 19695}, {15491, 32987}, {15513, 15696}, {15668, 23905}, {16055, 31125}, {16589, 17528}, {16986, 19690}, {16992, 33824}, {17004, 33260}, {17005, 33011}, {17006, 33022}, {17008, 32997}, {18403, 22120}, {18404, 23115}, {18841, 32532}, {19758, 37049}, {19780, 42097}, {19781, 42096}, {20065, 33279}, {20181, 26558}, {20998, 41238}, {21001, 37190}, {21043, 23899}, {21965, 24248}, {22110, 32831}, {22329, 33192}, {25639, 31449}, {29012, 40825}, {29323, 41412}, {30771, 34481}, {31430, 31441}, {31451, 31479}, {31456, 31493}, {31461, 31476}, {31463, 42273}, {31468, 31488}, {31644, 40879}, {31709, 42154}, {31710, 42155}, {32151, 43449}, {32459, 32970}, {32817, 33292}, {32820, 33290}, {32822, 33285}, {32829, 32984}, {32834, 33210}, {32838, 33215}, {32872, 42850}, {32963, 37647}, {32965, 37688}, {32989, 44381}, {33013, 42849}, {33023, 34229}, {33278, 37671}, {33802, 37902}, {34511, 37350}, {35287, 44401}, {35488, 39575}, {35955, 43459}, {36251, 42156}, {36252, 42153}, {37638, 39691}, {41406, 42431}, {41407, 42432}

X(44518) = midpoint of X(i) and X(j) for these {i,j}: {6392, 32006}, {42645, 42646}
X(44518) = reflection of X(i) in X(j) for these {i,j}: {3053, 3767}, {7776, 7825}
X(44518) = isotomic conjugate of the isogonal conjugate of X(34481)
X(44518) = polar conjugate of the isotomic conjugate of X(30771)
X(44518) = crosspoint of X(4) and X(2996)
X(44518) = crosssum of X(i) and X(j) for these (i,j): {3, 3053}, {6, 8780}, {577, 41619}
X(44518) = crossdifference of every pair of points on line {520, 6132}
X(44518) = areal center of cevian triangles of PU(4)
X(44518) = barycentric product X(i)*X(j) for these {i,j}: {4, 30771}, {76, 34481}
X(44518) = barycentric quotient X(i)/X(j) for these {i,j}: {30771, 69}, {34481, 6}
X(44518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 115, 13881}, {3, 13881, 37637}, {4, 5254, 6}, {4, 5286, 7745}, {4, 5523, 2207}, {4, 43448, 5254}, {5, 2549, 5013}, {5, 5013, 31489}, {12, 9598, 31477}, {20, 230, 5023}, {76, 7784, 599}, {76, 7841, 7784}, {76, 7911, 7879}, {115, 7748, 3}, {115, 7756, 7746}, {148, 5025, 1975}, {194, 7773, 9766}, {194, 14041, 7773}, {546, 15048, 2548}, {574, 39565, 1656}, {671, 7841, 34505}, {1506, 3851, 18584}, {1506, 18424, 3851}, {1975, 5025, 7778}, {2996, 32982, 69}, {3070, 3071, 6776}, {3091, 7738, 3815}, {3627, 5305, 7737}, {3734, 7861, 7866}, {3815, 7738, 22332}, {3830, 30435, 7747}, {3843, 9605, 5475}, {3850, 31406, 31415}, {3851, 5024, 1506}, {3934, 7872, 11287}, {5013, 31489, 31492}, {5024, 18424, 18584}, {5072, 31467, 7603}, {5210, 43619, 11742}, {5254, 7745, 5286}, {5286, 7745, 6}, {5309, 7747, 30435}, {5475, 7765, 9605}, {6337, 32972, 44377}, {7746, 7748, 7756}, {7746, 7756, 3}, {7772, 39590, 15484}, {7782, 14061, 33233}, {7784, 34505, 76}, {7816, 7844, 32954}, {7841, 7879, 7911}, {7841, 34505, 599}, {7842, 32457, 7751}, {7864, 16044, 11174}, {7872, 18546, 3934}, {7879, 7911, 7784}, {7885, 20081, 7788}, {7933, 17128, 7868}, {11648, 39563, 381}, {14064, 32815, 7789}, {23251, 23261, 36990}, {35830, 35831, 3}, {43291, 43619, 5210}

X(44519) = X(3)X(115)∩X(6)X(20)

Barycentrics    5*a^4 - 5*a^2*b^2 - 2*b^4 - 5*a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(44519) = 2 X[2548] - 3 X[5013], 5 X[2548] - 6 X[31406], 5 X[5013] - 4 X[31406], X[32826] - 3 X[33215]

X(44519) lies on these lines: {3, 115}, {4, 8719}, {5, 43619}, {6, 20}, {30, 2548}, {32, 3534}, {39, 1657}, {99, 7784}, {141, 33023}, {148, 33275}, {183, 33260}, {187, 15696}, {194, 6144}, {230, 3522}, {315, 8353}, {325, 32997}, {376, 5023}, {381, 37512}, {382, 574}, {548, 3767}, {550, 2549}, {566, 37196}, {599, 1975}, {626, 5077}, {1003, 7846}, {1078, 34505}, {1384, 7765}, {1506, 3830}, {1571, 28160}, {1656, 15515}, {2207, 35481}, {2996, 13468}, {3054, 15717}, {3055, 3832}, {3068, 9601}, {3146, 3815}, {3526, 8589}, {3528, 5585}, {3529, 7745}, {3530, 43620}, {3552, 7923}, {3589, 32981}, {3627, 31401}, {3763, 7791}, {3843, 18584}, {3853, 31415}, {3926, 33247}, {4296, 9594}, {4302, 16781}, {4324, 16502}, {5024, 7747}, {5054, 39565}, {5059, 7736}, {5063, 34622}, {5070, 18424}, {5073, 5475}, {5275, 37256}, {5286, 17538}, {5309, 15689}, {5691, 31443}, {5925, 32445}, {6337, 33272}, {6409, 26516}, {6410, 26521}, {6421, 42266}, {6422, 42267}, {6655, 7778}, {6658, 11174}, {6704, 11286}, {6781, 30435}, {7354, 31477}, {7503, 15109}, {7737, 15704}, {7739, 15686}, {7750, 33253}, {7753, 15685}, {7763, 19695}, {7773, 33256}, {7774, 33209}, {7777, 19691}, {7781, 7882}, {7782, 7841}, {7783, 7900}, {7787, 9855}, {7789, 32986}, {7790, 33235}, {7792, 33244}, {7797, 33268}, {7800, 8354}, {7802, 7905}, {7803, 33250}, {7808, 11159}, {7816, 7914}, {7839, 32480}, {7851, 13586}, {7861, 11288}, {7864, 33265}, {7872, 32456}, {7896, 15301}, {7898, 32821}, {7904, 20094}, {7913, 33242}, {7918, 33220}, {8252, 11294}, {8253, 11293}, {8553, 37198}, {8667, 33207}, {8725, 9605}, {8770, 10691}, {9300, 15683}, {9597, 15338}, {9598, 15326}, {9600, 35820}, {9604, 13346}, {9608, 12082}, {9609, 11413}, {9619, 28146}, {9655, 31451}, {9674, 13665}, {9722, 12976}, {10483, 31448}, {10992, 11646}, {11184, 33192}, {11257, 32469}, {11318, 22247}, {11648, 15513}, {12103, 15048}, {14064, 32459}, {15069, 32152}, {15075, 44249}, {15271, 32819}, {15491, 32979}, {15534, 20065}, {15682, 31404}, {15693, 39563}, {15700, 18362}, {16306, 16386}, {16310, 36748}, {16989, 33214}, {18480, 31422}, {19102, 42261}, {19105, 42260}, {21843, 33923}, {28168, 31430}, {28172, 31396}, {31295, 37661}, {31400, 33703}, {31403, 42414}, {31411, 42226}, {31457, 43457}, {31463, 42272}, {31829, 36751}, {32815, 33226}, {32826, 33215}, {32982, 44377}, {32996, 37647}, {34506, 41147}, {34873, 35930}, {42787, 43449}

X(44519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7748, 13881}, {4, 15815, 31489}, {99, 7910, 7881}, {99, 33234, 7784}, {376, 5254, 5023}, {382, 31467, 39590}, {548, 3767, 5210}, {550, 2549, 3053}, {574, 39590, 31467}, {3843, 31455, 18584}, {5024, 17800, 7747}, {6459, 6460, 33748}, {7872, 32456, 32954}, {7881, 7910, 7784}, {7881, 33234, 7910}, {32819, 32965, 15271}, {42258, 42259, 25406}

X(44520) = X(3)X(115)∩X(6)X(21)

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

X(44520) lies on these lines: {3, 115}, {6, 21}, {230, 4189}, {405, 31489}, {1006, 15815}, {2549, 5428}, {3053, 6914}, {3054, 4188}, {3055, 16859}, {3767, 7508}, {3815, 16865}, {5023, 6906}, {5254, 6875}, {5585, 37403}, {7737, 31649}, {7739, 28463}, {9604, 13323}, {11742, 37426}, {12104, 15048}

X(44520) = crossdifference of every pair of points on line {6132, 8672}

X(44521) = X(3)X(115)∩X(6)X(22)

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

X(44521) lies on these lines: {2, 13585}, {3, 115}, {6, 22}, {23, 3815}, {24, 15815}, {25, 31489}, {26, 5013}, {32, 13564}, {39, 2937}, {230, 6636}, {511, 9604}, {566, 10311}, {574, 2070}, {1194, 2965}, {1506, 18378}, {2548, 17714}, {2549, 7502}, {3054, 15246}, {3055, 13595}, {3767, 7525}, {5023, 10323}, {5024, 9699}, {5063, 37928}, {5116, 37123}, {5254, 7512}, {5309, 34006}, {5475, 5899}, {7492, 7735}, {7545, 7603}, {7555, 15048}, {7736, 37913}, {7738, 38435}, {7745, 12088}, {9590, 31443}, {9608, 22332}, {9712, 31490}, {9714, 31492}, {10314, 14806}, {11742, 21312}, {13621, 31455}, {15515, 43809}, {15574, 40341}, {15818, 36748}, {18472, 19220}, {18570, 43619}, {19165, 44453}, {31401, 37440}, {44180, 44377}

X(44521) = crossdifference of every pair of points on line {826, 6132}
X(44521) = {X(22),X(9609)}-harmonic conjugate of X(6)

X(44522) = X(3)X(115)∩X(6)X(23)

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

X(44522) lies on these lines: {3, 115}, {6, 23}, {22, 11063}, {25, 30537}, {230, 7492}, {576, 9604}, {1995, 9609}, {2549, 7575}, {3518, 22332}, {3767, 7555}, {3815, 14002}, {5013, 12106}, {5254, 7556}, {5306, 37913}, {7506, 31492}, {7737, 37967}, {7739, 37936}, {9699, 37923}, {10311, 41335}, {10985, 13351}, {12088, 22331}, {12105, 15048}, {15109, 40916}, {37950, 43619}

X(44522) = crossdifference of every pair of points on line {3906, 6132}

X(44523) = X(3)X(115)∩X(6)X(24)

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

X(44523) lies on these lines: {3, 115}, {6, 24}, {22, 1611}, {26, 3053}, {32, 2070}, {186, 5254}, {187, 2937}, {230, 7488}, {338, 1078}, {389, 9604}, {574, 43809}, {631, 15109}, {1609, 16195}, {1658, 3767}, {2548, 12106}, {2549, 37814}, {2931, 23128}, {3054, 37126}, {3518, 7745}, {5013, 6644}, {5206, 13564}, {5305, 7575}, {5306, 37940}, {5309, 37922}, {5475, 13621}, {6642, 31489}, {7525, 21843}, {7545, 39590}, {7556, 11063}, {7737, 37440}, {7747, 18378}, {8553, 9715}, {8770, 15818}, {9609, 15815}, {9697, 15087}, {9699, 30435}, {11648, 37955}, {18472, 38463}, {21844, 43448}

X(44523) = crossdifference of every pair of points on line {6132, 6368}
X(44523) = {X(9609),X(17928)}-harmonic conjugate of X(15815)

X(44524) = X(3)X(115)∩X(6)X(25)

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

X(44524) lies on these lines: {2, 9609}, {3, 115}, {6, 25}, {22, 230}, {23, 7735}, {24, 5254}, {26, 3767}, {32, 7517}, {39, 7506}, {50, 26283}, {172, 9658}, {183, 338}, {186, 43448}, {187, 12083}, {570, 10314}, {571, 1196}, {577, 34481}, {1184, 2965}, {1384, 5899}, {1609, 9909}, {1914, 9673}, {1995, 3815}, {2165, 15818}, {2548, 13861}, {2549, 6644}, {3053, 7387}, {3054, 7485}, {3518, 5286}, {3563, 15073}, {5013, 6642}, {5020, 31489}, {5023, 11414}, {5063, 40350}, {5210, 35243}, {5305, 37440}, {5309, 9699}, {6423, 35776}, {6424, 35777}, {6676, 9722}, {6781, 44457}, {7484, 15109}, {7502, 43291}, {7514, 43620}, {7530, 7737}, {7545, 15484}, {7736, 13595}, {7745, 10594}, {8573, 20850}, {8667, 15574}, {8770, 36748}, {9465, 26284}, {9605, 13621}, {9723, 44377}, {10046, 16781}, {12106, 15048}, {14002, 37665}, {14579, 21448}, {18378, 30435}, {20998, 40805}, {21397, 35481}, {33272, 34883}, {37689, 37913}, {38872, 44467}, {40801, 44439}

X(44524) = crossdifference of every pair of points on line {525, 6132}
X(44524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {22, 230, 8553}, {1971, 3981, 6}, {3518, 5286, 9608}, {7746, 9700, 3}, {9909, 34809, 1609}

X(44525) = X(3)X(115)∩X(6)X(26)

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

X(44525) lies on these lines: {3, 115}, {6, 26}, {22, 1184}, {23, 7745}, {24, 5013}, {32, 2937}, {39, 2070}, {52, 9604}, {140, 15109}, {187, 13564}, {195, 9697}, {230, 7512}, {1506, 13621}, {1658, 2549}, {1691, 2916}, {1971, 2917}, {2548, 37440}, {3518, 3815}, {3767, 7502}, {5210, 10323}, {5254, 7488}, {5286, 7556}, {5475, 18378}, {5899, 7747}, {6644, 15815}, {7506, 31489}, {7529, 18584}, {7555, 11063}, {7603, 18369}, {7735, 38435}, {7737, 17714}, {7753, 37956}, {9300, 37939}, {9603, 10282}, {9605, 9699}, {10117, 32445}, {10312, 37932}, {11250, 43619}, {12106, 31401}, {12107, 15048}, {19165, 38525}, {22159, 39537}, {33717, 42535}, {34864, 39565}, {37512, 43809}

X(44525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7748, 34866}, {24, 9609, 5013}

X(44526) = X(3)X(115)∩X(6)X(30)

Barycentrics    3*a^4 - 3*a^2*b^2 - 2*b^4 - 3*a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(44526) = 3 X[6] - 2 X[7737], 5 X[6] - 6 X[7739], 3 X[6] - 4 X[15048], 5 X[6] - 4 X[18907], 5 X[6] - 2 X[43618], X[6] + 2 X[43619], X[69] - 3 X[33272], 2 X[141] - 3 X[32986], 3 X[599] - 4 X[7761], 3 X[2549] - X[7737], 5 X[2549] - 3 X[7739], 3 X[2549] - 2 X[15048], 5 X[2549] - 2 X[18907], 5 X[2549] - X[43618], 4 X[3589] - 3 X[14033], 4 X[3631] - 3 X[32836], 4 X[3734] - 5 X[3763], 2 X[3734] - 3 X[11287], 5 X[3763] - 6 X[11287], 4 X[4045] - 3 X[11286], 3 X[5077] - 2 X[7761], 5 X[7737] - 9 X[7739], 5 X[7737] - 6 X[18907], 5 X[7737] - 3 X[43618], X[7737] + 3 X[43619], 9 X[7739] - 10 X[15048], 3 X[7739] - 2 X[18907], 3 X[7739] - X[43618], 3 X[7739] + 5 X[43619], 4 X[7798] - 3 X[15534], 4 X[7804] - 3 X[11159], 3 X[10516] - 4 X[37242], 4 X[14929] - 3 X[15533], 5 X[15048] - 3 X[18907], 10 X[15048] - 3 X[43618], 2 X[15048] + 3 X[43619], 2 X[18907] + 5 X[43619], X[32815] - 3 X[32986], X[43618] + 5 X[43619]

Let KA be the symmedian point of the A-anti-altimedial triangle, and define KB, KC cyclically. X(44526) = X(6)-of-KAKBKC. X(44526) is also X(69) of the reflection triangle of X(6). (Randy Hutson, September 30, 2021)

X(44526) lies on these lines: {2, 11147}, {3, 115}, {4, 3815}, {5, 15815}, {6, 30}, {20, 3053}, {32, 1657}, {39, 382}, {69, 33272}, {76, 33234}, {99, 7778}, {141, 32815}, {148, 183}, {187, 3534}, {194, 33256}, {230, 376}, {232, 44438}, {315, 19695}, {316, 9766}, {325, 8716}, {378, 9609}, {381, 574}, {385, 33264}, {538, 40341}, {543, 599}, {546, 31401}, {549, 43620}, {550, 3767}, {570, 18494}, {620, 11318}, {671, 7610}, {754, 6144}, {999, 9664}, {1003, 7790}, {1015, 9668}, {1285, 3529}, {1350, 15993}, {1384, 5309}, {1478, 31477}, {1500, 9655}, {1506, 3843}, {1513, 8719}, {1569, 38744}, {1571, 18480}, {1572, 28146}, {1611, 7667}, {1648, 15538}, {1656, 37512}, {1870, 9594}, {1975, 3314}, {2023, 10723}, {2207, 18560}, {2275, 12953}, {2276, 12943}, {2548, 3627}, {2782, 15069}, {3054, 3524}, {3055, 3545}, {3070, 12257}, {3071, 12256}, {3094, 36990}, {3146, 7738}, {3172, 40234}, {3258, 35901}, {3269, 18396}, {3295, 9651}, {3526, 15515}, {3543, 7736}, {3552, 7851}, {3585, 31448}, {3589, 14033}, {3631, 32836}, {3734, 3763}, {3785, 33247}, {3788, 35022}, {3830, 5024}, {3845, 31415}, {3849, 7798}, {3851, 31455}, {3853, 31406}, {3861, 31450}, {3926, 33238}, {4045, 11286}, {5017, 14532}, {5054, 8589}, {5055, 18424}, {5073, 7747}, {5076, 39590}, {5116, 35930}, {5206, 15696}, {5275, 17579}, {5304, 15683}, {5305, 15704}, {5306, 11001}, {5318, 44463}, {5321, 44459}, {5339, 22512}, {5340, 22513}, {5355, 15685}, {5459, 6772}, {5460, 6775}, {5523, 35481}, {5569, 36523}, {5585, 8703}, {5587, 31443}, {5895, 32445}, {5899, 9699}, {5913, 16063}, {6032, 31133}, {6284, 9597}, {6337, 32982}, {6421, 35821}, {6422, 35820}, {6423, 42267}, {6424, 42266}, {6564, 9600}, {6658, 7864}, {6759, 9603}, {6800, 36181}, {7354, 9598}, {7386, 8770}, {7514, 15109}, {7618, 37350}, {7620, 11168}, {7750, 32997}, {7753, 15684}, {7754, 7802}, {7763, 33229}, {7765, 17800}, {7770, 7847}, {7773, 7783}, {7774, 33192}, {7776, 7781}, {7782, 7887}, {7787, 19696}, {7788, 7898}, {7789, 32974}, {7791, 32819}, {7792, 33007}, {7793, 33267}, {7795, 8357}, {7797, 33257}, {7803, 19687}, {7804, 11159}, {7806, 33265}, {7816, 7866}, {7823, 19691}, {7828, 33235}, {7835, 33219}, {7844, 11288}, {7852, 33242}, {7861, 32954}, {7868, 7924}, {7874, 33241}, {7875, 19686}, {7879, 7910}, {7881, 7911}, {7885, 32821}, {7908, 15301}, {7913, 33237}, {7918, 33217}, {7919, 33220}, {8288, 37638}, {8352, 11184}, {8353, 8667}, {8354, 8556}, {8356, 11185}, {8553, 35243}, {8588, 15688}, {8597, 11163}, {8860, 41135}, {8976, 9674}, {8981, 9601}, {9112, 16965}, {9113, 16964}, {9220, 9818}, {9300, 15682}, {9541, 9602}, {9604, 13352}, {9606, 17578}, {9607, 14482}, {9619, 22793}, {9620, 28160}, {9650, 31461}, {9654, 31451}, {9745, 10989}, {9862, 39646}, {9880, 40248}, {9956, 31422}, {10311, 37196}, {10317, 19220}, {10418, 32216}, {10516, 37242}, {10722, 11257}, {11063, 33532}, {11165, 31173}, {11173, 19924}, {11174, 11361}, {11297, 43028}, {11298, 43029}, {11317, 42849}, {11359, 24275}, {11480, 44461}, {11481, 44465}, {12017, 22862}, {12605, 15075}, {14269, 43457}, {14614, 14712}, {14929, 15533}, {15491, 32983}, {15655, 15689}, {15693, 18362}, {15820, 34609}, {16041, 44377}, {16043, 32826}, {16989, 33193}, {16990, 33263}, {17008, 33207}, {18492, 31421}, {18561, 22121}, {18562, 22120}, {18563, 23115}, {18573, 44468}, {20065, 33271}, {30771, 40349}, {31152, 39602}, {31441, 38140}, {31463, 42284}, {31467, 31652}, {32828, 33226}, {33006, 37647}, {33008, 37688}, {33216, 44381}, {33418, 42491}, {33419, 42490}, {35490, 39575}, {35606, 36194}

X(44526) = midpoint of X(2549) and X(43619)
X(44526) = reflection of X(i) in X(j) for these {i,j}: {6, 2549}, {599, 5077}, {6144, 22253}, {7737, 15048}, {32815, 141}, {43618, 18907}
X(44526) = crossdifference of every pair of points on line {6132, 8675}
X(44526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 115, 37637}, {20, 5254, 3053}, {99, 7841, 7778}, {115, 37637, 13881}, {148, 183, 34505}, {148, 7833, 183}, {230, 376, 5210}, {316, 31859, 9766}, {376, 43448, 230}, {381, 574, 31489}, {381, 31489, 18584}, {550, 3767, 5023}, {671, 35955, 7610}, {1384, 15681, 6781}, {1975, 6655, 7784}, {2549, 7737, 15048}, {2549, 43618, 7739}, {3146, 7738, 7745}, {3314, 20094, 1975}, {3734, 11287, 3763}, {3830, 5024, 5475}, {5073, 9605, 7747}, {5309, 6781, 1384}, {6284, 9597, 16781}, {6655, 20094, 3314}, {6772, 11296, 16644}, {6775, 11295, 16645}, {7737, 15048, 6}, {7739, 18907, 6}, {7739, 43618, 18907}, {7748, 7756, 3}, {7781, 7842, 7776}, {7783, 33019, 7773}, {7816, 7872, 7866}, {7844, 32456, 11288}, {8356, 11185, 15271}, {8597, 32480, 11163}, {8703, 21843, 5585}, {8703, 43291, 21843}, {15515, 39565, 3526}, {22862, 22906, 12017}, {32815, 32986, 141}, {42154, 42155, 43273}

X(44527) = X(3)X(115)∩X(6)X(49)

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

X(44527) lies on these lines: {3, 115}, {6, 49}, {24, 3767}, {25, 32}, {26, 230}, {39, 6642}, {140, 9609}, {159, 1692}, {161, 19627}, {187, 7387}, {233, 31455}, {1384, 18378}, {1504, 8276}, {1505, 8277}, {1506, 5020}, {1598, 7747}, {1609, 9714}, {1658, 43291}, {1995, 2548}, {2241, 10046}, {2242, 10037}, {2549, 17928}, {2937, 8553}, {3053, 7517}, {3054, 7516}, {3124, 14585}, {3291, 26283}, {3517, 7755}, {3518, 7735}, {3964, 7888}, {5023, 12083}, {5058, 8908}, {5206, 11414}, {5254, 6644}, {5305, 9608}, {5475, 7529}, {6781, 39568}, {7503, 43620}, {7542, 9722}, {7737, 10594}, {7745, 13861}, {7780, 15574}, {8588, 37198}, {9604, 36753}, {9697, 11402}, {9818, 39565}, {10323, 21843}, {12963, 35777}, {12968, 35776}, {13621, 30435}, {15484, 18369}, {15513, 35243}, {18560, 21397}, {19459, 39764}, {20987, 40825}, {22467, 43448}, {32048, 32661}, {32977, 44180}, {33238, 34883}, {37951, 41361}

X(44527) = X(1969)-isoconjugate of X(44405)
X(44527) = crosssum of X(69) and X(6515)
X(44527) = crossdifference of every pair of points on line {3265, 6132}
X(44527) = barycentric product X(25)*X(3548)
X(44527) = barycentric quotient X(i)/X(j) for these {i,j}: {3548, 305}, {14575, 44405}
X(44527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5305, 12106, 9608}, {7749, 9700, 3}

X(44528) = X(3)X(115)∩X(6)X(64)

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

X(44528) lies on these lines: {3, 115}, {4, 9608}, {6, 64}, {20, 8553}, {230, 11413}, {338, 1975}, {378, 5254}, {2548, 31861}, {2549, 7526}, {3053, 12085}, {3520, 21397}, {3767, 12084}, {5013, 9818}, {5023, 21312}, {5286, 14865}, {5866, 32972}, {7395, 15815}, {7527, 7738}, {7735, 12086}, {7745, 35502}, {9603, 18451}, {9604, 11425}, {9609, 14118}, {9722, 31829}, {11325, 20987}, {11479, 31489}

X(44528) = crossdifference of every pair of points on line {6132, 8057}

X(44529) = X(3)X(115)∩X(6)X(67)

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

X(44529) lies on these lines: {2, 94}, {3, 115}, {6, 67}, {50, 230}, {111, 930}, {112, 1594}, {187, 7574}, {265, 32761}, {427, 38872}, {1368, 14910}, {1995, 9220}, {2088, 15061}, {2165, 40347}, {2549, 18580}, {3016, 14643}, {3018, 31489}, {3767, 18281}, {3815, 30789}, {5159, 16310}, {5254, 37118}, {5286, 37119}, {5475, 7579}, {5477, 32306}, {7778, 18375}, {8553, 26283}, {8889, 39176}, {9604, 17974}, {10254, 19220}, {10510, 15993}, {11646, 32233}, {15122, 43291}, {15133, 32661}, {16308, 19656}, {34844, 36751}

X(44529) = crossdifference of every pair of points on line {6132, 9517}
X(44529) = {X(230),X(858)}-harmonic conjugate of X(50)

X(44530) = X(3)X(115)∩X(6)X(76)

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

X(44530) lies on these lines: {3, 115}, {4, 2076}, {6, 76}, {32, 7697}, {230, 384}, {458, 21001}, {1613, 41231}, {1691, 6248}, {2549, 34873}, {3053, 10104}, {3054, 7824}, {3406, 39560}, {3734, 32189}, {3788, 31489}, {3815, 7836}, {3934, 11356}, {5013, 40108}, {5017, 10516}, {5023, 9756}, {5116, 5254}, {5162, 39565}, {6287, 7747}, {6655, 9478}, {7737, 32151}, {7745, 15993}, {7919, 11285}, {10358, 13330}, {12251, 15514}, {14061, 34885}, {17128, 39095}, {37334, 39663}, {37446, 38654}, {43291, 44224}

X(44530) = crossdifference of every pair of points on line {688, 6132}
X(44530) = {X(35830),X(35831)}-harmonic conjugate of X(38732)

X(44531) = X(3)X(115)∩X(6)X(98)

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

X(44531) lies on these lines: {2, 4048}, {3, 115}, {5, 43449}, {6, 98}, {32, 13111}, {39, 12188}, {99, 11285}, {114, 10516}, {140, 34873}, {147, 3815}, {148, 7824}, {183, 1916}, {230, 2076}, {385, 15514}, {574, 7697}, {599, 32458}, {1003, 9166}, {1506, 6287}, {1569, 5024}, {2782, 5013}, {3027, 31477}, {3053, 12042}, {3054, 37455}, {3934, 8178}, {4027, 11174}, {5017, 6034}, {5023, 34473}, {5149, 6722}, {5152, 7770}, {5254, 14651}, {5461, 11286}, {5475, 38744}, {5880, 5988}, {5939, 10352}, {5969, 8556}, {5976, 15271}, {5984, 7736}, {6421, 35824}, {6422, 35825}, {6721, 42786}, {6784, 17970}, {7745, 9862}, {7752, 35701}, {7844, 11356}, {7887, 38907}, {8288, 30789}, {8667, 36849}, {9300, 11177}, {9600, 35878}, {10053, 16781}, {10488, 42849}, {11184, 33683}, {11676, 39663}, {12215, 44377}, {13174, 31443}, {13586, 44401}, {14532, 38747}, {14614, 36864}, {14981, 31492}, {15597, 33273}, {16921, 32528}, {22332, 38664}, {35006, 39095}, {39839, 43843}

X(44531) = crossdifference of every pair of points on line {5113, 6132}
X(44531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 2023, 6}, {115, 7748, 38732}, {115, 38224, 13881}, {230, 5999, 2076}, {5152, 14061, 7770}, {5984, 7736, 12830}, {13873, 13926, 38224}, {14651, 37334, 38654}

X(44532) = X(3)X(115)∩X(6)X(99)

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

X(44532) lies on these lines: {3, 115}, {6, 99}, {32, 13188}, {39, 19910}, {98, 5023}, {148, 230}, {187, 12188}, {384, 3815}, {550, 43449}, {620, 11286}, {1285, 35951}, {1503, 2076}, {1569, 30435}, {1975, 39652}, {2023, 15815}, {2782, 3053}, {3054, 33273}, {3231, 35933}, {3552, 5989}, {4226, 20998}, {5013, 33813}, {5116, 35925}, {5149, 35022}, {5152, 33235}, {5210, 12042}, {5254, 13172}, {5306, 8591}, {6423, 35878}, {6424, 35879}, {7747, 38743}, {7816, 34870}, {8178, 32456}, {8290, 37665}, {8553, 39832}, {8719, 38642}, {9478, 32965}, {10089, 16781}, {11646, 38749}, {12176, 39560}, {12829, 22331}, {15452, 31477}, {21001, 35941}, {32528, 33257}, {34873, 44224}, {35369, 37689}

X(44532) = crossdifference of every pair of points on line {888, 6132}
X(44532) = {X(2023),X(21166)}-harmonic conjugate of X(15815)

X(44533) = X(3)X(115)∩X(6)X(110)

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

X(44533) lies on these lines: {2, 13582}, {3, 115}, {6, 110}, {23, 230}, {25, 1989}, {32, 7545}, {50, 3291}, {112, 10594}, {187, 37924}, {231, 37972}, {265, 15538}, {399, 15544}, {575, 9604}, {686, 39232}, {1611, 38872}, {1634, 5020}, {1990, 37777}, {2965, 10985}, {3003, 40350}, {3018, 34809}, {3053, 7530}, {3054, 7496}, {3511, 21513}, {3518, 27376}, {3767, 12106}, {3815, 16042}, {5007, 18369}, {5013, 44468}, {5023, 12082}, {5158, 34481}, {5159, 34988}, {5210, 33532}, {5306, 13595}, {5477, 32254}, {5585, 41463}, {5621, 7418}, {5941, 11646}, {7575, 43291}, {7735, 14002}, {7753, 21308}, {7755, 13621}, {8428, 15959}, {8749, 36616}, {8770, 14910}, {9166, 35936}, {9209, 15470}, {9609, 40916}, {9696, 15039}, {9698, 22462}, {10317, 38463}, {10413, 38724}, {10418, 11284}, {18859, 39563}, {35296, 44401}, {38997, 39078}

X(44533) = polar conjugate of the isotomic conjugate of X(39562)
X(44533) = crossdifference of every pair of points on line {690, 6132}
X(44533) = barycentric product X(4)*X(39562)
X(44533) = barycentric quotient X(39562)/X(69)
X(44533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {23, 230, 11063}, {115, 2079, 34866}, {1995, 33900, 2930}, {11141, 11142, 7669}

X(44534) = X(3)X(115)∩X(6)X(114)

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

X(44534) lies on these lines: {2, 694}, {3, 115}, {5, 34870}, {6, 114}, {32, 6033}, {39, 15561}, {98, 230}, {99, 5254}, {147, 7735}, {148, 16925}, {172, 12184}, {187, 38741}, {325, 5111}, {542, 40825}, {543, 11288}, {574, 38750}, {620, 5013}, {671, 35297}, {1384, 38744}, {1569, 5309}, {1609, 9861}, {1648, 30789}, {1914, 12185}, {2021, 11632}, {2493, 38652}, {2549, 33813}, {2679, 17970}, {2782, 3767}, {2794, 3053}, {3054, 37450}, {4027, 7806}, {5023, 38749}, {5025, 39652}, {5038, 7792}, {5149, 6680}, {5152, 7857}, {5206, 38742}, {5210, 38747}, {5306, 6054}, {5461, 11287}, {5465, 14605}, {5984, 37689}, {6393, 8781}, {6656, 14061}, {6721, 31489}, {6722, 7866}, {7737, 22505}, {7738, 7932}, {7763, 35700}, {7778, 32458}, {7779, 36859}, {8290, 16984}, {8295, 37455}, {8356, 9166}, {8363, 31268}, {9300, 23234}, {9478, 37688}, {9598, 15452}, {9604, 39805}, {9753, 13330}, {11060, 34365}, {12042, 43449}, {12176, 37446}, {13172, 43448}, {13862, 42535}, {14651, 38642}, {14901, 15545}, {15815, 38748}, {22332, 38751}, {23053, 33008}, {24975, 32740}, {30435, 38743}, {34212, 40347}, {37242, 43620}, {37243, 39565}, {37459, 43291}

X(44534) = complement of X(46236)
X(44534) = crosspoint of X(98) and X(8781)
X(44534) = crosssum of X(511) and X(1692)
X(44534) = crossdifference of every pair of points on line {5027, 6132}
X(44534) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {115, 7746, 38224}, {147, 7735, 12829}, {230, 1513, 1691}, {2023, 5976, 3094}, {13873, 13926, 3}

X(44535) = X(3)X(115)∩X(6)X(140)

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

X(44535) lies on these lines: {2, 3053}, {3, 115}, {4, 3054}, {5, 5023}, {6, 140}, {32, 3526}, {39, 5054}, {76, 7610}, {141, 32970}, {183, 7836}, {187, 1656}, {230, 631}, {325, 33000}, {381, 5206}, {382, 15513}, {487, 615}, {488, 590}, {524, 32829}, {549, 3767}, {550, 5585}, {574, 15720}, {599, 3788}, {618, 16644}, {619, 16645}, {632, 2548}, {639, 8252}, {640, 8253}, {1030, 19547}, {1078, 7778}, {1153, 6683}, {1384, 1506}, {1560, 8778}, {1588, 9602}, {1611, 7499}, {1657, 8588}, {1691, 3763}, {1968, 37453}, {1970, 26958}, {1971, 40686}, {1975, 17004}, {2076, 14693}, {2207, 10018}, {2549, 3530}, {3055, 3533}, {3523, 5254}, {3525, 3815}, {3546, 36748}, {3552, 17006}, {3589, 32978}, {3628, 7737}, {3630, 32825}, {3785, 32977}, {3843, 6781}, {3850, 43618}, {3851, 15655}, {3926, 13468}, {3934, 11288}, {5007, 31467}, {5017, 6680}, {5024, 7755}, {5055, 7747}, {5070, 5475}, {5073, 12815}, {5079, 39590}, {5215, 31239}, {5304, 9606}, {5305, 14869}, {5306, 15702}, {5309, 15701}, {5326, 9596}, {5355, 31457}, {5368, 9605}, {5433, 16781}, {5569, 7830}, {6144, 7764}, {6392, 23055}, {6642, 8553}, {7294, 9599}, {7494, 8770}, {7622, 32450}, {7735, 10303}, {7739, 11812}, {7762, 11184}, {7763, 8667}, {7769, 7877}, {7771, 7887}, {7774, 33204}, {7780, 7916}, {7789, 32989}, {7792, 33001}, {7793, 7941}, {7795, 8556}, {7807, 15271}, {7824, 7932}, {7831, 33218}, {7841, 43459}, {7851, 33004}, {7854, 31274}, {7857, 7859}, {7868, 33245}, {7881, 12829}, {7886, 11287}, {7935, 33240}, {8376, 8960}, {8591, 8860}, {9300, 15709}, {9604, 13336}, {9675, 13951}, {9698, 43136}, {9770, 32871}, {9771, 32884}, {10104, 12177}, {10299, 43448}, {10356, 37071}, {10516, 37466}, {11173, 25555}, {11174, 33015}, {11648, 15700}, {11793, 15575}, {12108, 15048}, {13846, 41490}, {13847, 41491}, {14033, 32867}, {14061, 33234}, {14064, 44381}, {14067, 43450}, {14093, 39563}, {14907, 33249}, {15484, 35007}, {15515, 15693}, {15597, 32838}, {15688, 18362}, {15694, 30435}, {15712, 43291}, {16239, 18907}, {16925, 37688}, {16989, 33188}, {16990, 33262}, {17008, 33206}, {19780, 43029}, {19781, 43028}, {20065, 37647}, {22831, 42093}, {22832, 42094}, {23053, 35287}, {27376, 35486}, {32828, 33216}, {32832, 35297}, {32883, 32983}, {33190, 41139}, {33215, 44401}, {33217, 42535}, {33272, 39143}, {33923, 43619}, {41406, 42936}, {41407, 42937}

X(44535) = midpoint of X(41975) and X(41976)
X(44535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7749, 37637}, {3, 37637, 13881}, {5, 21843, 5023}, {32, 3526, 31489}, {230, 631, 5013}, {549, 3767, 15815}, {1078, 7940, 7879}, {1078, 33233, 7778}, {3785, 32977, 44377}, {7815, 32954, 3763}, {7879, 7940, 7778}, {7879, 33233, 7940}, {8588, 39565, 1657}, {8860, 33274, 34505}, {15694, 30435, 31455}, {17004, 33259, 1975}, {32989, 34229, 7789}

X(44536) = X(3)X(115)∩X(6)X(147)

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

X(44536) lies on these lines: {2, 39091}, {3, 115}, {6, 147}, {45, 5988}, {98, 14458}, {99, 7943}, {114, 6034}, {148, 7851}, {230, 40236}, {671, 33220}, {1916, 7778}, {2023, 31489}, {3053, 43449}, {3124, 30789}, {3763, 5976}, {5306, 5984}, {7806, 11606}, {7868, 8782}, {7923, 20094}, {8178, 32954}, {9478, 15271}, {9862, 22331}, {11164, 33246}, {15561, 31492}, {32819, 33225}, {35006, 43460}, {36849, 40341}

X(44536) = {X(13873),X(13926)}-harmonic conjugate of X(38739)

X(44537) = X(3)X(115)∩X(6)X(156)

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

X(44537) lies on these lines: {3, 115}, {6, 156}, {230, 12088}, {1180, 1995}, {2207, 10312}, {3518, 5523}, {3767, 37440}, {5007, 7545}, {5306, 34484}, {5309, 13621}, {7530, 22331}, {7755, 18378}, {7772, 18369}, {10316, 38463}, {11063, 17714}, {11648, 43809}, {18362, 34864}

X(44538) = X(3)X(115)∩X(6)X(186)

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

X(44538) lies on these lines: {3, 115}, {6, 186}, {25, 18429}, {230, 10298}, {338, 7771}, {1658, 3053}, {2549, 15646}, {3767, 15331}, {5013, 37814}, {5023, 7488}, {5210, 7502}, {5254, 21844}, {5585, 6636}, {5913, 7492}, {6644, 31489}, {7575, 7737}, {9604, 11438}, {9608, 35479}, {9609, 15078}, {9699, 37922}, {15048, 18571}, {15815, 22467}, {34152, 43619}, {37936, 43618}

X(44538) = {X(3),X(2079)}-harmonic conjugate of X(37637)

X(44539) = X(3)X(115)∩X(6)X(194)

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

X(44539) lies on these lines: {3, 115}, {6, 194}, {32, 13108}, {230, 3552}, {382, 5162}, {401, 21001}, {1003, 14568}, {2076, 9873}, {2549, 44224}, {3054, 33004}, {3734, 34870}, {3815, 7891}, {5017, 15069}, {5023, 11676}, {5254, 35925}, {7769, 7770}, {7841, 34885}, {9863, 15993}, {11257, 39560}, {18440, 35374}

X(44539) = crossdifference of every pair of points on line {3221, 6132}

X(44540) = X(3)X(115)∩X(6)X(315)

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

X(44540) lies on these lines: {3, 115}, {6, 315}, {230, 7791}, {1656, 2021}, {1691, 10358}, {3053, 20576}, {3054, 16925}, {5286, 15993}, {6683, 7862}, {7735, 7904}, {7745, 20194}, {7817, 11287}, {7913, 13357}, {7933, 39095}, {15815, 37450}, {25555, 40825}

X(44541) = X(3)X(115)∩X(6)X(376)

Barycentrics    9*a^4 - 9*a^2*b^2 - 2*b^4 - 9*a^2*c^2 + 4*b^2*c^2 - 2*c^4 : :
X(44541) = 5 X[11742] + 4 X[31415], 4 X[31415] - 5 X[31489]

X(44541) lies on these lines: {3, 115}, {6, 376}, {20, 3815}, {30, 11742}, {39, 15696}, {99, 599}, {148, 7610}, {183, 20094}, {187, 15688}, {230, 5585}, {325, 33207}, {378, 15109}, {381, 8589}, {382, 7603}, {548, 3053}, {549, 43619}, {550, 5013}, {574, 3534}, {590, 26618}, {615, 26617}, {620, 5077}, {1384, 15695}, {1506, 17800}, {1587, 9601}, {1657, 37512}, {1975, 33275}, {2548, 12103}, {2549, 5210}, {3054, 15692}, {3055, 3543}, {3314, 33260}, {3522, 5023}, {3526, 39601}, {3528, 5254}, {3589, 35927}, {3763, 8356}, {3767, 33923}, {3830, 18584}, {5024, 6781}, {5054, 11614}, {5073, 31455}, {5275, 36004}, {5309, 15655}, {5475, 15681}, {6144, 31859}, {6409, 35944}, {6410, 35945}, {7738, 22331}, {7745, 17538}, {7761, 35022}, {7771, 34505}, {7773, 33267}, {7778, 7833}, {7782, 7784}, {7789, 33226}, {7792, 33208}, {7845, 11165}, {7851, 33276}, {7934, 33234}, {8252, 35949}, {8253, 35948}, {8556, 32815}, {8588, 14093}, {8716, 14907}, {8719, 36990}, {9112, 36968}, {9113, 36967}, {9300, 15697}, {9604, 37480}, {9766, 14976}, {9855, 42849}, {11063, 41463}, {11159, 14762}, {11174, 33265}, {11287, 32456}, {11299, 43028}, {11300, 43029}, {11480, 44250}, {12007, 41400}, {12100, 43620}, {15271, 33008}, {15326, 31477}, {15684, 43457}, {15686, 43618}, {15690, 18907}, {15694, 18424}, {15704, 31401}, {15706, 39563}, {15716, 18362}, {15993, 31884}, {16644, 35932}, {16645, 35931}, {16984, 33014}, {19103, 42261}, {19104, 42260}, {19708, 43448}, {21843, 34200}, {22332, 37665}, {26958, 35937}, {28168, 31441}, {32459, 32986}, {33192, 37647}, {33272, 44377}, {33751, 40825}, {36751, 44241}, {38749, 43273}, {39809, 40248}

X(44541) = midpoint of X(11742) and X(31489)
X(44541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 7756, 13881}, {230, 10304, 5585}, {2549, 8703, 5210}, {5024, 15689, 6781}, {8716, 14907, 40341}

X(44542) = X(3)X(115)∩X(6)X(474)

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

X(44542) lies on these lines: {3, 115}, {6, 474}, {21, 3054}, {187, 37251}, {230, 404}, {3053, 6911}, {3055, 17535}, {3149, 5023}, {3815, 17531}, {5124, 19514}, {5210, 6985}, {5254, 6940}, {5306, 36006}, {5585, 37426}, {6946, 7745}, {7735, 17572}, {8553, 37034}, {8588, 16117}, {11063, 16427}, {16408, 31489}, {25946, 37646}

X(44542) = crossdifference of every pair of points on line {4132, 6132}

X(44543) = EULER LINE INTERCEPT OF X(183)X(754)

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

See Kadir Altintas and Francisco Javier García Capitán, euclid 2127.

X(44543) lies on these lines: {2,3}, {6,14568}, {39,18546}, {76,9766}, {83,13881}, {115,11174}, {148,5024}, {183,754}, {316,15271}, {385,15484}, {538,11163}, {598,7610}, {599,7809}, {625,7868}, {671,2023}, {1506,1975}, {2548,7754}, {3734,7603}, {3763,7934}, {3814,20172}, {3815,11185}, {3934,7773}, {4045,18424}, {5031,21358}, {5050,14651}, {5182,7884}, {5640,12525}, {6179,12156}, {7615,7739}, {7622,11164}, {7745,13468}, {7752,7881}, {7753,14614}, {7775,7788}, {7778,18584}, {7786,15031}, {7799,11184}, {7808,7851}, {7811,8556}, {7812,8667}, {7817,18362}, {7875,14535}, {9745,23297}, {14561,22525}

X(44544) = X(5)X(6697)∩X(68)X(5663)

Barycentrics    a^2*((b^2+c^2)*a^12-2*(2*b^4+b^2*c^2+2*c^4)*a^10+(b^2+c^2)*(5*b^4-7*b^2*c^2+5*c^4)*a^8+2*(3*b^4-4*b^2*c^2+3*c^4)*b^2*c^2*a^6-(b^4-c^4)*(b^2-c^2)*(5*b^4+6*b^2*c^2+5*c^4)*a^4+4*(b^2-c^2)^2*(b^8+c^8+(b^4+3*b^2*c^2+c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3*(b^4+b^2*c^2+c^4)) : :
Barycentrics    (SB+SC)*((8*R^2-SB-SC)*S^2+(SA^2-SB*SC+2*SW^2-2*(4*R^2-2*SA+7*SW)*R^2)*SA) : :
X(44544) = X(3)-3*X(41715), 2*X(140)-3*X(41580), 3*X(154)-2*X(10627), 3*X(568)-X(12324), 3*X(1853)-4*X(10095), 3*X(2979)-5*X(14530), 3*X(3060)-X(34780), 3*X(5656)-X(18436), 3*X(5890)-X(13093), 3*X(5946)-2*X(6247), 5*X(10574)-3*X(35450), 3*X(11206)-X(37484), X(11412)-3*X(32063), 6*X(13363)-5*X(40686), 5*X(15026)-4*X(20299), 3*X(15067)-4*X(16252), 4*X(16982)-3*X(34751)

See Antreas Hatzipolakis and César Lozada, euclid 2143.

X(44544) lies on these lines: {3, 41715}, {5, 6697}, {64, 13630}, {68, 5663}, {140, 41580}, {143, 14216}, {154, 10627}, {185, 13488}, {468, 43896}, {568, 12324}, {1112, 11457}, {1154, 1498}, {1503, 10263}, {1619, 16266}, {1853, 10095}, {2781, 5609}, {2883, 5876}, {2979, 14530}, {3060, 34780}, {3357, 41589}, {3527, 5890}, {3627, 5446}, {5656, 18436}, {5889, 12315}, {5946, 6247}, {6225, 34783}, {6241, 44438}, {6243, 34781}, {7516, 34778}, {9833, 13391}, {9919, 32338}, {10574, 35450}, {11206, 37484}, {11412, 32063}, {13363, 40686}, {13491, 15311}, {15026, 20299}, {15067, 16252}, {16982, 34751}, {18952, 34944}, {22802, 22804}, {25738, 41736}

X(44544) = midpoint of X(i) and X(j) for these {i, j}: {5878, 6293}, {5889, 12315}, {6225, 34783}, {6243, 34781}
X(44544) = reflection of X(i) in X(j) for these (i, j): (64, 13630), (3357, 41589), (5876, 2883), (6101, 6759), (13491, 41725), (14216, 143)

X(44545) = X(5)X(10)∩X(6)X(19)

Barycentrics    a^3*b^2*(a+b)*c^2*(a+c)*(b+c)*(a^5*b+a^4*b^2-a*b^5-b^6+a^5*c+2*a^3*b^2*c-3*a*b^4*c+a^4*c^2+2*a^3*b*c^2+4*a*b^3*c^2+b^4*c^2+4*a*b^2*c^3-3*a*b*c^4+b^2*c^4-a*c^5-c^6) : :
X(44545) = 3*X(51)-X(1858), 3*X(5902)-X(18732)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2138.

X(44545) lies on these lines: {1,197}, {4,17869}, {5,10}, {6,19}, {8,15435}, {30,14453}, {51,1858}, {56,20276}, {72,10869}, {389,6001}, {518,14913}, {612,3057}, {912,12235}, {1325,1798}, {1385,39582}, {1610,17016}, {1824,1837}, {1828,1836}, {1854,17810}, {1953,28266}, {2183,2292}, {2339,19860}, {2771,12236}, {2778,7687}, {2933,37539}, {3185,3931}, {3556,7713}, {3666,23361}, {3753,19784}, {3812,6703}, {3868,9004}, {3869,14555}, {5902,18732}, {8231,30556}, {9957,30142}, {10327,14923}, {12709,14557}, {13323,34339}, {13750,18180}, {23846,37548}, {31811,39523}

X(44545) = midpoint of X(65) and X(1829)
X(44545) = reflection of X(37613) in X(3812)
X(44545) = complement of X(41600)
X(44545) = complement of the isogonal conjugate of X(40454)
X(44545) = complementary conjugate of the complement of X(40454)
X(44545) = crosssum of X(3) and X(960)
X(44545) = crosspoint of X(4) and X(961)
X(44545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (960,5836,5835), (2362,16232,478)

X(44546) = X(5)X(916)∩X(6)X(31)

Barycentrics    a^2*(a+b)*(a+c)*(b+c)*(a^2-b^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6*b^3-a^5*b^4-2*a^4*b^5+2*a^3*b^6+a^2*b^7-a*b^8+a^6*b^2*c-3*a^4*b^4*c+3*a^2*b^6*c-b^8*c+a^6*b*c^2-2*a^3*b^4*c^2-a^2*b^5*c^2+2*a*b^6*c^2+a^6*c^3-3*a^2*b^4*c^3+2*b^6*c^3-a^5*c^4-3*a^4*b*c^4-2*a^3*b^2*c^4-3*a^2*b^3*c^4-2*a*b^4*c^4-b^5*c^4-2*a^4*c^5-a^2*b^2*c^5-b^4*c^5+2*a^3*c^6+3*a^2*b*c^6+2*a*b^2*c^6+2*b^3*c^6+a^2*c^7-a*c^8-b*c^8) : :
X(44546) = 3*X(51)-X(1839), 3*X(3060)+X(20291)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2138.

X(44546) lies on these lines: {4,15320}, {5,916}, {6,31}, {51,1839}, {389,516}, {2772,7687}, {3060,20291}, {9028,14913}

X(44546) = midpoint of X(71) and X(14053)
X(44546) = crosssum of X(3) and X(34830)

X(44547) = X(1)X(6)∩X(4)X(65)

Barycentrics    a^3*b^2*(a+b)*c^2*(a+c)*(b+c)*(a^2-b^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)*(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^2-4*a^2*b^2*c^2-2*a*b^3*c^2+b^4*c^2-2*a^3*c^3-2*a*b^2*c^3+2*a^2*c^4+a*b*c^4+b^2*c^4+a*c^5-c^6) : :
X(44547) = 3*X(51)-X(1829),3*X(11245)+X(12135)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2138.

X(44547) lies on these lines: {1,6}, {3,1708}, {4,65}, {5,226}, {7,6835}, {8,11433}, {10,13567}, {11,12691}, {20,10394}, {25,40660}, {28,2182}, {33,1712}, {34,34032}, {40,9786}, {42,774}, {46,7580}, {51,1829}, {55,12710}, {56,12675}, {57,1071}, {63,11344}, {78,16465}, {165,1192}, {181,39591}, {184,11363}, {185,1902}, {201,14547}, {210,3085}, {224,37282}, {225,5721}, {227,37699}, {241,4303}, {255,4641}, {329,938}, {354,3086}, {355,39571}, {386,17102}, {387,7952}, {388,14872}, {389,517}, {440,10974}, {442,1737}, {443,5784}, {452,3869}, {497,5758}, {498,3740}, {499,3742}, {511,37613}, {515,12241}, {516,12432}, {569,24301}, {578,1385}, {581,1214}, {631,17603}, {758,6738}, {908,39772}, {910,2939}, {920,4640}, {943,11428}, {946,5173}, {962,5809}, {971,4292}, {974,2778}, {997,37244}, {1006,2646}, {1038,36746}, {1040,36745}, {1058,17642}, {1060,36742}, {1062,36754}, {1125,16193}, {1155,3651}, {1158,37541}, {1159,40266}, {1175,2074}, {1181,40658}, {1260,3811}, {1319,19365}, {1329,14454}, {1425,1876}, {1427,1745}, {1445,10884}, {1454,11502}, {1465,37732}, {1466,18238}, {1482,11432}, {1496,32912}, {1497,3744}, {1593,12262}, {1620,16192}, {1698,26958}, {1750,3339}, {1751,9895}, {1770,15726}, {1785,1834}, {1788,6908}, {1838,6354}, {1844,1901}, {1854,17822}, {1899,5090}, {1903,5746}, {2183,18673}, {2194,30733}, {2262,5802}, {2285,5776}, {2594,8758}, {2771,7687}, {2800,13601}, {2801,4298}, {2836,32246}, {2900,5687}, {3057,3488}, {3333,17625}, {3340,12672}, {3419,5836}, {3485,6846}, {3486,6987}, {3574,32331}, {3576,11425}, {3579,11438}, {3586,5903}, {3600,40269}, {3616,11427}, {3660,12005}, {3671,30329}, {3678,12564}, {3682,25091}, {3697,31434}, {3754,12446}, {3779,26939}, {3827,9969}, {3873,14986}, {3874,11019}, {3876,5703}, {3911,9940}, {3947,15064}, {4259,37179}, {4293,12680}, {4294,5759}, {4304,31793}, {4662,10039}, {4848,31788}, {5044,6675}, {5045,12242}, {5087,5570}, {5396,37565}, {5428,17010}, {5435,6962}, {5439,25525}, {5440,37308}, {5665,38271}, {5693,11529}, {5704,6933}, {5707,37696}, {5715,9581}, {5719,31835}, {5722,5812}, {5766,7671}, {5799,39579}, {5884,6260}, {5887,6913}, {5902,5927}, {6642,9928}, {6765,17658}, {6776,7718}, {6829,17606}, {6834,37566}, {6861,11374}, {6863,10202}, {6886,8232}, {6889,24914}, {6896,11023}, {6907,34339}, {6986,37787}, {6990,17605}, {7367,8886}, {7713,17810}, {8069,11517}, {8226,12047}, {8257,16410}, {9578,18908}, {9777,11396}, {9780,37643}, {9848,30305}, {9856,10392}, {9955,18388}, {10167,15803}, {10246,11426}, {10445,32118}, {10591,17604}, {10914,12625}, {10958,18838}, {11245,12135}, {11399,37538}, {11430,13624}, {11496,30223}, {11500,37550}, {11570,13257}, {11699,12227}, {12514,13615}, {12689,24248}, {13369,37582}, {13373,15325}, {13403,28160}, {14913,29957}, {15178,37505}, {15435,24476}, {15823,37306}, {16200,17622}, {16845,25917}, {17556,24473}, {18165,25516}, {18240,33709}, {18390,18480}, {18398,37692}, {18447,36750}, {18455,37509}, {18732,34371}, {18839,37722}, {20423,34643}, {20612,24982}, {22766,37249}, {24929,31837}, {31397,34790}, {31937,39542}, {33597,37583}, {35242,37487}, {36279,37411}, {37415,39598}, {37709,39779}

X(44547) = midpoint of X(i) and X(j) for these {i,j}: {65,1858}, {72,14054}, {185,1902}, {950,15556}
X(44547) = reflection of X(4292) in X(37544)
X(44547) = X(i)-complementary conjugate of X(j) for these (i,j): (943,34823), (1175,34851), (2982,18589)
X(44547) = X(513)-he conjugate of X(1054)
X(44547) = crossdifference of every pair of points on line X(513)-X(36054)
X(44547) = crosssum of X(i) and X(j) for these {i,j}: {1,3074}, {3,942}, {6,23207}
X(44547) = crosspoint of X(4)and X(943)
X(44547) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1728,405), (1,10398,10396), (1,10399,5728), (1,18397,72), (65,1837,7686), (65,1864,4), (65,1898,1836), (65,12688,4295), (72,405,960), (72,3555,11523), (72,5728,1), (201,14547,37528), (226,10395,5), (405,5729,1728), (920,11507,4640), (942,5777,226), (954,5728,5572), (1071,3149,9942), (1210,18389,942), (1708,10393,3), (1737,13750,3812), (3487,6832,11375), (3678,12564,13405), (3876,11020,5703), (5044,11018,13411), (5693,11529,12709), (7957,14100,4294), (10122,13411,11018), (10398,18412,5728), (10399,18397,1), (30329,31803,3671), (31789,37730,950)

X(44548) = X(6)X(41)∩X(389)X(515)

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2138.

X(44548) lies on these lines: {4,12930}, {5,34831}, {6,41}, {389,515}, {2779,7687}, {9786,15622}

X(44548) = midpoint of X(73) and X(14055)
X(44548) = crosssum of X(3)and X(34831)
X(44548) = {X(73),X(13738)}-harmonic conjugate of X(37836)

X(44549) = X(6)X(36894)∩X(187)X(441)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2144.

X(44549) lies on these lines: {6, 36894}, {187, 441}, {512, 11746}, {1503, 10151}

leftri

Centers and perspectors of ellipses [SCE, line] [SIE, line]: X(44550)-X(44555)

rightri

This preamble is contributed by Peter Moses, August 24-30, 2021.

Abbreviations: SCE = Steiner circumellipse; SIE = Steiner inellipse.

Suppose that L is a line given by u x + v y + w z = 0 (barycentrics). The SCE-inverse of L, denoted by [SCE, L], is given by

u (x^2 - y z) + v (y^2 - z x) + w (z^2 - x y) = 0,

and the SIE-inverse of L, denoted by [SIE, L], is given by

(2u + v + w) (x^2 - y z) + (u + 2v + w) (y^2 - z x) + (u + v + 2w) (z^2 - x y) = 0.

The center of [SCE, L] is -u + 2v + 2w : : = reflection of u : v : w in X(2).

The center of [SIE, L] is 2u + 5v + 5w : : = complement of midpoint of u : v : w and X(2).

The perspector of [SCE, L] is (2 v^2 + u w) (u v + 2 w^2) : : .

The perspector of [SIE, L] is (4 u^2 + 9 v^2 + 4 w^2 + 11 v w + 9 w u + 11 u v)*(4 u^2 + 4 v^2 + 9 w^2 + 11 v w + 11 w u + 9 u v) : :

The appearance of (L, k) in the following list means that X(k) is the center of the ellipse [SCE, L]:

(Brocard axis, 36900), (orthic axis, 1992), (anti-orthic axis, 4664), (Lemoine axis, 7757), (de Longchamps axis, 599), (Gergonne line, 3241), (Fermat line, 9979), (X(1)X(6), 31150), (Koiller line, 31169), (X(1)X(3), 44550), (Soddy line, 44551), (van Aubel line, 44552), (X(1)X(5), 44553), (Napoleon axis, 44554), (Hatzipolakis axis, 44555)

The appearance of (L, k) in the following list means that X(k) is the perspector of the ellipse [SCE, L]:

(orthic axis, 44556), (Lemoine aixs, 44557), (de Longchamps axis, 44558), (Gergonne line, 44559)

The appearance of (L, k) in the following list means that X(k) is the center of the ellipse [SIE, L]:

(orthic axis, 597), (anti-orthic axis, 4755), (de Logchamps axis, 20582), (Gergonne line, 551), (brocard axis, 44560), (X(1)X(3), 44561), (Lemoinne axis, 44562), (Soddy line, 44563), (Fermat line, 44564), (van Aubel line, 44565), (X(1)X(5), 44566), (X(1)X(6), 44567), (Napoleon axis, 44568), (Hatzipolakis axis, 44569), (Koiller line, 44570)

The appearance of (L, k) in the following list means that X(k) is the perspector of the ellipse [SIE, L]:

(orthic axis, 44571), (Gergonne line, 44572)

X(44550) = CENTER OF ELLIPSE [SCE, X(1)X(3)]

Barycentrics    (b - c)*(-2*a^3 + 2*a*b^2 - a*b*c + b^2*c + 2*a*c^2 + b*c^2) : :
X(44550) = 2 X[650] + X[21222], X[693] - 4 X[3960], 4 X[905] - X[4391], 2 X[905] + X[17496], 2 X[3669] + X[4560], 4 X[3669] - X[4801], 2 X[3762] - 5 X[31209], X[3904] + 2 X[4025], X[4391] + 2 X[17496], X[4462] - 4 X[14838], X[4474] - 4 X[25380], 2 X[4560] + X[4801], 2 X[17069] + X[30725], 2 X[31149] - 3 X[44429]

X(44550) lies on these lines: {2, 905}, {514, 1635}, {519, 1734}, {522, 14413}, {525, 1636}, {551, 8714}, {650, 21222}, {693, 3960}, {824, 30656}, {918, 14411}, {1946, 17549}, {2787, 31149}, {3227, 18821}, {3241, 3900}, {3669, 4560}, {3762, 31209}, {3904, 4025}, {3910, 14412}, {4462, 14838}, {4474, 25380}, {4664, 28898}, {4750, 28468}, {4776, 29148}, {4980, 23685}, {6002, 31147}, {13587, 22091}, {14415, 23880}, {16370, 22160}, {17069, 30725}, {20979, 28840}, {29126, 44435}, {29236, 36848}

X(44550) = midpoint of X(2) and X(17496)
X(44550) = reflection of X(i) in X(j) for these {i,j}: {2, 905}, {4391, 2}
X(44550) = anticomplement of X(45664)
X(44550) = crossdifference of every pair of points on line {2177, 34417}
X(44550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {905, 17496, 4391}, {3669, 4560, 4801}

X(44551) = CENTER OF ELLIPSE [SCE, SODDY LINE]

Barycentrics    (b - c)*(-5*a^2 + 2*a*b + 3*b^2 + 2*a*c - 2*b*c + 3*c^2) : :
X(44551) = 5 X[2] - X[25259], X[3239] + 2 X[4025], X[3239] - 4 X[7658], 5 X[3239] - 2 X[25259], 2 X[3676] + X[4765], X[3676] + 2 X[17069], X[3798] + 2 X[21212], X[4025] + 2 X[7658], 5 X[4025] + X[25259], 3 X[4453] + X[31150], 3 X[4750] + X[31147], X[4765] - 4 X[17069], 10 X[7658] - X[25259], X[16892] + 2 X[43061]

X(44551) lies on these lines: {2, 2400}, {514, 1635}, {522, 1638}, {525, 14345}, {599, 9031}, {652, 3928}, {658, 32040}, {2482, 35112}, {2786, 44432}, {3667, 4750}, {3676, 4762}, {3798, 4785}, {3828, 29212}, {4379, 28161}, {4728, 4962}, {4786, 44435}, {16892, 43061}, {20521, 23878}, {21183, 27486}

X(44551) = midpoint of X(i) and X(j) for these {i,j}: {2, 4025}, {4786, 44435}, {21183, 27486}
X(44551) = reflection of X(i) in X(j) for these {i,j}: {2, 7658}, {3239, 2}
X(44551) = anticomplement of X(45334)
X(44551) = crossdifference of every pair of points on line {2177, 41424}
X(44551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3676, 17069, 4765}, {4025, 7658, 3239}

X(44552) = CENTER OF ELLIPSE [SCE, VAN AUBEL LINE]

Barycentrics    (b^2 - c^2)*(-5*a^4 + 2*a^2*b^2 + 3*b^4 + 2*a^2*c^2 - 2*b^2*c^2 + 3*c^4) : :
X(44552) = 3 X[1637] - X[31174], X[2525] - 4 X[14341], X[3265] - 4 X[6587], X[3265] + 2 X[33294], 2 X[6587] + X[33294], 3 X[9979] + X[36900], 3 X[14420] + X[31176]

X(44552) lies on these lines: {2, 2419}, {351, 523}, {525, 1637}, {648, 31510}, {671, 42738}, {1992, 9007}, {2501, 23878}, {2525, 14341}, {2799, 9209}, {3566, 14420}, {14568, 41357}

X(44552) = midpoint of X(2) and X(33294)
X(44552) = reflection of X(i) in X(j) for these {i,j}: {2, 6587}, {3265, 2}
X(44552) = crossdifference of every pair of points on line {574, 26864}
X(44552) = {X(6587),X(33294)}-harmonic conjugate of X(3265)

X(44553) = CENTER OF ELLIPSE [SCE, LINE X(1)X(5)]

Barycentrics    (b - c)*(a^3 - 3*a^2*b - a*b^2 + 3*b^3 - 3*a^2*c + 5*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + 3*c^3) : :
X(44553) = 4 X[676] - 3 X[38314], X[3904] - 4 X[10015]

X(44553) lies on these lines: {2, 3904}, {514, 1635}, {525, 14391}, {664, 32040}, {676, 38314}, {903, 918}, {1638, 41802}, {3241, 6366}, {3679, 23887}, {3910, 14394}, {4391, 36038}, {4707, 21141}, {4762, 43052}, {9521, 34632}, {19293, 23184}, {21129, 28890}, {23884, 30565}, {28468, 31147}

X(44553) = reflection of X(i) in X(j) for these {i,j}: {2, 10015}, {3904, 2}
X(44553) = anticomplement of X(45341)
X(44553) = crossdifference of every pair of points on line {2177, 44109}

X(44554) = CENTER OF ELLIPSE [SCE, NAPOLEON AXIS]

Barycentrics    (b^2 - c^2)*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 5*b^2*c^2 + 3*c^4) : :
X(44554) = 3 X[3268] - 4 X[31174], 3 X[9979] - 2 X[36900], 4 X[12077] - X[41298]

X(44554) lies on these lines: {2, 12077}, {351, 523}, {3268, 31174}, {14446, 23872}, {14447, 23873}

X(44554) = reflection of X(i) in X(j) for these {i,j}: {2, 12077}, {41298, 2}

X(44555) = CENTER OF ELLIPSE [SCE, HATZIPOLAKIS AXIS]

Barycentrics    a^6 - 4*a^4*b^2 + 5*a^2*b^4 - 2*b^6 - 4*a^4*c^2 - 3*a^2*b^2*c^2 + 2*b^4*c^2 + 5*a^2*c^4 + 2*b^2*c^4 - 2*c^6 : :
X(44555) = 5 X[2] - 4 X[11064], X[23] - 4 X[41586], X[23] + 2 X[41724], 2 X[110] - 3 X[37907], 3 X[186] - 4 X[15361], X[323] - 4 X[3580], 5 X[323] - 8 X[11064], X[323] + 2 X[37779], 3 X[323] - 4 X[40112], 4 X[597] - 3 X[22151], 2 X[1992] - 3 X[37784], 5 X[3580] - 2 X[11064], 2 X[3580] + X[37779], 3 X[3580] - X[40112], 4 X[7426] - 3 X[35265], 3 X[8859] - 2 X[14999], 2 X[9143] - 3 X[35265], 4 X[11064] + 5 X[37779], 6 X[11064] - 5 X[40112], 4 X[13857] - 5 X[30745], X[14683] - 4 X[32269], X[14683] - 3 X[37909], 2 X[23061] - 5 X[30745], 2 X[23236] - 5 X[37953], 4 X[30714] - 7 X[37957], 4 X[32225] - 3 X[37907], 4 X[32269] - 3 X[37909], 4 X[35266] - 5 X[37760], 3 X[37779] + 2 X[40112], 2 X[41586] + X[41724]

X(44555) lies on these lines: {2, 6}, {23, 542}, {30, 3448}, {51, 25561}, {52, 7565}, {94, 671}, {110, 32225}, {186, 15361}, {381, 11002}, {399, 44266}, {511, 9140}, {525, 14391}, {530, 11092}, {531, 11078}, {539, 32263}, {547, 7605}, {576, 38397}, {1216, 43836}, {1503, 37901}, {2482, 35296}, {3410, 41588}, {3564, 7426}, {5169, 20423}, {5459, 33529}, {5460, 33530}, {5463, 11146}, {5464, 11145}, {5640, 11178}, {5642, 5965}, {5648, 8262}, {7464, 20126}, {7492, 43273}, {7570, 15019}, {7693, 18358}, {7703, 37517}, {7712, 39899}, {8288, 8586}, {8596, 40853}, {8724, 35298}, {9213, 36255}, {10020, 11271}, {10294, 15362}, {10296, 11564}, {10545, 43150}, {10706, 13754}, {11477, 31857}, {11645, 15107}, {11646, 13192}, {12383, 44265}, {12828, 37962}, {13857, 23061}, {14002, 15069}, {14683, 32269}, {14830, 37183}, {14918, 37765}, {15032, 44262}, {15052, 44275}, {16042, 34507}, {18387, 40909}, {18947, 37943}, {23236, 37953}, {24981, 32267}, {26869, 33884}, {30714, 37957}, {32515, 36194}, {35266, 37760}, {37126, 43573}

X(44555) = midpoint of X(i) and X(j) for these {i,j}: {2, 37779}, {15360, 41724}
X(44555) = reflection of X(i) in X(j) for these {i,j}: {2, 3580}, {23, 15360}, {110, 32225}, {323, 2}, {399, 44266}, {5648, 8262}, {7464, 20126}, {9143, 7426}, {9213, 36255}, {10989, 9140}, {12383, 44265}, {15360, 41586}, {23061, 13857}, {24981, 32267}
X(44555) = anticomplement of X(40112)
X(44555) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9060, 7192}, {34802, 4329}
X(44555) = crossdifference of every pair of points on line {512, 44109}
X(44555) = barycentric product X(3260)*X(39239)
X(44555) = barycentric quotient X(i)/X(j) for these {i,j}: {10294, 10295}, {15362, 381}, {39239, 74}
X(44555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 32225, 37907}, {323, 34545, 22151}, {323, 37784, 1994}, {3580, 6515, 37784}, {3580, 37779, 323}, {6792, 15993, 11580}, {7426, 9143, 35265}, {8115, 8116, 11004}, {37785, 37786, 41626}, {39107, 39108, 18122}, {41586, 41724, 23}

X(44556) = PERSPECTOR OF ELLIPSE [SCE, ORTHIC AXIS]

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

X(44556) lies on these lines: {4, 13479}, {30, 1351}, {125, 36889}, {523, 37643}, {1007, 3260}, {1990, 4232}, {2847, 35906}, {9308, 34285}, {16303, 35260}, {16326, 26869}, {34417, 39453}

X(44556) = isogonal conjugate of X(6090)
X(44556) = isotomic conjugate of X(32817)
X(44556) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6090}, {31, 32817}, {15144, 35200}
X(44556) = trilinear pole of line {1499, 1514}
X(44556) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32817}, {6, 6090}, {1990, 15144}

X(44557) = PERSPECTOR OF ELLIPSE [SCE, LEMOINE AXIS]

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

X(44557) lies on these lines: {110, 9515}, {187, 353}, {385, 30495}, {512, 9465}, {524, 3094}, {598, 3124}, {694, 3972}, {695, 6179}, {729, 34811}, {5640, 43950}, {7031, 40935}, {7937, 41259}, {11057, 20859}, {11654, 18872}, {30229, 35146}, {34154, 39024}

X(44557) = isogonal conjugate of X(3734)
X(44557) = isogonal conjugate of the anticomplement of X(4045)
X(44557) = isogonal conjugate of the complement of X(2549)
X(44557) = X(1)-isoconjugate of X(3734)
X(44557) = trilinear pole of line {351, 9009}
X(44557) = barycentric quotient X(6)/X(3734)

X(44558) = PERSPECTOR OF ELLIPSE [SCE, DE LONGCHAMPS AXIS]

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

X(44558) lies on these lines: {67, 7761}, {183, 9465}, {538, 599}, {625, 18575}, {626, 18375}, {2854, 34227}, {3001, 5094}, {3124, 9462}, {3314, 9464}, {3778, 4361}, {6322, 6325}, {7934, 18023}, {13377, 31173}

X(44558) = isotomic conjugate of X(3972)
X(44558) = isotomic conjugate of the anticomplement of X(7853)
X(44558) = isotomic conjugate of the complement of X(7898)
X(44558) = isotomic conjugate of the isogonal conjugate of X(30495)
X(44558) = X(7853)-cross conjugate of X(2)
X(44558) = X(i)-isoconjugate of X(j) for these (i,j): {31, 3972}, {163, 4108}
X(44558) = cevapoint of X(2) and X(7898)
X(44558) = trilinear pole of line {3906, 9148}
X(44558) = barycentric product X(76)*X(30495)
X(44558) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3972}, {523, 4108}, {30495, 6}

X(44559) = PERSPECTOR OF ELLIPSE [SCE, GERGONNE LINE]

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

X(44559) lies on these lines: {390, 527}, {1323, 5222}, {5328, 29627}, {26007, 36887}, {31189, 31190}

X(44559) = trilinear pole of line {1638, 6006}

X(44560) = CENTER OF ELLIPSE [SIE, BROCARD AXIS]

Barycentrics    (b^2 - c^2)*(5*a^4 - 5*a^2*b^2 - 5*a^2*c^2 + 2*b^2*c^2) : :
X(44560) = 5 X[2] - X[850], 13 X[2] - 5 X[31072], 7 X[2] - 5 X[31277], 7 X[2] + X[31296], 3 X[2] + X[36900], 4 X[2] + X[41300], 3 X[351] + X[31176], 5 X[647] + X[850], 2 X[647] + X[30476], 13 X[647] + 5 X[31072], 3 X[647] + X[31174], 7 X[647] + 5 X[31277], 7 X[647] - X[31296], 3 X[647] - X[36900], 4 X[647] - X[41300], 2 X[850] - 5 X[30476], 13 X[850] - 25 X[31072], 3 X[850] - 5 X[31174], 7 X[850] - 25 X[31277], 7 X[850] + 5 X[31296], 3 X[850] + 5 X[36900], 4 X[850] + 5 X[41300], X[19912] + 3 X[32232], 13 X[30476] - 10 X[31072], 3 X[30476] - 2 X[31174], 7 X[30476] - 10 X[31277], 7 X[30476] + 2 X[31296], 3 X[30476] + 2 X[36900], 2 X[30476] + X[41300], 15 X[31072] - 13 X[31174], 7 X[31072] - 13 X[31277], 35 X[31072] + 13 X[31296], 15 X[31072] + 13 X[36900], 20 X[31072] + 13 X[41300], 7 X[31174] - 15 X[31277], 7 X[31174] + 3 X[31296], 4 X[31174] + 3 X[41300], 5 X[31277] + X[31296], 15 X[31277] + 7 X[36900], 20 X[31277] + 7 X[41300], 3 X[31296] - 7 X[36900], 4 X[31296] - 7 X[41300], 4 X[36900] - 3 X[41300]

X(44560) lies on these lines: {2, 647}, {351, 31176}, {512, 11176}, {523, 44401}, {525, 14345}, {542, 22264}, {549, 30209}, {597, 8675}, {2799, 9209}, {4139, 4763}, {5996, 8644}, {6041, 11163}, {8651, 25423}, {8704, 9189}, {9030, 20582}, {9404, 19722}, {11633, 34290}, {11645, 42654}, {19912, 32232}

X(44560) = midpoint of X(i) and X(j) for these {i,j}: {2, 647}, {5996, 8644}, {31174, 36900}
X(44560) = reflection of X(30476) in X(2)
X(44560) = complement of X(31174)
X(44560) = crossdifference of every pair of points on line {237, 5210}
X(44560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 36900, 31174}, {647, 30476, 41300}, {647, 31174, 36900}, {647, 31277, 31296}

X(44561) = CENTER OF ELLIPSE [SIE, LINE X(1)X(3)]

Barycentrics    (b - c)*(5*a^3 - 5*a*b^2 - 2*a*b*c + 2*b^2*c - 5*a*c^2 + 2*b*c^2) : :
X(44561) = 5 X[2] - X[4391], 7 X[2] + X[17496], 5 X[905] + X[4391], 7 X[905] - X[17496], X[1734] + 3 X[25055], X[3960] + 2 X[31287], 7 X[4391] + 5 X[17496], 3 X[14419] + X[31149]

X(44561) lies on these lines: {2, 905}, {514, 14425}, {525, 14345}, {551, 3900}, {1734, 25055}, {1946, 16371}, {3960, 31287}, {4755, 28898}, {4762, 14838}, {5325, 20318}, {14419, 28475}, {16370, 22091}, {16417, 22160}, {29126, 44432}, {30234, 44429}

X(44561) = midpoint of X(i) and X(j) for these {i,j}: {2, 905}, {30234, 44429}
X(44561) = complement of X(45664)

X(44562) = CENTER OF ELLIPSE [SIE, LEMOINE AXIS]

Barycentrics    5*a^2*b^2 + 5*a^2*c^2 + 2*b^2*c^2 : :
X(44562) = 5 X[2] - X[76], 7 X[2] + X[194], 3 X[2] + X[7757], X[2] - 5 X[7786], 11 X[2] + X[11055], 7 X[2] - X[14711], 17 X[2] - X[20081], 31 X[2] + X[20105], 7 X[2] - 5 X[31239], 13 X[2] - 5 X[31276], 4 X[2] + X[32450], 7 X[5] + 5 X[32523], 5 X[39] + X[76], 7 X[39] - X[194], 2 X[39] + X[3934], X[39] + 2 X[6683], 3 X[39] - X[7757], X[39] + 5 X[7786], 3 X[39] + X[9466], 11 X[39] - X[11055], 7 X[39] + X[14711], 17 X[39] + X[20081], 31 X[39] - X[20105], 7 X[39] + 5 X[31239], 13 X[39] + 5 X[31276], 4 X[39] - X[32450], 7 X[76] + 5 X[194], 2 X[76] - 5 X[3934], X[76] - 10 X[6683], 3 X[76] + 5 X[7757], X[76] - 25 X[7786], 3 X[76] - 5 X[9466], 11 X[76] + 5 X[11055], 7 X[76] - 5 X[14711], 17 X[76] - 5 X[20081], 31 X[76] + 5 X[20105], 7 X[76] - 25 X[31239], 13 X[76] - 25 X[31276], 4 X[76] + 5 X[32450], 2 X[194] + 7 X[3934], X[194] + 14 X[6683], 3 X[194] - 7 X[7757], X[194] + 35 X[7786], 3 X[194] + 7 X[9466], 11 X[194] - 7 X[11055], 17 X[194] + 7 X[20081], 31 X[194] - 7 X[20105], X[194] + 5 X[31239], 13 X[194] + 35 X[31276], 4 X[194] - 7 X[32450], 3 X[262] + X[376], X[376] - 3 X[21163], X[381] + 3 X[11171], X[549] - 3 X[40108], 3 X[598] + X[33264], X[599] + 3 X[13331], 5 X[631] - X[33706]

X(44562) lies on these lines: {2, 39}, {3, 44422}, {5, 32523}, {6, 15482}, {30, 11272}, {37, 40479}, {83, 13586}, {140, 7829}, {183, 41748}, {187, 3329}, {262, 376}, {381, 11171}, {384, 31652}, {511, 549}, {524, 10007}, {543, 2023}, {547, 2782}, {551, 14839}, {574, 1003}, {598, 33264}, {599, 13331}, {620, 3589}, {625, 3815}, {626, 31406}, {631, 33706}, {726, 4755}, {730, 3828}, {732, 20582}, {736, 22110}, {754, 8359}, {1078, 5041}, {1506, 7861}, {1569, 14971}, {1656, 7902}, {1916, 41134}, {2021, 3849}, {2482, 6661}, {2548, 7842}, {2549, 32983}, {3055, 6722}, {3094, 7622}, {3095, 5054}, {3106, 11302}, {3107, 11301}, {3228, 39968}, {3524, 5188}, {3543, 22682}, {3545, 11257}, {3618, 22486}, {3734, 5024}, {3763, 7908}, {3819, 10191}, {3839, 32522}, {3972, 8589}, {5007, 7824}, {5008, 7771}, {5013, 7808}, {5052, 5569}, {5055, 6248}, {5066, 32516}, {5071, 7709}, {5149, 9888}, {5306, 34506}, {5319, 32978}, {5355, 37688}, {5475, 33017}, {5943, 44215}, {5971, 39389}, {5976, 9167}, {6179, 41940}, {6194, 15721}, {6292, 7895}, {6656, 9698}, {6704, 7789}, {7603, 7790}, {7618, 14039}, {7697, 15703}, {7736, 7761}, {7745, 8354}, {7747, 8353}, {7748, 33016}, {7751, 8556}, {7758, 32960}, {7759, 16043}, {7764, 7849}, {7765, 32992}, {7772, 7780}, {7774, 7848}, {7775, 11287}, {7777, 7853}, {7781, 22332}, {7787, 15513}, {7791, 7843}, {7798, 15271}, {7800, 7882}, {7805, 7815}, {7810, 41624}, {7811, 15810}, {7818, 11163}, {7821, 7876}, {7830, 8358}, {7831, 7845}, {7833, 14537}, {7840, 31168}, {7844, 31489}, {7847, 39590}, {7856, 33015}, {7858, 7873}, {7862, 31467}, {7864, 39565}, {7865, 9766}, {7878, 33004}, {7909, 16897}, {7919, 17005}, {7924, 31173}, {8176, 16041}, {8178, 12055}, {8703, 14881}, {9166, 11152}, {9771, 10150}, {9821, 15693}, {10124, 32515}, {11361, 32479}, {12040, 24256}, {12251, 15709}, {12263, 19883}, {12782, 25055}, {13354, 38064}, {14001, 31450}, {14033, 34504}, {14046, 39266}, {14093, 22728}, {14994, 21358}, {15048, 15491}, {15246, 37875}, {15692, 37809}, {15694, 15819}, {15699, 32448}, {15702, 22712}, {15713, 32521}, {16925, 31457}, {21356, 32451}, {22691, 37351}, {22692, 37352}, {22707, 37170}, {22708, 37171}, {25555, 37459}, {31407, 33025}, {31417, 32982}, {31492, 32954}, {32994, 39601}, {33013, 39563}, {33213, 44377}

X(44562) = midpoint of X(i) and X(j) for these {i,j}: {2, 39}, {3, 44422}, {194, 14711}, {262, 21163}, {5066, 32516}, {7745, 8354}, {7747, 8353}, {7753, 8356}, {7757, 9466}, {7810, 41624}, {7811, 41750}, {7833, 14537}, {8359, 9300}, {8703, 14881}, {15819, 32447}
X(44562) = reflection of X(i) in X(j) for these {i,j}: {2, 6683}, {3934, 2}, {7830, 8358}, {40344, 8359}
X(44562) = complement of X(9466)
X(44562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7757, 9466}, {39, 3934, 32450}, {39, 6683, 3934}, {39, 7786, 6683}, {39, 9466, 7757}, {39, 31239, 194}, {574, 7804, 32456}, {574, 11174, 7804}, {3329, 33273, 12150}, {3815, 4045, 625}, {5013, 7808, 7816}, {7764, 8362, 7849}, {7772, 11285, 7780}, {7803, 31455, 7886}, {7815, 9605, 7805}, {7858, 33021, 7873}, {7878, 33004, 35007}, {7919, 17005, 31275}, {8362, 9606, 7764}, {12150, 33273, 187}, {15810, 41750, 7811}

X(44563) = CENTER OF ELLIPSE [SIE, SODDY LINE]

Barycentrics    (b - c)*(-17*a^2 + 14*a*b + 3*b^2 + 14*a*c - 14*b*c + 3*c^2) : :
X(44563) = 5 X[2] - X[3239], 7 X[2] + X[4025], 17 X[2] - X[25259], 7 X[3239] + 5 X[4025], X[3239] + 5 X[7658], 17 X[3239] - 5 X[25259], X[4025] - 7 X[7658], 17 X[4025] + 7 X[25259], 17 X[7658] + X[25259]

X(44563) lies on these lines: {2, 2400}, {514, 14425}, {3667, 26275}, {9031, 20582}

X(44563) = midpoint of X(2) and X(7658)
X(44563) = complement of X(45334)

X(44564) = CENTER OF ELLIPSE [SIE, FERMAT LINE]

Barycentrics    (b^2 - c^2)*(-4*a^4 + 3*a^2*b^2 + b^4 + 3*a^2*c^2 - 4*b^2*c^2 + c^4) : :
Barycentrics    (cot B - cot C) (cot^2 A - 2 (cot A) (cot B + cot C) + 3 cot B cot C) : :
X(44564) = 5 X[2] - X[3268], 3 X[2] + X[9979], X[351] + 3 X[8371], X[351] - 3 X[9189], 5 X[1637] + X[3268], 3 X[1637] - X[9979], 3 X[1637] + X[14417], 3 X[3268] + 5 X[9979], 3 X[3268] - 5 X[14417], 3 X[5466] + X[9131], X[6587] + 2 X[14341], 2 X[6587] + X[30476], 3 X[8371] - X[9134], 3 X[9125] - X[9131], X[9134] + 3 X[9189], 3 X[10278] + X[14610], 3 X[11176] - X[14610], X[12075] + 2 X[44451], 4 X[14341] - X[30476], 5 X[31277] + X[33294]

X(44564) lies on these lines: {2, 1637}, {3, 9529}, {25, 25644}, {125, 14697}, {351, 2793}, {381, 44202}, {402, 5972}, {523, 44401}, {525, 3239}, {542, 42736}, {549, 44204}, {597, 9003}, {690, 5461}, {804, 10189}, {1196, 2507}, {1995, 42659}, {2492, 6719}, {2848, 35282}, {4763, 6089}, {5466, 9125}, {5943, 39469}, {9209, 23878}, {10278, 11176}, {12075, 44451}, {14273, 14977}, {14401, 15595}, {14582, 18883}, {23301, 32478}, {31277, 33294}, {33752, 42665}

X(44564) = midpoint of X(i) and X(j) for these {i,j}: {2, 1637}, {3, 44203}, {125, 14697}, {351, 9134}, {381, 44202}, {549, 44204}, {2492, 18310}, {5466, 9125}, {8371, 9189}, {9979, 14417}, {10278, 11176}, {14273, 14977}
X(44564) = complement of X(14417)
X(44564) = center of circle {{X(2),X(107),X(111),X(125)}}
X(44564) = X(i)-complementary conjugate of X(j) for these (i,j): {19, 5099}, {111, 34846}, {112, 16597}, {162, 126}, {691, 18589}, {823, 34517}, {897, 127}, {923, 15526}, {1973, 23992}, {8753, 8287}, {14908, 16595}, {17983, 21253}, {24019, 5181}, {32676, 2482}, {32729, 1214}, {32740, 16573}, {36045, 16051}, {36060, 122}, {36085, 1368}, {36115, 30739}, {36128, 125}, {36142, 3}
X(44564) = crossdifference of every pair of points on line {154, 5191}
X(44564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9979, 14417}, {351, 8371, 9134}, {1637, 14417, 9979}, {6587, 14341, 30476}, {9134, 9189, 351}

X(44565) = CENTER OF ELLIPSE [SIE, VAN AUBEL LINE]

Barycentrics    (b^2 - c^2)*(-17*a^4 + 14*a^2*b^2 + 3*b^4 + 14*a^2*c^2 - 14*b^2*c^2 + 3*c^4) : :
X(44565) = 5 X[2] - X[3265], 7 X[2] + X[33294], X[3265] + 5 X[6587], 7 X[3265] + 5 X[33294], 7 X[6587] - X[33294], 3 X[9209] + X[31174]

X(44565) lies on these lines: {2, 2419}, {523, 44401}, {597, 9007}, {9209, 31174}, {14341, 23878}, {30442, 37672}

X(44565) = midpoint of X(2) and X(6587)
X(44565) = crossdifference of every pair of points on line {5210, 42671}

X(44566) = CENTER OF ELLIPSE [SIE, LINE X(1)X(5)]

Barycentrics    (b - c)*(-2*a^3 - 3*a^2*b + 2*a*b^2 + 3*b^3 - 3*a^2*c + 8*a*b*c - 5*b^2*c + 2*a*c^2 - 5*b*c^2 + 3*c^3) : :
X(44566) = 5 X[2] - X[3904], X[3904] + 5 X[10015], X[24473] - 3 X[30691]

X(44566) lies on these lines: {2, 3904}, {514, 14425}, {519, 676}, {525, 1637}, {551, 6366}, {918, 21198}, {3828, 23887}, {4049, 6084}, {4528, 4745}, {4904, 21950}, {11125, 39472}, {24473, 30691}, {26275, 28294}, {28481, 31149}

X(44566) = midpoint of X(2) and X(10015)
X(44566) = reflection of X(4528) in X(4745)
X(44566) = complement of X(45341)

X(44567) = CENTER OF ELLIPSE [SIE, LINE X(1)X(6)]

Barycentrics    (b - c)*(5*a^2 - 5*a*b - 5*a*c + 2*b*c) : :
X(44567) = 5 X[2] - X[693], 7 X[2] + X[17494], 11 X[2] + 5 X[26777], 17 X[2] - X[26824], 13 X[2] - 5 X[26985], X[2] + 7 X[27115], 3 X[2] + X[31150], X[2] - 5 X[31209], 7 X[2] - 5 X[31250], 5 X[650] + X[693], 2 X[650] + X[4885], 7 X[650] - X[17494], 11 X[650] - 5 X[26777], 17 X[650] + X[26824], 13 X[650] + 5 X[26985], X[650] - 7 X[27115], 3 X[650] - X[31150], X[650] + 5 X[31209], 7 X[650] + 5 X[31250], X[650] + 2 X[31287], 2 X[693] - 5 X[4885], 7 X[693] + 5 X[17494], 11 X[693] + 25 X[26777], 17 X[693] - 5 X[26824], 13 X[693] - 25 X[26985], X[693] + 35 X[27115], 3 X[693] + 5 X[31150], X[693] - 25 X[31209], 7 X[693] - 25 X[31250], X[693] - 10 X[31287], 3 X[1635] + X[31147], 2 X[2516] + X[3835], 2 X[4394] + X[4940], X[4394] + 2 X[25666], 2 X[4521] + X[17069], 7 X[4885] + 2 X[17494], 11 X[4885] + 10 X[26777], 17 X[4885] - 2 X[26824], 13 X[4885] - 10 X[26985], X[4885] + 14 X[27115], 3 X[4885] + 2 X[31150], X[4885] - 10 X[31209], 7 X[4885] - 10 X[31250], X[4885] - 4 X[31287], 3 X[4893] + X[31148], X[4940] - 4 X[25666], 2 X[14838] + X[20317], 11 X[17494] - 35 X[26777], 17 X[17494] + 7 X[26824], 13 X[17494] + 35 X[26985], X[17494] - 49 X[27115], 3 X[17494] - 7 X[31150], X[17494] + 35 X[31209], X[17494] + 5 X[31250], X[17494] + 14 X[31287], 85 X[26777] + 11 X[26824], 13 X[26777] + 11 X[26985], 5 X[26777] - 77 X[27115], 15 X[26777] - 11 X[31150], X[26777] + 11 X[31209], 7 X[26777] + 11 X[31250], 5 X[26777] + 22 X[31287], 13 X[26824] - 85 X[26985], X[26824] + 119 X[27115], 3 X[26824] + 17 X[31150], X[26824] - 85 X[31209], 7 X[26824] - 85 X[31250], X[26824] - 34 X[31287], 5 X[26985] + 91 X[27115], 15 X[26985] + 13 X[31150], X[26985] - 13 X[31209], 7 X[26985] - 13 X[31250], 5 X[26985] - 26 X[31287], 21 X[27115] - X[31150], 7 X[27115] + 5 X[31209], 49 X[27115] + 5 X[31250], 7 X[27115] + 2 X[31287], X[31150] + 15 X[31209], 7 X[31150] + 15 X[31250], X[31150] + 6 X[31287], 7 X[31207] - X[43067], 7 X[31209] - X[31250], 5 X[31209] - 2 X[31287], 5 X[31250] - 14 X[31287]

X(44567) lies on these lines: {2, 650}, {30, 8142}, {513, 4763}, {514, 14425}, {523, 44401}, {528, 10006}, {549, 8760}, {551, 14077}, {597, 9001}, {1635, 6008}, {1639, 28898}, {2516, 3835}, {3035, 40540}, {3742, 9443}, {3828, 29066}, {3829, 15283}, {4394, 4785}, {4421, 8641}, {4521, 17069}, {4755, 4777}, {4893, 31148}, {4944, 27486}, {4948, 7662}, {4995, 11934}, {6084, 44432}, {9015, 20582}, {10196, 30520}, {10199, 35100}, {14727, 30610}, {14838, 20317}, {28840, 31286}, {31207, 43067}

X(44567) = midpoint of X(i) and X(j) for these {i,j}: {2, 650}, {4944, 27486}, {4948, 7662}
X(44567) = reflection of X(i) in X(j) for these {i,j}: {2, 31287}, {4885, 2}
X(44567) = complement of X(45320)
X(44567) = crossdifference of every pair of points on line {2223, 5210}
X(44567) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {650, 31209, 31287}, {650, 31250, 17494}, {650, 31287, 4885}, {4394, 25666, 4940}, {27115, 31209, 650}

X(44568) = CENTER OF ELLIPSE [SIE, NAPOLEON LINE]

Barycentrics    (b^2 - c^2)*(-2*a^4 - a^2*b^2 + 3*b^4 - a^2*c^2 - 8*b^2*c^2 + 3*c^4) : :
X(44568) = 5 X[2] - X[41298], 3 X[1637] - X[36900], 3 X[9134] - X[31176], 5 X[12077] + X[41298], 3 X[30474] - 5 X[31174]

X(44568) lies on these lines: {2, 12077}, {523, 44401}, {1637, 36900}, {2501, 23878}, {2799, 30474}, {9134, 31176}

X(44568) = midpoint of X(2) and X(12077)

X(44569) = CENTER OF ELLIPSE [SIE, HATZIPOLAKIS AXIS]

Barycentrics    2*a^6 + a^4*b^2 - 8*a^2*b^4 + 5*b^6 + a^4*c^2 + 12*a^2*b^2*c^2 - 5*b^4*c^2 - 8*a^2*c^4 - 5*b^2*c^4 + 5*c^6 : :
X(44569) = 5 X[2] - X[323], 7 X[2] + X[37779], 2 X[125] + X[32269], X[323] + 5 X[3580], 2 X[323] - 5 X[11064], 7 X[323] + 5 X[37779], 3 X[323] - 5 X[40112], 3 X[403] - X[10706], X[3292] - 4 X[37911], X[3448] + 2 X[15448], X[3448] + 3 X[37907], 2 X[3580] + X[11064], 7 X[3580] - X[37779], 3 X[3580] + X[40112], 2 X[5159] + X[41586], 7 X[11064] + 2 X[37779], 3 X[11064] - 2 X[40112], X[11160] + 3 X[37784], X[11799] - 3 X[15362], 3 X[15362] + X[20126], 2 X[15448] - 3 X[37907], X[16619] + 2 X[20379], 3 X[21356] + X[41617], 3 X[37779] + 7 X[40112]

X(44569) lies on these lines: {2, 6}, {23, 5621}, {30, 125}, {265, 44265}, {297, 671}, {373, 547}, {381, 20192}, {403, 10706}, {468, 542}, {511, 12099}, {525, 1637}, {541, 1514}, {549, 39242}, {858, 15360}, {1503, 7426}, {2482, 39021}, {3066, 3545}, {3292, 37911}, {3448, 15448}, {3543, 10606}, {3564, 5642}, {3830, 21970}, {3845, 34417}, {5094, 20423}, {5159, 13857}, {5943, 25565}, {6146, 34351}, {6791, 43291}, {7493, 43273}, {7542, 43573}, {7552, 26879}, {7565, 11745}, {8352, 41254}, {10264, 44266}, {10293, 11799}, {10989, 29181}, {11178, 35283}, {11179, 13394}, {11180, 35259}, {11284, 32599}, {11539, 22112}, {11579, 40114}, {11645, 32223}, {12310, 37955}, {13169, 32220}, {13754, 16227}, {14856, 16183}, {15471, 41720}, {16619, 20379}, {16657, 44218}, {18281, 41587}, {18390, 44285}, {21243, 25561}, {23294, 34613}, {23332, 31133}, {23410, 34826}, {35018, 44300}, {35303, 40710}, {35304, 40709}, {40920, 44456}

X(44569) = midpoint of X(i) and X(j) for these {i,j}: {2, 3580}, {125, 32225}, {265, 44265}, {858, 15360}, {7426, 9140}, {10264, 44266}, {11799, 20126}, {13169, 32220}, {13857, 41586}
X(44569) = reflection of X(i) in X(j) for these {i,j}: {11064, 2}, {13857, 5159}, {32269, 32225}, {35266, 468}, {37904, 32223}, {41720, 15471}
X(44569) = complement of X(40112)
X(44569) = isotomic conjugate of the polar conjugate of X(37984)
X(44569) = X(i)-complementary conjugate of X(j) for these (i,j): {9060, 4369}, {34802, 18589}
X(44569) = crosssum of X(32) and X(40114)
X(44569) = crossdifference of every pair of points on line {512, 26864}
X(44569) = barycentric product X(69)*X(37984)
X(44569) = barycentric quotient X(37984)/X(4)
X(44569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15362, 20126, 11799}, {37638, 37643, 37648}

X(44570) = CENTER OF ELLIPSE [SIE, KOILLER LINE]

Barycentrics    5*a^3*b - 10*a^2*b^2 + 5*a*b^3 + 5*a^3*c - 2*a^2*b*c - 5*a*b^2*c + 2*b^3*c - 10*a^2*c^2 - 5*a*b*c^2 - 4*b^2*c^2 + 5*a*c^3 + 2*b*c^3 : :
X(44570) = 5 X[2] - X[85], 7 X[2] + X[3177], 17 X[2] - X[20089], 3 X[2] + X[31169], X[2] - 5 X[31269], X[85] + 5 X[1212], 7 X[85] + 5 X[3177], 2 X[85] - 5 X[6706], 17 X[85] - 5 X[20089], 3 X[85] + 5 X[31169], X[85] - 25 X[31269], 7 X[1212] - X[3177], 2 X[1212] + X[6706], 17 X[1212] + X[20089], 3 X[1212] - X[31169], X[1212] + 5 X[31269], 2 X[3177] + 7 X[6706], 17 X[3177] + 7 X[20089], 3 X[3177] - 7 X[31169], X[3177] + 35 X[31269], 17 X[6706] - 2 X[20089], 3 X[6706] + 2 X[31169], X[6706] - 10 X[31269], 3 X[20089] + 17 X[31169], X[20089] - 85 X[31269], X[31169] + 15 X[31269]

X(44570) lies on these lines: {2, 85}, {518, 551}, {527, 10012}, {3739, 24036}, {3828, 28850}, {6666, 17044}, {16814, 24203}, {34625, 41313}

X(44570) = midpoint of X(2) and X(1212)
X(44570) = reflection of X(6706) in X(2)

X(44571) = PERSPECTOR OF ELLIPSE [SIE, KOILLER LINE]

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

X(44571) lies on these lines: {597, 754}, {3329, 31124}, {7792, 26235}

X(44571) = trilinear pole of line {12073, 14420}

X(44572) = PERSPECTOR OF ELLIPSE [SIE, GERGONNE LINE]

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

X(44572) lies on these lines: {524, 551}, {4031, 7181}, {6629, 26860}, {14210, 17023}, {16826, 27081}

X(44572) = trilinear pole of line {4750, 28209}

X(44573) = X(3)X(74)∩X(20)X(1986)

Barycentrics    a^2*(a^12*(b^2+c^2)-4*a^10*(b^4-b^2*c^2+c^4)-(b^2-c^2)^4*(b^6+4*b^4*c^2+4*b^2*c^4+c^6)+a^8*(5*b^6-8*b^4*c^2-8*b^2*c^4+5*c^6)-2*a^6*(5*b^6*c^2-18*b^4*c^4+5*b^2*c^6)+2*a^2*(b^2-c^2)^2*(2*b^8-b^6*c^2-8*b^4*c^4-b^2*c^6+2*c^8)-a^4*(5*b^10-23*b^8*c^2+22*b^6*c^4+22*b^4*c^6-23*b^2*c^8+5*c^10)) : :
Barycentrics    c^2*SA*(108 R^4 - 12 R^2 SA + SB SC + (-34 R^2 + 3 SA) SW + 2 SW^2) : :
X(44573) = 3*X(3)-X(7723),3*X(3)-2*X(13416),5*X(3)-X(22584),3*X(74)+X(15102),X(110)+3*X(15072),3*X(376)+X(7722),X(382)-3*X(16222),2*X(550)+X(13148),3*X(974)-2*X(11806),5*X(3522)-X(12219),3*X(5642)+X(17856),X(6241)+3*X(15035),2*X(6723)-3*X(16836),2*X(7723)-3*X(12358),5*X(7723)-3*X(22584),3*X(9730)-2*X(11746),3*X(9730)-X(12295),2*X(10113)-3*X(12099),5*X(10574)-X(10733),X(10721)-3*X(12824),X(11381)-3*X(36518),3*X(11459)-7*X(15036),X(11562)+3*X(14855),X(11806)-3*X(40647),3*X(12041)-X(15101),X(12111)-5*X(15051),X(12162)-3*X(38793),X(12270)+3*X(15055),3*X(12358)-4*X(13416),5*X(12358)-2*X(22584),X(12825)-3*X(15035),X(13202)-3*X(16223),10*X(13416)-3*X(22584),2*X(14641)+X(16105),3*X(14855)-X(16111),5*X(15021)-X(15100),5*X(15059)-9*X(20791),3*X(15072)-X(17854),3*X(17853)+X(24981),X(18439)-5*X(38794),X(21650)-3*X(38727),X(34783)+3*X(38723)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2147.

X(44573) lies on these lines: {2,12292}, {3,74}, {4,9826}, {5,12133}, {20,1986}, {30,1112}, {113,2883}, {125,15760}, {143,18565}, {146,6643}, {185,16163}, {265,15740}, {376,7722}, {382,16222}, {542,17855}, {550,13148}, {974,6146}, {1038,7727}, {1040,19470}, {1060,3024}, {1062,3028}, {1181,5504}, {1204,22109}, {1514,10297}, {1539,18404}, {2072,32111}, {2777,12605}, {2781,9967}, {2854,37511}, {3522,12219}, {3548,5656}, {3549,15061}, {5642,17856}, {5972,6000}, {6639,34128}, {6699,7542}, {6723,16836}, {6759,20771}, {6776,12121}, {6823,10264}, {7553,15473}, {7687,9729}, {7728,13203}, {9730,11746}, {10024,20304}, {10111,18914}, {10113,12099}, {10254,15088}, {10255,32137}, {10272,16196}, {10574,10733}, {10628,12363}, {10721,12824}, {10984,32607}, {10996,12317}, {11381,36518}, {11413,15463}, {11513,35826}, {11514,35827}, {11557,14641}, {11561,18563}, {11598,34116}, {11709,24301}, {12022,12236}, {12084,12228}, {12085,15472}, {12140,31833}, {12162,23328}, {12227,13346}, {12302,13198}, {12893,16165}, {13202,16223}, {13445,27866}, {13623,18125}, {13754,38726}, {15059,20791}, {15151,16003}, {15462,19149}, {17853,24981}, {18439,38794}, {18947,35513}, {19126,32305}, {19504,21312}, {20417,34002}, {20773,37814}, {21650,38727}, {27082,34783}, {31829,32423}

X(44573) = midpoint of X(i) and X(j) for these {i,j}: {20,1986}, {110,17854}, {113,10575}, {185,16163}, {1511,13491}, {6241,12825}, {11557,14641}, {11562,16111}
X(44573) = reflection of X(i) in X(j) for these (i,j): (4,9826), (265,16270), (974,40647), (1112,14708), (7553,15473), (7687,9729), (7723,13416), (10111,18914), (12133,5), (12140,31833), (12236,13630), (12295,11746), (12358,3), (15738,6699), (16003,15151), (16105,11557)
X(44573) = circumperp conjugate of X(12412)
X(44573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,7723,13416), (110,15072,17854), (6241,15035,12825), (7723,13416,12358), (9730,12295,11746), (11456,15072,13491), (11562,14855,16111)

X(44574) = X(2)X(10354)∩X(111)X(7610)

Barycentrics    4*a^12 - 17*a^10*b^2 - 16*a^8*b^4 + 24*a^6*b^6 + 10*a^4*b^8 - 7*a^2*b^10 + 2*b^12 - 17*a^10*c^2 + 106*a^8*b^2*c^2 - 15*a^6*b^4*c^2 - 86*a^4*b^6*c^2 + 40*a^2*b^8*c^2 - 12*b^10*c^2 - 16*a^8*c^4 - 15*a^6*b^2*c^4 + 42*a^4*b^4*c^4 - 25*a^2*b^6*c^4 + 6*b^8*c^4 + 24*a^6*c^6 - 86*a^4*b^2*c^6 - 25*a^2*b^4*c^6 + 40*b^6*c^6 + 10*a^4*c^8 + 40*a^2*b^2*c^8 + 6*b^4*c^8 - 7*a^2*c^10 - 12*b^2*c^10 + 2*c^12 : :
X(44574) = X[111] - 3 X[9829], 3 X[6031] + X[14360], 2 X[6719] - 3 X[10163], X[34792] - 3 X[38716]

See Antreas Hatzipolakis and Peter Moses, euclid 2152.

X(44574) lies on these lines: {2, 10354}, {111, 7610}, {126, 3849}, {376, 6031}, {1296, 12505}, {2854, 32311}, {5512, 31606}, {6719, 10163}, {7664, 9123}, {10355, 40727}, {12506, 40556}, {23699, 31729}, {31744, 33962}, {34792, 38716}

X(44574) = midpoint of X(1296) and X(12505)
X(44574) = reflection of X(i) in X(j) for these {i,j}: {5512, 31606}, {12506, 40556}

leftri

E-inverses of points on the Euler line, where E is a permutation ellipse: X(44575)-X(44580) and X(44549)-X(44653)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, September 1, 2021.

Three specific permutation ellipses are described in the preamble just before X(41133), as follows:

(1) The trisector ellipse passes through the points 0:1:2, 0:2:1, 1:0:2, 2:0:1, 1:2:0, 2:1:0. These six points trisect the sides of ABC. The ellipse is given by 2(x^2 + y^2 + z^2) - 5(y z + z x + x y) = 0.

(2) The unique self-dual permutation ellipse is given by x^2 + y^2 + z^2 - 4 (y z + z x + x y) = 0.

(3) The Steiner midway ellipse (SME), is, loosely speaking, the ellipse midway between the Steiner inellipse (SIE) and the Steiner circumellipse (SCE). That is, for each P on SCE, let P' be the intersection of the ray GP with SIE, and let P'' be the midpoint of PP'. Then SME is the set of all such midpoints. SME is given by 7(x^2 + y^2 + z^2) - 34(y z + z x + x y) = 0.

In general, the inverse of a point P = p : q : r in an ellipse j (x^2 + y^2 + z^2) - k (y z + z x + x y) = 0 is the point k*p^2 - j*p*q + j*q^2 - j*p*r - k*q*r + j*r^2 : : .

The center of every permutation ellipse E is X(2), so that the E-inverse of each point P on the Euler line is also on the Euler line. If P = X(2) + t*X(3) and E is give by h*(x^2 + y^2 + z^2) - k (y z + z x + x y) = 0, then

(E-inverse of P) = (4 (h - k) S^4 + 3 t ((2 h - k) S^4 - (2 h + k) SA SB SC SW)) X(2) - 4 (h - k) (1 + t) S^4 X(3).

The appearance of {i,j} in the following list means that X(j) = trisector-ellipse-inverse of X(i):

{3,44575}, {4,44576}, {5,44577}, {297,3545}, {376,44578}, {381,44579}, {401,5054}, {441,10304}, {448,15671}, {3524,40884}, {3839,44216}, {5055,40885}, {15699,40853}, {37907,40856}

The appearance of {i,j} in the following list means that X(j) = Steiner-midway-ellipse-inverse of X(i):

{297,3860}, {401,44580}, {441,15690}, {12101,44216}, {15685,44335}, {15759,40884}, {19710,44346}, {33699,44334}

The appearance of {i,j} in the following list means that X(j) = self-dual-ellipse-inverse of X(i):

{3,40884}, {4,44216}, {5,40885}, {20,44346}, {23, 44649}, {25, 44650}, {140, 44651}, {237, 44652}, {297,381}, {376,441}, {401,549}, {402, 44653}, {448,15670}, {547,40853}, {2454,2479}, {2455,2480}, {3534,44578}, {3543,44334}, {3830,44576}, {3845,44579}, {5066,44577}, {6660,6661}, {7426,40856}, {7924,21536}, {8703,44575}, {10684,44215}, {15013,44210}, {15014,44212}, {15158,44348}, {15159,44349}, {15677,44336}, {15683,44335}, {19686,44347}, {37901,44338}, {44211,44328}, {44213,44329}

The appearance of {i,j} in the following list means that X(j) = Steiner-circumellipse-inverse of X(i):

{3,401}, {4,297}, {5,40853}, {20,441}, {21,448}, {22,15013}, {23,40856}, {24,44328}, {25,15014}, {26,44329}, {27,447}, {28,44330}, {29,44331}, {237,10684}, {376,40884}, {381,40885}, {384,6660}, {427,40889}, {449,452}, {458,35474}, {468,40890}, {472,11093}, {473,11094}, {858,35923}, {1113,44332}, {1114,44333}, {2479,2479}, {2480,2480}, {3146,44334}, {3534,44575}, {3543,44216}, {3830,44579}, {3845,44577}, {4235,7473}, {4618,11355}, {5059,44335}, {6655,21536}, {6658,44347}, {7500,44340}, {8613,15781}, {11001,44578}, {14953,37045}, {15680,44336}, {15682,44576}, {15683,44346}, {20062,44337}, {20063,44338}, {31292,44342}, {31293,44343}, {31294,44344}, {31304,44339}, {31305,44341}, {37174,44228}, {37188,44252}

The appearance of {i,j} in the following list means that X(j) = Steiner-incircle-ellipse-inverse of X(i):

{3,441}, {4,44334}, {5,297}, {20,44335}, {21,44336}, {22,44337}, {23,44338}, {24,44339}, {25,44340}, {26,44341}, {27,44342}, {28,44343}, {29,44344}, {140,401}, {237,44345}, {376,44346}, {381,44216}, {384,44347}, {447,6678}, {448,6675}, {449,11108}, {468,40856}, {547,40885}, {549,40884}, {1113,44348}, {1114,44349}, {1375,37045}, {2454,2454}, {2455,2455}, {3628,40853}, {3845,44576}, {5066,44579}, {5159,35923}, {6656,21536}, {6660,7819}, {6676,15013}, {6677,15014}, {8703,44578}, {10020,44329}, {10109,44577}, {12100,44575}, {16238,44328}, {37911,40890}

X(44575) = TRISECTOR-ELLIPSE-INVERSE OF X(3)

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

X(44575) lies on these lines: {2, 3}, {385, 23967}, {1494, 3284}, {2966, 7840}, {5641, 7925}, {9410, 44436}, {14919, 31621}, {22110, 39359}

X(44576) = TRISECTOR-ELLIPSE-INVERSE OF X(4)

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

X(44576) lies on these lines: {2, 3}, {230, 5641}, {393, 36889}, {1494, 1990}, {2966, 44401}, {3284, 40477}, {7809, 11064}, {7884, 37648}, {14568, 44569}, {16080, 31621}, {22329, 35088}, {39358, 40996}

X(44577) = TRISECTOR-ELLIPSE-INVERSE OF X(5)

Barycentrics    a^8 - 7*a^6*b^2 + 6*a^4*b^4 + 5*a^2*b^6 - 5*b^8 - 7*a^6*c^2 + a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 11*b^6*c^2 + 6*a^4*c^4 - 5*a^2*b^2*c^4 - 12*b^4*c^4 + 5*a^2*c^6 + 11*b^2*c^6 - 5*c^8 : :

X(44577) lies on these lines: {2, 3}, {340, 18487}, {5641, 7779}, {9410, 14918}, {36435, 37766}, {36889, 40896}, {39359, 44367}

X(44578) = TRISECTOR-ELLIPSE-INVERSE OF X(376)

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

X(44578) lies on these lines: {2, 3}, {69, 36427}, {325, 23967}, {2966, 22110}, {5641, 44377}, {7799, 11064}, {9308, 36889}, {18487, 40477}

X(44579) = TRISECTOR-ELLIPSE-INVERSE OF X(381)

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

X(44579) lies on these lines: {2, 3}, {264, 36430}, {340, 3163}, {385, 5641}, {1494, 18487}, {1990, 39358}, {3580, 19570}, {7840, 35088}, {9410, 16080}, {11078, 41995}, {11092, 41996}, {14920, 36435}, {22329, 39359}, {39563, 41254}, {40506, 43768}

X(44580) = STEINER-MIDWAY-ELLIPSE-INVERSE OF X(401)

Barycentrics    34*a^4 - 41*a^2*b^2 + 7*b^4 - 41*a^2*c^2 - 14*b^2*c^2 + 7*c^4 : :
X(44580) = 7*X[2] + 9*X[3]

X(44580) lies on these lines: {2, 3}, {15, 42419}, {16, 42420}, {298, 33609}, {299, 33608}, {511, 41153}, {517, 41150}, {590, 42524}, {615, 42525}, {2782, 41151}, {3564, 41152}, {4745, 13624}, {4995, 37602}, {5351, 43107}, {5352, 43100}, {9541, 42640}, {10645, 43108}, {10646, 43109}, {11542, 42792}, {11543, 42791}, {11694, 38727}, {14711, 32516}, {15520, 21167}, {18487, 36422}, {23698, 41148}, {32787, 41962}, {32788, 41961}, {33614, 33618}, {33615, 33619}, {34380, 41149}, {41100, 42496}, {41101, 42497}, {41112, 42627}, {41113, 42628}, {41121, 42123}, {41122, 42122}, {41977, 42504}, {41978, 42505}, {42085, 43247}, {42086, 43246}, {42090, 43644}, {42091, 43649}, {42121, 42511}, {42124, 42510}, {42415, 43001}, {42416, 43000}, {42506, 43775}, {42507, 43776}, {42532, 42945}, {42533, 42944}, {42631, 43416}, {42632, 43417}, {42888, 43101}, {42889, 43104}, {42952, 42955}, {42953, 42954}, {43002, 43404}, {43003, 43403}, {43016, 43310}, {43017, 43311}, {43542, 43640}, {43543, 43639}

X(44581) = X(2)X(32638)∩X(4)X(195)

Barycentrics    -2 a^22+17 a^20 b^2-64 a^18 b^4+141 a^16 b^6-204 a^14 b^8+210 a^12 b^10-168 a^10 b^12+114 a^8 b^14-66 a^6 b^16+29 a^4 b^18-8 a^2 b^20+b^22+17 a^20 c^2-102 a^18 b^2 c^2+252 a^16 b^4 c^2-323 a^14 b^6 c^2+207 a^12 b^8 c^2-3 a^10 b^10 c^2-143 a^8 b^12 c^2+183 a^6 b^14 c^2-132 a^4 b^16 c^2+53 a^2 b^18 c^2-9 b^20 c^2-64 a^18 c^4+252 a^16 b^2 c^4-368 a^14 b^4 c^4+234 a^12 b^6 c^4-60 a^10 b^8 c^4+59 a^8 b^10 c^4-172 a^6 b^12 c^4+228 a^4 b^14 c^4-144 a^2 b^16 c^4+35 b^18 c^4+141 a^16 c^6-323 a^14 b^2 c^6+234 a^12 b^4 c^6-60 a^10 b^6 c^6-3 a^8 b^8 c^6+64 a^6 b^10 c^6-182 a^4 b^12 c^6+204 a^2 b^14 c^6-75 b^16 c^6-204 a^14 c^8+207 a^12 b^2 c^8-60 a^10 b^4 c^8-3 a^8 b^6 c^8-18 a^6 b^8 c^8+57 a^4 b^10 c^8-168 a^2 b^12 c^8+90 b^14 c^8+210 a^12 c^10-3 a^10 b^2 c^10+59 a^8 b^4 c^10+64 a^6 b^6 c^10+57 a^4 b^8 c^10+126 a^2 b^10 c^10-42 b^12 c^10-168 a^10 c^12-143 a^8 b^2 c^12-172 a^6 b^4 c^12-182 a^4 b^6 c^12-168 a^2 b^8 c^12-42 b^10 c^12+114 a^8 c^14+183 a^6 b^2 c^14+228 a^4 b^4 c^14+204 a^2 b^6 c^14+90 b^8 c^14-66 a^6 c^16-132 a^4 b^2 c^16-144 a^2 b^4 c^16-75 b^6 c^16+29 a^4 c^18+53 a^2 b^2 c^18+35 b^4 c^18-8 a^2 c^20-9 b^2 c^20+c^22 : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2168.

X(44581) lies on these lines: {2,32638}, {4,195}, {184,31675}, {578,12026}, {2937,6343}, {6243,13505}, {8494,13512}

leftri

Kenmotu-centers triangles: X(44582)-X(44648)

rightri

This preamble and centers X(44582)-X(44648) were contributed by César Eliud Lozada, September 1, 2021.

Let A', B', C' be the centers of the inner-Kenmotu squares, as showed in MathWorld's Kenmotu Point. Triangle A'B'C' is named here the 1st Kenmotu-centers triangle of ABC. As there exists an outer version of these squares with centers A", B", C", the triangle A"B"C" is refered here as the 2nd Kenmotu-centers triangle of ABC.

Barycentric coordinates of A' and A" are:

A' = a^2+2*S : b^2 : c^2

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

Both triangles are homothetic to ABC with homothetic center X(6). A'B'C' is the pedal triangle of X(371)-of-ABC with respect to the 1st Kenmotu-diagonals triangle and A"B"C" is the pedal triangle of X(372)-of-ABC with respect to the 2nd Kenmotu-diagonals triangle.

In the following list, (T, i, j) means that pairs of triangles (T, A'B'C'), (T, A"B"C") are perspective with perspectors X(i) and X(j), respectively:

(*ABC, 6, 6), (*ABC-X3 reflections, 1151, 1152), (*anti-Aquila, 7968, 7969), (*anti-Ara, 5412, 5413), (*anti-1st Auriga, 44582, 44583), (*anti-2nd Auriga, 44584, 44585), (*5th anti-Brocard, 44586, 44587), (*2nd anti-circumperp-tangential, 2067, 6502), (anti-Conway, 6, 6), (2nd anti-Conway, 6, 6), (*anti-Ehrmann-mid, 13665, 13785), (*anti-Euler, 1587, 1588), (2nd anti-extouch, 44588, 44589), (*anti-inner-Grebe, 6, 6), (*anti-outer-Grebe, 6, 6), (anti-Honsberger, 6, 6), (*anti-1st Kenmotu-free-vertices, 5062, 39), (*anti-2nd Kenmotu-free-vertices, 39, 5058), (*anti-Mandart-incircle, 44590, 44591), (anti-orthocentroidal, 44592, 44593), (*anti-3rd tri-squares-central, 44594, 44595), (*anti-4th tri-squares-central, 44596, 44597), (*anti-X3-ABC reflections, 372, 371), (*anti-inner-Yff, 19050, 19049), (*anti-outer-Yff, 19048, 19047), (*anticomplementary, 3068, 3069), (*Aquila, 18991, 18992), (*Ara, 44598, 44599), (*1st Auriga, 44600, 44601), (*2nd Auriga, 44602, 44603), (2nd Brocard, 6, 6), (*5th Brocard, 44604, 44605), (*2nd circumperp tangential, 44606, 44607), (circumsymmedial, 6, 6), (*Ehrmann-mid, 6564, 6565), (2nd Ehrmann, 6, 6), (*Euler, 3071, 3070), (1st excosine, 44608, 44609), (9th Fermat-Dao, 6, 6), (10th Fermat-Dao, 6, 6), (13th Fermat-Dao, 6, 6), (14th Fermat-Dao, 6, 6), (*outer-Garcia, 13911, 13973), (*Gemini 107, 13846, 13847), (*Gemini 109, 8253, 8252), (*Gemini 110, 615, 590), (*Gemini 111, 7585, 7586), (*Gossard, 44610, 44611), (*inner-Grebe, 6, 6), (*outer-Grebe, 6, 6), (Hatzipolakis-Moses, 44612, 44613), (3rd Hatzipolakis, 44614, 44615), (2nd Hyacinth, 44616, 44617), (*infinite-altitude, 3070, 3071), (*Johnson, 485, 486), (*inner-Johnson, 44618, 44619), (*outer-Johnson, 44620, 44621), (*1st Johnson-Yff, 31472, 44622), (*2nd Johnson-Yff, 44623, 44624), (*1st Kenmotu-centers, --, 6), (*2nd Kenmotu-centers, 6, --), (1st Kenmotu diagonals, 6, 6), (2nd Kenmotu diagonals, 6, 6), (*1st Kenmotu-free-vertices, 371, 6420), (*2nd Kenmotu-free-vertices, 6419, 372), (Lucas antipodal tangential, 44625, npt), (Lucas(-1) antipodal tangential, npt, 44626), (*Lucas homothetic, 44627, 44628), (*Lucas(-1) homothetic, 44629, 44630), (Mandart-excircles, 44631, 44632), (*Mandart-incircle, 2066, 5414), (*medial, 590, 615), (midheight, 44633, 44634), (*5th mixtilinear, 44635, 44636), (orthic, 44637, 44638), (orthocentroidal, 44639, 44640), (2nd orthosymmedial, 6, 6), (reflection, 44641, 44642), (symmedial, 6, 6), (tangential, 6, 6), (inner tri-equilateral, 6, 6), (outer tri-equilateral, 6, 6), (*3rd tri-squares-central, 590, 32787), (*4th tri-squares-central, 32788, 615), (Walsmith, 6, 6), (*X3-ABC reflections, 3311, 3312), (*inner-Yff, 1335, 1124), (*outer-Yff, 1124, 1335), (*inner-Yff tangents, 44643, 44644), (*outer-Yff tangents, 44645, 44646)

where an asterisk * is used for homothetic triangles and "npt" stands for "not perspective triangles". Also, in the above list, infinite-altitude triangle is inversely homothetic to both Kenmotu-centers triangles and these are inversely similar to the orthocentroidal and anti-orthocentroidal triangles.

X(44582) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-1ST AURIGA AND 1ST KENMOTU-CENTERS

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

X(44582) lies on these lines: {1, 44584}, {6, 5597}, {55, 44585}, {372, 26398}, {485, 26386}, {590, 26359}, {1151, 26290}, {1587, 26381}, {2066, 26351}, {2067, 26380}, {3068, 26394}, {3071, 26326}, {5412, 26371}, {7968, 26365}, {13665, 18496}, {13911, 26382}, {18991, 26296}, {19048, 26400}, {19050, 26399}, {26302, 44598}, {26310, 44604}, {26319, 44606}, {26379, 44586}, {26383, 44610}, {26387, 44623}, {26388, 31472}, {26389, 44620}, {26390, 44618}, {26393, 44590}, {26395, 44635}, {26396, 44594}, {26397, 44596}, {26401, 44645}, {26402, 44643}

X(44582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5597, 44583), (5597, 26385, 6)

X(44583) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-1ST AURIGA AND 2ND KENMOTU-CENTERS

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

X(44583) lies on these lines: {1, 44585}, {6, 5597}, {55, 44584}, {371, 26398}, {486, 26386}, {615, 26359}, {1152, 26290}, {1588, 26381}, {3069, 26394}, {3070, 26326}, {5413, 26371}, {5414, 26351}, {6502, 26380}, {7969, 26365}, {13785, 18496}, {13973, 26382}, {18992, 26296}, {19047, 26400}, {19049, 26399}, {26302, 44599}, {26310, 44605}, {26319, 44607}, {26379, 44587}, {26383, 44611}, {26387, 44624}, {26388, 44622}, {26389, 44621}, {26390, 44619}, {26393, 44591}, {26395, 44636}, {26396, 44595}, {26397, 44597}, {26401, 44646}, {26402, 44644}

X(44583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5597, 44582), (5597, 26384, 6)

X(44584) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-2ND AURIGA AND 1ST KENMOTU-CENTERS

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

X(44584) lies on these lines: {1, 44582}, {6, 5598}, {55, 44583}, {372, 26422}, {485, 26410}, {590, 26360}, {1151, 26291}, {1587, 26405}, {2066, 26352}, {2067, 26404}, {3068, 26418}, {3071, 26327}, {5412, 26372}, {7968, 26366}, {13665, 18498}, {13911, 26406}, {18991, 26297}, {19048, 26424}, {19050, 26423}, {26303, 44598}, {26311, 44604}, {26320, 44606}, {26403, 44586}, {26407, 44610}, {26411, 44623}, {26412, 31472}, {26413, 44620}, {26414, 44618}, {26417, 44590}, {26419, 44635}, {26420, 44594}, {26421, 44596}, {26425, 44645}, {26426, 44643}

X(44584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5598, 44585), (5598, 26409, 6)

X(44585) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-2ND AURIGA AND 2ND KENMOTU-CENTERS

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

X(44585) lies on these lines: {1, 44583}, {6, 5598}, {55, 44582}, {371, 26422}, {486, 26410}, {615, 26360}, {1152, 26291}, {1588, 26405}, {3069, 26418}, {3070, 26327}, {5413, 26372}, {5414, 26352}, {6502, 26404}, {7969, 26366}, {13785, 18498}, {13973, 26406}, {18992, 26297}, {19047, 26424}, {19049, 26423}, {26303, 44599}, {26311, 44605}, {26320, 44607}, {26403, 44587}, {26407, 44611}, {26411, 44624}, {26412, 44622}, {26413, 44621}, {26414, 44619}, {26417, 44591}, {26419, 44636}, {26420, 44595}, {26421, 44597}, {26425, 44646}, {26426, 44644}

X(44585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5598, 44584), (5598, 26408, 6)

X(44586) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 1ST KENMOTU-CENTERS

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

X(44586) lies on these lines: {3, 6}, {83, 590}, {98, 3071}, {485, 10796}, {486, 10104}, {615, 1078}, {1124, 10802}, {1335, 10801}, {1587, 10788}, {1971, 30427}, {2066, 10799}, {2067, 12835}, {3068, 7787}, {3069, 7793}, {3070, 12110}, {3224, 26461}, {3972, 19090}, {3981, 8576}, {5412, 11380}, {6179, 19089}, {6229, 13934}, {6561, 14880}, {6564, 18502}, {6813, 42535}, {7583, 32134}, {7780, 13983}, {7804, 8992}, {7808, 8253}, {7815, 8252}, {7968, 11364}, {7969, 12194}, {9540, 10359}, {10358, 42265}, {10789, 18991}, {10790, 44598}, {10791, 13911}, {10794, 44618}, {10795, 44620}, {10797, 31472}, {10798, 44623}, {10800, 44635}, {10803, 44643}, {10804, 44645}, {11294, 26430}, {11490, 44590}, {11837, 44600}, {11838, 44602}, {11839, 44610}, {11840, 44627}, {11841, 44629}, {12150, 32787}, {12203, 42258}, {13665, 18501}, {13938, 32788}, {19048, 26432}, {19050, 26431}, {22520, 44606}, {26379, 44582}, {26403, 44584}, {26429, 44594}

X(44586) = intersection, other than A,B,C, of conics {{A, B, C, X(83), X(5058)}} and {{A, B, C, X(98), X(43121)}}
X(44586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 32, 44587), (32, 5034, 43124), (32, 18994, 6), (371, 35766, 3398), (1207, 2965, 44587), (1342, 1343, 5058), (1687, 1688, 43121), (1691, 34870, 44587), (6421, 6424, 3311)

X(44587) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 2ND KENMOTU-CENTERS

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

X(44587) lies on these lines: {3, 6}, {83, 615}, {98, 3070}, {485, 10104}, {486, 10796}, {590, 1078}, {1124, 10801}, {1335, 10802}, {1588, 10788}, {1971, 30428}, {3068, 7793}, {3069, 7787}, {3071, 12110}, {3224, 26454}, {3972, 19089}, {3981, 8577}, {5413, 11380}, {5414, 10799}, {6179, 19090}, {6228, 13882}, {6502, 12835}, {6560, 14880}, {6565, 18502}, {6811, 42535}, {7584, 32134}, {7780, 8992}, {7804, 13983}, {7808, 8252}, {7815, 8253}, {7968, 12194}, {7969, 11364}, {10358, 42262}, {10359, 13935}, {10789, 18992}, {10790, 44599}, {10791, 13973}, {10794, 44619}, {10795, 44621}, {10797, 44622}, {10798, 44624}, {10800, 44636}, {10803, 44644}, {10804, 44646}, {11293, 26429}, {11490, 44591}, {11837, 44601}, {11838, 44603}, {11839, 44611}, {11840, 44628}, {11841, 44630}, {12150, 32788}, {12203, 42259}, {13785, 18501}, {13885, 32787}, {19047, 26432}, {19049, 26431}, {22520, 44607}, {26379, 44583}, {26403, 44585}, {26430, 44597}, {31463, 39387}

X(44587) = intersection, other than A,B,C, of conics {{A, B, C, X(83), X(5062)}} and {{A, B, C, X(98), X(43120)}}
X(44587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 32, 44586), (32, 5034, 43125), (32, 18993, 6), (83, 13938, 615), (372, 35767, 3398), (1207, 2965, 44586), (1342, 1343, 5062), (1687, 1688, 43120), (1691, 34870, 44586), (6422, 6423, 3312)

X(44588) = PERSPECTOR OF THESE TRIANGLES: 2ND ANTI-EXTOUCH AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(-a^2+b^2+c^2)*(8*(a^4+(b^2-c^2+2)*(b^2-c^2-2)+4)*S*b^2*c^2-(a^2+b^2-c^2)*(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(44588) lies on these lines: {3, 26945}, {25, 44633}, {125, 8943}, {184, 8939}, {371, 1181}, {372, 19360}, {1583, 10962}, {1593, 3070}, {1599, 18924}, {1899, 44589}, {3311, 19362}, {3312, 19361}, {3964, 11091}, {5406, 11514}, {6398, 19348}, {8573, 18998}, {44598, 44637}

X(44589) = PERSPECTOR OF THESE TRIANGLES: 2ND ANTI-EXTOUCH AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(-a^2+b^2+c^2)*(8*(a^4+(b^2-c^2+2)*(b^2-c^2-2)+4)*S*b^2*c^2+(a^2+b^2-c^2)*(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(44589) lies on these lines: {3, 26873}, {25, 44634}, {125, 8939}, {184, 8943}, {371, 19360}, {372, 1181}, {1584, 10960}, {1593, 3071}, {1600, 18923}, {1899, 44588}, {3311, 19361}, {3312, 19362}, {3964, 11090}, {5407, 11513}, {6221, 19348}, {8573, 18997}, {44599, 44638}

X(44590) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 1ST KENMOTU-CENTERS

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

X(44590) lies on these lines: {1, 8943}, {3, 7969}, {6, 31}, {21, 19065}, {35, 18991}, {56, 44635}, {100, 3068}, {197, 44598}, {371, 11248}, {372, 10267}, {404, 13902}, {405, 13973}, {485, 11499}, {590, 1376}, {615, 1001}, {1124, 11508}, {1151, 10310}, {1335, 11507}, {1486, 44599}, {1587, 11491}, {1621, 3069}, {1703, 10902}, {2067, 11509}, {2077, 9583}, {2178, 34121}, {2362, 37579}, {3070, 11500}, {3071, 11496}, {3295, 7968}, {3303, 19013}, {3311, 11849}, {3312, 37621}, {3746, 18992}, {3871, 19066}, {4413, 8253}, {4421, 32787}, {4423, 8252}, {4428, 13940}, {5217, 19014}, {5248, 13936}, {5259, 13947}, {5284, 32786}, {5412, 11383}, {5537, 9616}, {5687, 13911}, {6200, 35238}, {6221, 35000}, {6410, 8273}, {6419, 35773}, {6502, 11510}, {6564, 18491}, {7133, 8609}, {7583, 32141}, {8167, 32790}, {8715, 13883}, {8939, 8941}, {8983, 25440}, {10269, 35763}, {10534, 18621}, {10679, 35775}, {11249, 35641}, {11490, 44586}, {11492, 44600}, {11493, 44602}, {11494, 44604}, {11501, 31472}, {11502, 44623}, {11503, 44627}, {11504, 44629}, {11848, 44610}, {13665, 18524}, {13901, 19024}, {14882, 18996}, {19047, 35808}, {19049, 35809}, {26393, 44582}, {26417, 44584}, {26512, 44594}, {26513, 44596}, {35239, 35611}, {35642, 37622}, {35785, 35810}

X(44590) = crosspoint of X(1252) and X(6135)
X(44590) = crosssum of X(1086) and X(6364)
X(44590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 7969, 44606), (6, 55, 44591), (42, 606, 6), (55, 19000, 6), (371, 35772, 11248), (902, 41421, 6), (1376, 13887, 590), (3303, 19013, 44636), (5414, 5415, 6)

X(44591) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 2ND KENMOTU-CENTERS

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

X(44591) lies on these lines: {1, 8939}, {3, 7968}, {6, 31}, {21, 19066}, {35, 18992}, {56, 44636}, {100, 3069}, {197, 44599}, {371, 10267}, {372, 11248}, {404, 13959}, {405, 13911}, {486, 11499}, {590, 1001}, {615, 1376}, {1124, 11507}, {1152, 10310}, {1335, 11508}, {1486, 44598}, {1588, 11491}, {1621, 3068}, {1702, 10902}, {2067, 11510}, {2178, 34125}, {3070, 11496}, {3071, 11500}, {3295, 7969}, {3303, 19014}, {3311, 37621}, {3312, 11849}, {3746, 18991}, {3871, 19065}, {4413, 8252}, {4421, 32788}, {4423, 8253}, {4428, 13887}, {5217, 19013}, {5248, 13883}, {5259, 13893}, {5284, 32785}, {5413, 11383}, {5687, 13973}, {6396, 35238}, {6398, 35000}, {6409, 8273}, {6420, 35772}, {6502, 11509}, {6565, 18491}, {7584, 32141}, {8167, 32789}, {8609, 42013}, {8715, 13936}, {8943, 8945}, {9583, 34486}, {9616, 15931}, {10269, 35762}, {10533, 18621}, {10679, 35774}, {11249, 35642}, {11490, 44587}, {11492, 44601}, {11493, 44603}, {11494, 44605}, {11501, 44622}, {11502, 44624}, {11503, 44628}, {11504, 44630}, {11848, 44611}, {13785, 18524}, {13958, 19023}, {13971, 25440}, {14882, 18995}, {16232, 37579}, {19048, 35809}, {19050, 35808}, {26393, 44583}, {26417, 44585}, {26512, 44595}, {26513, 44597}, {35239, 35610}, {35641, 37622}, {35784, 35811}

X(44591) = crosspoint of X(1252) and X(6136)
X(44591) = crosssum of X(1086) and X(6365)
X(44591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 7968, 44607), (6, 55, 44590), (42, 605, 6), (55, 18999, 6), (372, 35773, 11248), (1376, 13940, 615), (2066, 5416, 6), (3303, 19014, 44635)

X(44592) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(-18*(b^2-c^2)^2*b^2*c^2*(-a^2+b^2+c^2)*S-a^12+3*(b^2+c^2)*a^10-18*b^2*c^2*a^8-(b^2+c^2)*(10*b^4-31*b^2*c^2+10*c^4)*a^6+3*(5*b^8+5*c^8-(b^4+11*b^2*c^2+c^4)*b^2*c^2)*a^4-9*(b^6-c^6)*(b^4-c^4)*a^2+(b^2-c^2)^4*(b^2+2*c^2)*(2*b^2+c^2)) : :

X(44592) lies on these lines: {74, 3070}, {125, 44593}, {371, 399}, {34417, 44639}

X(44593) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(-18*(b^2-c^2)^2*b^2*c^2*(-a^2+b^2+c^2)*S+a^12-3*(b^2+c^2)*a^10+18*b^2*c^2*a^8+(b^2+c^2)*(10*b^4-31*b^2*c^2+10*c^4)*a^6-3*(5*b^8+5*c^8-(b^4+11*b^2*c^2+c^4)*b^2*c^2)*a^4+9*(b^6-c^6)*(b^4-c^4)*a^2-(b^2-c^2)^4*(b^2+2*c^2)*(2*b^2+c^2)) : :

X(44593) lies on these lines: {74, 3071}, {125, 44592}, {372, 399}, {34417, 44640}

X(44594) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-3RD TRI-SQUARES-CENTRAL AND 1ST KENMOTU-CENTERS

Barycentrics    a^4+12*S*a^2+4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(44594) lies on these lines: {2, 6}, {20, 12962}, {39, 19064}, {371, 12124}, {372, 26516}, {393, 19042}, {485, 26468}, {1151, 26294}, {1285, 8375}, {1504, 1587}, {1588, 5475}, {2066, 26355}, {2067, 26435}, {3071, 26330}, {3087, 8035}, {3311, 18907}, {3524, 8376}, {5062, 9540}, {5412, 26375}, {5477, 19056}, {6419, 39660}, {6420, 31400}, {6422, 7581}, {6428, 31406}, {6564, 13651}, {6781, 9541}, {7968, 26369}, {8960, 19102}, {12963, 42522}, {12969, 31465}, {13665, 18539}, {13911, 26444}, {18991, 26300}, {19048, 26518}, {19050, 26517}, {22726, 44500}, {24244, 26460}, {26306, 44598}, {26314, 44604}, {26324, 44606}, {26396, 44582}, {26420, 44584}, {26429, 44586}, {26449, 44610}, {26473, 44623}, {26479, 31472}, {26485, 44620}, {26490, 44618}, {26512, 44590}, {26514, 44635}, {26519, 44645}, {26520, 44643}, {31401, 31483}, {31414, 44518}, {35822, 43448}, {43257, 43618}

X(44594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 230, 26456), (6, 3068, 44595), (6, 3815, 19053), (6, 7585, 44596), (6, 7736, 44597), (6, 31463, 7586), (6, 32787, 7735), (193, 7585, 3068), (230, 26456, 44595), (3068, 26361, 590), (3068, 26456, 230), (3068, 26462, 6), (8974, 13639, 44394), (31403, 44597, 7736)

X(44595) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-3RD TRI-SQUARES-CENTRAL AND 2ND KENMOTU-CENTERS

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

X(44595) lies on these lines: {2, 6}, {4, 6423}, {20, 12968}, {32, 1588}, {39, 13935}, {115, 23249}, {187, 9541}, {371, 19102}, {372, 5286}, {393, 5200}, {486, 26468}, {488, 33348}, {631, 6422}, {1152, 7738}, {1285, 23273}, {1384, 42215}, {1504, 9540}, {1505, 5319}, {1587, 3767}, {2165, 18819}, {2459, 35945}, {3053, 6459}, {3070, 26330}, {3087, 3127}, {3156, 8573}, {3312, 5305}, {3524, 9600}, {5023, 42638}, {5024, 35256}, {5218, 31459}, {5254, 6460}, {5280, 13963}, {5299, 13962}, {5411, 16318}, {5414, 26355}, {5420, 31400}, {6398, 15048}, {6424, 7582}, {6462, 32989}, {6502, 26435}, {6531, 24244}, {6560, 43448}, {6561, 41411}, {6564, 13834}, {6813, 39876}, {7584, 30435}, {7737, 23259}, {7745, 42561}, {7746, 31411}, {7747, 23263}, {7748, 43407}, {7969, 26369}, {8960, 19103}, {9575, 13971}, {9593, 13975}, {9605, 13966}, {10164, 31427}, {10577, 31404}, {10881, 41361}, {11293, 26429}, {11648, 43256}, {12969, 42523}, {13356, 13938}, {13357, 19089}, {13665, 43291}, {13711, 35822}, {13785, 18539}, {13881, 31412}, {13973, 26444}, {15484, 18762}, {18510, 21309}, {18992, 26300}, {19006, 34809}, {19047, 26518}, {19049, 26517}, {19105, 35770}, {26306, 44599}, {26314, 44605}, {26324, 44607}, {26396, 44583}, {26420, 44585}, {26449, 44611}, {26473, 44624}, {26479, 44622}, {26485, 44621}, {26490, 44619}, {26512, 44591}, {26514, 44636}, {26519, 44646}, {26520, 44644}, {30478, 31464}, {32497, 33364}

X(44595) = polar conjugate of the isotomic conjugate of X(12257)
X(44595) = barycentric product X(4)*X(12257)
X(44595) = trilinear product X(19)*X(12257)
X(44595) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(12257)}} and {{A, B, C, X(83), X(31403)}}
X(44595) = crosssum of X(6) and X(12305)
X(44595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6, 31403), (2, 8975, 590), (6, 230, 3068), (6, 615, 7736), (6, 3068, 44594), (6, 7586, 44597), (6, 7735, 44596), (230, 26456, 44594), (3068, 3069, 492), (3068, 26456, 6), (3767, 5062, 1587), (5304, 7586, 6), (5413, 41515, 26375)

X(44596) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-4TH TRI-SQUARES-CENTRAL AND 1ST KENMOTU-CENTERS

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

X(44596) lies on these lines: {2, 6}, {4, 6424}, {20, 12963}, {32, 1587}, {39, 9540}, {115, 23259}, {172, 31408}, {371, 5286}, {372, 19105}, {393, 5412}, {485, 26469}, {487, 33349}, {631, 6421}, {1151, 7738}, {1285, 23267}, {1384, 42216}, {1504, 5319}, {1505, 13935}, {1588, 3767}, {2066, 26356}, {2067, 26436}, {2165, 18820}, {2460, 35944}, {2549, 9541}, {3053, 6460}, {3071, 26331}, {3087, 3128}, {3155, 8573}, {3311, 5305}, {4386, 31413}, {5007, 31411}, {5023, 42637}, {5024, 35255}, {5254, 6459}, {5280, 13905}, {5299, 13904}, {5410, 16318}, {5418, 31400}, {6221, 15048}, {6423, 7581}, {6463, 32989}, {6531, 24243}, {6560, 41410}, {6561, 43448}, {6565, 13711}, {6811, 39875}, {7583, 30435}, {7737, 23249}, {7745, 31412}, {7747, 23253}, {7748, 43408}, {7968, 26370}, {8981, 9605}, {8983, 9575}, {9593, 13912}, {9600, 43509}, {9646, 31402}, {10576, 31404}, {10880, 41361}, {11294, 26430}, {11648, 43257}, {12962, 42522}, {13356, 13885}, {13357, 19090}, {13665, 18907}, {13785, 43291}, {13834, 35823}, {13881, 42561}, {13911, 26445}, {15484, 18538}, {18512, 21309}, {18991, 26301}, {19005, 34809}, {19048, 26523}, {19050, 26522}, {19102, 35771}, {26307, 44598}, {26315, 44604}, {26325, 44606}, {26397, 44582}, {26421, 44584}, {26450, 44610}, {26474, 44623}, {26480, 31472}, {26486, 44620}, {26491, 44618}, {26513, 44590}, {26515, 44635}, {26524, 44645}, {26525, 44643}, {32494, 33365}

X(44596) = polar conjugate of the isotomic conjugate of X(12256)
X(44596) = barycentric product X(4)*X(12256)
X(44596) = trilinear product X(19)*X(12256)
X(44596) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(12256)}} and {{A, B, C, X(325), X(24243)}}
X(44596) = crosssum of X(6) and X(12306)
X(44596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 13949, 615), (6, 230, 3069), (6, 590, 7736), (6, 3068, 31403), (6, 3069, 44597), (6, 7585, 44594), (6, 7735, 44595), (6, 31463, 37665), (230, 26463, 44597), (2549, 9675, 9541), (3068, 3069, 491), (3069, 26463, 6), (3767, 5058, 1588), (5304, 7585, 6), (5412, 41516, 26376), (8972, 37665, 31463)

X(44597) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-4TH TRI-SQUARES-CENTRAL AND 2ND KENMOTU-CENTERS

Barycentrics    a^4-12*S*a^2+4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(44597) lies on these lines: {2, 6}, {20, 12969}, {39, 19063}, {371, 26521}, {372, 12123}, {393, 19041}, {486, 26469}, {1152, 26295}, {1285, 8376}, {1505, 1588}, {1587, 5475}, {3070, 26331}, {3087, 8036}, {3312, 18907}, {3524, 8375}, {5058, 13935}, {5413, 26376}, {5414, 26356}, {5477, 19055}, {6419, 31400}, {6420, 39661}, {6421, 7582}, {6427, 31406}, {6502, 26436}, {6565, 13770}, {7969, 26370}, {12968, 42523}, {13785, 26438}, {13973, 26445}, {18992, 26301}, {19047, 26523}, {19049, 26522}, {22727, 44500}, {24243, 26455}, {26307, 44599}, {26315, 44605}, {26325, 44607}, {26397, 44583}, {26421, 44585}, {26430, 44587}, {26450, 44611}, {26474, 44624}, {26480, 44622}, {26486, 44621}, {26491, 44619}, {26513, 44591}, {26515, 44636}, {26524, 44646}, {26525, 44644}, {35823, 43448}, {43256, 43618}

X(44597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 230, 26463), (6, 3069, 44596), (6, 3815, 19054), (6, 7586, 44595), (6, 7736, 44594), (6, 32788, 7735), (193, 7586, 3069), (230, 26463, 44596), (3069, 26362, 615), (3069, 26457, 6), (3069, 26463, 230), (7736, 44594, 31403), (13759, 13950, 44392)

X(44598) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 1ST KENMOTU-CENTERS

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

X(44598) lies on these lines: {3, 485}, {6, 25}, {22, 3068}, {23, 7585}, {24, 1587}, {26, 7583}, {32, 8996}, {186, 23267}, {197, 44590}, {371, 7387}, {372, 6642}, {378, 23249}, {486, 7529}, {571, 26953}, {615, 5020}, {1124, 10046}, {1131, 14118}, {1151, 11414}, {1181, 12239}, {1335, 10037}, {1486, 44591}, {1588, 10594}, {1593, 23251}, {1597, 42284}, {1598, 3071}, {1600, 32806}, {1609, 3155}, {1995, 3069}, {2066, 10833}, {2067, 18954}, {2070, 18512}, {3103, 10960}, {3156, 31463}, {3311, 7517}, {3312, 7506}, {3518, 7581}, {3520, 23269}, {5062, 44527}, {5198, 23261}, {5899, 6199}, {6200, 35243}, {6221, 12083}, {6409, 37198}, {6417, 18378}, {6418, 13621}, {6419, 35777}, {6460, 17928}, {6561, 18534}, {6564, 9818}, {6636, 8972}, {6644, 42216}, {7393, 10576}, {7395, 42265}, {7484, 8253}, {7485, 32785}, {7503, 31412}, {7512, 13886}, {7514, 18538}, {7525, 13925}, {7530, 42215}, {7545, 18510}, {7582, 34484}, {7584, 13861}, {7586, 13595}, {7968, 11365}, {7969, 9798}, {8185, 18991}, {8190, 44600}, {8191, 44602}, {8192, 44635}, {8193, 13911}, {8194, 44627}, {8195, 44629}, {8252, 11284}, {8414, 19410}, {8573, 44193}, {8577, 26920}, {8854, 11513}, {8943, 44634}, {9540, 10323}, {9541, 12082}, {9658, 18996}, {9673, 19038}, {9694, 33524}, {9909, 32787}, {9937, 10665}, {10790, 44586}, {10828, 44604}, {10829, 44618}, {10830, 44620}, {10831, 31472}, {10832, 44623}, {10834, 44643}, {10835, 44645}, {11417, 26283}, {11427, 15187}, {11433, 15188}, {11479, 42273}, {11484, 42583}, {11853, 44610}, {12085, 35820}, {12087, 43512}, {13564, 13903}, {13567, 15200}, {13620, 43889}, {13884, 21213}, {13893, 37557}, {13943, 32788}, {14070, 35822}, {15199, 23292}, {15212, 26958}, {16064, 36549}, {16419, 32789}, {18535, 42283}, {18998, 19044}, {18999, 20988}, {19000, 20989}, {19048, 26309}, {19050, 26308}, {19117, 37440}, {21312, 42264}, {22654, 44606}, {23253, 35502}, {23275, 26863}, {26302, 44582}, {26303, 44584}, {26306, 44594}, {26307, 44596}, {31414, 38444}, {39568, 42258}, {42275, 44454}, {44588, 44637}

X(44598) = crosssum of X(371) and X(11514)
X(44598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 13889, 590), (6, 25, 44599), (25, 5595, 20987), (25, 19006, 6), (371, 35776, 7387), (3312, 7506, 8277), (6560, 9682, 3), (9694, 33524, 42638), (19459, 44524, 44599)

X(44599) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2ND KENMOTU-CENTERS

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

X(44599) lies on these lines: {3, 486}, {6, 25}, {22, 3069}, {23, 7586}, {24, 1588}, {26, 7584}, {186, 23273}, {197, 44591}, {371, 6642}, {372, 7387}, {378, 23259}, {485, 7529}, {590, 5020}, {1124, 10037}, {1132, 14118}, {1152, 11414}, {1181, 12240}, {1335, 10046}, {1486, 44590}, {1505, 8996}, {1587, 10594}, {1593, 23261}, {1597, 42283}, {1598, 3070}, {1599, 32805}, {1609, 3156}, {1995, 3068}, {2070, 18510}, {3102, 10962}, {3311, 7506}, {3312, 7517}, {3518, 7582}, {3520, 23275}, {5058, 8908}, {5063, 26953}, {5198, 23251}, {5414, 10833}, {5892, 9687}, {5899, 6395}, {6396, 35243}, {6398, 12083}, {6410, 37198}, {6417, 13621}, {6418, 18378}, {6420, 35776}, {6459, 17928}, {6502, 18954}, {6560, 18534}, {6565, 9818}, {6636, 13941}, {6644, 42215}, {7393, 10577}, {7395, 42262}, {7484, 8252}, {7485, 32786}, {7503, 42561}, {7512, 13939}, {7514, 18762}, {7525, 13993}, {7530, 42216}, {7545, 18512}, {7581, 34484}, {7583, 13861}, {7585, 13595}, {7968, 9798}, {7969, 11365}, {8185, 18992}, {8190, 44601}, {8191, 44603}, {8192, 44636}, {8193, 13973}, {8194, 44628}, {8195, 44630}, {8253, 11284}, {8406, 19411}, {8573, 44192}, {8576, 8911}, {8855, 11514}, {8939, 44633}, {9658, 18995}, {9673, 19037}, {9909, 32788}, {9937, 10666}, {10323, 13935}, {10790, 44587}, {10828, 44605}, {10829, 44619}, {10830, 44621}, {10831, 44622}, {10832, 44624}, {10834, 44644}, {10835, 44646}, {11418, 26283}, {11427, 15188}, {11433, 15187}, {11479, 42270}, {11484, 42582}, {11853, 44611}, {12085, 35821}, {12087, 43511}, {13564, 13961}, {13567, 15199}, {13620, 43890}, {13889, 32787}, {13937, 21213}, {13947, 37557}, {14070, 35823}, {15200, 23292}, {15211, 26958}, {16064, 36550}, {16419, 32790}, {18535, 42284}, {18997, 19043}, {18999, 20989}, {19000, 20988}, {19047, 26309}, {19049, 26308}, {19116, 37440}, {21312, 42263}, {22654, 44607}, {23263, 35502}, {23269, 26863}, {26302, 44583}, {26303, 44585}, {26306, 44595}, {26307, 44597}, {33524, 42637}, {39568, 42259}, {42276, 44454}, {44589, 44638}

X(44599) = crosssum of X(372) and X(11513)
X(44599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 13943, 615), (6, 25, 44598), (25, 5594, 20987), (25, 19005, 6), (372, 35777, 7387), (3311, 7506, 8276), (5420, 9683, 3), (19459, 44524, 44598)

X(44600) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST AURIGA AND 1ST KENMOTU-CENTERS

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

X(44600) lies on these lines: {1, 44603}, {6, 5597}, {55, 7969}, {371, 11252}, {485, 8200}, {590, 5599}, {1124, 11879}, {1151, 11822}, {1335, 11877}, {1587, 11843}, {2066, 11873}, {2067, 18955}, {3068, 5601}, {3070, 9834}, {3071, 8196}, {3311, 11875}, {5412, 11384}, {5598, 44635}, {6200, 35244}, {6419, 35781}, {6564, 18495}, {7583, 32146}, {7968, 11366}, {8186, 18991}, {8190, 44598}, {8197, 13911}, {8201, 44627}, {8202, 44629}, {11207, 32787}, {11253, 35641}, {11492, 44590}, {11493, 44606}, {11837, 44586}, {11861, 44604}, {11863, 44610}, {11865, 44618}, {11867, 44620}, {11869, 31472}, {11871, 44623}, {11881, 44643}, {11883, 44645}, {13944, 32788}, {19009, 44636}, {35245, 35611}, {35779, 35810}

X(44600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5597, 44601), (55, 7969, 44602), (371, 35778, 11252), (5597, 19008, 6), (5599, 13890, 590)

X(44601) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST AURIGA AND 2ND KENMOTU-CENTERS

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

X(44601) lies on these lines: {1, 44602}, {6, 5597}, {55, 7968}, {372, 11252}, {486, 8200}, {615, 5599}, {1124, 11877}, {1152, 11822}, {1335, 11879}, {1588, 11843}, {3069, 5601}, {3070, 8196}, {3071, 9834}, {3312, 11875}, {5413, 11384}, {5414, 11873}, {5598, 44636}, {6396, 35244}, {6420, 35778}, {6502, 18955}, {6565, 18495}, {7584, 32146}, {7969, 11366}, {8186, 18992}, {8190, 44599}, {8197, 13973}, {8201, 44628}, {8202, 44630}, {11207, 32788}, {11253, 35642}, {11492, 44591}, {11493, 44607}, {11837, 44587}, {11861, 44605}, {11863, 44611}, {11865, 44619}, {11867, 44621}, {11869, 44622}, {11871, 44624}, {11881, 44644}, {11883, 44646}, {13890, 32787}, {19010, 44635}, {35245, 35610}, {35780, 35811}

X(44601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5597, 44600), (55, 7968, 44603), (372, 35781, 11252), (5599, 13944, 615)

X(44602) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND AURIGA AND 1ST KENMOTU-CENTERS

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

X(44602) lies on these lines: {1, 44601}, {6, 5598}, {55, 7969}, {371, 11253}, {485, 8207}, {590, 5600}, {1124, 11880}, {1151, 11823}, {1335, 11878}, {1587, 11844}, {2066, 11874}, {2067, 18956}, {3068, 5602}, {3070, 9835}, {3071, 8203}, {3311, 11876}, {5412, 11385}, {5597, 44635}, {6200, 35245}, {6419, 35779}, {6564, 18497}, {7583, 32147}, {7968, 11367}, {8187, 18991}, {8191, 44598}, {8204, 13911}, {8208, 44627}, {8209, 44629}, {11208, 32787}, {11252, 35641}, {11492, 44606}, {11493, 44590}, {11838, 44586}, {11862, 44604}, {11864, 44610}, {11866, 44618}, {11868, 44620}, {11870, 31472}, {11872, 44623}, {11882, 44643}, {11884, 44645}, {13945, 32788}, {19007, 44636}, {35244, 35611}, {35781, 35810}

X(44602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5598, 44603), (55, 7969, 44600), (371, 35780, 11253), (5598, 19010, 6), (5600, 13891, 590)

X(44603) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND AURIGA AND 2ND KENMOTU-CENTERS

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

X(44603) lies on these lines: {1, 44600}, {6, 5598}, {55, 7968}, {372, 11253}, {486, 8207}, {615, 5600}, {1124, 11878}, {1152, 11823}, {1335, 11880}, {1588, 11844}, {3069, 5602}, {3070, 8203}, {3071, 9835}, {3312, 11876}, {5413, 11385}, {5414, 11874}, {5597, 44636}, {6396, 35245}, {6420, 35780}, {6502, 18956}, {6565, 18497}, {7584, 32147}, {7969, 11367}, {8187, 18992}, {8191, 44599}, {8204, 13973}, {8208, 44628}, {8209, 44630}, {11208, 32788}, {11252, 35642}, {11492, 44607}, {11493, 44591}, {11838, 44587}, {11862, 44605}, {11864, 44611}, {11866, 44619}, {11868, 44621}, {11870, 44622}, {11872, 44624}, {11882, 44644}, {11884, 44646}, {13891, 32787}, {19008, 44635}, {35244, 35610}, {35778, 35811}

X(44603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 5598, 44602), (55, 7968, 44601), (372, 35779, 11253), (5598, 19009, 6), (5600, 13945, 615)

X(44604) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 1ST KENMOTU-CENTERS

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

X(44604) lies on these lines: {3, 6}, {485, 9996}, {590, 3096}, {615, 7846}, {1124, 10047}, {1335, 10038}, {1587, 9862}, {2066, 10877}, {2067, 18957}, {2896, 3068}, {3069, 10583}, {3070, 9873}, {3071, 9993}, {3099, 18991}, {5412, 11386}, {6564, 18500}, {7583, 32151}, {7804, 13877}, {7811, 32787}, {7865, 13846}, {7914, 8253}, {7968, 11368}, {7969, 9941}, {9540, 10357}, {9857, 13911}, {9997, 44635}, {10356, 42265}, {10828, 44598}, {10871, 44618}, {10872, 44620}, {10873, 31472}, {10874, 44623}, {10875, 44627}, {10876, 44629}, {10878, 44643}, {10879, 44645}, {11494, 44590}, {11861, 44600}, {11862, 44602}, {11885, 44610}, {13665, 18503}, {13946, 32788}, {19048, 26318}, {19050, 26317}, {22744, 44606}, {26310, 44582}, {26311, 44584}, {26314, 44594}, {26315, 44596}, {35255, 42787}

X(44604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 32, 44605), (32, 9994, 2076), (32, 19012, 6), (371, 35782, 9821), (3096, 13892, 590)

X(44605) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 2ND KENMOTU-CENTERS

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

X(44605) lies on these lines: {3, 6}, {486, 9996}, {590, 7846}, {615, 3096}, {1124, 10038}, {1335, 10047}, {1588, 9862}, {2896, 3069}, {3068, 10583}, {3070, 9993}, {3071, 9873}, {3099, 18992}, {5413, 11386}, {5414, 10877}, {6502, 18957}, {6565, 18500}, {7584, 32151}, {7804, 13930}, {7811, 32788}, {7865, 13847}, {7914, 8252}, {7968, 9941}, {7969, 11368}, {9857, 13973}, {9997, 44636}, {10356, 42262}, {10357, 13935}, {10828, 44599}, {10871, 44619}, {10872, 44621}, {10873, 44622}, {10874, 44624}, {10875, 44628}, {10876, 44630}, {10878, 44644}, {10879, 44646}, {11494, 44591}, {11861, 44601}, {11862, 44603}, {11885, 44611}, {13785, 18503}, {13892, 32787}, {19047, 26318}, {19049, 26317}, {22744, 44607}, {26310, 44583}, {26311, 44585}, {26314, 44595}, {26315, 44597}, {35256, 42787}

X(44605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 32, 44604), (32, 9995, 2076), (32, 19011, 6), (372, 35783, 9821), (3096, 13946, 615)

X(44606) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND CIRCUMPERP TANGENTIAL AND 1ST KENMOTU-CENTERS

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

X(44606) lies on these lines: {1, 8939}, {3, 7969}, {6, 41}, {21, 13902}, {36, 18991}, {55, 44635}, {104, 1587}, {371, 11249}, {372, 10269}, {404, 19065}, {474, 13973}, {485, 22758}, {590, 958}, {605, 1201}, {615, 25524}, {956, 13911}, {993, 8983}, {999, 7968}, {1124, 22767}, {1151, 3428}, {1335, 22766}, {1470, 2362}, {1703, 37561}, {2066, 10966}, {2975, 3068}, {3069, 5253}, {3070, 12114}, {3071, 22753}, {3304, 18999}, {3311, 22765}, {3312, 37535}, {3556, 10533}, {5204, 19000}, {5258, 13893}, {5260, 32785}, {5412, 22479}, {5414, 22768}, {5416, 44646}, {5563, 18992}, {5584, 6409}, {6200, 35239}, {6419, 35785}, {6564, 18761}, {7583, 32153}, {8666, 13883}, {9583, 11012}, {10267, 35763}, {10680, 35775}, {11194, 32787}, {11248, 35641}, {11492, 44602}, {11493, 44600}, {13665, 26321}, {16232, 26437}, {18965, 19026}, {19003, 37587}, {19047, 35769}, {19049, 35768}, {22520, 44586}, {22654, 44598}, {22744, 44604}, {22755, 44610}, {22759, 31472}, {22760, 44623}, {22761, 44627}, {22762, 44629}, {22764, 32788}, {26319, 44582}, {26320, 44584}, {26324, 44594}, {26325, 44596}, {35238, 35611}, {35773, 35810}

X(44606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 7969, 44590), (6, 56, 44607), (56, 19014, 6), (371, 35784, 11249), (958, 22763, 590), (3304, 18999, 44636), (35773, 35810, 37622)

X(44607) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND CIRCUMPERP TANGENTIAL AND 2ND KENMOTU-CENTERS

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

X(44607) lies on these lines: {1, 8943}, {3, 7968}, {6, 41}, {21, 13959}, {36, 18992}, {55, 44636}, {104, 1588}, {371, 10269}, {372, 11249}, {404, 19066}, {474, 13911}, {486, 22758}, {590, 25524}, {606, 1201}, {615, 958}, {956, 13973}, {993, 13971}, {999, 7969}, {1124, 22766}, {1152, 3428}, {1335, 22767}, {1470, 16232}, {1702, 37561}, {2066, 22768}, {2362, 26437}, {2975, 3069}, {3068, 5253}, {3070, 22753}, {3071, 12114}, {3304, 19000}, {3311, 37535}, {3312, 22765}, {3556, 10534}, {5204, 18999}, {5258, 13947}, {5260, 32786}, {5413, 22479}, {5414, 10966}, {5415, 44645}, {5563, 18991}, {5584, 6410}, {5706, 41479}, {6396, 35239}, {6420, 35784}, {6565, 18761}, {7584, 32153}, {8666, 13936}, {10267, 35762}, {10680, 35774}, {11194, 32788}, {11248, 35642}, {11492, 44603}, {11493, 44601}, {13785, 26321}, {18966, 19025}, {19004, 37587}, {19048, 35768}, {19050, 35769}, {22520, 44587}, {22654, 44599}, {22744, 44605}, {22755, 44611}, {22759, 44622}, {22760, 44624}, {22761, 44628}, {22762, 44630}, {22763, 32787}, {26319, 44583}, {26320, 44585}, {26324, 44595}, {26325, 44597}, {35238, 35610}, {35772, 35811}

X(44607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 7968, 44591), (6, 56, 44606), (56, 19013, 6), (372, 35785, 11249), (958, 22764, 615), (35772, 35811, 37622)

X(44608) = PERSPECTOR OF THESE TRIANGLES: 1ST EXCOSINE AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(a^12-2*(b^2+c^2)*a^10-(5*b^2-c^2)*(b^2-5*c^2)*a^8+4*(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4)*a^6-16*(a^6+(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2))*S*b^2*c^2-(b^2-c^2)^2*(5*b^2-2*b*c+5*c^2)*(5*b^2+2*b*c+5*c^2)*a^4+2*(b^2-c^2)^2*a^2*(b^2+c^2)*(7*b^4+10*b^2*c^2+7*c^4)-(b^2-c^2)^4*(3*b^2+c^2)*(b^2+3*c^2)) : :

X(44608) lies on these lines: {64, 3070}, {154, 8939}, {371, 1498}, {1162, 17816}, {1853, 44609}, {10962, 17820}, {17810, 44633}, {17822, 19087}

X(44609) = PERSPECTOR OF THESE TRIANGLES: 1ST EXCOSINE AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(a^12-2*(b^2+c^2)*a^10-(5*b^2-c^2)*(b^2-5*c^2)*a^8+4*(b^2+c^2)*(5*b^4-14*b^2*c^2+5*c^4)*a^6+16*(a^6+(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2))*S*b^2*c^2-(b^2-c^2)^2*(5*b^2-2*b*c+5*c^2)*(5*b^2+2*b*c+5*c^2)*a^4+2*(b^2-c^2)^2*a^2*(b^2+c^2)*(7*b^4+10*b^2*c^2+7*c^4)-(b^2-c^2)^4*(3*b^2+c^2)*(b^2+3*c^2)) : :

X(44609) lies on these lines: {64, 3071}, {154, 8943}, {372, 1498}, {1163, 17815}, {1853, 44608}, {10960, 17819}, {17810, 44634}, {17822, 19088}

X(44610) = HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 1ST KENMOTU-CENTERS

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

X(44610) lies on these lines: {6, 402}, {30, 485}, {371, 11251}, {372, 26451}, {590, 1650}, {615, 15183}, {1124, 11913}, {1335, 11912}, {1587, 11845}, {1651, 32787}, {2066, 11909}, {2067, 18958}, {3068, 4240}, {3070, 12113}, {3071, 11897}, {3311, 11911}, {5412, 11832}, {6200, 35241}, {6419, 35791}, {6564, 18507}, {7583, 32162}, {7968, 11831}, {7969, 12438}, {8253, 15184}, {8998, 9033}, {11839, 44586}, {11848, 44590}, {11852, 18991}, {11853, 44598}, {11863, 44600}, {11864, 44602}, {11885, 44604}, {11900, 13911}, {11903, 44618}, {11904, 44620}, {11905, 31472}, {11906, 44623}, {11907, 44627}, {11908, 44629}, {11910, 44635}, {11914, 44643}, {11915, 44645}, {13665, 18508}, {13948, 32788}, {16190, 42259}, {16212, 19066}, {19048, 26453}, {19050, 26452}, {22755, 44606}, {26383, 44582}, {26407, 44584}, {26449, 44594}, {26450, 44596}

X(44610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 402, 44611), (371, 35790, 11251), (402, 19018, 6), (1650, 13894, 590)

X(44611) = HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 2ND KENMOTU-CENTERS

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

X(44611) lies on these lines: {6, 402}, {30, 486}, {371, 26451}, {372, 11251}, {590, 15183}, {615, 1650}, {1124, 11912}, {1335, 11913}, {1588, 11845}, {1651, 32788}, {3069, 4240}, {3070, 11897}, {3071, 12113}, {3312, 11911}, {5413, 11832}, {5414, 11909}, {6396, 35241}, {6420, 35790}, {6502, 18958}, {6565, 18507}, {7584, 32162}, {7968, 12438}, {7969, 11831}, {8252, 15184}, {9033, 13990}, {11839, 44587}, {11848, 44591}, {11852, 18992}, {11853, 44599}, {11863, 44601}, {11864, 44603}, {11885, 44605}, {11900, 13973}, {11903, 44619}, {11904, 44621}, {11905, 44622}, {11906, 44624}, {11907, 44628}, {11908, 44630}, {11910, 44636}, {11914, 44644}, {11915, 44646}, {13785, 18508}, {13894, 32787}, {16190, 42258}, {16212, 19065}, {19047, 26453}, {19049, 26452}, {22755, 44607}, {26383, 44583}, {26407, 44585}, {26449, 44595}, {26450, 44597}

X(44611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 402, 44610), (372, 35791, 11251), (402, 19017, 6), (1650, 13948, 615)

X(44612) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND 1ST KENMOTU-CENTERS

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

X(44612) lies on these lines: {371, 6102}, {3070, 13403}, {21640, 44637}

X(44613) = PERSPECTOR OF THESE TRIANGLES: HATZIPOLAKIS-MOSES AND 2ND KENMOTU-CENTERS

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

X(44613) lies on these lines: {372, 6102}, {3071, 13403}, {21641, 44638}

X(44614) = PERSPECTOR OF THESE TRIANGLES: 3RD HATZIPOLAKIS AND 1ST KENMOTU-CENTERS

Barycentrics    (3*(b^2+c^2)*a^10-(11*b^4+2*b^2*c^2+11*c^4)*a^8+(b^2+c^2)*(14*b^4-15*b^2*c^2+14*c^4)*a^6-(6*b^8+6*c^8+5*(-4*b^2*c^2+(b^2-c^2)^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(b^4-5*b^2*c^2+c^4)*a^2+(4*a^10-10*(b^2+c^2)*a^8+4*(b^4+7*b^2*c^2+c^4)*a^6+2*(2*b-c)*(b+2*c)*(b-2*c)*(2*b+c)*(b^2+c^2)*a^4-8*(b^2-c^2)^2*(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-6*b^2*c^2+2*c^4))*S+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4)*a^2 : :

X(44614) lies on these lines: {54, 22961}, {371, 13630}

X(44615) = PERSPECTOR OF THESE TRIANGLES: 3RD HATZIPOLAKIS AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(3*(b^2+c^2)*a^10-(11*b^4+2*b^2*c^2+11*c^4)*a^8+(b^2+c^2)*(14*b^4-15*b^2*c^2+14*c^4)*a^6-(6*b^8+6*c^8+5*(-4*b^2*c^2+(b^2-c^2)^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(b^4-5*b^2*c^2+c^4)*a^2-(4*a^10-10*(b^2+c^2)*a^8+4*(b^4+7*b^2*c^2+c^4)*a^6+2*(2*b-c)*(b+2*c)*(b-2*c)*(2*b+c)*(b^2+c^2)*a^4-8*(b^2-c^2)^2*(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-6*b^2*c^2+2*c^4))*S+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4) : :

X(44615) lies on these lines: {54, 22960}, {372, 13630}

X(44616) = PERSPECTOR OF THESE TRIANGLES: 2ND HYACINTH AND 1ST KENMOTU-CENTERS

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

X(44616) lies on these lines: {125, 10960}, {184, 10962}, {185, 371}, {216, 44617}, {1151, 17818}, {1885, 3070}, {1899, 6413}, {5412, 44633}, {6146, 10897}, {6414, 13198}, {6467, 11513}, {8939, 19355}, {10938, 18457}, {11417, 26441}, {15905, 19022}

X(44617) = PERSPECTOR OF THESE TRIANGLES: 2ND HYACINTH AND 2ND KENMOTU-CENTERS

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

X(44617) lies on these lines: {125, 10962}, {184, 8963}, {185, 372}, {216, 44616}, {1152, 17818}, {1885, 3071}, {1899, 6414}, {5413, 44634}, {6146, 10898}, {6413, 13198}, {6467, 11514}, {8943, 19356}, {8982, 11418}, {10938, 18459}, {15905, 19021}

X(44618) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 1ST KENMOTU-CENTERS

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

X(44618) lies on these lines: {5, 19048}, {6, 11}, {12, 44643}, {355, 485}, {371, 10525}, {372, 26492}, {377, 13906}, {496, 19047}, {590, 1376}, {1124, 10948}, {1151, 11826}, {1335, 10523}, {1587, 10785}, {1588, 10598}, {2066, 10947}, {2067, 18961}, {3068, 3434}, {3069, 10584}, {3070, 12114}, {3071, 10893}, {3311, 11928}, {5412, 11390}, {6200, 35249}, {6419, 35797}, {6564, 18516}, {7583, 10943}, {7741, 26465}, {7968, 11373}, {8983, 17647}, {10794, 44586}, {10826, 18991}, {10829, 44598}, {10871, 44604}, {10914, 13911}, {10944, 31472}, {10945, 44627}, {10946, 44629}, {10949, 19028}, {11235, 32787}, {11865, 44600}, {11866, 44602}, {11903, 44610}, {13665, 18519}, {13846, 34612}, {13898, 19000}, {13952, 32788}, {13973, 17619}, {26390, 44582}, {26414, 44584}, {26459, 37720}, {26490, 44594}, {26491, 44596}, {31584, 31586}, {37722, 44644}

X(44618) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 11, 44619), (11, 19024, 6), (371, 35796, 10525), (485, 7969, 44620), (1376, 13895, 590)

X(44619) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 2ND KENMOTU-CENTERS

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

X(44619) lies on these lines: {5, 19047}, {6, 11}, {12, 44644}, {355, 486}, {371, 26492}, {372, 10525}, {377, 13964}, {496, 19048}, {615, 1376}, {1124, 10523}, {1152, 11826}, {1335, 10948}, {1587, 10598}, {1588, 10785}, {3068, 10584}, {3069, 3434}, {3070, 10893}, {3071, 12114}, {3312, 11928}, {5413, 11390}, {5414, 10947}, {6396, 35249}, {6420, 35796}, {6502, 18961}, {6565, 18516}, {7584, 10943}, {7741, 26459}, {7969, 11373}, {10794, 44587}, {10826, 18992}, {10829, 44599}, {10871, 44605}, {10914, 13973}, {10944, 44622}, {10945, 44628}, {10946, 44630}, {10949, 19027}, {11235, 32788}, {11865, 44601}, {11866, 44603}, {11903, 44611}, {13785, 18519}, {13847, 34612}, {13895, 32787}, {13911, 17619}, {13955, 18999}, {13971, 17647}, {26390, 44583}, {26414, 44585}, {26465, 37720}, {26490, 44595}, {26491, 44597}, {31585, 31587}, {37722, 44643}

X(44619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 11, 44618), (11, 19023, 6), (372, 35797, 10525), (486, 7968, 44621), (1376, 13952, 615)

X(44620) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 1ST KENMOTU-CENTERS

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

X(44620) lies on these lines: {5, 19050}, {6, 12}, {11, 44645}, {72, 13911}, {355, 485}, {371, 10526}, {372, 26487}, {495, 19049}, {590, 958}, {1124, 10523}, {1151, 11827}, {1335, 10954}, {1587, 10786}, {1588, 10599}, {2066, 10953}, {2067, 18962}, {2478, 13907}, {3068, 3436}, {3069, 10585}, {3070, 11500}, {3071, 10894}, {3311, 11929}, {5412, 11391}, {6200, 35250}, {6253, 23251}, {6419, 35799}, {6564, 18517}, {7583, 10942}, {7951, 26464}, {7968, 11374}, {8253, 24953}, {10795, 44586}, {10827, 18991}, {10830, 44598}, {10872, 44604}, {10950, 44623}, {10951, 44627}, {10952, 44629}, {10955, 19030}, {11236, 32787}, {11867, 44600}, {11868, 44602}, {11904, 44610}, {13665, 18518}, {13846, 34606}, {13883, 21077}, {13893, 41229}, {13897, 19014}, {13953, 32788}, {15888, 44646}, {26389, 44582}, {26413, 44584}, {26458, 37719}, {26485, 44594}, {26486, 44596}

X(44620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12, 44621), (12, 19026, 6), (371, 35798, 10526), (485, 7969, 44618), (958, 13896, 590)

X(44621) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 2ND KENMOTU-CENTERS

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

X(44621) lies on these lines: {5, 19049}, {6, 12}, {11, 44646}, {72, 13973}, {355, 486}, {371, 26487}, {372, 10526}, {495, 19050}, {615, 958}, {1124, 10954}, {1152, 11827}, {1335, 10523}, {1587, 10599}, {1588, 10786}, {2478, 13965}, {3068, 10585}, {3069, 3436}, {3070, 10894}, {3071, 11500}, {3312, 11929}, {5413, 11391}, {5414, 10953}, {6253, 23261}, {6396, 35250}, {6420, 35798}, {6502, 18962}, {6565, 18517}, {7584, 10942}, {7951, 26458}, {7969, 11374}, {8252, 24953}, {10795, 44587}, {10827, 18992}, {10830, 44599}, {10872, 44605}, {10950, 44624}, {10951, 44628}, {10952, 44630}, {10955, 19029}, {11236, 32788}, {11867, 44601}, {11868, 44603}, {11904, 44611}, {13785, 18518}, {13847, 34606}, {13896, 32787}, {13936, 21077}, {13947, 41229}, {13954, 19013}, {15888, 44645}, {26389, 44583}, {26413, 44585}, {26464, 37719}, {26485, 44595}, {26486, 44597}

X(44621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12, 44620), (12, 19025, 6), (372, 35799, 10526), (486, 7968, 44619), (958, 13953, 615)

X(44622) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST JOHNSON-YFF AND 2ND KENMOTU-CENTERS

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

X(44622) lies on these lines: {1, 486}, {2, 2067}, {3, 9647}, {4, 5414}, {5, 1335}, {6, 12}, {10, 16232}, {11, 3298}, {35, 6561}, {36, 5420}, {55, 3071}, {56, 615}, {57, 13947}, {65, 13973}, {92, 1586}, {226, 2362}, {371, 498}, {372, 1478}, {388, 3069}, {390, 1132}, {442, 1377}, {485, 3301}, {492, 1909}, {495, 1124}, {496, 18762}, {497, 42561}, {499, 10577}, {590, 18996}, {908, 30557}, {999, 13951}, {1056, 13939}, {1151, 5432}, {1152, 7354}, {1378, 17757}, {1479, 6565}, {1504, 31476}, {1587, 10590}, {1588, 2066}, {1656, 9661}, {1702, 31434}, {1703, 9612}, {1773, 6203}, {2476, 31484}, {3068, 10588}, {3070, 10895}, {3102, 10063}, {3295, 13785}, {3297, 15888}, {3299, 31475}, {3304, 13955}, {3311, 9646}, {3312, 9654}, {3476, 13959}, {3485, 19065}, {3582, 42603}, {3583, 42268}, {3585, 6560}, {3592, 13901}, {3600, 13941}, {3614, 19030}, {4292, 13975}, {4293, 13935}, {4294, 23259}, {4299, 6396}, {4302, 35821}, {4317, 35813}, {4324, 42275}, {5010, 42260}, {5062, 9650}, {5083, 13976}, {5177, 31413}, {5217, 42258}, {5218, 6459}, {5219, 18991}, {5229, 6460}, {5252, 7968}, {5261, 7586}, {5413, 11392}, {5433, 8252}, {5434, 13847}, {5726, 19003}, {6221, 31499}, {6284, 23261}, {6347, 13388}, {6398, 9655}, {6410, 15326}, {6419, 13905}, {6420, 35800}, {6422, 31460}, {6429, 31500}, {6437, 9648}, {7288, 32786}, {7483, 9678}, {7582, 8164}, {7583, 10592}, {7741, 42274}, {7969, 11375}, {8253, 18965}, {8833, 26040}, {9578, 18992}, {10039, 35775}, {10055, 10666}, {10056, 35808}, {10106, 13971}, {10483, 42261}, {10573, 35789}, {10576, 13904}, {10797, 44587}, {10820, 18968}, {10831, 44599}, {10873, 44605}, {10896, 42270}, {10944, 44619}, {10956, 44644}, {10957, 44646}, {11237, 18995}, {11501, 44591}, {11869, 44601}, {11870, 44603}, {11905, 44611}, {11930, 44628}, {11931, 44630}, {12047, 35774}, {12647, 35642}, {12943, 42259}, {12953, 42283}, {13897, 32787}, {13898, 42582}, {13934, 18989}, {13962, 35769}, {13966, 18990}, {15338, 42263}, {15868, 35819}, {15950, 44635}, {18510, 31474}, {18513, 22644}, {19047, 26482}, {19049, 26481}, {22759, 44607}, {24987, 30556}, {25466, 31473}, {26388, 44583}, {26412, 44585}, {26479, 44595}, {26480, 44597}, {31464, 37661}, {31562, 42013}, {38235, 39542}

X(44622) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1586)}} and {{A, B, C, X(92), X(486)}}
X(44622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 486, 44624), (5, 1335, 44623), (6, 12, 31472), (12, 19027, 6), (56, 13954, 615), (226, 13936, 2362), (372, 35801, 1478), (388, 3069, 6502), (495, 7584, 1124), (1478, 13963, 372), (1588, 3085, 2066), (3298, 42262, 11), (3301, 7951, 485), (3311, 31479, 9646), (3614, 19030, 42265), (5261, 7586, 31408), (6565, 35809, 1479), (7354, 13958, 1152), (10577, 35768, 499), (10895, 19037, 3070), (15888, 19029, 3297)

X(44623) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND JOHNSON-YFF AND 1ST KENMOTU-CENTERS

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

X(44623) lies on these lines: {1, 485}, {2, 5414}, {3, 9661}, {4, 2067}, {5, 1335}, {6, 11}, {12, 3298}, {35, 5418}, {36, 6560}, {55, 590}, {56, 3070}, {227, 38487}, {273, 1659}, {350, 491}, {371, 1479}, {372, 499}, {382, 9647}, {388, 31412}, {390, 8972}, {486, 3301}, {495, 18538}, {496, 1124}, {497, 2066}, {498, 10576}, {615, 19037}, {946, 16232}, {950, 8983}, {999, 13665}, {1058, 13886}, {1131, 3600}, {1151, 6284}, {1152, 5433}, {1210, 2362}, {1377, 4187}, {1378, 24390}, {1478, 6564}, {1500, 31481}, {1587, 3086}, {1588, 10591}, {1697, 13893}, {1702, 9614}, {1737, 35774}, {1837, 7969}, {2276, 31463}, {2478, 31453}, {2961, 6204}, {3057, 13911}, {3058, 13846}, {3069, 10589}, {3071, 10896}, {3103, 10079}, {3295, 8976}, {3297, 19028}, {3299, 37720}, {3303, 13897}, {3311, 9669}, {3486, 13902}, {3583, 6561}, {3584, 42602}, {3585, 42269}, {3586, 9583}, {3594, 18966}, {3816, 31473}, {4293, 23249}, {4294, 9540}, {4299, 35820}, {4302, 6200}, {4309, 35812}, {4316, 42276}, {4330, 9680}, {5058, 9665}, {5204, 42259}, {5218, 32785}, {5225, 6459}, {5274, 7585}, {5393, 7133}, {5412, 11393}, {5432, 8253}, {6221, 9660}, {6409, 15338}, {6419, 35803}, {6420, 13962}, {6429, 9662}, {6460, 7288}, {6734, 30557}, {7173, 19027}, {7280, 42261}, {7354, 23251}, {7584, 10593}, {7951, 42277}, {7968, 11376}, {8252, 13958}, {8909, 12428}, {8960, 13905}, {8981, 15171}, {8988, 15558}, {9580, 9616}, {9581, 18991}, {9670, 31454}, {9678, 11113}, {10071, 10665}, {10072, 35769}, {10483, 22644}, {10573, 35641}, {10577, 13963}, {10624, 13912}, {10798, 44586}, {10819, 12896}, {10832, 44598}, {10874, 44604}, {10895, 42273}, {10950, 44620}, {10958, 44643}, {10959, 44645}, {11238, 19038}, {11502, 44590}, {11871, 44600}, {11872, 44602}, {11906, 44610}, {11932, 44627}, {11933, 44629}, {12053, 13883}, {12647, 35788}, {12943, 42284}, {12953, 42258}, {12958, 16502}, {13082, 13882}, {13889, 16541}, {13925, 15172}, {13954, 42583}, {13955, 32788}, {13973, 17606}, {14986, 31408}, {15325, 42216}, {15326, 42264}, {15867, 35816}, {18514, 22615}, {19048, 26476}, {19050, 26475}, {22760, 44606}, {24388, 42013}, {26387, 44582}, {26411, 44584}, {26473, 44594}, {26474, 44596}, {30384, 35775}, {30556, 41012}, {31584, 31588}

X(44623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 485, 31472), (5, 1335, 44622), (6, 11, 44624), (11, 19030, 6), (55, 13898, 590), (371, 35802, 1479), (496, 7583, 1124), (497, 3068, 2066), (1479, 13904, 371), (1587, 3086, 6502), (3295, 8976, 9646), (3297, 19028, 31475), (3298, 42265, 12), (3301, 7741, 486), (6221, 9668, 9660), (6284, 18965, 1151), (6564, 35768, 1478), (7173, 19027, 42262), (8960, 35808, 13905), (10576, 35809, 498), (10896, 18996, 3071), (19028, 37722, 3297)

X(44624) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND JOHNSON-YFF AND 2ND KENMOTU-CENTERS

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

X(44624) lies on these lines: {1, 486}, {2, 2066}, {3, 9660}, {4, 6502}, {5, 1124}, {6, 11}, {12, 3297}, {35, 5420}, {36, 6561}, {55, 615}, {56, 3071}, {273, 13390}, {350, 492}, {371, 499}, {372, 1479}, {388, 42561}, {390, 13941}, {485, 3299}, {495, 18762}, {496, 1335}, {497, 3069}, {498, 10577}, {590, 19038}, {946, 2362}, {950, 13971}, {999, 13785}, {1058, 13939}, {1132, 3600}, {1151, 5433}, {1152, 6284}, {1210, 16232}, {1377, 24390}, {1378, 4187}, {1478, 6565}, {1506, 31471}, {1587, 10591}, {1588, 2067}, {1656, 9646}, {1697, 13947}, {1698, 31432}, {1703, 9614}, {1737, 35775}, {1837, 7968}, {2886, 31473}, {2961, 6203}, {3057, 13973}, {3058, 13847}, {3068, 10589}, {3070, 10896}, {3091, 31408}, {3102, 10079}, {3295, 13951}, {3298, 19027}, {3301, 37720}, {3303, 13954}, {3311, 9661}, {3312, 9669}, {3486, 13959}, {3526, 31499}, {3583, 6560}, {3584, 42603}, {3585, 42268}, {3592, 18965}, {3815, 31459}, {4293, 23259}, {4294, 13935}, {4299, 35821}, {4302, 6396}, {4309, 35813}, {4316, 42275}, {5062, 9665}, {5204, 42258}, {5218, 32786}, {5225, 6460}, {5231, 31438}, {5274, 7586}, {5405, 7595}, {5413, 11393}, {5432, 8252}, {6398, 9668}, {6410, 15338}, {6419, 13904}, {6420, 35802}, {6437, 9663}, {6459, 7288}, {6734, 30556}, {7133, 24388}, {7173, 19028}, {7280, 42260}, {7354, 23261}, {7583, 10593}, {7951, 42274}, {7969, 11376}, {8253, 13901}, {9581, 18992}, {9616, 31231}, {9679, 13747}, {10071, 10666}, {10072, 35768}, {10483, 22615}, {10527, 31453}, {10573, 35642}, {10576, 13905}, {10624, 13975}, {10798, 44587}, {10820, 12896}, {10832, 44599}, {10874, 44605}, {10895, 42270}, {10950, 44621}, {10958, 44644}, {10959, 44646}, {11238, 19037}, {11502, 44591}, {11680, 31484}, {11871, 44601}, {11872, 44603}, {11906, 44611}, {11932, 44628}, {11933, 44630}, {12053, 13936}, {12647, 35789}, {12943, 42283}, {12953, 42259}, {12959, 16502}, {13081, 13934}, {13897, 42582}, {13898, 32787}, {13911, 17606}, {13943, 16541}, {13963, 35809}, {13966, 15171}, {13976, 15558}, {13993, 15172}, {15325, 42215}, {15326, 42263}, {15867, 35817}, {18514, 22644}, {19047, 26476}, {19049, 26475}, {22760, 44607}, {22791, 38235}, {26387, 44583}, {26411, 44585}, {26473, 44595}, {26474, 44597}, {30384, 35774}, {30557, 40490}, {31585, 31589}

X(44624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 486, 44622), (5, 1124, 31472), (6, 11, 44623), (11, 19029, 6), (55, 13955, 615), (372, 35803, 1479), (496, 7584, 1335), (497, 3069, 5414), (1124, 31472, 31475), (1479, 13962, 372), (1588, 3086, 2067), (3297, 42262, 12), (3299, 7741, 485), (6284, 18966, 1152), (6565, 35769, 1478), (7173, 19028, 42265), (10577, 35808, 498), (10896, 18995, 3070), (19027, 37722, 3298)

X(44625) = PERSPECTOR OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(3*a^10-(b^2+c^2)*a^8-10*(3*b^4+8*b^2*c^2+3*c^4)*a^6+2*(b^2+c^2)*(29*b^4+24*b^2*c^2+29*c^4)*a^4-37*(b^4-c^4)^2*a^2+8*(a^8-6*(b^2+c^2)*a^6+2*(4*b^4+b^2*c^2+4*c^4)*a^4-2*(b^2+c^2)*((b^2+c^2)^2-9*b^2*c^2)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2)*S+(b^2-c^2)^2*(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)) : :

X(44625) lies on these lines: {6, 19358}, {371, 18980}, {3070, 12257}, {10962, 19439}, {19410, 44633}

X(44626) = PERSPECTOR OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(3*a^10-(b^2+c^2)*a^8-10*(3*b^4+8*b^2*c^2+3*c^4)*a^6+2*(b^2+c^2)*(29*b^4+24*b^2*c^2+29*c^4)*a^4-37*(b^4-c^4)^2*a^2-8*(a^8-6*(b^2+c^2)*a^6+2*(4*b^4+b^2*c^2+4*c^4)*a^4-2*(b^2+c^2)*((b^2+c^2)^2-9*b^2*c^2)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2)*S+(b^2-c^2)^2*(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)) : :

X(44626) lies on these lines: {6, 19359}, {372, 18981}, {3071, 12256}, {10960, 19438}, {19411, 44634}

X(44627) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(3*a^8-32*(b^2+c^2)*a^6+2*(25*b^4+36*b^2*c^2+25*c^4)*a^4-8*(b^2+c^2)*(2*b^4-25*b^2*c^2+2*c^4)*a^2-4*(a^6+(b^2+c^2)*a^4-(17*b^4+74*b^2*c^2+17*c^4)*a^2+(b^2+c^2)*(7*b^4-26*b^2*c^2+7*c^4))*S-(5*b^4+58*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :

X(44627) lies on these lines: {6, 493}, {371, 10669}, {485, 8220}, {590, 8222}, {1124, 11953}, {1151, 11828}, {1152, 10238}, {1335, 11951}, {1587, 11846}, {2066, 11947}, {2067, 18963}, {3053, 8408}, {3068, 6392}, {3070, 9838}, {3071, 8212}, {3311, 11949}, {5412, 11394}, {6413, 6465}, {6419, 35807}, {6461, 44629}, {6564, 18520}, {7583, 32177}, {7968, 11377}, {7969, 12440}, {8188, 18991}, {8194, 44598}, {8201, 44600}, {8208, 44602}, {8210, 44635}, {8214, 13911}, {10875, 44604}, {10945, 44618}, {10951, 44620}, {11503, 44590}, {11840, 44586}, {11907, 44610}, {11930, 31472}, {11932, 44623}, {11955, 44643}, {11957, 44645}, {12152, 32787}, {12962, 17819}, {13956, 32788}, {22761, 44606}

X(44627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 493, 44628), (371, 35804, 10669), (493, 19032, 6), (8222, 13899, 590)

X(44628) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(-9*a^8+32*(b^2+c^2)*a^6-2*(19*b^4-20*b^2*c^2+19*c^4)*a^4+8*(b^2+c^2)*(2*b^4-11*b^2*c^2+2*c^4)*a^2+4*(3*a^6+7*(b^2+c^2)*a^4-(11*b^4+14*b^2*c^2+11*c^4)*a^2+(b^2+c^2)*(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2))*S+(b^2-c^2)^2*(-b^4+46*b^2*c^2-c^4)) : :

X(44628) lies on these lines: {6, 493}, {216, 44629}, {372, 10669}, {486, 8220}, {615, 8222}, {1124, 11951}, {1152, 11828}, {1335, 11953}, {1588, 11846}, {3069, 6462}, {3070, 8212}, {3071, 9838}, {3312, 11949}, {5013, 8408}, {5413, 11394}, {5414, 11947}, {6413, 8770}, {6420, 35804}, {6461, 44630}, {6465, 8911}, {6502, 18963}, {6565, 18520}, {7584, 32177}, {7968, 12440}, {7969, 11377}, {8188, 18992}, {8194, 44599}, {8201, 44601}, {8208, 44603}, {8210, 44636}, {8214, 13973}, {10875, 44605}, {10945, 44619}, {10951, 44621}, {11503, 44591}, {11840, 44587}, {11907, 44611}, {11930, 44622}, {11932, 44624}, {11955, 44644}, {11957, 44646}, {12152, 32788}, {13899, 32787}, {22761, 44607}

X(44628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 493, 44627), (372, 35807, 10669), (493, 19031, 6), (8222, 13956, 615)

X(44629) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND 1ST KENMOTU-CENTERS

Barycentrics    a^2*(-9*a^8+32*(b^2+c^2)*a^6-2*(19*b^4-20*b^2*c^2+19*c^4)*a^4+8*(b^2+c^2)*(2*b^4-11*b^2*c^2+2*c^4)*a^2-4*(3*a^6+7*(b^2+c^2)*a^4-(11*b^4+14*b^2*c^2+11*c^4)*a^2+(b^2+c^2)*(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2))*S+(b^2-c^2)^2*(-b^4+46*b^2*c^2-c^4)) : :

X(44629) lies on these lines: {6, 494}, {216, 44628}, {371, 10673}, {485, 8221}, {590, 8223}, {1124, 11954}, {1151, 11829}, {1335, 11952}, {1587, 11847}, {2066, 11948}, {2067, 18964}, {3068, 6463}, {3070, 9839}, {3071, 8213}, {3311, 11950}, {5013, 8420}, {5412, 11395}, {6414, 8770}, {6419, 35805}, {6461, 44627}, {6466, 26920}, {6564, 18522}, {7583, 32178}, {7968, 11378}, {7969, 12441}, {8189, 18991}, {8195, 44598}, {8202, 44600}, {8209, 44602}, {8211, 44635}, {8215, 13911}, {10876, 44604}, {10946, 44618}, {10952, 44620}, {11504, 44590}, {11841, 44586}, {11908, 44610}, {11931, 31472}, {11933, 44623}, {11956, 44643}, {11958, 44645}, {12153, 32787}, {13957, 32788}, {22762, 44606}

X(44629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 494, 44630), (371, 35806, 10673), (494, 19034, 6), (8223, 13900, 590)

X(44630) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND 2ND KENMOTU-CENTERS

Barycentrics    a^2*(3*a^8-32*(b^2+c^2)*a^6+2*(25*b^4+36*b^2*c^2+25*c^4)*a^4-8*(b^2+c^2)*(2*b^4-25*b^2*c^2+2*c^4)*a^2+4*(a^6+(b^2+c^2)*a^4-(17*b^4+74*b^2*c^2+17*c^4)*a^2+(b^2+c^2)*(7*b^4-26*b^2*c^2+7*c^4))*S-(5*b^4+58*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :

X(44630) lies on these lines: {6, 494}, {372, 10673}, {486, 8221}, {615, 8223}, {1124, 11952}, {1151, 10240}, {1152, 11829}, {1335, 11954}, {1588, 11847}, {3053, 8420}, {3069, 6392}, {3070, 8213}, {3071, 9839}, {3312, 11950}, {5413, 11395}, {5414, 11948}, {6414, 6466}, {6420, 35806}, {6461, 44628}, {6502, 18964}, {6565, 18522}, {7584, 32178}, {7968, 12441}, {7969, 11378}, {8189, 18992}, {8195, 44599}, {8202, 44601}, {8209, 44603}, {8211, 44636}, {8215, 13973}, {10876, 44605}, {10946, 44619}, {10952, 44621}, {11504, 44591}, {11841, 44587}, {11908, 44611}, {11931, 44622}, {11933, 44624}, {11956, 44644}, {11958, 44646}, {12153, 32788}, {12969, 17820}, {13900, 32787}, {22762, 44607}

X(44630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 494, 44629), (372, 35805, 10673), (494, 19033, 6), (8223, 13957, 615)

X(44631) = PERSPECTOR OF THESE TRIANGLES: MANDART-EXCIRCLES AND 1ST KENMOTU-CENTERS

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

X(44631) lies on this line: {65, 2067}

X(44632) = PERSPECTOR OF THESE TRIANGLES: MANDART-EXCIRCLES AND 2ND KENMOTU-CENTERS

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

X(44632) lies on this line: {65, 5416}

X(44633) = PERSPECTOR OF THESE TRIANGLES: MIDHEIGHT AND 1ST KENMOTU-CENTERS

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

X(44633) lies on these lines: {5, 371}, {6, 1585}, {25, 44588}, {51, 44637}, {125, 44638}, {389, 3070}, {393, 19040}, {577, 615}, {591, 19408}, {1151, 6809}, {1587, 11431}, {1588, 43841}, {3087, 19039}, {3093, 19360}, {3311, 8966}, {5412, 44616}, {6748, 13567}, {7687, 42283}, {8939, 44599}, {8969, 36412}, {17810, 44608}, {19410, 44625}

X(44633) = crosssum of X(3) and X(10960)
X(44633) = {X(6748), X(13567)}-harmonic conjugate of X(44634)

X(44634) = PERSPECTOR OF THESE TRIANGLES: MIDHEIGHT AND 2ND KENMOTU-CENTERS

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

X(44634) lies on these lines: {5, 372}, {6, 1586}, {25, 44589}, {51, 44638}, {125, 44637}, {389, 3071}, {393, 19039}, {577, 590}, {1152, 6810}, {1587, 43841}, {1588, 11431}, {1991, 19409}, {3087, 19040}, {3092, 19360}, {3284, 8969}, {3312, 13960}, {5413, 44617}, {6748, 13567}, {7687, 42284}, {8943, 44598}, {17810, 44609}, {19411, 44626}

X(44634) = crosssum of X(3) and X(10962)
X(44634) = {X(6748), X(13567)}-harmonic conjugate of X(44633)

X(44635) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH MIXTILINEAR AND 1ST KENMOTU-CENTERS

Barycentrics    a*(-2*(3*a^2+(b+c)*a+2*b^2+2*c^2)*S+(a+b+c)*(a^3-2*(b+c)*a^2-(3*b^2-4*b*c+3*c^2)*a+2*(b^2-c^2)*(b-c))) : :
Trilinears    2 r + R sin A : :
X(44635) = 4*X(8983)-3*X(13846) = 3*X(13846)-2*X(13911)

X(44635) lies on these lines: {1, 6}, {3, 35611}, {8, 590}, {10, 8253}, {36, 38235}, {40, 6409}, {55, 44606}, {56, 44590}, {145, 3068}, {175, 17365}, {176, 1086}, {230, 26369}, {355, 42265}, {371, 1482}, {372, 10246}, {485, 952}, {486, 5901}, {515, 23251}, {517, 1151}, {519, 8983}, {551, 13847}, {615, 3616}, {674, 6283}, {944, 3070}, {946, 23261}, {962, 42258}, {1125, 8252}, {1152, 1385}, {1317, 19028}, {1319, 2362}, {1377, 30144}, {1388, 6502}, {1483, 7583}, {1587, 7967}, {1588, 10595}, {1656, 35789}, {1702, 16200}, {1703, 6426}, {2066, 2098}, {2067, 2099}, {3069, 3622}, {3071, 5603}, {3083, 37679}, {3084, 37674}, {3241, 19066}, {3244, 13883}, {3303, 19014}, {3304, 19000}, {3311, 10247}, {3312, 35762}, {3576, 6410}, {3579, 6411}, {3592, 10222}, {3594, 15178}, {3617, 32785}, {3621, 8972}, {3623, 7585}, {3632, 13893}, {3633, 13888}, {3636, 13971}, {3878, 9678}, {3913, 22763}, {4000, 17805}, {4644, 17802}, {5412, 11396}, {5414, 34471}, {5418, 5690}, {5420, 38028}, {5550, 32790}, {5597, 44602}, {5598, 44600}, {5731, 42259}, {5790, 10576}, {5818, 42582}, {5844, 8981}, {5854, 13922}, {5886, 42262}, {6200, 12702}, {6221, 8148}, {6412, 13624}, {6419, 35811}, {6425, 7982}, {6429, 9585}, {6430, 30392}, {6431, 33179}, {6433, 9582}, {6437, 11278}, {6560, 34773}, {6561, 22791}, {6564, 18525}, {6565, 18493}, {7362, 8679}, {7584, 10283}, {7991, 9615}, {8192, 44598}, {8210, 44627}, {8211, 44629}, {8960, 35842}, {8976, 12645}, {9041, 24842}, {9053, 13910}, {9540, 12245}, {9617, 10141}, {9646, 12647}, {9661, 10573}, {9780, 32789}, {9812, 42271}, {9997, 44604}, {10671, 11707}, {10672, 11708}, {10800, 44586}, {10944, 31472}, {10950, 44620}, {11011, 16232}, {11910, 44610}, {12269, 13881}, {12454, 13891}, {12455, 13890}, {12699, 42263}, {13665, 18526}, {13912, 28234}, {13947, 25055}, {13959, 32788}, {15950, 44622}, {18357, 42277}, {18481, 42264}, {18538, 37705}, {19008, 44603}, {19010, 44601}, {19030, 37734}, {20035, 36581}, {20070, 42638}, {22615, 40273}, {22644, 28186}, {26395, 44582}, {26419, 44584}, {26514, 44594}, {26515, 44596}, {28174, 42260}, {38022, 42603}, {38034, 42268}

X(44635) = reflection of X(13911) in X(8983)
X(44635) = intersection, other than A,B,C, of conics {{A, B, C, X(56), X(3299)}} and {{A, B, C, X(81), X(44636)}}
X(44635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44636), (1, 3641, 5604), (1, 7969, 6), (1, 18991, 7968), (8, 13902, 590), (371, 35810, 1482), (1124, 19048, 6), (1125, 13973, 8252), (1335, 19050, 6), (1385, 35774, 1152), (3303, 19014, 44591), (3311, 10247, 35642), (3312, 37624, 35762), (3616, 19065, 615), (6221, 8148, 35610), (7968, 7969, 18991), (7968, 18991, 6), (8976, 12645, 35788), (8983, 13911, 13846), (10576, 35843, 5790), (35641, 35763, 3), (44643, 44645, 6)

X(44636) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 2ND KENMOTU-CENTERS

Barycentrics    a*(2*(3*a^2+(b+c)*a+2*b^2+2*c^2)*S+(a+b+c)*(a^3-2*(b+c)*a^2-(3*b^2-4*b*c+3*c^2)*a+2*(b^2-c^2)*(b-c))) : :
Trilinears    2 r - R sin A : :
X(44636) = 3*X(13847)-4*X(13971) = 3*X(13847)-2*X(13973)

X(44636) lies on these lines: {1, 6}, {3, 35610}, {8, 615}, {10, 8252}, {40, 6410}, {55, 44607}, {56, 44591}, {145, 3069}, {175, 1086}, {176, 17365}, {214, 9679}, {230, 26370}, {355, 42262}, {371, 10246}, {372, 1482}, {485, 5901}, {486, 952}, {515, 23261}, {517, 1152}, {519, 13847}, {551, 13846}, {590, 3616}, {674, 6405}, {944, 3071}, {946, 23251}, {962, 42259}, {1125, 8253}, {1151, 1385}, {1317, 19027}, {1319, 16232}, {1378, 30144}, {1388, 2067}, {1483, 7584}, {1587, 10595}, {1588, 7967}, {1656, 35788}, {1702, 6425}, {1703, 16200}, {2066, 34471}, {2098, 5414}, {2099, 6502}, {2362, 11011}, {3068, 3622}, {3070, 5603}, {3083, 37674}, {3084, 37679}, {3241, 19065}, {3244, 13936}, {3303, 19013}, {3304, 18999}, {3311, 35763}, {3312, 10247}, {3576, 6409}, {3579, 6412}, {3592, 15178}, {3594, 10222}, {3617, 32786}, {3621, 13941}, {3623, 7586}, {3632, 13947}, {3633, 13942}, {3636, 8983}, {3913, 22764}, {4000, 17802}, {4644, 17805}, {5413, 11396}, {5418, 38028}, {5420, 5690}, {5550, 32789}, {5597, 44603}, {5598, 44601}, {5731, 42258}, {5790, 10577}, {5818, 42583}, {5844, 13966}, {5854, 13991}, {5886, 42265}, {6396, 12702}, {6398, 8148}, {6411, 13624}, {6420, 35810}, {6426, 7982}, {6429, 9615}, {6430, 11531}, {6432, 33179}, {6438, 11278}, {6560, 22791}, {6561, 34773}, {6564, 18493}, {6565, 18525}, {7353, 8679}, {7583, 10283}, {8192, 44599}, {8210, 44628}, {8211, 44630}, {9041, 24843}, {9053, 13972}, {9616, 30389}, {9618, 10141}, {9780, 32790}, {9812, 42272}, {9997, 44605}, {10667, 11707}, {10668, 11708}, {10800, 44587}, {10944, 44619}, {10950, 44621}, {11009, 38235}, {11910, 44611}, {12245, 13935}, {12268, 13881}, {12454, 13945}, {12455, 13944}, {12645, 13951}, {12699, 42264}, {13384, 31432}, {13785, 18526}, {13893, 25055}, {13902, 32787}, {13975, 28234}, {15950, 31472}, {18357, 42274}, {18481, 42263}, {18762, 37705}, {19007, 44602}, {19009, 44600}, {19029, 37734}, {20035, 36582}, {20070, 42637}, {22615, 28186}, {22644, 40273}, {26395, 44583}, {26419, 44585}, {26514, 44595}, {26515, 44597}, {28174, 42261}, {38022, 42602}, {38034, 42269}

X(44636) = reflection of X(13973) in X(13971)
X(44636) = intersection, other than A,B,C, of conics {{A, B, C, X(56), X(3301)}} and {{A, B, C, X(81), X(44635)}}
X(44636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44635), (1, 3640, 5605), (1, 7968, 6), (1, 18992, 7969), (8, 13959, 615), (372, 35811, 1482), (1124, 19049, 6), (1125, 13911, 8253), (1335, 19047, 6), (1385, 35775, 1151), (3303, 19013, 44590), (3304, 18999, 44606), (3311, 37624, 35763), (3312, 10247, 35641), (3616, 19066, 590), (6398, 8148, 35611), (7968, 7969, 18992), (7969, 18992, 6), (10577, 35842, 5790), (12645, 13951, 35789), (13971, 13973, 13847), (35642, 35762, 3), (44644, 44646, 6)

X(44637) = PERSPECTOR OF THESE TRIANGLES: ORTHIC AND 1ST KENMOTU-CENTERS

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

X(44637) lies on these lines: {4, 371}, {6, 32588}, {25, 8939}, {51, 44633}, {113, 18539}, {125, 44634}, {155, 3092}, {185, 3070}, {193, 13428}, {230, 427}, {264, 44365}, {317, 492}, {393, 19042}, {1162, 26375}, {1249, 26462}, {2914, 12376}, {3071, 3574}, {3087, 3127}, {3535, 26361}, {3575, 13019}, {5895, 23251}, {8911, 8968}, {8966, 10132}, {13202, 42284}, {13748, 19355}, {21640, 44612}, {26456, 40065}, {26953, 36656}, {44588, 44598}

X(44637) = polar conjugate of the isotomic conjugate of X(641)
X(44637) = barycentric product X(i)*X(j) for these {i, j}: {4, 641}, {590, 1585}
X(44637) = barycentric quotient X(i)/X(j) for these (i, j): (590, 11090), (641, 69)
X(44637) = trilinear product X(19)*X(641)
X(44637) = trilinear quotient X(641)/X(63)
X(44637) = crosspoint of X(4) and X(1585)
X(44637) = crosssum of X(3) and X(6413)
X(44637) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (590, 11090), (641, 69)
X(44637) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3068, 41515), (427, 6748, 44638)

X(44638) = PERSPECTOR OF THESE TRIANGLES: ORTHIC AND 2ND KENMOTU-CENTERS

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

X(44638) lies on these lines: {4, 372}, {6, 32587}, {25, 8943}, {51, 44634}, {113, 26438}, {125, 44633}, {155, 3093}, {185, 3071}, {193, 13439}, {230, 427}, {264, 44364}, {317, 491}, {393, 19041}, {1163, 5200}, {1249, 26457}, {2914, 12375}, {3070, 3574}, {3087, 3128}, {3536, 26362}, {3575, 13020}, {5895, 23261}, {10133, 13960}, {13202, 42283}, {13749, 19356}, {21641, 44613}, {26463, 40065}, {44589, 44599}

X(44638) = polar conjugate of the isotomic conjugate of X(642)
X(44638) = barycentric product X(i)*X(j) for these {i, j}: {4, 642}, {615, 1586}
X(44638) = barycentric quotient X(i)/X(j) for these (i, j): (615, 11091), (642, 69)
X(44638) = trilinear product X(19)*X(642)
X(44638) = trilinear quotient X(642)/X(63)
X(44638) = crosspoint of X(4) and X(1586)
X(44638) = crosssum of X(3) and X(6414)
X(44638) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (615, 11091), (642, 69)
X(44638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3069, 41516), (427, 6748, 44637)

X(44639) = PERSPECTOR OF THESE TRIANGLES: ORTHOCENTROIDAL AND 1ST KENMOTU-CENTERS

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

X(44639) lies on these lines: {125, 44640}, {371, 381}, {3070, 5890}, {3071, 7699}, {34417, 44592}

X(44640) = PERSPECTOR OF THESE TRIANGLES: ORTHOCENTROIDAL AND 2ND KENMOTU-CENTERS

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

X(44640) lies on these lines: {125, 44639}, {372, 381}, {3070, 7699}, {3071, 5890}, {34417, 44593}

X(44641) = PERSPECTOR OF THESE TRIANGLES: REFLECTION AND 1ST KENMOTU-CENTERS

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

X(44641) lies on these lines: {371, 382}, {3070, 6241}

X(44642) = PERSPECTOR OF THESE TRIANGLES: REFLECTION AND 2ND KENMOTU-CENTERS

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

X(44642) lies on these lines: {372, 382}, {3071, 6241}

X(44643) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS AND 1ST KENMOTU-CENTERS

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

X(44643) lies on these lines: {1, 6}, {12, 44618}, {119, 42265}, {230, 26523}, {371, 10679}, {372, 16203}, {485, 10942}, {590, 5552}, {1151, 11248}, {1152, 10269}, {1587, 10805}, {1588, 10596}, {2066, 10965}, {2067, 11509}, {2077, 6409}, {3068, 10528}, {3069, 10586}, {3070, 12115}, {3071, 10531}, {3311, 12000}, {3592, 37622}, {5412, 11400}, {5414, 22768}, {6200, 35251}, {6256, 23251}, {6283, 12329}, {6410, 37561}, {6419, 35817}, {6564, 18542}, {7583, 32213}, {8252, 10200}, {8253, 26364}, {10803, 44586}, {10834, 44598}, {10878, 44604}, {10915, 13911}, {10955, 19030}, {10956, 31472}, {10958, 44623}, {11239, 32787}, {11881, 44600}, {11882, 44602}, {11914, 44610}, {11955, 44627}, {11956, 44629}, {13665, 18545}, {13904, 15867}, {13964, 32788}, {15888, 19024}, {23261, 26333}, {26402, 44582}, {26426, 44584}, {26520, 44594}, {26525, 44596}, {34339, 35774}, {37722, 44619}

X(44643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44644), (1, 19048, 6), (1, 26343, 10930), (1, 26465, 19047), (6, 3298, 44646), (6, 44635, 44645), (371, 35816, 10679), (1335, 7969, 6), (3301, 19050, 6), (5552, 13906, 590), (18991, 19049, 6), (19047, 19048, 26465), (19047, 26465, 6)

X(44644) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS AND 2ND KENMOTU-CENTERS

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

X(44644) lies on these lines: {1, 6}, {12, 44619}, {119, 42262}, {230, 26518}, {371, 16203}, {372, 10679}, {486, 10942}, {615, 5552}, {1151, 10269}, {1152, 11248}, {1587, 10596}, {1588, 10805}, {2066, 22768}, {2077, 6410}, {3068, 10586}, {3069, 10528}, {3070, 10531}, {3071, 12115}, {3312, 12000}, {3594, 37622}, {5413, 11400}, {5414, 10965}, {5554, 31473}, {6256, 23261}, {6396, 35251}, {6405, 12329}, {6409, 37561}, {6420, 35816}, {6502, 11509}, {6565, 18542}, {7584, 32213}, {8252, 26364}, {8253, 10200}, {10803, 44587}, {10834, 44599}, {10878, 44605}, {10915, 13973}, {10955, 19029}, {10956, 44622}, {10958, 44624}, {11239, 32788}, {11881, 44601}, {11882, 44603}, {11914, 44611}, {11955, 44628}, {11956, 44630}, {13785, 18545}, {13906, 32787}, {13962, 15867}, {15888, 19023}, {23251, 26333}, {26402, 44583}, {26426, 44585}, {26520, 44595}, {26525, 44597}, {34339, 35775}, {37722, 44618}

X(44644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44643), (1, 19047, 6), (1, 26350, 10929), (1, 26459, 19048), (6, 3297, 44645), (6, 44636, 44646), (372, 35817, 10679), (1124, 7968, 6), (3299, 19049, 6), (5552, 13964, 615), (18992, 19050, 6), (19047, 19048, 26459), (19048, 26459, 6)

X(44645) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS AND 1ST KENMOTU-CENTERS

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

X(44645) lies on these lines: {1, 6}, {11, 44620}, {230, 26522}, {371, 10680}, {372, 16202}, {485, 10943}, {590, 10527}, {1151, 11249}, {1152, 10267}, {1587, 10806}, {1588, 10597}, {2066, 10966}, {2067, 18967}, {3068, 10529}, {3069, 10587}, {3070, 12116}, {3071, 10532}, {3311, 12001}, {5412, 11401}, {5415, 44607}, {6200, 35252}, {6409, 11012}, {6410, 10902}, {6419, 35819}, {6426, 34486}, {6502, 11510}, {6564, 18544}, {7362, 22769}, {7583, 32214}, {8252, 10198}, {8253, 26363}, {10804, 44586}, {10835, 44598}, {10879, 44604}, {10916, 13911}, {10949, 19028}, {10957, 31472}, {10959, 44623}, {11240, 32787}, {11883, 44600}, {11884, 44602}, {11915, 44610}, {11957, 44627}, {11958, 44629}, {13665, 18543}, {13905, 15868}, {13965, 32788}, {15888, 44621}, {19026, 37722}, {23261, 26332}, {26401, 44582}, {26425, 44584}, {26470, 42265}, {26519, 44594}, {26524, 44596}

X(44645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44646), (1, 19050, 6), (1, 26342, 10932), (1, 26464, 19049), (6, 3297, 44644), (6, 44635, 44643), (371, 35818, 10680), (1124, 7969, 6), (3299, 19048, 6), (10527, 13907, 590), (18991, 19047, 6), (19049, 19050, 26464), (19049, 26464, 6)

X(44646) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS AND 2ND KENMOTU-CENTERS

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

X(44646) lies on these lines: {1, 6}, {11, 44621}, {230, 26517}, {371, 16202}, {372, 10680}, {486, 10943}, {615, 10527}, {1151, 10267}, {1152, 11249}, {1587, 10597}, {1588, 10806}, {2067, 11510}, {3068, 10587}, {3069, 10529}, {3070, 10532}, {3071, 12116}, {3312, 12001}, {5413, 11401}, {5414, 10966}, {5416, 44606}, {6396, 35252}, {6409, 10902}, {6410, 11012}, {6420, 35818}, {6425, 34486}, {6502, 18967}, {6565, 18544}, {7353, 22769}, {7584, 32214}, {8252, 26363}, {8253, 10198}, {10804, 44587}, {10835, 44599}, {10879, 44605}, {10916, 13973}, {10949, 19027}, {10957, 44622}, {10959, 44624}, {11240, 32788}, {11883, 44601}, {11884, 44603}, {11915, 44611}, {11957, 44628}, {11958, 44630}, {13785, 18543}, {13907, 32787}, {13963, 15868}, {15888, 44620}, {19025, 37722}, {23251, 26332}, {26401, 44583}, {26425, 44585}, {26470, 42262}, {26519, 44595}, {26524, 44597}

X(44646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 44645), (1, 19049, 6), (1, 26349, 10931), (1, 26458, 19050), (6, 3298, 44643), (6, 44636, 44644), (372, 35819, 10680), (1335, 7968, 6), (3301, 19047, 6), (10527, 13965, 615), (18992, 19048, 6), (19049, 19050, 26458), (19050, 26458, 6)

X(44647) = PERSPECTOR OF THESE TRIANGLES 1ST KENMOTU-CENTERS AND PEDAL-OF-X(371)

Barycentrics    (a^2+S)*(a^2+2*S) : :
X(44647) = X(32497)-3*X(32787)

X(44647) lies on these lines: {4, 44594}, {5, 6}, {30, 12962}, {32, 32421}, {187, 31454}, {193, 5490}, {230, 8960}, {371, 12961}, {395, 6304}, {396, 6305}, {488, 3068}, {491, 7881}, {524, 42009}, {574, 31483}, {590, 641}, {615, 6118}, {631, 8376}, {732, 13877}, {1124, 10084}, {1151, 12124}, {1335, 10068}, {1384, 31487}, {1504, 3070}, {1506, 13880}, {1587, 6422}, {1991, 3933}, {2066, 13082}, {2067, 18988}, {3055, 35813}, {3071, 6250}, {3311, 12602}, {3312, 31463}, {3592, 7737}, {3594, 31401}, {3815, 6420}, {5023, 22541}, {5024, 31465}, {5206, 41963}, {5210, 9680}, {5254, 35822}, {5412, 12148}, {6199, 22810}, {6337, 35306}, {6419, 7745}, {6421, 7581}, {6424, 7585}, {6427, 15484}, {6459, 12297}, {6460, 9600}, {6560, 44519}, {6561, 22646}, {6564, 22625}, {7603, 43880}, {7746, 43879}, {7968, 12269}, {7981, 44635}, {8253, 13771}, {8787, 13874}, {8981, 12968}, {9907, 18991}, {9922, 44598}, {9987, 44604}, {12159, 27088}, {12211, 44586}, {12344, 44590}, {12486, 44600}, {12487, 44602}, {12788, 13911}, {12800, 44610}, {12829, 13873}, {12929, 44618}, {12939, 44620}, {12949, 31472}, {12959, 44623}, {12969, 31406}, {13004, 44627}, {13005, 44629}, {13134, 44643}, {13135, 44645}, {13665, 32499}, {13886, 44595}, {14229, 14912}, {18362, 41952}, {19053, 31404}, {22623, 35685}, {22624, 44606}, {26460, 44637}, {26462, 31412}, {31414, 43448}, {32491, 44392}, {35815, 41411}, {35945, 37839}, {37512, 41946}

X(44647) = midpoint of X(371) and X(35832)
X(44647) = reflection of X(485) in X(7583)
X(44647) = barycentric product X(590)*X(3068)
X(44647) = barycentric quotient X(590)/X(5490)
X(44647) = X(590)-reciprocal conjugate of-X(5490)
X(44647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 2548, 44648), (6, 13881, 19102), (485, 19102, 13881), (488, 3068, 13882), (641, 13879, 590), (7581, 31403, 6421), (19105, 22592, 13881)

X(44648) = PERSPECTOR OF THESE TRIANGLES 2ND KENMOTU-CENTERS AND PEDAL-OF-X(372)

Barycentrics    (a^2-S)*(a^2-2*S) : :
X(44648) = X(32494)-3*X(32788)

X(44648) lies on these lines: {4, 44597}, {5, 6}, {30, 12969}, {32, 32419}, {193, 5491}, {372, 12966}, {395, 6300}, {396, 6301}, {487, 3069}, {492, 7881}, {524, 42060}, {590, 6119}, {591, 3933}, {615, 642}, {631, 8375}, {732, 13930}, {1124, 10067}, {1152, 12123}, {1335, 10083}, {1505, 3071}, {1506, 13921}, {1588, 6421}, {3055, 35812}, {3070, 6251}, {3311, 31467}, {3312, 12601}, {3592, 31401}, {3594, 7737}, {3815, 6419}, {5023, 19101}, {5206, 41964}, {5254, 35823}, {5413, 12147}, {5414, 13081}, {6337, 35305}, {6395, 22809}, {6417, 31463}, {6420, 7745}, {6422, 7582}, {6423, 7586}, {6428, 15484}, {6460, 12296}, {6502, 18989}, {6560, 22617}, {6561, 44519}, {6565, 22596}, {7603, 43879}, {7746, 43880}, {7969, 12268}, {7980, 44636}, {8252, 13650}, {8787, 13927}, {9906, 18992}, {9921, 44599}, {9986, 44605}, {12158, 27088}, {12210, 44587}, {12343, 44591}, {12484, 44601}, {12485, 44603}, {12787, 13973}, {12799, 44611}, {12829, 13926}, {12928, 44619}, {12938, 44621}, {12948, 44622}, {12958, 44624}, {12962, 31406}, {12963, 13966}, {13002, 44628}, {13003, 44630}, {13132, 44644}, {13133, 44646}, {13785, 32498}, {13939, 44596}, {14244, 14912}, {18362, 41951}, {19054, 31404}, {22594, 35684}, {22595, 44607}, {26455, 44638}, {26457, 42561}, {31454, 31455}, {32490, 44394}, {35814, 41410}, {37512, 41945}

X(44648) = midpoint of X(372) and X(35833)
X(44648) = reflection of X(486) in X(7584)
X(44648) = barycentric product X(615)*X(3069)
X(44648) = barycentric quotient X(615)/X(5491)
X(44648) = X(615)-reciprocal conjugate of-X(5491)
X(44648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 2548, 44647), (6, 13881, 19105), (486, 19104, 6), (486, 19105, 13881), (487, 3069, 13934), (642, 13933, 615), (13770, 19104, 486), (19102, 22591, 13881)

X(44649) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(23)

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

See the preamble just before X(44575).

X(44649) lies on these lines: {2, 3}, {525, 597}, {543, 40477}, {3734, 24975}, {5663, 41145}, {7804, 18122}

X(44650) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(25)

Barycentrics    4*a^10 - 3*a^8*b^2 - 4*a^6*b^4 + 2*a^4*b^6 + b^10 - 3*a^8*c^2 + 14*a^6*b^2*c^2 - 4*a^4*b^4*c^2 - 10*a^2*b^6*c^2 + 3*b^8*c^2 - 4*a^6*c^4 - 4*a^4*b^2*c^4 + 20*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 - 10*a^2*b^2*c^6 - 4*b^4*c^6 + 3*b^2*c^8 + c^10 : :

X(44650) lies on these lines: {2, 3}, {525, 14398}, {538, 3163}, {1494, 32695}, {6000, 41145}, {8749, 16076}

X(44651) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(140)

Barycentrics    5*a^8 - 11*a^6*b^2 + 6*a^4*b^4 + a^2*b^6 - b^8 - 11*a^6*c^2 + 5*a^4*b^2*c^2 - a^2*b^4*c^2 + 7*b^6*c^2 + 6*a^4*c^4 - a^2*b^2*c^4 - 12*b^4*c^4 + a^2*c^6 + 7*b^2*c^6 - c^8 : :

X(44651) lies on these lines: {2, 3}, {287, 19924}, {323, 20094}, {543, 40870}, {1272, 44363}, {6781, 41254}, {11645, 40867}, {18831, 43768}, {29317, 41145}, {39359, 41136}

X(44651) = anticomplement of X(40885)

X(44652) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(237)

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

X(44652) lies on these lines: {2, 3}, {525, 7757}, {2421, 32833}, {2966, 12150}, {8716, 34360}

X(44653) = SELF-DUAL-PERMUTATION-ELLIPSE-INVERSE OF X(402)

Barycentrics    5*a^16 - 10*a^14*b^2 - 10*a^12*b^4 + 35*a^10*b^6 - 16*a^8*b^8 - 16*a^6*b^10 + 14*a^4*b^12 - a^2*b^14 - b^16 - 10*a^14*c^2 + 50*a^12*b^2*c^2 - 45*a^10*b^4*c^2 - 61*a^8*b^6*c^2 + 98*a^6*b^8*c^2 - 18*a^4*b^10*c^2 - 19*a^2*b^12*c^2 + 5*b^14*c^2 - 10*a^12*c^4 - 45*a^10*b^2*c^4 + 159*a^8*b^4*c^4 - 82*a^6*b^6*c^4 - 84*a^4*b^8*c^4 + 63*a^2*b^10*c^4 - b^12*c^4 + 35*a^10*c^6 - 61*a^8*b^2*c^6 - 82*a^6*b^4*c^6 + 176*a^4*b^6*c^6 - 43*a^2*b^8*c^6 - 25*b^10*c^6 - 16*a^8*c^8 + 98*a^6*b^2*c^8 - 84*a^4*b^4*c^8 - 43*a^2*b^6*c^8 + 44*b^8*c^8 - 16*a^6*c^10 - 18*a^4*b^2*c^10 + 63*a^2*b^4*c^10 - 25*b^6*c^10 + 14*a^4*c^12 - 19*a^2*b^2*c^12 - b^4*c^12 - a^2*c^14 + 5*b^2*c^14 - c^16 : :

X(44653) lies on these lines: {2, 3}, {525, 39358}, {1494, 39062}, {3163, 23582}, {16076, 39352}

X(44654) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF 1ST KENMOTU-CENTERS TRIANGLE

Barycentrics    a^2*(-2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44654) = 3*X(6)-X(9733) = 3*X(6)-2*X(44476) = 3*X(182)-2*X(43141) = 3*X(576)-X(44471) = X(1160)-5*X(11482) = X(1161)+3*X(1351) = 3*X(15520)-X(42859) = X(44471)+3*X(44472) = 2*X(44471)-3*X(44655) = 2*X(44472)+X(44655)

The squared-radius of this circle is: ρ2 = (2+9*tan(ω)^2+6*tan(ω)+2*cot(ω)+cot(ω)^2)*R^2/4
Centers X(44654)-X(44657) were contributed by César Lozada, Sept. 01, 2021.

X(44654) lies on these lines: {3, 6}, {542, 13748}, {6289, 34507}, {14853, 32489}, {30428, 34117}

X(44654) = midpoint of X(i) and X(j) for these {i, j}: {576, 44472}, {9732, 11477}, {37517, 42858}
X(44654) = reflection of X(i) in X(j) for these (i, j): (3, 44475), (9733, 44476), (9739, 575), (44655, 576)
X(44654) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(44510)}} and {{A, B, C, X(4), X(39655)}}
X(44654) = Brocard circle-inverse of-X(44510)
X(44654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44510), (3, 8376, 43121), (3, 9974, 44486), (6, 9733, 44476), (6, 39679, 44657), (6, 43119, 575), (182, 576, 44502), (371, 372, 39655), (576, 44473, 5097), (576, 44485, 6), (576, 44486, 9974), (1351, 9975, 576)

X(44655) = CENTER OF LOZADA-LEMOINE-CIRCLE-1A OF 2ND KENMOTU-CENTERS TRIANGLE

Barycentrics    a^2*(2*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*S+(a^2+b^2+c^2)*((b^2+c^2)*a^2-b^4-c^4)) : :
X(44655) = 3*X(6)-X(9732) = 3*X(6)-2*X(44475) = 3*X(182)-2*X(43144) = 3*X(576)-X(44472) = X(1160)+3*X(1351) = X(1161)-5*X(11482) = 3*X(15520)-X(42858) = 3*X(44471)+X(44472) = 2*X(44471)+X(44654) = 2*X(44472)-3*X(44654)

The squared-radius of this circle is: ρ2 = (2+9*tan(ω)^2-6*tan(ω)-2*cot(ω)+cot(ω)^2)*R^2/4

X(44655) lies on these lines: {3, 6}, {542, 13749}, {6290, 34507}, {14853, 32488}, {30427, 34117}

X(44655) = midpoint of X(i) and X(j) for these {i, j}: {576, 44471}, {9733, 11477}, {37517, 42859}
X(44655) = reflection of X(i) in X(j) for these (i, j): (3, 44476), (9732, 44475), (9738, 575), (44654, 576)
X(44655) = Brocard circle-inverse of-X(44509)
X(44655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44509), (3, 8375, 43120), (6, 9732, 44475), (6, 39648, 44656), (6, 43118, 575), (182, 576, 44501), (371, 372, 39654), (576, 44474, 5097), (576, 44486, 6), (1351, 9974, 576)

X(44656) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF 1ST KENMOTU-CENTERS TRIANGLE

Barycentrics    a^2*(-2*(a^2+b^2+c^2)*S+2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4) : :
X(44656) = 3*X(6)+X(1151) = 3*X(6)-X(9974) = 3*X(182)-X(7690) = 3*X(597)-X(23311) = X(1151)-3*X(19145) = 3*X(5476)-X(22819) = X(9974)+3*X(19145) = 3*X(13910)-2*X(13925)

The squared-radius of this circle is: ρ2 = R^2*S^2*(9*S^2+12*SW*S+5*SW^2)/(2*SW*(SW+2*S))^2

X(44656) lies on these lines: {3, 6}, {114, 13638}, {485, 542}, {486, 25555}, {524, 641}, {538, 22623}, {590, 34507}, {597, 7584}, {1503, 6250}, {1587, 11179}, {3071, 5476}, {3299, 7362}, {3301, 6283}, {3564, 13879}, {3618, 13939}, {3818, 42273}, {5032, 42522}, {5418, 40107}, {5420, 10168}, {5965, 8995}, {6291, 44102}, {6459, 20423}, {7582, 12322}, {7583, 8550}, {7804, 22594}, {8541, 10880}, {8584, 41490}, {8681, 8909}, {8976, 15069}, {10576, 11178}, {11645, 23251}, {13935, 38064}, {14561, 39876}, {18553, 42265}, {19130, 42268}, {19924, 42260}, {22165, 43211}, {22644, 29012}, {31670, 43408}, {38317, 42583}

X(44656) = midpoint of X(i) and X(j) for these {i, j}: {6, 19145}, {576, 12974}, {1151, 9974}, {5023, 9975}
X(44656) = reflection of X(44510) in X(575)
X(44656) = Brocard circle-inverse of-X(44502)
X(44656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44502), (6, 371, 576), (6, 372, 44474), (6, 575, 44657), (6, 1151, 9974), (6, 3311, 44501), (6, 3592, 9975), (6, 5038, 1505), (6, 6422, 44499), (6, 6424, 44500), (6, 9975, 22330), (6, 35841, 15520), (6, 39648, 44655), (182, 576, 9739), (182, 44472, 3), (182, 44474, 372), (371, 5062, 9739), (371, 12968, 12974), (372, 1151, 7690), (575, 44475, 182), (575, 44482, 6), (575, 44483, 5050), (3364, 3389, 1504), (9974, 19145, 1151), (39561, 42833, 6)

X(44657) = CENTER OF LOZADA-LEMOINE-CIRCLE-1B OF 2ND KENMOTU-CENTERS TRIANGLE

Barycentrics    a^2*(2*(a^2+b^2+c^2)*S+2*a^4-3*(b^2+c^2)*a^2+b^4-4*b^2*c^2+c^4) : :
X(44657) = 3*X(6)+X(1152) = 3*X(6)-X(9975) = 3*X(182)-X(7692) = 3*X(597)-X(23312) = X(1152)-3*X(19146) = 3*X(5476)-X(22820) = X(9975)+3*X(19146) = 3*X(13972)-2*X(13993)

The squared-radius of this circle is: ρ2 = R^2*S^2*(9*S^2-12*SW*S+5*SW^2)/(2*SW*(SW-2*S))^2

X(44657) lies on these lines: {3, 6}, {114, 13758}, {485, 25555}, {486, 542}, {524, 642}, {538, 22594}, {597, 7583}, {615, 34507}, {1503, 6251}, {1588, 11179}, {3070, 5476}, {3299, 6405}, {3301, 7353}, {3564, 13880}, {3618, 13886}, {3818, 42270}, {5032, 42523}, {5418, 10168}, {5420, 40107}, {5965, 13986}, {6406, 44102}, {6460, 20423}, {7581, 12323}, {7584, 8550}, {7804, 22623}, {8541, 10881}, {8584, 41491}, {9540, 38064}, {10577, 11178}, {11645, 23261}, {13951, 15069}, {14561, 31412}, {18553, 42262}, {19130, 42269}, {19924, 42261}, {22165, 43212}, {22615, 29012}, {31670, 43407}, {38317, 42582}

X(44657) = midpoint of X(i) and X(j) for these {i, j}: {6, 19146}, {576, 12975}, {1152, 9975}, {5023, 9974}
X(44657) = reflection of X(44509) in X(575)
X(44657) = Brocard circle-inverse of-X(44501)
X(44657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 6, 44501), (6, 371, 44473), (6, 372, 576), (6, 575, 44656), (6, 1152, 9975), (6, 3312, 44502), (6, 3594, 9974), (6, 5038, 1504), (6, 6420, 44474), (6, 6421, 44499), (6, 6423, 44500), (6, 9974, 22330), (6, 35840, 15520), (6, 39679, 44654), (182, 44471, 3), (182, 44473, 371), (182, 44474, 39), (371, 1152, 7692), (372, 5058, 9738), (575, 44476, 182), (575, 44481, 6), (575, 44484, 5050), (3365, 3390, 1505), (9975, 19146, 1152), (22234, 44474, 6), (39561, 42832, 6)

X(44658) = X(468)X(31489)∩X(524)X(5050)

Barycentrics    9 a^8-36 a^6 (b^2+c^2)+a^4 (44 b^4+56 b^2 c^2+44 c^4)-20 a^2 (b^2-c^2)^2 (b^2+c^2)+(b^2-c^2)^2 (3 b^4-10 b^2 c^2+3 c^4) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2187.

X(44658) lies on these lines: {468,31489}, {524,5050}, {3266,34229}

X(44659) = X(1)X(15)∩X(30)X(511)

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

X(44659) lies on these lines: {1, 15}, {8, 621}, {10, 623}, {16, 5184}, {30, 511}, {40, 14538}, {65, 3639}, {187, 11708}, {238, 39150}, {298, 12781}, {355, 20428}, {396, 11705}, {944, 36993}, {946, 7684}, {1125, 6671}, {1385, 13350}, {1482, 5611}, {1698, 40334}, {1699, 41036}, {1757, 39151}, {3576, 21158}, {3579, 36755}, {4663, 44498}, {5011, 5240}, {5195, 36929}, {5691, 36992}, {5692, 11095}, {5903, 10651}, {5978, 12780}, {6109, 11706}, {6780, 7974}, {7975, 36967}, {9901, 36969}, {11739, 14138}, {12194, 36759}, {13178, 23004}, {15178, 21401}, {16475, 36757}, {22510, 38220}, {38221, 39555}

X(44659) = barycentric product X(29651)*X(32726)
X(44659) = barycentric quotient X(933)/X(7179)
X(44659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 15, 11707}, {1, 3179, 10648}

X(44660) = X(1)X(16)∩X(30)X(511)

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

X(44660) lies on these lines: {1, 16}, {8, 622}, {10, 624}, {15, 5184}, {30, 511}, {40, 14539}, {65, 3638}, {187, 11707}, {238, 39151}, {299, 12780}, {355, 20429}, {395, 11706}, {944, 36995}, {946, 7685}, {1125, 6672}, {1385, 13349}, {1482, 5615}, {1698, 40335}, {1699, 41037}, {1757, 39150}, {3576, 21159}, {3579, 36756}, {4663, 44497}, {5011, 5239}, {5195, 36928}, {5691, 36994}, {5692, 11096}, {5902, 16038}, {5903, 10652}, {5979, 12781}, {6108, 11705}, {6779, 7975}, {7974, 36968}, {9900, 36970}, {11740, 14139}, {12194, 36760}, {13178, 23005}, {15178, 21402}, {16475, 36758}, {22511, 38220}, {38221, 39554}

X(44660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16, 11708}, {1, 41225, 10647}

X(44661) = X(1)X(19)∩X(30)X(511)

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

X(44661) lies on these lines: {1, 19}, {2, 10158}, {6, 32118}, {8, 2893}, {10, 4523}, {30, 511}, {33, 1763}, {40, 30265}, {55, 25080}, {63, 20243}, {65, 1439}, {72, 1903}, {73, 1825}, {80, 43735}, {145, 20061}, {169, 25088}, {212, 1726}, {226, 1824}, {228, 16577}, {238, 1731}, {291, 16100}, {306, 4463}, {390, 25255}, {614, 40973}, {851, 18210}, {910, 8554}, {942, 3946}, {946, 1871}, {950, 1829}, {990, 7289}, {991, 18161}, {993, 34176}, {1001, 25081}, {1040, 21370}, {1043, 18719}, {1125, 9895}, {1214, 3198}, {1284, 4516}, {1331, 21368}, {1385, 44220}, {1419, 2263}, {1486, 3295}, {1495, 20129}, {1697, 2292}, {1698, 31257}, {1736, 2183}, {1754, 26934}, {1762, 2328}, {1830, 2635}, {1831, 2654}, {1834, 35650}, {1859, 40960}, {1864, 14557}, {1872, 6260}, {1876, 43035}, {1962, 10389}, {2000, 24611}, {2078, 2611}, {2195, 3512}, {2262, 5728}, {2357, 40152}, {2550, 18698}, {2658, 10571}, {2901, 22023}, {3100, 3220}, {3101, 5285}, {3242, 4016}, {3509, 38479}, {3576, 21160}, {3583, 5146}, {3663, 24476}, {3678, 3773}, {3679, 31154}, {3708, 18413}, {3754, 4085}, {3868, 3875}, {3869, 3886}, {3874, 32921}, {3876, 17286}, {3878, 32941}, {3932, 20714}, {3938, 4137}, {3939, 6211}, {3958, 5223}, {3988, 4535}, {4067, 4133}, {4084, 4780}, {4127, 4527}, {4184, 16585}, {4223, 16547}, {4292, 18732}, {4647, 5082}, {4743, 4757}, {4872, 7112}, {5014, 20896}, {5132, 25065}, {5732, 18725}, {5884, 24257}, {5908, 9942}, {6737, 41600}, {6738, 44545}, {6757, 18517}, {7015, 10570}, {7069, 21361}, {7290, 40977}, {10122, 18180}, {10267, 39475}, {10313, 32116}, {10373, 12688}, {11031, 18163}, {12432, 22300}, {13329, 16560}, {13405, 40635}, {14206, 14956}, {14213, 20242}, {15076, 24248}, {15904, 16272}, {16309, 32126}, {17447, 37575}, {17463, 20470}, {20075, 25254}, {20683, 20701}, {21319, 21807}, {21621, 34822}, {22076, 40661}, {23305, 31419}, {32850, 35550}, {35016, 44253}, {39772, 41723}, {40940, 40959}, {40998, 41581}

X(44661) = isogonal conjugate of X(26702)
X(44661) = crossdifference of every pair of points on line {6, 656}
X(44661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2939, 2360}, {2, 11221, 10158}, {65, 40965, 3755}, {228, 21318, 16577}, {851, 18210, 18593}, {1824, 17441, 226}, {3100, 7291, 3220}, {3678, 3773, 4538}

X(44662) = X(1)X(25)∩X(30)X(511)

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

X(44662) lies on these lines: {1, 25}, {8, 1370}, {10, 1368}, {30, 511}, {40, 21312}, {65, 222}, {72, 5928}, {145, 7500}, {197, 1060}, {355, 18531}, {392, 41581}, {495, 40635}, {551, 44212}, {607, 18596}, {859, 8758}, {942, 44545}, {944, 18533}, {946, 1596}, {958, 41340}, {960, 41883}, {1062, 22654}, {1125, 6677}, {1385, 6644}, {1386, 19136}, {1478, 1824}, {1479, 1828}, {1482, 18534}, {1660, 40660}, {1698, 31255}, {1871, 26332}, {1872, 6256}, {1878, 3583}, {1900, 3585}, {1902, 5691}, {2182, 22123}, {2333, 18671}, {2611, 28377}, {3488, 7717}, {3679, 31152}, {3751, 10602}, {3753, 11213}, {3877, 41717}, {4297, 44241}, {5089, 18669}, {5587, 16072}, {5882, 37458}, {5901, 44233}, {7530, 10222}, {8148, 44454}, {9590, 37917}, {9895, 25466}, {11720, 20772}, {12106, 15178}, {12528, 41733}, {15654, 17102}, {18639, 36907}, {18659, 20235}, {18725, 30503}, {19875, 32216}, {22793, 44276}, {23361, 37565}, {26255, 38314}

X(44662) = crossdifference of every pair of points on line {6, 2522}
X(44662) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1773, 27802}, {1, 7713, 11365}, {1, 8185, 11363}, {8192, 11396, 1}

X(44663) = X(2)X(65)∩X(30)X(511)

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

X(44663) lies on these lines: {1, 3052}, {2, 65}, {3, 4930}, {8, 1836}, {9, 18421}, {10, 3838}, {30, 511}, {35, 33595}, {40, 4421}, {46, 5730}, {56, 11682}, {57, 5289}, {63, 2099}, {72, 3679}, {75, 41846}, {78, 37567}, {100, 5183}, {145, 28646}, {214, 5122}, {221, 37672}, {354, 3877}, {376, 9943}, {381, 5887}, {392, 3742}, {484, 4867}, {549, 34339}, {551, 942}, {908, 5123}, {946, 3829}, {956, 25415}, {958, 3340}, {986, 4719}, {997, 16417}, {1001, 11529}, {1012, 7982}, {1071, 34620}, {1125, 4757}, {1149, 3999}, {1155, 4511}, {1159, 15254}, {1201, 42040}, {1319, 3218}, {1329, 4848}, {1376, 2093}, {1482, 11260}, {1698, 4004}, {1709, 6762}, {1737, 5087}, {1837, 11415}, {1858, 11114}, {2262, 37654}, {2646, 17549}, {2650, 37548}, {2975, 11011}, {3057, 3241}, {3058, 34695}, {3189, 20070}, {3207, 36643}, {3243, 9819}, {3246, 30117}, {3295, 12559}, {3303, 11520}, {3336, 17614}, {3339, 15829}, {3436, 41687}, {3476, 9965}, {3509, 6603}, {3543, 12688}, {3555, 3901}, {3556, 9909}, {3579, 22836}, {3612, 19704}, {3617, 4005}, {3621, 28647}, {3626, 4127}, {3649, 24987}, {3653, 10202}, {3654, 37562}, {3655, 12675}, {3656, 24474}, {3671, 5837}, {3678, 4745}, {3681, 4711}, {3683, 16858}, {3698, 3876}, {3714, 42034}, {3740, 3753}, {3746, 16126}, {3754, 3828}, {3811, 12702}, {3813, 4301}, {3830, 40266}, {3845, 16616}, {3848, 4744}, {3873, 5919}, {3874, 9957}, {3881, 31792}, {3884, 5045}, {3890, 17609}, {3895, 41711}, {3898, 5049}, {3913, 7580}, {3919, 10176}, {3922, 9780}, {3959, 21874}, {3988, 4691}, {4067, 4669}, {4134, 38098}, {4295, 5794}, {4323, 30478}, {4373, 41446}, {4520, 21808}, {4646, 42043}, {4652, 34471}, {4654, 12709}, {4677, 5904}, {4685, 22300}, {4723, 27795}, {4866, 11530}, {4906, 16483}, {4908, 21864}, {4921, 41723}, {4973, 5126}, {4980, 17164}, {5131, 35271}, {5176, 17484}, {5220, 9623}, {5221, 19861}, {5252, 5905}, {5288, 11280}, {5298, 18838}, {5300, 42378}, {5302, 19860}, {5330, 20323}, {5434, 34742}, {5493, 12437}, {5538, 17613}, {5563, 19525}, {5572, 15933}, {5690, 21077}, {5693, 34700}, {5710, 16403}, {5724, 41011}, {5734, 6974}, {5884, 31786}, {5885, 31838}, {6172, 7672}, {6284, 41575}, {6763, 11009}, {6767, 15570}, {6872, 37724}, {6907, 11362}, {6914, 8666}, {6923, 32537}, {7508, 15178}, {7957, 34607}, {8256, 21075}, {8261, 15670}, {9140, 10693}, {9589, 12625}, {9708, 15481}, {9961, 15683}, {10031, 17660}, {10157, 38076}, {10178, 10304}, {10273, 26446}, {10459, 42039}, {10707, 17638}, {10716, 34242}, {10916, 22791}, {10974, 16052}, {11010, 41696}, {11113, 44547}, {11235, 12672}, {11237, 31164}, {11278, 22837}, {11533, 37327}, {12047, 17530}, {12100, 40296}, {12245, 32049}, {12514, 16418}, {12532, 17636}, {12649, 12701}, {13253, 22560}, {13601, 15556}, {13750, 37298}, {14054, 34741}, {14872, 34627}, {14923, 31145}, {15071, 34628}, {15678, 17637}, {15934, 42819}, {17078, 34855}, {17274, 24471}, {17281, 21853}, {17781, 34606}, {18178, 41629}, {18227, 31142}, {18391, 24703}, {19925, 31821}, {20076, 37738}, {20347, 43037}, {21849, 42450}, {21969, 42448}, {22299, 35652}, {24440, 36634}, {25413, 34718}, {30144, 37582}, {30147, 31445}, {31141, 41538}, {31393, 42871}, {31788, 31806}, {31803, 34648}, {31837, 35004}, {34195, 37080}, {34625, 34640}, {34626, 34701}, {34772, 37568}, {37598, 42042}, {37623, 40257}, {41591, 44545}, {42054, 43222}

X(44663) = barycentric quotient X(15884)/X(24128)
X(44663) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3928, 11194}, {2, 3869, 31165}, {2, 31165, 960}, {10, 39542, 3838}, {65, 960, 3812}, {65, 3869, 960}, {65, 31165, 2}, {72, 5836, 4662}, {72, 5903, 5836}, {354, 3877, 10179}, {392, 5902, 3742}, {484, 4867, 5440}, {908, 40663, 5123}, {1125, 4757, 31794}, {3057, 3868, 34791}, {3241, 28610, 34610}, {3339, 15829, 25524}, {3340, 12526, 958}, {3671, 5837, 25466}, {3679, 28609, 11236}, {3753, 5692, 3740}, {3878, 4084, 942}, {3899, 5902, 392}, {3901, 5697, 3555}, {4301, 24391, 3813}, {6762, 11531, 10912}, {7982, 12513, 33895}, {7991, 11523, 3913}

X(44664) = X(2)X(85)∩X(30)X(511)

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

X(44664) lies on these lines: {1, 24352}, {2, 85}, {10, 25355}, {30, 511}, {190, 40872}, {220, 9312}, {664, 6603}, {668, 40883}, {672, 21139}, {910, 3732}, {1111, 43065}, {1146, 9436}, {1319, 9318}, {1323, 17044}, {1565, 5179}, {1944, 6610}, {2348, 9317}, {3175, 36854}, {3227, 14727}, {3290, 7200}, {3474, 28124}, {3673, 40133}, {3693, 30806}, {3928, 5792}, {4059, 17451}, {4363, 9623}, {4515, 16284}, {4659, 4915}, {4875, 20880}, {5123, 24318}, {5199, 40483}, {7223, 40131}, {7960, 18391}, {9263, 41794}, {10481, 21258}, {14206, 26011}, {15587, 21084}, {21605, 27523}, {26006, 43066}, {28609, 29573}, {28610, 42051}, {32100, 43984}, {34522, 40719}, {35110, 36905}, {39351, 40868}

X(44664) = crossdifference of every pair of points on line {6, 8641}
X(44664) = barycentric quotient X(42763)/X(38964)
X(44664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1212, 44570}, {2, 3177, 31169}, {2, 31169, 1212}, {85, 1212, 6706}, {85, 3177, 1212}, {85, 31169, 2}, {241, 30807, 34852}, {279, 30695, 6554}, {664, 10025, 6603}, {672, 21139, 43037}, {1323, 40869, 17044}, {3177, 20089, 85}, {3732, 5088, 910}, {6706, 44570, 2}, {9312, 30625, 220}, {10481, 41006, 21258}, {16284, 25242, 4515}, {18663, 18750, 1427}, {42048, 42050, 2}

X(44665) = X(3)X(68)∩X(30)X(511)

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

X(44665) lies on these lines: {1, 9931}, {2, 12022}, {3, 68}, {4, 155}, {5, 578}, {6, 18420}, {15, 10659}, {16, 10660}, {20, 11411}, {24, 41587}, {26, 19908}, {30, 511}, {40, 9896}, {49, 10024}, {52, 3575}, {54, 13160}, {55, 10055}, {56, 10071}, {69, 34801}, {74, 16386}, {110, 403}, {113, 10151}, {125, 10257}, {128, 32410}, {131, 12095}, {136, 21268}, {140, 5449}, {141, 7514}, {143, 11745}, {156, 15761}, {159, 23044}, {182, 19141}, {184, 15760}, {185, 12421}, {186, 2931}, {193, 40909}, {230, 32661}, {235, 10539}, {265, 2072}, {323, 3153}, {355, 9928}, {371, 12424}, {372, 12425}, {378, 11442}, {381, 3167}, {382, 9936}, {389, 10112}, {394, 18396}, {399, 1514}, {427, 13352}, {450, 6761}, {468, 30714}, {485, 8909}, {486, 12232}, {546, 5448}, {549, 12024}, {550, 7689}, {567, 37347}, {568, 38321}, {569, 7399}, {571, 41523}, {576, 9926}, {858, 15133}, {974, 10111}, {1069, 1479}, {1092, 11585}, {1160, 9930}, {1161, 9929}, {1181, 19458}, {1204, 44240}, {1216, 12362}, {1300, 5962}, {1351, 18494}, {1352, 9818}, {1353, 7706}, {1370, 37483}, {1385, 12259}, {1437, 37361}, {1478, 3157}, {1482, 9933}, {1495, 37971}, {1498, 12420}, {1511, 44452}, {1568, 3292}, {1593, 12166}, {1594, 34148}, {1596, 41619}, {1597, 18440}, {1619, 7387}, {1853, 37497}, {1885, 12162}, {2070, 12310}, {2071, 3448}, {2883, 32139}, {2888, 14118}, {3060, 7576}, {3070, 10665}, {3071, 10666}, {3146, 16659}, {3311, 19062}, {3312, 19061}, {3398, 12193}, {3410, 7527}, {3519, 43689}, {3530, 20191}, {3543, 16658}, {3547, 18925}, {3548, 35602}, {3549, 19357}, {3627, 15083}, {3628, 43575}, {3631, 33533}, {3830, 16654}, {3853, 16656}, {4549, 11898}, {4846, 6391}, {5055, 35283}, {5133, 15033}, {5159, 14156}, {5254, 23128}, {5318, 10661}, {5321, 10662}, {5446, 6756}, {5462, 9825}, {5480, 11818}, {5502, 34104}, {5562, 12605}, {5576, 6288}, {5889, 6240}, {5890, 38323}, {5891, 34664}, {5892, 43573}, {5893, 44279}, {5894, 34350}, {5907, 13403}, {5921, 11472}, {5943, 10127}, {6090, 16072}, {6102, 13568}, {6237, 6253}, {6238, 6284}, {6241, 12282}, {6247, 12084}, {6515, 18533}, {6642, 39571}, {6643, 18945}, {6644, 13567}, {6676, 18475}, {6677, 43586}, {6696, 11250}, {6699, 16976}, {6776, 41614}, {6803, 15805}, {6815, 36752}, {6823, 31804}, {6841, 41608}, {7352, 7354}, {7403, 11424}, {7499, 37513}, {7512, 12254}, {7528, 10982}, {7542, 13367}, {7568, 10610}, {7723, 25740}, {8548, 8550}, {8884, 19196}, {8907, 25715}, {8912, 13903}, {8981, 13909}, {9703, 10254}, {9730, 11245}, {9786, 18951}, {9821, 9923}, {9925, 31861}, {9934, 12419}, {10018, 11449}, {10020, 32171}, {10113, 23323}, {10116, 18914}, {10149, 12888}, {10192, 10201}, {10263, 11819}, {10264, 12901}, {10282, 13383}, {10295, 41724}, {10525, 12422}, {10526, 12423}, {10540, 11799}, {10605, 18917}, {10619, 34002}, {10620, 20725}, {10625, 11750}, {10627, 13470}, {10669, 12426}, {10673, 12427}, {10679, 12430}, {10680, 12431}, {11202, 34351}, {11248, 12328}, {11249, 22659}, {11251, 12418}, {11252, 12415}, {11253, 12416}, {11264, 13630}, {11412, 12225}, {11413, 11457}, {11430, 21243}, {11440, 35491}, {11456, 44440}, {11800, 32411}, {11801, 33547}, {11828, 26293}, {11829, 26292}, {12085, 14216}, {12111, 12271}, {12121, 44246}, {12160, 12173}, {12161, 12233}, {12295, 13473}, {12893, 15646}, {12897, 13488}, {12902, 18403}, {13346, 18381}, {13364, 23410}, {13392, 15350}, {13419, 13598}, {13434, 14788}, {13478, 37356}, {13561, 23336}, {13619, 15085}, {13861, 15873}, {13966, 13970}, {14644, 43572}, {14683, 32111}, {14790, 37498}, {14826, 18537}, {15014, 40867}, {15068, 34966}, {15072, 44458}, {15152, 43893}, {15153, 37938}, {15341, 22146}, {15454, 35235}, {15595, 44340}, {15958, 40631}, {16163, 21663}, {16266, 18569}, {16534, 37984}, {17834, 17845}, {17928, 18912}, {18281, 23332}, {18388, 34986}, {18405, 37672}, {18436, 18563}, {18519, 42461}, {18534, 31383}, {18559, 41628}, {18909, 18934}, {18918, 37669}, {18939, 19498}, {18940, 19499}, {18980, 19486}, {18981, 19487}, {19194, 19205}, {19376, 32306}, {19597, 32444}, {20080, 41465}, {22467, 26879}, {23293, 37118}, {23411, 40240}, {26926, 37511}, {28787, 34800}, {32110, 32263}, {32144, 32767}, {32272, 34802}, {33563, 37814}, {34005, 41730}, {34622, 35450}, {34826, 43394}, {35237, 35513}, {35243, 37485}, {35500, 43818}, {35921, 37636}, {36989, 37488}, {37458, 41588}, {37478, 44239}, {43604, 44247}, {43831, 43844}, {44214, 44569}

X(44665) = isogonal conjugate of X(1299)
X(44665) = isotomic conjugate of the polar conjugate of X(16310)
X(44665) = Thomson-isogonal conjugate of X(13398)
X(44665) = crossdifference of every pair of points on line {6, 6753}
X(44665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 68, 12359}, {3, 343, 44201}, {3, 9937, 9932}, {3, 12301, 9938}, {3, 12309, 9937}, {3, 12359, 44158}, {3, 12429, 68}, {3, 44076, 6146}, {4, 155, 22660}, {4, 6193, 155}, {4, 14516, 12134}, {5, 1147, 9820}, {5, 12370, 12241}, {5, 23307, 20302}, {5, 43595, 578}, {20, 11411, 12163}, {20, 34799, 34224}, {68, 12118, 3}, {131, 12095, 27087}, {143, 31830, 11745}, {155, 12293, 4}, {156, 15761, 16252}, {265, 5504, 23306}, {265, 22115, 2072}, {381, 3167, 5654}, {389, 10112, 13292}, {394, 18396, 18531}, {567, 37347, 37649}, {1147, 9927, 5}, {1568, 13851, 10297}, {1853, 37497, 44441}, {2072, 22115, 11064}, {3292, 13851, 1568}, {5449, 12038, 140}, {5562, 21659, 12605}, {5889, 12278, 6240}, {6193, 12293, 22660}, {6193, 12318, 15316}, {6288, 37472, 5576}, {6515, 18533, 37489}, {6756, 13142, 5446}, {9306, 18390, 5}, {9931, 19471, 1}, {9932, 9938, 3}, {9937, 12301, 3}, {10116, 40647, 18914}, {11264, 13630, 43588}, {11412, 12289, 12225}, {11818, 39522, 5480}, {12084, 32140, 6247}, {12118, 12429, 12359}, {12293, 15316, 22661}, {12301, 12309, 9932}, {12428, 18970, 1}, {13292, 31833, 389}, {13346, 18381, 23335}, {13352, 18474, 427}, {15033, 41171, 5133}, {15115, 15123, 5159}, {15133, 15136, 858}, {18914, 31829, 40647}, {19908, 32048, 26}, {22663, 43588, 32166}, {25739, 43574, 858}

X(44666) = X(4)X(15)∩X(30)X(511)

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

X(44666) lies on these lines: {2, 16652}, {3, 623}, {4, 15}, {5, 6671}, {14, 33389}, {16, 33518}, {20, 621}, {30, 511}, {40, 22896}, {55, 22904}, {56, 22905}, {98, 11602}, {125, 37974}, {147, 22508}, {187, 5321}, {298, 5473}, {316, 11132}, {382, 5611}, {383, 5479}, {396, 5478}, {485, 22921}, {486, 22922}, {546, 21401}, {550, 35725}, {624, 13449}, {631, 40334}, {636, 42675}, {944, 22912}, {946, 11707}, {1080, 9749}, {1352, 22737}, {1478, 22929}, {1479, 22930}, {1495, 37975}, {1546, 32111}, {1587, 19070}, {1588, 19071}, {1657, 5868}, {2076, 22512}, {3146, 22113}, {3389, 41018}, {3529, 22844}, {3575, 22482}, {3642, 44465}, {5073, 5869}, {5085, 11296}, {5334, 41406}, {5339, 19780}, {5349, 12815}, {5460, 38225}, {5474, 5978}, {5480, 42117}, {5691, 22652}, {5870, 22899}, {5871, 22898}, {5982, 5999}, {5995, 38943}, {6110, 6530}, {6111, 41204}, {6284, 22910}, {6670, 44223}, {6770, 36969}, {6771, 33560}, {6773, 36968}, {6776, 42086}, {6780, 36962}, {6781, 41071}, {6783, 23005}, {7354, 18973}, {7710, 22666}, {8259, 10613}, {8550, 42118}, {8594, 41043}, {8721, 22916}, {9117, 41061}, {9834, 22670}, {9835, 22674}, {9838, 22908}, {9839, 22909}, {9863, 22749}, {9873, 22746}, {9982, 32596}, {10516, 11295}, {10617, 42163}, {10645, 37463}, {10653, 14912}, {10654, 14853}, {10667, 23251}, {10671, 23261}, {11001, 36386}, {11092, 44466}, {11480, 41040}, {11500, 22558}, {11676, 41044}, {12110, 16964}, {12113, 22897}, {12114, 22772}, {12115, 22931}, {12116, 22932}, {12177, 22527}, {14369, 18863}, {14639, 22510}, {15640, 33626}, {15682, 36366}, {16809, 37464}, {16942, 37637}, {16965, 37007}, {19106, 22900}, {21159, 35932}, {21445, 22511}, {22892, 41016}, {22893, 41017}, {22894, 42099}, {22901, 36995}, {22925, 36656}, {22926, 36655}, {30560, 42814}, {35229, 38228}, {35820, 35848}, {35821, 35847}, {35931, 41042}, {36186, 41888}, {36252, 36760}, {36990, 42096}, {39838, 41070}, {39874, 42113}, {41035, 41056}, {41041, 42093}, {41055, 42108}

X(44666) = Thomson-isogonal conjugate of X(16806)
X(44666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16626, 629}, {3, 20428, 623}, {4, 15, 7684}, {4, 17, 22832}, {4, 22532, 17}, {4, 36993, 15}, {5, 13350, 6671}, {15, 17, 14138}, {15, 36992, 4}, {20, 621, 14538}, {20, 627, 22890}, {383, 36970, 5479}, {5321, 41034, 7685}, {14138, 31705, 17}, {22907, 42085, 22906}, {36761, 36961, 9749}, {36961, 36967, 1080}, {36992, 36993, 7684}, {39555, 41037, 38227}, {41021, 42100, 36995}

X(44667) = X(4)X(16)∩X(30)X(511)

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

X(44667) lies on these lines: {2, 16653}, {3, 624}, {4, 16}, {5, 6672}, {13, 33388}, {15, 33517}, {20, 622}, {30, 511}, {40, 22851}, {55, 22859}, {56, 22860}, {98, 11603}, {125, 37975}, {147, 22506}, {187, 5318}, {299, 5474}, {316, 11133}, {382, 5615}, {383, 9750}, {395, 5479}, {485, 22876}, {486, 22877}, {546, 21402}, {550, 35726}, {619, 44250}, {623, 13449}, {631, 40335}, {635, 42674}, {944, 22867}, {946, 11708}, {1080, 5478}, {1352, 22736}, {1478, 22884}, {1479, 22885}, {1495, 37974}, {1545, 32111}, {1587, 19072}, {1588, 19069}, {1657, 5869}, {2076, 22513}, {3146, 22114}, {3529, 22845}, {3575, 22481}, {3643, 44461}, {5073, 5868}, {5085, 11295}, {5335, 41407}, {5340, 19781}, {5350, 12815}, {5459, 38225}, {5473, 5979}, {5480, 42118}, {5691, 22651}, {5870, 22854}, {5871, 22853}, {5983, 5999}, {5994, 38944}, {6110, 41204}, {6111, 6530}, {6284, 22865}, {6770, 36967}, {6773, 36970}, {6774, 33561}, {6776, 42085}, {6779, 36961}, {6781, 41070}, {6782, 23004}, {7354, 18972}, {7710, 22665}, {8260, 10614}, {8550, 42117}, {8595, 41042}, {8721, 22871}, {9115, 41060}, {9834, 22669}, {9835, 22673}, {9838, 22863}, {9839, 22864}, {9863, 22748}, {9873, 22745}, {9981, 32597}, {10516, 11296}, {10616, 42166}, {10646, 37464}, {10653, 14853}, {10654, 14912}, {10668, 23251}, {10672, 23261}, {11001, 36388}, {11078, 44462}, {11481, 41041}, {11500, 22557}, {11676, 41045}, {12110, 16965}, {12113, 22852}, {12114, 22771}, {12115, 22886}, {12116, 22887}, {12177, 22526}, {14368, 18864}, {14639, 22511}, {15640, 33627}, {15682, 36368}, {16808, 37463}, {16943, 37637}, {16964, 37008}, {19107, 22856}, {21158, 35931}, {21445, 22510}, {22797, 44223}, {22847, 41016}, {22848, 41017}, {22850, 42100}, {22855, 36993}, {22880, 36656}, {22881, 36655}, {30559, 42813}, {34602, 41061}, {35230, 38228}, {35820, 35846}, {35821, 35849}, {35932, 41043}, {36185, 41887}, {36251, 36759}, {36761, 36781}, {36990, 42097}, {39838, 41071}, {39874, 42112}, {41034, 41057}, {41040, 42094}, {41054, 42109}

X(44667) = Thomson-isogonal conjugate of X(16807)
X(44667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 16627, 630}, {3, 20429, 624}, {4, 16, 7685}, {4, 18, 22831}, {4, 22531, 18}, {4, 36995, 16}, {5, 13349, 6672}, {16, 18, 14139}, {16, 36994, 4}, {20, 622, 14539}, {20, 628, 22843}, {1080, 36969, 5478}, {5318, 41035, 7684}, {14139, 31706, 18}, {22861, 42086, 22862}, {36962, 36968, 383}, {36962, 41458, 9750}, {36994, 36995, 7685}, {39554, 41036, 38227}, {41020, 42099, 36993}

X(44668) = X(6)X(24)∩X(30)X(511)

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

X(44668) lies on these lines: {2, 34751}, {3, 9972}, {4, 9973}, {5, 41579}, {6, 24}, {20, 7729}, {23, 3047}, {30, 511}, {51, 10192}, {52, 3629}, {66, 3519}, {67, 32353}, {69, 1225}, {141, 1209}, {143, 5097}, {154, 3060}, {159, 195}, {160, 30258}, {161, 1993}, {182, 9977}, {193, 31304}, {206, 576}, {235, 1843}, {265, 32299}, {323, 15139}, {343, 27365}, {389, 12007}, {468, 11746}, {546, 43129}, {575, 39125}, {577, 37813}, {597, 44211}, {599, 31180}, {611, 10066}, {613, 10082}, {858, 41673}, {974, 10295}, {1112, 1495}, {1176, 42059}, {1177, 43704}, {1216, 3631}, {1350, 7691}, {1352, 6288}, {1353, 36966}, {1386, 12266}, {1469, 18984}, {1533, 2883}, {1853, 2979}, {1971, 5111}, {2914, 12367}, {2916, 10203}, {2935, 43576}, {3043, 15140}, {3056, 13079}, {3094, 9985}, {3095, 15270}, {3098, 15578}, {3242, 7979}, {3313, 15583}, {3416, 12785}, {3448, 41732}, {3580, 32316}, {3589, 6153}, {3751, 9905}, {3763, 31282}, {3779, 32370}, {3818, 22804}, {3917, 23332}, {5050, 11216}, {5085, 15078}, {5092, 43615}, {5093, 19153}, {5095, 14049}, {5446, 16252}, {5462, 6329}, {5476, 44270}, {5562, 41362}, {5596, 11271}, {5876, 34786}, {5889, 17845}, {5891, 23324}, {5892, 44323}, {5893, 13598}, {5898, 15137}, {5921, 41726}, {5946, 11202}, {6101, 18381}, {6102, 34785}, {6239, 23251}, {6242, 6776}, {6243, 9833}, {6247, 10625}, {6293, 11008}, {6400, 23261}, {6467, 8550}, {6593, 11597}, {6696, 15644}, {6697, 14076}, {6746, 13367}, {6759, 10263}, {7464, 11598}, {7575, 12236}, {7731, 16176}, {8146, 35717}, {8254, 18583}, {8989, 11266}, {8995, 13910}, {9822, 32396}, {9924, 11477}, {9934, 37924}, {9969, 11808}, {9971, 14853}, {10114, 32317}, {10115, 32284}, {10117, 15107}, {10169, 38110}, {10182, 13363}, {10249, 31884}, {10250, 17508}, {10296, 12825}, {10313, 19165}, {10510, 25714}, {10516, 11188}, {10627, 20299}, {10752, 43580}, {11002, 35260}, {11243, 36978}, {11244, 36980}, {11412, 40341}, {11442, 41730}, {11459, 18405}, {11557, 41595}, {11574, 16196}, {11591, 18383}, {11663, 15800}, {11800, 32223}, {11805, 32271}, {11898, 32402}, {12167, 32333}, {12175, 19459}, {12208, 12212}, {12272, 15069}, {12283, 32339}, {12294, 32340}, {12300, 35490}, {12307, 33878}, {12316, 39879}, {12325, 32337}, {12329, 12341}, {12452, 12480}, {12453, 12481}, {12583, 12797}, {12586, 12926}, {12587, 12936}, {12588, 12946}, {12589, 12956}, {12590, 12998}, {12591, 12999}, {12594, 13121}, {12595, 13122}, {12824, 35265}, {12965, 35840}, {12971, 35841}, {13198, 21284}, {13365, 25555}, {13418, 15321}, {13565, 24206}, {13567, 41599}, {13972, 13986}, {14216, 37484}, {14561, 16776}, {14913, 32393}, {15060, 18376}, {15067, 23325}, {15116, 15124}, {15118, 32355}, {15448, 44084}, {15520, 23042}, {16105, 32111}, {16881, 20585}, {17847, 23061}, {18121, 39569}, {18440, 22815}, {18935, 32334}, {19127, 44259}, {19130, 44235}, {20424, 21850}, {21230, 23300}, {21849, 32267}, {21969, 34750}, {22769, 22781}, {23048, 38317}, {23315, 41603}, {23606, 35225}, {26543, 41592}, {26926, 32377}, {28343, 41363}, {32113, 32246}, {32142, 32767}, {32336, 39897}, {32357, 37491}, {32390, 39873}, {33563, 44322}, {37511, 40929}, {38898, 41731}, {39588, 44269}

X(44668) = crossdifference of every pair of points on line {6, 6368}
X(44668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 41713, 41578}, {5, 41714, 41579}, {6, 2917, 32344}, {6, 34787, 15577}, {52, 34782, 41589}, {54, 2917, 32391}, {54, 6152, 973}, {54, 13423, 6152}, {159, 1351, 34117}, {195, 9920, 32379}, {389, 32366, 12007}, {973, 11577, 54}, {2888, 12226, 41590}, {2888, 32346, 6145}, {3574, 11576, 11743}, {3629, 34782, 41729}, {5097, 10282, 41593}, {5480, 12061, 1843}, {6152, 12291, 11577}, {6153, 6689, 9827}, {6276, 6277, 54}, {6403, 15073, 6}, {6467, 19161, 8550}, {9924, 11477, 19149}, {9924, 19149, 15581}, {9935, 32379, 9920}, {9973, 44439, 4}, {11808, 40632, 12242}, {12272, 41716, 15069}, {12291, 13423, 54}, {13368, 15074, 9977}, {13433, 40632, 11808}, {15532, 32196, 1493}, {21969, 34750, 41580}, {32344, 32368, 6}, {32368, 34787, 32391}

X(44669) = X(8)X(21)∩X(30)X(511)

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

X(44669) lies on these lines: {1, 442}, {4, 12635}, {5, 22836}, {6, 24247}, {8, 21}, {10, 6675}, {11, 4511}, {12, 5086}, {30, 511}, {36, 10609}, {40, 44238}, {41, 40997}, {42, 5724}, {56, 12649}, {65, 16465}, {72, 10572}, {78, 1329}, {79, 3633}, {80, 6596}, {100, 5172}, {141, 24249}, {145, 388}, {165, 34701}, {191, 2136}, {200, 5727}, {214, 15325}, {224, 41552}, {243, 5081}, {355, 3811}, {390, 42014}, {496, 30144}, {497, 5289}, {551, 24386}, {938, 25524}, {942, 17647}, {944, 3428}, {950, 960}, {956, 37286}, {962, 36999}, {997, 3816}, {1001, 3488}, {1056, 6601}, {1125, 12433}, {1145, 32760}, {1146, 3684}, {1210, 6691}, {1317, 38460}, {1319, 26015}, {1320, 11604}, {1376, 18391}, {1385, 10916}, {1388, 10529}, {1479, 5730}, {1482, 13463}, {1483, 5499}, {1532, 6326}, {1697, 42012}, {1699, 34647}, {1737, 3035}, {1770, 4018}, {1824, 1891}, {2161, 3943}, {2321, 4544}, {2348, 3039}, {2550, 8255}, {2551, 20007}, {2646, 4999}, {2893, 41003}, {3036, 3689}, {3057, 31938}, {3058, 3877}, {3065, 12641}, {3158, 3679}, {3169, 17362}, {3174, 9623}, {3218, 15326}, {3241, 3475}, {3244, 11263}, {3255, 4900}, {3436, 20013}, {3476, 35990}, {3485, 5175}, {3582, 34123}, {3583, 4867}, {3584, 38058}, {3585, 41696}, {3586, 24703}, {3600, 20008}, {3601, 26066}, {3616, 31245}, {3617, 15674}, {3621, 11684}, {3625, 3647}, {3635, 6701}, {3648, 20053}, {3654, 44255}, {3665, 21285}, {3681, 34606}, {3699, 36926}, {3710, 42378}, {3812, 6738}, {3814, 12019}, {3822, 5719}, {3829, 5886}, {3847, 9581}, {3868, 7354}, {3869, 6284}, {3870, 5252}, {3872, 4863}, {3873, 5434}, {3874, 18990}, {3878, 15171}, {3884, 15172}, {3893, 17637}, {3898, 15170}, {3901, 10483}, {3932, 16086}, {3935, 5176}, {4187, 37702}, {4190, 5221}, {4297, 24391}, {4304, 4640}, {4314, 5837}, {4316, 4880}, {4361, 26130}, {4420, 21031}, {4421, 5657}, {4430, 34605}, {4662, 5795}, {4669, 15673}, {4677, 17525}, {4678, 15676}, {4693, 4953}, {4848, 41547}, {4855, 24914}, {4861, 37734}, {4881, 5298}, {4939, 4975}, {5080, 13272}, {5123, 6745}, {5173, 10106}, {5193, 41556}, {5227, 17299}, {5231, 13384}, {5428, 5690}, {5538, 37374}, {5542, 33558}, {5559, 6597}, {5603, 11235}, {5687, 8069}, {5691, 11523}, {5692, 11113}, {5694, 37290}, {5731, 11194}, {5761, 10894}, {5836, 8261}, {5880, 11529}, {5881, 6765}, {5882, 11260}, {5884, 31775}, {5901, 24387}, {5902, 11112}, {5905, 12943}, {6067, 30284}, {6224, 22560}, {6265, 14527}, {6599, 13143}, {6600, 9708}, {6668, 13411}, {6736, 33559}, {6762, 41338}, {6764, 33557}, {6842, 37733}, {6909, 9803}, {6913, 42843}, {7181, 17136}, {7198, 20247}, {7281, 42064}, {7424, 37783}, {7483, 37571}, {7701, 12703}, {7967, 34625}, {7972, 34600}, {8148, 18499}, {8583, 37723}, {8666, 34773}, {8728, 30143}, {9710, 19860}, {9711, 37721}, {9778, 34626}, {9812, 34706}, {10222, 33592}, {10404, 11520}, {10527, 34471}, {10544, 18178}, {10679, 12645}, {10698, 13271}, {10915, 16617}, {11011, 41550}, {11015, 15338}, {11220, 34742}, {11236, 25568}, {11246, 17579}, {11415, 12953}, {11519, 16143}, {11524, 12657}, {11680, 15950}, {11682, 12701}, {12245, 37000}, {12247, 13205}, {12436, 17706}, {12448, 16120}, {12541, 14450}, {12546, 16124}, {12629, 16132}, {12630, 16133}, {12633, 16135}, {12634, 16136}, {12638, 16144}, {12643, 16146}, {12644, 16147}, {12646, 16151}, {12913, 39777}, {13257, 41698}, {13464, 20288}, {14942, 17947}, {15677, 31145}, {15908, 21740}, {15933, 38053}, {15934, 25557}, {15954, 21147}, {16117, 18526}, {16155, 30323}, {16160, 37705}, {16200, 34640}, {17530, 37701}, {17533, 37718}, {17606, 27385}, {17648, 17653}, {17670, 30139}, {17728, 35262}, {18242, 37700}, {18407, 22791}, {18480, 21077}, {18481, 37584}, {18635, 24435}, {18755, 21965}, {19589, 32847}, {19914, 25438}, {20054, 31888}, {24389, 42819}, {24987, 37080}, {25416, 41702}, {25639, 37737}, {26446, 28465}, {28460, 34718}, {30115, 37715}, {30146, 33186}, {30147, 31419}, {31650, 38112}, {31789, 31806}, {31870, 37281}, {33667, 39776}, {33858, 37401}, {34619, 34700}, {35634, 35637}, {36846, 37738}, {36920, 41542}, {36974, 41014}, {37468, 37625}, {37662, 37717}, {41687, 41697}

X(44669) = crossdifference of every pair of points on line {6, 7180}
X(44669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 442, 11281}, {1, 3419, 2886}, {1, 5794, 25466}, {8, 21, 21677}, {8, 1043, 3704}, {8, 3189, 3913}, {8, 3486, 958}, {8, 4720, 4046}, {8, 10543, 18253}, {8, 12536, 3189}, {10, 24929, 6690}, {10, 35016, 6675}, {21, 3871, 31660}, {21, 21677, 18253}, {78, 1837, 1329}, {145, 2475, 34195}, {145, 3434, 2099}, {355, 3811, 12607}, {355, 37533, 7680}, {950, 6737, 960}, {997, 5722, 3816}, {1737, 5440, 3035}, {2475, 34195, 3649}, {2646, 6734, 4999}, {2900, 12625, 3419}, {3244, 21627, 33895}, {3633, 9613, 41863}, {4863, 37740, 3872}, {5086, 34772, 12}, {5687, 10573, 8256}, {5690, 8715, 32157}, {5731, 24477, 11194}, {5795, 6743, 4662}, {5881, 6765, 32049}, {6598, 39783, 11281}, {6675, 15174, 35016}, {9581, 25681, 3847}, {10543, 21677, 21}, {12649, 35979, 41574}, {19914, 25438, 32198}, {36930, 36931, 3943}

X(44670) = X(19)X(25)∩X(30)X(511)

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

X(44670) lies on these lines: {1, 15076}, {6, 12723}, {9, 21867}, {12, 1900}, {19, 25}, {30, 511}, {44, 21889}, {48, 4336}, {65, 4331}, {69, 12530}, {71, 22272}, {72, 5695}, {75, 1370}, {141, 18252}, {192, 7500}, {210, 17281}, {227, 1887}, {241, 17463}, {354, 17301}, {448, 1632}, {692, 2182}, {851, 8758}, {950, 44545}, {984, 5119}, {990, 22769}, {1108, 3941}, {1368, 2886}, {1386, 12722}, {1596, 7680}, {1633, 7291}, {1660, 10537}, {1699, 27471}, {1721, 7289}, {1742, 18161}, {1766, 12329}, {1770, 18732}, {1826, 22273}, {1828, 12953}, {1829, 6284}, {1836, 17441}, {1871, 11496}, {1872, 11500}, {1902, 6253}, {1953, 2293}, {2099, 2263}, {2161, 2195}, {2183, 2310}, {2223, 4516}, {2262, 14100}, {2267, 28125}, {2270, 4907}, {2294, 4343}, {2340, 21801}, {2667, 18674}, {2951, 18725}, {3000, 3942}, {3052, 40962}, {3059, 21871}, {3242, 12721}, {3262, 20556}, {3290, 39688}, {3419, 3696}, {3428, 21312}, {3668, 5173}, {3740, 17359}, {3742, 17382}, {3755, 32118}, {3779, 21853}, {3914, 40959}, {3923, 4523}, {3965, 22298}, {4061, 8804}, {4436, 25083}, {4456, 22283}, {4463, 32929}, {4640, 34176}, {4688, 31140}, {4698, 6677}, {4749, 41015}, {4755, 44212}, {5248, 9895}, {6644, 32613}, {7297, 16686}, {8053, 40937}, {9441, 16560}, {9627, 40985}, {9778, 27472}, {10164, 27473}, {10679, 18534}, {11018, 40646}, {12610, 24388}, {12675, 43160}, {12711, 43213}, {15569, 24929}, {16465, 18655}, {16548, 40910}, {17276, 17635}, {17872, 40934}, {18531, 37820}, {18533, 30273}, {21011, 22274}, {21059, 40968}, {21328, 22053}, {22300, 44547}, {24248, 24476}, {31238, 31245}, {36999, 44438}, {37528, 42440}, {44454, 44455}

X(44670) = isogonal conjugate of X(43363)
X(44670) = isogonal conjugate of the anticomplement of X(20623)
X(44670) = Thomson-isogonal conjugate of X(44059)
X(44670) = medial-isogonal conjugate of X(20623)
X(44670) = crossdifference of every pair of points on line {6, 905}
X(44670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 4319, 1486}, {55, 1824, 40635}, {2182, 41339, 692}, {2223, 4516, 8609}, {4436, 35552, 25083}, {12723, 40965, 6}, {15496, 15503, 15494}, {17872, 40934, 40941}

X(44671) = X(30)X(511)∩X(37)X(42)

Barycentrics    a*(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(44671) lies on these lines: {1, 24450}, {9, 4068}, {30, 511}, {37, 42}, {44, 3747}, {75, 3873}, {192, 4661}, {312, 25294}, {354, 4688}, {594, 22279}, {1045, 16696}, {1213, 4111}, {1441, 43915}, {1500, 22292}, {2234, 16726}, {2245, 4433}, {2294, 3059}, {2321, 21865}, {3158, 10434}, {3271, 4969}, {3555, 4647}, {3681, 4664}, {3684, 23398}, {3688, 17388}, {3689, 3724}, {3696, 3753}, {3739, 3741}, {3740, 4755}, {3743, 34790}, {3778, 21858}, {3779, 17299}, {3780, 4749}, {3789, 41312}, {3811, 42443}, {3842, 3956}, {3892, 24325}, {3896, 22275}, {3919, 4709}, {3943, 20683}, {3950, 22312}, {3958, 14100}, {3963, 22289}, {3968, 4732}, {3993, 4134}, {4016, 23668}, {4046, 40952}, {4053, 4516}, {4094, 42084}, {4361, 35892}, {4399, 17049}, {4430, 4740}, {4436, 18206}, {4515, 22317}, {4553, 6542}, {4698, 6685}, {4716, 38485}, {4735, 20691}, {15185, 18698}, {17362, 21746}, {17372, 17792}, {17386, 25279}, {18137, 25277}, {18147, 25291}, {19998, 41683}, {20016, 25048}, {20713, 40965}, {20715, 21889}, {21009, 35342}, {22316, 40504}, {25081, 40659}, {25255, 34784}, {31238, 31241}, {32843, 38390}

X(44671) = crossdifference of every pair of points on line {6, 1019}
X(44671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 22271, 40607}, {872, 22167, 37}, {1045, 24437, 16696}, {2321, 22277, 21865}, {2667, 3728, 37}, {3943, 20683, 40521}, {4111, 4890, 1213}

X(44672) = ISOGONAL CONJUGATE OF X(8592)

Barycentrics    a^2*(4*a^16-8*b^16-4*b^14*c^2-8*b^12*c^4-16*b^10*c^6+73*b^8*c^8-16*b^6*c^10-8*b^4*c^12-4*b^2*c^14-8*c^16-10*a^14*(b^2+c^2)+2*a^6*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)^2+a^12*(10*b^4+17*b^2*c^2+10*c^4)+a^10*(-37*b^6+24*b^4*c^2+24*b^2*c^4-37*c^6)-a^8*(2*b^8+26*b^6*c^2-51*b^4*c^4+26*b^2*c^6+2*c^8)+a^4*(40*b^12-30*b^10*c^2-48*b^8*c^4+71*b^6*c^6-48*b^4*c^8-30*b^2*c^10+40*c^12)+a^2*(8*b^14+38*b^12*c^2-93*b^10*c^4+46*b^8*c^6+46*b^6*c^8-93*b^4*c^10+38*b^2*c^12+8*c^14)) : :
Barycentrics    a^2*(9 S^4-6 S^2 (3 SB-4 SW) SW+SW^2 (36 SB SC+36 SC^2-6 SB SW-36 SC SW-SW^2))*(-9 S^4+6 S^2 SW (3 SC+2 SW)+SW^2 (36 SC^2-30 SC SW+SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2192.

X(44672) lies on this line: {840, 9830}

X(44672) = isogonal conjugate of X(8592)
X(44672) = X(6)-reciprocal conjugate of X(8592)
X(44672) = X(i)-vertex conjugate of X(j) for these (i,j): {6,5104}, {5104,6}

X(44673) = X(2)X(1568)∩X(3)X(2929)

Barycentrics    -2*a^10+4*a^8*(b^2+c^2)-7*a^4*(b^2-c^2)^2*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^6*(b^4-8*b^2*c^2+c^4)+a^2*(b^2-c^2)^2*(5*b^4+4*b^2*c^2+5*c^4) : :
Barycentrics    S^2 (28 R^2-SA-6 SW)-3 SB SC (4 R^2-SW) : :
X(44673) = X(23)+5*X(38729),X(74)+3*X(37943),3*X(125)-X(25739),3*X(186)+X(25739),X(265)+3*X(37955),2*X(468)+X(20417),X(2070)+3*X(15061),X(2071)-3*X(38727),X(3153)-5*X(15059),7*X(3526)+X(32608),2*X(6699)+X(32223),2*X(6723)+X(32110),X(7575)+2*X(20397),X(10264)+3*X(16532),X(13619)+3*X(14644),7*X(15057)+5*X(37760),5*X(15081)+3*X(35489),X(16163)-3*X(37941),X(18403)-3*X(23515),2*X(18571)+X(36253),X(18859)-5*X(38728),X(20379)+2*X(22249),X(22115)-3*X(38793),3*X(34128)-X(37938)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2193.

X(44673) lies on these lines: {2,1568}, {3,2929}, {4,11204}, {5,20191}, {6,5054}, {23,38729}, {24,11550}, {25,23329}, {30,6699}, {51,37118}, {54,13418}, {74,37943}, {125,186}, {140,389}, {143,5498}, {184,10182}, {185,10018}, {265,37955}, {378,10193}, {381,37487}, {382,1620}, {403,2777}, {468,6000}, {511,10257}, {539,1511}, {549,11430}, {567,32068}, {578,631}, {632,12233}, {1192,1656}, {1204,7505}, {1503,37935}, {1533,13445}, {1596,23328}, {1899,11202}, {2070,15061}, {2071,38727}, {2072,6723}, {3147,6759}, {3153,15059}, {3357,3542}, {3515,18381}, {3517,40686}, {3523,39571}, {3524,37643}, {3526,9786}, {3530,12241}, {3574,6143}, {3575,32767}, {3589,21851}, {3628,13568}, {5446,23336}, {5449,37814}, {5907,16238}, {5944,18128}, {5965,22115}, {5972,13754}, {6241,14862}, {6368,16810}, {6622,20427}, {6644,21243}, {6676,16836}, {6696,13474}, {6760,15526}, {6761,16080}, {7542,9729}, {7575,20397}, {10020,40647}, {10112,12038}, {10116,32171}, {10125,13630}, {10226,12897}, {10264,16532}, {10295,13851}, {10298,26913}, {10303,23061}, {10540,16003}, {10605,37453}, {11425,15720}, {11427,15702}, {11563,12041}, {11657,24930}, {11704,34797}, {11746,13391}, {11808,20376}, {12227,38794}, {13289,37917}, {13367,26879}, {13399,14157}, {13619,14644}, {13621,18488}, {14165,40664}, {14389,16226}, {14869,37505}, {15057,37760}, {15062,21451}, {15081,35489}, {15311,37942}, {15644,16196}, {15646,17702}, {15750,34785}, {16111,31726}, {16163,37941}, {16881,34421}, {17704,34002}, {17821,26944}, {18376,37196}, {18403,23515}, {18475,34477}, {18533,23325}, {18571,30522}, {18859,38728}, {18931,38282}, {20379,22249}, {21659,21844}, {23332,37458}, {25555,37473}, {34128,37938}, {35482,38848}, {37968,38726}

X(44673) = midpoint of X(i) and X(j) for these {i,j}: {125,186}, {403,21663}, {1533,13445}, {2072,32110}, {10295,13851}, {10540,16003}, {11563,12041}, {13399,14157}, {16111,31726}
X(44673) = reflection of X(i) in X(j) for these (i,j): (2072,6723), (14156,140), (38726,37968)
X(44673) = complement of X(1568)
X(44673) = X(i)-complementary conjugate of X(j) for these (i,j): (2190,113), (2616,16177)
X(44673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,11438,18388), (3,26958,18390), (24,20299,13419), (140,12006,6689), (549,13567,11430), (1899,35486,11202), (3147,26937,6759), (6696,21841,13474), (21844,26917,21659)

X(44674) = X(137)X(1157)∩X(231)X(2072)

Barycentrics    -2*a^16+12*a^14*(b^2+c^2)-(b^2-c^2)^6*(b^4-b^2*c^2+c^4)-10*a^12*(3*b^4+4*b^2*c^2+3*c^4)+a^2*(b^2-c^2)^4*(5*b^6-4*b^4*c^2-4*b^2*c^4+5*c^6)+a^6*(b^2-c^2)^2*(22*b^6+17*b^4*c^2+17*b^2*c^4+22*c^6)+a^10*(41*b^6+43*b^4*c^2+43*b^2*c^4+41*c^6)-a^4*(b^2-c^2)^2*(12*b^8-11*b^6*c^2-4*b^4*c^4-11*b^2*c^6+12*c^8)-a^8*(35*b^8+6*b^6*c^2+8*b^4*c^4+6*b^2*c^6+35*c^8) : :
Barycentrics    5S^4-SB SC (3 R^2-SW) (4 R^2-SW)+S^2 (-20 R^4-15 R^2 SA+SB SC+24 R^2 SW+6 SA SW-7 SW^2) : :
X(44674) = X(19552)-3*X(23516)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2193.

X(44674) lies on these lines: {5,24147}, {128,19553}, {137,1157}, {195,31376}, {231,2072}, {1154,34837}, {3627,17507}, {3628,10275}, {10615,13372}, {11539,33992}, {16337,40631}, {19552,23516}, {22051,23280}, {24385,25150}, {25044,33545}, {34598,36842}

X(44674) = midpoint of X(i) and X(j) for these {i,j}: {5,24147}, {128,19553}, {137,1157}, {16337,40631}
X(44674) = reflection of X(13372) in X(10615)
X(44674) = {X(22051),X(23280)}-harmonic conjugate of X(34768)

X(44675) = X(1)X(2)∩X(11)X(515)

Barycentrics    -2*a^4+3*a^2*(b-c)^2+a^3*(b+c)-a*(b-c)^2*(b+c)-(b^2-c^2)^2 : :
Barycentrics    3 S^2+a b SB-3 b c SB+a c SC-3 b c SC-SB SC : :
X(44675) = X(1)+3*X(3582),3*X(2)+X(38460),5*X(36)-X(15228),X(36)+3*X(16173),X(149)+3*X(4881),4*X(1125)-X(6745),X(1155)-3*X(5298),2*X(1387)+X(3911),X(1737)-3*X(3582),5*X(3616)-X(4511),5*X(3616)+X(26015),X(3814)-3*X(32557),X(5080)-9*X(32558),X(5176)-5*X(31272),X(5440)-3*X(34123),X(13528)-3*X(21154),X(15228)+5*X(30384),3*X(16173)-X(30384)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2193.

X(44675) lies on these lines: {1,2}, {3,10624}, {4,1420}, {5,10106}, {11,515}, {12,20323}, {20,9614}, {30,5126}, {35,12575}, {36,516}, {40,7288}, {46,4301}, {55,10165}, {56,946}, {57,5603}, {65,13464}, {77,24213}, {80,28236}, {84,10305}, {104,1519}, {105,2728}, {106,1309}, {140,9957}, {142,20270}, {149,4881}, {165,30305}, {226,999}, {244,1735}, {354,15950}, {376,9580}, {388,6939}, {390,30282}, {392,5745}, {484,28228}, {495,11230}, {496,950}, {497,3576}, {517,1387}, {518,33562}, {522,905}, {527,38026}, {529,5087}, {553,39542}, {595,3075}, {611,38049}, {631,1697}, {758,5570}, {859,17197}, {912,5083}, {942,5901}, {944,6969}, {952,25405}, {956,3452}, {962,5265}, {993,22767}, {1000,33994}, {1056,5219}, {1058,3601}, {1106,1777}, {1111,1323}, {1124,8983}, {1155,5298}, {1329,11260}, {1335,13971}, {1388,1837}, {1421,1870}, {1457,32486}, {1467,6847}, {1478,3817}, {1479,4297}, {1482,4848}, {1497,37522}, {1617,22753}, {1699,4293}, {1706,17567}, {1727,3338}, {1736,4694}, {1739,24028}, {1771,3915}, {1772,24168}, {1776,3333}, {1788,7982}, {1858,12005}, {1861,15500}, {1875,23711}, {1878,2840}, {2006,40437}, {2078,6905}, {2093,5435}, {2098,11362}, {2099,17728}, {2170,8074}, {2551,25522}, {2646,37722}, {2800,18838}, {2802,6681}, {2829,22835}, {2975,12572}, {3035,3880}, {3036,33956}, {3057,5433}, {3058,37600}, {3090,9578}, {3091,4308}, {3304,11375}, {3340,10595}, {3361,4295}, {3419,24386}, {3421,30827}, {3434,35262}, {3474,31162}, {3476,5587}, {3487,10396}, {3488,13384}, {3523,9785}, {3524,35445}, {3554,20262}, {3583,21578}, {3585,12571}, {3586,5274}, {3600,9612}, {3612,4314}, {3653,37606}, {3656,36279}, {3660,6001}, {3668,24179}, {3683,31157}, {3753,6692}, {3814,32557}, {3825,10523}, {3873,18397}, {3885,17566}, {3947,37692}, {4187,5795}, {4294,7987}, {4298,5563}, {4305,30389}, {4316,28158}, {4321,38037}, {4342,5119}, {4939,38462}, {5044,16215}, {5045,12242}, {5048,28234}, {5049,5719}, {5080,32558}, {5082,5438}, {5120,21068}, {5122,28174}, {5123,6667}, {5176,31272}, {5204,12701}, {5218,31393}, {5248,8071}, {5252,10175}, {5253,12436}, {5258,18250}, {5316,9708}, {5432,5919}, {5434,17605}, {5440,5853}, {5443,12577}, {5542,8545}, {5657,7962}, {5687,21627}, {5691,10591}, {5722,10246}, {5727,7967}, {5730,24391}, {5818,37709}, {5836,6691}, {5840,18857}, {5880,40726}, {6245,10785}, {6284,37429}, {6349,24177}, {6690,10179}, {6705,12672}, {6796,11510}, {6842,24927}, {6848,12650}, {6931,36977}, {6932,10572}, {6945,7741}, {7280,12512}, {7354,18483}, {7373,11374}, {7951,10171}, {7988,10590}, {8068,33709}, {8256,33895}, {8666,12527}, {9436,24203}, {9540,31432}, {9661,13883}, {9669,18481}, {9955,18990}, {10069,21636}, {10073,33337}, {10074,21635}, {10089,11599}, {10090,21630}, {10091,13605}, {10122,11281}, {10265,12740}, {10304,30332}, {10384,21151}, {10398,11038}, {10593,18480}, {10896,31673}, {10912,37828}, {10914,13747}, {10944,17606}, {10948,17647}, {10950,13607}, {11025,18412}, {11028,11726}, {11047,12686}, {11194,24703}, {11263,16152}, {11508,25440}, {11545,12735}, {11725,24472}, {11734,12016}, {12019,28204}, {12513,21075}, {12563,18398}, {12589,39870}, {12608,22760}, {12609,22766}, {12832,25485}, {13273,24042}, {13528,21154}, {13624,15171}, {14115,29008}, {15079,37707}, {15178,37730}, {15287,17073}, {15298,38059}, {15326,28150}, {15524,26932}, {16485,34231}, {17614,24390}, {18393,37587}, {22465,24225}, {22791,37582}, {23536,32577}, {23537,37094}, {24210,37617}, {24929,38028}, {30478,31435}, {34590,39756}, {37080,40270}, {37602,37701}, {37624,37739}, {37708,38155}

X(44675) = midpoint of X(i) and X(j) for these {i,j}: {1,1737}, {11,1319}, {36,30384}, {104,1519}, {1387,15325}, {3583,21578}, {4511,26015}, {5048,40663}, {5126,7743}, {6735,38460}, {11545,12735}
X(44675) = reflection of X(i) in X(j) for these (i,j): (3911,15325), (5123,6667), (5570,18240)
X(44675) = complement of X(6735)
X(44675) = X(21)-beth conjugate of X(22350)
X(44675) = X(i)-complementary conjugate of X(j) for these (i,j): (56,119), (104,1329), (909,3452), (1397,23980), (1408,34586)
X(44675) = X(649)-he conjugate of X(9359)
X(44675) = X(1537)-reciprocal conjugate of X(908)
X(44675) = crossdifference of every pair of points on line X(198)-X(649)
X(44675) = crosssum of X(55)and X(22356)
X(44675) = crosspoint of X(7)and X(6336)
X(44675) = barycentric product of X(1537) and X(34234)
X(44675) = barycentric quotient of X(1537) and X(908)
X(44675) = trilinear product of X(104) and X(1537)
X(44675) = trilinear quotient of X(1537) and X(517)
X(44675) = circumconic-centered-at-X(1)-inverse of X(6765)
X(44675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,31397), (1,499,10), (1,1125,13411), (1,3086,1210), (1,3582,1737), (1,3624,3085), (1,10072,11019), (1,10573,3244), (2,3872,10), (2,38460,6735), (3,11373,12053), (3,12053,10624), (4,1420,4311), (5,24928,10106), (8,3616,24558), (36,16173,30384), (56,946,4292), (56,11376,946), (496,1385,950), (496,6907,15845), (497,3576,4304), (499,3086,15866), (551,11019,1), (938,3622,1), (962,5265,15803), (999,5886,226), (1387,6713,15558), (1388,1837,5882), (1478,23708,3817), (1479,37618,4297), (1699,13462,4293), (2098,24914,11362), (3057,5433,6684), (3091,4308,9613), (3304,11375,21620), (3333,9624,3485), (3361,11522,4295), (3476,10589,5587), (3576,37704,497), (3616,4666,551), (3616,10527,19861), (3616,14986,1), (3616,24541,1125), (3636,6738,1), (3817,4315,1478), (4342,10164,5119), (5204,12701,31730), (5265,18220,962), (5274,5731,3586), (5563,12047,4298), (5563,37735,12047), (7962,31231,5657), (8666,21616,12527), (10527,19861,10), (10916,30144,6737), (12513,25681,21075), (21842,37720,10572)

X(44676) = X(515)X(1317)∩X(516)X(3025)

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

See Antreas Hatzipolakis and César Lozada, euclid 2204.

X(44676) lies on these lines: {514, 23152}, {515, 1317}, {516, 3025}, {1836, 34230}, {3667, 17660}, {30384, 34586}

X(44677) = X(30)X(5107)∩X(3815)X(5915)

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

See Antreas Hatzipolakis and César Lozada, euclid 2204.

X(44677) lies on these lines: {30, 5107}, {3815, 5915}, {9143, 34169}

X(44678) = X(2)X(187)∩X(4)X(754)

Barycentrics    7*a^4-5*b^4+6*b^2*c^2-5*c^4 : :
X(44678) = 13*X(2)-12*X(1153), 7*X(2)-6*X(5569), 5*X(2)-6*X(8176), 4*X(2)-3*X(8182), X(2)-3*X(23334), 5*X(4)-2*X(7751), 4*X(4)-X(14023), 3*X(4)-2*X(18546), 14*X(1153)-13*X(5569), 10*X(1153)-13*X(8176), 16*X(1153)-13*X(8182), 4*X(1153)-13*X(23334), 5*X(5569)-7*X(8176), 8*X(5569)-7*X(8182), 2*X(5569)-7*X(23334), 8*X(7751)-5*X(14023), 3*X(7751)-5*X(18546), 8*X(8176)-5*X(8182), 2*X(8176)-5*X(23334), X(8182)-4*X(23334)

See Antreas Hatzipolakis and César Lozada, euclid 2223.

X(44678) lies on these lines: {2, 187}, {4, 754}, {20, 7843}, {30, 8716}, {32, 16041}, {39, 33272}, {315, 11361}, {325, 43618}, {376, 7775}, {381, 13468}, {382, 7758}, {524, 3830}, {538, 3543}, {543, 10722}, {591, 6560}, {626, 14039}, {1007, 6781}, {1078, 33005}, {1285, 7844}, {1991, 6561}, {1992, 11648}, {2548, 8356}, {2549, 41624}, {3146, 7759}, {3363, 8556}, {3529, 7764}, {3534, 7618}, {3545, 12110}, {3767, 7823}, {3785, 39590}, {3832, 7780}, {3845, 7615}, {3860, 16509}, {5007, 32982}, {5032, 39593}, {5066, 7610}, {5071, 34506}, {5077, 9300}, {5319, 33229}, {5862, 33623}, {5863, 33625}, {6033, 9890}, {6179, 32996}, {7617, 41106}, {7619, 15719}, {7622, 19708}, {7739, 7812}, {7745, 11287}, {7747, 7795}, {7748, 41750}, {7750, 44543}, {7753, 32986}, {7757, 33192}, {7760, 33279}, {7763, 33265}, {7768, 14068}, {7772, 33238}, {7773, 35297}, {7774, 43619}, {7781, 33703}, {7785, 33264}, {7790, 12156}, {7796, 33280}, {7799, 33193}, {7802, 31401}, {7806, 33291}, {7809, 33007}, {7810, 32983}, {7811, 33016}, {7814, 33244}, {7821, 32981}, {7824, 31417}, {7825, 33285}, {7827, 33278}, {7837, 8597}, {7842, 33210}, {7845, 32815}, {7854, 32979}, {7855, 32826}, {7858, 32997}, {7860, 14035}, {7870, 33187}, {7873, 32971}, {7885, 14036}, {7888, 33239}, {7916, 32822}, {7922, 14031}, {7936, 33269}, {8352, 14614}, {8353, 11163}, {8358, 42849}, {8588, 34803}, {8703, 11184}, {9698, 33226}, {9761, 42510}, {9763, 42511}, {9770, 11001}, {9771, 15693}, {9993, 41099}, {11160, 14711}, {11165, 15685}, {11317, 37671}, {12040, 15690}, {12150, 33251}, {14568, 20065}, {14645, 39838}, {15640, 32479}, {15683, 34504}, {15687, 34505}, {18424, 37667}, {25486, 36521}, {32816, 35927}, {32972, 35007}, {34229, 43457}, {38737, 41400}

X(44678) = reflection of X(i) in X(j) for these (i, j): (376, 7775), (8667, 3845), (9890, 6033), (15683, 34504), (34505, 15687)
X(44678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14976, 14907), (3845, 8667, 7615), (7747, 7818, 14033), (7747, 32006, 7795), (7812, 33017, 7739), (7818, 14033, 7795), (14033, 32006, 7818)

X(44679) = X(3)X(64)∩X(66)X(182)

Barycentrics    (a^14-4*(b^2+c^2)*a^12+(5*b^4+8*b^2*c^2+5*c^4)*a^10-2*(b^2+c^2)*b^2*c^2*a^8-(5*b^8+5*c^8+2*(2*b^2-c^2)*(b^2-2*c^2)*b^2*c^2)*a^6+4*(b^4-c^4)^2*(b^2+c^2)*a^4-(b^4-c^4)^2*(b^4+4*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2)*a^2 : :
X(44679) = 3*X(6759)-4*X(40276)

See Antreas Hatzipolakis and César Lozada, euclid 2225.

X(44679) lies on these lines: {3, 64}, {26, 34146}, {49, 34780}, {54, 32064}, {66, 182}, {110, 34781}, {140, 206}, {161, 10625}, {184, 3541}, {427, 578}, {511, 9937}, {542, 12420}, {569, 1853}, {1092, 1370}, {1147, 1503}, {1614, 12324}, {2393, 9925}, {2883, 31833}, {3518, 41715}, {3546, 5596}, {3575, 22802}, {5157, 23042}, {5447, 15577}, {5462, 34117}, {5576, 23325}, {5651, 7383}, {5878, 18533}, {6225, 14157}, {6642, 19149}, {7399, 35283}, {7487, 43617}, {7506, 41580}, {8991, 9687}, {9920, 13340}, {9934, 10990}, {9968, 12106}, {10984, 32379}, {11438, 41725}, {11695, 19137}, {12112, 36983}, {12118, 13346}, {12134, 41602}, {12359, 44470}, {13336, 40686}, {13347, 25563}, {13382, 17818}, {15132, 36201}, {15805, 19153}, {16655, 44080}, {17538, 43813}, {19161, 34779}, {23329, 37515}, {25738, 41603}, {31099, 34148}, {31723, 34786}, {32345, 39242}, {34785, 37480}, {37440, 44544}

X(44679) = reflection of X(32321) in X(10282)
X(44679) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1498, 10539, 6759), (1498, 15139, 10539)

X(44680) = CENTER OF THIS CIRCLE: CIRCUMCIRCLE-INVERSE OF LINE X(5)X(6)

Barycentrics    a^2*(b^2 - c^2)*(2*a^6 - 4*a^4*b^2 + 2*a^2*b^4 - 4*a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4) : :
X(44680) = X[5926] + 2 X[39477], X[39228] + 2 X[39481]

X(44680) lies on these lines: {3, 3566}, {25, 42399}, {157, 18310}, {186, 523}, {249, 1576}, {297, 44451}, {512, 5926}, {525, 14270}, {669, 5940}, {1624, 7468}, {9494, 25423}, {10278, 35225}, {13558, 41357}

X(44680) = midpoint of X(i) and X(j) for these {i,j}: {3, 34952}, {5926, 39228}, {39477, 39481}
X(44680) = reflection of X(i) in X(j) for these {i,j}: {5926, 39481}, {39228, 39477}
X(44680) = circumcircle-inverse of X(18348)
X(44680) = crosspoint of X(54) and X(3565)
X(44680) = crosssum of X(i) and X(j) for these (i,j): {5, 3566}, {512, 13881}, {520, 30771}
X(44680) = crossdifference of every pair of points on line {216, 37637}

X(44681) = (name pending)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2234.

X(44681) lies on this line: {1658, 2979}

X(44682) = 63RD HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    4*S^2 + 11*a^2*(-a^2 + b^2 + c^2) : :
Barycentrics    12*a^4 - 13*a^2*b^2 + b^4 - 13*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(44682) = 3 X[2] + 11 X[3], 39 X[2] - 11 X[4], 18 X[2] - 11 X[5], 45 X[2] + 11 X[20], 15 X[2] - 22 X[140], 17 X[2] + 11 X[376], 25 X[2] - 11 X[381], 81 X[2] - 11 X[382], 57 X[2] - 22 X[546], 29 X[2] - 22 X[547], 27 X[2] + 22 X[548], 4 X[2] - 11 X[549], 24 X[2] + 11 X[550], 27 X[2] - 55 X[631], 48 X[2] - 55 X[632], 69 X[2] - 55 X[1656], 87 X[2] + 11 X[1657], 15 X[2] - 11 X[3090], 111 X[2] - 55 X[3091], 123 X[2] - 11 X[3146], 57 X[2] + 55 X[3522], 3 X[2] - 11 X[3523], 5 X[2] - 33 X[3524], 93 X[2] - 121 X[3525], 9 X[2] - 11 X[3526], 9 X[2] + 11 X[3528], 129 X[2] + 11 X[3529], 9 X[2] - 44 X[3530], 159 X[2] - 187 X[3533], 31 X[2] + 11 X[3534], 67 X[2] - 11 X[3543], 327 X[2] - 187 X[3544], 61 X[2] - 33 X[3545], 60 X[2] - 11 X[3627], 51 X[2] - 44 X[3628], 53 X[2] - 11 X[3830], 27 X[2] - 11 X[3832], 89 X[2] - 33 X[3839], 153 X[2] - 55 X[3843], 32 X[2] - 11 X[3845], 93 X[2] - 44 X[3850], 21 X[2] - 11 X[3851], 9 X[2] - 2 X[3853], 411 X[2] - 187 X[3854], 261 X[2] - 121 X[3855], 207 X[2] - 88 X[3856], 24 X[2] - 11 X[3857], 12 X[2] - 5 X[3858], 243 X[2] - 110 X[3859], 107 X[2] - 44 X[3860], 135 X[2] - 44 X[3861], 19 X[2] - 33 X[5054], 47 X[2] - 33 X[5055], 177 X[2] - 121 X[5056], 213 X[2] + 11 X[5059], 43 X[2] - 22 X[5066], 171 X[2] - 143 X[5067], 255 X[2] - 143 X[5068], 135 X[2] - 121 X[5070], 83 X[2] - 55 X[5071], 219 X[2] - 121 X[5072], 15 X[2] - X[5073], 237 X[2] - 55 X[5076], 213 X[2] - 143 X[5079], 243 X[2] - 187 X[7486], 10 X[2] + 11 X[8703], 65 X[2] - 44 X[10109], 37 X[2] - 44 X[10124], 3 X[2] - 143 X[10299], 87 X[2] - 143 X[10303], 23 X[2] + 33 X[10304], 73 X[2] + 11 X[11001], 26 X[2] - 33 X[11539], 67 X[2] - 88 X[11540], 375 X[2] - 11 X[11541], 79 X[2] - 44 X[11737], 23 X[2] - 44 X[11812], X[2] - 22 X[12100], 85 X[2] - 22 X[12101], 177 X[2] - 44 X[12102], 69 X[2] + 22 X[12103], 39 X[2] - 88 X[12108], 15 X[2] - 8 X[12811], 159 X[2] - 110 X[12812], 43 X[2] + 55 X[14093], 103 X[2] - 33 X[14269], 6 X[2] - 11 X[14869], 83 X[2] - 132 X[14890], 5 X[2] + 44 X[14891], 115 X[2] - 66 X[14892], 71 X[2] - 22 X[14893], 321 X[2] - 209 X[15022], 179 X[2] - 11 X[15640], 59 X[2] + 11 X[15681], 95 X[2] - 11 X[15682], 101 X[2] + 11 X[15683], 109 X[2] - 11 X[15684], 115 X[2] + 11 X[15685], 38 X[2] + 11 X[15686], 46 X[2] - 11 X[15687], 37 X[2] + 33 X[15688], 65 X[2] + 33 X[15689], 41 X[2] + 22 X[15690], 5 X[2] + 2 X[15691], X[2] + 55 X[15692], 13 X[2] - 55 X[15693], 41 X[2] - 55 X[15694], 71 X[2] + 55 X[15695]

See Antreas Hatzipolakis and Peter Moses, euclid 2240.

X(44682) lies on these lines: {2, 3}, {74, 22251}, {141, 30507}, {187, 9606}, {395, 43776}, {396, 43775}, {397, 42966}, {398, 42967}, {952, 9588}, {1152, 9680}, {1353, 5092}, {1483, 11362}, {2548, 5585}, {3053, 31450}, {3068, 6497}, {3069, 6496}, {3411, 5352}, {3412, 5351}, {3448, 15042}, {3579, 10283}, {3624, 28178}, {3819, 13491}, {3933, 43459}, {4297, 38138}, {4301, 31663}, {4309, 15325}, {4325, 5432}, {4330, 5433}, {4701, 5690}, {5010, 10386}, {5204, 31452}, {5206, 31406}, {5210, 31492}, {5267, 9711}, {5319, 15815}, {5334, 43634}, {5335, 43635}, {5349, 43204}, {5350, 43203}, {5355, 9607}, {5562, 11592}, {5735, 38111}, {5881, 38112}, {5888, 43613}, {5894, 10182}, {5901, 35242}, {5946, 13348}, {6053, 12041}, {6101, 16836}, {6102, 15606}, {6199, 9692}, {6200, 19116}, {6396, 19117}, {6409, 35256}, {6410, 35255}, {6411, 7584}, {6412, 7583}, {6451, 13935}, {6452, 9540}, {6456, 31487}, {6500, 9542}, {6519, 19053}, {6522, 19054}, {6684, 37705}, {7280, 15888}, {7586, 9693}, {7759, 12040}, {7765, 8589}, {7917, 14929}, {7987, 31425}, {7998, 31834}, {9541, 13993}, {9624, 16192}, {9698, 15513}, {10095, 36987}, {10164, 34773}, {10193, 34782}, {10627, 14531}, {10645, 16773}, {10646, 16772}, {10653, 42773}, {10654, 42774}, {10984, 40111}, {11464, 43903}, {11542, 42490}, {11543, 42491}, {11693, 38632}, {11694, 15054}, {11749, 38700}, {12512, 38034}, {13340, 16881}, {13369, 33575}, {13392, 15041}, {13421, 16226}, {13451, 15028}, {14073, 38710}, {14128, 14855}, {14449, 15045}, {14677, 38793}, {14810, 38110}, {15036, 15057}, {15048, 15515}, {15051, 23236}, {15655, 31400}, {16163, 20396}, {16808, 42492}, {16809, 42493}, {16960, 42685}, {16961, 42684}, {16964, 43253}, {20125, 38633}, {20379, 34153}, {21163, 32521}, {21167, 40107}, {22236, 42634}, {22238, 42633}, {23238, 38618}, {23302, 42433}, {23303, 42434}, {23324, 32903}, {26614, 38734}, {28186, 31423}, {29181, 42785}, {30282, 34753}, {31666, 43174}, {33416, 42144}, {33417, 42145}, {34754, 42686}, {34755, 42687}, {34783, 44324}, {35812, 42216}, {35813, 42215}, {36836, 42913}, {36843, 42912}, {36969, 42949}, {36970, 42948}, {38229, 38736}, {38723, 40685}, {40693, 42916}, {40694, 42917}, {42087, 42489}, {42088, 42488}, {42089, 43194}, {42090, 43102}, {42091, 43103}, {42092, 43193}, {42108, 43638}, {42109, 43643}, {42121, 42147}, {42122, 42153}, {42123, 42156}, {42124, 42148}, {42129, 43772}, {42132, 43771}, {42135, 43632}, {42136, 43327}, {42137, 43326}, {42138, 43633}, {42163, 42529}, {42166, 42528}, {42260, 43435}, {42261, 43434}, {42435, 42930}, {42436, 42931}, {42496, 43640}, {42497, 43639}, {42584, 43029}, {42585, 43028}, {42631, 43107}, {42632, 43100}, {42682, 43782}, {42683, 43781}, {42815, 43777}, {42816, 43778}, {42924, 43428}, {42925, 43429}, {42938, 43206}, {42939, 43205}, {42942, 43774}, {42943, 43773}

X(44682) = midpoint of X(i) and X(j) for these {i,j}: {3, 3523}, {550, 3857}, {3526, 3528}, {15698, 15700}
X(44682) = reflection of X(i) in X(j) for these {i,j}: {5, 3526}, {549, 19711}, {3090, 140}, {14869, 3523}, {15687, 41106}, {15700, 12100}, {19711, 15700}
X(44682) = anticomplement of X(44904)
X(44682) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3, 33923}, {2, 3853, 5}, {2, 5073, 12811}, {2, 15696, 3853}, {2, 15704, 3858}, {2, 33923, 15704}, {2, 35421, 33699}, {3, 4, 34200}, {3, 140, 8703}, {3, 381, 21735}, {3, 382, 21734}, {3, 549, 550}, {3, 631, 548}, {3, 1656, 10304}, {3, 1657, 19708}, {3, 3522, 15759}, {3, 3524, 140}, {3, 3526, 3528}, {3, 3530, 5}, {3, 3533, 41982}, {3, 5054, 3522}, {3, 7485, 11250}, {3, 10299, 12100}, {3, 12100, 15712}, {3, 15693, 4}, {3, 15700, 3523}, {3, 15706, 10299}, {3, 15707, 1657}, {3, 15708, 41981}, {3, 15712, 549}, {3, 15717, 3530}, {3, 15718, 1656}, {3, 15720, 376}, {3, 33923, 15714}, {3, 37126, 10226}, {4, 12108, 11539}, {4, 15693, 12108}, {4, 15705, 3}, {5, 20, 3627}, {5, 3530, 549}, {5, 3843, 38071}, {5, 3853, 3858}, {5, 8703, 20}, {5, 14869, 3526}, {5, 15687, 3856}, {5, 15704, 3853}, {5, 15712, 3530}, {20, 140, 5}, {20, 631, 5070}, {20, 5070, 3861}, {20, 15682, 17800}, {20, 15696, 15691}, {20, 15717, 3524}, {20, 33703, 15685}, {140, 548, 3861}, {140, 3627, 15699}, {140, 3861, 5070}, {140, 8703, 3627}, {140, 12101, 3628}, {140, 12103, 14892}, {140, 12108, 15721}, {140, 12811, 2}, {140, 14891, 3}, {140, 15691, 12811}, {140, 15699, 632}, {140, 15701, 14869}, {140, 33923, 5073}, {140, 44245, 381}, {376, 15713, 38071}, {376, 15720, 3628}, {376, 41983, 15713}, {381, 3524, 44580}, {381, 21735, 44245}, {382, 631, 16239}, {382, 7486, 3859}, {382, 16239, 5}, {382, 21734, 548}, {546, 3522, 15686}, {546, 5067, 5}, {546, 15759, 3522}, {547, 3855, 5}, {548, 631, 5}, {548, 3530, 631}, {548, 3861, 20}, {548, 16239, 382}, {549, 550, 632}, {549, 3627, 140}, {549, 8703, 15699}, {549, 15714, 35404}, {549, 17504, 15711}, {549, 19710, 11539}, {549, 38071, 15713}, {550, 632, 3845}, {550, 15699, 3627}, {550, 15711, 3}, {550, 35404, 15704}, {631, 3528, 3832}, {631, 3832, 3526}, {631, 5070, 140}, {631, 21734, 382}, {1656, 3856, 5}, {1656, 10304, 12103}, {1656, 12103, 15687}, {1656, 33703, 3856}, {1657, 10303, 547}, {1657, 15707, 10303}, {1657, 41990, 3627}, {2041, 2042, 5055}, {3090, 3523, 15701}, {3090, 3524, 3523}, {3090, 3528, 20}, {3090, 3627, 3857}, {3090, 15701, 140}, {3146, 15694, 35018}, {3146, 35018, 23046}, {3522, 5054, 546}, {3522, 5067, 17800}, {3522, 15715, 3}, {3523, 3528, 3526}, {3523, 3832, 631}, {3523, 14869, 549}, {3523, 15698, 3}, {3523, 15703, 12108}, {3524, 8703, 549}, {3524, 14891, 8703}, {3524, 15692, 15716}, {3524, 15705, 15689}, {3524, 15715, 15682}, {3524, 15716, 14891}, {3524, 15721, 15693}, {3525, 3534, 3850}, {3528, 3530, 14869}, {3528, 15700, 3530}, {3529, 14093, 41981}, {3530, 12100, 15717}, {3530, 15717, 15712}, {3533, 3830, 12812}, {3627, 5073, 35404}, {3627, 8703, 550}, {3628, 3843, 5}, {3628, 12101, 5068}, {3628, 15720, 15713}, {3628, 41983, 15720}, {3853, 15696, 15704}, {3853, 33923, 15696}, {3856, 12103, 33703}, {3856, 33703, 15687}, {3857, 14869, 632}, {3858, 15704, 35404}, {3858, 33923, 550}, {3859, 7486, 5}, {3859, 16239, 7486}, {3861, 5070, 5}, {5054, 15715, 15759}, {5054, 15759, 15686}, {5054, 17800, 5067}, {5054, 35403, 2}, {5056, 15681, 12102}, {5059, 5079, 14893}, {5059, 15709, 5079}, {5066, 41981, 3529}, {5067, 17800, 546}, {5068, 15720, 140}, {5073, 15691, 15704}, {5073, 15696, 20}, {5079, 15695, 5059}, {7987, 31425, 37727}, {8703, 10109, 19710}, {8703, 15713, 12101}, {8703, 17504, 14891}, {8703, 19711, 15701}, {10109, 12108, 140}, {10109, 15721, 11539}, {10109, 34200, 15689}, {10124, 15688, 33699}, {10299, 15692, 3}, {10303, 19708, 1657}, {10304, 11812, 15687}, {10304, 15718, 11812}, {11539, 15693, 549}, {11539, 17504, 15705}, {11539, 34200, 19710}, {11812, 12103, 1656}, {12100, 14891, 3524}, {12100, 15692, 17504}, {12100, 15698, 19711}, {12100, 15716, 8703}, {12100, 17504, 549}, {12101, 41986, 381}, {12108, 34200, 4}, {12811, 15691, 5073}, {12811, 15704, 3627}, {12811, 33923, 15691}, {14093, 15708, 5066}, {14869, 15712, 19711}, {14869, 19711, 3523}, {14891, 15712, 3627}, {14891, 15716, 17504}, {14892, 15685, 15687}, {15682, 15759, 8703}, {15687, 15718, 549}, {15688, 15719, 10124}, {15689, 15693, 15721}, {15689, 15701, 15703}, {15689, 15721, 10109}, {15689, 34200, 8703}, {15690, 15694, 23046}, {15690, 35018, 3146}, {15692, 15706, 12100}, {15693, 15705, 34200}, {15693, 34200, 11539}, {15694, 15710, 15690}, {15694, 35018, 41992}, {15695, 15709, 14893}, {15698, 15717, 3526}, {15700, 15701, 3524}, {15701, 15716, 15698}, {15704, 15714, 33923}, {15707, 19708, 547}, {15711, 15712, 632}, {15712, 17504, 3}, {15713, 41983, 549}, {15720, 35401, 3525}, {15721, 19710, 15699}, {21735, 44245, 8703}, {23046, 41992, 35018}, {34551, 34552, 15687}, {35231, 35232, 37947}, {36445, 36463, 35418}, {37298, 37307, 17563}, {44245, 44580, 140}

X(44683) = X(3)X(69)∩X(30)X(343)

Barycentrics    (a^2-b^2-c^2) (2 a^8+a^6 b^2-9 a^4 b^4+7 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2-7 a^2 b^4 c^2+4 b^6 c^2-9 a^4 c^4-7 a^2 b^2 c^4-6 b^4 c^4+7 a^2 c^6+4 b^2 c^6-c^8) : :
Barycentrics    SA (2 S^2+3 (4 R^2-SW) (SA-SW)) : :
X(44683) = X(11456)-3*X(44210)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2254.

X(44683) lies on these lines: {3,69}, {5,3066}, {30,343}, {140,5562}, {141,11438}, {185,16197}, {265,12605}, {389,3589}, {394,549}, {468,11459}, {524,11430}, {546,1531}, {547,1568}, {548,12041}, {550,32140}, {568,18583}, {578,3629}, {599,37487}, {1092,3530}, {1216,6699}, {1352,37458}, {1353,37506}, {1495,33591}, {1595,17834}, {1658,31831}, {2854,20417}, {3426,34621}, {3575,41171}, {3580,34664}, {3581,18358}, {3763,9786}, {3818,6756}, {4549,14852}, {4846,6823}, {5054,37669}, {5305,22416}, {5663,16618}, {5876,13383}, {5889,14389}, {5890,7499}, {5891,6677}, {5907,21841}, {6053,16252}, {6090,35486}, {6144,11425}, {6676,13754}, {6696,15644}, {7405,37490}, {7426,15052}, {7503,37644}, {7512,12317}, {7526,13142}, {7542,18436}, {7689,31829}, {9306,37935}, {9818,41588}, {10020,10272}, {10154,18451}, {10257,23039}, {10539,44277}, {10565,32063}, {10605,43653}, {11245,35921}, {11442,44239}, {11456,44210}, {11591,16238}, {12359,12362}, {13352,34380}, {14118,37779}, {14831,37649}, {15030,32269}, {15037,22151}, {15060,44233}, {15068,34351}, {15108,37941}, {15606,25563}, {15712,35602}, {16657,41586}, {16658,37899}, {18435,37971}, {18536,23291}, {23328,37480}, {32608,37347}, {34002,34783}, {34507,37934}, {34725,41465}, {35450,35513}, {41614,44218}

X(44683) = midpoint of X(11442) and X(44239)
X(44683) = reflection of X(6676) in X(44201)
X(44683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,11411,31804), (1216,44158,16196), (10519,18931,3)

X(44684) = (name pending)

Barycentrics    (a^2-b^2-c^2) (a^10-6 a^8 b^2+5 a^6 b^4+5 a^4 b^6-6 a^2 b^8+b^10-4 a^8 c^2+3 a^6 b^2 c^2+2 a^4 b^4 c^2+3 a^2 b^6 c^2-4 b^8 c^2+6 a^6 c^4+13 a^4 b^2 c^4+13 a^2 b^4 c^4+6 b^6 c^4-4 a^4 c^6-11 a^2 b^2 c^6-4 b^4 c^6+a^2 c^8+b^2 c^8) (a^10-4 a^8 b^2+6 a^6 b^4-4 a^4 b^6+a^2 b^8-6 a^8 c^2+3 a^6 b^2 c^2+13 a^4 b^4 c^2-11 a^2 b^6 c^2+b^8 c^2+5 a^6 c^4+2 a^4 b^2 c^4+13 a^2 b^4 c^4-4 b^6 c^4+5 a^4 c^6+3 a^2 b^2 c^6+6 b^4 c^6-6 a^2 c^8-4 b^2 c^8+c^10) : :
Barycentrics    SA (S^2 (12 R^2-2 SC-5 SW)+3 SC^2 (4 R^2-SW)) (2 S^2 (SA+SC-2 SW)+3 (4 R^2-SW) (SA-SW) (SC-SW)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2254.

