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This is PART 18: Centers X(34001) - X(36000)

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


X(34001) =  X(3)X(33708)∩X(99)X(5201)

Barycentrics    a^2*(a^8*b^2 + 7*a^6*b^4 - 7*a^4*b^6 - a^2*b^8 + a^8*c^2 + 24*a^6*b^2*c^2 + 16*a^4*b^4*c^2 - 20*a^2*b^6*c^2 - b^8*c^2 + 7*a^6*c^4 + 16*a^4*b^2*c^4 - 8*a^2*b^4*c^4 - 3*b^6*c^4 - 7*a^4*c^6 - 20*a^2*b^2*c^6 - 3*b^4*c^6 - a^2*c^8 - b^2*c^8 + Sqrt[a^2*b^2 + a^2*c^2 + b^2*c^2]*(5*a^6*b^2 - 5*a^2*b^6 + 5*a^6*c^2 + 32*a^4*b^2*c^2 - 12*a^2*b^4*c^2 - 5*b^6*c^2 - 12*a^2*b^2*c^4 + 2*b^4*c^4 - 5*a^2*c^6 - 5*b^2*c^6)) : :

X(34001) lies on the cubic K1126 and these lines: {3,33708}, {99,5201}


X(34002) =  38TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34002) lies on these lines: {2,3}, {17,11516}, {18,11515}, {68,3796}, {141,10539}, {216,7755}, {323,2889}, {397,10635}, {398,10634}, {511,12242}, {524,13431}, {567,13142}, {1147,13394}, {1176,3519}, {1209,1503}, {1506,22052}, {1614,31831}, {3917,9820}, {5012,13292}, {5447,11064}, {5462,32269}, {5596,14530}, {5882,24301}, {5891,16252}, {5907,14862}, {6000,32348}, {6696,14855}, {7746,10979}, {8550,19131}, {8960,11514}, {10625,23292}, {10984,12359}, {11574,25555}, {11649,13433}, {11695,32223}, {11803,12606}, {11898,19119}, {12241,18555}, {12358,16534}, {13336,13567}, {14864,21243}, {29317,32396}

X(34002) = complement of X(15559)


X(34003) =  39TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34003) lies on these lines: {2,3}, {216,14773}, {511,32078}, {3289,22052}, {3928,26900}, {3929,26901}, {13409,26907}, {15107,31626}


X(34004) =  40TH HATZIPOLAKIS-MOSES-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 - 2*a^6*b^2*c^2 + 7*a^4*b^4*c^2 + 3*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 7*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 2*b^6*c^4 + 4*a^4*c^6 + 3*a^2*b^2*c^6 + 2*b^4*c^6 - 4*a^2*c^8 - 3*b^2*c^8 + c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34004) lies on these lines: {2,3}, {511,8254}, {1216,15806}, {1503,14076}, {3796,18356}, {5012,32165}, {5663,32348}, {6689,13391}, {10272,11793}, {11592,14156}, {32205,32223}

X(34004) = complement of X(33332)


X(34005) =  41ST HATZIPOLAKIS-MOSES-EULER POINT

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 + 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 - 2*a^6*c^4 - 4*a^4*b^2*c^4 + 16*a^2*b^4*c^4 - 2*b^6*c^4 + 8*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 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34005) lies on these lines: {2,3}, {1503,15062}, {3580,13403}, {3629,5889}, {3796,5925}, {5878,6800}, {5894,6293}, {6146,11440}, {6329,13434}, {7689,12022}, {9628,15326}, {11557,16111}, {11793,16163}, {12383,31831}, {12825,24981}, {14810,26156}


X(34006) =  42ND HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34006) lies on these lines: {2,3}, {542,2916}, {1154,6030}, {1350,9703}, {10610,13482}, {13353,21849}, {14627,21969}, {15037,22352}, {15038,15107}, {15080,15087}


X(34007) =  43RD HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34007) lies on these lines: {2,3}, {49,12383}, {54,17702}, {146,12162}, {184,12278}, {185,3448}, {265,13630}, {316,26166}, {511,32338}, {542,9972}, {1199,12370}, {1204,23293}, {1235,13219}, {1531,11793}, {1994,12233}, {2777,14076}, {2888,13754}, {2892,29959}, {3410,12111}, {3519,18363}, {3521,5663}, {3567,7706}, {3580,13568}, {3818,11439}, {4846,11457}, {5012,21659}, {5562,15108}, {5890,9927}, {5894,13203}, {6000,32337}, {6241,18474}, {6800,17845}, {7592,12293}, {7730,15103}, {7748,26216}, {9539,11392}, {9545,12118}, {9705,30714}, {9729,13851}, {9827,22948}, {10113,12006}, {10574,18392}, {10733,13403}, {11003,19467}, {11064,22555}, {11440,21243}, {11550,12279}, {11746,22466}, {12254,30522}, {13391,15800}, {13579,31363}, {13585,13599}, {14516,14683}, {15035,19479}, {15043,18390}, {15055,19506}, {15072,18381}, {15305,22802}, {15740,18434}, {18376,20791}, {18380,20792}, {18382,25406}, {18387,18917}, {18553,32247}, {29012,32340}, {32273,33749}, {32353,32369}

X(34007) = anticomplement of X(14118)


X(34008) =  44TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34008) lies on these lines: {2,3}, {13,16463}, {15,1337}, {16,15080}, {61,3060}, {62,5012}, {110,14538}, {184,11126}, {216,11421}, {302,33801}, {396,11142}, {511,11127}, {524,14173}, {577,11420}, {616,2925}, {617,1606}, {621,2923}, {627,1607}, {1495,11131}, {1993,5864}, {2004,5238}, {2979,14540}, {3098,11130}, {3442,22113}, {3455,22570}, {3796,22238}, {5237,6030}, {5615,11003}, {5873,19779}, {6582,14181}, {9736,14169}, {10409,16642}, {11542,21310}, {14705,30560}, {22236,33586}

X(34008) = complement of X(10210)
X(34008) = {X(3),X(22)}-harmonic conjugate of X(34009)


X(34009) =  45TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29355.

X(34009) lies on these lines: {2,3}, {14,16464}, {15,15080}, {16,1338}, {61,5012}, {62,3060}, {110,14539}, {184,11127}, {216,11420}, {303,33801}, {395,11141}, {511,11126}, {524,14179}, {577,11421}, {616,1605}, {617,2926}, {622,2924}, {628,1608}, {1495,11130}, {1993,5865}, {2005,5237}, {2979,14541}, {3098,11131}, {3443,22114}, {3455,22568}, {3796,22236}, {5238,6030}, {5611,11003}, {5872,19778}, {6295,14177}, {9735,14170}, {10410,16643}, {11543,21311}, {14704,30559}, {22238,33586}

X(34009) = {X(3),X(22)}-harmonic conjugate of X(34008)


X(34010) =  MIDPOINT OF X(3) AND X(33900)

Barycentrics    a^2*(a^8 - 4*a^6*b^2 + 4*a^2*b^6 - b^8 - 4*a^6*c^2 + 15*a^4*b^2*c^2 - 9*a^2*b^4*c^2 + 5*b^6*c^2 - 9*a^2*b^2*c^4 - 6*b^4*c^4 + 4*a^2*c^6 + 5*b^2*c^6 - c^8) : :

X(34010) lies on these lines: {3,543}, {23,111}, {182,1511}, {1995,9172}, {2793,14270}, {2847,14657}, {2930,18800}, {6644,14702}, {6719,11284}, {7496,10717}, {7617,10204}, {12106,15562}, {15560,33695}, {15922,33980}, {23699,31861}

X(34010) = midpoint of X(3) and X(33900)
X(34010) = reflection of X(14662) in X(111)
X(34010) = circumcircle-inverse of X(11632)
X(34010) = crosssum of X(1648) and X(9023)
X(34010) = singular focus of the cubic K1126


X(34011) =  MIDPOINT OF X(3) AND X(33998)

Barycentrics    a^2*(a^14 - 8*a^12*b^2 + 6*a^10*b^4 + 15*a^8*b^6 - 15*a^6*b^8 - 6*a^4*b^10 + 8*a^2*b^12 - b^14 - 8*a^12*c^2 + 83*a^10*b^2*c^2 - 144*a^8*b^4*c^2 + 7*a^6*b^6*c^2 + 143*a^4*b^8*c^2 - 90*a^2*b^10*c^2 + 9*b^12*c^2 + 6*a^10*c^4 - 144*a^8*b^2*c^4 + 338*a^6*b^4*c^4 - 225*a^4*b^6*c^4 + 144*a^2*b^8*c^4 - 7*b^10*c^4 + 15*a^8*c^6 + 7*a^6*b^2*c^6 - 225*a^4*b^4*c^6 - 92*a^2*b^6*c^6 - b^8*c^6 - 15*a^6*c^8 + 143*a^4*b^2*c^8 + 144*a^2*b^4*c^8 - b^6*c^8 - 6*a^4*c^10 - 90*a^2*b^2*c^10 - 7*b^4*c^10 + 8*a^2*c^12 + 9*b^2*c^12 - c^14) : :

X(34011) lies on these lines: {3,9172}, {23,2696}, {6644,12584}, {28662,33851}

X(34011) = midpoint of X(3) and X(33998)
X(34011) = singular focus of the cubic K1127


X(34012) =  X(3)X(9027)∩X(1296)X(1995)

Barycentrics    a^2*(3*a^14 - 23*a^12*b^2 + 15*a^10*b^4 + 45*a^8*b^6 - 39*a^6*b^8 - 21*a^4*b^10 + 21*a^2*b^12 - b^14 - 23*a^12*c^2 + 266*a^10*b^2*c^2 - 581*a^8*b^4*c^2 - 60*a^6*b^6*c^2 + 599*a^4*b^8*c^2 - 206*a^2*b^10*c^2 + 5*b^12*c^2 + 15*a^10*c^4 - 581*a^8*b^2*c^4 + 2566*a^6*b^4*c^4 - 1986*a^4*b^6*c^4 + 235*a^2*b^8*c^4 + 7*b^10*c^4 + 45*a^8*c^6 - 60*a^6*b^2*c^6 - 1986*a^4*b^4*c^6 + 924*a^2*b^6*c^6 - 11*b^8*c^6 - 39*a^6*c^8 + 599*a^4*b^2*c^8 + 235*a^2*b^4*c^8 - 11*b^6*c^8 - 21*a^4*c^10 - 206*a^2*b^2*c^10 + 7*b^4*c^10 + 21*a^2*c^12 + 5*b^2*c^12 - c^14) : :

X(34012) lies on the cubics K1118 and K1127 and on these lines: {3,9027}, {1296,1995}


X(34013) =  X(3)X(67)∩X(23)X(99)

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

X(34013) lies on the cubics K903 and K1127 and on these lines: {2,13233}, {3,67}, {23,99}, {25,15300}, {39,32740}, {110,9177}, {114,31861}, {115,11284}, {187,9027}, {237,33972}, {543,1995}, {574,2502}, {671,16042}, {3148,33929}, {5026,8546}, {5162,33756}, {6054,7464}, {7530,10992}, {8591,14002}, {8723,9135}, {14915,18860}, {20897,33975}

X(34013) = isogonal conjugate of isotomic conjugate of X(36883)
X(34013) = circumcircle-inverse of X(5648)
X(34013) = crossdifference of every pair of points on line {2492, 9185}
X(34013) = X(2482),X(2936)}-harmonic conjugate of X(3455)


X(34014) =  X(3)X(33850)∩X(187)X(1296)

Barycentrics    a^2*(a^12 - 8*a^10*b^2 - 19*a^8*b^4 + 19*a^4*b^8 + 8*a^2*b^10 - b^12 - 8*a^10*c^2 + 121*a^8*b^2*c^2 - 78*a^6*b^4*c^2 + 112*a^4*b^6*c^2 - 98*a^2*b^8*c^2 + 15*b^10*c^2 - 19*a^8*c^4 - 78*a^6*b^2*c^4 - 174*a^4*b^4*c^4 + 26*a^2*b^6*c^4 - 75*b^8*c^4 + 112*a^4*b^2*c^6 + 26*a^2*b^4*c^6 + 250*b^6*c^6 + 19*a^4*c^8 - 98*a^2*b^2*c^8 - 75*b^4*c^8 + 8*a^2*c^10 + 15*b^2*c^10 - c^12) : :

X(34014) lies on the cubic K1127 and these lines: {3,33850}, {187,1296}, {353,5085}, {11580,14688}


X(34015) =  X(3)X(524)∩X(1296)X(9027)

Barycentrics    a^2*(a^12 - 2*a^10*b^2 - 7*a^8*b^4 + 7*a^4*b^8 + 2*a^2*b^10 - b^12 - 2*a^10*c^2 + 13*a^8*b^2*c^2 - 42*a^6*b^4*c^2 - 26*a^4*b^6*c^2 + 28*a^2*b^8*c^2 - 3*b^10*c^2 - 7*a^8*c^4 - 42*a^6*b^2*c^4 + 510*a^4*b^4*c^4 - 286*a^2*b^6*c^4 + 33*b^8*c^4 - 26*a^4*b^2*c^6 - 286*a^2*b^4*c^6 + 70*b^6*c^6 + 7*a^4*c^8 + 28*a^2*b^2*c^8 + 33*b^4*c^8 + 2*a^2*c^10 - 3*b^2*c^10 - c^12) : :

X(34015) lies on the cubic K1127 and these lines: {3,524}, {1296,9027}, {6082,9084}






leftri  Perspectors involving Talitha triangles: X(34016)-X(34088)  rightri
This preamble and centers X(34016)-X(34088) were contributed by Clark Kimberling and Peter Moses, August 19, 2019.

Suppose that P = p : q : r is a point. The P-Talitha triangle of X, denoted by TT(P,X), is the triangle A'B'C' is defined by its vertices:

A' = -p/(y - z) : q/(z + x) : -r/(x + y)
B' = -p/(y + z) : -q/(z - x) : r/(x + y)
C' = p/(y + z) : -q/(z + x) : -r/(x - y).

The triangle A'B'C' is inscribed in the circumconic -p y z + q z x + r x y = 0.

The name: Talitha is a star on the front paw of Ursa Major.

In the following table, contributed by César Lozada, August 21, 2019, the notation X(i) is abbreviated as i. For example,

TT(2,1)         excentral       17277

means that the X(2)-Talitha triangle of X(1) is perspective to the excentral triangle and that the perspector is X(17277). See here for definitions of named triangles.

Talitha triangle 2nd triangle perspector Talitha triangle 2nd triangle perspector Talitha triangle 2nd triangle perspector Talitha triangle 2nd triangle perspector
TT(2,1) excentral 17277 TT(6,7) 1st Sharygin 34027 TT(1,1) Gemini 1 21 TT(1,6) anticevian of X(33760) 1
TT(2,1) Steiner 100 TT(6,7) Honsberger 34028 TT(1,1) Gemini 8 21 TT(1,6) anticevian of X(75) 34065
TT(2,1) tangential of excentral 75 TT(6,7) 3rd Euler 34029 TT(1,1) Gemini 18 100 TT(1,6) anticevian of X(82) 34066
TT(2,1) Gemini 2 333 TT(6,7) 4th Euler 34030 TT(1,1) Gemini 19 34053 TT(1,6) anticevian of X(660) 34067
TT(2,1) Gemini 18 100 TT(6,7) 2nd Pamfilos-Zhou 34031 TT(1,1) Gemini 19 34053 TT(1,6) anticevian of X(1156) 34068
TT(2,1) Gemini 23 86 TT(6,7) 2nd extouch 34032 TT(1,6) cevian of X(1580) 34054 TT(1,6) anticevian of X(1492) 34069
TT(2,1) Gemini 90 34016 TT(6,7) 6th mixtilinear 34033 TT(1,6) cevian of X(3112) 34055 TT(1,6) anticevian of X(2236) 34070
TT(2,1) Gemini 92 34017 TT(6,7) outer tangential mid-arc 34034 TT(1,7) medial 3160 TT(1,6) anticevian of X(4598) 34071
TT(2,6) tangential 1078 TT(6,7) 1st Conway 34035 TT(1,6) anticomplementary 347 TT(1,6) anticevian of X(4599) 34072
TT(2,6) circummedial 1799 TT(6,7) incircle-inverse of ABC 34036 TT(1,6) 1st circumperp 934 TT(1,6) anticevian of X(4604) 34073
TT(2,6) Ara 5594 TT(6,7) outer Hutson 34038 TT(1,6) inner Conway 651 TT(1,6) anticevian of X(4606) 34074
TT(2,6) Gemini 44 1799 TT(6,7) T(-1,3) 34039 TT(1,6) Gemini 1 77 TT(1,6) anticevian of X(4607) 34075
TT(2,7) 1st circumperp 927 TT(6,7) Hutson intouch 34040 TT(1,6) Gemini 2 17080 TT(1,6) anticevian of X(8052) 34076
TT(2,7) Gemini 7 34018 TT(6,7) Atik 34041 TT(1,6) Gemini 7 340560 TT(1,6) anticevian of X(20332) 34077
TT(2,7) Gemini 30 (inner Conway) 658 TT(6,7) Ascella 34042 TT(1,6) Gemini 13 31600 TT(1,6) anticevian of X(20332) 34077
TT(2,7) Gemini 85 34019 TT(6,7) reflection of X(1) in BC,CA,AB 34043 TT(1,6) Gemini 30 651 TT(1,6) anticevian of X(23707) 34078
TT(2,7) Soddy 348 TT(6,7) 3rd Conway 34044 TT(1,6) Gemini 33 31526 TT(1,6) anticevian of X(24624) 34079
TT(2,75) excentral 34020 TT(6,7) /Conway-circle-inverse of ABC 34045 TT(1,6) Gemini 34 34057 TT(1,6) anticevian of X(27834) 34080
TT(2,75) Gemini 7 274 TT(6,7) incircle-circles 34046 TT(1,6) Gemini 60 34058 TT(1,6) anticevian of X(33295) 34081
TT(2,75) Gemini 17 1978 TT(6,7) 1st excosine 34047 TT(1,6) Gemini 61 34059 TT(1,6) anticevian of X(33764) 34082
TT(2,75) Gemini 19 34021 TT(6,7) 2nd Zaniah 34048 TT(1,6) Gemini 65 34060 TT(1,75) Soddy 3160
TT(2,75) Gemini 90 34022 TT(6,7) Ursa-Major 34049 TT(1,6) Gemini 75 34061 TT(1,75) tangential 34063
TT(2,75) Gemini 92 34023 TT(6,7) Wasat 34050 TT(1,6) Gemini 75 34061 TT(1,75) Steiner 190
TT(2,75) Gemini 104 31008 TT(6,7) Gemini 7 34051 TT(1,6) Gemini 100 34062 TT(1,75) Wasat 671
TT(2,514) medial 190 TT(6,7) Gemini 15 34052 TT(1,6) anticevian of X(31) 18042 TT(1,75) Gemini 1 86
TT(2,514) Gemini 110 34024 TT(6,75) excentral 32911 TT(1,6) anticevian of X(88) 9456 TT(1,75) Gemini 17 190
TT(6,1) excentral 572 TT(6,7) Steiner 190 TT(1,6) anticevian of X(100) 692 TT(1,75) Gemini 24 33296
TT(6,1) tangential of excentral 48 TT(6,7) Wasat 24624 TT(1,6) anticevian of X(162) 32676 TT(1,75) Gemini 105 34064
TT(6,6) anticomplementary 19121 TT(6,7) Gemini 2 81 TT(1,6) anticevian of X(190) 101 TT(1,514) incentral 100
TT(6,6) tangentail of anticomplementary 206 TT(6,7) Gemini 7 81 TT(1,6) anticevian of X(651) 1415 TT(75,1) anticomplementary 17143
TT(6,6) anti-Honsberger 1176 TT(6,7) Gemini 17 190 TT(1,6) anticevian of X(653) 32674 TT(75,1) 2nd Conway 76
TT(6,6) Gemini 43 1176 TT(6,7) Gemini 54 86 TT(1,6) anticevian of X(655) 32675 TT(75,1) /Conway-circle-inverse of ABC 314
TT(6,7) incentral 1419 TT(6,7) Gemini 68 27644 TT(1,6) anticevian of X(658) 1461 TT(75,1) Garcia reflection 314
TT(6,7) excentral 222 TT(6,7) Gemini 2 81 TT(1,6) anticevian of X(662) 163 TT(75,1) Gemini 8 314
TT(6,7) intouch 222 TT(6,7) Gemini 2 81 TT(1,6) anticevian of X(673) 1438 TT(75,1) Gemini 18 668
TT(6,7) 1st circumperp 109 TT(1,1) anticomplementary 2975 TT(1,6) anticevian of X(799) 662 TT(75,6) excentral 33764
TT(6,7) 2nd circumperp 10571 TT(1,1) tangential 1621 TT(1,6) anticevian of X(823) 24019 TT(75,7) 1st circumperp 36083
TT(6,7) hexyl 1394 TT(1,1) Feuerbach 2 TT(1,6) anticevian of X(897) 923 TT(75,7) Gemini 7 36084
TT(6,7) 2nd Conway 18623 TT(1,1) 2nd circumperp 21 TT(1,6) anticevian of X(1821) 1910 TT(75,7) Gemini 30 (inner Conway) 36085
TT(6,7) infinite altitude 221 TT(1,1) 1st Sharygin 21 TT(1,6) anticevian of X(2349) 2159 TT(75,75) anticomplementary 36086
TT(6,7) inner Conway 651 TT(1,1) Steiner 100 TT(1,6) anticevian of X(2580) 2576 TT(75,75) Wasat 36087
TT(6,7) Gemini 30 651 TT(1,1) 1st Conway 21 TT(1,6) anticevian of X(2581) 2577 TT(75,75) Gemini 17 1978
TT(6,7) inner tangential mid-arc 34025 TT(1,1) 2nd Conway 3 TT(1,6) anticevian of X(3257) 32665 TT(75,75) Gemini 105 36088
TT(6,7) Yff central 34026 TT(1,1) Garcia reflection 21 TT(1,6) anticevian of X(32680) 32678 TT(6,7) Hutson intouch 34040

underbar

X(34016) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(1)) AND GEMINI 90

Barycentrics    (a + b)*(a + c)*(a^2 - b^2 - b*c - c^2) : :
Barycentrics    csc A cot A' : :, where A'B'C' is the incentral triangle

X(34016) lies on these lines: {1,6626}, {2,1171}, {8,99}, {10,6629}, {21,33297}, {35,319}, {57,85}, {58,30966}, {60,7796}, {63,33935}, {69,261}, {75,267}, {76,799}, {81,17397}, {86,3624}, {90,314}, {190,33775}, {191,17762}, {348,4573}, {599,25536}, {662,17346}, {668,7095}, {671,23942}, {757,5224}, {763,1654}, {873,5278}, {1098,4592}, {1125,16480}, {1414,33298}, {1634,23851}, {1931,3661}, {1963,17248}, {2003,17095}, {3219,33939}, {3578,7799}, {3616,33770}, {3926,7058}, {4416,27691}, {4589,22116}, {4590,9273}, {4593,29534}, {4643,19623}, {4690,16702}, {4921,16712}, {5209,6376}, {5361,24587}, {5372,18135}, {5739,7763}, {5741,7769}, {7096,16551}, {7793,17343}, {14829,18140}, {16552,20371}, {16605,16756}, {16704,16705}, {16816,24617}, {16825,18827}, {16833,24378}, {16887,33295}, {17201,30941}, {17277,29459}, {17381,33766}, {19742,27162}, {21839,27954}, {24044,24074}, {29433,29479}, {29742,29764}

X(34016) = isotomic conjugate of X(8818)
X(34016) = trilinear pole of line X(2605)X(3268)


X(34017) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(1)) AND GEMINI 92

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

X(34017) lies on these lines: {2,594}, {75,29561}, {308,310}, {673,29437}, {17026,18044}, {17031,18082}, {17277,18046}, {18040,29447}, {29446,29456}, {29453,29455}


X(34018) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(7)) AND GEMINI 7

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

X(34018) lies on these lines: {1,85}, {2,4554}, {7,1002}, {37,31618}, {57,658}, {76,30701}, {81,1462}, {105,927}, {274,16699}, {277,348}, {278,13149}, {291,1738}, {294,1170}, {331,17905}, {955,14548}, {985,1416}, {1086,4569}, {1257,20911}, {1390,1441}, {1434,29775}, {1814,2982}, {1920,32017}, {4624,25430}, {5376,20568}, {8056,27829}, {9445,11019}, {15474,17076}, {17088,21907}, {18140,32019}, {20173,21609}

X(34018) = isotomic conjugate of X(3693)
X(34018) = trilinear pole of line X(7)X(885)


X(34019) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(7)) AND GEMINI 85

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

X(34019) lies on these lines: {2,31618}, {75,31627}, {144,4569}, {346,4554}, {348,17075}, {1229,7182}, {3160,33677}, {18135,20946}, {20935,31527}, {23062,30854}


X(34020) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(75)) AND EXCENTRAL

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

X(34020) lies on these lines: {1,6384}, {2,39}, {43,668}, {57,4554}, {75,6682}, {99,4203}, {100,18064}, {192,6383}, {334,29671}, {350,3663}, {561,4850}, {799,7304}, {811,4219}, {870,29650}, {1018,29391}, {1909,6685}, {1920,3666}, {1921,3752}, {1965,29821}, {1966,17596}, {1978,17147}, {3210,6382}, {3674,7196}, {3741,17143}, {3821,7018}, {4000,27314}, {4595,29708}, {4623,5035}, {6376,16569}, {6381,6686}, {7244,17593}, {8049,31002}, {10009,17490}, {10436,25502}, {10446,30962}, {14135,15488}, {14829,30940}, {16574,29557}, {17790,25350}, {18046,29470}, {18148,33296}, {20284,32453}, {23632,26974}, {24169,30631}, {29759,29764}, {30632,33125}


X(34021) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(75)) AND GEMINI 19

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

X(34021) lies on these lines: {37,670}, {76,10472}, {86,2665}, {141,18140}, {256,314}, {274,1107}, {334,20339}, {799,894}, {1966,6626}, {4554,27691}, {5283,6374}, {7199,27854}, {8033,10436}, {21246,29968}, {27269,32746}


X(34022) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(75)) AND GEMINI 90

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

X(34022) lies on these lines: {6,799}, {37,30938}, {86,87}, {192,670}, {274,4751}, {310,27164}, {873,15668}, {2669,6376}, {4625,6180}, {8053,18064}, {18140,18143}, {29742,29764}


X(34023) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(75)) AND GEMINI 92

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

X(34023) lies on these lines: {1,29447}, {2,1500}, {81,83}, {668,3780}, {5263,29484}, {16549,29764}, {17277,18046}, {21787,29486}, {24512,29757}, {29438,29454}, {29456,29476}


X(34024) =  PERSPECTOR OF THESE TRIANGLES: TT(X(2),X(514)) AND GEMINI 110

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

X(34024) lies on these lines: {190,514}, {664,30726}, {4422,6630}, {6632,14475}


X(34025) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND INNER TANGENTIAL MID-ARC

Barycentrics    a*(a + b - c)*(a - b + c)*((a - b - c)^2*(a + b - c)*(a - b + c) + 2*a*(a - b - c)*(a^2 - b^2 - c^2)*Sin[A/2] - 2*(a - c)*(a - b + c)*(a^2 - b^2 + c^2)*Sin[B/2] - 2*(a - b)*(a + b - c)*(a^2 + b^2 - c^2)*Sin[C/2]) : :

X(34025) lies on these lines: {1, 8095}, {57, 266}, {109, 8075}, {177, 8120}, {221, 8091}, {222, 2089}, {223, 8078}, {651, 11690}, {1394, 8081}, {1455, 18448}, {1456, 10503}, {8077, 10571}, {8093, 21147}, {9793, 18623}


X(34026) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND YFF CENTRAL

Barycentrics    a*(a + b - c)*(a - b + c)*(a*(a^2 - b^2 - c^2) + 2*b*(a - b - c)*c*Sin[A/2]) : :

X(34026) lies on these lines: {1, 12685}, {57, 289}, {109, 7589}, {173, 223}, {174, 222}, {177, 8120}, {221, 8351}, {651, 8126}, {1394, 7590}, {1455, 18454}, {1456, 10502}, {3157, 8130}, {7587, 10571}, {8125, 17074}, {11891, 18623}, {12445, 21147}


X(34027) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 1st SHARYGIN

Barycentrics    a*(a + b - c)*(a - b + c)*(a^6 - a^5*b + 2*a^3*b^3 - a^2*b^4 - a*b^5 - a^5*c + 3*a^4*b*c - 2*a^2*b^3*c - a*b^4*c + b^5*c + 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(34027) lies on these lines: {1, 12683}, {21, 10571}, {31, 1423}, {109, 4220}, {221, 9840}, {222, 1284}, {223, 846}, {651, 11688}, {1394, 8235}, {1455, 30285}, {1456, 17611}, {2292, 21147}, {3677, 11031}, {9791, 18623}


X(34028) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND HONSBERGER

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

X(34028) lies on these lines: {1, 12669}, {6, 279}, {7, 27}, {9, 77}, {48, 934}, {109, 7676}, {142, 17074}, {144, 394}, {219, 3160}, {221, 390}, {223, 1445}, {347, 23144}, {348, 2287}, {516, 3562}, {518, 4296}, {610, 7177}, {971, 6198}, {1255, 1422}, {1394, 7675}, {1409, 14189}, {1414, 7054}, {1439, 7291}, {1440, 24553}, {1449, 2124}, {1455, 30284}, {1456, 5572}, {2257, 4350}, {3157, 5759}, {3945, 6180}, {4318, 15185}, {5686, 9370}, {5762, 23070}, {5781, 10004}, {5817, 8757}, {7672, 21147}, {7677, 10571}, {8732, 24597}


X(34029) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 3rd EULER

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

X(34029) lies on these lines: {1, 4}, {2, 109}, {5, 221}, {7, 33107}, {11, 222}, {227, 12699}, {499, 603}, {608, 5747}, {651, 11680}, {664, 11185}, {859, 1470}, {908, 8270}, {1038, 21616}, {1214, 24703}, {1394, 8227}, {1455, 5886}, {1456, 17605}, {1465, 1836}, {1630, 3087}, {1707, 3911}, {1771, 6834}, {1777, 6833}, {1935, 26363}, {2003, 11269}, {2887, 26364}, {2969, 14593}, {3006, 28997}, {3434, 4551}, {3676, 7056}, {4318, 31053}, {5057, 17080}, {5226, 33112}, {7078, 15908}, {7737, 17966}, {8757, 26470}, {9370, 24390}, {9779, 18623}, {10200, 25490}, {12609, 19372}, {15601, 31231}, {17194, 26105}, {17625, 17721}, {25958, 28780}, {28018, 28034}, {28074, 28086}, {29007, 29664}


X(34030) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 4th EULER

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

X(34030) lies on these lines: {1, 6833}, {2, 10571}, {4, 109}, {5, 221}, {10, 1038}, {12, 222}, {34, 1737}, {40, 1076}, {46, 225}, {57, 5230}, {65, 5292}, {73, 498}, {119, 8757}, {223, 1698}, {227, 26446}, {278, 1714}, {355, 1455}, {478, 5816}, {499, 1457}, {603, 1478}, {651, 11681}, {664, 7763}, {1068, 1735}, {1118, 1782}, {1158, 1785}, {1214, 26066}, {1394, 5587}, {1409, 5747}, {1421, 28074}, {1456, 17606}, {1465, 24914}, {1726, 14257}, {1854, 15252}, {1877, 10826}, {3075, 26332}, {3086, 32486}, {3340, 11269}, {3911, 11512}, {4306, 17734}, {4551, 5552}, {4848, 33137}, {5711, 15844}, {5713, 19349}, {5930, 6684}, {6718, 10570}, {6734, 8270}, {7952, 14647}, {9316, 21935}, {9370, 17757}, {9780, 18623}, {9817, 12617}, {9948, 16870}, {12709, 17720}, {15253, 17054}, {17751, 28774}, {18838, 24159}, {24806, 26363}


X(34031) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 2nd PAMFILOS-ZHOU

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

X(34031) lies on these lines: {1, 12681}, {109, 8224}, {221, 7596}, {222, 8243}, {223, 8231}, {651, 11687}, {1394, 8234}, {1419, 7133}, {1456, 17610}, {1659, 7595}, {8225, 10571}, {9789, 18623}, {9808, 21147}