X(44684) lies on this line: {184, 44683}

X(44685) = CENTER OF PAVLOS-I CIRCLE

Barycentrics    a*(3*a^5*b - 3*a^4*b^2 - 6*a^3*b^3 + 6*a^2*b^4 + 3*a*b^5 - 3*b^6 + 3*a^5*c - 20*a^4*b*c + 23*a^3*b^2*c + 8*a^2*b^3*c - 26*a*b^4*c + 12*b^5*c - 3*a^4*c^2 + 23*a^3*b*c^2 - 58*a^2*b^2*c^2 + 23*a*b^3*c^2 + 3*b^4*c^2 - 6*a^3*c^3 + 8*a^2*b*c^3 + 23*a*b^2*c^3 - 24*b^3*c^3 + 6*a^2*c^4 - 26*a*b*c^4 + 3*b^2*c^4 + 3*a*c^5 + 12*b*c^5 - 3*c^6) : :
X(44685) = 3 X[13375] - 2 X[31794], 3 X[13375] - X[39777]

See Antreas Hatzipolakis and Peter Moses, euclid 2268.

X(44685) lies on these lines: {1, 6797}, {999, 11524}, {1317, 12009}, {3585, 5559}, {3614, 30384}, {5844, 10106}, {10914, 15179}, {13375, 31794}

X(44685) = reflection of X(39777) in X(31794)
X(44685) = {X(13375),X(39777)}-harmonic conjugate of X(31794)

X(44686) = CENTER OF PAVLOS-II CIRCLE

Barycentrics    2*a^16 - 3*a^14*b^2 - 7*a^12*b^4 + 13*a^10*b^6 + 5*a^8*b^8 - 17*a^6*b^10 + 3*a^4*b^12 + 7*a^2*b^14 - 3*b^16 - 3*a^14*c^2 + 20*a^12*b^2*c^2 - 6*a^10*b^4*c^2 - 64*a^8*b^6*c^2 + 61*a^6*b^8*c^2 + 24*a^4*b^10*c^2 - 44*a^2*b^12*c^2 + 12*b^14*c^2 - 7*a^12*c^4 - 6*a^10*b^2*c^4 + 86*a^8*b^4*c^4 - 28*a^6*b^6*c^4 - 123*a^4*b^8*c^4 + 90*a^2*b^10*c^4 - 12*b^12*c^4 + 13*a^10*c^6 - 64*a^8*b^2*c^6 - 28*a^6*b^4*c^6 + 192*a^4*b^6*c^6 - 53*a^2*b^8*c^6 - 12*b^10*c^6 + 5*a^8*c^8 + 61*a^6*b^2*c^8 - 123*a^4*b^4*c^8 - 53*a^2*b^6*c^8 + 30*b^8*c^8 - 17*a^6*c^10 + 24*a^4*b^2*c^10 + 90*a^2*b^4*c^10 - 12*b^6*c^10 + 3*a^4*c^12 - 44*a^2*b^2*c^12 - 12*b^4*c^12 + 7*a^2*c^14 + 12*b^2*c^14 - 3*c^16 : :
X(44686) = 3 X[4] + X[22549], 3 X[5] - X[22962], 9 X[381] - X[22550], X[2929] - 5 X[3091], 7 X[3832] + X[22555], 7 X[3832] - 3 X[22971], 3 X[5943] - 2 X[32050], 3 X[11245] - 4 X[32163], X[22549] - 3 X[23308], X[22555] + 3 X[22971], 3 X[22816] + X[22962]

See Antreas Hatzipolakis and Peter Moses, euclid 2268.

X(44686) lies on these lines: {4, 22549}, {5, 13293}, {381, 22550}, {542, 31985}, {895, 15044}, {2929, 3091}, {3627, 22978}, {3832, 22555}, {5943, 32050}, {10151, 22538}, {10297, 22834}, {11245, 32163}, {14913, 22968}, {22800, 22804}, {22960, 42273}, {22961, 42270}, {22970, 23047}, {22974, 42110}, {22975, 42107}, {43613, 43616}

X(44686) = midpoint of X(i) and X(j) for these {i,j}: {4, 23308}, {5, 22816}, {3627, 22978}, {22538, 22966}
X(44686) = X(23308)-of-Euler-triangle
X(44686) = {X(3832),X(22555)}-harmonic conjugate of X(22971)



leftri

Dao-conjugates: X(44687)-X(44729)

rightri

This preamble and centers X(44687)-X(44729) were contributed by César Eliud Lozada, September 8, 2021.

Let ABC be a triangle, P a point on its plane and and Ω an arbitrary circumconic of ABC. Lines AP, BP, CP cut again Ω at A', B', C', respectively, and parallel lines through these points to BC, CA, AB cut Ω again at A", B", C", respectively. Then lines AA", BB", CC" concur. (Dao Thanh Oai, May 12, 2021).

If barycentric coordinates of the center X of Ω are X = x : y : z and P = p: q : r, then D, the point of intersection of AA", BB", CC", is:

  D = D(X, P) = x*(x - y - z)*q*r : :

For a given X, this transformation is an involution. D(X, P) is introduced here as the X-Dao conjugate of P. More precisely, D(X, P) may be expressed as an isoconjugate:

  (X-Dao conjugate of W) = W-isoconjugate of P, where W = W(X)= a^3*(x - y + z)*(x + y - z)*y*z : :

The appearance of (m, n) in the following list means that (X(m)-Dao conjugate of P) = X(n)-isoconjugate of-P for all points P:

(1, 56), (2, 31), (3, 1), (4, 19614), (5, 2190), (6, 19), (8, 16945), (9, 6), (10, 58), (11, 109), (31, 7357), (37, 1333), (57, 2192), (66, 19616), (113, 36053), (114, 36051), (115, 163), (119, 36052), (124, 36050), (125, 162), (132, 293), (133, 35200), (136, 4575), (137, 36134), (141, 82), (206, 75), (214, 106), (216, 2148), (220, 17107), (223, 55), (226, 2299), (238, 30648), (244, 110), (478, 9), (512, 24037), (513, 765), (514, 1110), (520, 24021), (521, 24033), (522, 24027), (523, 1101), (626, 38847), (650, 2149), (661, 1252), (798, 5383), (960, 2363), (1015, 101), (1084, 662), (1086, 692), (1146, 1415), (1147, 158), (1193, 40453), (1209, 2216), (1213, 28615), (1214, 2194), (1249, 48), (1312, 1823), (1313, 1822), (1511, 36119), (1560, 36060), (1566, 36039), (2482, 923), (2679, 36036), (2883, 775), (2887, 40415), (3005, 24041), (3124, 4599), (3160, 41), (3161, 604), (3162, 63), (3163, 2159), (3258, 36034), (3259, 36037), (3341, 221), (3343, 204), (3647, 1126), (3789, 985), (3900, 24013), (4075, 849), (4370, 9456), (4858, 1576), (5139, 4592), (5190, 906), (5375, 649), (5452, 57), (5514, 36049), (5516, 36042), (5519, 36041), (5521, 1331), (5945, 1049), (6184, 1438), (6260, 1167), (6337, 1973), (6374, 560), (6376, 32), (6503, 1096), (6505, 25), (6523, 255), (6552, 1106), (6554, 7084), (6593, 897), (6600, 269), (6608, 7339), (6609, 200), (6615, 59), (6626, 213), (6631, 667), (6651, 1911), (7952, 603), (8054, 100), (8290, 1967), (8299, 741), (9296, 1919), (9410, 9406), (9428, 1924), (9460, 2251), (9467, 1966), (9470, 238), (10001, 3063), (10017, 36040), (11517, 34), (11597, 2166), (11672, 1910), (14092, 610), (14363, 2169), (14714, 934), (14993, 6149), (15166, 2576), (15167, 2577), (15259, 326), (15261, 18156), (15267, 1098), (15449, 34072), (15477, 14210), (15487, 7123), (15526, 32676), (15607, 36048), (15898, 36), (15899, 896), (16221, 36061), (16586, 34858), (16596, 32652), (17113, 1253), (17115, 7045), (17417, 32693), (17419, 8687), (17421, 32691), (17423, 811), (17761, 43076), (17793, 727), (18314, 23995), (18334, 32678), (18591, 2259), (18809, 36062), (19557, 292), (19576, 1581), (19584, 40746), (20532, 34077), (20619, 36058), (20620, 36059), (20621, 36057), (20622, 36056), (21975, 2964), (22391, 92), (23050, 7053), (23972, 911), (23980, 909), (23986, 32677), (23992, 36142), (24151, 3052), (24771, 1407), (25640, 1795), (26932, 8750), (31654, 36045), (31998, 798), (32664, 2), (32746, 34248), (33504, 36046), (33675, 9454), (33678, 21760), (34021, 1918), (34116, 91), (34452, 3112), (34467, 1897), (34544, 1989), (34586, 759), (34591, 112), (34853, 47), (34961, 4017), (35068, 18268), (35069, 34079), (35071, 24019), (35072, 32674), (35091, 36141), (35092, 32665), (35094, 32666), (35110, 34068), (35119, 34067), (35128, 32675), (35204, 1411), (35508, 1461), (35579, 36043), (35580, 36044), (35581, 36047), (36033, 4), (36103, 3), (36213, 43763), (36830, 661), (36896, 2173), (36899, 1755), (36906, 1914), (36908, 2328), (36911, 28607), (37836, 40430)

The X-Dao conjugate of P is the trilinear quotient P / W, and, among all known kinds of conjugaties, the X-Dao conjugate of P is:

This figure shows the locus of a point P such that for X = X(1), we have D(P,X) = D(X,P). Comparable loci result for other choices of X. In general, each such locus consists of three cubics, each of which passes through the vertices of the medial triangle, through X, and through the X(2)-Ceva conjugate of X. Also, the A-cubic passes through A, the B-cubic passes through B, and the C-cubic passes through C.

X(44687) = X(1)-DAO CONJUGATE OF X(5)

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

X(44687) lies on these lines: {10, 275}, {54, 72}, {78, 35196}, {95, 307}, {212, 318}, {1098, 1793}, {2328, 44040}, {7253, 35193}, {8806, 38808}, {27529, 37154}

X(44687) = isogonal conjugate of X(1393)
X(44687) = barycentric product X(i)*X(j) for these {i, j}: {8, 2167}, {9, 95}, {33, 34386}, {41, 34384}, {54, 312}, {78, 275}
X(44687) = barycentric quotient X(i)/X(j) for these (i, j): (3, 44708), (8, 14213), (9, 5), (21, 17167), (33, 53), (41, 51)
X(44687) = trilinear product X(i)*X(j) for these {i, j}: {8, 54}, {9, 2167}, {10, 35196}, {55, 95}, {78, 2190}, {97, 281}
X(44687) = trilinear quotient X(i)/X(j) for these (i, j): (3, 30493), (5, 41279), (8, 5), (9, 1953), (21, 18180), (33, 2181)
X(44687) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1385)}} and {{A, B, C, X(3), X(7012)}}
X(44687) = cevapoint of X(i) and X(j) for these (i, j): {1, 6796}, {9, 212}
X(44687) = X(95)-Ceva conjugate of-X(2167)
X(44687) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 30493}, {5, 56}, {7, 51}, {19, 44708}
X(44687) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 44708), (8, 14213), (9, 5), (21, 17167)

X(44688) = X(1)-DAO CONJUGATE OF X(13)

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

X(44688) lies on these lines: {1, 5362}, {8, 1251}, {9, 21}, {15, 42701}, {72, 11097}, {100, 1277}, {200, 36738}, {312, 44691}, {377, 30328}, {404, 1653}, {518, 10648}, {758, 3179}, {1250, 4420}, {1276, 3869}, {1652, 3868}, {3681, 11752}, {3936, 39152}, {3969, 40713}, {4511, 5240}, {5242, 27385}, {5243, 6734}, {5245, 6735}, {25440, 41225}, {27383, 30414}, {37834, 40714}

X(44688) = barycentric product X(i)*X(j) for these {i, j}: {9, 298}, {15, 312}, {78, 470}, {318, 44718}, {323, 44691}, {643, 23870}
X(44688) = barycentric quotient X(i)/X(j) for these (i, j): (9, 13), (15, 57), (33, 8737), (41, 3457), (55, 2153), (78, 40709)
X(44688) = trilinear product X(i)*X(j) for these {i, j}: {8, 15}, {55, 298}, {219, 470}, {281, 44718}, {312, 2151}, {345, 8739}
X(44688) = trilinear quotient X(i)/X(j) for these (i, j): (8, 13), (9, 2153), (15, 56), (55, 3457), (219, 36296), (281, 8737)
X(44688) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(44691)}} and {{A, B, C, X(15), X(284)}}
X(44688) = X(9)-Hirst inverse of-X(44689)
X(44688) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 3457}, {13, 56}, {57, 2153}, {222, 8737}
X(44688) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 13), (15, 57), (33, 8737), (41, 3457)
X(44688) = {X(9), X(78)}-harmonic conjugate of X(44689)

X(44689) = X(1)-DAO CONJUGATE OF X(14)

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

X(44689) lies on these lines: {1, 5367}, {8, 5240}, {9, 21}, {16, 42701}, {72, 11098}, {100, 1276}, {200, 36737}, {312, 44690}, {377, 30327}, {404, 1652}, {518, 10647}, {758, 41225}, {1277, 3869}, {1653, 3868}, {3179, 25440}, {3681, 11789}, {3936, 39153}, {3969, 40714}, {4420, 10638}, {4511, 5239}, {5242, 6734}, {5243, 27385}, {5246, 6735}, {27383, 30415}, {37831, 40713}

X(44689) = barycentric product X(i)*X(j) for these {i, j}: {9, 299}, {16, 312}, {78, 471}, {318, 44719}, {323, 44690}, {643, 23871}
X(44689) = barycentric quotient X(i)/X(j) for these (i, j): (9, 14), (16, 57), (33, 8738), (41, 3458), (55, 2154), (78, 40710)
X(44689) = trilinear product X(i)*X(j) for these {i, j}: {8, 16}, {55, 299}, {219, 471}, {281, 44719}, {312, 2152}, {345, 8740}
X(44689) = trilinear quotient X(i)/X(j) for these (i, j): (8, 14), (9, 2154), (16, 56), (55, 3458), (219, 36297), (281, 8738)
X(44689) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(44690)}} and {{A, B, C, X(16), X(284)}}
X(44689) = X(9)-Hirst inverse of-X(44688)
X(44689) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 3458}, {14, 56}, {57, 2154}, {222, 8738}
X(44689) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 14), (16, 57), (33, 8738), (41, 3458)
X(44689) = {X(9), X(78)}-harmonic conjugate of X(44688)

X(44690) = X(1)-DAO CONJUGATE OF X(15)

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

X(44690) lies on these lines: {8, 36933}, {9, 44691}, {13, 321}, {300, 349}, {312, 44689}

X(44690) = barycentric product X(i)*X(j) for these {i, j}: {9, 300}, {13, 312}, {94, 44689}, {318, 40709}
X(44690) = barycentric quotient X(i)/X(j) for these (i, j): (9, 15), (13, 57), (33, 8739), (41, 34394), (55, 2151), (78, 44718)
X(44690) = trilinear product X(i)*X(j) for these {i, j}: {8, 13}, {55, 300}, {281, 40709}, {312, 2153}, {345, 8737}, {645, 20578}
X(44690) = trilinear quotient X(i)/X(j) for these (i, j): (8, 15), (9, 2151), (13, 56), (55, 34394), (281, 8739), (300, 7)
X(44690) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(40713)}} and {{A, B, C, X(9), X(44689)}}
X(44690) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 34394}, {15, 56}, {57, 2151}, {222, 8739}
X(44690) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 15), (13, 57), (33, 8739), (41, 34394)

X(44691) = X(1)-DAO CONJUGATE OF X(16)

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

X(44691) lies on these lines: {8, 36932}, {9, 44690}, {14, 321}, {301, 349}, {312, 44688}, {7150, 15065}

X(44691) = barycentric product X(i)*X(j) for these {i, j}: {9, 301}, {14, 312}, {94, 44688}, {318, 40710}
X(44691) = barycentric quotient X(i)/X(j) for these (i, j): (9, 16), (14, 57), (33, 8740), (41, 34395), (55, 2152), (78, 44719)
X(44691) = trilinear product X(i)*X(j) for these {i, j}: {8, 14}, {55, 301}, {281, 40710}, {312, 2154}, {345, 8738}, {645, 20579}
X(44691) = trilinear quotient X(i)/X(j) for these (i, j): (8, 16), (9, 2152), (14, 56), (55, 34395), (281, 8740), (301, 7)
X(44691) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(40714)}} and {{A, B, C, X(9), X(44688)}}
X(44691) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 34395}, {16, 56}, {57, 2152}, {222, 8740}
X(44691) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 16), (14, 57), (33, 8740), (41, 34395)

X(44692) = X(1)-DAO CONJUGATE OF X(20)

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

Let A'B'C' and A"B"C" be the Hutson intouch and anti-Hutson intouch triangles. Let A* be the trilinear product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(44692). (Randy Hutson, September 30, 2021)

X(44692) lies on the cubic K033 and these lines: {1, 1073}, {4, 8806}, {8, 253}, {9, 7156}, {10, 459}, {40, 64}, {78, 7070}, {84, 1295}, {271, 1394}, {728, 2324}, {1902, 31942}, {3343, 44547}, {3346, 39130}, {3576, 14379}, {3679, 13157}, {3710, 5423}, {3718, 5931}, {3811, 15501}, {5587, 6526}, {6247, 7358}, {7982, 8798}, {9780, 14572}, {11589, 35242}

X(44692) = reflection of X(3176) in X(10)
X(44692) = isogonal conjugate of X(1394)
X(44692) = isotomic conjugate of X(33673)
X(44692) = polar conjugate of X(44697)
X(44692) = barycentric product X(i)*X(j) for these {i, j}: {8, 2184}, {9, 253}, {33, 34403}, {37, 5931}, {41, 41530}, {64, 312}
X(44692) = barycentric quotient X(i)/X(j) for these (i, j): (1, 18623), (4, 44697), (8, 18750), (9, 20), (19, 44696), (25, 3213)
X(44692) = trilinear product X(i)*X(j) for these {i, j}: {2, 30457}, {8, 64}, {9, 2184}, {33, 19611}, {42, 5931}, {55, 253}
X(44692) = trilinear quotient X(i)/X(j) for these (i, j): (2, 18623), (4, 44696), (8, 20), (9, 610), (10, 5930), (19, 3213)
X(44692) = trilinear pole of the line {4130, 8611}
X(44692) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(40)}} and {{A, B, C, X(3), X(15629)}}
X(44692) = cevapoint of X(i) and X(j) for these (i, j): {1, 7992}, {64, 41088}
X(44692) = crosspoint of X(280) and X(1034)
X(44692) = crosssum of X(221) and X(1035)
X(44692) = X(8)-Beth conjugate of-X(3176)
X(44692) = X(253)-Ceva conjugate of-X(2184)
X(44692) = X(i)-cross conjugate of-X(j) for these (i, j): (33, 9), (65, 8)
X(44692) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 44696}, {6, 18623}, {7, 154}, {20, 56}
X(44692) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 18623), (4, 44697), (8, 18750), (9, 20)
X(44692) = {X(253), X(19611)}-harmonic conjugate of X(8809)

X(44693) = X(1)-DAO CONJUGATE OF X(30)

Barycentrics    a*(-a+b+c)*(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)) : :

X(44693) lies on these lines: {8, 6739}, {10, 1897}, {72, 74}, {78, 643}, {200, 4736}, {307, 319}, {644, 3694}, {2159, 36147}, {3699, 3710}, {3718, 7257}, {8806, 10152}, {13138, 14919}, {35200, 36037}

X(44693) = barycentric product X(i)*X(j) for these {i, j}: {8, 2349}, {9, 1494}, {55, 33805}, {74, 312}, {75, 15627}, {318, 14919}
X(44693) = barycentric quotient X(i)/X(j) for these (i, j): (1, 6357), (8, 14206), (9, 30), (21, 18653), (33, 1990), (41, 1495)
X(44693) = trilinear product X(i)*X(j) for these {i, j}: {2, 15627}, {8, 74}, {9, 2349}, {41, 33805}, {55, 1494}, {78, 36119}
X(44693) = trilinear quotient X(i)/X(j) for these (i, j): (2, 6357), (8, 30), (30, 1354), (41, 9406), (55, 1495), (74, 56)
X(44693) = trilinear pole of the line {9, 8611}
X(44693) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3579)}} and {{A, B, C, X(8), X(319)}}
X(44693) = X(1043)-Beth conjugate of-X(6742)
X(44693) = X(1494)-Ceva conjugate of-X(2349)
X(44693) = X(758)-cross conjugate of-X(8)
X(44693) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 6357}, {7, 1495}, {30, 56}, {57, 2173}
X(44693) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 6357), (8, 14206), (9, 30), (21, 18653)

X(44694) = X(1)-DAO CONJUGATE OF X(98)

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

X(44694) lies on these lines: {1, 15988}, {2, 11031}, {8, 192}, {9, 21}, {10, 1733}, {33, 3719}, {37, 15984}, {38, 3705}, {40, 12530}, {63, 4220}, {72, 9840}, {100, 2708}, {200, 846}, {201, 7270}, {210, 17611}, {212, 4123}, {238, 4511}, {240, 297}, {244, 31280}, {312, 7069}, {318, 3718}, {329, 37443}, {333, 40967}, {345, 24430}, {511, 1959}, {518, 1284}, {522, 3717}, {614, 25894}, {726, 23690}, {756, 7081}, {758, 1756}, {908, 8229}, {942, 28402}, {956, 30285}, {960, 8240}, {971, 25083}, {982, 26132}, {1145, 12770}, {1259, 3145}, {1260, 20834}, {1281, 4712}, {1423, 3868}, {1580, 1757}, {1736, 3912}, {1737, 25010}, {1944, 27401}, {1958, 37237}, {2310, 3685}, {3057, 12642}, {3220, 37311}, {3508, 3930}, {3555, 11043}, {3679, 30358}, {3681, 11203}, {3695, 35194}, {3703, 24431}, {3729, 12725}, {3869, 6210}, {3872, 7174}, {3882, 44661}, {3932, 24433}, {4357, 5051}, {4425, 4847}, {4516, 35104}, {4645, 17950}, {4853, 11533}, {5220, 8424}, {5223, 8245}, {5692, 30366}, {5904, 30362}, {6743, 12579}, {7004, 32851}, {7080, 27544}, {7360, 24031}, {8238, 34784}, {8270, 34027}, {8679, 35552}, {8731, 27391}, {9852, 20007}, {9959, 34790}, {10544, 22198}, {11679, 35623}, {12526, 12713}, {13097, 17628}, {17353, 27385}, {19582, 26116}, {20556, 29054}, {21717, 25977}, {22220, 41531}, {24349, 26125}, {24443, 41886}, {25591, 26123}, {26685, 27383}, {26699, 34772}, {27388, 28950}, {27509, 27516}, {27529, 33159}, {28287, 37225}

X(44694) = reflection of X(1733) in X(10)
X(44694) = barycentric product X(i)*X(j) for these {i, j}: {8, 1959}, {9, 325}, {33, 6393}, {78, 297}, {212, 44132}, {219, 40703}
X(44694) = barycentric quotient X(i)/X(j) for these (i, j): (8, 1821), (9, 98), (33, 6531), (41, 1976), (55, 1910), (78, 287)
X(44694) = trilinear product X(i)*X(j) for these {i, j}: {8, 511}, {9, 1959}, {55, 325}, {78, 240}, {212, 40703}, {219, 297}
X(44694) = trilinear quotient X(i)/X(j) for these (i, j): (8, 98), (11, 43920), (55, 1976), (78, 293), (219, 248), (232, 608)
X(44694) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(4086)}} and {{A, B, C, X(21), X(257)}}
X(44694) = perspector of the circumconic {{A, B, C, X(312), X(643)}}
X(44694) = crossdifference of every pair of points on line {X(604), X(4017)}
X(44694) = crosspoint of X(i) and X(j) for these (i, j): {75, 36800}, {765, 36801}
X(44694) = X(8)-Beth conjugate of-X(1733)
X(44694) = X(325)-Ceva conjugate of-X(1959)
X(44694) = X(9)-Daleth conjugate of-X(33299)
X(44694) = X(i)-Hirst inverse of-X(j) for these (i, j): {9, 78}, {78, 9}
X(44694) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 1976}, {34, 293}, {56, 98}, {57, 1910}
X(44694) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 1821), (9, 98), (33, 6531), (41, 1976)
X(44694) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (210, 17611, 18235), (256, 984, 2292), (984, 4073, 8), (2292, 10868, 9791)

X(44695) = X(4)-DAO CONJUGATE OF X(7)

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

X(44695) lies on the cubic K709 and these lines: {1, 4}, {3, 40836}, {9, 7003}, {11, 17917}, {20, 1895}, {25, 1604}, {29, 4313}, {40, 1712}, {55, 281}, {64, 10365}, {92, 390}, {108, 7011}, {158, 4294}, {196, 516}, {200, 7007}, {204, 1249}, {207, 12565}, {219, 8805}, {273, 37104}, {318, 452}, {329, 1897}, {345, 36797}, {412, 938}, {440, 18678}, {461, 1856}, {610, 3079}, {653, 9778}, {1096, 4319}, {1118, 6284}, {1148, 6361}, {1210, 37417}, {1253, 7076}, {1753, 10396}, {1767, 10860}, {1783, 7074}, {1784, 4302}, {1859, 14100}, {1864, 5802}, {1885, 7103}, {2192, 5776}, {3100, 37185}, {3195, 22124}, {3198, 6525}, {3601, 7498}, {4304, 37028}, {4314, 39585}, {5175, 17555}, {5274, 17923}, {5809, 37279}, {6001, 17832}, {6056, 6062}, {6060, 27382}, {7071, 7102}, {7101, 7172}, {8899, 10377}, {10383, 37276}, {13405, 39531}, {17112, 28120}, {17718, 42387}, {37411, 38290}, {38271, 38272}

X(44695) = polar conjugate of the isotomic conjugate of X(27382)
X(44695) = barycentric product X(i)*X(j) for these {i, j}: {4, 27382}, {8, 1249}, {9, 1895}, {20, 281}, {29, 8804}, {33, 18750}
X(44695) = barycentric quotient X(i)/X(j) for these (i, j): (8, 34403), (9, 19611), (19, 8809), (33, 2184), (41, 19614), (55, 1073)
X(44695) = trilinear product X(i)*X(j) for these {i, j}: {2, 7156}, {4, 7070}, {8, 204}, {9, 1249}, {19, 27382}, {20, 33}
X(44695) = trilinear quotient X(i)/X(j) for these (i, j): (4, 8809), (8, 19611), (9, 1073), (20, 77), (33, 64), (41, 14642)
X(44695) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(20)}} and {{A, B, C, X(4), X(1895)}}
X(44695) = pole of the trilinear polar of X(27382) with respect to Mandart inellipse
X(44695) = cevapoint of X(33) and X(8802)
X(44695) = crosspoint of X(8) and X(27382)
X(44695) = X(8)-Ceva conjugate of-X(281)
X(44695) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 8809}, {7, 19614}, {34, 15394}, {56, 19611}
X(44695) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (8, 34403), (9, 19611), (19, 8809), (33, 2184)
X(44695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 3488, 34231), (4, 18283, 1490), (20, 1895, 44696), (55, 1857, 281), (243, 497, 278), (1785, 3586, 4), (7008, 40971, 9)

X(44696) = X(4)-DAO CONJUGATE OF X(8)

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

X(44696) lies on the cubic K654 and these lines: {1, 196}, {3, 108}, {4, 57}, {7, 29}, {8, 653}, {20, 1895}, {25, 7103}, {28, 56}, {34, 269}, {40, 1767}, {65, 3332}, {92, 3600}, {158, 4293}, {204, 1394}, {221, 32714}, {222, 3194}, {225, 7490}, {226, 7498}, {268, 40838}, {273, 4198}, {281, 388}, {282, 8894}, {318, 6904}, {342, 452}, {515, 3176}, {610, 1249}, {942, 34231}, {944, 1148}, {1068, 7501}, {1096, 4320}, {1106, 1430}, {1354, 7335}, {1359, 36127}, {1360, 6284}, {1398, 1851}, {1435, 1842}, {1466, 4219}, {1503, 10365}, {1617, 41227}, {1783, 9370}, {1784, 4299}, {1785, 15803}, {1838, 3361}, {1847, 3598}, {1857, 7354}, {1863, 1885}, {1870, 34489}, {1875, 37566}, {1876, 28076}, {3209, 38860}, {3485, 17188}, {3487, 7531}, {4200, 8732}, {4298, 39585}, {5125, 5435}, {5265, 17923}, {5657, 8762}, {5704, 7541}, {5708, 7510}, {5714, 7551}, {5744, 17555}, {6525, 40933}, {7011, 9122}, {7288, 17917}, {7338, 18623}, {7518, 21454}, {7524, 24470}, {9776, 11109}, {11023, 36123}, {14256, 36118}, {15500, 37531}, {17728, 42379}, {18541, 44225}, {28102, 36570}, {28108, 37055}, {37551, 40971}

X(44696) = reflection of X(3182) in X(40657)
X(44696) = polar conjugate of the isotomic conjugate of X(18623)
X(44696) = barycentric product X(i)*X(j) for these {i, j}: {1, 44697}, {4, 18623}, {7, 1249}, {19, 33673}, {20, 278}, {27, 5930}
X(44696) = barycentric quotient X(i)/X(j) for these (i, j): (7, 34403), (19, 44692), (20, 345), (25, 30457), (27, 5931), (34, 2184)
X(44696) = trilinear product X(i)*X(j) for these {i, j}: {2, 3213}, {4, 1394}, {6, 44697}, {7, 204}, {19, 18623}, {20, 34}
X(44696) = trilinear quotient X(i)/X(j) for these (i, j): (4, 44692), (7, 19611), (19, 30457), (20, 78), (56, 19614), (57, 1073)
X(44696) = Zosma transform of X(7992)
X(44696) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3341)}} and {{A, B, C, X(7), X(7338)}}
X(44696) = pole of the trilinear polar of X(18623) with respect to incircle
X(44696) = cevapoint of X(i) and X(j) for these (i, j): {204, 3213}, {207, 208}
X(44696) = crosspoint of X(7) and X(18623)
X(44696) = crosssum of X(55) and X(30457)
X(44696) = X(i)-Beth conjugate of-X(j) for these (i, j): (1, 6285), (29, 40836), (162, 3)
X(44696) = X(i)-Ceva conjugate of-X(j) for these (i, j): (7, 278), (29, 34)
X(44696) = X(204)-cross conjugate of-X(1249)
X(44696) = X(278)-Hirst inverse of-X(43045)
X(44696) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 44692}, {8, 19614}, {9, 1073}, {33, 15394}
X(44696) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (7, 34403), (19, 44692), (20, 345), (25, 30457)
X(44696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (20, 1895, 44695), (56, 1118, 278), (108, 14257, 7952), (388, 1940, 281), (1785, 15803, 37417)

X(44697) = X(4)-DAO CONJUGATE OF X(9)

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

X(44697) lies on these lines: {2, 342}, {4, 10429}, {7, 92}, {20, 1895}, {27, 57}, {63, 653}, {77, 41083}, {85, 31623}, {108, 7411}, {158, 4292}, {207, 10884}, {208, 412}, {223, 36118}, {226, 37448}, {278, 279}, {312, 18026}, {318, 377}, {393, 7365}, {1071, 1148}, {1249, 18623}, {1394, 44698}, {1445, 37279}, {1448, 8747}, {3176, 9799}, {3213, 33673}, {7103, 37392}, {7282, 37181}, {7952, 37108}, {15466, 18750}, {20207, 40616}, {32714, 34035}, {37434, 40836}

X(44697) = isotomic conjugate of the isogonal conjugate of X(3213)
X(44697) = polar conjugate of X(44692)
X(44697) = barycentric product X(i)*X(j) for these {i, j}: {4, 33673}, {7, 1895}, {20, 273}, {34, 14615}, {57, 15466}, {75, 44696}
X(44697) = barycentric quotient X(i)/X(j) for these (i, j): (4, 44692), (7, 19611), (19, 30457), (20, 78), (56, 19614), (57, 1073)
X(44697) = trilinear product X(i)*X(j) for these {i, j}: {2, 44696}, {4, 18623}, {7, 1249}, {19, 33673}, {20, 278}, {27, 5930}
X(44697) = trilinear quotient X(i)/X(j) for these (i, j): (4, 30457), (7, 1073), (20, 219), (34, 2155), (56, 14642), (57, 19614)
X(44697) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(41084)}} and {{A, B, C, X(20), X(27)}}
X(44697) = cevapoint of X(i) and X(j) for these (i, j): {196, 40837}, {1249, 44696}, {1394, 3213}
X(44697) = crosspoint of X(85) and X(33673)
X(44697) = X(648)-Beth conjugate of-X(63)
X(44697) = X(85)-Ceva conjugate of-X(273)
X(44697) = X(i)-cross conjugate of-X(j) for these (i, j): (1249, 1895), (1394, 33673)
X(44697) = pole wrt polar circle of trilinear polar of X(44692) (line X(4130)X(8611))
X(44697) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 30457}, {8, 14642}, {9, 19614}, {41, 19611}
X(44697) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 44692), (7, 19611), (19, 30457), (20, 78)

X(44698) = X(4)-DAO CONJUGATE OF X(10)

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

X(44698) lies on these lines: {4, 37666}, {6, 7513}, {20, 1249}, {21, 7149}, {27, 58}, {28, 279}, {29, 81}, {112, 4229}, {162, 283}, {204, 1097}, {273, 1453}, {297, 20077}, {648, 1043}, {653, 4296}, {1010, 2322}, {1172, 10429}, {1394, 44697}, {1724, 26003}, {1897, 7283}, {3945, 7498}, {4195, 9308}, {4340, 37448}, {5125, 24597}, {7338, 18623}, {7379, 16318}, {8743, 13727}, {14361, 24565}

X(44698) = barycentric product X(i)*X(j) for these {i, j}: {20, 27}, {21, 44697}, {28, 18750}, {29, 18623}, {58, 15466}, {81, 1895}
X(44698) = barycentric quotient X(i)/X(j) for these (i, j): (20, 306), (27, 253), (28, 2184), (58, 1073), (81, 19611), (86, 34403)
X(44698) = trilinear product X(i)*X(j) for these {i, j}: {20, 28}, {21, 44696}, {27, 610}, {29, 1394}, {58, 1895}, {81, 1249}
X(44698) = trilinear quotient X(i)/X(j) for these (i, j): (20, 72), (27, 2184), (28, 64), (29, 44692), (58, 19614), (81, 1073)
X(44698) = intersection, other than A,B,C, of conics {{A, B, C, X(20), X(27)}} and {{A, B, C, X(58), X(1394)}}
X(44698) = cevapoint of X(204) and X(1249)
X(44698) = X(86)-Ceva conjugate of-X(27)
X(44698) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 19614}, {37, 1073}, {42, 19611}, {64, 72}
X(44698) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (20, 306), (27, 253), (28, 2184), (58, 1073)
X(44698) = {X(58), X(8747)}-harmonic conjugate of X(27)

X(44699) = X(4)-DAO CONJUGATE OF X(11)

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

X(44699) lies on these lines: {59, 5080}, {1936, 7012}, {4564, 15742}

X(44699) = barycentric product X(i)*X(j) for these {i, j}: {59, 15466}, {765, 44697}, {1016, 44696}, {1249, 4998}, {1275, 44695}
X(44699) = barycentric quotient X(i)/X(j) for these (i, j): (20, 26932), (59, 1073), (154, 7117), (204, 2170), (610, 7004), (1249, 11)
X(44699) = trilinear product X(i)*X(j) for these {i, j}: {20, 7012}, {59, 1895}, {204, 4998}, {765, 44696}, {1016, 3213}, {1249, 4564}
X(44699) = trilinear quotient X(i)/X(j) for these (i, j): (20, 7004), (59, 19614), (204, 3271), (610, 7117), (1249, 2170), (1394, 3937)
X(44699) = X(i)-isoconjugate-of-X(j) for these {i, j}: {11, 19614}, {64, 7004}, {1073, 2170}
X(44699) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (20, 26932), (59, 1073), (154, 7117), (204, 2170)

X(44700) = X(4)-DAO CONJUGATE OF X(13)

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

X(44700) lies on these lines: {3, 36303}, {4, 11408}, {15, 470}, {20, 1249}, {30, 36302}, {53, 42147}, {393, 42119}, {471, 5334}, {472, 10654}, {1990, 42087}, {6111, 42099}, {6116, 16964}, {6530, 36993}, {15466, 44703}, {40138, 42120}

X(44700) = barycentric product X(i)*X(j) for these {i, j}: {15, 15466}, {20, 470}, {298, 1249}, {323, 44703}
X(44700) = barycentric quotient X(i)/X(j) for these (i, j): (15, 1073), (20, 40709), (154, 36296), (204, 2153), (298, 34403), (470, 253)
X(44700) = trilinear product X(i)*X(j) for these {i, j}: {15, 1895}, {204, 298}, {470, 610}, {1094, 44702}
X(44700) = trilinear quotient X(i)/X(j) for these (i, j): (15, 19614), (204, 3457), (298, 19611), (470, 2184), (610, 36296), (1249, 2153)
X(44700) = intersection, other than A,B,C, of conics {{A, B, C, X(15), X(15905)}} and {{A, B, C, X(20), X(470)}}
X(44700) = X(298)-Ceva conjugate of-X(470)
X(44700) = X(1249)-Hirst inverse of-X(44701)
X(44700) = X(i)-isoconjugate-of-X(j) for these {i, j}: {13, 19614}, {1073, 2153}
X(44700) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (15, 1073), (20, 40709), (154, 36296), (204, 2153)
X(44700) = {X(20), X(1249)}-harmonic conjugate of X(44701)

X(44701) = X(4)-DAO CONJUGATE OF X(14)

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

X(44701) lies on these lines: {3, 36302}, {4, 11409}, {16, 471}, {20, 1249}, {30, 36303}, {53, 42148}, {393, 42120}, {470, 5335}, {473, 10653}, {1990, 42088}, {6110, 42100}, {6117, 16965}, {6530, 36995}, {15466, 44702}, {40138, 42119}

X(44701) = barycentric product X(i)*X(j) for these {i, j}: {16, 15466}, {20, 471}, {299, 1249}, {323, 44702}
X(44701) = barycentric quotient X(i)/X(j) for these (i, j): (16, 1073), (20, 40710), (154, 36297), (204, 2154), (299, 34403), (471, 253)
X(44701) = trilinear product X(i)*X(j) for these {i, j}: {16, 1895}, {204, 299}, {471, 610}, {1095, 44703}
X(44701) = trilinear quotient X(i)/X(j) for these (i, j): (16, 19614), (204, 3458), (299, 19611), (471, 2184), (610, 36297), (1249, 2154)
X(44701) = intersection, other than A,B,C, of conics {{A, B, C, X(16), X(15905)}} and {{A, B, C, X(20), X(471)}}
X(44701) = X(299)-Ceva conjugate of-X(471)
X(44701) = X(1249)-Hirst inverse of-X(44700)
X(44701) = X(i)-isoconjugate-of-X(j) for these {i, j}: {14, 19614}, {1073, 2154}
X(44701) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (16, 1073), (20, 40710), (154, 36297), (204, 2154)
X(44701) = {X(20), X(1249)}-harmonic conjugate of X(44700)

X(44702) = X(4)-DAO CONJUGATE OF X(15)

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

X(44702) lies on these lines: {13, 473}, {1249, 44703}, {15466, 44701}, {16770, 37192}

X(44702) = barycentric product X(i)*X(j) for these {i, j}: {13, 15466}, {94, 44701}, {300, 1249}
X(44702) = barycentric quotient X(i)/X(j) for these (i, j): (13, 1073), (20, 44718), (204, 2151), (300, 34403), (1249, 15)
X(44702) = trilinear product X(i)*X(j) for these {i, j}: {13, 1895}, {204, 300}
X(44702) = trilinear quotient X(i)/X(j) for these (i, j): (13, 19614), (204, 34394), (300, 19611), (1249, 2151)
X(44702) = X(i)-isoconjugate-of-X(j) for these {i, j}: {15, 19614}, {1073, 2151}
X(44702) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (13, 1073), (20, 44718), (204, 2151), (300, 34403)

X(44703) = X(4)-DAO CONJUGATE OF X(16)

Barycentrics    1/a^2*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(-2*S+(a^2-b^2+c^2)*sqrt(3))*(-2*S+(a^2+b^2-c^2)*sqrt(3))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(44703) lies on these lines: {14, 472}, {1249, 44702}, {15466, 44700}, {16771, 37192}

X(44703) = barycentric product X(i)*X(j) for these {i, j}: {14, 15466}, {94, 44700}, {301, 1249}
X(44703) = barycentric quotient X(i)/X(j) for these (i, j): (14, 1073), (20, 44719), (204, 2152), (301, 34403), (1249, 16)
X(44703) = trilinear product X(i)*X(j) for these {i, j}: {14, 1895}, {204, 301}
X(44703) = trilinear quotient X(i)/X(j) for these (i, j): (14, 19614), (204, 34395), (301, 19611), (1249, 2152)
X(44703) = X(i)-isoconjugate-of-X(j) for these {i, j}: {16, 19614}, {1073, 2152}
X(44703) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (14, 1073), (20, 44719), (204, 2152), (301, 34403)

X(44704) = X(4)-DAO CONJUGATE OF X(98)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^2-b^4-c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(44704) = 3*X(297)-4*X(39569) = 5*X(3091)-2*X(40996) = 3*X(6530)-2*X(39569)

X(44704) lies on these lines: {2, 42352}, {3, 41371}, {4, 193}, {20, 1249}, {30, 15351}, {69, 10002}, {107, 11064}, {132, 325}, {147, 1529}, {182, 42873}, {235, 3186}, {264, 5480}, {297, 511}, {317, 11477}, {393, 37200}, {441, 15312}, {458, 14853}, {648, 1503}, {858, 35360}, {877, 6393}, {1105, 13568}, {1350, 15274}, {1368, 3168}, {1513, 2967}, {1515, 15063}, {1559, 8057}, {1941, 3575}, {1990, 29181}, {2322, 7379}, {3091, 32000}, {5702, 33748}, {5895, 34808}, {5999, 16318}, {6525, 37669}, {6528, 44137}, {6616, 32605}, {6747, 21969}, {7396, 14361}, {7774, 37074}, {8928, 42396}, {9753, 40801}, {10151, 38294}, {10516, 44134}, {10519, 11331}, {11547, 33586}, {14165, 32269}, {14249, 14615}, {14927, 15258}, {15069, 42854}, {18583, 37124}, {25406, 40138}, {31670, 33971}, {34380, 44228}, {37070, 37645}, {37443, 41083}, {41375, 41716}

X(44704) = reflection of X(297) in X(6530)
X(44704) = barycentric product X(i)*X(j) for these {i, j}: {20, 297}, {154, 44132}, {232, 14615}, {240, 18750}, {325, 1249}, {511, 15466}
X(44704) = barycentric quotient X(i)/X(j) for these (i, j): (20, 287), (154, 248), (204, 1910), (232, 64), (237, 14642), (240, 2184)
X(44704) = trilinear product X(i)*X(j) for these {i, j}: {20, 240}, {154, 40703}, {204, 325}, {232, 18750}, {297, 610}, {511, 1895}
X(44704) = trilinear quotient X(i)/X(j) for these (i, j): (20, 293), (204, 1976), (232, 2155), (240, 64), (297, 2184), (325, 19611)
X(44704) = intersection, other than A,B,C, of conics {{A, B, C, X(20), X(297)}} and {{A, B, C, X(325), X(36413)}}
X(44704) = X(325)-Ceva conjugate of-X(297)
X(44704) = X(20)-Daleth conjugate of-X(42459)
X(44704) = X(i)-Hirst inverse of-X(j) for these (i, j): {20, 1249}, {297, 15595}, {1249, 20}
X(44704) = X(i)-isoconjugate-of-X(j) for these {i, j}: {64, 293}, {98, 19614}, {248, 2184}, {287, 2155}
X(44704) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (20, 287), (154, 248), (204, 1910), (232, 64)
X(44704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 1351, 27377), (1350, 15274, 17907)

X(44705) = X(4)-DAO CONJUGATE OF X(99)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(b^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(44705) = 3*X(9209)-2*X(39201)

X(44705) lies on these lines: {4, 525}, {24, 39228}, {25, 9209}, {136, 5099}, {403, 39510}, {460, 512}, {523, 10151}, {1559, 8057}, {1593, 22089}, {2971, 38368}, {3566, 16230}, {3800, 14618}, {5139, 38970}, {6587, 42658}, {6755, 34983}, {7378, 30474}, {14018, 41800}, {23879, 39532}

X(44705) = reflection of X(i) in X(j) for these (i, j): (14618, 39533), (16229, 42399), (42658, 6587)
X(44705) = polar conjugate of X(44326)
X(44705) = pole wrt polar circle of trilinear polar of X(44326) (line X(20)X(64))
X(44705) = barycentric product X(i)*X(j) for these {i, j}: {4, 6587}, {19, 17898}, {20, 2501}, {107, 1562}, {122, 6529}, {154, 14618}
X(44705) = barycentric quotient X(i)/X(j) for these (i, j): (4, 44326), (20, 4563), (122, 4143), (125, 14638), (154, 4558), (204, 662)
X(44705) = trilinear product X(i)*X(j) for these {i, j}: {19, 6587}, {25, 17898}, {34, 14308}, {154, 24006}, {158, 42658}, {204, 523}
X(44705) = trilinear quotient X(i)/X(j) for these (i, j): (20, 4592), (92, 44326), (154, 4575), (204, 110), (512, 19614), (523, 19611)
X(44705) = Zosma transform of X(24018)
X(44705) = intersection, other than A,B,C, of conics {{A, B, C, X(20), X(460)}} and {{A, B, C, X(25), X(1559)}}
X(44705) = center of circle {{X(468), X(1112), X(13473)}}
X(44705) = perspector of the circumconic {{A, B, C, X(393), X(459)}}
X(44705) = polar circle-inverse of-X(18337)
X(44705) = pole of the trilinear polar of X(6529) with respect to Orthic inconic
X(44705) = crossdifference of every pair of points on line {X(394), X(1073)}
X(44705) = crosspoint of X(i) and X(j) for these (i, j): {4, 6529}, {523, 6587}
X(44705) = X(i)-Ceva conjugate of-X(j) for these (i, j): (4, 1562), (523, 2501)
X(44705) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 44326}, {64, 4592}, {99, 19614}, {110, 19611}
X(44705) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 44326), (20, 4563), (122, 4143), (125, 14638)

X(44706) = X(5)-DAO CONJUGATE OF X(1)

Barycentrics    a*((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
Trilinears    b^2 + c^2 - 4 R^2 : :
Trilinears    cos A cos(B - C) : :
Trilinears    cos 2B + cos 2C : :
Trilinears    tan B tan C + 1 : :
Trilinears    A-power of circle O(3,4) : :
Trilinears    X(4)-power of A-Apollonian circle : :
Trilinears    X(4)-power of A-power circle : :
Trilinears    b^2 + c^2 + (A-power of Grebe circle) : :
Trilinears    A-power of X(4)-altimedial circle : :
Trilinears    A'-power of circumcircle : :, where A'B'C' is the circumcevian polar triangle of X(3)

X(44706) is the intersection of the tangents at X(255) and X(1087) to the inellipse centered at X(23998). The trilinear polar of X(44706) passes through the crosssum of X(162) and X(163) and the crosspoint of X(2582) and X(2584). (Randy Hutson, September 30, 2021)

X(44706) lies on the conic {{A,B,C,X(47),X(48)}} and these lines: {1, 21}, {3, 201}, {4, 24430}, {5, 1393}, {8, 24028}, {9, 25062}, {10, 1074}, {12, 24431}, {33, 5709}, {34, 7330}, {46, 990}, {48, 1820}, {57, 7549}, {72, 17102}, {73, 912}, {75, 158}, {212, 1062}, {227, 14872}, {238, 28731}, {244, 499}, {271, 4853}, {293, 34055}, {307, 1210}, {386, 18397}, {411, 3465}, {447, 32939}, {498, 756}, {580, 33178}, {603, 1060}, {651, 3468}, {750, 17700}, {775, 36119}, {820, 2632}, {942, 40152}, {976, 8069}, {982, 3086}, {984, 3085}, {986, 18391}, {1064, 1858}, {1066, 8758}, {1071, 1214}, {1076, 6245}, {1087, 2181}, {1099, 36063}, {1103, 5223}, {1125, 24539}, {1158, 8270}, {1254, 1478}, {1329, 24433}, {1457, 5887}, {1465, 5777}, {1479, 2310}, {1712, 23052}, {1720, 42012}, {1733, 2085}, {1737, 21935}, {1745, 12528}, {1755, 17442}, {1762, 41227}, {1772, 18395}, {1776, 3073}, {1777, 4347}, {1782, 3220}, {1784, 20879}, {1785, 6734}, {1789, 2169}, {1838, 22464}, {1854, 3428}, {1870, 1935}, {1930, 17875}, {1936, 6198}, {1953, 2179}, {1973, 16567}, {2003, 8555}, {2074, 40602}, {2166, 2962}, {2172, 21374}, {2361, 9630}, {2599, 39271}, {2654, 24474}, {2947, 9960}, {3072, 7098}, {3074, 3219}, {3075, 3218}, {3157, 20277}, {3467, 38458}, {3582, 42040}, {3584, 42041}, {3666, 44547}, {3678, 24025}, {3708, 4020}, {3927, 7078}, {3938, 11508}, {3953, 44675}, {3958, 22134}, {4008, 17446}, {4100, 17438}, {4328, 7013}, {4392, 14986}, {4414, 11507}, {4642, 10573}, {4722, 16473}, {5492, 39542}, {5562, 30493}, {5693, 10571}, {5694, 34586}, {5791, 40967}, {5884, 37558}, {6051, 16193}, {7070, 9643}, {7100, 23070}, {10056, 42039}, {10072, 42038}, {10391, 37528}, {10703, 11014}, {11398, 24320}, {11399, 37581}, {13006, 33299}, {13369, 22053}, {14058, 42456}, {15623, 23981}, {15908, 38357}, {16697, 44709}, {17869, 26013}, {18210, 23154}, {18398, 21346}, {20223, 39585}, {21368, 37231}, {21406, 23665}, {22072, 31837}, {22097, 41340}, {23681, 24046}, {24434, 25466}, {26128, 28435}, {34831, 41013}, {35014, 40944}, {37532, 37696}, {39796, 41393}

X(44706) = reflection of X(i) in X(j) for these (i, j): (73, 37565), (41013, 34831)
X(44706) = isogonal conjugate of X(2190)
X(44706) = isotomic conjugate of X(40440)
X(44706) = barycentric product X(i)*X(j) for these {i, j}: {1, 343}, {3, 14213}, {5, 63}, {6, 18695}, {8, 44708}, {10, 16697}
X(44706) = barycentric quotient X(i)/X(j) for these (i, j): (1, 275), (3, 2167), (5, 92), (19, 8884), (31, 8882), (48, 54)
X(44706) = trilinear product X(i)*X(j) for these {i, j}: {2, 216}, {3, 5}, {4, 5562}, {6, 343}, {7, 44707}, {8, 30493}
X(44706) = trilinear quotient X(i)/X(j) for these (i, j): (2, 275), (3, 54), (4, 8884), (5, 4), (6, 8882), (20, 38808)
X(44706) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(1820)}} and {{A, B, C, X(3), X(44710)}}
X(44706) = perspector of the circumconic {{A, B, C, X(662), X(2617)}}
X(44706) = pole of the trilinear polar of X(1789) with respect to MacBeath circumconic
X(44706) = pole of the trilinear polar of X(16697) with respect to Johnson circumconic
X(44706) = cevapoint of X(216) and X(44707)
X(44706) = crosspoint of X(i) and X(j) for these (i, j): {1, 91}, {63, 75}
X(44706) = crosssum of X(i) and X(j) for these (i, j): {1, 47}, {19, 31}
X(44706) = X(i)-Ceva conjugate of-X(j) for these (i, j): (75, 14213), (811, 24018)
X(44706) = X(216)-cross conjugate of-X(44708)
X(44706) = X(63)-Daleth conjugate of-X(1959)
X(44706) = {P,U}-harmonic conjugate of X(661), where P and U are the 1st and 2nd bicentrics of X(47)
X(44706) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 8882}, {3, 8884}, {4, 54}, {6, 275}
X(44706) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 275), (3, 2167), (5, 92), (19, 8884)
X(44706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 63, 255), (1, 920, 31), (1, 1725, 774), (1, 1749, 2964), (3, 22457, 22342), (5, 35194, 7069), (38, 774, 1), (72, 17102, 22350), (201, 7004, 3), (255, 18477, 1), (1060, 24467, 603), (1062, 26921, 212), (1071, 1214, 4303), (1393, 7069, 5), (1736, 3670, 1210), (2172, 21374, 2312), (3708, 4020, 18671), (12528, 17080, 1745), (24430, 37591, 4), (30493, 44707, 5562)

X(44707) = X(5)-DAO CONJUGATE OF X(7)

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

X(44707) lies on these lines: {1, 7066}, {8, 1857}, {9, 11436}, {11, 26942}, {33, 26893}, {40, 6285}, {48, 3611}, {51, 1953}, {52, 35194}, {55, 219}, {63, 1364}, {71, 23207}, {72, 519}, {185, 201}, {212, 3270}, {497, 26872}, {511, 24430}, {517, 1859}, {610, 11190}, {916, 1214}, {1040, 3781}, {2318, 40945}, {2323, 11429}, {2875, 3185}, {3056, 5227}, {3271, 7082}, {3303, 7078}, {3682, 40946}, {3703, 7062}, {3917, 7004}, {3990, 14547}, {5045, 10207}, {5285, 10535}, {5562, 30493}, {5907, 37591}, {6062, 7065}, {6238, 26921}, {6254, 9121}, {7070, 11189}, {7072, 10833}, {7085, 19354}, {7144, 43819}, {8896, 21334}, {10544, 22760}, {18922, 26939}, {20683, 22131}, {21015, 26956}

X(44707) = reflection of X(39796) in X(1214)
X(44707) = barycentric product X(i)*X(j) for these {i, j}: {5, 219}, {8, 216}, {9, 44706}, {41, 18695}, {51, 345}, {53, 1259}
X(44707) = barycentric quotient X(i)/X(j) for these (i, j): (5, 331), (8, 276), (9, 40440), (41, 2190), (51, 278), (55, 275)
X(44707) = trilinear product X(i)*X(j) for these {i, j}: {3, 7069}, {5, 212}, {9, 216}, {33, 5562}, {41, 343}, {51, 78}
X(44707) = trilinear quotient X(i)/X(j) for these (i, j): (5, 273), (8, 40440), (9, 275), (33, 8884), (41, 8882), (51, 34)
X(44707) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(8021)}} and {{A, B, C, X(8), X(6056)}}
X(44707) = pole of the trilinear polar of X(219) with respect to Mandart inellipse
X(44707) = pole of the trilinear polar of X(44706) with respect to Johnson circumconic
X(44707) = crosspoint of X(8) and X(219)
X(44707) = crosssum of X(56) and X(278)
X(44707) = X(i)-isoconjugate-of-X(j) for these {i, j}: {7, 2190}, {54, 273}, {56, 40440}, {57, 275}
X(44707) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 331), (8, 276), (9, 40440), (41, 2190)
X(44707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 11435, 37993), (55, 219, 6056), (3270, 3690, 212), (5562, 44706, 30493)

X(44708) = X(5)-DAO CONJUGATE OF X(9)

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

X(44708) lies on these lines: {1, 36986}, {7, 6360}, {57, 77}, {65, 4303}, {73, 18732}, {85, 44129}, {196, 18677}, {222, 3942}, {223, 18725}, {226, 18726}, {278, 18161}, {651, 21367}, {1214, 22097}, {1393, 18180}, {1953, 42459}, {3340, 10884}, {3911, 18652}, {5437, 41081}, {5562, 30493}, {6358, 17880}, {6508, 26932}, {7146, 17170}, {18607, 40152}

X(44708) = isotomic conjugate of the polar conjugate of X(1393)
X(44708) = barycentric product X(i)*X(j) for these {i, j}: {5, 77}, {7, 44706}, {51, 7182}, {53, 7183}, {56, 18695}, {57, 343}
X(44708) = barycentric quotient X(i)/X(j) for these (i, j): (3, 44687), (5, 318), (7, 40440), (34, 8884), (51, 33), (56, 2190)
X(44708) = trilinear product X(i)*X(j) for these {i, j}: {2, 30493}, {5, 222}, {7, 216}, {51, 348}, {53, 1804}, {56, 343}
X(44708) = trilinear quotient X(i)/X(j) for these (i, j): (5, 281), (7, 275), (51, 607), (53, 1857), (56, 8882), (57, 2190)
X(44708) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(1817)}} and {{A, B, C, X(57), X(1393)}}
X(44708) = cevapoint of X(216) and X(30493)
X(44708) = crosspoint of X(77) and X(85)
X(44708) = crosssum of X(33) and X(41)
X(44708) = X(343)-Beth conjugate of-X(343)
X(44708) = X(216)-cross conjugate of-X(44706)
X(44708) = X(i)-isoconjugate-of-X(j) for these {i, j}: {8, 8882}, {9, 2190}, {19, 44687}, {33, 2167}
X(44708) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 44687), (5, 318), (7, 40440), (34, 8884)
X(44708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (57, 77, 7125), (3942, 37755, 222)

X(44709) = X(5)-DAO CONJUGATE OF X(10)

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

X(44709) lies on these lines: {1, 18662}, {36, 58}, {73, 1437}, {81, 1816}, {86, 8747}, {216, 217}, {283, 22350}, {602, 3286}, {1393, 18180}, {1451, 7125}, {1457, 37227}, {1459, 7100}, {1812, 3682}, {2193, 22063}, {3736, 37570}, {3784, 40679}, {16697, 44706}, {16887, 17216}, {18178, 31849}

X(44709) = isogonal conjugate of the polar conjugate of X(17167)
X(44709) = barycentric product X(i)*X(j) for these {i, j}: {1, 16697}, {3, 17167}, {5, 1790}, {21, 44708}, {27, 5562}, {51, 17206}
X(44709) = barycentric quotient X(i)/X(j) for these (i, j): (27, 8795), (51, 1826), (58, 275), (81, 40440), (86, 276), (216, 10)
X(44709) = trilinear product X(i)*X(j) for these {i, j}: {3, 18180}, {5, 1437}, {6, 16697}, {21, 30493}, {28, 5562}, {48, 17167}
X(44709) = trilinear quotient X(i)/X(j) for these (i, j): (5, 41013), (28, 8884), (51, 1824), (58, 2190), (81, 275), (86, 40440)
X(44709) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(4225)}} and {{A, B, C, X(36), X(73)}}
X(44709) = perspector of the circumconic {{A, B, C, X(4556), X(23181)}}
X(44709) = crosspoint of X(86) and X(1790)
X(44709) = crosssum of X(42) and X(1826)
X(44709) = X(86)-Ceva conjugate of-X(17167)
X(44709) = X(216)-cross conjugate of-X(16697)
X(44709) = X(1790)-Daleth conjugate of-X(17209)
X(44709) = X(i)-isoconjugate-of-X(j) for these {i, j}: {10, 2190}, {37, 275}, {42, 40440}, {54, 41013}
X(44709) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (27, 8795), (51, 1826), (58, 275), (81, 40440)

X(44710) = X(5)-DAO CONJUGATE OF X(11)

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

X(44710) lies on these lines: {59, 3286}, {765, 1259}

X(44710) = barycentric product X(i)*X(j) for these {i, j}: {5, 44717}, {59, 343}, {216, 4998}, {765, 44708}, {1016, 30493}, {1275, 44707}
X(44710) = barycentric quotient X(i)/X(j) for these (i, j): (51, 8735), (59, 275), (216, 11), (217, 3271), (343, 34387), (418, 7117)
X(44710) = trilinear product X(i)*X(j) for these {i, j}: {59, 44706}, {216, 4564}, {343, 2149}, {765, 30493}, {1252, 44708}, {1953, 44717}
X(44710) = trilinear quotient X(i)/X(j) for these (i, j): (59, 2190), (201, 8901), (216, 2170), (343, 4858), (1393, 2969), (1953, 8735)
X(44710) = X(i)-isoconjugate-of-X(j) for these {i, j}: {11, 2190}, {270, 8901}, {275, 2170}
X(44710) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (51, 8735), (59, 275), (216, 11), (217, 3271)

X(44711) = X(5)-DAO CONJUGATE OF X(13)

Barycentrics    a^2*(2*S+(-a^2+b^2+c^2)*3^(1/2))*((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2) : :

X(44711) lies on these lines: {3, 10662}, {5, 36300}, {14, 11140}, {15, 323}, {61, 10677}, {216, 217}, {298, 11094}, {343, 44714}, {621, 40853}, {622, 17035}, {1568, 44713}, {3104, 5334}, {6117, 14918}, {7691, 8839}, {8175, 22648}, {10633, 40580}, {10635, 36297}, {10661, 36296}, {14817, 42121}, {15091, 18468}

X(44711) = isogonal conjugate of the polar conjugate of X(33529)
X(44711) = barycentric product X(i)*X(j) for these {i, j}: {3, 33529}, {5, 44718}, {15, 343}, {216, 298}, {323, 44714}, {394, 6117}
X(44711) = barycentric quotient X(i)/X(j) for these (i, j): (15, 275), (51, 8737), (216, 13), (217, 3457), (298, 276), (343, 300)
X(44711) = trilinear product X(i)*X(j) for these {i, j}: {15, 44706}, {48, 33529}, {255, 6117}, {343, 2151}, {1094, 44713}, {1953, 44718}
X(44711) = trilinear quotient X(i)/X(j) for these (i, j): (15, 2190), (216, 2153), (298, 40440), (1953, 8737), (2151, 8882)
X(44711) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(11127)}} and {{A, B, C, X(5), X(11146)}}
X(44711) = perspector of the circumconic {{A, B, C, X(17402), X(23181)}}
X(44711) = crossdifference of every pair of points on line {X(8740), X(20578)}
X(44711) = crosspoint of X(298) and X(44718)
X(44711) = crosssum of X(4) and X(10633)
X(44711) = X(298)-Ceva conjugate of-X(33529)
X(44711) = X(216)-Hirst inverse of-X(44712)
X(44711) = X(i)-isoconjugate-of-X(j) for these {i, j}: {13, 2190}, {275, 2153}
X(44711) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (15, 275), (51, 8737), (216, 13), (217, 3457)
X(44711) = {X(216), X(5562)}-harmonic conjugate of X(44712)

X(44712) = X(5)-DAO CONJUGATE OF X(14)

Barycentrics    a^2*(-2*S+(-a^2+b^2+c^2)*3^(1/2))*((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2) : :

X(44712) lies on these lines: {3, 10661}, {5, 36301}, {13, 11140}, {16, 323}, {62, 10678}, {216, 217}, {299, 11093}, {343, 44713}, {621, 17035}, {622, 40853}, {1568, 44714}, {3105, 5335}, {6116, 14918}, {7691, 8837}, {8174, 22649}, {10632, 40581}, {10634, 36296}, {10662, 36297}, {14816, 42124}, {15091, 18470}

X(44712) = isogonal conjugate of the polar conjugate of X(33530)
X(44712) = barycentric product X(i)*X(j) for these {i, j}: {3, 33530}, {5, 44719}, {16, 343}, {216, 299}, {323, 44713}, {394, 6116}
X(44712) = barycentric quotient X(i)/X(j) for these (i, j): (16, 275), (51, 8738), (216, 14), (217, 3458), (299, 276), (343, 301)
X(44712) = trilinear product X(i)*X(j) for these {i, j}: {16, 44706}, {48, 33530}, {255, 6116}, {343, 2152}, {1095, 44714}, {1953, 44719}
X(44712) = trilinear quotient X(i)/X(j) for these (i, j): (16, 2190), (216, 2154), (299, 40440), (1953, 8738), (2152, 8882)
X(44712) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(11126)}} and {{A, B, C, X(5), X(11145)}}
X(44712) = perspector of the circumconic {{A, B, C, X(17403), X(23181)}}
X(44712) = crossdifference of every pair of points on line {X(8739), X(20579)}
X(44712) = crosspoint of X(299) and X(44719)
X(44712) = crosssum of X(4) and X(10632)
X(44712) = X(299)-Ceva conjugate of-X(33530)
X(44712) = X(216)-Hirst inverse of-X(44711)
X(44712) = X(i)-isoconjugate-of-X(j) for these {i, j}: {14, 2190}, {275, 2154}
X(44712) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (16, 275), (51, 8738), (216, 14), (217, 3458)
X(44712) = {X(216), X(5562)}-harmonic conjugate of X(44711)

X(44713) = X(5)-DAO CONJUGATE OF X(15)

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2)*(2*S+(a^2-b^2+c^2)*3^(1/2))*(2*S+(a^2+b^2-c^2)*3^(1/2)) : :

X(44713) lies on these lines: {2, 13}, {68, 36296}, {216, 44714}, {265, 10663}, {300, 18027}, {324, 6116}, {343, 44712}, {1568, 44711}, {8737, 18855}, {10217, 10661}

X(44713) = barycentric product X(i)*X(j) for these {i, j}: {5, 40709}, {13, 343}, {94, 44712}, {216, 300}, {265, 33530}, {311, 36296}
X(44713) = barycentric quotient X(i)/X(j) for these (i, j): (5, 470), (13, 275), (51, 8739), (216, 15), (217, 34394), (300, 276)
X(44713) = trilinear product X(i)*X(j) for these {i, j}: {13, 44706}, {343, 2153}, {1953, 40709}
X(44713) = trilinear quotient X(i)/X(j) for these (i, j): (13, 2190), (216, 2151), (300, 40440), (1087, 6117), (1953, 8739), (2153, 8882)
X(44713) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(5)}} and {{A, B, C, X(53), X(5335)}}
X(44713) = crosspoint of X(300) and X(40709)
X(44713) = X(1568)-cross conjugate of-X(44714)
X(44713) = X(i)-isoconjugate-of-X(j) for these {i, j}: {15, 2190}, {275, 2151}, {470, 2148}
X(44713) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 470), (13, 275), (51, 8739), (216, 15)

X(44714) = X(5)-DAO CONJUGATE OF X(16)

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2)*(-2*S+(a^2-b^2+c^2)*3^(1/2))*(-2*S+(a^2+b^2-c^2)*3^(1/2)) : :

X(44714) lies on these lines: {2, 14}, {68, 36297}, {216, 44713}, {265, 10664}, {301, 18027}, {324, 6117}, {343, 44711}, {1568, 44712}, {8738, 18855}, {10218, 10662}

X(44714) = barycentric product X(i)*X(j) for these {i, j}: {5, 40710}, {14, 343}, {94, 44711}, {216, 301}, {265, 33529}, {311, 36297}
X(44714) = barycentric quotient X(i)/X(j) for these (i, j): (5, 471), (14, 275), (51, 8740), (216, 16), (217, 34395), (301, 276)
X(44714) = trilinear product X(i)*X(j) for these {i, j}: {14, 44706}, {343, 2154}, {1953, 40710}
X(44714) = trilinear quotient X(i)/X(j) for these (i, j): (14, 2190), (216, 2152), (301, 40440), (1087, 6116), (1953, 8740)
X(44714) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(5)}} and {{A, B, C, X(53), X(5334)}}
X(44714) = crosspoint of X(301) and X(40710)
X(44714) = X(1568)-cross conjugate of-X(44713)
X(44714) = X(i)-isoconjugate-of-X(j) for these {i, j}: {16, 2190}, {275, 2152}, {471, 2148}
X(44714) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 471), (14, 275), (51, 8740), (216, 16)

X(44715) = X(5)-DAO CONJUGATE OF X(30)

Barycentrics    a^2*(a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*(a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))*((b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(44715) = 3*X(3)-2*X(40948) = 3*X(381)-2*X(34334) = 3*X(2972)-X(40948)

X(44715) lies on the Johnson circumconic and these lines: {3, 74}, {4, 6662}, {5, 35360}, {52, 8798}, {113, 15526}, {122, 16003}, {186, 32902}, {195, 3463}, {216, 1625}, {264, 339}, {265, 6334}, {382, 10152}, {1154, 36831}, {1304, 2070}, {1552, 31726}, {1624, 10628}, {1650, 10264}, {1656, 13599}, {2072, 12079}, {2394, 35098}, {2781, 43919}, {5502, 13289}, {5562, 23181}, {5627, 38896}, {7574, 17986}, {8749, 15851}, {10749, 13556}, {12113, 14677}, {12219, 15329}, {13115, 18534}, {13409, 15060}, {13754, 44436}, {14059, 34783}, {14128, 42441}, {14703, 17835}, {15354, 20126}, {15905, 18877}, {16077, 41208}, {16186, 21650}, {18403, 34150}, {18508, 20127}, {18531, 36875}, {22115, 34329}, {23039, 42487}, {23071, 35200}, {32110, 34147}, {36748, 39849}

X(44715) = midpoint of X(4) and X(44003)
X(44715) = reflection of X(i) in X(j) for these (i, j): (3, 2972), (35360, 5)
X(44715) = isotomic conjugate of X(43752)
X(44715) = barycentric product X(i)*X(j) for these {i, j}: {5, 14919}, {74, 343}, {216, 1494}, {311, 18877}, {525, 36831}, {1273, 11079}
X(44715) = barycentric quotient X(i)/X(j) for these (i, j): (3, 43768), (51, 1990), (74, 275), (216, 30), (217, 1495), (343, 3260)
X(44715) = trilinear product X(i)*X(j) for these {i, j}: {5, 35200}, {74, 44706}, {216, 2349}, {217, 33805}, {343, 2159}, {656, 36831}
X(44715) = trilinear quotient X(i)/X(j) for these (i, j): (5, 1784), (63, 43768), (74, 2190), (216, 2173), (217, 9406), (343, 14206)
X(44715) = trilinear pole of the line {216, 17434}
X(44715) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(5)}} and {{A, B, C, X(4), X(1614)}}
X(44715) = Johnson circumconic-antipode of-X(35360)
X(44715) = crossdifference of every pair of points on line {X(1637), X(39176)}
X(44715) = crosspoint of X(1494) and X(14919)
X(44715) = crosssum of X(1495) and X(1990)
X(44715) = X(1154)-cross conjugate of-X(3)
X(44715) = X(44003)-of-Euler-triangle
X(44715) = X(35360)-of-Johnson-triangle
X(44715) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 43768}, {30, 2190}, {54, 1784}, {275, 2173}
X(44715) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 43768), (51, 1990), (74, 275), (216, 30)
X(44715) = {X(12358), X(34333)}-harmonic conjugate of X(3)

X(44716) = X(5)-DAO CONJUGATE OF X(98)

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

X(44716) lies on these lines: {3, 1176}, {4, 28728}, {5, 311}, {52, 40981}, {69, 30258}, {155, 40947}, {216, 217}, {237, 511}, {262, 305}, {325, 2967}, {394, 6641}, {1216, 22062}, {1351, 3964}, {1352, 23635}, {1503, 3001}, {1568, 6368}, {1634, 44668}, {2972, 11064}, {3095, 3926}, {3564, 20975}, {5965, 18114}, {8573, 12160}, {9753, 34254}, {9967, 20775}, {10313, 39803}, {12251, 28441}, {13754, 22087}, {14570, 32428}, {15073, 20794}, {32191, 35222}, {37804, 38227}

X(44716) = barycentric product X(i)*X(j) for these {i, j}: {5, 36212}, {51, 6393}, {216, 325}, {237, 28706}, {297, 5562}, {311, 3289}
X(44716) = barycentric quotient X(i)/X(j) for these (i, j): (5, 16081), (51, 6531), (216, 98), (217, 1976), (232, 8884), (237, 8882)
X(44716) = trilinear product X(i)*X(j) for these {i, j}: {216, 1959}, {237, 18695}, {240, 5562}, {255, 39569}, {343, 1755}, {418, 40703}
X(44716) = trilinear quotient X(i)/X(j) for these (i, j): (5, 36120), (216, 1910), (240, 8884), (325, 40440), (343, 1821), (511, 2190)
X(44716) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(217)}} and {{A, B, C, X(262), X(14575)}}
X(44716) = center of circle {{X(110), X(3153), X(7468)}}
X(44716) = perspector of the circumconic {{A, B, C, X(343), X(2421)}}
X(44716) = crossdifference of every pair of points on line {X(2395), X(8882)}
X(44716) = crosspoint of X(325) and X(36212)
X(44716) = crosssum of X(4) and X(19128)
X(44716) = X(216)-Hirst inverse of-X(5562)
X(44716) = X(i)-isoconjugate-of-X(j) for these {i, j}: {54, 36120}, {98, 2190}, {275, 1910}, {293, 8884}
X(44716) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (5, 16081), (51, 6531), (216, 98), (217, 1976)
X(44716) = {X(3), X(19139)}-harmonic conjugate of X(14575)

X(44717) = X(6)-DAO CONJUGATE OF X(11)

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

X(44717) lies on these lines: {59, 3286}, {109, 29241}, {241, 1252}, {345, 1016}, {905, 906}, {1025, 1983}, {1259, 6065}, {4567, 4998}, {4587, 6517}, {8560, 17044}

X(44717) = isogonal conjugate of X(8735)
X(44717) = isotomic conjugate of the polar conjugate of X(59)
X(44717) = barycentric product X(i)*X(j) for these {i, j}: {3, 4998}, {59, 69}, {63, 4564}, {71, 4620}, {73, 4600}, {77, 765}
X(44717) = barycentric quotient X(i)/X(j) for these (i, j): (3, 11), (7, 2973), (8, 21666), (12, 2970), (48, 2170), (55, 42069)
X(44717) = trilinear product X(i)*X(j) for these {i, j}: {3, 4564}, {48, 4998}, {59, 63}, {69, 2149}, {73, 4567}, {77, 1252}
X(44717) = trilinear quotient X(i)/X(j) for these (i, j): (3, 2170), (9, 42069), (48, 3271), (57, 2969), (59, 19), (63, 11)
X(44717) = trilinear pole of the line {1331, 1813}
X(44717) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(241)}} and {{A, B, C, X(63), X(97)}}
X(44717) = cevapoint of X(i) and X(j) for these (i, j): {3, 906}, {109, 579}, {219, 1331}, {222, 1813}
X(44717) = X(i)-cross conjugate of-X(j) for these (i, j): (3, 6516), (63, 4558), (219, 1331), (222, 1813)
X(44717) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 2170}, {9, 2969}, {11, 19}, {25, 4858}
X(44717) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 11), (7, 2973), (8, 21666), (12, 2970)
X(44717) = {X(1252), X(1262)}-harmonic conjugate of X(4564)

X(44718) = X(6)-DAO CONJUGATE OF X(13)

Barycentrics    a^2*(2*S+(-a^2+b^2+c^2)*3^(1/2))*(-a^2+b^2+c^2) : :

X(44718) lies on these lines: {2, 17}, {3, 49}, {15, 323}, {16, 15066}, {22, 14540}, {25, 5864}, {61, 1993}, {69, 36297}, {110, 14538}, {298, 340}, {466, 40711}, {511, 3129}, {616, 5668}, {618, 2902}, {633, 19773}, {2979, 14541}, {3130, 9306}, {3165, 14369}, {5237, 11145}, {5463, 40112}, {5615, 5651}, {7799, 40156}, {8604, 44382}, {8838, 40334}, {10217, 11064}, {10646, 11130}, {11004, 41477}, {11083, 44383}, {11086, 34375}, {11092, 19778}, {11427, 37177}, {11459, 35470}, {14169, 33884}, {14919, 38414}, {16063, 41021}, {17811, 22238}, {21158, 34424}, {22236, 37672}, {23061, 38431}, {34755, 41478}, {35469, 43574}

X(44718) = isogonal conjugate of X(8737)
X(44718) = isotomic conjugate of the polar conjugate of X(15)
X(44718) = isotomic conjugate of isogonal conjugate of X(46112)
X(44718) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(11146)}} and {{A, B, C, X(3), X(15)}}
X(44718) = perspector of the circumconic {{A, B, C, X(4558), X(17402)}}
X(44718) = pole of the trilinear polar of X(36306) with respect to Stammler hyperbola
X(44718) = crossdifference of every pair of points on line {X(462), X(2501)}
X(44718) = X(298)-Ceva conjugate of-X(15)
X(44718) = X(3)-Hirst inverse of-X(44719)
X(44718) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 2153}, {13, 19}, {92, 3457}, {158, 36296}
X(44718) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 13), (14, 6344), (15, 4), (20, 44702)
X(44718) = barycentric product X(i)*X(j) for these {i, j}: {3, 298}, {15, 69}, {77, 44688}, {95, 44711}, {97, 33529}, {301, 22115}
X(44718) = barycentric quotient X(i)/X(j) for these (i, j): (3, 13), (14, 6344), (15, 4), (20, 44702), (48, 2153), (50, 8740)
X(44718) = trilinear product X(i)*X(j) for these {i, j}: {15, 63}, {48, 298}, {69, 2151}, {222, 44688}, {255, 470}, {304, 34394}
X(44718) = trilinear quotient X(i)/X(j) for these (i, j): (3, 2153), (15, 19), (48, 3457), (63, 13), (255, 36296), (298, 92)
X(44718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 634, 33530), (2, 11126, 62), (3, 394, 44719), (298, 470, 33529), (323, 11131, 15), (323, 11146, 11127), (2979, 34009, 14541), (3292, 36212, 44719), (11127, 11131, 11146), (11127, 11146, 15), (33529, 41887, 470)

X(44719) = X(6)-DAO CONJUGATE OF X(14)

Barycentrics    a^2*(-2*S+(-a^2+b^2+c^2)*3^(1/2))*(-a^2+b^2+c^2) : :

X(44719) lies on these lines: {2, 18}, {3, 49}, {15, 15066}, {16, 323}, {22, 14541}, {25, 5865}, {62, 1993}, {69, 36296}, {110, 14539}, {299, 340}, {465, 40712}, {511, 3130}, {617, 5669}, {619, 2903}, {634, 19772}, {1854, 14170}, {2979, 14540}, {3129, 9306}, {3166, 14368}, {5238, 11146}, {5464, 40112}, {5611, 5651}, {7799, 40157}, {8603, 44383}, {8836, 40335}, {10218, 11064}, {10645, 11131}, {11004, 41478}, {11078, 19779}, {11081, 34373}, {11088, 44382}, {11427, 37178}, {11459, 35469}, {14919, 38413}, {16063, 41020}, {17811, 22236}, {21159, 34425}, {22238, 37672}, {23061, 38432}, {34754, 41477}, {35470, 43574}

X(44719) = isogonal conjugate of X(8738)
X(44719) = isotomic conjugate of the polar conjugate of X(16)
X(44719) = isotomic conjugate of isogonal conjugate of X(46113)
X(44719) = barycentric product X(i)*X(j) for these {i, j}: {3, 299}, {16, 69}, {77, 44689}, {95, 44712}, {97, 33530}, {300, 22115}
X(44719) = barycentric quotient X(i)/X(j) for these (i, j): (3, 14), (13, 6344), (16, 4), (20, 44703), (48, 2154), (50, 8739)
X(44719) = trilinear product X(i)*X(j) for these {i, j}: {16, 63}, {48, 299}, {69, 2152}, {222, 44689}, {255, 471}, {304, 34395}
X(44719) = trilinear quotient X(i)/X(j) for these (i, j): (3, 2154), (16, 19), (48, 3458), (63, 14), (255, 36297), (299, 92)
X(44719) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(11145)}} and {{A, B, C, X(3), X(16)}}
X(44719) = perspector of the circumconic {{A, B, C, X(4558), X(17403)}}
X(44719) = pole of the trilinear polar of X(36309) with respect to Stammler hyperbola
X(44719) = crossdifference of every pair of points on line {X(463), X(2501)}
X(44719) = X(299)-Ceva conjugate of-X(16)
X(44719) = X(3)-Hirst inverse of-X(44718)
X(44719) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 2154}, {14, 19}, {92, 3458}, {158, 36297}
X(44719) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 14), (13, 6344), (16, 4), (20, 44703)
X(44719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 633, 33529), (2, 11127, 61), (3, 394, 44718), (299, 471, 33530), (323, 11130, 16), (323, 11145, 11126), (2979, 34008, 14540), (3292, 36212, 44718), (11126, 11130, 11145), (11126, 11145, 16), (33530, 41888, 471)

X(44720) = X(8)-DAO CONJUGATE OF X(1)

Barycentrics    (-a+b+c)*(3*a-b-c)/a : :
X(44720) = 4*X(10)-X(34860)

X(44720) lies on the cubic K971 and these lines: {1, 1120}, {2, 6552}, {8, 210}, {9, 30693}, {10, 982}, {69, 28616}, {75, 3617}, {85, 668}, {145, 4487}, {321, 4678}, {322, 35550}, {355, 21290}, {518, 30090}, {1089, 4668}, {1220, 39946}, {1222, 19861}, {1376, 9369}, {1706, 32939}, {2136, 30568}, {2885, 3756}, {3177, 40883}, {3210, 21896}, {3263, 10513}, {3421, 7270}, {3436, 11677}, {3596, 24393}, {3621, 4358}, {3626, 4066}, {3632, 3992}, {3673, 4986}, {3679, 4385}, {3698, 24349}, {3705, 21031}, {3717, 4073}, {3718, 4901}, {3871, 4571}, {3952, 14923}, {3985, 4050}, {3996, 4882}, {4125, 4746}, {4152, 37734}, {4397, 4926}, {4437, 26531}, {4515, 27523}, {4742, 20053}, {4745, 28612}, {4848, 4899}, {4855, 43290}, {4918, 44728}, {4936, 30720}, {5205, 12513}, {5686, 17787}, {5815, 33066}, {5836, 32937}, {6376, 40609}, {6790, 37727}, {7080, 32851}, {9458, 32577}, {10528, 33116}, {10529, 37758}, {10587, 17263}, {12607, 29641}, {12640, 44723}, {12649, 20946}, {16602, 26046}, {16610, 17480}, {16729, 17539}, {17158, 18135}, {17597, 25965}, {17751, 20923}, {17786, 24987}, {19788, 40603}, {19799, 33091}, {20936, 29226}, {20942, 31145}, {21041, 28096}, {24524, 30758}, {25082, 30730}, {33891, 40598}, {38200, 44139}

X(44720) = midpoint of X(8) and X(19582)
X(44720) = reflection of X(i) in X(j) for these (i, j): (2, 6553), (24174, 10), (34860, 24174)
X(44720) = isogonal conjugate of X(16945)
X(44720) = isotomic conjugate of X(19604)
X(44720) = barycentric product X(i)*X(j) for these {i, j}: {1, 44723}, {8, 18743}, {63, 44721}, {75, 3161}, {76, 3158}, {85, 6555}
X(44720) = barycentric quotient X(i)/X(j) for these (i, j): (1, 40151), (9, 3445), (55, 38266), (69, 27832), (75, 27818), (100, 38828)
X(44720) = trilinear product X(i)*X(j) for these {i, j}: {2, 3161}, {3, 44721}, {4, 44722}, {6, 44723}, {7, 6555}, {8, 145}
X(44720) = trilinear quotient X(i)/X(j) for these (i, j): (2, 40151), (8, 3445), (9, 38266), (76, 27818), (145, 56), (190, 38828)
X(44720) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3667)}} and {{A, B, C, X(2), X(8055)}}
X(44720) = cevapoint of X(8) and X(6552)
X(44720) = crosspoint of X(i) and X(j) for these (i, j): {75, 18743}, {145, 39701}
X(44720) = crosssum of X(i) and X(j) for these (i, j): {31, 38266}, {1357, 6363}
X(44720) = X(8)-Beth conjugate of-X(24174)
X(44720) = X(i)-Ceva conjugate of-X(j) for these (i, j): (75, 312), (668, 4462)
X(44720) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 40151}, {32, 27818}, {56, 3445}, {57, 38266}
X(44720) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 40151), (9, 3445), (55, 38266), (69, 27832)
X(44720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8, 341, 312), (8, 2551, 4514), (8, 3701, 4673), (8, 4723, 341), (8, 8055, 12541), (8, 27538, 3057), (145, 6555, 44722), (341, 4673, 3701), (3263, 16284, 21605), (3263, 25278, 16284), (3617, 4696, 75), (3701, 4673, 312), (3714, 4711, 8), (6556, 10005, 3617)

X(44721) = X(8)-DAO CONJUGATE OF X(3)

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

X(44721) lies on these lines: {4, 10744}, {264, 3264}, {280, 27506}, {318, 341}, {1222, 11109}, {4193, 35014}, {4397, 11681}, {44722, 44723}

X(44721) = polar conjugate of X(40151)
X(44721) = barycentric product X(i)*X(j) for these {i, j}: {4, 44723}, {92, 44720}, {145, 7017}, {264, 3161}, {318, 18743}, {331, 6555}
X(44721) = barycentric quotient X(i)/X(j) for these (i, j): (4, 40151), (19, 16945), (33, 38266), (75, 27832), (92, 19604), (145, 222)
X(44721) = trilinear product X(i)*X(j) for these {i, j}: {4, 44720}, {19, 44723}, {92, 3161}, {145, 318}, {158, 44722}, {264, 3158}
X(44721) = trilinear quotient X(i)/X(j) for these (i, j): (4, 16945), (76, 27832), (92, 40151), (145, 603), (264, 19604), (281, 38266)
X(44721) = intersection, other than A,B,C, of conics {{A, B, C, X(145), X(280)}} and {{A, B, C, X(341), X(44720)}}
X(44721) = X(264)-Ceva conjugate of-X(7017)
X(44721) = X(i)-isoconjugate-of-X(j) for these {i, j}: {3, 16945}, {32, 27832}, {48, 40151}, {184, 19604}
X(44721) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 40151), (19, 16945), (33, 38266), (75, 27832)

X(44722) = X(8)-DAO CONJUGATE OF X(4)

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

X(44722) lies on these lines: {3, 1811}, {8, 11}, {69, 3984}, {72, 3784}, {78, 345}, {145, 4487}, {219, 30681}, {312, 20007}, {344, 34772}, {348, 4561}, {519, 42020}, {944, 6790}, {1026, 10571}, {1043, 36624}, {1420, 4899}, {1791, 42469}, {1997, 12649}, {2646, 27549}, {2899, 44669}, {2975, 4578}, {3161, 44725}, {3189, 19582}, {3445, 9041}, {3486, 27538}, {3621, 6552}, {3622, 10005}, {3940, 21530}, {4126, 34471}, {4358, 20013}, {4437, 26658}, {6556, 31145}, {6789, 15637}, {8055, 12536}, {8834, 12541}, {11523, 18141}, {11851, 16610}, {12437, 30568}, {17336, 17576}, {17597, 25879}, {18738, 25278}, {20111, 40883}, {28370, 30614}, {44721, 44723}

X(44722) = isogonal conjugate of the polar conjugate of X(44723)
X(44722) = isotomic conjugate of the polar conjugate of X(3161)
X(44722) = barycentric product X(i)*X(j) for these {i, j}: {3, 44723}, {63, 44720}, {69, 3161}, {78, 18743}, {145, 345}, {287, 44728}
X(44722) = barycentric quotient X(i)/X(j) for these (i, j): (3, 40151), (48, 16945), (63, 19604), (69, 27818), (145, 278), (212, 38266)
X(44722) = trilinear product X(i)*X(j) for these {i, j}: {3, 44720}, {8, 4855}, {48, 44723}, {63, 3161}, {69, 3158}, {77, 6555}
X(44722) = trilinear quotient X(i)/X(j) for these (i, j): (3, 16945), (63, 40151), (69, 19604), (78, 3445), (145, 34), (219, 38266)
X(44722) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(4487)}} and {{A, B, C, X(78), X(145)}}
X(44722) = X(69)-Ceva conjugate of-X(345)
X(44722) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 16945}, {19, 40151}, {25, 19604}, {34, 3445}
X(44722) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 40151), (48, 16945), (63, 19604), (69, 27818)
X(44722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (78, 1265, 345), (145, 6555, 44720), (44725, 44726, 3161)

X(44723) = X(8)-DAO CONJUGATE OF X(6)

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

X(44723) lies on these lines: {2, 646}, {75, 24175}, {76, 4052}, {312, 2321}, {321, 3662}, {1230, 4671}, {1978, 6063}, {2092, 41839}, {3169, 30568}, {3264, 20942}, {3948, 40598}, {3950, 18743}, {3975, 4050}, {5105, 34064}, {12640, 44720}, {20691, 25125}, {30567, 40875}, {30710, 39696}, {36916, 42032}, {44721, 44722}

X(44723) = isotomic conjugate of X(40151)
X(44723) = polar conjugate of the isogonal conjugate of X(44722)
X(44723) = barycentric product X(i)*X(j) for these {i, j}: {69, 44721}, {75, 44720}, {76, 3161}, {145, 3596}, {264, 44722}, {290, 44728}
X(44723) = barycentric quotient X(i)/X(j) for these (i, j): (1, 16945), (8, 3445), (9, 38266), (75, 19604), (76, 27818), (145, 56)
X(44723) = trilinear product X(i)*X(j) for these {i, j}: {2, 44720}, {8, 18743}, {63, 44721}, {75, 3161}, {76, 3158}, {85, 6555}
X(44723) = trilinear quotient X(i)/X(j) for these (i, j): (2, 16945), (8, 38266), (76, 19604), (145, 604), (305, 27832), (312, 3445)
X(44723) = trilinear pole of the line {14284, 44728}
X(44723) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(4462)}} and {{A, B, C, X(8), X(3452)}}
X(44723) = X(76)-Ceva conjugate of-X(3596)
X(44723) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 16945}, {32, 19604}, {56, 38266}, {560, 27818}
X(44723) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 16945), (8, 3445), (9, 38266), (75, 19604)
X(44723) = {X(312), X(4110)}-harmonic conjugate of X(3452)

X(44724) = X(8)-DAO CONJUGATE OF X(11)

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

X(44724) lies on these lines: {4, 6073}, {59, 6049}, {190, 4962}, {238, 519}, {3667, 43290}, {5378, 36598}

X(44724) = barycentric product X(i)*X(j) for these {i, j}: {59, 44723}, {145, 1016}, {190, 43290}, {664, 30720}, {765, 18743}, {1275, 6555}
X(44724) = barycentric quotient X(i)/X(j) for these (i, j): (59, 40151), (145, 1086), (1016, 4373), (1110, 38266), (1252, 3445), (1743, 244)
X(44724) = trilinear product X(i)*X(j) for these {i, j}: {59, 44720}, {100, 43290}, {145, 765}, {651, 30720}, {1016, 1743}, {1252, 18743}
X(44724) = trilinear quotient X(i)/X(j) for these (i, j): (59, 16945), (145, 244), (765, 3445), (1252, 38266), (1420, 1357), (1743, 1015)
X(44724) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(145)}} and {{A, B, C, X(238), X(1743)}}
X(44724) = cevapoint of X(145) and X(43290)
X(44724) = X(145)-cross conjugate of-X(43290)
X(44724) = X(i)-isoconjugate-of-X(j) for these {i, j}: {11, 16945}, {244, 3445}, {764, 1293}, {1086, 38266}
X(44724) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (59, 40151), (145, 1086), (1016, 4373), (1110, 38266)
X(44724) = {X(765), X(4076)}-harmonic conjugate of X(1016)

X(44725) = X(8)-DAO CONJUGATE OF X(13)

Barycentrics    (2*S+(-a^2+b^2+c^2)*3^(1/2))*(-a+b+c)*(3*a-b-c) : :

X(44725) lies on this line: {3161, 44722}

X(44725) = barycentric product X(i)*X(j) for these {i, j}: {15, 44723}, {298, 3161}, {470, 44722}
X(44725) = barycentric quotient X(i)/X(j) for these (i, j): (15, 40151), (298, 27818)
X(44725) = trilinear product X(i)*X(j) for these {i, j}: {15, 44720}, {145, 44688}, {298, 3158}
X(44725) = trilinear quotient X(i)/X(j) for these (i, j): (15, 16945), (298, 19604)
X(44725) = X(13)-isoconjugate-of-X(16945)
X(44725) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (15, 40151), (298, 27818)
X(44725) = {X(3161), X(44722)}-harmonic conjugate of X(44726)

X(44726) = X(8)-DAO CONJUGATE OF X(14)

Barycentrics    (-2*S+(-a^2+b^2+c^2)*3^(1/2))*(-a+b+c)*(3*a-b-c) : :

X(44726) lies on this line: {3161, 44722}

X(44726) = barycentric product X(i)*X(j) for these {i, j}: {16, 44723}, {299, 3161}, {471, 44722}
X(44726) = barycentric quotient X(i)/X(j) for these (i, j): (16, 40151), (299, 27818)
X(44726) = trilinear product X(i)*X(j) for these {i, j}: {16, 44720}, {145, 44689}, {299, 3158}
X(44726) = trilinear quotient X(i)/X(j) for these (i, j): (16, 16945), (299, 19604)
X(44726) = X(14)-isoconjugate-of-X(16945)
X(44726) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (16, 40151), (299, 27818)
X(44726) = {X(3161), X(44722)}-harmonic conjugate of X(44725)

X(44727) = X(8)-DAO CONJUGATE OF X(30)

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

X(44727) lies on these lines: {74, 6079}, {3699, 3710}

X(44727) = barycentric product X(i)*X(j) for these {i, j}: {74, 44723}, {1494, 3161}
X(44727) = barycentric quotient X(i)/X(j) for these (i, j): (74, 40151), (145, 6357), (1494, 27818)
X(44727) = trilinear product X(i)*X(j) for these {i, j}: {74, 44720}, {145, 44693}, {1494, 3158}
X(44727) = trilinear quotient X(i)/X(j) for these (i, j): (74, 16945), (1494, 19604)
X(44727) = X(i)-isoconjugate-of-X(j) for these {i, j}: {30, 16945}, {1495, 19604}
X(44727) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (74, 40151), (145, 6357), (1494, 27818)

X(44728) = X(8)-DAO CONJUGATE OF X(98)

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

X(44728) lies on these lines: {3161, 44722}, {3699, 3712}, {4918, 44720}, {14284, 44729}

X(44728) = barycentric product X(i)*X(j) for these {i, j}: {297, 44722}, {325, 3161}, {511, 44723}
X(44728) = barycentric quotient X(i)/X(j) for these (i, j): (325, 27818), (511, 40151), (1755, 16945)
X(44728) = trilinear product X(i)*X(j) for these {i, j}: {145, 44694}, {240, 44722}, {325, 3158}, {511, 44720}, {1755, 44723}
X(44728) = trilinear quotient X(i)/X(j) for these (i, j): (325, 19604), (511, 16945)
X(44728) = X(98)-isoconjugate-of-X(16945)
X(44728) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (325, 27818), (511, 40151)

X(44729) = X(8)-DAO CONJUGATE OF X(99)

Barycentrics    (3*a-b-c)*(-a+b+c)*(b^2-c^2) : :
X(44729) = 4*X(10)-X(7178) = X(663)+2*X(4528) = 5*X(1698)-2*X(34958) = X(3700)+2*X(4041) = 4*X(4147)-X(21120) = X(4162)-4*X(4521) = X(4162)+2*X(4546) = X(4462)+2*X(4925) = 2*X(4521)+X(4546) = 4*X(4705)-X(4841) = X(4729)+2*X(14321) = X(4814)+2*X(4990) = 4*X(17072)-X(21104) = 2*X(20317)+X(44448) = 2*X(34790)+X(44410)

X(44729) lies on these lines: {10, 7178}, {210, 512}, {523, 14429}, {663, 4528}, {1639, 3900}, {1698, 34958}, {2512, 4705}, {3679, 28473}, {3700, 4041}, {3810, 4147}, {4162, 4521}, {4462, 4925}, {4729, 14321}, {4770, 6367}, {4777, 7628}, {4814, 4990}, {6362, 14430}, {8643, 14425}, {14284, 44728}, {14324, 21958}, {17072, 21104}, {20317, 44448}, {21031, 21051}, {34790, 44410}

X(44729) = reflection of X(8643) in X(14425)
X(44729) = barycentric product X(i)*X(j) for these {i, j}: {8, 14321}, {9, 4404}, {10, 4521}, {145, 3700}, {210, 4462}, {226, 4546}
X(44729) = barycentric quotient X(i)/X(j) for these (i, j): (42, 38828), (145, 4573), (210, 27834), (512, 40151), (523, 27818), (656, 27832)
X(44729) = trilinear product X(i)*X(j) for these {i, j}: {8, 4729}, {9, 14321}, {10, 4162}, {37, 4521}, {55, 4404}, {65, 4546}
X(44729) = trilinear quotient X(i)/X(j) for these (i, j): (37, 38828), (145, 1414), (210, 1293), (512, 16945), (523, 19604), (525, 27832)
X(44729) = crossdifference of every pair of points on line {X(1412), X(33628)}
X(44729) = crosspoint of X(i) and X(j) for these (i, j): {10, 30730}, {523, 14321}
X(44729) = X(200)-Beth conjugate of-X(512)
X(44729) = X(i)-Ceva conjugate of-X(j) for these (i, j): (10, 21950), (523, 3700)
X(44729) = X(i)-isoconjugate-of-X(j) for these {i, j}: {81, 38828}, {99, 16945}, {110, 19604}, {112, 27832}
X(44729) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (42, 38828), (145, 4573), (210, 27834), (512, 40151)
X(44729) = {X(4521), X(4546)}-harmonic conjugate of X(4162)

X(44730) = X(8)X(908)∩X(261)X(8822)

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

See Antreas Hatzipolakis and César Lozada, euclid 2304.

X(44730) lies on these lines: {8, 908}, {261, 8822}, {28626, 36640}

X(44730) = X(1213)-Dao conjugate of-X(5744)
X(44730) = X(1126)-isoconjugate-of-X(3576)
X(44730) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1100, 3576), (1125, 5744), (1839, 34231)
X(44730) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3916)}} and {{A, B, C, X(8), X(261)}}
X(44730) = barycentric quotient X(i)/X(j) for these (i, j): (1100, 3576), (1125, 5744), (1839, 34231)
X(44730) = trilinear product X(1125)*X(3577)
X(44730) = trilinear quotient X(1125)/X(3576)

X(44731) = ISOGONAL CONJUGATE OF X(5071)

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

See Antreas Hatzipolakis and César Lozada, euclid 2304.

X(44731) lies on Jerabek circumhyperbola and these lines: {4, 41450}, {25, 14491}, {68, 5070}, {69, 5054}, {74, 11402}, {184, 3531}, {265, 19709}, {381, 43699}, {578, 22334}, {895, 32609}, {1173, 9707}, {1176, 44456}, {1495, 3527}, {1597, 13603}, {3426, 44109}, {3532, 11430}, {4846, 15681}, {5079, 15077}, {5093, 43697}, {6199, 6414}, {6395, 6413}, {11425, 43691}, {11464, 34567}, {12017, 34817}, {12174, 16835}, {12308, 34802}, {13452, 15032}, {14483, 26864}, {14490, 17809}, {15037, 15316}, {15696, 15740}, {17040, 37935}, {21309, 43718}

X(44731) = isogonal conjugate of X(5071)
X(44731) = X(6)-vertex conjugate of-X(3527)
X(44731) = intersection, other than A,B,C, of conic {{A, B, C, X(2), X(11485)}} and Jerabek hyperbola
X(44731) = trilinear quotient X(1125)/X(3576)

X(44732) = X(3)X(324)∩X(4)X(51)

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

See Antreas Hatzipolakis and César Lozada, euclid 2304.

X(44732) lies on these lines: {2, 22270}, {3, 324}, {4, 51}, {24, 157}, {54, 8795}, {93, 186}, {107, 13597}, {125, 6750}, {140, 14978}, {232, 37121}, {235, 13381}, {264, 631}, {275, 1199}, {340, 12325}, {378, 41365}, {393, 570}, {436, 1614}, {450, 43598}, {467, 12359}, {847, 18533}, {1092, 3260}, {1105, 7464}, {1595, 14569}, {1629, 3518}, {2970, 3575}, {3090, 15466}, {3331, 27359}, {3538, 43999}, {3853, 34334}, {4994, 8794}, {5012, 37127}, {5446, 35360}, {5462, 30506}, {6143, 14165}, {6530, 15559}, {6747, 20299}, {6853, 31623}, {11411, 37192}, {11547, 37119}, {12006, 35719}, {13139, 21844}, {13409, 41481}, {14152, 41202}, {15464, 37118}, {18027, 44146}, {26166, 42368}, {26876, 42329}, {37124, 44142}, {37514, 41244}

X(44732) = polar conjugate of X(31626)
X(44732) = crosssum of X(418) and X(577)
X(44732) = X(i)-Dao conjugate of-X(j) for these (i, j): (136, 39180), (140, 5562), (233, 394), (1493, 1092)
X(44732) = X(i)-isoconjugate-of-X(j) for these {i, j}: {48, 31626}, {255, 1173}
X(44732) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (4, 31626), (140, 394), (233, 5562), (393, 1173)
X(44732) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(3567)}} and {{A, B, C, X(4), X(140)}}
X(44732) = barycentric product X(i)*X(j) for these {i, j}: {4, 40684}, {140, 2052}, {158, 20879}, {233, 8795}, {264, 6748}, {275, 14978}
X(44732) = barycentric quotient X(i)/X(j) for these (i, j): (4, 31626), (140, 394), (233, 5562), (393, 1173), (1093, 39284), (1232, 3926)
X(44732) = trilinear product X(i)*X(j) for these {i, j}: {19, 40684}, {92, 6748}, {140, 158}, {393, 20879}, {1096, 1232}, {2052, 17438}
X(44732) = trilinear quotient X(i)/X(j) for these (i, j): (92, 31626), (140, 255), (158, 1173), (1232, 326), (2190, 20574)
X(44732) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 1075, 3567), (4, 2052, 13450), (4, 3168, 9781), (140, 14978, 40684), (185, 8887, 4), (389, 42400, 4)

leftri

Points associated with Vijay ellipses: X(44733)-X(44743)

rightri

This preamble is based on notes contributed by Dasari Naga Vijay Krishna, September 6, 2021.

In the plane of a triangle ABC, let A'B'C' = medial triangle and A"B"C" = orthic triangle, and let Ea denote the ellipse that passes through A and has foci B' and C'; clearly Ea passes through A' and A". Define ellipses Eb and Ec cyclically. Names and barycentric equations for these ellipses follow:

Ea = Vijay A-ellipse: (a^2 - b^2 + c^2) y^2 + (a^2 + b^2 - c^2) z^2 - 2 c (b + c) x y - 2 b (b + c) y z = 0
Eb = Vijay B-ellipse: (b^2 - c^2 + a^2) z^2 + (b^2 + c^2 - a^2) x^2 - 2 a (c + a) y z - 2 c (c + a) z x = 0
Ec = Vijay C-ellipse: (c^2 - a^2 + b^2) x^2 + (c^2 + a^2 - b^2) y^2 - 2 b (a + b) z x - 2 a (c + b) x y = 0

See Vijay A-ellipse, Vijay B-ellipse, Vijay C-ellipse.

Let
Ab = the point, other than A, in Ea∩AB, and define Bc and Ca cyclically;
Ac = the point, other than A, in Ea∩AC, and define Ba and Cb cyclically;
A1 = AbCb∩AcBc, and define B1 and C1 cyclically;
A2 = BaBc∩CaCb, and define B2 and C2 cyclically;
A3 = AbBc∩AcCb, and define B3 and C3 cyclically;
A4 = AbCa∩AcBa, and define B4 and C4 cyclically;
A5 = the point in Eb∩Ec that is closer to A than the other point in Eb∩Ec, and define B5 and C5 cyclically;
A6 = the point in Eb∩Ec, other than A5, and define B6 and C6 cyclically;
A7 = BC∩A5A6, and define B7 and C7 cyclically.

Barycentrics for the nine points just above are as follows:

Ab = (a^2+c^2-b^2) : 2*c*(b+c) : 0

Ac = (a^2- c^2+b^2) : 0 : 2*b*(b+c)

A1 = -(a +b + c)*(3*a^3 + b^3 + c^3 + a*c*(a - c) + a*b*(a - b) - b*c*(b + c) + 2*a*b*c) : 2*(b + c)*(a + b - c)*(c*(a + b + c) + 2*a*b) : 2*(b + c)*(a - b + c)*(b*(a + b + c)+ 2*a*c)

A2 = b*c*(a +b + c)*(3*a^3 + b^3 + c^3 + a*c*(a - c) + a*b*(a - b) - b*c*(b + c) + 2*a*b*c) : a*(a + b - c)*(b^2 + c^2 - a^2)*(c*(a + b + c) + 2*a*b) : a*(a - b + c)*(c^2 + b^2 - a^2)*(b*(a + b + c)+ 2*a*c)

A3 = (a + b + c)*(a^2- b^2 + c^2)*(a^2 + b^2 - c^2)*(3*a^3 + b^3 + c^3 + a*c*(a - c) + a*b*(a - b) -b*c*(b + c) + 2*a*b*c) : 4*a*(b + a)*(b + c)*(a - b + c)*(a^2 + b^2 - c^2)*(b*(a + b + c)+ 2*a*c) : 4*a*(c + b)*(c + a)*(a + b - c)*(a^2 - b^2 + c^2)*(c*(a + b + c) + 2*a*b)

A4 = (b^7 + c^7 - a^7 + a^6*(b + c) - b^6*(c + a) - c^6*(a + b) + a^5*(b - c)^2 + b^5*(c^2+2*c*a - a^2) + c^5*(b^2 + 2*b*a - a^2) -a^4*(b + c)*(b - c)^2 + a^3*(b^4 + c^4 - 6*b^2*c^2) + 6*a^2*b^2*c^2*(b + c) - a^2*b*c*(b^3 + c^3) + 13*a*b^2*c^2*(b^2 + c^2) + 15*b^3*c^3*(b + c) + 20*a*b^3*c^3) : 2*c*(c + b)*(b^2 + c^2 - a^2)*(a^3 + 3*b^3 + c^3 + b*c*(b - c) + b*a*(b - a) - c*a*(c + a) + 2*a*b*c) : 2*b*(b + c)*(b^2 + c^2 -a^2)* (a^3 + b^3 + 3*c^3 + c*b*(c - b) + c*a*(c - a) - a*b*(a + b) + 2*a*b*c)

A5 = -(a + b + c)*(a^2 - b^2 + c^2)*(a^2 + b^2 - c^2) : (a^2 + b^2 - c^2)*(c*(a + b + c)*(a + b - c) -2*S*(a + b)) : (a^2 - b^2 + c^2)*(b*(a + b + c)*(a - b + c) - 2*S*(a + c))
     (by Francisco Javier García Capitán)

A6 = -(a + b + c)*(a^2 - b^2 + c^2)*(a^2 + b^2 - c^2) : (a^2 + b^2 - c^2)*(c*(a + b + c)*(a + b - c) + 2*S*(a + b)) : (a^2 - b^2 + c^2)*(b*(a + b + c)*(a - b + c) + 2*S*(a + c))
     (by Francisco Javier García Capitán)

A7 = 0 : (a + b)*(a^2 + b^2 - c^2) : (a + c)*(a^2 - b^2 + c^2)

Collinearities:

A2, A3, A4 are collinear.
A5, A6, A7 are collinear.
X(44733), X(44734), X(44739), X(44741) are collinear.
X(44733), X(44735), X(44738) are collinear.
X(44733), X(1659), X(44742) are collinear.
X(44733), X(13390), X(44743) are collinear.
X(2), X(4), X(44734), X(27) are collinear.
X(1659), X(13390), X(4) are collinear.
X(4), X(44742), X(44743) are collinear.

Perspectors:

X(2) = AA'∩BB'∩CC'
X(4) = AA'' ∩ BB'' ∩ CC'' = A5A6 ∩ B5B6 ∩ C5C6 = A5A7 ∩ B5B7∩ C5C7 = A6A7 ∩ B6B7∩ C6C7
X(27) = AA7 ∩ BB7∩ CC7
X(1659) = AA5∩ BB5∩ CC5
X(13390) = AA6∩ BB6∩ CC6
X(44733) = AA1 ∩ BB1∩ CC1 = AA2 ∩ BB2 ∩ CC2 = A1A2 ∩ B1B2∩ C1C2
X(44734) = AA3∩ BB3∩ CC3
X(44735) = AA4∩ BB4∩ CC4
X(44736) = A'A1∩ B'B1∩ C'C1
X(44737) = A'A2∩ B'B2∩ C'C2
X(44738) = A1A3∩ B1B3∩ C1C3
X(44739) = A1A4∩ B1B4∩ C1C4
X(44740) = A1A7∩ B1B7∩ C1C7
X(44741) = A2A3∩ B2B3∩ C2C3 = A2A4∩ B2B4∩ C2C4= A3A4∩ B3B4∩ C3C4;
X(44742) = A2A5∩ B2B5∩ C2C5
X(44743) = A2A6∩ B2B6∩ C2C6
X(44744) = A2A7∩ B2B7∩ C2C7
X(44745) = A3A5∩ B3B5∩ C3C5
X(44746) = A3A6∩ B3B6∩ C3C6

Barycentrics for A6, X(44743), X(47746) can be obtained by replacing S by -S in the barycentrics for A5, X(44742), X(44745) respectively.

The triangle AnBnCn is here named the nth Vijay triangle, for n = 1,2,3,4,5,6,7.

X(44733) = PERSPECTOR OF THESE TRIANGLES: ABC AND 1ST VIJAY

Barycentrics    (a - b + c)*(a + b - c)*(c*(a + b + c) + 2*a*b)*(b*(a + b + c)+ 2*a*c) : :
Trilinears    1/(b + c + a cos A) : :

X(44733) is also the perspector of ABC and the 2nd Vijay triangle and also the 1st and 2nd Vijay triangles.

See X(44733).

X(44733) lies on these lines: {1, 11233}, {2, 65}, {7, 941}, {27, 34}, {57, 86}, {75, 226}, {85, 310}, {273, 1848}, {312, 1240}, {553, 39704}, {604, 2185}, {673, 2258}, {675, 32693}, {871, 30545}, {903, 4654}, {986, 3671}, {1088, 3674}, {1246, 1465}, {1268, 5219}, {1403, 14621}, {1659, 44742}, {1999, 2099}, {2285, 2339}, {3210, 4373}, {3340, 11679}, {3911, 30598}, {3982, 39707}, {4384, 5665}, {4385, 10408}, {4850, 20028}, {5222, 45784}, {5226, 5936}, {5287, 24806}, {5323, 27174}, {5435, 28626}, {6384, 7146}, {7672, 22275}, {10401, 17778}, {10478, 44735}, {13390, 44743}, {18229, 18421}, {21233, 25525}, {21453, 37555}, {21454, 30712}, {27447, 27460}, {27483, 27489}, {27491, 27494}, {36588, 42051}, {41849, 43054}, {44734, 44739}

X(44733) = isogonal conjugate of X(2268)
X(44733) = isotomic conjugate of X(11679)
X(44733) = cevapoint of X(i) and X(j) for these {i,j}: {1, 2285}, {941, 959}

X(44734) = PERSPECTOR OF THESE TRIANGLES: ABC AND 3RD VIJAY

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

See X(44734).

X(44734) lies on these lines: {2, 3}, {19, 17185}, {58, 1957}, {81, 92}, {86, 278}, {162, 39673}, {240, 35623}, {281, 333}, {283, 5271}, {286, 1396}, {314, 1172}, {1444, 27339}, {1838, 25526}, {1844, 35637}, {1848, 17182}, {1865, 6703}, {1896, 36419}, {1999, 41013}, {2193, 6708}, {2287, 28950}, {2327, 27413}, {2360, 35635}, {3187, 3193}, {3687, 40950}, {5307, 10436}, {5333, 17923}, {17167, 30687}, {17188, 24210}, {17917, 25507}, {18344, 28939}, {44733, 44739}

X(44734) = polar conjugate of isogonal conjugate of {X(1805),X(1806)}-harmonic conjugate of X(6)

X(44735) = PERSPECTOR OF THESE TRIANGLES: ABC AND 4TH VIJAY

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

See X(44735).

X(44735) lies on these lines: {1, 75}, {2, 5831}, {4, 7}, {6, 1944}, {8, 24993}, {9, 25371}, {40, 10889}, {57, 10444}, {65, 10446}, {69, 18391}, {81, 92}, {85, 3664}, {91, 40438}, {145, 20895}, {158, 286}, {192, 25083}, {218, 27420}, {242, 37492}, {312, 17355}, {319, 10573}, {321, 7229}, {322, 3879}, {497, 12723}, {498, 28653}, {499, 17322}, {894, 20171}, {954, 16849}, {982, 3663}, {1014, 17134}, {1111, 4888}, {1210, 4357}, {1229, 5749}, {1441, 3945}, {1447, 19544}, {1449, 4858}, {1737, 5224}, {1743, 30854}, {1837, 10401}, {1999, 26625}, {2285, 6996}, {3086, 17321}, {3187, 26637}, {3333, 35635}, {3339, 10442}, {3596, 4737}, {3616, 24547}, {3618, 37788}, {3666, 27339}, {3668, 33673}, {3672, 14986}, {3729, 20173}, {3758, 20927}, {3868, 20245}, {3869, 17183}, {3877, 21273}, {3946, 44356}, {3991, 27544}, {4000, 20227}, {4374, 44409}, {4406, 21185}, {4461, 31325}, {4511, 24540}, {4862, 7264}, {4896, 24208}, {4967, 31397}, {5222, 20905}, {5249, 19788}, {5256, 18690}, {5271, 26638}, {5308, 25001}, {5564, 12647}, {7175, 24268}, {7179, 20256}, {7190, 24203}, {7191, 24566}, {10072, 17320}, {10478, 44733}, {10580, 31995}, {14534, 43683}, {16601, 26059}, {16706, 28738}, {17016, 24537}, {17181, 41003}, {17302, 20270}, {17316, 26665}, {17353, 20946}, {17378, 20930}, {17923, 19785}, {18655, 37422}, {18816, 36123}, {19790, 23681}, {19802, 25527}, {19804, 34258}, {21296, 26563}, {22134, 27644}, {24635, 25255}, {25065, 31225}, {25510, 25895}, {26538, 26626}, {30273, 37609}

X(44735) = polar conjugate of X(31626)

X(44736) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 1ST VIJAY

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

See X(44736).

X(44736) lies on these lines: {4, 1125}, {7, 941}, {20, 17056}, {39, 36698}, {165, 4035}, {464, 6509}, {573, 3928}, {610, 966}, {3772, 5731}, {4417, 6337}, {5739, 37781}, {6503, 11340}, {6999, 41849}, {8299, 37400}, {15349, 18673}, {30828, 40605}, {37642, 44740}

X(44737) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND 2ND VIJAY

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

See X(44737).

X(44737) lies on these lines: {75, 226}, {1210, 17063}, {1947, 40837}, {5794, 10449}, {16062, 34822}

X(44738) = PERSPECTOR OF THESE TRIANGLES: 1ST VIJAY AND 3RD VIJAY

Barycentrics    (5*b*c^11 + 4*a*c^11 + 4*b^2*c^10 + 4*a*b*c^10 + 4*a^2*c^10 - 15*b^3*c^9 - 4*a*b^2*c^9 - 5*a^2*b*c^9 - 8*a^3*c^9 - 16*b^4*c^8 - 52*a*b^3*c^8 - 80*a^2*b^2*c^8 - 36*a^3*b*c^8 - 8*a^4*c^8 + 10*b^5*c^7 - 48*a*b^4*c^7 - 68*a^2*b^3*c^7 - 84*a^3*b^2*c^7 - 54*a^4*b*c^7 + 24*b^6*c^6 + 96*a*b^5*c^6 + 76*a^2*b^4*c^6 - 28*a^3*b^3*c^6 - 4*a^4*b^2*c^6 + 12*a^5*b*c^6 + 10*b^7*c^5 + 96*a*b^6*c^5 + 146*a^2*b^5*c^5 + 156*a^3*b^4*c^5 + 54*a^4*b^3*c^5 + 4*a^5*b^2*c^5 + 38*a^6*b*c^5 + 8*a^7*c^5 - 16*b^8*c^4 - 48*a*b^7*c^4 + 76*a^2*b^6*c^4 + 156*a^3*b^5*c^4 + 56*a^4*b^4*c^4 + 48*a^5*b^3*c^4 + 36*a^6*b^2*c^4 + 4*a^7*b*c^4 + 8*a^8*c^4 - 15*b^9*c^3 - 52*a*b^8*c^3 - 68*a^2*b^7*c^3 - 28*a^3*b^6*c^3 + 54*a^4*b^5*c^3 + 48*a^5*b^4*c^3 + 44*a^6*b^3*c^3 + 68*a^7*b^2*c^3 + 17*a^8*b*c^3 - 4*a^9*c^3 + 4*b^10*c^2 - 4*a*b^9*c^2 - 80*a^2*b^8*c^2 - 84*a^3*b^7*c^2 - 4*a^4*b^6*c^2 + 4*a^5*b^5*c^2 + 36*a^6*b^4*c^2 + 68*a^7*b^3*c^2 + 48*a^8*b^2*c^2 + 16*a^9*b*c^2 - 4*a^10*c^2 + 5*b^11*c + 4*a*b^10*c - 5*a^2*b^9*c - 36*a^3*b^8*c - 54*a^4*b^7*c + 12*a^5*b^6*c + 38*a^6*b^5*c + 4*a^7*b^4*c + 17*a^8*b^3*c + 16*a^9*b^2*c - a^10*b*c + 4*a*b^11 + 4*a^2*b^10 - 8*a^3*b^9 - 8*a^4*b^8 + 8*a^7*b^5 + 8*a^8*b^4 - 4*a^9*b^3 - 4*a^10*b^2) : :

See X(44738).

X(44738) lies on these lines: {1010, 7987}, {10478, 44733}

X(44739) = PERSPECTOR OF THESE TRIANGLES: 1ST VIJAY AND 4TH VIJAY

Barycentrics    (b + a)*(c + a)*(c^3 - b*c^2 - a*c^2 - b^2*c + 2*a*b*c + a^2*c + b^3 - a*b^2 + a^2*b + 3*a^3)*(4*b*c^7 - a*c^7 + 8*b^2*c^6 + 5*a*b*c^6 + a^2*c^6 - 4*b^3*c^5 + 3*a*b^2*c^5 - 10*a^2*b*c^5 + a^3*c^5 - 16*b^4*c^4 - 23*a*b^3*c^4 - 21*a^2*b^2*c^4 + a^3*b*c^4 - a^4*c^4 - 4*b^5*c^3 - 23*a*b^4*c^3 - 20*a^2*b^3*c^3 - 14*a^3*b^2*c^3 + 4*a^4*b*c^3 + a^5*c^3 + 8*b^6*c^2 + 3*a*b^5*c^2 - 21*a^2*b^4*c^2 - 14*a^3*b^3*c^2 + 14*a^4*b^2*c^2 - 5*a^5*b*c^2 - a^6*c^2 + 4*b^7*c + 5*a*b^6*c - 10*a^2*b^5*c + a^3*b^4*c + 4*a^4*b^3*c - 5*a^5*b^2*c + 2*a^6*b*c - a^7*c - a*b^7 + a^2*b^6 + a^3*b^5 - a^4*b^4 + a^5*b^3 - a^6*b^2 - a^7*b + a^8) : :

See X(44739).

X(44739) lies on these lines: {44733, 44734}

X(44740) = PERSPECTOR OF THESE TRIANGLES: 1ST VIJAY AND 7TH VIJAY

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

See X(44740).

X(44740) lies on these lines: {57, 86}, {4292, 24161}, {37642, 44736}

X(44741) = PERSPECTOR OF THESE TRIANGLES: 2ND VIJAY AND 3RD VIJAY

Barycentrics    (c^2-b^2-a^2)*(c^2-b^2+a^2)*(c^3-b*c^2-a*c^2-b^2*c+2*a*b*c+a^2*c+b^3-a*b^2+a^2*b+3*a^3)*(b*c^8+2*a*c^8-b^2*c^7-3*a*b*c^7+b^3*c^6+6*a*b^2*c^6+a^2*b*c^6-6*a^3*c^6+15*b^4*c^5+23*a*b^3*c^5+9*a^2*b^2*c^5+5*a^3*b*c^5+15*b^5*c^4+24*a*b^4*c^4+18*a^2*b^3*c^4-2*a^3*b^2*c^4-5*a^4*b*c^4+6*a^5*c^4+b^6*c^3+23*a*b^5*c^3+18*a^2*b^4*c^3-14*a^3*b^3*c^3-11*a^4*b^2*c^3-a^5*b*c^3-b^7*c^2+6*a*b^6*c^2+9*a^2*b^5*c^2-2*a^3*b^4*c^2-11*a^4*b^3*c^2-2*a^5*b^2*c^2+3*a^6*b*c^2-2*a^7*c^2+b^8*c-3*a*b^7*c+a^2*b^6*c+5*a^3*b^5*c-5*a^4*b^4*c-a^5*b^3*c+3*a^6*b^2*c-a^7*b*c+2*a*b^8-6*a^3*b^6+6*a^5*b^4-2*a^7*b^2) : :

X(44741) is also the perspector of the 2nd and 4th Vijay triangles, and also the 3rd and 4th Vijay triangles.

See X(44741).

X(44741) lies on this line: {44733, 44734}

X(44742) = PERSPECTOR OF THESE TRIANGLES: 2ND VIJAY AND 5TH VIJAY

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^9*b^3 + a^8*b^4 - 3*a^7*b^5 - 3*a^6*b^6 + 3*a^5*b^7 + 3*a^4*b^8 - a^3*b^9 - a^2*b^10 + 2*a^9*b^2*c + a^8*b^3*c - 5*a^7*b^4*c - 3*a^6*b^5*c + 3*a^5*b^6*c + 3*a^4*b^7*c + a^3*b^8*c - a^2*b^9*c - a*b^10*c + 2*a^9*b*c^2 - 3*a^8*b^2*c^2 - 11*a^7*b^3*c^2 + a^6*b^4*c^2 + 12*a^5*b^5*c^2 + 2*a^4*b^6*c^2 - 3*a^3*b^7*c^2 + a^2*b^8*c^2 - b^10*c^2 + a^9*c^3 + a^8*b*c^3 - 11*a^7*b^2*c^3 - 8*a^6*b^3*c^3 + 26*a^5*b^4*c^3 + 25*a^4*b^5*c^3 - 3*a^3*b^6*c^3 - 2*a^2*b^7*c^3 + 3*a*b^8*c^3 + a^8*c^4 - 5*a^7*b*c^4 + a^6*b^2*c^4 + 26*a^5*b^3*c^4 + 38*a^4*b^4*c^4 + 22*a^3*b^5*c^4 + a*b^7*c^4 + 4*b^8*c^4 - 3*a^7*c^5 - 3*a^6*b*c^5 + 12*a^5*b^2*c^5 + 25*a^4*b^3*c^5 + 22*a^3*b^4*c^5 + 6*a^2*b^5*c^5 - 3*a*b^6*c^5 - 3*a^6*c^6 + 3*a^5*b*c^6 + 2*a^4*b^2*c^6 - 3*a^3*b^3*c^6 - 3*a*b^5*c^6 - 6*b^6*c^6 + 3*a^5*c^7 + 3*a^4*b*c^7 - 3*a^3*b^2*c^7 - 2*a^2*b^3*c^7 + a*b^4*c^7 + 3*a^4*c^8 + a^3*b*c^8 + a^2*b^2*c^8 + 3*a*b^3*c^8 + 4*b^4*c^8 - a^3*c^9 - a^2*b*c^9 - a^2*c^10 - a*b*c^10 - b^2*c^10 - 2*(a + b)*(a + c)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*b*c - 2*a^5*b^2*c - 2*a^4*b^3*c + 2*a^3*b^4*c + a^2*b^5*c + a^6*c^2 - 2*a^5*b*c^2 - 2*a^4*b^2*c^2 + 8*a^3*b^3*c^2 + 3*a^2*b^4*c^2 - 2*a*b^5*c^2 + 2*b^6*c^2 - 2*a^4*b*c^3 + 8*a^3*b^2*c^3 + 6*a^2*b^3*c^3 + 2*a*b^4*c^3 - 2*a^4*c^4 + 2*a^3*b*c^4 + 3*a^2*b^2*c^4 + 2*a*b^3*c^4 - 4*b^4*c^4 + a^2*b*c^5 - 2*a*b^2*c^5 + a^2*c^6 + 2*b^2*c^6)*S) : :

See X(44742).

X(44742) lies on these lines: {4, 3736}, {1659, 44733}

X(44743) = PERSPECTOR OF THESE TRIANGLES: 2ND VIJAY AND 6TH VIJAY

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^9*b^3 + a^8*b^4 - 3*a^7*b^5 - 3*a^6*b^6 + 3*a^5*b^7 + 3*a^4*b^8 - a^3*b^9 - a^2*b^10 + 2*a^9*b^2*c + a^8*b^3*c - 5*a^7*b^4*c - 3*a^6*b^5*c + 3*a^5*b^6*c + 3*a^4*b^7*c + a^3*b^8*c - a^2*b^9*c - a*b^10*c + 2*a^9*b*c^2 - 3*a^8*b^2*c^2 - 11*a^7*b^3*c^2 + a^6*b^4*c^2 + 12*a^5*b^5*c^2 + 2*a^4*b^6*c^2 - 3*a^3*b^7*c^2 + a^2*b^8*c^2 - b^10*c^2 + a^9*c^3 + a^8*b*c^3 - 11*a^7*b^2*c^3 - 8*a^6*b^3*c^3 + 26*a^5*b^4*c^3 + 25*a^4*b^5*c^3 - 3*a^3*b^6*c^3 - 2*a^2*b^7*c^3 + 3*a*b^8*c^3 + a^8*c^4 - 5*a^7*b*c^4 + a^6*b^2*c^4 + 26*a^5*b^3*c^4 + 38*a^4*b^4*c^4 + 22*a^3*b^5*c^4 + a*b^7*c^4 + 4*b^8*c^4 - 3*a^7*c^5 - 3*a^6*b*c^5 + 12*a^5*b^2*c^5 + 25*a^4*b^3*c^5 + 22*a^3*b^4*c^5 + 6*a^2*b^5*c^5 - 3*a*b^6*c^5 - 3*a^6*c^6 + 3*a^5*b*c^6 + 2*a^4*b^2*c^6 - 3*a^3*b^3*c^6 - 3*a*b^5*c^6 - 6*b^6*c^6 + 3*a^5*c^7 + 3*a^4*b*c^7 - 3*a^3*b^2*c^7 - 2*a^2*b^3*c^7 + a*b^4*c^7 + 3*a^4*c^8 + a^3*b*c^8 + a^2*b^2*c^8 + 3*a*b^3*c^8 + 4*b^4*c^8 - a^3*c^9 - a^2*b*c^9 - a^2*c^10 - a*b*c^10 - b^2*c^10 + 2*(a + b)*(a + c)*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6 + a^6*b*c - 2*a^5*b^2*c - 2*a^4*b^3*c + 2*a^3*b^4*c + a^2*b^5*c + a^6*c^2 - 2*a^5*b*c^2 - 2*a^4*b^2*c^2 + 8*a^3*b^3*c^2 + 3*a^2*b^4*c^2 - 2*a*b^5*c^2 + 2*b^6*c^2 - 2*a^4*b*c^3 + 8*a^3*b^2*c^3 + 6*a^2*b^3*c^3 + 2*a*b^4*c^3 - 2*a^4*c^4 + 2*a^3*b*c^4 + 3*a^2*b^2*c^4 + 2*a*b^3*c^4 - 4*b^4*c^4 + a^2*b*c^5 - 2*a*b^2*c^5 + a^2*c^6 + 2*b^2*c^6)*S) : :

See X(44743).

X(44743) lies on these lines: {4, 3736}, {13390, 44733}

X(44744) = PERSPECTOR OF THESE TRIANGLES: 2ND VIJAY AND 7TH VIJAY

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

See X(44744).

X(44744) lies on these lines: {4, 18165}, {273, 1848}, {6245, 23537}

X(44745) = PERSPECTOR OF THESE TRIANGLES: 3rd VIJAY AND 5th VIJAY

Barycentrics    ((18*(b+c)*a^12+12*(b^2-3*b*c+c^2)*a^11-8*(b+c)*(7*b^2+24*b*c+7*c^2)*a^10-4*(7*b^4+7*c^4+b*c*(49*b^2+188*b*c+49*c^2))*a^9+2*(b+c)*(25*b^4+25*c^4+2*b*c*(22*b^2-43*b*c+22*c^2))*a^8+8*(b^6+c^6+(29*b^4+29*c^4+b*c*(183*b^2+176*b*c+183*c^2))*b*c)*a^7+16*(b+c)*(b^6+c^6+(8*b^4+8*c^4+b*c*(33*b^2+112*b*c+33*c^2))*b*c)*a^6+8*(3*b^8+3*c^8-(11*b^6+11*c^6+(122*b^4+122*c^4+b*c*(27*b^2-10*b*c+27*c^2))*b*c)*b*c)*a^5-2*(b+c)*(29*b^8+29*c^8+2*(28*b^6+28*c^6+(74*b^4+74*c^4+b*c*(452*b^2+31*b*c+452*c^2))*b*c)*b*c)*a^4-4*(5*b^8+5*c^8-(15*b^6+15*c^6+(36*b^4+36*c^4-b*c*(159*b^2+286*b*c+159*c^2))*b*c)*b*c)*(b^2+c^2)*a^3+8*(b+c)*(5*b^10+5*c^10+(8*b^8+8*c^8-(3*b^6+3*c^6-2*(24*b^4+24*c^4-b*c*(21*b^2-32*b*c+21*c^2))*b*c)*b*c)*b*c)*a^2+4*(b^10+c^10+(5*b^8+5*c^8+(21*b^6+21*c^6-2*(28*b^4+28*c^4-b*c*(45*b^2-29*b*c+45*c^2))*b*c)*b*c)*b*c)*(b+c)^2*a-2*(b^4-c^4)*(b^2-c^2)^3*(b+c)*(5*b^2-12*b*c+5*c^2))*S+5*a^15-2*(b+c)*a^14-(25*b^2+42*b*c+25*c^2)*a^13+2*(b+c)*(6*b^2-17*b*c+6*c^2)*a^12+(49*b^4+49*c^4+6*(46*b^2+59*b*c+46*c^2)*b*c)*a^11-2*(b+c)*(11*b^4+11*c^4-2*b*c*(53*b^2+113*b*c+53*c^2))*a^10-(37*b^6+37*c^6+(614*b^4+614*c^4+b*c*(295*b^2-164*b*c+295*c^2))*b*c)*a^9+2*(b+c)*(4*b^6+4*c^6-(231*b^4+231*c^4+2*b*c*(288*b^2+139*b*c+288*c^2))*b*c)*a^8-(17*b^8+17*c^8-2*(268*b^6+268*c^6-(290*b^4+290*c^4+b*c*(892*b^2+923*b*c+892*c^2))*b*c)*b*c)*a^7+2*(b+c)*(9*b^8+9*c^8+2*(94*b^6+94*c^6+(314*b^4+314*c^4+b*c*(2*b^2+11*b*c+2*c^2))*b*c)*b*c)*a^6+(53*b^10+53*c^10-(118*b^8+118*c^8-(1081*b^6+1081*c^6+2*(1156*b^4+1156*c^4+b*c*(1033*b^2+878*b*c+1033*c^2))*b*c)*b*c)*b*c)*a^5-2*(b+c)*(10*b^10+10*c^10+(15*b^8+15*c^8+2*(141*b^6+141*c^6-(194*b^4+194*c^4+b*c*(226*b^2+75*b*c+226*c^2))*b*c)*b*c)*b*c)*a^4-(37*b^10+37*c^10-(30*b^8+30*c^8-(501*b^6+501*c^6-2*(24*b^4+24*c^4+b*c*(29*b^2+210*b*c+29*c^2))*b*c)*b*c)*b*c)*(b+c)^2*a^3+2*(3*b^10+3*c^10-(44*b^8+44*c^8-(95*b^6+95*c^6-2*(176*b^4+176*c^4-19*b*c*(13*b^2-12*b*c+13*c^2))*b*c)*b*c)*b*c)*(b+c)^3*a^2+(b^2-c^2)^2*(b+c)^2*(9*b^8+9*c^8-2*(6*b^6+6*c^6+(12*b^4+12*c^4-b*c*(14*b^2-65*b*c+14*c^2))*b*c)*b*c)*a+2*(b^4-c^4)*(b^2-c^2)^3*b*c*(b+c)*(7*b^2-12*b*c+7*c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(44745) lies on these lines: {4,142}, {1659,44734}, {14121,44735}, {44741,44742}

X(44746) = PERSPECTOR OF THESE TRIANGLES: 3rd VIJAY AND 6th VIJAY

Barycentrics    ((18*(b+c)*a^12+12*(b^2-3*b*c+c^2)*a^11-8*(b+c)*(7*b^2+24*b*c+7*c^2)*a^10-4*(7*b^4+7*c^4+b*c*(49*b^2+188*b*c+49*c^2))*a^9+2*(b+c)*(25*b^4+25*c^4+2*b*c*(22*b^2-43*b*c+22*c^2))*a^8+8*(b^6+c^6+(29*b^4+29*c^4+b*c*(183*b^2+176*b*c+183*c^2))*b*c)*a^7+16*(b+c)*(b^6+c^6+(8*b^4+8*c^4+b*c*(33*b^2+112*b*c+33*c^2))*b*c)*a^6+8*(3*b^8+3*c^8-(11*b^6+11*c^6+(122*b^4+122*c^4+b*c*(27*b^2-10*b*c+27*c^2))*b*c)*b*c)*a^5-2*(b+c)*(29*b^8+29*c^8+2*(28*b^6+28*c^6+(74*b^4+74*c^4+b*c*(452*b^2+31*b*c+452*c^2))*b*c)*b*c)*a^4-4*(5*b^8+5*c^8-(15*b^6+15*c^6+(36*b^4+36*c^4-b*c*(159*b^2+286*b*c+159*c^2))*b*c)*b*c)*(b^2+c^2)*a^3+8*(b+c)*(5*b^10+5*c^10+(8*b^8+8*c^8-(3*b^6+3*c^6-2*(24*b^4+24*c^4-b*c*(21*b^2-32*b*c+21*c^2))*b*c)*b*c)*b*c)*a^2+4*(b^10+c^10+(5*b^8+5*c^8+(21*b^6+21*c^6-2*(28*b^4+28*c^4-b*c*(45*b^2-29*b*c+45*c^2))*b*c)*b*c)*b*c)*(b+c)^2*a-2*(b^4-c^4)*(b^2-c^2)^3*(b+c)*(5*b^2-12*b*c+5*c^2))*S-5*a^15+2*(b+c)*a^14+(25*b^2+42*b*c+25*c^2)*a^13-2*(b+c)*(6*b^2-17*b*c+6*c^2)*a^12-(49*b^4+49*c^4+6*(46*b^2+59*b*c+46*c^2)*b*c)*a^11+2*(b+c)*(11*b^4+11*c^4-2*b*c*(53*b^2+113*b*c+53*c^2))*a^10+(37*b^6+37*c^6+(614*b^4+614*c^4+b*c*(295*b^2-164*b*c+295*c^2))*b*c)*a^9-2*(b+c)*(4*b^6+4*c^6-(231*b^4+231*c^4+2*b*c*(288*b^2+139*b*c+288*c^2))*b*c)*a^8+(17*b^8+17*c^8-2*(268*b^6+268*c^6-(290*b^4+290*c^4+b*c*(892*b^2+923*b*c+892*c^2))*b*c)*b*c)*a^7-2*(b+c)*(9*b^8+9*c^8+2*(94*b^6+94*c^6+(314*b^4+314*c^4+b*c*(2*b^2+11*b*c+2*c^2))*b*c)*b*c)*a^6-(53*b^10+53*c^10-(118*b^8+118*c^8-(1081*b^6+1081*c^6+2*(1156*b^4+1156*c^4+b*c*(1033*b^2+878*b*c+1033*c^2))*b*c)*b*c)*b*c)*a^5+2*(b+c)*(10*b^10+10*c^10+(15*b^8+15*c^8+2*(141*b^6+141*c^6-(194*b^4+194*c^4+b*c*(226*b^2+75*b*c+226*c^2))*b*c)*b*c)*b*c)*a^4+(37*b^10+37*c^10-(30*b^8+30*c^8-(501*b^6+501*c^6-2*(24*b^4+24*c^4+b*c*(29*b^2+210*b*c+29*c^2))*b*c)*b*c)*b*c)*(b+c)^2*a^3-2*(3*b^10+3*c^10-(44*b^8+44*c^8-(95*b^6+95*c^6-2*(176*b^4+176*c^4-19*b*c*(13*b^2-12*b*c+13*c^2))*b*c)*b*c)*b*c)*(b+c)^3*a^2-(b^2-c^2)^2*(b+c)^2*(9*b^8+9*c^8-2*(6*b^6+6*c^6+(12*b^4+12*c^4-b*c*(14*b^2-65*b*c+14*c^2))*b*c)*b*c)*a-2*(b^4-c^4)*(b^2-c^2)^3*b*c*(b+c)*(7*b^2-12*b*c+7*c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(44746) lies on these lines: {4,142}, {7090,44735}, {13390,44734}, {44741,44743}

X(44747) = X(3)X(10706)∩X(3845)X(11738)

Barycentrics    (7*a^4 - 5*a^2*b^2 - 2*b^4 - 5*a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(4*a^6 + 2*a^4*b^2 - 16*a^2*b^4 + 10*b^6 + 2*a^4*c^2 + 33*a^2*b^2*c^2 - 10*b^4*c^2 - 16*a^2*c^4 - 10*b^2*c^4 + 10*c^6) : :
X(44747) = 4 X[3845] - 5 X[18550], 5 X[7712] - 4 X[15690], 2 X[11738] - 5 X[18550]

Let P be a point on the Euler line and A'B'C' the pedal triangle of P. Let A1, B1, C1 be the orthogonal projections of A, B, C on PA', PB', PC', resp. Let A2, B2, C2 be the reflections of A1, B1, C1 in A, B, C, resp. The locus of the circumcenter of A2B2C2, as P moves on the Euler line, is a cubic here named Hatzipolakis-Moses cubic.

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44747) lies on the Hatzipolakis-Moses cubic (K1238) and these lines: {3, 10706}, {3845, 11738}, {7712, 15690}, {11645, 15534}, {13851, 38335}

X(44747) = reflection of X(11738) in X(3845)

X(44748) = X(3)X(10113)∩X(30)X(38942)

Barycentrics    (3*a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(4*a^6 - 6*a^4*b^2 + 2*b^6 - 6*a^4*c^2 + 9*a^2*b^2*c^2 - 2*b^4*c^2 - 2*b^2*c^4 + 2*c^6) : :
X(44748) = 3 X[3] - 2 X[11704], 3 X[3534] - X[11999]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44748) lies on the cubic K1238 and these lines: {3, 10113}, {30, 38942}, {382, 33556}, {550, 11270}, {1657, 5895}, {3534, 7689}, {12103, 34799}, {13619, 16880}, {15681, 32139}, {32767, 35495}

X(44748) = reflection of X(11270) in X(550)

X(44749) = X(3)X(24981)∩X(30)X(11487)

Barycentrics    (3*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 - 6*a^4*b^2 + 9*a^2*b^4 - 4*b^6 - 6*a^4*c^2 + 18*a^2*b^2*c^2 + 4*b^4*c^2 + 9*a^2*c^4 + 4*b^2*c^4 - 4*c^6) : :
X(44749) = X[1656] + 2 X[42021], X[3527] + 4 X[42021], 3 X[15694] - 2 X[15805]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44749) lies on the cubic K1238 and these lines: {3, 24981}, {30, 11487}, {631, 11402}, {632, 3618}, {1656, 3527}, {5644, 41586}, {7405, 40912}, {10602, 21230}, {15694, 15805}, {15712, 18913}

X(44749) = reflection of X(3527) in X(1656)

X(44750) = X(3)X(541)∩X(30)X(1351)

Barycentrics    (5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4)*(a^6 + 2*a^4*b^2 - 7*a^2*b^4 + 4*b^6 + 2*a^4*c^2 + 18*a^2*b^2*c^2 - 4*b^4*c^2 - 7*a^2*c^4 - 4*b^2*c^4 + 4*c^6) : :
X(44750) = 3 X[1597] - 4 X[5476], X[3426] - 4 X[4846], 2 X[4549] - 3 X[15688], 4 X[4550] - 5 X[15694], 3 X[5055] - 2 X[11472], 4 X[7706] - 3 X[14269], 4 X[8717] - 3 X[15689], X[13093] - 4 X[18431], 5 X[14093] - 4 X[35254], 7 X[15701] - 6 X[32620]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44750) lies on the cubic K1238 and these lines: {2, 35450}, {3, 541}, {30, 1351}, {51, 3830}, {376, 26864}, {381, 1514}, {1597, 5476}, {2777, 15303}, {3163, 38920}, {3167, 3534}, {3549, 43719}, {4549, 15688}, {4550, 15694}, {5055, 11472}, {5656, 44273}, {5663, 13169}, {6000, 29959}, {7552, 34469}, {7706, 14269}, {8717, 15689}, {10575, 34725}, {10605, 32225}, {10706, 32216}, {11179, 15471}, {11245, 15682}, {12315, 38323}, {12824, 15072}, {13093, 18431}, {14093, 35254}, {15311, 19153}, {15681, 18445}, {15684, 40909}, {15686, 41465}, {15701, 32620}, {19347, 34622}

X(44750) = reflection of X(i) in X(j) for these {i,j}: {381, 4846}, {3426, 381}, {15681, 35237}, {15684, 40909}, {41465, 15686}

X(44751) = X(3)X(9140)∩X(30)X(41398)

Barycentrics    (a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4)*(4*a^6 - 10*a^4*b^2 + 8*a^2*b^4 - 2*b^6 - 10*a^4*c^2 + 9*a^2*b^2*c^2 + 2*b^4*c^2 + 8*a^2*c^4 + 2*b^2*c^4 - 2*c^6) : :
X(44751) = X[3] + 2 X[38397], 4 X[140] - X[9716], 3 X[5054] - X[11935], 3 X[5055] - 2 X[7699], 7 X[15703] - 2 X[36852]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44751) lies on the cubic K1238 and these lines: {3, 9140}, {30, 41398}, {140, 9716}, {182, 599}, {376, 18387}, {381, 1531}, {549, 3431}, {1853, 3534}, {5055, 7699}, {5476, 13321}, {15066, 15087}, {15703, 36852}

X(44751) = midpoint of X(376) and X(18387)
X(44751) = reflection of X(3431) in X(549)
X(44751) = {X(4550),X(15362)}-harmonic conjugate of X(381)

X(44752) = X(3)X(974)∩X(30)X(155)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 4*b^6*c^2 - 2*a^2*b^2*c^4 - 6*b^4*c^4 + 2*a^2*c^6 + 4*b^2*c^6 - c^8) : :
X(44752) = 2 X[235] - 3 X[5654], 3 X[3167] - X[7517], 4 X[5449] - 5 X[31282], 4 X[12038] - 3 X[15078]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44752) lies on the cubic K1238 and these lines: {2, 33563}, {3, 974}, {6, 16238}, {24, 52}, {30, 155}, {49, 44259}, {68, 394}, {110, 32048}, {136, 34756}, {235, 5654}, {323, 6193}, {539, 31180}, {599, 8548}, {847, 2986}, {1181, 44240}, {1368, 12421}, {1370, 12420}, {1514, 22660}, {3167, 7517}, {3548, 19509}, {5422, 43839}, {5449, 15066}, {5925, 17838}, {6640, 15317}, {7592, 12038}, {8538, 20806}, {9786, 12161}, {9820, 36749}, {9925, 10510}, {9932, 22115}, {9938, 18436}, {11412, 19908}, {11413, 13754}, {11441, 17702}, {11477, 19139}, {12293, 15068}, {12364, 37483}, {14984, 38851}, {16196, 19458}, {18404, 44665}, {18911, 32166}, {19504, 20771}, {34148, 44269}, {37484, 41615}, {39522, 44235}, {44226, 44413}

X(44752) = midpoint of X(6193) and X(37444)
X(44752) = reflection of X(i) in X(j) for these {i,j}: {24, 1147}, {68, 11585}, {31725, 22660}
X(44752) = anticomplement of X(33563)
X(44752) = crosspoint of X(10665) and X(10666)
X(44752) = crosssum of X(13429) and X(13440)
X(44752) = barycentric product X(1993)*X(3548)
X(44752) = barycentric quotient X(i)/X(j) for these {i,j}: {3548, 5392}, {44527, 14593}
X(44752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {394, 15316, 68}, {16266, 34966, 155}

X(44753) = X(3)X(11559)∩X(30)X(6288)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44753) lies on the cubic K1238 and these lines: {3, 11559}, {30, 6288}, {74, 34577}, {2937, 33541}, {3520, 11591}, {3521, 10024}, {7552, 32138}, {11250, 15103}, {18350, 33282}, {18488, 31724}, {19362, 34783}

X(44753) = midpoint of X(2937) and X(33541)
X(44753) = reflection of X(i) in X(j) for these {i,j}: {3521, 10024}, {31724, 18488}

X(44754) = X(3)X(15738)∩X(30)X(599)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4)*(a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + 3*b^8 + 2*a^6*c^2 + 12*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 12*b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 18*b^4*c^4 - 2*a^2*c^6 - 12*b^2*c^6 + 3*c^8) : :
X(44754) = 2 X[18570] - 3 X[32620]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44754) lies on the cubic K1238 and these lines: {3, 15738}, {22, 14915}, {30, 599}, {64, 16618}, {378, 4550}, {541, 12827}, {3426, 12083}, {4846, 10605}, {5663, 41612}, {5895, 21230}, {7502, 10606}, {10510, 31861}, {13754, 41614}, {17811, 18570}, {21766, 38726}, {35480, 41594}, {40909, 44263}

X(44754) = midpoint of X(3426) and X(12083)
X(44754) = reflection of X(i) in X(j) for these {i,j}: {378, 4550}, {4846, 15760}, {35237, 7502}, {40909, 44263}

X(44755) = X(3)X(146)∩X(30)X(15801)

Barycentrics    (4*a^4 - 3*a^2*b^2 - b^4 - 3*a^2*c^2 + 2*b^2*c^2 - c^4)*(a^6 + a^4*b^2 - 5*a^2*b^4 + 3*b^6 + a^4*c^2 + 11*a^2*b^2*c^2 - 3*b^4*c^2 - 5*a^2*c^4 - 3*b^2*c^4 + 3*c^6) : :
X(44755) = 2 X[546] - 3 X[3521], 5 X[3091] - 3 X[33541], 3 X[3521] - X[16835], 4 X[3628] - 3 X[15062], 7 X[3857] - 6 X[18488], 3 X[8718] - 2 X[12103], 3 X[18442] - 4 X[44245]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44755) lies on the cubic K1238 and these lines: {3, 146}, {5, 15105}, {30, 15801}, {74, 12010}, {143, 185}, {546, 3521}, {550, 44110}, {2777, 11702}, {3091, 33541}, {3628, 15062}, {3629, 29012}, {3857, 18488}, {6000, 6153}, {8718, 12103}, {10254, 13452}, {11557, 13491}, {11558, 43806}, {15311, 32401}, {15704, 41597}, {18442, 44245}, {39884, 41579}

X(44755) = reflection of X(i) in X(j) for these {i,j}: {5, 43585}, {3627, 34563}, {16835, 546}
X(44755) = {X(3521),X(16835)}-harmonic conjugate of X(546)

X(44756) = X(3)X(5900)∩X(5)X(2889)

Barycentrics    (2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^6 - 5*a^4*b^2 + 7*a^2*b^4 - 3*b^6 - 5*a^4*c^2 + 11*a^2*b^2*c^2 + 3*b^4*c^2 + 7*a^2*c^4 + 3*b^2*c^4 - 3*c^6) : :
X(44756) = 5 X[140] - 2 X[34564], 5 X[632] - 3 X[15047], X[1173] + 3 X[34483], 2 X[3628] + 3 X[34483], X[15704] - 3 X[33542]

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44756) lies on the cubic K1238 and these lines: {3, 5900}, {5, 2889}, {140, 1493}, {546, 1216}, {632, 1994}, {1173, 3628}, {3589, 22330}, {12103, 35240}, {12108, 22115}, {13445, 44245}, {15704, 33542}, {34577, 40107}

X(44756) = midpoint of X(5) and X(2889)
X(44756) = reflection of X(1173) in X(3628)

X(44757) = X(6)X(25)∩X(1503)X(5001)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44757) lies on the cubic K1238 and these lines: {3, 34136}, {6, 25}, {22, 41198}, {110, 41199}, {1503, 5001}, {5000, 15577}, {5002, 14927}, {41197, 42671}

X(44757) = reflection of X(34135) in X(5001)
X(44757) = isogonal conjugate of the isotomic conjugate of X(5003)
X(44757) = X(i)-Ceva conjugate of X(j) for these (i,j): {1297, 41196}, {32619, 6}
X(44757) = X(75)-isoconjugate of X(34135)
X(44757) = barycentric product X(i)*X(j) for these {i,j}: {6, 5003}, {34239, 41196}
X(44757) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34135}, {5003, 76}

X(44758) = X(6)X(25)∩X(1503)X(5000)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2285.

X(44758) lies on the cubic K1238 and these lines: {3, 34135}, {6, 25}, {22, 41199}, {110, 41198}, {1503, 5000}, {5001, 15577}, {5003, 14927}, {41196, 42671}

X(44758) = reflection of X(34136) in X(5000)
X(44758) = isogonal conjugate of the isotomic conjugate of X(5002)
X(44758) = X(i)-Ceva conjugate of X(j) for these (i,j): {1297, 41197}, {32618, 6}
X(44758) = X(75)-isoconjugate of X(34136)
X(44758) = barycentric product X(i)*X(j) for these {i,j}: {6, 5002}, {34240, 41197}
X(44758) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 34136}, {5002, 76}

X(44759) = ISOGONAL CONJUGATE OF X(5881)

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

See Antreas Hatzipolakis and César Lozada, euclid 2291.

X(44759) lies on these lines: {36, 221}, {40, 4511}, {198, 2323}, {208, 1420}, {3615, 9624}, {9581, 40437}

X(44759) = isogonal conjugate of X(5881)
X(44759) = X(i)-vertex conjugate of-X(j) for these (i, j): {1, 945}, {84, 3445}
X(44759) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(36)}} and {{A, B, C, X(3), X(106)}}

X(44760) = ISOGONAL CONJUGATE OF X(9589)

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

See Antreas Hatzipolakis and César Lozada, euclid 2291.

X(44760) lies on these lines: {33, 3361}, {200, 4652}, {220, 35202}

X(44760) = isogonal conjugate of X(9589)
X(44760) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(33)}} and {{A, B, C, X(3), X(2163)}}

X(44761) = (name pending)

Barycentrics    a^2*(2*a^10-(6*b^2+7*c^2)*a^8+4*(b^4-b^2*c^2+3*c^4)*a^6+2*(b^2-c^2)*(2*b^4+13*b^2*c^2+7*c^4)*a^4-2*(b^2-c^2)*(3*b^6+5*c^6+(5*b^2+11*c^2)*b^2*c^2)*a^2+(2*b^4-b^2*c^2+3*c^4)*(b^2-c^2)^3)*(2*a^10-(7*b^2+6*c^2)*a^8+4*(3*b^4-b^2*c^2+c^4)*a^6-2*(b^2-c^2)*(7*b^4+13*b^2*c^2+2*c^4)*a^4+2*(b^2-c^2)*(5*b^6+3*c^6+(11*b^2+5*c^2)*b^2*c^2)*a^2-(3*b^4-b^2*c^2+2*c^4)*(b^2-c^2)^3) : :
Barycentrics    ((SB+3*SW)*S^4-2*SB*(SC+SA)^2*(3*S^2+2*SB^2))*((SC+3*SW)*S^4-2*SC*(SA+SB)^2*(3*S^2+2*SC^2))*(SB+SC) : :

See Antreas Hatzipolakis and César Lozada, euclid 2291.

X(44761) lies on this line: {3523, 6525}

X(44761) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(3523)}} and {{A, B, C, X(20), X(25)}}

X(44762) = X(4)X(6)∩X(64)X(3522)

Barycentrics    4*a^10-13*(b^2+c^2)*a^8+8*(2*b^4-b^2*c^2+2*c^4)*a^6-10*(b^4-c^4)*(b^2-c^2)*a^4+4*(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(44762) = X(4)-3*X(1498), 2*X(4)-3*X(2883), 5*X(4)-9*X(5656), 5*X(4)-6*X(5893), 11*X(4)-9*X(18405), X(4)+3*X(34781), 4*X(4)-3*X(41362), 5*X(1498)-3*X(5656), 5*X(1498)-2*X(5893), 11*X(1498)-3*X(18405), 4*X(1498)-X(41362), 5*X(2883)-6*X(5656), 5*X(2883)-4*X(5893), 11*X(2883)-6*X(18405), X(2883)+2*X(34781), 3*X(5656)-2*X(5893), 11*X(5656)-5*X(18405), 3*X(5656)+5*X(34781), 12*X(5656)-5*X(41362), 2*X(5893)+5*X(34781), 8*X(5893)-5*X(41362)

See Antreas Hatzipolakis and César Lozada, euclid 2291.

X(44762) lies on these lines: {4, 6}, {5, 14862}, {30, 15083}, {64, 3522}, {140, 6247}, {154, 3523}, {524, 39568}, {550, 1216}, {1595, 5465}, {1619, 3515}, {1656, 14216}, {1657, 9833}, {1853, 5056}, {3357, 33533}, {3533, 40686}, {3542, 15152}, {3629, 13598}, {3850, 18381}, {3851, 34780}, {3858, 23324}, {4314, 5882}, {5059, 6225}, {5068, 32064}, {5073, 5878}, {6293, 37900}, {7509, 15579}, {7715, 13382}, {8991, 10533}, {9914, 39879}, {10282, 15712}, {10299, 17821}, {10301, 41589}, {10323, 15582}, {10534, 13980}, {10606, 21735}, {10619, 11381}, {11414, 15581}, {11479, 31166}, {12164, 29181}, {12174, 13568}, {13464, 40658}, {13474, 31804}, {13567, 26883}, {14530, 15720}, {14683, 17812}, {14791, 40285}, {15018, 17823}, {15448, 26937}, {15647, 20417}, {15873, 18914}, {16534, 23315}, {18560, 32359}, {25406, 33537}, {30402, 42945}, {30403, 42944}, {35243, 42021}, {40660, 43174}

X(44762) = midpoint of X(i) and X(j) for these {i, j}: {1498, 34781}, {6225, 17845}, {9833, 12315}, {14683, 17812}
X(44762) = reflection of X(i) in X(j) for these (i, j): (2883, 1498), (5894, 34782), (6247, 6759), (12324, 6696), (14216, 16252), (14864, 14862), (15105, 550), (15583, 19149), (23332, 32063), (41362, 2883)
X(44762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (154, 12324, 6696), (550, 15105, 5894), (1181, 16621, 5480), (5656, 5893, 2883), (6247, 6759, 10192), (11456, 16655, 12233), (12174, 31383, 13568), (14216, 16252, 23332), (14216, 32063, 16252), (14862, 14864, 5), (15105, 34782, 550)

X(44763) = ISOGONAL CONJUGATE OF X(33703)

Barycentrics    a^2*(5*a^4-2*(5*b^2-2*c^2)*a^2+(b^2-c^2)*(5*b^2+9*c^2))*(5*a^4+2*(2*b^2-5*c^2)*a^2-(b^2-c^2)*(9*b^2+5*c^2)) : :

See Antreas Hatzipolakis and César Lozada, euclid 2291.

X(44763) lies on these lines: {3, 44108}, {4, 43903}, {24, 11738}, {30, 15749}, {54, 11410}, {68, 3534}, {69, 548}, {74, 12315}, {265, 17800}, {549, 15740}, {1204, 3527}, {1593, 14483}, {1598, 14490}, {3357, 22334}, {3431, 19347}, {3516, 34567}, {3521, 5055}, {3526, 4846}, {3532, 6759}, {3628, 31371}, {5504, 15041}, {6000, 43691}, {6391, 7689}, {6407, 6415}, {6408, 6416}, {6413, 6455}, {6414, 6456}, {10605, 43908}, {11270, 34469}, {12023, 18381}, {13093, 13452}, {15077, 15704}, {15684, 21400}, {16835, 35450}, {17821, 43713}, {21663, 43719}

X(44763) = isogonal conjugate of X(33703)
X(44763) = X(i)-vertex conjugate of-X(j) for these (i, j): {3, 43691}, {64, 43719}
X(44763) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(28171)}} and {{A, B, C, X(2), X(32875)}}

X(44764) = X(74)X(28291)∩X(110)X(284)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2298.

X(44764) lies on these lines: {30, 511}, {74, 28291}, {110, 284}, {125, 17052}, {2893, 3448}, {3024, 5580}, {5640, 30437}, {11720, 15746}.

X(44765) = ISOGONAL CONJUGATE OF X(6589)

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

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44765) lies on MacBeath circumconic and these lines: {2, 40626}, {6, 23978}, {37, 16731}, {110, 1897}, {145, 10570}, {190, 4558}, {287, 41233}, {522, 15386}, {651, 24035}, {895, 15232}, {1331, 3952}, {1332, 4033}, {1461, 17496}, {1797, 4080}, {1813, 2406}, {1814, 2995}, {1815, 17316}, {1978, 4563}, {1993, 2988}, {1999, 18359}, {2217, 41683}, {2990, 3187}, {7253, 8750}, {9056, 35183}, {23983, 23986}

X(44765) = anticomplement of X(40626)
X(44765) = isogonal conjugate of X(6589)
X(44765) = isotomic conjugate of the anticomplement of X(6332)
X(44765) = isotomic conjugate of polar conjugate of X(26704)
X(44765) = isotomic conjugate of trilinear pole of line X(124)X(34588)
X(44765) = isotomic conjugate of crossdifference of X(42) and X(184)
X(44765) = isotomic conjugate of Steiner-circumellipse pole of line X(1)X(4)
X(44765) = cevapoint of X(i) and X(j) for these (i, j): {6, 522}, {37, 521}, {513, 20227}, {514, 3772}
X(44765) = X(6)-cross conjugate of-X(15386)
X(44765) = X(i)-Dao conjugate of-X(j) for these (i, j): (9, 21189), (11, 38345), (1146, 124)
X(44765) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 21189}, {42, 16754}, {109, 38345}, {124, 1415}
X(44765) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 21189), (81, 16754), (100, 3869), (101, 573)
X(44765) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(653)}} and {{A, B, C, X(81), X(1461)}}
X(44765) = pole of the trilinear polar of X(4225) wrt Stammler hyperbola
X(44765) = trilinear pole of the line {3, 10}
X(44765) = orthocorrespondent of X(124)
X(44765) = barycentric product X(i)*X(j) for these {i, j}: {69, 26704}, {75, 36050}, {76, 32653}, {99, 15232}, {100, 2995}, {190, 13478}
X(44765) = barycentric quotient X(i)/X(j) for these (i, j): (1, 21189), (81, 16754), (100, 3869), (101, 573), (109, 10571), (110, 4225)
X(44765) = trilinear product X(i)*X(j) for these {i, j}: {2, 36050}, {63, 26704}, {75, 32653}, {100, 13478}, {101, 2995}, {190, 2217}
X(44765) = trilinear quotient X(i)/X(j) for these (i, j): (2, 21189), (86, 16754), (100, 573), (101, 3185), (190, 3869), (522, 38345)

X(44766) = ISOGONAL CONJUGATE OF X(2485)

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

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44766) lies on MacBeath circumconic and these lines: {6, 36793}, {66, 193}, {107, 33294}, {110, 1289}, {287, 1993}, {525, 15388}, {858, 34237}, {1331, 4568}, {1814, 40571}, {1916, 5986}, {2373, 38356}, {2986, 7754}, {2987, 6515}, {3580, 34138}, {4558, 4576}, {7760, 40421}, {8024, 40357}, {10766, 25053}, {14376, 14919}, {14999, 43755}, {18019, 40404}, {23974, 23976}, {40318, 41909}

X(44766) = isogonal conjugate of X(2485)
X(44766) = isotomic conjugate of X(33294)
X(44766) = anticomplement of X(2)-Ceva conjugate of X(3265)
X(44766) = cevapoint of X(i) and X(j) for these (i, j): {6, 525}, {39, 520}, {523, 3767}
X(44766) = X(i)-cross conjugate of-X(j) for these (i, j): (6, 15388), (1576, 99)
X(44766) = X(i)-Dao conjugate of-X(j) for these (i, j): (6, 8673), (39, 23881), (125, 38356), (1511, 14396)
X(44766) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 8673}, {22, 661}, {42, 16757}, {127, 32676}
X(44766) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 8673), (66, 523), (81, 16757), (86, 21178)
X(44766) = X(648)-vertex conjugate of-X(4630)
X(44766) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(107)}} and {{A, B, C, X(99), X(1916)}}
X(44766) = pole of the trilinear polar of X(22) wrt Stammler hyperbola
X(44766) = trilinear pole of the line {3, 66} (the complement of the van Aubel line)
X(44766) = orthocorrespondent of X(127)
X(44766) = barycentric product X(i)*X(j) for these {i, j}: {66, 99}, {69, 1289}, {110, 18018}, {525, 44183}, {648, 14376}, {670, 2353}
X(44766) = barycentric quotient X(i)/X(j) for these (i, j): (3, 8673), (66, 523), (81, 16757), (86, 21178), (99, 315), (100, 4463)
X(44766) = trilinear product X(i)*X(j) for these {i, j}: {63, 1289}, {66, 662}, {99, 2156}, {162, 14376}, {163, 18018}, {656, 44183}
X(44766) = trilinear quotient X(i)/X(j) for these (i, j): (63, 8673), (66, 661), (86, 16757), (99, 1760), (100, 4456), (110, 2172)

X(44767) = X(23)X(36849)∩X(110)X(33294)

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

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44767) lies on MacBeath circumconic and these lines: {23, 36849}, {110, 33294}, {287, 34137}, {297, 2987}, {4226, 43754}, {4558, 14588}, {19577, 31127}

X(44767) = isotomic conjugate of the anticomplement of X(6333)
X(44767) = cevapoint of X(i) and X(j) for these (i, j): {6, 2799}, {230, 525}, {511, 2485}
X(44767) = X(i)-isoconjugate-of-X(j) for these {i, j}: {656, 41363}, {661, 37183}
X(44767) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (110, 37183), (112, 41363)
X(44767) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(685)}} and {{A, B, C, X(76), X(935)}}
X(44767) = pole of the trilinear polar of X(37183) wrt Stammler hyperbola
X(44767) = trilinear pole of the line {3, 114}
X(44767) = orthocorrespondent of X(36471)
X(44767) = barycentric product X(670)*X(39644)
X(44767) = barycentric quotient X(i)/X(j) for these (i, j): (110, 37183), (112, 41363)
X(44767) = trilinear product X(799)*X(39644)
X(44767) = trilinear quotient X(i)/X(j) for these (i, j): (162, 41363), (662, 37183)

X(44768) = ISOGONAL CONJUGATE OF X(6132)

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

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44768) lies on MacBeath circumconic and these lines: {110, 2501}, {523, 4558}, {850, 4563}, {1331, 4024}, {1332, 4036}, {2395, 43754}, {2987, 3564}, {5159, 12079}, {8115, 39241}, {8116, 39240}, {15328, 43755}, {20578, 38414}, {20579, 38413}, {41724, 43756}

X(44768) = isogonal conjugate of X(6132)
X(44768) = cevapoint of X(i) and X(j) for these (i, j): {523, 3564}, {525, 44377}
X(44768) = X(114)-Dao conjugate of-X(38359)
X(44768) = X(661)-isoconjugate-of-X(35296)
X(44768) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (110, 35296), (230, 38359)
X(44768) = intersection, other than A,B,C, of conic {{A, B, C, X(4), X(18878)}} and MacBeath circumconic
X(44768) = pole of the trilinear polar of X(35296) wrt Stammler hyperbola X(44768) = trilinear pole of the line {3, 115}
X(44768) = orthocorrespondent of X(36472)
X(44768) = barycentric quotient X(i)/X(j) for these (i, j): (110, 35296), (230, 38359)
X(44768) = trilinear quotient X(i)/X(j) for these (i, j): (662, 35296), (1733, 38359)

X(44769) = ISOGONAL CONJUGATE OF X(1637)

Barycentrics    a^2*(a^4-(2*b^2-c^2)*a^2+(b^2-c^2)*(b^2+2*c^2))*(a^2-b^2)*(a^4+(b^2-2*c^2)*a^2-(b^2-c^2)*(2*b^2+c^2))*(a^2-c^2) : :
X(44769) = 2*X(110)-3*X(250)

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44769) lies on MacBeath circumconic and these lines: {2, 30528}, {3, 41433}, {6, 5649}, {74, 511}, {99, 32681}, {110, 250}, {249, 4558}, {287, 524}, {323, 3284}, {340, 687}, {525, 648}, {526, 14560}, {662, 35049}, {1331, 4570}, {1332, 4567}, {1351, 35908}, {1797, 4591}, {1992, 36890}, {2349, 37140}, {2394, 2407}, {2420, 14590}, {2421, 2433}, {2987, 8749}, {3564, 17986}, {4143, 4563}, {5467, 14380}, {5562, 38933}, {7473, 9007}, {8057, 30716}, {8115, 39298}, {8116, 39299}, {10411, 18879}, {10421, 44665}, {15291, 22151}, {17402, 38414}, {17403, 38413}, {17708, 34211}, {32717, 40352}, {37480, 39174}, {37779, 43768}, {40354, 41909}, {41174, 43187}

X(44769) = reflection of X(i) in X(j) for these (i, j): (323, 3284), (340, 3580)
X(44769) = isogonal conjugate of X(1637)
X(44769) = isotomic conjugate of X(41079)
X(44769) = cevapoint of X(i) and X(j) for these (i, j): {3, 1636}, {6, 526}, {74, 2433}, {110, 2420}, {520, 3284}
X(44769) = X(i)-cross conjugate of-X(j) for these (i, j): (6, 15395), (323, 249), (1636, 3)
X(44769) = X(i)-Dao conjugate of-X(j) for these (i, j): (5, 14391), (6, 9033), (9, 36035), (206, 14398)
X(44769) = X(i)-isoconjugate-of-X(j) for these {i, j}: {4, 2631}, {6, 36035}, {10, 14399}, {19, 9033}
X(44769) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 36035), (3, 9033), (32, 14398), (48, 2631)
X(44769) = X(648)-vertex conjugate of-X(32738)
X(44769) = X(46)-Zayin conjugate of-X(2631)
X(44769) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(2410)}} and {{A, B, C, X(6), X(2715)}}
X(44769) = pole of the trilinear polar of X(30) wrt Stammler hyperbola
X(44769) = trilinear pole of the tangent to the Parry circle at X(110)
X(44769) = trilinear pole of the line {3, 74}
X(44769) = 1st Saragossa point of X(14560)
X(44769) = orthocorrespondent of X(1304)
X(44769) = barycentric product X(i)*X(j) for these {i, j}: {3, 16077}, {69, 1304}, {74, 99}, {75, 36034}, {76, 32640}, {95, 36831}
X(44769) = barycentric quotient X(i)/X(j) for these (i, j): (1, 36035), (3, 9033), (32, 14398), (48, 2631), (58, 11125), (74, 523)
X(44769) = trilinear product X(i)*X(j) for these {i, j}: {2, 36034}, {48, 16077}, {63, 1304}, {69, 36131}, {74, 662}, {75, 32640}
X(44769) = trilinear quotient X(i)/X(j) for these (i, j): (2, 36035), (3, 2631), (21, 14400), (31, 14398), (47, 14397), (48, 9409)

X(44770) = X(23)X(1297)∩X(110)X(23964)

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

See Francisco Javier García Capitán and César Lozada, euclid 2310.

X(44770) lies on MacBeath circumconic and these lines: {23, 1297}, {107, 33294}, {110, 23964}, {250, 4558}, {287, 297}, {340, 35140}, {520, 15384}, {648, 8057}, {651, 36092}, {895, 15262}, {1331, 36046}, {1332, 5379}, {2409, 2419}, {4230, 34212}, {4240, 17708}, {4563, 18020}, {9517, 32695}, {11477, 39265}, {17974, 43952}

X(44770) = reflection of X(250) in X(40596)
X(44770) = isotomic conjugate of the isogonal conjugate of X(32649)
X(44770) = cevapoint of X(i) and X(j) for these (i, j): {6, 2881}, {25, 3569}, {110, 4230}, {112, 2445}
X(44770) = X(511)-cross conjugate of-X(250)
X(44770) = X(6)-Dao conjugate of-X(39473)
X(44770) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 39473}, {441, 661}, {523, 8766}, {525, 2312}
X(44770) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 39473), (110, 441), (112, 1503), (163, 8766)
X(44770) = X(i)-vertex conjugate of-X(j) for these (i, j): {3, 32695}, {685, 43754}
X(44770) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(2764)}} and {{A, B, C, X(4), X(10423)}}
X(44770) = pole of the trilinear polar of X(441) wrt Stammler hyperbola
X(44770) = trilinear pole of the line {3, 112} (the tangent to the Moses-Parry circle at X(112))
X(44770) = orthocorrespondent of X(32687)
X(44770) = barycentric product X(i)*X(j) for these {i, j}: {63, 36092}, {69, 32687}, {75, 36046}, {76, 32649}, {99, 43717}, {110, 6330}
X(44770) = barycentric quotient X(i)/X(j) for these (i, j): (3, 39473), (110, 441), (112, 1503), (163, 8766), (250, 34211), (648, 30737)
X(44770) = trilinear product X(i)*X(j) for these {i, j}: {2, 36046}, {3, 36092}, {63, 32687}, {75, 32649}, {110, 8767}, {162, 1297}
X(44770) = trilinear quotient X(i)/X(j) for these (i, j): (63, 39473), (110, 8766), (112, 2312), (162, 1503), (163, 8779), (662, 441)

X(44771) = MIDPOINT OF X(4207) AND X(8782)

Barycentrics    (a^2 - b*c)*(a^2 + b*c)*(-b^6 + a^2*b^2*c^2 - b^4*c^2 - b^2*c^4 - c^6) : :
X(44771) = 4 X[6390] - X[41747]

X(44771) lies on the cubic K1239 and these lines: {3, 32429}, {6, 8149}, {76, 141}, {385, 732}, {511, 6033}, {736, 2076}, {1916, 5031}, {3098, 37004}, {3564, 32521}, {3818, 40252}, {5207, 8782}, {5306, 9764}, {5969, 14041}, {6309, 6390}, {7792, 13331}, {7807, 32449}, {10007, 16897}, {13196, 32451}, {16044, 18906}

X(44771) = midpoint of X(5207) and X(8782)
X(44771) = reflection of X(i) in X(j) for these {i,j}: {1691, 5976}, {1916, 5031}, {3094, 6393}, {32451, 13196}
X(44771) = crossdifference of every pair of points on line {882, 9426}
X(44771) = barycentric product X(3978)*X(32452)
X(44771) = barycentric quotient X(i)/X(j) for these {i,j}: {14295, 30492}, {32452, 694}

X(44772) = MIDPOINT OF X(76) AND X(2896)

Barycentrics    (b^2 + c^2)*(-a^6 + a^4*b^2 + a^2*b^4 + a^4*c^2 + 5*a^2*b^2*c^2 + 2*b^4*c^2 + a^2*c^4 + 2*b^2*c^4) : :
X(44772) = X[83] - 3 X[42006], 2 X[3934] - 3 X[42006], 4 X[6683] - 5 X[31268], 4 X[6704] - 5 X[31239], 3 X[7697] - X[13111], 3 X[7811] - X[9990], X[12252] - 3 X[22712], X[20088] - 5 X[31276], 3 X[31168] - X[32476]

X(44772) lies on the cubic K1239 and these lines: {32, 24273}, {39, 141}, {69, 31982}, {76, 148}, {83, 385}, {183, 8150}, {187, 6308}, {298, 6297}, {299, 6296}, {325, 32190}, {511, 6287}, {538, 31168}, {599, 32452}, {626, 9478}, {754, 8370}, {1078, 8290}, {1916, 32027}, {3314, 32189}, {3620, 31981}, {3627, 6248}, {5008, 41650}, {5188, 29012}, {6683, 7909}, {6704, 7792}, {7697, 13111}, {7768, 9866}, {7801, 13086}, {7804, 12206}, {7811, 9990}, {7826, 24256}, {8149, 16990}, {12252, 22712}, {12263, 17766}, {12783, 14839}, {14023, 20088}, {14042, 39266}, {24206, 35437}

X(44772) = midpoint of X(76) and X(2896)
X(44772) = reflection of X(i) in X(j) for these {i,j}: {39, 6292}, {83, 3934}
X(44772) = anticomplement of the isogonal conjugate of X(39397)
X(44772) = X(39397)-anticomplementary conjugate of X(8)
X(44772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {83, 41755, 5007}, {83, 42006, 3934}, {8362, 41756, 39}

X(44773) = REFLECTION OF X(574) IN X(15810)

Barycentrics    (a^2 - 2*b^2 - 2*c^2)*(4*a^6 - 6*a^4*b^2 + b^6 - 6*a^4*c^2 - 15*a^2*b^2*c^2 - 3*b^4*c^2 - 3*b^2*c^4 + c^6) : :

X(44773) lies on the cubic K1239 and these lines: {32, 7610}, {76, 543}, {115, 11167}, {183, 3849}, {574, 599}, {2482, 11151}, {7771, 8592}, {7849, 7888}, {8722, 11645}, {8860, 14762}, {9774, 10991}, {11185, 40246}

X(44773) = reflection of X(574) in X(15810)

X(44774) = MIDPOINT OF X(69) AND X(31958)

Barycentrics    (a^4 - a^2*b^2 - a^2*c^2 - 2*b^2*c^2)*(2*a^2*b^4 - b^6 + 5*a^2*b^2*c^2 + b^4*c^2 + 2*a^2*c^4 + b^2*c^4 - c^6) : :
X(44774) = X[76] + 2 X[40107], X[182] + 2 X[14994], X[576] - 4 X[3934], 5 X[3763] - 2 X[44423], 2 X[5097] - 5 X[40332], X[9821] + 2 X[18553], 3 X[10516] - X[22728], 3 X[21358] - X[32447], 5 X[40330] - X[44434]

X(44774) lies on the cubic K1239 and these lines: {6, 32149}, {69, 31958}, {76, 22677}, {141, 11261}, {182, 183}, {262, 3314}, {298, 22714}, {299, 22715}, {381, 511}, {542, 8592}, {576, 3934}, {732, 40108}, {1352, 6194}, {3763, 44423}, {5097, 40332}, {7709, 32833}, {7815, 43157}, {8992, 44474}, {9821, 18553}, {11171, 39785}, {13983, 44473}, {17004, 32451}, {21358, 32447}, {22676, 29012}, {40330, 44434}

X(44774) = midpoint of X(i) and X(j) for these {i,j}: {69, 31958}, {76, 22677}, {1352, 6194}, {14994, 15819}
X(44774) = reflection of X(i) in X(j) for these {i,j}: {6, 32149}, {182, 15819}, {262, 24206}, {11261, 141}, {22677, 40107}

X(44775) = MIDPOINT OF X(7709) AND X(7779)

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

X(44775) lies on the cubic K1239 and these lines: {5, 76}, {385, 40108}, {511, 8724}, {524, 11171}, {599, 11261}, {736, 32519}, {2080, 39099}, {2782, 7840}, {3398, 22525}, {7709, 7779}, {12054, 21445}

X(44775) = midpoint of X(7709) and X(7779)
X(44775) = reflection of X(i) in X(j) for these {i,j}: {385, 40108}, {7697, 325}

X(44776) = MIDPOINT OF X(627) AND X(633)

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

X(44776) lies on the cubic K1239 and these lines: {2, 8259}, {3, 298}, {17, 299}, {61, 302}, {62, 42672}, {69, 22113}, {76, 11602}, {511, 16626}, {532, 7768}, {533, 37007}, {599, 5025}, {620, 25608}, {634, 16629}, {636, 22891}, {3105, 22894}, {5007, 37785}, {5487, 11122}, {7796, 35689}, {7814, 21359}, {7907, 9761}, {11129, 36782}, {11296, 33622}, {11309, 42989}, {14540, 44666}, {22685, 44362}, {22890, 41022}, {34540, 44029}, {35932, 36386}

X(44776) = midpoint of X(627) and X(633)
X(44776) = reflection of X(i) in X(j) for these {i,j}: {17, 635}, {61, 629}, {25608, 620}
X(44776) = isotomic conjugate of X(18813)
X(44776) = isogonal conjugate of vertex conjugate of X(17) and X(61)
X(44776) = anticomplement of X(8259)
X(44776) = barycentric product X(302)*X(34540)
X(44776) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18813}, {61, 34533}, {34540, 17}, {44033, 2004}
X(44776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {17, 40707, 44030}, {299, 44030, 17}

X(44777) = MIDPOINT OF X(628) AND X(634)

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

X(44777) lies on the cubic K1239 and these lines: {2, 8260}, {3, 299}, {18, 298}, {61, 42673}, {62, 303}, {69, 22114}, {76, 11603}, {511, 16627}, {532, 37008}, {533, 7768}, {599, 5025}, {620, 25609}, {633, 16628}, {635, 22846}, {3104, 22850}, {5007, 37786}, {5488, 11121}, {7796, 35688}, {7814, 21360}, {7907, 9763}, {11295, 33624}, {11310, 42988}, {14541, 44667}, {22683, 44361}, {22843, 41023}, {34541, 44031}, {35931, 36388}

X(44777) = midpoint of X(628) and X(634)
X(44777) = reflection of X(i) in X(j) for these {i,j}: {18, 636}, {62, 630}, {25609, 620}
X(44777) = isotomic conjugate of X(18814)
X(44777) = anticomplement of X(8260)
X(44777) = isogonal conjugate of vertex conjugate of X(18) and X(62)
X(44777) = barycentric product X(303)*X(34541)
X(44777) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 18814}, {62, 34534}, {34541, 18}, {44035, 2005}
X(44777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {18, 40706, 44032}, {298, 44032, 18}

X(44778) = ISOGONAL CONJUGATE OF X(42811)

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

X(44778) lies on the cubic K934 and these lines: {4, 32619}, {6, 25}, {182, 5001}, {237, 41196}, {263, 41201}, {419, 41194}, {511, 5000}, {1976, 41200}, {3148, 41197}, {9306, 41199}, {19128, 32618}, {34136, 34146}, {40894, 43576}

X(44778) = isogonal conjugate of X(42811)
X(44778) = isogonal conjugate of the isotomic conjugate of X(5000)
X(44778) = polar conjugate of the isotomic conjugate of X(41196)
X(44778) = X(i)-Ceva conjugate of X(j) for these (i,j): {5000, 41196}, {41200, 32}
X(44778) = crosspoint of X(i) and X(j) for these (i,j): {4, 41200}, {6, 34136}
X(44778) = crosssum of X(i) and X(j) for these (i,j): {2, 5002}, {3, 41198}
X(44778) = crossdifference of every pair of points on line {525, 41199}
X(44778) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42811}, {63, 41194}, {75, 32618}, {304, 41200}, {336, 5001}, {1821, 41199}
X(44778) = barycentric product X(i)*X(j) for these {i,j}: {4, 41196}, {6, 5000}, {25, 41198}, {232, 32619}, {237, 41195}, {511, 41201}, {2211, 42812}
X(44778) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 42811}, {25, 41194}, {32, 32618}, {237, 41199}, {1974, 41200}, {2211, 5001}, {5000, 76}, {9418, 41197}, {41195, 18024}, {41196, 69}, {41198, 305}, {41201, 290}
X(44778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 44757, 184}, {18374, 44757, 1495}

X(44779) = ISOGONAL CONJUGATE OF X(42812)

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

X(44779 lies on the cubic K934 and these lines: {4, 32618}, {6, 25}, {182, 5000}, {237, 41197}, {263, 41200}, {419, 41195}, {511, 5001}, {1976, 41201}, {3148, 41196}, {9306, 41198}, {19128, 32619}, {34135, 34146}, {40895, 43576}

X(44779) = isogonal conjugate of X(42812)
X(44779) = isogonal conjugate of the isotomic conjugate of X(5001)
X(44779) = polar conjugate of the isotomic conjugate of X(41197)
X(44779) = X(i)-Ceva conjugate of X(j) for these (i,j): {5001, 41197}, {41201, 32}
X(44779) = crosspoint of X(i) and X(j) for these (i,j): {4, 41201}, {6, 34135}
X(44779) = crosssum of X(i) and X(j) for these (i,j): {2, 5003}, {3, 41199}
X(44779) = crossdifference of every pair of points on line {525, 41198}
X(44779) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42812}, {63, 41195}, {75, 32619}, {304, 41201}, {336, 5000}, {1821, 41198}
X(44779) = barycentric product X(i)*X(j) for these {i,j}: {4, 41197}, {6, 5001}, {25, 41199}, {232, 32618}, {237, 41194}, {511, 41200}, {2211, 42811}
X(44779) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 42812}, {25, 41195}, {32, 32619}, {237, 41198}, {1974, 41201}, {2211, 5000}, {5001, 76}, {9418, 41196}, {41194, 18024}, {41197, 69}, {41199, 305}, {41200, 290}
X(44779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 44758, 184}, {18374, 44758, 1495}

X(44780) = ISOTOMIC CONJUGATE OF X(32618)

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

X(44780) lies on these lines: {2, 41195}, {4, 69}, {99, 40894}, {183, 5001}, {290, 32618}, {297, 41198}, {325, 5000}, {385, 41200}, {458, 41199}, {5002, 30737}, {14907, 40895}

X(44780) = isotomic conjugate of X(32618)
X(44780) = anticomplement of X(41196)
X(44780) = polar conjugate of X(41200)
X(44780) = anticomplement of the isogonal conjugate of X(41194)
X(44780) = isotomic conjugate of the isogonal conjugate of X(5000)
X(44780) = polar conjugate of the isogonal conjugate of X(41198)
X(44780) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1821, 5003}, {32618, 6360}, {41194, 8}, {41200, 192}, {42811, 4329}
X(44780) = cevapoint of X(5000) and X(41198)
X(44780) = X(i)-isoconjugate of X(j) for these (i,j): {31, 32618}, {48, 41200}, {560, 42811}, {1910, 41197}, {9247, 41194}
X(44780) = barycentric product X(i)*X(j) for these {i,j}: {76, 5000}, {264, 41198}, {297, 42812}, {325, 41195}, {18022, 41196}, {32619, 44132}
X(44780) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32618}, {4, 41200}, {76, 42811}, {264, 41194}, {297, 5001}, {325, 41199}, {511, 41197}, {5000, 6}, {32619, 248}, {41195, 98}, {41196, 184}, {41198, 3}, {41201, 1976}, {42812, 287}

X(44781) = ISOTOMIC CONJUGATE OF X(32619)

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

X(44781) lies on these lines: {2, 41194}, {4, 69}, {99, 40895}, {183, 5000}, {290, 32619}, {297, 41199}, {325, 5001}, {385, 41201}, {458, 41198}, {5003, 30737}, {14907, 40894}

X(44781) = isotomic conjugate of X(32619)
X(44781) = anticomplement of X(41197)
X(44781) = polar conjugate of X(41201)
X(44781) = anticomplement of the isogonal conjugate of X(41195)
X(44781) = isotomic conjugate of the isogonal conjugate of X(5001)
X(44781) = polar conjugate of the isogonal conjugate of X(41199)
X(44781) = cevapoint of X(5001) and X(41199)
X(44781) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1821, 5002}, {32619, 6360}, {41195, 8}, {41201, 192}, {42812, 4329}
X(44781) = X(i)-isoconjugate of X(j) for these (i,j): {31, 32619}, {48, 41201}, {560, 42812}, {1910, 41196}, {9247, 41195}
X(44781) = barycentric product X(i)*X(j) for these {i,j}: {76, 5001}, {264, 41199}, {297, 42811}, {325, 41194}, {18022, 41197}, {32618, 44132}
X(44781) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32619}, {4, 41201}, {76, 42812}, {264, 41195}, {297, 5000}, {325, 41198}, {511, 41196}, {5001, 6}, {32618, 248}, {41194, 98}, {41197, 184}, {41199, 3}, {41200, 1976}, {42811, 287}

X(44782) = MIDPOINT OF X(2475) AND X(3869)

Barycentrics    a*(a - b - c)*(b + c)*(a^4 - 2*a^2*b^2 + b^4 + a^2*b*c - a*b^2*c - 2*b^3*c - 2*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 - 2*b*c^3 + c^4) : :
X(47782) = X[191] - 3 X[5692], 3 X[210] - 2 X[21677], 3 X[210] - 4 X[40661], 3 X[354] - 4 X[11281], 3 X[354] - 2 X[39772], 3 X[392] - 2 X[35016], 2 X[942] - 3 X[26725], 4 X[960] - X[17637], X[3057] + 2 X[31938], 2 X[3649] + X[3962], 4 X[3812] - 5 X[31254], X[4018] - 4 X[6701], 4 X[6675] - 5 X[25917], 4 X[12447] - X[41551]

X(44782) lies on the cubics K1238 and K1240 and these lines: {1, 15910}, {2, 8261}, {3, 191}, {8, 11604}, {10, 12}, {21, 60}, {30, 5887}, {224, 34879}, {354, 11281}, {377, 14450}, {392, 35016}, {517, 37230}, {518, 15988}, {912, 33858}, {936, 1454}, {942, 26725}, {997, 37308}, {1155, 11684}, {1385, 22115}, {1836, 2475}, {2098, 4516}, {2245, 3958}, {2650, 40967}, {2975, 17660}, {3057, 31938}, {3065, 12786}, {3120, 41501}, {3647, 37600}, {3651, 6001}, {3812, 31254}, {3868, 28628}, {3876, 26066}, {3878, 6284}, {3899, 41869}, {3940, 11507}, {4190, 31888}, {5044, 13750}, {5499, 14988}, {5698, 15680}, {5784, 17768}, {5904, 16126}, {6175, 44663}, {6596, 6597}, {6675, 25917}, {6690, 20612}, {6831, 20117}, {6917, 16159}, {7483, 10176}, {9708, 41686}, {10543, 14100}, {11680, 20288}, {12005, 31157}, {12447, 41551}, {12514, 37286}, {12635, 37228}, {12671, 44238}, {15829, 30223}, {16117, 40266}, {16139, 31837}, {17647, 17653}, {18253, 27385}, {18389, 24953}, {24474, 33592}, {24987, 41550}, {26202, 35459}, {30144, 33667}, {31143, 41742}, {31806, 37468}, {33668, 37438}, {34790, 41684}

X(44782) = midpoint of X(i) and X(j) for these {i,j}: {2475, 3869}, {5693, 16132}, {5904, 16126}, {16117, 40266}
X(44782) = reflection of X(i) in X(j) for these {i,j}: {21, 960}, {65, 442}, {16139, 31837}, {17637, 21}, {17660, 39778}, {21677, 40661}, {24474, 33592}, {39772, 11281}
X(44782) = isogonal conjugate of vertex conjugate of X(21) and X(65)
X(44782) = isotomic conjugate of Brianchon point (perspector) of inconic centered at X(8261)
X(44782) = anticomplement of X(8261)
X(44782) = crosspoint of X(21) and X(6598)
X(44782) = crosssum of X(65) and X(37583)
X(44782) = barycentric product X(321)*X(37564)
X(44782) = barycentric quotient X(37564)/X(81)
X(44782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {191, 35204, 22937}, {960, 1858, 3683}, {1858, 33857, 17637}, {3678, 21031, 210}, {4511, 18259, 21}, {11281, 39772, 354}, {21677, 40661, 210}

X(44783) = MIDPOINT OF X(72) AND X(2894)

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

X(44783) lies on the cubic K1240 and these lines: {3, 5696}, {8, 24298}, {72, 2894}, {442, 942}, {943, 5044}, {1125, 5572}, {2949, 12664}, {5762, 5777}, {5842, 31793}, {6737, 9957}, {7686, 34790}

X(44783) = midpoint of X(72) and X(2894)
X(44783) = reflection of X(943) in X(5044)

X(44784) = X(8)X(210)∩X(214)X(519)

Barycentrics    (a - b - c)*(2*a - b - c)*(a^2 - a*b - 2*b^2 - a*c + 4*b*c - 2*c^2) : :
X(47784) = 3 X[3679] - X[41702]

X(47784) lies on the cubic K1240 and these lines: {1, 37829}, {3, 3632}, {8, 210}, {40, 36972}, {44, 21942}, {100, 33956}, {145, 17728}, {214, 519}, {354, 12648}, {515, 13996}, {517, 10742}, {521, 4768}, {952, 13528}, {1155, 38455}, {1320, 5123}, {1532, 28234}, {2098, 15347}, {2136, 30223}, {3244, 13747}, {3621, 24477}, {3625, 37568}, {3626, 4187}, {3679, 20196}, {3698, 17582}, {3922, 9776}, {3929, 4677}, {3962, 12245}, {4746, 15862}, {5044, 5559}, {5048, 5854}, {5541, 28204}, {5552, 33176}, {5687, 34880}, {5905, 32049}, {6154, 28236}, {6921, 20050}, {7743, 12653}, {8256, 20323}, {10107, 26842}, {10912, 17606}, {10915, 11011}, {10950, 12640}, {11113, 34641}, {14923, 32537}, {26015, 32426}, {33559, 41012}, {37566, 41687}

X(44784) = reflection of X(i) in X(j) for these {i,j}: {1319, 1145}, {1320, 5123}, {5048, 6735}, {12653, 7743}
X(44784) = X(2415)-Ceva conjugate of X(1639)
X(44784) = crosspoint of X(8) and X(12641)
X(44784) = crosssum of X(56) and X(5193)
X(44784) = barycentric product X(i)*X(j) for these {i,j}: {519, 30827}, {2098, 4358}, {2325, 4862}
X(44784) = barycentric quotient X(i)/X(j) for these {i,j}: {2098, 88}, {4358, 18811}, {4723, 34523}, {17424, 23345}, {30827, 903}, {34524, 1320}, {34543, 1417}

X(44785) = X(7)X(8)∩X(516)X(10609)

Barycentrics    (a - b - c)*(a^2 + a*b - 2*b^2 + a*c + 4*b*c - 2*c^2)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :
X(44785) = 5 X[1698] - 3 X[41700]

X(44785) lies on the cubic K1240 and these lines: {7, 8}, {516, 10609}, {527, 1155}, {908, 5851}, {1156, 5087}, {1698, 41700}, {3254, 15733}, {3689, 5856}, {4312, 4867}, {4511, 28534}, {4679, 17603}, {4860, 5231}, {5057, 15726}, {5698, 37600}, {11112, 30424}, {11662, 25440}

X(44785) = midpoint of X(4312) and X(4867)
X(44785) = reflection of X(i) in X(j) for these {i,j}: {1155, 10427}, {1156, 5087}, {6068, 6745}, {36920, 2550}
X(44785) = crosspoint of X(3254) and X(34919)
X(44785) = crosssum of X(2078) and X(37541)
X(44785) = barycentric product X(i)*X(j) for these {i,j}: {527, 5231}, {6173, 6745}, {30806, 34522}, {37780, 42014}
X(44785) = barycentric quotient X(i)/X(j) for these {i,j}: {4860, 34056}, {5231, 1121}, {17425, 23351}, {32578, 4845}, {34522, 1156}, {37780, 18810}, {42014, 41798}

X(44786) = REFLECTION OF X(20421) IN X(8703)

Barycentrics    (5*a^4 - a^2*b^2 - 4*b^4 - a^2*c^2 + 8*b^2*c^2 - 4*c^4)*(16*a^6 - 28*a^4*b^2 + 8*a^2*b^4 + 4*b^6 - 28*a^4*c^2 + 33*a^2*b^2*c^2 - 4*b^4*c^2 + 8*a^2*c^4 - 4*b^2*c^4 + 4*c^6) : :

X(47786) lies on the cubic K1238 and these lines: {3, 20396}, {3098, 15533}, {3534, 7712}, {8703, 20421}, {8717, 15689}, {11202, 15681}, {15688, 41398}, {44749, 44753}

X(44786) = reflection of X(20421) in X(8703)

X(44787) = REFLECTION OF X(3091) IN X(16254)

Barycentrics    (a^4 + 2*a^2*b^2 - 3*b^4 + 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*(9*a^6 - 24*a^4*b^2 + 21*a^2*b^4 - 6*b^6 - 24*a^4*c^2 + 22*a^2*b^2*c^2 + 6*b^4*c^2 + 21*a^2*c^4 + 6*b^2*c^4 - 6*c^6) : :

X(44787) lies on the cubic K1238 and these lines: {631, 14528}, {632, 12161}, {3091, 15741}, {3697, 15016}, {8567, 17538}, {14924, 43841}, {44752, 44756}, {44754, 44755}

X(44787) = reflection of X(i) in X(j) for these {i,j}: {3091, 16254}, {14528, 631}

X(44788) = REFLECTION OF X(3532) IN X(20)

Barycentrics    (5*a^4 - 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 6*b^2*c^2 - 3*c^4)*(9*a^6 - 12*a^4*b^2 - 3*a^2*b^4 + 6*b^6 - 12*a^4*c^2 + 22*a^2*b^2*c^2 - 6*b^4*c^2 - 3*a^2*c^4 - 6*b^2*c^4 + 6*c^6) : :
X(44788) = 3 X[20] - X[15077], 3 X[376] - 2 X[43592], X[3146] - 3 X[27082], 3 X[3532] - 2 X[15077]

X(44788) lies on the cubic K1238 and these lines: {3, 6723}, {20, 3532}, {376, 43592}, {1498, 3529}, {1514, 11541}, {3146, 15748}, {12163, 15704}, {14927, 20080}, {44752, 44755}, {44754, 44756}

X(44788) = reflection of X(3532) in X(20)

X(44789) = REFLECTION OF X(3518) IN X(1493)

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

X(44789) lies on the cubic K1238 and these lines: {2, 18368}, {3, 11806}, {30, 15801}, {49, 143}, {195, 6759}, {3519, 37452}, {3589, 9972}, {12291, 17714}, {21230, 44756}

X(44789) = reflection of X(i) in X(j) for these {i,j}: {3518, 1493}, {3519, 37452}

X(44790) = REFLECTION OF X(1593) IN X(33537)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 6*b^2*c^2 + c^4)*(a^8 + 4*a^6*b^2 - 6*a^4*b^4 - 4*a^2*b^6 + 5*b^8 + 4*a^6*c^2 + 28*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 20*b^6*c^2 - 6*a^4*c^4 + 4*a^2*b^2*c^4 + 30*b^4*c^4 - 4*a^2*c^6 - 20*b^2*c^6 + 5*c^8) : :

X(47790) lies on the cubic K1238 and these lines: {30, 11487}, {599, 5895}, {1593, 17811}, {3620, 11469}, {6823, 15740}, {8567, 16936}, {12173, 29181}, {21230, 44750}, {22334, 34817}

X(44790) = midpoint of X(11469) and X(37201)
X(44790) = reflection of X(i) in X(j) for these {i,j}: {1593, 33537}, {15740, 6823}

X(44791) = REFLECTION OF X(3580) IN X(7706)

Barycentrics    (a^2 - b^2 - c^2)*(4*a^6 - 7*a^4*b^2 + 2*a^2*b^4 + b^6 - 7*a^4*c^2 + 6*a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4 + c^6)*(a^8 + 2*a^6*b^2 - 4*a^4*b^4 - 2*a^2*b^6 + 3*b^8 + 2*a^6*c^2 + 12*a^4*b^2*c^2 + 2*a^2*b^4*c^2 - 12*b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 18*b^4*c^4 - 2*a^2*c^6 - 12*b^2*c^6 + 3*c^8) : :

X(47791) lies on the cubic K1238 and these lines: {3, 4549}, {30, 9970}, {541, 858}, {3580, 7706}, {10294, 10295}, {10297, 34802}, {11477, 40909}, {13754, 32275}

X(44791) = reflection of X(i) in X(j) for these {i,j}: {3580, 7706}, {4549, 11064}, {34802, 10297}

X(44792) = X(7)X(312)∩X(145)X(321)

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

See Francisco Javier García Capitán euclid 2327.

X(44792) lies on these lines: {7, 312}, {75, 2999}, {145, 321}, {329, 341}, {346, 1427}, {2064, 17378}, {4054, 19814}, {4389, 4656}

X(44793) = X(1)X(391)∩X(8)X(35671)

Barycentrics    -a^7 - 9 a^6 (b + c) + (b + c)^7 + 3 a (b + c)^4 (3 b^2 - 2 b c + 3 c^2) + a^3 (b + c)^2 (13 b^2 - 22 b c + 13 c^2) - a^5 (21 b^2 + 34 b c + 21 c^2) + a^4 (-13 b^3 + 25 b^2 c + 25 b c^2 - 13 c^3) + a^2 (21 b^5 - 23 b^4 c - 46 b^3 c^2 - 46 b^2 c^3 - 23 b c^4 + 21 c^5) : :

See Francisco Javier García Capitán euclid 2327.

X(44793) lies on this line: {1, 391}, {8, 35671}

X(44794) = X(1)X(3523)∩X(57)X(2347)

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

See Francisco Javier García Capitán euclid 2327.

X(44794) lies on these lines: {1, 3523}, {2, 39126}, {7, 8056}, {57, 2347}, {88, 21454}, {145, 40420}, {226, 39963}, {279, 3752}, {346, 32017}, {955, 11227}, {1022, 30723}, {1422, 37666}, {3911, 25430}, {34056, 37642}

X(44795) = X(4)X(94)∩X(24)X(125)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^12-3*a^10*(b^2+c^2)+(b^2-c^2)^4*(b^2+c^2)^2+a^8*(3*b^4+5*b^2*c^2+3*c^4)-2*a^6*(b^6+b^4*c^2+b^2*c^4+c^6)+a^4*(3*b^8-b^6*c^2-2*b^4*c^4-b^2*c^6+3*c^8)-3*a^2*(b^10-b^8*c^2-b^2*c^8+c^10)) : :
Barycentrics    SB SC (18 R^4-3 R^2 SA+SA^2-17 R^2 SW+3 SW^2) : :
X(44795) = X(265)+2*X(34514), 4*X(7687)-X(11456)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2332.

X(44795) lies on these lines: {4,94}, {24,125}, {25,38724}, {67,6403}, {74,6145}, {110,1594}, {113,7547}, {186,15061}, {235,11801}, {378,12827}, {399,7507}, {403,1503}, {427,15463}, {542,39588}, {974,11457}, {1511,37119}, {1593,12902}, {1870,12903}, {1974,20301}, {2777,11550}, {2904,10111}, {3043,23236}, {3047,12419}, {3518,15027}, {3520,6288}, {3541,12383}, {3542,15081}, {3574,12227}, {3575,10264}, {3818,32250}, {5094,32609}, {5504,14516}, {5576,12228}, {6143,38794}, {6152,33565}, {6198,12904}, {6639,20773}, {6699,32534}, {6800,23515}, {7505,20304}, {7544,9826}, {7576,9140}, {7577,14643}, {7687,11456}, {7723,18569}, {9927,12295}, {9934,16659}, {10018,15059}, {10295,15055}, {10539,33547}, {10594,36253}, {10620,12173}, {10733,15062}, {11473,35835}, {11474,35834}, {11564,15321}, {11572,19506}, {11744,12290}, {11746,18912}, {11818,16222}, {12041,35471}, {12162,19479}, {12167,32306}, {12294,32273}, {12300,36853}, {12358,37444}, {13198,32379}, {13211,41722}, {13619,38788}, {14216,17854}, {14591,44529}, {14864,17855}, {15035,37118}, {15041,37196}, {15472,15559}, {16163,35477}, {18386,38789}, {18400,32607}, {18559,20126}, {20127,34797}, {20300,38851}, {20397,35479}, {21243,22109}, {21844,38728}, {22584,31724}, {35472,38727}, {35473,38723}, {37932,44407}

X(44795) = reflection of X(i) in X(j) for these (i,j): {6800,23515}, {15463,427}, {22109,21243}
X(44795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4,3448,1986}, {125,12140,24}, {10113,12133,4}, {11572,21650,19506}, {12134,23306,110}

X(44796) = X(382)X(10606)∩X(548)X(44108)

Barycentrics    (-6 a^4+5 a^2 b^2+b^4+5 a^2 c^2-2 b^2 c^2+c^4) (6 a^6-3 a^4 b^2-12 a^2 b^4+9 b^6-3 a^4 c^2+22 a^2 b^2 c^2-9 b^4 c^2-12 a^2 c^4-9 b^2 c^4+9 c^6) : :
X(44796) = 5*X(7666)-9*X(15688)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2341.

X(44796) lies on these lines: {382, 10606}, {548, 44108}, {3528, 4549}, {7666, 15688}

X(44797) = X(7)X(312)∩X(75)X(2297)

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

See Francisco Javier García Capitán euclid 2349.

X(44797) lies on these lines: {7,312}, {75,2297}, {894,7050}, {1219,4461}, {3729,7091}

X(44798) = X(2)X(37)∩X(9)X(165)

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

See Francisco Javier García Capitán euclid 2349.

X(44798) lies on these lines: {1,4515}, {2,37}, {6,200}, {8,40133}, {9,165}, {10,1146}, {39,19868}, {44,4386}, {65,33299}, {72,16549}, {141,1418}, {169,9709}, {210,672}, {220,936}, {241,29611}, {292,1222}, {354,3930}, {392,1018}, {474,17742}, {518,17754}, {594,1108}, {728,8583}, {960,3501}, {997,6603}, {1100,3870}, {1107,5793}, {1125,3991}, {1155,5282}, {1319,4390}, {1334,25917}, {1536,12618}, {1574,16583}, {1593,7079}, {1698,16601}, {1766,19541}, {2262,22278}, {2280,3689}, {2321,11019}, {3057,39244}, {3061,5836}, {3242,39959}, {3452,17747}, {3555,4006}, {3617,4875}, {3679,43065}, {3683,41423}, {3684,36528}, {3691,3983}, {3694,5750}, {3697,16552}, {3698,17451}, {3723,4666}, {3729,24352}, {3730,5044}, {3744,33854}, {3820,5179}, {3935,16666}, {3965,5749}, {3970,5439}, {4187,21073}, {4253,34790}, {4361,24600}, {4363,40719}, {4413,40131}, {4513,19861}, {4533,17746}, {4662,21384}, {4670,14828}, {4712,20331}, {4851,14548}, {4891,10580}, {5231,8609}, {5440,16788}, {5743,21856}, {5835,20691}, {6745,17369}, {6762,7323}, {7484,40181}, {8301,15254}, {8582,21049}, {9623,34522}, {9780,25082}, {9843,21096}, {10025,17351}, {10582,16777}, {16284,27340}, {16728,30966}, {16851,31448}, {16970,37679}, {17233,31038}, {17267,30813}, {17299,36845}, {17308,25083}, {17345,40868}, {17355,20103}, {17606,21029}, {18228,41325}, {20672,30618}, {20719,34434}, {21896,41015}, {24047,31445}, {24982,40997}, {25244,26563}, {28594,37592}, {29604,37597}

X(44799) = REFLECTION OF X(21310) IN X(6771)

Barycentrics    Sqrt[3]*(2*a^12*b^2 - 9*a^10*b^4 + 15*a^8*b^6 - 10*a^6*b^8 + 3*a^2*b^12 - b^14 + 2*a^12*c^2 - 21*a^10*b^2*c^2 + 66*a^8*b^4*c^2 - 49*a^6*b^6*c^2 + 3*a^4*b^8*c^2 - 6*a^2*b^10*c^2 + 5*b^12*c^2 - 9*a^10*c^4 + 66*a^8*b^2*c^4 - 42*a^6*b^4*c^4 - 27*a^4*b^6*c^4 - 3*a^2*b^8*c^4 - 9*b^10*c^4 + 15*a^8*c^6 - 49*a^6*b^2*c^6 - 27*a^4*b^4*c^6 + 12*a^2*b^6*c^6 + 5*b^8*c^6 - 10*a^6*c^8 + 3*a^4*b^2*c^8 - 3*a^2*b^4*c^8 + 5*b^6*c^8 - 6*a^2*b^2*c^10 - 9*b^4*c^10 + 3*a^2*c^12 + 5*b^2*c^12 - c^14) + 2*(2*a^12 - 6*a^10*b^2 + 3*a^8*b^4 + 8*a^6*b^6 - 12*a^4*b^8 + 6*a^2*b^10 - b^12 - 6*a^10*c^2 + 39*a^8*b^2*c^2 + 6*a^6*b^4*c^2 - 27*a^4*b^6*c^2 - 18*a^2*b^8*c^2 + 6*b^10*c^2 + 3*a^8*c^4 + 6*a^6*b^2*c^4 - 66*a^4*b^4*c^4 + 12*a^2*b^6*c^4 - 15*b^8*c^4 + 8*a^6*c^6 - 27*a^4*b^2*c^6 + 12*a^2*b^4*c^6 + 20*b^6*c^6 - 12*a^4*c^8 - 18*a^2*b^2*c^8 - 15*b^4*c^8 + 6*a^2*c^10 + 6*b^2*c^10 - c^12)*S : :

See Elias M. Hagos and Peter Moses, euclid 2353.

X(44799) lies on this line: {3, 13}

X(44799) = reflection of X(21310) in X(6771)
X(44799) = {X(5473),X(21156)}-harmonic conjugate of X(6104)

X(44800) = REFLECTION OF X(21311) IN X(6774)

Barycentrics    Sqrt[3]*(2*a^12*b^2 - 9*a^10*b^4 + 15*a^8*b^6 - 10*a^6*b^8 + 3*a^2*b^12 - b^14 + 2*a^12*c^2 - 21*a^10*b^2*c^2 + 66*a^8*b^4*c^2 - 49*a^6*b^6*c^2 + 3*a^4*b^8*c^2 - 6*a^2*b^10*c^2 + 5*b^12*c^2 - 9*a^10*c^4 + 66*a^8*b^2*c^4 - 42*a^6*b^4*c^4 - 27*a^4*b^6*c^4 - 3*a^2*b^8*c^4 - 9*b^10*c^4 + 15*a^8*c^6 - 49*a^6*b^2*c^6 - 27*a^4*b^4*c^6 + 12*a^2*b^6*c^6 + 5*b^8*c^6 - 10*a^6*c^8 + 3*a^4*b^2*c^8 - 3*a^2*b^4*c^8 + 5*b^6*c^8 - 6*a^2*b^2*c^10 - 9*b^4*c^10 + 3*a^2*c^12 + 5*b^2*c^12 - c^14) - 2*(2*a^12 - 6*a^10*b^2 + 3*a^8*b^4 + 8*a^6*b^6 - 12*a^4*b^8 + 6*a^2*b^10 - b^12 - 6*a^10*c^2 + 39*a^8*b^2*c^2 + 6*a^6*b^4*c^2 - 27*a^4*b^6*c^2 - 18*a^2*b^8*c^2 + 6*b^10*c^2 + 3*a^8*c^4 + 6*a^6*b^2*c^4 - 66*a^4*b^4*c^4 + 12*a^2*b^6*c^4 - 15*b^8*c^4 + 8*a^6*c^6 - 27*a^4*b^2*c^6 + 12*a^2*b^4*c^6 + 20*b^6*c^6 - 12*a^4*c^8 - 18*a^2*b^2*c^8 - 15*b^4*c^8 + 6*a^2*c^10 + 6*b^2*c^10 - c^12)*S : :

See Elias M. Hagos and Peter Moses, euclid 2362.

X(44800) lies on this line: {3, 14}

X(44800) = reflection of X(21311) in X(6774)
X(44800) = {X(5474),X(21157)}-harmonic conjugate of X(6105)

X(44801) = X(6)X(3091)∩X(3532)X(3832)

Barycentrics    18*a^10 + 3*a^8*b^2 - 72*a^6*b^4 + 18*a^4*b^6 + 78*a^2*b^8 - 45*b^10 + 3*a^8*c^2 + 80*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 216*a^2*b^6*c^2 + 135*b^8*c^2 - 72*a^6*c^4 - 2*a^4*b^2*c^4 + 276*a^2*b^4*c^4 - 90*b^6*c^4 + 18*a^4*c^6 - 216*a^2*b^2*c^6 - 90*b^4*c^6 + 78*a^2*c^8 + 135*b^2*c^8 - 45*c^10 : :
X(44801) = 9 X[2] - X[44788], 3 X[381] + X[43592], 15 X[3091] + X[15077], 35 X[3091] - 3 X[32605], X[3532] + 7 X[3832], 19 X[15022] - 3 X[27082], 7 X[15077] + 9 X[32605]

See Antreas Hatzipolakis and Peter Moses, euclid 2382.

X(44801) lies on these lines: {2, 44788}, {6, 3091}, {381, 43592}, {3532, 3832}, {15022, 27082}

X(44802) = 64TH HATZIPOLAKIS-MOSES-EULER POINT

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

As a point on the Euler line, X(44802) has Shinagawa coefficients (E-4*F,2*E+4*F).

See Antreas Hatzipolakis and Peter Moses, euclid 2383.

X(44802) lies on these lines: {2, 3}, {6, 9545}, {32, 15355}, {49, 1199}, {51, 34148}, {52, 323}, {54, 5462}, {96, 11815}, {110, 389}, {143, 22115}, {156, 15032}, {161, 18928}, {182, 15028}, {184, 15043}, {185, 15053}, {195, 16881}, {390, 10046}, {511, 43811}, {567, 15026}, {569, 11464}, {575, 6467}, {578, 5640}, {1092, 3060}, {1125, 9590}, {1147, 1994}, {1173, 5504}, {1181, 35264}, {1204, 15305}, {1493, 11597}, {1495, 9729}, {1511, 10095}, {1588, 9682}, {1614, 9730}, {2079, 7745}, {2393, 43815}, {2888, 3580}, {2917, 37649}, {2929, 13568}, {2931, 9820}, {3043, 16222}, {3047, 20771}, {3066, 11425}, {3284, 8882}, {3292, 15801}, {3357, 11439}, {3410, 12359}, {3448, 12134}, {3574, 5972}, {3581, 11591}, {3600, 10037}, {3618, 15577}, {3620, 37488}, {3815, 44523}, {4993, 19185}, {5012, 10282}, {5254, 44537}, {5286, 44527}, {5422, 19357}, {5446, 38848}, {5449, 41171}, {5609, 11561}, {5622, 43130}, {5651, 11444}, {5731, 8185}, {5866, 32819}, {5889, 9306}, {5890, 10539}, {5907, 43614}, {5926, 10280}, {5943, 13367}, {5944, 13353}, {6000, 43601}, {6102, 18350}, {6146, 43816}, {6193, 37644}, {6403, 8538}, {6684, 9625}, {6696, 10117}, {6699, 18488}, {6759, 10574}, {6800, 37514}, {7585, 8276}, {7586, 8277}, {7592, 9544}, {7689, 15058}, {7691, 11793}, {7735, 9608}, {7738, 44524}, {7999, 37478}, {8907, 11427}, {9538, 11399}, {9541, 35777}, {9591, 10164}, {9626, 10165}, {9659, 10588}, {9672, 10589}, {9706, 13366}, {9707, 11003}, {9713, 30478}, {9723, 32835}, {9780, 15177}, {9781, 13352}, {9786, 11441}, {9833, 18911}, {9924, 15582}, {10314, 26216}, {10316, 10986}, {10540, 13630}, {10541, 35707}, {10545, 11430}, {10546, 11438}, {10601, 17821}, {10610, 32205}, {10982, 38942}, {10984, 26881}, {10985, 22401}, {11002, 36747}, {11004, 37493}, {11064, 11745}, {11202, 11451}, {11264, 23236}, {11440, 15030}, {11468, 16261}, {11477, 20806}, {11562, 14094}, {12038, 15033}, {12112, 13491}, {12278, 18390}, {12282, 41619}, {12300, 12358}, {12370, 12383}, {12584, 22330}, {13142, 22550}, {13289, 15059}, {13346, 34417}, {13347, 35268}, {13364, 43394}, {13392, 22051}, {13445, 13474}, {13472, 15317}, {13567, 14516}, {13598, 44106}, {13754, 43598}, {14061, 39854}, {14157, 40647}, {14627, 32609}, {14652, 34837}, {14671, 14675}, {14683, 18932}, {14915, 43804}, {15019, 37505}, {15045, 26882}, {15049, 43610}, {15054, 21650}, {15062, 21663}, {15066, 17834}, {15068, 37490}, {15072, 26883}, {15080, 37515}, {15107, 15644}, {15139, 41589}, {15531, 44489}, {15647, 32184}, {16194, 43604}, {17701, 41671}, {17704, 32237}, {17810, 35602}, {18381, 26913}, {18474, 26917}, {18475, 43651}, {18912, 34799}, {18931, 40914}, {19459, 33748}, {20190, 43129}, {20304, 22804}, {20791, 44082}, {20987, 25406}, {22800, 34563}, {22948, 44573}, {25739, 43817}, {30482, 30504}, {30522, 43821}, {31834, 32608}, {33884, 37486}, {34782, 37648}, {34835, 40604}

X(44802) = midpoint of X(13621) and X(43809)
X(44802) = reflection of X(34484) in X(13621)
X(44802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24, 7488), (2, 7487, 37444), (2, 7488, 37126), (2, 31304, 6643), (3, 4, 12086), (3, 5, 35500), (3, 378, 35494), (3, 546, 14865), (3, 1995, 3091), (3, 3091, 7527), (3, 3518, 23), (3, 3525, 7496), (3, 3627, 7464), (3, 3628, 7550), (3, 7506, 13861), (3, 7530, 3529), (3, 7545, 3627), (3, 10594, 3146), (3, 12086, 2071), (3, 12088, 16661), (3, 12106, 3518), (3, 13861, 4), (3, 18369, 546), (3, 31861, 35475), (3, 35494, 35497), (3, 35500, 14118), (3, 37440, 12088), (3, 37924, 12103), (4, 631, 44441), (4, 3548, 31074), (4, 6644, 22467), (4, 7506, 13595), (4, 22467, 2071), (5, 186, 14118), (5, 10018, 2), (5, 16532, 140), (5, 34330, 1656), (5, 38321, 4), (5, 44211, 10018), (23, 16661, 12088), (24, 6642, 2), (24, 7488, 37940), (24, 7509, 14070), (24, 10018, 186), (25, 17928, 20), (26, 631, 6636), (49, 5946, 1199), (54, 5462, 34545), (110, 16223, 40640), (140, 2070, 7512), (140, 7512, 15246), (140, 13163, 33332), (156, 37481, 15032), (186, 14118, 38448), (186, 35500, 3), (376, 7517, 12087), (378, 7529, 3832), (381, 37814, 3520), (403, 31833, 34007), (468, 9825, 13160), (550, 18378, 37925), (569, 15024, 15018), (631, 35482, 23336), (632, 12107, 3), (1147, 3567, 1994), (1511, 10095, 37472), (1594, 16238, 2), (1598, 11413, 3543), (1656, 1658, 35921), (2070, 16532, 186), (2072, 31830, 4), (3146, 14002, 10594), (3147, 7401, 2), (3147, 14940, 10018), (3292, 16625, 15801), (3515, 5020, 7503), (3515, 7503, 10298), (3518, 12088, 37440), (3520, 37814, 37941), (3525, 7556, 3), (3530, 37936, 13564), (3545, 21844, 7526), (3546, 37122, 7391), (3627, 7545, 26863), (3628, 7575, 3), (3843, 11250, 13596), (3850, 15646, 14130), (5004, 5005, 7667), (5020, 7503, 5056), (5056, 10298, 7503), (5576, 44452, 6143), (5640, 11449, 578), (5890, 10539, 43605), (5943, 13367, 13434), (5944, 13363, 13353), (6636, 44441, 2071),(6644, 7506, 4),(6644, 13595, 2071),(6644, 13861, 3),(6644, 38321, 186),(7464, 26863, 3627),(7528, 37119, 5169),(7542, 10127, 14788),(7542, 14788, 2), (7542, 44234, 10018), (7555, 14869, 3), (7667, 40132, 7570), (9707, 36752, 11003), (9714, 10323, 37913), (9786, 35259, 11441), (10024, 44232, 37943), (10303, 38435, 3), (10546, 11438, 15052), (11464, 15024, 569), (12086, 22467, 3), (12088, 37440, 23), (12134, 26879, 3448), (13595, 22467, 4), (14130, 21308, 3850), (14157, 43597, 40647), (15026, 32171, 567), (15717, 37913, 10323), (16881, 40111, 195), (21451, 34007, 403), (22462, 34864, 547), (22462, 37922, 34864), (26863, 35500, 18403), (37126, 37940, 7488), (38848, 43574, 5446)

X(44803) = 65TH HATZIPOLAKIS-MOSES-EULER POINT

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

As a point on the Euler line, X(44803) has Shinagawa coefficients (-2*F,5*E-2*F).

See Antreas Hatzipolakis and Peter Moses, euclid 2383.

X(44803) lies on these lines: {2, 3}, {113, 11817}, {133, 11792}, {389, 32111}, {1503, 43812}, {1514, 11745}, {1533, 9729}, {1614, 16657}, {2883, 3567}, {5254, 33885}, {5890, 15873}, {6152, 22970}, {11381, 26879}, {11439, 12359}, {11457, 15811}, {12022, 26883}, {12112, 18914}, {12241, 14157}, {12290, 13567}, {12300, 41598}, {13366, 14862}, {13568, 38848}, {15033, 16252}, {15063, 16625}, {15305, 41587}, {16621, 25739}, {16659, 18390}, {22660, 36852}, {22802, 34417}, {33880, 39565}

X(44803) = midpoint of X(4) and X(34484)
X(44803) = X(34484)-of-Euler-triangle
X(44803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 25, 18560), (4, 186, 13488), (4, 235, 1594), (4, 403, 15559), (4, 1598, 7576), (4, 3089, 378), (4, 3518, 1885), (4, 3542, 35502), (4, 6623, 7547), (4, 7487, 35490), (4, 7505, 1597), (4, 7577, 1907), (4, 10594, 6240), (4, 16868, 1595), (4, 26863, 6756), (4, 37119, 11403), (4, 37122, 35480), (235, 1594, 403), (235, 1595, 16868), (428, 44226, 4), (546, 11799, 13160), (1595, 16868, 1594), (1596, 1906, 4), (1885, 3518, 10295), (3542, 35502, 37118), (3843, 15761, 5133), (3861, 11563, 5576), (7487, 35490, 6240), (10594, 35490, 7487), (11403, 37119, 35484), (13861, 31725, 38323), (18535, 37197, 4)

X(44804) = 66TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^10 - 3*a^8*b^2 - 2*a^6*b^4 + 4*a^4*b^6 - b^10 - 3*a^8*c^2 + 24*a^6*b^2*c^2 - 24*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 + 48*a^2*b^4*c^4 - 2*b^6*c^4 + 4*a^4*c^6 - 24*a^2*b^2*c^6 - 2*b^4*c^6 + 3*b^2*c^8 - c^10 : :

See Antreas Hatzipolakis and Peter Moses, euclid 2385.

X(44804) lies on these lines: {2, 3}, {541, 11746}, {2777, 13570}, {6000, 32068}, {16194, 16657}, {18914, 32137}, {23327, 38136}

leftri

Centers of circumcircle-inverses of lines: X(X(44805)-X(44827)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, September 17, 2021.

The circumcircle-inverse of a line is a circle. (If the line, L, passes through X(3), the circle is has infinite radius; i.e. the inverse is the line L itself.)

Suppose that a line L is the trilinear polar of a point p : q : r. The circumcircle-inverse of L is the circle with center X and A-power given as follows:

X = a^2*(c^2*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4)*p*q + b^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*p*r - a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)*q*r) : :

A-power: a^2*b^2*c^2*q*r / (b^2*(a^2 - b^2 + c^2)*p*r + c^2*(a^2 + b^2 - c^2)*p*q - a^2*(a^2 - b^2 - c^2)*q*r)).

The appearance of {{i,j}, {n1, n2, ..., nk}, m} in the following list means that the circumcircle-inverse of the line X(i)X(j) passes through the points X(n1), X(n2), ..., X(nk) and has center X(m).

{{1,6}, {3,36,187,667,5144,11650,32625,32758},39227}
{{1,21}, {3,36,501,1325,1326,3733,5161,5197,5867},39210}
{{2,6}, {3,23,187,353,669,5866,5867,5937,5938,5939,5940,5941,6031,32531,34014},5926}
{{2,11}, {3,23,100,105,659,1155,5144,9471,14664,14667,19915,19916},44805}
{{2,98}, {3,23,98,110,2080,13558,14652,14673,19165,33900,42667,42668},14270}
{{2,99}, {3,23,99,111,2079,2930,5104,11641,11643,14669,14678,14682,15564,18773,18774,33998},11616}
{{2,112}, {3,23,112,1177,2373,8428,34107,39857},44806}
{{4,11}, {3,104,108,186,1319,11700,11713,14667},44807}
{{4,32}, {3,98,112,186,1691,2079,5621,14671,14675,15462,34131},25644}
{{4,110}, {3,110,186,1300,13557,14674,14703,15470,15478},44808}
{{5,49}, {3,110,1141,1157,2070,5961,8157,14097,14889},44809}
{{6,13}, {3,187,1511,2079,5961,6104,6105,7575,12042,14270,14702,14703,15550,34010,34217},39477}
{{6,74}, {3,74,112,187,3184,9409,10295,10991,34106,34109,38749},44810}
{{6,110}, {3,110,111,187,351,2482,6055,6091,7426,7600,9828,9829,19901,19902},9126}
{{7,21}, {3,1319,1325,3110,4367,5144,5937,32624},44811}
{{10,21}, {3,100,484,759,1054,1324,1325,2948,10260,12778},44812}
{{22,99}, {3,99,126,127,858,2373,5181,35522},44813}
{{22,110}, {3,110,114,122,684,858,1297,14981,18860,34147},8552}
{{23,110}, {2,3,110,842,8724,14649,14685,34291,35911},44814}
{{23,111}, {2,3,6,111,691,5653,9173,9174,9178,11579,11632,11637,11638,14174,14180,14699,14700,15546,15744,21732,21733,35905,36202},9175}
{{23,385}, {2,3,3111,5108,6789,9153,9828,32531,34583},549}
{{24,108}, {3,11,108,119,403,915,18341,18838,39534},44815}
{{24,110}, {3,110,131,136,403,1299,12095,15367},44816}
{{24,112}, {3,112,114,115,403,1692,3563,5622,15745,17994},44817}
{{25,98}, {3,98,107,468,5972,6130,20417,34841},44818}
{{25,105}, {3,105,108,468,676,3035,3660,20418},44819}
{{25,111}, {3,111,112,468,620,2030,2492,6593,11623},44820}
{{30,98}, {3,98,182,691,11643,15567,32305,33695,34010},44821}
{{30,99}, {3,99,842,3098,8723,12584,21395,34217},44822}
{{30,115}, {3,6,2079,5941,12188,16010,33900,38582},44823}
{{35,37}, {3,484,502,4705,5164,5948,30447,32758},44824}
{{36,238}, {1,3,1083,3109,3110,6789,13868,31866},1385}
{{39,83}, {3,99,733,2076,14691,15927,37896,39557},44825}
{{69,74}, {3,74,99,5866,7464,14368,14369,35002,35453},44826}
{{71,74}, {3,74,101,5672,5673,6326,12738,36026},44827}
{{187,237}, {3,6,1083,1316,5091,5108,6141,6142,6232,6322,6795,8429,9129,11650,13414,13415,13511,13515,13516,14685,18332,18338,22740,22742,24279,35901,43765}},182} (the Brocard circle)

X(44805) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(2)X(13)

Barycentrics    a*(b - c)*(2*a^5 - 2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a^4*c + 2*a^3*b*c + a^2*b^2*c - b^4*c - 2*a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 + 2*a^2*c^3 + b^2*c^3 - b*c^4) : :
X(44805) = X[1482] - 3 X[25569], 7 X[3526] - 5 X[30795], 3 X[3576] + X[21385], 3 X[4401] + X[15599], 3 X[10246] - X[21343], X[13266] + 3 X[34474]

X(44805) lies on these lines: {3, 659}, {36, 30725}, {100, 19916}, {105, 19915}, {140, 3837}, {514, 39227}, {517, 1960}, {523, 5926}, {667, 28473}, {891, 1385}, {900, 33814}, {1482, 25569}, {1483, 25574}, {2775, 38324}, {2804, 39200}, {2812, 35100}, {2821, 3579}, {3309, 4394}, {3526, 30795}, {3576, 21385}, {3887, 38327}, {4925, 25440}, {4977, 39210}, {6084, 38603}, {6550, 23961}, {8648, 10015}, {10246, 21343}, {13266, 34474}, {16408, 25926}, {19513, 27675}, {19921, 38572}, {26275, 42670}

X(44805) = midpoint of X(i) and X(j) for these {i,j}: {3, 659}, {100, 19916}, {105, 19915}, {19921, 38572}
X(44805) = reflection of X(3837) in X(140)

X(44806) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(2)X(112)

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

X(44806) lies on these lines: {523, 5926}, {2799, 39854}, {2881, 25644}, {9033, 15577}, {16230, 19165}, {37814, 44205}

X(44807) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(4)X(11)

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

X(44807) lies on these lines: {3, 2804}, {513, 17099}, {523, 15646}, {900, 38602}, {1385, 8677}, {2826, 39200}, {4188, 23678}, {6087, 38606}, {23961, 35013}

X(44808) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(4)X(110)

Barycentrics    a^4*(b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4) : :
X(44808) = X[8562] - X[14314]

X(44808) lies on these lines: {3, 40047}, {186, 14222}, {523, 15646}, {526, 1511}, {924, 12095}, {5961, 43088}, {6132, 15647}, {8553, 14273}, {8675, 39201}, {9033, 12893}, {11616, 37813}, {34291, 42659}, {36790, 38354}

X(44808) = midpoint of X(186) and X(15470)
X(44808) = crosssum of X(265) and X(10412)
X(44808) = crossdifference of every pair of points on line {1989, 2165}
X(44808) = X(i)-isoconjugate of X(j) for these (i,j): {68, 36129}, {91, 476}, {94, 36145}, {847, 36061}, {925, 2166}, {2165, 32680}, {5392, 32678}, {14560, 20571}
X(44808) = barycentric product X(i)*X(j) for these {i,j}: {24, 8552}, {47, 32679}, {50, 6563}, {323, 924}, {340, 30451}, {526, 1993}, {571, 3268}, {1147, 44427}, {2624, 44179}, {7763, 14270}, {7799, 34952}, {16186, 41679}, {34948, 42701}
X(44808) = barycentric quotient X(i)/X(j) for these {i,j}: {47, 32680}, {50, 925}, {186, 30450}, {526, 5392}, {563, 36061}, {571, 476}, {924, 94}, {1993, 35139}, {2624, 91}, {6563, 20573}, {6753, 6344}, {8552, 20563}, {14270, 2165}, {14397, 14254}, {19627, 32734}, {30451, 265}, {32679, 20571}, {34952, 1989}, {39013, 43088}

X(44809) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(5)X(49)

Barycentrics    a^4*(b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4) : :
X(44809) = X[14809] - 4 X[39477]

X(44809) lies on these lines: {186, 523}, {526, 1511}, {900, 6097}, {1510, 6150}, {6132, 42659}, {6140, 12011}, {13558, 14656}, {14385, 38897}, {14809, 39477}, {25149, 38618}

X(44809) = crosspoint of X(54) and X(1291)
X(44809) = crosssum of X(6368) and X(16336)
X(44809) = crossdifference of every pair of points on line {216, 1989}
X(44809) = X(i)-isoconjugate of X(j) for these (i,j): {93, 36061}, {94, 36148}, {476, 2962}, {930, 2166}, {2963, 32680}, {3519, 36129}, {11140, 32678}
X(44809) = barycentric product X(i)*X(j) for these {i,j}: {49, 44427}, {50, 41298}, {323, 1510}, {526, 1994}, {2964, 32679}, {2965, 3268}, {3518, 8552}, {7769, 14270}, {14165, 37084}, {25044, 41078}
X(44809) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 930}, {186, 38342}, {526, 11140}, {1510, 94}, {1994, 35139}, {2081, 25043}, {2624, 2962}, {2964, 32680}, {2965, 476}, {14270, 2963}, {19627, 32737}, {41298, 20573}, {44427, 20572}

X(44810) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(6)X(74)

Barycentrics    a^2*(b^2 - c^2)*(a^2 - b^2 - c^2)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 3*b^2*c^2 - c^4) : :
X(44810) = 3 X[3] - X[684], 2 X[684] - 3 X[8552], X[684] + 3 X[9409], 2 X[6132] - 3 X[9126], X[8552] + 2 X[9409], 3 X[9126] - 4 X[39477], X[16230] - 3 X[44202], 3 X[25644] - X[33752]

X(44810) lies on these lines: {3, 684}, {20, 41079}, {30, 6130}, {512, 5926}, {523, 44205}, {525, 42658}, {526, 12041}, {550, 2797}, {691, 23969}, {2799, 38747}, {2881, 25644}, {3804, 9125}, {9033, 38726}, {11615, 14270}, {16111, 32119}, {16230, 44202}, {20188, 37084}, {31953, 38741}

X(44810) = midpoint of X(i) and X(j) for these {i,j}: {3, 9409}, {20, 41079}, {16111, 32119}, {31953, 38741}
X(44810) = reflection of X(i) in X(j) for these {i,j}: {6132, 39477}, {8552, 3}, {11615, 14270}
X(44810) = crossdifference of every pair of points on line {6103, 37637}
X(44810) = barycentric product X(525)*X(35265)
X(44810) = barycentric quotient X(35265)/X(648)
X(44810) = {X(6132),X(39477)}-harmonic conjugate of X(9126)

X(44811) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(7)X(21)

Barycentrics    a*(b - c)*(2*a^5 - 2*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a^4*c + 4*a^3*b*c - a^2*b^2*c - 2*a*b^3*c + b^4*c - 2*a^3*c^2 - a^2*b*c^2 + 2*a*b^2*c^2 - b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 - b^2*c^3 + b*c^4) : :
X(44811) = X[1019] + 3 X[3576], X[4129] - 3 X[10165], X[4879] - 3 X[10246], 3 X[14419] - X[39212]

X(44811) lies on these lines: {3, 4367}, {36, 7178}, {56, 34958}, {104, 2752}, {140, 21051}, {182, 9040}, {512, 1385}, {513, 17099}, {514, 39227}, {523, 39210}, {667, 2826}, {1019, 3576}, {2646, 44410}, {4129, 10165}, {4807, 5882}, {4879, 10246}, {4977, 39225}, {5258, 44729}, {10269, 29126}, {14419, 39212}, {24929, 39541}, {37561, 39577}

X(44811) = midpoint of X(i) and X(j) for these {i,j}: {3, 4367}, {4807, 5882}
X(44811) = reflection of X(21051) in X(140)

X(44812) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(10)X(21)

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

X(44812) lies on these lines: {36, 30572}, {40, 2776}, {513, 10225}, {522, 39200}, {523, 39210}, {572, 21832}, {659, 2827}, {900, 33814}, {1385, 4145}, {3667, 4491}, {3738, 38327}, {4777, 39227}, {6089, 38612}

X(44813) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(22)X(99)

Barycentrics    (b^2 - c^2)*(a^10 - a^8*b^2 - a^6*b^4 + a^4*b^6 - a^8*c^2 - 3*a^6*b^2*c^2 + 8*a^4*b^4*c^2 - 5*a^2*b^6*c^2 + b^8*c^2 - a^6*c^4 + 8*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - b^6*c^4 + a^4*c^6 - 5*a^2*b^2*c^6 - b^4*c^6 + b^2*c^8) : :
X(44813) = l3 X[11176] - 2 X[11620]

X(44813) lies on these lines: {3, 35522}, {140, 2492}, {182, 9035}, {511, 24284}, {523, 7623}, {804, 11616}, {2793, 6131}, {2797, 18312}, {2799, 6036}, {11176, 11620}, {19510, 30209}

X(44813) = midpoint of X(3) and X(35522)
X(44813) = reflection of X(2492) in X(140)
X(44813) = X(i)-complementary conjugate of X(j) for these (i,j): {293, 14672}, {36084, 8542}

X(44814) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(23)X(110)

Barycentrics    a^2*(b^2 - c^2)*(2*a^2 - b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2) : :
X(44814) = 4 X[140] - X[23105], X[684] + 2 X[39477], 3 X[5055] - 2 X[39482], 2 X[8552] + X[14270]

X(44814) lies on these lines: {3, 512}, {32, 10567}, {39, 6041}, {140, 23105}, {182, 8675}, {351, 690}, {523, 549}, {526, 1511}, {543, 39528}, {574, 647}, {684, 39477}, {888, 7638}, {924, 11202}, {1116, 8029}, {2088, 16186}, {2780, 6132}, {2799, 11621}, {3906, 11171}, {5013, 8574}, {5055, 39482}, {5118, 15329}, {5664, 31378}, {7622, 23878}, {9137, 9168}, {9178, 37742}, {9734, 30209}, {14096, 17414}, {14355, 14385}, {14649, 35911}, {14685, 42654}, {20403, 33752}

X(44814) = midpoint of X(i) and X(j) for these {i,j}: {3, 34291}, {8552, 9126}, {14649, 35911}
X(44814) = reflection of X(i) in X(j) for these {i,j}: {8029, 1116}, {9178, 37742}, {14270, 9126}
X(44814) = isogonal conjugate of isotomic conjugate of X(45808)
X(44814) = tripolar centroid of X(323)
X(44814) = X(9213)-Ceva conjugate of X(526)
X(44814) = crosspoint of X(i) and X(j) for these (i,j): {526, 9213}, {5467, 9717}
X(44814) = crosssum of X(i) and X(j) for these (i,j): {476, 14559}, {523, 9140}, {5466, 9214}
X(44814) = crossdifference of every pair of points on line {111, 230}
X(44814) = X(i)-isoconjugate of X(j) for these (i,j): {94, 36142}, {111, 32680}, {476, 897}, {671, 32678}, {691, 2166}, {895, 36129}, {923, 35139}, {1989, 36085}, {16092, 36096}, {17983, 36061}, {23894, 39295}
X(44814) = barycentric product X(i)*X(j) for these {i,j}: {15, 9205}, {16, 9204}, {50, 35522}, {186, 14417}, {187, 3268}, {323, 690}, {351, 7799}, {468, 8552}, {524, 526}, {896, 32679}, {1648, 10411}, {2088, 5468}, {2482, 9213}, {2624, 14210}, {3266, 14270}, {3292, 44427}, {4235, 16186}, {5664, 9717}, {14419, 42701}
X(44814) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 691}, {187, 476}, {323, 892}, {351, 1989}, {524, 35139}, {526, 671}, {690, 94}, {896, 32680}, {922, 32678}, {1648, 10412}, {1649, 43084}, {2088, 5466}, {2624, 897}, {2642, 2166}, {3268, 18023}, {5467, 39295}, {6137, 36310}, {6138, 36307}, {6149, 36085}, {8552, 30786}, {9204, 301}, {9205, 300}, {9717, 39290}, {14270, 111}, {14273, 6344}, {14417, 328}, {14567, 14560}, {16186, 14977}, {18334, 9213}, {19627, 32729}, {21906, 15475}, {23200, 32662}, {34394, 9207}, {34395, 9206}, {35522, 20573}, {39689, 14559}

X(44815) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(24)X(108)

Barycentrics    (a - b - c)*(b - c)*(2*a^6 + a^5*b - 3*a^4*b^2 - 2*a^3*b^3 + a*b^5 + b^6 + a^5*c + 2*a^4*b*c - a*b^4*c - 2*b^5*c - 3*a^4*c^2 + 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^3*c^3 + 4*b^3*c^3 - a*b*c^4 - b^2*c^4 + a*c^5 - 2*b*c^5 + c^6) : :
X(44815) = 3 X[2] + X[44428], X[39478] + 3 X[39493]

X(44815) lies on these lines: {2, 44428}, {3, 39534}, {5, 900}, {513, 18856}, {521, 14837}, {523, 44452}, {676, 2804}, {5020, 26275}, {6087, 38606}, {6642, 39200}, {6644, 39478}, {7392, 44433}

X(44815) = midpoint of X(3) and X(39534)
X(44815) = X(i)-complementary conjugate of X(j) for these (i,j): {34, 15608}, {109, 42423}, {913, 26932}, {915, 124}, {6099, 34823}, {32655, 16596}, {32698, 3452}, {36052, 123}, {36106, 1329}
X(44815) = crossdifference of every pair of points on line {3197, 8553}
X(44815) = X(44929)-of-Johnson-triangle

X(44816) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(24)X(110)

Barycentrics    a^2*(b^2 - c^2)*(a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) : :
X(44816) = X[8552] - 4 X[8562]

X(44816) lies on these lines: {523, 44452}, {526, 1511}, {924, 44680}, {5972, 6132}, {34291, 42665}

X(44816) = X(44816)-complementary conjugate of X(21253)
X(44816) = crosspoint of X(10411) and X(14165)
X(44816) = crosssum of X(265) and X(43088)
X(44816) = crossdifference of every pair of points on line {1989, 2079}
X(44816) = X(i)-isoconjugate of X(j) for these (i,j): {254, 36061}, {476, 921}, {2166, 13398}, {6504, 32678}, {15316, 36129}
X(44816) = barycentric product X(i)*X(j) for these {i,j}: {155, 44427}, {526, 6515}, {920, 32679}, {1609, 3268}, {2624, 33808}, {3542, 8552}, {8883, 41078}
X(44816) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 13398}, {526, 6504}, {920, 32680}, {1609, 476}, {2081, 8800}, {2624, 921}, {6515, 35139}

X(44817) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(24)X(112)

Barycentrics    a^2*(b^2 - c^2)*(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 - 4*b^4*c^4 + 2*a^2*c^6 + b^2*c^6 - c^8) : :
X(44817) = 2 X[2492] + X[8552], X[6132] + 2 X[37742]

X(44817) lies on these lines: {3, 17994}, {5, 804}, {351, 5020}, {394, 14397}, {523, 44452}, {525, 2485}, {620, 2492}, {647, 14341}, {686, 10601}, {2510, 7656}, {2780, 9818}, {2881, 25644}, {3566, 7631}, {6036, 6132}, {6644, 9126}, {6677, 11176}, {7392, 9147}, {9023, 9813}, {9517, 15462}, {15646, 20403}, {16040, 30476}, {18537, 19912}

X(44817) = midpoint of X(i) and X(j) for these {i,j}: {3, 17994}, {6132, 7663}, {25644, 33752}
X(44817) = reflection of X(7663) in X(37742)
X(44817) = complement of the isotomic conjugate of X(32697)
X(44817) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 31842}, {1973, 36472}, {3563, 21253}, {32654, 34846}, {32697, 2887}, {36051, 127}, {36105, 626}
X(44817) = crosspoint of X(2) and X(32697)
X(44817) = crossdifference of every pair of points on line {159, 7669}
X(44817) = {X(2485),X(2507)}-harmonic conjugate of X(2508)

X(44818) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(25)X(98)

Barycentrics    (b^2 - c^2)*(3*a^8 - 6*a^6*b^2 + 3*a^4*b^4 - 6*a^6*c^2 + 7*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + b^6*c^2 + 3*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + b^2*c^6) : :
X(48818) = 3 X[2] + X[9409], 3 X[3] + X[41079], 3 X[549] - X[8552], 5 X[631] - X[684], X[850] + 3 X[39201], 3 X[6130] - X[41079], X[6334] + 3 X[44202], 3 X[11176] - X[21731], X[18312] + 3 X[25644], 5 X[31277] + 3 X[42658], X[31953] + 3 X[34473], X[32119] + 3 X[38727]

X(44818) lies on these lines: {2, 9409}, {3, 2797}, {140, 9517}, {512, 31286}, {526, 6699}, {549, 8552}, {631, 684}, {804, 12042}, {850, 39201}, {878, 31635}, {2881, 6720}, {5961, 10412}, {6086, 38605}, {6334, 44202}, {11176, 21731}, {31277, 42658}, {31953, 34473}, {32119, 38727}

X(44818) = midpoint of X(3) and X(6130)

X(44819) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(25)X(105)

Barycentrics    (b - c)*(6*a^6 - 7*a^5*b - 5*a^4*b^2 + 6*a^3*b^3 + a*b^5 - b^6 - 7*a^5*c + 10*a^4*b*c + 2*a^3*b^2*c - 4*a^2*b^3*c - 3*a*b^4*c + 2*b^5*c - 5*a^4*c^2 + 2*a^3*b*c^2 + 2*a*b^3*c^2 + b^4*c^2 + 6*a^3*c^3 - 4*a^2*b*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - 3*a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6) : :
X(44819) = 3 X[3576] + X[10015], X[4528] - 3 X[26446], X[14110] + 3 X[30691]

X(44819) lies on these lines: {3, 676}, {900, 6713}, {928, 9940}, {1385, 6366}, {2487, 3309}, {3576, 10015}, {4528, 26446}, {6084, 38603}, {6087, 38606}, {14110, 30691}, {29162, 39227}

X(44819) = midpoint of X(3) and X(676)

X(44820) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(25)X(111)

Barycentrics    a^2*(b^2 - c^2)*(3*a^8 - 4*a^6*b^2 - 2*a^4*b^4 + 4*a^2*b^6 - b^8 - 4*a^6*c^2 - 3*a^4*b^2*c^2 + 6*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 + 6*a^2*b^2*c^4 - 8*b^4*c^4 + 4*a^2*c^6 + b^2*c^6 - c^8) : :
X(44820) = 5 X[631] - X[35522], X[1350] + 3 X[14398], X[3569] + 3 X[5085], 3 X[9126] - X[11616], 3 X[9175] + X[11616], 7 X[10541] + X[39232], X[11620] - 3 X[11621], 3 X[25644] + X[33752]

X(44820) lies on these lines: {3, 2492}, {182, 526}, {631, 35522}, {804, 6036}, {1350, 14398}, {2780, 11620}, {2881, 25644}, {3569, 5085}, {6088, 9126}, {10541, 39232}, {32472, 39511}, {37742, 39477}

X(44820) = midpoint of X(i) and X(j) for these {i,j}: {3, 2492}, {9126, 9175}, {37742, 39477}

X(44821) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(30)X(98)

Barycentrics    a^2*(b^2 - c^2)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - a^6*c^2 + 2*a^2*b^4*c^2 - a^4*c^4 + 2*a^2*b^2*c^4 - 3*b^4*c^4 + a^2*c^6) : :
X(48821) = X[8723] - 3 X[17508], X[14270] - 3 X[25644]

X(44821) lies on these lines: {2, 669}, {3, 523}, {22, 10278}, {23, 8371}, {25, 10189}, {157, 18310}, {512, 5092}, {525, 8177}, {804, 12042}, {1649, 7496}, {2407, 41337}, {2451, 44453}, {2793, 39477}, {3049, 39560}, {5466, 7492}, {6636, 8029}, {7485, 10190}, {7525, 10279}, {8723, 17508}, {9175, 20403}, {10280, 17714}, {11123, 15246}, {11634, 35345}, {14824, 18105}, {18311, 37895}, {18313, 32472}, {30219, 33884}

X(44821) = reflection of X(39495) in X(5092)
X(44821) = reflection of X(39495) in the Brocard axis
X(44821) = crosssum of X(512) and X(6034)
X(44821) = crossdifference of every pair of points on line {3003, 3229}

X(44822) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(30)X(99)

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

X(44822) lies on these lines: {3, 523}, {22, 10190}, {23, 1649}, {376, 25423}, {511, 39495}, {512, 14810}, {526, 33851}, {669, 7492}, {804, 11616}, {2451, 39560}, {2799, 39477}, {3049, 44453}, {3098, 8723}, {3566, 15577}, {4226, 41337}, {5926, 7502}, {6636, 11123}, {7484, 10189}, {7485, 10278}, {7493, 44451}, {7496, 8371}, {7525, 32204}, {8029, 15246}, {11003, 30219}, {14824, 21006}, {15915, 34291}, {16063, 23301}, {18310, 35936}, {20403, 33752}

X(44822) = midpoint of X(3098) and X(8723)
X(44822) = crosssum of X(512) and X(11646)
X(44822) = {X(7492),X(9168)}-harmonic conjugate of X(669)

X(44823) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(30)X(115)

Barycentrics    a^2*(b^2 - c^2)*(a^8 - a^6*b^2 - a^4*b^4 + a^2*b^6 - a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 + b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 - 4*b^4*c^4 + a^2*c^6 + b^2*c^6) : :
X(44823) = 3 X[5085] - 2 X[39495], 3 X[9175] - X[33752], 3 X[9175] - 2 X[37742], X[11616] - 3 X[25644], 3 X[25644] - 2 X[39477]

X(44823) lies on these lines: {3, 523}, {22, 8029}, {23, 5466}, {25, 10278}, {26, 10279}, {83, 18105}, {98, 804}, {182, 512}, {381, 25423}, {526, 11579}, {669, 1995}, {685, 4230}, {690, 13233}, {691, 5467}, {850, 26233}, {882, 9467}, {1116, 39214}, {1177, 2492}, {1499, 31861}, {1649, 40916}, {1691, 2422}, {1968, 2489}, {2065, 23350}, {2395, 21525}, {2793, 14270}, {2799, 39831}, {2881, 7663}, {3050, 39560}, {5020, 10189}, {5027, 14271}, {5085, 39495}, {5092, 8723}, {5094, 23301}, {5169, 44445}, {5926, 6644}, {5967, 34290}, {7484, 10190}, {7485, 11123}, {7496, 9168}, {7516, 8151}, {7533, 31299}, {9134, 42659}, {9173, 42667}, {9174, 42668}, {10117, 12064}, {10280, 13861}, {11616, 25644}, {11643, 23288}, {13187, 39832}, {18311, 35936}

X(44823) = reflection of X(i) in X(j) for these {i,j}: {5027, 14271}, {8723, 5092}, {11616, 39477}, {33752, 37742}
X(44823) = crosspoint of X(98) and X(691)
X(44823) = crosssum of X(i) and X(j) for these (i,j): {511, 690}, {524, 24284}, {625, 2799}, {14961, 39469}
X(44823) = crossdifference of every pair of points on line {3003, 5181}
X(44823) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {669, 8371, 1995}, {9175, 33752, 37742}, {11616, 25644, 39477}

X(44824) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(35)X(37)

Barycentrics    a*(b - c)*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c - a^3*b*c + 4*a^2*b^2*c - a*b^3*c - b^4*c - 2*a^3*c^2 + 4*a^2*b*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 - a*b*c^3 + 2*b^2*c^3 + a*c^4 - b*c^4 - c^5) : :
X(44824) = 5 X[631] - X[17166], X[2533] - 3 X[26446]

X(44824) lies on these lines: {3, 4705}, {513, 10225}, {514, 6684}, {631, 17166}, {650, 1734}, {905, 28537}, {1053, 34460}, {2533, 26446}, {2775, 39212}, {5690, 29298}, {6003, 8043}, {8678, 39227}, {10279, 12071}, {14838, 28473}, {33814, 38611}

X(44824) = midpoint of X(3) and X(4705)
X(44824) = reflection of X(12071) in X(10279)

X(44825) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(39)X(83)

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

X(44825) lies on these lines: {512, 35375}, {804, 11616}, {887, 38749}, {9494, 32473}

X(44826) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(69)X(74)

Barycentrics    a^2*(b^2 - c^2)*(a^6 - 2*a^4*b^2 + a^2*b^4 - 2*a^4*c^2 + 5*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 - 2*b^2*c^4) : :
X(44826) = 3 X[3] - 2 X[39477], 3 X[14270] - 4 X[39477], 2 X[14271] - 3 X[17508]

X(44826) lies on these lines: {2, 39509}, {3, 690}, {23, 9191}, {74, 9161}, {99, 14295}, {378, 16230}, {512, 684}, {523, 14634}, {526, 12041}, {574, 2491}, {804, 11616}, {888, 9145}, {924, 14314}, {2524, 37085}, {2780, 6132}, {2797, 23105}, {2799, 39831}, {3520, 44427}, {3566, 5926}, {3906, 42660}, {5027, 9155}, {6367, 44408}, {7496, 9185}, {7638, 9737}, {9003, 32305}, {9033, 12302}, {9189, 40916}, {9208, 14096}, {9479, 25644}, {13293, 13496}, {14271, 17508}, {14424, 42659}, {18117, 30209}, {22091, 42653}, {26316, 39499}

X(44826) = reflection of X(14270) in X(3)
X(44826) = anticomplement of X(39509)
X(44826) = crosspoint of X(99) and X(32901)
X(44826) = crossdifference of every pair of points on line {2493, 7735}

X(44827) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(71)X(74)

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

X(44827) lies on these lines: {1, 30574}, {3, 2774}, {386, 42662}, {526, 12041}, {926, 38599}, {2785, 22836}, {3887, 38324}, {4707, 34772}, {8674, 11699}

X(44828) = X(194)X(17974) ∩ X(1988)X(10559)

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

In a triangle ABC with circumcircle W, let
LA = perpendicular bisector of side BC, and define LB and LC cyclically;
A1 = LA∩BC, and define B1 and C1 cyclically;
A2 = LC∩CA and define B2 and C2 cyclically;
U = circle {{A1B1C1}};
V = circle {{A2B2C2}}.
The circles U, V, and W are coaxial. (Dan Reznik, Sep. 14, 2021).
The radical axis common to these circles is the trlinear polar of X(44828). Note: neither of the triangles A1B1C1 and A2B2C2 is a central triangle. (César Lozada, Sep 17, 2021).

X(44828) lies on these lines: {194, 17974}, {1988, 10559}, {5504, 43710}

X(44828) = isogonal conjugate of the complement of X(22089)
X(44828) = barycentric product X(i)*X(j) for these {i, j}: {99, 1988}, {648, 40800}
X(44828) = barycentric quotient X(i)/X(j) for these (i, j): (110, 3164), (112, 3168), (1576, 32445), (1625, 42453), (1988, 523)
X(44828) = trilinear product X(i)*X(j) for these {i, j}: {162, 40800}, {662, 1988}
X(44828) = trilinear quotient X(i)/X(j) for these (i, j): (162, 3168), (163, 32445), (662, 3164), (1988, 661)
X(44828) = trilinear pole of the line {577, 1971}
X(44828) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(6528)}} and {{A, B, C, X(54), X(112)}}
X(44828) = pole of the tripolar of X(3164) wrt Stammler hyperbola
X(44828) = X(648)-cross conjugate of-X(110)
X(44828) = X(i)-isoconjugate-of-X(j) for these {i, j}: {656, 3168}, {661, 3164}, {1577, 32445}
X(44828) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (110, 3164), (112, 3168), (1576, 32445), (1625, 42453)

X(44829) = X(3)X(161)∩X(4)X(83)

Barycentrics    2*a^10-4*a^8*(b^2+c^2)-(b^2-c^2)^4*(b^2+c^2)+a^4*(b^2+c^2)^3+a^6*(b^4+c^4)+a^2*(b^2-c^2)^2*(b^4+c^4) : :
Barycentrics    SB SC (8 R^2-5 SW)+S^2 (-SA+2 SW) : :
X(44829) = 2*X(52)-3*X(11225),2*X(143)-3*X(43573),3*X(376)+X(12289),3*X(2979)+X(34799),5*X(3522)-X(12278),5*X(3567)-6*X(32068),3*X(3917)-X(14516),3*X(5892)-2*X(31830),3*X(5943)-2*X(6756),3*X(7667)-2*X(13348),3*X(11245)-2*X(16625),2*X(12897)-3*X(13403),X(12897)-6*X(13470),X(13403)-4*X(13470),3*X(15030)-X(16659),7*X(15056)+X(40241),X(16655)-3*X(34664),3*X(16836)-2*X(31833)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2398.

X(44829) lies on these lines: {3,161}, {4,83}, {5,13419}, {20,1204}, {26,32223}, {30,143}, {52,11225}, {113,14862}, {125,7488}, {184,37444}, {185,12225}, {265,13564}, {376,12289}, {382,36752}, {511,6146}, {539,6101}, {542,1205}, {548,30522}, {550,10264}, {569,31723}, {578,14790}, {858,13367}, {1147,14791}, {1154,10116}, {1181,44469}, {1199,33749}, {1350,12429}, {1368,34782}, {1370,13346}, {1498,34776}, {1503,5907}, {1568,1614}, {1657,37489}, {2070,43817}, {2777,10575}, {2883,6593}, {2979,34799}, {3146,15019}, {3153,43831}, {3448,7691}, {3522,12278}, {3529,18916}, {3548,11202} ,{3549,23325}, {3567,32068}, {3574,5012}, {3575,9729}, {3796,7507}, {3818,7395}, {3917,14516}, {5092,7399}, {5449,7502}, {5462,11819}, {5576,37513}, {5892,31830}, {5899,43821}, {5921,11821}, {5943,6756}, {5944,37938}, {5965,11412}, {5972,10282}, {6000,12605}, {6102,18128}, {6640,10182}, {6643,9306}, {6689,39504}, {6699,15331}, {6723,10018}, {6759,18531}, {6815,13347}, {6816,31383}, {6823,23300}, {7383,17508} ,{7387,18390}, {7391,11424}, {7503,11550}, {7509,24206}, {7512,25739}, {7542,32767}, {7544,43650}, {7553,10110}, {7555,36253}, {7556,26917}, {7667,13348}, {7687,15761}, {8550,31802}, {9707,31180}, {10117,22808}, {10224,44516}, {10619,34148}, {10620,18442}, {10625,44076}, {10627,32423}, {11245,16625}, {11414,18396}, {11425,34609}, {11430,23335}, {11440,13399}, {11449,31101}, {11479,36990}, {11565,13391}, {11572,13160}, {11645,16655}, {11793,12134}, {12022,29317}, {12118,37480}, {12242,32046}, {12254,43574}, {12293,35243}, {13142,29181}, {13371,18475}, {13376,43823}, {14156,32171}, {15030,16659}, {15056,40241}, {15100,41726}, {15126,16196}, {15595,28723}, {15760,18383}, {15800,43845}, {16063,43652}, {16195,26958}, {16657,29323}, {16836,31833}, {17506,38727}, {17710,31807}, {17814,18536}, {17821,30771}, {18282,20304}, {18388,18569}, {18420,37515}, {18428,34007}, {18494,37514}, {18911,31304}, {21841,32237}, {22467,41482}, {25738,37478}, {31152,35602}, {31305,39571}, {31804,34986}, {32332,32396}, {34725,43273}, {35452,43393}, {37949,43835}, {43394,43582}, {43604,44242}, {43865,43893}

X(44829) = midpoint of X(i) and X(j) for these {i,j}: {3,11750}, {20,21659}, {185,12225}, {5562,34224}, {10575,18563}, {10625,44076}
X(44829) = reflection of X(i) in X(j) for these (i,j): (550,17712), (3575,9729), (5907,12362), (6102,18128), (7553,10110), (10112,6146), (11819,5462), (12134,11793), (12140,6723), (13419,5), (13598,12241), (32332,32396)
X(44829) = X(13419)-of-Johnson-triangle
X(44829) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,18381,21243), (3,21243,32348), (1370,19467,13346), (5944,37938,43839), (6643,9833,9306), (10282,11585,5972), (11572,22352,13160), (13399,35240,11440)

X(44830) = X(110)X(924)∩X(512)X(7471)

Barycentrics    a^2*(-(a^12*(b^2+c^2)^2)-b^2*c^2*(b^2-c^2)^4*(b^4+c^4)+a^2*(b^2-c^2)^4*(b^6+5*b^4*c^2+5*b^2*c^4+c^6)+a^10*(5*b^6+3*b^4*c^2+3*b^2*c^4+5*c^6)+a^8*(-10*b^8+5*b^6*c^2-14*b^4*c^4+5*b^2*c^6-10*c^8)+2*a^6*(5*b^10-6*b^8*c^2+5*b^6*c^4+5*b^4*c^6-6*b^2*c^8+5*c^10)+a^4*(-5*b^12+6*b^10*c^2+7*b^8*c^4-20*b^6*c^6+7*b^4*c^8+6*b^2*c^10-5*c^12)) : :
Barycentrics    (SA-SB) (SA-SC) (SB+SC) (-36 R^4+S^2+6 R^2 SA+SB SC+10 R^2 SW-SA SW-SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2398.

X(44830) lies on these lines: {110,924}, {512,7471}, {520,14611}, {858,1568}, {1302,2713}, {1510,3233}, {3154,5663}, {8675,14480}

X(44830) = midpoint of X(110) and X(11751)

X(44831) =  REFLECTION OF X(4) IN X(22)

Barycentrics    3*a^10-5*(b^2+c^2)*a^8-2*(b^4+b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(3*b^4-2*b^2*c^2+3*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(44831) = 3*X(2)-4*X(7502), 9*X(2)-8*X(39504), 5*X(3)-4*X(44236), 3*X(4)-4*X(15760), 5*X(4)-8*X(16618), 3*X(22)-2*X(15760), 5*X(22)-4*X(16618), 3*X(376)-2*X(378), 3*X(376)-4*X(44239), 5*X(376)-4*X(44285), 5*X(378)-6*X(44285), 3*X(381)-4*X(25337), 4*X(427)-5*X(631), 4*X(549)-3*X(31105)

As a point on the Euler line, X(44831) has Shinagawa coefficients (2*E+4*F, -5*E-8*F).

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44831) lies on these lines: {2, 3}, {74, 1286}, {99, 44128}, {311, 14907}, {1176, 15033}, {1238, 32817}, {2781, 5596}, {4549, 15305}, {5654, 26881}, {6776, 9019}, {9833, 11412}, {10606, 34775}, {11442, 37478}, {11459, 31383}, {12022, 33586}, {12112, 41465}, {12118, 41482}, {14516, 37486}, {14853, 19127}, {17834, 34224}, {18388, 35268}, {41171, 43653}

X(44831) = midpoint of X(i) and X(j) for these {i, j}: {20, 20062}, {1657, 44457}
X(44831) = reflection of X(i) in X(j) for these (i, j): (4, 22), (378, 44239), (7391, 3), (11442, 37478), (31133, 44261), (31723, 7502), (33703, 35480), (35481, 20), (44288, 7555), (44440, 12083)
X(44831) = anticomplement of X(31723)
X(44831) = circumperp conjugate of X(37978)
X(44831) = intersection, other than A, B, C, of circumconics {{A, B, C, X(30), X(18124)}} and {{A, B, C, X(74), X(21213)}}
X(44831) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 11818, 2), (3, 14788, 631), (20, 18533, 376), (20, 31304, 3), (20, 31305, 4), (378, 44239, 376), (382, 13160, 4), (550, 31830, 3), (2070, 14791, 2), (3146, 3547, 4), (3537, 17538, 376), (7387, 12225, 4), (7502, 31723, 2), (7503, 7553, 4), (9818, 34603, 4), (10295, 21312, 376), (11001, 13619, 20), (31133, 44261, 3524)

X(44832) =  REFLECTION OF X(4) IN X(37990)

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6-9*b^2*c^2*a^4+2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2-(b^6-c^6)*(b^2-c^2)) : :
X(44832) = 4*X(3)-X(7550)

As a point on the Euler line, X(44832) has Shinagawa coefficients (6*E+4*F, -7*E-4*F).

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44832) lies on these lines: {2, 3}, {54, 13348}, {74, 7953}, {99, 1232}, {112, 22052}, {399, 44324}, {1199, 10625}, {1994, 13340}, {2979, 15032}, {3098, 5890}, {3455, 21166}, {3567, 13347}, {3819, 14157}, {5092, 15033}, {5446, 12834}, {5621, 12383}, {5891, 12112}, {5892, 15107}, {8573, 14482}, {8717, 15305}, {8718, 11793}, {10606, 35228}, {12325, 18914}, {13339, 13391}, {13366, 15644}, {14627, 33542}, {14907, 44180}, {15004, 37515}, {18445, 33884}, {18451, 21766}, {20417, 25714}, {20791, 37478}, {22352, 43574}

X(44832) = midpoint of X(20) and X(37349)
X(44832) = reflection of X(i) in X(j) for these (i, j): (4, 37990), (7550, 15246), (15246, 3), (34545, 13339)
X(44832) = circumperp conjugate of X(20063)
X(44832) = intersection, other than A, B, C, of circumconics {{A, B, C, X(25), X(13597)}} and {{A, B, C, X(30), X(41435)}}
X(44832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 22, 3524), (3, 3522, 3520), (3, 6636, 186), (3, 8703, 2071), (3, 10323, 631), (3, 13564, 3530), (3, 35243, 7485), (3, 37198, 7509), (186, 6636, 7512), (376, 35921, 7464), (3522, 18533, 376), (3528, 3537, 376), (3528, 35503, 3522), (7393, 33524, 4), (7484, 12082, 3545), (7485, 35243, 4), (7492, 15692, 6644), (7509, 37198, 3529), (15646, 15759, 3), (18534, 40916, 5071), (21312, 41463, 376)

X(44833) = X(2)X(1351)∩X(4)X(5447)

Barycentrics    a^6-7*(b^2+c^2)*a^4+(7*b^4+26*b^2*c^2+7*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(44833) = 3*X(2)+2*X(40912)

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44833) lies on these lines: {2, 1351}, {4, 5447}, {5, 3531}, {24, 16543}, {30, 40911}, {69, 5650}, {110, 3524}, {125, 21356}, {141, 16051}, {155, 631}, {376, 11472}, {443, 10597}, {524, 5646}, {1352, 3819}, {1353, 16419}, {1992, 22112}, {3523, 6090}, {3525, 6689}, {3528, 15062}, {3533, 14389}, {3537, 5891}, {3538, 11487}, {3619, 30785}, {3620, 30739}, {3917, 7392}, {4550, 41468}, {4563, 34229}, {4648, 29639}, {5878, 10996}, {5921, 43957}, {5972, 9970}, {6288, 6643}, {6515, 44299}, {6800, 10299}, {6803, 7999}, {7401, 32142}, {7494, 10192}, {7693, 33884}, {7728, 13416}, {9168, 18556}, {10519, 32269}, {13243, 14021}, {14912, 38396}, {15059, 32263}, {15709, 40112}, {15717, 26864}, {17040, 26869}, {17567, 26637}, {24981, 25406}, {32620, 35483}, {37643, 40107}

X(44833) = midpoint of X(5544) and X(40912)
X(44833) = reflection of X(i) in X(j) for these (i, j): (4, 18489), (3531, 5)
X(44833) = anticomplement of X(5544)
X(44833) = isotomic conjugate of the polar conjugate of X(39662)
X(44833) = pole of the line {5050, 44413} wrt Stammler hyperbola
X(44833) = barycentric product X(69)*X(39662)
X(44833) = trilinear product X(63)*X(39662)
X(44833) = X(3531)-of-Johnson-triangle

X(44834) = X(2)X(399)∩X(4)X(10627)

Barycentrics    a^10-5*(b^2+c^2)*a^8+(10*b^4+9*b^2*c^2+10*c^4)*a^6-2*(b^2+c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^4+(5*b^4+21*b^2*c^2+5*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(44834) = 7*X(3090)-4*X(5643), 9*X(3545)-4*X(14483)

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44834) lies on these lines: {2, 399}, {4, 10627}, {5, 12316}, {8, 6900}, {20, 33539}, {30, 18551}, {69, 1568}, {146, 15060}, {376, 4550}, {631, 5651}, {1154, 7693}, {1352, 32255}, {3090, 5449}, {3524, 7712}, {3533, 44516}, {3855, 11487}, {5054, 41450}, {5066, 15108}, {5067, 18917}, {5071, 37644}, {11444, 15436}, {14926, 32423}, {18383, 32346}, {18489, 41099}

X(44834) = reflection of X(20) in X(33544)

X(44835) = ISOGONAL CONJUGATE OF X(21161)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44835) lies on Jerabek circumhyperbola and these lines: {72, 18357}, {6866, 34259}, {7687, 10693}

X(44835) = isogonal conjugate of X(21161)
X(44835) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(4)}} and {{A, B, C, X(9), X(11529)}}

X(44836) = X(52)X(15077)∩X(66)X(21851)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44836) lies on Jerabek circumhyperbola and these lines: {52, 15077}, {66, 21851}, {68, 11572}, {69, 18474}, {1177, 7687}, {3431, 8889}, {4846, 11550}, {6146, 43908}, {6293, 38447}, {13202, 34802}, {14457, 18383}, {14542, 18381}, {15432, 43697}, {18390, 43726}, {33586, 34801}

X(44836) = isogonal conjugate of X(44837)
X(44836) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(4)}} and {{A, B, C, X(52), X(14531)}}

X(44837) =  ISOGONAL CONJUGATE OF X(44836)

Barycentrics    a^2*(3*a^8-6*(b^2+c^2)*a^6+2*b^2*c^2*a^4+2*(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(3*b^4+4*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :
X(44837) = X(2)+2*X(44261), 2*X(3)+X(22), 4*X(3)-X(378), X(3)+2*X(7502), 5*X(3)+4*X(7555), 8*X(3)+X(12082), 5*X(3)+X(12083), 5*X(3)-2*X(18570), 11*X(3)+X(44457), X(4)-4*X(6676), X(4)+2*X(44239), X(20)+2*X(15760), 2*X(20)+X(35480), 2*X(22)+X(378), X(22)-4*X(7502), 5*X(22)-8*X(7555), 4*X(22)-X(12082), 5*X(22)-2*X(12083), 5*X(22)+4*X(18570), 11*X(22)-2*X(44457)

As a point on the Euler line, X(44837) has Shinagawa coefficients (2*E+6*F, -3*E-6*F).

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44837) lies on these lines: {2, 3}, {54, 17834}, {64, 8718}, {74, 16165}, {154, 11459}, {155, 7691}, {159, 11180}, {182, 16226}, {394, 11464}, {519, 15177}, {539, 41594}, {542, 22109}, {578, 21969}, {599, 15577}, {671, 39828}, {754, 3425}, {1154, 34513}, {1327, 35776}, {1328, 35777}, {1350, 19127}, {1609, 14836}, {1992, 37488}, {1993, 18475}, {2482, 39854}, {2781, 15035}, {2931, 9140}, {3058, 9672}, {3060, 37506}, {3098, 20806}, {3431, 33878}, {3567, 37476}, {3796, 5890}, {3917, 11202}, {5012, 37489}, {5085, 9019}, {5422, 37513}, {5434, 9659}, {5562, 9707}, {5642, 13289}, {5891, 35264}, {6000, 35268}, {6030, 15072}, {6054, 39857}, {6055, 39825}, {6776, 16789}, {6800, 13754}, {7592, 14831}, {7712, 41450}, {7799, 9723}, {8717, 44754}, {8882, 36751}, {9143, 12412}, {9590, 19875}, {9625, 38021}, {9700, 11648}, {9704, 12307}, {9798, 34627}, {9876, 39803}, {10117, 10706}, {10610, 36749}, {10714, 14703}, {10717, 14657}, {10718, 19165}, {11412, 19357}, {11425, 13482}, {11438, 22352}, {11442, 44201}, {11605, 34841}, {12243, 39832}, {12254, 12429}, {12410, 34631}, {13445, 35237}, {14805, 39522}, {15033, 33586}, {15107, 44413}, {15513, 19220}, {15565, 38698}, {15811, 43613}, {17814, 26882}, {18451, 26881}, {22802, 35240}, {25739, 37638}, {26613, 32762}, {31158, 39475}, {34148, 37486}, {34177, 36989}, {34774, 35228}

X(44837) = reflection of X(37970) in X(37941)
X(44837) = isogonal conjugate of X(44836)
X(44837) = X(22)-Gibert-Moses centroid
X(44837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14070, 24), (3, 22, 378), (3, 3515, 631), (3, 7488, 24), (3, 7502, 22), (3, 9715, 4), (3, 11414, 3520), (3, 14070, 2), (3, 34006, 3534), (3, 35243, 2071), (3, 37922, 5054), (24, 35477, 1594), (2070, 7514, 1995), (3515, 7487, 24), (3522, 38448, 3), (3528, 17506, 3), (3545, 37939, 25), (5054, 37922, 6644), (6636, 10298, 3), (6676, 44239, 4), (7395, 16195, 3518), (7555, 12083, 22), (7556, 35921, 25), (9715, 44239, 22), (10304, 37941, 3), (14070, 44261, 22), (14118, 38435, 7387), (18475, 37478, 1993), (34006, 44262, 22), (35477, 35480, 378), (35921, 37939, 3545)

X(44838) = X(373)X(1992)∩X(381)X(5650)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44838) lies on these lines: {373, 1992}, {381, 5650}, {3167, 5544}, {3917, 7392}, {5642, 38110}, {15030, 44750}

X(44839) = X(6)X(376)∩X(98)X(5039)

Barycentrics    (7*a^2+b^2+c^2)*(a^4+4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(44839) = 2*X(14484)-3*X(14853)

See Antreas Hatzipolakis and César Lozada, Euclid 2399 .

X(44839) lies on these lines: {6, 376}, {98, 5039}, {193, 41623}, {511, 14930}, {524, 18842}, {5032, 8591}, {6776, 18907}, {7736, 10519}, {10304, 42852}, {11257, 44500}, {22486, 41622}, {30435, 39647}

X(44839) = reflection of X(14482) in X(6)

leftri

Perspectors involving 8th and 9th mixtilinear triangles: X(44840)-X(44859)

rightri

This preamble and centers X(44840)-X(44859) were contributed by César Eliud Lozada, September 18, 2021.

Dan Reznik defines 8th and 9th mixtilinear triangles as follows:

Barycentric coordinates of the A-vertices of these triangles are:

The appearance of (T, i) in the following list means that triangles 8th mixtilinear and T are perspective with perspector X(i):

(ABC, 55), (anti-Mandart-incircle, 55), (anti-tangential-midarc, 44840), (Apus, 55), (Bevan antipodal, 44841), (Gemini 35, 5284), (Gemini 40, 5274), (Gemini 76, 44842), (Gemini 81, 44843), (Mandart-incircle, 55), (mixtilinear, 6767), (3rd mixtilinear, 999), (4th mixtilinear*, 55), where * stands for homothetic triangles.

The appearance of (T, i) in the following list means that triangles 9th mixtilinear and T are perspective with perspector X(i):

(ABC, 56), (2nd anti-circumperp-tangential, 56), (anti-Mandart-incircle, 44844), (Apollonius, 44845), (Apus, 44846), (2nd circumperp tangential, 56), (excentral, 7991), (extangents, 5183), (Feuerbach, 44847), (Gemini 26, 44848), (Gemini 39, 8165), (Gemini 80, 44849), (incentral, 7962), (3rd Jenkins, 44850), (Mandart-excircles, 56), (2nd mixtilinear, 8158), (3rd mixtilinear*, 56), (4th mixtilinear, 6244), (5th mixtilinear, 44851), (Montesdeoca-Hung, 44852), (Paasche-Hutson, 44857), (1st Przybyłowski-Bollin, 44853), (2nd Przybyłowski-Bollin, 44854), (3rd Przybyłowski-Bollin, 44855), (4th Przybyłowski-Bollin, 44856), where * stands for homothetic triangles.

The following pairs of triangles are orthologic: (8th mixtilinear, 4th mixtilinear) and (9th mixtilinear, 3rd mixtilinear). No known triangle was found to be parallelogic to either A'B'C' or A"B"C". The 9th mixtilinear triangle is also directly similar to these triangles: excenters-midpoints, Garcia-reflection (named Gemini 8 too) and 2nd Schiffler.

X(44840) = PERSPECTOR OF THESE TRIANGLES: 8th MIXTILINEAR AND ANTI-TANGENTIAL-MIDARC

Barycentrics    a*(2*a^3-3*(b+c)*a^2-2*(b+c)^2*a+3*(b^2-c^2)*(b-c)) : :
X(44840) = 3*X(40)-5*X(7688) = 4*X(40)-5*X(7964) = X(3748)+2*X(5425) = 4*X(7688)-3*X(7964) = 5*X(13151)-4*X(13624) = 5*X(15931)-7*X(30389)

X(44840) lies on these lines: {1, 3}, {11, 4870}, {12, 6738}, {19, 16884}, {30, 11551}, {71, 3723}, {72, 30143}, {145, 28629}, {200, 4731}, {388, 37724}, {405, 3962}, {497, 15933}, {518, 8539}, {519, 3925}, {553, 15326}, {758, 3683}, {938, 10589}, {950, 3649}, {958, 11520}, {960, 5284}, {993, 24473}, {1001, 31165}, {1056, 37740}, {1100, 2170}, {1104, 2308}, {1125, 21677}, {1317, 38055}, {1358, 3664}, {1362, 34930}, {1411, 13404}, {1464, 14547}, {1621, 44663}, {1737, 5719}, {1776, 16140}, {1788, 18221}, {1836, 3488}, {1837, 3487}, {1859, 1870}, {1888, 6198}, {2218, 31503}, {2550, 3241}, {2654, 11553}, {2772, 3022}, {3059, 3872}, {3189, 3623}, {3328, 31524}, {3475, 5252}, {3476, 11038}, {3485, 5274}, {3486, 10404}, {3555, 30147}, {3671, 6284}, {3689, 3753}, {3698, 3811}, {3742, 4511}, {3812, 34772}, {3870, 8168}, {3871, 10107}, {3877, 11025}, {3880, 3957}, {3889, 11260}, {3901, 31445}, {3916, 35016}, {3922, 5687}, {3982, 28164}, {4004, 8715}, {4005, 11523}, {4018, 5248}, {4292, 10543}, {4304, 11246}, {4654, 12943}, {4658, 37816}, {4666, 5289}, {4880, 5426}, {5044, 41696}, {5249, 44669}, {5259, 16126}, {5326, 13411}, {5432, 38068}, {5434, 5542}, {5439, 22836}, {5440, 5883}, {5558, 6049}, {5572, 12755}, {5603, 39782}, {5703, 24914}, {5722, 17605}, {5727, 11237}, {5882, 6253}, {6147, 10572}, {6734, 11281}, {6744, 37722}, {6877, 8164}, {7269, 15337}, {8167, 12635}, {9613, 41870}, {9623, 41711}, {9654, 37721}, {9956, 37731}, {10179, 29817}, {10501, 18454}, {10502, 18456}, {10528, 37829}, {10699, 17015}, {10896, 37723}, {10950, 21620}, {11019, 15950}, {11374, 17606}, {11541, 43733}, {11552, 28146}, {12047, 12433}, {12649, 28628}, {12736, 41541}, {13374, 21740}, {13405, 40663}, {13407, 37730}, {14054, 44782}, {14497, 18490}, {14563, 31397}, {14828, 43037}, {15174, 24470}, {15823, 39772}, {15837, 30329}, {15935, 39542}, {16138, 17637}, {18393, 18527}, {18406, 28204}, {24391, 24953}, {25466, 41575}, {31870, 33597}

X(44840) = midpoint of X(1) and X(5425)
X(44840) = reflection of X(3748) in X(1)
X(44840) = intersection, other than A, B, C, of circumconics {{A, B, C, X(19), X(18421)}} and {{A, B, C, X(36), X(13404)}}
X(44840) = pole of the line {1, 3651} with respect to Feuerbach circumhyperbola
X(44840) = pole of the line {21, 3962} with respect to Stammler hyperbola
X(44840) = crosssum of X(1) and X(24929)
X(44840) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 354, 1319), (1, 942, 2646), (1, 2099, 5919), (1, 3333, 34471), (1, 3340, 3303), (1, 5045, 20323), (1, 5902, 24929), (1, 7962, 8162), (1, 10980, 13384), (1, 11009, 31792), (1, 11518, 56), (1, 11529, 55), (1, 15934, 354), (1, 18398, 1385), (1, 18421, 10389), (1, 25415, 6767), (1, 37602, 25405), (55, 65, 5183), (55, 11529, 65), (942, 2646, 32636), (942, 13624, 3337), (2099, 8162, 7962), (5045, 5570, 354), (5183, 37080, 55), (5902, 24929, 1155), (6767, 25415, 3057), (7962, 8162, 5919), (10980, 13384, 56), (11518, 13384, 10980), (12433, 16137, 12047), (25405, 37602, 20323)

X(44841) = PERSPECTOR OF THESE TRIANGLES: 8th MIXTILINEAR AND BEVAN ANTIPODAL

Barycentrics    a*(a^2-4*(b+c)*a+3*(b-c)^2) : :
X(44841) = 3*X(165)-5*X(10857) = 3*X(7308)-4*X(8167) = 2*X(8167)-3*X(10582)

X(44841) lies on these lines: {1, 3}, {2, 3243}, {7, 9580}, {9, 3873}, {31, 35227}, {42, 5573}, {63, 29817}, {105, 28148}, {142, 36845}, {200, 3742}, {226, 5274}, {388, 6744}, {390, 553}, {497, 4654}, {518, 7308}, {519, 26040}, {612, 17450}, {672, 3247}, {938, 9578}, {950, 11037}, {968, 17449}, {1001, 3929}, {1002, 3720}, {1056, 5727}, {1125, 41863}, {1149, 10460}, {1174, 9310}, {1210, 8164}, {1376, 15570}, {1419, 34036}, {1421, 11028}, {1423, 44843}, {1449, 7191}, {1621, 3928}, {1699, 20330}, {2280, 38849}, {2308, 7290}, {2346, 39669}, {2886, 31146}, {3058, 4312}, {3158, 3306}, {3242, 17022}, {3296, 4292}, {3305, 4430}, {3315, 5256}, {3434, 6173}, {3474, 30331}, {3475, 5219}, {3485, 21625}, {3486, 12577}, {3577, 7967}, {3600, 5558}, {3616, 11523}, {3621, 11530}, {3622, 5273}, {3623, 3680}, {3636, 12559}, {3646, 5904}, {3664, 19604}, {3723, 42316}, {3751, 29820}, {3812, 8168}, {3870, 5437}, {3874, 31435}, {3889, 6762}, {3911, 10578}, {3938, 15600}, {3982, 9812}, {4031, 9778}, {4295, 40270}, {4322, 7273}, {4328, 34855}, {4355, 6284}, {4423, 5223}, {4512, 42819}, {4514, 17298}, {4658, 5324}, {4659, 17140}, {4845, 14760}, {4847, 38053}, {4863, 38052}, {4864, 37674}, {4890, 4907}, {4923, 41915}, {5083, 11020}, {5226, 30318}, {5249, 24392}, {5326, 17728}, {5435, 11526}, {5439, 6765}, {5572, 16112}, {5853, 9776}, {6147, 9614}, {6738, 37709}, {7226, 16676}, {7322, 16496}, {7965, 11522}, {7971, 10595}, {8012, 17474}, {8090, 10502}, {8236, 21454}, {8423, 10501}, {8580, 41711}, {9581, 10590}, {9613, 12433}, {9814, 14100}, {10167, 43166}, {10177, 36973}, {10384, 10391}, {10385, 43179}, {11036, 12053}, {11551, 31162}, {12005, 12705}, {12047, 41870}, {13405, 31231}, {15601, 32912}, {16491, 43149}, {17051, 31249}, {18240, 37736}, {20195, 25006}, {20196, 25568}, {24600, 27475}, {25525, 26015}, {25527, 29843}, {26105, 31142}, {28039, 44842}, {30813, 38186}, {31140, 38024}, {38021, 41858}

X(44841) = reflection of X(7308) in X(10582)
X(44841) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(10389)}} and {{A, B, C, X(9), X(3748)}}
X(44841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 165, 3748), (1, 354, 57), (1, 942, 1697), (1, 3333, 3601), (1, 3339, 3303), (1, 10980, 55), (1, 11518, 3340), (1, 18193, 3750), (1, 18398, 40), (1, 18421, 5919), (1, 30343, 3304), (1, 30350, 354), (55, 8171, 2078), (55, 10980, 57), (165, 4860, 57), (354, 3748, 4860), (354, 17642, 942), (942, 6767, 2093), (942, 10383, 57), (999, 13384, 1420), (2093, 6767, 1697), (3306, 3957, 3158), (3513, 3514, 3339), (3742, 42871, 200), (3748, 4860, 165), (3873, 4666, 9), (4883, 17597, 1), (5045, 12915, 354), (5049, 15934, 1), (5597, 5598, 5217), (7962, 11529, 3340), (9812, 30340, 3982), (10383, 17642, 1697), (10580, 11038, 226), (16496, 26102, 7322), (18398, 36946, 1)

X(44842) = PERSPECTOR OF THESE TRIANGLES: 8th MIXTILINEAR AND GEMINI 76

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

X(44842) lies on these lines: {1, 28038}, {2, 11}, {36, 37328}, {10980, 28017}, {11529, 35667}, {13384, 15839}, {28039, 44841}

X(44843) = PERSPECTOR OF THESE TRIANGLES: 8th MIXTILINEAR AND GEMINI 81

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

X(44843) lies on these lines: {1, 28387}, {2, 11}, {999, 9840}, {1201, 13384}, {1423, 44841}, {2293, 28360}, {2646, 28370}, {3056, 29814}, {3601, 28352}, {3622, 8240}, {4392, 17611}, {4666, 28402}, {7419, 26357}, {7962, 10459}, {9812, 37575}, {33104, 37262}, {33106, 37400}

X(44844) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND ANTI-MANDART-INCIRCLE

Barycentrics    a^2*(-a+b+c)*(9*a^5-15*(b+c)*a^4+2*(3*b^2+32*b*c+3*c^2)*a^3+2*(b+c)*(3*b^2-28*b*c+3*c^2)*a^2-(15*b^4+15*c^4-2*(8*b^2+15*b*c+8*c^2)*b*c)*a+(b^2-c^2)*(b-c)*(9*b^2-6*b*c+9*c^2)) : :

X(44844) lies on these lines: {56, 3243}, {12329, 44846}

X(44845) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND APOLLONIUS

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

X(44845) lies on these lines: {10, 7173}, {43, 5128}, {56, 181}, {573, 44846}, {970, 3030}, {1682, 7962}, {4252, 23638}, {5183, 10822}, {6244, 9566}, {8158, 10823}, {8165, 9564}, {11758, 44853}, {11767, 44854}, {11776, 44855}, {11785, 44856}, {11989, 44852}

X(44846) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND APUS

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

X(44846) lies on these lines: {1, 3}, {4, 44847}, {20, 21031}, {84, 3711}, {516, 24954}, {573, 44845}, {1376, 3146}, {1399, 38293}, {3091, 4413}, {3522, 4421}, {3525, 8166}, {3532, 15625}, {3689, 9841}, {4423, 10303}, {4428, 15717}, {4995, 37108}, {5433, 35514}, {5493, 16371}, {6174, 37421}, {6425, 19000}, {6426, 18999}, {6848, 34630}, {6890, 31140}, {6922, 9671}, {6926, 11238}, {8169, 36002}, {9589, 16417}, {9670, 37364}, {10304, 34749}, {10896, 31777}, {11499, 15704}, {11500, 17538}, {11759, 44853}, {11768, 44854}, {11777, 44855}, {11786, 44856}, {11990, 44852}, {12327, 15034}, {12329, 44844}, {12512, 21075}, {12679, 31730}, {13204, 15021}, {17531, 26129}

X(44846) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5537, 3303), (3, 6244, 7991), (3, 7991, 56), (3, 35448, 10222), (8273, 11248, 55), (32622, 32623, 8171)

X(44847) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND FEUERBACH

Barycentrics    (3*b^2-2*b*c+3*c^2)*a^2-8*(b+c)*b*c*a-3*(b^2-c^2)^2 : :
X(44847) = 2*X(4187)+X(21031) = 4*X(4187)-X(37722) = 2*X(21031)+X(37722)

X(44847) lies on these lines: {2, 12}, {4, 44846}, {5, 3654}, {10, 7173}, {11, 3679}, {119, 549}, {381, 6244}, {496, 4677}, {498, 16857}, {519, 4187}, {551, 17757}, {993, 31235}, {1006, 20400}, {1698, 1836}, {2099, 5328}, {2476, 34501}, {2478, 4421}, {3035, 17549}, {3241, 3816}, {3452, 31165}, {3524, 18242}, {3617, 3847}, {3634, 3916}, {3649, 8582}, {3653, 10942}, {3655, 37725}, {3814, 3828}, {3817, 4731}, {3825, 4669}, {3829, 4193}, {3929, 24914}, {4428, 5552}, {4745, 24390}, {4995, 10958}, {5055, 7958}, {5071, 7680}, {5080, 36006}, {5154, 9710}, {5251, 5326}, {5252, 20196}, {5432, 16418}, {5947, 38109}, {6174, 11113}, {6690, 16861}, {6945, 34632}, {6963, 34627}, {7354, 16417}, {7508, 38763}, {7951, 19876}, {8171, 31479}, {9656, 17580}, {10072, 34689}, {10176, 34122}, {10198, 19536}, {10944, 24954}, {11238, 34720}, {11755, 44853}, {11764, 44854}, {11773, 44855}, {11782, 44856}, {11992, 44852}, {12607, 38314}, {15326, 16371}, {15888, 17527}, {15950, 30827}, {16370, 26364}, {16858, 27529}, {17556, 34612}, {17575, 19883}, {17619, 21677}, {18227, 18838}, {20487, 42056}, {21155, 28443}, {23513, 38112}, {24387, 38098}, {24982, 44663}, {25016, 25882}, {26470, 38100}, {28458, 32554}, {31650, 38070}, {34628, 37364}, {34648, 37374}, {38042, 38129}, {38092, 42356}

X(44847) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 3436, 40726), (2, 31141, 5434), (2, 34606, 5298), (2551, 31246, 5433), (3814, 3828, 17530), (3828, 17530, 3925), (4187, 21031, 37722)

X(44848) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND GEMINI 26

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

X(44848) lies on these lines: {1, 33994}, {2, 4345}, {8, 31190}, {9, 5128}, {10, 56}, {12, 38208}, {100, 950}, {392, 3634}, {553, 17757}, {946, 1698}, {1000, 3624}, {1001, 8582}, {1125, 1145}, {1478, 19875}, {1512, 5251}, {1788, 5223}, {3086, 11525}, {3243, 7080}, {3754, 41389}, {3814, 3828}, {4301, 31246}, {5836, 6667}, {6244, 6913}, {6692, 6735}, {6968, 10175}, {9588, 35514}, {9711, 15481}, {11518, 27525}, {17613, 19925}, {17614, 41554}, {20103, 40663}, {31339, 44849}, {34122, 41166}

X(44848) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1698, 5657, 5316), (5657, 8166, 7991)

X(44849) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND GEMINI 80

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

X(44849) lies on these lines: {2, 12}, {978, 7991}, {1193, 7962}, {5128, 27627}, {5183, 27625}, {6244, 19513}, {8158, 19549}, {27621, 28268}, {27626, 28257}, {27649, 28239}, {31339, 44848}

X(44850) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 3rd JENKINS

Barycentrics    (3*b+c)*(b+3*c)*(b+c)*a^10+(15*b^4+15*c^4+2*(31*b^2+45*b*c+31*c^2)*b*c)*a^9+(b+c)*(21*b^4+21*c^4+2*(31*b^2+52*b*c+31*c^2)*b*c)*a^8-(3*b^6+3*c^6-(26*b^4+26*c^4+(197*b^2+314*b*c+197*c^2)*b*c)*b*c)*a^7-(b+c)*(27*b^6+27*c^6-(2*b^4+2*c^4+(95*b^2+123*b*c+95*c^2)*b*c)*b*c)*a^6-(15*b^8+15*c^8+2*(23*b^6+23*c^6+(34*b^4+34*c^4-(69*b^2+163*b*c+69*c^2)*b*c)*b*c)*b*c)*a^5+(b+c)*(3*b^8+3*c^8-(38*b^6+38*c^6-(6*b^4+6*c^4+(107*b^2+44*b*c+107*c^2)*b*c)*b*c)*b*c)*a^4+(3*b^8+3*c^8+2*(11*b^4+11*c^4-(19*b^2-89*b*c+19*c^2)*b*c)*b^2*c^2)*(b+c)^2*a^3+(12*b^6+12*c^6-(24*b^4+24*c^4-7*(11*b^2-10*b*c+11*c^2)*b*c)*b*c)*(b+c)^3*b*c*a^2+4*(3*b^4+3*c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*(b+c)^4*b^2*c^2*a+3*(b^2-c^2)^2*(b+c)^3*b^3*c^3 : :

X(44850) lies on this line: {56, 37868}

X(44851) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 5th MIXTILINEAR

Barycentrics    a*(a^6-3*(b+c)*a^5-(4*b^2-21*b*c+4*c^2)*a^4+4*(b+c)*(b^2-7*b*c+c^2)*a^3+(3*b^4+3*c^4-2*(11*b^2-45*b*c+11*c^2)*b*c)*a^2-(b+c)*(b^4+c^4-6*(4*b^2-9*b*c+4*c^2)*b*c)*a-3*(b^2-c^2)^2*b*c) : :

X(44851) lies on these lines: {56, 3241}, {644, 11260}, {8834, 12513}

X(44852) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND MONTESDEOCA-HUNG

Barycentrics    a*(4*a^9+8*(b+c)*a^8+4*(b^2+8*b*c+c^2)*a^7+40*(b+c)*b*c*a^6-3*(b^4+c^4-2*(2*b^2+11*b*c+2*c^2)*b*c)*a^5-(3*b+c)*(b+3*c)*(b+c)*(b^2-6*b*c+c^2)*a^4+2*(b^6+c^6+(4*b^4+4*c^4+(5*b^2+12*b*c+5*c^2)*b*c)*b*c)*a^3-2*(b^4+c^4+2*(3*b^2-2*b*c+3*c^2)*b*c)*(b+c)^3*a^2-(7*b^6+7*c^6+(6*b^4+6*c^4-(7*b^2-20*b*c+7*c^2)*b*c)*b*c)*(b+c)^2*a+(b^4-c^4)^2*(-3*b-3*c)) : :

X(44852) lies on these lines: {56, 6042}, {5183, 11991}, {6043, 7991}, {6244, 11995}, {7962, 11993}, {8158, 11994}, {11761, 44853}, {11770, 44854}, {11779, 44855}, {11788, 44856}, {11989, 44845}, {11990, 44846}, {11992, 44847}

X(44853) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 1st PRZYBYŁOWSKI-BOLLIN

Barycentrics    a*(4*S*(-a+b+c)*(a^2-2*(b+c)*a-3*(b-c)^2)*sqrt(SW+sqrt(3)*S)-(2*S+(-a^2+b^2+c^2)*sqrt(3))*(3*a^3-3*(b+c)*a^2-(3*b^2-14*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c))*a) : :

X(44853) lies on these lines: {15, 7962}, {56, 11753}, {1379, 44856}, {1380, 44855}, {5183, 11756}, {6244, 11760}, {7991, 11754}, {8158, 11757}, {11755, 44847}, {11758, 44845}, {11759, 44846}, {11761, 44852}

X(44853) = {X(15), X(7962)}-harmonic conjugate of X(44854)

X(44854) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 2nd PRZYBYŁOWSKI-BOLLIN

Barycentrics    a*(4*S*(-a+b+c)*(a^2-2*(b+c)*a-3*(b-c)^2)*sqrt(SW+sqrt(3)*S)+(2*S+(-a^2+b^2+c^2)*sqrt(3))*(3*a^3-3*(b+c)*a^2-(3*b^2-14*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c))*a) : :

X(44854) lies on these lines: {15, 7962}, {56, 11762}, {1379, 44855}, {1380, 44856}, {5183, 11765}, {6244, 11769}, {7991, 11763}, {8158, 11766}, {11764, 44847}, {11767, 44845}, {11768, 44846}, {11770, 44852}

X(44854) = {X(15), X(7962)}-harmonic conjugate of X(44853)

X(44855) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 3rd PRZYBYŁOWSKI-BOLLIN

Barycentrics    a*(-4*S*(-a+b+c)*(a^2-2*(b+c)*a-3*(b-c)^2)*sqrt(SW-sqrt(3)*S)+(2*S-(-a^2+b^2+c^2)*sqrt(3))*(3*a^3-3*(b+c)*a^2-(3*b^2-14*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c))*a) : :

X(44855) lies on these lines: {16, 7962}, {56, 11771}, {1379, 44854}, {1380, 44853}, {5183, 11774}, {6244, 11778}, {7991, 11772}, {8158, 11775}, {11773, 44847}, {11776, 44845}, {11777, 44846}, {11779, 44852}

X(44855) = {X(16), X(7962)}-harmonic conjugate of X(44856)

X(44856) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND 4th PRZYBYŁOWSKI-BOLLIN

Barycentrics    a*(4*S*(-a+b+c)*(a^2-2*(b+c)*a-3*(b-c)^2)*sqrt(SW-sqrt(3)*S)+(2*S-(-a^2+b^2+c^2)*sqrt(3))*(3*a^3-3*(b+c)*a^2-(3*b^2-14*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c))*a) : :

X(44856) lies on these lines: {16, 7962}, {56, 11780}, {1379, 44853}, {1380, 44854}, {5183, 11783}, {6244, 11787}, {7991, 11781}, {8158, 11784}, {11782, 44847}, {11785, 44845}, {11786, 44846}, {11788, 44852}

X(44856) = {X(16), X(7962)}-harmonic conjugate of X(44855)

X(44857) = PERSPECTOR OF THESE TRIANGLES: 9th MIXTILINEAR AND PAASCHE-HUTSON

Barycentrics    a*(2*(7*a^8-12*(2*b^2-3*b*c+2*c^2)*a^6+16*(b+c)*b*c*a^5+2*(15*b^4+15*c^4-2*(4*b^2-25*b*c+4*c^2)*b*c)*a^4-8*(3*b+c)*(b+3*c)*(b+c)*b*c*a^3-4*(b^2-c^2)^2*(4*b^2+5*b*c+4*c^2)*a^2+8*(b^2-c^2)^2*(b+c)*b*c*a+3*(b^2-c^2)^4)*S*b*c-(a^11+(b+c)*a^10-(5*b^2-6*b*c+5*c^2)*a^9-(b+c)*(5*b^2-4*b*c+5*c^2)*a^8+2*(5*b^4+5*c^4-8*(b^2-b*c+c^2)*b*c)*a^7+2*(b+c)*(5*b^4+5*c^4-4*(b^2-3*b*c+c^2)*b*c)*a^6-2*(5*b^6+5*c^6-3*(b+2*c)*(2*b+c)*(b^2-4*b*c+c^2)*b*c)*a^5-2*(b+c)*(5*b^6+5*c^6+3*(7*b^2+4*b*c+7*c^2)*b^2*c^2)*a^4+(5*b^8+5*c^8+2*(4*b^4+4*c^4+3*(32*b^2+17*b*c+32*c^2)*b*c)*b^2*c^2)*a^3+(b^2-c^2)^2*(b+c)*(5*b^4+5*c^4+2*(4*b^2+21*b*c+4*c^2)*b*c)*a^2-(b^2-c^2)^2*(b^6+c^6+(2*b^4+2*c^4+3*(b^2+28*b*c+c^2)*b*c)*b*c)*a+(b^2-c^2)^3*(b-c)*(-b^4-c^4-2*(3*b^2+11*b*c+3*c^2)*b*c))*a) : :

X(44857) lies on these lines: {56, 1123}, {7962, 38003}, {7991, 38004}

X(44858) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 8th MIXTILINEAR TO 1st MIXTILINEAR

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

The reciprocal cyclologic center of these triangles is X(44859).

X(44858) lies on these lines: {1, 651}, {6, 101}, {36, 59}, {55, 103}, {56, 20958}, {58, 1066}, {116, 4648}, {118, 5274}, {150, 3945}, {221, 38502}, {244, 1282}, {269, 2093}, {386, 11505}, {388, 5733}, {495, 10708}, {497, 10710}, {517, 6610}, {518, 6510}, {537, 40865}, {544, 1056}, {595, 3157}, {673, 4649}, {942, 1126}, {1253, 5010}, {1279, 25405}, {1391, 39148}, {1411, 13404}, {1419, 2823}, {1421, 11028}, {1456, 5048}, {1471, 37587}, {1718, 18413}, {1743, 28345}, {2195, 24436}, {2263, 25415}, {2293, 2772}, {2784, 4349}, {2808, 6767}, {2809, 11529}, {3022, 8162}, {3041, 8167}, {3295, 38668}, {3600, 33520}, {4038, 17719}, {4658, 17197}, {5399, 33771}, {5723, 38055}, {6710, 37650}, {7290, 11712}, {10571, 38560}, {10964, 34040}, {17245, 31273}, {38599, 42314}

X(44858) = reflection of X(101) in X(36942)
X(44858) = intersection, other than A, B, C, of circumconics {{A, B, C, X(59), X(103)}} and {{A, B, C, X(101), X(1391)}}
X(44858) = perspector of the circumconic {{A, B, C, X(901), X(4619)}}
X(44858) = crossdifference of every pair of points on line {X(900), X(42462)}
X(44858) = X(59)-Daleth conjugate of-X(36)
X(44858) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 15730, 15735), (1, 34931, 4845), (202, 203, 101), (4845, 34931, 38666)

X(44859) = CYCLOLOGIC CENTER OF THESE TRIANGLES: 1st MIXTILINEAR TO 8th MIXTILINEAR

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

The reciprocal cyclologic center of these triangles is X(44858).

X(44859) lies on the mixtilinear circle and these lines: {55, 101}, {105, 3576}, {10058, 15015}, {25532, 26105}

X(44859) = {X(55), X(34068)}-harmonic conjugate of X(101)

X(44860) = (name pending)

Barycentrics    2 a^18 (b^2-c^2)^2+b^4 c^4 (b^2-c^2)^6 (b^2+c^2)+a^16 (-9 b^6+13 b^4 c^2+13 b^2 c^4-9 c^6)-a^2 b^2 c^2 (b^2-c^2)^4 (2 b^8+3 b^6 c^2-6 b^4 c^4+3 b^2 c^6+2 c^8)+a^14 (14 b^8-5 b^6 c^2-44 b^4 c^4-5 b^2 c^6+14 c^8)+a^4 (b^2-c^2)^4 (b^10+12 b^8 c^2+9 b^6 c^4+9 b^4 c^6+12 b^2 c^8+c^10)-a^12 (5 b^10+30 b^8 c^2-47 b^6 c^4-47 b^4 c^6+30 b^2 c^8+5 c^10)+a^8 (b^2-c^2)^2 (13 b^10+3 b^8 c^2-19 b^6 c^4-19 b^4 c^6+3 b^2 c^8+13 c^10)-a^6 (b^2-c^2)^2 (6 b^12+17 b^10 c^2-18 b^8 c^4+14 b^6 c^6-18 b^4 c^8+17 b^2 c^10+6 c^12)-a^10 (10 b^12-48 b^10 c^2+25 b^8 c^4+30 b^6 c^6+25 b^4 c^8-48 b^2 c^10+10 c^12) : :

See X(44828) and Antreas Hatzipolakis and Angel Montesdeoca, Euclid 2416 .

X(44860) lies on this line: {2, 3}

X(44861) = ISOGONAL CONJUGATE OF X(9940)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2411 .

X(44861) lies on Feuerbach circumhyperbola and these lines: {7, 3149}, {8, 6913}, {79, 6260}, {84, 41562}, {104, 44547}, {1466, 10305}, {3091, 43740}, {3255, 3651}, {6601, 38037}, {7091, 18446}, {10308, 12664}, {10309, 37541}, {10310, 34919}

X(44861) = isogonal conjugate of X(9940)
X(44861) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(4)}} and {{A, B, C, X(3), X(14493)}}

X(44862) = X(3)X(2929)∩X(5)X(5092)

Barycentrics    2*a^10-4*(b^2+c^2)*a^8+(b^4-4*b^2*c^2+c^4)*a^6+(b^2+c^2)*(b^4+10*b^2*c^2+c^4)*a^4+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(44862) = 3*X(3)+X(13403), X(52)-3*X(32068), 3*X(140)+X(13470), 15*X(631)+X(12289), 5*X(1216)+3*X(11232), 3*X(1216)+X(32358), 5*X(1656)-X(13419), 7*X(3523)+X(21659), 7*X(3526)+X(11750), 3*X(3819)+X(6146), 3*X(3917)+X(10112), 9*X(5650)-X(14516), X(6101)+3*X(43573), 3*X(6688)-X(6756), 3*X(7667)+X(13598), 3*X(8703)+X(12897), 3*X(9729)-X(13568), 9*X(11232)-5*X(32358), 3*X(11793)-X(31831), 3*X(12362)+X(13568)

See Antreas Hatzipolakis and César Lozada, Euclid 2411 .

X(44862) lies on these lines: {3, 2929}, {4, 13347}, {5, 5092}, {20, 34417}, {30, 11695}, {52, 32068}, {68, 40107}, {125, 32348}, {140, 13470}, {159, 7393}, {182, 6643}, {382, 16936}, {539, 32142}, {542, 11793}, {546, 17712}, {550, 15873}, {631, 12289}, {1216, 5965}, {1594, 32396}, {1656, 13419}, {3098, 39571}, {3523, 21659}, {3524, 27082}, {3526, 11750}, {3530, 17702}, {3547, 17508}, {3628, 44407}, {3819, 6146}, {3917, 10112}, {5446, 19924}, {5650, 14516}, {6030, 21451}, {6101, 43573}, {6688, 6756}, {6689, 37938}, {6723, 7542}, {7386, 13346}, {7509, 21243}, {7512, 32223}, {7514, 20299}, {7544, 22112}, {7667, 13598}, {8703, 12897}, {9729, 12362}, {10110, 29317}, {10116, 15067}, {10610, 14156}, {10691, 12241}, {11225, 11412}, {11264, 44324}, {11424, 16063}, {11591, 18128}, {12108, 30522}, {12161, 33749}, {12242, 13353}, {12605, 16836}, {13292, 15606}, {13336, 18388}, {14790, 19130}, {14791, 19136}, {15807, 44245}, {18531, 37515}, {18536, 37514}, {19481, 41673}, {20304, 34004}, {23061, 43838}, {27355, 34603}, {34005, 37853}, {35240, 43601}, {35921, 43608}, {37444, 43650}, {37452, 37513}, {41586, 43816}

X(44862) = midpoint of X(i) and X(j) for these {i, j}: {546, 17712}, {9729, 12362}, {11591, 18128}, {12241, 13348}, {13292, 15606}, {15807, 44245}, {19481, 41673}
X(44862) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (125, 37126, 32348), (7393, 18381, 24206), (10691, 12241, 13348)

X(44863) = X(4)X(4846)∩X(5)X(3819)

Barycentrics    (a^8*(b^2+c^2))-a^2*(b^2-c^2)^2*(b^4-10*b^2*c^2+c^4)-a^6*(3*b^4+8*b^2*c^2+3*c^4)-a^4*(-3*b^6+5*b^4*c^2+5*b^2*c^4-3*c^6) : :
Barycentrics    (SB+SC) (6 S^2-SA (14 R^2-5 SA+SW)) : :
X(44863) = 7*X(4)+9*X(5640),5*X(4)+3*X(9730),7*X(4)+X(10575),9*X(4)+7*X(15043),3*X(4)+X(40647),7*X(5)-3*X(3819),3*X(5)-X(5447),3*X(5)+X(13598),5*X(5)-X(15644),X(20)-9*X(14845),3*X(51)+5*X(3843),9*X(51)-X(34783),X(52)+7*X(3832),3*X(52)+5*X(15058),7*X(140)-9*X(12045),X(143)+3*X(546),X(143)-3*X(10110),5*X(143)-9*X(13451),X(143)+9*X(13570),5*X(143)-3*X(16625),9*X(373)-X(1657),3*X(381)+X(5446),9*X(381)-X(5562),7*X(381)+X(21969),X(382)+3*X(5892),X(389)+3*X(3845),5*X(546)+3*X(13451),X(546)-3*X(13570),5*X(546)+X(16625),3*X(547)-X(13348),X(548)-3*X(6688),X(1216)-5*X(3091),9*X(3545)-X(10625),5*X(3567)+3*X(16194),X(3627)+3*X(5943),9*X(3819)-7*X(5447),9*X(3819)+7*X(13598),3*X(3830)+X(14641),9*X(3839)+7*X(9781),9*X(3839)-X(12162),2*X(3850)+X(12002),3*X(3850)-X(14128),7*X(3851)-3*X(10170),X(3853)+3*X(13364),7*X(3857)+X(10263),5*X(3858)-X(5907),5*X(3859)-X(11591),3*X(5066)-X(11793),3*X(5446)+X(5562),5*X(5446)-X(6243),7*X(5446)-3*X(21969),5*X(5447)-3*X(15644),7*X(5462)-9*X(5640),5*X(5462)-3*X(9730),7*X(5462)-X(10575),9*X(5462)-7*X(15043),3*X(5462)-X(40647),3*X(5480)+X(43130),5*X(5562)+3*X(6243),7*X(5562)+9*X(21969),9*X(5640)-X(10575),X(5876)+3*X(21849),X(5876)-9*X(23046),3*X(5946)+X(13474),X(6101)-9*X(38071),X(9729)-3*X(13364),9*X(9730)-X(12279),9*X(9730)-5*X(40647),7*X(9781)+X(12162),5*X(10110)-3*X(13451),X(10110)+3*X(13570),5*X(10110)-X(16625),3*X(10219)-4*X(12046),3*X(10219)-2*X(12108),3*X(10575)-7*X(40647),X(11381)-9*X(14269),5*X(11695)-3*X(17704),5*X(11695)-6*X(32205),3*X(11737)-X(32142),3*X(12002)+2*X(14128),X(12279)-5*X(40647),X(13403)+3*X(13490),X(13451)+5*X(13570),3*X(13451)-X(16625),5*X(13598)+3*X(15644),X(13630)+3*X(14893),3*X(14855)+5*X(17578),5*X(15026)+3*X(15687),7*X(15043)-3*X(40647),X(15074)-9*X(38136),X(16655)+3*X(43573),3*X(17704)-10*X(18874),5*X(18874)-3*X(32205),X(21849)+3*X(23046),3*X(32062)+5*X(37481)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2417.

X(44863) lies on these lines: {3,27355}, {4,4846}, {5,3819}, {20,14845}, {30,11695}, {51,3843}, {52,3832}, {140,12045}, {143,546}, {373,1657}, {381,5446}, {382,5892}, {389,3845}, {511,3850}, {541,15465}, {547,13348}, {548,6688}, {1154,3856}, {1216,3091}, {3545,10625}, {3567,16194}, {3627,5943}, {3830,14641}, {3839,9781}, {3851,10170}, {3853,9729}, {3854,11412}, {3855,5891}, {3857,10263}, {3858,5907}, {3859,11591}, {3861,6000}, {3917,5072}, {5059,11465}, {5066,11793}, {5070,36987}, {5448,5480}, {5449,15873}, {5876,21849}, {5889,41099}, {5946,13474}, {6101,38071}, {6689,37971}, {7529,35602}, {7545,13367}, {7564,9969}, {7689,17810}, {8717,15805}, {10219,12046}, {10594,18475}, {10610,32237}, {11017,40247}, {11381,14269}, {11444,41106}, {11451,33703}, {11737,32142}, {12006,12102}, {12038,13861}, {12811,13391}, {13382,32137}, {13403,13490}, {13434,26863}, {13630,14893}, {14855,15024}, {15026,15687}, {15028,15682}, {15060,41991}, {15074,38136}, {15761,19130}, {16621,18128}, {16655,43573}, {18555,41171}, {23841,40273}, {32062,37481}, {32110,38848}, {35502,43604}, {39522,41597}, {43839,44233}

X(44863) = midpoint of X(i) and X(j) for these {i,j}: {4,5462}, {546,10110}, {3853,9729}, {3861,10095}, {5447,13598}, {12006,12102}, {13382,32137}, {16621,18128}, {23841,40273}
X(44863) = reflection of X(i) in X(j) for these (i,j): (11695,18874), (12108,12046), (17704,32205), (40247,11017)
X(44863) = X(5462)-of-Euler-triangle
X(44863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5640,10575), (5,13598,5447), (3839,9781,12162), (3853,13364,9729), (9730,12279,40647), (10110,13570,546), (10110,16625,13451), (12046,12108,10219), (15024,17578,14855)

X(44864) = X(916)X(5806)∩X(946)X(9052)

Barycentrics    a^2*(2*a^4*b^2*c^2*(b+c)^2+a^8*(b^2+c^2)+2*a*(b-c)^2*(b+c)^5*(b^2-3*b*c+c^2)-2*a^7*(b^3+c^3)-2*a^6*(b^4+5*b^2*c^2+c^4)-(b-c)^4*(b+c)^2*(b^4+2*b^3*c-4*b^2*c^2+2*b*c^3+c^4)+2*a^5*(3*b^5+5*b^3*c^2+5*b^2*c^3+3*c^5)+a^3*(-6*b^7+4*b^5*c^2+2*b^4*c^3+2*b^3*c^4+4*b^2*c^5-6*c^7)+2*a^2*(b^8-b^6*c^2+4*b^5*c^3-8*b^4*c^4+4*b^3*c^5-b^2*c^6+c^8)) : :
Barycentrics    (SB+SC) (S^2 (3 a b+3 a c+3 b c+8 R^2-5 SW)-SA (8 a b R^2+8 a c R^2+8 b c R^2-2 a b SA-3 a c SA-3 b c SA+a b SC-a c SC+a c SW-8 R^2 SW+4 SA SW-SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2417.

X(44864) lies on these lines: {4,29957}, {916,5806}, {946,9052}, {2808,5805}

X(44864) = midpoint of X(4) and X(29957)
X(44864) = X(29957)-of-Euler-triangle

X(44865) = X(375)X(9943)∩X(389)X(2808)

Barycentrics    a^2 (-2 a^5 b c (b+c)+a^6 (b^2+c^2)+4 a^3 b c (b^3+c^3)+a^4 (-3 b^4+2 b^3 c-4 b^2 c^2+2 b c^3-3 c^4)-(b^2-c^2)^2 (b^4-2 b^3 c-4 b^2 c^2-2 b c^3+c^4)-2 a b c (b^5-b^4 c-b c^4+c^5)+a^2 (3 b^6-4 b^5 c-3 b^4 c^2-3 b^2 c^4-4 b c^5+3 c^6)) : :
Barycentrics    (SB+SC) (-3 S^4+a (b-c) SA^2 SC+S^2 (-4 a b R^2+4 b c R^2+a c SA-b c SA+4 R^2 SA-2 SA^2+a b SW-b c SW+SA SW)) : :
X(44865) = 3*X(51)+X(12528),3*X(375)-X(9943),X(1071)-3*X(5943),5*X(3091)-X(23154),2*X(5806)-3*X(13570),X(5907)-3*X(5927),3*X(6688)-2*X(9940),3*X(10157)-X(11573),3*X(16625)-2*X(31819)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2417.

X(44865) lies on these lines: {4,29958}, {51,12528}, {72,13598}, {375,9943}, {389,2808}, {511,5777}, {912,10110}, {916,16625}, {946,2810}, {970,7330}, {971,9729}, {1071,5943}, {2807,31871}, {2818,18480}, {3091,23154}, {3937,6915}, {5044,13348}, {5752,5779}, {5806,13570}, {5811,10441}, {5907,5927}, {6001,23841}, {6688,9940}, {9911,43146}, {10157,11573}, {11695,13369}, {13257,18180}, {15488,37822}

X(44865) = midpoint of X(i) and X(j) for these {i,j}: {4,29958}, {72,13598}, {389,40263}
X(44865) = reflection of X(i) in X(j) for these (i,j): (12109,10110), (13348,5044), (13369,11695)
X(44865) = X(29958)-of-Euler-triangle

X(44866) = X(6)X(382)∩X(20)X(1147)

Barycentrics    (4*a^4-3*a^2*b^2-b^4-3*a^2*c^2+2*b^2*c^2-c^4)*(a^4*b^2-2*a^2*b^4+b^6+a^4*c^2+6*a^2*b^2*c^2-b^4*c^2-2*a^2*c^4-b^2*c^4+c^6) : :
Barycentrics    (3 S^2-5 SB SC) (10 R^2-SA-SW) : :
X(44866) = 4*X(5)-3*X(18488),X(20)-3*X(8718),X(382)-3*X(3521),4*X(548)-3*X(35240),5*X(631)-3*X(15062),7*X(3526)-3*X(33541),7*X(3528)-9*X(6030),3*X(8718)+X(43599),3*X(15072)-X(43846),5*X(15696)-3*X(18442)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2417.

X(44866) lies on these lines: {2,16835}, {3,14862}, {4,12834}, {5,10575}, {6,382}, {20,1147}, {30,1493}, {113,37452}, {141,12162}, {206,5878}, {541,7512}, {548,1511}, {550,44110}, {631,4550}, {942,17705}, {1209,6000}, {1533,13630}, {1906,9730}, {2777,11597}, {2883,14855}, {3526,33537}, {3528,6030}, {3534,22333}, {4846,37122}, {6593,12605}, {7493,43689}, {9936,37201}, {11598,16252}, {13202,13470}, {13445,44516}, {13488,44102}, {14861,18369}, {14915,15559}, {15072,43817}, {15311,32391}, {15696,18442}, {15760,16254}, {16003,17854}, {16196,38795}, {20427,34472}, {22834,44573}, {22948,40647}, {32223,43807}, {32478,41167}

X(44866) = midpoint of X(20) and X(43599)
X(44866) = reflection of X(i) in X(j) for these (i,j): (22948,40647), (34563,43585)
X(44866) = complement of X(16835)
X(44866) = complementary conjugate of X(546)
X(44866) = X(i)-complementary conjugate of X(j) for these (i,j): (1,546), (550,10)
X(44866) = {X(8718),X(43599)}-harmonic conjugate of X(20)

X(44867) = X(51)X(40448)∩X(3567)X(17401)

Barycentrics    a^2*(a^18*b^2-8*a^16*b^4+29*a^14*b^6-63*a^12*b^8+91*a^10*b^10-91*a^8*b^12+63*a^6*b^14-29*a^4*b^16+8*a^2*b^18-b^20+a^18*c^2-14*a^16*b^2*c^2+54*a^14*b^4*c^2-93*a^12*b^6*c^2+66*a^10*b^8*c^2+29*a^8*b^10*c^2-106*a^6*b^12*c^2+101*a^4*b^14*c^2-47*a^2*b^16*c^2+9*b^18*c^2-8*a^16*c^4+54*a^14*b^2*c^4-114*a^12*b^4*c^4+83*a^10*b^6*c^4+5*a^8*b^8*c^4+20*a^6*b^10*c^4-120*a^4*b^12*c^4+115*a^2*b^14*c^4-35*b^16*c^4+29*a^14*c^6-93*a^12*b^2*c^6+83*a^10*b^4*c^6-14*a^8*b^6*c^6+23*a^6*b^8*c^6+35*a^4*b^10*c^6-143*a^2*b^12*c^6+80*b^14*c^6-63*a^12*c^8+66*a^10*b^2*c^8+5*a^8*b^4*c^8+23*a^6*b^6*c^8+26*a^4*b^8*c^8+67*a^2*b^10*c^8-124*b^12*c^8+91*a^10*c^10+29*a^8*b^2*c^10+20*a^6*b^4*c^10+35*a^4*b^6*c^10+67*a^2*b^8*c^10+142*b^10*c^10-91*a^8*c^12-106*a^6*b^2*c^12-120*a^4*b^4*c^12-143*a^2*b^6*c^12-124*b^8*c^12+63*a^6*c^14+101*a^4*b^2*c^14+115*a^2*b^4*c^14+80*b^6*c^14-29*a^4*c^16-47*a^2*b^2*c^16-35*b^4*c^16+8*a^2*c^18+9*b^2*c^18-c^20) : :
Barycentrics    (SB+SC) (5 S^4+S^2 (2 SA^2+12 R^2 SW-3 SA SW-3 SW^2)+SA (4 R^2-SW) (16 R^4-8 R^2 SW+2 SA SW-SW^2)) : :
X(44867) = 3*X(51)-X(40448),5*X(3567)-X(17401)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2417.

X(44867) lies on these lines: {51,40448}, {3567,17401}

X(44868) = X(4)X(44702)∩X(5478)X(5663)

Barycentrics    a^2*(3*b^2*c^2*(3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2))-S*sqrt(3)*((b^2+c^2)*a^6-3*(b^2+c^2)^2*a^4+(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*((b^2+c^2)^2-9*b^2*c^2))) : :
X(44868) = 3*X(51)-X(40448),5*X(3567)-X(17401)

See Antreas Hatzipolakis and César Lozada, euclid 2418.

X(44868) lies on these lines: {4, 44702}, {5478, 5663}, {5480, 14984}, {7684, 13391}, {11624, 36961}, {36978, 41036}

X(44869) = X(4)X(44703)∩X(5479)X(5663)

Barycentrics    a^2*(3*b^2*c^2*(3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2))+S*sqrt(3)*((b^2+c^2)*a^6-3*(b^2+c^2)^2*a^4+(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*((b^2+c^2)^2-9*b^2*c^2))) : :
X(44869) = 3*X(11626)+X(36962), X(36980)+3*X(41037)

See Antreas Hatzipolakis and César Lozada, euclid 2418.

X(44869) lies on these lines: {4, 44703}, {5479, 5663}, {5480, 14984}, {7685, 13391}, {11626, 36962}, {36980, 41037}

X(44870) = X(4)X(69)∩X(5)X(2883)

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4-4*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2) : :
Barycentrics    SA^2 (SB + SC) - SB SC (SB + SC - 16 R^2) : :
Barycentrics    sec^2 A (tan B + tan C) + sec^2 B (2 tan A + tan C) + sec^2 C (2 tan A + tan B) - 2 (2 tan A + tan B + tan C) sec A sec B sec C : :
X(44870) = 3*X(2)+X(11381), 3*X(2)+5*X(11439), 9*X(2)-X(12279), X(3)+3*X(16194), 3*X(4)+X(5562), 7*X(4)+X(11412), 5*X(4)+3*X(11459), 3*X(4)-X(13598), X(4)+3*X(15030), 3*X(4)+5*X(15058), X(4)-9*X(16261), 5*X(3818)-X(41714), 3*X(3818)-X(43130), X(11381)-5*X(11439), 3*X(11381)+X(12279), X(11381)+2*X(17704), 15*X(11439)+X(12279), 5*X(11439)+2*X(17704), X(12279)-6*X(17704), X(13474)-3*X(16194)

See Antreas Hatzipolakis and César Lozada, euclid 2418.

Let A'B'C' be the anticomplementary triangle. Let OA be the circle centered at A' and tangent to line BC, and define OB and OC cyclically. X(44870) is the radical center of circles OA, OB, OC. (Randy Hutson, September 30, 2021)

X(44870) lies on these lines: {2, 11381}, {3, 13474}, {4, 69}, {5, 2883}, {20, 3819}, {26, 4550}, {30, 5447}, {51, 3832}, {52, 3843}, {64, 5020}, {113, 5576}, {140, 14915}, {143, 546}, {182, 1498}, {185, 3091}, {186, 43613}, {216, 38297}, {235, 21243}, {343, 1906}, {373, 5068}, {381, 389}, {382, 5891}, {394, 11403}, {517, 6743}, {541, 23410}, {542, 12241}, {549, 14641}, {550, 10170}, {575, 1181}, {576, 12164}, {578, 18451}, {631, 11455}, {916, 5806}, {924, 32181}, {942, 2808}, {970, 19541}, {1092, 35502}, {1147, 31861}, {1154, 3861}, {1199, 14094}, {1204, 1995}, {1209, 11799}, {1216, 3627}, {1495, 14118}, {1568, 15559}, {1593, 9306}, {1597, 13346}, {1656, 10575}, {1853, 36982}, {2070, 33539}, {2071, 43614}, {2072, 18488}, {2777, 31833}, {2807, 19925}, {2818, 31937}, {2979, 17578}, {3090, 10219}, {3098, 39568}, {3146, 3917}, {3149, 15489}, {3357, 6642}, {3516, 35259}, {3521, 16623}, {3522, 5650}, {3523, 15082}, {3526, 14855}, {3543, 11444}, {3544, 15045}, {3545, 6241}, {3628, 11017}, {3830, 10625}, {3839, 5889}, {3845, 5446}, {3850, 5462}, {3851, 9730}, {3853, 11591}, {3854, 5640}, {3855, 5890}, {3856, 10095}, {3857, 5946}, {3858, 6102}, {3859, 13364}, {3860, 16881}, {5056, 15072}, {5059, 7998}, {5066, 13630}, {5076, 23039}, {5092, 7395}, {5097, 10982}, {5133, 43831}, {5448, 20302}, {5651, 11413}, {5656, 38317}, {5878, 7401}, {5893, 9822}, {5927, 29958}, {5965, 13142}, {5972, 12133}, {6053, 12242}, {6101, 15687}, {6225, 7392}, {6243, 14269}, {6285, 19372}, {6288, 22816}, {6677, 6696}, {6759, 9818}, {6823, 24206}, {7355, 9817}, {7398, 11469}, {7403, 18388}, {7486, 20791}, {7488, 32237}, {7503, 26883}, {7526, 10282}, {7527, 13367}, {7529, 11438}, {7550, 8718}, {7592, 15516}, {7687, 22530}, {7689, 13861}, {7723, 11807}, {7999, 33703}, {9781, 14831}, {9815, 18920}, {9825, 15311}, {10019, 44084}, {10112, 16657}, {10151, 13446}, {10539, 11430}, {10564, 18551}, {10601, 12174}, {10606, 32602}, {10984, 20190}, {11250, 43586}, {11414, 14810}, {11424, 11441}, {11440, 13595}, {11484, 13093}, {11574, 36990}, {11645, 16655}, {11737, 32205}, {11800, 12825}, {12002, 14449}, {12006, 12811}, {12046, 41989}, {12102, 13391}, {12134, 13403}, {12233, 19130}, {12280, 43581}, {12292, 36518}, {12315, 37514}, {12362, 16621}, {12605, 13419}, {13292, 40240}, {13366, 43605}, {13376, 23323}, {13433, 22815}, {13621, 32110}, {13731, 33536}, {14157, 35500}, {14216, 18537}, {14788, 32111}, {14845, 37481}, {14865, 43598}, {14893, 31834}, {15026, 38071}, {15033, 43844}, {15035, 35478}, {15052, 34148}, {15062, 21663}, {15063, 20301}, {15074, 39884}, {15083, 39522}, {15738, 38791}, {15807, 32423}, {16226, 41106}, {16238, 25563}, {16654, 29323}, {17834, 18535}, {17855, 23515}, {18420, 22802}, {18445, 37505}, {20191, 44232}, {21734, 44299}, {22076, 36002}, {22967, 32184}, {28258, 33811}, {30443, 40132}, {31831, 44804}, {31942, 40801}, {32063, 37476}, {37434, 37521}, {38444, 44082}, {43839, 44236}, {44158, 44233}

X(44870) = midpoint of X(i) and X(j) for these {i, j}: {3, 13474}, {4, 5907}, {140, 32137}, {382, 15644}, {389, 12162}, {1216, 3627}, {3819, 32062}, {3853, 11591}, {5446, 5876}, {5562, 13598}, {5943, 15305}, {5972, 12133}, {7723, 11807}, {11574, 36990}, {11800, 12825}, {12134, 13403}, {12294, 14913}, {12362, 16621}, {12605, 13419}, {13433, 22815}, {15738, 38791}
X(44870) = reflection of X(i) in X(j) for these (i, j): (185, 15012), (1216, 40247), (3628, 11017), (5447, 14128), (5462, 3850), (9729, 5), (10095, 3856), (10110, 546), (12006, 12811), (12109, 5806), (13292, 40240), (13348, 11793), (13376, 23323), (13382, 5462), (13446, 10151), (14449, 12002), (15606, 11591), (16625, 10110), (22967, 32184), (23841, 19925), (40647, 11695)
X(44870) = complement of the complement of X(11381)
X(44870) = anticomplement of X(17704)
X(44870) = X(5907)-of-Euler-triangle
X(44870) = X(9729)-of-Johnson-triangle
X(44870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11439, 11381), (3, 16194, 13474), (4, 5562, 13598), (4, 15030, 5907), (4, 15058, 5562), (5, 9729, 6688), (5, 13491, 5892), (5, 40647, 11695), (185, 3091, 5943), (185, 5943, 15012), (185, 27355, 15043), (381, 12162, 389), (382, 5891, 15644), (546, 10110, 13570), (1216, 15060, 40247), (1498, 11479, 182), (3091, 15043, 27355), (3091, 15305, 185), (3627, 15060, 1216), (3832, 12111, 51), (3843, 18435, 52), (5068, 10574, 373), (5447, 14128, 11793), (5562, 15030, 15058), (5562, 15058, 5907), (5907, 13598, 5562), (6642, 11472, 3357), (11695, 40647, 9729), (13570, 16625, 10110), (15043, 27355, 5943), (15811, 33537, 3)

X(44871) = X(4)X(5892)∩X(143)X(546)

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4+16*b^2*c^2+3*c^4)*a^4+3*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^4-22*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(44871) = 5*X(4)+3*X(5892), 17*X(4)+15*X(11451), 7*X(4)+X(14641), 7*X(4)+9*X(14845), 13*X(4)+3*X(14855), 19*X(4)+13*X(15028), 17*X(5)-9*X(15082), X(52)+15*X(41099), X(143)+7*X(546), 3*X(143)-7*X(10110), 11*X(143)-7*X(16625), 3*X(546)+X(10110), 13*X(546)+3*X(13451), X(546)+3*X(13570), 11*X(546)+X(16625), 7*X(5892)-15*X(14845), 13*X(5892)-5*X(14855), 13*X(10110)-9*X(13451), X(10110)-9*X(13570), 11*X(10110)-3*X(16625), X(14641)-9*X(14845)

See Antreas Hatzipolakis and César Lozada, euclid 2418.

X(44871) lies on these lines: {4, 5892}, {5, 15082}, {30, 12046}, {52, 41099}, {143, 546}, {185, 3843}, {381, 1216}, {511, 3856}, {3091, 5447}, {3567, 3839}, {3832, 5446}, {3845, 5462}, {3850, 32142}, {3854, 10625}, {3855, 10170}, {3857, 13598}, {3858, 10263}, {3859, 11793}, {3861, 12006}, {5076, 27355}, {5907, 23046}, {9729, 14893}, {10219, 44245}, {11695, 12102}, {15644, 38071}, {32137, 41987}

X(44871) = midpoint of X(11695) and X(12102)
X(44871) = {X(4), X(14845)}-harmonic conjugate of X(14641)

X(44872) = X(4)X(11270)∩X(143)X(546)

Barycentrics    4*a^10-3*(b^2+c^2)*a^8-(9*b^4-20*b^2*c^2+9*c^4)*a^6+5*(b^4-c^4)*(b^2-c^2)*a^4+3*(b^2-c^2)^2*(3*b^4-4*b^2*c^2+3*c^4)*a^2-6*(b^4-c^4)*(b^2-c^2)^3 : :
X(44872) = 3*X(4)+X(21663), 9*X(381)-X(22115), X(1514)+5*X(7687), X(1514)-5*X(10151), 3*X(13851)+X(14157)

See Antreas Hatzipolakis and César Lozada, euclid 2418.

X(44872) lies on these lines: {4, 11270}, {30, 6723}, {143, 546}, {184, 35488}, {381, 9306}, {511, 23323}, {974, 1514}, {3843, 17810}, {5480, 23046}, {6623, 18376}, {7505, 32903}, {10019, 13403}, {13198, 13851}, {13473, 44673}, {14269, 15432}, {18383, 37197}, {18400, 37984}, {22802, 23291}, {29323, 43893}

X(44872) = midpoint of X(i) and X(j) for these {i, j}: {7687, 10151}, {13473, 44673}

X(44873) = X(3)X(6079)∩X(4)X(5516)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2428 .

Let LA be the reflection of line X(2254)X(3030) in BC, and define LB and LC cyclically. Let A" = LB∩LC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(44873). (Randy Hutson, September 30, 2021)

X(44873) lies on the circumcircle and these lines: {3, 6079}, {4, 5516}, {56, 44046}, {100, 4487}, {106, 3667}, {145, 901}, {519, 1293}, {2743, 8715}, {5205, 9104}, {6551, 44724}, {24813, 28233}

X(44873) = reflection of X(i) in X(j) for these (i, j): (4, 5516), (6079, 3)
X(44873) = X(519)-Dao conjugate of-X(14507)
X(44873) = antipode of X(6079) in circumcircle
X(44873) = trilinear pole of the line {6, 14425}
X(44873) = Collings transform of X(5516)
X(44873) = V-transform of X(6079)

X(44874) = X(3)X(6080)∩X(110)X(6760)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2428 .

X(44874) lies on the circumcircle and these lines: {3, 6080}, {4, 35579}, {107, 6000}, {110, 6760}, {476, 12279}, {520, 1294}, {1301, 14157}, {1304, 6759}, {6241, 22239}, {22456, 36893}

X(44874) = reflection of X(i) in X(j) for these (i, j): (4, 35579), (6080, 3)
X(44874) = Collings transform of X(35579)
X(44874) = V-transform of X(6080)

X(44875) = X(110)X(8595)∩X(530)X(9202)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2428 .

X(44875) lies on the circumcircle and these lines: {110, 8595}, {530, 9202}, {2378, 27551}, {5980, 9080}

X(44875) = trilinear pole of the line {6, 9195}
X(44875) = Λ(X(15), X(2502))

X(44876) = X(3)X(675)∩X(103)X(376)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2428 .

X(44876) lies on the circumcircle and these lines: {2, 25642}, {3, 675}, {4, 5513}, {40, 29068}, {55, 33966}, {56, 6025}, {98, 7430}, {102, 5759}, {103, 376}, {104, 30273}, {105, 1006}, {107, 4249}, {110, 4237}, {111, 6998}, {190, 29241}, {378, 917}, {476, 36032}, {514, 39640}, {516, 38884}, {631, 31380}, {842, 36026}, {1297, 30266}, {1302, 4243}, {1311, 7420}, {2291, 5657}, {2688, 7464}, {4223, 9061}, {4241, 9064}, {4245, 9083}, {4250, 9107}, {7431, 39438}, {7437, 9058}, {7453, 9084}, {7460, 9056}, {7479, 9060}, {15344, 36009}, {21312, 41905}, {24808, 36028}, {26703, 36029}, {26708, 35921}, {32656, 32682}

X(44876) = reflection of X(i) in X(j) for these (i, j): (4, 5513), (675, 3)
X(44876) = anticomplement of X(25642)
X(44876) = isogonal conjugate of the complementary conjugate of X(25642)
X(44876) = circumperp conjugate of X(675)
X(44876) = circumnormal-isogonal conjugate of X(674)
X(44876) = X(78)-Gimel conjugate of-X(29068)
X(44876) = X(675)-reciprocal conjugate of-X(2412)
X(44876) = antipode of X(675) in circumcircle
X(44876) = trilinear pole of the line {6, 2438}
X(44876) = Collings transform of X(5513)
X(44876) = Λ(normal to hyperbola {{A,B,C,X(1),X(6)}} at X(6))
X(44876) = Λ(line of degenerate pedal triangle of X(675))
X(44876) = V-transform of X(674)
X(44876) = barycentric quotient X(675)/X(2412)

X(44877) = ISOGONAL CONJUGATE OF X(40135)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2446 .

X(44877) lies on Kiepert circumhyperbola and these lines: {2, 34570}, {4, 5972}, {98, 5159}, {459, 648}, {2052, 36789}, {2394, 41077}, {2996, 41254}, {3424, 30769}, {11064, 16080}, {14492, 44212}, {32831, 44326}, {37669, 38253}

X(44877) = isogonal conjugate of X(40135)
X(44877) = isotomic conjugate of the complement of X(11064)
X(44877) = polar conjugate of X(10151)
X(44877) = tripole of the tangent to Euler-Gergonne-Soddy circle at X(20)
X(44877) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(249), X(14919)}}
X(44877) = trilinear pole of the line {20, 523}
X(44877) = barycentric product X(76)*X(34570)
X(44877) = barycentric quotient X(i)/X(j) for these (i, j): (3, 21663), (4, 10151), (30, 13202), (69, 40996), (75, 18699), (110, 5502)
X(44877) = trilinear product X(75)*X(34570)
X(44877) = trilinear quotient X(i)/X(j) for these (i, j): (63, 21663), (76, 18699), (92, 10151), (304, 40996), (662, 5502)

X(44878) =  EULER LINE INTERCEPT OF X(74)X(41447)

Barycentrics    (7*a^4-14*(b^2+c^2)*a^2+7*b^4+4*b^2*c^2+7*c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*a^2 : :
Barycentrics    (5 S^2-9 SA^2) SB SC (SB+SC) : :

As a point on the Euler line, X(44878) has Shinagawa coefficients (14*F, -5*E-14*F).

See Antreas Hatzipolakis, Ercole Suppa and César Lozada, Euclid 2445 and Euclid 2446 .

X(44878) lies on these lines: {2, 3}, {74, 41447}, {1192, 26882}, {1620, 12290}, {3431, 9777}, {5102, 19128}, {5210, 33885}, {5412, 6481}, {5413, 6480}, {6431, 10881}, {6432, 10880}, {8739, 34754}, {8740, 34755}, {11202, 13366}, {11438, 44110}, {11455, 41424}, {11464, 17809}, {11738, 35450}, {12834, 37506}, {14157, 37487}, {14483, 17810}, {15034, 37672}, {15035, 33586}, {32110, 35264}, {37517, 44102}, {38725, 44795}

X(44878) = reflection of X(33184) in X(33830)
X(44878) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(5066)}} and {{A, B, C, X(54), X(7486)}}
X(44878) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3,25,13596), (24,186,378), (24,3515,35479), (24,32534,10594), (24,35472,25), (24,35477,3518), (24,35479,32534), (24,35502,3517), (25,186,35472), (25,35472,378), (186,378,32534), (186,44272,37970), (378,35479,186), (1597,35477,378), (2070,15078,12082), (3517,21844,35502), (3518,15750,35477), (11410,35502,378), (21844,35478,3)

X(44879) =  EULER LINE INTERCEPT OF X(74)X(13452)

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

As a point on the Euler line, X(44879) has Shinagawa coefficients (8*F, -3*E-8*F).

See Antreas Hatzipolakis, Ercole Suppa and César Lozada, Euclid 2445 and Euclid 2446 .

X(44879) lies on these lines: {2, 3}, {39, 10986}, {52, 11449}, {54, 11202}, {61, 10633}, {62, 10632}, {74, 13452}, {107, 20480}, {110, 15083}, {182, 11663}, {185, 26882}, {232, 35007}, {389, 11423}, {393, 11063}, {568, 1493}, {575, 6403}, {576, 19128}, {578, 1173}, {944, 9590}, {1147, 15801}, {1192, 11456}, {1199, 19357}, {1204, 14157}, {1249, 41758}, {1495, 6241}, {1511, 6243}, {1609, 33630}, {1614, 11438}, {1620, 41447}, {1843, 20190}, {2914, 32609}, {2916, 33750}, {2931, 6193}, {3043, 15034}, {3053, 8744}, {3060, 12038}, {3098, 43811}, {3357, 11270}, {3567, 13367}, {3574, 10182}, {5007, 39575}, {5237, 10641}, {5238, 10642}, {5286, 44523}, {5412, 6454}, {5413, 6453}, {5609, 7722}, {5889, 41597}, {5890, 10282}, {5944, 11003}, {5946, 6152}, {5963, 14593}, {6102, 9544}, {6361, 9625}, {6419, 10881}, {6420, 10880}, {6696, 16658}, {6776, 15582}, {7592, 17821}, {7731, 17701}, {7745, 44538}, {7771, 44142}, {7772, 10312}, {8164, 9659}, {8537, 22234}, {8743, 22331}, {8753, 11643}, {9707, 9786}, {9716, 34397}, {9781, 11430}, {10117, 12250}, {10541, 39588}, {10984, 43597}, {10985, 37512}, {11002, 37472}, {11381, 11468}, {11424, 38848}, {11431, 19468}, {11444, 43586}, {11550, 43608}, {11576, 13363}, {11704, 18383}, {11750, 26913}, {12111, 32110}, {12112, 37487}, {12140, 20397}, {12163, 35264}, {12168, 32605}, {12254, 18912}, {12279, 43604}, {12290, 21663}, {12300, 15067}, {13200, 15562}, {13289, 14216}, {13336, 43584}, {13346, 15035}, {13382, 44110}, {14644, 34786}, {14651, 39854}, {14656, 14703}, {14671, 34131}, {14805, 15026}, {14810, 44091}, {14853, 35228}, {14912, 15577}, {15020, 15463}, {15028, 37513}, {15043, 18475}, {15178, 41722}, {15581, 39874}, {16655, 43607}, {18400, 26917}, {19123, 21851}, {20421, 22334}, {23061, 44077}, {23294, 44673}, {26879, 34782}, {26881, 40647}, {32139, 35265}, {34513, 37471}, {35268, 43804}

X(44879) = intersection, other than A, B, C, of circumconics {{A, B, C, X(5), X(13472)}} and {{A, B, C, X(6), X(3851)}}
X(44879) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,35475), (3,23,3529), (3,24,3518), (3,25,35502), (3,1995,35500), (3,2070,17714), (3,3515,35479), (3,3517,5198), (3,3518,4), (3,5198,378), (3,7530,12086), (3,10594,14865), (3,12088,17538), (3,12106,3091), (3,12107,38435), (3,13154,37126), (3,13861,7527), (3,17714,20), (3,35475,35473), (3,35479,186), (3,35502,3520), (3,37440,3146), (4,186,21844), (4,3147,14940), (4,21844,35473), (4,23040,378), (4,35489,20), (5,18559,4), (24,186,4), (24,378,3517), (24,3515,186), (24,15750,34484), (24,32534,25), (24,35479,3), (25,3520,4), (25,32534,3520), (25,35502,26863), (26,22467,376), (186,2070,35489), (186,3517,23040), (186,3518,3), (186,3520,32534), (186,17506,15750), (186,18559,10298), (186,34484,17506), (378,3517,34484), (378,15750,17506), (378,17506,23040), (378,34484,4), (403,34797,4), (468,6240,16868), (568,32171,9545), (1598,13596,4), (1598,35477,13596), (2070,37814,20), (2071,7517,33703), (3089,35481,4), (3517,15750,378), (3517,17506,4), (3518,3520,26863), (3518,14865,10594), (3518,16868,14002), (3518,26863,25), (3520,26863,35502), (3542,35471,4), (3542,37460,35471), (3575,7577,4), (3575,10018,7577), (3575,37935,10018), (3627,18571,3), (5198,15750,3), (5944,37481,11003), (6240,16868,4), (6642,35921,5067), (6642,38444,35921), (6644,7488,631), (7487,35486,37119), (7487,37119,4), (7502,43809,3523), (7505,18533,4), (7506,14118,3545), (7506,18324,14118), (7512,17928,3524), (7526,13595,3855), (9707,9786,15032), (9714,11413,37925), (10594,14865,4), (11250,18378,3543), (12086,37952,3), (12107,38435,7556), (13595,38448,7526), (13621,18570,3832), (14070,17928,7512), (15078,37939,11001), (15191,15208,3536), (15192,15207,3535), (15750,23040,21844), (15750,34484,23040), (17506,34484,378), (17714,37814,3), (18378,37955,11250), (21841,37931,18560), (21844,35475,3) ,(22467,37940,26), (26863,35502,4), (32534,35502,3), (35489,37814,21844), (35732,42282,18404), (37936,43615,1657)

X(44880) =  EULER LINE INTERCEPT OF X(1199)X(11202)

Barycentrics    (5*a^4+5*b^4+3*b^2*c^2+5*c^4-10*(b^2+c^2)*a^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*a^2 : :
Barycentrics    (7 S^2-13 SA^2) SB SC (SB+SC) : :

As a point on the Euler line, X(44880) has Shinagawa coefficients (20*F, -7*E-20*F).

See Antreas Hatzipolakis, Ercole Suppa and César Lozada, Euclid 2445 and Euclid 2446 .

X(44880) lies on these lines: {2, 3}, {1199, 11202}, {8537, 15516}, {10282, 43602}, {11468, 44082}, {14531, 15034}, {15107, 43898}

X(44880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,11410,14865), (4,35479,186), (24,186,3520), (24,3520,3518), (24,15750,4), (24,21844,34484), (24,32534,1598), (24,35479,15750), (186,3518,17506), (186,14865,32534), (186,34484,21844), (1598,23040,14865), (1598,32534,23040), (2070,43615,12087), (3517,35473,26863), (3518,17506,13596), (10594,35501,4), (14865,23040,3520), (18378,18571,35497), (21844,34484,3520), (37814,37940,12088), (44232,44265,20)

X(44881) = X(1824)X(1893)∩X(1830)X(1859)

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

See Ivan Pavlov and César Lozada, Euclid 2447 .

X(44881) lies on these lines: {1824, 1893}, {1830, 1859}, {1876, 5089}, {1902, 5185}

X(44881) = Zosma transform of X(43672)

X(44882) = MIDPOINT OF X(6) AND X(20)

Barycentrics    4*a^6 - a^4*b^2 - 2*a^2*b^4 - b^6 - a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 - c^6 : :
X(44882) = 3 X[2] + X[14927], 3 X[3] - X[1352], 5 X[3] - X[18440], 4 X[3] - 3 X[21167], X[4] - 3 X[5085], X[6] - 3 X[25406], 7 X[6] - 9 X[33748], X[20] + 3 X[25406], 7 X[20] + 9 X[33748], X[67] - 3 X[15055], X[69] - 5 X[3522], X[69] - 3 X[31884], 2 X[140] - 3 X[17508], 3 X[141] - 2 X[1352], 5 X[141] - 2 X[18440], 2 X[141] - 3 X[21167], 3 X[154] - X[41735], 3 X[165] - X[3416], 4 X[182] - 3 X[597], 5 X[182] - 3 X[5476], 3 X[182] - 2 X[18583], 3 X[186] - 2 X[32218], 3 X[376] - X[1350], 3 X[376] + X[6776], X[382] - 5 X[12017], X[382] - 3 X[14561], 2 X[546] - 3 X[38317], 8 X[548] - X[3630], 3 X[549] - 2 X[24206], 3 X[549] - X[39884], 3 X[550] + X[1353], 4 X[550] + X[3629], 2 X[550] + X[8550], 2 X[575] + X[15704], X[576] + 2 X[12103], 5 X[597] - 4 X[5476], 3 X[597] - 2 X[5480], 9 X[597] - 8 X[18583], 3 X[599] - X[5921], X[599] - 3 X[10304], 5 X[631] - 3 X[10516], 5 X[631] - 9 X[33750], 5 X[631] - 4 X[34573], X[962] - 3 X[38315], X[1350] + 3 X[43273], X[1351] + 3 X[3534], 2 X[1351] - 3 X[8584], X[1351] - 3 X[11179], 5 X[1352] - 3 X[18440], 4 X[1352] - 9 X[21167], 4 X[1353] - 3 X[3629], 2 X[1353] - 3 X[8550], X[1657] + 3 X[5050], 4 X[3098] - X[3630], X[3146] - 5 X[3618], X[3146] - 7 X[10541], X[3242] - 3 X[5731], X[3448] - 3 X[5621], 5 X[3522] - 3 X[31884], 7 X[3523] - 5 X[3763], 3 X[3524] - 2 X[20582], 9 X[3524] - 5 X[40330], 7 X[3528] - 2 X[3631], 7 X[3528] - 3 X[10519], 7 X[3528] - X[15069], 7 X[3528] + X[39874], X[3529] + 4 X[6329], X[3529] + 3 X[14853], 2 X[3534] + X[8584], 3 X[3534] + 2 X[12007], 2 X[3589] - 3 X[5085], 5 X[3618] - 7 X[10541], 7 X[3619] - 11 X[15717], 5 X[3620] - 13 X[21734], X[3627] - 4 X[20190], X[3627] - 3 X[38110], 2 X[3631] - 3 X[10519], 2 X[3631] + X[39874], X[3818] - 3 X[17508], X[3830] - 3 X[38064], 2 X[3844] - 3 X[10164], 3 X[5050] - X[31670], 2 X[5097] + 3 X[15686], 3 X[5103] - 2 X[13449], 6 X[5476] - 5 X[5480], 9 X[5476] - 10 X[18583], 3 X[5480] - 4 X[18583], X[5691] - 3 X[38047], X[5895] - 5 X[19132], X[5921] - 9 X[10304], 3 X[6034] - X[10723], 4 X[6329] - 3 X[14853], 2 X[6698] - 3 X[38727], X[6776] - 3 X[43273], X[7728] - 3 X[15462], 3 X[8584] - 4 X[12007], 3 X[8703] - 2 X[14810], 4 X[8703] - X[22165], 3 X[8703] - 4 X[33751], X[9589] - 5 X[16491], 3 X[9730] - 2 X[32191], X[9967] + 3 X[14855], X[9971] - 3 X[20791], 2 X[10007] - 3 X[21163], 3 X[10167] - X[24476], X[10516] - 3 X[33750], 3 X[10516] - 4 X[34573], 3 X[10519] - X[15069], 3 X[10519] + X[39874], 3 X[11179] - 2 X[12007], X[11180] - 5 X[19708], X[11477] - 3 X[14912], X[11477] + 5 X[17538], 3 X[11539] - 2 X[25561], X[11646] - 3 X[34473], X[11898] - 9 X[15688], 5 X[12017] - 3 X[14561], X[12583] - 3 X[16190], X[12699] - 3 X[38029], 8 X[14810] - 3 X[22165], 3 X[14848] + X[15685], 3 X[14912] + 5 X[17538], 3 X[14912] - 2 X[32455], X[14982] - 3 X[15035], 5 X[15051] - X[41737], 3 X[15072] + X[41716], 4 X[15516] + 3 X[19710], X[15534] + 5 X[15697], 4 X[15578] - 3 X[23328], X[15682] - 3 X[38072], 5 X[15692] - 3 X[21358], 5 X[15696] - X[33878], 5 X[15712] - 2 X[18553], 2 X[16111] + X[25329], 8 X[16239] - 7 X[42786], 2 X[16252] - 3 X[23041], 3 X[16386] + X[32220], 5 X[17538] + 2 X[32455], 4 X[18440] - 15 X[21167], X[18525] - 3 X[38116], 2 X[19130] - 3 X[38110], 4 X[20190] - 3 X[38110], 4 X[20300] - 3 X[23324], 6 X[20582] - 5 X[40330], 3 X[22165] - 16 X[33751], 3 X[22676] + X[32451], X[22802] - 3 X[23042], 3 X[23046] - 4 X[25565], 7 X[25406] - 3 X[33748], 4 X[25555] - 3 X[38136], X[31671] - 3 X[38115], X[31672] - 3 X[38117], X[31673] - 3 X[38118], X[32113] - 3 X[44280], X[33697] - 3 X[38167], X[33699] - 3 X[38079], 9 X[33750] - 4 X[34573], 4 X[33923] - X[34507], 5 X[35242] - X[39885], X[36883] - 3 X[38716], 3 X[38035] - X[41869]

X(44882) lies on the cubic K1241 and these lines: {2, 14927}, {3, 66}, {4, 3589}, {5, 5092}, {6, 20}, {22, 13567}, {23, 37648}, {30, 182}, {40, 5846}, {64, 5596}, {67, 15055}, {69, 3522}, {74, 32233}, {98, 13468}, {125, 44210}, {140, 3818}, {154, 7386}, {165, 3416}, {184, 7667}, {185, 3313}, {186, 32218}, {206, 2883}, {230, 37182}, {287, 35937}, {343, 3448}, {373, 10301}, {376, 524}, {382, 12017}, {411, 5096}, {427, 22352}, {428, 43650}, {468, 35268}, {511, 550}, {516, 1386}, {518, 4297}, {542, 8703}, {546, 38317}, {548, 3098}, {549, 11645}, {575, 15704}, {576, 12103}, {599, 5921}, {611, 4299}, {613, 4302}, {631, 10516}, {698, 11257}, {732, 5188}, {742, 30271}, {858, 13394}, {944, 9053}, {962, 38315}, {1071, 9021}, {1176, 12225}, {1204, 26926}, {1351, 3534}, {1368, 5972}, {1370, 3796}, {1428, 6284}, {1469, 15326}, {1495, 30739}, {1629, 37873}, {1657, 5050}, {1691, 5254}, {1692, 7756}, {1843, 16622}, {1853, 7494}, {1885, 1974}, {1890, 5834}, {1906, 44091}, {2330, 7354}, {2393, 44241}, {2549, 40825}, {2550, 5792}, {2777, 6593}, {2781, 9967}, {2794, 5026}, {2854, 15151}, {2916, 2929}, {3056, 15338}, {3070, 13910}, {3071, 13972}, {3146, 3618}, {3242, 5731}, {3357, 34776}, {3398, 8725}, {3523, 3763}, {3524, 20582}, {3528, 3631}, {3529, 6329}, {3530, 18358}, {3547, 34775}, {3575, 3867}, {3580, 7492}, {3619, 15717}, {3620, 21734}, {3627, 19130}, {3815, 5116}, {3827, 9943}, {3830, 38064}, {3844, 10164}, {3845, 10168}, {3917, 24981}, {4189, 26543}, {4220, 5743}, {4265, 6909}, {5033, 7748}, {5052, 6781}, {5097, 15686}, {5103, 13449}, {5182, 8353}, {5189, 14389}, {5204, 12589}, {5217, 12588}, {5227, 37551}, {5305, 41412}, {5422, 20062}, {5473, 35696}, {5474, 35692}, {5622, 44458}, {5640, 37900}, {5651, 43957}, {5663, 35254}, {5691, 38047}, {5732, 5845}, {5847, 12512}, {5848, 38759}, {5870, 36703}, {5871, 36701}, {5894, 11574}, {5895, 19132}, {5907, 44762}, {5969, 38738}, {5984, 37671}, {6030, 15059}, {6034, 10723}, {6146, 10323}, {6393, 7782}, {6403, 8705}, {6467, 21652}, {6560, 19145}, {6561, 19146}, {6643, 16252}, {6676, 23332}, {6698, 38727}, {6699, 32274}, {6703, 26118}, {6748, 37200}, {6756, 37515}, {6770, 33458}, {6773, 33459}, {6800, 11064}, {6823, 23300}, {6995, 17825}, {7289, 9841}, {7379, 17398}, {7385, 17337}, {7387, 15873}, {7390, 17259}, {7391, 37649}, {7395, 16621}, {7417, 32525}, {7422, 18122}, {7470, 12252}, {7484, 31383}, {7499, 11550}, {7500, 10601}, {7509, 16655}, {7525, 12359}, {7540, 13339}, {7550, 16658}, {7553, 13336}, {7580, 36741}, {7710, 7778}, {7715, 11695}, {7728, 15462}, {7737, 14532}, {7750, 12215}, {7792, 40236}, {7833, 39141}, {8142, 9001}, {8177, 39646}, {8357, 36997}, {8920, 30736}, {9019, 19161}, {9024, 24466}, {9055, 30273}, {9306, 10691}, {9541, 39875}, {9589, 16491}, {9606, 12055}, {9729, 9969}, {9730, 32191}, {9771, 9774}, {9822, 17704}, {9825, 13347}, {9970, 20127}, {9971, 20791}, {10007, 21163}, {10167, 24476}, {10565, 26958}, {10575, 15105}, {10752, 41595}, {10984, 12233}, {10991, 14928}, {10996, 17845}, {11178, 12100}, {11180, 19708}, {11206, 17811}, {11414, 12241}, {11477, 14912}, {11479, 16656}, {11539, 25561}, {11579, 12121}, {11646, 34473}, {11745, 31305}, {11898, 15688}, {12082, 16657}, {12177, 38741}, {12220, 37473}, {12256, 12305}, {12257, 12306}, {12383, 16010}, {12583, 16190}, {12699, 38029}, {13196, 13355}, {13564, 41587}, {13748, 21737}, {13860, 15491}, {14227, 36702}, {14242, 36717}, {14688, 23699}, {14826, 44833}, {14848, 15685}, {14982, 15035}, {15018, 20063}, {15048, 20194}, {15051, 41737}, {15072, 16775}, {15122, 34513}, {15311, 19149}, {15516, 19710}, {15534, 15697}, {15583, 19126}, {15598, 22712}, {15644, 31804}, {15681, 20423}, {15682, 38072}, {15692, 21358}, {15696, 33878}, {15712, 18553}, {15760, 20300}, {15988, 37256}, {16197, 18381}, {16239, 42786}, {16264, 37124}, {16386, 32220}, {16623, 41435}, {16776, 16836}, {16789, 21663}, {17702, 25328}, {17712, 23335}, {17768, 24728}, {17800, 43621}, {17810, 34608}, {17928, 20987}, {18525, 38116}, {18560, 19128}, {19467, 37198}, {22676, 32451}, {22802, 23042}, {23046, 25565}, {24273, 37455}, {24309, 29207}, {25555, 38136}, {25964, 37254}, {26005, 35988}, {31671, 38115}, {31672, 38117}, {31673, 38118}, {31730, 39870}, {31805, 34381}, {32110, 44261}, {32113, 44280}, {32237, 44212}, {32271, 34584}, {32305, 32423}, {32429, 32515}, {32467, 35427}, {33697, 38167}, {33699, 38079}, {33923, 34507}, {34168, 40357}, {34380, 44245}, {34417, 37899}, {34938, 37476}, {35242, 39885}, {35243, 37488}, {35283, 40916}, {35471, 39588}, {36740, 37022}, {36757, 42158}, {36758, 42157}, {36883, 38716}, {36988, 42329}, {37201, 41256}, {37450, 43460}, {37479, 42534}, {37511, 40929}, {37931, 39871}, {38035, 41869}, {40132, 41424}, {41257, 44440}, {41464, 43601}

X(44882) = midpoint of X(i) and X(j) for these {i,j}: {6, 20}, {64, 5596}, {74, 32233}, {185, 3313}, {376, 43273}, {1350, 6776}, {1657, 31670}, {3357, 34776}, {3534, 11179}, {5894, 34774}, {7737, 14532}, {9970, 20127}, {10991, 14928}, {11579, 12121}, {12177, 38741}, {12220, 37473}, {12383, 16010}, {14927, 36990}, {15069, 39874}, {15681, 20423}, {15704, 21850}, {17800, 43621}, {17845, 36851}, {19467, 37485}, {31730, 39870}
X(44882) = reflection of X(i) in X(j) for these {i,j}: {4, 3589}, {5, 5092}, {66, 6696}, {141, 3}, {1351, 12007}, {2883, 206}, {3098, 548}, {3627, 19130}, {3629, 8550}, {3818, 140}, {3845, 10168}, {5480, 182}, {8584, 11179}, {9822, 17704}, {9969, 9729}, {10752, 41595}, {11178, 12100}, {11477, 32455}, {14810, 33751}, {15069, 3631}, {16776, 16836}, {18358, 3530}, {19130, 20190}, {21850, 575}, {32274, 6699}, {33851, 38726}, {39884, 24206}, {40929, 37511}, {41362, 23300}
X(44882) = complement of X(36990)
X(44882) = circumtangential-isogonal conjugate of complement of X(45063)
X(44882) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14927, 36990}, {3, 141, 21167}, {3, 8721, 7789}, {4, 5085, 3589}, {20, 25406, 6}, {22, 18911, 32269}, {69, 3522, 31884}, {182, 5480, 597}, {376, 6776, 1350}, {382, 12017, 14561}, {549, 39884, 24206}, {631, 10516, 34573}, {858, 15080, 13394}, {1350, 43273, 6776}, {1351, 11179, 12007}, {1351, 12007, 8584}, {1370, 3796, 23292}, {1657, 5050, 31670}, {3528, 39874, 10519}, {3575, 19124, 3867}, {3627, 38110, 19130}, {3818, 17508, 140}, {6800, 16063, 11064}, {6823, 44829, 41362}, {10519, 15069, 3631}, {10519, 39874, 15069}, {11477, 14912, 32455}, {14810, 33751, 8703}, {15585, 15812, 141}, {18911, 32269, 13567}, {19130, 20190, 38110}, {31305, 37514, 11745}, {42147, 42148, 9607}, {42329, 44252, 36988}

X(44883) = MIDPOINT OF X(6) AND X(34778)

Barycentrics    a^2*(a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 + 4*a^6*b^2*c^2 - 3*b^8*c^2 - 2*a^6*c^4 - 2*a^2*b^4*c^4 + 4*b^6*c^4 + 2*a^4*c^6 + 4*b^4*c^6 + a^2*c^8 - 3*b^2*c^8 - c^10) : :
X(44883) = 3 X[3] - X[159], X[3] + 2 X[15579], 7 X[3] - 2 X[15580], 4 X[3] - X[15581], 5 X[3] - 2 X[15582], 3 X[3] - 2 X[35228], 5 X[3] - X[39879], X[6] - 3 X[10249], X[6] + 3 X[10606], X[64] + 3 X[5085], 3 X[64] + 5 X[19132], X[141] - 3 X[23328], 2 X[159] - 3 X[15577], X[159] - 6 X[15578], X[159] + 6 X[15579], 7 X[159] - 6 X[15580], 4 X[159] - 3 X[15581], 5 X[159] - 6 X[15582], 5 X[159] - 3 X[39879], 3 X[182] - X[34779], 3 X[182] - 2 X[41593], 3 X[376] + X[36851], 5 X[631] - X[41735], X[1350] - 5 X[8567], X[1498] - 3 X[23041], 3 X[1853] - X[34775], X[2935] + 3 X[5621], X[3098] - 3 X[11204], 2 X[3357] + X[34117], 3 X[3357] + X[34779], 3 X[3357] + 2 X[41593], X[3818] - 3 X[23329], 9 X[5085] - 5 X[19132], 3 X[5085] - X[19149], 4 X[6696] - X[34118], 2 X[6697] - 3 X[23329], X[6759] - 3 X[17508], X[8549] + 5 X[8567], X[9968] - 4 X[20190], 3 X[10249] + X[34778], 3 X[10250] - X[37517], 3 X[10250] - 2 X[39125], 3 X[10606] - X[34778], 3 X[11216] - X[44456], 5 X[12017] - 3 X[19153], 5 X[12017] + 3 X[35450], 3 X[14561] + X[20427], X[15577] - 4 X[15578], X[15577] + 4 X[15579], 7 X[15577] - 4 X[15580], 5 X[15577] - 4 X[15582], 3 X[15577] - 4 X[35228], 5 X[15577] - 2 X[39879], 7 X[15578] - X[15580], 8 X[15578] - X[15581], 5 X[15578] - X[15582], 3 X[15578] - X[35228], 10 X[15578] - X[39879], 7 X[15579] + X[15580], 8 X[15579] + X[15581], 5 X[15579] + X[15582], 3 X[15579] + X[35228], 10 X[15579] + X[39879], 8 X[15580] - 7 X[15581], 5 X[15580] - 7 X[15582], 3 X[15580] - 7 X[35228], 10 X[15580] - 7 X[39879], 5 X[15581] - 8 X[15582], 3 X[15581] - 8 X[35228], 5 X[15581] - 4 X[39879], 3 X[15582] - 5 X[35228], 5 X[19132] - 3 X[19149], X[22802] - 3 X[38317], 3 X[23327] - X[31670], 3 X[31884] - X[34787], 2 X[32138] + X[44469], 9 X[33750] - X[34781], 3 X[34117] - 2 X[34779], 3 X[34117] - 4 X[41593], 10 X[35228] - 3 X[39879], X[36990] - 5 X[40686]

X(44883) lies on the cubic K1241 and these lines: {2, 32125}, {3, 66}, {4, 20300}, {6, 74}, {22, 1853}, {24, 35217}, {25, 23332}, {26, 20299}, {30, 18382}, {64, 1176}, {69, 2071}, {140, 32321}, {154, 5888}, {161, 6636}, {182, 3357}, {186, 20987}, {206, 4550}, {376, 36851}, {381, 9919}, {511, 7689}, {542, 12901}, {631, 41735}, {1204, 19124}, {1350, 7691}, {1351, 11999}, {1498, 7509}, {1593, 5480}, {1619, 7484}, {1660, 3819}, {1843, 21663}, {1995, 7703}, {2393, 3098}, {2697, 2867}, {2777, 7706}, {2854, 12302}, {2883, 7395}, {2916, 44837}, {3516, 8550}, {3520, 6776}, {3564, 9938}, {3566, 44821}, {3579, 3827}, {3589, 4846}, {3618, 7527}, {3763, 35219}, {3818, 6644}, {5000, 34135}, {5001, 34136}, {5020, 40920}, {5050, 14130}, {5094, 13171}, {5116, 32445}, {5169, 13203}, {5596, 35921}, {5656, 7550}, {5893, 9914}, {5925, 35502}, {5969, 39812}, {6403, 11468}, {6759, 7516}, {6800, 15139}, {7393, 16252}, {7488, 14927}, {7496, 35260}, {7499, 41602}, {7530, 23325}, {7716, 37487}, {7729, 20806}, {9786, 32184}, {9968, 20190}, {9969, 11438}, {10250, 37517}, {10516, 17928}, {10594, 11704}, {10620, 15141}, {11180, 37948}, {11188, 15055}, {11206, 15246}, {11216, 44456}, {11410, 19459}, {11414, 41362}, {11440, 41716}, {11550, 21213}, {11645, 18324}, {12017, 19153}, {12082, 18405}, {12085, 29181}, {12250, 35500}, {12292, 38851}, {12324, 37126}, {13093, 34864}, {13293, 32305}, {13445, 19121}, {13861, 32767}, {14118, 25406}, {14561, 20427}, {14853, 14865}, {14912, 35475}, {15583, 21312}, {15647, 35259}, {16013, 35477}, {18383, 29323}, {18534, 23324}, {18859, 33878}, {19125, 40928}, {19596, 35472}, {20376, 32357}, {21397, 35902}, {22802, 38317}, {22978, 34507}, {23327, 31670}, {24206, 25563}, {31884, 34787}, {32062, 44091}, {32138, 44469}, {32600, 34776}, {33750, 34781}, {35473, 39874}, {36753, 43814}, {37473, 39588}, {37514, 41589}, {37814, 39884}, {41580, 43650}

X(44883) = midpoint of X(i) and X(j) for these {i,j}: {6, 34778}, {64, 19149}, {182, 3357}, {1350, 8549}, {5480, 5894}, {10249, 10606}, {10620, 15141}, {12085, 37488}, {13293, 32305}, {14216, 36989}, {15578, 15579}, {19153, 35450}, {33878, 34777}
X(44883) = reflection of X(i) in X(j) for these {i,j}: {3, 15578}, {4, 20300}, {159, 35228}, {206, 5092}, {3818, 6697}, {15577, 3}, {15581, 15577}, {18382, 23300}, {24206, 25563}, {32191, 32184}, {34117, 182}, {34779, 41593}, {37517, 39125}, {39879, 15582}
X(44883) = crosssum of X(9411) and X(39008)
X(44883) = crossdifference of every pair of points on line {2485, 9033}
X(44883) = X(9)-of-Trinh-triangle if ABC is acute
X(44883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 159, 35228}, {6, 10606, 34778}, {64, 5085, 19149}, {159, 35228, 15577}, {182, 34779, 41593}, {1204, 19124, 19161}, {1619, 7484, 10192}, {3818, 23329, 6697}, {6636, 32064, 161}, {8567, 32345, 11413}, {9914, 11479, 5893}, {10249, 34778, 6}, {10250, 37517, 39125}, {34779, 41593, 34117}

X(44884) = X(3)X(3162)∩X(4)X(37801)

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

X(44884) lies on the cubic K1241 and these lines: {3, 3162}, {4, 37801}, {182, 3357}, {550, 38608}, {7503, 40358}, {13335, 31833}

X(44885) = X(3)X(206)∩X(251)X(1297)

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

X(44885) lies on the cubic K1241 and these lines: {3, 206}, {251, 1297}, {1350, 22135}, {3313, 34137}, {5157, 8743}

leftri

Line conjugates on the Euler line: X(44886)-X(44896)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, September 25, 2021.

See the preamble just before X(237) for an introduction to line conjugates. If P and U are points on the Euler line then the P-line conjugate of U is identical to the X(2)-line conjugate of U, and if U = X(2) + k X(3), then the X(2)-line conjugate of U is given by the combo

3*a^2*b^2*c^2*(3*a^2*b^2*c^2*(3 + J^2)*k + 4*(a^2 + b^2 + c^2)*S^2)*X(2) - 4*S^2*(3*a^2*b^2*c^2*(a^2 + b^2 + c^2)*k + 4*(b^2*c^2 + a^2*(b^2 + c^2))*S^2)*X(3).

X(44886) = X(2)-LINE CONJUGATE OF X(5)

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

X(44886) lies on these lines: {2, 3}, {50, 647}, {373, 35222}, {511, 23181}, {1495, 1624}, {1634, 3292}, {1993, 23158}, {2970, 32428}, {5201, 41586}, {6146, 16035}, {6746, 42441}, {10540, 44830}, {14165, 19189}, {14965, 23584}, {16030, 23195}, {18381, 31381}, {23061, 36829}, {32111, 43919}, {34834, 38987}

X(44887) = X(2)-LINE CONJUGATE OF X(22)

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

X(44887) lies on these lines: {2, 3}, {125, 42671}, {230, 9475}, {647, 826}, {3589, 23635}, {5972, 36212}, {13567, 34396}, {15526, 44102}, {19118, 20208}, {26156, 37893}

X(44888) = X(2)-LINE CONJUGATE OF X(24)

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

X(44888) lies on these lines: {2, 3}, {343, 23606}, {647, 6368}, {1209, 37081}, {2972, 11064}, {3284, 41586}, {3292, 15526}, {3564, 35442}, {5642, 34147}, {5972, 44436}, {6090, 20208}, {6394, 37804}, {6800, 18437}, {13394, 42353}, {13409, 23292}, {14389, 30258}

X(44889) = X(2)-LINE CONJUGATE OF X(30)

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

X(44889) lies on these lines: {2, 3}, {6, 647}, {51, 16186}, {74, 43919}, {125, 1624}, {184, 5502}, {373, 44114}, {878, 5967}, {1112, 2972}, {1634, 5642}, {1986, 44715}, {2421, 6090}, {3066, 14687}, {5012, 30510}, {5201, 32225}, {5467, 5651}, {5622, 40352}, {5640, 33927}, {5890, 14264}, {5943, 18114}, {5946, 14670}, {5972, 23181}, {7740, 11430}, {9155, 40283}, {9475, 44467}, {9826, 34333}, {10605, 34329}, {12099, 20975}, {12133, 40948}, {12828, 15526}, {13171, 14673}, {14385, 15033}, {14703, 19457}, {15920, 26864}, {32235, 35357}, {44084, 44436}

X(44889) = crossdifference of every pair of points on line X(30)X(647)

X(44890) = X(2)-LINE CONJUGATE OF X(140)

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

X(44890) lies on these lines: {2, 3}, {51, 16030}, {107, 19189}, {373, 41328}, {647, 11063}, {1634, 41586}, {3292, 5201}, {5651, 8266}, {10282, 16035}, {13450, 19173}, {23061, 38987}, {23181, 32223}

X(44891) = X(2)-LINE CONJUGATE OF X(186)

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

X(44891) lies on these lines: {2, 3}, {95, 42308}, {110, 35442}, {216, 647}, {2972, 5972}, {3284, 32225}, {5502, 10192}, {5642, 15526}, {6334, 14697}, {6389, 14685}, {6699, 40948}, {7740, 10182}, {9717, 41005}, {10272, 44715}, {13394, 35912}, {13409, 16186}, {18475, 39170}, {41172, 41939}, {42441, 44516}

X(44892) = X(2)-LINE CONJUGATE OF X(378)

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

X(44892) lies on these lines: {2, 3}, {647, 9033}, {2972, 32269}, {3284, 5642}, {10317, 32227}, {14919, 15360}, {15526, 32225}, {32223, 44436}, {34147, 41586}

X(44893) = X(2)-LINE CONJUGATE OF X(418)

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

X(44893) lies on these lines: {2, 3}, {216, 44131}, {217, 13567}, {287, 19128}, {338, 11062}, {647, 14165}, {2052, 43679}, {3003, 37778}, {9476, 32649}, {14570, 44138}, {26958, 32445}, {32085, 42330}, {36212, 44146}, {39575, 40814}

X(44894) = X(2)-LINE CONJUGATE OF X(427)

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

X(44894) lies on these lines: {2, 3}, {160, 31267}, {216, 35222}, {251, 15861}, {647, 8673}, {1624, 42671}, {1634, 3284}, {3167, 22135}, {5201, 15526}, {9306, 10316}, {10192, 23208}, {14965, 36213}, {20794, 38292}, {22120, 23158}, {23181, 36212}, {40981, 41005}

X(44895) = X(2)-LINE CONJUGATE OF X(858)

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

X(44895) lies on these lines: {2, 3}, {32, 647}, {1576, 38851}, {1624, 5191}, {1634, 6593}, {2493, 9475}, {3003, 35370}, {3455, 15359}, {5201, 8262}, {5502, 34396}, {9155, 23181}, {16776, 23635}, {20975, 32246}, {41167, 44127}

X(44896) = X(2)-LINE CONJUGATE OF X(1368)

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

X(44896) lies on these lines: {2, 3}, {647, 41336}, {1634, 44102}, {1843, 35222}, {3229, 35325}, {3511, 44090}, {34811, 36879}, {36212, 44084}, {40981, 41584}

X(44897) = X(4)X(9)∩X(514)X(2490)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Euclid 2476 .

X(44897) lies on these lines: {4,9}, {514,2490}, {1146,28236}, {1323,31183}, {2391,40483}, {4301,27541}, {23972,35092}

X(44898) = 2ND HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    2 a^7+2 a^6 b-a^5 b^2-a^4 b^3-2 a^3 b^4-2 a^2 b^5+a b^6+b^7+2 a^6 c+2 a^5 b c-a^4 b^2 c-a^3 b^3 c-2 a^2 b^4 c-a b^5 c+b^6 c-a^5 c^2-a^4 b c^2+4 a^3 b^2 c^2+4 a^2 b^3 c^2-a b^4 c^2-b^5 c^2-a^4 c^3-a^3 b c^3+4 a^2 b^2 c^3+2 a b^3 c^3-b^4 c^3-2 a^3 c^4-2 a^2 b c^4-a b^2 c^4-b^3 c^4-2 a^2 c^5-a b c^5-b^2 c^5+a c^6+b c^6+c^7 : :
Barycentrics    S^2*(r*R-12*R^2+3*SW)-SB*SC*(3*r*R + SW) : :
X(44898) = 3*X(2)+X(1325),5*X(631)-X(36001),3*X(7478)+X(36154),3*X(13587)+X(36175),5*X(37760)-X(37919)

As a point on the Euler line, X(44898) has Shinagawa coefficients (12*r^4+44*r^3*R+48*r^2*R^2-3*S^2,-4*r^4-4*r^3*R+S^2).

See Antreas Hatzipolakis, Francisco Javier García Capitán and Ercole Suppa, Euclid 2479 and Euclid 2551 .

X(44898) lies on these lines: {2,3}, {36,5520}, {517,5972}, {523,8043}, {1718,35466}, {3833,6703}, {5433,39751}, {6713,22104}, {6714,40544}

X(44898) = midpoint of X(i) and X(j) for these {i,j}: {36,5520}, {1325,30447}, {3109,36195}
X(44898) = complement of X(30447)
X(44898) = complementary conjugate of the complement of X(43700)
X(44898) = X(523)-vertex conjugate of X(20831)
X(44898) = crossdifference of every pair of points on line X(647)-X(1030)
X(44898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1325,30447), (2,11116,37346), (1113,1114,20831), (24904,37158,442)

X(44899) = X(4)X(52)∩X(136)X(34382)

Barycentrics    ((b^2+c^2)*a^10-(3*b^4+8*b^2*c^2+3*c^4)*a^8+2*(b^2+c^2)*(b^4+7*b^2*c^2+c^4)*a^6+2*(b^8+c^8-(9*b^4+4*b^2*c^2+9*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(3*b^8+3*c^8-2*(9*b^4-11*b^2*c^2+9*c^4)*b^2*c^2)*a^2+(b^8+c^8-2*(2*b^4+b^2*c^2+2*c^4)*b^2*c^2)*(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*a^2 : :
Barycentrics    SB*SC*(SB+SC)*((2*R^2*(6*R^2-3*SA-5*SW)-SB*SC+2*SW^2)*S^2-(2*R^2-SW)*SA^2*SW) : :

See Antreas Hatzipolakis, Francisco Javier García Capitán and César Lozada Euclid 2502 and Euclid 2570 .

X(44899) lies on these lines: {4, 52}, {136, 34382}, {858, 16221}, {924, 2501}, {3291, 10311}, {6642, 12095}, {7529, 13557}

X(44899) = crossdifference of every pair of points on line {X(155), X(30451)}
X(44899) = perspector of the circumconic {{A, B, C, X(254), X(30450)}}
X(44899) = inverse of X(6515) in polar circle

X(44900) =  EULER LINE INTERCEPT OF X(34155)X(34380)

Barycentrics    4 a^10-10 a^8 b^2+4 a^6 b^4+8 a^4 b^6-8 a^2 b^8+2 b^10-10 a^8 c^2+10 a^6 b^2 c^2-9 a^4 b^4 c^2+15 a^2 b^6 c^2-6 b^8 c^2+4 a^6 c^4-9 a^4 b^2 c^4-14 a^2 b^4 c^4+4 b^6 c^4+8 a^4 c^6+15 a^2 b^2 c^6+4 b^4 c^6-8 a^2 c^8-6 b^2 c^8+2 c^10 : :
Barycentrics    S^2 (49 R^2 - 12 SW) - SB SC (3 R^2 - 4 SW) : :
X(44900) = 3*X(2)+X(37936),X(20)-5*X(15646),3*X(20)+5*X(31726),X(140)+5*X(468),3*X(140)+5*X(10096),7*X(140)-5*X(10257),7*X(140)+5*X(25338),9*X(140)+5*X(37971),X(140)-5*X(44234),3*X(140)-5*X(44452),5*X(186)+3*X(381),X(186)+3*X(44282),X(381)-5*X(44282),5*X(403)-X(3627),X(403)+3*X(16532),3*X(468)-X(10096),7*X(468)+X(10257),7*X(468)-X(25338),9*X(468)-X(37971),3*X(468)+X(44452),5*X(1656)+3*X(37940),5*X(2072)-9*X(15699),7*X(3090)+5*X(7575),X(3153)+3*X(7575),9*X(3524)-5*X(34152),7*X(3526)+X(37925),X(3627)+5*X(18571),X(3861)+5*X(22249),9*X(5054)-X(37944),X(5073)-5*X(44283),X(5899)-9*X(37907),3*X(8703)+5*X(11563),9*X(8703)-5*X(16386),3*X(8703)-5*X(37968),X(8703)-5*X(44214),7*X(10096)+3*X(10257),7*X(10096)-3*X(25338),3*X(10096)-X(37971),X(10096)+3*X(44234),3*X(10109)-5*X(15350),5*X(10151)-3*X(12101),X(10151)+3*X(18579),9*X(10257)+7*X(37971),X(10257)-7*X(44234),3*X(10257)-7*X(44452),5*X(11558)+3*X(15691),3*X(11563)+X(16386),X(11563)+3*X(44214),X(12101)+5*X(18579),2*X(12811)+5*X(37935),3*X(15646)+X(31726),X(15646)+3*X(37943),7*X(15701)+5*X(44266),2*X(16239)+X(37897),X(16386)-3*X(37968),X(16386)-9*X(44214),3*X(16532)-X(18571),X(18572)+3*X(37922),X(18859)+3*X(44266),9*X(25338)-7*X(37971),X(25338)+7*X(44234),3*X(25338)+7*X(44452),5*X(30745)+3*X(37956),X(31726)-9*X(37943),5*X(37760)-X(37947),2*X(37911)+X(44264),3*X(37941)+X(44267),3*X(37955)+X(44283),X(37968)-3*X(44214),X(37971)+9*X(44234),X(37971)+3*X(44452),3*X(44234)-X(44452)

As a point on the Euler line, X(44900) has Shinagawa coefficients (E - 48*F, 13*E + 16*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2480 .

X(44900) lies on these lines: {2,3}, {34155,34380}

X(44900) = midpoint of X(i) and X(j) for these {i,j}: {403,18571}, {468,44234} ,{546,37931}, {10096,44452}, {10257,25338}, {11563,37968}, {12105,37938}, {16531,37942}
X(44900) = complement of the complement of X(37936)
X(44900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (403,16532,18571), (468,44452,10096), (5498,21841,12102), (10096,44234,44452), (10125,44232,3850), (11563,44214,37968), (12106,34330,13413), (12106,37453,34330), (13413,34330,3628), (16238,18282,3530), (31681,31682,3859)

X(44901) = X(1)X(4)∩X(522)X(676)

Barycentrics    2 a^6-a^5 b-a^4 b^2+2 a^3 b^3-4 a^2 b^4-a b^5+3 b^6-a^5 c+2 a^4 b c-2 a^3 b^2 c-4 a^2 b^3 c+11 a b^4 c-6 b^5 c-a^4 c^2-2 a^3 b c^2+16 a^2 b^2 c^2-10 a b^3 c^2-3 b^4 c^2+2 a^3 c^3-4 a^2 b c^3-10 a b^2 c^3+12 b^3 c^3-4 a^2 c^4+11 a b c^4-3 b^2 c^4-a c^5-6 b c^5+3 c^6 : :
X(44901) = X(1)+3*X(1785),X(1155)+3*X(3326)

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2487 .

X(44901) lies on these lines: {1,4}, {516,15252}, {522,676}, {1155,1360}, {1538,10271}, {5550,10538}, {10863,37800}, {18613,34142}, {34050,38357}

X(44901) = midpoint of X(34050) and X(38357)
X(44901) = X(652)-he conjugate of X(2636)
X(44901) = radical trace of incircle and polar circle
X(44901) = crossdifference of every pair of points on line X(652)-X(3207)

X(44902) = COMPLEMENT OF X(1639)

Barycentrics    (b-c)*(4*a^2-3*(b+c)*a-b^2+4*b*c-c^2) : :
X(44902) = 3*X(2)+X(4453), 5*X(2)-X(30565), 2*X(676)+X(4925), X(676)+2*X(25380), X(1635)+3*X(14475), 3*X(1638)+X(1639), 3*X(1638)-X(4453), 5*X(1638)+X(30565), 5*X(1639)-3*X(30565), 2*X(2487)+X(3835), 2*X(2490)+X(3776), X(3004)+5*X(24924), X(3676)+2*X(31287), 5*X(4453)+3*X(30565), X(4885)+2*X(7658), 2*X(4885)+X(17069), X(4925)-4*X(25380), X(4927)-3*X(14475), 4*X(7658)-X(17069), 4*X(44563)-X(44567)

See Antreas Hatzipolakis and César Lozada, Euclid 2493 .

X(44902) lies on these lines: {2, 918}, {210, 30704}, {514, 14425}, {522, 676}, {900, 4928}, {926, 3742}, {1086, 24228}, {1635, 4927}, {2487, 3835}, {2490, 3776}, {2976, 4940}, {2977, 4802}, {3004, 24924}, {3035, 6366}, {3310, 16610}, {3676, 31287}, {3740, 42341}, {3756, 4904}, {3910, 41800}, {3960, 21198}, {4025, 4944}, {4369, 4977}, {4728, 4984}, {4750, 4958}, {4763, 6084}, {4773, 21297}, {4897, 30835}, {4976, 26985}, {6544, 21115}, {14315, 34824}, {14321, 28906}, {17063, 24462}, {17278, 24141}, {21104, 31209}, {21196, 28187}, {21199, 23876}, {21212, 28863}, {22086, 37520}, {23729, 27013}, {25666, 28855}, {26275, 36848}, {28882, 31286}, {28898, 44551}, {30725, 31227}

X(44902) = midpoint of X(i) and X(j) for these {i, j}: {2, 1638}, {210, 30704}, {1635, 4927}, {1639, 4453}, {3960, 21198}, {4025, 4944}, {4763, 21204}, {4773, 21297}, {26275, 36848}
X(44902) = complement of X(1639)
X(44902) = crossdifference of every pair of points on line {X(3207), X(21781)}
X(44902) = X(i)-complementary conjugate of-X(j) for these (i, j): (57, 3259), (88, 124), (106, 26932), (109, 16594)
X(44902) = perspector of the circumconic {{A, B, C, X(4480), X(10405)}}
X(44902) = barycentric product X(i)*X(j) for these {i, j}: {86, 21952}, {190, 23766}, {514, 4480}
X(44902) = trilinear product X(i)*X(j) for these {i, j}: {81, 21952}, {100, 23766}, {513, 4480}
X(44902) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 4453, 1639), (676, 25380, 4925), (1635, 14475, 4927), (1638, 1639, 4453), (4885, 7658, 17069)

X(44903) = EULER LINE INTERCEPT OF X(551)X(28154)

Barycentrics    28 a^4-17 a^2 b^2-11 b^4-17 a^2 c^2+22 b^2 c^2-11 c^4 : :
Barycentrics    17 S^2 - 39 SB SC : :
X(44903) = 8*X(2)-7*X(3857), 5*X(2)-4*X(3861), 7*X(2)-5*X(5076), 5*X(2)-X(11541), 5*X(3)-4*X(10109), 3*X(3)-2*X(14893), 9*X(3)-5*X(35434), 4*X(3)-3*X(38071), 7*X(3)-5*X(41099), 3*X(4)-4*X(10124), 3*X(4)-5*X(14093), 5*X(4)-6*X(14892), 5*X(4)-7*X(15701), 2*X(4)-3*X(17504), 3*X(4)-X(35400), 6*X(5)-7*X(549), 4*X(5)-7*X(550), 7*X(5)-10*X(3522), 5*X(5)-6*X(3524), X(5)+2*X(3529), 3*X(5)-2*X(3543), 10*X(5)-7*X(3627), 8*X(5)-7*X(3845), 5*X(5)-2*X(5073), 5*X(5)-7*X(8703), 7*X(5)-8*X(11812), 5*X(5)-4*X(12101), 7*X(5)-6*X(14269), 3*X(5)-7*X(15686), 9*X(5)-7*X(15687), 9*X(5)-10*X(15694), X(5)-7*X(15704), 4*X(5)-5*X(15711), 2*X(5)-7*X(19710), 3*X(5)-4*X(34200), 5*X(5)-8*X(44245), 5*X(20)-2*X(140), 9*X(20)-5*X(376), 3*X(20)-X(381), 8*X(20)-5*X(550), X(20)+5*X(1657), 7*X(20)-3*X(3524), 7*X(20)+5*X(3529), 7*X(20)-5*X(3534), 4*X(20)-X(3627), 7*X(20)-X(5073), X(20)-5*X(11001), 7*X(20)-2*X(12101), 9*X(20)-4*X(14891), 3*X(20)-5*X(15681), 5*X(20)-X(15682), 3*X(20)+5*X(15683), 6*X(20)-5*X(15686), 5*X(20)-3*X(15689), 3*X(20)-2*X(15691), 8*X(20)-3*X(15699), 2*X(20)-5*X(15704), 4*X(20)-5*X(19710), 7*X(20)-4*X(44245), 6*X(140)-5*X(381), 8*X(140)-5*X(3627), 4*X(140)-5*X(8703), 7*X(140)-5*X(12101), 9*X(140)-10*X(14891), 2*X(140)+5*X(15685), 2*X(140)-3*X(15689), 3*X(140)-5*X(15691), 7*X(140)-10*X(44245), 5*X(376)-3*X(381), 3*X(376)-2*X(547),4*X(376)-3*X(549), 8*X(376)-9*X(550), X(376)+9*X(1657), 7*X(376)+9*X(3529), 7*X(376)-9*X(3534), 7*X(376)-3*X(3543), 10*X(376)-9*X(8703), X(376)-9*X(11001), 5*X(376)-4*X(14891), X(376)-3*X(15681), X(376)+3*X(15683), 3*X(376)-X(15684), 5*X(376)+9*X(15685), 2*X(376)-3*X(15686), 5*X(376)-6*X(15691), 7*X(376)-5*X(15694), 9*X(376)-7*X(15700), 2*X(376)-9*X(15704), 6*X(376)-5*X(15714), 4*X(376)-9*X(19710), 7*X(376)-6*X(34200), 8*X(376)-3*X(35404), 9*X(381)-10*X(547), 4*X(381)-5*X(549), 7*X(381)-9*X(3524), 7*X(381)-5*X(3543), 4*X(381)-3*X(3627), 7*X(381)-3*X(5073), 2*X(381)-3*X(8703), 7*X(381)-6*X(12101), 3*X(381)-4*X(14891), X(381)-5*X(15681), 5*X(381)-3*X(15682), X(381)+5*X(15683), 9*X(381)-5*X(15684), X(381)+3*X(15685), 2*X(381)-5*X(15686), 6*X(381)-5*X(15687), 5*X(381)-9*X(15689), 8*X(381)-9*X(15699), 7*X(381)-10*X(34200), 8*X(381)-5*X(35404), 9*X(381)-8*X(41988), 3*X(382)-5*X(5071), 5*X(382)-8*X(12811), 2*X(382)-3*X(23046), 3*X(382)+8*X(35413), 2*X(546)-3*X(10304), 6*X(546)-7*X(15703), 4*X(546)-5*X(15713), 3*X(546)-X(35408), 5*X(546)-6*X(41984), 8*X(547)-9*X(549), 5*X(547)-6*X(14891), 2*X(547)-9*X(15681), 2*X(547)+9*X(15683), 4*X(547)-9*X(15686), 4*X(547)-3*X(15687), 5*X(547)-9*X(15691), 6*X(547)-7*X(15700), 4*X(547)-5*X(15714), 7*X(547)-9*X(34200), 5*X(547)-4*X(41988), 10*X(548)-7*X(3090), 8*X(548)-5*X(3858), 2*X(548)+X(5059), 4*X(548)-3*X(11539), 3*X(548)-2*X(11737), 6*X(548)-5*X(15692), 5*X(548)-3*X(41987), 2*X(549)-3*X(550), 7*X(549)-4*X(3543), 5*X(549)-3*X(3627), 4*X(549)-3*X(3845), 5*X(549)-6*X(8703), X(549)-4*X(15681), X(549)+4*X(15683), 9*X(549)-4*X(15684), 3*X(549)-2*X(15687), 5*X(549)-8*X(15691), 10*X(549)-9*X(15699), X(549)-6*X(15704), 9*X(549)-10*X(15714), X(549)-3*X(19710), 7*X(549)-8*X(34200), 8*X(550)-5*X(632), X(550)+8*X(1657), 7*X(550)+8*X(3529), 7*X(550)-8*X(3534), 5*X(550)-2*X(3627), 5*X(550)-4*X(8703), X(550)-8*X(11001), 3*X(550)-8*X(15681), 3*X(550)+8*X(15683), 5*X(550)+8*X(15685), 3*X(550)-4*X(15686), 9*X(550)-4*X(15687), 5*X(550)-3*X(15699), X(550)-4*X(15704), 7*X(550)-5*X(15711), 3*X(550)-X(35404), 5*X(631)-4*X(3860), 5*X(631)-6*X(41982), 5*X(632)-4*X(3845), 7*X(632)-8*X(15711), 7*X(1657)-X(3529)

See ER JKH and Ercole Suppa, Euclid 2521 .

X(44903) lies on these lines: {2,3}, {551,28154}, {1353,19924}, {1483,28198}, {3241,28216}, {3579,38081}, {3655,28178}, {3679,28190}, {4746,37705}, {5334,43647}, {5335,43648}, {6200,41952}, {6396,41951}, {6419,43786}, {6420,43785}, {6435,42216}, {6436,42215}, {6468,43316}, {6469,43317}, {6484,43340}, {6485,43341}, {6494,19054}, {6495,19053}, {6498,43257}, {6499,43256}, {7756,34571}, {9540,42538}, {10283,28150}, {10653,42585}, {10654,42584}, {10721,11694}, {12816,42945}, {12817,42944}, {13935,42537}, {14075,15048}, {16241,43873}, {16242,43874}, {16962,43773}, {16963,43774}, {16964,42899}, {16965,42898}, {23251,42639}, {23261,42640}, {28160,34638}, {28164,38176}, {28168,38138}, {28174,34628}, {28182,31162}, {28202,34773}, {28224,34632}, {34754,43231}, {34755,43230}, {36969,43642}, {36970,43641}, {37640,43639}, {37641,43640}, {38079,42785}, {41100,43776}, {41101,43775}, {41943,42941}, {41944,42940}, {41945,42226}, {41946,42225}, {42085,42634}, {42086,42633}, {42087,42922}, {42088,42923}, {42090,42916}, {42091,42917}, {42096,42913}, {42097,42912}, {42099,43631}, {42100,43630}, {42108,42528}, {42109,42529}, {42112,42625}, {42113,42626}, {42121,43402}, {42124,43401}, {42144,42430}, {42145,42429}, {42266,43209}, {42267,43210}, {42494,42514}, {42495,42515}, {42543,43228}, {42544,43229}, {42629,42777}, {42630,42778}, {42692,43200}, {42693,43199}, {42888,43404}, {42889,43403}, {43030,43203}, {43031,43204}, {43100,43247}, {43107,43246}

X(44903) = midpoint of X(i) and X(j) for these {i,j}: {2,17800}, {20,15685}, {1657,11001}, {3529,3534}, {3830,5059}, {15681,15683}
X(44903) = reflection of X(i) in X(j) for these (i,j): (2,12103), (4,15690), (5,3534), (381,15691), (382,12100), (549,15686), (550,19710), (3146,5066), (3543,34200), (3627,8703), (3830,548), (3845,550), (5073,12101), (8703,20), (10721,11694), (12101,44245), (15640,3853), (15682,140), (15684,547), (15686,15681), (15687,376), (15704,11001), (19710,15704), (33699,3), (35404,549)
X(44903) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,382,3854), (4,14093,10124), (5,5073,3627), (5,8703,3524), (5,34200,549), (20,3524,3534), (20,3529,5073), (20,3627,550), (20,5059,3090), (20,17800,3861), (140,15689,8703), (376,14891,8703), (376,15684,547), (376,15687,549), (381,5073,3543), (381,8703,549), (381,15681,20), (381,15691,8703), (381,15721,547), (546,3533,5), (547,15714,549), (547,41988,381), (549,3627,381), (549,15686,550), (549,35404,3845), (550,15699,8703), (550,35404,549), (1314,7488,7480), (1314,31050,5177), (1315,16349,1982), (1315,26683,11348), (1650,33021,1315), (3146,15688,5066), (3524,3543,381), (3524,12101,5), (3524,44245,8703), (3534,5073,3524), (3534,12101,8703), (3534,14269,3522), (3534,15694,376), (3543,15683,3529), (3543,34200,5), (3627,15699,3845), (3830,11539,3858), (3839,15695,3530), (3860,41982,631), (4204,26058,1315), (4218,36689,1314), (5054,15640,3853), (5070,35403,381), (5073,44245,5), (6866,37147,1314), (6885,19258,1314), (7388,16444,1315), (7714,16906,1314), (8361,37025,1314), (8703,33699,10109), (10109,14893,381), (10124,17504,549), (11539,15692,549), (11539,41990,3090), (11737,41987,381), (11812,14269,5), (11812,33217,1315), (12101,44245,3524), (14093,35400,4), (14869,15718,549), (14891,15691,376), (14891,41988,547), (15218,16050,1315), (15640,17538,5054), (15681,15685,381), (15682,15689,140), (15683,15704,549), (15686,15687,376), (15687,15714,547), (15690,15701,8703), (15697,33703,5055), (15702,15712,549), (15702,35403,5066), (16066,27938,1314), (16066,27938,1315), (17568,21898,1314), (17568,21898,1315), (19264,19670,1315), (19708,38335,3628), (25791,37913,1314), (27126,33229,1315), (27212,31186,1314), (27212,31186,1315), (28987,36651,1315), (33732,36190,1314), (33827,37983,1314), (36439,36457,3530), (37025,37047,1315), (41987,41990,3858)

X(44904) =  EULER LINE INTERCEPT OF X(17)X(42143)

Barycentrics    2*a^4-15*(b^2+c^2)*a^2+13*(b^2-c^2)^2 : :
X(44904) = 3*X(2)+11*X(5), 18*X(2)-11*X(140), 17*X(2)+11*X(381), 4*X(2)-11*X(547), 3*X(2)-11*X(3090), 15*X(2)-11*X(3526), 9*X(2)+11*X(3851), 6*X(2)+X(3853), 15*X(2)+11*X(3857), 9*X(2)+5*X(3858), 10*X(2)+11*X(5066), 3*X(2)+4*X(12811), 8*X(2)-X(15691), 19*X(2)-11*X(15702), 7*X(2)-11*X(15703), 15*X(2)-X(15704), 19*X(2)-5*X(15714), 9*X(2)-2*X(33923), 13*X(2)+X(35404), 13*X(2)+11*X(41106)

As a point on the Euler line, X(44904) has Shinagawa coefficients (15, 11).

See Antreas Hatzipolakis, César Lozada and Peter Moses, Euclid 2524 and Euclid 2537 .

X(44904) lies on these lines: {2, 3}, {17, 42143}, {18, 42146}, {389, 12046}, {397, 42628}, {398, 42627}, {1151, 43378}, {1152, 43379}, {3311, 43412}, {3312, 43411}, {3316, 43377}, {3317, 43376}, {3590, 6417}, {3591, 6418}, {3592, 43343}, {3594, 43342}, {4701, 13464}, {5318, 43300}, {5321, 43301}, {5418, 6439}, {5420, 6440}, {5690, 30315}, {5734, 38081}, {5892, 11017}, {5943, 31834}, {6425, 43433}, {6426, 43432}, {6427, 42639}, {6428, 42640}, {6435, 41948}, {6436, 41947}, {6441, 42274}, {6442, 42277}, {6476, 41963}, {6477, 41964}, {6478, 42270}, {6479, 42273}, {7583, 41949}, {7584, 41950}, {7745, 12815}, {7989, 28224}, {8960, 18762}, {10110, 44324}, {10170, 14449}, {10171, 13607}, {10172, 40273}, {10187, 42813}, {10188, 42814}, {10194, 42216}, {10195, 42215}, {10619, 20584}, {10627, 12002}, {10653, 43422}, {10654, 43423}, {11488, 42690}, {11489, 42691}, {11522, 38042}, {11542, 43302}, {11543, 43303}, {11793, 13421}, {11803, 13565}, {12007, 18358}, {13363, 13382}, {13431, 22051}, {15067, 27355}, {15088, 16534}, {16241, 43634}, {16242, 43635}, {16808, 42971}, {16809, 42970}, {16960, 43774}, {16961, 43773}, {16964, 42590}, {16965, 42591}, {16966, 42925}, {16967, 42924}, {18583, 43150}, {22235, 42818}, {22237, 42817}, {23302, 41973}, {23303, 41974}, {35814, 42583}, {35815, 42582}, {36969, 42793}, {36970, 42794}, {37714, 38022}, {37832, 43426}, {37835, 43427}, {40693, 42474}, {40694, 42475}, {42107, 42936}, {42110, 42937}, {42121, 42921}, {42122, 42949}, {42123, 42948}, {42124, 42920}, {42126, 42492}, {42127, 42493}, {42129, 42494}, {42132, 42495}, {42135, 43238}, {42136, 42684}, {42137, 42685}, {42138, 43239}, {42144, 42773}, {42145, 42774}, {42147, 43483}, {42148, 43484}, {42153, 42496}, {42156, 42497}, {42157, 42687}, {42158, 42686}, {42262, 43430}, {42265, 43431}, {42268, 43339}, {42269, 43338}, {42488, 43417}, {42489, 43416}, {42580, 43104}, {42581, 43101}, {42598, 42993}, {42599, 42992}, {42692, 43486}, {42693, 43485}, {42795, 42959}, {42796, 42958}, {42912, 42934}, {42913, 42935}, {42918, 42945}, {42919, 42944}

X(44904) = midpoint of X(i) and X(j) for these {i, j}: {5, 3090}, {3526, 3857}, {3832, 14869}, {3845, 15700}
X(44904) = reflection of X(i) in X(j) for these (i, j): (3526, 3628), (12103, 3528), (19711, 10124), (34200, 15701), (41106, 11737)
X(44904) = complement of X(44682)
X(44904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 33923, 140), (4, 140, 548), (4, 3628, 140), (5, 547, 546), (5, 12812, 547), (5, 35018, 140), (140, 5066, 4), (140, 35018, 547), (548, 5066, 546), (1656, 3850, 140), (1656, 5068, 550), (1656, 5073, 2), (3526, 3851, 4), (3529, 35382, 5), (3545, 15697, 381), (3628, 3856, 549), (3628, 5066, 548), (3854, 10304, 4), (3858, 15704, 4), (3859, 12101, 546), (5055, 5066, 547), (5055, 15022, 5), (5056, 15022, 4), (5071, 5079, 5), (5072, 7486, 549), (10124, 15720, 140), (12103, 15759, 548), (15699, 15714, 2), (15712, 16239, 140), (16239, 41989, 381)

X(44905) = EULER LINE INTERCEPT OF X(12028)X(23956)

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

See Antreas Hatzipolakis and César Lozada, Euclid 2524 .

X(44905) lies on these lines: {2, 3}, {12028, 23956}, {13851, 15463}

X(44905) = inverse of X(44242) in 1st Droz-Farny circle
X(44905) = {X(4), X(2072)}-harmonic conjugate of X(186)

X(44906) = EULER LINE INTERCEPT OF X(5972)X(6001)

Barycentrics    (a^2-b^2-c^2)*(2*a^8-3*a^6*b^2-a^4*b^4+3*a^2*b^6-b^8-2*a^6*b*c-2*a^5*b^2*c+a^4*b^3*c+a^3*b^4*c+3*a^2*b^5*c+3*a*b^6*c-3*a^6*c^2-2*a^5*b*c^2+6*a^4*b^2*c^2+a^3*b^3*c^2-3*a^2*b^4*c^2+3*a*b^5*c^2+4*b^6*c^2+a^4*b*c^3+a^3*b^2*c^3-6*a^2*b^3*c^3-6*a*b^4*c^3-a^4*c^4+a^3*b*c^4-3*a^2*b^2*c^4-6*a*b^3*c^4-6*b^4*c^4+3*a^2*b*c^5+3*a*b^2*c^5+3*a^2*c^6+3*a*b*c^6+4*b^2*c^6-c^8) : :
X(44906) = 5*X(631)-X(37979)

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44906) lies on these lines: {2,3}, {5972,6001}

X(44906) = complement of X(37982)

X(44907) =  MIDPOINT OF X(858) AND X(37932)

Barycentrics    (a^2-b^2-c^2) (2 a^14-5 a^12 b^2+9 a^8 b^6-6 a^6 b^8-3 a^4 b^10+4 a^2 b^12-b^14-5 a^12 c^2+8 a^10 b^2 c^2-5 a^8 b^4 c^2+4 a^6 b^6 c^2+5 a^4 b^8 c^2-12 a^2 b^10 c^2+5 b^12 c^2-5 a^8 b^2 c^4-4 a^6 b^4 c^4-2 a^4 b^6 c^4+12 a^2 b^8 c^4-9 b^10 c^4+9 a^8 c^6+4 a^6 b^2 c^6-2 a^4 b^4 c^6-8 a^2 b^6 c^6+5 b^8 c^6-6 a^6 c^8+5 a^4 b^2 c^8+12 a^2 b^4 c^8+5 b^6 c^8-3 a^4 c^10-12 a^2 b^2 c^10-9 b^4 c^10+4 a^2 c^12+5 b^2 c^12-c^14) : :
Barycentrics    SA*(2*S^2*(4 R^2-SW)+(SA-SW)(6*R^4+8*R^2*SA-6*R^2*SW-2*SA*SW+SW^2)) : :
X(44907) = 2*X(6676)+X(10257),X(15760)+2*X(16976),X(16387)+2*X(44452)

As a point on the Euler line, X(44907) has Shinagawa coefficients (E^2-4*E*F-24*F^2,-E^2+4*E*F+8*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44907) lies on this line: {2, 3}

X(44907) = midpoint of X(858) and X(37932)
X(44907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140,6676,15760), (140,16976,10257), (468,10257,2072), (2072,44246,7574), (5159,12105,10297), (6640,44259,427), (6676,16196,25337), (6676,16977,16387), (7542,10257,44452), (16196,25337,44249), (16977,44452,10257)

X(44908) =  EULER LINE INTERCEPT OF X(125)X(18653)

Barycentrics    2 a^6+2 a^5 b-a^4 b^2-a^3 b^3-2 a^2 b^4-a b^5+b^6+2 a^5 c+2 a^4 b c-a^3 b^2 c-a^2 b^3 c-a b^4 c-b^5 c-a^4 c^2-a^3 b c^2+4 a^2 b^2 c^2+2 a b^3 c^2-b^4 c^2-a^3 c^3-a^2 b c^3+2 a b^2 c^3+2 b^3 c^3-2 a^2 c^4-a b c^4-b^2 c^4-a c^5-b c^5+c^6 : :
Barycentrics    4 r^4*S^2+16*r^3*R*S^2-96*r^2*R^2*S^2+S^4-12*r^4*SB*SC-48*r^3*R*SB*SC-3*S^2*SB*SC+24*r^2*S^2*SW-8*r^2*SB*SC*SW : :
X(44908) = 3*X(2)+X(5196),5*X(631)-X(36026),3*X(1325)+X(36154),X(1544)-3*X(36518),X(36154)-3*X(36195)

As a point on the Euler line, X(44908) has Shinagawa coefficients (a*b+a*c+b*c+6*F,-3*a*b-3*a*c-3*b*c-2*E-2*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44908) lies on these lines: {2,3}, {125,18653}, {516,5972}, {523,2487}, {1054,35466}, {1544,36518}, {4892,44387}, {5087,40539}, {6703,24169}, {6712,22104}, {7286,17779}, {13411,40655}, {17070,44378}, {31380,40544}

X(44908) = midpoint of X(i) and X(j) for these {i,j}: {125,18653}, {1325,36195}, {5196,33329}, {14953,37167}
X(44908) = reflection of X(23) in X(523)X(13246)
X(44908) = complement of X(33329)
X(44908) = X(523)-vertex conjugate of X(20834)
X(44908) = crossdifference of every pair of points on line X(647)-X(18755)
X(44908) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5196,33329), (1113,1114,20834), (1312,1313,34119)

X(44909) = X(4)X(6)∩X(187)X(3184)

Barycentrics    2*a^10-(b^2+c^2)*a^8+4*(b^2-c^2)^2*a^6-10*(b^4-c^4)*(b^2-c^2)*a^4+2*(b^2-c^2)^2*(b^4+6*b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)^3 : :
X(44909) = X(5523)+3*X(6794), 3*X(5523)+X(13509), 5*X(5523)-X(15340), 9*X(6794)-X(13509), 15*X(6794)+X(15340), 3*X(6794)-X(15341), 5*X(13509)+3*X(15340), X(13509)-3*X(15341), X(15340)+5*X(15341)

See Antreas Hatzipolakis and César Lozada, Euclid 2570 .

X(44909) lies on these lines: {4, 6}, {187, 3184}, {525, 3239}, {1562, 15311}, {5179, 23982}, {5305, 13568}, {13202, 15639}, {15048, 23292}, {18017, 33636}, {37487, 37689}

X(44909) = midpoint of X(i) and X(j) for these {i, j}: {1562, 16318}, {5523, 15341}
X(44909) = crossdifference of every pair of points on line {X(154), X(520)}
X(44909) = crosssum of X(6) and X(34147)
X(44909) = complement of isotomic conjugate of isogonal conjugate of X(34147)
X(44909) = perspector of the circumconic {{A, B, C, X(107), X(253)}}
X(44909) = inverse of X(1249) in polar circle
X(44909) = {X(5523), X(6794)}-harmonic conjugate of X(15341)

X(44910) =  EULER LINE INTERCEPT OF X(36)X(5954)

Barycentrics    2 a^7-3 a^5 b^2-a^3 b^4-a^2 b^5+2 a b^6+b^7-2 a^5 b c-2 a^4 b^2 c+a^3 b^3 c+a b^5 c+2 b^6 c-3 a^5 c^2-2 a^4 b c^2+6 a^3 b^2 c^2+3 a^2 b^3 c^2-2 a b^4 c^2+a^3 b c^3+3 a^2 b^2 c^3-2 a b^3 c^3-3 b^4 c^3-a^3 c^4-2 a b^2 c^4-3 b^3 c^4-a^2 c^5+a b c^5+2 a c^6+2 b c^6+c^7 : :
Barycentrics    S^2*(a*b*c+36*a*R^2+12*b*R^2+24*c*R^2+a*SA+b*SB+c*SC-8*a*SW-2*b*SW-5*c*SW)-SB*SC*(3*a*b*c-3*b*SC+3*c*SC+a*SW+4*b*SW+c*SW) : :
X(44910) = 3*X(2)+X(7424),X(1558)-3*X(36518)

As a point on the Euler line, X(44910) has Shinagawa coefficients (28*r^4+96*r^3*R+96*r^2*R^2-5*S^2,-20*r^4-32*r^3*R-S^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44910) lies on these lines: {2,3}, {36,5954}, {515,5972}, {523,8045}, {1558,36518}, {6711,22104}, {11545,21054}, {14873,37816}

X(44910) = midpoint of X(i) and X(j) for these {i,j}: {1325,33329}, {3109,30447}, {7424,36195}
X(44910) = X(523)-vertex conjugate of X(20836)
X(44910) = crossdifference of every pair of points on line X(647)-X(2305)
X(44910) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7424,36195), (1113,1114,20836)

X(44911) =  EULER LINE INTERCEPT OF X(5449)X(31831)

Barycentrics    2*a^10-7*a^8*(b^2+c^2)+3*(b^2-c^2)^4*(b^2+c^2)-8*a^2*(b^2-c^2)^2*(b^4+c^4)+a^6*(6*b^4+8*b^2*c^2+6*c^4)+4*a^4*(b^6-2*b^4*c^2-2*b^2*c^4+c^6) : :
Barycentrics    S^2*(22*R^2-5*SW)+SB*SC*(6*R^2-SW) : :
X(44911) = 3*X(2)+X(403),9*X(2)-X(2071),3*X(5)+X(15646),3*X(5)-X(23323),X(5)+2*X(37911),2*X(140)+X(37984),X(186)+7*X(3090),3*X(381)-X(13473),3*X(381)+X(44246),3*X(403)+X(2071),X(468)+5*X(1656),3*X(468)-X(2070),5*X(468)+X(7574),6*X(547)+X(37935),3*X(547)+X(44234),3*X(549)+X(44283),5*X(631)-X(16386),5*X(632)-X(34152),3*X(858)+X(37925),X(858)+3*X(37943),5*X(1656)-X(2072),X(2070)+3*X(2072),5*X(2070)+3*X(7574),X(2071)-3*X(10257),5*X(2072)-X(7574),7*X(3090)-X(10297),5*X(3091)+3*X(37941),7*X(3526)+X(31726),3*X(3545)+X(44280),4*X(3628)-X(5159),4*X(3628)+X(37942),9*X(5055)-X(18403),9*X(5055)+X(37931),3*X(5055)+X(44214),X(5159)+4*X(15350),3*X(5943)-X(32411),3*X(11799)+X(35452),5*X(12812)+X(18571),4*X(15350)-X(37942),X(15646)-6*X(37911),X(15646)-3*X(44452),9*X(15699)-X(37938),3*X(15699)+X(44282),X(16531)+4*X(35018),3*X(16532)+X(18572),X(18323)+3*X(37955),X(18403)+3*X(44214),X(21663)+3*X(36518),X(23323)+6*X(37911),X(23323)+3*X(44452),8*X(35018)+X(37934),X(37925)-9*X(37943),X(37925)-3*X(37971),X(37931)-3*X(44214),3*X(37938)+X(37947),X(37938)+3*X(44282),3*X(37943)-X(37971),X(37947)-9*X(44282)

As a point on the Euler line, X(44911) has Shinagawa coefficients (E-10*F, E-2*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44911) lies on these lines: {2,3}, {5449,31831}, {5654,26958}, {5891,16227}, {5893,43604}, {5943,32411}, {6000,6723}, {6699,15311}, {6720,44381}, {9813,38317}, {9820,13292}, {9822,11649}, {9826,12900}, {9827,32396}, {10149,37696}, {10249,39884}, {10272,12827}, {11204,18418}, {11692,40670}, {11704,14516}, {12228,40111}, {12241,43839}, {15088,30522}, {15448,44407}, {19137,42786}, {21663,36518}, {34840,40557}

X(44911) = midpoint of X(i) and X(j) for these {i,j}: {3,10151}, {5,44452}, {186,10297}, {403,10257}, {468,2072}, {546,37968}, {858,37971}, {3628,15350}, {5159,37942}, {5891,16227}, {11563,15122}, {13473,44246}, {15646,23323}, {16976,37984}, {18403,37931}
X(44911) = reflection of X(i) in X(j) for these (i,j): (16976,140), (37897,10096), (37934,16531), (37935,44234), (44452,37911)
X(44911) = complement of X(10257)
X(44911) = X(523)-vertex conjugate of X(9714)
X(44911) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,403,10257), (2,15760,140), (5,632,7526), (5,6677,10127), (5,15646,23323), (5,16238,31833), (5,37814,546), (5,44263,5066), (140,546,10226), (140,13406,31829), (381,44246,13473), (403,9818,23323), (468,10151,37951), (547,3628,11548), (547,6677,5), (858,37943,37971), (1113,1114,9714), (1312,1313,23335), (1315,34007,21737), (1368,10201,16618), (1656,37439,547), (3090,6642,5), (5055,18420,5), (7505,11585,13383), (9825,35018,5), (10224,44232,6756), (18403,44214,37931), (18531,37453,34351), (23323,44452,15646), (23336,44235,13488), (34559,34562,34331)

X(44912) =  EULER LINE INTERCEPT OF X(6723)X(15311)

Barycentrics    2 a^10-11 a^8 b^2+14 a^6 b^4+4 a^4 b^6-16 a^2 b^8+7 b^10-11 a^8 c^2+8 a^6 b^2 c^2-12 a^4 b^4 c^2+36 a^2 b^6 c^2-21 b^8 c^2+14 a^6 c^4-12 a^4 b^2 c^4-40 a^2 b^4 c^4+14 b^6 c^4+4 a^4 c^6+36 a^2 b^2 c^6+14 b^4 c^6-16 a^2 c^8-21 b^2 c^8+7 c^10 : :
Barycentrics    S^2*(40*R^2-9 SW)+SB*SC*(24*R^2-5*SW) : :
X(44912) = 3*X(2)+X(10151),9*X(2)-X(16386),7*X(5)+X(15646),5*X(5)-X(23323),2*X(5)+X(37911),3*X(5)+X(44452),X(403)+7*X(3090),7*X(403)+X(7464),3*X(468)+X(3153),7*X(468)-3*X(37940),5*X(1656)-X(10257),5*X(1656)+X(37984),X(2072)-9*X(5055),7*X(2072)+X(5899),3*X(2072)+X(37971),7*X(3090)-X(5159),5*X(3091)-X(13473),7*X(3153)+9*X(37940),7*X(3851)+X(44246),9*X(5055)+X(37942),3*X(5066)+X(37968),7*X(5159)-X(7464),X(5899)-7*X(37942),3*X(5899)-7*X(37971),7*X(10096)-3*X(12105),3*X(10109)+X(15350),3*X(10151)+X(16386),3*X(10257)+X(31726),7*X(10297)+5*X(37958),5*X(15646)+7*X(23323),2*X(15646)-7*X(37911),3*X(15646)-7*X(44452),9*X(15699)-X(34152),X(16386)-3*X(16976),2*X(23323)+5*X(37911),3*X(23323)+5*X(44452),X(31726)-3*X(37984),X(37897)-3*X(37943),3*X(37911)-2*X(44452),7*X(37935)-5*X(37958),3*X(37942)-X(37971)

As a point on the Euler line, X(44912) has Shinagawa coefficients (E-9*F,E-5*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2551 .

X(44912) lies on these lines: {2,3}, {6723,15311}, {9817,10149}, {9822,13376}, {18418,23328}, {22973,32396}, {32605,43594}

X(44912) = midpoint of X(i) and X(j) for these {i,j}: {403,5159}, {2072,37942}, {10151,16976}, {10257,37984}, {10297,37935}, {18403,37934}
X(44912) = complement of X(16976)
X(44912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10151,16976), (2,44241,140), (5,6644,5066), (5,15699,9818), (5,16238,3850), (5,31833,12811), (3628,3850,5498), (5020,5071,5)

X(44913) = X(1)X(22461)∩X(58)X(7073)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2550.

X(44913) lies on these lines: {1, 22461}, {58, 7073}, {229, 2646}, {1834, 30436}, {2392, 5048}, {24929, 33642}

X(44913) = midpoint of X(1) and X(22461)

X(44914) = X(140)X(389)∩X(632)X(6509)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(5*a^8 - 14*a^6*b^2 + 13*a^4*b^4 - 4*a^2*b^6 - 14*a^6*c^2 + 13*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 3*b^6*c^2 + 13*a^4*c^4 + 4*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 3*b^2*c^6) : :
X(44914) = 5 X[2] - X[11197], 3 X[2] + X[32078], 11 X[3525] + X[42441], 5 X[10184] - 3 X[11197], X[10184] + 3 X[12012], 13 X[10303] - X[31388], X[11197] + 5 X[12012], 3 X[11197] + 5 X[32078], 3 X[12012] - X[32078]

See Antreas Hatzipolakis and Peter Moses, Euclid 2577 .

X(44914) lies on these lines: {2, 10184}, {3, 14635}, {51, 10003}, {140, 389}, {632, 6509}, {3525, 42441}, {5647, 15366}, {10303, 31388}, {26907, 42862}, {38283, 42556}

X(44914) = midpoint of X(i) and X(j) for these {i,j}: {2, 12012}, {3, 14635}, {10184, 32078}
X(44914) = complement of X(10184)
X(44914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 32078, 10184}, {10184, 12012, 32078}

X(44915) = X(2)X(6)∩X(67)X(42007)

Barycentrics    (2 a^2-b^2-c^2)(2 a^6-2 a^4 (b^2+c^2)-a^2 (b^4-4 b^2 c^2+c^4)+3 (b^2-c^2)^2 (b^2+c^2)) : :
X(44915) = X(8030) - 2 X(599), 2 X(6) -3 X(1648)

See Angel Montesdeoca, Euclid 2593 .

X(44915) lies on these lines: {2,6}, {67,42007}, {111,32244}, {187,14357}, {542,14444}, {1640,33915}, {5104,41721}, {5181,39689}, {8869,11416}, {11646,13169}

leftri

Hatzipolakis-Lozada circles: X(44916)-X(44934)

rightri

This preamble and centers X(44916)-X(44934) were contributed by César Eliud Lozada, October 2, 2021.

Let ABC be a triangle, A'B'C' the orthic triangle of ABC, P a point and A"B"C" the circumcevian triangle of P. Then the circumcircles of AA'A", BB'B", CC'C" are coaxial and their radical axis is the line X(4)P. (Antreas Hatzipolakis, euclid 2510, Sept. 27, 2021)

Assume O and H are the circumcenter and orthocenter of ABC, respectively.

  1. For P = x : y : z, the radical trace T(P) of the three circles is:
     T(P)= SA*x*(y*SB^2+SC^2*z)+SB*SC*(y^2*SB+z^2*SC-2*SA*x*(y+z))+(SB+SC)*(SA^2*x^2-SB*SC*y*z) : :
  2. If P lies on the nine-point circle of ABC then T(P) = P.
  3. If P lies in the infinite, then T(P) lies on the circle with diameter OH. The appearance of (i, j) in the following list means that T(X(i))=X(j):
    (30, 3), (511, 31848), (512, 31850), (513, 31849), (514, 31851), (515, 31866), (516, 31852), (517, 31847), (518, 31865), (519, 31853), (523, 4), (524, 15098), (525, 43389), (542, 31854), (900, 18341), (952, 18342), (1503, 18338), (2574, 43396), (2575, 43395), (3564, 18348), (3566, 18347), (5965, 31864)
  4. The appearance of (i, j) in the following partial list means that T(X(i))=X(j), for P other than specified in (2) and (3):
    (1, 16869), (2, 5159), (3, 10257), (5, 2072), (6, 15341), (7, 31894), (8, 31895), (9, 31896), (10, 31897), (13, 44933), (14, 44934), (19, 8074), (20, 16976), (22, 16977), (23, 6676), (24, 44452), (25, 468), (28, 44898), (33, 16870), (53, 5523), (107, 24930), (186, 140), (225, 1785), (232, 34235), (235, 403), (278, 44901), (281, 5199), (297, 11007), (378, 15122), (381, 10297), (403, 5), (407, 36195), (427, 858), (428, 23), (429, 30447), (430, 33329), (431, 37982), (460, 1316), (462, 32460), (463, 32461), (468, 2), (497, 20130), (546, 18403), (847, 16269), (858, 1368), (860, 36155), (868, 37987), ...
  5. When P moves on a line L through H, T(P) moves also on the line L. In particular, if P is a point on the Euler line of ABC and such that OP/OH = t, then OT/OH = (4*R^2-SW)/(2*OH*(1-t)).
  6. When P moves on a line L* no passing through H, the locus of T(P) is a circle Ω(L*), here named the Hatzipolakis-Lozada circle of line L*. Ω(L*) passes through H and, if P* is the trilinear pole of L*, then the center O* of Ω(L*) is the dilation of X(3091) with center the polar-conjugate-of-P and ratio 5/4.
  7. If L* is the orthic axis X(231)X(232) of ABC then O*=X(381); if L* is the Hatzipolakis axis X(5)X(523) of ABC then O*=X(23323). For other named or notable lines of ABC, O* are given in X(44916) to X(44932).
  8. The transformation L* → Ω(L*) is not an inversion in a circle.

A generalization follows: Let t be a real number and let A'B'C' be the circumcevian triangle of a point P. Let A" be the point given by the vector equation PA" = t*PA', and define B" and C" cyclically. Denote the triangle A"B"C" by T(P,t). Suppose that U is a point other than P and k is a real number, and let A'''B'''C''' = T(U,k). The circles (AA"A'''), (BB''B'''), (CC''C''') are coaxial, and their radical axis is the line PU. In the construction above, P is arbitrary, U = X(4), t = 1, and k = 1/2. (Vu Thanh Tung, October 4, 2021)

X(44916) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(44)X(513) (ANTIORTHIC AXIS)

Barycentrics    (b+c)*a^9-2*b*c*a^8+2*(b+c)*b*c*a^7-2*(b+c)*(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^5+4*(b^2-c^2)^2*b*c*a^4+2*(b^2-c^2)*(b-c)*(4*b^4+4*c^4+(3*b^2+2*b*c+3*c^2)*b*c)*a^3-3*(b^4-c^4)*(b^2-c^2)^2*(b-c)*a-2*(b^2-c^2)^4*b*c : :
X(44916) = X(92)-5*X(3091) = 3*X(381)-X(39529) = 7*X(3832)+X(6360)

X(44916) lies on these lines: {4, 1214}, {5, 10}, {40, 7532}, {92, 3091}, {127, 25640}, {381, 39529}, {1103, 19372}, {1532, 1848}, {1902, 3142}, {2883, 12616}, {3832, 6360}, {5603, 37695}, {5786, 15836}, {5908, 34831}, {6261, 37696}, {6907, 18589}, {7535, 11496}, {9817, 24806}, {10395, 12233}, {11479, 23853}, {24316, 37822}

X(44916) = midpoint of X(4) and X(1214)
X(44916) = reflection of X(6708) in X(5)
X(44916) = X(1214)-of-Euler-triangle
X(44916) = X(6708)-of-Johnson-triangle

X(44917) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(181) (APOLLONIUS LINE)

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

X(44917) lies on these lines: {522, 44923}, {8672, 44918}

X(44918) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(3)X(6) (BROCARD AXIS)

Barycentrics    (a^8+(b^2+c^2)*a^6-(5*b^4-2*b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2)*(b^2-c^2) : :
X(44918) = 3*X(381)-X(16229) = 5*X(3091)-X(14618) = X(44921)-3*X(44926) = X(44921)-4*X(44931) = 3*X(44926)-4*X(44931) = 3*X(44926)-2*X(44932)

X(44918) lies on these lines: {5, 512}, {381, 16229}, {523, 44923}, {525, 44921}, {2506, 2548}, {3091, 14618}, {4151, 44928}, {5996, 7392}, {8672, 44917}, {9517, 39510}, {23878, 44919}, {32472, 44817}

X(44918) = reflection of X(i) in X(j) for these (i, j): (44921, 44932), (44932, 44931)
X(44918) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (44921, 44926, 44932), (44931, 44932, 44926)

X(44919) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(39)X(512) (BROCARD LINE)

Barycentrics    (b^2+c^2)^2*a^12+(b^2+c^2)*b^2*c^2*a^10-2*(3*b^4+5*b^2*c^2+3*c^4)*(b^4-b^2*c^2+c^4)*a^8+2*(b^2+c^2)*(4*b^8+4*c^8-(7*b^4-10*b^2*c^2+7*c^4)*b^2*c^2)*a^6-(b^2-c^2)^4*(3*b^4+2*b^2*c^2+3*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2*a^2-2*(b^2-c^2)^4*b^4*c^4 : :
X(44919) = 5*X(3091)-X(17984)

X(44919) lies on these lines: {5, 141}, {3091, 17984}, {11479, 24729}, {15980, 39569}, {23878, 44918}

X(44920) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(325)X(523) (DE LONGCHAMPS LINE)

Barycentrics    2*a^10-(b^2+c^2)*a^8-2*(3*b^4-4*b^2*c^2+3*c^4)*a^6+4*(b^2+c^2)*(b^4+c^4)*a^4+4*(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :
X(44920) = 3*X(2)+X(44438) = X(4)+3*X(16072) = 3*X(4)+X(21312) = 3*X(5)-X(6644) = X(20)-5*X(31255) = X(25)-5*X(3091) = 3*X(381)-X(1596) = 3*X(381)+X(18531) = 9*X(381)-X(18534)

X(44920) lies on these lines: {2, 3}, {113, 21637}, {182, 18418}, {1568, 16657}, {1660, 41362}, {3564, 18390}, {3589, 36201}, {3818, 15583}, {5480, 9813}, {5893, 9729}, {6696, 44686}, {7687, 14913}, {8263, 10516}, {9822, 13570}, {10643, 42101}, {10644, 42102}, {10961, 42283}, {10963, 42284}, {11416, 39871}, {12241, 34986}, {17814, 34966}, {18388, 18583}, {18440, 18918}, {19925, 44662}, {20772, 36518}

X(44920) = midpoint of X(i) and X(j) for these {i, j}: {4, 1368}, {1596, 18531}, {1660, 41362}, {44241, 44438}
X(44920) = reflection of X(6677) in X(5)
X(44920) = complement of X(44241)
X(44920) = X(1368)-of-Euler-triangle
X(44920) = X(6677)-of-Johnson-triangle
X(44920) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 16976, 140), (3, 381, 6623), (4, 3545, 7398), (4, 16072, 1368), (5, 3627, 6642), (5, 7526, 3628), (381, 18531, 1596), (381, 18537, 5), (403, 34664, 6676), (3628, 10226, 140), (3851, 7404, 5), (5020, 21312, 6644), (6816, 37197, 6823), (12362, 13487, 235)

X(44921) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(6)X(13) (FERMAT AXIS)

Barycentrics    (2*a^8-(b^2+c^2)*a^6-(5*b^4-6*b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :
X(44921) = 3*X(381)-X(16230) = X(684)+3*X(42733) = 5*X(3091)-X(44427) = 3*X(14566)-X(44810) = 2*X(44918)-3*X(44926) = 3*X(44918)-4*X(44931) = 9*X(44926)-8*X(44931) = 3*X(44926)-4*X(44932) = 2*X(44931)-3*X(44932)

X(44921) lies on these lines: {4, 6334}, {5, 690}, {381, 16230}, {525, 44918}, {684, 42733}, {2799, 39491}, {3091, 44427}, {6370, 44929}, {6644, 44826}, {7526, 14270}, {9033, 10113}, {9191, 40132}, {11620, 44817}, {14566, 44810}, {18570, 39477}, {18576, 43083}

X(44921) = midpoint of X(4) and X(6334)
X(44921) = reflection of X(44918) in X(44932)
X(44921) = X(6334)-of-Euler-triangle
X(44921) = {X(44918), X(44932)}-harmonic conjugate of X(44926)

X(44922) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(241)X(514) (GERGONNE LINE)

Barycentrics    2*a^9-(b+c)*a^8+(b^2+c^2)*a^7-(b^2-c^2)*(b-c)*a^6-(7*b^4-6*b^2*c^2+7*c^4)*a^5+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^3+(b^2-c^2)*(b-c)*(5*b^4-2*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^4*a+(b^2-c^2)^3*(b-c)*(-4*b^2-2*b*c-4*c^2) : :
X(44922) = X(281)-5*X(3091) = X(347)+7*X(3832) = 3*X(381)-X(39531)

X(44922) lies on these lines: {4, 17073}, {5, 516}, {281, 3091}, {347, 3832}, {381, 39529}, {1699, 6708}, {9779, 37695}, {11479, 36641}

X(44922) = midpoint of X(4) and X(17073)
X(44922) = X(17073)-of-Euler-triangle

X(44923) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(3) (I-O LINE)

Barycentrics    (a^7+(b^2+c^2)*a^5-(5*b^4+5*c^4+2*(b^2-b*c+c^2)*b*c)*a^3+2*(b+c)*(b^2+c^2)*b*c*a^2+(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2*a-2*(b^2-c^2)^2*(b+c)*b*c)*(b-c) : :
X(44923) = 3*X(381)-X(16228) = 5*X(3091)-X(44426) = 2*X(44928)+X(44929)

X(44923) lies on these lines: {5, 33528}, {381, 16228}, {522, 44917}, {523, 44918}, {3091, 44426}, {3667, 44815}, {7392, 44429}

X(44923) = midpoint of X(44925) and X(44928)
X(44923) = reflection of X(44929) in X(44925)

X(44924) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(187)X(237) (LEMOINE AXIS)

Barycentrics    (b^2+c^2)*a^10+2*b^2*c^2*a^8-2*(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^6+8*(b^6-c^6)*(b^2-c^2)*a^4-3*(b^4-c^4)*(b^2-c^2)^3*a^2-2*(b^2-c^2)^4*b^2*c^2 : :
X(44924) = 3*X(4)+X(42329) = 3*X(216)-X(42329) = X(264)-5*X(3091) = 3*X(381)+X(30258) = 3*X(381)-X(39530) = X(3164)+7*X(3832) = 17*X(3854)-X(40896) = 9*X(5055)-5*X(40329)

X(44924) lies on these lines: {4, 216}, {5, 141}, {23, 4993}, {30, 10003}, {132, 5133}, {253, 264}, {381, 30258}, {546, 32428}, {3164, 3832}, {3850, 42862}, {3854, 40896}, {5055, 40329}, {5158, 33971}, {5943, 40641}, {6530, 36412}, {10314, 13860}, {12294, 37988}, {18583, 23583}, {34146, 34845}, {37871, 39243}, {40684, 44003}

X(44924) = midpoint of X(i) and X(j) for these {i, j}: {4, 216}, {30258, 39530}
X(44924) = reflection of X(i) in X(j) for these (i, j): (14767, 5), (42487, 11793), (42862, 3850)
X(44924) = X(216)-of-Euler-triangle
X(44924) = X(14767)-of-Johnson-triangle
X(44924) = {X(381), X(30258)}-harmonic conjugate of X(39530)

X(44925) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(2) (NAGEL LINE)

Barycentrics    (3*a^7-(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-(11*b^4+11*c^4+2*(2*b^2-3*b*c+2*c^2)*b*c)*a^3+(b+c)*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2+(7*b^2+4*b*c+7*c^2)*(b^2-c^2)^2*a-(b^2-c^2)^2*(b+c)*(b^2+4*b*c+c^2))*(b-c) : :
X(44925) = 3*X(381)-X(16231) = 5*X(3091)-X(7649) = 7*X(3832)+X(20294) = X(44928)+2*X(44929)

X(44925) lies on these lines: {4, 20315}, {5, 3667}, {381, 16231}, {522, 44917}, {523, 44931}, {3091, 7649}, {3832, 20294}, {4057, 11479}, {9817, 43924}, {9818, 39225}, {19372, 42312}

X(44925) = midpoint of X(i) and X(j) for these {i, j}: {4, 20315}, {44923, 44929}
X(44925) = reflection of X(44928) in X(44923)
X(44925) = X(20315)-of-Euler-triangle

X(44926) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(6)X(17) (NAPOLEON AXIS)

Barycentrics    (4*a^8+(b^2+c^2)*a^6-5*(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(11*b^4-16*b^2*c^2+11*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :
X(44926) = 2*X(44918)+X(44921) = X(44918)-4*X(44931) = X(44918)+2*X(44932) = X(44921)+8*X(44931) = X(44921)-4*X(44932) = 2*X(44931)+X(44932)

X(44926) lies on these lines: {5, 32478}, {525, 44918}

X(44926) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (44918, 44932, 44921), (44931, 44932, 44918)

X(44927) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(3259)X(3326) (SHERMAN LINE)

Barycentrics    2*a^16-4*(b+c)*a^15-2*(b^2-8*b*c+c^2)*a^14+2*(b+c)*(5*b^2-12*b*c+5*c^2)*a^13-(9*b^4+9*c^4+4*b*c*(4*b^2-13*b*c+4*c^2))*a^12+2*(b^2-c^2)*(b-c)*(2*b^2+21*b*c+2*c^2)*a^11+2*(4*b^4+4*c^4-b*c*(17*b^2+46*b*c+17*c^2))*(b-c)^2*a^10-2*(b^2-c^2)*(b-c)*(15*b^4+15*c^4-b*c*(7*b^2+38*b*c+7*c^2))*a^9+(25*b^6+25*c^6+(96*b^4+96*c^4-b*c*(15*b^2+124*b*c+15*c^2))*b*c)*(b-c)^2*a^8+4*(b^2-c^2)*(b-c)*(5*b^6+5*c^6-(29*b^4+29*c^4-b*c*(3*b^2+22*b*c+3*c^2))*b*c)*a^7-2*(21*b^8+21*c^8+2*(2*b^6+2*c^6-(42*b^4+42*c^4+b*c*(2*b^2-47*b*c+2*c^2))*b*c)*b*c)*(b-c)^2*a^6+2*(b^2-c^2)*(b-c)^3*(7*b^6+7*c^6+(50*b^4+50*c^4+b*c*(13*b^2-28*b*c+13*c^2))*b*c)*a^5+(b^2-c^2)^2*(b-c)^2*(17*b^6+17*c^6-(66*b^4+66*c^4+b*c*(27*b^2-64*b*c+27*c^2))*b*c)*a^4-2*(b^2-c^2)^3*(b-c)*(10*b^6+10*c^6-(9*b^4+9*c^4+2*b*c*(13*b^2-21*b*c+13*c^2))*b*c)*a^3+2*(b^2-c^2)^4*(b-c)^2*(2*b^4+2*c^4+5*b*c*(3*b^2-2*b*c+3*c^2))*a^2+2*(b^2-c^2)^5*(b-c)*(3*b^4+3*c^4-b*c*(9*b^2-10*b*c+9*c^2))*a-(b^2-c^2)^6*(3*b^4+3*c^4-2*b*c*(3*b^2-4*b*c+3*c^2)) : :
X(44927) = 3*X(4)+X(2734) = 3*X(381)-X(39535) = X(1309)-5*X(3091) = X(2734)-3*X(10017)

X(44927) lies on these lines: {4, 2734}, {5, 38617}, {381, 39535}, {515, 11727}, {946, 39546}, {1309, 3091}

X(44927) = midpoint of X(4) and X(10017)
X(44927) = reflection of X(40558) in X(5)
X(44927) = X(10017)-of-Euler-triangle
X(44927) = X(40558)-of-Johnson-triangle
X(44927) = complement of circumperp conjugate of X(10016)

X(44928) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(7) (SODDY LINE)

Barycentrics    (a^6+3*(b^2+c^2)*a^4-4*(b+c)*(b^2+c^2)*a^3-(5*b^4+5*c^4-2*b*c*(b+2*c)*(2*b+c))*a^2+4*(b^2-c^2)^2*(b+c)*a+(b^2-4*b*c+c^2)*(b^2-c^2)^2)*(b-c) : :
X(44928) = 3*X(381)-X(39532) = 3*X(42756)+X(44448) = X(44408)-3*X(44432) = 3*X(44923)-X(44929) = 3*X(44925)-2*X(44929)

X(44928) lies on these lines: {5, 514}, {381, 39532}, {522, 44917}, {663, 19372}, {4151, 44918}, {4449, 9817}, {4843, 44931}, {5020, 44408}, {6006, 44815}, {6642, 39476}, {42756, 44448}

X(44928) = reflection of X(44925) in X(44923)

X(44929) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(5)

Barycentrics    (2*a^7-(b+c)*a^6+(b+c)*(b^2+c^2)*a^4-2*(3*b^4+3*c^4+(b-c)^2*b*c)*a^3+(b+c)*(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^2+2*(b^2-c^2)^2*(2*b^2+b*c+2*c^2)*a-(b^2-c^2)^2*(b+c)^3)*(b-c) : :
X(44929) = 3*X(381)-X(39534) = 5*X(3091)-X(44428) = 3*X(44923)-2*X(44928) = 3*X(44925)-X(44928)

X(44929) lies on these lines: {5, 900}, {381, 39534}, {522, 44917}, {523, 23323}, {3091, 44428}, {6370, 44921}, {6677, 30792}, {7392, 31131}, {7526, 39478}, {9818, 39200}

X(44929) = reflection of X(i) in X(j) for these (i, j): (44815, 5), (44923, 44925)
X(44929) = X(44815)-of-Johnson-triangle

X(44930) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(1)X(6)

Barycentrics    (a^9-2*(b+c)*a^8+2*(b^2+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6-2*(2*b^2+3*b*c+2*c^2)*(b-c)^2*a^5+2*(b+c)*(5*b^4+5*c^4-b*c*(b+c)^2)*a^4-2*(b^4+c^4)*(b+c)^2*a^3-2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+4*b*c*(b^2+b*c+c^2))*a^2+(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))*(b^2-c^2)^2*a-2*(b^2-c^2)^3*(b-c)*b*c)*(b-c) : :
X(44930) = 3*X(381)-X(39536) = 5*X(3091)-X(17924)

X(44930) lies on these lines: {5, 3309}, {381, 39536}, {522, 44917}, {525, 44918}, {3091, 17924}, {4162, 37695}, {8226, 18344}

X(44931) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(2)X(6)

Barycentrics    (5*a^8+2*(b^2+c^2)*a^6-4*(5*b^4-3*b^2*c^2+5*c^4)*a^4+2*(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4)*a^2-(b^2-c^2)^2*(b^4+10*b^2*c^2+c^4))*(b^2-c^2) : :
X(44931) = 3*X(381)-X(39533) = X(2501)-5*X(3091) = 7*X(3832)+X(6563) = 3*X(44918)+X(44921) = X(44918)+3*X(44926) = X(44921)-9*X(44926) = X(44921)-3*X(44932) = 3*X(44926)-X(44932)

X(44931) lies on these lines: {5, 1499}, {381, 39533}, {523, 44925}, {525, 44918}, {669, 11479}, {2501, 3091}, {3832, 6563}, {4843, 44928}, {5926, 9818}

X(44931) = midpoint of X(44918) and X(44932)
X(44931) = reflection of X(14341) in X(5)
X(44931) = perspector of the circumconic {{A, B, C, X(9227), X(42352)}}
X(44931) = X(14341)-of-Johnson-triangle
X(44931) = {X(44918), X(44926)}-harmonic conjugate of X(44932)

X(44932) = CENTER OF THE HATZIPOLAKIS-LOZADA CIRCLE OF LINE X(5)X(6)

Barycentrics    (3*a^8-2*(5*b^4-4*b^2*c^2+5*c^4)*a^4+4*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :
X(44932) = X(44918)-3*X(44926) = X(44921)+3*X(44926) = X(44921)+2*X(44931) = 3*X(44926)-2*X(44931)

X(44932) lies on these lines: {5, 3566}, {523, 23323}, {525, 44918}, {546, 42399}, {6334, 44705}, {6587, 39510}, {7526, 44680}, {9818, 34952}

X(44932) = midpoint of X(i) and X(j) for these {i, j}: {6334, 44705}, {44918, 44921}
X(44932) = reflection of X(i) in X(j) for these (i, j): (6587, 39510), (42399, 546), (44918, 44931)
X(44932) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (44918, 44926, 44931), (44921, 44926, 44918)

X(44933) = X(4)X(13) ∩ X(5459)X(33499)

Barycentrics    9*(b^2+c^2)*a^10-18*(b^4+b^2*c^2+c^4)*a^8-12*(b^2-c^2)^2*b^2*c^2*a^4+6*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^6-2*sqrt(3)*(-a^2+b^2+c^2)*(4*a^6-(b^2+c^2)*a^4+2*(b^4-3*b^2*c^2+c^4)*a^2-5*(b^4-c^4)*(b^2-c^2))*S*a^2+9*(b^8-c^8)*(b^2-c^2)*a^2-6*(b^6-c^6)*(b^2-c^2)^3 : :
X(44933) = 3*X(13)+X(5669) = 3*X(5459)-X(33499)

X(44933) lies on these lines: {4, 13}, {5459, 33499}, {5994, 33517}, {6334, 23870}

X(44933) = midpoint of X(5994) and X(33517)

X(44934) = X(4)X(14) ∩ X(5460)X(33501)

Barycentrics    9*(b^2+c^2)*a^10-18*(b^4+b^2*c^2+c^4)*a^8-12*(b^2-c^2)^2*b^2*c^2*a^4+6*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^6+2*sqrt(3)*(-a^2+b^2+c^2)*(4*a^6-(b^2+c^2)*a^4+2*(b^4-3*b^2*c^2+c^4)*a^2-5*(b^4-c^4)*(b^2-c^2))*S*a^2+9*(b^8-c^8)*(b^2-c^2)*a^2-6*(b^6-c^6)*(b^2-c^2)^3 : :
X(44934) = 3*X(14)+X(5668) = 3*X(5460)-X(33501)

X(44934) lies on these lines: {4, 14}, {5460, 33501}, {5995, 33518}, {6334, 23871}

X(44934) = midpoint of X(5995) and X(33518)

X(44935) = X(4)X(18853)∩X(30)X(568)

Barycentrics    2 a^10-7 a^8 (b^2+c^2)+2 a^6 (5 b^4+18 b^2 c^2+5 c^4)-8 a^4 (b^6+b^4 c^2+b^2 c^4+c^6)+4 a^2 (b^2-c^2)^2 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^4 (b^2+c^2) : :
X(44935) = X(7667) - 2 X(16657)

See Angel Montesdeoca, Euclid 2601 .

X(44935) lies on these lines: {4,18853}, {30,568}, {524,32062}, {1906,37498}, {3146,13142}, {3543,3564}, {3575,13598}, {5073,13292}, {5076,31831}, {7553,30522}, {7667,16657}, {9777,35513}, {11202,37904}, {11402,34621}, {13570,35283}, {15305,34380}, {15311,21969}, {18914,33703}, {36987,43957}

X(44935) = reflection of X(7667) in X(16657)

X(44936) = 67TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    (-2*a^2 + b^2 + c^2 - Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4])*(3*a^2*(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)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - a^2*b^2*c^2*J^2 - 2*Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S)*(3*a^2*(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)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] - a^2*b^2*c^2*J^2 + Sqrt[-2*a^8 + 3*a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - 2*b^8 + 3*a^6*c^2 - 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 3*b^6*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - 2*c^8 + 2*a^2*b^2*c^2*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]*J^2]*S) : :
Barycentrics    (-2*a^2 + b^2 + c^2 - Rt1)*(3*a^2*(a^2 - b^2)*(a^2 - c^2) - a^2*b^2*c^2*J^2 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Rt1 - 2*Rt2*S)*(3*a^2*(a^2 - b^2)*(a^2 - c^2) - a^2*b^2*c^2*J^2 - (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*Rt1 + Rt2*S) : : , where Rt1 = Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4], Rt2 = Sqrt[-2 a^8+3 a^6 b^2-2 a^4 b^4+3 a^2 b^6-2 b^8+3 a^6 c^2-2 a^4 b^2 c^2-2 a^2 b^4 c^2+3 b^6 c^2-2 a^4 c^4-2 a^2 b^2 c^4-2 b^4 c^4+3 a^2 c^6+3 b^2 c^6-2 c^8+2 a^2 b^2 c^2 Rt1 J^2] and J = |OH|/R

See Antreas Hatzipolakis and Peter Moses, euclid 2603.

X(44936) lies on the curve Q001 and these lines: {2 ,3}, {476, 32443}

X(44936) = antigonal image of X(39158)
X(44936) = symgonal image of X(39163)
X(44936) = barycentric product X(6190)*X(40989)
X(44936) = barycentric quotient X(i)/X(j) for these {i,j}: {1379, 40991},44937) {40989, 3413}

leftri

Hatzipolakis-Suppa circle: X(44937)-X(44966)

rightri

This preamble and centers X(44937)-X(44966) were contributed by Antreas Hatzipolakis and Ercole Suppa, October 4, 2021.

Let P be a point in the plane of a triangle ABC, and let

H = X(4) = orthocenter of ABC;
A'B'C' = pedal triangle of H;
A"B"C" be the pedal triangle P;
(Oa) = circumcircle of, AA'A", and define (Ob) and (Oc) cyclically;
Ra = radical axis of (Ob) and (Oc), and define Rb and Rc cyclically;
(O'a) = reflection of (Oa) in AA', and define (O'b) and (O'c) cyclically;
R'a = radical axis of (O'b) and (O'c), and R'b and R'c cyclically;
R'1 = reflection of R'a in BC, and define R'2 and R'3 cyclically.

The locus of P such that the parallels to R'1, R'2, R'3 through A', B', C', resp. concur is the union of the Euler line and the circle(X(382), R), where R = circumradius of ABC.

The circle (X(382), R) is here named the Hatzipolakis-Suppa Circle, or HS-circle.

Let Q=Q(P) be the point in which the three parallels concur. The following properties hold:

1. (X(382), R) is the reflection of the circumcircle in the orthocenter H.

2. The appearance of {i, j} in the following list means that P=X(i) lies on Euler line and Q(P)=X(j): {4,235}, {30,403}, {382,4}, {3146,5}, {3543,1596}, {5073,16868}, {10296,37984}, {10736,1312}, {10737,1313}, {12173,3089}, {15682,427}, {15684,7577}, {35480,25}, {35490,37197}, {44438,6623}

3. The appearance of {i, j} in the following list means that P=X(i) lies on HS-circle and Q(P)=X(j): {10152,133}, {10721,125}, {10722,115}, {10723,114}, {10724,119}, {10725,118}, {10726,124}, {10727,116}, {10728,11}, {10729,5511}, {10730,5510}, {10731,25640}, {10732,117}, {10733,113}, {10734,5512}, {10735,132}, {10736,1312}, {10737,1313}, {14989,3258}

4. If P lies on HS-circle then Q(P) is the image of P under the homothety with center H and factor k = -1/2.

5. When P moves on the Euler line the locus of point Q(P) is the Euler line.

The function that maps the Euler line to the Euler line is here named the 1st Hatziplakis-Suppa transform (or 1st HS transform).

6. When P moves on HS-circle the locus of Q(P) is the nine-point circle of ABC.

The function that maps the HS-circle to the nine-point circle is here named the "2nd Hatzipolakis-Suppa" transform (or 2nd HS transform).

7. Let L1 = radical axis of (HS-circle, circumcircle); then
      X(16080) = trilinear pole (L1)
      X(4) = radical trace (HS-circle,circumcircle)

8. Let L2 = radical axis of (HS-circle, ninepoint-circle); then
      X(38253) = trilinear pole of L2
      X(13473) = radical trace of (HS-circle,ninepoint-circle).

9. Other points on HS-circle or on the nine-point circle are X(44937)-X(44946) and X(44967)-X(44992).

10. Other points on the Euler line are X(44947)-X(44956).

X(44937) = 1ST HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    2 a^10 b^6-2 a^6 b^10+3 a^12 b^2 c^2-4 a^10 b^4 c^2-4 a^8 b^6 c^2+10 a^6 b^8 c^2+a^4 b^10 c^2-2 a^2 b^12 c^2-4 a^10 b^2 c^4+2 a^8 b^4 c^4-4 a^6 b^6 c^4-2 a^4 b^8 c^4+6 a^2 b^10 c^4+2 a^10 c^6-4 a^8 b^2 c^6-4 a^6 b^4 c^6+5 a^4 b^6 c^6-6 a^2 b^8 c^6-b^10 c^6+10 a^6 b^2 c^8-2 a^4 b^4 c^8-6 a^2 b^6 c^8+2 b^8 c^8-2 a^6 c^10+a^4 b^2 c^10+6 a^2 b^4 c^10-b^6 c^10-2 a^2 b^2 c^12 : :
Barycentrics    2 SB SC SW^6+S^6 (-48 R^4+4 R^2 SA+SB SC+20 R^2 SW-SA SW)-S^2 SW^3 (48 R^2 SB SC-5 SB SC SW-SB SW^2-SC SW^2+SW^3)+2 S^4 (144 R^4 SB SC-60 R^2 SB SC SW-6 R^2 SB SW^2-6 R^2 SC SW^2+2 SB SC SW^2+8 R^2 SW^3+SB SW^3+SC SW^3-SW^4) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44937) lies on the HS Circle and these lines: {4,689}, {20,35971}, {30,37888}, {6026,12953}, {7334,12943}

X(44937) = reflection of X(i) in X(j) for these (i,j): (20,35971), (689,4)

X(44938) = 2ND HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44938) lies on the HS Circle and these lines: {4,727}, {20,20551}, {30,28469}, {10725,36997}

X(44938) = reflection of X(i) in X(j) for these (i,j): (20,20551), (727,4)

X(44939) = 3RD HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics   9*a^8*b^2*c^2*(b^2+c^2)-4*b^4*c^4*(b^2-c^2)^2*(b^2+c^2)+3*a^10*(2*b^4-5*b^2*c^2+2*c^4)+a^2*b^2*c^2*(b^2-c^2)^2*(10*b^4-3*b^2*c^2+10*c^4)-a^6*(2*b^8+3*b^6*c^2+17*b^4*c^4+3*b^2*c^6+2*c^8)+a^4*(-4*b^10+7*b^8*c^2+3*b^6*c^4+3*b^4*c^6+7*b^2*c^8-4*c^10) : :
Barycentrics    S^6 (-12 R^2+SA-SW)+4 SB SC SW^5+2 S^4 (27 R^2 SB SC+9 R^2 SB SW+9 R^2 SC SW+2 SB SC SW+3 R^2 SW^2-SB SW^2-SC SW^2)-S^2 SW^2 (108 R^2 SB SC-8 SB SC SW+SB SW^2+SC SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44939) lies on the HS Circle and these lines: {4,729}, {30,39639}, {5969,10723}, {6022,12953}, {7333,12943}, {10733,36997}

X(44939) = reflection of X(729) in X(4)

X(44940) = 4TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44940) lies on the HS Circle and these lines: {4,741}, {30,6010}, {1356,12943}, {5691,10723}

X(44940) = reflection of X(741) in X(4)

X(44941) = 5TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    2 a^8 b^4-2 a^4 b^8+3 a^10 b c-4 a^8 b^3 c-3 a^7 b^4 c-a^6 b^5 c+4 a^5 b^6 c+2 a^4 b^7 c+a^3 b^8 c-2 a b^10 c+4 a^4 b^6 c^2-4 a^8 b c^3+2 a^6 b^3 c^3-2 a^4 b^5 c^3-2 a^3 b^6 c^3+6 a b^8 c^3+2 a^8 c^4-3 a^7 b c^4-a^4 b^4 c^4+a^3 b^5 c^4-2 a^2 b^6 c^4+2 a b^7 c^4-b^8 c^4-a^6 b c^5-2 a^4 b^3 c^5+a^3 b^4 c^5-6 a b^6 c^5+4 a^5 b c^6+4 a^4 b^2 c^6-2 a^3 b^3 c^6-2 a^2 b^4 c^6-6 a b^5 c^6+2 b^6 c^6+2 a^4 b c^7+2 a b^4 c^7-2 a^4 c^8+a^3 b c^8+6 a b^3 c^8-b^4 c^8-2 a b c^10 : :
X(44941) = 5*X(3091)-4*X(40545)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44941) lies on the HS Circle and these lines: {4,789}, {3091,40545}

X(44941) = reflection of X(789) in X(4)

X(44942) = 6TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^12-2 a^10 b^2-3 a^9 b^3-a^8 b^4+a^5 b^7+2 a^4 b^8+2 a^3 b^9-2 b^12-2 a^10 c^2+2 a^8 b^2 c^2+4 a^7 b^3 c^2-2 a^5 b^5 c^2-6 a^3 b^7 c^2+4 b^10 c^2-3 a^9 c^3+4 a^7 b^2 c^3+3 a^6 b^3 c^3+a^5 b^4 c^3-4 a^4 b^5 c^3-2 a^3 b^6 c^3-a^2 b^7 c^3+2 b^9 c^3-a^8 c^4+a^5 b^3 c^4+6 a^3 b^5 c^4-2 b^8 c^4-2 a^5 b^2 c^5-4 a^4 b^3 c^5+6 a^3 b^4 c^5+2 a^2 b^5 c^5-2 b^7 c^5-2 a^3 b^3 c^6+a^5 c^7-6 a^3 b^2 c^7-a^2 b^3 c^7-2 b^5 c^7+2 a^4 c^8-2 b^4 c^8+2 a^3 c^9+2 b^3 c^9+4 b^2 c^10-2 c^12 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44942) lies on the HS Circle and these lines: {4,825}, {30,28844}

X(44942) = reflection of X(825) in X(4)

X(44943) = 7TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^14-2 a^12 b^2-4 a^10 b^4+a^6 b^8+4 a^4 b^10-2 b^14-2 a^12 c^2+2 a^10 b^2 c^2+4 a^8 b^4 c^2-2 a^6 b^6 c^2-6 a^4 b^8 c^2+4 b^12 c^2-4 a^10 c^4+4 a^8 b^2 c^4+5 a^6 b^4 c^4-a^2 b^8 c^4-2 a^6 b^2 c^6+2 a^2 b^6 c^6-2 b^8 c^6+a^6 c^8-6 a^4 b^2 c^8-a^2 b^4 c^8-2 b^6 c^8+4 a^4 c^10+4 b^2 c^12-2 c^14 : :
Barycentrics    S^4 (2 R^2-SW)+SB SC (50 R^2-13 SW) SW^2+S^2 (-9 R^2 SB SC+5 R^2 SA SW+5 SB SC SW-10 R^2 SW^2+2 SB SW^2+2 SC SW^2+SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44943) lies on the HS Circle and these lines: {4,827}, {30,1287}, {3627,6321}, {9479,10723}

X(44943) = reflection of X(827) in X(4)

X(44944) = 8TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^10+3 a^9 b-2 a^8 b^2-3 a^7 b^3-2 a^6 b^4+3 a^5 b^5+4 a^4 b^6-a^3 b^7-a^2 b^8-2 a b^9-2 b^10+3 a^9 c+3 a^8 b c-7 a^7 b^2 c-7 a^6 b^3 c+3 a^5 b^4 c+3 a^4 b^5 c+3 a^3 b^6 c+3 a^2 b^7 c-2 a b^8 c-2 b^9 c-2 a^8 c^2-7 a^7 b c^2+2 a^6 b^2 c^2+2 a^5 b^3 c^2+3 a^4 b^4 c^2+5 a^3 b^5 c^2+8 a b^7 c^2+5 b^8 c^2-3 a^7 c^3-7 a^6 b c^3+2 a^5 b^2 c^3+8 a^4 b^3 c^3-7 a^3 b^4 c^3-7 a^2 b^5 c^3+8 a b^6 c^3+6 b^7 c^3-2 a^6 c^4+3 a^5 b c^4+3 a^4 b^2 c^4-7 a^3 b^3 c^4-6 a^2 b^4 c^4-12 a b^5 c^4-3 b^6 c^4+3 a^5 c^5+3 a^4 b c^5+5 a^3 b^2 c^5-7 a^2 b^3 c^5-12 a b^4 c^5-8 b^5 c^5+4 a^4 c^6+3 a^3 b c^6+8 a b^3 c^6-3 b^4 c^6-a^3 c^7+3 a^2 b c^7+8 a b^2 c^7+6 b^3 c^7-a^2 c^8-2 a b c^8+5 b^2 c^8-2 a c^9-2 b c^9-2 c^10 : :
X(44944) = 5*X(3091)-4*X(40547)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44944) lies on the HS Circle and these lines: {4,835}, {20,5515}, {3091,40547}

X(44944) = reflection of X(i) in X(j) for these (i,j): (20,5515), (835,4)

X(44945) = 9TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^14+7 a^12 b^2-16 a^10 b^4-12 a^8 b^6+19 a^6 b^8+7 a^4 b^10-6 a^2 b^12-2 b^14+7 a^12 c^2+11 a^10 b^2 c^2-26 a^8 b^4 c^2+16 a^6 b^6 c^2-3 a^4 b^8 c^2-3 a^2 b^10 c^2-2 b^12 c^2-16 a^10 c^4-26 a^8 b^2 c^4+74 a^6 b^4 c^4-36 a^4 b^6 c^4+50 a^2 b^8 c^4+18 b^10 c^4-12 a^8 c^6+16 a^6 b^2 c^6-36 a^4 b^4 c^6-82 a^2 b^6 c^6-14 b^8 c^6+19 a^6 c^8-3 a^4 b^2 c^8+50 a^2 b^4 c^8-14 b^6 c^8+7 a^4 c^10-3 a^2 b^2 c^10+18 b^4 c^10-6 a^2 c^12-2 b^2 c^12-2 c^14 : :
Barycentrics    2 SB SC (32 R^2-7 SW) SW^2+S^4 (4 R^2+SW)+S^2 (-18 R^2 SB SC-8 R^2 SA SW-5 SB SC SW-8 R^2 SW^2-SA SW^2+3 SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44945) lies on the HS Circle and these lines: {4,907}, {30,29180}

X(44945) = reflection of X(907) in X(4)

X(44946) = 10TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^14-14 a^12 b^2+14 a^10 b^4-16 a^8 b^6+11 a^6 b^8-4 a^4 b^10+8 a^2 b^12-2 b^14-14 a^12 c^2+70 a^10 b^2 c^2-70 a^8 b^4 c^2+28 a^6 b^6 c^2+10 a^4 b^8 c^2-52 a^2 b^10 c^2+12 b^12 c^2+14 a^10 c^4-70 a^8 b^2 c^4+85 a^6 b^4 c^4-46 a^4 b^6 c^4+113 a^2 b^8 c^4-20 b^10 c^4-16 a^8 c^6+28 a^6 b^2 c^6-46 a^4 b^4 c^6-122 a^2 b^6 c^6+10 b^8 c^6+11 a^6 c^8+10 a^4 b^2 c^8+113 a^2 b^4 c^8+10 b^6 c^8-4 a^4 c^10-52 a^2 b^2 c^10-20 b^4 c^10+8 a^2 c^12+12 b^2 c^12-2 c^14 : :
Barycentrics    9 S^6 (30 R^2-SA-3 SW)+6 SA (SA-SW) SW^5-S^2 SW^2 (162 R^2 SA^2-162 R^2 SA SW-6 SA^2 SW+7 SA SW^2-5 SW^3)+6 S^4 (54 R^2 SA^2-45 R^2 SA SW-6 SA^2 SW-24 R^2 SW^2+5 SA SW^2+SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44946) lies on the HS Circle and these lines: {4,2709}, {30,843}, {511,10734}, {524,10722}, {1499,10723}

X(44946) = reflection of X(2709) in X(4)

X(44947) = 1ST HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (a^4 b^4-a^2 b^6+a^2 b^4 c^2+a^4 c^4+a^2 b^2 c^4-2 b^4 c^4-a^2 c^6) (a^6 b^2+a^4 b^4+a^6 c^2-2 a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2+a^4 c^4-a^2 b^2 c^4-2 b^4 c^4+b^2 c^6) : :
Barycentrics    (S^4+SA SW^3+S^2 (-12 R^2 SA+2 SA^2+SA SW-SW^2)) (2 S^4+(SA-SW) SW^3+S^2 (-12 R^2 SA+2 SA^2+12 R^2 SW+SA SW-SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44947) lies on the nine-point circle and these lines: {4,689}, {5,35971}, {11,7334}, {12,6026}, {125,6656}, {316,2679}

X(44947) = midpoint of X(4) and X(689)
X(44947) = reflection of X(35971) in X(5)
X(44947) = complement of the circumperp conjugate of X(689)
X(44947) = X(44937)-image under 2nd HS transform
X(44947) = X(689)-of-Euler-triangle
X(44947) = X(35971)-of-Johnson-triangle

X(44948) = 2ND HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44948) lies on the nine-point circle and these lines: {2,28469}, {4,727}, {5,20551}, {114,7683}, {121,19130}, {3271,5518}, {5513,29641}

X(44948) = midpoint of X(4) and X(727)
X(44948) = reflection of X(20551) in X(5)
X(44948) = complement of X(28469)
X(44948) = complementary conjugate of X(28470)
X(44948) = X(i)-complementary conjugate of X(j) for these (i,j): (1,28470), (6,43051), (513,3705)
X(44948) = X(4)-Ceva conjugate of X(28470)
X(44948) = orthic-isogonal conjugate of X(28470)
X(44948) = X(727)-of-Euler-triangle
X(44948) = X(20551)-of-Johnson-triangle
X(44948) = X(44938)-image under 2nd HS transform

X(44949) = 3RD HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (b^2-c^2)^2 (-3 a^4-a^2 b^2-a^2 c^2+2 b^2 c^2) (2 a^4 b^2-2 a^2 b^4+2 a^4 c^2+a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4) : :
Barycentrics    (b^2-c^2)^2 (S^2+(3 SA-2 SW) SW) (2 S^2 SA-SW (3 SB SC+SA SW)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44949) lies on the nine-point circle and these lines: {2,39639}, {4,729}, {11,7333}, {12,6022}, {114,5480}, {126,13518}, {511,9152}, {6388,9151}, {9993,9998}

X(44949) = midpoint of X(4) and X(729)
X(44949) = complement of X(39639)
X(44949) = complementary conjugate of X(32472)
X(44949) = X(1)-complementary conjugate of X(32472)
X(44949) = X(4)-Ceva conjugate of X(32472)
X(44949) = orthic-isogonal conjugate of X(32472)
X(44949) = X(729)-of-Euler-triangle
X(44949) = X(44939)-image under 2nd HS transform

X(44950) = 4TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (b-c)^2 (-a^3-a^2 b-a^2 c-a b c+b^2 c+b c^2) (-a^3 b^2+a b^4-a^3 b c+a^2 b^2 c-a^3 c^2+a^2 b c^2-a b^2 c^2-b^3 c^2-b^2 c^3+a c^4) : :
X(44950) = 3*X(1699)+X(5539)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44950) lies on the nine-point circle and these lines: {2,6010}, {4,741}, {11,1356}, {114,946}, {115,17197}, {118,7683}, {1699,5539}, {2051,5213}, {2886,3037}, {6002,16613}, {10478,38477}, {15611,17761}, {24220,30448}

X(44950) = midpoint of X(4) and X(741)
X(44950) = complement of X(6010)
X(44950) = complementary conjugate of X(6002)
X(44950) = X(i)-complementary conjugate of X(j) for these (i,j): (1,6002), (6,7180), (513,3687), (649,28244), (661,34528)
X(44950) = X(4)-Ceva conjugate of X(6002)
X(44950) = orthic-isogonal conjugate of X(6002)
X(44950) = X(741)-of-Euler-triangle
X(44950) = X(44940)-image under 2nd HS transform

X(44951) = 5TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44951) lies on the nine-point circle and these lines: {3,40545}, {4,789}, {11,37596}, {116,3821}, {125,26601}

X(44951) = midpoint of X(4) and X(789)
X(44951) = reflection of X(3) in X(40545)
X(44951) = complement of the circumperp conjugate of X(789)
X(44951) = complement of circumcircle antipode of X(789)
X(44951) = complement of areal center of pedal triangles of PU(8)
X(44951) = X(789)-of-Euler-triangle
X(44951) = X(44941)-image under 2nd HS transform

X(44952) = 6TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44952) lies on the nine-point circle and these lines: {2,28844}, {4,825}, {11,5244}, {1890,5190}

X(44952) = midpoint of X(4) and X(825)
X(44952) = complement of X(28844)
X(44952) = complementary conjugate of X(28845)
X(44952) = X(1)-complementary conjugate of X(28845)
X(44952) = X(4)-Ceva conjugate of X(28845)
X(44952) = orthic-isogonal conjugate of X(28845)
X(44952) = X(825)-of-Euler-triangle
X(44952) = X(44942)-image under 2nd HS transform

X(44953) = 7TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (2*a^6-(b^2-c^2)^2*(b^2+c^2)-a^2*(b^4+c^4))*(b^8-b^6*c^2-b^2*c^6+c^8-a^4*(b^2-c^2)^2+a^6*(b^2+c^2)-a^2*(b^6+c^6)) : :
Barycentrics    (S^2 (SA-2 SW)+5 SB SC SW) (-SB SC-10 R^2 SW+SA SW+2 SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44953) lies on the nine-point circle and these lines: {2,29011}, {4,827}, {23,3258}, {114,9479}, {115,546}, {122,6676}, {125,5133}, {127,3934}, {136,428}, {137,37349}, {1196,35971}, {1556,29012}, {2967,39216}, {5099,11563}, {5103,36471}, {7797,33695}, {7859,35500}, {14675,18403}, {33504,34137}

X(44953) = midpoint of X(4) and X(827)
X(44953) = complement of X(29011)
X(44953) = complementary conjugate of X(29012)
X(44953) = X(1)-complementary conjugate of X(29012)
X(44953) = X(4)-Ceva conjugate of X(29012)
X(44953) = orthic-isogonal conjugate of X(29012)
X(44953) = X(827)-of-Euler-triangle
X(44953) = X(44943)-image under 2nd HS transform

X(44954) = 8TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44954) lies on the nine-point circle and these lines: {3,40547}, {4,835}, {5,5515}, {11,37592}, {37,5517}, {113,24828}, {116,4657}, {124,958}, {125,4205}, {132,21664}, {1867,5521}

X(44954) = midpoint of X(4) and X(835)
X(44954) = reflection of X(i) in X(j) for these (i,j): (3,40547), (5515,5)
X(44954) = complement of the circumperp conjugate of X(835)
X(44954) = complement of X(45136)
X(44954) = complementary conjugate of isogonal conjugate of X(45136)
X(44954) = complementary conjugate of ctic conjugate of X(45136)
X(44954) = X(835)-of-Euler-triangle
X(44954) = X(5515)-of-Euler-triangle
X(44954) = X(44944)-image under 2nd HS transform

X(44955) = 9TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (2 a^6+3 a^4 b^2-4 a^2 b^4-b^6+3 a^4 c^2+b^4 c^2-4 a^2 c^4+b^2 c^4-c^6) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2+8 a^4 b^2 c^2-3 a^2 b^4 c^2+2 b^6 c^2-a^4 c^4-3 a^2 b^2 c^4-6 b^4 c^4-a^2 c^6+2 b^2 c^6+c^8) : :
Barycentrics    (SB SC-8 R^2 SW+2 SA SW+SW^2) (-4 SB SC SW+S^2 (SA+SW)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44955) lies on the nine-point circle and these lines: {2,15613}, {4,907}, {115,147}, {122,7386}, {125,37439}, {127,6823}, {133,2967}, {136,5064}, {1906,5139}

X(44955) = midpoint of X(4) and X(907)
X(44955) = complement of X(29180)
X(44955) = complementary conjugate of X(29181)
X(44955) = X(1)-complementary conjugate of X(29181)
X(44955) = X(4)-Ceva conjugate of X(29181)
X(44955) = orthic-isogonal conjugate of X(29181)
X(44955) = X(907)-of-Euler-triangle
X(44955) = X(44945)-image under 2nd HS transform

X(44956) = 10TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics   (2 a^4-2 a^2 b^2-b^4-2 a^2 c^2+4 b^2 c^2-c^4) (a^4 b^2-4 a^2 b^4+b^6+a^4 c^2+4 a^2 b^2 c^2-4 a^2 c^4+c^6) : :
Barycentrics    (2 S^2-2 SA^2+SB^2-6 SB SC+SC^2) (-3 S^2 (6 R^2-SA)-SW (3 SA^2-SA SW-SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2616.

X(44956) lies on the nine-point circle and these lines: {2,843}, {4,2709}, {114,1499}, {115,524}, {125,625}, {126,512}, {141,5099}, {316,352}, {511,5512}, {543,35586}, {3258,5108}, {3849,9127}, {5103,20389}, {5461,16341}, {5465,40553}, {7603,9151}, {8176,9169}, {8704,11569}, {11182,31655}, {11511,38971}

X(44956) = midpoint of X(i) and X(j) for these {i,j}: {4,2709}, {316,352}
X(44956) = reflection of X(16341) in X(5461)
X(44956) = complement of X(843)
X(44956) = complementary conjugate of X(543)
X(44956) = X(i)-complementary conjugate of X(j) for these (i,j): (1,543), (543,10), (661,44398), (1641,16597), (2502,37)
X(44956) = X(4)-Ceva conjugate of X(543)
X(44956) = orthic-isogonal conjugate of X(543)
X(44956) = X(2709)-of-Euler-triangle
X(44956) = X(44946)-image under 2nd HS transform

X(44957) =  1ST HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (5 a^4 b^2-10 a^2 b^4+5 b^6+5 a^4 c^2+16 a^2 b^2 c^2-5 b^4 c^2-10 a^2 c^4-5 b^2 c^4+5 c^6) : :
Barycentrics    SB SC (36 R^2-5 SA-5 SW) : :
X(44957) = 3*X(4)+5*X(6353), X(4)-5*X(6623), 7*X(4)+5*X(37460), 3*X(381)+X(20850), 9*X(5055)-5*X(30771),X(6353)+3*X(6623) ,7*X(6353)-3*X(37460), 7*X(6623)+X(37460)

As a point on the Euler line, X(44957) has Shinagawa coefficients (5*F,-4*E+5*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44957) lies on these lines: {2,3}, {113,41588}, {5663,44079}, {12024,18390}, {21850,41585}

X(44957) = complement of the circumperp conjugate of X(37984)
X(44957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,35501), (4,3542,15750), (4,37942,549), (25,37984,3845), (235,403,1596), (403,1596,5), (546,3089,7715), (3542,44226,550), (3542,44438,37935), (6353,6995,20850), (10151,37458,15687), (21841,37197,3627), (37935,44226,44438), (37935,44438,550)
X(44957) = X(2)-image under 1st HS transform

X(44958) =  2ND HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+3 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : :
Barycentrics    SB SC (7 R^2-SA-SW) : :

As a point on the Euler line, X(44958) has Shinagawa coefficients (4*F,-3*E+4*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44958) lies on these lines: {2,3}, {6,43891}, {113,5889}, {125,12290}, {154,12254}, {155,2914}, {847,15424}, {1093,6344}, {1181,22533}, {1495,12289}, {1614,18390}, {1870,37720}, {2883,26879}, {3060,5448}, {3567,43831}, {3574,13423}, {3767,8744}, {4351,7741}, {4354,7951}, {5449,15305}, {5894,11270}, {6000,26917}, {6198,37719}, {7687,18394}, {7699,12063}, {7722,15063}, {7747,10986}, {7765,39575}, {7796,44146}, {9544,12370}, {9656,11399}, {9671,11398}, {9705,22750}, {9781,18388}, {9934,14644}, {10540,34799}, {10575,26913}, {10632,16964}, {10633,16965}, {10641,42814}, {10642,42813}, {10985,39590}, {11381,23294}, {11455,11704}, {11456,43808}, {11457,12112}, {11464,13403}, {11475,42488}, {11476,42489}, {12022,16252}, {12133,20396}, {12168,22550}, {12292,20379}, {12293,35264}, {12300,15060}, {13452,15105}, {14249,18349}, {14577,18353}, {15032,39571}, {15072,43817}, {15112,25641}, {15752,18405}, {16837,36809}, {18555,41597}, {19357,43818}, {21659,26882}, {22802,43599}, {25739,26883}, {32062,32767}, {34786,44082}

X(44958) = barycentric product of X(2052) and X(34783)
X(44958) = trilinear product of X(158) and X(34783)
X(44958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,14865), (4,24,34797), (4,403,16868), (4,3089,34484), (4,3147,35481), (4,3518,18559), (4,3542,186), (4,6143,1593), (4,6353,35471), (4,6622,7505), (4,7505,3520), (4,13619,35490), (4,14940,378), (4,16868,7577), (4,21844,18560), (4,35473,1885), (4,35482,1597), (4,37119,13596), (4,37943,3), (5,1596,1907), (5,1906,15559), (5,1907,1594), (5,3843,5169), (5,11799,20), (24,10594,18378), (24,37197,4), (25,35488,4), (235,403,4), (235,13406,10594), (381,10594,4), (381,18378,18377), (403,7576,10254), (403,11563,37943), (468,18560,21844), (468,44226,18560), (546,7576,4), (1594,1596,4), (1598,7547,4), (1657,35479,35489), (1885,10018,35473), (1885,37942,10018), (1906,15559,4), (2041,2042,2071), (3060,18504,5448), (3089,3832,37122), (3091,37349,7564), (3147,35481,17506), (3515,35490,13619), (3542,6623,4), (3627,10255,31074), (3832,37122,4), (6240,10151,4), (6640,44276,12086), (10151,21841,6240), (11455,11704,20299), (11563,44235,3), (18560,44226,4), (37943,44235,16868)
X(44958) = X(3)-image under 1st HS transform

X(44959) =  3RD HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^4 b^2-6 a^2 b^4+3 b^6+3 a^4 c^2+10 a^2 b^2 c^2-3 b^4 c^2-6 a^2 c^4-3 b^2 c^4+3 c^6) : :
Barycentrics    SB SC (22 R^2-3 SA-3 SW) : :

As a point on the Euler line, X(44959) has Shinagawa coefficients (6*F,-5*E+6*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

XX(44959) lies on these lines: {2,3}, {3574,13433}, {8550,19123}, {10294,13431}, {10641,44016}, {10642,44015}, {13598,36518}, {14644,16655}, {16534,18555}, {23294,43592}

X(44959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,403,35487), (4,468,35491), (4,3517,6240), (4,3542,32534), (4,7505,3516), (4,32534,18560), (4,35487,1594), (403,15559,16868), (1596,16868,15559), (3089,35488,7576), (3517,37197,4), (3858,10301,4), (10018,35492,3523)
X(44959) = X(5)-image under 1st HS transform

X(44960) =  4TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^4 b^2-6 a^2 b^4+3 b^6+3 a^4 c^2+8 a^2 b^2 c^2-3 b^4 c^2-6 a^2 c^4-3 b^2 c^4+3 c^6) : :
Barycentrics    SB SC (20 R^2-3 SA-3 SW) : :
X(44960) = X(4)+3*X(3542),X(4)-3*X(37197),3*X(381)+X(9714),5*X(1656)-3*X(3548),X(3515)-3*X(3542),X(3515)+3*X(37197),7*X(3523)-3*X(30552)

As a point on the Euler line, X(44960) has Shinagawa coefficients (3*F,-2*E+3*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44960) lies on these lines: {2,3}, {33,10592}, {34,10593}, {49,22750}, {53,39565}, {113,16879}, {1204,1514}, {1829,38034}, {1843,38136}, {1902,38042}, {1974,39884}, {2207,43291}, {3092,18538}, {3093,18762}, {3532,20427}, {5448,21847}, {5654,13142}, {5656,26944}, {5878,26958}, {5893,11438}, {6102,44084}, {6247,43592}, {7687,15647}, {8550,41593}, {10192,13403}, {10641,42135}, {10642,42138}, {12131,38229}, {12135,38138}, {12162,44079}, {12250,43719}, {12300,21357}, {12315,23291}, {13382,13567}, {13419,23324}, {13474,23332}, {14530,18945}, {14644,16659}, {15105,20417}, {15448,34785}, {15873,18388}, {16252,18390}, {16621,23325}, {22660,41588}, {26917,32111}, {35764,42270}, {35765,42273}, {36518,41673}

X(44960) = midpoint of X(i) and X(j) for these {i,j}: {4,3515}, {3542,37197}
X(44960) = X(3515)-of-Euler-triangle
X(44960) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6622,37942), (3,6623,44226), (4,468,550), (4,3542,3515), (4,7505,35477), (4,10019,3858), (4,14940,35478), (4,17506,18560), (4,21841,37458), (4,35477,1885), (4,37460,5073), (4,37943,17506), (5,235,1596), (5,1596,1595), (5,15761,6823), (24,10151,3627), (235,403,5), (381,3089,6756), (382,21974,3), (427,16868,5), (1885,7505,549), (3147,44438,548), (3515,37197,4), (3575,35488,3845), (3858,7715,4), (6622,6623,3), (7403,10254,5), (13488,21313,37458), (14813,14814,16196), (15761,44235,5), (16252,18390,31804), (21841,37984,4), (37942,44226,3)
X(44960) = X(20)-image under 1st HS transform

X(44961) =  5TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    2 a^8 b^2-4 a^6 b^4+4 a^2 b^8-2 b^10+2 a^8 c^2+6 a^6 b^2 c^2-a^4 b^4 c^2-13 a^2 b^6 c^2+6 b^8 c^2-4 a^6 c^4-a^4 b^2 c^4+18 a^2 b^4 c^4-4 b^6 c^4-13 a^2 b^2 c^6-4 b^4 c^6+4 a^2 c^8+6 b^2 c^8-2 c^10 : :
Barycentrics    S^2 (15 R^2-4 SW)+SB SC (27 R^2-4 SW) : :
X(44961) = 3*X(2)+X(18325),X(3)-3*X(44282),3*X(4)+5*X(37760),X(5)-3*X(403),3*X(5)-X(858),5*X(5)-3*X(2072),X(5)+3*X(11563),7*X(5)-3*X(37938),5*X(5)+3*X(43893),X(20)-3*X(15646),X(20)+3*X(31726),X(20)-9*X(37943),X(23)+3*X(381),X(23)-3*X(44266),3*X(113)+X(41586),3*X(140)-4*X(37911),3*X(186)+X(382),3*X(381)-X(18572),X(382)-3*X(44283),9*X(403)-X(858),5*X(403)-X(2072),3*X(403)+X(11799),7*X(403)-X(37938),5*X(403)+X(43893),5*X(468)-3*X(18579),3*X(468)-2*X(22249),2*X(546)+X(12105),3*X(546)+2*X(37897),3*X(547)-2*X(5159),X(548)+3*X(11558),X(548)-6*X(37942),2*X(548)-3*X(37968),X(548)-3*X(44234),X(550)-3*X(44214),5*X(631)-3*X(34152),5*X(858)-9*X(2072),X(858)+9*X(11563),X(858)+3*X(11799),7*X(858)-9*X(37938),5*X(858)+9*X(43893),X(1533)+3*X(23515),5*X(1656)-X(7464),X(1657)-5*X(37952),3*X(2070)+5*X(3843),3*X(2070)+X(10296),3*X(2071)-7*X(3526),X(2072)+5*X(11563),3*X(2072)+5*X(11799),7*X(2072)-5*X(37938),5*X(3091)-X(7574),5*X(3091)+X(37967),2*X(3530)-3*X(44452),9*X(3545)-X(5189),X(3830)+3*X(37907),3*X(3830)+5*X(37958),7*X(3832)-3*X(18403),7*X(3832)+3*X(37936),5*X(3843)-X(10296),3*X(3845)-X(18323),2*X(3850)+X(16619),6*X(3850)+X(37899),7*X(3851)+X(37924),X(3853)+3*X(10096),3*X(3853)+5*X(22248),4*X(3856)-3*X(23323),4*X(3856)+3*X(37971),2*X(3860)+X(37904),2*X(3861)-3*X(10151),9*X(5055)-5*X(30745),9*X(5055)-X(35001),5*X(5076)+7*X(37957),3*X(5476)-X(15826),3*X(5655)+X(41724),3*X(7426)+X(18323),3*X(7574)+X(20063),3*X(7575)-5*X(37760),9*X(10096)-5*X(22248),3*X(10151)+2*X(44264),3*X(10257)-4*X(16239),3*X(10297)+X(37899),X(10706)+3*X(15362),X(10989)-5*X(19709),X(11558)+2*X(37942),2*X(11558)+X(37968),3*X(11563)-X(11799),7*X(11563)+X(37938),5*X(11563)-X(43893),7*X(11799)+3*X(37938),5*X(11799)-3*X(43893),3*X(12105)-4*X(37897),X(12105)+4*X(37984),X(12112)+3*X(38724),X(12902)+3*X(35265),3*X(13619)+5*X(17578),9*X(15046)-X(37496),3*X(15350)-2*X(16239),X(15646)-3*X(37943),5*X(15696)-9*X(37941),X(15704)-3*X(44280),3*X(16532)-X(44246),3*X(16619)-X(37899),X(17800)-9*X(37955),5*X(18571)-6*X(18579),3*X(18571)-4*X(22249),X(18572)+3*X(44266),9*X(18579)-10*X(22249),X(20063)-3*X(37967),3*X(25338)-2*X(37897),X(25338)+2*X(37984),5*X(30745)-X(35001),X(31726)+3*X(37943),X(37897)+3*X(37984),X(37900)+9*X(38071),X(37901)+7*X(41106),9*X(37907)-5*X(37958),3*X(37909)+5*X(41099),5*X(37938)+7*X(43893),4*X(37942)-X(37968),X(44267)+3*X(44282)

As a point on the Euler line, X(44961) has Shinagawa coefficients (E+16*F,-11*E+16*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44961) lies on these lines: {2,3}, {113,41586}, {115,16308}, {523,19918}, {524,25566}, {1495,10113}, {1503,11801}, {1533,23515}, {1539,32110}, {3564,19140}, {3818,32217}, {5160,7951}, {5448,14449}, {5476,15826}, {5655,41724}, {6000,20379}, {6243,18504}, {7286,7741}, {8262,32271}, {8705,16511}, {10149,37719}, {10264,32111}, {10706,15362}, {12112,38724}, {12902,35265}, {13451,18388}, {14915,20304}, {15046,37496}, {15807,44516}, {16306,43291}, {16881,43831}, {18435,38397}, {21850,32113}

X(44961) = midpoint of X(i) and X(j) for these {i,j}: {3,44267}, {4,7575}, {5,11799}, {23,18572}, {186,44283}, {381,44266}, {403,11563}, {546,25338}, {1495,10113}, {1539,32110}, {2072,43893}, {3153,37947}, {3627,10295}, {3818,32217}, {3845,7426}, {3861,44264}, {7574,37967}, {8262,32271}, {10264,32111}, {10297,16619}, {11558,44234}, {15646,31726}, {15687,44265}, {18325,37950}, {18403,37936}, {21850,32113}, {23323,37971}, {37969,44288}
X(44961) = reflection of X(i) in X(j) for these (i,j): (546,37984), (10257,15350), (10297,3850), (12105,25338), (15122,3628), (18571,468), (37968,44234), (44234,37942)
X(44961) = complement of X(37950)
X(44961) = Ehrmann-side-to-orthic similarity image of X(18572)
X(44961) = X(7575)-of-Euler-triangle
X(44961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,18325,37950), (5,3845,5169), (5,5169,13413), (5,11563,11799), (23,381,18572), (235,403,37984), (235,13406,546), (403,7426,10254), (403,10295,16868), (403,11799,5), (1312,1313,3845), (1596,39504,14893), (2070,3843,10296), (3845,10254,13413), (5055,35001,30745), (5169,10254,5), (6756,37984,10151), (7505,44271,10226), (7575,44281,15331), (11558,37942,37968), (11563,13406,25338), (15761,44235,140), (18567,35488,546), (18572,44266,23), (25338,44234,13383), (31681,31682,6677), (31726,37943,15646), (35488,37440,18567), (44267,44282,3)
X(44961) = X(23)-image under 1st HS transform

X(44962) =  6TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (7 a^4 b^2-14 a^2 b^4+7 b^6+7 a^4 c^2+22 a^2 b^2 c^2-7 b^4 c^2-14 a^2 c^4-7 b^2 c^4+7 c^6) : :
Barycentrics    SB SC (50 R^2-7 SA-7 SW) : :

As a point on the Euler line, X(44962) has Shinagawa coefficients (14*F,-11*E+14*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44962) lies on these lines: {2,3}, {10641,42904}, {10642,42905}

X(44962) = X(140)-image under 1st HS transform

X(44963) =  7TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^8 b^2-3 a^6 b^4+2 a^4 b^6-3 a^2 b^8+2 b^10+2 a^8 c^2+8 a^6 b^2 c^2+3 a^4 b^4 c^2+4 a^2 b^6 c^2-b^8 c^2-3 a^6 c^4+3 a^4 b^2 c^4-6 a^2 b^4 c^4-b^6 c^4+2 a^4 c^6+4 a^2 b^2 c^6-b^4 c^6-3 a^2 c^8-b^2 c^8+2 c^10) : :
Barycentrics    SB SC (S^2 (20 R^2-3 SA-3 SW)-(36 R^2-5 SA-5 SW) SW^2) : :

As a point on the Euler line, X(44963) has Shinagawa coefficients (F*(5*E^2+10*E*F+5*F^2-3*S^2),-4*E^3-3*E^2*F+5*F^3-3*F*S^2+2*E(3*F^2+S^2)).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44963) lies on this line: {2,3}

X(44963) = X(384)-image under 1st HS transform

X(44964) =  8TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (21 a^8 b^2-64 a^6 b^4+86 a^4 b^6-64 a^2 b^8+21 b^10+21 a^8 c^2+20 a^6 b^2 c^2-42 a^4 b^4 c^2+108 a^2 b^6 c^2-43 b^8 c^2-64 a^6 c^4-42 a^4 b^2 c^4-104 a^2 b^4 c^4+22 b^6 c^4+86 a^4 c^6+108 a^2 b^2 c^6+22 b^4 c^6-64 a^2 c^8-43 b^2 c^8+21 c^10) : :
Barycentrics    SB SC (16 S^2 (7 R^2-SA-SW)-(36 R^2-5 SA-5 SW) SW^2) : :

As a point on the Euler line, X(44964) has Shinagawa coefficients (F*(5E*^2+10*E*F*+5*F^2-16*S^2),-4*E^3-3*E^2*F+6*E*F*^2+5*F^3+12*E*S^2-16*F*S^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44964) lies on this line: {2,3}

X(44964) = X(439)-image under 1st HS transform

X(44965) =  9TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^8 b^2-7 a^6 b^4+8 a^4 b^6-7 a^2 b^8+3 b^10+3 a^8 c^2+7 a^6 b^2 c^2-a^4 b^4 c^2+11 a^2 b^6 c^2-4 b^8 c^2-7 a^6 c^4-a^4 b^2 c^4-12 a^2 b^4 c^4+b^6 c^4+8 a^4 c^6+11 a^2 b^2 c^6+b^4 c^6-7 a^2 c^8-4 b^2 c^8+3 c^10) : :
Barycentrics    SB SC (S^2 (48 R^2-7 SA-7 SW)-(36 R^2-5 SA-5 SW) SW^2) : :

As a point on the Euler line, X(44965) has Shinagawa coefficients (F*(5*E^2+10*E*F+5*F^2-7*S^2),-4*E^3-3*E^2*F+6*E*F^2+5*F^3+5*E*S^2-7*F*S^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44965) lies on this line: {2,3}

X(44965) = X(1003)-image under 1st HS transform

X(44966) =  10TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^8 b^2-10 a^6 b^4+14 a^4 b^6-10 a^2 b^8+3 b^10+3 a^8 c^2+2 a^6 b^2 c^2-8 a^4 b^4 c^2+18 a^2 b^6 c^2-7 b^8 c^2-10 a^6 c^4-8 a^4 b^2 c^4-16 a^2 b^4 c^4+4 b^6 c^4+14 a^4 c^6+18 a^2 b^2 c^6+4 b^4 c^6-10 a^2 c^8-7 b^2 c^8+3 c^10) : :
Barycentrics    SB SC (S^2 (36 R^2-5 SA-5 SW)-(8 R^2-SA-SW) SW^2) : :

As a point on the Euler line, X(44966) has Shinagawa coefficients (F*(E*^2+2*E*F+F^2-5*S^2),-E^3-E^2*F+E*F^2+F^3+4*E*S^2-5*F*S^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2617 .

X(44966) = lies on these lines: {2,3}, {2971,39663}

X(44966) = X(1513)-image under 1st HS transform

X(44967) = REFLECTION OF X(476) IN X(4)

Barycentrics    3*a^16 - 8*a^14*b^2 + 2*a^12*b^4 + 7*a^10*b^6 + 5*a^8*b^8 - 18*a^6*b^10 + 8*a^4*b^12 + 3*a^2*b^14 - 2*b^16 - 8*a^14*c^2 + 30*a^12*b^2*c^2 - 25*a^10*b^4*c^2 - 30*a^8*b^6*c^2 + 51*a^6*b^8*c^2 - 5*a^4*b^10*c^2 - 22*a^2*b^12*c^2 + 9*b^14*c^2 + 2*a^12*c^4 - 25*a^10*b^2*c^4 + 69*a^8*b^4*c^4 - 35*a^6*b^6*c^4 - 45*a^4*b^8*c^4 + 48*a^2*b^10*c^4 - 14*b^12*c^4 + 7*a^10*c^6 - 30*a^8*b^2*c^6 - 35*a^6*b^4*c^6 + 84*a^4*b^6*c^6 - 29*a^2*b^8*c^6 + 7*b^10*c^6 + 5*a^8*c^8 + 51*a^6*b^2*c^8 - 45*a^4*b^4*c^8 - 29*a^2*b^6*c^8 - 18*a^6*c^10 - 5*a^4*b^2*c^10 + 48*a^2*b^4*c^10 + 7*b^6*c^10 + 8*a^4*c^12 - 22*a^2*b^2*c^12 - 14*b^4*c^12 + 3*a^2*c^14 + 9*b^2*c^14 - 2*c^16 : :
X(44967) = 3 X[4] - 2 X[25641], 4 X[5] - 3 X[38700], 2 X[20] - 3 X[38701], 3 X[376] - 4 X[31379], 3 X[381] - 2 X[38609], 3 X[476] - 4 X[25641], 2 X[477] - 3 X[34312], 5 X[3091] - 4 X[22104], 2 X[3146] + X[38678], 4 X[3154] - 3 X[15055], 4 X[3258] - 3 X[38701], 3 X[3543] - X[34193], 4 X[3627] - X[38677], 3 X[3830] - X[38580], 3 X[5627] - 4 X[10113], 8 X[12052] - 7 X[15043], 3 X[14993] - 4 X[21316], 7 X[15020] - 8 X[31945], 4 X[20957] - 3 X[34312]

See Peter Moses, Euclid 2600 .

X(44967) lies on the HS circle and these lines: {4, 476}, {5, 38700}, {20, 3258}, {30, 110}, {74, 36184}, {146, 39139}, {376, 31379}, {381, 38609}, {382, 14264}, {523, 10733}, {1531, 36188}, {1539, 36193}, {1657, 38610}, {2777, 14508}, {3091, 22104}, {3146, 14731}, {3154, 15055}, {3448, 32417}, {3543, 34193}, {3627, 38677}, {3830, 38580}, {3853, 18319}, {5073, 38581}, {5627, 10113}, {5889, 16978}, {10296, 10722}, {12052, 15043}, {12943, 33964}, {12953, 33965}, {13202, 36172}, {14480, 17702}, {14993, 21316}, {15020, 31945}, {15468, 34584}, {16340, 20127}

X(44967) = midpoint of X(i) and X(j) for these {i,j}: {3146, 14731}, {5073, 38581}
X(44967) = reflection of X(i) in X(j) for these {i,j}: {20, 3258}, {74, 36184}, {476, 4}, {477, 20957}, {1657, 38610}, {5889, 16978}, {14508, 17511}, {14989, 382}, {18319, 3853}, {20127, 16340}, {36172, 13202}, {36188, 1531}, {36193, 1539}, {38678, 14731}
X(44967) = reflection of X(10733) in the Euler line
X(44967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20, 3258, 38701}, {477, 20957, 34312}

X(44968) = REFLECTION OF X(675) IN X(4)

Barycentrics    3*a^10 - 3*a^9*b - 6*a^8*b^2 + 11*a^7*b^3 - 6*a^6*b^4 - 3*a^5*b^5 + 8*a^4*b^6 - 7*a^3*b^7 + 3*a^2*b^8 + 2*a*b^9 - 2*b^10 - 3*a^9*c + 3*a^8*b*c + a^7*b^2*c - a^6*b^3*c + 5*a^5*b^4*c - 5*a^4*b^5*c - a^3*b^6*c + a^2*b^7*c - 2*a*b^8*c + 2*b^9*c - 6*a^8*c^2 + a^7*b*c^2 + 2*a^6*b^2*c^2 - 2*a^5*b^3*c^2 - 7*a^4*b^4*c^2 + 13*a^3*b^5*c^2 - 4*a^2*b^6*c^2 - 4*a*b^7*c^2 + 7*b^8*c^2 + 11*a^7*c^3 - a^6*b*c^3 - 2*a^5*b^2*c^3 + 8*a^4*b^3*c^3 - 5*a^3*b^4*c^3 - 5*a^2*b^5*c^3 + 4*a*b^6*c^3 - 10*b^7*c^3 - 6*a^6*c^4 + 5*a^5*b*c^4 - 7*a^4*b^2*c^4 - 5*a^3*b^3*c^4 + 10*a^2*b^4*c^4 - 5*b^6*c^4 - 3*a^5*c^5 - 5*a^4*b*c^5 + 13*a^3*b^2*c^5 - 5*a^2*b^3*c^5 + 16*b^5*c^5 + 8*a^4*c^6 - a^3*b*c^6 - 4*a^2*b^2*c^6 + 4*a*b^3*c^6 - 5*b^4*c^6 - 7*a^3*c^7 + a^2*b*c^7 - 4*a*b^2*c^7 - 10*b^3*c^7 + 3*a^2*c^8 - 2*a*b*c^8 + 7*b^2*c^8 + 2*a*c^9 + 2*b*c^9 - 2*c^10 : :
X(44968) = 3 X[4] - 2 X[25642], 3 X[675] - 4 X[25642], 5 X[3091] - 4 X[31380]

See Peter Moses, Euclid 2600 .

X(44968) lies on the HS circle and these lines: {4, 675}, {20, 5513}, {30, 44876}, {152, 544}, {3091, 31380}, {6025, 12953}, {10732, 36991}, {12943, 33966}

X(44968) = reflection of X(i) in X(j) for these {i,j}: {20, 5513}, {675, 4}

X(44969) = REFLECTION OF X(691) IN X(4)

Barycentrics    3*a^14 - 8*a^12*b^2 + 4*a^10*b^4 + 8*a^8*b^6 - 11*a^6*b^8 + 2*a^4*b^10 + 4*a^2*b^12 - 2*b^14 - 8*a^12*c^2 + 26*a^10*b^2*c^2 - 26*a^8*b^4*c^2 + 16*a^6*b^6*c^2 + 2*a^4*b^8*c^2 - 18*a^2*b^10*c^2 + 8*b^12*c^2 + 4*a^10*c^4 - 26*a^8*b^2*c^4 + 9*a^6*b^4*c^4 - 6*a^4*b^6*c^4 + 35*a^2*b^8*c^4 - 12*b^10*c^4 + 8*a^8*c^6 + 16*a^6*b^2*c^6 - 6*a^4*b^4*c^6 - 42*a^2*b^6*c^6 + 6*b^8*c^6 - 11*a^6*c^8 + 2*a^4*b^2*c^8 + 35*a^2*b^4*c^8 + 6*b^6*c^8 + 2*a^4*c^10 - 18*a^2*b^2*c^10 - 12*b^4*c^10 + 4*a^2*c^12 + 8*b^2*c^12 - 2*c^14 : :
X(44969) = 3 X[4] - 2 X[16188], 4 X[5] - 3 X[38702], 2 X[20] - 3 X[38704], 4 X[113] - 3 X[249], 3 X[376] - 4 X[16760], 3 X[381] - 2 X[38611], 4 X[625] - 3 X[2071], 3 X[691] - 4 X[16188], 5 X[3091] - 4 X[40544], 2 X[3146] + X[38680], 4 X[3627] - X[38679], 3 X[3830] - X[38582], 4 X[5099] - 3 X[38704], 2 X[12383] - 3 X[33803], 4 X[14120] - 3 X[34473]

See Peter Moses, Euclid 2600 .

X(44969) lies on the HS circle and these lines: {4, 691}, {5, 38702}, {20, 5099}, {30, 99}, {113, 249}, {376, 16760}, {381, 38611}, {511, 10721}, {512, 10733}, {523, 10723}, {625, 2071}, {1657, 38613}, {2080, 44267}, {2794, 36174}, {3091, 40544}, {3146, 38680}, {3627, 38679}, {3830, 38582}, {5073, 38583}, {6023, 12943}, {6027, 12953}, {7464, 13449}, {10296, 10734}, {12383, 33803}, {14120, 34473}, {14989, 15682}, {36173, 39838}, {40890, 42426}

X(44969) = midpoint of X(5073) and X(38583)
X(44969) = reflection of X(i) in X(j) for these {i,j}: {20, 5099}, {691, 4}, {1657, 38613}, {2080, 44267}, {7464, 13449}, {36173, 39838}, {38679, 38953}, {38953, 3627}
X(44969) = reflection of X(10723) in the Euler line
X(44969) = {X(20),X(5099)}-harmonic conjugate of X(38704)

X(44970) = REFLECTION OF X(759) IN X(4)

Barycentrics    3*a^10 - 3*a^9*b - 4*a^8*b^2 + 7*a^7*b^3 - 4*a^6*b^4 - 3*a^5*b^5 + 6*a^4*b^6 - 3*a^3*b^7 + a^2*b^8 + 2*a*b^9 - 2*b^10 - 3*a^9*c + 2*a^8*b*c - a^7*b^2*c - 2*a^6*b^3*c + 6*a^5*b^4*c - 4*a^4*b^5*c - a^3*b^6*c + 2*a^2*b^7*c - a*b^8*c + 2*b^9*c - 4*a^8*c^2 - a^7*b*c^2 + 15*a^6*b^2*c^2 - 6*a^5*b^3*c^2 - 7*a^4*b^4*c^2 + 10*a^3*b^5*c^2 - 10*a^2*b^6*c^2 - 3*a*b^7*c^2 + 6*b^8*c^2 + 7*a^7*c^3 - 2*a^6*b*c^3 - 6*a^5*b^2*c^3 + 10*a^4*b^3*c^3 - 6*a^3*b^4*c^3 - 2*a^2*b^5*c^3 + 3*a*b^6*c^3 - 8*b^7*c^3 - 4*a^6*c^4 + 6*a^5*b*c^4 - 7*a^4*b^2*c^4 - 6*a^3*b^3*c^4 + 18*a^2*b^4*c^4 - a*b^5*c^4 - 4*b^6*c^4 - 3*a^5*c^5 - 4*a^4*b*c^5 + 10*a^3*b^2*c^5 - 2*a^2*b^3*c^5 - a*b^4*c^5 + 12*b^5*c^5 + 6*a^4*c^6 - a^3*b*c^6 - 10*a^2*b^2*c^6 + 3*a*b^3*c^6 - 4*b^4*c^6 - 3*a^3*c^7 + 2*a^2*b*c^7 - 3*a*b^2*c^7 - 8*b^3*c^7 + a^2*c^8 - a*b*c^8 + 6*b^2*c^8 + 2*a*c^9 + 2*b*c^9 - 2*c^10 : :
X(44970) = 3 X[4] - 2 X[42425], 3 X[381] - 2 X[38612], 3 X[759] - 4 X[42425], 3 X[3830] - X[14663]

See Peter Moses, Euclid 2600 .

X(44970) lies on the HS circle and these lines: {4, 759}, {20, 31845}, {30, 6011}, {381, 38612}, {1365, 12943}, {3830, 14663}, {5691, 10733}, {10724, 34789}, {10732, 12688}, {12953, 34194}

X(44970) = reflection of X(i) in X(j) for these {i,j}: {20, 31845}, {759, 4}

X(44971) = REFLECTION OF X(805) IN X(4)

Barycentrics    a^2*(-(a^10*b^6) + 3*a^8*b^8 - 3*a^6*b^10 + a^4*b^12 + 3*a^12*b^2*c^2 - 7*a^10*b^4*c^2 + 10*a^8*b^6*c^2 - 4*a^6*b^8*c^2 - 3*a^4*b^10*c^2 + 3*a^2*b^12*c^2 - 2*b^14*c^2 - 7*a^10*b^2*c^4 + 8*a^8*b^4*c^4 - 11*a^6*b^6*c^4 + 13*a^4*b^8*c^4 - 7*a^2*b^10*c^4 + 8*b^12*c^4 - a^10*c^6 + 10*a^8*b^2*c^6 - 11*a^6*b^4*c^6 - 3*a^4*b^6*c^6 + 2*a^2*b^8*c^6 - 15*b^10*c^6 + 3*a^8*c^8 - 4*a^6*b^2*c^8 + 13*a^4*b^4*c^8 + 2*a^2*b^6*c^8 + 18*b^8*c^8 - 3*a^6*c^10 - 3*a^4*b^2*c^10 - 7*a^2*b^4*c^10 - 15*b^6*c^10 + a^4*c^12 + 3*a^2*b^2*c^12 + 8*b^4*c^12 - 2*b^2*c^14) : :
X(44971) = 3 X[4] - 2 X[33330], 4 X[5] - 3 X[38703], 3 X[805] - 4 X[33330], 5 X[3091] - 4 X[22103]

See Peter Moses, Euclid 2600 .

X(44971) lies on the HS circle and these lines: {4, 805}, {5, 38703}, {20, 2679}, {30, 2698}, {511, 10722}, {512, 10723}, {2794, 14510}, {3091, 22103}, {5889, 16979}, {12953, 44042}, {14509, 23698}

X(44971) = reflection of X(i) in X(j) for these {i,j}: {20, 2679}, {805, 4}, {5889, 16979}, {14510, 31513}

X(44972) = REFLECTION OF X(842) IN X(4)

Barycentrics    3*a^14 - 6*a^12*b^2 + 2*a^10*b^4 + 4*a^8*b^6 - 7*a^6*b^8 + 4*a^4*b^10 + 2*a^2*b^12 - 2*b^14 - 6*a^12*c^2 + 14*a^10*b^2*c^2 - 10*a^8*b^4*c^2 + 4*a^6*b^6*c^2 + 2*a^4*b^8*c^2 - 12*a^2*b^10*c^2 + 8*b^12*c^2 + 2*a^10*c^4 - 10*a^8*b^2*c^4 + 9*a^6*b^4*c^4 - 6*a^4*b^6*c^4 + 19*a^2*b^8*c^4 - 12*b^10*c^4 + 4*a^8*c^6 + 4*a^6*b^2*c^6 - 6*a^4*b^4*c^6 - 18*a^2*b^6*c^6 + 6*b^8*c^6 - 7*a^6*c^8 + 2*a^4*b^2*c^8 + 19*a^2*b^4*c^8 + 6*b^6*c^8 + 4*a^4*c^10 - 12*a^2*b^2*c^10 - 12*b^4*c^10 + 2*a^2*c^12 + 8*b^2*c^12 - 2*c^14 : :
X(44972) = 3 X[4] - 2 X[5099], 4 X[5] - 3 X[38704], 2 X[20] - 3 X[38702], 3 X[249] - 2 X[12121], 3 X[376] - 4 X[40544], 3 X[381] - 2 X[38613], 3 X[842] - 4 X[5099], 5 X[3091] - 4 X[16760], 2 X[3146] + X[38679], 4 X[3627] - X[38680], 3 X[3830] - X[38583], 2 X[10295] - 3 X[38227], 3 X[14639] - 2 X[36166], 4 X[16188] - 3 X[38702], 3 X[21166] - 4 X[36170]

See Peter Moses, Euclid 2600 .

X(44972) lies on the HS circle and these lines: {4, 842}, {5, 38704}, {20, 16188}, {30, 98}, {249, 12121}, {316, 18323}, {376, 40544}, {381, 38613}, {511, 10296}, {512, 10721}, {523, 10722}, {1551, 12117}, {1657, 38611}, {2394, 14989}, {3091, 16760}, {3146, 38679}, {3543, 34193}, {3627, 38680}, {3830, 38583}, {5073, 38582}, {6023, 12953}, {6027, 12943}, {10152, 17986}, {10295, 38227}, {14639, 36166}, {18572, 35002}, {21166, 36170}, {23698, 36173}, {36174, 39809}

X(44972) = midpoint of X(5073) and X(38582)
X(44972) = reflection of X(i) in X(j) for these {i,j}: {20, 16188}, {316, 18323}, {691, 38953}, {842, 4}, {1657, 38611}, {12117, 1551}, {35002, 18572}, {36174, 39809}
X(44972) = reflection of X(10722) in the Euler line
X(44972) = {X(20),X(16188)}-harmonic conjugate of X(38702)

X(44973) = REFLECTION OF X(901) IN X(4)

Barycentrics    3*a^10 - 6*a^9*b - 5*a^8*b^2 + 14*a^7*b^3 - a^6*b^4 - 6*a^5*b^5 + 3*a^4*b^6 - 6*a^3*b^7 + 2*a^2*b^8 + 4*a*b^9 - 2*b^10 - 6*a^9*c + 24*a^8*b*c - 12*a^7*b^2*c - 30*a^6*b^3*c + 26*a^5*b^4*c - 8*a^4*b^5*c + 8*a^3*b^6*c + 10*a^2*b^7*c - 16*a*b^8*c + 4*b^9*c - 5*a^8*c^2 - 12*a^7*b*c^2 + 41*a^6*b^2*c^2 - 12*a^5*b^3*c^2 - 13*a^4*b^4*c^2 + 20*a^3*b^5*c^2 - 29*a^2*b^6*c^2 + 4*a*b^7*c^2 + 6*b^8*c^2 + 14*a^7*c^3 - 30*a^6*b*c^3 - 12*a^5*b^2*c^3 + 32*a^4*b^3*c^3 - 22*a^3*b^4*c^3 - 10*a^2*b^5*c^3 + 44*a*b^6*c^3 - 16*b^7*c^3 - a^6*c^4 + 26*a^5*b*c^4 - 13*a^4*b^2*c^4 - 22*a^3*b^3*c^4 + 54*a^2*b^4*c^4 - 36*a*b^5*c^4 - 4*b^6*c^4 - 6*a^5*c^5 - 8*a^4*b*c^5 + 20*a^3*b^2*c^5 - 10*a^2*b^3*c^5 - 36*a*b^4*c^5 + 24*b^5*c^5 + 3*a^4*c^6 + 8*a^3*b*c^6 - 29*a^2*b^2*c^6 + 44*a*b^3*c^6 - 4*b^4*c^6 - 6*a^3*c^7 + 10*a^2*b*c^7 + 4*a*b^2*c^7 - 16*b^3*c^7 + 2*a^2*c^8 - 16*a*b*c^8 + 6*b^2*c^8 + 4*a*c^9 + 4*b*c^9 - 2*c^10 : :
X(44973) = 3 X[4] - 2 X[31841], 4 X[5] - 3 X[38705], 2 X[20] - 3 X[38707], 3 X[381] - 2 X[38614], 3 X[901] - 4 X[31841], 5 X[3091] - 4 X[22102], 2 X[3146] + X[38682], 4 X[3259] - 3 X[38707], 3 X[3830] - X[38584]

See Peter Moses, Euclid 2600 .

X(44973) lies on the HS circle and these lines: {4, 901}, {5, 38705}, {20, 3259}, {23, 39479}, {30, 953}, {381, 38614}, {513, 10724}, {517, 10728}, {1657, 38617}, {2829, 14511}, {3025, 12953}, {3091, 22102}, {3146, 38682}, {3586, 33645}, {3627, 38954}, {3830, 38584}, {5073, 38586}, {5840, 14513}, {9579, 24201}, {12943, 13756}

X(44973) = midpoint of X(5073) and X(38586)
X(44973) = reflection of X(i) in X(j) for these {i,j}: {20, 3259}, {901, 4}, {953, 40100}, {1657, 38617}, {14511, 31512}, {38954, 3627}
X(44973) = {X(20),X(3259)}-harmonic conjugate of X(38707)

X(44974) = REFLECTION OF X(925) IN X(4)

Barycentrics    3*a^16 - 11*a^14*b^2 + 15*a^12*b^4 - 13*a^10*b^6 + 15*a^8*b^8 - 13*a^6*b^10 + a^4*b^12 + 5*a^2*b^14 - 2*b^16 - 11*a^14*c^2 + 33*a^12*b^2*c^2 - 31*a^10*b^4*c^2 - 3*a^8*b^6*c^2 + 19*a^6*b^8*c^2 + 7*a^4*b^10*c^2 - 25*a^2*b^12*c^2 + 11*b^14*c^2 + 15*a^12*c^4 - 31*a^10*b^2*c^4 + 36*a^8*b^4*c^4 - 14*a^6*b^6*c^4 - 25*a^4*b^8*c^4 + 45*a^2*b^10*c^4 - 26*b^12*c^4 - 13*a^10*c^6 - 3*a^8*b^2*c^6 - 14*a^6*b^4*c^6 + 34*a^4*b^6*c^6 - 25*a^2*b^8*c^6 + 37*b^10*c^6 + 15*a^8*c^8 + 19*a^6*b^2*c^8 - 25*a^4*b^4*c^8 - 25*a^2*b^6*c^8 - 40*b^8*c^8 - 13*a^6*c^10 + 7*a^4*b^2*c^10 + 45*a^2*b^4*c^10 + 37*b^6*c^10 + a^4*c^12 - 25*a^2*b^2*c^12 - 26*b^4*c^12 + 5*a^2*c^14 + 11*b^2*c^14 - 2*c^16 : :
X(44974) = 3 X[4] - 2 X[131], 2 X[20] - 3 X[38718], 4 X[131] - 3 X[925], 4 X[136] - 3 X[38718], 3 X[376] - 4 X[34840], 5 X[3091] - 4 X[34844], 2 X[21667] + X[33703]

See Peter Moses, Euclid 2600 .

X(44974) lies on the HS circle and these lines: {4, 131}, {20, 136}, {23, 5961}, {30, 1300}, {146, 3146}, {376, 34840}, {3091, 34844}, {7391, 39118}, {12086, 13496}, {21667, 33703}

X(44974) = reflection of X(i) in X(j) for these {i,j}: {20, 136}, {925, 4}, {1300, 13556}
X(44974) = {X(20),X(136)}-harmonic conjugate of X(38718)

X(44975) = REFLECTION OF X(927) IN X(4)

Barycentrics    3*a^12 - 6*a^11*b + a^10*b^2 + 4*a^8*b^4 + 2*a^6*b^6 - 4*a^5*b^7 - 5*a^4*b^8 + 6*a^3*b^9 - 3*a^2*b^10 + 4*a*b^11 - 2*b^12 - 6*a^11*c + 12*a^10*b*c + 2*a^9*b^2*c - 10*a^8*b^3*c - 2*a^7*b^4*c - 6*a^6*b^5*c + 14*a^5*b^6*c - 2*a^4*b^7*c + 2*a^2*b^9*c - 8*a*b^10*c + 4*b^11*c + a^10*c^2 + 2*a^9*b*c^2 - 9*a^8*b^2*c^2 + 10*a^7*b^3*c^2 - 2*a^6*b^4*c^2 - 18*a^5*b^5*c^2 + 30*a^4*b^6*c^2 - 18*a^3*b^7*c^2 + 9*a^2*b^8*c^2 - 8*a*b^9*c^2 + 3*b^10*c^2 - 10*a^8*b*c^3 + 10*a^7*b^2*c^3 + 8*a^6*b^3*c^3 + 8*a^5*b^4*c^3 - 18*a^4*b^5*c^3 - 6*a^3*b^6*c^3 - 4*a^2*b^7*c^3 + 20*a*b^8*c^3 - 8*b^9*c^3 + 4*a^8*c^4 - 2*a^7*b*c^4 - 2*a^6*b^2*c^4 + 8*a^5*b^3*c^4 - 10*a^4*b^4*c^4 + 18*a^3*b^5*c^4 - 22*a^2*b^6*c^4 + 8*a*b^7*c^4 - 2*b^8*c^4 - 6*a^6*b*c^5 - 18*a^5*b^2*c^5 - 18*a^4*b^3*c^5 + 18*a^3*b^4*c^5 + 36*a^2*b^5*c^5 - 16*a*b^6*c^5 + 4*b^7*c^5 + 2*a^6*c^6 + 14*a^5*b*c^6 + 30*a^4*b^2*c^6 - 6*a^3*b^3*c^6 - 22*a^2*b^4*c^6 - 16*a*b^5*c^6 + 2*b^6*c^6 - 4*a^5*c^7 - 2*a^4*b*c^7 - 18*a^3*b^2*c^7 - 4*a^2*b^3*c^7 + 8*a*b^4*c^7 + 4*b^5*c^7 - 5*a^4*c^8 + 9*a^2*b^2*c^8 + 20*a*b^3*c^8 - 2*b^4*c^8 + 6*a^3*c^9 + 2*a^2*b*c^9 - 8*a*b^2*c^9 - 8*b^3*c^9 - 3*a^2*c^10 - 8*a*b*c^10 + 3*b^2*c^10 + 4*a*c^11 + 4*b*c^11 - 2*c^12 : :
X(44975) = 3 X[4] - 2 X[33331], 3 X[927] - 4 X[33331], 5 X[3091] - 4 X[40554]

See Peter Moses, Euclid 2600 .

X(44975) lies on the HS circle and these lines: {4, 927}, {20, 1566}, {30, 2724}, {514, 10725}, {516, 3732}, {3091, 40554}, {3146, 14732}, {12953, 44043}

X(44975) = midpoint of X(3146) and X(14732)
X(44975) = reflection of X(i) in X(j) for these {i,j}: {{20, 1566}, {927, 4}

X(44976) = REFLECTION OF X(930) IN X(4)

Barycentrics    3*a^16 - 14*a^14*b^2 + 28*a^12*b^4 - 33*a^10*b^6 + 25*a^8*b^8 - 8*a^6*b^10 - 6*a^4*b^12 + 7*a^2*b^14 - 2*b^16 - 14*a^14*c^2 + 42*a^12*b^2*c^2 - 47*a^10*b^4*c^2 + 22*a^8*b^6*c^2 - 7*a^6*b^8*c^2 + 23*a^4*b^10*c^2 - 32*a^2*b^12*c^2 + 13*b^14*c^2 + 28*a^12*c^4 - 47*a^10*b^2*c^4 + 29*a^8*b^4*c^4 - 3*a^6*b^6*c^4 - 23*a^4*b^8*c^4 + 54*a^2*b^10*c^4 - 38*b^12*c^4 - 33*a^10*c^6 + 22*a^8*b^2*c^6 - 3*a^6*b^4*c^6 + 12*a^4*b^6*c^6 - 29*a^2*b^8*c^6 + 67*b^10*c^6 + 25*a^8*c^8 - 7*a^6*b^2*c^8 - 23*a^4*b^4*c^8 - 29*a^2*b^6*c^8 - 80*b^8*c^8 - 8*a^6*c^10 + 23*a^4*b^2*c^10 + 54*a^2*b^4*c^10 + 67*b^6*c^10 - 6*a^4*c^12 - 32*a^2*b^2*c^12 - 38*b^4*c^12 + 7*a^2*c^14 + 13*b^2*c^14 - 2*c^16 : :
X(44976) = 3 X[4] - 2 X[128], 4 X[5] - 3 X[38706], 2 X[20] - 3 X[38710], 2 X[74] - 3 X[34308], 4 X[128] - 3 X[930], 4 X[137] - 3 X[38710], 3 X[376] - 4 X[34837], 3 X[381] - 2 X[38615], 2 X[548] - 3 X[25147], 5 X[631] - 6 X[23516], 3 X[1141] - 4 X[1263], 5 X[3091] - 4 X[13372], 2 X[3146] + X[38683], 3 X[3627] - X[14073], 4 X[3627] - X[38681], 3 X[3830] - X[13512], 3 X[3845] - 2 X[6592], 4 X[3861] - 3 X[23237], 2 X[14073] - 3 X[31656], 4 X[14073] - 3 X[38681], 7 X[14869] - 8 X[25339]

See Peter Moses, Euclid 2600 .

X(44976) lies on the HS circle and these lines: {4, 128}, {5, 38706}, {20, 137}, {23, 23320}, {30, 1141}, {74, 34308}, {376, 34837}, {381, 38615}, {382, 25150}, {548, 25147}, {631, 23516}, {1595, 14769}, {1657, 38618}, {3091, 13372}, {3146, 11671}, {3327, 12953}, {3627, 14073}, {3830, 13512}, {3845, 6592}, {3853, 14072}, {3861, 23237}, {5073, 38587}, {7159, 12943}, {7604, 10285}, {10721, 16659}, {12026, 15704}, {12087, 14652}, {12121, 43966}, {14869, 25339}, {34418, 37925}

X(44976) = midpoint of X(i) and X(j) for these {i,j}: {3146, 11671}, {5073, 38587}
X(44976) = reflection of X(i) in X(j) for these {i,j}: {{20, 137}, {930, 4}, {1657, 38618}, {12121, 43966}, {14072, 3853}, {15704, 12026}, {31656, 3627}, {38681, 31656}, {38683, 11671}
X(44976) = {X(20),X(137)}-harmonic conjugate of X(38710)

X(44977) = REFLECTION OF X(933) IN X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^18 - 14*a^16*b^2 + 25*a^14*b^4 - 22*a^12*b^6 + 13*a^10*b^8 - 8*a^8*b^10 - a^6*b^12 + 10*a^4*b^14 - 8*a^2*b^16 + 2*b^18 - 14*a^16*c^2 + 48*a^14*b^2*c^2 - 58*a^12*b^4*c^2 + 24*a^10*b^6*c^2 + 6*a^8*b^8*c^2 - 22*a^4*b^12*c^2 + 24*a^2*b^14*c^2 - 8*b^16*c^2 + 25*a^14*c^4 - 58*a^12*b^2*c^4 + 49*a^10*b^4*c^4 - 16*a^8*b^6*c^4 - 11*a^6*b^8*c^4 + 24*a^4*b^10*c^4 - 23*a^2*b^12*c^4 + 10*b^14*c^4 - 22*a^12*c^6 + 24*a^10*b^2*c^6 - 16*a^8*b^4*c^6 + 24*a^6*b^6*c^6 - 12*a^4*b^8*c^6 + 4*a^2*b^10*c^6 - 2*b^12*c^6 + 13*a^10*c^8 + 6*a^8*b^2*c^8 - 11*a^6*b^4*c^8 - 12*a^4*b^6*c^8 + 6*a^2*b^8*c^8 - 2*b^10*c^8 - 8*a^8*c^10 + 24*a^4*b^4*c^10 + 4*a^2*b^6*c^10 - 2*b^8*c^10 - a^6*c^12 - 22*a^4*b^2*c^12 - 23*a^2*b^4*c^12 - 2*b^6*c^12 + 10*a^4*c^14 + 24*a^2*b^2*c^14 + 10*b^4*c^14 - 8*a^2*c^16 - 8*b^2*c^16 + 2*c^18) : :
X(44977) = 3 X[4] - 2 X[18402], 3 X[381] - 2 X[38616], 3 X[933] - 4 X[18402], 3 X[3830] - X[38585]

See Peter Moses, Euclid 2600 .

X(44977) lies on the HS circle and these lines: {4, 137}, {20, 20625}, {30, 18401}, {381, 38616}, {562, 14989}, {3520, 35467}, {3830, 38585}, {6242, 10628}, {10722, 16264}, {11587, 13596}, {13506, 34786}

X(44977) = reflection of X(i) in X(j) for these {i,j}: {20, 20625}, {933, 4}

X(44978) = REFLECTION OF X(934) IN X(4)

Barycentrics    3*a^11 - 3*a^10*b - 10*a^9*b^2 + 10*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 5*a^3*b^8 + 5*a^2*b^9 + 2*a*b^10 - 2*b^11 - 3*a^10*c + 19*a^9*b*c - 6*a^8*b^2*c - 28*a^7*b^3*c + 2*a^6*b^4*c + 18*a^5*b^5*c - 4*a^4*b^6*c + 4*a^3*b^7*c + 9*a^2*b^8*c - 13*a*b^9*c + 2*b^10*c - 10*a^9*c^2 - 6*a^8*b*c^2 + 32*a^7*b^2*c^2 + 8*a^6*b^3*c^2 - 36*a^5*b^4*c^2 + 12*a^4*b^5*c^2 + 16*a^3*b^6*c^2 - 24*a^2*b^7*c^2 - 2*a*b^8*c^2 + 10*b^9*c^2 + 10*a^8*c^3 - 28*a^7*b*c^3 + 8*a^6*b^2*c^3 + 36*a^5*b^3*c^3 - 8*a^4*b^4*c^3 - 20*a^3*b^5*c^3 - 32*a^2*b^6*c^3 + 44*a*b^7*c^3 - 10*b^8*c^3 + 10*a^7*c^4 + 2*a^6*b*c^4 - 36*a^5*b^2*c^4 - 8*a^4*b^3*c^4 + 10*a^3*b^4*c^4 + 42*a^2*b^5*c^4 - 20*b^7*c^4 - 10*a^6*c^5 + 18*a^5*b*c^5 + 12*a^4*b^2*c^5 - 20*a^3*b^3*c^5 + 42*a^2*b^4*c^5 - 62*a*b^5*c^5 + 20*b^6*c^5 - 4*a^4*b*c^6 + 16*a^3*b^2*c^6 - 32*a^2*b^3*c^6 + 20*b^5*c^6 + 4*a^3*b*c^7 - 24*a^2*b^2*c^7 + 44*a*b^3*c^7 - 20*b^4*c^7 - 5*a^3*c^8 + 9*a^2*b*c^8 - 2*a*b^2*c^8 - 10*b^3*c^8 + 5*a^2*c^9 - 13*a*b*c^9 + 10*b^2*c^9 + 2*a*c^10 + 2*b*c^10 - 2*c^11 : :
X(44978) = 5 X[3091] - 4 X[40555]

See Peter Moses, Euclid 2600 .

X(44978) lies on the HS circle and these lines: {4, 934}, {20, 5514}, {30, 972}, {1360, 12943}, {3091, 40555}, {5691, 10727}, {6366, 10724}, {10728, 12247}, {10729, 37001}

X(44978) = reflection of X(i) in X(j) for these {i,j}: {20, 5514}, {934, 4}

X(44979) = REFLECTION OF X(953) IN X(4)

Barycentrics    3*a^10 - 6*a^9*b - 3*a^8*b^2 + 14*a^7*b^3 - 7*a^6*b^4 - 6*a^5*b^5 + 9*a^4*b^6 - 6*a^3*b^7 + 4*a*b^9 - 2*b^10 - 6*a^9*c + 24*a^8*b*c - 20*a^7*b^2*c - 22*a^6*b^3*c + 42*a^5*b^4*c - 24*a^4*b^5*c + 18*a^2*b^7*c - 16*a*b^8*c + 4*b^9*c - 3*a^8*c^2 - 20*a^7*b*c^2 + 61*a^6*b^2*c^2 - 36*a^5*b^3*c^2 - 25*a^4*b^4*c^2 + 52*a^3*b^5*c^2 - 39*a^2*b^6*c^2 + 4*a*b^7*c^2 + 6*b^8*c^2 + 14*a^7*c^3 - 22*a^6*b*c^3 - 36*a^5*b^2*c^3 + 80*a^4*b^3*c^3 - 46*a^3*b^4*c^3 - 18*a^2*b^5*c^3 + 44*a*b^6*c^3 - 16*b^7*c^3 - 7*a^6*c^4 + 42*a^5*b*c^4 - 25*a^4*b^2*c^4 - 46*a^3*b^3*c^4 + 78*a^2*b^4*c^4 - 36*a*b^5*c^4 - 4*b^6*c^4 - 6*a^5*c^5 - 24*a^4*b*c^5 + 52*a^3*b^2*c^5 - 18*a^2*b^3*c^5 - 36*a*b^4*c^5 + 24*b^5*c^5 + 9*a^4*c^6 - 39*a^2*b^2*c^6 + 44*a*b^3*c^6 - 4*b^4*c^6 - 6*a^3*c^7 + 18*a^2*b*c^7 + 4*a*b^2*c^7 - 16*b^3*c^7 - 16*a*b*c^8 + 6*b^2*c^8 + 4*a*c^9 + 4*b*c^9 - 2*c^10 : :
X(44979) = 3 X[4] - 2 X[3259], 4 X[5] - 3 X[38707], 2 X[20] - 3 X[38705], 3 X[376] - 4 X[22102], 3 X[381] - 2 X[38617], 3 X[953] - 4 X[3259], 4 X[3627] - X[38682], 3 X[3830] - X[38586], 4 X[31841] - 3 X[38705]

See Peter Moses, Euclid 2600 .

X(44979) lies on the HS circle and these lines: {4, 953}, {5, 38707}, {20, 31841}, {30, 901}, {376, 22102}, {378, 10016}, {381, 38617}, {513, 10728}, {517, 10724}, {1657, 38614}, {2222, 14127}, {3025, 12943}, {3520, 39479}, {3586, 24201}, {3627, 38682}, {3830, 38586}, {5073, 38584}, {6073, 13199}, {6075, 12248}, {9579, 33645}, {10738, 14511}, {10742, 14513}, {12953, 13756}

X(44979) = midpoint of X(5073) and X(38584)
X(44979) = reflection of X(i) in X(j) for these {i,j}: {20, 31841}, {901, 38954}, {953, 4}, {1657, 38614}, {12248, 6075}, {13199, 6073}, {14511, 10738}, {14513, 10742}, {38682, 40100}, {40100, 3627}
X(44979) = {X(20),X(31841)}-harmonic conjugate of X(38705)

X(44980) = REFLECTION OF X(972) IN X(4)

Barycentrics    3*a^11 - 3*a^10*b - 10*a^9*b^2 + 10*a^8*b^3 + 10*a^7*b^4 - 10*a^6*b^5 - 5*a^3*b^8 + 5*a^2*b^9 + 2*a*b^10 - 2*b^11 - 3*a^10*c + 23*a^9*b*c - 10*a^8*b^2*c - 22*a^7*b^3*c - 10*a^6*b^4*c + 8*a^5*b^5*c + 16*a^4*b^6*c + 6*a^3*b^7*c + 5*a^2*b^8*c - 15*a*b^9*c + 2*b^10*c - 10*a^9*c^2 - 10*a^8*b*c^2 + 24*a^7*b^2*c^2 + 20*a^6*b^3*c^2 - 12*a^5*b^4*c^2 - 8*a^3*b^6*c^2 - 20*a^2*b^7*c^2 + 6*a*b^8*c^2 + 10*b^9*c^2 + 10*a^8*c^3 - 22*a^7*b*c^3 + 20*a^6*b^2*c^3 + 8*a^5*b^3*c^3 - 16*a^4*b^4*c^3 - 6*a^3*b^5*c^3 - 20*a^2*b^6*c^3 + 36*a*b^7*c^3 - 10*b^8*c^3 + 10*a^7*c^4 - 10*a^6*b*c^4 - 12*a^5*b^2*c^4 - 16*a^4*b^3*c^4 + 26*a^3*b^4*c^4 + 30*a^2*b^5*c^4 - 8*a*b^6*c^4 - 20*b^7*c^4 - 10*a^6*c^5 + 8*a^5*b*c^5 - 6*a^3*b^3*c^5 + 30*a^2*b^4*c^5 - 42*a*b^5*c^5 + 20*b^6*c^5 + 16*a^4*b*c^6 - 8*a^3*b^2*c^6 - 20*a^2*b^3*c^6 - 8*a*b^4*c^6 + 20*b^5*c^6 + 6*a^3*b*c^7 - 20*a^2*b^2*c^7 + 36*a*b^3*c^7 - 20*b^4*c^7 - 5*a^3*c^8 + 5*a^2*b*c^8 + 6*a*b^2*c^8 - 10*b^3*c^8 + 5*a^2*c^9 - 15*a*b*c^9 + 10*b^2*c^9 + 2*a*c^10 + 2*b*c^10 - 2*c^11 : :
X(44980) = 3 X[4] - 2 X[5514], 3 X[376] - 4 X[40555], 3 X[972] - 4 X[5514]

See Peter Moses, Euclid 2600 .

X(44980) lies on the HS circle and these lines: {4, 972}, {30, 934}, {376, 40555}, {1360, 12953}, {6366, 10728}, {9803, 10724}, {10725, 41869}, {10731, 36999}

X(44980) = reflection of X(972) in X(4)

X(44981) = REFLECTION OF X(1141) IN X(4)

Barycentrics    3*a^16 - 12*a^14*b^2 + 18*a^12*b^4 - 13*a^10*b^6 + 5*a^8*b^8 + 2*a^6*b^10 - 8*a^4*b^12 + 7*a^2*b^14 - 2*b^16 - 12*a^14*c^2 + 30*a^12*b^2*c^2 - 23*a^10*b^4*c^2 + 2*a^8*b^6*c^2 + a^6*b^8*c^2 + 17*a^4*b^10*c^2 - 26*a^2*b^12*c^2 + 11*b^14*c^2 + 18*a^12*c^4 - 23*a^10*b^2*c^4 + 13*a^8*b^4*c^4 - 3*a^6*b^6*c^4 - 15*a^4*b^8*c^4 + 36*a^2*b^10*c^4 - 26*b^12*c^4 - 13*a^10*c^6 + 2*a^8*b^2*c^6 - 3*a^6*b^4*c^6 + 12*a^4*b^6*c^6 - 17*a^2*b^8*c^6 + 37*b^10*c^6 + 5*a^8*c^8 + a^6*b^2*c^8 - 15*a^4*b^4*c^8 - 17*a^2*b^6*c^8 - 40*b^8*c^8 + 2*a^6*c^10 + 17*a^4*b^2*c^10 + 36*a^2*b^4*c^10 + 37*b^6*c^10 - 8*a^4*c^12 - 26*a^2*b^2*c^12 - 26*b^4*c^12 + 7*a^2*c^14 + 11*b^2*c^14 - 2*c^16 : :
X(44981) = 3 X[4] - 2 X[137], 4 X[5] - 3 X[38710], 2 X[20] - 3 X[38706], 4 X[128] - 3 X[38706], 4 X[137] - 3 X[1141], 3 X[376] - 4 X[13372], 3 X[381] - 2 X[38618], 2 X[548] - 3 X[23237], 3 X[930] - 4 X[14072], 5 X[3091] - 4 X[34837], 2 X[3146] + X[38681], 3 X[3543] - X[11671], 4 X[3627] - X[38683], 3 X[3830] - X[38587], 7 X[3832] - 6 X[23516], 3 X[3845] - 2 X[12026], 4 X[3861] - 3 X[25147], 4 X[10113] - 3 X[34308], X[13504] - 3 X[15305], 2 X[14072] - 3 X[31656], 8 X[25339] - 9 X[38071]

See Peter Moses, Euclid 2600 .

X(44981) lies on the HS circle and these lines: {4, 137}, {5, 38710}, {20, 128}, {30, 930}, {376, 13372}, {378, 15959}, {381, 38618}, {382, 25150}, {403, 23319}, {548, 23237}, {1263, 3853}, {1593, 15960}, {1657, 38615}, {3091, 34837}, {3146, 38681}, {3327, 12943}, {3520, 23320}, {3543, 11671}, {3627, 7728}, {3830, 38587}, {3832, 23516}, {3845, 12026}, {3861, 25147}, {5073, 13512}, {6592, 15704}, {7159, 12953}, {7527, 14652}, {7604, 33545}, {10113, 34308}, {10152, 18349}, {12290, 13505}, {13504, 15305}, {14769, 15760}, {14865, 34418}, {15619, 25148}, {25339, 38071}, {32410, 34224}

X(44981) = midpoint of X(i) and X(j) for these {i,j}: {5073, 13512}, {12290, 13505}
X(44981) = reflection of X(i) in X(j) for these {i,j}: {20, 128}, {930, 31656}, {1141, 4}, {1263, 3853}, {1657, 38615}, {15704, 6592}, {34224, 32410}
X(44981) = {X(20),X(128)}-harmonic conjugate of X(38706)

X(44982) = REFLECTION OF X(1290) IN X(4)

Barycentrics    3*a^13 - 3*a^12*b - 7*a^11*b^2 + 7*a^10*b^3 + 10*a^7*b^6 - 10*a^6*b^7 - 5*a^5*b^8 + 5*a^4*b^9 - 3*a^3*b^10 + 3*a^2*b^11 + 2*a*b^12 - 2*b^13 - 3*a^12*c + 7*a^11*b*c + 3*a^10*b^2*c - 9*a^9*b^3*c + 2*a^8*b^4*c - 10*a^7*b^5*c + 4*a^6*b^6*c + 14*a^5*b^7*c - 9*a^4*b^8*c + 3*a^3*b^9*c + a^2*b^10*c - 5*a*b^11*c + 2*b^12*c - 7*a^11*c^2 + 3*a^10*b*c^2 + 19*a^9*b^2*c^2 - 13*a^8*b^3*c^2 - 14*a^7*b^4*c^2 + 18*a^6*b^5*c^2 - 7*a^5*b^6*c^2 - a^4*b^7*c^2 + 15*a^3*b^8*c^2 - 15*a^2*b^9*c^2 - 6*a*b^10*c^2 + 8*b^11*c^2 + 7*a^10*c^3 - 9*a^9*b*c^3 - 13*a^8*b^2*c^3 + 35*a^7*b^3*c^3 - 8*a^6*b^4*c^3 - 16*a^5*b^5*c^3 + 19*a^4*b^6*c^3 - 25*a^3*b^7*c^3 + 3*a^2*b^8*c^3 + 15*a*b^9*c^3 - 8*b^10*c^3 + 2*a^8*b*c^4 - 14*a^7*b^2*c^4 - 8*a^6*b^3*c^4 + 24*a^5*b^4*c^4 - 14*a^4*b^5*c^4 - 12*a^3*b^6*c^4 + 26*a^2*b^7*c^4 + 6*a*b^8*c^4 - 10*b^9*c^4 - 10*a^7*b*c^5 + 18*a^6*b^2*c^5 - 16*a^5*b^3*c^5 - 14*a^4*b^4*c^5 + 44*a^3*b^5*c^5 - 18*a^2*b^6*c^5 - 10*a*b^7*c^5 + 10*b^8*c^5 + 10*a^7*c^6 + 4*a^6*b*c^6 - 7*a^5*b^2*c^6 + 19*a^4*b^3*c^6 - 12*a^3*b^4*c^6 - 18*a^2*b^5*c^6 - 4*a*b^6*c^6 - 10*a^6*c^7 + 14*a^5*b*c^7 - a^4*b^2*c^7 - 25*a^3*b^3*c^7 + 26*a^2*b^4*c^7 - 10*a*b^5*c^7 - 5*a^5*c^8 - 9*a^4*b*c^8 + 15*a^3*b^2*c^8 + 3*a^2*b^3*c^8 + 6*a*b^4*c^8 + 10*b^5*c^8 + 5*a^4*c^9 + 3*a^3*b*c^9 - 15*a^2*b^2*c^9 + 15*a*b^3*c^9 - 10*b^4*c^9 - 3*a^3*c^10 + a^2*b*c^10 - 6*a*b^2*c^10 - 8*b^3*c^10 + 3*a^2*c^11 - 5*a*b*c^11 + 8*b^2*c^11 + 2*a*c^12 + 2*b*c^12 - 2*c^13 : :
X(44982) = 3 X[4] - 2 X[42422], 4 X[5] - 3 X[38711], 3 X[1290] - 4 X[42422], 3 X[3830] - X[38588]

See Peter Moses, Euclid 2600 .

X(44982) lies on the HS circle and these lines: {4, 1290}, {5, 38711}, {20, 5520}, {30, 100}, {513, 10733}, {517, 10721}, {523, 10724}, {2829, 36175}, {3830, 38588}, {10296, 10729}, {12943, 31524}, {12953, 31522}

X(44982) = reflection of X(i) in X(j) for these {i,j}: {20, 5520}, {1290, 4}
X(44982) = reflection of X(10724) in the Euler line

X(44983) = REFLECTION OF X(1292) IN X(4)

Barycentrics    3*a^8 - 6*a^7*b + 5*a^6*b^2 - 4*a^5*b^3 - a^4*b^4 + 6*a^3*b^5 - 5*a^2*b^6 + 4*a*b^7 - 2*b^8 - 6*a^7*c + a^6*b*c + 9*a^5*b^2*c - 12*a^4*b^3*c + 6*a^3*b^4*c - a^2*b^5*c - a*b^6*c + 4*b^7*c + 5*a^6*c^2 + 9*a^5*b*c^2 + 6*a^4*b^2*c^2 - 8*a^3*b^3*c^2 + a^2*b^4*c^2 - 13*a*b^5*c^2 - 4*a^5*c^3 - 12*a^4*b*c^3 - 8*a^3*b^2*c^3 + 10*a^2*b^3*c^3 + 10*a*b^4*c^3 - 4*b^5*c^3 - a^4*c^4 + 6*a^3*b*c^4 + a^2*b^2*c^4 + 10*a*b^3*c^4 + 4*b^4*c^4 + 6*a^3*c^5 - a^2*b*c^5 - 13*a*b^2*c^5 - 4*b^3*c^5 - 5*a^2*c^6 - a*b*c^6 + 4*a*c^7 + 4*b*c^7 - 2*c^8 : :
X(44983) = 3 X[4] - 2 X[120], 4 X[5] - 3 X[38712], 2 X[20] - 3 X[38694], 4 X[120] - 3 X[1292], 3 X[376] - 4 X[6714], 3 X[381] - 2 X[38619], 3 X[3146] + X[20097], 2 X[3146] + X[38670], 3 X[3543] - X[20344], 4 X[3627] - X[38684], 3 X[3830] - X[38589], 4 X[5511] - 3 X[38694], 3 X[10712] - 2 X[38589], X[20097] - 3 X[34547], 2 X[20097] - 3 X[38670]

See Peter Moses, Euclid 2600 .

X(44983) lies on the HS circle and these lines: {4, 120}, {5, 38712}, {20, 5511}, {30, 105}, {376, 6714}, {381, 38619}, {382, 10729}, {528, 10728}, {1358, 12953}, {1657, 38603}, {2775, 10733}, {2788, 10723}, {2795, 10722}, {2809, 10727}, {2814, 10732}, {2820, 10725}, {2826, 10724}, {2835, 10726}, {2836, 10721}, {3021, 12943}, {3146, 20097}, {3543, 20344}, {3627, 10743}, {3830, 10712}, {5073, 38575}, {9519, 10730}, {9520, 10152}, {9521, 10731}, {9522, 10734}, {9523, 10735}, {10699, 12699}, {10760, 31670}

X(44983) = midpoint of X(i) and X(j) for these {i,j}: {3146, 34547}, {5073, 38575}
X(44983) = reflection of X(i) in X(j) for these {i,j}: {20, 5511}, {105, 15521}, {1292, 4}, {1657, 38603}, {10699, 12699}, {10712, 3830}, {10729, 382}, {10743, 3627}, {10760, 31670}, {38670, 34547}, {38684, 10743}
X(44983) = {X(20),X(5511)}-harmonic conjugate of X(38694)

X(44984) = REFLECTION OF X(1293) IN X(4)

Barycentrics    3*a^7 - 6*a^6*b - 8*a^5*b^2 + 4*a^4*b^3 + a^3*b^4 + 4*a^2*b^5 + 4*a*b^6 - 2*b^7 - 6*a^6*c + 27*a^5*b*c + a^4*b^2*c - 9*a^3*b^3*c + a^2*b^4*c - 18*a*b^5*c + 4*b^6*c - 8*a^5*c^2 + a^4*b*c^2 - 4*a^3*b^2*c^2 - a^2*b^3*c^2 - 4*a*b^4*c^2 + 8*b^5*c^2 + 4*a^4*c^3 - 9*a^3*b*c^3 - a^2*b^2*c^3 + 36*a*b^3*c^3 - 10*b^4*c^3 + a^3*c^4 + a^2*b*c^4 - 4*a*b^2*c^4 - 10*b^3*c^4 + 4*a^2*c^5 - 18*a*b*c^5 + 8*b^2*c^5 + 4*a*c^6 + 4*b*c^6 - 2*c^7 : :
X(44984) = 3 X[4] - 2 X[121], 4 X[5] - 3 X[38713], 2 X[20] - 3 X[38695], 4 X[121] - 3 X[1293], 3 X[376] - 4 X[6715], 3 X[381] - 2 X[38620], 3 X[3146] + X[20098], 2 X[3146] + X[38671], 3 X[3543] - X[21290], 4 X[3627] - X[38685], 3 X[3830] - X[38590], 4 X[5510] - 3 X[38695], 3 X[10713] - 2 X[38590], X[20098] - 3 X[34548], 2 X[20098] - 3 X[38671]

See Peter Moses, Euclid 2600 .

X(44984) lies on the HS circle and these lines: {4, 121}, {5, 38713}, {20, 5510}, {30, 106}, {376, 6715}, {381, 38620}, {382, 10730}, {1357, 12953}, {1657, 38604}, {2776, 10733}, {2789, 10723}, {2796, 10722}, {2802, 10728}, {2810, 10727}, {2815, 10732}, {2821, 10725}, {2827, 10724}, {2841, 10726}, {2842, 10721}, {3146, 20098}, {3543, 21290}, {3627, 10744}, {3830, 10713}, {5073, 38576}, {6018, 12943}, {9519, 10729}, {9524, 10152}, {9525, 10731}, {9526, 10734}, {9527, 10735}, {10700, 12699}, {10761, 31670}

X(44984) = midpoint of X(i) and X(j) for these {i,j}: {3146, 34548}, {5073, 38576}
X(44984) = reflection of X(i) in X(j) for these {i,j}: {20, 5510}, {106, 15522}, {1293, 4}, {1657, 38604}, {10700, 12699}, {10713, 3830}, {10730, 382}, {10744, 3627}, {10761, 31670}, {38671, 34548}, {38685, 10744}
X(44984) = {X(20),X(5510)}-harmonic conjugate of X(38695)

X(44985) = REFLECTION OF X(1294) IN X(4)

Barycentrics    3*a^16 - 3*a^14*b^2 - 21*a^12*b^4 + 47*a^10*b^6 - 25*a^8*b^8 - 13*a^6*b^10 + 13*a^4*b^12 + a^2*b^14 - 2*b^16 - 3*a^14*c^2 + 45*a^12*b^2*c^2 - 47*a^10*b^4*c^2 - 85*a^8*b^6*c^2 + 127*a^6*b^8*c^2 - 13*a^4*b^10*c^2 - 29*a^2*b^12*c^2 + 5*b^14*c^2 - 21*a^12*c^4 - 47*a^10*b^2*c^4 + 220*a^8*b^4*c^4 - 114*a^6*b^6*c^4 - 129*a^4*b^8*c^4 + 81*a^2*b^10*c^4 + 10*b^12*c^4 + 47*a^10*c^6 - 85*a^8*b^2*c^6 - 114*a^6*b^4*c^6 + 258*a^4*b^6*c^6 - 53*a^2*b^8*c^6 - 53*b^10*c^6 - 25*a^8*c^8 + 127*a^6*b^2*c^8 - 129*a^4*b^4*c^8 - 53*a^2*b^6*c^8 + 80*b^8*c^8 - 13*a^6*c^10 - 13*a^4*b^2*c^10 + 81*a^2*b^4*c^10 - 53*b^6*c^10 + 13*a^4*c^12 - 29*a^2*b^2*c^12 + 10*b^4*c^12 + a^2*c^14 + 5*b^2*c^14 - 2*c^16 : :
X(44985) = 3 X[4] - 2 X[122], 4 X[5] - 3 X[38714], 2 X[20] - 3 X[23239], 3 X[107] - 2 X[23240], 4 X[122] - 3 X[1294], X[122] - 3 X[38956], 4 X[133] - 3 X[23239], 3 X[376] - 4 X[6716], 3 X[381] - 2 X[38621], X[1294] - 4 X[38956], 5 X[3091] - 4 X[34842], 2 X[3146] + X[38672], 3 X[3543] - X[34186], 4 X[3627] - X[38686], 3 X[3830] - X[38591], 7 X[3832] - 6 X[36520], 3 X[10714] - 2 X[38591], 3 X[22337] - X[23240], X[23241] + 2 X[33703]

See Peter Moses, Euclid 2600 .

X(44985) lies on the HS circle and these lines: {4, 122}, {5, 38714}, {20, 133}, {30, 107}, {376, 6716}, {381, 38621}, {382, 10152}, {1657, 38605}, {2777, 3146}, {2790, 10723}, {2797, 10722}, {2803, 10728}, {2811, 10727}, {2816, 10732}, {2822, 10725}, {2828, 10724}, {2846, 10726}, {3091, 34842}, {3184, 3529}, {3324, 12953}, {3518, 40082}, {3543, 34186}, {3627, 10745}, {3830, 10714}, {3832, 36520}, {5073, 38577}, {5667, 23241}, {7158, 12943}, {9033, 10721}, {9520, 10729}, {9524, 10730}, {9528, 10731}, {9529, 10734}, {9530, 10735}, {10701, 12699}, {10762, 31670}, {33897, 34286}

X(44985) = midpoint of X(i) and X(j) for these {i,j}: {{3146, 34549}, {5073, 38577}, {5667, 33703}
X(44985) = reflection of X(i) in X(j) for these {i,j}: {4, 38956}, {20, 133}, {107, 22337}, {1294, 4}, {1657, 38605}, {3529, 3184}, {10152, 382}, {10701, 12699}, {10714, 3830}, {10745, 3627}, {10762, 31670}, {23241, 5667}, {33897, 34286}, {38672, 34549}, {38686, 10745}
X(44985) = {X(20),X(133)}-harmonic conjugate of X(23239)

X(44986) = REFLECTION OF X(1295) IN X(4)

Barycentrics    3*a^13 - 3*a^12*b - 7*a^11*b^2 + 7*a^10*b^3 + 10*a^7*b^6 - 10*a^6*b^7 - 5*a^5*b^8 + 5*a^4*b^9 - 3*a^3*b^10 + 3*a^2*b^11 + 2*a*b^12 - 2*b^13 - 3*a^12*c + 17*a^11*b*c - 7*a^10*b^2*c - 23*a^9*b^3*c + 28*a^8*b^4*c - 22*a^7*b^5*c - 14*a^6*b^6*c + 34*a^5*b^7*c - 11*a^4*b^8*c + 5*a^3*b^9*c + 5*a^2*b^10*c - 11*a*b^11*c + 2*b^12*c - 7*a^11*c^2 - 7*a^10*b*c^2 + 46*a^9*b^2*c^2 - 28*a^8*b^3*c^2 - 42*a^7*b^4*c^2 + 62*a^6*b^5*c^2 - 24*a^5*b^6*c^2 - 4*a^4*b^7*c^2 + 25*a^3*b^8*c^2 - 31*a^2*b^9*c^2 + 2*a*b^10*c^2 + 8*b^11*c^2 + 7*a^10*c^3 - 23*a^9*b*c^3 - 28*a^8*b^2*c^3 + 108*a^7*b^3*c^3 - 38*a^6*b^4*c^3 - 50*a^5*b^5*c^3 + 60*a^4*b^6*c^3 - 68*a^3*b^7*c^3 + 7*a^2*b^8*c^3 + 33*a*b^9*c^3 - 8*b^10*c^3 + 28*a^8*b*c^4 - 42*a^7*b^2*c^4 - 38*a^6*b^3*c^4 + 90*a^5*b^4*c^4 - 50*a^4*b^5*c^4 - 22*a^3*b^6*c^4 + 70*a^2*b^7*c^4 - 26*a*b^8*c^4 - 10*b^9*c^4 - 22*a^7*b*c^5 + 62*a^6*b^2*c^5 - 50*a^5*b^3*c^5 - 50*a^4*b^4*c^5 + 126*a^3*b^5*c^5 - 54*a^2*b^6*c^5 - 22*a*b^7*c^5 + 10*b^8*c^5 + 10*a^7*c^6 - 14*a^6*b*c^6 - 24*a^5*b^2*c^6 + 60*a^4*b^3*c^6 - 22*a^3*b^4*c^6 - 54*a^2*b^5*c^6 + 44*a*b^6*c^6 - 10*a^6*c^7 + 34*a^5*b*c^7 - 4*a^4*b^2*c^7 - 68*a^3*b^3*c^7 + 70*a^2*b^4*c^7 - 22*a*b^5*c^7 - 5*a^5*c^8 - 11*a^4*b*c^8 + 25*a^3*b^2*c^8 + 7*a^2*b^3*c^8 - 26*a*b^4*c^8 + 10*b^5*c^8 + 5*a^4*c^9 + 5*a^3*b*c^9 - 31*a^2*b^2*c^9 + 33*a*b^3*c^9 - 10*b^4*c^9 - 3*a^3*c^10 + 5*a^2*b*c^10 + 2*a*b^2*c^10 - 8*b^3*c^10 + 3*a^2*c^11 - 11*a*b*c^11 + 8*b^2*c^11 + 2*a*c^12 + 2*b*c^12 - 2*c^13 : :
X(44986) = 3 X[4] - 2 X[123], 4 X[5] - 3 X[38715], 2 X[20] - 3 X[38696], 4 X[123] - 3 X[1295], 3 X[376] - 4 X[6717], 3 X[381] - 2 X[38622], 2 X[3146] + X[38673], 3 X[3543] - X[34188], 4 X[3627] - X[38687], 3 X[3830] - X[38592], 3 X[10715] - 2 X[38592], 4 X[25640] - 3 X[38696]

See Peter Moses, Euclid 2600 .

X(44986) lies on the HS circle and these lines: {4, 123}, {5, 38715}, {20, 25640}, {30, 108}, {149, 2829}, {376, 6717}, {381, 38622}, {382, 10731}, {1359, 12953}, {1657, 38606}, {2778, 10733}, {2791, 10723}, {2798, 10722}, {2804, 10728}, {2812, 10727}, {2817, 10732}, {2823, 10725}, {2849, 10726}, {2850, 10721}, {3318, 12943}, {3543, 34188}, {3627, 10746}, {3830, 10715}, {5073, 38578}, {9521, 10729}, {9525, 10730}, {9528, 10152}, {9531, 10734}, {10702, 12699}, {10763, 31670}

X(44986) = midpoint of X(i) and X(j) for these {i,j}: {3146, 34550}, {5073, 38578}
X(44986) = reflection of X(i) in X(j) for these {i,j}: {20, 25640}, {108, 33566}, {1295, 4}, {1657, 38606}, {10702, 12699}, {10715, 3830}, {10731, 382}, {10746, 3627}, {10763, 31670}, {38673, 34550}, {38687, 10746}
X(44986) = {X(20),X(25640)}-harmonic conjugate of X(38696)

X(44987) = REFLECTION OF X(1296) IN X(4)

Barycentrics    3*a^10 - 11*a^8*b^2 - 9*a^6*b^4 + 13*a^4*b^6 + 6*a^2*b^8 - 2*b^10 - 11*a^8*c^2 + 69*a^6*b^2*c^2 - 37*a^4*b^4*c^2 - 47*a^2*b^6*c^2 + 10*b^8*c^2 - 9*a^6*c^4 - 37*a^4*b^2*c^4 + 98*a^2*b^4*c^4 - 8*b^6*c^4 + 13*a^4*c^6 - 47*a^2*b^2*c^6 - 8*b^4*c^6 + 6*a^2*c^8 + 10*b^2*c^8 - 2*c^10 : :
X(44987) = 3 X[4] - 2 X[126], 4 X[5] - 3 X[38716], 2 X[20] - 3 X[38698], 7 X[111] - 6 X[14666], 4 X[126] - 3 X[1296], 4 X[140] - 3 X[38798], 3 X[376] - 4 X[6719], 3 X[381] - 2 X[38623], 4 X[548] - 5 X[38806], 2 X[550] - 3 X[38796], 5 X[631] - 4 X[38803], X[1657] - 3 X[38799], 5 X[3091] - 4 X[40556], 3 X[3146] + X[20099], 2 X[3146] + X[38675], 5 X[3522] - 6 X[38804], X[3529] - 4 X[38801], 3 X[3543] - X[14360], 4 X[3627] - X[38688], 3 X[3830] - X[38593], 5 X[3843] - 4 X[40340], 4 X[5512] - 3 X[38698], 3 X[10717] - 2 X[38593], X[10734] + 2 X[38800], 2 X[14650] - 3 X[38799], 3 X[14666] - 7 X[22338], 2 X[20099] - 3 X[38675], 3 X[38716] - 2 X[38797]

See Peter Moses, Euclid 2600 .

X(44987) lies on the HS circle and these lines: {2, 38805}, {4, 126}, {5, 38716}, {20, 5512}, {30, 111}, {140, 38798}, {376, 6719}, {381, 38623}, {382, 10734}, {543, 10722}, {548, 38806}, {550, 38796}, {631, 38803}, {1657, 14650}, {2780, 10733}, {2793, 10723}, {2805, 10728}, {2813, 10727}, {2819, 10732}, {2824, 10725}, {2830, 10724}, {2852, 10726}, {2854, 10721}, {3048, 13352}, {3091, 40556}, {3146, 20099}, {3325, 12953}, {3522, 38804}, {3529, 38801}, {3543, 14360}, {3627, 10748}, {3830, 10717}, {3843, 40340}, {5073, 11258}, {6019, 12943}, {6564, 11835}, {6565, 11836}, {9172, 11001}, {9522, 10729}, {9526, 10730}, {9529, 10152}, {9531, 10731}, {9541, 11833}, {10113, 35447}, {10704, 12699}, {10765, 31670}, {14654, 33703}, {15684, 32424}

X(44987) = midpoint of X(i) and X(j) for these {i,j}: {382, 38800}, {5073, 11258}, {14654, 33703}
X(44987) = reflection of X(i) in X(j) for these {i,j}: {20, 5512}, {111, 22338}, {1296, 4}, {1657, 14650}, {10704, 12699}, {10717, 3830}, {10734, 382}, {10748, 3627}, {10765, 31670}, {11001, 9172}, {35447, 10113}, {38688, 10748}, {38797, 5}
X(44987) = anticomplement of X(38805)
X(44987) = X(38797)-of-Johnson-triangle
X(44987) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 38797, 38716}, {20, 5512, 38698}, {1657, 38799, 14650}

X(44988) = REFLECTION OF X(1297) IN X(4)

Barycentrics    3*a^14 - 3*a^12*b^2 - 2*a^10*b^4 - a^6*b^8 + 5*a^4*b^10 - 2*b^14 - 3*a^12*c^2 + 7*a^10*b^2*c^2 - 2*a^6*b^6*c^2 - 3*a^4*b^8*c^2 - 5*a^2*b^10*c^2 + 6*b^12*c^2 - 2*a^10*c^4 + 6*a^6*b^4*c^4 - 2*a^4*b^6*c^4 + 4*a^2*b^8*c^4 - 6*b^10*c^4 - 2*a^6*b^2*c^6 - 2*a^4*b^4*c^6 + 2*a^2*b^6*c^6 + 2*b^8*c^6 - a^6*c^8 - 3*a^4*b^2*c^8 + 4*a^2*b^4*c^8 + 2*b^6*c^8 + 5*a^4*c^10 - 5*a^2*b^2*c^10 - 6*b^4*c^10 + 6*b^2*c^12 - 2*c^14 : :
X(44988) = 3 X[4] - 2 X[127], 3 X[4] - X[12253], 4 X[5] - 3 X[38717], 2 X[20] - 3 X[38699], 4 X[127] - 3 X[1297], 4 X[132] - 3 X[38699], 3 X[376] - 4 X[6720], 3 X[381] - 2 X[38624], 3 X[1297] - 2 X[12253], 3 X[1699] - 2 X[12265], 5 X[3091] - 4 X[34841], 2 X[3146] + X[38676], 3 X[3543] - X[13219], 4 X[3627] - X[38689], 3 X[3830] - X[13115], 3 X[3830] - 2 X[19163], 5 X[10574] - 6 X[16224], 3 X[10718] - 2 X[13115], 3 X[10718] - 4 X[19163]

See Peter Moses, Euclid 2600 .

X(44988) lies on the HS circle and these lines: {3, 19160}, {4, 127}, {5, 38717}, {20, 132}, {23, 34217}, {30, 112}, {148, 2794}, {253, 317}, {376, 6720}, {381, 38624}, {382, 10735}, {515, 13099}, {516, 12784}, {1593, 12413}, {1657, 38608}, {1699, 12265}, {2781, 10733}, {2799, 10722}, {2806, 10728}, {2825, 10725}, {2831, 10724}, {2853, 10726}, {3070, 19094}, {3071, 19093}, {3091, 34841}, {3320, 12953}, {3529, 14689}, {3583, 13117}, {3585, 13116}, {3627, 10749}, {3830, 10718}, {5073, 13310}, {6020, 12943}, {6253, 12935}, {6256, 13118}, {6284, 12945}, {6560, 19114}, {6561, 19115}, {7354, 12955}, {7500, 9157}, {9517, 10721}, {9518, 10727}, {9523, 10729}, {9527, 10730}, {9532, 10732}, {9541, 13923}, {10483, 13312}, {10574, 16224}, {10705, 12699}, {10766, 31670}, {12145, 12173}, {13166, 44438}, {13200, 33703}, {13280, 31673}, {13918, 31412}, {13985, 42561}, {18559, 18876}

X(44988) = midpoint of X(i) and X(j) for these {i,j}: {3146, 12384}, {5073, 13310}, {13200, 33703}
X(44988) = reflection of X(i) in X(j) for these {i,j}: {3, 19160}, {20, 132}, {112, 12918}, {1297, 4}, {1657, 38608}, {3529, 14689}, {10705, 12699}, {10718, 3830}, {10735, 382}, {10749, 3627}, {10766, 31670}, {12253, 127}, {13115, 19163}, {13280, 31673}, {38676, 12384}, {38689, 10749}
X(44988) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 12253, 127}, {20, 132, 38699}, {127, 12253, 1297}, {3830, 13115, 19163}, {13115, 19163, 10718}

X(44989) = REFLECTION OF X(1298) IN X(4)

Barycentrics    a^2*(a^18*b^6 - 7*a^16*b^8 + 21*a^14*b^10 - 35*a^12*b^12 + 35*a^10*b^14 - 21*a^8*b^16 + 7*a^6*b^18 - a^4*b^20 + 3*a^20*b^2*c^2 - 13*a^18*b^4*c^2 + 16*a^16*b^6*c^2 + 6*a^14*b^8*c^2 - 22*a^12*b^10*c^2 - 6*a^10*b^12*c^2 + 38*a^8*b^14*c^2 - 26*a^6*b^16*c^2 - a^4*b^18*c^2 + 7*a^2*b^20*c^2 - 2*b^22*c^2 - 13*a^18*b^2*c^4 + 48*a^16*b^4*c^4 - 63*a^14*b^6*c^4 + 29*a^12*b^8*c^4 + 20*a^10*b^10*c^4 - 50*a^8*b^12*c^4 + 29*a^6*b^14*c^4 + 25*a^4*b^16*c^4 - 37*a^2*b^18*c^4 + 12*b^20*c^4 + a^18*c^6 + 16*a^16*b^2*c^6 - 63*a^14*b^4*c^6 + 83*a^12*b^6*c^6 - 49*a^10*b^8*c^6 + 25*a^8*b^10*c^6 - a^6*b^12*c^6 - 63*a^4*b^14*c^6 + 80*a^2*b^16*c^6 - 29*b^18*c^6 - 7*a^16*c^8 + 6*a^14*b^2*c^8 + 29*a^12*b^4*c^8 - 49*a^10*b^6*c^8 + 16*a^8*b^8*c^8 - 9*a^6*b^10*c^8 + 68*a^4*b^12*c^8 - 88*a^2*b^14*c^8 + 34*b^16*c^8 + 21*a^14*c^10 - 22*a^12*b^2*c^10 + 20*a^10*b^4*c^10 + 25*a^8*b^6*c^10 - 9*a^6*b^8*c^10 - 56*a^4*b^10*c^10 + 38*a^2*b^12*c^10 - 17*b^14*c^10 - 35*a^12*c^12 - 6*a^10*b^2*c^12 - 50*a^8*b^4*c^12 - a^6*b^6*c^12 + 68*a^4*b^8*c^12 + 38*a^2*b^10*c^12 + 4*b^12*c^12 + 35*a^10*c^14 + 38*a^8*b^2*c^14 + 29*a^6*b^4*c^14 - 63*a^4*b^6*c^14 - 88*a^2*b^8*c^14 - 17*b^10*c^14 - 21*a^8*c^16 - 26*a^6*b^2*c^16 + 25*a^4*b^4*c^16 + 80*a^2*b^6*c^16 + 34*b^8*c^16 + 7*a^6*c^18 - a^4*b^2*c^18 - 37*a^2*b^4*c^18 - 29*b^6*c^18 - a^4*c^20 + 7*a^2*b^2*c^20 + 12*b^4*c^20 - 2*b^2*c^22) : :
X(44989) = 3 X[4] - 2 X[130], 4 X[130] - 3 X[1298], 3 X[376] - 4 X[34839], 5 X[3091] - 4 X[34838], 3 X[3830] - X[38594]

See Peter Moses, Euclid 2600 .

X(44989) lies on the HS circle and these lines: {4, 130}, {20, 129}, {30, 1303}, {376, 34839}, {1593, 22551}, {3091, 34838}, {3830, 38594}, {6000, 21661}, {10152, 32062}, {10733, 32438}

X(44989) = reflection of X(i) in X(j) for these {i,j}: {20, 129}, {1298, 4}

X(44990) = REFLECTION OF X(1300) IN X(4)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^12 - 9*a^10*b^2 + 8*a^8*b^4 - 2*a^6*b^6 + 3*a^4*b^8 - 5*a^2*b^10 + 2*b^12 - 9*a^10*c^2 + 23*a^8*b^2*c^2 - 16*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 13*a^2*b^8*c^2 - 5*b^10*c^2 + 8*a^8*c^4 - 16*a^6*b^2*c^4 + 18*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 2*b^8*c^4 - 2*a^6*c^6 - 6*a^4*b^2*c^6 - 8*a^2*b^4*c^6 + 2*b^6*c^6 + 3*a^4*c^8 + 13*a^2*b^2*c^8 + 2*b^4*c^8 - 5*a^2*c^10 - 5*b^2*c^10 + 2*c^12) : :
X(44990) = 3 X[4] - 2 X[136], 4 X[5] - 3 X[38718], 4 X[136] - 3 X[1300], 3 X[376] - 4 X[34844], 5 X[3091] - 4 X[34840], 5 X[17578] - 2 X[21667]

See Peter Moses, Euclid 2600 .

X(44990) lies on the HS circle and these lines: {4, 110}, {5, 38718}, {20, 131}, {30, 925}, {186, 13496}, {376, 34844}, {378, 13558}, {847, 12293}, {2970, 12902}, {3091, 34840}, {3520, 5961}, {3627, 13556}, {10152, 14264}, {10723, 20774}, {12121, 35235}, {17578, 21667}

X(44990) = reflection of X(i) in X(j) for these {i,j}: {20, 131}, {1300, 4}, {13556, 3627}
X(44990) = polar circle inverse of X(12295)

X(44991) = REFLECTION OF X(1303) IN X(4)

Barycentrics    a^2*(-(a^18*b^6) + 7*a^16*b^8 - 21*a^14*b^10 + 35*a^12*b^12 - 35*a^10*b^14 + 21*a^8*b^16 - 7*a^6*b^18 + a^4*b^20 + 3*a^20*b^2*c^2 - 13*a^18*b^4*c^2 + 18*a^16*b^6*c^2 - 22*a^12*b^10*c^2 + 14*a^10*b^12*c^2 + 8*a^8*b^14*c^2 - 8*a^6*b^16*c^2 - 5*a^4*b^18*c^2 + 7*a^2*b^20*c^2 - 2*b^22*c^2 - 13*a^18*b^2*c^4 + 48*a^16*b^4*c^4 - 59*a^14*b^6*c^4 + 11*a^12*b^8*c^4 + 52*a^10*b^10*c^4 - 78*a^8*b^12*c^4 + 41*a^6*b^14*c^4 + 23*a^4*b^16*c^4 - 37*a^2*b^18*c^4 + 12*b^20*c^4 - a^18*c^6 + 18*a^16*b^2*c^6 - 59*a^14*b^4*c^6 + 75*a^12*b^6*c^6 - 49*a^10*b^8*c^6 + 41*a^8*b^10*c^6 - 17*a^6*b^12*c^6 - 59*a^4*b^14*c^6 + 78*a^2*b^16*c^6 - 27*b^18*c^6 + 7*a^16*c^8 + 11*a^12*b^4*c^8 - 49*a^10*b^6*c^8 + 16*a^8*b^8*c^8 - 9*a^6*b^10*c^8 + 84*a^4*b^12*c^8 - 82*a^2*b^14*c^8 + 22*b^16*c^8 - 21*a^14*c^10 - 22*a^12*b^2*c^10 + 52*a^10*b^4*c^10 + 41*a^8*b^6*c^10 - 9*a^6*b^8*c^10 - 88*a^4*b^10*c^10 + 34*a^2*b^12*c^10 + 13*b^14*c^10 + 35*a^12*c^12 + 14*a^10*b^2*c^12 - 78*a^8*b^4*c^12 - 17*a^6*b^6*c^12 + 84*a^4*b^8*c^12 + 34*a^2*b^10*c^12 - 36*b^12*c^12 - 35*a^10*c^14 + 8*a^8*b^2*c^14 + 41*a^6*b^4*c^14 - 59*a^4*b^6*c^14 - 82*a^2*b^8*c^14 + 13*b^10*c^14 + 21*a^8*c^16 - 8*a^6*b^2*c^16 + 23*a^4*b^4*c^16 + 78*a^2*b^6*c^16 + 22*b^8*c^16 - 7*a^6*c^18 - 5*a^4*b^2*c^18 - 37*a^2*b^4*c^18 - 27*b^6*c^18 + a^4*c^20 + 7*a^2*b^2*c^20 + 12*b^4*c^20 - 2*b^2*c^22) : :
X(44991) = 3 X[4] - 2 X[129], 4 X[129] - 3 X[1303], 3 X[376] - 4 X[34838], 5 X[3091] - 4 X[34839]

See Peter Moses, Euclid 2600 .

X(44991) lies on the HS circle and these lines: {4, 129}, {20, 130}, {30, 1298}, {376, 34838}, {3091, 34839}, {5073, 38594}, {10721, 32438}, {10722, 11381}, {13598, 21661}

X(44991) = midpoint of X(5073) and X(38594)
X(44991) = reflection of X(i) in X(j) for these {i,j}: {20, 130}, {1303, 4}, {21661, 13598}

X(44992) = REFLECTION OF X(1304) IN X(4)

Barycentrics   (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(3*a^18 - 8*a^16*b^2 - 7*a^14*b^4 + 44*a^12*b^6 - 47*a^10*b^8 + 2*a^8*b^10 + 23*a^6*b^12 - 8*a^4*b^14 - 4*a^2*b^16 + 2*b^18 - 8*a^16*c^2 + 48*a^14*b^2*c^2 - 62*a^12*b^4*c^2 - 58*a^10*b^6*c^2 + 168*a^8*b^8*c^2 - 84*a^6*b^10*c^2 - 30*a^4*b^12*c^2 + 30*a^2*b^14*c^2 - 4*b^16*c^2 - 7*a^14*c^4 - 62*a^12*b^2*c^4 + 229*a^10*b^4*c^4 - 172*a^8*b^6*c^4 - 115*a^6*b^8*c^4 + 172*a^4*b^10*c^4 - 35*a^2*b^12*c^4 - 10*b^14*c^4 + 44*a^12*c^6 - 58*a^10*b^2*c^6 - 172*a^8*b^4*c^6 + 352*a^6*b^6*c^6 - 134*a^4*b^8*c^6 - 66*a^2*b^10*c^6 + 34*b^12*c^6 - 47*a^10*c^8 + 168*a^8*b^2*c^8 - 115*a^6*b^4*c^8 - 134*a^4*b^6*c^8 + 150*a^2*b^8*c^8 - 22*b^10*c^8 + 2*a^8*c^10 - 84*a^6*b^2*c^10 + 172*a^4*b^4*c^10 - 66*a^2*b^6*c^10 - 22*b^8*c^10 + 23*a^6*c^12 - 30*a^4*b^2*c^12 - 35*a^2*b^4*c^12 + 34*b^6*c^12 - 8*a^4*c^14 + 30*a^2*b^2*c^14 - 10*b^4*c^14 - 4*a^2*c^16 - 4*b^2*c^16 + 2*c^18) : :
X(44992) = 3 X[4] - 2 X[18809], 4 X[5] - 3 X[38719], 3 X[381] - 2 X[38625], 3 X[1304] - 4 X[18809], 5 X[3091] - 4 X[40557], 3 X[3830] - X[38595]

See Peter Moses, Euclid 2600 .

X(44992) lies on the HS circle and these lines: {4, 477}, {5, 38719}, {20, 16177}, {30, 107}, {250, 36172}, {381, 38625}, {520, 10733}, {523, 10152}, {3091, 40557}, {3830, 38595}, {6000, 7722}, {13997, 18383}, {15384, 38956}, {18507, 20957}

X(44992) = reflection of X(i) in X(j) for these {i,j}: {20, 16177}, {1304, 4}, {13997, 18383}
X(44992) = reflection of X(10152) in the Euler line

X(44993) = X(2)X(972)∩X(11)X(57)

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

See Ercole Suppa, euclid 2620.

X(44993) = lies on the nine-point circle and these lines: {2,972}, {3,40555}, {4,934}, {5,5514}, {11,57} ,{116,946}, {119,6366}, {123,2886}, {124,3817}, {223,13612}, {693,21665}, {971,1543} ,{1851,5521}, {5511,7681}, {5520,16309}, {5806,15607}, {6831,15725}, {10739,37726}

X(44993) = midpoint of X(4) and X(934)
X(44993) = reflection of X(i) in X(j) for these (i,j): (3,40555), (5514,5)
X(44993) = reflection of X(11) in X(5)X(6366)
X(44993) = complement of X(972)
X(44993)= complementary conjugate of X(971)
X(44993) = X(4)-Ceva conjugate of X(971)
X(44993)= X(i)-complementary conjugate of X(j) for these (i,j): (1,971), (6,43035), (971,10), (2272,2)
X(44993) = X(934)-of-Euler-triangle
X(44993) = X(5514)-of-Johnson-triangle
X(44993) = X(44978)-image under 2nd HS transform

X(44994) = (name pending)

Barycentrics    a*(a^7 - 5*a^5*b^2 + 4*a^4*b^3 + 3*a^3*b^4 - 4*a^2*b^5 + a*b^6 + 7*a^5*b*c + a^4*b^2*c - 10*a^3*b^3*c - 10*a^2*b^4*c + 11*a*b^5*c + b^6*c - 5*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 14*a^2*b^3*c^2 - 5*a*b^4*c^2 - 3*b^5*c^2 + 4*a^4*c^3 - 10*a^3*b*c^3 + 14*a^2*b^2*c^3 - 14*a*b^3*c^3 + 2*b^4*c^3 + 3*a^3*c^4 - 10*a^2*b*c^4 - 5*a*b^2*c^4 + 2*b^3*c^4 - 4*a^2*c^5 + 11*a*b*c^5 - 3*b^2*c^5 + a*c^6 + b*c^6 + 2*(2*a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + 2*a*b^4 + 2*a^4*c + 6*a^3*b*c - 11*a^2*b^2*c + 3*b^4*c - 2*a^3*c^2 - 11*a^2*b*c^2 - 4*a*b^2*c^2 - 3*b^3*c^2 - 2*a^2*c^3 - 3*b^2*c^3 + 2*a*c^4 + 3*b*c^4)*S + 2*b*(a + b - c)*c*(a - b + c)*((7*a^2*b - 6*a*b^2 - b^3 + 7*a^2*c - 10*a*b*c + b^2*c - 6*a*c^2 + b*c^2 - c^3)*Cos[A/2] + (4*a^3 + a^2*b - 3*a*b^2 - 2*b^3 + a^2*c - 8*a*b*c - 8*b^2*c - 3*a*c^2 - 8*b*c^2 - 2*c^3)*Sin[A/2]) + 2*a*(a - b - c)*(a + b - c)*c*((2*a^3 + 3*a^2*b - 5*b^3 - a^2*c + 8*a*b*c - 5*b^2*c - 2*a*c^2 + 9*b*c^2 + c^3)*Cos[B/2] + (4*a^2*b + a*b^2 - 5*b^3 + 3*a^2*c + 12*a*b*c - 2*b^2*c + 4*a*c^2 + 6*b*c^2 + c^3)*Sin[B/2]) + 2*a*b*(a - b - c)*(a - b + c)*((2*a^3 - a^2*b - 2*a*b^2 + b^3 + 3*a^2*c + 8*a*b*c + 9*b^2*c - 5*b*c^2 - 5*c^3)*Cos[C/2] + (3*a^2*b + 4*a*b^2 + b^3 + 4*a^2*c + 12*a*b*c + 6*b^2*c + a*c^2 - 2*b*c^2 - 5*c^3)*Sin[C/2])) : :

Let A'B'C' be the Incentral triangle. The incenters of the triangles IAB', IB'C, ICA', IA'B, IBC' and IC'A lie on an ellipse with center X(44994)

See Antreas Hatzipolakis and Peter Moses, euclid 2622.

X(44994) = lies on these lines: { }

X(44995) =  11TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (11 a^4 b^2-22 a^2 b^4+11 b^6+11 a^4 c^2+36 a^2 b^2 c^2-11 b^4 c^2-22 a^2 c^4-11 b^2 c^4+11 c^6) : :
Barycentrics    SB SC (80 R^2-11 SA-11 SW) : :

As a point on the Euler line, X(44995) has Shinagawa coefficients (11*F,-9*E+11*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(44995) = lies on this line: {2,3}

X(44995) = X(3090)-image under 1st HS transform

X(44996) =  12TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (7 a^4 b^2-14 a^2 b^4+7 b^6+7 a^4 c^2+24 a^2 b^2 c^2-7 b^4 c^2-14 a^2 c^4-7 b^2 c^4+7 c^6) : :
Barycentrics    52 R^2-7 SA-7 SW : :

As a point on the Euler line, X(44996) has Shinagawa coefficients (7*F,-6*E+7*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(44996) = lies on this line: {2,3}

X(44996) = X(3091)-image under 1st HS transform

X(44997) =  13TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (7 a^8 b^2-48 a^6 b^4+82 a^4 b^6-48 a^2 b^8+7 b^10+7 a^8 c^2-42 a^6 b^2 c^2-82 a^4 b^4 c^2+94 a^2 b^6 c^2-41 b^8 c^2-48 a^6 c^4-82 a^4 b^2 c^4-76 a^2 b^4 c^4+34 b^6 c^4+82 a^4 c^6+94 a^2 b^2 c^6+34 b^4 c^6-48 a^2 c^8-41 b^2 c^8+7 c^10) : :
Barycentrics    SB SC (6 S^2 (15 R^2-2 SA-2 SW)+(36 R^2-5 SA-5 SW) SW^2) : :

As a point on the Euler line, X(44997) has Shinagawa coefficients (2*F*(5*E^2+10*E*F+5*F^2+12*S^2),-8*E^3-6*E^2*F+12*E*F^2+10*F^3-21*E*S^2+24*F*S^2F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(44997) = lies on this line: {2,3}

X(44997) = X(3363)-image under 1st HS transform

X(44998) =  14TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (11 a^4 b^2-22 a^2 b^4+11 b^6+11 a^4 c^2+32 a^2 b^2 c^2-11 b^4 c^2-22 a^2 c^4-11 b^2 c^4+11 c^6) : :
Barycentrics    SB SC (76 R^2-11 SA-11 SW) : :

As a point on the Euler line, X(44998) has Shinagawa coefficients (11*F,-8*E+11*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(44998) = lies on this line: {2,3}

X(44998) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (235,403,1595), (235,16868,1596)
X(44998) = X(3522)-image under 1st HS transform

X(44999) =  15TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (13 a^4 b^2-26 a^2 b^4+13 b^6+13 a^4 c^2+40 a^2 b^2 c^2-13 b^4 c^2-26 a^2 c^4-13 b^2 c^4+13 c^6) : :
Barycentrics    SB SC (92 R^2-13 SA-13 SW) : :

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(44999) = lies on this line: {2,3}

X(44999) = X(3523)-image under 1st HS transform

X(45000) =  16TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (17 a^4 b^2-34 a^2 b^4+17 b^6+17 a^4 c^2+52 a^2 b^2 c^2-17 b^4 c^2-34 a^2 c^4-17 b^2 c^4+17 c^6) : :
Barycentrics    SB SC (120 R^2-17 SA-17 SW) : :

As a point on the Euler line, X(45000) has Shinagawa coefficients (17*F,-13*E+17*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45000) = lies on this line: {2,3}

X(45000) = {X(1596),X(16868)}-harmonic conjugate of X(427)
X(45000) = X(3524)-image under 1st HS transform

X(45001) =  17TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (19 a^4 b^2-38 a^2 b^4+19 b^6+19 a^4 c^2+60 a^2 b^2 c^2-19 b^4 c^2-38 a^2 c^4-19 b^2 c^4+19 c^6): : :
Barycentrics    SB SC (136 R^2-19 SA-19 SW) : :

As a point on the Euler line, X(45001) has Shinagawa coefficients (19*F,-15*E+19*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45001) = lies on this line: {2,3}

X(45001) = X(3525)-image under 1st HS transform

X(45002) =  18TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (6 a^4 b^2-12 a^2 b^4+6 b^6+6 a^4 c^2+19 a^2 b^2 c^2-6 b^4 c^2-12 a^2 c^4-6 b^2 c^4+6 c^6) : :
Barycentrics    SB SC (43 R^2-6 SA-6 SW) : :

As a point on the Euler line, X(45002) has Shinagawa coefficients (24*F,-19*E+24*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45002) = lies on this line: {2,3}

X(45002) = {X(403),X(1907)}-harmonic conjugate of X(16868)
X(45002) = X(3526)-image under 1st HS transform

X(45003) =  19TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (15 a^4 b^2-30 a^2 b^4+15 b^6+15 a^4 c^2+44 a^2 b^2 c^2-15 b^4 c^2-30 a^2 c^4-15 b^2 c^4+15 c^6) : :
Barycentrics    SB SC (104 R^2-15 SA-15 SW) : :

As a point on the Euler line, X(45003) has Shinagawa coefficients (15*F,-11*E+15*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45003) = lies on this line: {2,3}

X(45003) = {X(235),X(403)}-harmonic conjugate of X(1907)
X(45003) = X(3528)-image under 1st HS transform

X(45004) =  20TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (5 a^4 b^2-10 a^2 b^4+5 b^6+5 a^4 c^2+12 a^2 b^2 c^2-5 b^4 c^2-10 a^2 c^4-5 b^2 c^4+5 c^6) : :
Barycentrics    SB SC (32 R^2-5 SA-5 SW) : :

As a point on the Euler line, X(45004) has Shinagawa coefficients (5*F,-3*E+5*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45004) = lies on this line: {2,3}

X(45004) = midpoint of X(4) and X(35479)
X(45004) = crosssum of X(184)and X(33636)
X(45004) = X(35479)-of-Euler-triangle
X(45004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,11410,1885), (5,235,1906), (5,1596,15559), (5,1906,427), (3089,23047,10301), (6622,37197,468), (15559,16868,5), (15750,37197,4)
X(45004) = X(3529)-image under 1st HS transform

X(45005) =  21ST HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (15 a^4 b^2-30 a^2 b^4+15 b^6+15 a^4 c^2+46 a^2 b^2 c^2-15 b^4 c^2-30 a^2 c^4-15 b^2 c^4+15 c^6) : :
Barycentrics    SB SC (106 R^2-15 SA-15 SW) : :

As a point on the Euler line, X(45005) has Shinagawa coefficients (30*F,-23*E+30*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45005) = lies on this line: {2,3}

X(45005) = X(3530)-image under 1st HS transform

X(45006) =  22ND HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (29 a^4 b^2-58 a^2 b^4+29 b^6+29 a^4 c^2+92 a^2 b^2 c^2-29 b^4 c^2-58 a^2 c^4-29 b^2 c^4+29 c^6) : :
Barycentrics    SB SC (208 R^2-29 SA-29 SW) : :

As a point on the Euler line, X(45006) has Shinagawa coefficients (29*F,-23*E+29*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45006) = lies on this line: {2,3}

X(45006) = X(3533)-image under 1st HS transform

X(45007) =  23RD HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (4 a^4 b^2-8 a^2 b^4+4 b^6+4 a^4 c^2+11 a^2 b^2 c^2-4 b^4 c^2-8 a^2 c^4-4 b^2 c^4+4 c^6) : :
Barycentrics    SB SC (27 R^2-4 SA-4 SW) : :

As a point on the Euler line, X(45007) has Shinagawa coefficients (16*F,-11*E+16*F).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45007) = lies on this line: {2,3}

X(45007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (235,403,7577), (403,1596,16868), (6623,37943,4), (35480,44281,35481)
X(45007) = X(3534)-image under 1st HS transform

X(45008) =  24TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (-40 a^6 b^4 c^2+80 a^4 b^6 c^2-40 a^2 b^8 c^2-40 a^6 b^2 c^4-128 a^4 b^4 c^4+40 a^2 b^6 c^4+80 a^4 b^2 c^6+40 a^2 b^4 c^6-40 a^2 b^2 c^8+a^8 b^2 S-4 a^6 b^4 S+6 a^4 b^6 S-4 a^2 b^8 S+b^10 S+a^8 c^2 S-6 a^4 b^4 c^2 S+8 a^2 b^6 c^2 S-3 b^8 c^2 S-4 a^6 c^4 S-6 a^4 b^2 c^4 S-8 a^2 b^4 c^4 S+2 b^6 c^4 S+6 a^4 c^6 S+8 a^2 b^2 c^6 S+2 b^4 c^6 S-4 a^2 c^8 S-3 b^2 c^8 S+c^10 S) : :
Barycentrics    SB SC (8 R^2 (36 R^2-5 SA-5 SW)+S (8 R^2-SA-SW)) : :

As a point on the Euler line, X(45008) has Shinagawa coefficients (F*(10*E+S),-8*E^2+10*E*F-E*S+F*S).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45008) = lies on this line: {2,3}

X(45008) = X(3539)-image under 1st HS transform

X(45009) =  25TH HATZIPOLAKIS-SUPPA-EULER POINT

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (40 a^6 b^4 c^2-80 a^4 b^6 c^2+40 a^2 b^8 c^2+40 a^6 b^2 c^4+128 a^4 b^4 c^4-40 a^2 b^6 c^4-80 a^4 b^2 c^6-40 a^2 b^4 c^6+40 a^2 b^2 c^8+a^8 b^2 S-4 a^6 b^4 S+6 a^4 b^6 S-4 a^2 b^8 S+b^10 S+a^8 c^2 S-6 a^4 b^4 c^2 S+8 a^2 b^6 c^2 S-3 b^8 c^2 S-4 a^6 c^4 S-6 a^4 b^2 c^4 S-8 a^2 b^4 c^4 S+2 b^6 c^4 S+6 a^4 c^6 S+8 a^2 b^2 c^6 S+2 b^4 c^6 S-4 a^2 c^8 S-3 b^2 c^8 S+c^10 S) : :
Barycentrics    SB SC (-8 R^2 (36 R^2 - 5 SA - 5 SW) + S (8 R^2 - SA - SW)) : :

As a point on the Euler line, X(45009) has Shinagawa coefficients (F*(10*E-S),-8*E^2+10*E*F+E*S-F*S).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2629 .

X(45009) = lies on this line: {2,3}

X(45009) = X(3540)-image under 1st HS transform

leftri

1st and 2nd Aubert points: X(45010)-X(45124)

rightri

This preamble and centers X(45010)-X(45124) were contributed by César Eliud Lozada, October 6, 2021, based on a proposal by Ivan Pavlov.

A complete quadrilateral Q formed by four lines ℓ1, ℓ2, ℓ3, ℓ4 is denoted here as Q = [ℓ1, ℓ2, ℓ3, ℓ4], or also as Q = [ABCD], if it is composed by lines AB, BC, CD, DA.

In every complete quadrilateral Q there are four component triangles, each bounded by three different lines of Q. It is well known that the orthocenters of these triangles are collinear on a line called the Aubert line (or Steiner line) of Q.

  1. Let T '=A'B'C' and T"=A"B"C" be two non-homothetic distinct triangles and let ℓa, ℓb, ℓc be the Aubert lines of quadrilaterals [B'C'B"C"], [C'A'C"A"] and [A'B'A"B"], respectively. Then ℓa, ℓb, ℓc concur in a point here named the 1st Aubert point of T' and T". (Note: If T' and T" are homothetic, the described Aubert lines bound a triangle homothetic to T and T").

    In the following list, (T, i) means that the 1st Aubert point of ABC and triangle T is X(i) and a double dash -- means a not calculated center:

    (1st Altintas-isodynamic, --), (2nd Altintas-isodynamic, --), (Andromeda, --), (anti-Artzt, 2), (anti-Ascella, 45010), (anti-Atik, 45011), (1st anti-Brocard, 9772), (4th anti-Brocard, 45012), (6th anti-Brocard, 45013), (1st anti-circumperp, 2888), (anti-Conway, 19362), (2nd anti-Conway, 3527), (3rd anti-Euler, 45014), (4th anti-Euler, 7699), (anti-excenters-reflections, 4), (2nd anti-extouch, 45015), (anti-inner-Garcia, 6265), (anti-Honsberger, 45016), (anti-Hutson intouch, 11472), (anti-incircle-circles, 1351), (anti-inverse-in-incircle, 4), (anti-McCay, 11152), (6th anti-mixtilinear, 11487), (anti-orthocentroidal, 45019), (1st anti-orthosymmedial, 45020), (anti-1st Parry, 110), (anti-2nd Parry, 111), (1st anti-Sharygin, 45021), (anti-tangential-midarc, 45022), (3rd anti-tri-squares, 45023), (4th anti-tri-squares, 45024), (anti-Ursa minor, 18488), (anti-Wasat, 3574), (antiAOA, 45025), (Antlia, --), (AOA, 45026), (Apus, 45027), (Aries, 3), (Artzt, 2), (Ascella, 11023), (Atik, --), (Ayme, 45028), (Bankoff, --), (BCI, 10230), (Bevan antipodal, 34498), (1st Brocard-reflected, 98), (1st Brocard, 262), (2nd Brocard, 6800), (3rd Brocard, 182), (4th Brocard, 7737), (6th Brocard, 45029), (7th Brocard, 45030), (9th Brocard, 45031), (circummedial, 2), (circumnormal, --), (circumorthic, 4), (1st circumperp, 191), (2nd circumperp, 1), (circumsymmedial, 6), (circumtangential, --), (inner-Conway, 8), (Conway, 2894), (2nd Conway, 11024), (3rd Conway, 45032), (4th Conway, 45033), (5th Conway, --), (Ehrmann-side, 6288), (Ehrmann-vertex, 4), (1st Ehrmann, 895), (2nd Ehrmann, 45034), (2nd Euler, 1352), (3rd Euler, 45035), (4th Euler, 17057), (5th Euler, 2), (excenters-midpoints, 45036), (excenters-reflections, 1), (excentral, 1), (1st excosine, 33537), (2nd excosine, 45037), (extangents, 45038), (extouch, 4), (2nd extouch, 45039), (3rd extouch, --), (4th extouch, --), (5th extouch, 45040), (inner-Fermat, 17), (outer-Fermat, 18), (1st Fermat-Dao, --), (2nd Fermat-Dao, --), (3rd Fermat-Dao, --), (4th Fermat-Dao, --), (5th Fermat-Dao, --), (6th Fermat-Dao, --), (7th Fermat-Dao, --), (8th Fermat-Dao, --), (10th Fermat-Dao, --), (11th Fermat-Dao, 2), (12th Fermat-Dao, 2), (13th Fermat-Dao, --), (14th Fermat-Dao, --), (15th Fermat-Dao, --), (16th Fermat-Dao, --), (1st inner-Fermat-Dao-Nhi, --), (2nd inner-Fermat-Dao-Nhi, --), (3rd inner-Fermat-Dao-Nhi, 36386), (4th inner-Fermat-Dao-Nhi, --), (1st outer-Fermat-Dao-Nhi, --), (2nd outer-Fermat-Dao-Nhi, --), (3rd outer-Fermat-Dao-Nhi, 36388), (4th outer-Fermat-Dao-Nhi, --), (Feuerbach, 45041), (Fuhrmann, 5903), (2nd Fuhrmann, 5904), (inner-Garcia, 355), (Garcia-reflection, 65), (1st half-diamonds-central, --), (2nd half-diamonds-central, --), (1st half-diamonds, --), (2nd half-diamonds, 33387), (1st half-squares, 33364), (2nd half-squares, 33365), (Hatzipolakis-Moses, --), (1st Hatzipolakis, --), (2nd Hatzipolakis, 4), (3rd Hatzipolakis, 45042), (hexyl, 9), (Honsberger, 45043), (Hung-Feuerbach, --), (Hutson extouch, 3296), (inner-Hutson, --), (Hutson intouch, 1000), (outer-Hutson, --), (1st Hyacinth, 45044), (2nd Hyacinth, 45045), (incentral, 4), (incircle-circles, 7), (intangents, 45046), (intouch, 4), (inverse-in-Conway, 1), (inverse-in-excircles, 45047), (inverse-in-incircle, 1), (1st isodynamic-Dao, 6777), (2nd isodynamic-Dao, 6778), (3rd isodynamic-Dao, --), (4th isodynamic-Dao, --), (Jenkins-contact, --), (Jenkins-tangential, --), (1st Jenkins, 45048), (2nd Jenkins, 10), (3rd Jenkins, --), (K798e, 149), (K798i, 2475), (1st Kenmotu diagonals, 45049), (2nd Kenmotu diagonals, 45050), (2nd Kenmotu-free-vertices, --), (Kosnita, 110), (Largest-circumscribed-equilateral, 5463), (Lemoine, 4), (1st Lemoine-Dao, --), (2nd Lemoine-Dao, --), (inner-Le Viet An, 3), (outer-Le Viet An, 3), (Lucas antipodal, 1151), (Lucas(-1) antipodal, 1152), (Lucas antipodal tangents, --), (Lucas(-1) antipodal tangents, --), (Lucas Brocard, --), (Lucas(-1) Brocard, --), (Lucas central, 45051), (Lucas(-1) central, 45052), (Lucas inner, 1151), (Lucas(-1) inner, 1152), (Lucas inner tangential, 45053), (Lucas(-1) inner tangential, 45054), (Lucas reflection, --), (Lucas(-1) reflection, --), (Lucas secondary central, 45055), (Lucas(-1) secondary central, 45056), (Lucas 1st secondary tangents, 45057), (Lucas(-1) 1st secondary tangents, 45058), (Lucas 2nd secondary tangents, 45059), (Lucas(-1) 2nd secondary tangents, 45060), (Lucas tangents, 371), (Lucas(-1) tangents, 372), (Macbeath, 4), (Malfatti, --), (Mandart-excircles, 45061), (McCay, 598), (midarc, 45086), (2nd midarc, 45087), (midheight, 15740), (mixtilinear, 222), (2nd mixtilinear, 218), (3rd mixtilinear, 56), (4th mixtilinear, 55), (6th mixtilinear, 4866), (7th mixtilinear, --), (8th mixtilinear, 55), (9th mixtilinear, 56), (Montesdeoca-Hung, --), (1st Morley-midpoint, --), (2nd Morley-midpoint, --), (3rd Morley-midpoint, --), (1st Morley, 5628), (2nd Morley, 5630), (3rd Morley, 5632), (1st Morley-adjunct midpoint, --), (2nd Morley-adjunct midpoint, --), (3rd Morley-adjunct midpoint, --), (1st Morley-adjunct, 5629), (2nd Morley-adjunct, 5631), (3rd Morley-adjunct, 5633), (Moses-Hung, --), (Moses-Soddy, 1), (Moses-Steiner osculatory, 45017), (Moses-Steiner reflection, 45018), (inner-Napoleon, 13), (outer-Napoleon, 14), (1st Neuberg, 83), (2nd Neuberg, 76), (orthic, 4), (orthic axes, 45062), (orthocentroidal, 4846), (1st orthosymmedial, 45063), (2nd orthosymmedial, 45064), (1st Pamfilos-Zhou, --), (2nd Pamfilos-Zhou, --), (1st Parry, 110), (2nd Parry, 111), (3rd Parry, --), (Pelletier, 1), (1st Przybyłowski-Bollin, --), (3rd Przybyłowski-Bollin, --), (4th Przybyłowski-Bollin, --), (reflection, 68), (Roussel, --), (1st Savin, 1), (2nd Savin, --), (1st Schiffler, 45065), (2nd Schiffler, 1319), (Schroeter, 5), (1st Sharygin, 45066), (2nd Sharygin, 45067), (Soddy, 45068), (inner-Soddy, --), (2nd inner-Soddy, 45069), (outer-Soddy, --), (2nd outer-Soddy, 45070), (inner-squares, 45071), (outer-squares, 45072), (Stammler, --), (Steiner, 4), (submedial, 45073), (symmedial, 4), (tangential, 155), (tangential-midarc, --), (2nd tangential-midarc, --), (inner tri-equilateral, 45074), (outer tri-equilateral, 45075), (1st tri-squares-central, 45076), (2nd tri-squares-central, 45077), (1st tri-squares, 485), (2nd tri-squares, 486), (3rd tri-squares, 45078), (4th tri-squares, 45079), (Trinh, 15062), (Ursa-major, 45080), (Ursa-minor, 45081), (inner-Vecten, 485), (2nd inner-Vecten, 486), (3er inner-Vecten, --), (outer-Vecten, 486), (2nd outer-Vecten, 485), (3rd outer-Vecten, --), (Vu-Dao-X(15)-isodynamic, --), (Vu-Dao-X(16)-isodynamic, --), (Walsmith, 45082), (Wasat, 442), (X-parabola-tangential, 31990), (Yff central, --), (Yff contact, 4), (Yiu, 45083), (Yiu tangents, 35720), (1st Zaniah, 45084), (2nd Zaniah, 45085)
  2. Let T '=A'B'C' and T"=A"B"C" be two triangles and let ℓa, ℓb, ℓc be the Aubert lines of quadrilaterals [C'A'B'A"], [A'B'C'B"], [B'C'A'C"], respectively. Let A* = ℓb∩ℓc, B* = ℓc∩ℓa and C* = ℓa∩ℓb. Then, for some pairs of triangles T ' and T", the lines A*A', B*B' and C*C' concur in a point here introduced as the 2nd Aubert point of T' to T".

    In the following list, (T, i) means that the 2nd Aubert point of ABC to triangle T is X(i):

    (ABC-X3 reflections, 1), (anti-Atik, 45088), (1st anti-circumperp, 5), (2nd anti-Conway, 3531), (anti-Ehrmann-mid, 523), (2nd anti-extouch, 45089), (anti-tangential-midarc, --), (anti-X3-ABC reflections, 45090), (anticomplementary, 1), (Aries, 4846), (BCI, --), (Bevan antipodal, 45091), (1st Brocard-reflected, 14458), (1st Brocard, 14492), (3rd Brocard, 45092), (6th Brocard, 45093), (7th Brocard, 45094), (9th Brocard, 4), (1st circumperp, 45095), (Conway, 6734), (3rd Conway, 10), (1st Ehrmann, 45096), (excenters-midpoints, --), (excentral, 43672), (1st excosine, --), (2nd excosine, --), (extangents, --), (extouch, 1), (inner-Fermat, 13), (outer-Fermat, 14), (Garcia-reflection, 79), (1st half-squares, 485), (2nd half-squares, 486), (2nd Hatzipolakis, 1), (hexyl, 226), (Hutson extouch, 45097), (Hutson intouch, 45098), (2nd Hyacinth, 45099), (incentral, 1), (incircle-circles, 45100), (intouch, 1), (Lemoine, 1), (Lucas antipodal, 45101), (Lucas(-1) antipodal, 45102), (Lucas central, --), (Lucas(-1) central, --), (Lucas tangents, --), (Lucas(-1) tangents, --), (Macbeath, 1), (McCay, 45103), (medial, 1), (midheight, 14484), (mixtilinear, 45104), (2nd mixtilinear, --), (6th mixtilinear, 17758), (1st Morley, --), (2nd Morley, --), (3rd Morley, --), (1st Morley-adjunct, --), (2nd Morley-adjunct, --), (3rd Morley-adjunct, --), (inner-Napoleon, 12816), (outer-Napoleon, 12817), (1st Neuberg, 598), (2nd Neuberg, 671), (orthic, 1), (orthic axes, 45105), (orthocentroidal, 14492), (reflection, 98), (2nd inner-Soddy, --), (2nd outer-Soddy, --), (inner-squares, 45106), (outer-squares, 45107), (Steiner, 1), (symmedial, 1), (tangential, 43917), (inner-Vecten, 1), (2nd inner-Vecten, 1), (outer-Vecten, 1), (2nd outer-Vecten, 1), (X3-ABC reflections, 45108), (Yff contact, 1)

    Similarly, in the below list, (T, i) indicates that the 2nd Aubert point of triangle T to ABC is X(i):

    (ABC-X3 reflections, 1), (1st anti-Brocard, 45109), (6th anti-Brocard, --), (1st anti-circumperp, --), (anti-Conway, 45110), (anti-Hutson intouch, --), (anti-incircle-circles, --), (anti-McCay, 45111), (6th anti-mixtilinear, 45112), (anti-orthocentroidal, 45113), (anti-X3-ABC reflections, 45114), (anticomplementary, 1), (Atik, 45115), (Bevan antipodal, 1), (1st circumperp, 3579), (2nd Conway, 45116), (Ehrmann-mid, 523), (Ehrmann-side, 45117), (2nd Euler, 45118), (excenters-midpoints, --), (excentral, 1), (extouch, 45119), (2nd extouch, 45120), (15th Fermat-Dao, --), (16th Fermat-Dao, --), (Garcia-reflection, 45121), (2nd half-diamonds, 618), (1st half-squares, 1), (2nd half-squares, 1), (2nd Hyacinth, --), (intouch, 45122), (1st isodynamic-Dao, --), (2nd isodynamic-Dao, --), (3rd isodynamic-Dao, --), (4th isodynamic-Dao, --), (Kosnita, --), (Largest-circumscribed-equilateral, 1), (Lucas antipodal, --), (Lucas(-1) antipodal, --), (Lucas tangents, --), (Lucas(-1) tangents, --), (medial, 1), (midarc, 12908), (2nd midarc, --), (1st Morley, --), (2nd Morley, --), (3rd Morley, --), (1st Morley-adjunct, --), (2nd Morley-adjunct, --), (3rd Morley-adjunct, --), (Moses-Soddy, 1), (orthic, 45123), (Pelletier, 1), (Schroeter, 1), (Soddy, 1), (inner-Soddy, --), (2nd inner-Soddy, --), (outer-Soddy, --), (2nd outer-Soddy, --), (tangential, 1), (tangential-midarc, --), (Trinh, --), (inner-Vecten, 3), (2nd inner-Vecten, --), (outer-Vecten, 3), (2nd outer-Vecten, --), (X-parabola-tangential, 1), (X3-ABC reflections, 45124), (Yff central, --), (Yiu tangents, 1), (2nd Zaniah, --)

    From these two last lists, it can be seen that the existence of the 2nd Aubert point T' to T" does not guarantee the existence of the 2nd Aubert point T" to T'.

X(45010) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ANTI-ASCELLA

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

X(45010) lies on these lines: {3, 1112}, {4, 13292}, {24, 38942}, {25, 1147}, {51, 45045}, {195, 1598}, {235, 1351}, {389, 1593}, {427, 3527}, {576, 5198}, {1597, 33541}, {1885, 11432}, {1993, 22750}, {2888, 35488}, {3515, 13346}, {3516, 43604}, {3517, 7666}, {5020, 43823}, {5480, 7507}, {9716, 10594}, {10151, 12164}, {10274, 26864}, {10602, 34117}, {12160, 22660}, {12165, 38791}, {12174, 18396}, {12241, 44438}, {12315, 13473}, {15010, 43652}, {18534, 35603}, {19118, 44469}, {26883, 34777}, {37498, 44084}

X(45010) = crosssum of X(3) and X(19360)

X(45011) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ANTI-ATIK

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

X(45011) lies on the Jerabek circumhyperbola and these lines: {3, 11433}, {5, 6391}, {6, 3089}, {54, 6353}, {64, 12241}, {68, 18537}, {69, 6804}, {74, 18916}, {185, 35512}, {389, 15740}, {578, 18928}, {1596, 3527}, {3426, 12324}, {4846, 5446}, {6145, 18918}, {6677, 11426}, {9820, 11427}, {15077, 18390}, {16657, 18910}, {21841, 43908}

X(45011) = isogonal conjugate of the complement of X(6816)
X(45011) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(1093)}} and Jerabek hyperbola
X(45011) = Cevapoint of X(1587) and X(1588)

X(45012) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 4th ANTI-BROCARD

Barycentrics    a^2*(a^8+(b^2+c^2)*a^6-3*(b^4+11*b^2*c^2+c^4)*a^4-(b^2+c^2)*(5*b^4-44*b^2*c^2+5*c^4)*a^2-2*b^8+(19*b^4-66*b^2*c^2+19*c^4)*b^2*c^2-2*c^8) : :

X(45012) lies on these lines: {2, 99}, {6, 33962}, {25, 10355}, {32, 38688}, {39, 38675}, {187, 1296}, {1384, 38593}, {3055, 38796}, {5024, 11258}, {5107, 10765}, {5210, 38623}, {5512, 31415}, {8588, 38716}, {8589, 38698}, {10787, 11152}, {23699, 43619}, {32424, 44526}

X(45012) = reflection of X(99) in the line X(2793)X(9966)
X(45012) = X(7737)-of-4th anti-Brocard triangle

X(45013) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 6th ANTI-BROCARD

Barycentrics    a^16-4*(b^2+c^2)*a^14+5*(b^4+c^4)*a^12-2*(b^2+c^2)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^10-2*(b^8+c^8+(7*b^4-2*b^2*c^2+7*c^4)*b^2*c^2)*a^8+(b^2+c^2)*(3*b^8+3*c^8-(2*b^4+7*b^2*c^2+2*c^4)*b^2*c^2)*a^6-(2*b^12+2*c^12-(3*b^8+3*c^8+(9*b^4-5*b^2*c^2+9*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(b^8+c^8-(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*b^2*c^2)*a^2+(b^2-c^2)^2*(b^8+c^8-(b^4-b^2*c^2+c^4)*b^2*c^2)*b^2*c^2 : :

X(45013) lies on these lines: {83, 14639}, {98, 3406}, {99, 3095}, {114, 45029}, {5182, 5476}, {6033, 40239}, {8295, 10352}, {10350, 21166}, {12177, 35377}

X(45013) = reflection of X(i) in X(j) for these (i, j): (98, 3406), (45029, 114)
X(45013) = X(45029)-of-6th anti-Brocard triangle

X(45014) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd ANTI-EULER

Barycentrics    5*(b^2+c^2)*a^8-(13*b^4-3*b^2*c^2+13*c^4)*a^6+3*(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^4+(b^4-9*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(45014) = 4*X(5)-3*X(11704) = X(20)-3*X(38942) = 5*X(631)-3*X(11270) = 7*X(3526)-3*X(11999) = X(17800)-3*X(44748)

X(45014) lies on these lines: {3, 43585}, {4, 41597}, {5, 5890}, {20, 10282}, {110, 382}, {113, 5889}, {195, 3843}, {381, 1173}, {399, 18394}, {548, 35257}, {631, 4846}, {858, 36983}, {1568, 12279}, {1656, 43603}, {1906, 21850}, {3060, 22660}, {3091, 11431}, {3153, 40241}, {3526, 11440}, {3818, 3832}, {3853, 12278}, {3855, 11442}, {3861, 14516}, {5067, 45073}, {5070, 14926}, {5446, 36852}, {5448, 15305}, {5655, 18377}, {6583, 10883}, {7703, 12162}, {8549, 41737}, {10706, 12084}, {11439, 15559}, {11446, 37719}, {11452, 16964}, {11453, 16965}, {12270, 15063}, {13754, 18504}, {15044, 15752}, {15056, 24206}, {15072, 37452}, {16534, 34797}, {17578, 32605}, {17800, 44748}, {19139, 45034}, {19367, 37720}, {25712, 33703}, {32364, 41726}

X(45014) = reflection of X(5889) in X(16880)
X(45014) = X(45035)-of-3rd-anti-Euler-triangle if ABC is acute

X(45015) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd ANTI-EXTOUCH

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

X(45015) lies on the Stammler hyperbola and these lines: {2, 15438}, {3, 12233}, {4, 159}, {5, 9937}, {6, 5562}, {24, 43841}, {25, 2917}, {155, 578}, {184, 1498}, {195, 11426}, {378, 12250}, {389, 7393}, {1352, 11479}, {1597, 9833}, {2916, 11414}, {2918, 7387}, {2931, 5972}, {2935, 3516}, {3851, 5898}, {5622, 7592}, {6146, 32621}, {7484, 9786}, {7503, 11427}, {7509, 11821}, {7526, 9908}, {9605, 40675}, {11430, 12085}, {11432, 15047}, {13367, 38396}, {16936, 21312}, {31802, 37488}, {36747, 45118}, {39568, 43621}

X(45015) = touchpoint of the line {7503, 11427} and Stammler hyperbola
X(45015) = tangential-isogonal conjugate-of-X(9715)
X(45015) = X(45039)-of-2nd-anti-extouch-triangle if ABC is acute

X(45016) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ANTI-HONSBERGER

Barycentrics    a^2*(a^10-3*(b^2+c^2)*a^8+(2*b^4+b^2*c^2+2*c^4)*a^6+2*(b^6+c^6)*a^4-(3*b^8+3*c^8-(3*b^4-4*b^2*c^2+3*c^4)*b^2*c^2)*a^2+(b^8-c^8)*(b^2-c^2)) : :
X(45016) = X(3)-4*X(6593) = X(3)+2*X(9970) = 2*X(5)+X(11061) = 4*X(5)-X(32306) = 2*X(6)+X(399) = 5*X(6)-2*X(9976) = X(6)+2*X(19140) = X(6)-4*X(25556) = 4*X(113)-X(18440) = X(381)+2*X(34319) = 5*X(399)+4*X(9976) = X(399)-4*X(19140) = X(399)+8*X(25556) = X(399)+4*X(34155) = X(5655)+2*X(15303) = 2*X(6593)+X(9970) = X(9976)+5*X(19140) = X(9976)-10*X(25556) = X(9976)-5*X(34155) = 4*X(9976)-5*X(39562) = 2*X(11061)+X(32306) = X(15141)+2*X(34117)

X(45016) lies on the cubic K938 and these lines: {2, 15106}, {3, 1177}, {4, 45034}, {5, 9512}, {6, 13}, {25, 110}, {30, 22151}, {67, 1656}, {69, 10201}, {74, 12017}, {112, 38551}, {125, 10601}, {140, 32247}, {146, 40640}, {155, 5095}, {156, 15073}, {182, 5621}, {193, 2914}, {195, 576}, {206, 11597}, {323, 7426}, {382, 15140}, {394, 5642}, {403, 3564}, {511, 2070}, {568, 19136}, {575, 12162}, {597, 15037}, {599, 44493}, {895, 3527}, {999, 32289}, {1147, 11470}, {1181, 15063}, {1350, 15040}, {1352, 10254}, {1482, 32278}, {1498, 38791}, {1503, 18403}, {1511, 10752}, {1614, 15074}, {1974, 2931}, {1986, 12168}, {1992, 44275}, {1994, 9143}, {1995, 15135}, {2080, 21419}, {2393, 10540}, {2771, 16475}, {2854, 5093}, {2892, 11585}, {2904, 32240}, {2935, 34779}, {2937, 34116}, {3066, 32235}, {3091, 32341}, {3295, 32290}, {3448, 5133}, {3534, 19379}, {3618, 10264}, {3751, 11699}, {3851, 32274}, {5020, 41670}, {5050, 5622}, {5070, 6698}, {5085, 15041}, {5181, 44492}, {5422, 9140}, {5480, 12902}, {5899, 9019}, {5972, 17811}, {6642, 15132}, {6759, 8538}, {7394, 14683}, {7506, 38851}, {7517, 40949}, {7529, 32246}, {7576, 12383}, {7592, 14094}, {7669, 18114}, {7728, 19149}, {7731, 19121}, {8537, 32248}, {8548, 11441}, {9517, 17994}, {9654, 32243}, {9669, 32297}, {9821, 23133}, {9909, 16165}, {9956, 32261}, {9967, 10117}, {9974, 32292}, {9975, 32291}, {10255, 34118}, {10510, 37924}, {10516, 15046}, {10602, 41743}, {10628, 19131}, {10661, 45075}, {10662, 45074}, {10665, 45050}, {10666, 45049}, {10706, 11456}, {11416, 14157}, {11477, 12584}, {11479, 15738}, {11579, 12308}, {11649, 19596}, {11818, 14853}, {12006, 43894}, {12134, 14627}, {12164, 35603}, {12165, 12825}, {12167, 12596}, {12228, 12412}, {12294, 12302}, {12645, 32298}, {13248, 39879}, {13289, 19132}, {13392, 44213}, {13417, 44078}, {13630, 43815}, {13754, 44102}, {14530, 38885}, {14561, 38724}, {14687, 35357}, {15018, 38079}, {15045, 43578}, {15047, 25555}, {15068, 41614}, {15069, 41731}, {15091, 41583}, {15118, 36753}, {15128, 26944}, {15131, 30771}, {15133, 32239}, {15136, 37933}, {15137, 37923}, {15342, 18348}, {15805, 20397}, {16003, 32300}, {16105, 39568}, {16176, 34507}, {16776, 21308}, {17702, 18494}, {17814, 32275}, {18386, 39588}, {18436, 44470}, {18531, 41719}, {18564, 38790}, {19128, 37954}, {19129, 41593}, {19154, 38898}, {19361, 32140}, {19457, 21637}, {20301, 25335}, {20417, 37514}, {20771, 44456}, {21970, 32227}, {23236, 36749}, {30714, 36747}, {31479, 32307}, {31860, 40291}, {32139, 41737}, {32249, 43598}, {32257, 38795}, {32284, 43844}, {35463, 41336}, {35876, 42265}, {35877, 42262}, {43704, 43726}

X(45016) = midpoint of X(i) and X(j) for these {i, j}: {399, 39562}, {9970, 15462}, {10516, 25331}, {10540, 18449}, {11416, 14157}, {19140, 34155}
X(45016) = reflection of X(i) in X(j) for these (i, j): (3, 15462), (6, 34155), (2070, 18374), (5621, 182), (10620, 5621), (15041, 5085), (15462, 6593), (34155, 25556), (38724, 14561), (39562, 6)
X(45016) = reflection of X(115) in the line X(690)X(44817)
X(45016) = perspector of the circumconic {{A, B, C, X(476), X(32697)}}
X(45016) = inverse of X(5477) in 2nd Lemoine (or cosine) circle
X(45016) = inverse of X(14675) in 1st Lemoine circle
X(45016) = intersection, other than A, B, C, of circumconics {{A, B, C, X(115), X(35372)}} and {{A, B, C, X(265), X(2987)}}
X(45016) = X(5050)-of-anti-orthocentroidal triangle
X(45016) = X(15462)-of-X3-ABC reflections triangle
X(45016) = X(45043)-of-anti-Honsberger-triangle if ABC is acute
X(45016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 11061, 32306), (6, 5476, 15038), (6, 19140, 399), (25, 110, 45082), (110, 1112, 12310), (110, 12824, 25), (110, 20772, 8780), (895, 5609, 32254), (1511, 10752, 33878), (5655, 18445, 399), (6053, 12227, 17838), (6053, 17838, 399), (6593, 9970, 3), (11482, 32254, 895), (12168, 19118, 19138), (15140, 32233, 44469), (15303, 19140, 18445), (17814, 32276, 32275), (19140, 25556, 6), (32233, 32271, 382)

X(45017) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND MOSES-STEINER OSCULATORY

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

X(45017) lies on these lines: {2, 7748}, {3, 10847}, {6, 3552}, {20, 6033}, {99, 5206}, {187, 20105}, {194, 32456}, {376, 7929}, {384, 14535}, {550, 7897}, {2482, 7912}, {2996, 20094}, {3522, 5921}, {3534, 7947}, {3616, 24248}, {5059, 18860}, {5237, 22689}, {5238, 22687}, {6337, 7900}, {6390, 33268}, {6453, 22623}, {6454, 22594}, {6680, 32480}, {7492, 15652}, {7754, 13586}, {7766, 33235}, {7779, 33254}, {7802, 35022}, {7815, 33022}, {7871, 14976}, {7891, 33264}, {7906, 8598}, {7928, 44541}, {7939, 15696}, {9939, 36521}, {11160, 33208}, {16508, 35287}, {16509, 33274}, {18501, 35951}, {18840, 33008}, {19693, 31400}, {20088, 35927}, {20190, 31958}, {32831, 33214}, {33252, 37668}

X(45017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (99, 33014, 20081), (6337, 33265, 7900)

X(45018) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND MOSES-STEINER REFLECTION

Barycentrics    5*a^6-4*(b^2+c^2)*a^4+(b^4-b^2*c^2+c^4)*a^2-(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2) : :
X(45018) = 4*X(6)-3*X(671) = 2*X(6)-3*X(8593) = 5*X(6)-6*X(8787) = X(6)-3*X(10488) = 2*X(69)-3*X(99) = 3*X(99)-4*X(14928) = 4*X(141)-3*X(11161) = 8*X(141)-9*X(41134) = 5*X(671)-8*X(8787) = X(671)-4*X(10488) = 6*X(2482)-5*X(3620) = 8*X(3589)-9*X(5182) = 4*X(3589)-3*X(11646) = 16*X(3589)-15*X(14061) = 3*X(5182)-2*X(11646) = 6*X(5182)-5*X(14061) = 5*X(8593)-4*X(8787) = 2*X(8787)-5*X(10488) = 2*X(11161)-3*X(41134) = 4*X(11646)-5*X(14061)

X(45018) lies on these lines: {6, 598}, {69, 74}, {98, 35705}, {110, 30786}, {141, 11161}, {147, 3424}, {148, 5477}, {183, 9774}, {193, 543}, {302, 9760}, {303, 9762}, {316, 1503}, {575, 15031}, {2482, 3620}, {2782, 39899}, {3448, 7664}, {3564, 23235}, {3589, 5182}, {3618, 9166}, {3629, 10754}, {3763, 5026}, {3818, 12177}, {4563, 14360}, {5092, 19905}, {5921, 14981}, {5939, 6054}, {5969, 6144}, {6776, 11185}, {7771, 43273}, {7782, 15069}, {7844, 39141}, {7931, 8289}, {8588, 19911}, {8591, 20080}, {9140, 35356}, {9877, 17008}, {9884, 16496}, {10553, 24981}, {10723, 10753}, {11160, 15300}, {11606, 33686}, {11645, 39099}, {12117, 33878}, {14645, 20094}, {18358, 23234}, {20774, 32002}, {21395, 35707}

X(45018) = reflection of X(i) in X(j) for these (i, j): (69, 14928), (148, 5477), (671, 8593), (5921, 14981), (8593, 10488), (10723, 10753), (11160, 15300), (38664, 6776)
X(45018) = reflection of X(99) in the line X(690)X(11061)
X(45018) = (6th anti-Brocard)-anticomplement-of-X(11646)
X(45018) = (anti-1st Parry)-anticomplement-of-X(69)
X(45018) = X(1351)-of-anti-McCay triangle
X(45018) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (69, 14928, 99), (5182, 11646, 14061)

X(45019) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ANTI-ORTHOCENTROIDAL

Barycentrics    a^2*(a^14-2*(b^2+c^2)*a^12-3*(b^4-5*b^2*c^2+c^4)*a^10+(b^2+c^2)*(10*b^4-13*b^2*c^2+10*c^4)*a^8-(5*b^8+5*c^8+2*b^2*c^2*(28*b^4-39*b^2*c^2+28*c^4))*a^6-6*(b^2+c^2)*(b^8+c^8-4*b^2*c^2*(3*b^4-5*b^2*c^2+3*c^4))*a^4+(7*b^8+7*c^8-b^2*c^2*(b^4+60*b^2*c^2+c^4))*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)^3*(-2*b^4-11*b^2*c^2-2*c^4)) : :
X(45019) = 2*X(8717)-3*X(15035) = X(10293)-3*X(14643) = X(11820)-3*X(32609) = 2*X(12041)-3*X(32620)

X(45019) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2, 74}, {6, 5663}, {30, 5648}, {110, 841}, {154, 1511}, {399, 3167}, {974, 5644}, {1514, 3581}, {1539, 40909}, {2777, 3098}, {2780, 8552}, {2781, 44754}, {2914, 13596}, {5544, 10620}, {5609, 37497}, {5643, 9730}, {5646, 12041}, {5653, 30230}, {5654, 15063}, {5655, 11064}, {5656, 16534}, {5888, 15055}, {6000, 19140}, {6030, 15051}, {7706, 10545}, {7712, 8717}, {9716, 13352}, {10293, 14643}, {10721, 12380}, {10752, 13754}, {11438, 38791}, {11820, 32609}, {12379, 15122}, {14924, 37475}, {17702, 41737}, {20126, 37648}, {20127, 35254}

X(45019) = midpoint of X(399) and X(3426)
X(45019) = reflection of X(i) in X(j) for these (i, j): (74, 4550), (4846, 113), (11579, 31861), (20127, 35254), (34802, 11472), (35237, 1511), (40909, 1539)
X(45019) = inverse of X(10564) in Stammler hyperbola
X(45019) = (anti-orthocentroidal)-isogonal conjugate-of-X(11456)
X(45019) = X(4846)-of-anti-orthocentroidal triangle

X(45020) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st ANTI-ORTHOSYMMEDIAL

Barycentrics    (a^22-2*(b^2+c^2)*a^20+(b^4+8*b^2*c^2+c^4)*a^18-2*(b^6+c^6)*a^16+(2*b^8+2*c^8+(3*b^4+11*b^2*c^2+3*c^4)*b^2*c^2)*a^14+(b^2+c^2)*(4*b^8+4*c^8-(19*b^4-34*b^2*c^2+19*c^4)*b^2*c^2)*a^12-(6*b^12+6*c^12+(21*b^8+21*c^8+(31*b^4-20*b^2*c^2+31*c^4)*b^2*c^2)*b^2*c^2)*a^10+(b^2+c^2)*(4*b^12+4*c^12+(17*b^8+17*c^8-10*(3*b^4-5*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*a^8-(b^2-c^2)^2*(3*b^12+3*c^12-(11*b^8+11*c^8+2*(17*b^4+14*b^2*c^2+17*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)*(2*b^6+5*b^4*c^2+c^6)*(b^6+5*b^2*c^4+2*c^6)*a^4+(5*b^12+5*c^12+(13*b^8+13*c^8+4*(8*b^4+11*b^2*c^2+8*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^8-c^8)*(b^2-c^2)^3*(2*b^8+2*c^8+7*(b^2+c^2)^2*b^2*c^2))*a^2 : :

X(45020) lies on these lines: {132, 45063}, {141, 1297}, {427, 14983}, {11610, 41413}

X(45020) = reflection of X(45063) in X(132)
X(45020) = X(45063)-of-1st anti-orthosymmedial triangle

X(45021) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st ANTI-SHARYGIN

Barycentrics    1/a^2*(a^18-5*(b^2+c^2)*a^16+(8*b^4+13*b^2*c^2+8*c^4)*a^14-2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^12-(b^2-c^2)^2*(4*b^4+3*b^2*c^2+4*c^4)*a^10-2*(b^4-c^4)*(b^2-c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^8+(20*b^8+20*c^8+(7*b^4+10*b^2*c^2+7*c^4)*b^2*c^2)*(b^2-c^2)^2*a^6-18*(b^8-c^8)*(b^2-c^2)^3*a^4+(7*b^4+9*b^2*c^2+7*c^4)*(b^2-c^2)^6*a^2-(b^2+c^2)*(b^2-c^2)^8)*(a^2-b^2+c^2)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2)*(a^2+b^2-c^2)*(a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2) : :

X(45021) lies on these lines: {195, 275}, {6750, 8795}, {8884, 32345}, {14130, 19176}

X(45022) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ANTI-TANGENTIAL-MIDARC

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

X(45022) lies on these lines: {1, 15622}, {4, 35097}, {11, 65}, {46, 23383}, {57, 33810}, {109, 31849}, {113, 13999}, {225, 1830}, {273, 4566}, {517, 1457}, {900, 11570}, {942, 2654}, {1111, 24471}, {1439, 2823}, {1478, 12586}, {1845, 8677}, {3028, 3326}, {3216, 4674}, {3812, 25493}, {5902, 43915}, {22272, 22277}

X(45022) = reflection of X(i) in X(j) for these (i, j): (7004, 942), (24028, 45122)
X(45022) = trilinear product X(1769)*X(7451)
X(45022) = crosssum of X(55) and X(2250)
X(45022) = X(273)-Ceva conjugate of-X(1465)
X(45022) = (intouch)-isogonal conjugate-of-X(1464)

X(45023) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd ANTI-TRI-SQUARES

Barycentrics    6*a^6-5*(b^2+c^2)*a^4+2*(b^4+c^4)*a^2-2*(3*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S-3*(b^2+c^2)*(b^2-c^2)^2 : :
X(45023) = 5*X(485)-4*X(13921) = X(486)-3*X(1327) = 5*X(486)-6*X(13932) = 4*X(486)-3*X(22616) = X(487)+3*X(33456) = 5*X(1327)-2*X(13932) = 4*X(1327)-X(22616) = 8*X(13932)-5*X(22616)

X(45023) lies on these lines: {4, 22617}, {30, 22645}, {371, 13674}, {381, 486}, {485, 13921}, {487, 33456}, {642, 11147}, {3564, 22646}, {5013, 45024}, {6290, 6560}, {6300, 36400}, {6301, 36396}, {6460, 45079}, {6561, 13749}, {6564, 12256}, {8982, 9758}, {12123, 42276}, {12221, 23253}, {12601, 42284}, {12602, 22592}, {13711, 43791}, {13713, 42272}, {22485, 32419}, {23249, 35830}, {42268, 44648}

X(45023) = midpoint of X(22592) and X(22644)
X(45023) = (4th anti-tri-squares)-complement-of-X(486)

X(45024) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 4th ANTI-TRI-SQUARES

Barycentrics    6*a^6-5*(b^2+c^2)*a^4+2*(b^4+c^4)*a^2+2*(3*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S-3*(b^2+c^2)*(b^2-c^2)^2 : :
X(45024) = X(485)-3*X(1328) = 5*X(485)-6*X(13850) = 4*X(485)-3*X(22645) = 5*X(486)-4*X(13880) = X(488)+3*X(33457) = 5*X(1328)-2*X(13850) = 4*X(1328)-X(22645) = 8*X(13850)-5*X(22645)

X(45024) lies on these lines: {4, 22646}, {30, 22616}, {372, 13794}, {381, 485}, {486, 13880}, {488, 33457}, {641, 11147}, {3564, 22617}, {5013, 45023}, {6289, 6561}, {6304, 36401}, {6305, 36397}, {6459, 45078}, {6560, 13748}, {6565, 12257}, {9757, 26441}, {12124, 42275}, {12222, 23263}, {12601, 22591}, {12602, 42283}, {13834, 43792}, {13836, 42271}, {22484, 32421}, {23259, 35831}, {42269, 44647}

X(45024) = midpoint of X(22591) and X(22615)
X(45024) = (3rd anti-tri-squares)-complement-of-X(485)

X(45025) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND AAOA

Barycentrics    (4*a^18-14*(b^2+c^2)*a^16+(8*b^4+45*b^2*c^2+8*c^4)*a^14+8*(b^2+c^2)*(3*b^4-8*b^2*c^2+3*c^4)*a^12-(32*b^8+32*c^8+(22*b^4-85*b^2*c^2+22*c^4)*b^2*c^2)*a^10-(b^2+c^2)*(4*b^8+4*c^8-(73*b^4-131*b^2*c^2+73*c^4)*b^2*c^2)*a^8+(24*b^12+24*c^12-(47*b^8+47*c^8+(27*b^4-104*b^2*c^2+27*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)*(2*b^4+7*b^2*c^2+2*c^4)*(4*b^4-7*b^2*c^2+4*c^4)*a^4-2*(b^2-c^2)^4*(2*b^8+2*c^8-(4*b^4+9*b^2*c^2+4*c^4)*b^2*c^2)*a^2+(b^2-c^2)^6*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4))*(-a^2+b^2+c^2)*a^2 : :

X(45025) lies on these lines: {3, 43720}, {30, 19374}, {1351, 2931}, {1352, 1511}, {1495, 7728}, {6288, 12038}, {7575, 9970}, {7579, 17702}, {7703, 44795}, {13289, 19149}, {15132, 15331}, {32315, 45082}

X(45026) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND AOA

Barycentrics    3*a^22-7*(b^2+c^2)*a^20-(10*b^4-39*b^2*c^2+10*c^4)*a^18+2*(b^2+c^2)*(17*b^4-40*b^2*c^2+17*c^4)*a^16+2*(3*b^8+3*c^8-(41*b^4-100*b^2*c^2+41*c^4)*b^2*c^2)*a^14-2*(b^2+c^2)*(35*b^8+35*c^8-2*(65*b^4-97*b^2*c^2+65*c^4)*b^2*c^2)*a^12+4*(7*b^12+7*c^12+(3*b^8+3*c^8-2*(37*b^4-58*b^2*c^2+37*c^4)*b^2*c^2)*b^2*c^2)*a^10+4*(b^2+c^2)*(15*b^12+15*c^12-(81*b^8+81*c^8-b^2*c^2*(197*b^4-258*b^2*c^2+197*c^4))*b^2*c^2)*a^8-(b^2-c^2)^2*(49*b^12+49*c^12-(48*b^8+48*c^8+b^2*c^2*(97*b^4-224*b^2*c^2+97*c^4))*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)^3*(11*b^8+11*c^8-2*b^2*c^2*(32*b^4-49*b^2*c^2+32*c^4))*a^4+(22*b^4-27*b^2*c^2+22*c^4)*(b^2+c^2)^2*(b^2-c^2)^6*a^2-6*(b^2+c^2)^3*(b^2-c^2)^8 : :

X(45026) lies on these lines: {182, 7687}, {265, 41729}, {11562, 44084}, {13491, 16270}, {19506, 32223}, {32227, 37981}

X(45027) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND APUS

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

X(45027) lies on this line: {55, 474}

X(45028) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND AYME

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

X(45028) lies on these lines: {3, 9958}, {17749, 19547}, {19544, 34466}

X(45029) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 6th BROCARD

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

X(45029) lies on these lines: {2, 3398}, {4, 8782}, {20, 5207}, {114, 45013}, {147, 5152}, {194, 1352}, {315, 6194}, {2896, 36998}, {3399, 9996}, {3767, 10336}, {6033, 40253}, {7785, 9753}, {7893, 37446}, {9863, 10998}, {10345, 24206}, {20065, 35379}

X(45029) = reflection of X(45013) in X(114)
X(45029) = anticomplement of X(3406)
X(45029) = anticomplementary conjugate of X(12251)
X(45029) = X(1)-anticomplementary conjugate of-X(12251)
X(45029) = X(45013)-of-6th Brocard triangle

X(45030) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 7th BROCARD

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

X(45030) lies on these lines: {3, 230}, {6, 2967}, {24, 39646}, {25, 1503}, {26, 14880}, {98, 157}, {185, 44099}, {523, 878}, {1975, 17928}, {2782, 2936}, {3148, 9756}, {5085, 10329}, {7503, 7851}, {10311, 19161}, {13860, 34845}, {19165, 30549}, {34117, 44089}

X(45030) = isogonal conjugate of the cyclocevian conjugate of X(40815)
X(45030) = (tangential)-isogonal conjugate-of-X(1350)
X(45030) = {X(9755), X(37930)}-harmonic conjugate of X(3425)

X(45031) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 9th BROCARD

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

X(45031) lies on these lines: {4, 32}, {20, 6394}, {64, 290}, {287, 297}, {458, 9756}, {1976, 19124}, {3424, 37174}, {6759, 17974}, {32696, 44228}

X(45031) = barycentric product X(i)*X(j) for these {i, j}: {287, 10002}, {1350, 16081}, {1529, 9476}
X(45031) = barycentric quotient X(i)/X(j) for these (i, j): (98, 42287), (1350, 36212), (1529, 15595)
X(45031) = trilinear product X(i)*X(j) for these {i, j}: {293, 10002}, {1350, 36120}
X(45031) = trilinear quotient X(1821)/X(42287)
X(45031) = inverse of X(6531) in Kiepert circumhyperbola
X(45031) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(6330)}} and {{A, B, C, X(32), X(64)}}
X(45031) = X(98)-reciprocal conjugate of-X(42287)
X(45031) = {X(4), X(98)}-harmonic conjugate of X(6531)

X(45032) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd CONWAY

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

X(45032) lies on the Feuerbach circumhyperbola and these lines: {1, 2049}, {4, 12435}, {7, 10447}, {9, 3714}, {10, 941}, {21, 11679}, {40, 43739}, {80, 5814}, {84, 1764}, {104, 10882}, {952, 10890}, {2321, 2335}, {3296, 11021}, {3577, 11521}, {3679, 43073}, {5558, 10453}, {7091, 35613}, {10435, 10441}, {10456, 35629}, {16343, 18229}

X(45032) = barycentric quotient X(9)/X(19767)
X(45032) = trilinear quotient X(8)/X(19767)
X(45032) = touchpoint of the line {10480, 45032} and Feuerbach circumhyperbola
X(45032) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(3), X(12435)}}
X(45032) = X(1)-Dao conjugate of-X(19767)
X(45032) = X(56)-isoconjugate-of-X(19767)
X(45032) = X(9)-reciprocal conjugate of-X(19767)

X(45033) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 4th CONWAY

Barycentrics    a*((3*b+4*c)*(4*b+3*c)*a^7+3*(9*b^2+20*b*c+9*c^2)*a^5*b*c+12*(b+c)*(2*b^2+3*b*c+2*c^2)*a^6-8*(b+c)*(b^2+c^2)*(3*b^2+4*b*c+3*c^2)*a^4-4*(b+c)*(9*b^4+9*c^4+(19*b^2+30*b*c+19*c^2)*b*c)*a^2*b*c-4*(b^2+c^2)*(b+c)^3*b^2*c^2-(12*b^6+12*c^6+(81*b^4+81*c^4+2*(88*b^2+103*b*c+88*c^2)*b*c)*b*c)*a^3-(3*b^2+2*b*c+3*c^2)*(b^2+8*b*c+c^2)*(b+c)^2*a*b*c) : :

X(45033) lies on these lines: {}

X(45034) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd EHRMANN

Barycentrics    a^2*(a^10-3*(b^2+c^2)*a^8+(b^2+2*c^2)*(2*b^2+c^2)*a^6+2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^4-(3*b^4-5*b^2*c^2+3*c^4)*(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4)) : :
X(45034) = 4*X(576)+X(33541) = 2*X(1176)-3*X(5050) = 3*X(1351)+2*X(15062) = 5*X(12017)-2*X(41464) = 2*X(15321)+X(39899)

X(45034) lies on these lines: {3, 9019}, {4, 45016}, {6, 382}, {51, 37928}, {143, 5622}, {155, 18440}, {182, 2916}, {195, 542}, {381, 44469}, {511, 12307}, {546, 22151}, {575, 15038}, {576, 16010}, {1176, 3527}, {1350, 32600}, {1351, 1593}, {1657, 44480}, {1843, 12038}, {2393, 37472}, {3564, 11271}, {3830, 34117}, {5422, 6030}, {6287, 23133}, {6776, 19362}, {7530, 27085}, {7689, 19124}, {8549, 34780}, {8550, 14627}, {9704, 15581}, {9715, 12017}, {9977, 19924}, {10095, 43815}, {10249, 37490}, {10982, 14848}, {11405, 22948}, {11482, 39522}, {12280, 14984}, {13621, 15462}, {15033, 15074}, {15106, 31857}, {15135, 31133}, {15317, 15321}, {18125, 32306}, {18442, 44492}, {18488, 36747}, {18560, 21850}, {18562, 31670}, {19128, 43823}, {19139, 45014}, {21308, 43811}, {22115, 43130}, {34545, 37900}, {34726, 36752}, {43273, 44494}

X(45034) = reflection of X(i) in X(j) for these (i, j): (1350, 32600), (2916, 182), (32306, 18125)

X(45035) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd EULER

Barycentrics    (b+c)*a^3+(2*b-c)*(b-2*c)*a^2-(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(45035) = 3*X(1)-2*X(39781) = 2*X(10)-3*X(7705) = X(145)-3*X(1392) = 3*X(5560)+2*X(39781)

X(45035) lies on these lines: {1, 381}, {4, 11715}, {5, 45081}, {10, 3877}, {11, 5903}, {35, 474}, {36, 4333}, {56, 16118}, {79, 10072}, {80, 145}, {354, 31828}, {496, 3649}, {497, 5443}, {499, 5442}, {515, 7704}, {517, 15079}, {946, 5902}, {1058, 37731}, {1125, 41862}, {1319, 18514}, {1479, 2475}, {1482, 37718}, {1656, 37563}, {1699, 5563}, {2802, 5154}, {3338, 11372}, {3582, 12699}, {3583, 11376}, {3585, 11373}, {3586, 24926}, {3632, 5087}, {3635, 37706}, {3636, 10129}, {3646, 5119}, {3746, 8227}, {3817, 37719}, {3884, 17057}, {3898, 5141}, {4188, 32557}, {4294, 5444}, {4668, 11525}, {4857, 5886}, {5426, 20288}, {5445, 10589}, {5542, 10394}, {5559, 5818}, {5561, 31776}, {5603, 37702}, {5691, 22835}, {5692, 24387}, {5904, 11813}, {6958, 14217}, {7288, 15228}, {7580, 34890}, {7951, 12053}, {8236, 37701}, {9581, 11009}, {9612, 37602}, {9668, 37616}, {9671, 10246}, {10483, 44675}, {10593, 18395}, {10624, 19878}, {10707, 22836}, {10785, 34789}, {11681, 21630}, {12260, 37692}, {13463, 17533}, {14804, 22753}, {15174, 15950}, {15803, 31671}, {16155, 26363}, {18513, 24928}, {19861, 31159}, {19877, 30305}, {22837, 37375}, {23800, 24457}, {25055, 34706}, {32558, 37256}, {37722, 38034}

X(45035) = midpoint of X(1) and X(5560)
X(45035) = inverse of X(5903) in Feuerbach circumhyperbola
X(45035) = (anti-Aquila)-isogonal conjugate-of-X(21842)
X(45035) = X(5560)-of-anti-Aquila triangle
X(45035) = X(11999)-of-Hutson intouch triangle
X(45035) = X(16880)-of-2nd circumperp triangle
X(45035) = X(45014)-of-3rd Euler triangle
X(45035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (496, 18393, 18398), (499, 6361, 5442), (946, 37720, 5902), (1479, 3616, 5441), (1479, 37735, 37525), (3582, 12699, 37524), (3583, 11376, 21842), (3616, 5441, 37525), (4857, 5886, 37571), (5441, 37735, 3616), (5442, 6361, 37572), (7741, 30384, 5697), (9614, 23708, 35), (11238, 18493, 1)

X(45036) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND EXCENTERS-MIDPOINTS

Barycentrics    a*(3*a-b-c)*(3*a^2-3*b^2+2*b*c-3*c^2) : :
X(45036) = 5*X(3616)-3*X(18220)

X(45036) lies on these lines: {1, 4004}, {2, 7319}, {3, 7971}, {9, 3207}, {10, 631}, {36, 11517}, {40, 214}, {55, 22754}, {56, 3243}, {57, 34195}, {78, 6594}, {100, 3680}, {119, 6922}, {142, 390}, {145, 1420}, {200, 37605}, {404, 13384}, {442, 3586}, {936, 13624}, {997, 3647}, {999, 8000}, {1145, 3632}, {1279, 7963}, {1319, 2136}, {1376, 7990}, {1385, 1706}, {1394, 14812}, {1449, 2092}, {1519, 6934}, {2093, 19537}, {2646, 5437}, {3086, 34701}, {3247, 34261}, {3340, 4188}, {3486, 31190}, {3577, 6924}, {3635, 11041}, {3749, 15839}, {3922, 34471}, {3928, 3962}, {3929, 5303}, {4018, 15803}, {4297, 30827}, {4864, 33804}, {5126, 6765}, {5128, 13587}, {5265, 12437}, {5289, 16192}, {5440, 6762}, {5534, 18857}, {5698, 43182}, {5727, 6921}, {5836, 30392}, {5837, 15717}, {6260, 6987}, {6830, 18492}, {7080, 34716}, {7280, 35204}, {7288, 12625}, {7290, 41886}, {8583, 15346}, {9614, 34123}, {9708, 31666}, {9841, 37837}, {10470, 10472}, {10543, 31249}, {10882, 40600}, {11506, 38289}, {11682, 37307}, {12639, 13100}, {17057, 17619}, {17502, 31424}, {17614, 30282}, {18221, 20057}, {21214, 35338}, {30144, 35242}

X(45036) = complement of X(7319)
X(45036) = complementary conjugate of the complement of X(5204)
X(45036) = barycentric product X(i)*X(j) for these {i, j}: {145, 3928}, {1743, 21296}
X(45036) = barycentric quotient X(1743)/X(7319)
X(45036) = trilinear product X(i)*X(j) for these {i, j}: {145, 5204}, {1743, 3928}
X(45036) = trilinear quotient X(i)/X(j) for these (i, j): (145, 7319), (1743, 41441)
X(45036) = center of the circumconic {{A, B, C, X(100), X(43290)}}
X(45036) = crosspoint of X(2) and X(21296)
X(45036) = X(2)-Ceva conjugate of-X(1743)
X(45036) = X(i)-complementary conjugate of-X(j) for these (i, j): (31, 1743), (41, 27541)
X(45036) = (medial)-isotomic conjugate-of-X(1743)
X(45036) = X(7319)-of-medial triangle
X(45036) = X(16879)-of-Wasat triangle
X(45036) = X(27082)-of-1st circumperp triangle
X(45036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1420, 4855, 3158), (4855, 4881, 1420)

X(45037) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd EXCOSINE

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(3*a^12-10*(b^2+c^2)*a^10+(13*b^4-10*b^2*c^2+13*c^4)*a^8-12*(b^4-c^4)*(b^2-c^2)*a^6+(13*b^4+54*b^2*c^2+13*c^4)*(b^2-c^2)^2*a^4-2*(b^4-c^4)*(b^2-c^2)*(5*b^4+6*b^2*c^2+5*c^4)*a^2+3*(b^2-c^2)^6) : :
X(45037) = 5*X(4)-4*X(33531)

X(45037) lies on these lines: {4, 20208}, {20, 6525}, {30, 3183}, {64, 3146}, {382, 33893}, {1033, 39568}, {3543, 35711}, {7396, 18288}, {9541, 22838}, {13155, 15005}, {17037, 44762}

X(45037) = reflection of X(i) in X(j) for these (i, j): (20, 34286), (33893, 382)

X(45038) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND EXTANGENTS

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

X(45038) lies on these lines: {1, 8021}, {10, 12}, {40, 15624}, {92, 3868}, {517, 18673}, {942, 2260}, {1071, 8680}, {1243, 43708}, {1762, 18180}, {1859, 14054}, {1871, 15762}, {3191, 15443}, {5496, 37080}, {6237, 37826}, {10902, 42443}, {26934, 37536}

X(45038) = trilinear product X(i)*X(j) for these {i, j}: {442, 580}, {942, 3191}
X(45038) = trilinear quotient X(580)/X(1175)
X(45038) = intersection, other than A, B, C, of circumconics {{A, B, C, X(10), X(3191)}} and {{A, B, C, X(12), X(2294)}}
X(45038) = X(92)-Ceva conjugate of-X(40937)

X(45039) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd EXTOUCH

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

X(45039) lies on these lines: {1, 24389}, {2, 224}, {4, 960}, {8, 226}, {9, 20}, {10, 1490}, {72, 2550}, {149, 5175}, {329, 2475}, {405, 4305}, {442, 18391}, {443, 5784}, {517, 18255}, {519, 12654}, {936, 6847}, {938, 25525}, {954, 3189}, {997, 6846}, {1376, 3651}, {1708, 6904}, {1788, 37240}, {2886, 3487}, {3036, 6937}, {3486, 37224}, {4301, 5715}, {5044, 6851}, {5436, 5809}, {5705, 5768}, {5745, 9799}, {5746, 9534}, {5777, 6259}, {5811, 18254}, {6895, 18228}, {6916, 12664}, {6987, 17647}, {8232, 20007}, {18446, 19843}

X(45039) = X(45015)-of-2nd extouch triangle

X(45040) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 5th EXTOUCH

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

X(45040) lies on these lines: {20, 21147}, {28, 3474}, {46, 3089}, {57, 946}, {109, 962}, {14450, 14648}

X(45041) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND FEUERBACH

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

X(45041) lies on this line: {3634, 3925}

X(45042) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd HATZIPOLAKIS

Barycentrics    (a^14-2*(b^2+c^2)*a^12-(3*b^4-19*b^2*c^2+3*c^4)*a^10+(b^2+c^2)*(10*b^4-27*b^2*c^2+10*c^4)*a^8-5*(b^2-c^2)^2*(b^4+12*b^2*c^2+c^4)*a^6-2*(b^2+c^2)*(3*b^8+3*c^8-(49*b^4-116*b^2*c^2+49*c^4)*b^2*c^2)*a^4+(7*b^8+7*c^8-(35*b^4+24*b^2*c^2+35*c^4)*b^2*c^2)*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)^3*(-2*b^4+b^2*c^2-2*c^4))*a^2 : :

X(45042) lies on these lines: {6, 9694}, {20, 15053}, {185, 15052}, {5890, 12164}, {5894, 37648}, {6804, 7689}, {12022, 43597}, {15012, 15801}, {22467, 40673}

X(45043) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND HONSBERGER

Barycentrics    (a^5-(b+c)*a^4+b*c*a^3-(b^2+3*b*c+c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)^3)*(-a+b+c) : :
X(45043) = X(7)+2*X(80) = X(8)+2*X(3254) = 4*X(11)-X(390) = 2*X(11)+X(20119) = 2*X(100)-5*X(40333) = 4*X(142)-X(6224) = X(149)+2*X(2550) = X(390)+2*X(20119) = X(1156)-4*X(12019) = 4*X(1387)-X(12730) = 2*X(5542)+X(9897) = 3*X(5686)-4*X(38211) = X(5759)-4*X(12619) = 2*X(5805)+X(12247) = 4*X(6594)-7*X(9780) = 4*X(6797)-X(7672) = 3*X(11038)-2*X(14151) = 3*X(11038)-4*X(38055) = 4*X(12736)-X(12755) = 3*X(38092)-4*X(38202)

X(45043) lies on these lines: {1, 37771}, {2, 11}, {3, 30312}, {4, 653}, {5, 8543}, {7, 80}, {8, 3254}, {9, 11604}, {10, 2894}, {30, 30295}, {33, 33131}, {46, 37433}, {65, 6894}, {104, 8732}, {119, 6843}, {142, 6224}, {377, 10427}, {381, 30311}, {388, 18221}, {443, 10609}, {484, 516}, {496, 6946}, {515, 18450}, {518, 5176}, {527, 5080}, {950, 5528}, {952, 1056}, {954, 6829}, {997, 21630}, {1006, 7676}, {1320, 6601}, {1387, 6854}, {1445, 10265}, {1479, 5445}, {1484, 6911}, {1738, 3100}, {1788, 6895}, {1837, 2475}, {1864, 20292}, {1936, 33139}, {2310, 24715}, {2346, 6881}, {2478, 18231}, {2551, 6068}, {2607, 14543}, {3475, 41701}, {3487, 12738}, {3585, 30424}, {3600, 38669}, {3601, 26060}, {4312, 18513}, {4511, 5853}, {4514, 28930}, {5046, 5698}, {5086, 5784}, {5226, 5660}, {5261, 37725}, {5265, 20418}, {5270, 43180}, {5435, 11219}, {5542, 9897}, {5587, 8545}, {5657, 36976}, {5686, 5856}, {5691, 8544}, {5722, 7671}, {5727, 6173}, {5759, 12619}, {5766, 6594}, {5775, 6172}, {5805, 6797}, {5829, 21933}, {5840, 6987}, {5851, 13273}, {5881, 30318}, {6067, 22560}, {6246, 36991}, {6326, 21617}, {6702, 18230}, {6827, 10738}, {6830, 7678}, {6858, 38752}, {6883, 38121}, {6884, 11507}, {6901, 37730}, {6905, 7677}, {6947, 38131}, {6957, 38159}, {6963, 9669}, {7069, 33100}, {7080, 34894}, {7675, 38052}, {8236, 16173}, {8257, 9581}, {9778, 41166}, {9802, 15558}, {9817, 33134}, {10058, 37106}, {10090, 43161}, {10572, 37163}, {10578, 41553}, {10826, 13729}, {10883, 37541}, {10950, 25557}, {11041, 38073}, {11526, 38036}, {12433, 15901}, {12560, 21635}, {12747, 31657}, {12773, 28452}, {13243, 24465}, {13384, 38093}, {14986, 37726}, {15015, 38204}, {15254, 17606}, {21180, 30574}, {22760, 37256}, {24430, 33102}, {32558, 38316}, {32850, 37788}, {34578, 34930}, {37797, 44425}, {38037, 39692}, {38454, 40663}, {41798, 43960}

X(45043) = reflection of X(i) in X(j) for these (i, j): (8236, 16173), (14151, 38055), (15015, 38204), (18450, 30379), (37787, 1737)
X(45043) = inverse of X(149) in Feuerbach circumhyperbola
X(45043) = intersection, other than A, B, C, of circumconics {{A, B, C, X(80), X(28071)}} and {{A, B, C, X(105), X(3254)}}
X(45043) = X(19128)-of-Fuhrmann triangle
X(45043) = X(45016)-of-Honsberger triangle
X(45043) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 9803, 12755), (80, 12736, 9803), (1837, 5880, 10394), (14151, 38055, 11038), (40565, 40566, 149)

X(45044) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st HYACINTH

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

X(45044) lies on these lines: {4, 11536}, {49, 24572}, {54, 3448}, {68, 184}, {2918, 32358}, {3146, 12289}, {5889, 12254}, {6776, 7689}, {12161, 17824}, {12902, 19362}, {19467, 34783}, {36253, 43838}

X(45044) = reflection of X(4) in X(11536)
X(45044) = X(11536)-of-anti-Euler triangle

X(45045) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd HYACINTH

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

X(45045) lies on the Stammler hyperbola and these lines: {2, 15438}, {3, 12241}, {6, 1092}, {24, 159}, {25, 185}, {51, 45010}, {125, 1593}, {155, 389}, {195, 11432}, {399, 7506}, {578, 15805}, {631, 31521}, {1192, 11414}, {1598, 5878}, {1619, 18909}, {2916, 9715}, {2917, 3515}, {2918, 14070}, {3167, 22955}, {3527, 11746}, {3567, 15141}, {5020, 9815}, {5198, 34563}, {5890, 22535}, {6240, 18948}, {6644, 9937}, {7387, 11438}, {7393, 32348}, {7395, 26958}, {7503, 37643}, {9826, 12161}, {9914, 10605}, {11425, 43650}, {11426, 15047}, {11433, 17928}, {12085, 18390}, {12421, 18951}, {13730, 33810}, {17814, 40917}, {17821, 19459}, {19357, 32621}, {23361, 37310}, {37198, 37487}

X(45045) = complement of X(15438)
X(45045) = isogonal conjugate of the cyclocevian conjugate of X(8796)
X(45045) = inverse of X(1593) in Jerabek circumhyperbola
X(45045) = touchpoint of the line {11433, 17928} and Stammler hyperbola
X(45045) = (tangential)-isogonal conjugate-of-X(11414)
X(45045) = X(15438)-of-medial triangle

X(45046) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND INTANGENTS

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

X(45046) lies on these lines: {52, 1866}, {65, 2779}, {73, 500}, {515, 3057}, {521, 41013}, {1364, 6831}, {1439, 3664}, {1479, 12586}, {3075, 34462}, {3746, 9440}, {6238, 37826}, {7282, 17220}, {7414, 36059}, {15622, 32760}, {37468, 40944}

X(45047) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND INVERSE-IN-EXCIRCLES

Barycentrics    a*(3*a^3-3*(b+c)*a^2-(7*b^2-10*b*c+7*c^2)*a-(b+c)*(b^2-6*b*c+c^2)) : :
X(45047) = X(1)-4*X(8688) = X(6552)-3*X(25567)

X(45047) lies on these lines: {1, 474}, {3, 11506}, {36, 13737}, {56, 23511}, {57, 5666}, {88, 11682}, {106, 12629}, {165, 21214}, {978, 1400}, {988, 3731}, {997, 24167}, {1054, 7991}, {1125, 4779}, {1420, 16610}, {1722, 13462}, {2292, 8583}, {2999, 5253}, {3008, 5265}, {3616, 24175}, {3624, 24178}, {3646, 37599}, {3756, 12625}, {4220, 5272}, {4512, 28352}, {4853, 32577}, {5121, 5691}, {5930, 7288}, {6552, 25567}, {7982, 25580}, {8572, 16602}, {8686, 39123}, {9746, 16020}, {10165, 24779}, {16469, 37608}, {16667, 37607}, {24161, 28617}, {26718, 44669}, {30478, 31183}, {34039, 43048}, {36603, 44663}

X(45047) = midpoint of X(8056) and X(24174)
X(45047) = barycentric product X(57)*X(28661)
X(45047) = trilinear product X(56)*X(28661)
X(45047) = (1st circumperp)-isogonal conjugate-of-X(3913)
X(45047) = X(21)-Beth conjugate of-X(7963)
X(45047) = X(9)-Zayin conjugate of-X(1743)
X(45047) = X(3183)-of-2nd circumperp triangle
X(45047) = X(3346)-of-1st circumperp triangle
X(45047) = X(6523)-of-excentral triangle
X(45047) = X(33546)-of-hexyl triangle
X(45047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10563, 10912), (1, 11512, 8056), (978, 3361, 1743), (2136, 3445, 1), (5836, 15839, 1), (7963, 8056, 1)

X(45048) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st JENKINS

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

X(45048) lies on these lines: {10, 17601}, {333, 993}, {345, 5657}, {573, 32851}, {3596, 3977}, {3687, 17346}, {5273, 37175}, {5745, 34258}

X(45049) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st KENMOTU DIAGONALS