X(34032) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 2nd EXTOUCH

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

X(34032) lies on these lines: {1, 12664}, {4, 221}, {6, 278}, {9, 223}, {10, 10361}, {40, 15498}, {57, 2182}, {72, 9370}, {108, 154}, {109, 7580}, {196, 3197}, {218, 10402}, {222, 226}, {225, 5706}, {329, 394}, {405, 10571}, {1020, 7011}, {1035, 1745}, {1060, 5777}, {1068, 1181}, {1260, 4551}, {1394, 1490}, {1419, 1422}, {1425, 4185}, {1435, 2261}, {1455, 18446}, {1456, 1864}, {1465, 1708}, {1498, 7952}, {2122, 6260}, {3157, 5812}, {3330, 17056}, {3713, 26942}, {5759, 7074}, {5905, 23144}, {5930, 7078}, {12848, 32911}, {17903, 23982}, {19354, 23710}, {23986, 32714}


X(34033) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 6th MIXTILINEAR

Barycentrics    a*(a + b - c)*(a - b + c)*(5*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(34033) lies on these lines: {1, 84}, {31, 269}, {33, 3062}, {34, 3339}, {35, 1035}, {40, 22117}, {47, 15803}, {55, 1419}, {57, 1456}, {77, 4512}, {81, 12560}, {109, 165}, {200, 651}, {204, 32714}, {278, 4312}, {478, 1743}, {516, 18623}, {603, 3361}, {1406, 1467}, {1407, 7290}, {1457, 13462}, {1461, 2187}, {1707, 5018}, {1935, 5234}, {2093, 9572}, {2951, 7070}, {2999, 9316}, {3052, 6610}, {3157, 6769}, {3562, 12651}, {4296, 12526}, {4337, 30282}, {4882, 9370}, {5223, 8270}, {5269, 6180}, {7987, 10571}, {7991, 21147}, {9364, 23511}, {10582, 17074}, {18594, 30456}


X(34034) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 2nd (OUTER) TANGENTIAL MID-ARC

Barycentrics    a*(a + b - c)*(a - b + c)*((a - b - c)^2*(a + b - c)*(a - b + c) - 2*a*(a - b - c)*(a^2 - b^2 - c^2)*Sin[A/2] + 2*(a - c)*(a - b + c)*(a^2 - b^2 + c^2)*Sin[B/2] + 2*(a - b)*(a + b - c)*(a^2 + b^2 - c^2)*Sin[C/2]) : :

X(34034) lies on these lines: {1, 8095}, {109, 8076}, {174, 222}, {221, 8092}, {223, 258}, {651, 8125}, {1394, 8082}, {1455, 18456}, {1456, 10501}, {3157, 8129}, {7588, 10571}, {8094, 21147}, {8126, 17074}, {9795, 18623}


X(34035) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 1st CONWAY

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

X(34035) lies on these lines: {1, 9960}, {7, 27}, {20, 221}, {21, 10571}, {57, 7147}, {63, 223}, {109, 7411}, {285, 3616}, {347, 394}, {411, 3561}, {934, 7125}, {1071, 1870}, {1394, 10884}, {1455, 18444}, {1456, 10391}, {1465, 32911}, {1993, 9965}, {3562, 5930}, {3868, 21147}, {4318, 16465}, {5249, 17074}, {5712, 6180}, {16049, 19367}


X(34036) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND INCIRCLE-INVERSE OF ABC

Barycentrics    a*(a + b - c)*(a - b + c)*(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(34036) lies on these lines: {1, 4}, {2, 4318}, {6, 5173}, {7, 7191}, {9, 25885}, {10, 19372}, {31, 57}, {46, 602}, {55, 1465}, {56, 1448}, {65, 16466}, {77, 4666}, {78, 5300}, {79, 15176}, {169, 5452}, {221, 942}, {222, 354}, {227, 3295}, {238, 1708}, {255, 12704}, {269, 2191}, {516, 1040}, {517, 7074}, {603, 3338}, {612, 5219}, {651, 3873}, {664, 21609}, {975, 11375}, {990, 1836}, {997, 2887}, {998, 1411}, {999, 1455}, {1001, 1214}, {1038, 1125}, {1042, 28082}, {1060, 5886}, {1062, 12699}, {1193, 4332}, {1254, 3915}, {1279, 1427}, {1394, 3333}, {1407, 3660}, {1420, 4320}, {1423, 28356}, {1441, 24552}, {1442, 29814}, {1621, 17080}, {1709, 7004}, {1718, 25415}, {1722, 4848}, {1758, 8616}, {1854, 9856}, {1943, 10453}, {2000, 11680}, {2262, 22124}, {2285, 16470}, {2807, 11436}, {3006, 28776}, {3100, 9812}, {3555, 9370}, {3616, 4296}, {3669, 11193}, {3677, 11031}, {3772, 15253}, {3811, 4865}, {3817, 9817}, {3827, 21370}, {3870, 4551}, {3911, 5272}, {3920, 5226}, {3947, 30145}, {3952, 28996}, {3966, 26942}, {4138, 30144}, {4298, 30148}, {4319, 9580}, {4323, 17016}, {4327, 4654}, {4449, 23615}, {5018, 29820}, {5262, 8900}, {5435, 7292}, {5901, 32047}, {6180, 14523}, {6505, 30985}, {7226, 29007}, {7672, 32911}, {7956, 15252}, {8225, 13388}, {8543, 28606}, {8743, 20613}, {9364, 17063}, {10436, 24566}, {10580, 18623}, {11496, 17102}, {11723, 19469}, {11735, 19505}, {12514, 25540}, {12703, 24028}, {13389, 31546}, {14594, 18743}, {16823, 27339}, {17140, 28968}, {17165, 28997}, {18447, 18493}, {19861, 24984}, {20588, 23693}, {24806, 25496}, {28780, 29667}


X(34037) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND INNER HUTSON

Barycentrics    a*(a + b - c)*(a - b + c)*(-5*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 + 2*a*(a^2 - b^2 - c^2)*Sin[A/2] + 2*(a - c)*(a^2 - b^2 + c^2)*Sin[B/2] + 2*(a - b)*(a^2 + b^2 - c^2)*Sin[C/2]) : :

X(34037) lies on these lines: {1, 12673}, {109, 8107}, {221, 9836}, {222, 8113}, {223, 363}, {651, 11685}, {1394, 8111}, {1456, 17607}, {8109, 10571}, {9783, 18623}, {9805, 21147}


X(34038) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND OUTER HUTSON

Barycentrics    a*(a + b - c)*(a - b + c)*(5*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 + 2*a*(a^2 - b^2 - c^2)*Sin[A/2] + 2*(a - c)*(a^2 - b^2 + c^2)*Sin[B/2] + 2*(a - b)*(a^2 + b^2 - c^2)*Sin[C/2]) : :

X(34038) lies on these lines: {1, 12674}, {109, 8108}, {168, 223}, {221, 9837}, {222, 8114}, {651, 11686}, {1394, 8112}, {1456, 17608}, {8110, 10571}, {9787, 18623}, {9806, 21147}


X(34039) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND T(-1,3)

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

The triangle T(-1,3) is defined in TCCT, article 6.42.

X(34039) lies on these lines: {1, 4}, {40, 1455}, {109, 7991}, {145, 18623}, {221, 7982}, {222, 3340}, {227, 3576}, {280, 1219}, {517, 1394}, {603, 2093}, {610, 14571}, {651, 11682}, {664, 21605}, {1035, 22770}, {1038, 9623}, {1419, 7201}, {1420, 1465}, {1448, 15955}, {1456, 2098}, {1854, 10864}, {2800, 2956}, {4853, 8270}, {5048, 10964}, {20076, 22464}


X(34040) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND HUTSON INTOUCH

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

X(34040) lies on these lines: {1, 84}, {3, 945}, {6, 3340}, {19, 22124}, {33, 9856}, {34, 517}, {40, 1465}, {55, 10571}, {56, 106}, {57, 1191}, {65, 16466}, {73, 1480}, {81, 4323}, {145, 651}, {218, 4559}, {219, 608}, {223, 1697}, {225, 12699}, {226, 5710}, {227, 5119}, {255, 22770}, {278, 412}, {355, 1877}, {392, 1038}, {394, 11682}, {405, 24806}, {519, 9370}, {603, 999}, {611, 2647}, {952, 8757}, {956, 1935}, {960, 8270}, {990, 17634}, {995, 1466}, {1042, 1617}, {1106, 1149}, {1201, 9316}, {1203, 18421}, {1214, 5250}, {1319, 1406}, {1399, 26437}, {1407, 1420}, {1421, 17054}, {1456, 3057}, {1464, 11510}, {1470, 18360}, {1482, 3157}, {1718, 5903}, {1771, 22753}, {1943, 4673}, {2099, 15955}, {2390, 22654}, {3256, 4255}, {3339, 5315}, {3445, 5193}, {3485, 5711}, {3622, 17074}, {3660, 28011}, {3753, 19372}, {3877, 4296}, {3878, 4347}, {3913, 4551}, {4383, 4848}, {4696, 28997}, {5930, 10624}, {6180, 10106}, {6244, 22072}, {6357, 30305}, {7074, 7991}, {8148, 23071}, {8158, 22117}, {9364, 21214}, {9785, 18623}, {10247, 23070}, {10306, 22350}, {11520, 23144}, {12053, 16388}, {14594, 19582}, {15829, 17811}


X(34041) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND ATIK

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

X(34041) lies on these lines: {1, 9948}, {8, 1943}, {33, 3062}, {109, 7070}, {221, 9856}, {222, 3745}, {223, 8580}, {612, 1419}, {651, 11678}, {1038, 12447}, {1394, 10864}, {1455, 30283}, {1456, 17604}, {2124, 28043}, {7955, 8916}, {8583, 10571}


X(34042) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND ASCELLA

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

X(34042) lies on these lines: {1, 9942}, {3, 102}, {6, 57}, {226, 2050}, {227, 5709}, {277, 5723}, {278, 940}, {394, 17080}, {651, 5744}, {942, 21147}, {1035, 4303}, {1214, 17811}, {1394, 8726}, {1406, 1466}, {1413, 9940}, {1455, 18443}, {1456, 17603}, {1480, 24929}, {1498, 17102}, {3218, 23144}, {4617, 14256}, {5718, 6180}, {6611, 7125}, {8732, 24597}, {9776, 17074}, {17086, 26625}, {17810, 20122}


X(34043) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND REFLECTION OF X(1) IN SIDES BC,CA,AB

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

X(34043) lies on these lines: {1, 84}, {6, 3339}, {10, 651}, {31, 4306}, {34, 5902}, {35, 73}, {36, 47}, {40, 1419}, {46, 223}, {56, 2163}, {57, 1203}, {58, 1042}, {65, 267}, {77, 12514}, {79, 225}, {81, 3671}, {165, 7078}, {191, 1214}, {227, 484}, {386, 9316}, {394, 12526}, {501, 4565}, {516, 3562}, {517, 23070}, {595, 1458}, {758, 4296}, {942, 1456}, {995, 1106}, {1035, 11507}, {1038, 5692}, {1044, 1754}, {1046, 1409}, {1060, 5693}, {1125, 17074}, {1191, 13462}, {1399, 1464}, {1407, 3361}, {1427, 15932}, {1428, 17114}, {1442, 3743}, {1457, 5563}, {1465, 3336}, {1466, 5313}, {1735, 3468}, {1745, 1771}, {1768, 17102}, {1770, 5930}, {1781, 30456}, {1870, 5884}, {1935, 5251}, {1943, 4647}, {2594, 3256}, {2771, 18447}, {3062, 15811}, {3194, 32714}, {3216, 9364}, {3220, 14529}, {3340, 16474}, {3428, 23072}, {3579, 23071}, {3679, 9370}, {3868, 4347}, {3869, 22128}, {3874, 4318}, {4295, 18623}, {4303, 15931}, {4312, 5706}, {5228, 5586}, {5258, 24806}, {5290, 5711}, {5584, 22117}, {5587, 8757}, {5903, 21147}, {5904, 8270}, {6357, 11552}, {7098, 18593}, {11011, 16490}, {11700, 21740}


X(34044) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 3rd CONWAY

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

X(34044) lies on these lines: {1, 84}, {40, 23131}, {57, 16700}, {77, 17185}, {109, 10434}, {223, 1764}, {478, 2003}, {603, 1412}, {651, 11679}, {1407, 2300}, {1456, 10473}, {1943, 10447}, {3929, 4559}, {5307, 10435}, {10446, 18623}, {10476, 20744}, {10571, 10882}, {12435, 21147}, {15479, 17811}


X(34045) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND CONWAY-CIRCLE-INVERSE OF ABC

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

X(34045) lies on these lines: {1, 4}, {56, 16700}, {109, 1764}, {221, 10441}, {222, 10473}, {664, 18138}, {1394, 10476}, {1402, 1465}, {1456, 21334}, {1999, 4318}, {4347, 17733}, {4417, 4551}, {8270, 11679}, {8543, 25058}, {10453, 18623}


X(34046) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND INCIRCLE-CIRCLES

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

The incircle-circles triangle is defined at X(11034).

X(34046) lies on these lines: {1, 84}, {2, 9370}, {3, 947}, {6, 41}, {8, 17074}, {21, 23144}, {31, 4322}, {33, 12136}, {34, 354}, {42, 1106}, {55, 603}, {57, 227}, {58, 1617}, {65, 1407}, {81, 3600}, {104, 1181}, {109, 3295}, {154, 22654}, {155, 32153}, {212, 8273}, {220, 20752}, {223, 3333}, {225, 1889}, {388, 940}, {394, 2975}, {474, 4551}, {478, 8581}, {518, 1038}, {608, 4327}, {651, 3616}, {942, 21147}, {958, 17811}, {993, 3173}, {999, 10571}, {1001, 1935}, {1035, 14547}, {1124, 32556}, {1191, 1319}, {1203, 13462}, {1335, 32555}, {1385, 3157}, {1388, 1616}, {1393, 4860}, {1399, 3052}, {1406, 2099}, {1409, 2256}, {1415, 4258}, {1420, 2003}, {1448, 5173}, {1449, 7091}, {1456, 17609}, {1457, 3304}, {1464, 26437}, {1465, 3338}, {1470, 2594}, {1745, 22753}, {2551, 25934}, {2646, 19349}, {2654, 15811}, {3075, 11500}, {3197, 14597}, {3298, 7133}, {3339, 16474}, {3428, 4303}, {3476, 5710}, {3485, 6180}, {3555, 8270}, {3562, 5731}, {3576, 7078}, {3702, 28968}, {3742, 19372}, {3745, 9850}, {3869, 22129}, {3873, 4296}, {3881, 4347}, {3889, 4318}, {4265, 10831}, {4293, 5706}, {4298, 5930}, {4383, 7288}, {5253, 10601}, {5265, 32911}, {5313, 13370}, {5484, 26625}, {5584, 22053}, {5707, 18990}, {5711, 10106}, {5886, 8757}, {7335, 20986}, {7373, 15306}, {10246, 23070}, {11037, 18623}, {12513, 24806}, {17814, 22758}, {17825, 25524}, {18451, 26321}, {20306, 26871}, {22776, 22972}

X(34056) = isogonal conjugate of X(6603)


X(34047) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 1st EXCOSINE

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10 + 10*a^8*b*c - 4*a^7*b^2*c - 12*a^6*b^3*c - 4*a^5*b^4*c - 8*a^4*b^5*c + 20*a^3*b^6*c + 12*a^2*b^7*c - 12*a*b^8*c - 2*b^9*c - 5*a^8*c^2 - 4*a^7*b*c^2 + 12*a^6*b^2*c^2 + 4*a^5*b^3*c^2 + 2*a^4*b^4*c^2 + 4*a^3*b^5*c^2 - 20*a^2*b^6*c^2 - 4*a*b^7*c^2 + 11*b^8*c^2 - 12*a^6*b*c^3 + 4*a^5*b^2*c^3 + 32*a^4*b^3*c^3 - 24*a^3*b^4*c^3 - 12*a^2*b^5*c^3 + 20*a*b^6*c^3 - 8*b^7*c^3 + 10*a^6*c^4 - 4*a^5*b*c^4 + 2*a^4*b^2*c^4 - 24*a^3*b^3*c^4 + 30*a^2*b^4*c^4 - 4*a*b^5*c^4 - 10*b^6*c^4 - 8*a^4*b*c^5 + 4*a^3*b^2*c^5 - 12*a^2*b^3*c^5 - 4*a*b^4*c^5 + 20*b^5*c^5 - 10*a^4*c^6 + 20*a^3*b*c^6 - 20*a^2*b^2*c^6 + 20*a*b^3*c^6 - 10*b^4*c^6 + 12*a^2*b*c^7 - 4*a*b^2*c^7 - 8*b^3*c^7 + 5*a^2*c^8 - 12*a*b*c^8 + 11*b^2*c^8 - 2*b*c^9 - c^10) : :

The 1st excosine triangle is defined at X(17807).

X(34047) lies on these lines: {6, 32714}, {221, 1419}, {222, 7152}, {2124, 7959}, {2192, 14522}


X(34048) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND 2nd ZANIAH

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

The 2nd Zaniah triangle is defined at X(18214).

X(34048) lies on these lines: {1, 1864}, {2, 222}, {3, 1745}, {4, 7078}, {5, 3157}, {6, 226}, {7, 32911}, {9, 223}, {10, 221}, {12, 5711}, {19, 14557}, {33, 5927}, {34, 72}, {40, 5909}, {44, 1427}, {45, 16577}, {55, 4551}, {56, 1724}, {57, 1122}, {63, 1465}, {73, 405}, {77, 3305}, {81, 5226}, {109, 1376}, {142, 17825}, {196, 1783}, {198, 21361}, {201, 15650}, {210, 1456}, {212, 2635}, {218, 948}, {219, 278}, {227, 12514}, {238, 1617}, {255, 3149}, {312, 1943}, {321, 28997}, {344, 28979}, {381, 23071}, {388, 16466}, {394, 908}, {440, 3330}, {442, 19349}, {474, 603}, {478, 1211}, {516, 7074}, {604, 28387}, {607, 5813}, {608, 5739}, {611, 26098}, {613, 15253}, {614, 17625}, {748, 1458}, {899, 9316}, {916, 11436}, {936, 1394}, {940, 2003}, {942, 19372}, {954, 14547}, {956, 1457}, {958, 10571}, {960, 21147}, {971, 1040}, {997, 1455}, {1012, 22350}, {1014, 27643}, {1020, 6611}, {1038, 5044}, {1103, 12705}, {1106, 27627}, {1124, 1659}, {1191, 10106}, {1203, 5290}, {1260, 23693}, {1335, 13390}, {1396, 2287}, {1406, 24914}, {1407, 3911}, {1413, 6700}, {1419, 7308}, {1421, 17597}, {1441, 26223}, {1466, 3216}, {1476, 28370}, {1498, 6260}, {1656, 23070}, {1709, 9371}, {1750, 7070}, {1758, 7262}, {1763, 2182}, {1777, 10310}, {1838, 5812}, {1848, 22132}, {1854, 31803}, {1877, 3419}, {1936, 19541}, {1944, 20921}, {1993, 31053}, {2050, 23131}, {2183, 11347}, {2192, 16870}, {2256, 4656}, {2265, 26934}, {2267, 21483}, {2323, 28609}, {2911, 6354}, {3075, 6918}, {3091, 3562}, {3219, 17080}, {3452, 17811}, {3476, 16483}, {3660, 5272}, {3678, 4347}, {3681, 4318}, {3751, 5173}, {3782, 5723}, {3784, 16434}, {3876, 4296}, {3936, 28776}, {3955, 19544}, {4358, 28996}, {4359, 28968}, {4559, 6358}, {4654, 5228}, {5120, 27659}, {5236, 22131}, {5249, 10601}, {5256, 8545}, {5307, 22134}, {5422, 31019}, {5706, 9612}, {5710, 9578}, {5741, 28774}, {5748, 23140}, {5752, 7066}, {5779, 24430}, {5811, 7952}, {5928, 18588}, {5930, 12572}, {6127, 14793}, {6357, 31018}, {7069, 20277}, {7082, 8758}, {7175, 27623}, {7330, 17102}, {7354, 31832}, {7535, 19366}, {8543, 17018}, {9363, 21214}, {9364, 16569}, {9708, 24806}, {9817, 10157}, {11433, 16608}, {12586, 17111}, {13257, 19354}, {14594, 27538}, {14997, 21454}, {15066, 27131}, {16435, 22097}, {16885, 18593}, {17277, 27339}, {17776, 28966}, {17781, 22464}, {18139, 28741}, {18228, 18623}, {18743, 28965}, {19540, 22161}, {20122, 25514}, {20266, 26005}, {22128, 30852}, {23125, 30076}, {23151, 33066}, {25934, 30827}, {26668, 27540}, {28606, 29007}, {28780, 32782}, {31835, 32047}

X(34048) = complement of X(26871)


X(34049) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND URSA-MAJOR

Barycentrics    ursa-major, see X(17603) at a*(a + b - c)*(a - b + c)*(2*a^6 - 3*a^5*b - a^4*b^2 + 2*a^3*b^3 + a*b^5 - b^6 - 3*a^5*c + 10*a^4*b*c - 4*a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c - a^4*c^2 - 4*a^3*b*c^2 + 4*a^2*b^2*c^2 + b^4*c^2 + 2*a^3*c^3 - 2*a^2*b*c^3 - a*b*c^4 + b^2*c^4 + a*c^5 - c^6) : :

The Ursa-Major triangle is defined at X(17603).

X(34049) lies on these lines: {1, 84}, {11, 1456}, {105, 2720}, {109, 2739}, {223, 1376}, {521, 2254}, {651, 17615}, {1054, 1465}, {1829, 3450}, {3434, 18623}, {3955, 31788}, {5930, 11826}, {7952, 12676}, {8270, 17658}, {10571, 17614}, {10914, 21147}


X(34050) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND WASAT

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

The Wasat triangle is defined at X(21616).

X(34050) lies on these lines: {1, 3427}, {2, 77}, {3, 5930}, {4, 1394}, {10, 1038}, {11, 1456}, {19, 57}, {34, 1210}, {73, 13411}, {84, 7952}, {108, 3220}, {109, 516}, {117, 515}, {221, 946}, {222, 226}, {225, 603}, {227, 6684}, {241, 514}, {255, 1076}, {306, 28774}, {307, 1150}, {347, 5744}, {601, 1070}, {651, 908}, {664, 32851}, {971, 15252}, {1035, 3149}, {1079, 10320}, {1106, 23536}, {1125, 10571}, {1214, 5745}, {1407, 3772}, {1410, 27622}, {1413, 6260}, {1419, 4648}, {1433, 20264}, {1445, 24597}, {1448, 5292}, {1457, 32486}, {1458, 3011}, {1715, 8803}, {1737, 4351}, {1738, 9364}, {1785, 1795}, {1818, 4551}, {1838, 3075}, {1854, 9948}, {1935, 12572}, {1943, 3687}, {2263, 11269}, {3218, 22464}, {3452, 17811}, {3660, 15253}, {3717, 14594}, {3914, 9316}, {3977, 4552}, {4054, 28968}, {4296, 6734}, {4318, 26015}, {4320, 5230}, {4322, 28027}, {4334, 29658}, {4347, 10916}, {4565, 18653}, {4847, 8270}, {5018, 33140}, {5249, 17074}, {5316, 25878}, {5435, 18624}, {5717, 15844}, {5812, 23072}, {6001, 12016}, {6705, 17102}, {7004, 23710}, {7011, 15509}, {8581, 17602}, {9370, 21075}, {9843, 19372}, {12053, 16388}, {15832, 26066}, {17086, 24627}, {30379, 33129}

X(34050) = isogonal conjugate of X(15629)
X(34050) = trilinear pole of line X(1359)X(6087)
X(34050) = crossdifference of every pair of points on line X(55)X(2432)


X(34051) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND GEMINI 7

Barycentrics    a*(a + b - c)*(a - 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(34051) lies on these lines: {1, 104}, {2, 222}, {28, 1408}, {56, 957}, {57, 909}, {73, 6940}, {77, 16586}, {81, 4565}, {88, 1443}, {105, 2720}, {108, 3937}, {223, 8056}, {241, 32641}, {274, 4573}, {277, 5723}, {278, 1086}, {279, 4617}, {291, 9364}, {1257, 1809}, {1319, 10428}, {1462, 2423}, {1797, 24029}, {2224, 32669}, {2250, 25430}, {2401, 21786}, {2982, 14578}, {2990, 3218}, {3157, 6961}, {6357, 21907}, {6648, 18816}, {6977, 19349}, {6981, 8757}, {15474, 18623}, {22129, 26611}

X(34051) = trilinear pole of line X(56)X(513)


X(34052) =  PERSPECTOR OF THESE TRIANGLES: TT(X(6),X(7)) AND GEMINI 15

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

X(34052) lies on these lines: {1, 10309}, {2, 77}, {57, 909}, {222, 3666}, {278, 4341}, {651, 6505}, {1394, 6906}, {1413, 17102}, {1419, 18675}, {1763, 7125}, {5930, 6850}, {21147, 23555}


X(34053) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(1)) AND GEMINI 19

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

X(34053) lies on these lines: {2, 1029}, {21, 25354}, {37, 662}, {99, 27705}, {229, 409}, {551, 759}, {1001, 11101}, {5333, 25361}, {6626, 27707}


X(34054) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND CEVIAN TRIANGLE OF X(1580)

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

X(34054) lies on these lines: {1, 82}, {75, 18834}, {83, 3821}, {896, 4599}, {1966, 4593}, {2085, 33760}, {8852, 20872}


X(34055) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND CEVIAN TRIANGLE OF X(3112)

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

The trilinear polar of X(34055) passes through X(656).

X(34055) lies on these lines: {1, 82}, {3, 22367}, {8, 26270}, {38, 1932}, {48, 304}, {63, 9247}, {72, 1176}, {75, 2172}, {83, 226}, {92, 1973}, {251, 17011}, {306, 1799}, {662, 1930}, {827, 26702}, {1214, 28713}, {1580, 9236}, {1910, 4593}, {1959, 2167}, {2349, 4599}, {4020, 4592}, {10548, 17086}

X(34055) = isogonal conjugate of X(17442)
X(34055) = isotomic conjugate of X(20883)


X(34056) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 7

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

X(34056) lies on these lines: {1, 651}, {2, 664}, {57, 934}, {81, 1414}, {88, 241}, {89, 5228}, {105, 1319}, {274, 4625}, {277, 3160}, {278, 8735}, {279, 1086}, {955, 15934}, {1002, 2099}, {1025, 1320}, {1170, 18889}, {1323, 15727}, {1462, 2087}, {1642, 5376}, {3960, 23893}, {8056, 17080}


X(34057) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 34

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

X(34057) lies on these lines: {6, 3212}, {7, 3056}, {9, 3177}, {77, 614}, {348, 24752}


X(34058) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 60

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

X(34058) lies on these lines: {1441, 27340}, {2170, 3212}, {3160, 17077}


X(34059) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 61

Barycentrics    (a + b - c)*(a - b + c)*(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(34059) lies on these lines: {1, 1446}, {2, 3160}, {7, 950}, {8, 253}, {20, 14256}, {29, 34}, {40, 4566}, {57, 3188}, {78, 664}, {223, 27413}, {226, 31042}, {269, 17863}, {278, 25935}, {279, 938}, {348, 6734}, {411, 934}, {651, 30625}, {653, 7156}, {728, 4552}, {1119, 18650}, {1210, 1323}, {1439, 18655}, {1445, 2082}, {1847, 10884}, {3668, 5738}, {3673, 4350}, {3875, 20008}, {4384, 17080}, {5088, 7177}, {6604, 22464}, {7176, 24268}, {9436, 12649}, {20007, 25718}, {27382, 30695}


X(34060) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 65

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

X(34060) lies on these lines: {2, 3160}, {7, 738}, {8, 23244}, {20, 934}, {75, 280}, {77, 3616}, {78, 5932}, {279, 3673}, {516, 17106}, {664, 7080}, {962, 7177}, {1323, 3086}, {2898, 7176}, {8232, 17084}, {8732, 14189}, {11240, 22464}, {18623, 26006}


X(34061) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 75

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

X(34061) lies on these lines: {6, 3212}, {57, 17209}, {651, 27994}


X(34062) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(7)) AND GEMINI 100

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

X(34062) lies on these lines: {2, 92}, {7, 28369}, {77, 30097}, {223, 30076}, {651, 20348}, {1423, 22464}, {3160, 30032}, {3212, 3959}, {17787, 28739}, {30090, 33673}


X(34063) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(75)) AND TANGENTIAL

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

X(34063) lies on these lines: {1, 75}, {2, 21025}, {6, 330}, {8, 27162}, {42, 25303}, {43, 6384}, {56, 664}, {69, 20036}, {76, 995}, {99, 595}, {101, 7760}, {145, 30962}, {190, 194}, {192, 16969}, {239, 3752}, {320, 24215}, {350, 1201}, {385, 21008}, {519, 24170}, {668, 3216}, {742, 7187}, {899, 25280}, {903, 14260}, {940, 4393}, {978, 6376}, {1015, 17034}, {1107, 16827}, {1191, 1975}, {1193, 1909}, {1268, 19853}, {1575, 17752}, {2238, 21226}, {2275, 17033}, {2975, 33295}, {3009, 32926}, {3207, 14614}, {3230, 25264}, {3552, 21793}, {3730, 7757}, {3759, 6647}, {3780, 9263}, {4352, 4389}, {4511, 20436}, {4713, 20081}, {7176, 24471}, {7769, 17734}, {7783, 17735}, {9534, 32025}, {10027, 20691}, {14974, 31859}, {16678, 16689}, {16706, 30038}, {16826, 25130}, {17137, 33947}, {17148, 27644}, {17152, 18600}, {17262, 32107}, {17283, 29960}, {17285, 27248}, {17314, 26106}, {17316, 24654}, {17318, 32095}, {17350, 32005}, {17377, 20018}, {17489, 33946}, {17790, 30054}, {18170, 23485}, {19701, 24670}, {20037, 21281}, {20040, 30941}, {20154, 31490}, {21214, 30963}, {22199, 24519}, {24068, 33948}, {24652, 31028}, {26752, 26972}, {27195, 29438}

X(34063) = anticomplement of X(21025)


X(34064) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(75)) AND GEMINI 105

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

X(34064) lies on these lines: {1, 312}, {2, 594}, {37, 333}, {42, 3699}, {63, 4664}, {75, 5287}, {81, 190}, {86, 321}, {100, 27804}, {142, 19796}, {145, 14555}, {171, 3993}, {192, 940}, {226, 664}, {292, 21883}, {306, 17315}, {335, 20362}, {519, 4886}, {612, 3996}, {726, 4038}, {740, 1961}, {894, 3175}, {903, 26842}, {968, 3769}, {1010, 2901}, {1100, 27064}, {1126, 4075}, {1211, 6542}, {1258, 32095}, {1449, 30568}, {1482, 9535}, {1897, 31623}, {1962, 17763}, {2321, 19808}, {2999, 30829}, {3159, 4658}, {3187, 17277}, {3210, 17318}, {3241, 5289}, {3247, 11679}, {3305, 3759}, {3666, 17319}, {3672, 18141}, {3685, 3745}, {3703, 29837}, {3706, 16830}, {3720, 32922}, {3757, 15569}, {3782, 17300}, {3840, 17600}, {3875, 17022}, {3879, 4656}, {3891, 29814}, {3896, 5297}, {3912, 19786}, {3943, 6703}, {3971, 4649}, {3989, 32919}, {4001, 17258}, {4078, 33118}, {4358, 17011}, {4359, 17021}, {4383, 4393}, {4384, 25430}, {4413, 4734}, {4415, 17390}, {4425, 32846}, {4431, 19797}, {4641, 17261}, {4645, 4854}, {4648, 30699}, {4670, 22034}, {4671, 19684}, {4682, 32932}, {4687, 5271}, {4851, 27184}, {4970, 17122}, {5249, 17317}, {5256, 17393}, {5263, 5311}, {5284, 17150}, {5294, 17264}, {5333, 31025}, {5712, 29585}, {5737, 16672}, {5739, 17377}, {5743, 17388}, {5846, 20069}, {5905, 17378}, {6541, 32780}, {7308, 16834}, {8055, 20057}, {9345, 17155}, {9347, 32929}, {14828, 20173}, {14829, 28606}, {14996, 32933}, {16685, 31035}, {16704, 33761}, {16826, 25507}, {16828, 25431}, {17184, 17297}, {17234, 19785}, {17240, 19812}, {17242, 29841}, {17244, 24789}, {17246, 26840}, {17263, 26723}, {17273, 32863}, {17283, 32774}, {17295, 32782}, {17305, 33172}, {17316, 18134}, {17321, 34255}, {17592, 29649}, {17602, 29839}, {18139, 33155}, {18206, 32026}, {19701, 24670}, {19732, 27268}, {19738, 25417}, {19820, 24199}, {20176, 27065}, {21933, 28811}, {24051, 24053}, {24210, 33073}, {25527, 29573}, {25529, 33133}, {25531, 29821}, {26102, 32921}, {27191, 33150}, {29635, 33092}, {29645, 33158}, {29653, 33135}, {29816, 32943}, {29829, 32862}, {29833, 33157}, {29845, 32848}, {29847, 33156}, {29854, 33128}, {30950, 32924}