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

X(45049) lies on these lines: {6, 36656}, {1351, 10665}, {3093, 44413}, {10666, 45016}

X(45050) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd KENMOTU DIAGONALS

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

X(45050) lies on these lines: {6, 36655}, {1351, 10666}, {3092, 44413}, {3312, 26918}, {10665, 45016}

X(45051) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS CENTRAL

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6-22*b^2*c^2*a^4+2*(b^2+c^2)*(b^4+12*b^2*c^2+c^4)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2+S*(a^6-11*(b^2+c^2)*a^4+(11*b^4+18*b^2*c^2+11*c^4)*a^2-(b^2+c^2)*(b^4-10*b^2*c^2+c^4))) : :

X(45051) lies on these lines: {3, 1587}, {1151, 26454}

X(45052) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS(-1) CENTRAL

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6-22*b^2*c^2*a^4+2*(b^2+c^2)*(b^4+12*b^2*c^2+c^4)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2-S*(a^6-11*(b^2+c^2)*a^4+(11*b^4+18*b^2*c^2+11*c^4)*a^2-(b^2+c^2)*(b^4-10*b^2*c^2+c^4))) : :

X(45052) lies on these lines: {3, 1588}, {1152, 26461}

X(45053) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS INNER TANGENTIAL

Barycentrics    a^2*((8*a^6-280*(b^2+c^2)*a^4+8*(59*b^4+78*b^2*c^2+59*c^4)*a^2-8*(b^2+c^2)*(25*b^4-58*b^2*c^2+25*c^4))*S+103*a^8-308*(b^2+c^2)*a^6+2*(141*b^4+58*b^2*c^2+141*c^4)*a^4-4*(b^2+c^2)*(13*b^4-129*b^2*c^2+13*c^4)*a^2-(b^2-c^2)^2*(25*b^4+226*b^2*c^2+25*c^4)) : :

X(45053) lies on this line: {485, 3627}

X(45054) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS(-1) INNER TANGENTIAL

Barycentrics    a^2*(-(8*a^6-280*(b^2+c^2)*a^4+8*(59*b^4+78*b^2*c^2+59*c^4)*a^2-8*(b^2+c^2)*(25*b^4-58*b^2*c^2+25*c^4))*S+103*a^8-308*(b^2+c^2)*a^6+2*(141*b^4+58*b^2*c^2+141*c^4)*a^4-4*(b^2+c^2)*(13*b^4-129*b^2*c^2+13*c^4)*a^2-(b^2-c^2)^2*(25*b^4+226*b^2*c^2+25*c^4)) : :

X(45054) lies on this line: {486, 3627}

X(45055) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS SECONDARY CENTRAL

Barycentrics    a^2*(-4*(19*a^4-20*(b^2+c^2)*a^2+17*b^4-16*b^2*c^2+17*c^4)*S+a^6-11*(b^2+c^2)*a^4+(27*b^4-14*b^2*c^2+27*c^4)*a^2-(b^2+c^2)*(17*b^4-26*b^2*c^2+17*c^4)) : :

X(45055) lies on this line: {381, 1588}

X(45056) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS(-1) SECONDARY CENTRAL

Barycentrics    a^2*(4*(19*a^4-20*(b^2+c^2)*a^2+17*b^4-16*b^2*c^2+17*c^4)*S+a^6-11*(b^2+c^2)*a^4+(27*b^4-14*b^2*c^2+27*c^4)*a^2-(b^2+c^2)*(17*b^4-26*b^2*c^2+17*c^4)) : :

X(45056) lies on this line: {381, 1587}

X(45057) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS 1st SECONDARY TANGENTS

Barycentrics    a^2*(2*S*(2*a^6-7*(b^2+c^2)*a^4+2*(5*b^4-3*b^2*c^2+5*c^4)*a^2-(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4))+11*a^8-37*(b^2+c^2)*a^6+(51*b^4+22*b^2*c^2+51*c^4)*a^4-7*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^2*(10*b^4+4*b^2*c^2+10*c^4)) : :

X(45057) lies on this line: {485, 1132}

X(45058) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS(-1) 1st SECONDARY TANGENTS

Barycentrics    a^2*(-2*S*(2*a^6-7*(b^2+c^2)*a^4+2*(5*b^4-3*b^2*c^2+5*c^4)*a^2-(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4))+11*a^8-37*(b^2+c^2)*a^6+(51*b^4+22*b^2*c^2+51*c^4)*a^4-7*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+(b^2-c^2)^2*(10*b^4+4*b^2*c^2+10*c^4)) : :

X(45058) lies on this line: {486, 1131}

X(45059) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS 2nd SECONDARY TANGENTS

Barycentrics    a^2*(-2*(6*a^6-43*(b^2+c^2)*a^4+2*(19*b^4+25*b^2*c^2+19*c^4)*a^2-(b^2+c^2)*(b^4-32*b^2*c^2+c^4))*S+a^8-39*(b^2+c^2)*a^6+(81*b^4+506*b^2*c^2+81*c^4)*a^4-(b^2+c^2)*(49*b^4+338*b^2*c^2+49*c^4)*a^2+(b^2-c^2)^2*(6*b^4+52*b^2*c^2+6*c^4)) : :

X(45059) lies on this line: {372, 3523}

X(45060) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND LUCAS(-1) 2nd SECONDARY TANGENTS

Barycentrics    a^2*(2*(6*a^6-43*(b^2+c^2)*a^4+2*(19*b^4+25*b^2*c^2+19*c^4)*a^2-(b^2+c^2)*(b^4-32*b^2*c^2+c^4))*S+a^8-39*(b^2+c^2)*a^6+(81*b^4+506*b^2*c^2+81*c^4)*a^4-(b^2+c^2)*(49*b^4+338*b^2*c^2+49*c^4)*a^2+2*(3*b^4+26*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :

X(45060) lies on this line: {371, 3523}

X(45061) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND MANDART-EXCIRCLES

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

X(45061) lies on these lines: {1, 1473}, {37, 169}, {55, 474}, {595, 4253}, {1086, 12912}, {1621, 17526}, {3303, 3743}, {3744, 27802}, {3746, 5268}, {3749, 11365}, {5710, 43220}, {11508, 28077}, {11517, 23404}, {16486, 41455}, {37500, 37547}

X(45062) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND ORTHIC AXES

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

X(45062) lies on these lines: {3, 275}, {4, 154}, {6, 1075}, {54, 19212}, {107, 3527}, {378, 31381}, {436, 10982}, {458, 1092}, {578, 8887}, {631, 3087}, {2052, 11426}, {3526, 43462}, {3541, 41770}, {7592, 19180}, {10110, 37070}, {11424, 41372}, {11431, 41425}, {36747, 37127}

X(45062) = X(275)-Ceva conjugate of-X(3087)

X(45063) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st ORTHOSYMMEDIAL

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

X(45063) lies on these lines: {6, 12384}, {20, 32}, {132, 45020}, {7391, 11002}, {12918, 40281}

X(45063) = reflection of X(45020) in X(132)
X(45063) = anticomplement of the circumtangential-isogonal conjugate of X(44882)
X(45063) = X(45020)-of-1st orthosymmedial triangle

X(45064) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd ORTHOSYMMEDIAL

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

X(45064) lies on these lines: {25, 5012}, {373, 21419}

X(45065) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st SCHIFFLER

Barycentrics    a*(a-b+c)*(a+b-c)*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)^3)*(2*a+b+c) : :
X(45065) = 2*X(3649)-3*X(4870)

X(45065) lies on these lines: {1, 13465}, {21, 1319}, {30, 10039}, {55, 7701}, {65, 191}, {79, 22937}, {553, 1125}, {758, 11011}, {896, 11553}, {912, 3652}, {942, 1749}, {2646, 2771}, {3057, 13743}, {3065, 3746}, {3485, 31888}, {3579, 16118}, {3648, 4640}, {3651, 41695}, {4995, 41543}, {5183, 16139}, {5217, 16143}, {5221, 16857}, {5252, 15680}, {5441, 26202}, {6149, 8143}, {6681, 11263}, {6841, 16142}, {10021, 33593}, {11375, 14450}, {11684, 44663}, {13089, 14882}, {13405, 13995}, {15178, 33856}, {15694, 37524}, {16132, 37600}, {16159, 17605}, {17768, 21617}, {18244, 37731}, {18253, 24982}, {19919, 44840}, {21816, 36075}, {28443, 37605}, {35016, 41554}, {41389, 44782}

X(45065) = reflection of X(3916) in X(3647)
X(45065) = barycentric quotient X(1100)/X(10266)
X(45065) = trilinear product X(1125)*X(14882)
X(45065) = trilinear quotient X(1125)/X(10266)
X(45065) = X(1126)-isoconjugate-of-X(10266)
X(45065) = X(1100)-reciprocal conjugate of-X(10266)
X(45065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (79, 22937, 1155), (3647, 3649, 41542), (3647, 41546, 3649), (3649, 41542, 32636), (3916, 4870, 32636)

X(45066) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st SHARYGIN

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

X(45066) lies on these lines: {256, 3743}, {846, 7701}, {2292, 2475}

X(45066) = X(45021)-of-1st Sharygin triangle

X(45067) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd SHARYGIN

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

X(45067) lies on these lines: {100, 2340}, {291, 2801}, {511, 5536}, {7004, 17719}, {17664, 20358}

X(45068) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND SODDY

Barycentrics    21*a^7-23*(b+c)*a^6-(7*b^2-38*b*c+7*c^2)*a^5-27*(b^2-c^2)*(b-c)*a^4+(47*b^2+58*b*c+47*c^2)*(b-c)^2*a^3-5*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a^2+(b+3*c)*(3*b+c)*(b-c)^4*a+(b^2-c^2)*(b-c)^3*(-9*b^2-6*b*c-9*c^2) : :
X(45068) = 2*X(32056)-3*X(32079) = 3*X(32079)-4*X(32199)

X(45068) lies on these lines: {1, 10136}, {1158, 1445}, {32056, 32079}

X(45068) = reflection of X(32056) in X(32199)
X(45068) = (Soddy)-isogonal conjugate-of-X(32056)
X(45068) = X(4)-of-Soddy triangle
X(45068) = {X(32056), X(32199)}-harmonic conjugate of X(32079)

X(45069) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd INNER-SODDY

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

X(45069) lies on these lines: {176, 9778}, {946, 30380}

X(45070) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd OUTER-SODDY

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

X(45070) lies on these lines: {175, 9778}, {946, 30381}

X(45071) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND INNER-SQUARES

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

X(45071) lies on these lines: {20, 371}, {64, 485}, {2883, 36656}

X(45071) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(1270)}} and {{A, B, C, X(20), X(485)}}

X(45072) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND OUTER-SQUARES

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

X(45072) lies on these lines: {20, 372}, {64, 486}, {2883, 36655}

X(45072) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(1271)}} and {{A, B, C, X(20), X(486)}}

X(45073) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND SUBMEDIAL

Barycentrics    a^10+3*(b^2+c^2)*a^8-2*(7*b^4-10*b^2*c^2+7*c^4)*a^6+2*(b^2+c^2)*(7*b^4-26*b^2*c^2+7*c^4)*a^4-3*(b^2-c^2)^4*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

X(45073) lies on these lines: {2, 12163}, {3, 3618}, {4, 5892}, {5, 6225}, {20, 15805}, {30, 35253}, {69, 37481}, {113, 3090}, {140, 43841}, {381, 15740}, {578, 19142}, {631, 6689}, {962, 5804}, {1539, 13203}, {2888, 18916}, {3147, 43584}, {3525, 36852}, {3529, 5643}, {3537, 5446}, {3544, 7703}, {3545, 18488}, {3818, 7401}, {5067, 45014}, {5071, 18504}, {5462, 10996}, {5878, 6688}, {5890, 11487}, {5925, 17825}, {6696, 7404}, {6803, 9730}, {6815, 15045}, {7392, 40647}, {7405, 18913}, {7528, 40280}, {7728, 31371}, {8549, 34782}, {9815, 16836}, {10127, 34781}, {10128, 12315}, {11433, 12006}, {11695, 18537}, {13347, 33751}, {14786, 18931}, {15024, 37201}, {16270, 18909}, {18420, 22804}, {18928, 22808}, {18933, 18948}, {33748, 36752}

X(45073) = anticomplement of X(33540)
X(45073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6803, 9730, 11411), (9815, 16836, 34938)

X(45074) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND INNER TRI-EQUILATERAL

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

X(45074) lies on these lines: {6, 7684}, {1351, 10661}, {10662, 45016}

X(45075) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND OUTER TRI-EQUILATERAL

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

X(45075) lies on these lines: {6, 7685}, {1351, 10662}, {10661, 45016}

X(45076) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st TRI-SQUARES-CENTRAL

Barycentrics    2*(8*a^4-17*(b^2+c^2)*a^2+(b^2-c^2)^2)*S+11*a^6-12*(b^2+c^2)*a^4-(b^4+30*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2) : :
X(45076) = 3*X(485)-2*X(1131)

X(45076) lies on these lines: {20, 485}, {371, 5860}, {486, 11315}, {1384, 19103}, {3316, 6250}, {3592, 13771}, {5418, 13835}, {6221, 6278}, {6449, 36734}, {7583, 12124}, {12123, 35255}, {13651, 31454}, {13882, 42258}, {32495, 42882}, {40288, 42260}

X(45076) = {X(12297), X(13924)}-harmonic conjugate of X(485)

X(45077) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd TRI-SQUARES-CENTRAL

Barycentrics    -2*(8*a^4-17*(b^2+c^2)*a^2+(b^2-c^2)^2)*S+11*a^6-12*(b^2+c^2)*a^4-(b^4+30*b^2*c^2+c^4)*a^2+2*(b^4-c^4)*(b^2-c^2) : :
X(45077) = 3*X(486)-2*X(1132)

X(45077) lies on these lines: {20, 486}, {372, 5861}, {485, 11316}, {1384, 19104}, {3317, 6251}, {3594, 13650}, {5420, 13712}, {6281, 6398}, {6450, 36718}, {7584, 12123}, {12124, 35256}, {13934, 42259}, {32492, 42883}, {40289, 42261}

X(45077) = {X(12296), X(32209)}-harmonic conjugate of X(486)

X(45078) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 3rd TRI-SQUARES

Barycentrics    9*a^6-11*(b^2+c^2)*a^4-(b^4+30*b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)+4*S*(3*a^4-8*(b^2+c^2)*a^2+(b^2-c^2)^2) : :
X(45078) = 4*X(485)-3*X(14241)

X(45078) lies on these lines: {4, 13882}, {371, 22591}, {376, 485}, {631, 3103}, {641, 7582}, {3068, 12510}, {3564, 8912}, {5418, 35830}, {6118, 33365}, {6459, 45024}, {6462, 39387}, {8972, 12602}, {8981, 12222}, {9540, 32497}, {12124, 23267}, {12257, 14227}, {13886, 26516}, {13925, 22810}, {26441, 33344}, {32499, 33703}, {32785, 35831}

X(45079) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 4th TRI-SQUARES

Barycentrics    9*a^6-11*(b^2+c^2)*a^4-(b^4+30*b^2*c^2+c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)-4*S*(3*a^4-8*(b^2+c^2)*a^2+(b^2-c^2)^2) : :
X(45079) = 4*X(486)-3*X(14226)

X(45079) lies on these lines: {4, 13934}, {372, 22592}, {376, 486}, {631, 3102}, {642, 7581}, {3069, 12509}, {5420, 35831}, {6119, 33364}, {6460, 45023}, {6463, 39388}, {8982, 33345}, {12123, 23273}, {12221, 13966}, {12256, 14242}, {12601, 13941}, {13935, 32494}, {13939, 26521}, {13993, 22809}, {32498, 33703}, {32786, 35830}

X(45080) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND URSA MAJOR

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

X(45080) lies on these lines: {8, 56}, {10, 1532}, {11, 5836}, {84, 355}, {936, 5690}, {1145, 17614}, {1210, 10914}, {2136, 33995}, {2550, 16112}, {2886, 7705}, {3035, 45081}, {3698, 25973}, {4848, 17658}, {5784, 24393}, {5795, 17613}, {6736, 17625}, {6979, 45085}, {8582, 17622}, {10912, 14986}, {11019, 17648}, {15863, 17647}, {17615, 21677}, {17665, 24390}, {19861, 37828}, {26492, 40587}

X(45080) = midpoint of X(8) and X(1476)
X(45080) = X(22538)-of-Wasat triangle
X(45080) = X(22549)-of-2nd Zaniah triangle

X(45081) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND URSA MINOR

Barycentrics    2*(b+c)*a^3-(b^2+10*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(45081) = 4*X(1)-X(39777) = 3*X(1389)-5*X(10595) = 5*X(1698)-X(11524) = 5*X(2346)-3*X(8236) = 3*X(3649)-4*X(13407) = 4*X(5559)+X(39777) = X(5690)-3*X(34352) = X(13143)-3*X(16173) = 2*X(13407)-3*X(15888)

X(45081) lies on these lines: {1, 140}, {2, 10912}, {5, 45035}, {8, 344}, {10, 3893}, {11, 9956}, {12, 946}, {21, 38455}, {30, 37563}, {40, 5434}, {55, 944}, {56, 41824}, {65, 5542}, {79, 28212}, {80, 15172}, {145, 18231}, {354, 11362}, {355, 3058}, {388, 20070}, {392, 10915}, {404, 32157}, {442, 2802}, {495, 5697}, {496, 15079}, {517, 3649}, {519, 15670}, {523, 2292}, {758, 12267}, {938, 8162}, {952, 3746}, {958, 12648}, {962, 11237}, {1000, 1389}, {1056, 37567}, {1125, 1145}, {1317, 2646}, {1329, 3890}, {1385, 4995}, {1388, 5218}, {1482, 10056}, {1483, 37571}, {1697, 5252}, {1698, 11524}, {1737, 31792}, {1837, 31393}, {2136, 34720}, {2476, 13463}, {2886, 3885}, {3035, 45080}, {3245, 24470}, {3295, 7489}, {3304, 5657}, {3338, 3654}, {3476, 5217}, {3576, 31436}, {3584, 5901}, {3601, 37738}, {3614, 30384}, {3616, 8256}, {3628, 13143}, {3632, 10389}, {3646, 3679}, {3872, 24953}, {3877, 12607}, {3880, 24987}, {3884, 17757}, {3895, 5794}, {3898, 4187}, {3925, 10914}, {4298, 5183}, {4309, 18525}, {4330, 28186}, {4333, 5119}, {4421, 27870}, {4534, 16601}, {4668, 37723}, {4848, 17609}, {4857, 18357}, {4861, 6690}, {4999, 38460}, {5048, 13411}, {5250, 32049}, {5270, 28174}, {5289, 10528}, {5298, 6684}, {5441, 28224}, {5719, 11009}, {5724, 37588}, {5818, 11238}, {5836, 13996}, {6154, 17647}, {6174, 17614}, {6361, 9657}, {6736, 25917}, {6738, 36920}, {6767, 10573}, {6842, 10284}, {7173, 12053}, {7294, 44675}, {7320, 9780}, {7483, 22837}, {7962, 11218}, {7982, 17718}, {7991, 10404}, {8582, 37829}, {9578, 9819}, {9581, 30337}, {9656, 9812}, {9785, 10896}, {10106, 12512}, {10179, 24982}, {10246, 31452}, {10247, 31480}, {10502, 12908}, {10895, 30305}, {11010, 18990}, {11011, 13405}, {11239, 12635}, {11246, 12702}, {11260, 31157}, {11374, 30323}, {11376, 31434}, {11518, 38122}, {12260, 18391}, {12433, 41684}, {12735, 24926}, {14923, 25466}, {15170, 37702}, {15171, 37710}, {15908, 23340}, {16118, 28216}, {21154, 24927}, {22791, 37719}, {24387, 38058}, {24541, 33895}, {24564, 34501}, {24929, 30538}, {26066, 36846}, {26087, 38033}, {32636, 43174}, {34689, 41229}, {34710, 38314}, {34753, 37602}, {37712, 41864}, {37720, 38042}

X(45081) = midpoint of X(1) and X(5559)
X(45081) = reflection of X(i) in X(j) for these (i, j): (3649, 15888), (10543, 3746)
X(45081) = inverse of X(9957) in Feuerbach circumhyperbola
X(45081) = (anti-Aquila)-isogonal conjugate-of-X(20323)
X(45081) = X(5559)-of-anti-Aquila triangle
X(45081) = X(8718)-of-intouch triangle
X(45081) = X(15062)-of-Hutson intouch triangle
X(45081) = X(18488)-of-Ursa-minor triangle
X(45081) = X(22948)-of-2nd circumperp triangle
X(45081) = X(34563)-of-Wasat triangle
X(45081) = X(43599)-of-4th Euler triangle
X(45081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 5919, 37722), (392, 10915, 21031), (1000, 3085, 2098), (1697, 5252, 6284), (2098, 3085, 15950), (3057, 31397, 12), (3632, 10389, 37724), (5250, 32049, 34606), (6684, 20323, 5298), (9578, 9819, 12701), (10106, 37568, 15326)

X(45082) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND WALSMITH

Barycentrics    a^2*(-a^2+b^2+c^2)*(3*a^8-2*(b^2+c^2)*a^6-(4*b^4-7*b^2*c^2+4*c^4)*a^4+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(45082) = 2*X(110)+X(12310) = X(399)+2*X(2931) = 4*X(1147)-7*X(15039) = 5*X(1656)-2*X(15133) = 4*X(5609)-X(12164) = X(7387)+2*X(15132) = 4*X(10272)-X(12319) = X(10620)-4*X(12893) = 2*X(12302)-5*X(15040) = X(12412)-4*X(20773) = X(12429)+2*X(23236) = 2*X(15136)+X(37924) = 2*X(15139)+X(37928)

X(45082) lies on these lines: {3, 15738}, {6, 40291}, {22, 15106}, {25, 110}, {113, 18494}, {125, 3796}, {154, 542}, {184, 39562}, {351, 9517}, {381, 5642}, {399, 1495}, {403, 12383}, {1147, 7545}, {1511, 9818}, {1656, 15133}, {2502, 32661}, {2781, 9909}, {2854, 19153}, {2930, 8681}, {3517, 25711}, {3531, 5504}, {3564, 7426}, {5020, 15462}, {5050, 12099}, {5609, 12164}, {5651, 12302}, {5663, 14070}, {6800, 9140}, {7387, 15132}, {7493, 32306}, {9019, 15141}, {10201, 32423}, {10254, 12902}, {10272, 11818}, {10620, 12893}, {10733, 18386}, {11061, 37897}, {11064, 18403}, {12412, 20773}, {12429, 23236}, {12827, 18440}, {12828, 21970}, {13394, 38724}, {15041, 35268}, {15131, 34609}, {15136, 37924}, {15139, 37928}, {17508, 44321}, {17810, 34155}, {19138, 26864}, {19140, 41424}, {22146, 35901}, {25556, 31860}, {32315, 45025}, {35266, 44665}

X(45082) = midpoint of X(3167) and X(12310)
X(45082) = reflection of X(i) in X(j) for these (i, j): (3167, 110), (34609, 15131)
X(45082) = inverse of X(32609) in Thomson-Gibert-Moses hyperbola
X(45082) = X(12824)-of-Ara triangle
X(45082) = {X(25), X(110)}-harmonic conjugate of X(45016)

X(45083) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND YIU

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

X(45083) lies on these lines: {3, 49}, {1141, 13150}, {1157, 5961}, {1510, 6150}, {1994, 25044}, {2070, 14656}, {31626, 40441}

X(45083) = barycentric product X(i)*X(j) for these {i, j}: {231, 44180}, {539, 1994}
X(45083) = barycentric quotient X(i)/X(j) for these (i, j): (231, 93), (539, 11140)
X(45083) = trilinear product X(539)*X(2964)
X(45083) = trilinear quotient X(539)/X(2962)
X(45083) = perspector of the circumconic {{A, B, C, X(1994), X(4558)}}
X(45083) = crossdifference of every pair of points on line {X(2501), X(2963)}
X(45083) = X(128)-Dao conjugate of-X(93)
X(45083) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (231, 93), (539, 11140)

X(45084) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 1st ZANIAH

Barycentrics    a^7+3*(b+c)*a^6-(5*b^2-18*b*c+5*c^2)*a^5-(b+c)*(7*b^2-10*b*c+7*c^2)*a^4+(7*b^4+7*c^4-2*(10*b^2+3*b*c+10*c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(5*b^2-2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*a-(b^2-c^2)^3*(b-c) : :

X(45084) lies on these lines: {1, 376}, {2, 18243}, {377, 9799}, {631, 1768}, {1158, 21151}, {2550, 13369}, {2551, 40296}, {2771, 45085}, {3646, 6857}, {4189, 22775}, {4294, 24465}, {4866, 10305}, {5071, 41862}, {5221, 37427}, {5758, 10178}, {5805, 6851}, {5817, 6260}, {5850, 37560}, {6684, 36996}, {6745, 41852}, {6764, 12675}, {6899, 10044}, {7171, 28629}, {7701, 17552}, {9843, 34919}, {10310, 37105}, {12005, 35514}, {16127, 17559}, {18483, 38073}, {34619, 44284}

X(45085) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd ZANIAH

Barycentrics    a^4-4*(b+c)*a^3+8*b*c*a^2+4*(b+c)*(b^2+b*c+c^2)*a-(b^2-c^2)^2 : :
X(45085) = 4*X(3626)-X(31509) = 2*X(10390)-3*X(38053) = X(13867)-5*X(25917)

X(45085) lies on these lines: {1, 4878}, {2, 11281}, {4, 10176}, {8, 3740}, {9, 4297}, {10, 3090}, {11, 3617}, {20, 16112}, {69, 41875}, {145, 4423}, {355, 2551}, {376, 7701}, {388, 3876}, {443, 5692}, {497, 5178}, {519, 3646}, {631, 6326}, {758, 17582}, {936, 6261}, {958, 38669}, {960, 962}, {966, 3061}, {997, 6878}, {1000, 3626}, {1001, 20007}, {1056, 3678}, {1211, 27686}, {1483, 9708}, {2771, 45084}, {3035, 18231}, {3189, 31435}, {3305, 3486}, {3452, 7989}, {3475, 3984}, {3523, 18253}, {3528, 3647}, {3600, 5220}, {3649, 37436}, {3679, 21627}, {3757, 44722}, {3820, 6971}, {3869, 26040}, {3962, 9776}, {4301, 38200}, {4313, 15254}, {4323, 9780}, {4679, 5175}, {5084, 37702}, {5129, 44669}, {5229, 31018}, {5250, 34607}, {5284, 20013}, {5302, 5731}, {5325, 7987}, {5438, 18249}, {5686, 12513}, {5730, 19855}, {5794, 18228}, {5795, 7990}, {5837, 8580}, {5901, 19843}, {6737, 7308}, {6764, 10179}, {6850, 16128}, {6863, 38752}, {6864, 16134}, {6897, 16116}, {6916, 16127}, {6979, 45080}, {7705, 26129}, {7738, 21879}, {7958, 9710}, {8583, 24477}, {9534, 25661}, {9782, 37462}, {10165, 31446}, {10248, 24703}, {10283, 31494}, {10390, 11523}, {10806, 15863}, {11024, 44663}, {12563, 20195}, {15174, 16860}, {16145, 37256}, {16845, 22836}, {17561, 37571}, {18061, 32022}, {34610, 41229}

X(45085) = midpoint of X(8) and X(7320)
X(45085) = reflection of X(11530) in X(10)
X(45085) = complement of X(18221)
X(45085) = X(8)-Beth conjugate of-X(11530)
X(45085) = X(11530)-of-outer-Garcia triangle
X(45085) = X(18221)-of-medial triangle
X(45085) = X(33537)-of-2nd Zaniah triangle
X(45085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (8, 25917, 26105), (3984, 24564, 3475)

X(45086) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND MIDARC

Barycentrics    a*((b-c)*(a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+(a^2+(3*b-4*c)*a-(b-c)*c)*(a-b+c)*sin(B/2)-(a^2-(4*b-3*c)*a+(b-c)*b)*(a+b-c)*sin(C/2)-(b-c)*(-a+b+c)*(3*a-b-c)) : :

X(45086) lies on these lines: {1, 168}, {55, 13444}, {57, 16012}, {177, 7371}, {258, 503}, {260, 13443}, {3062, 8089}, {8112, 22753}

X(45086) = (tangential-midarc)-isogonal conjugate-of-X(11534)

X(45087) = 1st AUBERT POINT OF THESE TRIANGLES: ABC AND 2nd MIDARC

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

X(45087) lies on these lines: {1, 8084}, {266, 8090}, {8076, 18291}

X(45087) = (2nd tangential-midarc)-isogonal conjugate-of-X(11899)

X(45088) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO ANTI-ATIK

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

X(45088) lies on the Jerabek circumhyperbola and these lines: {3, 16657}, {6, 1596}, {51, 4846}, {54, 3089}, {64, 13488}, {66, 18390}, {68, 15030}, {69, 5891}, {74, 12828}, {113, 895}, {381, 6391}, {389, 43695}, {974, 10293}, {1093, 31366}, {1514, 3531}, {1899, 3426}, {3431, 6353}, {3519, 15739}, {5504, 5642}, {5505, 7687}, {5654, 10982}, {6804, 7998}, {9730, 15740}, {10110, 14542}, {11433, 35512}, {13851, 44836}, {14216, 22334}, {14528, 21841}, {15585, 43725}, {16835, 18912}, {20791, 35513}, {38260, 45089}

X(45088) = isotomic conjugate of the anticomplement of X(33871)
X(45088) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(1596)}} and Jerabek hyperbola

X(45089) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 2nd ANTI-EXTOUCH

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

X(45089) lies on these lines: {2, 17834}, {3, 37649}, {4, 6}, {5, 51}, {20, 37476}, {24, 11745}, {26, 13394}, {30, 569}, {54, 7576}, {68, 381}, {132, 38965}, {140, 37478}, {141, 11412}, {154, 37122}, {155, 7528}, {156, 13490}, {161, 10594}, {184, 6756}, {185, 1595}, {235, 10110}, {262, 13599}, {378, 13568}, {389, 427}, {394, 7401}, {403, 9781}, {428, 6759}, {511, 7399}, {546, 13292}, {550, 37513}, {567, 40441}, {568, 5576}, {575, 44829}, {578, 3575}, {597, 43651}, {858, 15043}, {1092, 9825}, {1173, 6145}, {1205, 16105}, {1216, 7405}, {1350, 7383}, {1352, 12160}, {1370, 37514}, {1495, 7715}, {1593, 20427}, {1594, 3567}, {1596, 43831}, {1853, 18916}, {1899, 11432}, {1907, 6000}, {1993, 7544}, {1994, 14516}, {2888, 41628}, {2917, 3518}, {2965, 8883}, {3060, 13160}, {3088, 10605}, {3091, 6515}, {3541, 9786}, {3542, 17810}, {3547, 33586}, {3549, 32269}, {3589, 7509}, {3627, 11750}, {3629, 41171}, {3796, 31305}, {3845, 18379}, {5064, 14216}, {5093, 12429}, {5097, 10112}, {5133, 5889}, {5422, 37444}, {5446, 15760}, {5448, 12235}, {5462, 11585}, {5476, 34664}, {5654, 7529}, {5878, 11403}, {5890, 6247}, {5894, 14865}, {5907, 19130}, {5946, 13371}, {6240, 15033}, {6243, 37347}, {6353, 43841}, {6642, 11064}, {6643, 10601}, {6750, 6755}, {6815, 37498}, {6823, 9967}, {6997, 17814}, {7378, 18909}, {7394, 11441}, {7395, 14561}, {7403, 13754}, {7487, 11427}, {7488, 14389}, {7506, 9820}, {7507, 9777}, {7514, 31815}, {7566, 11442}, {7667, 37515}, {7699, 35487}, {8254, 12107}, {8718, 34613}, {8799, 36412}, {8800, 40449}, {8887, 14569}, {8905, 41169}, {9730, 23335}, {9833, 11402}, {10282, 12242}, {10323, 29181}, {11188, 12271}, {11245, 18381}, {11414, 31670}, {11425, 18533}, {11426, 18494}, {11431, 18950}, {11444, 37990}, {11550, 16198}, {11572, 34565}, {11695, 30739}, {11793, 37439}, {11818, 12134}, {11819, 32046}, {12225, 13434}, {12362, 18583}, {13352, 31833}, {13366, 31804}, {13367, 37458}, {13406, 13451}, {13598, 44479}, {14627, 44076}, {14790, 36752}, {14848, 34725}, {15030, 38136}, {15038, 31724}, {15311, 35502}, {15897, 40645}, {16222, 23306}, {16625, 21243}, {16881, 39504}, {17500, 27356}, {18390, 23047}, {18400, 37505}, {18420, 36747}, {19347, 31383}, {21841, 34417}, {23332, 26879}, {27354, 44716}, {27377, 43995}, {31723, 36753}, {31830, 34116}, {32263, 36518}, {33537, 38072}, {34514, 43588}, {34864, 35254}, {35482, 43607}, {36749, 44665}, {37349, 43605}, {37472, 38321}, {37490, 44158}, {38260, 45088}, {42353, 42441}, {43839, 44211}

X(45089) = midpoint of X(4) and X(7592)
X(45089) = barycentric product X(i)*X(j) for these {i, j}: {5, 11427}, {324, 19357}, {343, 7487}
X(45089) = barycentric quotient X(53)/X(18855)
X(45089) = perspector of the circumconic {{A, B, C, X(107), X(14570)}}
X(45089) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(343)}} and {{A, B, C, X(5), X(393)}}
X(45089) = crossdifference of every pair of points on line {X(520), X(2623)}
X(45089) = crosssum of X(3) and X(569)
X(45089) = X(53)-reciprocal conjugate of-X(18855)
X(45089) = X(7592)-of-Euler triangle
X(45089) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6, 6146), (4, 1181, 16655), (4, 1199, 34224), (4, 1498, 16654), (4, 10982, 16657), (4, 11456, 16621), (4, 12022, 41362), (4, 14853, 10982), (4, 15032, 16659), (4, 41371, 41365), (5, 52, 343), (5, 143, 41587), (5, 31802, 5562), (51, 52, 973), (51, 3574, 5), (54, 7576, 34782), (381, 37493, 68), (403, 9781, 15873), (546, 13292, 18474), (568, 5576, 12359), (1199, 34224, 8550), (1594, 3567, 13567), (5462, 11585, 37648), (5480, 12233, 4), (5890, 15559, 6247), (7487, 11427, 19357), (7507, 9777, 39571), (10110, 18388, 235), (11412, 14788, 141), (11745, 23292, 24), (16198, 18914, 11550)

X(45090) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO ANTI-X3-ABC REFLECTIONS

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

X(45090) lies on these lines: {3, 45124}, {232, 5064}, {325, 37353}, {511, 1656}, {4994, 10594}, {7485, 39668}, {37988, 45108}

X(45090) = midpoint of X(3) and X(45124)
X(45090) = trilinear pole of the line {3569, 7950}
X(45090) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(3763)}} and {{A, B, C, X(3), X(13351)}}
X(45090) = X(45124)-of-anti-X3-ABC reflections triangle

X(45091) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO BEVAN ANTIPODAL

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

X(45091) lies on these lines: {117, 374}, {946, 38357}, {8227, 21228}

X(45092) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 3rd BROCARD

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

X(45092) lies on the Kiepert circumhyperbola and these lines: {76, 40951}, {10551, 40016}

X(45093) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 6th BROCARD

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

X(45093) lies on these lines: {2, 32}, {262, 305}, {308, 325}, {427, 6331}, {1007, 17500}, {3815, 16890}, {3978, 5133}, {7778, 18092}, {8781, 30505}, {8840, 39287}, {14096, 16275}

X(45093) = barycentric product X(308)*X(3095)
X(45093) = barycentric quotient X(83)/X(3406)
X(45093) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(20025)}} and {{A, B, C, X(32), X(262)}}
X(45093) = X(83)-reciprocal conjugate of-X(3406)
X(45093) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 8878, 8623), (2, 20022, 1799)

X(45094) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 7th BROCARD

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

X(45094) lies on these lines: {51, 287}, {3095, 3926}

X(45094) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(287)}} and {{A, B, C, X(51), X(44716)}}

X(45095) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 1st CIRCUMPERP

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

X(45095) lies on these lines: {1, 15232}, {10, 908}, {42, 80}, {594, 17757}, {1089, 23555}, {1220, 5251}, {1577, 42768}, {1785, 1826}, {3293, 5587}, {3822, 26580}, {3971, 4013}, {4705, 42757}, {5219, 24806}, {8068, 44411}, {18082, 31160}

X(45095) = barycentric product X(321)*X(994)
X(45095) = barycentric quotient X(i)/X(j) for these (i, j): (10, 1150), (37, 993), (42, 2278), (994, 81)
X(45095) = trilinear product X(10)*X(994)
X(45095) = trilinear quotient X(i)/X(j) for these (i, j): (10, 993), (37, 2278), (321, 1150), (994, 58)
X(45095) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(3869)}} and {{A, B, C, X(4), X(2476)}}
X(45095) = X(i)-Dao conjugate of-X(j) for these (i, j): (10, 993), (37, 1150)
X(45095) = X(i)-isoconjugate-of-X(j) for these {i, j}: {58, 993}, {81, 2278}, {1150, 1333}
X(45095) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (10, 1150), (37, 993), (42, 2278), (994, 81)

X(45096) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 1st EHRMANN

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

X(45096) lies on these lines: {32, 9076}, {141, 858}, {194, 31125}, {427, 1180}, {1236, 8024}, {5169, 7790}, {9463, 20021}, {15523, 21017}

X(45096) = isogonal conjugate of X(19127)
X(45096) = isotomic conjugate of X(26233)
X(45096) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(66)}} and {{A, B, C, X(4), X(850)}}
X(45096) = X(251)-vertex conjugate of-X(1176)

X(45097) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO HUTSON EXTOUCH

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

X(45097) lies on the Kiepert circumhyperbola and these lines: {4, 5022}, {10, 10186}, {76, 36682}, {226, 5817}, {275, 461}, {1446, 5704}, {2051, 5756}, {2996, 36652}, {3090, 17758}, {5395, 13727}, {6625, 36660}, {8727, 45100}, {13576, 35514}, {14853, 45098}

X(45097) = polar conjugate of the anticomplement of X(25932)
X(45097) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(10980)}} and Kiepert hyperbola

X(45098) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO HUTSON INTOUCH

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

X(45098) lies on the Kiepert circumhyperbola and these lines: {4, 4255}, {5, 43533}, {10, 3090}, {76, 7402}, {83, 7397}, {226, 4862}, {275, 7490}, {321, 5748}, {469, 8796}, {631, 43531}, {1132, 2048}, {2996, 7377}, {4052, 42049}, {5395, 6996}, {5397, 6880}, {6625, 36698}, {6835, 13583}, {7382, 13579}, {7406, 18845}, {9776, 30588}, {10478, 14554}, {10563, 11522}, {14853, 45097}, {30828, 40012}

X(45098) = polar conjugate of the anticomplement of X(25876)
X(45098) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(6557)}} and Kiepert hyperbola

X(45099) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO 2nd HYACINTH

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

X(45099) lies on these lines: {51, 394}, {10982, 22401}

X(45100) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO INCIRCLE-CIRCLES

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

X(45100) lies on the Kiepert circumhyperbola and these lines: {2, 37499}, {10, 962}, {20, 43531}, {83, 7406}, {226, 3672}, {275, 6994}, {321, 20921}, {459, 469}, {1751, 37681}, {2048, 3317}, {3832, 14555}, {4052, 28313}, {4869, 40012}, {6996, 18841}, {7377, 18840}, {7384, 32022}, {8727, 45097}, {10478, 17758}, {13478, 37666}

X(45100) = isotomic conjugate of X(37655)
X(45100) = trilinear pole of the line {523, 7661}
X(45100) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(5), X(6994)}}

X(45101) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO LUCAS ANTIPODAL

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

X(45101) lies on the Kiepert circumhyperbola and these lines: {2, 9733}, {4, 6422}, {5, 5490}, {6, 14229}, {98, 1587}, {262, 6201}, {275, 5200}, {486, 14853}, {1588, 14234}, {2052, 3127}, {2996, 6290}, {3070, 14244}, {3406, 44596}, {5480, 45102}, {5491, 10983}, {5871, 14458}, {6202, 14231}, {6776, 14232}, {10514, 42023}, {10783, 14228}, {14238, 23249}

X(45101) = isogonal conjugate of X(43119)
X(45101) = isotomic conjugate of the anticomplement of X(31463)
X(45101) = touchpoint of the line {31463, 45101} and Kiepert circumhyperbola
X(45101) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(3), X(3127)}}

X(45102) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO LUCAS(-1) ANTIPODAL

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

X(45102) lies on the Kiepert circumhyperbola and these lines: {2, 9732}, {4, 6421}, {5, 5491}, {6, 14244}, {83, 21737}, {98, 1588}, {262, 6202}, {485, 14853}, {486, 21736}, {1587, 14238}, {2052, 3128}, {2996, 6289}, {3071, 14229}, {3406, 44595}, {5480, 45101}, {5490, 10983}, {5870, 14458}, {6201, 14245}, {6776, 14237}, {10515, 42024}, {10784, 14243}, {14234, 23259}

X(45102) = isogonal conjugate of X(43118)
X(45102) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(3), X(3128)}}

X(45103) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO MCCAY

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

X(45103) lies on the Kiepert circumhyperbola and these lines: {2, 6781}, {4, 22234}, {5, 10185}, {6, 17503}, {30, 7608}, {76, 11317}, {83, 8352}, {98, 3845}, {99, 8786}, {262, 3830}, {316, 10302}, {381, 7607}, {671, 8584}, {2996, 7812}, {3534, 11669}, {3839, 43537}, {5066, 38230}, {5475, 10484}, {5503, 35705}, {7612, 41099}, {7790, 18842}, {7841, 43527}, {7883, 18840}, {8370, 10159}, {8587, 9166}, {8594, 40706}, {8595, 40707}, {8597, 44562}, {9855, 39590}, {10153, 14061}, {10155, 11001}, {10187, 11303}, {10188, 11304}, {11054, 43676}, {11361, 43529}, {11606, 12156}, {11668, 19709}, {12101, 14492}, {14041, 43528}, {14042, 39785}, {14494, 15682}, {14537, 43535}, {15300, 35005}

X(45103) = reflection of X(99) in X(8786)
X(45103) = isogonal conjugate of X(8589)
X(45103) = isotomic conjugate of X(22165)
X(45103) = antigonal conjugate of the anticomplement of X(8786)
X(45103) = symgonal image of X(8786)
X(45103) = trilinear pole of the line {523, 8859}
X(45103) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(3), X(22234)}}
X(45103) = Cevapoint of X(i) and X(j) for these (i, j): {2, 15534}, {6, 14002}
X(45103) = X(45111)-of-McCay triangle

X(45104) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO MIXTILINEAR

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

X(45104) lies on these lines: {946, 982}, {3061, 20262}, {3705, 23528}

X(45104) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(262)}} and {{A, B, C, X(4), X(946)}}

X(45105) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO ORTHIC AXES

Barycentrics    (a^2-b^2+c^2)*(5*a^8-4*(5*b^2+6*c^2)*a^6+2*(13*b^4+10*b^2*c^2+19*c^4)*a^4-4*(b^2-c^2)*(3*b^4-b^2*c^2-6*c^4)*a^2+(b^4-10*b^2*c^2+5*c^4)*(b^2-c^2)^2)*(a^2+b^2-c^2)*(5*a^8-4*(6*b^2+5*c^2)*a^6+2*(19*b^4+10*b^2*c^2+13*c^4)*a^4-4*(b^2-c^2)*(6*b^4+b^2*c^2-3*c^4)*a^2+(5*b^4-10*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(45105) lies on the Kiepert circumhyperbola and the line {6748, 7612}

X(45106) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO INNER-SQUARES

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

X(45106) lies on the Kiepert circumhyperbola and these lines: {485, 15883}, {1132, 6201}, {1328, 5480}, {1352, 42023}, {2996, 6281}, {3070, 14232}, {3424, 23249}, {5490, 6250}, {6202, 43561}, {6776, 22541}, {7612, 13711}, {13687, 41895}, {14228, 23269}, {14237, 23251}

X(45107) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO OUTER-SQUARES

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

X(45107) lies on the Kiepert circumhyperbola and these lines: {486, 15884}, {1131, 6202}, {1327, 5480}, {1352, 42024}, {2996, 6278}, {3071, 14237}, {3424, 23259}, {5491, 6251}, {6201, 43560}, {6776, 19101}, {7612, 13834}, {13807, 41895}, {14232, 23261}, {14243, 23275}

X(45108) = 2nd AUBERT POINT OF THESE TRIANGLES: ABC TO X3-ABC REFLECTIONS

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

X(45108) lies on these lines: {3, 45114}, {140, 143}, {232, 428}, {252, 1173}, {262, 34845}, {325, 1232}, {338, 43458}, {2980, 7772}, {3329, 6636}, {3518, 19189}, {5968, 7533}, {6530, 15559}, {6664, 7764}, {14765, 37900}, {37988, 45090}

X(45108) = reflection of X(3) in X(45114)
X(45108) = isotomic conjugate of the complement of X(13571)
X(45108) = trilinear pole of the line {3569, 7927}
X(45108) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(428)}} and {{A, B, C, X(3), X(5421)}}
X(45108) = Cevapoint of X(i) and X(j) for these (i, j): {2, 13571}, {39, 51}
X(45108) = X(45114)-of-X3-ABC reflections triangle

X(45109) = 2nd AUBERT POINT OF THESE TRIANGLES: 1st ANTI-BROCARD TO ABC

Barycentrics    a^12+7*(b^2+c^2)*a^10+4*(b^2+2*c^2)*(2*b^2+c^2)*a^8-(b^2+c^2)*(11*b^4+7*b^2*c^2+11*c^4)*a^6-(b^8+c^8+3*(5*b^4+7*b^2*c^2+5*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(2*b^8+2*c^8-(13*b^4+8*b^2*c^2+13*c^4)*b^2*c^2)*a^2+(b^6-c^6)*(b^2-c^2)*(-2*b^4-7*b^2*c^2-2*c^4) : :
X(45109) = 3*X(6054)-X(14492)

X(45109) lies on these lines: {2, 7711}, {30, 9866}, {147, 1350}, {542, 8290}, {1916, 6054}, {4353, 5988}, {5999, 7799}, {7840, 19924}, {8291, 36776}, {8295, 39096}, {8724, 22498}

X(45109) = (1st anti-Brocard)-isogonal conjugate-of-X(12042)
X(45109) = X(9765)-of-anti-McCay triangle
X(45109) = X(14492)-of-1st anti-Brocard triangle

X(45110) = 2nd AUBERT POINT OF THESE TRIANGLES: ANTI-CONWAY TO ABC

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

X(45110) lies on these lines: {26, 52}, {54, 32392}, {185, 18570}, {389, 10274}, {576, 19459}, {578, 1181}, {1154, 13367}, {1614, 2393}, {3542, 41593}, {3564, 21637}, {3574, 6146}, {3627, 21659}, {5446, 11536}, {6776, 11423}, {11422, 18925}, {12160, 19357}, {13353, 43590}, {14448, 32226}, {18396, 37505}, {18534, 19347}, {19125, 44470}, {40441, 45118}

X(45110) = {X(32341), X(34397)}-harmonic conjugate of X(389)

X(45111) = 2nd AUBERT POINT OF THESE TRIANGLES: ANTI-MCCAY TO ABC

Barycentrics    23*a^8-103*(b^2+c^2)*a^6+3*(33*b^4+46*b^2*c^2+33*c^4)*a^4-(b^2+c^2)*(22*b^4+29*b^2*c^2+22*c^4)*a^2-4*b^8+(59*b^4-117*b^2*c^2+59*c^4)*b^2*c^2-4*c^8 : :

X(45111) lies on these lines: {2, 13188}, {99, 8587}, {385, 15300}, {543, 8786}, {8591, 15534}, {8592, 14537}, {9855, 10992}

X(45111) = X(45103)-of-anti-McCay triangle

X(45112) = 2nd AUBERT POINT OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO ABC

Barycentrics    a^2*(-a^2+b^2+c^2)*((b^2+c^2)^2*a^8-(b^2+c^2)^3*a^6-(b^8+c^8+(b^4+8*b^2*c^2+c^4)*b^2*c^2)*a^4+(b^4-c^4)^2*(b^2+c^2)*a^2+(b^4-c^4)^2*b^2*c^2) : :
X(45112) = X(217)+3*X(3917) = 3*X(3819)-X(34850)

X(45112) lies on these lines: {2, 27370}, {3, 3202}, {217, 3917}, {511, 40645}, {1503, 11793}, {3819, 34850}, {3934, 11574}

X(45112) = complement of X(27370)

X(45113) = 2nd AUBERT POINT OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL TO ABC

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

X(45113) lies on these lines: {3, 13210}, {6, 9984}, {39, 74}, {110, 5092}, {113, 7859}, {125, 43460}, {542, 8290}, {1691, 12192}, {5116, 5621}, {5663, 12054}, {6699, 7832}, {9138, 14660}, {12041, 35002}

X(45113) = X(14492)-of-anti-orthocentroidal triangle

X(45114) = 2nd AUBERT POINT OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS TO ABC

Barycentrics    4*(b^2+c^2)*a^10-(5*b^2+3*c^2)*(3*b^2+5*c^2)*a^8+(b^2+c^2)*(21*b^4+10*b^2*c^2+21*c^4)*a^6-(13*b^8+13*c^8-(13*b^4+48*b^2*c^2+13*c^4)*b^2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4)*a^2+(b^2-c^2)^4*b^2*c^2 : :

X(45114) lies on these lines: {3, 45108}, {511, 3530}, {7488, 41328}

X(45114) = midpoint of X(3) and X(45108)
X(45114) = X(45108)-of-anti-X3-ABC reflections triangle

X(45115) = 2nd AUBERT POINT OF THESE TRIANGLES: ATIK TO ABC

Barycentrics    3*(b+c)*a^6-4*(b^2+3*b*c+c^2)*a^5-5*(b+c)*(b^2-10*b*c+c^2)*a^4+4*(2*b^4+2*c^4-(3*b^2+38*b*c+3*c^2)*b*c)*a^3+(b+c)*(b^4+c^4-6*(8*b^2-37*b*c+8*c^2)*b*c)*a^2-4*(b^2-c^2)^2*(b^2-6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

X(45115) lies on these lines: {8, 3306}, {10, 12448}, {1000, 8580}, {1145, 3921}, {3062, 3679}, {4711, 15587}, {9856, 11362}

X(45115) = X(45088)-of-Atik triangle

X(45116) = 2nd AUBERT POINT OF THESE TRIANGLES: 2nd CONWAY TO ABC

Barycentrics    a^4+2*(b+c)*a^3-20*b*c*a^2-2*(b+c)*(b^2-26*b*c+c^2)*a-(b^2-c^2)^2 : :
X(45116) = 3*X(8)+2*X(18490) = 4*X(3626)+X(11034)

X(45116) lies on these lines: {7, 3679}, {8, 4002}, {10, 12541}, {962, 3617}, {1145, 9802}, {3626, 11034}, {4678, 11024}, {4745, 18228}, {5252, 41824}, {5815, 14450}, {31672, 35514}

X(45116) = reflection of X(36835) in X(10)
X(45116) = X(8)-Beth conjugate of-X(36835)
X(45116) = X(3531)-of-2nd Conway triangle
X(45116) = X(36835)-of-outer-Garcia triangle

X(45117) = 2nd AUBERT POINT OF THESE TRIANGLES: EHRMANN-SIDE TO ABC

Barycentrics    (-a^2+b^2+c^2)^2*((b^2+c^2)^2*a^18-3*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^16+2*(7*b^8+7*c^8+(7*b^4+11*b^2*c^2+7*c^4)*b^2*c^2)*a^14+3*(2*b^8+2*c^8-(7*b^4-5*b^2*c^2+7*c^4)*b^2*c^2)*b^2*c^2*a^10-2*(b^2+c^2)*(7*b^8+7*c^8-(3*b^4-8*b^2*c^2+3*c^4)*b^2*c^2)*a^12+2*(b^8-c^8)*(b^2-c^2)*(7*b^4-4*b^2*c^2+7*c^4)*a^8-(b^2-c^2)^2*(14*b^12+14*c^12-(6*b^8+6*c^8+5*(2*b^4-b^2*c^2+2*c^4)*b^2*c^2)*b^2*c^2)*a^6+6*(b^4+b^2*c^2+c^4)*(b^2-c^2)^6*(b^2+c^2)*a^4-(b^2-c^2)^8*(b^2+c^2)*b^2*c^2-(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^6*(b^4+b^2*c^2+c^4)*a^2)*a^2 : :

X(45117) lies on these lines: {12162, 18562}, {18436, 31388}

X(45118) = 2nd AUBERT POINT OF THESE TRIANGLES: 2nd EULER TO ABC

Barycentrics    a^2*(-a^2+b^2+c^2)*((b^2+c^2)*a^10-(3*b^4+4*b^2*c^2+3*c^4)*a^8+2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^6+2*(b^4+c^4+(b^2-c^2)*b*c)*(b^4+c^4-(b^2-c^2)*b*c)*a^4-3*(b^8-c^8)*(b^2-c^2)*a^2+(b^4-c^4)^2*(b^2-c^2)^2) : :
X(45118) = 3*X(5891)-X(18474) = X(11442)-5*X(11444) = X(18445)+3*X(23039)

X(45118) lies on these lines: {3, 49}, {5, 31807}, {52, 3549}, {130, 131}, {159, 18451}, {161, 17814}, {381, 1843}, {389, 7542}, {511, 15760}, {542, 11574}, {1154, 6676}, {1352, 2393}, {1368, 15067}, {2072, 10170}, {3546, 7999}, {3547, 11412}, {3574, 5446}, {3819, 10257}, {5462, 6639}, {5907, 12134}, {6000, 44249}, {6101, 6823}, {6643, 11442}, {7723, 24981}, {9833, 12162}, {10201, 44084}, {10297, 11649}, {10627, 31829}, {11585, 11793}, {11591, 12362}, {12058, 35243}, {12294, 18534}, {13348, 44240}, {13416, 44324}, {15038, 18449}, {15068, 15818}, {15087, 19129}, {16196, 32142}, {18435, 34750}, {36747, 45015}, {40441, 45110}

X(45118) = midpoint of X(184) and X(5562)
X(45118) = reflection of X(21243) in X(11793)
X(45118) = complement of 2nd-anti-Conway-to-orthic similarity image of X(18388)
X(45118) = {X(5891), X(9967)}-harmonic conjugate of X(18531)

X(45119) = 2nd AUBERT POINT OF THESE TRIANGLES: EXTOUCH TO ABC

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

X(45119) lies on this line: {9954, 13227}

X(45120) = 2nd AUBERT POINT OF THESE TRIANGLES: 2nd EXTOUCH TO ABC

Barycentrics    a*((b+c)*a^5-(b^2+4*b*c+c^2)*a^4-2*(b+c)^3*a^3+2*(b^2+c^2)*(b+c)^2*a^2+(b+c)^5*a-(b^4-c^4)*(b^2-c^2)) : :
X(45120) = X(377)-5*X(3876) = X(3868)-5*X(31259) = 3*X(5692)+X(41229)

X(45120) lies on these lines: {1, 6}, {4, 210}, {30, 5777}, {63, 37282}, {65, 19855}, {78, 20835}, {191, 9943}, {226, 3824}, {329, 377}, {354, 16845}, {442, 3740}, {452, 3681}, {936, 37270}, {943, 3683}, {950, 34790}, {1005, 4420}, {1006, 12675}, {1071, 3929}, {1260, 12514}, {1490, 37426}, {1708, 3927}, {1770, 15587}, {1829, 3690}, {1864, 4005}, {1868, 1889}, {2318, 37528}, {3059, 4294}, {3419, 4662}, {3487, 25917}, {3678, 6743}, {3715, 41538}, {3811, 13615}, {3868, 31259}, {3877, 6764}, {3940, 10393}, {4640, 11517}, {4866, 15104}, {5584, 6001}, {5715, 31142}, {5759, 12688}, {5812, 44229}, {6172, 12528}, {6832, 13374}, {6872, 20007}, {6987, 14872}, {7085, 40660}, {7686, 21677}, {8273, 18446}, {10391, 31445}, {10916, 14022}, {12262, 26935}, {16299, 37597}, {26878, 44782}, {26915, 41722}, {28609, 44217}, {37398, 41609}

X(45120) = midpoint of X(72) and X(405)
X(45120) = reflection of X(8728) in X(5044)
X(45120) = X(45089)-of-2nd extouch triangle
X(45120) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (9, 72, 44547), (72, 392, 11523), (72, 5728, 5904)

X(45121) = 2nd AUBERT POINT OF THESE TRIANGLES: GARCIA-REFLECTION TO ABC

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

X(45121) lies on the Feuerbach circumhyperbola and these lines: {7, 38455}, {84, 11525}, {1000, 3820}, {1476, 5836}, {3452, 12641}, {15179, 39779}

X(45122) = 2nd AUBERT POINT OF THESE TRIANGLES: INTOUCH TO ABC

Barycentrics    a*(b+c)*(a^3-(b^2-3*b*c+c^2)*a-(b+c)*b*c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*(a-b+c)*(a+b-c) : :
X(45122) = 3*X(3753)-X(38955)

X(45122) lies on these lines: {12, 17705}, {65, 3293}, {517, 1457}, {942, 952}, {1439, 1441}, {1455, 35059}, {2810, 24471}, {2818, 5909}, {3028, 40663}, {3040, 36949}, {3754, 20617}, {3812, 34589}, {4552, 22306}

X(45122) = midpoint of X(i) and X(j) for these {i, j}: {65, 4551}, {24028, 45022}
X(45122) = reflection of X(34589) in X(3812)

X(45123) = 2nd AUBERT POINT OF THESE TRIANGLES: ORTHIC TO ABC

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

X(45123) lies on these lines: {5, 15897}, {51, 324}, {52, 1625}, {143, 27370}, {182, 17409}, {232, 511}, {389, 7745}, {428, 542}, {9517, 11557}, {12294, 44422}

X(45123) = midpoint of X(52) and X(1625)
X(45123) = crosspoint of X(4) and X(19128)
X(45123) = X(4)-Ceva conjugate of-X(39569)
X(45123) = (orthic)-isogonal conjugate-of-X(39569)

X(45124) = 2nd AUBERT POINT OF THESE TRIANGLES: X3-ABC REFLECTIONS TO ABC

Barycentrics    10*(b^2+c^2)*a^10-(18*b^4+19*b^2*c^2+18*c^4)*a^8-2*(b^2+c^2)*(3*b^4+19*b^2*c^2+3*c^4)*a^6+(26*b^8+26*c^8+(19*b^4-30*b^2*c^2+19*c^4)*b^2*c^2)*a^4-6*(b^4-c^4)*(b^2-c^2)*(2*b^2-c^2)*(b^2-2*c^2)*a^2-8*(b^2-c^2)^4*b^2*c^2 : :

X(45124) lies on these lines: {3, 45090}, {511, 3843}

X(45124) = reflection of X(3) in X(45090)
X(45124) = X(45090)-of-X3-ABC reflections triangle

X(45125) = (name pending)

Barycentrics    a/((a + b + c)*(a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - 4*a^2*b^4 + a*b^5 + a^5*c + 9*a^4*b*c - 10*a^3*b^2*c - 10*a^2*b^3*c + 9*a*b^4*c + b^5*c - 4*a^4*c^2 - 10*a^3*b*c^2 + 28*a^2*b^2*c^2 - 10*a*b^3*c^2 - 4*b^4*c^2 + 6*a^3*c^3 - 10*a^2*b*c^3 - 10*a*b^2*c^3 + 6*b^3*c^3 - 4*a^2*c^4 + 9*a*b*c^4 - 4*b^2*c^4 + a*c^5 + b*c^5 + 2*(3*a^3*b - 6*a^2*b^2 + 3*a*b^3 + 3*a^3*c - 2*a^2*b*c - 3*a*b^2*c + 2*b^3*c - 6*a^2*c^2 - 3*a*b*c^2 - 4*b^2*c^2 + 3*a*c^3 + 2*b*c^3)*S) + 2*b*(a + b - c)*c*(a - b + c)*((6*a^3 + 3*a^2*b - 8*a*b^2 - b^3 + 3*a^2*c - 12*a*b*c + b^2*c - 8*a*c^2 + b*c^2 - c^3)*Cos[A/2] + (a + b + c)*(5*a^2 - 4*a*b - b^2 - 4*a*c - 4*b*c - c^2)*Sin[A/2]) + 2*a*(a - b - c)*(a + b - c)*c*((2*a^3 + 7*a^2*b - 4*a*b^2 - 5*b^3 - a^2*c + 8*a*b*c - 3*b^2*c - 2*a*c^2 + 7*b*c^2 + c^3)*Cos[B/2] + (a + b + c)*(2*a^2 + 3*a*b - 5*b^2 + 6*a*c + 3*b*c + 2*c^2)*Sin[B/2]) + 2*a*b*(a - b - c)*(a - b + c)*((2*a^3 - a^2*b - 2*a*b^2 + b^3 + 7*a^2*c + 8*a*b*c + 7*b^2*c - 4*a*c^2 - 3*b*c^2 - 5*c^3)*Cos[C/2] + (a + b + c)*(2*a^2 + 6*a*b + 2*b^2 + 3*a*c + 3*b*c - 5*c^2)*Sin[C/2])) : :

X(45125) is the perspector of the ellipse described in X(44994)

See Antreas Hatzipolakis and Peter Moses, euclid 2638.

X(45125) = lies on these lines: { }

X(45126) = X(1)X(4)∩X(2)X(914)

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

X(45126) lies on the cubic K1242 and these lines: {1, 4}, {2, 914}, {6, 1214}, {7, 17011}, {9, 16577}, {37, 34048}, {42, 8270}, {48, 1763}, {57, 77}, {63, 1993}, {65, 37063}, {109, 17594}, {198, 14557}, {221, 3931}, {222, 3666}, {227, 5711}, {239, 27339}, {269, 553}, {326, 3687}, {371, 8978}, {372, 13388}, {386, 1038}, {500, 1062}, {572, 10319}, {612, 4551}, {651, 28606}, {940, 1465}, {975, 37694}, {991, 1040}, {997, 1211}, {1060, 5396}, {1100, 1427}, {1158, 1181}, {1386, 1617}, {1419, 34035}, {1421, 4666}, {1429, 16787}, {1439, 6611}, {1441, 19684}, {1443, 17013}, {1456, 37593}, {1458, 17017}, {1726, 2261}, {1741, 21482}, {2006, 31266}, {2262, 11347}, {2269, 40152}, {2317, 26934}, {2331, 18676}, {2360, 7713}, {2594, 3811}, {2910, 5811}, {2999, 3911}, {3554, 40940}, {3677, 5083}, {3982, 4328}, {3995, 28997}, {4114, 7271}, {4254, 7011}, {4296, 19767}, {4318, 17018}, {4383, 43065}, {4417, 44179}, {4552, 26223}, {4654, 7190}, {4886, 17095}, {5219, 5287}, {5226, 17019}, {5262, 34489}, {5435, 17012}, {5437, 43048}, {5530, 34030}, {5928, 41007}, {6180, 20182}, {6349, 11433}, {6350, 11427}, {6510, 17811}, {7078, 37528}, {7210, 41245}, {7308, 25930}, {8257, 10601}, {10884, 33178}, {13567, 17073}, {14100, 20581}, {16609, 37523}, {17043, 41883}, {17056, 37695}, {17086, 17778}, {17102, 36746}, {17147, 28968}, {17599, 17625}, {17859, 20238}, {18161, 21370}, {18623, 34492}, {18652, 20266}, {19763, 22341}, {20122, 37516}, {25430, 34056}, {26012, 37699}, {26132, 26639}, {27186, 37771}, {28780, 32858}, {28996, 31035}, {34040, 37548}, {34046, 37592}, {36742, 37565}

X(45126) = isotomic conjugate of the polar conjugate of X(1452)
X(45126) = X(44733)-Ceva conjugate of X(57)
X(45126) = X(36744)-cross conjugate of X(12514)
X(45126) = cevapoint of X(2286) and X(11509)
X(45126) = crosssum of X(2310) and X(17412)
X(45126) = crossdifference of every pair of points on line {652, 4041}
X(45126) = barycentric product X(i)*X(j) for these {i,j}: {7, 12514}, {57, 5739}, {69, 1452}, {77, 406}, {85, 36744}, {226, 27174}, {1412, 42707}, {2285, 14258}, {7182, 44086}
X(45126) = barycentric quotient X(i)/X(j) for these {i,j}: {406, 318}, {1452, 4}, {5739, 312}, {12514, 8}, {27174, 333}, {36744, 9}, {42707, 30713}, {44086, 33}
X(45126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 223, 226}, {1, 581, 10393}, {1, 1079, 13407}, {6, 1214, 1708}, {77, 5256, 57}, {81, 17080, 57}, {278, 5712, 226}, {581, 8555, 1}, {1100, 1427, 37543}, {1659, 13390, 12047}, {4850, 17074, 57}, {14547, 20277, 1}

X(45127) = X(2)X(1172)∩X(3)X(1779)

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

X(45127) lies on the cubic K1242 and these lines: {2, 1172}, {3, 1779}, {6, 1214}, {48, 3173}, {212, 3682}, {219, 3998}, {394, 2193}, {997, 2192}, {1217, 6827}, {1751, 18589}, {1812, 3926}, {2219, 37543}, {3346, 6987}, {5783, 32777}, {7522, 18588}, {13395, 32726}, {17073, 19716}, {36023, 40397}

X(45127) = X(i)-isoconjugate of X(j) for these (i,j): {19, 377}, {27, 43214}, {92, 37538}, {281, 1448}
X(45127) = trilinear pole of line {520, 1946}
X(45127) = barycentric product X(521)*X(13395)
X(45127) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 377}, {184, 37538}, {228, 43214}, {603, 1448}, {13395, 18026}

X(45128) = X(1)X(1751)∩X(3)X(81)

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

X(45128) lies on the cubic K1242 and these lines: {1, 1751}, {3, 81}, {6, 943}, {379, 17011}, {581, 5757}, {1449, 2215}

X(45128) = barycentric quotient X(i)/X(j) for these {i,j}: {584, 405}, {5248, 5271}, {5278, 44140}

X(45129) = X(6)X(1780)∩X(25)X(584)

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

X(45129) lies on the the circumconic {{A,B,C,X(2),X(6)}}, the cubic K1242, and these lines: {6, 1780}, {25, 584}, {37, 2271}, {42, 2911}, {967, 1333}, {1100, 1427}, {1171, 7054}, {1400, 36744}, {2350, 36743}, {3444, 4289}, {4258, 14553}, {5110, 39966}, {5153, 39951}

X(45129) = isogonal conjugate of the anticomplement of X(19732)
X(45129) = X(63)-isoconjugate of X(14018)
X(45129) = barycentric quotient X(25)/X(14018)

X(45130) = X(3)X(2164)∩X(4)X(2911)

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

X(45130) lies on the cubic K1242 and these lines: {3, 2164}, {4, 2911}, {6, 23604}, {81, 4000}, {1751, 18589}

X(45131) = X(1)X(318)∩X(72)X(519)

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

See Antreas Hatzipolakis and Angel Montesdeoca, euclid 2662.

X(45131) lies on these lines: {1,318}, {10,25091}, {29,3469}, {72,519}, {73,38462}, {496,31680}, {515,1872}, {517,42456}, {522,5884}, {2654,41013}, {3075,10538}, {5722,34588}

X(45132) = ISOGONAL CONJUGATE OF X(37557)

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

See Antreas Hatzipolakis and César Lozada, euclid 2664.

X(45132) lies on the Feuerbach circumhyperbola and these lines: {8, 7391}, {9, 16545}, {21, 19836}, {989, 3585}

X(45132) = isogonal conjugate of X(37557)
X(45132) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(10), X(19836)}}

X(45133) = X(8)X(14790)∩X(9)X(18597)

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

See Antreas Hatzipolakis and César Lozada, euclid 2664.

X(45133) lies on the Feuerbach circumhyperbola and these lines: {8, 14790}, {9, 18597}

X(45133) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(28), X(14790)}}

X(45134) = X(1)X(18426)∩X(8)X(3153)

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

See Antreas Hatzipolakis and César Lozada, euclid 2664.

X(45134) lies on the Feuerbach circumhyperbola and these lines: {1, 18426}, {8, 3153}

X(45134) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(28), X(3153)}}

X(45135) = X(4)X(134)∩X(52)X(925)

Barycentrics    a^2*(c^2*a^10+(b^4-2*b^2*c^2-4*c^4)*a^8-2*(2*b^6-3*c^6-(2*b^2+c^2)*b^2*c^2)*a^6+2*(3*b^8-2*c^8-(5*b^4-b^2*c^2-c^4)*b^2*c^2)*a^4-(b^2-c^2)^3*(4*b^4+b^2*c^2+c^4)*a^2+(b^2-c^2)^4*b^4)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2)*(b^2*a^10-(4*b^4+2*b^2*c^2-c^4)*a^8+2*(3*b^6-2*c^6+(b^2+2*c^2)*b^2*c^2)*a^6-2*(2*b^8-3*c^8-(b^4+b^2*c^2-5*c^4)*b^2*c^2)*a^4+(b^4+b^2*c^2+4*c^4)*(b^2-c^2)^3*a^2+(b^2-c^2)^4*c^4)*(a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2) : :
Barycentrics    (SB+SC)*(S^2+SA*SB)*(S^2+SA*SC)*(S^2-4*R^2*(3*R^2+SC-2*SW)-2*SA*SB-SW^2)*(S^2-4*R^2*(3*R^2+SB-2*SW)-2*SA*SC-SW^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 2666.

X(45135) lies on the circumcircle and these lines: {4, 134}, {52, 925}, {54, 13398}, {110, 3133}, {571, 32692}, {930, 5889}, {13352, 20185}

X(45135) = reflection of X(4) in X(134)
X(45135) = circumperp conjugate of the anticomplement of X(134)
X(45135) = circumnormal-isogonal conjugate of the complementary conjugate of X(134)
X(45135) = Collings transform of X(134)
X(45135) = X(107)-of-circumorthic-triangle

X(45136) = X(3)X(835)∩X(109)X(388)

Barycentrics    (a^5+b*a^4-(b^2-c^2)*a^3-(b^3+2*b*c^2-c^3)*a^2-(b^2-c^2)*(b+c)*c^2)*(a^5+c*a^4+(b^2-c^2)*a^3+(b^3-2*b^2*c-c^3)*a^2+(b^2-c^2)*(b+c)*b^2) : :
X(45136) = 5*X(631)-4*X(40547)

See Antreas Hatzipolakis and César Lozada, euclid 2666.

X(45136) lies on the circumcircle and these lines: {2, 44954}, {3, 835}, {4, 5515}, {30, 44944}, {74, 24813}, {75, 1310}, {99, 44154}, {100, 4385}, {101, 2345}, {109, 388}, {110, 1010}, {376, 28477}, {631, 40547}, {7427, 9078}, {9057, 36025}

X(45136) = reflection of X(i) in X(j) for these (i, j): (4, 5515), (835, 3)
X(45136) = anticomplement of X(44954)
X(45136) = isogonal conjugate of the complementary conjugate of X(44954)
X(45136) = circumperp conjugate of X(835)
X(45136) = circumnormal-isogonal conjugate of X(834)
X(45136) = circumtangential-isogonal conjugate of the complementary conjugate of X(44954)
X(45136) = antipode of X(835) in circumcircle
X(45136) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(5264)}} and {{A, B, C, X(3), X(7087)}}
X(45136) = trilinear pole of the line {6, 6590}
X(45136) = Collings transform of X(5515)
X(45136) = V-transform of X(834)

X(45137) = X(3)X(1310)∩X(4)X(5517)

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

See Antreas Hatzipolakis and César Lozada, euclid 2666.

X(45137) lies on the circumcircle and these lines: {3, 1310}, {4, 5517}, {31, 32691}, {40, 28477}, {100, 5739}, {101, 12514}, {107, 4206}, {108, 1460}, {109, 17594}, {110, 27174}, {835, 30273}

X(45137) = reflection of X(i) in X(j) for these (i, j): (4, 5517), (1310, 3)
X(45137) = isogonal conjugate of the circumnormal-isogonal conjugate of X(1310)
X(45137) = circumperp conjugate of X(1310)
X(45137) = circumnormal-isogonal conjugate of X(8678)
X(45137) = circumtangential-isogonal conjugate of the circumnormal-isogonal conjugate of X(1310)
X(45137) = X(78)-Gimel conjugate of-X(28477)
X(45137) = antipode of X(1310) in circumcircle
X(45137) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(17594)}} and {{A, B, C, X(3), X(31)}}
X(45137) = Collings transform of X(5517)
X(45137) = V-transform of X(1310)
X(45137) = Cundy-Parry Phi transform of X(14258)

X(45138) = X(4)X(5522)∩X(110)X(631)

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

See Antreas Hatzipolakis and César Lozada, euclid 2666.

X(45138) lies on the circumcircle and these lines: {3, 43351}, {4, 5522}, {99, 44149}, {107, 10594}, {110, 631}, {112, 3087}, {476, 37925}, {925, 11414}, {1302, 5020}, {1304, 40630}, {7422, 29180}, {9060, 37897}

X(45138) = reflection of X(i) in X(j) for these (i, j): (4, 5522), (43351, 3)
X(45138) = isogonal conjugate of the circumnormal-isogonal conjugate of X(43351)
X(45138) = circumperp conjugate of X(43351)
X(45138) = circumnormal-isogonal conjugate of the isogonal conjugate of X(43351)
X(45138) = circumtangential-isogonal conjugate of the circumnormal-isogonal conjugate of X(43351)
X(45138) = antipode of X(43351) in circumcircle
X(45138) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(1179)}} and {{A, B, C, X(4), X(95)}}
X(45138) = Collings transform of X(5522)
X(45138) = V-transform of X(43351)

X(45139) = (name pending)

Barycentrics    (3 b^2 c^2 (b^2-c^2)^4+a^10 (b^2+c^2)-2 a^8 (2 b^4+b^2 c^2+2 c^4)+a^2 (b^2-c^2)^2 (b^6-8 b^4 c^2-8 b^2 c^4+c^6)+a^6 (6 b^6-3 b^4 c^2-3 b^2 c^4+6 c^6)+a^4 (-4 b^8+11 b^6 c^2+4 b^4 c^4+11 b^2 c^6-4 c^8)) (-8 a^38 (b^2+c^2)+8 a^36 (11 b^4+16 b^2 c^2+11 c^4)-2 a^34 (203 b^6+346 b^4 c^2+346 b^2 c^4+203 c^6)+b^2 c^2 (b^2-c^2)^14 (b^8-7 b^6 c^2+2 b^4 c^4-7 b^2 c^6+c^8)+a^32 (911 b^8+1504 b^6 c^2+1546 b^4 c^4+1504 b^2 c^6+911 c^8)+a^30 (-375 b^10+379 b^8 c^2+1770 b^6 c^4+1770 b^4 c^6+379 b^2 c^8-375 c^10)-a^28 (3951 b^12+9642 b^10 c^2+14415 b^8 c^4+16400 b^6 c^6+14415 b^4 c^8+9642 b^2 c^10+3951 c^12)-a^2 (b^2-c^2)^12 (b^14+14 b^12 c^2-28 b^10 c^4-19 b^8 c^6-19 b^6 c^8-28 b^4 c^10+14 b^2 c^12+c^14)+a^26 (13871 b^14+23294 b^12 c^2+25244 b^10 c^4+25107 b^8 c^6+25107 b^6 c^8+25244 b^4 c^10+23294 b^2 c^12+13871 c^14)+a^4 (b^2-c^2)^10 (11 b^16+48 b^14 c^2+80 b^12 c^4-75 b^10 c^6-137 b^8 c^8-75 b^6 c^10+80 b^4 c^12+48 b^2 c^14+11 c^16)-a^24 (26377 b^16+25133 b^14 c^2+9558 b^12 c^4-3005 b^10 c^6-7662 b^8 c^8-3005 b^6 c^10+9558 b^4 c^12+25133 b^2 c^14+26377 c^16)-a^6 (b^2-c^2)^8 (53 b^18-121 b^16 c^2+731 b^14 c^4+284 b^12 c^6-633 b^10 c^8-633 b^8 c^10+284 b^6 c^12+731 b^4 c^14-121 b^2 c^16+53 c^18)+a^22 (34177 b^18+3035 b^16 c^2-25859 b^14 c^4-40867 b^12 c^6-48642 b^10 c^8-48642 b^8 c^10-40867 b^6 c^12-25859 b^4 c^14+3035 b^2 c^16+34177 c^18)+a^20 (-32175 b^20+28798 b^18 c^2+32897 b^16 c^4+27070 b^14 c^6+28874 b^12 c^8+29376 b^10 c^10+28874 b^8 c^12+27070 b^6 c^14+32897 b^4 c^16+28798 b^2 c^18-32175 c^20)+a^8 (b^2-c^2)^6 (133 b^20-1385 b^18 c^2+2047 b^16 c^4+1943 b^14 c^6-629 b^12 c^8-3316 b^10 c^10-629 b^8 c^12+1943 b^6 c^14+2047 b^4 c^16-1385 b^2 c^18+133 c^20)-a^10 (b^2-c^2)^4 (75 b^22-4422 b^20 c^2+3258 b^18 c^4+4600 b^16 c^6+2121 b^14 c^8-6304 b^12 c^10-6304 b^10 c^12+2121 b^8 c^14+4600 b^6 c^16+3258 b^4 c^18-4422 b^2 c^20+75 c^22)+a^18 (22451 b^22-42042 b^20 c^2+3938 b^18 c^4+11404 b^16 c^6+1037 b^14 c^8+890 b^12 c^10+890 b^10 c^12+1037 b^8 c^14+11404 b^6 c^16+3938 b^4 c^18-42042 b^2 c^20+22451 c^22)+a^16 (-11531 b^24+28883 b^22 c^2-33799 b^20 c^4-3962 b^18 c^6+12622 b^16 c^8+1863 b^14 c^10+2128 b^12 c^12+1863 b^10 c^14+12622 b^8 c^16-3962 b^6 c^18-33799 b^4 c^20+28883 b^2 c^22-11531 c^24)-a^12 (b^2-c^2)^2 (837 b^24+5536 b^22 c^2-1348 b^20 c^4-7401 b^18 c^6-7496 b^16 c^8+7215 b^14 c^10+8320 b^12 c^12+7215 b^10 c^14-7496 b^8 c^16-7401 b^6 c^18-1348 b^4 c^20+5536 b^2 c^22+837 c^24)+a^14 (4147 b^26-7663 b^24 c^2+19280 b^22 c^4-19939 b^20 c^6-10560 b^18 c^8+13150 b^16 c^10+2071 b^14 c^12+2071 b^12 c^14+13150 b^10 c^16-10560 b^8 c^18-19939 b^6 c^20+19280 b^4 c^22-7663 b^2 c^24+4147 c^26)) : :
Barycentrics    131072 S^16 (10 R^4+3 S^2-2 R^2 SA+2 SA^2-8 R^2 SW+SW^2) (36 S^8+2 S^6 (1463 R^4-45 R^2 SA+9 SA^2-351 R^2 SW+9 SA SW-8 SW^2)+S^4 (11904 R^8+1985 R^6 SA+463 R^4 SA^2-11101 R^6 SW-1797 R^4 SA SW+294 R^2 SA^2 SW+2394 R^4 SW^2-158 R^2 SA SW^2-106 SA^2 SW^2+120 R^2 SW^3+138 SA SW^3-68 SW^4)+S^2 (-16100 R^12-4610 R^10 SA-948 R^8 SA^2+43370 R^10 SW+11922 R^8 SA SW+3877 R^6 SA^2 SW-48142 R^8 SW^2-13864 R^6 SA SW^2-3040 R^4 SA^2 SW^2+27511 R^6 SW^3+7314 R^4 SA SW^3+728 R^2 SA^2 SW^3-8444 R^4 SW^4-1574 R^2 SA SW^4-50 SA^2 SW^4+1318 R^2 SW^5+110 SA SW^5-80 SW^6)-SA (SA-SW) (1750 R^12-3350 R^10 SW+1890 R^8 SW^2+159 R^6 SW^3-457 R^4 SW^4+126 R^2 SW^5-10 SW^6)) : :

Let P be a point and A'B'C', A"B"C" the cevian triangle and pedal triangle of P, resp. Let A* be the other than midpoint of AP intersection of the nine-point circles of APB', APC'. Define B*, C* cyclically. The triangles A"B"C" and A*B*C* are cyclologic. For P= X(1), the cyclologic center (A*B*C*, A"B"C") is X(1319) and the cyclologic center (A"B"C", A*B*C*) is X(3649). For P = X(5), the cyclologic center (A*B*C*, A"B"C") is X(30482) and the cyclologic center (A"B"C", A*B*C*) is X(45139)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2667.

X(45139) = lies on these lines: { }

X(45140) = X(3)X(106)∩X(6)X(649)

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

X(45140) lies on the cubic K297 and these lines: {3, 106}, {6, 649}, {55, 43922}, {183, 903}, {901, 3052}, {956, 4674}, {1376, 36814}, {1616, 17109}, {16434, 38531}, {21448, 42316}

X(45140) = X(4358)-isoconjugate of X(17222)
X(45140) = crosssum of X(519) and X(12035)
X(45140) = crossdifference of every pair of points on line {519, 14425}
X(45140) = barycentric product X(i)*X(j) for these {i,j}: {106, 17132}, {2226, 12035}
X(45140) = barycentric quotient X(i)/X(j) for these {i,j}: {12035, 36791}, {17132, 3264}

X(45141) = X(2)X(253)∩X(3)X(112)

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

X(45141) lies on the cubic K297 and these lines: {2, 253}, {3, 112}, {4, 9605}, {5, 41361}, {6, 25}, {19, 3752}, {22, 15905}, {24, 30435}, {30, 41370}, {32, 3515}, {39, 1593}, {53, 5064}, {74, 38920}, {132, 15274}, {182, 40801}, {183, 648}, {186, 1384}, {193, 37187}, {216, 1033}, {230, 37453}, {235, 5286}, {262, 33971}, {264, 11174}, {297, 7774}, {317, 41624}, {325, 17907}, {378, 5024}, {381, 5523}, {383, 36302}, {393, 427}, {428, 3087}, {458, 3329}, {468, 2452}, {574, 11410}, {577, 17409}, {607, 1193}, {608, 9316}, {612, 2336}, {614, 1108}, {672, 3195}, {800, 1184}, {956, 1783}, {1080, 36303}, {1191, 41320}, {1212, 36103}, {1285, 37460}, {1370, 42459}, {1398, 2275}, {1513, 41371}, {1529, 7710}, {1560, 32216}, {1609, 8792}, {1656, 41366}, {1829, 9575}, {1885, 7738}, {1902, 9593}, {1968, 3516}, {1990, 3815}, {1995, 15262}, {2276, 7071}, {2332, 4255}, {2548, 7507}, {2549, 44438}, {3018, 8791}, {3053, 15750}, {3199, 5198}, {3314, 11331}, {3424, 41374}, {3517, 10312}, {3541, 31406}, {3542, 5305}, {4232, 5702}, {5020, 15355}, {5050, 41363}, {5065, 36417}, {5158, 11284}, {5254, 37197}, {5280, 11399}, {5299, 11398}, {5304, 6353}, {5475, 18386}, {6000, 41376}, {6103, 37637}, {6529, 41372}, {6530, 9744}, {6531, 36822}, {6591, 42758}, {6995, 14930}, {7124, 8192}, {7387, 22120}, {7499, 8879}, {7539, 13854}, {7737, 37196}, {7745, 12173}, {7783, 37199}, {7794, 40187}, {8749, 9717}, {8750, 42316}, {8755, 29639}, {8889, 33630}, {8962, 15210}, {9715, 10316}, {9753, 42873}, {9756, 15576}, {9909, 10313}, {10151, 43448}, {10317, 14070}, {10542, 11470}, {10565, 36413}, {11163, 37765}, {11380, 13356}, {11414, 23115}, {11479, 26216}, {12083, 22121}, {12174, 32445}, {12315, 41367}, {13509, 32063}, {13526, 13613}, {13860, 41204}, {13884, 44596}, {13937, 44595}, {14248, 17980}, {14961, 21312}, {15014, 31859}, {15487, 40943}, {15655, 35472}, {16252, 41369}, {16308, 21284}, {18533, 18907}, {18535, 22246}, {19544, 41364}, {21058, 22057}, {22253, 44146}, {23050, 44798}, {31467, 37119}, {34266, 37360}, {35325, 40805}, {37182, 44704}, {37689, 38282}

X(45141) = isogonal conjugate of X(42287)
X(45141) = isogonal conjugate of the polar conjugate of X(10002)
X(45141) = polar conjugate of the isotomic conjugate of X(1350)
X(45141) = X(i)-Ceva conjugate of X(j) for these (i,j): {40801, 25}, {42330, 4}
X(45141) = crosspoint of X(i) and X(j) for these (i,j): {4, 42373}, {1350, 40813}
X(45141) = crosssum of X(525) and X(12037)
X(45141) = crossdifference of every pair of points on line {525, 42658}
X(45141) = X(i)-isoconjugate of X(j) for these (i,j): {1, 42287}, {63, 3424}
X(45141) = barycentric product X(i)*X(j) for these {i,j}: {1, 23052}, {3, 10002}, {4, 1350}, {25, 37668}, {511, 45031}, {1249, 40813}, {1297, 1529}, {1301, 14343}, {7710, 40801}, {12037, 23964}
X(45141) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 42287}, {25, 3424}, {1301, 35571}, {1350, 69}, {1529, 30737}, {10002, 264}, {12037, 36793}, {23052, 75}, {37668, 305}, {40813, 34403}, {45031, 290}
X(45141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1249, 16318}, {3, 3172, 8778}, {3, 8743, 3172}, {6, 154, 8779}, {6, 232, 25}, {25, 44090, 19118}, {39, 2207, 1593}, {393, 7736, 427}, {1968, 5013, 3516}, {2548, 27376, 7507}, {3517, 43136, 10312}, {5410, 5411, 19118}, {6530, 9744, 37074}, {6995, 14930, 40065}, {7710, 10002, 1529}, {8743, 39575, 3}, {9909, 38292, 10313}

X(45142) = X(3)X(741)∩X(6)X(798)

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

X(45142) lies on the cubic K297 and these lines: {3, 741}, {6, 798}, {183, 18827}, {291, 956}, {292, 3445}, {813, 12029}, {1376, 36817}, {4383, 4584}

X(45142) = X(3570)-isoconjugate of X(23835)
X(45142) = barycentric product X(876)*X(23831)
X(45142) = barycentric quotient X(i)/X(j) for these {i,j}: {875, 23835}, {23831, 874}

X(45143) = X(2)X(36877)∩X(3)X(111)

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

X(45143) lies on the cubic K297 and these lines: {2, 36877}, {3, 111}, {6, 512}, {183, 671}, {381, 34169}, {691, 1384}, {892, 22253}, {5024, 5968}, {7841, 31125}, {8371, 35606}, {9145, 45012}, {9214, 15048}, {9605, 14246}, {11284, 41936}, {11318, 30786}, {11580, 11634}, {15899, 40126}, {21309, 41404}

X(45143) = X(23889)-isoconjugate of X(43674)
X(45143) = crosssum of X(524) and X(12036)
X(45143) = crossdifference of every pair of points on line {524, 9125}
X(45143) = barycentric product X(10630)*X(12036)
X(45143) = barycentric quotient X(i)/X(j) for these {i,j}: {9178, 43674}, {12036, 36792}
X(45143) = {X(14609),X(17964)}-harmonic conjugate of X(6)

X(45144) = X(3)X(101)∩X(6)X(657)

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

X(45144) lies on the cubic K297 and these lines: {3, 101}, {6, 657}, {183, 18025}, {953, 40116}, {1023, 5440}, {3052, 32642}, {3446, 37519}, {4604, 36101}, {22356, 23344}

X(45144) = X(i)-isoconjugate of X(j) for these (i,j): {88, 516}, {106, 30807}, {676, 3257}, {903, 910}, {1022, 2398}, {1320, 43035}, {1456, 4997}, {4674, 14953}, {9456, 35517}, {23345, 42719}, {26006, 36125}
X(45144) = trilinear pole of line {902, 22086}
X(45144) = crossdifference of every pair of points on line {516, 676}
X(45144) = barycentric product X(i)*X(j) for these {i,j}: {44, 36101}, {103, 519}, {677, 900}, {902, 18025}, {911, 4358}, {1815, 8756}, {2338, 3911}, {2400, 23344}, {2424, 17780}, {3689, 43736}, {3762, 36039}, {4528, 24016}, {5440, 36122}, {9503, 14439}, {36056, 38462}
X(45144) = barycentric quotient X(i)/X(j) for these {i,j}: {44, 30807}, {103, 903}, {519, 35517}, {677, 4555}, {902, 516}, {911, 88}, {1023, 42719}, {1404, 43035}, {1960, 676}, {2251, 910}, {2338, 4997}, {2424, 6548}, {3285, 14953}, {22086, 39470}, {22356, 26006}, {23344, 2398}, {32642, 901}, {32657, 1797}, {36039, 3257}, {36101, 20568}

X(45145) = X(3)X(8)∩X(6)X(650)

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

X(45145) lies on the cubic K297 and these lines: {3, 8}, {6, 650}, {183, 18816}, {381, 38952}, {1309, 29348}, {1376, 15635}, {2250, 42316}, {3658, 16704}, {4009, 23343}, {22753, 31849}

X(45145) = X(30583)-cross conjugate of X(23343)
X(45145) = trilinear pole of line {3230, 4526}
X(45145) = crossdifference of every pair of points on line {517, 3310}
X(45145) = X(i)-isoconjugate of X(j) for these (i,j): {517, 37129}, {739, 908}, {859, 41683}, {898, 1769}, {1457, 36798}, {2183, 3227}, {2397, 23892}, {3310, 4607}, {10015, 34075}, {14260, 36872}, {32718, 36038}
X(45145) = barycentric product X(i)*X(j) for these {i,j}: {104, 536}, {891, 13136}, {899, 34234}, {909, 6381}, {2401, 23343}, {2423, 41314}, {3230, 18816}, {4009, 34051}, {4728, 36037}, {14430, 37136}, {34858, 35543}, {36816, 36819}
X(45145) = barycentric quotient X(i)/X(j) for these {i,j}: {104, 3227}, {536, 3262}, {890, 3310}, {891, 10015}, {899, 908}, {909, 37129}, {1646, 42753}, {2250, 41683}, {2423, 43928}, {3230, 517}, {3768, 1769}, {4526, 2804}, {4728, 36038}, {13136, 889}, {14434, 42764}, {14437, 23757}, {19945, 42754}, {23343, 2397}, {32641, 898}, {34234, 31002}, {34858, 739}, {36037, 4607}

X(45146) = X(3)X(805)∩X(6)X(694)

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

X(45146) lies on the cubic K297 and these lines: {3, 805}, {6, 694}, {183, 18829}, {381, 38947}, {1916, 11170}, {4230, 44090}, {5050, 34238}, {5085, 9467}, {5968, 9154}, {6234, 32524}, {11171, 16068}

X(45146) = X(1580)-isoconjugate of X(43532)
X(45146) = barycentric product X(i)*X(j) for these {i,j}: {694, 39099}, {1916, 2080}
X(45146) = barycentric quotient X(i)/X(j) for these {i,j}: {694, 43532}, {2080, 385}, {39099, 3978}
X(45146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9468, 18872, 6}, {18872, 40810, 36214}

X(45147) = X(30)X(511)∩X(110)X(930)

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

See Antreas Hatzipolakis and Peter Moses, euclid 2674.

X(45147) lies on these lines: {2, 42737}, {3, 44809}, {4, 14050}, {5, 43966}, {30, 511}, {74, 1141}, {99, 39448}, {110, 930}, {113, 128}, {125, 137}, {246, 338}, {351, 14698}, {399, 6069}, {879, 34437}, {1109, 14101}, {1176, 22105}, {1263, 10264}, {1291, 43965}, {1511, 38615}, {2081, 14582}, {2914, 27423}, {3024, 3327}, {3028, 7159}, {3448, 11671}, {3521, 15453}, {5653, 32312}, {5972, 13372}, {6132, 14417}, {6140, 6592}, {6699, 34837}, {7728, 31656}, {7731, 13505}, {9138, 13291}, {9140, 11117}, {9148, 13448}, {9919, 15960}, {10117, 15959}, {10278, 36255}, {10620, 38587}, {10721, 44981}, {10733, 44976}, {11559, 15328}, {11615, 44816}, {11702, 14071}, {12041, 38618}, {13201, 13504}, {13289, 23320}, {13856, 34770}, {14094, 38681}, {14106, 23290}, {14220, 16835}, {14270, 14695}, {14424, 34291}, {14446, 23284}, {14447, 23283}, {14559, 35345}, {14652, 14809}, {14677, 43083}, {14697, 15366}, {14857, 43958}, {15035, 38706}, {15054, 38683}, {15055, 38710}, {15321, 35909}, {15543, 20126}, {18125, 35364}, {18310, 39905}, {18311, 38359}, {18781, 44427}, {23315, 23319}, {23515, 23516}, {38638, 38640}, {40048, 43689}, {44808, 44826}

X(45147) = isogonal conjugate of X(1291)
X(45147) = isotomic conjugate of the isogonal conjugate of X(6140)
X(45147) = Thomson isogonal conjugate of X(14979)
X(45147) = crossdifference of every pair of points on line {6, 3200}
X(45147) = circumorthic-isogonal conjugate of X(14979)
X(45147) = 1st-circumperp-isogonal conjugate of X(1291)
X(45147) = 2nd-circumperp-isogonal conjugate of X(14979)
X(45147) = Lucas-isogonal conjugate of X(14979)
X(45147) = Napoleon-Feuerbach-isogonal conjugate of X(14979)
X(45147) = crosspoint of X(94) and X(99)
X(45147) = crosssum of X(50) and X(512)
X(45147) = X(1510)-of-orthocentroidal-triangle
X(45147) = X(1510)-of-anti-orthocentroidal-triangle
X(45147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 13290, 14610}, {9138, 13291, 32193}

X(45148) = 11TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^18-8 a^16 b^2+9 a^14 b^4-6 a^12 b^6+a^10 b^8+2 a^8 b^10-a^6 b^12-2 a^4 b^14+4 a^2 b^16-2 b^18-8 a^16 c^2+16 a^14 b^2 c^2-12 a^12 b^4 c^2+10 a^10 b^6 c^2-2 a^8 b^8 c^2-6 a^6 b^10 c^2+6 a^4 b^12 c^2-12 a^2 b^14 c^2+8 b^16 c^2+9 a^14 c^4-12 a^12 b^2 c^4-3 a^10 b^4 c^4-2 a^8 b^6 c^4+9 a^6 b^8 c^4+13 a^2 b^12 c^4-14 b^14 c^4-6 a^12 c^6+10 a^10 b^2 c^6-2 a^8 b^4 c^6-4 a^6 b^6 c^6-4 a^4 b^8 c^6-8 a^2 b^10 c^6+14 b^12 c^6+a^10 c^8-2 a^8 b^2 c^8+9 a^6 b^4 c^8-4 a^4 b^6 c^8+6 a^2 b^8 c^8-6 b^10 c^8+2 a^8 c^10-6 a^6 b^2 c^10-8 a^2 b^6 c^10-6 b^8 c^10-a^6 c^12+6 a^4 b^2 c^12+13 a^2 b^4 c^12+14 b^6 c^12-2 a^4 c^14-12 a^2 b^2 c^14-14 b^4 c^14+4 a^2 c^16+8 b^2 c^16-2 c^18 : :
Barycentrics    S^6 (6 R^2+SA-3 SW)-4 SB SC (4 R^2-SW) SW^4-2 S^4 (18 R^2 SB SC-3 R^2 SA SW-7 SB SC SW+7 R^2 SW^2+SA SW^2-2 SW^3)+S^2 SW^2 (66 R^2 SB SC-4 R^2 SA SW-18 SB SC SW+4 R^2 SW^2+SA SW^2-SW^3) : :
X(45148) = 3*X(14639)-2*X(41175)

For the definition of the HS circle see the preamble before X(44937)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45148) lies on the HS Circle and these lines: {4,2715}, {20,36471}, {30,2710}, {512,10735}, {525,10723}, {1503,5111}, {14639,41175}

X(45148) = reflection of X(i) in X(j) for these (i,j): (20,36471), (2715,4)

X(45149) = 12TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^16-14 a^14 b^2+30 a^12 b^4-27 a^10 b^6+19 a^8 b^8-6 a^6 b^10-10 a^4 b^12+7 a^2 b^14-2 b^16-14 a^14 c^2+50 a^12 b^2 c^2-91 a^10 b^4 c^2+50 a^8 b^6 c^2-23 a^6 b^8 c^2+67 a^4 b^10 c^2-36 a^2 b^12 c^2+13 b^14 c^2+30 a^12 c^4-91 a^10 b^2 c^4+161 a^8 b^4 c^4-83 a^6 b^6 c^4-135 a^4 b^8 c^4+68 a^2 b^10 c^4-38 b^12 c^4-27 a^10 c^6+50 a^8 b^2 c^6-83 a^6 b^4 c^6+248 a^4 b^6 c^6-47 a^2 b^8 c^6+63 b^10 c^6+19 a^8 c^8-23 a^6 b^2 c^8-135 a^4 b^4 c^8-47 a^2 b^6 c^8-72 b^8 c^8-6 a^6 c^10+67 a^4 b^2 c^10+68 a^2 b^4 c^10+63 b^6 c^10-10 a^4 c^12-36 a^2 b^2 c^12-38 b^4 c^12+7 a^2 c^14+13 b^2 c^14-2 c^16 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45149) lies on the HS Circle and this line: {4,2858}

X(45149) = reflection of X(2858) in X(4)

X(45150) = 13TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^12 b^4-9 a^10 b^6+3 a^8 b^8+5 a^6 b^10-2 a^4 b^12-6 a^12 b^2 c^2+9 a^10 b^4 c^2+11 a^8 b^6 c^2-21 a^6 b^8 c^2-a^4 b^10 c^2+4 a^2 b^12 c^2+3 a^12 c^4+9 a^10 b^2 c^4-31 a^8 b^4 c^4+21 a^6 b^6 c^4+23 a^4 b^8 c^4-13 a^2 b^10 c^4-2 b^12 c^4-9 a^10 c^6+11 a^8 b^2 c^6+21 a^6 b^4 c^6-51 a^4 b^6 c^6+11 a^2 b^8 c^6+9 b^10 c^6+3 a^8 c^8-21 a^6 b^2 c^8+23 a^4 b^4 c^8+11 a^2 b^6 c^8-14 b^8 c^8+5 a^6 c^10-a^4 b^2 c^10-13 a^2 b^4 c^10+9 b^6 c^10-2 a^4 c^12+4 a^2 b^2 c^12-2 b^4 c^12 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45150) lies on the HS Circle and this line: {4,3222}

X(45150) = reflection of X(3222) in X(4)
X(45158) = orthic-isogonal conjugate of X(2794)
X(45158) = X(2715)-of-Euler-triangle
X(45158) = X(36471)-of-Johnson-triangle

X(45151) = 14TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^14-12 a^12 b^2+10 a^10 b^4+12 a^8 b^6-19 a^6 b^8+2 a^4 b^10+6 a^2 b^12-2 b^14-12 a^12 c^2+46 a^10 b^2 c^2-48 a^8 b^4 c^2+22 a^6 b^6 c^2+18 a^4 b^8 c^2-38 a^2 b^10 c^2+12 b^12 c^2+10 a^10 c^4-48 a^8 b^2 c^4+21 a^6 b^4 c^4-20 a^4 b^6 c^4+79 a^2 b^8 c^4-24 b^10 c^4+12 a^8 c^6+22 a^6 b^2 c^6-20 a^4 b^4 c^6-94 a^2 b^6 c^6+14 b^8 c^6-19 a^6 c^8+18 a^4 b^2 c^8+79 a^2 b^4 c^8+14 b^6 c^8+2 a^4 c^10-38 a^2 b^2 c^10-24 b^4 c^10+6 a^2 c^12+12 b^2 c^12-2 c^14 : :

Barycentrics    5 S^4 (5 R^2-SW)+3 SB SC SW^2 (R^2+SW)+S^2 (-150 R^2 SB SC-5 R^2 SA SW+25 SB SC SW+R^2 SW^2+2 SA SW^2-SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45151) lies on the HS Circle and these lines: {4,5966}, {20,31843}, {3627,6033}, {10733,32062}

X(45151) = reflection of X(i) in X(j) for these (i,j): (20,31843), (5966,4)

X(45152) = 15TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^8 b^3+4 a^7 b^4-2 a^5 b^6-3 a^4 b^7-2 a^3 b^8-3 a^9 b c+4 a^7 b^3 c-a^6 b^4 c-2 a^5 b^5 c+a^4 b^6 c-a^3 b^7 c+2 a b^9 c-6 a^7 b^2 c^2-9 a^6 b^3 c^2+9 a^4 b^5 c^2+10 a^3 b^6 c^2+4 a^2 b^7 c^2+3 a^8 c^3+4 a^7 b c^3-9 a^6 b^2 c^3-3 a^5 b^3 c^3+4 a^4 b^4 c^3+2 a^3 b^5 c^3+3 a^2 b^6 c^3-6 a b^7 c^3-2 b^8 c^3+4 a^7 c^4-a^6 b c^4+4 a^4 b^3 c^4-14 a^3 b^4 c^4-11 a^2 b^5 c^4-2 b^7 c^4-2 a^5 b c^5+9 a^4 b^2 c^5+2 a^3 b^3 c^5-11 a^2 b^4 c^5+8 a b^5 c^5+4 b^6 c^5-2 a^5 c^6+a^4 b c^6+10 a^3 b^2 c^6+3 a^2 b^3 c^6+4 b^5 c^6-3 a^4 c^7-a^3 b c^7+4 a^2 b^2 c^7-6 a b^3 c^7-2 b^4 c^7-2 a^3 c^8-2 b^3 c^8+2 a b c^9 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45152) lies on the HS Circle and these lines: {4,6010}, {30,741}, {1356,12953}, {10722,41869}

X(45152) = reflection of X(6010) in X(4)

X(45153) = 16TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    12 a^16-38 a^14 b^2-6 a^12 b^4+92 a^10 b^6-32 a^8 b^8-70 a^6 b^10+34 a^4 b^12+16 a^2 b^14-8 b^16-38 a^14 c^2+131 a^12 b^2 c^2-97 a^10 b^4 c^2-137 a^8 b^6 c^2+227 a^6 b^8 c^2-34 a^4 b^10 c^2-92 a^2 b^12 c^2+40 b^14 c^2-6 a^12 c^4-97 a^10 b^2 c^4+261 a^8 b^4 c^4-149 a^6 b^6 c^4-145 a^4 b^8 c^4+168 a^2 b^10 c^4-48 b^12 c^4+92 a^10 c^6-137 a^8 b^2 c^6-149 a^6 b^4 c^6+306 a^4 b^6 c^6-92 a^2 b^8 c^6-40 b^10 c^6-32 a^8 c^8+227 a^6 b^2 c^8-145 a^4 b^4 c^8-92 a^2 b^6 c^8+112 b^8 c^8-70 a^6 c^10-34 a^4 b^2 c^10+168 a^2 b^4 c^10-40 b^6 c^10+34 a^4 c^12-92 a^2 b^2 c^12-48 b^4 c^12+16 a^2 c^14+40 b^2 c^14-8 c^16 : :
Barycentrics    S^4 (36 R^2-11 SW) (9 R^2-2 SW)+2 SB SC (27 R^2-7 SW) SW^3+S^2 (-1458 R^4 SB SC+783 R^2 SB SC SW+27 R^2 SA SW^2-102 SB SC SW^2-21 R^2 SW^3+8 SB SW^3+8 SC SW^3-2 SW^4) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45153) lies on the HS Circle and these lines: {4,6236}, {20,34113}, {30,6325}, {3830,10734}, {10733,32228}, {11594,14989}

X(45153) = reflection of X(i) in X(j) for these (i,j): (20,34113), (6236,4)

X(45154) = 17TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    2 a^14 b^6+4 a^12 b^8-4 a^8 b^12-2 a^6 b^14+3 a^16 b^2 c^2-a^14 b^4 c^2-5 a^12 b^6 c^2+2 a^10 b^8 c^2+11 a^8 b^10 c^2+9 a^6 b^12 c^2-a^4 b^14 c^2-2 a^2 b^16 c^2-a^14 b^2 c^4-3 a^12 b^4 c^4-10 a^10 b^6 c^4-4 a^8 b^8 c^4+3 a^6 b^10 c^4+3 a^4 b^12 c^4+4 a^2 b^14 c^4+2 a^14 c^6-5 a^12 b^2 c^6-10 a^10 b^4 c^6-2 a^8 b^6 c^6-2 a^6 b^8 c^6+4 a^4 b^10 c^6-2 a^2 b^12 c^6-b^14 c^6+4 a^12 c^8+2 a^10 b^2 c^8-4 a^8 b^4 c^8-2 a^6 b^6 c^8-8 a^2 b^10 c^8+11 a^8 b^2 c^10+3 a^6 b^4 c^10+4 a^4 b^6 c^10-8 a^2 b^8 c^10+2 b^10 c^10-4 a^8 c^12+9 a^6 b^2 c^12+3 a^4 b^4 c^12-2 a^2 b^6 c^12-2 a^6 c^14-a^4 b^2 c^14+4 a^2 b^4 c^14-b^6 c^14-2 a^2 b^2 c^16 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45154) lies on the HS Circle and this line: {4,6572}

X(45154) = reflection of X(6572) in X(4)

X(45155) = 18TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    3 a^14+4 a^12 b^2-12 a^10 b^4-8 a^8 b^6+13 a^6 b^8+6 a^4 b^10-4 a^2 b^12-2 b^14+4 a^12 c^2+2 a^10 b^2 c^2-6 a^8 b^4 c^2+4 a^6 b^6 c^2-6 a^4 b^8 c^2+2 a^2 b^10 c^2-12 a^10 c^4-6 a^8 b^2 c^4+41 a^6 b^4 c^4-18 a^4 b^6 c^4+19 a^2 b^8 c^4+12 b^10 c^4-8 a^8 c^6+4 a^6 b^2 c^6-18 a^4 b^4 c^6-34 a^2 b^6 c^6-10 b^8 c^6+13 a^6 c^8-6 a^4 b^2 c^8+19 a^2 b^4 c^8-10 b^6 c^8+6 a^4 c^10+2 a^2 b^2 c^10+12 b^4 c^10-4 a^2 c^12-2 c^14 : :
Barycentrics    SB SC (98 R^2-23 SW) SW^2+S^4 (2 R^2+SW)+S^2 (-9 R^2 SB SC-7 R^2 SA SW-5 SB SC SW-14 R^2 SW^2+2 SB SW^2+2 SC SW^2+3 SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45155) lies on the HS Circle and these lines: {4,7953}, {30,29316}

X(45155) = reflection of X(7953) in X(4)

X(45156) = 19TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    48 a^14-188 a^12 b^2+218 a^10 b^4-40 a^8 b^6-46 a^6 b^8-64 a^4 b^10+104 a^2 b^12-32 b^14-188 a^12 c^2+559 a^10 b^2 c^2-559 a^8 b^4 c^2+241 a^6 b^6 c^2+193 a^4 b^8 c^2-430 a^2 b^10 c^2+168 b^12 c^2+218 a^10 c^4-559 a^8 b^2 c^4+301 a^6 b^4 c^4-217 a^4 b^6 c^4+737 a^2 b^8 c^4-308 b^10 c^4-40 a^8 c^6+241 a^6 b^2 c^6-217 a^4 b^4 c^6-806 a^2 b^6 c^6+172 b^8 c^6-46 a^6 c^8+193 a^4 b^2 c^8+737 a^2 b^4 c^8+172 b^6 c^8-64 a^4 c^10-430 a^2 b^2 c^10-308 b^4 c^10+104 a^2 c^12+168 b^2 c^12-32 c^14 : :
Barycentrics    27 S^6 (18 R^2+3 SA-5 SW)-6 SB SC SW^5-6 S^4 (486 R^2 SB SC+27 R^2 SA SW-99 SB SC SW-3 R^2 SW^2+3 SB SW^2+3 SC SW^2-2 SW^3)+S^2 SW^2 (162 R^2 SB SC+12 SB SC SW-SB SW^2-SC SW^2+2 SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45156) lies on the HS Circle and these lines: {4,8600}, {20,16938}

X(45156) = reflection of X(i) in X(j) for these (i,j): (20,16938), (8600,4)

X(45157) = 20TH HATZIPOLAKIS-SUPPA POINT ON THE HS CIRCLE

Barycentrics    12 a^10+6 a^9 b-20 a^8 b^2-14 a^7 b^3-4 a^6 b^4+6 a^5 b^5+12 a^4 b^6+6 a^3 b^7+8 a^2 b^8-4 a b^9-8 b^10+6 a^9 c-9 a^8 b c-3 a^7 b^2 c+10 a^6 b^3 c-6 a^5 b^4 c+3 a^4 b^5 c-3 a^3 b^6 c+6 a b^8 c-4 b^9 c-20 a^8 c^2-3 a^7 b c^2+59 a^6 b^2 c^2+12 a^5 b^3 c^2-22 a^4 b^4 c^2-15 a^3 b^5 c^2-41 a^2 b^6 c^2+6 a b^7 c^2+24 b^8 c^2-14 a^7 c^3+10 a^6 b c^3+12 a^5 b^2 c^3-22 a^4 b^3 c^3+12 a^3 b^4 c^3-14 a b^6 c^3+16 b^7 c^3-4 a^6 c^4-6 a^5 b c^4-22 a^4 b^2 c^4+12 a^3 b^3 c^4+66 a^2 b^4 c^4+6 a b^5 c^4-16 b^6 c^4+6 a^5 c^5+3 a^4 b c^5-15 a^3 b^2 c^5+6 a b^4 c^5-24 b^5 c^5+12 a^4 c^6-3 a^3 b c^6-41 a^2 b^2 c^6-14 a b^3 c^6-16 b^4 c^6+6 a^3 c^7+6 a b^2 c^7+16 b^3 c^7+8 a^2 c^8+6 a b c^8+24 b^2 c^8-4 a c^9-4 b c^9-8 c^10 : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45157) lies on the HS Circle and these lines: {4,8652}, {30,28145}

X(45157) = reflection of X(8652) in X(4)

X(45158) = 11TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (2 a^8-2 a^6 b^2+a^4 b^4-b^8-2 a^6 c^2+2 b^6 c^2+a^4 c^4-2 b^4 c^4+2 b^2 c^6-c^8) (a^8 b^2-2 a^6 b^4+2 a^4 b^6-2 a^2 b^8+b^10+a^8 c^2-a^4 b^4 c^2+2 a^2 b^6 c^2-2 b^8 c^2-2 a^6 c^4-a^4 b^2 c^4+b^6 c^4+2 a^4 c^6+2 a^2 b^2 c^6+b^4 c^6-2 a^2 c^8-2 b^2 c^8+c^10) : :
Barycentrics    S^4+2 SB SC SW^2+S^2 (-3 SB SC-SB SW-SC SW)) (S^2 (-6 R^2+SA+SW)+SW (SB SC+4 R^2 SW-SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45158) lies on the nine-point circle and these lines: {2,2710}, {4,2715}, {5,36471}, {39,38974}, {114,525}, {115,1503}, {122,441}, {125,2450}, {127,511}, {132,512}, {133,39533}, {136,460}, {182,38971}, {1316,3258}, {1640,38975}, {2679,31850}, {2794,41175}, {5099,5480}, {6794,9744}, {18338,33504}

X(45158) = midpoint of X(4) and X(2715)
X(45158) = reflection of X(36471) in X(5)
X(45158) = complement of X(2710)
X(45158) = complementary conjugate of X(2794)
X(45158) = X(4)-Ceva conjugate of X(2794)
X(45158) = X(i)-complementary conjugate of X(j) for these (i,j): (1,2794), (2794,10)
X(45158) = X(45148)-image under 2nd HS transform

X(45159) = 12TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (a^6 b^2-a^4 b^4+a^2 b^6-b^8+a^6 c^2-6 a^4 b^2 c^2+4 a^2 b^4 c^2+3 b^6 c^2-a^4 c^4+4 a^2 b^2 c^4-8 b^4 c^4+a^2 c^6+3 b^2 c^6-c^8) (2 a^8-4 a^6 b^2+9 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2-8 a^4 b^2 c^2+2 a^2 b^4 c^2-2 b^6 c^2+9 a^4 c^4+2 a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-2 b^2 c^6+c^8) : :
Barycentrics    2 S^4-(SA-SW) SW^3+S^2 (18 R^2 SA-3 SA^2-18 R^2 SW+SA SW)) (4 S^4+SA SW^3+S^2 (-18 R^2 SA+3 SA^2-SA SW-SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45159) lies on the nine-point circle and these lines: {4,2858}, {5099,7818}

X(45159) = midpoint of X(4) and X(2858)
X(45159) = complement of the circumperp conjugate of X(2858)
X(45159) = X(2858)-of-Euler-triangle
X(45159) = X(45149)-image under 2nd HS transform

X(45160) = 13TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    ((3*b^4-4*b^2*c^2+3*c^4)*a^4-(b^6+c^6)*a^2+(b^2-c^2)^2*b^2*c^2)*((b^2-c^2)^2*a^4-(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2+b^2*c^2*(b^4-4*b^2*c^2+c^4)) : :
Barycentrics    2 S^4+(SA-SW) SW^3+S^2 (-24 R^2 SA+4 SA^2+24 R^2 SW+SA SW-3 SW^2)) (3 S^4+SA SW^3+S^2 (-24 R^2 SA+4 SA^2+SA SW-SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45160) lies on the nine-point circle and these lines: {4,3222}, {125,5025}, {264,5139}

X(45160) = midpoint of X(4) and X(3222)
X(45160) = complement of the circumperp conjugate of X(3222)
X(45160) = X(3222)-of-Euler-triangle
X(45160) = X(45150)-image under 2nd HS transform

X(45161) = 14TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (b^2-c^2)^2 (-4 a^2+b^2+c^2) (-a^4+b^4-3 b^2 c^2+c^4) : :
Barycentrics    (b^2-c^2)^2 (S^2+SA^2+4 SB SC) (5 SA-3 SW) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

Let A'B'C' be the orthic triangle. The Napoleon axes of triangles AB'C', BC'A', CA'B' bound a triangle perspective to A'B'C' at X(45161). (Randy Hutson, November 30, 2021)

X(45161) lies on the nine-point circle and these lines: {4,5966}, {5,31843}, {23,31655}, {113,43129}, {114,546}, {120,30446}, {126,5133}, {128,381}, {428,1560}, {1596,18402}, {10024,31842}, {11563,16188}, {14671,26863}

X(45161) = midpoint of X(4) and X(5966)
X(45161) = reflection of X(31843) in X(5)
X(45161) = complement of the isogonal conjugate of X(32478)
X(45161) = complementary conjugate of X(32478)
X(45161) = X(4)-Ceva conjugate of X(32478)
X(45161) = X(i)-complementary conjugate of X(j) for these (i,j): (1,32478), (661,3631), (3629,4369)
X(45161) = orthic-isogonal conjugate of X(32478)
X(45161) = X(5966)-of-Euler-triangle
X(45161) = X(31843)-of-Johnson-triangle
X(45161) = X(45151)-image under 2nd HS transform

X(45162) = 15TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (b+c)^2 (a^2-b c) (a^2 b^2+a b^3-a^2 b c-a b^2 c+a^2 c^2-a b c^2-b^2 c^2+a c^3) : :
X(45162) = 5*X(1698)-X(5539)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45162) lies on the nine-point circle and these lines: {2,741}, {4,6010}, {10,115}, {11,1211}, {12,1356}, {114,27929}, {116,3454}, {125,2887}, {126,27854}, {442,5518}, {1329,3037}, {1698,5539}, {3124,43534}, {3842,16587}, {5224,19643}, {5515,17755}, {10479,38477}, {16592,40533}, {19950,21431}, {19951,21604}, {20337,20340}, {20541,20546}, {21257,34528}

X(45162) = midpoint of X(4) and X(6010)
X(45162) = complement of X(741)
X(45162) = complementary conjugate of X(740)
X(45162) = X(4)-Ceva conjugate of X(740)
X(45162) = X(i)-complementary conjugate of X(j) for these (i,j): (1,740), (10,3836), (37,3912), (42,1575), (65,1738)
X(45162) = crosspoint of X(2)and X(35544)
X(45162) = barycentric product X(86)*X(20658)
X(45162) = trilinear product X(81)*X(20658)
X(45162) = orthic-isogonal conjugate of X(740)
X(45162) = X(6010)-of-Euler-triangle
X(45162) = X(45152)-image under 2nd HS transform

X(45163) = 16TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (4 a^6-a^4 b^2-3 a^2 b^4+2 b^6-a^4 c^2+4 a^2 b^2 c^2-2 b^4 c^2-3 a^2 c^4-2 b^2 c^4+2 c^6) (-2 a^4 b^2+2 b^6-2 a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-b^2 c^4+2 c^6) : :
Barycentrics    S^2 (18 R^2-7 SW)-SA (3 SA-4 SW) SW) (4 S^2 (9 R^2-2 SW)+(SA-SW) (3 SA-SW) SW : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45163) lies on the nine-point circle and these lines: {2,6325}, {4,6236}, {5,34113}, {113,32228}, {115,5169}, {125,597}, {127,10163}, {542,12494}, {690,13234}, {2781,12624}, {3258,11594}, {3845,5512}, {3849,5099}, {5139,10301}, {8704,16188}, {9517,13249}, {10254,14672}

X(45163) = midpoint of X(4) and X(6236)
X(45163) = reflection of X(34113) in X(5)
X(45163) = complement of X(6325)
X(45163) = complementary conjugate of X(8705)
X(45163) = X(4)-Ceva conjugate of X(8705)
X(45163) = X(1)-complementary conjugate of X(8705)
X(45163) = orthic-isogonal conjugate of X(8705)
X(45163) = X(6236)-of-Euler-triangle
X(45163) = X(34113)-of-Johnson-triangle
X(45163) = X(45153)-image under 2nd HS transform

X(45164) = 17TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (a^6 b^4-a^2 b^8+a^4 b^4 c^2+a^2 b^6 c^2+a^6 c^4+a^4 b^2 c^4-2 b^6 c^4+a^2 b^2 c^6-2 b^4 c^6-a^2 c^8) (a^8 b^2+2 a^6 b^4+a^4 b^6+a^8 c^2-2 a^6 b^2 c^2-a^4 b^4 c^2-a^2 b^6 c^2+b^8 c^2+2 a^6 c^4-a^4 b^2 c^4-2 a^2 b^4 c^4-b^6 c^4+a^4 c^6-a^2 b^2 c^6-b^4 c^6+b^2 c^8) : :
Barycentrics    SA SW^4+S^4 (2 R^2+SW)-S^2 SW (14 R^2 SA-2 SA^2-SA SW+SW^2)) (-2 S^4 (2 R^2-SW)+(SA-SW) SW^4-S^2 SW (14 R^2 SA-2 SA^2-14 R^2 SW-SA SW+SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45164) lies on the nine-point circle and this line: {4,6572}

X(45164) = midpoint of X(4) and X(6572)
X(45164) = complement of the circumperp conjugate of X(6572)
X(45164) = X(6572)-of-Euler-triangle
X(45164) = X(45154)-image under 2nd HS transform

X(45165) = 18TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (2 a^6+2 a^4 b^2-3 a^2 b^4-b^6+2 a^4 c^2+b^4 c^2-3 a^2 c^4+b^2 c^4-c^6) (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2+6 a^4 b^2 c^2-2 a^2 b^4 c^2+b^6 c^2-a^4 c^4-2 a^2 b^2 c^4-4 b^4 c^4-a^2 c^6+b^2 c^6+c^8) : :
Barycentrics    (SB SC-14 R^2 SW+3 SA SW+2 SW^2) (-7 SB SC SW+S^2 (SA+2 SW)) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45165) lies on the nine-point circle and these lines: {2,29316}, {4,7953}, {115,3850}, {122,10691}, {125,6688}, {127,7849}, {3258,5189}, {5099,43893}

X(45165) = midpoint of X(4) and X(7953)
X(45165) = complement of X(29316)
X(45165) = complementary conjugate of X(29317)
X(45165) = X(4)-Ceva conjugate of X(29317)
X(45165) = X(1)-complementary conjugate of X(29317)
X(45165) = orthic-isogonal conjugate of X(29317)
X(45165) = X(7953)-of-Euler-triangle
X(45165) = X(45155)-image under 2nd HS transform

X(45166) = 19TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

Barycentrics    (2 a^2-b^2-c^2) (2 a^2 b^2-4 b^4+2 a^2 c^2+7 b^2 c^2-4 c^4) (8 a^4-5 a^2 b^2-4 b^4-5 a^2 c^2+10 b^2 c^2-4 c^4) : :
Barycentrics    (8 S^2-2 SA^2+SB^2-24 SB SC+SC^2) (3 SA-SW) (6 S^2+9 SB SC-SB SW-SC SW-SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45166) lies on the nine-point circle and these lines: {4,8600}, {5,16938}, {115,8584}

X(45166) = midpoint of X(4) and X(8600)
X(45166) = reflection of X(16938) in X(5)
X(45166) = complement of the isogonal conjugate of X(32479)
X(45166) = complementary conjugate of X(32479)
X(45166) = X(4)-Ceva conjugate of X(32479)
X(45166) = X(1)-complementary conjugate of X(32479)
X(45166) = orthic-isogonal conjugate of X(32479)
X(45166) = X(8600)-of-Euler-triangle
X(45166) = X(16938)-of-Johnson-triangle
X(45166) = X(45156)-image under 2nd HS transform

X(45167) = 20TH HATZIPOLAKIS-SUPPA POINT ON THE NINE-POINT CIRCLE

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2679.

X(45167) lies on the nine-point circle and these lines: {2,28145}, {4,8652}, {11,6147}

X(45167) = midpoint of X(4) and X(8652)
X(45167) = complement of X(28145)
X(45167) = complementary conjugate of X(28146)
X(45167) = X(4)-Ceva conjugate of X(28146)
X(45167) = X(1)-complementary conjugate of X(28146)
X(45167) = orthic-isogonal conjugate of X(28146)
X(45167) = X(8652)-of-Euler-triangle
X(45167) = X(45157)-image under 2nd HS transform

X(45168) =  (name pending)

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

As a point on the Euler line, X(45168) has Shinagawa coefficients ((r+R)((r+2*R)^2-s^2),r^3+5*r^2*R+3*R^3-R*s^2-r*(-8*R^2+s^2)).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45168) lies on this line: {2,3}

X(45168) = {X(403),X(1594)}-harmonic conjugate of X(429)

X(45169) =  EULER LINE INTERCEPT OF X(1175)X(5504)

Barycentrics    a^2 (a^11-a^10 b-3 a^9 b^2+3 a^8 b^3+2 a^7 b^4-2 a^6 b^5+2 a^5 b^6-2 a^4 b^7-3 a^3 b^8+3 a^2 b^9+a b^10-b^11-a^10 c+a^8 b^2 c+2 a^6 b^4 c-2 a^4 b^6 c-a^2 b^8 c+b^10 c-3 a^9 c^2+a^8 b c^2+9 a^7 b^2 c^2-5 a^6 b^3 c^2-9 a^5 b^4 c^2+9 a^4 b^5 c^2+3 a^3 b^6 c^2-7 a^2 b^7 c^2+2 b^9 c^2+3 a^8 c^3-5 a^6 b^2 c^3+2 a^5 b^3 c^3-a^4 b^4 c^3+5 a^2 b^6 c^3-2 a b^7 c^3-2 b^8 c^3+2 a^7 c^4+2 a^6 b c^4-9 a^5 b^2 c^4-a^4 b^3 c^4+8 a^3 b^4 c^4-a b^6 c^4-b^7 c^4-2 a^6 c^5+9 a^4 b^2 c^5+4 a b^5 c^5+b^6 c^5+2 a^5 c^6-2 a^4 b c^6+3 a^3 b^2 c^6+5 a^2 b^3 c^6-a b^4 c^6+b^5 c^6-2 a^4 c^7-7 a^2 b^2 c^7-2 a b^3 c^7-b^4 c^7-3 a^3 c^8-a^2 b c^8-2 b^3 c^8+3 a^2 c^9+2 b^2 c^9+a c^10+b c^10-c^11) : :
Barycentrics    (SB+SC) (S^2 (-a b c+4 a R^2-4 b R^2+b SC-c SC-a SW)+SA (4 a b c R^2-a b c SA-16 a R^2 SA+16 b R^2 SA+16 b R^2 SC-16 c R^2 SC+b SA SC-c SA SC-16 b R^2 SW+3 a SA SW-4 b SA SW-4 b SC SW+4 c SC SW+4 b SW^2)) : :

As a point on the Euler line, X(45169) has Shinagawa coefficients ((E-4*F)*(b*SB+a*(E+F-SB-SC)+c*SC),4*(2*R^3*S+b*F*SB+a*F*(E+F-SB-SC)+c*F*SC)).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45169) lies on these lines: {2,3}, {1175,5504}

X(45169) = {X(3),X(14017)}-harmonic conjugate of X(6869)

X(45170) =  EULER LINE INTERCEPT OF X(1176)X(5504)

Barycentrics   a^2 (a^2-b^2-c^2) (a^12-2 a^10 b^2-a^8 b^4+4 a^6 b^6-a^4 b^8-2 a^2 b^10+b^12-2 a^10 c^2+3 a^8 b^2 c^2-2 a^6 b^4 c^2+4 a^2 b^8 c^2-3 b^10 c^2-a^8 c^4-2 a^6 b^2 c^4-6 a^4 b^4 c^4-2 a^2 b^6 c^4+3 b^8 c^4+4 a^6 c^6-2 a^2 b^4 c^6-2 b^6 c^6-a^4 c^8+4 a^2 b^2 c^8+3 b^4 c^8-2 a^2 c^10-3 b^2 c^10+c^12) : :
Barycentrics    (SA (SB+SC) (-24 R^4+S^2-4 R^2 SA+SA^2+24 R^2 SW-4 SW^2) : :

As a point on the Euler line, X(45170) has Shinagawa coefficients (E^2-2*E*F-8*F^2,-E^2+ 4*E*F+8*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45170) lies on these lines: {2,3}, {1176,5504}, {9932,18909}, {10282,22955}, {12236,19129}, {18945,32048}, {22533,33563}

X(45170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,24,3546), (3,26,6643), (3,2070,1368), (3,2937,12362), (3,44259,18531), (186,7512,22), (6636,22467,2071), (18531,44259,23)

X(45171) =  EULER LINE INTERCEPT OF X(511)X(1177)

Barycentrics    a^2 (a^14-3 a^12 b^2+a^10 b^4+5 a^8 b^6-5 a^6 b^8-a^4 b^10+3 a^2 b^12-b^14-3 a^12 c^2+9 a^10 b^2 c^2-8 a^8 b^4 c^2-2 a^6 b^6 c^2+9 a^4 b^8 c^2-7 a^2 b^10 c^2+2 b^12 c^2+a^10 c^4-8 a^8 b^2 c^4+6 a^6 b^4 c^4-4 a^4 b^6 c^4+5 a^2 b^8 c^4+5 a^8 c^6-2 a^6 b^2 c^6-4 a^4 b^4 c^6-2 a^2 b^6 c^6-b^8 c^6-5 a^6 c^8+9 a^4 b^2 c^8+5 a^2 b^4 c^8-b^6 c^8-a^4 c^10-7 a^2 b^2 c^10+3 a^2 c^12+2 b^2 c^12-c^14) : :
Barycentrics    (SB+SC) (S^2 (6 R^2-SW)+SA (36 R^4+6 R^2 SA-28 R^2 SW-SA SW+4 SW^2)) : :
X(45171) = 3*X(186)-4*X(1658),3*X(2070)-5*X(16195),3*X(2070)-X(37972),3*X(2071)-2*X(12084),3*X(5899)-7*X(10244),2*X(7575)-3*X(14070),3*X(9909)-X(37924),3*X(10154)-2*X(25338),8*X(10226)-9*X(37948),9*X(10245)-5*X(37923),8*X(12107)-5*X(37953),4*X(13371)-5*X(30745),4*X(15122)-3*X(44441),5*X(16195)-X(37972),4*X(17714)-X(37946),6*X(18324)-5*X(37952),4*X(20773)-3*X(35265),3*X(33591)-2*X(44264),3*X(37907)-4*X(44213)

As a point on the Euler line, X(45171) has Shinagawa coefficients (E^2-16*F^2,-3*E^2+4*E*F+16*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45171) lies on these lines: {2,3}, {511,1177}, {1154,41615}, {1216,22955}, {1495,22109}, {1503,2931}, {2781,15136}, {3564,12412}, {5160,9645}, {11649,34788}, {12893,14915}, {16227,36752}, {18449,41616}, {19138,41744}, {19154,34513}, {20773,35265}, {41603,44407}

X(45171) = midpoint of X(i) and X(j) for these {i,j}: {3,37928}, {5189,31305}, {35001,39568}
X(45171) = reflection of X(i) in X(j) for these (i,j): (23,26), (12085,37950), (14790,858), (41744,19138)
X(45171) = isogonal conjugate of the antigonal conjugate of X(18532)
X(45171) = circumcircle-inverse of X(18531)
X(45171) = X(523)-vertex conjugate of X(18531)
X(45171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,2070,468), (3,34622,11250), (23,2071,4), (23,7488,186), (26,38450,4), (186,7512,37978), (186,37929,3), (1113,1114,18531), (1658,12084,22467), (2070,2072,37951), (2071,37951,2072), (5159,44272,6642), (7464,37980,7574), (7517,18859,18323), (7575,15122,6644), (10298,35485,3), (16195,37972,2070)

X(45172) =  EULER LINE INTERCEPT OF X(52)X(5504)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+9 a^6 b^2 c^2-9 a^4 b^4 c^2+3 a^2 b^6 c^2+2 a^6 c^4-9 a^4 b^2 c^4+4 a^2 b^4 c^4-b^6 c^4+2 a^4 c^6+3 a^2 b^2 c^6-b^4 c^6-3 a^2 c^8+c^10) : :
Barycentrics    SB SC (SB+SC) (-SA^2 (14 R^2-3 SW)+S^2 (6 R^2-SW)) : :

As a point on the Euler line, X(45172) has Shinagawa coefficients (2*E*F-8*F^2,-E^2+8*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45172) lies on these lines: {2,3}, {52,5504}, {74,22750}, {1112,37495}, {1300,13450}, {2052,21396}, {2079,27376}, {3003,41758}, {5562,22955}, {6193,18532}, {6403,11458}, {10282,43392}, {12294,43811}, {13289,17855}, {14157,36982}, {15472,41671}, {32171,34397}

X(45172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,24,3542), (3,26,37201), (3,1885,3520), (4,24,37951), (4,186,22467), (24,6240,3518), (25,12084,4), (186,7512,32534), (1658,3515,186), (2072,3575,4), (3548,38450,2071), (6644,38450,3548), (7512,21735,6636), (14865,37777,35488)

X(45173) =  EULER LINE INTERCEPT OF X(53)X(2079)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+9 a^6 b^2 c^2-9 a^4 b^4 c^2+3 a^2 b^6 c^2+2 a^6 c^4-9 a^4 b^2 c^4-b^6 c^4+2 a^4 c^6+3 a^2 b^2 c^6-b^4 c^6-3 a^2 c^8+c^10) : :
Barycentrics    SB SC (SB+SC) (-3 SA^2 (4 R^2-SW)+S^2 (8 R^2-SW)) : :

As a point on the Euler line, X(45173) has Shinagawa coefficients (-E*F+4*F^2,E^2-4*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45173) lies on these lines: {2,3}, {53,2079}, {800,8749}, {1843,5622}, {1974,11202}, {2929,6247}, {5504,11557}, {5890,44080}, {6403,37784}, {7716,10249}, {9545,35603}, {11430,44079}, {11464,44077}, {12162,22750}, {12412,18932}, {15033,44084}, {18475,19128}, {18532,37644}, {18913,32321}

X(45173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,2070,10154), (4,37951,13595), (25,378,6623), (7575,13490,6644)

X(45174) =  (name pending)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+9 a^6 b^2 c^2+2 a^5 b^3 c^2-9 a^4 b^4 c^2-4 a^3 b^5 c^2+3 a^2 b^6 c^2+2 a b^7 c^2+2 a^5 b^2 c^3+2 a^4 b^3 c^3-4 a^3 b^4 c^3-4 a^2 b^5 c^3+2 a b^6 c^3+2 b^7 c^3+2 a^6 c^4-9 a^4 b^2 c^4-4 a^3 b^3 c^4-4 a b^5 c^4-b^6 c^4-4 a^3 b^2 c^5-4 a^2 b^3 c^5-4 a b^4 c^5-4 b^5 c^5+2 a^4 c^6+3 a^2 b^2 c^6+2 a b^3 c^6-b^4 c^6+2 a b^2 c^7+2 b^3 c^7-3 a^2 c^8+c^10) : :
Barycentrics    SB SC (SB+SC) (S^4+4 r^2 SA^2 (r^2+4 r R-12 R^2+3 SW)+S^2 (4 r^4+16 r^3 R+32 r^2 R^2+SA^2-4 r^2 SW)) : :

As a point on the Euler line, X(45174) has Shinagawa coefficients (-E*F+4*F^2,(a*b+a*c+b*c)*E+E^2-4*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45174) lies on this line: {2,3}

X(45174) = {X(28),X(4219)}-harmonic conjugate of X(403)

X(45175) =  (name pending)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+9 a^6 b^2 c^2-9 a^4 b^4 c^2+3 a^2 b^6 c^2+2 a^4 b^3 c^3-2 a^3 b^4 c^3-2 a^2 b^5 c^3+2 a b^6 c^3+2 a^6 c^4-9 a^4 b^2 c^4-2 a^3 b^3 c^4+4 a^2 b^4 c^4-2 a b^5 c^4-b^6 c^4-2 a^2 b^3 c^5-2 a b^4 c^5+2 a^4 c^6+3 a^2 b^2 c^6+2 a b^3 c^6-b^4 c^6-3 a^2 c^8+c^10) : :
Barycentrics    SB SC (SB+SC) (S^4 (a b+a c+12 R^2-2 SW)+a (b-c) SA^2 SC SW+S^2 (-4 a b R^2 SA+4 b c R^2 SA+a b SA^2+a c SA^2-28 R^2 SA^2-4 a b R^2 SC+4 a c R^2 SC+6 SA^2 SW+a b SC SW-a c SC SW)) : :

As a point on the Euler line, X(45175) has Shinagawa coefficients (2*F*(-E+4*F)*S^2,(E^2-8*F^2)*S^2+b*c*E*(S^2-SB*SC)+a*E*(b*(E+F-SC)*SC+c*(S^2-SA*SC))).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45175) lies on this line: {2,3}

X(45176) =  MIDPOINT OF X(28) AND X(7412)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+9 a^6 b^2 c^2-2 a^5 b^3 c^2-9 a^4 b^4 c^2+4 a^3 b^5 c^2+3 a^2 b^6 c^2-2 a b^7 c^2-2 a^5 b^2 c^3+2 a^4 b^3 c^3+2 a b^6 c^3-2 b^7 c^3+2 a^6 c^4-9 a^4 b^2 c^4+8 a^2 b^4 c^4-b^6 c^4+4 a^3 b^2 c^5+4 b^5 c^5+2 a^4 c^6+3 a^2 b^2 c^6+2 a b^3 c^6-b^4 c^6-2 a b^2 c^7-2 b^3 c^7-3 a^2 c^8+c^10) : :
Barycentrics    SB SC (SB+SC) (S^4 (-b c+4 R^2-SW)+a (b-c) SA^2 SC SW+S^2 (-4 a b R^2 SA+4 b c R^2 SA-b c SA^2-16 R^2 SA^2-4 a b R^2 SC+4 a c R^2 SC+3 SA^2 SW+a b SC SW-a c SC SW)) : :

As a point on the Euler line, X(45176) has Shinagawa coefficients (F*(-E+4*F)*S^2,-4*F^2*S^2-b*c*E*SB*SC+a*E*(-c*SA*SC-b*(S^2+SC*(-E-F+SC)))).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2688 .

X(45176) lies on this line: {2,3}

X(45176) = midpoint of X(28) and X(7412)

X(45177) =  EULER LINE INTERCEPT OF X(6145)X(20303)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2+2 a^6 b^4 c^2-4 a^4 b^6 c^2+5 a^2 b^8 c^2-2 b^10 c^2-3 a^8 c^4+2 a^6 b^2 c^4-2 a^2 b^6 c^4-b^8 c^4+2 a^6 c^6-4 a^4 b^2 c^6-2 a^2 b^4 c^6+4 b^6 c^6+2 a^4 c^8+5 a^2 b^2 c^8-b^4 c^8-3 a^2 c^10-2 b^2 c^10+c^12) : :
Barycentrics    SB SC (14 R^4-4 R^2 SA-8 R^2 SW+SA SW+SW^2) : :

As a point on the Euler line, X(45177) has Shinagawa coefficients (8*F^2,-E^2+8*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45177) lies on this line: {2,3}, {6145,20303}, {6247,17854}, {11441,32123}, {11457,40285}, {12140,33547}, {12825,22660}, {14516,20302}

X(45177) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24,378,38450), (235,427,3627), (235,13371,6240), (403,1594,3), (403,18560,15761), (427,15761,18560), (1594,6240,13371), (1594,13160,37119), (7577,16868,3090), (16868,37119,13160)

X(45178) =  EULER LINE INTERCEPT OF X(6247)X(43904)

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2-2 a^6 b^4 c^2+4 a^4 b^6 c^2+a^2 b^8 c^2-2 b^10 c^2-3 a^8 c^4-2 a^6 b^2 c^4+2 a^2 b^6 c^4-b^8 c^4+2 a^6 c^6+4 a^4 b^2 c^6+2 a^2 b^4 c^6+4 b^6 c^6+2 a^4 c^8+a^2 b^2 c^8-b^4 c^8-3 a^2 c^10-2 b^2 c^10+c^12) : :
Barycentrics    SB SC (6 R^4-2 R^2 SA-6 R^2 SW+SA SW+SW^2) : :

As a point on the Euler line, X(45178) has Shinagawa coefficients (4*E*F+8*F^2,-E^2+4*E*F+8*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45178) lies on these lines: {2,3}, {6247,43904}, {12140,18475}, {18374,20300}, {18381,44078}, {20303,34224}, {22661,25715}

X(45178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,14940,7512), (24,7566,4), (403,1594,25), (403,5133,4), (403,35487,10254), (5133,13160,9818), (7503,35488,4), (9818,10254,13160), (37347,37981,378)

X(45179) =  EULER LINE INTERCEPT OF X(52)X(33563)

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2+2 a^6 b^4 c^2-4 a^4 b^6 c^2+5 a^2 b^8 c^2-2 b^10 c^2-3 a^8 c^4+2 a^6 b^2 c^4-4 a^4 b^4 c^4-2 a^2 b^6 c^4-b^8 c^4+2 a^6 c^6-4 a^4 b^2 c^6-2 a^2 b^4 c^6+4 b^6 c^6+2 a^4 c^8+5 a^2 b^2 c^8-b^4 c^8-3 a^2 c^10-2 b^2 c^10+c^12) : :
Barycentrics    SB SC (12 R^4-4 R^2 SA-8 R^2 SW+SA SW+SW^2) : :
X(45179) = X(11413)-5*X(31236)

As a point on the Euler line, X(45179) has Shinagawa coefficients (4*F^2,-E^2+4*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45179) lies on these lines: {2,3}, {52,33563}, {2904,13292}, {5622,23300}, {10605,32125}, {11456,41602}, {12134,20302}, {12140,20771}, {14516,23307}, {15473,32743}, {18451,32123}, {18912,35603}, {39113,44131}

X(45179) = midpoint of X(i) and X(j) for these {i,j}: {4,44269}, {235,427}, {18570,44271}, {31723,44259}, {37814,44288}
X(45179) = harmonic center of nine-point circle and incircle of orthic triangle if ABC is acute
X(45179) = X(44269)-of-Euler-triangle
X(45179) = X(8069)-of-orthic-triangle if ABC is acute
X(45179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24,1594,11585), (24,35490,31304), (235,11585,24), (378,403,15760), (403,1594,2), (427,15760,378), (1595,1596,15687), (7577,16868,5071), (18560,37970,44249), (44226,44235,235)

X(45180) = X(30)X(137)∩X(50)X(115)

Barycentrics   (2 a^10-5 a^8 b^2+4 a^6 b^4-2 a^4 b^6+2 a^2 b^8-b^10-5 a^8 c^2+6 a^6 b^2 c^2-a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+4 a^6 c^4-a^4 b^2 c^4+2 a^2 b^4 c^4-2 b^6 c^4-2 a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-4 a^8 b^2 c^2+5 a^6 b^4 c^2-6 a^4 b^6 c^2+8 a^2 b^8 c^2-4 b^10 c^2-3 a^8 c^4+5 a^6 b^2 c^4+2 a^4 b^4 c^4-5 a^2 b^6 c^4+7 b^8 c^4+2 a^6 c^6-6 a^4 b^2 c^6-5 a^2 b^4 c^6-8 b^6 c^6+2 a^4 c^8+8 a^2 b^2 c^8+7 b^4 c^8-3 a^2 c^10-4 b^2 c^10+c^12) : :
Barycentrics    (-SB SC (9 R^2-4 SW)+S^2 (3 R^2+2 SA-2 SW)) (-18 R^4+3 R^2 SA+2 SB SC+11 R^2 SW-2 SW^2) : :
X(45180) = 3*X(15392)-X(38587)

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45180) lies on the nine-point circle and these lines: {2,14979}, {3,14980}, {4,1291}, {30,137}, {50,115}, {113,1510}, {125,1154}, {128,523}, {136,186}, {140,3258}, {1157,3153}, {1594,16221}, {2072,16336}, {3574,35591}, {3575,16178}, {5099,37347}, {15392,38587}, {16177,37452}, {30714,43969}

X(45180) = midpoint of X(i) and X(j) for these {i,j}: {3,14980}, {4,1291}, {1157,3153}
X(45180) = reflection of X(i) in X(j) for these (i,j): (186,10615), (16336,2072), (30714,43969)
X(45180) = complement of X(14979)
X(45180) = complementary conjugate of X(32423)
X(45180) = X(4)-Ceva conjugate of X(32423)
X(45180) = X(1)-complementary conjugate of X(32423)
X(45180) = reflection of X(128) in Euler line
X(45180) = orthic-isogonal conjugate of X(32423)
X(45180) = X(1291)-of-Euler-triangle
X(45180) = X(930)-of-reflection-of-Euler-triangle-in-Euler-line

X(45181) =  EULER LINE INTERCEPT OF X(1273)X(44138)

Barycentrics   (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2+3 a^2 b^8 c^2-2 b^10 c^2-3 a^8 c^4-2 a^4 b^4 c^4-b^8 c^4+2 a^6 c^6+4 b^6 c^6+2 a^4 c^8+3 a^2 b^2 c^8-b^4 c^8-3 a^2 c^10-2 b^2 c^10+c^12) : :
Barycentrics    SB SC (9 R^4-3 R^2 SA-7 R^2 SW+SA SW+SW^2) : :
X(45181) = 3*X(403)+X(15559),3*X(5576)+X(11799),9*X(7565)-X(10296)

As a point on the Euler line, X(45181) has Shinagawa coefficients (4*E*F+16*F^2,-3*E^2+4*E*F+16*F^2).

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45181) lies on these lines: {2,3}, {1273,44138}, {2914,3564}, {6344,6530}, {16227,26879}, {20301,44102}

X(45181) = nine-point-circle-inverse of X(7576)
X(45181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,37943,23), (403,1594,468), (403,37981,4), (468,10151,6756), (1312,1313,7576)

X(45182) = MIDPOINT OF X(3) AND X(22472)

Barycentrics   (b^2-c^2)^2 (3 a^12-8 a^10 b^2+3 a^8 b^4+8 a^6 b^6-7 a^4 b^8+b^12-8 a^10 c^2+20 a^8 b^2 c^2-16 a^6 b^4 c^2+2 a^4 b^6 c^2+4 a^2 b^8 c^2-2 b^10 c^2+3 a^8 c^4-16 a^6 b^2 c^4+22 a^4 b^4 c^4-4 a^2 b^6 c^4-b^8 c^4+8 a^6 c^6+2 a^4 b^2 c^6-4 a^2 b^4 c^6+4 b^6 c^6-7 a^4 c^8+4 a^2 b^2 c^8-b^4 c^8-2 b^2 c^10+c^12) (-a^14+3 a^12 b^2-a^10 b^4-5 a^8 b^6+5 a^6 b^8+a^4 b^10-3 a^2 b^12+b^14+3 a^12 c^2-12 a^10 b^2 c^2+11 a^8 b^4 c^2+8 a^6 b^6 c^2-15 a^4 b^8 c^2+4 a^2 b^10 c^2+b^12 c^2-a^10 c^4+11 a^8 b^2 c^4-36 a^6 b^4 c^4+22 a^4 b^6 c^4+13 a^2 b^8 c^4-9 b^10 c^4-5 a^8 c^6+8 a^6 b^2 c^6+22 a^4 b^4 c^6-28 a^2 b^6 c^6+7 b^8 c^6+5 a^6 c^8-15 a^4 b^2 c^8+13 a^2 b^4 c^8+7 b^6 c^8+a^4 c^10+4 a^2 b^2 c^10-9 b^4 c^10-3 a^2 c^12+b^2 c^12+c^14) : :
Barycentrics    (b^2-c^2)^2 (54 R^4+9 R^2 SA-29 R^2 SW-2 SA SW+4 SW^2) (S^2 (9 R^2-2 SW)-SA (108 R^4-9 R^2 SA-40 R^2 SW+2 SA SW+4 SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

XX(45182) lies on the nine-point circle and this line: {3, 22472}

X(45182) = midpoint of X(3) and X(22472)
X(45182) = complement of the antigonal conjugate of X(4)

X(45183) = X(125)X(6102)∩X(136)X(6240)

Barycentrics   (2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2+6 a^6 b^2 c^2-2 a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4-2 a^4 b^2 c^4+4 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10) (a^10 b^2-3 a^8 b^4+2 a^6 b^6+2 a^4 b^8-3 a^2 b^10+b^12+a^10 c^2-2 a^8 b^2 c^2+2 a^6 b^4 c^2-5 a^4 b^6 c^2+7 a^2 b^8 c^2-3 b^10 c^2-3 a^8 c^4+2 a^6 b^2 c^4+4 a^4 b^4 c^4-4 a^2 b^6 c^4+3 b^8 c^4+2 a^6 c^6-5 a^4 b^2 c^6-4 a^2 b^4 c^6-2 b^6 c^6+2 a^4 c^8+7 a^2 b^2 c^8+3 b^4 c^8-3 a^2 c^10-3 b^2 c^10+c^12) : :
Barycentrics    (S^2 (5 R^2+SA-2 SW)-5 SB SC (3 R^2-SW)) (30 R^4-5 R^2 SA-SB SC-15 R^2 SW+SA SW+2 SW^2) : :

See Antreas Hatzipolakis and Ercole Suppa, Euclid 2702 .

X(45183) lies on the nine-point circle and these lines: {2,22751}, {3,22752} ,{4,12092}, {5,14103}, {125,6102}, {136,6240}, {3258,15646}, {18388,21662}

X(45183) = midpoint of X(i) and X(j) for these {i,j}: {3,22752}, {4,12092}
X(45183) = reflection of X(14103) in X(5)
X(45183) = complement of X(22751)
X(45183) = complementary conjugate of X(30522)
X(45183) = X(4)-Ceva conjugate of X(30522)
X(45183) = X(1)-complementary conjugate of X(30522)
X(45183) = nine-point-circle-antipode of X(14103)
X(45183) = X(12092)-of-Euler-triangle
X(45183) = X(14103)-of-Johnson-triangle

X(45184) = X(155)X(195)∩X(156)X(5965)

Barycentrics    (a^2-b^2-c^2) (4 a^8-10 a^6 b^2+9 a^4 b^4-4 a^2 b^6+b^8-10 a^6 c^2-4 a^4 b^2 c^2+4 a^2 b^4 c^2-4 b^6 c^2+9 a^4 c^4+4 a^2 b^2 c^4+6 b^4 c^4-4 a^2 c^6-4 b^2 c^6+c^8) : :
Barycentrics    SA (4 S^2-(SB+SC) (2 R^2+3 SB+3 SC)) : :
X(45184) = 7*X(155)-3*X(381), 3*X(155)-X(9927), 5*X(155)-X(12429), 6*X(381)-7*X(5448), 9*X(381)-7*X(9927), 5*X(631)-7*X(1147), 5*X(631)+7*X(9936), 3*X(1147)-X(11411), 3*X(1147)-2*X(20191), X(3146)+7*X(6193), X(3146)-7*X(15083), 3*X(3167)-2*X(43839), 8*X(3628)-7*X(5449), 6*X(3628)-7*X(9820), 4*X(3628)-7*X(41597), 3*X(5448)-2*X(9927), 5*X(5448)-2*X(12429), 3*X(5449)-4*X(9820), 7*X(7689)-9*X(10304), 2*X(9820)-3*X(41597), 5*X(9927)-3*X(12429), 3*X(9936)+X(11411), 3*X(9936)+2*X(20191), 3*X(11001)-7*X(12118), 9*X(11539)-7*X(12359), 7*X(12038)-6*X(12100), 7*X(12164)+X(17800)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2737.

X(45184) lies on these lines: {68,5056}, {146,3146}, {155,195}, {156,5965}, {394,18128}, {550,6101}, {575,3564}, {631,1147}, {912,32900}, {3060,13431}, {3167,43839}, {3853,16656}, {6243,24981}, {7689,10304}, {11001,12118}, {11539,12359}, {12038,12100}, {12061,34382}, {12164,17800}, {19588,32048}, {23411,32455}

X(45184) = midpoint of X(i) and X(j) for these {i,j}: {1147,9936}, {6193,15083}
X(45184) = reflection of X(i) in X(j) for these (i,j): (5448,155), (5449,41597), (11411,20191)
X(45184) = {X(1147),X(11411)}-harmonic conjugate of X(20191)

X(45185) = X(4)X(54)∩X(26)X(542)

Barycentrics    4 a^10-10 a^8 b^2+7 a^6 b^4-a^4 b^6+a^2 b^8-b^10-10 a^8 c^2+4 a^6 b^2 c^2+a^4 b^4 c^2+2 a^2 b^6 c^2+3 b^8 c^2+7 a^6 c^4+a^4 b^2 c^4-6 a^2 b^4 c^4-2 b^6 c^4-a^4 c^6+2 a^2 b^2 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10 : :
Barycentrics    3 S^2 (4 R^2+SA-2 SW)-SB SC (4 R^2-7 SW) : :
X(45185) = X(4)-3*X(6759), X(4)+3*X(9833), X(4)-9*X(11206), 5*X(4)-3*X(34786), 8*X(140)-9*X(10182), 2*X(140)-3*X(10282), 4*X(140)-3*X(20299), 9*X(154)-5*X(1656), 3*X(154)-X(18381), 3*X(159)-X(34507), 3*X(206)-2*X(25555), 5*X(550)-3*X(5894), 3*X(550)-X(15105), 5*X(550)-6*X(32903), X(550)-3*X(34782), X(576)-3*X(31166), 3*X(1498)+X(1657), 5*X(1656)-3*X(18381), X(1657)-3*X(34785), 3*X(1660)-X(14791), 3*X(3357)-5*X(3522), X(3519)+3*X(32359), 5*X(3522)+3*X(34781), 7*X(3523)-9*X(11202), 7*X(3523)-3*X(14216), 7*X(3523)-6*X(25563), 2*X(3850)-3*X(16252), 4*X(3850)-3*X(18383), 5*X(3858)-3*X(41362), X(5059)+3*X(5878), X(5073)+3*X(17845), X(5073)-3*X(22802), X(5073)-9*X(32063), 3*X(5449)-4*X(18282), 9*X(5894)-5*X(15105), X(5894)-5*X(34782), 2*X(6247)-3*X(10193), 3*X(6247)-5*X(15712), X(6759)-3*X(11206), 3*X(6759)-2*X(14862), 5*X(6759)-X(34786), X(9833)+3*X(11206), 3*X(9833)+2*X(14862), 5*X(9833)+X(34786), 3*X(10182)-4*X(10282), 9*X(10182)-4*X(14864), 3*X(10182)-2*X(20299), 3*X(10192)-2*X(32767), 9*X(10193)-10*X(15712), 3*X(10282)-X(14864), 3*X(11202)-X(14216), 3*X(11202)-2*X(25563), 3*X(11204)-X(12324), 9*X(11206)-2*X(14862), X(13432)+3*X(17846), 10*X(14862)-3*X(34786), 2*X(14864)-3*X(20299), X(15105)-9*X(34782), 5*X(17821)-3*X(23329), 5*X(17821)-X(34780), X(17845)+3*X(32063), X(22802)-3*X(32063), 3*X(23042)-X(36851), 3*X(23329)-X(34780), 2*X(32903)-5*X(34782), X(34788)-3*X(41719)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2730.

X(45185) lies on these lines: {4,54}, {17,30402}, {18,30403}, {26,542}, {30,41597}, {64,26861}, {125,26882}, {140,1503}, {154,1656}, {156,44407}, {159,2918}, {206,25555}, {399,1498}, {428,37505}, {539,17714}, {550,1216}, {576,31166}, {1147,29012}, {1487,11816}, {1495,34224}, {1594,44110}, {1598,40240}, {1660,14791}, {1971,7755}, {2937,3519}, {3098,42021}, {3357,3522}, {3517,20987}, {3523,11202}, {3533,32064}, {3567,34564}, {3850,16252}, {3851,14530}, {3858,41362}, {4857,10535}, {5056,23325}, {5059,5878}, {5073,17845}, {5270,26888}, {5449,18282}, {5462,43129}, {6243,13431}, {6247,10193}, {7517,10112}, {7525,15582}, {7553,34986}, {7715,8550}, {8779,41366}, {8960,10533}, {9707,11550}, {9920,10117}, {9935,15107}, {9968,19924}, {10110,31804}, {10116,37440}, {10192,32767}, {10540,11750}, {10675,42158}, {10676,42157}, {10990,12281}, {11204,12324}, {11412,24981}, {11430,16655}, {11645,23335}, {12106,18128}, {12134,34002}, {12279,16163}, {13202,40242}, {13367,16659}, {13382,37458}, {13399,21844}, {13432,17846}, {15152,44226}, {15720,17821}, {16266,29317}, {16534,40276}, {17826,42988}, {17827,42989}, {18555,44076}, {18912,44082}, {19130,32046}, {20417,32534}, {23042,36851}, {25738,32223}, {32237,41587}, {34514,44516}, {34788,41719}, {37460,43616}

X(45185) = midpoint of X(i) and X(j) for these {i,j}: {1498,34785}, {3357,34781}, {6759,9833}, {17845,22802}, {34776,39879}
X(45185) = reflection of X(i) in X(j) for these (i,j): (4,14862), (5894,32903), (14216,25563), (14864,140), (18383,16252), (20299,10282), (40107,15582)
X(45185) = crosssum of X(1510)and X(2972)
X(45185) = crosspoint of X(930)and X(32230)
X(45185) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,184,12242), (4,6759,14862), (140,14864,20299), (9833,11206,6759), (10282,14864,140), (10282,20299,10182), (11202,14216,25563), (12242,13419,4), (17821,34780,23329), (17845,32063,22802)

X(45186) = X(3)X(51)∩X(4)X(69)

Barycentrics    a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-6 a^4 b^2 c^2+a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4+a^2 b^2 c^4-6 b^4 c^4+3 a^2 c^6+4 b^2 c^6-c^8) : :
Barycentrics    (SB+SC) (S^2-4 R^2 SA+SB SC) : :
X(45186) = 3*X(2)-4*X(10110), 2*X(3)-3*X(51), 3*X(3)-4*X(5462), 5*X(3)-6*X(5892), 4*X(3)-3*X(36987), 3*X(4)-2*X(5907), 3*X(4)-X(11412), 5*X(4)-3*X(11459), 4*X(4)-3*X(15030), 7*X(4)-5*X(15058), 4*X(5)-3*X(3917), 3*X(5)-2*X(10627), 5*X(5)-4*X(32142), X(20)-3*X(3060), 3*X(20)-5*X(10574), 3*X(51)-4*X(5446), 9*X(51)-8*X(5462), 5*X(51)-4*X(5892), 3*X(52)-2*X(6102), 3*X(52)-X(10575), 5*X(52)-2*X(13491), 3*X(52)-4*X(14449), 4*X(52)-3*X(14831), 2*X(52)-3*X(21969), X(64)-3*X(34751), 8*X(140)-9*X(373), 4*X(143)-3*X(9730), 3*X(185)-4*X(6102), X(185)-4*X(10263), 3*X(185)-2*X(10575), 5*X(185)-4*X(13491), 3*X(185)-8*X(14449), 2*X(185)-3*X(14831), X(185)-3*X(21969), 3*X(376)-5*X(3567), 3*X(376)-4*X(9729), 2*X(376)-3*X(16226), 3*X(381)-2*X(1216), 3*X(381)-X(37484), 2*X(382)+X(14531), 3*X(382)-X(18439), 2*X(389)-3*X(3060), 6*X(389)-5*X(10574), 4*X(546)-3*X(5891), 2*X(548)-3*X(5946), 3*X(549)-4*X(10095), 2*X(550)-3*X(9730), 3*X(568)-X(1657), 3*X(568)-2*X(40647), 4*X(576)-3*X(40673), 5*X(631)-6*X(5943), 5*X(631)-7*X(9781), 5*X(631)-4*X(13348), 5*X(632)-6*X(13364), 4*X(1112)-3*X(16223), 3*X(1351)-2*X(32284)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2736.

X(45186) lies on these lines: {2,10110}, {3,51}, {4,69}, {5,3917}, {6,10984}, {20,389}, {22,578}, {23,10282}, {24,13346}, {25,1092}, {26,13352}, {30,52}, {49,5899}, {54,12088}, {64,34751}, {68,11550}, {74,11800}, {110,11807}, {125,23335}, {140,373}, {143,550}, {155,18534}, {181,601}, {182,10323}, {184,7387}, {186,15010}, {193,34781}, {195,37924}, {235,1568}, {343,1595}, {376,3567}, {381,1216}, {382,6243}, {394,1598}, {500,7420}, {517,16980}, {524,16621}, {542,16659}, {546,5891}, {548,5946}, {549,10095}, {567,13564}, {568,1657}, {569,22352}, {576,7592}, {602,3271}, {631,5943}, {632,13364}, {674,14872}, {970,6906}, {1012,5752}, {1064,10475}, {1112,16163}, {1147,1495}, {1154,3627}, {1181,1351}, {1199,5097}, {1204,12085}, {1350,7395}, {1370,39571}, {1498,2393}, {1519,35631}, {1532,37536}, {1533,2883}, {1593,17834}, {1614,34986}, {1656,5447}, {1899,34938}, {1907,11576}, {1941,8884}, {1986,16879}, {1993,6759}, {1994,12087}, {2070,12038}, {2777,21649}, {2781,41362}, {2794,39817}, {2807,41869}, {2937,18475}, {2979,3091}, {3090,3819}, {3098,7509}, {3133,23217}, {3146,5889}, {3149,37482}, {3199,3289}, {3292,7530}, {3313,5480}, {3448,14864}, {3515,37497}, {3517,35602}, {3518,43574}, {3522,11002}, {3523,5640}, {3524,15024}, {3525,6688}, {3528,15045}, {3529,5890}, {3530,13451}, {3534,37481}, {3538,18928}, {3543,12111}, {3545,7999}, {3560,22076}, {3564,16655}, {3574,12363}, {3580,20299}, {3628,14845}, {3796,11426}, {3830,18436}, {3832,11444}, {3839,15056}, {3843,23039}, {3845,11591}, {3850,15067}, {3851,10170}, {3853,5876}, {3855,13570}, {3858,14128}, {3861,15060}, {3937,37532}, {4297,31757}, {5012,37505}, {5056,7998}, {5059,13382}, {5068,33884}, {5073,14915}, {5076,18435}, {5092,43651}, {5102,32366}, {5188,27375}, {5198,17814}, {5422,37515}, {5448,11799}, {5651,7529}, {5657,23841}, {5663,13421}, {5706,37516}, {5709,26892}, {5751,9122}, {5925,31978}, {5965,16658}, {6153,12307}, {6193,31383}, {6241,33703}, {6515,14216}, {6636,13434}, {6642,34417}, {6823,9967}, {6834,37521}, {6907,18180}, {6950,15489}, {6979,33852}, {7330,26893}, {7383,14561}, {7391,18381}, {7400,11574}, {7404,43653}, {7409,33523}, {7488,11430}, {7500,9833}, {7506,44106}, {7512,15033}, {7525,37513}, {7526,37478}, {7527,7691}, {7555,10610}, {8276,9686}, {8541,44492}, {8703,12006}, {8718,15032}, {9019,12233}, {9306,10594}, {9545,26881}, {9715,11425}, {9777,37198}, {9786,21312}, {9818,37486}, {9820,37971}, {9822,10519}, {9909,19357}, {9914,34777}, {9927,11572}, {9973,15811}, {10018,32223}, {10112,29012}, {10303,11451}, {10531,35645}, {10540,41597}, {10564,37814}, {10619,43595}, {10628,10733}, {11017,23046}, {11064,21841}, {11250,32110}, {11387,14826}, {11413,11438}, {11456,37517}, {11479,33878}, {11539,11592}, {11557,12121}, {11692,18859}, {11746,38727}, {11806,20127}, {11826,22300}, {12022,29317}, {12058,18390}, {12083,13366}, {12084,21663}, {12099,38729}, {12102,31834}, {12103,16881}, {12107,43394}, {12163,12235}, {12164,34382}, {12225,13403}, {12236,16111}, {12237,12256}, {12238,12257}, {12239,42258}, {12240,42259}, {12241,19161}, {12280,13433}, {12290,15682}, {12295,21650}, {12359,41586}, {12362,16657}, {12811,44324}, {12897,18563}, {13321,15696}, {13358,14677}, {13363,15712}, {13417,17702}, {13419,14516}, {13446,37943}, {13621,37496}, {13630,14855}, {13851,18569}, {14110,42450}, {14520,36012}, {14627,44111}, {14641,17800}, {14689,16225}, {14788,19130}, {14869,32205}, {14984,15063}, {15004,35243}, {15012,17538}, {15028,15717}, {15035,41671}, {15038,37471}, {15073,34621}, {15305,17578}, {15472,22109}, {15559,21243}, {15705,40284}, {15800,22815}, {15801,43605}, {16197,37649}, {16222,38726}, {16386,32411}, {16654,34380}, {16661,34545}, {17080,34956}, {17712,43573}, {17928,37480}, {18281,32225}, {18378,22115}, {18483,31738}, {18925,34608}, {19124,37488}, {19467,31305}, {19925,31737}, {20423,44479}, {22416,33843}, {23154,24474}, {23698,39846}, {25739,43896}, {26882,32237}, {28150,31728}, {28164,31732}, {30531,44325}, {31074,32767}, {31304,34785}, {31730,31760}, {31810,44544}, {31817,31871}, {32171,37936}, {34565,36753}, {34782,41580}, {36518,41673}, {36742,40952}, {36978,42147}, {36979,42432}, {36980,42148}, {36981,42431}, {37200,41365}, {37406,39271}, {37437,41723}, {37474,40954}, {38281,44436}, {38738,39835}, {38749,39806}, {44076,44407}, {44102,44469}

X(45186) = midpoint of X(i) and X(j) for these {i,j}: {382,6243}, {3146,5889}, {5073,34783}, {6241,33703}, {11381,14531}
X(45186) = reflection of X(i) in X(j) for these (i,j): (3,5446), (4,13598), (20,389), (52,10263), (74,11800), (110,11807), (185,52), (376,21849), (550,143), (1350,9969), (1657,40647), (3313,5480), (4297,31757), (5188,27375), (5447,12002), (5562,4), (5876,3853), (5925,31978), (6101,546), (6102,14449), (6146,13142), (6467,1351), (7691,11808), (9967,21850), (10575,6102), (10625,5), (11381,382), (11412,5907), (11750,12370), (11826,22300), (12058,18390), (12103,16881), (12111,13474), (12121,11557), (12162,3627), (12163,12235), (12225,13403), (12256,12237), (12257,12238), (12280,13433), (12294,31670), (12307,6153), (13630,16982), (14110,42450), (14516,13419), (14531,6243), (14677,13358), (14831,21969), (15063,16105), (15644,10110), (15704,13630), (16111,12236), (16163,1112), (16386,32411), (17800,14641), (18563,12897), (18859,11692), (20127,11806), (21650,12295), (23154,24474), (31730,31760), (31737,19925), (31738,18483), (31817,31871), (31834,12102), (34224,10112), (36987,51), (37484,1216), (38738,39835), (38749,39806), (43581,15800)
X(45186) = anticomplement of X(15644)
X(45186) = crosssum of X(3) and X(11411)
X(45186) = anticomplement of X(3) wrt orthic triangle
X(45186) = X(10625)-of-Johnson-triangle
X(45186) = barycentric product X(343)*X(19173)
X(45186) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3527,10601), (3,5446,51), (3,44413,11424), (4,5562,15030), (4,11412,5907), (5,10625,3917), (6,11414,10984), (20,3060,389), (23,34148,10282), (25,37498,1092), (26,13352,13367), (52,185,14831), (52,10263,21969), (52,10575,6102), (143,550,9730), (155,18534,26883), (185,21969,52), (376,3567,9729), (376,21849,16226), (381,37484,1216), (546,6101,5891), (568,1657,40647), (631,9781,5943), (1112,16163,16223), (1147,7517,1495), (1351,39568,1181), (1498,11477,12160), (1656,5447,5650), (1656,13340,5447), (1993,6759,43844), (2070,37495,12038), (2937,37472,18475), (2979,3091,11793), (3522,11002,15043), (3522,15043,16836), (3523,5640,11695), (3528,15045,17704), (3530,13451,15026), (3543,12111,13474), (3567,9729,16226), (3627,12162,32062), (3853,5876,16194), (5650,27355,1656), (5907,11412,5562), (5943,13348,631), (6102,10263,14449), (6102,10575,185), (6102,14449,52), (6642,37483,43652), (7387,36747,184), (7530,16266,10539), (9729,21849,3567), (9777,37198,37514), (9927,31723,11572), (10110,15644,2), (10539,16266,3292), (12002,13340,27355), (12085,37489,1204), (13630,15704,14855), (14516,34603,13419), (23335,41587,125), (34417,43652,6642)

X(45187) = X(3)X(49)∩X(373)X(389)

Barycentrics    a^2 (a^2-b^2-c^2) (3 a^4 b^2-6 a^2 b^4+3 b^6+3 a^4 c^2+4 a^2 b^2 c^2-3 b^4 c^2-6 a^2 c^4-3 b^2 c^4+3 c^6) : :
Barycentrics    SA (SB+SC) (-16 R^2+3 SA+3 SW) : :
X(45187) = 9*X(2)-8*X(15012), 4*X(3)-3*X(185), 5*X(3)-6*X(1216), 8*X(3)-9*X(3917), 2*X(3)-3*X(5562), X(3)-3*X(18436), 7*X(3)-9*X(23039), 5*X(3)-3*X(34783), 7*X(3)-6*X(40647), 4*X(4)-3*X(21969), 4*X(5)-3*X(14831), 9*X(51)-10*X(3091), 3*X(51)-2*X(5889), 3*X(51)-4*X(5907), 9*X(51)-8*X(16625), 3*X(52)-4*X(546), 2*X(52)-3*X(15030), 6*X(143)-7*X(3857), 3*X(154)-2*X(32392), 5*X(185)-8*X(1216), 2*X(185)-3*X(3917), X(185)-4*X(18436), 5*X(185)-4*X(34783), 7*X(185)-8*X(40647), 9*X(373)-8*X(389), 3*X(373)-4*X(11459), 3*X(376)-4*X(15606), 6*X(389)-7*X(3090), 2*X(389)-3*X(11459), 3*X(389)-4*X(40247), 2*X(546)-3*X(5876), 8*X(546)-9*X(15030), 5*X(631)-4*X(13382), 10*X(632)-9*X(9730), 5*X(632)-6*X(11591), 4*X(1216)-5*X(5562), 2*X(1216)-5*X(18436), 7*X(1216)-5*X(40647), 10*X(1656)-9*X(16226), 7*X(3090)-9*X(11459), 7*X(3090)-8*X(40247), 5*X(3091)-3*X(5889), 5*X(3091)-6*X(5907), 5*X(3091)-4*X(16625), 2*X(3146)-3*X(11381)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2744.

X(45187) lies on these lines: {2,15012}, {3,49}, {4,14531}, {5,14831}, {51,3091}, {52,546}, {68,13851}, {69,31371}, {143,3857}, {154,32392}, {235,41586}, {323,11440}, {343,43831}, {373,389}, {376,15606}, {511,3146}, {524,1885}, {539,18563}, {542,12225}, {568,5072}, {576,40318}, {631,13382}, {632,9730}, {852,8798}, {916,23154}, {1154,3627}, {1192,6090}, {1350,12174}, {1495,11441}, {1531,9927}, {1568,12359}, {1656,16226}, {2781,36982}, {2807,7991}, {3516,37672}, {3525,5890}, {3529,6000}, {3542,32225}, {3564,21659}, {3567,27355}, {3628,5891}, {3819,10574}, {3832,21849}, {4550,36749}, {5076,6243}, {5079,5462}, {5446,18435}, {5609,12107}, {5642,13148}, {5650,9729}, {5651,9786}, {5663,10625}, {5895,40341}, {5943,15022}, {5946,12812}, {6101,10575}, {6241,15644}, {6467,41716}, {7488,44110}, {7503,13366}, {7527,15801}, {7556,41725}, {7691,43605}, {7722,15034}, {7723,21649}, {7998,17704}, {7999,16836}, {8567,22967}, {9777,33537}, {9924,34146}, {9934,10628}, {10110,15058}, {10170,37481}, {10263,12102}, {10564,32138}, {10627,14855}, {11403,11477}, {11424,12160}, {11442,11572}, {11479,15004}, {=11541,12290}, {11800,15044}, {12002,14269}, {12086,23061}, {12108,15067}, {12173,15069}, {12219,15054}, {12239,43879}, {12240,43880}, {12358,38729}, {12811,15060}, {12825,13417}, {13340,14641}, {13348,15072}, {13421,15687}, {13434,44111}, {13491,44245}, {13598,15305}, {13630,14869}, {14118,34986}, {14683,41482}, {14845,16881}, {14864,43895}, {14915,37484}, {15010,16879}, {15029,41671}, {16223,38795}, {17506,43572}, {17834,26883}, {18350,32608}, {21650,21651}, {24981,34782}, {30714,44242}, {31504,34003}, {32139,37478}, {32352,41578}, {35500,37505}, {38734,39817}, {38745,39846}

X(45187) = reflection of X(i) in X(j) for these (i,j): (52,5876), (185,5562), (5562,18436), (5889,5907), (6102,31834), (6241,15644), (6467,41716), (10575,6101), (11381,12111), (13417,12825), (14531,4), (21649,7723), (34783,1216), (43581,41726)
X(45187) = crosssum of X(i)and X(j) for these {i,j}: {4,3515}, {25,33630}
X(45187) = crosspoint of X(i) and X(j) for these {i,j}: {3,15077}, {69,36609}
X(45187) = barycentric product X(3926)*X(15010)
X(45187) = trilinear product and X(326)*X(15010)
X(45187) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,15083,43844), (4,14531,21969), (52,5876,15030), (185,5562,3917), (389,40247,3090), (1092,12163,21663), (3090,11459,40247), (3091,5889,16625), (3091,16625,51), (5889,5907,51), (5907,16625,3091), (6101,10575,36987), (6102,31834,5891), (9729,11444,5650)

leftri

V-perspeconics: X(45188)-X(45194)

rightri

This preamble and centers X(45188)-X(45194) were contributed by César Eliud Lozada, October 15, 2021.

Let T' = A'B'C' and T" = A"B"C" be two perspective triangles, neither inscribed in the other. Denote A'b = A'B" ∩ B'C' and A'c = A'C" ∩ B'C', and cyclically B'c, B'a, C'a, C'b. Then these six points lie on a conic, here introduced as the V-perspeconic of T' to T". A reciprocal V-perspeconic T" to T' is found by swapping T' and T" in the previous construction.

The appearance of (T, i, j) in the following list means that the centers of the V-perspeconics of ABC-to-T and T-to-ABC are X(i) and X(j), respectively:

(ABC-X3 reflections, 17807, 45188), (Ehrmann-mid, 45191, 45192), (outer-Garcia, 3588, 45189), (Gemini 107, 2, 2), (Gemini 109, 2, 2), (Gemini 110, 2, 2), (Gemini 111, 2, 2), (Johnson, 31353, 45190), (5th mixtilinear, 45193, 45194)

X(45188) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ABC

Barycentrics    a^2*(a^22-(b^2+c^2)*a^20-(3*b^2+2*b*c-3*c^2)*(3*b^2-2*b*c-3*c^2)*a^18+17*(b^4-c^4)*(b^2-c^2)*a^16+2*(b^2-c^2)^2*(5*b^4-38*b^2*c^2+5*c^4)*a^14-2*(b^4-c^4)*(b^2-c^2)*(21*b^4-22*b^2*c^2+21*c^4)*a^12+2*(b^2-c^2)^2*(7*b^8+7*c^8+2*(54*b^4+37*b^2*c^2+54*c^4)*b^2*c^2)*a^10+2*(b^4-c^4)*(b^2-c^2)*(17*b^8+17*c^8-2*(52*b^4+113*b^2*c^2+52*c^4)*b^2*c^2)*a^8-(b^2-c^2)^2*(27*b^12+27*c^12+(102*b^8+102*c^8-(715*b^4+876*b^2*c^2+715*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)*(5*b^12+5*c^12-(166*b^8+166*c^8-(299*b^4+236*b^2*c^2+299*c^4)*b^2*c^2)*b^2*c^2)*a^4+(11*b^12+11*c^12-(22*b^8+22*c^8+(171*b^4+148*b^2*c^2+171*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^2-c^2)^6*(b^2+c^2)*(3*b^8+3*c^8+2*(6*b^4+b^2*c^2+6*c^4)*b^2*c^2)) : :

The center of the reciprocal V-perspeconic of these triangles is X(17807).

X(45188) lies on these lines: {2, 31362}, {3, 14091}, {4, 31367}, {20, 31369}, {1350, 5925}

X(45188) = midpoint of X(20) and X(31369)
X(45188) = reflection of X(i) in X(j) for these (i, j): (4, 31367), (17807, 3)
X(45188) = anticomplement of X(31362)
X(45188) = X(17807)-of-ABC-X3 reflections triangle
X(45188) = X(31367)-of-anti-Euler triangle

X(45189) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: GARCIA OUTER TO ABC

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

The center of the reciprocal V-perspeconic of these triangles is X(3588).

X(45189) lies on these lines: {1, 9551}, {4, 43739}, {10, 3588}, {11, 1402}, {442, 4026}, {3696, 10914}, {3931, 14749}, {37715, 42837}

X(45189) = reflection of X(3588) in X(10)
X(45189) = X(3588)-of-outer-Garcia triangle
X(45189) = X(8)-Beth conjugate of-X(3588)

X(45190) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: JOHNSON TO ABC

Barycentrics    ((b^2+c^2)*a^22-(7*b^4+8*b^2*c^2+7*c^4)*a^20+(b^2+c^2)*(22*b^4+5*b^2*c^2+22*c^4)*a^18-(42*b^8+42*c^8+(49*b^4+38*b^2*c^2+49*c^4)*b^2*c^2)*a^16+(b^2+c^2)*(56*b^8+56*c^8-3*(b^2-3*c^2)*(3*b^2-c^2)*b^2*c^2)*a^14-(56*b^12+56*c^12+(b^2-c^2)^2*(3*b^2-b*c+3*c^2)*(3*b^2+b*c+3*c^2)*b^2*c^2)*a^12+(b^4-c^4)*(b^2-c^2)*(42*b^8+42*c^8+5*(b^4+10*b^2*c^2+c^4)*b^2*c^2)*a^10-(b^2-c^2)^2*(22*b^12+22*c^12-(3*b^4+3*c^4+(3*b^2+2*b*c+3*c^2)*b*c)*(3*b^4+3*c^4-(3*b^2-2*b*c+3*c^2)*b*c)*b^2*c^2)*a^8+(b^4-c^4)*(b^2-c^2)^3*(7*b^8+7*c^8-(15*b^4-4*b^2*c^2+15*c^4)*b^2*c^2)*a^6-(b^2-c^2)^6*(b^8+c^8-(7*b^4+11*b^2*c^2+7*c^4)*b^2*c^2)*a^4-2*(b^2-c^2)^8*(b^2+c^2)*b^2*c^2*a^2-(b^2-c^2)^8*b^4*c^4)*(-a^2+b^2+c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

The center of the reciprocal V-perspeconic of these triangles is X(31353).

X(45190) lies on these lines: {5, 31353}, {12162, 39530}

X(45190) = reflection of X(31353) in X(5)
X(45190) = X(31353)-of-Johnson triangle

X(45191) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: ABC TO EHRMANN-MID

Barycentrics    a^2*(2*a^14-5*(b^2+c^2)*a^12-(3*b^4-16*b^2*c^2+3*c^4)*a^10+(b^2+c^2)*(20*b^4-33*b^2*c^2+20*c^4)*a^8-4*(5*b^8+5*c^8+2*(b^4-5*b^2*c^2+c^4)*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)*(3*b^4+20*b^2*c^2+3*c^4)*a^4+(5*b^4+12*b^2*c^2+5*c^4)*(b^2-c^2)^4*a^2-(b^4-c^4)*(b^2-c^2)^3*(b^2+2*c^2)*(2*b^2+c^2))*(-a^2+b^2+c^2)*(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

The center of the reciprocal V-perspeconic of these triangles is X(45192).

X(45191) lies on these lines: {185, 15860}, {3163, 3574}, {13202, 15816}

X(45192) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: EHRMANN-MID TO ABC

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

The center of the reciprocal V-perspeconic of these triangles is X(45191).

X(45192) lies on these lines: {30, 141}, {265, 11079}

X(45193) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: ABC TO MIXTILINEAR5

Barycentrics    a*(-a+b+c)^2*(a^5-(b+c)*a^4-2*(b^2-6*b*c+c^2)*a^3+2*(b+c)*(b^2-8*b*c+c^2)*a^2+(b^4+c^4-2*(10*b^2-27*b*c+10*c^2)*b*c)*a-(b^2-c^2)*(b-c)*(b^2-6*b*c+c^2)) : :

The center of the reciprocal V-perspeconic of these triangles is X(45194).

X(45193) lies on these lines: {1, 45194}, {200, 4907}, {3161, 10580}

X(45193) = reflection of X(45194) in X(1)
X(45193) = X(45194)-of-5th mixtilinear triangle
X(45193) = X(145)-Ceva conjugate of-X(200)

X(45194) = CENTER OF THE V-PERSPECONIC OF THESE TRIANGLES: MIXTILINEAR5 TO ABC

Barycentrics    a*(a^8-6*(b+c)*a^7+2*(5*b^2+22*b*c+5*c^2)*a^6+2*(b+c)*(b^2-52*b*c+c^2)*a^5-4*(5*b^4+5*c^4-(23*b^2+64*b*c+23*c^2)*b*c)*a^4+2*(b+c)*(7*b^4+7*c^4+2*(4*b^2-79*b*c+4*c^2)*b*c)*a^3+2*(3*b^6+3*c^6-(62*b^4+62*c^4-(189*b^2-68*b*c+189*c^2)*b*c)*b*c)*a^2-2*(b+c)*(b^2+c^2)*(b^2-6*b*c+c^2)*(5*b^2-14*b*c+5*c^2)*a+(3*b^4+3*c^4-2*(6*b^2-b*c+6*c^2)*b*c)*(b^2-c^2)^2) : :

The center of the reciprocal V-perspeconic of these triangles is X(45193).

X(45194) lies on these lines: {1, 45193}, {3242, 8583}

X(45194) = reflection of X(45193) in X(1)
X(45194) = X(45193)-of-5th mixtilinear triangle

X(45195) = X(30)X(52)∩X(125)X(6662)

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

See Antreas Hatzipolakis and Ercole Suppa, euclid 2765.

X(45195) lies on the Euler hyperbola and these lines: {24,16035}, {30,52}, {74,13489}, {125,6662}, {184,6663}, {195,2452}, {800,1990}, {3260,11585}, {14254,44235}, {15454,37814}, {16238,40678}, {31804,42453}

X(45195) = anticomplement of the complementary conjugate of X(43817)
X(45195) = isogonal conjugate of the anticomplement of X(43817)

leftri

Perspectors of Electra(B) triangles and cevian triangles: X(45196)-X(45215)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 17, 2021.

In this paragraph (but not the next), all coordinates are trilinears. Suppose that P = p : q : r and U = u : v : w, where neither P nor U lies on a sideline of ABC. Let L(p,q,r,u,v,w) = p^2 * (v^2 + w^2) - u*(q v + r w), so that (as in the Glossary), the point

L(u,v,w) : L(v,w,u) : L(w,u,v)

is the P-line conjugate of U, which lies on this line: PU. Let A'B'C' be the anticevian triangle of U, so that A' = -u : w : w, and B' and C' are determined cyclically. Let

A" = P-line conjugate of A', and define B" and C" cyclically. The triangle A"B"C" is here named the (P,U)-Electra triangle. If P = X(1) = 1:1:1, then the locus of a point X = x : y : z such that the cevian triangle of X is perspective to the (X(1),U)-Electra triangle is given by

v (u + w) (v^2 + w^2 + u v - u w) y^2 z - w (u + v) (v^2 + w^2 + u w - v w) y z^2 + (cyclic) - (v - w)(w - u)(u - v)(u + v + w) x y z = 0

Next, suppose that all coordinates in the preceding paragraph are barycentrics. The triangle A"B"C" is then the (X(1),U)-Electra(B) triangle. The appearance of (i,j,k)) in the following list means that the (X(1),X(i))-Electra(B) triangle is perspective to the cevian triangle of X(j), and the perspector is X(k):

(1,2,4357), (1,69,3687), (1,85,45196), (1,86,2), (1,1909,10), (1,6384,45197), (1,28660,76)
(3,2,45198), (3,95,2), (3,9291,5))
(4,2,41005), (4,264,2), (4,1975,3), (4,6340,45199), (4,6527,46200)
(6,2,6656), (6,83,2), (6,315,46201), (6,9230,141)
(7,2,41006), (7,85,2}, (7,3729,9), (7,6557,45202), (7,30695,45203)
(8,2,3663), (8,75,2), (8,4452,45204), (8,9312,1), (8,27818,45205)
(10,2,17322), (10,1268,2), (10,4360,17011)
(63,2,45206), (63,333,2), (63,1947,226)
(69,2,5254}, (69,76,2), (69,459,45207), (69,6392,40326), (69,9308,6), (69,21447,393)
(75,2,1107}, (75,57,45208), (75,87,45209), (75,274,2), (75,894,37), (75,27644,6)
(76,2,45210), (76,25,45211), (76,308,2), (76,384,39}, (76,38834,32)
(99,2,45212), (99,148,115), (99,4590,2)
(141,2,7859}, (141,7760,34482), (141,10159,2), (141,40000,3589)
(190,2,45213), (190,1016,2), (190,4440,1086)
(511,2,290), (511,290,2), (511,14382,98), (511,16081,264), (511,39058,511)
(514,2,190), (514,190,2), (514,1016,1016), (514,6631,514)
(518,2,2481), (518,2481,2), (518,33675,518), (518,34018,85), (518,36796,75)
(523,2,99), (523, 99,2), (523, 4590,4590), (523, 31998,523), (523,37880,9293)
(668,2,45214), (668,9263,1015), (668,31625,2)
(850,2,45215), (850,6331,2), (850,9514,647)

X(45196) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(1))-ELECTRA(B) AND CEVIAN OF X(85)

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

X(45196) lies on these lines: {2, 18632}, {7, 1330}, {9, 20914}, {10, 307}, {57, 85}, {76, 20236}, {226, 1231}, {348, 18593}, {349, 6358}, {1038, 9312}, {1111, 3670}, {1447, 16817}, {1930, 7179}, {2476, 23581}, {3212, 9534}, {3674, 3687}, {3704, 18697}, {4552, 28594}, {4858, 17062}, {14210, 17084}, {16611, 41785}, {16732, 21965}, {17306, 21588}, {17308, 40702}, {17877, 25639}, {17887, 24026}, {18140, 21586}, {21207, 23555}, {30077, 30097}, {30545, 33935}

X(45196) = X(i)-Ceva conjugate of X(j) for these (i,j): {85, 3674}, {4572, 4077}
X(45196) = X(1211)-cross conjugate of X(18697)
X(45196) = X(i)-isoconjugate of X(j) for these (i,j): {41, 2363}, {55, 1169}, {607, 1798}, {1791, 2204}, {2175, 14534}, {2194, 2298}, {2299, 2359}, {7252, 32736}, {8687, 21789}, {9448, 40827}
X(45196) = cevapoint of X(1211) and X(41003)
X(45196) = crosspoint of X(85) and X(349)
X(45196) = barycentric product X(i)*X(j) for these {i,j}: {7, 18697}, {12, 16739}, {57, 1228}, {75, 41003}, {85, 1211}, {226, 20911}, {313, 24471}, {321, 3674}, {349, 3666}, {429, 7182}, {1088, 3704}, {1231, 1848}, {1441, 4357}, {1446, 3687}, {2092, 20567}, {2292, 6063}, {3725, 41283}, {4509, 4552}, {4554, 21124}, {6358, 16705}
X(45196) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 2363}, {57, 1169}, {77, 1798}, {85, 14534}, {226, 2298}, {307, 1791}, {349, 30710}, {429, 33}, {960, 2328}, {1020, 8687}, {1193, 2194}, {1211, 9}, {1214, 2359}, {1228, 312}, {1441, 1220}, {1829, 2299}, {1848, 1172}, {2092, 41}, {2292, 55}, {2354, 2204}, {3004, 3737}, {3666, 284}, {3668, 961}, {3674, 81}, {3687, 2287}, {3704, 200}, {3725, 2175}, {3882, 5546}, {3910, 1021}, {4077, 4581}, {4357, 21}, {4509, 4560}, {4551, 32736}, {4552, 36147}, {4566, 36098}, {4918, 3158}, {6042, 40966}, {6358, 14624}, {16705, 2185}, {16739, 261}, {17185, 7054}, {17420, 21789}, {18697, 8}, {20567, 40827}, {20653, 210}, {20911, 333}, {21033, 220}, {21124, 650}, {21810, 1334}, {22076, 212}, {22097, 2193}, {24471, 58}, {27697, 2329}, {40153, 2150}, {40966, 1253}, {41003, 1}, {44092, 2212}

X(45197) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(1))-ELECTRA(B) AND CEVIAN OF X(6384)

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

X(45197) lies on these lines: {2, 256}, {10, 3728}, {58, 87}, {330, 1654}, {444, 2354}, {932, 38453}, {1193, 27455}, {1245, 23493}, {1400, 2238}, {2053, 8615}, {2228, 25144}, {2292, 27697}, {3123, 20891}, {3705, 17065}, {6378, 21796}, {6384, 30966}, {7015, 25838}, {16569, 20984}, {27424, 31339}, {27678, 28269}, {28256, 28258}

X(45197) = X(1211)-cross conjugate of X(2292)
X(45197) = crosspoint of X(87) and X(42027)
X(45197) = crosssum of X(43) and X(38832)
X(45197) = X(i)-isoconjugate of X(j) for these (i,j): {43, 2363}, {192, 1169}, {1220, 38832}, {2176, 14534}, {2298, 27644}, {8687, 27527}, {8707, 16695}, {17217, 32736}, {18197, 36147}
X(45197) = barycentric product X(i)*X(j) for these {i,j}: {10, 27455}, {87, 1211}, {330, 2292}, {932, 21124}, {1228, 7121}, {2092, 6384}, {2162, 18697}, {2319, 41003}, {3666, 42027}, {3704, 7153}, {3725, 6383}, {4357, 16606}, {6378, 16739}, {7148, 16705}, {7209, 40966}, {20911, 23493}
X(45197) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 14534}, {1193, 27644}, {1211, 6376}, {2092, 43}, {2162, 2363}, {2292, 192}, {2300, 38832}, {3666, 33296}, {3704, 4110}, {3725, 2176}, {4357, 31008}, {6371, 18197}, {6384, 40827}, {7121, 1169}, {7148, 14624}, {15373, 1798}, {16606, 1220}, {17420, 27527}, {18697, 6382}, {21033, 27538}, {21124, 20906}, {21810, 3971}, {22076, 22370}, {22381, 2359}, {23493, 2298}, {27455, 86}, {27697, 41318}, {40966, 3208}, {41003, 30545}, {42027, 30710}, {42661, 21834}

X(45198) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(3))-ELECTRA(B) AND CEVIAN OF X(2)

Barycentrics    a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 3*a^2*b^4*c^2 + 2*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 2*b^2*c^6 - c^8 : :
Barycentrics    (csc A) (cos A - cos 2A cos(B - C)) : :
Barycentrics    csc A sin(A + T) : :, T as at X(389)

X(45198) lies on these lines: {2, 6}, {3, 317}, {4, 20477}, {5, 264}, {30, 32002}, {32, 26205}, {39, 26155}, {53, 3164}, {76, 7399}, {95, 140}, {98, 26926}, {114, 14913}, {216, 297}, {233, 14767}, {253, 5056}, {286, 6831}, {311, 13160}, {315, 7395}, {316, 34664}, {339, 37347}, {401, 6748}, {403, 44131}, {441, 36794}, {458, 6389}, {547, 1494}, {577, 27377}, {631, 32001}, {637, 6809}, {638, 6810}, {1232, 1238}, {1235, 14788}, {1513, 1843}, {1656, 40995}, {1975, 6815}, {2450, 23635}, {2893, 7567}, {2897, 6960}, {3087, 37188}, {3090, 32000}, {3091, 6527}, {3186, 37446}, {3628, 40996}, {3926, 6803}, {3933, 44149}, {3964, 7763}, {5133, 30737}, {6677, 44096}, {6776, 20792}, {6804, 32816}, {6816, 7773}, {7503, 7750}, {7549, 21287}, {7752, 14615}, {7769, 22468}, {7811, 38434}, {8795, 16089}, {9722, 41760}, {9723, 17928}, {9744, 19459}, {10024, 44138}, {10979, 35937}, {13409, 34965}, {16264, 18437}, {19166, 26879}, {22467, 44180}, {26154, 26216}, {36998, 44200}

X(45198) = isotomic conjugate of X(40448)
X(45198) = polar conjugate of X(40402)
X(45198) = complement of the isotomic conjugate of X(9290)
X(45198) = isotomic conjugate of the isogonal conjugate of X(389)
X(45198) = X(i)-complementary conjugate of X(j) for these (i,j): {661, 130}, {1303, 4369}, {9251, 141}, {9290, 2887}
X(45198) = X(42405)-Ceva conjugate of X(525)
X(45198) = cevapoint of X(34836) and X(42441)
X(45198) = crosspoint of X(i) and X(j) for these (i,j): {2, 9290}, {95, 264}
X(45198) = crosssum of X(i) and X(j) for these (i,j): {6, 1970}, {32, 44088}, {51, 184}
X(45198) = pole wrt polar circle of trilinear polar of X(40402) (line X(2501)X(39201))
X(45198) = X(i)-isoconjugate of X(j) for these (i,j): {31, 40448}, {48, 40402}, {560, 42333}
X(45198) = barycentric product X(i)*X(j) for these {i,j}: {76, 389}, {95, 34836}, {276, 42441}, {311, 19170}, {6750, 34386}
X(45198) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40448}, {4, 40402}, {76, 42333}, {389, 6}, {6750, 53}, {19170, 54}, {34836, 5}, {42441, 216}
X(45198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 40680, 20477}, {5, 30258, 6530}, {5, 41005, 264}, {69, 39113, 325}, {95, 340, 41008}, {140, 41008, 95}, {233, 15526, 14767}, {302, 303, 11064}, {401, 17035, 6748}, {491, 492, 394}, {1232, 1273, 1238}, {6748, 34828, 401}

X(45199) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(4))-ELECTRA(B) AND CEVIAN OF X(6340)

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

X(45199) lies on these lines: {2, 9307}, {3, 6391}, {232, 8770}, {1093, 6622}, {2996, 3164}, {6340, 40032}, {27364, 43917}

X(45199) = X(6509)-cross conjugate of X(185)
X(45199) = crosspoint of X(6391) and X(34208)
X(45199) = crosssum of X(i) and X(j) for these (i,j): {193, 37199}, {3167, 6353}
X(45199) = X(i)-isoconjugate of X(j) for these (i,j): {775, 6353}, {821, 3167}, {1105, 1707}
X(45199) = barycentric product X(i)*X(j) for these {i,j}: {185, 2996}, {800, 6340}, {6391, 13567}, {6508, 8769}, {6509, 34208}, {8770, 41005}, {19180, 27364}
X(45199) = barycentric quotient X(i)/X(j) for these {i,j}: {185, 193}, {235, 21447}, {417, 10607}, {800, 6353}, {6340, 40830}, {6391, 801}, {6508, 18156}, {6509, 6337}, {8770, 1105}, {40319, 41890}

X(45200) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(4))-ELECTRA(B) AND CEVIAN OF X(6527)

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

X(45200) lies on these lines: {2, 3}, {69, 1073}, {343, 44436}, {459, 6527}, {800, 6509}, {1503, 33581}, {1624, 41602}, {1661, 41735}, {5065, 23292}, {6389, 26958}, {6716, 36988}, {7011, 27509}, {14361, 42458}, {15466, 20207}, {15905, 37669}, {17811, 41008}, {18750, 40616}, {20208, 37877}, {41883, 44360}

X(45200) = isotomic conjugate of the polar conjugate of X(2883)
X(45200) = X(13380)-complementary conjugate of X(20305)
X(45200) = X(i)-Ceva conjugate of X(j) for these (i,j): {69, 185}, {4558, 20580}
X(45200) = X(2883)-cross conjugate of X(41005)
X(45200) = crosspoint of X(15466) and X(37669)
X(45200) = crosssum of X(14642) and X(41489)
X(45200) = X(i)-isoconjugate of X(j) for these (i,j): {775, 41489}, {821, 14642}, {1105, 2155}
X(45200) = barycentric product X(i)*X(j) for these {i,j}: {20, 41005}, {69, 2883}, {185, 14615}, {6508, 18750}, {6509, 15466}, {13567, 37669}, {18603, 42699}, {20580, 41678}, {35602, 44131}
X(45200) = barycentric quotient X(i)/X(j) for these {i,j}: {20, 1105}, {185, 64}, {235, 6526}, {417, 14379}, {800, 41489}, {820, 19614}, {1624, 1301}, {1895, 821}, {2883, 4}, {6508, 2184}, {6509, 1073}, {13567, 459}, {15905, 41890}, {36982, 39268}, {37669, 801}, {41005, 253}
X(45200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6617, 441}, {6509, 13567, 41005}, {20207, 42459, 15466}

X(45201) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(6))-ELECTRA(B) AND CEVIAN OF X(315)

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

X(45201) lies on these lines: {2, 6}, {3, 34254}, {5, 40022}, {22, 7750}, {23, 1369}, {25, 315}, {30, 16275}, {76, 427}, {99, 7667}, {264, 15809}, {305, 1368}, {316, 428}, {468, 7768}, {626, 1196}, {858, 8024}, {1078, 7499}, {1092, 37450}, {1194, 6656}, {1370, 1975}, {1513, 5562}, {1799, 6676}, {3785, 7494}, {3917, 4121}, {3926, 7386}, {5020, 7776}, {5064, 11185}, {5133, 39998}, {5305, 30785}, {6390, 10691}, {6995, 32006}, {6997, 7773}, {7391, 32819}, {7392, 32816}, {7396, 32830}, {7484, 7763}, {7539, 32832}, {7664, 42052}, {7745, 8878}, {7752, 37439}, {7789, 16951}, {7795, 11324}, {7796, 30739}, {7799, 43957}, {7811, 44210}, {7818, 34481}, {7860, 10301}, {8267, 31107}, {9464, 31101}, {10154, 14929}, {14376, 28719}, {14994, 21243}, {15437, 32971}, {16063, 32820}, {17279, 39249}, {23115, 28427}, {23295, 34384}, {26235, 37990}, {31152, 32833}, {32815, 44442}

X(45201) = midpoint of X(16275) and X(16276)
X(45201) = isotomic conjugate of the isogonal conjugate of X(11574)
X(45201) = isotomic conjugate of the polar conjugate of X(6656)
X(45201) = X(76)-Ceva conjugate of X(23642)
X(45201) = X(i)-cross conjugate of X(j) for these (i,j): {11574, 6656}, {22424, 11574}
X(45201) = crosspoint of X(305) and X(1799)
X(45201) = crosssum of X(i) and X(j) for these (i,j): {25, 11380}, {1843, 1974}
X(45201) = barycentric product X(i)*X(j) for these {i,j}: {69, 6656}, {76, 11574}, {304, 17446}, {305, 1194}, {308, 22424}, {1799, 21248}, {16735, 20336}, {21424, 34055}
X(45201) = barycentric quotient X(i)/X(j) for these {i,j}: {305, 1241}, {1194, 25}, {2514, 2489}, {6656, 4}, {11574, 6}, {16735, 28}, {17446, 19}, {21248, 427}, {21336, 17442}, {21424, 20883}, {22424, 39}, {23642, 1843}
X(45201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5359, 7792}, {1194, 21248, 6656}, {1368, 3933, 305}, {1611, 7778, 2}, {1799, 37804, 6676}, {3917, 4121, 6393}, {6676, 7767, 1799}, {8878, 16950, 7745}

X(45202) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(7))-ELECTRA(B) AND CEVIAN OF X(6557)

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

X(45202) lies on these lines: {2, 9311}, {9, 3057}, {105, 11260}, {241, 8056}, {2348, 36846}, {3177, 4373}, {3290, 3445}, {3924, 38266}, {4534, 6736}, {6557, 30854}, {23062, 27818}

X(45202) = crosspoint of X(3680) and X(27818)
X(45202) = X(3052)-isoconjugate of X(23618)
X(45202) = barycentric product X(i)*X(j) for these {i,j}: {1200, 40014}, {3680, 11019}, {4373, 14100}, {6557, 40133}, {8056, 41006}
X(45202) = barycentric quotient X(i)/X(j) for these {i,j}: {1200, 1743}, {8056, 23618}, {11019, 39126}, {14100, 145}, {20978, 1420}, {40133, 5435}, {41006, 18743}

X(45203) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(7))-ELECTRA(B) AND CEVIAN OF X(30695)

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

X(45203) lies on these lines: {2, 7}, {8, 19605}, {200, 23244}, {220, 20103}, {516, 11051}, {5574, 24477}, {5853, 41795}, {11019, 21049}, {13609, 21060}, {24393, 41796}, {30695, 36620}

X(45203) = X(8)-Ceva conjugate of X(14100)
X(45203) = X(43182)-cross conjugate of X(41006)
X(45203) = barycentric product X(i)*X(j) for these {i,j}: {8, 43182}, {144, 41006}, {14100, 16284}
X(45203) = barycentric quotient X(i)/X(j) for these {i,j}: {144, 23618}, {1200, 11051}, {11019, 36620}, {14100, 3062}, {41006, 10405}, {43182, 7}

X(45204) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(8))-ELECTRA(B) AND CEVIAN OF X(4452)

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

X(45204) lies on these lines: {1, 2}, {6, 6692}, {7, 8056}, {142, 16602}, {222, 3911}, {226, 16610}, {238, 10164}, {908, 24177}, {982, 21060}, {988, 18250}, {1266, 4052}, {1323, 36636}, {1402, 28239}, {1453, 17567}, {1708, 43068}, {1738, 3817}, {1743, 5435}, {1997, 3875}, {3175, 16594}, {3210, 27130}, {3306, 10900}, {3452, 3663}, {3600, 45047}, {3664, 5437}, {3666, 5316}, {3671, 24174}, {3755, 3816}, {3756, 4849}, {3947, 24178}, {3950, 18743}, {3977, 26688}, {4000, 30827}, {4072, 30861}, {4298, 11512}, {4301, 24440}, {4308, 7963}, {4349, 17122}, {4417, 21255}, {4452, 6557}, {4641, 43055}, {4656, 4850}, {4859, 5226}, {4887, 28609}, {4929, 6555}, {4997, 19796}, {5249, 37651}, {5542, 17063}, {5573, 25568}, {5717, 16408}, {5745, 37679}, {5748, 23681}, {5850, 18193}, {6244, 19517}, {6510, 37646}, {10171, 17064}, {15654, 37269}, {17056, 31197}, {21077, 24171}, {21363, 28272}, {21627, 21896}, {22128, 32911}, {24597, 31224}, {28346, 44858}, {30424, 33096}, {31190, 37642}, {31227, 41629}, {33111, 38204}, {41886, 43182}

X(45204) = complement of X(30567)
X(45204) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 3057}, {651, 30719}, {30610, 31182}
X(45204) = X(12640)-cross conjugate of X(3663)
X(45204) = X(i)-isoconjugate of X(j) for these (i,j): {1222, 38266}, {1261, 40151}, {3445, 23617}, {3451, 3680}
X(45204) = crosspoint of X(5435) and X(18743)
X(45204) = barycentric product X(i)*X(j) for these {i,j}: {7, 12640}, {145, 3663}, {664, 14284}, {1122, 44720}, {1420, 20895}, {1743, 26563}, {3057, 39126}, {3452, 5435}, {3667, 21272}, {3752, 18743}, {3950, 18600}, {4394, 21580}, {4415, 41629}, {4462, 21362}, {4848, 17183}, {25268, 30719}
X(45204) = barycentric quotient X(i)/X(j) for these {i,j}: {145, 1222}, {1122, 19604}, {1201, 3445}, {1420, 1476}, {1743, 23617}, {3057, 3680}, {3158, 1261}, {3452, 6557}, {3663, 4373}, {3752, 8056}, {3756, 40451}, {4415, 4052}, {5435, 40420}, {6736, 6556}, {12640, 8}, {14284, 522}, {18743, 32017}, {20228, 38266}, {21362, 27834}, {23845, 1293}, {26563, 40014}, {28006, 27831}, {43290, 8706}
X(45204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2999, 39595}, {2, 23511, 3008}, {43, 5121, 11019}, {226, 16610, 24175}, {3452, 3752, 3663}, {16569, 24239, 10}, {16602, 37662, 142}, {16610, 37663, 226}

X(45205) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(8))-ELECTRA(B) AND CEVIAN OF X(27818)

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

X(45205) lies on these lines: {1, 16079}, {85, 5226}, {738, 40151}, {1358, 12053}, {3674, 10029}, {4373, 17480}, {27834, 30618}

X(45205) = X(3752)-cross conjugate of X(1122)
X(45205) = X(i)-isoconjugate of X(j) for these (i,j): {1261, 1743}, {1476, 4936}, {3158, 23617}, {3451, 6555}
X(45205) = barycentric product X(i)*X(j) for these {i,j}: {1122, 4373}, {3663, 19604}, {3752, 27818}, {26563, 40151}
X(45205) = barycentric quotient X(i)/X(j) for these {i,j}: {1122, 145}, {1201, 3158}, {2347, 4936}, {3057, 6555}, {3445, 1261}, {3663, 44720}, {3752, 3161}, {6363, 4162}, {6615, 4546}, {19604, 1222}, {21362, 30720}, {26563, 44723}, {27818, 32017}, {40151, 23617}, {42336, 8643}

X(45206) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(63))-ELECTRA(B) AND CEVIAN OF X(2)

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

X(45206) lies on these lines: {1, 406}, {2, 914}, {8, 2900}, {9, 306}, {10, 581}, {37, 41883}, {58, 34851}, {63, 6515}, {77, 20266}, {92, 226}, {124, 24210}, {261, 284}, {281, 5712}, {386, 34823}, {440, 9119}, {464, 1741}, {950, 5081}, {958, 5814}, {960, 41014}, {991, 34822}, {1125, 8555}, {1146, 6708}, {1158, 18909}, {1211, 40937}, {1214, 13567}, {1427, 16608}, {1708, 6350}, {1826, 10478}, {1944, 17778}, {2262, 19542}, {2323, 40571}, {2635, 21911}, {2968, 10391}, {3061, 3452}, {3089, 15836}, {3187, 28796}, {3666, 26932}, {3931, 20306}, {4357, 16579}, {4858, 5249}, {5273, 33077}, {5325, 17346}, {5713, 39585}, {6349, 37643}, {8257, 18928}, {8896, 15830}, {17019, 28836}, {17073, 26958}, {17080, 26540}, {17316, 27539}, {17442, 23619}, {18161, 21621}, {18228, 32858}, {18249, 21081}, {24239, 34589}, {25525, 25935}, {28950, 31034}

X(45206) = complement of X(1943)
X(45206) = complement of the isogonal conjugate of X(7106)
X(45206) = complement of the isotomic conjugate of X(7108)
X(45206) = isotomic conjugate of the isogonal conjugate of X(1195)
X(45206) = X(i)-complementary conjugate of X(j) for these (i,j): {7016, 18589}, {7105, 141}, {7106, 10}, {7107, 3}, {7108, 2887}
X(45206) = X(811)-Ceva conjugate of X(522)
X(45206) = crosspoint of X(i) and X(j) for these (i,j): {2, 7108}, {92, 333}
X(45206) = crosssum of X(i) and X(j) for these (i,j): {6, 1950}, {48, 1400}, {73, 9316}
X(45206) = barycentric product X(i)*X(j) for these {i,j}: {21, 18692}, {75, 1858}, {76, 1195}, {332, 431}
X(45206) = barycentric quotient X(i)/X(j) for these {i,j}: {431, 225}, {1195, 6}, {1858, 1}, {18692, 1441}
X(45206) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1146, 17056, 6708}, {4035, 41006, 3452}, {6350, 11433, 1708}

X(45207) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(69))-ELECTRA(B) AND CEVIAN OF X(459)

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

X(45207) lies on these lines: {2, 9289}, {6, 64}, {194, 253}, {235, 1562}, {459, 40814}, {1073, 36212}, {3148, 33581}, {8779, 11413}, {15341, 16196}

X(45207) = X(1196)-cross conjugate of X(6467)
X(45207) = crosspoint of X(64) and X(34403)
X(45207) = crosssum of X(20) and X(3172)
X(45207) = X(i)-isoconjugate of X(j) for these (i,j): {204, 40405}, {610, 40413}
X(45207) = barycentric product X(i)*X(j) for these {i,j}: {64, 1368}, {253, 6467}, {459, 22401}, {682, 41530}, {1073, 5254}, {1196, 34403}, {2155, 21406}, {2184, 18671}, {17872, 19611}
X(45207) = barycentric quotient X(i)/X(j) for these {i,j}: {64, 40413}, {253, 683}, {682, 154}, {1073, 40405}, {1196, 1249}, {1368, 14615}, {5254, 15466}, {6467, 20}, {17872, 1895}, {18671, 18750}, {22401, 37669}, {40325, 6525}

X(45208) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(75))-ELECTRA(B) AND CEVIAN OF X(57)

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

X(45208) lies on these lines: {1, 20594}, {2, 257}, {7, 1655}, {12, 21025}, {37, 65}, {56, 40750}, {57, 40773}, {85, 6385}, {226, 3948}, {604, 2303}, {959, 30571}, {978, 16611}, {1107, 22065}, {1193, 2170}, {1953, 2277}, {2099, 16969}, {3125, 40986}, {3142, 21044}, {3674, 29968}, {3691, 15984}, {3728, 27880}, {3930, 4095}, {4051, 4771}, {5011, 35206}, {16583, 22197}, {16605, 22173}, {17443, 28244}, {17452, 37598}, {18055, 18743}, {18176, 35102}, {18726, 21246}, {20271, 20284}, {20891, 30097}, {21769, 28082}, {29988, 43040}, {30011, 41006}

X(45208) = X(4554)-Ceva conjugate of X(4017)
X(45208) = X(21838)-cross conjugate of X(3728)
X(45208) = X(i)-isoconjugate of X(j) for these (i,j): {21, 1258}, {55, 40409}, {284, 40418}, {1221, 2194}, {6064, 40525}
X(45208) = cevapoint of X(21838) and X(39780)
X(45208) = crosspoint of X(65) and X(85)
X(45208) = crosssum of X(21) and X(41)
X(45208) = barycentric product X(i)*X(j) for these {i,j}: {7, 3728}, {12, 18169}, {37, 30097}, {57, 21024}, {65, 3741}, {75, 39780}, {85, 21838}, {226, 1107}, {349, 1197}, {1014, 21713}, {1400, 20891}, {1434, 22206}, {1441, 2309}, {2171, 16738}, {4554, 40627}, {7249, 27880}, {22065, 40149}
X(45208) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 40409}, {65, 40418}, {226, 1221}, {1107, 333}, {1197, 284}, {1400, 1258}, {2309, 21}, {3728, 8}, {3741, 314}, {18169, 261}, {20891, 28660}, {21024, 312}, {21700, 210}, {21713, 3701}, {21838, 9}, {22065, 1812}, {22206, 2321}, {22389, 283}, {23212, 212}, {23473, 3794}, {27880, 7081}, {30097, 274}, {39780, 1}, {40627, 650}
X(45208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {37, 20719, 1334}, {20707, 21808, 37}

X(45209) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(75))-ELECTRA(B) AND CEVIAN OF X(87)

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

X(45209) lies on these lines: {6, 904}, {37, 23493}, {86, 87}, {1333, 7121}, {1918, 8022}, {2162, 20992}, {3248, 4022}

X(45209) = X(7121)-Ceva conjugate of X(1197)
X(45209) = X(i)-isoconjugate of X(j) for these (i,j): {192, 40409}, {1221, 27644}, {1258, 31008}, {33296, 40418}
X(45209) = crosspoint of X(87) and X(21759)
X(45209) = crosssum of X(43) and X(31008)
X(45209) = barycentric product X(i)*X(j) for these {i,j}: {87, 21838}, {932, 40627}, {1107, 23493}, {1197, 42027}, {2162, 3728}, {2309, 16606}, {2319, 39780}, {3741, 21759}, {6378, 18169}, {7121, 21024}
X(45209) = barycentric quotient X(i)/X(j) for these {i,j}: {1197, 33296}, {2309, 31008}, {3728, 6382}, {7121, 40409}, {21759, 40418}, {21838, 6376}, {23212, 22370}, {23493, 1221}, {39780, 30545}, {40627, 20906}

X(45210) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(76))-ELECTRA(B) AND CEVIAN OF X(2)

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

X(45210) lies on these lines: {2, 2998}, {3, 6}, {37, 17793}, {69, 3117}, {76, 8264}, {115, 44947}, {141, 706}, {183, 1196}, {308, 3934}, {385, 1194}, {538, 33769}, {689, 17965}, {710, 6656}, {733, 40000}, {1015, 17049}, {1084, 3589}, {1176, 14602}, {1180, 7766}, {2275, 18194}, {2309, 20868}, {3224, 17042}, {3228, 39968}, {3329, 37875}, {3511, 14913}, {3763, 34811}, {3764, 23643}, {3778, 20862}, {3917, 18899}, {5976, 37891}, {6377, 16706}, {6378, 17289}, {8041, 32748}, {8149, 40073}, {8623, 22062}, {9233, 19127}, {9462, 34816}, {9468, 40708}, {9865, 40035}, {16584, 27633}, {17322, 21827}, {20464, 23632}, {20859, 23635}, {20975, 23642}, {21327, 26971}, {21838, 37678}, {23432, 23633}, {23484, 24443}, {24530, 37686}, {31276, 36648}

X(45210) = complement of X(9230)
X(45210) = Moses-circle-inverse of X(14962)
X(45210) = complement of the isotomic conjugate of X(695)
X(45210) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 37890}, {32, 19564}, {560, 37891}, {695, 2887}, {1580, 39082}, {9229, 21235}, {9236, 2}, {9239, 40379}, {9285, 626}, {9288, 141}
X(45210) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 37890}, {689, 512}
X(45210) = crosspoint of X(i) and X(j) for these (i,j): {2, 695}, {6, 308}, {251, 3224}
X(45210) = crosssum of X(i) and X(j) for these (i,j): {2, 3051}, {6, 384}, {141, 194}
X(45210) = crossdifference of every pair of points on line {523, 9491}
X(45210) = barycentric product X(695)*X(37890)
X(45210) = barycentric quotient X(37890)/X(9230)
X(45210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 8266, 41331}, {6, 41328, 33875}, {141, 8265, 3229}, {2028, 2029, 14962}, {8266, 41331, 187}

X(45211) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(76))-ELECTRA(B) AND CEVIAN OF X(25)

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

X(45211) lies on these lines: {2, 3186}, {6, 41293}, {39, 1843}, {264, 44161}, {427, 35540}, {12144, 32476}

X(45211) = crosspoint of X(264) and X(1843)
X(45211) = crosssum of X(184) and X(1799)

X(45212) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(99))-ELECTRA(B) AND CEVIAN OF X(2)

Barycentrics    (b - c)^2*(b + c)^2*(3*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 + b^2*c^2 + c^4) : :
X(45212) = 9 X[2] - X[31372], 3 X[2] + X[35511], X[115] + 2 X[23992], 5 X[115] - 4 X[31644], X[115] - 4 X[44398], 2 X[620] + X[44373], X[892] - 4 X[6722], 3 X[2482] - 2 X[14588], 2 X[5461] + X[18823], 6 X[5461] - 5 X[40429], 9 X[9167] - 8 X[36953], 2 X[9182] - 5 X[31274], X[9182] - 4 X[40486], 5 X[14061] + X[39356], 3 X[18823] + 5 X[40429], 4 X[22247] - X[44372], 5 X[23991] - 2 X[31644], 5 X[23992] + 2 X[31644], X[23992] + 2 X[44398], 5 X[31274] - 8 X[40486], X[31372] - 3 X[31998], X[31372] + 3 X[35511], X[31644] - 5 X[44398]

X(45212) lies on these lines: {2, 31372}, {44, 10026}, {115, 523}, {230, 37911}, {385, 37804}, {513, 41180}, {520, 14113}, {524, 1692}, {543, 33799}, {620, 4590}, {892, 6722}, {1506, 18122}, {2482, 14588}, {2514, 20975}, {2679, 3221}, {5461, 14728}, {6537, 24348}, {7746, 36207}, {7749, 40879}, {8057, 41181}, {8287, 35080}, {9182, 31274}, {14061, 39356}, {15525, 35088}, {15526, 35078}, {16316, 40350}, {22247, 44372}

X(45212) = complement of X(31998)
X(45212) = midpoint of X(i) and X(j) for these {i,j}: {4590, 44373}, {23991, 23992}, {31998, 35511}
X(45212) = reflection of X(i) in X(j) for these {i,j}: {115, 23991}, {4590, 620}, {23991, 44398}
X(45212) = complement of the isotomic conjugate of X(9293)
X(45212) = X(i)-complementary conjugate of X(j) for these (i,j): {798, 31998}, {9217, 4369}, {9293, 2887}, {9395, 512}, {9396, 141}, {19610, 8287}, {35511, 42327}
X(45212) = crosspoint of X(i) and X(j) for these (i,j): {2, 9293}, {523, 4590}
X(45212) = crosssum of X(i) and X(j) for these (i,j): {6, 9218}, {110, 3124}
X(45212) = crossdifference of every pair of points on line {5467, 39138}
X(45212) = barycentric product X(523)*X(10190)
X(45212) = barycentric quotient X(10190)/X(99)
X(45212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 35511, 31998}, {9182, 40486, 31274}, {23992, 44398, 115}

X(45213) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(190))-ELECTRA(B) AND CEVIAN OF X(2)

Barycentrics    (b - c)^2*(3*a^2 - 3*a*b + b^2 - 3*a*c + b*c + c^2) : :
X(45213) = 3 X[2] + X[6630], 7 X[1086] - 4 X[6549], 5 X[1086] - 4 X[31647], X[1086] + 2 X[35092], 3 X[4370] - 2 X[32094], X[4555] - 4 X[40480], 7 X[6547] - 2 X[6549], 5 X[6547] - 2 X[31647], 5 X[6549] - 7 X[31647], 2 X[6549] + 7 X[35092], X[6633] - 4 X[40488], 5 X[27191] + X[39349], 2 X[31647] + 5 X[35092]

X(45213) lies on these lines: {2, 6630}, {11, 21051}, {141, 35957}, {239, 26629}, {514, 1086}, {519, 1279}, {545, 32028}, {594, 36230}, {1015, 14838}, {1016, 4422}, {1146, 20317}, {3756, 9260}, {3912, 25125}, {4083, 38989}, {4361, 30225}, {4370, 32094}, {4530, 27918}, {4534, 21120}, {4555, 40480}, {6166, 27929}, {6542, 37680}, {6633, 40488}, {17245, 36226}, {17366, 24281}, {17724, 36236}, {27191, 39349}, {35094, 40621}

X(45213) = midpoint of X(i) and X(j) for these {i,j}: {6547, 35092}, {6630, 6631}
X(45213) = reflection of X(i) in X(j) for these {i,j}: {1016, 4422}, {1086, 6547}
X(45213) = complement of X(6631)
X(45213) = complement of the isogonal conjugate of X(9262)
X(45213) = complement of the isotomic conjugate of X(42555)
X(45213) = X(i)-complementary conjugate of X(j) for these (i,j): {667, 6631}, {6164, 141}, {6630, 21260}, {9262, 10}, {9282, 3835}, {42555, 2887}
X(45213) = X(i)-Ceva conjugate of X(j) for these (i,j): {4473, 10196}, {10196, 24129}
X(45213) = X(24129)-cross conjugate of X(10196)
X(45213) = crosspoint of X(i) and X(j) for these (i,j): {2, 42555}, {514, 1016}, {4473, 10196}
X(45213) = crosssum of X(i) and X(j) for these (i,j): {6, 41405}, {101, 1015}
X(45213) = barycentric product X(i)*X(j) for these {i,j}: {190, 24129}, {514, 10196}, {1086, 4473}
X(45213) = barycentric quotient X(i)/X(j) for these {i,j}: {4473, 1016}, {10196, 190}, {24129, 514}
X(45213) = {X(2),X(6630)}-harmonic conjugate of X(6631)

X(45214) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(668))-ELECTRA(B) AND CEVIAN OF X(2)

Barycentrics    a^2*(b - c)^2*(a^2*b^2 + a^2*b*c - 3*a*b^2*c + a^2*c^2 - 3*a*b*c^2 + 3*b^2*c^2) : :
X(45214) = 3 X[2] + X[9295], X[889] - 4 X[40479], 5 X[1015] - 4 X[31645], X[1015] + 2 X[39011], 2 X[31645] + 5 X[39011]

X(45214) lies on these lines: {2, 9295}, {39, 24338}, {513, 1015}, {536, 21830}, {889, 40479}, {1086, 42327}, {1575, 2325}, {3709, 6377}, {9294, 35119}, {9402, 38978}, {17053, 24289}, {27076, 31625}

X(45214) = midpoint of X(9295) and X(9296)
X(45214) = reflection of X(31625) in X(27076)
X(45214) = complement of X(9296)
X(45214) = complement of the isogonal conjugate of X(9299)
X(45214) = complement of the isotomic conjugate of X(9267)
X(45214) = X(i)-complementary conjugate of X(j) for these (i,j): {1919, 9296}, {9265, 3835}, {9267, 2887}, {9295, 21262}, {9299, 10}, {9361, 21260}
X(45214) = crosspoint of X(i) and X(j) for these (i,j): {2, 9267}, {513, 31625}
X(45214) = crosssum of X(i) and X(j) for these (i,j): {6, 9266}, {100, 1977}
X(45214) = {X(2),X(9295)}-harmonic conjugate of X(9296)

X(45215) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(850))-ELECTRA(B) AND CEVIAN OF X(2)

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

X(45215) lies on these lines: {2, 11794}, {110, 647}, {125, 5661}, {570, 44529}, {1196, 44534}, {1624, 1625}, {2021, 3229}, {3448, 38352}, {5972, 11672}, {6331, 30476}, {14961, 15131}, {14966, 23181}, {43188, 44560}

X(45215) = complement of X(46247)
X(45215) = X(41174)-Ceva conjugate of X(511)
X(45215) = crosspoint of X(110) and X(6331)
X(45215) = crosssum of X(i) and X(j) for these (i,j): {6, 9512}, {523, 3049}
X(45215) = trilinear pole of line {22416, 23635}
X(45215) = crossdifference of every pair of points on line {868, 17423}
X(45215) = barycentric product X(i)*X(j) for these {i,j}: {99, 23635}, {100, 18175}, {110, 21243}, {163, 17865}, {648, 22416}, {670, 40951}, {4576, 10551}, {23181, 23295}
X(45215) = barycentric quotient X(i)/X(j) for these {i,j}: {17865, 20948}, {18175, 693}, {21243, 850}, {22416, 525}, {23635, 523}, {40951, 512}
X(45215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {125, 23584, 5661}, {647, 35319, 110}, {23181, 35325, 14966}

leftri

Perspectors of Electra(B) triangles and cevian triangles: X(45216)-X(45236)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 18, 2021.

Let A"B"C" be the (X(1),U)-Electra triangle, defined (in terms of trilinear coordinates) in the preamble just before X(45196). In barycentric coordinates, the locus of X = x : y : z such that the cevian triangle of X is perspective to the (X(1),U)-Electra triangle is the cubic given by the following equation:

a^3 c v (c u + a w) (a c^2 v^2 + a b^2 w^2 + b c^2 u v - b^2 c u w) y^2 z
- a^3 b w (b u + a v) (a c^2 v^2 + a b^2 w^2 + b^2 c u w - b c^2 v w) y z^2
+ (cyclic) - a b c (c v - b w)(a w - c u)(b u - a v)(b c u + c a v + a b w) x y z = 0.

The appearance of (i,j,k)) in the following list means that the (X(1),X(i))-Electra triangle is perspective to the cevian triangle of X(j), and the perspector is X(k):

(2,1,2309), (2,56,39780), (2,86,1), (2,171,42), (2,171,42), (2,38832,31)
(3,1,1858), (3,21,1), (3,1940,65)
(6,1,3666), (6,7,41003), (6,63,960), (6,81,1), (6,314,75), (6,894,k37), (6,4296,1214)
(9,1,3752), (9,2,1), (9,6180,6)
(31,1,17446), (31,82,1), (31,19645,38)
(37,1,17011), (37,1255,1), (37,1963,1100), (37,32911,1203)
(42,2668,3720), (42,17277,16819), (42,40433,1)
(48,2167,1), (48,9252,1953)
(55,1,5572), (55,2346,1), (55,9446,354)
(57,1,14100), (57,7,3), (57,1376,55)
(65,409,2646), (65,17097,1)
(75,1582,1964), (75,3112,1)
(100,765,1), (100,1054,244)
(101,4564,1), (101,9317,2170)
(512,3903,1), (512,9425,4367)
(523,2612,2605), (523,6742,1)
(649,1,190), (649,190,1), (649,7035,7035), (649,9362,649j)
(672,1,673), (672,673,1), (672,6654,105), (672,14942,2), (672,33674,239)

X(45216) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(2))-ELECTRA AND CEVIAN OF X(1740)

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

X(45216) lies on these lines: {1, 2}, {6, 23493}, {213, 9490}, {330, 1740}, {904, 3684}, {1107, 2309}, {1575, 7148}, {1964, 17448}, {2176, 2209}, {2238, 4161}, {2269, 16514}, {2308, 23525}, {3010, 20667}, {3727, 4093}, {3747, 20967}, {3780, 20464}, {4116, 23652}, {7032, 23457}, {20963, 23532}, {20969, 23548}, {21384, 22343}, {25277, 36857}

X(45217) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(2))-ELECTRA AND CEVIAN OF X(2162)

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

X(45217) lies on these lines: {31, 7104}, {42, 21759}, {81, 330}, {1185, 2053}, {1206, 34252}, {1977, 23632}

X(45218) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(6))-ELECTRA AND CEVIAN OF X(330)

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

X(45218) lies on these lines: {1, 893}, {37, 714}, {87, 846}, {330, 37870}, {1333, 2162}, {1402, 3747}, {2053, 44115}, {2281, 21759}, {3121, 22230}, {3666, 27455}, {3741, 6377}, {4687, 27439}, {16969, 23853}, {17053, 29634}

X(45219) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(9))-ELECTRA AND CEVIAN OF X(23511)

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

X(45219) lies on these lines: {1, 6}, {31, 20323}, {36, 15854}, {56, 23205}, {57, 3445}, {65, 1149}, {106, 37582}, {145, 4487}, {165, 8572}, {386, 31792}, {517, 24046}, {595, 24928}, {603, 1319}, {614, 2098}, {899, 3893}, {902, 37605}, {978, 3880}, {995, 4646}, {1086, 4301}, {1155, 32577}, {1193, 5919}, {1201, 3057}, {1385, 40091}, {1420, 3052}, {1722, 10912}, {1738, 13463}, {2099, 28011}, {2275, 21872}, {3214, 17460}, {3621, 30861}, {3656, 24159}, {3666, 3890}, {3680, 23511}, {3698, 28352}, {3756, 4848}, {3869, 21342}, {3884, 37592}, {3898, 3931}, {3924, 5048}, {3976, 44663}, {4018, 4694}, {4255, 31393}, {4358, 20041}, {4383, 36846}, {4731, 28257}, {4853, 37679}, {4883, 31503}, {5121, 8256}, {5573, 11531}, {5836, 16602}, {7982, 17054}, {7987, 21000}, {8158, 15287}, {10107, 17063}, {10222, 30117}, {10475, 18613}, {11011, 28082}, {11682, 17597}, {12577, 17365}, {14923, 16610}, {15950, 28027}, {16948, 18211}, {17480, 28582}, {17614, 37610}, {19861, 37542}, {20036, 20923}, {20284, 20287}, {23072, 37817}, {28018, 40663}, {28074, 41687}

X(45220) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(31))-ELECTRA AND CEVIAN OF X(1760)

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

X(45220) lies on these lines: {1, 21}, {69, 18730}, {75, 17442}, {304, 18671}, {1760, 1973}, {1930, 17865}, {1953, 18049}, {1958, 18596}, {2329, 28731}, {3061, 27509}, {16735, 17446}, {17868, 18715}, {20235, 29960}, {25083, 37613}, {26158, 33864}

X(45221) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(37))-ELECTRA AND CEVIAN OF X(86)

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

X(45221) lies on these lines: {10, 86}, {1100, 4359}, {1203, 17322}, {1790, 42028}, {3873, 17394}, {5564, 42437}, {18133, 19717}

X(45222) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(37))-ELECTRA AND CEVIAN OF X(43993)

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

X(45222) lies on these lines: {1, 2}, {6, 17147}, {75, 19717}, {81, 17495}, {99, 1171}, {190, 41823}, {238, 27804}, {321, 4852}, {553, 4982}, {894, 19743}, {1086, 42045}, {1100, 4359}, {1126, 43993}, {1213, 41820}, {1230, 20970}, {1386, 3896}, {1962, 4974}, {2308, 4427}, {2895, 17302}, {3175, 41241}, {3210, 37685}, {3218, 18163}, {3219, 17121}, {3578, 4969}, {3589, 3969}, {3666, 16704}, {3751, 20068}, {3759, 19742}, {3821, 20290}, {3875, 26223}, {3946, 17184}, {3952, 32928}, {3995, 4360}, {4001, 4856}, {4085, 28599}, {4272, 26971}, {4357, 43990}, {4361, 19684}, {4363, 19738}, {4383, 31035}, {4395, 37631}, {4464, 31011}, {4649, 17140}, {4716, 17163}, {4734, 17126}, {4753, 42039}, {4850, 37639}, {4886, 27081}, {5278, 20182}, {5564, 6539}, {7839, 25257}, {10601, 25243}, {14953, 18732}, {14996, 17490}, {14997, 41839}, {15668, 30562}, {16475, 32929}, {16477, 32936}, {16666, 42051}, {17045, 41809}, {17117, 19741}, {17118, 19739}, {17119, 19722}, {17145, 17598}, {17165, 32921}, {17301, 32859}, {17319, 27065}, {17366, 18139}, {17377, 33172}, {17379, 20174}, {17380, 32782}, {17475, 21753}, {17483, 42754}, {17491, 33145}, {17600, 32864}, {17772, 32781}, {17778, 33150}, {18046, 40603}, {18089, 18092}, {19785, 31034}, {19786, 31037}, {19821, 27793}, {20086, 26840}, {20090, 26842}, {20475, 41328}, {24589, 37595}, {26806, 41819}, {28604, 41821}, {31017, 32774}, {34064, 37680}, {41011, 44006}

X(45223) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(42))-ELECTRA AND CEVIAN OF X(1)

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

X(45223) lies on these lines: {1, 6}, {11, 41877}, {42, 17277}, {43, 17259}, {55, 23375}, {86, 310}, {87, 10013}, {171, 8053}, {239, 2667}, {308, 18091}, {673, 4343}, {740, 20174}, {869, 4687}, {872, 17260}, {894, 24425}, {940, 20992}, {985, 5301}, {1045, 3739}, {1125, 3736}, {1213, 3783}, {1500, 17049}, {1621, 1918}, {1740, 15668}, {1964, 16826}, {2274, 3616}, {2276, 4446}, {2664, 4698}, {2669, 9403}, {2999, 20156}, {3248, 39916}, {3286, 37607}, {3622, 20146}, {3741, 27164}, {3995, 17142}, {4022, 40773}, {4038, 18166}, {4093, 17011}, {4261, 17065}, {4360, 21352}, {4364, 25421}, {4423, 27623}, {5132, 23383}, {5145, 12263}, {5156, 5248}, {5256, 20154}, {5284, 27644}, {5287, 20131}, {5311, 20159}, {7032, 17394}, {7184, 17392}, {7191, 17000}, {10436, 24696}, {15320, 33095}, {16738, 29824}, {17012, 20138}, {17018, 17349}, {17019, 20132}, {17021, 20137}, {17022, 20135}, {17032, 20140}, {17234, 30969}, {17243, 40790}, {17264, 21803}, {17316, 20139}, {17319, 17445}, {17379, 29814}, {17446, 21808}, {17470, 18714}, {20134, 29570}, {20141, 29580}, {20147, 29817}, {20151, 29597}, {21238, 27020}, {24210, 34830}, {24688, 25538}, {24732, 30963}, {25368, 29820}, {27252, 29830}, {27270, 33173}, {29437, 32918}, {29767, 32919}, {30116, 32941}

X(45224) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(48))-ELECTRA AND CEVIAN OF X(1)

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

X(45224) lies on these lines: {1, 21}, {48, 1748}, {92, 1953}, {2167, 17438}, {6507, 44179}, {14213, 17858}, {17451, 27287}, {17859, 20879}, {17923, 37755}, {18041, 18750}, {18161, 26871}

X(45225) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(48))-ELECTRA AND CEVIAN OF X(1748)

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

X(45225) lies on these lines: {1, 19}, {33, 11491}, {92, 44706}, {158, 774}, {255, 1748}, {564, 1725}, {920, 1957}, {1735, 1838}, {1871, 17102}, {5174, 24028}, {14571, 44547}

X(45226) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(55))-ELECTRA AND CEVIAN OF X2)

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

X(45226) lies on these lines: {2, 220}, {142, 354}, {1040, 10582}, {1088, 40593}, {2140, 8226}, {3838, 43959}, {4000, 10580}, {4904, 13405}, {5249, 39063}, {5437, 30813}, {6173, 30623}, {6608, 17069}, {6706, 25006}, {7056, 17113}, {11019, 24181}, {17044, 29817}, {24600, 24789}, {25964, 40719}, {41918, 42048}

X(45227) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(55))-ELECTRA AND CEVIAN OF X(1445)

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

X(45227) lies on these lines: {1, 3}, {2, 15853}, {6, 4350}, {7, 1212}, {85, 4875}, {170, 14100}, {220, 1445}, {279, 1418}, {910, 38859}, {948, 24789}, {1104, 1462}, {1170, 38459}, {1427, 5222}, {1434, 16699}, {1475, 34855}, {3160, 17092}, {3693, 6604}, {4515, 32003}, {6180, 16572}, {6762, 21446}, {10481, 24181}, {21454, 24635}, {25082, 32098}, {25242, 39126}, {26964, 37780}, {27000, 33765}

X(45228) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(57))-ELECTRA AND CEVIAN OF X(2951)

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

X(45228) lies on these lines: {1, 3}, {170, 43064}, {1212, 10939}, {1419, 15856}, {5918, 43044}, {11260, 28071}, {14100, 20978}, {15853, 40967}

X(45229) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(57))-ELECTRA AND CEVIAN OF X(3680)

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

X(45229) lies on these lines: {1, 9309}, {55, 2347}, {390, 3680}, {479, 19604}, {1458, 3445}, {1742, 8056}

X(45230) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(65))-ELECTRA AND CEVIAN OF X(1)

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

X(45230) lies on these lines: {1, 4}, {3, 7098}, {11, 11281}, {21, 60}, {35, 31806}, {36, 22347}, {46, 6876}, {55, 3869}, {56, 18444}, {65, 411}, {78, 5218}, {390, 26837}, {920, 3612}, {942, 33858}, {997, 6857}, {1001, 10394}, {1046, 22361}, {1737, 6853}, {1788, 6988}, {1837, 2476}, {1854, 19765}, {1898, 6912}, {1936, 2650}, {2098, 3957}, {2651, 9399}, {2886, 5086}, {3085, 37700}, {3086, 37615}, {3100, 5327}, {3218, 37564}, {3474, 12520}, {3601, 12514}, {3868, 26357}, {3873, 10966}, {4292, 16132}, {4294, 37533}, {4295, 6869}, {4305, 6868}, {4313, 11415}, {4316, 18244}, {4861, 37740}, {5057, 10543}, {5176, 10955}, {5572, 20323}, {5698, 7675}, {5720, 10588}, {5727, 30147}, {5730, 37284}, {5887, 12711}, {6326, 13411}, {6825, 18391}, {6828, 11375}, {6841, 37737}, {6842, 37730}, {6873, 37692}, {6874, 10826}, {6905, 13750}, {6932, 37724}, {6942, 17700}, {7082, 16865}, {7288, 18443}, {8543, 14100}, {9581, 30143}, {10448, 24430}, {10953, 31053}, {11012, 18389}, {11114, 34647}, {12739, 17638}, {13384, 30144}, {14793, 26877}, {17015, 33177}, {17097, 36002}, {37734, 38460}

X(45231) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(65))-ELECTRA AND CEVIAN OF X(411)

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

X(45231) lies on these lines: {1, 3}, {21, 40950}, {33, 20846}, {73, 3561}, {225, 411}, {283, 6514}, {1068, 6876}, {2193, 2323}, {2269, 4288}, {6734, 34851}, {9817, 11344}

X(45232) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(75))-ELECTRA AND CEVIAN OF X(1)

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

X(45232) lies on these lines: {1, 1965}, {31, 48}, {42, 17475}, {75, 33782}, {1580, 9236}, {1926, 3112}, {1966, 23489}, {2234, 33764}, {3747, 23371}, {4117, 17469}, {7032, 23538}, {7191, 38986}, {16685, 23396}, {17872, 23491}, {20372, 23429}

X(45233) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(100))-ELECTRA AND CEVIAN OF X(1)

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

X(45233) lies on these lines: {1, 6163}, {31, 39026}, {38, 24482}, {44, 3930}, {100, 1052}, {238, 4511}, {244, 513}, {748, 36278}, {756, 16482}, {765, 3722}, {1086, 39386}, {1757, 40091}, {2170, 21834}, {2310, 34949}, {2605, 3248}, {3123, 6615}, {3271, 4475}, {3315, 39343}, {7208, 28855}, {9458, 30721}, {21330, 36294}, {28082, 36280}

X(45234) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(101))-ELECTRA AND CEVIAN OF X(1)

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

X(45234) lies on these lines: {1, 9323}, {514, 1111}, {1429, 1959}, {3911, 26006}, {4564, 17439}, {4879, 17463}, {6547, 42770}, {9318, 9322}, {21921, 25342}, {24318, 39244}, {38375, 40629}

X(45235) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(523))-ELECTRA AND CEVIAN OF X(1)

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

X(45235) lies on these lines: {1, 5}, {109, 39633}, {110, 1101}, {523, 2617}, {2613, 7137}, {3737, 14985}

X(45236) = PERSPECTOR OF THESE TRIANGLES: (X(1),X(523))-ELECTRA AND CEVIAN OF X(2617)

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

X(45236) lies on these lines: {1, 523}, {12, 2617}, {49, 5396}, {110, 2594}, {229, 1464}, {500, 501}, {1634, 29958}, {18165, 44709}, {34586, 37816}

X(45237) = X(4)X(94)∩X(6)X(110)

Barycentrics    a^2*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10 + a^8*c^2 - 2*a^6*b^2*c^2 + 3*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 4*b^8*c^2 - 2*a^6*c^4 + 3*a^4*b^2*c^4 + 6*a^2*b^4*c^4 - 3*b^6*c^4 - 6*a^2*b^2*c^6 - 3*b^4*c^6 + 2*a^2*c^8 + 4*b^2*c^8 - c^10) : :
X(45237) = X[20] - 4 X[16270], X[52] + 2 X[36253], X[110] - 3 X[5640], X[110] - 4 X[11746], X[125] + 2 X[11800], 2 X[265] + X[1986], X[265] + 2 X[12236], 3 X[373] - 2 X[5972], X[895] + 2 X[32246], 2 X[974] + X[10733], 2 X[1112] + X[3448], 2 X[1112] - 3 X[11002], X[1986] - 4 X[12236], 5 X[3091] - 8 X[15465], X[3448] + 3 X[11002], 5 X[3567] - 2 X[25711], 2 X[5446] + X[16003], 4 X[5462] - X[30714], X[5609] - 4 X[10095], 3 X[5640] - 4 X[11746], 3 X[5640] - 2 X[41670], 3 X[5650] - 4 X[6723], X[5889] + 2 X[15738], X[6101] - 4 X[20396], X[6243] + 5 X[15027], 4 X[7687] - X[12825], 2 X[7687] + X[21649], X[7723] - 4 X[11801], 3 X[7998] - 5 X[15059], 3 X[7998] - 2 X[41673], 7 X[9781] - X[14094], 4 X[9822] - X[32114], 4 X[9826] - X[12383], 4 X[9827] - X[25714], 4 X[10110] - X[15063], 4 X[10113] - X[12292], X[10113] + 2 X[13358], 2 X[10170] - 3 X[23515], X[10263] + 2 X[20379], X[10625] - 4 X[20397], X[10990] + 2 X[13598], 5 X[11444] - 11 X[15025], X[11459] - 3 X[14644], 2 X[11806] + X[12295], 4 X[11806] - X[17854], 4 X[12052] - X[14611], 2 X[12079] + X[16978], X[12111] - 7 X[15044], X[12121] - 3 X[40280], X[12284] + 3 X[16261], X[12292] + 8 X[13358], 2 X[12295] + X[17854], 2 X[12358] - 5 X[15081], X[12825] + 2 X[21649], X[12902] + 2 X[14708], X[13340] - 3 X[15061], 4 X[13416] - 3 X[33884], X[14448] - 4 X[16625], 7 X[15020] - 13 X[15028], 11 X[15024] - 5 X[15034], X[15054] + 2 X[16105], 5 X[15059] - 2 X[41673], 2 X[15644] - 5 X[38729], X[24981] - 4 X[41671], 2 X[25328] + X[40949]

See Antreas Hatzipolakis and Peter Moses, euclid 2783.

X(45237) lies on these lines: {2, 12099}, {4, 94}, {6, 110}, {20, 16270}, {22, 5622}, {25, 39562}, {51, 542}, {52, 36253}, {74, 37489}, {125, 511}, {323, 18449}, {373, 5972}, {526, 9979}, {567, 1511}, {974, 10733}, {1154, 44555}, {1177, 26284}, {1209, 10170}, {1351, 15106}, {1640, 9517}, {1853, 2781}, {2393, 7426}, {2931, 15035}, {3047, 34397}, {3091, 15465}, {3260, 36901}, {3567, 25711}, {3581, 7464}, {5012, 16165}, {5050, 12310}, {5093, 19504}, {5097, 32235}, {5102, 17847}, {5133, 12827}, {5422, 15462}, {5446, 16003}, {5462, 30714}, {5609, 10095}, {5621, 33586}, {5642, 5943}, {5650, 6723}, {5889, 15738}, {6090, 12596}, {6101, 20396}, {6243, 15027}, {6642, 12228}, {6800, 13198}, {7493, 18919}, {7495, 15118}, {7687, 12825}, {7723, 10297}, {7998, 15059}, {8549, 10117}, {8705, 32269}, {9019, 15360}, {9545, 15026}, {9730, 12022}, {9777, 45016}, {9781, 14094}, {9822, 32114}, {9826, 12383}, {9827, 25714}, {9976, 34417}, {10110, 15063}, {10263, 20379}, {10625, 20397}, {10721, 40909}, {10990, 13598}, {11402, 45082}, {11413, 15055}, {11444, 15025}, {11459, 14644}, {11597, 13365}, {11649, 32225}, {11806, 12295}, {12052, 14611}, {12079, 16978}, {12085, 15041}, {12111, 15044}, {12121, 40280}, {12201, 41330}, {12284, 16261}, {12358, 15081}, {12893, 39242}, {12902, 14708}, {13340, 15061}, {13416, 16051}, {13595, 20772}, {14448, 16625}, {15004, 34155}, {15020, 15028}, {15024, 15034}, {15054, 16105}, {15067, 20304}, {15085, 32620}, {15132, 36749}, {15133, 18912}, {15329, 20975}, {15644, 38729}, {16163, 16836}, {16222, 32423}, {16981, 31099}, {18114, 34834}, {24981, 41671}, {25328, 40949}, {25556, 44107}, {26283, 32251}, {27085, 41613}, {27866, 43815}, {32227, 40132}, {32274, 41724}, {34603, 36201}, {34990, 36829}, {37784, 37980}, {38260, 38534}, {40291, 44109}, {43605, 43823}

X(45237) = midpoint of X(i) and X(j) for these {i,j}: {265, 568}, {895, 11188}, {3060, 9140}, {10733, 15072}, {15030, 21649}
X(45237) = reflection of X(i) in X(j) for these {i,j}: {2, 12099}, {110, 41670}, {568, 12236}, {1511, 13363}, {1986, 568}, {5642, 5943}, {11188, 32246}, {12824, 51}, {12825, 15030}, {15030, 7687}, {15067, 20304}, {15072, 974}, {16163, 16836}, {41670, 11746}
X(45237) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 5640, 41670}, {265, 12236, 1986}, {2931, 37506, 15035}, {5640, 11188, 1995}, {7687, 21649, 12825}, {11002, 37644, 568}, {11746, 41670, 5640}, {11806, 12295, 17854}

leftri

Vu-Miquel points: X(45238)-X(45255)

rightri

This preamble and centers X(45238)-X(45255) were contributed by César Eliud Lozada, October 20, 2021.

Let ABC be a triangle and A', B', C' three any points on the sidelines BC, CA, AB, respectively. Miquel theorem states that three circles {{AB'C'}}, {{BC'A'}}, {{CA'B'}} have a common point.

Consider now two points P1 and P2. Then the three conics ({AB'C'P1P2}}, {{BC'A'P1P2}} and {{CA'B'P1P2}} have a common point other than P1 and P2. (Vu Thanh Tung, Euclid 2699, October 12, 2021).

The common intersection of that three conics is named here the Vu-Miquel point of (P1, P2) with respect to {A', B', C'}.

This section deals with Vu-Miquel points of (P1, P2) with respect to {A', B', C'} when A', B', C' are the cevian traces of a point P. Assume A'B'C' is the cevian triangle of a point P = U : V : W (trilinear coordinates used here), P1 = u1 : v1 : w1 and P2 = u2 : v2 : w2. Then the common point of the three conics is:

   Q(P, P1, P2) = ((U*du + V*dv + W*dw)*U*V*W*su + (V^2*W^2*u1*u2 - (V^2*w1*w2 + W^2*v1*v2)*U^2)*du)*u1*u2/(V*dv + W*dw) : :

where du, dv, dw are the cyclic differences:

   du = v1*w2 - v2*w1,  dv = w1*u2 - w2*u1,   dw = u1*v2 - u2*v1

and su, sv, sw are the cyclic sums:

   su = v1*w2 + v2*w1,  sv = w1*u2 + w2*u1,   sw = u1*v2 + u2*v1

In the following list, (i, j, k, n) means that the Vu-Miquel point of (X(i), X(j)) with respect to the cevian traces of X(k) is X(n):

(1, 2, 1, 192), (1, 3, 1, 3157), (1, 4, 1, 1148), (1, 5, 1, 45238), (1, 6, 1, 55), (1, 7, 1, 31526), (1, 8, 1, 19582), (1, 9, 1, 3158), (1, 10, 1, 3159), (1, 11, 1, 523), (1, 15, 1, 202), (1, 16, 1, 203), (1, 19, 1, 204), (1, 20, 1, 45239), (2, 6, 1, 45240), (2, 7, 1, 45241), (2, 8, 1, 45242), (3, 6, 1, 45243), (6, 7, 1, 45244), (6, 9, 1, 6), (11, 12, 1, 2618), (15, 16, 1, 214), (2, 3, 2, 6), (2, 4, 2, 1249), (2, 5, 2, 216), (2, 6, 2, 3), (2, 7, 2, 3160), (2, 9, 2, 1), (2, 10, 2, 37), (2, 11, 2, 650), (2, 13, 2, 40578), (2, 14, 2, 41889), (2, 15, 2, 40580), (2, 16, 2, 40581), (2, 19, 2, 36103), (2, 20, 2, 45245), (3, 4, 2, 3), (3, 5, 2, 5), (3, 6, 2, 2), (3, 7, 2, 45246), (3, 8, 2, 45247), (3, 9, 2, 57), (3, 10, 2, 65), (3, 15, 2, 30471), (3, 16, 2, 30472), (3, 20, 2, 45248), (4, 5, 2, 14363), (4, 6, 2, 3343), (4, 9, 2, 223), (4, 20, 2, 4), (5, 6, 2, 343), (5, 20, 2, 45249), (6, 7, 2, 45250), (6, 9, 2, 9), (6, 20, 2, 45251), (7, 8, 2, 45252), (7, 9, 2, 9), (8, 9, 2, 24151), (9, 10, 2, 226), (9, 19, 2, 478), (10, 12, 2, 10), (10, 19, 2, 36908), (13, 14, 2, 9410), (3, 4, 3, 1075), (3, 5, 3, 15912), (3, 6, 3, 154), (3, 7, 3, 45253), (3, 8, 3, 45254), (3, 15, 3, 3165), (3, 16, 3, 3166), (3, 20, 3, 45255), (15, 16, 3, 11587), (4, 5, 4, 52), (4, 6, 4, 25), (4, 7, 4, 2898), (4, 8, 4, 2899), (4, 9, 4, 2900), (4, 10, 4, 2901), (4, 11, 4, 513), (4, 12, 4, 15443), (4, 13, 4, 15441), (4, 14, 4, 15442), (4, 15, 4, 2902), (4, 16, 4, 2903), (4, 17, 4, 15444), (4, 18, 4, 15445), (4, 19, 4, 33), (4, 20, 4, 32605), (6, 19, 4, 19), (9, 19, 4, 2911), (10, 19, 4, 209), (5, 11, 5, 8819), (6, 7, 6, 31604), (6, 9, 6, 3169), (6, 10, 6, 22024), (6, 15, 6, 3170), (6, 16, 6, 3171), (15, 16, 6, 14972), (7, 8, 7, 8055), (7, 9, 7, 3174), (7, 10, 7, 3175), (7, 11, 7, 514), (8, 9, 8, 200), (8, 10, 8, 72), (8, 11, 8, 522), (9, 10, 9, 22027), (9, 11, 9, 6362), (10, 11, 10, 17420), (10, 12, 10, 201), (13, 15, 13, 62), (13, 16, 13, 5612), (14, 15, 14, 5616), (14, 16, 14, 61)

X(45238) = VU-MIQUEL POINT OF (X(1), X(5)) WRT CEVIAN TRACES OF X(1)

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*((b+c)*a^5-(b+c)*(2*b^2-b*c+2*c^2)*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(45238) lies on these lines: {1, 2595}, {5, 1087}, {52, 517}, {54, 655}, {216, 7561}, {324, 41013}, {523, 5399}, {2596, 3460}, {17479, 22457}

X(45238) = barycentric product X(i)*X(j) for these {i, j}: {5, 17479}, {311, 21768}, {324, 22457}
X(45238) = trilinear product X(1953)*X(17479)
X(45238) = X(1)-Ceva conjugate of-X(5)
X(45238) = {X(1087), X(2599)}-harmonic conjugate of X(5)

X(45239) = VU-MIQUEL POINT OF (X(1), X(20)) WRT CEVIAN TRACES OF X(1)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*((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(45239) lies on these lines: {1, 6359}, {20, 1097}, {64, 658}, {354, 1858}, {1398, 37258}, {4300, 31526}, {20277, 37523}, {40133, 40941}

X(45239) = barycentric product X(i)*X(j) for these {i, j}: {20, 18663}, {1044, 18750}
X(45239) = barycentric quotient X(i)/X(j) for these (i, j): (610, 7038), (1044, 2184)
X(45239) = trilinear product X(i)*X(j) for these {i, j}: {20, 1044}, {610, 18663}
X(45239) = trilinear quotient X(i)/X(j) for these (i, j): (20, 7038), (1044, 64)
X(45239) = crosspoint of X(1) and X(1044)
X(45239) = crosssum of X(1) and X(7038)
X(45239) = X(1)-Ceva conjugate of-X(20)
X(45239) = X(64)-isoconjugate-of-X(7038)
X(45239) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (610, 7038), (1044, 2184)
X(45239) = {X(1097), X(40933)}-harmonic conjugate of X(20)

X(45240) = VU-MIQUEL POINT OF (X(2), X(6)) WRT CEVIAN TRACES OF X(1)

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

X(45240) lies on the cubic K1017 and these lines: {2, 257}, {3, 41882}, {6, 904}, {192, 39917}, {194, 40849}, {694, 2275}, {984, 40792}, {1193, 40729}, {1934, 18055}, {3208, 3903}, {6376, 27805}, {7260, 27891}, {9310, 29055}, {17103, 37134}

X(45240) = barycentric product X(i)*X(j) for these {i, j}: {256, 17792}, {893, 17760}, {1581, 8844}
X(45240) = barycentric quotient X(256)/X(18299)
X(45240) = trilinear product X(i)*X(j) for these {i, j}: {257, 18758}, {694, 8844}, {893, 17792}, {904, 17760}
X(45240) = trilinear quotient X(257)/X(18299)
X(45240) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(39919)}} and {{A, B, C, X(2), X(2053)}}
X(45240) = X(172)-isoconjugate-of-X(18299)
X(45240) = X(256)-reciprocal conjugate of-X(18299)

X(45241) = VU-MIQUEL POINT OF (X(2), X(7)) WRT CEVIAN TRACES OF X(1)

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

X(45241) lies on these lines: {7, 27498}, {3208, 30610}, {5222, 45252}

X(45241) = X(1)-Ceva conjugate of-X(9311)

X(45242) = VU-MIQUEL POINT OF (X(2), X(8)) WRT CEVIAN TRACES OF X(1)

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

X(45242) lies on these lines: {2, 17786}, {8, 3056}, {312, 17452}, {322, 35544}, {646, 3169}, {668, 1423}, {3596, 21246}, {3728, 44720}, {3759, 4033}, {4385, 43677}, {19581, 39467}, {20348, 40875}, {24524, 28369}, {28358, 30473}

X(45242) = isotomic conjugate of the isogonal conjugate of X(32468)
X(45242) = barycentric product X(76)*X(32468)
X(45242) = trilinear product X(75)*X(32468)
X(45242) = X(1)-Ceva conjugate of-X(312)
X(45242) = {X(4110), X(27424)}-harmonic conjugate of X(17787)

X(45243) = VU-MIQUEL POINT OF (X(3), X(6)) WRT CEVIAN TRACES OF X(1)

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

X(45243) lies on these lines: {3, 7100}, {6, 7072}

X(45243) = barycentric product X(63)*X(41504)
X(45243) = trilinear product X(i)*X(j) for these {i, j}: {3, 41504}, {1770, 8606}
X(45243) = trilinear quotient X(1770)/X(7282)

X(45244) = VU-MIQUEL POINT OF (X(6), X(7)) WRT CEVIAN TRACES OF X(1)

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

X(45244) lies on this line: {7, 2310}

X(45245) = VU-MIQUEL POINT OF (X(2), X(20)) WRT CEVIAN TRACES OF X(2)

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

X(45245) lies on these lines: {2, 15851}, {6, 3091}, {7, 23681}, {20, 1249}, {193, 15595}, {216, 3523}, {233, 7486}, {253, 648}, {393, 3163}, {441, 17037}, {938, 1449}, {1033, 2071}, {1119, 24604}, {1196, 5304}, {1560, 30769}, {3146, 33630}, {3162, 7396}, {3529, 33636}, {3839, 40065}, {5056, 5702}, {5177, 40582}, {6525, 23608}, {7129, 9538}, {7398, 40179}, {7585, 8968}, {9308, 11348}, {10304, 36748}, {10565, 40938}, {14091, 14961}, {17558, 40937}, {18311, 20580}, {18594, 18624}, {18848, 31361}, {31099, 40583}, {32974, 37895}, {33198, 37891}, {37174, 39081}, {40133, 40941}

X(45245) = complement of X(35510)
X(45245) = polar conjugate of the isotomic conjugate of X(27082)
X(45245) = (medial)-isotomic conjugate-of-X(20)
X(45245) = barycentric product X(i)*X(j) for these {i, j}: {4, 27082}, {20, 3146}
X(45245) = barycentric quotient X(i)/X(j) for these (i, j): (3, 15400), (20, 35510), (154, 3532), (1249, 38253)
X(45245) = trilinear product X(i)*X(j) for these {i, j}: {19, 27082}, {20, 18594}, {610, 3146}, {1895, 38292}
X(45245) = trilinear quotient X(i)/X(j) for these (i, j): (63, 15400), (610, 3532), (1097, 40170)
X(45245) = center of the circumconic {{A, B, C, X(648), X(658)}}
X(45245) = crosssum of X(6) and X(3532)
X(45245) = X(2)-Ceva conjugate of-X(20)
X(45245) = X(31)-complementary conjugate of-X(20)
X(45245) = X(i)-Dao conjugate of-X(j) for these (i, j): (4, 38253), (6, 15400), (20, 2)
X(45245) = X(441)-Hirst inverse of-X(17037)
X(45245) = X(19)-isoconjugate-of-X(15400)
X(45245) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (3, 15400), (20, 35510), (154, 3532), (1249, 38253)
X(45245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1249, 36413, 20), (33630, 38292, 3146)

X(45246) = VU-MIQUEL POINT OF (X(3), X(7)) WRT CEVIAN TRACES OF X(2)

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

X(45246) lies on the cubic K1060 and these lines: {3, 11051}, {7, 2886}, {55, 43344}

X(45247) = VU-MIQUEL POINT OF (X(3), X(8)) WRT CEVIAN TRACES OF X(2)

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

X(45247) lies on the cubic K259 and these lines: {1, 764}, {3, 106}, {6, 40595}, {8, 11}, {55, 1318}, {56, 901}, {220, 2316}, {999, 16944}, {1149, 17109}, {1168, 24864}, {1479, 38950}, {1482, 5516}, {1616, 39026}, {2226, 41436}, {3756, 14026}, {3880, 23705}, {4674, 5697}, {7952, 36125}, {9456, 16781}, {11236, 19634}, {11238, 36590}, {19636, 30305}

X(45247) = barycentric product X(i)*X(j) for these {i, j}: {88, 3880}, {312, 17109}, {1022, 23705}, {1149, 4997}, {1266, 2316}, {1318, 16594}
X(45247) = barycentric quotient X(i)/X(j) for these (i, j): (1149, 3911), (1320, 36805), (1878, 37790)
X(45247) = trilinear product X(i)*X(j) for these {i, j}: {8, 17109}, {106, 3880}, {1149, 1320}, {1318, 17460}
X(45247) = trilinear quotient X(i)/X(j) for these (i, j): (1149, 1319), (1320, 1120), (1878, 1877)
X(45247) = perspector of the circumconic {{A, B, C, X(88), X(4582)}}
X(45247) = center of the circumconic {{A, B, C, X(901), X(1320)}}
X(45247) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(1293)}} and {{A, B, C, X(3), X(44722)}}
X(45247) = crosspoint of X(1318) and X(1320)
X(45247) = crosssum of X(i) and X(j) for these (i, j): {900, 14027}, {1317, 1319}
X(45247) = X(i)-Ceva conjugate of-X(j) for these (i, j): (901, 6085), (1320, 3880)
X(45247) = X(1149)-complementary conjugate of-X(119)
X(45247) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1120, 1319}, {1404, 36805}
X(45247) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1149, 3911), (1320, 36805)
X(45247) = {X(1), X(14260)}-harmonic conjugate of X(34230)

X(45248) = VU-MIQUEL POINT OF (X(3), X(20)) WRT CEVIAN TRACES OF X(2)

Barycentrics    a^2*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(3*a^4-6*(b^2+c^2)*a^2+3*b^4+2*b^2*c^2+3*c^4) : :
X(45248) = X(20)-3*X(27082) = X(20)+3*X(32605) = 9*X(3545)-8*X(44801) = X(14531)-3*X(16879) = 3*X(15035)-2*X(39084)

X(45248) lies on the cubic K426 and these lines: {2, 14528}, {3, 3532}, {5, 11425}, {6, 14914}, {20, 154}, {24, 6593}, {30, 44788}, {49, 37475}, {64, 110}, {113, 382}, {141, 631}, {155, 1511}, {184, 17818}, {206, 1092}, {381, 7666}, {394, 7691}, {548, 4549}, {960, 1071}, {1147, 9786}, {1151, 10962}, {1152, 10960}, {1181, 9705}, {1192, 3167}, {1209, 3526}, {1493, 6644}, {1498, 41427}, {1620, 12164}, {2070, 17713}, {3053, 11672}, {3292, 15750}, {3515, 14531}, {3528, 9707}, {3545, 44801}, {3796, 15717}, {3832, 35259}, {4306, 23072}, {4550, 12038}, {5070, 37506}, {5102, 15887}, {5562, 34472}, {5642, 37197}, {6759, 11820}, {6800, 21734}, {8542, 32621}, {8567, 11441}, {8780, 15811}, {9306, 33537}, {9706, 17809}, {9714, 37498}, {10619, 31255}, {10639, 11481}, {10640, 11480}, {10938, 11793}, {11064, 17845}, {11202, 15606}, {11597, 12316}, {12163, 40111}, {13367, 17811}, {13754, 33556}, {15035, 39084}, {15116, 32233}, {15696, 18442}, {17578, 35264}, {17810, 34148}, {17834, 22115}, {22236, 40580}, {22238, 40581}, {32534, 43572}, {37499, 40589}, {38448, 41398}

X(45248) = midpoint of X(27082) and X(32605)
X(45248) = reflection of X(i) in X(j) for these (i, j): (3532, 3), (15077, 43592)
X(45248) = complement of X(15077)
X(45248) = anticomplement of X(43592)
X(45248) = complementary conjugate of the complement of X(3515)
X(45248) = (ABC-X3 reflections)-isogonal conjugate-of-X(3529)
X(45248) = (medial)-isotomic conjugate-of-X(15905)
X(45248) = barycentric product X(i)*X(j) for these {i, j}: {20, 37672}, {154, 32831}, {394, 34286}
X(45248) = barycentric quotient X(32)/X(33585)
X(45248) = trilinear product X(i)*X(j) for these {i, j}: {255, 34286}, {610, 37672}
X(45248) = trilinear quotient X(31)/X(33585)
X(45248) = inverse of X(64) in Stammler hyperbola
X(45248) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(27082)}} and {{A, B, C, X(6), X(32605)}}
X(45248) = crosspoint of X(2) and X(32001)
X(45248) = X(2)-Ceva conjugate of-X(15905)
X(45248) = X(31)-complementary conjugate of-X(15905)
X(45248) = X(206)-Dao conjugate of-X(33585)
X(45248) = X(75)-isoconjugate-of-X(33585)
X(45248) = X(32)-reciprocal conjugate of-X(33585)
X(45248) = X(3532)-of-ABC-X3 reflections triangle
X(45248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 15077, 43592), (1092, 17821, 1350)

X(45249) = VU-MIQUEL POINT OF (X(5), X(20)) WRT CEVIAN TRACES OF X(2)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^8-3*(b^2+c^2)*a^6+(3*b^4-2*b^2*c^2+3*c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+4*(b^2-c^2)^2*b^2*c^2) : :
X(45249) = X(20)-5*X(45255)

X(45249) lies on these lines: {20, 6525}, {381, 389}, {3090, 14767}, {8798, 35360}, {14363, 41481}

X(45249) = complement of the isogonal conjugate of X(26883)
X(45249) = complementary conjugate of the complement of X(26883)
X(45249) = (medial)-isotomic conjugate-of-X(42459)
X(45249) = center of the circumconic {{A, B, C, X(1302), X(35360)}}
X(45249) = X(2)-Ceva conjugate of-X(42459)
X(45249) = X(31)-complementary conjugate of-X(42459)

X(45250) = VU-MIQUEL POINT OF (X(6), X(7)) WRT CEVIAN TRACES OF X(2)

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

X(45250) lies on the cubic K220 and these lines: {6, 2338}, {7, 281}, {9, 521}, {219, 677}, {268, 6600}, {15905, 39026}

X(45250) = barycentric quotient X(2272)/X(43035)
X(45250) = trilinear product X(971)*X(2338)
X(45250) = trilinear quotient X(i)/X(j) for these (i, j): (971, 43035), (2272, 1456)
X(45250) = center of the circumconic {{A, B, C, X(677), X(36101)}}
X(45250) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(268)}} and {{A, B, C, X(9), X(653)}}
X(45250) = crosssum of X(910) and X(1360)
X(45250) = X(972)-isoconjugate-of-X(43035)

X(45251) = VU-MIQUEL POINT OF (X(6), X(20)) WRT CEVIAN TRACES OF X(2)

Barycentrics    a^2*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(-a^2+b^2+c^2)*(3*a^8-12*(b^2+c^2)*a^6+2*(9*b^4-10*b^2*c^2+9*c^4)*a^4-12*(b^4-c^4)*(b^2-c^2)*a^2+(3*b^4+26*b^2*c^2+3*c^4)*(b^2-c^2)^2)*(a^4+2*(3*b^2-c^2)*a^2-(b^2-c^2)*(7*b^2+c^2))*(a^4-2*(b^2-3*c^2)*a^2+(b^2-c^2)*(b^2+7*c^2)) : :

X(45251) lies on these lines: {6, 36609}, {20, 20207}

X(45252) = VU-MIQUEL POINT OF (X(7), X(8)) WRT CEVIAN TRACES OF X(2)

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

X(45252) lies on these lines: {5222, 45241}, {9309, 11019}

X(45253) = VU-MIQUEL POINT OF (X(3), X(7)) WRT CEVIAN TRACES OF X(3)

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

X(45253) lies on these lines: {7, 39796}, {1442, 20277}, {2898, 5929}, {3188, 3212}

X(45253) = X(3)-Ceva conjugate of-X(7)
X(45253) = X(331)-Dao conjugate of-X(264)

X(45254) = VU-MIQUEL POINT OF (X(3), X(8)) WRT CEVIAN TRACES OF X(3)

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

X(45254) lies on these lines: {8, 40944}, {329, 2899}, {4511, 19582}, {7335, 13136}, {27382, 27397}

X(45254) = X(3)-Ceva conjugate of-X(8)

X(45255) = VU-MIQUEL POINT OF (X(3), X(20)) WRT CEVIAN TRACES OF X(3)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^10-(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^8+2*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^6-4*(b^4-c^4)^2*a^4+(b^4-c^4)*(b^2-c^2)*(b^4+4*b^2*c^2+c^4)*a^2-(b^2-c^2)^4*b^2*c^2) : :
X(45255) = X(20)+4*X(45249)

X(45255) lies on these lines: {2, 185}, {3, 3168}, {4, 40948}, {20, 6525}, {216, 3523}, {417, 3164}, {631, 12012}, {648, 14379}, {17928, 40947}, {20277, 37523}, {37106, 40946}

X(45255) = barycentric quotient X(20)/X(35061)
X(45255) = X(3)-Ceva conjugate of-X(20)
X(45255) = X(20)-reciprocal conjugate of-X(35061)

leftri

1st and 2nd MacBeath-Simmons circles: X(45256)-X(45265)

rightri

This preamble and centers X(45256)-X(45265) were contributed by César Eliud Lozada, October 21, 2021.

In a triangle ABC, let Ω1 and Ω2 be two circles, both passing through X(125) and with centers X(45256) and X(45257), respectively. Their respective squared radius are:

  ρ12 = (sqrt(3)*(S^2+(4*R^2-SW)^2)*S-(16*R^2-5*SW)*S^2+(4*R^2-SW)*(16*R^4-3*SW*(4*R^2-SW)))/(2*(SW+sqrt(3)*S-R^2))^2

  ρ22 = (-sqrt(3)*(S^2+(4*R^2-SW)^2)*S-(16*R^2-5*SW)*S^2+(4*R^2-SW)*(16*R^4-3*SW*(4*R^2-SW)))/(2*(SW-sqrt(3)*S-R^2))^2

Some properties of these circles are:

Ω1 and Ω2 are named here the 1st MacbBeath-Simmons circle and the 2nd MacbBeath-Simmons circle, respectively.

Note: Simmons conics were introduced in the preamble just before X(41993).

X(45256) = CENTER OF 1st MACBEATH-SIMMONS CIRCLE

Barycentrics    4*a^10+2*a^2*(b^2-c^2)^4-11*a^8*(b^2+c^2)-4*a^4*(b^2-c^2)*(b^4-c^4)-(b^4-c^4)*(b^2-c^2)^3+2*a^6*(5*b^4+6*b^2*c^2+5*c^4)+8*sqrt(3)*((b^2+c^2)*a^2-(b^2-c^2)^2)*S^3 : :
Barycentrics    (4*R^2+2*SA-3*SW)*S^2-(8*R^2-3*SW)*SB*SC-S*sqrt(3)*(S^2+SB*SC) : :

X(45256) lies on these lines: {5, 49}, {13, 8837}, {17, 3166}, {20, 16629}, {125, 45262}, {137, 45264}, {3575, 10632}, {5318, 32585}, {5894, 23721}, {6116, 35717}, {7691, 11144}, {17845, 42156}, {18582, 19467}, {32348, 33530}

X(45256) = midpoint of X(i) and X(j) for these {i, j}: {125, 45262}, {137, 45264}

X(45257) = CENTER OF 2nd MACBEATH-SIMMONS CIRCLE

Barycentrics    4*a^10+2*a^2*(b^2-c^2)^4-11*a^8*(b^2+c^2)-4*a^4*(b^2-c^2)*(b^4-c^4)-(b^4-c^4)*(b^2-c^2)^3+2*a^6*(5*b^4+6*b^2*c^2+5*c^4)-8*sqrt(3)*((b^2+c^2)*a^2-(b^2-c^2)^2)*S^3 : :
Barycentrics    (4*R^2+2*SA-3*SW)*S^2-(8*R^2-3*SW)*SB*SC+S*sqrt(3)*(S^2+SB*SC) : :

X(45257) lies on these lines: {5, 49}, {14, 8839}, {18, 3165}, {20, 16628}, {125, 45263}, {137, 45265}, {3575, 10633}, {5321, 32586}, {5894, 23722}, {6117, 35717}, {7691, 11143}, {17845, 42153}, {18581, 19467}, {32348, 33529}

X(45257) = midpoint of X(i) and X(j) for these {i, j}: {125, 45263}, {137, 45265}

X(45258) = RADICAL TRACE OF 1st AND 2nd MACBEATH-SIMMONS CIRCLES

Barycentrics    (b^2-c^2)^2*(a^14-4*(b^2+c^2)*a^12+(7*b^4+12*b^2*c^2+7*c^4)*a^10-2*(b^2+c^2)*(5*b^4+2*b^2*c^2+5*c^4)*a^8+3*(5*b^8+5*c^8+(b^4+3*b^2*c^2+c^4)*b^2*c^2)*a^6-(b^2+c^2)*(16*b^8+16*c^8-(31*b^4-38*b^2*c^2+31*c^4)*b^2*c^2)*a^4+(9*b^8+9*c^8-(b^4-3*b^2*c^2+c^4)*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3*(2*b^4-b^2*c^2+2*c^4)) : :
X(45258) = X(110)+3*X(34308) = X(113)-3*X(23516) = X(128)-3*X(23515) = X(930)-5*X(15059) = X(1141)+3*X(14644) = X(10264)+3*X(25147) = X(10733)+3*X(38710) = 11*X(15025)+X(38683) = 3*X(15055)+X(44976) = 3*X(25147)-X(43966) = 3*X(34128)-X(38615)

X(45258) lies on these lines: {5, 49}, {113, 23516}, {125, 137}, {128, 23515}, {930, 15059}, {6723, 13372}, {10113, 38618}, {10264, 25147}, {10412, 14225}, {10733, 38710}, {14652, 15329}, {15025, 38683}, {15055, 44976}, {17702, 34837}, {20304, 25150}, {34128, 38615}

X(45258) = midpoint of X(i) and X(j) for these {i, j}: {125, 137}, {10113, 38618}, {10264, 43966}, {11801, 12026}
X(45258) = reflection of X(13372) in X(6723)
X(45258) = inverse of X(8901) in nine-point circle
X(45258) = crossdifference of every pair of points on line {X(2081), X(35324)}
X(45258) = {X(10264), X(25147)}-harmonic conjugate of X(43966)

X(45259) = RADICAL CENTER OF CIRCUMCIRCLE AND 1st AND 2nd MACBEATH-SIMMONS CIRCLES

Barycentrics    (b^2-c^2)*(a^8-3*(b^2+c^2)*a^6+(2*b^4+b^2*c^2+2*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^4-b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(45259) = X(4)+3*X(42731) = X(684)+3*X(9979) = 3*X(1637)-X(6130) = 3*X(1637)+X(16230) = 5*X(1656)-X(41078)

X(45259) lies on these lines: {4, 42731}, {5, 24978}, {98, 5966}, {107, 933}, {125, 137}, {230, 231}, {526, 11746}, {684, 9979}, {1656, 41078}, {2799, 6721}, {7608, 14223}, {9479, 31953}, {13558, 14656}, {14316, 41167}, {32119, 32226}

X(45259) = midpoint of X(i) and X(j) for these {i, j}: {5, 24978}, {6130, 16230}, {14316, 41167}
X(45259) = barycentric product X(523)*X(40853)
X(45259) = trilinear product X(661)*X(40853)
X(45259) = perspector of the circumconic {{A, B, C, X(4), X(39183)}}
X(45259) = inverse of X(12077) in Dao-Moses-Telv circle
X(45259) = intersection, other than A, B, C, of circumconics {{A, B, C, X(107), X(12077)}} and {{A, B, C, X(232), X(5966)}}
X(45259) = crossdifference of every pair of points on line {X(3), X(35324)}
X(45259) = X(523)-Hirst inverse of-X(12077)
X(45259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1637, 16230, 6130), (24007, 24008, 12077)

X(45260) = RADICAL CENTER OF INCIRCLE AND 1st AND 2nd MACBEATH-SIMMONS CIRCLES

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

X(45260) lies on these lines: {11, 244}, {12, 2292}, {30, 551}, {125, 137}, {523, 42759}, {1621, 13589}, {3884, 25031}, {3944, 24429}, {4425, 6690}, {8070, 14755}, {24871, 37735}, {36250, 37730}

X(45260) = reflection of X(5) in the line X(20304)X(45147)
X(45260) = perspector of the circumconic {{A, B, C, X(514), X(39183)}}
X(45260) = crossdifference of every pair of points on line {X(101), X(35324)}
X(45260) = crosssum of X(692) and X(4282)

X(45261) = RADICAL CENTER OF ANTICOMPLEMENTARY CIRCLE AND 1st AND 2nd MACBEATH-SIMMONS CIRCLES

Barycentrics    (a^8-(b^2+c^2)*a^6-(2*b^4+15*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(3*b^4+4*b^2*c^2+3*c^4)*a^2-(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :

X(45261) lies on these lines: {125, 137}, {523, 2525}

X(45261) = perspector of the circumconic {{A, B, C, X(18840), X(39183)}}
X(45261) = crossdifference of every pair of points on line {X(30435), X(35324)}

X(45262) = ANTIPODE OF X(125) IN 1st MACBEATH-SIMMONS CIRCLE

Barycentrics    8*S^3*sqrt(3)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))+8*a^16-30*(b^2+c^2)*a^14+5*(7*b^4+18*b^2*c^2+7*c^4)*a^12-2*(b^2+c^2)*(b^4+41*b^2*c^2+c^4)*a^10-(25*b^8+25*c^8-2*(7*b^4+43*b^2*c^2+7*c^4)*b^2*c^2)*a^8+22*(b^6-c^6)*(b^4-c^4)*a^6-3*(b^2-c^2)^2*(5*b^4-2*b^2*c^2+5*c^4)*(b^4+c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)^3*(5*b^4-4*b^2*c^2+5*c^4)*a^2-3*(b^2+c^2)^2*(b^2-c^2)^6 : :

X(45262) lies on these lines: {113, 137}, {125, 45256}, {45147, 45264}

X(45262) = reflection of X(125) in X(45256)

X(45263) = ANTIPODE OF X(125) IN 2nd MACBEATH-SIMMONS CIRCLE

Barycentrics    -8*S^3*sqrt(3)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*((b^2+c^2)*a^4-2*(b^4-b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))+8*a^16-30*(b^2+c^2)*a^14+5*(7*b^4+18*b^2*c^2+7*c^4)*a^12-2*(b^2+c^2)*(b^4+41*b^2*c^2+c^4)*a^10-(25*b^8+25*c^8-2*(7*b^4+43*b^2*c^2+7*c^4)*b^2*c^2)*a^8+22*(b^6-c^6)*(b^4-c^4)*a^6-3*(b^2-c^2)^2*(5*b^4-2*b^2*c^2+5*c^4)*(b^4+c^4)*a^4+2*(b^4-c^4)*(b^2-c^2)^3*(5*b^4-4*b^2*c^2+5*c^4)*a^2-3*(b^2+c^2)^2*(b^2-c^2)^6 : :

X(45263) lies on these lines: {113, 137}, {125, 45257}, {45147, 45265}

X(45263) = reflection of X(125) in X(45257)

X(45264) = ANTIPODE OF X(137) IN 1st MACBEATH-SIMMONS CIRCLE

Barycentrics    8*S^3*sqrt(3)*((-a^2+b^2+c^2)^2-b^2*c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^8-4*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4)+8*a^22-54*(b^2+c^2)*a^20+4*(41*b^4+66*b^2*c^2+41*c^4)*a^18-(b^2+c^2)*(305*b^4+226*b^2*c^2+305*c^4)*a^16+2*(201*b^8+201*c^8+2*b^2*c^2*(137*b^4+159*b^2*c^2+137*c^4))*a^14-(b^2+c^2)*(406*b^8+406*c^8-b^2*c^2*(163*b^4-480*b^2*c^2+163*c^4))*a^12+2*(161*b^12+161*c^12-(59*b^8+59*c^8-b^2*c^2*(23*b^4+38*b^2*c^2+23*c^4))*b^2*c^2)*a^10-(b^4-c^4)*(b^2-c^2)*(204*b^8+204*c^8-b^2*c^2*(97*b^4-208*b^2*c^2+97*c^4))*a^8+2*(b^2-c^2)^2*(55*b^12+55*c^12-2*(21*b^8+21*c^8-b^2*c^2*(2*b^2-c^2)*(b^2-2*c^2))*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)^3*(52*b^8+52*c^8-b^2*c^2*(61*b^4-62*b^2*c^2+61*c^4))*a^4+2*(b^2-c^2)^6*(9*b^8+9*c^8+b^2*c^2*(3*b^4+b^2*c^2+3*c^4))*a^2-(b^2-c^2)^8*(b^2+c^2)*(3*b^4-b^2*c^2+3*c^4) : :

X(45264) lies on these lines: {125, 128}, {137, 45256}, {45147, 45262}

X(45264) = reflection of X(137) in X(45256)

X(45265) = ANTIPODE OF X(137) IN 2nd MACBEATH-SIMMONS CIRCLE

Barycentrics    -8*S^3*sqrt(3)*((-a^2+b^2+c^2)^2-b^2*c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)*(2*a^8-4*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4)+8*a^22-54*(b^2+c^2)*a^20+4*(41*b^4+66*b^2*c^2+41*c^4)*a^18-(b^2+c^2)*(305*b^4+226*b^2*c^2+305*c^4)*a^16+2*(201*b^8+201*c^8+2*b^2*c^2*(137*b^4+159*b^2*c^2+137*c^4))*a^14-(b^2+c^2)*(406*b^8+406*c^8-b^2*c^2*(163*b^4-480*b^2*c^2+163*c^4))*a^12+2*(161*b^12+161*c^12-(59*b^8+59*c^8-b^2*c^2*(23*b^4+38*b^2*c^2+23*c^4))*b^2*c^2)*a^10-(b^4-c^4)*(b^2-c^2)*(204*b^8+204*c^8-b^2*c^2*(97*b^4-208*b^2*c^2+97*c^4))*a^8+2*(b^2-c^2)^2*(55*b^12+55*c^12-2*(21*b^8+21*c^8-b^2*c^2*(2*b^2-c^2)*(b^2-2*c^2))*b^2*c^2)*a^6-(b^4-c^4)*(b^2-c^2)^3*(52*b^8+52*c^8-b^2*c^2*(61*b^4-62*b^2*c^2+61*c^4))*a^4+2*(b^2-c^2)^6*(9*b^8+9*c^8+b^2*c^2*(3*b^4+b^2*c^2+3*c^4))*a^2-(b^2-c^2)^8*(b^2+c^2)*(3*b^4-b^2*c^2+3*c^4) : :

X(45265) lies on these lines: {125, 128}, {137, 45257}, {45147, 45263}

X(45265) = reflection of X(137) in X(45257)

leftri

(B)-line conjugates: X(45266)-X(45285)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 21, 2021.

If points P and U, given by trilinears P = p : q : r and U = u : v : w are distinct, and neither lies on a sideline BC, CA, AB, then the P-line conjugate of U is the point given by trilinears

p(v^2 + w^2) - u(qv + rw) : q(w^2 + u^2) - v(rw + pu) : r(u^2 + v^2) - w(pu + qv).

Now suppose that P and Q have barycentrics p : q : r and u : v : w, and define the P-(B)line conjugate of U as the point F(P,U) given by barycentrics

p(v^2 + w^2) - u(qv + rw) : q(w^2 + u^2) - v(rw + pu) : r(u^2 + v^2) - w(pu + qv).

The points X(45266) - X(45385) are (B)line conjugates.

The appearance of (i,j) in the following list means that X(i) and X(j) are on the Euler line and F(X(i)) = X(j), so that also, F(X(j)) = X(i):

(2,30), {3,297), (4,441), (5,401), (20,44334), (30,2), (140,40853), (297,3), (376,44216), (381,40884), (384,21536), (401,5), (427,15013), (440,447), (441,4), (442,448), (443,449), (447,440), (448,442), (449,443), (458,44231), (465,11094), (466,11093), (468,35923), (547,44651), (549,40885), (857,37045), (858,40856), (1312,44333), (1313,44332), (1368,15014), (1370,44340), (2454,2480), (2455,2479), (2475,44336), (2479,2455), (2480,2454), (3146,44335), (3151,44342), (3152,44344), (3534,44576), (3543,44346), (3830,44578), (3845,44575), (4235,37987), (5159,40890), (5189,44338), (6655,44347), (6656,6660), (6660,6656), (6676,40889), (7391,44337), (8703,44579), (10684,21531), (10989,44649), (11049,44653), (11093,466), (11094,465), (11585,44328), (12100,44577), (13371,44329), (14790,44341), (14807,44348), (14808,44349), (14957,44345), (15013,427), (15014,1368), (18641,44331), (21530,44330), (21531,10684), (21536,384), (31152,44650), (35923,468), (37045,857), (37188,44228), (37444,44339), (37987,4235), (40853,140), (40856,858), (40884,381), (40885,549), (40889,6676), (40890,5159), (44216,376), (44228,37188), (44231,458), (44328,11585), (44329,13371), (44330,21530), (44331,18641), (44332,1313), (44333,1312), (44334,20), (44335,3146), (44336,2475), (44337,7391), (44338,5189), (44339,37444), (44340,1370), (44341,14790), (44342,3151), (44344,3152), (44345,14957), (44346,3543), (44347,6655), (44348,14807), (44349,14808), (44575,3845), (44576,3534), (44577,12100), (44578,3830), (44579,8703), (44649,10989), (44650,31152), (44651,547), (44653,11049)

See also the preambles just before X(44886) and X(45196).

X(45266) = X(1)-(B)LINE CONJUGATE OF X(3)

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

X(45266) lies on these lines: {1, 3}, {240, 916}, {297, 525}, {1748, 22130}, {3663, 18179}, {26155, 26158}, {26205, 26208}

X(45266) = polar conjugate of X(39429)
X(45266) = X(48)-isoconjugate of X(39429)
X(45266) = crossdifference of every pair of points on line {184, 650}
X(45266) = barycentric quotient X(4)/X(39429)

X(45267) = X(1)-(B)LINE CONJUGATE OF X(55)

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

X(45267) lies on these lines: {1, 3}, {918, 3261}, {3125, 43037}, {3665, 3721}, {5435, 30884}, {26526, 26527}, {28734, 28735}, {28964, 28969}

X(45267) = X(41)-isoconjugate of X(767)
X(45267) = crossdifference of every pair of points on line {650, 2175}
X(45267) = barycentric product X(i)*X(j) for these {i,j}: {7, 35552}, {766, 6063}, {8629, 41283}
X(45267) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 767}, {766, 55}, {8629, 2175}, {35552, 8}

X(45268) = X(1)-(B)LINE CONJUGATE OF X(56)

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

X(45268) lies on these lines: {1, 3}, {3700, 3910}, {3912, 38345}, {17452, 18179}, {26575, 26576}, {28789, 28790}, {28919, 28924}

X(45268) = crossdifference of every pair of points on line {650, 1397}

X(45269) = X(1)-(B)LINE CONJUGATE OF X(57)

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

X(45269) lies on these lines: {1, 3}, {10, 7069}, {11, 1739}, {201, 43174}, {255, 40256}, {515, 24028}, {516, 1877}, {519, 7004}, {522, 3717}, {774, 4848}, {950, 4642}, {1149, 15558}, {1393, 4301}, {1455, 17613}, {1457, 24025}, {1736, 40663}, {1772, 30384}, {1837, 3987}, {1854, 5687}, {1858, 3293}, {2310, 4695}, {2654, 3754}, {2800, 22350}, {3679, 24430}, {3814, 35015}, {3878, 22072}, {4511, 10703}, {4551, 6001}, {4646, 12711}, {4868, 14547}, {4972, 24982}, {5176, 18340}, {5552, 27504}, {9581, 24440}, {11362, 44706}, {12053, 24443}, {17757, 38357}, {27380, 27385}, {34587, 38981}

X(45269) = reflection of X(i) in X(j) for these {i,j}: {1457, 24025}, {38462, 10}
X(45269) = X(56)-isoconjugate of X(2370)
X(45269) = crosspoint of X(318) and X(1320)
X(45269) = crosssum of X(603) and X(1319)
X(45269) = crossdifference of every pair of points on line {604, 650}
X(45269) = barycentric product X(i)*X(j) for these {i,j}: {9, 3007}, {312, 2390}
X(45269) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 2370}, {2390, 57}, {3007, 85}, {20619, 37790}

X(45270) = X(1)-(B)LINE CONJUGATE OF X(65)

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

X(45270) lies on these lines: {1, 3}, {86, 14749}, {141, 22071}, {239, 11998}, {650, 3975}, {1959, 38345}, {2234, 2310}, {2269, 16696}, {2330, 39957}, {5745, 16699}, {21226, 24499}

X(45270) = crossdifference of every pair of points on line {650, 1402}

X(45271) = X(1)-(B)LINE CONJUGATE OF X(4)

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

X(45271) lies on these lines: {1, 4}, {48, 3663}, {441, 525}, {516, 8766}, {3008, 34591}, {3946, 22063}, {6001, 43045}, {6508, 40940}, {20902, 24209}, {22130, 22131}, {28409, 28425}, {28698, 28717}, {37755, 39595}

X(45271) = isotomic conjugate of the polar conjugate of X(2385)
X(45271) = X(19)-isoconjugate of X(2365)
X(45271) = crossdifference of every pair of points on line {25, 652}
X(45271) = barycentric product X(69)*X(2385)
X(45271) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2365}, {2385, 4}

X(45272) = X(1)-(B)LINE CONJUGATE OF X(226)

Barycentrics    a*(a - b - c)*(a^5 - a^3*b^2 + a^2*b^3 - b^5 + a^3*b*c - a^2*b^2*c - a*b^3*c + b^4*c - a^3*c^2 - a^2*b*c^2 + 2*a*b^2*c^2 + a^2*c^3 - a*b*c^3 + b*c^4 - c^5) : :
X(45372) = 4 X[16869] - X[38945], 4 X[16870] - X[18340]

X(45272) lies on these the cubics K591 and K685 and on these lines: {1, 4}, {3, 1854}, {8, 27504}, {11, 30117}, {30, 38357}, {35, 37116}, {36, 1725}, {37, 1630}, {55, 1324}, {58, 1858}, {78, 27379}, {102, 517}, {104, 35187}, {109, 2733}, {113, 10017}, {124, 5081}, {191, 22361}, {212, 5692}, {255, 5693}, {318, 10570}, {501, 2360}, {522, 663}, {603, 15071}, {758, 1936}, {774, 37583}, {960, 2328}, {968, 3601}, {971, 1455}, {976, 1697}, {993, 24430}, {997, 1040}, {999, 35455}, {1038, 12520}, {1062, 18339}, {1193, 33178}, {1319, 37815}, {1385, 37806}, {1731, 2341}, {1735, 6905}, {1935, 31803}, {1951, 8558}, {2361, 5127}, {2390, 38507}, {2689, 26701}, {2701, 26702}, {2716, 36082}, {2732, 26700}, {3074, 20117}, {3075, 5884}, {3326, 5497}, {3430, 41600}, {3924, 9581}, {3938, 7962}, {3940, 7074}, {4303, 16132}, {5011, 38345}, {5048, 10700}, {5251, 7069}, {5440, 9371}, {6326, 22350}, {9785, 36565}, {10950, 15955}, {11012, 44706}, {11363, 41401}, {12711, 37539}, {14629, 15898}, {14874, 38336}, {16086, 27542}, {17102, 37837}, {19861, 24538}, {30223, 37817}

X(45272) = midpoint of X(1) and X(3465)
X(45272) = reflection of X(i) in X(j) for these {i,j}: {1785, 16869}, {5081, 124}, {38945, 1785}
X(45272) = incircle-inverse of X(950)
X(45272) = Conway-circle inverse of X(10454)
X(45272) = polar-circle inverse of X(225)
X(45272) = X(65)-isoconjugate of X(39435)
X(45272) = crosssum of X(i) and X(j) for these (i,j): {36, 34043}, {65, 1455}
X(45272) = crossdifference of every pair of points on line {652, 1400}
X(45272) = X(1324)-of-Mandart-incircle-triangle
X(45272) = barycentric quotient X(284)/X(39435)
X(45272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1490, 21147}, {1, 6261, 10571}, {6198, 21740, 1}, {10950, 33177, 15955}

X(45273) = X(1)-(B)LINE CONJUGATE OF X(11)

Barycentrics    (a - b)*(a - c)*(a + b - c)*(a - 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(45273) lies on these lines: {1, 5}, {101, 10015}, {190, 644}, {1086, 9317}, {1110, 28473}, {4564, 7178}, {26653, 26656}, {28961, 28964}, {35110, 39047}

X(45273) = crossdifference of every pair of points on line {654, 3271}
X(45273) = X(2316)-isoconjugate of X(39155)
X(45273) = barycentric quotient X(i)/X(j) for these {i,j}: {1319, 39155}, {5170, 7252}, {39154, 1320}
X(45273) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {651, 664, 30725}

X(45274) = X(1)-(B)LINE CONJUGATE OF X(952)

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

X(45274) lies on these lines: {1, 5}, {2, 3904}, {1086, 36949}, {1565, 43043}, {3960, 5662}, {34460, 43048}

X(45275) = X(1)-(B)LINE CONJUGATE OF X(7)

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

X(45275) lies on these lines: {1, 7}, {11, 17067}, {522, 650}, {527, 41339}, {1040, 40998}, {1266, 14942}, {1854, 6737}, {1861, 8582}, {2310, 3008}, {2324, 14522}, {3270, 29353}, {3946, 14100}, {4000, 4907}, {4012, 4901}, {5218, 16676}, {5537, 40577}, {7071, 24309}, {9950, 28739}, {15726, 43035}, {34919, 42317}, {37374, 44901}

X(45275) = crossdifference of every pair of points on line {56, 657}
X(45275) = X(57)-isoconjugate of X(2371)
X(45275) = barycentric product X(8)*X(2391)
X(45275) = barycentric quotient X(i)/X(j) for these {i,j}: {55, 2371}, {2391, 7}
X(45275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3000, 3012, 1323}, {9371, 16870, 6745}

X(45276) = X(1)-(B)LINE CONJUGATE OF X(516)

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

X(45276) lies on these lines: {1, 7}, {2, 2400}, {37, 44357}, {86, 37045}, {658, 15252}, {1565, 24813}, {4000, 17435}, {4419, 35091}, {4626, 28344}, {7056, 23586}, {9357, 17596}, {13609, 23587}, {35150, 35169}

X(45276) = midpoint of X(1) and X(35031)
X(45276) = {X(7),X(3160)}-harmonic conjugate of X(23973)

X(45277) = X(3)-(B)LINE CONJUGATE OF X(74)

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

X(45277) lies on these lines: {3, 74}, {115, 127}, {1625, 34990}, {3269, 8552}, {4558, 22146}

X(45277) = crossdifference of every pair of points on line {1576, 1637}
X(45277) = {X(6334),X(15526)}-harmonic conjugate of X(339)

X(45278) = X(3)-(B)LINE CONJUGATE OF X(1511)

Barycentrics    a^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)*(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) : :

X(45278) lies on these lines: {3, 74}, {94, 2394}, {11079, 44769}, {14061, 16080}

X(45278) = X(2420)-isoconjugate of X(2627)
X(45278) = barycentric product X(1494)*X(18114)
X(45278) = barycentric quotient X(i)/X(j) for these {i,j}: {5627, 39430}, {18114, 30}

X(45279) = X(4)-(B)LINE CONJUGATE OF X(6)

Barycentrics    a^6*b^2 + a^4*b^4 - a^2*b^6 - b^8 + a^6*c^2 - 4*a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + 2*b^2*c^6 - c^8 : :
X(45379) = 3 X[6530] - 4 X[18121]

X(45279) lies on these lines: {4, 6}, {66, 9308}, {98, 16310}, {159, 17907}, {264, 23300}, {297, 2393}, {317, 34777}, {325, 523}, {441, 1632}, {458, 23327}, {685, 31636}, {895, 44767}, {1513, 3003}, {1995, 7792}, {2450, 20975}, {2790, 44231}, {2794, 3284}, {6033, 10297}, {7464, 12253}, {7774, 31099}, {7778, 16051}, {11585, 14059}, {16303, 43460}, {20806, 41757}, {23583, 42671}, {31133, 41624}, {32113, 38361}, {34146, 44704}, {39129, 40009}

X(45279) = reflection of X(i) in X(j) for these {i,j}: {1632, 441}, {42671, 23583}
X(45279) = isotomic conjugate of X(2366)
X(45279) = isotomic conjugate of the isogonal conjugate of X(2386)
X(45279) = X(31)-isoconjugate of X(2366)
X(45279) = crossdifference of every pair of points on line {32, 520}
X(45279) = (de Longchamps line)∩(van Aubel line)
X(45279) = barycentric product X(76)*X(2386)
X(45279) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 2366}, {2386, 6}

X(45280) = X(4)-(B)LINE CONJUGATE OF X(1503)

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

X(45280) lies on these lines: {2, 2419}, {4, 6}, {122, 23591}, {2549, 39008}, {3767, 41172}, {9530, 23976}, {15048, 40856}

X(45280) = crossdifference of every pair of points on line {520, 42671}
X(45280) = {X(4),X(1249)}-harmonic conjugate of X(23977)

X(45281) = X(4)-(B)LINE CONJUGATE OF X(9)

Barycentrics    a^4*b - 2*a^3*b^2 + 2*a^2*b^3 - 2*a*b^4 + b^5 + a^4*c + 2*a^3*b*c - 2*a^2*b^2*c - b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 + 2*a^2*c^3 - 2*a*c^4 - b*c^4 + c^5 : :
X(45281) = 2 X[10] - 3 X[1861], X[145] - 3 X[4318], 3 X[3100] - 5 X[3616]

X(45281) lies on these lines: {4, 9}, {7, 24388}, {141, 17668}, {145, 4318}, {522, 693}, {551, 9611}, {1699, 34822}, {1737, 24715}, {2323, 28849}, {3100, 3616}, {3834, 10427}, {3912, 35338}, {4659, 4847}, {4907, 18634}, {6708, 7965}, {9801, 27509}, {10177, 21258}, {14100, 16608}, {15726, 26932}, {17059, 30379}, {20872, 37034}, {24537, 24564}, {24983, 25011}, {31638, 36086}, {36949, 41339}

X(45281) = reflection of X(41339) in X(36949)
X(45281) = crossdifference of every pair of points on line {41, 1459}

X(45282) = X(4)-(B)LINE CONJUGATE OF X(516)

Barycentrics    a^6 - a^5*b - a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 + b^6 - a^5*c + a^4*b*c + a^3*b^2*c - a^2*b^3*c + a^3*b*c^2 - 2*a^2*b^2*c^2 + 2*a*b^3*c^2 - b^4*c^2 - a^3*c^3 - a^2*b*c^3 + 2*a*b^2*c^3 + 2*a^2*c^4 - b^2*c^4 - 2*a*c^5 + c^6 : :
X(45282) = 5 X[1698] - X[35031]

X(45282) lies on these lines: {2, 2400}, {4, 9}, {1698, 35031}, {7433, 26244}, {15252, 23970}, {17354, 35158}, {24275, 35082}, {28780, 41352}

X(45283) = X(3)-(B)LINE CONJUGATE OF X(32)

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

X(45283) lies on these lines: {3, 6}, {625, 44114}, {826, 850}, {18796, 33802}, {22151, 39840}

X(45283) = crossdifference of every pair of points on line {523, 1501}

X(45284) = X(3)-(B)LINE CONJUGATE OF X(187)

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

X(45284) lies on these lines: {3, 6}, {76, 850}, {512, 38523}, {538, 36194}, {690, 36165}, {7817, 32225}, {7827, 15360}

X(45284) = midpoint of X(38523) and X(38526)
X(45284) = 2nd-Brocard-circle inverse of X(5467)
X(45284) = crossdifference of every pair of points on line {523, 14567}
X(45284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1670, 1671, 5467}

X(45285) = X(3)-(B)LINE CONJUGATE OF X(566)

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

X(45285) lies on these lines: {3, 6}, {526, 850}, {2387, 3014}, {3580, 7668}, {13240, 37638}

X(45286) = X(4)X(110)∩X(52)X(539)

Barycentrics    2 a^10-4 a^8 b^2+a^6 b^4+a^4 b^6+a^2 b^8-b^10-4 a^8 c^2+4 a^6 b^2 c^2-3 a^4 b^4 c^2+3 b^8 c^2+a^6 c^4-3 a^4 b^2 c^4-2 a^2 b^4 c^4-2 b^6 c^4+a^4 c^6-2 b^4 c^6+a^2 c^8+3 b^2 c^8-c^10 : :
Barycentrics    S^2 (6 R^2+SA-2 SW)-5 SB SC (2 R^2-SW) : :
X(45286) = 3*X(3)-2*X(17712),3*X(4)+X(12278),3*X(51)-X(44076),X(52)-3*X(7576),5*X(52)-3*X(41628),X(185)-3*X(38321),3*X(381)-X(21659),3*X(389)-2*X(43588),3*X(546)-2*X(15807),5*X(3091)-X(12289),5*X(3567)-X(34799),3*X(5446)-2*X(13142),4*X(5462)-3*X(43573),3*X(5891)-X(12225),3*X(5892)-4*X(9825),X(6102)-3*X(38322),2*X(6146)-3*X(43573),3*X(6756)-X(13142),3*X(7576)+X(14516),5*X(7576)-X(41628),3*X(9730)-2*X(18128),3*X(9730)-X(34224),2*X(10110)-3*X(13490),X(10116)-4*X(31830),3*X(10116)-4*X(43588),3*X(10170)-2*X(12362),X(10575)-3*X(38323),3*X(11225)-2*X(11264),3*X(11225)-4*X(16881),X(12111)+3*X(18559),2*X(12278)+3*X(12897),X(12370)-3*X(13490),4*X(13163)-3*X(13364),3*X(13364)-2*X(43575),3*X(13403)-4*X(15807),5*X(14516)+3*X(41628),3*X(15030)-X(18563),3*X(15305)+X(34797),3*X(16194)-X(18560),9*X(16261)-X(40242),X(16659)+3*X(38323),9*X(20791)-X(40241),3*X(31830)-X(43588)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2804.

X(45286) lies on these lines: {2,11750}, {3,2918}, {4,110}, {5,5944}, {24,5449}, {25,9927}, {30,1216}, {49,3574}, {51,44076}, {52,539}, {68,7487}, {141,550}, {143,10112}, {155,18494}, {156,18388}, {185,38321}, {186,20191}, {195,23236}, {265,13621}, {381,19357}, {382,394}, {389,10116}, {427,12038}, {511,11819}, {541,18439}, {542,6102}, {546,13403}, {567,10619}, {569,7544}, {578,11818}, {1092,31723}, {1181,7706}, {1209,7488}, {1495,10024}, {1503,31833}, {1568,18350}, {1594,43839}, {1596,23307}, {1598,12293}, {1657,11472}, {1658,21243}, {2070,6288}, {2072,11572}, {3091,12289}, {3153,43598}, {3357,34118}, {3517,14852}, {3520,18488}, {3567,34799}, {3575,12134}, {3628,13470}, {3818,7526}, {4846,34781}, {5446,6756}, {5462,6146}, {5480,43595}, {5562,32332}, {5576,13367}, {5891,12225}, {5892,9825}, {5972,10224}, {6000,43577}, {6153,41589}, {6240,12162}, {6247,43604}, {6644,18381}, {6699,20299}, {6746,25711}, {7528,19467}, {7529,18396}, {7547,35264}, {7575,34826}, {7687,18379}, {7689,18533}, {9306,18569}, {9730,18128}, {9818,17845}, {9833,18420}, {10110,12370}, {10170,12362}, {10255,12900}, {10274,18428}, {10540,43831}, {10575,16659}, {11225,11264}, {11250,38726}, {11438,32140}, {11565,32205}, {11585,43586}, {11745,13292}, {12085,36990}, {12106,36253}, {12111,18559}, {12163,18440}, {12164,40909}, {12173,18451}, {12254,13434}, {12359,37458}, {13163,13364}, {13368,38898}, {13371,14156}, {13619,15062}, {13861,18390}, {14157,34007}, {14533,36412}, {14641,31829}, {14644,21451}, {14788,37513}, {14915,16655}, {15030,18563}, {15305,34797}, {15646,25563}, {16194,18560}, {16261,40242}, {16625,32358}, {17823,37514}, {18369,43821}, {20127,33541}, {20397,23294}, {20771,33547}, {20791,40241}, {25739,43817}, {27371,32661}, {31304,37478}, {31857,38942}, {32171,39504}, {32354,40441}, {32415,32767}, {33332,43394}, {35921,41482}, {40276,44263}, {41714,41725}

X(45286) = midpoint of X(i) and X(j) for these {i,j}: {52,14516}, {3575,12134}, {6240,12162}, {10575,16659}
X(45286) = reflection of X(i) in X(j) for these (i,j): (389,31830), (5446,6756), (6146,5462), (10112,143), (10116,389), (11264,16881), (11565,32205), (12370,10110), (12897,4), (13292,11745), (13403,546), (13470,3628), (14641,31829), (32358,16625), (34224,18128), (40647,31833), (43575,13163)
X(45286) = complement of X(11750)
X(45286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,10539,5448), (5,10282,44516), (5,18475,6689), (5,34782,18475), (24,18474,5449), (1568,32340,31724), (3818,34785,7526), (5448,10539,16534), (5462,6146,43573), (7488,41171,1209), (7576,14516,52), (9730,34224,18128), (11264,16881,11225), (12370,13490,10110), (13163,43575,13364), (16659,38323,10575), (18350,31724,1568), (18379,44235,7687), (20299,37814,6699), (34514,37814,20299)

X(45287) = X(1)X(4)∩X(8)X(46)

Barycentrics    2 a^4-a^3 b-a^2 b^2+a b^3-b^4-a^3 c+4 a^2 b c-a b^2 c-a^2 c^2-a b c^2+2 b^2 c^2+a c^3-c^4 : :
X(45287) = 3*X(1)-2*X(950),9*X(1)-8*X(40270),3*X(65)-4*X(24470),3*X(354)-2*X(37730),2*X(942)-3*X(5434),X(950)-3*X(10106),4*X(950)-3*X(10572),3*X(950)-4*X(40270),5*X(1698)-4*X(5795),X(1770)+2*X(10944),3*X(3058)-4*X(31792),X(3868)-3*X(34605),5*X(3890)-3*X(11114),X(4084)-3*X(34637),2*X(4292)+X(37707),4*X(4298)-3*X(5902),4*X(4298)-X(37706),4*X(5044)-3*X(34606),3*X(5434)-X(10950),3*X(5692)-2*X(12527),2*X(5836)-3*X(11112),3*X(5902)-X(37706),3*X(5919)-2*X(15171),4*X(6738)-5*X(18398),4*X(10106)-X(10572),9*X(10106)-4*X(40270),3*X(11246)-4*X(31776),4*X(12433)-5*X(17609),X(14923)-3*X(17579),3*X(18990)-2*X(24470)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2804.

X(45287) lies on these lines: {1,4}, {2,10827}, {3,5252}, {5,1319}, {8,46}, {10,36}, {11,18480}, {12,1385}, {20,5119}, {21,14798}, {30,3057}, {35,4297}, {40,4299},{55,18481},{56,355},{57,4317},{65,952},{72,529},{79,1320},{80,1210},{100,10915},{104,10057},{106,28018},{140,37605},{145,4295}, {150,7176}, {191,5837}, {214,27385}, {229,6740}, {354,28224}, {381,11376}, {382,12701}, {484,4325}, {495,2646}, {496,20323}, {498,3576}, {499,1420}, {516,5697}, {517,1770}, {519,3868}, {535,3878}, {546,1387}, {550,37568}, {551,5443}, {664,4911}, {908,30144}, {912,18970}, {938,37721}, {942,5434}, {956,5794}, {958,7742}, {962,30323}, {993,24987}, {997,3436}, {999,1837}, {1000,3529}, {1012,11508}, {1071,13375}, {1076,30115}, {1125,3897}, {1145,32537}, {1155,5690}, {1158,37002}, {1222,5100}, {1317,10222}, {1329,17614}, {1376,40293}, {1388,5886}, {1470,11499}, {1482,1836}, {1483,11011}, {1697,4302}, {1698,5795}, {1709,10043}, {1756,3883}, {1777,37610}, {2093,3632}, {2098,12699}, {2099,9657}, {2475,4861}, {2829,12672}, {3058,26088}, {3075,9363}, {3085,3612}, {3086,4308}, {3091,23708}, {3146,30305}, {3149,22767}, {3179,5246}, {3304,5722}, {3319,23842}, {3333,5727}, {3336,4848}, {3337,9897}, {3338,3600}, {3339,18452}, {3361,37712}, {3419,12513}, {3434,36977}, {3474,12245}, {3560,11510}, {3579,15326}, {3584,37616}, {3601,10056}, {3614,11230}, {3616,5187}, {3626,37524}, {3633,4312}, {3649,37734}, {3655,11237}, {3674,7272}, {3679,15803}, {3746,4304}, {3751,39901}, {3817,37735}, {3822,24541}, {3839,18220}, {3874,41575}, {3890,11114}, {3911,18395}, {3914,15955}, {3947,24926}, {4018,5855}, {4294,9800}, {4298,5902}, {4303,10459}, {4309,31393}, {4314,5441}, {4316,11010}, {4324,37563}, {4333,6361}, {4511,20060}, {4847,5288}, {4881,27529}, {5044,34606}, {5048,22791}, {5057,5330}, {5080,21616}, {5083,31870}, {5086,10916}, {5123,13747}, {5126,5433}, {5131,38127}, {5175,34625}, {5179,9310}, {5204,26446}, {5226,31410}, {5245,41225}, {5249,30147}, {5432,13624}, {5440,12607}, {5493,5559}, {5530,44039}, {5533,6246}, {5541,12640}, {5570,7686}, {5687,32049}, {5692,12527}, {5724,37592}, {5726,30389}, {5766,15298}, {5790,24914}, {5818,7288}, {5836,11112}, {5840,23340}, {5842,12709}, {5901,17605}, {5904,6737}, {5919,15171}, {6147,37728}, {6224,34772}, {6265,12763}, {6284,9957}, {6647,17046}, {6684,7280}, {6691,17619}, {6734,8666}, {6735,25440}, {6738,18398}, {6796,14793}, {6826,34489}, {6842,10957}, {6906,32760}, {6918,41426}, {6924,34880}, {6943,13411}, {6945,7741}, {6973,8227}, {6985,10966}, {7163,10570}, {7173,38140}, {7681,10948}, {7962,41869}, {7982,9579}, {7987,31434}, {8068,11715}, {8069,12114}, {8071,11500}, {9369,16086}, {9581,10072}, {9583,13905}, {9596,9619}, {9597,9620}, {9654,10246}, {9834,26417}, {9835,26393}, {9850,20420}, {9864,10089}, {9955,25405}, {10069,13178}, {10081,13211}, {10087,12119}, {10090,12751}, {10091,12368}, {10404,11551}, {10592,38028}, {10624,28164}, {10738,20586}, {10742,12740}, {10896,11373}, {10914,38455}, {10954,15844}, {11019,37006}, {11246,31776}, {11260,24390}, {11502,18518}, {11545,34753}, {11827,31786}, {12247,26877}, {12433,17609}, {12573,18412}, {12645,36279}, {12675,13750}, {12679,40267}, {12711,39779}, {12735,12743}, {12737,13273}, {12784,13312}, {13117,13280}, {13405,37571}, {13462,37714}, {14526,33858}, {15178,15950}, {15325,17606}, {15446,17010}, {15888,24929}, {15934,37724}, {16371,37828}, {17742,24247}, {18519,22760}, {18646,25526}, {18961,37820}, {18962,37700}, {22758,37579}, {23675,30117}, {24954,35272}, {26364,35262}, {28470,28591}, {30282,31452}, {31436,31508}, {32636,37705}, {33593,37230}, {35762,35801}, {35763,35800}, {36845,41709}, {37281,37566}, {37572,43174}, {37582,40663}, {37826,40271}

X(45287) = midpoint of X(i) and X(j) for these {i,j}: {5697,10483}, {5903,37707}, {7354,10944}
X(45287) = reflection of X(i) in X(j) for these (i,j): (1,10106), (8,17647), (65,18990), (1770,7354), (5903,4292), (5904,6737), (6284,9957), (10572,1), (10950,942), (11827,31786), (12743,12735), (18412,12573), (41575,3874)
X(45287) = X(652)-he conjugate of X(2636)
X(45287) = (1,4,30384), (1,388,13407), (1,1478,12047), (1,3583,12053), (1,3585,946), (1,5270,226), (1,5691,1479), (1,9613,1478), (1,18393,13464), (3,5252,10039), (4,3476,1), (8,4293,46), (10,4311,36), (35,36975,4297), (36,37710,10), (40,37709,12647), (46,37708,8), (56,355,1737), (57,5881,10573), (65,17660,24475), (79,7972,11009), (80,5563,1210), (145,4295,25415), (226,5882,1), (388,944,1), (404,5176,10), (495,34773,2646), (999,18525,1837), (1056,3486,1), (1210,4315,5563), (1388,10895,5886), (1420,5587,499), (1482,9655,1836), (1483,39542,11011), (1836,37738,1482), (2098,12943,12699), (3085,5731,3612), (3336,41684,4848), (3338,37711,18391), (3485,7967,1), (3576,9578,498), (3600,18391,3338), (3616,10590,37692), (3655,11374,34471), (4297,31397,35), (4299,12647,40), (4316,11010,31730), (4317,10573,57), (4511,20060,21077), (5126,9956,5433), (5434,10950,942), (6246,41554,5533), (7951,21842,1125), (7972,11009,3244), (9654,10246,11375), (10039,21578,3), (10827,37618,2), (11237,34471,11374), (12053,31673,3583), (12645,36279,41687), (15325,18357,17606), (18480,24928,11), (37525,37719,13411)

X(45288) = X(1)X(1399)∩X(10)X(12)

Barycentrics    a (a+b-c) (a-b+c) (b+c) (a^3-a^2 b-a b^2+b^3-a^2 c+2 a b c-a c^2+c^3) : :
X(45288) = 5*X(65)-4*X(12432),3*X(65)-2*X(15556),6*X(12432)-5*X(15556)

See Antreas Hatzipolakis and Ercole Suppa, euclid 2804.

X(45288) lies on these lines: {1,1399}, {8,18961}, {10,12}, {11,5887}, {35,11571}, {56,3218}, {80,16153}, {388,17483}, {517,1770}, {551,13751}, {912,10950}, {942,15950}, {960,5433}, {994,9552}, {1071,1317}, {1155,31806}, {1319,3878}, {1385,11570}, {1388,3877}, {1408,18417}, {1409,4016}, {1411,1935}, {1420,3899}, {1470,5730}, {1479,40266}, {1728,11529}, {1737,5694}, {1836,37625}, {1837,5693}, {2098,12114}, {2099,3868}, {2594,4424}, {2646,5884}, {2771,10572}, {3256,41696}, {3340,3901}, {3585,16159}, {3721,4559}, {3825,20118}, {3827,39897}, {3874,11011}, {3884,5083}, {3924,7299}, {4187,12832}, {4324,5697}, {5086,13273}, {5252,5270}, {5298,31165}, {5425,16140}, {5432,34339}, {5434,34742}, {5690,26482}, {5692,24914}, {5787,12701}, {5794,7702}, {5883,20107}, {5902,11375}, {5904,41687}, {5905,18962}, {6001,6284}, {7288,18419}, {7294,25917}, {7962,10085}, {9652,14529}, {10039,35004}, {10052,12647}, {10914,41559}, {10949,12672}, {10956,37562}, {10957,24474}, {11501,37567}, {11509,12635}, {12053,17638}, {12666,37001}, {12711,41537}, {12736,17606}, {13601,34689}, {14110,15326}, {14740,37829}, {14882,34772}, {17605,31870}, {17619,18254}, {18395,31835}, {21740,22775}, {26481,39542}, {30144,34880}, {37563,37736}

X(45288) = barycentric product X(i)*X(j) for these (i,j): (226,30144), (321,34880)
X(45288) = trilinear product X(i)*X(j) for these (i,j): (10,34880), (65,30144)
X(45288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65,72,40663), (65,3962,41538), (65,12709,3649), (226,4084,65), (960,18838,5433), (3057,17660,5882), (4018,12709,65), (12736,20117,17606)

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12th Vijay transforms: X(45289)-X(45297)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 24, 2021.

The Vijay 12th parallel transform, V12, is defined in the preamble just before X(43970) as follows: if U = u : v : w (barycentrics), then

V12(U) = v*w^2 - 3*u*w^2 + v^2*w + 2*u*v*w + u^2*w - 3*u*v^2 + u^2*v : :

Let KK denote the cubic that is the isotomic conjugate of K015. If X lies on infinity line or the Steiner circumellipse, then V12(X) lies on KK. The appearance of (i,j) in the following examples means that V12(X(i)) = X(j):

(30,45289), (512,44007), (513,44008), (514,44009), (519,20042), (522,45290), (523,44010), (524,45291), (525,45292), (527,45293), (690,45294), (900,48295).

X(45289) = 12TH VIJAY TRANSFORM OF X(30)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - a^6*b^2 + a^4*b^4 - 3*a^2*b^6 + 2*b^8 - a^6*c^2 - a^4*b^2*c^2 + 3*a^2*b^4*c^2 - b^6*c^2 + a^4*c^4 + 3*a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 - b^2*c^6 + 2*c^8) : :
X(45289) = 9 X[2] - 8 X[402], 3 X[2] - 4 X[1650], 5 X[2] - 4 X[1651], 7 X[2] - 4 X[3081], 7 X[2] - 8 X[11049], 21 X[2] - 20 X[15183], 15 X[2] - 16 X[15184], 11 X[2] - 8 X[34582], 3 X[20] - 4 X[35241], 3 X[376] - 2 X[18508], 2 X[402] - 3 X[1650], 10 X[402] - 9 X[1651], 14 X[402] - 9 X[3081], 4 X[402] - 3 X[4240], 7 X[402] - 9 X[11049], 4 X[402] - 9 X[11050], 14 X[402] - 15 X[15183], 5 X[402] - 6 X[15184], 11 X[402] - 9 X[34582], 5 X[631] - 4 X[32162], 5 X[1650] - 3 X[1651], 7 X[1650] - 3 X[3081], 7 X[1650] - 6 X[11049], 2 X[1650] - 3 X[11050], 7 X[1650] - 5 X[15183], 5 X[1650] - 4 X[15184], 11 X[1650] - 6 X[34582], 7 X[1651] - 5 X[3081], 6 X[1651] - 5 X[4240], 7 X[1651] - 10 X[11049], 2 X[1651] - 5 X[11050], 21 X[1651] - 25 X[15183], 3 X[1651] - 4 X[15184], 11 X[1651] - 10 X[34582], 6 X[3081] - 7 X[4240], 2 X[3081] - 7 X[11050], 3 X[3081] - 5 X[15183], 15 X[3081] - 28 X[15184], 11 X[3081] - 14 X[34582], 7 X[3090] - 6 X[11911], 5 X[3091] - 4 X[11251], 5 X[3522] - 4 X[12113], 7 X[3523] - 6 X[11845], 3 X[3524] - 2 X[20128], 3 X[3543] - 4 X[18507], 5 X[3617] - 4 X[12438], 5 X[3620] - 4 X[12583], 7 X[3622] - 6 X[16212], 5 X[3623] - 4 X[12626], 7 X[4240] - 12 X[11049], X[4240] - 3 X[11050], 7 X[4240] - 10 X[15183], 5 X[4240] - 8 X[15184], 11 X[4240] - 12 X[34582], 13 X[5068] - 12 X[11897], 7 X[9780] - 6 X[11852], 13 X[10303] - 12 X[26451], 4 X[11049] - 7 X[11050], 6 X[11049] - 5 X[15183], 15 X[11049] - 14 X[15184], 11 X[11049] - 7 X[34582], 21 X[11050] - 10 X[15183], 15 X[11050] - 8 X[15184], 11 X[11050] - 4 X[34582], 25 X[15183] - 28 X[15184], 55 X[15183] - 42 X[34582], 22 X[15184] - 15 X[34582], 11 X[15717] - 8 X[15774], 12 X[16190] - 13 X[21734]

X(45289) lies on the cubic K(pending) and these lines: {2, 3}, {145, 11900}, {390, 11906}, {1294, 10733}, {1297, 31127}, {1503, 33988}, {2693, 44967}, {3448, 9033}, {3600, 11905}, {3617, 12438}, {3620, 12583}, {3621, 11910}, {3622, 16212}, {3623, 12626}, {5261, 18958}, {5274, 11909}, {8972, 44610}, {9780, 11852}, {10714, 14919}, {13941, 44611}, {14721, 44010}, {16075, 39358}, {35912, 43768}

X(45289) = reflection of X(i) in X(j) for these {i,j}: {2, 11050}, {3081, 11049}, {4240, 1650}, {39358, 16075}
X(45289) = anticomplement of X(4240)
X(45289) = de Longchamps circle inverse of X(7471)
X(45289) = anticomplement of the isogonal conjugate of X(14380)
X(45289) = anticomplement of the isotomic conjugate of X(34767)
X(45289) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {74, 7253}, {656, 146}, {810, 39358}, {1494, 21300}, {2159, 525}, {2349, 850}, {2394, 21270}, {2433, 5905}, {4575, 14611}, {14380, 8}, {14919, 7192}, {18808, 5906}, {18877, 4560}, {34767, 6327}, {35200, 523}, {36034, 110}, {36061, 41512}, {36119, 520}, {36131, 648}, {40352, 17498}
X(45289) = X(34767)-Ceva conjugate of X(2)
X(45289) = crosspoint of X(1494) and X(23582)
X(45289) = crosssum of X(1495) and X(3269)
X(45289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {550, 28144, 44891}, {1650, 1651, 15184}, {1650, 3081, 15183}, {1650, 4240, 2}, {1650, 15183, 11049}, {3448, 34186, 44003}, {3448, 44003, 44004}, {4240, 11050, 1650}

X(45290) = 12TH VIJAY TRANSFORM OF X(522)

Barycentrics    (a - b - c)*(b - c)*(7*a^4 - 7*a^3*b - 3*a^2*b^2 - a*b^3 + 4*b^4 - 7*a^3*c + 13*a^2*b*c + a*b^2*c - 7*b^3*c - 3*a^2*c^2 + a*b*c^2 + 6*b^2*c^2 - a*c^3 - 7*b*c^3 + 4*c^4) : :
X(45290) = 5 X[2] - 4 X[14476], 7 X[2] - 4 X[23615], 7 X[14476] - 5 X[23615]

X(45290) lies on the cubic K(pending) and these lines: {2, 522}, {145, 4025}, {3177, 17494}, {6366, 39357}, {29212, 34619}

X(45290) = anticomplement of the isogonal conjugate of X(23346)
X(45290) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {109, 5057}, {1055, 37781}, {1155, 33650}, {1323, 21293}, {1415, 527}, {2149, 30565}, {6610, 150}, {23346, 8}, {23890, 69}, {24027, 6366}, {36141, 1121}

X(45291) = 12TH VIJAY TRANSFORM OF X(524)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 3*b^2*c^2 + 2*c^4) : :
X(45291) = 5 X[2] - 4 X[1641], 3 X[2] - 4 X[1648], 7 X[2] - 4 X[8030], 9 X[2] - 8 X[11053], 11 X[2] - 8 X[38239], 3 X[1641] - 5 X[1648], 6 X[1641] - 5 X[5468], 7 X[1641] - 5 X[8030], 9 X[1641] - 10 X[11053], 11 X[1641] - 10 X[38239], 7 X[1648] - 3 X[8030], 3 X[1648] - 2 X[11053], 11 X[1648] - 6 X[38239], 7 X[5468] - 6 X[8030], 3 X[5468] - 4 X[11053], 11 X[5468] - 12 X[38239], 9 X[8030] - 14 X[11053], 11 X[8030] - 14 X[38239], 11 X[11053] - 9 X[38239], 4 X[14444] - X[20094], X[20080] - 4 X[44915]

X(45291) lies on the cubics K241 and K(pending) and these lines: {2, 6}, {148, 690}, {754, 35606}, {3564, 7417}, {5477, 7664}, {5642, 10552}, {5967, 41586}, {6787, 11002}, {6791, 40915}, {7665, 9143}, {8352, 36877}, {9140, 10754}, {10787, 20099}, {14061, 39024}, {14444, 20094}, {22253, 36194}, {31127, 36849}, {32244, 41909}, {33912, 42344}, {33921, 39356}, {34763, 35511}

X(45291) = reflection of X(5468) in X(1648)
X(45291) = anticomplement of X(5468)
X(45291) = orthoptic-circle of the Steiner circumellipe inverse of X(5912)
X(45291) = anticomplement of the isogonal conjugate of X(9178)
X(45291) = anticomplement of the isotomic conjugate of X(5466)
X(45291) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {31, 44010}, {111, 7192}, {661, 14360}, {671, 17217}, {691, 21295}, {798, 8591}, {897, 512}, {923, 523}, {5466, 6327}, {8753, 7253}, {9178, 8}, {10097, 4329}, {17983, 21300}, {18023, 21305}, {23894, 69}, {24006, 34518}, {32729, 6758}, {32740, 4560}, {36045, 9146}, {36060, 6563}, {36085, 4576}, {36128, 850}, {36142, 99}, {43926, 17140}
X(45291) = X(i)-Ceva conjugate of X(j) for these (i,j): {671, 31644}, {892, 19598}, {5466, 2}, {42349, 2482}
X(45291) = X(i)-isoconjugate of X(j) for these (i,j): {923, 36953}, {14052, 36060}, {36142, 36955}
X(45291) = crosspoint of X(671) and X(4590)
X(45291) = crosssum of X(187) and X(3124)
X(45291) = crossdifference of every pair of points on line {512, 20976}
X(45291) = barycentric product X(i)*X(j) for these {i,j}: {524, 14061}, {690, 33799}, {2642, 33809}, {3266, 39024}, {14060, 44146}, {33803, 35522}
X(45291) = barycentric quotient X(i)/X(j) for these {i,j}: {468, 14052}, {524, 36953}, {690, 36955}, {14060, 895}, {14061, 671}, {33799, 892}, {33803, 691}, {39024, 111}
X(45291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 20080, 38940}, {69, 6792, 2}, {1648, 5468, 2}, {6189, 6190, 44398}, {9140, 10754, 31125}, {39365, 39366, 44373}

X(45292) = 12TH VIJAY TRANSFORM OF X(525)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(7*a^8 - 7*a^6*b^2 - 3*a^4*b^4 - a^2*b^6 + 4*b^8 - 7*a^6*c^2 + 13*a^4*b^2*c^2 + a^2*b^4*c^2 - 7*b^6*c^2 - 3*a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 7*b^2*c^6 + 4*c^8) : :
X(45292) = 3 X[2] - 4 X[14401], 7 X[2] - 4 X[23616], 11 X[2] - 8 X[38240], 7 X[14401] - 3 X[23616], 11 X[14401] - 6 X[38240], 6 X[23616] - 7 X[34767], 11 X[23616] - 14 X[38240], 11 X[34767] - 12 X[38240]

X(45292) lies on the cubic K(pending) and these lines: {2, 525}, {193, 9007}, {520, 11002}, {3164, 7712}, {7665, 14697}, {7693, 17035}, {9033, 9143}, {9979, 40867}

X(45292) = reflection of X(34767) in X(14401)
X(45292) = anticomplement of X(34767)
X(45292) = anticomplement of the isogonal conjugate of X(23347)
X(45292) = anticomplement of the isotomic conjugate of X(4240)
X(45292) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1990, 21294}, {2173, 13219}, {2420, 4329}, {4240, 6327}, {9406, 39352}, {14581, 21221}, {23347, 8}, {24001, 315}, {24019, 340}, {32676, 30}, {36131, 1494}
X(45292) = X(4240)-Ceva conjugate of X(2)
X(45292) = crosspoint of X(648) and X(31621)
X(45292) = crosssum of X(647) and X(9408)
X(45292) = barycentric quotient X(2420)/X(15410)
X(45292) = {X(14401),X(34767)}-harmonic conjugate of X(2)

X(45293) = 12TH VIJAY TRANSFORM OF X(527)

Barycentrics    (2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2)*(a^4 - a^3*b + a^2*b^2 - 3*a*b^3 + 2*b^4 - a^3*c - a^2*b*c + 3*a*b^2*c - b^3*c + a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 - 3*a*c^3 - b*c^3 + 2*c^4) : :
X(45293) = 5 X[2] - 4 X[14477], 3 X[2] - 4 X[33573], 3 X[14477] - 5 X[33573]

X(45293) lies on the cubic K(pending) and these lines: {2, 7}, {149, 6366}, {14732, 44009}, {17036, 20042}, {24703, 43989}, {31058, 36101}

X(45293) = anticomplement of the isogonal conjugate of X(23351)
X(45293) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1156, 21302}, {2291, 693}, {3063, 39357}, {4845, 513}, {18889, 514}, {23351, 8}, {23893, 69}, {34068, 522}, {35348, 3434}, {36141, 664}, {41798, 20295}
X(45293) = X(1121)-Ceva conjugate of X(31648)
X(45293) = X(34068)-isoconjugate of X(36956)
X(45293) = crosspoint of X(1121) and X(1275)
X(45293) = crosssum of X(1055) and X(14936)
X(45293) = barycentric product X(527)*X(31640)
X(45293) = barycentric quotient X(i)/X(j) for these {i,j}: {527, 36956}, {31640, 1121}

X(45294) = 12TH VIJAY TRANSFORM OF X(690)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(7*a^8 - 14*a^6*b^2 + 9*a^4*b^4 - 2*a^2*b^6 + 4*b^8 - 14*a^6*c^2 + 24*a^4*b^2*c^2 - 12*a^2*b^4*c^2 - 14*b^6*c^2 + 9*a^4*c^4 - 12*a^2*b^2*c^4 + 27*b^4*c^4 - 2*a^2*c^6 - 14*b^2*c^6 + 4*c^8) : :

X(45294) lies on the cubic K(pending) and these lines: {2, 690}, {523, 5468}, {524, 5466}, {1641, 9168}, {33919, 44010}

X(45294) = reflection of X(9168) in X(1641)
X(45294) = anticomplement of X(34763)
X(45294) = anticomplement of the isogonal conjugate of X(23348)
X(45294) = anticomplement of the isotomic conjugate of X(34760)
X(45294) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {923, 44373}, {17948, 21294}, {17955, 3448}, {17964, 21221}, {23348, 8}, {34760, 6327}, {36142, 543}
X(45294) = X(34760)-Ceva conjugate of X(2)

X(45295) = 12TH VIJAY TRANSFORM OF X(900)

Barycentrics    (2*a - b - c)*(b - c)*(7*a^4 - 14*a^3*b + 9*a^2*b^2 - 2*a*b^3 + 4*b^4 - 14*a^3*c + 24*a^2*b*c - 12*a*b^2*c - 14*b^3*c + 9*a^2*c^2 - 12*a*b*c^2 + 27*b^2*c^2 - 2*a*c^3 - 14*b*c^3 + 4*c^4) : :

X(45295) lies on the cubic K(pending) and these lines: {2, 900}, {514, 17780}, {519, 6548}, {1644, 31992}, {6550, 44009}, {20042, 33922}

X(45295) = reflection of X(31992) in X(1644)
X(45295) = anticomplement of X(34764)
X(45295) = anticomplement of the isotomic conjugate of X(34762)
X(45295) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {32665, 545}, {34762, 6327}
X(45295) = X(34762)-Ceva conjugate of X(2)

X(45296) = 12TH VIJAY TRANSFORM OF X(6189)

Barycentrics    (b^2 - c^2)*(a^4 - a^2*b^2 + 4*b^4 - a^2*c^2 - 7*b^2*c^2 + 4*c^4 + 4*(2*a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : :
X(45296) = 3 X[2] - 4 X[13636]

X(45296) lies on the cubic K(pending) and these lines: {2, 1340}, {148, 690}, {523, 39366}

X(45296) = reflection of X(30508) in X(13636)
X(45296) = anticomplement of X(30508)
X(45296) = anticomplement of the isogonal conjugate of X(41880)
X(45296) = anticomplement of the isotomic conjugate of X(30509)
X(45296) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 6190}, {2028, 21221}, {30509, 6327}, {39023, 21294}, {41880, 8}
X(45296) = X(30509)-Ceva conjugate of X(2)
X(45296) = crosssum of X(2028) and X(5639)
X(45296) = crossdifference of every pair of points on line {5638, 20976}
X(45296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13636, 30508, 2}, {39302, 39303, 35914}, {39306, 39307, 1349}

X(45297) = 12TH VIJAY TRANSFORM OF X(6190)

Barycentrics    (b^2-c^2)*(a^4-(b^2+c^2)*a^2+4*b^4-7*b^2*c^2+4*c^4-4*(2*a^2-b^2-c^2)*sqrt(a^4-b^2*a^2-c^2*a^2+b^4-b^2*c^2+c^4)) : :
X(45297) = 3 X[2] - 4 X[13722]

X(45297) lies on the cubic K(pending) and these lines: {2, 1341}, {148, 690}, {523, 39365}

X(45297) = reflection of X(30509) in X(13722)
X(45297) = anticomplement of X(30509)
X(45297) = anticomplement of the isogonal conjugate of X(41881)
X(45297) = anticomplement of the isotomic conjugate of X(30508)
X(45297) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 6189}, {2029, 21221}, {30508, 6327}, {39022, 21294}, {41881, 8}
X(45297) = X(30508)-Ceva conjugate of X(2)
X(45297) = crosssum of X(2029) and X(5638)
X(45297) = crossdifference of every pair of points on line {5639, 20976}
X(45297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13722, 30509, 2}, {39158, 39159, 31863}, {39300, 39301, 35913}, {39304, 39305, 1348}

X(45298) = X(2)X(3167)∩X(5)X(1181)

Barycentrics    2*a^6-3*(b^2+c^2)*a^4+2*(b^4-4*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(45298) = 2*X(3)+X(13142), 2*X(5)+X(18914), 2*X(140)+X(13292), 3*X(373)-2*X(10128), 2*X(389)+X(12362), X(428)-3*X(5640), 5*X(632)+X(32358), X(1885)+5*X(10574), X(2979)-3*X(43957), X(3575)-7*X(15043), 4*X(3589)-X(13562), 5*X(3618)+X(26926), 4*X(3628)-X(31831), 2*X(3628)+X(43588), 4*X(5462)-X(6756), X(6146)+2*X(9825), 3*X(7998)+X(41628), 2*X(9729)+X(12241), 4*X(9729)-X(31829), X(10691)+4*X(32068), X(31831)+2*X(43588)

See Antreas Hatzipolakis and César Lozada, euclid 2819.

X(45298) lies on these lines: {2, 3167}, {3, 11433}, {5, 1181}, {6, 1368}, {20, 44935}, {30, 51}, {54, 22663}, {68, 15805}, {69, 16419}, {125, 37649}, {140, 343}, {154, 11179}, {182, 6676}, {184, 6677}, {373, 10128}, {381, 5644}, {389, 12362}, {394, 1353}, {427, 5422}, {428, 5640}, {440, 37510}, {441, 3398}, {468, 5012}, {511, 10691}, {524, 3819}, {542, 6688}, {546, 11550}, {549, 37506}, {550, 33586}, {567, 10257}, {575, 5159}, {576, 10300}, {578, 16196}, {597, 23332}, {632, 32358}, {858, 34545}, {1351, 7386}, {1352, 17825}, {1370, 9777}, {1503, 5943}, {1853, 14561}, {1885, 10574}, {1993, 30739}, {2072, 15037}, {2979, 43957}, {3060, 7667}, {3066, 31383}, {3448, 37990}, {3527, 34938}, {3530, 41586}, {3546, 11426}, {3575, 15043}, {3580, 7499}, {3589, 11548}, {3618, 23291}, {3628, 15806}, {3740, 5849}, {3742, 5848}, {3796, 10154}, {3917, 7734}, {5020, 6776}, {5064, 38136}, {5133, 15018}, {5268, 39897}, {5272, 39873}, {5305, 42295}, {5462, 6756}, {5544, 5921}, {5576, 15047}, {5890, 34664}, {5892, 43573}, {6146, 9825}, {6353, 39871}, {6515, 7484}, {6642, 31804}, {6643, 11432}, {6803, 12429}, {6804, 12164}, {6823, 37514}, {6997, 39884}, {7392, 18440}, {7393, 18951}, {7395, 18916}, {7398, 39874}, {7399, 18912}, {7404, 26944}, {7405, 25738}, {7485, 37644}, {7494, 12017}, {7514, 44683}, {7542, 13353}, {7998, 41628}, {8263, 32621}, {8550, 9306}, {8780, 40132}, {8964, 43118}, {9729, 12241}, {9909, 25406}, {10109, 23515}, {10127, 13363}, {10565, 21970}, {11064, 13366}, {11427, 30771}, {11430, 16976}, {11442, 18358}, {11479, 18909}, {11585, 36753}, {11745, 44829}, {12006, 30522}, {12007, 34986}, {12022, 15045}, {12100, 39242}, {12605, 37481}, {13160, 43816}, {13336, 16197}, {13340, 43934}, {13361, 35283}, {13488, 40647}, {13568, 15012}, {13857, 34566}, {14389, 26913}, {14516, 15028}, {14788, 43808}, {14826, 39899}, {14853, 34609}, {15024, 34224}, {15053, 37931}, {16238, 32046}, {16239, 32165}, {16625, 44862}, {18475, 37935}, {19717, 37050}, {20965, 21531}, {21530, 37509}, {21849, 29181}, {22352, 32269}, {23195, 44221}, {23293, 37454}, {26005, 37527}, {26879, 43651}, {31255, 37645}, {32166, 43839}, {34002, 37471}, {35266, 44108}, {37475, 44241}

X(45298) = midpoint of X(i) and X(j) for these {i, j}: {2, 11245}, {20, 44935}, {3060, 7667}, {3819, 11225}, {5890, 34664}, {5892, 43573}
X(45298) = reflection of X(i) in X(j) for these (i, j): (3917, 7734), (10127, 13363)
X(45298) = isogonal conjugate of X(45299)
X(45298) = complement of complement of X(45968)
X(45298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14912, 3167), (3, 11433, 41588), (182, 13567, 6676), (184, 37648, 6677), (343, 43650, 140), (427, 5422, 18583), (1370, 9777, 21850), (1899, 10601, 5), (3589, 21243, 11548), (3628, 43588, 31831), (5422, 18911, 427), (6643, 11432, 31802), (6776, 18928, 5020), (9729, 12241, 31829), (11442, 37439, 18358), (13336, 41587, 16197), (37514, 39571, 6823)

X(45299) = ISOGONAL CONJUGATE OF X(45298)

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

See Antreas Hatzipolakis and César Lozada, euclid 2819.

X(45299) lies on this line: {1351, 3567}

X(45299) = isogonal conjugate of X(45298)
X(45299) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(26206)}} and {{A, B, C, X(6), X(1007)}}

X(45300) = ISOGONAL CONJUGATE OF X(9729)

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

See Antreas Hatzipolakis and César Lozada, euclid 2819.

Let A'B'C' be the midheight triangle. Let BA and CA be the orthogonal projections of B' and C' on line BC, resp. Let (OA) be the circle with segment BACA as diameter, and define (OB) and (OC) cyclically. X(45300) is the radical center of circles (OA), (OB), (OC). (Randy Hutson, November 30, 2021)

X(45300) lies on Kiepert circumhyperbola and these lines: {2, 11424}, {3, 37874}, {4, 5065}, {5, 801}, {6, 13380}, {76, 11479}, {83, 6823}, {98, 12241}, {235, 275}, {485, 6814}, {486, 6812}, {1593, 2052}, {3091, 43670}, {3424, 18945}, {10982, 13599}, {11413, 34289}, {14853, 31363}, {16657, 40448}

X(45300) = isogonal conjugate of X(9729)
X(45300) = intersection, other than A, B, C, of circumconics Kiepert hyperbola and {{A, B, C, X(3), X(1593)}}

X(45301) = ISOGONAL CONJUGATE OF X(12241)

Barycentrics    a^2*(a^10-(3*b^2+2*c^2)*a^8+2*(b^4+4*b^2*c^2+c^4)*a^6+2*(b^6-2*c^6-(6*b^2+c^2)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(3*b^2-5*c^2)*a^2+(b^4+b^2*c^2+2*c^4)*(b^2-c^2)^3)*(a^10-(2*b^2+3*c^2)*a^8+2*(b^4+4*b^2*c^2+c^4)*a^6-2*(2*b^6-c^6+(b^2+6*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(5*b^2-3*c^2)*a^2-(2*b^4+b^2*c^2+c^4)*(b^2-c^2)^3) : :
Barycentrics    (SB+SC)*((SA+SB)*S^4-2*SA^2*SB^2*SC)*((SA+SC)*S^4-2*SA^2*SB*SC^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 2819.

X(45301) lies on these lines: {97, 22467}, {394, 5889}, {6803, 14376}, {9729, 17974}, {14118, 31626}

X(45301) = isogonal conjugate of X(12241)
X(45301) = Cevapoint of X(3) and X(389)
X(45301) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(3)}} and {{A, B, C, X(4), X(17928)}}

X(45302) = ISOGONAL CONJUGATE OF X(31829)

Barycentrics    a^2*(a^8-(4*b^2-c^2)*a^6+(6*b^4-b^2*c^2-3*c^4)*a^4-(4*b^6+c^6+(b^2-14*c^2)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(b^2+2*c^2))*(a^2+b^2-c^2)*(a^8+(-4*c^2+b^2)*a^6-(3*b^4+b^2*c^2-6*c^4)*a^4-(b^6+4*c^6-(14*b^2-c^2)*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) : :
Barycentrics    SB*SC*(SB+SC)*(S^4+(SA*SC+SB^2-SW^2)*S^2-4*SA^2*SC^2) *(S^4+(SA*SB+SC^2-SW^2)*S^2-4*SA^2*SB^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 2819.

X(45302) lies on Jerabek circumhyperbola and these lines: {68, 12897}, {265, 13488}, {1177, 22967}, {1593, 6391}, {1596, 3521}, {3089, 4846}, {6353, 15740}, {11744, 13568}, {14861, 21841}, {38263, 39588}

X(45302) = isogonal conjugate of X(31829)
X(45302) = intersection, other than A, B, C, of circumconics Jerabek hyperbola and {{A, B, C, X(20), X(3425)}}

X(45303) = X(2)X(154)∩X(5)X(113)

Barycentrics    (b^2+c^2)*a^4+2*(b^4+c^4)*a^2-3*(b^4-c^4)*(b^2-c^2) : :
X(45303) = 2*X(140)+X(34514), X(343)+2*X(427), X(343)-4*X(21243), X(427)+2*X(21243), 7*X(3090)-X(11456), 2*X(6676)+X(11550), 2*X(6800)-3*X(13394), X(11442)+2*X(23292), X(11442)+5*X(31236), 2*X(23292)-5*X(31236), X(31723)+2*X(44201), 4*X(34573)-X(35707)

See Antreas Hatzipolakis and César Lozada, euclid 2823.

X(45303) lies on these lines: {2, 154}, {3, 44836}, {4, 32269}, {5, 113}, {51, 38136}, {140, 34513}, {141, 858}, {182, 37454}, {343, 427}, {381, 20192}, {394, 8889}, {403, 16261}, {468, 3818}, {546, 34417}, {568, 5576}, {597, 9140}, {1209, 23335}, {1235, 36793}, {1350, 31099}, {1352, 5094}, {1368, 5650}, {1370, 31884}, {1495, 39884}, {1514, 11472}, {1594, 11459}, {1899, 5050}, {2854, 12827}, {3066, 3091}, {3090, 11456}, {3146, 15431}, {3448, 8550}, {3541, 37497}, {3549, 16655}, {3580, 5169}, {3589, 18911}, {3628, 13336}, {3629, 41724}, {3630, 23061}, {3815, 8288}, {3843, 21970}, {3845, 32225}, {4550, 10297}, {5068, 43592}, {5079, 5544}, {5102, 6515}, {5133, 5640}, {5159, 5651}, {5254, 39691}, {5449, 7403}, {5894, 13203}, {5972, 18553}, {6146, 37506}, {6247, 13160}, {6388, 39565}, {6676, 11550}, {6697, 15126}, {6815, 40686}, {6997, 26958}, {7378, 33586}, {7399, 16836}, {7493, 36990}, {7495, 44882}, {7499, 17508}, {7552, 16658}, {7566, 11745}, {7569, 11457}, {7574, 35254}, {7730, 11660}, {10170, 11585}, {10193, 44268}, {10301, 32223}, {10601, 23291}, {11245, 39561}, {11440, 16775}, {11442, 23292}, {11548, 43650}, {12310, 14130}, {13371, 15067}, {14118, 41362}, {14561, 26869}, {14788, 23294}, {14852, 16657}, {14915, 15760}, {15035, 37118}, {15055, 23328}, {15059, 35904}, {15069, 37645}, {15082, 24206}, {16051, 40330}, {16264, 41203}, {17702, 44218}, {18474, 39242}, {18531, 32620}, {21850, 41586}, {22804, 44242}, {23300, 41603}, {23315, 23330}, {23325, 34664}, {26913, 37990}, {29012, 44210}, {29181, 31133}, {31074, 33884}, {31723, 44201}, {34573, 35707}, {34609, 43653}, {34826, 41587}, {37347, 40280}, {38727, 44273}, {40709, 41034}, {40710, 41035}, {43608, 43804}

X(45303) = midpoint of X(i) and X(j) for these {i, j}: {11550, 35268}, {15035, 44795}, {18474, 39242}, {34513, 34514}
X(45303) = reflection of X(i) in X(j) for these (i, j): (13394, 2), (34513, 140), (35268, 6676)
X(45303) = complement of X(6800)
X(45303) = center of circle {{X(12037), X(33504), X(35594)}}
X(45303) = X(16165)-of-orthocentroidal triangle
X(45303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 10516, 35283), (2, 32064, 3796), (4, 37638, 32269), (5, 125, 37648), (381, 44569, 20192), (427, 21243, 343), (1352, 5094, 11064), (3091, 37643, 3066), (3448, 14389, 8550), (3580, 5169, 5480), (5133, 23293, 13567), (5159, 18358, 5651), (5576, 12359, 45089), (11442, 31236, 23292)

X(45304) = X(2)X(3659)∩X(4)X(7597)

Barycentrics    Sin[B]^2/(Csc[A/2] - Csc[C/2]) - Sin[C]^2/(Csc[A/2] - Csc[B/2]) : :

See Er Jkh and Peter Moses, euclid 2860.

X(45304) lies on the nine-point circle and these lines: {2, 3659}, {4, 7597}, {5, 21633}, {11, 10501}, {12, 10506}, {178, 12622}, {226, 12814}, {8085, 13559}, {8086, 13560}

X(45304) = midpoint of X(4) and X(7597)
X(45304) = complement of X(3659)
X(45304) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 10492}, {10492, 34849}, {16011, 6728}

X(45305) = X(2)X(1742)∩X(4)X(9)

Barycentrics    a^3*b^2 - 2*a^2*b^3 + a*b^4 - 2*a^3*b*c + a^2*b^2*c + b^4*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 2*a^2*c^3 - b^2*c^3 + a*c^4 + b*c^4 : :
X(45305) = 3 X[1699] - X[10446], 3 X[3817] - 2 X[24220], 3 X[10164] - 2 X[41430]

See Er Jkh and Peter Moses, euclid 2860.

X(45305) lies on these lines: {1, 26125}, {2, 1742}, {4, 9}, {5, 3836}, {37, 28850}, {141, 42356}, {226, 21746}, {238, 13727}, {256, 2481}, {511, 946}, {515, 31394}, {551, 1064}, {639, 31555}, {640, 31556}, {752, 36722}, {894, 9355}, {971, 24325}, {990, 16825}, {991, 1125}, {1210, 41246}, {1211, 7965}, {1215, 5927}, {1441, 2310}, {1699, 3741}, {1709, 3980}, {1716, 3008}, {1721, 4384}, {2347, 13576}, {2792, 6033}, {2808, 17049}, {2887, 8226}, {2951, 16832}, {3000, 17077}, {3062, 25590}, {3091, 3831}, {3146, 31339}, {3454, 12558}, {3664, 11019}, {3667, 24182}, {3686, 28849}, {3739, 15726}, {3755, 4263}, {3817, 3840}, {3846, 8727}, {3912, 21299}, {3914, 23659}, {3993, 29016}, {4090, 15064}, {4192, 10164}, {4297, 9840}, {4301, 29311}, {4363, 16112}, {4388, 4416}, {4655, 5805}, {4871, 7988}, {5779, 32935}, {5851, 7228}, {9440, 14942}, {9441, 17277}, {9778, 26037}, {9779, 30942}, {9812, 31330}, {9956, 29229}, {10175, 29349}, {10868, 24547}, {10883, 25760}, {10916, 17770}, {11362, 29309}, {11495, 17259}, {12723, 16609}, {14058, 25639}, {15587, 34852}, {16850, 38059}, {17050, 43168}, {17330, 28854}, {17331, 25006}, {17332, 38454}, {17362, 28870}, {17364, 26015}, {17447, 23774}, {19541, 32916}, {20258, 20544}, {20540, 20547}, {21084, 30807}, {21620, 39543}, {24333, 29655}, {31211, 43151}, {32784, 36652}, {32917, 36002}, {36991, 39581}, {37193, 40998}, {40718, 43672}

X(45305) = midpoint of X(4) and X(6210)
X(45305) = reflection of X(991) in X(1125)
X(45305) = complement of X(1742)
X(45305) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 40593}, {43750, 2886}
X(45305) = barycentric product X(10)*X(26802)
X(45305) = barycentric quotient X(26802)/X(86)

X(45306) = 3RD HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    a (a^18-b a^17-c a^17-4 b^2 a^16-4 c^2 a^16+4 b^3 a^15+4 c^3 a^15+3 b c^2 a^15+3 b^2 c a^15+4 b^4 a^14+4 c^4 a^14+2 b^2 c^2 a^14-4 b^5 a^13-4 c^5 a^13-b c^4 a^13+2 b^2 c^3 a^13+2 b^3 c^2 a^13-b^4 c a^13+4 b^6 a^12+4 c^6 a^12+20 b^2 c^4 a^12+20 b^4 c^2 a^12-4 b^7 a^11-4 c^7 a^11-5 b c^6 a^11-24 b^2 c^5 a^11-24 b^3 c^4 a^11-24 b^4 c^3 a^11-24 b^5 c^2 a^11-5 b^6 c a^11-10 b^8 a^10-10 c^8 a^10-23 b^2 c^6 a^10-53 b^4 c^4 a^10-23 b^6 c^2 a^10+10 b^9 a^9+10 c^9 a^9+5 b c^8 a^9+19 b^2 c^7 a^9+24 b^3 c^6 a^9+42 b^4 c^5 a^9+42 b^5 c^4 a^9+24 b^6 c^3 a^9+19 b^7 c^2 a^9+5 b^8 c a^9+4 b^10 a^8+4 c^10 a^8-9 b^2 c^8 a^8+49 b^4 c^6 a^8+49 b^6 c^4 a^8-9 b^8 c^2 a^8-4 b^11 a^7-4 c^11 a^7+b c^10 a^7+19 b^2 c^9 a^7-9 b^3 c^8 a^7-34 b^4 c^7 a^7-19 b^5 c^6 a^7-19 b^6 c^5 a^7-34 b^7 c^4 a^7-9 b^8 c^3 a^7+19 b^9 c^2 a^7+b^10 c a^7+4 b^12 a^6+4 c^12 a^6+20 b^2 c^10 a^6-31 b^4 c^8 a^6-6 b^6 c^6 a^6-31 b^8 c^4 a^6+20 b^10 c^2 a^6-4 b^13 a^5-4 c^13 a^5-3 b c^12 a^5-24 b^2 c^11 a^5+9 b^3 c^10 a^5+42 b^4 c^9 a^5-b^5 c^8 a^5-19 b^6 c^7 a^5-19 b^7 c^6 a^5-b^8 c^5 a^5+42 b^9 c^4 a^5+9 b^10 c^3 a^5-24 b^11 c^2 a^5-3 b^12 c a^5-4 b^14 a^4-4 c^14 a^4-6 b^2 c^12 a^4+8 b^4 c^10 a^4+2 b^6 c^8 a^4+2 b^8 c^6 a^4+8 b^10 c^4 a^4-6 b^12 c^2 a^4+4 b^15 a^3+4 c^15 a^3+b c^14 a^3+2 b^2 c^13 a^3-7 b^3 c^12 a^3-24 b^4 c^11 a^3+9 b^5 c^10 a^3+24 b^6 c^9 a^3-9 b^7 c^8 a^3-9 b^8 c^7 a^3+24 b^9 c^6 a^3+9 b^10 c^5 a^3-24 b^11 c^4 a^3-7 b^12 c^3 a^3+2 b^13 c^2 a^3+b^14 c a^3+b^16 a^2+c^16 a^2+b^2 c^14 a^2-2 b^4 c^12 a^2-17 b^6 c^10 a^2+34 b^8 c^8 a^2-17 b^10 c^6 a^2-2 b^12 c^4 a^2+b^14 c^2 a^2-b^17 a-c^17 a+3 b^2 c^15 a+b^3 c^14 a-b^4 c^13 a-3 b^5 c^12 a-5 b^6 c^11 a+b^7 c^10 a+5 b^8 c^9 a+5 b^9 c^8 a+b^10 c^7 a-5 b^11 c^6 a-3 b^12 c^5 a-b^13 c^4 a+b^14 c^3 a+3 b^15 c^2 a-b^2 c^16+5 b^4 c^14-9 b^6 c^12+5 b^8 c^10+5 b^10 c^8-9 b^12 c^6+5 b^14 c^4-b^16 c^2)-a (a-b-c) (a+b-c) (a-b+c) (b+c) (a+b+c) (a^4-b^2 a^2-c^2 a^2+b c a^2-b c^3+2 b^2 c^2-b^3 c) (a^6-b^2 a^4-c^2 a^4-b^4 a^2-c^4 a^2-9 b^2 c^2 a^2+b^6+c^6-b^2 c^4-b^4 c^2) T : : , where T = 2S·OH

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2885.

X(45306) lies on this line: {2, 3}

X(45307) = 4TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    a (a^18-b a^17-c a^17-4 b^2 a^16-4 c^2 a^16+4 b^3 a^15+4 c^3 a^15+3 b c^2 a^15+3 b^2 c a^15+4 b^4 a^14+4 c^4 a^14+2 b^2 c^2 a^14-4 b^5 a^13-4 c^5 a^13-b c^4 a^13+2 b^2 c^3 a^13+2 b^3 c^2 a^13-b^4 c a^13+4 b^6 a^12+4 c^6 a^12+20 b^2 c^4 a^12+20 b^4 c^2 a^12-4 b^7 a^11-4 c^7 a^11-5 b c^6 a^11-24 b^2 c^5 a^11-24 b^3 c^4 a^11-24 b^4 c^3 a^11-24 b^5 c^2 a^11-5 b^6 c a^11-10 b^8 a^10-10 c^8 a^10-23 b^2 c^6 a^10-53 b^4 c^4 a^10-23 b^6 c^2 a^10+10 b^9 a^9+10 c^9 a^9+5 b c^8 a^9+19 b^2 c^7 a^9+24 b^3 c^6 a^9+42 b^4 c^5 a^9+42 b^5 c^4 a^9+24 b^6 c^3 a^9+19 b^7 c^2 a^9+5 b^8 c a^9+4 b^10 a^8+4 c^10 a^8-9 b^2 c^8 a^8+49 b^4 c^6 a^8+49 b^6 c^4 a^8-9 b^8 c^2 a^8-4 b^11 a^7-4 c^11 a^7+b c^10 a^7+19 b^2 c^9 a^7-9 b^3 c^8 a^7-34 b^4 c^7 a^7-19 b^5 c^6 a^7-19 b^6 c^5 a^7-34 b^7 c^4 a^7-9 b^8 c^3 a^7+19 b^9 c^2 a^7+b^10 c a^7+4 b^12 a^6+4 c^12 a^6+20 b^2 c^10 a^6-31 b^4 c^8 a^6-6 b^6 c^6 a^6-31 b^8 c^4 a^6+20 b^10 c^2 a^6-4 b^13 a^5-4 c^13 a^5-3 b c^12 a^5-24 b^2 c^11 a^5+9 b^3 c^10 a^5+42 b^4 c^9 a^5-b^5 c^8 a^5-19 b^6 c^7 a^5-19 b^7 c^6 a^5-b^8 c^5 a^5+42 b^9 c^4 a^5+9 b^10 c^3 a^5-24 b^11 c^2 a^5-3 b^12 c a^5-4 b^14 a^4-4 c^14 a^4-6 b^2 c^12 a^4+8 b^4 c^10 a^4+2 b^6 c^8 a^4+2 b^8 c^6 a^4+8 b^10 c^4 a^4-6 b^12 c^2 a^4+4 b^15 a^3+4 c^15 a^3+b c^14 a^3+2 b^2 c^13 a^3-7 b^3 c^12 a^3-24 b^4 c^11 a^3+9 b^5 c^10 a^3+24 b^6 c^9 a^3-9 b^7 c^8 a^3-9 b^8 c^7 a^3+24 b^9 c^6 a^3+9 b^10 c^5 a^3-24 b^11 c^4 a^3-7 b^12 c^3 a^3+2 b^13 c^2 a^3+b^14 c a^3+b^16 a^2+c^16 a^2+b^2 c^14 a^2-2 b^4 c^12 a^2-17 b^6 c^10 a^2+34 b^8 c^8 a^2-17 b^10 c^6 a^2-2 b^12 c^4 a^2+b^14 c^2 a^2-b^17 a-c^17 a+3 b^2 c^15 a+b^3 c^14 a-b^4 c^13 a-3 b^5 c^12 a-5 b^6 c^11 a+b^7 c^10 a+5 b^8 c^9 a+5 b^9 c^8 a+b^10 c^7 a-5 b^11 c^6 a-3 b^12 c^5 a-b^13 c^4 a+b^14 c^3 a+3 b^15 c^2 a-b^2 c^16+5 b^4 c^14-9 b^6 c^12+5 b^8 c^10+5 b^10 c^8-9 b^12 c^6+5 b^14 c^4-b^16 c^2)-a (a-b-c) (a+b-c) (a-b+c) (b+c) (a+b+c) (a^4-b^2 a^2-c^2 a^2+b c a^2-b c^3+2 b^2 c^2-b^3 c) (a^6-b^2 a^4-c^2 a^4-b^4 a^2-c^4 a^2-9 b^2 c^2 a^2+b^6+c^6-b^2 c^4-b^4 c^2) (-T) : : , where T = 2S·OH

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2885.

X(45307) lies on this line: {2, 3}

X(45308) = 5TH HATZIPOLAKIS-GARCÍA CAPITÁN-EULER POINT

Barycentrics    a^2 (a^8 - 2 a^6 b^2 + 2 a^2 b^6 - b^8 - 2 a^6 c^2 - 9 a^4 b^2 c^2 + 12 a^2 b^4 c^2 - b^6 c^2 + 12 a^2 b^2 c^4 + 4 b^4 c^4 + 2 a^2 c^6 - b^2 c^6 - c^8) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 2885.

X(45308) lies on these lines: {2, 3}, {323, 5447}, {524, 2889}, {1181, 21766}, {1199, 10627}, {1994, 13336}, {2979, 37515}, {3098, 15043}, {5092, 34148}, {5562, 41462}, {5889, 13347}, {7592, 33884}, {7691, 16836}, {7998, 10984}, {7999, 43605}, {8718, 10170}, {10625, 34545}, {11002, 15805}, {11592, 22115}, {11695, 15107}, {12112, 14128}, {13348, 13434}, {15574, 32835}, {17508, 43652}

X(45308) = midpoint of X(45306) and X(45307)

X(45309) = MIDPOINT OF X(3) AND X(14854)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^20 - 8*a^18*b^2 + 8*a^16*b^4 + 8*a^14*b^6 - 22*a^12*b^8 + 20*a^10*b^10 - 22*a^8*b^12 + 32*a^6*b^14 - 28*a^4*b^16 + 12*a^2*b^18 - 2*b^20 - 8*a^18*c^2 + 34*a^16*b^2*c^2 - 48*a^14*b^4*c^2 + 15*a^12*b^6*c^2 + 13*a^10*b^8*c^2 + 30*a^8*b^10*c^2 - 102*a^6*b^12*c^2 + 115*a^4*b^14*c^2 - 63*a^2*b^16*c^2 + 14*b^18*c^2 + 8*a^16*c^4 - 48*a^14*b^2*c^4 + 84*a^12*b^4*c^4 - 49*a^10*b^6*c^4 - 39*a^8*b^8*c^4 + 132*a^6*b^10*c^4 - 181*a^4*b^12*c^4 + 135*a^2*b^14*c^4 - 42*b^16*c^4 + 8*a^14*c^6 + 15*a^12*b^2*c^6 - 49*a^10*b^4*c^6 + 68*a^8*b^6*c^6 - 62*a^6*b^8*c^6 + 137*a^4*b^10*c^6 - 147*a^2*b^12*c^6 + 72*b^14*c^6 - 22*a^12*c^8 + 13*a^10*b^2*c^8 - 39*a^8*b^4*c^8 - 62*a^6*b^6*c^8 - 86*a^4*b^8*c^8 + 63*a^2*b^10*c^8 - 84*b^12*c^8 + 20*a^10*c^10 + 30*a^8*b^2*c^10 + 132*a^6*b^4*c^10 + 137*a^4*b^6*c^10 + 63*a^2*b^8*c^10 + 84*b^10*c^10 - 22*a^8*c^12 - 102*a^6*b^2*c^12 - 181*a^4*b^4*c^12 - 147*a^2*b^6*c^12 - 84*b^8*c^12 + 32*a^6*c^14 + 115*a^4*b^2*c^14 + 135*a^2*b^4*c^14 + 72*b^6*c^14 - 28*a^4*c^16 - 63*a^2*b^2*c^16 - 42*b^4*c^16 + 12*a^2*c^18 + 14*b^2*c^18 - 2*c^20) : :
X(45309) = 3 X[15061] + X[34310]

See Kadir Altintas and Peter Moses, euclid 2902.

X(45309) lies on this line: {3, 125}

X(45309) = midpoint of X(3) and X(14854)



leftri

MIDPOINTS OF X(2) AND OTHER POINTS: X(45310)-X(45344) AND X(45657)-X(45693)

rightri

This preamble is contributed by Clark Kimberling and Peter Moses, October 31, 2021.

Suppose that P = p : q : r and U = u : v : w (barycentrics). Then the P-(B)line conjugate of the line L at infinity is the point

a*(v^2 + w^2) + (v + w)*(b*v + c*w) : :

The locus of this point as U ranges through L is the ellipse, E(P,L), given by

(q + r) x^2 + (r + p) y^2 + (p + q) z^2 - (q + r) y z - (r + p) z x - (p + q) x y = 0.

The center of E(P,L) is the midpoint of X(2) and P, with barycentrics

4 p + q + r : p + 4 q + r : p + q + 4r.

The perspector E(P,L) is 1/(p^2 + 2 q^2 + 2 r^2 + 5 q r + p q + p r) : : .

The major axis of E(P,L) is parallel to the major axis of the Steiner circumellipse, and the minor axis of E(P,L) is parallel to the minor axis of the Steiner circumellipse. Thus, the ellipse E(P,L) is simply and translation and dilation of the Steiner circumellipse.

X(45310) = MIDPOINT OF X(2) AND X(11)

Barycentrics    2*a^3 - 2*a^2*b - 5*a*b^2 + 5*b^3 - 2*a^2*c + 12*a*b*c - 5*b^2*c - 5*a*c^2 - 5*b*c^2 + 5*c^3 : :
X(45310) = X[1] - 3 X[38026], 5 X[2] - X[100], 7 X[2] + X[149], 11 X[2] - X[6154], 3 X[2] + X[10707], 17 X[2] - X[20095], 7 X[2] - 5 X[31235], X[2] - 5 X[31272], 7 X[2] - 2 X[35023], X[3] - 3 X[38069], X[4] - 3 X[38077], 2 X[5] + X[20418], X[5] - 3 X[38084], 4 X[5] - X[38757], X[6] - 3 X[38090], X[7] - 3 X[38095], X[8] - 3 X[38099], X[9] - 3 X[38102], X[10] - 3 X[38104], 5 X[11] + X[100], 7 X[11] - X[149], 2 X[11] + X[3035], 11 X[11] + X[6154], 3 X[11] + X[6174], X[11] + 2 X[6667], 3 X[11] - X[10707], 17 X[11] + X[20095], 7 X[11] + 5 X[31235], X[11] + 5 X[31272], 7 X[11] + 2 X[35023], X[12] - 3 X[38106], X[80] + 3 X[25055], 7 X[100] + 5 X[149], 2 X[100] - 5 X[3035], 11 X[100] - 5 X[6154], 3 X[100] - 5 X[6174], X[100] - 10 X[6667], 3 X[100] + 5 X[10707], 17 X[100] - 5 X[20095], 7 X[100] - 25 X[31235], X[100] - 25 X[31272], 7 X[100] - 10 X[35023], X[104] + 3 X[3545], X[119] - 3 X[5055], 2 X[149] + 7 X[3035], 11 X[149] + 7 X[6154], 3 X[149] + 7 X[6174], X[149] + 14 X[6667], 3 X[149] - 7 X[10707], 17 X[149] + 7 X[20095], X[149] + 5 X[31235], X[149] + 35 X[31272], X[149] + 2 X[35023], X[214] - 3 X[19883], X[376] - 3 X[21154], X[381] - 3 X[23513], 2 X[547] - 3 X[38319], 8 X[547] - 3 X[38758], X[549] - 3 X[34126], 3 X[551] - X[11274], X[551] - 3 X[32557],

X(45310) lies on these lines: {1, 38026}, {2, 11}, {3, 38069}, {4, 38077}, {5, 10199}, {6, 38090}, {7, 38095}, {8, 38099}, {9, 38102}, {10, 38104}, {12, 38106}, {30, 6713}, {80, 25055}, {104, 3545}, {119, 5055}, {214, 19883}, {376, 21154}, {381, 2829}, {499, 3847}, {519, 1387}, {527, 5087}, {529, 3582}, {535, 15325}, {545, 24318}, {547, 551}, {549, 5840}, {597, 5848}, {900, 4928}, {1125, 12019}, {1145, 19875}, {1317, 38314}, {1484, 10197}, {1537, 38021}, {1647, 37691}, {1656, 20400}, {2787, 5461}, {2801, 3742}, {2802, 3828}, {3086, 11236}, {3090, 37725}, {3241, 32558}, {3452, 38216}, {3524, 24466}, {3526, 10993}, {3543, 38693}, {3584, 5533}, {3616, 10031}, {3624, 10609}, {3628, 38629}, {3655, 38032}, {3679, 5854}, {3817, 38207}, {3825, 4999}, {3830, 38761}, {3845, 38602}, {3911, 28534}, {4193, 34606}, {4666, 41701}, {4677, 25416}, {4762, 10006}, {4870, 20118}, {4996, 16858}, {5054, 10738}, {5056, 38669}, {5067, 38665}, {5070, 38763}, {5071, 10711}, {5084, 13272}, {5121, 17070}, {5154, 34605}, {5219, 17051}, {5272, 9639}, {5298, 37375}, {5433, 11114}, {5851, 6173}, {6691, 7741}, {6931, 11240}, {7173, 17577}, {7288, 34620}, {8256, 34640}, {8703, 22938}, {9024, 20582}, {9780, 13996}, {10058, 16371}, {10090, 16370}, {10172, 10179}, {10200, 17528}, {10304, 10724}, {10427, 38093}, {10591, 34706}, {10593, 17564}, {10728, 41099}, {10742, 19709}, {10755, 21356}, {10769, 41134}, {11179, 38119}, {11539, 33814}, {11604, 15671}, {11681, 34749}, {12248, 41106}, {12736, 44663}, {12831, 25557}, {13199, 15709}, {14269, 38753}, {15017, 30291}, {15171, 20107}, {15684, 38754}, {15687, 38141}, {15694, 38636}, {15702, 34474}, {15703, 38752}, {16174, 28194}, {17100, 36006}, {17392, 17717}, {17530, 39692}, {17728, 31164}, {18861, 28461}, {19907, 38022}, {22110, 41144}, {22799, 38071}, {24841, 26136}, {25377, 40480}, {26127, 31260}, {30308, 34789}, {30827, 38211}, {31162, 38038}, {34123, 37718}, {34627, 38156}, {34641, 38213}, {34648, 38161}, {34718, 38128}, {35113, 35509}, {41553, 42819}

X(45310) = midpoint of X(i) and X(j) for these {i,j}: {2, 11}, {3582, 17533}, {3830, 38761}, {3845, 38602}, {4677, 25416}, {5298, 37375}, {6174, 10707}, {8703, 22938}, {16173, 34122}, {34123, 37718}
X(45310) = reflection of X(i) in X(j) for these {i,j}: {2, 6667}, {3035, 2}
X(45310) = complement of X(6174)
X(45310) = X(i)-complementary conjugate of X(j) for these (i,j): {88, 31844}, {106, 10427}, {1156, 121}, {2291, 16594}, {9456, 35110}, {34068, 4370}, {35348, 3259}
X(45310) = crossdifference of every pair of points on line {665, 21781}
X(45310) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 10707, 6174}, {5, 20418, 38757}, {11, 6174, 10707}, {11, 6667, 3035}, {11, 31235, 149}, {11, 31272, 6667}, {149, 31235, 35023}, {1387, 6702, 3036}, {1656, 37726, 20400}, {6702, 33709, 1387}, {31235, 35023, 3035}

X(45311) = MIDPOINT OF X(2) AND X(125)

Barycentrics    2*a^6 - 2*a^4*b^2 - 5*a^2*b^4 + 5*b^6 - 2*a^4*c^2 + 12*a^2*b^2*c^2 - 5*b^4*c^2 - 5*a^2*c^4 - 5*b^2*c^4 + 5*c^6 : :
X(45311) = 5 X[2] - X[110], 7 X[2] + X[3448], 3 X[2] + X[9140], 9 X[2] - X[9143], 17 X[2] - X[14683], X[2] - 5 X[15059], 11 X[2] - X[24981], X[4] + 5 X[38729], X[5] + 2 X[20397], 2 X[5] + X[20417], 4 X[5] - X[38791], X[20] + 11 X[15025], X[67] + 2 X[32300], X[74] + 3 X[3545], X[110] + 5 X[125], 7 X[110] + 5 X[3448], 3 X[110] - 5 X[5642], 2 X[110] - 5 X[5972], X[110] - 10 X[6723], 3 X[110] + 5 X[9140], 9 X[110] - 5 X[9143], 17 X[110] - 5 X[14683], X[110] - 25 X[15059], 11 X[110] - 5 X[24981], X[113] - 3 X[5055], 7 X[125] - X[3448], 3 X[125] + X[5642], 2 X[125] + X[5972], X[125] + 2 X[6723], 3 X[125] - X[9140], 9 X[125] + X[9143], 17 X[125] + X[14683], X[125] + 5 X[15059], 11 X[125] + X[24981], X[140] + 2 X[20396], 2 X[140] + X[36253], X[265] + 3 X[5054], 3 X[373] - X[12824], X[376] + 3 X[14644], X[376] - 3 X[38727], 5 X[381] + 3 X[15041], X[381] + 3 X[15061], X[381] - 3 X[23515], 2 X[547] + 3 X[38725], 8 X[547] - 3 X[38792], X[549] - 3 X[34128], X[895] + 3 X[21356], X[1511] - 3 X[11539], X[1539] - 3 X[38071], 5 X[1656] - X[5655], 5 X[1656] + X[16003], X[1986] - 3 X[16226], 7 X[3090] - X[15063], 5 X[3091] + X[10990], 5 X[3091] + 7 X[15057], 3 X[3448] + 7 X[5642], 2 X[3448] + 7 X[5972], X[3448] + 14 X[6723], 3 X[3448] - 7 X[9140], 9 X[3448] + 7 X[9143], 17 X[3448] + 7 X[14683], X[3448] + 35 X[15059], 11 X[3448] + 7 X[24981], 5 X[3522] + 7 X[15044], 3 X[3524] + 5 X[15081], 3 X[3524] - X[16163], 7 X[3526] + 5 X[15027], 7 X[3526] - X[30714], 17 X[3533] - 5 X[15034], X[3534] - 5 X[38728], X[3543] + 3 X[15055], 5 X[3618] - X[41720], 7 X[3619] - X[32114], 5 X[3628] + X[13393], 4 X[3628] - X[16534], 2 X[3628] + X[20379], 10 X[3628] - X[38632], 5 X[3763] - X[5648], 7 X[3832] + 5 X[15021], 3 X[3839] - X[13202], 3 X[5032] + X[32244], 3 X[5055] + X[20126], 11 X[5056] + X[15054], 13 X[5067] - X[14094], 11 X[5070] - 5 X[38795], 5 X[5071] - X[10706], 5 X[5071] - 3 X[36518], X[5181] - 3 X[21358], 2 X[5449] + X[15115], X[5465] - 3 X[14971], 3 X[5622] + X[11180], 2 X[5642] - 3 X[5972], X[5642] - 6 X[6723], 3 X[5642] - X[9143], 17 X[5642] - 3 X[14683], X[5642] - 15 X[15059], 11 X[5642] - 3 X[24981], X[5907] + 2 X[16270], X[5972] - 4 X[6723], 3 X[5972] + 2 X[9140], 9 X[5972] - 2 X[9143], 17 X[5972] - 2 X[14683], X[5972] - 10 X[15059], 11 X[5972] - 2 X[24981],

X(45311) lies on these lines: {2, 98}, {4, 38729}, {5, 541}, {6, 40920}, {20, 15025}, {30, 6699}, {67, 15303}, {74, 3545}, {113, 5055}, {140, 20396}, {265, 5054}, {373, 12824}, {376, 14644}, {381, 2777}, {427, 20192}, {468, 11645}, {511, 12099}, {519, 11735}, {524, 5159}, {543, 11007}, {547, 5663}, {549, 17702}, {690, 5461}, {858, 19924}, {895, 21356}, {1511, 11539}, {1539, 38071}, {1656, 5655}, {1986, 16226}, {2781, 5943}, {2794, 34094}, {2799, 42736}, {2854, 15082}, {3090, 15063}, {3091, 10990}, {3522, 15044}, {3524, 15081}, {3526, 15027}, {3533, 15034}, {3534, 12295}, {3543, 15055}, {3580, 13857}, {3589, 10173}, {3618, 41720}, {3619, 32114}, {3628, 13393}, {3763, 5648}, {3819, 14984}, {3830, 16111}, {3832, 15021}, {3839, 13202}, {3845, 12041}, {3917, 45237}, {5020, 5621}, {5032, 32244}, {5056, 15054}, {5066, 15088}, {5067, 14094}, {5070, 38795}, {5071, 10706}, {5085, 45082}, {5092, 16165}, {5094, 5476}, {5095, 13169}, {5181, 21358}, {5449, 15115}, {5465, 14971}, {5644, 15131}, {5907, 16270}, {5965, 40112}, {6034, 6388}, {6053, 10264}, {6143, 43836}, {6640, 10112}, {6688, 41670}, {6722, 32525}, {7426, 29012}, {7667, 41674}, {7706, 39484}, {7728, 19709}, {8288, 10418}, {8703, 10113}, {9033, 11049}, {9144, 14061}, {9166, 11006}, {9530, 24930}, {9729, 15738}, {9880, 41254}, {10124, 32423}, {10304, 10733}, {10601, 15106}, {10721, 41099}, {10819, 43254}, {10820, 43255}, {10989, 29317}, {11284, 32305}, {11477, 31856}, {11656, 38224}, {11693, 38794}, {11695, 25711}, {11720, 19883}, {11746, 21849}, {11800, 13416}, {11801, 12100}, {12121, 15693}, {12244, 41106}, {12383, 15709}, {12902, 15701}, {13148, 15012}, {13211, 25055}, {13212, 15354}, {13392, 41984}, {13598, 15465}, {13851, 44280}, {14269, 20127}, {14448, 15043}, {14643, 15703}, {14677, 23046}, {14830, 35282}, {14893, 34584}, {15035, 15702}, {15036, 15719}, {15051, 15708}, {15078, 32607}, {15305, 17853}, {15684, 38788}, {15694, 38638}, {15700, 38723}, {15713, 34153}, {15723, 32609}, {16280, 20398}, {17811, 39562}, {17825, 45016}, {17838, 19348}, {18400, 44214}, {20301, 30739}, {20423, 30775}, {22264, 23878}, {23332, 36201}, {23698, 36194}, {25561, 45303}, {30745, 41586}, {31105, 34417}, {31523, 38026}, {31860, 38335}, {32216, 37638}, {34331, 43573}, {34351, 44829}, {41672, 41939}

X(45311) = midpoint of X(i) and X(j) for these {i,j}: {2, 125}, {67, 15303}, {113, 20126}, {858, 32225}, {3534, 12295}, {3580, 13857}, {3830, 16111}, {3845, 12041}, {3917, 45237}, {5095, 13169}, {5465, 15357}, {5642, 9140}, {5655, 16003}, {8703, 10113}, {11006, 16278}, {11801, 12100}, {13212, 15354}, {13851, 44280}, {14644, 38727}, {15061, 23515}, {15305, 17853}, {38724, 38793}
X(45311) = reflection of X(i) in X(j) for these {i,j}: {2, 6723}, {5066, 15088}, {5972, 2}, {15303, 32300}, {21849, 11746}, {32267, 468}, {38726, 12100}, {41670, 6688}
X(45311) = complement of X(5642)
X(45311) = orthoptic-circle-of-Steiner-inellipe inverse of X(11177)
X(45311) = complement of the isogonal conjugate of X(9139)
X(45311) = psi-transform of X(3543)
X(45311) = X(i)-complementary conjugate of X(j) for these (i,j): {74, 16597}, {897, 113}, {923, 3163}, {2159, 2482}, {2349, 126}, {9139, 10}, {36034, 1649}, {36119, 5181}, {36131, 18311}, {36142, 5664}
X(45311) = crossdifference of every pair of points on line {3569, 9412}
X(45311) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9140, 5642}, {5, 20397, 20417}, {5, 20417, 38791}, {125, 5642, 9140}, {125, 6723, 5972}, {125, 15059, 6723}, {140, 20396, 36253}, {3091, 15057, 10990}, {3526, 15027, 30714}, {3628, 20379, 16534}, {5055, 20126, 113}, {5071, 10706, 36518}, {6698, 15118, 32257}, {6699, 7687, 37853}, {6699, 20304, 7687}, {9166, 11006, 16278}, {10264, 12900, 6053}, {10601, 15106, 34155}, {14971, 15357, 5465}, {20304, 40685, 6699}, {20379, 38632, 13393}, {30775, 37643, 20423}

X(45312) = MIDPOINT OF X(2) AND X(340)

Barycentrics    2*a^8 - 5*a^6*b^2 + 7*a^2*b^6 - 4*b^8 - 5*a^6*c^2 + 8*a^4*b^2*c^2 - 7*a^2*b^4*c^2 + 4*b^6*c^2 - 7*a^2*b^2*c^4 + 7*a^2*c^6 + 4*b^2*c^6 - 4*c^8 : :
X(45312) = 2 X[340] + X[3284], X[648] - 3 X[44579], 2 X[23583] - 3 X[44576], 3 X[37765] - X[39358], X[39352] + 3 X[44577], 4 X[40484] - 3 X[44578]

X(45312) lies on these lines: {2, 340}, {30, 15526}, {297, 18487}, {338, 39563}, {381, 511}, {520, 31174}, {524, 3163}, {648, 44579}, {754, 44650}, {1494, 1972}, {1651, 32225}, {3631, 36412}, {7810, 44218}, {11331, 15860}, {15980, 32257}, {16334, 39838}, {20128, 32110}, {23583, 44576}, {37765, 39358}, {39352, 44577}, {40484, 44578}

X(45312) = midpoint of X(i) and X(j) for these {i,j}: {2, 340}, {1494, 40885}
X(45312) = reflection of X(i) in X(j) for these {i,j}: {3163, 44216}, {3284, 2}, {18487, 297}

X(45313) = MIDPOINT OF X(2) AND X(649)

Barycentrics    (b - c)*(4*a^2 - a*b - a*c + b*c) : :
X(45313) = 5 X[2] - X[20295], 13 X[2] - 5 X[26798], 7 X[2] + X[26853], X[2] - 5 X[27013], 11 X[2] - 7 X[27138], 7 X[2] - 5 X[30835], 5 X[2] - 7 X[31207], 2 X[649] + X[3835], 5 X[649] + X[20295], 13 X[649] + 5 X[26798], 7 X[649] - X[26853], X[649] + 5 X[27013], 11 X[649] + 7 X[27138], 7 X[649] + 5 X[30835], 3 X[649] + X[31147], 5 X[649] + 7 X[31207], X[649] + 2 X[31286], 2 X[650] + X[4932], 3 X[1635] + X[31148], 3 X[1635] - X[31150], 4 X[2487] - X[3776], 2 X[2527] + X[17069], X[3798] + 2 X[43061], 5 X[3835] - 2 X[20295], 13 X[3835] - 10 X[26798], 7 X[3835] + 2 X[26853], X[3835] - 10 X[27013], 11 X[3835] - 14 X[27138], 7 X[3835] - 10 X[30835], 3 X[3835] - 2 X[31147], 5 X[3835] - 14 X[31207], X[3835] - 4 X[31286], X[4369] + 2 X[4394], X[4380] + 5 X[24924], 3 X[4763] - 2 X[44567], 2 X[4782] + X[24720], X[4790] + 2 X[25666], X[4813] - 7 X[27115], X[4979] + 5 X[31209], 13 X[20295] - 25 X[26798], 7 X[20295] + 5 X[26853], X[20295] - 25 X[27013], 11 X[20295] - 35 X[27138], 7 X[20295] - 25 X[30835], 3 X[20295] - 5 X[31147], X[20295] - 7 X[31207], X[20295] - 10 X[31286], 35 X[26798] + 13 X[26853], X[26798] - 13 X[27013], 55 X[26798] - 91 X[27138], 7 X[26798] - 13 X[30835], 15 X[26798] - 13 X[31147], 25 X[26798] - 91 X[31207], 5 X[26798] - 26 X[31286], X[26853] + 35 X[27013], 11 X[26853] + 49 X[27138], X[26853] + 5 X[30835], 3 X[26853] + 7 X[31147], 5 X[26853] + 49 X[31207], X[26853] + 14 X[31286], 55 X[27013] - 7 X[27138], 7 X[27013] - X[30835], 15 X[27013] - X[31147], 25 X[27013] - 7 X[31207], 5 X[27013] - 2 X[31286], 49 X[27138] - 55 X[30835], 21 X[27138] - 11 X[31147], 5 X[27138] - 11 X[31207], 7 X[27138] - 22 X[31286], 15 X[30835] - 7 X[31147], 25 X[30835] - 49 X[31207], 5 X[30835] - 14 X[31286], 5 X[31147] - 21 X[31207], X[31147] - 6 X[31286], 7 X[31207] - 10 X[31286], 3 X[44432] - 4 X[44563]

X(45313) lies on these lines: {2, 649}, {512, 11176}, {513, 4763}, {514, 1635}, {551, 14474}, {597, 9002}, {650, 4932}, {788, 38238}, {1638, 28882}, {1639, 28867}, {2487, 3776}, {2527, 17069}, {2786, 4786}, {3667, 31131}, {3798, 43061}, {4369, 4394}, {4380, 24924}, {4421, 23865}, {4448, 6006}, {4521, 5325}, {4685, 7234}, {4750, 30519}, {4782, 24720}, {4790, 25666}, {4813, 27115}, {4928, 6008}, {4979, 31209}, {6174, 37998}, {8650, 26249}, {9283, 35119}, {9294, 41142}, {10196, 28846}, {18200, 41629}, {21052, 28525}, {23655, 42043}, {28906, 30565}, {29126, 44566}, {44432, 44563}

X(45313) = midpoint of X(i) and X(j) for these {i,j}: {2, 649}, {31148, 31150}
X(45313) = reflection of X(i) in X(j) for these {i,j}: {2, 31286}, {3835, 2}
X(45313) = complement of X(31147)
X(45313) = crossdifference of every pair of points on line {2177, 3009}
X(45313) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 27013, 31286}, {649, 30835, 26853}, {649, 31207, 20295}, {649, 31286, 3835}, {1635, 31148, 31150}

X(45314) = MIDPOINT OF X(2) AND X(659)

Barycentrics    (2*a - b - c)*(b - c)*(2*a^2 + a*b + a*c - b*c) : :
X(45314) = 7 X[2] - 5 X[30795], 2 X[659] + X[3837], 7 X[659] + 5 X[30795], X[3241] - 3 X[25569], 7 X[3837] - 10 X[30795], 2 X[4401] + X[21051], 2 X[4782] + X[4806], X[21343] - 3 X[38314], X[21385] + 3 X[25055]

X(45314) lies on these lines: {2, 659}, {30, 44805}, {351, 523}, {513, 4763}, {514, 14422}, {519, 1960}, {549, 2826}, {551, 891}, {900, 1635}, {1638, 4977}, {3241, 25569}, {4401, 21051}, {4759, 4782}, {4762, 4874}, {4809, 6546}, {5298, 30725}, {21343, 38314}, {21385, 25055}, {29240, 44566}

X(45314) = midpoint of X(i) and X(j) for these {i,j}: {2, 659}, {1635, 4448}, {4809, 6546}
X(45314) = reflection of X(i) in X(j) for these {i,j}: {3837, 2}, {28602, 14425}
X(45314) = tripolar centroid of X(4393)
X(45314) = X(i)-isoconjugate of X(j) for these (i,j): {88, 43077}, {4555, 40735}, {27494, 32665}
X(45314) = crossdifference of every pair of points on line {106, 574}
X(45314) = barycentric product X(i)*X(j) for these {i,j}: {514, 4759}, {519, 4785}, {900, 4393}, {1635, 30963}, {1960, 10009}, {3762, 16468}, {4358, 4782}, {4806, 16704}, {14429, 31912}
X(45314) = barycentric quotient X(i)/X(j) for these {i,j}: {900, 27494}, {902, 43077}, {4120, 34475}, {4393, 4555}, {4759, 190}, {4782, 88}, {4785, 903}, {4806, 4080}, {14408, 40780}, {16468, 3257}, {21793, 901}, {34476, 4591}

X(45315) = MIDPOINT OF X(2) AND X(661)

Barycentrics    (b - c)*(a^2 - 4*a*b - 4*a*c + b*c) : :
X(45315) = 5 X[2] - X[7192], 7 X[2] - 5 X[24924], 7 X[2] + X[31290], 2 X[661] + X[4369], 5 X[661] + X[7192], 7 X[661] + 5 X[24924], X[661] + 2 X[25666], 3 X[661] + X[31148], 7 X[661] - X[31290], 5 X[3835] - 2 X[23813], 5 X[4369] - 2 X[7192], 7 X[4369] - 10 X[24924], X[4369] - 4 X[25666], 3 X[4369] - 2 X[31148], 7 X[4369] + 2 X[31290], X[4761] - 3 X[19875], 3 X[4763] - 4 X[44567], 3 X[4776] - X[31147], 3 X[4776] + X[31150], 2 X[4806] + X[4913], X[4813] + 5 X[31209], 3 X[4893] + X[31147], 3 X[4893] - X[31150], X[4932] - 4 X[31287], X[4979] - 7 X[27115], 7 X[7192] - 25 X[24924], X[7192] - 10 X[25666], 3 X[7192] - 5 X[31148], 7 X[7192] + 5 X[31290], 2 X[14321] + X[21196], 5 X[24924] - 14 X[25666], 15 X[24924] - 7 X[31148], 5 X[24924] + X[31290], 6 X[25666] - X[31148], 14 X[25666] + X[31290], 7 X[31148] + 3 X[31290]

X(45315) lies on these lines: {2, 661}, {513, 4763}, {514, 1639}, {551, 4160}, {597, 9013}, {650, 4785}, {812, 4776}, {1638, 28855}, {3835, 4762}, {4010, 4948}, {4370, 35089}, {4453, 28871}, {4664, 24577}, {4761, 19875}, {4806, 4913}, {4813, 31209}, {4842, 24622}, {4932, 31287}, {4979, 27115}, {8672, 44560}, {9279, 10180}, {14321, 21196}, {14838, 21894}, {18829, 27805}, {28846, 44551}, {28863, 30565}, {28878, 44432}, {28890, 44435}, {28902, 44902}, {29051, 31149}, {40459, 41140}

X(45315) = midpoint of X(i) and X(j) for these {i,j}: {2, 661}, {4010, 4948}, {4776, 4893}, {31147, 31150}
X(45315) = reflection of X(i) in X(j) for these {i,j}: {2, 25666}, {4369, 2}
X(45315) = complement of X(31148)
X(45315) = anticomplement of X(45663)
X(45315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {661, 24924, 31290}, {661, 25666, 4369}, {4776, 31150, 31147}, {4893, 31147, 31150}

X(45316) = MIDPOINT OF X(2) AND X(663)

Barycentrics    (b - c)*(4*a^3 - 3*a^2*b - a*b^2 - 3*a^2*c - a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(45316) = 5 X[2] - X[21302], 2 X[663] + X[17072], 5 X[663] + X[21302], 2 X[1125] + X[4794], 4 X[1125] - X[24720], 2 X[1960] + X[3835], 5 X[3616] + X[4724], X[4040] + 3 X[25055], X[4449] - 3 X[38314], X[4775] + 2 X[31286], 2 X[4794] + X[24720], X[4814] - 7 X[27115], X[4895] + 5 X[31209], 3 X[8643] + X[31147], 5 X[8656] + X[20295], 5 X[17072] - 2 X[21302]

X(45316) lies on these lines: {2, 663}, {512, 11176}, {514, 551}, {519, 4147}, {522, 14414}, {597, 9029}, {667, 4785}, {1125, 4794}, {1960, 3835}, {3309, 44561}, {3616, 4724}, {3667, 14419}, {3900, 44567}, {4040, 25055}, {4449, 38314}, {4775, 31286}, {4814, 27115}, {4895, 31209}, {8643, 31147}, {8656, 20295}, {26275, 28468}, {28470, 31149}, {28473, 44566}, {40726, 44408}

X(45316) = midpoint of X(2) and X(663)
X(45316) = reflection of X(17072) in X(2)
X(45316) = {X(1125),X(4794)}-harmonic conjugate of X(24720)

X(45317) = MIDPOINT OF X(2) AND X(669)

Barycentrics    (b - c)*(b + c)*(4*a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2) : :
X(45317) = 7 X[2] - 5 X[31279], 7 X[2] + X[31299], 5 X[2] - X[44445], X[351] - 3 X[15724], 3 X[351] - X[36900], 2 X[669] + X[23301], 3 X[669] + X[31176], 7 X[669] + 5 X[31279], 7 X[669] - X[31299], 5 X[669] + X[44445], X[669] + 2 X[44451], X[4108] + 3 X[15724], 3 X[4108] + X[36900], 2 X[6134] + X[22105], 3 X[8644] + X[31174], 3 X[9131] + X[44554], 3 X[11176] - 2 X[44560], 3 X[15699] - 2 X[39511], 9 X[15724] - X[36900], 3 X[23301] - 2 X[31176], 7 X[23301] - 10 X[31279], 7 X[23301] + 2 X[31299], 5 X[23301] - 2 X[44445], X[23301] - 4 X[44451], 7 X[31176] - 15 X[31279], 7 X[31176] + 3 X[31299], 5 X[31176] - 3 X[44445], X[31176] - 6 X[44451], 5 X[31279] + X[31299], 25 X[31279] - 7 X[44445], 5 X[31279] - 14 X[44451], 5 X[31299] + 7 X[44445], X[31299] + 14 X[44451], 3 X[32193] - 2 X[44552], X[44445] - 10 X[44451]

X(45317) lies on these lines: {2, 669}, {30, 5926}, {351, 523}, {512, 11176}, {549, 1499}, {597, 9009}, {804, 8644}, {5652, 11186}, {6134, 22105}, {8651, 23878}, {9006, 38237}, {10190, 44210}, {10278, 44212}, {14824, 33274}, {15699, 39511}, {18829, 23356}

X(45317) = midpoint of X(i) and X(j) for these {i,j}: {2, 669}, {351, 4108}, {5652, 11186}
X(45317) = reflection of X(i) in X(j) for these {i,j}: {2, 44451}, {23301, 2}
X(45317) = complement of X(31176)
X(45317) = crossdifference of every pair of points on line {574, 3229}
X(45317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {669, 31279, 31299}, {669, 44451, 23301}, {4108, 15724, 351}

X(45318) = MIDPOINT OF X(2) AND X(676)

Barycentrics    (b - c)*(10*a^3 - 3*a^2*b - 4*a*b^2 - 3*b^3 - 3*a^2*c - 4*a*b*c + 7*b^2*c - 4*a*c^2 + 7*b*c^2 - 3*c^3) : :
X(45318) = X[4528] - 3 X[19875], X[10015] + 3 X[25055], 3 X[30691] + X[31165]

X(45318) lies on these lines: {2, 676}, {30, 44819}, {522, 44563}, {523, 44401}, {549, 9521}, {551, 6366}, {900, 4928}, {4528, 19875}, {6362, 44561}, {10015, 25055}, {30691, 31165}

X(45318) = midpoint of X(i) and X(j) for these {i,j}: {2, 676}, {551, 44566}
X(45318) = X(26716)-complementary conjugate of X(16594)
X(45318) = crossdifference of every pair of points on line {5210, 21781}

X(45319) = MIDPOINT OF X(2) AND X(684)

Barycentrics    (b^2 - c^2)*(a^8 - 6*a^6*b^2 + 9*a^4*b^4 - 4*a^2*b^6 - 6*a^6*c^2 + 9*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + b^6*c^2 + 9*a^4*c^4 - 4*a^2*b^2*c^4 - 2*b^4*c^4 - 4*a^2*c^6 + b^2*c^6) : :
X(45319) = 2 X[684] + X[6130], 3 X[3524] - X[9409], 3 X[5054] - 2 X[44818], 3 X[5055] - X[41079], 3 X[23234] - X[31953]

X(45319) lies on these lines: {2, 684}, {30, 8552}, {381, 2797}, {520, 44560}, {523, 7625}, {526, 5642}, {549, 9517}, {804, 6054}, {3524, 9409}, {5054, 44818}, {5055, 41079}, {6086, 10714}, {9033, 11049}, {12100, 44810}, {13857, 32112}, {23234, 31953}

X(45319) = midpoint of X(i) and X(j) for these {i,j}: {2, 684}, {13857, 32112}
X(45319) = reflection of X(i) in X(j) for these {i,j}: {6130, 2}, {44810, 12100}
X(45319) = anticomplement of X(45682)
X(45319) = X(36034)-complementary conjugate of X(15819)
X(45319) = crossdifference of every pair of points on line {1384, 9412}

X(45320) = MIDPOINT OF X(2) AND X(693)

Barycentrics    (b - c)*(a^2 - a*b - a*c + 4*b*c) : :
X(45320) = 5 X[2] - X[17494], 13 X[2] - 5 X[26777], 7 X[2] + X[26824], X[2] - 5 X[26985], 11 X[2] - 7 X[27115], 7 X[2] - 5 X[31209], 4 X[2] - 5 X[31250], 5 X[2] - 4 X[31287], X[649] + 2 X[23813], X[650] + 2 X[693], X[650] - 4 X[4885], 5 X[650] - 2 X[17494], 13 X[650] - 10 X[26777], 7 X[650] + 2 X[26824], X[650] - 10 X[26985], 11 X[650] - 14 X[27115], 3 X[650] - 2 X[31150], 7 X[650] - 10 X[31209], 2 X[650] - 5 X[31250], 5 X[650] - 8 X[31287], 3 X[650] - 4 X[44567], X[693] + 2 X[4885], 5 X[693] + X[17494], 13 X[693] + 5 X[26777], 7 X[693] - X[26824], X[693] + 5 X[26985], 11 X[693] + 7 X[27115], 3 X[693] + X[31150], 7 X[693] + 5 X[31209], 4 X[693] + 5 X[31250], 5 X[693] + 4 X[31287], 3 X[693] + 2 X[44567], X[905] + 2 X[4823], 2 X[1577] + X[3669], 3 X[1638] - 2 X[44551], 4 X[2516] - 7 X[31207], X[2526] - 4 X[3837], X[2526] + 2 X[7662], 2 X[3239] + X[21104], 2 X[3676] + X[3700], 2 X[3835] + X[43067], 2 X[3837] + X[7662], 2 X[4010] + X[7659], 2 X[4025] + X[4820], X[4106] + 2 X[4369], 2 X[4106] + X[4790], 4 X[4369] - X[4790], 3 X[4379] + X[31147], 3 X[4379] - X[31148], X[4382] + 2 X[4394], X[4382] + 5 X[24924], 2 X[4394] - 5 X[24924], X[4500] + 2 X[21212], X[4664] + 3 X[4828], 3 X[4728] - X[31147], 3 X[4728] + X[31148]

X(45320) lies on these lines: {2, 650}, {11, 35094}, {210, 9443}, {381, 8760}, {513, 4379}, {514, 1639}, {522, 1638}, {523, 7625}, {536, 4411}, {551, 29066}, {597, 9015}, {599, 9001}, {649, 23813}, {824, 21204}, {905, 4823}, {918, 4944}, {1577, 3669}, {1938, 31165}, {2516, 31207}, {2526, 3837}, {3239, 21104}, {3676, 3700}, {3679, 14077}, {3835, 28840}, {4010, 7659}, {4025, 4820}, {4106, 4369}, {4382, 4394}, {4453, 28898}, {4500, 21212}, {4554, 32041}, {4664, 4828}, {4688, 4777}, {4750, 4926}, {4789, 28894}, {4801, 20317}, {4802, 30601}, {4940, 7192}, {4948, 30795}, {4976, 7658}, {4979, 7653}, {6008, 21297}, {6182, 31140}, {6545, 30520}, {8142, 10304}, {8678, 31149}, {9373, 31141}, {9404, 25924}, {10072, 30235}, {11238, 11934}, {13466, 35121}, {14415, 23880}, {15584, 34612}, {20942, 21611}, {21438, 42034}, {23878, 43051}, {23882, 44561}, {29033, 30234}, {35073, 35089}

X(45320) = midpoint of X(i) and X(j) for these {i,j}: {2, 693}, {4379, 4728}, {4789, 44435}, {31147, 31148}
X(45320) = reflection of X(i) in X(j) for these {i,j}: {2, 4885}, {650, 2}, {31150, 44567}
X(45320) = complement of X(31150)
X(45320) = anticomplement of X(44567)
X(45320) = crossdifference of every pair of points on line {1384, 2223}
X(45320) = barycentric product X(i)*X(j) for these {i,j}: {514, 4659}, {693, 4413}
X(45320) = barycentric quotient X(i)/X(j) for these {i,j}: {4413, 100}, {4659, 190}
X(45320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31150, 44567}, {650, 4885, 31250}, {693, 4885, 650}, {693, 26985, 4885}, {693, 30024, 18071}, {693, 30061, 29739}, {693, 31209, 26824}, {3837, 7662, 2526}, {4106, 4369, 4790}, {4379, 31147, 31148}, {4382, 24924, 4394}, {4728, 31148, 31147}, {17494, 31287, 650}, {29427, 29808, 693}, {29488, 29739, 30061}, {31150, 44567, 650}

X(45321) = MIDPOINT OF X(2) AND X(879)

Barycentrics    (b^2 - c^2)*(-a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2)*(-2*a^6 + 2*a^4*b^2 - a^2*b^4 + b^6 + 2*a^4*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(45321) = X[879] + 2 X[40550], 2 X[879] + X[41167], 4 X[40550] - X[41167]

X(45321) lies on these lines: {2, 879}, {182, 23878}, {381, 512}, {520, 599}, {523, 597}, {525, 549}, {542, 18312}, {690, 6055}, {1640, 6041}, {3906, 11171}, {9033, 41145}, {33754, 38317}

X(45321) = midpoint of X(2) and X(879)
X(45321) = reflection of X(i) in X(j) for these {i,j}: {2, 40550}, {41167, 2}
X(45321) = X(36885)-Ceva conjugate of X(542)
X(45321) = X(i)-isoconjugate of X(j) for these (i,j): {2186, 5649}, {3402, 6035}
X(45321) = crosspoint of X(542) and X(36885)
X(45321) = crossdifference of every pair of points on line {842, 5104}
X(45321) = barycentric product X(i)*X(j) for these {i,j}: {182, 18312}, {183, 1640}, {542, 23878}, {6041, 20023}
X(45321) = barycentric quotient X(i)/X(j) for these {i,j}: {182, 5649}, {183, 6035}, {1640, 262}, {3288, 842}, {5191, 26714}, {6041, 263}, {6784, 14998}, {18312, 327}, {23878, 5641}, {23967, 36885}, {34369, 6037}
X(45321) = {X(879),X(40550)}-harmonic conjugate of X(41167)

X(45322) = MIDPOINT OF X(2) AND X(885)

Barycentrics    (b - c)*(-a^2 + a*b + a*c + 2*b*c)*(-2*a^3 + 2*a^2*b - a*b^2 + b^3 + 2*a^2*c - b^2*c - a*c^2 - b*c^2 + c^3) : :
X(45322) = 2 X[885] + X[3126], X[885] + 2 X[40551], X[3126] - 4 X[40551]

X(45322) lies on these lines: {2, 885}, {381, 3309}, {513, 6173}, {514, 551}, {1001, 4762}, {1642, 1643}, {2826, 14419}, {3679, 3900}, {3887, 4800}, {6362, 15670}, {11193, 31140}, {11247, 17528}, {14077, 30583}

X(45322) = midpoint of X(2) and X(885)
X(45322) = reflection of X(i) in X(j) for these {i,j}: {2, 40551}, {3126, 2}
X(45322) = X(i)-isoconjugate of X(j) for these (i,j): {840, 37138}, {8693, 37131}
X(45322) = crossdifference of every pair of points on line {840, 8693}
X(45322) = barycentric product X(i)*X(j) for these {i,j}: {528, 4762}, {1643, 4441}
X(45322) = barycentric quotient X(i)/X(j) for these {i,j}: {528, 32041}, {1643, 1002}, {2246, 37138}, {4724, 37131}, {4762, 18821}
X(45322) = {X(885),X(40551)}-harmonic conjugate of X(3126)

X(45323) = MIDPOINT OF X(2) AND X(1491)

Barycentrics    (b - c)*(a^3 - 4*a*b^2 - 4*a*b*c + b^2*c - 4*a*c^2 + b*c^2) : :
X(45323) = 2 X[1491] + X[4874], X[31150] + 3 X[44429]

X(45323) lies on these lines: {2, 1491}, {513, 4763}, {514, 3828}, {523, 7625}, {597, 9014}, {693, 4948}, {814, 31149}, {3837, 4762}, {4705, 19870}, {4777, 4928}, {4785, 9508}, {4893, 36848}, {8678, 44561}, {14425, 28209}, {25380, 28840}, {29324, 44550}, {29328, 31147}, {29362, 31150}

X(45323) = midpoint of X(i) and X(j) for these {i,j}: {2, 1491}, {693, 4948}, {4893, 36848}
X(45323) = reflection of X(4874) in X(2)
X(45323) = crossdifference of every pair of points on line {1384, 8624}

X(45324) = MIDPOINT OF X(2) AND X(1577)

Barycentrics    (b - c)*(a^3 - a*b^2 - a*b*c + 4*b^2*c - a*c^2 + 4*b*c^2) : :
X(45324) = 5 X[2] - X[4560], 5 X[1577] + X[4560], 2 X[1577] + X[14838], 5 X[1698] + X[4804], 4 X[3634] - X[4913], X[3679] - 3 X[21052], X[3762] + 5 X[26985], X[3960] + 2 X[4791], X[3960] - 4 X[4885], X[4041] - 3 X[19875], 2 X[4560] - 5 X[14838], X[4791] + 2 X[4885], 3 X[5055] - X[39212]

X(45324) lies on these lines: {2, 1577}, {514, 1639}, {551, 3907}, {830, 31149}, {1698, 4804}, {3634, 4913}, {3679, 21052}, {3762, 26985}, {3828, 4151}, {3910, 44566}, {3960, 4791}, {4041, 19875}, {4129, 28840}, {4160, 14431}, {4762, 4823}, {5055, 39212}, {13466, 35085}, {15309, 31148}, {15455, 35139}, {16857, 21789}, {23879, 31174}, {23880, 44561}, {23882, 44567}, {40459, 41144}

X(45324) = midpoint of X(2) and X(1577)
X(45324) = reflection of X(14838) in X(2)
X(45324) = complement of X(45671)
X(45324) = {X(4791),X(4885)}-harmonic conjugate of X(3960)

X(45325) = MIDPOINT OF X(2) AND X(1636)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-3*a^8 + 4*a^6*b^2 + a^4*b^4 - 2*a^2*b^6 + 4*a^6*c^2 - 7*a^4*b^2*c^2 + 2*a^2*b^4*c^2 + b^6*c^2 + a^4*c^4 + 2*a^2*b^2*c^4 - 2*b^4*c^4 - 2*a^2*c^6 + b^2*c^6) : :
X(45325) = X[9979] - 3 X[14401]

X(45325) lies on these lines: {2, 1636}, {402, 5972}, {441, 525}, {520, 44560}, {2421, 36841}, {2799, 14345}, {2881, 10192}, {5642, 9517}, {9035, 14396}, {9979, 14401}, {11176, 39469}, {16534, 42651}, {38240, 39473}, {43665, 44877}

X(45325) = midpoint of X(2) and X(1636)
X(45325) = X(i)-complementary conjugate of X(j) for these (i,j): {26717, 34846}, {32725, 226}, {36139, 5}
X(45325) = complement of tripolar centroid of X(264)

X(45326) = MIDPOINT OF X(2) AND X(1639)

Barycentrics    (b - c)*(4*a^2 - 5*a*b + b^2 - 5*a*c + 4*b*c + c^2) : :
X(45326) = 5 X[2] - X[4453], 3 X[2] + X[30565], X[1638] + 3 X[1639], 5 X[1638] - 3 X[4453], 2 X[1638] - 3 X[44902], 5 X[1639] + X[4453], 3 X[1639] - X[30565], 2 X[1639] + X[44902], 2 X[2490] + X[3835], 2 X[3239] + X[17069], X[3239] + 2 X[31287], X[3700] + 5 X[31209], 2 X[3716] + X[4925], 3 X[4453] + 5 X[30565], 2 X[4453] - 5 X[44902], X[4468] + 5 X[31250], 2 X[4521] + X[4885], X[4728] + 3 X[6544], X[4897] - 7 X[31207], X[4940] + 2 X[43061], X[4976] - 7 X[27115], X[14321] + 2 X[31286], X[14392] + 3 X[14476], 3 X[14430] + X[30573], X[17069] - 4 X[31287], X[23729] - 7 X[27138], X[27486] - 5 X[31209], 2 X[30565] + 3 X[44902]

X(45326) lies on these lines: {2, 918}, {354, 30700}, {513, 30792}, {514, 4521}, {522, 44567}, {654, 3305}, {812, 14425}, {900, 3035}, {926, 3740}, {2490, 3835}, {3239, 17069}, {3310, 44307}, {3700, 27486}, {3742, 42341}, {4369, 28902}, {4468, 31250}, {4728, 6009}, {4897, 31207}, {4927, 6546}, {4928, 6084}, {4940, 43061}, {4976, 27115}, {6139, 15254}, {6370, 44564}, {13609, 26932}, {14321, 28867}, {14392, 14476}, {14430, 30573}, {17279, 24141}, {23729, 27138}, {23884, 44566}, {24720, 28209}, {30520, 44432}, {30855, 31992}

X(45326) = midpoint of X(i) and X(j) for these {i,j}: {2, 1639}, {354, 30700}, {1638, 30565}, {3700, 27486}, {4927, 6546}, {4928, 10196}
X(45326) = reflection of X(44902) in X(2)
X(45326) = complement of X(1638)
X(45326) = X(i)-complementary conjugate of X(j) for these (i,j): {41, 35091}, {100, 31844}, {101, 10427}, {692, 35110}, {1121, 21252}, {1156, 116}, {2291, 11}, {4845, 26932}, {14733, 142}, {18889, 1146}, {32728, 3752}, {34056, 17059}, {34068, 1086}, {35157, 17046}, {36141, 1}, {37139, 2886}, {41798, 124}
X(45326) = crossdifference of every pair of points on line {3052, 9259}
X(45326) = barycentric product X(75)*X(23057)
X(45326) = barycentric quotient X(23057)/X(1)
X(45326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 30565, 1638}, {1638, 1639, 30565}, {3239, 31287, 17069}

X(45327) = MIDPOINT OF X(2) AND X(1640)

Barycentrics    (b^2 - c^2)*(3*a^6 - 2*a^4*b^2 - b^6 - 2*a^4*c^2 - a^2*b^2*c^2 + 2*b^4*c^2 + 2*b^2*c^4 - c^6) : :
X(45327) = X[8371] - X[34290], X[14316] + 2 X[24284]

X(45327) lies on these lines: {2, 525}, {184, 39500}, {381, 1499}, {458, 2501}, {512, 5943}, {523, 597}, {524, 18310}, {690, 5461}, {826, 10190}, {868, 39491}, {2444, 5466}, {2799, 14316}, {3288, 15018}, {3566, 10189}, {3800, 5644}, {3906, 44560}, {5892, 30209}, {9140, 32313}, {9172, 11656}, {9517, 12099}, {12073, 44568}, {37457, 39228}

X(45327) = midpoint of X(i) and X(j) for these {i,j}: {2, 1640}, {5652, 8029}, {9140, 32313}
X(45327) = reflection of X(11182) in X(10189)
X(45327) = X(i)-complementary conjugate of X(j) for these (i,j): {897, 36471}, {923, 35088}, {1910, 5099}, {2715, 16597}, {9154, 21253}, {32729, 16591}, {36084, 126}, {36104, 5181}, {36142, 114}
X(45327) = crossdifference of every pair of points on line {1495, 5104}

X(45328) = MIDPOINT OF X(2) AND X(2254)

Barycentrics    (b - c)*(a^3 + 3*a^2*b - 4*a*b^2 + 3*a^2*c - a*b*c + b^2*c - 4*a*c^2 + b*c^2) : :
X(45328) = 2 X[2254] + X[3716], X[2254] + 2 X[25380], X[3241] - 3 X[14413], X[3716] - 4 X[25380], X[3762] - 3 X[19875], X[4458] + 2 X[4925], X[4830] - 4 X[9508], X[4895] - 3 X[38314], X[4913] + 2 X[24720], 3 X[30574] - X[44553], X[31147] - 3 X[44429]

X(45328) lies on these lines: {2, 2254}, {513, 4763}, {519, 3960}, {522, 1638}, {551, 3887}, {812, 36848}, {900, 4928}, {1491, 28840}, {2826, 44566}, {3241, 14413}, {3309, 44561}, {3667, 26275}, {3738, 6174}, {3762, 19875}, {3907, 44550}, {4458, 4925}, {4750, 31131}, {4762, 4913}, {4830, 9508}, {4895, 38314}, {4948, 21146}, {6002, 31149}, {8648, 13587}, {14419, 28521}, {30574, 44553}, {31147, 44429}

X(45328) = midpoint of X(i) and X(j) for these {i,j}: {2, 2254}, {4750, 31131}, {4948, 21146}
X(45328) = reflection of X(i) in X(j) for these {i,j}: {2, 25380}, {3716, 2}
X(45328) = X(i)-complementary conjugate of X(j) for these (i,j): {901, 3789}, {1002, 3259}, {8693, 16594}, {37138, 121}
X(45328) = {X(2254),X(25380)}-harmonic conjugate of X(3716)

X(45329) = MIDPOINT OF X(2) AND X(2395)

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

X(45329) lies on these lines: {2, 647}, {338, 1084}, {351, 2793}, {523, 597}, {804, 9208}, {1640, 2799}, {4108, 5466}, {6041, 22329}, {7817, 23105}, {8029, 44568}, {10097, 11159}, {10189, 44565}, {10278, 44212}, {23967, 35087}, {32472, 34290}, {34291, 42849}

X(45329) = midpoint of X(i) and X(j) for these {i,j}: {2, 2395}, {10097, 11159}
X(45329) = crossdifference of every pair of points on line {237, 5104}
X(45329) = barycentric product X(523)*X(5939)
X(45329) = barycentric quotient X(5939)/X(99)

X(45330) = MIDPOINT OF X(2) AND X(2396)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^4*b^2 - 2*a^2*b^4 + a^4*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4) : :
X(45330) = X[2396] + 2 X[11052]

X(45330) lies on these lines: {2, 39}, {32, 19663}, {187, 5468}, {351, 690}, {524, 18872}, {574, 5108}, {1648, 7813}, {3228, 9468}, {5106, 5969}, {6390, 11053}, {7810, 16186}, {7818, 36194}, {7820, 41939}, {7863, 15000}, {7897, 40877}, {7998, 14916}, {8030, 40517}, {9888, 35279}, {11182, 35077}, {11672, 35073}, {13586, 17941}, {27088, 38239}, {32456, 34245}

X(45330) = midpoint of X(2) and X(2396)
X(45330) = reflection of X(2) in X(11052)
X(45330) = tripolar centroid of X(14607)
X(45330) = X(i)-isoconjugate of X(j) for these (i,j): {897, 5970}, {923, 35146}, {14606, 36085}
X(45330) = crosssum of X(6) and X(14898)
X(45330) = crossdifference of every pair of points on line {111, 669}
X(45330) = barycentric product X(i)*X(j) for these {i,j}: {524, 5969}, {690, 14607}, {3266, 5106}, {5468, 11182}
X(45330) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 5970}, {351, 14606}, {524, 35146}, {5106, 111}, {5969, 671}, {11182, 5466}, {14607, 892}
X(45330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {39, 3229, 1645}

X(45331) = MIDPOINT OF X(2) AND X(2407)

Barycentrics    4*a^8 - 6*a^6*b^2 + a^4*b^4 + b^8 - 6*a^6*c^2 + 10*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 3*b^6*c^2 + a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 - 3*b^2*c^6 + c^8 : :
X(45331) = X[2407] + 2 X[24975]

X(45331) lies on these lines: {2, 6}, {30, 5467}, {114, 15303}, {530, 18776}, {531, 18777}, {543, 3018}, {671, 1989}, {1990, 4235}, {2482, 2799}, {3284, 31173}, {4226, 9214}, {4590, 22254}, {5461, 6128}, {5642, 9003}, {5649, 30528}, {5968, 7426}, {5972, 12583}, {6054, 34319}, {6793, 12036}, {9175, 9177}, {13169, 30789}, {14590, 37765}, {16092, 23348}, {16303, 27088}, {18573, 33274}, {23967, 35087}, {33928, 35266}, {34288, 37809}

X(45331) = midpoint of X(i) and X(j) for these {i,j}: {2, 2407}, {4226, 9214}, {5467, 14995}
X(45331) = reflection of X(2) in X(24975)
X(45331) = crossdifference of every pair of points on line {512, 2378}
X(45331) = barycentric product X(530)*X(531)
X(45331) = barycentric quotient X(i)/X(j) for these {i,j}: {530, 43092}, {531, 43091}, {18776, 36317}, {18777, 36316}
X(45331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {395, 396, 1648}, {39107, 39108, 14999}

X(45332) = MIDPOINT OF X(2) AND X(2533)

Barycentrics    (b - c)*(a^3 + 3*a^2*b - a*b^2 + 3*a^2*c - a*b*c + 4*b^2*c - a*c^2 + 4*b*c^2) : :
X[4705] - 3 X[19875], X[4774] + 5 X[24924], X[4824] - 7 X[9780], 3 X[21052] + X[31148]

X(45332) lies on these lines: {2, 2533}, {513, 14431}, {514, 3828}, {551, 29298}, {4049, 29128}, {4705, 19875}, {4774, 24924}, {4824, 9780}, {9422, 13466}, {21051, 28840}, {21052, 31148}, {29142, 44566}

X(45332) = midpoint of X(2) and X(2533)

X(45333) = MIDPOINT OF X(2) AND X(3005)

Barycentrics    (b^2 - c^2)*(a^4 - 4*a^2*b^2 - 4*a^2*c^2 + b^2*c^2) : :
X(45333) = 3 X[5996] - X[31176], 3 X[5996] + X[36900], 3 X[8029] - X[44554], 3 X[10278] - 2 X[44568], 3 X[11176] - 4 X[44560], 3 X[17414] + X[31176], 3 X[17414] - X[36900]

X(45333) lies on these lines: {2, 881}, {512, 11176}, {523, 7625}, {597, 9012}, {647, 25423}, {804, 5996}, {8029, 44554}, {9005, 14406}, {9191, 31950}, {10278, 44568}, {23301, 23878}, {32193, 32473}

X(45333) = midpoint of X(i) and X(j) for these {i,j}: {2, 3005}, {5996, 17414}, {31176, 36900}
X(45333) = crosssum of X(688) and X(34811)
X(45333) = crossdifference of every pair of points on line {1384, 8623}
X(45333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5996, 36900, 31176}, {17414, 31176, 36900}

X(45334) = MIDPOINT OF X(2) AND X(3239)

Barycentrics    (b - c)*(7*a^2 - 10*a*b + 3*b^2 - 10*a*c + 10*b*c + 3*c^2) : :
X(45334) = 5 X[2] - X[4025], 7 X[2] + X[25259], 5 X[3239] + X[4025], 2 X[3239] + X[7658], 7 X[3239] - X[25259], 3 X[3239] + X[44551], 3 X[3239] + 2 X[44563], 2 X[4025] - 5 X[7658], 7 X[4025] + 5 X[25259], 3 X[4025] - 5 X[44551], 3 X[4025] - 10 X[44563], X[4369] + 2 X[14350], 3 X[6332] + X[44553], 7 X[7658] + 2 X[25259], 3 X[7658] - 2 X[44551], 3 X[7658] - 4 X[44563], 3 X[25259] + 7 X[44551], 3 X[25259] + 14 X[44563]

X(45334) lies on these lines: {2, 2400}, {514, 1639}, {522, 44567}, {525, 44565}, {597, 9031}, {3667, 31131}, {3679, 14476}, {4369, 14350}, {4521, 4762}, {4763, 4962}, {4785, 43061}, {6332, 44553}, {28143, 29594}

X(45334) = midpoint of X(2) and X(3239)
X(45334) = reflection of X(i) in X(j) for these {i,j}: {7658, 2}, {44551, 44563}
X(45334) = complement of X(44551)
X(45334) = anticomplement of X(44563)
X(45334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44551, 44563}, {44551, 44563, 7658}

X(45335) = MIDPOINT OF X(2) AND X(3288)

Barycentrics    (b^2 - c^2)*(4*a^6 - 3*a^4*b^2 - a^2*b^4 - 3*a^4*c^2 - 7*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4) : :
X(45335) = X[599] + 3 X[3050], X[1992] - 3 X[3049], X[21733] - 3 X[32232]

X(45335) lies on these lines: {2, 3288}, {351, 1499}, {512, 11176}, {523, 597}, {525, 1636}, {599, 3050}, {1992, 3049}, {2422, 11163}, {5027, 25423}, {5652, 17414}, {5996, 9135}, {7927, 44568}, {12073, 44564}, {21733, 32232}, {23878, 24284}, {30217, 31176}, {37809, 42660}

X(45335) = midpoint of X(i) and X(j) for these {i,j}: {2, 3288}, {5652, 17414}, {5996, 9135}
X(45335) = crossdifference of every pair of points on line {5104, 8585}

X(45336) = MIDPOINT OF X(2) AND X(3569)

Barycentrics    (b^2 - c^2)*(a^6 + 3*a^4*b^2 - 4*a^2*b^4 + 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 - 4*a^2*c^4 + b^2*c^4) : :
X(45336) = X[1992] - 3 X[14398], 2 X[3569] + X[24284], 3 X[9210] - X[36900], 3 X[21358] - X[35522]

X(45336) lies on these lines: {2, 3569}, {351, 14084}, {512, 11176}, {524, 2492}, {525, 1637}, {526, 597}, {599, 9035}, {690, 5461}, {804, 9208}, {826, 44568}, {1499, 9189}, {1640, 39905}, {1992, 14398}, {2780, 9172}, {3566, 44565}, {5642, 9517}, {5653, 9169}, {5996, 11186}, {9210, 36900}, {9420, 11328}, {10162, 32228}, {19912, 21733}, {21358, 35522}

X(45336) = midpoint of X(i) and X(j) for these {i,j}: {2, 3569}, {351, 34290}, {1640, 39905}, {5653, 36255}, {5996, 11186}, {9208, 11182}, {19912, 21733}
X(45336) = reflection of X(24284) in X(2)
X(45336) = X(i)-complementary conjugate of X(j) for these (i,j): {2186, 5099}, {3402, 23992}, {26714, 16597}, {36142, 15819}

X(45337) = MIDPOINT OF X(2) AND X(3716)

Barycentrics    (b - c)*(5*a^3 - 3*a^2*b - 2*a*b^2 - 3*a^2*c - 5*a*b*c + 5*b^2*c - 2*a*c^2 + 5*b*c^2) : :
X(45337) = 5 X[2] - X[2254], X[2254] + 5 X[3716], 2 X[2254] - 5 X[25380], X[3241] + 3 X[14430], 2 X[3716] + X[25380], X[3762] + 3 X[25055], X[3960] - 3 X[19883], 3 X[14432] + X[44553], 3 X[23057] + X[31145]

X(45337) lies on these lines: {2, 2254}, {522, 44567}, {2785, 44566}, {3241, 14430}, {3667, 30792}, {3762, 25055}, {3828, 3887}, {3960, 19883}, {4448, 4928}, {4763, 4800}, {4874, 28840}, {8648, 16858}, {14432, 44553}, {23057, 31145}

X(45337) = midpoint of X(i) and X(j) for these {i,j}: {2, 3716}, {4448, 4928}, {4763, 4800}
X(45337) = reflection of X(25380) in X(2)

X(45338) = MIDPOINT OF X(2) AND X(3766)

Barycentrics    (b - c)*(a^2 - a*b - a*c - 2*b*c)*(a*b + a*c - 2*b*c) : :
X(45338) = X[665] + 2 X[3766], X[3709] + 2 X[4408], X[4664] + 3 X[21606], X[4740] - 3 X[21433]

X(45338) lies on these lines: {2, 665}, {512, 9148}, {514, 1639}, {536, 4526}, {891, 4728}, {900, 4688}, {928, 10708}, {1960, 4508}, {3709, 4408}, {4375, 8658}, {4435, 16833}, {4664, 21606}, {4740, 21433}, {6372, 31148}

X(45338) = midpoint of X(i) and X(j) for these {i,j}: {2, 3766}, {4728, 14433}
X(45338) = reflection of X(665) in X(2)
X(45338) = tripolar centroid of X(4441)
X(45338) = X(i)-isoconjugate of X(j) for these (i,j): {739, 37138}, {898, 2279}, {1002, 34075}, {8693, 37129}, {27475, 32718}
X(45338) = crossdifference of every pair of points on line {739, 8693}
X(45338) = barycentric product X(i)*X(j) for these {i,j}: {536, 4762}, {891, 4441}, {3768, 21615}, {4384, 4728}, {4724, 6381}, {14430, 40719}
X(45338) = barycentric quotient X(i)/X(j) for these {i,j}: {536, 32041}, {891, 1002}, {899, 37138}, {1001, 898}, {2280, 34075}, {3230, 8693}, {3768, 2279}, {4384, 4607}, {4441, 889}, {4526, 40779}, {4724, 37129}, {4728, 27475}, {4762, 3227}, {4804, 41683}

X(45339) = MIDPOINT OF X(2) AND X(3835)

Barycentrics    (b - c)*(2*a^2 - 5*a*b - 5*a*c + 5*b*c) : :
X(45339) = 5 X[2] - X[649], 7 X[2] + X[20295], 11 X[2] + 5 X[26798], 17 X[2] - X[26853], 13 X[2] - 5 X[27013], X[2] + 7 X[27138], X[2] - 5 X[30835], 3 X[2] + X[31147], 11 X[2] - 7 X[31207], X[649] + 5 X[3835], 7 X[649] + 5 X[20295], 11 X[649] + 25 X[26798], 17 X[649] - 5 X[26853], 13 X[649] - 25 X[27013], X[649] + 35 X[27138], X[649] - 25 X[30835], 3 X[649] + 5 X[31147], 11 X[649] - 35 X[31207], 2 X[649] - 5 X[31286], 7 X[3835] - X[20295], 11 X[3835] - 5 X[26798], 17 X[3835] + X[26853], 13 X[3835] + 5 X[27013], X[3835] - 7 X[27138], X[3835] + 5 X[30835], 3 X[3835] - X[31147], 11 X[3835] + 7 X[31207], 2 X[3835] + X[31286], X[4664] + 3 X[27485], 3 X[4728] + X[31150], 3 X[4776] + X[31148], 11 X[20295] - 35 X[26798], 17 X[20295] + 7 X[26853], 13 X[20295] + 35 X[27013], X[20295] - 49 X[27138], X[20295] + 35 X[30835], 3 X[20295] - 7 X[31147], 11 X[20295] + 49 X[31207], 2 X[20295] + 7 X[31286], 85 X[26798] + 11 X[26853], 13 X[26798] + 11 X[27013], 5 X[26798] - 77 X[27138], X[26798] + 11 X[30835], 15 X[26798] - 11 X[31147], 5 X[26798] + 7 X[31207], 10 X[26798] + 11 X[31286], 13 X[26853] - 85 X[27013], X[26853] + 119 X[27138], X[26853] - 85 X[30835], 3 X[26853] + 17 X[31147], 11 X[26853] - 119 X[31207], 2 X[26853] - 17 X[31286], 5 X[27013] + 91 X[27138], X[27013] - 13 X[30835], 15 X[27013] + 13 X[31147], 55 X[27013] - 91 X[31207], 10 X[27013] - 13 X[31286], 7 X[27138] + 5 X[30835], 21 X[27138] - X[31147], 11 X[27138] + X[31207], 14 X[27138] + X[31286], 15 X[30835] + X[31147], 55 X[30835] - 7 X[31207], 10 X[30835] - X[31286], 11 X[31147] + 21 X[31207], 2 X[31147] + 3 X[31286], 14 X[31207] - 11 X[31286], 3 X[44432] - X[44551]

X(45339) lies on these lines: {2, 649}, {514, 1639}, {812, 44567}, {1638, 28906}, {2786, 44432}, {3667, 26275}, {3828, 29350}, {4664, 27485}, {4728, 31150}, {4762, 25666}, {4776, 31148}, {4885, 28840}, {6002, 44561}, {9002, 20582}, {9294, 41144}, {24749, 42043}, {28468, 44566}, {28470, 31149}, {28867, 44902}

X(45339) = midpoint of X(2) and X(3835)
X(45339) = reflection of X(31286) in X(2)
X(45339) = {X(27138),X(30835)}-harmonic conjugate of X(3835)

X(45340) = MIDPOINT OF X(2) AND X(3837)

Barycentrics    (b - c)*(2*a^3 - 5*a*b^2 - 2*a*b*c + 5*b^2*c - 5*a*c^2 + 5*b*c^2) : :
X(45340) = 5 X[2] - X[659], X[2] - 5 X[30795], X[659] + 5 X[3837], X[659] - 25 X[30795], X[1960] - 3 X[19883], X[3837] + 5 X[30795], 3 X[11539] - X[44805]

X(45340) lies on these lines: {2, 659}, {523, 7625}, {547, 2826}, {814, 44561}, {891, 3828}, {900, 4928}, {1960, 19883}, {3679, 25574}, {4927, 28602}, {11539, 44805}, {29078, 44551}, {29362, 44567}

X(45340) = X(45340) = midpoint of X(i) and X(j) for these {i,j}: {2, 3837}, {4927, 28602}
X(45340) = X(i)-complementary conjugate of X(j) for these (i,j): {32665, 27481}, {40735, 35092}, {43077, 16594}
X(45340) = crossdifference of every pair of points on line {1384, 21781}

X(45341) = MIDPOINT OF X(2) AND X(3904)

Barycentrics    (b - c)*(4*a^3 - 3*a^2*b - 4*a*b^2 + 3*b^3 - 3*a^2*c + 2*a*b*c + b^2*c - 4*a*c^2 + b*c^2 + 3*c^3) : :
X(45341) = 2 X[676] - 3 X[25055], 3 X[905] - 2 X[44551], 2 X[3904] + X[10015], 3 X[3904] + X[44553], 3 X[3904] + 2 X[44566], 3 X[10015] - 2 X[44553], 3 X[10015] - 4 X[44566], 3 X[14837] - 4 X[44563], 3 X[41800] - 4 X[44561]

X(45341) lies on these lines: {2, 3904}, {514, 1639}, {525, 1636}, {551, 23887}, {644, 25272}, {676, 25055}, {900, 30580}, {905, 44551}, {918, 1642}, {928, 31165}, {1638, 23884}, {2826, 14432}, {3679, 6366}, {4528, 4677}, {9521, 31162}, {14837, 44563}, {19251, 23184}, {19254, 23220}, {28294, 31131}, {29126, 31147}, {30565, 31171}, {41800, 44561}

X(45341) = midpoint of X(2) and X(3904)
X(45341) = reflection of X(i) in X(j) for these {i,j}: {4677, 4528}, {10015, 2}, {44553, 44566}
X(45341) = complement of X(44553)
X(45341) = anticomplement of X(44566)
X(45341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44553, 44566}, {44553, 44566, 10015}

X(45342) = MIDPOINT OF X(2) AND X(4010)

Barycentrics    (b - c)*(a^3 - 3*a^2*b - a*b^2 - 3*a^2*c - a*b*c + 4*b^2*c - a*c^2 + 4*b*c^2) : :
X(45342) = X[3241] + 3 X[30709], X[3679] - 3 X[14431], 2 X[4010] + X[9508], X[4730] - 3 X[19875], X[4922] - 3 X[38314]

X(45342) lies on these lines: {2, 4010}, {513, 4379}, {551, 2787}, {900, 4928}, {3241, 30709}, {3679, 14431}, {4106, 26248}, {4448, 21297}, {4486, 4762}, {4730, 19875}, {4785, 4874}, {4802, 30565}, {4804, 4948}, {4806, 28840}, {4922, 38314}, {4926, 31131}, {4944, 29204}, {8674, 9140}

X(45342) = midpoint of X(i) and X(j) for these {i,j}: {2, 4010}, {4448, 21297}, {4728, 4800}, {4804, 4948}
X(45342) = reflection of X(9508) in X(2)
X(45342) = anticomplement of X(45691)
X(45342) = X(i)-complementary conjugate of X(j) for these (i,j): {28841, 16594}, {30571, 3259}, {32665, 31336}

X(45343) = MIDPOINT OF X(2) AND X(4024)

Barycentrics    (b - c)*(a^2 - a*b + 3*b^2 - a*c + 7*b*c + 3*c^2) : :
X(45343) = 5 X[2] - X[17161], 5 X[4024] + X[17161], 2 X[4024] + X[21196], 3 X[4789] - X[31148], 3 X[4931] + X[31148], 3 X[10196] - 2 X[31150], 2 X[17161] - 5 X[21196]

X(45343) lies on these lines: {2, 4024}, {514, 4120}, {824, 21204}, {2786, 4789}, {3700, 28840}, {3828, 8371}, {4370, 35087}, {4500, 4762}, {4755, 4777}, {4763, 28183}, {4785, 6590}, {10196, 31150}, {18154, 42029}, {20942, 25667}, {23879, 31174}

X(45343) = midpoint of X(i) and X(j) for these {i,j}: {2, 4024}, {4789, 4931}
X(45343) = reflection of X(21196) in X(2)

X(45344) = MIDPOINT OF X(2) AND X(4088)

Barycentrics    (b - c)*(a^3 - 4*a*b^2 + 3*b^3 - 7*a*b*c + 4*b^2*c - 4*a*c^2 + 4*b*c^2 + 3*c^3) : :
X(45344) = X[3241] - 3 X[14432], 2 X[4088] + X[4458], X[4707] - 3 X[19875], 3 X[14430] - X[44553]

X(45344) lies on these lines: {2, 4088}, {514, 31149}, {522, 14392}, {2785, 3679}, {3241, 14432}, {3738, 30700}, {3807, 32041}, {4122, 4948}, {4522, 4762}, {4707, 19875}, {4728, 28147}, {4778, 31131}, {4800, 28161}, {14430, 44553}

X(45344) = midpoint of X(i) and X(j) for these {i,j}: {2, 4088}, {4122, 4948}
X(45344) = reflection of X(4458) in X(2)
X(45344) = anticomplement of X(45668)

leftri

Inverse triangles and anti- triangles: X(45345)-X(45656)

rightri

This preamble and centers X(45345)-X(45656) were contributed by César Eliud Lozada, October 30, 2021.

Inverse triangles were introduced by Peter Moses in the preamble just before X(42005) with more or less these terms: Suppose that T = A'B'C' is a triangle with vertices A', B', C' represented in normalized barycentric coordinates. Let M be the matrix representation of A'B'C', and let M-1 denote the inverse of M. Then the rows of M-1, interpreted as vertices of a triangle T-1, define the inverse triangle of A'B'C'.

Matrices M and M-1 algebraically behave as expected, it is to say, products of matrices M.M-1 = M-1.M = (3x3)-identity matrix, which represents ABC. Then it should be reasonable to expect that triangles T-of-T-1 and T-1-of-T should be both ABC, in other words, that T-1 is the anti-triangle-of-T and vice-versa, but actually, this only occurs when T and ABC are similar or homothetic and, in both cases, T ' and T are homothetic triangles.

The following properties can be deduced:

Because of the preceeding, inverse triangles allow to determine the anti-triangles of triangles T similar or homothetic to ABC. Similar triangles to ABC in the index of triangles are: (1st and 6th Brocard, inner-Garcia, orthocentroidal, 1st orthosymmedial and 1st and 2nd Parry) and homothetic triangles are:

(ABC-X3 reflections, anticomplementary, Aquila, Ara, 1st Auriga, 2nd Auriga, 5th Brocard, 2nd circumperp tangential, Ehrmann-mid, Euler, outer-Garcia, Gemini 107, Gemini 109, Gemini 110, Gemini 111, Gossard, inner-Grebe, outer-Grebe, infinite-altitude, Johnson, inner-Johnson, outer-Johnson, 1st Johnson-Yff, 2nd Johnson-Yff, 1st Kenmotu-centers, 2nd Kenmotu-centers, 1st Kenmotu-free-vertices, 2nd Kenmotu-free-vertices, Lucas(-1) homothetic, Lucas(+1) homothetic, Mandart-incircle, medial, 5th mixtilnear, 3rd tri-squares-central, 4th tri-squares-central, X3-ABC reflections, 1st Vijay-Paasche-Hutson, 3rd Vijay-Paasche-Hutson, 4th Vijay-Paasche-Hutson, 6th Vijay-Paasche-Hutson, 8th Vijay-Paasche-Hutson, 13th Vijay-Paasche-Hutson, 14th Vijay-Paasche-Hutson, 19th Vijay-Paasche-Hutson, 29th Vijay-Paasche-Hutson, 31th Vijay-Paasche-Hutson, Vijay-Paasche-midpoints, inner-Yff, outer-Yff, inner-Yff tangents, outer-Yff tangents).

New anti-triangles in ETC, found from definition of inverse-triangles, are showed in the following table:

45345
Triangle T anti-triangle of T (T-1)
A-vertex barycentric coordinates		 Triangle T anti-triangle of T (T-1)
A-vertex barycentric coordinates
ABC-X3 reflections ABC-X3 reflections 1st Kenmotu-centers 1st anti-Kenmotu-centers
-(b^2+c^2+2*S) : b^2 : c^2
1st Auriga 1st anti-Auriga
S*sqrt(R*(4*R+r))*a+(a+b+c)*((b^2+c^2)*a-(b+c)*(b-c)^2) :
(S*sqrt(R*(4*R+r))-b*(a+b+c)*(a-b+c))*b :
(S*sqrt(R*(4*R+r))-c*(a+b+c)*(a+b-c))*c		 2nd Kenmotu-centers 2nd anti-Kenmotu-centers
-(b^2+c^2-2*S) : b^2 : c^2
2nd Auriga 2nd anti-Auriga
-S*sqrt(R*(4*R+r))*a+(a+b+B3)*(a*(b^2+c^2)-(b+c)*(b-c)^2) :
(-S*sqrt(R*(4*R+r))+b*(a+b+c)*(a-b+c))*b :
(-S*sqrt(R*(4*R+r))+c*(a+b+c)*(a-c+b))*c 		1st Kenmotu-free-vertices 1st anti-Kenmotu-free-vertices
(SA+S)*(SB+SC+2*S) : (SB-S)*b^2 : (SC-S)*c^2
Ehrmann-mid anti-Ehrmann-mid
3*a^2*(-a^2+b^2+c^2) :
2*a^4-(b^2+4*c^2)*a^2-(b^2-c^2)*(b^2+2*c^2) :
2*a^4-(c^2+4*b^2)*a^2-(c^2-b^2)*(c^2+2*b^2)		 2nd Kenmotu-free-vertices 2nd anti-Kenmotu-free-vertices
(SA-S)*(SB+SC-2*S) : (SB+S)*b^2 : (SC+S)*c^2
inner-Garcia anti-inner-Garcia
-(a^2-b^2+b*c-c^2)*a^2 :
(a*(a*b-c^2)-(b^2-c^2)*(b-c))*b :
(a*(a*c-b^2)-(c^2-b^2)*(c-b))*c		 Lucas(-1) homothetic anti-Lucas(-1) homothetic
S*(2*S^2-(SA+SW)*S+SA^2) :
b^2*(2*S^2-(SB+SW)*S+SB^2) :
c^2*(2*S^2-(SC+SW)*S+SC^2)
outer-Garcia outer-Garcia Lucas(+1) homothetic anti-Lucas(+1) homothetic
-S*(2*S^2+(SA+SW)*S+SA^2) :
b^2*(2*S^2+(SB+SW)*S+SB^2) :
c^2*(2*S^2+(SC+SW)*S+SC^2)
Gemini 107 anti-Gemini 107
-1 : 2 : 2		 1st Parry 1st anti-Parry
-(SA-SB)*(SA-SC)*(2*S^2-(SB-SC)^2) :
b^2*(SA-SB)*(SB-SC)*(2*SW-3*SC) :
c^2*(SA-SC)*(SC-SB)*(2*SW-3*SB)
Gemini 109 anti-Gemini 109
-3 : 2 : 2		 2nd Parry 2nd anti-Parry
2*S^2*(5*S^2-6*R^2*(3*SA-SW)+5*SA^2-2*SB*SC-2*SW^2)/(3*SA-SW) :
(S^2-3*SA*SB)*b^2 :
(S^2-3*SA*SC)*c^2
Gemini 110 anti-Gemini 117
-3 : 1 : 1		 3rd tri-squares central 3rd anti-tri-squares central
-b^2-c^2-3*S : b^2+S : c^2+S
Gemini 111 anti-Gemini 111
2 : 1 : 1		 4th tri-squares central 4th anti-tri-squares central
-b^2-c^2+3*S : b^2-S : c^2-S
Gossard Gossard X(3)-ABC-reflections anti-X(3)-ABC-reflections
3*S^2-SB*SC : S^2-SA*SC : S^2-SA*SB
Johnson Johnson inner-Yff anti-inner-Yff
(a^4-2*(b^2+c^2)*a^2 - 2*(b+c)*b*c*a+(b^2-c^2)^2)/(2*a*b*c) : b : c
inner-Johnson 2nd Johnson-Yff outer-Yff anti-outer-Yff
(a^4-2*(b^2+c^2)*a^2 + 2*(b+c)*b*c*a+(b^2-c^2)^2)/(2*a*b*c) : -b : -c
outer-Johnson 1st Johnson-Yff inner-Yff tangents outer-Yff tangents
1st Johnson-Yff outer-Johnson outer-Yff tangents inner-Yff tangents
2nd Johnson-Yff inner-Johnson

X(45345) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 1st ANTI-KENMOTU CENTERS

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

X(45345) lies on these lines: {1, 45347}, {3, 45349}, {6, 5597}, {492, 26394}, {3102, 45360}, {6289, 26386}, {12305, 26290}, {18496, 45375}, {26296, 45426}, {26302, 45428}, {26310, 45434}, {26319, 45436}, {26326, 45440}, {26351, 45470}, {26359, 45472}, {26365, 45398}, {26371, 45400}, {26379, 45402}, {26380, 45404}, {26381, 45406}, {26382, 45444}, {26383, 45446}, {26387, 45460}, {26388, 45458}, {26389, 45456}, {26390, 45454}, {26391, 45415}, {26392, 45412}, {26393, 45416}, {26395, 45476}, {26397, 45421}, {26398, 43119}, {26399, 45422}, {26400, 45424}, {26401, 45496}, {26402, 45494}, {45352, 45411}, {45354, 45432}, {45355, 45438}, {45357, 45462}, {45361, 45464}, {45362, 45467}, {45365, 45484}, {45366, 45487}, {45369, 45488}, {45371, 45490}, {45373, 45492}

X(45346) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 2nd ANTI-KENMOTU CENTERS

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

X(45346) lies on these lines: {1, 45348}, {3, 45350}, {6, 5598}, {491, 26418}, {3103, 45359}, {6290, 26410}, {12306, 26291}, {18498, 45376}, {26297, 45427}, {26303, 45429}, {26311, 45435}, {26320, 45437}, {26327, 45441}, {26352, 45471}, {26360, 45473}, {26366, 45399}, {26372, 45401}, {26403, 45403}, {26404, 45405}, {26405, 45407}, {26406, 45445}, {26407, 45447}, {26411, 45461}, {26412, 45459}, {26413, 45457}, {26414, 45455}, {26415, 45413}, {26416, 45414}, {26417, 45417}, {26419, 45477}, {26420, 45420}, {26422, 43118}, {26423, 45423}, {26424, 45425}, {26425, 45497}, {26426, 45495}, {45351, 45410}, {45353, 45431}, {45356, 45439}, {45358, 45463}, {45363, 45466}, {45364, 45465}, {45367, 45485}, {45368, 45486}, {45370, 45489}, {45372, 45491}, {45374, 45493}

X(45347) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 1st ANTI-KENMOTU CENTERS

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

X(45347) lies on these lines: {1, 45345}, {3, 45351}, {6, 5598}, {492, 26418}, {3102, 45358}, {6289, 26410}, {12305, 26291}, {18498, 45375}, {26297, 45426}, {26303, 45428}, {26311, 45434}, {26320, 45436}, {26327, 45440}, {26352, 45470}, {26360, 45472}, {26366, 45398}, {26372, 45400}, {26403, 45402}, {26404, 45404}, {26405, 45406}, {26406, 45444}, {26407, 45446}, {26411, 45460}, {26412, 45458}, {26413, 45456}, {26414, 45454}, {26415, 45415}, {26416, 45412}, {26417, 45416}, {26419, 45476}, {26421, 45421}, {26422, 43119}, {26423, 45422}, {26424, 45424}, {26425, 45496}, {26426, 45494}, {45350, 45411}, {45353, 45430}, {45356, 45438}, {45359, 45462}, {45363, 45464}, {45364, 45467}, {45367, 45487}, {45368, 45484}, {45370, 45488}, {45372, 45490}, {45374, 45492}

X(45348) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 2nd ANTI-KENMOTU CENTERS

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

X(45348) lies on these lines: {1, 45346}, {3, 45352}, {6, 5597}, {491, 26394}, {3103, 45357}, {6290, 26386}, {12306, 26290}, {18496, 45376}, {26296, 45427}, {26302, 45429}, {26310, 45435}, {26319, 45437}, {26326, 45441}, {26351, 45471}, {26359, 45473}, {26365, 45399}, {26371, 45401}, {26379, 45403}, {26380, 45405}, {26381, 45407}, {26382, 45445}, {26383, 45447}, {26387, 45461}, {26388, 45459}, {26389, 45457}, {26390, 45455}, {26391, 45413}, {26392, 45414}, {26393, 45417}, {26395, 45477}, {26396, 45420}, {26398, 43118}, {26399, 45423}, {26400, 45425}, {26401, 45497}, {26402, 45495}, {45349, 45410}, {45354, 45433}, {45355, 45439}, {45360, 45463}, {45361, 45466}, {45362, 45465}, {45365, 45486}, {45366, 45485}, {45369, 45489}, {45371, 45491}, {45373, 45493}

X(45349) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 1st ANTI-KENMOTU-FREE-VERTICES

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

X(45349) lies on these lines: {1, 45351}, {3, 45345}, {39, 44583}, {182, 26398}, {372, 5597}, {641, 26359}, {5062, 44582}, {18496, 45377}, {26290, 45498}, {26296, 45530}, {26302, 45532}, {26310, 45538}, {26319, 45540}, {26326, 45544}, {26334, 45550}, {26344, 45553}, {26351, 45570}, {26365, 45500}, {26371, 45502}, {26379, 45504}, {26380, 45506}, {26381, 45510}, {26382, 45546}, {26383, 45548}, {26384, 45512}, {26385, 45515}, {26386, 45554}, {26387, 45562}, {26388, 45560}, {26389, 45558}, {26390, 45556}, {26391, 45519}, {26392, 45516}, {26393, 45520}, {26394, 45508}, {26395, 45572}, {26396, 45522}, {26397, 45525}, {26399, 45526}, {26400, 45528}, {26401, 45586}, {26402, 45584}, {45348, 45410}, {45354, 45536}, {45355, 45542}, {45360, 45565}, {45361, 45566}, {45362, 45569}, {45365, 45574}, {45366, 45577}, {45369, 45578}, {45371, 45580}, {45373, 45582}

X(45349) = X(35778)-of-1st anti-Kenmotu-free-vertices triangle

X(45350) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 2nd ANTI-KENMOTU-FREE-VERTICES

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

X(45350) lies on these lines: {1, 45352}, {3, 45346}, {39, 44584}, {182, 26422}, {371, 5598}, {642, 26360}, {5058, 44585}, {18498, 45378}, {26291, 45499}, {26297, 45531}, {26303, 45533}, {26311, 45539}, {26320, 45541}, {26327, 45545}, {26335, 45552}, {26345, 45551}, {26352, 45571}, {26366, 45501}, {26372, 45503}, {26403, 45505}, {26404, 45507}, {26405, 45511}, {26406, 45547}, {26407, 45549}, {26408, 45514}, {26409, 45513}, {26410, 45555}, {26411, 45563}, {26412, 45561}, {26413, 45559}, {26414, 45557}, {26415, 45517}, {26416, 45518}, {26417, 45521}, {26418, 45509}, {26419, 45573}, {26420, 45524}, {26421, 45523}, {26423, 45527}, {26424, 45529}, {26425, 45587}, {26426, 45585}, {45347, 45411}, {45353, 45535}, {45356, 45543}, {45359, 45564}, {45363, 45568}, {45364, 45567}, {45367, 45575}, {45368, 45576}, {45370, 45579}, {45372, 45581}, {45374, 45583}

X(45351) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 1st ANTI-KENMOTU-FREE-VERTICES

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

X(45351) lies on these lines: {1, 45349}, {3, 45347}, {39, 44585}, {182, 26422}, {372, 5598}, {641, 26360}, {5062, 44584}, {18498, 45377}, {26291, 45498}, {26297, 45530}, {26303, 45532}, {26311, 45538}, {26320, 45540}, {26327, 45544}, {26335, 45550}, {26345, 45553}, {26352, 45570}, {26366, 45500}, {26372, 45502}, {26403, 45504}, {26404, 45506}, {26405, 45510}, {26406, 45546}, {26407, 45548}, {26408, 45512}, {26409, 45515}, {26410, 45554}, {26411, 45562}, {26412, 45560}, {26413, 45558}, {26414, 45556}, {26415, 45519}, {26416, 45516}, {26417, 45520}, {26418, 45508}, {26419, 45572}, {26420, 45522}, {26421, 45525}, {26423, 45526}, {26424, 45528}, {26425, 45586}, {26426, 45584}, {45346, 45410}, {45353, 45534}, {45356, 45542}, {45358, 45565}, {45363, 45566}, {45364, 45569}, {45367, 45577}, {45368, 45574}, {45370, 45578}, {45372, 45580}, {45374, 45582}

X(45351) = X(35780)-of-1st anti-Kenmotu-free-vertices triangle

X(45352) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 2nd ANTI-KENMOTU-FREE-VERTICES

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

X(45352) lies on these lines: {1, 45350}, {3, 45348}, {39, 44582}, {182, 26398}, {371, 5597}, {642, 26359}, {5058, 44583}, {18496, 45378}, {26290, 45499}, {26296, 45531}, {26302, 45533}, {26310, 45539}, {26319, 45541}, {26326, 45545}, {26334, 45552}, {26344, 45551}, {26351, 45571}, {26365, 45501}, {26371, 45503}, {26379, 45505}, {26380, 45507}, {26381, 45511}, {26382, 45547}, {26383, 45549}, {26384, 45514}, {26385, 45513}, {26386, 45555}, {26387, 45563}, {26388, 45561}, {26389, 45559}, {26390, 45557}, {26391, 45517}, {26392, 45518}, {26393, 45521}, {26394, 45509}, {26395, 45573}, {26396, 45524}, {26397, 45523}, {26399, 45527}, {26400, 45529}, {26401, 45587}, {26402, 45585}, {45345, 45411}, {45354, 45537}, {45355, 45543}, {45357, 45564}, {45361, 45568}, {45362, 45567}, {45365, 45576}, {45366, 45575}, {45369, 45579}, {45371, 45581}, {45373, 45583}

X(45353) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 1st AURIGA

Barycentrics    a*(16*S^2*R*(4*R+r)-4*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*S*sqrt(R*(4*R+r))-a*(a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))) : :

X(45353) lies on these lines: {1, 3}, {1450, 8201}, {5599, 26360}, {5601, 26418}, {8190, 26303}, {8196, 26327}, {8197, 26406}, {8198, 26335}, {8199, 26345}, {8200, 26410}, {8202, 45363}, {11384, 26372}, {11837, 26403}, {11843, 26405}, {11861, 26311}, {11863, 26407}, {11865, 26414}, {11867, 26413}, {11869, 26412}, {11871, 26411}, {13890, 45368}, {13944, 45367}, {18495, 45356}, {18498, 45379}, {19007, 26408}, {19008, 26409}, {26415, 45589}, {26416, 45588}, {35778, 45359}, {35781, 45358}, {44584, 44600}, {44585, 44601}, {45346, 45431}, {45347, 45430}, {45350, 45535}, {45351, 45534}

X(45353) = X(45354)-of-1st Auriga triangle

X(45354) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 2nd AURIGA

Barycentrics    a*(16*S^2*R*(4*R+r)+4*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))*S*sqrt(R*(4*R+r))-a*(a+b+c)*(-a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))) : :

X(45354) lies on these lines: {1, 3}, {5600, 26359}, {5602, 26394}, {8191, 26302}, {8203, 26326}, {8204, 26382}, {8205, 26334}, {8206, 26344}, {8207, 26386}, {8208, 45362}, {8209, 45361}, {11385, 26371}, {11838, 26379}, {11844, 26381}, {11862, 26310}, {11864, 26383}, {11866, 26390}, {11868, 26389}, {11870, 26388}, {11872, 26387}, {13891, 45365}, {13945, 45366}, {18496, 45380}, {18497, 45355}, {19009, 26384}, {19010, 26385}, {26391, 45591}, {26392, 45590}, {35779, 45360}, {35780, 45357}, {44582, 44602}, {44583, 44603}, {45345, 45432}, {45348, 45433}, {45349, 45536}, {45352, 45537}

X(45354) = X(45353)-of-2nd Auriga triangle

X(45355) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND EHRMANN-MID

Barycentrics    -4*a*(a+b-c)*(-a+b+c)*(a-b+c)*S*sqrt(R*(4*R+r))+2*a^7-2*(b+c)*a^6-2*(b^2+c^2)*a^5+2*(b^3+c^3)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(45355) = 3*X(381)-X(45379)

X(45355) lies on these lines: {1, 18407}, {4, 26371}, {5, 26398}, {30, 26359}, {381, 5597}, {382, 26290}, {546, 26326}, {1478, 26387}, {1479, 26388}, {3091, 26381}, {3583, 26351}, {3585, 26380}, {3843, 45369}, {5598, 18499}, {5842, 26422}, {6564, 44582}, {6565, 44583}, {9818, 26302}, {9955, 26365}, {10895, 45371}, {10896, 45373}, {12699, 26382}, {13665, 26385}, {13785, 26384}, {18491, 26393}, {18492, 26296}, {18497, 45354}, {18500, 26310}, {18502, 26379}, {18507, 26383}, {18509, 26334}, {18511, 26344}, {18516, 26390}, {18517, 26389}, {18520, 45362}, {18522, 45361}, {18525, 26395}, {18538, 45365}, {18542, 26402}, {18544, 26401}, {18761, 26319}, {18762, 45366}, {26391, 45593}, {26392, 45592}, {26399, 45630}, {26400, 45631}, {35786, 45357}, {35787, 45360}, {45345, 45438}, {45348, 45439}, {45349, 45542}, {45352, 45543}

X(45355) = X(45379)-of-Ehrmann-mid triangle
X(45355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 18407, 45356), (381, 18496, 5597)

X(45356) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND EHRMANN-MID

Barycentrics    4*a*(a+b-c)*(-a+b+c)*(a-b+c)*S*sqrt(R*(4*R+r))+2*a^7-2*(b+c)*a^6-2*(b^2+c^2)*a^5+2*(b^3+c^3)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(45356) = 3*X(381)-X(45380)

X(45356) lies on these lines: {1, 18407}, {4, 26372}, {5, 26422}, {30, 26360}, {381, 5598}, {382, 26291}, {546, 26327}, {1478, 26411}, {1479, 26412}, {3091, 26405}, {3583, 26352}, {3585, 26404}, {3843, 45370}, {5597, 18499}, {5842, 26398}, {6564, 44584}, {6565, 44585}, {9818, 26303}, {9955, 26366}, {10895, 45372}, {10896, 45374}, {12699, 26406}, {13665, 26409}, {13785, 26408}, {18491, 26417}, {18492, 26297}, {18495, 45353}, {18500, 26311}, {18502, 26403}, {18507, 26407}, {18509, 26335}, {18511, 26345}, {18516, 26414}, {18517, 26413}, {18520, 45364}, {18522, 45363}, {18525, 26419}, {18538, 45368}, {18542, 26426}, {18544, 26425}, {18761, 26320}, {18762, 45367}, {26415, 45593}, {26416, 45592}, {26423, 45630}, {26424, 45631}, {35786, 45359}, {35787, 45358}, {45346, 45439}, {45347, 45438}, {45350, 45543}, {45351, 45542}

X(45356) = X(45380)-of-Ehrmann-mid triangle
X(45356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 18407, 45355), (381, 18498, 5598)

X(45357) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 1st KENMOTU-FREE-VERTICES

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

X(45357) lies on these lines: {1, 45359}, {6, 45360}, {371, 44582}, {372, 5597}, {485, 26394}, {3103, 45348}, {6200, 26290}, {6396, 26398}, {6419, 26385}, {6420, 44583}, {6560, 26381}, {6564, 26386}, {6565, 26326}, {10576, 26359}, {10679, 45358}, {18496, 23251}, {26296, 35774}, {26302, 35776}, {26310, 35782}, {26319, 35784}, {26334, 35792}, {26344, 35794}, {26351, 35808}, {26365, 35762}, {26371, 35764}, {26379, 35766}, {26380, 35768}, {26382, 35788}, {26383, 35790}, {26384, 35770}, {26387, 35802}, {26388, 35800}, {26389, 35798}, {26390, 35796}, {26391, 45601}, {26392, 45599}, {26393, 35772}, {26395, 35810}, {26397, 39661}, {26399, 45640}, {26400, 45642}, {26401, 35818}, {26402, 35816}, {35769, 45373}, {35780, 45354}, {35786, 45355}, {35804, 45362}, {35806, 45361}, {35809, 45371}, {35812, 45365}, {35814, 45366}, {45345, 45462}, {45352, 45564}

X(45357) = X(45534)-of-1st Kenmotu-free-vertices triangle
X(45357) = {X(6), X(45369)}-harmonic conjugate of X(45360)

X(45358) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 2nd KENMOTU-FREE-VERTICES

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

X(45358) lies on these lines: {1, 45360}, {6, 45359}, {371, 5598}, {372, 44585}, {486, 26418}, {3102, 45347}, {6200, 26422}, {6396, 26291}, {6419, 44584}, {6420, 26408}, {6561, 26405}, {6564, 26327}, {6565, 26410}, {10577, 26360}, {10679, 45357}, {18498, 23261}, {26297, 35775}, {26303, 35777}, {26311, 35783}, {26320, 35785}, {26335, 35795}, {26345, 35793}, {26352, 35809}, {26366, 35763}, {26372, 35765}, {26403, 35767}, {26404, 35769}, {26406, 35789}, {26407, 35791}, {26409, 35771}, {26411, 35803}, {26412, 35801}, {26413, 35799}, {26414, 35797}, {26415, 45600}, {26416, 45602}, {26417, 35773}, {26419, 35811}, {26420, 39660}, {26423, 45641}, {26424, 45643}, {26425, 35819}, {26426, 35817}, {35768, 45374}, {35781, 45353}, {35787, 45356}, {35805, 45363}, {35807, 45364}, {35808, 45372}, {35813, 45367}, {35815, 45368}, {45346, 45463}, {45351, 45565}

X(45358) = X(45537)-of-2nd Kenmotu-free-vertices triangle
X(45358) = {X(6), X(45370)}-harmonic conjugate of X(45359)

X(45359) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 1st KENMOTU-FREE-VERTICES

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

X(45359) lies on these lines: {1, 45357}, {6, 45358}, {371, 44584}, {372, 5598}, {485, 26418}, {3103, 45346}, {6200, 26291}, {6396, 26422}, {6419, 26409}, {6420, 44585}, {6560, 26405}, {6564, 26410}, {6565, 26327}, {10576, 26360}, {10679, 45360}, {18498, 23251}, {26297, 35774}, {26303, 35776}, {26311, 35782}, {26320, 35784}, {26335, 35792}, {26345, 35794}, {26352, 35808}, {26366, 35762}, {26372, 35764}, {26403, 35766}, {26404, 35768}, {26406, 35788}, {26407, 35790}, {26408, 35770}, {26411, 35802}, {26412, 35800}, {26413, 35798}, {26414, 35796}, {26415, 45601}, {26416, 45599}, {26417, 35772}, {26419, 35810}, {26421, 39661}, {26423, 45640}, {26424, 45642}, {26425, 35818}, {26426, 35816}, {35769, 45374}, {35778, 45353}, {35786, 45356}, {35804, 45364}, {35806, 45363}, {35809, 45372}, {35812, 45368}, {35814, 45367}, {45347, 45462}, {45350, 45564}

X(45359) = X(45536)-of-1st Kenmotu-free-vertices triangle
X(45359) = {X(6), X(45370)}-harmonic conjugate of X(45358)

X(45360) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 2nd KENMOTU-FREE-VERTICES

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

X(45360) lies on these lines: {1, 45358}, {6, 45357}, {371, 5597}, {372, 44583}, {486, 26394}, {3102, 45345}, {6200, 26398}, {6396, 26290}, {6419, 44582}, {6420, 26384}, {6561, 26381}, {6564, 26326}, {6565, 26386}, {10577, 26359}, {10679, 45359}, {18496, 23261}, {26296, 35775}, {26302, 35777}, {26310, 35783}, {26319, 35785}, {26334, 35795}, {26344, 35793}, {26351, 35809}, {26365, 35763}, {26371, 35765}, {26379, 35767}, {26380, 35769}, {26382, 35789}, {26383, 35791}, {26385, 35771}, {26387, 35803}, {26388, 35801}, {26389, 35799}, {26390, 35797}, {26391, 45600}, {26392, 45602}, {26393, 35773}, {26395, 35811}, {26396, 39660}, {26399, 45641}, {26400, 45643}, {26401, 35819}, {26402, 35817}, {35768, 45373}, {35779, 45354}, {35787, 45355}, {35805, 45361}, {35807, 45362}, {35808, 45371}, {35813, 45366}, {35815, 45365}, {45348, 45463}, {45349, 45565}

X(45360) = X(45535)-of-2nd Kenmotu-free-vertices triangle
X(45360) = {X(6), X(45369)}-harmonic conjugate of X(45357)

X(45361) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND LUCAS(-1) HOMOTHETIC

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

X(45361) lies on these lines: {1, 45363}, {494, 5597}, {6461, 45362}, {6463, 26394}, {8189, 26296}, {8195, 26302}, {8209, 45354}, {8211, 26395}, {8213, 26326}, {8215, 26382}, {8217, 26334}, {8219, 26344}, {8221, 26386}, {8223, 26359}, {10876, 26310}, {10946, 26390}, {10952, 26389}, {11378, 26365}, {11395, 26371}, {11504, 26393}, {11829, 26290}, {11841, 26379}, {11847, 26381}, {11908, 26383}, {11931, 26388}, {11933, 26387}, {11948, 26351}, {11950, 45369}, {11952, 45371}, {11954, 45373}, {11956, 26402}, {11958, 26401}, {13900, 45365}, {13957, 45366}, {18496, 45382}, {18522, 45355}, {18964, 26380}, {19033, 26384}, {19034, 26385}, {22762, 26319}, {26391, 45603}, {26398, 45624}, {26399, 45644}, {26400, 45646}, {35805, 45360}, {35806, 45357}, {44582, 44629}, {44583, 44630}, {45345, 45464}, {45348, 45466}, {45349, 45566}, {45352, 45568}

X(45361) = X(45588)-of-Lucas(-1) homothetic triangle

X(45362) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND LUCAS(+1) HOMOTHETIC

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

X(45362) lies on these lines: {1, 45364}, {493, 5597}, {6461, 45361}, {6462, 26394}, {8188, 26296}, {8194, 26302}, {8208, 45354}, {8210, 26395}, {8212, 26326}, {8214, 26382}, {8216, 26334}, {8218, 26344}, {8220, 26386}, {8222, 26359}, {10875, 26310}, {10945, 26390}, {10951, 26389}, {11377, 26365}, {11394, 26371}, {11503, 26393}, {11828, 26290}, {11840, 26379}, {11846, 26381}, {11907, 26383}, {11930, 26388}, {11932, 26387}, {11947, 26351}, {11949, 45369}, {11951, 45371}, {11953, 45373}, {11955, 26402}, {11957, 26401}, {13899, 45365}, {13956, 45366}, {18496, 45381}, {18520, 45355}, {18963, 26380}, {19031, 26384}, {19032, 26385}, {22761, 26319}, {26392, 45604}, {26398, 45623}, {26399, 45645}, {26400, 45647}, {35804, 45357}, {35807, 45360}, {44582, 44627}, {44583, 44628}, {45345, 45467}, {45348, 45465}, {45349, 45569}, {45352, 45567}

X(45362) = X(45589)-of-Lucas(+1) homothetic triangle

X(45363) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND LUCAS(-1) HOMOTHETIC

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

X(45363) lies on these lines: {1, 45361}, {494, 5598}, {6461, 45364}, {6463, 26418}, {8189, 26297}, {8195, 26303}, {8202, 45353}, {8211, 26419}, {8213, 26327}, {8215, 26406}, {8217, 26335}, {8219, 26345}, {8221, 26410}, {8223, 26360}, {10876, 26311}, {10946, 26414}, {10952, 26413}, {11378, 26366}, {11395, 26372}, {11504, 26417}, {11829, 26291}, {11841, 26403}, {11847, 26405}, {11908, 26407}, {11931, 26412}, {11933, 26411}, {11948, 26352}, {11950, 45370}, {11952, 45372}, {11954, 45374}, {11956, 26426}, {11958, 26425}, {13900, 45368}, {13957, 45367}, {18498, 45382}, {18522, 45356}, {18964, 26404}, {19033, 26408}, {19034, 26409}, {22762, 26320}, {26415, 45603}, {26422, 45624}, {26423, 45644}, {26424, 45646}, {35805, 45358}, {35806, 45359}, {44584, 44629}, {44585, 44630}, {45346, 45466}, {45347, 45464}, {45350, 45568}, {45351, 45566}

X(45363) = X(45590)-of-Lucas(-1) homothetic triangle

X(45364) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND LUCAS(+1) HOMOTHETIC

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

X(45364) lies on these lines: {1, 45362}, {493, 5598}, {1450, 8201}, {6461, 45363}, {6462, 26418}, {8188, 26297}, {8194, 26303}, {8210, 26419}, {8212, 26327}, {8214, 26406}, {8216, 26335}, {8218, 26345}, {8220, 26410}, {8222, 26360}, {10875, 26311}, {10945, 26414}, {10951, 26413}, {11377, 26366}, {11394, 26372}, {11503, 26417}, {11828, 26291}, {11840, 26403}, {11846, 26405}, {11907, 26407}, {11930, 26412}, {11932, 26411}, {11947, 26352}, {11949, 45370}, {11951, 45372}, {11953, 45374}, {11955, 26426}, {11957, 26425}, {13899, 45368}, {13956, 45367}, {18498, 45381}, {18520, 45356}, {18963, 26404}, {19031, 26408}, {19032, 26409}, {22761, 26320}, {26416, 45604}, {26422, 45623}, {26423, 45645}, {26424, 45647}, {35804, 45359}, {35807, 45358}, {44584, 44627}, {44585, 44628}, {45346, 45465}, {45347, 45467}, {45350, 45567}, {45351, 45569}

X(45364) = X(45591)-of-Lucas(+1) homothetic triangle

X(45365) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 3rd TRI-SQUARES-CENTRAL

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

X(45365) lies on these lines: {1, 45368}, {2, 26385}, {6, 45366}, {371, 26326}, {590, 26359}, {3068, 5597}, {6690, 45367}, {7583, 26398}, {7585, 26384}, {8972, 26394}, {8974, 26334}, {8975, 26344}, {8976, 26386}, {9540, 26290}, {13883, 26365}, {13884, 26371}, {13885, 26379}, {13886, 26381}, {13887, 26393}, {13888, 26296}, {13889, 26302}, {13891, 45354}, {13892, 26310}, {13893, 26382}, {13894, 26383}, {13895, 26390}, {13896, 26389}, {13897, 26388}, {13898, 26387}, {13899, 45362}, {13900, 45361}, {13901, 26351}, {13902, 26395}, {13903, 45369}, {13904, 45371}, {13905, 45373}, {13906, 26402}, {13907, 26401}, {18496, 45384}, {18538, 45355}, {18965, 26380}, {22763, 26319}, {26391, 45607}, {26392, 45605}, {26399, 45650}, {26400, 45652}, {32787, 44583}, {35812, 45357}, {35815, 45360}, {45345, 45484}, {45348, 45486}, {45349, 45574}, {45352, 45576}

X(45365) = {X(590), X(44582)}-harmonic conjugate of X(26359)

X(45366) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND 4th TRI-SQUARES-CENTRAL

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

X(45366) lies on these lines: {1, 45367}, {2, 26384}, {6, 45365}, {372, 26326}, {615, 26359}, {3069, 5597}, {6690, 45368}, {7584, 26398}, {7586, 26385}, {13935, 26290}, {13936, 26365}, {13937, 26371}, {13938, 26379}, {13939, 26381}, {13940, 26393}, {13941, 26394}, {13942, 26296}, {13943, 26302}, {13945, 45354}, {13946, 26310}, {13947, 26382}, {13948, 26383}, {13949, 26334}, {13950, 26344}, {13951, 26386}, {13952, 26390}, {13953, 26389}, {13954, 26388}, {13955, 26387}, {13956, 45362}, {13957, 45361}, {13958, 26351}, {13959, 26395}, {13961, 45369}, {13962, 45371}, {13963, 45373}, {13964, 26402}, {13965, 26401}, {18496, 45385}, {18762, 45355}, {18966, 26380}, {22764, 26319}, {26391, 45606}, {26392, 45608}, {26399, 45651}, {26400, 45653}, {32788, 44582}, {35813, 45360}, {35814, 45357}, {45345, 45487}, {45348, 45485}, {45349, 45577}, {45352, 45575}

X(45366) = {X(615), X(44583)}-harmonic conjugate of X(26359)

X(45367) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 4th TRI-SQUARES-CENTRAL

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

X(45367) lies on these lines: {1, 45366}, {2, 26408}, {6, 45368}, {372, 26327}, {615, 26360}, {3069, 5598}, {6690, 45365}, {7584, 26422}, {7586, 26409}, {13935, 26291}, {13936, 26366}, {13937, 26372}, {13938, 26403}, {13939, 26405}, {13940, 26417}, {13941, 26418}, {13942, 26297}, {13943, 26303}, {13944, 45353}, {13946, 26311}, {13947, 26406}, {13948, 26407}, {13949, 26335}, {13950, 26345}, {13951, 26410}, {13952, 26414}, {13953, 26413}, {13954, 26412}, {13955, 26411}, {13956, 45364}, {13957, 45363}, {13958, 26352}, {13959, 26419}, {13961, 45370}, {13962, 45372}, {13963, 45374}, {13964, 26426}, {13965, 26425}, {18498, 45385}, {18762, 45356}, {18966, 26404}, {22764, 26320}, {26415, 45606}, {26416, 45608}, {26423, 45651}, {26424, 45653}, {32788, 44584}, {35813, 45358}, {35814, 45359}, {45346, 45485}, {45347, 45487}, {45350, 45575}, {45351, 45577}

X(45367) = {X(615), X(44585)}-harmonic conjugate of X(26360)

X(45368) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND 3rd TRI-SQUARES-CENTRAL

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

X(45368) lies on these lines: {1, 45365}, {2, 26409}, {6, 45367}, {371, 26327}, {590, 26360}, {3068, 5598}, {6690, 45366}, {7583, 26422}, {7585, 26408}, {8972, 26418}, {8974, 26335}, {8975, 26345}, {8976, 26410}, {9540, 26291}, {13883, 26366}, {13884, 26372}, {13885, 26403}, {13886, 26405}, {13887, 26417}, {13888, 26297}, {13889, 26303}, {13890, 45353}, {13892, 26311}, {13893, 26406}, {13894, 26407}, {13895, 26414}, {13896, 26413}, {13897, 26412}, {13898, 26411}, {13899, 45364}, {13900, 45363}, {13901, 26352}, {13902, 26419}, {13903, 45370}, {13904, 45372}, {13905, 45374}, {13906, 26426}, {13907, 26425}, {18498, 45384}, {18538, 45356}, {18965, 26404}, {22763, 26320}, {26415, 45607}, {26416, 45605}, {26423, 45650}, {26424, 45652}, {32787, 44585}, {35812, 45359}, {35815, 45358}, {45346, 45486}, {45347, 45484}, {45350, 45576}, {45351, 45574}

X(45368) = {X(590), X(44584)}-harmonic conjugate of X(26360)

X(45369) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND X3-ABC REFLECTIONS

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

X(45369) lies on these lines: {1, 3}, {4, 18496}, {5, 26394}, {6, 45357}, {30, 26381}, {381, 26326}, {1598, 26371}, {1656, 26359}, {3311, 44582}, {3312, 44583}, {3843, 45355}, {5790, 26382}, {6417, 26385}, {6418, 26384}, {7517, 26302}, {9301, 26310}, {9654, 26388}, {9669, 26387}, {11842, 26379}, {11911, 26383}, {11916, 26334}, {11917, 26344}, {11928, 26390}, {11929, 26389}, {11949, 45362}, {11950, 45361}, {13903, 45365}, {13961, 45366}, {26391, 45610}, {26392, 45609}, {45345, 45488}, {45348, 45489}, {45349, 45578}, {45352, 45579}

X(45369) = reflection of X(3) in X(45620)
X(45369) = X(11366)-of-anti-outer-Yff triangle
X(45369) = X(45620)-of-X3-ABC reflections triangle
X(45369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10679, 45370), (5597, 26290, 26398), (10202, 16203, 45370), (26290, 26398, 3), (26326, 26386, 381), (26351, 45371, 3295), (26380, 45373, 999), (45357, 45360, 6)

X(45370) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND X3-ABC REFLECTIONS

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

X(45370) lies on these lines: {1, 3}, {4, 18498}, {5, 26418}, {6, 45358}, {30, 26405}, {381, 26327}, {1598, 26372}, {1656, 26360}, {3311, 44584}, {3312, 44585}, {3843, 45356}, {5790, 26406}, {6417, 26409}, {6418, 26408}, {7517, 26303}, {9301, 26311}, {9654, 26412}, {9669, 26411}, {11842, 26403}, {11911, 26407}, {11916, 26335}, {11917, 26345}, {11928, 26414}, {11929, 26413}, {11949, 45364}, {11950, 45363}, {13903, 45368}, {13961, 45367}, {26415, 45610}, {26416, 45609}, {45346, 45489}, {45347, 45488}, {45350, 45579}, {45351, 45578}

X(45370) = reflection of X(3) in X(45621)
X(45370) = X(11367)-of-anti-outer-Yff triangle
X(45370) = X(45621)-of-X3-ABC reflections triangle
X(45370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10679, 45369), (5598, 26291, 26422), (10202, 16203, 45369), (26291, 26422, 3), (26327, 26410, 381), (26352, 45372, 3295), (26404, 45374, 999), (45358, 45359, 6)

X(45371) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND INNER-YFF

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

X(45371) lies on these lines: {1, 3}, {5, 26387}, {12, 26386}, {388, 26381}, {495, 26388}, {498, 26359}, {1124, 44583}, {1335, 44582}, {1479, 26326}, {3085, 26394}, {3299, 26384}, {3301, 26385}, {9654, 18496}, {10037, 26302}, {10038, 26310}, {10039, 26382}, {10040, 26334}, {10041, 26344}, {10523, 26390}, {10801, 26379}, {10895, 45355}, {10954, 26389}, {11398, 26371}, {11912, 26383}, {11951, 45362}, {11952, 45361}, {13904, 45365}, {13962, 45366}, {26391, 45612}, {26392, 45611}, {35808, 45360}, {35809, 45357}, {45345, 45490}, {45348, 45491}, {45349, 45580}, {45352, 45581}

X(45371) = X(45625)-of-inner-Yff triangle
X(45371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45374), (1, 5597, 45373), (3295, 45369, 26351), (5570, 11507, 45372), (5597, 26395, 26399)

X(45372) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND INNER-YFF

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

X(45372) lies on these lines: {1, 3}, {5, 26411}, {12, 26410}, {388, 26405}, {495, 26412}, {498, 26360}, {1124, 44585}, {1335, 44584}, {1479, 26327}, {3085, 26418}, {3299, 26408}, {3301, 26409}, {9654, 18498}, {10037, 26303}, {10038, 26311}, {10039, 26406}, {10040, 26335}, {10041, 26345}, {10523, 26414}, {10801, 26403}, {10895, 45356}, {10954, 26413}, {11398, 26372}, {11912, 26407}, {11951, 45364}, {11952, 45363}, {13904, 45368}, {13962, 45367}, {26415, 45612}, {26416, 45611}, {35808, 45358}, {35809, 45359}, {45346, 45491}, {45347, 45490}, {45350, 45581}, {45351, 45580}

X(45372) = X(45626)-of-inner-Yff triangle
X(45372) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45373), (1, 5598, 45374), (3295, 45370, 26352), (5570, 11507, 45371), (5598, 26419, 26423)

X(45373) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-AURIGA AND OUTER-YFF

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

X(45373) lies on these lines: {1, 3}, {5, 26388}, {11, 26386}, {496, 26387}, {497, 26381}, {499, 26359}, {1124, 44582}, {1335, 44583}, {1478, 26326}, {1737, 26382}, {3086, 26394}, {3299, 26385}, {3301, 26384}, {9669, 18496}, {10046, 26302}, {10047, 26310}, {10048, 26334}, {10049, 26344}, {10523, 26389}, {10802, 26379}, {10896, 45355}, {10948, 26390}, {11399, 26371}, {11913, 26383}, {11953, 45362}, {11954, 45361}, {13905, 45365}, {13963, 45366}, {26391, 45614}, {26392, 45613}, {35768, 45360}, {35769, 45357}, {45345, 45492}, {45348, 45493}, {45349, 45582}, {45352, 45583}

X(45373) = X(45627)-of-outer-Yff triangle
X(45373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45372), (1, 5597, 45371), (1, 32760, 5598), (999, 45369, 26380), (5597, 26395, 26400)

X(45374) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-AURIGA AND OUTER-YFF

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

X(45374) lies on these lines: {1, 3}, {5, 26412}, {11, 26410}, {496, 26411}, {497, 26405}, {499, 26360}, {1124, 44584}, {1335, 44585}, {1478, 26327}, {1737, 26406}, {3086, 26418}, {3299, 26409}, {3301, 26408}, {9669, 18498}, {10046, 26303}, {10047, 26311}, {10048, 26335}, {10049, 26345}, {10523, 26413}, {10802, 26403}, {10896, 45356}, {10948, 26414}, {11399, 26372}, {11913, 26407}, {11953, 45364}, {11954, 45363}, {13905, 45368}, {13963, 45367}, {26415, 45614}, {26416, 45613}, {35768, 45358}, {35769, 45359}, {45346, 45493}, {45347, 45492}, {45350, 45583}, {45351, 45582}

X(45374) = X(45628)-of-outer-Yff triangle
X(45374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45371), (1, 5598, 45372), (1, 32760, 5597), (999, 45370, 26404), (5598, 26419, 26424)

X(45375) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 1st ANTI-KENMOTU CENTERS

Barycentrics    3*a^6-2*(b^2+c^2)*a^4+(b^2+c^2)^2*a^2+(2*a^4+2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S-2*(b^4-c^4)*(b^2-c^2) : :
X(45375) = 3*X(381)-2*X(6564)

X(45375) lies on these lines: {3, 639}, {4, 43133}, {5, 45378}, {6, 13}, {30, 492}, {371, 45542}, {382, 9733}, {591, 3830}, {999, 45460}, {1656, 43119}, {1657, 12305}, {3070, 22625}, {3095, 12601}, {3102, 23261}, {3295, 45458}, {3618, 13794}, {3843, 6251}, {3845, 26438}, {5870, 26468}, {6421, 37243}, {6560, 13771}, {6811, 45579}, {9654, 45490}, {9655, 45404}, {9668, 45470}, {9669, 45492}, {12702, 45444}, {13758, 18762}, {13951, 39679}, {15069, 44654}, {18480, 45426}, {18493, 45398}, {18494, 45400}, {18496, 45345}, {18498, 45347}, {18501, 45402}, {18503, 45434}, {18508, 45446}, {18518, 45456}, {18519, 45454}, {18521, 45415}, {18523, 45412}, {18524, 45416}, {18526, 45476}, {18542, 45424}, {18543, 45496}, {18544, 45422}, {18545, 45494}, {18553, 44485}, {22806, 32787}, {22819, 35820}, {23251, 45462}, {26321, 45436}, {33878, 36733}, {36656, 45489}, {45379, 45430}, {45380, 45432}, {45381, 45467}, {45382, 45464}, {45384, 45484}, {45385, 45487}

X(45375) = reflection of X(i) in X(j) for these (i, j): (3, 45554), (45511, 5), (45579, 6811)
X(45375) = X(6564)-of-anti-Ehrmann-mid triangle
X(45375) = X(45511)-of-Johnson triangle
X(45375) = X(45554)-of-X3-ABC reflections triangle
X(45375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 45406, 45411), (6, 6033, 45376), (6, 45438, 381), (381, 18440, 45376), (381, 26346, 18440), (381, 39899, 13665), (3818, 6565, 381), (6289, 13748, 3), (13785, 18511, 381)

X(45376) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 2nd ANTI-KENMOTU CENTERS

Barycentrics    3*a^6-2*(b^2+c^2)*a^4+(b^2+c^2)^2*a^2-(2*a^4+2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S-2*(b^4-c^4)*(b^2-c^2) : :
X(45376) = 3*X(381)-2*X(6565)

X(45376) lies on these lines: {3, 640}, {4, 43134}, {5, 45377}, {6, 13}, {30, 491}, {372, 45543}, {382, 9732}, {999, 45461}, {1656, 43118}, {1657, 12306}, {1991, 3830}, {3071, 22596}, {3095, 12602}, {3103, 23251}, {3295, 45459}, {3618, 13674}, {3843, 6250}, {3845, 18539}, {5871, 26469}, {6422, 37243}, {6561, 13650}, {6813, 45578}, {8976, 39648}, {9654, 45491}, {9655, 45405}, {9668, 45471}, {9669, 45493}, {12702, 45445}, {13638, 18538}, {15069, 44655}, {18480, 45427}, {18493, 45399}, {18494, 45401}, {18496, 45348}, {18498, 45346}, {18501, 45403}, {18503, 45435}, {18508, 45447}, {18518, 45457}, {18519, 45455}, {18521, 45413}, {18523, 45414}, {18524, 45417}, {18526, 45477}, {18542, 45425}, {18543, 45497}, {18544, 45423}, {18545, 45495}, {18553, 44486}, {22807, 32788}, {22820, 35821}, {23261, 45463}, {26321, 45437}, {31463, 37345}, {33878, 36719}, {36655, 45488}, {45379, 45431}, {45380, 45433}, {45381, 45465}, {45382, 45466}, {45384, 45486}, {45385, 45485}

X(45376) = reflection of X(i) in X(j) for these (i, j): (3, 45555), (45510, 5), (45578, 6813)
X(45376) = X(6565)-of-anti-Ehrmann-mid triangle
X(45376) = X(45510)-of-Johnson triangle
X(45376) = X(45555)-of-X3-ABC reflections triangle
X(45376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 45407, 45410), (6, 6033, 45375), (6, 45439, 381), (381, 18440, 45375), (381, 26336, 18440), (381, 39899, 13785), (3818, 6564, 381), (6290, 13749, 3), (13665, 18509, 381)

X(45377) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 1st ANTI-KENMOTU-FREE-VERTICES

Barycentrics    3*a^6-2*(b^2+c^2)*a^4+(b^2+c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)-2*S*(a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2) : :
X(45377) = 3*X(381)-2*X(35786)

X(45377) lies on these lines: {3, 639}, {4, 45578}, {5, 45376}, {30, 45508}, {39, 13785}, {182, 1656}, {372, 381}, {382, 9739}, {542, 10576}, {550, 18539}, {999, 45562}, {1152, 45438}, {1657, 45498}, {3295, 45560}, {3526, 26346}, {3534, 7690}, {3818, 10577}, {3830, 41490}, {3843, 45544}, {3850, 26438}, {5062, 13665}, {5079, 26336}, {6033, 43118}, {6287, 6399}, {6565, 7748}, {6811, 7774}, {8976, 39899}, {9654, 45580}, {9655, 45506}, {9668, 45570}, {9669, 45582}, {9744, 45411}, {12702, 45546}, {14848, 44474}, {15069, 44475}, {15484, 18993}, {15884, 37243}, {18358, 32490}, {18480, 45530}, {18493, 45500}, {18494, 45502}, {18496, 45349}, {18498, 45351}, {18501, 45504}, {18503, 45538}, {18508, 45548}, {18510, 45512}, {18512, 45515}, {18518, 45558}, {18519, 45556}, {18521, 45519}, {18523, 45516}, {18524, 45520}, {18526, 45572}, {18542, 45528}, {18543, 45586}, {18544, 45526}, {18545, 45584}, {22625, 42259}, {22806, 41946}, {22819, 42267}, {23261, 45565}, {26321, 45540}, {31481, 39893}, {36656, 45488}, {45379, 45534}, {45380, 45536}, {45381, 45569}, {45382, 45566}, {45384, 45574}, {45385, 45577}

X(45377) = X(35786)-of-these triangles: {anti-Ehrmann-mid, 1st anti-Kenmotu-free-vertices}
X(45377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 45510, 45410), (372, 45542, 381), (1656, 18440, 45378)

X(45378) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 2nd ANTI-KENMOTU-FREE-VERTICES

Barycentrics    3*a^6-2*(b^2+c^2)*a^4+(b^2+c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)+2*S*(a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2) : :
X(45378) = 3*X(381)-2*X(35787)

X(45378) lies on these lines: {3, 640}, {4, 45579}, {5, 45375}, {30, 45509}, {39, 13665}, {182, 1656}, {371, 381}, {382, 9738}, {542, 10577}, {550, 26438}, {999, 45563}, {1151, 45439}, {1657, 45499}, {3295, 45561}, {3526, 26336}, {3534, 7692}, {3818, 10576}, {3830, 41491}, {3843, 45545}, {3850, 18539}, {5058, 13785}, {5079, 26346}, {6033, 43119}, {6222, 6287}, {6564, 7748}, {6813, 7774}, {8960, 19130}, {9654, 45581}, {9655, 45507}, {9668, 45571}, {9669, 45583}, {9744, 45410}, {12702, 45547}, {13951, 39899}, {14848, 44473}, {15069, 44476}, {15484, 18994}, {15883, 37243}, {18358, 32491}, {18480, 45531}, {18493, 45501}, {18494, 45503}, {18496, 45352}, {18498, 45350}, {18501, 45505}, {18503, 45539}, {18508, 45549}, {18510, 45514}, {18512, 45513}, {18518, 45559}, {18519, 45557}, {18521, 45517}, {18523, 45518}, {18524, 45521}, {18526, 45573}, {18542, 45529}, {18543, 45587}, {18544, 45527}, {18545, 45585}, {22596, 42258}, {22807, 41945}, {22820, 42266}, {23251, 45564}, {26321, 45541}, {36655, 45489}, {45379, 45535}, {45380, 45537}, {45381, 45567}, {45382, 45568}, {45384, 45576}, {45385, 45575}

X(45378) = X(35787)-of-anti-Ehrmann-mid triangle
X(45378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 45511, 45411), (371, 45543, 381), (1656, 18440, 45377)

X(45379) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 1st AURIGA

Barycentrics    (3*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+2*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2)*sqrt(R*(4*R+r))+2*S*(-a+b+c)*a^2 : :
X(45379) = 3*X(381)-2*X(45355)

X(45379) lies on these lines: {3, 5599}, {4, 11875}, {5, 11843}, {30, 5601}, {55, 5441}, {381, 5597}, {382, 11252}, {399, 12466}, {952, 11876}, {999, 11871}, {1656, 45620}, {1657, 11822}, {3295, 11869}, {3534, 35244}, {3830, 11207}, {3843, 8196}, {5598, 18526}, {5602, 37705}, {8148, 12454}, {8186, 18480}, {8187, 28204}, {8197, 12702}, {8198, 26336}, {8199, 26346}, {8201, 45381}, {8202, 45382}, {8203, 10247}, {9654, 11877}, {9655, 18955}, {9668, 11873}, {9669, 11879}, {11253, 12645}, {11366, 18493}, {11384, 18494}, {11492, 18524}, {11493, 26321}, {11837, 18501}, {11844, 28224}, {11861, 18503}, {11863, 18508}, {11865, 18519}, {11867, 18518}, {11881, 18545}, {11883, 18543}, {12164, 12415}, {12179, 38744}, {12365, 38790}, {12452, 18440}, {12462, 38756}, {12902, 13208}, {13176, 38733}, {13665, 44600}, {13785, 44601}, {13890, 45384}, {13944, 45385}, {18498, 45353}, {18510, 19007}, {18512, 19008}, {18521, 45589}, {18523, 45588}, {18542, 45627}, {18544, 45625}, {23251, 35778}, {23261, 35781}, {43835, 43850}, {45375, 45430}, {45376, 45431}, {45377, 45534}, {45378, 45535}

X(45379) = X(45355)-of-these triangles: {anti-Ehrmann-mid, 1st Auriga}
X(45379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 32146, 11875), (55, 18525, 45380), (5597, 18495, 381), (8200, 9834, 3)

X(45380) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 2nd AURIGA

Barycentrics    -(3*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+2*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2)*sqrt(R*(4*R+r))+2*S*(-a+b+c)*a^2 : :
X(45380) = 3*X(381)-2*X(45356)

X(45380) lies on these lines: {3, 5600}, {4, 11876}, {5, 11844}, {30, 5602}, {55, 5441}, {381, 5598}, {382, 11253}, {399, 12467}, {952, 11875}, {999, 11872}, {1656, 45621}, {1657, 11823}, {3295, 11870}, {3534, 35245}, {3830, 11208}, {3843, 8203}, {5597, 18526}, {5601, 37705}, {8148, 12455}, {8186, 28204}, {8187, 18480}, {8196, 10247}, {8204, 12702}, {8205, 26336}, {8206, 26346}, {8208, 45381}, {8209, 45382}, {9654, 11878}, {9655, 18956}, {9668, 11874}, {9669, 11880}, {11252, 12645}, {11367, 18493}, {11385, 18494}, {11492, 26321}, {11493, 18524}, {11838, 18501}, {11843, 28224}, {11862, 18503}, {11864, 18508}, {11866, 18519}, {11868, 18518}, {11882, 18545}, {11884, 18543}, {12164, 12416}, {12180, 38744}, {12366, 38790}, {12453, 18440}, {12463, 38756}, {12902, 13209}, {13177, 38733}, {13665, 44602}, {13785, 44603}, {13891, 45384}, {13945, 45385}, {18496, 45354}, {18510, 19009}, {18512, 19010}, {18521, 45591}, {18523, 45590}, {18542, 45628}, {18544, 45626}, {23251, 35780}, {23261, 35779}, {43835, 43851}, {45375, 45432}, {45376, 45433}, {45377, 45536}, {45378, 45537}

X(45380) = X(45356)-of-these triangles: {anti-Ehrmann-mid, 2nd Auriga}
X(45380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 32147, 11876), (55, 18525, 45379), (5598, 18497, 381), (8207, 9835, 3)

X(45381) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND LUCAS(+1) HOMOTHETIC

Barycentrics    (-a^2+b^2+c^2)*(-2*(a^6-3*(b^2+c^2)*a^4+(b^2-3*c^2)*(3*b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2))*a^2+(a^6+12*(b^2+c^2)*a^4-(3*b^4-2*b^2*c^2+3*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2))*S) : :
X(45381) = 3*X(381)-2*X(45593)

X(45381) lies on these lines: {3, 8194}, {4, 11949}, {5, 11846}, {30, 6462}, {381, 493}, {382, 10669}, {399, 12894}, {999, 11932}, {1656, 45623}, {1657, 11828}, {3295, 11930}, {3830, 12152}, {3843, 8212}, {6461, 45382}, {8148, 12636}, {8188, 18480}, {8201, 45379}, {8208, 45380}, {8210, 18526}, {8214, 12702}, {8216, 26336}, {8218, 26346}, {9654, 11951}, {9655, 18963}, {9668, 11947}, {9669, 11953}, {10875, 18503}, {10945, 18519}, {10951, 18518}, {11377, 18493}, {11394, 18494}, {11503, 18524}, {11840, 18501}, {11907, 18508}, {11955, 18545}, {11957, 18543}, {12164, 12426}, {12186, 38744}, {12377, 38790}, {12440, 18525}, {12590, 18440}, {12765, 38756}, {12902, 13215}, {13184, 38733}, {13665, 44627}, {13785, 44628}, {13899, 45384}, {13956, 45385}, {18496, 45362}, {18498, 45364}, {18510, 19031}, {18512, 19032}, {18523, 45604}, {18542, 45647}, {18544, 45645}, {22761, 26321}, {23251, 35804}, {23261, 35807}, {45375, 45467}, {45376, 45465}, {45377, 45569}, {45378, 45567}

X(45381) = X(45593)-of-these triangles: {anti-Ehrmann-mid, Lucas(+1) homothetic}
X(45381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 32177, 11949), (493, 18520, 381), (8220, 9838, 3)

X(45382) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND LUCAS(-1) HOMOTHETIC

Barycentrics    (-a^2+b^2+c^2)*(-2*(a^6-3*(b^2+c^2)*a^4+(b^2-3*c^2)*(3*b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2))*a^2-(a^6+12*(b^2+c^2)*a^4-(3*b^4-2*b^2*c^2+3*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2))*S) : :
X(45382) = 3*X(381)-2*X(45592)

X(45382) lies on these lines: {3, 8195}, {4, 11950}, {5, 11847}, {30, 6463}, {381, 494}, {382, 10673}, {399, 12895}, {999, 11933}, {1656, 45624}, {1657, 11829}, {3295, 11931}, {3830, 12153}, {3843, 8213}, {6461, 45381}, {8148, 12637}, {8189, 18480}, {8202, 45379}, {8209, 45380}, {8211, 18526}, {8215, 12702}, {8217, 26336}, {8219, 26346}, {9654, 11952}, {9655, 18964}, {9668, 11948}, {9669, 11954}, {10876, 18503}, {10946, 18519}, {10952, 18518}, {11378, 18493}, {11395, 18494}, {11504, 18524}, {11841, 18501}, {11908, 18508}, {11956, 18545}, {11958, 18543}, {12164, 12427}, {12187, 38744}, {12378, 38790}, {12441, 18525}, {12591, 18440}, {12766, 38756}, {12902, 13216}, {13185, 38733}, {13665, 44629}, {13785, 44630}, {13900, 45384}, {13957, 45385}, {18496, 45361}, {18498, 45363}, {18510, 19033}, {18512, 19034}, {18521, 45603}, {18542, 45646}, {18544, 45644}, {22762, 26321}, {23251, 35806}, {23261, 35805}, {45375, 45464}, {45376, 45466}, {45377, 45566}, {45378, 45568}

X(45382) = X(45592)-of-these triangles: {anti-Ehrmann-mid, Lucas(-1) homothetic}
X(45382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 32178, 11950), (494, 18522, 381), (8221, 9839, 3)

X(45383) = PERSPECTOR OF THESE TRIANGLES: ANTI-EHRMANN-MID AND ORTHIC AXES

Barycentrics    (3*a^12-(b^2+c^2)*a^10-2*(11*b^4+6*b^2*c^2+11*c^4)*a^8+2*(b^2+c^2)*(19*b^4-16*b^2*c^2+19*c^4)*a^6-(17*b^8+17*c^8-2*(14*b^4+13*b^2*c^2+14*c^4)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(5*b^2+c^2)*(b^2+5*c^2)*a^2+4*(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(45383) lies on these lines: {53, 381}, {3830, 41244}

X(45383) = X(4)-Waw conjugate of-X(18494)

X(45384) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    -8*S*a^2+a^4-5*(b^2+c^2)*a^2+4*(b^2-c^2)^2 : :
X(45384) = X(6445)-4*X(8972) = 7*X(6445)+4*X(43507) = 3*X(6445)-4*X(43509) = 7*X(8972)+X(43507) = 3*X(8972)-X(43509) = 3*X(43507)+7*X(43509)

X(45384) lies on these lines: {2, 6395}, {3, 485}, {4, 13903}, {5, 6417}, {6, 5055}, {20, 42540}, {30, 6445}, {140, 3316}, {371, 3843}, {372, 42558}, {381, 3068}, {382, 6407}, {399, 13915}, {486, 6500}, {547, 7586}, {548, 23269}, {549, 23267}, {550, 1131}, {615, 15703}, {999, 13898}, {1132, 12811}, {1151, 5073}, {1327, 15685}, {1587, 3526}, {1588, 5072}, {1656, 3069}, {1657, 9540}, {3071, 31487}, {3090, 19117}, {3091, 42604}, {3295, 13897}, {3311, 3851}, {3312, 5070}, {3534, 23249}, {3628, 7581}, {3830, 6221}, {3845, 43508}, {3853, 43512}, {3855, 42522}, {3857, 23275}, {5054, 6446}, {5056, 19116}, {5066, 23273}, {5079, 7584}, {5420, 41948}, {6200, 15681}, {6390, 32806}, {6396, 15701}, {6398, 8253}, {6411, 15695}, {6412, 15718}, {6425, 35786}, {6427, 42262}, {6428, 10577}, {6447, 35821}, {6449, 17800}, {6450, 43378}, {6451, 15689}, {6452, 15707}, {6455, 35820}, {6460, 15720}, {6472, 9681}, {6473, 41962}, {6474, 42271}, {6496, 42267}, {6501, 13951}, {6519, 42266}, {6561, 14269}, {6767, 44623}, {7373, 31472}, {7486, 13993}, {8148, 13911}, {8277, 22462}, {8703, 14241}, {8854, 18459}, {8912, 12293}, {8974, 26336}, {8975, 26346}, {8980, 38744}, {8983, 18525}, {8994, 38790}, {8997, 38733}, {8998, 12902}, {9542, 15682}, {9605, 31481}, {9654, 13904}, {9655, 18965}, {9668, 13901}, {9669, 13905}, {9680, 42272}, {9690, 15684}, {9691, 42258}, {10195, 42600}, {11539, 43510}, {11541, 43560}, {11812, 43889}, {12164, 13909}, {12601, 13650}, {12702, 13893}, {13785, 19709}, {13847, 43882}, {13883, 18493}, {13884, 18494}, {13885, 18501}, {13887, 18524}, {13888, 18480}, {13890, 45379}, {13891, 45380}, {13892, 18503}, {13894, 18508}, {13895, 18519}, {13896, 18518}, {13899, 45381}, {13900, 45382}, {13902, 18526}, {13906, 18545}, {13907, 18543}, {13910, 18440}, {13913, 38756}, {13939, 35018}, {13941, 15699}, {14869, 43511}, {15688, 42226}, {15692, 43374}, {15708, 43517}, {15712, 43376}, {15716, 43256}, {15722, 42572}, {18496, 45365}, {18498, 45368}, {18521, 45607}, {18523, 45605}, {18542, 45652}, {18544, 45650}, {18586, 42818}, {18587, 42817}, {18762, 19054}, {19030, 31479}, {21308, 44599}, {22644, 41963}, {22763, 26321}, {23261, 35815}, {31454, 42269}, {32788, 42527}, {35400, 43318}, {35402, 43321}, {35403, 41945}, {35734, 42224}, {36439, 43543}, {36452, 42984}, {36457, 43542}, {36469, 42985}, {38335, 42225}, {41950, 41951}, {41954, 41967}, {41955, 42273}, {42140, 42200}, {42141, 42202}, {42526, 43415}, {42570, 43407}, {42601, 42606}, {42608, 43380}, {43506, 43884}, {43835, 43863}, {45375, 45484}, {45376, 45486}, {45377, 45574}, {45378, 45576}

X(45384) = crosspoint of X(485) and X(43568)
X(45384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 18512, 6395), (2, 43536, 42639), (4, 13925, 13903), (5, 7585, 18510), (6, 5055, 45385), (381, 3068, 6199), (382, 8981, 6407), (485, 590, 13665), (485, 8976, 3), (485, 43879, 8976), (590, 13665, 3), (1656, 7583, 6418), (3068, 18538, 381), (3071, 43430, 31487), (3311, 42265, 3851), (3628, 7581, 13961), (5054, 42216, 6446), (6221, 6564, 3830), (6398, 8253, 15694), (6449, 23251, 17800), (6564, 13846, 6221), (7585, 18510, 6417), (8253, 35822, 6398), (8960, 42265, 3311), (8976, 13665, 590), (8981, 31412, 382), (15694, 43881, 8253), (23249, 35255, 3534), (23251, 35812, 6449), (32785, 42216, 5054), (32787, 42277, 13785)

X(45385) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EHRMANN-MID AND 4th TRI-SQUARES-CENTRAL

Barycentrics    8*S*a^2+a^4-5*(b^2+c^2)*a^2+4*(b^2-c^2)^2 : :
X(45385) = X(6446)-4*X(13941) = 7*X(6446)+4*X(43508) = 3*X(6446)-4*X(43510) = 7*X(13941)+X(43508) = 3*X(13941)-X(43510) = 3*X(43508)+7*X(43510)

X(45385) lies on these lines: {2, 6199}, {3, 486}, {4, 13961}, {5, 6418}, {6, 5055}, {20, 42539}, {30, 6446}, {140, 3317}, {371, 42557}, {372, 3843}, {381, 3069}, {382, 6408}, {399, 13979}, {485, 6501}, {547, 7585}, {548, 23275}, {549, 23273}, {550, 1132}, {590, 15703}, {999, 13955}, {1131, 12811}, {1152, 5073}, {1328, 15685}, {1587, 5072}, {1588, 3526}, {1656, 3068}, {1657, 13935}, {3070, 43431}, {3090, 19116}, {3091, 42605}, {3295, 13954}, {3311, 5070}, {3312, 3851}, {3525, 9542}, {3534, 23259}, {3628, 7582}, {3830, 6398}, {3845, 43507}, {3853, 43511}, {3855, 42523}, {3857, 23269}, {5054, 6445}, {5056, 19117}, {5066, 23267}, {5079, 7583}, {5418, 41947}, {6200, 15701}, {6221, 8252}, {6390, 32805}, {6396, 15681}, {6411, 15718}, {6412, 15695}, {6426, 35787}, {6427, 10576}, {6428, 42265}, {6448, 35820}, {6449, 43379}, {6450, 17800}, {6451, 15707}, {6452, 15689}, {6456, 35821}, {6459, 15720}, {6472, 41961}, {6474, 9680}, {6475, 42272}, {6497, 42266}, {6500, 8976}, {6522, 42267}, {6560, 14269}, {6767, 44624}, {7373, 44622}, {7486, 13925}, {8148, 13973}, {8276, 22462}, {8703, 14226}, {8855, 18457}, {8912, 43839}, {8972, 15699}, {9541, 15693}, {9654, 13962}, {9655, 18966}, {9668, 13958}, {9669, 13963}, {9690, 42527}, {10194, 42601}, {11539, 43509}, {11541, 43561}, {11812, 43890}, {12164, 13970}, {12602, 13771}, {12702, 13947}, {12902, 13990}, {13665, 19709}, {13846, 43881}, {13886, 35018}, {13936, 18493}, {13937, 18494}, {13938, 18501}, {13940, 18524}, {13942, 18480}, {13944, 45379}, {13945, 45380}, {13946, 18503}, {13948, 18508}, {13949, 26336}, {13950, 26346}, {13952, 18519}, {13953, 18518}, {13956, 45381}, {13957, 45382}, {13959, 18526}, {13964, 18545}, {13965, 18543}, {13967, 38744}, {13969, 38790}, {13971, 18525}, {13972, 18440}, {13977, 38756}, {13989, 38733}, {14869, 43512}, {15684, 41951}, {15688, 42225}, {15692, 43375}, {15708, 43518}, {15712, 43377}, {15716, 43257}, {15722, 42573}, {17851, 42276}, {18496, 45366}, {18498, 45367}, {18521, 45606}, {18523, 45608}, {18538, 19053}, {18542, 45653}, {18544, 45651}, {18586, 42817}, {18587, 42818}, {19029, 31479}, {21308, 44598}, {22615, 41964}, {22764, 26321}, {23251, 35814}, {32787, 42526}, {35400, 43319}, {35402, 43320}, {35403, 41946}, {35734, 42221}, {36439, 43542}, {36453, 42985}, {36457, 43543}, {36470, 42984}, {38335, 42226}, {41949, 41952}, {41953, 41968}, {41956, 42270}, {42140, 42199}, {42141, 42201}, {42571, 43408}, {42600, 42607}, {42609, 43381}, {43505, 43883}, {43835, 43864}, {45375, 45487}, {45376, 45485}, {45377, 45577}, {45378, 45575}

X(45385) = crosspoint of X(486) and X(43569)
X(45385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 18510, 6199), (4, 13993, 13961), (5, 7586, 18512), (6, 5055, 45384), (381, 3069, 6395), (382, 13966, 6408), (486, 615, 13785), (486, 13951, 3), (486, 43880, 13951), (615, 13785, 3), (1656, 7584, 6417), (3069, 18762, 381), (3311, 10577, 5070), (3312, 42262, 3851), (3628, 7582, 13903), (5054, 42215, 6445), (6221, 8252, 15694), (6398, 6565, 3830), (6450, 23261, 17800), (6452, 42263, 15689), (6565, 13847, 6398), (7586, 18512, 6418), (8252, 35823, 6221), (13785, 13951, 615), (13966, 42561, 382), (15694, 43882, 8252), (23259, 35256, 3534), (23261, 35813, 6450), (32786, 42215, 5054), (32788, 42274, 13665)

X(45386) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND ANTI-HUTSON INTOUCH

Barycentrics    a^2*(a^10-(5*b^2+b*c+5*c^2)*a^8+7*(b+c)*b*c*a^7+(10*b^4+10*c^4-(5*b^2-14*b*c+5*c^2)*b*c)*a^6-(b+c)*(13*b^2+8*b*c+13*c^2)*b*c*a^5-(10*b^6+10*c^6-(13*b^4+13*c^4-2*(6*b-c)*(b-6*c)*b*c)*b*c)*a^4+(b+c)*(5*b^4+5*c^4+2*(8*b^2-25*b*c+8*c^2)*b*c)*b*c*a^3+(5*b^4+5*c^4+(13*b^2+24*b*c+13*c^2)*b*c)*(b-c)^4*a^2+(b^4-c^4)*(b-c)*(b^2-6*b*c+c^2)*b*c*a+(b^2-c^2)^2*(b-c)^2*(-b^4-c^4-2*(b+c)^2*b*c)) : :

X(45386) lies on these lines: {3, 1387}, {100, 189}

X(45386) = (anti-inner-Garcia)-isogonal conjugate-of-X(5531)
X(45386) = (anti-Hutson intouch)-isogonal conjugate-of-X(36)
X(45386) = X(963)-of-anti-inner-Garcia triangle

X(45387) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND ANTI-INCIRCLE-CIRCLES

Barycentrics    a^2*(a^11-5*(b+c)*a^10+(b^2+35*b*c+c^2)*a^9+(b+c)*(19*b^2-66*b*c+19*c^2)*a^8-2*(7*b^4+7*c^4+5*(7*b^2-16*b*c+7*c^2)*b*c)*a^7-2*(b+c)*(13*b^4+13*c^4-6*(16*b^2-21*b*c+16*c^2)*b*c)*a^6+2*(13*b^6+13*c^6-(143*b^2-142*b*c+143*c^2)*b^2*c^2)*a^5+2*(b+c)*(7*b^6+7*c^6-(90*b^4+90*c^4-(221*b^2-240*b*c+221*c^2)*b*c)*b*c)*a^4-(19*b^8+19*c^8-2*(35*b^6+35*c^6+(44*b^4+44*c^4-(127*b^2-99*b*c+127*c^2)*b*c)*b*c)*b*c)*a^3-(b^2-c^2)*(b-c)*(b^6+c^6-(46*b^4+46*c^4-(111*b^2-28*b*c+111*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^2*(5*b^6+5*c^6-(35*b^4+35*c^4-(47*b^2-30*b*c+47*c^2)*b*c)*b*c)*a+(b^2-c^2)^3*(b-c)*(-b^4-c^4+2*(2*b^2+b*c+2*c^2)*b*c)) : :

X(45387) lies on these lines: {3, 1317}, {26285, 45388}

X(45387) = (anti-inner-Garcia)-isogonal conjugate-of-X(6264)
X(45387) = (anti-incircle-circles)-isogonal conjugate-of-X(36)

X(45388) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND 6th ANTI-MIXTILINEAR

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

X(45388) lies on these lines: {35, 392}, {3666, 34442}, {9912, 17594}, {10902, 45396}, {26285, 45387}

X(45388) = (6th anti-mixtilinear)-isogonal conjugate-of-X(36)
X(45388) = X(1791)-of-anti-inner-Garcia triangle

X(45389) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND EHRMANN-SIDE

Barycentrics    a^2*(a^14-2*(b+c)*a^13-(3*b^2-8*b*c+3*c^2)*a^12+2*(b+c)*(4*b^2-5*b*c+4*c^2)*a^11+(b^2-21*b*c+c^2)*(b^2-b*c+c^2)*a^10-(b^3+c^3)*(10*b^2-21*b*c+10*c^2)*a^9+(5*b^6+5*c^6+(9*b^4+9*c^4-(37*b^2-59*b*c+37*c^2)*b*c)*b*c)*a^8-(b+c)*(24*b^4+24*c^4-(65*b^2-88*b*c+65*c^2)*b*c)*b*c*a^7-(5*b^8+5*c^8-(24*b^6+24*c^6-(48*b^2-73*b*c+48*c^2)*b^2*c^2)*b*c)*a^6+(b+c)*(10*b^8+10*c^8-(14*b^6+14*c^6+(29*b^2-25*b*c+29*c^2)*(b-c)^2*b*c)*b*c)*a^5-(b^10+c^10+(26*b^8+26*c^8-(37*b^6+37*c^6+(5*b^4+5*c^4-(41*b^2-58*b*c+41*c^2)*b*c)*b*c)*b*c)*b*c)*a^4-(b^2-c^2)*(b-c)*(8*b^8+8*c^8-(b^2+c^2)*(10*b^4+10*c^4+(15*b^2-16*b*c+15*c^2)*b*c)*b*c)*a^3+(3*b^8+3*c^8+(6*b^6+6*c^6-(17*b^4+17*c^4-10*(2*b^2-3*b*c+2*c^2)*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^2+(b^2-c^2)^3*(b-c)*(2*b^6+2*c^6-(5*b^4+5*c^4-(2*b^2-9*b*c+2*c^2)*b*c)*b*c)*a-(b^3-c^3)*(b-c)*(b^4-c^4)*(b^2-c^2)^3) : :

X(45389) lies on this line: {26285, 45396}

X(45389) = (Ehrmann-side)-isogonal conjugate-of-X(36)

X(45390) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND 2nd EULER

Barycentrics    a^2*(a^14-2*(b+c)*a^13-(3*b^2-8*b*c+3*c^2)*a^12+(b+c)*(8*b^2-11*b*c+8*c^2)*a^11+(b^4+c^4-3*(7*b^2-9*b*c+7*c^2)*b*c)*a^10-(b+c)*(10*b^4+10*c^4-7*(5*b^2-7*b*c+5*c^2)*b*c)*a^9-2*(b^2-c^2)*(b-c)*(15*b^2-14*b*c+15*c^2)*a^7*b*c+(5*b^6+5*c^6+(5*b^4+5*c^4-(45*b^2-68*b*c+45*c^2)*b*c)*b*c)*a^8-(5*b^8+5*c^8-2*(15*b^6+15*c^6-(32*b^2-49*b*c+32*c^2)*b^2*c^2)*b*c)*a^6+2*(b+c)*(5*b^8+5*c^8-(5*b^6+5*c^6+(25*b^4+25*c^4-(63*b^2-82*b*c+63*c^2)*b*c)*b*c)*b*c)*a^5-(b^10+c^10+(30*b^8+30*c^8-(45*b^6+45*c^6+2*(3*b^4+3*c^4-2*(15*b^2-22*b*c+15*c^2)*b*c)*b*c)*b*c)*b*c)*a^4-(b^2-c^2)*(b-c)*(8*b^8+8*c^8-(9*b^6+9*c^6+(9*b^2-4*b*c+9*c^2)*(2*b^2-b*c+2*c^2)*b*c)*b*c)*a^3+(3*b^8+3*c^8+(7*b^6+7*c^6-3*(7*b^4+7*c^4-3*(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*b*c)*(b^2-c^2)^2*a^2+(b^2-c^2)^3*(b-c)*(2*b^6+2*c^6-(5*b^2+2*b*c+5*c^2)*(b^2-b*c+c^2)*b*c)*a-(b^3-c^3)*(b-c)*(b^4-c^4)*(b^2-c^2)^3) : :

X(45390) lies on these lines: {35, 45396}, {2932, 4855}, {5840, 37361}, {9912, 11248}

X(45390) = (2nd Euler)-isogonal conjugate-of-X(36)

X(45391) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND EXCENTERS-REFLECTIONS

Barycentrics    a*(9*a^6-18*(b+c)*a^5-(9*b^2-59*b*c+9*c^2)*a^4+3*(b+c)*(12*b^2-23*b*c+12*c^2)*a^3-(9*b^4+9*c^4+(47*b^2-76*b*c+47*c^2)*b*c)*a^2-(b+c)*(18*b^4+18*c^4-(69*b^2-94*b*c+69*c^2)*b*c)*a+3*(3*b^2-4*b*c+3*c^2)*(b^2-c^2)^2) : :

X(45391) lies on these lines: {1, 31235}, {56, 16126}, {100, 7982}, {2077, 22775}, {4511, 6264}, {5048, 12653}, {6326, 12650}

X(45391) = X(43734)-of-anti-inner-Garcia triangle

X(45392) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND FUHRMANN

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

In the plane of a triangle ABC, let
I = incenter = x(1)
Na = Nagel point = X(8)
Sp = Spieker point, X(10)
MaMbMc = medial triangle
Pa = line through Na parallel to MaSp
D,D' = points of intersection of circle {{I,B,C}} and Pa
E,F = diagonal points, other than BC∩DD', of the complete quadrilateral BCDD'
La = line EF, and define Lb and Lc cyclically
A' = Lbcap;Lc, and define B' and C' cyclically
The triangles ABC and A'"B'C' are perspective. Their perspector is X(37741). The (finite) fixed point of the affine transformation that maps ABC onto A'B'C' is X(45392). (Angel Montesdeoca, Octover 30, 2022)

X(45392) lies on these lines: {1, 4996}, {3, 3417}, {8, 35}, {20, 2077}, {21, 1837}, {46, 27086}, {55, 3885}, {72, 33862}, {78, 191}, {100, 355}, {145, 32760}, {326, 7279}, {390, 10527}, {404, 5880}, {480, 1259}, {997, 37293}, {1158, 4855}, {1442, 9723}, {1482, 19920}, {1621, 11373}, {1770, 27385}, {2475, 7951}, {2975, 32613}, {3218, 36152}, {3434, 6892}, {3576, 34758}, {3601, 31660}, {3616, 11023}, {3868, 5172}, {3871, 37740}, {3872, 11524}, {3876, 12745}, {3877, 37564}, {4188, 4295}, {5086, 6914}, {5176, 32141}, {5218, 10522}, {5248, 37720}, {5440, 5694}, {5450, 6224}, {6845, 18407}, {6876, 44447}, {8666, 14795}, {10093, 34772}, {10094, 14793}, {11509, 37300}, {11517, 38901}, {11849, 14923}, {12532, 37700}, {13587, 34647}, {14792, 30144}, {17010, 41575}, {21537, 28922}, {34474, 37713}, {44179, 44180}

X(45392) = reflection of X(1482) in X(19920)
X(45392) = circumnormal-isogonal conjugate of X(26286)
X(45392) = X(643)-Beth conjugate of-X(355)
X(45392) = X(1411)-isoconjugate-of-X(33599)
X(45392) = (anti-inner-Garcia)-isogonal conjugate-of-X(6224)
X(45392) = X(15446)-of-anti-inner-Garcia triangle
X(45392) = X(16000)-of-Fuhrmann triangle
X(45392) = X(16013)-of-2nd circumperp triangle
X(45392) = X(30538)-of-anti-Mandart-incircle triangle

X(45393) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND GARCIA-REFLECTION

Barycentrics    a*(-a+b+c)*(a^4-b*a^3-(b^2-b*c+2*c^2)*a^2+(b^2+c^2)*b*a+(b^2-c^2)*(b-c)*c)*(a^4-c*a^3-(2*b^2-b*c+c^2)*a^2+(b^2+c^2)*c*a+(b^2-c^2)*(b-c)*b) : :
Barycentrics    Cos[(5 A)/2] Csc[A/2] : : (Peter Moses, November 1, 2021)

X(45393) lies on the Feuerbach circumhyperbola and these lines: {1, 1331}, {3, 5553}, {4, 100}, {7, 1470}, {8, 4571}, {9, 4587}, {11, 1259}, {55, 30513}, {63, 15528}, {78, 90}, {79, 27385}, {80, 6735}, {84, 224}, {104, 912}, {214, 7284}, {404, 7702}, {908, 2077}, {943, 45395}, {1000, 1621}, {1145, 10679}, {1172, 5546}, {1389, 23340}, {1809, 43728}, {1810, 35355}, {1811, 23836}, {1896, 36797}, {1937, 23693}, {2057, 38271}, {2298, 32655}, {2327, 43729}, {3035, 5555}, {3427, 6224}, {3577, 12703}, {5248, 7162}, {5251, 5559}, {5692, 15446}, {5854, 10965}, {6068, 42885}, {6598, 10395}, {6909, 34256}, {8068, 26364}, {9809, 10309}, {10087, 10915}, {10305, 38693}, {10707, 24298}, {10956, 18962}, {20118, 41710}, {24465, 37282}, {34474, 37713}, {36106, 36121}

X(45393) = reflection of X(i) in X(j) for these (i, j): (100, 11517), (43740, 11)
X(45393) = isogonal conjugate of X(18838)
X(45393) = antigonal conjugate of X(43740)
X(45393) = barycentric product X(i)*X(j) for these {i, j}: {8, 2990}, {78, 37203}, {312, 36052}, {345, 915}, {645, 3657}, {913, 3718}
X(45393) = barycentric quotient X(i)/X(j) for these (i, j): (9, 1737), (44, 12832), (55, 8609), (78, 914), (212, 2252), (219, 912)
X(45393) = trilinear product X(i)*X(j) for these {i, j}: {8, 36052}, {9, 2990}, {78, 915}, {219, 37203}, {312, 32655}, {345, 913}
X(45393) = trilinear quotient X(i)/X(j) for these (i, j): (8, 1737), (9, 8609), (78, 912), (219, 2252), (345, 914), (519, 12832)
X(45393) = symgonal image of X(11517)
X(45393) = trilinear pole of the line {219, 650}
X(45393) = antipode of X(43740) in Feuerbach circumhyperbola
X(45393) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(3), X(11248)}}
X(45393) = Cevapoint of X(1) and X(2077)
X(45393) = X(643)-Beth conjugate of-X(12775)
X(45393) = X(i)-Dao conjugate of-X(j) for these (i, j): (1, 1737), (214, 12832), (1145, 119)
X(45393) = X(i)-isoconjugate-of-X(j) for these {i, j}: {34, 912}, {56, 1737}, {57, 8609}, {106, 12832}
X(45393) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (9, 1737), (44, 12832), (55, 8609), (78, 914)
X(45393) = X(90)-of-anti-inner-Garcia triangle

X(45394) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND INVERSE-IN-CONWAY

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

X(45394) lies on these lines: {100, 10441}, {4511, 35638}, {6326, 35635}, {35204, 35637}

X(45395) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND INVERSE-IN-INCIRCLE

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

X(45395) lies on these lines: {100, 942}, {149, 946}, {943, 45393}, {1387, 4511}, {2077, 3218}, {3870, 9803}, {3957, 14563}, {10122, 35204}, {13243, 18238}, {18220, 22836}

X(45396) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND KOSNITA

Barycentrics    a^2*(a^11-2*(b+c)*a^10-2*(b^2-4*b*c+c^2)*a^9+(b+c)*(7*b^2-12*b*c+7*c^2)*a^8-2*(b^4+c^4+(8*b^2-11*b*c+8*c^2)*b*c)*a^7-(b+c)*(8*b^4+8*c^4-3*(11*b^2-14*b*c+11*c^2)*b*c)*a^6+(8*b^6+8*c^6-(31*b^2-41*b*c+31*c^2)*b^2*c^2)*a^5+(b+c)*(2*b^2-b*c+2*c^2)*(b^4+c^4-(13*b^2-23*b*c+13*c^2)*b*c)*a^4-(7*b^8+7*c^8-(16*b^6+16*c^6+(4*b^4+4*c^4-(29*b^2-36*b*c+29*c^2)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(7*b^4+7*c^4-(12*b^2-13*b*c+12*c^2)*b*c)*b*c)*a^2+(2*b^6+2*c^6-(8*b^4+8*c^4-(11*b^2-12*b*c+11*c^2)*b*c)*b*c)*(b^2-c^2)^2*a+(b^2-c^2)^3*(b-c)*(-b^4-c^4+(b^2-b*c+c^2)*b*c)) : :

X(45396) lies on these lines: {3, 80}, {35, 45390}, {1324, 12751}, {4996, 37812}, {7488, 17100}, {10902, 45388}, {10913, 42620}, {26285, 45389}

X(45396) = (anti-inner-Garcia)-isogonal conjugate-of-X(6265)
X(45396) = (Kosnita)-isogonal conjugate-of-X(36)
X(45396) = X(3417)-of-anti-inner-Garcia triangle

X(45397) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-GARCIA AND TRINH

Barycentrics    a^2*(a^11-4*(b^2+c^2)*a^9-(b+c)*(b^2-4*b*c+c^2)*a^8+2*(3*b^4+5*b^2*c^2+3*c^4)*a^7+(b+c)*(b^2+c^2)*(4*b^2-13*b*c+4*c^2)*a^6-(4*b^6+4*c^6+(11*b^2-9*b*c+11*c^2)*b^2*c^2)*a^5-3*(b^4-c^4)*(b^2-c^2)*a*b^2*c^2-(b+c)*(6*b^6+6*c^6-(15*b^4+15*c^4-(13*b^2-25*b*c+13*c^2)*b*c)*b*c)*a^4+(b^8+c^8+(8*b^4+8*c^4-3*(3*b^2+4*b*c+3*c^2)*b*c)*b^2*c^2)*a^3+(b^2-c^2)*(b-c)*(4*b^6+4*c^6+(b^4+c^4+(4*b^2-7*b*c+4*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^3*(b-c)*(-b^4-c^4-(b^2+3*b*c+c^2)*b*c)) : :

X(45397) lies on this line: {3, 14217}

X(45397) = (anti-inner-Garcia)-isogonal conjugate-of-X(12738)
X(45397) = (Trinh)-isogonal conjugate-of-X(36)
X(45397) = X(10623)-of-anti-inner-Garcia triangle

X(45398) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-AQUILA

Barycentrics    a*(2*a^2+b^2+c^2+(b+c)*a+2*S) : :
X(45398) = 3*X(1)+X(5589) = X(5589)-3*X(18991)

X(45398) lies on these lines: {1, 6}, {2, 45444}, {3, 45500}, {492, 3616}, {515, 45440}, {517, 43119}, {551, 591}, {946, 13748}, {999, 45436}, {1125, 45472}, {1319, 45404}, {1385, 9733}, {1482, 45411}, {1659, 17061}, {2646, 45470}, {3084, 3745}, {3102, 35763}, {3295, 45416}, {3576, 12305}, {3594, 45530}, {4667, 31569}, {5262, 30386}, {5603, 45406}, {5886, 6289}, {6420, 45572}, {9955, 45438}, {10246, 45488}, {10800, 35641}, {11315, 45546}, {11363, 45400}, {11364, 45402}, {11365, 45428}, {11366, 45430}, {11367, 45432}, {11368, 45434}, {11373, 45454}, {11374, 45456}, {11375, 45458}, {11376, 45460}, {11377, 45467}, {11378, 45464}, {11831, 45446}, {13758, 19065}, {13883, 45484}, {13936, 45487}, {18493, 45375}, {26365, 45345}, {26366, 45347}, {26367, 45415}, {26368, 45412}, {26370, 38314}, {35762, 45462}, {35774, 39679}

X(45398) = midpoint of X(1) and X(18991)
X(45398) = X(18991)-of-anti-Aquila triangle
X(45398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1386, 45399), (1, 3751, 5604), (1, 16475, 7968), (1, 19004, 3640), (1, 45426, 45476), (1, 45427, 3242), (6, 45476, 45426), (3640, 19004, 4663), (38315, 44635, 1)

X(45399) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-AQUILA

Barycentrics    a*(2*a^2+b^2+c^2+(b+c)*a-2*S) : :
X(45399) = 3*X(1)+X(5588) = X(5588)-3*X(18992)

X(45399) lies on these lines: {1, 6}, {2, 45445}, {3, 45501}, {491, 3616}, {515, 45441}, {517, 43118}, {551, 1991}, {946, 13749}, {999, 45437}, {1125, 45473}, {1319, 45405}, {1385, 9732}, {1482, 45410}, {2646, 45471}, {3083, 3745}, {3103, 35762}, {3295, 45417}, {3576, 12306}, {3592, 45531}, {4667, 31570}, {5262, 30385}, {5603, 45407}, {5886, 6290}, {6419, 45573}, {9955, 45439}, {10246, 45489}, {10800, 35642}, {11316, 45547}, {11363, 45401}, {11364, 45403}, {11365, 45429}, {11366, 45431}, {11367, 45433}, {11368, 45435}, {11373, 45455}, {11374, 45457}, {11375, 45459}, {11376, 45461}, {11377, 45465}, {11378, 45466}, {11831, 45447}, {13390, 17061}, {13638, 19066}, {13883, 45486}, {13936, 45485}, {18493, 45376}, {26365, 45348}, {26366, 45346}, {26367, 45413}, {26368, 45414}, {26369, 38314}, {35763, 45463}, {35775, 39648}

X(45399) = midpoint of X(1) and X(18992)
X(45399) = X(18992)-of-anti-Aquila triangle
X(45399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1386, 45398), (1, 3751, 5605), (1, 11370, 1386), (1, 16475, 7969), (1, 19003, 3641), (1, 45426, 3242), (1, 45427, 45477), (6, 45477, 45427), (3641, 19003, 4663), (38315, 44636, 1)

X(45400) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-ARA

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

X(45400) lies on these lines: {3, 45502}, {4, 488}, {6, 25}, {24, 43119}, {33, 45470}, {34, 45404}, {235, 45440}, {371, 9922}, {427, 45472}, {428, 591}, {1162, 1165}, {1593, 6291}, {1598, 45488}, {3069, 5200}, {3102, 35765}, {3517, 45411}, {3575, 13748}, {5090, 45444}, {5198, 6406}, {6460, 6466}, {7487, 45406}, {7713, 45426}, {7714, 26376}, {8946, 10594}, {11363, 45398}, {11380, 45402}, {11383, 45416}, {11384, 45430}, {11385, 45432}, {11386, 45434}, {11390, 45454}, {11391, 45456}, {11392, 45458}, {11393, 45460}, {11394, 45467}, {11395, 45464}, {11396, 45476}, {11398, 45490}, {11399, 45492}, {11400, 45494}, {11401, 45496}, {11474, 40325}, {11832, 45446}, {12147, 37122}, {13884, 45484}, {13937, 45487}, {18494, 45375}, {19443, 19447}, {19446, 26373}, {22479, 45436}, {26371, 45345}, {26372, 45347}, {26374, 45412}, {26377, 45422}, {26378, 45424}, {35764, 45462}, {45452, 45474}

X(45400) = isogonal conjugate of the isotomic conjugate of X(3127)
X(45400) = polar conjugate of the isotomic conjugate of X(6422)
X(45400) = barycentric product X(i)*X(j) for these {i, j}: {4, 6422}, {6, 3127}, {25, 5590}
X(45400) = trilinear product X(i)*X(j) for these {i, j}: {19, 6422}, {31, 3127}, {1973, 5590}
X(45400) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(5490)}} and {{A, B, C, X(25), X(3127)}}
X(45400) = X(44598)-of-anti-Ara triangle
X(45400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (25, 1843, 45401), (25, 5411, 1974), (25, 11389, 1843), (25, 12167, 5412), (492, 26375, 45478), (1162, 1165, 3127)

X(45401) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-ARA

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

X(45401) lies on these lines: {3, 45503}, {4, 487}, {6, 25}, {24, 43118}, {33, 45471}, {34, 45405}, {235, 45441}, {372, 9921}, {427, 45473}, {428, 1991}, {1163, 1164}, {1593, 6406}, {1598, 45489}, {3103, 35764}, {3148, 26953}, {3517, 45410}, {3575, 13749}, {5090, 45445}, {5198, 6291}, {6459, 6465}, {6620, 31403}, {7487, 45407}, {7713, 45427}, {7714, 26375}, {8948, 10594}, {11363, 45399}, {11380, 45403}, {11383, 45417}, {11384, 45431}, {11385, 45433}, {11386, 45435}, {11390, 45455}, {11391, 45457}, {11392, 45459}, {11393, 45461}, {11394, 45465}, {11395, 45466}, {11396, 45477}, {11398, 45491}, {11399, 45493}, {11400, 45495}, {11401, 45497}, {11473, 40325}, {11832, 45447}, {12148, 37122}, {13884, 45486}, {13937, 45485}, {18494, 45376}, {19442, 19446}, {19447, 26374}, {22479, 45437}, {26371, 45348}, {26372, 45346}, {26373, 45413}, {26377, 45423}, {26378, 45425}, {35765, 45463}, {45453, 45475}

X(45401) = isogonal conjugate of the isotomic conjugate of X(3128)
X(45401) = polar conjugate of the isotomic conjugate of X(6421)
X(45401) = barycentric product X(i)*X(j) for these {i, j}: {4, 6421}, {6, 3128}, {25, 5591}
X(45401) = trilinear product X(i)*X(j) for these {i, j}: {19, 6421}, {31, 3128}
X(45401) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(5491)}} and {{A, B, C, X(25), X(3128)}}
X(45401) = X(44599)-of-anti-Ara triangle
X(45401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (25, 1843, 45400), (25, 5410, 1974), (25, 11388, 1843), (25, 12167, 5413), (491, 26376, 45479), (1163, 1164, 3128)

X(45402) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 5th ANTI-BROCARD

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

X(45402) lies on these lines: {3, 6}, {83, 45472}, {98, 45440}, {492, 7787}, {591, 12150}, {6289, 10796}, {10788, 45406}, {10789, 45426}, {10790, 45428}, {10791, 45444}, {10794, 45454}, {10795, 45456}, {10797, 45458}, {10798, 45460}, {10799, 45470}, {10800, 45476}, {10801, 45490}, {10802, 45492}, {10803, 45494}, {10804, 45496}, {11364, 45398}, {11380, 45400}, {11490, 45416}, {11837, 45430}, {11838, 45432}, {11839, 45446}, {11840, 45467}, {11841, 45464}, {12110, 13748}, {12835, 45404}, {13885, 45484}, {13938, 45487}, {18501, 45375}, {18502, 45438}, {22520, 45436}, {26379, 45345}, {26403, 45347}, {26427, 45415}, {26428, 45412}, {26430, 45421}, {26431, 45422}, {26432, 45424}

X(45402) = X(44604)-of-5th anti-Brocard triangle
X(45402) = X(45402)-of-circumsymmedial triangle
X(45402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12968, 45434), (32, 5039, 44586), (32, 12212, 45403), (32, 18993, 1691), (3102, 5007, 6)

X(45403) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 5th ANTI-BROCARD

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

X(45403) lies on these lines: {3, 6}, {83, 45473}, {98, 45441}, {491, 7787}, {1991, 12150}, {6290, 10796}, {10788, 45407}, {10789, 45427}, {10790, 45429}, {10791, 45445}, {10794, 45455}, {10795, 45457}, {10797, 45459}, {10798, 45461}, {10799, 45471}, {10800, 45477}, {10801, 45491}, {10802, 45493}, {10803, 45495}, {10804, 45497}, {11364, 45399}, {11380, 45401}, {11490, 45417}, {11837, 45431}, {11838, 45433}, {11839, 45447}, {11840, 45465}, {11841, 45466}, {12110, 13749}, {12835, 37256}, {13885, 45486}, {13938, 45485}, {18501, 45376}, {18502, 45439}, {22520, 45437}, {26379, 45348}, {26403, 45346}, {26427, 45413}, {26428, 45414}, {26429, 45420}, {26431, 45423}, {26432, 45425}

X(45403) = X(44605)-of-5th anti-Brocard triangle
X(45403) = X(45403)-of-circumsymmedial triangle
X(45403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12963, 45435), (32, 5039, 44587), (32, 10792, 12212), (32, 12212, 45402), (32, 18994, 1691), (3103, 5007, 6)

X(45404) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

X(45404) lies on these lines: {1, 9733}, {3, 45490}, {4, 45460}, {6, 41}, {11, 45440}, {12, 45472}, {34, 45400}, {36, 43119}, {55, 7362}, {57, 8945}, {65, 26495}, {325, 45459}, {388, 492}, {591, 5434}, {999, 45488}, {1319, 45398}, {1470, 45424}, {1478, 6289}, {2099, 45476}, {3095, 45493}, {3102, 35769}, {3304, 7353}, {3585, 45438}, {4293, 45406}, {4317, 18989}, {5194, 35809}, {5252, 45444}, {7354, 13748}, {9655, 45375}, {11509, 45416}, {12314, 45582}, {12835, 45402}, {18954, 45428}, {18955, 45430}, {18956, 45432}, {18957, 45434}, {18958, 45446}, {18961, 45454}, {18962, 45456}, {18963, 45467}, {18964, 45464}, {18965, 45484}, {18966, 45487}, {18967, 45496}, {19370, 26433}, {19371, 19443}, {19475, 26292}, {26380, 45345}, {26404, 45347}, {26434, 45412}, {26436, 45421}, {26437, 45422}, {35768, 45462}, {36656, 45562}, {45411, 45507}

X(45404) = barycentric product X(7)*X(31459)
X(45404) = trilinear product X(57)*X(31459)
X(45404) = X(44606)-of-2nd anti-circumperp-tangential triangle
X(45404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9733, 45470), (56, 1469, 45405), (56, 18995, 1428), (388, 492, 45458), (999, 45488, 45492)

X(45405) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

X(45405) lies on these lines: {1, 9732}, {3, 45491}, {4, 45461}, {6, 41}, {11, 45441}, {12, 45473}, {34, 45401}, {36, 43118}, {55, 7353}, {57, 8941}, {65, 26504}, {325, 45458}, {388, 491}, {999, 45489}, {1319, 45399}, {1470, 45425}, {1478, 6290}, {1991, 5434}, {2099, 45477}, {3095, 45492}, {3103, 35768}, {3304, 7362}, {3585, 45439}, {4293, 45407}, {4317, 18988}, {5194, 35808}, {5252, 45445}, {7354, 13749}, {9655, 45376}, {11509, 45417}, {12313, 45583}, {12835, 37256}, {18954, 45429}, {18955, 45431}, {18956, 45433}, {18957, 45435}, {18958, 45447}, {18961, 45455}, {18962, 45457}, {18963, 45465}, {18964, 45466}, {18965, 45486}, {18966, 45485}, {18967, 45497}, {19370, 19442}, {19371, 26434}, {19476, 26293}, {26380, 45348}, {26404, 45346}, {26433, 45413}, {26435, 45420}, {26437, 45423}, {35769, 45463}, {36655, 45563}, {45410, 45506}

X(45405) = X(44607)-of-2nd anti-circumperp-tangential triangle
X(45405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9732, 45471), (56, 1469, 45404), (56, 18959, 1469), (56, 18996, 1428), (388, 491, 45459), (999, 45489, 45493)

X(45406) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-EULER

Barycentrics    2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S-(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2) : :
X(45406) = 3*X(35948)-4*X(43121) = 3*X(45421)-2*X(45488)

X(45406) lies on these lines: {2, 6222}, {3, 489}, {4, 6}, {5, 45375}, {20, 6463}, {22, 13428}, {24, 45428}, {30, 8982}, {98, 486}, {104, 45436}, {182, 7389}, {184, 1585}, {262, 14232}, {325, 487}, {371, 6811}, {372, 36998}, {376, 591}, {388, 45490}, {485, 14234}, {488, 1975}, {490, 2782}, {497, 45492}, {515, 45426}, {542, 44485}, {631, 45472}, {638, 3564}, {639, 45553}, {1132, 14244}, {1152, 44392}, {1328, 14238}, {1352, 7388}, {1505, 2794}, {1513, 6424}, {1586, 1899}, {1589, 26905}, {1614, 30427}, {2043, 6773}, {2044, 6770}, {3069, 39679}, {3085, 45458}, {3086, 45460}, {3091, 45438}, {3102, 6561}, {3127, 11427}, {3128, 32064}, {3311, 36656}, {3398, 37342}, {3424, 45102}, {3536, 23291}, {3839, 13674}, {4293, 45404}, {4294, 45470}, {5200, 11433}, {5418, 43461}, {5603, 45398}, {5657, 45444}, {5890, 6239}, {6033, 33431}, {6250, 35822}, {6272, 6280}, {6290, 32488}, {6398, 45525}, {6400, 12283}, {6420, 45544}, {6459, 21736}, {6560, 45462}, {6565, 22587}, {6806, 14826}, {7376, 40330}, {7487, 45400}, {7584, 36714}, {7774, 9732}, {7967, 45476}, {8414, 9540}, {8981, 45524}, {9739, 35947}, {9766, 26289}, {9862, 45434}, {10519, 36701}, {10533, 44633}, {10785, 45454}, {10786, 45456}, {10788, 45402}, {10805, 45494}, {10806, 45496}, {11179, 44510}, {11293, 43118}, {11313, 26348}, {11402, 32588}, {11431, 45474}, {11491, 45416}, {11824, 32419}, {11843, 45430}, {11844, 45432}, {11845, 45446}, {11846, 45467}, {11847, 45464}, {11916, 36733}, {12115, 45424}, {12116, 45422}, {12117, 12124}, {12123, 32433}, {12298, 40673}, {12322, 21737}, {12510, 26288}, {13022, 44197}, {13757, 14830}, {13785, 36655}, {13886, 45484}, {13935, 32494}, {13939, 45487}, {14231, 14237}, {14245, 19103}, {18510, 36712}, {23514, 45543}, {26381, 45345}, {26405, 45347}, {26439, 45415}, {26440, 45412}, {32492, 35821}, {33750, 36702}, {35831, 39894}, {35948, 43121}, {36709, 42215}, {39387, 43120}, {41020, 42235}, {41021, 42236}

X(45406) = reflection of X(i) in X(j) for these (i, j): (4, 3071), (489, 3), (36714, 7584), (45407, 39646)
X(45406) = crosssum of X(3) and X(45488)
X(45406) = X(489)-of-ABC-X3 reflections triangle
X(45406) = X(3071)-of-anti-Euler triangle
X(45406) = X(13748)-of-2nd Kenmotu-centers triangle
X(45406) = X(42060)-of-2nd half-squares triangle
X(45406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6776, 45407), (4, 7582, 14853), (4, 10784, 6776), (4, 14912, 1587), (4, 39874, 5871), (6, 13748, 4), (1588, 5870, 4), (3070, 14233, 4), (5871, 23259, 4), (6146, 33971, 45407), (6289, 43119, 2), (8550, 14233, 3070), (12322, 25406, 21737), (13749, 23261, 4), (14227, 23273, 4), (14230, 42283, 4), (14239, 42284, 4), (14242, 23275, 4), (26441, 45510, 3), (36990, 45441, 4), (43120, 45554, 39387), (45375, 45411, 5)

X(45407) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-EULER

Barycentrics    2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S+(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2) : :
X(45407) = 3*X(35949)-4*X(43120) = 3*X(45420)-2*X(45489)

X(45407) lies on these lines: {2, 6290}, {3, 490}, {4, 6}, {5, 45376}, {20, 6462}, {22, 13439}, {24, 45429}, {30, 26441}, {69, 21737}, {98, 485}, {104, 45437}, {182, 7388}, {184, 1586}, {262, 14237}, {325, 488}, {371, 36998}, {372, 6813}, {376, 1991}, {388, 45491}, {486, 14238}, {487, 1975}, {489, 2782}, {497, 45493}, {515, 45427}, {542, 44486}, {631, 45473}, {637, 3564}, {640, 45552}, {1131, 14229}, {1151, 44394}, {1327, 14234}, {1352, 7389}, {1504, 2794}, {1513, 6423}, {1585, 1899}, {1590, 26905}, {1614, 30428}, {2043, 6770}, {2044, 6773}, {3068, 21736}, {3085, 45459}, {3086, 45461}, {3091, 45439}, {3103, 6560}, {3127, 32064}, {3128, 11427}, {3312, 36655}, {3398, 37343}, {3424, 45101}, {3535, 23291}, {3839, 13794}, {4293, 45405}, {4294, 45471}, {5200, 11206}, {5420, 43461}, {5603, 45399}, {5657, 45445}, {5890, 6400}, {6033, 33430}, {6221, 45524}, {6239, 12283}, {6251, 35823}, {6273, 6279}, {6289, 32489}, {6419, 45545}, {6561, 45463}, {6564, 22618}, {6805, 14826}, {7375, 40330}, {7487, 45401}, {7583, 36709}, {7774, 9733}, {7967, 45477}, {8406, 13935}, {9540, 32497}, {9738, 35946}, {9766, 26288}, {9862, 45435}, {10519, 36703}, {10534, 44634}, {10785, 45455}, {10786, 45457}, {10788, 45403}, {10805, 45495}, {10806, 45497}, {11179, 44509}, {11294, 43119}, {11314, 26341}, {11402, 32587}, {11431, 45475}, {11491, 45417}, {11825, 32421}, {11843, 45431}, {11844, 45433}, {11845, 45447}, {11846, 45465}, {11847, 45466}, {11917, 36719}, {12115, 45425}, {12116, 45423}, {12117, 12123}, {12124, 32436}, {12299, 40673}, {12323, 25406}, {12509, 26289}, {13021, 44198}, {13637, 14830}, {13665, 36656}, {13886, 45486}, {13939, 45485}, {13966, 45525}, {14231, 19104}, {14232, 14245}, {18512, 36711}, {23514, 45542}, {26381, 45348}, {26405, 45346}, {26439, 45413}, {26440, 45414}, {31463, 37334}, {32495, 35820}, {33750, 36717}, {35830, 39893}, {35949, 43120}, {36714, 42216}, {39388, 43121}, {41020, 42237}, {41021, 42238}

X(45407) = reflection of X(i) in X(j) for these (i, j): (4, 3070), (490, 3), (36709, 7583), (45406, 39646)
X(45407) = crosssum of X(3) and X(45489)
X(45407) = X(490)-of-ABC-X3 reflections triangle
X(45407) = X(3070)-of-anti-Euler triangle
X(45407) = X(13749)-of-1st Kenmotu-centers triangle
X(45407) = X(42060)-of-1st half-squares triangle
X(45407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6776, 45406), (4, 7581, 14853), (4, 10783, 6776), (4, 14912, 1588), (4, 39874, 5870), (6, 13749, 4), (1587, 5871, 4), (5870, 23249, 4), (6290, 43118, 2), (8982, 45511, 3), (13748, 23251, 4), (14227, 23269, 4), (14233, 42284, 4), (14235, 42283, 4), (14242, 23267, 4), (36990, 45440, 4), (43121, 45555, 39388), (45376, 45410, 5)

X(45408) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd ANTI-EXTOUCH

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

X(45408) lies on these lines: {25, 45474}, {184, 44196}, {1181, 9733}, {1593, 13748}, {19360, 43119}, {19361, 45411}, {19362, 45488}, {45428, 45478}

X(45409) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd ANTI-EXTOUCH

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

X(45409) lies on these lines: {25, 45475}, {184, 44199}, {1181, 9732}, {1593, 13749}, {19360, 43118}, {19361, 45410}, {19362, 45489}, {45429, 45479}

X(45410) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 1st ANTI-KENMOTU-FREE-VERTICES

Barycentrics    a^2*(3*a^4-4*b^2*c^2-4*(b^2+c^2)*a^2+(b^2-c^2)^2+2*(-a^2+b^2+c^2)*S) : :
X(45410) = X(3)+2*X(6420) = 5*X(3)-4*X(43126) = 3*X(3)+4*X(43145) = 5*X(6420)+2*X(43126) = 3*X(6420)-2*X(43145) = 3*X(43126)+5*X(43145) = 2*X(43126)-5*X(45552) = 2*X(43145)+3*X(45552)

X(45410) lies on these lines: {3, 6}, {5, 45376}, {140, 491}, {381, 13749}, {382, 45441}, {549, 45420}, {641, 3526}, {999, 45491}, {1385, 45427}, {1482, 45399}, {1584, 11402}, {1656, 6119}, {1991, 5054}, {3155, 5422}, {3156, 5012}, {3295, 45493}, {3517, 45401}, {3618, 12256}, {3851, 45439}, {5408, 43650}, {5409, 13366}, {5418, 44394}, {6036, 31481}, {6316, 22727}, {6776, 37342}, {6811, 16989}, {7506, 45429}, {7583, 13638}, {7586, 21737}, {7709, 35939}, {9540, 26521}, {9744, 45378}, {10133, 10601}, {11291, 14912}, {12601, 18510}, {15047, 44199}, {16202, 45423}, {16203, 45425}, {18583, 36709}, {19361, 45409}, {26507, 43908}, {31479, 45459}, {35945, 43511}, {37535, 45437}, {37621, 45417}, {37624, 45477}, {38224, 42265}, {45346, 45351}, {45348, 45349}, {45405, 45506}, {45413, 45519}, {45414, 45516}, {45431, 45534}, {45433, 45536}, {45445, 45546}, {45447, 45548}, {45455, 45556}, {45457, 45558}, {45461, 45562}, {45465, 45569}, {45466, 45566}, {45471, 45570}, {45485, 45577}, {45486, 45574}, {45495, 45584}, {45497, 45586}

X(45410) = midpoint of X(6420) and X(45552)
X(45410) = reflection of X(3) in X(45552)
X(45410) = inverse of X(45489) in Brocard circle
X(45410) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(45489)}} and {{A, B, C, X(54), X(12963)}}
X(45410) = X(6420)-of-1st anti-Kenmotu-free-vertices triangle
X(45410) = X(45410)-of-circumsymmedial triangle
X(45410) = X(45552)-of-X3-ABC reflections triangle
X(45410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5093, 1161), (3, 6418, 1351), (6, 1152, 3103), (6, 39648, 3311), (6, 43118, 3), (182, 372, 3), (371, 43121, 3), (372, 5062, 3312), (372, 45550, 182), (372, 45553, 9739), (575, 43121, 371), (575, 44486, 6), (1152, 43119, 3), (1161, 6428, 5093), (3312, 6398, 8416), (3312, 26341, 3), (3594, 5085, 9733), (5085, 9733, 3), (5092, 11825, 3), (6200, 12975, 3), (6396, 43120, 3), (6398, 26348, 3), (6501, 11916, 11482), (9738, 39561, 6419), (9739, 44475, 9732), (9739, 45553, 3), (11824, 35770, 576), (12017, 12314, 3), (35423, 43156, 3), (44656, 45512, 6417)

X(45411) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd ANTI-KENMOTU-FREE-VERTICES

Barycentrics    a^2*(3*a^4-4*b^2*c^2-4*(b^2+c^2)*a^2+(b^2-c^2)^2-2*(-a^2+b^2+c^2)*S) : :
X(45411) = X(3)+2*X(6419) = 5*X(3)-4*X(43127) = 3*X(3)+4*X(43143) = 5*X(6419)+2*X(43127) = 3*X(6419)-2*X(43143) = 3*X(43127)+5*X(43143) = 2*X(43127)-5*X(45553) = 2*X(43143)+3*X(45553)

X(45411) lies on these lines: {3, 6}, {5, 45375}, {140, 492}, {381, 13748}, {382, 45440}, {549, 45421}, {591, 5054}, {642, 3526}, {999, 45490}, {1385, 45426}, {1482, 45398}, {1583, 11402}, {1656, 6118}, {3155, 5012}, {3156, 5422}, {3295, 45492}, {3517, 45400}, {3618, 12257}, {3851, 45438}, {5408, 13366}, {5409, 43650}, {5420, 44392}, {6312, 22726}, {6776, 37343}, {6813, 16989}, {7506, 45428}, {7584, 13758}, {7709, 35938}, {8908, 10963}, {9744, 45377}, {10132, 10601}, {11292, 14912}, {12602, 18512}, {13935, 26516}, {15047, 44196}, {16202, 45422}, {16203, 45424}, {18583, 36714}, {19361, 45408}, {26498, 43908}, {31479, 45458}, {35944, 43512}, {37535, 45436}, {37621, 45416}, {37624, 45476}, {38224, 42262}, {45345, 45352}, {45347, 45350}, {45404, 45507}, {45412, 45518}, {45415, 45517}, {45430, 45535}, {45432, 45537}, {45444, 45547}, {45446, 45549}, {45454, 45557}, {45456, 45559}, {45460, 45563}, {45464, 45568}, {45467, 45567}, {45470, 45571}, {45484, 45576}, {45487, 45575}, {45494, 45585}, {45496, 45587}

X(45411) = midpoint of X(6419) and X(45553)
X(45411) = reflection of X(3) in X(45553)
X(45411) = inverse of X(45488) in Brocard circle
X(45411) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(45488)}} and {{A, B, C, X(54), X(12968)}}
X(45411) = X(45411)-of-circumsymmedial triangle
X(45411) = X(45553)-of-X3-ABC reflections triangle
X(45411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5093, 1160), (3, 6417, 1351), (6, 1151, 3102), (6, 39679, 3312), (6, 43119, 3), (182, 371, 3), (371, 5058, 3311), (371, 45551, 182), (371, 45552, 9738), (372, 43120, 3), (575, 43120, 372), (575, 44485, 6), (1151, 43118, 3), (1160, 6427, 5093), (3311, 6221, 8396), (3311, 26348, 3), (3592, 5085, 9732), (5085, 9732, 3), (5092, 11824, 3), (6200, 43121, 3), (6221, 26341, 3), (6396, 12974, 3), (6500, 11917, 11482), (9738, 44476, 9733), (9738, 45552, 3), (9739, 39561, 6420), (11825, 35771, 576), (12017, 12313, 3), (35423, 43155, 3), (44657, 45513, 6418)

X(45412) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-LUCAS(-1) HOMOTHETIC

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

X(45412) lies on these lines: {3, 45516}, {6, 494}, {492, 26503}, {3102, 45602}, {5491, 32838}, {6289, 26467}, {6464, 45415}, {12305, 26293}, {18523, 45375}, {26299, 45426}, {26305, 45428}, {26313, 45434}, {26323, 45436}, {26329, 45440}, {26354, 45470}, {26368, 45398}, {26374, 45400}, {26392, 45345}, {26416, 45347}, {26428, 45402}, {26434, 45404}, {26440, 45406}, {26443, 45444}, {26448, 45446}, {26472, 45460}, {26478, 45458}, {26484, 45456}, {26489, 45454}, {26502, 45416}, {26504, 45476}, {26506, 45421}, {26507, 43119}, {26508, 45422}, {26509, 45424}, {26510, 45496}, {26511, 45494}, {42022, 45462}, {45411, 45518}, {45430, 45588}, {45432, 45590}, {45438, 45592}, {45467, 45604}, {45484, 45605}, {45487, 45608}, {45488, 45609}, {45490, 45611}, {45492, 45613}

X(45412) = barycentric product X(494)*X(26361)
X(45412) = {X(6), X(45464)}-harmonic conjugate of X(19443)

X(45413) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-LUCAS(+1) HOMOTHETIC

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

X(45413) lies on these lines: {3, 45517}, {6, 493}, {491, 26494}, {3103, 45601}, {3663, 26496}, {5490, 32838}, {6290, 26466}, {6464, 45414}, {12306, 26292}, {18521, 45376}, {26298, 45427}, {26304, 45429}, {26312, 45435}, {26322, 45437}, {26328, 45441}, {26353, 45471}, {26367, 45399}, {26373, 45401}, {26391, 45348}, {26415, 45346}, {26427, 45403}, {26433, 45405}, {26439, 45407}, {26442, 45445}, {26447, 45447}, {26471, 45461}, {26477, 45459}, {26483, 45457}, {26488, 45455}, {26493, 45417}, {26495, 45477}, {26498, 43118}, {26499, 45423}, {26500, 45425}, {26501, 45497}, {45410, 45519}, {45431, 45589}, {45433, 45591}, {45439, 45593}, {45463, 45600}, {45466, 45603}, {45485, 45606}, {45486, 45607}, {45489, 45610}, {45491, 45612}, {45493, 45614}, {45495, 45615}

X(45413) = barycentric product X(493)*X(26362)
X(45413) = {X(6), X(45465)}-harmonic conjugate of X(19442)

X(45414) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-LUCAS(-1) HOMOTHETIC

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

X(45414) lies on these lines: {3, 45518}, {6, 494}, {53, 24243}, {372, 42022}, {491, 19421}, {3103, 45599}, {3926, 5491}, {6290, 26467}, {6464, 45413}, {8946, 44199}, {9723, 19442}, {9732, 18981}, {12306, 13022}, {13749, 19499}, {18415, 45439}, {18523, 45376}, {19371, 26434}, {19419, 45489}, {19435, 26354}, {19441, 26507}, {19447, 26374}, {26299, 45427}, {26305, 45429}, {26313, 45435}, {26323, 45437}, {26329, 45441}, {26368, 45399}, {26392, 45348}, {26416, 45346}, {26428, 45403}, {26440, 45407}, {26443, 45445}, {26448, 45447}, {26472, 45461}, {26478, 45459}, {26484, 45457}, {26489, 45455}, {26502, 45417}, {26504, 45477}, {26505, 45420}, {26508, 45423}, {26509, 45425}, {26510, 45497}, {26511, 45495}, {45410, 45516}, {45431, 45588}, {45433, 45590}, {45463, 45602}, {45465, 45604}, {45485, 45608}, {45486, 45605}, {45491, 45611}, {45493, 45613}

X(45414) = isotomic conjugate of the polar conjugate of X(26374)
X(45414) = barycentric product X(i)*X(j) for these {i, j}: {69, 26374}, {494, 5591}
X(45414) = trilinear product X(63)*X(26374)
X(45414) = {X(8943), X(12591)}-harmonic conjugate of X(19443)

X(45415) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-LUCAS(+1) HOMOTHETIC

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

X(45415) lies on these lines: {3, 45519}, {6, 493}, {53, 24244}, {492, 19420}, {3102, 45600}, {3926, 5490}, {6289, 26466}, {6464, 45412}, {8948, 44196}, {9723, 19443}, {9733, 18980}, {12305, 13021}, {13748, 19498}, {18414, 45438}, {18521, 45375}, {19370, 26433}, {19418, 45488}, {19434, 26353}, {19440, 26498}, {19446, 26373}, {26298, 45426}, {26304, 45428}, {26312, 45434}, {26322, 45436}, {26328, 45440}, {26367, 45398}, {26391, 45345}, {26415, 45347}, {26427, 45402}, {26439, 45406}, {26442, 45444}, {26447, 45446}, {26471, 45460}, {26477, 45458}, {26483, 45456}, {26488, 45454}, {26493, 45416}, {26495, 45476}, {26497, 45421}, {26499, 45422}, {26500, 45424}, {26501, 45496}, {45411, 45517}, {45430, 45589}, {45432, 45591}, {45462, 45601}, {45464, 45603}, {45484, 45607}, {45487, 45606}, {45490, 45612}, {45492, 45614}, {45494, 45615}

X(45415) = isotomic conjugate of the polar conjugate of X(26373)
X(45415) = barycentric product X(i)*X(j) for these {i, j}: {69, 26373}, {493, 5590}
X(45415) = trilinear product X(63)*X(26373)
X(45415) = {X(8939), X(12590)}-harmonic conjugate of X(19442)

X(45416) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-MANDART-INCIRCLE

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

X(45416) lies on these lines: {3, 45436}, {6, 31}, {35, 45426}, {56, 45476}, {100, 492}, {197, 45428}, {591, 4421}, {1376, 45454}, {1631, 34125}, {3102, 35773}, {3295, 45398}, {4557, 34121}, {5687, 45444}, {6289, 11499}, {9733, 11248}, {10267, 43119}, {10310, 12305}, {11383, 45400}, {11490, 45402}, {11491, 45406}, {11492, 45430}, {11493, 45432}, {11494, 45434}, {11496, 45440}, {11500, 13748}, {11501, 45458}, {11502, 45460}, {11503, 45467}, {11504, 45464}, {11507, 45490}, {11508, 45492}, {11509, 45404}, {11510, 45496}, {11848, 45446}, {11849, 45488}, {13887, 45484}, {13940, 45487}, {15624, 44197}, {18491, 45438}, {18524, 45375}, {19215, 44196}, {26393, 45345}, {26417, 45347}, {26493, 45415}, {26502, 45412}, {26513, 45421}, {35772, 45462}, {37621, 45411}

X(45416) = X(2066)-of-anti-Mandart-incircle triangle
X(45416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (55, 11498, 12329), (55, 12329, 45417)

X(45417) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-MANDART-INCIRCLE

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

X(45417) lies on these lines: {3, 45437}, {6, 31}, {35, 45427}, {56, 45477}, {100, 491}, {197, 45429}, {1376, 45455}, {1631, 34121}, {1991, 4421}, {3103, 35772}, {3295, 45399}, {4557, 34125}, {5687, 45445}, {6290, 11499}, {9732, 11248}, {10267, 43118}, {10310, 12306}, {11383, 45401}, {11490, 45403}, {11491, 45407}, {11492, 45431}, {11493, 45433}, {11494, 45435}, {11496, 45441}, {11500, 13749}, {11501, 45459}, {11502, 45461}, {11503, 45465}, {11504, 45466}, {11507, 45491}, {11508, 45493}, {11509, 45405}, {11510, 45497}, {11848, 45447}, {11849, 45489}, {13887, 45486}, {13940, 45485}, {15624, 44198}, {18491, 45439}, {18524, 45376}, {19216, 44199}, {26393, 45348}, {26417, 45346}, {26493, 45413}, {26502, 45414}, {26512, 45420}, {35773, 45463}, {37621, 45410}

X(45417) = X(5414)-of-anti-Mandart-incircle triangle
X(45417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (55, 11497, 12329), (55, 12329, 45416)

X(45418) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-ORTHOCENTROIDAL

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

X(45418) lies on these lines: {74, 13748}, {399, 9733}, {34417, 44592}

X(45419) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-ORTHOCENTROIDAL

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

X(45419) lies on these lines: {74, 13749}, {399, 9732}, {34417, 44593}

X(45420) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 3rd ANTI-TRI-SQUARES-CENTRAL

Barycentrics    5*a^2-b^2-c^2+2*S : :
X(45420) = 4*X(371)-X(490) = X(637)-4*X(7583) = 2*X(3070)+X(43134) = 4*X(32787)-X(32808) = X(45407)+2*X(45489)

X(45420) lies on these lines: {2, 6}, {3, 45524}, {4, 33457}, {30, 26441}, {32, 35305}, {99, 12159}, {371, 490}, {372, 41491}, {376, 9732}, {485, 14568}, {487, 7581}, {488, 45078}, {489, 1587}, {519, 26300}, {549, 45410}, {576, 6813}, {598, 12158}, {637, 7583}, {638, 3311}, {640, 35771}, {641, 35815}, {642, 35770}, {648, 1585}, {671, 1327}, {754, 1504}, {1151, 43133}, {1351, 45511}, {1353, 45510}, {3070, 43134}, {3103, 7757}, {3312, 45509}, {3524, 26516}, {3543, 13749}, {3545, 6290}, {3592, 11294}, {3663, 26496}, {3839, 26330}, {3845, 18539}, {5097, 45555}, {5875, 36723}, {6179, 35685}, {6321, 33431}, {6419, 7388}, {6420, 39388}, {6422, 8356}, {6423, 35297}, {6427, 11314}, {6428, 11316}, {6459, 12222}, {6460, 26618}, {6462, 35927}, {6561, 22485}, {7389, 7760}, {7714, 26375}, {7752, 19102}, {7754, 44647}, {7763, 45515}, {7780, 31483}, {7812, 39660}, {7824, 31465}, {7894, 19105}, {8981, 45508}, {8982, 12313}, {9768, 13873}, {10304, 12306}, {10385, 26355}, {10754, 13640}, {11055, 13669}, {11239, 26518}, {11240, 26517}, {11315, 31487}, {11916, 36719}, {12156, 13789}, {12221, 31412}, {12962, 20065}, {13644, 22253}, {13651, 42023}, {13678, 26615}, {13681, 22616}, {15682, 33456}, {22722, 32451}, {26288, 39648}, {26306, 45429}, {26314, 45435}, {26324, 45437}, {26369, 38314}, {26396, 45348}, {26420, 45346}, {26429, 45403}, {26435, 45405}, {26444, 45445}, {26449, 45447}, {26473, 45461}, {26479, 45459}, {26485, 45457}, {26490, 45455}, {26505, 45414}, {26512, 45417}, {26514, 45477}, {26519, 45497}, {26520, 45495}, {32419, 35822}, {32798, 35578}, {34208, 45479}, {34511, 35306}, {41099, 45439}

X(45420) = reflection of X(i) in X(j) for these (i, j): (2, 32787), (490, 35949), (32808, 2), (35949, 371), (45421, 14614)
X(45420) = isotomic conjugate of X(42023)
X(45420) = barycentric product X(i)*X(j) for these {i, j}: {76, 41411}, {492, 21463}
X(45420) = trilinear product X(75)*X(41411)
X(45420) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(21463)}} and {{A, B, C, X(4), X(5860)}}
X(45420) = X(1991)-of-1st Kenmotu-centers triangle
X(45420) = X(32787)-of-3rd anti-tri-squares-central triangle
X(45420) = X(41491)-of-1st Kenmotu-free-vertices triangle
X(45420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 193, 5860), (2, 1991, 491), (2, 1992, 45421), (2, 5032, 19053), (2, 5860, 492), (2, 5861, 32809), (2, 45421, 13757), (6, 1991, 2), (6, 41624, 45421), (6, 44394, 13638), (193, 3068, 492), (591, 13846, 2), (1991, 15534, 7774), (1992, 13637, 13757), (1992, 13639, 13637), (3068, 5860, 2), (3068, 26339, 193), (5861, 19054, 2), (7774, 44394, 491), (8860, 41149, 45421), (13637, 45421, 2), (13846, 15534, 591), (15534, 22329, 45421), (17346, 41629, 45421), (37640, 37641, 44596), (37785, 37786, 13637)

X(45421) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 4th ANTI-TRI-SQUARES-CENTRAL

Barycentrics    5*a^2-b^2-c^2-2*S : :
X(45421) = 4*X(372)-X(489) = X(638)-4*X(7584) = 2*X(3071)+X(43133) = 4*X(32788)-X(32809) = X(45406)+2*X(45488)

X(45421) lies on these lines: {2, 6}, {3, 45525}, {4, 33456}, {30, 8982}, {32, 35306}, {99, 12158}, {371, 41490}, {372, 489}, {376, 9733}, {486, 14568}, {487, 45079}, {488, 7582}, {490, 1588}, {519, 26301}, {549, 45411}, {576, 6811}, {598, 12159}, {637, 3312}, {638, 7584}, {639, 35770}, {641, 35771}, {642, 35814}, {648, 1586}, {671, 1328}, {754, 1505}, {1152, 43134}, {1351, 45510}, {1353, 45511}, {3071, 43133}, {3102, 7757}, {3311, 45508}, {3524, 26521}, {3543, 13748}, {3545, 6289}, {3594, 11293}, {3839, 26331}, {3845, 26438}, {5097, 45554}, {5874, 36726}, {6179, 35684}, {6321, 33430}, {6419, 39387}, {6420, 7389}, {6421, 8356}, {6423, 35953}, {6424, 35297}, {6427, 11315}, {6428, 11313}, {6459, 26617}, {6460, 12221}, {6463, 35927}, {6560, 22484}, {7388, 7760}, {7714, 26376}, {7752, 19105}, {7754, 44648}, {7763, 45514}, {7812, 39661}, {7894, 19102}, {9767, 13926}, {10304, 12305}, {10385, 26356}, {10754, 13760}, {11055, 13789}, {11239, 26523}, {11240, 26522}, {11917, 36733}, {12156, 13669}, {12222, 42561}, {12314, 26441}, {12969, 20065}, {13763, 22253}, {13770, 42024}, {13798, 26616}, {13801, 22645}, {13966, 45509}, {15682, 33457}, {22723, 32451}, {26289, 39679}, {26307, 45428}, {26315, 45434}, {26325, 45436}, {26370, 38314}, {26397, 45345}, {26421, 45347}, {26430, 45402}, {26436, 45404}, {26445, 45444}, {26450, 45446}, {26474, 45460}, {26480, 45458}, {26486, 45456}, {26491, 45454}, {26497, 45415}, {26506, 45412}, {26513, 45416}, {26515, 45476}, {26524, 45496}, {26525, 45494}, {32421, 35823}, {32797, 35578}, {34208, 45478}, {34511, 35305}, {41099, 45438}

X(45421) = reflection of X(i) in X(j) for these (i, j): (2, 32788), (489, 35948), (32809, 2), (35948, 372), (45420, 14614)
X(45421) = isotomic conjugate of X(42024)
X(45421) = barycentric product X(i)*X(j) for these {i, j}: {76, 41410}, {491, 21464}
X(45421) = trilinear product X(75)*X(41410)
X(45421) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(21464)}} and {{A, B, C, X(4), X(5861)}}
X(45421) = X(591)-of-2nd Kenmotu-centers triangle
X(45421) = X(41490)-of-2nd Kenmotu-free-vertices triangle
X(45421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 193, 5861), (2, 591, 492), (2, 1992, 45420), (2, 5032, 19054), (2, 5860, 32808), (2, 5861, 491), (2, 45420, 13637), (6, 591, 2), (6, 41624, 45420), (193, 3069, 491), (1991, 13847, 2), (1992, 13757, 13637), (1992, 13759, 13757), (3069, 5861, 2), (3069, 26340, 193), (5860, 19053, 2), (7774, 44392, 492), (8584, 11163, 45420), (8860, 41149, 45420), (13757, 45420, 2), (17346, 41629, 45420), (37640, 37641, 44595), (37785, 37786, 13757)

X(45422) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-INNER-YFF

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

X(45422) lies on these lines: {1, 6}, {3, 45526}, {5, 45456}, {371, 45586}, {492, 10527}, {3102, 45641}, {6289, 26470}, {6734, 45444}, {9733, 11249}, {10267, 43119}, {10680, 45488}, {10943, 45454}, {11012, 12305}, {11240, 26522}, {12116, 45406}, {13758, 13965}, {16202, 45411}, {18544, 45375}, {26308, 45428}, {26317, 45434}, {26332, 45440}, {26357, 45470}, {26363, 45472}, {26377, 45400}, {26399, 45345}, {26423, 45347}, {26431, 45402}, {26437, 45404}, {26452, 45446}, {26475, 45460}, {26481, 45458}, {26499, 45415}, {26508, 45412}, {45430, 45625}, {45432, 45626}, {45438, 45630}, {45462, 45640}, {45464, 45644}, {45467, 45645}, {45484, 45650}, {45487, 45651}

X(45422) = reflection of X(45425) in X(1335)
X(45422) = X(1335)-of-anti-inner-Yff triangle
X(45422) = X(45425)-of-outer-Yff tangents triangle
X(45422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 45476, 45490), (6, 45496, 1), (10932, 19049, 1), (12595, 44646, 1)

X(45423) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-INNER-YFF

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

X(45423) lies on these lines: {1, 6}, {3, 45527}, {5, 45457}, {372, 45587}, {491, 10527}, {3103, 45640}, {6290, 26470}, {6734, 45445}, {9732, 11249}, {10267, 43118}, {10680, 45489}, {10943, 45455}, {11012, 12306}, {11240, 26517}, {12116, 45407}, {13638, 13907}, {16202, 45410}, {18544, 45376}, {26308, 45429}, {26317, 45435}, {26332, 45441}, {26357, 45471}, {26363, 45473}, {26377, 45401}, {26399, 45348}, {26423, 45346}, {26431, 45403}, {26437, 45405}, {26452, 45447}, {26475, 45461}, {26481, 45459}, {26499, 45413}, {26508, 45414}, {45431, 45625}, {45433, 45626}, {45439, 45630}, {45463, 45641}, {45465, 45645}, {45466, 45644}, {45485, 45651}, {45486, 45650}

X(45423) = reflection of X(45424) in X(1124)
X(45423) = X(1124)-of-anti-inner-Yff triangle
X(45423) = X(45424)-of-outer-Yff tangents triangle
X(45423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 45477, 45491), (6, 45497, 1), (10931, 19050, 1), (12595, 44645, 1)

X(45424) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ANTI-OUTER-YFF

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

X(45424) lies on these lines: {1, 6}, {3, 45528}, {5, 45454}, {119, 6289}, {371, 45584}, {492, 5552}, {1470, 45404}, {2077, 12305}, {3102, 45643}, {6256, 13748}, {6735, 45444}, {9733, 11248}, {10269, 43119}, {10679, 45488}, {10942, 45456}, {11239, 26523}, {12115, 45406}, {13758, 13964}, {16203, 45411}, {18542, 45375}, {26309, 45428}, {26318, 45434}, {26333, 45440}, {26358, 45470}, {26364, 45472}, {26378, 45400}, {26400, 45345}, {26424, 45347}, {26432, 45402}, {26453, 45446}, {26476, 45460}, {26482, 45458}, {26500, 45415}, {26509, 45412}, {45430, 45627}, {45432, 45628}, {45438, 45631}, {45462, 45642}, {45464, 45646}, {45467, 45647}, {45484, 45652}, {45487, 45653}

X(45424) = reflection of X(45423) in X(1124)
X(45424) = X(1124)-of-anti-outer-Yff triangle
X(45424) = X(45423)-of-inner-Yff tangents triangle
X(45424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 45476, 45492), (6, 45494, 1), (10930, 19047, 1), (12594, 44644, 1)

X(45425) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ANTI-OUTER-YFF

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

X(45425) lies on these lines: {1, 6}, {3, 45529}, {5, 45455}, {119, 6290}, {372, 45585}, {491, 5552}, {1470, 45405}, {2077, 12306}, {3103, 45642}, {6256, 13749}, {6735, 45445}, {9732, 11248}, {10269, 43118}, {10679, 45489}, {10942, 45457}, {11239, 26518}, {12115, 45407}, {13638, 13906}, {16203, 45410}, {18542, 45376}, {26309, 45429}, {26318, 45435}, {26333, 45441}, {26358, 45471}, {26364, 45473}, {26378, 45401}, {26400, 45348}, {26424, 45346}, {26432, 45403}, {26453, 45447}, {26476, 45461}, {26482, 45459}, {26500, 45413}, {26509, 45414}, {45431, 45627}, {45433, 45628}, {45439, 45631}, {45463, 45643}, {45465, 45647}, {45466, 45646}, {45485, 45653}, {45486, 45652}

X(45425) = reflection of X(45422) in X(1335)
X(45425) = X(1335)-of-anti-outer-Yff triangle
X(45425) = X(45422)-of-inner-Yff tangents triangle
X(45425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 45477, 45493), (6, 45495, 1), (10929, 19048, 1), (12594, 44643, 1)

X(45426) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND AQUILA

Barycentrics    a*(-a^2+b^2+c^2-2*(b+c)*a+2*S) : :
X(45426) = 3*X(1)-2*X(5605) = X(5605)-3*X(7968)

X(45426) lies on these lines: {1, 6}, {3, 45530}, {10, 492}, {35, 45416}, {36, 45436}, {40, 9733}, {43, 13388}, {57, 8945}, {165, 12305}, {291, 16232}, {481, 1738}, {515, 45406}, {517, 45488}, {519, 26301}, {591, 3679}, {760, 1505}, {1385, 45411}, {1697, 45470}, {1698, 45472}, {1699, 45440}, {3008, 31569}, {3099, 45434}, {3102, 12782}, {3576, 43119}, {5587, 6289}, {5691, 13748}, {5881, 9906}, {6419, 45572}, {6420, 45500}, {7713, 45400}, {8185, 45428}, {8186, 45430}, {8187, 45432}, {8188, 45467}, {8189, 45464}, {8414, 9615}, {9441, 31563}, {9578, 45458}, {9581, 45460}, {10789, 45402}, {10826, 45454}, {10827, 45456}, {11852, 45446}, {12787, 35684}, {13389, 32913}, {13390, 33137}, {13758, 13971}, {13888, 45484}, {13942, 45487}, {13973, 44392}, {18480, 45375}, {18492, 45438}, {24440, 34495}, {24715, 30425}, {26296, 45345}, {26297, 45347}, {26298, 45415}, {26299, 45412}, {35774, 45462}

X(45426) = reflection of X(1) in X(7968)
X(45426) = intersection, other than A, B, C, of circumconics {{A, B, C, X(238), X(16232)}} and {{A, B, C, X(291), X(30557)}}
X(45426) = X(7968)-of-Aquila triangle
X(45426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1757, 30557), (1, 3751, 45427), (1, 5588, 3751), (1, 19003, 16475), (6, 45476, 45398), (1386, 5604, 1), (3242, 45399, 1), (3640, 18992, 1), (45398, 45476, 1)

X(45427) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND AQUILA

Barycentrics    a*(-a^2+b^2+c^2-2*(b+c)*a-2*S) : :
X(45427) = 3*X(1)-2*X(5604) = X(5604)-3*X(7969)

X(45427) lies on these lines: {1, 6}, {3, 45531}, {10, 491}, {35, 45417}, {36, 45437}, {40, 9732}, {43, 13389}, {57, 8941}, {165, 12306}, {291, 2362}, {482, 1738}, {515, 45407}, {517, 45489}, {519, 26300}, {760, 1504}, {1385, 45410}, {1659, 33137}, {1697, 45471}, {1698, 45473}, {1699, 45441}, {1991, 3679}, {3008, 31570}, {3099, 45435}, {3103, 12782}, {3576, 43118}, {5587, 6290}, {5691, 13749}, {5881, 9907}, {6419, 45501}, {6420, 45573}, {7613, 21169}, {7713, 45401}, {8185, 45429}, {8186, 45431}, {8187, 45433}, {8188, 45465}, {8189, 45466}, {8983, 13638}, {9441, 31564}, {9578, 45459}, {9581, 45461}, {9583, 39648}, {10789, 45403}, {10826, 45455}, {10827, 45457}, {11852, 45447}, {12788, 35685}, {13388, 32913}, {13888, 45486}, {13911, 44394}, {13942, 45485}, {18480, 45376}, {18492, 45439}, {24440, 34494}, {24715, 30426}, {26296, 45348}, {26297, 45346}, {26298, 45413}, {26299, 45414}, {35775, 45463}

X(45427) = reflection of X(1) in X(7969)
X(45427) = intersection, other than A, B, C, of circumconics {{A, B, C, X(238), X(2362)}} and {{A, B, C, X(291), X(30556)}}
X(45427) = X(7969)-of-Aquila triangle
X(45427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 1757, 30556), (1, 3751, 45426), (1, 5589, 3751), (1, 19004, 16475), (6, 45477, 45399), (1386, 5605, 1), (3242, 45398, 1), (3641, 18991, 1), (45399, 45477, 1)

X(45428) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ARA

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

X(45428) lies on these lines: {3, 639}, {6, 25}, {22, 492}, {24, 45406}, {157, 615}, {160, 3155}, {197, 45416}, {237, 44193}, {486, 2353}, {591, 9909}, {1586, 41761}, {1598, 45440}, {3070, 12148}, {3102, 35777}, {5938, 13785}, {6398, 11641}, {6642, 43119}, {7387, 9733}, {7506, 45411}, {7517, 8996}, {8185, 45426}, {8190, 45430}, {8191, 45432}, {8192, 45476}, {8193, 45444}, {8194, 45467}, {8195, 45464}, {8277, 39679}, {9714, 9921}, {9818, 45438}, {10037, 45490}, {10046, 45492}, {10790, 45402}, {10828, 45434}, {10829, 45454}, {10830, 45456}, {10831, 45458}, {10832, 45460}, {10833, 45470}, {10834, 45494}, {10835, 45496}, {11365, 45398}, {11414, 12305}, {11853, 45446}, {13889, 45484}, {13943, 33582}, {18954, 45404}, {22654, 45436}, {26302, 45345}, {26303, 45347}, {26304, 45415}, {26305, 45412}, {26307, 45421}, {26308, 45422}, {26309, 45424}, {35776, 45462}, {40947, 44192}, {45408, 45478}

X(45428) = X(5412)-of-Ara triangle
X(45428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (25, 159, 45429), (25, 5594, 159), (25, 19459, 44598)

X(45429) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ARA

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

X(45429) lies on these lines: {3, 640}, {6, 25}, {22, 491}, {24, 45407}, {157, 590}, {160, 3156}, {197, 45417}, {237, 44192}, {485, 2353}, {1585, 41761}, {1598, 45441}, {1991, 9909}, {3071, 12147}, {3103, 35776}, {3148, 31463}, {5938, 13665}, {6221, 11641}, {6642, 43118}, {7387, 9732}, {7506, 45410}, {7517, 45489}, {8185, 45427}, {8190, 45431}, {8191, 45433}, {8192, 45477}, {8193, 45445}, {8194, 45465}, {8195, 45466}, {8276, 39648}, {8969, 13889}, {9714, 9922}, {9818, 45439}, {10037, 45491}, {10046, 45493}, {10790, 45403}, {10828, 45435}, {10829, 45455}, {10830, 45457}, {10831, 45459}, {10832, 45461}, {10833, 45471}, {10834, 45495}, {10835, 45497}, {11365, 45399}, {11414, 12306}, {11853, 45447}, {13943, 45485}, {18954, 45405}, {22654, 45437}, {26302, 45348}, {26303, 45346}, {26304, 45413}, {26305, 45414}, {26306, 45420}, {26308, 45423}, {26309, 45425}, {35777, 45463}, {40947, 44193}, {45409, 45479}

X(45429) = X(5413)-of-Ara triangle
X(45429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (25, 159, 45428), (25, 5595, 159), (25, 19459, 44599), (8996, 45533, 3)

X(45430) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 1st AURIGA

Barycentrics    a*(-4*(2*S-(b+c)*a+c^2+b^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2+2*S)) : :

X(45430) lies on these lines: {3, 45534}, {6, 5597}, {55, 45432}, {492, 5601}, {591, 11207}, {3102, 35781}, {5598, 45476}, {5599, 45472}, {6289, 8200}, {8186, 45426}, {8190, 45428}, {8196, 45440}, {8197, 45444}, {8201, 45467}, {8202, 45464}, {9733, 11252}, {9834, 13748}, {11366, 45398}, {11384, 45400}, {11492, 45416}, {11493, 45436}, {11822, 12305}, {11837, 45402}, {11843, 45406}, {11861, 45434}, {11863, 45446}, {11865, 45454}, {11867, 45456}, {11869, 45458}, {11871, 45460}, {11873, 45470}, {11875, 45488}, {11877, 45490}, {11879, 45492}, {11881, 45494}, {11883, 45496}, {13890, 45484}, {13944, 45487}, {18495, 45438}, {18955, 45404}, {35778, 45462}, {43119, 45620}, {45347, 45353}, {45375, 45379}, {45411, 45535}, {45412, 45588}, {45415, 45589}, {45422, 45625}, {45424, 45627}

X(45430) = X(44582)-of-1st Auriga triangle
X(45430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5597, 8199, 12452), (5597, 12452, 45431)

X(45431) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 1st AURIGA

Barycentrics    a*(-4*(-2*S-(b+c)*a+c^2+b^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2-2*S)) : :

X(45431) lies on these lines: {3, 45535}, {6, 5597}, {55, 45433}, {491, 5601}, {1991, 11207}, {3103, 35778}, {5598, 45477}, {5599, 45473}, {6290, 8200}, {8186, 45427}, {8190, 45429}, {8196, 45441}, {8197, 45445}, {8201, 45465}, {8202, 45466}, {9732, 11252}, {9834, 13749}, {11366, 45399}, {11384, 45401}, {11492, 45417}, {11493, 45437}, {11822, 12306}, {11837, 45403}, {11843, 45407}, {11861, 45435}, {11863, 45447}, {11865, 45455}, {11867, 45457}, {11869, 45459}, {11871, 45461}, {11873, 45471}, {11875, 45489}, {11877, 45491}, {11879, 45493}, {11881, 45495}, {11883, 45497}, {13890, 45486}, {13944, 45485}, {18495, 45439}, {18955, 45405}, {35781, 45463}, {43118, 45620}, {45346, 45353}, {45376, 45379}, {45410, 45534}, {45413, 45589}, {45414, 45588}, {45423, 45625}, {45425, 45627}

X(45431) = X(44583)-of-1st Auriga triangle
X(45431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5597, 8198, 12452), (5597, 12452, 45430)

X(45432) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd AURIGA

Barycentrics    a*(4*(2*S-(b+c)*a+c^2+b^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2+2*S)) : :

X(45432) lies on these lines: {3, 45536}, {6, 5598}, {55, 45430}, {492, 5602}, {591, 11208}, {3102, 35779}, {5597, 45476}, {5600, 45472}, {6289, 8207}, {8187, 45426}, {8191, 45428}, {8203, 45440}, {8204, 45444}, {8208, 45467}, {8209, 45464}, {9733, 11253}, {9835, 13748}, {11367, 45398}, {11385, 45400}, {11492, 45436}, {11493, 45416}, {11823, 12305}, {11838, 45402}, {11844, 45406}, {11862, 45434}, {11864, 45446}, {11866, 45454}, {11868, 45456}, {11870, 45458}, {11872, 45460}, {11874, 45470}, {11876, 45488}, {11878, 45490}, {11880, 45492}, {11882, 45494}, {11884, 45496}, {13891, 45484}, {13945, 45487}, {18497, 45438}, {18956, 45404}, {35780, 45462}, {43119, 45621}, {45345, 45354}, {45375, 45380}, {45411, 45537}, {45412, 45590}, {45415, 45591}, {45422, 45626}, {45424, 45628}

X(45432) = X(44584)-of-2nd Auriga triangle
X(45432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5598, 8206, 12453), (5598, 12453, 45433)

X(45433) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd AURIGA

Barycentrics    a*(4*(-2*S-(b+c)*a+c^2+b^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(a^2+b^2+c^2-2*S)) : :

X(45433) lies on these lines: {3, 45537}, {6, 5598}, {55, 45431}, {491, 5602}, {1991, 11208}, {3103, 35780}, {5597, 45477}, {5600, 45473}, {6290, 8207}, {8187, 45427}, {8191, 45429}, {8203, 45441}, {8204, 45445}, {8208, 45465}, {8209, 45466}, {9732, 11253}, {9835, 13749}, {11367, 45399}, {11385, 45401}, {11492, 45437}, {11493, 45417}, {11823, 12306}, {11838, 45403}, {11844, 45407}, {11862, 45435}, {11864, 45447}, {11866, 45455}, {11868, 45457}, {11870, 45459}, {11872, 45461}, {11874, 45471}, {11876, 45489}, {11878, 45491}, {11880, 45493}, {11882, 45495}, {11884, 45497}, {13891, 45486}, {13945, 45485}, {18497, 45439}, {18956, 45405}, {35779, 45463}, {43118, 45621}, {45348, 45354}, {45376, 45380}, {45410, 45536}, {45413, 45591}, {45414, 45590}, {45423, 45626}, {45425, 45628}

X(45433) = X(44585)-of-2nd Auriga triangle
X(45433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5598, 8205, 12453), (5598, 12453, 45432)

X(45434) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 5th BROCARD

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

X(45434) lies on these lines: {3, 6}, {486, 43449}, {490, 13758}, {492, 2896}, {591, 7811}, {3096, 45472}, {3099, 45426}, {6289, 9987}, {6312, 13881}, {9857, 45444}, {9862, 45406}, {9873, 13748}, {9993, 45440}, {9997, 45476}, {10038, 45490}, {10047, 45492}, {10801, 12840}, {10802, 12841}, {10828, 45428}, {10871, 45454}, {10872, 45456}, {10873, 45458}, {10874, 45460}, {10875, 45467}, {10876, 45464}, {10877, 45470}, {10878, 45494}, {10879, 45496}, {11368, 45398}, {11386, 45400}, {11494, 45416}, {11861, 45430}, {11862, 45432}, {11885, 45446}, {13892, 45484}, {13946, 45487}, {18500, 45438}, {18503, 45375}, {18957, 45404}, {22744, 45436}, {26310, 45345}, {26311, 45347}, {26312, 45415}, {26313, 45412}, {26315, 45421}, {26317, 45422}, {26318, 45424}

X(45434) = X(44586)-of-5th Brocard triangle
X(45434) = X(45434)-of-circumsymmedial triangle
X(45434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12968, 45402), (32, 3094, 45435), (32, 9995, 3094)

X(45435) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 5th BROCARD

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

X(45435) lies on these lines: {3, 6}, {485, 43449}, {489, 13638}, {491, 2896}, {1991, 7811}, {3096, 45473}, {3099, 45427}, {6290, 9986}, {6316, 13881}, {9857, 45445}, {9862, 45407}, {9873, 13749}, {9993, 45441}, {9997, 45477}, {10038, 45491}, {10047, 45493}, {10801, 12841}, {10802, 12840}, {10828, 45429}, {10871, 45455}, {10872, 45457}, {10873, 45459}, {10874, 45461}, {10875, 45465}, {10876, 45466}, {10877, 45471}, {10878, 45495}, {10879, 45497}, {11368, 45399}, {11386, 45401}, {11494, 45417}, {11861, 45431}, {11862, 45433}, {11885, 45447}, {13892, 45486}, {13946, 45485}, {18500, 45439}, {18503, 45376}, {18957, 45405}, {22744, 45437}, {26310, 45348}, {26311, 45346}, {26312, 45413}, {26313, 45414}, {26314, 45420}, {26317, 45423}, {26318, 45425}

X(45435) = X(44587)-of-5th Brocard triangle
X(45435) = X(45435)-of-circumsymmedial triangle
X(45435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 12963, 45403), (32, 3094, 45434), (32, 9994, 3094)

X(45436) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd CIRCUMPERP TANGENTIAL

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

X(45436) lies on these lines: {1, 44196}, {3, 45416}, {6, 41}, {36, 45426}, {55, 45476}, {104, 45406}, {492, 2975}, {591, 11194}, {956, 45444}, {958, 45456}, {999, 45398}, {3102, 35785}, {3428, 12305}, {5135, 45580}, {6289, 22624}, {9733, 11249}, {10269, 43119}, {10966, 45470}, {11492, 45432}, {11493, 45430}, {12114, 13748}, {18761, 45438}, {22479, 45400}, {22520, 45402}, {22654, 45428}, {22744, 45434}, {22753, 45440}, {22755, 45446}, {22759, 45458}, {22760, 45460}, {22761, 45467}, {22762, 45464}, {22763, 45484}, {22764, 45487}, {22765, 45488}, {22766, 45490}, {22767, 45492}, {22768, 45494}, {26319, 45345}, {26320, 45347}, {26321, 45375}, {26322, 45415}, {26323, 45412}, {26325, 45421}, {35784, 45462}, {37535, 45411}

X(45436) = X(2067)-of-2nd circumperp tangential triangle
X(45436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (56, 22757, 22769), (56, 22769, 45437)

X(45437) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd CIRCUMPERP TANGENTIAL

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

X(45437) lies on these lines: {1, 44199}, {3, 45417}, {6, 41}, {36, 45427}, {55, 45477}, {104, 45407}, {491, 2975}, {956, 45445}, {958, 45457}, {999, 45399}, {1991, 11194}, {3103, 35784}, {3428, 12306}, {5135, 45581}, {6290, 22595}, {9732, 11249}, {10269, 43118}, {10966, 45471}, {11492, 45433}, {11493, 45431}, {12114, 13749}, {18761, 45439}, {22479, 45401}, {22520, 45403}, {22654, 45429}, {22744, 45435}, {22753, 45441}, {22755, 45447}, {22759, 45459}, {22760, 45461}, {22761, 45465}, {22762, 45466}, {22763, 45486}, {22764, 45485}, {22765, 45489}, {22766, 45491}, {22767, 45493}, {22768, 45495}, {26319, 45348}, {26320, 45346}, {26321, 45376}, {26322, 45413}, {26323, 45414}, {26324, 45420}, {35785, 45463}, {37535, 45410}

X(45437) = X(6502)-of-2nd circumperp tangential triangle
X(45437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (56, 22756, 22769), (56, 22769, 45436)

X(45438) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND EHRMANN-MID

Barycentrics    a^6+2*b^2*c^2*a^2+(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45438) = 3*X(381)-X(13665) = 3*X(381)+X(26346)

X(45438) lies on these lines: {2, 22806}, {3, 45542}, {4, 488}, {5, 13748}, {6, 13}, {30, 45472}, {382, 12305}, {546, 45440}, {591, 3845}, {1152, 45377}, {1478, 45460}, {1479, 45458}, {1995, 13654}, {3091, 45406}, {3098, 36733}, {3102, 35787}, {3583, 45470}, {3585, 45404}, {3832, 22596}, {3843, 45488}, {3851, 45411}, {6561, 18539}, {8414, 10576}, {9732, 36656}, {9818, 45428}, {9955, 45398}, {10895, 45490}, {10896, 45492}, {12699, 45444}, {14233, 37343}, {18414, 45415}, {18415, 19443}, {18491, 45416}, {18492, 45426}, {18495, 45430}, {18497, 45432}, {18500, 45434}, {18502, 45402}, {18507, 45446}, {18516, 45454}, {18517, 45456}, {18520, 45467}, {18522, 45464}, {18525, 45476}, {18538, 45484}, {18542, 45494}, {18544, 45496}, {18553, 44654}, {18761, 45436}, {18762, 36726}, {22807, 41106}, {35786, 45462}, {39679, 42262}, {41099, 45421}, {42284, 44392}, {45345, 45355}, {45347, 45356}, {45412, 45592}, {45422, 45630}, {45424, 45631}

X(45438) = midpoint of X(13665) and X(26346)
X(45438) = X(13665)-of-Ehrmann-mid triangle
X(45438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6289, 9733), (5, 13748, 43119), (381, 3818, 45439), (381, 13785, 19130), (381, 18440, 6564), (381, 18511, 3818), (381, 26346, 13665), (381, 45375, 6)

X(45439) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND EHRMANN-MID

Barycentrics    a^6+2*b^2*c^2*a^2-(a^4+(b^2+c^2)*a^2-2*(b^2-c^2)^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45439) = 3*X(381)-X(13785) = 3*X(381)+X(26336)

X(45439) lies on these lines: {2, 22807}, {3, 45543}, {4, 487}, {5, 13749}, {6, 13}, {30, 45473}, {382, 12306}, {546, 45441}, {1151, 45378}, {1478, 45461}, {1479, 45459}, {1991, 3845}, {1995, 13774}, {3091, 45407}, {3098, 36719}, {3103, 35786}, {3583, 45471}, {3585, 45405}, {3832, 22625}, {3843, 45489}, {3851, 45410}, {6560, 26438}, {8406, 10577}, {9733, 36655}, {9818, 45429}, {9955, 45399}, {10895, 45491}, {10896, 45493}, {12699, 45445}, {13692, 32807}, {14230, 37342}, {18414, 19442}, {18415, 45414}, {18491, 45417}, {18492, 45427}, {18495, 45431}, {18497, 45433}, {18500, 45435}, {18502, 45403}, {18507, 45447}, {18516, 45455}, {18517, 45457}, {18520, 45465}, {18522, 45466}, {18525, 45477}, {18538, 36723}, {18542, 45495}, {18544, 45497}, {18553, 44655}, {18761, 45437}, {18762, 45485}, {21736, 32813}, {22806, 41106}, {35787, 45463}, {39648, 42265}, {41099, 45420}, {42283, 44394}, {45346, 45356}, {45348, 45355}, {45413, 45593}, {45423, 45630}, {45425, 45631}

X(45439) = midpoint of X(13785) and X(26336)
X(45439) = X(13785)-of-Ehrmann-mid triangle
X(45439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6290, 9732), (5, 13749, 43118), (381, 3818, 45438), (381, 13665, 19130), (381, 18440, 6565), (381, 18509, 3818), (381, 26336, 13785), (381, 45376, 6)

X(45440) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND EULER

Barycentrics    3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2+2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45440) = 3*X(4)+X(10783) = 3*X(1587)-X(10783) = 3*X(14853)-X(39876)

X(45440) lies on these lines: {2, 12305}, {3, 45544}, {4, 6}, {5, 9733}, {11, 45404}, {12, 45470}, {30, 43119}, {51, 12298}, {98, 45402}, {132, 11474}, {235, 45400}, {262, 14245}, {371, 13644}, {372, 36656}, {381, 591}, {382, 45411}, {485, 36714}, {486, 45106}, {492, 3091}, {515, 45398}, {546, 45438}, {637, 11477}, {638, 10516}, {639, 1160}, {1131, 14484}, {1152, 6811}, {1350, 7389}, {1478, 45492}, {1479, 45490}, {1585, 17810}, {1598, 45428}, {1699, 45426}, {3095, 6290}, {3102, 6564}, {3127, 13567}, {3574, 45478}, {3592, 26441}, {3594, 45510}, {3839, 26331}, {3843, 6251}, {5200, 23292}, {5476, 44510}, {5587, 45444}, {5603, 45476}, {6215, 36726}, {6460, 7374}, {6560, 36709}, {6565, 45462}, {6813, 42265}, {7000, 31412}, {7683, 36679}, {7699, 45480}, {7778, 23312}, {8196, 45430}, {8203, 45432}, {8212, 45467}, {8213, 45464}, {8414, 42258}, {8416, 45577}, {9753, 12968}, {9766, 26469}, {9993, 45434}, {10531, 45494}, {10532, 45496}, {10893, 45454}, {10894, 45456}, {10895, 45458}, {10896, 45460}, {11293, 12306}, {11313, 11825}, {11315, 45498}, {11496, 45416}, {11897, 45446}, {11916, 32419}, {12257, 32787}, {12314, 45554}, {12818, 45107}, {13665, 36712}, {13687, 14269}, {14231, 14240}, {14238, 14492}, {21736, 42259}, {21737, 29181}, {22646, 42268}, {22753, 45436}, {26326, 45345}, {26327, 45347}, {26328, 45415}, {26329, 45412}, {26332, 45422}, {26333, 45424}, {26341, 36733}, {31463, 45101}, {36658, 42216}, {37342, 45473}, {41034, 42257}, {41035, 42256}, {42270, 44392}

X(45440) = midpoint of X(4) and X(1587)
X(45440) = crosspoint of X(4) and X(45101)
X(45440) = crosssum of X(3) and X(43119)
X(45440) = X(1587)-of-Euler triangle
X(45440) = X(13748)-of-anti-outer-Grebe triangle
X(45440) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6, 13748), (4, 1588, 14233), (4, 3070, 13749), (4, 5480, 45441), (4, 6201, 5480), (4, 7581, 5870), (4, 14853, 3071), (4, 23249, 14230), (4, 23253, 14235), (4, 23259, 14239), (4, 23267, 5871), (4, 45407, 36990), (5, 9733, 45472), (381, 45488, 6289), (5870, 7581, 8550), (6289, 45488, 591)

X(45441) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND EULER

Barycentrics    3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45441) = 3*X(4)+X(10784) = 3*X(1588)-X(10784) = 3*X(14853)-X(39875)

X(45441) lies on these lines: {2, 12306}, {3, 45545}, {4, 6}, {5, 9732}, {11, 45405}, {12, 45471}, {30, 43118}, {51, 12299}, {98, 45403}, {132, 11473}, {235, 45401}, {262, 14231}, {371, 36655}, {372, 13763}, {381, 1991}, {382, 45410}, {485, 45107}, {486, 36709}, {491, 3091}, {515, 45399}, {546, 45439}, {615, 21736}, {637, 10516}, {638, 11477}, {640, 1161}, {1132, 14484}, {1151, 6813}, {1350, 7388}, {1478, 45493}, {1479, 45491}, {1586, 17810}, {1598, 45429}, {1699, 45427}, {3095, 6289}, {3103, 6565}, {3128, 13567}, {3574, 45479}, {3589, 21737}, {3592, 45511}, {3594, 8982}, {3839, 26330}, {3843, 6250}, {5476, 44509}, {5587, 45445}, {5603, 45477}, {6214, 36723}, {6459, 7000}, {6561, 36714}, {6564, 45463}, {6811, 42262}, {7374, 42561}, {7683, 36678}, {7699, 45481}, {7778, 23311}, {8196, 45431}, {8203, 45433}, {8212, 45465}, {8213, 45466}, {8396, 45576}, {8406, 42259}, {9753, 12963}, {9766, 26468}, {9993, 45435}, {10192, 19219}, {10531, 45495}, {10532, 45497}, {10893, 45455}, {10894, 45457}, {10895, 45459}, {10896, 45461}, {11294, 12305}, {11314, 11824}, {11316, 45499}, {11496, 45417}, {11897, 45447}, {11917, 32421}, {12256, 32788}, {12313, 45555}, {12819, 45106}, {13785, 36711}, {13807, 14269}, {14234, 14492}, {14236, 14245}, {22617, 42269}, {22753, 45437}, {26326, 45348}, {26327, 45346}, {26328, 45413}, {26329, 45414}, {26332, 45423}, {26333, 45425}, {26348, 36719}, {36657, 42215}, {37343, 45472}, {41034, 42255}, {41035, 42254}, {42273, 44394}

X(45441) = midpoint of X(4) and X(1588)
X(45441) = crosspoint of X(4) and X(45102)
X(45441) = crosssum of X(3) and X(43118)
X(45441) = X(1588)-of-Euler triangle
X(45441) = X(13749)-of-anti-inner-Grebe triangle
X(45441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 6, 13749), (4, 1587, 14230), (4, 3071, 13748), (4, 5480, 45440), (4, 6202, 5480), (4, 7582, 5871), (4, 14853, 3070), (4, 23249, 14235), (4, 23259, 14233), (4, 23263, 14239), (4, 23273, 5870), (4, 45406, 36990), (5, 9732, 45473), (381, 45489, 6290), (5871, 7582, 8550), (6290, 45489, 1991)

X(45442) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 1st EXCOSINE

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

X(45442) lies on these lines: {64, 13748}, {154, 19408}, {1498, 9733}, {17810, 44608}, {31382, 45443}

X(45443) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 1st EXCOSINE

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

X(45443) lies on these lines: {64, 13749}, {154, 19409}, {1498, 9732}, {17810, 44609}, {31382, 45442}

X(45444) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND OUTER-GARCIA

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

X(45444) lies on these lines: {1, 45472}, {2, 45398}, {3, 45546}, {6, 10}, {8, 492}, {40, 13748}, {65, 45458}, {72, 45456}, {355, 9733}, {372, 4769}, {515, 12305}, {517, 6289}, {518, 12949}, {519, 45476}, {591, 3679}, {956, 45436}, {1737, 45492}, {1837, 45470}, {3057, 45460}, {3102, 35789}, {3932, 7090}, {3966, 6348}, {5090, 45400}, {5252, 45404}, {5587, 45440}, {5657, 45406}, {5687, 45416}, {5790, 45488}, {6734, 45422}, {6735, 45424}, {8193, 45428}, {8197, 45430}, {8204, 45432}, {8214, 45467}, {8215, 45464}, {9766, 26301}, {9857, 45434}, {10039, 45490}, {10791, 45402}, {10914, 45454}, {10915, 45494}, {10916, 45496}, {11313, 45500}, {11900, 45446}, {12699, 45438}, {12702, 45375}, {13688, 28198}, {13893, 45484}, {13947, 45487}, {26382, 45345}, {26406, 45347}, {26442, 45415}, {26443, 45412}, {26445, 45421}, {26446, 43119}, {35788, 45462}, {45411, 45547}

X(45444) = reflection of X(13911) in X(10)
X(45444) = X(8)-Beth conjugate of-X(13911)
X(45444) = X(13911)-of-outer-Garcia triangle
X(45444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 3416, 45445), (10, 5688, 3416), (10, 13936, 38047)

X(45445) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND OUTER-GARCIA

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

X(45445) lies on these lines: {1, 45473}, {2, 45399}, {3, 45547}, {6, 10}, {8, 491}, {40, 13749}, {65, 45459}, {72, 45457}, {355, 9732}, {371, 4769}, {515, 12306}, {517, 6290}, {518, 12948}, {519, 45477}, {956, 45437}, {1737, 45493}, {1837, 45471}, {1991, 3679}, {3057, 45461}, {3103, 35788}, {3932, 14121}, {3966, 6347}, {5090, 45401}, {5252, 45405}, {5587, 45441}, {5657, 45407}, {5687, 45417}, {5790, 45489}, {6734, 45423}, {6735, 45425}, {8193, 45429}, {8197, 45431}, {8204, 45433}, {8214, 45465}, {8215, 45466}, {9766, 26300}, {9857, 45435}, {10039, 45491}, {10791, 45403}, {10914, 45455}, {10915, 45495}, {10916, 45497}, {11314, 45501}, {11900, 45447}, {12699, 45439}, {12702, 45376}, {13808, 28198}, {13893, 45486}, {13947, 45485}, {26382, 45348}, {26406, 45346}, {26442, 45413}, {26443, 45414}, {26444, 45420}, {26446, 43118}, {35789, 45463}, {45410, 45546}

X(45445) = reflection of X(13973) in X(10)
X(45445) = X(8)-Beth conjugate of-X(13973)
X(45445) = X(13973)-of-outer-Garcia triangle
X(45445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (10, 3416, 45444), (10, 5689, 3416), (10, 13883, 38047)

X(45446) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND GOSSARD

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

X(45446) lies on these lines: {3, 45548}, {6, 402}, {30, 6289}, {492, 4240}, {591, 1651}, {1650, 45472}, {3102, 35791}, {9733, 11251}, {11831, 45398}, {11832, 45400}, {11839, 45402}, {11845, 45406}, {11848, 45416}, {11852, 45426}, {11853, 45428}, {11863, 45430}, {11864, 45432}, {11885, 45434}, {11897, 45440}, {11900, 45444}, {11903, 45454}, {11904, 45456}, {11905, 45458}, {11906, 45460}, {11907, 45467}, {11908, 45464}, {11909, 45470}, {11910, 45476}, {11911, 45488}, {11912, 45490}, {11913, 45492}, {11914, 45494}, {11915, 45496}, {12113, 13748}, {13894, 45484}, {13948, 45487}, {18507, 45438}, {18508, 45375}, {18958, 45404}, {22755, 45436}, {26383, 45345}, {26407, 45347}, {26447, 45415}, {26448, 45412}, {26450, 45421}, {26451, 43119}, {26452, 45422}, {26453, 45424}, {35790, 45462}, {45411, 45549}

X(45446) = reflection of X(44610) in X(402)
X(45446) = X(44610)-of-Gossard triangle
X(45446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (402, 11902, 12583), (402, 12583, 45447)

X(45447) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND GOSSARD

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

X(45447) lies on these lines: {3, 45549}, {6, 402}, {30, 6290}, {491, 4240}, {1650, 45473}, {1651, 1991}, {3103, 35790}, {9732, 11251}, {11831, 45399}, {11832, 45401}, {11839, 45403}, {11845, 45407}, {11848, 45417}, {11852, 45427}, {11853, 45429}, {11863, 45431}, {11864, 45433}, {11885, 45435}, {11897, 45441}, {11900, 45445}, {11903, 45455}, {11904, 45457}, {11905, 45459}, {11906, 45461}, {11907, 45465}, {11908, 45466}, {11909, 45471}, {11910, 45477}, {11911, 45489}, {11912, 45491}, {11913, 45493}, {11914, 45495}, {11915, 45497}, {12113, 13749}, {13894, 45486}, {13948, 45485}, {18507, 45439}, {18508, 45376}, {18958, 45405}, {22755, 45437}, {26383, 45348}, {26407, 45346}, {26447, 45413}, {26448, 45414}, {26449, 45420}, {26451, 43118}, {26452, 45423}, {26453, 45425}, {35791, 45463}, {45410, 45548}

X(45447) = reflection of X(44611) in X(402)
X(45447) = X(44611)-of-Gossard triangle
X(45447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (402, 11901, 12583), (402, 12583, 45446)

X(45448) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 3rd HATZIPOLAKIS

Barycentrics    a^2*(-2*((b^2+c^2)*a^8-2*(b^4+10*b^2*c^2+c^4)*a^6+27*(b^2+c^2)*b^2*c^2*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^6+c^6)*(b^2-c^2)^2)*S+a^12-5*(b^2+c^2)*a^10+(10*b^4+13*b^2*c^2+10*c^4)*a^8-10*(b^2+c^2)*(b^4-b^2*c^2+c^4)*a^6+(5*b^8+5*c^8-2*(2*b^4+41*b^2*c^2+2*c^4)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2-(b^2-c^2)^4*b^2*c^2) : :

X(45448) lies on these lines: {6, 32328}, {9733, 13630}

X(45449) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 3rd HATZIPOLAKIS

Barycentrics    a^2*(2*((b^2+c^2)*a^8-2*(b^4+10*b^2*c^2+c^4)*a^6+27*(b^2+c^2)*b^2*c^2*a^4+2*(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^6+c^6)*(b^2-c^2)^2)*S+a^12-5*(b^2+c^2)*a^10+(10*b^4+13*b^2*c^2+10*c^4)*a^8-10*(b^2+c^2)*(b^4-b^2*c^2+c^4)*a^6+(5*b^8+5*c^8-2*(2*b^4+41*b^2*c^2+2*c^4)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2-(b^2-c^2)^4*b^2*c^2) : :

X(45449) lies on these lines: {6, 32327}, {9732, 13630}

X(45450) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND HATZIPOLAKIS-MOSES

Barycentrics    a^2*(-2*((b^2+c^2)*a^8-2*(b^4+6*b^2*c^2+c^4)*a^6+15*(b^2+c^2)*b^2*c^2*a^4+2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-(b^6+c^6)*(b^2-c^2)^2)*S+a^12-5*(b^2+c^2)*a^10+(10*b^4+13*b^2*c^2+10*c^4)*a^8-2*(b^2+c^2)*(5*b^4-3*b^2*c^2+5*c^4)*a^6+(5*b^8+5*c^8-2*(2*b^4+13*b^2*c^2+2*c^4)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^2+c^2)*(b^4+c^4)*a^2-(b^2-c^2)^4*b^2*c^2) : :

X(45450) lies on these lines: {6102, 9733}, {13403, 13748}

X(45451) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND HATZIPOLAKIS-MOSES

Barycentrics    a^2*(2*((b^2+c^2)*a^8-2*(b^4+6*b^2*c^2+c^4)*a^6+15*(b^2+c^2)*b^2*c^2*a^4+2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-(b^6+c^6)*(b^2-c^2)^2)*S+a^12-5*(b^2+c^2)*a^10+(10*b^4+13*b^2*c^2+10*c^4)*a^8-2*(b^2+c^2)*(5*b^4-3*b^2*c^2+5*c^4)*a^6+(5*b^8+5*c^8-2*(2*b^4+13*b^2*c^2+2*c^4)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^2+c^2)*(b^4+c^4)*a^2-(b^2-c^2)^4*b^2*c^2) : :

X(45451) lies on these lines: {6102, 9732}, {13403, 13749}

X(45452) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd HYACINTH

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

X(45452) lies on these lines: {185, 9733}, {1885, 13748}, {12305, 17818}, {45400, 45474}

X(45453) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd HYACINTH

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

X(45453) lies on these lines: {185, 9732}, {1885, 13749}, {12306, 17818}, {45401, 45475}

X(45454) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND INNER-JOHNSON

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

X(45454) lies on these lines: {3, 45556}, {5, 45424}, {6, 11}, {12, 45494}, {355, 6289}, {492, 3434}, {591, 11235}, {1376, 45416}, {3102, 35797}, {7969, 12959}, {9733, 10525}, {10523, 45490}, {10785, 45406}, {10794, 45402}, {10826, 45426}, {10829, 45428}, {10871, 45434}, {10893, 45440}, {10914, 45444}, {10943, 45422}, {10944, 45458}, {10945, 45467}, {10946, 45464}, {10947, 45470}, {10948, 45492}, {10949, 45496}, {11373, 45398}, {11390, 45400}, {11826, 12305}, {11865, 45430}, {11866, 45432}, {11903, 45446}, {11928, 45488}, {12114, 13748}, {13895, 45484}, {13952, 45487}, {17627, 44670}, {18516, 45438}, {18519, 45375}, {18961, 45404}, {26390, 45345}, {26414, 45347}, {26488, 45415}, {26489, 45412}, {26491, 45421}, {26492, 43119}, {35796, 45462}, {45411, 45557}

X(45454) = X(44623)-of-inner-Johnson triangle
X(45454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (11, 10920, 12586), (11, 12586, 45455)

X(45455) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND INNER-JOHNSON

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

X(45455) lies on these lines: {3, 45557}, {5, 45425}, {6, 11}, {12, 45495}, {355, 6290}, {491, 3434}, {1376, 45417}, {1991, 11235}, {3103, 35796}, {7968, 12958}, {9732, 10525}, {10523, 45491}, {10785, 45407}, {10794, 45403}, {10826, 45427}, {10829, 45429}, {10871, 45435}, {10893, 45441}, {10914, 45445}, {10943, 45423}, {10944, 45459}, {10945, 45465}, {10946, 45466}, {10947, 45471}, {10948, 45493}, {10949, 45497}, {11373, 45399}, {11390, 45401}, {11826, 12306}, {11865, 45431}, {11866, 45433}, {11903, 45447}, {11928, 45489}, {12114, 13749}, {13895, 45486}, {13952, 45485}, {18516, 45439}, {18519, 45376}, {18961, 45405}, {26390, 45348}, {26414, 45346}, {26488, 45413}, {26489, 45414}, {26490, 45420}, {26492, 43118}, {35797, 45463}, {45410, 45556}

X(45455) = X(44624)-of-inner-Johnson triangle
X(45455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (11, 10919, 12586), (11, 12586, 45454)

X(45456) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND OUTER-JOHNSON

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

X(45456) lies on these lines: {3, 45558}, {5, 45422}, {6, 12}, {11, 45496}, {72, 45444}, {355, 6289}, {492, 3436}, {591, 11236}, {958, 45436}, {3102, 35799}, {7969, 12949}, {9733, 10526}, {10523, 45492}, {10786, 45406}, {10795, 45402}, {10827, 45426}, {10830, 45428}, {10872, 45434}, {10894, 45440}, {10942, 45424}, {10950, 45460}, {10951, 45467}, {10952, 45464}, {10953, 45470}, {10954, 45490}, {10955, 45494}, {11374, 45398}, {11391, 45400}, {11500, 13748}, {11827, 12305}, {11867, 45430}, {11868, 45432}, {11904, 45446}, {11929, 45488}, {13896, 45484}, {13953, 45487}, {18517, 45438}, {18518, 45375}, {18962, 45404}, {26389, 45345}, {26413, 45347}, {26483, 45415}, {26484, 45412}, {26486, 45421}, {26487, 43119}, {35798, 45462}, {45411, 45559}

X(45456) = X(31472)-of-outer-Johnson triangle
X(45456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (12, 10922, 12587), (12, 12587, 45457)

X(45457) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND OUTER-JOHNSON

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

X(45457) lies on these lines: {3, 45559}, {5, 45423}, {6, 12}, {11, 45497}, {72, 45445}, {355, 6290}, {491, 3436}, {958, 45437}, {1991, 11236}, {3103, 35798}, {7968, 12948}, {9732, 10526}, {10523, 45493}, {10786, 45407}, {10795, 45403}, {10827, 45427}, {10830, 45429}, {10872, 45435}, {10894, 45441}, {10942, 45425}, {10950, 45461}, {10951, 45465}, {10952, 45466}, {10953, 45471}, {10954, 45491}, {10955, 45495}, {11374, 45399}, {11391, 45401}, {11500, 13749}, {11827, 12306}, {11867, 45431}, {11868, 45433}, {11904, 45447}, {11929, 45489}, {13896, 45486}, {13953, 45485}, {18517, 45439}, {18518, 45376}, {18962, 45405}, {26389, 45348}, {26413, 45346}, {26483, 45413}, {26484, 45414}, {26485, 45420}, {26487, 43118}, {35799, 45463}, {45410, 45558}

X(45457) = X(44622)-of-outer-Johnson triangle
X(45457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (12, 10921, 12587), (12, 12587, 45456)

X(45458) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 1st JOHNSON-YFF

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

X(45458) lies on these lines: {1, 6289}, {3, 45560}, {4, 45470}, {5, 45492}, {6, 12}, {55, 13748}, {56, 45472}, {65, 45444}, {325, 45405}, {388, 492}, {495, 45490}, {498, 43119}, {591, 11237}, {1478, 9733}, {1479, 45438}, {3085, 45406}, {3102, 35801}, {3295, 45375}, {7354, 12305}, {9578, 45426}, {9654, 45488}, {10797, 45402}, {10831, 45428}, {10873, 45434}, {10895, 45440}, {10944, 45454}, {10956, 45494}, {10957, 45496}, {11375, 45398}, {11392, 45400}, {11501, 45416}, {11869, 45430}, {11870, 45432}, {11905, 45446}, {11930, 45467}, {11931, 45464}, {12948, 37719}, {13897, 45484}, {13954, 45487}, {22759, 45436}, {26388, 45345}, {26412, 45347}, {26477, 45415}, {26478, 45412}, {26480, 45421}, {26481, 45422}, {26482, 45424}, {31479, 45411}, {35800, 45462}

X(45458) = X(44620)-of-1st Johnson-Yff triangle
X(45458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6289, 45460), (388, 492, 45404)

X(45459) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 1st JOHNSON-YFF

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

X(45459) lies on these lines: {1, 6290}, {3, 45561}, {4, 45471}, {5, 45493}, {6, 12}, {55, 13749}, {56, 45473}, {65, 45445}, {325, 45404}, {388, 491}, {495, 45491}, {498, 43118}, {1478, 9732}, {1479, 45439}, {1991, 11237}, {3085, 45407}, {3103, 35800}, {3295, 45376}, {7354, 12306}, {9578, 45427}, {9646, 39648}, {9654, 45489}, {10797, 45403}, {10831, 45429}, {10873, 45435}, {10895, 45441}, {10944, 45455}, {10956, 45495}, {10957, 45497}, {11375, 45399}, {11392, 45401}, {11501, 45417}, {11869, 45431}, {11870, 45433}, {11905, 45447}, {11930, 45465}, {11931, 45466}, {12949, 37719}, {13897, 45486}, {13954, 45485}, {22759, 45437}, {26388, 45348}, {26412, 45346}, {26477, 45413}, {26478, 45414}, {26479, 45420}, {26481, 45423}, {26482, 45425}, {31479, 45410}, {35801, 45463}

X(45459) = X(44621)-of-1st Johnson-Yff triangle
X(45459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6290, 45461), (12, 10923, 12588), (12, 39897, 44622), (388, 491, 45405)

X(45460) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 2nd JOHNSON-YFF

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

X(45460) lies on these lines: {1, 6289}, {3, 45562}, {4, 45404}, {5, 45490}, {6, 11}, {55, 45472}, {56, 13748}, {325, 45471}, {492, 497}, {496, 45492}, {499, 43119}, {591, 11238}, {999, 45375}, {1478, 45438}, {1479, 9733}, {3057, 45444}, {3086, 45406}, {3102, 35803}, {6284, 12305}, {9581, 45426}, {9669, 45488}, {10798, 45402}, {10832, 45428}, {10874, 45434}, {10896, 45440}, {10950, 45456}, {10958, 45494}, {10959, 45496}, {11376, 45398}, {11393, 45400}, {11502, 45416}, {11871, 45430}, {11872, 45432}, {11906, 45446}, {11932, 45467}, {11933, 45464}, {12958, 37720}, {13898, 45484}, {13955, 45487}, {22760, 45436}, {26387, 45345}, {26411, 45347}, {26471, 45415}, {26472, 45412}, {26474, 45421}, {26475, 45422}, {26476, 45424}, {35802, 45462}, {45411, 45563}

X(45460) = X(44618)-of-2nd Johnson-Yff triangle
X(45460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6289, 45458), (11, 10926, 12589), (11, 12589, 45461), (11, 39873, 44623), (492, 497, 45470)

X(45461) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd JOHNSON-YFF

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

X(45461) lies on these lines: {1, 6290}, {3, 45563}, {4, 45405}, {5, 45491}, {6, 11}, {55, 45473}, {56, 13749}, {325, 45470}, {491, 497}, {496, 45493}, {499, 43118}, {999, 45376}, {1478, 45439}, {1479, 9732}, {1991, 11238}, {3057, 45445}, {3086, 45407}, {3103, 35802}, {6284, 12306}, {9581, 45427}, {9661, 39648}, {9669, 45489}, {10798, 45403}, {10832, 45429}, {10874, 45435}, {10896, 45441}, {10950, 45457}, {10958, 45495}, {10959, 45497}, {11376, 45399}, {11393, 45401}, {11502, 45417}, {11871, 45431}, {11872, 45433}, {11906, 45447}, {11932, 45465}, {11933, 45466}, {12959, 37720}, {13898, 45486}, {13955, 45485}, {22760, 45437}, {26387, 45348}, {26411, 45346}, {26471, 45413}, {26472, 45414}, {26473, 45420}, {26475, 45423}, {26476, 45425}, {35803, 45463}, {45410, 45562}

X(45461) = X(44619)-of-2nd Johnson-Yff triangle
X(45461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6290, 45459), (11, 10925, 12589), (11, 12589, 45460), (11, 39873, 44624), (491, 497, 45471)

X(45462) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 1st KENMOTU-FREE-VERTICES

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

X(45462) lies on these lines: {3, 6}, {4, 3440}, {5, 44392}, {485, 492}, {486, 14245}, {588, 32568}, {591, 7775}, {1587, 7785}, {1993, 8577}, {3093, 8948}, {5408, 8956}, {5420, 7857}, {6289, 6564}, {6560, 45406}, {6565, 45440}, {7599, 11004}, {7755, 13967}, {7760, 19055}, {7763, 35685}, {7774, 13926}, {7781, 35878}, {7812, 39661}, {7862, 10576}, {8035, 15234}, {12110, 19108}, {12221, 20088}, {13748, 35820}, {18548, 35868}, {19102, 39660}, {23251, 45375}, {26468, 32499}, {33006, 35702}, {35762, 45398}, {35764, 45400}, {35768, 45404}, {35769, 45492}, {35772, 45416}, {35774, 45426}, {35776, 45428}, {35778, 45430}, {35780, 45432}, {35784, 45436}, {35786, 45438}, {35788, 45444}, {35790, 45446}, {35796, 45454}, {35798, 45456}, {35800, 45458}, {35802, 45460}, {35804, 45467}, {35806, 45464}, {35808, 45470}, {35809, 45490}, {35810, 45476}, {35812, 45484}, {35814, 45487}, {35816, 45494}, {35818, 45496}, {42022, 45412}, {45345, 45357}, {45347, 45359}, {45415, 45601}, {45422, 45640}, {45424, 45642}

X(45462) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(2459)}} and {{A, B, C, X(83), X(35767)}}
X(45462) = X(5062)-of-1st Kenmotu-free-vertices triangle
X(45462) = X(45462)-of-circumsymmedial triangle
X(45462) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(2459)
X(45462) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 1152, 45411), (6, 3095, 45463), (6, 9733, 371), (6, 45488, 3102), (32, 576, 45463), (371, 372, 2459), (371, 6420, 45515), (372, 35794, 35840), (372, 35840, 3103), (372, 45564, 1152), (1160, 19145, 40274), (1342, 1343, 35767), (1351, 3312, 6423), (1351, 6423, 371), (3371, 3372, 43120), (3385, 3386, 5062), (3557, 3558, 371), (5111, 44587, 1504), (6395, 15884, 372), (6418, 8416, 19146), (6422, 12314, 45565), (8416, 19146, 372), (9975, 30435, 6419), (12968, 45489, 2460), (44476, 44502, 6)

X(45463) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 2nd KENMOTU-FREE-VERTICES

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

X(45463) lies on these lines: {3, 6}, {4, 32492}, {5, 44394}, {485, 14231}, {486, 491}, {589, 32575}, {1588, 7785}, {1991, 7775}, {1993, 8576}, {3092, 8946}, {5418, 7857}, {6290, 6565}, {6561, 45407}, {6564, 45441}, {7598, 11004}, {7755, 8980}, {7760, 19056}, {7763, 35684}, {7774, 13873}, {7781, 35879}, {7812, 39660}, {7862, 10577}, {8036, 15233}, {12110, 19109}, {12222, 20088}, {13749, 35821}, {18548, 35869}, {19105, 39661}, {23261, 45376}, {26469, 32498}, {33006, 35703}, {35763, 45399}, {35765, 45401}, {35768, 45493}, {35769, 45405}, {35773, 45417}, {35775, 45427}, {35777, 45429}, {35779, 45433}, {35781, 45431}, {35785, 45437}, {35787, 45439}, {35789, 45445}, {35791, 45447}, {35797, 45455}, {35799, 45457}, {35801, 45459}, {35803, 45461}, {35805, 45466}, {35807, 45465}, {35808, 45491}, {35809, 45471}, {35811, 45477}, {35813, 45485}, {35815, 45486}, {35817, 45495}, {35819, 45497}, {45346, 45358}, {45348, 45360}, {45413, 45600}, {45414, 45602}, {45423, 45641}, {45425, 45643}

X(45463) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(2460)}} and {{A, B, C, X(83), X(35766)}}
X(45463) = X(5058)-of-2nd Kenmotu-free-vertices triangle
X(45463) = X(45463)-of-circumsymmedial triangle
X(45463) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(2460)
X(45463) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 1151, 45410), (6, 3095, 45462), (6, 9732, 372), (6, 45489, 3103), (32, 576, 45462), (371, 372, 2460), (371, 35795, 35841), (371, 35841, 3102), (371, 45565, 1151), (372, 6419, 45514), (1161, 19146, 40275), (1342, 1343, 35766), (1351, 3311, 6424), (1351, 6424, 372), (3371, 3372, 5058), (3385, 3386, 43121), (3557, 3558, 372), (5111, 44586, 1505), (6199, 15883, 371), (6417, 8396, 19145), (6421, 12313, 45564), (8396, 19145, 371), (9974, 30435, 6420), (12963, 45488, 2459), (44475, 44501, 6)

X(45464) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND LUCAS(-1) HOMOTHETIC

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

X(45464) lies on these lines: {3, 45566}, {6, 494}, {492, 6463}, {591, 12153}, {3102, 35805}, {3964, 45465}, {6289, 8221}, {6458, 6466}, {6461, 45467}, {8189, 45426}, {8195, 45428}, {8202, 45430}, {8209, 45432}, {8211, 45476}, {8213, 45440}, {8215, 45444}, {8223, 45472}, {9733, 10673}, {9839, 13748}, {10876, 45434}, {10946, 45454}, {10952, 45456}, {11378, 45398}, {11395, 45400}, {11504, 45416}, {11829, 12305}, {11841, 45402}, {11847, 45406}, {11908, 45446}, {11931, 45458}, {11933, 45460}, {11948, 45470}, {11950, 45488}, {11952, 45490}, {11954, 45492}, {11956, 45494}, {11958, 45496}, {13900, 45484}, {13957, 45487}, {18522, 45438}, {18964, 45404}, {22762, 45436}, {35806, 45462}, {43119, 45624}, {45345, 45361}, {45347, 45363}, {45375, 45382}, {45411, 45568}, {45415, 45603}, {45422, 45644}, {45424, 45646}

X(45464) = X(45595)-of-Lucas(-1) homothetic triangle
X(45464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 8219, 12591), (494, 12591, 45466), (19443, 45412, 6)

X(45465) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND LUCAS(+1) HOMOTHETIC

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

X(45465) lies on these lines: {3, 45567}, {6, 493}, {491, 6462}, {1991, 12152}, {3103, 35804}, {3964, 45464}, {6290, 8220}, {6457, 6465}, {6461, 45466}, {8188, 45427}, {8194, 45429}, {8201, 45431}, {8208, 45433}, {8210, 45477}, {8212, 45441}, {8214, 45445}, {8222, 45473}, {9732, 10669}, {9838, 13749}, {10875, 45435}, {10945, 45455}, {10951, 45457}, {11377, 45399}, {11394, 45401}, {11503, 45417}, {11828, 12306}, {11840, 45403}, {11846, 45407}, {11907, 45447}, {11930, 45459}, {11932, 45461}, {11947, 45471}, {11949, 45489}, {11951, 45491}, {11953, 45493}, {11955, 45495}, {11957, 45497}, {13899, 45486}, {13956, 45485}, {18520, 45439}, {18963, 45405}, {22761, 45437}, {35807, 45463}, {43118, 45623}, {45346, 45364}, {45348, 45362}, {45376, 45381}, {45410, 45569}, {45414, 45604}, {45423, 45645}, {45425, 45647}

X(45465) = X(45596)-of-Lucas(+1) homothetic triangle
X(45465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 8216, 12590), (493, 12590, 45467), (19442, 45413, 6)

X(45466) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(a^6+(b^2+c^2)*a^4-5*(b^2+c^2)^2*a^2-(3*a^4-6*(b^2+c^2)*a^2-b^4-18*b^2*c^2-c^4)*S+(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2)) : :

X(45466) lies on these lines: {3, 45568}, {6, 494}, {491, 6463}, {1991, 12153}, {3103, 35806}, {6290, 8221}, {6461, 45465}, {6466, 8406}, {8189, 45427}, {8195, 45429}, {8202, 45431}, {8209, 45433}, {8211, 45477}, {8213, 45441}, {8215, 45445}, {8223, 45473}, {9732, 10673}, {9839, 13749}, {10876, 45435}, {10946, 45455}, {10952, 45457}, {11378, 45399}, {11395, 45401}, {11504, 45417}, {11829, 12306}, {11841, 45403}, {11847, 45407}, {11908, 45447}, {11931, 45459}, {11933, 45461}, {11948, 45471}, {11950, 45489}, {11952, 45491}, {11954, 45493}, {11956, 45495}, {11958, 45497}, {13900, 45486}, {13957, 45485}, {18522, 45439}, {18964, 45405}, {22762, 45437}, {35805, 45463}, {43118, 45624}, {45346, 45363}, {45348, 45361}, {45376, 45382}, {45410, 45566}, {45413, 45603}, {45423, 45644}, {45425, 45646}

X(45466) = X(45598)-of-Lucas(-1) homothetic triangle
X(45466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 8217, 12591), (494, 12591, 45464)

X(45467) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(a^6+(b^2+c^2)*a^4-5*(b^2+c^2)^2*a^2+(3*a^4-6*(b^2+c^2)*a^2-b^4-18*b^2*c^2-c^4)*S+(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2)) : :

X(45467) lies on these lines: {3, 45569}, {6, 493}, {492, 6462}, {591, 12152}, {3102, 35807}, {6289, 8220}, {6461, 45464}, {6465, 8414}, {8188, 45426}, {8194, 45428}, {8201, 45430}, {8208, 45432}, {8210, 45476}, {8212, 45440}, {8214, 45444}, {8222, 45472}, {9733, 10669}, {9838, 13748}, {10875, 45434}, {10945, 45454}, {10951, 45456}, {11377, 45398}, {11394, 45400}, {11503, 45416}, {11828, 12305}, {11840, 45402}, {11846, 45406}, {11907, 45446}, {11930, 45458}, {11932, 45460}, {11947, 45470}, {11949, 45488}, {11951, 45490}, {11953, 45492}, {11955, 45494}, {11957, 45496}, {13899, 45484}, {13956, 45487}, {18520, 45438}, {18963, 45404}, {22761, 45436}, {35804, 45462}, {43119, 45623}, {45345, 45362}, {45347, 45364}, {45375, 45381}, {45411, 45567}, {45412, 45604}, {45422, 45645}, {45424, 45647}

X(45467) = X(45597)-of-Lucas(+1) homothetic triangle
X(45467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 8218, 12590), (493, 12590, 45465)

X(45468) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND MANDART-EXCIRCLES

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

X(45468) lies on this line: {65, 26495}

X(45469) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND MANDART-EXCIRCLES

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

X(45469) lies on this line: {65, 26504}

X(45470) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND MANDART-INCIRCLE

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

X(45470) lies on these lines: {1, 9733}, {3, 45492}, {4, 45458}, {6, 31}, {11, 45472}, {12, 45440}, {33, 45400}, {35, 43119}, {56, 6283}, {325, 45461}, {492, 497}, {591, 3058}, {1479, 6289}, {1697, 45426}, {1837, 45444}, {2098, 45476}, {2646, 45398}, {3057, 8211}, {3095, 45491}, {3102, 35809}, {3295, 45488}, {3303, 6405}, {3583, 45438}, {4294, 45406}, {4309, 13081}, {5148, 35769}, {6284, 13748}, {9668, 45375}, {10385, 26356}, {10799, 45402}, {10833, 45428}, {10877, 45434}, {10947, 45454}, {10953, 45456}, {10965, 45494}, {10966, 45436}, {11873, 45430}, {11874, 45432}, {11909, 45446}, {11947, 45467}, {11948, 45464}, {12314, 45580}, {13043, 26292}, {13901, 45484}, {13958, 45487}, {19434, 26353}, {19435, 19443}, {26351, 45345}, {26352, 45347}, {26354, 45412}, {26357, 45422}, {26358, 45424}, {35808, 45462}, {36656, 45560}, {45411, 45571}

X(45470) = X(44590)-of-Mandart-incircle triangle
X(45470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9733, 45404), (55, 3056, 45471), (55, 10928, 3056), (55, 19037, 2330), (492, 497, 45460), (3295, 45488, 45490)

X(45471) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND MANDART-INCIRCLE

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

X(45471) lies on these lines: {1, 9732}, {3, 45493}, {4, 45459}, {6, 31}, {11, 45473}, {12, 45441}, {33, 45401}, {35, 43118}, {56, 6405}, {325, 45460}, {491, 497}, {1479, 6290}, {1697, 45427}, {1837, 45445}, {1991, 3058}, {2098, 45477}, {2646, 45399}, {3057, 8210}, {3095, 45490}, {3103, 35808}, {3295, 45489}, {3303, 6283}, {3583, 45439}, {4294, 45407}, {4309, 13082}, {5148, 35768}, {6284, 13749}, {9668, 45376}, {10385, 26355}, {10799, 45403}, {10833, 45429}, {10877, 45435}, {10947, 45455}, {10953, 45457}, {10965, 45495}, {10966, 45437}, {11873, 45431}, {11874, 45433}, {11909, 45447}, {11947, 45465}, {11948, 45466}, {12313, 45581}, {13044, 26293}, {13901, 45486}, {13958, 45485}, {19434, 19442}, {19435, 26354}, {26351, 45348}, {26352, 45346}, {26353, 45413}, {26357, 45423}, {26358, 45425}, {35809, 45463}, {36655, 45561}, {45410, 45570}

X(45471) = X(44591)-of-Mandart-incircle triangle
X(45471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9732, 45405), (55, 3056, 45470), (55, 10927, 3056), (55, 19038, 2330), (491, 497, 45461), (3295, 45489, 45491)

X(45472) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND MEDIAL

Barycentrics    b^2+c^2+2*S : :
X(45472) = 3*X(2)+X(1270) = 5*X(2)-X(13639) = 7*X(2)-2*X(13664) = 5*X(1270)+3*X(13639) = 2*X(1270)+3*X(13663) = 7*X(1270)+6*X(13664) = 5*X(3068)-3*X(13639) = 2*X(3068)-3*X(13663) = 7*X(3068)-6*X(13664) = 2*X(13639)-5*X(13663) = 7*X(13639)-10*X(13664) = 7*X(13663)-4*X(13664)

X(45472) lies on these lines: {1, 45444}, {2, 6}, {3, 639}, {4, 12305}, {5, 9733}, {8, 45476}, {10, 31534}, {11, 45470}, {12, 45404}, {20, 33364}, {30, 45438}, {55, 45460}, {56, 45458}, {83, 45402}, {140, 43119}, {315, 12968}, {338, 34391}, {371, 3788}, {372, 626}, {427, 45400}, {485, 32491}, {486, 7795}, {487, 8414}, {488, 3070}, {489, 6409}, {490, 23251}, {498, 45490}, {499, 45492}, {594, 5391}, {620, 6200}, {623, 18587}, {624, 18586}, {625, 6564}, {631, 45406}, {637, 1151}, {638, 42265}, {640, 1656}, {642, 3526}, {732, 13647}, {754, 13644}, {958, 45436}, {1086, 1267}, {1125, 45398}, {1152, 7389}, {1350, 6811}, {1376, 45416}, {1504, 7888}, {1505, 7867}, {1583, 44599}, {1589, 34828}, {1591, 11090}, {1650, 45446}, {1698, 45426}, {2854, 13641}, {3071, 7789}, {3090, 23312}, {3095, 13827}, {3096, 45434}, {3102, 3934}, {3317, 18840}, {3534, 13701}, {3661, 32792}, {3662, 32791}, {3734, 6565}, {3926, 5490}, {3964, 8939}, {4364, 6352}, {4422, 30412}, {4657, 5405}, {4665, 32794}, {4851, 5393}, {5058, 7874}, {5062, 7821}, {5085, 45510}, {5210, 35306}, {5420, 7800}, {5491, 32838}, {5552, 45494}, {5599, 45430}, {5600, 45432}, {5969, 13653}, {6033, 13692}, {6118, 8976}, {6119, 35684}, {6221, 32419}, {6228, 32432}, {6292, 13931}, {6351, 17243}, {6389, 24246}, {6396, 7761}, {6398, 41490}, {6410, 11293}, {6412, 35948}, {6421, 7866}, {6423, 7776}, {6424, 32954}, {6425, 43134}, {6698, 13774}, {6813, 10516}, {6814, 33537}, {7263, 32793}, {7374, 29181}, {7375, 13935}, {7388, 42262}, {7764, 45512}, {7804, 13763}, {7807, 12963}, {7815, 11316}, {7816, 35821}, {7825, 35820}, {7834, 45513}, {7842, 42267}, {7849, 35813}, {7862, 10576}, {7880, 35823}, {8222, 45467}, {8223, 45464}, {9540, 42838}, {9605, 45577}, {10303, 33365}, {10515, 36714}, {10527, 45496}, {11294, 23261}, {11825, 36656}, {12322, 42258}, {12323, 42273}, {13665, 32421}, {13712, 42276}, {13821, 15701}, {13924, 42882}, {14233, 26468}, {14907, 35953}, {15069, 45511}, {16043, 32494}, {17227, 32803}, {17228, 32804}, {17332, 30413}, {19443, 23299}, {23259, 26619}, {23267, 26288}, {23313, 26292}, {26289, 43509}, {26359, 45345}, {26360, 45347}, {26363, 45422}, {26364, 45424}, {32588, 45478}, {34507, 44510}, {35247, 36733}, {35949, 42263}, {37343, 45441}, {40107, 44654}

X(45472) = midpoint of X(1270) and X(3068)
X(45472) = reflection of X(i) in X(j) for these (i, j): (13663, 2), (45473, 37690)
X(45472) = complement of X(3068)
X(45472) = isotomic conjugate of the isogonal conjugate of X(1504)
X(45472) = complementary conjugate of the complement of X(493)
X(45472) = barycentric product X(i)*X(j) for these {i, j}: {69, 32588}, {76, 1504}, {485, 42009}
X(45472) = barycentric quotient X(i)/X(j) for these (i, j): (485, 18819), (1504, 6)
X(45472) = trilinear product X(i)*X(j) for these {i, j}: {63, 32588}, {75, 1504}
X(45472) = trilinear quotient X(1504)/X(31)
X(45472) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(32588)}} and {{A, B, C, X(6), X(1504)}}
X(45472) = crosspoint of X(2) and X(5490)
X(45472) = crosssum of X(6) and X(6423)
X(45472) = X(2)-Ceva conjugate of-X(13882)
X(45472) = X(i)-complementary conjugate of-X(j) for these (i, j): (31, 13882), (48, 24246), (493, 10), (1306, 4369)
X(45472) = X(485)-reciprocal conjugate of-X(18819)
X(45472) = X(3534)-of-1st tri-squares triangle
X(45472) = X(6561)-of-outer-Vecten triangle
X(45472) = X(13653)-of-1st Brocard triangle
X(45472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 69, 590), (2, 491, 8253), (2, 492, 6), (2, 1270, 3068), (2, 1271, 32785), (2, 3069, 3589), (2, 3593, 3069), (2, 5590, 141), (2, 32805, 615), (2, 32806, 32789), (2, 32807, 8252), (2, 32808, 13846), (2, 32810, 32787), (2, 32814, 8972), (6, 492, 591), (6, 5590, 15835), (69, 590, 1991), (141, 44390, 492), (590, 44392, 6), (599, 8253, 491), (639, 641, 3), (1991, 44400, 13783), (3619, 32812, 2), (3763, 8252, 2), (5590, 26361, 2), (5590, 32790, 15834), (5590, 32807, 15271), (5860, 7585, 3629), (5861, 32814, 3630), (7389, 45508, 1152), (7778, 44390, 591), (8972, 32814, 5861)

X(45473) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND MEDIAL

Barycentrics    b^2+c^2-2*S : :
X(45473) = 3*X(2)+X(1271) = 5*X(2)-X(13759) = 7*X(2)-2*X(13784) = 5*X(1271)+3*X(13759) = 2*X(1271)+3*X(13783) = 7*X(1271)+6*X(13784) = 5*X(3069)-3*X(13759) = 2*X(3069)-3*X(13783) = 7*X(3069)-6*X(13784) = 2*X(13759)-5*X(13783) = 7*X(13759)-10*X(13784) = 7*X(13783)-4*X(13784)

X(45473) lies on these lines: {1, 45445}, {2, 6}, {3, 640}, {4, 12306}, {5, 9732}, {8, 45477}, {10, 31535}, {11, 45471}, {12, 45405}, {20, 33365}, {30, 45439}, {55, 45461}, {56, 45459}, {83, 45403}, {140, 43118}, {315, 12963}, {338, 34392}, {371, 626}, {372, 3788}, {427, 45401}, {485, 7795}, {486, 32490}, {487, 3071}, {488, 8406}, {489, 23261}, {490, 6410}, {498, 45491}, {499, 45493}, {594, 1267}, {620, 6396}, {623, 18586}, {624, 18587}, {625, 6565}, {631, 45407}, {637, 42262}, {638, 1152}, {639, 1656}, {641, 3526}, {732, 13766}, {754, 13763}, {958, 45437}, {1086, 5391}, {1125, 45399}, {1151, 7388}, {1350, 6813}, {1376, 45417}, {1504, 7867}, {1505, 7888}, {1584, 44598}, {1590, 34828}, {1592, 11091}, {1650, 45447}, {1698, 45427}, {2854, 13774}, {3070, 7789}, {3090, 23311}, {3095, 13707}, {3096, 45435}, {3103, 3934}, {3316, 18840}, {3534, 13821}, {3661, 32791}, {3662, 32792}, {3734, 6564}, {3926, 5491}, {3964, 8943}, {4364, 6351}, {4422, 30413}, {4657, 5393}, {4665, 32793}, {4851, 5405}, {5058, 7821}, {5062, 7874}, {5085, 45511}, {5210, 35305}, {5418, 7800}, {5490, 32838}, {5552, 45495}, {5599, 45431}, {5600, 45433}, {5969, 13773}, {6033, 13812}, {6118, 35685}, {6119, 13951}, {6200, 7761}, {6221, 41491}, {6229, 32435}, {6292, 13878}, {6352, 17243}, {6389, 24245}, {6398, 32421}, {6409, 11294}, {6411, 35949}, {6422, 7866}, {6423, 32954}, {6424, 7776}, {6426, 43133}, {6698, 13654}, {6811, 10516}, {6812, 33537}, {7000, 29181}, {7263, 32794}, {7376, 9540}, {7389, 42265}, {7764, 45513}, {7804, 13644}, {7807, 12968}, {7815, 11315}, {7816, 35820}, {7818, 9675}, {7822, 31481}, {7825, 35821}, {7834, 45512}, {7842, 42266}, {7849, 35812}, {7862, 10577}, {7869, 8960}, {7880, 35822}, {7935, 9674}, {8222, 45465}, {8223, 45466}, {9600, 11287}, {9605, 45576}, {10303, 33364}, {10514, 36709}, {10527, 45497}, {10961, 26875}, {11293, 23251}, {11824, 36655}, {12322, 42270}, {12323, 42259}, {13701, 15701}, {13785, 32419}, {13835, 42275}, {13935, 42840}, {14230, 26469}, {15069, 45510}, {16043, 32497}, {17227, 32804}, {17228, 32803}, {17332, 30412}, {19442, 23298}, {23249, 26620}, {23273, 26289}, {23314, 26293}, {26288, 43510}, {26359, 45348}, {26360, 45346}, {26363, 45423}, {26364, 45425}, {31465, 33221}, {32209, 42883}, {32587, 45479}, {34507, 44509}, {35246, 36719}, {35948, 42264}, {37342, 45440}, {40107, 44655}

X(45473) = midpoint of X(1271) and X(3069)
X(45473) = reflection of X(i) in X(j) for these (i, j): (13783, 2), (45472, 37690)
X(45473) = complement of X(3069)
X(45473) = isotomic conjugate of the isogonal conjugate of X(1505)
X(45473) = complementary conjugate of the complement of X(494)
X(45473) = barycentric product X(i)*X(j) for these {i, j}: {69, 32587}, {76, 1505}, {486, 42060}
X(45473) = barycentric quotient X(i)/X(j) for these (i, j): (486, 18820), (1505, 6)
X(45473) = trilinear product X(i)*X(j) for these {i, j}: {63, 32587}, {75, 1505}
X(45473) = trilinear quotient X(1505)/X(31)
X(45473) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(32587)}} and {{A, B, C, X(6), X(1505)}}
X(45473) = crosspoint of X(2) and X(5491)
X(45473) = crosssum of X(6) and X(6424)
X(45473) = X(2)-Ceva conjugate of-X(13934)
X(45473) = X(i)-complementary conjugate of-X(j) for these (i, j): (31, 13934), (48, 24245), (494, 10), (1307, 4369)
X(45473) = X(486)-reciprocal conjugate of-X(18820)
X(45473) = X(3534)-of-2nd tri-squares triangle
X(45473) = X(6561)-of-inner-Vecten triangle
X(45473) = X(13773)-of-1st Brocard triangle
X(45473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 69, 615), (2, 141, 45472), (2, 491, 6), (2, 1270, 32786), (2, 1271, 3069), (2, 3068, 3589), (2, 3595, 3068), (2, 3620, 32805), (2, 5591, 141), (2, 32805, 32790), (2, 32806, 590), (2, 32809, 13847), (2, 32811, 32788), (2, 32813, 32789), (6, 491, 1991), (6, 5591, 15834), (6, 7778, 45472), (69, 615, 591), (69, 44377, 45472), (141, 44391, 491), (591, 44393, 13663), (599, 8253, 37688), (615, 44394, 6), (640, 642, 3), (3619, 32813, 2), (3763, 8253, 2), (5591, 26362, 2), (5591, 32789, 15835), (5861, 7586, 3629), (7388, 45509, 1151), (7778, 44391, 1991), (44377, 44391, 44394)

X(45474) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND MIDHEIGHT

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

X(45474) lies on these lines: {5, 9733}, {6, 1162}, {25, 45408}, {51, 45478}, {389, 13748}, {11431, 45406}, {13889, 44196}, {17810, 44608}, {45400, 45452}

X(45475) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND MIDHEIGHT

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

X(45475) lies on these lines: {5, 9732}, {6, 1163}, {25, 45409}, {51, 45479}, {389, 13749}, {11431, 45407}, {13943, 44199}, {17810, 44609}, {45401, 45453}

X(45476) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 5th MIXTILINEAR

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

X(45476) lies on these lines: {1, 6}, {3, 45572}, {8, 45472}, {55, 45436}, {56, 45416}, {145, 492}, {517, 12305}, {519, 45444}, {591, 3241}, {674, 7362}, {760, 12968}, {944, 13748}, {952, 6289}, {1482, 9733}, {2098, 45470}, {2099, 45404}, {3102, 35811}, {3594, 45500}, {4675, 31569}, {5597, 45432}, {5598, 45430}, {5603, 45440}, {6283, 8679}, {6426, 45530}, {7967, 45406}, {8192, 45428}, {8210, 45467}, {8211, 45464}, {9997, 45434}, {10246, 43119}, {10247, 45488}, {10800, 45402}, {10944, 45454}, {10950, 45456}, {11396, 45400}, {11910, 45446}, {13902, 45484}, {13959, 45487}, {18525, 45438}, {18526, 45375}, {26395, 45345}, {26419, 45347}, {26495, 45415}, {26504, 45412}, {26515, 45421}, {35762, 39679}, {35810, 45462}, {37624, 45411}

X(45476) = reflection of X(44635) in X(1)
X(45476) = X(44635)-of-5th mixtilinear triangle
X(45476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3242, 45477), (1, 3640, 7969), (1, 5604, 3242), (1, 7968, 38315), (1, 45426, 45398), (45398, 45426, 6), (45422, 45490, 6), (45424, 45492, 6), (45494, 45496, 6)

X(45477) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 5th MIXTILINEAR

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

X(45477) lies on these lines: {1, 6}, {3, 45573}, {8, 45473}, {55, 45437}, {56, 45417}, {145, 491}, {517, 12306}, {519, 45445}, {674, 7353}, {760, 12963}, {944, 13749}, {952, 6290}, {1482, 9732}, {1991, 3241}, {2098, 45471}, {2099, 45405}, {3103, 35810}, {3592, 45501}, {4675, 31570}, {5597, 45433}, {5598, 45431}, {5603, 45441}, {6405, 8679}, {6425, 45531}, {7967, 45407}, {8192, 45429}, {8210, 45465}, {8211, 45466}, {9997, 45435}, {10246, 43118}, {10247, 45489}, {10800, 45403}, {10944, 45455}, {10950, 45457}, {11396, 45401}, {11910, 45447}, {13902, 45486}, {13959, 45485}, {18525, 45439}, {18526, 45376}, {26395, 45348}, {26419, 45346}, {26495, 45413}, {26504, 45414}, {26514, 45420}, {35763, 39648}, {35811, 45463}, {37624, 45410}

X(45477) = reflection of X(44636) in X(1)
X(45477) = X(44636)-of-5th mixtilinear triangle
X(45477) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3242, 45476), (1, 3641, 7968), (1, 5605, 3242), (1, 7969, 38315), (1, 45427, 45399), (45399, 45427, 6), (45423, 45491, 6), (45425, 45493, 6), (45495, 45497, 6)

X(45478) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ORTHIC

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

X(45478) lies on these lines: {4, 488}, {6, 6219}, {25, 8939}, {51, 45474}, {185, 6291}, {428, 9766}, {1162, 5860}, {3068, 5200}, {3127, 26361}, {3574, 45440}, {18539, 40909}, {32588, 45472}, {34208, 45421}, {45408, 45428}

X(45478) = polar conjugate of the isotomic conjugate of X(13882)
X(45478) = barycentric product X(4)*X(13882)
X(45478) = trilinear product X(19)*X(13882)
X(45478) = crosspoint of X(4) and X(5200)
X(45478) = X(8939)-of-anti-Ara triangle
X(45478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 488, 13051), (492, 26375, 45400)

X(45479) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ORTHIC

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

X(45479) lies on these lines: {4, 487}, {6, 6220}, {25, 8943}, {51, 45475}, {185, 6406}, {428, 9766}, {1163, 5861}, {3069, 10133}, {3128, 26362}, {3574, 45441}, {26438, 40909}, {32587, 45473}, {34208, 45420}, {45409, 45429}

X(45479) = polar conjugate of the isotomic conjugate of X(13934)
X(45479) = barycentric product X(4)*X(13934)
X(45479) = trilinear product X(19)*X(13934)
X(45479) = X(4)-Ceva conjugate of-X(32587)
X(45479) = X(8943)-of-anti-Ara triangle
X(45479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 487, 13052), (491, 26376, 45401)

X(45480) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND ORTHOCENTROIDAL

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

X(45480) lies on these lines: {381, 9733}, {5890, 13748}, {7699, 45440}, {34417, 44592}

X(45481) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND ORTHOCENTROIDAL

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

X(45481) lies on these lines: {381, 9732}, {5890, 13749}, {7699, 45441}, {34417, 44593}

X(45482) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND REFLECTION

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

X(45482) lies on these lines: {184, 44641}, {382, 9733}, {3155, 44196}, {6241, 13748}

X(45483) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND REFLECTION

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

X(45483) lies on these lines: {184, 44642}, {382, 9732}, {3156, 44199}, {6241, 13749}

X(45484) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 3rd TRI-SQUARES-CENTRAL

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

X(45484) lies on these lines: {2, 6}, {3, 45574}, {371, 13644}, {381, 40286}, {485, 7694}, {1151, 35947}, {3070, 8414}, {3102, 35815}, {5062, 13882}, {6289, 8976}, {7583, 43119}, {8981, 9733}, {9540, 12305}, {11315, 45515}, {13657, 13660}, {13883, 45398}, {13884, 45400}, {13885, 45402}, {13886, 45406}, {13887, 45416}, {13888, 45426}, {13889, 45428}, {13890, 45430}, {13891, 45432}, {13892, 45434}, {13893, 45444}, {13894, 45446}, {13895, 45454}, {13896, 45456}, {13897, 45458}, {13898, 45460}, {13899, 45467}, {13900, 45464}, {13901, 45470}, {13902, 45476}, {13903, 45488}, {13904, 45490}, {13905, 45492}, {13906, 45494}, {13907, 45496}, {18538, 45438}, {18965, 45404}, {19102, 32491}, {22763, 45436}, {31465, 32494}, {35812, 45462}, {45345, 45365}, {45347, 45368}, {45375, 45384}, {45411, 45576}, {45412, 45605}, {45415, 45607}, {45422, 45650}, {45424, 45652}

X(45484) = X(44594)-of-3rd tri-squares-central triangle
X(45484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 590, 45472), (597, 7736, 45485), (3068, 8975, 13910), (3068, 13638, 13846), (3068, 13910, 45486), (13758, 19054, 6), (13846, 37637, 590)

X(45485) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 4th TRI-SQUARES-CENTRAL

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

X(45485) lies on these lines: {2, 6}, {3, 45575}, {372, 13763}, {381, 40287}, {486, 7694}, {1152, 35946}, {3071, 8406}, {3103, 35814}, {5058, 13934}, {6290, 13933}, {7584, 43118}, {9732, 13966}, {11316, 45514}, {12306, 13935}, {13777, 13780}, {13936, 45399}, {13937, 45401}, {13938, 45403}, {13939, 45407}, {13940, 45417}, {13942, 45427}, {13943, 45429}, {13944, 45431}, {13945, 45433}, {13946, 45435}, {13947, 45445}, {13948, 45447}, {13952, 45455}, {13953, 45457}, {13954, 45459}, {13955, 45461}, {13956, 45465}, {13957, 45466}, {13958, 45471}, {13959, 45477}, {13961, 45489}, {13962, 45491}, {13963, 45493}, {13964, 45495}, {13965, 45497}, {18762, 45439}, {18966, 45405}, {19105, 32490}, {22764, 45437}, {35813, 45463}, {45346, 45367}, {45348, 45366}, {45376, 45385}, {45410, 45577}, {45413, 45606}, {45414, 45608}, {45423, 45651}, {45425, 45653}

X(45485) = X(44597)-of-4th tri-squares-central triangle
X(45485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 615, 45473), (597, 7736, 45484), (3069, 13949, 13972), (3069, 13972, 45487), (13638, 19053, 6), (13847, 37637, 615)

X(45486) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND 3rd TRI-SQUARES-CENTRAL