X(34065) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(75)

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

X(34065) lies on these lines: {1, 82}, {48, 18156}, {75, 1973}, {304, 662}, {1910, 18832}, {1933, 23478}, {4593, 33778}, {16568, 17442}, {16788, 18747}, {20931, 21593}


X(34066) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4418)

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

X(34066) lies on these lines: {1, 82}, {662, 20932}


X(34067) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(660)

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

X(34067) is the barycentric product of the circumcircle intercepts of line X(1)X(39). As the barycentric product of circumcircle-X(1)-antipodes, X(34067) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31). (Randy Hutson, October 8, 2019)

X(34067) lies on these lines: {3, 22116}, {36, 291}, {99, 4583}, {100, 4562}, {101, 667}, {163, 1110}, {292, 1438}, {335, 30908}, {659, 666}, {660, 662}, {669, 1252}, {692, 1919}, {875, 32665}, {876, 2283}, {890, 901}, {909, 7077}, {911, 2196}, {932, 8684}, {1910, 17798}, {1911, 9456}, {1922, 2251}, {2210, 14598}, {3252, 15624}, {3573, 5378}, {4998, 24533}, {17938, 32736}, {18266, 18267}, {23344, 23349}

X(34067) = isogonal conjugate of X(3766)
X(34067) = trilinear pole of line X(31)X(1911)


X(34068) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(1156)

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

X(34068) lies on these lines: {21, 662}, {25, 32674}, {31, 1415}, {41, 692}, {55, 101}, {56, 1461}, {105, 1319}, {163, 2194}, {884, 1438}, {909, 23351}, {911, 8648}, {1121, 4586}, {1155, 10426}, {1470, 3423}, {2204, 32676}, {2223, 32665}, {2251, 32666}, {3207, 16686}, {6187, 32675}, {32669, 32728}

X(34068) = isogonal conjugate of X(30806)


X(34069) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(1492)

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

X(34069) lies on these lines: {101, 825}, {163, 14574}, {662, 1492}, {789, 827}, {911, 20986}, {923, 19626}, {985, 2224}, {1438, 5332}, {1910, 14601}, {2206, 18268}, {4586, 33904}, {8022, 14945}, {29018, 30670}


X(34070) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(2236)

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

X(34070) lies on these lines: {1, 82}, {662, 19559}, {1959, 19578}


X(34071) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4598)

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

X(34071) lies on these lines: {6, 20467}, {41, 17105}, {87, 572}, {101, 932}, {171, 18269}, {190, 1919}, {284, 18268}, {330, 2224}, {662, 4598}, {727, 21762}, {909, 2319}, {911, 23086}, {923, 23493}, {2162, 2278}, {3888, 20981}, {4107, 4572}, {4251, 7121}, {4499, 8632}, {4586, 18830}, {6384, 30896}


X(34072) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4599)

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

X(34072) lies on these lines: {82, 1910}, {101, 827}, {662, 4599}, {692, 4630}, {4577, 4586}, {4593, 18062}


X(34073) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4604)

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

X(34073) lies on these lines: {32, 9456}, {58, 28658}, {89, 2224}, {101, 4588}, {109, 32675}, {662, 4604}, {692, 1983}, {825, 8695}, {909, 2364}, {1438, 2163}, {4262, 19654}, {4291, 32677}, {4586, 4597}, {13486, 32678}, {32665, 32739}


X(34074) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4606)

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

X(34074) lies on these lines: {101, 8694}, {644, 662}, {909, 4287}, {1438, 2334}, {1461, 4559}, {2224, 25430}, {5545, 8693}, {9456, 21742}

X(34074) = isogonal conjugate of X(4801)


X(34075) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(4607)

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

X(34075) is the trilinear product X(100)*X(739). As the trilinear product of circumcircle-X(6)-antipodes, X(36134) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31). (Randy Hutson, January 17, 2020)

X(34075) lies on these lines: {59, 1415}, {101, 765}, {163, 4570}, {662, 4600}, {677, 911}, {692, 1252}, {739, 901}, {889, 4586}, {1438, 5053}, {1461, 7045}, {2224, 3227}, {3257, 8632}, {3285, 18268}, {7012, 32674}, {7113, 19621}, {9268, 23892}, {23344, 23349}

X(34075) = isogonal conjugate of X(4728)
X(34075) = trilinear pole of line X(31)X(101)
X(34075) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 4728}, {2, 891}, {75, 3768}, {76, 890}, {244, 23891}, {513, 536}, {514, 899}, {649, 6381}, {668, 1646}, {693, 3230}, {1086, 23343}
X(34075) = trilinear product X(i)*X(j) for these {i,j}: {2, 32718}, {6, 898}, {31, 4607}, {32, 889}, {100, 739}, {667, 5381}, {692, 3227}, {765, 23892}, {1016, 23349}, {31002, 32739}
X(34075) = trilinear quotient X(i)/X(j) for these (i,j): (1, 4728), (6, 891), (31, 3768), (32, 890), (100, 536), (101, 899), (190, 6381), (667, 1646), (692, 3230), (739, 513), (765, 23891), (889, 76), (898, 2), (1252, 23343), (3227, 693), (4607, 75), (5381, 668), (23343, 13466), (23349, 1015), (23892, 244), (31002, 3261), (32718, 6)
X(34075) = barycentric product X(i)*X(j) for these {i,j}: {1, 898}, {6, 4607}, {31, 889}, {75, 32718}, {101, 3227}, {190, 739}, {692, 31002}, {1016, 23892}, {7035, 23349}
X(34075) = barycentric quotient X(i)/X(j) for these (i,j): (6, 4728), (31, 891), (32, 3768), (100, 6381), (101, 536), (190, 35543), (560, 890), (692, 899), (739, 514), (889, 561), (898, 75), (1110, 23343), (1252, 23891), (1919, 1646), (3227, 3261), (4607, 76), (23349, 244), (23892, 1086), (32718, 1), (32739, 3230)


X(34076) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(8052)

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

X(34076) lies on these lines: {101, 21383}, {662, 8052}, {692, 23861}


X(34077) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(20332)

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

X(34077) lies on these lines: {1, 20600}, {6, 20467}, {32, 101}, {560, 692}, {662, 1333}, {713, 8709}, {1438, 23355}, {2210, 14598}, {3226, 4586}, {4386, 17743}, {4593, 32020}


X(34078) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(23707)

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

X(34078) lies on these lines: {31, 32674}, {101, 212}, {184, 1415}, {283, 662}, {603, 1461}, {2299, 24019}, {32667, 32727}


X(34079) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(24624)

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

X(34079) is the barycentric product of the circumcircle intercepts of line X(1)X(523). As the barycentric product of circumcircle-X(1)-antipodes, X(34079) lies on conic {{A,B,C,X(101),X(163)}} with center X(32664) and perspector X(31). (Randy Hutson, October 8, 2019)

X(34079) lies on these lines: {2, 662}, {6, 163}, {25, 32676}, {37, 101}, {42, 692}, {58, 28658}, {110, 17455}, {308, 4593}, {393, 24019}, {584, 941}, {859, 3285}, {923, 9178}, {1333, 1400}, {1411, 1474}, {1412, 1427}, {1576, 3271}, {1790, 16736}, {1793, 4280}, {1910, 2395}, {1989, 32678}, {2159, 2433}, {2183, 19622}, {2189, 8882}, {2224, 30927}, {2350, 5035}, {2576, 8106}, {2577, 8105}, {2605, 14998}, {2998, 30933}, {3108, 5109}, {3572, 18268}, {4271, 7054}, {4586, 14616}, {6740, 14624}, {7113, 30117}, {30878, 30903}

X(34079) = isogonal conjugate of X(3936)
X(34079) = trilinear pole of line X(31)X(512)
X(34079) = crossdifference of every pair of points on line X(4707)X(4736)
X(34079) = barycentric product X(1)*X(759)


X(34080) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(24834)

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

X(34080) lies on these lines: {19, 17465}, {41, 16945}, {48, 9456}, {101, 1293}, {198, 909}, {610, 16561}, {644, 4394}, {662, 27834}, {906, 32645}, {1438, 3207}, {2224, 8056}, {4373, 30907}

X(34080) = isogonal conjugate of X(4462)


X(34081) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(33295)

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

X(34081) lies on these lines: {1, 82}, {662, 18157}, {12150, 16712}, {16876, 25050}


X(34082) =  PERSPECTOR OF THESE TRIANGLES: TT(X(1),X(6)) AND ANTICEVIAN OF X(33764)

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

X(34082) lies on these lines: {1, 82}, {662, 18051}


X(34083) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(7)) AND 1st CIRCUMPERP

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

X(34083) lies on these lines: {100, 4572}, {101, 4554}, {109, 4569}, {110, 4625}, {664, 29052}, {20567, 26236}


X(34084) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(7)) AND GEMINI 7

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

X(34084) lies on these lines: {1, 4554}, {2, 4572}, {57, 4569}, {81, 4625}, {1170, 30627}


X(34085) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(7)) AND GEMINI 30 (INNER CONWAY)

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

X(34085) lies on these lines: {100, 693}, {190, 3261}, {651, 666}, {660, 883}, {662, 4620}, {673, 10030}, {1110, 19594}, {1156, 2481}, {1458, 33674}, {1462, 20332}, {1492, 32735}, {6063, 24596}, {6654, 30545}, {14189, 33675}, {23707, 31637}, {30988, 32578}

X(34085) = trilinear pole of line X(1)X(85)
X(34085) = isotomic conjugate of isogonal conjugate of X(36146)


X(34086) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(75)) AND ANTICOMPLEMENTARY

Barycentrics    b^2*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(34086) lies on these lines: {2, 1221}, {7, 4572}, {38, 75}, {76, 17236}, {192, 1978}, {308, 17305}, {668, 25277}, {670, 4360}, {1278, 6382}, {1502, 4389}, {1921, 20892}, {2296, 3223}, {3123, 24732}, {3663, 18891}, {3978, 6646}, {4772, 10009}, {6386, 18133}, {9230, 17302}, {10436, 27663}, {17137, 17153}, {17246, 30736}, {17273, 33769}, {18152, 20891}, {19565, 28367}

X(34086) = anticomplement of X(6378)


X(34087) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(75)) AND WASAT

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

X(34087) lies on these lines: {2, 670}, {4, 6331}, {10, 1978}, {76, 3124}, {83, 689}, {98, 9150}, {111, 880}, {226, 4572}, {262, 11059}, {321, 6386}, {598, 14608}, {671, 886}, {1648, 18896}, {1916, 3266}, {5466, 14295}, {5485, 20023}, {9211, 14458}

X(34087) = isogonal conjugate of X(33875)
X(34087) = isotomic conjugate of X(3231)
X(34087) = trilinear pole of line X(76)X(523) (the line through X(76) parallel to the trilinear polar of X(76))
X(34087) = crossdifference of every pair of points on line X(887)X(14406)
X(34087) = cevapoint of X(i) and X(j) for these (i,j): {76, 30736}, {3124, 9148}
X(34087) = crosssum of X(887) and X(1645)


X(34088) =  PERSPECTOR OF THESE TRIANGLES: TT(X(75),X(75)) AND GEMINI 105

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

X(34088) lies on these lines: {2, 1221}, {76, 6539}, {226, 4572}, {668, 25294}, {1978, 3995}, {27184, 31055}


X(34089) =  ISOGONAL CONJUGATE OF X(6417)

Barycentrics    (a^2+b^2-c^2+8 S) (a^2-b^2+c^2+8 S) : :
Barycentrics    (4 S+SB) (4 S+SC) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29359.

X(34089) lies on these lines: {2,6418}, {4,6409}, {76,32813}, {2996,7376}, {3068,10194}, {5395,7375}, {7388,18845}

X(34089) = isogonal conjugate of X(6417)
X(34089) = isotomic conjugate of X(32812)
X(34089) = barycentric quotient of X(i) and X(j) for these {i,j}: {2,32812}, {6,6417}
X(34089) = trilinear quotient of X(i) and X(j) for these {i,j}: {1,6417}, {75,32812}


X(34090) =  (name pending)

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+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) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4+8 a^2 S-8 b^2 S-8 c^2 S) : :
Barycentrics    (16 R^2-4 SW)S^3 + (-256 R^4 SB-256 R^4 SC-32 R^2 SB SC+256 R^4 SW+128 R^2 SB SW+128 R^2 SC SW+8 SB SC SW-128 R^2 SW^2-16 SB SW^2-16 SC SW^2+16 SW^3)S -128 R^4 SB SC-S^2 SB SC+48 R^2 SB SC SW-4 SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29359.

X(34090) lies on this line: {4,253}


X(34091) =  ISOGONAL CONJUGATE OF X(6418)

Barycentrics    (a^2+b^2-c^2-8 S) (a^2-b^2+c^2-8 S) : :
Barycentrics    (4 S-SB) (4 S-SC) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29359.

X(34091) lies on these lines: {2,6417}, {4,6410}, {76,32812}, {2996,7375}, {3069,10195}, {5395,7376}, {5491,32807}, {7389,18845}

X(34091) = isogonal conjugate of X(6418)
X(34091) = isotomic conjugate of X(32813)
X(34091) = barycentric quotient of X(i) and X(j) for these {i,j}: {2,32813}, {6,6418}
X(34091) = trilinear quotient of X(i) and X(j) for these {i,j}: {1,6418}, {75,32813}


X(34092) =  (name pending)

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+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) (3 a^4-2 a^2 b^2-b^4-2 a^2 c^2+2 b^2 c^2-c^4-8 a^2 S+8 b^2 S+8 c^2 S) : :
Barycentrics    (16 R^2-4 SW)S^3 + (-256 R^4 SB-256 R^4 SC-32 R^2 SB SC+256 R^4 SW+128 R^2 SB SW+128 R^2 SC SW+8 SB SC SW-128 R^2 SW^2-16 SB SW^2-16 SC SW^2+16 SW^3)S + 128 R^4 SB SC+S^2 SB SC-48 R^2 SB SC SW+4 SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29359.

X(34092) lies on this line: {4,253}


X(34093) =  46TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29365 and Antreas Hatzipolakis and Peter Moses, Euclid 33 .

X(34093) lies on these lines: {2,3}, {51,523}, {275,1304}, {1624,8901}, {1994,14611}, {2452,9777}, {2453,17810}, {2790,12099}, {5097,30221}, {6795,10601}, {10412,14583}, {12079,13567}, {16319,23292}


X(34094) =  47TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29369.

X(34094) lies on these lines: {2,3}, {476,7698}, {523,597}, {543,5972}, {1648,18907}, {2782,5642}, {3734,11053}, {4045,10160}, {7606,16324}, {7792,16092}, {7804,22104}, {10796,11657}, {11174,16316}, {14561,16279}, {15048,31945}

X(34094) = circumcircle-inverse of X(37914)


X(34095) =  X(2)X(51)∩X(110)X(8722)

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

X(34095 lies on these lines: {2,51}, {110,8722}, {237,15066}, {512,14907}, {543,12149}, {574,8623}, {3051,7485}, {4576,20023}, {6310,32997}, {6784,17008}, {7833,9879}, {8352,12525}, {11185,14957}, {14096,21766}, {16063,20021}

X(34095) = anticomplement of isotomic conjugate of polar conjugate of X(34096)
X(34095) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(22503)
X(34095) = {X(7998),X(11673)}-harmonic conjugate of X(2)


X(34096) =  X(4)X(39)∩X(1351)X(1625)

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

X(34096) lies on these lines: {4,39}, {1351,1625}, {2211,8541}

X(34096) = barycentric product X(25)*X(7697)
X(34096) = barycentric quotient X(7697)/X(305)
X(34096) = polar conjugate of isotomic conjugate of complement of X(34095)


X(34097) =  X(3)X(6)∩X(251)X(9462)

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

X(34097) lies on these lines: {3,6}, {251,9462}


X(34098) =  X(2)X(3)∩X(51)X(9465)

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

X(34098) lies on the cubic K1125 and these lines: {2,3}, {51,9465}, {111,263}, {154,20965}, {353,1495}, {1383,5191}, {1634,9770}, {1843,15355}, {1992,9149}, {2493,9971}, {2502,33876}, {3051,17810}, {3231,31860}, {9917,32834}, {10097,11186}, {10546,33873}, {23208,31404}

X(34098) = X(14485)-Ceva conjugate of X(6)
X(34098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{25, 3148, 23}, {5020, 7467, 2}, {11284, 14096, 2}


X(34099) =  X(2)X(9743)∩X(6)X(160)

Barycentrics    a^2*(10*a^6*b^4 - 12*a^4*b^6 + 2*a^2*b^8 + 19*a^6*b^2*c^2 - 9*a^4*b^4*c^2 - 11*a^2*b^6*c^2 + b^8*c^2 + 10*a^6*c^4 - 9*a^4*b^2*c^4 - 6*a^2*b^4*c^4 - b^6*c^4 - 12*a^4*c^6 - 11*a^2*b^2*c^6 - b^4*c^6 + 2*a^2*c^8 + b^2*c^8) : :

X(34099) lies on the cubic K1125, the Thomson-Gibert-Moses hyperbola, and these lines: {2,9743}, {6,160}, {110,8722}, {5544,11328}, {5646,14096}, {5652,8644}, {9155,33876}

X(34099) = Thomson-isogonal conjugate of X(13860)


X(34100) =  (name pending)

Barycentrics    a^2*(4*a^12*b^4 - 24*a^8*b^8 + 32*a^6*b^10 - 12*a^4*b^12 + 7*a^12*b^2*c^2 + 9*a^10*b^4*c^2 - 2*a^8*b^6*c^2 - 6*a^6*b^8*c^2 + 11*a^4*b^10*c^2 - 19*a^2*b^12*c^2 + 4*a^12*c^4 + 9*a^10*b^2*c^4 + 52*a^8*b^4*c^4 - 2*a^6*b^6*c^4 - 8*a^4*b^8*c^4 - 47*a^2*b^10*c^4 - 8*b^12*c^4 - 2*a^8*b^2*c^6 - 2*a^6*b^4*c^6 + 18*a^4*b^6*c^6 + 66*a^2*b^8*c^6 - 32*b^10*c^6 - 24*a^8*c^8 - 6*a^6*b^2*c^8 - 8*a^4*b^4*c^8 + 66*a^2*b^6*c^8 + 80*b^8*c^8 + 32*a^6*c^10 + 11*a^4*b^2*c^10 - 47*a^2*b^4*c^10 - 32*b^6*c^10 - 12*a^4*c^12 - 19*a^2*b^2*c^12 - 8*b^4*c^12) : :

X(34100) lies on the cubic K1125 and this line: {1350,14532}


X(34101) =  X(3)X(14111)∩X(5)X(51)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29372.

X(34101) lies on these lines: {3,14111}, {5,51}


X(34102) =  X(5)X(22101)∩X(137)X(32638)

Barycentrics    2*a^22 - 21*a^20*b^2 + 98*a^18*b^4 - 267*a^16*b^6 + 468*a^14*b^8 - 546*a^12*b^10 + 420*a^10*b^12 - 198*a^8*b^14 + 42*a^6*b^16 + 7*a^4*b^18 - 6*a^2*b^20 + b^22 - 21*a^20*c^2 + 150*a^18*b^2*c^2 - 430*a^16*b^4*c^2 + 583*a^14*b^6*c^2 - 227*a^12*b^8*c^2 - 405*a^10*b^10*c^2 + 635*a^8*b^12*c^2 - 355*a^6*b^14*c^2 + 52*a^4*b^16*c^2 + 27*a^2*b^18*c^2 - 9*b^20*c^2 + 98*a^18*c^4 - 430*a^16*b^2*c^4 + 624*a^14*b^4*c^4 - 218*a^12*b^6*c^4 - 94*a^10*b^8*c^4 - 353*a^8*b^10*c^4 + 694*a^6*b^12*c^4 - 346*a^4*b^14*c^4 - 10*a^2*b^16*c^4 + 35*b^18*c^4 - 267*a^16*c^6 + 583*a^14*b^2*c^6 - 218*a^12*b^4*c^6 - 100*a^10*b^6*c^6 - 39*a^8*b^8*c^6 - 360*a^6*b^10*c^6 + 656*a^4*b^12*c^6 - 180*a^2*b^14*c^6 - 75*b^16*c^6 + 468*a^14*c^8 - 227*a^12*b^2*c^8 - 94*a^10*b^4*c^8 - 39*a^8*b^6*c^8 - 42*a^6*b^8*c^8 - 369*a^4*b^10*c^8 + 528*a^2*b^12*c^8 + 90*b^14*c^8 - 546*a^12*c^10 - 405*a^10*b^2*c^10 - 353*a^8*b^4*c^10 - 360*a^6*b^6*c^10 - 369*a^4*b^8*c^10 - 718*a^2*b^10*c^10 - 42*b^12*c^10 + 420*a^10*c^12 + 635*a^8*b^2*c^12 + 694*a^6*b^4*c^12 + 656*a^4*b^6*c^12 + 528*a^2*b^8*c^12 - 42*b^10*c^12 - 198*a^8*c^14 - 355*a^6*b^2*c^14 - 346*a^4*b^4*c^14 - 180*a^2*b^6*c^14 + 90*b^8*c^14 + 42*a^6*c^16 + 52*a^4*b^2*c^16 - 10*a^2*b^4*c^16 - 75*b^6*c^16 + 7*a^4*c^18 + 27*a^2*b^2*c^18 + 35*b^4*c^18 - 6*a^2*c^20 - 9*b^2*c^20 + c^22 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29372.

X(34102) lies on these lines: {5,22101}, {137,32638}, {547,1209}, {18583,20413}


X(34103) =  X(2)X(22100)∩X(524)X(547)

Barycentrics    16*a^10 - 62*a^8*b^2 + 100*a^6*b^4 - 95*a^4*b^6 + 46*a^2*b^8 - 5*b^10 - 62*a^8*c^2 + 22*a^6*b^2*c^2 + 135*a^4*b^4*c^2 - 197*a^2*b^6*c^2 + 22*b^8*c^2 + 100*a^6*c^4 + 135*a^4*b^2*c^4 + 270*a^2*b^4*c^4 - 17*b^6*c^4 - 95*a^4*c^6 - 197*a^2*b^2*c^6 - 17*b^4*c^6 + 46*a^2*c^8 + 22*b^2*c^8 - 5*c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29372.

X(34103) lies on these lines: {2,22100}, {524,547}


X(34104) =  KIRIKAMI-EULER IMAGE OF X(113)

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

See Peter Moses, Hyacinthos 29373.

X(34104) lies on these lines: {2,3}, {3258,23097}, {3580,14264}, {15454,16319}

X(34104) = isogonal conjugate of X(39379)
X(34104) = anticomplement of X(39234)
X(34104) = trilinear product X(i)*X(j) for these {i,j}: {113, 1725}, {1784, 34333}
X(34104) = X(403)-Ceva conjugate of X(113)
X(34104) = crossdifference of every pair of points on line {647, 15470}
X(34104) = barycentric product X(113)*X(3580)
X(34104) = barycentric quotient X(i)/X(j) for these {i,j}: {113, 2986}, {3003, 10419}


X(34105) =  X(3)X(14900)∩X(23)X(935)

Barycentrics    a^2*(a^14 - 4*a^12*b^2 + 2*a^10*b^4 + 7*a^8*b^6 - 7*a^6*b^8 - 2*a^4*b^10 + 4*a^2*b^12 - b^14 - 4*a^12*c^2 + 11*a^10*b^2*c^2 - 12*a^8*b^4*c^2 + 3*a^6*b^6*c^2 + 11*a^4*b^8*c^2 - 14*a^2*b^10*c^2 + 5*b^12*c^2 + 2*a^10*c^4 - 12*a^8*b^2*c^4 + 10*a^6*b^4*c^4 - 9*a^4*b^6*c^4 + 8*a^2*b^8*c^4 + b^10*c^4 + 7*a^8*c^6 + 3*a^6*b^2*c^6 - 9*a^4*b^4*c^6 + 4*a^2*b^6*c^6 - 5*b^8*c^6 - 7*a^6*c^8 + 11*a^4*b^2*c^8 + 8*a^2*b^4*c^8 - 5*b^6*c^8 - 2*a^4*c^10 - 14*a^2*b^2*c^10 + b^4*c^10 + 4*a^2*c^12 + 5*b^2*c^12 - c^14) : :

X(34105) lies on these lines: {3, 14900}, {23, 935}, {12584, 13289}


X(34106) =  X(3)X(2854)∩X(25)X(111)

Barycentrics    a^2*(a^8 - 6*a^6*b^2 - 2*a^4*b^4 + 6*a^2*b^6 + b^8 - 6*a^6*c^2 + 25*a^4*b^2*c^2 - 11*a^2*b^4*c^2 - 2*a^4*c^4 - 11*a^2*b^2*c^4 - 2*b^4*c^4 + 6*a^2*c^6 + c^8) : :

X(34106) lies on the cubic K1128 and these lines: {3, 2854}, {25, 111}, {187, 5938}, {543, 3534}, {1296, 33977}, {1576, 21309}, {2080, 11258}, {2780, 9409}, {2813, 15621}, {3455, 5210}, {5024, 10765}, {9215, 9486}, {10295, 14654}, {15922, 33980}, {33861, 33962}

X(34106) = reflection of X(11258) in X(33900)
X(34106) = {X(5191),X(14908)}-harmonic conjugate of X(1384)


X(34107) =  X(3)X(66)∩X(112)X(2393)

Barycentrics    a^2*(a^16 - 2*a^12*b^4 + 2*a^4*b^12 - b^16 + 3*a^12*b^2*c^2 - 4*a^10*b^4*c^2 - 3*a^8*b^6*c^2 + 4*a^6*b^8*c^2 + a^4*b^10*c^2 - b^14*c^2 - 2*a^12*c^4 - 4*a^10*b^2*c^4 + 14*a^8*b^4*c^4 - 4*a^6*b^6*c^4 - 2*a^4*b^8*c^4 - 8*a^2*b^10*c^4 + 6*b^12*c^4 - 3*a^8*b^2*c^6 - 4*a^6*b^4*c^6 - 2*a^4*b^6*c^6 + 8*a^2*b^8*c^6 + b^10*c^6 + 4*a^6*b^2*c^8 - 2*a^4*b^4*c^8 + 8*a^2*b^6*c^8 - 10*b^8*c^8 + a^4*b^2*c^10 - 8*a^2*b^4*c^10 + b^6*c^10 + 2*a^4*c^12 + 6*b^4*c^12 - b^2*c^14 - c^16) : :

X(34107) lies on the cubic K1128 and these lines: {3, 66}, {112, 2393}, {206, 13509}, {2373, 2867}

X(34107) = reflection of X(13509) in X(206)


X(34108) =  X(3)X(2393)∩X(112)X(1995)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(5*a^12 - 12*a^10*b^2 - 11*a^8*b^4 + 24*a^6*b^6 + 7*a^4*b^8 - 12*a^2*b^10 - b^12 - 12*a^10*c^2 + 46*a^8*b^2*c^2 - 40*a^6*b^4*c^2 - 52*a^4*b^6*c^2 + 52*a^2*b^8*c^2 + 6*b^10*c^2 - 11*a^8*c^4 - 40*a^6*b^2*c^4 + 90*a^4*b^4*c^4 - 40*a^2*b^6*c^4 + b^8*c^4 + 24*a^6*c^6 - 52*a^4*b^2*c^6 - 40*a^2*b^4*c^6 - 12*b^6*c^6 + 7*a^4*c^8 + 52*a^2*b^2*c^8 + b^4*c^8 - 12*a^2*c^10 + 6*b^2*c^10 - c^12) : :

X(34108) lies on the cubics K1118 and K1128 and on these lines: {3, 2393}, {112, 1995}


X(34109) =  X(3)X(64)∩X(74)X(2764)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(2*a^12 + a^10*b^2 - 19*a^8*b^4 + 26*a^6*b^6 - 4*a^4*b^8 - 11*a^2*b^10 + 5*b^12 + a^10*c^2 + 34*a^8*b^2*c^2 - 26*a^6*b^4*c^2 - 56*a^4*b^6*c^2 + 49*a^2*b^8*c^2 - 2*b^10*c^2 - 19*a^8*c^4 - 26*a^6*b^2*c^4 + 120*a^4*b^4*c^4 - 38*a^2*b^6*c^4 - 37*b^8*c^4 + 26*a^6*c^6 - 56*a^4*b^2*c^6 - 38*a^2*b^4*c^6 + 68*b^6*c^6 - 4*a^4*c^8 + 49*a^2*b^2*c^8 - 37*b^4*c^8 - 11*a^2*c^10 - 2*b^2*c^10 + 5*c^12) : :
X(34109) = 5 X[3] - 3 X[6760],2 X[3] - 3 X[11589],4 X[3] - 3 X[12096],3 X[2693] - X[7464],X[3529] + 3 X[6761],2 X[6760] - 5 X[11589],4 X[6760] - 5 X[12096],2 X[10297] - 3 X[16177]

X(34109) lies on the cubic K1128 and these lines: {3, 64}, {74, 2764}, {112, 2693}, {520, 9409}, {1503, 3184}, {1514, 13611}, {1515, 6716}, {3146, 6523}, {3529, 6761}, {10297, 16177}, {11457, 33553}

X(34109) = reflection of X(i) in X(j) for these {i,j}: {1515, 6716}, {12096, 11589}
X(34109) = circumcircle-inverse of X(10606)


X(34110) =  X(5)X(20189)∩X(1173)X(3628)

Barycentrics    (a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-3 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-3 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2+2 c^4) : :
Barycentrics    15 S^4 +(-4 R^4+7 R^2 SB+7 R^2 SC+3 SB SC-3 R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 +2 R^4 SB SC+3 R^2 SB SC SW+SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29378.

X(34110) lies on these lines: {5,20189}, {1173,3628}, {1656,26862}

X(34110) = isogonal conjugate of X(36153)
X(34110) = barycentric quotient X(115)/X(11792)
X(34110) = trilinear quotient X(1109)/X(11792)


X(34111) =  MIDPOINT OF X(4) AND X(32749)

Barycentrics    (b - c)^2*(b + c)^2*(-a^12 + 2*a^10*b^2 + 3*a^8*b^4 - 12*a^6*b^6 + 13*a^4*b^8 - 6*a^2*b^10 + b^12 + 2*a^10*c^2 + 4*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 12*a^2*b^8*c^2 - 4*b^10*c^2 + 3*a^8*c^4 - 10*a^6*b^2*c^4 - 3*a^4*b^4*c^4 - 6*a^2*b^6*c^4 + 7*b^8*c^4 - 12*a^6*c^6 - 4*a^4*b^2*c^6 - 6*a^2*b^4*c^6 - 8*b^6*c^6 + 13*a^4*c^8 + 12*a^2*b^2*c^8 + 7*b^4*c^8 - 6*a^2*c^10 - 4*b^2*c^10 + c^12)*(-a^14 + 5*a^12*b^2 - 11*a^10*b^4 + 15*a^8*b^6 - 15*a^6*b^8 + 11*a^4*b^10 - 5*a^2*b^12 + b^14 + 5*a^12*c^2 - 16*a^10*b^2*c^2 + 21*a^8*b^4*c^2 - 12*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 - 11*a^10*c^4 + 21*a^8*b^2*c^4 - 11*a^6*b^4*c^4 - 3*a^4*b^6*c^4 - 5*a^2*b^8*c^4 + 9*b^10*c^4 + 15*a^8*c^6 - 12*a^6*b^2*c^6 - 3*a^4*b^4*c^6 - 4*a^2*b^6*c^6 - 5*b^8*c^6 - 15*a^6*c^8 - 5*a^4*b^2*c^8 - 5*a^2*b^4*c^8 - 5*b^6*c^8 + 11*a^4*c^10 + 12*a^2*b^2*c^10 + 9*b^4*c^10 - 5*a^2*c^12 - 5*b^2*c^12 + c^14) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29381.

X(34111) lies on the nine-point circle and this line: {4, 32749}

X(34111) = midpoint of X(4) and X(32749)


X(34112) =  CENTER OF 1-ERHMANN CIRCLE

Barycentrics    a^2*(b^4 - 2*b^3*c - 2*b*c^3 + c^4 + 2*a*b*c*(b + c) - a^2*(b^2 + c^2) + (a - b)*(-a + c)*S) : :
X(34112) = (a^2 + b^2 + c^2) X[6] - (a^2 + b^2 + c^2 + S) X[101]

On page 18 of Jean-Pierre Ehrmann, Congruent Inscribed Rectangles, a circle is described. For any μ > 0, the circle, which passes through points X(100), X(106) and a certain point P(μ), is here named the μ-Ehrmann circle. Its center is given by

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

This center lies on the line {3,2827} and is given by the combos (r/μ - s) X[3] + s X[4491] = ((a + b + c)^2 - 2 S/μ) X[3] - (a + b + c)^2 X[4491]. X(34112) is obtained by putting μ = 1. (Peter Moses, July 15, 2019)

X(34112) lies on these lines: {6, 101}, {103, 11825}, {116, 5590}, {118, 6201}, {150, 1270}, {544, 5860}, {1160, 2808}, {1282, 5588}, {1362, 18960}, {2772, 7726}, {2774, 7733}, {2784, 6226}, {2786, 6320}, {2801, 12754}, {2809, 3640}, {2825, 12806}, {3022, 10928}, {3887, 13270}, {5185, 11389}, {5604, 10695}, {6214, 10739}, {9518, 13283}, {11371, 11712}


X(34113) =  MIDPOINT OF X(4) AND X(6325)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29386.

X(34113) lies on the nine-point circle and these lines: {2, 6236}, {4, 6325}, {125, 32228}, {126, 549}, {381, 15922}, {542, 13234}, {690, 12494}, {1560, 6032}, {2781, 13249}, {3849, 16188}, {5099, 8704}, {9517, 12624}, {11594, 25641}

X(34113) = midpoint of X(4) and X(6325)
X(34113) = complement of X(6236)
X(34113) = orthocentroidal circle inverse of X(15922)


X(34114) =  MIDPOINT OF X(3) AND X(2904)

Barycentrics    a^2 (a^14-4 a^12 b^2+5 a^10 b^4-5 a^6 b^8+4 a^4 b^10-a^2 b^12-4 a^12 c^2+10 a^10 b^2 c^2-7 a^8 b^4 c^2+2 a^6 b^6 c^2-4 a^4 b^8 c^2+4 a^2 b^10 c^2-b^12 c^2+5 a^10 c^4-7 a^8 b^2 c^4+2 a^6 b^4 c^4+2 a^4 b^6 c^4-5 a^2 b^8 c^4+3 b^10 c^4+2 a^6 b^2 c^6+2 a^4 b^4 c^6+4 a^2 b^6 c^6-2 b^8 c^6-5 a^6 c^8-4 a^4 b^2 c^8-5 a^2 b^4 c^8-2 b^6 c^8+4 a^4 c^10+4 a^2 b^2 c^10+3 b^4 c^10-a^2 c^12-b^2 c^12) : :
Barycentrics    (11 R^4-4 R^2 SB-4 R^2 SC-7 R^2 SW+SB SW+SC SW+SW^2)S^2 -5 R^4 SB SC+5 R^2 SB SC SW-SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29387.

X(34114) lies on these lines: {2,24572}, {3,1986}, {5,156}, {49,18912}, {54,6644}, {70,13353}, {110,11704}, {389,12038}, {575,32366}, {578,7706}, {1147,32358}, {1614,18504}, {5012,6241}, {5449,5972}, {5944,15074}, {6642,15047}, {6689,14076}, {6759,23323}, {8538,19154}, {8548,15462}, {9826,32171}, {12827,18356}, {13198,32139}, {18570,32392}

X(34114) = midpoint of X(i) and X(j) for these {i,j}: {3,2904}, {8907,15317}


X(34115) =  MIDPOINT OF X(3) AND X(70)

Barycentrics    a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-2 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^2-6 a^4 b^10 c^2+9 a^2 b^12 c^2-4 b^14 c^2-5 a^12 c^4+3 a^10 b^2 c^4-6 a^8 b^4 c^4+6 a^6 b^6 c^4-5 a^4 b^8 c^4+3 a^2 b^10 c^4+4 b^12 c^4+9 a^10 c^6-4 a^8 b^2 c^6+6 a^6 b^4 c^6+4 a^4 b^6 c^6-7 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^8-7 a^2 b^6 c^8-10 b^8 c^8-5 a^6 c^10-6 a^4 b^2 c^10+3 a^2 b^4 c^10+4 b^6 c^10+9 a^4 c^12+9 a^2 b^2 c^12+4 b^4 c^12-5 a^2 c^14-4 b^2 c^14+c^16 : :
Barycentrics    (11 R^4+3 R^2 SB+3 R^2 SC-11 R^2 SW-SB SW-SC SW+2 SW^2)S^2 -5 R^4 SB SC+3 R^2 SB SC SW : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29387.

X(34115) lies on these lines: {3,70}, {52,125}, {141,5944}, {5895,11472}, {6101,12359}, {6240,20127}, {6247,12041}, {7568,9306}, {15317,16266}, {18281,19360}

X(34115) = midpoint of X(3) and X(70)


X(34116) =  COMPLEMENT OF X(70)

Barycentrics    a^4 (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+2 b^6 c^2-2 b^4 c^4+2 a^2 c^6+2 b^2 c^6-c^8) : :
Barycentrics    (10 R^4-3 R^2 SB-3 R^2 SC-7 R^2 SW+SB SW+SC SW+SW^2)S^2 -2 R^4 SB SC+3 R^2 SB SC SW-SB SC SW^2 : :
X(34116) = 3*X[2]-X[70]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29387.

X(34116) lies on these lines: {2,70}, {3,6293}, {5,156}, {6,49}, {24,52}, {54,7544}, {68,110}, {113,6759}, {141,7542}, {206,9967}, {343,10020}, {578,18428}, {960,24301}, {973,1493}, {1092,1511}, {1209,6639}, {2883,12605}, {2917,11597}, {3548,15116}, {3575,13352}, {4550,7503}, {6145,10255}, {6515,11271}, {6593,8538}, {7547,15432}, {7689,11562}, {8780,9704}, {9714,18374}, {10316,11672}, {10897,10962}, {10898,10960}, {10984,13491}, {12363,32391}, {15136,15750}, {17834,22115}, {18377,26883}, {18531,32379}

X(34116) = midpoint of X(2904) and X(8907)
X(34116) = complement of X(70)
X(34116) = complementary conjugate of X(13371)
X(34116) = barycentric product of X(i) and X(j) for these {i,j}: {26,1993}, {8746,9723}
X(34116) = barycentric quotient of X(i) and X(j) for these {i,j}: {26,5392}, {571,70}, {1993,20564}, {8746,847}
X(34116) = trilinear product X(26)*X(47)
X(34116) = trilinear quotient of X(i) and X(j) for these {i,j}: {26,91}, {571,2158}
X(34116) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {569,10539,18474}


X(34117) =  MIDPOINT OF X(159) AND X(1351)

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^10-3 a^8 c^2+4 a^2 b^6 c^2-b^8 c^2+2 a^6 c^4-2 a^2 b^4 c^4+2 a^4 c^6+4 a^2 b^2 c^6-3 a^2 c^8-b^2 c^8+c^10) : :
Barycentrics    (3 R^2 SB+3 R^2 SC-SB SW-SC SW)S^2 + 4 R^2 SB SC SW : :
X(34117) = 3*X[182]-X[3357], 3*X[597]-X[6247], X[3098]-3*X[23042], 9*X[5050]-X[13093], 9*X[5085]-5*X[8567], 3*X[5102]+X[9924], 3*X[5476]-X[18381], X[5878]+3*X[11179], X[9934]+X[13248], 3*X[10168]-2*X[25563], 3*X[10250]-5*X[22234], 7*X[10541]-3*X[10606], 3*X[11216]-5*X[11482], X[14216]-3*X[23327], 2*X[18583]-X[23300], X[20299]-2*X[25555]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29388.

X(34117) lies on these lines: {3,1177}, {4,6}, {20,22151}, {23,154}, {24,18374}, {25,15135}, {26,206}, {64,7527}, {66,5576}, {141,3549}, {155,524}, {156,14984}, {157,30258}, {159,195}, {161,3060}, {182,3357}, {184,11470}, {378,15138}, {381,18432}, {389,19136}, {394,7493}, {542,8548}, {575,6000}, {576,2393}, {597,6247}, {599,7552}, {1350,7488}, {1352,10024}, {1598,19362}, {1614,2904}, {1619,11402}, {1853,5169}, {1971,13330}, {1974,19161}, {1994,7519}, {1995,15139}, {2777,25556}, {2888,11061}, {3098,23042}, {3172,28343}, {3564,15761}, {3589,15805}, {3818,9977}, {3843,32369}, {5050,13093}, {5085,8567}, {5102,9924}, {5198,11743}, {5476,18381}, {5622,6241}, {5878,11179}, {6145,7566}, {6293,7503}, {6403,20987}, {6800,10117}, {7387,9019}, {7529,16776}, {7540,9833}, {7556,17821}, {7564,19130}, {7691,19121}, {8537,14157}, {8540,26888}, {8541,26883}, {8547,15074}, {8721,22120}, {9715,32391}, {9934,13248}, {9971,10594}, {10168,25563}, {10250,22234}, {10510,12082}, {10535,19369}, {10541,10606}, {10601,23332}, {10752,11464}, {11216,11482}, {11799,18445}, {11819,21850}, {12294,21637}, {14216,23327}, {15066,17847}, {15068,16534}, {15647,19504}, {18583,23300}, {18911,32125}, {19164,22240}, {19347,32621}, {20299,25555}, {22802,32271}, {32046,32321}, {32251,32274}

X(34117) = midpoint of X(i) and X(j) for these {i,j}: {159,1351}, {576,6759}, {9934,13248}, {11216,32063}
X(34117) = reflection of X(i) in X(j) for these {i,j}: {64,15579}, {66,20300}, {9924,15580}, {15577,206}, {15581,6759}, {18382,5480}, {20299,25555}, {23300,18583}
X(34117) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,1181,8550}, {6,1498,8549}, {64,10249,15579}, {66,14561,20300}, {1350,19132,23041}, {5480,8550,12241}, {8549,19149,1498}


X(34118) =  MIDPOINT OF X(66) AND X(1352)

Barycentrics    a^12-2 a^10 b^2+a^8 b^4-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 b^10 c^2+a^8 c^4-2 a^6 b^2 c^4+2 a^4 b^4 c^4-2 a^2 b^6 c^4+b^8 c^4-2 a^2 b^4 c^6-4 b^6 c^6-a^4 c^8+b^4 c^8+2 a^2 c^10+2 b^2 c^10-c^12 : :
Barycentrics    (3 R^2 SB+3 R^2 SC-4 R^2 SW-SB SW-SC SW+SW^2)S^2 -2 R^2 SB SC SW+SB SC SW^2 : :
X(34118) = X[206]-2*X[24206], 5*X[1656]-3*X[19153], 5*X[3763]-3*X[23041], 3*X[3818]-X[22802], X[6759]-3*X[11178], X[11477]-3*X[23049], 3*X[14810]-2*X[32903], 5*X[17821]-9*X[21358]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29388.

X(34118) lies on these lines: {2,15139}, {3,66}, {4,67}, {6,70}, {68,524}, {69,1225}, {154,7495}, {182,6689}, {195,15141}, {206,24206}, {343,26283}, {394,858}, {511,9927}, {542,1147}, {575,18952}, {576,12585}, {631,32337}, {924,18312}, {1216,2393}, {1350,12225}, {1656,19153}, {1899,5094}, {2854,15133}, {3410,16063}, {3448,9716}, {3549,19127}, {3564,13371}, {3763,23041}, {3818,22802}, {3827,5694}, {5480,7507}, {5596,7558}, {5622,23294}, {5921,28419}, {5925,6240}, {6000,18431}, {6293,7544}, {6640,15462}, {6759,11178}, {7505,18374}, {7528,16776}, {7545,32262}, {7729,14982}, {9019,14790}, {9630,12589}, {10249,11457}, {10295,10606}, {10516,13160}, {11470,32285}, {11477,23049}, {14003,20021}, {14810,32903}, {15073,25739}, {15106,23315}, {17821,21358}, {18475,18580}, {18909,32184}, {18919,22533}, {21284,31383}, {31670,31724}

X(34118) = midpoint of X(i) and X(j) for these {i,j}: {66,1352}, {8549,15069}, {15141,32306}
X(34118) = reflection of X(i) in X(j) for these {i,j}: {6,20300}, {182,6697}, {206,24206}, {575,32767}, {9833,15582}, {15577,141}
X(34118) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1853,15069,8549}


X(34119) =  COMPLEMENT OF X(199)

Barycentrics    a^4 b^2+a^3 b^3-a b^5-b^6+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 b^3 c^2+b^4 c^2+a^3 c^3+a^2 b c^3+2 a b^2 c^3+2 b^3 c^3-a b c^4+b^2 c^4-a c^5-b c^5-c^6 : :
Barycentrics   (2 p^2-8 R^2+SW)S^2 + 2 p^2 SB SC+SB SC SW : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29391.

X(34119) lies on these lines: {2,3}, {11,7073}, {12,1961}, {58,14873}, {86,8044}, {125,17167}, {495,5311}, {496,17017}, {1211,25688}, {2886,27798}, {8287,18165}, {15523,21682}, {19737,19755}, {20531,21098}, {21243,24220}, {23304,29644}, {23922,32778}, {26481,29657}

X(34119) = complement of X(199)


X(34120) =  COMPLEMENT OF X(406)

Barycentrics    (a^2-b^2-c^2) (a^5+a^4 b-a b^4-b^5+a^4 c+2 a^3 b c-b^4 c+2 a b^2 c^2+2 b^3 c^2+2 b^2 c^3-a c^4-b c^4-c^5) : :
Barycentrics    R S^2 + (-4 a R^2-4 b R^2-4 c R^2+a SW+b SW+c SW)S -R SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29391.

X(34120) lies on these lines: {2,3}, {8,18447}, {10,1060}, {69,20746}, {123,958}, {222,26955}, {255,21912}, {499,17102}, {942,20266}, {1038,1698}, {1040,3624}, {1062,1125}, {1214,19854}, {1437,1899}, {2968,10527}, {3616,18455}, {3739,6389}, {3767,16716}, {5275,23115}, {5276,22120}, {5277,10316}, {5283,14961}, {5714,28836}, {8227,25915}, {10267,25968}, {15668,18642}, {16589,22401}, {17614,24301}, {18592,31198}

X(34120) = complement of X(406)


X(34121) =  ISOGONAL CONJUGATE OF X(13386)

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

See Ercole Suppa, Hyacinthos 29397 and Peter Moses, Hyacinthos 29399.

See also X(34125).

X(34121) lies on the conics {{A,B,C,X(3),X(28)}}, {{A,B,C,X(6),X(1805)}}, {{A,B,C,X(19),X(6213)}}, {{A,B,C,X(37),X(1123)}}, {{A,B,C,X(55),X(606)}}, the cubic K171, and on these lines: {3, 6213}, {4, 16027}, {19, 25}, {28, 1123}, {41, 5416}, {48, 5414}, {56, 2362}, {104, 6135}, {480, 11498}, {603, 606}, {607, 5413}, {608, 5412}, {958, 7090}, {1335, 1437}, {1400, 5415}, {1436, 13456}, {1444, 5391}, {1598, 6212}, {1633, 30296}, {1659, 13887}, {1950, 26953}, {2066, 2183}, {2082, 19005}, {2285, 19006}, {2333, 8948}, {3069, 30385}, {3207, 30336}, {3542, 16033}, {6204, 13889}, {7289, 13388}, {7348, 13943}, {16036, 19173}, {19215, 19588}

X(34121) = isogonal conjugate of X(13386)
X(34121) = isogonal conjugate of the anticomplement of X(13388)
X(34121) = isogonal conjugate of the isotomic conjugate of X(13387)
X(34121) = isogonal conjugate of the polar conjugate of X(1123)
X(34121) = polar conjugate of the isotomic conjugate of X(1335)
X(34121) = X(i)-Ceva conjugate of X(j) for these (i,j): {7133, 6}, {13387, 1335}
X(34121) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13386}, {2, 6212}, {4, 3083}, {19, 1267}, {33, 13453}, {34, 13425}, {63, 1336}, {77, 13426}, {78, 13459}, {92, 1124}, {264, 605}, {345, 13460}, {348, 13427}, {1897, 6364}, {4025, 6136}, {6213, 13424}, {13389, 14121}, {13390, 30556}
X(34121) = crosspoint of X(1123) and X(13387)
X(34121) = barycentric product X(i)*X(j) for these {i,j}: {1, 6213}, {3, 1123}, {4, 1335}, {6, 13387}, {19, 3084}, {25, 5391}, {77, 13456}, {78, 13438}, {92, 606}, {219, 13437}, {222, 13454}, {607, 13436}, {608, 13458}, {905, 6135}, {1659, 5414}, {1783, 6365}, {2067, 7090}, {2362, 30557}, {6413, 13457}, {7133, 13388}
X(34121) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 1267}, {6, 13386}, {25, 1336}, {31, 6212}, {48, 3083}, {184, 1124}, {219, 13425}, {222, 13453}, {606, 63}, {607, 13426}, {608, 13459}, {1123, 264}, {1335, 69}, {1395, 13460}, {2212, 13427}, {3084, 304}, {3937, 22107}, {5391, 305}, {6135, 6335}, {6213, 75}, {6365, 15413}, {9247, 605}, {13387, 76}, {13437, 331}, {13438, 273}, {13454, 7017}, {13456, 318}, {22383, 6364}
X(34121) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 25, 34125}, {37, 197, 34125}, {55, 198, 34125}, {910, 1486, 34125}, {4386, 20876, 34125}, {5275, 23381, 34125}, {11434, 15494, 34125}


X(34122) =  MIDPOINT OF X(80) AND X(15015)

Barycentrics    2*a^3*b - 3*a^2*b^2 - 2*a*b^3 + 3*b^4 + 2*a^3*c - 2*a^2*b*c + 4*a*b^2*c - 3*a^2*c^2 + 4*a*b*c^2 - 6*b^2*c^2 - 2*a*c^3 + 3*c^4 : :
X(34122) = X[1] + 2 X[3036],X[1] - 4 X[6667],7 X[2] - X[10031],4 X[5] - X[1537],X[8] + 2 X[1387],2 X[8] + X[25416],X[8] + 5 X[31272],X[8] + 3 X[32558],2 X[10] + X[11],4 X[10] - X[1145],X[10] + 2 X[6702],10 X[10] - X[13996],5 X[10] + X[21630],2 X[11] + X[1145],X[11] - 4 X[6702],5 X[11] + X[13996],5 X[11] - 2 X[21630],X[65] + 2 X[18254],X[72] + 2 X[12736],X[80] + 5 X[1698],X[80] + 2 X[3035],2 X[80] + X[10609],X[100] - 7 X[9780],X[100] + 2 X[12019],2 X[100] + X[12690],X[104] + 5 X[5818],X[119] - 4 X[9956],X[119] + 2 X[12619],4 X[119] - X[13257],2 X[149] + X[12732],X[153] + 2 X[13226],X[214] - 4 X[3634],2 X[214] - 5 X[31235],X[355] + 2 X[6713],4 X[1125] - X[1317],2 X[1125] + X[15863],X[1145] + 8 X[6702],5 X[1145] - 2 X[13996],5 X[1145] + 4 X[21630],X[1317] + 2 X[15863],X[1320] + 5 X[3617],4 X[1387] - X[25416],2 X[1387] - 5 X[31272],2 X[1387] - 3 X[32558],5 X[1656] - 2 X[11729],5 X[1656] + X[19914],5 X[1698] - 2 X[3035],10 X[1698] - X[10609],5 X[1698] - X[15015],X[1737] + 2 X[5123],2 X[1737] + X[17757],4 X[3035] - X[10609],X[3036] + 2 X[6667],7 X[3090] - X[10698],X[3555] - 4 X[18240],5 X[3616] + X[12531],5 X[3616] - 2 X[12735],7 X[3624] - X[7972],X[3626] + 2 X[33709],4 X[3628] - X[19907],8 X[3634] - 5 X[31235],5 X[3697] - 2 X[14740],5 X[3698] + X[17638],4 X[3812] - X[11570],4 X[3822] - X[12831],4 X[3826] - X[10427],4 X[3828] - X[6174],X[4511] + 2 X[11545],5 X[4668] + X[26726],2 X[5044] + X[6797],2 X[5083] - 5 X[5439],4 X[5123] - X[17757],X[5176] + 2 X[15325],2 X[5836] + X[12758],X[6224] - 13 X[19877],X[6246] + 2 X[6684],2 X[6246] + X[24466],X[6326] - 4 X[20400],4 X[6684] - X[24466],20 X[6702] + X[13996],10 X[6702] - X[21630],4 X[6723] - X[31525],7 X[9780] + 2 X[12019],14 X[9780] + X[12690],2 X[9956] + X[12619],16 X[9956] - X[13257],X[10265] + 5 X[31399],X[10914] + 2 X[15558],X[11362] + 2 X[16174],2 X[11729] + X[19914],4 X[12019] - X[12690],X[12119] - 7 X[31423],X[12531] + 2 X[12735],8 X[12619] + X[13257],X[12751] + 2 X[20418],X[13996] + 2 X[21630],X[14872] + 2 X[15528],5 X[15017] - 17 X[30315],X[17636] + 5 X[25917],5 X[18230] + X[20119],4 X[19878] - X[33812],X[25416] - 10 X[31272],X[25416] - 6 X[32558],5 X[31272] - 3 X[32558]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29407.

X(34122) lies on these lines: {1,3036}, {2,952}, {5,1537}, {8,1387}, {10,11}, {12,5883}, {21,33814}, {65,18254}, {72,12736}, {80,1698}, {100,405}, {104,474}, {119,125}, {149,5084}, {153,443}, {214,3634}, {355,6713}, {377,10742}, {404,18357}, {406,1862}, {452,13199}, {475,12138}, {515,21154}, {517,17533}, {518,1737}, {519,32557}, {528,19875}, {632,3897}, {900,14431}, {958,10090}, {1001,10087}, {1125,1317}, {1320,3617}, {1329,5692}, {1376,10058}, {1482,6931}, {1484,17527}, {1656,5554}, {1772,24433}, {1788,24465}, {2475,22799}, {2478,10738}, {2800,3753}, {2804,14429}, {2829,5587}, {3090,10698}, {3555,18240}, {3614,3754}, {3616,12531}, {3624,7972}, {3626,33709}, {3628,19907}, {3679,5854}, {3697,14740}, {3698,17638}, {3756,24222}, {3812,11570}, {3813,15079}, {3816,5533}, {3822,12831}, {3826,10427}, {3828,6174}, {3847,5697}, {4193,5690}, {4205,9978}, {4208,13243}, {4511,11545}, {4668,26726}, {4857,32157}, {4881,28224}, {4996,5260}, {5044,6797}, {5046,22938}, {5083,5439}, {5129,20095}, {5151,11105}, {5154,22791}, {5176,15325}, {5187,12702}, {5541,31435}, {5657,17556}, {5817,17532}, {5836,12758}, {5840,11113}, {6224,19877}, {6246,6684}, {6264,8583}, {6265,19860}, {6326,20400}, {6723,31525}, {6788,17724}, {6831,32554}, {6856,9952}, {6857,9945}, {6904,12248}, {6910,12747}, {6913,12775}, {6921,18525}, {7741,8256}, {8165,15650}, {8582,10265}, {8728,11698}, {8988,13936}, {9809,11024}, {9963,17558}, {10039,10179}, {10074,25524}, {10592,12532}, {10593,14923}, {10728,26062}, {10956,20118}, {11108,12331}, {11362,16174}, {11715,17614}, {12119,31423}, {12737,19861}, {12751,20418}, {12773,16408}, {13587,28186}, {13883,13976}, {13893,19077}, {13947,19078}, {14193,26073}, {14439,21044}, {14872,15528}, {15017,30315}, {17100,19525}, {17575,24987}, {17768,31160}, {18230,20119}, {19878,33812}, {25490,26029}, {25491,26030}, {25513,26046}

X(34122) = midpoint of X(i) and X(j) for these {i,j}: {80, 15015}, {3679, 16173}
X(34122) = reflection of X(i) in X(j) for these {i,j}: {10609, 15015}, {15015, 3035}
X(34122) = centroid of X(1)X(8)X(11)
X(34122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 1387, 25416}, {8, 31272, 1387}, {10, 11, 1145}, {10, 6702, 11}, {10, 17606, 24390}, {10, 17619, 4187}, {11, 13996, 21630}, {80, 1698, 3035}, {80, 3035, 10609}, {100, 12019, 12690}, {214, 3634, 31235}, {1125, 15863, 1317}, {1656, 19914, 11729}, {1737, 5123, 17757}, {3036, 6667, 1}, {3616, 12531, 12735}, {3753, 10175, 17530}, {6246, 6684, 24466}, {7705, 25005, 5}, {9956, 12619, 119}, {9956, 24982, 442}


X(34123) =  X(1)X(1145)∩X(2)X(952)

Barycentrics    4*a^4 - 2*a^3*b - 5*a^2*b^2 + 2*a*b^3 + b^4 - 2*a^3*c + 6*a^2*b*c - 5*a^2*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4 : :
X(34123) = 2 X[1] + X[1145],X[1] + 2 X[3035],4 X[1] - X[25416],7 X[1] - X[26726],5 X[2] + X[10031],2 X[3] + X[1537],X[3] + 2 X[11729],X[8] + 2 X[12735],2 X[10] + X[1317],2 X[10] - 5 X[31235],X[11] + 2 X[214],X[11] - 4 X[1125],2 X[11] + X[10609],4 X[11] - X[12690],5 X[11] - 8 X[33709],X[72] + 2 X[5083],X[80] - 7 X[3624],X[80] - 4 X[6667],X[100] + 2 X[1387],X[100] + 5 X[3616],5 X[100] + X[9802],4 X[100] - X[12732],2 X[104] + X[13257],X[119] + 2 X[1385],2 X[140] + X[19907],X[149] + 2 X[9945],X[214] + 2 X[1125],4 X[214] - X[10609],8 X[214] + X[12690],5 X[214] + 4 X[33709],2 X[551] + X[6174],5 X[631] + X[10698],X[908] + 2 X[5126],2 X[946] + X[24466],2 X[960] + X[11570],2 X[993] + X[12831],2 X[1001] + X[10427],8 X[1125] + X[10609],16 X[1125] - X[12690],5 X[1125] - 2 X[33709],X[1145] - 4 X[3035],2 X[1145] + X[25416],7 X[1145] + 2 X[26726],X[1317] + 5 X[31235],2 X[1319] + X[17757],X[1320] - 7 X[3622],2 X[1387] - 5 X[3616],10 X[1387] - X[9802],8 X[1387] + X[12732],X[1537] - 4 X[11729],5 X[1698] - 2 X[3036],5 X[1698] + X[7972],8 X[3035] + X[25416],14 X[3035] + X[26726],2 X[3036] + X[7972],7 X[3526] - X[19914],X[3555] + 2 X[14740],25 X[3616] - X[9802],20 X[3616] + X[12732],7 X[3624] - 4 X[6667],4 X[3634] - X[15863],2 X[3634] + X[33812],8 X[3636] + X[13996],2 X[3828] + X[11274],X[4511] + 2 X[15325],2 X[5087] + X[21578],5 X[5439] - 2 X[12736],2 X[5542] + X[6068],11 X[5550] + X[6224],11 X[5550] - 2 X[12019],11 X[5550] - 5 X[31272],2 X[5901] + X[33814],2 X[5972] + X[31525],X[6154] + 14 X[15808],X[6154] + 2 X[21630],X[6224] + 2 X[12019],X[6224] + 5 X[31272],X[6265] + 2 X[6713],X[6326] + 2 X[20418],2 X[6684] + X[25485],2 X[6702] - 5 X[19862],2 X[6702] + X[33337],X[6735] + 2 X[25405],5 X[8227] + X[12119],7 X[9624] - X[14217],7 X[9780] - X[12531],4 X[9802] + 5 X[12732],2 X[10609] + X[12690],X[10609] + 4 X[32557],5 X[10609] + 16 X[33709],X[10707] - 3 X[32558],2 X[11813] + X[15326],2 X[12019] - 5 X[31272],X[12611] + 2 X[13624],X[12665] + 2 X[12675],X[12690] - 8 X[32557],5 X[12690] - 32 X[33709],X[12743] + 2 X[17647],X[12751] - 4 X[20400],X[12832] + 2 X[30144],X[15015] + 3 X[25055],5 X[15017] + 7 X[30389],2 X[15254] + X[25558],7 X[15808] - X[21630],X[15863] + 2 X[33812],X[16173] - 3 X[25055],X[17660] + 2 X[18254],X[17660] + 5 X[25917],2 X[18254] - 5 X[25917],5 X[19862] + X[33337],7 X[25416] - 4 X[26726],5 X[32557] - 4 X[33709]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29407.

X(34123) lies on these lines: {1,1145}, {2,952}, {3,1537}, {8,12735}, {10,1317}, {11,214}, {21,33860}, {30,4881}, {36,17768}, {72,5083}, {80,3624}, {100,474}, {104,405}, {106,17724}, {119,1385}, {140,19907}, {149,443}, {153,5084}, {377,10738}, {392,2800}, {404,5901}, {406,12138}, {452,12248}, {475,1862}, {515,17533}, {528,15015}, {549,3877}, {551,2802}, {631,10698}, {758,5298}, {900,14419}, {908,5126}, {946,24466}, {954,5856}, {958,10074}, {960,11570}, {962,19537}, {993,12831}, {997,12739}, {1001,10058}, {1319,10956}, {1320,3622}, {1329,21842}, {1375,24559}, {1376,10087}, {1388,26364}, {1482,6921}, {1483,25005}, {1484,8728}, {1621,17100}, {1698,3036}, {1768,31435}, {2475,22938}, {2478,10742}, {2771,5642}, {2804,11125}, {2829,3576}, {2886,5533}, {3109,25533}, {3303,25438}, {3485,24465}, {3526,19914}, {3555,14740}, {3634,15863}, {3636,13996}, {3828,11274}, {3897,11698}, {3898,4995}, {4188,22791}, {4190,18493}, {4208,9963}, {4511,15325}, {4855,11373}, {4996,5253}, {5046,22799}, {5087,21578}, {5250,12515}, {5433,12832}, {5439,12736}, {5440,5853}, {5444,6690}, {5542,6068}, {5550,6224}, {5603,16371}, {5686,14151}, {5690,17566}, {5730,7288}, {5731,17556}, {5794,10073}, {5840,5886}, {5882,17619}, {5972,31525}, {6154,15808}, {6265,6713}, {6326,8583}, {6684,25485}, {6702,19862}, {6735,25405}, {6767,13278}, {6857,13226}, {6904,13199}, 6931,18525}, {6933,12747}, {7968,13922}, {7969,13991}, {8068,25466}, {8227,12119}, {9155,9978}, {9624,14217}, {9778,19705}, {9780,12531}, {9809,19526}, {10090,11507}, {10176,31157}, {10283,17564}, {10707,32558}, {10728,26129}, {11108,12773}, {11230,17530}, {11715,17575}, {11813,15326}, {12331,16408}, {12611,13624}, {12619,24987}, {12665,12675}, {12737,19860}, {12738,17590}, {12751,20400}, {13243,17558}, {13279,16410}, {13587,28174}, {13902,19112}, {13959,19113}, {15017,25522}, {15178,24982}, {15251,16377}, {15254,25558}, {16203,25875}, {16370,21151}, {16383,28915}, {17044,25532}, {17529,22935}, {17580,20095}, {17660,18254}, {18224,22766}, {18861,19525}, {24928,27385}

X(34123) = midpoint of X(i) and X(j) for these {i,j}: {214, 32557}, {5686, 14151}, {15015, 16173}
X(34123) = reflection of X(i) in X(j) for these {i,j}: {11, 32557}, {21154, 10165}, {23513, 11230}, {32557, 1125}
X(34123) = QA-P34 (Euler-Poncelet Point of the Centroid Quadrangle) of quadrangle ABCX(1)
X(34123) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1145, 25416}, {1, 3035, 1145}, {3, 11729, 1537}, {11, 214, 10609}, {11, 10609, 12690}, {80, 3624, 6667}, {100, 3616, 1387}, {214, 1125, 11}, {1125, 17614, 442}, {1317, 31235, 10}, {1698, 7972, 3036}, {3634, 33812, 15863}, {5550, 6224, 31272}, {6224, 31272, 12019}, {15015, 25055, 16173}, {17660, 25917, 18254}, {19862, 33337, 6702}


X(34124) =  X(105)X(474)∩X(115)X(120)

Barycentrics    2*a^4*b - 5*a^3*b^2 + 3*a^2*b^3 - 3*a*b^4 + 3*b^5 + 2*a^4*c - 6*a^3*b*c + 7*a^2*b^2*c - 6*a*b^3*c - 5*b^4*c - 5*a^3*c^2 + 7*a^2*b*c^2 + 10*a*b^2*c^2 + 2*b^3*c^2 + 3*a^2*c^3 - 6*a*b*c^3 + 2*b^2*c^3 - 3*a*c^4 - 5*b*c^4 + 3*c^5 : :
X(34124) = 2 X[10] + X[1358],4 X[1125] - X[3021],5 X[1698] - 2 X[3039]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29407.