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

X(45486) lies on these lines: {2, 6}, {3, 45576}, {371, 36655}, {485, 6222}, {640, 30435}, {642, 9605}, {3103, 35812}, {3311, 45555}, {6033, 13670}, {6221, 36719}, {6290, 8976}, {6564, 13644}, {7583, 43118}, {8969, 13889}, {8981, 9732}, {9540, 12306}, {11316, 45513}, {13654, 32303}, {13883, 45399}, {13884, 45401}, {13885, 45403}, {13886, 45407}, {13887, 45417}, {13888, 45427}, {13890, 45431}, {13891, 45433}, {13892, 45435}, {13893, 45445}, {13894, 45447}, {13895, 45455}, {13896, 45457}, {13897, 45459}, {13898, 45461}, {13899, 45465}, {13900, 45466}, {13901, 45471}, {13902, 45477}, {13903, 45489}, {13904, 45491}, {13905, 45493}, {13906, 45495}, {13907, 45497}, {13924, 35685}, {18538, 36723}, {18965, 45405}, {22763, 45437}, {35815, 45463}, {45346, 45368}, {45348, 45365}, {45376, 45384}, {45410, 45574}, {45413, 45607}, {45414, 45605}, {45423, 45650}, {45425, 45652}

X(45486) = X(44595)-of-3rd tri-squares-central triangle
X(45486) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 590, 45473), (141, 7735, 45487), (485, 39648, 13749), (590, 32787, 31463), (3068, 13638, 6), (3068, 13910, 45484)

X(45487) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND 4th TRI-SQUARES-CENTRAL

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

X(45487) lies on these lines: {2, 6}, {3, 45577}, {372, 36656}, {486, 6399}, {639, 30435}, {641, 9605}, {3102, 35813}, {3312, 45554}, {6033, 13790}, {6289, 13880}, {6398, 36733}, {6565, 13763}, {7584, 43119}, {9733, 13966}, {11315, 45512}, {12305, 13935}, {13774, 32304}, {13936, 45398}, {13937, 45400}, {13938, 45402}, {13939, 45406}, {13940, 45416}, {13942, 45426}, {13943, 33582}, {13944, 45430}, {13945, 45432}, {13946, 45434}, {13947, 45444}, {13948, 45446}, {13952, 45454}, {13953, 45456}, {13954, 45458}, {13955, 45460}, {13956, 45467}, {13957, 45464}, {13958, 45470}, {13959, 45476}, {13961, 45488}, {13962, 45490}, {13963, 45492}, {13964, 45494}, {13965, 45496}, {18762, 36726}, {18966, 45404}, {22764, 45436}, {32209, 35684}, {35814, 45462}, {45345, 45366}, {45347, 45367}, {45375, 45385}, {45411, 45575}, {45412, 45608}, {45415, 45606}, {45422, 45651}, {45424, 45653}

X(45487) = X(44596)-of-4th tri-squares-central triangle
X(45487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 615, 45472), (141, 7735, 45486), (486, 39679, 13748), (3069, 13758, 6), (3069, 13950, 13972), (3069, 13972, 45485)

X(45488) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND X3-ABC REFLECTIONS

Barycentrics    a^2*(a^4-4*(b^2+c^2)*a^2+2*(-a^2+b^2+c^2)*S+3*b^4-2*b^2*c^2+3*c^4) : :
X(45488) = 3*X(3)-2*X(11824) = 3*X(3)-4*X(43121) = 3*X(6)-2*X(44485) = 3*X(372)-X(11824) = 3*X(372)-2*X(43121) = 4*X(640)-5*X(1656) = 9*X(5050)-8*X(44483) = 3*X(5093)-2*X(35841) = 5*X(11482)-4*X(44501) = 3*X(12968)-2*X(12974) = X(45406)-3*X(45421)

X(45488) lies on these lines: {3, 6}, {4, 43133}, {5, 492}, {30, 8982}, {51, 5408}, {69, 37342}, {195, 44196}, {381, 591}, {382, 12601}, {486, 44392}, {488, 14853}, {517, 45426}, {640, 1656}, {999, 45404}, {1583, 9777}, {1598, 45400}, {1992, 12257}, {1993, 3156}, {2070, 8989}, {3060, 3155}, {3295, 45470}, {3564, 36714}, {3843, 45438}, {5790, 45444}, {6202, 26288}, {6230, 6315}, {6290, 35684}, {6321, 23261}, {6813, 7774}, {7517, 8996}, {8825, 12161}, {8950, 13023}, {9654, 45458}, {9669, 45460}, {10133, 33586}, {10246, 45398}, {10247, 45476}, {10673, 12164}, {10679, 45424}, {10680, 45422}, {10788, 35938}, {10852, 44434}, {11315, 20576}, {11849, 45416}, {11875, 45430}, {11876, 45432}, {11911, 45446}, {11928, 45454}, {11929, 45456}, {11949, 45467}, {11950, 45464}, {12000, 45494}, {12001, 45496}, {13713, 38335}, {13758, 13966}, {13903, 45484}, {13961, 45487}, {18440, 36712}, {18586, 20425}, {18587, 20426}, {19362, 45408}, {19418, 45415}, {19419, 19443}, {21850, 36709}, {22765, 45436}, {23312, 44390}, {23698, 35833}, {35944, 43511}, {36655, 45376}, {36656, 45377}, {45345, 45369}, {45347, 45370}, {45412, 45609}

X(45488) = midpoint of X(4) and X(43133)
X(45488) = reflection of X(i) in X(j) for these (i, j): (3, 372), (638, 5), (11824, 43121)
X(45488) = inverse of X(45411) in Brocard circle
X(45488) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(45411)}} and {{A, B, C, X(4), X(12968)}}
X(45488) = X(372)-of-X3-ABC reflections triangle
X(45488) = X(638)-of-Johnson triangle
X(45488) = X(9733)-of-2nd Kenmotu-centers triangle
X(45488) = X(9739)-of-2nd Kenmotu-free-vertices triangle
X(45488) = X(43133)-of-Euler triangle
X(45488) = X(45488)-of-circumsymmedial triangle
X(45488) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(12968)
X(45488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5093, 3311), (3, 11917, 1351), (3, 44456, 1161), (6, 9733, 3), (182, 11825, 3), (371, 372, 12968), (371, 9739, 3), (372, 35841, 32), (372, 40275, 1152), (576, 9739, 371), (576, 44476, 6), (1152, 9732, 3), (1160, 3312, 3), (1161, 6398, 3), (1350, 43118, 3), (1351, 12314, 3), (3098, 45552, 3), (3102, 45462, 6), (5017, 43156, 3), (6200, 43141, 3), (6289, 45440, 381), (6396, 9738, 3), (6420, 11825, 182), (9605, 43140, 3), (11824, 43121, 3), (12305, 43119, 3), (12975, 45499, 3), (35770, 45553, 575), (39658, 43137, 3), (42859, 45513, 3), (43120, 45498, 3), (45404, 45492, 999), (45470, 45490, 3295), (45489, 45578, 3)

X(45489) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND X3-ABC REFLECTIONS

Barycentrics    a^2*(a^4-4*(b^2+c^2)*a^2-2*(-a^2+b^2+c^2)*S+3*b^4-2*b^2*c^2+3*c^4) : :
X(45489) = 3*X(3)-2*X(11825) = 3*X(3)-4*X(43120) = 3*X(6)-2*X(44486) = 3*X(371)-X(11825) = 3*X(371)-2*X(43120) = 4*X(639)-5*X(1656) = 9*X(5050)-8*X(44484) = 3*X(5093)-2*X(35840) = 5*X(11482)-4*X(44502) = 3*X(12963)-2*X(12975) = X(45407)-3*X(45420)

X(45489) lies on these lines: {3, 6}, {4, 43134}, {5, 491}, {30, 26441}, {51, 5409}, {69, 37343}, {193, 21736}, {195, 44199}, {381, 1991}, {382, 12602}, {485, 44394}, {487, 14853}, {517, 45427}, {639, 1656}, {999, 45405}, {1584, 9777}, {1598, 45401}, {1992, 12256}, {1993, 3155}, {3060, 3156}, {3295, 45471}, {3564, 36709}, {3843, 45439}, {5790, 45445}, {6201, 26289}, {6231, 6311}, {6289, 35685}, {6321, 23251}, {6811, 7774}, {7517, 45429}, {7585, 21737}, {8981, 13638}, {9654, 45459}, {9669, 45461}, {9681, 12124}, {10132, 33586}, {10246, 45399}, {10247, 45477}, {10669, 12164}, {10679, 45425}, {10680, 45423}, {10788, 35939}, {10851, 44434}, {11316, 20576}, {11849, 45417}, {11875, 45431}, {11876, 45433}, {11911, 45447}, {11928, 45455}, {11929, 45457}, {11949, 45465}, {11950, 45466}, {12000, 45495}, {12001, 45497}, {13024, 26293}, {13836, 38335}, {13903, 45486}, {13961, 45485}, {18440, 36711}, {18586, 20426}, {18587, 20425}, {19362, 45409}, {19418, 19442}, {19419, 45414}, {21850, 36714}, {22765, 45437}, {23311, 44391}, {23698, 35832}, {35945, 43512}, {36655, 45378}, {36656, 45375}, {45346, 45370}, {45348, 45369}, {45413, 45610}

X(45489) = midpoint of X(4) and X(43134)
X(45489) = reflection of X(i) in X(j) for these (i, j): (3, 371), (637, 5), (11825, 43120)
X(45489) = inverse of X(2460) in Kenmotu circle
X(45489) = inverse of X(45410) in Brocard circle
X(45489) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(45410)}} and {{A, B, C, X(4), X(12963)}}
X(45489) = X(371)-of-X3-ABC reflections triangle
X(45489) = X(637)-of-Johnson triangle
X(45489) = X(9732)-of-1st Kenmotu-centers triangle
X(45489) = X(9738)-of-1st Kenmotu-free-vertices triangle
X(45489) = X(43134)-of-Euler triangle
X(45489) = X(45489)-of-circumsymmedial triangle
X(45489) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(12963)
X(45489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5093, 3312), (3, 11916, 1351), (3, 44456, 1160), (6, 9732, 3), (32, 35840, 3312), (182, 11824, 3), (371, 372, 12963), (371, 1504, 3311), (371, 35840, 32), (371, 40274, 1151), (576, 44475, 6), (1151, 9733, 3), (1160, 6221, 3), (1161, 3311, 3), (1350, 43119, 3), (1351, 12313, 3), (3098, 45553, 3), (3103, 45463, 6), (5017, 43155, 3), (6200, 9739, 3), (6290, 45441, 381), (6396, 43144, 3), (6419, 11824, 182), (9605, 43137, 3), (11825, 43120, 3), (12306, 43118, 3), (12974, 45498, 3), (35771, 45552, 575), (39649, 43140, 3), (42858, 45512, 3), (43121, 45499, 3), (45405, 45493, 999), (45471, 45491, 3295), (45488, 45579, 3)

X(45490) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND INNER-YFF

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

X(45490) lies on these lines: {1, 6}, {3, 45404}, {5, 45460}, {12, 6289}, {35, 12305}, {55, 9733}, {56, 43119}, {388, 45406}, {492, 3085}, {495, 45458}, {498, 45472}, {591, 10056}, {999, 45411}, {1478, 13748}, {1479, 45440}, {2330, 7362}, {3095, 45471}, {3102, 35808}, {3295, 45470}, {3312, 45582}, {6283, 19369}, {6502, 39679}, {9654, 45375}, {10037, 45428}, {10038, 45434}, {10039, 45444}, {10067, 15888}, {10084, 19030}, {10523, 45454}, {10801, 45402}, {10802, 12838}, {10895, 45438}, {10954, 45456}, {11398, 45400}, {11507, 45416}, {11877, 45430}, {11878, 45432}, {11912, 45446}, {11951, 45467}, {11952, 45464}, {12314, 45570}, {13904, 45484}, {13962, 45487}, {22766, 45436}, {31474, 45583}, {35809, 45462}, {45345, 45371}, {45347, 45372}, {45412, 45611}, {45415, 45612}

X(45490) = midpoint of X(10930) and X(19050)
X(45490) = X(10930)-of-outer-Yff triangle
X(45490) = X(19050)-of-inner-Yff triangle
X(45490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45492), (1, 611, 45491), (1, 1124, 45493), (1, 3299, 613), (6, 45476, 45422), (3295, 45488, 45470)

X(45491) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND INNER-YFF

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

X(45491) lies on these lines: {1, 6}, {3, 45405}, {5, 45461}, {12, 6290}, {35, 12306}, {55, 9732}, {56, 43118}, {388, 45407}, {491, 3085}, {495, 45459}, {498, 45473}, {999, 45410}, {1478, 13749}, {1479, 45441}, {1991, 10056}, {2067, 39648}, {2330, 7353}, {3095, 45470}, {3103, 35809}, {3295, 45471}, {3311, 45583}, {6405, 19369}, {9654, 45376}, {10037, 45429}, {10038, 45435}, {10039, 45445}, {10068, 15888}, {10083, 19029}, {10523, 45455}, {10801, 45403}, {10802, 12839}, {10895, 45439}, {10954, 45457}, {11398, 45401}, {11507, 45417}, {11877, 45431}, {11878, 45433}, {11912, 45447}, {11951, 45465}, {11952, 45466}, {12313, 45571}, {13904, 45486}, {13962, 45485}, {22766, 45437}, {35808, 45463}, {45346, 45372}, {45348, 45371}, {45413, 45612}, {45414, 45611}

X(45491) = midpoint of X(10929) and X(19049)
X(45491) = X(10929)-of-outer-Yff triangle
X(45491) = X(19049)-of-inner-Yff triangle
X(45491) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45493), (1, 611, 45490), (1, 1335, 45492), (1, 3301, 613), (6, 45477, 45423), (3295, 45489, 45471)

X(45492) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND OUTER-YFF

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

X(45492) lies on these lines: {1, 6}, {3, 45470}, {5, 45458}, {11, 6289}, {36, 12305}, {55, 43119}, {56, 9733}, {492, 3086}, {496, 45460}, {497, 45406}, {499, 45472}, {591, 10072}, {999, 45404}, {1428, 6283}, {1478, 45440}, {1479, 13748}, {1737, 45444}, {3095, 45405}, {3102, 35768}, {3295, 45411}, {3312, 45580}, {5414, 39679}, {7362, 8540}, {9669, 45375}, {10046, 45428}, {10047, 45434}, {10068, 19028}, {10083, 37722}, {10523, 45456}, {10801, 12839}, {10802, 45402}, {10896, 45438}, {10948, 45454}, {11399, 45400}, {11508, 45416}, {11879, 45430}, {11880, 45432}, {11913, 45446}, {11953, 45467}, {11954, 45464}, {12314, 45506}, {13905, 45484}, {13963, 45487}, {22767, 45436}, {35769, 45462}, {45345, 45373}, {45347, 45374}, {45412, 45613}, {45415, 45614}

X(45492) = midpoint of X(10932) and X(19048)
X(45492) = X(10932)-of-inner-Yff triangle
X(45492) = X(19048)-of-outer-Yff triangle
X(45492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45490), (1, 613, 45493), (1, 1335, 45491), (1, 3301, 611), (6, 45476, 45424), (999, 45488, 45404)

X(45493) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND OUTER-YFF

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

X(45493) lies on these lines: {1, 6}, {3, 45471}, {5, 45459}, {11, 6290}, {36, 12306}, {55, 43118}, {56, 9732}, {491, 3086}, {496, 45461}, {497, 45407}, {499, 45473}, {999, 45405}, {1428, 6405}, {1478, 45441}, {1479, 13749}, {1737, 45445}, {1991, 10072}, {2066, 39648}, {3095, 45404}, {3103, 35769}, {3295, 45410}, {3311, 45581}, {7353, 8540}, {9669, 45376}, {10046, 45429}, {10047, 45435}, {10067, 19027}, {10084, 37722}, {10523, 45457}, {10801, 12838}, {10802, 45403}, {10896, 45439}, {10948, 45455}, {11399, 45401}, {11508, 45417}, {11879, 45431}, {11880, 45433}, {11913, 45447}, {11953, 45465}, {11954, 45466}, {12313, 45507}, {13905, 45486}, {13963, 45485}, {22767, 45437}, {35768, 45463}, {45346, 45374}, {45348, 45373}, {45413, 45614}, {45414, 45613}

X(45493) = midpoint of X(10931) and X(19047)
X(45493) = X(10931)-of-inner-Yff triangle
X(45493) = X(19047)-of-outer-Yff triangle
X(45493) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45491), (1, 613, 45492), (1, 1124, 45490), (1, 3299, 611), (1, 10048, 613), (6, 45477, 45425), (999, 45489, 45405)

X(45494) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND INNER-YFF TANGENTS

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

X(45494) lies on these lines: {1, 6}, {3, 45584}, {12, 45454}, {492, 10528}, {591, 11239}, {1152, 45528}, {3102, 35817}, {5552, 45472}, {6289, 10942}, {7362, 12329}, {9733, 10679}, {10531, 45440}, {10803, 45402}, {10805, 45406}, {10834, 45428}, {10878, 45434}, {10915, 45444}, {10955, 45456}, {10956, 45458}, {10958, 45460}, {10965, 45470}, {11248, 12305}, {11400, 45400}, {11509, 45404}, {11881, 45430}, {11882, 45432}, {11914, 45446}, {11955, 45467}, {11956, 45464}, {12000, 45488}, {12115, 13748}, {13906, 45484}, {13964, 45487}, {16203, 43119}, {18542, 45438}, {18545, 45375}, {22768, 45436}, {26402, 45345}, {26426, 45347}, {26511, 45412}, {26525, 45421}, {35816, 45462}, {45411, 45585}, {45415, 45615}

X(45494) = X(44645)-of-inner-Yff tangents triangle
X(45494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10930, 12594), (1, 12594, 45495), (1, 26350, 19048), (1, 45424, 6), (6, 45476, 45496), (1124, 5604, 12595), (3242, 3297, 45497)

X(45495) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND INNER-YFF TANGENTS

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

X(45495) lies on these lines: {1, 6}, {3, 45585}, {12, 45455}, {491, 10528}, {1151, 45529}, {1991, 11239}, {3103, 35816}, {5552, 45473}, {6290, 10942}, {7353, 12329}, {9732, 10679}, {10531, 45441}, {10803, 45403}, {10805, 45407}, {10834, 45429}, {10878, 45435}, {10915, 45445}, {10955, 45457}, {10956, 45459}, {10958, 45461}, {10965, 45471}, {11248, 12306}, {11400, 45401}, {11509, 45405}, {11881, 45431}, {11882, 45433}, {11914, 45447}, {11955, 45465}, {11956, 45466}, {12000, 45489}, {12115, 13749}, {13906, 45486}, {13964, 45485}, {16203, 43118}, {18542, 45439}, {18545, 45376}, {22768, 45437}, {26402, 45348}, {26426, 45346}, {26511, 45414}, {26520, 45420}, {35817, 45463}, {45410, 45584}, {45413, 45615}

X(45495) = X(44646)-of-inner-Yff tangents triangle
X(45495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10929, 12594), (1, 12594, 45494), (1, 26343, 19047), (1, 45425, 6), (6, 45477, 45497), (1335, 5605, 12595), (3242, 3298, 45496)

X(45496) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS AND OUTER-YFF TANGENTS

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

X(45496) lies on these lines: {1, 6}, {3, 45586}, {11, 45456}, {492, 10529}, {591, 11240}, {1152, 45526}, {3102, 35819}, {6283, 22769}, {6289, 10943}, {9733, 10680}, {10527, 45472}, {10532, 45440}, {10804, 45402}, {10806, 45406}, {10835, 45428}, {10879, 45434}, {10916, 45444}, {10949, 45454}, {10957, 45458}, {10959, 45460}, {10966, 45436}, {11249, 12305}, {11401, 45400}, {11510, 45416}, {11883, 45430}, {11884, 45432}, {11915, 45446}, {11957, 45467}, {11958, 45464}, {12001, 45488}, {12116, 13748}, {13907, 45484}, {13965, 45487}, {16202, 43119}, {18543, 45375}, {18544, 45438}, {18967, 45404}, {26401, 45345}, {26425, 45347}, {26501, 45415}, {26510, 45412}, {26524, 45421}, {35818, 45462}, {45411, 45587}

X(45496) = X(44643)-of-outer-Yff tangents triangle
X(45496) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10932, 12595), (1, 12595, 45497), (1, 26349, 19050), (1, 45422, 6), (6, 45476, 45494), (1335, 5604, 12594), (3242, 3298, 45495)

X(45497) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS AND OUTER-YFF TANGENTS

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

X(45497) lies on these lines: {1, 6}, {3, 45587}, {11, 45457}, {491, 10529}, {1151, 45527}, {1991, 11240}, {3103, 35818}, {6290, 10943}, {6405, 22769}, {9732, 10680}, {10527, 45473}, {10532, 45441}, {10804, 45403}, {10806, 45407}, {10835, 45429}, {10879, 45435}, {10916, 45445}, {10949, 45455}, {10957, 45459}, {10959, 45461}, {10966, 45437}, {11249, 12306}, {11401, 45401}, {11510, 45417}, {11883, 45431}, {11884, 45433}, {11915, 45447}, {11957, 45465}, {11958, 45466}, {12001, 45489}, {12116, 13749}, {13907, 45486}, {13965, 45485}, {16202, 43118}, {18543, 45376}, {18544, 45439}, {18967, 45405}, {26401, 45348}, {26425, 45346}, {26501, 45413}, {26510, 45414}, {26519, 45420}, {35819, 45463}, {45410, 45586}

X(45497) = X(44644)-of-outer-Yff tangents triangle
X(45497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 10931, 12595), (1, 12595, 45496), (1, 26342, 19049), (1, 45423, 6), (6, 45477, 45495), (1124, 5605, 12594), (3242, 3297, 45494)

X(45498) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ABC-X3 REFLECTIONS

Barycentrics    a^2*(a^4+2*(b^2+c^2)*a^2-3*b^4-2*b^2*c^2-3*c^4-6*S*(-a^2+b^2+c^2)) : :
X(45498) = 3*X(3)-X(45579) = 3*X(6200)-2*X(45579)

X(45498) lies on these lines: {2, 45106}, {3, 6}, {4, 641}, {20, 3593}, {23, 35299}, {30, 45548}, {35, 45580}, {36, 45582}, {55, 45506}, {56, 45570}, {98, 13708}, {110, 5408}, {165, 45530}, {376, 13786}, {382, 45542}, {488, 5921}, {490, 7763}, {492, 35947}, {515, 45546}, {517, 45572}, {591, 38747}, {1007, 9757}, {1296, 32422}, {1583, 3066}, {1593, 8948}, {1600, 21766}, {1657, 45377}, {2077, 45528}, {3156, 5651}, {3428, 45540}, {3522, 12123}, {3543, 13666}, {3576, 45500}, {6281, 32841}, {6284, 45562}, {6312, 33370}, {6561, 35945}, {7354, 45560}, {9540, 45574}, {10304, 13759}, {10310, 45520}, {11012, 45526}, {11248, 45584}, {11249, 45586}, {11315, 45440}, {11414, 45532}, {11822, 45534}, {11823, 45536}, {11826, 45556}, {11827, 45558}, {11828, 45569}, {11829, 45566}, {13935, 45577}, {15066, 35300}, {21735, 26295}, {21737, 42261}, {26290, 45349}, {26291, 45351}, {26292, 45519}, {26293, 45516}, {32421, 45511}, {35949, 45545}, {36703, 42637}

X(45498) = midpoint of X(492) and X(35947)
X(45498) = reflection of X(6200) in X(3)
X(45498) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45515)}} and {{A, B, C, X(6), X(45106)}}
X(45498) = X(6200)-of-these triangles: {ABC-X3 reflections, 1st anti-Kenmotu-free-vertices}
X(45498) = X(45498)-of-circumsymmedial triangle
X(45498) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 1160, 1151), (3, 1161, 6409), (3, 1350, 45499), (3, 6398, 5085), (3, 9733, 371), (3, 9739, 372), (3, 11825, 11824), (3, 12305, 11825), (3, 12314, 43119), (3, 35247, 31884), (3, 43140, 39648), (3, 45488, 43120), (3, 45489, 12974), (3, 45578, 182), (6, 6566, 372), (39, 1152, 372), (182, 45578, 372), (371, 2459, 41411), (371, 6396, 2459), (1350, 45499, 11824), (6396, 45565, 372), (6412, 31884, 3), (6431, 8409, 371), (7690, 9739, 3), (9737, 9739, 9733), (11825, 45499, 1350), (12314, 43119, 6420), (12974, 45489, 6453), (14538, 14539, 11825), (43120, 45488, 6419)

X(45499) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ABC-X3 REFLECTIONS

Barycentrics    a^2*(a^4+2*(b^2+c^2)*a^2-3*b^4-2*b^2*c^2-3*c^4+6*S*(-a^2+b^2+c^2)) : :
X(45499) = 3*X(3)-X(45578) = 3*X(6396)-2*X(45578)

X(45499) lies on these lines: {2, 45107}, {3, 6}, {4, 642}, {20, 3595}, {23, 35300}, {30, 45549}, {35, 45581}, {36, 45583}, {55, 45507}, {56, 45571}, {98, 13828}, {110, 5409}, {165, 45531}, {376, 13666}, {382, 45543}, {487, 5921}, {489, 7763}, {491, 35946}, {515, 45547}, {517, 45573}, {1007, 9758}, {1296, 32420}, {1584, 3066}, {1593, 8946}, {1599, 21766}, {1657, 45378}, {1991, 38747}, {2077, 45529}, {3155, 5651}, {3428, 45541}, {3522, 12124}, {3543, 13786}, {3576, 45501}, {5418, 21737}, {6278, 32841}, {6284, 45563}, {6316, 33371}, {6560, 35944}, {7354, 45561}, {9540, 45576}, {10304, 13639}, {10310, 45521}, {11012, 45527}, {11248, 45585}, {11249, 45587}, {11316, 45441}, {11414, 45533}, {11822, 45535}, {11823, 45537}, {11826, 45557}, {11827, 45559}, {11828, 45567}, {11829, 45568}, {13935, 45575}, {15066, 35299}, {17811, 21097}, {21735, 26294}, {21736, 40330}, {26290, 45352}, {26291, 45350}, {26292, 45517}, {26293, 45518}, {32419, 45510}, {35948, 45544}, {36701, 42638}

X(45499) = midpoint of X(491) and X(35946)
X(45499) = reflection of X(6396) in X(3)
X(45499) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45514)}} and {{A, B, C, X(6), X(45107)}}
X(45499) = X(6396)-of-ABC-X3 reflections triangle
X(45499) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 1161, 1152), (3, 9732, 372), (3, 9738, 371), (3, 45579, 182), (6, 6567, 371), (39, 1151, 371), (182, 45579, 371), (6200, 45564, 371), (6411, 31884, 3), (6432, 8401, 372), (7692, 9738, 3), (11824, 45498, 1350)

X(45500) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-AQUILA

Barycentrics    a*(2*(2*a^2+(b+c)*a+b^2+c^2)*S+(a+b+c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))) : :
X(45500) = 3*X(16475)-X(35775)

X(45500) lies on these lines: {1, 372}, {2, 45546}, {3, 45398}, {11, 5405}, {39, 7969}, {40, 11371}, {182, 517}, {515, 45544}, {551, 41490}, {641, 1125}, {999, 45540}, {1319, 45506}, {1385, 9739}, {1482, 45399}, {2646, 45570}, {3295, 45520}, {3576, 45498}, {3594, 45476}, {3616, 45508}, {4663, 44474}, {5062, 7968}, {5603, 45510}, {5886, 45554}, {6420, 45426}, {7690, 13624}, {7982, 11370}, {9955, 45542}, {10246, 45578}, {11313, 45444}, {11363, 45502}, {11364, 45504}, {11365, 45532}, {11366, 45534}, {11367, 45536}, {11368, 45538}, {11373, 45556}, {11374, 45558}, {11375, 45560}, {11376, 45562}, {11377, 45569}, {11378, 45566}, {11831, 45548}, {12268, 13464}, {13883, 45574}, {13936, 45577}, {16475, 35775}, {18493, 45377}, {18991, 45512}, {18992, 45515}, {26365, 45349}, {26366, 45351}, {26367, 45519}, {26368, 45516}, {26369, 45522}, {26370, 45525}, {35763, 45565}

X(45500) = midpoint of X(1) and X(35774)
X(45500) = reflection of X(45501) in X(1386)
X(45500) = X(35774)-of-these triangles: {anti-Aquila, 1st anti-Kenmotu-free-vertices}
X(45500) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45530, 45572), (372, 45572, 45530)

X(45501) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-AQUILA

Barycentrics    a*(2*(2*a^2+(b+c)*a+b^2+c^2)*S-(a+b+c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c))) : :
X(45501) = 3*X(16475)-X(35774)

X(45501) lies on these lines: {1, 371}, {2, 45547}, {3, 45399}, {11, 5393}, {39, 7968}, {40, 11370}, {182, 517}, {515, 45545}, {551, 41491}, {642, 1125}, {999, 45541}, {1319, 45507}, {1385, 9738}, {1482, 45398}, {2646, 45571}, {3295, 45521}, {3576, 45499}, {3592, 45477}, {3616, 45509}, {4663, 44473}, {5058, 7969}, {5603, 45511}, {5886, 45555}, {6419, 45427}, {7692, 13624}, {7982, 11371}, {9955, 45543}, {10246, 45579}, {11314, 45445}, {11363, 45503}, {11364, 45505}, {11365, 45533}, {11366, 45535}, {11367, 45537}, {11368, 45539}, {11373, 45557}, {11374, 45559}, {11375, 45561}, {11376, 45563}, {11377, 45567}, {11378, 45568}, {11831, 45549}, {12269, 13464}, {13883, 45576}, {13936, 45575}, {16475, 35774}, {18493, 45378}, {18991, 45514}, {18992, 45513}, {26365, 45352}, {26366, 45350}, {26367, 45517}, {26368, 45518}, {26369, 45524}, {26370, 45523}, {35762, 45564}

X(45501) = midpoint of X(1) and X(35775)
X(45501) = reflection of X(45500) in X(1386)
X(45501) = X(35775)-of-anti-Aquila triangle
X(45501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45531, 45573), (371, 45573, 45531)

X(45502) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-ARA

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

X(45502) lies on these lines: {3, 45400}, {4, 9739}, {24, 182}, {25, 372}, {33, 45570}, {34, 45506}, {39, 5413}, {232, 18993}, {235, 45544}, {378, 6291}, {427, 641}, {428, 41490}, {1593, 8948}, {1598, 45578}, {1974, 10881}, {3088, 26375}, {3515, 11389}, {3517, 45401}, {5062, 5412}, {5090, 45546}, {5200, 13935}, {5410, 45515}, {5411, 45512}, {6406, 10594}, {6566, 11474}, {7487, 45510}, {7713, 45530}, {8541, 10880}, {11363, 45500}, {11380, 45504}, {11383, 45520}, {11384, 45534}, {11385, 45536}, {11386, 45538}, {11388, 45550}, {11390, 45556}, {11391, 45558}, {11392, 45560}, {11393, 45562}, {11394, 45569}, {11396, 45572}, {11398, 45580}, {11399, 45582}, {11400, 45584}, {11401, 45586}, {11832, 45548}, {13884, 45574}, {13937, 45577}, {18494, 45377}, {22479, 45540}, {26371, 45349}, {26372, 45351}, {26373, 45519}, {26376, 45525}, {26377, 45526}, {26378, 45528}, {35765, 45565}, {44102, 44474}

X(45502) = X(35776)-of-these triangles: {anti-Ara, 1st anti-Kenmotu-free-vertices}
X(45502) = {X(24), X(1843)}-harmonic conjugate of X(45503)

X(45503) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-ARA

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

X(45503) lies on these lines: {3, 45401}, {4, 9738}, {24, 182}, {25, 371}, {33, 45571}, {34, 45507}, {39, 5412}, {232, 18994}, {235, 45545}, {378, 6406}, {427, 642}, {428, 41491}, {1593, 8946}, {1598, 45579}, {1974, 10880}, {3088, 26376}, {3515, 11388}, {3517, 45400}, {5058, 5413}, {5090, 45547}, {5410, 45513}, {5411, 45514}, {6291, 10594}, {6567, 11473}, {7487, 45511}, {7713, 45531}, {8541, 10881}, {11363, 45501}, {11380, 45505}, {11383, 45521}, {11384, 45535}, {11385, 45537}, {11386, 45539}, {11389, 45551}, {11390, 45557}, {11391, 45559}, {11392, 45561}, {11393, 45563}, {11395, 45568}, {11396, 45573}, {11398, 45581}, {11399, 45583}, {11400, 45585}, {11401, 45587}, {11832, 45549}, {13884, 45576}, {13937, 45575}, {18494, 45378}, {22479, 45541}, {26371, 45352}, {26372, 45350}, {26374, 45518}, {26375, 45524}, {26377, 45527}, {26378, 45529}, {35764, 45564}, {44102, 44473}

X(45503) = X(35777)-of-anti-Ara triangle
X(45503) = {X(24), X(1843)}-harmonic conjugate of X(45502)

X(45504) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 5th ANTI-BROCARD

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

X(45504) lies on these lines: {3, 6}, {83, 641}, {98, 45544}, {7787, 45508}, {10359, 26429}, {10788, 45510}, {10789, 45530}, {10790, 45532}, {10791, 45546}, {10794, 45556}, {10795, 45558}, {10796, 45554}, {10797, 45560}, {10798, 45562}, {10799, 45570}, {10800, 45572}, {10801, 45580}, {10802, 45582}, {10803, 45584}, {10804, 45586}, {11364, 45500}, {11380, 45502}, {11490, 45520}, {11837, 45534}, {11838, 45536}, {11839, 45548}, {11840, 45569}, {11841, 45566}, {12150, 41490}, {12835, 45506}, {13885, 45574}, {13938, 45577}, {18501, 45377}, {18502, 45542}, {22520, 45540}, {26379, 45349}, {26403, 45351}, {26427, 45519}, {26428, 45516}, {26430, 45525}, {26431, 45526}, {26432, 45528}

X(45504) = X(35782)-of-these triangles: {5th anti-Brocard, 1st anti-Kenmotu-free-vertices}
X(45504) = X(45504)-of-circumsymmedial triangle
X(45504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (372, 35766, 2461), (2080, 12212, 45505), (11842, 45578, 182)

X(45505) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 5th ANTI-BROCARD

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

X(45505) lies on these lines: {3, 6}, {83, 642}, {98, 45545}, {7787, 45509}, {10359, 26430}, {10788, 45511}, {10789, 45531}, {10790, 45533}, {10791, 45547}, {10794, 45557}, {10795, 45559}, {10796, 45555}, {10797, 45561}, {10798, 45563}, {10799, 45571}, {10800, 45573}, {10801, 45581}, {10802, 45583}, {10803, 45585}, {10804, 45587}, {11364, 45501}, {11380, 45503}, {11490, 45521}, {11837, 45535}, {11838, 45537}, {11839, 45549}, {11840, 45567}, {11841, 45568}, {12150, 41491}, {12835, 45507}, {13885, 45576}, {13938, 45575}, {18501, 45378}, {18502, 45543}, {22520, 45541}, {26379, 45352}, {26403, 45350}, {26427, 45517}, {26428, 45518}, {26429, 45524}, {26431, 45527}, {26432, 45529}

X(45505) = X(35783)-of-5th anti-Brocard triangle
X(45505) = X(45505)-of-circumsymmedial triangle
X(45505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (32, 9675, 1691), (2080, 12212, 45504), (11842, 45579, 182)

X(45506) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

X(45506) lies on these lines: {1, 9739}, {3, 45404}, {4, 45562}, {11, 45544}, {12, 641}, {34, 45502}, {35, 7362}, {36, 182}, {39, 6502}, {55, 45498}, {56, 372}, {57, 45530}, {388, 45508}, {999, 45578}, {1319, 45500}, {1470, 45528}, {1478, 45554}, {2067, 5062}, {2099, 45572}, {2275, 18993}, {3085, 26435}, {3585, 45542}, {4293, 45510}, {5204, 18960}, {5252, 45546}, {5414, 6566}, {5434, 41490}, {5563, 7353}, {7763, 45561}, {9655, 45377}, {9737, 45571}, {11509, 45520}, {12314, 45492}, {12835, 45504}, {18954, 45532}, {18955, 45534}, {18956, 45536}, {18957, 45538}, {18958, 45548}, {18959, 45550}, {18961, 45556}, {18962, 45558}, {18963, 45569}, {18964, 45566}, {18965, 45574}, {18966, 45577}, {18967, 45586}, {18995, 45512}, {18996, 45515}, {19369, 44656}, {26380, 45349}, {26404, 45351}, {26433, 45519}, {26434, 45516}, {26436, 45525}, {26437, 45526}, {35769, 45565}, {45405, 45410}

X(45506) = X(35784)-of-these triangles: {2nd anti-circumperp-tangential, 1st anti-Kenmotu-free-vertices}
X(45506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9739, 45570), (36, 1469, 45507), (388, 45508, 45560), (999, 45578, 45582)

X(45507) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

X(45507) lies on these lines: {1, 9738}, {3, 45405}, {4, 45563}, {11, 45545}, {12, 642}, {34, 45503}, {35, 7353}, {36, 182}, {39, 2067}, {55, 45499}, {56, 371}, {57, 45531}, {388, 45509}, {999, 45579}, {1319, 45501}, {1470, 45529}, {1478, 45555}, {2066, 6567}, {2099, 45573}, {2275, 18994}, {3085, 26436}, {3585, 45543}, {4293, 45511}, {5058, 6502}, {5204, 18959}, {5252, 45547}, {5434, 41491}, {5563, 7362}, {7763, 45560}, {9655, 45378}, {9737, 45570}, {11509, 45521}, {12313, 45493}, {12835, 45505}, {18954, 45533}, {18955, 45535}, {18956, 45537}, {18957, 45539}, {18958, 45549}, {18960, 45551}, {18961, 45557}, {18962, 45559}, {18963, 45567}, {18964, 45568}, {18965, 45576}, {18966, 45575}, {18967, 45587}, {18995, 45514}, {18996, 45513}, {19369, 44657}, {26380, 45352}, {26404, 45350}, {26433, 45517}, {26434, 45518}, {26435, 45524}, {26437, 45527}, {35768, 45564}, {45404, 45411}

X(45507) = X(35785)-of-2nd anti-circumperp-tangential triangle
X(45507) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9738, 45571), (36, 1469, 45506), (388, 45509, 45561), (999, 45579, 45583)

X(45508) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTICOMPLEMENTARY

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

X(45508) lies on these lines: {2, 372}, {3, 489}, {4, 9739}, {5, 490}, {6, 39387}, {8, 45546}, {10, 45530}, {20, 3593}, {22, 45532}, {30, 45377}, {39, 3069}, {54, 69}, {100, 45520}, {140, 491}, {141, 39388}, {145, 45572}, {193, 9540}, {302, 2041}, {303, 2042}, {315, 43121}, {325, 43118}, {376, 7690}, {388, 45506}, {486, 11294}, {487, 1270}, {497, 45562}, {549, 32808}, {591, 1151}, {615, 5254}, {625, 6460}, {639, 6396}, {671, 3317}, {1152, 7389}, {1271, 10303}, {1585, 5406}, {1992, 44656}, {2896, 45538}, {2975, 45540}, {3068, 5062}, {3071, 35949}, {3085, 45580}, {3086, 45582}, {3090, 12323}, {3091, 45544}, {3096, 7375}, {3311, 45421}, {3312, 11315}, {3434, 45556}, {3436, 45558}, {3524, 32810}, {3525, 32806}, {3533, 32813}, {3616, 45500}, {3629, 31454}, {3785, 26521}, {3788, 5591}, {3839, 13678}, {4240, 45548}, {5054, 32809}, {5490, 14234}, {5491, 7607}, {5552, 45528}, {5590, 7800}, {5601, 45534}, {5602, 45536}, {6200, 43134}, {6398, 11313}, {6410, 35948}, {6421, 13758}, {6459, 7618}, {6462, 45569}, {6463, 45566}, {6560, 32489}, {6656, 15884}, {6811, 9733}, {7376, 7803}, {7581, 7857}, {7584, 13757}, {7585, 45515}, {7586, 45512}, {7736, 18993}, {7761, 15886}, {7778, 8406}, {7787, 45504}, {7807, 19146}, {8400, 11287}, {8972, 45574}, {8981, 45420}, {9541, 12221}, {9744, 12256}, {9883, 9894}, {9995, 13653}, {10527, 45526}, {10528, 45584}, {10529, 45586}, {10577, 32488}, {10784, 34473}, {12975, 14907}, {13430, 16391}, {13701, 35823}, {13712, 42268}, {13748, 35947}, {13938, 31401}, {13939, 26619}, {13941, 45577}, {15702, 32811}, {23311, 42259}, {23312, 32790}, {26394, 45349}, {26418, 45351}, {26494, 45519}, {26503, 45516}, {26617, 43408}, {32491, 42216}, {32786, 43620}, {33440, 42254}, {33442, 42255}, {44364, 45523}

X(45508) = anticomplement of X(10576)
X(45508) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(8035)}} and {{A, B, C, X(54), X(8577)}}
X(45508) = X(10576)-of-1st anti-Kenmotu-free-vertices triangle
X(45508) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 488, 638), (2, 43133, 485), (3, 492, 637), (69, 631, 45509), (182, 7763, 45509), (372, 641, 2), (490, 32807, 5), (639, 6396, 11293), (641, 41490, 372), (1152, 45472, 7389), (1270, 3523, 487), (3069, 33364, 11292), (9739, 45554, 4), (12323, 32812, 3090), (45506, 45560, 388), (45510, 45522, 3), (45562, 45570, 497)

X(45509) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTICOMPLEMENTARY

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

X(45509) lies on these lines: {2, 371}, {3, 490}, {4, 9738}, {5, 489}, {6, 39388}, {8, 45547}, {10, 45531}, {20, 3595}, {22, 45533}, {30, 45378}, {39, 3068}, {54, 69}, {100, 45521}, {140, 492}, {141, 39387}, {145, 45573}, {193, 13935}, {302, 2042}, {303, 2041}, {315, 43120}, {325, 43119}, {376, 7692}, {388, 45507}, {485, 11293}, {488, 1271}, {497, 45563}, {549, 32809}, {590, 5254}, {625, 6459}, {640, 6200}, {671, 3316}, {1151, 7388}, {1152, 1991}, {1270, 10303}, {1586, 5407}, {1992, 44657}, {2896, 45539}, {2975, 45541}, {3069, 5058}, {3070, 35948}, {3085, 45581}, {3086, 45583}, {3090, 12322}, {3091, 45545}, {3096, 7376}, {3311, 11316}, {3312, 45420}, {3434, 45557}, {3436, 45559}, {3524, 32811}, {3525, 32805}, {3526, 32807}, {3533, 32812}, {3589, 31454}, {3616, 45501}, {3785, 26516}, {3788, 5590}, {3839, 13798}, {4240, 45549}, {5054, 32808}, {5490, 7607}, {5491, 14238}, {5552, 45529}, {5591, 7800}, {5601, 45535}, {5602, 45537}, {6221, 11314}, {6396, 43133}, {6409, 35949}, {6422, 13638}, {6460, 7618}, {6462, 45567}, {6463, 45568}, {6561, 32488}, {6656, 15883}, {6813, 9732}, {7375, 7803}, {7582, 7857}, {7583, 13637}, {7585, 45513}, {7586, 45514}, {7736, 18994}, {7761, 15885}, {7778, 8414}, {7787, 45505}, {7807, 19145}, {8407, 11287}, {8972, 45576}, {8997, 9994}, {9744, 12257}, {9882, 9892}, {10527, 45527}, {10528, 45585}, {10529, 45587}, {10576, 32489}, {10783, 34473}, {12974, 14907}, {13441, 16391}, {13749, 35946}, {13821, 35822}, {13835, 42269}, {13885, 31401}, {13886, 26620}, {13941, 45575}, {13966, 45421}, {15702, 32810}, {23311, 32789}, {23312, 42258}, {26394, 45352}, {26418, 45350}, {26494, 45517}, {26503, 45518}, {26618, 43407}, {32490, 42215}, {32785, 43620}, {33441, 42256}, {33443, 42257}, {44365, 45522}

X(45509) = anticomplement of X(10577)
X(45509) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(8036)}} and {{A, B, C, X(54), X(8576)}}
X(45509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 487, 637), (3, 491, 638), (69, 631, 45508), (182, 7763, 45508), (371, 642, 2), (640, 6200, 11294), (642, 41491, 371), (1151, 45473, 7388), (1271, 3523, 488), (3068, 33365, 11291), (9738, 45555, 4), (12322, 32813, 3090), (45507, 45561, 388), (45511, 45523, 3), (45563, 45571, 497)

X(45510) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-EULER

Barycentrics    2*(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2)-16*S^3 : :

X(45510) lies on these lines: {2, 98}, {3, 489}, {4, 372}, {5, 45376}, {6, 6811}, {20, 9739}, {24, 45532}, {30, 13757}, {39, 1588}, {76, 488}, {104, 45540}, {262, 22719}, {376, 13786}, {388, 45580}, {487, 7763}, {491, 3564}, {497, 45582}, {515, 45530}, {590, 8550}, {591, 1350}, {615, 1503}, {631, 641}, {639, 45552}, {1132, 14231}, {1152, 13748}, {1181, 6809}, {1270, 10519}, {1351, 45421}, {1353, 45420}, {1498, 6814}, {1513, 13758}, {1585, 10133}, {1586, 33971}, {1587, 3767}, {1589, 26870}, {1692, 44596}, {3068, 14912}, {3085, 45560}, {3086, 45562}, {3090, 10514}, {3091, 45542}, {3312, 36656}, {3317, 14237}, {3522, 7690}, {3594, 45440}, {4293, 45506}, {4294, 45570}, {5071, 13674}, {5085, 45472}, {5200, 43976}, {5480, 32788}, {5603, 45500}, {5657, 45546}, {5870, 13935}, {6146, 6810}, {6215, 32490}, {6280, 6316}, {6289, 7389}, {6561, 45565}, {6808, 34781}, {6812, 12241}, {7000, 13941}, {7374, 7586}, {7388, 39646}, {7487, 45502}, {7581, 45515}, {7582, 45512}, {7584, 36709}, {7585, 44656}, {7710, 39888}, {7735, 39875}, {7736, 39876}, {7967, 45572}, {9540, 45524}, {9737, 12221}, {9738, 43134}, {9755, 13638}, {9862, 45538}, {10518, 36717}, {10574, 12276}, {10706, 13762}, {10753, 13760}, {10785, 45556}, {10786, 45558}, {10788, 45504}, {10805, 45584}, {10806, 45586}, {10846, 18440}, {11257, 11294}, {11293, 36998}, {11313, 26341}, {11315, 26348}, {11491, 45520}, {11843, 45534}, {11844, 45536}, {11845, 45548}, {11846, 45569}, {11847, 45566}, {11898, 32809}, {12007, 32787}, {12115, 45528}, {12116, 45526}, {12509, 26295}, {13749, 42262}, {13834, 23267}, {13847, 36990}, {13886, 45574}, {13951, 36655}, {13961, 36712}, {13966, 36714}, {13993, 36657}, {14226, 45107}, {14230, 42270}, {14233, 42259}, {14239, 42272}, {15069, 45473}, {18424, 23249}, {22700, 33435}, {23259, 43619}, {25406, 32805}, {26381, 45349}, {26405, 45351}, {26439, 45519}, {26440, 45516}, {26521, 39647}, {30399, 41770}, {32419, 45499}, {32786, 39874}, {35823, 45545}, {39387, 43119}

X(45510) = reflection of X(i) in X(j) for these (i, j): (4, 6565), (6813, 615), (45376, 5)
X(45510) = anticomplement of X(45555)
X(45510) = intersection, other than A, B, C, of circumconics {{A, B, C, X(98), X(41516)}} and {{A, B, C, X(287), X(486)}}
X(45510) = X(6565)-of-these triangles: {anti-Euler, 1st anti-Kenmotu-free-vertices}
X(45510) = X(45376)-of-Johnson triangle
X(45510) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6776, 45511), (3, 45406, 26441), (3, 45508, 45522), (4, 12256, 8982), (98, 1352, 45511), (182, 9744, 45511), (182, 45554, 2), (631, 10784, 12257), (641, 45553, 631), (6289, 43118, 7389), (7374, 7586, 14853), (11179, 43461, 45511), (45377, 45410, 5)

X(45511) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-EULER

Barycentrics    2*(-a^2+b^2+c^2)*(3*a^4+(b^2-c^2)^2)+16*S^3 : :

X(45511) lies on these lines: {2, 98}, {3, 490}, {4, 371}, {5, 45375}, {6, 6813}, {20, 9738}, {24, 45533}, {30, 13637}, {39, 1587}, {76, 487}, {104, 45541}, {262, 22717}, {376, 13666}, {388, 45581}, {488, 7763}, {492, 3564}, {497, 45583}, {515, 45531}, {590, 1503}, {615, 8550}, {631, 642}, {640, 45553}, {1131, 14245}, {1151, 13749}, {1181, 6810}, {1271, 10519}, {1350, 1991}, {1351, 45420}, {1353, 45421}, {1498, 6812}, {1513, 13638}, {1585, 33971}, {1586, 10132}, {1588, 3767}, {1590, 26870}, {1692, 44595}, {3069, 14912}, {3085, 45561}, {3086, 45563}, {3090, 10515}, {3091, 45543}, {3311, 36655}, {3316, 14227}, {3522, 7692}, {3592, 45441}, {4293, 45507}, {4294, 45571}, {5071, 13794}, {5085, 45473}, {5480, 32787}, {5603, 45501}, {5657, 45547}, {5871, 9540}, {6146, 6809}, {6214, 32491}, {6279, 6312}, {6290, 7388}, {6560, 45564}, {6567, 9541}, {6807, 34781}, {6814, 12241}, {7000, 7585}, {7374, 8972}, {7389, 39646}, {7487, 45503}, {7581, 45513}, {7582, 45514}, {7583, 36714}, {7586, 44657}, {7710, 39887}, {7735, 39876}, {7736, 39875}, {7967, 45573}, {8976, 36656}, {8981, 36709}, {9737, 12222}, {9739, 43133}, {9755, 13758}, {9862, 45539}, {10517, 36702}, {10574, 12277}, {10706, 13643}, {10753, 13640}, {10785, 45557}, {10786, 45559}, {10788, 45505}, {10805, 45585}, {10806, 45587}, {10845, 18440}, {11257, 11293}, {11294, 36998}, {11314, 26348}, {11316, 26341}, {11491, 45521}, {11843, 45535}, {11844, 45537}, {11845, 45549}, {11846, 45567}, {11847, 45568}, {11898, 32808}, {12007, 32788}, {12115, 45529}, {12116, 45527}, {12510, 26294}, {13711, 23273}, {13748, 42265}, {13846, 36990}, {13903, 36711}, {13925, 36658}, {13935, 45525}, {13939, 45575}, {14230, 42258}, {14233, 42273}, {14235, 42271}, {14241, 45106}, {15069, 45472}, {18424, 23259}, {22699, 33434}, {23249, 43619}, {25406, 32806}, {26381, 45352}, {26405, 45350}, {26439, 45517}, {26440, 45518}, {26516, 39647}, {30398, 41770}, {32421, 45498}, {32489, 43142}, {32785, 39874}, {32807, 39899}, {35822, 45544}, {39388, 43118}

X(45511) = reflection of X(i) in X(j) for these (i, j): (4, 6564), (6811, 590), (45375, 5)
X(45511) = anticomplement of X(45554)
X(45511) = intersection, other than A, B, C, of circumconics {{A, B, C, X(98), X(41515)}} and {{A, B, C, X(287), X(485)}}
X(45511) = X(6564)-of-anti-Euler triangle
X(45511) = X(45375)-of-Johnson triangle
X(45511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6776, 45510), (3, 45407, 8982), (3, 45509, 45523), (4, 12257, 26441), (98, 1352, 45510), (182, 9744, 45510), (182, 45555, 2), (631, 10783, 12256), (642, 45552, 631), (5871, 9540, 21736), (6290, 43119, 7388), (7000, 7585, 14853), (45378, 45411, 5)

X(45512) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-INNER-GREBE

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

X(45512) lies on these lines: {2, 45577}, {3, 6}, {232, 35765}, {485, 5286}, {486, 7736}, {487, 7585}, {491, 7803}, {493, 39951}, {590, 5305}, {615, 31406}, {641, 3069}, {642, 13638}, {1194, 8854}, {1285, 42638}, {1572, 35610}, {1587, 45544}, {1588, 7374}, {2023, 35825}, {2066, 5299}, {2067, 5280}, {2275, 35769}, {2276, 35809}, {2548, 6565}, {2549, 35820}, {3068, 7375}, {3070, 15048}, {3071, 36658}, {3090, 13834}, {3092, 45141}, {3108, 8576}, {3299, 45580}, {3301, 45582}, {3767, 10576}, {3815, 10577}, {5254, 6564}, {5304, 9540}, {5309, 31481}, {5319, 31465}, {5411, 45502}, {5413, 39575}, {5418, 7735}, {5420, 31400}, {5475, 35787}, {6560, 7738}, {7581, 12257}, {7582, 45510}, {7584, 45554}, {7586, 45508}, {7737, 42266}, {7739, 31411}, {7745, 35821}, {7764, 45472}, {7834, 45473}, {7968, 45572}, {8252, 31467}, {8743, 11473}, {8855, 8956}, {9300, 35823}, {9575, 35775}, {9593, 35774}, {9596, 35801}, {9599, 35803}, {9606, 35813}, {9619, 35762}, {9620, 35641}, {11315, 45487}, {13785, 45542}, {13880, 32807}, {13936, 45546}, {15484, 23261}, {16502, 31459}, {18510, 45377}, {18907, 42258}, {18991, 45500}, {18995, 45506}, {18999, 45520}, {19003, 45530}, {19005, 45532}, {19007, 45534}, {19009, 45536}, {19013, 45540}, {19017, 45548}, {19023, 45556}, {19025, 45558}, {19027, 45560}, {19029, 45562}, {19031, 45569}, {19033, 45566}, {19037, 45570}, {19047, 45584}, {19049, 45586}, {19053, 41490}, {19054, 26620}, {26384, 45349}, {26408, 45351}, {26454, 45519}, {26455, 45516}, {26456, 45522}, {26457, 45525}, {26458, 45526}, {26459, 45528}, {31404, 42274}, {35786, 44518}, {42269, 43448}, {42582, 43291}

X(45512) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45553)}} and {{A, B, C, X(54), X(11825)}}
X(45512) = X(35792)-of-these triangles: {anti-inner-Grebe, 1st anti-Kenmotu-free-vertices}
X(45512) = X(45512)-of-circumsymmedial triangle
X(45512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 45489, 42858), (6, 39, 372), (6, 1151, 30435), (6, 1504, 6419), (6, 1505, 35770), (6, 5013, 6423), (6, 6421, 6420), (6, 6422, 371), (6, 6444, 1384), (6, 10542, 9974), (6, 12962, 5058), (6, 12963, 5007), (371, 3103, 11824), (371, 40275, 6200), (1151, 30435, 41410), (1504, 7772, 6), (3311, 9732, 371), (3312, 9739, 372), (3371, 3372, 19145), (3385, 3386, 39679), (3594, 6566, 372), (5013, 6423, 6396), (5041, 5058, 6), (5058, 9994, 371), (5286, 31403, 485), (6417, 45410, 44656), (6420, 45565, 372), (6421, 18993, 372), (6423, 9995, 372), (9739, 44474, 3312), (31400, 44595, 5420), (44502, 45488, 35794)

X(45513) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-OUTER-GREBE

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

X(45513) lies on these lines: {2, 45576}, {3, 6}, {232, 35764}, {485, 7736}, {486, 5286}, {488, 7586}, {492, 7803}, {494, 39951}, {590, 31406}, {615, 5305}, {641, 13758}, {642, 3068}, {1194, 8855}, {1285, 42637}, {1572, 35611}, {1587, 7000}, {1588, 45545}, {2023, 35824}, {2275, 35768}, {2276, 35808}, {2548, 6564}, {2549, 35821}, {3069, 7376}, {3070, 36657}, {3071, 15048}, {3090, 13711}, {3093, 45141}, {3108, 8577}, {3299, 45583}, {3301, 45581}, {3767, 10577}, {3815, 10576}, {5254, 6565}, {5280, 6502}, {5299, 5414}, {5304, 13935}, {5319, 35813}, {5410, 45503}, {5412, 39575}, {5418, 31400}, {5420, 7735}, {5475, 35786}, {6561, 7738}, {7581, 45511}, {7582, 12256}, {7583, 45555}, {7585, 45509}, {7737, 42267}, {7739, 35823}, {7745, 35820}, {7764, 45473}, {7834, 45472}, {7969, 45573}, {8253, 31467}, {8743, 11474}, {8960, 31463}, {9300, 35822}, {9575, 35774}, {9593, 35775}, {9596, 35800}, {9599, 35802}, {9606, 35812}, {9619, 35763}, {9620, 35642}, {11316, 45486}, {13665, 45543}, {13883, 45547}, {15484, 23251}, {16502, 35809}, {18512, 45378}, {18907, 42259}, {18992, 45501}, {18996, 45507}, {19000, 45521}, {19004, 45531}, {19006, 45533}, {19008, 45535}, {19010, 45537}, {19014, 45541}, {19018, 45549}, {19024, 45557}, {19026, 45559}, {19028, 45561}, {19030, 45563}, {19032, 45567}, {19034, 45568}, {19038, 45571}, {19048, 45585}, {19050, 45587}, {19053, 26619}, {19054, 41491}, {26385, 45352}, {26409, 45350}, {26460, 45517}, {26461, 45518}, {26462, 45524}, {26463, 45523}, {26464, 45527}, {26465, 45529}, {31404, 42277}, {35787, 44518}, {42268, 43448}, {42583, 43291}

X(45513) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45552)}} and {{A, B, C, X(54), X(11824)}}
X(45513) = X(35793)-of-anti-outer-Grebe triangle
X(45513) = X(45513)-of-circumsymmedial triangle
X(45513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 45488, 42859), (6, 39, 371), (6, 371, 45514), (6, 1152, 30435), (6, 1504, 35771), (6, 1505, 6420), (6, 5013, 6424), (6, 6421, 372), (6, 6422, 6419), (6, 6443, 1384), (6, 12968, 5007), (6, 12969, 5062), (372, 3102, 11825), (372, 40274, 6396), (1152, 30435, 41411), (1505, 7772, 6), (3311, 9738, 371), (3312, 9733, 372), (3371, 3372, 39648), (3385, 3386, 19146), (3592, 6567, 371), (5013, 6424, 6200), (5041, 5062, 6), (5062, 9995, 372), (6418, 45411, 44657), (6419, 45564, 371), (6422, 18994, 371), (6424, 9994, 371), (9738, 44473, 3311), (31400, 44596, 5418), (44501, 45489, 35795)

X(45514) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-INNER-GREBE

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

X(45514) lies on these lines: {2, 45575}, {3, 6}, {4, 19105}, {230, 44648}, {486, 44596}, {591, 3788}, {642, 3069}, {1587, 45545}, {1991, 7838}, {2548, 8960}, {3291, 8855}, {3299, 45581}, {3301, 45583}, {3543, 19100}, {3767, 35823}, {3815, 35812}, {5304, 9757}, {5410, 8946}, {5411, 45503}, {7582, 45511}, {7584, 45555}, {7585, 45576}, {7586, 45509}, {7745, 35822}, {7763, 45421}, {7803, 45574}, {7968, 45573}, {11316, 45485}, {12222, 19103}, {13711, 42561}, {13770, 13939}, {13785, 45543}, {13936, 45547}, {18510, 45378}, {18991, 45501}, {18995, 45507}, {18999, 45521}, {19003, 45531}, {19005, 45533}, {19007, 45535}, {19009, 45537}, {19013, 45541}, {19017, 45549}, {19023, 45557}, {19025, 45559}, {19027, 45561}, {19029, 45563}, {19031, 45567}, {19033, 45568}, {19037, 45571}, {19047, 45585}, {19049, 45587}, {19053, 41491}, {26384, 45352}, {26408, 45350}, {26454, 45517}, {26455, 45518}, {26456, 45524}, {26457, 45523}, {26458, 45527}, {26459, 45529}, {31406, 31454}, {31463, 35815}

X(45514) = barycentric product X(589)*X(13933)
X(45514) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45499)}} and {{A, B, C, X(251), X(6420)}}
X(45514) = X(35795)-of-anti-inner-Grebe triangle
X(45514) = X(45514)-of-circumsymmedial triangle
X(45514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 44501, 35793), (4, 26463, 19105), (6, 371, 45513), (6, 3311, 45512), (6, 3592, 9605), (6, 5058, 371), (6, 6423, 35770), (6, 6424, 372), (6, 12962, 7772), (6, 12963, 1505), (6, 30435, 45515), (182, 3311, 371), (187, 12969, 6454), (371, 372, 45499), (372, 6419, 45463), (372, 6424, 41410), (1505, 12963, 6396), (1692, 5058, 6424), (3385, 3386, 15884), (5013, 8375, 6453), (5039, 34571, 45515), (6427, 43136, 6)

X(45515) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-OUTER-GREBE

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

X(45515) lies on these lines: {2, 45574}, {3, 6}, {4, 19102}, {230, 8960}, {485, 44595}, {591, 7838}, {641, 3068}, {1588, 45544}, {1991, 3788}, {3291, 8854}, {3299, 45582}, {3301, 45580}, {3543, 19099}, {3767, 35822}, {3815, 35813}, {5304, 9758}, {5410, 45502}, {5411, 8948}, {5420, 31403}, {7581, 45510}, {7583, 45554}, {7585, 45508}, {7586, 45577}, {7745, 35823}, {7763, 45420}, {7803, 45575}, {7969, 45572}, {10576, 31411}, {11315, 45484}, {12221, 19104}, {13651, 13886}, {13665, 45542}, {13834, 31412}, {13883, 45546}, {18512, 45377}, {18992, 45500}, {18996, 45506}, {19000, 45520}, {19004, 45530}, {19006, 45532}, {19008, 45534}, {19010, 45536}, {19014, 45540}, {19018, 45548}, {19024, 45556}, {19026, 45558}, {19028, 45560}, {19030, 45562}, {19032, 45569}, {19034, 45566}, {19038, 45570}, {19048, 45584}, {19050, 45586}, {19054, 41490}, {26385, 45349}, {26409, 45351}, {26460, 45519}, {26461, 45516}, {26462, 45522}, {26463, 45525}, {26464, 45526}, {26465, 45528}

X(45515) = barycentric product X(588)*X(13879)
X(45515) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(45498)}} and {{A, B, C, X(251), X(6419)}}
X(45515) = X(35794)-of-these triangles: {anti-outer-Grebe, 1st anti-Kenmotu-free-vertices}
X(45515) = X(45515)-of-circumsymmedial triangle
X(45515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 44502, 35792), (4, 26456, 19102), (6, 32, 6419), (6, 372, 45512), (6, 3312, 45513), (6, 3594, 9605), (6, 5062, 372), (6, 6423, 371), (6, 6424, 35771), (6, 12968, 1504), (6, 12969, 7772), (182, 3312, 372), (187, 12962, 6453), (230, 44647, 8960), (371, 372, 45498), (371, 6420, 45462), (371, 6423, 41411), (1504, 12968, 6200), (1692, 5062, 6423), (3371, 3372, 15883), (5013, 8376, 6454), (5039, 34571, 45514), (6428, 43136, 6)

X(45516) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(-2*(a^6-11*(b^2+c^2)*a^4+(15*b^4+22*b^2*c^2+15*c^4)*a^2-(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4))*S+5*a^8-16*(b^2+c^2)*a^6+2*(11*b^4-4*b^2*c^2+11*c^4)*a^4-16*(-b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2+(5*b^4-14*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :

X(45516) lies on these lines: {3, 45412}, {25, 372}, {39, 45598}, {182, 26507}, {641, 5491}, {5062, 45595}, {6464, 45519}, {18523, 45377}, {26293, 45498}, {26299, 45530}, {26313, 45538}, {26323, 45540}, {26329, 45544}, {26338, 45553}, {26354, 45570}, {26368, 45500}, {26392, 45349}, {26416, 45351}, {26428, 45504}, {26434, 45506}, {26440, 45510}, {26443, 45546}, {26448, 45548}, {26455, 45512}, {26461, 45515}, {26467, 45554}, {26472, 45562}, {26478, 45560}, {26484, 45558}, {26489, 45556}, {26502, 45520}, {26503, 45508}, {26504, 45572}, {26505, 45522}, {26506, 45525}, {26508, 45526}, {26509, 45528}, {26510, 45586}, {26511, 45584}, {45410, 45414}, {45534, 45588}, {45536, 45590}, {45542, 45592}, {45550, 45594}, {45565, 45602}, {45569, 45604}, {45574, 45605}, {45577, 45608}, {45578, 45609}, {45580, 45611}, {45582, 45613}

X(45516) = X(35806)-of-1st anti-Kenmotu-free-vertices triangle

X(45517) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(2*(a^6-11*(b^2+c^2)*a^4+(15*b^4+22*b^2*c^2+15*c^4)*a^2-(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4))*S+5*a^8-16*(b^2+c^2)*a^6+2*(11*b^4-4*b^2*c^2+11*c^4)*a^4-16*(-b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2+(5*b^4-14*b^2*c^2+5*c^4)*(b^2-c^2)^2) : :

X(45517) lies on these lines: {3, 45413}, {25, 371}, {39, 45597}, {182, 26498}, {642, 5490}, {5058, 45596}, {6464, 45518}, {18521, 45378}, {26292, 45499}, {26298, 45531}, {26312, 45539}, {26322, 45541}, {26328, 45545}, {26337, 45552}, {26347, 45551}, {26353, 45571}, {26367, 45501}, {26391, 45352}, {26415, 45350}, {26427, 45505}, {26433, 45507}, {26439, 45511}, {26442, 45547}, {26447, 45549}, {26454, 45514}, {26460, 45513}, {26466, 45555}, {26471, 45563}, {26477, 45561}, {26483, 45559}, {26488, 45557}, {26493, 45521}, {26494, 45509}, {26495, 45573}, {26496, 45524}, {26497, 45523}, {26499, 45527}, {26500, 45529}, {26501, 45587}, {45411, 45415}, {45535, 45589}, {45537, 45591}, {45543, 45593}, {45564, 45601}, {45568, 45603}, {45575, 45606}, {45576, 45607}, {45579, 45610}, {45581, 45612}, {45583, 45614}, {45585, 45615}

X(45518) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(-2*(a^6-11*(b^2+c^2)*a^4+(15*b^4+22*b^2*c^2+15*c^4)*a^2-(b^2+c^2)*(5*b^4-22*b^2*c^2+5*c^4))*S+3*a^8-8*(b^2+c^2)*a^6+2*(b^4-12*b^2*c^2+c^4)*a^4+8*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^2+(b^2-c^2)^2*(-5*b^4-18*b^2*c^2-5*c^4)) : :

X(45518) lies on these lines: {3, 45414}, {39, 45595}, {182, 26507}, {371, 494}, {485, 642}, {1152, 42022}, {5058, 45598}, {6464, 45517}, {8943, 12966}, {12303, 19493}, {18523, 45378}, {23309, 32494}, {26293, 45499}, {26299, 45531}, {26305, 45533}, {26313, 45539}, {26323, 45541}, {26329, 45545}, {26338, 45551}, {26354, 45571}, {26368, 45501}, {26374, 45503}, {26392, 45352}, {26416, 45350}, {26428, 45505}, {26434, 45507}, {26440, 45511}, {26443, 45547}, {26448, 45549}, {26455, 45514}, {26461, 45513}, {26467, 45555}, {26472, 45563}, {26478, 45561}, {26484, 45559}, {26489, 45557}, {26502, 45521}, {26503, 45509}, {26504, 45573}, {26505, 45524}, {26506, 45523}, {26508, 45527}, {26509, 45529}, {26510, 45587}, {26511, 45585}, {45411, 45412}, {45535, 45588}, {45537, 45590}, {45543, 45592}, {45552, 45594}, {45564, 45599}, {45567, 45604}, {45575, 45608}, {45576, 45605}, {45579, 45609}, {45581, 45611}, {45583, 45613}

X(45519) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(2*(a^6-11*(b^2+c^2)*a^4+(15*b^4+22*b^2*c^2+15*c^4)*a^2-(b^2+c^2)*(5*b^4-22*b^2*c^2+5*c^4))*S+3*a^8-8*(b^2+c^2)*a^6+2*(b^4-12*b^2*c^2+c^4)*a^4+8*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^2+(b^2-c^2)^2*(-5*b^4-18*b^2*c^2-5*c^4)) : :

X(45519) lies on these lines: {3, 45415}, {39, 45596}, {182, 26498}, {372, 493}, {486, 641}, {5062, 45597}, {6464, 45516}, {8939, 12961}, {12304, 19492}, {18521, 45377}, {23310, 32497}, {26292, 45498}, {26298, 45530}, {26304, 45532}, {26312, 45538}, {26322, 45540}, {26328, 45544}, {26337, 45550}, {26347, 45553}, {26353, 45570}, {26367, 45500}, {26373, 45502}, {26391, 45349}, {26415, 45351}, {26427, 45504}, {26433, 45506}, {26439, 45510}, {26442, 45546}, {26447, 45548}, {26454, 45512}, {26460, 45515}, {26466, 45554}, {26471, 45562}, {26477, 45560}, {26483, 45558}, {26488, 45556}, {26493, 45520}, {26494, 45508}, {26495, 45572}, {26496, 45522}, {26497, 45525}, {26499, 45526}, {26500, 45528}, {26501, 45586}, {45410, 45413}, {45534, 45589}, {45536, 45591}, {45542, 45593}, {45565, 45600}, {45566, 45603}, {45574, 45607}, {45577, 45606}, {45578, 45610}, {45580, 45612}, {45582, 45614}, {45584, 45615}

X(45519) = X(35804)-of-1st anti-Kenmotu-free-vertices triangle

X(45520) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-MANDART-INCIRCLE

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

X(45520) lies on these lines: {3, 45416}, {35, 45530}, {39, 44591}, {55, 372}, {56, 45572}, {100, 45508}, {182, 9052}, {197, 45532}, {641, 1376}, {3295, 45500}, {4421, 41490}, {5062, 44590}, {5687, 45546}, {7690, 35238}, {9739, 11248}, {10310, 45498}, {11383, 45502}, {11490, 45504}, {11491, 45510}, {11492, 45534}, {11493, 45536}, {11494, 45538}, {11496, 45544}, {11497, 45550}, {11498, 45553}, {11499, 45554}, {11500, 45558}, {11501, 45560}, {11502, 45562}, {11503, 45569}, {11504, 45566}, {11507, 45580}, {11508, 45582}, {11509, 45506}, {11510, 45586}, {11848, 45548}, {11849, 45578}, {13887, 45574}, {13940, 45577}, {18491, 45542}, {18524, 45377}, {18999, 45512}, {19000, 45515}, {26393, 45349}, {26417, 45351}, {26493, 45519}, {26502, 45516}, {26512, 45522}, {26513, 45525}, {35773, 45565}, {37621, 45410}

X(45520) = X(35808)-of-these triangles: {1st anti-Kenmotu-free-vertices, anti-Mandart-incircle}
X(45520) = {X(10267), X(12329)}-harmonic conjugate of X(45521)

X(45521) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-MANDART-INCIRCLE

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

X(45521) lies on these lines: {3, 45417}, {35, 45531}, {39, 44590}, {55, 371}, {56, 45573}, {100, 45509}, {182, 9052}, {197, 45533}, {642, 1376}, {3295, 45501}, {4421, 41491}, {5058, 44591}, {5687, 45547}, {7692, 35238}, {9738, 11248}, {10310, 45499}, {11383, 45503}, {11490, 45505}, {11491, 45511}, {11492, 45535}, {11493, 45537}, {11494, 45539}, {11496, 45545}, {11497, 45552}, {11498, 45551}, {11499, 45555}, {11500, 45559}, {11501, 45561}, {11502, 45563}, {11503, 45567}, {11504, 45568}, {11507, 45581}, {11508, 45583}, {11509, 45507}, {11510, 45587}, {11848, 45549}, {11849, 45579}, {13887, 45576}, {13940, 45575}, {18491, 45543}, {18524, 45378}, {18999, 45514}, {19000, 45513}, {26393, 45352}, {26417, 45350}, {26493, 45517}, {26502, 45518}, {26512, 45524}, {26513, 45523}, {35772, 45564}, {37621, 45411}

X(45521) = X(35809)-of-anti-Mandart-incircle triangle
X(45521) = {X(10267), X(12329)}-harmonic conjugate of X(45520)

X(45522) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 3rd ANTI-TRI-SQUARES-CENTRAL

Barycentrics    -2*(5*a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2)*S+(-a^2+b^2+c^2)*(a^4+4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(45522) = 5*X(631)-2*X(35812)

X(45522) lies on these lines: {2, 9739}, {3, 489}, {4, 641}, {20, 7690}, {39, 13935}, {140, 45578}, {182, 193}, {316, 21737}, {372, 631}, {487, 7771}, {488, 7763}, {549, 45410}, {550, 18539}, {944, 26444}, {1007, 8982}, {1587, 6566}, {3085, 26435}, {3086, 26355}, {3088, 26375}, {3090, 26330}, {3146, 45542}, {3524, 5860}, {4293, 26479}, {4294, 26473}, {5062, 9540}, {5218, 45580}, {5420, 39660}, {6289, 35947}, {6684, 26300}, {6811, 12305}, {7288, 45582}, {9733, 39387}, {10323, 26306}, {10357, 26314}, {10359, 26429}, {10517, 26339}, {11293, 43141}, {12221, 33362}, {12245, 26514}, {12510, 22646}, {12974, 43134}, {15884, 37450}, {18993, 31400}, {26324, 45540}, {26369, 45500}, {26396, 45349}, {26420, 45351}, {26449, 45548}, {26456, 45512}, {26462, 45515}, {26485, 45558}, {26490, 45556}, {26496, 45519}, {26505, 45516}, {26512, 45520}, {26517, 45526}, {26518, 45528}, {26519, 45586}, {26520, 45584}, {42523, 44474}, {44365, 45509}

X(45522) = X(35812)-of-these triangles: {1st anti-Kenmotu-free-vertices, 3rd anti-tri-squares-central}
X(45522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 492, 26441), (3, 45508, 45510), (193, 3523, 26516), (193, 26516, 45524), (641, 45498, 4), (3523, 10519, 45523), (7690, 45554, 20), (26294, 26361, 4)

X(45523) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 4th ANTI-TRI-SQUARES-CENTRAL

Barycentrics    2*(5*a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2)*S+(-a^2+b^2+c^2)*(a^4+4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(45523) = 5*X(631)-2*X(35813)

X(45523) lies on these lines: {2, 9738}, {3, 490}, {4, 642}, {20, 7692}, {39, 9540}, {140, 45579}, {182, 193}, {371, 631}, {487, 7763}, {488, 7771}, {549, 45411}, {550, 26438}, {944, 26445}, {1007, 26441}, {1588, 6567}, {3085, 26436}, {3086, 26356}, {3088, 26376}, {3090, 26331}, {3146, 45543}, {3524, 5861}, {4293, 26480}, {4294, 26474}, {5058, 13935}, {5218, 45581}, {5418, 39661}, {6290, 35946}, {6684, 26301}, {6813, 12306}, {7288, 45583}, {9732, 39388}, {10323, 26307}, {10357, 26315}, {10359, 26430}, {10518, 26340}, {11294, 43144}, {12222, 33363}, {12245, 26515}, {12509, 22617}, {12975, 43133}, {15883, 37450}, {18994, 31400}, {26325, 45541}, {26370, 45501}, {26397, 45352}, {26421, 45350}, {26450, 45549}, {26457, 45514}, {26463, 45513}, {26486, 45559}, {26491, 45557}, {26497, 45517}, {26506, 45518}, {26513, 45521}, {26522, 45527}, {26523, 45529}, {26524, 45587}, {26525, 45585}, {42522, 44473}, {44364, 45508}

X(45523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 491, 8982), (3, 45509, 45511), (193, 3523, 26521), (193, 26521, 45525), (642, 45499, 4), (3523, 10519, 45522), (7692, 45555, 20), (26295, 26362, 4)

X(45524) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 3rd ANTI-TRI-SQUARES-CENTRAL

Barycentrics    7*a^6-11*(b^2+c^2)*a^4+5*(b^2-c^2)^2*a^2+2*(5*a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45524) = X(4)-6*X(35815)

X(45524) lies on these lines: {3, 45420}, {4, 371}, {39, 19064}, {140, 492}, {182, 193}, {550, 45579}, {631, 5860}, {642, 3533}, {1151, 8982}, {1588, 7746}, {3522, 9738}, {3592, 6813}, {3850, 18539}, {3854, 45543}, {5056, 26468}, {5058, 44595}, {5882, 26300}, {6221, 45407}, {6437, 13749}, {6776, 33372}, {6811, 31454}, {7374, 43883}, {8550, 41963}, {8981, 45406}, {8997, 33430}, {9540, 45510}, {10299, 26339}, {12256, 13650}, {12974, 43133}, {14853, 42522}, {21735, 26294}, {26306, 45533}, {26314, 45539}, {26324, 45541}, {26355, 45571}, {26369, 45501}, {26375, 45503}, {26396, 45352}, {26420, 45350}, {26429, 45505}, {26435, 45507}, {26444, 45547}, {26449, 45549}, {26456, 45514}, {26462, 45513}, {26473, 45563}, {26479, 45561}, {26485, 45559}, {26490, 45557}, {26496, 45517}, {26505, 45518}, {26512, 45521}, {26514, 45573}, {26517, 45527}, {26518, 45529}, {26519, 45587}, {26520, 45585}

X(45524) = X(35815)-of-3rd anti-tri-squares-central triangle
X(45524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (193, 26516, 45522), (41491, 45551, 631)

X(45525) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 4th ANTI-TRI-SQUARES-CENTRAL

Barycentrics    7*a^6-11*(b^2+c^2)*a^4+5*(b^2-c^2)^2*a^2-2*(5*a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2)*S-(b^4-c^4)*(b^2-c^2) : :
X(45525) = X(4)-6*X(35814)

X(45525) lies on these lines: {3, 45421}, {4, 372}, {39, 19063}, {140, 491}, {182, 193}, {550, 45578}, {631, 5861}, {641, 3533}, {1152, 26441}, {1587, 7746}, {3522, 9739}, {3594, 6811}, {3850, 26438}, {3854, 45542}, {5056, 26469}, {5062, 44596}, {5882, 26301}, {6398, 45406}, {6438, 13748}, {6776, 33373}, {7000, 43884}, {8550, 41964}, {10299, 26340}, {12257, 13771}, {12975, 43134}, {13935, 45511}, {13966, 45407}, {13989, 33431}, {14853, 42523}, {21735, 26295}, {26307, 45532}, {26315, 45538}, {26325, 45540}, {26356, 45570}, {26370, 45500}, {26376, 45502}, {26397, 45349}, {26421, 45351}, {26430, 45504}, {26436, 45506}, {26445, 45546}, {26450, 45548}, {26457, 45512}, {26463, 45515}, {26474, 45562}, {26480, 45560}, {26486, 45558}, {26491, 45556}, {26497, 45519}, {26506, 45516}, {26513, 45520}, {26515, 45572}, {26522, 45526}, {26523, 45528}, {26524, 45586}, {26525, 45584}

X(45525) = X(35814)-of-1st anti-Kenmotu-free-vertices triangle
X(45525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (193, 26521, 45523), (41490, 45550, 631)

X(45526) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-INNER-YFF

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

X(45526) lies on these lines: {1, 372}, {3, 45422}, {5, 45558}, {39, 19049}, {182, 9052}, {641, 26363}, {1152, 45496}, {5062, 19050}, {6734, 45546}, {9739, 11249}, {10527, 45508}, {10680, 45578}, {10902, 26349}, {10943, 45556}, {11012, 45498}, {12116, 45510}, {16202, 45410}, {18544, 45377}, {26308, 45532}, {26317, 45538}, {26332, 45544}, {26342, 34486}, {26357, 45570}, {26377, 45502}, {26399, 45349}, {26423, 45351}, {26431, 45504}, {26437, 45506}, {26452, 45548}, {26458, 45512}, {26464, 45515}, {26470, 45554}, {26475, 45562}, {26481, 45560}, {26499, 45519}, {26508, 45516}, {26517, 45522}, {26522, 45525}, {45534, 45625}, {45536, 45626}, {45542, 45630}, {45565, 45641}, {45566, 45644}, {45569, 45645}, {45574, 45650}, {45577, 45651}

X(45526) = X(35809)-of-these triangles: {1st anti-Kenmotu-free-vertices, anti-inner-Yff}
X(45526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (372, 45572, 45580), (372, 45586, 1)

X(45527) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-INNER-YFF

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

X(45527) lies on these lines: {1, 371}, {3, 45423}, {5, 45559}, {39, 19050}, {182, 9052}, {642, 26363}, {1151, 45497}, {5058, 19049}, {6734, 45547}, {9738, 11249}, {10527, 45509}, {10680, 45579}, {10902, 26342}, {10943, 45557}, {11012, 45499}, {12116, 45511}, {16202, 45411}, {18544, 45378}, {26308, 45533}, {26317, 45539}, {26332, 45545}, {26349, 34486}, {26357, 45571}, {26377, 45503}, {26399, 45352}, {26423, 45350}, {26431, 45505}, {26437, 45507}, {26452, 45549}, {26458, 45514}, {26464, 45513}, {26470, 45555}, {26475, 45563}, {26481, 45561}, {26499, 45517}, {26508, 45518}, {26517, 45524}, {26522, 45523}, {45535, 45625}, {45537, 45626}, {45543, 45630}, {45564, 45640}, {45567, 45645}, {45568, 45644}, {45575, 45651}, {45576, 45650}

X(45527) = X(35808)-of-anti-inner-Yff triangle
X(45527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (371, 45573, 45581), (371, 45587, 1)

X(45528) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ANTI-OUTER-YFF

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

X(45528) lies on these lines: {1, 372}, {3, 45424}, {5, 45556}, {39, 19047}, {119, 45554}, {182, 2810}, {641, 26364}, {1152, 45494}, {1470, 45506}, {2077, 45498}, {5062, 19048}, {5552, 45508}, {6735, 45546}, {9739, 11248}, {10679, 45578}, {10942, 45558}, {12115, 45510}, {16203, 45410}, {18542, 45377}, {26309, 45532}, {26318, 45538}, {26333, 45544}, {26343, 45550}, {26350, 37561}, {26358, 45570}, {26378, 45502}, {26400, 45349}, {26424, 45351}, {26432, 45504}, {26453, 45548}, {26459, 45512}, {26465, 45515}, {26476, 45562}, {26482, 45560}, {26500, 45519}, {26509, 45516}, {26518, 45522}, {26523, 45525}, {45534, 45627}, {45536, 45628}, {45542, 45631}, {45565, 45643}, {45566, 45646}, {45569, 45647}, {45574, 45652}, {45577, 45653}

X(45528) = X(35769)-of-these triangles: {1st anti-Kenmotu-free-vertices, anti-outer-Yff}
X(45528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (372, 45572, 45582), (372, 45584, 1)

X(45529) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ANTI-OUTER-YFF

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

X(45529) lies on these lines: {1, 371}, {3, 45425}, {5, 45557}, {39, 19048}, {119, 45555}, {182, 2810}, {642, 26364}, {1151, 45495}, {1470, 45507}, {2077, 45499}, {5058, 19047}, {5552, 45509}, {6735, 45547}, {9738, 11248}, {10679, 45579}, {10942, 45559}, {12115, 45511}, {16203, 45411}, {18542, 45378}, {26309, 45533}, {26318, 45539}, {26333, 45545}, {26343, 37561}, {26350, 45551}, {26358, 45571}, {26378, 45503}, {26400, 45352}, {26424, 45350}, {26432, 45505}, {26453, 45549}, {26459, 45514}, {26465, 45513}, {26476, 45563}, {26482, 45561}, {26500, 45517}, {26509, 45518}, {26518, 45524}, {26523, 45523}, {45535, 45627}, {45537, 45628}, {45543, 45631}, {45564, 45642}, {45567, 45647}, {45568, 45646}, {45575, 45653}, {45576, 45652}

X(45529) = X(35768)-of-anti-outer-Yff triangle
X(45529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (371, 45573, 45583), (371, 45585, 1)

X(45530) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND AQUILA

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

X(45530) lies on these lines: {1, 372}, {3, 45426}, {10, 45508}, {35, 45520}, {36, 45540}, {39, 18992}, {40, 9739}, {57, 45506}, {165, 45498}, {182, 3576}, {515, 45510}, {517, 45578}, {641, 1698}, {1385, 45410}, {1697, 45570}, {1699, 45544}, {3099, 45538}, {3594, 45398}, {3679, 41490}, {5062, 18991}, {5587, 45554}, {5588, 7987}, {5589, 30389}, {5882, 26301}, {6426, 45476}, {6684, 26300}, {7690, 35242}, {7713, 45502}, {8185, 45532}, {8186, 45534}, {8187, 45536}, {8188, 45569}, {8189, 45566}, {9575, 18993}, {9578, 45560}, {9581, 45562}, {10789, 45504}, {10826, 45556}, {10827, 45558}, {11852, 45548}, {13888, 45574}, {13942, 45577}, {18480, 45377}, {18492, 45542}, {19003, 45512}, {19004, 45515}, {26296, 45349}, {26297, 45351}, {26298, 45519}, {26299, 45516}, {35775, 45565}

X(45530) = reflection of X(1) in X(35762)
X(45530) = X(35762)-of-these triangles: {1st anti-Kenmotu-free-vertices, Aquila}
X(45530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (372, 45572, 45500), (3576, 3751, 45531), (45500, 45572, 1)

X(45531) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND AQUILA

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

X(45531) lies on these lines: {1, 371}, {3, 45427}, {10, 45509}, {35, 45521}, {36, 45541}, {39, 18991}, {40, 9738}, {57, 45507}, {165, 45499}, {182, 3576}, {515, 45511}, {517, 45579}, {642, 1698}, {1385, 45411}, {1697, 45571}, {1699, 45545}, {3099, 45539}, {3592, 45399}, {3641, 9615}, {3679, 41491}, {5058, 18992}, {5587, 45555}, {5588, 30389}, {5589, 7987}, {5882, 26300}, {6425, 45477}, {6567, 9616}, {6684, 26301}, {7692, 35242}, {7713, 45503}, {8185, 45533}, {8186, 45535}, {8187, 45537}, {8188, 45567}, {8189, 45568}, {9575, 18994}, {9578, 45561}, {9581, 45563}, {10789, 45505}, {10826, 45557}, {10827, 45559}, {11852, 45549}, {13888, 45576}, {13942, 45575}, {18480, 45378}, {18492, 45543}, {19003, 45514}, {19004, 45513}, {26296, 45352}, {26297, 45350}, {26298, 45517}, {26299, 45518}, {35774, 45564}

X(45531) = reflection of X(1) in X(35763)
X(45531) = X(35763)-of-Aquila triangle
X(45531) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (371, 45573, 45501), (3576, 3751, 45530), (45501, 45573, 1)

X(45532) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND ARA

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

X(45532) lies on these lines: {3, 639}, {22, 45508}, {24, 45510}, {25, 372}, {39, 44599}, {159, 182}, {197, 45520}, {1598, 45544}, {3155, 23195}, {3156, 5420}, {3517, 9921}, {5062, 44527}, {5594, 45553}, {5595, 45550}, {7387, 9739}, {7506, 45410}, {7517, 45578}, {7690, 35243}, {8185, 45530}, {8190, 45534}, {8191, 45536}, {8192, 45572}, {8193, 45546}, {8194, 45569}, {8276, 19459}, {9818, 45542}, {9909, 41490}, {10037, 45580}, {10046, 45582}, {10323, 26306}, {10790, 45504}, {10828, 45538}, {10829, 45556}, {10830, 45558}, {10831, 45560}, {10832, 45562}, {10833, 45570}, {10834, 45584}, {10835, 45586}, {11365, 45500}, {11414, 45498}, {11853, 45548}, {13889, 45574}, {13943, 45577}, {18954, 45506}, {19005, 45512}, {19006, 45515}, {22654, 45540}, {26302, 45349}, {26303, 45351}, {26304, 45519}, {26307, 45525}, {26308, 45526}, {26309, 45528}, {32621, 44656}, {35777, 45565}

X(45532) = X(35764)-of-these triangles: {1st anti-Kenmotu-free-vertices, Ara}
X(45532) = {X(159), X(6642)}-harmonic conjugate of X(45533)

X(45533) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND ARA

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

X(45533) lies on these lines: {3, 640}, {22, 45509}, {24, 45511}, {25, 371}, {39, 44598}, {159, 182}, {197, 45521}, {1598, 45545}, {3155, 5418}, {3156, 23195}, {3517, 9922}, {5058, 8908}, {5594, 45551}, {5595, 45552}, {7387, 9738}, {7506, 45411}, {7517, 45579}, {7692, 35243}, {8185, 45531}, {8190, 45535}, {8191, 45537}, {8192, 45573}, {8193, 45547}, {8195, 45568}, {8277, 19459}, {9818, 45543}, {9909, 41491}, {10037, 45581}, {10046, 45583}, {10323, 26307}, {10790, 45505}, {10828, 45539}, {10829, 45557}, {10830, 45559}, {10831, 45561}, {10832, 45563}, {10833, 45571}, {10834, 45585}, {10835, 45587}, {11365, 45501}, {11414, 45499}, {11853, 45549}, {13889, 45576}, {13943, 45575}, {18954, 45507}, {19005, 45514}, {19006, 45513}, {22654, 45541}, {26302, 45352}, {26303, 45350}, {26305, 45518}, {26306, 45524}, {26308, 45527}, {26309, 45529}, {32621, 44657}, {35776, 45564}

X(45533) = X(35765)-of-Ara triangle
X(45533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 45429, 8996), (159, 6642, 45532)

X(45534) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 1st AURIGA

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

X(45534) lies on these lines: {3, 45430}, {39, 44601}, {55, 45536}, {182, 12452}, {372, 5597}, {641, 5599}, {5062, 44600}, {5598, 45572}, {5601, 45508}, {7690, 35244}, {8186, 45530}, {8190, 45532}, {8196, 45544}, {8197, 45546}, {8198, 45550}, {8199, 45553}, {8200, 45554}, {8201, 45569}, {8202, 45566}, {9739, 11252}, {11207, 41490}, {11366, 45500}, {11384, 45502}, {11492, 45520}, {11493, 45540}, {11822, 45498}, {11837, 45504}, {11843, 45510}, {11861, 45538}, {11863, 45548}, {11865, 45556}, {11867, 45558}, {11869, 45560}, {11871, 45562}, {11873, 45570}, {11875, 45578}, {11877, 45580}, {11879, 45582}, {11881, 45584}, {11883, 45586}, {13890, 45574}, {13944, 45577}, {18495, 45542}, {18955, 45506}, {19007, 45512}, {19008, 45515}, {35781, 45565}, {45351, 45353}, {45377, 45379}, {45410, 45431}, {45516, 45588}, {45519, 45589}, {45526, 45625}, {45528, 45627}

X(45534) = X(45357)-of-these triangles: {1st anti-Kenmotu-free-vertices, 1st Auriga}
X(45534) = {X(12452), X(45620)}-harmonic conjugate of X(45535)

X(45535) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 1st AURIGA

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

X(45535) lies on these lines: {3, 45431}, {39, 44600}, {55, 45537}, {182, 12452}, {371, 5597}, {642, 5599}, {5058, 44601}, {5598, 45573}, {5601, 45509}, {7692, 35244}, {8186, 45531}, {8190, 45533}, {8196, 45545}, {8197, 45547}, {8198, 45552}, {8199, 45551}, {8200, 45555}, {8201, 45567}, {8202, 45568}, {9738, 11252}, {11207, 41491}, {11366, 45501}, {11384, 45503}, {11492, 45521}, {11493, 45541}, {11822, 45499}, {11837, 45505}, {11843, 45511}, {11861, 45539}, {11863, 45549}, {11865, 45557}, {11867, 45559}, {11869, 45561}, {11871, 45563}, {11873, 45571}, {11875, 45579}, {11877, 45581}, {11879, 45583}, {11881, 45585}, {11883, 45587}, {13890, 45576}, {13944, 45575}, {18495, 45543}, {18955, 45507}, {19007, 45514}, {19008, 45513}, {35778, 45564}, {45350, 45353}, {45378, 45379}, {45411, 45430}, {45517, 45589}, {45518, 45588}, {45527, 45625}, {45529, 45627}

X(45535) = X(45360)-of-1st Auriga triangle
X(45535) = {X(12452), X(45620)}-harmonic conjugate of X(45534)

X(45536) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 2nd AURIGA

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

X(45536) lies on these lines: {3, 45432}, {39, 44603}, {55, 45534}, {182, 12453}, {372, 5598}, {641, 5600}, {5062, 44602}, {5597, 45572}, {5602, 45508}, {7690, 35245}, {8187, 45530}, {8191, 45532}, {8203, 45544}, {8204, 45546}, {8205, 45550}, {8206, 45553}, {8207, 45554}, {8208, 45569}, {8209, 45566}, {9739, 11253}, {11208, 41490}, {11367, 45500}, {11385, 45502}, {11492, 45540}, {11493, 45520}, {11823, 45498}, {11838, 45504}, {11844, 45510}, {11862, 45538}, {11864, 45548}, {11866, 45556}, {11868, 45558}, {11870, 45560}, {11872, 45562}, {11874, 45570}, {11876, 45578}, {11878, 45580}, {11880, 45582}, {11882, 45584}, {11884, 45586}, {13891, 45574}, {13945, 45577}, {18497, 45542}, {18956, 45506}, {19009, 45512}, {19010, 45515}, {35779, 45565}, {45349, 45354}, {45377, 45380}, {45410, 45433}, {45516, 45590}, {45519, 45591}, {45526, 45626}, {45528, 45628}

X(45536) = X(45359)-of-these triangles: {1st anti-Kenmotu-free-vertices, 2nd Auriga}
X(45536) = {X(12453), X(45621)}-harmonic conjugate of X(45537)

X(45537) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 2nd AURIGA

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

X(45537) lies on these lines: {3, 45433}, {39, 44602}, {55, 45535}, {182, 12453}, {371, 5598}, {642, 5600}, {5058, 44603}, {5597, 45573}, {5602, 45509}, {7692, 35245}, {8187, 45531}, {8191, 45533}, {8203, 45545}, {8204, 45547}, {8205, 45552}, {8206, 45551}, {8207, 45555}, {8208, 45567}, {8209, 45568}, {9738, 11253}, {11208, 41491}, {11367, 45501}, {11385, 45503}, {11492, 45541}, {11493, 45521}, {11823, 45499}, {11838, 45505}, {11844, 45511}, {11862, 45539}, {11864, 45549}, {11866, 45557}, {11868, 45559}, {11870, 45561}, {11872, 45563}, {11874, 45571}, {11876, 45579}, {11878, 45581}, {11880, 45583}, {11882, 45585}, {11884, 45587}, {13891, 45576}, {13945, 45575}, {18497, 45543}, {18956, 45507}, {19009, 45514}, {19010, 45513}, {35780, 45564}, {45352, 45354}, {45378, 45380}, {45411, 45432}, {45517, 45591}, {45518, 45590}, {45527, 45626}, {45529, 45628}

X(45537) = X(45358)-of-2nd Auriga triangle
X(45537) = {X(12453), X(45621)}-harmonic conjugate of X(45536)

X(45538) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 5th BROCARD

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

X(45538) lies on these lines: {3, 6}, {641, 3096}, {2896, 45508}, {3099, 45530}, {5420, 33370}, {7811, 41490}, {9857, 45546}, {9862, 45510}, {9993, 45544}, {9996, 45554}, {9997, 45572}, {10038, 45580}, {10047, 45582}, {10357, 26314}, {10828, 45532}, {10871, 45556}, {10872, 45558}, {10873, 45560}, {10874, 45562}, {10875, 45569}, {10876, 45566}, {10877, 45570}, {10878, 45584}, {10879, 45586}, {11368, 45500}, {11386, 45502}, {11494, 45520}, {11861, 45534}, {11862, 45536}, {11885, 45548}, {13892, 45574}, {13946, 45577}, {18500, 45542}, {18503, 45377}, {18957, 45506}, {22744, 45540}, {26310, 45349}, {26311, 45351}, {26312, 45519}, {26313, 45516}, {26315, 45525}, {26317, 45526}, {26318, 45528}

X(45538) = X(35766)-of-these triangles: {1st anti-Kenmotu-free-vertices, 5th Brocard}
X(45538) = X(45538)-of-circumsymmedial triangle
X(45538) = {X(3094), X(26316)}-harmonic conjugate of X(45539)

X(45539) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 5th BROCARD

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

X(45539) lies on these lines: {3, 6}, {642, 3096}, {2896, 45509}, {3099, 45531}, {5418, 33371}, {7811, 41491}, {9857, 45547}, {9862, 45511}, {9993, 45545}, {9996, 45555}, {9997, 45573}, {10038, 45581}, {10047, 45583}, {10357, 26315}, {10828, 45533}, {10871, 45557}, {10872, 45559}, {10873, 45561}, {10874, 45563}, {10875, 45567}, {10876, 45568}, {10877, 45571}, {10878, 45585}, {10879, 45587}, {11368, 45501}, {11386, 45503}, {11494, 45521}, {11861, 45535}, {11862, 45537}, {11885, 45549}, {13892, 45576}, {13946, 45575}, {18500, 45543}, {18503, 45378}, {18957, 45507}, {22744, 45541}, {26310, 45352}, {26311, 45350}, {26312, 45517}, {26313, 45518}, {26314, 45524}, {26317, 45527}, {26318, 45529}

X(45539) = X(35767)-of-5th Brocard triangle
X(45539) = X(45539)-of-circumsymmedial triangle
X(45539) = {X(3094), X(26316)}-harmonic conjugate of X(45538)

X(45540) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 2nd CIRCUMPERP TANGENTIAL

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

X(45540) lies on these lines: {3, 45416}, {36, 45530}, {39, 44607}, {55, 45572}, {56, 372}, {104, 45510}, {182, 2810}, {641, 958}, {956, 45546}, {999, 45500}, {2975, 45508}, {3428, 45498}, {5062, 44606}, {7690, 35239}, {9739, 11249}, {10966, 45570}, {11194, 41490}, {11492, 45536}, {11493, 45534}, {12114, 45556}, {18761, 45542}, {19013, 45512}, {19014, 45515}, {22479, 45502}, {22520, 45504}, {22654, 45532}, {22744, 45538}, {22753, 45544}, {22755, 45548}, {22756, 45550}, {22757, 45553}, {22758, 45554}, {22759, 45560}, {22760, 45562}, {22761, 45569}, {22762, 45566}, {22763, 45574}, {22764, 45577}, {22765, 45578}, {22766, 45580}, {22767, 45582}, {22768, 45584}, {26319, 45349}, {26320, 45351}, {26321, 45377}, {26322, 45519}, {26323, 45516}, {26324, 45522}, {26325, 45525}, {35785, 45565}, {37535, 45410}

X(45540) = X(35768)-of-these triangles: {1st anti-Kenmotu-free-vertices, 2nd circumperp tangential}
X(45540) = {X(10269), X(22769)}-harmonic conjugate of X(45541)

X(45541) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 2nd CIRCUMPERP TANGENTIAL

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

X(45541) lies on these lines: {3, 45417}, {36, 45531}, {39, 44606}, {55, 45573}, {56, 371}, {104, 45511}, {182, 2810}, {642, 958}, {956, 45547}, {999, 45501}, {2975, 45509}, {3428, 45499}, {5058, 44607}, {7692, 35239}, {9738, 11249}, {10966, 45571}, {11194, 41491}, {11492, 45537}, {11493, 45535}, {12114, 45557}, {18761, 45543}, {19013, 45514}, {19014, 45513}, {22479, 45503}, {22520, 45505}, {22654, 45533}, {22744, 45539}, {22753, 45545}, {22755, 45549}, {22756, 45552}, {22757, 45551}, {22758, 45555}, {22759, 45561}, {22760, 45563}, {22761, 45567}, {22762, 45568}, {22763, 45576}, {22764, 45575}, {22765, 45579}, {22766, 45581}, {22767, 45583}, {22768, 45585}, {26319, 45352}, {26320, 45350}, {26321, 45378}, {26322, 45517}, {26323, 45518}, {26324, 45524}, {26325, 45523}, {35784, 45564}, {37535, 45411}

X(45541) = X(35769)-of-2nd circumperp tangential triangle
X(45541) = {X(10269), X(22769)}-harmonic conjugate of X(45540)

X(45542) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND EHRMANN-MID

Barycentrics    2*S*((b^2+c^2)*a^2-(b^2-c^2)^2)+a^6+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2) : :
X(45542) = 3*X(381)-X(23251) = X(22625)-3*X(22806) = 6*X(22806)-X(22819)

X(45542) lies on these lines: {3, 45438}, {4, 9739}, {5, 182}, {30, 641}, {39, 6565}, {114, 489}, {371, 45375}, {372, 381}, {382, 45498}, {485, 542}, {486, 19130}, {511, 6289}, {538, 6311}, {546, 22820}, {639, 3098}, {640, 11178}, {1478, 45562}, {1479, 45560}, {1656, 18511}, {2043, 22796}, {2044, 22797}, {3091, 45510}, {3146, 45522}, {3367, 37333}, {3392, 37332}, {3583, 45570}, {3585, 45506}, {3788, 7692}, {3843, 45578}, {3845, 41490}, {3850, 22596}, {3851, 45410}, {3854, 45525}, {5062, 6564}, {5072, 18509}, {5092, 11313}, {5475, 18993}, {5476, 7584}, {5965, 6278}, {6000, 9823}, {6214, 34507}, {6290, 18553}, {6566, 35820}, {7374, 26468}, {8976, 26346}, {9737, 36709}, {9818, 45532}, {9955, 45500}, {10515, 24206}, {10895, 45580}, {10896, 45582}, {11550, 15234}, {11737, 22807}, {12699, 45546}, {13665, 45515}, {13748, 43120}, {13785, 45512}, {18358, 23312}, {18440, 42265}, {18491, 45520}, {18492, 45530}, {18495, 45534}, {18497, 45536}, {18500, 45538}, {18502, 45504}, {18507, 45548}, {18516, 45556}, {18517, 45558}, {18520, 45569}, {18522, 45566}, {18525, 45572}, {18538, 45574}, {18539, 42258}, {18542, 45584}, {18544, 45586}, {18761, 45540}, {18762, 45577}, {23514, 45407}, {35787, 45565}, {45349, 45355}, {45351, 45356}, {45516, 45592}, {45519, 45593}, {45526, 45630}, {45528, 45631}

X(45542) = midpoint of X(6289) and X(36656)
X(45542) = reflection of X(i) in X(j) for these (i, j): (7690, 641), (22819, 22625)
X(45542) = X(23251)-of-these triangles: {1st anti-Kenmotu-free-vertices, Ehrmann-mid}
X(45542) = X(35684)-of-outer-Vecten triangle
X(45542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 45554, 9739), (5, 3818, 45543), (381, 45377, 372)

X(45543) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND EHRMANN-MID

Barycentrics    -2*S*((b^2+c^2)*a^2-(b^2-c^2)^2)+a^6+2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2) : :
X(45543) = 3*X(381)-X(23261) = X(22596)-3*X(22807) = 6*X(22807)-X(22820)

X(45543) lies on these lines: {3, 45439}, {4, 9738}, {5, 182}, {30, 642}, {39, 6564}, {114, 490}, {371, 381}, {372, 45376}, {382, 45499}, {485, 19130}, {486, 542}, {511, 6290}, {538, 6315}, {546, 22819}, {639, 11178}, {640, 3098}, {1478, 45563}, {1479, 45561}, {1656, 18509}, {2043, 22797}, {2044, 22796}, {3091, 45511}, {3146, 45523}, {3366, 37333}, {3391, 37332}, {3583, 45571}, {3585, 45507}, {3788, 7690}, {3843, 45579}, {3845, 41491}, {3850, 22625}, {3851, 45411}, {3854, 45524}, {5058, 6565}, {5072, 18511}, {5092, 11314}, {5475, 18994}, {5476, 7583}, {5965, 6281}, {6000, 9824}, {6215, 34507}, {6289, 18553}, {6567, 35821}, {7000, 26469}, {9737, 36714}, {9818, 45533}, {9955, 45501}, {9994, 31481}, {10514, 24206}, {10895, 45581}, {10896, 45583}, {11550, 15233}, {11737, 22806}, {12699, 45547}, {13665, 45513}, {13749, 43121}, {13785, 45514}, {13951, 26336}, {18358, 23311}, {18440, 42262}, {18491, 45521}, {18492, 45531}, {18495, 45535}, {18497, 45537}, {18500, 45539}, {18502, 45505}, {18507, 45549}, {18516, 45557}, {18517, 45559}, {18520, 45567}, {18522, 45568}, {18525, 45573}, {18538, 45576}, {18542, 45585}, {18544, 45587}, {18761, 45541}, {18762, 45575}, {23514, 45406}, {26438, 42259}, {35786, 45564}, {45350, 45356}, {45352, 45355}, {45517, 45593}, {45518, 45592}, {45527, 45630}, {45529, 45631}

X(45543) = midpoint of X(6290) and X(36655)
X(45543) = reflection of X(i) in X(j) for these (i, j): (7692, 642), (22820, 22596)
X(45543) = X(23261)-of-Ehrmann-mid triangle
X(45543) = X(35684)-of-inner-Vecten triangle
X(45543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 45555, 9738), (5, 3818, 45542), (381, 45378, 371)

X(45544) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND EULER

Barycentrics    2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*S+3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2) : :
X(45544) = X(6561)-3*X(14853)

X(45544) lies on these lines: {2, 45106}, {3, 45440}, {4, 372}, {5, 641}, {11, 45506}, {12, 45570}, {20, 6201}, {30, 182}, {39, 3070}, {98, 45504}, {140, 7690}, {235, 45502}, {371, 45574}, {381, 13687}, {382, 45410}, {485, 45101}, {488, 10515}, {515, 45500}, {546, 22820}, {615, 6566}, {639, 7784}, {640, 7795}, {642, 9737}, {1152, 36656}, {1327, 14494}, {1351, 32419}, {1352, 32421}, {1478, 45582}, {1479, 45580}, {1503, 42216}, {1587, 45512}, {1588, 45515}, {1598, 45532}, {1699, 45530}, {3071, 5062}, {3090, 26330}, {3091, 45508}, {3146, 6202}, {3156, 8968}, {3312, 13748}, {3788, 23312}, {3843, 45377}, {5254, 18993}, {5587, 45546}, {5603, 45572}, {6281, 12222}, {6289, 12314}, {6290, 12602}, {6396, 6811}, {6419, 26441}, {6420, 45406}, {6561, 14853}, {6564, 6813}, {7000, 23249}, {7389, 11825}, {7584, 14233}, {7684, 15765}, {7685, 18585}, {7761, 42859}, {7803, 45552}, {7825, 23311}, {8196, 45534}, {8203, 45536}, {8212, 45569}, {8213, 45566}, {8550, 44474}, {9993, 45538}, {10514, 12323}, {10531, 45584}, {10532, 45586}, {10893, 45556}, {10894, 45558}, {10895, 45560}, {10896, 45562}, {11292, 12124}, {11293, 11824}, {11313, 12305}, {11496, 45520}, {11897, 45548}, {13749, 36712}, {13807, 15687}, {14230, 36657}, {21736, 42261}, {22716, 37343}, {22753, 45540}, {23251, 36655}, {26326, 45349}, {26327, 45351}, {26328, 45519}, {26329, 45516}, {26332, 45526}, {26333, 45528}, {32788, 39838}, {35822, 45511}, {35948, 45499}, {36709, 42259}, {42215, 44656}, {42284, 43457}

X(45544) = midpoint of X(4) and X(6560)
X(45544) = reflection of X(45545) in X(5480)
X(45544) = X(6560)-of-these triangles: {1st anti-Kenmotu-free-vertices, Euler}
X(45544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 9739, 641), (381, 45578, 45554), (45554, 45578, 41490)

X(45545) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND EULER

Barycentrics    -2*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*S+3*(b^2+c^2)*a^4-2*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2) : :
X(45545) = X(6560)-3*X(14853)

X(45545) lies on these lines: {2, 45107}, {3, 45441}, {4, 371}, {5, 642}, {11, 45507}, {12, 45571}, {20, 6202}, {30, 182}, {39, 3071}, {98, 45505}, {136, 8968}, {140, 7692}, {235, 45503}, {372, 45575}, {381, 13807}, {382, 45411}, {486, 21736}, {487, 10514}, {515, 45501}, {546, 22819}, {590, 6567}, {639, 7795}, {640, 7784}, {641, 9737}, {1151, 36655}, {1328, 14494}, {1351, 32421}, {1352, 32419}, {1478, 45583}, {1479, 45581}, {1503, 42215}, {1587, 45514}, {1588, 45513}, {1598, 45533}, {1699, 45531}, {3070, 5058}, {3090, 26331}, {3091, 45509}, {3146, 6201}, {3311, 13749}, {3788, 23311}, {3843, 45378}, {5254, 18994}, {5587, 45547}, {5603, 45573}, {6200, 6813}, {6278, 12221}, {6289, 12601}, {6290, 12313}, {6419, 45407}, {6420, 8982}, {6560, 14853}, {6565, 6811}, {7000, 9541}, {7374, 23259}, {7388, 11824}, {7583, 14230}, {7684, 18585}, {7685, 15765}, {7761, 42858}, {7803, 45553}, {7825, 23312}, {8196, 45535}, {8203, 45537}, {8212, 45567}, {8213, 45568}, {8550, 44473}, {9993, 45539}, {10515, 12322}, {10531, 45585}, {10532, 45587}, {10893, 45557}, {10894, 45559}, {10895, 45561}, {10896, 45563}, {11291, 12123}, {11294, 11825}, {11314, 12306}, {11496, 45521}, {11897, 45549}, {13687, 15687}, {13748, 36711}, {14233, 36658}, {22718, 37342}, {22753, 45541}, {23261, 36656}, {26326, 45352}, {26327, 45350}, {26328, 45517}, {26329, 45518}, {26332, 45527}, {26333, 45529}, {32787, 39838}, {35823, 45510}, {35949, 45498}, {36714, 42258}, {42216, 44657}, {42283, 43457}

X(45545) = midpoint of X(4) and X(6561)
X(45545) = reflection of X(45544) in X(5480)
X(45545) = X(6561)-of-Euler triangle
X(45545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 9738, 642), (381, 45579, 45555), (45555, 45579, 41491)

X(45546) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND OUTER-GARCIA

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

X(45546) lies on these lines: {1, 641}, {2, 45500}, {3, 45444}, {8, 45508}, {10, 372}, {39, 13973}, {65, 45560}, {72, 45558}, {182, 3416}, {355, 9739}, {515, 45498}, {517, 45554}, {519, 45572}, {944, 26444}, {956, 45540}, {1737, 45582}, {1837, 45570}, {3057, 45562}, {3679, 41490}, {4769, 43121}, {5062, 13911}, {5090, 45502}, {5252, 45506}, {5587, 45544}, {5657, 45510}, {5687, 45520}, {5688, 6684}, {5689, 45550}, {5790, 45578}, {6734, 45526}, {6735, 45528}, {7690, 18481}, {8193, 45532}, {8197, 45534}, {8204, 45536}, {8214, 45569}, {8215, 45566}, {9857, 45538}, {10039, 45580}, {10791, 45504}, {10914, 45556}, {10915, 45584}, {10916, 45586}, {11315, 45398}, {11900, 45548}, {12699, 45542}, {12702, 45377}, {13688, 31162}, {13883, 45515}, {13893, 45574}, {13936, 45512}, {13947, 45577}, {26382, 45349}, {26406, 45351}, {26442, 45519}, {26443, 45516}, {26445, 45525}, {35789, 45565}, {45410, 45445}

X(45546) = reflection of X(35788) in X(10)
X(45546) = X(8)-Beth conjugate of-X(35788)
X(45546) = X(35788)-of-these triangles: {1st anti-Kenmotu-free-vertices, outer-Garcia}
X(45546) = {X(3416), X(26446)}-harmonic conjugate of X(45547)

X(45547) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND OUTER-GARCIA

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

X(45547) lies on these lines: {1, 642}, {2, 45501}, {3, 45445}, {8, 45509}, {10, 371}, {39, 13911}, {65, 45561}, {72, 45559}, {182, 3416}, {355, 9738}, {515, 45499}, {517, 45555}, {519, 45573}, {944, 26445}, {956, 45541}, {1737, 45583}, {1837, 45571}, {3057, 45563}, {3679, 41491}, {4769, 43120}, {5058, 13973}, {5090, 45503}, {5252, 45507}, {5587, 45545}, {5657, 45511}, {5687, 45521}, {5688, 45551}, {5689, 6684}, {5790, 45579}, {6734, 45527}, {6735, 45529}, {7692, 18481}, {8193, 45533}, {8197, 45535}, {8204, 45537}, {8214, 45567}, {8215, 45568}, {9857, 45539}, {10039, 45581}, {10791, 45505}, {10914, 45557}, {10915, 45585}, {10916, 45587}, {11316, 45399}, {11900, 45549}, {12699, 45543}, {12702, 45378}, {13808, 31162}, {13883, 45513}, {13893, 45576}, {13936, 45514}, {13947, 45575}, {26382, 45352}, {26406, 45350}, {26442, 45517}, {26443, 45518}, {26444, 45524}, {35788, 45564}, {45411, 45444}

X(45547) = reflection of X(35789) in X(10)
X(45547) = X(8)-Beth conjugate of-X(35789)
X(45547) = X(35789)-of-outer-Garcia triangle
X(45547) = {X(3416), X(26446)}-harmonic conjugate of X(45546)

X(45548) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND GOSSARD

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(-2*a^12+5*(b^2+c^2)*a^10+(b^4-16*b^2*c^2+c^4)*a^8-(b^2+c^2)*(14*b^4-31*b^2*c^2+14*c^4)*a^6+(16*b^4+25*b^2*c^2+16*c^4)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*(7*b^4+5*b^2*c^2+7*c^4)*a^2+2*((b^2+c^2)*a^8-4*b^2*c^2*a^6-(b^2+c^2)*(4*(b^2-c^2)^2-b^2*c^2)*a^4+4*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4))*S+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4) : :

X(45548) lies on these lines: {3, 45446}, {30, 45498}, {39, 44611}, {182, 12583}, {372, 402}, {641, 1650}, {1651, 41490}, {4240, 45508}, {5062, 44610}, {7690, 35241}, {9739, 11251}, {11831, 45500}, {11832, 45502}, {11839, 45504}, {11845, 45510}, {11848, 45520}, {11852, 45530}, {11853, 45532}, {11863, 45534}, {11864, 45536}, {11885, 45538}, {11897, 45544}, {11900, 45546}, {11901, 45550}, {11902, 45553}, {11903, 45556}, {11904, 45558}, {11905, 45560}, {11906, 45562}, {11907, 45569}, {11908, 45566}, {11909, 45570}, {11910, 45572}, {11911, 45578}, {11912, 45580}, {11913, 45582}, {11914, 45584}, {11915, 45586}, {13894, 45574}, {13948, 45577}, {18507, 45542}, {18508, 45377}, {18958, 45506}, {19017, 45512}, {19018, 45515}, {22755, 45540}, {26383, 45349}, {26407, 45351}, {26447, 45519}, {26448, 45516}, {26449, 45522}, {26450, 45525}, {26452, 45526}, {26453, 45528}, {35791, 45565}, {45410, 45447}

X(45548) = reflection of X(35790) in X(402)
X(45548) = X(35790)-of-these triangles: {1st anti-Kenmotu-free-vertices, Gossard}
X(45548) = {X(12583), X(26451)}-harmonic conjugate of X(45549)

X(45549) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND GOSSARD

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(-2*a^12+5*(b^2+c^2)*a^10+(b^4-16*b^2*c^2+c^4)*a^8-(b^2+c^2)*(14*b^4-31*b^2*c^2+14*c^4)*a^6+(16*b^4+25*b^2*c^2+16*c^4)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*(7*b^4+5*b^2*c^2+7*c^4)*a^2-2*((b^2+c^2)*a^8-4*b^2*c^2*a^6-(b^2+c^2)*(4*(b^2-c^2)^2-b^2*c^2)*a^4+4*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4))*S+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4) : :

X(45549) lies on these lines: {3, 45447}, {30, 45499}, {39, 44610}, {182, 12583}, {371, 402}, {642, 1650}, {1651, 41491}, {4240, 45509}, {5058, 44611}, {7692, 35241}, {9738, 11251}, {11831, 45501}, {11832, 45503}, {11839, 45505}, {11845, 45511}, {11848, 45521}, {11852, 45531}, {11853, 45533}, {11863, 45535}, {11864, 45537}, {11885, 45539}, {11897, 45545}, {11900, 45547}, {11901, 45552}, {11902, 45551}, {11903, 45557}, {11904, 45559}, {11905, 45561}, {11906, 45563}, {11907, 45567}, {11908, 45568}, {11909, 45571}, {11910, 45573}, {11911, 45579}, {11912, 45581}, {11913, 45583}, {11914, 45585}, {11915, 45587}, {13894, 45576}, {13948, 45575}, {18507, 45543}, {18508, 45378}, {18958, 45507}, {19017, 45514}, {19018, 45513}, {22755, 45541}, {26383, 45352}, {26407, 45350}, {26447, 45517}, {26448, 45518}, {26449, 45524}, {26450, 45523}, {26452, 45527}, {26453, 45529}, {35790, 45564}, {45411, 45446}

X(45549) = reflection of X(35791) in X(402)
X(45549) = X(35791)-of-Gossard triangle
X(45549) = {X(12583), X(26451)}-harmonic conjugate of X(45548)

X(45550) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND INNER-GREBE

Barycentrics    a^2*(2*(-a^2+b^2+c^2)*S+5*a^4-6*(b^2+c^2)*a^2+b^4-10*b^2*c^2+c^4) : :
X(45550) = 2*X(3)+3*X(35770)

X(45550) lies on these lines: {2, 6281}, {3, 6}, {487, 33748}, {597, 36709}, {631, 5861}, {632, 5875}, {641, 3525}, {1271, 10303}, {3090, 10514}, {3091, 5871}, {3146, 6202}, {3545, 13810}, {3628, 6215}, {3746, 10048}, {3788, 15834}, {5056, 13690}, {5072, 18509}, {5079, 26336}, {5563, 10040}, {5589, 30389}, {5595, 45532}, {5605, 15178}, {5689, 45546}, {6278, 7375}, {7725, 15021}, {7732, 15034}, {7786, 22699}, {7829, 36656}, {7982, 11370}, {8198, 45534}, {8205, 45536}, {8216, 45569}, {8217, 45566}, {8974, 45574}, {10517, 26339}, {10919, 45556}, {10921, 45558}, {10923, 45560}, {10925, 45562}, {10927, 45570}, {10929, 45584}, {10931, 45586}, {11179, 37342}, {11388, 45502}, {11497, 45520}, {11901, 45548}, {12803, 15027}, {13949, 45577}, {18959, 45506}, {19053, 36703}, {22623, 33347}, {22756, 45540}, {26334, 45349}, {26335, 45351}, {26337, 45519}, {26342, 34486}, {26343, 45528}, {36712, 43273}, {45516, 45594}

X(45550) = X(35770)-of-these triangles: {1st anti-Kenmotu-free-vertices, inner-Grebe}
X(45550) = X(45550)-of-circumsymmedial triangle
X(45550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 575, 6419), (3, 6428, 11477), (6, 26341, 45552), (6, 45552, 11824), (61, 62, 45513), (182, 372, 45553), (182, 45410, 372), (372, 45553, 45498), (631, 45525, 41490), (3312, 5085, 11825), (3594, 10541, 3), (5050, 43118, 371), (43121, 45411, 6200)

X(45551) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND OUTER-GREBE

Barycentrics    a^2*(-2*(-a^2+b^2+c^2)*S+5*a^4-6*(b^2+c^2)*a^2+b^4-10*b^2*c^2+c^4) : :
X(45551) = 2*X(3)+3*X(35771)

X(45551) lies on these lines: {2, 6278}, {3, 6}, {488, 33748}, {597, 36714}, {631, 5860}, {632, 5874}, {642, 3525}, {1270, 10303}, {3090, 10515}, {3091, 5870}, {3146, 6201}, {3545, 13691}, {3628, 6214}, {3746, 10049}, {3788, 15835}, {5056, 13811}, {5072, 18511}, {5079, 26346}, {5563, 10041}, {5588, 30389}, {5594, 45533}, {5604, 15178}, {5688, 45547}, {6281, 7376}, {7726, 15021}, {7733, 15034}, {7786, 22700}, {7829, 36655}, {7982, 11371}, {8199, 45535}, {8206, 45537}, {8218, 45567}, {8219, 45568}, {8975, 45576}, {9681, 35944}, {10518, 26340}, {10920, 45557}, {10922, 45559}, {10924, 45561}, {10926, 45563}, {10928, 45571}, {10930, 45585}, {10932, 45587}, {11179, 37343}, {11389, 45503}, {11498, 45521}, {11902, 45549}, {12804, 15027}, {13950, 45575}, {18960, 45507}, {19054, 36701}, {22594, 33346}, {22757, 45541}, {26338, 45518}, {26344, 45352}, {26345, 45350}, {26347, 45517}, {26349, 34486}, {26350, 45529}, {36711, 43273}

X(45551) = X(35771)-of-outer-Grebe triangle
X(45551) = X(45551)-of-circumsymmedial triangle
X(45551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 575, 6420), (3, 6427, 11477), (6, 26348, 45553), (6, 45553, 11825), (61, 62, 45512), (182, 371, 45552), (182, 45411, 371), (371, 45552, 45499), (631, 45524, 41491), (3311, 5085, 11824), (3592, 10541, 3), (5050, 43119, 372), (43120, 45410, 6396)

X(45552) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND INNER-GREBE

Barycentrics    a^2*(3*a^4-b^4-6*b^2*c^2-c^4-2*(b^2+c^2)*a^2+2*(-a^2+b^2+c^2)*S) : :
X(45552) = 2*X(3)+X(6420) = 3*X(3)-2*X(43126) = 5*X(3)+2*X(43145) = 3*X(6420)+4*X(43126) = 5*X(6420)-4*X(43145) = 5*X(43126)+3*X(43145) = 2*X(43126)+3*X(45410) = 2*X(43145)-5*X(45410)

X(45552) lies on these lines: {2, 5871}, {3, 6}, {20, 6202}, {35, 10048}, {36, 10040}, {40, 11370}, {140, 6215}, {165, 12697}, {486, 21737}, {488, 1271}, {549, 5875}, {620, 15834}, {631, 642}, {639, 45510}, {640, 45407}, {641, 9744}, {1385, 5605}, {1584, 3796}, {1587, 8974}, {1588, 13949}, {1656, 18509}, {3069, 36703}, {3155, 43650}, {3156, 22352}, {3515, 11388}, {3524, 5861}, {3526, 26336}, {3576, 3641}, {3589, 36709}, {5012, 5409}, {5204, 18959}, {5217, 10927}, {5408, 7485}, {5422, 13616}, {5432, 10923}, {5433, 10925}, {5589, 7987}, {5595, 45533}, {5657, 12627}, {5689, 6684}, {6227, 34473}, {6267, 10606}, {6270, 21156}, {6271, 21157}, {6273, 22712}, {6275, 6312}, {6319, 21166}, {6459, 12123}, {6460, 36701}, {7375, 10515}, {7484, 10133}, {7725, 15055}, {7732, 15035}, {7803, 45544}, {7834, 36656}, {8198, 45535}, {8205, 45537}, {8216, 45567}, {8217, 45568}, {8311, 33997}, {10267, 10931}, {10269, 10929}, {10299, 26339}, {10846, 15271}, {10902, 26342}, {10919, 45557}, {10921, 45559}, {11257, 22718}, {11291, 25406}, {11314, 13749}, {11497, 45521}, {11901, 45549}, {12753, 38693}, {12803, 15061}, {12805, 38717}, {13269, 34474}, {13282, 38699}, {13690, 15702}, {13786, 43257}, {22716, 33347}, {22756, 45541}, {26334, 45352}, {26335, 45350}, {26337, 45517}, {26343, 37561}, {35944, 42260}, {36714, 44882}, {45518, 45594}

X(45552) = midpoint of X(3) and X(45410)
X(45552) = reflection of X(6420) in X(45410)
X(45552) = inverse of X(11824) in Brocard circle
X(45552) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(11824)}} and {{A, B, C, X(4), X(45513)}}
X(45552) = X(6420)-of-inner-Grebe triangle
X(45552) = X(45410)-of-anti-X3-ABC reflections triangle
X(45552) = X(45552)-of-circumsymmedial triangle
X(45552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 5871, 10514), (3, 6, 11824), (3, 182, 371), (3, 372, 11825), (3, 3312, 1350), (3, 5050, 9732), (3, 26341, 6), (3, 26348, 1151), (3, 43118, 372), (3, 43119, 6200), (3, 43121, 6396), (3, 45411, 9738), (3, 45488, 3098), (3, 45579, 7692), (6, 1151, 8396), (372, 6200, 3102), (1151, 5058, 371), (1151, 8402, 6443), (1151, 8404, 8375), (1151, 8406, 6421), (1152, 6423, 372), (3594, 31884, 1160), (5050, 9732, 6419), (5092, 43121, 3), (6421, 8406, 372), (9738, 44471, 1161), (9738, 45411, 371), (9739, 17508, 3), (11824, 45550, 6), (45499, 45551, 371)

X(45553) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND OUTER-GREBE

Barycentrics    a^2*(3*a^4-b^4-6*b^2*c^2-c^4-2*(b^2+c^2)*a^2-2*(-a^2+b^2+c^2)*S) : :
X(45553) = 2*X(3)+X(6419) = 3*X(3)-2*X(43127) = 5*X(3)+2*X(43143) = 3*X(6419)+4*X(43127) = 5*X(6419)-4*X(43143) = 5*X(43127)+3*X(43143) = 2*X(43127)+3*X(45411) = 2*X(43143)-5*X(45411)

X(45553) lies on these lines: {2, 5870}, {3, 6}, {20, 6201}, {35, 10049}, {36, 10041}, {40, 11371}, {140, 6214}, {165, 12698}, {487, 1270}, {549, 5874}, {620, 15835}, {631, 641}, {639, 45406}, {640, 45511}, {642, 9744}, {1385, 5604}, {1583, 3796}, {1587, 8975}, {1588, 13950}, {1656, 18511}, {3068, 36701}, {3155, 22352}, {3156, 43650}, {3515, 11389}, {3524, 5860}, {3526, 26346}, {3576, 3640}, {3589, 36714}, {5012, 5408}, {5204, 18960}, {5217, 10928}, {5409, 7485}, {5422, 13617}, {5432, 10924}, {5433, 10926}, {5481, 26922}, {5588, 7987}, {5594, 45532}, {5657, 12628}, {5688, 6684}, {6226, 34473}, {6266, 10606}, {6268, 21156}, {6269, 21157}, {6272, 22712}, {6274, 6316}, {6320, 21166}, {6459, 36703}, {6460, 12124}, {6561, 21737}, {7376, 10514}, {7484, 10132}, {7726, 15055}, {7733, 15035}, {7803, 45545}, {7834, 36655}, {8199, 45534}, {8206, 45536}, {8218, 45569}, {8219, 45566}, {8310, 33997}, {9541, 12123}, {10267, 10932}, {10269, 10930}, {10299, 26340}, {10845, 15271}, {10902, 26349}, {10920, 45556}, {10922, 45558}, {11257, 22716}, {11292, 25406}, {11313, 13748}, {11498, 45520}, {11902, 45548}, {12754, 38693}, {12804, 15061}, {12806, 38717}, {13270, 34474}, {13283, 38699}, {13666, 43256}, {13811, 15702}, {22718, 33346}, {22757, 45540}, {26338, 45516}, {26344, 45349}, {26345, 45351}, {26347, 45519}, {26350, 37561}, {34112, 38690}, {35945, 42261}, {36709, 44882}

X(45553) = midpoint of X(3) and X(45411)
X(45553) = reflection of X(6419) in X(45411)
X(45553) = inverse of X(11825) in Brocard circle
X(45553) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(11825)}} and {{A, B, C, X(4), X(45512)}}
X(45553) = X(6419)-of-these triangles: {1st anti-Kenmotu-free-vertices, outer-Grebe}
X(45553) = X(45411)-of-anti-X3-ABC reflections triangle
X(45553) = X(45553)-of-circumsymmedial triangle
X(45553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 5870, 10515), (3, 6, 11825), (3, 182, 372), (3, 371, 11824), (3, 3311, 1350), (3, 5050, 9733), (3, 26341, 1152), (3, 26348, 6), (3, 43118, 6396), (3, 43119, 371), (3, 43120, 6200), (3, 45410, 9739), (3, 45489, 3098), (3, 45578, 7690), (6, 1152, 8416), (371, 6396, 3103), (1151, 6424, 371), (1152, 5062, 372), (1152, 8410, 6444), (1152, 8412, 8376), (1152, 8414, 6422), (3592, 31884, 1161), (5050, 9733, 6420), (5092, 43120, 3), (6422, 8414, 371), (9738, 17508, 3), (9739, 44472, 1160), (9739, 45410, 372), (11825, 45551, 6), (45498, 45550, 372)

X(45554) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND JOHNSON

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

X(45554) lies on these lines: {1, 45560}, {2, 98}, {3, 639}, {4, 9739}, {5, 372}, {11, 45582}, {12, 45580}, {20, 7690}, {30, 45498}, {39, 486}, {119, 45528}, {140, 6214}, {194, 33372}, {355, 45556}, {381, 13687}, {485, 5062}, {489, 43144}, {491, 34507}, {492, 511}, {517, 45546}, {590, 3564}, {591, 1351}, {637, 7763}, {638, 32832}, {952, 45572}, {1270, 44472}, {1353, 32787}, {1478, 45506}, {1479, 45570}, {1586, 39530}, {1656, 6119}, {1991, 11898}, {2043, 20428}, {2044, 20429}, {2045, 37824}, {2046, 37825}, {2047, 37823}, {2548, 18993}, {3068, 44656}, {3069, 14561}, {3312, 45487}, {3317, 36664}, {3593, 7374}, {3628, 6215}, {3818, 6813}, {5055, 13783}, {5056, 26469}, {5097, 45421}, {5406, 32588}, {5418, 6278}, {5420, 10515}, {5476, 13757}, {5587, 45530}, {5874, 8981}, {5886, 45500}, {5907, 6809}, {6248, 7388}, {6560, 6566}, {6565, 45565}, {6814, 44870}, {7389, 43121}, {7583, 45515}, {7584, 45512}, {7586, 44474}, {8200, 45534}, {8207, 45536}, {8220, 45569}, {8221, 45566}, {8252, 10516}, {8253, 15069}, {8976, 45574}, {8982, 32489}, {9733, 36656}, {9737, 21736}, {9758, 22727}, {9996, 45538}, {10356, 13935}, {10796, 45504}, {10942, 45584}, {10943, 45586}, {11293, 12975}, {11313, 43118}, {11315, 43119}, {11499, 45520}, {12314, 45440}, {12974, 26441}, {13812, 15694}, {13951, 45577}, {16626, 42282}, {16627, 35732}, {18358, 32790}, {18424, 42274}, {18583, 32788}, {18906, 33340}, {19146, 37071}, {22758, 45540}, {26386, 45349}, {26410, 45351}, {26466, 45519}, {26467, 45516}, {26470, 45526}, {32419, 45579}, {35840, 44392}, {39387, 43120}, {42262, 44518}

X(45554) = midpoint of X(i) and X(j) for these {i, j}: {3, 45375}, {492, 6811}
X(45554) = reflection of X(6564) in X(5)
X(45554) = complement of X(45511)
X(45554) = X(492)-of-McCay triangle
X(45554) = X(6564)-of-these triangles: {1st anti-Kenmotu-free-vertices, Johnson}
X(45554) = X(6811)-of-1st Brocard triangle
X(45554) = X(45375)-of-anti-X3-ABC reflections triangle
X(45554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 1352, 45555), (2, 45510, 182), (4, 45508, 9739), (20, 45522, 7690), (114, 182, 45555), (381, 45578, 45544), (5420, 10515, 37342), (9739, 45542, 4), (12177, 43461, 45555), (39387, 45406, 43120), (41490, 45544, 45578), (45560, 45562, 1)

X(45555) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND JOHNSON

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

X(45555) lies on these lines: {1, 45561}, {2, 98}, {3, 640}, {4, 9738}, {5, 371}, {11, 45583}, {12, 45581}, {20, 7692}, {30, 45499}, {39, 485}, {119, 45529}, {140, 6215}, {194, 33373}, {355, 45557}, {381, 13807}, {486, 5058}, {490, 43141}, {491, 511}, {492, 34507}, {517, 45547}, {591, 11898}, {615, 3564}, {623, 35742}, {637, 32832}, {638, 7763}, {952, 45573}, {1271, 44471}, {1351, 1991}, {1353, 32788}, {1478, 45507}, {1479, 45571}, {1585, 39530}, {1656, 6118}, {2043, 20429}, {2044, 20428}, {2045, 37825}, {2046, 37824}, {2548, 18994}, {3068, 14561}, {3069, 44657}, {3311, 45486}, {3316, 36665}, {3595, 7000}, {3628, 6214}, {3818, 6811}, {5055, 13663}, {5056, 26468}, {5097, 45420}, {5407, 32587}, {5418, 10514}, {5420, 6281}, {5476, 13637}, {5587, 45531}, {5875, 13966}, {5886, 45501}, {5907, 6810}, {6248, 7389}, {6561, 6567}, {6564, 45564}, {6812, 44870}, {7388, 43120}, {7583, 45513}, {7584, 45514}, {7585, 44473}, {8200, 45535}, {8207, 45537}, {8220, 45567}, {8221, 45568}, {8252, 15069}, {8253, 10516}, {8976, 45576}, {8982, 12975}, {9540, 10356}, {9732, 36655}, {9757, 22726}, {9996, 45539}, {10796, 45505}, {10942, 45585}, {10943, 45587}, {11294, 12974}, {11314, 43119}, {11316, 43118}, {11499, 45521}, {12313, 45441}, {13692, 15694}, {13951, 45575}, {16626, 35732}, {16627, 42282}, {18358, 32789}, {18424, 42277}, {18583, 32787}, {18906, 33341}, {19145, 37071}, {22758, 45541}, {26386, 45352}, {26410, 45350}, {26441, 32488}, {26466, 45517}, {26467, 45518}, {26470, 45527}, {32421, 45578}, {32807, 43150}, {35841, 44394}, {39388, 43121}, {42265, 44518}

X(45555) = midpoint of X(i) and X(j) for these {i, j}: {3, 45376}, {491, 6813}
X(45555) = reflection of X(6565) in X(5)
X(45555) = complement of X(45510)
X(45555) = X(6565)-of-Johnson triangle
X(45555) = X(45376)-of-anti-X3-ABC reflections triangle
X(45555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 1352, 45554), (2, 45511, 182), (4, 45509, 9738), (20, 45523, 7692), (114, 182, 45554), (381, 45579, 45545), (5418, 10514, 37343), (9738, 45543, 4), (12177, 43461, 45554), (39388, 45407, 43121), (41491, 45545, 45579), (45561, 45563, 1)

X(45556) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND INNER-JOHNSON

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

X(45556) lies on these lines: {3, 45454}, {5, 45528}, {11, 372}, {12, 45584}, {39, 44619}, {182, 12586}, {355, 45554}, {641, 1376}, {3434, 45508}, {5062, 44618}, {7690, 35249}, {9739, 10525}, {10523, 45580}, {10785, 45510}, {10794, 45504}, {10826, 45530}, {10829, 45532}, {10871, 45538}, {10893, 45544}, {10914, 45546}, {10919, 45550}, {10920, 45553}, {10943, 45526}, {10944, 45560}, {10945, 45569}, {10946, 45566}, {10947, 45570}, {10948, 45582}, {10949, 45586}, {11235, 41490}, {11373, 45500}, {11390, 45502}, {11826, 45498}, {11865, 45534}, {11866, 45536}, {11903, 45548}, {11928, 45578}, {12114, 45540}, {13895, 45574}, {13952, 45577}, {18516, 45542}, {18519, 45377}, {18961, 45506}, {19023, 45512}, {19024, 45515}, {26390, 45349}, {26414, 45351}, {26488, 45519}, {26489, 45516}, {26490, 45522}, {26491, 45525}, {35797, 45565}, {45410, 45455}

X(45556) = X(35802)-of-these triangles: {1st anti-Kenmotu-free-vertices, inner-Johnson}
X(45556) = {X(12586), X(26492)}-harmonic conjugate of X(45557)

X(45557) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND INNER-JOHNSON

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

X(45557) lies on these lines: {3, 45455}, {5, 45529}, {11, 371}, {12, 45585}, {39, 44618}, {182, 12586}, {355, 45555}, {642, 1376}, {3434, 45509}, {5058, 44619}, {7692, 35249}, {9738, 10525}, {10523, 45581}, {10785, 45511}, {10794, 45505}, {10826, 45531}, {10829, 45533}, {10871, 45539}, {10893, 45545}, {10914, 45547}, {10919, 45552}, {10920, 45551}, {10943, 45527}, {10944, 45561}, {10945, 45567}, {10946, 45568}, {10947, 45571}, {10948, 45583}, {10949, 45587}, {11235, 41491}, {11373, 45501}, {11390, 45503}, {11826, 45499}, {11865, 45535}, {11866, 45537}, {11903, 45549}, {11928, 45579}, {12114, 45541}, {13895, 45576}, {13952, 45575}, {18516, 45543}, {18519, 45378}, {18961, 45507}, {19023, 45514}, {19024, 45513}, {26390, 45352}, {26414, 45350}, {26488, 45517}, {26489, 45518}, {26490, 45524}, {26491, 45523}, {35796, 45564}, {45411, 45454}

X(45557) = X(35803)-of-inner-Johnson triangle
X(45557) = {X(12586), X(26492)}-harmonic conjugate of X(45556)

X(45558) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND OUTER-JOHNSON

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

X(45558) lies on these lines: {3, 45456}, {5, 45526}, {11, 45586}, {12, 372}, {39, 44621}, {72, 45546}, {182, 12587}, {355, 45554}, {641, 958}, {3436, 45508}, {5062, 44620}, {7690, 35250}, {9739, 10526}, {10523, 45582}, {10786, 45510}, {10795, 45504}, {10827, 45530}, {10830, 45532}, {10872, 45538}, {10894, 45544}, {10921, 45550}, {10922, 45553}, {10942, 45528}, {10950, 45562}, {10951, 45569}, {10952, 45566}, {10953, 45570}, {10954, 45580}, {10955, 45584}, {11236, 41490}, {11374, 45500}, {11391, 45502}, {11500, 45520}, {11827, 45498}, {11867, 45534}, {11868, 45536}, {11904, 45548}, {11929, 45578}, {13896, 45574}, {13953, 45577}, {18517, 45542}, {18518, 45377}, {18962, 45506}, {19025, 45512}, {19026, 45515}, {26389, 45349}, {26413, 45351}, {26483, 45519}, {26484, 45516}, {26485, 45522}, {26486, 45525}, {35799, 45565}, {45410, 45457}

X(45558) = X(35800)-of-these triangles: {1st anti-Kenmotu-free-vertices, outer-Johnson}
X(45558) = {X(12587), X(26487)}-harmonic conjugate of X(45559)

X(45559) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND OUTER-JOHNSON

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

X(45559) lies on these lines: {3, 45457}, {5, 45527}, {11, 45587}, {12, 371}, {39, 44620}, {72, 45547}, {182, 12587}, {355, 45555}, {642, 958}, {3436, 45509}, {5058, 44621}, {7692, 35250}, {9738, 10526}, {10523, 45583}, {10786, 45511}, {10795, 45505}, {10827, 45531}, {10830, 45533}, {10872, 45539}, {10894, 45545}, {10921, 45552}, {10922, 45551}, {10942, 45529}, {10950, 45563}, {10951, 45567}, {10952, 45568}, {10953, 45571}, {10954, 45581}, {10955, 45585}, {11236, 41491}, {11374, 45501}, {11391, 45503}, {11500, 45521}, {11827, 45499}, {11867, 45535}, {11868, 45537}, {11904, 45549}, {11929, 45579}, {13896, 45576}, {13953, 45575}, {18517, 45543}, {18518, 45378}, {18962, 45507}, {19025, 45514}, {19026, 45513}, {26389, 45352}, {26413, 45350}, {26483, 45517}, {26484, 45518}, {26485, 45524}, {26486, 45523}, {35798, 45564}, {45411, 45456}

X(45559) = X(35801)-of-outer-Johnson triangle
X(45559) = {X(12587), X(26487)}-harmonic conjugate of X(45558)

X(45560) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 1st JOHNSON-YFF

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

X(45560) lies on these lines: {1, 45554}, {3, 45458}, {4, 45570}, {5, 45582}, {12, 372}, {39, 44622}, {56, 641}, {65, 45546}, {114, 45563}, {182, 498}, {388, 45506}, {495, 45580}, {1478, 9739}, {1479, 45542}, {3085, 45510}, {3295, 45377}, {4293, 26479}, {4299, 7690}, {5062, 31472}, {5432, 10924}, {7354, 45498}, {7763, 45507}, {9578, 45530}, {9596, 18993}, {9646, 39897}, {9654, 45578}, {10797, 45504}, {10831, 45532}, {10873, 45538}, {10895, 45544}, {10923, 45550}, {10944, 45556}, {10956, 45584}, {10957, 45586}, {11237, 41490}, {11238, 13695}, {11375, 45500}, {11392, 45502}, {11501, 45520}, {11869, 45534}, {11870, 45536}, {11905, 45548}, {11930, 45569}, {11931, 45566}, {13897, 45574}, {13905, 44656}, {13954, 45577}, {19027, 45512}, {19028, 45515}, {22759, 45540}, {26388, 45349}, {26412, 45351}, {26477, 45519}, {26478, 45516}, {26480, 45525}, {26481, 45526}, {26482, 45528}, {31479, 45410}, {35801, 45565}, {36656, 45470}

X(45560) = X(35798)-of-these triangles: {1st anti-Kenmotu-free-vertices, 1st Johnson-Yff}
X(45560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45554, 45562), (388, 45508, 45506), (498, 12588, 45561)

X(45561) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 1st JOHNSON-YFF

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

X(45561) lies on these lines: {1, 45555}, {3, 45459}, {4, 45571}, {5, 45583}, {12, 371}, {39, 31472}, {56, 642}, {65, 45547}, {114, 45562}, {182, 498}, {388, 45507}, {495, 45581}, {1478, 9738}, {1479, 45543}, {3085, 45511}, {3295, 45378}, {4293, 26480}, {4299, 7692}, {5058, 44622}, {5432, 10923}, {7354, 45499}, {7763, 45506}, {9578, 45531}, {9596, 18994}, {9654, 45579}, {10797, 45505}, {10831, 45533}, {10873, 45539}, {10895, 45545}, {10924, 45551}, {10944, 45557}, {10956, 45585}, {10957, 45587}, {11237, 41491}, {11238, 13815}, {11375, 45501}, {11392, 45503}, {11501, 45521}, {11869, 45535}, {11870, 45537}, {11905, 45549}, {11930, 45567}, {11931, 45568}, {13897, 45576}, {13954, 45575}, {13963, 44657}, {19027, 45514}, {19028, 45513}, {22759, 45541}, {26388, 45352}, {26412, 45350}, {26477, 45517}, {26478, 45518}, {26479, 45524}, {26481, 45527}, {26482, 45529}, {31479, 45411}, {35800, 45564}, {36655, 45471}

X(45561) = X(35799)-of-1st Johnson-Yff triangle
X(45561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45555, 45563), (388, 45509, 45507), (498, 12588, 45560)

X(45562) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 2nd JOHNSON-YFF

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

X(45562) lies on these lines: {1, 45554}, {3, 45460}, {4, 45506}, {5, 45580}, {11, 372}, {39, 44624}, {55, 641}, {114, 45561}, {182, 499}, {496, 45582}, {497, 45508}, {999, 45377}, {1478, 45542}, {1479, 9739}, {3057, 45546}, {3086, 45510}, {4294, 26473}, {4302, 7690}, {5062, 44623}, {5433, 10926}, {6284, 45498}, {7763, 45571}, {9581, 45530}, {9599, 18993}, {9661, 39873}, {9669, 45578}, {10798, 45504}, {10832, 45532}, {10874, 45538}, {10896, 45544}, {10925, 45550}, {10950, 45558}, {10958, 45584}, {10959, 45586}, {11237, 13696}, {11238, 41490}, {11376, 45500}, {11393, 45502}, {11502, 45520}, {11871, 45534}, {11872, 45536}, {11906, 45548}, {11932, 45569}, {11933, 45566}, {13898, 45574}, {13904, 44656}, {13955, 45577}, {19029, 45512}, {19030, 45515}, {22760, 45540}, {26387, 45349}, {26411, 45351}, {26471, 45519}, {26472, 45516}, {26474, 45525}, {26475, 45526}, {26476, 45528}, {35803, 45565}, {36656, 45404}, {45410, 45461}

X(45562) = X(35796)-of-these triangles: {1st anti-Kenmotu-free-vertices, 2nd Johnson-Yff}
X(45562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45554, 45560), (497, 45508, 45570), (499, 12589, 45563)

X(45563) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 2nd JOHNSON-YFF

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

X(45563) lies on these lines: {1, 45555}, {3, 45461}, {4, 45507}, {5, 45581}, {11, 371}, {39, 44623}, {55, 642}, {114, 45560}, {182, 499}, {496, 45583}, {497, 45509}, {999, 45378}, {1478, 45543}, {1479, 9738}, {3057, 45547}, {3086, 45511}, {4294, 26474}, {4302, 7692}, {5058, 44624}, {5433, 10925}, {6284, 45499}, {7763, 45570}, {9581, 45531}, {9599, 18994}, {9669, 45579}, {10798, 45505}, {10832, 45533}, {10874, 45539}, {10896, 45545}, {10926, 45551}, {10950, 45559}, {10958, 45585}, {10959, 45587}, {11237, 13816}, {11238, 41491}, {11376, 45501}, {11393, 45503}, {11502, 45521}, {11871, 45535}, {11872, 45537}, {11906, 45549}, {11932, 45567}, {11933, 45568}, {13898, 45576}, {13955, 45575}, {13962, 44657}, {19029, 45514}, {19030, 45513}, {22760, 45541}, {26387, 45352}, {26411, 45350}, {26471, 45517}, {26472, 45518}, {26473, 45524}, {26475, 45527}, {26476, 45529}, {35802, 45564}, {36655, 45405}, {45411, 45460}

X(45563) = X(35797)-of-2nd Johnson-Yff triangle
X(45563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45555, 45561), (497, 45509, 45571), (499, 12589, 45562)

X(45564) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 1st KENMOTU-FREE-VERTICES

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

X(45564) lies on these lines: {3, 6}, {485, 11293}, {638, 32965}, {639, 7763}, {640, 7791}, {642, 7389}, {1584, 8956}, {5418, 39661}, {5422, 32575}, {6560, 45511}, {6564, 45555}, {6565, 6811}, {7485, 32568}, {7841, 32432}, {10839, 22725}, {13637, 35822}, {23251, 45378}, {31403, 35944}, {32421, 33008}, {35762, 45501}, {35764, 45503}, {35768, 45507}, {35769, 45583}, {35772, 45521}, {35774, 45531}, {35776, 45533}, {35778, 45535}, {35780, 45537}, {35784, 45541}, {35786, 45543}, {35787, 36656}, {35788, 45547}, {35790, 45549}, {35796, 45557}, {35798, 45559}, {35800, 45561}, {35802, 45563}, {35804, 45567}, {35806, 45568}, {35808, 45571}, {35809, 45581}, {35810, 45573}, {35812, 45576}, {35814, 45575}, {35816, 45585}, {35818, 45587}, {45350, 45359}, {45352, 45357}, {45517, 45601}, {45518, 45599}, {45527, 45640}, {45529, 45642}

X(45564) = reflection of X(45565) in X(574)
X(45564) = X(1667)-of-2nd Brocard triangle
X(45564) = X(45564)-of-circumsymmedial triangle
X(45564) = X(45565)-of-1st Kenmotu-free-vertices triangle
X(45564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 3103, 372), (6, 45579, 371), (39, 9738, 371), (39, 18994, 45513), (371, 6396, 182), (371, 6420, 5058), (371, 45499, 6200), (371, 45513, 6419), (1152, 45462, 372), (3371, 3372, 45410), (3385, 3386, 12963), (5013, 9732, 3102), (6396, 35840, 372), (6421, 12313, 45463), (9737, 9738, 43155)

X(45565) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 2nd KENMOTU-FREE-VERTICES

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

X(45565) lies on these lines: {3, 6}, {486, 11294}, {637, 32965}, {639, 7791}, {640, 7763}, {641, 7388}, {5420, 39660}, {5422, 32568}, {6561, 45510}, {6564, 6813}, {6565, 45554}, {7485, 32575}, {7841, 32435}, {10840, 22724}, {13757, 35823}, {23261, 45377}, {32419, 33008}, {35763, 45500}, {35765, 45502}, {35768, 45582}, {35769, 45506}, {35773, 45520}, {35775, 45530}, {35777, 45532}, {35779, 45536}, {35781, 45534}, {35785, 45540}, {35786, 36655}, {35787, 45542}, {35789, 45546}, {35791, 45548}, {35797, 45556}, {35799, 45558}, {35801, 45560}, {35803, 45562}, {35805, 45566}, {35807, 45569}, {35808, 45580}, {35809, 45570}, {35811, 45572}, {35813, 45577}, {35815, 45574}, {35817, 45584}, {35819, 45586}, {45349, 45360}, {45351, 45358}, {45516, 45602}, {45519, 45600}, {45526, 45641}, {45528, 45643}

X(45565) = reflection of X(45564) in X(574)
X(45565) = X(1666)-of-2nd Brocard triangle
X(45565) = X(45564)-of-these triangles: {1st anti-Kenmotu-free-vertices, 2nd Kenmotu-free-vertices}
X(45565) = X(45565)-of-circumsymmedial triangle
X(45565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 3102, 371), (6, 45578, 372), (39, 9739, 372), (39, 18993, 45512), (372, 6200, 182), (372, 6419, 5062), (372, 45498, 6396), (372, 45512, 6420), (1151, 45463, 371), (3371, 3372, 12968), (3385, 3386, 45411), (5013, 9733, 3103), (6200, 35841, 371), (6422, 12314, 45462), (9737, 9739, 43156)

X(45566) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND LUCAS(-1) HOMOTHETIC

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

X(45566) lies on these lines: {3, 45464}, {25, 372}, {39, 44630}, {182, 12591}, {641, 8223}, {5062, 44629}, {6461, 45569}, {6463, 45508}, {8189, 45530}, {8202, 45534}, {8209, 45536}, {8211, 45572}, {8213, 45544}, {8215, 45546}, {8217, 45550}, {8219, 45553}, {8221, 45554}, {9739, 10673}, {10876, 45538}, {10946, 45556}, {10952, 45558}, {11378, 45500}, {11504, 45520}, {11829, 45498}, {11841, 45504}, {11847, 45510}, {11908, 45548}, {11931, 45560}, {11933, 45562}, {11948, 45570}, {11950, 45578}, {11952, 45580}, {11954, 45582}, {11956, 45584}, {11958, 45586}, {12153, 41490}, {13900, 45574}, {13957, 45577}, {18522, 45542}, {18964, 45506}, {19033, 45512}, {19034, 45515}, {22762, 45540}, {35805, 45565}, {45349, 45361}, {45351, 45363}, {45377, 45382}, {45410, 45466}, {45519, 45603}, {45526, 45644}, {45528, 45646}

X(45566) = X(45599)-of-these triangles: {1st anti-Kenmotu-free-vertices, Lucas(-1) homothetic}
X(45566) = {X(12591), X(45624)}-harmonic conjugate of X(45568)

X(45567) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND LUCAS(+1) HOMOTHETIC

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

X(45567) lies on these lines: {3, 45465}, {25, 371}, {39, 44627}, {182, 12590}, {642, 8222}, {5058, 44628}, {6461, 45568}, {6462, 45509}, {8188, 45531}, {8201, 45535}, {8208, 45537}, {8210, 45573}, {8212, 45545}, {8214, 45547}, {8216, 45552}, {8218, 45551}, {8220, 45555}, {9738, 10669}, {10875, 45539}, {10945, 45557}, {10951, 45559}, {11377, 45501}, {11503, 45521}, {11828, 45499}, {11840, 45505}, {11846, 45511}, {11907, 45549}, {11930, 45561}, {11932, 45563}, {11947, 45571}, {11949, 45579}, {11951, 45581}, {11953, 45583}, {11955, 45585}, {11957, 45587}, {12152, 41491}, {13899, 45576}, {13956, 45575}, {18520, 45543}, {18963, 45507}, {19031, 45514}, {19032, 45513}, {22761, 45541}, {35804, 45564}, {45350, 45364}, {45352, 45362}, {45378, 45381}, {45411, 45467}, {45518, 45604}, {45527, 45645}, {45529, 45647}

X(45567) = X(45600)-of-Lucas(+1) homothetic triangle
X(45567) = {X(12590), X(45623)}-harmonic conjugate of X(45569)

X(45568) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*(3*a^8-12*(b^2+c^2)*a^6+2*(7*b^4-8*b^2*c^2+7*c^4)*a^4-4*(b^2+c^2)*(b^4-16*b^2*c^2+c^4)*a^2+2*(3*a^6+3*(b^2+c^2)*a^4-11*(b^2+c^2)^2*a^2+(b^2+c^2)*(5*b^4-22*b^2*c^2+5*c^4))*S-(b^4+18*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(45568) lies on these lines: {3, 45466}, {39, 44629}, {182, 12591}, {371, 494}, {642, 8223}, {5058, 44630}, {6461, 45567}, {6463, 45509}, {8189, 45531}, {8195, 45533}, {8202, 45535}, {8209, 45537}, {8211, 45573}, {8213, 45545}, {8215, 45547}, {8217, 45552}, {8219, 45551}, {8221, 45555}, {9738, 10673}, {10876, 45539}, {10946, 45557}, {10952, 45559}, {11378, 45501}, {11395, 45503}, {11504, 45521}, {11829, 45499}, {11841, 45505}, {11847, 45511}, {11908, 45549}, {11931, 45561}, {11933, 45563}, {11948, 45571}, {11950, 45579}, {11952, 45581}, {11954, 45583}, {11956, 45585}, {11958, 45587}, {12153, 41491}, {13900, 45576}, {13957, 45575}, {18522, 45543}, {18964, 45507}, {19033, 45514}, {19034, 45513}, {22762, 45541}, {35806, 45564}, {45350, 45363}, {45352, 45361}, {45378, 45382}, {45411, 45464}, {45517, 45603}, {45527, 45644}, {45529, 45646}

X(45568) = X(45602)-of-Lucas(-1) homothetic triangle
X(45568) = {X(12591), X(45624)}-harmonic conjugate of X(45566)

X(45569) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(3*a^8-12*(b^2+c^2)*a^6+2*(7*b^4-8*b^2*c^2+7*c^4)*a^4-4*(b^2+c^2)*(b^4-16*b^2*c^2+c^4)*a^2-2*(3*a^6+3*(b^2+c^2)*a^4-11*(b^2+c^2)^2*a^2+(b^2+c^2)*(5*b^4-22*b^2*c^2+5*c^4))*S-(b^4+18*b^2*c^2+c^4)*(b^2-c^2)^2) : :

X(45569) lies on these lines: {3, 45467}, {39, 44628}, {182, 12590}, {372, 493}, {641, 8222}, {5062, 44627}, {6461, 45566}, {6462, 45508}, {8188, 45530}, {8194, 45532}, {8201, 45534}, {8208, 45536}, {8210, 45572}, {8212, 45544}, {8214, 45546}, {8216, 45550}, {8218, 45553}, {8220, 45554}, {9739, 10669}, {10875, 45538}, {10945, 45556}, {10951, 45558}, {11377, 45500}, {11394, 45502}, {11503, 45520}, {11828, 45498}, {11840, 45504}, {11846, 45510}, {11907, 45548}, {11930, 45560}, {11932, 45562}, {11947, 45570}, {11949, 45578}, {11951, 45580}, {11953, 45582}, {11955, 45584}, {11957, 45586}, {12152, 41490}, {13899, 45574}, {13956, 45577}, {18520, 45542}, {18963, 45506}, {19031, 45512}, {19032, 45515}, {22761, 45540}, {35807, 45565}, {45349, 45362}, {45351, 45364}, {45377, 45381}, {45410, 45465}, {45516, 45604}, {45526, 45645}, {45528, 45647}

X(45569) = X(45601)-of-these triangles: {1st anti-Kenmotu-free-vertices, Lucas(+1) homothetic}
X(45569) = {X(12590), X(45623)}-harmonic conjugate of X(45567)

X(45570) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND MANDART-INCIRCLE

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

X(45570) lies on these lines: {1, 9739}, {3, 45470}, {4, 45560}, {11, 641}, {12, 45544}, {33, 45502}, {35, 182}, {36, 6283}, {39, 5414}, {55, 372}, {56, 45498}, {497, 45508}, {1479, 45554}, {1697, 45530}, {1837, 45546}, {2066, 5062}, {2098, 45572}, {2276, 18993}, {2646, 45500}, {3058, 41490}, {3086, 26355}, {3295, 45578}, {3583, 45542}, {3746, 6405}, {4294, 45510}, {5217, 10928}, {6502, 6566}, {7763, 45563}, {8540, 44656}, {9668, 45377}, {9737, 45507}, {10799, 45504}, {10833, 45532}, {10877, 45538}, {10927, 45550}, {10947, 45556}, {10953, 45558}, {10965, 45584}, {10966, 45540}, {11873, 45534}, {11874, 45536}, {11909, 45548}, {11947, 45569}, {11948, 45566}, {12314, 45490}, {13901, 45574}, {13958, 45577}, {19037, 45512}, {19038, 45515}, {26351, 45349}, {26352, 45351}, {26353, 45519}, {26354, 45516}, {26356, 45525}, {26357, 45526}, {26358, 45528}, {35809, 45565}, {45410, 45471}

X(45570) = X(35772)-of-these triangles: {1st anti-Kenmotu-free-vertices, Mandart-incircle}
X(45570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9739, 45506), (35, 3056, 45571), (497, 45508, 45562), (3295, 45578, 45580)

X(45571) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND MANDART-INCIRCLE

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

X(45571) lies on these lines: {1, 9738}, {3, 45471}, {4, 45561}, {11, 642}, {12, 45545}, {33, 45503}, {35, 182}, {36, 6405}, {39, 2066}, {55, 371}, {56, 45499}, {497, 45509}, {1479, 45555}, {1697, 45531}, {1837, 45547}, {2067, 6567}, {2098, 45573}, {2276, 18994}, {2646, 45501}, {3058, 41491}, {3086, 26356}, {3295, 45579}, {3583, 45543}, {3746, 6283}, {4294, 45511}, {5058, 5414}, {5217, 10927}, {7763, 45562}, {8540, 44657}, {9668, 45378}, {9737, 45506}, {10799, 45505}, {10833, 45533}, {10877, 45539}, {10928, 45551}, {10947, 45557}, {10953, 45559}, {10965, 45585}, {10966, 45541}, {11873, 45535}, {11874, 45537}, {11909, 45549}, {11947, 45567}, {11948, 45568}, {12313, 45491}, {13901, 45576}, {13958, 45575}, {19037, 45514}, {19038, 45513}, {26351, 45352}, {26352, 45350}, {26353, 45517}, {26354, 45518}, {26355, 45524}, {26357, 45527}, {26358, 45529}, {35808, 45564}, {45411, 45470}

X(45571) = X(35773)-of-Mandart-incircle triangle
X(45571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 9738, 45507), (35, 3056, 45570), (497, 45509, 45563), (3295, 45579, 45581)

X(45572) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 5th MIXTILINEAR

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

X(45572) lies on these lines: {1, 372}, {3, 45476}, {8, 641}, {39, 44636}, {55, 45540}, {56, 45520}, {145, 45508}, {182, 3242}, {517, 45498}, {518, 35763}, {519, 45546}, {952, 45554}, {1385, 5604}, {1482, 9739}, {2098, 45570}, {2099, 45506}, {3241, 41490}, {5062, 44635}, {5597, 45536}, {5598, 45534}, {5603, 45544}, {5605, 15178}, {6419, 45426}, {6420, 45398}, {7690, 12702}, {7967, 45510}, {7968, 45512}, {7969, 45515}, {8192, 45532}, {8210, 45569}, {8211, 45566}, {9997, 45538}, {10247, 45578}, {10800, 45504}, {10944, 45556}, {10950, 45558}, {11396, 45502}, {11910, 45548}, {12245, 26514}, {13702, 34627}, {13902, 45574}, {13959, 45577}, {18525, 45542}, {18526, 45377}, {26395, 45349}, {26419, 45351}, {26495, 45519}, {26504, 45516}, {26515, 45525}, {35811, 45565}, {37624, 45410}

X(45572) = reflection of X(35810) in X(1)
X(45572) = X(35810)-of-these triangles: {1st anti-Kenmotu-free-vertices, 5th mixtilinear}
X(45572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45530, 45500), (3242, 10246, 45573), (45500, 45530, 372), (45526, 45580, 372), (45528, 45582, 372), (45584, 45586, 372)

X(45573) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 5th MIXTILINEAR

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

X(45573) lies on these lines: {1, 371}, {3, 45477}, {8, 642}, {39, 44635}, {55, 45541}, {56, 45521}, {145, 45509}, {182, 3242}, {517, 45499}, {518, 35762}, {519, 45547}, {952, 45555}, {1385, 5605}, {1482, 9738}, {2098, 45571}, {2099, 45507}, {3241, 41491}, {5058, 44636}, {5597, 45537}, {5598, 45535}, {5603, 45545}, {5604, 15178}, {6419, 45399}, {6420, 45427}, {7692, 12702}, {7967, 45511}, {7968, 45514}, {7969, 45513}, {8192, 45533}, {8210, 45567}, {8211, 45568}, {9997, 45539}, {10247, 45579}, {10800, 45505}, {10944, 45557}, {10950, 45559}, {11396, 45503}, {11910, 45549}, {12245, 26515}, {13822, 34627}, {13902, 45576}, {13959, 45575}, {18525, 45543}, {18526, 45378}, {26395, 45352}, {26419, 45350}, {26495, 45517}, {26504, 45518}, {26514, 45524}, {35810, 45564}, {37624, 45411}

X(45573) = reflection of X(35811) in X(1)
X(45573) = X(35811)-of-5th mixtilinear triangle
X(45573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45531, 45501), (3242, 10246, 45572), (45501, 45531, 371), (45527, 45581, 371), (45529, 45583, 371), (45585, 45587, 371)

X(45574) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 3rd TRI-SQUARES-CENTRAL

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

X(45574) lies on these lines: {2, 45515}, {3, 45484}, {5, 44656}, {6, 639}, {39, 13821}, {69, 19103}, {182, 7583}, {371, 45544}, {372, 631}, {485, 6776}, {487, 7585}, {590, 641}, {597, 7775}, {640, 31411}, {1587, 8975}, {1656, 40286}, {1692, 32432}, {3592, 13644}, {7690, 35255}, {7763, 13637}, {7803, 45514}, {8960, 13638}, {8972, 45508}, {8974, 45550}, {8976, 45554}, {8981, 9739}, {9540, 45498}, {13846, 41490}, {13880, 32491}, {13883, 45500}, {13884, 45502}, {13885, 45504}, {13886, 45510}, {13887, 45520}, {13888, 45530}, {13889, 45532}, {13890, 45534}, {13891, 45536}, {13892, 45538}, {13893, 45546}, {13894, 45548}, {13895, 45556}, {13896, 45558}, {13897, 45560}, {13898, 45562}, {13899, 45569}, {13900, 45566}, {13901, 45570}, {13902, 45572}, {13903, 45578}, {13904, 45580}, {13905, 45582}, {13906, 45584}, {13907, 45586}, {14227, 31412}, {18538, 45542}, {18965, 45506}, {22763, 45540}, {35815, 45565}, {45349, 45365}, {45351, 45368}, {45377, 45384}, {45410, 45486}, {45516, 45605}, {45519, 45607}, {45526, 45650}, {45528, 45652}

X(45574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (590, 5062, 641), (7583, 13910, 45576), (8960, 13638, 13921)

X(45575) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 4th TRI-SQUARES-CENTRAL

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

X(45575) lies on these lines: {2, 45514}, {3, 45485}, {5, 44657}, {6, 640}, {39, 13701}, {69, 19104}, {182, 7584}, {371, 631}, {372, 45545}, {486, 6776}, {488, 7586}, {597, 7775}, {615, 642}, {1588, 13949}, {1656, 40287}, {1692, 32435}, {3594, 13763}, {7692, 35256}, {7763, 13757}, {7803, 45515}, {9738, 13966}, {13758, 13880}, {13847, 41491}, {13921, 32490}, {13935, 45499}, {13936, 45501}, {13937, 45503}, {13938, 45505}, {13939, 45511}, {13940, 45521}, {13941, 45509}, {13942, 45531}, {13943, 45533}, {13944, 45535}, {13945, 45537}, {13946, 45539}, {13947, 45547}, {13948, 45549}, {13950, 45551}, {13951, 45555}, {13952, 45557}, {13953, 45559}, {13954, 45561}, {13955, 45563}, {13956, 45567}, {13957, 45568}, {13958, 45571}, {13959, 45573}, {13961, 45579}, {13962, 45581}, {13963, 45583}, {13964, 45585}, {13965, 45587}, {14242, 42561}, {18762, 45543}, {18966, 45507}, {22764, 45541}, {35814, 45564}, {45350, 45367}, {45352, 45366}, {45378, 45385}, {45411, 45487}, {45517, 45606}, {45518, 45608}, {45527, 45651}, {45529, 45653}

X(45575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (615, 5058, 642), (7584, 13972, 45577)

X(45576) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND 3rd TRI-SQUARES-CENTRAL

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

X(45576) lies on these lines: {2, 45513}, {3, 45486}, {4, 371}, {6, 640}, {39, 590}, {141, 5305}, {182, 7583}, {193, 19102}, {230, 641}, {372, 13638}, {486, 40330}, {490, 41411}, {491, 7803}, {492, 13880}, {615, 7755}, {639, 3767}, {1504, 44394}, {1587, 8974}, {1588, 10514}, {1692, 44647}, {2909, 30398}, {3311, 6290}, {3618, 5491}, {5058, 7753}, {5254, 9994}, {5306, 19011}, {6281, 39876}, {6423, 32421}, {6567, 31454}, {7584, 34507}, {7585, 45514}, {7692, 35255}, {8396, 45441}, {8968, 40938}, {8970, 18289}, {8972, 45509}, {8975, 45551}, {8976, 45555}, {8981, 9738}, {9540, 45499}, {9605, 45473}, {11294, 41410}, {13644, 23251}, {13720, 13908}, {13846, 41491}, {13883, 45501}, {13884, 45503}, {13885, 45505}, {13887, 45521}, {13888, 45531}, {13889, 45533}, {13890, 45535}, {13891, 45537}, {13892, 45539}, {13893, 45547}, {13894, 45549}, {13895, 45557}, {13896, 45559}, {13897, 45561}, {13898, 45563}, {13899, 45567}, {13900, 45568}, {13901, 45571}, {13902, 45573}, {13903, 45579}, {13904, 45581}, {13905, 45583}, {13906, 45585}, {13907, 45587}, {18538, 45543}, {18965, 45507}, {19117, 44657}, {22763, 45541}, {23311, 43291}, {35812, 45564}, {45350, 45368}, {45352, 45365}, {45378, 45384}, {45411, 45484}, {45517, 45607}, {45518, 45605}, {45527, 45650}, {45529, 45652}

X(45576) = X(39660)-of-3rd tri-squares-central triangle
X(45576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 590, 642), (141, 5305, 45577), (3068, 8960, 13879), (7583, 13910, 45574)

X(45577) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND 4th TRI-SQUARES-CENTRAL

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

X(45577) lies on these lines: {2, 45512}, {3, 45487}, {4, 372}, {6, 639}, {39, 615}, {141, 5305}, {182, 7584}, {193, 19105}, {230, 642}, {371, 13758}, {485, 40330}, {489, 41410}, {491, 13921}, {492, 7803}, {590, 7755}, {640, 3767}, {1505, 44392}, {1587, 10515}, {1588, 13950}, {1692, 44648}, {2909, 30399}, {3312, 6289}, {3618, 5490}, {5062, 7753}, {5254, 9995}, {5306, 19012}, {6118, 31463}, {6278, 39875}, {6424, 32419}, {7583, 34507}, {7586, 45515}, {7690, 35256}, {8416, 45440}, {9605, 45472}, {9739, 13966}, {11293, 41411}, {13763, 23261}, {13843, 13968}, {13847, 41490}, {13935, 45498}, {13936, 45500}, {13937, 45502}, {13938, 45504}, {13940, 45520}, {13941, 45508}, {13942, 45530}, {13943, 45532}, {13944, 45534}, {13945, 45536}, {13946, 45538}, {13947, 45546}, {13948, 45548}, {13949, 45550}, {13951, 45554}, {13952, 45556}, {13953, 45558}, {13954, 45560}, {13955, 45562}, {13956, 45569}, {13957, 45566}, {13958, 45570}, {13959, 45572}, {13961, 45578}, {13962, 45580}, {13963, 45582}, {13964, 45584}, {13965, 45586}, {18762, 45542}, {18966, 45506}, {19116, 44656}, {22764, 45540}, {23312, 43291}, {35813, 45565}, {45349, 45366}, {45351, 45367}, {45377, 45385}, {45410, 45485}, {45516, 45608}, {45519, 45606}, {45526, 45651}, {45528, 45653}

X(45577) = X(39661)-of-these triangles: {1st anti-Kenmotu-free-vertices, 4th tri-squares-central}
X(45577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 615, 641), (141, 5305, 45576), (7584, 13972, 45575)

X(45578) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND X3-ABC REFLECTIONS

Barycentrics    a^2*(6*(-a^2+b^2+c^2)*S+a^4-4*(b^2+c^2)*a^2+3*b^4-2*b^2*c^2+3*c^4) : :
X(45578) = 3*X(3)-2*X(45499) = 3*X(6396)-X(45499)

X(45578) lies on these lines: {3, 6}, {4, 45377}, {5, 490}, {30, 13757}, {110, 3156}, {140, 45522}, {381, 13687}, {488, 37342}, {517, 45530}, {550, 45525}, {641, 1656}, {999, 45506}, {1598, 45502}, {1657, 12601}, {3066, 5406}, {3292, 35300}, {3295, 45570}, {3843, 45542}, {5054, 13663}, {5408, 5651}, {5790, 45546}, {5999, 10852}, {6321, 13847}, {6813, 45376}, {7517, 45532}, {7586, 35945}, {7766, 10851}, {9654, 45560}, {9669, 45562}, {10246, 45500}, {10247, 45572}, {10679, 45528}, {10680, 45526}, {11829, 13024}, {11849, 45520}, {11875, 45534}, {11876, 45536}, {11911, 45548}, {11928, 45556}, {11929, 45558}, {11949, 45569}, {11950, 45566}, {12000, 45584}, {12001, 45586}, {12256, 14927}, {13903, 45574}, {13961, 45577}, {16001, 42178}, {16002, 42176}, {21737, 43511}, {22765, 45540}, {31481, 35832}, {32421, 45555}, {34417, 35299}, {36714, 39884}, {45349, 45369}, {45351, 45370}, {45516, 45609}, {45519, 45610}

X(45578) = reflection of X(i) in X(j) for these (i, j): (3, 6396), (45376, 6813)
X(45578) = X(6396)-of-these triangles: {1st anti-Kenmotu-free-vertices, X3-ABC reflections}
X(45578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 1152, 2459), (39, 372, 3312), (182, 45498, 3), (371, 43141, 3), (372, 9739, 3), (372, 45498, 182), (372, 45565, 6), (1152, 9733, 3), (1160, 6450, 3), (1161, 6456, 3), (6410, 9732, 3), (7690, 45553, 3), (11824, 12975, 3), (11825, 43121, 3), (12305, 43118, 3), (39658, 43140, 3), (45488, 45579, 1351)

X(45579) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND X3-ABC REFLECTIONS

Barycentrics    a^2*(-6*(-a^2+b^2+c^2)*S+a^4-4*(b^2+c^2)*a^2+3*b^4-2*b^2*c^2+3*c^4) : :
X(45579) = 3*X(3)-2*X(45498) = 3*X(6200)-X(45498)

X(45579) lies on these lines: {3, 6}, {4, 45378}, {5, 489}, {30, 13637}, {110, 3155}, {140, 45523}, {381, 13807}, {487, 37343}, {517, 45531}, {550, 45524}, {642, 1656}, {999, 45507}, {1598, 45503}, {1657, 12602}, {3066, 5407}, {3167, 21097}, {3292, 35299}, {3295, 45571}, {3843, 45543}, {5054, 13783}, {5409, 5651}, {5790, 45547}, {5921, 21736}, {5999, 10851}, {6321, 13846}, {6811, 45375}, {7517, 45533}, {7585, 35944}, {7766, 10852}, {9654, 45561}, {9669, 45563}, {10246, 45501}, {10247, 45573}, {10679, 45529}, {10680, 45527}, {11828, 13023}, {11849, 45521}, {11875, 45535}, {11876, 45537}, {11911, 45549}, {11928, 45557}, {11929, 45559}, {11949, 45567}, {11950, 45568}, {12000, 45585}, {12001, 45587}, {12257, 14927}, {13903, 45576}, {13961, 45575}, {16001, 42177}, {16002, 42175}, {22765, 45541}, {32419, 45554}, {34417, 35300}, {36709, 39884}, {45350, 45370}, {45352, 45369}, {45517, 45610}, {45518, 45609}

X(45579) = reflection of X(i) in X(j) for these (i, j): (3, 6200), (45375, 6811)
X(45579) = X(6200)-of-X3-ABC reflections triangle
X(45579) = X(45579)-of-circumsymmedial triangle
X(45579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 371, 45411), (3, 3311, 45410), (3, 5093, 6398), (3, 6199, 5050), (3, 11916, 12314), (3, 12313, 45489), (3, 45489, 45488), (6, 1151, 2460), (39, 371, 3311), (182, 45499, 3), (182, 45505, 11842), (371, 6567, 6221), (371, 9738, 3), (371, 11825, 43142), (371, 45499, 182), (371, 45564, 6), (372, 43144, 3), (1151, 9732, 3), (1160, 6455, 3), (1161, 6449, 3), (1351, 45578, 45488), (6409, 9733, 3), (6425, 12306, 43119), (6453, 11824, 43120), (7692, 45552, 3), (11824, 43120, 3), (11825, 12974, 3), (12306, 43119, 3), (39649, 43137, 3), (45489, 45578, 1351), (45507, 45583, 999), (45545, 45555, 381), (45571, 45581, 3295)

X(45580) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND INNER-YFF

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

X(45580) lies on these lines: {1, 372}, {3, 45404}, {5, 45562}, {12, 45554}, {35, 45498}, {36, 10041}, {39, 1124}, {55, 9739}, {56, 182}, {388, 45510}, {495, 45560}, {498, 641}, {613, 18995}, {999, 45410}, {1335, 2242}, {1479, 45544}, {3085, 45508}, {3295, 45570}, {3297, 31477}, {3299, 45512}, {3301, 45515}, {3312, 45492}, {5135, 45436}, {5217, 7690}, {5218, 45522}, {5563, 10040}, {9654, 45377}, {10037, 45532}, {10038, 45538}, {10039, 45546}, {10056, 41490}, {10523, 45556}, {10801, 45504}, {10895, 45542}, {10954, 45558}, {11398, 45502}, {11507, 45520}, {11877, 45534}, {11878, 45536}, {11912, 45548}, {11951, 45569}, {11952, 45566}, {12314, 45470}, {13904, 45574}, {13962, 45577}, {16502, 18993}, {18996, 44656}, {22766, 45540}, {35808, 45565}, {45349, 45371}, {45351, 45372}, {45516, 45611}, {45519, 45612}

X(45580) = X(45640)-of-these triangles: {1st anti-Kenmotu-free-vertices, inner-Yff}
X(45580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 372, 45582), (56, 611, 45581), (372, 45572, 45526), (3295, 45578, 45570)

X(45581) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND INNER-YFF

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

X(45581) lies on these lines: {1, 371}, {3, 45405}, {5, 45563}, {12, 45555}, {35, 45499}, {36, 10040}, {39, 1335}, {55, 9738}, {56, 182}, {388, 45511}, {495, 45561}, {498, 642}, {613, 18996}, {999, 45411}, {1124, 2242}, {1479, 45545}, {3085, 45509}, {3295, 45571}, {3298, 31477}, {3299, 45514}, {3301, 45513}, {3311, 45493}, {5135, 45437}, {5217, 7692}, {5218, 45523}, {5563, 10041}, {9654, 45378}, {10037, 45533}, {10038, 45539}, {10039, 45547}, {10056, 41491}, {10523, 45557}, {10801, 45505}, {10895, 45543}, {10954, 45559}, {11398, 45503}, {11507, 45521}, {11877, 45535}, {11878, 45537}, {11912, 45549}, {11951, 45567}, {11952, 45568}, {12313, 45471}, {13904, 45576}, {13962, 45575}, {16502, 18994}, {18995, 44657}, {22766, 45541}, {35809, 45564}, {45350, 45372}, {45352, 45371}, {45517, 45612}, {45518, 45611}

X(45581) = X(45641)-of-inner-Yff triangle
X(45581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 371, 45583), (56, 611, 45580), (371, 45573, 45527), (3295, 45579, 45571)

X(45582) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND OUTER-YFF

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

X(45582) lies on these lines: {1, 372}, {3, 45470}, {5, 45560}, {11, 45554}, {35, 10049}, {36, 45498}, {39, 1335}, {55, 182}, {56, 9739}, {496, 45562}, {497, 45510}, {499, 641}, {611, 19037}, {999, 45506}, {1124, 2241}, {1478, 45544}, {1737, 45546}, {3086, 45508}, {3295, 45410}, {3299, 45515}, {3301, 45512}, {3312, 45490}, {3746, 10048}, {5204, 7690}, {5416, 16475}, {7288, 45522}, {9669, 45377}, {10046, 45532}, {10047, 45538}, {10072, 41490}, {10523, 45558}, {10802, 45504}, {10896, 45542}, {10948, 45556}, {11399, 45502}, {11508, 45520}, {11879, 45534}, {11880, 45536}, {11913, 45548}, {11953, 45569}, {11954, 45566}, {12314, 45404}, {13905, 45574}, {13963, 45577}, {19038, 44656}, {22767, 45540}, {35768, 45565}, {45349, 45373}, {45351, 45374}, {45516, 45613}, {45519, 45614}

X(45582) = X(45642)-of-these triangles: {1st anti-Kenmotu-free-vertices, outer-Yff}
X(45582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 372, 45580), (55, 613, 45583), (372, 45572, 45528), (999, 45578, 45506)

X(45583) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND OUTER-YFF

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

X(45583) lies on these lines: {1, 371}, {3, 45471}, {5, 45561}, {11, 45555}, {35, 10048}, {36, 45499}, {39, 1124}, {55, 182}, {56, 9738}, {496, 45563}, {497, 45511}, {499, 642}, {611, 19038}, {999, 45507}, {1335, 2241}, {1478, 45545}, {1737, 45547}, {3086, 45509}, {3295, 45411}, {3299, 45513}, {3301, 45514}, {3311, 45491}, {3746, 10049}, {5204, 7692}, {5415, 16475}, {7288, 45523}, {9669, 45378}, {10046, 45533}, {10047, 45539}, {10072, 41491}, {10523, 45559}, {10802, 45505}, {10896, 45543}, {10948, 45557}, {11399, 45503}, {11508, 45521}, {11879, 45535}, {11880, 45537}, {11913, 45549}, {11953, 45567}, {11954, 45568}, {12313, 45405}, {13905, 45576}, {13963, 45575}, {19037, 44657}, {22767, 45541}, {31474, 45490}, {35769, 45564}, {45350, 45374}, {45352, 45373}, {45517, 45614}, {45518, 45613}

X(45583) = X(45643)-of-outer-Yff triangle
X(45583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 371, 45581), (55, 613, 45582), (371, 45573, 45529), (999, 45579, 45507)

X(45584) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND INNER-YFF TANGENTS

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

X(45584) lies on these lines: {1, 372}, {3, 45494}, {12, 45556}, {39, 44644}, {182, 12594}, {371, 45424}, {641, 5552}, {5062, 44643}, {7690, 35251}, {9739, 10679}, {10269, 10930}, {10528, 45508}, {10531, 45544}, {10803, 45504}, {10805, 45510}, {10834, 45532}, {10878, 45538}, {10915, 45546}, {10929, 45550}, {10942, 45554}, {10955, 45558}, {10956, 45560}, {10958, 45562}, {10965, 45570}, {11239, 41490}, {11248, 45498}, {11400, 45502}, {11509, 45506}, {11881, 45534}, {11882, 45536}, {11914, 45548}, {11955, 45569}, {11956, 45566}, {12000, 45578}, {13906, 45574}, {13964, 45577}, {18542, 45542}, {18545, 45377}, {19047, 45512}, {19048, 45515}, {22768, 45540}, {26402, 45349}, {26426, 45351}, {26511, 45516}, {26520, 45522}, {26525, 45525}, {35817, 45565}, {45410, 45495}, {45519, 45615}

X(45584) = X(35818)-of-these triangles: {1st anti-Kenmotu-free-vertices, inner-Yff tangents}
X(45584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45528, 372), (372, 45572, 45586), (12594, 16203, 45585)

X(45585) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND INNER-YFF TANGENTS

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

X(45585) lies on these lines: {1, 371}, {3, 45495}, {12, 45557}, {39, 44643}, {182, 12594}, {372, 45425}, {642, 5552}, {5058, 44644}, {7692, 35251}, {9738, 10679}, {10269, 10929}, {10528, 45509}, {10531, 45545}, {10803, 45505}, {10805, 45511}, {10834, 45533}, {10878, 45539}, {10915, 45547}, {10930, 45551}, {10942, 45555}, {10955, 45559}, {10956, 45561}, {10958, 45563}, {10965, 45571}, {11239, 41491}, {11248, 45499}, {11400, 45503}, {11509, 45507}, {11881, 45535}, {11882, 45537}, {11914, 45549}, {11955, 45567}, {11956, 45568}, {12000, 45579}, {13906, 45576}, {13964, 45575}, {18542, 45543}, {18545, 45378}, {19047, 45514}, {19048, 45513}, {22768, 45541}, {26402, 45352}, {26426, 45350}, {26511, 45518}, {26520, 45524}, {26525, 45523}, {35816, 45564}, {45411, 45494}, {45517, 45615}

X(45585) = X(35819)-of-inner-Yff tangents triangle
X(45585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45529, 371), (371, 45573, 45587), (12594, 16203, 45584)

X(45586) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES AND OUTER-YFF TANGENTS

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

X(45586) lies on these lines: {1, 372}, {3, 45496}, {11, 45558}, {39, 44646}, {182, 12595}, {371, 45422}, {641, 10527}, {5062, 44645}, {7690, 35252}, {9739, 10680}, {10267, 10932}, {10529, 45508}, {10532, 45544}, {10804, 45504}, {10806, 45510}, {10835, 45532}, {10879, 45538}, {10916, 45546}, {10931, 45550}, {10943, 45554}, {10949, 45556}, {10957, 45560}, {10959, 45562}, {10966, 45540}, {11240, 41490}, {11249, 45498}, {11401, 45502}, {11510, 45520}, {11883, 45534}, {11884, 45536}, {11915, 45548}, {11957, 45569}, {11958, 45566}, {12001, 45578}, {13907, 45574}, {13965, 45577}, {18543, 45377}, {18544, 45542}, {18967, 45506}, {19049, 45512}, {19050, 45515}, {26401, 45349}, {26425, 45351}, {26501, 45519}, {26510, 45516}, {26519, 45522}, {26524, 45525}, {35819, 45565}, {45410, 45497}

X(45586) = X(35816)-of-these triangles: {1st anti-Kenmotu-free-vertices, outer-Yff tangents}
X(45586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45526, 372), (372, 45572, 45584), (12595, 16202, 45587)

X(45587) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES AND OUTER-YFF TANGENTS

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

X(45587) lies on these lines: {1, 371}, {3, 45497}, {11, 45559}, {39, 44645}, {182, 12595}, {372, 45423}, {642, 10527}, {5058, 44646}, {7692, 35252}, {9738, 10680}, {10267, 10931}, {10529, 45509}, {10532, 45545}, {10804, 45505}, {10806, 45511}, {10835, 45533}, {10879, 45539}, {10916, 45547}, {10932, 45551}, {10943, 45555}, {10949, 45557}, {10957, 45561}, {10959, 45563}, {10966, 45541}, {11240, 41491}, {11249, 45499}, {11401, 45503}, {11510, 45521}, {11883, 45535}, {11884, 45537}, {11915, 45549}, {11957, 45567}, {11958, 45568}, {12001, 45579}, {13907, 45576}, {13965, 45575}, {18543, 45378}, {18544, 45543}, {18967, 45507}, {19049, 45514}, {19050, 45513}, {26401, 45352}, {26425, 45350}, {26501, 45517}, {26510, 45518}, {26519, 45524}, {26524, 45523}, {35818, 45564}, {45411, 45496}

X(45587) = X(35817)-of-outer-Yff tangents triangle
X(45587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 45527, 371), (371, 45573, 45585), (12595, 16202, 45586)

X(45588) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 1st AURIGA

Barycentrics    a*(-4*((a^4-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a-4*b^2*c^2+(b^2-c^2)^2)*S+(b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a-(b^4-c^4)*(b^2-c^2))*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*((a^4-6*(b^2+c^2)*a^2-4*b^2*c^2+(b^2-c^2)^2)*S-a^6+(b^2+c^2)*a^4+(b^4+10*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2))) : :

X(45588) lies on these lines: {55, 45590}, {494, 5597}, {5491, 5599}, {5598, 26504}, {5601, 26503}, {6464, 45589}, {8186, 26299}, {8190, 26305}, {8196, 26329}, {8197, 26443}, {8198, 45594}, {8199, 26338}, {8200, 26467}, {8201, 45604}, {11366, 26368}, {11384, 26374}, {11492, 26502}, {11493, 26323}, {11822, 26293}, {11837, 26428}, {11843, 26440}, {11861, 26313}, {11863, 26448}, {11865, 26489}, {11867, 26484}, {11869, 26478}, {11871, 26472}, {11873, 26354}, {11875, 45609}, {11877, 45611}, {11879, 45613}, {11881, 26511}, {11883, 26510}, {13890, 45605}, {13944, 45608}, {18495, 45592}, {18523, 45379}, {18955, 26434}, {19007, 26455}, {19008, 26461}, {26416, 45353}, {26507, 45620}, {26508, 45625}, {26509, 45627}, {35778, 45599}, {35781, 45602}, {44600, 45595}, {44601, 45598}, {45412, 45430}, {45414, 45431}, {45516, 45534}, {45518, 45535}

X(45588) = X(45361)-of-1st Auriga triangle

X(45589) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 1st AURIGA

Barycentrics    a*(-4*(-(a^4-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a-4*b^2*c^2+(b^2-c^2)^2)*S+(b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a-(b^4-c^4)*(b^2-c^2))*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(-(a^4-6*(b^2+c^2)*a^2-4*b^2*c^2+(b^2-c^2)^2)*S-a^6+(b^2+c^2)*a^4+(b^4+10*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2))) : :

X(45589) lies on these lines: {55, 45591}, {493, 5597}, {5490, 5599}, {5598, 26495}, {5601, 26494}, {6464, 45588}, {8186, 26298}, {8190, 26304}, {8196, 26328}, {8197, 26442}, {8198, 26337}, {8199, 26347}, {8200, 26466}, {8202, 45603}, {11366, 26367}, {11384, 26373}, {11492, 26493}, {11493, 26322}, {11822, 26292}, {11837, 26427}, {11843, 26439}, {11861, 26312}, {11863, 26447}, {11865, 26488}, {11867, 26483}, {11869, 26477}, {11871, 26471}, {11873, 26353}, {11875, 45610}, {11877, 45612}, {11879, 45614}, {11881, 45615}, {11883, 26501}, {13890, 45607}, {13944, 45606}, {18495, 45593}, {18521, 45379}, {18955, 26433}, {19007, 26454}, {19008, 26460}, {26415, 45353}, {26498, 45620}, {26499, 45625}, {26500, 45627}, {35778, 45601}, {35781, 45600}, {44600, 45597}, {44601, 45596}, {45413, 45431}, {45415, 45430}, {45517, 45535}, {45519, 45534}

X(45589) = X(45362)-of-1st Auriga triangle

X(45590) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 2nd AURIGA

Barycentrics    a*(4*((a^4-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a-4*b^2*c^2+(b^2-c^2)^2)*S+(b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a-(b^4-c^4)*(b^2-c^2))*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*((a^4-6*(b^2+c^2)*a^2-4*b^2*c^2+(b^2-c^2)^2)*S-a^6+(b^2+c^2)*a^4+(b^4+10*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2))) : :

X(45590) lies on these lines: {55, 45588}, {494, 5598}, {5491, 5600}, {5597, 26504}, {5602, 26503}, {6464, 45591}, {8187, 26299}, {8191, 26305}, {8203, 26329}, {8204, 26443}, {8205, 45594}, {8206, 26338}, {8207, 26467}, {8208, 45604}, {11367, 26368}, {11385, 26374}, {11492, 26323}, {11493, 26502}, {11823, 26293}, {11838, 26428}, {11844, 26440}, {11862, 26313}, {11864, 26448}, {11866, 26489}, {11868, 26484}, {11870, 26478}, {11872, 26472}, {11874, 26354}, {11876, 45609}, {11878, 45611}, {11880, 45613}, {11882, 26511}, {11884, 26510}, {13891, 45605}, {13945, 45608}, {18497, 45592}, {18523, 45380}, {18956, 26434}, {19009, 26455}, {19010, 26461}, {26392, 45354}, {26507, 45621}, {26508, 45626}, {26509, 45628}, {35779, 45602}, {35780, 45599}, {44602, 45595}, {44603, 45598}, {45412, 45432}, {45414, 45433}, {45516, 45536}, {45518, 45537}

X(45590) = X(45363)-of-2nd Auriga triangle

X(45591) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 2nd AURIGA

Barycentrics    a*(4*(-(a^4-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a-4*b^2*c^2+(b^2-c^2)^2)*S+(b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a-(b^4-c^4)*(b^2-c^2))*S*sqrt(R*(4*R+r))+a*(a+b+c)*(-a+b+c)*(-(a^4-6*(b^2+c^2)*a^2-4*b^2*c^2+(b^2-c^2)^2)*S-a^6+(b^2+c^2)*a^4+(b^4+10*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2))) : :

X(45591) lies on these lines: {55, 45589}, {493, 5598}, {5490, 5600}, {5597, 26495}, {5602, 26494}, {6464, 45590}, {8187, 26298}, {8191, 26304}, {8203, 26328}, {8204, 26442}, {8205, 26337}, {8206, 26347}, {8207, 26466}, {8209, 45603}, {11367, 26367}, {11385, 26373}, {11492, 26322}, {11493, 26493}, {11823, 26292}, {11838, 26427}, {11844, 26439}, {11862, 26312}, {11864, 26447}, {11866, 26488}, {11868, 26483}, {11870, 26477}, {11872, 26471}, {11874, 26353}, {11876, 45610}, {11878, 45612}, {11880, 45614}, {11882, 45615}, {11884, 26501}, {13891, 45607}, {13945, 45606}, {18497, 45593}, {18521, 45380}, {18956, 26433}, {19009, 26454}, {19010, 26460}, {26391, 45354}, {26498, 45621}, {26499, 45626}, {26500, 45628}, {35779, 45600}, {35780, 45601}, {44602, 45597}, {44603, 45596}, {45413, 45433}, {45415, 45432}, {45517, 45537}, {45519, 45536}

X(45591) = X(45364)-of-2nd Auriga triangle

X(45592) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND EHRMANN-MID

Barycentrics    -(a^8-9*(b^2+c^2)*a^6+(b^2+c^2)^2*a^4+(b^2+c^2)*((3*b^2-3*c^2)^2-4*b^2*c^2)*a^2-2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2)*S+2*a^10-4*(b^2+c^2)*a^8+2*(b^4-4*b^2*c^2+c^4)*a^6-2*(b^2+c^2)*(b^4+c^4)*a^4+4*(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(45592) = 3*X(381)-X(45382)

X(45592) lies on these lines: {4, 26374}, {5, 26507}, {30, 5491}, {381, 494}, {382, 26293}, {546, 26329}, {1478, 26472}, {1479, 26478}, {3091, 26440}, {3583, 26354}, {3585, 26434}, {3843, 45609}, {6464, 45593}, {6564, 45595}, {6565, 45598}, {8946, 18494}, {9818, 26305}, {9955, 26368}, {10895, 45611}, {10896, 45613}, {12699, 26443}, {13665, 26461}, {13785, 26455}, {18415, 45414}, {18491, 26502}, {18492, 26299}, {18495, 45588}, {18497, 45590}, {18500, 26313}, {18502, 26428}, {18507, 26448}, {18509, 45594}, {18511, 26338}, {18516, 26489}, {18517, 26484}, {18520, 45604}, {18525, 26504}, {18538, 45605}, {18542, 26511}, {18544, 26510}, {18761, 26323}, {18762, 45608}, {24243, 26438}, {26392, 45355}, {26416, 45356}, {26508, 45630}, {26509, 45631}, {35786, 45599}, {35787, 45602}, {42022, 44665}, {45412, 45438}, {45516, 45542}, {45518, 45543}

X(45592) = X(45382)-of-Ehrmann-mid triangle
X(45592) = {X(381), X(18523)}-harmonic conjugate of X(494)

X(45593) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND EHRMANN-MID

Barycentrics    (a^8-9*(b^2+c^2)*a^6+(b^2+c^2)^2*a^4+(b^2+c^2)*((3*b^2-3*c^2)^2-4*b^2*c^2)*a^2-2*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2)*S+2*a^10-4*(b^2+c^2)*a^8+2*(b^4-4*b^2*c^2+c^4)*a^6-2*(b^2+c^2)*(b^4+c^4)*a^4+4*(b^2-c^2)^2*(b^4+4*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(45593) = 3*X(381)-X(45381)

X(45593) lies on these lines: {4, 26373}, {5, 26498}, {30, 5490}, {381, 493}, {382, 26292}, {546, 26328}, {1478, 26471}, {1479, 26477}, {3091, 26439}, {3583, 26353}, {3585, 26433}, {3843, 45610}, {6464, 45592}, {6564, 45597}, {6565, 45596}, {8948, 18494}, {9818, 26304}, {9955, 26367}, {10895, 45612}, {10896, 45614}, {12699, 26442}, {13665, 26460}, {13785, 26454}, {18414, 45415}, {18491, 26493}, {18492, 26298}, {18495, 45589}, {18497, 45591}, {18500, 26312}, {18502, 26427}, {18507, 26447}, {18509, 26337}, {18511, 26347}, {18516, 26488}, {18517, 26483}, {18522, 45603}, {18525, 26495}, {18538, 45607}, {18539, 24244}, {18542, 45615}, {18544, 26501}, {18761, 26322}, {18762, 45606}, {26391, 45355}, {26415, 45356}, {26499, 45630}, {26500, 45631}, {35786, 45601}, {35787, 45600}, {45413, 45439}, {45517, 45543}, {45519, 45542}

X(45593) = X(45381)-of-Ehrmann-mid triangle
X(45593) = {X(381), X(18521)}-harmonic conjugate of X(493)

X(45594) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND INNER-GREBE

Barycentrics    a^2*(b^2+c^2)*(a^4-4*b^2*c^2-2*(b^2+c^2)*(a^2-2*S)+(b^2-c^2)^2) : :
Barycentrics    (B-power of A-Lucas(-1) circle) + (C-power of A-Lucas(-1) circle) : :

X(45594) lies on these lines: {6, 494}, {76, 5491}, {755, 1307}, {1271, 26503}, {2353, 6458}, {5589, 26299}, {5595, 26305}, {5605, 26504}, {5689, 26443}, {5861, 26506}, {6202, 26329}, {6215, 26467}, {6464, 26337}, {8041, 26347}, {8198, 45588}, {8205, 45590}, {8216, 45604}, {8974, 45605}, {9994, 26313}, {10040, 45611}, {10048, 45613}, {10783, 26440}, {10792, 26428}, {10919, 26489}, {10921, 26484}, {10923, 26478}, {10925, 26472}, {10927, 26354}, {10929, 26511}, {10931, 26510}, {11370, 26368}, {11388, 26374}, {11497, 26502}, {11824, 26293}, {11901, 26448}, {11916, 45609}, {13949, 45608}, {18509, 45592}, {18523, 26336}, {18959, 26434}, {19359, 32575}, {22756, 26323}, {26334, 26392}, {26335, 26416}, {26339, 26505}, {26341, 26507}, {26342, 26508}, {26343, 26509}, {35792, 45599}, {35795, 45602}, {45516, 45550}, {45518, 45552}

X(45594) = barycentric product X(i)*X(j) for these {i, j}: {39, 5491}, {141, 494}, {826, 1307}
X(45594) = barycentric quotient X(i)/X(j) for these (i, j): (39, 3069), (494, 83), (1307, 4577)
X(45594) = trilinear product X(i)*X(j) for these {i, j}: {38, 494}, {1930, 26461}, {1964, 5491}
X(45594) = trilinear quotient X(i)/X(j) for these (i, j): (38, 3069), (494, 82), (1307, 4599), (1964, 6424)
X(45594) = crosspoint of X(494) and X(5491)
X(45594) = X(141)-Dao conjugate of-X(3069)
X(45594) = X(82)-isoconjugate-of-X(3069)
X(45594) = X(19033)-of-inner-Grebe triangle
X(45594) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (39, 3069), (494, 83), (1307, 4577)
X(45594) = endo-homothetic center of these triangles: anti-inner-Grebe and Lucas(-1) homothetic

X(45595) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 1st KENMOTU-CENTERS

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

X(45595) lies on these lines: {6, 494}, {39, 45518}, {230, 24243}, {371, 45599}, {372, 26507}, {485, 26467}, {590, 5491}, {1124, 45613}, {1151, 26293}, {1335, 45611}, {1587, 26440}, {1609, 1611}, {2066, 26354}, {2067, 26434}, {3053, 8946}, {3068, 26503}, {3071, 26329}, {3311, 45609}, {5062, 45516}, {5412, 26374}, {6419, 45602}, {6423, 42022}, {6464, 45597}, {6564, 45592}, {7968, 26368}, {8770, 44193}, {13665, 18523}, {13911, 26443}, {18991, 26299}, {19048, 26509}, {19050, 26508}, {26305, 44598}, {26313, 44604}, {26323, 44606}, {26392, 44582}, {26416, 44584}, {26428, 44586}, {26448, 44610}, {26472, 44623}, {26478, 31472}, {26484, 44620}, {26489, 44618}, {26502, 44590}, {26504, 44635}, {26505, 44594}, {26506, 44596}, {26510, 44645}, {26511, 44643}, {32788, 45608}, {44600, 45588}, {44602, 45590}, {44627, 45604}

X(45595) = isogonal conjugate of X(26494)
X(45595) = polar conjugate of the isotomic conjugate of X(42022)
X(45595) = barycentric product X(i)*X(j) for these {i, j}: {4, 42022}, {6, 26503}, {488, 8946}, {494, 3068}
X(45595) = barycentric quotient X(i)/X(j) for these (i, j): (32, 45596), (494, 5490)
X(45595) = trilinear product X(19)*X(42022)
X(45595) = trilinear quotient X(31)/X(45596)
X(45595) = crosssum of X(i) and X(j) for these (i, j): {6, 12978}, {493, 45603}
X(45595) = X(206)-Dao conjugate of-X(45596)
X(45595) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 45596}, {487, 19218}
X(45595) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 45596), (494, 5490)
X(45595) = X(45464)-of-1st Kenmotu-centers triangle
X(45595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 494, 45598), (494, 26461, 6), (1609, 1611, 45596), (5491, 45605, 590)

X(45596) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 2nd KENMOTU-CENTERS

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

X(45596) lies on these lines: {6, 493}, {39, 45519}, {230, 24244}, {371, 26498}, {372, 45600}, {486, 26466}, {615, 5490}, {1124, 45612}, {1152, 26292}, {1335, 45614}, {1588, 26439}, {1609, 1611}, {3053, 8948}, {3069, 26494}, {3070, 26328}, {3312, 45610}, {5058, 45517}, {5413, 26373}, {5414, 26353}, {6420, 45601}, {6464, 45598}, {6502, 26433}, {6565, 45593}, {7969, 26367}, {8770, 44192}, {13785, 18521}, {13973, 26442}, {18992, 26298}, {19047, 26500}, {19049, 26499}, {26304, 44599}, {26312, 44605}, {26322, 44607}, {26391, 44583}, {26415, 44585}, {26427, 44587}, {26447, 44611}, {26471, 44624}, {26477, 44622}, {26483, 44621}, {26488, 44619}, {26493, 44591}, {26495, 44636}, {26496, 44595}, {26497, 44597}, {26501, 44646}, {32787, 45607}, {44601, 45589}, {44603, 45591}, {44630, 45603}, {44644, 45615}

X(45596) = isogonal conjugate of X(26503)
X(45596) = barycentric product X(i)*X(j) for these {i, j}: {6, 26494}, {487, 8948}, {493, 3069}
X(45596) = barycentric quotient X(i)/X(j) for these (i, j): (32, 45595), (184, 42022), (493, 5491)
X(45596) = trilinear product X(31)*X(26494)
X(45596) = trilinear quotient X(i)/X(j) for these (i, j): (31, 45595), (48, 42022)
X(45596) = crosssum of X(i) and X(j) for these (i, j): {6, 12979}, {494, 45604}
X(45596) = X(206)-Dao conjugate of-X(45595)
X(45596) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 45595}, {92, 42022}, {488, 19217}
X(45596) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (32, 45595), (184, 42022), (493, 5491)
X(45596) = X(45465)-of-2nd Kenmotu-centers triangle
X(45596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 493, 45597), (493, 26454, 6), (1609, 1611, 45595), (5490, 45606, 615)

X(45597) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 1st KENMOTU-CENTERS

Barycentrics    a^2*((3*a^4-10*(b^2+c^2)*a^2-12*b^2*c^2+3*(b^2-c^2)^2)*S+a^6+(b^2+c^2)*a^4-(5*b^4+18*b^2*c^2+5*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)) : :

X(45597) lies on these lines: {6, 493}, {39, 45517}, {371, 45601}, {372, 26498}, {485, 26466}, {590, 5490}, {1124, 45614}, {1151, 26292}, {1335, 45612}, {1504, 13889}, {1587, 26439}, {2066, 26353}, {2067, 26433}, {3068, 26494}, {3071, 26328}, {3311, 45610}, {5062, 45519}, {5412, 26373}, {6419, 45600}, {6464, 45595}, {6564, 45593}, {7968, 26367}, {10318, 45598}, {13665, 18521}, {13911, 26442}, {18991, 26298}, {19048, 26500}, {19050, 26499}, {19422, 35840}, {26304, 44598}, {26312, 44604}, {26322, 44606}, {26391, 44582}, {26415, 44584}, {26427, 44586}, {26447, 44610}, {26471, 44623}, {26477, 31472}, {26483, 44620}, {26488, 44618}, {26493, 44590}, {26495, 44635}, {26496, 44594}, {26497, 44596}, {26501, 44645}, {32788, 45606}, {44600, 45589}, {44602, 45591}, {44629, 45603}, {44643, 45615}

X(45597) = barycentric product X(493)*X(32785)
X(45597) = X(45467)-of-1st Kenmotu-centers triangle
X(45597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 493, 45596), (493, 26460, 6), (5490, 45607, 590)

X(45598) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 2nd KENMOTU-CENTERS

Barycentrics    a^2*(-(3*a^4-10*(b^2+c^2)*a^2-12*b^2*c^2+3*(b^2-c^2)^2)*S+a^6+(b^2+c^2)*a^4-(5*b^4+18*b^2*c^2+5*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)) : :

X(45598) lies on these lines: {6, 494}, {39, 45516}, {371, 26507}, {372, 45602}, {486, 26467}, {615, 5491}, {1124, 45611}, {1152, 26293}, {1335, 45613}, {1505, 13943}, {1588, 26440}, {3069, 26503}, {3070, 26329}, {3312, 45609}, {5058, 45518}, {5413, 26374}, {5414, 26354}, {6420, 45599}, {6464, 45596}, {6502, 26434}, {6565, 45592}, {7969, 26368}, {10318, 45597}, {13785, 18523}, {13973, 26443}, {18992, 26299}, {19047, 26509}, {19049, 26508}, {19423, 35841}, {26305, 44599}, {26313, 44605}, {26323, 44607}, {26392, 44583}, {26416, 44585}, {26428, 44587}, {26448, 44611}, {26472, 44624}, {26478, 44622}, {26484, 44621}, {26489, 44619}, {26502, 44591}, {26504, 44636}, {26505, 44595}, {26506, 44597}, {26510, 44646}, {26511, 44644}, {32787, 45605}, {44601, 45588}, {44603, 45590}, {44628, 45604}

X(45598) = barycentric product X(494)*X(32786)
X(45598) = X(45466)-of-2nd Kenmotu-centers triangle
X(45598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 494, 45595), (494, 26455, 6), (5491, 45608, 615)

X(45599) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 1st KENMOTU-FREE-VERTICES

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

X(45599) lies on these lines: {6, 45602}, {25, 372}, {371, 45595}, {485, 26503}, {3103, 45414}, {5491, 10576}, {6200, 26293}, {6396, 26507}, {6419, 26461}, {6420, 45598}, {6464, 45601}, {6560, 26440}, {6564, 26467}, {6565, 26329}, {12978, 44629}, {18523, 23251}, {26299, 35774}, {26313, 35782}, {26323, 35784}, {26338, 35794}, {26354, 35808}, {26368, 35762}, {26392, 45357}, {26416, 45359}, {26428, 35766}, {26434, 35768}, {26443, 35788}, {26448, 35790}, {26455, 35770}, {26472, 35802}, {26478, 35800}, {26484, 35798}, {26489, 35796}, {26502, 35772}, {26504, 35810}, {26506, 39661}, {26508, 45640}, {26509, 45642}, {26510, 35818}, {26511, 35816}, {35769, 45613}, {35778, 45588}, {35780, 45590}, {35786, 45592}, {35792, 45594}, {35804, 45604}, {35809, 45611}, {35812, 45605}, {35814, 45608}, {42022, 45412}, {45518, 45564}

X(45599) = X(45566)-of-1st Kenmotu-free-vertices triangle
X(45599) = {X(6), X(45609)}-harmonic conjugate of X(45602)

X(45600) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 2nd KENMOTU-FREE-VERTICES

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

X(45600) lies on these lines: {6, 45601}, {25, 371}, {372, 45596}, {486, 26494}, {3102, 45415}, {5490, 10577}, {6200, 26498}, {6396, 26292}, {6419, 45597}, {6420, 26454}, {6464, 45602}, {6561, 26439}, {6564, 26328}, {6565, 26466}, {12979, 44628}, {18521, 23261}, {26298, 35775}, {26312, 35783}, {26322, 35785}, {26337, 35795}, {26347, 35793}, {26353, 35809}, {26367, 35763}, {26391, 45360}, {26415, 45358}, {26427, 35767}, {26433, 35769}, {26442, 35789}, {26447, 35791}, {26460, 35771}, {26471, 35803}, {26477, 35801}, {26483, 35799}, {26488, 35797}, {26493, 35773}, {26495, 35811}, {26496, 39660}, {26499, 45641}, {26500, 45643}, {26501, 35819}, {35768, 45614}, {35779, 45591}, {35781, 45589}, {35787, 45593}, {35805, 45603}, {35808, 45612}, {35813, 45606}, {35815, 43592}, {35817, 45615}, {45413, 45463}, {45519, 45565}

X(45600) = X(45567)-of-2nd Kenmotu-free-vertices triangle
X(45600) = {X(6), X(45610)}-harmonic conjugate of X(45601)

X(45601) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 1st KENMOTU-FREE-VERTICES

Barycentrics    a^2*(-2*(3*a^6-(b^2+c^2)*a^4-(11*b^4+46*b^2*c^2+11*c^4)*a^2+(b^2+c^2)*(9*b^4-14*b^2*c^2+9*c^4))*S+3*a^8-20*(b^2+c^2)*a^6+2*(19*b^4+16*b^2*c^2+19*c^4)*a^4-4*(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)*a^2+(7*b^4-26*b^2*c^2+7*c^4)*(b^2-c^2)^2) : :

X(45601) lies on these lines: {6, 45600}, {371, 45597}, {372, 493}, {485, 26494}, {3103, 45413}, {5490, 10576}, {6200, 26292}, {6396, 26498}, {6419, 26460}, {6420, 45596}, {6464, 45599}, {6560, 26439}, {6564, 26466}, {6565, 26328}, {18521, 23251}, {26298, 35774}, {26304, 35776}, {26312, 35782}, {26322, 35784}, {26337, 35792}, {26347, 35794}, {26353, 35808}, {26367, 35762}, {26373, 35764}, {26391, 45357}, {26415, 45359}, {26427, 35766}, {26433, 35768}, {26442, 35788}, {26447, 35790}, {26454, 35770}, {26471, 35802}, {26477, 35800}, {26483, 35798}, {26488, 35796}, {26493, 35772}, {26495, 35810}, {26497, 39661}, {26499, 45640}, {26500, 45642}, {26501, 35818}, {35769, 45614}, {35778, 45589}, {35780, 45591}, {35786, 45593}, {35806, 45603}, {35809, 45612}, {35812, 45607}, {35814, 45606}, {35816, 45615}, {45415, 45462}, {45517, 45564}

X(45601) = X(45569)-of-1st Kenmotu-free-vertices triangle
X(45601) = {X(6), X(45610)}-harmonic conjugate of X(45600)

X(45602) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 2nd KENMOTU-FREE-VERTICES

Barycentrics    a^2*(2*(3*a^6-(b^2+c^2)*a^4-(11*b^4+46*b^2*c^2+11*c^4)*a^2+(b^2+c^2)*(9*b^4-14*b^2*c^2+9*c^4))*S+3*a^8-20*(b^2+c^2)*a^6+2*(19*b^4+16*b^2*c^2+19*c^4)*a^4-4*(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)*a^2+(7*b^4-26*b^2*c^2+7*c^4)*(b^2-c^2)^2) : :

X(45602) lies on these lines: {6, 45599}, {371, 494}, {372, 45598}, {486, 26503}, {3102, 45412}, {5491, 10577}, {6200, 26507}, {6396, 26293}, {6419, 45595}, {6420, 26455}, {6464, 45600}, {6561, 26440}, {6564, 26329}, {6565, 26467}, {18523, 23261}, {26299, 35775}, {26305, 35777}, {26313, 35783}, {26323, 35785}, {26338, 35793}, {26354, 35809}, {26368, 35763}, {26374, 35765}, {26392, 45360}, {26416, 45358}, {26428, 35767}, {26434, 35769}, {26443, 35789}, {26448, 35791}, {26461, 35771}, {26472, 35803}, {26478, 35801}, {26484, 35799}, {26489, 35797}, {26502, 35773}, {26504, 35811}, {26505, 39660}, {26508, 45641}, {26509, 45643}, {26510, 35819}, {26511, 35817}, {35768, 45613}, {35779, 45590}, {35781, 45588}, {35787, 45592}, {35795, 45594}, {35807, 45604}, {35808, 45611}, {35813, 45608}, {35815, 45605}, {45414, 45463}, {45516, 45565}

X(45602) = X(45568)-of-2nd Kenmotu-free-vertices triangle
X(45602) = {X(6), X(45609)}-harmonic conjugate of X(45599)

X(45603) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND LUCAS(-1) HOMOTHETIC

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

X(45603) lies on these lines: {39, 493}, {5490, 6504}, {6463, 26494}, {8189, 26298}, {8195, 26304}, {8202, 45589}, {8209, 45591}, {8211, 26495}, {8213, 26328}, {8215, 26442}, {8217, 26337}, {8219, 26347}, {8221, 26466}, {10876, 26312}, {10946, 26488}, {10952, 26483}, {11378, 26367}, {11395, 26373}, {11504, 26493}, {11829, 26292}, {11841, 26427}, {11847, 26439}, {11908, 26447}, {11931, 26477}, {11933, 26471}, {11948, 26353}, {11950, 45610}, {11952, 45612}, {11954, 45614}, {11956, 45615}, {11958, 26501}, {13900, 45607}, {13957, 45606}, {18521, 45382}, {18522, 45593}, {18964, 26433}, {19033, 26454}, {19034, 26460}, {22762, 26322}, {26391, 45361}, {26415, 45363}, {26498, 45624}, {26499, 45644}, {26500, 45646}, {35805, 45600}, {35806, 45601}, {44629, 45597}, {44630, 45596}, {45413, 45466}, {45415, 45464}, {45517, 45568}, {45519, 45566}

X(45603) = reflection of X(45604) in X(6503)
X(45603) = X(45604)-of-Lucas(-1) homothetic triangle
X(45603) = X(494)-Dao conjugate of-X(26503)

X(45604) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND LUCAS(+1) HOMOTHETIC

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

X(45604) lies on these lines: {39, 493}, {5491, 6504}, {6462, 26503}, {8188, 26299}, {8194, 26305}, {8201, 45588}, {8208, 45590}, {8210, 26504}, {8212, 26329}, {8214, 26443}, {8216, 45594}, {8218, 26338}, {8220, 26467}, {10875, 26313}, {10945, 26489}, {10951, 26484}, {11377, 26368}, {11394, 26374}, {11503, 26502}, {11828, 26293}, {11840, 26428}, {11846, 26440}, {11907, 26448}, {11930, 26478}, {11932, 26472}, {11947, 26354}, {11949, 45609}, {11951, 45611}, {11953, 45613}, {11955, 26511}, {11957, 26510}, {13899, 45605}, {13956, 45608}, {18520, 45592}, {18523, 45381}, {18963, 26434}, {19031, 26455}, {19032, 26461}, {22761, 26323}, {26392, 45362}, {26416, 45364}, {26507, 45623}, {26508, 45645}, {26509, 45647}, {35804, 45599}, {35807, 45602}, {44627, 45595}, {44628, 45598}, {45412, 45467}, {45414, 45465}, {45516, 45569}, {45518, 45567}

X(45604) = reflection of X(45603) in X(6503)
X(45604) = X(45603)-of-Lucas(+1) homothetic triangle
X(45604) = X(493)-Dao conjugate of-X(26494)

X(45605) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 3rd TRI-SQUARES-CENTRAL

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

X(45605) lies on these lines: {2, 26461}, {6, 45608}, {371, 26329}, {494, 3068}, {590, 5491}, {6464, 45607}, {7583, 26507}, {7585, 26455}, {8972, 26503}, {8974, 45594}, {8975, 26338}, {8976, 26467}, {9540, 26293}, {13883, 26368}, {13884, 26374}, {13885, 26428}, {13886, 26440}, {13887, 26502}, {13888, 26299}, {13889, 26305}, {13890, 45588}, {13891, 45590}, {13892, 26313}, {13893, 26443}, {13894, 26448}, {13895, 26489}, {13896, 26484}, {13897, 26478}, {13898, 26472}, {13899, 45604}, {13901, 26354}, {13902, 26504}, {13903, 45609}, {13904, 45611}, {13905, 45613}, {13906, 26511}, {13907, 26510}, {18523, 45384}, {18538, 45592}, {18965, 26434}, {22763, 26323}, {26392, 45365}, {26416, 45368}, {26508, 45650}, {26509, 45652}, {32787, 45598}, {35812, 45599}, {35815, 45602}, {45412, 45484}, {45414, 45486}, {45516, 45574}, {45518, 45576}

X(45605) = {X(590), X(45595)}-harmonic conjugate of X(5491)

X(45606) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 4th TRI-SQUARES-CENTRAL

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

X(45606) lies on these lines: {2, 26454}, {6, 45607}, {372, 26328}, {493, 3069}, {615, 5490}, {6464, 45608}, {7584, 26498}, {7586, 26460}, {13935, 26292}, {13936, 26367}, {13937, 26373}, {13938, 26427}, {13939, 26439}, {13940, 26493}, {13941, 26494}, {13942, 26298}, {13943, 26304}, {13944, 45589}, {13945, 45591}, {13946, 26312}, {13947, 26442}, {13948, 26447}, {13949, 26337}, {13950, 26347}, {13951, 26466}, {13952, 26488}, {13953, 26483}, {13954, 26477}, {13955, 26471}, {13957, 45603}, {13958, 26353}, {13959, 26495}, {13961, 45610}, {13962, 45612}, {13963, 45614}, {13964, 45615}, {13965, 26501}, {18521, 45385}, {18762, 45593}, {18966, 26433}, {22764, 26322}, {26391, 45366}, {26415, 45367}, {26499, 45651}, {26500, 45653}, {32788, 45597}, {35813, 45600}, {35814, 45601}, {45413, 45485}, {45415, 45487}, {45517, 45575}, {45519, 45577}

X(45606) = {X(615), X(45596)}-harmonic conjugate of X(5490)

X(45607) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    3*a^8+8*(b^2+c^2)*a^6-2*(13*b^4+36*b^2*c^2+13*c^4)*a^4+8*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2+4*(3*a^6-9*(b^2+c^2)*a^4+(b^4-22*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*S-(b^2-c^2)^2*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2) : :

X(45607) lies on these lines: {2, 26460}, {6, 45606}, {371, 26328}, {393, 493}, {485, 18926}, {590, 5490}, {6464, 45605}, {7583, 26498}, {7585, 26454}, {8972, 26494}, {8974, 26337}, {8975, 26347}, {8976, 26466}, {9540, 26292}, {13883, 26367}, {13884, 26373}, {13885, 26427}, {13886, 26439}, {13887, 26493}, {13888, 26298}, {13889, 26304}, {13890, 45589}, {13891, 45591}, {13892, 26312}, {13893, 26442}, {13894, 26447}, {13895, 26488}, {13896, 26483}, {13897, 26477}, {13898, 26471}, {13900, 45603}, {13901, 26353}, {13902, 26495}, {13903, 45610}, {13904, 45612}, {13905, 45614}, {13906, 45615}, {13907, 26501}, {18521, 45384}, {18538, 45593}, {18965, 26433}, {22763, 26322}, {26391, 45365}, {26415, 45368}, {26499, 45650}, {26500, 45652}, {32787, 45596}, {35812, 45601}, {35815, 43592}, {45413, 45486}, {45415, 45484}, {45517, 45576}, {45519, 45574}

X(45607) = {X(590), X(45597)}-harmonic conjugate of X(5490)

X(45608) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND 4th TRI-SQUARES-CENTRAL

Barycentrics    3*a^8+8*(b^2+c^2)*a^6-2*(13*b^4+36*b^2*c^2+13*c^4)*a^4+8*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-4*(3*a^6-9*(b^2+c^2)*a^4+(b^4-22*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*S-(b^2-c^2)^2*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2) : :

X(45608) lies on these lines: {2, 26455}, {6, 45605}, {372, 26329}, {393, 494}, {486, 18927}, {615, 5491}, {6464, 45606}, {7584, 26507}, {7586, 26461}, {13935, 26293}, {13936, 26368}, {13937, 26374}, {13938, 26428}, {13939, 26440}, {13940, 26502}, {13941, 26503}, {13942, 26299}, {13943, 26305}, {13944, 45588}, {13945, 45590}, {13946, 26313}, {13947, 26443}, {13948, 26448}, {13949, 45594}, {13950, 26338}, {13951, 26467}, {13952, 26489}, {13953, 26484}, {13954, 26478}, {13955, 26472}, {13956, 45604}, {13958, 26354}, {13959, 26504}, {13961, 45609}, {13962, 45611}, {13963, 45613}, {13964, 26511}, {13965, 26510}, {18523, 45385}, {18762, 45592}, {18966, 26434}, {22764, 26323}, {26392, 45366}, {26416, 45367}, {26508, 45651}, {26509, 45653}, {32788, 45595}, {35813, 45602}, {35814, 45599}, {45412, 45487}, {45414, 45485}, {45516, 45577}, {45518, 45575}

X(45608) = {X(615), X(45598)}-harmonic conjugate of X(5491)

X(45609) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND X3-ABC REFLECTIONS

Barycentrics    (a^8-6*(b^2+c^2)*a^6+4*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(5*b^4-12*b^2*c^2+5*c^4)*a^2+(a^6+(b^2+c^2)*a^4-(9*b^4+26*b^2*c^2+9*c^4)*a^2+(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4))*S+(b^2-c^2)^2*(b^2-3*c^2)*(3*b^2-c^2))*a^2 : :

X(45609) lies on these lines: {3, 494}, {4, 18523}, {5, 26503}, {6, 45599}, {30, 26440}, {381, 26329}, {517, 26299}, {999, 26434}, {1598, 26374}, {1649, 11849}, {1656, 5491}, {3295, 26354}, {3311, 45595}, {3312, 45598}, {3843, 45592}, {5790, 26443}, {6417, 26461}, {6418, 26455}, {6464, 45610}, {7517, 26305}, {9301, 26313}, {9654, 26478}, {9669, 26472}, {10246, 26368}, {10247, 26504}, {10679, 26509}, {10680, 26508}, {11842, 26428}, {11875, 45588}, {11876, 45590}, {11911, 26448}, {11916, 45594}, {11917, 26338}, {11928, 26489}, {11929, 26484}, {11949, 45604}, {12000, 26511}, {12001, 26510}, {13903, 45605}, {13961, 45608}, {19419, 45414}, {22765, 26323}, {26392, 45369}, {26416, 45370}, {45412, 45488}, {45516, 45578}, {45518, 45579}

X(45609) = reflection of X(3) in X(45624)
X(45609) = X(45624)-of-X3-ABC reflections triangle
X(45609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 26293, 26507), (26293, 26507, 3), (26329, 26467, 381), (26354, 45611, 3295), (26434, 45613, 999), (45599, 45602, 6)

X(45610) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND X3-ABC REFLECTIONS

Barycentrics    a^2*(a^8-6*(b^2+c^2)*a^6+4*(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(b^2+c^2)*(5*b^4-12*b^2*c^2+5*c^4)*a^2-(a^6+(b^2+c^2)*a^4-(9*b^4+26*b^2*c^2+9*c^4)*a^2+(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4))*S+(b^2-c^2)^2*(b^2-3*c^2)*(3*b^2-c^2)) : :

X(45610) lies on these lines: {3, 493}, {4, 18521}, {5, 26494}, {6, 45600}, {30, 26439}, {381, 26328}, {517, 26298}, {999, 26433}, {1598, 26373}, {1656, 5490}, {3295, 26353}, {3311, 45597}, {3312, 45596}, {3843, 45593}, {5790, 26442}, {6417, 26460}, {6418, 26454}, {6464, 45609}, {7517, 26304}, {9301, 26312}, {9654, 26477}, {9669, 26471}, {10246, 26367}, {10247, 26495}, {10679, 26500}, {10680, 26499}, {11842, 26427}, {11849, 26493}, {11875, 45589}, {11876, 45591}, {11911, 26447}, {11916, 26337}, {11917, 26347}, {11928, 26488}, {11929, 26483}, {11950, 45603}, {12000, 45615}, {12001, 26501}, {13903, 45607}, {13961, 45606}, {19418, 45415}, {22765, 26322}, {26391, 45369}, {26415, 45370}, {45413, 45489}, {45517, 45579}, {45519, 45578}

X(45610) = reflection of X(3) in X(45623)
X(45610) = X(45623)-of-X3-ABC reflections triangle
X(45610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 26292, 26498), (26292, 26498, 3), (26328, 26466, 381), (26353, 45612, 3295), (26433, 45614, 999), (45600, 45601, 6)

X(45611) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND INNER-YFF

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

X(45611) lies on these lines: {1, 494}, {3, 26434}, {5, 26472}, {12, 26467}, {35, 26293}, {56, 26507}, {388, 26440}, {495, 26478}, {498, 5491}, {1124, 45598}, {1335, 45595}, {1479, 26329}, {3085, 26503}, {3295, 26354}, {3299, 26455}, {3301, 26461}, {6464, 45612}, {9654, 18523}, {10037, 26305}, {10038, 26313}, {10039, 26443}, {10040, 45594}, {10041, 26338}, {10318, 45614}, {10523, 26489}, {10801, 26428}, {10895, 45592}, {10954, 26484}, {11398, 26374}, {11507, 26502}, {11877, 45588}, {11878, 45590}, {11912, 26448}, {11951, 45604}, {13904, 45605}, {13962, 45608}, {22766, 26323}, {26392, 45371}, {26416, 45372}, {35808, 45602}, {35809, 45599}, {45412, 45490}, {45414, 45491}, {45516, 45580}, {45518, 45581}

X(45611) = X(45644)-of-inner-Yff triangle
X(45611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 494, 45613), (494, 26504, 26508), (3295, 45609, 26354)

X(45612) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND INNER-YFF

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

X(45612) lies on these lines: {1, 493}, {3, 26433}, {5, 26471}, {12, 26466}, {35, 26292}, {56, 26498}, {388, 26439}, {495, 26477}, {498, 5490}, {1124, 45596}, {1335, 45597}, {1479, 26328}, {3085, 26494}, {3295, 26353}, {3299, 26454}, {3301, 26460}, {6464, 45611}, {9654, 18521}, {10037, 26304}, {10038, 26312}, {10039, 26442}, {10040, 26337}, {10041, 26347}, {10318, 45613}, {10523, 26488}, {10801, 26427}, {10895, 45593}, {10954, 26483}, {11398, 26373}, {11507, 26493}, {11877, 45589}, {11878, 45591}, {11912, 26447}, {11952, 45603}, {13904, 45607}, {13962, 45606}, {22766, 26322}, {26391, 45371}, {26415, 45372}, {35808, 45600}, {35809, 45601}, {45413, 45491}, {45415, 45490}, {45517, 45581}, {45519, 45580}

X(45612) = X(45645)-of-inner-Yff triangle
X(45612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 493, 45614), (493, 26495, 26499), (3295, 45610, 26353)

X(45613) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC AND OUTER-YFF

Barycentrics    a^2*(a^8-2*(2*b^2-b*c+2*c^2)*a^6+2*(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^4-2*(2*b^6+2*c^6+(b^2-4*b*c+c^2)*(b-c)^2*b*c)*a^2+(2*(2*b^2-b*c+2*c^2)*a^4-4*(2*b^2-3*b*c+2*c^2)*(b^2+c^2)*a^2+2*(2*b^3-b^2*c+c^3)*(b^3-b*c^2+2*c^3))*S+(b^2-c^2)^2*(b-c)^2*(b^2+4*b*c+c^2)) : :

X(45613) lies on these lines: {1, 494}, {3, 26354}, {5, 26478}, {11, 26467}, {36, 26293}, {55, 26507}, {496, 26472}, {497, 26440}, {499, 5491}, {999, 26434}, {1124, 45595}, {1335, 45598}, {1478, 26329}, {1737, 26443}, {3086, 26503}, {3299, 26461}, {3301, 26455}, {6464, 45614}, {9669, 18523}, {10046, 26305}, {10047, 26313}, {10048, 45594}, {10049, 26338}, {10318, 45612}, {10523, 26484}, {10802, 26428}, {10896, 45592}, {10948, 26489}, {11399, 26374}, {11508, 26502}, {11879, 45588}, {11880, 45590}, {11913, 26448}, {11953, 45604}, {13905, 45605}, {13963, 45608}, {22767, 26323}, {26392, 45373}, {26416, 45374}, {35768, 45602}, {35769, 45599}, {45412, 45492}, {45414, 45493}, {45516, 45582}, {45518, 45583}

X(45613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 494, 45611), (494, 26504, 26509), (999, 45609, 26434)

X(45614) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND OUTER-YFF

Barycentrics    a^2*(a^8-2*(2*b^2-b*c+2*c^2)*a^6+2*(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^4-2*(2*b^6+2*c^6+(b^2-4*b*c+c^2)*(b-c)^2*b*c)*a^2-(2*(2*b^2-b*c+2*c^2)*a^4-4*(2*b^2-3*b*c+2*c^2)*(b^2+c^2)*a^2+2*(2*b^3-b^2*c+c^3)*(b^3-b*c^2+2*c^3))*S+(b^2-c^2)^2*(b-c)^2*(b^2+4*b*c+c^2)) : :

X(45614) lies on these lines: {1, 493}, {3, 26353}, {5, 26477}, {11, 26466}, {36, 26292}, {55, 26498}, {496, 26471}, {497, 26439}, {499, 5490}, {999, 26433}, {1124, 45597}, {1335, 45596}, {1478, 26328}, {1737, 26442}, {3086, 26494}, {3299, 26460}, {3301, 26454}, {6464, 45613}, {9669, 18521}, {10046, 26304}, {10047, 26312}, {10048, 26337}, {10049, 26347}, {10318, 45611}, {10523, 26483}, {10802, 26427}, {10896, 45593}, {10948, 26488}, {11399, 26373}, {11508, 26493}, {11879, 45589}, {11880, 45591}, {11913, 26447}, {11954, 45603}, {13905, 45607}, {13963, 45606}, {22767, 26322}, {26391, 45373}, {26415, 45374}, {35768, 45600}, {35769, 45601}, {45413, 45493}, {45415, 45492}, {45517, 45583}, {45519, 45582}

X(45614) = X(45647)-of-outer-Yff triangle
X(45614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 493, 45612), (493, 26495, 26500), (999, 45610, 26433)

X(45615) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC AND INNER-YFF TANGENTS

Barycentrics    a^2*(4*((b^2+b*c+c^2)*a^4-2*(b^2+c^2)*(b+c)^2*a^2+2*(b+c)*(b^2+c^2)*b*c*a+b^6+c^6+(b^4+c^4-b*c*(b^2+6*b*c+c^2))*b*c)*S+(-a+b+c)*(a^7+(b+c)*a^6-3*(b^2+c^2)*a^5-(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b+c)*(3*b^4+3*c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^2-(b^2-4*b*c+c^2)*(b^2+c^2)^2*a+(b^2-c^2)*(b-c)*(-b^4-c^4+2*b*c*(2*b^2+3*b*c+2*c^2)))) : :

X(45615) lies on these lines: {1, 493}, {12, 26488}, {5490, 5552}, {6464, 26511}, {8948, 26378}, {10528, 26494}, {10531, 26328}, {10803, 26427}, {10805, 26439}, {10834, 26304}, {10878, 26312}, {10915, 26442}, {10929, 26337}, {10930, 26347}, {10942, 26466}, {10955, 26483}, {10956, 26477}, {10958, 26471}, {10965, 26353}, {11248, 26292}, {11400, 26373}, {11509, 26433}, {11881, 45589}, {11882, 45591}, {11914, 26447}, {11956, 45603}, {12000, 45610}, {13906, 45607}, {13964, 45606}, {16203, 26498}, {18521, 18545}, {18542, 45593}, {19047, 26454}, {19048, 26460}, {22768, 26322}, {24244, 26518}, {26391, 26402}, {26415, 26426}, {26496, 26520}, {26497, 26525}, {35816, 45601}, {35817, 45600}, {44643, 45597}, {44644, 45596}, {45413, 45495}, {45415, 45494}, {45517, 45585}, {45519, 45584}

X(45615) = X(11957)-of-inner-Yff tangents triangle
X(45615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 26500, 493), (493, 26495, 26501)

X(45616) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-PARRY AND 1st ANTI-CIRCUMPERP

Barycentrics    (a^12-5*(b^2+c^2)*a^10+2*(5*b^4+6*b^2*c^2+5*c^4)*a^8-(b^2+c^2)*(17*b^4-18*b^2*c^2+17*c^4)*a^6+11*(b^2-c^2)^2*(2*b^4+b^2*c^2+2*c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*(7*b^4-13*b^2*c^2+7*c^4)*a^2+3*(b^4-b^2*c^2+c^4)*(b^2-c^2)^4)*(a^2-c^2)*(a^2-b^2) : :

X(45616) lies on these lines: {3, 45618}, {7488, 11419}

X(45616) = X(45618)-of-ABC-X3 reflections triangle
X(45616) = reflection of X(45618) in X(3)

X(45617) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-PARRY AND ARIES

Barycentrics    a^2*(a^10-5*(b^2+c^2)*a^8+4*(b^4+6*b^2*c^2+c^4)*a^6+4*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^4-5*(b^2-c^2)^2*(b^4-10*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-10*b^2*c^2+c^4))*(a^2-c^2)*(a^2-b^2) : :

X(45617) lies on these lines: {110, 20187}, {1296, 20186}, {6759, 10836}

X(45617) = reflection of X(10836) in X(6759)

X(45618) = PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-PARRY AND CIRCUMORTHIC

Barycentrics    (a^12-4*(b^2+c^2)*a^10+(10*b^4+b^2*c^2+10*c^4)*a^8-(b^2+c^2)*(13*b^4-15*b^2*c^2+13*c^4)*a^6+(7*b^8+7*c^8+2*(4*b^4-11*b^2*c^2+4*c^4)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)*(b^4+11*b^2*c^2+c^4)*a^2+3*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b^2*c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(45618) lies on these lines: {3, 45616}, {4, 22100}

X(45618) = X(45616)-of-ABC-X3 reflections triangle
X(45618) = reflection of X(45616) in X(3)

X(45619) = PERSPECTOR OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND AAOA

Barycentrics    a^2*(a^20-3*(b^2+c^2)*a^18-2*(b^4-4*b^2*c^2+c^4)*a^16+(b^2+c^2)*(4*b^2+7*b*c+4*c^2)*(4*b^2-7*b*c+4*c^2)*a^14-(14*b^8+14*c^8+(14*b^4-11*b^2*c^2+14*c^4)*b^2*c^2)*a^12-(b^2+c^2)*(14*b^8+14*c^8-(27*b^4-34*b^2*c^2+27*c^4)*b^2*c^2)*a^10+2*(14*b^12+14*c^12-(b^2-c^2)^2*(b^4+11*b^2*c^2+c^4)*b^2*c^2)*a^8-(b^2+c^2)*(8*b^12+8*c^12-(5*b^8+5*c^8+6*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^2*(11*b^12+11*c^12+2*(b^4+b^2*c^2+c^4)*(8*b^4+b^2*c^2+8*c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^3*(9*b^8+9*c^8+(21*b^4+16*b^2*c^2+21*c^4)*b^2*c^2)*a^2-2*(b^4+3*b^2*c^2+c^4)*(b^2+c^2)^2*(b^2-c^2)^6) : :

X(45619) lies on these lines: {5, 15142}, {67, 32423}, {74, 7495}, {125, 3581}, {182, 10628}, {542, 40919}, {1154, 10510}, {2931, 19376}, {4550, 38789}, {6145, 34115}, {7550, 25711}, {7555, 34802}, {7691, 9140}, {9919, 11472}, {11597, 19381}, {12041, 15138}, {15089, 15137}, {19402, 22109}

X(45620) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND 1st AURIGA

Barycentrics    a*((2*a^3-(b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*sqrt(R*(4*R+r))+S*(-a+b+c)*a) : :
X(45620) = 2*X(11366)+X(35245)

X(45620) lies on these lines: {1, 3}, {2, 8200}, {5, 9834}, {24, 11384}, {30, 8196}, {125, 12466}, {140, 5599}, {182, 12452}, {371, 44601}, {372, 44600}, {498, 11869}, {499, 11871}, {515, 18497}, {549, 11207}, {631, 5601}, {944, 8207}, {952, 5600}, {1483, 12455}, {1511, 13208}, {1656, 45379}, {2080, 11837}, {3311, 19007}, {3312, 19008}, {3357, 12468}, {5602, 7967}, {5690, 12454}, {5901, 8203}, {6200, 35781}, {6396, 35778}, {6642, 8190}, {6771, 12472}, {6774, 12470}, {7583, 13890}, {7584, 13944}, {8197, 26446}, {8198, 26341}, {8199, 26348}, {8201, 45623}, {8202, 45624}, {8204, 37727}, {9835, 34773}, {10610, 12480}, {11861, 26316}, {11863, 26451}, {11865, 26492}, {11867, 26487}, {12041, 12365}, {12042, 12179}, {12359, 12415}, {12456, 34862}, {12460, 12619}, {12462, 38602}, {12463, 19907}, {12478, 38624}, {13176, 33813}, {13228, 33814}, {13229, 38608}, {22668, 40108}, {26498, 45589}, {26507, 45588}, {35762, 44603}, {35763, 44602}, {43118, 45431}, {43119, 45430}, {43394, 43850}

X(45620) = midpoint of X(3) and X(45369)
X(45620) = X(45369)-of-these triangles: {anti-X3-ABC reflections, 1st Auriga}
X(45620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11843, 8200), (3, 5597, 11252), (3, 11252, 35244), (3, 11875, 11822), (5, 9834, 18495), (55, 1385, 45621), (55, 1388, 11880), (140, 32146, 5599), (5597, 11822, 11875), (11822, 11875, 11252), (45534, 45535, 12452)

X(45621) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND 2nd AURIGA

Barycentrics    a*(-(2*a^3-(b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*sqrt(R*(4*R+r))+S*(-a+b+c)*a) : :
X(45621) = 2*X(11367)+X(35244)

X(45621) lies on these lines: {1, 3}, {2, 8207}, {5, 9835}, {24, 11385}, {30, 8203}, {125, 12467}, {140, 5600}, {182, 12453}, {371, 44603}, {372, 44602}, {498, 11870}, {499, 11872}, {515, 18495}, {549, 11208}, {631, 5602}, {944, 8200}, {952, 5599}, {1483, 12454}, {1511, 13209}, {1656, 45380}, {2080, 11838}, {3311, 19009}, {3312, 19010}, {3357, 12469}, {5601, 7967}, {5690, 12455}, {5901, 8196}, {6200, 35779}, {6396, 35780}, {6642, 8191}, {6771, 12473}, {6774, 12471}, {7583, 13891}, {7584, 13945}, {8197, 37727}, {8204, 26446}, {8205, 26341}, {8206, 26348}, {8208, 45623}, {8209, 45624}, {9834, 34773}, {10610, 12481}, {11862, 26316}, {11864, 26451}, {11866, 26492}, {11868, 26487}, {12041, 12366}, {12042, 12180}, {12359, 12416}, {12457, 34862}, {12461, 12619}, {12462, 19907}, {12463, 38602}, {12479, 38624}, {13177, 33813}, {13230, 33814}, {13231, 38608}, {22672, 40108}, {26498, 45591}, {26507, 45590}, {35762, 44601}, {35763, 44600}, {43118, 45433}, {43119, 45432}, {43394, 43851}

X(45621) = midpoint of X(3) and X(45370)
X(45621) = X(45370)-of-these triangles: {anti-X3-ABC reflections, 2nd Auriga}
X(45621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11844, 8207), (3, 5598, 11253), (3, 11253, 35245), (3, 11876, 11823), (5, 9835, 18497), (55, 1385, 45620), (55, 1388, 11879), (140, 32147, 5600), (5598, 11823, 11876), (11823, 11876, 11253)

X(45622) = PERSPECTOR OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND EHRMANN-SIDE

Barycentrics    ((b^2+c^2)*a^6-b^2*c^2*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
X(45622) = 5*X(3091)+X(13452)

X(45622) lies on these lines: {2, 6288}, {3, 13851}, {4, 12041}, {5, 6241}, {30, 11704}, {125, 5448}, {140, 12278}, {182, 1656}, {265, 6640}, {381, 3357}, {568, 10224}, {1147, 15089}, {1594, 34115}, {2072, 12359}, {3091, 3521}, {3548, 22647}, {3850, 7703}, {3851, 11472}, {5055, 15805}, {5449, 22815}, {5462, 7579}, {5655, 15114}, {5946, 12300}, {6102, 20396}, {6699, 18565}, {7577, 37481}, {10157, 34339}, {11250, 14644}, {11585, 18438}, {12121, 27082}, {12429, 15123}, {13093, 15126}, {13413, 15024}, {13561, 18435}, {14643, 32140}, {15027, 15128}, {15059, 16013}, {15646, 18394}, {18379, 34128}, {18392, 43615}, {18404, 18442}, {18445, 19360}, {18563, 35257}, {18564, 20191}, {20299, 23515}, {23323, 43607}, {23325, 32345}, {30744, 37495}, {31283, 37472}, {34786, 37955}, {35475, 43865}, {37119, 43821}, {37484, 37938}, {41362, 44214}, {41587, 43592}

X(45622) = midpoint of X(3) and X(21400)
X(45622) = reflection of X(18504) in X(5)
X(45622) = inverse of X(34783) in Jerabek circumhyperbola
X(45622) = X(18504)-of-Johnson triangle
X(45622) = X(21400)-of-anti-X3-ABC reflections triangle
X(45622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 23294, 18439), (125, 10255, 34783)

X(45623) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(-(a^6+5*(b^2+c^2)*a^4-9*(b^2+c^2)^2*a^2+(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2))*S+a^8-4*(b^2+c^2)*a^6+2*(3*b^4-4*b^2*c^2+3*c^4)*a^4-4*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2)*(b^2+c^2)*a^2+(b^2-c^2)^2*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2)) : :

X(45623) lies on these lines: {2, 8220}, {3, 493}, {5, 9838}, {24, 11394}, {30, 8212}, {35, 11947}, {36, 18963}, {55, 11953}, {56, 11951}, {125, 12894}, {140, 8222}, {182, 12590}, {371, 44628}, {372, 44627}, {498, 11930}, {499, 11932}, {517, 11377}, {549, 12152}, {631, 6462}, {1385, 12440}, {1511, 13215}, {1656, 45381}, {2080, 11840}, {3311, 19031}, {3312, 19032}, {3357, 12986}, {3576, 8188}, {3579, 22841}, {5690, 12636}, {6200, 35807}, {6396, 35804}, {6461, 45624}, {6642, 8194}, {6771, 12990}, {6774, 12988}, {7583, 13899}, {7584, 13956}, {8201, 45620}, {8208, 45621}, {8210, 10246}, {8214, 26446}, {8216, 26341}, {8218, 26348}, {10267, 11503}, {10269, 22761}, {10610, 12998}, {10875, 26316}, {10945, 26492}, {10951, 26487}, {11907, 26451}, {11955, 16203}, {11957, 16202}, {12041, 12377}, {12042, 12186}, {12359, 12426}, {12619, 12741}, {12765, 38602}, {12996, 38624}, {13184, 33813}, {13275, 33814}, {13298, 38608}, {18245, 34862}, {22709, 40108}, {26398, 45362}, {26422, 45364}, {26507, 45604}, {43118, 45465}, {43119, 45467}

X(45623) = midpoint of X(3) and X(45610)
X(45623) = X(45610)-of-these triangles: {anti-X3-ABC reflections, Lucas(+1) homothetic}
X(45623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11846, 8220), (3, 493, 10669), (3, 11949, 11828), (3, 26498, 11198), (5, 9838, 18520), (140, 32177, 8222), (493, 11828, 11949), (11828, 11949, 10669), (45567, 45569, 12590)

X(45624) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-X3-ABC REFLECTIONS AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*((a^6+5*(b^2+c^2)*a^4-9*(b^2+c^2)^2*a^2+(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2))*S+a^8-4*(b^2+c^2)*a^6+2*(3*b^4-4*b^2*c^2+3*c^4)*a^4-4*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2)*(b^2+c^2)*a^2+(b^2-c^2)^2*(b^2-2*b*c-c^2)*(b^2+2*b*c-c^2)) : :

X(45624) lies on these lines: {2, 8221}, {3, 494}, {5, 9839}, {24, 11395}, {30, 8213}, {35, 11948}, {36, 18964}, {55, 11954}, {56, 11952}, {125, 12895}, {140, 8223}, {182, 12591}, {371, 44630}, {372, 44629}, {498, 11931}, {499, 11933}, {517, 11378}, {549, 12153}, {631, 6463}, {1385, 12441}, {1511, 13216}, {1656, 45382}, {2080, 11841}, {3311, 19033}, {3312, 19034}, {3357, 12987}, {3576, 8189}, {3579, 22842}, {5690, 12637}, {6200, 35805}, {6396, 35806}, {6461, 45623}, {6642, 8195}, {6771, 12991}, {6774, 12989}, {7583, 13900}, {7584, 13957}, {8202, 45620}, {8209, 45621}, {8211, 10246}, {8215, 26446}, {8217, 26341}, {8219, 26348}, {10267, 11504}, {10269, 22762}, {10610, 12999}, {10876, 26316}, {10946, 26492}, {10952, 26487}, {11908, 26451}, {11956, 16203}, {11958, 16202}, {12041, 12378}, {12042, 12187}, {12359, 12427}, {12619, 12742}, {12766, 38602}, {12997, 38624}, {13185, 33813}, {13276, 33814}, {13299, 38608}, {18246, 34862}, {22710, 40108}, {26398, 45361}, {26422, 45363}, {26498, 45603}, {43118, 45466}, {43119, 45464}

X(45624) = midpoint of X(3) and X(45609)
X(45624) = X(45609)-of-these triangles: {anti-X3-ABC reflections, Lucas(-1) homothetic}
X(45624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11847, 8221), (3, 494, 10673), (3, 11950, 11829), (3, 26507, 32077), (5, 9839, 18522), (140, 32178, 8223), (494, 11829, 11950), (11829, 11950, 10673), (45566, 45568, 12591)

X(45625) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 1st AURIGA

Barycentrics    a*(4*(a+b-c)*(a-b+c)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c)))*(-a+b+c) : :

X(45625) lies on these lines: {1, 3}, {5, 11867}, {952, 11868}, {2886, 26387}, {5599, 26363}, {5601, 10527}, {5842, 26412}, {6734, 8197}, {8190, 26308}, {8196, 26332}, {8198, 26342}, {8199, 26349}, {8200, 26470}, {8201, 45645}, {8202, 45644}, {10943, 11865}, {11384, 26377}, {11837, 26431}, {11843, 12116}, {11861, 26317}, {11863, 26452}, {11869, 26481}, {11871, 26475}, {13890, 45650}, {13944, 45651}, {18495, 45630}, {18544, 45379}, {19007, 26458}, {19008, 26464}, {19049, 44601}, {19050, 44600}, {26499, 45589}, {26508, 45588}, {35778, 45640}, {35781, 45641}, {45422, 45430}, {45423, 45431}, {45526, 45534}, {45527, 45535}

X(45625) = X(45371)-of-these triangles: {anti-inner-Yff, 1st Auriga}
X(45625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45626), (55, 5597, 45627), (55, 11366, 26351), (55, 26352, 45628), (5597, 5598, 11877), (5597, 11883, 1), (18839, 26357, 45626)

X(45626) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 2nd AURIGA

Barycentrics    a*(-4*(a+b-c)*(a-b+c)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c)))*(-a+b+c) : :

X(45626) lies on these lines: {1, 3}, {5, 11868}, {952, 11867}, {2886, 26411}, {5600, 26363}, {5602, 10527}, {5842, 26388}, {6734, 8204}, {8191, 26308}, {8203, 26332}, {8205, 26342}, {8206, 26349}, {8207, 26470}, {8208, 45645}, {8209, 45644}, {10943, 11866}, {11385, 26377}, {11838, 26431}, {11844, 12116}, {11862, 26317}, {11864, 26452}, {11870, 26481}, {11872, 26475}, {13891, 45650}, {13945, 45651}, {18497, 45630}, {18544, 45380}, {19009, 26458}, {19010, 26464}, {19049, 44603}, {19050, 44602}, {26499, 45591}, {26508, 45590}, {35779, 45641}, {35780, 45640}, {45422, 45432}, {45423, 45433}, {45526, 45536}, {45527, 45537}

X(45626) = X(45372)-of-these triangles: {anti-inner-Yff, 2nd Auriga}
X(45626) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45625), (55, 5598, 45628), (55, 11367, 26352), (55, 26351, 45627), (5597, 5598, 11878), (5598, 11884, 1), (18839, 26357, 45625)

X(45627) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 1st AURIGA

Barycentrics    a*(4*(a+b-c)*(a-b+c)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)))*(-a+b+c) : :

X(45627) lies on these lines: {1, 3}, {5, 11865}, {119, 8200}, {528, 26411}, {952, 11866}, {5552, 5601}, {5599, 26364}, {6256, 9834}, {6735, 8197}, {7680, 26388}, {8190, 26309}, {8196, 26333}, {8198, 26343}, {8199, 26350}, {8201, 45647}, {8202, 45646}, {10942, 11867}, {11384, 26378}, {11837, 26432}, {11843, 12115}, {11861, 26318}, {11863, 26453}, {11869, 26482}, {11871, 26476}, {12460, 12751}, {13228, 25438}, {13890, 45652}, {13944, 45653}, {18495, 45631}, {18542, 45379}, {19007, 26459}, {19008, 26465}, {19047, 44601}, {19048, 44600}, {26500, 45589}, {26509, 45588}, {35778, 45642}, {35781, 45643}, {45424, 45430}, {45425, 45431}, {45528, 45534}, {45529, 45535}

X(45627) = X(45373)-of-these triangles: {anti-outer-Yff, 1st Auriga}
X(45627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45628), (55, 5597, 45625), (55, 11366, 26352), (55, 26351, 45626), (5597, 5598, 11879), (5597, 11881, 1)

X(45628) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 2nd AURIGA

Barycentrics    a*(-4*(a+b-c)*(a-b+c)*S*sqrt(R*(4*R+r))+a*(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)))*(-a+b+c) : :

X(45628) lies on these lines: {1, 3}, {5, 11866}, {119, 8207}, {528, 26387}, {952, 11865}, {5552, 5602}, {5600, 26364}, {6256, 9835}, {6735, 8204}, {7680, 26412}, {8191, 26309}, {8203, 26333}, {8205, 26343}, {8206, 26350}, {8208, 45647}, {8209, 45646}, {10942, 11868}, {11385, 26378}, {11838, 26432}, {11844, 12115}, {11862, 26318}, {11864, 26453}, {11870, 26482}, {11872, 26476}, {12461, 12751}, {13230, 25438}, {13891, 45652}, {13945, 45653}, {18497, 45631}, {18542, 45380}, {19009, 26459}, {19010, 26465}, {19047, 44603}, {19048, 44602}, {26500, 45591}, {26509, 45590}, {35779, 45643}, {35780, 45642}, {45424, 45432}, {45425, 45433}, {45528, 45536}, {45529, 45537}

X(45628) = X(45374)-of-these triangles: {anti-outer-Yff, 2nd Auriga}
X(45628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 55, 45627), (55, 5598, 45626), (55, 11367, 26351), (55, 26352, 45625), (5597, 5598, 11880), (5598, 11882, 1)

X(45629) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND 3rd CONWAY

Barycentrics    (b+2*c)*(2*b+c)*a^11+3*(b+c)*(2*b^2+b*c+2*c^2)*a^10+2*(b^2+12*b*c+c^2)*b*c*a^9-8*(b+c)*(2*b^4+2*c^4-3*(b^2+c^2)*b*c)*a^8-2*(6*b^6+6*c^6-(5*b^4+5*c^4-2*(3*b-c)*(b-3*c)*b*c)*b*c)*a^7+2*(b+c)*(6*b^6+6*c^6-(17*b^4+17*c^4-4*(b^2-3*b*c+c^2)*b*c)*b*c)*a^6+4*(4*b^8+4*c^8-(5*b^6+5*c^6+2*(b^4+c^4+3*(2*b^2+b*c+2*c^2)*b*c)*b*c)*b*c)*a^5+4*(b+c)*(4*b^6+4*c^6-(5*b^4+5*c^4+(b^2+c^2)*b*c)*b*c)*b*c*a^4-(b^2-c^2)^2*(b-c)^2*(6*b^4+6*c^4+(b+3*c)*(3*b+c)*b*c)*a^3-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6+(9*b^4+9*c^4-2*(3*b^2-7*b*c+3*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)^2*(3*b^4+3*c^4-2*(b^2-b*c+c^2)*b*c)*b*c*a-4*(b^2-c^2)^4*(b+c)*b^2*c^2 : :

X(45629) lies on these lines: {6256, 12547}, {11021, 45655}, {12551, 12751}

X(45630) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND EHRMANN-MID

Barycentrics    a^7-(b+c)*a^6-2*b*c*a^5-(3*b^4+3*c^4-2*(b^2-b*c+c^2)*b*c)*a^3+(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(45630) = 3*X(381)-X(9654)

X(45630) lies on these lines: {1, 381}, {3, 18407}, {4, 2975}, {5, 1001}, {11, 44229}, {30, 26363}, {35, 18499}, {56, 37230}, {153, 3832}, {355, 908}, {376, 32633}, {382, 11012}, {497, 6866}, {499, 28452}, {527, 10916}, {546, 7956}, {944, 7548}, {952, 10894}, {1385, 25525}, {1478, 26475}, {1479, 6841}, {1621, 6873}, {1656, 10902}, {1699, 3901}, {2886, 35239}, {3091, 10587}, {3579, 5705}, {3583, 26357}, {3585, 18519}, {3614, 34746}, {3656, 41575}, {3830, 35252}, {3839, 10529}, {3843, 10680}, {3845, 34739}, {3851, 16202}, {3855, 10806}, {3858, 32214}, {5072, 34486}, {5707, 29287}, {5709, 5789}, {5840, 6847}, {5842, 6862}, {5881, 11929}, {6253, 6863}, {6564, 19050}, {6565, 19049}, {6713, 6885}, {6734, 12699}, {6796, 20104}, {6827, 19855}, {6830, 11499}, {6831, 11248}, {6844, 7080}, {6849, 10591}, {6851, 31418}, {6859, 31659}, {6860, 37000}, {6870, 10531}, {6903, 33108}, {6917, 10269}, {6953, 23513}, {6985, 25639}, {7308, 9956}, {7680, 37622}, {7701, 37005}, {7741, 18406}, {7951, 18518}, {8227, 24299}, {8727, 10525}, {9657, 12773}, {9818, 26308}, {10268, 11231}, {10742, 18967}, {10786, 38109}, {11235, 22791}, {11236, 37705}, {11240, 41099}, {11269, 13408}, {11401, 18386}, {12702, 31140}, {12704, 18540}, {12943, 26321}, {12953, 13743}, {13465, 22798}, {13624, 17528}, {13665, 26464}, {13785, 26458}, {15934, 33592}, {17532, 18481}, {17556, 24987}, {18495, 45625}, {18497, 45626}, {18500, 26317}, {18502, 26431}, {18507, 26452}, {18509, 26342}, {18511, 26349}, {18520, 45645}, {18522, 45644}, {18538, 45650}, {18762, 45651}, {19854, 28459}, {26399, 45355}, {26423, 45356}, {26492, 37281}, {26499, 45593}, {26508, 45592}, {31159, 41869}, {31295, 38761}, {31447, 38121}, {35238, 37356}, {35786, 45640}, {35787, 45641}, {36152, 37251}, {45422, 45438}, {45423, 45439}, {45526, 45542}, {45527, 45543}

X(45630) = reflection of X(26487) in X(5)
X(45630) = X(9654)-of-these triangles: {anti-inner-Yff, Ehrmann-mid}
X(45630) = X(26487)-of-Johnson triangle
X(45630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 26470, 11249), (5, 18517, 18491), (381, 9669, 9955), (381, 18480, 45631), (381, 18525, 10895), (381, 18544, 1), (546, 10943, 26332), (6831, 37820, 11248)

X(45631) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND EHRMANN-MID

Barycentrics    a^7-(b+c)*a^6+6*b*c*a^5-4*(b+c)*b*c*a^4-(3*b^4+3*c^4-2*(b^2+3*b*c+c^2)*b*c)*a^3+3*(b^2-c^2)^2*(b+c)*a^2+2*(b^2-c^2)^2*(b^2-4*b*c+c^2)*a-2*(b^2-c^2)^3*(b-c) : :
X(45631) = 3*X(381)-X(9669)

X(45631) lies on these lines: {1, 381}, {3, 31246}, {4, 100}, {5, 6256}, {30, 26364}, {56, 10742}, {145, 10711}, {355, 1519}, {382, 2077}, {546, 10894}, {952, 10893}, {1329, 35238}, {1470, 3585}, {1478, 26476}, {1479, 26482}, {1482, 12611}, {1532, 11249}, {1656, 37561}, {1699, 11929}, {2829, 6959}, {3091, 10586}, {3359, 9956}, {3583, 18518}, {3830, 35251}, {3832, 10531}, {3839, 10528}, {3843, 10679}, {3845, 34706}, {3851, 16203}, {3855, 10805}, {3858, 32213}, {3913, 11698}, {4188, 10728}, {5450, 20107}, {5554, 17577}, {5587, 31937}, {5841, 6848}, {5881, 11928}, {6564, 19048}, {6565, 19047}, {6713, 6981}, {6735, 12699}, {6837, 38109}, {6842, 19854}, {6871, 30513}, {6921, 38761}, {6924, 22799}, {6929, 10267}, {6930, 31659}, {6941, 22758}, {6968, 26470}, {6973, 12667}, {7173, 34697}, {7681, 38757}, {7741, 18519}, {7951, 37234}, {7989, 18540}, {8227, 24927}, {9818, 26309}, {10270, 11231}, {10598, 37726}, {10785, 23513}, {10826, 18838}, {10915, 18483}, {10958, 44229}, {11235, 37705}, {11236, 22791}, {11239, 41099}, {11400, 18386}, {11508, 12764}, {11509, 37230}, {12608, 19925}, {12664, 31828}, {12702, 31141}, {12943, 37251}, {12953, 18524}, {13665, 26465}, {13785, 26459}, {15528, 38161}, {17532, 24982}, {17556, 18481}, {18495, 45627}, {18497, 45628}, {18499, 18514}, {18500, 26318}, {18502, 26432}, {18507, 26453}, {18509, 26343}, {18511, 26350}, {18520, 45647}, {18522, 45646}, {18538, 45652}, {18762, 45653}, {22938, 25438}, {26400, 45355}, {26424, 45356}, {26500, 45593}, {26509, 45592}, {31160, 41869}, {32612, 40267}, {34918, 40587}, {35239, 37406}, {35786, 45642}, {35787, 45643}, {45424, 45438}, {45425, 45439}, {45528, 45542}, {45529, 45543}

X(45631) = reflection of X(26492) in X(5)
X(45631) = X(9669)-of-these triangles: {anti-outer-Yff, Ehrmann-mid}
X(45631) = X(26492)-of-Johnson triangle
X(45631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 119, 11248), (5, 6256, 10269), (5, 18516, 18761), (381, 9654, 9955), (381, 18480, 45630), (381, 18525, 10896), (381, 18542, 1), (546, 10942, 26333), (1532, 37821, 11249), (6929, 18242, 10267), (10942, 26333, 37622)

X(45632) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND HEXYL

Barycentrics    a*(a^9-(b+c)*a^8-4*(b^2-b*c+c^2)*a^7+4*(b^2-c^2)*(b-c)*a^6+2*(3*b^4+3*c^4-2*(b-c)^2*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4-8*(b^2-b*c+c^2)*b*c)*a^4-4*(b^4+c^4-(b^2-b*c+c^2)*b*c)*(b+c)^2*a^3+4*(b^6-c^6)*(b-c)*a^2+(b^2-c^2)^2*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a-(b^2-c^2)^4*(b+c)) : :
X(45632) = 2*X(224)-3*X(3576) = 3*X(5587)-4*X(10395)

X(45632) lies on these lines: {1, 90}, {9, 2302}, {40, 12625}, {56, 41703}, {57, 10042}, {84, 10431}, {200, 10902}, {224, 3576}, {1158, 12649}, {1709, 24474}, {1728, 17857}, {1768, 5709}, {1836, 10943}, {3333, 45654}, {3338, 26470}, {3683, 16202}, {5534, 14798}, {5587, 10395}, {5698, 10806}, {5732, 11012}, {6264, 6762}, {6284, 24467}, {6734, 11919}, {7701, 39772}, {7966, 36922}, {10052, 45638}, {10085, 11249}, {10267, 41229}, {10680, 12688}, {10916, 45633}, {12651, 37625}, {12703, 41575}, {14054, 37569}, {17437, 41699}, {26363, 41540}, {36999, 37532}, {37561, 41559}

X(45632) = X(90)-of-anti-inner-Yff triangle
X(45632) = X(70)-of-hexyl triangle
X(45632) = X(34115)-of-6th mixtilinear triangle

X(45633) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND HEXYL

Barycentrics    a*(a^9-(b+c)*a^8-4*(b^2-3*b*c+c^2)*a^7+4*(b^2-c^2)*(b-c)*a^6+2*(3*b^4+3*c^4-2*(7*b^2-10*b*c+7*c^2)*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4-8*(b^2-b*c+c^2)*b*c)*a^4-4*(b^6+c^6-(5*b^4+5*c^4-4*(2*b^2-b*c+2*c^2)*b*c)*b*c)*a^3+4*(b^6-c^6)*(b-c)*a^2+(b^2-c^2)^2*(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a-(b^2-c^2)^4*(b+c)) : :

X(45633) lies on these lines: {1, 1406}, {40, 37002}, {57, 10050}, {84, 377}, {1519, 6173}, {1709, 10269}, {1768, 3359}, {2077, 5732}, {3333, 45655}, {3358, 21164}, {3576, 12686}, {5119, 38761}, {6841, 37534}, {7330, 16209}, {7966, 10860}, {8583, 37561}, {9841, 16132}, {10085, 37562}, {10916, 45632}, {11525, 39776}, {11920, 12704}, {15016, 18219}, {37704, 45639}

X(45633) = X(7284)-of-anti-outer-Yff triangle

X(45634) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND HUTSON INTOUCH

Barycentrics    (-a+b+c)*(a^9-3*(b+c)*a^8+8*b*c*a^7+2*(b+c)*(4*b^2-5*b*c+4*c^2)*a^6-6*(b^3+c^3)*(b+c)*a^5-2*(b+c)*(3*b^2-2*b*c+c^2)*(b^2-2*b*c+3*c^2)*a^4+4*(2*b^4+2*c^4+(b^2+c^2)*b*c)*(b-c)^2*a^3-2*(b^2-c^2)*(b-c)^3*b*c*a^2-(b^2-c^2)^2*(b-c)^2*(3*b^2-4*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3) : :

X(45634) lies on these lines: {1, 6825}, {65, 45654}, {390, 12649}, {2098, 26481}, {7962, 10043}, {10543, 26357}, {10947, 12672}, {24474, 45638}

X(45635) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND HUTSON INTOUCH

Barycentrics    (-a+b+c)*(a^9-3*(b+c)*a^8+16*b*c*a^7+2*(b+c)*(4*b^2-15*b*c+4*c^2)*a^6-6*(b^4+c^4+3*(b^2-4*b*c+c^2)*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4-2*(14*b^2-27*b*c+14*c^2)*b*c)*a^4+4*(2*b^4+2*c^4+(b^2-16*b*c+c^2)*b*c)*(b-c)^2*a^3-22*(b^2-c^2)*(b-c)^3*b*c*a^2-(b^2-c^2)^2*(b-c)^2*(3*b^2-8*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3) : :

X(45635) lies on these lines: {1, 6891}, {65, 45655}, {390, 12648}, {1519, 10590}, {1837, 17648}, {2550, 39776}, {3057, 41389}, {5252, 6256}, {5554, 18220}, {5587, 10043}, {7962, 10051}, {10543, 26358}, {12647, 12751}, {18838, 45637}, {37562, 45639}

X(45636) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND INCIRCLE-CIRCLES

Barycentrics    a^10+2*(b+c)*a^9-(9*b^2-2*b*c+9*c^2)*a^8-2*(b+c)*(2*b^2+9*b*c+2*c^2)*a^7+2*(11*b^4+14*b^2*c^2+11*c^4)*a^6+2*b*c*(b+c)*(23*b^2-14*b*c+23*c^2)*a^5-2*(11*b^6+11*c^6+(2*b^4+2*c^4-7*b*c*(b^2+4*b*c+c^2))*b*c)*a^4+2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-5*b*c*(3*b^2+2*b*c+3*c^2))*a^3+9*(b^2-c^2)^4*a^2-2*(b^2-c^2)^3*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^4*(b-c)^2 : :

X(45636) lies on these lines: {942, 5794}, {2894, 36845}, {3296, 5231}, {5045, 45654}, {5083, 12757}, {10980, 11045}, {11037, 39772}

X(45637) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND INCIRCLE-CIRCLES

Barycentrics    a^10+2*(b+c)*a^9-(9*b^2-10*b*c+9*c^2)*a^8-2*(b+c)*(2*b^2-b*c+2*c^2)*a^7+2*(11*b^4+11*c^4-2*(10*b^2-23*b*c+10*c^2)*b*c)*a^6+2*(b+c)*(b^2-18*b*c+c^2)*b*c*a^5-2*(11*b^6+11*c^6-(26*b^4+26*c^4-7*(7*b^2-12*b*c+7*c^2)*b*c)*b*c)*a^4+2*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-(b^2-18*b*c+c^2)*b*c)*a^3+3*(b^2-c^2)^2*(b-c)^2*(3*b^2-2*b*c+3*c^2)*a^2-2*(b^2-c^2)^2*(b-c)^2*(b^3+c^3)*a-(b^2-c^2)^4*(b-c)^2 : :

X(45637) lies on these lines: {1519, 36996}, {5045, 45655}, {5083, 12751}, {5542, 8257}, {5722, 6256}, {10980, 11046}, {18240, 26333}, {18838, 45635}

X(45638) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND INTOUCH

Barycentrics    (a^4-2*(b^2+c^2)*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2)*(a^4-2*(b+c)^2*a^2+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

X(45638) lies on these lines: {1, 6868}, {7, 7183}, {57, 10044}, {65, 3419}, {354, 45654}, {553, 45639}, {954, 37579}, {1071, 1836}, {1466, 10427}, {1537, 18967}, {1858, 26332}, {3085, 37550}, {3600, 39778}, {3649, 26437}, {10052, 45632}, {10529, 12913}, {12647, 13375}, {24474, 45634}

X(45639) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND INTOUCH

Barycentrics    (a^4-2*(b^2+c^2)*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2)*(a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :

X(45639) lies on these lines: {1, 6948}, {7, 5552}, {56, 1537}, {57, 10052}, {65, 44784}, {354, 45655}, {553, 45638}, {1071, 1837}, {1470, 3649}, {1519, 3086}, {5553, 10598}, {7702, 26333}, {10573, 11570}, {11248, 24465}, {12913, 21454}, {28074, 39771}, {37562, 45635}, {37704, 45633}

X(45640) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 1st KENMOTU-FREE-VERTICES

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4+c^4-2*(2*b-c)*(b-2*c)*b*c)*a+2*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)) : :

X(45640) lies on these lines: {1, 372}, {3, 44645}, {5, 35798}, {6, 10680}, {371, 11249}, {485, 10527}, {486, 10532}, {1124, 10966}, {1151, 35252}, {1152, 16202}, {1335, 18967}, {1587, 10529}, {3069, 10597}, {3070, 10943}, {3093, 11401}, {3103, 45423}, {3312, 12001}, {5418, 13907}, {5709, 35610}, {6200, 11012}, {6396, 10267}, {6419, 26464}, {6420, 19049}, {6460, 10806}, {6560, 12116}, {6564, 26470}, {6565, 26332}, {6734, 35788}, {10576, 26363}, {10587, 13935}, {12704, 35775}, {18544, 23251}, {24474, 35642}, {26308, 35776}, {26317, 35782}, {26342, 35792}, {26349, 35794}, {26357, 35808}, {26377, 35764}, {26399, 45357}, {26423, 45359}, {26431, 35766}, {26437, 35768}, {26452, 35790}, {26458, 35770}, {26475, 35802}, {26481, 35800}, {26499, 45601}, {26508, 45599}, {26522, 39661}, {32214, 42216}, {35778, 45625}, {35780, 45626}, {35786, 45630}, {35804, 45645}, {35806, 45644}, {35812, 45650}, {35814, 45651}, {35817, 44591}, {35857, 37726}, {45422, 45462}, {45527, 45564}

X(45640) = X(45580)-of-these triangles: {anti-inner-Yff, 1st Kenmotu-free-vertices}
X(45640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10680, 45641), (372, 35641, 45642), (372, 35810, 35809), (372, 35816, 5414), (372, 35818, 1), (3312, 12001, 44646), (6420, 35819, 19049), (6502, 35774, 372), (11249, 19050, 371)

X(45641) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 2nd KENMOTU-FREE-VERTICES

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4+c^4-2*(2*b-c)*(b-2*c)*b*c)*a-2*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c))*S-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)) : :

X(45641) lies on these lines: {1, 371}, {3, 44646}, {5, 35799}, {6, 10680}, {372, 11249}, {485, 10532}, {486, 10527}, {1124, 18967}, {1151, 16202}, {1152, 35252}, {1335, 10966}, {1588, 10529}, {3068, 10597}, {3071, 10943}, {3092, 11401}, {3102, 45422}, {3311, 12001}, {5420, 13965}, {5709, 35611}, {6200, 10267}, {6396, 11012}, {6419, 19050}, {6420, 26458}, {6459, 10806}, {6561, 12116}, {6564, 26332}, {6565, 26470}, {6734, 35789}, {9540, 10587}, {10577, 26363}, {12704, 35774}, {18544, 23261}, {24474, 35641}, {26308, 35777}, {26317, 35783}, {26342, 35795}, {26349, 35793}, {26357, 35809}, {26377, 35765}, {26399, 45360}, {26423, 45358}, {26431, 35767}, {26437, 35769}, {26452, 35791}, {26464, 35771}, {26475, 35803}, {26481, 35801}, {26499, 45600}, {26508, 45602}, {26517, 39660}, {32214, 42215}, {35779, 45626}, {35781, 45625}, {35787, 45630}, {35805, 45644}, {35807, 45645}, {35813, 45651}, {35815, 45650}, {35816, 44590}, {35856, 37726}, {45423, 45463}, {45526, 45565}

X(45641) = X(45581)-of-these triangles: {anti-inner-Yff, 2nd Kenmotu-free-vertices}
X(45641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10680, 45640), (371, 35642, 45643), (371, 35811, 35808), (371, 35817, 2066), (371, 35819, 1), (2067, 35775, 371), (3311, 12001, 44645), (6419, 35818, 19050), (11249, 19049, 372)

X(45642) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 1st KENMOTU-FREE-VERTICES

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*a+2*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*S-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)) : :

X(45642) lies on these lines: {1, 372}, {3, 44643}, {5, 35796}, {6, 10679}, {119, 6564}, {371, 11248}, {485, 5552}, {486, 10531}, {1124, 10965}, {1151, 35251}, {1152, 16203}, {1335, 11509}, {1470, 35768}, {1587, 10528}, {2077, 6200}, {3069, 10596}, {3070, 10942}, {3093, 11400}, {3103, 45425}, {3312, 12000}, {5415, 6419}, {5418, 13906}, {6256, 35820}, {6396, 10269}, {6420, 19047}, {6460, 10805}, {6560, 12115}, {6565, 26333}, {6735, 35788}, {10576, 26364}, {10586, 13935}, {12703, 35775}, {12751, 35852}, {18542, 23251}, {23340, 35642}, {25438, 35882}, {26309, 35776}, {26318, 35782}, {26343, 35792}, {26350, 35794}, {26358, 35808}, {26378, 35764}, {26400, 45357}, {26424, 45359}, {26432, 35766}, {26453, 35790}, {26459, 35770}, {26476, 35802}, {26482, 35800}, {26500, 45601}, {26509, 45599}, {26523, 39661}, {32213, 42216}, {35611, 37562}, {35778, 45627}, {35780, 45628}, {35786, 45631}, {35804, 45647}, {35806, 45646}, {35812, 45652}, {35814, 45653}, {35819, 44607}, {45424, 45462}, {45529, 45564}

X(45642) = X(45582)-of-these triangles: {anti-outer-Yff, 1st Kenmotu-free-vertices}
X(45642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10679, 45643), (372, 35641, 45640), (372, 35810, 35769), (372, 35816, 1), (372, 35818, 6502), (3312, 12000, 44644), (5414, 35774, 372), (6420, 35817, 19047), (11248, 19048, 371), (19047, 37622, 35817)

X(45643) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 2nd KENMOTU-FREE-VERTICES

Barycentrics    a^2*(a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*a-2*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c))*S-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)) : :

X(45643) lies on these lines: {1, 371}, {3, 44644}, {5, 35797}, {6, 10679}, {119, 6565}, {372, 11248}, {485, 10531}, {486, 5552}, {1124, 11509}, {1151, 16203}, {1152, 35251}, {1335, 10965}, {1470, 35769}, {1588, 10528}, {2077, 6396}, {3068, 10596}, {3071, 10942}, {3092, 11400}, {3102, 45424}, {3311, 12000}, {5416, 6420}, {5420, 13964}, {6200, 10269}, {6256, 35821}, {6419, 19048}, {6459, 10805}, {6561, 12115}, {6564, 26333}, {6735, 35789}, {9540, 10586}, {10577, 26364}, {12703, 35774}, {12751, 35853}, {13373, 31439}, {18542, 23261}, {23340, 35641}, {25438, 35883}, {26309, 35777}, {26318, 35783}, {26343, 35795}, {26350, 35793}, {26358, 35809}, {26378, 35765}, {26400, 45360}, {26424, 45358}, {26432, 35767}, {26453, 35791}, {26465, 35771}, {26476, 35803}, {26482, 35801}, {26500, 45600}, {26509, 45602}, {26518, 39660}, {32213, 42215}, {35610, 37562}, {35779, 45628}, {35781, 45627}, {35787, 45631}, {35805, 45646}, {35807, 45647}, {35813, 45653}, {35815, 45652}, {35818, 44606}, {45425, 45463}, {45528, 45565}

X(45643) = X(45583)-of-these triangles: {anti-outer-Yff, 2nd Kenmotu-free-vertices}
X(45643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10679, 45642), (371, 35642, 45641), (371, 35811, 35768), (371, 35817, 1), (371, 35819, 2067), (2066, 35775, 371), (3311, 12000, 44643), (6419, 35816, 19048), (11248, 19047, 372), (19048, 37622, 35816)

X(45644) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*((2*(b+c)*a^9+4*(3*b^2+2*b*c+3*c^2)*a^8-4*(b+c)*(2*b^2+b*c+2*c^2)*a^7-4*(12*b^4+12*c^4+b*c*(11*b^2+28*b*c+11*c^2))*a^6+4*(b+c)*(3*b^4+3*c^4-2*b*c*(3*b^2+b*c+3*c^2))*a^5+8*(9*b^6+9*c^6+(9*b^4+9*c^4+b*c*(19*b^2+12*b*c+19*c^2))*b*c)*a^4-4*(b+c)*(b^2+c^2)*(2*b^4+2*c^4-b*c*(15*b^2+16*b*c+15*c^2))*a^3-4*(12*b^8+12*c^8+(11*b^6+11*c^6-5*b^2*c^2*(3*b^2+8*b*c+3*c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*(b^4+c^4-2*b*c*(8*b^2+5*b*c+8*c^2))*a+4*(b^2-c^2)^2*(3*b^6+3*c^6+(2*b^4+2*c^4-3*(b^2+c^2)*b*c)*b*c))*S-(-a+b+c)*(a+b+c)*(3*a^10-(15*b^2+2*b*c+15*c^2)*a^8-(b+c)*(2*b^2+9*b*c+2*c^2)*a^7+(30*b^4+30*c^4-b*c*(2*b^2-11*b*c+2*c^2))*a^6+(b+c)*(6*b^4+6*c^4+b*c*(23*b^2+2*b*c+23*c^2))*a^5-(30*b^6+30*c^6-(22*b^4+22*c^4+b*c*(57*b^2+2*b*c+57*c^2))*b*c)*a^4-(b+c)*(b^2+4*b*c+c^2)*(6*b^4+6*c^4-b*c*(5*b^2+8*b*c+5*c^2))*a^3+(15*b^8+15*c^8-(26*b^6+26*c^6+(63*b^4+63*c^4-2*b*c*(39*b^2+2*b*c+39*c^2))*b*c)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*(2*b^4+2*c^4+b*c*(b^2-12*b*c+c^2))*a-(3*b^8+3*c^8-(2*b^6+2*c^6+(25*b^4+25*c^4-2*b*c*(b^2+20*b*c+c^2))*b*c)*b*c)*(b-c)^2)) : :

X(45644) lies on these lines: {1, 494}, {5, 10952}, {5709, 22842}, {6461, 45645}, {6463, 10527}, {6734, 8215}, {8195, 26308}, {8202, 45625}, {8209, 45626}, {8213, 26332}, {8217, 26342}, {8219, 26349}, {8221, 26470}, {8223, 26363}, {10267, 11504}, {10673, 11249}, {10680, 11950}, {10876, 26317}, {10943, 10946}, {11012, 11829}, {11395, 26377}, {11841, 26431}, {11847, 12116}, {11908, 26452}, {11931, 26481}, {11933, 26475}, {11948, 26357}, {13900, 45650}, {13957, 45651}, {18522, 45630}, {18544, 45382}, {18964, 26437}, {19033, 26458}, {19034, 26464}, {19049, 44630}, {19050, 44629}, {26399, 45361}, {26423, 45363}, {26499, 45603}, {35805, 45641}, {35806, 45640}, {45422, 45464}, {45423, 45466}, {45526, 45566}, {45527, 45568}

X(45644) = X(45611)-of-these triangles: {anti-inner-Yff, Lucas(-1) homothetic}
X(45644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 8211, 11952), (494, 11958, 1), (494, 12441, 45646)

X(45645) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(-(2*(b+c)*a^9+4*(3*b^2+2*b*c+3*c^2)*a^8-4*(b+c)*(2*b^2+b*c+2*c^2)*a^7-4*(12*b^4+12*c^4+b*c*(11*b^2+28*b*c+11*c^2))*a^6+4*(b+c)*(3*b^4+3*c^4-2*b*c*(3*b^2+b*c+3*c^2))*a^5+8*(9*b^6+9*c^6+(9*b^4+9*c^4+b*c*(19*b^2+12*b*c+19*c^2))*b*c)*a^4-4*(b+c)*(b^2+c^2)*(2*b^4+2*c^4-b*c*(15*b^2+16*b*c+15*c^2))*a^3-4*(12*b^8+12*c^8+(11*b^6+11*c^6-5*b^2*c^2*(3*b^2+8*b*c+3*c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*(b^4+c^4-2*b*c*(8*b^2+5*b*c+8*c^2))*a+4*(b^2-c^2)^2*(3*b^6+3*c^6+(2*b^4+2*c^4-3*(b^2+c^2)*b*c)*b*c))*S-(-a+b+c)*(a+b+c)*(3*a^10-(15*b^2+2*b*c+15*c^2)*a^8-(b+c)*(2*b^2+9*b*c+2*c^2)*a^7+(30*b^4+30*c^4-b*c*(2*b^2-11*b*c+2*c^2))*a^6+(b+c)*(6*b^4+6*c^4+b*c*(23*b^2+2*b*c+23*c^2))*a^5-(30*b^6+30*c^6-(22*b^4+22*c^4+b*c*(57*b^2+2*b*c+57*c^2))*b*c)*a^4-(b+c)*(b^2+4*b*c+c^2)*(6*b^4+6*c^4-b*c*(5*b^2+8*b*c+5*c^2))*a^3+(15*b^8+15*c^8-(26*b^6+26*c^6+(63*b^4+63*c^4-2*b*c*(39*b^2+2*b*c+39*c^2))*b*c)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*(2*b^4+2*c^4+b*c*(b^2-12*b*c+c^2))*a-(3*b^8+3*c^8-(2*b^6+2*c^6+(25*b^4+25*c^4-2*b*c*(b^2+20*b*c+c^2))*b*c)*b*c)*(b-c)^2)) : :

X(45645) lies on these lines: {1, 493}, {5, 10951}, {5709, 22841}, {6461, 45644}, {6462, 10527}, {6734, 8214}, {8194, 26308}, {8201, 45625}, {8208, 45626}, {8212, 26332}, {8216, 26342}, {8218, 26349}, {8220, 26470}, {8222, 26363}, {10267, 11503}, {10669, 11249}, {10680, 11949}, {10875, 26317}, {10943, 10945}, {11012, 11828}, {11394, 26377}, {11840, 26431}, {11846, 12116}, {11907, 26452}, {11930, 26481}, {11932, 26475}, {11947, 26357}, {13899, 45650}, {13956, 45651}, {18520, 45630}, {18544, 45381}, {18963, 26437}, {19031, 26458}, {19032, 26464}, {19049, 44628}, {19050, 44627}, {26399, 45362}, {26423, 45364}, {26508, 45604}, {35804, 45640}, {35807, 45641}, {45422, 45467}, {45423, 45465}, {45526, 45569}, {45527, 45567}

X(45645) = X(45612)-of-these triangles: {anti-inner-Yff, Lucas(+1) homothetic}
X(45645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 8210, 11951), (493, 11957, 1), (493, 12440, 45647)

X(45646) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND LUCAS(-1) HOMOTHETIC

Barycentrics    a^2*((2*(b+c)*a^9-4*(3*b^2-2*b*c+3*c^2)*a^8-4*(b+c)*(2*b^2-b*c+2*c^2)*a^7+4*(12*b^4+12*c^4-b*c*(9*b^2-28*b*c+9*c^2))*a^6+4*(b+c)*(3*b^4+3*c^4-2*b*c*(5*b^2-b*c+5*c^2))*a^5-8*(b^2-b*c+c^2)*(9*b^4+9*c^4+2*b*c*(b^2+6*b*c+c^2))*a^4-4*(b+c)*(b^2+c^2)*(2*b^4+2*c^4-17*(b^2+c^2)*b*c)*a^3+4*(12*b^8+12*c^8-(9*b^4+9*c^4-2*b*c*(9*b^2-19*b*c+9*c^2))*(b+c)^2*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*(b^4+c^4-2*b*c*(8*b^2+b*c+8*c^2))*a-4*(b^2-c^2)^2*(3*b^6+3*c^6-(2*b^4+2*c^4+3*(b^2+c^2)*b*c)*b*c))*S+(a+b+c)*(-a+b+c)*(3*a^10-(15*b^2-14*b*c+15*c^2)*a^8+(b+c)*(2*b^2+7*b*c+2*c^2)*a^7+(30*b^4+30*c^4-(46*b^2-43*b*c+46*c^2)*b*c)*a^6-(b+c)*(6*b^4+6*c^4+b*c*(17*b^2-10*b*c+17*c^2))*a^5-(30*b^6+30*c^6-(58*b^4+58*c^4+b*c*(b^2+42*b*c+c^2))*b*c)*a^4+(b+c)*(6*b^6+6*c^6+(13*b^4+13*c^4-22*b*c*(b^2+b*c+c^2))*b*c)*a^3+(15*b^8+15*c^8-(30*b^6+30*c^6+(47*b^4+47*c^4-2*b*c*(41*b^2-14*b*c+41*c^2))*b*c)*b*c)*a^2-(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(7*b^4+7*c^4+2*b*c*(b^2-25*b*c+c^2))*b*c)*a-(3*b^8+3*c^8+(2*b^6+2*c^6-(25*b^4+25*c^4+2*(b^2-20*b*c+c^2)*b*c)*b*c)*b*c)*(b-c)^2)) : :

X(45646) lies on these lines: {1, 494}, {5, 10946}, {119, 8221}, {1470, 18964}, {2077, 11829}, {5552, 6463}, {6256, 9839}, {6461, 45647}, {6735, 8215}, {8195, 26309}, {8202, 45627}, {8209, 45628}, {8213, 26333}, {8217, 26343}, {8219, 26350}, {8223, 26364}, {10269, 22762}, {10673, 11248}, {10679, 11950}, {10876, 26318}, {10942, 10952}, {11395, 26378}, {11841, 26432}, {11847, 12115}, {11908, 26453}, {11931, 26482}, {11933, 26476}, {11948, 26358}, {12742, 12751}, {13276, 25438}, {13900, 45652}, {13957, 45653}, {18522, 45631}, {18542, 45382}, {19033, 26459}, {19034, 26465}, {19047, 44630}, {19048, 44629}, {26400, 45361}, {26424, 45363}, {26500, 45603}, {35805, 45643}, {35806, 45642}, {45424, 45464}, {45425, 45466}, {45528, 45566}, {45529, 45568}

X(45646) = X(45613)-of-these triangles: {anti-outer-Yff, Lucas(-1) homothetic}
X(45646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 8211, 11954), (494, 11956, 1), (494, 12441, 45644)

X(45647) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND LUCAS(+1) HOMOTHETIC

Barycentrics    a^2*(-(2*(b+c)*a^9-4*(3*b^2-2*b*c+3*c^2)*a^8-4*(b+c)*(2*b^2-b*c+2*c^2)*a^7+4*(12*b^4+12*c^4-b*c*(9*b^2-28*b*c+9*c^2))*a^6+4*(b+c)*(3*b^4+3*c^4-2*b*c*(5*b^2-b*c+5*c^2))*a^5-8*(b^2-b*c+c^2)*(9*b^4+9*c^4+2*b*c*(b^2+6*b*c+c^2))*a^4-4*(b+c)*(b^2+c^2)*(2*b^4+2*c^4-17*(b^2+c^2)*b*c)*a^3+4*(12*b^8+12*c^8-(9*b^4+9*c^4-2*b*c*(9*b^2-19*b*c+9*c^2))*(b+c)^2*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*(b^4+c^4-2*b*c*(8*b^2+b*c+8*c^2))*a-4*(b^2-c^2)^2*(3*b^6+3*c^6-(2*b^4+2*c^4+3*(b^2+c^2)*b*c)*b*c))*S+(a+b+c)*(-a+b+c)*(3*a^10-(15*b^2-14*b*c+15*c^2)*a^8+(b+c)*(2*b^2+7*b*c+2*c^2)*a^7+(30*b^4+30*c^4-(46*b^2-43*b*c+46*c^2)*b*c)*a^6-(b+c)*(6*b^4+6*c^4+b*c*(17*b^2-10*b*c+17*c^2))*a^5-(30*b^6+30*c^6-(58*b^4+58*c^4+b*c*(b^2+42*b*c+c^2))*b*c)*a^4+(b+c)*(6*b^6+6*c^6+(13*b^4+13*c^4-22*b*c*(b^2+b*c+c^2))*b*c)*a^3+(15*b^8+15*c^8-(30*b^6+30*c^6+(47*b^4+47*c^4-2*b*c*(41*b^2-14*b*c+41*c^2))*b*c)*b*c)*a^2-(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(7*b^4+7*c^4+2*b*c*(b^2-25*b*c+c^2))*b*c)*a-(3*b^8+3*c^8+(2*b^6+2*c^6-(25*b^4+25*c^4+2*(b^2-20*b*c+c^2)*b*c)*b*c)*b*c)*(b-c)^2)) : :

X(45647) lies on these lines: {1, 493}, {5, 10945}, {119, 8220}, {1470, 18963}, {2077, 11828}, {5552, 6462}, {6256, 9838}, {6461, 45646}, {6735, 8214}, {8194, 26309}, {8201, 45627}, {8208, 45628}, {8212, 26333}, {8216, 26343}, {8218, 26350}, {8222, 26364}, {10269, 22761}, {10669, 11248}, {10679, 11949}, {10875, 26318}, {10942, 10951}, {11394, 26378}, {11840, 26432}, {11846, 12115}, {11907, 26453}, {11930, 26482}, {11932, 26476}, {11947, 26358}, {12741, 12751}, {13275, 25438}, {13899, 45652}, {13956, 45653}, {18520, 45631}, {18542, 45381}, {19031, 26459}, {19032, 26465}, {19047, 44628}, {19048, 44627}, {26400, 45362}, {26424, 45364}, {26509, 45604}, {35804, 45642}, {35807, 45643}, {45424, 45467}, {45425, 45465}, {45528, 45569}, {45529, 45567}

X(45647) = X(45614)-of-these triangles: {anti-outer-Yff, Lucas(+1) homothetic}
X(45647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 8210, 11953), (493, 11955, 1), (493, 12440, 45645)

X(45648) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND 6th MIXTILINEAR

Barycentrics    a*(a^9+(b+c)*a^8-8*(b^2-b*c+c^2)*a^7-12*(b+c)*b*c*a^6+2*(9*b^4+9*c^4-4*(b+c)^2*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4-10*(b^2+c^2)*b*c)*a^4-4*(4*b^6+4*c^6+(2*b^2-3*b*c+2*c^2)*(b-c)^2*b*c)*a^3+4*(b^3+c^3)*(b-c)^2*(2*b+c)*(b+2*c)*a^2+(b^2-c^2)^2*(b+c)^2*(5*b^2-2*b*c+5*c^2)*a-(b^2-c^2)^3*(b-c)*(b+3*c)*(3*b+c)) : :

X(45648) lies on these lines: {1, 1898}, {10045, 10980}, {10940, 10970}, {30304, 45649}

X(45648) = X(36599)-of-anti-inner-Yff triangle

X(45649) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND 6th MIXTILINEAR

Barycentrics    a*(a^9+(b+c)*a^8-8*(b-c)^2*a^7+4*(b+c)*b*c*a^6+2*(9*b^4+9*c^4-8*(3*b^2-4*b*c+3*c^2)*b*c)*a^5-2*(b+c)*(3*b^4+3*c^4+2*b*c*(3*b^2-8*b*c+3*c^2))*a^4-4*(4*b^6+4*c^6-3*b*c*(4*b^2+5*b*c+4*c^2)*(b-c)^2)*a^3+4*(b^2-c^2)*(b-c)*(2*b^4+2*c^4+b*c*(7*b^2-b*c+7*c^2))*a^2+(b^2-c^2)^2*(5*b^4+5*c^4-2*b*c*(8*b^2+5*b*c+8*c^2))*a-(b^2-c^2)^3*(b-c)*(b+3*c)*(3*b+c)) : :

X(45649) lies on these lines: {3062, 6870}, {4668, 12751}, {6256, 7992}, {10092, 10980}, {10941, 10971}, {12686, 16208}, {30304, 45648}

X(45650) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    (a^4-2*(b+c)^2*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^2*(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c)) : :

X(45650) lies on these lines: {1, 1336}, {2, 26464}, {5, 13896}, {6, 10198}, {371, 26332}, {405, 19026}, {590, 1378}, {1587, 10902}, {5709, 13912}, {6734, 13893}, {7583, 10267}, {7585, 26458}, {8972, 10527}, {8974, 26342}, {8975, 26349}, {8976, 26470}, {8981, 11249}, {9540, 11012}, {10680, 13903}, {10943, 13895}, {12116, 13886}, {13846, 44645}, {13884, 26377}, {13885, 26431}, {13889, 26308}, {13890, 45625}, {13891, 45626}, {13892, 26317}, {13894, 26452}, {13897, 26481}, {13898, 26475}, {13899, 45645}, {13900, 45644}, {13901, 26357}, {13965, 19054}, {18538, 45630}, {18544, 45384}, {18965, 26437}, {18991, 24987}, {19028, 37579}, {19049, 32787}, {26399, 45365}, {26423, 45368}, {26499, 45607}, {26508, 45605}, {31408, 37583}, {35812, 45640}, {35815, 45641}, {45422, 45484}, {45423, 45486}, {45526, 45574}, {45527, 45576}

X(45650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10198, 45651), (590, 19050, 26363)

X(45651) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-YFF AND 4th TRI-SQUARES-CENTRAL

Barycentrics    -(a^4-2*(b+c)^2*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^2*(a+b+c)*(a^3-(b+c)*a^2-(b^2+c^2)*a+(b^2-c^2)*(b-c)) : :

X(45651) lies on these lines: {1, 1123}, {2, 26458}, {5, 13953}, {6, 10198}, {372, 26332}, {405, 19025}, {615, 1377}, {1588, 10902}, {5709, 13975}, {6734, 13947}, {7584, 10267}, {7586, 26464}, {10527, 13941}, {10680, 13961}, {10943, 13952}, {11012, 13935}, {11249, 13966}, {12116, 13939}, {13847, 44646}, {13907, 19053}, {13937, 26377}, {13938, 26431}, {13943, 26308}, {13944, 45625}, {13945, 45626}, {13946, 26317}, {13948, 26452}, {13949, 26342}, {13950, 26349}, {13951, 26470}, {13954, 26481}, {13955, 26475}, {13956, 45645}, {13957, 45644}, {13958, 26357}, {18544, 45385}, {18762, 45630}, {18966, 26437}, {18992, 24987}, {19027, 37579}, {19050, 32788}, {26399, 45366}, {26423, 45367}, {26499, 45606}, {26508, 45608}, {35813, 45641}, {35814, 45640}, {45422, 45487}, {45423, 45485}, {45526, 45577}, {45527, 45575}

X(45651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10198, 45650), (615, 19049, 26363), (3069, 13959, 13962), (3069, 13965, 1), (3069, 13971, 45653)

X(45652) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 3rd TRI-SQUARES-CENTRAL

Barycentrics    (a^4-2*(b-c)^2*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^2*(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

X(45652) lies on these lines: {1, 1336}, {2, 26465}, {5, 13895}, {6, 10200}, {119, 8976}, {371, 26333}, {474, 19024}, {485, 6256}, {590, 1377}, {1470, 18965}, {1587, 37561}, {2077, 9540}, {5193, 31408}, {5552, 8972}, {6735, 13893}, {7583, 10269}, {7585, 26459}, {8974, 26343}, {8975, 26350}, {8981, 11248}, {8988, 12751}, {10679, 13903}, {10942, 13896}, {12115, 13886}, {13846, 44643}, {13884, 26378}, {13885, 26432}, {13889, 26309}, {13890, 45627}, {13891, 45628}, {13892, 26318}, {13894, 26453}, {13897, 26482}, {13898, 26476}, {13899, 45647}, {13900, 45646}, {13901, 26358}, {13922, 25438}, {13964, 19054}, {18538, 45631}, {18542, 45384}, {18991, 24982}, {19047, 32787}, {26400, 45365}, {26424, 45368}, {26500, 45607}, {26509, 45605}, {31412, 41698}, {35812, 45642}, {35815, 45643}, {45424, 45484}, {45425, 45486}, {45528, 45574}, {45529, 45576}

X(45652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10200, 45653), (590, 19048, 26364)

X(45653) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-YFF AND 4th TRI-SQUARES-CENTRAL

Barycentrics    -(a^4-2*(b-c)^2*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2)*S+a^2*(a+b+c)*(a^3-(b+c)*a^2-(b^2-4*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

X(45653) lies on these lines: {1, 1123}, {2, 26459}, {5, 13952}, {6, 10200}, {119, 13951}, {372, 26333}, {474, 19023}, {486, 6256}, {615, 1378}, {1470, 18966}, {1588, 37561}, {2077, 13935}, {5552, 13941}, {6735, 13947}, {7584, 10269}, {7586, 26465}, {10679, 13961}, {10942, 13953}, {11248, 13940}, {12115, 13939}, {12751, 13976}, {13847, 44644}, {13906, 19053}, {13937, 26378}, {13938, 26432}, {13943, 26309}, {13944, 45627}, {13945, 45628}, {13946, 26318}, {13948, 26453}, {13949, 26343}, {13950, 26350}, {13954, 26482}, {13955, 26476}, {13956, 45647}, {13957, 45646}, {13958, 26358}, {13991, 25438}, {18542, 45385}, {18762, 45631}, {18992, 24982}, {19048, 32788}, {26400, 45366}, {26424, 45367}, {26500, 45606}, {26509, 45608}, {35813, 45643}, {35814, 45642}, {41698, 42561}, {45424, 45487}, {45425, 45485}, {45528, 45577}, {45529, 45575}

X(45653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 10200, 45652), (615, 19047, 26364), (3069, 13959, 13963), (3069, 13964, 1), (3069, 13971, 45651)

X(45654) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND 1st ZANIAH

Barycentrics    a^10-(5*b^2+2*b*c+5*c^2)*a^8-6*(b+c)*b*c*a^7+2*(5*b^2-6*b*c+5*c^2)*(b+c)^2*a^6+2*(b+c)*(7*b^2+6*b*c+7*c^2)*b*c*a^5-2*(5*b^6+5*c^6+(4*b^4+4*c^4-(13*b^2+8*b*c+13*c^2)*b*c)*b*c)*a^4-2*(b^2-c^2)*(b-c)*(5*b^2+14*b*c+5*c^2)*b*c*a^3+5*(b^2-c^2)^4*a^2+2*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*(b-c)^2 : :

X(45654) lies on these lines: {1, 5758}, {65, 45634}, {142, 26363}, {354, 45638}, {497, 10122}, {999, 11281}, {3333, 45632}, {5045, 45636}, {6734, 11024}, {9614, 18224}, {10045, 10980}, {11021, 45656}, {12677, 26332}

X(45655) = PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF AND 1st ZANIAH

Barycentrics    a^10-(5*b^2-6*b*c+5*c^2)*a^8+6*(b+c)*b*c*a^7+2*(5*b^2-2*b*c+5*c^2)*(b-c)^2*a^6-2*(b+c)*(7*b^2-10*b*c+7*c^2)*b*c*a^5-2*(b^2+c^2)*(5*b^4+5*c^4-2*(8*b^2-7*b*c+8*c^2)*b*c)*a^4+2*(b^2-c^2)*(b-c)*(5*b^2-2*b*c+5*c^2)*b*c*a^3+(b^2-c^2)^2*(b-c)^2*(5*b^2-6*b*c+5*c^2)*a^2-2*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*(b-c)^2 : :

X(45655) lies on these lines: {65, 45635}, {142, 26364}, {354, 45639}, {388, 12736}, {942, 6256}, {1387, 11496}, {3333, 45633}, {5045, 45637}, {6735, 11024}, {9612, 18223}, {10092, 10980}, {10590, 11023}, {11021, 45629}, {12676, 18238}

X(45656) = PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF AND 3rd CONWAY

Barycentrics    (b+2*c)*(2*b+c)*a^11+3*(b+c)*(2*b^2+b*c+2*c^2)*a^10-2*(7*b^2+8*b*c+7*c^2)*b*c*a^9-8*(b+c)*(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*a^8-2*(6*b^6+6*c^6+(11*b^4+11*c^4+2*(17*b^2+22*b*c+17*c^2)*b*c)*b*c)*a^7+2*(b+c)*(6*b^6+6*c^6-(b^4+c^4+4*(b^2+b*c+c^2)*b*c)*b*c)*a^6+4*(4*b^6+4*c^6-(b^2-5*b*c+c^2)*(b^2-b*c+c^2)*b*c)*(b+c)^2*a^5+4*(8*b^4+8*c^4-(9*b^2-5*b*c+9*c^2)*b*c)*(b+c)^3*b*c*a^4-(b^2-c^2)^2*(6*b^6+6*c^6-(9*b^4+9*c^4+2*(23*b^2+39*b*c+23*c^2)*b*c)*b*c)*a^3-(b^2-c^2)^2*(b+c)*(2*b^6+2*c^6+(9*b^4+9*c^4-2*(11*b^2+17*b*c+11*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)^2*(3*b^4+3*c^4-2*(b^2+3*b*c+c^2)*b*c)*b*c*a-4*(b^2-c^2)^4*(b+c)*b^2*c^2 : :

X(45656) lies on this line: {11021, 45654}

X(45657) = MIDPOINT OF X(2) AND X(876)

Barycentrics    (b - c)*(a^2 + 2*a*b + 2*a*c + b*c)*(a^2*b - 2*a*b^2 + a^2*c + b^2*c - 2*a*c^2 + b*c^2) : :
X(45657) = X[876] + 2 X[40549]

X(45657) lies on these lines: {2, 876}, {512, 551}, {513, 4755}, {514, 3828}, {523, 4688}, {3842, 28840}, {4784, 16826}, {4806, 29571}, {14405, 35123}, {29580, 38348}, {30665, 45328}, {31148, 45333}, {45338, 45340}

X(45657) = midpoint of X(2) and X(876)
X(45657) = reflection of X(i) in X(j) for these {i,j}: {2, 40549}, {45338, 45340}
X(45657) = crossdifference of every pair of points on line {2382, 28841}
X(45657) = barycentric product X(i)*X(j) for these {i,j}: {537, 28840}, {16826, 36848}
X(45657) = barycentric quotient X(i)/X(j) for these {i,j}: {28840, 18822}, {36848, 27483}

X(45658) = MIDPOINT OF X(2) AND X(3250)

Barycentrics    (b - c)*(a^3*b - 4*a^2*b^2 + a^3*c - 4*a^2*b*c - 4*a^2*c^2 + b^2*c^2) : :

X(45658) lies on these lines: {2, 3250}, {512, 11176}, {513, 4755}, {514, 1639}, {597, 9011}, {665, 4785}, {784, 45343}, {4083, 44567}, {4763, 29350}, {4817, 16831}, {30665, 45323}

X(45658) = midpoint of X(2) and X(3250)
X(45658) = reflection of X(45338) in X(45339)

X(45659) = MIDPOINT OF X(2) AND X(3261)

Barycentrics    (b - c)*(a^3*b - a^2*b^2 + a^3*c - a^2*b*c - a^2*c^2 + 4*b^2*c^2) : :
X(45659) = 5 X[2] - X[21225], 2 X[3261] + X[6586], 5 X[3261] + X[21225], X[4408] + 2 X[17066], 5 X[6586] - 2 X[21225]

X(45659) lies on these lines: {2, 2412}, {513, 45338}, {522, 4688}, {523, 7625}, {536, 20907}, {599, 9000}, {824, 45324}, {4408, 17066}, {9466, 30519}, {28623, 44551}, {35073, 35085}

X(45659) = midpoint of X(2) and X(3261)
X(45659) = reflection of X(6586) in X(2)
X(45659) = crossdifference of every pair of points on line {1384, 8618}

X(45660) = MIDPOINT OF X(2) AND X(4086)

Barycentrics    (b - c)*(a^4 + a^3*b - a^2*b^2 - a*b^3 + a^3*c - a^2*b*c - 4*a*b^2*c + 4*b^3*c - a^2*c^2 - 4*a*b*c^2 + 8*b^2*c^2 - a*c^3 + 4*b*c^3) : :
X(45660) = X[656] - 3 X[19875]

X(45660) lies on these lines: {2, 4086}, {513, 28603}, {519, 8062}, {523, 45324}, {656, 19875}, {3679, 35057}, {4139, 45342}, {8058, 45334}, {8672, 45332}, {28147, 45320}, {31174, 45344}

X(45660) = midpoint of X(2) and X(4086)

X(45661) = MIDPOINT OF X(2) AND X(4120)

Barycentrics    (b - c)*(a^2 - 3*a*b + b^2 - 3*a*c + 3*b*c + c^2) : :
X(45661) = 2 X[3239] + X[3835], 2 X[3700] + X[21196], X[3700] + 2 X[25666], 3 X[4120] + X[4750], 2 X[4129] + X[8045], X[4369] + 2 X[14321], X[4458] + 2 X[18004], 3 X[14431] + X[30605], X[16892] - 7 X[27138], X[21196] - 4 X[25666], 2 X[21212] + X[25259], 2 X[21212] - 5 X[30835], 5 X[24924] + X[44449], X[25259] + 5 X[30835], X[45313] - 4 X[45334], 2 X[45315] + X[45343], 2 X[45342] + X[45344]

X(45661) lies on these lines: {2, 2786}, {514, 661}, {522, 4800}, {523, 45315}, {646, 3807}, {690, 5461}, {812, 1639}, {824, 4944}, {900, 3035}, {918, 4928}, {1638, 37691}, {2785, 14431}, {2789, 14432}, {3667, 31131}, {3700, 21196}, {4049, 4945}, {4369, 14321}, {4375, 6544}, {4379, 28855}, {4458, 18004}, {4522, 4777}, {4784, 6006}, {4786, 31286}, {4926, 44567}, {4927, 28890}, {6546, 21297}, {14426, 30665}, {16892, 27138}, {21098, 24086}, {21212, 25259}, {21894, 44307}, {21959, 32778}, {24924, 44449}, {26275, 45337}, {28851, 45320}, {30519, 45339}, {30792, 45328}, {36800, 36805}

X(45661) = midpoint of X(i) and X(j) for these {i,j}: {2, 4120}, {661, 4789}, {4728, 30565}, {6546, 21297}, {14432, 30709}
X(45661) = reflection of X(i) in X(j) for these {i,j}: {4763, 45326}, {4786, 31286}, {10196, 1639}, {21204, 4928}, {26275, 45337}, {45328, 30792}
X(45661) = complement of X(4750)
X(45661) = X(i)-complementary conjugate of X(j) for these (i,j): {37, 5099}, {100, 126}, {101, 16597}, {111, 11}, {213, 23992}, {671, 21252}, {691, 3739}, {692, 2482}, {892, 21240}, {897, 116}, {923, 1086}, {1783, 5181}, {5380, 141}, {5547, 26932}, {6335, 34517}, {7316, 4904}, {32672, 41311}, {32729, 3666}, {32740, 1015}, {36060, 2968}, {36085, 3741}, {36142, 1125}
X(45661) = X(101)-isoconjugate of X(7312)
X(45661) = crossdifference of every pair of points on line {31, 5168}
X(45661) = barycentric product X(693)*X(5524)
X(45661) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 7312}, {5524, 100}
X(45661) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3700, 25666, 21196}, {25259, 30835, 21212}

X(45662) = MIDPOINT OF X(2) AND X(4226)

Barycentrics    (2*a^2 - b^2 - c^2)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :
X(45662) = X[868] + 2 X[4226], 3 X[3524] - X[7422], 3 X[23234] - X[34174]

X(45662) lies on these lines: {2, 3}, {99, 22254}, {110, 8724}, {187, 1648}, {230, 35606}, {325, 34245}, {351, 690}, {524, 5467}, {542, 5191}, {574, 41939}, {1384, 6792}, {1494, 6394}, {1576, 34319}, {1640, 6041}, {2080, 15360}, {3163, 9475}, {3292, 8030}, {5108, 14685}, {5468, 6390}, {5648, 9145}, {5652, 34291}, {6054, 35278}, {6090, 14916}, {7777, 40871}, {7799, 17941}, {9140, 14830}, {9169, 37809}, {9216, 16508}, {9486, 10418}, {11053, 32459}, {13857, 18860}, {14653, 32227}, {14687, 20423}, {14995, 24975}, {23234, 34174}, {30789, 38741}, {33927, 40112}

X(45662) = midpoint of X(2) and X(4226)
X(45662) = reflection of X(i) in X(j) for these {i,j}: {868, 2}, {14995, 24975}
X(45662) = orthoptic-circle-of-Steiner-inellipse inverse of X(1551)
X(45662) = tripolar centroid of X(14999)
X(45662) = X(i)-Ceva conjugate of X(j) for these (i,j): {7473, 32313}, {16092, 542}
X(45662) = X(i)-isoconjugate of X(j) for these (i,j): {842, 897}, {923, 5641}, {5649, 23894}, {9213, 36096}, {14223, 36142}, {14998, 36085}
X(45662) = crosspoint of X(542) and X(16092)
X(45662) = crossdifference of every pair of points on line {111, 647}
X(45662) = center of circle {{X(2),X(110),X(476),X(4226)}}
X(45662) = intersection of tangents to circle {{X(2),X(110),X(2770),X(5463),X(5464)}} at X(2) and X(110)
X(45662) = barycentric product X(i)*X(j) for these {i,j}: {524, 542}, {648, 39474}, {690, 14999}, {1640, 5468}, {2247, 14210}, {2482, 16092}, {3266, 5191}, {5467, 18312}, {6103, 6390}, {7473, 14417}, {17708, 32313}
X(45662) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 842}, {351, 14998}, {524, 5641}, {542, 671}, {690, 14223}, {1640, 5466}, {2247, 897}, {5191, 111}, {5467, 5649}, {5468, 6035}, {5477, 34174}, {6041, 9178}, {6103, 17983}, {14999, 892}, {23967, 16092}, {32313, 9979}, {34369, 9154}, {39474, 525}
X(45662) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 376, 36194}, {2, 7426, 14694}, {2, 10304, 35922}, {3, 441, 1650}, {3, 15329, 237}, {549, 34094, 2}, {2482, 5642, 9155}, {2482, 35282, 5642}, {7426, 37461, 237}, {11183, 44814, 1649}

X(45663) = MIDPOINT OF X(2) AND X(4369)

Barycentrics    (b - c)*(5*a^2 - 2*a*b - 2*a*c + 5*b*c) : :
X(45663) = 5 X[2] - X[661], 7 X[2] + X[7192], X[2] - 5 X[24924], 3 X[2] + X[31148], 17 X[2] - X[31290], X[661] + 5 X[4369], 7 X[661] + 5 X[7192], X[661] - 25 X[24924], 2 X[661] - 5 X[25666], 3 X[661] + 5 X[31148], 17 X[661] - 5 X[31290], 3 X[661] - 5 X[45315], 2 X[2516] - 5 X[31286], 7 X[4369] - X[7192], X[4369] + 5 X[24924], 2 X[4369] + X[25666], 3 X[4369] - X[31148], 17 X[4369] + X[31290], 3 X[4369] + X[45315], 3 X[4379] + X[31150], X[4761] + 3 X[25055], 3 X[4763] - X[31150], 3 X[4928] - X[31147], X[4932] + 5 X[31250], X[7192] + 35 X[24924], 2 X[7192] + 7 X[25666], 3 X[7192] - 7 X[31148], 17 X[7192] + 7 X[31290], 3 X[7192] + 7 X[45315], 10 X[24924] - X[25666], 15 X[24924] + X[31148], 85 X[24924] - X[31290], 15 X[24924] - X[45315], 3 X[25666] + 2 X[31148], 17 X[25666] - 2 X[31290], 3 X[25666] - 2 X[45315], 17 X[31148] + 3 X[31290], 3 X[31290] - 17 X[45315]

X(45663) lies on these lines: {2, 661}, {513, 45337}, {514, 14425}, {812, 45313}, {824, 44551}, {1638, 28863}, {1639, 28871}, {2516, 4762}, {3828, 4160}, {3907, 45332}, {4379, 4763}, {4761, 25055}, {4785, 4885}, {4928, 31147}, {4932, 31250}, {6002, 45324}, {9013, 20582}, {28846, 45334}, {28855, 45326}, {28859, 44432}, {40459, 41141}

X(45663) = midpoint of X(i) and X(j) for these {i,j}: {2, 4369}, {4379, 4763}, {31148, 45315}, {45313, 45320}
X(45663) = reflection of X(25666) in X(2)
X(45663) = complement of X(45315)
X(45663) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 31148, 45315}, {4369, 45315, 31148}

X(45664) = MIDPOINT OF X(2) AND X(4391)

Barycentrics    (b - c)*(a^3 - a*b^2 - 4*a*b*c + 4*b^2*c - a*c^2 + 4*b*c^2) : :
X(45664) = 5 X[2] - X[17496], X[650] + 2 X[4791], X[905] + 2 X[4391], 5 X[905] - 2 X[17496], 3 X[905] - 2 X[44550], 3 X[905] - 4 X[44561], X[1577] + 2 X[20317], X[1734] - 3 X[19875], 2 X[3239] + X[10015], X[3762] + 2 X[4885], 2 X[3960] - 5 X[31250], 5 X[4391] + X[17496], 3 X[4391] + X[44550], 3 X[4391] + 2 X[44561], X[4944] + 2 X[21198], 3 X[14431] - X[31149], 3 X[17496] - 5 X[44550], 3 X[17496] - 10 X[44561], X[21385] + 2 X[23813], 3 X[41800] - 2 X[44551], 4 X[45334] - X[45341]

X(45664) lies on these lines: {2, 905}, {513, 14431}, {514, 1639}, {525, 1637}, {650, 4791}, {814, 45314}, {1577, 4762}, {1734, 19875}, {1946, 16418}, {2787, 30234}, {3239, 10015}, {3309, 21052}, {3679, 3900}, {3762, 4885}, {3828, 8714}, {3907, 45316}, {3960, 31250}, {4083, 45342}, {4106, 29512}, {4688, 28898}, {4944, 21198}, {6002, 45313}, {13466, 35113}, {14077, 14430}, {14432, 28537}, {16417, 22091}, {16857, 22160}, {18145, 40495}, {21191, 28840}, {21385, 23813}, {23685, 42034}, {23877, 45344}, {23880, 44567}, {23882, 31150}, {28475, 30709}, {41800, 44551}

X(45664) = midpoint of X(2) and X(4391)
X(45664) = reflection of X(i) in X(j) for these {i,j}: {905, 2}, {44550, 44561}, {45316, 45337}, {45320, 45324}
X(45664) = complement of X(44550)
X(45664) = anticomplement of X(44561)
X(45664) = crossdifference of every pair of points on line {3941, 26864}
X(45664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 44550, 44561}, {44550, 44561, 905}

X(45665) = MIDPOINT OF X(2) AND X(4444)

Barycentrics    (b - c)*(a^4 + a^3*b - a^2*b^2 - 4*a*b^3 + a^3*c + 2*a^2*b*c - a*b^2*c + b^3*c - a^2*c^2 - a*b*c^2 + 5*b^2*c^2 - 4*a*c^3 + b*c^3) : :
X(45665) = 2 X[4444] + X[27929]

X(45665) = X(456) lies on these lines: {2, 661}, {115, 17205}, {381, 6002}, {514, 3828}, {551, 40459}, {812, 36848}, {824, 21204}, {918, 45340}, {2786, 45342}, {4120, 4453}, {4762, 25381}, {9466, 45324}, {28846, 45339}, {35121, 35123}

X(45665) = midpoint of X(2) and X(4444)
X(45665) = reflection of X(27929) in X(2)

X(45666) = MIDPOINT OF X(2) AND X(4448)

Barycentrics    (b - c)*(3*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(45666) = X[764] - 7 X[3624], 5 X[1698] + X[6161], 2 X[3716] + X[9508], 3 X[4448] + X[36848], 2 X[13246] + X[18004], X[14421] - 3 X[25055], 5 X[19862] - 2 X[19947], 4 X[19878] - X[23814], X[45314] + 2 X[45337], 2 X[45314] + X[45342], 4 X[45337] - X[45342]

X(45666) lies on these lines: {1, 9260}, {2, 513}, {37, 650}, {392, 4083}, {514, 1125}, {522, 28602}, {523, 10180}, {551, 9269}, {659, 4728}, {667, 5251}, {764, 3624}, {812, 45314}, {900, 3035}, {1635, 4800}, {1639, 26275}, {1647, 16507}, {1698, 6161}, {1960, 28603}, {3251, 3679}, {3309, 11231}, {4369, 6707}, {4375, 4782}, {4775, 36531}, {4802, 6546}, {4806, 25381}, {4809, 30565}, {4977, 21204}, {6006, 6666}, {6370, 32193}, {6545, 28195}, {6548, 30598}, {7662, 28151}, {13246, 18004}, {14028, 33920}, {14296, 21606}, {14421, 25055}, {14429, 28114}, {14430, 25569}, {14475, 28220}, {19862, 19947}, {19878, 23814}, {27855, 30963}, {29188, 45332}

X(45666) = midpoint of X(i) and X(j) for these {i,j}: {1, 30583}, {2, 4448}, {659, 4728}, {1635, 4800}, {1639, 26275}, {1960, 28603}, {3251, 3679}, {3716, 4763}, {4809, 30565}, {14429, 28114}, {14430, 25569}
X(45666) = reflection of X(i) in X(j) for these {i,j}: {9269, 551}, {9508, 4763}
X(45666) = complement of X(36848)
X(45666) = X(i)-complementary conjugate of X(j) for these (i,j): {692, 35123}, {2382, 11}, {18822, 21252}
X(45666) = crossdifference of every pair of points on line {36, 3230}
X(45666) = {X(45314),X(45337)}-harmonic conjugate of X(45342)

X(45667) = MIDPOINT OF X(2) AND X(4449)

Barycentrics    (b - c)*(4*a^3 - 3*a^2*b - a*b^2 - 3*a^2*c + 5*a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(45667) = 2 X[1] + X[24720], X[663] - 3 X[38314], 7 X[3622] - X[4724], 4 X[3636] - X[4794], X[4147] + 2 X[4449], 3 X[9269] + X[45332], 3 X[14413] - X[44550], X[21343] + 2 X[31286]

X(45667) lies on these lines: {1, 24720}, {2, 4147}, {514, 551}, {519, 17072}, {522, 14413}, {663, 38314}, {3622, 4724}, {3636, 4794}, {3900, 45328}, {3907, 45320}, {4083, 45313}, {4367, 4785}, {4428, 44408}, {9269, 45332}, {14077, 44561}, {21343, 24768}, {22090, 42042}, {23877, 45341}, {29226, 45314}, {29324, 45342}

X(45667) = midpoint of X(2) and X(4449)
X(45667) = reflection of X(i) in X(j) for these {i,j}: {4147, 2}, {45316, 551}

X(45668) = MIDPOINT OF X(2) AND X(4458)

Barycentrics    (b - c)*(5*a^3 - 2*a*b^2 - 3*b^3 + a*b*c + 2*b^2*c - 2*a*c^2 + 2*b*c^2 - 3*c^3) : :
X(45668) = 5 X[2] - X[4088], X[551] + 3 X[21181], 3 X[1638] - X[45328], X[3241] + 3 X[30574], X[4088] + 5 X[4458], 3 X[4088] - 5 X[45344], 3 X[4458] + X[45344], X[4707] + 3 X[25055], 3 X[14413] + X[44553]

X(45668) lies on these lines: {2, 4088}, {514, 14422}, {522, 1638}, {551, 2785}, {918, 45318}, {2786, 45342}, {3241, 30574}, {4707, 25055}, {4763, 28147}, {4778, 26275}, {4809, 21204}, {14413, 44553}, {23877, 44561}, {29037, 45324}

X(45668) = midpoint of X(i) and X(j) for these {i,j}: {2, 4458}, {4809, 21204}
X(45668) = reflection of X(45337) in X(45318)
X(45668) = complement of X(45344)

X(45669) = MIDPOINT OF X(2) AND X(4467)

Barycentrics    (b - c)*(4*a^2 - a*b - 3*b^2 - a*c - 2*b*c - 3*c^2) : :
X(45669) = 3 X[1638] - 4 X[44551], 3 X[1638] - 2 X[45320], 3 X[1639] - 4 X[44567], 4 X[2487] - X[4024], X[3700] + 2 X[4467], X[3700] - 4 X[17069], 2 X[4025] + X[4976], 4 X[4025] - X[21104], X[4467] + 2 X[17069], 3 X[4750] - X[31148], X[4820] - 4 X[7658], X[4841] + 2 X[4897], X[4841] - 4 X[21196], X[4897] + 2 X[21196], 3 X[4944] - 4 X[45334], 2 X[4976] + X[21104], X[7178] - 4 X[21192], 3 X[27486] - X[31150], 3 X[41800] - 2 X[45324]

X(45669) lies on these lines: {2, 3700}, {514, 4773}, {522, 1638}, {523, 4750}, {525, 14395}, {553, 17094}, {824, 45313}, {900, 31147}, {918, 27486}, {1021, 3928}, {1639, 28898}, {2487, 4024}, {2786, 45315}, {3004, 4785}, {3910, 14412}, {3929, 9404}, {4025, 4762}, {4379, 28183}, {4688, 31174}, {4728, 28221}, {4786, 28894}, {4820, 7658}, {4841, 4897}, {4944, 45334}, {7178, 21192}, {8714, 45338}, {20508, 23878}, {23876, 45341}, {29037, 44729}, {29078, 45323}, {29232, 31149}, {41800, 45324}

X(45669) = midpoint of X(2) and X(4467)
X(45669) = reflection of X(i) in X(j) for these {i,j}: {2, 17069}, {3700, 2}, {45320, 44551}
X(45669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4025, 4976, 21104}, {4467, 17069, 3700}, {4897, 21196, 4841}, {44551, 45320, 1638}

X(45670) = MIDPOINT OF X(2) AND X(4468)

Barycentrics    (b - c)*(5*a^2 - 8*a*b + 3*b^2 - 8*a*c + 2*b*c + 3*c^2) : :
X(45670) = 3 X[1638] - 4 X[44563], 3 X[1639] - X[45320], 3 X[1639] - 2 X[45334], X[3676] + 2 X[4468], X[3676] - 4 X[4521], X[4106] - 4 X[14350], X[4468] + 2 X[4521], X[4897] - 4 X[31182], 3 X[6546] + X[31147], 3 X[10196] - X[45313], 3 X[30565] + X[31150]

X(45670) lies on these lines: {2, 3676}, {514, 1639}, {522, 14392}, {649, 3929}, {918, 44551}, {1121, 35168}, {1638, 44563}, {3239, 4762}, {3679, 28292}, {4106, 14350}, {4785, 11068}, {4897, 31182}, {6006, 6172}, {6546, 31142}, {10196, 28846}, {28878, 31148}, {28890, 44432}, {29350, 31165}

X(45670) = midpoint of X(2) and X(4468)
X(45670) = reflection of X(i) in X(j) for these {i,j}: {2, 4521}, {3676, 2}, {44551, 44567}, {45320, 45334}
X(45670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1639, 45320, 45334}, {4468, 4521, 3676}

X(45671) = MIDPOINT OF X(2) AND X(4560)

Barycentrics    (b - c)*(-2*a^3 + 2*a*b^2 + 2*a*b*c + b^2*c + 2*a*c^2 + b*c^2) : :
X(45671) = X[1] + 2 X[4913], 4 X[650] - X[3762], 4 X[905] - X[4978], 4 X[1125] - X[4804], X[1577] + 2 X[4560], X[1577] - 4 X[14838], 3 X[1577] - 4 X[45324], 4 X[3828] - 3 X[21052], 2 X[3960] + X[17494], X[4086] - 4 X[8043], X[4560] + 2 X[14838], 3 X[4560] + 2 X[45324], X[4707] - 4 X[17069], X[4761] - 4 X[9508], 2 X[4770] + X[4922], 2 X[4791] - 5 X[31209], X[4815] - 4 X[31947], 3 X[14838] - X[45324], X[21222] + 5 X[26777]

X(45671) lies on these lines: {1, 4913}, {2, 1577}, {30, 39212}, {514, 1635}, {519, 4041}, {522, 14414}, {523, 14419}, {525, 14395}, {551, 4151}, {650, 3762}, {798, 1019}, {812, 14405}, {814, 31149}, {824, 14402}, {905, 4762}, {1125, 4804}, {2482, 35113}, {3227, 35153}, {3679, 3907}, {3828, 21052}, {3910, 45341}, {3960, 17494}, {4086, 8043}, {4367, 4948}, {4707, 17069}, {4761, 9508}, {4770, 4922}, {4776, 29178}, {4785, 14349}, {4791, 31209}, {4815, 31947}, {4858, 35128}, {4893, 29148}, {6002, 45315}, {6362, 15670}, {16370, 21789}, {21222, 26777}, {23876, 27486}, {23879, 36900}, {23880, 44567}, {23882, 44561}, {29013, 31147}, {29033, 44429}, {29037, 45344}, {29051, 45328}, {40459, 41142}

X(45671) = midpoint of X(i) and X(j) for these {i,j}: {2, 4560}, {4367, 4948}, {31150, 44550}
X(45671) = reflection of X(i) in X(j) for these {i,j}: {2, 14838}, {1577, 2}, {31149, 45323}, {45320, 44561}
X(45671) = anticomplement of X(45324)
X(45671) = crossdifference of every pair of points on line {2177, 2667}
X(45671) = {X(4560),X(14838)}-harmonic conjugate of X(1577)

X(45672) = MIDPOINT OF X(2) AND X(4576)

Barycentrics    (2*a^2 - b^2 - c^2)*(a^2*b^2 + a^2*c^2 - 2*b^2*c^2) : :
X(45672) = 5 X[2] - X[25047], X[3124] + 2 X[4576], 5 X[3124] - 2 X[25047], 5 X[4576] + X[25047], 3 X[21356] + X[25052]

X(45672) lies on these lines: {2, 694}, {6, 9146}, {99, 2502}, {126, 1648}, {141, 8288}, {323, 12151}, {351, 690}, {524, 3266}, {538, 3231}, {599, 2854}, {620, 41939}, {888, 6786}, {1350, 9775}, {3005, 35077}, {4563, 5182}, {5026, 5468}, {5104, 5971}, {5650, 9466}, {7192, 35068}, {7664, 11053}, {7757, 35275}, {8030, 18800}, {8627, 13586}, {9143, 14916}, {11059, 13410}, {11646, 14360}, {21356, 25052}, {21358, 42008}, {22165, 25325}, {34537, 35146}

X(45672) = midpoint of X(i) and X(j) for these {i,j}: {2, 4576}, {22165, 25325}
X(45672) = reflection of X(3124) in X(2)
X(45672) = tripolar centroid of X(23342)
X(45672) = X(14608)-Ceva conjugate of X(524)
X(45672) = X(i)-isoconjugate of X(j) for these (i,j): {111, 37132}, {729, 897}, {923, 3228}, {9178, 36133}, {23894, 32717}
X(45672) = crosspoint of X(524) and X(14608)
X(45672) = crosssum of X(i) and X(j) for these (i,j): {111, 14609}, {729, 41309}
X(45672) = crossdifference of every pair of points on line {111, 729}
X(45672) = barycentric product X(i)*X(j) for these {i,j}: {187, 30736}, {524, 538}, {690, 23342}, {2234, 14210}, {3231, 3266}, {5118, 35522}, {5468, 9148}, {14608, 35073}, {14609, 36792}, {21839, 30938}
X(45672) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 729}, {524, 3228}, {538, 671}, {888, 9178}, {896, 37132}, {2234, 897}, {2482, 14608}, {3231, 111}, {3266, 34087}, {5118, 691}, {5467, 32717}, {5468, 9150}, {6786, 5968}, {9148, 5466}, {14424, 35366}, {14609, 10630}, {23342, 892}, {23889, 36133}, {30736, 18023}, {33875, 32740}, {36822, 9154}, {39689, 41309}
X(45672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 5108, 2502}, {2482, 12036, 1641}, {2482, 45330, 9155}, {5026, 5468, 39689}, {5468, 31128, 5026}

X(45673) = MIDPOINT OF X(2) AND X(4724)

Barycentrics    (b - c)*(4*a^3 - 3*a^2*b - a*b^2 - 3*a^2*c - 7*a*b*c + b^2*c - a*c^2 + b*c^2) : :
X(45673) = 2 X[551] - 3 X[45316], X[661] + 2 X[8689], 3 X[663] - X[3241], X[3679] + 3 X[4040], 2 X[3679] - 3 X[4147], 4 X[3828] - 3 X[17072], 2 X[4040] + X[4147], 2 X[4724] + X[24720]

X(45673) lies on these lines: {2, 4724}, {513, 4763}, {514, 551}, {519, 4794}, {522, 14392}, {659, 4785}, {661, 8689}, {663, 3241}, {1635, 6006}, {3679, 4040}, {3716, 4762}, {3828, 17072}, {4778, 31148}, {4977, 45318}, {8672, 45317}, {26275, 28851}, {28209, 44902}, {28225, 44563}, {29186, 45324}, {29246, 45332}, {29362, 45342}, {45320, 45337}

X(45673) = midpoint of X(2) and X(4724)
X(45673) = reflection of X(i) in X(j) for these {i,j}: {24720, 2}, {45313, 45314}, {45320, 45337}, {45328, 44567}

X(45674) = MIDPOINT OF X(2) AND X(4750)

Barycentrics    (b - c)*(3*a^2 - a*b - b^2 - a*c + b*c - c^2) : :
X(45674) = X[64974] + 2 X[21212], 3 X[1635] + X[21115], 3 X[1638] + X[4773], 3 X[1638] - X[4927], X[2254] + 2 X[13246], 4 X[2487] - X[4369], 2 X[2487] + X[17069], 8 X[2487] + X[21196], X[3776] + 2 X[4394], 2 X[3798] + X[3835], X[3798] + 2 X[7658], X[3835] - 4 X[7658], X[4025] + 2 X[31286], X[4120] + 3 X[4750], X[4369] + 2 X[17069], 2 X[4369] + X[21196], 3 X[4453] - X[21115], X[4458] + 2 X[9508], X[4467] + 5 X[24924], 3 X[4763] - 2 X[14425], 2 X[4773] + 3 X[21204], X[4897] + 2 X[25666], 2 X[4927] - 3 X[21204], X[4931] - 5 X[24924], X[4984] + 3 X[14475], X[8045] + 2 X[21192], 3 X[10196] - 4 X[14425], 3 X[14419] - X[30580], 3 X[14475] - X[21297], X[16892] + 5 X[27013], 4 X[17069] - X[21196], X[25259] - 7 X[31207], 2 X[44551] + X[45313]

X(45674) lies on these lines: {2, 2786}, {88, 4049}, {142, 13277}, {244, 17761}, {514, 1635}, {522, 4809}, {523, 2487}, {620, 690}, {649, 21212}, {650, 28851}, {659, 4778}, {812, 1638}, {900, 4928}, {905, 28468}, {918, 4763}, {1054, 2789}, {1637, 21209}, {2254, 3667}, {2785, 14419}, {2826, 13226}, {3310, 3960}, {3776, 4394}, {4025, 26248}, {4379, 27486}, {4467, 4931}, {4500, 28205}, {4707, 30577}, {4785, 4786}, {4850, 21828}, {4885, 4926}, {4893, 28855}, {4897, 25666}, {4984, 14475}, {5075, 17596}, {6002, 41800}, {6548, 14435}, {6712, 31380}, {7653, 28199}, {8045, 21192}, {9283, 21138}, {9511, 28292}, {11219, 34311}, {14431, 24287}, {16610, 21894}, {16892, 27013}, {17494, 21116}, {21129, 21222}, {25259, 31207}, {28602, 45344}

X(45674) = midpoint of X(i) and X(j) for these {i,j}: {2, 4750}, {649, 44435}, {1635, 4453}, {2254, 44433}, {3798, 44432}, {4379, 27486}, {4467, 4931}, {4773, 4927}, {4984, 21297}, {6548, 14435}, {17494, 21116}, {21129, 21222}
X(45674) = reflection of X(i) in X(j) for these {i,j}: {3835, 44432}, {4928, 44902}, {10196, 4763}, {21204, 1638}, {44432, 7658}, {44433, 13246}, {44435, 21212}, {45344, 28602}
X(45674) = complement of X(4120)
X(45674) = complement of the isogonal conjugate of X(4591)
X(45674) = complement of the isotomic conjugate of X(4615)
X(45674) = psi-transform of X(38941)
X(45674) = X(i)-complementary conjugate of X(j) for these (i,j): {81, 3259}, {88, 125}, {106, 8287}, {110, 16594}, {163, 4370}, {593, 34590}, {662, 121}, {901, 1211}, {903, 21253}, {1333, 35092}, {1417, 17058}, {1797, 34846}, {3257, 3454}, {4555, 21245}, {4556, 34587}, {4565, 1145}, {4591, 10}, {4615, 2887}, {4622, 141}, {4634, 626}, {5376, 31946}, {9268, 4129}, {9456, 115}, {32659, 16573}, {32665, 1213}, {32719, 16589}, {36058, 15526}
X(45674) = crosspoint of X(2) and X(4615)
X(45674) = crosssum of X(6) and X(14407)
X(45674) = crossdifference of every pair of points on line {2177, 5168}
X(45674) = barycentric product X(514)*X(20072)
X(45674) = barycentric quotient X(i)/X(j) for these {i,j}: {20072, 190}, {23166, 1331}
X(45674) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1638, 4773, 4927}, {2487, 17069, 4369}, {3798, 7658, 3835}, {4369, 17069, 21196}, {4984, 14475, 21297}

X(45675) = MIDPOINT OF X(2) AND X(4763)

Barycentrics    (b - c)*(5*a^2 - 4*a*b - 4*a*c + 3*b*c) : :
X(45675) = 3 X[2] + X[1635], 5 X[2] - X[4728], 9 X[2] - X[21297], 5 X[1635] + 3 X[4728], X[1635] - 3 X[4763], 3 X[1635] + X[21297], 2 X[2490] + X[21212], X[4369] - 7 X[31207], X[4369] + 5 X[31209], X[4453] + 3 X[6544], X[4728] + 5 X[4763], 3 X[4728] - 5 X[4928], 9 X[4728] - 5 X[21297], 3 X[4763] + X[4928], 9 X[4763] + X[21297], X[4830] + 5 X[30795], X[4893] + 7 X[31207], X[4893] - 5 X[31209], 3 X[4928] - X[21297], X[14408] + 3 X[27344], X[21115] + 3 X[31992], 5 X[24924] + 7 X[27115], X[25666] + 2 X[31286], X[25666] - 4 X[31287], 7 X[31207] + 5 X[31209], X[31286] + 2 X[31287]

X(45675) lies on these lines: {2, 812}, {351, 27798}, {513, 6687}, {514, 14425}, {676, 28161}, {740, 11176}, {891, 40479}, {900, 45337}, {1022, 31227}, {1638, 10196}, {2490, 21212}, {2786, 45326}, {2977, 28147}, {3035, 3887}, {4148, 14438}, {4369, 4893}, {4448, 45328}, {4453, 6544}, {4830, 30795}, {4925, 4962}, {6008, 45339}, {6684, 38328}, {14408, 27344}, {14426, 38238}, {21115, 31992}, {21758, 37680}, {22314, 43223}, {24924, 27115}, {28882, 44432}

X(45675) = midpoint of X(i) and X(j) for these {i,j}: {2, 4763}, {351, 27798}, {1635, 4928}, {1638, 10196}, {4369, 4893}, {4448, 45328}, {14425, 44902}, {14426, 38238}
X(45675) = complement of X(4928)
X(45675) = crosssum of X(6) and X(23650)
X(45675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1635, 4928}, {4763, 4928, 1635}, {31207, 31209, 4369}, {31286, 31287, 25666}

X(45676) = MIDPOINT OF X(2) AND X(4824)

Barycentrics    (b - c)*(a^3 - 3*a^2*b - 7*a*b^2 - 3*a^2*c - 13*a*b*c - 2*b^2*c - 7*a*c^2 - 2*b*c^2) : :
X(45676) = X[3679] - 3 X[4705], 4 X[3828] - 3 X[45332]

X(45676) lies on these lines: {2, 4824}, {513, 14404}, {514, 3828}, {523, 45315}, {661, 4948}, {3679, 4705}, {4120, 4777}, {4802, 30601}, {4928, 28179}, {4977, 45328}, {9508, 28840}, {28147, 45339}, {28175, 45340}, {29298, 34641}

X(45676) = midpoint of X(i) and X(j) for these {i,j}: {2, 4824}, {661, 4948}
X(45676) = reflection of X(45342) in X(45315)

X(45677) = MIDPOINT OF X(2) AND X(4927)

Barycentrics    (b - c)*(2*a^2 - 3*a*b - b^2 - 3*a*c + 6*b*c - c^2) : :
X(45677) = X[676] + 2 X[3837], 3 X[1638] - X[4750], X[1638] - 3 X[14475], 2 X[2487] + X[4106], 2 X[2490] - 5 X[31250], 2 X[2527] + X[23729], 2 X[2527] - 5 X[24924], X[3004] + 5 X[26985], 2 X[3676] + X[14321], 3 X[4728] + X[4750], X[4728] + 3 X[14475], X[4750] - 9 X[14475], X[4789] - 5 X[26985], 2 X[4927] + X[14425], 3 X[6548] + X[30565], 2 X[7658] + X[23813], X[21104] + 5 X[30835], X[23729] + 5 X[24924], X[23770] + 5 X[30795]

X(45677) lies on these lines: {2, 4927}, {11, 244}, {514, 4521}, {523, 7625}, {551, 28294}, {812, 44902}, {918, 4928}, {1639, 6545}, {2487, 4106}, {2490, 31250}, {2527, 23729}, {2826, 45310}, {3004, 4789}, {3676, 14321}, {4379, 4977}, {4762, 44432}, {4763, 6009}, {4776, 28902}, {4778, 45337}, {4874, 28209}, {4926, 44551}, {4997, 6548}, {7658, 23813}, {21104, 30835}, {23770, 30795}, {23888, 44566}, {26275, 45318}, {28851, 45339}, {31147, 39386}

X(45677) = midpoint of X(i) and X(j) for these {i,j}: {2, 4927}, {1638, 4728}, {1639, 6545}, {3004, 4789}, {4106, 4786}, {4928, 21204}
X(45677) = reflection of X(i) in X(j) for these {i,j}: {4786, 2487}, {14425, 2}, {26275, 45318}, {30792, 45340}
X(45677) = tripolar centroid of X(4373)
X(45677) = X(100)-isoconjugate of X(17222)
X(45677) = crossdifference of every pair of points on line {101, 1384}
X(45677) = barycentric product X(i)*X(j) for these {i,j}: {514, 17132}, {6548, 12035}
X(45677) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 17222}, {12035, 17780}, {17132, 190}, {45140, 901}
X(45677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4728, 14475, 1638}, {23729, 24924, 2527}

X(45678) = MIDPOINT OF X(2) AND X(4928)

Barycentrics    (b - c)*(3*a^2 - 4*a*b - 4*a*c + 5*b*c) : :
X(45678) = 5 X[2] - X[1635], 3 X[2] + X[4728], 7 X[2] + X[21297], 3 X[1635] + 5 X[4728], 3 X[1635] - 5 X[4763], X[1635] + 5 X[4928], 7 X[1635] + 5 X[21297], X[3716] + 5 X[30795], 5 X[3835] + X[4790], X[3835] + 5 X[31250], 5 X[4369] + X[4813], X[4369] + 5 X[30835], X[4728] - 3 X[4928], 7 X[4728] - 3 X[21297], X[4763] + 3 X[4928], 7 X[4763] + 3 X[21297], 5 X[4776] - X[4813], X[4776] - 5 X[30835], X[4790] - 25 X[31250], X[4813] - 25 X[30835], 2 X[4885] + X[25666], 7 X[4928] - X[21297], 3 X[14475] + X[30565], 5 X[24924] + 7 X[27138], 5 X[30795] - X[36848], X[45337] + 2 X[45340]

X(45678) lies on these lines: {2, 812}, {513, 45337}, {514, 4521}, {522, 30792}, {824, 44432}, {900, 6667}, {1638, 37691}, {1639, 21204}, {2786, 44902}, {2820, 10171}, {3716, 30795}, {3835, 4790}, {3887, 45310}, {4369, 4776}, {4379, 45315}, {4508, 29578}, {4800, 45328}, {4927, 10196}, {4962, 44563}, {6008, 31286}, {9148, 10180}, {14475, 30565}, {18743, 20908}, {24924, 27138}, {27929, 30865}

X(45678) = midpoint of X(i) and X(j) for these {i,j}: {2, 4928}, {1639, 21204}, {3716, 36848}, {4369, 4776}, {4379, 45315}, {4728, 4763}, {4800, 45328}, {4927, 10196}, {9148, 10180}
X(45678) = complement of X(4763)
X(45678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4728, 4763}, {4763, 4928, 4728}

X(45679) = MIDPOINT OF X(2) AND X(4984)

Barycentrics    (b - c)*(-5*a^2 + a*b + b^2 + a*c + b*c + c^2) : :
X(45679) = 2 X[649] + X[21196], 3 X[1635] - X[30565], X[1638] + 3 X[4773], 5 X[1638] - 3 X[4927], 4 X[1638] - 3 X[21204], X[4380] + 2 X[21212], X[4750] + 3 X[14435], 3 X[4763] - 2 X[45326], 2 X[4765] + X[4932], 5 X[4773] + X[4927], 4 X[4773] + X[21204], 4 X[4927] - 5 X[21204], 3 X[10196] - 2 X[30565], 4 X[45313] - X[45343]

X(45679) lies on these lines: {2, 4984}, {239, 514}, {522, 45313}, {650, 28867}, {812, 1638}, {900, 3035}, {1491, 6006}, {1635, 2786}, {4155, 38238}, {4380, 21212}, {4394, 28898}, {4728, 24183}, {4897, 28871}, {6544, 41841}, {28217, 45315}, {28855, 31150}

X(45679) = midpoint of X(i) and X(j) for these {i,j}: {2, 4984}, {649, 27486}
X(45679) = reflection of X(i) in X(j) for these {i,j}: {10196, 1635}, {21196, 27486}
X(45679) = X(8700)-complementary conjugate of X(11)
X(45679) = crossdifference of every pair of points on line {42, 9259}

X(45680) = MIDPOINT OF X(2) AND X(5027)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(-2*a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2) : :
X(45680) = 3 X[11176] - X[45336]

X(45680) lies on these lines: {2, 5027}, {351, 690}, {512, 11176}, {597, 888}, {826, 14403}, {1637, 7927}, {2793, 10168}, {4108, 9189}, {5476, 11616}, {5652, 9208}, {9135, 11182}, {14610, 45327}

X(45680) = midpoint of X(i) and X(j) for these {i,j}: {2, 5027}, {351, 11183}, {5476, 11616}, {5652, 9208}, {9135, 11182}, {14610, 45327}, {45317, 45335}
X(45680) = tripolar centroid of X(7766)
X(45680) = X(i)-isoconjugate of X(j) for these (i,j): {897, 25424}, {36142, 43688}
X(45680) = crossdifference of every pair of points on line {111, 12149}
X(45680) = barycentric product X(i)*X(j) for these {i,j}: {351, 41259}, {524, 25423}, {690, 7766}, {22105, 32449}
X(45680) = barycentric quotient X(i)/X(j) for these {i,j}: {187, 25424}, {690, 43688}, {7766, 892}, {25423, 671}

X(45681) = MIDPOINT OF X(2) AND X(5664)

Barycentrics    (b^2 - c^2)*(3*a^8 - 8*a^6*b^2 + 6*a^4*b^4 - b^8 - 8*a^6*c^2 + 11*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 2*b^6*c^2 + 6*a^4*c^4 - 5*a^2*b^2*c^4 - 2*b^4*c^4 + 2*b^2*c^6 - c^8) : :
X(45681) = 5 X[2] - X[2394], X[2394] + 5 X[5664], 2 X[2394] - 5 X[14566], 3 X[3524] - X[18556], 7 X[3526] - X[5489], 3 X[5055] - X[42733], 2 X[5664] + X[14566], 2 X[8552] + X[24978], X[14223] + 3 X[41134]

X(45681) lies on these lines: {2, 525}, {5, 39491}, {520, 5892}, {523, 549}, {620, 2492}, {2797, 44204}, {3524, 18556}, {3526, 5489}, {5055, 42733}, {6587, 15048}, {8724, 42738}, {9517, 44202}, {14223, 41134}, {15421, 35361}, {18310, 23878}, {35481, 44705}

X(45681) = midpoint of X(i) and X(j) for these {i,j}: {2, 5664}, {8724, 42738}
X(45681) = reflection of X(i) in X(j) for these {i,j}: {14566, 2}, {39491, 5}
X(45681) = complement of the isotomic conjugate of X(30528)
X(45681) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 25641}, {477, 21253}, {30528, 2887}, {32663, 34846}, {36047, 34827}, {36062, 127}, {36151, 125}
X(45681) = crosspoint of X(2) and X(30528)
X(45681) = crossdifference of every pair of points on line {1495, 7669}
X(45681) = barycentric product X(850)*X(32609)
X(45681) = barycentric quotient X(32609)/X(110)

X(45682) = MIDPOINT OF X(2) AND X(6130)

Barycentrics    (b^2 - c^2)*(5*a^8 - 12*a^6*b^2 + 9*a^4*b^4 - 2*a^2*b^6 - 12*a^6*c^2 + 9*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 5*b^6*c^2 + 9*a^4*c^4 - 2*a^2*b^2*c^4 - 10*b^4*c^4 - 2*a^2*c^6 + 5*b^2*c^6) : :
X(45682) = 5 X[2] - X[684], X[684] + 5 X[6130], 3 X[684] - 5 X[45319], 3 X[3545] + X[9409], 3 X[5054] + X[41079], 3 X[6130] + X[45319], X[8552] - 3 X[11539]

X(45682) lies on these lines: {2, 684}, {30, 44818}, {523, 44401}, {526, 45311}, {547, 9517}, {549, 2797}, {804, 6055}, {3545, 9409}, {3845, 44810}, {5054, 41079}, {8552, 11539}, {9033, 40477}, {45321, 45336}

X(45682) = complement of X(45319)
X(45682) = midpoint of X(i) and X(j) for these {i,j}: {2, 6130}, {3845, 44810}, {45321, 45336}

X(45683) = MIDPOINT OF X(2) AND X(6332)

Barycentrics    (b - c)*(5*a^3 - 3*a^2*b - 5*a*b^2 + 3*b^3 - 3*a^2*c - 2*a*b*c + 5*b^2*c - 5*a*c^2 + 5*b*c^2 + 3*c^3) : :
X(45683) = 2 X[6332] + X[14837], 3 X[19883] - X[20517], 3 X[41800] - 4 X[44563], 2 X[45334] + X[45341]

X(45683) lies on these lines: {2, 2399}, {514, 1639}, {519, 4163}, {522, 14414}, {525, 14345}, {905, 4130}, {3762, 29005}, {3810, 45337}, {3910, 44567}, {3929, 4091}, {14432, 28292}, {19883, 20517}, {28478, 45313}, {29082, 45340}, {41800, 44563}

X(45683) = midpoint of X(2) and X(6332)
X(45683) = reflection of X(i) in X(j) for these {i,j}: {14837, 2}, {44551, 44561}

X(45684) = MIDPOINT OF X(2) AND X(6544)

Barycentrics    (b - c)*(7*a^2 - 7*a*b + b^2 - 7*a*c + 5*b*c + c^2) : :
X(45684) = 7 X[2] - X[6545], 5 X[2] + X[6546], 5 X[2] - X[6548], 2 X[2] + X[10196], 4 X[2] - X[21204], 3 X[2] + X[31992], 7 X[2] + X[44009], 7 X[3624] + 2 X[32212], 8 X[3634] + X[5592], 8 X[4422] + X[42555], X[4763] + 2 X[45326], X[4928] + 2 X[14425], 7 X[6544] + X[6545], 5 X[6544] - X[6546], 5 X[6544] + X[6548], 3 X[6544] + X[14475], 4 X[6544] + X[21204], 3 X[6544] - X[31992], 7 X[6544] - X[44009], 5 X[6545] + 7 X[6546], 5 X[6545] - 7 X[6548], 2 X[6545] + 7 X[10196], 3 X[6545] - 7 X[14475], 4 X[6545] - 7 X[21204], 3 X[6545] + 7 X[31992], 2 X[6546] - 5 X[10196], 3 X[6546] + 5 X[14475], 4 X[6546] + 5 X[21204], 3 X[6546] - 5 X[31992], 7 X[6546] - 5 X[44009], 2 X[6548] + 5 X[10196], 3 X[6548] - 5 X[14475], 4 X[6548] - 5 X[21204], 3 X[6548] + 5 X[31992], 7 X[6548] + 5 X[44009], 3 X[10196] + 2 X[14475], 2 X[10196] + X[21204], 3 X[10196] - 2 X[31992], 7 X[10196] - 2 X[44009], 4 X[14475] - 3 X[21204], 7 X[14475] + 3 X[44009], X[21196] - 10 X[31209], 3 X[21204] + 4 X[31992], 7 X[21204] + 4 X[44009], 7 X[31992] - 3 X[44009], 13 X[34595] - 4 X[44315], 8 X[44567] + X[45343]

X(45684) lies on these lines: {2, 514}, {649, 35595}, {900, 3035}, {1644, 34764}, {3624, 32212}, {3634, 5592}, {3835, 5316}, {4448, 6006}, {4755, 4777}, {4776, 25381}, {4928, 6009}, {21196, 31209}, {25377, 30583}, {27929, 28886}, {28209, 45315}, {34595, 44315}

X(45684) = midpoint of X(i) and X(j) for these {i,j}: {2, 6544}, {1644, 34764}, {6545, 44009}, {6546, 6548}, {14475, 31992}
X(45684) = reflection of X(10196) in X(6544)
X(45684) = complement of X(14475)
X(45684) = X(i)-complementary conjugate of X(j) for these (i,j): {692, 35121}, {2384, 11}, {35168, 21252}
X(45684) = crossdifference of every pair of points on line {902, 9259}
X(45684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 10196, 21204}, {2, 31992, 14475}, {6544, 14475, 31992}

X(45685) = MIDPOINT OF X(2) AND X(6590)

Barycentrics    (b - c)*(5*a^2 - 2*a*b + 3*b^2 - 2*a*c + 8*b*c + 3*c^2) : :
X(45685) = X[4500] + 2 X[43061], 3 X[4789] + X[31150], 3 X[4927] - 5 X[45320]

X(45685) lies on these lines: {2, 6590}, {514, 1639}, {522, 45313}, {523, 44401}, {824, 44551}, {3239, 28840}, {4500, 43061}, {4762, 11068}, {4763, 28161}, {4786, 4931}, {4789, 31150}, {4802, 45326}, {28846, 31148}, {28894, 44432}

X(45685) = midpoint of X(i) and X(j) for these {i,j}: {2, 6590}, {4786, 4931}, {45313, 45343}
X(45685) = reflection of X(45315) in X(45334)

X(45686) = MIDPOINT OF X(2) AND X(7253)

Barycentrics    (b - c)*(2*a^4 - a^3*b - 2*a^2*b^2 + a*b^3 - a^3*c - 2*a^2*b*c + a*b^2*c + 2*b^3*c - 2*a^2*c^2 + a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3) : :
X[656] + 2 X[7253], X[656] - 4 X[8062], X[4064] + 2 X[44409], X[7253] + 2 X[8062]

X(45686) lies on these lines: {2, 656}, {381, 30212}, {513, 4379}, {519, 4086}, {522, 14414}, {523, 14432}, {525, 11125}, {661, 24506}, {832, 31149}, {3679, 35057}, {4064, 44409}, {4171, 17281}, {6003, 45324}, {8674, 21052}, {8768, 31158}, {9253, 31163}, {16370, 23226}, {16418, 23189}, {17378, 18160}, {28623, 44550}

X(45686) = midpoint of X(2) and X(7253)
X(45686) = reflection of X(i) in X(j) for these {i,j}: {2, 8062}, {656, 2}
X(45686) = {X(7253),X(8062)}-harmonic conjugate of X(656)

X(45687) = MIDPOINT OF X(2) AND X(9131)

Barycentrics    (b^2 - c^2)*(4*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 + c^4) : :
X(45687) = 3 X[351] - X[14420], X[1637] - 3 X[9125], 2 X[1637] - 3 X[9189], 3 X[1649] - X[9148], X[3268] + 3 X[9123], X[3268] - 3 X[9168], 2 X[5027] + X[6333], 2 X[6131] + X[14273], 4 X[6132] - X[16230], X[6563] + 2 X[8651], 3 X[9123] - X[9147], 3 X[9125] - 2 X[11176], 2 X[9131] + X[9134], X[9147] + 3 X[9168], 3 X[9189] - 4 X[11176], 3 X[11123] + X[14420], X[11615] + 2 X[32204], X[12077] - 4 X[44451], X[14417] + 2 X[14610]

X(45687) is the intersection of lines X(115)X(125) of the inner and outer Vecten triangles.