X(34124) lies on these lines: {2,28915}, {10,1358}, {105,474}, {115,120}, {377,10743}, {405,1292}, {443,20344}, {528,15015}, {1125,3021}, {1698,3039}, {2478,15521}, {2809,3753}, {4187,5511}, {6714,13747}, {11716,17614}, {17580,20097}


X(34125) =  ISOGONAL CONJUGATE OF X(13387)

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

See also X(34121).

X(34125) lies on the conics {{A,B,C,X(3),X(28)}}, {{A,B,C,X(6),X(1806)}}, {{A,B,C,X(19),X(6212)}}, {{A,B,C,X(37),X(1124)}}, {{A,B,C,X(55),X(605)}}, the cubic K171, and on these lines: {3, 6212}, {4, 16033}, {19, 25}, {28, 1336}, {41, 5415}, {48, 2066}, {56, 7968}, {104, 6136}, {480, 11497}, {603, 605}, {607, 5412}, {608, 5413}, {958, 14121}, {1124, 1437}, {1267, 1444}, {1400, 5416}, {1436, 13427}, {1598, 6213}, {1633, 30297}, {1951, 26953}, {2082, 19006}, {2183, 5414}, {2285, 19005}, {2333, 8946}, {3068, 30386}, {3207, 30335}, {3542, 16027}, {6203, 13943}, {7289, 13389}, {7347, 13889}, {13390, 13940}, {16031, 19173}, {19216, 19588}

X(34125) = isogonal conjugate of X(13387)
X(34125) = isogonal conjugate of the anticomplement of X(13389)
X(34125) = isogonal conjugate of the isotomic conjugate of X(13386)
X(34125) = isogonal conjugate of the polar conjugate of X(1336)
X(34125) = polar conjugate of the isotomic conjugate of X(1124)
X(34125) = X(13386)-Ceva conjugate of X(1124)
X(34125) = perspector of ABC and unary cofactor triangle of 2nd Pamfilos-Zhou triangle
X(34125) = crosspoint of X(1336) and X(13386)
X(34125) = crosssum of X(1335) and X(34121)
X(34125) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13387}, {2, 6213}, {4, 3084}, {19, 5391}, {33, 13436}, {34, 13458}, {63, 1123}, {75, 34121}, {77, 13454}, {78, 13437}, {92, 1335}, {264, 606}, {345, 13438}, {348, 13456}, {1659, 30557}, {1897, 6365}, {4025, 6135}, {6212, 13435}, {7090, 13388}
X(34125) = barycentric product X(i)*X(j) for these {i,j}: {1, 6212}, {3, 1336}, {4, 1124}, {6, 13386}, {19, 3083}, {25, 1267}, {77, 13427}, {78, 13460}, {92, 605}, {219, 13459}, {222, 13426}, {607, 13453}, {608, 13425}, {905, 6136}, {1783, 6364}, {2066, 13390}, {6502, 14121}, {13424, 34121}, {16232, 30556}
X(34125) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 5391}, {6, 13387}, {25, 1123}, {31, 6213}, {32, 34121}, {48, 3084}, {184, 1335}, {219, 13458}, {222, 13436}, {605, 63}, {607, 13454}, {608, 13437}, {1124, 69}, {1267, 305}, {1336, 264}, {1395, 13438}, {2212, 13456}, {3083, 304}, {3937, 22106}, {5413, 13457}, {6136, 6335}, {6212, 75}, {6364, 15413}, {9247, 606}, {13386, 76}, {13426, 7017}, {13427, 318}, {13459, 331}, {13460, 273}, {22383, 6365}, {34121, 13435}
X(34125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {19, 25, 34121}, {37, 197, 34121}, {55, 198, 34121}, {910, 1486, 34121}, {4386, 20876, 34121}, {5275, 23381, 34121}, {11434, 15494, 34121}


X(34126) =  X(2)X(952)∩X(11)X(35)

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+8 a^5 b c-13 a^3 b^3 c+4 a^2 b^4 c+5 a b^5 c-2 b^6 c-6 a^5 c^2+8 a^3 b^2 c^2+2 a^2 b^3 c^2+2 a b^4 c^2-6 b^5 c^2+6 a^4 c^3-13 a^3 b c^3+2 a^2 b^2 c^3-10 a b^3 c^3+6 b^4 c^3+6 a^3 c^4+4 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 : :
Barycentrics    (4 a R^2-4 b R^2-4 a SB+4 b SB-c SB-4 a SC-b SC+4 c SC+4 a SW+b SW)S^2 -13 R S^3+3 R S SB SC+b SB SC^2-c SB SC^2-b SB SC SW : :
X(34126) = 5*X[3]+X[10724], X[100]-7*X[3526], X[104]+5*X[1656], X[119]-4*X[3628], X[149]+11*X[3525], X[153]-13*X[5067], 5*X[631]+X[10738], 5*X[632]+X[1484], 2*X[1125]+X[12619], X[1385]+2*X[6702], X[1483]+2*X[3036], 5*X[1698]+X[12737], 7*X[3090]-X[10742], 4*X[3530]-X[24466], X[3579]+2*X[16174], 5*X[3616]+X[19914], 7*X[3624]-X[6265], 7*X[3851]-X[10728], 11*X[5056]+X[12248], 11*X[5070]+X[12773], 11*X[5550]+X[12247], X[5657]+3*X[32558], X[6174]-4*X[10124], X[6246]+2*X[13624], X[6684]+2*X[33709], 2*X[6701]+X[33856], X[6797]+2*X[31838], 5*X[8227]+X[12515], 2*X[9956]+X[11715], X[10265]+5*X[19862], 13*X[10303]-X[13199], X[10707]+5*X[15694], X[10711]-7*X[15703], X[11698]+2*X[20418], 2*X[15178]+X[15863], X[16173]+X[26446], 8*X[16239]-5*X[31235], 2*X[18254]+X[24475], X[19916]+5*X[30795]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29414.

X(34126) lies on these lines: {2,952}, {3,10724}, {5,2829}, {11,35}, {30,21154}, {100,3526}, {104,1656}, {119,3628}, {149,3525}, {153,5067}, {498,12735}, {499,1387}, {517,32557}, {528,11539}, {549,5840}, {631,10738}, {632,1484}, {1125,12619}, {1385,6702}, {1483,3036}, {1537,6952}, {1698,12737}, {2800,3833}, {2802,11231}, {3090,10742}, {3530,24466}, {3579,16174}, {3582,5844}, {3616,19914}, {3624,6265}, {3851,10728}, {5056,12248}, {5070,12773}, {5432,5533}, {5433,8068}, {5550,12247}, {5657,32558}, {6174,10124}, {6246,13624}, {6684,33709}, {6701,33856}, {6797,31838}, {6861,13226}, {6958,22791}, {7489,18861}, {7505,12138}, {7583,13977}, {7584,13913}, {8227,12515}, {8976,19081}, {9956,11715}, {10199,10283}, {10200,11729}, {10265,19862}, {10303,13199}, {10707,15694}, {10711,15703}, {11698,20418}, {13951,19082}, {15178,15863}, {16173,26446}, {16239,31235}, {18254,24475}, {19916,30795}

X(34126) = midpoint of X(i) and X(j) for these {i,j}: {16173,26446}, {21154,23513}
X(34126) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {11,140,33814}, {632,1484,3035}, {1125,12619,19907}, {6667,6713,5}
X(34126) = complement of X(38752)
X(34126) = centroid of X(3)X(5)X(11)
X(34126) = center of the Vu pedal-centroidal circle of X(104)


X(34127) =  COMPLEMENT OF X(15561)

Barycentrics    2 a^8-5 a^6 b^2+7 a^4 b^4-6 a^2 b^6+2 b^8-5 a^6 c^2+3 a^2 b^4 c^2-7 b^6 c^2+7 a^4 c^4+3 a^2 b^2 c^4+10 b^4 c^4-6 a^2 c^6-7 b^2 c^6+2 c^8 : :
Barycentrics    (3 SB SC-2 SB SW-2 SC SW-3 SW^2)S^2 + 11 S^4+SB SC SW^2 : :
X(34127) = 5*X[3]+X[10723], X[4]-4*X[15092], X[98]+5*X[1656], X[99]-7*X[3526], X[114]-4*X[3628], X[115]+2*X[140], X[147]-13*X[5067], X[148]+11*X[3525], 2*X[547]+X[6055], X[549]+2*X[5461], 2*X[620]-5*X[632], 5*X[631]+X[6321], X[671]+5*X[15694], X[1511]+2*X[15359], X[2482]-4*X[10124], 7*X[3090]-X[6033], 7*X[3851]-X[10722], X[5054]+X[9166], 11*X[5056]+X[9862], 5*X[5071]+X[14830], X[5690]+2*X[11725], 2*X[5972]+X[15535], X[6054]-7*X[15703], 2*X[6721]+X[11623], 2*X[6723]+X[33511], X[9880]+2*X[12100], 2*X[9956]+X[11710], 13*X[10303]-X[13172], X[12117]-7*X[15701], 2*X[14693]+X[15980], 5*X[15059]+X[18332], 8*X[16239]-5*X[31274]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29414.

X(34127) lies on these lines: {2,2782}, {3,10723}, {4,15092}, {5,2794}, {30,5215}, {98,1656}, {99,3526}, {114,3628}, {115,140}, {147,5067}, {148,3525}, {542,15699}, {543,11539}, {547,6055}, {549,5461}, {620,632}, {631,6321}, {671,15694}, {1506,12829}, {1511,15359}, {2023,7746}, {2482,10124}, {2784,10172}, {3090,6033}, {3398,32967}, {3851,10722}, {4027,16922}, {5054,9166}, {5056,9862}, {5070,7943}, {5071,14830}, {5690,11725}, {5972,15535}, {6054,15703}, {6721,11623}, {6723,33511}, {7505,12131}, {7583,13967}, {7607,7934}, {7828,11272}, {7887,10104}, {7901,12176}, {7940,13108}, {9753,14881}, {9880,12100}, {9956,11710}, {10303,13172}, {12117,15701}, {14693,15980}, {15059,18332}, {16239,31274}

X(34127) = midpoint of X(i) and X(j) for these {i,j}: {3,14639}, {5054,9166}, {14651,15561}
X(34127) = reflection of X(22515) in X(14639)
X(34127) = complement of X(15561)
X(34127) = center of the Vu pedal-centroidal circle of X(98)
X(34127) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,14651,15561}, {5,6036,12042}, {5,12042,22505}, {115,140,33813}, {547,6055,22566}, {6036,6722,5}


X(34128) =  COMPLEMENT OF X(14643)

Barycentrics    2 a^10-5 a^8 b^2+a^6 b^4+7 a^4 b^6-7 a^2 b^8+2 b^10-5 a^8 c^2+12 a^6 b^2 c^2-10 a^4 b^4 c^2+9 a^2 b^6 c^2-6 b^8 c^2+a^6 c^4-10 a^4 b^2 c^4-4 a^2 b^4 c^4+4 b^6 c^4+7 a^4 c^6+9 a^2 b^2 c^6+4 b^4 c^6-7 a^2 c^8-6 b^2 c^8+2 c^10 : :
Barycentrics    (39 R^2+SB+SC-9 SW)S^2 + SB SC SW -9 R^2 SB SC : :
X(34128) = 7*X[2]-X[5655], 2*X[3]+X[10113], X[4]-4*X[15088], 4*X[5]-X[1539], X[74]+5*X[1656], X[110]-7*X[3526], X[113]-4*X[3628], X[146]-13*X[5067], X[182]+2*X[6698], X[265]+5*X[631], X[381]+X[15055], 2*X[546]+X[16111], 2*X[548]+X[12295], X[550]+2*X[7687], 2*X[620]+X[15535], 10*X[632]-X[5609], 2*X[974]+X[5876], 2*X[1112]-5*X[15026], X[1353]+2*X[32257], 7*X[3090]-X[7728], 5*X[3091]+X[20127], X[3448]+11*X[3525], 7*X[3523]-X[12121], 2*X[3530]+X[11801], X[3581]+5*X[30745], 5*X[3763]+X[11579], 4*X[3850]-X[13202], 7*X[3851]-X[10721], 3*X[5054]-X[15035], 3*X[5055]+X[15041], 11*X[5056]+X[12244], 11*X[5070]+X[10620], 13*X[5079]+5*X[15021], 2*X[5092]+X[32274], 2*X[5447]+X[11800], 4*X[5498]-X[25487], X[5642]-4*X[10124], X[5690]+2*X[11735], 2*X[6684]+X[12261], 4*X[6689]-X[11702], X[7723]+2*X[13630], X[7731]-13*X[15028], X[9140]+5*X[15694], 2*X[9956]+X[11709], 4*X[10125]-X[20773], X[10263]-4*X[11746], 2*X[10272]+X[16003], 13*X[10303]-X[12383], 5*X[10574]+X[22584], X[10706]-3*X[15046], 4*X[11540]-X[11694], X[11557]-4*X[11695], 2*X[11793]+X[11806], X[11804]+2*X[32348], X[11805]-4*X[32396], X[12778]-7*X[31423], X[12825]-4*X[14128], 2*X[12900]+X[20417], X[12902]+5*X[15051], X[13201]+11*X[15024], X[13358]+2*X[32142], 2*X[13363]-X[16222], 4*X[13392]-X[24981], X[14651]+X[14850], X[15101]+2*X[25711], 2*X[15359]+X[33813], X[15647]+2*X[20299], X[16340]+2*X[22104], 2*X[20191]+X[33547], 2*X[20301]+X[33851], 2*X[32156]+X[32311]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29414.

X(34128) lies on these lines: {2,5655}, {3,10113}, {4,15088}, {5,1539}, {30,23515}, {74,1656}, {110,3526}, {113,3628}, {125,128}, {146,5067}, {182,6698}, {265,631}, {371,13979}, {372,13915}, {381,15055}, {541,15699}, {542,11539}, {546,16111}, {548,12295}, {549,17702}, {550,7687}, {620,15535}, {632,5609}, {974,5876}, {1112,15026}, {1353,32257}, {1493,26879}, {1986,6143}, {2931,7516}, {3043,13353}, {3090,7728}, {3091,20127}, {3448,3525}, {3523,12121}, {3530,11801}, {3548,6101}, {3581,30745}, {3763,11579}, {3850,13202}, {3851,10721}, {5054,15035}, {5055,15041}, {5056,12244}, {5070,10620}, {5079,15021}, {5092,32274}, {5447,11800}, {5498,25487}, {5642,10124}, {5690,11735}, {5892,10628}, {5944,10182}, {5965,14156}, {6102,6640}, {6684,12261}, {6689,11702}, {7505,12133}, {7583,13969}, {7723,13630}, {7731,15028}, {8976,19059}, {9140,15694}, {9540,19051}, {9826,15131}, {9956,11709}, {10125,20773}, {10263,11746}, {10272,16003}, {10303,12383}, {10574,22584}, {10706,15046}, {11540,11694}, {11557,11695}, {11793,11806}, {11804,32348}, {11805,32396}, {12292,14940}, {12778,31423}, {12825,14128}, {12900,20417}, {12902,15051}, {13201,15024}, {13358,32142}, {13363,16222} ,{13392,24981}, {13935,19052}, {13951,19060}, {14651,14850}, {15101,25711}, {15359,33813}, {15647,20299}, {15805,17847}, {16340,22104}, {16657,23336}, {20191,33547}, {20301,33851}, {22804,32767}, {32156,32311}

X(34128) = midpoint of X(i) and X(j) for these {i,j}: {2,15061}, {3,14644}, {381,15055}, {9140,32609}, {14651,14850}
X(34128) = reflection of X(i) in X(j) for these {i,j}: {10113,14644}, {14644,20304}, {16222,13363}
X(34128) = complement of X(14643)
X(34128) = center of the Vu pedal-centroidal circle of X(74)
X(34128) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,15059,20304} ,{3,20304,10113}, {5,6699,12041}, {5,12041,1539}, {125,140,1511}, {632,10264,5972}, {632,20397,5609}, {3523,15081,12121}, {3530,11801,16163}, {5972,10264,5609}, {5972,20397,10264}, {6699,6723,5}, {12236,13416,6101}, {12902,15720,15051}


X(34129) =  X(3)X(132)∩X(112)X(7750)

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

X(34129) lies on the cubi K1129 and on these lines: {3, 132}, {112, 7750}, {127, 2207}, {297, 11610}, {394, 3162}, {1297, 6530}, {1503, 17974}, {2794, 20993}

X(34129) = X(2508)-isoconjugate of X(4592)
X(34129) = cevapoint of X(i) and X(j) for these (i,j): {427, 16318}, {868, 2489}
X(34129) = trilinear pole of line {520, 1843}
X(34129) = barycentric quotient X(2489)/X(2508)


X(34130) =  ISOGONAL CONJUGATE OF X(147)

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

The trilinear polar of X(34130) passes through X(3569).

X(34130) lies on the cubics K270, K422, K785, K1001, K1129, and on these lines: {3, 3493}, {147, 325}, {232, 1691}, {264, 14382}, {419, 6530}, {511, 3506}, {1351, 18873}, {5085, 5968}, {9474, 9475}

X(34130) = isogonal conjugate of X(147)
X(34130) = isogonal conjugate of the anticomplement of X(98)
X(34130) = isogonal conjugate of the complement of X(5984)
X(34130) = isogonal conjugate of the isotomic conjugate of X(9473)
X(34130) = X(i)-cross conjugate of X(j) for these (i,j): {1297, 64}, {1976, 6}, {17980, 3224}
X(34130) = X(511)-vertex conjugate of X(511)
X(34130) = X(i)-isoconjugate of X(j) for these (i,j): {1, 147}, {2, 16559}
X(34130) = barycentric product X(6)*X(9473)
X(34130) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 147}, {31, 16559}, {9473, 76}


X(34131) =  CIRCUMCIRCLE-INVERSE OF X(132)

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

X(34131) lies on the the tangential circle, the cubic K1129, and these lines: {3, 132}, {22, 107}, {24, 98}, {25, 125}, {232, 1691}, {1297, 32713}, {1637, 8428}, {2079, 3515}, {2931, 32119}, {2967, 15462}, {8743, 13195}, {13558, 21213}, {15141, 23347}, {16230, 19165}

X(34131) = circumcircle-inverse of X(132)
X(34131) = tangential-isogonal conjugate of X(34146)
X(34131) = Dao-Moses-Telv-circle-inverse of X(8428)


X(34132) =  CIRCUMCIRCLE-INVERSE OF X(9467)

Barycentrics    a^2*(-b^2 + a*c)*(b^2 + a*c)*(a*b - c^2)*(a*b + c^2)*(a^16*b^2 - 2*a^14*b^4 + a^12*b^6 - a^10*b^8 + 2*a^8*b^10 - a^6*b^12 + a^16*c^2 - a^14*b^2*c^2 + 2*a^10*b^6*c^2 + a^6*b^10*c^2 - 3*a^4*b^12*c^2 - 2*a^14*c^4 - a^8*b^6*c^4 - a^6*b^8*c^4 + 3*a^4*b^10*c^4 + b^14*c^4 + a^12*c^6 + 2*a^10*b^2*c^6 - a^8*b^4*c^6 - 3*a^6*b^6*c^6 + a^4*b^8*c^6 + a^2*b^10*c^6 - a^10*c^8 - a^6*b^4*c^8 + a^4*b^6*c^8 - 2*a^2*b^8*c^8 - b^10*c^8 + 2*a^8*c^10 + a^6*b^2*c^10 + 3*a^4*b^4*c^10 + a^2*b^6*c^10 - b^8*c^10 - a^6*c^12 - 3*a^4*b^2*c^12 + b^4*c^14) : :

X(34132) lies on the cubic K1129 and these lines: {3, 3493}, {98, 446}, {14251, 15920}

X(34132) = circumcircle-inverse of X(9467)


X(34133) =  CIRCUMCIRCLE-INVERSE OF X(5404)

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

X(34133) lies on the cubic K1129 and these lines: {3, 1677}, {1343, 27375}

X(34133) = circumcircle-inverse of X(5404)


X(34134) =  CIRCUMCIRCLE-INVERSE OF X(5403)

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

X(34134) lies on the cubic K1129 and these lines: {3, 1676}, {1342, 27375}

X(34134) = circumcircle-inverse of X(5403)


X(34135) =  ISOGONAL CONJUGATE OF X(5003)

Barycentrics    1/((-a^4 + b^4 + c^4)*S - 4*SA*Sqrt[SA*SB*SC*SW]) : :

X(34135) lies on the Jerabek circumhyperbola, the cubic K1129, and these lines: {25, 125}, {69, 5002}, {1503, 5001}, {2781, 32618}

X(34135) = isogonal conjugate of X(5003)
X(34135) = isogonal conjugate of the anticomplement of X(5001)
X(34135) = X(1)-isoconjugate of X(5003)
X(34135) = barycentric quotient X(6)/X(5003)


X(34136) =  ISOGONAL CONJUGATE OF X(5002)

Barycentrics    1/((-a^4 + b^4 + c^4)*S + 4*SA*Sqrt[SA*SB*SC*SW]) : :

X(34136) lies on the Jerabek circumhyperbola, the cubic K1129, and these lines: {25, 125}, {69, 5003}, {1503, 5000}, {2781, 32619}

X(34136) = isogonal conjugate of X(5002)
X(34136) = isogonal conjugate of the anticomplement of X(5000)
X(34136) = X(1)-isoconjugate of X(5002)
X(34136) = barycentric quotient X(6)/X(5002)


X(34137) =  X(4)X(6)∩X(22)X(22135)

Barycentrics    a^2*(-a^2 + b^2 + c^2)*(-a^8 + b^8 + a^4*b^2*c^2 - b^6*c^2 - b^2*c^6 + c^8) : :
X(34137) = 2 X[6] + X[13509],2 X[8779] + X[10766]

X(34137) lies on these lines: {4, 6}, {22, 22135}, {69, 23128}, {182, 26216}, {184, 1180}, {511, 1297}, {525, 3049}, {1562, 29012}, {1691, 3269}, {1692, 5622}, {3094, 14585}, {3564, 22146}, {3926, 20806}, {5028, 15073}, {5305, 26926}, {5938, 9407}, {6636, 22075}, {7797, 17035}, {10312, 19161}, {14885, 21637}, {19139, 22120}, {26204, 26206}

X(34137) = anticomplement of X(34138)
X(34137) = antigonal conjugate of X(34237)
X(34137) = second-Lemoine-circle-inverse of X(6776)
X(34137) = polar-circle-inverse of X(27376)
X(34137) = anticomplement of the isogonal conjugate of X(11610)
X(34137) = anticomplement of the isotomic conjugate of X(31636)
X(34137) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {98, 17492}, {1821, 66}, {1910, 7391}, {1976, 17481}, {2172, 147}, {11610, 8}, {31636, 6327}
X(34137) = X(31636)-Ceva conjugate of X(2)
X(34137) = crosssum of X(i) and X(j) for these (i,j): {427, 16318}, {868, 2489}
X(34137) = crossdifference of every pair of points on line {520, 1843}
X(34137) = barycentric product X(2508)*X(4563)
X(34137) = barycentric quotient X(2508)/X(2501)
X(34137) = {X(6),X(19149)}-harmonic conjugate of X(8743)


X(34138) =  ISOGONAL CONJUGATE OF X(11610)

Barycentrics    (a^4 + b^4 - c^4)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^4 - b^4 + c^4) : :
X(34138) = 7 X[3619] - X[13509]

X(34138) lies on the cubic K570 and these lines: {3, 66}, {76, 5523}, {95, 7832}, {98, 15407}, {132, 511}, {249, 15388}, {343, 13854}, {525, 23285}, {1289, 2710}, {3619, 13509}, {8891, 21243}, {9289, 26154}, {24206, 27372}

X(34138) = isogonal conjugate of X(11610)
X(34138) = isotomic conjugate of X(31636)
X(34138) = complement of X(34137)
X(34138) = circumcircle-inverse of X(2353)
X(34138) = X(237)-cross conjugate of X(325)
X(34138) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11610}, {22, 1910}, {31, 31636}, {98, 2172}, {206, 1821}, {290, 17453}, {293, 8743}, {336, 17409}, {1760, 1976}, {14601, 20641}
X(34138) = crosssum of X(8779) and X(22391)
X(34138) = crossdifference of every pair of points on line {206, 2485}
X(34138) = barycentric product X(i)*X(j) for these {i,j}: {66, 325}, {297, 14376}, {511, 18018}, {1289, 6333}, {6393, 13854}
X(34138) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 31636}, {6, 11610}, {66, 98}, {232, 8743}, {237, 206}, {297, 17907}, {325, 315}, {511, 22}, {684, 8673}, {1289, 685}, {1755, 2172}, {1959, 1760}, {2156, 1910}, {2211, 17409}, {2353, 1976}, {2421, 4611}, {2799, 33294}, {3289, 10316}, {3569, 2485}, {9417, 17453}, {9418, 20968}, {13854, 6531}, {14376, 287}, {18018, 290}


X(34139) =  CIRCUMCIRCLE-INVERSE OF X(11715)

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

X(34139) lies on the cubic K1130 and these lines: {3, 2802}, {35, 13541}, {36, 1054}, {55, 10700}, {56, 106}, {121, 958}, {182, 2810}, {993, 11814}, {999, 11717}, {1001, 11731}, {1293, 3428}, {2776, 22583}, {2789, 22504}, {2796, 22514}, {2827, 22775}, {2840, 22654}, {2842, 22586}, {2844, 19162}, {2975, 21290}, {5510, 22753}, {6018, 10966}, {6715, 25524}, {9527, 19159}, {10744, 22758}

X(34139) = circumcircle-inverse of X(11715)


X(34140) =  X(3)X(2802)∩X(5537)X(8679)

Barycentrics    a^2*(a^7 - 4*a^6*b + 2*a^5*b^2 + 7*a^4*b^3 - 7*a^3*b^4 - 2*a^2*b^5 + 4*a*b^6 - b^7 - 4*a^6*c + 21*a^5*b*c - 28*a^4*b^2*c - 6*a^3*b^3*c + 31*a^2*b^4*c - 15*a*b^5*c + b^6*c + 2*a^5*c^2 - 28*a^4*b*c^2 + 70*a^3*b^2*c^2 - 43*a^2*b^3*c^2 - 9*a*b^4*c^2 + 8*b^5*c^2 + 7*a^4*c^3 - 6*a^3*b*c^3 - 43*a^2*b^2*c^3 + 48*a*b^3*c^3 - 8*b^4*c^3 - 7*a^3*c^4 + 31*a^2*b*c^4 - 9*a*b^2*c^4 - 8*b^3*c^4 - 2*a^2*c^5 - 15*a*b*c^5 + 8*b^2*c^5 + 4*a*c^6 + b*c^6 - c^7) : :

X(34140) lies on the cubic K1130 and these lines: {3, 2802}, {5537, 8679}


X(34141) =  X(3)X(8679)∩X(109)X(1995)

Barycentrics    a^2*(a^9 + a^8*b - 8*a^7*b^2 - 2*a^6*b^3 + 18*a^5*b^4 - 16*a^3*b^6 + 2*a^2*b^7 + 5*a*b^8 - b^9 + a^8*c - 8*a^7*b*c + 26*a^6*b^2*c - 14*a^5*b^3*c - 42*a^4*b^4*c + 44*a^3*b^5*c + 10*a^2*b^6*c - 22*a*b^7*c + 5*b^8*c - 8*a^7*c^2 + 26*a^6*b*c^2 - 48*a^5*b^2*c^2 + 50*a^4*b^3*c^2 - 46*a^2*b^5*c^2 + 32*a*b^6*c^2 - 6*b^7*c^2 - 2*a^6*c^3 - 14*a^5*b*c^3 + 50*a^4*b^2*c^3 - 56*a^3*b^3*c^3 + 34*a^2*b^4*c^3 - 6*a*b^5*c^3 - 6*b^6*c^3 + 18*a^5*c^4 - 42*a^4*b*c^4 + 34*a^2*b^3*c^4 - 18*a*b^4*c^4 + 8*b^5*c^4 + 44*a^3*b*c^5 - 46*a^2*b^2*c^5 - 6*a*b^3*c^5 + 8*b^4*c^5 - 16*a^3*c^6 + 10*a^2*b*c^6 + 32*a*b^2*c^6 - 6*b^3*c^6 + 2*a^2*c^7 - 22*a*b*c^7 - 6*b^2*c^7 + 5*a*c^8 + 5*b*c^8 - c^9) : :

X(34141) lies on the cubics K1118 and K1130, and on these lines: {3, 8679}, {109, 1995}


X(34142) =  X(3)X(10)∩X(109)X(8679)

Barycentrics    a^2*(a^9*b - a^8*b^2 - 2*a^7*b^3 + 3*a^6*b^4 - 3*a^4*b^6 + 2*a^3*b^7 + a^2*b^8 - a*b^9 + a^9*c - 4*a^8*b*c + 6*a^7*b^2*c - 11*a^5*b^4*c + 11*a^4*b^5*c - 6*a^2*b^7*c + 4*a*b^8*c - b^9*c - a^8*c^2 + 6*a^7*b*c^2 - 14*a^6*b^2*c^2 + 13*a^5*b^3*c^2 + 3*a^4*b^4*c^2 - 16*a^3*b^5*c^2 + 12*a^2*b^6*c^2 - 3*a*b^7*c^2 - 2*a^7*c^3 + 13*a^5*b^2*c^3 - 22*a^4*b^3*c^3 + 14*a^3*b^4*c^3 + 2*a^2*b^5*c^3 - 9*a*b^6*c^3 + 4*b^7*c^3 + 3*a^6*c^4 - 11*a^5*b*c^4 + 3*a^4*b^2*c^4 + 14*a^3*b^3*c^4 - 18*a^2*b^4*c^4 + 9*a*b^5*c^4 + 11*a^4*b*c^5 - 16*a^3*b^2*c^5 + 2*a^2*b^3*c^5 + 9*a*b^4*c^5 - 6*b^5*c^5 - 3*a^4*c^6 + 12*a^2*b^2*c^6 - 9*a*b^3*c^6 + 2*a^3*c^7 - 6*a^2*b*c^7 - 3*a*b^2*c^7 + 4*b^3*c^7 + a^2*c^8 + 4*a*b*c^8 - a*c^9 - b*c^9) : :

X(34142) lies on the cubic K1130 and on these lines: {3, 10}, {109, 8679}, {1309, 1311}


X(34143) =  X(3)X(9)∩X(109)X(5537)

Barycentrics    a^2*(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10 + 5*a^8*b*c + 4*a^7*b^2*c - 20*a^6*b^3*c - 4*a^5*b^4*c + 22*a^4*b^5*c - 4*a^3*b^6*c - 4*a^2*b^7*c + 4*a*b^8*c - 3*b^9*c - 5*a^8*c^2 + 4*a^7*b*c^2 + 8*a^6*b^2*c^2 + 8*a^5*b^3*c^2 - 30*a^4*b^4*c^2 + 4*a^3*b^5*c^2 + 24*a^2*b^6*c^2 - 16*a*b^7*c^2 + 3*b^8*c^2 - 20*a^6*b*c^3 + 8*a^5*b^2*c^3 + 36*a^4*b^3*c^3 - 28*a^2*b^5*c^3 - 8*a*b^6*c^3 + 12*b^7*c^3 + 10*a^6*c^4 - 4*a^5*b*c^4 - 30*a^4*b^2*c^4 + 6*a^2*b^4*c^4 + 20*a*b^5*c^4 - 2*b^6*c^4 + 22*a^4*b*c^5 + 4*a^3*b^2*c^5 - 28*a^2*b^3*c^5 + 20*a*b^4*c^5 - 18*b^5*c^5 - 10*a^4*c^6 - 4*a^3*b*c^6 + 24*a^2*b^2*c^6 - 8*a*b^3*c^6 - 2*b^4*c^6 - 4*a^2*b*c^7 - 16*a*b^2*c^7 + 12*b^3*c^7 + 5*a^2*c^8 + 4*a*b*c^8 + 3*b^2*c^8 - 3*b*c^9 - c^10) : :

Let A'B'C' be the X(4)-Brocard triangle. Let LA, LB, LC be lines through A', B', C', respectively, parallel to the Euler line. Let L'A be the reflection of LA in sideline BC, and define L'B and L'C cyclically. The lines L'A, L'B, L'C concur in X(34143). (Randy Hutson, October 8, 2019)

X(34143) lies on the cubic K1130 and these lines: {3, 9}, {109, 5537}


X(34144) =  SINGULAR FOCUS OF CUBIC K1130

Barycentrics    a^2*(a^10 - a^9*b - 4*a^8*b^2 + 2*a^7*b^3 + 7*a^6*b^4 - 7*a^4*b^6 - 2*a^3*b^7 + 4*a^2*b^8 + a*b^9 - b^10 - a^9*c + 3*a^8*b*c + 4*a^7*b^2*c - 8*a^6*b^3*c - 8*a^5*b^4*c + 8*a^4*b^5*c + 8*a^3*b^6*c - 4*a^2*b^7*c - 3*a*b^8*c + b^9*c - 4*a^8*c^2 + 4*a^7*b*c^2 - 2*a^6*b^2*c^2 + 6*a^5*b^3*c^2 + 2*a^4*b^4*c^2 - 5*a^3*b^5*c^2 - a^2*b^6*c^2 - 5*a*b^7*c^2 + 5*b^8*c^2 + 2*a^7*c^3 - 8*a^6*b*c^3 + 6*a^5*b^2*c^3 - 4*a^4*b^3*c^3 - a^3*b^4*c^3 - 3*a^2*b^5*c^3 + 11*a*b^6*c^3 - 3*b^7*c^3 + 7*a^6*c^4 - 8*a^5*b*c^4 + 2*a^4*b^2*c^4 - a^3*b^3*c^4 + 8*a^2*b^4*c^4 - 4*a*b^5*c^4 - 4*b^6*c^4 + 8*a^4*b*c^5 - 5*a^3*b^2*c^5 - 3*a^2*b^3*c^5 - 4*a*b^4*c^5 + 4*b^5*c^5 - 7*a^4*c^6 + 8*a^3*b*c^6 - a^2*b^2*c^6 + 11*a*b^3*c^6 - 4*b^4*c^6 - 2*a^3*c^7 - 4*a^2*b*c^7 - 5*a*b^2*c^7 - 3*b^3*c^7 + 4*a^2*c^8 - 3*a*b*c^8 + 5*b^2*c^8 + a*c^9 + b*c^9 - c^10) : :
X(34144) = J^2 X[23] - 9 X[1311]

X(34144) lies on this line: {23, 1311}
X(34144) = singular focus of cubic K1130


X(34145) =  X(30)X(74)∩X(1994)X(14731)

Barycentrics    2 a^18-7 a^16 b^2+8 a^14 b^4-2 a^12 b^6-4 a^10 b^8+8 a^8 b^10-8 a^6 b^12+2 a^4 b^14+2 a^2 b^16-b^18-7 a^16 c^2+20 a^14 b^2 c^2-20 a^12 b^4 c^2+10 a^10 b^6 c^2-8 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+5 b^16 c^2+8 a^14 c^4-20 a^12 b^2 c^4+14 a^10 b^4 c^4-3 a^8 b^6 c^4+12 a^6 b^8 c^4-23 a^4 b^10 c^4+22 a^2 b^12 c^4-10 b^14 c^4-2 a^12 c^6+10 a^10 b^2 c^6-3 a^8 b^4 c^6-20 a^6 b^6 c^6+15 a^4 b^8 c^6-12 a^2 b^10 c^6+10 b^12 c^6-4 a^10 c^8-8 a^8 b^2 c^8+12 a^6 b^4 c^8+15 a^4 b^6 c^8-4 b^10 c^8+8 a^8 c^10+6 a^6 b^2 c^10-23 a^4 b^4 c^10-12 a^2 b^6 c^10-4 b^8 c^10-8 a^6 c^12+6 a^4 b^2 c^12+22 a^2 b^4 c^12+10 b^6 c^12+2 a^4 c^14-12 a^2 b^2 c^14-10 b^4 c^14+2 a^2 c^16+5 b^2 c^16-c^18 : :
Barycentrics    (7 R^2-2 SW)S^4+(108 R^6+18 R^4 SB+18 R^4 SC-21 R^2 SB SC-39 R^4 SW-4 R^2 SB SW-4 R^2 SC SW+6 SB SC SW-5 R^2 SW^2+2 SW^3)S^2-63 R^4 SB SC SW+39 R^2 SB SC SW^2-6 SB SC SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29421.

X(34145) lies on these lines: {30,74}, {1994,14731}, {3258,23292}, {3575,16221}, {7667,16188}, {12370,16168}


X(34146) =  X(3)X(206)∩X(4)X(66)

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

X(34146) lies on these lines: {3,206}, {4,66}, {5,6697}, {6,64}, {20,3313}, {30,511}, {51,1853}, {52,14216}, {67,11744}, {69,6225}, {74,1177}, {112,34137}, {113,15116}, {125,15126}, {127,34138}, {141,2883}, {146,2892}, {154,3917}, {159,1350}, {182,3357}, {219,3556}, {221,10387}, {222,7169}, {376,31166}, {389,1595}, {394,1619}, {568,23049}, {575,15579}, {611,10060}, {613,10076}, {631,31267}, {974,15118}, {1073,1661}, {1147,32321}, {1176,14118}, {1204,1974}, {1205,32264}, {1216,3098}, {1297,19158}, {1351,8549}, {1352,5878}, {1386,12262}, {1469,6285}, {1495,10117}, {1514,15738}, {1843,5895}, {1885,26926}, {1971,2076}, {2211,3269}, {2935,15138}, {2979,11206}, {3056,7355}, {3060,32064}, {3094,12502}, {3146,20079}, {3242,7973}, {3292,17847}, {3416,12779}, {3516,19125}, {3589,6696}, {3618,10574}, {3619,15056}, {3751,9899}, {3818,22802}, {3819,10192}, {5050,10249}, {5085,10606}, {5092,15578}, {5102,11216}, {5157,7503}, {5181,12825}, {5188,15270}, {5446,18381}, {5447,7525}, {5462,19130}, {5502,11853}, {5656,10519}, {5889,12324}, {5890,14853}, {5892,23329}, {5893,9822}, {5894,11574}, {5943,23332}, {5972,16977}, {6102,21850}, {6145,15321}, {6241,6776}, {6403,12290}, {6467,30443}, {6593,11598}, {7716,15811}, {7722,10752}, {7730,32337}, {8550,15105}, {8567,19132}, {8889,15011}, {8991,13910}, {9730,14561}, {9833,10625}, {9967,10575}, {9970,11562}, {9971,32062}, {10110,16198}, {10250,15520}, {10516,15030}, {10605,19136}, {11061,12270}, {11204,17508}, {11413,20806}, {11440,19121}, {11472,12039}, {11557,32271}, {11793,16197}, {12084,19139}, {12133,32246}, {12145,16318}, {12174,19459}, {12202,12212}, {12220,12279}, {12235,32140}, {12281,32247}, {12298,23251}, {12299,23261}, {12315,18436}, {12329,12335}, {12367,17812}, {12452,12468}, {12453,12469}, {12583,12791}, {12586,12920}, {12587,12930}, {12588,12940}, {12589,12950}, {12590,12986}, {12591,12987}, {12594,13094}, {12595,13095}, {13293,19140}, {13445,22151}, {13474,21851}, {13562,31829}, {13630,18583}, {13972,13980}, {15012,32184}, {15063,19510}, {15072,25406}, {15305,29959}, {15407,32696}, {16836,23328}, {17835,32262}, {18374,21663}, {18383,21852}, {18439,18440}, {19154,32138}, {20987,26883}, {22769,22778}, {23039,32063}, {32127,32276}, {33522,33523}

X(34146) = isogonal conjugate of X(34168)
X(34146) = crosssum of X(i) and X(j) for these (i,j): {3, 1503}, {5002, 5003}
X(34146) = crossdifference of every pair of points on line X(6)X(8057)
X(34146) = crosspoint of X(i) and X(j) for these {i,j}: {4, 1297}, {34135, 34136}
X(34146) = Lucas-isogonal conjugate of X(1289)
X(34146) = Thomson-isogonal conjugate of X(1289)
X(34146) = tangential-isogonal conjugate of X(34131)


X(34147) =  X(2)X(15258)∩X(3)X(64)

Barycentrics    a^2*(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 + 10*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 + 10*b^4*c^4 + 5*a^2*c^6 - 4*b^2*c^6 - c^8) : :
X(34147) = X[3] - 3 X[6760],4 X[3] - 3 X[11589],2 X[3] - 3 X[12096],X[23] - 3 X[1304],7 X[3090] - 3 X[6761],4 X[5159] - 3 X[16177],4 X[6760] - X[11589],6 X[6760] - X[34109],3 X[11589] - 2 X[34109],3 X[12096] - X[34109]

X(34147) lies on the curves K903, K1095, Q071, and these lines: {2,15258}, {3,64}, {23,1297}, {114,5159}, {122,1503}, {216,5651}, {441,14981}, {468,15526}, {520,647}, {577,6090}, {1495,2972}, {2764,13509}, {3090,6761}, {3628,33549}, {5158,11284}, {10163,26269}, {12324,31377}, {17845,33546}

X(34147) = midpoint of X(2764) and X(13509)
X(34147) = reflection of X(i) in X(j) for these {i,j}: {11589, 12096}, {12096, 6760}, {34109, 3}
X(34147) = X(2764)-Ceva conjugate of X(520)
X(34147) = crossdifference of every pair of points on line {4, 6587}
X(34147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1073, 33924}, {3, 34109, 11589}, {154, 33924, 3}, {852, 3292, 3284}, {12096, 34109, 3}


X(34148) =  X(3)X(54)∩X(4)X(110)

Barycentrics    a^2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - 3*a^6*c^2 + 5*a^4*b^2*c^2 - a^2*b^4*c^2 - b^6*c^2 + 3*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - a^2*c^6 - b^2*c^6) : :
Barycentrics    SA (S^2 + SB SC)^2 - SB (S^2 + SC SA)^2 - SC (S^2 + SA SB)^2 : :
X(34148) = 2 X[3] + (J^2 - 4) X[54], 2 X[4] - J^2 X[110], X[20] + (J^2 - 3) X[184], 2 X[52] - J^2 X[186]

X(34148) lies on these lines: {1, 9637}, {2, 578}, {3, 54}, {4, 110}, {5, 15033}, {6, 2929}, {20, 184}, {22, 19357}, {23, 10282}, {24, 3060}, {26, 11464}, {30, 49}, {52, 186}, {55, 9653}, {56, 9666}, {60, 581}, {68, 23293}, {74, 11250}, {125, 10112}, {140, 567}, {143, 1511}, {155, 378}, {156, 382}, {182, 193}, {185, 2071}, {194, 17974}, {215, 7354}, {249, 31850}, {265, 10224}, {323, 5562}, {381, 13482}, {389, 1994}, {394, 7503}, {403, 9820}, {411, 1437}, {427, 14516}, {436, 1941}, {468, 13142}, {485, 9676}, {511, 7488}, {546, 18350}, {549, 13353}, {568, 12228}, {569, 631}, {576, 15020}, {858, 6146}, {1069, 11446}, {1175, 5751}, {1176, 1350}, {1181, 11413}, {1199, 9730}, {1204, 13198}, {1351, 3515}, {1370, 18925}, {1495, 13598}, {1533, 14862}, {1568, 13403}, {1593, 3167}, {1657, 8718}, {1658, 6243}, {1970, 3289}, {1995, 10982}, {2070, 10263}, {2072, 12370}, {2477, 6284}, {2777, 3047}, {2794, 3044}, {2829, 3045}, {2888, 21243}, {2914, 11562}, {2937, 5944}, {3091, 9306}, {3146, 6759}, {3153, 21659}, {3157, 19367}, {3200, 16964}, {3201, 16965}, {3203, 12203}, {3292, 5907}, {3357, 9716}, {3448, 20299}, {3516, 12164}, {3518, 5446}, {3520, 11440}, {3522, 10984}, {3524, 13336}, {3541, 6193}, {3543, 26883}, {3546, 18911}, {3548, 18912}, {3567, 6644}, {3581, 15331}, {3627, 10540}, {5056, 5651}, {5094, 12429}, {5198, 8780}, {5422, 11426}, {5449, 6143}, {5462, 15019}, {5609, 32137}, {5622, 15057}, {5640, 6642}, {5643, 11465}, {5691, 9586}, {5876, 14130}, {5921, 19124}, {5946, 14627}, {6000, 12086}, {6090, 11479}, {6241, 12084}, {6403, 8907}, {6636, 15644}, {6640, 15059}, {6746, 8537}, {6800, 11414}, {6815, 11427}, {7387, 9707}, {7391, 9833}, {7395, 15066}, {7399, 14389}, {7464, 10575}, {7506, 9781}, {7509, 7998}, {7512, 10625}, {7514, 7999}, {7517, 26882}, {7525, 13340}, {7526, 11459}, {7529, 10546}, {7547, 12293}, {7575, 14449}, {7577, 9927}, {7722, 12901}, {7745, 9603}, {7747, 9696}, {7756, 9697}, {8909, 11447}, {9140, 18281}, {9652, 12943}, {9667, 12953}, {9729, 13366}, {9786, 15078}, {9818, 15056}, {10110, 13595}, {10255, 14644}, {10257, 13292}, {10264, 15089}, {10312, 32661}, {10313, 14585}, {10323, 15080}, {10564, 15032}, {11004, 11438}, {11064, 12241}, {11245, 16196}, {11416, 15073}, {11439, 18451}, {11454, 12163}, {11456, 12085}, {11585, 12022}, {11597, 20424}, {11694, 20193}, {11695, 15018}, {11750, 12254}, {11800, 17701}, {12106, 15034}, {12134, 15559}, {12162, 14865}, {12219, 32607}, {12270, 12302}, {12271, 15316}, {12289, 18569}, {12290, 32139}, {12308, 33541}, {12359, 15136}, {12834, 15024}, {12902, 18379}, {13160, 23292}, {13339, 15712}, {13347, 15692}, {13348, 22352}, {13358, 15002}, {13371, 25739}, {13491, 18859}, {13621, 32609}, {14059, 14919}, {14094, 18439}, {15026, 15038}, {15058, 15068}, {15100, 17847}, {17506, 32110}, {17821, 33586}, {18381, 31074}, {18388, 34007}, {18436, 18570}, {18504, 22750}, {18560, 22660}, {18882, 32379}, {19347, 21312}, {23336, 32358}, {30522, 31724}, {32329, 34114}

X(34148) = reflection of X(i) in X(j) for these {i,j}: {110, 3043}, {1614, 49}, {2937, 5944}, {7488, 13367}, {11440, 3520}
X(34148) = X(2190)-anticomplementary conjugate of X(2888)
X(34148) = crosspoint of X(249) and X(18831)
X(34148) = crosssum of X(115) and X(15451)
X(34148) = crossdifference of every pair of points on line {686, 12077}
X(34148) = pole of Euler line wrt conic {{X(3), X(6), X(24), X(60), X(143), X(1511), X(1986)}}
X(34148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 578, 13434}, {3, 54, 5012}, {3, 195, 6102}, {3, 1993, 5889}, {3, 7592, 10574}, {3, 11412, 7691}, {3, 12161, 5890}, {3, 12316, 32608}, {3, 15087, 13630}, {3, 16266, 11412}, {4, 1147, 110}, {4, 12118, 12278}, {6, 17928, 15043}, {20, 9545, 184}, {52, 12038, 186}, {113, 12897, 4}, {155, 378, 12111}, {156, 382, 14157}, {156, 9703, 9705}, {184, 9545, 9706}, {184, 13346, 20}, {323, 14118, 5562}, {378, 12111, 15062}, {382, 9703, 156}, {389, 22467, 15053}, {394, 7503, 11444}, {394, 11425, 7503}, {578, 1092, 2}, {1147, 13352, 4}, {1181, 11413, 15072}, {1493, 13630, 15087}, {1593, 3167, 11441}, {1593, 11441, 15305}, {1657, 11935, 9704}, {1993, 5889, 15801}, {1994, 22467, 389}, {3060, 11449, 24}, {3146, 9544, 6759}, {3522, 11003, 10984}, {3541, 6193, 11442}, {3548, 18912, 26913}, {5504, 15463, 110}, {5562, 11430, 14118}, {5654, 13352, 15472}, {6241, 12084, 13445}, {6640, 26917, 15059}, {7387, 9707, 26881}, {7547, 12293, 18392}, {7691, 23061, 11412}, {9306, 11424, 3091}, {9705, 14157, 156}, {10257, 13292, 26879}, {10263, 32171, 2070}, {10574, 11422, 7592}, {10610, 10627, 3}, {10625, 18475, 7512}, {11412, 16266, 23061}, {11456, 12085, 12279}, {12084, 18445, 6241}, {12228, 15035, 27866}, {18281, 25738, 23294}, {23294, 25738, 9140}


X(34149) =  X(54)X(143)∩X(546)X(6346)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4-2 a^2 c^2-3 b^2 c^2+c^4) (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+2 a^6 b^2 c^2+a^4 b^4 c^2+5 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4+a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+5 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10) : :
Barycentrics    (15 R^4+19 R^2 SB+19 R^2 SC+3 R^2 SW-6 SB SW-6 SC SW-2 SW^2)S^2-R^4 SB SC+R^2 SB SC SW+2 SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29424.

X(34149) lies on these lines: {54,143}, {546,6346}, {1209,10272}, {9969,15516}, {10095,11817}, {10610,11557}, {11561,12041}, {13367,14449}, {13491,33541}


X(34150) =  ISOGONAL CONJUGATE OF X(15469)

Barycentrics    (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) (2 a^8-2 a^6 b^2-a^4 b^4+b^8-2 a^6 c^2+4 a^4 b^2 c^2-4 b^6 c^2-a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : :
Barycentrics    S^4 + (-18 R^2 SB-18 R^2 SC-3 SB SC+12 R^2 SW+4 SB SW+4 SC SW-3 SW^2)S^2 -324 R^4 SB SC+144 R^2 SB SC SW-15 SB SC SW^2 : :
X(34150) = X[74] - 3 X[5627], 2 X[140] - 3 X[21315], 3 X[403] - 2 X[16319], X[477] - 3 X[14644], 2 X[3154] - 3 X[14644], 3 X[5627] - 2 X[12079], 3 X[5627] + X[14989], 3 X[9140] - X[14508], 3 X[10706] + X[31874], 4 X[12068] - 3 X[15035], 2 X[12079] + X[14989], 4 X[12900] - 3 X[31378], X[14934] - 4 X[21316], 3 X[23515] - 2 X[31379].

See Antreas Hatzipolakis, Ercole Suppa and Peter Moses, Hyacinthos 29424 and Hyacinthos 29428.

X(34150) lies on the cubic K025 and these lines: {4, 523}, {5, 14385}, {30, 74}, {113, 14611}, {140, 21315}, {230, 32640}, {316, 1494}, {381, 9717}, {403, 1300}, {477, 3154}, {542, 1553}, {546, 3470}, {548, 21317}, {671, 9139}, {1263, 11558}, {2777, 6070}, {3233, 12383}, {3258, 7687}, {5523, 8749}, {5962, 10152}, {7471, 15468}, {10113, 16168}, {10295, 11657}, {10297, 14919}, {10706, 31874}, {11801, 16340}, {12068, 15035}, {12900, 31378}, {13202, 32417}, {16163, 22104}, {16243, 25338}, {22265, 32111}, {23515, 31379}

X(34150) = midpoint of X(i) and X(j) for these {i,j}: {74, 14989}, {476, 10733}
X(34150) = reflection of X(i) in X(j) for these {i,j}: {5, 21316}, {74, 12079}, {477, 3154}, {3258, 7687}, {7471, 25641}, {10295, 11657}, {12383, 3233}, {14611, 113}, {14934, 5}, {16163, 22104}, {16340, 11801}, {21317, 548}
X(34150) = isogonal conjugate of X(15469)
X(34150) = antigonal image of X(7471)
X(34150) = symgonal image of X(3154)
X(34150) = X(1)-isoconjugate of X(15469)
X(34150) = reflection of X(74) in its Simson line (line X(125)X(523))
X(34150) = barycentric product X(i)*X(j) for these {i,j}: {94, 15468}, {1494, 3018}, {2394, 7471}, {16080, 17702}
X(34150) = barycentric quotient X (i)/X(j) for these {i,j}: {6, 15469}, {2433, 15453}, {3018, 30}, {7471, 2407}, {8749, 32710}, {15468, 323}, {17702, 11064}
X(34150) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {74, 5627, 12079}, {477, 14644, 3154}, {5627, 14989, 74}


X(34151) =  X(4)X(8)∩X(59)X(108)

Barycentrics    a (a-b) (a-c) (a^4 b^2-2 a^2 b^4+b^6-2 a^3 b^2 c+2 a^2 b^3 c+2 a b^4 c-2 b^5 c+a^4 c^2-2 a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+2 a^2 b c^3-2 a b^2 c^3+4 b^3 c^3-2 a^2 c^4+2 a b c^4-b^2 c^4-2 b c^5+c^6) : :
X(34151) = 3*X[2]-2*X[14115], X[3025]-2*X[3035], X[3937]-2*X[22102], 3*X[10707]+X[31877], 5*X[31272]-4*X[33646]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29424.

X(34151) lies on these lines: {2,14115}, {4,8}, {59,108}, {100,513}, {912,18341}, {1331,2222}, {2810,6075}, {2818,6073}, {3025,3035}, {3937,22102}, {3952,20293}, {5375,14298}, {5854,13756}, {10707,31877}, {11681,31849}, {15313,15343}, {31272,33646}

X(34151) = reflection of X(i) in X(j) for these {i,j}: {100,15632}, {3025,3035}, {3937,22102}
X(34151) = anticomplement of X(14115)
X(34151) = reflection of X(100) in its Simson line (line X(119)X(517))


X(34152) =  48TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    a^2*(2*a^8 - 4*a^6*b^2 + 4*a^2*b^6 - 2*b^8 - 4*a^6*c^2 + 12*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - b^6*c^2 - 7*a^2*b^2*c^4 + 6*b^4*c^4 + 4*a^2*c^6 - b^2*c^6 - 2*c^8) : :
X(34152) = 7 X[3] - X[23], 3 X[3] - X[186], 5 X[3] - X[2070], 9 X[3] - X[5899], 5 X[3] + X[7464], 4 X[3] - X[7575], 11 X[3] - 2 X[12105], 3 X[5] - 2 X[10151], 3 X[23] - 7 X[186], 5 X[23] - 7 X[2070], X[23] + 7 X[2071], 9 X[23] - 7 X[5899], 5 X[23] + 7 X[7464], 4 X[23] - 7 X[7575], 11 X[23] - 14 X[12105], 2 X[23] - 7 X[15646], 3 X[140] - X[11558], 3 X[140] - 2 X[15350], 5 X[186] - 3 X[2070], X[186] + 3 X[2071], 3 X[186] - X[5899], 5 X[186] + 3 X[7464], 4 X[186] - 3 X[7575], 11 X[186] - 6 X[12105], 2 X[186] - 3 X[15646], 3 X[376] + X[3153], 3 X[403] - 2 X[11558], 3 X[403] - 4 X[15350], 2 X[548] + X[858], 3 X[549] - X[11563], X[550] + 2 X[15122], X[2070] + 5 X[2071], 9 X[2070] - 5 X[5899], 4 X[2070] - 5 X[7575], 11 X[2070] - 10 X[12105], 2 X[2070] - 5 X[15646], 9 X[2071] + X[5899], 5 X[2071] - X[7464], 4 X[2071] + X[7575], 11 X[2071] + 2 X[12105], 2 X[2071] + X[15646], 5 X[3522] + X[7574], 5 X[3522] - X[13619], 4 X[3530] - X[11799], X[3627] - 4 X[5159], 5 X[5899] + 9 X[7464], 4 X[5899] - 9 X[7575], 11 X[5899] - 18 X[12105], 2 X[5899] - 9 X[15646], X[7426] - 4 X[14891], 4 X[7464] + 5 X[7575], 11 X[7464] + 10 X[12105], 2 X[7464] + 5 X[15646], 11 X[7575] - 8 X[12105], X[10096] - 3 X[12100], X[10151] - 3 X[10257], X[10296] + 5 X[15696], 2 X[10297] + X[15704], X[10540] - 3 X[15035], X[10989] + 5 X[14093], 2 X[11064] + X[14677], 4 X[12105] - 11 X[15646], X[13445] + 3 X[15035], X[14157] - 5 X[15051]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29426.

X(34152) lies on these lines: {2, 3}, {36, 10149}, {74, 22115}, {539, 20417}, {974, 1154}, {1092, 32138}, {1493, 13382}, {1511, 6000}, {1568, 16111}, {2693, 6760}, {2777, 14156}, {3098, 10250}, {3564, 5621}, {5562, 32210}, {9590, 28190}, {9625, 28182}, {10263, 32411}, {10264, 12901}, {10540, 13445}, {10575, 32171}, {10606, 15068}, {10979, 16328}, {11064, 14677}, {11440, 31834}, {11454, 23039}, {11468, 18436}, {11649, 14810}, {11695, 13446}, {11793, 22966}, {11809, 14794}, {12038, 13491}, {12041, 13754}, {12121, 25739}, {13293, 15311}, {13366, 13630}, {13391, 32110}, {13399, 30714}, {13568, 20424}, {14157, 15051}, {14805, 20791}, {15515, 16308}, {16163, 30522}, {21230, 22978}, {22549, 30507}

X(34152) = midpoint of X(i) and X(j) for these {i,j}: {3, 2071}, {74, 22115}, {1568, 16111}, {2070, 7464}, {2693, 6760}, {7574, 13619}, {10540, 13445}, {10564, 21663}, {12121, 25739}, {13399, 30714}
X(34152) = reflection of X(i) in X(j) for these {i,j}: {5, 10257}, {403, 140}, {7575, 15646}, {10263, 32411}, {11558, 15350}, {13446, 11695}, {15646, 3}
X(34152) = circumcircle-inverse of X(1657)
X(34152) = orthoptic circle of the Steiner inellipse inverse of X(31101)
X(34152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 20, 15331}, {3,186,37968}, {3, 1657, 21844}, {3, 2937, 17506}, {3, 3520, 140}, {3, 3534, 10298}, {3, 7464, 18571}, {3, 11250, 5}, {3, 11410, 7514}, {3, 11413, 1658}, {3, 14118, 3530}, {3, 15246, 14891}, {3, 18570, 549}, {3, 18859, 186}, {3, 21312, 18324}, {140, 1885, 5}, {140, 11558, 15350}, {186, 378, 10151}, {186, 2071, 18859}, {550, 15122, 18572}, {1113, 1114, 1657}, {1657, 21844, 12107}, {1658, 11413, 15704}, {3522, 23040, 3}, {3534, 10298, 7555}, {7514, 11410, 18570}, {11558, 15350, 403}, {13445, 15035, 10540}


X(34153) =  COMPLEMENT OF X(12902)

Barycentrics    4*a^10 - 9*a^8*b^2 + 4*a^6*b^4 + 2*a^4*b^6 - b^10 - 9*a^8*c^2 + 16*a^6*b^2*c^2 - 7*a^4*b^4*c^2 - 3*a^2*b^6*c^2 + 3*b^8*c^2 + 4*a^6*c^4 - 7*a^4*b^2*c^4 + 6*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 + 3*b^2*c^8 - c^10 : :
X(34153) = 3 X[2] - 5 X[15040], 3 X[3] - X[3448], X[4] - 3 X[32609], 3 X[5] - 4 X[5972], 5 X[5] - 4 X[7687], 3 X[5] - 2 X[10113], 7 X[5] - 8 X[12900], 3 X[5] - 5 X[22251], 5 X[110] - 3 X[5655], 3 X[110] - X[7728], 7 X[110] - 3 X[10706], 5 X[110] - X[10721], 2 X[125] - 3 X[549], 2 X[140] - 3 X[15035], 6 X[140] - 5 X[15059], 2 X[143] - 3 X[16223], X[265] - 3 X[15035], 3 X[265] - 5 X[15059], 3 X[376] - X[10620], 3 X[376] + X[14683], 2 X[546] - 3 X[14643], 10 X[546] - 13 X[15029], 2 X[546] - 5 X[15034], 3 X[546] - 7 X[22250], 2 X[548] + X[23236], 5 X[550] - 2 X[10990], 3 X[550] - 2 X[16111], 3 X[550] + 2 X[24981], X[550] + 2 X[30714], 5 X[632] - 4 X[20304], 3 X[1350] + X[25336], 3 X[1511] - 2 X[5972], 5 X[1511] - 2 X[7687], 3 X[1511] - X[10113], 7 X[1511] - 4 X[12900], 6 X[1511] - 5 X[22251], 3 X[2979] + X[15102], X[3146] - 7 X[15039], X[3146] - 5 X[20125], 2 X[3448] - 3 X[10264], X[3448] + 3 X[12383], 5 X[3522] - X[12317], 5 X[3522] - 3 X[15041], 7 X[3526] - 5 X[15081], 4 X[3530] - 5 X[15051], 4 X[3530] - 3 X[15061], 3 X[3534] - X[12244], 3 X[3534] + X[12308], 4 X[3628] - 3 X[14644], 4 X[3628] - 7 X[15020], 7 X[3832] - 9 X[15046], 3 X[3845] - 2 X[12295], 2 X[5609] + X[15704], 3 X[5642] - X[12295], 9 X[5655] - 5 X[7728], 7 X[5655] - 5 X[10706], 3 X[5655] - X[10721], 3 X[5655] + 5 X[12121], 3 X[5946] - 2 X[11800], 5 X[5972] - 3 X[7687], 7 X[5972] - 6 X[12900], 4 X[5972] - 5 X[22251], 4 X[6699] - 5 X[15712], 8 X[6723] - 9 X[11539], 3 X[7575] - 2 X[32269], 6 X[7687] - 5 X[10113], 7 X[7687] - 10 X[12900], 12 X[7687] - 25 X[22251], 7 X[7728] - 9 X[10706], 5 X[7728] - 3 X[10721], X[7728] + 3 X[12121], 3 X[8703] - 2 X[12041], 3 X[9140] - 7 X[15036], 3 X[9143] + X[12244], 3 X[9143] - X[12308], 3 X[9730] - 2 X[13358], 7 X[10113] - 12 X[12900], 2 X[10113] - 5 X[22251], X[10264] + 2 X[12383], 2 X[10272] - 3 X[32609], 3 X[10519] - X[32306], 15 X[10706] - 7 X[10721], 3 X[10706] + 7 X[12121], X[10721] + 5 X[12121], X[10733] - 4 X[13392], X[10733] - 3 X[14643], 5 X[10733] - 13 X[15029], X[10733] - 5 X[15034], 3 X[10733] - 14 X[22250], 4 X[10990] - 5 X[14677], 3 X[10990] - 5 X[16111], X[10990] - 5 X[16163], 3 X[10990] + 5 X[24981], X[10990] + 5 X[30714], 2 X[11801] - 5 X[15040], 5 X[12017] - 3 X[25320], 4 X[12068] - 3 X[21315], 6 X[12100] - 7 X[15036], 2 X[12103] + X[14094], 8 X[12108] - 5 X[15027], X[12281] - 3 X[23039], X[12317] - 3 X[15041], X[12407] - 3 X[26446], 10 X[12812] - 7 X[15044], 24 X[12900] - 35 X[22251], X[12902] - 5 X[15040], X[13201] - 3 X[13340], 4 X[13392] - 3 X[14643], 20 X[13392] - 13 X[15029], 4 X[13392] - 5 X[15034], 6 X[13392] - 7 X[22250], 15 X[14643] - 13 X[15029], 3 X[14643] - 5 X[15034], 9 X[14643] - 14 X[22250], 3 X[14644] - 7 X[15020], 3 X[14677] - 4 X[16111], X[14677] - 4 X[16163], 3 X[14677] + 4 X[24981], X[14677] + 4 X[30714], 7 X[14869] - 6 X[34128], 4 X[14934] - 3 X[33855], 13 X[15029] - 25 X[15034], 39 X[15029] - 70 X[22250], 15 X[15034] - 14 X[22250], 9 X[15035] - 5 X[15059], 7 X[15039] - 5 X[20125], 13 X[15042] - 11 X[15717], 5 X[15051] - 3 X[15061], 3 X[15055] - 4 X[33923], 8 X[15088] - 9 X[15699], 3 X[15462] - 2 X[18583], X[15545] - 3 X[21166], X[16111] - 3 X[16163], X[16111] + 3 X[30714], 3 X[16163] + X[24981], X[24981] - 3 X[30714]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29426.