X(45687) lies on these lines: {2, 9131}, {99, 110}, {114, 126}, {136, 5139}, {230, 231}, {351, 2799}, {669, 32473}, {804, 10190}, {826, 14403}, {2374, 3563}, {3800, 45336}, {6563, 8651}, {7665, 9185}, {8029, 44564}, {9003, 32114}, {9126, 44202}, {9191, 9485}, {9204, 13304}, {9205, 13305}, {11615, 32204}, {15475, 18883}, {38749, 42663}

X(45687) = midpoint of X(i) and X(j) for these {i,j}: {2, 9131}, {351, 11123}, {3268, 9147}, {9123, 9168}, {9185, 44010}, {9191, 9485}, {9204, 13304}, {9205, 13305}, {10190, 14610}
X(45687) = reflection of X(i) in X(j) for these {i,j}: {1637, 11176}, {8029, 44564}, {9134, 2}, {9189, 9125}, {14417, 10190}, {44202, 9126}
X(45687) = anticomplement of X(45688)
X(45687) = complement of the isotomic conjugate of X(2858)
X(45687) = X(2858)-complementary conjugate of X(2887)
X(45687) = X(662)-isoconjugate of X(14498)
X(45687) = crosspoint of X(2) and X(2858)
X(45687) = crosssum of X(6) and X(2872)
X(45687) = crossdifference of every pair of points on line {3, 3124}
X(45687) = barycentric product X(523)*X(35297)
X(45687) = barycentric quotient X(i)/X(j) for these {i,j}: {512, 14498}, {35297, 99}
X(45687) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1637, 9125, 11176}, {1637, 11176, 9189}, {3268, 9123, 9147}, {6131, 6132, 41360}, {9147, 9168, 3268}, {14273, 41360, 16230}

X(45688) = MIDPOINT OF X(2) AND X(9134)

Barycentrics    (b^2 - c^2)*(2*a^4 - 3*a^2*b^2 - b^4 - 3*a^2*c^2 + 6*b^2*c^2 - c^4) : :
X(45688) = 5 X[2] - X[9131], X[1637] - 3 X[8371], 3 X[1637] - X[14420], X[3268] + 3 X[5466], 3 X[8371] + X[9148], 9 X[8371] - X[14420], X[8651] - 4 X[14341], X[9131] + 5 X[9134], X[9147] - 3 X[9189], 3 X[9148] + X[14420], X[12075] + 2 X[30476], X[12077] + 5 X[31279]

X(45688) lies on these lines: {2, 9131}, {115, 125}, {523, 4885}, {804, 10189}, {826, 31174}, {1196, 7656}, {2793, 6036}, {2799, 10278}, {3268, 5466}, {8029, 14417}, {8651, 14341}, {9147, 9189}, {12077, 31279}, {14277, 44529}, {23301, 32473}

X(45688) = midpoint of X(i) and X(j) for these {i,j}: {2, 9134}, {1637, 9148}, {8029, 14417}
X(45688) = reflection of X(44564) in X(10189)
X(45688) = complement of X(45687)
X(45688) = tripolar centroid of X(2996)
X(45688) = X(14498)-complementary conjugate of X(8287)
X(45688) = crossdifference of every pair of points on line {110, 3053}
X(45688) = {X(8371),X(9148)}-harmonic conjugate of X(1637)

X(45689) = MIDPOINT OF X(2) AND X(9148)

Barycentrics    (b^2 - c^2)*(a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 3*b^2*c^2) : :
X(45689) = 5 X[2] - X[9147], 5 X[351] - 3 X[9147], X[351] + 3 X[9148], 2 X[351] - 3 X[11176], X[850] + 5 X[31279], 3 X[1649] - X[9131], X[3005] + 5 X[31072], 4 X[3628] - X[11615], 3 X[5055] - X[19912], 3 X[8371] - X[9979], 3 X[8371] + X[14424], X[8644] - 5 X[31277], X[8663] - 7 X[27138], X[9138] - 5 X[15059], X[9147] + 5 X[9148], 2 X[9147] - 5 X[11176], 2 X[9148] + X[11176], X[9162] - 5 X[40334], X[9163] - 5 X[40335], 3 X[9191] + X[9979], 3 X[9191] - X[14424], X[9213] - 5 X[30745], 3 X[15061] - X[19902], X[17414] - 5 X[31279], X[23301] + 2 X[30476], 2 X[31174] + X[45333], 5 X[31277] - 2 X[44451]

X(45689) lies on these lines: {2, 351}, {5, 2780}, {30, 16235}, {125, 526}, {126, 6088}, {140, 9126}, {141, 9023}, {512, 625}, {523, 7625}, {690, 5461}, {782, 10191}, {850, 17414}, {858, 20403}, {888, 36950}, {1637, 9479}, {1649, 9131}, {2793, 14610}, {2799, 10278}, {3005, 31072}, {3111, 31858}, {3268, 8029}, {3589, 9188}, {3628, 11615}, {4108, 31176}, {4155, 4928}, {5055, 19912}, {5466, 42010}, {7664, 11631}, {8371, 9191}, {8644, 31277}, {8663, 27138}, {8675, 19510}, {8889, 17994}, {9035, 9171}, {9138, 15059}, {9151, 35078}, {9162, 40334}, {9163, 40335}, {9178, 30786}, {9213, 30745}, {10162, 31772}, {11183, 22566}, {14277, 18311}, {14279, 18310}, {15061, 19902}, {17993, 31125}, {25126, 30864}, {32472, 45317}, {42659, 44821}

X(45689) = midpoint of X(i) and X(j) for these {i,j}: {2, 9148}, {850, 17414}, {3268, 8029}, {4108, 31176}, {4928, 27798}, {8371, 9191}, {9134, 14417}, {9178, 35522}, {9979, 14424}, {14277, 18311}, {14279, 18310}
X(45689) = reflection of X(i) in X(j) for these {i,j}: {1637, 10189}, {8644, 44451}, {9126, 140}, {9188, 3589}, {11176, 2}, {32193, 44564}
X(45689) = isotomic conjugate of cevapoint of X(2) and X(351)
X(45689) = isotomic conjugate of trilinear pole of line X(194)X(1992)
X(45689) = complement of X(351)
X(45689) = complement of the isogonal conjugate of X(892)
X(45689) = medial isogonal conjugate of X(23992)
X(45689) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 23992}, {75, 5099}, {99, 16597}, {111, 16592}, {662, 2482}, {671, 8287}, {691, 37}, {799, 126}, {811, 5181}, {892, 10}, {895, 16573}, {897, 115}, {923, 1084}, {4599, 7664}, {5380, 1213}, {5466, 24040}, {7316, 16613}, {18023, 21253}, {23894, 23991}, {24041, 1649}, {30786, 34846}, {32680, 13162}, {32729, 16584}, {34574, 16611}, {36085, 2}, {36128, 6388}, {36142, 39}, {36827, 16587}
X(45689) = X(i)-isoconjugate of X(j) for these (i,j): {162, 38279}, {163, 9227}
X(45689) = crossdifference of every pair of points on line {1384, 1613}
X(45689) = barycentric product X(i)*X(j) for these {i,j}: {525, 38294}, {850, 9225}
X(45689) = barycentric quotient X(i)/X(j) for these {i,j}: {523, 9227}, {647, 38279}, {9225, 110}, {38294, 648}
X(45689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8371, 14424, 9979}, {9191, 9979, 14424}

X(45690) = MIDPOINT OF X(2) AND X(9171)

Barycentrics    (b^2 - c^2)*(3*a^6 - 2*a^4*b^2 - 2*a^2*b^4 - 2*a^4*c^2 + 3*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4) : :
X(45690) = 5 X[3618] - X[5652], 5 X[3618] + X[22260], X[9147] - 3 X[14428], X[9148] + 3 X[14398], 3 X[14561] + X[21732], 3 X[14848] + X[21733]

X(45690) lies on these lines: {2, 9171}, {6, 11182}, {512, 597}, {523, 3589}, {690, 2492}, {888, 11176}, {1499, 18583}, {3221, 16776}, {3329, 5996}, {3618, 5652}, {3906, 24256}, {5476, 9175}, {8371, 41939}, {9147, 14428}, {9148, 14398}, {9178, 11183}, {10545, 15724}, {11053, 33921}, {14561, 21732}, {14848, 21733}, {19130, 32472}, {33511, 39492}, {33919, 45327}

X(45690) = midpoint of X(i) and X(j) for these {i,j}: {2, 9171}, {6, 11182}, {5476, 9175}, {5652, 22260}, {9178, 11183}
X(45690) = X(i)-complementary conjugate of X(j) for these (i,j): {923, 9151}, {32717, 16597}, {36133, 126}, {36142, 35073}, {37132, 5099}
X(45690) = crossdifference of every pair of points on line {2076, 2930}

X(45691) = MIDPOINT OF X(2) AND X(9508)

Barycentrics    (b - c)*(5*a^3 + 3*a^2*b - 5*a*b^2 + 3*a^2*c - 5*a*b*c + 2*b^2*c - 5*a*c^2 + 2*b*c^2) : :
X(45691) = 5 X[2] - X[4010], X[3679] + 3 X[14419], X[4010] + 5 X[9508], 3 X[4010] - 5 X[45342], X[4730] + 3 X[25055], 3 X[4763] - X[45314], 3 X[4763] + X[45328], 3 X[9508] + X[45342], 3 X[14431] - 7 X[19876], 3 X[28602] - X[45344]

X(45691) lies on these lines: {2, 4010}, {513, 4763}, {812, 45340}, {900, 45337}, {1638, 4802}, {2787, 3828}, {3679, 14419}, {4083, 44561}, {4132, 45333}, {4730, 25055}, {4926, 26275}, {5642, 6174}, {9260, 14422}, {14431, 19876}, {28602, 45344}, {29328, 45339}

X(45691) = midpoint of X(i) and X(j) for these {i,j}: {2, 9508}, {45313, 45323}, {45314, 45328}
X(45691) = complement of X(45342)
X(45691) = {X(4763),X(45328)}-harmonic conjugate of X(45314)

X(45692) = MIDPOINT OF X(2) AND X(11182)

Barycentrics    (b^2 - c^2)*(a^6 + a^4*b^2 - 3*a^2*b^4 + a^4*c^2 - a^2*b^2*c^2 + 2*b^4*c^2 - 3*a^2*c^4 + 2*b^2*c^4) : :
X(45692) = 5 X[2] - X[5652], 3 X[2] + X[34290], 5 X[3763] + X[22260], 3 X[5055] + X[21733], X[5652] + 5 X[11182], 3 X[5652] - 5 X[11183], 3 X[5652] + 5 X[34290], X[9178] + 3 X[21358], 3 X[11182] + X[11183], 3 X[11182] - X[34290]

X(45692) lies on these lines: {2, 512}, {523, 20582}, {525, 10189}, {547, 1499}, {599, 9171}, {690, 5461}, {804, 5113}, {826, 10278}, {868, 39482}, {3221, 29959}, {3763, 22260}, {3906, 8371}, {5055, 21733}, {7927, 10190}, {7950, 8029}, {9148, 9208}, {9175, 11178}, {9178, 21358}, {12073, 45333}, {18310, 33919}, {20186, 23329}, {25561, 32472}

X(45692) = midpoint of X(i) and X(j) for these {i,j}: {2, 11182}, {599, 9171}, {9148, 9208}, {9175, 11178}, {11183, 34290}
X(45692) = complement of X(11183)
X(45692) = X(i)-complementary conjugate of X(j) for these (i,j): {691, 19563}, {805, 16597}, {897, 2679}, {923, 35078}, {1581, 5099}, {1967, 23992}, {36085, 39080}, {36142, 5976}, {37134, 126}
X(45692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 34290, 11183}, {11182, 11183, 34290}

X(45693) = MIDPOINT OF X(2) AND X(11183)

Barycentrics    (b^2 - c^2)*(3*a^6 - a^4*b^2 - a^2*b^4 - a^4*c^2 - 3*a^2*b^2*c^2 + 2*b^4*c^2 - a^2*c^4 + 2*b^2*c^4) : :
X(45693) = 3 X[2] + X[5652], 5 X[2] - X[34290], X[1640] + 3 X[1649], X[5113] + 2 X[24284], X[5652] - 3 X[11183], 5 X[5652] + 3 X[34290], X[11182] + 3 X[11183], 5 X[11182] - 3 X[34290], 5 X[11183] + X[34290], 5 X[15694] - X[21733]

X(45693) lies on these lines: {2, 512}, {39, 647}, {140, 1499}, {523, 3589}, {620, 690}, {826, 10190}, {3800, 10189}, {5027, 9148}, {5092, 32472}, {5466, 43527}, {6704, 12073}, {7927, 10278}, {7950, 11123}, {8704, 32149}, {15694, 21733}, {34291, 45321}

X(45693) = complement of X(11182)
X(45693) = midpoint of X(i) and X(j) for these {i,j}: {2, 11183}, {5027, 9148}, {5652, 11182}, {10190, 45327}, {11176, 24284}, {34291, 45321}
X(45693) = reflection of X(5113) in X(11176)
X(45693) = X(i)-complementary conjugate of X(j) for these (i,j): {163, 35077}, {662, 9152}, {5970, 8287}, {14606, 24040}, {35146, 21253}
X(45693) = crossdifference of every pair of points on line {23, 2076}
X(45693) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5652, 11182}, {11182, 11183, 5652}

X(45694) = CENTROID OF {A, B, C, X(1138)}

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - a^6*c^2 + 7*a^4*b^2*c^2 - 5*a^2*b^4*c^2 + 8*b^6*c^2 - 3*a^4*c^4 - 5*a^2*b^2*c^4 - 12*b^4*c^4 + 5*a^2*c^6 + 8*b^2*c^6 - 2*c^8) : :
Barycentrics    sin(A)*(2*cos(A)+1)*(2*cos(A)-1)*(2*cos(A)-cos(B-C))*(7+2*cos(2*A)+2*cos(2*B)+2*cos(2*C)-4*cos(2*(B-C))) : :
X(45694) = 3 X[2] + X[1138], X[1511] + 2 X[3258], X[1539] + 2 X[38610], X[5609] + 2 X[16340], X[10113] + 2 X[14934], X[12041] - 4 X[31379], 4 X[12900] - X[18319], X[14480] + 2 X[20379], X[14731] + 5 X[38794], 7 X[15039] - X[31876]

Let D = X(1138) and U = X(45694). Then U (that is, U-of- ABC) = U(DBC) = U(ADC) = U(ABD). This property of D is analogous to X(5) when D is X(4), and to X(6699) when D is X(74). (Blue, the Trigonographer, 23 October, 2021. See Mathematics Stack Exchange.

X(45694) lies on these lines: {2, 1138}, {5, 33855}, {30, 113}, {140, 20393}, {381, 14385}, {523, 34128}, {5609, 16340}, {5627, 20304}, {10113, 14934}, {10264, 18285}, {10615, 38605}, {12041, 31379}, {12900, 18319}, {14480, 20379}, {14731, 38794}, {15039, 31876}, {34334, 37943}, {34584, 38701}

X(45694) = midpoint of X(i) and X(j) for these {i,j}: {5, 33855}, {140, 20393}, {1138, 14993}, {3258, 31378}, {10264, 18285}
X(45694) = reflection of X(i) in X(j) for these {i,j}: {1511, 31378}, {5627, 20304}
X(45694) = complement of X(14993)
X(45694) = X(6149)-complementary conjugate of X(14993)
X(45694) = crosspoint of X(36210) and X(36211)
X(45694) = crosssum of X(36208) and X(36209)
X(45694) = {X(2),X(1138)}-harmonic conjugate of X(14993)

leftri

Ortho-perspective and para-perspective triangles: X(45695)-X(45729)

rightri

This preamble and centers X(45695)-X(45729) were contributed by César Eliud Lozada, November 1, 2021.

(1) Let T1=A1B1C1 and T2=A2B2C2 be two non-orthologic triangles. Denote as T' = A'B'C' the triangle bounded by the lines through A1, B1, C1 perpendicular to B2C2, C2A2, A2B2, respectively, and denote as T" = A"B"C" the triangle bounded by the lines through A2, B2, C2 perpendicular to B1C1, C1A1, A1B1, respectively. Then for, some pairs of triangles T1 and T2, T1 and T' are perspective if, and only if, T2 and T" are perspective.

(2) Let T1=A1B1C1 and T2=A2B2C2 be two non-parallelogic triangles. Denote as T' = A'B'C' the triangle bounded by the lines through A1, B1, C1 parallel to B2C2, C2A2, A2B2, respectively, and denote as T" = A"B"C" the triangle bounded by the lines through A2, B2, C2 parallel to B1C1, C1A1, A1B1, respectively. Then for, some pairs of triangles T1 and T2, T1 and T' are perspective if, and only if, T2 and T" are perspective.

In case (1), triangles T1 and T2 are said to be ortho-perspective and perspectors (T1, T') and (T2, T") are named here the ortho-perspector T1 to T2 and ortho-perspector T2 to T1, respectively.

In case (2), triangles T1 and T2 are said to be para-perspective and perspectors (T1, T') and (T2, T") are introduced here as the para-perspector T1 to T2 and para-perspector T2 to T1, respectively.

Reference: Vu Thanh Tung, Euclid 2834 (Message slightly modified by César Lozada).

The appearance of (T, i, j) in the following lists means that triangles ABC and T are ortho-perspective with ortho-perspectors X(i) and X(j):

(1st anti-Parry, 2, 99), (2nd anti-Parry, 2, 98), (1st Parry, 2, 9123), (2nd Parry, 2, 9185), (2nd Sharygin, 1, 45695)

The appearance of (T, i, j) in the following lists means that triangles ABC and T are para-perspective with para-perspectors X(i) and X(j):

(ABC-X3 reflections, 2, 376), (anti-Aquila, 2, 551), (anti-Ara, 2, 428), (anti-Ascella, 6, 1593), (anti-Atik, 6, 18909), (1st anti-Auriga, 2, 45696), (2nd anti-Auriga, 2, 45697), (5th anti-Brocard, 2, 12150), (2nd anti-circumperp-tangential, 2, 5434), (1st anti-circumperp, 6, 20), (anti-Conway, 6, 578), (2nd anti-Conway, 6, 389), (anti-Ehrmann-mid, 2, 3830), (anti-Euler, 2, 376), (3rd anti-Euler, 6, 12111), (4th anti-Euler, 6, 6241), (anti-excenters-reflections, 6, 4), (2nd anti-extouch, 6, 1181), (anti-inner-Grebe, 2, 19053), (anti-outer-Grebe, 2, 19054), (anti-Honsberger, 6, 182), (anti-Hutson intouch, 6, 3), (anti-incircle-circles, 6, 3), (anti-inverse-in-incircle, 6, 4), (1st anti-Kenmotu centers, 2, 591), (2nd anti-Kenmotu centers, 2, 1991), (1st anti-Kenmotu-free-vertices, 2, 41490), (2nd anti- Kenmotu-free-vertices, 2, 41491), (anti-Lucas(+1) homothetic, 2, 45699), (anti-Lucas(-1) homothetic, 2, 45698), (anti-Mandart-incircle, 2, 4421), (6th anti-mixtilinear, 6, 3), (1st anti-Sharygin, 6, 8884), (anti-tangential-midarc, 6, 1), (3rd anti-tri-squares-central, 2, 5860), (4th anti-tri-squares-central, 2, 5861), (anti-Ursa minor, 6, 5), (anti-Wasat, 6, 185), (anti-X3-ABC reflections, 2, 549), (anti-inner-Yff, 2, 45700), (anti-outer-Yff, 2, 45701), (anticomplementary, 2, 2), (Aquila, 2, 3679), (Ara, 2, 9909), (Ascella, 1, 142), (Atik, 1, 15587), (1st Auriga, 2, 11207), (2nd Auriga, 2, 11208), (Ayme, 37, 10), (Bevan antipodal, 57, 223), (5th Brocard, 2, 7811), (circumorthic, 6, 4), (2nd circumperp tangential, 2, 11194), (1st circumperp, 1, 11495), (2nd circumperp, 1, 1001), (inner-Conway, 1, 144), (Conway, 1, 7), (2nd Conway, 1, 7), (3rd Conway, 1, 10442), (Ehrmann-mid, 2, 3845), (Ehrmann-side, 6, 3), (Ehrmann-vertex, 6, 4), (2nd Ehrmann, 6, 576), (Euler, 2, 381), (2nd Euler, 6, 3), (3rd Euler, 1, 42356), (4th Euler, 1, 3826), (excenters-reflections, 1, 3243), (excentral, 1, 9), (1st excosine, 6, 1498), (2nd excosine, 393, 3183), (extangents, 6, 40), (extouch, 9, 8), (2nd extouch, 1, 9), (3rd extouch, 57, 223), (outer-Garcia, 2, 3679), (Gossard, 2, 1651), (inner-Grebe, 2, 5861), (outer-Grebe, 2, 5860), (2nd Hatzipolakis, 17054, 1119), (hexyl, 1, 5732), (Honsberger, 1, 7), (inner-Hutson, 1, 45702), (Hutson intouch, 1, 390), (outer-Hutson, 1, 45703), (incentral, 37, 1), (incircle-circles, 1, 5542), (intangents, 6, 1), (intouch, 1, 7), (inverse-in-Conway, 1, 35892), (inverse-in-incircle, 1, 5572), (Johnson, 2, 381), (inner-Johnson, 2, 11235), (outer-Johnson, 2, 11236), (1st Johnson-Yff, 2, 11237), (2nd Johnson-Yff, 2, 11238), (1st Kenmotu-centers, 2, 32787), (2nd Kenmotu-centers, 2, 32788), (1st Kenmotu diagonals, 6, 371), (2nd Kenmotu diagonals, 6, 372), (1st Kenmotu-free-vertices, 2, 35822), (2nd Kenmotu-free-vertices, 2, 35823), (Kosnita, 6, 3), (Lemoine, 597, 598), (Lucas antipodal tangents, 6, 18980), (Lucas(-1) antipodal tangents, 6, 18981), (Lucas(+1) homothetic, 2, 12152), (Lucas(-1) homothetic, 2, 12153), (Macbeath, 5, 264), (Mandart-incircle, 2, 3058), (medial, 2, 2), (midheight, 3, 6), (5th mixtilinear, 2, 3241), (6th mixtilinear, 1, 2951), (7th mixtilinear, 279, 15913), (Moses-Soddy, 514, 1086), (orthic, 6, 4), (2nd Pamfilos-Zhou, 1, 45704), (Pelletier, 650, 11), (Schroeter, 523, 115), (1st Sharygin, 1, 45705), (Soddy, 7, 3160), (inner-squares, 485, 8966), (outer-squares, 486, 13960), (Steiner, 523, 99), (submedial, 6, 5), (symmedial, 39, 6), (tangential, 6, 3), (tangential-midarc, 1, 45706), (2nd tangential-midarc, 1, 45707), (inner tri-equilateral, 6, 15), (outer tri-equilateral, 6, 16), (3rd tri-squares-central, 2, 13846), (4th tri-squares-central, 2, 13847), (Trinh, 6, 3), (Ursa-major, 1, 17668), (Ursa-minor, 1, 14100), (Wasat, 1, 142), (X-parabola-tangential, 115, 523), (X3-ABC reflections, 2, 381), (Yff central, 1, 45708), (Yff contact, 514, 190), (inner-Yff, 2, 10056), (outer-Yff, 2, 10072), (inner-Yff tangents, 2, 11239), (outer-Yff tangents, 2, 11240), (1st Zaniah, 9, 1), (2nd Zaniah, 1, 9)

The appearance of (T, i, j) in the following lists means that triangles excentral and T are ortho-perspective with para-perspectors X(i) and X(j):

(1st anti-Parry, 9, 45709), (2nd anti-Parry, 9, 45710), (1st Parry, 9, 9810), (2nd Parry, 9, 9811), (2nd Sharygin, 165, 1635)

The appearance of (T, i, j) in the following lists means that triangles excentral and T are para-perspective with para-perspectors X(i) and X(j):

(ABC, 9, 1), (ABC-X3 reflections, 9, 40), (anti-Aquila, 9, 1), (anti-Ara, 9, 1829), (1st anti-Auriga, 9, 45711), (2nd anti-Auriga, 9, 45712), (5th anti-Brocard, 9, 12194), (2nd anti-circumperp-tangential, 9, 65), (anti-Ehrmann-mid, 9, 18525), (anti-Euler, 9, 944), (anti-inner-Grebe, 9, 18992), (anti-outer-Grebe, 9, 18991), (1st anti-Kenmotu centers, 9, 45713), (2nd anti-Kenmotu centers, 9, 45714), (1st anti-Kenmotu-free-vertices, 9, 45715), (2nd anti- Kenmotu-free-vertices, 9, 45716), (anti-Lucas(+1) homothetic, 9, 45718), (anti-Lucas(-1) homothetic, 9, 45717), (anti-Mandart-incircle, 9, 3), (3rd anti-tri-squares-central, 9, 45719), (4th anti-tri-squares-central, 9, 45720), (anti-X3-ABC reflections, 9, 1385), (anti-inner-Yff, 9, 1), (anti-outer-Yff, 9, 1), (anticomplementary, 9, 8), (Aquila, 9, 1), (Ara, 9, 9798), (Ascella, 165, 11227), (Atik, 165, 5927), (1st Auriga, 9, 55), (2nd Auriga, 9, 55), (Bevan antipodal, 610, 1394), (5th Brocard, 9, 9941), (2nd circumperp tangential, 9, 3), (1st circumperp, 165, 165), (2nd circumperp, 165, 3576), (inner-Conway, 165, 3681), (Conway, 165, 11220), (2nd Conway, 165, 9812), (3rd Conway, 165, 10439), (Ehrmann-mid, 9, 18480), (Euler, 9, 946), (3rd Euler, 165, 3817), (4th Euler, 165, 10175), (excenters-midpoints, 40, 9), (excenters-reflections, 165, 11224), (2nd extouch, 165, 5927), (3rd extouch, 610, 15498), (outer-Garcia, 9, 8), (Garcia-reflection, 40, 7), (Gossard, 9, 12438), (inner-Grebe, 9, 3641), (outer-Grebe, 9, 3640), (hexyl, 165, 3576), (Honsberger, 165, 7671), (inner-Hutson, 165, 11222), (Hutson intouch, 165, 5919), (outer-Hutson, 165, 11223), (incircle-circles, 165, 5049), (intouch, 165, 354), (inverse-in-Conway, 165, 10439), (inverse-in-incircle, 165, 354), (1st Jenkins, 573, 10), (Johnson, 9, 355), (inner-Johnson, 9, 355), (outer-Johnson, 9, 355), (1st Johnson-Yff, 9, 5252), (2nd Johnson-Yff, 9, 1837), (1st Kenmotu-centers, 9, 7969), (2nd Kenmotu-centers, 9, 7968), (1st Kenmotu-free-vertices, 9, 35641), (2nd Kenmotu-free-vertices, 9, 35642), (Lucas(+1) homothetic, 9, 12440), (Lucas(-1) homothetic, 9, 12441), (Mandart-incircle, 9, 3057), (medial, 9, 10), (midarc, 164, 8083), (2nd midarc, 164, 10968), (5th mixtilinear, 9, 1), (6th mixtilinear, 165, 165), (7th mixtilinear, 45721, 8916), (Moses-Soddy, 649, 11), (2nd Pamfilos-Zhou, 165, 11211), (Pelletier, 513, 3022), (1st Savin, 573, 1), (2nd Schiffler, 40, 1156), (1st Sharygin, 165, 11203), (tangential-midarc, 165, 11192), (2nd tangential-midarc, 165, 11217), (3rd tri-squares-central, 9, 8983), (4th tri-squares-central, 9, 13971), (Ursa-major, 165, 5927), (Ursa-minor, 165, 354), (Wasat, 165, 3817), (X3-ABC reflections, 9, 1482), (Yff central, 165, 11195), (Yff contact, 649, 100), (inner-Yff, 9, 1), (outer-Yff, 9, 1), (inner-Yff tangents, 9, 1), (outer-Yff tangents, 9, 1), (2nd Zaniah, 165, 3740)

The appearance of (T, i, j) in the following lists means that triangles orthic and T are ortho-perspective with para-perspectors X(i) and X(j):

(1st anti-Parry, 4, 45722), (2nd anti-Parry, 4, 45723), (1st Parry, 4, 9135), (2nd Parry, 4, 3569)

The appearance of (T, i, j) in the following lists means that triangles orthic and T are para-perspective with para-perspectors X(i) and X(j):

(ABC, 4, 6), (ABC-X3 reflections, 4, 1350), (anti-Aquila, 4, 1386), (anti-Ara, 4, 1843), (anti-Ascella, 51, 11402), (anti-Atik, 51, 18950), (1st anti-Auriga, 4, 45724), (2nd anti-Auriga, 4, 45725), (5th anti-Brocard, 4, 12212), (2nd anti-circumperp-tangential, 4, 1469), (1st anti-circumperp, 51, 2979), (anti-Conway, 51, 11402), (2nd anti-Conway, 51, 51), (anti-Ehrmann-mid, 4, 18440), (anti-Euler, 4, 6776), (3rd anti-Euler, 51, 2979), (4th anti-Euler, 51, 5890), (anti-excenters-reflections, 51, 32062), (2nd anti-extouch, 51, 11402), (anti-inner-Grebe, 4, 6), (anti-outer-Grebe, 4, 6), (anti-Honsberger, 51, 19153), (anti-Hutson intouch, 51, 10606), (anti-incircle-circles, 51, 32063), (anti-inverse-in-incircle, 51, 32064), (1st anti-Kenmotu centers, 4, 6), (2nd anti-Kenmotu centers, 4, 6), (1st anti-Kenmotu-free-vertices, 4, 182), (2nd anti- Kenmotu-free-vertices, 4, 182), (anti-Lucas(+1) homothetic, 4, 45727), (anti-Lucas(-1) homothetic, 4, 45726), (anti-Mandart-incircle, 4, 12329), (6th anti-mixtilinear, 51, 3819), (1st anti-Sharygin, 51, 19209), (anti-tangential-midarc, 51, 32065), (3rd anti-tri-squares-central, 4, 193), (4th anti-tri-squares-central, 4, 193), (anti-Ursa minor, 51, 23332), (anti-Wasat, 51, 51), (anti-X3-ABC reflections, 4, 182), (anti-inner-Yff, 4, 45728), (anti-outer-Yff, 4, 45729), (anticomplementary, 4, 69), (Aquila, 4, 3751), (Ara, 4, 159), (Aries, 6, 1498), (1st Auriga, 4, 12452), (2nd Auriga, 4, 12453), (Ayme, 1859, 40635), (5th Brocard, 4, 3094), (circumorthic, 51, 5890), (2nd circumperp tangential, 4, 22769), (Ehrmann-mid, 4, 3818), (Ehrmann-side, 51, 18435), (Ehrmann-vertex, 51, 18376), (2nd Ehrmann, 51, 11216), (Euler, 4, 5480), (2nd Euler, 51, 5891), (1st excosine, 51, 154), (2nd excosine, 6525, 17833), (extangents, 51, 11190), (outer-Garcia, 4, 3416), (Gossard, 4, 12583), (inner-Grebe, 4, 6), (outer-Grebe, 4, 6), (2nd Hyacinth, 6, 185), (incentral, 1859, 55), (intangents, 51, 11189), (Johnson, 4, 1352), (inner-Johnson, 4, 12586), (outer-Johnson, 4, 12587), (1st Johnson-Yff, 4, 12588), (2nd Johnson-Yff, 4, 12589), (1st Kenmotu-centers, 4, 6), (2nd Kenmotu-centers, 4, 6), (1st Kenmotu diagonals, 51, 11241), (2nd Kenmotu diagonals, 51, 11242), (1st Kenmotu-free-vertices, 4, 35840), (2nd Kenmotu-free-vertices, 4, 35841), (Kosnita, 51, 11202), (Lucas antipodal tangents, 51, 32070), (Lucas(-1) antipodal tangents, 51, 32071), (Lucas(+1) homothetic, 4, 12590), (Lucas(-1) homothetic, 4, 12591), (Mandart-incircle, 4, 3056), (medial, 4, 141), (5th mixtilinear, 4, 3242), (orthic axes, 6748, 4), (Pelletier, 18344, 3270), (Schroeter, 2501, 125), (Steiner, 2501, 110), (submedial, 51, 6688), (tangential, 51, 154), (inner tri-equilateral, 51, 11243), (outer tri-equilateral, 51, 11244), (3rd tri-squares-central, 4, 13910), (4th tri-squares-central, 4, 13972), (Trinh, 51, 11204), (X3-ABC reflections, 4, 1351), (inner-Yff, 4, 611), (outer-Yff, 4, 613), (inner-Yff tangents, 4, 12594), (outer-Yff tangents, 4, 12595), (Yiu tangents, 6748, 550)

X(45695) = ORTHO-PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN TO ABC

Barycentrics    a*(b-c)*(a^4-2*(b+c)*a^3+(b^2-b*c+c^2)*a^2+(b^2+c^2)*b*c) : :
X(45695) = X(3126)-3*X(14419)

The reciprocal ortho-perspector of these triangles is X(1).

X(45695) lies on these lines: {1, 665}, {3, 8638}, {56, 43042}, {100, 14074}, {104, 105}, {214, 3126}, {514, 667}, {663, 905}, {900, 1001}, {1960, 24286}, {2530, 42325}, {2787, 40551}, {3766, 16823}, {3798, 8641}, {3800, 8639}, {3900, 9508}, {4025, 8642}, {4477, 31286}, {4712, 38379}, {4750, 8645}, {6366, 8301}, {11715, 15746}, {14077, 30234}, {15747, 22775}, {16158, 28473}, {16466, 22155}, {16850, 24287}, {17069, 23865}

X(45695) = perspector of the circumconic {{A, B, C, X(14621), X(39273)}}
X(45695) = crossdifference of every pair of points on line {X(2276), X(40131)}
X(45695) = X(21)-Beth conjugate of-X(665)
X(45695) = (2nd Sharygin)-isogonal conjugate-of-X(1635)
X(45695) = center of circle {{X(1), X(5088), X(5526)}}
X(45695) = X(6)-of-2nd Sharygin triangle
X(45695) = X(24284)-of-hexyl triangle
X(45695) = X(43042)-of-2nd circumperp tangential triangle
X(45695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (13245, 13246, 659), (31596, 31597, 659)

X(45696) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-AURIGA TO ABC

Barycentrics    12*a*S*sqrt(R*(4*R+r))+(a+b+c)*(2*a^3-2*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : :

The reciprocal para-perspector of these triangles is X(2).

X(45696) lies on these lines: {1, 528}, {2, 5597}, {376, 26290}, {381, 26326}, {428, 26371}, {519, 45711}, {524, 45724}, {549, 26398}, {551, 26365}, {591, 45345}, {1651, 26383}, {1991, 45348}, {3058, 26351}, {3241, 26395}, {3679, 26296}, {3830, 18496}, {3845, 45355}, {4421, 26393}, {5434, 26380}, {5860, 26344}, {5861, 26334}, {7811, 26310}, {9909, 26302}, {10056, 45371}, {10072, 45373}, {11194, 26319}, {11208, 45354}, {11235, 26390}, {11236, 26389}, {11237, 26388}, {11238, 26387}, {11239, 26402}, {11240, 26401}, {11366, 26360}, {12150, 26379}, {12152, 45362}, {12153, 45361}, {13846, 45365}, {13847, 45366}, {19053, 26384}, {19054, 26385}, {26391, 45699}, {26392, 45698}, {26399, 45700}, {26400, 45701}, {26413, 34706}, {32787, 44582}, {32788, 44583}, {35822, 45357}, {35823, 45360}, {41490, 45349}, {41491, 45352}

X(45696) = reflection of X(45697) in X(1)
X(45696) = (1st anti-Auriga)-isogonal conjugate-of-X(45724)
X(45696) = X(45697)-of-5th mixtilinear triangle
X(45696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5597, 26394, 26359), (26386, 45369, 26326)

X(45697) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-AURIGA TO ABC

Barycentrics    -12*a*S*sqrt(R*(4*R+r))+(a+b+c)*(2*a^3-2*(b+c)*a^2+(b^2+c^2)*a-(b^2-c^2)*(b-c)) : :

The reciprocal para-perspector of these triangles is X(2).

X(45697) lies on these lines: {1, 528}, {2, 5598}, {376, 26291}, {381, 26327}, {428, 26372}, {519, 45712}, {524, 45725}, {549, 26422}, {551, 26366}, {591, 45347}, {1651, 26407}, {1991, 45346}, {3058, 26352}, {3241, 26419}, {3679, 26297}, {3830, 18498}, {3845, 45356}, {4421, 26417}, {5434, 26404}, {5860, 26345}, {5861, 26335}, {7811, 26311}, {9909, 26303}, {10056, 45372}, {10072, 45374}, {11194, 26320}, {11207, 45353}, {11235, 26414}, {11236, 26413}, {11237, 26412}, {11238, 26411}, {11239, 26426}, {11240, 26425}, {11367, 26359}, {12150, 26403}, {12152, 45364}, {12153, 45363}, {13846, 45368}, {13847, 45367}, {19053, 26408}, {19054, 26409}, {26389, 34706}, {26415, 45699}, {26416, 45698}, {26423, 45700}, {26424, 45701}, {32787, 44584}, {32788, 44585}, {35822, 45359}, {35823, 45358}, {41490, 45351}, {41491, 45350}

X(45697) = reflection of X(45696) in X(1)
X(45697) = (2nd anti-Auriga)-isogonal conjugate-of-X(45725)
X(45697) = X(45696)-of-5th mixtilinear triangle
X(45697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5598, 26418, 26360), (26410, 45370, 26327)

X(45698) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC TO ABC

Barycentrics    (2*a^2+S)*(a^4-4*b^2*c^2+(b^2-c^2)^2-2*(b^2+c^2)*(a^2-2*S)) : :

The reciprocal para-perspector of these triangles is X(2).

X(45698) lies on these lines: {2, 494}, {376, 26293}, {381, 26329}, {428, 26374}, {519, 45717}, {524, 45726}, {549, 26507}, {551, 26368}, {591, 45412}, {1651, 26448}, {1991, 45414}, {3058, 26354}, {3241, 26504}, {3679, 26299}, {3830, 18523}, {3845, 45592}, {4421, 26502}, {5434, 26434}, {5860, 26338}, {5861, 26506}, {6464, 45699}, {7714, 8946}, {7811, 26313}, {9909, 26305}, {10056, 45611}, {10072, 45613}, {11194, 26323}, {11207, 45588}, {11208, 45590}, {11235, 26489}, {11236, 26484}, {11237, 26478}, {11238, 26472}, {11239, 26511}, {11240, 26510}, {12150, 26428}, {12152, 45604}, {13846, 45605}, {13847, 45608}, {19053, 26455}, {19054, 26461}, {24243, 45421}, {26392, 45696}, {26416, 45697}, {26508, 45700}, {26509, 45701}, {32787, 45595}, {32788, 45598}, {35822, 45599}, {35823, 45602}, {41490, 45516}, {41491, 45518}

X(45698) = isogonal conjugate of X(41486)
X(45698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 26503, 5491), (26467, 45609, 26329)

X(45699) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC TO ABC

Barycentrics    (2*a^2-S)*(a^4-4*b^2*c^2+(b^2-c^2)^2-2*(b^2+c^2)*(a^2+2*S)) : :

The reciprocal para-perspector of these triangles is X(2).

X(45699) lies on these lines: {2, 493}, {376, 26292}, {381, 26328}, {428, 26373}, {519, 45718}, {524, 45727}, {549, 26498}, {551, 26367}, {591, 45415}, {1651, 26447}, {1991, 45413}, {3058, 26353}, {3241, 26495}, {3679, 26298}, {3830, 18521}, {3845, 45593}, {4421, 26493}, {5434, 26433}, {5860, 26347}, {5861, 26337}, {6464, 45698}, {7714, 8948}, {7811, 26312}, {9909, 26304}, {10056, 45612}, {10072, 45614}, {11194, 26322}, {11207, 45589}, {11208, 45591}, {11235, 26488}, {11236, 26483}, {11237, 26477}, {11238, 26471}, {11239, 45615}, {11240, 26501}, {12150, 26427}, {12153, 45603}, {13846, 45607}, {13847, 45606}, {19053, 26454}, {19054, 26460}, {24244, 45420}, {26391, 45696}, {26415, 45697}, {26499, 45700}, {26500, 45701}, {32787, 45597}, {32788, 45596}, {35822, 45601}, {35823, 45600}, {41490, 45519}, {41491, 45517}

X(45699) = isogonal conjugate of X(41485)
X(45699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 26494, 5490), (26466, 45610, 26328)

X(45700) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF TO ABC

Barycentrics    a^4-2*(b-c)^2*a^2-2*(b+c)*b*c*a+(b^2-c^2)^2 : :
X(45700) = X(1)+2*X(10916) = X(8)+2*X(22837) = 4*X(1125)-X(3811) = 5*X(1698)-2*X(10915) = 5*X(1698)+X(12629) = 5*X(3616)-2*X(22836) = 7*X(3624)-X(6765) = 11*X(5550)+X(6764) = 2*X(10915)+X(12629)

The reciprocal para-perspector of these triangles is X(2).

X(45700) lies on these lines: {1, 2}, {3, 528}, {4, 535}, {5, 11236}, {9, 37704}, {11, 956}, {12, 34749}, {30, 10525}, {35, 34719}, {36, 3434}, {40, 10785}, {55, 37298}, {56, 11112}, {63, 30384}, {69, 24202}, {72, 11376}, {99, 12357}, {104, 37430}, {140, 3913}, {149, 4302}, {329, 11813}, {355, 11260}, {376, 11012}, {377, 5563}, {381, 529}, {388, 25639}, {405, 31458}, {428, 26377}, {442, 3304}, {474, 33925}, {475, 23710}, {496, 958}, {497, 993}, {515, 24386}, {517, 34640}, {518, 5886}, {524, 45728}, {527, 946}, {536, 34511}, {549, 4421}, {591, 45422}, {599, 12595}, {631, 8715}, {671, 13190}, {758, 5603}, {903, 24848}, {908, 23708}, {952, 34700}, {960, 11373}, {962, 5536}, {999, 2886}, {1012, 34742}, {1056, 3822}, {1058, 5248}, {1319, 3419}, {1329, 45310}, {1376, 15325}, {1387, 5289}, {1420, 17647}, {1478, 11680}, {1479, 2975}, {1483, 26487}, {1484, 22560}, {1651, 26452}, {1656, 12607}, {1991, 45423}, {2078, 5082}, {2094, 4295}, {2136, 31423}, {2323, 37654}, {2475, 4317}, {2478, 5258}, {2551, 3825}, {2800, 5770}, {2802, 5657}, {3058, 10959}, {3149, 34746}, {3193, 4921}, {3254, 5698}, {3295, 4999}, {3303, 7483}, {3333, 6173}, {3421, 3814}, {3428, 37428}, {3436, 5288}, {3474, 4973}, {3475, 3892}, {3485, 3874}, {3487, 3881}, {3524, 10806}, {3534, 18543}, {3545, 10532}, {3555, 11375}, {3576, 24392}, {3585, 20076}, {3600, 31418}, {3653, 24299}, {3656, 24474}, {3678, 38216}, {3680, 6967}, {3697, 24954}, {3746, 6910}, {3753, 17728}, {3767, 17448}, {3816, 9708}, {3830, 18544}, {3845, 34739}, {3880, 26446}, {3916, 12701}, {3928, 12704}, {3929, 18232}, {3976, 24159}, {4189, 4309}, {4294, 5267}, {4297, 37427}, {4301, 6847}, {4325, 31295}, {4345, 5775}, {4361, 17043}, {4428, 15170}, {4479, 32833}, {4742, 33113}, {4857, 6872}, {4863, 5440}, {4870, 14054}, {4975, 17776}, {4995, 34699}, {5010, 20075}, {5045, 28628}, {5054, 16202}, {5055, 12001}, {5064, 11401}, {5071, 10597}, {5218, 25439}, {5270, 6871}, {5291, 9599}, {5298, 10949}, {5315, 24597}, {5328, 33709}, {5433, 5687}, {5434, 10957}, {5463, 13107}, {5464, 13106}, {5537, 6966}, {5540, 26258}, {5553, 45632}, {5690, 10912}, {5692, 16173}, {5694, 34647}, {5709, 6705}, {5719, 42871}, {5734, 6888}, {5735, 37434}, {5744, 21630}, {5790, 38455}, {5794, 24928}, {5844, 34710}, {5850, 38037}, {5853, 10165}, {5854, 38128}, {5855, 10247}, {5860, 26349}, {5861, 26342}, {5881, 6834}, {5882, 6825}, {5901, 12635}, {5904, 37735}, {5905, 18393}, {6054, 12190}, {6067, 42884}, {6684, 21627}, {6690, 6767}, {6691, 9709}, {6713, 25438}, {6762, 8227}, {6763, 11415}, {6824, 13464}, {6832, 9624}, {6833, 7982}, {6837, 11522}, {6862, 10222}, {6863, 37727}, {6889, 12625}, {6890, 7991}, {6891, 11362}, {6906, 34629}, {6914, 34741}, {6926, 43174}, {6932, 38669}, {6933, 37719}, {6941, 10711}, {6953, 37714}, {6958, 33895}, {6989, 12437}, {7373, 25466}, {7743, 24703}, {7757, 13110}, {7763, 17144}, {7811, 26317}, {7865, 10879}, {8609, 17281}, {8703, 34626}, {9140, 13218}, {9710, 16408}, {9909, 26308}, {9956, 32049}, {9957, 26066}, {10164, 34639}, {10246, 44669}, {10303, 12632}, {10526, 34688}, {10585, 31262}, {10706, 12382}, {10718, 13314}, {10914, 24914}, {10966, 11113}, {11180, 39903}, {11207, 45625}, {11208, 45626}, {11237, 17530}, {11355, 19762}, {11374, 34791}, {12047, 31164}, {12053, 12514}, {12150, 26431}, {12152, 45645}, {12153, 45644}, {12699, 28534}, {12702, 13463}, {12709, 24473}, {12750, 21842}, {13113, 31168}, {13271, 35249}, {13587, 36152}, {13712, 13717}, {13835, 13840}, {13846, 44645}, {13847, 44646}, {15346, 25557}, {15347, 38066}, {15446, 43740}, {15485, 27317}, {15621, 19550}, {16483, 35466}, {16486, 31187}, {16845, 38025}, {16853, 31494}, {16864, 34501}, {17301, 37592}, {17533, 31141}, {17549, 34611}, {17580, 38092}, {19049, 32788}, {19050, 32787}, {19053, 26458}, {19054, 26464}, {20691, 31401}, {21616, 31142}, {22712, 22732}, {22758, 26333}, {22765, 37820}, {22835, 37822}, {24524, 32832}, {25524, 31419}, {25681, 34790}, {26399, 45696}, {26423, 45697}, {26499, 45699}, {26508, 45698}, {28609, 38021}, {30389, 37112}, {31484, 35769}, {32310, 34319}, {33141, 37617}, {34630, 37022}, {34744, 37625}, {35822, 45640}, {35823, 45641}, {36977, 37710}, {37449, 37546}, {38055, 42014}, {41012, 41229}, {41490, 45526}, {41491, 45527}, {43836, 43862}

X(45700) = midpoint of X(i) and X(j) for these {i, j}: {2, 34625}, {4, 34610}, {3576, 24392}, {3830, 34740}, {3928, 31162}, {5603, 24477}, {11194, 11235}, {11236, 12513}, {34620, 34706}
X(45700) = reflection of X(i) in X(j) for these (i, j): (381, 3829), (4421, 549), (11236, 5), (34610, 8666), (34626, 8703), (34739, 3845), (45701, 2)
X(45700) = complement of X(34619)
X(45700) = isogonal conjugate of X(41487)
X(45700) = (anti-inner-Yff)-isogonal conjugate-of-X(45728)
X(45700) = center of circle {{X(100), X(1292), X(30720)}}
X(45700) = X(2)-of-anti-inner-Yff triangle
X(45700) = X(11112)-of-2nd circumperp tangential triangle
X(45700) = X(11236)-of-Johnson triangle
X(45700) = X(11240)-of-anti-outer-Yff triangle
X(45700) = X(17556)-of-inner-Johnson triangle
X(45700) = X(34610)-of-Euler triangle
X(45700) = X(45701)-of-outer-Yff tangents triangle
X(45700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 5231, 10), (1, 10527, 26363), (1, 26363, 10198), (2, 3086, 10199), (2, 3241, 10056), (2, 10199, 10200), (2, 10529, 11240), (2, 11239, 3584), (2, 11240, 1), (8, 499, 26364), (10, 3086, 10200), (10, 10199, 2), (11, 34606, 17556), (56, 31140, 11112), (956, 17556, 34606), (1058, 30478, 5248), (1698, 12629, 10915), (3086, 5231, 26363), (3582, 3679, 2), (4847, 44675, 997), (4853, 5231, 6734), (5258, 37720, 2478), (5288, 7741, 3436), (8666, 24387, 4), (10527, 10529, 1), (10527, 11240, 2), (10680, 26470, 26332), (10707, 11114, 1479), (11112, 24390, 31140), (11194, 34706, 34620), (11235, 34620, 34706), (14986, 19843, 1125), (17577, 34605, 1478)

X(45701) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF TO ABC

Barycentrics    a^4-2*(b+c)^2*a^2+2*(b+c)*b*c*a+(b^2-c^2)^2 : :
X(45701) = X(1)+2*X(10915) = X(8)+2*X(22836) = 2*X(10)+X(3811) = 5*X(1698)+X(6765) = 5*X(1698)-2*X(10916) = 5*X(3616)-2*X(22837) = 7*X(3624)-X(12629) = X(6764)-13*X(19877) = X(6765)+2*X(10916) = X(11519)-13*X(34595)

The reciprocal para-perspector of these triangles is X(2).

X(45701) lies on these lines: {1, 2}, {3, 529}, {4, 8715}, {5, 3913}, {11, 34699}, {12, 5687}, {21, 31452}, {30, 4421}, {35, 3436}, {36, 34690}, {40, 10786}, {55, 11113}, {99, 12356}, {100, 1478}, {119, 381}, {140, 12513}, {214, 3476}, {281, 15065}, {346, 4125}, {355, 33596}, {376, 535}, {377, 37719}, {388, 25440}, {405, 21031}, {428, 26378}, {474, 15888}, {484, 5905}, {495, 1376}, {497, 3814}, {517, 34647}, {518, 10202}, {524, 45729}, {527, 3359}, {549, 10269}, {591, 45424}, {599, 12594}, {631, 8666}, {671, 13189}, {752, 23693}, {758, 5657}, {903, 24847}, {908, 5119}, {942, 37828}, {952, 34717}, {956, 5432}, {993, 3421}, {999, 3035}, {1001, 3820}, {1012, 34697}, {1058, 3825}, {1145, 2099}, {1259, 10954}, {1329, 3295}, {1385, 32049}, {1470, 5434}, {1479, 3871}, {1483, 26492}, {1512, 37569}, {1519, 12703}, {1651, 26453}, {1656, 3813}, {1697, 21616}, {1706, 12609}, {1788, 3874}, {1991, 45425}, {2136, 8227}, {2478, 3746}, {2550, 3822}, {2551, 5248}, {2801, 14647}, {2802, 5603}, {2886, 31479}, {3058, 10958}, {3090, 24387}, {3149, 34687}, {3158, 5587}, {3161, 21087}, {3189, 5818}, {3303, 4187}, {3304, 13747}, {3419, 3689}, {3434, 7951}, {3475, 5883}, {3487, 3754}, {3524, 10805}, {3534, 18545}, {3543, 41698}, {3545, 10531}, {3555, 24914}, {3576, 34716}, {3583, 20075}, {3653, 24927}, {3654, 37562}, {3656, 7686}, {3680, 6983}, {3694, 17281}, {3731, 27522}, {3753, 17718}, {3767, 20691}, {3816, 6767}, {3829, 5055}, {3830, 18542}, {3833, 38053}, {3845, 34706}, {3880, 5886}, {3895, 30384}, {3918, 28629}, {3956, 38057}, {3992, 17776}, {4188, 4317}, {4190, 5270}, {4293, 34637}, {4299, 20060}, {4301, 6848}, {4302, 5080}, {4309, 5046}, {4313, 5828}, {4386, 31409}, {4654, 41540}, {4662, 5791}, {4692, 17740}, {4695, 33127}, {4723, 33113}, {4737, 32851}, {4848, 12559}, {4855, 45287}, {4857, 5187}, {4866, 31446}, {4930, 34718}, {4995, 10955}, {5054, 16203}, {5056, 12632}, {5064, 11400}, {5071, 10596}, {5082, 10588}, {5123, 5722}, {5252, 5440}, {5258, 6910}, {5298, 34749}, {5463, 13105}, {5464, 13104}, {5493, 37421}, {5525, 26258}, {5534, 12616}, {5537, 6925}, {5541, 18393}, {5563, 6921}, {5690, 12635}, {5734, 6979}, {5748, 11813}, {5790, 44669}, {5836, 11374}, {5844, 34743}, {5853, 10175}, {5854, 10247}, {5855, 38129}, {5860, 26350}, {5861, 26343}, {5881, 6833}, {5882, 6891}, {5901, 10912}, {6054, 12189}, {6668, 31493}, {6684, 37534}, {6690, 9708}, {6691, 7373}, {6762, 31423}, {6824, 12437}, {6825, 11362}, {6830, 38665}, {6832, 12625}, {6834, 7982}, {6837, 37714}, {6838, 7991}, {6884, 12536}, {6887, 31399}, {6889, 11523}, {6905, 34617}, {6908, 43174}, {6924, 34688}, {6931, 37720}, {6944, 12640}, {6947, 34486}, {6953, 11522}, {6958, 32537}, {6959, 10222}, {6989, 24391}, {7280, 20076}, {7483, 31458}, {7580, 34618}, {7757, 13109}, {7763, 24524}, {7795, 25102}, {7811, 26318}, {7865, 10878}, {7952, 18676}, {8703, 34620}, {9140, 13217}, {9578, 17647}, {9588, 37112}, {9709, 25466}, {9711, 11108}, {9909, 26309}, {9957, 25681}, {10164, 21164}, {10172, 24386}, {10246, 38455}, {10306, 18242}, {10310, 37429}, {10525, 34741}, {10584, 31263}, {10590, 17784}, {10706, 12381}, {10707, 13278}, {10711, 12775}, {10718, 13313}, {10914, 11375}, {10965, 11238}, {11010, 11415}, {11036, 33815}, {11112, 11237}, {11180, 39902}, {11207, 45627}, {11208, 45628}, {11499, 26332}, {11501, 11517}, {11698, 13205}, {11849, 37821}, {12150, 26432}, {12152, 45647}, {12153, 45646}, {12514, 21075}, {12608, 28194}, {12641, 14497}, {12678, 17613}, {12702, 32157}, {12751, 34627}, {13112, 31168}, {13463, 18493}, {13587, 34605}, {13712, 13716}, {13835, 13839}, {13846, 44643}, {13847, 44644}, {14740, 18389}, {15175, 30513}, {16483, 37663}, {16608, 17313}, {16973, 31398}, {16975, 31497}, {17144, 32832}, {17243, 40483}, {17448, 31401}, {17530, 31140}, {17564, 40726}, {18838, 24473}, {19047, 32788}, {19048, 32787}, {19053, 26459}, {19054, 26465}, {20612, 41686}, {21842, 36977}, {22712, 22731}, {22768, 37298}, {24159, 24440}, {24820, 30448}, {25522, 37556}, {26066, 34790}, {26400, 45696}, {26424, 45697}, {26500, 45699}, {26509, 45698}, {30827, 31393}, {31165, 41389}, {31410, 37435}, {32309, 34319}, {35822, 45642}, {35823, 45643}, {35996, 37546}, {41490, 45528}, {41491, 45529}, {43836, 43861}

X(45701) = midpoint of X(i) and X(j) for these {i, j}: {2, 34619}, {4, 34607}, {40, 28609}, {3158, 5587}, {3830, 34707}, {3913, 11235}, {4421, 11236}, {4930, 34718}, {5657, 25568}, {34626, 34739}
X(45701) = reflection of X(i) in X(j) for these (i, j): (11194, 549), (11235, 5), (24386, 10172), (28609, 21077), (34607, 8715), (34620, 8703), (34706, 3845), (45700, 2)
X(45701) = complement of X(34625)
X(45701) = isogonal conjugate of X(41442)
X(45701) = intersection, other than A, B, C, of circumconics {{A, B, C, X(78), X(36910)}} and {{A, B, C, X(158), X(499)}}
X(45701) = (anti-outer-Yff)-isogonal conjugate-of-X(45729)
X(45701) = center of circle {{X(4), X(10773), X(34607)}}
X(45701) = X(2)-of-anti-outer-Yff triangle
X(45701) = X(11113)-of-anti-Mandart-incircle triangle
X(45701) = X(11235)-of-Johnson triangle
X(45701) = X(11239)-of-anti-inner-Yff triangle
X(45701) = X(17532)-of-outer-Johnson triangle
X(45701) = X(34607)-of-Euler triangle
X(45701) = X(41715)-of-4th Euler triangle
X(45701) = X(45700)-of-inner-Yff tangents triangle
X(45701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 5552, 26364), (1, 26364, 10200), (2, 3085, 10197), (2, 3241, 10072), (2, 10197, 10198), (2, 10528, 11239), (2, 11239, 1), (2, 11240, 3582), (8, 498, 26363), (10, 3085, 10198), (10, 10197, 2), (12, 34612, 17532), (55, 31141, 11113), (145, 27529, 499), (200, 31434, 10), (1698, 6765, 10916), (3085, 7080, 10), (3421, 5218, 993), (3584, 3679, 2), (3814, 25439, 497), (3871, 11681, 1479), (4421, 34739, 34626), (5432, 34689, 31157), (5552, 10528, 1), (5552, 11239, 2), (5687, 17532, 34612), (6745, 31397, 997), (10942, 11248, 6256), (11113, 17757, 31141), (11236, 34626, 34739), (31157, 34689, 956)

X(45702) = PARA-PERSPECTOR OF THESE TRIANGLES: INNER-HUTSON TO ABC

Barycentrics    a*(4*(a+b-c)*(a-b+c)*b*c*sin(A/2)-2*(-a+b+c)*(a+b-c)^2*c*sin(B/2)-2*(-a+b+c)*(a-b+c)^2*b*sin(C/2)-4*(b+c)*a*(a^2+(b-c)^2)+a^4+2*(3*b^2-2*b*c+3*c^2)*a^2+(b^2+6*b*c+c^2)*(b-c)^2) : :

The reciprocal para-perspector of these triangles is X(1).

X(45702) lies on these lines: {7, 8113}, {9, 363}, {142, 11854}, {144, 11685}, {166, 16016}, {188, 9807}, {390, 8390}, {516, 9836}, {518, 9805}, {527, 12880}, {528, 12733}, {971, 12488}, {1001, 8109}, {2801, 12759}, {2951, 8140}, {3243, 11527}, {3826, 8380}, {5542, 11039}, {5572, 11026}, {5732, 8111}, {5851, 13260}, {5853, 12633}, {6732, 45707}, {8107, 11495}, {8133, 45706}, {8377, 42356}, {8391, 45705}, {10442, 11892}, {11222, 15726}, {11856, 15587}, {11922, 27990}, {11923, 45708}, {14100, 17607}, {16135, 17768}, {17621, 17668}, {35618, 35892}

X(45702) = reflection of X(45703) in X(2951)
X(45702) = (inner-Hutson)-isogonal conjugate-of-X(11222)
X(45702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 8385, 8113), (363, 5934, 22993), (9783, 11886, 8113), (12561, 12574, 9836)

X(45703) = PARA-PERSPECTOR OF THESE TRIANGLES: OUTER-HUTSON TO ABC

Barycentrics    a*(-4*(a+b-c)*(a-b+c)*b*c*sin(A/2)+2*(-a+b+c)*(a+b-c)^2*c*sin(B/2)+2*(-a+b+c)*(a-b+c)^2*b*sin(C/2)-4*(b+c)*a*(a^2+(b-c)^2)+a^4+2*(3*b^2-2*b*c+3*c^2)*a^2+(b^2+6*b*c+c^2)*(b-c)^2) : :

The reciprocal para-perspector of these triangles is X(1).

X(45703) lies on these lines: {7, 174}, {9, 164}, {142, 11855}, {144, 11686}, {167, 16016}, {390, 8392}, {516, 9837}, {518, 9806}, {527, 12885}, {528, 12734}, {971, 12489}, {1001, 8110}, {2801, 12760}, {2951, 8140}, {3243, 11528}, {3826, 8381}, {5542, 11040}, {5572, 11027}, {5732, 8112}, {5851, 13261}, {5853, 12634}, {6731, 11691}, {8108, 11495}, {8135, 45706}, {8378, 42356}, {10442, 11893}, {11223, 15726}, {11857, 15587}, {11925, 45704}, {11926, 45705}, {14100, 17608}, {16136, 17768}, {17623, 17668}, {35619, 35892}

X(45703) = reflection of X(45702) in X(2951)
X(45703) = (outer-Hutson)-isogonal conjugate-of-X(11223)
X(45703) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 8386, 8114), (168, 5935, 22994), (9787, 11887, 8114), (12562, 12576, 9837)

X(45704) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO ABC

Barycentrics    2*(2*a^5-(b+c)*a^4+2*(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2+2*(b-c)^2*b*c*a-(b^4-c^4)*(b-c))*S+(b+c)*a^6-2*(b-c)^2*a^5+(b+c)^3*a^4-2*(3*b^2+2*b*c+3*c^2)*b*c*a^3-(b^4-c^4)*(b-c)*a^2+2*(b^3-c^3)*(b-c)^3*a-(b^4-c^4)*(b-c)^3 : :

The reciprocal para-perspector of these triangles is X(1).

X(45704) lies on these lines: {3, 142}, {7, 1659}, {9, 7595}, {55, 30381}, {144, 11687}, {390, 8239}, {482, 1418}, {497, 30276}, {518, 9808}, {527, 1991}, {528, 12744}, {971, 12490}, {1836, 30380}, {2801, 12768}, {2951, 8244}, {3243, 11532}, {3474, 30277}, {3826, 8230}, {3946, 45399}, {5542, 11042}, {5572, 11030}, {5698, 33365}, {5732, 8234}, {5851, 13262}, {5853, 12638}, {6213, 7289}, {8228, 42356}, {8246, 45705}, {8247, 45706}, {8248, 45707}, {10442, 10891}, {10867, 15587}, {11211, 15726}, {11922, 27990}, {11925, 45703}, {11996, 45708}, {14100, 17610}, {16144, 17768}, {16432, 34125}, {17627, 17668}, {30556, 31551}, {30557, 31550}, {35622, 35892}

X(45704) = complement of isogonal conjugate of X(46377)
X(45704) = (2nd Pamfilos-Zhou)-isogonal conjugate-of-X(11211)
X(45704) = X(6)-of-2nd Pamfilos-Zhou triangle
X(45704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 31590, 31534), (7, 8237, 8243), (946, 31541, 31535), (8224, 8225, 1486), (8225, 31575, 7596), (8231, 8233, 18234), (9789, 10885, 8243), (12566, 12578, 7596)

X(45705) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN TO ABC

Barycentrics    a*((b+c)*(a^2-3*b*c)*a-(b^2+c^2)*(a^2+b*c)) : :

The reciprocal para-perspector of these triangles is X(1).

X(45705) lies on these lines: {7, 21}, {9, 43}, {37, 4447}, {55, 17257}, {142, 4425}, {144, 11688}, {171, 25421}, {390, 8240}, {405, 24248}, {516, 9840}, {518, 2292}, {524, 4068}, {527, 13097}, {528, 12746}, {971, 9959}, {992, 24696}, {1213, 4436}, {1334, 17792}, {1376, 5296}, {1400, 4640}, {1423, 4512}, {1469, 5250}, {1580, 16503}, {1621, 6646}, {1654, 4433}, {2550, 26045}, {2795, 18698}, {2801, 12770}, {2951, 8245}, {3059, 44694}, {3062, 30363}, {3243, 11533}, {3683, 28287}, {3747, 28369}, {3816, 17077}, {3826, 5051}, {3923, 16850}, {3941, 41312}, {3986, 41430}, {4204, 4418}, {4220, 11495}, {4312, 30362}, {4357, 8299}, {4364, 8053}, {4679, 29965}, {4754, 25124}, {4890, 18206}, {5218, 27282}, {5259, 32857}, {5284, 26806}, {5542, 11043}, {5572, 11031}, {5732, 8235}, {5851, 13265}, {5853, 12642}, {5880, 37225}, {7676, 36528}, {8229, 42356}, {8246, 45704}, {8249, 45706}, {8250, 45707}, {8391, 45702}, {8425, 45708}, {10442, 10892}, {10868, 15587}, {11203, 15726}, {11926, 45703}, {14100, 17611}, {15254, 27627}, {15972, 29057}, {16601, 18252}, {16684, 17246}, {16696, 39688}, {17185, 39780}, {17247, 23407}, {17258, 21320}, {17291, 44304}, {17628, 17668}, {18412, 30358}, {26626, 36635}, {28363, 45047}, {30097, 40998}, {32776, 37319}, {35623, 35892}

X(45705) = barycentric product X(256)*X(27968)
X(45705) = trilinear product X(893)*X(27968)
X(45705) = perspector of the circumconic {{A, B, C, X(3903), X(4573)}}
X(45705) = intersection, other than A, B, C, of circumconics {{A, B, C, X(9), X(17103)}} and {{A, B, C, X(43), X(27968)}}
X(45705) = crossdifference of every pair of points on line {X(3709), X(4367)}
X(45705) = (1st Sharygin)-isogonal conjugate-of-X(11203)
X(45705) = X(6)-of-1st Sharygin triangle
X(45705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 8238, 1284), (21, 9791, 1284), (846, 4199, 18235), (846, 41856, 4199), (8238, 30359, 7), (12567, 12579, 9840), (30360, 30361, 1284), (30364, 30365, 1284), (31576, 31577, 9840)

X(45706) = PARA-PERSPECTOR OF THESE TRIANGLES: TANGENTIAL-MIDARC TO ABC

Barycentrics    a*(-4*b*c*sin(A/2)+2*c*(a+b-c)*sin(B/2)+2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2) : :

The reciprocal para-perspector of these triangles is X(1).

X(45706) lies on these lines: {1, 45707}, {7, 1488}, {9, 173}, {142, 8733}, {144, 11690}, {174, 6732}, {390, 8241}, {516, 8091}, {518, 8093}, {527, 8101}, {528, 8097}, {971, 8095}, {1001, 8077}, {2801, 12771}, {2951, 8089}, {3062, 30371}, {3243, 11534}, {3826, 8087}, {4312, 30370}, {5542, 11044}, {5572, 11032}, {5732, 8081}, {5851, 8103}, {5853, 12643}, {8075, 11495}, {8085, 42356}, {8133, 45702}, {8135, 45703}, {8247, 45704}, {8249, 45705}, {8390, 11038}, {8581, 10506}, {9836, 11372}, {10442, 11894}, {10503, 14100}, {11192, 15726}, {11858, 15587}, {16146, 17768}, {17629, 17668}, {18399, 18412}, {21624, 39121}, {35624, 35892}

X(45706) = reflection of X(45707) in X(1)
X(45706) = (tangential-midarc)-isogonal conjugate-of-X(11192)
X(45706) = X(12329)-of-Hutson intouch triangle
X(45706) = X(22769)-of-intouch triangle
X(45706) = X(45707)-of-5th mixtilinear triangle
X(45706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 8387, 2089), (8387, 30367, 7), (12568, 12580, 8091), (30368, 30369, 2089), (30372, 30373, 2089), (31578, 31579, 8091)

X(45707) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO ABC

Barycentrics    a*(4*b*c*sin(A/2)-2*c*(a+b-c)*sin(B/2)-2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2) : :

The reciprocal para-perspector of these triangles is X(1).

X(45707) lies on these lines: {1, 45706}, {7, 174}, {9, 258}, {142, 236}, {144, 8125}, {177, 10390}, {188, 3174}, {259, 1488}, {390, 8242}, {516, 8092}, {518, 8094}, {527, 8102}, {528, 8098}, {971, 8096}, {1001, 7588}, {2089, 7670}, {2801, 12772}, {2951, 8090}, {3062, 30395}, {3243, 8422}, {3826, 8088}, {4312, 30420}, {5542, 8351}, {5572, 11033}, {5732, 8082}, {5762, 8129}, {5851, 8104}, {5853, 12644}, {6731, 7048}, {6732, 45702}, {8076, 11495}, {8086, 42356}, {8127, 21168}, {8130, 31657}, {8248, 45704}, {8250, 45705}, {9837, 32183}, {10442, 11895}, {10501, 14100}, {11038, 11924}, {11217, 15726}, {11859, 15587}, {16147, 17768}, {17630, 17668}, {18409, 18412}, {35625, 35892}

X(45707) = reflection of X(45706) in X(1)
X(45707) = (2nd tangential-midarc)-isogonal conjugate-of-X(11217)
X(45707) = X(12329)-of-intouch triangle
X(45707) = X(22769)-of-Hutson intouch triangle
X(45707) = X(45706)-of-5th mixtilinear triangle
X(45707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 174, 45708), (7, 8388, 174), (8388, 30405, 7), (8734, 21623, 236), (9795, 11889, 174), (12569, 12581, 8092), (30418, 30419, 174), (30421, 30422, 174), (31580, 31581, 8092)

X(45708) = PARA-PERSPECTOR OF THESE TRIANGLES: YFF CENTRAL TO ABC

Barycentrics    4*(b-c)*b*c*sin(A/2)+2*a*(a^2-(2*b-c)*a+(b-c)*(b-2*c))*sin(B/2)-2*(a^2+(b-2*c)*a+(2*b-c)*(b-c))*a*sin(C/2)-(b-c)*(a^2-2*(b+c)*a+(b-c)^2) : :
Barycentrics    a^2 (1 - sin(A/2)) + 2 a (b + c) sin(A/2) - (b - c)^2 (1 + sin(A/2)) : :
Barycentrics    (a^2 + b^2 + c^2 - 2 a b - 2 a c - 2 b c) sin(A/2) - 2 (a + b - c) (a - b + c) : :

The reciprocal para-perspector of these triangles is X(1).

X(45708) lies on these lines: {7, 174}, {9, 173}, {142, 7028}, {144, 8126}, {259, 2089}, {390, 11924}, {516, 8351}, {518, 12445}, {527, 13098}, {528, 12748}, {971, 12491}, {1001, 7587}, {2801, 12774}, {2951, 8423}, {3062, 30394}, {3243, 11535}, {3826, 8382}, {4312, 30408}, {5542, 8092}, {5572, 8083}, {5732, 7590}, {5762, 8130}, {5851, 13267}, {5853, 12646}, {6731, 7057}, {7589, 11495}, {8128, 21168}, {8129, 31657}, {8242, 11038}, {8379, 42356}, {8425, 45705}, {10442, 11896}, {10502, 14100}, {11195, 15726}, {11860, 15587}, {11923, 45702}, {11996, 45704}, {16151, 17768}, {17631, 17668}, {18408, 18412}, {35627, 35892}

X(45708) = Yff-central-isogonal conjugate-of-X(11195)
X(45708) = X(6)-of-Yff-central-triangle
X(45708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (7, 174, 45707), (7, 8389, 174), (173, 7593, 236), (173, 41855, 7593), (8389, 30404, 7), (8729, 21624, 7028), (11890, 11891, 174), (30406, 30407, 174), (30409, 30410, 174), (31592, 31593, 8351)

X(45709) = ORTHO-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-PARRY TO EXCENTRAL

Barycentrics    a*(a^3-2*(b^2+c^2)*a+3*(b^2-c^2)*(b-c))*(a^2-b^2)*(a^2-c^2) : :

The reciprocal ortho-perspector of these triangles is X(9).

X(45709) lies on these lines: {3, 45710}, {99, 2705}, {110, 6011}, {931, 1293}, {1296, 8691}, {3309, 45722}, {9216, 9811}

X(45709) = reflection of X(45710) in X(3)
X(45709) = X(45710)-of-ABC-X3 reflections triangle
X(45709) = X(1)-of-1st anti-Parry triangle
X(45709) = inverse of X(9810) in Stammler hyperbola

X(45710) = ORTHO-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-PARRY TO EXCENTRAL

Barycentrics    a*(a^7-(b^2+c^2)*a^5-3*(b^2-c^2)*(b-c)*a^4+(2*b^4-3*b^2*c^2+2*c^4)*a^3+3*(b^4-c^4)*(b-c)*a^2-2*(b^4-c^4)*(b^2-c^2)*a-3*(b^2-c^2)*(b-c)*b^2*c^2) : :

The reciprocal ortho-perspector of these triangles is X(9).

X(45710) lies on these lines: {3, 45709}, {74, 759}, {98, 2712}, {111, 9811}, {3309, 45723}, {9215, 9810}, {37508, 41430}

X(45710) = X(45709)-of-ABC-X3 reflections triangle
X(45710) = X(1)-of-2nd anti-Parry triangle
X(45710) = reflection of X(45709) in X(3)

X(45711) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-AURIGA TO EXCENTRAL

Barycentrics    a*(4*S*sqrt(R*(4*R+r))+(b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

The reciprocal para-perspector of these triangles is X(9).

X(45711) lies on these lines: {1, 3}, {8, 26382}, {10, 26359}, {355, 26386}, {518, 45724}, {519, 45696}, {944, 26381}, {946, 26326}, {1829, 26371}, {1837, 26387}, {3434, 26406}, {3640, 26344}, {3641, 26334}, {5252, 26388}, {7968, 44583}, {7969, 44582}, {8983, 45365}, {9798, 26302}, {9941, 26310}, {12194, 26379}, {12438, 26383}, {12440, 45362}, {12441, 45361}, {12699, 26413}, {13971, 45366}, {18480, 45355}, {18496, 18525}, {18991, 26385}, {18992, 26384}, {26391, 45718}, {26392, 45717}, {26396, 45719}, {26397, 45720}, {35641, 45357}, {35642, 45360}, {45345, 45713}, {45348, 45714}, {45349, 45715}, {45352, 45716}

X(45711) = reflection of X(45712) in X(1)
X(45711) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(26425)}} and {{A, B, C, X(8), X(26423)}}
X(45711) = X(45712)-of-5th mixtilinear triangle
X(45711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 40, 26423), (1, 46, 26425), (1, 55, 26366), (1, 5119, 5598), (1, 5597, 26365), (1, 8186, 24929), (1, 12703, 26424), (1, 25415, 26419), (1, 26296, 5597), (8, 26394, 26382), (65, 15934, 45712), (354, 1159, 45712), (2099, 11366, 1), (3057, 6767, 45712), (5045, 18421, 45712), (5597, 26395, 1), (5597, 26401, 45373), (5597, 26402, 45371), (5597, 45354, 55), (9819, 31792, 45712), (9957, 31393, 45712), (26296, 26395, 26365), (26319, 26393, 3), (26389, 26390, 26386), (26399, 26400, 5597)

X(45712) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-AURIGA TO EXCENTRAL

Barycentrics    a*(-4*S*sqrt(R*(4*R+r))+(b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

The reciprocal para-perspector of these triangles is X(9).

X(45712) lies on these lines: {1, 3}, {8, 26406}, {10, 26360}, {355, 26410}, {518, 45725}, {519, 45697}, {944, 26405}, {946, 26327}, {1829, 26372}, {1837, 26411}, {3434, 26382}, {3640, 26345}, {3641, 26335}, {5252, 26412}, {7968, 44585}, {7969, 44584}, {8983, 45368}, {9798, 26303}, {9941, 26311}, {12194, 26403}, {12438, 26407}, {12440, 45364}, {12441, 45363}, {12699, 26389}, {13971, 45367}, {18480, 45356}, {18498, 18525}, {18991, 26409}, {18992, 26408}, {26415, 45718}, {26416, 45717}, {26420, 45719}, {26421, 45720}, {35641, 45359}, {35642, 45358}, {45346, 45714}, {45347, 45713}, {45350, 45716}, {45351, 45715}

X(45712) = reflection of X(45711) in X(1)
X(45712) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(26401)}} and {{A, B, C, X(8), X(26399)}}
X(45712) = X(45711)-of-5th mixtilinear triangle
X(45712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 40, 26399), (1, 46, 26401), (1, 55, 26365), (1, 5119, 5597), (1, 5598, 26366), (1, 8187, 24929), (1, 12703, 26400), (1, 25415, 26395), (1, 26297, 5598), (8, 26418, 26406), (65, 15934, 45711), (354, 1159, 45711), (942, 11529, 45711), (2099, 11367, 1), (3057, 6767, 45711), (5045, 18421, 45711), (5598, 26419, 1), (5598, 26425, 45374), (5598, 26426, 45372), (5598, 45353, 55), (9819, 31792, 45711), (26297, 26419, 26366), (26320, 26417, 3), (26413, 26414, 26410), (26423, 26424, 5598)

X(45713) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU CENTERS TO EXCENTRAL

Barycentrics    a*((b+c)*a-b^2-c^2-2*S) : :
X(45713) = 3*X(1)-X(3641) = 3*X(3241)-X(45720) = 3*X(3640)+X(3641) = 2*X(3640)+X(45714) = 2*X(3641)-3*X(45714) = 3*X(10246)-2*X(45716)

The reciprocal para-perspector of these triangles is X(9).

Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically (as at X(1327)). Let OA be the circumcenter of AABAC, and define OB and OC cyclically. Triangle OAOBOC is homothetic to ABC at X(6), and X(45713) is the incenter of OAOBOC. (Randy Hutson, November 30, 2021)

X(45713) lies on these lines: {1, 6}, {3, 45416}, {8, 492}, {10, 31534}, {40, 12305}, {55, 45430}, {65, 26495}, {142, 31569}, {175, 2550}, {210, 3084}, {325, 45445}, {354, 3083}, {355, 6289}, {371, 45572}, {372, 760}, {377, 10911}, {481, 5880}, {515, 13748}, {517, 9733}, {519, 591}, {944, 45406}, {946, 45440}, {1152, 45530}, {1376, 13388}, {1385, 43119}, {1482, 45488}, {1829, 45400}, {1837, 45460}, {2809, 8225}, {2886, 13390}, {3057, 8211}, {3102, 14839}, {3241, 45421}, {3312, 45500}, {3752, 8945}, {3779, 7362}, {4849, 8941}, {5252, 45458}, {5698, 30334}, {5794, 31532}, {8983, 45484}, {9798, 45428}, {9941, 45434}, {10068, 44620}, {10084, 44618}, {10246, 45411}, {11495, 31563}, {12194, 45402}, {12438, 45446}, {12440, 45467}, {12441, 45464}, {13758, 13959}, {13911, 26300}, {13971, 45487}, {18480, 45438}, {18525, 45375}, {25557, 30342}, {35641, 45462}, {45345, 45711}, {45347, 45712}, {45412, 45717}, {45415, 45718}

X(45713) = midpoint of X(i) and X(j) for these {i, j}: {1, 3640}, {8, 45719}
X(45713) = reflection of X(i) in X(j) for these (i, j): (3, 45715), (45714, 1)
X(45713) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(34215)}} and {{A, B, C, X(8), X(26495)}}
X(45713) = X(3640)-of-anti-Aquila triangle
X(45713) = X(13749)-of-2nd circumperp triangle
X(45713) = X(14230)-of-excentral triangle
X(45713) = X(14233)-of-excenters-reflections triangle
X(45713) = X(45398)-of-outer-Grebe triangle
X(45713) = X(45714)-of-5th mixtilinear triangle
X(45713) = X(45715)-of-X3-ABC reflections triangle
X(45713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45398), (1, 3751, 7969), (1, 5588, 18991), (1, 7968, 45399), (1, 16496, 5605), (1, 18992, 1386), (1, 19003, 11371), (1, 30556, 1001), (1, 45426, 6), (1, 45427, 44635), (6, 45476, 1), (6, 45494, 45490), (6, 45496, 45492), (8, 492, 45444), (3242, 44636, 1), (5588, 18991, 4663), (5604, 7968, 1), (7174, 11260, 45714), (10930, 19049, 611), (10932, 19047, 613), (12594, 44646, 45491), (12595, 44644, 45493), (45416, 45436, 3), (45422, 45424, 6), (45426, 45476, 45398), (45430, 45432, 55), (45454, 45456, 6289)

X(45714) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU CENTERS TO EXCENTRAL

Barycentrics    a*((b+c)*a-b^2-c^2+2*S) : :
X(45714) = 3*X(1)-X(3640) = 3*X(3241)-X(45719) = X(3640)+3*X(3641) = 2*X(3640)-3*X(45713) = 2*X(3641)+X(45713) = 3*X(10246)-2*X(45715)

The reciprocal para-perspector of these triangles is X(9).

Let BBACAC be the internal square on side BC, and define CCBABA and AACBCB cyclically (as at X(1328)). Let OA be the circumcenter of AABAC, and define OB and OC cyclically. Triangle OAOBOC is homothetic to ABC at X(6), and X(45714) is the incenter of OAOBOC. (Randy Hutson, November 30, 2021)

X(45714) lies on these lines: {1, 6}, {3, 45417}, {8, 491}, {10, 31535}, {40, 12306}, {55, 45431}, {65, 26504}, {142, 31570}, {176, 2550}, {210, 3083}, {325, 45444}, {354, 3084}, {355, 6290}, {371, 760}, {372, 45573}, {377, 10910}, {482, 5880}, {515, 13749}, {517, 9732}, {519, 1991}, {944, 45407}, {946, 45441}, {1151, 45531}, {1376, 13389}, {1377, 8978}, {1378, 8953}, {1385, 43118}, {1482, 45489}, {1659, 2886}, {1829, 45401}, {1837, 45461}, {2809, 31546}, {3057, 8210}, {3103, 14839}, {3241, 45420}, {3311, 45501}, {3752, 8941}, {3779, 7353}, {4849, 8945}, {5252, 45459}, {5698, 30333}, {5794, 31533}, {8983, 45486}, {9798, 45429}, {9941, 45435}, {10067, 44621}, {10083, 44619}, {10246, 45410}, {11495, 31564}, {12194, 45403}, {12438, 45447}, {12440, 45465}, {12441, 45466}, {13638, 13902}, {13971, 45485}, {13973, 26301}, {18480, 45439}, {18525, 45376}, {25557, 30341}, {35642, 45463}, {45346, 45712}, {45348, 45711}, {45413, 45718}, {45414, 45717}

X(45714) = midpoint of X(i) and X(j) for these {i, j}: {1, 3641}, {8, 45720}
X(45714) = reflection of X(i) in X(j) for these (i, j): (3, 45716), (45713, 1)
X(45714) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(34216)}} and {{A, B, C, X(8), X(26504)}}
X(45714) = X(3641)-of-anti-Aquila triangle
X(45714) = X(13748)-of-2nd circumperp triangle
X(45714) = X(14230)-of-excenters-reflections triangle
X(45714) = X(14233)-of-excentral triangle
X(45714) = X(45399)-of-inner-Grebe triangle
X(45714) = X(45713)-of-5th mixtilinear triangle
X(45714) = X(45716)-of-X3-ABC reflections triangle
X(45714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6, 45399), (1, 3751, 7968), (1, 5589, 18992), (1, 7969, 45398), (1, 16496, 5604), (1, 18991, 1386), (1, 19004, 11370), (1, 30557, 1001), (1, 45426, 44636), (1, 45427, 6), (6, 45477, 1), (6, 45495, 45491), (6, 45497, 45493), (8, 491, 45445), (3242, 44635, 1), (5589, 18992, 4663), (5605, 7969, 1), (7174, 11260, 45713), (10929, 19050, 611), (10931, 19048, 613), (12594, 44645, 45490), (12595, 44643, 45492), (45417, 45437, 3), (45423, 45425, 6), (45427, 45477, 45399), (45431, 45433, 55), (45455, 45457, 6290)

X(45715) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-KENMOTU-FREE-VERTICES TO EXCENTRAL

Barycentrics    a*(2*((b+c)*a-b^2-c^2)*S+(a+b+c)*(2*a^3-(b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))) : :
X(45715) = 3*X(3576)+X(3640) = 3*X(5657)+X(45719) = 3*X(10246)-X(45714)

The reciprocal para-perspector of these triangles is X(9).

X(45715) lies on these lines: {1, 372}, {3, 45416}, {8, 45508}, {10, 641}, {39, 7968}, {40, 45498}, {55, 45534}, {65, 45506}, {182, 518}, {355, 45554}, {371, 45426}, {517, 9739}, {519, 41490}, {760, 43121}, {944, 45510}, {946, 45544}, {1152, 45476}, {1482, 45578}, {1829, 45502}, {1837, 45562}, {3057, 45570}, {3312, 45398}, {3576, 3640}, {3579, 7690}, {3641, 45550}, {4663, 44656}, {5062, 7969}, {5252, 45560}, {5588, 9583}, {5657, 45522}, {7763, 45547}, {8983, 45574}, {9798, 45532}, {9941, 45538}, {10246, 45410}, {12194, 45504}, {12438, 45548}, {12440, 45569}, {12441, 45566}, {13971, 45577}, {18480, 45542}, {18525, 45377}, {18991, 45515}, {18992, 45512}, {30556, 31546}, {31534, 31557}, {31541, 31569}, {31545, 31563}, {35642, 45565}, {35763, 45427}, {45349, 45711}, {45351, 45712}, {45516, 45717}, {45519, 45718}, {45525, 45720}

X(45715) = midpoint of X(3) and X(45713)
X(45715) = reflection of X(45716) in X(1385)
X(45715) = center of circle {{X(3), X(1083), X(45713)}}
X(45715) = X(1)-of-1st anti-Kenmotu-free-vertices triangle
X(45715) = X(45713)-of-anti-X3-ABC reflections triangle
X(45715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 372, 45500), (1, 45530, 372), (8, 45508, 45546), (372, 45572, 1), (372, 45584, 45580), (372, 45586, 45582), (45520, 45540, 3), (45526, 45528, 372), (45530, 45572, 45500), (45534, 45536, 55), (45556, 45558, 45554)

X(45716) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-KENMOTU-FREE-VERTICES TO EXCENTRAL

Barycentrics    a*(-2*((b+c)*a-b^2-c^2)*S+(a+b+c)*(2*a^3-(b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))) : :
X(45716) = 3*X(3576)+X(3641) = 3*X(5657)+X(45720) = 3*X(10246)-X(45713)

The reciprocal para-perspector of these triangles is X(9).

X(45716) lies on these lines: {1, 371}, {3, 45417}, {8, 45509}, {10, 642}, {39, 7969}, {40, 45499}, {55, 45535}, {65, 45507}, {182, 518}, {355, 45555}, {372, 45427}, {517, 9738}, {519, 41491}, {760, 43120}, {944, 45511}, {946, 45545}, {1151, 45477}, {1482, 45579}, {1829, 45503}, {1837, 45563}, {3057, 45571}, {3311, 45399}, {3576, 3641}, {3579, 7692}, {3640, 45551}, {4663, 44657}, {5058, 7968}, {5252, 45561}, {5657, 45523}, {7763, 45546}, {8225, 30557}, {8983, 45576}, {9798, 45533}, {9941, 45539}, {10246, 45411}, {12194, 45505}, {12438, 45549}, {12440, 45567}, {12441, 45568}, {13971, 45575}, {18480, 45543}, {18525, 45378}, {18991, 45513}, {18992, 45514}, {31535, 31558}, {31540, 31570}, {31544, 31564}, {31550, 31575}, {35641, 45564}, {35762, 45426}, {45350, 45712}, {45352, 45711}, {45517, 45718}, {45518, 45717}, {45524, 45719}

X(45716) = midpoint of X(3) and X(45714)
X(45716) = reflection of X(45715) in X(1385)
X(45716) = center of circle {{X(3), X(1083), X(45714)}}
X(45716) = X(45427)-of-1st anti-Kenmotu-free-vertices triangle
X(45716) = X(45714)-of-anti-X3-ABC reflections triangle
X(45716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 371, 45501), (1, 45531, 371), (8, 45509, 45547), (371, 45573, 1), (371, 45585, 45581), (371, 45587, 45583), (45521, 45541, 3), (45527, 45529, 371), (45531, 45573, 45501), (45535, 45537, 55), (45557, 45559, 45555)

X(45717) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC TO EXCENTRAL

Barycentrics    a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a+(a^4-4*b^2*c^2-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*S-(b^2+c^2)*(b^2-c^2)^2) : :

The reciprocal para-perspector of these triangles is X(9).

X(45717) lies on these lines: {1, 494}, {3, 26323}, {8, 26443}, {10, 1267}, {40, 26293}, {55, 45588}, {65, 26434}, {355, 26467}, {518, 45726}, {519, 45698}, {944, 26440}, {946, 26329}, {1385, 26507}, {1482, 45609}, {1829, 26374}, {1837, 26472}, {3057, 26354}, {3640, 26338}, {3641, 45594}, {5252, 26478}, {6464, 45718}, {7713, 8946}, {7968, 45598}, {7969, 45595}, {8983, 45605}, {9798, 26305}, {9941, 26313}, {10318, 26367}, {12194, 26428}, {12438, 26448}, {12440, 45604}, {13971, 45608}, {18480, 45592}, {18523, 18525}, {18991, 26461}, {18992, 26455}, {24243, 26301}, {26392, 45711}, {26416, 45712}, {26505, 45719}, {26506, 45720}, {35641, 45599}, {35642, 45602}, {45412, 45713}, {45414, 45714}, {45516, 45715}, {45518, 45716}

X(45717) = trilinear product X(494)*X(6351)
X(45717) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 494, 26368), (1, 26299, 494), (8, 26503, 26443), (494, 26504, 1), (494, 26510, 45613), (494, 26511, 45611), (26299, 26504, 26368), (26323, 26502, 3), (26484, 26489, 26467), (26508, 26509, 494), (45588, 45590, 55)

X(45718) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC TO EXCENTRAL

Barycentrics    a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+2*(b^2+c^2)^2*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a-(a^4-4*b^2*c^2-2*(b^2+c^2)*a^2+4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*S-(b^2+c^2)*(b^2-c^2)^2) : :

The reciprocal para-perspector of these triangles is X(9).

X(45718) lies on these lines: {1, 493}, {3, 26322}, {8, 26442}, {10, 5391}, {40, 26292}, {55, 45589}, {65, 26433}, {355, 26466}, {518, 45727}, {519, 45699}, {944, 26439}, {946, 26328}, {1385, 26498}, {1482, 45610}, {1829, 26373}, {1837, 26471}, {3057, 26353}, {3640, 26347}, {3641, 26337}, {5252, 26477}, {6464, 45717}, {7713, 8948}, {7968, 45596}, {7969, 45597}, {8983, 45607}, {9798, 26304}, {9941, 26312}, {10318, 26368}, {12194, 26427}, {12438, 26447}, {12441, 45603}, {13971, 45606}, {18480, 45593}, {18521, 18525}, {18991, 26460}, {18992, 26454}, {24244, 26300}, {26391, 45711}, {26415, 45712}, {26496, 45719}, {26497, 45720}, {35641, 45601}, {35642, 45600}, {45413, 45714}, {45415, 45713}, {45517, 45716}, {45519, 45715}

X(45718) = trilinear product X(493)*X(6352)
X(45718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 493, 26367), (1, 26298, 493), (8, 26494, 26442), (493, 26495, 1), (493, 26501, 45614), (493, 45615, 45612), (26298, 26495, 26367), (26322, 26493, 3), (26483, 26488, 26466), (26499, 26500, 493), (45589, 45591, 55)

X(45719) = PARA-PERSPECTOR OF THESE TRIANGLES: 3rd ANTI-TRI-SQUARES-CENTRAL TO EXCENTRAL

Barycentrics    -(3*a-b-c)*S+a*((b+c)*a-b^2-c^2) : :
X(45719) = 3*X(3241)-2*X(45714) = 3*X(5657)-4*X(45715)

The reciprocal para-perspector of these triangles is X(9).

X(45719) lies on these lines: {1, 1336}, {3, 26324}, {8, 492}, {10, 26361}, {40, 26294}, {65, 26435}, {144, 145}, {230, 26370}, {355, 26468}, {371, 7981}, {519, 3640}, {944, 26441}, {946, 26330}, {1007, 26445}, {1385, 26516}, {1829, 26375}, {1837, 26473}, {3241, 45420}, {3244, 3641}, {5252, 26479}, {5657, 45522}, {7968, 44595}, {7969, 44594}, {9798, 26306}, {9907, 31412}, {9941, 26314}, {12194, 26429}, {12269, 13886}, {12438, 26449}, {13667, 14241}, {18525, 18539}, {18991, 26462}, {18992, 26456}, {19065, 45426}, {24244, 26495}, {26396, 45711}, {26420, 45712}, {26496, 45718}, {26505, 45717}, {35642, 39660}, {45524, 45716}

X(45719) = reflection of X(i) in X(j) for these (i, j): (8, 45713), (3641, 3244), (45720, 145)
X(45719) = X(1)-of-3rd anti-tri-squares-central triangle
X(45719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3068, 26369), (1, 26300, 3068), (8, 492, 26444), (3068, 26514, 1), (26300, 26514, 26369), (26324, 26512, 3), (26485, 26490, 26468), (26517, 26518, 3068)

X(45720) = PARA-PERSPECTOR OF THESE TRIANGLES: 4th ANTI-TRI-SQUARES-CENTRAL TO EXCENTRAL

Barycentrics    (3*a-b-c)*S+a*((b+c)*a-b^2-c^2) : :
X(45720) = 3*X(3241)-2*X(45713) = 3*X(5657)-4*X(45716)

The reciprocal para-perspector of these triangles is X(9).

X(45720) lies on these lines: {1, 1123}, {3, 26325}, {8, 491}, {10, 26362}, {40, 26295}, {65, 26436}, {144, 145}, {230, 26369}, {355, 26469}, {372, 7980}, {519, 3641}, {944, 8982}, {946, 26331}, {1007, 26444}, {1385, 26521}, {1829, 26376}, {1837, 26474}, {3241, 45421}, {3244, 3640}, {5252, 26480}, {5657, 45523}, {7968, 44597}, {7969, 44596}, {9798, 26307}, {9906, 42561}, {9941, 26315}, {12194, 26430}, {12268, 13939}, {12438, 26450}, {13787, 14226}, {18525, 26438}, {18991, 26463}, {18992, 26457}, {19066, 45427}, {24243, 26504}, {26397, 45711}, {26421, 45712}, {26497, 45718}, {26506, 45717}, {35641, 39661}, {45525, 45715}

X(45720) = reflection of X(i) in X(j) for these (i, j): (8, 45714), (3640, 3244), (45719, 145)
X(45720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 3069, 26370), (1, 26301, 3069), (8, 491, 26445), (3069, 26515, 1), (26301, 26515, 26370), (26325, 26513, 3), (26486, 26491, 26469), (26522, 26523, 3069)

X(45721) = PARA-PERSPECTOR OF THESE TRIANGLES: EXCENTRAL TO 7th MIXTILINEAR

Barycentrics    a*(a^5+3*(b+c)*a^4-2*(7*b^2-10*b*c+7*c^2)*a^3+14*(b^2-c^2)*(b-c)*a^2-3*(b^2+6*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3) : :
X(45721) = 3*X(165)-X(8917)

The reciprocal para-perspector of these triangles is X(8916).

X(45721) lies on the cubic K1044 and these lines: {1, 11051}, {9, 5658}, {55, 32625}, {57, 279}, {165, 220}, {934, 8830}, {971, 19605}, {1743, 4650}, {3207, 31508}

X(45721) = intersection, other than A, B, C, of circumconics {{A, B, C, X(220), X(2124)}} and {{A, B, C, X(279), X(2125)}}
X(45721) = crossdifference of every pair of points on line {X(4105), X(17427)}
X(45721) = X(188)-Aleph conjugate of-X(3062)
X(45721) = X(144)-Ceva conjugate of-X(1)
X(45721) = (excentral)-isogonal conjugate-of-X(3062)
X(45721) = X(165)-Zayin conjugate of-X(2124)

X(45722) = ORTHO-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-PARRY TO ORTHIC

Barycentrics    a^2*(a^4-(b^2+c^2)*a^2+4*b^4-16*b^2*c^2+4*c^4)*(a^2-c^2)*(a^2-b^2) : :

The reciprocal ortho-perspector of these triangles is X(4).

X(45722) lies on these lines: {3, 2502}, {6, 33962}, {30, 12151}, {99, 1499}, {110, 1296}, {182, 14609}, {511, 9966}, {512, 9124}, {541, 38738}, {691, 6233}, {1350, 2930}, {3124, 11258}, {3288, 32583}, {3309, 45709}, {3569, 9216}, {8717, 8722}, {9181, 30209}, {9871, 11580}, {11842, 32463}, {12074, 39639}, {14650, 20998}, {14915, 18860}, {14928, 23699}, {38593, 39689}

X(45722) = reflection of X(45723) in X(3)
X(45722) = barycentric product X(i)*X(j) for these {i, j}: {99, 22111}, {110, 40727}
X(45722) = trilinear product X(i)*X(j) for these {i, j}: {163, 40727}, {662, 22111}
X(45722) = inverse of X(9135) in Stammler hyperbola
X(45722) = crossdifference of every pair of points on line {X(6791), X(9189)}
X(45722) = (1st anti-Parry)-isogonal conjugate-of-X(99)
X(45722) = reflection of X(99) in the line X(524)X(9966)
X(45722) = X(6)-of-1st anti-Parry triangle
X(45722) = X(45723)-of-these triangles: {ABC-X3 reflections, circumsymmedial}
X(45722) = {X(9202), X(9203)}-harmonic conjugate of X(9145)

X(45723) = ORTHO-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-PARRY TO ORTHIC

Barycentrics    a^2*(a^8-(5*b^4-7*b^2*c^2+5*c^4)*a^4+(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-2*(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^2) : :

The reciprocal ortho-perspector of these triangles is X(4).

X(45723) lies on the cubic K909 and these lines: {3, 2502}, {6, 5663}, {30, 5104}, {74, 111}, {98, 843}, {115, 541}, {187, 14915}, {248, 1384}, {352, 7464}, {381, 8288}, {399, 39689}, {511, 9186}, {512, 2378}, {574, 3016}, {690, 5916}, {755, 14388}, {842, 6323}, {1350, 33962}, {1648, 20126}, {2030, 6000}, {3124, 10620}, {3231, 40115}, {3309, 45710}, {4549, 43619}, {4846, 43620}, {5107, 13754}, {5210, 35237}, {5655, 41939}, {7706, 18424}, {7728, 39691}, {8585, 37470}, {8588, 8717}, {9135, 9215}, {9225, 37950}, {11820, 15655}, {12041, 20998}, {12308, 20976}, {22265, 32228}, {32305, 32740}

X(45723) = reflection of X(45722) in X(3)
X(45723) = perspector of the circumconic {{A, B, C, X(9060), X(9139)}}
X(45723) = crossdifference of every pair of points on line {X(5642), X(9003)}
X(45723) = (2nd anti-Parry)-isogonal conjugate-of-X(98)
X(45723) = reflection of X(98) in the line X(524)X(13233)
X(45723) = X(6)-of-2nd anti-Parry triangle
X(45723) = X(45722)-of-these triangles: {ABC-X3 reflections, circumsymmedial}
X(45723) = {X(2378), X(2379)}-harmonic conjugate of X(9142)

X(45724) = PARA-PERSPECTOR OF THESE TRIANGLES: 1st ANTI-AURIGA TO ORTHIC

Barycentrics    a*(2*(a^2+b^2+c^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*((b^2+c^2)*a-b^3-c^3)) : :

The reciprocal para-perspector of these triangles is X(4).

X(45724) lies on these lines: {1, 674}, {6, 5597}, {69, 26394}, {141, 26359}, {159, 26302}, {182, 26398}, {193, 26396}, {518, 45711}, {524, 45696}, {611, 45371}, {613, 45373}, {1350, 26290}, {1351, 45369}, {1352, 26386}, {1386, 26365}, {1469, 26380}, {1843, 26371}, {3056, 26351}, {3094, 26310}, {3242, 26395}, {3416, 26382}, {3751, 26296}, {3818, 45355}, {5480, 26326}, {6776, 26381}, {12212, 26379}, {12329, 26393}, {12453, 45354}, {12583, 26383}, {12586, 26390}, {12587, 26389}, {12588, 26388}, {12589, 26387}, {12590, 45362}, {12591, 45361}, {12594, 26402}, {12595, 26401}, {13910, 45365}, {13972, 45366}, {18440, 18496}, {22769, 26319}, {26391, 45727}, {26392, 45726}, {26399, 45728}, {26400, 45729}, {35840, 45357}, {35841, 45360}

X(45724) = reflection of X(45725) in X(1)
X(45724) = (1st anti-Auriga)-isogonal conjugate-of-X(45696)
X(45724) = X(45725)-of-5th mixtilinear triangle
X(45724) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5597, 26334, 45348), (45345, 45348, 5597), (45349, 45352, 26398)

X(45725) = PARA-PERSPECTOR OF THESE TRIANGLES: 2nd ANTI-AURIGA TO ORTHIC

Barycentrics    a*(-2*(a^2+b^2+c^2)*S*sqrt(R*(4*R+r))+a*(a+b+c)*((b^2+c^2)*a-b^3-c^3)) : :

The reciprocal para-perspector of these triangles is X(4).

X(45725) lies on these lines: {1, 674}, {6, 5598}, {69, 26418}, {141, 26360}, {159, 26303}, {182, 26422}, {193, 26420}, {518, 45712}, {524, 45697}, {611, 45372}, {613, 45374}, {1350, 26291}, {1351, 45370}, {1352, 26410}, {1386, 26366}, {1469, 26404}, {1843, 26372}, {3056, 26352}, {3094, 26311}, {3242, 26419}, {3416, 26406}, {3751, 26297}, {3818, 45356}, {5480, 26327}, {6776, 26405}, {12212, 26403}, {12329, 26417}, {12452, 45353}, {12583, 26407}, {12586, 26414}, {12587, 26413}, {12588, 26412}, {12589, 26411}, {12590, 45364}, {12591, 45363}, {12594, 26426}, {12595, 26425}, {13910, 45368}, {13972, 45367}, {18440, 18498}, {22769, 26320}, {26415, 45727}, {26416, 45726}, {26423, 45728}, {26424, 45729}, {35840, 45359}, {35841, 45358}

X(45725) = reflection of X(45724) in X(1)
X(45725) = (2nd anti-Auriga)-isogonal conjugate-of-X(45697)
X(45725) = X(45724)-of-5th mixtilinear triangle
X(45725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5598, 26335, 45346), (5598, 26345, 45347), (26335, 26345, 5598), (45346, 45347, 5598), (45350, 45351, 26422)

X(45726) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(-1) HOMOTHETIC TO ORTHIC

Barycentrics    a^2*(4*(b^2+c^2)*b^2*c^2-(a^4-2*(b^2+c^2)*a^2+5*b^4+2*b^2*c^2+5*c^4)*S) : :

The reciprocal para-perspector of these triangles is X(4).

X(45726) lies on these lines: {6, 494}, {69, 26503}, {141, 5490}, {159, 26305}, {182, 26507}, {193, 26505}, {518, 45717}, {524, 45698}, {611, 45611}, {613, 45613}, {1152, 13068}, {1350, 26293}, {1351, 45609}, {1352, 26467}, {1386, 26368}, {1469, 26434}, {1843, 26374}, {3056, 26354}, {3094, 26313}, {3242, 26504}, {3416, 26443}, {3751, 26299}, {3818, 45592}, {5480, 26329}, {6776, 26440}, {7716, 8946}, {12212, 26428}, {12329, 26502}, {12452, 45588}, {12453, 45590}, {12583, 26448}, {12586, 26489}, {12587, 26484}, {12588, 26478}, {12589, 26472}, {12590, 45604}, {12594, 26511}, {12595, 26510}, {13910, 45605}, {13972, 45608}, {18440, 18523}, {22769, 26323}, {26392, 45724}, {26416, 45725}, {26508, 45728}, {26509, 45729}, {35840, 45599}, {35841, 45602}

X(45726) = barycentric product X(494)*X(5590)
X(45726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (494, 26338, 45412), (494, 45594, 45414), (26338, 45594, 494), (45412, 45414, 494), (45516, 45518, 26507)

X(45727) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-LUCAS(+1) HOMOTHETIC TO ORTHIC

Barycentrics    a^2*(4*(b^2+c^2)*b^2*c^2+(a^4-2*(b^2+c^2)*a^2+5*b^4+2*b^2*c^2+5*c^4)*S) : :

The reciprocal para-perspector of these triangles is X(4).

X(45727) lies on these lines: {6, 493}, {69, 26494}, {141, 5490}, {159, 26304}, {182, 26498}, {193, 26496}, {518, 45718}, {524, 45699}, {611, 45612}, {613, 45614}, {1151, 12977}, {1350, 26292}, {1351, 45610}, {1352, 26466}, {1386, 26367}, {1469, 26433}, {1843, 26373}, {3056, 26353}, {3094, 26312}, {3242, 26495}, {3416, 26442}, {3751, 26298}, {3818, 45593}, {5480, 26328}, {6776, 26439}, {7716, 8948}, {12212, 26427}, {12329, 26493}, {12452, 45589}, {12453, 45591}, {12583, 26447}, {12586, 26488}, {12587, 26483}, {12588, 26477}, {12589, 26471}, {12591, 45603}, {12594, 45615}, {12595, 26501}, {13910, 45607}, {13972, 45606}, {18440, 18521}, {22769, 26322}, {26391, 45724}, {26415, 45725}, {26499, 45728}, {26500, 45729}, {35840, 45601}, {35841, 45600}

X(45727) = barycentric product X(493)*X(5591)
X(45727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (493, 26337, 45413), (493, 26347, 45415), (26337, 26347, 493), (45413, 45415, 493), (45517, 45519, 26498)

X(45728) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-INNER-YFF TO ORTHIC

Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2-2*(b+c)*b*c*a+b^4+c^4+2*(b^2-b*c+c^2)*b*c) : :
X(45728) = 5*X(12017)-2*X(41454)

The reciprocal para-perspector of these triangles is X(4).

X(45728) lies on these lines: {1, 6}, {3, 674}, {4, 39903}, {5, 12587}, {55, 5135}, {56, 1813}, {69, 10527}, {81, 24477}, {141, 26363}, {155, 34381}, {159, 26308}, {182, 9052}, {193, 10529}, {210, 10601}, {283, 3477}, {354, 394}, {511, 11249}, {519, 44414}, {524, 45700}, {575, 43146}, {576, 2810}, {895, 13218}, {940, 29639}, {946, 9028}, {1350, 9047}, {1351, 8679}, {1352, 5849}, {1353, 32214}, {1428, 3779}, {1456, 23144}, {1469, 26437}, {1471, 1818}, {1486, 7193}, {1843, 26377}, {1992, 11240}, {1993, 3873}, {1994, 4430}, {2886, 37543}, {3011, 4383}, {3056, 26357}, {3094, 26317}, {3157, 3874}, {3173, 34036}, {3193, 17584}, {3416, 6734}, {3564, 10943}, {3589, 10198}, {3681, 5422}, {3740, 17825}, {3742, 17811}, {3811, 36754}, {3818, 45630}, {3827, 19149}, {3844, 5705}, {4331, 34253}, {4661, 34545}, {5039, 10804}, {5050, 9049}, {5085, 10902}, {5093, 9026}, {5096, 36152}, {5204, 33844}, {5228, 5880}, {5480, 26332}, {5707, 5847}, {5820, 10957}, {5848, 37726}, {6210, 18162}, {6776, 12116}, {7289, 12704}, {8593, 12357}, {8758, 21494}, {9020, 36750}, {9037, 11477}, {9039, 11482}, {10532, 14853}, {10752, 12382}, {10753, 12190}, {10754, 13190}, {10755, 13279}, {10759, 12776}, {10766, 13314}, {10806, 14912}, {10835, 19459}, {10959, 39873}, {10966, 37516}, {10982, 14872}, {11269, 30960}, {11365, 42463}, {11401, 12167}, {12017, 41454}, {12212, 26431}, {12452, 45625}, {12453, 45626}, {12583, 26452}, {12588, 26481}, {12589, 26475}, {12590, 45645}, {12591, 45644}, {12675, 37498}, {13110, 32451}, {13374, 17814}, {13910, 45650}, {13972, 45651}, {18440, 18544}, {18543, 39899}, {20986, 37581}, {24299, 38029}, {24848, 32029}, {24987, 38047}, {25568, 26228}, {26399, 45724}, {26423, 45725}, {26499, 45727}, {26508, 45726}, {26889, 37577}, {33878, 35252}, {34377, 36742}, {34893, 41487}, {35840, 45640}, {35841, 45641}, {37659, 38053}, {42450, 42461}, {44590, 45582}, {44591, 45583}

X(45728) = reflection of X(i) in X(j) for these (i, j): (12329, 182), (12587, 5), (22769, 43149), (43146, 575), (45729, 6)
X(45728) = (anti-inner-Yff)-isogonal conjugate-of-X(45700)
X(45728) = X(6)-of-anti-inner-Yff triangle
X(45728) = X(4259)-of-2nd circumperp tangential triangle
X(45728) = X(12587)-of-Johnson triangle
X(45728) = X(12595)-of-anti-outer-Yff triangle
X(45728) = X(45729)-of-outer-Yff tangents triangle
X(45728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 26342, 45423), (1, 26349, 45422), (6, 1191, 16791), (6, 3242, 611), (6, 12595, 1), (1428, 3779, 36741), (3299, 3640, 45424), (3301, 3641, 45425), (10931, 19049, 1), (10932, 19050, 1), (26342, 26349, 1), (26517, 26522, 10529), (44645, 45496, 1), (44646, 45497, 1), (45422, 45423, 1), (45526, 45527, 10267)

X(45729) = PARA-PERSPECTOR OF THESE TRIANGLES: ANTI-OUTER-YFF TO ORTHIC

Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2+2*(b+c)*b*c*a+b^4+c^4-2*(b^2+b*c+c^2)*b*c) : :
X(45729) = 2*X(41454)+X(44456)

The reciprocal para-perspector of these triangles is X(4).

X(45729) lies on these lines: {1, 6}, {3, 8679}, {4, 39902}, {5, 12586}, {10, 3157}, {12, 5820}, {55, 1331}, {69, 5552}, {81, 25568}, {119, 1352}, {141, 26364}, {159, 26309}, {182, 2810}, {193, 10528}, {197, 3955}, {200, 2003}, {210, 394}, {221, 5836}, {222, 1376}, {354, 10601}, {375, 5020}, {511, 11248}, {517, 44413}, {524, 45701}, {575, 43149}, {576, 9052}, {651, 2550}, {674, 1351}, {692, 24320}, {758, 44414}, {895, 13217}, {991, 3939}, {1181, 14872}, {1259, 2594}, {1329, 41344}, {1332, 27549}, {1350, 2077}, {1353, 32213}, {1469, 1470}, {1480, 2802}, {1503, 6256}, {1706, 34043}, {1843, 26378}, {1992, 11239}, {1993, 3681}, {1994, 4661}, {2002, 18838}, {2330, 36740}, {2551, 3562}, {2886, 34029}, {3056, 26358}, {3094, 26318}, {3359, 34371}, {3416, 6735}, {3556, 29958}, {3564, 10942}, {3589, 10200}, {3740, 17811}, {3742, 17825}, {3779, 19369}, {3811, 36742}, {3818, 45631}, {3827, 8549}, {3873, 5422}, {4259, 11509}, {4430, 34545}, {4640, 7074}, {5039, 10803}, {5050, 9026}, {5085, 37561}, {5093, 9049}, {5135, 22768}, {5480, 26333}, {5554, 15988}, {5707, 21077}, {5710, 32049}, {5711, 12607}, {5784, 23144}, {5794, 9370}, {5847, 10915}, {5880, 6180}, {6211, 18161}, {6776, 12115}, {7485, 23155}, {8593, 12356}, {8759, 30513}, {9004, 10202}, {9024, 25438}, {9047, 11477}, {9708, 23071}, {9709, 23070}, {9928, 34339}, {10531, 14853}, {10752, 12381}, {10753, 12189}, {10754, 13189}, {10755, 13278}, {10759, 12775}, {10766, 13313}, {10805, 14912}, {10834, 19459}, {10956, 39897}, {10958, 39873}, {11400, 12167}, {12212, 26432}, {12410, 42450}, {12452, 45627}, {12453, 45628}, {12583, 26453}, {12588, 26482}, {12589, 26476}, {12590, 45647}, {12591, 45646}, {12675, 37514}, {13109, 32451}, {13373, 15805}, {13910, 45652}, {13972, 45653}, {15624, 37474}, {18440, 18542}, {18545, 39899}, {20588, 45126}, {22141, 37606}, {24477, 32911}, {24847, 32029}, {24927, 38029}, {24982, 38047}, {26400, 45724}, {26424, 45725}, {26500, 45727}, {26509, 45726}, {26892, 37577}, {33878, 35251}, {34893, 41442}, {35840, 45642}, {35841, 45643}, {36990, 41698}, {37659, 38057}, {41454, 44456}, {44606, 45580}, {44607, 45581}

X(45729) = reflection of X(i) in X(j) for these (i, j): (12329, 43146), (12586, 5), (22769, 182), (43149, 575), (45728, 6)
X(45729) = (anti-outer-Yff)-isogonal conjugate-of-X(45701)
X(45729) = X(6)-of-anti-outer-Yff triangle
X(45729) = X(5820)-of-outer-Johnson triangle
X(45729) = X(12586)-of-Johnson triangle
X(45729) = X(12594)-of-anti-inner-Yff triangle
X(45729) = X(37516)-of-anti-Mandart-incircle triangle
X(45729) = X(45728)-of-inner-Yff tangents triangle
X(45729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 23693, 42843), (1, 26343, 45425), (1, 26350, 45424), (6, 3242, 613), (6, 12594, 1), (3299, 3641, 45423), (3301, 3640, 45422), (10929, 19047, 1), (10930, 19048, 1), (26343, 26350, 1), (26518, 26523, 10528), (44643, 45494, 1), (44644, 45495, 1), (45424, 45425, 1), (45528, 45529, 10269)

X(45730) = X(52)X(10116)∩X(542)X(5891)

Barycentrics    4*a^10-11*(b^2+c^2)*a^8+(11*b^4+6*b^2*c^2+11*c^4)*a^6-(b^2+c^2)*(7*b^4-12*b^2*c^2+7*c^4)*a^4+(5*b^4+4*b^2*c^2+5*c^4)*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (5*SA-2*R^2-4*SW)*S^2-(6*R^2-7*SW)*SB*SC : :
X(45730) = X(52)-4*X(10116), X(52)+2*X(34224), 4*X(6146)-X(12162), 2*X(10116)+X(34224), X(10575)+2*X(44076), 4*X(11264)-X(45186), 4*X(13470)-X(45187), X(14516)-4*X(18128), 3*X(14845)-4*X(43573), X(34799)+2*X(40647)

See Antreas Hatzipolakis and César Lozada, euclid 2953.

X(45730) lies on these lines: {52, 10116}, {185, 30522}, {381, 39561}, {542, 5891}, {3060, 11232}, {3448, 18475}, {6146, 12162}, {6776, 18474}, {7540, 11225}, {9704, 32767}, {10575, 44076}, {11264, 45186}, {11442, 37513}, {12022, 16194}, {12134, 45298}, {12289, 34796}, {13366, 34514}, {13470, 45187}, {14516, 18128}, {14845, 43573}, {14855, 44665}, {14864, 37472}, {15432, 18388}, {34799, 40647}, {39504, 44109}

X(45730) = midpoint of X(12289) and X(34796)
X(45730) = reflection of X(i) in X(j) for these (i, j): (3060, 11232), (7540, 11225), (12134, 45298), (16194, 12022)
X(45730) = {X(10116), X(34224)}-harmonic conjugate of X(52)

X(45731) = X(3)X(2888)∩X(5)X(156)

Barycentrics    2*a^10-5*(b^2+c^2)*a^8+4*(b^4+b^2*c^2+c^4)*a^6-(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(b^2-c^2)^2*(2*b^4+b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (R^2-2*SB-2*SC)*S^2-(7*R^2-4*SW)*SB*SC : :
X(45731) = 3*X(5)-2*X(12134), 3*X(381)-4*X(43575), 4*X(389)-3*X(38322), 2*X(546)-3*X(12022), 3*X(568)-4*X(32165), 3*X(3845)-4*X(12241), 7*X(3857)-12*X(12024), 9*X(5640)-8*X(13163), 3*X(5946)-2*X(45286), 3*X(6146)-X(12134), 3*X(7576)-4*X(16881), 3*X(11232)-2*X(16625), 3*X(11245)-2*X(31830), 4*X(12102)-3*X(16658), 5*X(15026)-6*X(43573), 3*X(15687)-2*X(16655), 4*X(15807)-3*X(16194)

See Antreas Hatzipolakis and César Lozada, euclid 2953.

X(45731) lies on these lines: {3, 2888}, {4, 13585}, {5, 156}, {30, 5889}, {49, 10224}, {52, 11264}, {54, 39504}, {68, 7502}, {125, 32171}, {140, 11449}, {185, 30522}, {265, 1614}, {381, 43575}, {389, 38322}, {539, 6101}, {542, 5876}, {546, 11423}, {550, 7689}, {568, 32165}, {576, 1353}, {578, 33332}, {1147, 37938}, {1154, 11750}, {1181, 44263}, {1498, 44271}, {1594, 18432}, {1658, 25738}, {1899, 37814}, {2883, 44283}, {3410, 34864}, {3574, 32136}, {3575, 43588}, {3580, 12107}, {3581, 41482}, {3845, 12241}, {3850, 18504}, {3853, 16659}, {3857, 12024}, {5012, 6288}, {5449, 5944}, {5498, 23294}, {5562, 13470}, {5640, 13163}, {5663, 21659}, {5946, 45286}, {6102, 10116}, {6193, 14791}, {6759, 11563}, {7564, 11402}, {7575, 34782}, {7576, 16881}, {7577, 9704}, {8548, 12293}, {9544, 10255}, {9833, 37440}, {10096, 26882}, {10112, 10263}, {10114, 38898}, {10125, 11464}, {10226, 16013}, {10274, 11804}, {10540, 44235}, {10602, 11061}, {10610, 21243}, {11232, 16625}, {11245, 31830}, {11250, 11457}, {11422, 22051}, {11425, 44287}, {11456, 44279}, {11565, 11591}, {11585, 40111}, {11801, 16868}, {11819, 13292}, {12102, 16658}, {12106, 18912}, {12161, 17824}, {12236, 41589}, {13353, 41171}, {13367, 13561}, {13491, 17702}, {14940, 38724}, {15026, 43573}, {15101, 15532}, {15687, 16655}, {15807, 16194}, {16003, 32210}, {18377, 18445}, {18379, 43831}, {18381, 36966}, {18396, 32139}, {18403, 43605}, {18475, 34826}, {18570, 19467}, {18572, 22660}, {20299, 43394}, {26917, 44234}, {34781, 44276}, {37936, 41587}

X(45731) = midpoint of X(i) and X(j) for these {i, j}: {3, 34799}, {12289, 34783}, {34224, 44076}
X(45731) = reflection of X(i) in X(j) for these (i, j): (5, 6146), (52, 11264), (3575, 43588), (3627, 12370), (5562, 13470), (6101, 44829), (6102, 10116), (10263, 10112), (11591, 11565), (11819, 13292), (14516, 140), (16659, 3853), (38898, 10114)
X(45731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (49, 25739, 10224), (265, 1614, 13406), (578, 34514, 33332), (3448, 12254, 3), (7577, 9704, 15806), (12161, 44288, 20424), (18474, 32046, 5), (19467, 32140, 18570)

X(45732) = X(140)X(542)∩X(143)X(1503)

Barycentrics    2*a^10-6*(b^2+c^2)*a^8+(7*b^4+2*b^2*c^2+7*c^4)*a^6-(b^2+c^2)*(5*b^4-9*b^2*c^2+5*c^4)*a^4+3*(b^6-c^6)*(b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (3*R^2-3*SA+2*SW)*S^2-(R^2+3*SW)*SB*SC : :
X(45732) = 2*X(3850)-3*X(43573), 9*X(5943)-8*X(23409), 2*X(10095)-3*X(11245), X(10112)-3*X(10116), 2*X(10112)-3*X(11264), 3*X(11232)-X(13598), 3*X(12022)-2*X(15807), 3*X(13630)-2*X(31833), 3*X(14893)-4*X(40240), 3*X(18914)-X(31833)

See Antreas Hatzipolakis and César Lozada, euclid 2953.

X(45732) lies on these lines: {5, 17713}, {30, 10112}, {140, 542}, {143, 1503}, {156, 1899}, {184, 13561}, {185, 30522}, {427, 32136}, {539, 548}, {3410, 37471}, {3448, 34826}, {3564, 10627}, {3850, 43573}, {5133, 36153}, {5446, 32165}, {5663, 6146}, {5943, 23409}, {5965, 17712}, {6102, 34224}, {6153, 27552}, {6776, 32046}, {7592, 34514}, {9704, 23294}, {10095, 11245}, {10264, 13367}, {10540, 21451}, {10619, 16003}, {11232, 13598}, {11565, 12605}, {11579, 40441}, {12006, 12134}, {12022, 15807}, {12241, 32137}, {12289, 34798}, {13366, 33332}, {13391, 32358}, {13419, 16881}, {13470, 13754}, {13491, 44076}, {13564, 41724}, {13630, 18914}, {14449, 29012}, {14862, 44961}, {14893, 40240}, {15806, 32767}, {16266, 39899}, {18951, 39874}, {19467, 32138}, {21230, 22352}, {23060, 25337}, {32423, 40647}, {34780, 39522}, {37938, 43844}

X(45732) = midpoint of X(i) and X(j) for these {i, j}: {6102, 34224}, {12289, 34798}, {13491, 44076}
X(45732) = reflection of X(i) in X(j) for these (i, j): (140, 18128), (143, 43588), (5446, 32165), (6153, 27552), (11264, 10116), (12134, 12006), (12605, 11565), (13419, 16881), (13630, 18914), (32137, 12241)
X(45732) = {X(6776), X(32140)}-harmonic conjugate of X(32046)

X(45733) = ISOGONAL CONJUGATE OF X(37939)

Barycentrics    (3*a^8-(7*b^2+6*c^2)*a^6+2*(4*b^2+c^2)*b^2*a^4-(b^2-c^2)*(7*b^4+5*b^2*c^2+6*c^4)*a^2+3*(b^4-c^4)*(b^2-c^2)^2)*(3*a^8-(6*b^2+7*c^2)*a^6+2*(b^2+4*c^2)*c^2*a^4+(b^2-c^2)*(6*b^4+5*b^2*c^2+7*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)^2) : :
Barycentrics    (7*S^2+(12*R^2-12*SW+7*SB)*SB)*(7*S^2+(12*R^2-12*SW+7*SC)*SC) : :

See Antreas Hatzipolakis and César Lozada, euclid 2953.

X(45733) lies on Jerabek circumhyperbola and these lines: {125, 43908}, {3532, 34785}, {5663, 31371}, {6243, 32533}, {14528, 26944}, {19155, 40330}

X(45733) = reflection of X(43908) in X(125)
X(45733) = isogonal conjugate of X(37939)
X(45733) = antigonal conjugate of X(43908)
X(45733) = antipode of X(43908) in Jerabek circumhyperbola

X(45734) = X(185)X(30522)∩X(9825)X(12006)

Barycentrics    6*a^10-16*(b^2+c^2)*a^8+5*(3*b^4+2*b^2*c^2+3*c^4)*a^6-3*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^4+(7*b^4+5*b^2*c^2+7*c^4)*(b^2-c^2)^2*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (R^2-7*SA+6*SW)*S^2+(13*R^2-11*SW)*SB*SC : :
X(45734) = 6*X(9825)-7*X(12006), X(10263)-3*X(11264), X(10263)+3*X(34224), 3*X(13470)-X(18436)

See Antreas Hatzipolakis and César Lozada, euclid 2953.

X(45734) lies on these lines: {185, 30522}, {9825, 12006}, {10263, 11264}, {13470, 18436}, {13561, 19357}

X(45734) = midpoint of X(11264) and X(34224)

X(45735) = EULER LINE INTERCEPT OF X(49)X(389)

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+3*b^2*c^2*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2) : :
X(45735) = X(3)+2*X(3518), 15*X(381)-16*X(13487)

As a point on the Euler line, X(45735) has Shinagawa coefficients (E-8*F, 3*E+8*F).

See Antreas Hatzipolakis and César Lozada, euclid 2955.

X(45735) lies on these lines: {2, 3}, {6, 16867}, {32, 2079}, {35, 9629}, {39, 44523}, {49, 389}, {51, 12038}, {52, 22115}, {54, 5946}, {110, 6102}, {125, 45286}, {143, 1511}, {155, 37490}, {156, 5890}, {182, 9973}, {184, 37481}, {185, 10540}, {195, 568}, {399, 2929}, {567, 5462}, {569, 2917}, {574, 44525}, {576, 7666}, {578, 15038}, {1092, 6243}, {1192, 18451}, {1204, 10620}, {1385, 9590}, {1495, 40647}, {1614, 13630}, {1994, 16881}, {2916, 17508}, {3053, 9699}, {3167, 9932}, {3311, 9682}, {3357, 10117}, {3432, 14367}, {3567, 11449}, {3574, 43839}, {3579, 9625}, {3581, 5562}, {5012, 5944}, {5050, 15577}, {5446, 37495}, {5448, 12893}, {5449, 6288}, {5876, 43598}, {5892, 37471}, {5907, 32110}, {5926, 10279}, {6241, 43807}, {6403, 11255}, {6455, 9683}, {6746, 9826}, {7592, 9704}, {7689, 18435}, {7691, 15067}, {8718, 43804}, {8778, 21397}, {8907, 11935}, {9306, 18436}, {9591, 31663}, {9608, 30435}, {9626, 13624}, {9642, 11399}, {9659, 31479}, {9700, 15815}, {9703, 12161}, {9706, 32136}, {9730, 10282}, {9786, 18445}, {9919, 20427}, {9971, 45034}, {10095, 15033}, {10112, 30714}, {10182, 23358}, {10263, 43574}, {10545, 18874}, {10546, 15058}, {10564, 13598}, {10574, 26882}, {10610, 13363}, {10619, 43573}, {10937, 19353}, {10984, 40280}, {10986, 22121}, {11002, 38638}, {11179, 15582}, {11197, 34292}, {11381, 43604}, {11439, 11468}, {11464, 15037}, {11557, 17701}, {11694, 11803}, {11695, 37513}, {11999, 40914}, {12026, 34418}, {12041, 32137}, {12118, 12310}, {12163, 35259}, {12254, 43816}, {12278, 12902}, {12280, 43704}, {12307, 23039}, {12412, 32140}, {12897, 16163}, {13289, 15061}, {13321, 36749}, {13340, 43652}, {13352, 15040}, {13353, 18475}, {13434, 15026}, {13470, 41482}, {13491, 14157}, {13567, 44076}, {13754, 18350}, {14561, 35228}, {14644, 18379}, {14674, 14703}, {14831, 41597}, {15035, 38848}, {15060, 43614}, {15062, 32210}, {15074, 43815}, {15139, 41725}, {15305, 32138}, {15561, 39825}, {15567, 19165}, {15801, 43572}, {17821, 36752}, {18400, 43817}, {18430, 19457}, {18488, 25563}, {19357, 36753}, {19908, 32539}, {21659, 43821}, {22109, 38794}, {22337, 40082}, {23294, 34514}, {23325, 32345}, {23843, 35220}, {26917, 38724}, {30481, 30504}, {32139, 35264}, {34826, 41171}, {37473, 45016}, {37477, 45186}, {37512, 44521}, {38224, 39854}

X(45735) = midpoint of X(i) and X(j) for these {i, j}: {3, 18378}, {3518, 22467}
X(45735) = reflection of X(i) in X(j) for these (i, j): (3, 22467), (18378, 3518)
X(45735) = isogonal conjugate of X(45736)
X(45735) = perspector of the circumconic {{A, B, C, X(648), X(12092)}}
X(45735) = inverse of X(10224) in: orthocentroidal circle, Yff hyperbola
X(45735) = inverse of X(11563) in circumcircle
X(45735) = inverse of X(37922) in tangential circle
X(45735) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(16867)}} and {{A, B, C, X(5), X(15002)}}
X(45735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 1658, 3), (4, 37814, 3), (5, 186, 3), (24, 6644, 3), (26, 17928, 3), (140, 7488, 3), (549, 7512, 3), (568, 1147, 195), (631, 7502, 3), (2070, 43809, 3), (2071, 43615, 3), (3515, 6642, 3), (3520, 15646, 3), (3523, 7525, 3), (3530, 6636, 3), (5462, 13367, 567), (5944, 12006, 5012), (5946, 32171, 54), (7503, 18324, 3), (7514, 38444, 3), (7516, 44837, 3), (7526, 32534, 3), (7527, 18571, 3), (9818, 15750, 3), (10226, 37941, 3), (10539, 34783, 399), (12084, 15078, 3), (13620, 16239, 3), (14118, 15331, 3), (14130, 37955, 3), (14157, 43601, 13491), (15041, 33541, 3357), (16196, 45170, 3), (16238, 45172, 3), (18570, 21844, 3), (20478, 20479, 2), (35220, 35221, 23843), (35497, 37968, 3)

X(45736) = ISOGONAL CONJUGATE OF X(45735)

Barycentrics    (a^8-2*(b^2+c^2)*a^6+(2*b^2+3*c^2)*b^2*a^4-(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^2)*(a^8-2*(b^2+c^2)*a^6+(3*b^2+2*c^2)*c^2*a^4+(b^2-c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^2) : :
Barycentrics    (SB*(SB-2*SW+5*R^2)+S^2)*(SC*(SC-2*SW+5*R^2)+S^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 2955.

X(45736) lies on Jerabek circumhyperbola and these lines: {2, 16867}, {3, 43808}, {5, 15002}, {6, 16868}, {54, 14940}, {74, 21659}, {265, 5876}, {1173, 18388}, {1176, 43812}, {1614, 10114}, {1899, 11270}, {2888, 11804}, {3426, 35490}, {3431, 18912}, {3448, 11559}, {3521, 6102}, {3527, 35488}, {5449, 5504}, {7552, 40441}, {7577, 42059}, {7722, 11744}, {11442, 32533}, {11457, 13452}, {11738, 14216}, {13406, 15087}, {13472, 39571}, {15032, 32165}, {17505, 18379}, {18390, 43891}, {18550, 44279}

X(45736) = isogonal conjugate of X(45735)
X(45736) = antigonal conjugate of the isogonal conjugate of X(11563)
X(45736) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(16868)}} and Jerabek hyperbola

X(45737) = X(14385)X(43809)∩X(17701)X(22115)

Barycentrics    a^2*(a^12-2*(b^2+2*c^2)*a^10-(b^2-7*c^2)*(b^2+c^2)*a^8+2*(2*b^6-4*c^6-(b^2+2*c^2)*b^2*c^2)*a^6-(b^8-7*c^8+(2*b^4-5*b^2*c^2+6*c^4)*b^2*c^2)*a^4-2*(b^2-c^2)^2*(b^6+2*c^6-(b^2+c^2)*b^2*c^2)*a^2+(b^4+c^4)*(b^2-c^2)^4)*(a^12-2*(2*b^2+c^2)*a^10+(7*b^2-c^2)*(b^2+c^2)*a^8-2*(4*b^6-2*c^6+(2*b^2+c^2)*b^2*c^2)*a^6+(7*b^8-c^8-(6*b^4-5*b^2*c^2+2*c^4)*b^2*c^2)*a^4-2*(b^2-c^2)^2*(2*b^6+c^6-(b^2+c^2)*b^2*c^2)*a^2+(b^4+c^4)*(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
Barycentrics    (S^2-SB*SC)*(R^2*(27*R^2-6*SB-14*SW)+2*SB^2+2*SW^2)*(R^2*(27*R^2-6*SC-14*SW)+2*SC^2+2*SW^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 2955.

X(45737) lies on these lines: {14385, 43809}, {17701, 22115}

X(45737) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(50)}} and {{A, B, C, X(30), X(43809)}}

leftri

Perspectors involving the Dao triangle: X(45738)-X(45744)

rightri

This preamble, based on notes from Dao Thanh Oai, is contributed by Clark Kimberling and Peter Moses, Novermber 2, 2021, with amendments by Randy Hutson, November 7, 2021.

Let (a) denote the A-Soddy ellipse, through A that has foci B and C. Define (b) and (c) cyclically. (The A-Soddy ellipse is defined by Randy Hutson at X(6349). The A-Soddy hyperbola also passes through A and has foci B and C; see Paul Yiu, Introduction to the Geometry of the Triangle, p. 143). . Let

d(a) = (b)∩(c); d(b) = (c)∩(a); d(c) = (a)∩(b); the lines d(a), d(b), d(c) concur in X(20).

A' = d(a)∩BC; B' = d(b)∩CA; C' = d(c)∩AB; the lines AA", BB", CC" concur in X(1043).

The triangle A'B'C', here named the Dao triangle, is perspective to certain other triangles, with perspectors X(45738)-X(45744). (Actually, the Dao triangle is the cevian triangle of X(1043); Randy Hutson, November 7, 2021)

The conic (a),

2*(a^2 + b^2 + 2*b*c + c^2)*y*z + 4*(b + c)*x*(c*y + b*z) - (a - b - c)*(a + b + c)*(y^2 + z^2) = 0,

passes through the A-vertex of these triangles: anticomplementary, orthic-of-anticomplementary (also called the Gemini 113 triangle, the dual of the orthic triangle, the 1st anti-circumperp triangle, and the circumanticevian triangle of X(2)).

The line d(a) is given by

(b - c)*(a + b + c)^2*x + (a + b - c)^2*(a + c)*y + (-a - b)*(a - b + c)^2*z = 0. The A-vertex of the Dao triangles is

A' = 0 : (a + b)*(a - b + c)^2 : (a + b - c)^2*(a + c),

The Dao triangle is perspective to every anticevian triangle; specifically, if P = p : q : r, then the perspector is

p ((a + b - c)^2*(a - b + c)^2*(b + c)*p - (a - b - c)^2*(a + b - c)^2*(a + c)*q - (a + b)*(a - b - c)^2*(a - b + c)^2*r) : :

If P lies on the cubic K004, then A'B'C' is perspective to the antipedal triangle of P.

The conics (b) and (c) meet in two real points, assumed in the discussion above, and also two nonreal points, considered here. In this case, the points d(a), d(b), d(c) are nonreal, and the vertices A', B', C', given by

A' = 0 : -a - b : a + c, B' = b + a, 0, -b - c, C' = -c - a, c + b, 0,

are collinear on this line:

(b + c) x + (c + a) y + (a + b) z = 0,

which passes through X(i) for these i: 239, 514, 649, 1019, 1021, 3218, 3798, 4025, 4063, 4091, 4107, 4498, 4560, 4707, 4750, 4765, 4786, 4932, 4960, 4988, 5011, 5088, 5773, 6629, 7192, 7291, 14280, 14351, 14435, 14953, 14964, 16704, 16892, 17209, 17494, 17495, 17496, 17497, 17498, 17729, 18197, 18206, 18653, 18668, 20367, 21124, 21192, 21196, 21222, 21385, 21832, 23572, 23755, 23829, 24148, 27486, 28586, 31348, 42744, 45679, 45745, 45746, 45747, 45748, 45749, 45750, 45751, 45751, 45752, 45753, 45754, 45755.

X(45738) = PERSPECTOR OF THESE TRIANGLES: DAO AND ANTICOMPLEMENTARY

Barycentrics    a^5 - 2*a^4*b + 2*a^2*b^3 - a*b^4 - 2*a^4*c + 2*b^4*c + 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 - a*c^4 + 2*b*c^4 : :
X(45739) = {3 X[21160] - 2 X[43160]}

X(45739) lies on these lines: {1, 25255}, {2, 3668}, {7, 25935}, {8, 144}, {9, 1441}, {19, 27}, {57, 20905}, {77, 1944}, {78, 25252}, {189, 4001}, {190, 322}, {192, 3870}, {193, 41575}, {223, 28950}, {253, 306}, {269, 26651}, {281, 307}, {344, 908}, {347, 26006}, {519, 20110}, {534, 17781}, {610, 14543}, {653, 7013}, {857, 41010}, {894, 2263}, {1214, 27413}, {1707, 26000}, {1723, 17861}, {1763, 22001}, {1826, 24316}, {2257, 17863}, {2324, 4552}, {3305, 20921}, {3436, 3717}, {3827, 43216}, {3869, 3886}, {4032, 40131}, {4331, 17257}, {4652, 21160}, {5813, 10445}, {6180, 44664}, {6335, 18749}, {6508, 18674}, {6554, 25019}, {6765, 24068}, {7079, 20914}, {7101, 33672}, {7291, 10444}, {7359, 17073}, {7719, 20235}, {9312, 37659}, {10436, 24635}, {14953, 18594}, {17080, 27411}, {17139, 18713}, {17263, 30852}, {17264, 31158}, {17363, 39351}, {18634, 41804}, {20348, 30059}, {21084, 28043}, {21371, 36796}, {22370, 33677}, {22464, 27509}, {23058, 25000}, {24554, 40719}, {24612, 25727}, {25930, 27420}, {26668, 43035}, {28921, 34050}, {30620, 38454}

X(45738) = reflection of X(i) in X(j) for these {i,j}: {4329, 8804}, {18655, 19}
X(45738) = anticomplement of X(3668)
X(45738) = anticomplement of the isogonal conjugate of X(2328)
X(45738) = anticomplement of the isotomic conjugate of X(1043)
X(45738) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {9, 2893}, {21, 3434}, {41, 17778}, {55, 2475}, {58, 36845}, {60, 3873}, {81, 6604}, {110, 3900}, {112, 17896}, {163, 4025}, {200, 1330}, {212, 3152}, {219, 2897}, {220, 2895}, {284, 7}, {314, 21280}, {333, 21285}, {346, 21287}, {593, 17158}, {643, 21302}, {657, 21221}, {1021, 150}, {1043, 6327}, {1098, 17135}, {1175, 16465}, {1253, 1654}, {1333, 4452}, {1792, 1370}, {1802, 3151}, {2150, 3875}, {2185, 20244}, {2193, 347}, {2194, 145}, {2203, 11851}, {2204, 30699}, {2206, 17480}, {2287, 69}, {2299, 12649}, {2322, 21270}, {2326, 17220}, {2327, 4329}, {2328, 8}, {2332, 5905}, {3239, 21294}, {3900, 3448}, {4183, 4}, {4282, 41803}, {4612, 4374}, {5546, 693}, {6061, 3869}, {6065, 3909}, {7054, 75}, {7058, 17137}, {7253, 21293}, {7256, 21301}, {7258, 21304}, {7259, 20295}, {8021, 2894}, {8641, 148}, {14827, 1655}, {21789, 149}, {23609, 2975}, {35192, 41808}
X(45738) = X(1043)-Ceva conjugate of X(2)
X(45738) = X(7580)-cross conjugate of X(34059)
X(45738) = barycentric product X(i)*X(j) for these {i,j}: {8, 34059}, {75, 7580}
X(45738) = barycentric quotient X(i)/X(j) for these {i,j}: {7580, 1}, {34059, 7}, {38388, 2310}
X(45738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {63, 92, 5271}, {144, 5942, 4416}, {190, 322, 3692}, {347, 27382, 26006}, {3729, 30625, 144}, {14543, 17134, 610}

X(45739) = PERSPECTOR OF THESE TRIANGLES: DAO AND TANGENTIAL

Barycentrics    a^2*(a^6*b - a^5*b^2 - 2*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 - a*b^6 + a^6*c + 5*a^4*b^2*c - 5*a^2*b^4*c - b^6*c - a^5*c^2 + 5*a^4*b*c^2 - 4*a^3*b^2*c^2 + a*b^4*c^2 - b^5*c^2 - 2*a^4*c^3 + 2*b^4*c^3 + 2*a^3*c^4 - 5*a^2*b*c^4 + a*b^2*c^4 + 2*b^3*c^4 + a^2*c^5 - b^2*c^5 - a*c^6 - b*c^6) : :

X(45739) lies on these lines: {3, 9948}, {55, 1615}, {197, 1604}, {228, 15837}, {333, 1610}, {610, 3185}, {910, 1402}, {2155, 12329}, {2933, 15625}, {10460, 22088}, {10537, 35327}, {15509, 20470}

X(45739) = X(1043)-Ceva conjugate of X(6)
X(45739) = crosssum of X(512) and X(13609)

X(45740) = PERSPECTOR OF THESE TRIANGLES: DAO AND MOSES-SODDY

Barycentrics    (b - c)^2*(2*a^5 - 3*a^4*b + 2*a^2*b^3 - 2*a*b^4 + b^5 - 3*a^4*c + 2*a^3*b*c + 4*a^2*b^2*c - 2*a*b^3*c - b^4*c + 4*a^2*b*c^2 + 2*a^2*c^3 - 2*a*b*c^3 - 2*a*c^4 - b*c^4 + c^5) : :

X(45740) lies on these lines: {116, 24775}, {244, 2968}, {905, 17213}, {997, 4859}, {1086, 34591}, {4534, 38374}, {4653, 24780}, {14714, 17761}, {14837, 21950}, {17205, 17216}

X(45740) = X(951)-complementary conjugate of X(20317)
X(45740) = X(1043)-Ceva conjugate of X(514)

X(45741) = PERSPECTOR OF THESE TRIANGLES: DAO AND SCHROETER

Barycentrics    (b - c)^2*(b + c)*(3*a^6 - 3*a^5*b - 4*a^4*b^2 + 6*a^3*b^3 - a^2*b^4 - 3*a*b^5 + 2*b^6 - 3*a^5*c - a^4*b*c + 6*a^3*b^2*c + 2*a^2*b^3*c - 3*a*b^4*c - b^5*c - 4*a^4*c^2 + 6*a^3*b*c^2 + 10*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*b^4*c^2 + 6*a^3*c^3 + 2*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - 3*a*b*c^4 - 2*b^2*c^4 - 3*a*c^5 - b*c^5 + 2*c^6) : :

X(45741) lies on these lines: {11, 3138}, {115, 13609}, {1086, 2968}, {1834, 15252}

X(45741) = X(1043)-Ceva conjugate of X(523)

X(45742) = PERSPECTOR OF THESE TRIANGLES: DAO AND SODDY

Barycentrics    (a + b - c)*(a - b + c)*(a^7 - 2*a^6*b + 3*a^5*b^2 - 4*a^4*b^3 - a^3*b^4 + 6*a^2*b^5 - 3*a*b^6 - 2*a^6*c + 6*a^4*b^2*c - 6*a^2*b^4*c + 2*b^6*c + 3*a^5*c^2 + 6*a^4*b*c^2 - 6*a^3*b^2*c^2 + 3*a*b^4*c^2 - 6*b^5*c^2 - 4*a^4*c^3 + 4*b^4*c^3 - a^3*c^4 - 6*a^2*b*c^4 + 3*a*b^2*c^4 + 4*b^3*c^4 + 6*a^2*c^5 - 6*b^2*c^5 - 3*a*c^6 + 2*b*c^6) : :

X(45742) lies on these lines: {1, 9446}, {77, 17194}, {314, 5931}, {347, 4847}, {658, 7070}, {2898, 3668}, {2947, 41353}, {4384, 17080}, {7056, 40960}

X(45742) = X(1043)-Ceva conjugate of X(7)

X(45743) = PERSPECTOR OF THESE TRIANGLES: DAO AND (VERTEX TRIANGLE OF ABC AND INTANGENTS TRIANGLE)

Barycentrics    a*(a - b - c)*(b - c)^2*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c - a^2*b*c - 3*a*b^2*c + b^3*c - a^2*c^2 - 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + b*c^3) : :

X(45743) lies on these lines: {210, 4578}, {513, 21945}, {650, 3122}, {2310, 24012}, {3271, 4081}, {4459, 18191}, {16726, 17418}

X(45743) = X(1043)-Ceva conjugate of X(650)
X(45743) = crosspoint of X(9) and X(4581)
X(45743) = crosssum of X(i) and X(j) for these (i,j): {65, 21362}, {1402, 35326}
X(45743) = barycentric product X(3239)*X(34496)
X(45743) = barycentric quotient X(34496)/X(658)

X(45744) = PERSPECTOR OF THESE TRIANGLES: DAO AND GEMINI 17

Barycentrics    (b + c)*(-a^4 + a^3*b + a^2*b^2 - a*b^3 + a^3*c - a^2*b*c - a*b^2*c + b^3*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 - a*c^3 + b*c^3) : :

X(45744) lies on these lines: {2, 37}, {7, 22021}, {8, 25252}, {9, 37076}, {10, 25255}, {20, 72}, {69, 40905}, {78, 990}, {100, 11683}, {149, 31122}, {190, 2287}, {193, 20013}, {253, 6360}, {306, 3668}, {307, 2321}, {322, 40903}, {894, 5764}, {936, 3159}, {938, 2901}, {965, 17262}, {1043, 1257}, {1441, 3694}, {1446, 4515}, {2198, 27624}, {2257, 3187}, {2895, 18666}, {3242, 17148}, {3661, 25241}, {3695, 25015}, {3930, 4032}, {3943, 18635}, {3949, 8680}, {3965, 30807}, {4019, 17751}, {4319, 32929}, {4363, 5736}, {4431, 6734}, {4442, 21955}, {4463, 17784}, {4651, 21039}, {4665, 5742}, {4873, 18634}, {5227, 17134}, {5279, 14953}, {5703, 7229}, {5738, 17314}, {5773, 21061}, {6646, 25257}, {15936, 17390}, {16517, 31036}, {17164, 42289}, {17248, 25261}, {17257, 17676}, {17587, 40571}, {20927, 27108}, {21029, 27691}, {21049, 25004}, {21346, 29824}, {22012, 41246}, {25000, 40997}

X(45744) = anticomplement of X(17863)
X(45744) = anticomplement of the isotomic conjugate of X(1257)
X(45744) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {951, 3434}, {1257, 6327}, {2983, 69}, {29163, 20295}, {40431, 20242}, {40445, 11442}
X(45744) = X(i)-Ceva conjugate of X(j) for these (i,j): {1043, 306}, {1257, 2}
X(45744) = barycentric product X(i)*X(j) for these {i,j}: {321, 37659}, {4219, 20336}, {27808, 44408}
X(45744) = barycentric quotient X(i)/X(j) for these {i,j}: {4219, 28}, {37659, 81}, {44408, 3733}
X(45744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {75, 27396, 2}, {321, 3998, 2}

X(45745) = X(230)X(231)∩X(239)X(514)

Barycentrics    (b - c)*(-a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2) : :
X(45745) = 7 X[649] - 8 X[14351], 7 X[649] - 9 X[14435], 5 X[650] - 4 X[2490], 7 X[650] - 6 X[14425], 7 X[661] - 3 X[4958], 8 X[2490] - 5 X[6590], 14 X[2490] - 15 X[14425], 2 X[2529] - 3 X[4394], 2 X[3239] - 3 X[4893], 2 X[3798] - 3 X[27486], X[4024] - 3 X[4893], 3 X[4379] - 4 X[7658], 4 X[4521] - X[4838], X[4608] - 5 X[27013], 2 X[4765] + X[4988], 7 X[4765] - 4 X[14351], 14 X[4765] - 9 X[14435], 3 X[4786] - 2 X[4932], 3 X[4789] - 5 X[31209], 7 X[4988] + 8 X[14351], 7 X[4988] + 9 X[14435], 7 X[6590] - 12 X[14425], X[7192] - 3 X[27486], 2 X[11068] - 3 X[31150], 8 X[14351] - 9 X[14435], 3 X[21183] - 4 X[21212], X[26824] - 3 X[44435], 5 X[26985] - 6 X[44432]

X(45745) lies on these lines: {230, 231}, {239, 514}, {513, 4841}, {522, 661}, {693, 24622}, {824, 4468}, {1635, 4458}, {1639, 28165}, {2516, 28151}, {2522, 29142}, {2527, 28179}, {2529, 4394}, {3004, 4762}, {3121, 24186}, {3239, 4024}, {3261, 29771}, {3667, 4813}, {3700, 4777}, {4041, 7654}, {4079, 24089}, {4151, 42664}, {4379, 7658}, {4467, 28846}, {4500, 25666}, {4521, 4838}, {4608, 27013}, {4770, 21719}, {4773, 28195}, {4778, 4979}, {4789, 25594}, {4790, 4977}, {4820, 14321}, {4944, 28187}, {4984, 28225}, {5558, 23893}, {6545, 41926}, {9001, 13401}, {11068, 31150}, {17069, 43067}, {17161, 25259}, {20974, 44311}, {21183, 21212}, {25667, 35519}, {26824, 44435}, {26985, 44432}, {28155, 43061}

X(45745) = midpoint of X(i) and X(j) for these {i,j}: {649, 4988}, {4841, 4976}, {17161, 25259}
X(45745) = reflection of X(i) in X(j) for these {i,j}: {649, 4765}, {4024, 3239}, {4025, 21196}, {4500, 25666}, {4820, 14321}, {6590, 650}, {7192, 3798}, {43067, 17069}
X(45745) = X(31)-complementary conjugate of X(38960)
X(45745) = X(2)-Ceva conjugate of X(38960)
X(45745) = X(i)-isoconjugate of X(j) for these (i,j): {100, 967}, {101, 969}
X(45745) = crosssum of X(649) and X(16466)
X(45745) = crossdifference of every pair of points on line {3, 42}
X(45745) = barycentric product X(i)*X(j) for these {i,j}: {1, 7650}, {514, 966}, {522, 3485}, {523, 11110}, {693, 968}, {2271, 3261}, {4025, 4207}, {4288, 14618}
X(45745) = barycentric quotient X(i)/X(j) for these {i,j}: {513, 969}, {649, 967}, {966, 190}, {968, 100}, {2271, 101}, {3485, 664}, {4207, 1897}, {4288, 4558}, {7650, 75}, {11110, 99}
X(45745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 14435, 14351}, {4024, 4893, 3239}, {7192, 27486, 3798}

X(45746) = X(239)X(514)∩X(325)X(523)

Barycentrics    (b - c)*(a*b + b^2 + a*c + b*c + c^2) : :
X(45746) = 5 X[2] - 4 X[45685], 2 X[649] - 3 X[27486], 5 X[649] - 6 X[45679], 5 X[693] - 6 X[4927], 2 X[693] - 3 X[44435], 5 X[3004] - 3 X[4927], 4 X[3004] - 3 X[44435], 4 X[3676] - X[4608], 2 X[3700] - 3 X[4776], 3 X[4379] - 4 X[21212], 3 X[4453] - 2 X[43067], 2 X[4500] - 3 X[4728], 3 X[4728] - X[4838], 3 X[4750] - 2 X[4932], 3 X[4789] - 4 X[4885], 4 X[4927] - 5 X[44435], 5 X[6590] - 6 X[45685], 4 X[11068] - 5 X[26777], 4 X[21196] - 3 X[27486], 5 X[21196] - 3 X[45679], 5 X[27486] - 4 X[45679]

X(45746) lies on these lines: {2, 6590}, {239, 514}, {325, 523}, {513, 4467}, {522, 17161}, {647, 27648}, {650, 16757}, {661, 824}, {784, 3766}, {835, 37215}, {918, 4841}, {2254, 4818}, {2786, 4813}, {2978, 30665}, {3250, 27647}, {3676, 4608}, {3700, 4776}, {3805, 20983}, {3835, 4024}, {4106, 4777}, {4379, 21212}, {4380, 4976}, {4453, 4802}, {4500, 4728}, {4509, 14208}, {4785, 23731}, {4789, 4885}, {4820, 4940}, {4897, 4977}, {4979, 28859}, {5936, 6548}, {6545, 25381}, {6586, 24948}, {11068, 26777}, {14349, 23879}, {16751, 43060}, {17094, 24002}, {17198, 18210}, {18155, 20949}, {20526, 29821}, {20950, 29771}, {20952, 25667}, {21183, 28155}, {21297, 28161}, {21348, 27674}, {23813, 28165}, {26985, 30763}, {28846, 31290}, {28898, 44449}, {29116, 43041}

X(45746) = midpoint of X(i) and X(j) for these {i,j}: {4988, 16892}, {17161, 20295}
X(45746) = reflection of X(i) in X(j) for these {i,j}: {649, 21196}, {693, 3004}, {2254, 4818}, {4024, 3835}, {4380, 4976}, {4820, 4940}, {4838, 4500}, {7192, 4025}, {25259, 661}
X(45746) = isotomic conjugate of X(835)
X(45746) = anticomplement of X(6590)
X(45746) = anticomplement of the isotomic conjugate of X(37215)
X(45746) = isotomic conjugate of the anticomplement of X(5515)
X(45746) = isotomic conjugate of the isogonal conjugate of X(834)
X(45746) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1036, 37781}, {1245, 21221}, {1310, 69}, {1472, 4440}, {2221, 149}, {2281, 148}, {2339, 33650}, {32691, 5905}, {36099, 4}, {37215, 6327}
X(45746) = X(i)-Ceva conjugate of X(j) for these (i,j): {33948, 5224}, {37215, 2}
X(45746) = X(5515)-cross conjugate of X(2)
X(45746) = crosspoint of X(5224) and X(33948)
X(45746) = crosssum of X(213) and X(8646)
X(45746) = crossdifference of every pair of points on line {32, 42}
X(45746) = X(i)-isoconjugate of X(j) for these (i,j): {31, 835}, {32, 37218}, {101, 2214}, {692, 43531}, {1110, 43927}
X(45746) = barycentric product X(i)*X(j) for these {i,j}: {75, 14349}, {76, 834}, {86, 23879}, {310, 42664}, {386, 3261}, {469, 4025}, {513, 33935}, {514, 5224}, {522, 33949}, {693, 28606}, {1019, 42714}, {1086, 33948}, {1502, 8637}, {1509, 23282}, {3876, 24002}, {5515, 37215}
X(45746) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 835}, {75, 37218}, {386, 101}, {469, 1897}, {513, 2214}, {514, 43531}, {834, 6}, {1086, 43927}, {3876, 644}, {5224, 190}, {5515, 6590}, {8637, 32}, {14349, 1}, {23282, 594}, {23879, 10}, {28606, 100}, {33935, 668}, {33948, 1016}, {33949, 664}, {39016, 8637}, {42664, 42}, {42714, 4033}, {44103, 8750}
X(45746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 21196, 27486}, {693, 3004, 44435}, {693, 20906, 850}, {4728, 4838, 4500}

X(45747) = X(6)X(13)∩X(239)X(514)

Barycentrics    (a + b)*(a + c)*(2*a^5 - a^4*b - a^3*b^2 - a^2*b^3 + 2*a*b^4 - b^5 - a^4*c + 2*a^2*b^2*c - b^4*c - a^3*c^2 + 2*a^2*b*c^2 - 4*a*b^2*c^2 + 2*b^3*c^2 - a^2*c^3 + 2*b^2*c^3 + 2*a*c^4 - b*c^4 - c^5) : :

X(45747) lies on these lines: {6, 13}, {86, 116}, {163, 6740}, {239, 514}, {996, 15628}, {1150, 24636}

X(45747) = crossdifference of every pair of points on line {42, 526}

X(45748) = X(4)X(6)∩X(239)X(514)

Barycentrics    (a + b)*(a + c)*(2*a^5 - a^4*b - 2*a^2*b^3 + 2*a*b^4 - b^5 - a^4*c + 2*a^2*b^2*c - b^4*c + 2*a^2*b*c^2 - 4*a*b^2*c^2 + 2*b^3*c^2 - 2*a^2*c^3 + 2*b^2*c^3 + 2*a*c^4 - b*c^4 - c^5) : :

X(45748) lies on these lines: {4, 6}, {239, 514}, {284, 5768}, {956, 36017}, {1146, 37168}, {1150, 24580}, {14529, 17911}, {14544, 18669}, {14955, 39351}, {20222, 21374}, {31014, 31054}

X(45748) = crossdifference of every pair of points on line {42, 520}

X(45749) = X(1)X(5)∩X(239)X(514)

Barycentrics    2*a^5 - a^4*b - 2*a^3*b^2 + a^2*b^3 - a^4*c + 2*a^3*b*c + a*b^3*c - 2*b^4*c - 2*a^3*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + a^2*c^3 + a*b*c^3 + 2*b^2*c^3 - 2*b*c^4 : :

X(45749) lies on these lines: {1, 5}, {239, 514}, {544, 16834}, {664, 24618}, {1404, 24209}, {3629, 5845}, {4384, 24581}, {4957, 17455}, {17753, 25049}

X(45749) = crossdifference of every pair of points on line {42, 654}

X(45750) = X(239)X(514)∩X(3259)X(3326)

Barycentrics    (b - c)*(-(a^2*b) + b^3 - a^2*c + 2*a*b*c - b^2*c - b*c^2 + c^3)*(-2*a^5 + a^4*b + 3*a^3*b^2 - 2*a^2*b^3 - a*b^4 + b^5 + a^4*c - 4*a^3*b*c + a^2*b^2*c + 2*a*b^3*c + 3*a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 2*a^2*c^3 + 2*a*b*c^3 - b^2*c^3 - a*c^4 + c^5) : :

X(45750) lies on these lines: {239, 514}, {3259, 3326}

X(45750) = crossdifference of every pair of points on line {42, 32641}

X(45751) = X(1)X(6)∩X(239)X(514)

Barycentrics    a*(a^2*b - a*b^2 + a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2) : :

X(45751) lies on these lines: {1, 6}, {8, 4253}, {10, 1475}, {36, 3684}, {39, 3293}, {41, 8666}, {43, 23632}, {57, 7223}, {63, 16834}, {76, 29742}, {100, 5030}, {145, 3730}, {149, 5134}, {169, 17736}, {200, 40956}, {214, 38980}, {239, 514}, {244, 16611}, {274, 20448}, {330, 39797}, {519, 672}, {573, 5731}, {579, 5839}, {583, 17362}, {668, 37686}, {758, 2170}, {894, 16829}, {942, 4875}, {993, 2280}, {995, 37657}, {996, 2279}, {999, 37658}, {1015, 2238}, {1026, 24496}, {1111, 35102}, {1125, 3691}, {1150, 3306}, {1197, 18192}, {1206, 18169}, {1334, 3244}, {1400, 4315}, {1573, 24512}, {1575, 31855}, {1759, 2082}, {1909, 29433}, {2183, 4700}, {2223, 3689}, {2245, 4969}, {2260, 3686}, {2269, 4856}, {2275, 3216}, {2350, 4651}, {2975, 4251}, {3208, 3633}, {3214, 23649}, {3219, 14751}, {3290, 4694}, {3305, 29597}, {3496, 6763}, {3501, 3632}, {3509, 5540}, {3670, 41015}, {3678, 39244}, {3679, 17754}, {3711, 21010}, {3739, 18164}, {3743, 39247}, {3759, 16574}, {3761, 17026}, {3871, 24047}, {3874, 17451}, {3881, 21808}, {3930, 24036}, {3948, 29769}, {3953, 16583}, {3975, 29456}, {3985, 4975}, {4006, 25066}, {4051, 5903}, {4520, 31792}, {4551, 43039}, {4662, 25068}, {5021, 5264}, {5022, 5687}, {5179, 26015}, {5541, 24578}, {6376, 29438}, {7272, 24694}, {7308, 37869}, {7719, 40985}, {9263, 40859}, {9336, 21214}, {10469, 19853}, {14210, 17755}, {14963, 20974}, {15983, 28402}, {16679, 22271}, {16738, 16819}, {16826, 35595}, {16832, 37660}, {16887, 26965}, {17034, 21226}, {17149, 29557}, {17151, 44421}, {17366, 29812}, {17499, 26801}, {17753, 20059}, {17761, 20347}, {18135, 29748}, {18140, 29750}, {18152, 29758}, {18785, 39697}, {19765, 31468}, {20435, 32092}, {20605, 32922}, {21070, 26770}, {21369, 32912}, {21373, 40131}, {23393, 40638}, {23427, 45216}, {24215, 24790}, {24524, 29381}, {25278, 29400}, {25280, 29375}, {25286, 29502}, {25287, 29503}, {25296, 29510}, {25298, 29511}, {25303, 29383}, {25439, 41423}, {27065, 29580}, {27109, 40006}, {29459, 31997}, {29824, 40614}, {30109, 30941}, {31466, 37693}, {33792, 34063}, {34253, 43059}

X(45751) = reflection of X(i) in X(j) for these {i,j}: {1018, 672}, {3930, 24036}, {20347, 17761}
X(45751) = X(3227)-Ceva conjugate of X(1)
X(45751) = X(44671)-cross conjugate of X(29824)
X(45751) = crosspoint of X(i) and X(j) for these (i,j): {81, 37129}, {765, 4607}
X(45751) = crosssum of X(i) and X(j) for these (i,j): {37, 899}, {244, 3768}, {649, 16507}
X(45751) = crossdifference of every pair of points on line {42, 513}
X(45751) = barycentric product X(i)*X(j) for these {i,j}: {1, 29824}, {86, 44671}, {3227, 40614}
X(45751) = barycentric quotient X(i)/X(j) for these {i,j}: {29824, 75}, {40614, 536}, {44671, 10}
X(45751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 16552, 3294}, {1, 21384, 16552}, {6, 956, 16788}, {6, 16975, 1}, {8, 4253, 16549}, {36, 3684, 35342}, {37, 16971, 1}, {39, 3780, 3293}, {213, 17448, 1}, {239, 18206, 20367}, {1107, 20963, 1}, {1212, 3555, 3970}, {1757, 24727, 20372}, {3509, 5540, 21372}, {3691, 17474, 1125}, {5288, 17745, 2329}, {6762, 16572, 17742}, {16975, 21384, 21061}

X(45752) = X(1)X(181)∩X(239)X(514)

Barycentrics    a*(a^5*b^2 + a^4*b^3 - a^3*b^4 - a^2*b^5 + 4*a^4*b^2*c + 2*a^3*b^3*c - 4*a^2*b^4*c - 2*a*b^5*c + a^5*c^2 + 4*a^4*b*c^2 - 2*a*b^4*c^2 + b^5*c^2 + a^4*c^3 + 2*a^3*b*c^3 + 4*a*b^3*c^3 - b^4*c^3 - a^3*c^4 - 4*a^2*b*c^4 - 2*a*b^2*c^4 - b^3*c^4 - a^2*c^5 - 2*a*b*c^5 + b^2*c^5) : :

X(45752) lies on these lines: {1, 181}, {239, 514}

X(45753) = X(6)X(17)∩X(239)X(514)

Barycentrics    (a + b)*(a + c)*(2*a^5 - a^4*b - 3*a^3*b^2 + a^2*b^3 + 2*a*b^4 - b^5 - a^4*c + 2*a^2*b^2*c - b^4*c - 3*a^3*c^2 + 2*a^2*b*c^2 - 4*a*b^2*c^2 + 2*b^3*c^2 + a^2*c^3 + 2*b^2*c^3 + 2*a*c^4 - b*c^4 - c^5) : :

X(45753) lies on these lines: {6, 17}, {239, 514}

X(45753) = crossdifference of every pair of points on line {42, 1510}

X(45754) = X(5)X(523)∩X(239)X(514)

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 + a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + b^5*c + a^4*c^2 - 2*a^3*b*c^2 - a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - 2*a^2*c^4 + a*b*c^4 - b^2*c^4 + a*c^5 + b*c^5 + c^6) : :

X(45754) lies on these lines: {5, 523}, {239, 514}

X(45754) = crossdifference of every pair of points on line {42, 50}

X(45755) = X(9)X(522)∩X(239)X(514)

Barycentrics    a*(a - b - c)*(b - c)*(a^2 - a*b - a*c - 2*b*c) : :
X(45755) = 2 X[9] - 3 X[657]

X(45755) lies on these lines: {9, 522}, {239, 514}, {513, 4827}, {521, 21127}, {650, 663}, {652, 4976}, {661, 3309}, {665, 4449}, {812, 30804}, {1100, 1459}, {1635, 14077}, {1643, 3126}, {2320, 23893}, {3063, 17418}, {3239, 28058}, {3709, 4501}, {3887, 4893}, {4148, 4391}, {4367, 4394}, {4777, 22108}, {4813, 42325}, {6586, 16777}, {16667, 21173}, {17069, 43042}, {21390, 28161}, {25943, 25955}

X(45755) = reflection of X(43042) in X(17069)
X(45755) = X(i)-Ceva conjugate of X(j) for these (i,j): {4762, 4724}, {32040, 1}
X(45755) = X(i)-isoconjugate of X(j) for these (i,j): {7, 8693}, {56, 32041}, {57, 37138}, {100, 42290}, {109, 27475}, {651, 1002}, {664, 2279}, {934, 40779}, {4551, 42302}, {9436, 36138}, {32724, 40704}
X(45755) = crosspoint of X(190) and X(39959)
X(45755) = crosssum of X(649) and X(7290)
X(45755) = crossdifference of every pair of points on line {42, 57}
X(45755) = barycentric product X(i)*X(j) for these {i,j}: {8, 4724}, {9, 4762}, {21, 4804}, {513, 3886}, {514, 37658}, {522, 1001}, {649, 28809}, {650, 4384}, {663, 4441}, {1471, 4397}, {2280, 4391}, {3063, 21615}, {3064, 23151}, {3239, 5228}, {3696, 3737}, {3900, 40719}, {4025, 28044}, {4044, 7252}, {4130, 42309}, {4702, 23838}, {7253, 42289}, {8611, 31926}
X(45755) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 32041}, {41, 8693}, {55, 37138}, {649, 42290}, {650, 27475}, {657, 40779}, {663, 1002}, {1001, 664}, {1471, 934}, {2280, 651}, {3063, 2279}, {3886, 668}, {4384, 4554}, {4441, 4572}, {4724, 7}, {4762, 85}, {4804, 1441}, {5228, 658}, {7252, 42302}, {28044, 1897}, {28809, 1978}, {37658, 190}, {40719, 4569}, {42289, 4566}, {42309, 36838}
X(45755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {649, 21832, 4498}, {650, 4435, 663}, {1021, 4765, 649}, {3709, 4501, 42312}

X(45756) = X(110)X(9811)∩X(239)X(514)

Barycentrics    a*(a + b)*(b - c)*(a + c)*(a^5 - a^3*b^2 - a^2*b^3 + b^5 + a^3*b*c - 2*a^2*b^2*c + a*b^3*c - a^3*c^2 - 2*a^2*b*c^2 + 4*a*b^2*c^2 - b^3*c^2 - a^2*c^3 + a*b*c^3 - b^2*c^3 + c^5) : :

X(45756) lies on these lines: {63, 3268}, {110, 9811}, {239, 514}, {2986, 4444}, {14392, 35997}

X(45757) =  COMPLEMENT OF X(41983)

Barycentrics    10*a^4-41*(b^2+c^2)*a^2+31*(b^2-c^2)^2 : :
X(45757) = 5*X(2)+7*X(5), 13*X(2)-7*X(140), 17*X(2)+7*X(381), X(2)-7*X(547), 7*X(2)-X(548), 19*X(2)-7*X(549), 9*X(2)+7*X(3545), 11*X(2)+X(3627), 4*X(2)-7*X(3628), 19*X(2)+5*X(3843), 2*X(2)+X(3850), 20*X(2)+7*X(3860), 15*X(2)-7*X(5054), X(2)+7*X(5055), 11*X(2)+7*X(5066), 13*X(2)+11*X(5072), 2*X(2)+7*X(10109), 10*X(2)-7*X(10124), 11*X(2)-7*X(11539), 8*X(2)+7*X(11737), 16*X(2)-7*X(11812), 5*X(2)-2*X(12108), 19*X(2)+14*X(12811), X(2)+5*X(12812), 4*X(2)-X(14891), 5*X(2)+X(14893), 13*X(2)-X(15686), 9*X(2)-X(15689), 3*X(2)-7*X(15699), 11*X(2)-3*X(15706), 17*X(2)-5*X(15712), 17*X(2)-14*X(16239), 3*X(2)+X(23046), X(2)+14*X(35018), 13*X(2)+7*X(38071), 7*X(2)+X(38335), 8*X(2)-7*X(41984), 6*X(2)-7*X(41985), 9*X(2)+14*X(41986)

As a point on the Euler line, X(45757) has Shinagawa coefficients (41, 21).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45757) lies on these lines: {2, 3}, {1327, 42569}, {1328, 42568}, {16241, 42692}, {16242, 42693}, {16267, 42497}, {16268, 42496}, {16962, 42143}, {16963, 42146}, {16966, 43644}, {16967, 43649}, {18538, 42557}, {18762, 42558}, {20585, 32396}, {38083, 38127}, {41100, 42591}, {41101, 42590}, {41121, 42436}, {41122, 42435}, {42121, 42474}, {42124, 42475}, {42149, 43246}, {42152, 43247}, {42936, 43108}, {42937, 43109}

X(45757) = midpoint of X(i) and X(j) for these {i, j}: {2, 14892}, {4, 41982}, {140, 38071}, {546, 3524}, {547, 5055}, {548, 38335}, {3839, 12100}, {3850, 14890}, {5066, 11539}, {11737, 41984}, {12101, 15688}, {17504, 41987}, {23046, 41983}
X(45757) = reflection of X(i) in X(j) for these (i, j): (3524, 11540), (3530, 11539), (3545, 41986), (3850, 14892), (5055, 35018), (10109, 5055), (11812, 41984), (12102, 3839), (14890, 2), (14891, 14890), (41982, 44580), (41984, 3628), (41985, 15699)
X(45757) = complement of X(41983)
X(45757) = anticomplement of X(45758)
X(45757) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 3843, 549), (2, 15686, 140), (5, 376, 5066), (5, 10124, 3860), (5, 12108, 3850), (5, 15699, 5054), (376, 3832, 3830), (547, 5066, 1656), (547, 10109, 3628), (547, 12812, 2), (1656, 3530, 3628), (1657, 15703, 2), (3523, 5079, 5), (3530, 3850, 3627), (3628, 14891, 2), (3830, 5071, 5), (3843, 12811, 3850), (3850, 14893, 3860), (5054, 14269, 376), (10109, 10124, 5), (11737, 11812, 3861), (11737, 33923, 3860), (12100, 14893, 1657), (12103, 44904, 5), (15684, 41990, 546), (15699, 23046, 2), (15699, 41985, 3628), (15759, 41989, 381), (19709, 34200, 3856), (34559, 34562, 631)

X(45758) =  COMPLEMENT OF X(45757)

Barycentrics    62*a^4-103*(b^2+c^2)*a^2+41*(b^2-c^2)^2 : :
X(45758) = 5*X(2)+7*X(140), 19*X(2)-7*X(547), 11*X(2)+X(548), 17*X(2)+7*X(549), 13*X(2)-7*X(3628), 7*X(2)-X(3850), 9*X(2)+7*X(5054), X(2)-7*X(10124), X(2)+7*X(11539), 2*X(2)+7*X(11540), 11*X(2)+7*X(11812), 2*X(2)+X(12108), 17*X(2)-5*X(12812), 5*X(2)+X(14891), 5*X(2)-X(14892), 13*X(2)-X(14893), 15*X(2)+X(15689), 15*X(2)-7*X(15699), 13*X(2)+3*X(15706), 19*X(2)+5*X(15712), 4*X(2)-7*X(16239), 9*X(2)-X(23046), 17*X(2)-X(38335), 3*X(2)+X(41983), 5*X(2)-7*X(41984), 9*X(2)-7*X(41985), 20*X(2)+7*X(44580)

As a point on the Euler line, X(45758) has Shinagawa coefficients (103, -21).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45758) lies on these lines: {2, 3}, {42419, 42978}, {42420, 42979}, {42498, 42973}, {42499, 42972}, {42594, 43416}, {42595, 43417}, {43100, 43103}, {43102, 43107}

X(45758) = midpoint of X(i) and X(j) for these {i, j}: {2, 14890}, {140, 41984}, {3524, 10109}, {3530, 5055}, {3860, 41982}, {5054, 41985}, {10124, 11539}, {14891, 14892}, {15759, 38071}
X(45758) = reflection of X(i) in X(j) for these (i, j): (11540, 11539), (12108, 14890), (41986, 15699)
X(45758) = complement of X(45757)
X(45758) = complement of the complement of X(41983)
X(45758) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14893, 3628), (2, 15702, 1657), (2, 15712, 547), (3530, 3628, 3858), (3545, 10304, 382), (8703, 15694, 140), (8703, 15699, 3545), (10109, 14891, 3627), (10124, 11812, 632), (12101, 15721, 3530), (12108, 35018, 548), (12812, 21735, 3861), (15686, 19708, 548), (15699, 17504, 381), (15713, 41991, 549)

X(45759) =  COMPLEMENT OF X(38335)

Barycentrics    20*a^4-19*(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(45759) = X(2)-7*X(3), 19*X(2)-7*X(4), 10*X(2)-7*X(5), 17*X(2)+7*X(20), 11*X(2)-14*X(140), 5*X(2)+7*X(376), 13*X(2)-7*X(381), 17*X(2)-14*X(547), X(2)+2*X(548), 4*X(2)-7*X(549), 8*X(2)+7*X(550), 5*X(2)+X(1657), 3*X(2)-7*X(3524), 11*X(2)+7*X(3534), 11*X(2)-7*X(3545), 4*X(2)-X(3627), 15*X(2)-7*X(3839), 11*X(2)-5*X(3843), 16*X(2)-7*X(3845), 7*X(2)-4*X(3850), 5*X(2)-7*X(5054), 9*X(2)-7*X(5055), 17*X(2)-11*X(5072), 2*X(2)+7*X(8703), X(2)+7*X(10304), 6*X(2)-7*X(11539), 5*X(2)-14*X(12100), 5*X(2)-8*X(12108), 13*X(2)-10*X(12812), X(2)+5*X(14093), 17*X(2)-7*X(14269), 3*X(2)-4*X(14890), X(2)-4*X(14891), 5*X(2)-2*X(14893), 7*X(2)-X(15684), 2*X(2)+X(15686), 3*X(2)+7*X(15688), 13*X(2)+14*X(15690), 19*X(2)+14*X(15691), 8*X(2)-7*X(15699), X(2)-3*X(15706), 2*X(2)-5*X(15712), 5*X(2)-11*X(15718), 2*X(2)-7*X(17504), 7*X(2)+5*X(17538), 20*X(2)+7*X(19710), X(2)+11*X(21735), 13*X(2)-X(33703), X(2)+14*X(34200), 5*X(2)+11*X(35418), 12*X(2)-7*X(38071), 3*X(2)+14*X(41982)

As a point on the Euler line, X(45759) has Shinagawa coefficients (19, -21).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45759) lies on these lines: {2, 3}, {13, 43631}, {14, 43630}, {61, 42792}, {62, 42791}, {395, 42923}, {396, 42922}, {397, 42631}, {398, 42632}, {515, 38081}, {516, 38022}, {1353, 14810}, {1483, 31663}, {3098, 32455}, {3579, 3635}, {3625, 34773}, {3633, 3655}, {3653, 28174}, {3654, 16192}, {4691, 37705}, {4745, 31447}, {5122, 15935}, {5204, 15170}, {5210, 7739}, {5304, 15603}, {5306, 15513}, {5351, 42436}, {5352, 42435}, {5418, 42639}, {5420, 42640}, {5642, 14677}, {5655, 15036}, {6144, 11179}, {6431, 43526}, {6432, 43525}, {6453, 42524}, {6454, 42525}, {6455, 19054}, {6456, 19053}, {6560, 43211}, {6561, 43212}, {6564, 42566}, {6565, 42567}, {7280, 10386}, {7583, 42568}, {7584, 42569}, {7691, 20585}, {7799, 14929}, {7865, 32459}, {8588, 15048}, {8589, 18907}, {9300, 15515}, {9681, 43802}, {10164, 28208}, {10165, 28202}, {10283, 17502}, {10575, 11592}, {10645, 42928}, {10646, 42929}, {11480, 42633}, {11481, 42634}, {11542, 42625}, {11543, 42626}, {11645, 21167}, {11694, 15051}, {12820, 43240}, {12821, 43241}, {13391, 16226}, {13392, 15042}, {13482, 37471}, {15072, 44324}, {16267, 42118}, {16268, 42117}, {16962, 42943}, {16963, 42942}, {19106, 42500}, {19107, 42501}, {19116, 41945}, {19117, 41946}, {19875, 28186}, {19883, 28146}, {19924, 38110}, {20053, 34718}, {22266, 34648}, {26614, 38229}, {28160, 38068}, {28164, 38083}, {28168, 38076}, {28178, 38021}, {28198, 38028}, {28204, 38127}, {28212, 38314}, {28224, 38066}, {29181, 38079}, {32516, 33706}, {33416, 43402}, {33417, 43401}, {33606, 42890}, {33607, 42891}, {33750, 34380}, {33751, 39884}, {36836, 42510}, {36843, 42511}, {36967, 42121}, {36968, 42124}, {37712, 38112}, {37832, 42145}, {37835, 42144}, {38742, 41134}, {40693, 43109}, {40694, 43108}, {41107, 42945}, {41108, 42944}, {41119, 42490}, {41120, 42491}, {41121, 43491}, {41122, 43492}, {42087, 42972}, {42088, 42973}, {42090, 43417}, {42091, 43416}, {42107, 42430}, {42110, 42429}, {42119, 42497}, {42120, 42496}, {42122, 42917}, {42123, 42916}, {42258, 42573}, {42259, 42572}, {42263, 43255}, {42264, 43254}, {42488, 43246}, {42489, 43247}, {42492, 43104}, {42493, 43101}, {42686, 43500}, {42687, 43499}, {42793, 42991}, {42794, 42990}, {42956, 43025}, {42957, 43024}, {42968, 43777}, {42969, 43778}, {42970, 43032}, {42971, 43033}, {42988, 43635}, {42989, 43634}, {43230, 43467}, {43231, 43468}, {43244, 43483}, {43245, 43484}, {43291, 44541}, {43481, 43869}, {43482, 43870}, {43550, 43633}, {43551, 43632}

X(45759) = midpoint of X(i) and X(j) for these {i, j}: {2, 15689}, {3, 10304}, {20, 14269}, {376, 5054}, {548, 41983}, {550, 15699}, {3524, 15688}, {3534, 3545}, {8703, 17504}, {15686, 23046}, {15718, 35418}, {38742, 41134}
X(45759) = reflection of X(i) in X(j) for these (i, j): (2, 41983), (5, 5054), (546, 41985), (549, 17504), (3543, 41987), (3545, 140), (3627, 23046), (3845, 15699), (5054, 12100), (8703, 10304), (10304, 34200), (11539, 3524), (14269, 547), (14892, 14890), (15686, 15689), (15687, 3545), (15688, 41982), (15689, 548), (15699, 549), (17504, 3), (23046, 2), (23047, 7509), (33699, 14269), (38071, 11539), (38229, 26614), (38335, 14892), (41983, 14891), (41985, 44580), (41987, 3628), (43893, 37907)
X(45759) = complement of X(38335)
X(45759) = anticomplement of X(14892)
X(45759) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 14893, 5), (3, 3528, 140), (3, 14093, 2), (3, 33923, 5), (5, 8703, 376), (140, 3146, 5), (376, 12100, 5), (376, 41106, 20), (547, 41106, 5), (548, 14891, 2), (549, 3627, 2), (549, 19710, 5), (1657, 12108, 5), (1657, 15718, 2), (3523, 12103, 5), (3534, 15692, 140), (3545, 15707, 140), (3830, 10124, 5), (3860, 15703, 5), (8703, 15714, 3), (10304, 15710, 3), (11812, 15691, 4), (12108, 14893, 2), (14890, 14892, 2), (15686, 15712, 2), (15687, 19711, 140), (15689, 15706, 2), (15695, 15700, 4), (15705, 35418, 2), (15759, 34200, 3)

X(45760) =  COMPLEMENT OF X(12812)

Barycentrics    18*a^4-29*(b^2+c^2)*a^2+11*(b^2-c^2)^2 : :
X(45760) = 3*X(2)+7*X(140), 17*X(2)-7*X(547), 9*X(2)+X(548), 13*X(2)+7*X(549), 9*X(2)+7*X(631), 3*X(2)-7*X(632), 15*X(2)-7*X(1656), 18*X(2)+7*X(3530), 12*X(2)-7*X(3628), 9*X(2)-X(3843), 6*X(2)-X(3850), 2*X(2)-7*X(10124), X(2)+14*X(11540), 8*X(2)+7*X(11812), 3*X(2)+2*X(12108), 7*X(2)+X(14093), 2*X(2)+3*X(14890), 4*X(2)+X(14891), 13*X(2)-3*X(14892), 11*X(2)-X(14893), 19*X(2)+X(15686), 17*X(2)+7*X(15693), X(2)+7*X(15694), 3*X(2)+X(15712), 5*X(2)+7*X(15713), 9*X(2)-14*X(16239), 15*X(2)+X(17538), 7*X(2)+3*X(41983)

As a point on the Euler line, X(45760) has Shinagawa coefficients (29, -7).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45760) lies on these lines: {2, 3}, {395, 42802}, {396, 42801}, {3054, 31457}, {3625, 11231}, {3633, 38028}, {5346, 9606}, {6144, 38110}, {6433, 43341}, {6434, 43340}, {6499, 31487}, {6688, 11592}, {7294, 15172}, {9680, 42601}, {9698, 34571}, {10165, 22266}, {11542, 43873}, {11543, 43874}, {12045, 12046}, {13392, 20379}, {13630, 15082}, {14075, 31455}, {14531, 44324}, {15606, 16881}, {16772, 16961}, {16773, 16960}, {19862, 28212}, {19878, 28232}, {20053, 38112}, {22236, 42513}, {22238, 42512}, {28186, 31253}, {28224, 31399}, {32455, 40107}, {32889, 34229}, {33416, 42435}, {33417, 42436}, {35814, 42566}, {35815, 42567}, {42122, 43644}, {42123, 43649}, {42415, 42687}, {42416, 42686}, {42433, 42683}, {42434, 42682}, {42496, 42936}, {42497, 42937}, {42500, 42925}, {42501, 42924}, {42516, 42989}, {42517, 42988}, {42520, 43107}, {42521, 43100}, {42580, 43492}, {42581, 43491}, {42590, 42944}, {42591, 42945}, {42777, 42990}, {42778, 42991}, {42813, 42928}, {42814, 42929}, {42912, 42948}, {42913, 42949}, {42938, 42955}, {42939, 42954}

X(45760) = midpoint of X(i) and X(j) for these {i, j}: midpoint of X(i) and X(j) for these {i, j}: {140, 632}, {546, 3522}, {547, 15693}, {548, 3843}, {5066, 15714}, {5071, 12100}, {12812, 15712}
X(45760) = reflection of X(i) in X(j) for these (i, j): (3091, 35018), (3530, 631), (3850, 12812), (12102, 3858), (15694, 11540), (15712, 12108), (41989, 1656)
X(45760) = complement of X(12812)
X(45760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 3850, 3628), (2, 17538, 1656), (5, 549, 3528), (5, 17800, 546), (140, 10124, 3628), (140, 16239, 3530), (548, 3853, 1657), (548, 12108, 3530), (631, 3526, 632), (631, 5067, 3522), (632, 15694, 140), (632, 15712, 2), (632, 15713, 1656), (1656, 15695, 3091), (1656, 17578, 5), (3526, 5070, 3533), (3526, 15694, 631), (3528, 5070, 5), (3530, 3850, 548), (3530, 16239, 3628), (3843, 15712, 548), (3861, 11812, 3530), (5054, 15682, 549), (5067, 17800, 5), (5067, 44682, 546), (10124, 14890, 2), (14869, 15709, 140), (15689, 15723, 2), (33923, 41984, 3628), (35018, 41985, 3628)

X(45761) =  EULER LINE INTERCEPT OF X(42496)X(42685)

Barycentrics    142*a^4-(155*(b^2+c^2))*a^2+13*(b^2-c^2)^2 : :
X(45761) = 15*X(3)+13*X(549), 11*X(3)+13*X(3523), 8*X(3)+13*X(3530), 5*X(3)+2*X(11540), 6*X(3)+X(11737), X(3)+13*X(12100), 13*X(3)+X(12101), 6*X(3)-13*X(14891), 5*X(3)-13*X(15698), 3*X(3)+13*X(15700), 19*X(3)+13*X(15701), 20*X(3)-13*X(15759), 7*X(3)+13*X(19711), 7*X(3)+X(41106), X(3)-13*X(44682), 5*X(3)+X(44904), X(4)-15*X(41983), 15*X(140)-X(15684), 3*X(140)+11*X(15715), 19*X(140)-5*X(41099)

As a point on the Euler line, X(45761) has Shinagawa coefficients (155, -129).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45761) lies on these lines: {2, 3}, {42496, 42685}, {42497, 42684}, {42928, 43000}, {42929, 43001}

X(45761) = midpoint of X(i) and X(j) for these {i, j}: {3851, 15690}, {12100, 44682}, {15702, 34200}
X(45761) = reflection of X(i) in X(j) for these (i, j): (11812, 3523), (14869, 44580), (44904, 11540)
X(45761) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (549, 10304, 547), (549, 15684, 140), (549, 15714, 4), (632, 15686, 381), (10304, 15701, 3857), (10304, 19710, 548), (11540, 23046, 3628), (11737, 14891, 3), (15683, 15718, 549), (15698, 15700, 549), (15700, 15703, 3524), (15712, 44245, 3530)

X(45762) =  EULER LINE INTERCEPT OF X(35786)X(43434)

Barycentrics    58*a^4+13*(b^2+c^2)*a^2-71*(b^2-c^2)^2 : :
X(45762) = 6*X(3)+X(35408), 5*X(3)-12*X(41986), 15*X(4)+13*X(381), 8*X(4)+13*X(546), 11*X(4)+13*X(3832), X(4)+13*X(3845), 17*X(4)+13*X(3857), 5*X(4)+2*X(10109), 11*X(4)+3*X(11539), 20*X(4)-13*X(12101), 17*X(4)+4*X(12108), 6*X(4)-13*X(14893), 3*X(4)+X(15703), 13*X(4)+X(19710), 6*X(4)+X(34200), 19*X(4)+13*X(41106), 5*X(4)+X(44682), 19*X(5)-5*X(15697), 9*X(5)+5*X(35434), 3*X(20)-10*X(10124), X(20)-15*X(23046), 8*X(20)-15*X(41982)

As a point on the Euler line, X(45762) has Shinagawa coefficients (13, 129).

See Antreas Hatzipolakis and César Lozada, Euclid 2983 .

X(45762) lies on these lines: {2, 3}, {35786, 43434}, {35787, 43435}, {42692, 43781}, {42693, 43782}, {42898, 43776}, {42899, 43775}

X(45762) = midpoint of X(i) and X(j) for these {i, j}: {3528, 33699}, {3627, 15701}
X(45762) = reflection of X(i) in X(j) for these (i, j): (3851, 3860), (5066, 3832), (12100, 44904), (12103, 15698), (15690, 14869), (15702, 11737), (34200, 15703), (44682, 10109)
X(45762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 12108, 3853), (376, 5066, 547), (381, 5073, 5071), (381, 35403, 20), (546, 15691, 381), (1656, 3832, 3857), (1656, 15684, 376), (3544, 15717, 1656), (3856, 15706, 5066), (10109, 15689, 140), (12101, 41987, 546), (14893, 34200, 4), (14893, 41987, 381)

X(45763) = X(1)X(2)∩X(106)X(518)

Barycentrics    a*( a^3-2*(b+c)*a^2-(2*b^2-3*b*c+2*c^2)*a+(b+c)*(b^2+c^2)) : :
X(45763) = X(1)+3*X(5529), X(5524)-3*X(5529)

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45763) lies on these lines: {1, 2}, {6, 35272}, {58, 17614}, {101, 2718}, {106, 518}, {214, 238}, {244, 4867}, {392, 4256}, {501, 13624}, {517, 38671}, {595, 37589}, {741, 2759}, {748, 37525}, {758, 18201}, {759, 2705}, {849, 16948}, {902, 15015}, {960, 37599}, {2712, 8691}, {2726, 29067}, {2758, 29055}, {3576, 7609}, {3667, 3737}, {3685, 34587}, {3722, 16489}, {3927, 8572}, {3987, 5330}, {4257, 35262}, {4262, 39254}, {4561, 17160}, {5181, 34586}, {5440, 40091}, {5692, 36263}, {5730, 24046}, {5904, 32577}, {7312, 17770}, {10176, 37617}, {10246, 37679}, {16173, 33136}, {32912, 37587}, {34123, 35466}

X(45763) = midpoint of X(i) and X(j) for these {i, j}: {1, 5524}, {5211, 6790}
X(45763) = reflection of X(i) in X(j) for these (i, j): (5205, 6789), (6788, 5121)
X(45763) = X(21)-Beth conjugate of-X(7292)
X(45763) = inverse of X(29639) in orthoptic circle of Steiner inellipse
X(45763) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(2718)}} and {{A, B, C, X(104), X(37762)}}
X(45763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 5529, 5524), (1, 7292, 30117), (995, 997, 30115), (4511, 7292, 1)

X(45764) = X(1)X(5)∩X(21)X(214)

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^3-(b^4+c^4+b*c*(4*b^2-7*b*c+4*c^2))*a^2-2*(b^3+c^3)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)) : :
X(45764) = 3*X(15015)-2*X(17100), 3*X(37718)-4*X(39692)

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45764) lies on these lines: {1, 5}, {21, 214}, {35, 17638}, {36, 1749}, {78, 5541}, {100, 3878}, {104, 3467}, {149, 22836}, {191, 4996}, {484, 2800}, {499, 9803}, {900, 3737}, {960, 14794}, {997, 5010}, {1125, 39778}, {1389, 13143}, {1768, 6261}, {2801, 7677}, {3336, 10090}, {3337, 11570}, {3583, 4511}, {3585, 21635}, {3746, 41541}, {3811, 12653}, {4299, 9809}, {5046, 6224}, {5538, 5840}, {5563, 17660}, {5697, 12331}, {5887, 14792}, {5903, 37251}, {5904, 22560}, {6667, 38410}, {6702, 7504}, {6763, 12532}, {6839, 18393}, {6949, 12247}, {6979, 10573}, {7489, 37525}, {7491, 12119}, {10265, 21740}, {10483, 16128}, {10609, 13146}, {10698, 11280}, {11009, 17636}, {11014, 19914}, {12611, 37230}, {12758, 37563}, {12773, 21842}, {13624, 33856}, {14800, 16767}, {16132, 38602}, {16143, 38761}, {21630, 34772}, {34789, 37468}

X(45764) = reflection of X(1768) in X(18861)
X(45764) = X(21)-Beth conjugate of-X(38458)
X(45764) = inverse of X(12433) in incircle
X(45764) = inverse of X(37700) in hexyl circle
X(45764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 6326, 41689), (11, 1317, 12433), (80, 5443, 11), (80, 6265, 1), (80, 7972, 10950), (960, 33598, 35204), (6264, 6326, 37700), (7972, 12740, 1), (10090, 11571, 3336), (12738, 12740, 7972), (12739, 16173, 1), (17638, 22935, 35), (37733, 37735, 1)

X(45765) = X(1)X(6)∩X(105)X(519)

Barycentrics    a*(a^4-(b+c)*a^3+(b^2+b*c+c^2)*a^2-(b+c)*(b^2-3*b*c+c^2)*a-2*b*c*(b^2+c^2)) : :
X(45765) = 3*X(2725)-2*X(5144)

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45765) lies on these lines: {1, 6}, {21, 6629}, {36, 2482}, {55, 21509}, {100, 2725}, {105, 519}, {516, 38684}, {674, 3908}, {759, 2753}, {840, 29351}, {1019, 3309}, {1486, 17296}, {1621, 29574}, {2691, 26702}, {2751, 14074}, {2752, 8691}, {2754, 26700}, {3220, 4966}, {3674, 16133}, {3685, 5088}, {4378, 30605}, {4413, 17284}, {4511, 11712}, {4869, 24309}, {5773, 29824}, {16686, 17374}, {17294, 26241}, {25495, 38473}, {25514, 35613}, {27169, 27248}, {31138, 38530}

X(45765) = midpoint of X(1) and X(5525)
X(45765) = reflection of X(i) in X(j) for these (i, j): (5526, 1083), (8692, 3973)
X(45765) = X(21)-Beth conjugate of-X(16784)
X(45765) = inverse of X(10477) in Conway circle
X(45765) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(2725)}} and {{A, B, C, X(105), X(16784)}}
X(45765) = {X(1), X(238)}-harmonic conjugate of X(16784)

X(45766) = X(1)X(4)∩X(3)X(318)

Barycentrics    (a^6-(2*b^2+b*c+2*c^2)*a^4+2*(b+c)*b*c*a^3+(b^2-c^2)^2*a^2-2*(b^2-c^2)*(b-c)*b*c*a+(b^2-c^2)^2*b*c)*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45766) lies on these lines: {1, 4}, {3, 318}, {24, 1324}, {29, 33597}, {30, 21664}, {40, 42456}, {104, 1309}, {108, 2734}, {125, 860}, {158, 11500}, {208, 1148}, {228, 7412}, {355, 17555}, {406, 10786}, {412, 1071}, {475, 10785}, {517, 1897}, {915, 2731}, {917, 2730}, {952, 5081}, {1319, 36123}, {1385, 11109}, {1593, 35455}, {1784, 44425}, {1871, 37390}, {1887, 12675}, {1895, 3149}, {1940, 6796}, {2222, 32706}, {2695, 30250}, {2716, 32704}, {2723, 26706}, {2733, 26704}, {5088, 18026}, {5657, 7046}, {6906, 38870}, {6927, 40836}, {6934, 14257}, {9940, 37278}, {14004, 39529}, {30273, 33971}, {32613, 37295}, {35013, 44428}

X(45766) = midpoint of X(1897) and X(37420)
X(45766) = reflection of X(i) in X(j) for these (i, j): (4, 1785), (10538, 3), (45272, 31866)
X(45766) = crosssum of X(1364) and X(8677)
X(45766) = inverse of X(946) in polar circle
X(45766) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(2734)}} and {{A, B, C, X(34), X(8884)}}
X(45766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 7967, 34231), (4, 18283, 944), (7046, 37410, 5657)

X(45767) = X(1)X(5)∩X(104)X(26713)

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-(6*b^2-b*c+6*c^2)*b*c*a^6-(b+c)*(3*b^4+3*c^4-(7*b^2-3*b*c+7*c^2)*b*c)*a^5+(2*b^6+2*c^6+(5*b^4+5*c^4-(7*b^2+4*b*c+7*c^2)*b*c)*b*c)*a^4+(b^2-c^2)*(b-c)*(b^4+c^4-(6*b^2+7*b*c+6*c^2)*b*c)*a^3-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2*a^2+3*(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^4*b*c : :

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45767) lies on these lines: {1, 5}, {104, 26713}, {7488, 17100}, {18861, 35921}, {34864, 35451}

X(45768) = X(5)X(6)∩X(2070)X(2936)

Barycentrics    a^14-2*(b^2+c^2)*a^12+(b^4-b^2*c^2+c^4)*a^10+(b^2+c^2)*b^2*c^2*a^8-(b^8+c^8-(7*b^4+4*b^2*c^2+7*c^4)*b^2*c^2)*a^6+(b^2+c^2)*(2*b^8+2*c^8-3*(5*b^4-6*b^2*c^2+5*c^4)*b^2*c^2)*a^4-(b^2-c^2)^2*(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2-2*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2 : :

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45768) lies on these lines: {5, 6}, {2070, 2936}, {5866, 7488}, {34864, 35463}, {40118, 44061}

X(45768) = {X(5), X(41205)}-harmonic conjugate of X(38463)

X(45769) = X(5)X(6)∩X(187)X(399)

Barycentrics    a^2*(a^8-4*(b^2+c^2)*a^6+(6*b^4+5*b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(4*b^4-3*b^2*c^2+4*c^4)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(45769) = 8*X(9721)-9*X(15538)

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985.

X(45769) lies on these lines: {5, 6}, {187, 399}, {195, 39565}, {323, 13580}, {353, 5866}, {394, 44526}, {755, 10425}, {1181, 44535}, {2079, 13754}, {3053, 11441}, {3054, 15032}, {3288, 3566}, {5023, 32139}, {5210, 11456}, {5562, 44525}, {5609, 11063}, {5891, 9604}, {9697, 34864}, {14901, 22121}, {15038, 39601}, {15067, 15109}, {18436, 44523}, {18445, 37637}, {22115, 34866}, {23039, 44521}

X(45769) = crossdifference of every pair of points on line {X(924), X(5943)}

X(45770) = X(1)X(5)∩X(3)X(960)

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b+c)*(2*b^2-3*b*c+2*c^2)*a^3-(b^4+c^4+2*(2*b^2-3*b*c+2*c^2)*b*c)*a^2-2*(b^3+c^3)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)) : :
X(45770) = 2*X(496)-3*X(5886), 3*X(3576)-X(10085), 3*X(5587)-X(37711), 2*X(32612)-3*X(35262)

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985 and euclid 2989 .

X(45770) lies on these lines: {1, 5}, {2, 21740}, {3, 960}, {4, 4511}, {8, 6834}, {9, 12687}, {10, 6863}, {30, 12679}, {36, 5693}, {40, 3899}, {46, 6924}, {56, 912}, {63, 5694}, {65, 6911}, {72, 11249}, {78, 517}, {104, 12528}, {145, 6953}, {214, 5450}, {224, 1012}, {382, 35459}, {392, 10267}, {405, 1385}, {474, 34339}, {515, 6928}, {518, 10680}, {758, 37532}, {908, 10526}, {936, 26446}, {944, 2478}, {946, 12437}, {993, 20117}, {999, 44547}, {1001, 24299}, {1060, 10571}, {1062, 18339}, {1071, 10269}, {1125, 6861}, {1259, 41389}, {1319, 14872}, {1376, 37562}, {1420, 1728}, {1455, 8757}, {1482, 3811}, {1490, 18481}, {1519, 10525}, {1737, 6959}, {1788, 6970}, {1836, 41688}, {1858, 22766}, {1898, 2646}, {2771, 32612}, {2800, 25440}, {2801, 15297}, {2900, 7956}, {2932, 4855}, {3306, 5885}, {3338, 24475}, {3359, 5438}, {3428, 31837}, {3485, 6826}, {3486, 6893}, {3576, 7330}, {3612, 6914}, {3616, 6832}, {3622, 6886}, {3652, 5428}, {3679, 11014}, {3869, 6905}, {3870, 10222}, {3877, 11491}, {3878, 6796}, {3885, 38665}, {3895, 10284}, {3913, 23340}, {3940, 22770}, {4295, 6885}, {4305, 6930}, {4420, 12245}, {4679, 34773}, {4867, 37625}, {5086, 6941}, {5087, 18525}, {5119, 32141}, {5250, 32613}, {5289, 11500}, {5440, 11248}, {5538, 41869}, {5603, 6835}, {5657, 6962}, {5692, 11012}, {5704, 18467}, {5731, 5811}, {5770, 7288}, {5778, 40937}, {5779, 15254}, {5780, 9708}, {5790, 31493}, {5794, 6842}, {5817, 30284}, {5818, 6933}, {5843, 41707}, {5884, 9946}, {5927, 18761}, {6583, 11520}, {6675, 8583}, {6824, 45230}, {6881, 28628}, {6882, 25681}, {6883, 25917}, {6896, 10595}, {6898, 7967}, {6913, 10393}, {6917, 12047}, {6923, 12608}, {6929, 10572}, {6934, 11415}, {6944, 18391}, {6958, 12616}, {6961, 14647}, {6985, 14110}, {7491, 24703}, {7681, 44669}, {7701, 34600}, {8200, 26351}, {8207, 26352}, {9947, 15178}, {9956, 19860}, {9961, 37403}, {10175, 30147}, {10202, 25524}, {10395, 44675}, {10698, 14923}, {10884, 13624}, {11496, 33596}, {11928, 22835}, {12001, 34791}, {12114, 40263}, {12247, 25005}, {12635, 22753}, {12645, 33956}, {12650, 18528}, {12651, 12699}, {12675, 16203}, {15071, 37561}, {15325, 34489}, {15696, 43178}, {15704, 41860}, {16617, 33857}, {17556, 28204}, {19540, 30986}, {22791, 37569}, {22837, 24386}, {24987, 26487}, {28452, 34647}, {31806, 37584}, {31835, 41229}, {32153, 37618}, {33862, 35258}, {37281, 39542}

X(45770) = midpoint of X(i) and X(j) for these {i, j}: {1, 17857}, {3149, 5730}, {6934, 11415}
X(45770) = reflection of X(i) in X(j) for these (i, j): (46, 6924), (1837, 5), (2932, 22935), (6928, 21616), (20586, 19907), (32214, 5901)
X(45770) = X(1837)-of-Johnson-triangle
X(45770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 5534, 37727), (1, 5720, 355), (1, 6326, 37700), (3, 40266, 1158), (5, 37737, 5886), (36, 5693, 24467), (214, 31803, 5450), (355, 5886, 26470), (355, 6265, 1), (392, 33597, 10267), (946, 22836, 37533), (960, 37837, 3), (997, 6261, 3), (1071, 17614, 10269), (1385, 5777, 22758), (1490, 37611, 18481), (5438, 7971, 3359), (5440, 12672, 11248), (5587, 37692, 5), (5692, 11012, 26921), (5694, 26286, 63), (5886, 37733, 1), (10943, 11729, 11376), (11376, 12739, 1), (12608, 17647, 6923), (12635, 22753, 24474), (12738, 37727, 5534), (12740, 37738, 1), (18446, 19861, 1385), (22935, 26285, 4855)

X(45771) =  EULER LINE INTERCEPT OF X(1147)X(32601)

Barycentrics    15*a^10-27*(b^2+c^2)*a^8-6*((b^2+c^2)^2-16*b^2*c^2)*a^6+2*(b^2+c^2)*(15*b^4-38*b^2*c^2+15*c^4)*a^4-(b^2-c^2)^2*(9*b^4+38*b^2*c^2+9*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2)^3 : :

As a point on the Euler line, X(45771) has Shinagawa coefficients (E-6*F, -2*E+9*F).

See Jayendra Jha, Sankalp Savaran and César Lozada, euclid 2985 and euclid 2989 .

X(45771) lies on these lines: {2, 3}, {1147, 32601}, {6000, 25712}, {12324, 16163}, {15311, 45248}, {32603, 34382}

X(45771) = reflection of X(25712) in X(27082)
X(45771) = {X(3522), X(34005)}-harmonic conjugate of X(3538)

X(45772) =  X(20)X(2793)∩X(69)X(74)

Barycentrics    (a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)*(-3*a^12 + 8*a^10*b^2 - 8*a^8*b^4 + 5*a^6*b^6 - 2*a^4*b^8 - a^2*b^10 + b^12 + 8*a^10*c^2 - 18*a^8*b^2*c^2 + 13*a^6*b^4*c^2 - 6*a^4*b^6*c^2 + 6*a^2*b^8*c^2 - 3*b^10*c^2 - 8*a^8*c^4 + 13*a^6*b^2*c^4 - 3*a^4*b^4*c^4 - 3*a^2*b^6*c^4 + 2*b^8*c^4 + 5*a^6*c^6 - 6*a^4*b^2*c^6 - 3*a^2*b^4*c^6 - 2*a^4*c^8 + 6*a^2*b^2*c^8 + 2*b^4*c^8 - a^2*c^10 - 3*b^2*c^10 + c^12) : :

The osculating circle of the Steiner circumellipse at X(99) has squared radius (a^6 - a^4*b^2 - a^2*b^4 + b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6)^3/(4*(a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)^3*S^2) = (-9*SA*SB*SC + S^2*SW)^3/(4*S^2*(-3*S^2 + SW^2)^3) and passes through X(99) and X(892). Its center is X(45772)

See Randy Hutson and Peter Moses, euclid 2988.

X(45772) lies on these lines: {20, 2793}, {30, 9182}, {69, 74}, {325, 36166}, {3268, 38940}, {6054, 7664}, {6055, 30786}, {11177, 14360}, {16093, 23698}

X(45772) = {X(616),X(617)}-harmonic conjugate of X(11006)

X(45773) = ISOGONAL CONJUGATE OF X(33919)

Barycentrics    a^2/((b^2 - c^2)^3 (2 a^2 - b^2 - c^2)) : :

X(45773) is the circumcircle intercept, other than X(99), of the osculating circle of the Steiner circumellipse at X(99), centered at X(45772). (Randy Hutson, November 5, 2021)

X(45773) lies on the circumcircle, the osculating circle of the Steiner circumellipse at X(99), and these lines: {98, 30786}, {99, 11123}, {111, 34539}, {249, 843}, {250, 40119}, {476, 892}, {805, 32729}, {1304, 42743}, {2770, 4590}, {2868, 34537}, {5468, 20404}, {5970, 23357}, {9150, 42370}, {18020, 40118}, {32583, 32694}

X(45773) = isogonal conjugate of X(33919)
X(45773) = trilinear pole of line X(6)X(249)
X(45773) = Ψ(X(6), X(249))
X(45773) = Ψ(X(351), X(110))
X(45773) = Λ(X(1648), X(8029)) (line X(1648)X(8029) is the trilinear polar of X(115))
X(45773) = Λ(X(2682), X(14443)) (line X(2682)X(14443) is the trilinear polar of X(1648))
X(45773) = trilinear quotient X(661)/X(42344)

X(45774) = REFLECTION OF X(99) IN X(45772)

Barycentrics    5 a^12 - 12 a^10 (b^2 + c^2) + a^8 (9 b^4 + 28 b^2 c^2 + 9 c^4) - a^6 (b^2 + c^2) (3 b^4 + 16 b^2 c^2 + 3 c^4) + 3 a^4 (3 b^6 c^2 + b^4 c^4 + 3 b^2 c^6) + a^2 (b^2 + c^2) (3 b^8 - 14 b^6 c^2 + 20 b^4 c^4 - 14 b^2 c^6 + 3 c^8) - (b^2 - c^2)^2 (2 b^8 - b^6 c^2 - 3 b^4 c^4 - b^2 c^6 + 2 c^8) : :

X(45774) is the antipode of X(99) in the osculating circle of the Steiner circumellipse at X(99), centered at X(45772). (Randy Hutson, November 5, 2021)

X(45774) lies on the osculating circle of the Steiner circumellipse at X(99), and these lines: {30, 892}, {69, 74}, {98, 30786}, {147, 7664}, {671, 2407}, {1361, 5334}, {20159, 37227}, {32146, 39580}

X(45774) = reflection of X(99) in X(45772)

X(45775) = X(896)X(2644)∩X(2642)X(2643)

Barycentrics    a (b^2 - c^2)^4 (2 a^2 - b^2 - c^2) : :
Trilinears    A-power of osculating circle of Steiner circumellipse at X(99) : : (See note below,)

The trilinears shown indicate that the signed distances of X(45775) from the sidelines BC, CA, AB, are respectively proportional to the A-, B-, C- powers; the actual A-power is

(b^2 - c^2)^4*(2*a^2 - b^2 - c^2)/SF, where SF is the symmetric function (a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4)^2,

and likewise for the B- and C- powers. (Randy Hutson, November 5, 2021).

X(45775) lies on these lines: {896, 2644}, {2642, 2643}

X(45775) = barycentric product X(1)*X(42344)

X(45776) = X(1)X(84)∩X(5)X(10)

Barycentrics    a*((b+c)*a^5-(b^2+4*b*c+c^2)*a^4-2*(b+c)*(b^2-4*b*c+c^2)*a^3+2*(b^2+4*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-6*b*c+c^2)*a-(b^4-c^4)*(b^2-c^2)) : :
X(45776) = 3*X(1)-X(1071), 5*X(1)-X(15071), X(20)-5*X(3890), X(40)-3*X(392), X(65)-3*X(5603), 3*X(946)-2*X(5806), X(960)+4*X(26200), 3*X(960)-2*X(31837), X(1071)+3*X(12672), 2*X(1071)-3*X(12675), 5*X(1071)-3*X(15071), 3*X(3740)-2*X(5690), 3*X(5603)-2*X(13374), 4*X(5806)-3*X(7686), 2*X(12672)+X(12675), 5*X(12672)+X(15071), 5*X(12675)-2*X(15071), 3*X(12705)-X(17650), 3*X(12709)+X(17650), 6*X(26200)+X(31837)

See Antreas Hatzipolakis and César Lozada, euclid 3004.

X(45776) lies on these lines: {1, 84}, {3, 22754}, {4, 1000}, {5, 10}, {8, 6957}, {12, 1519}, {20, 3890}, {40, 392}, {55, 37837}, {65, 3086}, {72, 4853}, {104, 15179}, {145, 14872}, {210, 12245}, {354, 10595}, {355, 3880}, {377, 962}, {404, 13528}, {495, 12608}, {496, 12616}, {515, 9856}, {516, 3884}, {518, 1351}, {519, 5777}, {551, 9940}, {912, 10222}, {942, 1387}, {944, 5919}, {952, 31937}, {971, 5882}, {997, 10306}, {999, 1158}, {1058, 10866}, {1064, 37548}, {1125, 31788}, {1193, 42078}, {1210, 13601}, {1319, 6906}, {1385, 8717}, {1490, 31393}, {1532, 10039}, {1537, 12047}, {1697, 11500}, {1699, 5697}, {1836, 10532}, {1837, 10531}, {1858, 10698}, {1883, 1902}, {1898, 37740}, {1953, 9119}, {2077, 17614}, {2099, 44547}, {2771, 25485}, {2801, 3635}, {2802, 19925}, {2829, 10106}, {3090, 3698}, {3091, 14923}, {3149, 5119}, {3241, 12528}, {3244, 31803}, {3295, 6261}, {3303, 18446}, {3359, 25524}, {3428, 5250}, {3488, 9848}, {3555, 5693}, {3577, 4866}, {3579, 31838}, {3616, 6966}, {3625, 15064}, {3632, 18908}, {3656, 24474}, {3680, 38308}, {3742, 5901}, {3746, 33597}, {3753, 8227}, {3812, 5886}, {3868, 5734}, {3869, 6837}, {3898, 4297}, {3913, 5720}, {3918, 10171}, {4293, 17634}, {4313, 12671}, {4342, 21628}, {4640, 11249}, {4861, 6912}, {5045, 5884}, {5049, 12005}, {5176, 13729}, {5231, 37625}, {5270, 34789}, {5289, 37531}, {5290, 39779}, {5439, 9624}, {5450, 24928}, {5587, 10893}, {5657, 6983}, {5687, 12703}, {5694, 11278}, {5715, 17532}, {5730, 37569}, {5761, 34647}, {5785, 43166}, {5805, 37585}, {5842, 10624}, {5881, 5927}, {5903, 11522}, {5904, 11224}, {6265, 33596}, {6361, 6955}, {6684, 31798}, {6769, 15829}, {6797, 16174}, {6831, 30384}, {6854, 7957}, {6905, 37568}, {6923, 12699}, {6935, 37566}, {6944, 37828}, {6950, 37605}, {7330, 12513}, {7966, 30337}, {7967, 12680}, {7978, 10693}, {9842, 12640}, {9942, 24929}, {9947, 12448}, {10107, 18493}, {10165, 31787}, {10178, 13624}, {10199, 35004}, {10247, 40266}, {10283, 13373}, {10284, 18480}, {10310, 19861}, {10404, 10597}, {10679, 45770}, {10805, 12678}, {11012, 19525}, {11260, 22758}, {11372, 12650}, {12115, 12679}, {12514, 22770}, {12665, 25416}, {12740, 12775}, {12751, 17652}, {13369, 15178}, {13607, 15174}, {14647, 14986}, {15726, 18481}, {15908, 24987}, {16173, 17654}, {17605, 25414}, {17613, 37561}, {18242, 31397}, {18483, 44685}, {18839, 45288}, {20117, 28234}, {21669, 41695}, {21740, 37080}, {28194, 31793}, {28236, 31871}, {31165, 45120}, {37563, 44425}, {37582, 40256}, {37615, 42819}, {37622, 37700}, {37727, 40263}, {38028, 40296}, {41611, 41722}

X(45776) = midpoint of X(i) and X(j) for these {i, j}: {1, 12672}, {4, 3057}, {72, 7982}, {145, 14872}, {355, 23340}, {944, 12688}, {962, 14110}, {1482, 5887}, {1537, 12758}, {3244, 31803}, {3555, 5693}, {3878, 4301}, {5694, 11278}, {5777, 13600}, {7978, 10693}, {9856, 9957}, {10284, 18480}, {10698, 17638}, {12665, 25416}, {12705, 12709}, {12751, 17652}, {37727, 40263}
X(45776) = reflection of X(i) in X(j) for these (i, j): (65, 13374), (942, 13464), (3579, 31838), (5836, 5), (5882, 31792), (5884, 5045), (6797, 16174), (7686, 946), (9942, 40257), (9943, 1385), (11362, 5044), (12675, 1), (13369, 15178), (25413, 10107), (31786, 3884), (31788, 1125), (31798, 6684), (34339, 5901), (34790, 20117), (34791, 10222), (37562, 3812)
X(45776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 12705, 12114), (5, 946, 22835), (10, 946, 7681), (65, 5603, 13374), (946, 11362, 7682), (962, 3877, 14110), (5603, 6833, 11376), (5693, 16200, 3555), (5886, 37562, 3812), (5901, 34339, 3742), (5919, 12688, 944), (9943, 10179, 1385), (11011, 17638, 1858), (12672, 17622, 12114), (12709, 17622, 1)

X(45777) = X(10)X(12915)∩X(942)X(3822)

Barycentrics    a*((b+c)*a^5-(b^2+c^2)*a^4-2*(b+c)*(b^2-5*b*c+c^2)*a^3+2*(b^4+c^4-5*(b^2+c^2)*b*c)*a^2+(b+c)*(b^4+c^4-10*(b^2-b*c+c^2)*b*c)*a-(b^2-10*b*c+c^2)*(b^2-c^2)^2) : :
X(45777) = 7*X(4002)+3*X(34625), 9*X(4731)+X(12629), 4*X(24387)+X(31798)

See Antreas Hatzipolakis and César Lozada, euclid 3004.

X(45777) lies on these lines: {10, 12915}, {518, 31399}, {942, 3822}, {960, 45310}, {4002, 34625}, {4731, 12629}, {9710, 10916}, {10265, 31788}, {24387, 31798}

X(45778) = FOCUS, OTHER THAN X(13), OF SIMMONS CIRCUMCONIC WITH CENTER X(40578)

Barycentrics    2*a^12 + 7*a^10*b^2 - 32*a^8*b^4 + 33*a^6*b^6 - 7*a^4*b^8 - 4*a^2*b^10 + b^12 + 7*a^10*c^2 - 35*a^8*b^2*c^2 + 14*a^6*b^4*c^2 + 17*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 8*b^10*c^2 - 32*a^8*c^4 + 14*a^6*b^2*c^4 - 4*a^4*b^4*c^4 - a^2*b^6*c^4 + 23*b^8*c^4 + 33*a^6*c^6 + 17*a^4*b^2*c^6 - a^2*b^4*c^6 - 32*b^6*c^6 - 7*a^4*c^8 + 5*a^2*b^2*c^8 + 23*b^4*c^8 - 4*a^2*c^10 - 8*b^2*c^10 + c^12 + 2*Sqrt[3]*(4*a^10 - 9*a^8*b^2 + a^6*b^4 + 8*a^4*b^6 - 3*a^2*b^8 - b^10 - 9*a^8*c^2 + a^6*b^2*c^2 + 2*a^4*b^4*c^2 + 3*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 + 2*a^4*b^2*c^4 - 2*b^6*c^4 + 8*a^4*c^6 + 3*a^2*b^2*c^6 - 2*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 - c^10)*S : :
X(45778) = 3 X[13] - 4 X[10217], 3 X[470] - 4 X[6671], X[621] - 3 X[19772], 2 X[10217] - 3 X[40578]

X(45778) lies on the cubic K313 and these lines: {13, 15}, {20, 8919}, {143, 30439}, {376, 36299}, {470, 6671}, {531, 36308}, {618, 19776}, {621, 19772}, {1294, 5995}, {5473, 36788}, {5623, 36839}, {5961, 6105}, {8452, 16163}, {11481, 18777}, {15768, 40158}, {36306, 44700}

X(45778) = reflection of X(i) in X(j) for these {i,j}: {13, 40578}, {19776, 618}
X(45778) = isogonal conjugate of X(13) wrt anticevian triangle of X(13)
X(45778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {15, 11586, 15441}, {15, 15441, 13}, {15, 37851, 16962}

X(45779) = FOCUS, OTHER THAN X(14), OF SIMMONS CIRCUMCONIC WITH CENTER X(40579)

Barycentrics    2*a^12 + 7*a^10*b^2 - 32*a^8*b^4 + 33*a^6*b^6 - 7*a^4*b^8 - 4*a^2*b^10 + b^12 + 7*a^10*c^2 - 35*a^8*b^2*c^2 + 14*a^6*b^4*c^2 + 17*a^4*b^6*c^2 + 5*a^2*b^8*c^2 - 8*b^10*c^2 - 32*a^8*c^4 + 14*a^6*b^2*c^4 - 4*a^4*b^4*c^4 - a^2*b^6*c^4 + 23*b^8*c^4 + 33*a^6*c^6 + 17*a^4*b^2*c^6 - a^2*b^4*c^6 - 32*b^6*c^6 - 7*a^4*c^8 + 5*a^2*b^2*c^8 + 23*b^4*c^8 - 4*a^2*c^10 - 8*b^2*c^10 + c^12 - Sqrt[3]*(8*a^10 - 18*a^8*b^2 + 2*a^6*b^4 + 16*a^4*b^6 - 6*a^2*b^8 - 2*b^10 - 18*a^8*c^2 + 2*a^6*b^2*c^2 + 4*a^4*b^4*c^2 + 6*a^2*b^6*c^2 + 6*b^8*c^2 + 2*a^6*c^4 + 4*a^4*b^2*c^4 - 4*b^6*c^4 + 16*a^4*c^6 + 6*a^2*b^2*c^6 - 4*b^4*c^6 - 6*a^2*c^8 + 6*b^2*c^8 - 2*c^10)*S : :
X(45779) = 3 X[14] - 4 X[10218], 3 X[471] - 4 X[6672], X[622] - 3 X[19773], 2 X[10218] - 3 X[40579]

X(45779) lies on the cubic K313 and these lines: {14, 16}, {20, 8918}, {143, 30440}, {376, 36298}, {471, 6672}, {530, 36311}, {619, 19777}, {622, 19773}, {1294, 5994}, {5624, 36840}, {5961, 6104}, {8462, 16163}, {11480, 18776}, {15769, 40159}, {36309, 44701}

X(45779) = reflection of X(i) in X(j) for these {i,j}: {14, 40579}, {19777, 619}
X(45779) = isogonal conjugate of X(14) wrt anticevian triangle of X(14)
X(45779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16, 15442, 14}, {16, 15743, 15442}, {16, 37852, 16963}

X(45780) = X(24)X(52)∩X(68)X(70)

Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*(-a^2+b^2+c^2)*((b^2+c^2)*a^6-(b^4+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

See Antreas Hatzipolakis and César Lozada, euclid 3015.

X(45780) lies on these lines: {3, 15317}, {24, 52}, {30, 511}, {68, 70}, {143, 9820}, {155, 6243}, {184, 44259}, {186, 5504}, {235, 5446}, {343, 1216}, {389, 12038}, {2931, 18127}, {3060, 5654}, {5447, 16196}, {5462, 16238}, {5562, 9927}, {5889, 12118}, {5961, 12095}, {5962, 39118}, {5972, 32411}, {6101, 12359}, {6193, 31304}, {7565, 41713}, {7689, 10625}, {9729, 43615}, {9730, 15078}, {9818, 44439}, {9932, 16266}, {9936, 12271}, {9937, 15139}, {9967, 41614}, {10110, 44235}, {10257, 41673}, {10263, 22660}, {10627, 44158}, {11557, 20771}, {11746, 44911}, {11800, 33547}, {11818, 41714}, {12162, 35490}, {12163, 37484}, {12236, 44452}, {12259, 31738}, {12293, 18436}, {12301, 15138}, {12364, 15107}, {12420, 31305}, {12901, 21649}, {13352, 44269}, {13557, 13558}, {13598, 44271}, {14852, 23039}, {15085, 37477}, {15115, 44673}, {15316, 17834}, {18381, 32539}, {18474, 27365}, {19458, 37486}, {21268, 22823}, {21849, 44270}, {31725, 45186}, {37440, 41597}, {40647, 44240}

X(45780) = isogonal conjugate of X(45781)
X(45780) = circumnormal-isogonal conjugate of X(44062)
X(45780) = X(186)-Ceva conjugate of-X(12095)
X(45780) = X(2072)-Dao conjugate of-X(5962)
X(45780) = X(91)-isoconjugate-of-X(38534)
X(45780) = X(571)-reciprocal conjugate of-X(38534)
X(45780) = barycentric product X(1993)*X(2072)
X(45780) = barycentric quotient X(i)/X(j) for these (i, j): (571, 38534), (2072, 5392)
X(45780) = trilinear product X(47)*X(2072)
X(45780) = trilinear quotient X(i)/X(j) for these (i, j): (47, 38534), (2072, 91)

X(45781) = ISOGONAL CONJUGATE OF X(45780)

Barycentrics    (a^4-2*c^2*a^2+(b^2-c^2)^2)*(a^2+b^2-c^2)*(a^8-(4*b^2+c^2)*a^6+(2*b^2+c^2)*(3*b^2-c^2)*a^4-(4*b^6-b^4*c^2-c^6)*a^2+(b^4-c^4)*(b^2-c^2)*b^2)*(a^2-b^2+c^2)*(a^4-2*b^2*a^2+(b^2-c^2)^2)*(a^8-(b^2+4*c^2)*a^6-(b^2-3*c^2)*(b^2+2*c^2)*a^4+(b^6+b^2*c^4-4*c^6)*a^2+(b^4-c^4)*(b^2-c^2)*c^2) : :
X(45781) = 5*X(631)-4*X(15240)

See Antreas Hatzipolakis and César Lozada, euclid 3015.

X(45781) lies on the circumcircle and these lines: {3, 44062}, {4, 15241}, {24, 1288}, {26, 925}, {68, 110}, {96, 933}, {99, 20563}, {107, 847}, {112, 2165}, {631, 15240}, {2072, 5962}, {3565, 44831}, {13398, 32132}

X(45781) = reflection of X(i) in X(j) for these (i, j): (4, 15241), (44062, 3)
X(45781) = isogonal conjugate of X(45780)
X(45781) = X(403)-cross conjugate of-X(847)
X(45781) = X(47)-isoconjugate-of-X(2072)
X(45781) = antipode of X(44062) in circumcircle
X(45781) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(7505)}} and {{A, B, C, X(24), X(26)}}
X(45781) = Collings transform of X(15241)
X(45781) = V-transform of X(44062)
X(45781) = barycentric quotient X(2165)/X(2072)
X(45781) = trilinear product X(91)*X(38534)
X(45781) = trilinear quotient X(91)/X(2072)

X(45782) = X(1)X(27455)∩X(2)X(256)

Barycentrics    a*(a*b - a*c - b*c)*(a*b - a*c + b*c)*(b^2 + b*c + c^2) : :

X(45782) lies on the cubics K1012 and K1243 and these lines: {1, 27455}, {2, 256}, {3, 2053}, {6, 43}, {8, 291}, {10, 27424}, {57, 16571}, {75, 982}, {238, 19591}, {350, 4941}, {749, 4685}, {751, 2228}, {753, 932}, {984, 3116}, {986, 37148}, {1193, 23493}, {1469, 3783}, {1581, 3061}, {1698, 27432}, {1755, 7262}, {1964, 17716}, {2206, 13588}, {2236, 4650}, {2276, 3117}, {3123, 30998}, {3736, 3795}, {4446, 25350}, {4492, 17290}, {5329, 23086}, {5363, 7236}, {6376, 43225}, {7146, 19602}, {7209, 9436}, {7241, 24437}, {7242, 16720}, {9780, 27430}, {16569, 20368}, {16744, 24662}, {17063, 33891}, {17592, 45218}, {18830, 43096}, {23462, 31028}, {24456, 30963}, {26037, 27438}, {26042, 32780}, {26102, 28358}, {27450, 29576}

X(45782) = X(87)-Ceva conjugate of X(40783)
X(45782) = X(i)-cross conjugate of X(j) for these (i,j): {3094, 7146}, {3661, 984}
X(45782) = X(i)-isoconjugate of X(j) for these (i,j): {43, 985}, {192, 40746}, {789, 8640}, {825, 3835}, {870, 2209}, {1423, 2344}, {1492, 4083}, {2176, 14621}, {3123, 5384}, {3407, 20284}, {4586, 20979}, {4613, 16695}, {5388, 21762}, {16468, 40756}, {18898, 33890}, {20906, 34069}, {24533, 30670}, {27644, 40747}, {38832, 40718}
X(45782) = trilinear pole of line {3250, 3805}
X(45782) = barycentric product X(i)*X(j) for these {i,j}: {87, 3661}, {330, 984}, {561, 40736}, {824, 932}, {869, 6383}, {1469, 27424}, {1491, 4598}, {2162, 33931}, {2276, 6384}, {2319, 7179}, {3250, 18830}, {3790, 7153}, {3807, 43931}, {3864, 39914}, {4475, 5383}, {4517, 7209}, {7146, 7155}, {16606, 30966}, {27447, 40790}, {27494, 40783}, {40773, 42027}
X(45782) = barycentric quotient X(i)/X(j) for these {i,j}: {87, 14621}, {330, 870}, {788, 20979}, {824, 20906}, {869, 2176}, {932, 4586}, {984, 192}, {1469, 1423}, {1491, 3835}, {2053, 2344}, {2162, 985}, {2276, 43}, {3094, 41886}, {3116, 20284}, {3250, 4083}, {3661, 6376}, {3736, 27644}, {3781, 22370}, {3790, 4110}, {3799, 4595}, {3807, 36863}, {3862, 41531}, {3864, 40848}, {4475, 21138}, {4481, 17217}, {4517, 3208}, {4598, 789}, {6383, 871}, {7121, 40746}, {7146, 3212}, {7179, 30545}, {16606, 40718}, {18830, 37133}, {23493, 40747}, {30966, 31008}, {33931, 6382}, {34071, 1492}, {40728, 2209}, {40736, 31}, {40773, 33296}, {40783, 4393}, {40790, 17752}, {43931, 4817}
X(45782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2227, 41886}, {87, 2319, 34252}, {330, 7148, 7275}, {350, 24458, 4941}, {1575, 17792, 43}

X(45783) = X(1)X(1581)∩X(6)X(662)

Barycentrics    a*(a + b)*(a + c)*(-b^2 + a*c)*(a*b - c^2)*(a^2 - a*b - b^2 - a*c - b*c - c^2) : :

X(45783) lies on the cubics K135 and K1243 and these lines: {1, 1581}, {6, 662}, {81, 19557}, {239, 9278}, {291, 8935}, {672, 1931}, {741, 1193}, {799, 39367}, {2643, 24437}, {4589, 30669}, {6626, 21879}, {17260, 36800}, {17680, 40017}, {26801, 27189}

X(45783) = X(291)-Ceva conjugate of X(37128)
X(45783) = X(i)-isoconjugate of X(j) for these (i,j): {242, 15377}, {740, 2248}, {2054, 39922}, {2238, 13610}, {3747, 6625}, {3948, 18757}
X(45783) = crosspoint of X(4589) and X(39292)
X(45783) = crosssum of X(2086) and X(4455)
X(45783) = barycentric product X(i)*X(j) for these {i,j}: {291, 6626}, {335, 38814}, {741, 17762}, {846, 18827}, {1654, 37128}, {4584, 21196}, {18755, 40017}
X(45783) = barycentric quotient X(i)/X(j) for these {i,j}: {741, 13610}, {846, 740}, {1654, 3948}, {1931, 39922}, {2196, 15377}, {6626, 350}, {17762, 35544}, {18268, 2248}, {18755, 2238}, {21879, 4037}, {37128, 6625}, {38814, 239}

X(45784) = X(6)X(21)∩X(57)X(959)

Barycentrics    a*(a*b + b^2 + 2*a*c + b*c)*(3*a^2 + 2*a*b - b^2 + 2*a*c + 2*b*c - c^2)*(2*a*b + a*c + b*c + c^2) : :

X(45784) lies on the cubic K1243 and these lines: {6, 21}, {57, 959}, {386, 37400}, {3616, 31359}, {5222, 44733}, {37666, 37870}

X(45784) = X(i)-isoconjugate of X(j) for these (i,j): {958, 5665}, {1468, 43533}
X(45784) = barycentric product X(i)*X(j) for these {i,j}: {941, 3945}, {959, 5273}, {3601, 44733}, {4252, 34258}, {7490, 34259}
X(45784) = barycentric quotient X(i)/X(j) for these {i,j}: {941, 43533}, {3601, 11679}, {3945, 34284}, {4252, 940}
X(45784) = {X(5331),X(34259)}-harmonic conjugate of X(941)

X(45785) = X(2)X(981)∩X(6)X(980)

Barycentrics    a^3*(a^2*b + b^3 + a^2*c + a*b*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c + a*b*c + b^2*c + c^3) : :

X(45785) lies on the cubic K1243 and these lines: {2, 981}, {6, 980}, {39, 213}, {83, 274}, {572, 1178}, {992, 17353}, {1918, 1964}, {1974, 5019}, {2207, 4185}, {3225, 34020}, {5035, 32740}, {6184, 7109}, {16782, 17023}, {20228, 21759}

X(45785) = isogonal conjugate of the isotomic conjugate of X(39957)
X(45785) = X(i)-isoconjugate of X(j) for these (i,j): {2, 5263}, {8, 41245}, {75, 5276}, {522, 14612}
X(45785) = barycentric product X(i)*X(j) for these {i,j}: {6, 39957}, {31, 39712}
X(45785) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 5263}, {32, 5276}, {604, 41245}, {1415, 14612}, {39712, 561}, {39957, 76}

X(45786) = X(1)X(607)∩X(2)X(1172)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - b^4 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - 2*b^2*c^2 - c^4) : :

X(45786) lies on the cubic K1243 and these lines: {1, 607}, {2, 1172}, {3, 2204}, {4, 33854}, {6, 25}, {19, 614}, {22, 2189}, {28, 8743}, {57, 608}, {270, 19310}, {468, 5275}, {672, 2212}, {1193, 1973}, {1249, 37394}, {1333, 17409}, {1395, 21764}, {1829, 16502}, {1848, 24789}, {1914, 11383}, {2207, 4185}, {2275, 22479}, {2280, 40976}, {2326, 37090}, {3172, 37245}, {4329, 24605}, {5019, 36417}, {5276, 6353}, {5299, 7713}, {11396, 16781}, {13730, 23115}, {14017, 39575}, {15988, 37187}, {16318, 37432}, {17442, 28082}, {20831, 22120}, {22131, 37547}, {37675, 38282}

X(45786) = polar conjugate of the isotomic conjugate of X(36740)
X(45786) = barycentric product X(4)*X(36740)
X(45786) = barycentric quotient X(36740)/X(69)

X(45787) = X(2)X(314)∩X(6)X(1402)

Barycentrics    a^2*(a*b + b^2 + 2*a*c + b*c)*(2*a*b + a*c + b*c + c^2)*(a^2*b^2 + a*b^3 + a^2*b*c + a*b^2*c + b^3*c + a^2*c^2 + a*b*c^2 + a*c^3 + b*c^3) : :

X(45787) lies on the cubic K1243 and these lines: {2, 314}, {6, 1402}, {39, 34259}, {2277, 10473}, {5019, 39673}, {5283, 31359}

X(45787) = X(981)-isoconjugate of X(10436)
X(45787) = crossdifference of every pair of points on line {8639, 23880}
X(45787) = barycentric product X(i)*X(j) for these {i,j}: {941, 980}, {959, 35628}, {2274, 31359}
X(45787) = barycentric quotient X(i)/X(j) for these {i,j}: {980, 34284}, {2274, 10436}

X(45788) = ISOGONAL CONJUGATE OF X(35471)

Barycentrics    a^2*(a^6-(b^2-c^2)*a^4-(b^4-4*b^2*c^2+5*c^4)*a^2+(b^2+3*c^2)*(b^2-c^2)^2)*(-a^2+b^2+c^2)*(a^6+(b^2-c^2)*a^4-(5*b^4-4*b^2*c^2+c^4)*a^2+(3*b^2+c^2)*(b^2-c^2)^2) : :
Barycentrics    (7*R^2+SB-2*SW)*(7*R^2+SC-2*SW)*(S^2-SB*SC) : :

See Antreas Hatzipolakis and César Lozada, euclid 3018.

X(45788) lies on Jerabek circumhyperbola and these lines: {4, 34796}, {6, 34783}, {23, 13452}, {26, 74}, {30, 70}, {49, 14528}, {54, 7526}, {64, 7517}, {66, 20427}, {67, 1657}, {68, 18563}, {155, 16867}, {265, 12163}, {382, 6145}, {1173, 31861}, {3431, 14118}, {3521, 10605}, {3549, 15740}, {4846, 10024}, {5504, 7723}, {5576, 14542}, {5663, 38534}, {6000, 34438}, {7488, 11270}, {7527, 13472}, {7530, 16835}, {10575, 34436}, {10620, 11744}, {11799, 43695}, {13093, 34207}, {13754, 15317}, {15316, 18436}, {15321, 33541}, {18323, 32533}, {18434, 37489}, {18439, 18532}, {19151, 34864}, {33565, 37494}, {36747, 42059}

X(45788) = isogonal conjugate of X(35471)
X(45788) = X(3)-vertex conjugate of-X(34438)
X(45788) = intersection, other than A, B, C, of circumconics Jerabek hyperbola and {{A, B, C, X(5), X(7526)}}

X(45789) = X(1)X(32093)∩X(2)X(7)

Barycentrics    3*a^2 - 2*a*b - 5*b^2 - 2*a*c + 6*b*c - 5*c^2 : :

The barycentrics are proportional to the A-, B-, C- powers of the Spieker circle.

The trilinear polar of X(45789) passes through X(7657).

X(45789) lies on these lines: {1, 32093}, {2, 7}, {8, 4373}, {10, 4902}, {69, 3621}, {75, 4678}, {141, 4454}, {145, 3663}, {320, 3623}, {346, 17231}, {391, 1086}, {519, 32105}, {903, 42696}, {1122, 3869}, {1266, 20052}, {3161, 21255}, {3616, 4888}, {3617, 4887}, {3620, 4437}, {3622, 3664}, {3782, 37655}, {3875, 20014}, {3945, 4389}, {3973, 31189}, {4000, 17345}, {4310, 4655}, {4416, 24599}, {4419, 4869}, {4488, 17284}, {4644, 17235}, {4648, 7238}, {4657, 4747}, {5059, 10444}, {5232, 17273}, {5484, 41920}, {5843, 36682}, {7222, 17237}, {7271, 19861}, {7274, 19860}, {8822, 17539}, {9812, 43172}, {11038, 24723}, {14552, 33146}, {15829, 19604}, {17014, 17364}, {17151, 31145}, {17232, 20073}, {17247, 29624}, {17288, 29616}, {17298, 29621}, {17347, 37681}, {17361, 20049}, {19824, 31303}, {20036, 24214}, {26563, 39126}

X(45789) = isotomic conjugate of isogonal conjugate of X(8572)
X(45789) = X(6571)-anticomplementary conjugate of X(693)
X(45789) = crosspoint of X(4373) and X(30712)
X(45789) = isotomic conjugate of trilinear pole of line X(522)X(2490) (the radical axis of circumcircle and Spieker circle)
X(45789) = trilinear product X(i)*X(j) for these {i,j}: {75, 8572}, {662, 7657}
X(45789) = trilinear quotient X(i)/X(j) for these (i,j): (7657, 661), (8572, 31)
X(45789) = barycentric product X(i)*X(j) for these {i,j}: {76, 8572}, {99, 7657}
X(45789) = barycentric quotient X(i)/X(j) for these {i,j}: {7657, 523}, {8572, 6}
X(45789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 33800, 7}, {8, 4862, 4373}, {69, 4346, 4452}, {69, 4452, 3621}, {144, 3662, 2}, {3161, 21255, 30833}, {3616, 4888, 30712}, {3620, 4440, 4461}, {3663, 21296, 145}, {4419, 7232, 4869}, {4887, 17272, 31995}, {7238, 17255, 4648}, {9965, 17184, 2}, {17272, 31995, 3617}, {17273, 42697, 5232}, {21454, 27184, 2}, {26125, 27170, 2}

X(45790) = X(69)X(41298)∩X(95)X(5888)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2)*(-a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 + 3*a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6) : :

The barycentrics are proportional to the A-, B-, C- powers of the Lester circle.

X(45790) lies on these lines: {69, 41298}, {95, 5888}, {525, 37636}, {526, 3268}, {826, 34290}, {850, 1232}

X(45790) = isotomic conjugate of isogonal conjugate of X(8562)
X(45790) = X(850)-Ceva conjugate of X(3268)
X(45790) = isotomic conjugate of trilinear pole of line X(231)X(1989) (the radical axis of circumcircle and Lester circle)
X(45790) = X(i)-isoconjugate of X(j) for these (i,j): {163, 11071}, {14579, 32678}, {15392, 32676}
X(45790) = barycentric product X(i)*X(j) for these {i,j}: {76, 8562}, {850, 40604}, {2914, 3267}, {3268, 37779}, {7799, 45147}
X(45790) = barycentric quotient X(i)/X(j) for these {i,j}: {323, 1291}, {523, 11071}, {525, 15392}, {526, 14579}, {1749, 32678}, {2914, 112}, {3268, 13582}, {5612, 5995}, {5616, 5994}, {5664, 3471}, {6140, 11060}, {8552, 43704}, {8562, 6}, {10272, 41392}, {10413, 15475}, {11063, 14560}, {14566, 1117}, {37779, 476}, {40604, 110}, {41078, 1263}, {43958, 12077}, {45147, 1989}

X(45791) = X(2)X(37)∩X(200)X(220)

Barycentrics    a*(a - b - c)^3*(a*b - b^2 + a*c + 2*b*c - c^2) : :

The barycentrics are proportional to the A-, B-, C- powers of the Adams circle.

The trilinear polar of X(45791) passes through X(6607).

X(45791) lies on these lines: {2, 37}, {8, 15853}, {9, 42015}, {200, 220}, {480, 28070}, {1088, 25242}, {1190, 3174}, {1212, 4847}, {1536, 3695}, {3059, 8012}, {3692, 37658}, {3694, 40869}, {3870, 4513}, {3991, 13405}, {4082, 23970}, {4130, 23615}, {7580, 17742}, {8226, 21073}, {11019, 21096}, {25006, 40997}, {36845, 40133}

X(45791) = isotomic conjugate of isogonal conjugate of X(8551)
X(45791) = crosspoint of X(346) and X(728)
X(45791) = crosssum of X(738) and X(1407)
X(45791) = crossdifference of every pair of points on line {667, 43932}
X(45791) = X(i)-isoconjugate of X(j) for these (i,j): {56, 10509}, {269, 1170}, {479, 1174}, {604, 42311}, {738, 2346}, {1106, 31618}, {1119, 1803}, {1407, 21453}, {1435, 40443}, {7023, 32008}
X(45791) = trilinear product X(i)*X(j) for these {i,j}: {8, 8012}, {9, 3059}, {75, 8551}, {142, 480}, {200, 1212}, {220, 4847}, {341, 20229}, {346, 2293}, {354, 728}, {1229, 1253}, {1475, 5423}, {6602, 20880}
X(45791) = trilinear quotient X(i)/X(j) for these (i,j): (8, 10509), (142, 479), (200, 1170), (341, 31618), (346, 21453), (354, 738), (480, 1174), (728, 2346), (1212, 269), (1229, 1088), (1475, 7023), (2293, 1407), (3059, 57), (4847, 279), (5423, 32008), (8012, 56), (8551, 31), (20229, 1106), (20880, 23062)
X(45791) = barycentric product X(i)*X(j) for these {i,j}: {8, 3059}, {76, 8551}, {142, 728}, {200, 4847}, {220, 1229}, {312, 8012}, {341, 2293}, {346, 1212}, {354, 5423}, {480, 20880}, {646, 10581}, {668, 6607}, {1043, 21039}, {1233, 6602}, {1265, 1827}, {1475, 30693}, {1855, 3692}, {3239, 35341}, {3699, 6608}, {4082, 17194}, {4163, 35338}, {4515, 16713}, {4578, 6362}, {6558, 21127}
X(45791) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 42311}, {9, 10509}, {142, 23062}, {200, 21453}, {220, 1170}, {346, 31618}, {354, 479}, {480, 2346}, {728, 32008}, {1212, 279}, {1260, 40443}, {1475, 738}, {1802, 1803}, {1827, 1119}, {1855, 1847}, {2293, 269}, {2488, 43932}, {3059, 7}, {4578, 6606}, {4847, 1088}, {6602, 1174}, {6607, 513}, {6608, 3676}, {8012, 57}, {8551, 6}, {10581, 3669}, {20229, 1407}, {21039, 3668}, {21795, 1427}, {22079, 7053}, {35326, 4617}, {35338, 4626}, {35341, 658}
X(45791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {200, 24771, 220}

X(45792) = X(69)X(6333)∩X(525)X(3267)

Barycentrics    (b^2 - c^2)*(-a^2 + b^2 + c^2)*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2) : :
Barycentrics    csc A cot A sin(B - C) (1 + 2 cos 2A) : :
X(45792) =X[69] + 2 X[6333], X[3267] - 4 X[4143]

The barycentrics are proportional to the A-, B-, C- powers of the Dao-Moses-Telv circle.

X(45792) lies on these lines: {69, 6333}, {525, 3267}, {526, 3268}, {4558, 4563}, {12215, 39474}, {35522, 45147}, {36790, 36792}

X(45792) = isotomic conjugate of isogonal conjugate of X(8552)
X(45792) = isotomic conjugate of polar conjugate of X(3268)
X(45792) = X(8552)-cross conjugate of X(3268)
X(45792) = crosssum of X(i) and X(j) for these (i,j): {32, 14398}, {2489, 14581}
X(45792) = crossdifference of every pair of points on line {1974, 2971}
X(45792) = isotomic conjugate of trilinear pole of line X(25)X(1989) (the radical axis of circumcircle and Dao-Moses-Telv circle)
X(45792) = X(i)-isoconjugate of X(j) for these (i,j): {19, 14560}, {25, 32678}, {32, 36129}, {162, 11060}, {163, 18384}, {476, 1973}, {1096, 32662}, {1974, 32680}, {1989, 32676}, {2207, 36061}, {14583, 36131}
X(45792) = barycentric product X(i)*X(j) for these {i,j}: {69, 3268}, {76, 8552}, {304, 32679}, {305, 526}, {323, 3267}, {339, 10411}, {340, 3265}, {525, 7799}, {670, 16186}, {2624, 40364}, {3926, 44427}, {4143, 14165}, {6148, 34767}, {14270, 40050}, {14590, 36793}, {14918, 15414}, {15413, 42701}, {22115, 44173}, {34386, 41078}
X(45792) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 14560}, {63, 32678}, {69, 476}, {75, 36129}, {125, 15475}, {186, 32713}, {298, 36309}, {299, 36306}, {304, 32680}, {305, 35139}, {323, 112}, {326, 36061}, {339, 10412}, {340, 107}, {394, 32662}, {523, 18384}, {525, 1989}, {526, 25}, {647, 11060}, {850, 6344}, {1273, 35360}, {1511, 23347}, {2081, 3199}, {2088, 2489}, {2624, 1973}, {3265, 265}, {3267, 94}, {3268, 4}, {4563, 39295}, {5664, 1990}, {6148, 4240}, {6149, 32676}, {6333, 14356}, {6390, 14559}, {7799, 648}, {8552, 6}, {9033, 14583}, {9213, 8753}, {10411, 250}, {11064, 41392}, {14165, 6529}, {14208, 2166}, {14270, 1974}, {14314, 34417}, {14355, 32696}, {14380, 40355}, {14385, 32715}, {14590, 23964}, {14591, 41937}, {15526, 14582}, {16186, 512}, {22115, 1576}, {23870, 8738}, {23871, 8737}, {23965, 44427}, {32662, 23966}, {32679, 19}, {34767, 5627}, {36793, 14592}, {39495, 44089}, {41078, 53}, {42701, 1783}, {43083, 14595}, {44173, 18817}, {44427, 393}, {44718, 5994}, {44719, 5995}, {44808, 44077}, {44814, 44102}

X(45793) = X(2)X(94)∩X(6)X(5392)

Barycentrics    b^2*c^2*(-(a^2*b^2) + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)^2 : :
Barycentrics    (S^2 + SB SC)^2/(SB + SC) : :
Barycentrics    cos^2(B - C) : :
Barycentrics    1 + cos(2B - 2C) : :
Barycentrics    A'-power of circumcircle : :, where A'B'C' is the X(4)-altimedial triangle
Barycentrics    A-power of circumcircle of circumcevian polar triangle of X(3) : :
Barycentrics    A-power of circumcircle of circumcevian polar triangle of X(4) : br>

The barycentrics are proportional to the A-, B-, C- powers of the Hung circle.

Let P and Q be the points on the circumcircle whose Steiner lines are tangent to the circumcircle. Then X(45793) is the isogonal conjugate of the barycentric product P*Q. (Randy Hutson, November 30, 2021)

X(45793) lies on these lines: {2, 94}, {6, 5392}, {53, 311}, {76, 39284}, {143, 25043}, {264, 394}, {1352, 14593}, {1501, 42354}, {2052, 37638}, {2970, 21243}, {5562, 14978}, {10601, 41760}, {11064, 40684}, {11140, 11538}, {11197, 44716}, {18022, 36793}

X(45793) = isotomic conjugate of isogonal conjugate of X(36412)
X(45793) = X(14570)-Ceva conjugate of X(18314)
X(45793) = cevapoint of X(6) and X(1601)
X(45793) = isotomic conjugate of trilinear pole of line X(526)X(23286) (the radical axis of circumcircle and Hung circle)
X(45793) = polar conjugate of isogonal conjugate of complement of X(34148)
X(45793) = polar conjugate of isogonal conjugate of trilinear product of vertices of Johnson triangle
X(45793) = barycentric square of X(14213)
X(45793) = X(i)-isoconjugate of X(j) for these (i,j): {54, 2148}, {2169, 8882}, {2190, 14533}, {2616, 14586}, {2623, 36134}
X(45793) = barycentric product X(i)*X(j) for these {i,j}: {5, 311}, {53, 28706}, {75, 1087}, {76, 36412}, {324, 343}, {1225, 40449}, {1625, 15415}, {3596, 41279}, {14213, 14213}, {14570, 18314}, {23607, 34384}
X(45793) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 54}, {53, 8882}, {143, 25044}, {216, 14533}, {311, 95}, {324, 275}, {343, 97}, {1087, 1}, {1625, 14586}, {1953, 2148}, {2617, 36134}, {2618, 2616}, {3078, 13366}, {5562, 19210}, {6368, 23286}, {6663, 1614}, {10216, 143}, {12077, 2623}, {13450, 8884}, {14213, 2167}, {14570, 18315}, {18314, 15412}, {23181, 15958}, {23607, 51}, {24862, 20975}, {25043, 252}, {28706, 34386}, {31610, 288}, {34520, 15109}, {34836, 19170}, {34983, 39201}, {35360, 933}, {36412, 6}, {39019, 3269}, {39569, 19189}, {40449, 1166}, {40981, 14573}, {41212, 34980}, {41222, 39013}, {41279, 56}, {41587, 8883}, {42453, 26887}, {42459, 33629}, {44706, 2169}
X(45793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {311, 324, 343}

X(45794) = X(2)X(6)∩X(4)X(93)

Barycentrics    a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6 : :
Barycentrics    SA - R^2 : :
Barycentrics    cos 2A - cos 2B - cos 2C : :
Barycentrics    A-power of 1st Droz-Farny circle : :
Barycentrics    A'-power of 2nd Droz-Farny circle : :, where A'B'C' is the Johnson triangle
X(45794) = 3 X[2] - 4 X[343], 9 X[2] - 8 X[23292], 3 X[343] - 2 X[23292], 3 X[1993] - 4 X[23292], 7 X[3523] - 8 X[44201], 3 X[7391] - 4 X[11550], 3 X[11442] - 2 X[11550], 3 X[25321] - 4 X[41612]

The barycentrics are proportional to the A-, B-, C- powers of the 1st Droz-Farny circle.

X(45794) lies on these lines: {2, 6}, {3, 32358}, {4, 93}, {15, 40712}, {16, 40711}, {20, 11411}, {22, 3564}, {25, 11898}, {51, 34507}, {52, 7544}, {68, 70}, {94, 6504}, {110, 41674}, {184, 5965}, {194, 41655}, {317, 324}, {340, 2052}, {376, 18917}, {401, 7893}, {427, 34380}, {467, 9308}, {511, 7391}, {539, 37478}, {631, 13353}, {637, 13439}, {638, 13428}, {1092, 44673}, {1216, 18912}, {1238, 1609}, {1351, 5133}, {1352, 3060}, {1353, 7499}, {1370, 3448}, {1503, 20062}, {1614, 9936}, {1899, 2979}, {1995, 41588}, {2781, 41736}, {2904, 7505}, {2918, 38435}, {3522, 18909}, {3523, 18916}, {3529, 30522}, {3818, 21969}, {3917, 18911}, {3926, 35296}, {5012, 43653}, {5056, 11487}, {5059, 5925}, {5064, 44456}, {5093, 7539}, {5189, 32064}, {5392, 13579}, {5562, 18390}, {5921, 7500}, {6101, 25738}, {6193, 7488}, {6636, 6776}, {6997, 11002}, {7386, 33884}, {7426, 8780}, {7485, 11245}, {7492, 33522}, {7493, 9544}, {7494, 11003}, {7495, 11402}, {7509, 13292}, {7519, 15069}, {7558, 12161}, {7571, 18583}, {7691, 19467}, {7754, 41237}, {7762, 41231}, {7768, 40814}, {7855, 36212}, {9306, 41586}, {9777, 37990}, {10117, 11206}, {10519, 15246}, {10594, 31831}, {10625, 11457}, {11001, 12317}, {11143, 18581}, {11144, 18582}, {11225, 40107}, {11444, 39571}, {11547, 14918}, {12087, 34781}, {12160, 13160}, {12225, 12429}, {13595, 14826}, {13754, 44440}, {14516, 17834}, {14788, 37493}, {14853, 37353}, {15004, 24206}, {15107, 31383}, {16266, 37119}, {16981, 37349}, {18440, 34603}, {18925, 19468}, {18947, 38282}, {20078, 37781}, {20081, 40853}, {21849, 43150}, {22112, 32068}, {23061, 23293}, {23291, 31101}, {25321, 41612}, {32001, 37192}, {32140, 37484}, {32269, 35264}, {34118, 34751}, {34565, 38317}, {37068, 41008}, {37494, 44831}, {40684, 44134}, {40916, 45298}

X(45794) = reflection of X(i) in X(j) for these {i,j}: {193, 41614}, {1993, 343}, {7391, 11442}, {44831, 37494}
X(45794) = isotomic conjugate of X(13579)
X(45794) = anticomplement of X(1993)
X(45794) = anticomplement of the isogonal conjugate of X(2165)
X(45794) = anticomplement of the isotomic conjugate of X(5392)
X(45794) = isotomic conjugate of the isogonal conjugate of X(8553)
X(45794) = isotomic conjugate of the polar conjugate of X(7505)
X(45794) = anticomplementary isogonal conjugate of X(40697)
X(45794) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 40697}, {19, 6193}, {68, 4329}, {91, 69}, {96, 21271}, {847, 21270}, {925, 7192}, {1820, 20}, {2165, 8}, {2168, 3}, {2351, 6360}, {5392, 6327}, {14593, 5905}, {20571, 315}, {30450, 21300}, {32734, 4560}, {36145, 523}, {41271, 17479}
X(45794) = X(i)-Ceva conjugate of X(j) for these (i,j): {5392, 2}, {44128, 37444}
X(45794) = X(8553)-cross conjugate of X(7505)
X(45794) = crosssum of X(3124) and X(34952)
X(45794) = polar conjugate of X(24)-cross conjugate of X(4)
X(45794) = pole wrt polar circle of line X(1510)X(2501) (the orthic axis of the orthic triangle)
X(45794) = X(i)-isoconjugate of X(j) for these (i,j): {19, 15317}, {31, 13579}, {2148, 27361}
X(45794) = barycentric product X(i)*X(j) for these {i,j}: {69, 7505}, {76, 8553}, {2904, 20563}
X(45794) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13579}, {3, 15317}, {5, 27361}, {155, 18126}, {2904, 24}, {7505, 4}, {8553, 6}, {34853, 15242}
X(45794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 193, 1994}, {2, 6515, 37644}, {2, 11004, 11427}, {2, 37779, 6515}, {6, 37636, 2}, {68, 11412, 37444}, {69, 6515, 2}, {69, 37779, 37644}, {141, 5422, 2}, {343, 1993, 2}, {394, 3580, 2}, {1352, 3060, 7394}, {1899, 2979, 16063}, {2895, 5361, 5739}, {2979, 41724, 1899}, {3519, 6152, 2888}, {6189, 6190, 44388}, {6288, 31815, 4}, {11225, 40107, 43650}, {13567, 15066, 2}, {14516, 17834, 31304}, {14683, 37913, 11206}, {19778, 19779, 4}, {26609, 26625, 2}, {34224, 37486, 20}, {37636, 41628, 6}, {39365, 39366, 44363}, {41198, 41199, 28408}

X(45795) = X(3)X(69)∩X(311)X(32833)

Barycentrics    a^2*b^2*c^2*(a^2 - b^2 - c^2)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 + 4*a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6) : :

The barycentrics are proportional to the A-, B-, C- powers of the nine-point circle of the orthic triangle.

X(45795) lies on these lines: {3, 69}, {311, 32833}, {325, 7394}, {1007, 37990}, {1232, 7763}, {1272, 30698}, {7796, 44128}, {7799, 44149}, {7871, 32002}, {18354, 32817}, {20062, 37668}, {32821, 39113}, {32840, 44135}

X(45795) = isotomic conjugate of trilinear pole of line X(2501)X(20188) (the radical axis of circumcircle and nine-point circle of orthic triangle)
X(45795) = barycentric product X(3926)*X(37119)
X(45795) = barycentric quotient X(37119)/X(393)
X(45795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1238, 3964, 69}, {3933, 9723, 69}

X(45796) = X(2)X(42011)∩X(69)X(7618)

Barycentrics    4*a^6 - 18*a^4*b^2 + 24*a^2*b^4 - 8*b^6 - 18*a^4*c^2 + 21*a^2*b^2*c^2 + 3*b^4*c^2 + 24*a^2*c^4 + 3*b^2*c^4 - 8*c^6 : :

The barycentrics are proportional to the A-, B-, C- powers of the McCay circle.

X(45796) lies on these lines: {2, 42011}, {69, 7618}, {76, 338}, {524, 10485}, {3523, 11160}, {7607, 40107}, {7771, 15533}, {7782, 10488}, {7840, 22712}, {8591, 34507}, {8787, 11149}, {21356, 43620}

X(45796) = isotomic conjugate of isogonal conjugate of X(8566)
X(45796) = barycentric product X(76)*X(8566)
X(45796) = barycentric quotient X(8566)/X(6)

X(45797) = X(75)X(225)∩X(320)X(18690)

Barycentrics    b*c*(a^6 + a^5*b - a^4*b^2 - 2*a^3*b^3 - a^2*b^4 + a*b^5 + b^6 + a^5*c + a^4*b*c - a^3*b^2*c - a^2*b^3*c - a^4*c^2 - a^3*b*c^2 - a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 - a^2*b*c^3 - a*b^2*c^3 - a^2*c^4 - b^2*c^4 + a*c^5 + c^6) : :

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the extangents triangle.

X(45797) lies on these lines: {75, 225}, {320, 18690}, {333, 2160}, {4360, 37782}, {17095, 18695}, {17305, 20905}

X(45797) = isotomic conjugate of isogonal conjugate of X(8555)
X(45797) = barycentric product X(76)*X(8555)
X(45797) = barycentric quotient X(8555)/X(6)

X(45798) = X(75)X(26006)∩X(76)X(85)

Barycentrics    b*c*(-a + b + c)*(a^5 - a^3*b^2 - a^2*b^3 + b^5 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c - a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 - a^2*c^3 - a*b*c^3 - b*c^4 + c^5) : :

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the intangents triangle.

X(45798) lies on these lines: {75, 26006}, {76, 85}, {86, 26165}, {345, 28757}, {346, 28753}, {350, 17875}, {1043, 3615}, {3064, 6332}, {3260, 23978}, {7101, 20930}, {17877, 24781}, {17880, 37774}, {18025, 30807}, {23983, 37796}, {28793, 28808}, {37788, 44327}

X(45798) = isotomic conjugate of the isogonal conjugate of X(8558)
X(45798) = isotomic conjugate of trilinear pole of line X(73)X(663) (the radical axis of circumcircle and intangents circle)
X(45798) = barycentric product X(76)*X(8558)
X(45798) = barycentric quotient X(8558)/X(6)

X(45799) = X(2)X(1225)∩X(54)X(69)

Barycentrics    a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 + 5*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*b^6*c^2 + 6*a^4*c^4 + a^2*b^2*c^4 + 2*b^4*c^4 - 4*a^2*c^6 - 2*b^2*c^6 + c^8 : :
Barycentrics    (csc^2 A) (4 cos 2A + cot^2 A - cot A cot ω) : :
Barycentrics    (cot^2 A) (cot^2 B + cot^2 C - 6) - cot^2 B cot^2 C + 9 : :

The barycentrics are proportional to the A-, B-, C- powers of the reflection circumcircle.

X(45799) lies on these lines: {2, 1225}, {5, 18354}, {54, 69}, {76, 43666}, {99, 33643}, {140, 1238}, {183, 40002}, {264, 1272}, {311, 3459}, {317, 44879}, {325, 1369}, {1007, 1370}, {1232, 7799}, {3432, 7488}, {3618, 42406}, {3926, 22268}, {5169, 11671}, {6337, 7401}, {6527, 32835}, {6644, 9723}, {19583, 34803}, {21395, 44128}, {32829, 40697}, {34229, 41916}, {34254, 41927}

X(45799) = isotomic conjugate of X(3459)
X(45799) = anticomplement of X(2963)
X(45799) = anticomplement of the isogonal conjugate of X(1994)
X(45799) = anticomplement of the isotomic conjugate of X(7769)
X(45799) = isotomic conjugate of the anticomplement of X(21975)
X(45799) = isotomic conjugate of the isogonal conjugate of X(195)
X(45799) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {49, 6360}, {162, 18314}, {662, 1510}, {1510, 21221}, {1994, 8}, {2167, 6101}, {2216, 11140}, {2964, 2}, {2965, 192}, {3518, 5905}, {7769, 6327}, {25044, 17479}, {32002, 21270}, {41298, 21294}, {44180, 4329}
X(45799) = X(i)-Ceva conjugate of X(j) for these (i,j): {311, 69}, {7769, 2}
X(45799) = X(21975)-cross conjugate of X(2)
X(45799) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34433}, {31, 3459}
X(45799) = barycentric product X(i)*X(j) for these {i,j}: {76, 195}, {7769, 21975}
X(45799) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3459}, {3, 34433}, {195, 6}, {930, 39419}, {10615, 231}, {14367, 14579}, {15770, 11063}, {15787, 2965}, {21975, 2963}, {34302, 11071}, {37779, 11584}
X(45799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {95, 1273, 69}, {627, 628, 11271}

X(45800) = X(6)X(17)∩X(49)X(216)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^10 - 3*a^8*b^2 + 4*a^6*b^4 - 4*a^4*b^6 + 3*a^2*b^8 - b^10 - 3*a^8*c^2 + 5*a^6*b^2*c^2 - 2*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 2*a^4*b^2*c^4 - 2*b^6*c^4 - 4*a^4*c^6 - 3*a^2*b^2*c^6 - 2*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 - c^10) : :
Barycentrics    ((3 R^2 - SW) S^2 + (2 R^2 - SW) SA^2 - (2 R^2 SW + S^2 - SW^2) SA) (SB + SC) SA : :

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the MacBeath triangle.

X(45800) lies on these lines: {3, 14533}, {6, 17}, {49, 216}, {394, 20208}, {570, 22115}, {571, 18436}, {577, 22146}, {1154, 8882}, {1249, 35311}, {1993, 4993}, {2165, 15367}, {7691, 33629}, {8745, 15068}, {14576, 18350}, {15905, 23128}

X(45800) = isotomic conjugate of isogonal conjugate of X(8565)
X(45800) = isogonal conjugate of the polar conjugate of X(2888)
X(45800) = X(343)-Ceva conjugate of X(3)
X(45800) = crosssum of X(137) and X(2501)
X(45800) = X(92)-isoconjugate of X(3432)
X(45800) = isotomic conjugate of polar conjugate of X(5)-Ceva conjugate of X(6)
X(45800) = crossdifference of every pair of points on the line through X(1510) and the X(2)-Ceva conjugate of X(139)
X(45800) = barycentric product X(i)*X(j) for these {i,j}: {3, 2888}, {76, 8565}
X(45800) = barycentric quotient X(i)/X(j) for these {i,j}: {184, 3432}, {418, 6798}, {2888, 264}, {8565, 6}, {14533, 40140}
X(45800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14533, 42445, 3}

X(45801) = X(6)X(523)∩X(141)X(525)

Barycentrics    (b^2 - c^2)*(-a^6 + a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 + c^6) : :
X(45801) = X[6] - 3 X[1640], 2 X[141] - 3 X[18310], X[193] + 3 X[14977], 2 X[3589] - 3 X[45327], 5 X[3618] - 3 X[18311]

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the Steiner triangle.

In the plane of a triangle ABC, let
M = X(115), the center of the Kiepert hyperbola;
DEF = tangential triangle of the Kiepert hyperbola;
D' = reflection of D in BC, and define B' and F' cyclically; A' = AM∩E'F', and define B' and C' cyclically. The lines A'D', B'E', C'F' concur in X(45801). (Angel Montesdeoca, August 17, 2022)

X(45801) lies on these lines: {6, 523}, {53, 2501}, {66, 3566}, {111, 13291}, {141, 525}, {157, 878}, {193, 14977}, {512, 9969}, {570, 647}, {577, 38401}, {1249, 18808}, {1648, 10278}, {2165, 2433}, {3054, 9209}, {3569, 45147}, {3589, 45327}, {3618, 18311}, {5489, 6794}, {6368, 40550}, {6388, 12064}, {6792, 8029}, {8553, 14809}, {8574, 21203}, {10190, 41939}, {41760, 43665}

X(45801) = isogonal conjugate of X(40173)
X(45801) = isotomic conjugate of the isogonal conjugate of X(8574)
X(45801) = isotomic conjugate of trilinear pole of line X(23)X(325) (the radical axis of circumcircle and 2nd Steiner circle)
X(45801) = polar conjugate of isogonal conjugate of X(125)-Ceva conjugate of X(647)
X(45801) = polar conjugate of isogonal conjugate of X(2)-of-tangential-triangle-of-Jerabek-hyperbola
X(45801) = polar conjugate of trilinear pole of line X(297)X(323)
X(45801) = pole wrt polar circle of line X(297)X(323)
X(45801) = X(i)-Ceva conjugate of X(j) for these (i,j): {338, 523}, {30716, 7669}
X(45801) = X(7669)-cross conjugate of X(523)
X(45801) = crosspoint of X(41173) and X(43665)
X(45801) = crosssum of X(14966) and X(41167)
X(45801) = crossdifference of every pair of points on line {511, 2070}
X(45801) = X(i)-isoconjugate of X(j) for these (i,j): {1, 40173}, {163, 13485}, {662, 3447}
X(45801) = barycentric product X(i)*X(j) for these {i,j}: {10, 21203}, {76, 8574}, {125, 30716}, {338, 36830}, {514, 21092}, {523, 3448}, {661, 20941}, {850, 7669}, {1577, 16562}, {14366, 23105}, {14618, 22146}, {34349, 43665}
X(45801) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 40173}, {512, 3447}, {523, 13485}, {3448, 99}, {7669, 110}, {8029, 6328}, {8574, 6}, {16562, 662}, {20941, 799}, {21092, 190}, {21203, 86}, {22146, 4558}, {30716, 18020}, {34349, 2421}, {36830, 249}

X(45802) = X(2)X(6)∩X(521)X(3239)

Barycentrics    (a - b - c)*(2*a^6 - a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + 2*a*b^5 + b^6 + 2*a^3*b^2*c - 2*a*b^4*c - a^4*c^2 + 2*a^3*b*c^2 + 4*a^2*b^2*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*c^4 - 2*a*b*c^4 - b^2*c^4 + 2*a*c^5 + c^6) : :

The barycentrics are proportional to the A-, B-, C- powers of the Euler-Gergonne-Soddy circle.

X(45802) lies on these lines: {2, 6}, {521, 3239}, {607, 20306}, {1146, 5081}, {1503, 39690}, {1714, 41369}, {1901, 2322}, {2968, 8558}, {3686, 9119}, {12241, 16552}

X(45802) = isotomic conjugate of isogonal conjugate of X(8554)
X(45802) = crosspoint of X(8) and X(37202)
X(45802) = crosssum of X(56) and X(39690)
X(45802) = crossdifference of every pair of points on line {221, 512}
X(45802) = isotomic conjugate of trilinear pole of line X(347)X(523) (the radical axis of circumcircle and Euler-Gergonne-Soddy circle)
X(45802) = barycentric product X(76)*X(8554)
X(45802) = barycentric quotient X(8554)/X(6)
X(45802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {391, 5739, 37658}

X(45803) = X(2)X(33693)∩X(76)X(141)

Barycentrics    a^6*b^4 + a^2*b^8 + a^6*b^2*c^2 - a^4*b^4*c^2 + a^6*c^4 - a^4*b^2*c^4 - a^2*b^4*c^4 - b^6*c^4 - b^4*c^6 + a^2*c^8 : :
X(45803) = 4 X[2024] - 3 X[5182], 2 X[13196] - 3 X[13331]

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the 1st Neuberg triangle.

X(45803) lies on these lines: {2, 33693}, {6, 38907}, {39, 12215}, {69, 31981}, {76, 141}, {98, 385}, {194, 1352}, {736, 5207}, {1691, 5152}, {2024, 5182}, {2458, 8178}, {3095, 3564}, {3229, 40708}, {3767, 18806}, {5306, 22486}, {5969, 14568}, {7828, 24256}, {7832, 10007}, {7942, 40332}, {10000, 42534}, {10335, 11261}, {12216, 39101}, {13196, 13331}, {35375, 39652}

X(45803) = reflection of X(i) in X(j) for these {i,j}: {12215, 39}, {44771, 141}
X(45803) = isotomic conjugate of the isogonal conjugate of X(8569)
X(45803) = isotomic conjugate of trilinear pole of line X(669)X(7467) (the radical axis of circumcircle and 1st Neuberg circle)
X(45803) = crosssum of X(32) and X(2458)
X(45803) = barycentric product X(76)*X(8569)
X(45803) = barycentric quotient X(8569)/X(6)

X(45804) = X(6)X(76)∩X(69)X(31982)

Barycentrics    2*a^6*b^4 + 5*a^6*b^2*c^2 + 6*a^4*b^4*c^2 + 2*a^2*b^6*c^2 + b^8*c^2 + 2*a^6*c^4 + 6*a^4*b^2*c^4 + 5*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^2*b^2*c^6 + 2*b^4*c^6 + b^2*c^8 : :

The barycentrics are proportional to the A-, B-, C- powers of the circumcircle of the 2nd Neuberg triangle, and also to the A-, B-, C- powers of the circumcircle of the Montesdeoca-Lemoine triangle.

X(45804) lies on these lines: {6, 76}, {69, 31982}, {182, 8290}, {262, 3314}, {511, 2896}, {754, 22486}, {2458, 8150}, {2549, 18906}, {3589, 44771}, {3618, 8149}, {5039, 9983}, {6292, 32452}, {6309, 7820}, {6776, 31958}, {7779, 14994}, {7788, 9765}, {7836, 44423}, {7846, 13331}, {9865, 10336}, {12252, 13354}, {14561, 31276}

X(45804) = reflection of X(i) in X(j) for these {i,j}: {69, 44772}, {12252, 13354}, {24273, 24256}
X(45804) = isotomic conjugate of the isogonal conjugate of X(8570)
X(45804) = barycentric product X(76)*X(8570)
X(45804) = barycentric quotient X(8570)/X(6)

X(45805) = X(2)X(39)∩X(264)X(3127)

Barycentrics    b^2*c^2*(a^6 - 3*a^4*b^2 + a^2*b^4 + b^6 - 3*a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 + c^6 + 2*(a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 - c^4)*S) : :
Barycentrics    (1 + Cot[A])*Csc[A]^2 : : (Peter Moses, November 8, 2021)
Barycentrics    (SA + SB) (SA + SC) (SA + S) : :
Barycentrics    (csc^3 A) (cos A + sin A) : :
Barycentrics    (csc^2 A) (1 + cot A) : :

The barycentrics are proportional to the A-, B-, C- powers of the Kenmotu inner circle.

X(45805) lies on these lines: {2, 39}, {264, 3127}, {491, 40073}, {1502, 34392}, {1975, 3156}, {1991, 35549}, {4176, 6806}, {8956, 13877}, {16037, 34384}, {44166, 45473}

X(45805) = isotomic conjugate of X(8577)
X(45805) = isotomic conjugate of the isogonal conjugate of X(492)
X(45805) = X(42009)-cross conjugate of X(34391)
X(45805) = polar conjugate of isogonal conjugate of isotomic conjugate of X(41515)
X(45805) = {X(76),X(305)}-harmonic conjugate of X(45806)
X(45805) = X(i)-isoconjugate of X(j) for these (i,j): {31, 8577}, {485, 560}, {798, 39383}, {1397, 13455}, {1917, 34391}, {1973, 6413}, {9247, 41515}
X(45805) = barycentric product X(i)*X(j) for these {i,j}: {76, 492}, {305, 1585}, {371, 1502}, {5408, 18022}, {5413, 40050}, {7763, 34392}, {8911, 44161}
X(45805) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8577}, {69, 6413}, {76, 485}, {99, 39383}, {264, 41515}, {305, 11090}, {312, 13455}, {317, 5412}, {371, 32}, {491, 44192}, {492, 6}, {641, 5062}, {1502, 34391}, {1585, 25}, {5408, 184}, {5413, 1974}, {7763, 372}, {8911, 14575}, {9723, 26920}, {11091, 2351}, {13428, 8576}, {13430, 6414}, {16037, 41271}, {34384, 16032}, {34392, 2165}, {39387, 6423}, {42009, 1504}

X(45806) = X(2)X(39)∩X(264)X(3128)

Barycentrics    b^2*c^2*(a^6 - 3*a^4*b^2 + a^2*b^4 + b^6 - 3*a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 + c^6 - 2*(a^4 + 2*a^2*b^2 - b^4 + 2*a^2*c^2 - c^4)*S) : :
Barycentrics    (SA + SB) (SA + SC) (SA - S) : :
Barycentrics    (csc^3 A) (cos A - sin A) : :
Barycentrics    (csc^2 A) (1 - cot A) : :
Barycentrics    (1 - Cot[A])*Csc[A]^2 : : (Peter Moses, November 8, 2021)

The barycentrics are proportional to the A-, B-, C- powers of the Kenmotu outer circle.

X(45806) lies on these lines: {2, 39}, {264, 3128}, {492, 40073}, {591, 35549}, {1502, 34391}, {1975, 3155}, {4176, 6805}, {16032, 34384}, {44166, 45472}

X(45806) = isotomic conjugate of X(8576)
X(45806) = isotomic conjugate of the isogonal conjugate of X(491)
X(45806) = X(42060)-cross conjugate of X(34392)
X(45806) = polar conjugate of isogonal conjugate of isotomic conjugate of X(41516)
X(45806) = {X(76),X(305)}-harmonic conjugate of X(45805)
X(45806) = X(i)-isoconjugate of X(j) for these (i,j): {31, 8576}, {486, 560}, {798, 39384}, {1917, 34392}, {1973, 6414}, {9247, 41516}
X(45806) = barycentric product X(i)*X(j) for these {i,j}: {76, 491}, {305, 1586}, {372, 1502}, {5409, 18022}, {5412, 40050}, {6063, 13461}, {7763, 34391}, {26920, 44161}
X(45806) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 8576}, {69, 6414}, {76, 486}, {99, 39384}, {264, 41516}, {305, 11091}, {317, 5413}, {372, 32}, {491, 6}, {492, 44193}, {642, 5058}, {1502, 34392}, {1586, 25}, {3926, 26922}, {5409, 184}, {5412, 1974}, {7763, 371}, {9723, 8911}, {11090, 2351}, {13439, 8577}, {13441, 6413}, {13461, 55}, {16032, 41271}, {26920, 14575}, {34384, 16037}, {34391, 2165}, {39388, 6424}, {42060, 1505}

X(45807) = X(69)X(9517)∩X(76)X(14223)

Barycentrics    b^2*(b^2 - c^2)*c^2*(-2*a^2 + b^2 + c^2)*(-a^2 + b^2 + c^2) : :
Barycentrics    csc A cot A sin(B - C) (2 cot A - cot B - cot C) : :
Barycentrics    (csc^2 A) (tan B - tan C) (2 cot A - cot B - cot C) : :

The barycentrics are proportional to the A-, B-, C- powers of the Moses-Parry circle.

X(45807) lies on these lines: {69, 9517}, {76, 14223}, {125, 339}, {525, 3267}, {684, 3933}, {690, 5181}, {3785, 44810}, {3906, 23285}, {3926, 8552}, {4563, 24284}, {7767, 9409}, {23107, 41077}

X(45807) = isotomic conjugate of the isogonal conjugate of X(14417)
X(45807) = isotomic conjugate of the polar conjugate of X(35522)
X(45807) = isogonal conjugate of polar conjugate of isotomic conjugate of X(32729)
X(45807) = isotomic conjugate of trilinear pole of line X(25)X(111) (the radical axis of circumcircle and Moses-Parry circle)
X(45807) = polar conjugate of trilinear pole of line X(2207)X(8753)
X(45807) = pole wrt polar circle of line X(2207)X(8753)
X(45807) = X(14417)-cross conjugate of X(35522)
X(45807) = crosssum of X(2489) and X(14580)
X(45807) = crossdifference of every pair of points on line {1974, 32740}
X(45807) = X(i)-isoconjugate of X(j) for these (i,j): {19, 32729}, {25, 36142}, {111, 32676}, {112, 923}, {162, 32740}, {163, 8753}, {691, 1973}, {811, 19626}, {1576, 36128}, {1974, 36085}, {14908, 24019}, {19136, 36115}, {32713, 36060}
X(45807) = barycentric product X(i)*X(j) for these {i,j}: {69, 35522}, {76, 14417}, {305, 690}, {339, 5468}, {351, 40050}, {524, 3267}, {525, 3266}, {850, 6390}, {2642, 40364}, {3265, 44146}, {3292, 44173}, {4143, 37778}, {4235, 36793}, {4750, 40071}, {14208, 14210}, {14977, 36792}, {15413, 42713}, {20902, 24039}
X(45807) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 32729}, {63, 36142}, {69, 691}, {125, 9178}, {304, 36085}, {305, 892}, {339, 5466}, {351, 1974}, {468, 32713}, {520, 14908}, {523, 8753}, {524, 112}, {525, 111}, {647, 32740}, {656, 923}, {690, 25}, {850, 17983}, {896, 32676}, {1565, 43926}, {1577, 36128}, {1648, 2489}, {1649, 44102}, {2642, 1973}, {3049, 19626}, {3265, 895}, {3266, 648}, {3267, 671}, {3292, 1576}, {3933, 36827}, {4062, 8750}, {4235, 23964}, {4750, 1474}, {5468, 250}, {5486, 32709}, {5642, 23347}, {5967, 32696}, {6333, 5968}, {6390, 110}, {7813, 35325}, {9204, 8739}, {9205, 8740}, {9717, 32715}, {10097, 41936}, {11183, 44089}, {14208, 897}, {14210, 162}, {14273, 2207}, {14417, 6}, {14419, 2203}, {14424, 1843}, {14432, 2299}, {14977, 10630}, {15526, 10097}, {17094, 7316}, {18311, 8744}, {20336, 5380}, {20902, 23894}, {23200, 14574}, {24018, 36060}, {30786, 34574}, {33919, 2971}, {34767, 9139}, {35282, 2445}, {35522, 4}, {36792, 4235}, {36793, 14977}, {36890, 1304}, {37778, 6529}, {39474, 5191}, {40709, 9206}, {40710, 9207}, {42713, 1783}, {42721, 5379}, {44146, 107}, {44814, 34397}

X(45808) = X(69)X(523)∩X(141)X(1640)

Barycentrics    (b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(-a^2 + b^2 - b*c + c^2)*(-a^2 + b^2 + b*c + c^2) : :
X(45808) = 2 X[18310] - 3 X[21356], 3 X[21358] - 2 X[45327]

The barycentrics are proportional to the A-, B-, C- powers of the Hutson-Parry circle.

X(45808) lies on these lines: {69, 523}, {141, 1640}, {183, 30474}, {524, 18311}, {525, 599}, {526, 3268}, {690, 5181}, {850, 44148}, {1350, 1499}, {1649, 5467}, {2088, 5664}, {3906, 14994}, {6333, 9003}, {7778, 9209}, {9168, 38940}, {10411, 39495}, {11183, 33906}, {18310, 21356}, {21358, 45327}

X(45808) = reflection of X(1640) in X(141)
X(45808) = isotomic conjugate of the isogonal conjugate of X(44814)
X(45808) = crosspoint of X(i) and X(j) for these (i,j): {99, 9141}, {5468, 36890}
X(45808) = crosssum of X(512) and X(9142)
X(45808) = crossdifference of every pair of points on line {1692, 11060}
X(45808) = isotomic conjugate of trilinear pole of line X(111)X(230) (the radical axis of circumcircle and Hutson-Parry circle)
X(45808) = X(i)-isoconjugate of X(j) for these (i,j): {111, 32678}, {476, 923}, {897, 14560}, {1989, 36142}, {2153, 9207}, {2154, 9206}, {2166, 32729}, {8753, 36061}, {11060, 36085}, {14908, 36129}, {32662, 36128}, {32680, 32740}
X(45808) = barycentric product X(i)*X(j) for these {i,j}: {76, 44814}, {298, 9205}, {299, 9204}, {323, 35522}, {340, 14417}, {524, 3268}, {526, 3266}, {690, 7799}, {5664, 36890}, {6390, 44427}, {8552, 44146}, {9213, 36792}, {14210, 32679}, {14559, 23965}
X(45808) = barycentric quotient X(i)/X(j) for these {i,j}: {15, 9207}, {16, 9206}, {50, 32729}, {187, 14560}, {323, 691}, {351, 11060}, {524, 476}, {526, 111}, {690, 1989}, {896, 32678}, {1648, 15475}, {2088, 9178}, {2482, 14559}, {2624, 923}, {3266, 35139}, {3268, 671}, {3292, 32662}, {5468, 39295}, {5642, 41392}, {5664, 9214}, {6149, 36142}, {7799, 892}, {8552, 895}, {9204, 14}, {9205, 13}, {9213, 10630}, {14210, 32680}, {14270, 32740}, {14273, 18384}, {14417, 265}, {14559, 23588}, {16186, 10097}, {23870, 36310}, {23871, 36307}, {32679, 897}, {35522, 94}, {36890, 39290}, {42701, 5380}, {44427, 17983}, {44814, 6}, {45662, 23968}

X(45809) = X(2)X(6094)∩X(67)X(315)

Barycentrics    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) : :

The barycentrics are proportional to the A-, B-, C- powers of the Parry isodynamic circle.

The trilinear polar of X(45809) passes through X(8371).

X(45809) lies on these lines: {2, 6094}, {67, 315}, {76, 338}, {99, 2453}, {290, 35179}, {325, 523}, {2782, 23342}, {9464, 36882}, {11059, 11184}, {11185, 36883}, {18896, 40826}

X(45809) = isotomic conjugate of X(843)
X(45809) = isotomic conjugate of the anticomplement of X(44956)
X(45809) = isotomic conjugate of the isogonal conjugate of X(543)
X(45809) = X(44956)-cross conjugate of X(2)
X(45809) = polar conjugate of trilinear pole of line X(25)X(9135)
X(45809) = pole wrt polar circle of line X(25)X(9135)
X(45809) = X(i)-isoconjugate of X(j) for these (i,j): {31, 843}, {560, 18823}, {1924, 9170}
X(45809) = barycentric product X(i)*X(j) for these {i,j}: {76, 543}, {670, 8371}, {850, 9182}, {1502, 2502}, {1641, 18023}, {3266, 17948}, {4609, 9171}, {9181, 44173}, {34760, 35522}
X(45809) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 843}, {76, 18823}, {543, 6}, {670, 9170}, {850, 9180}, {1641, 187}, {2502, 32}, {8371, 512}, {9171, 669}, {9181, 1576}, {9182, 110}, {17948, 111}, {17955, 923}, {17964, 32740}, {18007, 9178}, {23348, 32729}, {33921, 351}, {34760, 691}, {35087, 2502}, {35522, 34763}
X(45809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 670, 36792}, {76, 18023, 338}, {338, 36792, 76}, {670, 18023, 76}, {3260, 30736, 35549}

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CT-Perspectors: X(45810)-X(45841)

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This preamble and centers X(45810)-X(45841) were contributed by César Eliud Lozada, November 7, 2021.

Let T1=A1B1C1 and T2=A2B2C2 be two triangles.
 Tangents from A1, B1, C1 to circles (A1B2C2), (B1C2A2), (C1A2B2) bound a triangle T'=A'B'C'.
 Tangents from A2, B2, C2 to circles (A2B1C1), (B2C1A1), (C2A1B1) bound a triangle T"=A"B"C".
Then the following statements are equivalent:
 1. Triangles A1B1C1 and A"B"C" are perspective.
 2. Triangles A2B2C2 and A'B'C' are perspective.
 3. (B1A2/C1A2)*.(C1B2/A1B2)*(A1C2/B1C2)=1.

Reference: Vu Thanh Tung, Euclid 3001 (Message slightly modified by César Lozada).

If any of the above conditions is fulfilled, perspectors in (1) and (2) are named here the CT-perspector T1 to T2 and the CT-perspector T2 to T1, respectively. (CT stands for Cyclology and Tangents.)