X(34153) lies on these lines: {2, 11801}, {3, 2888}, {4, 7666}, {5, 1511}, {20, 399}, {30, 110}, {52, 11561}, {74, 548}, {113, 3627}, {125, 549}, {140, 265}, {143, 16223}, {146, 1657}, {376, 10620}, {381, 11694}, {495, 18968}, {496, 12896}, {511, 25329}, {516, 11699}, {541, 15686}, {542, 8703}, {546, 10733}, {550, 5562}, {567, 27866}, {632, 20304}, {952, 12778}, {974, 31804}, {1154, 11562}, {1350, 25336}, {1352, 12302}, {1353, 14708}, {1503, 12584}, {1539, 16534}, {1595, 12140}, {2771, 4297}, {2777, 5609}, {2929, 2931}, {2935, 9833}, {2948, 18481}, {2979, 15102}, {3043, 6240}, {3146, 15039}, {3521, 9705}, {3522, 12317}, {3526, 15081}, {3530, 15051}, {3534, 9143}, {3564, 32233}, {3575, 15463}, {3580, 18571}, {3589, 32273}, {3628, 14644}, {3832, 15046}, {3845, 5642}, {4325, 6126}, {4330, 7343}, {5012, 15089}, {5092, 25328}, {5159, 32227}, {5844, 12898}, {5946, 11800}, {6101, 10628}, {6247, 25564}, {6593, 21850}, {6644, 12310}, {6699, 15712}, {6723, 11539}, {6756, 15472}, {7471, 18319}, {7487, 11566}, {7575, 32269}, {7583, 10819}, {7584, 10820}, {7722, 10295}, {7727, 15338}, {7978, 28212}, {9140, 12100}, {9730, 13358}, {9820, 19479}, {10088, 18990}, {10091, 15171}, {10224, 12278}, {10226, 14516}, {10263, 11557}, {10519, 32306}, {11061, 33878}, {11064, 18572}, {11449, 13406}, {11591, 21650}, {11597, 20424}, {11720, 22791}, {12017, 25320}, {12068, 21315}, {12084, 12168}, {12103, 14094}, {12108, 15027}, {12227, 13568}, {12228, 31833}, {12270, 15332}, {12281, 23039}, {12368, 28186}, {12407, 26446}, {12812, 15044}, {12893, 15646}, {12904, 15325}, {13201, 13340}, {13391, 13417}, {13605, 13624}, {13630, 21649}, {14869, 34128}, {15042, 15717}, {15055, 33923}, {15088, 15699}, {15326, 19470}, {15462, 18583}, {15545, 21166}, {15806, 34007}, {16160, 16164}, {17701, 21659}, {18400, 23315}, {18533, 19504}, {19140, 29181}, {22584, 31834}, {23335, 25487}

X(34153) = midpoint of X(i) and X(j) for these {i,j}: {3, 12383}, {20, 399}, {74, 23236}, {110, 12121}, {146, 1657}, {2931, 12118}, {2935, 9833}, {2948, 18481}, {3534, 9143}, {5898, 12254}, {10620, 14683}, {11061, 33878}, {12244, 12308}, {12270, 18436}, {14094, 20127}, {16111, 24981}, {16163, 30714}
X(34153) = reflection of X(i) in X(j) for these {i,j}: {4, 10272}, {5, 1511}, {52, 11561}, {74, 548}, {265, 140}, {381, 11694}, {546, 13392}, {550, 16163}, {1539, 16534}, {3580, 18571}, {3627, 113}, {3845, 5642}, {6247, 25564}, {9140, 12100}, {10113, 5972}, {10263, 11557}, {10264, 3}, {10733, 546}, {12902, 11801}, {13605, 13624}, {14677, 550}, {16160, 16164}, {18319, 7471}, {18572, 11064}, {19479, 9820}, {20127, 12103}, {20424, 11597}, {21649, 13630}, {21650, 11591}, {21850, 6593}, {22584, 31834}, {22791, 11720}, {23306, 12038}, {23335, 25487}, {25328, 5092}, {32273, 3589}
X(34153) = complement of X(12902)
X(34153) = anticomplement of X(11801)
X(34153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 12902, 11801}, {4, 32609, 10272}, {5, 22251, 5972}, {110, 10721, 5655}, {265, 15035, 140}, {376, 14683, 10620}, {546, 13392, 14643}, {1511, 5972, 22251}, {1511, 10113, 5972}, {3522, 12317, 15041}, {3534, 12308, 12244}, {5972, 10113, 5}, {9143, 12244, 12308}, {10733, 14643, 546}, {10733, 15034, 14643}, {12902, 15040, 2}, {14643, 15034, 13392}, {15051, 15061, 3530}, {16111, 30714, 24981}, {16163, 24981, 16111}


X(34154) =  X(5)X(524)∩X(51)X(187)

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

X(34154) lies on the cubic K941 and these lines: {5, 524}, {51, 187}, {249, 5640}, {598, 15019}, {3972, 12834}, {5475, 8035}, {9515, 13410}, {21807, 21839}

X(34154) = X(i)-isoconjugate of X(j) for these (i,j): {75, 11422}, {304, 10986}
X(34154) = barycentric product X(21448)*X(22100)
X(34154) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 11422}, {1974, 10986}, {22100, 11059}


X(34155) =  X(6)X(13)∩X(51)X(110)

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 - 3*a^8*c^2 + 3*a^6*b^2*c^2 - 2*a^4*b^4*c^2 + 4*a^2*b^6*c^2 - 2*b^8*c^2 + 2*a^6*c^4 - 2*a^4*b^2*c^4 - 4*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 - 2*b^2*c^8 + c^10) : :
X(34155) = 5 X[6] + X[399], 4 X[6] - X[9976], 2 X[6] + X[19140], X[6] + 2 X[25556], X[67] - 4 X[25555], X[110] + 2 X[5097], 5 X[182] - 2 X[12041], 4 X[399] + 5 X[9976], 2 X[399] - 5 X[19140], X[399] - 10 X[25556], 2 X[575] + X[9970], 10 X[575] - X[15054], 4 X[575] - X[32305], X[576] + 2 X[6593], 2 X[576] + X[12584], X[895] - 4 X[22330], 5 X[1656] + X[16176], X[2930] + 5 X[11482], X[3629] + 2 X[10272], 3 X[5050] - X[5621], 2 X[5092] + X[10752], X[5476] + 2 X[15303], 5 X[5622] - X[15054], 4 X[6329] - X[10264], 4 X[6593] - X[12584], 5 X[9970] + X[15054], 2 X[9970] + X[32305], X[9976] + 2 X[19140], X[9976] + 8 X[25556], 4 X[10095] - X[32299], X[11061] + 2 X[20301], X[11579] - 4 X[15516], 3 X[15035] - 5 X[15462], 2 X[15054] - 5 X[32305], 2 X[18553] + X[32234], 2 X[18583] + X[25329], X[19140] - 4 X[25556]

X(34155) lies on the cubics K062 and K941 and these lines: {3, 15140}, {5, 9977}, {6, 13}, {51, 110}, {67, 25555}, {125, 5422}, {143, 576}, {182, 2781}, {184, 12824}, {185, 575}, {186, 249}, {323, 32225}, {394, 5972}, {578, 25711}, {895, 1173}, {1112, 16165}, {1199, 14094}, {1576, 18114}, {1656, 16176}, {1993, 5642}, {2393, 11692}, {2854, 15520}, {2904, 5095}, {2930, 11482}, {3098, 25487}, {3629, 10272}, {5050, 5621}, {5092, 10752}, {5102, 32609}, {5354, 9759}, {5449, 32317}, {5462, 15132}, {5965, 14643}, {6329, 10264}, {6723, 17825}, {7503, 14448}, {7592, 15063}, {9969, 12596}, {10095, 11536}, {10168, 19379}, {10601, 15106}, {10706, 15032}, {11061, 18912}, {11557, 12228}, {11561, 12901}, {11579, 15516}, {11649, 18374}, {12161, 16534}, {12370, 18428}, {13413, 18583}, {13417, 22352}, {14561, 25321}, {14627, 23236}, {15037, 20126}, {15141, 15805}, {18553, 32234}, {19129, 32600}, {22802, 32271}

X(34155) = midpoint of X(i) and X(j) for these {i,j}: {5102, 32609}, {5622, 9970}, {14561, 25321}, {18374, 18449}
X(34155) = reflection of X(i) in X(j) for these {i,j}: {5622, 575}, {32305, 5622}
X(34155) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 19140, 9976}, {6, 25556, 19140}, {575, 9970, 32305}, {576, 6593, 12584}, {11557, 12228, 13289}


X(34156) =  X(3)X(525)∩X(4)X(32)

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

X(34156) lies on the cubic K009 and these lines: {2, 9476}, {3, 525}, {4, 32}, {20, 2966}, {682, 32540}, {685, 15258}, {1147, 14376}, {2715, 8721}, {3926, 17932}, {5967, 8550}, {6337, 6394}, {14003, 20021}, {14357, 14385}, {14378, 25044}

X(34156) = isogonal conjugate of X(39265)
X(34156) = Cundy-Parry Phi transform of X(525)
X(34156) = Cundy-Parry Psi transform of X(112)
X(34156) = X(8766)-complementary conjugate of X(31842)
X(34156) = X(98)-Ceva conjugate of X(1503)
X(34156) = X(i)-isoconjugate of X(j) for these (i,j): {240, 1297}, {511, 8767}, {1755, 6330}
X(34156) = crosssum of X(511) and X(2967)
X(34156) = crossdifference of every pair of points on line {232, 684}
X(34156) = barycentric product X(i)*X(j) for these {i,j}: {98, 441}, {248, 30737}, {287, 1503}, {290, 8779}, {336, 2312}, {1821, 8766}, {6394, 16318}
X(34156) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 6330}, {248, 1297}, {441, 325}, {1503, 297}, {1910, 8767}, {2312, 240}, {8766, 1959}, {8779, 511}, {9475, 2967}, {16318, 6530}, {23976, 132}, {32696, 32687}
X(34156) = trilinear product X(i)*X(j) for these {i,j}: {98, 8766}, {287, 2312}, {293, 1503}, {441, 1910}, {1821, 8779}, {15407, 24023}
X(34156) = {X(98),X(32545)}-harmonic conjugate of X(4)


X(34157) =  X(3)X(512)∩X(4)X(99)

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

X(34157) lies on the cubic K009 and these lines: {3, 512}, {4, 99}, {32, 1147}, {182, 2065}, {237, 23098}, {263, 576}, {327, 14382}, {2211, 14966}, {2698, 5171}, {3095, 27375}, {6787, 9734}, {14379, 15261}

X(34157) = isogonal conjugate of X(14265)
X(34157) = X(3563)-Ceva conjugate of X(511)
X(34157) = X(237)-cross conjugate of X(32654)
X(34157) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14265}, {98, 1733}, {230, 1821}, {290, 8772}, {336, 460}
X(34157) = cevapoint of X(237) and X(11672)
X(34157) = crosssum of X(2974) and X(3564)
X(34157) = trilinear pole of line {2491, 3289}
X(34157) = Cundy-Parry Phi transform of X(512)
X(34157) = Cundy-Parry Psi transform of X(99)
X(34157) = barycentric product X(i)*X(j) for these {i,j}: {237, 8781}, {325, 32654}, {511, 2987}, {684, 32697}, {1755, 8773}, {3569, 10425}
X(34157) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14265}, {237, 230}, {1755, 1733}, {2211, 460}, {2987, 290}, {3289, 3564}, {3563, 16081}, {8781, 18024}, {9417, 8772}, {9418, 1692}, {11672, 114}, {14966, 4226}, {32654, 98}, {32697, 22456}
X(34157) = {X(3563),X(14253)}-harmonic conjugate of X(9737)


X(34158) =  X(3)X(647)∩X(4)X(111)

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

X(34158) lies on the cubic K009 and these lines: {3, 647}, {4, 111}, {32, 1084}, {206, 15477}, {574, 14357}, {5254, 14609}, {6337, 14376}, {28407, 30786}

X(34158) = X(111)-Ceva conjugate of X(2393)
X(34158) = barycentric product X(i)*X(j) for these {i,j}: {111, 14961}, {858, 14908}, {895, 2393}
X(34158) = barycentric quotient X(i)/X(j) for these {i,j}: {14908, 2373}, {14961, 3266}


X(34159) =  X(3)X(667)∩X(4)X(120)

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

X(34159) lies on the cubic K009 and these lines: {3, 667}, {4, 120}, {32, 218}, {55, 22116}, {56, 6337}, {2223, 23102}

X(34159) = isogonal conjugate of X(14267)
X(34159) = X(15344)-Ceva conjugate of X(518)
X(34159) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14267}, {105, 1738}, {673, 3290}, {6185, 17464}, {16752, 18785}
X(34159) = cevapoint of X(2223) and X(6184)
X(34159) = crosssum of X(3290) and X(20455)
X(34159) = crossdifference of every pair of points on line {3290, 23770}
X(34159) = barycentric product X(i)*X(j) for these {i,j}: {518, 2991}, {4437, 15382}, {15344, 25083}
X(34159) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14267}, {665, 23770}, {672, 1738}, {2223, 3290}, {2991, 2481}, {3286, 16752}, {4712, 20431}, {6184, 120}, {15382, 6185}, {20683, 21956}, {20776, 20728}


X(34160) =  X(3)X(905)∩X(4)X(105)

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

X(34160) lies on the cubic K009 and these lines: {3, 905}, {4, 105}, {32, 56}, {3556, 32666}, {11517, 14376}

X(34160) = X(105)-Ceva conjugate of X(3827)
X(34160) = X(1861)-isoconjugate of X(26703)
X(34160) = barycentric product X(1814)*X(3827)
X(34160) = barycentric quotient X(32658)/X(26703)


X(34161) =  X(3)X(669)∩X(4)X(126)

Barycentrics    (2*a^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(34161) lies on the cubic K009 and these lines: {3, 669}, {4, 126}, {32, 1992}, {187, 23106}, {439, 23357}, {1147, 13608}, {1383, 3552}, {3926, 23992}, {5467, 15471}, {5970, 7793}, {7618, 15387}, {7664, 16925}, {8030, 14567}, {14214, 14385}, {14376, 15261}

X(34161) = isogonal conjugate of X(14263)
X(34161) = X(2374)-Ceva conjugate of X(524)
X(34161) = X(i)-isoconjugate of X(j) for these (i,j): {1, 14263}, {897, 3291}, {10630, 17466}, {11634, 23894}
X(34161) = cevapoint of X(i) and X(j) for these (i,j): {187, 2482}, {14417, 23992}
X(34161) = trilinear pole of line {3292, 9125}
X(34161) = barycentric product X(2374)*X(6390)
X(34161) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 14263}, {187, 3291}, {690, 9134}, {2374, 17983}, {2482, 126}, {3292, 8681}, {5467, 11634}, {15387, 10630}, {16702, 16756}


X(34162) =  X(2)X(271)∩X(4)X(189)

Barycentrics    (a - b - c)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c + 2*a*b*c + b^2*c - a*c^2 - b*c^2 - c^3)*(a^3 + a^2*b - a*b^2 - b^3 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + b*c^2 + c^3)*(a^9 + 3*a^8*b - 8*a^6*b^3 - 6*a^5*b^4 + 6*a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 - b^9 + 3*a^8*c + 4*a^6*b^2*c - 14*a^4*b^4*c + 4*a^2*b^6*c + 3*b^8*c + 4*a^6*b*c^2 + 12*a^5*b^2*c^2 + 8*a^4*b^3*c^2 - 8*a^3*b^4*c^2 - 12*a^2*b^5*c^2 - 4*a*b^6*c^2 - 8*a^6*c^3 + 8*a^4*b^2*c^3 + 8*a^2*b^4*c^3 - 8*b^6*c^3 - 6*a^5*c^4 - 14*a^4*b*c^4 - 8*a^3*b^2*c^4 + 8*a^2*b^3*c^4 + 14*a*b^4*c^4 + 6*b^5*c^4 + 6*a^4*c^5 - 12*a^2*b^2*c^5 + 6*b^4*c^5 + 8*a^3*c^6 + 4*a^2*b*c^6 - 4*a*b^2*c^6 - 8*b^3*c^6 - 3*a*c^8 + 3*b*c^8 - c^9) : :
X(34162) = 3 X[2] - 4 X[20210]

X(34162) lies on the cubic K007 and these lines: {2, 271}, {4, 189}, {7, 253}, {8, 1032}, {20, 3353}, {329, 14365}

X(34162) = reflection of X(3342) in X(20210)
X(34162) = isogonal conjugate of X(34167)
X(34162) = isotomic conjugate of the isogonal conjugate of X(28784)
X(34162) = anticomplement of X(3342)
X(34162) = anticomplement of the isogonal conjugate of X(3341)
X(34162) = isotomic conjugate of the anticomplement of X(3351)
X(34162) = anticomplementary-isogonal conjugate of X(1034)
X(34162) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 1034}, {84, 9799}, {1490, 6223}, {2192, 20212}, {3197, 20211}, {3341, 8}, {8885, 1895}
X(34162) = X(69)-Ceva conjugate of X(189)
X(34162) = X(3351)-cross conjugate of X(2)
X(34162) = X(221)-isoconjugate of X(3347)
X(34162) = cevapoint of X(3341) and X(3353)
X(34162) = perspector of ABC and pedal triangle of X(3353)
X(34162) = cyclocevian conjugate of perspector of ABC and pedal triangle of X(3354)
X(34162) = barycentric product X(76)*X(28784)
X(34162) = barycentric quotient X(i)/X(j) for these {i,j}: {282, 3347}, {3182, 223}, {3341, 3352}, {3351, 3342}, {8802, 2331}, {8894, 7952}, {28784, 6}
X(34162) = {X(3342),X(20210)}-harmonic conjugate of X(2)


X(34163) =  X(2)X(112)∩X(4)X(67)

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

X(34163) lies on the cubic K008 and these lines: {2, 112}, {4, 67}, {136, 8753}, {186, 13200}, {315, 648}, {317, 671}, {524, 5523}, {2072, 13310}, {2794, 18533}, {3153, 12384}, {7784, 8743}, {10749, 18420}, {11641, 21213}, {14649, 14900}

X(34163) = anticomplement of X(18876)
X(34163) = polar-circle inverse of X(32246)
X(34163) = circumcircle-of-anticomplementary-triangle-inverse of X(2892)
X(34163) = anticomplement of the isogonal conjugate of X(5523)
X(34163) = isotomic conjugate of the isogonal conjugate of X(8428)
X(34163) = polar conjugate of the isogonal conjugate of X(15141)
X(34163) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {19, 23}, {92, 2393}, {858, 4329}, {897, 2373}, {2393, 6360}, {5523, 8}, {14580, 192}, {17172, 20243}, {18669, 20}, {20884, 1370}, {21459, 17165}
X(34163) = X(316)-Ceva conjugate of X(4)
X(34163) = cevapoint of X(8428) and X(15141)
X(34163) = barycentric product X(i)*X(j) for these {i,j}: {76, 8428}, {264, 15141}
X(34163) = barycentric quotient X(i)/X(j) for these {i,j}: {8428, 6}, {15141, 3}, {19330, 895}
X(34163) = {X(11605),X(20410)}-harmonic conjugate of X(4)


X(34164) =  X(2)X(10354)∩X(69)X(671)

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

X(34164) lies on the cubic K008 and these lines: {2, 10354}, {69, 671}, {524, 13492}, {599, 14262}, {858, 13574}, {1296, 2482}, {2418, 8591}

X(34164) = isogonal conjugate of X(38533)
X(34164) = anticomplement of X(34581)
X(34164) = anticomplement of the isogonal conjugate of X(13492)
X(34164) = isotomic conjugate of the isogonal conjugate of X(10355)
X(34164) = X(13492)-anticomplementary conjugate of X(8)
X(34164) = barycentric product X(76)*X(10355)
X(34164) = barycentric quotient X(10355)/X(6)


X(34165) =  X(2)X(12505)∩X(4)X(524)

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

X(34165) lies on the cubics K008 and K617 and these lines: {2, 12505}, {4, 524}, {5, 21448}, {20, 1296}, {2418, 3926}, {3767, 17968}, {7841, 17952}, {10354, 10748}

X(34165) = isotomic conjugate of isogonal conjugate of X(38532)
X(34165) = anticomplement of X(13608)
X(34165) = anticomplement of the isogonal conjugate of X(14262)
X(34165) = X(14262)-anticomplementary conjugate of X(8)
X(34165) = barycentric product X(5485)*X(7493)
X(34165) = barycentric quotient X(i)/X(j) for these {i,j}: {7493, 1992}, {19153, 1384}
X(34165) = {X(4),X(5485)}-harmonic conjugate of X(14262)


X(34166) =  X(2)X(10354)∩X(4)X(10748)

Barycentrics    (a^2 - 3*a*b + b^2 + c^2)*(a^2 + 3*a*b + b^2 + c^2)*(a^2 + b^2 - 3*a*c + c^2)*(a^2 + b^2 + 3*a*c + c^2)*(a^4 - b^4 + 4*b^2*c^2 - c^4) : :
X(34166) = 4 X[6719] - 3 X[20481]

X(34166) lies on the cubic K008 and these lines: {2, 10354}, {4, 10748}, {111, 524}, {126, 8176}, {1995, 13493}, {5512, 14262}, {6719, 20481}, {10355, 11165}, {11148, 20099}

X(34166) = isotomic conjugate of X(39157)
X(34166) = reflection of X(14262) in X(5512)
X(34166) = anticomplement of X(10354)
X(34166) = antigonal image of X(14262)
X(34166) = isotomic conjugate of the isogonal conjugate of X(13493)
X(34166) = barycentric product X(76)*X(13493)
X(34166) = barycentric quotient X(i)/X(j) for these {i,j}: {1995, 11580}, {8542, 9872}, {11185, 11054}, {13493, 6}, {14262, 13492}


X(34167) =  X(3)X(3341)∩X(55)X(28785)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(a^9 - 3*a^8*b + 8*a^6*b^3 - 6*a^5*b^4 - 6*a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 + b^9 + 3*a^8*c + 4*a^6*b^2*c - 14*a^4*b^4*c + 4*a^2*b^6*c + 3*b^8*c - 4*a^6*b*c^2 + 12*a^5*b^2*c^2 - 8*a^4*b^3*c^2 - 8*a^3*b^4*c^2 + 12*a^2*b^5*c^2 - 4*a*b^6*c^2 - 8*a^6*c^3 + 8*a^4*b^2*c^3 + 8*a^2*b^4*c^3 - 8*b^6*c^3 - 6*a^5*c^4 + 14*a^4*b*c^4 - 8*a^3*b^2*c^4 - 8*a^2*b^3*c^4 + 14*a*b^4*c^4 - 6*b^5*c^4 + 6*a^4*c^5 - 12*a^2*b^2*c^5 + 6*b^4*c^5 + 8*a^3*c^6 - 4*a^2*b*c^6 - 4*a*b^2*c^6 + 8*b^3*c^6 - 3*a*c^8 - 3*b*c^8 - c^9)*(a^9 + 3*a^8*b - 8*a^6*b^3 - 6*a^5*b^4 + 6*a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 - b^9 - 3*a^8*c - 4*a^6*b^2*c + 14*a^4*b^4*c - 4*a^2*b^6*c - 3*b^8*c + 4*a^6*b*c^2 + 12*a^5*b^2*c^2 + 8*a^4*b^3*c^2 - 8*a^3*b^4*c^2 - 12*a^2*b^5*c^2 - 4*a*b^6*c^2 + 8*a^6*c^3 - 8*a^4*b^2*c^3 - 8*a^2*b^4*c^3 + 8*b^6*c^3 - 6*a^5*c^4 - 14*a^4*b*c^4 - 8*a^3*b^2*c^4 + 8*a^2*b^3*c^4 + 14*a*b^4*c^4 + 6*b^5*c^4 - 6*a^4*c^5 + 12*a^2*b^2*c^5 - 6*b^4*c^5 + 8*a^3*c^6 + 4*a^2*b*c^6 - 4*a*b^2*c^6 - 8*b^3*c^6 - 3*a*c^8 + 3*b*c^8 + c^9) : :

X(34167) lies on the cubic K172 and these lines: {3, 3341}, {55, 28785}, {56, 28782}, {154, 1035}, {198, 1033}

X(34167) = isogonal conjugate of X(34162)
X(34167) = isogonal conjugate of the anticomplement of X(3342)
X(34167) = X(3352)-Ceva conjugate of X(6)
X(34167) = X(25)-cross conjugate of X(198)
X(34167) = X(i)-isoconjugate of X(j) for these (i,j): {75, 28784}, {280, 3182}
X(34167) = crosspoint of X(3342) and X(3354)
X(34167) = crosssum of X(3341) and X(3353)
X(34167) = barycentric product X(i)*X(j) for these {i,j}: {223, 3347}, {3342, 3352}
X(34167) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 28784}, {2199, 3182}, {3195, 8894}


X(34168) =  X(2)X(1301)∩X(3)X(1289)

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

X(34168) lies on these lines: {2, 1301}, {3, 1289}, {20, 112}, {22, 107}, {23, 22239}, {25, 30249}, {30, 10423}, {99, 11413}, {110, 1370}, {378, 30251}, {691, 16386}, {827, 12225}, {858, 1304}, {935, 2071}, {1299, 7422}, {1302, 26283}, {2867, 10229}, {3565, 30552}, {6636, 20626}, {7493, 9064}, {9060, 16387}, {9107, 26253}, {13526, 15740}, {14944, 32687}, {21312, 30247}, {26706, 30267}

X(34168) = reflection of X(1289) in X(3)
X(34168) = isogonal conjugate of X(34146)
X(34168) = isotomic conjugate of the anticomplement of X(16318)
X(34168) = de-Longchamps-circle-inverse of X(12384)
X(34168) = Thomson-isogonal conjugate of X(8673)
X(34168) = X(16318)-cross conjugate of X(2)
X(34168) = cevapoint of X(i) and X(j) for these (i,j): {3, 1503}, {5002, 5003}
X(34168) = trilinear pole of line {6, 8057}
X(34168) = Λ(X(4), X(66))
X(34168) = Λ(X(6), X(64))
X(34168) = Lucas-isogonal conjugate of X(8673)
X(34168) = orthoptic-circle-of-Steiner-inellipse-inverse of X(35968)


X(34169) =  X(4)X(1499)∩X(30)X(111)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(2*a^6 - 2*a^4*b^2 - 3*a^2*b^4 + b^6 - 2*a^4*c^2 + 8*a^2*b^2*c^2 - b^4*c^2 - 3*a^2*c^4 - b^2*c^4 + c^6) : :
X(34169) = X[5971] - 3 X[14041]

X(34169) lies on the cubic K025 and these lines: {4, 1499}, {30, 111}, {115, 5912}, {230, 691}, {316, 524}, {2549, 5968}, {2770, 14120}, {5108, 7841}, {5203, 11605}, {5254, 14246}, {5380, 21956}, {5475, 14609}, {5523, 8753}, {5971, 14041}, {7472, 10418}, {8370, 32525}, {9169, 11317}, {30786, 33228}

X(34169) = reflection of X(i) in X(j) for these {i,j}: {2770, 14120}, {5912, 115}, {7472, 31655}
X(34169) = antigonal image of X(7472)
X(34169) = symgonal image of X(14120)
X(34169) = crosssum of X(187) and X(9177)
X(34169) = barycentric product X(i)*X(j) for these {i,j}: {671, 10418}, {5466, 7472}
X(34169) = barycentric quotient X(i)/X(j) for these {i,j}: {7472, 5468}, {10418, 524}
X(34169) = {X(115),X(17964)}-harmonic conjugate of X(16092)


X(34170) =  MIDPOINT OF X(4) AND X(6761)

Barycentrics    (a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 7*a^4*b^2*c^2 - 4*a^2*b^4*c^2 - b^6*c^2 - 4*a^2*b^2*c^4 + 4*b^4*c^4 + 2*a^2*c^6 - b^2*c^6 - c^8) : :
X(34170) = 5 X[3091] - 2 X[34147], X[3146] + 2 X[34109]

X(34170) lies on the cubic K025 and these lines: {2, 12096}, {3, 21396}, {4, 51}, {5, 6760}, {20, 6526}, {30, 107}, {133, 18400}, {137, 18809}, {275, 16657}, {316, 6528}, {403, 1300}, {520, 16229}, {1503, 1559}, {1596, 1629}, {2071, 16177}, {3091, 34147}, {3146, 6523}, {3153, 13573}, {3543, 6525}, {3839, 10002}, {5523, 6529}, {6530, 10151}, {6623, 11547}, {6624, 11206}, {10152, 15311}, {10540, 11251}

X(34170) = midpoint of X(4) and X(6761)
X(34170) = reflection of X(i) in X(j) for these {i,j}: {20, 11589}, {1304, 403}, {2071, 16177}, {6760, 5}
X(34170) = anticomplement of X(12096)
X(34170) = circumcircle-inverse of X(21396)
X(34170) = polar-circle-inverse of X(185)
X(34170) = circumcircle-of-anticomplementary-triangle-inverse of X(12324)
X(34170) = antigonal image of X(2071)
X(34170) = symgonal image of X(403)
X(34170) = polar conjugate of the isogonal conjugate of X(15262)
X(34170) = X(255)-isoconjugate of X(11744)
X(34170) = cevapoint of X(403) and X(15311)
X(34170) = barycentric product X(i)*X(j) for these {i,j}: {264, 15262}, {2052, 2071}
X(34170) = barycentric quotient X(i)/X(j) for these {i,j}: {393, 11744}, {2071, 394}, {6529, 22239}, {15262, 3}
X(34170) = {X(4),X(1075)}-harmonic conjugate of X(22802)


X(34171) =  X(4)X(2780)∩X(30)X(1296)

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

X(34171) lies on the cubic K025 and these lines: {4, 2780}, {30, 1296}, {111, 3143}, {126, 11634}, {316, 670}, {804, 14948}, {5203, 5523}, {7418, 23699}, {9134, 14263}

X(34171) = reflection of X(i) in X(j) for these {i,j}: {111, 3143}, {11634, 126}
X(34171) = antigonal image of X(11634)
X(34171) = symgonal image of X(3143)
X(34171) = barycentric quotient X(3291)/X(2854)


X(34172) =  X(4)X(6003)∩X(30)X(759)

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

X(34172) lies on the cubic K025 and these lines: {4, 6003}, {30, 759}, {80, 758}, {316, 14616}, {2161, 5134}


X(34173) =  X(4)X(885)∩X(30)X(105)

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

X(34173) lies on the cubic K025 and these lines: {4, 885}, {30, 105}, {316, 2481}, {518, 20556}, {666, 671}, {5134, 18785}, {5523, 8751}

X(34173) = antigonal image of X(7475)


X(34174) =  X(4)X(690)∩X(30)X(99)

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

X(34174) lies on the cubics K025 and K596 and on these lines: {4, 690}, {30, 99}, {98, 868}, {114, 4226}, {265, 671}, {804, 23350}, {1300, 11605}, {2794, 7422}, {5523, 5962}

X(34174) = reflection of X(i) in X(j) for these {i,j}: {98, 868}, {4226, 114}
X(34174) = antigonal image of X(4226)
X(34174) = symgonal image of X(868)
X(34174) = X(i)-isoconjugate of X(j) for these (i,j): {2247, 2987}, {5191, 8773}
X(34174) = barycentric product X(i)*X(j) for these {i,j}: {230, 5641}, {4226, 14223}
X(34174) = barycentric quotient X(i)/X(j) for these {i,j}: {230, 542}, {460, 6103}, {842, 2987}, {1692, 5191}, {4226, 14999}, {5641, 8781}, {5649, 10425}, {8772, 2247}


X(34175) =  X(4)X(512)∩X(30)X(98)

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

X(34175) lies on the cubic K025 and these lines: {4, 512}, {30, 98}, {265, 290}, {460, 685}, {511, 14957}, {567, 7827}, {1300, 2715}, {2395, 8430}, {5203, 10152}, {5476, 5967}, {5962, 11605}, {7468, 16188}, {13449, 13851}, {14221, 14984}

X(34175) = reflection of X(7468) in X(16188)
X(34175) = antigonal image of X(7468)
X(34175) = barycentric product X(i)*X(j) for these {i,j}: {290, 2493}, {2395, 14221}, {14984, 16081}
X(34175) = barycentric quotient X(i)/X(j) for these {i,j}: {2493, 511}, {7468, 2421}, {14221, 2396}


X(34176) =  MIDPOINT OF X(19) AND X(63)

Barycentrics    a*(a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-(b^2-c^2)^2*a^3+(b+c)*(b^4+c^4-2*(b^2+c^2)*b*c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)*(b-c)*(b^4+c^4)) : :
X(34176) = X(18446)-3*X(21160), 3*X(21165)-X(30265)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29439.