The appearance of (T, i, j) in the following lists means that the CT-perspector ABC to T is X(i) and the CT-perspector T to ABC is X(j) (Note: A double-dash -- means a not calculated perspector):

(ABC-X3 reflections, 64, 1498), (anti-Ara, --, 4), (anti-Ascella, 45810, 25), (1st anti-Brocard, 804, 99), (4th anti-Brocard, 6088, --), (1st anti-circumperp, 3, 22), (anti-Ehrmann-mid, 3, 45811), (anti-excenters-reflections, --, --), (anti-inner-Garcia, 45812, 45396), (anti-Hutson intouch, 1593, 45813), (anti-incircle-circles, 1598, 45814), (anti-inverse-in-incircle, --, --), (anti-McCay, 45815, 8593), (6th anti-mixtilinear, 2, 45816), (anti-orthocentroidal, 526, --), (anti-X3-ABC reflections, --, --), (anticomplementary, 4, 1), (Ara, 3, --), (Ascella, 57, 45817), (Bankoff, 3390, --), (BCI, --, --), (Bevan antipodal, 45818, 1), (1st Brocard, 6, 512), (2nd Brocard, --, --), (3rd Brocard, --, --), (4th Brocard, 45819, 523), (1st Brocard-reflected, 6, 3734), (circummedial, 25, 22), (circumorthic, 3, 24), (1st circumperp, 55, 3), (2nd circumperp, 56, 3), (circumsymmedial, 6, 6), (3rd Conway, 45820, 1764), (Ehrmann-mid, 45821, 4), (Ehrmann-side, 381, 45822), (Ehrmann-vertex, --, --), (1st Ehrmann, 1995, 10870), (2nd Euler, 5, 45823), (5th Euler, --, --), (excenters-midpoints, 45824, 45825), (excenters-reflections, --, --), (excentral, 3062, 1), (extouch, 1, 34488), (inner-Fermat, 6, 45826), (outer-Fermat, 6, 45827), (15th Fermat-Dao, --, 36766), (16th Fermat-Dao, --, --), (Fuhrmann, 56, 513), (2nd Fuhrmann, 55, 517), (inner-Garcia, 3417, 45828), (outer-Garcia, 4, 40), (Garcia-reflection, 4, 45829), (Gossard, --, --), (1st half-diamonds, --, --), (2nd half-diamonds, --, --), (1st half-squares, --, --), (2nd half-squares, --, --), (1st Hatzipolakis, --, --), (2nd Hatzipolakis, 1, --), (hexyl, 5665, 3), (Hutson extouch, --, --), (Hutson intouch, 5665, 1697), (incentral, 1, 38814), (incircle-circles, 45830, 3333), (intouch, 1, 57), (inverse-in-Conway, --, --), (inverse-in-incircle, --, --), (1st isodynamic-Dao, --, 23006), (2nd isodynamic-Dao, --, 23013), (3rd isodynamic-Dao, --, 36771), (4th isodynamic-Dao, --, --), (2nd Jenkins, --, --), (Johnson, 3, 4), (K798e, 1, --), (K798i, 1, --), (Kosnita, 24, 45831), (Lemoine, 1, --), (Lucas inner, 6425, 6429), (Lucas(-1) inner, 6426, 6430), (Lucas tangents, 3, 1151), (Lucas(-1) tangents, 3, 1152), (Macbeath, 1, 45832), (McCay, 6, --), (medial, 1, 3), (midheight, 45833, 4), (2nd mixtilinear, --, --), (3rd mixtilinear, 3445, 1616), (4th mixtilinear, 11051, 1615), (5th mixtilinear, --, --), (6th mixtilinear, 45834, 165), (8th mixtilinear, --, --), (9th mixtilinear, --, --), (1st Morley, 1134, --), (2nd Morley, 357, --), (3rd Morley, 1136, --), (1st Morley-adjunct, --, --), (2nd Morley-adjunct, --, --), (3rd Morley-adjunct, --, --), (Moses-Soddy, --, 1), (Moses-Steiner reflection, 45835, 69), (inner-Napoleon, 6, --), (outer-Napoleon, 6, --), (1st Neuberg, 6, 45836), (2nd Neuberg, 6, --), (orthic, 1, 193), (orthocentroidal, 18361, 523), (Pelletier, 45837, 1), (reflection, 45838, 35727), (Schroeter, --, 1), (Soddy, --, 1), (2nd inner-Soddy, --, --), (2nd outer-Soddy, --, --), (inner-squares, --, 8854), (outer-squares, --, 8855), (Steiner, 1, 4558), (symmedial, 1, 38817), (tangential, 25, 1), (Trinh, 378, 45839), (inner-Vecten, 6, 45840), (2nd inner-Vecten, 14236, 32488), (outer-Vecten, 6, 45841), (2nd outer-Vecten, 14240, 32489), (X-parabola-tangential, --, 1), (X3-ABC reflections, --, --), (Yff contact, 1, 1331)

X(45810) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO ANTI-ASCELLA

Barycentrics    a^2*(3*a^4-10*c^2*a^2+3*c^4-3*b^4)*(3*a^4-10*b^2*a^2+3*b^4-3*c^4) : :

The reciprocal CT-perspector of these triangles is X(25)

X(45810) lies on these lines: {3, 20080}, {98, 3516}, {184, 5023}, {2351, 40321}, {3425, 15750}, {5013, 40319}, {9605, 14908}

X(45810) = isogonal conjugate of the anticomplement of X(15815)
X(45810) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(3053)}} and {{A, B, C, X(3), X(25)}}

X(45811) = CT-PERSPECTOR OF THESE TRIANGLES: ANTI-EHRMANN-MID TO ABC

Barycentrics    a^2*(2*a^10-(b^2+c^2)*a^8-(7*b^4-6*b^2*c^2+7*c^4)*a^6+(b^2+c^2)*(7*b^4-13*b^2*c^2+7*c^4)*a^4+(b^8+c^8-2*b^2*c^2*(b^4-4*b^2*c^2+c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4+b^2*c^2-2*c^4)) : :
X(45811) = 3*X(381)-2*X(45821)

The reciprocal CT-perspector of these triangles is X(3)

X(45811) lies on these lines: {3, 34798}, {23, 385}, {381, 45821}, {3581, 14264}, {15107, 40352}, {20897, 40914}

X(45811) = X(45821)-of-anti-Ehrmann-mid triangle

X(45812) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO ANTI-INNER-GARCIA

Barycentrics    a^2*(a^2-b^2+b*c-c^2)*(a^6-(b+2*c)*a^5-(b^2-5*b*c+c^2)*a^4+(b-2*c)*(2*b^2+b*c-2*c^2)*a^3-(b^2+6*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b^3-2*c^3-5*(b-c)*b*c)*a+(b^2-c^2)*(b-c)*(b^3+c^3))*(a^6-(2*b+c)*a^5-(b^2-5*b*c+c^2)*a^4+(2*b-c)*(2*b^2-b*c-2*c^2)*a^3-(b^2+6*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(2*b^3-c^3-5*(b-c)*b*c)*a+(b^2-c^2)*(b-c)*(b^3+c^3)) : :

The reciprocal CT-perspector of these triangles is X(45396)

X(45812) lies on the cubic K274 and these lines: {36, 2800}, {859, 35451}

X(45812) = isogonal conjugate of X(38954)
X(45812) = X(45828)-of-anti-inner-Garcia triangle
X(45812) = X(4)-vertex conjugate of-X(952)

X(45813) = CT-PERSPECTOR OF THESE TRIANGLES: ANTI-HUTSON INTOUCH TO ABC

Barycentrics    a^2*(a^14-3*(b^2+c^2)*a^12+(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*a^10+(b^2+c^2)*(5*b^4+22*b^2*c^2+5*c^4)*a^8-(5*b^8+5*c^8-14*b^2*c^2*(2*b^4-b^2*c^2+2*c^4))*a^6-(b^2+c^2)*(b^8+c^8+2*b^2*c^2*(30*b^4-29*b^2*c^2+30*c^4))*a^4+3*(b^4-c^4)^2*(b^4+6*b^2*c^2+c^4)*a^2-(b^2-c^2)^6*(b^2+c^2)) : :

The reciprocal CT-perspector of these triangles is X(1593)

X(45813) lies on the cubic K1109 and these lines: {3, 12233}, {20, 36851}, {22, 6225}, {24, 40917}, {25, 33537}, {64, 161}, {185, 1350}, {550, 12301}, {1593, 35240}, {5889, 32621}, {6642, 33540}, {7387, 11472}, {7393, 40909}, {7488, 32605}, {7506, 14926}, {7517, 33539}, {9715, 10117}, {10984, 17809}, {12083, 33541}, {12163, 35243}, {12302, 37853}, {13730, 33536}, {15622, 37577}, {15644, 33543}, {15740, 43725}, {21312, 32345}, {33522, 33524}

X(45813) = (anti-Hutson intouch)-isogonal conjugate-of-X(3516)
X(45813) = X(45015)-of-ABC-X3 reflections triangle
X(45813) = reflection of X(45015) in X(3)

X(45814) = CT-PERSPECTOR OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ABC

Barycentrics    a^2*(a^14-3*(b^2+c^2)*a^12+(b^2-8*b*c+c^2)*(b^2+8*b*c+c^2)*a^10+(b^2+c^2)*(5*b^4+178*b^2*c^2+5*c^4)*a^8-(5*b^8+5*c^8+2*(82*b^4-185*b^2*c^2+82*c^4)*b^2*c^2)*a^6-(b^2+c^2)*(b^8+c^8-2*b^2*c^2*(30*b^4-187*b^2*c^2+30*c^4))*a^4+3*(b^2-c^2)^2*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*(b^4+6*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)) : :

The reciprocal CT-perspector of these triangles is X(1598)

X(45814) lies on these lines: {1216, 11820}, {12315, 15644}, {18535, 37514}

X(45815) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO ANTI-MCCAY

Barycentrics    (5*a^4-2*(b^2+c^2)*a^2+(2*b^2-c^2)*(b^2-2*c^2))*(2*a^6-3*(b^2+c^2)*a^4+3*(2*b^2-c^2)*(b^2+c^2)*a^2+2*c^6+3*(2*b^2-c^2)*b^2*c^2-7*b^6)*(2*a^6-3*(b^2+c^2)*a^4-3*(b^2-2*c^2)*(b^2+c^2)*a^2-7*c^6-3*(b^2-2*c^2)*b^2*c^2+2*b^6) : :

The reciprocal CT-perspector of these triangles is X(8593)

X(45815) lies on the line {325, 543}

X(45816) = CT-PERSPECTOR OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR TO ABC

Barycentrics    (-a^2+b^2+c^2)*(3*a^6+11*(b^2+c^2)*a^4+3*(3*b^4+2*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal CT-perspector of these triangles is X(2)

X(45816) lies on these lines: {2, 41464}, {3, 3618}, {69, 10691}, {141, 1853}, {193, 2979}, {511, 11431}, {1040, 2293}, {1176, 33750}, {1216, 6776}, {1974, 3522}, {3448, 15812}, {3555, 37613}, {3564, 11850}, {3785, 22078}, {5894, 19149}, {5907, 34781}, {6643, 33540}, {6699, 28438}, {7400, 19130}, {10519, 12359}, {11487, 18440}, {11821, 12164}, {15462, 37853}, {20806, 35254}

X(45816) = (6th anti-mixtilinear)-isogonal conjugate-of-X(7494)

X(45817) = CT-PERSPECTOR OF THESE TRIANGLES: ASCELLA TO ABC

Barycentrics    a*(9*a^6-(39*b^2-38*b*c+39*c^2)*a^4+8*(b+c)*(3*b^2-5*b*c+3*c^2)*a^3+(27*b^4+27*c^4-10*b*c*(2*b-c)*(b-2*c))*a^2-24*(b^3-c^3)*(b^2-c^2)*a+(b+3*c)*(3*b+c)*(b-c)^4) : :

The reciprocal CT-perspector of these triangles is X(57)

X(45817) lies on these lines: {3, 3062}, {3022, 3601}, {5745, 23058}

X(45817) = X(45810)-of-Ascella triangle

X(45818) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO BEVAN ANTIPODAL

Barycentrics    a*(a^3+3*(b-c)*a^2+(3*b+c)*(b-c)*a+(b^2-c^2)*(b-3*c))*(a^3-3*(b-c)*a^2-(b+3*c)*(b-c)*a+(b^2-c^2)*(3*b-c)) : :

The reciprocal CT-perspector of these triangles is X(1)

X(45818) lies on these lines: {46, 1743}, {145, 515}, {1420, 1455}

X(45818) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(145)}} and {{A, B, C, X(4), X(269)}}

X(45819) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO 4th BROCARD

Barycentrics    (a^4-(b^2-3*c^2)*a^2-(b^2-c^2)*c^2)*(a^4+(3*b^2-c^2)*a^2+(b^2-c^2)*b^2) : :

The reciprocal CT-perspector of these triangles is X(523)

X(45819) lies on these lines: {30, 182}, {183, 1995}, {523, 34417}, {1576, 10788}, {1990, 10301}, {2452, 9971}, {2782, 37827}, {3613, 33578}, {5640, 32224}, {6325, 13239}, {7737, 32740}, {7798, 25322}, {40981, 45838}

X(45819) = isogonal conjugate of X(7998)
X(45819) = isotomic conjugate of the anticomplement of X(5355)
X(45819) = 1st Saragossa point of X(30542)
X(45819) = trilinear pole of the line {1637, 3288}
X(45819) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(597)}} and {{A, B, C, X(3), X(33872)}}

X(45820) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO 3rd CONWAY

Barycentrics    a*((2*b^2+5*b*c+4*c^2)*a^3+2*(2*b^3+2*c^3+(4*b+3*c)*b*c)*a^2+(b+c)*(2*b^2+3*b*c+5*c^2)*b*a+2*(b+c)^2*b^2*c)*((4*b^2+5*b*c+2*c^2)*a^3+2*(2*b^3+2*c^3+(3*b+4*c)*b*c)*a^2+(b+c)*(5*b^2+3*b*c+2*c^2)*c*a+2*(b+c)^2*b*c^2) : :

The reciprocal CT-perspector of these triangles is X(1764)

X(45820) lies on the Feuerbach circumhyperbola and these lines: {2298, 10434}, {3666, 10435}, {3931, 45032}

X(45821) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO EHRMANN-MID

Barycentrics    (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-c^2*a^4-(3*b^4-b^2*c^2+c^4)*a^2+(2*b^2+c^2)*(b^2-c^2)^2)*(a^6-b^2*a^4-(b^4-b^2*c^2+3*c^4)*a^2+(b^2+2*c^2)*(b^2-c^2)^2) : :
X(45821) = 3*X(381)-X(45811)

The reciprocal CT-perspector of these triangles is X(4)

X(45821) lies on these lines: {4, 3581}, {316, 14387}, {381, 45811}, {477, 7574}, {523, 14314}, {1138, 3153}, {1531, 14254}, {5627, 18403}

X(45821) = isotomic conjugate of the anticomplement of X(15816)
X(45821) = trilinear quotient X(1784)/X(18559)
X(45821) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(30)}} and {{A, B, C, X(265), X(340)}}
X(45821) = X(45811)-of-Ehrmann-mid triangle
X(45821) = X(133)-Dao conjugate of-X(18559)

X(45822) = CT-PERSPECTOR OF THESE TRIANGLES: EHRMANN-SIDE TO ABC

Barycentrics    (-a^2+b^2+c^2)*(3*a^14-4*(b^2+c^2)*a^12-(8*b^4-11*b^2*c^2+8*c^4)*a^10+(b^2+c^2)*(13*b^4-18*b^2*c^2+13*c^4)*a^8+3*(b^2+b*c-c^2)^2*(b^2-b*c-c^2)^2*a^6-2*(b^4-c^4)*(b^2-c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^4+(2*b^4+7*b^2*c^2+2*c^4)*(b^2-c^2)^4*a^2+(b^2+c^2)*(b^2-c^2)^6) : :

The reciprocal CT-perspector of these triangles is X(381)

X(45822) lies on these lines: {3, 34798}, {30, 41466}, {265, 18564}, {1147, 18442}, {6288, 18562}, {12383, 22584}, {18439, 34785}, {18441, 19402}, {18882, 44249}

X(45822) = (Ehrmann-side)-isogonal conjugate-of-X(10254)

X(45823) = CT-PERSPECTOR OF THESE TRIANGLES: 2nd EULER TO ABC

Barycentrics    (-a^2+b^2+c^2)*(3*a^14-3*(b^2+c^2)*a^12-(11*b^4+6*b^2*c^2+11*c^4)*a^10+3*(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^8+(5*b^8+5*c^8-2*b^2*c^2*(2*b^4-15*b^2*c^2+2*c^4))*a^6-(b^4-c^4)*(b^2-c^2)*(13*b^4-2*b^2*c^2+13*c^4)*a^4+3*(b^4-c^4)^2*(b^2-c^2)^2*a^2+(b^2+c^2)*(b^2-c^2)^6) : :

The reciprocal CT-perspector of these triangles is X(5)

X(45823) lies on these lines: {3, 12233}, {68, 12362}, {125, 6643}, {155, 4549}, {184, 35240}, {1209, 18531}, {1352, 12605}, {5562, 6193}, {7723, 23236}, {9833, 12162}, {9967, 32366}, {11898, 44076}, {20427, 44249}, {31807, 45044}, {31829, 33543}

X(45823) = (2nd Euler)-isogonal conjugate-of-X(3549)

X(45824) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO EXCENTERS-MIDPOINTS

Barycentrics    a*(2*a-b-c)*(a^3-(b+c)*a^2-(3*b^2-6*b*c+c^2)*a+(b-c)*(3*b^2-c^2))*(a^3-(b+c)*a^2-(b^2-6*b*c+3*c^2)*a+(b-c)*(b^2-3*c^2)) : :

The reciprocal CT-perspector of these triangles is X(45825)

X(45824) lies on these lines: {518, 2098}, {13252, 30198}

X(45824) = barycentric quotient X(900)/X(29005)
X(45824) = intersection, other than A, B, C, of circumconics {{A, B, C, X(8), X(1319)}} and {{A, B, C, X(44), X(518)}}
X(45824) = X(900)-reciprocal conjugate of-X(29005)

X(45825) = CT-PERSPECTOR OF THESE TRIANGLES: EXCENTERS-MIDPOINTS TO ABC

Barycentrics    a*(4*a^4-10*(b+c)*a^3+(8*b^2+21*b*c+8*c^2)*a^2-(b+c)*(2*b^2+11*b*c+2*c^2)*a+4*b*c*(b^2+c^2)) : :

The reciprocal CT-perspector of these triangles is X(45824)

X(45825) lies on these lines: {1, 27819}, {3679, 4595}, {5732, 16192}

X(45825) = (excenters-midpoints)-isogonal conjugate-of-X(2)
X(45825) = X(20587)-of-1st circumperp triangle

X(45826) = CT-PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT TO ABC

Barycentrics    (-2*(2*a^6+3*(b^2+c^2)*a^4+b^2*c^2*a^2-(b^2+c^2)*(5*b^4-4*b^2*c^2+5*c^4))*S+((b^2+c^2)*a^6-(3*b^4-13*b^2*c^2+3*c^4)*a^4+3*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4+10*b^2*c^2+c^4)*(b^2-c^2)^2)*sqrt(3))*a^2 : :

The reciprocal CT-perspector of these triangles is X(6)

X(45826) lies on these lines: {30, 14541}, {10646, 34008}

X(45827) = CT-PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT TO ABC

Barycentrics    a^2*(2*(2*a^6+3*(b^2+c^2)*a^4+b^2*c^2*a^2-(b^2+c^2)*(5*b^4-4*b^2*c^2+5*c^4))*S+((b^2+c^2)*a^6-(3*b^4-13*b^2*c^2+3*c^4)*a^4+3*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4+10*b^2*c^2+c^4)*(b^2-c^2)^2)*sqrt(3)) : :

The reciprocal CT-perspector of these triangles is X(6)

X(45827) lies on these lines: {30, 14540}, {10645, 34009}

X(45828) = CT-PERSPECTOR OF THESE TRIANGLES: INNER-GARCIA TO ABC

Barycentrics    a^2*(a^11-2*(b+c)*a^10-(2*b^2-9*b*c+2*c^2)*a^9+(b+c)*(7*b^2-13*b*c+7*c^2)*a^8-(2*b^4+2*c^4+b*c*(19*b^2-25*b*c+19*c^2))*a^7-(b+c)*(8*b^4+8*c^4-b*c*(37*b^2-42*b*c+37*c^2))*a^6+(8*b^6+8*c^6+(3*b^4+3*c^4-11*b*c*(4*b^2-3*b*c+4*c^2))*b*c)*a^5+(b+c)*(2*b^6+2*c^6-(33*b^4+33*c^4-b*c*(65*b^2-59*b*c+65*c^2))*b*c)*a^4-(7*b^8+7*c^8-(15*b^6+15*c^6+(21*b^4+21*c^4-2*b*c*(16*b^2-b*c+16*c^2))*b*c)*b*c)*a^3+(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(11*b^4+11*c^4-b*c*(12*b^2+7*b*c+12*c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^6+c^6-(4*b^4+4*c^4-b*c*(2*b^2+b*c+2*c^2))*b*c)*a-(b^2-c^2)^3*(b-c)*(b^4-b^2*c^2+c^4)) : :

The reciprocal CT-perspector of these triangles is X(3417)

X(45828) lies on the line {100, 515}

X(45828) = X(45812)-of-inner-Garcia triangle

X(45829) = CT-PERSPECTOR OF THESE TRIANGLES: GARCIA-REFLECTION TO ABC

Barycentrics    a*((2*b-c)*(b-2*c)*a^3+2*(b+c)*b*c*a^2-(2*b^2-3*b*c+2*c^2)*(b+c)^2*a+4*(b^2-c^2)*(b-c)*b*c) : :
X(45829) = 3*X(1699)-2*X(37521)

The reciprocal CT-perspector of these triangles is X(4)

X(45829) lies on these lines: {1, 33551}, {165, 2108}, {200, 38389}, {354, 4888}, {382, 517}, {516, 4685}, {1293, 9350}, {1357, 21267}, {1699, 15310}, {3667, 17155}, {3681, 9519}, {3817, 4871}, {9812, 10439}, {30827, 38390}

X(45829) = X(13157)-of-Ursa-minor triangle
X(45829) = reflection of X(10439) in X(9812)

X(45830) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO INCIRCLE-CIRCLES

Barycentrics    a*(a^3-(b-3*c)*a^2-(b^2+14*b*c+c^2)*a+(b^2-c^2)*(b+3*c))*(a^3+(3*b-c)*a^2-(b^2+14*b*c+c^2)*a-(b^2-c^2)*(3*b+c)) : :

The reciprocal CT-perspector of these triangles is X(3333)

X(45830) lies on the Feuerbach circumhyperbola and these lines: {4, 9819}, {7, 7991}, {8, 30337}, {65, 45834}, {84, 9851}, {165, 7091}, {1000, 40270}, {1476, 7987}, {1697, 3062}, {2093, 5557}, {3296, 3339}, {3679, 15998}, {3890, 45121}, {4866, 31393}, {4900, 9957}, {5435, 30343}, {5558, 10980}, {5665, 11531}, {6601, 12640}, {6738, 7320}, {11224, 17097}, {13462, 15179}, {15909, 45081}, {30326, 33576}

X(45830) = touchpoint of the line {37556, 45830} and Feuerbach circumhyperbola
X(45830) = X(45814)-of-incircle-circles triangle
X(45830) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(3), X(9819)}}

X(45831) = CT-PERSPECTOR OF THESE TRIANGLES: KOSNITA TO ABC

Barycentrics    a^2*(a^14-3*(b^2+c^2)*a^12+(b^4+7*b^2*c^2+c^4)*a^10+(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^8-(5*b^8+5*c^8+b^2*c^2*(5*b^4-7*b^2*c^2+5*c^4))*a^6-(b^2+c^2)*(b^8+c^8-b^2*c^2*(9*b^4-14*b^2*c^2+9*c^4))*a^4+3*(b^2-c^2)^2*(b^8-b^4*c^4+c^8)*a^2-(b^4-c^4)*(b^2-c^2)^3*(b^4+b^2*c^2+c^4)) : :

The reciprocal CT-perspector of these triangles is X(24)

X(45831) lies on these lines: {3, 12278}, {20, 13293}, {22, 5925}, {24, 40909}, {54, 15053}, {68, 5963}, {110, 1658}, {182, 11443}, {186, 1147}, {1092, 32338}, {2888, 10298}, {2937, 38790}, {3520, 7703}, {6101, 19381}, {6759, 7488}, {7502, 8718}, {7512, 8717}, {7689, 12270}, {8907, 18324}, {9932, 32534}, {11935, 12161}, {14118, 18392}, {15069, 15577}, {15646, 44076}, {18571, 32358}, {19467, 22533}, {22550, 37954}, {38397, 38448}

X(45831) = reflection of X(i) in X(j) for these (i, j): (16000, 24572), (16013, 3)
X(45831) = circumnormal-isogonal conjugate of X(11250)
X(45831) = (Kosnita)-isogonal conjugate-of-X(4)
X(45831) = X(16013)-of-ABC-X3 reflections triangle
X(45831) = inverse of X(1658) in Stammler hyperbola

X(45832) = CT-PERSPECTOR OF THESE TRIANGLES: MACBEATH TO ABC

Barycentrics    a^2*(a^14-4*(b^2+c^2)*a^12+3*(2*b^4+3*b^2*c^2+2*c^4)*a^10-(b^2+c^2)*(5*b^4+b^2*c^2+5*c^4)*a^8+(5*b^8+2*b^4*c^4+5*c^8)*a^6-6*(b^8-c^8)*(b^2-c^2)*a^4+(4*b^8+4*c^8-(b^2+c^2)^2*b^2*c^2)*(b^2-c^2)^2*a^2-(b^6+c^6)*(b^2-c^2)^4)*(a^4-(b^2+2*c^2)*a^2-(b^2-c^2)*c^2)*(a^4-(2*b^2+c^2)*a^2+(b^2-c^2)*b^2) : :

The reciprocal CT-perspector of these triangles is X(1)

X(45832) lies on these lines: {3, 2120}, {97, 37636}, {1994, 14586}, {3575, 8883}, {8795, 44375}, {11077, 13567}

X(45832) = X(264)-Ceva conjugate of-X(54)

X(45833) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO MIDHEIGHT

Barycentrics    (a^4-2*(5*b^2-c^2)*a^2+(b^2-c^2)*(b^2+3*c^2))*(a^4+2*(b^2-5*c^2)*a^2-(b^2-c^2)*(3*b^2+c^2)) : :

The reciprocal CT-perspector of these triangles is X(4)

X(45833) lies on these lines: {193, 1350}, {5304, 6353}, {10002, 21447}

X(45833) = isotomic conjugate of the anticomplement of X(7738)
X(45833) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(193)}} and {{A, B, C, X(4), X(3522)}}

X(45834) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO 6th MIXTILINEAR

Barycentrics    a*(a^2-2*(b+3*c)*a+(b-c)*(b-5*c))*(a^2-2*(3*b+c)*a+(5*b-c)*(b-c)) : :

The reciprocal CT-perspector of these triangles is X(165)

X(45834) lies on the Feuerbach circumhyperbola and these lines: {7, 30350}, {8, 5542}, {9, 3742}, {21, 17207}, {65, 45830}, {84, 24644}, {165, 2346}, {294, 16667}, {354, 3062}, {516, 5558}, {885, 28225}, {942, 4866}, {943, 3361}, {1000, 18421}, {1100, 42317}, {1156, 18240}, {1476, 12560}, {2951, 10390}, {3243, 31509}, {3296, 4312}, {3339, 7160}, {3680, 42871}, {4321, 17097}, {4328, 43736}, {4900, 11529}, {5223, 32635}, {5541, 12868}, {5556, 43180}, {5728, 33576}, {7091, 30343}, {7319, 30340}, {7320, 11038}, {7707, 8090}, {9814, 11025}, {14100, 31507}, {16673, 40779}, {18398, 38271}, {20059, 40998}, {30308, 41857}

X(45834) = barycentric quotient X(57)/X(5543)
X(45834) = trilinear quotient X(7)/X(5543)
X(45834) = trilinear pole of the line {650, 23738}
X(45834) = touchpoint of the line {44841, 45834} and Feuerbach circumhyperbola
X(45834) = intersection, other than A, B, C, of circumconics Feuerbach hyperbola and {{A, B, C, X(55), X(30350)}}
X(45834) = X(223)-Dao conjugate of-X(5543)
X(45834) = X(55)-isoconjugate-of-X(5543)
X(45834) = X(45816)-of-6th mixtilinear triangle
X(45834) = X(57)-reciprocal conjugate of-X(5543)

X(45835) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO MOSES-STEINER REFLECTION

Barycentrics    (-a^2+b^2+c^2)*(a^6-(b^2-2*c^2)*a^4-(b^4-2*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^6+(2*b^2-c^2)*a^4+(2*b^4-c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

The reciprocal CT-perspector of these triangles is X(69)

X(45835) lies on the Jerabek circumhyperbola and these lines: {6, 31133}, {67, 11188}, {74, 3818}, {858, 19151}, {1177, 5169}, {1995, 34437}, {3619, 20421}, {5189, 5486}, {8549, 42059}, {12220, 13622}, {15328, 44445}, {17711, 43130}, {34439, 43608}, {39899, 43704}, {41737, 43578}

X(45835) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(31133)}} and Jerabek hyperbola

X(45836) = CT-PERSPECTOR OF THESE TRIANGLES: 1st NEUBERG TO ABC

Barycentrics    a^2*((2*b^4+b^2*c^2+2*c^4)*a^10-3*(b^2+c^2)*b^2*c^2*a^8+(2*b^8+2*c^8+3*b^2*c^2*(b^4+c^4))*a^6-(b^2+c^2)*(b^4+c^4)*(4*b^4-3*b^2*c^2+4*c^4)*a^4+2*(b^2-c^2)^2*b^4*c^4*a^2-2*b^4*c^4*(b^2+c^2)^3) : :

The reciprocal CT-perspector of these triangles is X(6)

X(45836) lies on these lines: {30, 7801}, {3098, 5106}

X(45837) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO PELLETIER

Barycentrics    (-a+b+c)*(a^6-2*b*a^5-(b-2*c)*b*a^4+2*(b-c)*(2*b^2+c^2)*a^3-(b-c)*(b^2+b*c+2*c^2)*b*a^2-2*(b^2-c^2)*(b-c)^2*b*a+(b^2-c^2)^2*(b-c)^2)*(a^6-2*c*a^5+(2*b-c)*c*a^4-2*(b-c)*(b^2+2*c^2)*a^3+(b-c)*(2*b^2+b*c+c^2)*c*a^2+2*(b^2-c^2)*(b-c)^2*c*a+(b^2-c^2)^2*(b-c)^2) : :

The reciprocal CT-perspector of these triangles is X(1)

X(45837) lies on these lines: {971, 3583}, {34529, 35604}

X(45837) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(971)}} and {{A, B, C, X(33), X(3583)}}

X(45838) = CT-PERSPECTOR OF THESE TRIANGLES: ABC TO REFLECTION

Barycentrics    (a^4-(b^2+c^2)*a^2-(b^2-c^2)*c^2)*(a^4-(b^2+c^2)*a^2+(b^2-c^2)*b^2) : :

The reciprocal CT-perspector of these triangles is X(35727)

X(45838) lies on these lines: {3, 39910}, {24, 157}, {32, 3613}, {98, 160}, {140, 141}, {183, 1232}, {230, 427}, {237, 2980}, {577, 7668}, {590, 26920}, {615, 8911}, {1078, 1502}, {1485, 2351}, {1899, 14533}, {2165, 32654}, {2965, 37988}, {5961, 7502}, {7846, 40425}, {7857, 40416}, {9756, 14486}, {10691, 13468}, {13860, 14495}, {15004, 45108}, {15109, 39906}, {15523, 21012}, {16063, 17008}, {16321, 37935}, {35894, 37688}, {37119, 41770}, {40981, 45819}

X(45838) = isogonal conjugate of X(3060)
X(45838) = isotomic conjugate of X(7752)
X(45838) = barycentric quotient X(i)/X(j) for these (i, j): (1, 18041), (115, 34981)
X(45838) = trilinear quotient X(i)/X(j) for these (i, j): (2, 18041), (1109, 34981)
X(45838) = trilinear pole of the line {826, 3288}
X(45838) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(66)}} and {{A, B, C, X(3), X(24)}}
X(45838) = Cevapoint of X(i) and X(j) for these (i, j): {2, 7793}, {125, 39201}
X(45838) = X(i)-Dao conjugate of-X(j) for these (i, j): (9, 18041), (523, 34981)
X(45838) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 18041}, {211, 3112}, {1101, 34981}
X(45838) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (1, 18041), (115, 34981)
X(45838) = X(6)-vertex conjugate of-X(2980)

X(45839) = CT-PERSPECTOR OF THESE TRIANGLES: TRINH TO ABC

Barycentrics    a^2*(a^10-(b^2+c^2)*a^8-(2*b^4-b^2*c^2+2*c^4)*a^6+2*(b^6+c^6)*a^4+(b^4+b^2*c^2+c^4)*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4)) : :

The reciprocal CT-perspector of these triangles is X(378)

X(45839) lies on these lines: {2, 32600}, {3, 34798}, {20, 32401}, {22, 1853}, {23, 7703}, {74, 7502}, {110, 35228}, {146, 10298}, {184, 323}, {186, 4550}, {376, 12319}, {550, 16013}, {1176, 19151}, {1658, 15062}, {3357, 7488}, {3448, 7492}, {3522, 22978}, {5012, 18438}, {7493, 13203}, {7512, 7689}, {7525, 7691}, {7998, 37978}, {9938, 10323}, {11443, 12220}, {12041, 19402}, {18324, 40914}, {18570, 35257}, {22109, 41398}, {22467, 33879}, {32598, 35268}, {38448, 45014}

X(45839) = reflection of X(19402) in X(12041)
X(45839) = Gibert-circumtangential conjugate of X(3818)
X(45839) = circumtangential-isogonal conjugate of X(18570)
X(45839) = (Trinh)-isogonal conjugate-of-X(35473)
X(45839) = inverse of X(35228) in Stammler hyperbola

X(45840) = CT-PERSPECTOR OF THESE TRIANGLES: INNER-VECTEN TO ABC

Barycentrics    a^2*(-2*(a^6+(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2))*S+(b^2+c^2)*a^6-3*(b^2-c^2)^2*a^4+(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2) : :

The reciprocal CT-perspector of these triangles is X(6)

X(45840) lies on these lines: {30, 6215}, {371, 6406}, {372, 30427}, {1495, 3156}, {6561, 8383}, {9733, 18451}, {16194, 45841}

X(45841) = CT-PERSPECTOR OF THESE TRIANGLES: OUTER-VECTEN TO ABC

Barycentrics    a^2*(2*(a^6+(b^4+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2))*S+(b^2+c^2)*a^6-3*(b^2-c^2)^2*a^4+(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2-(b^4+8*b^2*c^2+c^4)*(b^2-c^2)^2) : :

The reciprocal CT-perspector of these triangles is X(6)

X(45841) lies on these lines: {30, 6214}, {371, 30428}, {372, 6291}, {1495, 3155}, {6560, 8384}, {9732, 18451}, {16194, 45840}

X(45842) = EULER LINE INTERCEPT OF X(185)X(2055)

Barycentrics    a^2*(-a^2+b^2+c^2)*(a^12-3*(b^2+c^2)*a^10+(3*b^4+11*b^2*c^2+3*c^4)*a^8-2*(b^2+c^2)^3*a^6+(3*b^4-4*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^4-3*(b^4-c^4)*(b^2-c^2)^3*a^2+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^4) : :
Barycentrics    (S^2+8*R^2*(10*R^2-SA-4*SW)+2*SA^2+3*SW^2)*(S^2-SB*SC) : :

See Antreas Hatzipolakis and César Lozada, euclid 3039.

X(45842) lies on these lines: {2, 3}, {97, 15062}, {185, 2055}, {577, 3357}, {1941, 41481}, {3284, 13382}, {5663, 19193}, {6000, 14152}, {6241, 23606}, {12118, 18437}, {14059, 43652}, {14915, 37081}, {17702, 36245}

X(45842) = {X(418), X(3520)}-harmonic conjugate of X(3)

X(45843) = X(2)X(6)∩X(51)X(2056)

Barycentrics    a^2*(a^4+b^4-b^2*c^2+c^4-3*(b^2+c^2)*a^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 3039.

X(45843) lies on these lines: {2, 6}, {51, 2056}, {184, 13330}, {195, 3499}, {251, 20976}, {511, 10329}, {575, 3787}, {576, 3981}, {1194, 5111}, {1196, 5097}, {1501, 11422}, {1627, 35006}, {1691, 13366}, {1915, 5052}, {2076, 5012}, {2979, 5116}, {3291, 34565}, {3917, 5038}, {4271, 35216}, {5017, 11402}, {5104, 22352}, {5943, 9225}, {8041, 23061}, {13196, 16951}, {15514, 20859}, {15544, 33704}

X(45843) = reflection of X(10329) in X(14153)
X(45843) = crosssum of X(1084) and X(20188)
X(45843) = X(1506)-Ceva conjugate of-X(15109)
X(45843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 21001, 5422), (51, 2056, 20998), (1994, 3051, 6), (5052, 34986, 1915)

X(45844) = X(20)X(64)∩X(2060)X(5895)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(8*a^18-17*(b^2+c^2)*a^16-8*(4*b^4-15*b^2*c^2+4*c^4)*a^14+4*(b^2+c^2)*(27*b^4-58*b^2*c^2+27*c^4)*a^12-8*(b^2-c^2)^2*(4*b^4+47*b^2*c^2+4*c^4)*a^10-2*(b^4-c^4)*(b^2-c^2)*(79*b^4-254*b^2*c^2+79*c^4)*a^8+8*(b^2-c^2)^2*(24*b^8+24*c^8+(3*b-c)*(b-3*c)*(b+3*c)*(3*b+c)*b^2*c^2)*a^6-4*(b^4-c^4)*(b^2-c^2)^3*(17*b^4+82*b^2*c^2+17*c^4)*a^4-8*(b^2-c^2)^4*(b^8+c^8-(11*b^4+28*b^2*c^2+11*c^4)*b^2*c^2)*a^2+(b^2-c^2)^6*(b^2+c^2)*(7*b^4+18*b^2*c^2+7*c^4)) : :
Barycentrics    (S^2-2*SB*SC)*((8*R^2-SA-SW)*S^2+4*(4*R^2-SW)*(8*R^2*(20*R^2-7*SW)-SA^2+5*SW^2)) : :

See Antreas Hatzipolakis and César Lozada, euclid 3039.

X(45844) lies on these lines: {20, 64}, {2060, 5895}, {2883, 27089}, {3079, 5925}

leftri

Points associated with Vijay polar medial circles and Vijay polar medial triangles: X(45845)-X(45873)

rightri

This preamble is contributed by Clark Kimberling (Nov 3, 2021), based on notes from Dasari Naga Vijay Krishna, Nov 3, 2021.

In the plane of a triangle ABC, let
(O) = circumcircle of triangle ABC
A'B'C' = medial triangle;
(O)a = circle with B'C' as diameter, and define (O)b and (O)c cyclically;
Ab = AB ∩ (O)a, and define Bc and Ca cyclically;
Ac = AC ∩ (O)a, and define Ba and Cb cyclically;
A1 = AbB' ∩ AcC', and define B1 and C1 cyclically (here A1, B1, C1 lies on the nine point circle of triangle ABC) ;
Ka = polar of A wrt Oa (which passes through A1), and define Kb and Kc cyclically;
K'a = polar of A' wrt (O)a, and define K'b and K'c cyclically;
K''a= polar of A1 wrt (O)a (which passes through A), and define K''b and K''c cyclically;
Kab = polar of Ab wrt (O)a respectively, and define Kbc and Kca cyclically;
Kac = polar of Ac wrt (O)a respectively, and define Kba and Kcb cyclically;
(O)'a = the circle with BcCb as diameter, and define (O)'b and (O)'c cyclically;
La = polar of A wrt (O)'a, and define Lb and Lc cyclically;
L'a = polar of A' wrt (O)'a, and define L'b and L'c cyclically;
L''a= polar of A1 wrt (O)'a, and define L''b and L''c cyclically;
Lab = polar of Ab wrt (O)'c, and define Lbc and Lca cyclically;
Lac = polar of Ac wrt (O)'b, and define Lba and Lcb cyclically;
(O)''a = circle with AbAc as diameter, and define (O)''b and (O)''c cyclically;
Ta = polar of A wrt (O)''a, and define Tb and Tc cyclically;
T'a = polar of A' wrt (O)''a, and define T'b and T'c cyclically;
T''a= polar of A1 wrt (O)''a, and define T''b and T''c cyclically;
Tab = polar of Ab wrt (O)''a, and define Tbc and Tca cyclically;
Tac = polar of Ac wrt (O)''a, and define Tba and Tcb cyclically;
Ma = perpendicular bisector of AbAc, and define Mb, Mc cyclically;
M'a = Perpendicular bisector of BcCb or B'C', and define M'b, M'c cyclically;
Na = polar of A11 wrt to (O)a = line passing through A11 parallel to BC, and define Nb and Nc cyclically;
N'a = polar of A12 wrt to (O)a = line passing through A12 parallel to BC, and define N'b and N'c cyclically;
Pa = polar of A13 wrt to (O)'a = line passing through A13 parallel to BC, and define Pb and Pc cyclically;
P'a = polar of A14 wrt to (O)'a = line passing through A14 parallel to BC, and define P'b and P'c cyclically;
Qa = line passing through A15 parallel to BC, and define Qb and Qc cyclically;
Q'a = line passing through A16 parallel to BC, and define Q'b and Q'c cyclically;
VPMC1 = circumcircle of 23rd Vijay polar medial triangle;
VPMC2 = circumcircle of 24th Vijay polar medial triangle;
VPMC3 = circumcircle of 25th Vijay polar medial triangle;
VPMC4 = circumcircle of 26th Vijay polar medial triangle;

Define 26 points by the following intersections:

A2 = Kb ∩ Kc, and define B2 and C2 cyclically;
A3 = K''b ∩ K''c, and define B3 and C3 cyclically;
A4 = Lb ∩ Lc, and define B4 and C4 cyclically;
A5 = L'b ∩ L'c, and define B5 and C5 cyclically;
A6 = L''b ∩ L''c, and define B6 and C6 cyclically;
A7 = Kab ∩ Kac = center of the circle passes through the points {A, Ab, Ac, A1} = midpoint of A and A1, and define B7 and C7 cyclically;
A8 = Tb ∩ Tc, and define B8 and C8 cyclically;
A9 = T'b ∩ T'c, and define B9 and C9 cyclically;
A10 = T''b ∩ T''c, and define B10 and C10 cyclically;
A11 = the point in M'a ∩ (O)a that is closer to A than other point in M'a ∩ (O)a, and define B11 and C11 cyclically;
A12 = the point in M'a ∩ (O)a other that A11, and define B12 and C12 cyclically;
A14 = the point in M'a ∩ (O)'a that is closer to A than other point in M'a ∩ (O)'a, and define B14 and C14 cyclically;
A13 = the point in M'a ∩ (O)'a other than A14, and define B13 and C13 cyclically;
A15 = the point in Ma ∩ (O)''a that is closer to A than other point in Ma ∩ (O)''a, and define B15 and C15 cyclically;
A16 = the point in Ma ∩ (O)''a other that A15, and define B16 and C16 cyclically;
A17 = Nb ∩ Nc, and define B17 and C17 cyclically;
A18 = N'b ∩ N'c, and define B18 and C18 cyclically;
A19 = Pb ∩ Pc, and define B19 and C19 cyclically;
A20 = P'b ∩ P'c, and define B20 and C20 cyclically;
A21 = Qb ∩ Qc, and define B21 and C21 cyclically;
A22 = Q'b ∩ Q'c, and define B22 and C22 cyclically;
A23 = Lbc ∩ Lca = B1BcC' ∩ C1CaA',
B23 = Lca ∩ Lab = C1CaA' ∩ A1AbB',
C23 = Lab ∩ Lbc = A1AbB' ∩ B1BcC' ;
A24 = = Lcb ∩ Lac = B'C1Cb ∩ C'A1Ac,
B24 = Lac ∩ Lba = C'A1Ac ∩ A'B1Ba,
C24 = Lba ∩ Lcb = A'B1Ba ∩ B'C1Cb;
A25 = Tbc ∩ Tca,
B25 = Tca ∩ Tab,
C25 = Tab ∩ Tbc;
A26 = Tba ∩ Tcb,
B26 = Tcb ∩ Tac,
C26 = Tac ∩ Tba;

Related circles are here named as follows:

(O)a = 1st vijay A-orthic medial circle;
(O)b = 1st vijay B-orthic medial circle;
(O)c = 1st vijay C-orthic medial circle;
(O)'a = 2nd Vijay A-orthic medial circle;
(O)'b = 2nd Vijay B-orthic medial circle;
(O)'c = 2nd Vijay C-orthic medial circle;
(O)''a = 3rd Vijay A-orthic medial circle;
(O)''b = 3rd Vijay B-orthic medial circle;
(O)''c = 3rd Vijay C-orthic medial circle
VPMC1 = 1st Vijay polar medial circle;
VPMC2 = 2nd Vijay polar medial circle;
VPMC3 = 3rd Vijay polar medial circle;
VPMC4 = 4th Vijay polar medial circle ;

Barycentric equations for above defined circles and lines:

(O)a: (b^2 + c^2 - a^2)*x^2 + (a^2 + 3*c^2 - b^2)*y^2 + (a^2 + 3*b^2 - c^2)*z^2 + 2*y*z*(b^2 + c^2 - 3*a^2) - 4*b^2*x*z - 4*c^2*x*y = 0;
(O)b: (b^2 + 3*c^2 - a^2)*x^2 + (a^2 - b^2 + c^2)*y^2 + (b^2 + 3*a^2 - c^2)*z^2 + 2*z*x*(c^2 + a^2 - 3*b^2) - 4*a^2*y*z - 4*c^2*x*y = 0;
(O)c: (c^2 + 3*b^2 - a^2)*x^2 + (c^2 +3*a^2 - b^2)*y^2 + (a^2 + b^2 - c^2)*z^2 + 2*x*y*(b^2 + c^2 - 3a^2) - 4*b^2*x*z - 4*a^2*y*z = 0;
(O)'a: (8*a^2*b^2 + 8*a^2*c^2 + 6*b^2*c^2 - 5*a^4 -3*b^4 - 3*c^4)*x^2 + (a^2 - b^2 + c^2)*(3*a^2 - b^2 + c^2)*y^2 + ( a^2 + b^2 - c^2)*(3*a^2 + b^2 - c^2)*z^2 - 2*(a^2 - b^2 + c^2)^2*x*y - 2*(a^2 + b^2 - c^2)^2*x*z + 2*(b^4 - 2*b^2*c^2 + c^4 - 5*a^4)*y*z = 0;
(O)'b: (8*a^2*b^2 + 8*b^2*c^2 + 6*a^2*c^2 - 5*b^4 - 3*a^4 - 3*c^4)*y^2 + (b^2 - a^2 + c^2)*(3*b^2 - a^2 + c^2)*x^2 + ( a^2 + b^2 - c^2)*(3*b^2 + a^2 - c^2)*z^2 - 2*(b^2 - a^2 + c^2)^2*x*y - 2*(a^2 + b^2 - c^2)^2*y*z + 2*(a^4 - 2*a^2*c^2 + c^4 - 5*b^4)*x*z = 0;
(O)'c: (b^2 - a^2 + c^2)*(3*b^2 - a^2 + c^2)*x^2+ (a^2 - b^2 + c^2)*(3*a^2 - b^2 + c^2)*y^2 + (8*c^2*b^2 + 8*a^2*c^2 + 6*b^2*a^2 - 5*c^4 - 3*b^4 - 3*a^4) + 2*(b^4 - 2*b^2*a^2 + a^4 - 5*c^4)*y*x - 2*(c^2 - b^2 + a^2)^2*z*y - 2*(c^2 + b^2 - a^2)^2*x*z = 0;
(O)''a: (c^2 + b^2 - a^2)^3*x^2 + (a^2 - b^2 + 3*c^2)*(2*a^2*b^2 + 2*a^2*c^2 + 6*b^2*c^2 - a^4 - b^4 - c^4)*y^2 + (a^2 + 3*b^2 - c^2)*(2*a^2*b^2 + 2*a^2*c^2 + 6*b^2*c^2 - a^4 - b^4 - c^4)*z^2 + 2*x*y*(c^2 + b^2 - a^2)*(a^4 + b^4 - c^4 - 4*b^2*c^2 - 2*a^2*b^2) + 2*z*x*(c^2 + b^2 - a^2)*(a^4 - b^4 + c^4 - 4*b^2*c^2 - 2*a^2*c^2) +2*y*z*(c^2 + b^2 - a^2)*(a^4 - b^4 - c^4 + 6*b^2*c^2) = 0;
(O)''b: (b^2 - a^2 + 3*c^2)*(2*a^2*b^2 + 2*b^2*c^2 + 6*c^2*a^2 - a^4 - b^4 - c^4)*x^2 + (a^2+ c^2 - b^2)^3*y^2+ (b^2 - c^2 + 3*a^2)*(2*a^2*b^2 + 2*b^2*c^2 + 6*c^2*a^2 - a^4 - b^4 - c^4)*z^2 + 2*x*y*(c^2 + a^2 - b^2)*(a^4 + b^4 - c^4 - 4*a^2*c^2 - 2*a^2*b^2)+ 2*z*y*(c^2 + a^2 - b^2)*(c^4 + b^4 - a^4 - 4*a^2*c^2 - 2*c^2*b^2) + 2*x*z*(c^2 + a^2 - b^2)*(b^4 - a^4 - c^4 + 6*a^2*c^2) = 0;
(O)''c: (a^2 + b^2 - c^2)^3*z^2 + (c^2 - b^2 + 3*a^2)*(2*c^2*b^2 + 2*a^2*c^2 + 6*b^2*a^2 - a^4 - b^4 - c^4)*y^2 + (c^2 + 3*b^2 - a^2)*(2*c^2*b^2 + 2*a^2*c^2 + 6*b^2*a^2 - a^4 - b^4 - c^4)*x^2 + 2*z*y*(a^2 + b^2 - c^2)*(c^4 + b^4 - a^4 - 4*b^2*a^2 - 2*c^2*b^2) + 2*z*x*(a^2 + b^2 - c^2)*(a^4 - b^4 + c^4 - 4*b^2*a^2 - 2*a^2*c^2) +2*y*x*(a^2 + b^2 - c^2)*(c^4 - b^4 - a^4 + 6*b^2*a^2) = 0;
VPMC1: 16*S^2*(a^2*y*z + b^2*z*x + c^2*x*y) + (x + y + z)*(( c^6 + b^6 - a^6 + 5*a^2*b^2*(a^2 - b^2) + a^2*c^2*(a^2 - c^2) + b^2*c^2*(c^2 - 3*b^2))*x + ( c^6 + a^6 - b^6 + 5*c^2*b^2*(b^2 - c^2) + a^2*b^2*(b^2 - a^2) + a^2*c^2*(a^2 - 3*c^2))*y + (a^6 + b^6 - c^6 + 5*a^2*c^2*(c^2 - a^2) + b^2*c^2*(c^2 - b^2) + b^2*a^2*(b^2 - 3*a^2))*z) = 0;
VPMC2: 16*S^2*(a^2*y*z + b^2*z*x + c^2*x*y) + (x + y + z)*(( c^6 + b^6 - a^6 + 5*a^2*c^2*(a^2 - c^2) + a^2*b^2*(a^2 - b^2) + b^2*c^2*(b^2 - 3*c^2))*x + ( c^6 + a^6 - b^6 + 5*a^2*b^2*(b^2 - a^2) + c^2*b^2*(b^2 - c^2) + a^2*c^2*(c^2 - 3*a^2))*y + (a^6 + b^6 - c^6 + 5*b^2*c^2*(c^2 - b^2) + a^2*c^2*(c^2 - a^2) + b^2*a^2*(a^2 - 3*b^2))*z) = 0;
VPMC3: 64*S^2*(a^2 - b^2 + c^2)*(b^2 + c^2 - a^2)*(a^2 + b^2 - c^2)*(a^2*y*z + b^2*z*x + c^2*x*y) - (x + y + z)*( (b^2 + c^2 - a^2)*( 3*c^10 - 11*b^2*c^8 - 13*a^2*c^8 + 12*b^4*c^6 + 18*a^2*b^2*c^6 + 16*a^4*c^6 + 2*a^2*b^4*c^4 -10*a^4*b^2*c^4 - 4*a^6*c^4 - 7*b^8*c^2 - 6*a^2*b^6*c^2 + 6*a^4*b^4*c^2 + 10*a^6*b^2*c^2 - 3*a^8*c^2 + 3*b^10 - a^2*b^8 - 12*a^4*b^6 + 16*a^6*b^4 - 7*a^8*b^2 + a^10)*x + (a^2 + c^2 - b^2)*(3*c^10 - b^2*c^8 - 7*a^2*c^8 - 12*b^4*c^6 - 6*a^2*b^2*c^6 + 16*b^6*c^4 + 6*a^2*b^4*c^4 + 2*a^4*b^2*c^4 + 12*a^6*c^4 - 7*b^8*c^2 + 10*a^2*b^6*c^2 - 10*a^4*b^4*c^2 + 18*a^6*b^2*c^2 - 11*a^8*c^2 + b^10 - 3*a^2*b^8 - 4*a^4*b^6 + 16*a^6*b^4 - 13*a^8*b^2 + 3*a^10)*y + (a^2 + b^2 - c^2)*(c^10 - 3*b^2*c^8 - 7*a^2*c^8 - 4*b^4*c^6 + 10*a^2*b^2*c^6 + 16*a^4*c^6 + 16*b^6*c^4 - 10*a^2*b^4*c^4 + 6*a^4*b^2*c^4 - 12*a^6*c^4 - 13*b^8*c^2 + 18*a^2*b^6*c^2 + 2*a^4*b^4*c^2 - 6*a^6*b^2*c^2 - a^8*c^2 + 3*b^10 - 11*a^2*b^8 + 12*a^4*b^6 - 7*a^8*b^2 + 3*a^10)*z) = 0;
VPMC4: 64*S^2*(a^2 - b^2 + c^2)*(b^2 + c^2 - a^2)*(a^2 + b^2 - c^2)*(a^2*y*z + b^2*z*x + c^2*x*y) - (x + y + z)*( (b^2 + c^2 - a^2)* (3*c^10 - 7*b^2*c^8 - a^2*c^8 - 6*a^2*b^2*c^6 - 12*a^4*c^6 + 12*b^6*c^4 + 2*a^2*b^4*c^4 + 6*a^4*b^2*c^4 + 16*a^6*c^4 - 11*b^8*c^2 + 18*a^2*b^6*c^2 - 10*a^4*b^4*c^2 + 10*a^6*b^2*c^2 - 7*a^8*c^2 + 3*b^10 - 13*a^2*b^8 + 16*a^4*b^6 - 4*a^6*b^4 - 3*a^8*b^2 + a^10)*x + (a^2 + c^2 - b^2)*(3*c^10 - 13*b^2*c^8 - 11*a^2*c^8 + 16*b^4*c^6 + 18*a^2*b^2*c^6 + 12*a^4*c^6 - 4*b^6*c^4 - 10*a^2*b^4*c^4 + 2*a^4*b^2*c^4 - 3*b^8*c^2 + 10*a^2*b^6*c^2 + 6*a^4*b^4*c^2 - 6*a^6*b^2*c^2 - 7*a^8*c^2 + b^10 - 7*a^2*b^8 + 16*a^4*b^6 - 12*a^6*b^4 - a^8*b^2 + 3*a^10)*y + (a^2 + b^2 - c^2)*(c^10 - 7*b^2*c^8 - 3*a^2*c^8 + 16*b^4*c^6 + 10*a^2*b^2*c^6 - 4*a^4*c^6 - 12*b^6*c^4 + 6*a^2*b^4*c^4 - 10*a^4*b^2*c^4 + 16*a^6*c^4 - b^8*c^2 - 6*a^2*b^6*c^2 + 2*a^4*b^4*c^2 + 18*a^6*b^2*c^2 - 13*a^8*c^2 + 3*b^10 - 7*a^2*b^8 + 12*a^6*b^4 - 11*a^8*b^2 + 3*a^10)*z) = 0;
Ma : (b - c)*(c + b)*(c^2 + b^2 - a^2)*x + (3*c^4 - 4*b^2*c^2 - 3*a^2*c^2 + b^4 - a^2*b^2)*y + (-c^4 + 4*b^2*c^2 + a^2*c^2 - 3*b^4 + 3*a^2*b^2)*z = 0;
M'a : x*(c^2 - b^2) - y*(2*a^2 + c^2 - b^2) + z*(2*a^2 + b^2 - c^2) = 0;

Related triangles are here named as follows:

The triangle AnBnCn is here named the nth Vijay polar medial triangle, for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 .

Barycentrics for points defined above:

Ab = 3c^2 + a^2 - b^2 : b^2 + c^2 - a^2 : 0 ;
Bc = 0 : 3a^2 + b^2 - c^2 : a^2 + c^2 - b^2;
Ca = a^2 + b^2 - c^2 : 0 : 3b^2 + c^2 - a^2;
Ac = 3b^2 + a^2 - c^2 : 0 : b^2 + c^2 - a^2;
Ba = a^2 +c^2 - b^2 : 3c^2 + b^2 - a^2 : 0;
Cb = 0 : a^2 + b^2 - c^2 : 3a^2 + c^2 - b^2;
A1 = 2*a^2*(b^2 + c^2) - 2*(b^2 - c^2)^2 : (b^2 + c^2 - a^2)*(a^2 + b^2 - c^2) : (b^2 + c^2 - a^2)*(a^2 - b^2 + c^2) ;
A2 = -(3*a^4 + b^4 + c^4 - 2*b^2*c^2) : 2*(2*a^2*b^2 + c^2*(a^2 + b^2 - c^2)) : 2*(2*a^2*c^2 + b^2( a^2 + c^2 - b^2)) ;
A3 = - (a^2 - b^2 + c^2)*(a^2 + b^2 - c^2) : (b^2 + c^2 - a^2)*(b^2 - c^2 + a^2) : (c^2 + b^2 - a^2)*(c^2 - b^2 + a^2) ;
A4 = (7*c^8 - 30*b^2*c^6 - 27*a^2*c^6 + 46*b^4*c^4 + 27*a^2*b^2*c^4 + 37*a^4*c^4 - 30*b^6*c^2 + 27*a^2*b^4*c^2 + 24*a^4*b^2*c^2 - 21*a^6*c^2 + 7*b^8 - 27*a^2*b^6 + 37*a^4*b^4 - 21*a^6*b^2 + 4*a^8) : (c^2 + b^2 - a^2)^2*(3*c^4 - 5*b^2*c^2 - 5*a^2*c^2 + 2*b^4 - 2*a^2*b^2 + 2*a^4) : (c^2 + b^2 - a^2)^2*(2*c^4 - 5*b^2*c^2 - 2*a^2*c^2 + 3*b^4 - 5*a^2*b^2 + 2*a^4) ;
A5 = (c^2 + b^2 - a^2)*(c^6 - b^2*c^4 - 2*a^2*c^4 - b^4*c^2 + 2*a^2*b^2*c^2 + a^4*c^2 + b^6 -2*a^2*b^4 + a^4*b^2) : (3*b^2*c^6 - a^2*c^6 - b^4*c^4 - 4*a^2*b^2*c^4 + 3*a^4*c^4 - b^6*c^2 + a^2*b^4*c^2 + 3*a^4*b^2*c^2 - 3*a^6*c^2 - b^8 + 2*a^2*b^6 - 2*a^6*b^2 + a^8) : (-c^8 - b^2*c^6 + 2*a^2*c^6 - b^4*c^4 + a^2*b^2*c^4 + 3*b^6*c^2 - 4*a^2*b^4*c^2 + 3*a^4*b^2*c^2 - 2*a^6*c^2 - a^2*b^6 + 3*a^4*b^4 - 3*a^6*b^2 + a^8) ;
A6 = (c^8 - 6*b^2*c^6 - 9*a^2*c^6 + 10*b^4*c^4 + 9*a^2*b^2*c^4 + 19*a^4*c^4 - 6*b^6*c^2 + 9*a^2*b^4*c^2 + 12*a^4*b^2*c^2 - 15*a^6*c^2 + b^8 - 9*a^2*b^6 + 19*a^4*b^4 - 15*a^6*b^2 + 4*a^8) : (c^2 + b^2 - a^2)^2*(b^2*c^2 + a^2*c^2 - b^4 + 4*a^2*b^2 - a^4) : (c^2 + b^2 - a^2)^2*( -c^4 + b^2*c^2 + 4*a^2*c^2 + a^2*b^2 - a^4) ;
A7 = -2*(2*c^4 - 4*b^2*c^2 - 3*a^2*c^2 + 2*b^4 - 3*a^2*b^2 + a^4) : (b^2 + c^2 - a^2)*(a^2 + b^2 - c^2) : (b^2 + c^2 - a^2)*(a^2 - b^2 + c^2) ;
A8 = 2*a^2*(c^4 - 2*b^2*c^2 - a^2*c^2 + b^4 - a^2*b^2) : (c^6 - 3*b^2*c^4 - 3*a^2*c^4 + 3*b^4*c^2 + 4*a^2*b^2*c^2 + 3*a^4*c^2 - b^6 -a^2*b^4 + 3*a^4*b^2 - a^6) : (-c^6 + 3*b^2*c^4 - a^2*c^4 - 3*b^4*c^2 + 4*a^2*b^2*c^2 + 3*a^4*c^2 + b^6 - 3*a^2*b^4 + 3*a^4*b^2 - a^6) ;
A9 = a^2*(c^10 - 3*b^2*c^8 + 4*a^2*c^8 + 2*b^4*c^6 - 12*a^2*b^2*c^6 + 2*b^6*c^4 + 16*a^2*b^4*c^4 - 18*a^6*c^4 - 3*b^8*c^2 - 12*a^2*b^6*c^2 - 64*a^6*b^2*c^2 + 15*a^8*c^2 + b^10 + 4*a^2*b^8 - 18*a^6*b^4 + 15*a^8*b^2 - 2*a^10) : (-c^12 + 4*b^2*c^10 + 9*a^2*c^10 - 5*b^4*c^8 - 14*a^2*b^2*c^8 - 22*a^4*c^8 - 4*a^2*b^4*c^6 - 16*a^4*b^2*c^6 + 22*a^6*c^6 + 5*b^8*c^4 + 6*a^2*b^6*c^4 + 34*a^4*b^4*c^4 + 44*a^6*b^2*c^4 - 9*a^8*c^4 - 4*b^10*c^2 + 11*a^2*b^8*c^2 + 4*a^4*b^6*c^2 + 72*a^6*b^4*c^2 - 20*a^8*b^2*c^2 + a^10*c^2 + b^12 + 8*a^2*b^10 + 22*a^6*b^6 -17*a^8*b^4 + 2*a^10*b^2) : (c^12 - 4*b^2*c^10 - 8*a^2*c^10 + 5*b^4*c^8 + 11*a^2*b^2*c^8 + 6*a^2*b^4*c^6 + 4*a^4*b^2*c^6 + 22*a^6*c^6 - 5*b^8*c^4 - 4*a^2*b^6*c^4 + 34*a^4*b^4*c^4 + 72*a^6*b^2*c^4 - 17*a^8*c^4 + 4*b^10*c^2 -14*a^2*b^8*c^2 - 16*a^4*b^6*c^2 + 44*a^6*b^4*c^2 - 20*a^8*b^2*c^2 + 2*a^10*c^2-b^12 + 9*a^2*b^10 - 22*a^4*b^8 + 22*a^6*b^6 -9*a^8*b^4 + a^10*b^2) ;
A10 = -a^2*(c^4 -2*b^2*c^2 + a^2*c^2 + b^4 + a^2*b^2 - 2*a^4) : (c^2 - b^2 + a^2)*(c^4 - 2*b^2*c^2 - a^2*c^2 + b^4 + 2*a^2*b^2) : (-c^2 + b^2 + a^2)*(c^4 - 2*b^2*c^2 + 2*a^2*c^2 + b^4 - a^2*b^2) ;
A11 = 2(4*S^2 - a^4) : 2*S*(2*a^2 + b^2 - c^2) + a^2*(3*b^2 + c^2) - (b^2 - c^2)^2 : 2*S*(2*a^2 - b^2 + c^2) + a^2*(b^2 + 3*c^2) - (b^2 - c^2)^2 ;
A12 = 2(a^4 - 4*S^2) : 2*S*(2*a^2 + b^2 - c^2) -a^2*(3*b^2 + c^2) + (b^2 - c^2)^2 : 2*S*(2*a^2 - b^2 + c^2) - a^2*(b^2 + 3*c^2) + (b^2 - c^2)^2 ;
A13 = -2*a^4 : 2*S*(2*a^2 + b^2 - c^2) + a^2*(a^2 + b^2 - c^2) : 2*S*(2*a^2 - b^2 + c^2) + a^2*(a^2 - b^2 + c^2) ;
A14 = 2*a^4 : 2*S*(2*a^2 + b^2 - c^2) - a^2*(a^2 + b^2 - c^2) : 2*S*(2*a^2 - b^2 + c^2) - a^2*(a^2 - b^2 + c^2) ;
A15 = -(c^12 - 8*b^2*c^10 - 4*a^2*c^10 + 31*b^4*c^8 + 20*a^2*b^2*c^8 + 6*a^4*c^8 - 48*b^6*c^6 - 32*a^2*b^4*c^6 - 14*a^4*b^2*c^6 -4*a^6*c^6 + 31*b^8*c^4 - 32*a^2*b^6*c^4 + a^8*c^4 - 8*b^10*c^2 + 20*a^2*b^8*c^2 - 14*a^4*b^6*c^2 + 2*a^8*b^2*c^2 + b^12 - 4*a^2*b^10 + 6*a^4*b^8 - 4*a^6*b^6 + a^8*b^4) : b^2*(c^2 + b^2 - a^2)*(-3*b^2*c^6 - a^2*c^6 + 2*S*c^6 + 7*b^4*c^4 + 5*a^2*b^2*c^4 - 8*S*b^2*c^4 + 2*a^4*c^4 - 2*S*a^2*c^4 - 5*b^6*c^2 + 7*a^2*b^4*c^2 + 6*S*b^4*c^2 - a^4*b^2*c^2 - 6*S*a^2*b^2*c^2 - a^6*c^2 + b^8 -3*a^2*b^6 + 3*a^4*b^4 - a^6*b^2) : c^2*(c^2 + b^2 - a^2)*(c^8 - 5*b^2*c^6 - 3*a^2*c^6 + 7*b^4*c^4 + 7*a^2*b^2*c^4 + 6*S*b^2*c^4 + 3*a^4*c^4 - 3*b^6*c^2 + 5*a^2*b^4*c^2 - 8*S*b^4*c^2 - a^4*b^2*c^2 - 6*S*a^2*b^2*c^2 -a^6*c^2 - a^2*b^6 + 2*S*b^6 + 2*a^4*b^4 -2*S*a^2*b^4 - a^6*b^2) ;
A16 = -(c^12 - 8*b^2*c^10 - 4*a^2*c^10 + 31*b^4*c^8 + 20*a^2*b^2*c^8 + 6*a^4*c^8 - 48*b^6*c^6 - 32*a^2*b^4*c^6 - 14*a^4*b^2*c^6 -4*a^6*c^6 + 31*b^8*c^4 - 32*a^2*b^6*c^4 + a^8*c^4 - 8*b^10*c^2 + 20*a^2*b^8*c^2 - 14*a^4*b^6*c^2 + 2*a^8*b^2*c^2 + b^12 - 4*a^2*b^10 + 6*a^4*b^8 - 4*a^6*b^6 + a^8*b^4) : b^2*(c^2 + b^2 - a^2)*(-3*b^2*c^6 - a^2*c^6 -2*S*c^6 + 7*b^4*c^4 + 5*a^2*b^2*c^4 + 8*S*b^2*c^4 + 2*a^4*c^4 + 2*S*a^2*c^4 - 5*b^6*c^2 + 7*a^2*b^4*c^2 - 6*S*b^4*c^2 - a^4*b^2*c^2 + 6*S*a^2*b^2*c^2 - a^6*c^2 + b^8 -3*a^2*b^6 + 3*a^4*b^4 - a^6*b^2) : c^2*(c^2 + b^2 - a^2)*(c^8 - 5*b^2*c^6 - 3*a^2*c^6 + 7*b^4*c^4 + 7*a^2*b^2*c^4 - 6*S*b^2*c^4 + 3*a^4*c^4 - 3*b^6*c^2 + 5*a^2*b^4*c^2 + 8*S*b^4*c^2 - a^4*b^2*c^2 + 6*S*a^2*b^2*c^2 -a^6*c^2 - a^2*b^6 - 2*S*b^6 + 2*a^4*b^4 +2*S*a^2*b^4 - a^6*b^2) ;
A17 = (b^2 + c^2) : (2*S - b^2) : (2*S - c^2) ;
A18 = -(b^2 + c^2) : (2*S + b^2) : (2*S + c^2) ;
A19 = -(c^2 + b^2 + 4*S) : b^2 : c^2 ;
A20 = -(c^2 + b^2 - 4*S) : b^2 : c^2 ;
A21 = (a^2*b^2*c^18 + a^4*c^18 - 12*a^2*b^4*c^16 + 4*S*b^4*c^16 - 16*a^4*b^2*c^16 + 6*S*a^2*b^2*c^16 - 12*a^6*c^16 + 2*S*a^4*c^16 + 52*a^2*b^6*c^14 - 24*S*b^6*c^14 + 67*a^4*b^4*c^14 - 34*S*a^2*b^4*c^14 + 43*a^6*b^2*c^14 - 28*S*a^4*b^2*c^14 + 52*a^8*c^14 - 10*S*a^6*c^14 - 116*a^2*b^8*c^12 + 60*S*b^8*c^12 - 115*a^4*b^6*c^12 + 66*S*a^2*b^6*c^12 - 76*a^6*b^4*c^12 + 36*S*a^4*b^4*c^12 + 31*a^8*b^2*c^12 + 32*S*a^6*b^2*c^12 - 116*a^10*c^12 + 18*S*a^8*c^12 + 150*a^2*b^10*c^10 - 80*S*b^10*c^10 + 63*a^4*b^8*c^10 - 38*S*a^2*b^8*c^10 + 157*a^6*b^6*c^10 + 76*S*a^4*b^6*c^10 + 127*a^8*b^4*c^10 + 52*S*a^6*b^4*c^10 - 287*a^10*b^2*c^10 + 16*S*a^8*b^2*c^10 + 150*a^12*c^10 - 10*S*a^10*c^10 - 116*a^2*b^12*c^8 + 60*S*b^12*c^8 + 63*a^4*b^10*c^8 - 38*S*a^2*b^10*c^8 - 224*a^6*b^8*c^8 - 172*S*a^4*b^8*c^8 - 210*a^8*b^6*c^8 - 74*S*a^6*b^6*c^8 - 552*a^10*b^4*c^8 - 38*S*a^8*b^4*c^8 + 483*a^12*b^2*c^8 - 32*S*a^10*b^2*c^8 - 116*a^14*c^8 - 10*S*a^12*c^8 + 52*a^2*b^14*c^6 - 24*S*b^14*c^6 - 115*a^4*b^12*c^6 + 66*S*a^2*b^12*c^6 + 157*a^6*b^10*c^6 + 76*S*a^4*b^10*c^6 - 210*a^8*b^8*c^6 - 74*S*a^6*b^8*c^6 - 442*a^10*b^6*c^6 + 8*S*a^8*b^6*c^6 + 881*a^12*b^4*c^6 - 42*S*a^10*b^4*c^6 - 375*a^14*b^2*c^6 - 28*S*a^12*b^2*c^6 + 52*a^16*c^6 + 18*S*a^14*c^6 - 12*a^2*b^16*c^4 + 4*S*b^16*c^4 + 67*a^4*b^14*c^4 - 34*S*a^2*b^14*c^4 - 76*a^6*b^12*c^4 + 36*S*a^4*b^12*c^4 + 127*a^8*b^10*c^4 + 52*S*a^6*b^10*c^4 - 552*a^10*b^8*c^4 -38*S*a^8*b^8*c^4 + 881*a^12*b^6*c^4 - 42*S*a^10*b^6*c^4 - 564*a^14*b^4*c^4 -24*S*a^12*b^4*c^4 + 141*a^16*b^2*c^4 + 56*S*a^14*b^2*c^4 - 12*a^18*c^4 - 10*S*a^16*c^4 + a^2*b^18*c^2 -16*a^4*b^16*c^2 + 6*S*a^2*b^16*c^2 + 43*a^6*b^14*c^2 - 28*S*a^4*b^14*c^2 + 31*a^8*b^12*c^2 + 32*S*a^6*b^12*c^2 - 287*a^10*b^10*c^2 + 16*S*a^8*b^10*c^2 + 483*a^12*b^8*c^2 - 32*S*a^10*b^8*c^2 - 375*a^14*b^6*c^2 - 28*S*a^12*b^6*c^2 + 141*a^16*b^4*c^2 + 56*S*a^14*b^4*c^2 - 22*a^18*b^2*c^2 - 24*S*a^16*b^2*c^2 + a^20*c^2 + 2*S*a^18*c^2 + a^4*b^18 - 12*a^6*b^16 + 2*S*a^4*b^16 + 52*a^8*b^14 - 10*S*a^6*b^14 - 116*a^10*b^12 + 18*S*a^8*b^12 + 150*a^12*b^10 - 10*S*a^10*b^10 - 116*a^14*b^8 - 10*S*a^12*b^8 + 52*a^16*b^6 + 18*S*a^14*b^6 - 12*a^18*b^4 - 10*S*a^16*b^4 + a^20*b^2 + 2*S*a^18*b^2) : b^2*(b^2*c^6 + a^2*c^6 - 3*b^4*c^4 + 2*a^2*b^2*c^4 - 2*S*b^2*c^4 - 3*a^4*c^4 - 2*S*a^2*c^4 + 3*b^6*c^2 - 11*a^2*b^4*c^2 + 4*S*b^4*c^2 - 11*a^4*b^2*c^2 + 3*a^6*c^2 + 4*S*a^4*c^2 - b^8 + 8*a^2*b^6 - 2*S*b^6 - 14*a^4*b^4 + 2*S*a^2*b^4 + 8*a^6*b^2 + 2*S*a^4*b^2 - a^8 - 2*S*a^6)*(c^12 - 4*b^2*c^10 - 8*a^2*c^10 + 6*b^4*c^8 + 20*a^2*b^2*c^8 + 31*a^4*c^8 - 4*b^6*c^6 - 14*a^2*b^4*c^6 - 32*a^4*b^2*c^6 - 48*a^6*c^6 + b^8*c^4 - 32*a^6*b^2*c^4 + 31*a^8*c^4 + 2*a^2*b^8*c^2 - 14*a^6*b^4*c^2 + 20*a^8*b^2*c^2 - 8*a^10*c^2 + a^4*b^8 - 4*a^6*b^6 + 6*a^8*b^4 - 4*a^10*b^2 + a^12) : c^2*(-c^8 + 3*b^2*c^6 + 8*a^2*c^6 - 2*S*c^6 - 3*b^4*c^4 - 11*a^2*b^2*c^4 + 4*S*b^2*c^4 - 14*a^4*c^4 + 2*S*a^2*c^4 + b^6*c^2 + 2*a^2*b^4*c^2 - 2*S*b^4*c^2 - 11*a^4*b^2*c^2 + 8*a^6*c^2 + 2*S*a^4*c^2 + a^2*b^6 - 3*a^4*b^4 - 2*S*a^2*b^4 + 3*a^6*b^2 + 4*S*a^4*b^2 - a^8 - 2*S*a^6)*(b^4*c^8 + 2*a^2*b^2*c^8 + a^4*c^8 - 4*b^6*c^6 - 4*a^6*c^6 + 6*b^8*c^4 - 14*a^2*b^6*c^4 - 14*a^6*b^2*c^4 + 6*a^8*c^4 -4*b^10*c^2 + 20*a^2*b^8*c^2 - 32*a^4*b^6*c^2 -32*a^6*b^4*c^2 + 20*a^8*b^2*c^2 -4*a^10*c^2 + b^12 - 8*a^2*b^10 + 31*a^4*b^8 - 48*a^6*b^6 + 31*a^8*b^4 - 8*a^10*b^2 + a^12) ;
A22 = (a^2*b^2*c^18 + a^4*c^18 - 12*a^2*b^4*c^16 - 4*S*b^4*c^16 - 16*a^4*b^2*c^16 - 6*S*a^2*b^2*c^16 - 12*a^6*c^16 - 2*S*a^4*c^16 + 52*a^2*b^6*c^14 + 24*S*b^6*c^14 + 67*a^4*b^4*c^14 + 34*S*a^2*b^4*c^14 + 43*a^6*b^2*c^14 + 28*S*a^4*b^2*c^14 + 52*a^8*c^14 + 10*S*a^6*c^14 - 116*a^2*b^8*c^12 - 60*S*b^8*c^12 - 115*a^4*b^6*c^12 - 66*S*a^2*b^6*c^12 - 76*a^6*b^4*c^12 - 36*S*a^4*b^4*c^12 + 31*a^8*b^2*c^12 - 32*S*a^6*b^2*c^12 - 116*a^10*c^12 - 18*S*a^8*c^12 + 150*a^2*b^10*c^10 + 80*S*b^10*c^10 + 63*a^4*b^8*c^10 + 38*S*a^2*b^8*c^10 + 157*a^6*b^6*c^10 - 76*S*a^4*b^6*c^10 + 127*a^8*b^4*c^10 - 52*S*a^6*b^4*c^10 - 287*a^10*b^2*c^10 -16*S*a^8*b^2*c^10 + 150*a^12*c^10 + 10*S*a^10*c^10 - 116*a^2*b^12*c^8 - 60*S*b^12*c^8 + 63*a^4*b^10*c^8 + 38*S*a^2*b^10*c^8 - 224*a^6*b^8*c^8 + 172*S*a^4*b^8*c^8 - 210*a^8*b^6*c^8 + 74*S*a^6*b^6*c^8 - 552*a^10*b^4*c^8 + 38*S*a^8*b^4*c^8 + 483*a^12*b^2*c^8 + 32*S*a^10*b^2*c^8 - 116*a^14*c^8 + 10*S*a^12*c^8 + 52*a^2*b^14*c^6 + 24*S*b^14*c^6 - 115*a^4*b^12*c^6 - 66*S*a^2*b^12*c^6 + 157*a^6*b^10*c^6 - 76*S*a^4*b^10*c^6 - 210*a^8*b^8*c^6 + 74*S*a^6*b^8*c^6 - 442*a^10*b^6*c^6 - 8*S*a^8*b^6*c^6 + 881*a^12*b^4*c^6 + 42*S*a^10*b^4*c^6 - 375*a^14*b^2*c^6 + 28*S*a^12*b^2*c^6 + 52*a^16*c^6 - 18*S*a^14*c^6 - 12*a^2*b^16*c^4 - 4*S*b^16*c^4 + 67*a^4*b^14*c^4 + 34*S*a^2*b^14*c^4 - 76*a^6*b^12*c^4 - 36*S*a^4*b^12*c^4 + 127*a^8*b^10*c^4 - 52*S*a^6*b^10*c^4 - 552*a^10*b^8*c^4 + 38*S*a^8*b^8*c^4 + 881*a^12*b^6*c^4 + 42*S*a^10*b^6*c^4 - 564*a^14*b^4*c^4 + 24*S*a^12*b^4*c^4 + 141*a^16*b^2*c^4 - 56*S*a^14*b^2*c^4 - 12*a^18*c^4 + 10*S*a^16*c^4 + a^2*b^18*c^2 -16*a^4*b^16*c^2 - 6*S*a^2*b^16*c^2 + 43*a^6*b^14*c^2 + 28*S*a^4*b^14*c^2 + 31*a^8*b^12*c^2 - 32*S*a^6*b^12*c^2 - 287*a^10*b^10*c^2 - 16*S*a^8*b^10*c^2 + 483*a^12*b^8*c^2 + 32*S*a^10*b^8*c^2 - 375*a^14*b^6*c^2 + 28*S*a^12*b^6*c^2 + 141*a^16*b^4*c^2 - 56*S*a^14*b^4*c^2 - 22*a^18*b^2*c^2 + 24*S*a^16*b^2*c^2 + a^20*c^2 - 2*S*a^18*c^2 + a^4*b^18 - 12*a^6*b^16 - 2*S*a^4*b^16 + 52*a^8*b^14 + 10*S*a^6*b^14 - 116*a^10*b^12 - 18*S*a^8*b^12 + 150*a^12*b^10 + 10*S*a^10*b^10 - 116*a^14*b^8 + 10*S*a^12*b^8 + 52*a^16*b^6 - 18*S*a^14*b^6 - 12*a^18*b^4 + 10*S*a^16*b^4 + a^20*b^2 - 2*S*a^18*b^2) : b^2*(b^2*c^6 + a^2*c^6 - 3*b^4*c^4 + 2*a^2*b^2*c^4 + 2*S*b^2*c^4 - 3*a^4*c^4 + 2*S*a^2*c^4 + 3*b^6*c^2 - 11*a^2*b^4*c^2 - 4*S*b^4*c^2 - 11*a^4*b^2*c^2 + 3*a^6*c^2 - 4*S*a^4*c^2 - b^8 + 8*a^2*b^6 + 2*S*b^6 - 14*a^4*b^4 - 2*S*a^2*b^4 + 8*a^6*b^2 - 2*S*a^4*b^2 - a^8 + 2*S*a^6)*(c^12 - 4*b^2*c^10 - 8*a^2*c^10 + 6*b^4*c^8 + 20*a^2*b^2*c^8 + 31*a^4*c^8 - 4*b^6*c^6 - 14*a^2*b^4*c^6 - 32*a^4*b^2*c^6 - 48*a^6*c^6 + b^8*c^4 - 32*a^6*b^2*c^4 + 31*a^8*c^4 + 2*a^2*b^8*c^2 - 14*a^6*b^4*c^2 + 20*a^8*b^2*c^2 - 8*a^10*c^2 + a^4*b^8 - 4*a^6*b^6 + 6*a^8*b^4 - 4*a^10*b^2 + a^12) : c^2*(-c^8 + 3*b^2*c^6 + 8*a^2*c^6 + 2*S*c^6 - 3*b^4*c^4 - 11*a^2*b^2*c^4 - 4*S*b^2*c^4 - 14*a^4*c^4 - 2*S*a^2*c^4 + b^6*c^2 + 2*a^2*b^4*c^2 + 2*S*b^4*c^2 - 11*a^4*b^2*c^2 + 8*a^6*c^2 - 2*S*a^4*c^2 + a^2*b^6 - 3*a^4*b^4 + 2*S*a^2*b^4 + 3*a^6*b^2 - 4*S*a^4*b^2 - a^8 + 2*S*a^6)*(b^4*c^8 + 2*a^2*b^2*c^8 + a^4*c^8 - 4*b^6*c^6 - 4*a^6*c^6 + 6*b^8*c^4 - 14*a^2*b^6*c^4 - 14*a^6*b^2*c^4 + 6*a^8*c^4 -4*b^10*c^2 + 20*a^2*b^8*c^2 - 32*a^4*b^6*c^2 -32*a^6*b^4*c^2 + 20*a^8*b^2*c^2 -4*a^10*c^2 + b^12 - 8*a^2*b^10 + 31*a^4*b^8 - 48*a^6*b^6 + 31*a^8*b^4 - 8*a^10*b^2 + a^12) ;
A23 = (a^2 + b^2 - c^2)^2 : (c^4 - 2*a^2*c^2 - b^4 - 4*a^2*b^2 + a^4) : -2*b^2*(c^2 - b^2 + a^2) ;
B23 = -2*c^2*(a^2 + b^2 - c^2) : (c^2 + b^2 - a^2)^2 : (-c^4 - 4*b^2*c^2 + b^4 - 2*a^2*b^2 + a^4) ;
C23 = (c^4 - 2*b^2*c^2 - 4*a^2*c^2 + b^4 - a^4) : -2*a^2*(c^2 + b^2 - a^2) : (c^2 - b^2 + a^2)^2 ;
A24 = (c^4 - 2*b^2*c^2 + b^4 - 4*a^2*b^2 - a^4) : (a^2 + b^2 - c^2)^2 : -2*a^2*(c^2 + b^2 - a^2) ;
B24 = -2*b^2*(c^2 - b^2 + a^2) : (c^4 - 4*b^2*c^2 - 2*a^2*c^2 - b^4 + a^4) : (c^2 + b^2 - a^2)^2 ;
C24 = (c^2 - b^2 + a^2)^2 : -2*c^2*(-c^2 + b^2 + a^2) : (-c^4 - 4*a^2*c^2 + b^4 - 2*a^2*b^2 + a^4) ;
A25 = 2*a^2*(c^2 + b^2 - a^2)*(-c^4 + b^2*c^2 + 2*a^2*c^2 + a^2*b^2 - a^4) : (c^2 - b^2 + a^2)*(-c^6 + b^2*c^4 + 3*a^2*c^4 - b^4*c^2 + 2*a^2*b^2*c^2 - 3*a^4*c^2 + b^6 + a^2*b^4 - 3*a^4*b^2 + a^6) : (c^8 - 2*b^2*c^6 -4*a^2*c^6 + 2*b^4*c^4 + 4*a^2*b^2*c^4 + 6*a^4*c^4 - 2*b^6*c^2 + 4*a^2*b^4*c^2 + 2*a^4*b^2*c^2 - 4*a^6*c^2 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2 + a^8) ;
B25 = (c^8 - 4*b^2*c^6 - 2*a^2*c^6 + 6*b^4*c^4 + 4*a^2*b^2*c^4 + 2*a^4*c^4 - 4*b^6*c^2 + 2*a^2*b^4*c^2 + 4*a^4*b^2*c^2 - 2*a^6*c^2 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2 + a^8) : 2*b^2*(c^2 - b^2 + a^2)*(b^2*c^2 + a^2*c^2 - b^4 + 2*a^2*b^2 - a^4) : (-c^2 + b^2 + a^2)*(c^6 + b^2*c^4 - a^2*c^4 - 3*b^4*c^2 + 2*a^2*b^2*c^2 + a^4*c^2 + b^6 - 3*a^2*b^4 + 3*a^4*b^2 - a^6) ;
C25 = (c^2 + b^2 - a^2)*(c^6 - 3*b^2*c^4 - 3*a^2*c^4 + 3*b^4*c^2 + 2*a^2*b^2*c^2 + a^4*c^2 - b^6 + a^2*b^4 - a^4*b^2 + a^6) : (c^8 - 4*b^2*c^6 - 4*a^2*c^6 + 6*b^4*c^4 + 2*a^2*b^2*c^4 + 6*a^4*c^4 - 4*b^6*c^2 + 4*a^2*b^4*c^2 + 4*a^4*b^2*c^2 - 4*a^6*c^2 + b^8 - 2*a^2*b^6 + 2*a^4*b^4 - 2*a^6*b^2 + a^8) : 2*c^2*(c^2 - b^2 - a^2)*(c^4 - 2*b^2*c^2 - a^2*c^2 + b^4 - a^2*b^2) ;
A26 = 2*a^2*(c^2 + b^2 - a^2)*(b^2*c^2 + a^2*c^2 - b^4 + 2*a^2*b^2 - a^4) : (c^8 - 2*b^2*c^6 - 4*a^2*c^6 + 2*b^4*c^4 + 4*a^2*b^2*c^4 + 6*a^4*c^4 - 2*b^6*c^2 + 4*a^2*b^4*c^2 + 2*a^4*b^2*c^2 - 4*a^6*c^2 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2 + a^8) : (-c^2 + b^2 + a^2)*(c^6 - b^2*c^4 + a^2*c^4 + b^4*c^2 + 2*a^2*b^2*c^2 - 3*a^4*c^2 - b^6 + 3*a^2*b^4 - 3*a^4*b^2 + a^6) ;
B26 = (c^2 + b^2 - a^2)*(-c^6 + 3*b^2*c^4 + a^2*c^4 - 3*b^4*c^2 + 2*a^2*b^2*c^2 - a^4*c^2 + b^6 - 3*a^2*b^4 + a^4*b^2 + a^6) : 2*b^2*(c^2 - b^2 + a^2)*(-c^4 + 2*b^2*c^2 + a^2*c^2 - b^4 + a^2*b^2) : (c^8 - 4*b^2*c^6 - 2*a^2*c^6 + 6*b^4*c^4 + 4*a^2*b^2*c^4 + 2*a^4*c^4 - 4*b^6*c^2 + 2*a^2*b^4*c^2 + 4*a^4*b^2*c^2 - 2*a^6*c^2 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2 + a^8) ;
C26 = (c^8 - 4*b^2*c^6 - 4*a^2*c^6 + 6*b^4*c^4 + 2*a^2*b^2*c^4 + 6*a^4*c^4 - 4*b^6*c^2 + 4*a^2*b^4*c^2 + 4*a^4*b^2*c^2 - 4*a^6*c^2 + b^8 - 2*a^2*b^6 + 2*a^4*b^4 - 2*a^6*b^2 + a^8) : (c^2 - b^2 + a^2)*(c^6 - 3*b^2*c^4 - 3*a^2*c^4 + b^4*c^2 + 2*a^2*b^2*c^2 + 3*a^4*c^2 + b^6 - a^2*b^4 + a^4*b^2 - a^6) : 2*c^2*(-c^2 + b^2 + a^2)*(-c^4 + b^2*c^2 + 2*a^2*c^2 + a^2*b^2 - a^4) ;

Perspectors :

X(2) = AA' ∩ BB' ∩ CC'
X(3) = K'a ∩ K'b ∩ K'c = radical center of (O)a, (O)b, (O)c = A'A8 ∩ B'B8 ∩ C'C8
X(4) = AA1A3A7 ∩ BB1B3B7 ∩ CC1C3C7
X(5) = M'a ∩ M'b ∩ M'c = A'A1 ∩ B'B1 ∩ C'C1 = A11A12A13A14 ∩ B11B12B13B14 ∩ C11C12C13C14 = A23A24 ∩ B23B24 ∩ C23C24 = midpoint of A23A24 or midpoint of B23B24 or midpoint of C23C24
X(6) = AA19A20 ∩ BB19B20 ∩ CC19C20
X(113) = A'A5 ∩ B'B5 ∩ C'C5
X(127) = A1A5 ∩ B1B5 ∩ C1C5
X(140) = Ma ∩ Mb ∩ Mc = A7A15A16 ∩ B7B15B16 ∩ C7C15C16 = A25A26 ∩ B25B26 ∩ C25C26 = midpoint of A25A26 or midpoint of B25B26 or midpoint of C25C26
X(141) = A'A17A18 ∩ B'B17B18 ∩ C'C17C18
X(262) = AA2 ∩ BB2 ∩ CC2
X(590) = AA18 ∩ BB18 ∩ CC18 = (2*S + a^2) : (2*S + b^2) : (2*S + c^2)
X(615) = AA17 ∩ BB17 ∩ CC17 = (2*S - a^2) : (2*S - b^2) : (2*S - c^2)
X(641) = A'A12 ∩ B'B12 ∩ C'C12= (a^2 + 2*S)*(c^2 + b^2 - a^2 + 2*S) : (b^2 + 2*S)*(c^2 - b^2 + a^2 + 2*S) : (c^2 + 2*S)*(-c^2 + b^2 + a^2 + 2*S)
X(642) = A'A11 ∩ B'B11∩ C'C∩ = (a^2 - 2*S)*(c^2 + b^2 - a^2 - 2*S) : (b^2 - 2*S)*(c^2 - b^2 + a^2 - 2*S) : (c^2 - 2*S)*(-c^2 + b^2 + a^2 - 2*S)
X(1249) = A'A3 ∩ B'B3 ∩ C'C3
X(1656) = A'A7 ∩ B'B7 ∩ C'C7
X(7710) = A'A2 ∩ B'B2 ∩ C'C2
X(45845) = radical center of (O)'a, (O)'b, (O)'c
X(45846) = radical center of (O)''a, (O)''b, (O)''c
X(45847) = radical center of circumcircle, VPMC1, VPMC2
X(45848) = radical center of nine-point circle, VPMC1, VPMC2
X(45849) = radical center of incircle, VPMC1, VPMC2
X(45850) = radical center of Vijay excentral circle, VPMC1, VPMC2
X(45851) = radical center of Brocard circle, VPMC1, VPMC2
X(45852) = radical center of circumcircle, VPMC3, VPMC4
X(45853) = radical center of nine-point circle, VPMC3, VPMC4
X(45854) = radical center of incircle, VPMC3, VPMC4
X(45855) = radical center of Vijay excentral circle, VPMC3, VPMC4
X(45856) = radical center of Brocard circle, VPMC3, VPMC4
X(45857) = AA6 ∩ BB6 ∩ CC6
X(45858) = AA21 ∩ BB21 ∩ CC21
X(45859) = AA22 ∩ BB22 ∩ CC22
X(45860) = A1A17 ∩ B1B17 ∩ C1C17
X(45861) = A1A18 ∩ B1B18 ∩ C1C18
X(45862) = A1A19 ∩ B1B19 ∩ C1C19
X(45863) = A1A20 ∩ B1B20 ∩ C1C20
X(45864) = A2A3 ∩ B2B3 ∩ C2C3
X(45865) = A2A11 ∩ B2B11 ∩ C2C11
X(45866) = A2A12 ∩ B2B12 ∩ C2C12
X(45867) = A3A6 ∩ B3B6 ∩ C3C6
X(45868) = A7A17 ∩ B7B17 ∩ C7C17
X(45869) = A7A18 ∩ B7B18 ∩ C7C18
X(45870) = A7A20 ∩ B7B20 ∩ C7C20
X(45871) = A11A18A19 ∩ B11B18B19 ∩ C11C18C19
X(45872) = A12A17A20 ∩ B12B17B20 ∩ C12C17C20
X(45873) = A21A22 ∩ B21B22 ∩ C21C22

X(45845) = RADICAL CENTER OF THESE CIRCLES: 2ND VIJAY A-ORTHIC , B-ORTHIC, C-ORTHIC MEDIAL CIRCLES OF ABC

Barycentrics    (3*b^2*c^6 - 7*a^2*c^6 - 6*b^4*c^4 + 7*a^2*b^2*c^4 + 19*a^4*c^4 + 3*b^6*c^2 + 7*a^2*b^4*c^2 + 7*a^4*b^2*c^2 - 17*a^6*c^2 - 7*a^2*b^6 + 19*a^4*b^4 - 17*a^6*b^2 + 5*a^8) : :
X(45845) = 6*X(2)-5*X(40410)

X(45845) lies on these lines: {2, 10979}, {648, 22052}, {3098, 3528}

X(45846) = RADICAL CENTER OF THESE CIRCLES: 3ND VIJAY A-ORTHIC , B-ORTHIC, C-ORTHIC MEDIAL CIRCLES OF ABC

Barycentrics    (2*c^12 - 29*b^2*c^10 + 11*a^2*c^10 + 50*b^4*c^8 - 37*a^2*b^2*c^8 + 67*a^4*c^8 - 46*b^6*c^6 + 26*a^2*b^4*c^6 + 50*a^4*b^2*c^6 - 194*a^6*c^6 + 50*b^8*c^4 + 26*a^2*b^6*c^4 - 282*a^4*b^4*c^4 + 166*a^6*b^2*c^4 + 136*a^8*c^4 - 29*b^10*c^2 - 37*a^2*b^8*c^2 + 50*a^4*b^6*c^2 + 166*a^6*b^4*c^2 - 125*a^8*b^2*c^2 - 25*a^10*c^2 + 2*b^12 + 11*a^2*b^10 + 67*a^4*b^8 - 194*a^6*b^6 + 136*a^8*b^4 - 25*a^10*b^2 + 3*a^12) : :

X(45846) lies on these lines: {}

X(45847) = RADICAL CENTER OF THESE CIRCLES: CIRCUMCIRCLE, 1ST AND 2ND VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (b^2*c^10 - 4*b^4*c^8 - 2*a^2*b^2*c^8 - a^4*c^8 + 6*b^6*c^6 + 2*a^2*b^4*c^6 + a^4*b^2*c^6 + 3*a^6*c^6 - 4*b^8*c^4 + 2*a^2*b^6*c^4 - 4*a^4*b^4*c^4 + 3*a^6*b^2*c^4 - 3*a^8*c^4 + b^10*c^2 - 2*a^2*b^8*c^2 + a^4*b^6*c^2 + 3*a^6*b^4*c^2 - 4*a^8*b^2*c^2 + a^10*c^2 - a^4*b^8 + 3*a^6*b^6 - 3*a^8*b^4 + a^10*b^2) : :

X(45847) lies on these lines: {3,44145}, {5,141}, {523,44452}, {3003,16238}, {3260,6642}, {32428,44221}, {34845,41714}, {37813,37814}

X(45847) = midpoint of X(3) and X(44145) X(45847) = crossdifference of every pair of points on line {3050, 8553}

X(45848) = RADICAL CENTER OF THESE CIRCLES: NINE-POINT CIRCLE, 1ST AND 2ND VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (b^2*c^10 - 4*b^4*c^8 - 2*a^2*b^2*c^8 - a^4*c^8 + 6*b^6*c^6 + 2*a^2*b^4*c^6 + 2*a^4*b^2*c^6 + 3*a^6*c^6 - 4*b^8*c^4 + 2*a^2*b^6*c^4 - 4*a^4*b^4*c^4 + 2*a^6*b^2*c^4 - 3*a^8*c^4 + b^10*c^2 - 2*a^2*b^8*c^2 + 2*a^4*b^6*c^2 + 2*a^6*b^4*c^2 - 4*a^8*b^2*c^2 + a^10*c^2 - a^4*b^8 + 3*a^6*b^6 - 3*a^8*b^4 + a^10*b^2) : :

X(45848) lies on these lines: {5, 141}, {186, 30716}, {523, 31667}, {14940, 18114}, {39912, 41760}

X(45849) = RADICAL CENTER OF THESE CIRCLES: INCIRCLE, 1ST AND 2ND VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (b^2*c^10 - a^2*c^10 + 2*a*b^2*c^9 + 2*a^2*b*c^9 - 4*b^4*c^8 - 2*a*b^3*c^8 - a^2*b^2*c^8 + a^4*c^8 - 6*a*b^4*c^7 - 4*a^2*b^3*c^7 - 4*a^3*b^2*c^7 - 6*a^4*b*c^7 + 6*b^6*c^6 + 6*a*b^5*c^6 + 2*a^2*b^4*c^6 + 4*a^3*b^3*c^6 + 7*a^4*b^2*c^6 + 3*a^6*c^6 + 6*a*b^6*c^5 + 4*a^2*b^5*c^5 - 2*a^4*b^3*c^5 + 2*a^5*b^2*c^5 + 6*a^6*b*c^5 - 4*b^8*c^4 - 6*a*b^7*c^4 + 2*a^2*b^6*c^4 - 4*a^4*b^4*c^4 - 2*a^5*b^3*c^4 - 3*a^6*b^2*c^4 - 5*a^8*c^4 - 2*a*b^8*c^3 - 4*a^2*b^7*c^3 + 4*a^3*b^6*c^3 - 2*a^4*b^5*c^3 - 2*a^5*b^4*c^3 + 8*a^6*b^3*c^3 - 2*a^8*b*c^3 + b^10*c^2 + 2*a*b^9*c^2 - a^2*b^8*c^2 - 4*a^3*b^7*c^2 + 7*a^4*b^6*c^2 + 2*a^5*b^5*c^2 - 3*a^6*b^4*c^2 - 6*a^8*b^2*c^2 + 2*a^10*c^2 + 2*a^2*b^9*c - 6*a^4*b^7*c + 6*a^6*b^5*c - 2*a^8*b^3*c - a^2*b^10 + a^4*b^8 + 3*a^6*b^6 - 5*a^8*b^4 + 2*a^10*b^2) : :

X(45849) lies on this line: {5, 141}

X(45850) = RADICAL CENTER OF THESE CIRCLES: VIJAY EXCENTRAL CIRCLE, 1ST AND 2ND VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (2*b^2*c^10 - a*b^2*c^9 + a^3*c^9 - 8*b^4*c^8 + a*b^3*c^8 - 4*a^2*b^2*c^8 + a^3*b*c^8 - 2*a^4*c^8 + 3*a*b^4*c^7 - 3*a^5*c^7 + 12*b^6*c^6 - 3*a*b^5*c^6 + 4*a^2*b^4*c^6 - 4*a^3*b^3*c^6 + 4*a^4*b^2*c^6 - 3*a^5*b*c^6 + 6*a^6*c^6 - 3*a*b^6*c^5 + 2*a^3*b^4*c^5 - 2*a^5*b^2*c^5 + 3*a^7*c^5 - 8*b^8*c^4 + 3*a*b^7*c^4 + 4*a^2*b^6*c^4 + 2*a^3*b^5*c^4 - 8*a^4*b^4*c^4 + 4*a^6*b^2*c^4 + 3*a^7*b*c^4 - 6*a^8*c^4 + a*b^8*c^3 - 4*a^3*b^6*c^3 + 4*a^7*b^2*c^3 - a^9*c^3 + 2*b^10*c^2 - a*b^9*c^2 - 4*a^2*b^8*c^2 + 4*a^4*b^6*c^2 - 2*a^5*b^5*c^2 + 4*a^6*b^4*c^2 + 4*a^7*b^3*c^2 - 8*a^8*b^2*c^2 - a^9*b*c^2 + 2*a^10*c^2 + a^3*b^8*c - 3*a^5*b^6*c + 3*a^7*b^4*c - a^9*b^2*c + a^3*b^9 - 2*a^4*b^8 - 3*a^5*b^7 + 6*a^6*b^6 + 3*a^7*b^5 - 6*a^8*b^4 - a^9*b^3 + 2*a^10*b^2) : :

X(45850) lies on this line:{5,141}

X(45851) = RADICAL CENTER OF THESE CIRCLES: BROCARD CIRCLE, 1ST AND 2ND VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (b^2*c^12 - 3*b^4*c^10 + a^2*b^2*c^10 - 3*a^4*c^10 + 2*b^6*c^8 - 12*a^2*b^4*c^8 - 4*a^4*b^2*c^8 + 8*a^6*c^8 + 2*b^8*c^6 + 22*a^2*b^6*c^6 + 3*a^4*b^4*c^6 + 17*a^6*b^2*c^6 - 6*a^8*c^6 - 3*b^10*c^4 - 12*a^2*b^8*c^4 + 3*a^4*b^6*c^4 + 2*a^6*b^4*c^4 - 18*a^8*b^2*c^4 + b^12*c^2 + a^2*b^10*c^2 - 4*a^4*b^8*c^2 + 17*a^6*b^6*c^2 - 18*a^8*b^4*c^2 + 2*a^10*b^2*c^2 + a^12*c^2 - 3*a^4*b^10 + 8*a^6*b^8 - 6*a^8*b^6 + a^12*b^2) : :

X(45851) lies on these lines: {5, 141}, {3800, 6132}

X(45852) = RADICAL CENTER OF THESE CIRCLES: CIRCUMCIRCLE, 3RD AND 4TH VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    a^2*(c^16 - 7*b^2*c^14 - 7*a^2*c^14 + 24*b^4*c^12 + 24*a^2*b^2*c^12 + 17*a^4*c^12 - 49*b^6*c^10 - 36*a^2*b^4*c^10 - 18*a^4*b^2*c^10 - 15*a^6*c^10 + 62*b^8*c^8 + 19*a^2*b^6*c^8 + a^4*b^4*c^8 - 11*a^6*b^2*c^8 - 5*a^8*c^8 - 49*b^10*c^6 + 19*a^2*b^8*c^6 + 8*a^6*b^4*c^6 + 3*a^8*b^2*c^6 + 19*a^10*c^6 + 24*b^12*c^4 - 36*a^2*b^10*c^4 + a^4*b^8*c^4 + 8*a^6*b^6*c^4 - 8*a^8*b^4*c^4 + 24*a^10*b^2*c^4 - 13*a^12*c^4 - 7*b^14*c^2 + 24*a^2*b^12*c^2 - 18*a^4*b^10*c^2 - 11*a^6*b^8*c^2 + 3*a^8*b^6*c^2 + 24*a^10*b^4*c^2 - 18*a^12*b^2*c^2 + 3*a^14*c^2 + b^16 - 7*a^2*b^14 + 17*a^4*b^12 - 15*a^6*b^10 - 5*a^8*b^8 + 19*a^10*b^6 - 13*a^12*b^4 + 3*a^14*b^2) : :

X(45852) lies on these lines: {140, 6709}, {44056, 44668}

X(45853) = RADICAL CENTER OF THESE CIRCLES: NINE-POINT CIRCLE, 3RD AND 4TH VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (2*b^2*c^16 + a^2*c^16 - 10*b^4*c^14 - 11*a^2*b^2*c^14 - 5*a^4*c^14 + 18*b^6*c^12 + 28*a^2*b^4*c^12 + 20*a^4*b^2*c^12 + 9*a^6*c^12 - 10*b^8*c^10 - 29*a^2*b^6*c^10 - 24*a^4*b^4*c^10 - 14*a^6*b^2*c^10 - 5*a^8*c^10 - 10*b^10*c^8 + 22*a^2*b^8*c^8 + 9*a^4*b^6*c^8 + 5*a^6*b^4*c^8 + 9*a^8*b^2*c^8 - 5*a^10*c^8 + 18*b^12*c^6 - 29*a^2*b^10*c^6 + 9*a^4*b^8*c^6 + 14*a^8*b^4*c^6 - 21*a^10*b^2*c^6 + 9*a^12*c^6 - 10*b^14*c^4 + 28*a^2*b^12*c^4 - 24*a^4*b^10*c^4 + 5*a^6*b^8*c^4 + 14*a^8*b^6*c^4 - 32*a^10*b^4*c^4 + 24*a^12*b^2*c^4 - 5*a^14*c^4 + 2*b^16*c^2 - 11*a^2*b^14*c^2 + 20*a^4*b^12*c^2 - 14*a^6*b^10*c^2 + 9*a^8*b^8*c^2 - 21*a^10*b^6*c^2 + 24*a^12*b^4*c^2 - 10*a^14*b^2*c^2 + a^16*c^2 + a^2*b^16 - 5*a^4*b^14 + 9*a^6*b^12 - 5*a^8*b^10 - 5*a^10*b^8 + 9*a^12*b^6 - 5*a^14*b^4 + a^16*b^2) : :

X(45853) lies on these lines: {140, 6709}, {7514, 19189}, {13565, 24206}

X(45854) = RADICAL CENTER OF THESE CIRCLES: INCIRCLE, 3RD AND 4TH VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (4*b^2*c^16 + a^2*c^16 - 8*b^3*c^15 - 4*a*b^2*c^15 - 4*a^3*c^15 - 20*b^4*c^14 + 4*a*b^3*c^14 - 15*a^2*b^2*c^14 + 4*a^3*b*c^14 - 3*a^4*c^14 + 48*b^5*c^13 + 16*a*b^4*c^13 + 24*a^2*b^3*c^13 + 24*a^3*b^2*c^13 - 8*a^4*b*c^13 + 24*a^5*c^13 + 36*b^6*c^12 - 16*a*b^5*c^12 + 24*a^2*b^4*c^12 - 24*a^3*b^3*c^12 + 16*a^4*b^2*c^12 - 24*a^5*b*c^12 + a^6*c^12 - 120*b^7*c^11 - 20*a*b^6*c^11 - 72*a^2*b^5*c^11 - 32*a^3*b^4*c^11 - 40*a^5*b^2*c^11 + 40*a^6*b*c^11 - 60*a^7*c^11 - 20*b^8*c^10 + 20*a*b^7*c^10 + 23*a^2*b^6*c^10 + 32*a^3*b^5*c^10 - 4*a^4*b^4*c^10 + 40*a^5*b^3*c^10 - 18*a^6*b^2*c^10 + 60*a^7*b*c^10 + 5*a^8*c^10 + 160*b^9*c^9 + 48*a^2*b^7*c^9 + 4*a^3*b^6*c^9 + 8*a^4*b^5*c^9 + 8*a^5*b^4*c^9 - 40*a^6*b^3*c^9 + 4*a^7*b^2*c^9 - 80*a^8*b*c^9 + 80*a^9*c^9 - 20*b^10*c^8 - 66*a^2*b^8*c^8 - 4*a^3*b^7*c^8 - 9*a^4*b^6*c^8 - 8*a^5*b^5*c^8 + 17*a^6*b^4*c^8 - 4*a^7*b^3*c^8 + 53*a^8*b^2*c^8 - 80*a^9*b*c^8 - 5*a^10*c^8 - 120*b^11*c^7 + 20*a*b^10*c^7 + 48*a^2*b^9*c^7 - 4*a^3*b^8*c^7 - 8*a^8*b^3*c^7 + 44*a^9*b^2*c^7 + 80*a^10*b*c^7 - 60*a^11*c^7 + 36*b^12*c^6 - 20*a*b^11*c^6 + 23*a^2*b^10*c^6 + 4*a^3*b^9*c^6 - 9*a^4*b^8*c^6 + 20*a^8*b^4*c^6 - 44*a^9*b^3*c^6 - 69*a^10*b^2*c^6 + 60*a^11*b*c^6 - a^12*c^6 + 48*b^13*c^5 - 16*a*b^12*c^5 - 72*a^2*b^11*c^5 + 32*a^3*b^10*c^5 + 8*a^4*b^9*c^5 - 8*a^5*b^8*c^5 - 16*a^8*b^5*c^5 + 72*a^10*b^3*c^5 - 32*a^11*b^2*c^5 - 40*a^12*b*c^5 + 24*a^13*c^5 - 20*b^14*c^4 + 16*a*b^13*c^4 + 24*a^2*b^12*c^4 - 32*a^3*b^11*c^4 - 4*a^4*b^10*c^4 + 8*a^5*b^9*c^4 + 17*a^6*b^8*c^4 + 20*a^8*b^6*c^4 - 72*a^10*b^4*c^4 + 32*a^11*b^3*c^4 + 32*a^12*b^2*c^4 - 24*a^13*b*c^4 + 3*a^14*c^4 - 8*b^15*c^3 + 4*a*b^14*c^3 + 24*a^2*b^13*c^3 - 24*a^3*b^12*c^3 + 40*a^5*b^10*c^3 - 40*a^6*b^9*c^3 - 4*a^7*b^8*c^3 - 8*a^8*b^7*c^3 - 44*a^9*b^6*c^3 + 72*a^10*b^5*c^3 + 32*a^11*b^4*c^3 - 48*a^12*b^3*c^3 + 8*a^14*b*c^3 - 4*a^15*c^3 + 4*b^16*c^2 - 4*a*b^15*c^2 - 15*a^2*b^14*c^2 + 24*a^3*b^13*c^2 + 16*a^4*b^12*c^2 - 40*a^5*b^11*c^2 - 18*a^6*b^10*c^2 + 4*a^7*b^9*c^2 + 53*a^8*b^8*c^2 + 44*a^9*b^7*c^2 - 69*a^10*b^6*c^2 - 32*a^11*b^5*c^2 + 32*a^12*b^4*c^2 - 2*a^14*b^2*c^2 + 4*a^15*b*c^2 - a^16*c^2 + 4*a^3*b^14*c - 8*a^4*b^13*c - 24*a^5*b^12*c + 40*a^6*b^11*c + 60*a^7*b^10*c - 80*a^8*b^9*c - 80*a^9*b^8*c + 80*a^10*b^7*c + 60*a^11*b^6*c - 40*a^12*b^5*c - 24*a^13*b^4*c + 8*a^14*b^3*c + 4*a^15*b^2*c + a^2*b^16 - 4*a^3*b^15 - 3*a^4*b^14 + 24*a^5*b^13 + a^6*b^12 - 60*a^7*b^11 + 5*a^8*b^10 + 80*a^9*b^9 - 5*a^10*b^8 - 60*a^11*b^7 - a^12*b^6 + 24*a^13*b^5 + 3*a^14*b^4 - 4*a^15*b^3 - a^16*b^2) : :

X(45854) lies on this line: {140,6709}

X(45855) = RADICAL CENTER OF THESE CIRCLES: VIJAY EXCENTRAL CIRCLE, 3RD AND 4TH VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    (2*b^2*c^16 + a^2*c^16 + 2*a*b^2*c^15 - 2*a^3*c^15 - 10*b^4*c^14 + 6*a*b^3*c^14 - 11*a^2*b^2*c^14 + 2*a^3*b*c^14 - 5*a^4*c^14 - 16*a*b^4*c^13 + 16*a^5*c^13 + 18*b^6*c^12 - 32*a*b^5*c^12 + 28*a^2*b^4*c^12 - 24*a^3*b^3*c^12 + 20*a^4*b^2*c^12 - 8*a^5*b*c^12 + 9*a^6*c^12 + 50*a*b^6*c^11 + 20*a^3*b^4*c^11 - 20*a^5*b^2*c^11 - 50*a^7*c^11 - 10*b^8*c^10 + 70*a*b^7*c^10 - 29*a^2*b^6*c^10 + 52*a^3*b^5*c^10 - 24*a^4*b^4*c^10 + 20*a^5*b^3*c^10 - 14*a^6*b^2*c^10 + 10*a^7*b*c^10 - 5*a^8*c^10 - 80*a*b^8*c^9 - 22*a^3*b^6*c^9 + 22*a^7*b^2*c^9 + 80*a^9*c^9 - 10*b^10*c^8 - 80*a*b^9*c^8 + 22*a^2*b^8*c^8 - 26*a^3*b^7*c^8 + 9*a^4*b^6*c^8 - 8*a^5*b^5*c^8 + 5*a^6*b^4*c^8 + 18*a^7*b^3*c^8 + 9*a^8*b^2*c^8 - 5*a^10*c^8 + 70*a*b^10*c^7 - 26*a^3*b^8*c^7 + 26*a^9*b^2*c^7 - 70*a^11*c^7 + 18*b^12*c^6 + 50*a*b^11*c^6 - 29*a^2*b^10*c^6 - 22*a^3*b^9*c^6 + 9*a^4*b^8*c^6 + 14*a^8*b^4*c^6 - 18*a^9*b^3*c^6 - 21*a^10*b^2*c^6 - 10*a^11*b*c^6 + 9*a^12*c^6 - 32*a*b^12*c^5 + 52*a^3*b^10*c^5 - 8*a^5*b^8*c^5 + 8*a^9*b^4*c^5 - 52*a^11*b^2*c^5 + 32*a^13*c^5 - 10*b^14*c^4 - 16*a*b^13*c^4 + 28*a^2*b^12*c^4 + 20*a^3*b^11*c^4 - 24*a^4*b^10*c^4 + 5*a^6*b^8*c^4 + 14*a^8*b^6*c^4 + 8*a^9*b^5*c^4 - 32*a^10*b^4*c^4 - 20*a^11*b^3*c^4 + 24*a^12*b^2*c^4 + 8*a^13*b*c^4 - 5*a^14*c^4 + 6*a*b^14*c^3 - 24*a^3*b^12*c^3 + 20*a^5*b^10*c^3 + 18*a^7*b^8*c^3 - 18*a^9*b^6*c^3 - 20*a^11*b^4*c^3 + 24*a^13*b^2*c^3 - 6*a^15*c^3 + 2*b^16*c^2 + 2*a*b^15*c^2 - 11*a^2*b^14*c^2 + 20*a^4*b^12*c^2 - 20*a^5*b^11*c^2 - 14*a^6*b^10*c^2 + 22*a^7*b^9*c^2 + 9*a^8*b^8*c^2 + 26*a^9*b^7*c^2 - 21*a^10*b^6*c^2 - 52*a^11*b^5*c^2 + 24*a^12*b^4*c^2 + 24*a^13*b^3*c^2 - 10*a^14*b^2*c^2 - 2*a^15*b*c^2 + a^16*c^2 + 2*a^3*b^14*c - 8*a^5*b^12*c + 10*a^7*b^10*c - 10*a^11*b^6*c + 8*a^13*b^4*c - 2*a^15*b^2*c + a^2*b^16 - 2*a^3*b^15 - 5*a^4*b^14 + 16*a^5*b^13 + 9*a^6*b^12 - 50*a^7*b^11 - 5*a^8*b^10 + 80*a^9*b^9 - 5*a^10*b^8 - 70*a^11*b^7 + 9*a^12*b^6 + 32*a^13*b^5 - 5*a^14*b^4 - 6*a^15*b^3 + a^16*b^2) : :

X(45855) lies on this line: {140,6709}

X(45856) = RADICAL CENTER OF THESE CIRCLES: BROCARD CIRCLE, 3RD AND 4TH VIJAY POLAR MEDIAL CIRCLES OF ABC

Barycentrics    a^2*(c^18 - 2*b^2*c^16 - 10*a^2*c^16 - 3*b^4*c^14 + 14*a^2*b^2*c^14 + 42*a^4*c^14 + 11*b^6*c^12 + 4*a^2*b^4*c^12 - 33*a^4*b^2*c^12 - 98*a^6*c^12 - 7*b^8*c^10 - 6*a^2*b^6*c^10 - 13*a^4*b^4*c^10 + 20*a^6*b^2*c^10 + 140*a^8*c^10 - 7*b^10*c^8 - 4*a^2*b^8*c^8 + 4*a^4*b^6*c^8 + 34*a^6*b^4*c^8 + 39*a^8*b^2*c^8 - 126*a^10*c^8 + 11*b^12*c^6 - 6*a^2*b^10*c^6 + 4*a^4*b^8*c^6 + 16*a^6*b^6*c^6 - 17*a^8*b^4*c^6 - 78*a^10*b^2*c^6 + 70*a^12*c^6 - 3*b^14*c^4 + 4*a^2*b^12*c^4 - 13*a^4*b^10*c^4 + 34*a^6*b^8*c^4 - 17*a^8*b^6*c^4 - 40*a^10*b^4*c^4 + 57*a^12*b^2*c^4 - 22*a^14*c^4 - 2*b^16*c^2 + 14*a^2*b^14*c^2 - 33*a^4*b^12*c^2 + 20*a^6*b^10*c^2 + 39*a^8*b^8*c^2 - 78*a^10*b^6*c^2 + 57*a^12*b^4*c^2 - 20*a^14*b^2*c^2 + 3*a^16*c^2 + b^18 - 10*a^2*b^16 + 42*a^4*b^14 - 98*a^6*b^12 + 140*a^8*b^10 - 126*a^10*b^8 + 70*a^12*b^6 - 22*a^14*b^4 + 3*a^16*b^2) : :

X(45856) lies on this line: {140, 6709}

X(45857) = PERSPECTOR OF THESE TRIANGLES: ABC AND 6TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (b^2*(a^2 + c^2) + 4*a^2*c^2 - a^4 - c^4)*(c^2*(a^2 + b^2) + 4*a^2*b^2 - a^4 - b^4) : :

X(45857) lies on these lines: {160,45819}, {182,193}, {183,16419}, {262,14486}, {570,11596}, {1593,11257}, {5013,9307}, {5065,6353}, {7738,34208}, {20775,32085}, {37893,41624}

X(45857) = isogonal conjugate of X(5943)
X(45857) = isogonal conjugate of the complement of X(3917)
X(45857) = isotomic conjugate of the complement of X(7783)
X(45857) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5943}, {6, 17868}
X(45857) = cevapoint of X(i) and X(j) for these (i,j): {2, 7783}, {6, 20775}, {6337, 37894}
X(45857) = trilinear pole of line {3288, 3566}
X(45857) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17868}, {6, 5943}, {2056, 31989}

X(45858) = PERSPECTOR OF THESE TRIANGLES: ABC AND 21ST VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    a^2*((a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) - 2*(a^2*b^2 - b^4 + a^2*c^2 + 6*b^2*c^2 - c^4)*S) : :

X(45858) lies on these lines: {216, 373}, {493, 13943}, {1583, 26894}, {3069, 8956}, {3090, 8954}, {3103, 32786}, {5020, 6413}, {5413, 15215}, {5422, 10962}, {6414, 10601}, {6688, 8963}, {8911, 10963}, {15024, 32589}, {26950, 37343}

X(45858) = {X(216),X(373)}-harmonic conjugate of X(45859)

X(45859) = PERSPECTOR OF THESE TRIANGLES: ABC AND 22ND VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    a^2*((a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) + 2*(a^2*b^2 - b^4 + a^2*c^2 + 6*b^2*c^2 - c^4)*S) : :

X(45859) lies on these lines: {216, 373}, {494, 13889}, {1584, 26919}, {3090, 32589}, {3102, 32785}, {5020, 6414}, {5412, 15216}, {5422, 10960}, {5943, 8963}, {6413, 10601}, {8954, 15024}, {10961, 26920}, {26886, 43118}, {26951, 37342}

X(45859) = {X(216),X(373)}-harmonic conjugate of X(45858)

X(45860) = PERSPECTOR OF THESE TRIANGLES: 1ST AND 17TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    ( c^6 - b^2*c^4 + 2*a^2*c^4 - 4*S*c^4 - b^4*c^2 - 4*a^2*b^2*c^2 + 8*S*b^2*c^2 - 3*a^4*c^2 + 4*S*a^2*c^2 + b^6 + 2*a^2*b^4 - 4*S*b^4 - 3*a^4*b^2 + 4*S*a^2*b^2) : :
X(45860) = 3X[381]+X[12314] = 3X[597]-2X[44509] = 5X[3091]-X[12323] = 5X[6251]-3X[13807] = X[12256]-3X[13847] = 3X[12963]-X[26441]

X(45860) lies on these lies on these lines: {2,12306}, {3,45863}, {4,615}, {5,141}, {11,7353}, {12,6405}, {20,26331}, {30,6251}, {140,7692}, {230,3071}, {235,6406}, {371,45870}, {372,14230}, {381,12314}, {403,6400}, {427,12299}, {485,9975}, {486,1503}, {492,3091}, {524,6290}, {546,22820}, {597,44509}, {1132,43537}, {3069,13749}, {3070,12969}, {3090,6202}, {3367,41034}, {3392,41035}, {3545,6201}, {3564,45543}, {3850,6250}, {3860,13687}, {5023,23261}, {5318,10672}, {5321,10671}, {5871,13939}, {6560,14235}, {6565,14233}, {6759,30399}, {6811,42583}, {6823,12361}, {7000,13748}, {7584,8550}, {8252,21736}, {9733,44390}, {10024,22812}, {10577,36709}, {11824,32490}, {12172,37197}, {12232,12241}, {12233,12240}, {12256,13847}, {12604,15760}, {13966,32209}, {14231,38227}, {14239,42268}, {14853,42265}, {18762,36657}, {21656,43831}, {32788,45407}, {36656,42274}, {42582,45869}, {43880,45510}

X(45860) = midpoint of X(i) and X(j) for these {i,j}: {4, 1152}, {486, 36655}
X(45860) = reflection of X(i) in X(j) for these {i,j}: {7692, 140}, {8550, 44657}, {22820, 546}, {23312, 5}
X(45860) = complement of X(12306)
X(45860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 5480, 45861}, {6565, 36714, 14233}, {7000, 42561, 13748}

X(45861) = PERSPECTOR OF THESE TRIANGLES: 1ST AND 18TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    ( c^6 - b^2*c^4 + 2*a^2*c^4 + 4*S*c^4 - b^4*c^2 - 4*a^2*b^2*c^2 - 8*S*b^2*c^2 - 3*a^4*c^2 - 4*S*a^2*c^2 + b^6 + 2*a^2*b^4 + 4*S*b^4 - 3*a^4*b^2 - 4*S*a^2*b^2 ) : :
X(45861) = 3X[381]+X[12313] = 3X[597]-2X[44510] = 5X[3091]-X[12322] = 5X[6250]-3X[13687] = X[8982]-3X[12968] = X[12257]-3X[13846]

X(45861) lies on these lies on these lines: {2,12305}, {3,45862}, {4,590}, {5,141}, {11,7362}, {12,6283}, {20,26330}, {30,6250}, {140,7690}, {230,3070}, {235,6291}, {371,14233}, {381,12313}, {403,6239}, {427,12298}, {485,1503}, {486,9974}, {491,3091}, {524,6289}, {546,22819}, {597,44510}, {1131,43537}, {3068,13748}, {3071,12962}, {3090,6201}, {3366,41034}, {3391,41035}, {3545,6202}, {3564,45542}, {3850,6251}, {3860,13807}, {5023,21736}, {5318,10668}, {5321,10667}, {5870,13886}, {6561,14239}, {6564,14230}, {6565,45870}, {6759,30398}, {6813,42582}, {6823,12360}, {7374,13638}, {7583,8550}, {8968,10192}, {8981,13924}, {9732,44391}, {10024,22811}, {10576,36714}, {11825,32491}, {12171,37197}, {12231,12241}, {12233,12239}, {12257,13846}, {12603,15760}, {14235,42269}, {14245,38227}, {14853,42262}, {18538,36658}, {21655,43831}, {26441,31454}, {32787,45406}, {36655,42277}, {42583,45868}, {43879,45511}

X(45861) = midpoint of X(i) and X(j) for these {i,j}: {4, 1151}, {485, 36656}, {21736, 23251}
X(45861) = reflection of X(i) in X(j) for these {i,j}: {7690, 140}, {8550, 44656}, {22819, 546}, {23311, 5}
X(45861) = complement of X(12305)
X(45861) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 5480, 45860}, {6564, 36709, 14230}, {7374, 31412, 13749}

X(45862) = PERSPECTOR OF THESE TRIANGLES: 1ST AND 19th VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^6 - b^2*c^4 + 2*a^2*c^4 + 4*S*c^4 - b^4*c^2 - 4*a^2*b^2*c^2 - 8*S*b^2*c^2 - 3*a^4*c^2 + b^6 + 2*a^2*b^4 + 4*S*b^4 - 3*a^4*b^2 - 4*S*a^4) : :
X(45862) = 3X[4]+X[45407] = X[490]-5X[3091] = 3X[3070]-X[45407] = X[26441]-3X[32787]

X(45862) lies on these lies on these lines: {2,26294}, {3,45861}, {4,6}, {5,641}, {30,43120}, {262,14240}, {381,12314}, {485,15883}, {486,32495}, {490,3091}, {576,22819}, {578,30428}, {591,26468}, {1352,12602}, {3127,23332}, {3156,13021}, {3627,45545}, {3845,6251}, {3861,43145}, {5102,12221}, {5200,10192}, {5418,45869}, {6214,32421}, {6239,44439}, {6400,9971}, {6560,36656}, {6564,36714}, {6811,42259}, {6813,42273}, {7789,23312}, {9733,23311}, {10516,12323}, {11292,12297}, {11315,12124}, {11477,12322}, {12222,15069}, {12298,19161}, {12601,20423}, {13019,23292}, {13567,32588}, {13663,26516}, {14484,43560}, {18539,32455}, {21736,42264}, {26441,32787}, {32491,45498}, {32497,42265}, {35820,36709}, {36655,42269}, {41034,42238}, {41035,42237}, {42274,45868}

X(45862) = midpoint of X(i) and X(j) for these {i,j}: {4, 3070}, {35820, 36709}
X(45862) = orthosymmedial-circle-inverse of X(14233)
X(45862) = crosspoint of X(4) and X(14245)
X(45862) = crosssum of X(3) and X(43120)
X(45862) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6, 14233}, {4, 1587, 13748}, {4, 3071, 14239}, {4, 5480, 45863}, {4, 6201, 45441}, {4, 14853, 23261}, {4, 23249, 13749}, {4, 23251, 14230}, {4, 23267, 5870}, {4, 23269, 5871}, {4, 42284, 14235}, {4, 45440, 5480}, {1587, 13748, 8550}, {6201, 45441, 5480}, {6250, 45544, 5}, {14853, 23261, 45870}, {45440, 45441, 6201}

X(45863) = PERSPECTOR OF THESE TRIANGLES: 1ST AND 20TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    3*a^4*b^2 - 2*a^2*b^4 - b^6 + 3*a^4*c^2 + 4*a^2*b^2*c^2 + b^4*c^2 - 2*a^2*c^4 + b^2*c^4 - c^6 - 4*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S : :
X(45863) = 3X[4]+X[45406] = X[489]-5X[3091] = 3X[3071]-X[45406] = X[8982]-3X[32788]

X(45863) lies on these lies on these lines: {2,26295}, {3,45860}, {4,6}, {5,642}, {30,43121}, {262,14236}, {381,12313}, {485,32492}, {486,15884}, {489,3091}, {576,22820}, {578,30427}, {1352,12601}, {1991,26469}, {3128,23332}, {3155,13022}, {3627,45544}, {3845,6250}, {3861,43143}, {5102,12222}, {5420,45868}, {6215,32419}, {6239,9971}, {6400,44439}, {6561,36655}, {6565,36709}, {6811,42270}, {6813,42258}, {7789,23311}, {8982,32788}, {9732,23312}, {10516,12322}, {11291,12296}, {11316,12123}, {11477,12323}, {12221,15069}, {12299,19161}, {12602,20423}, {13020,23292}, {13567,32587}, {13783,26521}, {14484,43561}, {21736,32494}, {26438,32455}, {32490,45499}, {35821,36714}, {36656,42268}, {41034,42236}, {41035,42235}, {42277,45869}

X(45863) = midpoint of X(i) and X(j) for these {i,j}: {4, 3071}, {35821, 36714}
X(45863) = orthosymmedial-circle-inverse of X(14230)
X(45863) = crosspoint of X(4) and X(14231)
X(45863) = crosssum of X(3) and X(43121)
X(45863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6, 14230}, {4, 1588, 13749}, {4, 3070, 14235}, {4, 5480, 45862}, {4, 6202, 45440}, {4, 14853, 23251}, {4, 23259, 13748}, {4, 23261, 14233}, {4, 23273, 5871}, {4, 23275, 5870}, {4, 42283, 14239}, {4, 45441, 5480}, {1588, 13749, 8550}, {6202, 45440, 5480}, {6251, 45545, 5}, {14230, 45870, 6}, {45440, 45441, 6202}

X(45864) = PERSPECTOR OF THESE TRIANGLES: 2ND AND 3RD VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (-c^2 + b^2 + a^2)*(c^2 - b^2 + a^2)*(9*c^10 - 11*b^2*c^8 - 7*a^2*c^8 + 2*b^4*c^6 - 4*a^2*b^2*c^6 + 2*a^4*c^6 + 2*b^6*c^4 + 22*a^2*b^4*c^4 - 2*a^4*b^2*c^4 - 22*a^6*c^4 - 11*b^8*c^2 - 4*a^2*b^6*c^2 - 2*a^4*b^4*c^2 - 4*a^6*b^2*c^2 + 21*a^8*c^2 + 9*b^10 - 7*a^2*b^8 + 2*a^4*b^6 - 22*a^6*b^4 + 21*a^8*b^2 - 3*a^10) : :

X(45864) lies on these lies on these lines: {2,15312}, {4,3172}, {132,1249}, {262,459}, {376,38699}, {1529,41374}, {1853,14853}, {3079,38918}, {6353,6525}, {7714,9753}, {9747,14249}, {9756,10002}, {15428,41361}

X(45864) = X(10002)-Ceva conjugate of X(4)

X(45865) = PERSPECTOR OF THESE TRIANGLES: 2ND AND 11TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    ( 3*c^12 - 11*b^2*c^10 - 22*a^2*c^10 + 18*S*c^10 + 37*b^4*c^8 + 59*a^2*b^2*c^8 - 26*S*b^2*c^8 + 48*a^4*c^8 - 4*S*a^2*c^8 - 58*b^6*c^6 - 61*a^2*b^4*c^6 + 8*S*b^4*c^6 + 5*a^4*b^2*c^6 - 90*S*a^2*b^2*c^6 - 42*a^6*c^6 - 26*S*a^4*c^6 + 37*b^8*c^4 - 61*a^2*b^6*c^4 + 8*S*b^6*c^4 - 94*a^4*b^4*c^4 + 68*S*a^2*b^4*c^4 - 87*a^6*b^2*c^4 + 14*S*a^4*b^2*c^4 + 13*a^8*c^4 + 30*S*a^6*c^4 - 11*b^10*c^2 + 59*a^2*b^8*c^2 - 26*S*b^8*c^2 + 5*a^4*b^6*c^2 - 90*S*a^2*b^6*c^2 - 87*a^6*b^4*c^2 + 14*S*a^4*b^4*c^2 + 34*a^8*b^2*c^2 - 18*S*a^6*b^2*c^2 - 24*S*a^8*c^2 + 3*b^12 - 22*a^2*b^10 + 18*S*b^10 + 48*a^4*b^8 - 4*S*a^2*b^8 - 42*a^6*b^6 - 26*S*a^4*b^6 + 13*a^8*b^4 + 30*S*a^6*b^4 - 24*S*a^8*b^2 + 6*S*a^10 ) : :

X(45866) lies on these lines: {}

X(45866) = PERSPECTOR OF THESE TRIANGLES: 2ND AND 12TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    ( 3*c^12 - 11*b^2*c^10 - 22*a^2*c^10 - 18*S*c^10 + 37*b^4*c^8 + 59*a^2*b^2*c^8 + 26*S*b^2*c^8 + 48*a^4*c^8 + 4*S*a^2*c^8 - 58*b^6*c^6 - 61*a^2*b^4*c^6 - 8*S*b^4*c^6 + 5*a^4*b^2*c^6 + 90*S*a^2*b^2*c^6 - 42*a^6*c^6 + 26*S*a^4*c^6 + 37*b^8*c^4 - 61*a^2*b^6*c^4 - 8*S*b^6*c^4 - 94*a^4*b^4*c^4 - 68*S*a^2*b^4*c^4 - 87*a^6*b^2*c^4 - 14*S*a^4*b^2*c^4 + 13*a^8*c^4 - 30*S*a^6*c^4 - 11*b^10*c^2 + 59*a^2*b^8*c^2 + 26*S*b^8*c^2 + 5*a^4*b^6*c^2 + 90*S*a^2*b^6*c^2 - 87*a^6*b^4*c^2 - 14*S*a^4*b^4*c^2 + 34*a^8*b^2*c^2 + 18*S*a^6*b^2*c^2 + 24*S*a^8*c^2 + 3*b^12 - 22*a^2*b^10 - 18*S*b^10 + 48*a^4*b^8 + 4*S*a^2*b^8 - 42*a^6*b^6 + 26*S*a^4*b^6 + 13*a^8*b^4 - 30*S*a^6*b^4 + 24*S*a^8*b^2 - 6*S*a^10 ) : :

X(45867) lies on these lines: {}

X(45867) = PERSPECTOR OF THESE TRIANGLES: 3RD AND 6TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (-c^2 + b^2 + a^2)*(c^2 - b^2 + a^2)*(c^8 - 3*b^2*c^6 - 7*a^2*c^6 + 4*b^4*c^4 + 7*a^2*b^2*c^4 + 14*a^4*c^4 - 3*b^6*c^2 + 7*a^2*b^4*c^2 + 7*a^4*b^2*c^2 - 11*a^6*c^2 + b^8 - 7*a^2*b^6 + 14*a^4*b^4 - 11*a^6*b^2 + 3*a^8) : :

X(45867) lies on these lies on these lines: {4,1173}, {436,6749}, {3087,3168}, {5065,6353}

X(45867) = X(6748)-Ceva conjugate of X(4)

X(45868) = PERSPECTOR OF THESE TRIANGLES: 7TH AND 17TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^6 - b^2*c^4 + 4*a^2*c^4 - 8*S*c^4 - b^4*c^2 - 8*a^2*b^2*c^2 + 16*S*b^2*c^2 - 7*a^4*c^2 + 12*S*a^2*c^2 + b^6 + 4*a^2*b^4 - 8*S*b^4 - 7*a^4*b^2 + 12*S*a^2*b^2 + 2*a^6 - 4*S*a^4) : :
X(45868) = X[4]+3X[13935] = 7X[3523]-3X[26295]

X(45868) lies on these lies on these lines: {4,615}, {5,32421}, {140,9738}, {141,1351}, {230,45524}, {492,5056}, {1503,13951}, {3424,3591}, {3523,26295}, {5420,45863}, {5480,10577}, {6251,35256}, {6398,14235}, {6813,43880}, {7388,44400}, {8550,13972}, {13749,13941}, {13783,21737}, {13966,14230}, {14233,18762}, {32786,45441}, {36656,42603}, {42274,45862}, {42583,45861}

X(45869) = PERSPECTOR OF THESE TRIANGLES: 7TH AND 18TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^6 - b^2*c^4 + 4*a^2*c^4 + 8*S*c^4 - b^4*c^2 - 8*a^2*b^2*c^2 - 16*S*b^2*c^2 - 7*a^4*c^2 - 12*S*a^2*c^2 + b^6 + 4*a^2*b^4 + 8*S*b^4 - 7*a^4*b^2 - 12*S*a^2*b^2 + 2*a^6 + 4*S*a^4) : :
X(45869) = X[4]+3X[9540] = 7X[3523]-3X[26294]

X(45869) lies on these lies on these lines: {4,590}, {5,32419}, {140,9739}, {141,1351}, {230,45525}, {491,5056}, {1503,8976}, {3424,3590}, {3523,26294}, {5418,45862}, {5480,10576}, {6221,14239}, {6250,35255}, {6811,43879}, {7389,44393}, {8550,8960}, {8972,13748}, {8981,14233}, {14230,18538}, {32785,45440}, {36655,42602}, {42277,45863}, {42582,45860}

X(45870) = PERSPECTOR OF THESE TRIANGLES: 7TH AND 20TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^6 - b^2*c^4 + 4*a^2*c^4 - 4*S*c^4 - b^4*c^2 - 8*a^2*b^2*c^2 + 8*S*b^2*c^2 - 7*a^4*c^2 + b^6 + 4*a^2*b^4 - 4*S*b^4 - 7*a^4*b^2 + 2*a^6 + 4*S*a^4 ) : :
X(45870) = X[4]+3X[1588] = 5X[4]+3X[10784] = X[4]-3X[45441] = 5X[1588]-X[10784] = X[10784]+5X[45441]

X(45870) lies on these lies on these lines: {4,6}, {5,32419}, {140,9738}, {371,45860}, {550,43121}, {1656,12313}, {3858,22625}, {5056,32806}, {5102,12323}, {6251,7583}, {6565,45861}, {7584,45545}, {7585,26331}, {10516,12221}, {12601,14561}, {15520,22820}, {23312,45489}, {32788,45525}, {35823,36709}

X(45870) = midpoint of X(1588) and X(45441)
X(45870) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 45863, 14230}, {3071, 5480, 14233}, {6202, 23273, 13748}, {7582, 13749, 12007}, {14853, 23261, 45862}, {23259, 45440, 14239}

X(45871) = PERSPECTOR OF THESE TRIANGLES: 11TH , 18TH AND 19TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^2 + b^2 + 2*a^2 + 4*S) : :
X(45871) = 9X[2]-X[1270] = 3X[2]+X[3068] = 7X[2]+X[13639] = 4X[2]+X[13664] = X[1270]+3X[3068] = 7X[1270]+9X[13639] = X[1270]+9X[13663] = 4X[1270]+9X[13664] = X[1270]-3X[45472] = 7X[3068]-3X[13639] = X[3068]-3X[13663] = 4X[3068]-3X[13664] = X[13639]-7X[13663] = 4X[13639]-7X[13664] = 3X[13639]+7X[45472] = 4X[13663]-X[13664] = 3X[13663]+X[45472] = 3X[13664]+4X[45472]

X(45871) lies on these lies on these lines: {2,6}, {3,45861}, {4,42838}, {5,6118}, {140,9739}, {371,23311}, {485,11315}, {489,41963}, {637,31454}, {638,43879}, {639,7886}, {640,6680}, {641,7583}, {642,7834}, {1267,4395}, {1583,44192}, {3070,39387}, {3526,12314}, {3767,13882}, {3788,45576}, {5391,7227}, {5393,17243}, {5418,11313}, {6425,12322}, {6671,34552}, {6672,34551}, {6722,42215}, {6811,44882}, {7388,42582}, {7808,43125}, {7844,35255}, {8550,45554}, {9306,30398}, {10576,23312}, {11292,42265}, {11294,42273}, {12124,15293}, {17366,32791}, {17367,32803}, {17368,32804}, {17369,32792}, {18841,34089}, {32489,42258}, {35949,42284}, {37343,45439}

X(45871) = midpoint of X(i) and X(j) for these {i,j}: {2, 13663}, {3068, 45472}, {37689, 45473}
X(45871) = complement of X(45472)
X(45871) = crosssum of X(6) and X(1504)
X(45871) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 590, 141}, {2, 3068, 45472}, {2, 3589, 45872}, {2, 3618, 8252}, {2, 7585, 26361}, {2, 8972, 5590}, {2, 32785, 45473}, {2, 32806, 3763}, {2, 45473, 34573}, {5, 43120, 14233}, {6, 44381, 45872}, {230, 590, 13910}, {371, 32491, 23311}, {492, 32787, 3629}, {591, 7585, 32455}, {1991, 5590, 3631}, {5590, 8972, 1991}, {7585, 26361, 591}, {13638, 32785, 590}, {13663, 45472, 3068}

X(45872) = PERSPECTOR OF THESE TRIANGLES: 12TH , 17TH AND 20TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    (c^2 + b^2 + 2*a^2 - 4*S) : :
X(45872) = 9X[2]-X[1271] = 3X[2]+X[3069] = 7X[2]+X[13759] = 4X[2]+X[13784] = X[1271]+3X[3069] = 7X[1271]+9X[13759] = X[1271]+9X[13783] = 4X[1271]+9X[13784] = X[1271]-3X[45473] = 7X[3069]-3X[13759] = X[3069]-3X[13783] = 4X[3069]-3X[13784] = X[13759]-7X[13783] = 4X[13759]-7X[13784] = 3X[13759]+7X[45473] = 4X[13783]-X[13784] = 3X[13783]+X[45473] = 3X[13784]+4X[45473]

X(45872) lies on these lies on these lines: {2,6}, {3,45860}, {4,42840}, {5,6119}, {140,9738}, {372,23312}, {486,11316}, {490,41964}, {637,43880}, {639,6680}, {640,7886}, {641,7834}, {642,7584}, {1267,7227}, {1584,44193}, {3071,39388}, {3526,12313}, {3767,13934}, {3788,45577}, {4395,5391}, {5405,17243}, {5420,11314}, {6426,12323}, {6671,34551}, {6672,34552}, {6722,42216}, {6813,44882}, {7389,42583}, {7808,43124}, {7844,35256}, {8550,45555}, {9306,30399}, {10577,23311}, {11291,42262}, {11293,42270}, {12123,15294}, {17366,32792}, {17367,32804}, {17368,32803}, {17369,32791}, {18841,34091}, {32488,42259}, {35948,42283}, {37342,45438}

X(45872) = midpoint of X(i) and X(j) for these {i,j}: {2, 13783}, {3069, 45473}, {37689, 45472}
X(45872) = complement of X(45473)
X(45872) = crosssum of X(6) and X(1505)
X(45872) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 615, 141}, {2, 3069, 45473}, {2, 3589, 45871}, {2, 3618, 8253}, {2, 7586, 26362}, {2, 13941, 5591}, {2, 32786, 45472}, {2, 32805, 3763}, {2, 45472, 34573}, {5, 43121, 14230}, {6, 44381, 45871}, {230, 615, 13972}, {372, 32490, 23312}, {491, 32788, 3629}, {591, 5591, 3631}, {1991, 7586, 32455}, {5591, 13941, 591}, {7586, 26362, 1991}, {13758, 32786, 615}, {13783, 45473, 3069}

X(45873) = PERSPECTOR OF THESE TRIANGLES: 20TH AND 21TH VIJAY POLAR MEDIAL TRIANGLE

Barycentrics    a^2*(b^2*c^10 + a^2*c^10 - 8*b^4*c^8 - 3*a^2*b^2*c^8 - 3*a^4*c^8 + 14*b^6*c^6 + 2*a^2*b^4*c^6 + a^4*b^2*c^6 + 3*a^6*c^6 - 8*b^8*c^4 + 2*a^2*b^6*c^4 + 4*a^4*b^4*c^4 + 3*a^6*b^2*c^4 - a^8*c^4 + b^10*c^2 - 3*a^2*b^8*c^2 + a^4*b^6*c^2 + 3*a^6*b^4*c^2 - 2*a^8*b^2*c^2 + a^2*b^10 - 3*a^4*b^8 + 3*a^6*b^6 - a^8*b^4) : :
X(45873) = 5X[216]-9X[373] = 5X[264]+3X[3060] = 3X[3819]-5X[14767] = X[12162]-5X[39530]

X(45873) lies on these lies on these lines: {216,373}, {264,3060}, {511,546}, {520,6748}, {3819,10184}, {12006,32428}, {12162,39530}

X(45874) = X(100)X(6733)∩X(104)X(12771)

Barycentrics    Sin[A]^2/(Sin[B/2] - Sin[C/2]) : :

X(45874) lies on the circumcircle and these lines: {100, 6733}, {104, 12771}, {105, 41799}, {258, 10497}, {1311, 2090}

X(45874) = X(6733)-Ceva conjugate of X(3659)
X(45874) = crosspoint of X(109) and X(6733)
X(45874) = crosssum of X(522) and X(6728)
X(45874) = incircle-inverse of X(124)-of-intouch-triangle
X(45874) = X(i)-isoconjugate of X(j) for these (i,j): {236, 10492}, {7057, 10495}
X(45874) = barycentric product X(i)*X(j) for these {i,j}: {100, 41799}, {109, 2090}, {174, 3659}, {258, 43192}, {651, 15997}, {6733, 16015}, {7028, 13444}
X(45874) = barycentric quotient X(i)/X(j) for these {i,j}: {2090, 35519}, {3659, 556}, {15997, 4391}, {41799, 693}

X(45875) = TRILINEAR POLE OF X(1)X(164)

Barycentrics    Sin[A]/(Sin[B/2] - Sin[C/2]) : :

X(45875) lies on the curve CC9 and these lines: {56, 361}, {88, 41799}, {100, 6733}, {174, 289}, {259, 1488}, {1156, 15997}, {2090, 34234}, {3659, 13444}, {7028, 15495}

X(45875) = X(10495)-cross conjugate of X(1488)
X(45875) = cevapoint of X(i) and X(j) for these (i,j): {259, 10495}, {10492, 16015}
X(45875) = crosssum of X(650) and X(6729)
X(45875) = trilinear pole of line {1, 164} (line IO of the intouch triangle)
X(45875) = X(173)-isoconjugate of X(10495)
X(45875) = barycentric product X(i)*X(j) for these {i,j}: {190, 41799}, {651, 2090}, {664, 15997}, {3659, 4146}, {7048, 43192}
X(45875) = barycentric quotient X(i)/X(j) for these {i,j}: {289, 10492}, {2090, 4391}, {3659, 188}, {13444, 2089}, {15997, 522}, {16011, 6728}, {41799, 514}, {43192, 7057}

X(45876) = TRILINEAR POLE OF X(2)X(174)

Barycentrics    1/(Sin[B/2] - Sin[C/2]) : :

X(45876) lies on the Steiner circumellipse and these lines: {57, 16018}, {556, 15495}, {1121, 2090}, {1488, 4146}, {2481, 15997}, {3227, 41799}, {7057, 21456}

X(45876) = cevapoint of X(8422) and X(10492)
X(45876) = trilinear pole of line {2, 174}
X(45876) = X(10495)-isoconjugate of X(42622)
X(45876) = barycentric product X(i)*X(j) for these {i,j}: {664, 2090}, {668, 41799}, {4554, 15997}
X(45876) = barycentric quotient X(i)/X(j) for these {i,j}: {258, 10495}, {1488, 10492}, {2090, 522}, {3659, 259}, {10492, 6732}, {15997, 650}, {16011, 6729}, {16015, 6728}, {41799, 513}, {42017, 6730}, {43192, 173}

X(45877) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(1)X(164)

Barycentrics    Sin[A]*(Sin[B/2] - Sin[C/2]) : :

X(45877) lies on these lines: {44, 513}, {6728, 6729}

X(45877) = X(i)-Ceva conjugate of X(j) for these (i,j): {174, 6732}, {10492, 10495}, {10496, 55}
X(45877) = cevapoint of X(650) and X(6729)
X(45877) = crosspoint of X(174) and X(43192)
X(45877) = crosssum of X(i) and X(j) for these (i,j): {259, 10495}, {10492, 16015}
X(45877) = crossdifference of every pair of points on line {1, 164}
X(45877) = X(i)-isoconjugate of X(j) for these (i,j): {100, 41799}, {109, 2090}, {174, 3659}, {258, 43192}, {651, 15997}, {6733, 16015}, {7028, 13444}
X(45877) = barycentric product X(i)*X(j) for these {i,j}: {236, 10492}, {7057, 10495}
X(45877) = barycentric quotient X(i)/X(j) for these {i,j}: {649, 41799}, {650, 2090}, {663, 15997}, {6729, 16015}, {10495, 7048}, {42622, 43192}

X(45878) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(2)X(174)

Barycentrics    Sin[A]^2*(Sin[B/2] - Sin[C/2]) : :

X(45878) lies on this line: {187, 237}

X(45878) = crosssum of X(8422) and X(10492)
X(45878) = crossdifference of every pair of points on line {2, 174}
X(45878) = X(i)-isoconjugate of X(j) for these (i,j): {190, 41799}, {651, 2090}, {664, 15997}, {3659, 4146}, {7048, 43192}
X(45878) = barycentric product X(173)*X(10495)
X(45878) = barycentric quotient X(i)/X(j) for these {i,j}: {663, 2090}, {667, 41799}, {3063, 15997}

X(45879) = MIDPOINT OF X(2) AND X(15)

Barycentrics    Sqrt[3]*(4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*(4*a^2 + b^2 + c^2)*S : :
Barycentrics    4 a sin(A + π/3) + b sin(B + π/3) + c sin(C + π/3) : :
X(45879) = 5 X[2] - X[621], 7 X[2] - 5 X[40334], X[5] + 2 X[21401], 5 X[15] + X[621], 2 X[15] + X[623], X[15] + 2 X[6671], 7 X[15] + 5 X[40334], X[16] - 3 X[26613], X[376] - 3 X[21158], X[618] + 2 X[42912], 2 X[621] - 5 X[623], X[621] - 10 X[6671], 7 X[621] - 25 X[40334], X[623] - 4 X[6671], 7 X[623] - 10 X[40334], X[671] - 3 X[22510], X[1992] - 3 X[36757], 3 X[3524] - X[14538], X[3543] - 3 X[41036], 3 X[3545] + X[36993], 3 X[3839] - X[36992], 3 X[5054] + X[5611], 3 X[5055] - X[20428], 3 X[5215] - 2 X[6672], X[5463] + 3 X[16962], 2 X[6669] + X[42942], 14 X[6671] - 5 X[40334], X[6780] + 3 X[22490], X[7684] + 2 X[13350], 3 X[9166] - X[23004], 5 X[16960] - X[35752], 3 X[16962] - X[37786], 3 X[22489] - 2 X[33560], 3 X[22489] + X[36967], X[22493] - 5 X[36770], 2 X[33560] + X[36967], X[34315] - 3 X[37907], 4 X[35019] - X[42941]

X(45879) lies on these lines: {2, 14}, {5, 21401}, {6, 13084}, {13, 35931}, {16, 26613}, {23, 34314}, {30, 5459}, {61, 37785}, {187, 396}, {302, 22496}, {316, 43483}, {351, 9194}, {376, 21158}, {381, 44666}, {511, 549}, {519, 11707}, {524, 618}, {532, 5463}, {533, 10613}, {551, 44659}, {599, 11301}, {616, 22495}, {624, 3849}, {629, 22236}, {636, 7810}, {671, 22510}, {691, 34316}, {1992, 36757}, {3104, 33274}, {3524, 14538}, {3543, 41036}, {3545, 36993}, {3643, 9763}, {3839, 36992}, {5054, 5611}, {5055, 20428}, {5215, 6672}, {5238, 11303}, {6582, 22580}, {6669, 31693}, {6673, 16964}, {6694, 37341}, {7619, 33474}, {8584, 42633}, {8587, 43538}, {8598, 40671}, {8838, 11586}, {9166, 23004}, {9761, 11485}, {9762, 36764}, {9855, 23005}, {9885, 16508}, {10617, 43229}, {10645, 35932}, {10667, 13847}, {10671, 13846}, {11295, 16644}, {11296, 11480}, {11299, 39555}, {11304, 35229}, {11305, 36836}, {11306, 43238}, {11488, 22492}, {12100, 36755}, {14830, 33388}, {16267, 30560}, {16942, 42154}, {16960, 35752}, {18334, 40695}, {22489, 33560}, {22493, 36770}, {22573, 35917}, {22576, 36772}, {22997, 36766}, {23284, 23872}, {23302, 31694}, {31173, 37351}, {33480, 44215}, {33608, 36352}, {34315, 37907}, {34509, 37172}, {35019, 42941}, {37640, 41406}, {42913, 44498}, {43544, 45103}

X(45879) = midpoint of X(i) and X(j) for these {i,j}: {2, 15}, {13, 35931}, {14, 8594}, {23, 34314}, {396, 35304}, {616, 22495}, {691, 34316}, {5463, 37786}, {9855, 23005}, {31693, 42942}
X(45879) = reflection of X(i) in X(j) for these {i,j}: {2, 6671}, {623, 2}, {31693, 6669}, {36755, 12100}
X(45879) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13083, 619}, {15, 6671, 623}, {549, 597, 45880}, {5463, 16962, 37786}, {22490, 33417, 2}, {37172, 42152, 34509}

X(45880) = MIDPOINT OF X(2) AND X(16)

Barycentrics    Sqrt[3]*(4*a^4 - 5*a^2*b^2 + b^4 - 5*a^2*c^2 - 2*b^2*c^2 + c^4) + 2*(4*a^2 + b^2 + c^2)*S : :
Barycentrics    4 a sin(A - π/3) + b sin(B - π/3) + c sin(C - π/3) : :
X(45880) = 5 X[2] - X[622], 7 X[2] - 5 X[40335], X[5] + 2 X[21402], X[15] - 3 X[26613], 5 X[16] + X[622], 2 X[16] + X[624], X[16] + 2 X[6672], 7 X[16] + 5 X[40335], X[376] - 3 X[21159], X[619] + 2 X[42913], 2 X[622] - 5 X[624], X[622] - 10 X[6672], 7 X[622] - 25 X[40335], X[624] - 4 X[6672], 7 X[624] - 10 X[40335], X[671] - 3 X[22511], X[1992] - 3 X[36758], 3 X[3524] - X[14539], X[3543] - 3 X[41037], 3 X[3545] + X[36995], 3 X[3839] - X[36994], 3 X[5054] + X[5615], 3 X[5055] - X[20429], 3 X[5215] - 2 X[6671], X[5464] + 3 X[16963], 2 X[6670] + X[42943], 14 X[6672] - 5 X[40335], X[6779] + 3 X[22489], X[7685] + 2 X[13349], 3 X[9166] - X[23005], 5 X[16961] - X[36330], 3 X[16963] - X[37785], 3 X[22490] - 2 X[33561], 3 X[22490] + X[36968], 2 X[33561] + X[36968], X[34316] - 3 X[37907], 4 X[35020] - X[42940]

X(45880) lies on these lines: {2, 13}, {5, 21402}, {6, 13083}, {14, 35932}, {15, 26613}, {23, 34313}, {30, 5460}, {62, 37786}, {187, 395}, {303, 22495}, {316, 43484}, {351, 9195}, {376, 21159}, {381, 44667}, {511, 549}, {519, 11708}, {524, 619}, {532, 10614}, {533, 5464}, {542, 44223}, {551, 44660}, {599, 11302}, {617, 22496}, {623, 3849}, {630, 22238}, {635, 7810}, {671, 22511}, {691, 34315}, {1992, 36758}, {3105, 33274}, {3524, 14539}, {3543, 41037}, {3545, 36995}, {3642, 9761}, {3839, 36994}, {5054, 5615}, {5055, 20429}, {5215, 6671}, {5237, 11304}, {6295, 22579}, {6670, 31694}, {6674, 16965}, {6695, 37340}, {7619, 33475}, {8584, 42634}, {8587, 43539}, {8598, 40672}, {8836, 15743}, {9166, 23005}, {9763, 11486}, {9855, 23004}, {9886, 16508}, {10616, 43228}, {10646, 35931}, {10668, 13847}, {10672, 13846}, {11295, 11481}, {11296, 16645}, {11300, 39554}, {11303, 35230}, {11305, 43239}, {11306, 36843}, {11489, 22491}, {12100, 36756}, {14830, 33389}, {16268, 30559}, {16943, 42155}, {16961, 36330}, {18334, 40696}, {22490, 33561}, {22574, 35918}, {23283, 23873}, {23303, 31693}, {31173, 37352}, {33481, 44215}, {33609, 36346}, {34316, 37907}, {34508, 37173}, {35020, 42940}, {37641, 41407}, {41023, 44219}, {42912, 44497}, {43545, 45103}

X(45880) = midpoint of X(i) and X(j) for these {i,j}: {2, 16}, {13, 8595}, {14, 35932}, {23, 34313}, {395, 35303}, {617, 22496}, {691, 34315}, {5464, 37785}, {9855, 23004}, {31694, 42943}
X(45880) = reflection of X(i) in X(j) for these {i,j}: {2, 6672}, {624, 2}, {31694, 6670}, {36756, 12100}
X(45880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13084, 618}, {16, 6672, 624}, {549, 597, 45879}, {5464, 16963, 37785}, {22489, 33416, 2}, {37173, 42149, 34508}

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Intersections of remarkable lines: X(45881)-X(45955)

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This preamble and centers X(45881)-X(45955) were contributed by César Eliud Lozada, November 10, 2021.

This section deals with the intersections of these remarkable lines in a triangle:

Intersections of these lines are showed in the following list:

(antiorthic axis, Apollonius line, 45881), (antiorthic axis, Brocard axis, 2245), (antiorthic axis, Brocard line, 45882), (antiorthic axis, De Longchamps line, 1491), (antiorthic axis, Euler line, 851), (antiorthic axis, Fermat axis, 3013), (antiorthic axis, Gergonne line, 650), (antiorthic axis, G-K line, 2238), (antiorthic axis, Hatzipolakis axis, 18116), (antiorthic axis, I-H line, 2635), (antiorthic axis, I-K line, 44), (antiorthic axis, I-N line, 45885), (antiorthic axis, I-O line, 1155), (antiorthic axis, Lemoine axis, 649), (antiorthic axis, line at infinity, 513), (antiorthic axis, Nagel line, 899), (antiorthic axis, Napoleon axis, 45883), (antiorthic axis, N-K line, 45886), (antiorthic axis, orthic axis, 650), (antiorthic axis, Sherman line, 45884), (antiorthic axis, Soddy line, 3000), (antiorthic axis, van Aubel line, 3330), (Apollonius line, Brocard axis, 970), (Apollonius line, Brocard line, 45887), (Apollonius line, De Longchamps line, 45888), (Apollonius line, Euler line, 13731), (Apollonius line, Fermat axis, 45889), (Apollonius line, Gergonne line, 45890), (Apollonius line, G-K line, 45897), (Apollonius line, Hatzipolakis axis, 45896), (Apollonius line, I-H line, 1), (Apollonius line, I-K line, 1), (Apollonius line, I-N line, 1), (Apollonius line, I-O line, 1), (Apollonius line, Lemoine axis, 45891), (Apollonius line, line at infinity, 45955), (Apollonius line, Nagel line, 1), (Apollonius line, Napoleon axis, 45892), (Apollonius line, N-K line, 45898), (Apollonius line, orthic axis, 45893), (Apollonius line, Sherman line, 45894), (Apollonius line, Soddy line, 1), (Apollonius line, van Aubel line, 45895), (Brocard axis, Brocard line, 39), (Brocard axis, De Longchamps line, 3001), (Brocard axis, Euler line, 3), (Brocard axis, Fermat axis, 6), (Brocard axis, Gergonne line, 3002), (Brocard axis, G-K line, 6), (Brocard axis, Hatzipolakis axis, 18114), (Brocard axis, I-H line, 581), (Brocard axis, I-K line, 6), (Brocard axis, I-N line, 5396), (Brocard axis, I-O line, 3), (Brocard axis, Lemoine axis, 187), (Brocard axis, line at infinity, 511), (Brocard axis, Nagel line, 386), (Brocard axis, Napoleon axis, 6), (Brocard axis, N-K line, 6), (Brocard axis, orthic axis, 3003), (Brocard axis, Sherman line, 45899), (Brocard axis, Soddy line, 991), (Brocard axis, van Aubel line, 6), (Brocard line, De Longchamps line, 17415), (Brocard line, Euler line, 45900), (Brocard line, Fermat axis, 45901), (Brocard line, Gergonne line, 45902), (Brocard line, G-K line, 45914), (Brocard line, Hatzipolakis axis, 45911), (Brocard line, I-H line, 45904), (Brocard line, I-K line, 45913), (Brocard line, I-N line, 45912), (Brocard line, I-O line, 45903), (Brocard line, Lemoine axis, 512), (Brocard line, line at infinity, 512), (Brocard line, Nagel line, 45905), (Brocard line, Napoleon axis, 45906), (Brocard line, N-K line, 45915), (Brocard line, orthic axis, 45907), (Brocard line, Sherman line, 45908), (Brocard line, Soddy line, 45909), (Brocard line, van Aubel line, 45910), (De Longchamps line, Euler line, 858), (De Longchamps line, Fermat axis, 3014), (De Longchamps line, Gergonne line, 3004), (De Longchamps line, G-K line, 325), (De Longchamps line, Hatzipolakis axis, 523), (De Longchamps line, I-H line, 45917), (De Longchamps line, I-K line, 35552), (De Longchamps line, I-N line, 45920), (De Longchamps line, I-O line, 45916), (De Longchamps line, Lemoine axis, 3005), (De Longchamps line, line at infinity, 523), (De Longchamps line, Nagel line, 3006), (De Longchamps line, Napoleon axis, 45918), (De Longchamps line, N-K line, 45921), (De Longchamps line, orthic axis, 523), (De Longchamps line, Sherman line, 45919), (De Longchamps line, Soddy line, 3007), (De Longchamps line, van Aubel line, 45279), (Euler line, Fermat axis, 381), (Euler line, Gergonne line, 1375), (Euler line, G-K line, 2), (Euler line, Hatzipolakis axis, 5), (Euler line, I-H line, 4), (Euler line, I-K line, 405), (Euler line, I-N line, 5), (Euler line, I-O line, 3), (Euler line, Lemoine axis, 237), (Euler line, line at infinity, 30), (Euler line, Nagel line, 2), (Euler line, Napoleon axis, 1656), (Euler line, N-K line, 5), (Euler line, orthic axis, 468), (Euler line, Sherman line, 45922), (Euler line, Soddy line, 20), (Euler line, van Aubel line, 4), (Fermat axis, Gergonne line, 3015), (Fermat axis, G-K line, 6), (Fermat axis, Hatzipolakis axis, 14356), (Fermat axis, I-H line, 45924), (Fermat axis, I-K line, 6), (Fermat axis, I-N line, 45926), (Fermat axis, I-O line, 45923), (Fermat axis, Lemoine axis, 3016), (Fermat axis, line at infinity, 542), (Fermat axis, Nagel line, 3017), (Fermat axis, Napoleon axis, 6), (Fermat axis, N-K line, 6), (Fermat axis, orthic axis, 3018), (Fermat axis, Sherman line, 45925), (Fermat axis, Soddy line, 3019), (Fermat axis, van Aubel line, 6), (Gergonne line, G-K line, 35466), (Gergonne line, Hatzipolakis axis, 18118), (Gergonne line, I-H line, 43035), (Gergonne line, I-K line, 43065), (Gergonne line, I-N line, 5723), (Gergonne line, I-O line, 241), (Gergonne line, Lemoine axis, 665), (Gergonne line, line at infinity, 514), (Gergonne line, Nagel line, 3008), (Gergonne line, Napoleon axis, 45927), (Gergonne line, N-K line, 45930), (Gergonne line, orthic axis, 650), (Gergonne line, Sherman line, 45928), (Gergonne line, Soddy line, 1323), (Gergonne line, van Aubel line, 45929), (G-K line, Hatzipolakis axis, 18122), (G-K line, I-H line, 5712), (G-K line, I-K line, 6), (G-K line, I-N line, 5718), (G-K line, I-O line, 940), (G-K line, Lemoine axis, 3231), (G-K line, line at infinity, 524), (G-K line, Nagel line, 2), (G-K line, Napoleon axis, 6), (G-K line, N-K line, 6), (G-K line, orthic axis, 230), (G-K line, Sherman line, 45952), (G-K line, Soddy line, 3945), (G-K line, van Aubel line, 6), (Hatzipolakis axis, I-H line, 45934), (Hatzipolakis axis, I-K line, 45954), (Hatzipolakis axis, I-N line, 5), (Hatzipolakis axis, I-O line, 18115), (Hatzipolakis axis, Lemoine axis, 18117), (Hatzipolakis axis, line at infinity, 523), (Hatzipolakis axis, Nagel line, 18120), (Hatzipolakis axis, Napoleon axis, 45943), (Hatzipolakis axis, N-K line, 5), (Hatzipolakis axis, orthic axis, 523), (Hatzipolakis axis, Sherman line, 45949), (Hatzipolakis axis, Soddy line, 18119), (Hatzipolakis axis, van Aubel line, 18121), (I-H line, I-K line, 1), (I-H line, I-N line, 1), (I-H line, I-O line, 1), (I-H line, Lemoine axis, 45932), (I-H line, line at infinity, 515), (I-H line, Nagel line, 1), (I-H line, Napoleon axis, 45933), (I-H line, N-K line, 5713), (I-H line, orthic axis, 23710), (I-H line, Sherman line, 35015), (I-H line, Soddy line, 1), (I-H line, van Aubel line, 4), (I-K line, I-N line, 1), (I-K line, I-O line, 1), (I-K line, Lemoine axis, 3230), (I-K line, line at infinity, 518), (I-K line, Nagel line, 1), (I-K line, Napoleon axis, 6), (I-K line, N-K line, 6), (I-K line, orthic axis, 8609), (I-K line, Sherman line, 45951), (I-K line, Soddy line, 1), (I-K line, van Aubel line, 6), (I-N line, I-O line, 1), (I-N line, Lemoine axis, 45937), (I-N line, line at infinity, 952), (I-N line, Nagel line, 1), (I-N line, Napoleon axis, 45944), (I-N line, N-K line, 5), (I-N line, orthic axis, 45946), (I-N line, Sherman line, 45950), (I-N line, Soddy line, 1), (I-N line, van Aubel line, 5721), (I-O line, Lemoine axis, 2223), (I-O line, line at infinity, 517), (I-O line, Nagel line, 1), (I-O line, Napoleon axis, 45931), (I-O line, N-K line, 5707), (I-O line, orthic axis, 8758), (I-O line, Sherman line, 35014), (I-O line, Soddy line, 1), (I-O line, van Aubel line, 5706), (Lemoine axis, line at infinity, 512), (Lemoine axis, Nagel line, 3009), (Lemoine axis, Napoleon axis, 45935), (Lemoine axis, N-K line, 45938), (Lemoine axis, orthic axis, 647), (Lemoine axis, Sherman line, 45936), (Lemoine axis, Soddy line, 3010), (Lemoine axis, van Aubel line, 3331), (line at infinity, Nagel line, 519), (line at infinity, Napoleon axis, 5965), (line at infinity, N-K line, 3564), (line at infinity, orthic axis, 523), (line at infinity, Sherman line, 35013), (line at infinity, Soddy line, 516), (line at infinity, van Aubel line, 1503), (Nagel line, Napoleon axis, 45939), (Nagel line, N-K line, 5292), (Nagel line, orthic axis, 3011), (Nagel line, Sherman line, 45940), (Nagel line, Soddy line, 1), (Nagel line, van Aubel line, 387), (Napoleon axis, N-K line, 6), (Napoleon axis, orthic axis, 231), (Napoleon axis, Sherman line, 45941), (Napoleon axis, Soddy line, 45942), (Napoleon axis, van Aubel line, 6), (N-K line, orthic axis, 16310), (N-K line, Sherman line, 45953), (N-K line, Soddy line, 5733), (N-K line, van Aubel line, 6), (orthic axis, Sherman line, 45945), (orthic axis, Soddy line, 3012), (orthic axis, van Aubel line, 1990), (Sherman line, Soddy line, 45947), (Sherman line, van Aubel line, 45948), (Soddy line, van Aubel line, 3332)

X(45881) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND APOLLONIUS LINE

Barycentrics    a*((b+c)*a^4+2*(b+c)^2*a^3-(b+c)*(b^2+b*c+c^2)*a^2-2*(b^4+c^4+b*c*(b-c)^2)*a+(b^2-c^2)*(b-c)*b*c) : :

X(45881) lies on these lines: {1, 181}, {2, 10406}, {44, 513}, {1329, 2476}, {1402, 10440}, {1698, 10407}, {1836, 26037}, {3474, 26038}, {5204, 16451}

X(45881) = perspector of the circumconic {{A, B, C, X(1), X(43069)}}
X(45881) = X(i)-line conjugate of-X(j) for these (i, j): (181, 1), (970, 1)
X(45881) = {X(181), X(21363)}-harmonic conjugate of X(21321)

X(45882) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND BROCARD LINE

Barycentrics    a^2*(b^3-c^3)*(a^2+b*c) : :

X(45882) lies on these lines: {39, 512}, {44, 513}, {669, 25862}, {786, 16892}, {838, 23657}, {1019, 7255}, {2276, 30671}, {2295, 4367}, {4369, 4374}, {4705, 21763}, {4874, 27242}, {4913, 21832}, {7234, 20981}, {8678, 23572}, {9286, 24534}, {21053, 24718}, {21056, 21259}

X(45882) = reflection of X(14296) in X(4369)
X(45882) = barycentric product X(i)*X(j) for these {i, j}: {1, 3805}, {171, 1491}, {172, 824}, {335, 30654}, {385, 30671}, {513, 40790}
X(45882) = barycentric quotient X(i)/X(j) for these (i, j): (31, 30670), (171, 789), (172, 4586), (649, 40738), (667, 40763), (788, 256)
X(45882) = trilinear product X(i)*X(j) for these {i, j}: {6, 3805}, {171, 3250}, {172, 1491}, {291, 30654}, {649, 40790}, {661, 40731}
X(45882) = trilinear quotient X(i)/X(j) for these (i, j): (6, 30670), (171, 4586), (172, 1492), (513, 40738), (649, 40763), (788, 893)
X(45882) = perspector of the circumconic {{A, B, C, X(1), X(172)}}
X(45882) = crossdifference of every pair of points on line {X(1), X(257)}
X(45882) = crosssum of X(1491) and X(3727)
X(45882) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2, 30670}, {100, 40738}, {190, 40763}, {256, 4586}
X(45882) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (31, 30670), (171, 789), (172, 4586), (649, 40738)
X(45882) = (1st circumperp)-isotomic conjugate-of-X(30670)
X(45882) = X(i)-Zayin conjugate of-X(j) for these (i, j): (9, 30670), (39, 4613), (512, 14621), (513, 40738)

X(45883) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND NAPOLEON AXIS

Barycentrics    a*(a^6-2*(b^2+c^2)*a^4+(b^4+c^4+b*c*(b^2-b*c+c^2))*a^2-(b^2-c^2)^2*b*c)*(b+c) : :

X(45883) lies on these lines: {6, 17}, {44, 513}

X(45883) = barycentric product X(37)*X(24148)
X(45883) = trilinear product X(42)*X(24148)
X(45883) = perspector of the circumconic {{A, B, C, X(1), X(930)}}
X(45883) = crossdifference of every pair of points on line {X(1), X(1510)}
X(45883) = X(1141)-Ceva conjugate of-X(55)
X(45883) = X(1154)-Zayin conjugate of-X(57)
X(45883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2238, 45886, 2245), (2245, 45886, 3013)

X(45884) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND SHERMAN LINE

Barycentrics    a*((b+c)*a^4-2*(b^2+c^2)*a^3+(b+c)*b*c*a^2+2*(b^3-c^3)*(b-c)*a-(b^2-c^2)*(b^3-c^3))*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :
X(45884) = 3*X(42756)-2*X(42763)

X(45884) lies on these lines: {44, 513}, {514, 1512}, {1308, 40116}, {1465, 1769}, {1566, 3126}, {2222, 2720}, {3259, 3326}, {3738, 44425}

X(45884) = trilinear product X(1769)*X(2801)
X(45884) = trilinear quotient X(1769)/X(2717)
X(45884) = perspector of the circumconic {{A, B, C, X(1), X(10015)}}
X(45884) = crossdifference of every pair of points on line {X(1), X(32641)}
X(45884) = Stevanovic-circle-inverse of X(46393)
X(45884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (42755, 42762, 42756), (42755, 42769, 42768), (42758, 42762, 42768), (42758, 42772, 42756), (42769, 42772, 42762)

X(45885) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND I-N LINE

Barycentrics    a*((b+c)*a^4-(b+c)^2*a^3-(b+c)*(b^2-3*b*c+c^2)*a^2+(b^2+3*b*c+c^2)*(b-c)^2*a-2*(b^2-c^2)*(b-c)*b*c) : :

X(45885) lies on these lines: {1, 5}, {42, 17605}, {43, 1836}, {44, 513}, {73, 17606}, {118, 38964}, {244, 17660}, {386, 10407}, {1086, 12831}, {1212, 3119}, {1362, 4860}, {1376, 35281}, {1443, 37757}, {1464, 1737}, {1745, 24914}, {1772, 2771}, {2361, 44425}, {2801, 43048}, {2841, 3030}, {3214, 22300}, {3216, 7354}, {3240, 13576}, {4009, 23691}, {4337, 11231}, {4792, 14260}, {5204, 15654}, {5228, 34931}, {6174, 35338}, {6796, 7299}, {9780, 26028}, {11502, 34048}, {15064, 24431}, {16586, 24433}, {17638, 24028}, {23832, 29349}, {23845, 38389}, {27627, 28238}, {30950, 39046}, {37578, 37679}

X(45885) = perspector of the circumconic {{A, B, C, X(1), X(655)}}
X(45885) = X(i)-line conjugate of-X(j) for these (i, j): (5, 1), (11, 1), (12, 1), (44, 654)
X(45885) = reflection of X(11) in the line X(900)X(43048)
X(45885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (80, 6127, 34586), (899, 2635, 1155), (4551, 5400, 11)

X(45886) = INTERSECTION OF THESE LINES: ANTIORTHIC AXIS AND N-K LINE

Barycentrics    a*(b+c)*(a^6-(2*b^2-b*c+2*c^2)*a^4+(b^4+c^4)*a^2-(b^2-c^2)^2*b*c) : :

X(45886) lies on these lines: {5, 6}, {37, 2594}, {44, 513}, {218, 1901}, {692, 862}, {966, 26028}, {992, 4268}, {1030, 6097}, {1865, 2911}, {16453, 36743}, {21011, 21741}, {21853, 21874}

X(45886) = perspector of the circumconic {{A, B, C, X(1), X(925)}}
X(45886) = crossdifference of every pair of points on line {X(1), X(924)}
X(45886) = X(1300)-Ceva conjugate of-X(55)
X(45886) = X(i)-complementary conjugate of-X(j) for these (i, j): (42, 42423), (913, 3739), (915, 3741)
X(45886) = center of circle {{X(115), X(5164), X(36472)}}
X(45886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2238, 3330, 2245), (2245, 3013, 3330), (2245, 45883, 2238), (3013, 45883, 2245)

X(45887) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND BROCARD LINE

Barycentrics    a^2*((b^2+c^2)*b*c*a^6+(b^3+c^3)*(b+c)^2*a^5-(3*b^4+2*b^2*c^2+3*c^4)*b*c*a^4-(b+c)*(b^6+4*b^3*c^3+c^6)*a^3+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*b*c*a^2+(b+c)*(b^6+c^6+b*c*(b^2-b*c+c^2)*(b-c)^2)*b*c*a+(b^2-c^2)^2*b^3*c^3) : :

X(45887) lies on these lines: {1, 181}, {39, 512}

X(45887) = perspector of the circumconic {{A, B, C, X(694), X(43069)}}

X(45888) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND DE LONGCHAMPS LINE

Barycentrics    a*((b^4+c^4)*a^4+(b^3+c^3)*(b+c)^2*a^3-(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-(b+c)*(b^6+c^6+b*c*(b^2-b*c+c^2)*(b-c)^2)*a-(b^2-c^2)^2*b^2*c^2) : :

X(45888) lies on these lines: {1, 181}, {325, 523}

X(45888) = perspector of the circumconic {{A, B, C, X(76), X(43069)}}

X(45889) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND FERMAT AXIS

Barycentrics    a*((b+c)*a^8+2*(b+c)^2*a^7-(b+c)*(b^2-3*b*c+c^2)*a^6-(b^2+c^2)*(4*b^2+9*b*c+4*c^2)*a^5-(b+c)*(b^4+c^4+2*b*c*(4*b^2+3*b*c+4*c^2))*a^4+(2*b^6+2*c^6+(3*b^4+3*c^4-8*b*c*(b+c)^2)*b*c)*a^3+(b+c)*(b^6+c^6+b*c*(4*b^2-9*b*c+4*c^2)*(b+c)^2)*a^2+2*(b^2-c^2)^2*(b^2+3*b*c+c^2)*b*c*a+(b^4-c^4)*(b^2-c^2)*b*c*(b+c)) : :

X(45889) lies on these lines: {1, 181}, {5, 39523}, {6, 13}

X(45889) = perspector of the circumconic {{A, B, C, X(476), X(43069)}}
X(45889) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (970, 45898, 45892), (45895, 45898, 970)

X(45890) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND GERGONNE LINE

Barycentrics    a*(a+b-c)*(a-b+c)*((b^2+c^2)*a^3+4*(b+c)*b*c*a^2-(b^4+c^4+2*(b^2+c^2)*b*c)*a-2*(b+c)*b^2*c^2) : :

X(45890) lies on these lines: {1, 181}, {241, 514}, {948, 27339}, {4383, 10571}, {5228, 19731}

X(45890) = perspector of the circumconic {{A, B, C, X(7), X(43069)}}

X(45891) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND LEMOINE AXIS

Barycentrics    a^2*(2*b*c*a^4+(b+c)^3*a^3-3*(b^2+c^2)*b*c*a^2-(b+c)*(b^4+c^4)*a+(b^2-c^2)^2*b*c) : :

X(45891) lies on these lines: {1, 181}, {187, 237}, {213, 20959}, {2176, 20986}, {4511, 20683}

X(45891) = perspector of the circumconic {{A, B, C, X(6), X(43069)}}
X(45891) = crosssum of X(1) and X(45752)

X(45892) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND NAPOLEON AXIS

Barycentrics    a*((b+c)*a^8+2*(b+c)^2*a^7-(b+c)*(b^2-3*b*c+c^2)*a^6-(4*b^4+4*c^4+b*c*(7*b^2+4*b*c+7*c^2))*a^5-(b+c)*(b^4+c^4+6*b*c*(b^2+b*c+c^2))*a^4+(2*b^6+2*c^6+(b^4+c^4-4*b*c*(3*b^2+4*b*c+3*c^2))*b*c)*a^3+(b+c)*(b^6+c^6+(2*b^4+2*c^4-b*c*(b^2+10*b*c+c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^2+3*b*c+c^2)*b*c*a+(b^4-c^4)*(b^2-c^2)*b*c*(b+c)) : :

X(45892) lies on these lines: {1, 181}, {6, 17}, {13323, 19543}

X(45892) = perspector of the circumconic {{A, B, C, X(930), X(43069)}}
X(45892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (970, 45898, 45889), (45897, 45898, 970)

X(45893) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND ORTHIC AXIS

Barycentrics    a*((b^2+c^2)*a^6+(b+c)^3*a^5-2*(b^4+b^2*c^2+c^4)*a^4-2*(b+c)*(b^4+c^4+b*c*(b^2-3*b*c+c^2))*a^3+(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^4+c^4+2*(b^2+c^2)*b*c)*a+2*(b^2-c^2)^2*b^2*c^2) : :

X(45893) lies on these lines: {1, 181}, {230, 231}

X(45893) = perspector of the circumconic {{A, B, C, X(4), X(43069)}}

X(45894) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND SHERMAN LINE

Barycentrics    a*(b-c)^2*(-a+b+c)*((b+c)*a^7+2*b*c*a^6-3*(b+c)*(b^2+c^2)*a^5-3*(b-c)^2*b*c*a^4+(b+c)*(3*b^4+3*c^4+b*c*(b^2-b*c+c^2))*a^3+(b^2-4*b*c+c^2)*(b^2-b*c+c^2)*b*c*a^2-(b+c)*(b^6+c^6+b*c*(b-c)^4)*a-(b^2-c^2)^2*b^2*c^2)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45894) lies on these lines: {1, 181}, {3259, 3326}

X(45894) = perspector of the circumconic {{A, B, C, X(10015), X(43069)}}

X(45895) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND VAN AUBEL LINE

Barycentrics    a*((b+c)*a^8+2*(b+c)^2*a^7-(b+c)*(b^2-3*b*c+c^2)*a^6-2*(2*b^4+2*c^4+5*b*c*(b^2+b*c+c^2))*a^5-(b+c)*(b^4+c^4+3*b*c*(3*b^2+2*b*c+3*c^2))*a^4+2*(b^4-4*b^2*c^2+c^4)*(b+c)^2*a^3+(b^2-c^2)^2*(b+c)*(b^2+5*b*c+c^2)*a^2+2*(b^2-c^2)^2*(b^2+3*b*c+c^2)*b*c*a+(b^4-c^4)*(b^2-c^2)*b*c*(b+c)) : :

X(45895) lies on these lines: {1, 181}, {4, 6}, {5, 33137}, {1754, 13323}, {3931, 5755}, {5743, 19843}, {19766, 36745}

X(45895) = perspector of the circumconic {{A, B, C, X(107), X(43069)}}
X(45895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (970, 45889, 45898), (970, 45898, 45897)

X(45896) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND HATZIPOLAKIS AXIS

Barycentrics    a*((b^4+c^4)*a^8+(b^3+c^3)*(b+c)^2*a^7-(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^6-(b+c)*(3*b^6+3*c^6+b*c*(3*b^2+2*b*c+3*c^2)*(b-c)^2)*a^5+3*(b^8+c^8)*a^4+(b+c)*(3*b^8+3*c^8+(3*b^6+3*c^6-b*c*(3*b^2-5*b*c+3*c^2)*(3*b^2+4*b*c+3*c^2))*b*c)*a^3-(b^6+c^6)*(b^2-c^2)^2*a^2-(b^2-c^2)^2*(b+c)*(b^6+c^6+b*c*(b^2-b*c+c^2)*(b-c)^2)*a-(b^2-c^2)^4*b^2*c^2) : :

X(45896) lies on these lines: {1, 181}, {5, 523}

X(45896) = perspector of the circumconic {{A, B, C, X(94), X(43069)}}

X(45897) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND G-K LINE

Barycentrics    a*((b+c)*a^4+2*(b+c)^2*a^3+(b+c)*(b^2+3*b*c+c^2)*a^2+2*(b^2+3*b*c+c^2)*b*c*a+b*c*(b+c)*(b^2+c^2)) : :

X(45897) lies on these lines: {1, 181}, {2, 6}, {5, 5711}, {171, 4192}, {172, 41243}, {226, 24211}, {312, 2295}, {497, 5710}, {959, 37614}, {960, 22275}, {1400, 3666}, {2051, 39595}, {2277, 5256}, {2339, 21371}, {3664, 40687}, {3780, 4886}, {4199, 5156}, {4274, 17185}, {5835, 17751}, {10458, 30944}, {13323, 19513}, {14547, 37539}, {17012, 28249}, {30710, 41232}, {34466, 37594}, {37554, 37732}, {37559, 37693}

X(45897) = perspector of the circumconic {{A, B, C, X(99), X(43069)}}
X(45897) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(43074)}} and {{A, B, C, X(81), X(43070)}}
X(45897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 32911, 992), (970, 45892, 45898), (970, 45898, 45895)

X(45898) = INTERSECTION OF THESE LINES: APOLLONIUS LINE AND N-K LINE

Barycentrics    a*((b+c)*a^8+2*(b+c)^2*a^7-(b+c)*(b^2-3*b*c+c^2)*a^6-2*(2*b^4+2*c^4+b*c*(4*b^2+3*b*c+4*c^2))*a^5-(b+c)*(b^4+c^4+b*c*(7*b^2+6*b*c+7*c^2))*a^4+2*(b^6+c^6+(b^4+c^4-b*c*(5*b^2+8*b*c+5*c^2))*b*c)*a^3+(b+c)*(b^6+c^6+(3*b^4+3*c^4-b*c*(b^2+10*b*c+c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^2+3*b*c+c^2)*b*c*a+(b^4-c^4)*(b^2-c^2)*b*c*(b+c)) : :

X(45898) lies on these lines: {1, 181}, {5, 6}, {4192, 13323}, {5396, 37837}, {5743, 26363}

X(45898) = perspector of the circumconic {{A, B, C, X(925), X(43069)}}
X(45898) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (970, 45889, 45895), (970, 45892, 45897), (45889, 45892, 970), (45895, 45897, 970)

X(45899) = INTERSECTION OF THESE LINES: BROCARD AXIS AND SHERMAN LINE

Barycentrics    a^2*(b-c)^2*(-a^2+b^2+c^2)*((b+c)*a^4-2*(b^2+b*c+c^2)*a^3+2*(b+c)*b*c*a^2+2*(b^4-b^2*c^2+c^4)*a-(b+c)*(b^4-b^2*c^2+c^4))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45899) lies on these lines: {3, 6}, {3259, 3326}

X(45899) = perspector of the circumconic {{A, B, C, X(110), X(10015)}}
X(45899) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(42761)}} and {{A, B, C, X(6), X(42759)}}
X(45899) = crossdifference of every pair of points on line {X(523), X(32641)}
X(45899) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (35014, 42769, 42761), (45925, 45941, 45953), (45948, 45952, 45953), (45948, 45953, 45925), (45952, 45953, 45941)

X(45900) = INTERSECTION OF THESE LINES: BROCARD LINE AND EULER LINE

Barycentrics    a^2*((b^4+c^4)*a^6-(b^2+c^2)*(b^4+c^4)*a^4+(b^4+c^4)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

X(45900) lies on these lines: {2, 3}, {32, 2086}, {39, 512}, {99, 18896}, {115, 21444}, {141, 41337}, {148, 3511}, {230, 32518}, {669, 41939}, {2387, 32452}, {2882, 20975}, {3095, 38523}, {5254, 32540}, {7875, 32531}, {23098, 31848}, {23635, 41172}

X(45900) = perspector of the circumconic {{A, B, C, X(648), X(694)}}
X(45900) = inverse of X(1316) in 2nd Brocard circle
X(45900) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(882)}} and {{A, B, C, X(25), X(881)}}
X(45900) = crossdifference of every pair of points on line {X(385), X(647)}
X(45900) = center of circle {{X(6), X(2453), X(3094)}}
X(45900) = X(36213)-of-6th Brocard triangle
X(45900) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 11332, 865), (3, 4, 446), (2554, 2555, 1316), (10684, 10694, 1316)

X(45901) = INTERSECTION OF THESE LINES: BROCARD LINE AND FERMAT AXIS

Barycentrics    a^2*((b^4+c^4)*a^8-(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^6+(b^8+4*b^4*c^4+c^8)*a^4-(b^6+c^6)*b^2*c^2*a^2+(b^2-c^2)^2*b^2*c^2*(b^4+c^4)) : :

X(45901) lies on these lines: {6, 13}, {39, 512}, {182, 33330}, {323, 22254}, {754, 2421}, {2387, 44114}, {5028, 37841}, {5116, 14811}, {15920, 32761}

X(45901) = perspector of the circumconic {{A, B, C, X(476), X(694)}}
X(45901) = crossdifference of every pair of points on line {X(385), X(526)}
X(45901) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 45915, 45906), (45910, 45915, 39)

X(45902) = INTERSECTION OF THESE LINES: BROCARD LINE AND GERGONNE LINE

Barycentrics    a^2*(b-c)*((b^2+c^2)*a^2-(b^3+c^3)*a+(b^2+c^2)*b*c) : :

X(45902) lies on these lines: {1, 22229}, {2, 22222}, {39, 512}, {241, 514}, {647, 25861}, {649, 22092}, {667, 6373}, {1107, 21051}, {2275, 4367}, {2276, 4879}, {3063, 22090}, {3709, 3805}, {5299, 39577}, {14758, 44410}

X(45902) = barycentric product X(i)*X(j) for these {i, j}: {513, 17792}, {522, 41350}, {649, 17760}, {650, 28391}, {693, 18758}, {876, 8844}
X(45902) = barycentric quotient X(i)/X(j) for these (i, j): (513, 18299), (663, 39924)
X(45902) = trilinear product X(i)*X(j) for these {i, j}: {514, 18758}, {649, 17792}, {650, 41350}, {663, 28391}, {667, 17760}
X(45902) = trilinear quotient X(i)/X(j) for these (i, j): (514, 18299), (650, 39924)
X(45902) = perspector of the circumconic {{A, B, C, X(7), X(694)}}
X(45902) = crossdifference of every pair of points on line {X(55), X(192)}
X(45902) = crosspoint of X(6) and X(37137)
X(45902) = crosssum of X(i) and X(j) for these (i, j): {2, 3287}, {6, 24533}
X(45902) = X(904)-complementary conjugate of-X(5518)
X(45902) = X(1015)-Dao conjugate of-X(18299)
X(45902) = X(i)-isoconjugate-of-X(j) for these {i, j}: {101, 18299}, {651, 39924}
X(45902) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (513, 18299), (663, 39924)

X(45903) = INTERSECTION OF THESE LINES: BROCARD LINE AND I-O LINE

Barycentrics    a^2*((b^3+c^3)*a^4-(b^4+c^4)*a^3-(b+c)*b^2*c^2*a^2+(b^2+c^2)*b^2*c^2*a+(b^4-c^4)*b*c*(b-c)) : :

X(45903) lies on these lines: {1, 3}, {39, 512}, {766, 32452}, {5025, 20544}, {6655, 20556}, {7976, 38521}, {16696, 41337}, {19974, 32773}

X(45903) = perspector of the circumconic {{A, B, C, X(651), X(694)}}
X(45903) = inverse of X(5091) in 2nd Brocard circle
X(45903) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {6, 3094, 38530}, {99, 2703, 32454}
X(45903) = {X(2556), X(2557)}-harmonic conjugate of X(5091)

X(45904) = INTERSECTION OF THESE LINES: BROCARD LINE AND I-H LINE

Barycentrics    a^2*((b^4+c^4)*a^6-(b+c)*(b^4+c^4-b*c*(b^2-b*c+c^2))*a^5-(b^2-b*c+c^2)*(b^4+c^4)*a^4+(b^3+c^3)*(b^4-b^2*c^2+c^4)*a^3-(b^4+c^4+b*c*(b^2+b*c+c^2))*(b-c)^2*b*c*a^2+(b^2-c^2)*(b-c)*b^3*c^3*a+(b^2-c^2)*(b-c)*b^2*c^2*(b^3+c^3)) : :

X(45904) lies on these lines: {1, 4}, {39, 512}

X(45904) = perspector of the circumconic {{A, B, C, X(653), X(694)}}
X(45904) = {X(45903), X(45912)}-harmonic conjugate of X(45905)

X(45905) = INTERSECTION OF THESE LINES: BROCARD LINE AND NAGEL LINE

Barycentrics    a^2*((b^4+c^4)*a^3-(b^3+c^3)*b*c*a^2-(b^2+c^2)*b^2*c^2*a+b^2*c^2*(b^3+c^3)) : :

X(45905) lies on these lines: {1, 2}, {39, 512}, {2086, 20970}, {2170, 20862}, {5360, 20456}

X(45905) = perspector of the circumconic {{A, B, C, X(190), X(694)}}
X(45905) = intersection, other than A, B, C, of circumconics {{A, B, C, X(10), X(882)}} and {{A, B, C, X(42), X(881)}}
X(45905) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (45903, 45904, 45909), (45903, 45912, 45904)

X(45906) = INTERSECTION OF THESE LINES: BROCARD LINE AND NAPOLEON AXIS

Barycentrics    a^2*((b^4+c^4)*a^8-(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^6+(b^8+c^8+2*b^2*c^2*(b^4+4*b^2*c^2+c^4))*a^4-3*(b^6+c^6)*b^2*c^2*a^2+(b^2-c^2)^2*b^2*c^2*(b^4+c^4)) : :

X(45906) lies on these lines: {6, 17}, {39, 512}, {2421, 7838}

X(45906) = perspector of the circumconic {{A, B, C, X(694), X(930)}}
X(45906) = crossdifference of every pair of points on line {X(385), X(1510)}
X(45906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 45915, 45901), (45914, 45915, 39)

X(45907) = INTERSECTION OF THESE LINES: BROCARD LINE AND ORTHIC AXIS

Barycentrics    a^2*((b^2+c^2)*a^4-(b^4-b^2*c^2+c^4)*a^2+b^2*c^2*(b^2+c^2))*(b^2-c^2) : :
X(45907) = 4*X(39)-3*X(2524)

X(45907) lies on these lines: {39, 512}, {230, 231}, {669, 2451}, {688, 3804}, {888, 8644}, {4108, 9465}, {5996, 15302}, {14295, 30476}, {18573, 34291}, {22240, 33569}

X(45907) = reflection of X(i) in X(j) for these (i, j): (647, 2491), (14295, 30476)
X(45907) = complement of the isotomic conjugate of X(25424)
X(45907) = barycentric product X(i)*X(j) for these {i, j}: {512, 18906}, {523, 11328}, {661, 19591}, {804, 6234}
X(45907) = barycentric quotient X(512)/X(19222)
X(45907) = trilinear product X(i)*X(j) for these {i, j}: {512, 19591}, {661, 11328}, {798, 18906}
X(45907) = trilinear quotient X(661)/X(19222)
X(45907) = perspector of the circumconic {{A, B, C, X(4), X(694)}}
X(45907) = crossdifference of every pair of points on line {X(3), X(194)}
X(45907) = crosspoint of X(i) and X(j) for these (i, j): {2, 25424}, {112, 18898}
X(45907) = crosssum of X(i) and X(j) for these (i, j): {2, 3288}, {6, 25423}, {525, 3314}
X(45907) = X(1084)-Dao conjugate of-X(19222)
X(45907) = X(662)-isoconjugate-of-X(19222)
X(45907) = center of circle {{X(3), X(3095), X(11799)}}
X(45907) = X(512)-reciprocal conjugate of-X(19222)

X(45908) = INTERSECTION OF THESE LINES: BROCARD LINE AND SHERMAN LINE

Barycentrics    a^4*((b^4+c^4)*a^4-2*(b+c)*(b^4+c^4-b*c*(b^2-b*c+c^2))*a^3+2*(b^3-c^3)*(b-c)*b*c*a^2+2*(b+c)*(b^6+c^6-(2*b^4+2*c^4-b*c*(2*b^2-b*c+2*c^2))*b*c)*a-b^8-c^8+2*(b^3-c^3)*(b-c)*b^2*c^2)*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45908) lies on these lines: {39, 512}, {3259, 3326}

X(45908) = perspector of the circumconic {{A, B, C, X(694), X(10015)}}
X(45908) = crossdifference of every pair of points on line {X(385), X(32641)}

X(45909) = INTERSECTION OF THESE LINES: BROCARD LINE AND SODDY LINE

Barycentrics    a^2*((b^4+c^4)*a^5-(b+c)*(2*b^4+2*c^4-3*b*c*(b^2-b*c+c^2))*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^3-(b+c)*(b^4+c^4-b*c*(2*b^2-b*c+2*c^2))*b*c*a^2+(b^2-c^2)^2*b^2*c^2*a+(b^3+c^3)*(b-c)^2*b^2*c^2) : :

X(45909) lies on these lines: {1, 7}, {39, 512}

X(45909) = perspector of the circumconic {{A, B, C, X(658), X(694)}}

X(45910) = INTERSECTION OF THESE LINES: BROCARD LINE AND VAN AUBEL LINE

Barycentrics    a^2*((b^4+c^4)*a^8-2*(b^6+c^6)*a^6+(b^4+c^4)*(b^4-b^2*c^2+c^4)*a^4+(b^2-c^2)^2*b^2*c^2*(b^4+c^4)) : :

X(45910) lies on these lines: {4, 6}, {32, 446}, {39, 512}, {2086, 3767}, {2421, 7750}, {2715, 33695}, {2794, 9419}, {2967, 3269}, {3094, 38525}, {3288, 15000}, {3491, 40810}, {5661, 31848}, {8789, 38947}, {13330, 38520}, {32542, 37334}, {40951, 44114}, {44127, 44162}

X(45910) = perspector of the circumconic {{A, B, C, X(107), X(694)}}
X(45910) = crossdifference of every pair of points on line {X(385), X(520)}
X(45910) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 45901, 45915), (39, 45915, 45914)

X(45911) = INTERSECTION OF THESE LINES: BROCARD LINE AND HATZIPOLAKIS AXIS

Barycentrics    a^4*((b^4+b^2*c^2+c^4)*a^4-(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2+b^8+c^8-(b^4-3*b^2*c^2+c^4)*b^2*c^2)*(b^2-c^2) : :
X(45911) = X(23099)-3*X(34347)

X(45911) lies on these lines: {5, 523}, {32, 2491}, {39, 512}, {688, 42444}, {690, 3095}, {804, 14881}, {2799, 8149}, {7638, 9737}, {7752, 14295}, {18333, 23108}

X(45911) = reflection of X(i) in X(j) for these (i, j): (14270, 2491), (44826, 7638)
X(45911) = perspector of the circumconic {{A, B, C, X(94), X(694)}}
X(45911) = intersection, other than A, B, C, of circumconics {{A, B, C, X(32), X(18333)}} and {{A, B, C, X(83), X(45901)}}
X(45911) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 4, 3095}, {194, 38527, 38528}, {691, 2698, 6321}, {805, 842, 1916}
X(45911) = crossdifference of every pair of points on line {X(50), X(385)}

X(45912) = INTERSECTION OF THESE LINES: BROCARD LINE AND I-N LINE

Barycentrics    a^2*((b^4+c^4)*a^6-(b+c)*(b^4+c^4)*a^5-(b^4+c^4)*(b-c)^2*a^4+(b+c)*(b^6+c^6-(b^2-c^2)^2*b*c)*a^3-(b^2-b*c+c^2)*(b^4+c^4)*b*c*a^2-(b^3+c^3)*(b-c)^2*b^2*c^2*a+(b^2-c^2)*(b-c)*b^2*c^2*(b^3+c^3)) : :

X(45912) lies on these lines: {1, 5}, {39, 512}

X(45912) = perspector of the circumconic {{A, B, C, X(655), X(694)}}
X(45912) = center of circle {{X(3), X(3095), X(18342)}}
X(45912) = {X(45904), X(45905)}-harmonic conjugate of X(45903)

X(45913) = INTERSECTION OF THESE LINES: BROCARD LINE AND I-K LINE

Barycentrics    a^2*((b^3+c^3)*a^3-(b^2+c^2)*b*c*a^2-(b^3+c^3)*b*c*a+b*c*(b^4+c^4)) : :

X(45913) lies on these lines: {1, 6}, {39, 512}, {2086, 16589}, {2170, 20457}, {23639, 44114}, {24524, 35274}

X(45913) = perspector of the circumconic {{A, B, C, X(100), X(694)}}
X(45913) = intersection, other than A, B, C, of circumconics {{A, B, C, X(37), X(882)}} and {{A, B, C, X(213), X(881)}}
X(45913) = center of circle {{X(3), X(3095), X(14661)}}
X(45913) = crossdifference of every pair of points on line {X(385), X(5990)}

X(45914) = INTERSECTION OF THESE LINES: BROCARD LINE AND G-K LINE

Barycentrics    a^2*((b^4+c^4)*a^4-2*(b^2+c^2)*b^2*c^2*a^2+b^2*c^2*(b^4+c^4)) : :

X(45914) lies on these lines: {1, 9424}, {2, 6}, {39, 512}, {620, 9427}, {706, 3266}, {729, 41134}, {865, 44127}, {1084, 6786}, {2142, 14822}, {3124, 34383}, {3224, 7891}, {3499, 13571}, {6787, 30229}, {7764, 44164}, {7807, 9490}, {7906, 33786}, {8789, 10352}, {20859, 44114}, {21762, 26629}

X(45914) = perspector of the circumconic {{A, B, C, X(99), X(694)}}
X(45914) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(882)}} and {{A, B, C, X(6), X(881)}}
X(45914) = crossdifference of every pair of points on line {X(385), X(512)}
X(45914) = X(512)-Daleth conjugate of-X(14824)
X(45914) = X(i)-Hirst inverse of-X(j) for these (i, j): {39, 512}, {512, 39}
X(45914) = X(i)-line conjugate of-X(j) for these (i, j): (2, 385), (6, 385), (39, 512), (69, 385)
X(45914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 6, 2086), (6, 2421, 3051), (39, 45906, 45915), (39, 45915, 45910)

X(45915) = INTERSECTION OF THESE LINES: BROCARD LINE AND N-K LINE

Barycentrics    a^2*((b^4+c^4)*a^8-2*(b^2+c^2)*(b^4+c^4)*a^6+(b^8+c^8+b^2*c^2*(b^4+6*b^2*c^2+c^4))*a^4-2*(b^6+c^6)*b^2*c^2*a^2+(b^2-c^2)^2*b^2*c^2*(b^4+c^4)) : :

X(45915) lies on these lines: {5, 6}, {39, 512}, {1625, 12830}, {2086, 7746}, {2421, 7762}, {4173, 44114}

X(45915) = perspector of the circumconic {{A, B, C, X(694), X(925)}}
X(45915) = crossdifference of every pair of points on line {X(385), X(924)}
X(45915) = center of circle {{X(3), X(3095), X(18348)}}
X(45915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (39, 45901, 45910), (39, 45906, 45914), (45901, 45906, 39), (45910, 45914, 39)

X(45916) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND I-O LINE

Barycentrics    a*((b^3+c^3)*a^2-(b^2+c^2)*b*c*a-(b^4-c^4)*(b-c)) : :

X(45916) lies on these lines: {1, 3}, {11, 23690}, {325, 523}, {518, 18210}, {2611, 33136}, {2886, 21318}, {3120, 21326}, {3782, 21333}, {3838, 21807}, {5057, 24488}, {5087, 42753}, {5094, 29857}, {7495, 26230}, {16063, 29832}, {16790, 21493}, {17463, 26015}, {17718, 31395}, {18477, 30269}, {20684, 21329}, {20739, 21374}, {21325, 29690}, {25091, 41581}

X(45916) = perspector of the circumconic {{A, B, C, X(76), X(651)}}
X(45916) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(35519)}} and {{A, B, C, X(3), X(35518)}}
X(45916) = center of circle {{X(99), X(316), X(2703)}}
X(45916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3006, 3007, 45917), (3006, 45917, 45920)

X(45917) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND I-H LINE

Barycentrics    (b^2+c^2)*a^4-(b+c)*b*c*a^3+(b-c)^2*b*c*a^2-(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)*(b-c)*(b^3+c^3) : :

X(45917) lies on these lines: {1, 4}, {325, 523}, {860, 44662}, {1995, 26230}, {2834, 33305}, {3827, 23541}, {4466, 26013}, {6357, 29207}, {8229, 8758}, {16051, 29857}, {17904, 18596}, {18732, 23518}, {29832, 31099}, {29855, 40132}, {39572, 45946}

X(45917) = perspector of the circumconic {{A, B, C, X(76), X(653)}}
X(45917) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(35518)}} and {{A, B, C, X(4), X(35519)}}
X(45917) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3006, 3007, 45916), (45916, 45920, 3006)

X(45918) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND NAPOLEON AXIS

Barycentrics    (b^2+c^2)*a^6-2*(b^2+c^2)^2*a^4+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :

X(45918) lies on these lines: {6, 17}, {114, 5201}, {325, 523}, {6033, 37924}, {9220, 13108}, {11063, 15561}, {18121, 44716}

X(45918) = perspector of the circumconic {{A, B, C, X(76), X(930)}}
X(45918) = crossdifference of every pair of points on line {X(32), X(1510)}
X(45918) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (325, 45921, 3001), (3001, 45921, 3014)

X(45919) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND SHERMAN LINE

Barycentrics    a^2*((b^2+c^2)*a^2-2*(b+c)*b*c*a-b^4-c^4+2*(b^2+c^2)*b*c)*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45919) lies on these lines: {325, 523}, {513, 9371}, {764, 23101}, {901, 4638}, {2183, 3310}, {3259, 3326}

X(45919) = trilinear product X(1769)*X(2810)
X(45919) = perspector of the circumconic {{A, B, C, X(76), X(10015)}}
X(45919) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 7574, 38584}, {4, 953, 7464}
X(45919) = crossdifference of every pair of points on line {X(32), X(32641)}

X(45920) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND I-N LINE

Barycentrics    (b^2+c^2)*a^4-(b+c)*(b^2+c^2)*a^3+2*(b^2-b*c+c^2)*b*c*a^2+(b^3+c^3)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^3+c^3) : :

X(45920) lies on these lines: {1, 5}, {20, 27542}, {120, 29857}, {325, 523}, {528, 867}, {1329, 16594}, {2886, 32771}, {3303, 36561}, {3436, 28829}, {3782, 24816}, {3816, 29638}, {3925, 29861}, {4997, 11681}, {5169, 29832}, {5270, 30449}, {5434, 30448}, {5513, 5701}, {7796, 40365}, {9710, 36568}, {10883, 14942}, {20531, 29643}, {29829, 37632}, {29830, 30993}, {34501, 36499}, {36513, 37614}

X(45920) = perspector of the circumconic {{A, B, C, X(76), X(655)}}
X(45920) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 7574, 18342}, {4, 7464, 18341}
X(45920) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (11, 12, 37691), (3006, 45917, 45916)

X(45921) = INTERSECTION OF THESE LINES: DE LONGCHAMPS LINE AND N-K LINE

Barycentrics    (b^2+c^2)*a^6-(b^4+4*b^2*c^2+c^4)*a^4+(b^2+c^2)*(b^4+c^4)*a^2-(b^4+c^4)*(b^2-c^2)^2 : :

X(45921) lies on these lines: {2, 44197}, {5, 6}, {20, 41761}, {69, 23333}, {114, 3003}, {115, 45159}, {157, 41757}, {297, 1634}, {325, 523}, {427, 9766}, {524, 2450}, {1513, 5201}, {2854, 34827}, {5133, 41624}, {5169, 7774}, {6033, 11799}, {7778, 30739}, {7796, 40073}, {9132, 25328}, {11063, 37459}, {13338, 20576}, {20975, 44388}, {31636, 43754}

X(45921) = perspector of the circumconic {{A, B, C, X(76), X(925)}}
X(45921) = inverse of X(23128) in MacBeath circumconic
X(45921) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(6563)}} and {{A, B, C, X(68), X(3267)}}
X(45921) = crossdifference of every pair of points on line {X(32), X(924)}
X(45921) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 7574, 18348}, {4, 7464, 18347}
X(45921) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (325, 45279, 3001), (3001, 3014, 45279), (3001, 45918, 325), (3014, 45918, 3001)

X(45922) = INTERSECTION OF THESE LINES: EULER LINE AND SHERMAN LINE

Barycentrics    (b-c)^2*(2*a^5-2*(b+c)*a^4+2*b*c*a^3+(b^2-c^2)*(b-c)*a^2-2*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c))*(-a^2+b^2+c^2)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45922) lies on these lines: {2, 3}, {3259, 3326}, {10776, 20999}

X(45922) = perspector of the circumconic {{A, B, C, X(648), X(10015)}}
X(45922) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(42761)}} and {{A, B, C, X(4), X(42759)}}
X(45922) = center of circle {{X(4), X(953), X(43655)}}
X(45922) = crossdifference of every pair of points on line {X(647), X(32641)}

X(45923) = INTERSECTION OF THESE LINES: FERMAT AXIS AND I-O LINE

Barycentrics    a*(a^6+(b+c)*a^5-(2*b^2+b*c+2*c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+(b^4+c^4-b*c*(b+c)^2)*a^2+(b^2-c^2)^2*(b+c)*a+2*(b^2-c^2)^2*b*c) : :

X(45923) lies on these lines: {1, 3}, {4, 1029}, {5, 24883}, {6, 13}, {30, 81}, {42, 18524}, {58, 13743}, {79, 8614}, {140, 24936}, {191, 8143}, {222, 18541}, {226, 23071}, {323, 6175}, {376, 14996}, {382, 36742}, {386, 37251}, {387, 44229}, {394, 17528}, {403, 44097}, {405, 35193}, {442, 3193}, {500, 4658}, {547, 37680}, {549, 37633}, {567, 44085}, {1046, 5492}, {1172, 15762}, {1203, 9955}, {1468, 26321}, {1656, 24880}, {1657, 36746}, {1717, 17637}, {1725, 7073}, {1834, 13408}, {1963, 38430}, {1993, 17532}, {1994, 17577}, {2194, 10540}, {2915, 41723}, {3090, 24898}, {3194, 44225}, {3526, 36745}, {3562, 6147}, {3564, 30444}, {3651, 5453}, {3743, 16139}, {3877, 16430}, {4259, 23039}, {4260, 5891}, {4383, 5055}, {4653, 28443}, {5054, 37674}, {5070, 24902}, {5071, 14997}, {5127, 5398}, {5422, 17556}, {5499, 26131}, {5587, 39523}, {5754, 36558}, {5769, 33142}, {5790, 21696}, {5800, 18531}, {7562, 10539}, {9818, 44094}, {10895, 16473}, {10896, 16472}, {11246, 39751}, {12083, 36740}, {13632, 24512}, {13754, 40952}, {15033, 38555}, {15066, 44217}, {15696, 37501}, {15699, 37687}, {15988, 16052}, {16150, 24851}, {16159, 36250}, {16466, 18493}, {16474, 28204}, {16948, 31649}, {18180, 20831}, {18494, 44105}, {18534, 37492}, {19276, 26625}, {34545, 37375}, {38530, 38588}

X(45923) = reflection of X(3) in X(37527)
X(45923) = perspector of the circumconic {{A, B, C, X(476), X(651)}}
X(45923) = intersection, other than A, B, C, of circumconics {{A, B, C, X(35), X(2341)}} and {{A, B, C, X(65), X(1989)}}
X(45923) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {98, 2699, 22265}, {110, 1290, 15342}
X(45923) = X(37527)-of-X3-ABC reflections triangle
X(45923) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5707, 45931), (442, 3193, 22136), (1046, 5492, 13465), (1834, 13408, 37230), (3017, 3019, 45924), (3017, 45924, 45926), (5706, 5707, 3), (45924, 45926, 381)

X(45924) = INTERSECTION OF THESE LINES: FERMAT AXIS AND I-H LINE

Barycentrics    a^7-(b^2+b*c+c^2)*a^5-(b+c)*(b^2+b*c+c^2)*a^4-(b^3+c^3)*(b+c)*a^3+(b^2-c^2)*(b-c)*(2*b^2+3*b*c+2*c^2)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

X(45924) lies on these lines: {1, 4}, {3, 37816}, {5, 580}, {6, 13}, {20, 24936}, {30, 4653}, {40, 21674}, {42, 18406}, {58, 6841}, {79, 1725}, {212, 7951}, {284, 15762}, {386, 44229}, {403, 2299}, {407, 2360}, {442, 2328}, {573, 37584}, {860, 17188}, {968, 27577}, {1104, 9955}, {1451, 7741}, {1724, 6990}, {1754, 6829}, {2476, 35193}, {3073, 12558}, {3090, 24902}, {3091, 24883}, {3216, 6900}, {3332, 6843}, {4256, 28452}, {5068, 24898}, {5292, 6866}, {5341, 18453}, {5453, 44258}, {5492, 16159}, {5902, 19470}, {6828, 37530}, {6831, 37634}, {6845, 37522}, {6881, 13329}, {6894, 37732}, {7078, 10895}, {8727, 37469}, {15852, 22793}, {16125, 24851}, {16485, 38021}, {16499, 34617}, {23692, 37381}, {26131, 37433}, {28194, 33109}, {29661, 37400}, {31162, 33104}

X(45924) = perspector of the circumconic {{A, B, C, X(476), X(653)}}
X(45924) = intersection, other than A, B, C, of circumconics {{A, B, C, X(225), X(1989)}} and {{A, B, C, X(226), X(265)}}
X(45924) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (4, 5713, 581), (381, 45923, 45926), (581, 5713, 45933), (3017, 3019, 45923), (6841, 13408, 58), (45923, 45926, 3017)

X(45925) = INTERSECTION OF THESE LINES: FERMAT AXIS AND SHERMAN LINE

Barycentrics    (2*a^9-3*(b+c)*a^8-2*(b-c)^2*a^7+4*(b^3+c^3)*a^6-2*(b^2-3*b*c+c^2)*b*c*a^5+(b+c)*(2*b^2-7*b*c+2*c^2)*b*c*a^4-2*(b^6+c^6+(b^2-c^2)^2*b*c)*a^3+(b+c)*(2*b^4+2*c^4+b*c*(b^2-4*b*c+c^2))*b*c*a^2+2*(b^2-c^2)^2*(b^4+c^4)*a-(b^2-c^2)^2*(b+c)*(b^4+c^4))*(b-c)^2*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45925) lies on these lines: {6, 13}, {3259, 3326}

X(45925) = perspector of the circumconic {{A, B, C, X(476), X(10015)}}
X(45925) = crossdifference of every pair of points on line {X(526), X(32641)}
X(45925) = center of circle {{X(4), X(953), X(11005)}}
X(45925) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (45899, 45953, 45941), (45948, 45953, 45899)

X(45926) = INTERSECTION OF THESE LINES: FERMAT AXIS AND I-N LINE

Barycentrics    (a^2-c*a+c^2-b^2)*((b+c)*a^2+2*b*c*a-(b^2-c^2)*(b-c))*(a^2-b*a+b^2-c^2) : :

X(45926) lies on the cubic K383 and these lines: {1, 5}, {2, 6739}, {3, 759}, {4, 162}, {6, 13}, {20, 24898}, {30, 35466}, {31, 18407}, {58, 37230}, {140, 24885}, {273, 2973}, {387, 6866}, {403, 44113}, {405, 1793}, {442, 500}, {517, 33136}, {582, 1714}, {1046, 16159}, {1054, 34311}, {1168, 35457}, {1191, 18544}, {1210, 24235}, {1243, 45022}, {1480, 31140}, {1616, 18543}, {1766, 2161}, {1834, 6841}, {1865, 15762}, {2166, 14254}, {2222, 41345}, {2361, 3583}, {2611, 18115}, {2650, 33592}, {2771, 3120}, {3028, 5902}, {3052, 18499}, {3090, 24936}, {3652, 24851}, {3654, 32865}, {3656, 33141}, {3945, 6843}, {4192, 24892}, {5046, 35193}, {5127, 34172}, {5224, 14616}, {5230, 18517}, {5292, 44229}, {5492, 36250}, {5620, 10265}, {5742, 36910}, {6187, 13731}, {6702, 16578}, {6827, 37650}, {6830, 37651}, {6873, 19767}, {6881, 17245}, {6996, 24899}, {8727, 33810}, {9355, 16128}, {9956, 21674}, {12515, 24715}, {12764, 36052}, {16052, 26543}, {16466, 45630}, {17734, 18524}, {18480, 21935}, {19540, 24896}, {21161, 31204}, {21675, 40937}, {24161, 33858}, {25683, 40534}, {26446, 36815}, {27577, 32167}, {28452, 37646}, {29689, 40172}, {37251, 45939}

X(45926) = barycentric product X(i)*X(j) for these {i, j}: {80, 5249}, {94, 500}, {265, 445}, {328, 44095}, {442, 24624}, {942, 18359}
X(45926) = barycentric quotient X(i)/X(j) for these (i, j): (80, 40435), (442, 3936), (445, 340), (500, 323), (942, 3218), (1411, 2982)
X(45926) = trilinear product X(i)*X(j) for these {i, j}: {80, 942}, {265, 1844}, {442, 759}, {500, 2166}, {1411, 6734}
X(45926) = trilinear quotient X(i)/X(j) for these (i, j): (80, 943), (442, 758), (500, 6149), (759, 1175), (942, 36), (1234, 35550)
X(45926) = perspector of the circumconic {{A, B, C, X(476), X(655)}}
X(45926) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(270)}} and {{A, B, C, X(2), X(15762)}}
X(45926) = crosspoint of X(80) and X(2166)
X(45926) = crosssum of X(36) and X(6149)
X(45926) = X(442)-Dao conjugate of-X(4511)
X(45926) = X(i)-isoconjugate-of-X(j) for these {i, j}: {36, 943}, {758, 1175}
X(45926) = X(i)-reciprocal conjugate of-X(j) for these (i, j): (80, 40435), (442, 3936), (445, 340), (500, 323)
X(45926) = X(284)-Zayin conjugate of-X(2245)
X(45926) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {4, 11005, 18341}, {1054, 6788, 10774}
X(45926) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 5396, 45944), (5, 5721, 5396), (80, 2006, 1807), (381, 45923, 45924), (3017, 45924, 45923)

X(45927) = INTERSECTION OF THESE LINES: GERGONNE LINE AND NAPOLEON AXIS

Barycentrics    2*a^7-4*(b^2+c^2)*a^5+3*(b^4+c^4)*a^3-(b^4-c^4)*(b-c)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

X(45927) lies on these lines: {6, 17}, {241, 514}

X(45927) = perspector of the circumconic {{A, B, C, X(7), X(930)}}
X(45927) = crossdifference of every pair of points on line {X(55), X(1510)}
X(45927) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3002, 45930, 3015), (35466, 45930, 3002)

X(45928) = INTERSECTION OF THESE LINES: GERGONNE LINE AND SHERMAN LINE

Barycentrics    (2*a^5-2*(b+c)*a^4-(3*b^2-8*b*c+3*c^2)*a^3+3*(b^2-c^2)*(b-c)*a^2+(b-c)^4*a-(b-c)*(b^4-c^4))*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45928) lies on these lines: {241, 514}, {3259, 3326}

X(45928) = perspector of the circumconic {{A, B, C, X(7), X(10015)}}
X(45928) = center of circle {{X(3326), X(3638), X(3639)}}
X(45928) = crossdifference of every pair of points on line {X(55), X(32641)}

X(45929) = INTERSECTION OF THESE LINES: GERGONNE LINE AND VAN AUBEL LINE

Barycentrics    2*a^7-(b^2+c^2)*a^5-3*(b^2-c^2)*(b-c)*a^4+2*(b^4-c^4)*(b-c)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

X(45929) lies on these lines: {4, 6}, {221, 17905}, {241, 514}, {282, 7100}, {651, 38948}, {860, 45802}, {1146, 1870}, {1449, 5722}, {1886, 6001}, {2193, 44244}, {5011, 15725}, {5718, 30808}, {14256, 37800}, {17904, 20306}, {24597, 24604}, {25088, 40937}, {27317, 27325}, {31185, 31187}

X(45929) = perspector of the circumconic {{A, B, C, X(7), X(107)}}
X(45929) = intersection, other than A, B, C, of circumconics {{A, B, C, X(4), X(17094)}} and {{A, B, C, X(282), X(14838)}}
X(45929) = crossdifference of every pair of points on line {X(55), X(520)}
X(45929) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3002, 3015, 45930), (3002, 45930, 35466)

X(45930) = INTERSECTION OF THESE LINES: GERGONNE LINE AND N-K LINE

Barycentrics    2*a^7-3*(b^2+c^2)*a^5-(b^2-c^2)*(b-c)*a^4+2*(b^4+c^4)*a^3-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

X(45930) lies on these lines: {5, 6}, {163, 1884}, {241, 514}, {379, 24597}, {1723, 1729}, {2160, 39943}, {5829, 30424}, {6675, 34522}, {8229, 45748}, {27300, 27317}, {31186, 31187}

X(45930) = perspector of the circumconic {{A, B, C, X(7), X(925)}}
X(45930) = crossdifference of every pair of points on line {X(55), X(924)}
X(45930) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (485, 486, 32594), (3002, 3015, 45929), (3002, 45927, 35466), (3015, 45927, 3002), (5723, 35466, 1375), (35466, 45929, 3002)

X(45931) = INTERSECTION OF THESE LINES: I-O LINE AND NAPOLEON AXIS

Barycentrics    a*(a^6+(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4-2*(b+c)*(b^2+c^2)*a^3+(b^2+b*c+c^2)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2*(b+c)*a+2*(b^2-c^2)^2*b*c) : :

X(45931) lies on these lines: {1, 3}, {2, 22136}, {4, 14996}, {5, 81}, {6, 17}, {58, 7489}, {140, 37633}, {226, 23070}, {381, 36742}, {382, 36746}, {411, 5453}, {602, 9345}, {991, 16117}, {1203, 11230}, {1216, 40952}, {1396, 7546}, {1698, 39523}, {1994, 7504}, {2049, 26625}, {2194, 18350}, {2906, 7537}, {3090, 37685}, {3193, 7483}, {3526, 36754}, {3534, 37501}, {3562, 5719}, {3628, 32911}, {3945, 6825}, {4265, 13564}, {4340, 6923}, {4383, 5070}, {4658, 5396}, {5054, 36745}, {5138, 10539}, {5287, 26921}, {5712, 6863}, {5820, 25738}, {6853, 37635}, {6861, 37642}, {6887, 37666}, {6918, 34465}, {6924, 19767}, {6960, 41819}, {7393, 44094}, {7505, 44097}, {7517, 36740}, {7529, 37492}, {11374, 23071}, {13353, 44085}, {14627, 37693}, {14969, 18491}, {16863, 25934}, {17018, 32141}, {17021, 26878}, {17074, 24470}, {18166, 19648}, {18391, 32128}, {18524, 37698}, {19540, 19714}

X(45931) = perspector of the circumconic {{A, B, C, X(651), X(930)}}
X(45931) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 5707, 45923), (5, 81, 36750), (940, 5707, 3), (36754, 37674, 3526), (37527, 37536, 3), (45933, 45939, 45944), (45939, 45942, 45933), (45939, 45944, 1656)

X(45932) = INTERSECTION OF THESE LINES: I-H LINE AND LEMOINE AXIS

Barycentrics    a^2*((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) : :

X(45932) lies on these lines: {1, 4}, {187, 237}, {213, 217}, {1766, 20753}, {1964, 12723}, {1987, 43694}, {2176, 10537}, {2808, 43034}, {16969, 38297}, {20732, 21375}, {21352, 37988}, {27248, 37186}

X(45932) = perspector of the circumconic {{A, B, C, X(6), X(653)}}
X(45932) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(1946)}} and {{A, B, C, X(4), X(663)}}
X(45932) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2223, 45937, 3009), (3009, 3010, 2223)

X(45933) = INTERSECTION OF THESE LINES: I-H LINE AND NAPOLEON AXIS

Barycentrics    a^7+2*(b+c)*a^6-(b^2-b*c+c^2)*a^5-(b+c)*(5*b^2-b*c+5*c^2)*a^4-(b^2+b*c+c^2)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(4*b^2+5*b*c+4*c^2)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

X(45933) lies on these lines: {1, 4}, {5, 4658}, {6, 17}, {140, 580}, {573, 37532}, {5733, 6825}, {6831, 37631}, {6900, 22392}, {6922, 17392}, {7356, 18398}, {26131, 37163}

X(45933) = perspector of the circumconic {{A, B, C, X(653), X(930)}}
X(45933) = intersection, other than A, B, C, of circumconics {{A, B, C, X(225), X(2963)}} and {{A, B, C, X(226), X(3519)}}
X(45933) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (581, 5713, 45924), (5712, 5713, 581), (45931, 45944, 45939), (45939, 45942, 45931)

X(45934) = INTERSECTION OF THESE LINES: I-H LINE AND HATZIPOLAKIS AXIS

Barycentrics    (b^2+c^2)*a^8-(b+c)*b*c*a^7-(2*b^4+2*c^4-(b^2+c^2)*b*c)*a^6+(b^3+c^3)*b*c*a^5-(b^3-c^3)*(b-c)*b*c*a^4+(b^3-c^3)*b*c*(b^2-c^2)*a^3+(2*b^4+2*c^4-b*c*(b^2+b*c+c^2))*(b^2-c^2)^2*a^2-(b^2-c^2)^3*(b-c)*b*c*a-(b^2-c^2)^3*(b-c)*(b^3+c^3) : :

X(45934) lies on these lines: {1, 4}, {5, 523}

X(45934) = perspector of the circumconic {{A, B, C, X(94), X(653)}}
X(45934) = intersection, other than A, B, C, of circumconics {{A, B, C, X(34), X(43082)}} and {{A, B, C, X(73), X(43083)}}
X(45934) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {11, 3326, 42422}, {102, 477, 7728}, {109, 265, 476}, {146, 38573, 38581}, {1539, 38600, 38610}, {3448, 38579, 38580}, {10113, 38607, 38609}
X(45934) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 18115, 18120), (18119, 18120, 18115)

X(45935) = INTERSECTION OF THESE LINES: LEMOINE AXIS AND NAPOLEON AXIS

Barycentrics    a^2*((b^2+c^2)*a^6-2*(b^2+c^2)^2*a^4+(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2) : :

X(45935) lies on these lines: {6, 17}, {187, 237}, {1613, 5475}, {3051, 7603}, {5104, 15564}, {5309, 40805}, {9463, 31415}, {11060, 38463}

X(45935) = perspector of the circumconic {{A, B, C, X(6), X(930)}}
X(45935) = crossdifference of every pair of points on line {X(2), X(1510)}
X(45935) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (187, 45938, 3016), (3231, 45938, 187)

X(45936) = INTERSECTION OF THESE LINES: LEMOINE AXIS AND SHERMAN LINE

Barycentrics    a^2*((b^2+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^2-b*c+c^2)*b*c*a^2+2*(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2))*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45936) lies on these lines: {187, 237}, {3259, 3326}

X(45936) = trilinear product X(1769)*X(2875)
X(45936) = trilinear quotient X(1769)/X(2861)
X(45936) = perspector of the circumconic {{A, B, C, X(6), X(10015)}}
X(45936) = intersection, other than A, B, C, of circumconics {{A, B, C, X(187), X(42760)}} and {{A, B, C, X(237), X(42751)}}
X(45936) = center of circle {{X(4), X(953), X(11674)}}
X(45936) = crossdifference of every pair of points on line {X(2), X(32641)}

X(45937) = INTERSECTION OF THESE LINES: LEMOINE AXIS AND I-N LINE

Barycentrics    a^2*((b^2+c^2)*a^4-(b+c)*(b^2+c^2)*a^3-(b^4+c^4-2*(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(45937) lies on these lines: {1, 5}, {187, 237}

X(45937) = perspector of the circumconic {{A, B, C, X(6), X(655)}}
X(45937) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(8648)}} and {{A, B, C, X(12), X(42666)}}
X(45937) = crosssum of X(1) and X(45749)
X(45937) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 2080, 18342}, {4, 11674, 18341}
X(45937) = {X(3009), X(45932)}-harmonic conjugate of X(2223)

X(45938) = INTERSECTION OF THESE LINES: LEMOINE AXIS AND N-K LINE

Barycentrics    a^2*((b^2+c^2)*a^6-2*(b^4+b^2*c^2+c^4)*a^4+(b^2+c^2)*(b^4+c^4)*a^2-2*(b^2-c^2)^2*b^2*c^2) : :

X(45938) lies on these lines: {5, 6}, {115, 1568}, {187, 237}, {217, 7746}, {230, 1625}, {231, 16534}, {570, 10170}, {1613, 7737}, {2549, 40805}, {3051, 5475}, {3787, 33842}, {5033, 40643}, {5104, 11675}, {7603, 20965}, {11646, 18371}, {14806, 15827}, {15066, 41237}, {21001, 21843}

X(45938) = perspector of the circumconic {{A, B, C, X(6), X(925)}}
X(45938) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(34952)}} and {{A, B, C, X(68), X(647)}}
X(45938) = crossdifference of every pair of points on line {X(2), X(924)}
X(45938) = X(6)-vertex conjugate of-X(34952)
X(45938) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 2080, 18348}, {4, 11674, 18347}
X(45938) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (187, 3016, 3331), (187, 45935, 3231), (3016, 45935, 187), (3231, 3331, 187), (38463, 45769, 6)

X(45939) = INTERSECTION OF THESE LINES: NAGEL LINE AND NAPOLEON AXIS

Barycentrics    a^4+(b+c)*a^3-(b^2-b*c+c^2)*a^2+(b^2-c^2)^2 : :

X(45939) lies on these lines: {1, 2}, {4, 4257}, {5, 58}, {6, 17}, {11, 595}, {31, 7741}, {36, 21935}, {56, 39270}, {81, 7504}, {115, 33863}, {140, 1834}, {171, 25639}, {230, 4251}, {238, 3825}, {381, 4252}, {442, 37634}, {496, 40091}, {579, 37532}, {580, 6882}, {581, 6863}, {759, 37259}, {849, 24624}, {902, 4857}, {942, 24160}, {991, 6825}, {1086, 34753}, {1089, 33119}, {1393, 2006}, {1468, 7951}, {1724, 4193}, {1754, 6943}, {2271, 37637}, {2476, 37522}, {2650, 37701}, {2901, 32851}, {3052, 9669}, {3090, 37642}, {3120, 3336}, {3192, 7505}, {3286, 19648}, {3454, 14829}, {3526, 4255}, {3628, 37662}, {3670, 33133}, {3673, 29477}, {3767, 4253}, {3772, 24046}, {3814, 5247}, {3822, 37607}, {3841, 17122}, {3874, 17719}, {3911, 23537}, {3915, 37720}, {4066, 33167}, {4187, 35466}, {4192, 4278}, {4276, 13731}, {4642, 5445}, {4646, 11231}, {4653, 7483}, {4658, 5718}, {4999, 37715}, {5019, 32431}, {5021, 13881}, {5030, 5254}, {5132, 37621}, {5172, 16453}, {5255, 24387}, {5264, 11680}, {5358, 33849}, {5398, 6971}, {5400, 6979}, {5432, 33771}, {5713, 6859}, {5740, 17189}, {5883, 24161}, {6693, 13740}, {6830, 37530}, {6842, 37469}, {6922, 13329}, {6931, 24597}, {6949, 37732}, {6959, 34465}, {7486, 37666}, {7749, 18755}, {7769, 33296}, {7828, 37686}, {8666, 37716}, {8715, 33141}, {9581, 37817}, {9665, 21793}, {11108, 31187}, {13741, 41806}, {16287, 37564}, {16549, 17737}, {16574, 24895}, {16948, 37375}, {17124, 41859}, {17536, 31204}, {17596, 36250}, {18046, 25471}, {18178, 34466}, {18398, 33127}, {19540, 19762}, {19543, 26286}, {24167, 33147}, {31262, 33105}, {33094, 37572}, {37251, 45926}, {37536, 41329}

X(45939) = perspector of the circumconic {{A, B, C, X(190), X(930)}}
X(45939) = intersection, other than A, B, C, of circumconics {{A, B, C, X(10), X(2963)}} and {{A, B, C, X(306), X(3519)}}
X(45939) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 1714, 17749), (2, 5292, 386), (2, 10479, 24931), (2, 24883, 3216), (5, 37646, 58), (81, 7504, 37693), (140, 1834, 4256), (386, 5292, 3017), (499, 5230, 995), (1656, 45931, 45944), (27529, 33142, 3293), (31262, 37559, 33105), (45931, 45933, 45942), (45931, 45944, 45933)

X(45940) = INTERSECTION OF THESE LINES: NAGEL LINE AND SHERMAN LINE

Barycentrics    (b-c)^2*(3*a^4-3*(b+c)*a^3-(2*b^2-3*b*c+2*c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-(b^3+c^3)*(b+c))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45940) lies on these lines: {1, 2}, {3259, 3326}

X(45940) = perspector of the circumconic {{A, B, C, X(190), X(10015)}}
X(45940) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(42753)}} and {{A, B, C, X(2), X(42754)}}
X(45940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (35014, 35015, 45947), (35014, 45950, 35015)

X(45941) = INTERSECTION OF THESE LINES: NAPOLEON AXIS AND SHERMAN LINE

Barycentrics    (2*a^9-(b+c)*a^8-6*(b^2+c^2)*a^7+2*(b+c)*(b^2+c^2)*a^6+2*(b^2+b*c+c^2)*(4*b^2-3*b*c+4*c^2)*a^5-(b+c)*(2*b^4+2*c^4+b*c*(b+2*c)*(2*b+c))*a^4-2*(3*b^6+3*c^6+(b^2-c^2)^2*b*c)*a^3+(b+c)*(2*b^6+2*c^6+(2*b^4+2*c^4+b*c*(b^2-4*b*c+c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4)*a-(b^2-c^2)^2*(b+c)*(b^4+c^4))*(b-c)^2*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45941) lies on these lines: {6, 17}, {3259, 3326}

X(45941) = perspector of the circumconic {{A, B, C, X(930), X(10015)}}
X(45941) = crossdifference of every pair of points on line {X(1510), X(32641)}
X(45941) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (45899, 45953, 45925), (45952, 45953, 45899)

X(45942) = INTERSECTION OF THESE LINES: NAPOLEON AXIS AND SODDY LINE

Barycentrics    3*a^6+(b+c)*a^5-(6*b^2-b*c+6*c^2)*a^4-3*(b+c)*(b^2+c^2)*a^3+(4*b^2+5*b*c+4*c^2)*(b-c)^2*a^2+2*(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*(b-c)^2 : :

X(45942) lies on these lines: {1, 7}, {6, 17}, {549, 13329}, {3525, 4648}, {4658, 5721}, {6286, 18398}, {16239, 17245}

X(45942) = perspector of the circumconic {{A, B, C, X(658), X(930)}}
X(45942) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (991, 5733, 3019), (3945, 5733, 991), (45931, 45933, 45939)

X(45943) = INTERSECTION OF THESE LINES: NAPOLEON AXIS AND HATZIPOLAKIS AXIS

Barycentrics    (b^2+c^2)*a^10-2*(2*b^4+b^2*c^2+2*c^4)*a^8+7*(b^6+c^6)*a^6-(b^4+b^2*c^2+c^4)*(7*b^4-12*b^2*c^2+7*c^4)*a^4+4*(b^6+c^6)*(b^2-c^2)^2*a^2-(b^4+c^4)*(b^2-c^2)^4 : :
X(45943) = 5*X(1656)-2*X(10414)

X(45943) lies on these lines: {5, 523}, {6, 17}, {76, 1273}, {114, 9149}, {3628, 24975}, {24640, 37701}

X(45943) = perspector of the circumconic {{A, B, C, X(94), X(930)}}
X(45943) = crossdifference of every pair of points on line {X(50), X(1510)}
X(45943) = center of circle {{X(98), X(14979), X(31656)}}
X(45943) = PU(5)-harmonic conjugate of X(15475)
X(45943) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 18114, 14356), (5, 18122, 18114)

X(45944) = INTERSECTION OF THESE LINES: NAPOLEON AXIS AND I-N LINE

Barycentrics    (b+c)*a^6+(b^2+c^2)*a^5-(b+c)*(3*b^2-b*c+3*c^2)*a^4-2*(b^4+b^2*c^2+c^4)*a^3+3*(b^3-c^3)*(b^2-c^2)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

X(45944) lies on these lines: {1, 5}, {2, 5397}, {6, 17}, {500, 6831}, {517, 33105}, {970, 994}, {1594, 44113}, {3332, 6825}, {3628, 35466}, {4192, 29678}, {4648, 6978}, {5067, 24597}, {5070, 31187}, {5535, 21363}, {5712, 6859}, {5713, 6863}, {5755, 37532}, {6874, 19767}, {6881, 37662}, {6882, 17056}, {6902, 24936}, {6952, 26131}, {13731, 26286}, {18398, 18984}, {19648, 37621}, {24470, 43056}, {24880, 37509}, {24892, 39523}, {26446, 33111}, {31245, 44414}

X(45944) = perspector of the circumconic {{A, B, C, X(655), X(930)}}
X(45944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 34465, 5396), (5, 5396, 45926), (5, 5718, 5396), (1656, 45931, 45939), (45933, 45939, 45931)

X(45945) = INTERSECTION OF THESE LINES: ORTHIC AXIS AND SHERMAN LINE

Barycentrics    (2*a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b^4-c^4)*(b-c)*a-(b^4-c^4)*(b^2-c^2))*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45945) lies on these lines: {230, 231}, {1309, 2405}, {1455, 30725}, {1465, 2804}, {3259, 3326}

X(45945) = perspector of the circumconic {{A, B, C, X(4), X(10015)}}
X(45945) = center of circle {{X(4), X(953), X(10295)}}
X(45945) = crossdifference of every pair of points on line {X(3), X(32641)}

X(45946) = INTERSECTION OF THESE LINES: ORTHIC AXIS AND I-N LINE

Barycentrics    2*a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3-2*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)*b*c*a+(b^2-c^2)^2*(b-c)^2 : :

X(45946) lies on these lines: {1, 5}, {24, 108}, {105, 1995}, {225, 1852}, {230, 231}, {354, 17061}, {528, 30684}, {651, 12831}, {1072, 34664}, {1155, 22464}, {1658, 36152}, {2342, 38357}, {3035, 16586}, {3176, 7505}, {3911, 22465}, {5573, 17728}, {6718, 44311}, {6742, 18883}, {6831, 9630}, {7503, 26357}, {7542, 37565}, {7575, 11809}, {9628, 37447}, {9639, 37358}, {20277, 33127}, {23047, 40950}, {23972, 43960}, {29639, 37454}, {39572, 45917}

X(45946) = complement of the isotomic conjugate of X(2861)
X(45946) = perspector of the circumconic {{A, B, C, X(4), X(655)}}
X(45946) = intersection, other than A, B, C, of circumconics {{A, B, C, X(80), X(3064)}} and {{A, B, C, X(108), X(8609)}}
X(45946) = crossdifference of every pair of points on line {X(3), X(41155)}
X(45946) = crosspoint of X(2) and X(2861)
X(45946) = crosssum of X(6) and X(2875)
X(45946) = center of circles {{ X(i), X(j), X(k) }} for these {i, j, k}: {3, 11799, 18342}, {4, 10295, 18341}
X(45946) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (1, 2006, 11), (3011, 23710, 8758), (8758, 23710, 16272), (15252, 15253, 11)

X(45947) = INTERSECTION OF THESE LINES: SHERMAN LINE AND SODDY LINE

Barycentrics    (b-c)^2*(a^4-(b+c)*a^3-(2*b-c)*(b-2*c)*a^2+(b^2-c^2)*(b-c)*a+(b^2-b*c+c^2)*(b-c)^2)*(-a+b+c)^2*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45947) lies on these lines: {1, 7}, {3259, 3326}

X(45947) = perspector of the circumconic {{A, B, C, X(658), X(10015)}}
X(45947) = intersection, other than A, B, C, of circumconics {{A, B, C, X(7), X(35015)}} and {{A, B, C, X(77), X(35014)}}
X(45947) = {X(35014), X(35015)}-harmonic conjugate of X(45940)

X(45948) = INTERSECTION OF THESE LINES: SHERMAN LINE AND VAN AUBEL LINE

Barycentrics    (b-c)^2*(2*a^9-4*(b+c)*a^8+6*b*c*a^7+(b+c)*(5*b^2-6*b*c+5*c^2)*a^6-4*(b^4+c^4+b*c*(b^2-b*c+c^2))*a^5+(b+c)*(b^4+c^4+4*b*c*(b-c)^2)*a^4-2*(b^2-c^2)^2*b*c*a^3-(b^2-c^2)^3*(b-c)*a^2+2*(b^2-c^2)^2*(b^4+c^4)*a-(b^2-c^2)^2*(b+c)*(b^4+c^4))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45948) lies on these lines: {4, 6}, {3259, 3326}

X(45948) = perspector of the circumconic {{A, B, C, X(107), X(10015)}}
X(45948) = crossdifference of every pair of points on line {X(520), X(32641)}
X(45948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (35015, 42755, 42761), (45899, 45925, 45953), (45899, 45953, 45952)

X(45949) = INTERSECTION OF THESE LINES: SHERMAN LINE AND HATZIPOLAKIS AXIS

Barycentrics    a^2*(a^2-b^2+b*c-c^2)*((b^2+c^2)*a^4-2*(b+c)*b*c*a^3-(2*b^4+2*c^4-b*c*(b^2+4*b*c+c^2))*a^2+2*(b^2-c^2)*(b-c)*b*c*a+(b^2-c^2)*(b-c)*(b^3+c^3))*(b-c)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45949) lies on these lines: {5, 523}, {3259, 3326}, {8677, 34586}

X(45949) = perspector of the circumconic {{A, B, C, X(94), X(10015)}}
X(45949) = crossdifference of every pair of points on line {X(50), X(32641)}
X(45949) = center of circle {{X(3), X(4), X(953)}}
X(45949) = {X(42753), X(42757)}-harmonic conjugate of X(42750)

X(45950) = INTERSECTION OF THESE LINES: SHERMAN LINE AND I-N LINE

Barycentrics    (b-c)^2*(-a+b+c)*(a^2-b^2+b*c-c^2)*(2*a^4-2*(b+c)*a^3-(b^2-4*b*c+c^2)*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45950) lies on these lines: {1, 5}, {3259, 3326}

X(45950) = perspector of the circumconic {{A, B, C, X(655), X(3904)}}
X(45950) = crossdifference of every pair of points on line {X(32641), X(32675)}
X(45950) = X(952)-Ceva conjugate of-X(35013)
X(45950) = X(654)-reciprocal conjugate of-X(35011)
X(45950) = center of circle {{X(4), X(953), X(18341)}}
X(45950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (23757, 35015, 3326), (35015, 45940, 35014)

X(45951) = INTERSECTION OF THESE LINES: SHERMAN LINE AND I-K LINE

Barycentrics    a*(b-c)^2*(a^5-3*(b+c)*a^4+(b+2*c)*(2*b+c)*a^3+2*(b^2-c^2)*(b-c)*a^2-(3*b^4+3*c^4+(b^2+c^2)*b*c)*a+(b+c)*(b^4+c^4))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45951) lies on these lines: {1, 6}, {3259, 3326}

X(45951) = perspector of the circumconic {{A, B, C, X(100), X(10015)}}
X(45951) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(42754)}} and {{A, B, C, X(6), X(42753)}}

X(45952) = INTERSECTION OF THESE LINES: SHERMAN LINE AND G-K LINE

Barycentrics    (b-c)^2*(2*a^5-2*(2*b^2+b*c+2*c^2)*a^3+(b+c)^3*a^2+2*(b^4+c^4)*a-(b+c)*(b^4+c^4))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45952) lies on these lines: {2, 6}, {3259, 3326}

X(45952) = perspector of the circumconic {{A, B, C, X(99), X(10015)}}
X(45952) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(42759)}} and {{A, B, C, X(6), X(42752)}}
X(45952) = crossdifference of every pair of points on line {X(512), X(32641)}
X(45952) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (45899, 45941, 45953), (45899, 45953, 45948)

X(45953) = INTERSECTION OF THESE LINES: SHERMAN LINE AND N-K LINE

Barycentrics    (b-c)^2*(2*a^9-2*(b+c)*a^8-2*(2*b^2-b*c+2*c^2)*a^7+(b+c)*(3*b^2-2*b*c+3*c^2)*a^6+4*(b^2+c^2)^2*a^5-(b+c)*(b^4+6*b^2*c^2+c^4)*a^4-2*(2*b^6+2*c^6+(b^2-c^2)^2*b*c)*a^3+(b+c)*(b^6+c^6+(2*b^4+2*c^4+b*c*(b^2-4*b*c+c^2))*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4)*a-(b^2-c^2)^2*(b+c)*(b^4+c^4))*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :

X(45953) lies on these lines: {5, 6}, {3259, 3326}

X(45953) = perspector of the circumconic {{A, B, C, X(925), X(10015)}}
X(45953) = crossdifference of every pair of points on line {X(924), X(32641)}
X(45953) = center of circle {{X(4), X(953), X(18347)}}
X(45953) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (45899, 45925, 45948), (45899, 45941, 45952), (45925, 45941, 45899), (45948, 45952, 45899)

X(45954) = INTERSECTION OF THESE LINES: HATZIPOLAKIS AXIS AND I-K LINE

Barycentrics    a*((b^3+c^3)*a^5-(b^4+c^4)*a^4-(b+c)*(2*b^4+2*c^4-b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^3+c^3)*(b^2-c^2)^2*a-(b^4+c^4)*(b^2-c^2)^2) : :

X(45954) lies on these lines: {1, 6}, {5, 523}, {6675, 24975}, {13746, 14194}

X(45954) = perspector of the circumconic {{A, B, C, X(94), X(100)}}
X(45954) = center of circle {{X(3), X(4), X(14661)}}
X(45954) = intersection, other than A, B, C, of circumconics {{A, B, C, X(6), X(43082)}} and {{A, B, C, X(37), X(10412)}}

X(45955) = INTERSECTION OF THESE LINES: LINE AT INFINITY AND APOLLONIUS LINE

Barycentrics    a^2*((b^2+c^2)*a^3+(b+c)^3*a^2-(b^4+4*b^2*c^2+c^4)*a-(b+c)*(b^4+c^4+2*b*c*(b-c)^2)) : :

X(45955) lies on these lines: {1, 181}, {2, 39550}, {3, 33771}, {8, 10441}, {10, 35631}, {30, 511}, {40, 32913}, {42, 37620}, {51, 3877}, {182, 44414}, {355, 15488}, {375, 960}, {392, 5943}, {551, 10440}, {575, 39523}, {995, 9567}, {1385, 15489}, {1480, 37517}, {1482, 5752}, {2093, 3784}, {3030, 5529}, {3293, 19513}, {3576, 42042}, {3632, 10825}, {3679, 10439}, {3753, 3819}, {3781, 9623}, {3869, 16980}, {3872, 26893}, {3878, 15049}, {4757, 23157}, {5180, 38389}, {5255, 10544}, {5657, 37521}, {5690, 37536}, {5887, 44865}, {5901, 34466}, {9729, 31786}, {10625, 25413}, {10974, 15955}, {11695, 31838}, {12245, 31785}, {12672, 13598}, {12702, 37482}, {13348, 31788}, {15644, 37562}, {17015, 40952}, {18391, 35645}, {31779, 34790}, {37508, 41434}, {38474, 41684}

X(45956) = X(3)X(323)∩X(5)X(113)

Barycentrics    a^2*(2*(b^2+c^2)*a^6-2*a^4*(3*b^4-2*b^2*c^2+3*c^4)+(3*b^2-2*c^2)*(2*b^2-3*c^2)*(b^2+c^2)*a^2-(2*b^4+3*b^2*c^2+2*c^4)*(b^2-c^2)^2) : :
X(45956) = 5*X(3)-3*X(33884), X(5)+2*X(185), 5*X(5)-6*X(373), 19*X(5)-16*X(11017), 5*X(5)-8*X(12006), 5*X(5)-2*X(12162), 3*X(5)-4*X(13363), X(5)-4*X(13630), 3*X(5)-2*X(15030), 13*X(5)-16*X(32205), 5*X(185)+3*X(373), 19*X(185)+8*X(11017), 5*X(185)+4*X(12006), 5*X(185)+X(12162), 3*X(185)+2*X(13363), X(185)+2*X(13630), 3*X(185)+X(15030), 13*X(185)+8*X(32205), 3*X(373)-5*X(9730), 3*X(373)-4*X(12006), 3*X(373)-X(12162), 9*X(373)-10*X(13363), 3*X(373)-10*X(13630), 9*X(373)-5*X(15030)

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45956) lies on these lines: {3, 323}, {4, 15003}, {5, 113}, {30, 568}, {49, 15035}, {51, 15687}, {52, 15704}, {54, 10226}, {74, 567}, {110, 43596}, {140, 10574}, {143, 10575}, {184, 15646}, {382, 11002}, {389, 3627}, {397, 30439}, {398, 30440}, {511, 550}, {546, 5640}, {547, 15045}, {548, 5889}, {549, 3819}, {569, 32138}, {631, 31834}, {632, 5876}, {1154, 8703}, {1181, 37814}, {1204, 32046}, {1216, 44682}, {1503, 38322}, {1657, 14449}, {1658, 6800}, {1899, 44263}, {1986, 14677}, {1994, 18859}, {2071, 15087}, {2777, 40928}, {2782, 27779}, {2807, 10283}, {2979, 34200}, {3357, 39561}, {3520, 15041}, {3524, 44324}, {3529, 16981}, {3530, 7998}, {3543, 13321}, {3567, 3853}, {3581, 7555}, {3628, 12111}, {3830, 13451}, {3845, 5946}, {3850, 15043}, {3856, 11439}, {3857, 15012}, {3858, 5462}, {3861, 12290}, {3917, 17504}, {5050, 7526}, {5066, 15305}, {5085, 12163}, {5093, 12085}, {5254, 15544}, {5562, 15712}, {5609, 43586}, {5650, 11591}, {5656, 44275}, {5891, 11539}, {5892, 10219}, {5943, 38071}, {6243, 12103}, {6247, 33332}, {6636, 32608}, {6644, 8780}, {7527, 10620}, {7545, 12112}, {7575, 11438}, {7592, 11250}, {7706, 34514}, {7722, 37118}, {9781, 12102}, {9786, 37440}, {10095, 11381}, {10192, 16532}, {10540, 15053}, {10605, 18570}, {10628, 23328}, {10937, 11188}, {11412, 33923}, {11430, 12041}, {11433, 44276}, {11440, 13353}, {11444, 12108}, {11451, 11737}, {11454, 14805}, {11455, 14893}, {11456, 12106}, {11464, 18571}, {11465, 44904}, {11563, 13567}, {12086, 14627}, {12100, 20791}, {12105, 32124}, {12161, 37497}, {12174, 13861}, {12308, 15052}, {12811, 15024}, {12812, 15028}, {13352, 37950}, {13364, 16194}, {13391, 14831}, {13406, 26879}, {13434, 43806}, {13596, 15038}, {14094, 43584}, {14641, 16625}, {15058, 35018}, {15061, 34331}, {15062, 43600}, {15068, 37475}, {15579, 44494}, {16776, 39884}, {17714, 37490}, {17855, 38898}, {18350, 43597}, {18390, 44283}, {18445, 40111}, {18560, 43575}, {18564, 34796}, {18580, 18931}, {18912, 44279}, {18914, 45731}, {20424, 23335}, {22467, 32609}, {22584, 40685}, {22948, 44755}, {23300, 38136}, {31728, 34773}, {31829, 32358}, {31978, 44544}, {32139, 35259}, {32142, 45187}, {32423, 38323}, {32620, 37514}, {35265, 45735}, {35498, 38633}, {36979, 42088}, {36981, 42087}, {37484, 44245}, {39571, 44271}, {41715, 44804}, {43394, 43604}, {43598, 43603}, {43599, 43837}, {43605, 43809}, {43903, 43904}

X(45956) = midpoint of X(i) and X(j) for these {i, j}: {185, 9730}, {568, 15072}, {5889, 13340}, {11459, 34783}, {14831, 14855}, {18564, 34796}
X(45956) = reflection of X(i) in X(j) for these (i, j): (5, 9730), (2979, 34200), (3830, 13451), (3845, 5946), (5876, 10170), (9730, 13630), (10170, 9729), (11455, 14893), (11459, 140), (13340, 548), (15030, 13363), (15060, 5892), (15067, 16836), (15305, 5066), (15686, 14855), (15687, 51), (16194, 13364), (18435, 547), (23039, 12100), (23046, 16226), (39884, 16776)
X(45956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (49, 43601, 43615), (185, 13630, 5), (373, 9730, 12006), (389, 13491, 3627), (5876, 9729, 632), (5890, 15072, 568), (5892, 15060, 15699), (6102, 40647, 550), (6241, 37481, 546), (9730, 12162, 373), (9730, 15030, 13363), (10574, 11459, 40280), (10574, 34783, 140), (10620, 15037, 7527), (11459, 40280, 140), (12006, 12162, 5), (13363, 15030, 5), (13364, 16194, 23046), (13382, 40647, 6102), (15043, 18439, 3850), (15045, 18435, 547), (15067, 16836, 549), (16194, 16226, 13364), (20791, 23039, 12100), (34783, 40280, 11459), (43395, 43396, 20379), (43601, 43602, 49), (43602, 43611, 43601), (43611, 43906, 43602), (43807, 43845, 3520)

X(45957) = X(5)X(113)∩X(49)X(74)

Barycentrics    a^2*(2*(b^2+c^2)*a^6-2*(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(6*b^4-11*b^2*c^2+6*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :
X(45957) = 3*X(3)-2*X(31834), 7*X(3)-6*X(44324), 7*X(4)-9*X(13321), 5*X(4)-6*X(13451), 3*X(4)-4*X(16881), 17*X(5)-18*X(373), 5*X(5)-6*X(9730), 17*X(5)-16*X(11017), 7*X(5)-8*X(12006), 3*X(5)-2*X(12162), 11*X(5)-12*X(13363), 3*X(5)-4*X(13630), 7*X(5)-6*X(15030), 15*X(5)-16*X(32205), 17*X(185)-9*X(373), 5*X(185)-3*X(9730), 17*X(185)-8*X(11017), 7*X(185)-4*X(12006), 3*X(185)-X(12162), 11*X(185)-6*X(13363), 3*X(185)-2*X(13630), 7*X(185)-3*X(15030), 15*X(185)-8*X(32205), 15*X(13321)-14*X(13451), 9*X(13451)-10*X(16881), 7*X(31834)-9*X(44324)

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45957) lies on these lines: {3, 9544}, {4, 13321}, {5, 113}, {26, 12174}, {30, 5889}, {49, 74}, {51, 32137}, {54, 15054}, {64, 12161}, {110, 43615}, {140, 12111}, {143, 11381}, {146, 43808}, {156, 1204}, {184, 32138}, {195, 12086}, {389, 3845}, {399, 22467}, {541, 13403}, {542, 40929}, {546, 5890}, {547, 15058}, {548, 15072}, {549, 5876}, {550, 6101}, {567, 15062}, {568, 3853}, {632, 5907}, {1147, 34152}, {1154, 10575}, {1181, 18570}, {1353, 34146}, {1425, 32168}, {1483, 2807}, {1498, 37440}, {1511, 43604}, {1614, 15331}, {1658, 11456}, {1885, 43588}, {2777, 10116}, {2883, 11563}, {2979, 44245}, {3146, 14449}, {3270, 32143}, {3520, 10620}, {3521, 11564}, {3530, 11459}, {3567, 3861}, {3627, 5446}, {3628, 10574}, {3850, 15305}, {3856, 5640}, {3857, 5462}, {3858, 5946}, {3859, 16261}, {5066, 15043}, {5447, 45759}, {5498, 12281}, {5562, 8703}, {5878, 44271}, {5891, 14869}, {5894, 10628}, {6225, 18951}, {6288, 43895}, {6293, 13292}, {6759, 7575}, {6823, 21230}, {7502, 12163}, {7527, 43845}, {7530, 12315}, {7722, 18560}, {8550, 43392}, {9704, 35473}, {9705, 15055}, {9729, 15060}, {9781, 14893}, {10095, 16194}, {10109, 15028}, {10263, 14915}, {10605, 32139}, {10625, 15686}, {10627, 14855}, {10733, 43599}, {11250, 18445}, {11412, 12103}, {11444, 12100}, {11449, 37968}, {11455, 12102}, {11457, 18377}, {11539, 14128}, {11591, 15712}, {12038, 12041}, {12088, 32608}, {12108, 20791}, {12112, 18378}, {12233, 33332}, {12270, 14516}, {12308, 43809}, {12316, 35001}, {12317, 34007}, {12370, 15311}, {12412, 43905}, {12606, 44249}, {13364, 41991}, {13367, 32210}, {13434, 43596}, {13445, 37495}, {13568, 38322}, {13596, 14627}, {13598, 35404}, {14094, 18350}, {14130, 15032}, {14216, 44288}, {14641, 19710}, {14677, 17854}, {14831, 33699}, {14865, 15087}, {15026, 38071}, {15041, 35497}, {15045, 35018}, {15056, 16239}, {15067, 44682}, {15101, 23328}, {15806, 37118}, {16226, 18874}, {17504, 32142}, {17853, 43907}, {17855, 30507}, {17856, 34584}, {18400, 34798}, {18565, 34799}, {18567, 25739}, {20126, 34331}, {20417, 43839}, {21659, 45732}, {21663, 32171}, {22660, 37938}, {22802, 44283}, {22949, 34199}, {23039, 33923}, {25738, 44279}, {26879, 44235}, {32140, 44263}, {34153, 44240}, {40928, 40932}, {41587, 43893}, {43584, 43611}

X(45957) = midpoint of X(i) and X(j) for these {i, j}: {6241, 34783}, {6243, 12279}, {18565, 34799}
X(45957) = reflection of X(i) in X(j) for these (i, j): (5, 185), (550, 13491), (1885, 43588), (3146, 14449), (3627, 6102), (5876, 40647), (11381, 143), (11412, 12103), (12111, 140), (12162, 13630), (12290, 3853), (14677, 17854), (15704, 10575), (18436, 548), (18439, 546), (21659, 45732), (33699, 14831), (44544, 32392), (45187, 10627)
X(45957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (49, 74, 10226), (143, 11381, 15687), (156, 1204, 15646), (185, 12162, 13630), (373, 11017, 5), (399, 43807, 22467), (568, 12290, 3853), (5876, 40647, 549), (5890, 18439, 546), (6225, 18951, 44276), (10574, 18435, 3628), (10605, 32139, 37814), (12006, 15030, 5), (12162, 13630, 5), (13561, 43831, 5), (13630, 32205, 9730), (14094, 43601, 18350), (14627, 33541, 13596), (14855, 45187, 10627), (15026, 44870, 38071), (15056, 40280, 16239), (15062, 43602, 567), (15072, 18436, 548), (15305, 37481, 3850), (16003, 43831, 13561), (43395, 43396, 20304)

X(45958) = X(5)X(113)∩X(30)X(5447)

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4-b^2*c^2+3*c^4)*a^2-(b^2-c^2)^2*(b^4+7*b^2*c^2+c^4)) : :
X(45958) = 3*X(3)+5*X(11439), 5*X(3)+3*X(11455), 3*X(4)+X(6101), X(4)+3*X(15060), 5*X(4)+3*X(23039), 7*X(4)+X(37484), 5*X(5)-X(185), 13*X(5)-9*X(373), 7*X(5)-3*X(9730), 5*X(5)-4*X(12046), 3*X(5)+X(12162), 5*X(5)-3*X(13363), 3*X(5)-X(13630), X(5)+3*X(15030), 3*X(5)-2*X(32205), X(6101)-3*X(11591), X(6101)-9*X(15060), 5*X(6101)-9*X(23039), 7*X(6101)-3*X(37484), 5*X(11439)-3*X(32137), 3*X(11455)-5*X(32137), X(11591)-3*X(15060), 5*X(11591)-3*X(23039), 7*X(11591)-X(37484), 5*X(15060)-X(23039)

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45958) lies on these lines: {3, 11439}, {4, 2889}, {5, 113}, {30, 5447}, {49, 15052}, {51, 3857}, {52, 3858}, {54, 5609}, {143, 381}, {156, 9818}, {382, 15056}, {389, 5066}, {399, 13434}, {511, 3861}, {542, 43575}, {546, 1154}, {547, 40647}, {548, 10170}, {549, 11381}, {550, 11592}, {568, 3855}, {632, 10575}, {1209, 11563}, {1216, 3853}, {1511, 14130}, {1539, 34007}, {1568, 33332}, {1656, 13491}, {1658, 4550}, {2979, 5076}, {3090, 18439}, {3091, 6102}, {3520, 12133}, {3526, 12290}, {3530, 14915}, {3545, 15026}, {3627, 5891}, {3628, 6000}, {3818, 18377}, {3819, 12103}, {3830, 11444}, {3832, 18436}, {3839, 6243}, {3843, 10263}, {3845, 5562}, {3850, 10095}, {3851, 5946}, {3856, 10110}, {3860, 44863}, {5054, 12279}, {5055, 6241}, {5068, 37481}, {5070, 15072}, {5072, 5890}, {5073, 7999}, {5079, 10574}, {5462, 12811}, {5892, 12812}, {6143, 12292}, {6288, 10113}, {6642, 32138}, {6644, 32210}, {7387, 33533}, {7486, 40280}, {7514, 33537}, {7526, 32171}, {7527, 18350}, {7998, 17800}, {9729, 35018}, {10024, 13565}, {10109, 11695}, {10124, 17704}, {10226, 43586}, {10540, 10610}, {10620, 22462}, {10625, 15687}, {10628, 20584}, {11479, 32139}, {12041, 15062}, {12100, 14641}, {12102, 40247}, {12270, 15046}, {12308, 15047}, {12606, 18403}, {12825, 13358}, {13340, 17578}, {13470, 34664}, {13568, 23410}, {13598, 14893}, {14094, 43845}, {14157, 34864}, {14855, 14869}, {14984, 18553}, {15012, 41989}, {15043, 19709}, {15101, 38789}, {15647, 32401}, {15704, 32062}, {15800, 37349}, {15806, 16534}, {15807, 44665}, {16042, 38626}, {16252, 20376}, {17814, 31861}, {18451, 32046}, {18488, 37938}, {18537, 32140}, {21243, 44235}, {22831, 44869}, {22832, 44868}, {23236, 43818}, {24981, 36966}, {30507, 38727}, {32143, 37696}, {32168, 37697}, {32608, 38848}, {38632, 43605}, {41991, 45187}, {43584, 43807}

X(45958) = midpoint of X(i) and X(j) for these {i, j}: {3, 32137}, {4, 11591}, {143, 5876}, {546, 5907}, {548, 13474}, {1216, 3853}, {3627, 10627}, {5446, 31834}, {12162, 13630}, {12825, 13358}, {13364, 18435}
X(45958) = reflection of X(i) in X(j) for these (i, j): (5, 11017), (389, 18874), (550, 11592), (5462, 12811), (9729, 35018), (10095, 3850), (10110, 3856), (12006, 5), (13630, 32205), (32142, 14128)
X(45958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3, 33539, 43613), (4, 15060, 11591), (5, 185, 13363), (5, 12162, 13630), (5, 13363, 12046), (5, 13630, 32205), (185, 12046, 12006), (381, 5876, 143), (381, 15058, 5876), (382, 15056, 15067), (389, 5066, 18874), (546, 31834, 5446), (1656, 15305, 13491), (3091, 6102, 13364), (3091, 18435, 6102), (3545, 34783, 15026), (3627, 5891, 10627), (3843, 11459, 10263), (3851, 12111, 5946), (5446, 5907, 31834), (7527, 18350, 43394), (10170, 13474, 548), (10540, 35500, 10610), (10620, 22462, 43597), (12308, 15047, 43602), (13630, 32205, 12006), (14130, 43598, 1511), (15056, 16261, 382), (15062, 43809, 12041), (43613, 43614, 3)

X(45959) = X(4)X(93)∩X(5)X(113)

Barycentrics    a^2*((b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+3*(b^6+c^6)*a^2-(b^4+5*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(45959) = 3*X(2)+X(18439), 5*X(3)-X(12279), 3*X(3)+X(12290), 3*X(3)-7*X(15056), X(3)-5*X(15058), X(3)-3*X(15060), X(3)+3*X(15305), 3*X(12279)+5*X(12290), X(12279)-10*X(14128), X(12279)-15*X(15060), X(12279)+15*X(15305), X(12290)+6*X(14128), X(12290)+7*X(15056), X(12290)+15*X(15058), X(12290)+9*X(15060), X(12290)-9*X(15305), 6*X(14128)-7*X(15056), 2*X(14128)-5*X(15058), 2*X(14128)-3*X(15060), 2*X(14128)+3*X(15305)

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45959) lies on these lines: {2, 13491}, {3, 6030}, {4, 93}, {5, 113}, {20, 15067}, {30, 1216}, {49, 5609}, {51, 3858}, {52, 3845}, {54, 399}, {74, 43614}, {110, 14130}, {140, 6000}, {143, 546}, {146, 3521}, {155, 31861}, {156, 7526}, {265, 43865}, {381, 3567}, {382, 6101}, {389, 3850}, {403, 34826}, {511, 3853}, {547, 9729}, {548, 11793}, {549, 10575}, {550, 5891}, {567, 43605}, {568, 3832}, {1493, 15033}, {1498, 7514}, {1503, 13470}, {1511, 3520}, {1539, 7723}, {1568, 18488}, {1593, 15068}, {1597, 16266}, {1614, 10610}, {1656, 6241}, {1657, 11444}, {1986, 43823}, {2772, 5885}, {2781, 18553}, {2807, 18357}, {2979, 5073}, {3091, 5946}, {3146, 23039}, {3448, 43821}, {3526, 15072}, {3530, 10170}, {3534, 7999}, {3543, 37484}, {3545, 37481}, {3581, 34484}, {3627, 5562}, {3628, 40647}, {3819, 14641}, {3830, 11412}, {3843, 5889}, {3851, 5890}, {3856, 16881}, {3861, 5446}, {3917, 15704}, {4550, 6759}, {4846, 16623}, {5055, 10574}, {5066, 5462}, {5067, 40280}, {5072, 15043}, {5079, 15045}, {5447, 12103}, {5448, 39504}, {5449, 44235}, {5498, 5972}, {5621, 43811}, {5650, 44682}, {5892, 35018}, {5899, 7691}, {5943, 12811}, {5944, 10540}, {6143, 14643}, {6146, 45734}, {6644, 32138}, {6688, 44904}, {7393, 12315}, {7502, 26883}, {7687, 13358}, {7689, 12106}, {7699, 45014}, {7727, 43819}, {7728, 34007}, {7998, 15696}, {8703, 11592}, {9306, 11250}, {9818, 19347}, {9820, 44236}, {9927, 22816}, {10113, 12825}, {10116, 43575}, {10272, 43839}, {10539, 18570}, {10620, 43601}, {10625, 32062}, {11264, 12241}, {11440, 45735}, {11472, 12084}, {11565, 34224}, {11572, 18572}, {11577, 21659}, {11695, 12812}, {11699, 43822}, {11801, 44686}, {11802, 20584}, {12041, 22467}, {12102, 13598}, {12112, 37126}, {12133, 18560}, {12134, 30522}, {12163, 13861}, {12164, 39522}, {12281, 38789}, {12307, 15107}, {12308, 43845}, {12316, 18551}, {12317, 43816}, {12358, 34584}, {12363, 18564}, {12370, 15807}, {13142, 44804}, {13340, 33703}, {13348, 44324}, {13403, 32423}, {13406, 21243}, {13416, 44240}, {13434, 14094}, {13488, 31831}, {13565, 14076}, {13596, 37495}, {14449, 14893}, {14683, 43818}, {14831, 23046}, {14855, 15712}, {14865, 22115}, {15024, 19709}, {15032, 36153}, {15035, 35498}, {15037, 43602}, {15054, 43597}, {15073, 18440}, {15132, 43894}, {15687, 45186}, {16010, 43810}, {16160, 39271}, {16239, 16836}, {16657, 32358}, {16982, 45187}, {17854, 34128}, {17855, 40685}, {18356, 18390}, {18358, 34146}, {18379, 18474}, {18404, 34514}, {18445, 32136}, {18537, 18952}, {19470, 43820}, {20191, 44234}, {21230, 43893}, {22462, 43584}, {22584, 38898}, {28146, 31752}, {28160, 31751}, {31728, 38140}, {31738, 33697}, {31850, 31864}, {32143, 37729}, {32210, 37814}, {33534, 33543}, {34553, 42278}, {34555, 42279}, {34798, 38321}, {35450, 36983}, {40930, 43846}, {43586, 43615}, {44158, 44232}

X(45959) = midpoint of X(i) and X(j) for these {i, j}: {4, 5876}, {5, 12162}, {146, 15101}, {382, 6101}, {550, 11381}, {1216, 13474}, {1511, 12292}, {1539, 7723}, {3627, 5562}, {3853, 31834}, {6102, 12111}, {10113, 12825}, {10263, 18436}, {11591, 32137}, {13488, 31831}, {13491, 18439}, {15060, 15305}, {22584, 38898}, {31738, 33697}, {32196, 41726}
X(45959) = reflection of X(i) in X(j) for these (i, j): (3, 14128), (143, 546), (185, 12006), (389, 3850), (546, 44870), (548, 11793), (550, 32142), (974, 15088), (5446, 3861), (5447, 40247), (6102, 10095), (10116, 43575), (10627, 11591), (11264, 12241), (11591, 5907), (11802, 20584), (12006, 11017), (12103, 5447), (12370, 15807), (13358, 7687), (13598, 12102), (13630, 5), (14641, 33923), (16625, 44863), (16881, 3856), (17855, 40685), (34224, 11565), (40647, 3628), (43846, 40930)
X(45959) = complement of X(13491)
X(45959) = complementary conjugate of the complement of X(13489)
X(45959) = X(5)-of-X(4)-Brocard-triangle
X(45959) = X(20)-of-X(5)-altimedial-triangle
X(45959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 18439, 13491), (3, 15058, 15060), (3, 15060, 14128), (3, 33541, 13445), (4, 18435, 5876), (4, 18436, 10263), (5, 185, 12006), (5, 9730, 32205), (5, 10264, 43817), (5, 13561, 20304), (5, 13630, 13363), (74, 43614, 43809), (110, 14130, 43394), (110, 43613, 14130), (185, 12006, 13630), (381, 12111, 6102), (546, 13451, 44863), (3520, 18350, 1511), (5562, 16194, 3627), (5876, 10263, 18436), (5889, 16261, 3843), (5891, 11381, 550), (5907, 13474, 1216), (5907, 32137, 10627), (7526, 18451, 156), (10540, 14118, 5944), (11017, 12006, 5), (11439, 11459, 382), (11444, 11455, 1657), (12162, 15030, 5), (12290, 15056, 3), (12290, 15058, 15056), (13451, 16625, 143), (15056, 15305, 12290), (15058, 15305, 3), (15062, 43598, 3), (16625, 44863, 13451)

X(45960) = X(51)X(974)∩X(133)X(389)

Barycentrics    a^2*((b^2+c^2)*a^14-2*(2*b^4+b^2*c^2+2*c^4)*a^12+3*(b^2+c^2)*(b^4+c^4)*a^10+2*(5*b^8+5*c^8-4*b^2*c^2*(2*b^4-b^2*c^2+2*c^4))*a^8-(b^4-c^4)*(b^2-c^2)*(5*b^2-8*b*c+5*c^2)*(5*b^2+8*b*c+5*c^2)*a^6+2*(b^2-c^2)^2*(12*b^8+12*c^8+b^2*c^2*(3*b^4-2*b^2*c^2+3*c^4))*a^4-(b^4-c^4)*(b^2-c^2)^3*(11*b^4+12*b^2*c^2+11*c^4)*a^2+2*(b^2-c^2)^4*(b^8+c^8+2*(b^4+c^4)*b^2*c^2))*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :
Barycentrics    (SB+SC)*(S^2-3*SB*SC)*(S^8-3*SA*SW*S^6+(9*SB*SC+SW^2)*SA^2*S^4-(SA^2+4*SB*SC-SW^2)*SA^4*S^2-3*(SB+SC)^2*SA^6) : :

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45960) lies on these lines: {51, 974}, {133, 389}

X(45961) = X(24)X(122)∩X(25)X(2929)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*(a^14-2*(b^2+c^2)*a^12-2*(2*b^2-c^2)*(b^2-2*c^2)*a^10+15*(b^4-c^4)*(b^2-c^2)*a^8-(b^2-c^2)^2*(15*b^4+26*b^2*c^2+15*c^4)*a^6+4*(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4)*a^4+2*(b^6-c^6)*(b^2-c^2)^3*a^2-(b^4-c^4)^3*(b^2-c^2)) : :
Barycentrics    SB*SC*(S^2-2*SB*SC)*((8*R^2-SW)*S^2+(SB*SC+5*SW^2-SA^2+4*(3*SA-11*SW)*R^2+64*R^4)*SA) : :

See Antreas Hatzipolakis and César Lozada, euclid 3067.

X(45961) lies on these lines: {4, 11589}, {24, 122}, {25, 2929}, {1093, 6525}, {15427, 16655}

X(45962) = X(2)X(6)∩X(4)X(16747)

Barycentrics    a^4 - b^4 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 - 2*b^2*c^2 - c^4 : :

X(45962) lies on the cubic K1244 and these lines: {2, 6}, {4, 16747}, {7, 4388}, {8, 304}, {21, 3926}, {22, 1444}, {75, 1370}, {76, 2478}, {261, 34254}, {274, 315}, {305, 314}, {310, 37193}, {319, 10327}, {332, 37090}, {348, 4296}, {404, 3785}, {405, 3933}, {442, 7776}, {452, 32830}, {474, 7767}, {497, 4441}, {612, 3879}, {614, 4357}, {1078, 6921}, {1479, 20888}, {1655, 33029}, {1909, 3436}, {1975, 6872}, {2263, 3883}, {2475, 32006}, {2476, 32816}, {2550, 20553}, {2893, 26052}, {2896, 27318}, {2897, 7396}, {3290, 4643}, {3664, 32946}, {3757, 7179}, {4189, 6337}, {4190, 7750}, {4193, 32828}, {4352, 26117}, {4360, 19993}, {4362, 24241}, {4416, 40131}, {5084, 18135}, {5271, 18651}, {5272, 17272}, {5277, 14023}, {5283, 7758}, {5286, 17550}, {6390, 16370}, {6604, 17152}, {6646, 26274}, {6856, 32823}, {6857, 32818}, {6871, 7773}, {6910, 7763}, {6919, 32834}, {6931, 32832}, {6933, 7752}, {7191, 17321}, {7762, 11321}, {7768, 37462}, {7855, 16589}, {7879, 17670}, {7882, 36812}, {7893, 16917}, {7900, 33030}, {7906, 33047}, {7929, 17565}, {7941, 33045}, {8024, 44140}, {9464, 31106}, {9599, 21264}, {10446, 26118}, {11106, 32840}, {11111, 32817}, {11114, 32815}, {13725, 16705}, {14376, 28722}, {16849, 41014}, {16913, 20088}, {16915, 20065}, {17129, 33046}, {17137, 39581}, {17170, 20911}, {17206, 19310}, {17211, 24159}, {17257, 26242}, {17275, 30748}, {17314, 31087}, {17377, 20020}, {17577, 32827}, {17676, 18600}, {17685, 20081}, {18133, 41916}, {20539, 20628}, {23115, 28432}, {24165, 33869}, {25590, 33109}, {25907, 41005}, {25947, 41008}, {26035, 26099}, {26101, 29960}, {26227, 33864}, {31130, 33090}, {31156, 32833}, {33824, 40908}, {39998, 44147}, {40022, 44139}

X(45962) = anticomplement of X(5275)
X(45962) = isotomic conjugate of the isogonal conjugate of X(36740)
X(45962) = barycentric product X(i)*X(j) for these {i,j}: {76, 36740}, {305, 45786}
X(45962) = barycentric quotient X(i)/X(j) for these {i,j}: {36740, 6}, {45786, 25}
X(45962) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 193, 5276}, {2, 17002, 7735}, {69, 14548, 30941}, {274, 315, 377}, {319, 30758, 10327}, {325, 16992, 2}, {3314, 17000, 2}, {31130, 33090, 42696}

X(45963) = X(1)X(185)∩X(4)X(7)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + a^3*b*c - a*b^3*c + a^3*c^2 - b^3*c^2 - a^2*c^3 - a*b*c^3 - b^2*c^3 - a*c^4 + c^5) : :

X(45963) lies on the cubic K1244 and these lines: {1, 185}, {4, 7}, {6, 41}, {25, 222}, {33, 39796}, {34, 39791}, {51, 223}, {57, 851}, {65, 4331}, {77, 511}, {208, 30493}, {226, 1985}, {241, 4259}, {307, 10477}, {581, 1410}, {603, 3145}, {611, 1037}, {651, 4223}, {1011, 40152}, {1214, 26893}, {1362, 40131}, {1423, 13724}, {1435, 37993}, {1445, 4260}, {1463, 28074}, {1745, 3338}, {1804, 37474}, {1837, 20617}, {1854, 2654}, {2003, 5320}, {2263, 21746}, {2635, 4860}, {3142, 15844}, {3173, 26885}, {3660, 37367}, {3784, 4220}, {3868, 17950}, {5208, 30975}, {5732, 22440}, {7177, 14520}, {7248, 36570}, {11435, 37755}, {11436, 20277}, {11573, 37320}, {14524, 43215}, {17077, 37154}, {22277, 41712}, {23144, 24320}, {25466, 25964}, {25557, 39063}, {26927, 37195}, {36059, 44117}, {37194, 41344}, {37225, 37523}

X(45963) = barycentric product X(1446)*X(36020)
X(45963) = barycentric quotient X(36020)/X(2287)
X(45963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {222, 20122, 26892}, {1439, 5728, 1876}, {2067, 6502, 1951}

X(45964) = X(2)X(4259)∩X(10)X(17451)

Barycentrics    (a^2*b^2 - b^4 + a^2*b*c + a*b^2*c + 2*a^2*c^2 + a*b*c^2 + b^2*c^2)*(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 - c^4) : :

X(45964) lies on the Kiepert circumhyperbola, the cubic K1244, and these lines: {2, 4259}, {10, 17451}, {21, 83}, {38, 226}, {76, 2476}, {98, 5276}, {321, 2886}, {427, 40149}, {598, 11114}, {671, 17577}, {672, 4672}, {1446, 3665}, {2051, 8229}, {2276, 13576}, {2996, 6871}, {4049, 44429}, {4080, 10129}, {5395, 6872}, {5397, 6998}, {6539, 31079}, {6856, 18840}, {6857, 18841}, {7466, 40395}, {11111, 18842}, {12047, 40515}, {13478, 18169}, {25557, 30588}, {37149, 43531}, {40016, 40072}

X(45964) = isogonal conjugate of X(5135)
X(45964) = isotomic conjugate of X(37670)
X(45964) = isotomic conjugate of the anticomplement of X(37661)
X(45964) = X(37661)-cross conjugate of X(2)
X(45964) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5135}, {31, 37670}
X(45964) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 37670}, {6, 5135}

X(45965) = X(2)X(3781)∩X(3)X(14621)

Barycentrics    (a^3*b^2 - a*b^4 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c + 2*a^3*c^2 + 2*a^2*b*c^2 + a*b^2*c^2 + 2*a^2*c^3 + a*b*c^3 + b^2*c^3)*(2*a^3*b^2 + 2*a^2*b^3 + a^3*b*c + 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 - a*c^4 - b*c^4) : :

X(45965) lies on the conic {{A,B,C,X(2),X(7)}}, the cubic K1244, and these lines: {2, 3781}, {3, 14621}, {27, 3736}, {75, 29967}, {86, 4269}, {272, 27644}, {273, 7146}, {304, 871}, {310, 30985}, {1246, 4261}, {6384, 18134}, {15467, 30545}, {17241, 40010}, {24310, 40418}

X(45966) = X(2)X(3786)∩X(6)X(33718)

Barycentrics    a^2*(a^2*b^2 - b^4 + 2*a^2*b*c + 2*a*b^2*c + 2*a^2*c^2 + 2*a*b*c^2 + b^2*c^2)*(2*a^2*b^2 + 2*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2 - c^4) : :

X(45966) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1244, and these lines: {2, 3786}, {6, 33718}, {25, 1185}, {37, 3779}, {42, 5364}, {69, 1218}, {251, 5320}, {308, 44140}, {869, 1400}, {967, 20845}, {1427, 1469}, {1974, 40570}, {4259, 39957}, {8770, 21779}, {19767, 39971}

X(45966) = isogonal conjugate of X(16992)
X(45966) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16992}, {63, 11341}, {75, 5138}
X(45966) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 16992}, {25, 11341}, {32, 5138}

X(45967) = X(5)X(1199)∩X(6)X(2072)

Barycentrics    2*a^10-7*(b^2+c^2)*a^8+2*(5*b^4+3*b^2*c^2+5*c^4)*a^6-2*(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4+4*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(45967) = 5*X(3)+4*X(13142), X(3)-4*X(45298), 5*X(5)+4*X(32165), 4*X(51)-X(7540), X(51)+2*X(43573), X(381)+2*X(11245), 8*X(389)+X(18563), 5*X(1656)+4*X(13292), X(1657)+2*X(44935), 7*X(3090)+2*X(32358), 5*X(3091)+4*X(43588), 5*X(3843)+4*X(18914), 13*X(5079)-4*X(31831), 8*X(5462)+X(44076), X(5891)+2*X(11225), 2*X(5946)+X(12022), 4*X(5946)-X(38321), X(7540)+8*X(43573), 2*X(12022)+X(38321), X(13142)+5*X(45298)

See Antreas Hatzipolakis and César Lozada, euclid 3076.

X(45967) lies on these lines: 3, 11433}, {5, 1199}, {6, 2072}, {30, 11002}, {51, 7540}, {235, 43845}, {373, 539}, {381, 11245}, {389, 18563}, {427, 15038}, {542, 14845}, {567, 13567}, {1173, 43837}, {1656, 13292}, {1657, 44935}, {3090, 32358}, {3091, 43588}, {3431, 16531}, {3518, 43838}, {3564, 5055}, {3627, 43835}, {3843, 18914}, {5079, 31831}, {5422, 37347}, {5462, 44076}, {5576, 18912}, {5891, 11225}, {5946, 12022}, {6240, 43575}, {6640, 11426}, {6677, 9703}, {7399, 15047}, {7514, 37644}, {8550, 10540}, {9730, 32068}, {9777, 31723}, {10024, 36753}, {10095, 34224}, {10575, 40240}, {11432, 18404}, {11585, 14627}, {12225, 16881}, {12233, 43821}, {12241, 37481}, {12370, 15043}, {12834, 41171}, {13353, 41587}, {13371, 43816}, {13451, 34603}, {13621, 31804}, {14516, 15026}, {15019, 25739}, {15037, 15760}, {15067, 41628}, {16226, 17702}, {18911, 39522}, {22115, 37648}, {25321, 38724}, {36749, 37452}, {37505, 43817}, {43595, 43809}

X(45967) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5946, 12022, 38321), (36753, 39571, 10024)

X(45968) = X(5)X(1199)∩X(6)X(5133)

Barycentrics    2*a^6-3*(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) : :
X(45968) = 3*X(2)-4*X(45298), X(3)+2*X(32358), X(3)-4*X(43588), X(4)-4*X(13292), X(5)-4*X(32165), X(20)-4*X(18914), X(52)+2*X(10116), 2*X(52)+X(34224), X(68)-4*X(32166), X(185)+2*X(10112), X(193)+2*X(26926), 3*X(373)-4*X(32068), 4*X(389)-X(14516), 2*X(428)-3*X(11002), X(1986)+2*X(10111), 7*X(3090)-4*X(31831), X(3146)-4*X(13142), 4*X(10116)-X(34224), 3*X(11245)-2*X(45298), X(32358)+2*X(43588)

See Antreas Hatzipolakis and César Lozada, euclid 3076.

Let A'B'C' be the orthic triangle. Let A" be the reflection of A in B'C', and define B" and C" cyclically. X(45968) is the centroid of A"B"C". (Randy Hutson, November 30, 2021)

Let A'B'C' be the reflection triangle. Let AB = reflection of A' in CA, and define BC and CA cyclically. Let AC = reflection of A' in AB, and define BA and CB cyclically. Then X(45968) is the centroid of ABACBCBACACB. (Randy Hutson, November 30, 2021)

X(45968) lies on these lines: {2, 3167}, {3, 32358}, {4, 13292}, {5, 1199}, {6, 5133}, {20, 18914}, {22, 6515}, {23, 41588}, {24, 18951}, {25, 37644}, {30, 34796}, {49, 10018}, {51, 542}, {52, 10116}, {54, 12359}, {68, 7592}, {69, 7485}, {97, 26905}, {110, 13567}, {125, 34986}, {140, 9545}, {154, 7426}, {155, 18912}, {182, 37636}, {184, 3580}, {185, 10112}, {193, 1370}, {195, 13371}, {235, 43605}, {323, 1368}, {343, 5012}, {373, 32068}, {378, 18917}, {389, 14516}, {394, 18911}, {403, 18445}, {427, 1353}, {428, 11002}, {467, 41204}, {468, 9544}, {511, 41628}, {524, 2979}, {539, 9730}, {550, 32608}, {568, 7576}, {576, 11550}, {858, 1899}, {1147, 26879}, {1351, 7391}, {1352, 5422}, {1493, 13561}, {1503, 3060}, {1591, 5874}, {1592, 5875}, {1594, 12161}, {1614, 41587}, {1986, 10111}, {1992, 11216}, {1995, 11433}, {2888, 7399}, {2895, 7465}, {3090, 31831}, {3146, 13142}, {3519, 37471}, {3520, 43595}, {3567, 12134}, {3575, 34799}, {3618, 7571}, {3630, 41462}, {3681, 5849}, {3818, 15004}, {3845, 38789}, {3873, 5848}, {3917, 5965}, {3920, 39897}, {5064, 5093}, {5159, 9716}, {5446, 16659}, {5576, 18356}, {5889, 6146}, {5890, 38323}, {5891, 43573}, {5921, 6997}, {6102, 6240}, {6193, 12421}, {6636, 37779}, {6676, 11003}, {6995, 39871}, {7191, 39873}, {7378, 11405}, {7394, 9777}, {7484, 11898}, {7488, 31804}, {7496, 15108}, {7500, 39874}, {7503, 11411}, {7544, 11432}, {7667, 34380}, {7762, 14957}, {8584, 41720}, {9140, 15131}, {9143, 44212}, {9637, 26956}, {9703, 44452}, {9704, 10020}, {9820, 26917}, {10114, 21649}, {10169, 25320}, {10263, 45732}, {10601, 15069}, {10605, 16386}, {10625, 18128}, {10691, 33884}, {11004, 31074}, {11064, 26913}, {11179, 43653}, {11232, 12022}, {11413, 18909}, {11422, 23292}, {11427, 31236}, {11441, 39571}, {11457, 36747}, {11585, 43808}, {11812, 44751}, {12007, 37649}, {12111, 12241}, {12160, 37444}, {12163, 34005}, {12278, 13568}, {12317, 13596}, {12370, 18560}, {12429, 22663}, {13321, 13490}, {13366, 14389}, {13428, 45407}, {13439, 45406}, {13595, 14683}, {14531, 44829}, {14788, 36753}, {14831, 18400}, {15018, 37439}, {15032, 15760}, {15140, 25328}, {15305, 16657}, {15331, 36966}, {15559, 32140}, {17809, 37638}, {18583, 37353}, {18925, 38444}, {19130, 34565}, {19161, 27365}, {19481, 21650}, {21969, 29012}, {22467, 32334}, {23291, 30744}, {26881, 32269}, {32136, 34826}, {32223, 44110}, {32225, 44108}, {32423, 38321}, {33586, 37900}, {34507, 43650}, {37349, 39884}, {41597, 43817}, {44210, 44555}

X(45968) = midpoint of X(52) and X(45730)
X(45968) = reflection of X(i) in X(j) for these (i, j): (2, 11245), (51, 11225), (3146, 44935), (5891, 43573), (7576, 568), (15305, 16657), (34224, 45730), (34603, 3060), (38323, 5890), (44935, 13142), (45730, 10116)
X(45968) = anticomplement of the anticomplement of X(45298)
X(45968) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (6, 11442, 5133), (52, 10116, 34224), (68, 7592, 13160), (343, 5012, 7495), (343, 8550, 5012), (427, 1353, 1994), (1352, 5422, 37990), (1899, 1993, 858), (1994, 3448, 427), (3167, 26869, 2), (3410, 34545, 5), (5012, 41724, 343), (5889, 6146, 12225), (6102, 11264, 44076), (6102, 44076, 6240), (6193, 18916, 17928), (6515, 6776, 22), (8550, 41724, 7495), (9777, 18440, 7394), (11422, 23293, 23292), (12161, 25738, 1594), (12370, 34783, 18560), (13366, 21243, 14389), (23291, 37645, 30744), (32140, 36749, 15559), (32358, 43588, 3)

X(45969) = X(5)X(1199)∩X(6)X(39504)

Barycentrics    2*a^10-7*(b^2+c^2)*a^8+2*(5*b^4+3*b^2*c^2+5*c^4)*a^6-(b^2+c^2)*(8*b^4-15*b^2*c^2+8*c^4)*a^4+(b^2-c^2)^2*(2*b^2-3*b*c+2*c^2)*(2*b^2+3*b*c+2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(45969) = X(5)+2*X(32165), 3*X(51)+X(45730), X(140)+2*X(13292), X(546)+2*X(43588), 5*X(3567)+X(45731), 2*X(3628)+X(32358), X(3853)+2*X(18914), 2*X(5462)+X(11264), X(6102)+2*X(43575), X(6146)+2*X(16881), 2*X(10095)+X(10116), 2*X(10110)+X(45732), X(10112)+2*X(12006), 15*X(11245)+X(44935), X(11561)+2*X(19481), X(12103)+2*X(13142), 5*X(12812)-2*X(31831), X(13470)+2*X(16625), X(20424)+2*X(27552), X(32137)-4*X(40240)

See Antreas Hatzipolakis and César Lozada, euclid 3076.

X(45969) lies on these lines: {2, 21357}, {3, 43838}, {5, 1199}, {6, 39504}, {30, 568}, {51, 45730}, {54, 10125}, {140, 343}, {143, 44407}, {184, 10096}, {195, 43816}, {389, 30522}, {524, 44324}, {539, 13363}, {542, 13364}, {546, 12233}, {547, 597}, {549, 44555}, {1154, 11225}, {1353, 22151}, {1493, 43817}, {1503, 13451}, {1994, 37938}, {2888, 15047}, {3448, 15038}, {3567, 45731}, {3628, 11422}, {3853, 18914}, {5093, 31181}, {5449, 8254}, {5462, 11264}, {5498, 26879}, {5943, 11232}, {5946, 16223}, {6102, 43575}, {6146, 16881}, {7502, 37644}, {7555, 41588}, {7592, 13406}, {8550, 25338}, {9703, 13392}, {10095, 10116}, {10110, 45732}, {10112, 12006}, {10201, 14912}, {10224, 18912}, {10264, 15033}, {11250, 18916}, {11423, 12010}, {11433, 12106}, {11561, 19481}, {11563, 15032}, {11801, 18388}, {12103, 13142}, {12812, 31831}, {13154, 44756}, {13371, 15135}, {13470, 16625}, {13561, 37505}, {13567, 44234}, {13630, 31985}, {14627, 43808}, {14683, 21308}, {15004, 34514}, {15053, 34153}, {15121, 32144}, {15801, 43837}, {15806, 32136}, {18281, 18950}, {19362, 22533}, {20424, 27552}, {21230, 43651}, {32046, 34577}, {32137, 40240}, {36966, 45735}, {39571, 44235}, {43595, 43615}

X(45969) = midpoint of X(i) and X(j) for these {i, j}: {5943, 11232}, {11225, 43573}, {13292, 45298}
X(45969) = reflection of X(i) in X(j) for these (i, j): (140, 45298), (13363, 32068)

X(45970) = X(5)X(49)∩X(30)X(52)

Barycentrics    2*a^10-5*(b^2+c^2)*a^8+2*(2*b^4+3*b^2*c^2+2*c^4)*a^6-(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4+(2*b^4-b^2*c^2+2*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(45970) = X(5)-3*X(12022), 3*X(5)-X(14516), 3*X(381)+X(34799), X(548)-6*X(12024), 3*X(568)+X(12289), 5*X(3567)-3*X(38322), 2*X(3861)-3*X(16657), 2*X(6756)-3*X(13451), X(11381)+3*X(45730), 2*X(12006)-3*X(43573), 9*X(12022)-X(14516), 3*X(12022)-2*X(43575), 3*X(12022)+X(44076), 3*X(12101)-2*X(16621), X(12278)-5*X(37481), X(13474)+2*X(45734), X(14516)-6*X(43575), X(14516)+3*X(44076), X(21659)+2*X(32165), 2*X(43575)+X(44076)

See Antreas Hatzipolakis and César Lozada, euclid 3076.

X(45970) lies on these lines: {3, 43808}, {4, 13585}, {5, 49}, {20, 43807}, {30, 52}, {113, 43865}, {125, 5498}, {140, 5449}, {143, 11262}, {156, 18390}, {184, 13406}, {195, 3153}, {381, 34799}, {389, 30522}, {399, 43835}, {511, 13470}, {539, 11591}, {542, 45959}, {546, 12241}, {548, 12024}, {565, 8146}, {568, 12289}, {578, 39504}, {1154, 10112}, {1181, 44279}, {1503, 3853}, {1511, 43817}, {1614, 11563}, {1658, 19467}, {1899, 11250}, {1994, 20424}, {2070, 12254}, {2888, 34864}, {3448, 14130}, {3520, 10264}, {3521, 10733}, {3564, 31834}, {3567, 38322}, {3575, 16881}, {3627, 34224}, {3850, 12134}, {3861, 16657}, {5663, 10114}, {5944, 10619}, {6000, 45732}, {6143, 38724}, {6756, 13451}, {7514, 12429}, {7526, 18356}, {7564, 11426}, {7592, 44263}, {7722, 18560}, {8550, 32273}, {9545, 10255}, {9704, 16868}, {9927, 32046}, {10088, 43833}, {10091, 43832}, {10095, 45286}, {10096, 10282}, {10113, 43831}, {10125, 13367}, {10201, 18925}, {10226, 32607}, {11264, 13754}, {11381, 45730}, {11422, 18394}, {11423, 18392}, {11424, 34514}, {11430, 13561}, {11456, 44271}, {11536, 32365}, {11565, 13391}, {12006, 43573}, {12084, 43905}, {12101, 16621}, {12102, 16655}, {12106, 39571}, {12107, 41587}, {12118, 18952}, {12121, 43601}, {12140, 43823}, {12161, 18377}, {12201, 43828}, {12261, 43822}, {12278, 37481}, {12307, 37779}, {12334, 43847}, {12383, 43809}, {12407, 43830}, {12412, 43829}, {12466, 43850}, {12467, 43851}, {12501, 43854}, {12605, 32358}, {12778, 43827}, {12790, 43849}, {12803, 43852}, {12804, 43853}, {12889, 43859}, {12890, 43860}, {12893, 22962}, {12896, 43819}, {12898, 43824}, {12902, 34007}, {12903, 43857}, {12904, 43858}, {12905, 43861}, {12906, 43862}, {13358, 13630}, {13371, 43595}, {13474, 45734}, {13915, 43863}, {13979, 43864}, {14845, 23409}, {15033, 33332}, {15037, 43838}, {15646, 26879}, {15687, 16659}, {15761, 31804}, {16252, 44961}, {18383, 22051}, {18475, 34577}, {18476, 22815}, {18569, 18945}, {18570, 25738}, {18909, 34350}, {18912, 37814}, {18968, 43820}, {19051, 43825}, {19052, 43826}, {19177, 35887}, {19478, 43848}, {20127, 43806}, {20304, 43839}, {20379, 25563}, {21230, 35921}, {22467, 34153}, {22951, 38405}, {25739, 37472}, {27868, 44581}, {32171, 44234}, {32233, 43810}, {34148, 37938}, {34209, 36159}, {35834, 43867}, {35835, 43868}, {36749, 44288}, {38794, 43866}, {43584, 43837}

X(45970) = midpoint of X(i) and X(j) for these {i, j}: {4, 45731}, {5, 44076}, {3627, 34224}, {6102, 21659}, {6146, 12370}, {10116, 13403}, {10263, 11750}, {12605, 32358}, {18560, 45957}
X(45970) = reflection of X(i) in X(j) for these (i, j): (5, 43575), (546, 12241), (3575, 16881), (6102, 32165), (12134, 3850), (13358, 19481), (16655, 12102), (44829, 11565), (45286, 10095)
X(45970) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (5, 12022, 43575), (5, 36966, 49), (54, 265, 5), (110, 43821, 5), (125, 43394, 5498), (156, 18390, 44235), (1994, 31724, 20424), (3448, 43818, 14130), (6288, 13434, 5), (10733, 43602, 3521), (11801, 15806, 5), (12022, 44076, 5), (12161, 18396, 18377), (12383, 43816, 43809), (12902, 43845, 34007), (18379, 18388, 546), (18379, 32136, 18388)

X(45971) = EULER LINE INTERCEPT OF X(143)X(11561)

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4-3*b^2*c^2+c^4)*a^6+(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4-(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

See Antreas Hatzipolakis and César Lozada, euclid 3076.

Let triangle HaHbHc be as described in Anapolis #2471 (March 19, 2015, Antreas Hatzipolakis). Then X(45971) = X(5)-of-HaHbHc. (Randy Hutson, November 30, 2021)

X(45971) lies on these lines: {2, 3}, {143, 11561}, {195, 12383}, {389, 30522}, {568, 12278}, {1199, 15002}, {1204, 34514}, {2777, 32137}, {2888, 32608}, {3521, 14157}, {3574, 43394}, {5448, 10272}, {5640, 40242}, {5663, 45286}, {5890, 45731}, {5946, 21659}, {6102, 11562}, {6592, 23702}, {6746, 14708}, {7706, 32046}, {9729, 13470}, {10095, 13403}, {10733, 38848}, {11597, 20424}, {11745, 13451}, {11801, 18379}, {12110, 14675}, {12162, 34798}, {12244, 33541}, {12254, 43845}, {12289, 37481}, {12370, 16881}, {13365, 44829}, {13382, 45732}, {13419, 43577}, {13630, 18128}, {15038, 43818}, {15087, 36966}, {15100, 18439}, {15101, 21650}, {15102, 34783}, {15800, 43574}, {15806, 18388}, {16266, 40909}, {18430, 26917}, {32110, 34826}, {32165, 44076}, {34224, 45956}

X(45971) = midpoint of X(i) and X(j) for these {i, j}: {5, 6240}, {12162, 34798}, {13419, 43577}
X(45971) = reflection of X(i) in X(j) for these (i, j): (140, 31833), (546, 31830), (1885, 3861), (3853, 6756), (12103, 31829), (12370, 16881), (12605, 3628), (13403, 10095), (13470, 9729), (21659, 43575), (44076, 32165), (45732, 13382)

X(45972) = X(2)X(43704)∩X(3)X(32165)

Barycentrics    (a^8-2*(b^2+2*c^2)*a^6+(5*b^2+6*c^2)*c^2*a^4+(b^2-c^2)*(2*b^4-b^2*c^2+4*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^2)*(a^8-2*(2*b^2+c^2)*a^6+(6*b^2+5*c^2)*b^2*a^4-(b^2-c^2)*(4*b^4-b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 3076.

X(45972) lies on Jerabek circumhyperbola and these lines: {2, 43704}, {3, 32165}, {6, 37943}, {54, 10182}, {74, 10114}, {265, 3410}, {895, 24206}, {1899, 5900}, {3519, 15067}, {3521, 10263}, {3527, 44958}, {6102, 14861}, {6776, 34437}, {7722, 10293}, {11559, 12317}, {13418, 18912}, {14483, 18388}, {14912, 19151}, {16835, 21659}, {18911, 42021}

X(45972) = antigonal conjugate of the isogonal conjugate of X(25338)
X(45972) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(37779)}} and Jerabek hyperbola
X(45972) = trilinear pole of the line {647, 45147}

X(45973) = X(2)X(43704)∩X(5)X(323)

Barycentrics    a^2*(a^6-3*(b^2+c^2)*a^4+(3*b^4+5*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*(a^8-2*(b^2+c^2)*a^6+3*b^2*c^2*a^4+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-(b^2-c^2)^4) : :

See Antreas Hatzipolakis and César Lozada, euclid 3076.

X(45973) lies on these lines: {2, 43704}, {5, 323}, {1147, 15037}, {1493, 13363}, {1511, 5012}, {4550, 23039}, {5651, 14627}, {22333, 36752}

X(45974) = X(3)X(41)∩X(8)X(304)

Barycentrics    a^2*(b^2 + b*c + c^2)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - 2*a*b*c - b^2*c + a*c^2 + b*c^2 - c^3)*(a^3 - a^2*b + a*b^2 - b^3 - a^2*c - 2*a*b*c + b^2*c - a*c^2 - b*c^2 + c^3) :

X(45974) lies on the cubic K1244 and these lines: {3, 41}, {8, 304}, {7077, 22116}, {45963, 45966}

X(45974) = X(40732)-cross conjugate of X(2276)
X(45974) = X(i)-isoconjugate of X(j) for these (i,j): {870, 37580}, {948, 2344}, {985, 2550}, {14621, 40131}, {16054, 40747}
X(45974) = barycentric product X(i)*X(j) for these {i,j}: {949, 7179}, {984, 39273}, {3423, 3661}
X(45974) = barycentric quotient X(i)/X(j) for these {i,j}: {869, 40131}, {1469, 948}, {2276, 2550}, {3423, 14621}, {3736, 16054}, {39273, 870}, {40728, 37580}

X(45975) = X(6)X(8152)∩X(20)X(1352)

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6-(4*b^4-7*b^2*c^2+4*c^4)*a^4+(b^2+c^2)*(7*b^4-9*b^2*c^2+7*c^4)*a^2-2*b^8-2*c^8-2*b^2*c^2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)) : :

See Dan Reznik and César Lozada, euclid 3094.

X(45975) lies on these lines: {6, 8152}, {20, 1352}, {576, 34795}, {4550, 37890}, {11438, 37348}, {33018, 34417}, {33022, 35277}

X(45976) = EULER LINE INTERCEPT OF X(1)X(6797)

Barycentrics    a (a^6-a^5 (b+c)+a^4 (-2 b^2+3 b c-2 c^2)+2 a^3 (b^3+c^3)+a^2 (b^4-5 b^3 c+2 b^2 c^2-5 b c^3+c^4)-a (b^5-b^4 c-b c^4+c^5)+2 b c (b^2-c^2)^2) : :

See Antreas Hatzipolakis and Angel Montesdeoca, euclid 3114.

X(45976) lies on these lines: {1,6797}, {2,3}, {10,22765}, {12,10090}, {35,11230}, {36,9956}, {55,37735}, {56,5790}, {100,5901}, {104,18357}, {119,35451}, {191,41347}, {195,3216}, {355,10265}, {936,37532}, {946,35000}, {952,5253}, {999,10573}, {1125,37621}, {1376,1482}, {1385,18524}, {1437,13353}, {1470,9654}, {1698,26286}, {2077,9955}, {2975,38042}, {3075,23071}, {3218,31835}, {3306,37700}, {3336,5694}, {3616,32141}, {3624,32613}, {3754,6265}, {3833,10246}, {3871,10283}, {4265,38317}, {4413,11249}, {5096,24206}, {5259,33862}, {5267,10172}, {5277,34460}, {5288,38176}, {5398,17749}, {5433,41345}, {5437,37615}, {5438,37533}, {5443,14882}, {5587,26321}, {5687,10247}, {5720,37612}, {5780,37545}, {5818,32153}, {5883,37733}, {5885,6326}, {5886,11849}, {6127,8614}, {6691,26470}, {7173,10058}, {7676,38043}, {7686,35459}, {8071,31479}, {8227,26285}, {8256,9709}, {8715,34640}, {9655,40293}, {9851,18528}, {10200,37820}, {10269,18525}, {10827,34880}, {10966,17662}, {11012,11231}, {11248,18493}, {11491,38028}, {12005,12738}, {12702,22753}, {12776,19914}, {13151,40262}, {13465,20117}, {14131,34461}, {15047,34465}, {16203,18526}, {18480,37561}, {23015,36865}, {23070,37694}, {24475,27003}, {26201,35010}, {31272,38722}, {33539,33811}, {33814,38038}, {36750,37522}

X(45976) = midpoint of X(6915) and X(6940)
X(45976) = reflection of X(i) in X(j), for these {i, j}: {3,6940}, {13729,5}

X(45977) = X(1)X(1389)∩X(4)X(496)

Barycentrics    a*(a^6-(b+c)*a^5-(2*b-c)*(b-2*c)*a^4+2*(b+c)*(b^2-3*b*c+c^2)*a^3+(b-c)^2*(b^2-4*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a+(b^2-c^2)^2*b*c) : :

See Antreas Hatzipolakis and Angel Montesdeoca, euclid 3114.

X(45977) lies on these lines: {1,1389}, {2,10597}, {3,3622}, {4,496}, {5,20060}, {8,6946}, {11,38163}, {20,10596}, {21,5901}, {36,13464}, {40,3898}, {56,4295}, {58,32486}, {65,10698}, {79,104}, {100,10222}, {145,6911}, {354,21740}, {388,6941}, {404,1482}, {411,10246}, {474,12245}, {495,6949}, {517,5253}, {546,12773}, {551,11012}, {631,22770}, {943,34485}, {944,3304}, {952,6915}, {953,18180}, {956,3090}, {962,10269}, {993,9624}, {1006,3616}, {1056,6834}, {1058,6934}, {1149,3072}, {1210,12776}, {1319,13374}, {1385,3651}, {1483,37251}, {1484,37230}, {1519,4298}, {1537,24470}, {1621,26286}, {2800,3337}, {2975,5886}, {3086,6830}, {3149,7373}, {3241,11499}, {3244,38665}, {3295,6942}, {3421,6983}, {3436,6975}, {3485,22767}, {3576,12511}, {3585,10074}, {3636,10902}, {3656,32612}, {3832,18519}, {3871,6924}, {3877,37532}, {3881,6326}, {3884,5535}, {3889,37700}, {4188,10679}, {4220,29823}, {4231,7191}, {4292,5193}, {4293,10531}, {4301,37561}, {4308,5804}, {4317,26333}, {4881,33596}, {5049,33597}, {5067,9708}, {5154,11929}, {5229,10598}, {5260,11230}, {5265,6977}, {5288,10175}, {5434,7681}, {5450,11522}, {5657,25524}, {5690,17531}, {5734,11248}, {5818,12513}, {5882,37602}, {5883,11014}, {6265,6583}, {6826,10529}, {6827,10586}, {6829,10527}, {6839,10943}, {6845,10785}, {6848,10805}, {6893,20076}, {6901,24390}, {6909,22791}, {6912,18493}, {6938,41426}, {6952,15325}, {6954,10587}, {6970,10528}, {6979,10942}, {6986,38028}, {7677,20330}, {7682,15179}, {7686,20323}, {8227,8666}, {9657,10893}, {9779,18761}, {10072,26332}, {10090,11009}, {10267,38314}, {10310,40726}, {10589,10599}, {11036,37302}, {11849,13587}, {12515,26200}, {12672,26877}, {13279,34772}, {14497,32141}, {16200,25440}, {16417,34631}, {16861,38022}, {17609,37837}, {18391,18967}, {18398,40257}, {18480,38669}, {19649,29831}, {20067,37290}, {22768,30305}, {24299,29817}, {24558,37249}, {26321,38034}, {27003,37562}, {28452,32214}, {33668,33860}, {34605,37821}, {34773,36002}, {35252,37106}

X(45977) = reflection of X(6940) in X(5253)

X(45978) = X(128)X(11275)∩X(526)X(11557)

Barycentrics    a^2*(a^18*b^2 - 6*a^16*b^4 + 15*a^14*b^6 - 21*a^12*b^8 + 21*a^10*b^10 - 21*a^8*b^12 + 21*a^6*b^14 - 15*a^4*b^16 + 6*a^2*b^18 - b^20 + a^18*c^2 - 10*a^16*b^2*c^2 + 29*a^14*b^4*c^2 - 34*a^12*b^6*c^2 + 8*a^10*b^8*c^2 + 29*a^8*b^10*c^2 - 55*a^6*b^12*c^2 + 56*a^4*b^14*c^2 - 31*a^2*b^16*c^2 + 7*b^18*c^2 - 6*a^16*c^4 + 29*a^14*b^2*c^4 - 38*a^12*b^4*c^4 + 8*a^10*b^6*c^4 + 7*a^8*b^8*c^4 + 24*a^6*b^10*c^4 - 65*a^4*b^12*c^4 + 63*a^2*b^14*c^4 - 22*b^16*c^4 + 15*a^14*c^6 - 34*a^12*b^2*c^6 + 8*a^10*b^4*c^6 + 12*a^8*b^6*c^6 - 8*a^6*b^8*c^6 + 28*a^4*b^10*c^6 - 63*a^2*b^12*c^6 + 42*b^14*c^6 - 21*a^12*c^8 + 8*a^10*b^2*c^8 + 7*a^8*b^4*c^8 - 8*a^6*b^6*c^8 - 8*a^4*b^8*c^8 + 25*a^2*b^10*c^8 - 57*b^12*c^8 + 21*a^10*c^10 + 29*a^8*b^2*c^10 + 24*a^6*b^4*c^10 + 28*a^4*b^6*c^10 + 25*a^2*b^8*c^10 + 62*b^10*c^10 - 21*a^8*c^12 - 55*a^6*b^2*c^12 - 65*a^4*b^4*c^12 - 63*a^2*b^6*c^12 - 57*b^8*c^12 + 21*a^6*c^14 + 56*a^4*b^2*c^14 + 63*a^2*b^4*c^14 + 42*b^6*c^14 - 15*a^4*c^16 - 31*a^2*b^2*c^16 - 22*b^4*c^16 + 6*a^2*c^18 + 7*b^2*c^18 - c^20) : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 3127.

X(45978) lies on these lines: {128,11275}, {526,11557}, {1141,15537}, {6153,10095}

leftri

Points associated with Vijay orthic medial circles and Vijay orthic medial triangles: X(45979)

rightri

This preamble is contributed by Clark Kimberling (Nov 15, 2021), based on notes from Dasari Naga Vijay Krishna, Nov 15, 2021.

In the plane of a triangle ABC, let
A'B'C' = medial triangle ;
ApBpCp = orthic triangle ;
(O)a = circle with B'C' as diameter, and define (O)b and (O)c cyclically;
[Here (O)a, (O)b, (O)c are known as 1st Vijay A, B, C orthic medial circles; see the preamble just before X(45845).]
Ab = AB ∩ Oa, and define Bc and Ca cyclically;
Ac = AC ∩ Oa, and define Ba and Cb cyclically;

Define 8 points by the following intersections:

A1 = midpoint of AbAc, and define B1 and C1 cyclically;
A2 = midpoint of BcCa, and define B2 and C2 cyclically;
A3 = the point in (O)b ∩ (O)c other than A' and define B3 and C3 cyclically;
A4 = BcBa ∩ CaCb, and define B4 and C4 cyclically;
A5 = AbBc ∩ AcCb, and define B5 and C5 cyclically;
A6 = AbCa ∩ AcBa, and define B6 and C6 cyclically;
A7 = BcCa ∩ CbBa, and define B7 and C7 cyclically;
A8 = AbCb ∩ AcBc, and define B8 and C8 cyclically;
A9 = reflection of A3 wrt to BC, and define B9 and C9 cyclically;
A10 = midpoint of AAp and define B10 and C10 cyclically;

Related conic here named as follows:

There is a conic passes through the six points {A3, B3, C3, A10, B10, C10} here named as Vijay orthic medial conic.
This conic also contains the X(6720)[ and this is the only ETC center present on this conic] .
The barycentric equation of this conic is given by (5*c^6 - 5*b^2*c^4 - 5*a^2*c^4 - 5*b^4*c^2 + 2*a^2*b^2*c^2 + 3*a^4*c^2 + 5*b^6 - 5*a^2*b^4 + 3*a^4*b^2 - 3*a^6)*x^2 + (5*c^6 - 5*b^2*c^4 - 5*a^2*c^4 + 3*b^4*c^2 + 2*a^2*b^2*c^2 - 5*a^4*c^2 - 3*b^6 + 3*a^2*b^4 - 5*a^4*b^2 + 5*a^6)*y^2 + (-3*c^6 + 3*b^2*c^4 + 3*a^2*c^4 - 5*b^4*c^2 + 2*a^2*b^2*c^2 - 5*a^4*c^2 + 5*b^6 - 5*a^2*b^4 - 5*a^4*b^2 + 5*a^6)*z^2 + 2*(-c^2 + b^2 + a^2)*(3*c^4 + b^4 - 2*a^2*b^2 + a^4)*x*y + 2*(c^2 + b^2 - a^2)*(c^4 - 2*b^2*c^2 + b^4 + 3*a^4)*y*z + 2*(c^2 - b^2 + a^2)*(c^4 - 2*a^2*c^2 + 3*b^4 + a^4)*z*x = 0.
The Vijay orthic medial conic is an ellipse, a parabola, or a hyperbola according V is positive, zero, or negative,
where V = (c^12 - 2*b^2*c^10 - 2*a^2*c^10 + 3*b^4*c^8 + 2*a^2*b^2*c^8 + 3*a^4*c^8 - 4*b^6*c^6 - 4*a^6*c^6 + 3*b^8*c^4 - 6*a^4*b^4*c^4 + 3*a^8*c^4 - 2*b^10*c^2 + 2*a^2*b^8*c^2 + 2*a^8*b^2*c^2 - 2*a^10*c^2 + b^12 - 2*a^2*b^10 + 3*a^4*b^8 - 4*a^6*b^6 + 3*a^8*b^4 - 2*a^10*b^2 + a^12).

Related triangles are here named as follows:

The triangle AnBnCn is here named the nth Vijay orthic medial triangle, for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Barycentrics for points defined above:

Ab = 3c^2 + a^2 - b^2 : b^2 + c^2 - a^2 : 0
Bc = 0 : 3a^2 + b^2 - c^2 : a^2 + c^2 - b^2
Ca = a^2 + b^2 - c^2 : 0 : 3b^2 + c^2 - a^2
Ac = 3b^2 + a^2 - c^2 : 0 : b^2 + c^2 - a^2
Ba = a^2 +c^2 - b^2 : 3c^2 + b^2 - a^2 : 0
Cb = 0 : a^2 + b^2 - c^2 : 3a^2 + c^2 - b^2
A1 = -(b^4 + c^4 - 6*b^2*c^2 - a^2*b^2 - a^2*c^2) : b^2*(b^2 + c^2 - a^2) : c^2*(b^2 + c^2 - a^2)
A2 = 0 : 2*a^2 + b^2 - c^2 : 2*a^2 - b^2 + c^2
A3 = 2*a^2 : (a^2 - b^2 + c^2) : (a^2 + b^2 - c^2)
A4 = 2*a^4 : (c^4 - 2*a^2*c^2 - b^4 - 2*a^2*b^2 + a^4) : (-c^4 - 2*a^2*c^2 + b^4 - 2*a^2*b^2 + a^4)
A5 = 2*a^4*(-c^2 + 3*b^2 + a^2)*(3*c^2 - b^2 + a^2) : (-c^2 + b^2 + a^2)*(c^2 + b^2 - a^2)*(c^4 - 2*b^2*c^2 - 2*a^2*c^2 + b^4 - a^4) : (c^2 - b^2 + a^2)*(c^2 + b^2 - a^2)*(c^4 - 2*b^2*c^2 + b^4 - 2*a^2*b^2 - a^4)
A6 = (-c^8 + 2*a^2*c^6 + 10*b^4*c^4 - a^4*c^4 - b^8 + 2*a^2*b^6 - a^4*b^4) : b^4*(c^2 + b^2 - a^2)*(3*c^2 + b^2 - a^2) : c^4*(c^2 + b^2 - a^2)*(c^2 + 3*b^2 - a^2)
A7 = -2*a^4*(-c^2 + b^2 + a^2)*(c^2 - b^2 + a^2) : (c^2 - b^2 - 3*a^2)*(-c^2 + b^2 + a^2)*(c^4 + 2*a^2*c^2 - b^4 + 2*a^2*b^2 - a^4) : (c^2 - b^2 + a^2)*(c^2 - b^2 + 3*a^2)*(c^4 - 2*a^2*c^2 - b^4 - 2*a^2*b^2 + a^4)
A8 = -2*a^4*(a^2 + 3*b^2 - c^2)*(a^2 + 3*c^2 - b^2) : (b^2 + c^2 -a^2)*(3*a^2 + b^2 - c^2)*(c^4 - 2*b^2*c^2 + b^4 - 2*a^2*b^2 - a^4) : (b^2 + c^2 - a^2)*(3*a^2 - b^2 + c^2)*(c^4 - 2*b^2*c^2 - 2*a^2*c^2 + b^4 - a^4)
A9 = -2*a^2 : (3*a^2 + b^2 - c^2) : (3*a^2 - b^2 + c^2)
A10 = 2*a^2 : (a^2 - c^2 + b^2) : (a^2 + c^2 - b^2)

Perspectors :

X(2) = AA' ∩ BB' ∩ CC' = ApA3 ∩ BpB3 ∩ CpC3
X(5) = A2A4 ∩ B2B4 ∩ C2C4
X(6) = AA1 ∩ BB1 ∩ CC1 = A'A10 ∩ B'B10 ∩ C'C10
X(25) = ApA4 ∩ BpB4 ∩ CpC4
X(69) = AA3 ∩ BB3 ∩ CC3
X(193) = ApA9 ∩ BpB9 ∩ CpC9
X(206) = A'A4 ∩ B'B4 ∩ C'C4
X(5943) = A1A9 ∩ B1B9 ∩ C1C9
X(6676) = A3A4 ∩ B3B4 ∩ C3C4
X(9969) = A1A10 ∩ B1B10 ∩ C1C10 = (a^2*(c^6 - b^2*c^4 - b^4*c^2 - 2*a^2*b^2*c^2 - a^4*c^2 + b^6 - a^4*b^2) : b^2*(c^6 - a^2*c^4 - b^4*c^2 - 2*a^2*b^2*c^2 - a^4*c^2 - a^2*b^4 + a^6) : c^2*(-b^2*c^4 - a^2*c^4 - 2*a^2*b^2*c^2 + b^6 - a^2*b^4 - a^4*b^2 + a^6) )
X(45979) = center of Vijay orthic medial conic

X(45979) = CENTROID OF 4TH VIJAY ORTHIC MEDIAL TRIANLGE

Barycentrics    a^2*(c^10 - 3*b^2*c^8 - 2*a^2*c^8 + 2*b^4*c^6 + 6*a^2*b^2*c^6 + 2*b^6*c^4 - 8*a^2*b^4*c^4 + 4*a^4*b^2*c^4 + 2*a^6*c^4 -3*b^8*c^2 + 6*a^2*b^6*c^2 + 4*a^4*b^4*c^2 - 6*a^6*b^2*c^2 - a^8*c^2 + b^10 - 2*a^2*b^8 + 2*a^6*b^4 - a^8*b^2) : :

X(45979) lies on these lines: {}

X(45980) = X(4)X(524)∩X(30)X(31748)

Barycentrics    -10 a^10 - 3 a^8 b^2 + 88 a^6 b^4 - 82 a^4 b^6 - 6 a^2 b^8 + 13 b^10 - 3 a^8 c^2 + 28 a^6 b^2 c^2 + 42 a^4 b^4 c^2 + 12 a^2 b^6 c^2 - 47 b^8 c^2 + 88 a^6 c^4 + 42 a^4 b^2 c^4 - 12 a^2 b^4 c^4 + 34 b^6 c^4 - 82 a^4 c^6 + 12 a^2 b^2 c^6 + 34 b^4 c^6 - 6 a^2 c^8 - 47 b^2 c^8 + 13 c^10 : :

See Antreas Hatzipolakis and Francisco Javier García Capitán, euclid 3136.

X(45980) lies on these lines: {4, 524}, {30, 31748}

X(45981) = MIDPOINT OF X(11) AND X(39144)

Barycentrics    -(2*a^6-2*(b+c)*a^5-(9*b^2-20*b*c+9*c^2)*a^4+14*(b^2-c^2)*(b-c)*a^3+2*(b^2-11*b*c+c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(3*b^2-5*b*c+3*c^2)*a+5*(b^2-c^2)^2*(b-c)^2)*(a+b+c)*sqrt(r*(4*R+r))+S*(6*a^6-6*(b+c)*a^5-(11*b^2-28*b*c+11*c^2)*a^4+10*(b^2-c^2)*(b-c)*a^3+2*(3*b^2+7*b*c+3*c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(45981) = 3*X(11)-X(39145), 3*X(39144)+X(39145)

See Antreas Hatzipolakis and César Lozada, euclid 3139.

X(45981) lies on these lines: {11, 57}, {952, 40566}

X(45981) = midpoint of X(11) and X(39144)
X(45981) = reflection of X(45982) in X(11)

X(45982) = MIDPOINT OF X(11) AND X(39145)

Barycentrics    (2*a^6-2*(b+c)*a^5-(9*b^2-20*b*c+9*c^2)*a^4+14*(b^2-c^2)*(b-c)*a^3+2*(b^2-11*b*c+c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(3*b^2-5*b*c+3*c^2)*a+5*(b^2-c^2)^2*(b-c)^2)*(a+b+c)*sqrt(r*(4*R+r))+S*(6*a^6-6*(b+c)*a^5-(11*b^2-28*b*c+11*c^2)*a^4+10*(b^2-c^2)*(b-c)*a^3+2*(3*b^2+7*b*c+3*c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(45982) = 3*X(11)-X(39144), X(39144)+3*X(39145)

See Antreas Hatzipolakis and César Lozada, euclid 3139.

X(45982) lies on these lines: {11, 57}, {952, 40565}

X(45982) = midpoint of X(11) and X(39145)
X(45982) = reflection of X(45981) in X(11)

X(45983) = X(3)X(30022)∩X(6)X(76)

Barycentrics    b^2*c^2*(-a^7 - a^6*b - a^5*b^2 + a^3*b^4 - a^6*c + a^3*b^3*c + a^2*b^4*c - a^5*c^2 + a*b^4*c^2 + a^3*b*c^3 + a*b^3*c^3 + b^4*c^3 + a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + b^3*c^4) : :

X(45983) lies on the cubic K1247 and these lines: {3, 30022}, {6, 76}, {57, 18033}, {384, 28660}, {985, 987}, {1921, 3496}, {3978, 37086}

X(45984) = X(2)X(2199)∩X(4)X(171)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4 + 2*a^5*c - a^4*b*c - 2*a*b^4*c + b^5*c + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 - 2*a*b*c^4 + b*c^5) : :

X(45984) lies on the cubic K1247 and these lines: {2, 2199}, {4, 171}, {6, 57}, {83, 8808}, {1214, 3496}, {1460, 13737}, {1935, 19732}, {4209, 18623}, {5329, 37310}, {20368, 37413}

X(45985) = X(4)X(6)∩X(29)X(3172)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6 + a^5*b - 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 + a^5*c - a^4*b*c - a*b^4*c + b^5*c - 2*a^3*c^3 - 2*b^3*c^3 - 2*a^2*c^4 - a*b*c^4 + a*c^5 + b*c^5) : :

X(45985) lies on the cubic K1247 and these lines: {4, 6}, {29, 3172}, {32, 37055}, {57, 3500}, {112, 7531}, {238, 17920}, {281, 5255}, {412, 45141}, {985, 40836}, {7498, 19761}

X(45986) = X(3)X(256)∩X(6)X(1432)

Barycentrics    a*(a + b - c)*(a - b + c)*(b^2 + a*c)*(a*b + c^2)*(a^5 + a^2*b^3 + a^2*b^2*c + a^2*b*c^2 - a*b^2*c^2 + b^3*c^2 + a^2*c^3 + b^2*c^3) : :

X(45986) lies on the cubic K1248 and these lines: {3, 256}, {6, 1432}, {83, 3405}, {1400, 27982}, {1423, 27958}, {40432, 41526}

X(45987) = X(2)X(2305)∩X(6)X(20836)

Barycentrics    a^2*(a^3 + 2*a^2*b + 2*a*b^2 + b^3 + 2*a^2*c + a*b*c + 2*b^2*c - c^3)*(a^3 + 2*a^2*b - b^3 + 2*a^2*c + a*b*c + 2*a*c^2 + 2*b*c^2 + c^3) : :

X(45987) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1247, and these lines: {2, 2305}, {6, 20836}, {37, 3496}, {42, 23868}, {2248, 4275}, {33863, 39798}, {36744, 39967}

X(45987) = isogonal conjugate of X(37653)
X(45987) = X(4264)-cross conjugate of X(6)
X(45987) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37653}, {75, 5110}
X(45987) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 37653}, {32, 5110}

X(45988) = X(25)X(2305)∩X(37)X(986)

Barycentrics    a^2*(a^2*b - a*b^2 - 2*b^3 + a^2*c + a*b*c - b^2*c + a*c^2 + b*c^2)*(a^2*b + a*b^2 + a^2*c + a*b*c + b^2*c - a*c^2 - b*c^2 - 2*c^3) : :

X(45988) lies on the conic {{A,B,C,X(2),X(6)}}, the cubic K1248, and these lines: {25, 2305}, {37, 986}, {42, 2277}, {393, 3144}, {583, 39966}, {992, 39956}, {4261, 39967}, {4269, 21001}, {4286, 39982}, {5110, 37282}, {8770, 37500}, {9780, 14624}, {16606, 28244}, {18755, 45129}, {27623, 37128}, {27644, 39952}

X(45988) = isogonal conjugate of X(37652)
X(45988) = isotomic conjugate of X(30022)
X(45988) = isogonal conjugate of the anticomplement of X(18134)
X(45988) = X(579)-cross conjugate of X(6)
X(45988) = X(i)-isoconjugate of X(j) for these (i,j): {1, 37652}, {31, 30022}, {63, 37055}
X(45988) = trilinear pole of line {512, 23655}
X(45988) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 30022}, {6, 37652}, {25, 37055}, {21857, 38407}

X(45989) = X(1)X(1401)∩X(3)X(983)

Barycentrics    a*(a*b^2 + b^3 + a^2*c - a*b*c + b^2*c + a*c^2)*(a^2*b + a*b^2 - a*b*c + a*c^2 + b*c^2 + c^3) : :

X(45989) lies on the Feuerbach circumhyperbola, the cubic K1247, and these lines: {1, 1401}, {3, 983}, {4, 982}, {7, 3976}, {8, 38}, {9, 39}, {21, 1201}, {56, 987}, {57, 989}, {79, 3953}, {80, 3670}, {256, 37592}, {294, 23640}, {314, 3663}, {392, 8421}, {899, 32635}, {981, 41264}, {1000, 37598}, {1042, 1476}, {1125, 36798}, {2298, 37607}, {2344, 16502}, {2346, 37573}, {2481, 24214}, {3616, 38251}, {3666, 43073}, {3831, 13161}, {4202, 37716}, {4352, 41527}, {4424, 5559}, {4694, 5557}, {4862, 10435}, {4866, 6048}, {5255, 37328}, {7160, 17594}, {26094, 33123}

X(45989) = isogonal conjugate of X(5255)
X(45989) = X(32922)-cross conjugate of X(291)
X(45989) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5255}, {6, 27064}
X(45989) = trilinear pole of line {650, 21123}
X(45989) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 27064}, {6, 5255}, {3663, 24994}

X(45990) = X(4)X(69)∩X(9)X(29958)

Barycentrics    a^2*(a^4*b^3 + a^3*b^4 - a^2*b^5 - a*b^6 + a^4*b^2*c + a^3*b^3*c - a*b^5*c - b^6*c + a^4*b*c^2 - a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 + a^4*c^3 + a^3*b*c^3 - a^2*b^2*c^3 + 2*b^4*c^3 + a^3*c^4 + a*b^2*c^4 + 2*b^3*c^4 - a^2*c^5 - a*b*c^5 + b^2*c^5 - a*c^6 - b*c^6) : :

X(45990) lies on the cubic K1248 and these lines: {4, 69}, {9, 29958}, {43, 57}, {51, 17778}, {141, 1329}, {518, 22300}, {894, 23154}, {942, 24173}, {970, 16574}, {2299, 9306}, {3685, 42448}, {3786, 3917}, {5712, 5943}, {9017, 24476}

X(45990) = crossdifference of every pair of points on line {3049, 4435}
X(45990) = {X(1469),X(37676)}-harmonic conjugate of X(4260)

X(45991) = X(4)X(6)∩X(57)X(3497)

Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^5*b + a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - a*b^5 - b^6 - a^5*c + a^4*b*c + a*b^4*c - b^5*c + a^4*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 2*a^3*c^3 + 2*b^3*c^3 - a^2*c^4 + a*b*c^4 + b^2*c^4 - a*c^5 - b*c^5 - c^6) : :

X(45991) lies on the cubic K1248 and these lines: {4, 6}, {57, 3497}, {281, 986}, {291, 40836}, {412, 16318}, {3144, 3767}, {7498, 19758}, {7551, 39575}, {13161, 40942}

X(45992) = X(4)X(240)∩X(6)X(1432)

Barycentrics    a*(a + b - c)*(a - b + c)*(b^2 + a*c)*(a*b + c^2)*(a^5 - a^3*b^2 + a^2*b^3 - b^5 - a^3*b*c + 2*a^2*b^2*c - a*b^3*c - a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + b^3*c^2 + a^2*c^3 - a*b*c^3 + b^2*c^3 - c^5) : :

X(45992) lies on the cubics K354 and K1248 and on these lines: {4, 240}, {6, 1432}, {39, 3497}, {291, 36897}, {694, 24479}, {1916, 39930}

X(45992) = X(1916)-Ceva conjugate of X(1432)
X(45992) = X(2329)-isoconjugate of X(7351)
X(45992) = barycentric product X(6211)*X(7249)
X(45992) = barycentric quotient X(i)/X(j) for these {i,j}: {1431, 7351}, {6211, 7081}

X(45993) = X(3)X(11058)∩X(4)X(7712)

Barycentrics    (2*a^6-(2*b^2+3*c^2)*a^4-(2*b^2+c^2)*b^2*a^2+(2*b^2+c^2)*(b^2-c^2)^2)*(2*a^6-(3*b^2+2*c^2)*a^4-(b^2+2*c^2)*c^2*a^2+(b^2+2*c^2)*(b^2-c^2)^2)*(7*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 3158.

X(45993) lies on the cubic K618 and these lines: {3, 11058}, {4, 7712}

X(45994) = EULER LINE INTERCEPT OF X(1503)X(43395)

Barycentrics    -4*OH*a*b*c*((b^2+c^2)*a^2-(b^2-c^2)^2)*S+(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, euclid 3158.

X(45994) lies on these lines: {2, 3}, {1503, 43395}, {2574, 25711}, {2575, 15738}, {32550, 44407}

X(45995) = EULER LINE INTERCEPT OF X(1503)X(43396)

Barycentrics    4*OH*a*b*c*((b^2+c^2)*a^2-(b^2-c^2)^2)*S+(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

See Antreas Hatzipolakis and César Lozada, euclid 3158.

X(45995) lies on these lines: {2, 3}, {1503, 43396}, {2574, 15738}, {2575, 25711}, {32549, 44407}

X(45996) = X(3)X(11058)∩X(3628)X(16836)

Barycentrics    8*a^16+4*(b^2+c^2)*a^14-18*(7*b^4-11*b^2*c^2+7*c^4)*a^12+2*(b^2+c^2)*(139*b^4-292*b^2*c^2+139*c^4)*a^10-(200*b^8+200*c^8+(257*b^4-900*b^2*c^2+257*c^4)*b^2*c^2)*a^8-3*(b^4-c^4)*(b^2-c^2)*(16*b^4-187*b^2*c^2+16*c^4)*a^6+(146*b^8+146*c^8+(13*b^4-264*b^2*c^2+13*c^4)*b^2*c^2)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)^3*(74*b^4+149*b^2*c^2+74*c^4)*a^2+6*(b^2-c^2)^6*(b^2+2*c^2)*(2*b^2+c^2) : :

See Antreas Hatzipolakis and César Lozada, euclid 3158.

X(45996) lies on these lines: {3, 11058}, {3628, 16836}

X(45997) = X(3)X(275)∩X(5)X(51)

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^12-2*(b^2+c^2)*a^10-(2*b^4-3*b^2*c^2+2*c^4)*a^8+2*(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^6-(b^2-c^2)^2*(7*b^4+10*b^2*c^2+7*c^4)*a^4+2*(b^6+c^6)*(b^2-c^2)^2*a^2+(b^2-c^2)^4*b^2*c^2) : :
Barycentrics    (S^2+(4*R^2-SW)*(8*R^2+2*SA-3*SW))*(S^2+SB*SC) : :
X(45997) = 2*X(3)-3*X(12012), 4*X(5)-3*X(10184), 2*X(5)-3*X(14635), X(20)-3*X(32078), 3*X(381)-X(14978), 5*X(631)-6*X(44914), 5*X(3091)-3*X(11197)

See Antreas Hatzipolakis and César Lozada, euclid 3159.

X(45997) lies on these lines: {2, 31388}, {3, 275}, {4, 3164}, {5, 51}, {20, 32078}, {30, 35717}, {54, 37846}, {195, 19176}, {381, 14249}, {546, 35719}, {631, 44914}, {1593, 23709}, {3091, 11197}, {4993, 7691}, {5907, 44924}, {8799, 39530}, {8887, 42453}, {9825, 36212}, {18570, 34292}

X(45997) = midpoint of X(4) and X(42441)
X(45997) = reflection of X(i) in X(j) for these (i, j): (10184, 14635), (35719, 546)
X(45997) = complement of X(31388)
X(45997) = {X(3574), X(34836)}-harmonic conjugate of X(5)

X(45998) = X(2)X(104)∩X(6)X(517)

Barycentrics    a*(a^3 - a*b^2 + 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2)*(a^3 + a^2*b + a*b^2 + b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c - b^2*c + a*c^2 - b*c^2 + c^3) : :

X(45998) lies on the cubics K281, K386, K1243, K1245 and these lines: {2, 104}, {6, 517}, {999, 17720}, {4221, 35238}, {18516, 30444}

X(45998) = X(957)-isoconjugate of X(997)
X(45998) = barycentric quotient X(i)/X(j) for these {i,j}: {956, 17740}, {2267, 997}

X(45999) = X(4)X(572)∩X(57)X(1150)

Barycentrics    (a^2 + a*b + b^2 + a*c + b*c)*(a^2 + a*b + a*c + b*c + c^2)*(a^4 - b^4 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 + 2*b^2*c^2 - c^4) : :

X(45999) lies on the cubic K1245 and these lines: {4, 572}, {57, 1150}, {958, 19285}, {2214, 5880}

X(45999) = barycentric product X(377)*X(43531)
X(45999) = barycentric quotient X(i)/X(j) for these {i,j}: {377, 5224}, {37538, 386}

X(46000) = X(4)X(58)∩X(6)X(1437)

Barycentrics    a^2*(a^8 + a^7*b - a^6*b^2 - a^5*b^3 - a^4*b^4 - a^3*b^5 + a^2*b^6 + a*b^7 + a^7*c + 3*a^6*b*c - a^5*b^2*c - 5*a^4*b^3*c - a^3*b^4*c + a^2*b^5*c + a*b^6*c + b^7*c - a^6*c^2 - a^5*b*c^2 + 2*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + a*b^5*c^2 + 2*b^6*c^2 - a^5*c^3 - 5*a^4*b*c^3 + 2*a^2*b^3*c^3 - 3*a*b^4*c^3 - b^5*c^3 - a^4*c^4 - a^3*b*c^4 - 3*a^2*b^2*c^4 - 3*a*b^3*c^4 - 4*b^4*c^4 - a^3*c^5 + a^2*b*c^5 + a*b^2*c^5 - b^3*c^5 + a^2*c^6 + a*b*c^6 + 2*b^2*c^6 + a*c^7 + b*c^7) : :

X(46000) lies on the cubic K1245 and these lines: {4, 58}, {6, 1437}, {9, 255}, {57, 961}, {603, 21370}, {1150, 24537}, {1399, 40974}, {1737, 3075}, {37404, 37469}

This is the end of PART 23: Centers X(44001) - X(46000)

PART 1: Introduction and Centers X(1) - X(1000) PART 2: Centers X(1001) - X(3000) PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000) PART 5: Centers X(7001) - X(10000) PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000) PART 8: Centers X(14001) - X(16000) PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000) PART 11: Centers X(20001) - X(22000) PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000) PART 14: Centers X(26001) - X(28000) PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000) PART 17: Centers X(32001) - X(34000) PART 18: Centers X(34001) - X(36000)
PART 19: Centers X(36001) - X(38000) PART 20: Centers X(38001) - X(40000) PART 21: Centers X(40001) - X(42000)
PART 22: Centers X(42001) - X(44000) PART 23: Centers X(44001) - X(46000) PART 24: Centers X(46001) - X(48000)
PART 25: Centers X(48001) - X(50000) PART 26: Centers X(50001) - X(52000) PART 27: Centers X(52001) - X(54000)
PART 28: Centers X(54001) - X(56000) PART 29: Centers X(56001) - X(58000) PART 30: Centers X(58001) - X(60000)
PART 31: Centers X(60001) - X(62000) PART 32: Centers X(62001) - X(64000) PART 33: Centers X(64001) - X(66000)
PART 34: Centers X(66001) - X(68000) PART 35: Centers X(68001) - X(70000) PART 36: Centers X(70001) - X(72000)