X(34176) lies on these lines: {1, 16599}, {9, 20305}, {19, 27}, {515, 15941}, {2083, 16560}, {4331, 7098}, {5745, 18589}, {6518, 18161}, {7291, 24683}, {16565, 21374}, {18446, 21160}, {18805, 23998}, {21165, 30265}

X(34176) = midpoint of X(19) and X(63)
X(34176) = reflection of X(18589) in X(5745)


X(34177) =  MIDPOINT OF X(22) AND X(66)

Barycentrics    (b^2+c^2)*a^12-2*(b^4+c^4)*a^10-(b^4-c^4)*(b^2-c^2)*a^8+4*(b^8+c^8)*a^6-(b^4-c^4)^2*(b^2+c^2)*a^4-2*(b^4-c^4)^2*(b^4+c^4)*a^2+(b^4-c^4)^3*(b^2-c^2) : :
Barycentrics    (2*(7*SW+3*SA)*R^4-(5*SA+8*SW)*SW*R^2+(SA+SW)*SW^2)*S^2+R^2*SB*SC*SW^2 : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29439.

X(34177) lies on these lines: {5, 2781}, {22, 66}, {26, 34118}, {141, 206}, {343, 2393}, {427, 6697}, {1503, 7502}, {3580, 23327}, {9019, 23300}

X(34177) = midpoint of X(22) and X(66)
X(34177) = reflection of X(i) in X(j) for these (i,j): (206, 6676), (427, 6697)






leftri  P-vertex conjugate of P, for P on the line at infinity: X(34178) - X(34193)  rightri

Let P be a point on the line at infinity. The locus of the P-vertex conjugate of P, as P varies, is a quartic which is the isogonal conjugate of the anticomplementary circle. This quartic passes through the vertices of ABC and the tangential triangle, the circular points at infinity, and centers X(3446), X(3447), X(9217), X(22259), X(34130), X(34178), X(34179), X(34180), X(34181), X(34182), X(34183), X(34184), X(34185), X(34187), X(34189), X(34190), X(34191) and X(34192). If ABC is acute, the quartic is a closed curve. If ABC is obtuse, it meets the line at infinity at the isogonal conjugates of the anticomplements of PU(4) (the circumcircle intercepts of the anticomplementary circle).

Peter Moses gives the equation for the quartic as:

c^6 x^2 y^2 + b^2 c^2 (a^2 + b^2 + c^2) x^2 y z + a^2 c^2 (a^2 + b^2 + c^2) x y^2 z + b^6 x^2 z^2 + a^2 b^2 (a^2 + b^2 + c^2) x y z^2 + a^6 y^2 z^2 = 0

The quartic is not currently listed in Bernard Gibert's CTC.

See Hyacinthos 29443.

Contributed by Randy Hutson, August 31, 2019.

underbar



X(34178) = X(30)-VERTEX CONJUGATE OF X(30)

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

X(34178) lies on these lines: {3, 14993}, {378, 14385}, {399, 2935}, {1272, 2071}, {5621, 8675}, {5899, 14703}

X(34178) = isogonal conjugate of X(146)
X(34178) = X(30)-vertex conjugate of X(30)


X(34179) = X(514)-VERTEX CONJUGATE OF X(514)

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

X(34179) lies on these lines: {36, 5018}, {514, 20999}, {2361, 17798}

X(34179) = isogonal conjugate of X(150)
X(34179) = X(514)-vertex conjugate of X(514)
X(34179) = X(92)-isoconjugate of X(22145)


X(34180) = X(515)-VERTEX CONJUGATE OF X(515)

Barycentrics    a^2/(a^12 + a^11 (b + c) + a^10 b c + a^9 (3 b^3 - b^2 c - b c^2 + 3 c^3) - a^8 (3 b^4 + b^3 c + b c^3 + 3 c^4) - 2 a^7 (7 b^5 + b^4 c - 4 b^3 c^2 - 4 b^2 c^3 + b c^4 + 7 c^5) - 2 a^6 b c (b^4 - 2 b^3 c - 6 b^2 c^2 - 2 b c^3 + c^4) + 2 a^5 (7 b^7 + b^6 c - 5 b^5 c^2 - 3 b^4 c^3 - 3 b^3 c^4 - 5 b^2 c^5 + b c^6 + 7 c^7) + a^4 (3 b^8 + 2 b^7 c - 4 b^6 c^2 - 2 b^5 c^3 + 2 b^4 c^4 - 2 b^3 c^5 - 4 b^2 c^6 + 2 b c^7 + 3 c^8) - a^3 (3 b^9 - b^8 c + 8 b^6 c^3 - 10 b^5 c^4 - 10 b^4 c^5 + 8 b^3 c^6 - b c^8 + 3 c^9) + a^2 b c (b^8 - 4 b^7 c - 12 b^6 c^2 + 4 b^5 c^3 + 22 b^4 c^4 + 4 b^3 c^5 - 12 b^2 c^6 - 4 b c^7 + c^8) - a (b^11 + b^10 c - 3 b^9 c^2 - 3 b^8 c^3 + 2 b^7 c^4 + 2 b^6 c^5 + 2 b^5 c^6 + 2 b^4 c^7 - 3 b^3 c^8 - 3 b^2 c^9 + b c^10 + c^11) - b^12 - b^11 c + 4 b^10 c^2 + 3 b^9 c^3 - 7 b^8 c^4 - 2 b^7 c^5 + 8 b^6 c^6 - 2 b^5 c^7 - 7 b^4 c^8 + 3 b^3 c^9 + 4 b^2 c^10 - b c^11 - c^12) : :

X(34180) lies on these lines: {}

X(34180) = isogonal conjugate of X(151)
X(34180) = X(515)-vertex conjugate of X(515)


X(34181) = X(516)-VERTEX CONJUGATE OF X(516)

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

X(34181) lies on these lines: {}

X(34181) = isogonal conjugate of X(152)
X(34181) = X(516)-vertex conjugate of X(516)


X(34182) = X(517)-VERTEX CONJUGATE OF X(517)

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

X(34182) lies on these lines: {1319, 10016}, {2077, 2932}

X(34182) = isogonal conjugate of X(153)
X(34182) = X(517)-vertex conjugate of X(517)


X(34183) = X(518)-VERTEX CONJUGATE OF X(518)

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

X(34183) lies on these lines: {518, 3220}, {672, 5096}, {3252, 4265}

X(34183) = isogonal conjugate of X(20344)
X(34183) = X(518)-vertex conjugate of X(518)


X(34184) = X(519)-VERTEX CONJUGATE OF X(519)

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

X(34184) lies on these lines: {995, 16944}, {2718, 20999}

X(34184) = isogonal conjugate of X(21290)
X(34184) = X(519)-vertex conjugate of X(519)
X(34184) = X(92)-isoconjugate of X(23135)


X(34185) = X(520)-VERTEX CONJUGATE OF X(520)

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

X(34185) lies on these lines: {1503, 6761}, {6000, 15781}

X(34185) = isogonal conjugate of X(34186)
X(34185) = X(520)-vertex conjugate of X(520)


X(34186) = ISOGONAL CONJUGATE OF X(34185)

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

X(34186) lies on the anticomplementary circle and these lines: {2, 107}, {3, 3462}, {4, 2972}, {20, 110}, {22, 14673}, {40, 151}, {100, 2806}, {133, 3091}, {145, 10701}, {147, 1370}, {148, 2797}, {149, 2803}, {150, 2811}, {152, 2947}, {153, 2828}, {193, 10762}, {376, 23240}, {388, 3324}, {497, 7158}, {684, 6086}, {2409, 12253}, {2475, 9528}, {2693, 32417}, {2833, 20344}, {2839, 21290}, {2845, 34188}, {2846, 33650}, {2847, 6527}, {2848, 3268}, {2967, 7386}, {3146, 3346}, {3153, 34193}, {3164, 16063}, {3184, 3522}, {3448, 9033}, {3523, 23239}, {3616, 11718}, {3622, 11732}, {6723, 14847}, {7488, 14703}, {13611, 16080}, {14361, 33892}, {14538, 19772}, {14539, 19773}, {15059, 24930}, {15318, 18381}

X(34186) = reflection of X(20) in X(1294)
X(34186) = reflection of X(34549) in X(4)
X(34186) = isogonal conjugate of X(34185)
X(34186) = anticomplement of X(107)
X(34186) = anticomplementary conjugate of X(520)
X(34186) = anticomplementary-circle-antipode of X(34549)
X(34186) = de-Longchamps-circle-inverse of X(110)
X(34186) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(1297)


X(34187) = X(521)-VERTEX CONJUGATE OF X(521)

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

X(34187) lies on these lines: {}

X(34187) = isogonal conjugate of X(34188)
X(34187) = X(521)-vertex conjugate of X(521)


X(34188) = ISOGONAL CONJUGATE OF X(34187)

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

X(34188) lies on the anticomplementary circle and these lines: {2, 108}, {4, 280}, {8, 151}, {20, 100}, {92, 12384}, {145, 10702}, {146, 2778}, {147, 2791}, {148, 2798}, {149, 2804}, {150, 2812}, {152, 329}, {193, 10763}, {388, 1359}, {497, 3318}, {908, 3100}, {1370, 2834}, {2840, 21290}, {2845, 34186}, {2849, 33650}, {2850, 3448}, {2851, 14360}, {3091, 25640}, {3146, 10731}, {3616, 11719}, {3622, 11733}, {5080, 10538}

X(34188) = reflection of X(20) in X(1295)
X(34188) = reflection of X(34550) in X(4)
X(34188) = isogonal conjugate of X(34187)
X(34188) = anticomplement of X(108)
X(34188) = anticomplementary conjugate of X(521)
X(34188) = anticomplementary-circle-antipode of X(34550)
X(34188) = de-Longchamps-circle-inverse of X(100)
X(34188) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(26703)


X(34189) = X(522)-VERTEX CONJUGATE OF X(522)

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

X(34189) lies on these lines: {859, 14192}, {1324, 2077}, {20999, 23224}

X(34189) = isogonal conjugate of X(33650)
X(34189) = X(522)-vertex conjugate of X(522)


X(34190) = X(525)-VERTEX CONJUGATE OF X(525)

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

X(34190) lies on these lines: {525, 10117}, {1503, 2071}, {5938, 14961}

X(34190) = isogonal conjugate of X(13219)
X(34190) = X(525)-vertex conjugate of X(525)


X(34191) = X(526)-VERTEX CONJUGATE OF X(526)

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

X(34191) lies on these lines: {542, 14993}, {1272, 9143}

X(34191) = isogonal conjugate of X(14731)
X(34191) = X(526)-vertex conjugate of X(526)


X(34192) = X(5663)-VERTEX CONJUGATE OF X(5663)

Barycentrics    a^2/(a^16 - 2 a^12 (6 b^4 - 7 b^2 c^2 + 6 c^4) + 2 a^10 (13 b^6 - 8 b^4 c^2 - 8 b^2 c^4 + 13 c^6) - a^8 (20 b^8 + 28 b^6 c^2 - 81 b^4 c^4 + 28 b^2 c^6 + 20 c^8) + 2 a^6 (2 b^10 + 19 b^8 c^2 - 20 b^6 c^4 - 20 b^4 c^6 + 19 b^2 c^8 + 2 c^10) + a^4 b^2 c^2 (4 b^8 - 51 b^6 c^2 + 94 b^4 c^4 - 51 b^2 c^6 + 4 c^8) + 2 a^2 (b^14 - 7 b^12 c^2 + 15 b^10 c^4 - 9 b^8 c^6 - 9 b^6 c^8 + 15 b^4 c^10 - 7 b^2 c^12 + c^14) - b^16 + 2 b^14 c^2 + 8 b^12 c^4 - 34 b^10 c^6 + 50 b^8 c^8 - 34 b^6 c^10 + 8 b^4 c^12 + 2 b^2 c^14 - c^16) : :

X(34192) lies on these lines: {}

X(34192) = isogonal conjugate of X(34193)
X(34192) = X(5663)-vertex conjugate of X(5663)


X(34193) = ISOGONAL CONJUGATE OF X(34192)

Barycentrics    a^16 - 2 a^12 (6 b^4 - 7 b^2 c^2 + 6 c^4) + 2 a^10 (13 b^6 - 8 b^4 c^2 - 8 b^2 c^4 + 13 c^6) - a^8 (20 b^8 + 28 b^6 c^2 - 81 b^4 c^4 + 28 b^2 c^6 + 20 c^8) + 2 a^6 (2 b^10 + 19 b^8 c^2 - 20 b^6 c^4 - 20 b^4 c^6 + 19 b^2 c^8 + 2 c^10) + a^4 b^2 c^2 (4 b^8 - 51 b^6 c^2 + 94 b^4 c^4 - 51 b^2 c^6 + 4 c^8) + 2 a^2 (b^14 - 7 b^12 c^2 + 15 b^10 c^4 - 9 b^8 c^6 - 9 b^6 c^8 + 15 b^4 c^10 - 7 b^2 c^12 + c^14) - b^16 + 2 b^14 c^2 + 8 b^12 c^4 - 34 b^10 c^6 + 50 b^8 c^8 - 34 b^6 c^10 + 8 b^4 c^12 + 2 b^2 c^14 - c^16 : :

X(34193) lies on the anticomplementary circle and these lines: {2, 477}, {3, 18319}, {4, 14731}, {20, 476}, {30, 3448}, {146, 523}, {388, 33965}, {497, 33964}, {546, 11749}, {1553, 14480}, {3091, 3258}, {3146, 14989}, {3153, 34186}, {3523, 22104}, {6070, 14508}, {10296, 13219}, {12041, 14993}, {14360, 30474}, {14851, 20304}, {15081, 16340}

X(34193) = reflection of X(20) in X(476)
X(34193) = reflection of X(14731) in X(4)
X(34193) = isogonal conjugate of X(34192)
X(34193) = anticomplement of X(477)
X(34193) = anticomplementary conjugate of X(5663)
X(34193) = anticomplementary circle antipode of X(14731)
X(34193) = de-Longchamps-circle inverse of X(2693)
X(34193) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(9060)


X(34194) =  REFLECTION OF X(1365) IN X(1)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29448.

X(34194) lies on the incircle and these lines: {1,1365}, {11,214}, {36,14027}, {55,759}, {56,6011}, {942,1357}, {1283,8240}, {1356,21746}, {1358,3664}, {1364,10544}, {1682,6044}, {1697,21381}, {3022,11997}, {3024,3057}, {3295,14663}, {3323,5088}, {3326,5497}, {3685,4542}, {4092,6740}, {4313,19642}, {5048,31522}, {5441,12896}

X(34194) = reflection of X(1365) in X(1)


X(34195) =  REFLECTION OF X(21) IN X(1)

Barycentrics    a*(a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c - a*b*c - b^2*c - a*c^2 - b*c^2 + 2*c^3) : :
X(34195) = 2 X[1] - X[21], 3 X[1] - X[191], 5 X[1] - 2 X[3647], 5 X[1] - 3 X[5426], 4 X[1] - X[11684], 3 X[2] - 4 X[11281], X[8] - 4 X[16137], 2 X[10] - 3 X[26725], 4 X[10] - 5 X[31254], 3 X[21] - 2 X[191], 5 X[21] - 4 X[3647], 5 X[21] - 6 X[5426], X[21] + 2 X[16126], X[79] + 2 X[3244], X[145] + 2 X[3649], 5 X[191] - 6 X[3647], 5 X[191] - 9 X[5426], 4 X[191] - 3 X[11684], X[191] + 3 X[16126], 3 X[354] - 2 X[8261], 4 X[551] - 3 X[15671], 4 X[1385] - 3 X[21161], 3 X[3241] + X[14450], 2 X[3243] + X[16133], 5 X[3616] - 4 X[6675], 7 X[3622] - 5 X[15674], 7 X[3622] - 4 X[18253], 5 X[3623] - 2 X[10543], 5 X[3623] - X[15680], X[3632] - 4 X[6701], 4 X[3635] - X[5441], 2 X[3647] - 3 X[5426], 8 X[3647] - 5 X[11684], 2 X[3647] + 5 X[16126], X[3648] - 4 X[15174], X[3648] - 7 X[20057], 2 X[3652] - 3 X[28461], X[3652] - 4 X[33179], 12 X[5426] - 5 X[11684], 3 X[5426] + 5 X[16126], 2 X[5428] - 3 X[10246], 3 X[5603] - 2 X[6841], 3 X[6175] - 4 X[11263], 2 X[7982] + X[33557], 2 X[10021] - 3 X[10283], 3 X[10032] - 2 X[31888], 6 X[10222] - X[16138], 4 X[10222] - X[21669], 3 X[10247] - X[13743], 3 X[11224] + X[16143], X[11684] + 4 X[16126], X[12531] - 4 X[33593], 4 X[13607] - X[16113], 4 X[15174] - 7 X[20057], 5 X[15674] - 4 X[18253], 3 X[15677] - X[31888], 2 X[16138] - 3 X[21669], 2 X[16139] - 3 X[21161], 2 X[19919] - 3 X[28453], 6 X[26725] - 5 X[31254], 3 X[28461] - 8 X[33179]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29448.

X(34195) lies on these lines: {1,21}, {2,11281}, {3,31660}, {8,442}, {10,5425}, {30,944}, {35,4084}, {56,27086}, {65,100}, {72,5260}, {78,11529}, {79,1320}, {104,24475}, {145,388}, {214,3337}, {226,5086}, {354,8261}, {355,33592}, {390,2098}, {404,5902}, {484,4757}, {517,3651}, {518,15988}, {519,5178}, {551,15671}, {644,3970}, {908,6738}, {938,26129}, {942,4511}, {946,10707}, {950,5057}, {952,1389}, {960,5284}, {1043,17164}, {1125,4867}, {1159,5687}, {1201,3315}, {1210,31272}, {1280,1432}, {1317,12913}, {1385,16139}, {1388,5427}, {1392,10308}, {1434,17136}, {1466,18419}, {1476,5083}, {1770,11015}, {1837,31053}, {1844,17519}, {1999,10474}, {2136,3340}, {2287,2294}, {2550,20013}, {2646,3218}, {2771,7984}, {2795,7983}, {3057,3957}, {3219,3962}, {3242,28369}, {3243,16133}, {3336,13587}, {3339,4855}, {3485,11680}, {3486,5905}, {3488,11415}, {3555,4861}, {3616,5730}, {3622,5289}, {3632,6701}, {3635,5441}, {3648,15174}, {3652,28461}, {3671,20292}, {3681,11523}, {3689,10107}, {3753,4420}, {3812,9342}, {3871,5903}, {3872,8000}, {3885,25415}, {3924,32911}, {3935,5836}, {3940,9780}, {4018,24929}, {4067,5251}, {4134,32635}, {4188,5221}, {4360,17220}, {4430,12513}, {4463,17016}, {4647,4720}, {4666,15829}, {4673,20929}, {4880,5267}, {4881,32636}, {4930,15670}, {5047,5692}, {5048,17637}, {5180,15171}, {5204,23958}, {5249,6737}, {5428,10246}, {5440,31794}, {5499,5844}, {5506,17547}, {5554,25568}, {5603,6841}, {5691,31164}, {5693,6912}, {5694,6920}, {5794,31019}, {5835,33175}, {5855,15888}, {5883,17531}, {5884,6909}, {5885,6940}, {5904,30147}, {6224,18990}, {6265,6583}, {6326,6915}, {6831,9803}, {6986,31806}, {7354,17483}, {7966,7982}, {7971,10864}, {7990,11224}, {8148,16117}, {8422,16147}, {9528,10701}, {9963,11552}, {10021,10283}, {10032,15677}, {10057,12531}, {10176,17536}, {10247,13743}, {10523,11681}, {10572,16155}, {10595,11240}, {10609,24470}, {10890,16124}, {10950,20060}, {11239,12245}, {11518,19861}, {11551,17647}, {12560,25722}, {12653,13146}, {12709,16465}, {13465,31649}, {13607,16113}, {14110,18444}, {14563,21075}, {15178,22937}, {16141,33176}, {17541,30139}, {17605,20288}, {18398,30144}, {19919,28453}, {20040,32922}, {20586,33667}, {21285,33949}, {21740,24474}, {23345,28217}, {23536,26729}, {24391,24541}, {27714,31247}, {28212,31651}, {31663,33595}

X(34195) = midpoint of X(i) and X(j) for these {i,j}: {1, 16126}, {145, 2475}, {7982, 16132}, {8148, 16117}, {12653, 13146}
X(34195) = reflection of X(i) in X(j) for these {i,j}: {{8, 442}, {21, 1}, {355, 33592}, {442, 16137}, {2475, 3649}, {3651, 33858}, {10032, 15677}, {11684, 21}, {13465, 31649}, {15680, 10543}, {16139, 1385}, {21677, 11281}, {22937, 15178}, {33557, 16132}
X(34195) = anticomplement of X(21677)
X(34195) = X(17097)-anticomplementary conjugate of X(1330)
X(34195) = barycentric product X(81)*X(27690)
X(34195) = barycentric quotient X (27690)/X(321)
X(34195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2650, 81}, {1, 3868, 2975}, {1, 3869, 1621}, {1, 3894, 8666}, {1, 3901, 993}, {1, 11520, 3873}, {1, 11533, 1962}, {1, 11682, 3890}, {1, 12559, 3868}, {10, 26725, 31254}, {942, 4511, 5253}, {1385, 16139, 21161}, {2646, 3218, 5303}, {3244, 11009, 1320}, {3339, 4855, 9352}, {3340, 3870, 14923}, {3485, 12649, 11680}, {3647, 5426, 21}, {3648, 20057, 15174}, {5692, 30143, 5047}, {5730, 15934, 3616}, {5902, 22836, 404}, {6326, 31870, 6915}, {6737, 12563, 5249}, {11281, 21677, 2}, {11523, 19860, 3681}


X(34196) =  REFLECTION OF X(1) IN X(6011)

Barycentrics    a*(a^9 - 3*a^7*b^2 + a^6*b^3 + 3*a^5*b^4 - 3*a^4*b^5 - a^3*b^6 + 3*a^2*b^7 - b^9 - 8*a^7*b*c + 7*a^6*b^2*c + 8*a^5*b^3*c - 14*a^4*b^4*c + 4*a^3*b^5*c + 3*a^2*b^6*c - 4*a*b^7*c + 4*b^8*c - 3*a^7*c^2 + 7*a^6*b*c^2 + 3*a^5*b^2*c^2 + a^4*b^3*c^2 - a^3*b^4*c^2 - 7*a^2*b^5*c^2 + a*b^6*c^2 - b^7*c^2 + a^6*c^3 + 8*a^5*b*c^3 + a^4*b^2*c^3 - 16*a^3*b^3*c^3 + 9*a^2*b^4*c^3 + 4*a*b^5*c^3 - 11*b^6*c^3 + 3*a^5*c^4 - 14*a^4*b*c^4 - a^3*b^2*c^4 + 9*a^2*b^3*c^4 - 2*a*b^4*c^4 + 9*b^5*c^4 - 3*a^4*c^5 + 4*a^3*b*c^5 - 7*a^2*b^2*c^5 + 4*a*b^3*c^5 + 9*b^4*c^5 - a^3*c^6 + 3*a^2*b*c^6 + a*b^2*c^6 - 11*b^3*c^6 + 3*a^2*c^7 - 4*a*b*c^7 - b^2*c^7 + 4*b*c^8 - c^9) : :
X(34196) = 3 X[165] - 2 X[759], 3 X[1699] - 4 X[31845]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29448.

X(34196) lies on the Bevan circle and these lines: {1,6011}, {3,1054}, {20,1768}, {40,21381}, {165,759}, {573,5540}, {1365,1697}, {1695,6044}, {1699,31845}, {1742,5539}, {3579,14663}, {5535,16528}, {7991,9904}

X(34196) = reflection of X(i) in X(j) for these {i,j}: {1, 6011}, {14663, 3579}, {21381, 40}
X(34196) = excentral isogonal conjugate of X (758)


X(34197) =  EULER LINE INTERCEPT OF X(5961)X(25739)

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

See Tran Quang Hung and Peter Moses, Hyacinthos 29450.

X(34197) lies on these lines: {2, 3}, {5961, 25739}, {12219, 13557}

X(34197) = {X(186),X(2071)}-harmonic conjugate of X(17511)


X(34198) =  MIDPOINT OF X(946) AND X(5506)

Barycentrics    2 a^7-a^6 b-6 a^5 b^2+3 a^4 b^3+6 a^3 b^4-3 a^2 b^5-2 a b^6+b^7-a^6 c-6 a^5 b c+14 a^4 b^2 c+29 a^3 b^3 c-12 a^2 b^4 c-23 a b^5 c-b^6 c-6 a^5 c^2+14 a^4 b c^2-22 a^3 b^2 c^2+15 a^2 b^3 c^2+2 a b^4 c^2-3 b^5 c^2+3 a^4 c^3+29 a^3 b c^3+15 a^2 b^2 c^3+46 a b^3 c^3+3 b^4 c^3+6 a^3 c^4-12 a^2 b c^4+2 a b^2 c^4+3 b^3 c^4-3 a^2 c^5-23 a b c^5-3 b^2 c^5-2 a c^6-b c^6+c^7 : :
Barycentrics    (52 a R^2-52 b R^2+8 a SB-4 b SB-13 c SB+8 a SC-13 b SC-4 c SC-8 a SW+13 b SW)S^2 -58 R S^3-34 R S SB SC+13 b SB SC^2-13 c SB SC^2-13 b SB SC SW : :
X(34198) = X[946]+X[5506]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29452.

X(34198) lies on these lines: {946,5506}, {1158,3306}, {3628,6684}, {19919,33592}

X(34198) = midpoint of X(946) and X(5506)


X(34199) =  (name pending)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 29455 and HG270819.

X(34199) lies on this line: {2,3}


X(34200) =  EULER LINE INTERCEPT OF X(36)X(15170)

Barycentrics    14 a^4- 13 a^2 (b^2 + c^2) - (b^2 - c^2)^2 : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 29455 and HG270819.

X(34200) lies on these lines: {2,3}, {36,15170}, {143,17704}, {146,15042}, {165,3655}, {511,20583}, {519,8688}, {524,14810}, {541,11694}, {542,3631}, {551,17502}, {597,17508}, {962,32533}, {1125,28202}, {1992,33750}, {2549,5585}, {3058,7280}, {3098,3629}, {3163,22052}, {3241,4935}, {3244,3579}, {3576,28212}, {3582,15338}, {3584,15326}, {3626,28204}, {3632,3654}, {3636,13624}, {3653,22791}, {3656,7987}, {3679,28224}, {3828,28160}, {3982,5719}, {4031,24929}, {4995,18990}, {5010,5434}, {5023,7739}, {5092,6329}, {5204,15172}, {5206,5306}, {5210,15048}, {5298,15171}, {5305,15513}, {5309,8588}, {5447,31834}, {5493,31666}, {5655,13392}, {5657,17063}, {5690,16192}, {5892,13451}, {5901,12512}, {5907,11592}, {6154,7688}, {6390,7811}, {6449,19054}, {6450,19053}, {6452,9541}, {6455,19117}, {6456,19116}, {6459,6497}, {6460,6496}, {6500,9543}, {6684,28208}, {7753,8589}, {7880,32459}, {8182,8716}, {9143,15041}, {9167,22505}, {9729,14449}, {9778,10283}, {9880,26614}, {10164,28186}, {10165,28178}, {10168,29181}, {10178,14988}, {10263,16226}, {10990,11693}, {11178,21167}, {11179,31884}, {11230,28182}, {11231,28190}, {11645,33751}, {12006,21849}, {12041,24981}, {12244,22251}, {12702,20057}, {13340,20791}, {13348,13630}, {13353,13482}, {13391,16836}, {13421,15012}, {14128,14641}, {14830,21166}, {14855,15067}, {15040,22250}, {15808,31730}, {16881,21969}, {18583,19924}, {19883,22793}, {20582,29012}, {28216,31162}, {28228,31662}, {31805,31835}, {32006,32887}, {32448,33706}, {32450,32516}

X(34200) = midpoint of X(i) and X(j) for these {i,j}: {2, 550}, {3, 8703}, {4, 19710}, {5, 3534}, {20, 3845}, {140, 15690}, {376, 549}, {381, 15686}, {547, 15691}, {548, 12100}, {632, 15697}, {1657, 33699}, {3522, 15711}, {3594, 17827}, {3627, 11001}, {3830, 15704}, {5066, 12103}, {5655, 14677}, {9778, 10283}, {11539, 15689}, {14093, 15714}, {14855, 15067}, {15681, 15687}, {15688, 17504}, {15695, 15712}, {15696, 15713}, {15759, 33923}, {16190, 29318}, {23745, 550}, {32448, 33706}
X(34200) = reflection of X(i) in X(j) for these {i,j}: {2, 3530}, {3, 15759}, {4, 10109}, {5, 11812}, {140, 12100}, {381, 10124}, {546, 2}, {547, 549}, {548, 8703}, {549, 14891}, {3545, 14890}, {3627, 3860}, {3830, 3850}, {3845, 3628}, {3850, 11540}, {3853, 5066}, {3860, 16239}, {5066, 140}, {5655, 13392}, {8703, 33923}, {10109, 12108}, {12100, 3}, {12101, 5}, {12103, 15690}, {13451, 5892}, {14892, 5054}, {14893, 547}, {15682, 12102}, {15687, 11737}, {15690, 548}, {15691, 376}, {21849, 12006}, {21969, 16881}, {25338, 18579}, {33591, 18324}, {33699, 3861}
X(34200) = complement of X(15687)
X(34200) = anticomplement of X(11737)


X(34201) =  X(3)X(1129)∩X(35)X(259)

Barycentrics    Sin[A] (1 + 2 Cos[A / 2] + 2 Cos[A]) : :
Trilinears    sin(5A/4) csc(A/4) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29458.

X(34201) lies on these lines: {3,1129}, {35,259}, {55,10232}

X(34201) = isogonal conjugate of X(34202)
X(34201) = Hofstadter 5/4 point


X(34202) =  ISOGONAL CONJUGATE OF X(34201)

Barycentrics    Sin[A] / (1 + 2 Cos[A / 2] + 2 Cos[A]) : :
Trilinears    sin(A/4) csc(5A/4) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29458.

X(34202) lies on the conic {A,B,C,H,X(1127)} and these lines: {}

X(34202) = isogonal conjugate of X(34201)
X(34202) = Hofstadter -1/4 point


X(34203) =  X(2)X(6)∩X(6088)X(12149)

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

X(34203) lies on the cubics K089 and K686 and on these lines: {2,6}, {6088,12149}, {9146,9148}

X(34203) = isogonal conjugate of X(34204)
X(34203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14607, 23342, 2421}.


X(34204) =  ISOGONAL CONJUGATE OF X(34203)

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

X(34204) lies on the circumconic {{A,B,C,X(2),X(6)}}, the cubics K408 and K686, and on these lines: {2,8644}, {669,21448}, {694,9135}

X(34204) = isogonal conjugate of X(34203)


X(34205) =  X(2)X(187)∩X(99)X(523)

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

X(34205) lies on the cubic K088 and these lines: {2, 187}, {99, 523}, {843, 5182}, {5468, 9181}, {5914, 9166}, {9146, 9218}

X(34205) = crossdifference of every pair of points on line {17414, 21906}
X(34205) = {X(7472),X(9182)}-harmonic conjugate of X(99)


X(34206) =  X(2)X(23287)∩X(115)X(23288)

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

X(34206) lies on the cubic K088 and these lines: {2, 23287}, {115, 23288}, {599, 690}, {5094, 14273}, {10130, 22105}

X(34206) = isotomic conjugate of X(34207)
X(34206) = trilinear pole of line {1648, 3906}


X(34207) =  X(22)X(69)∩X(25)X(66)

Barycentrics    a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - b^2*c^4 - c^6)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 + b^2*c^4 + c^6) : :
Barycentrics    a^2/((a^2 + b^2 + c^2) sin 2A + (c^2 - b^2 - a^2) sin 2B + (b^2 - c^2 - a^2) sin 2C) : :

For a construction see Francisco Javier García Capitán, Euclid 777

X(34207) lies on the Jerabek circumhyperbola, the cubics K161, K174, K178, and these lines: {3, 206}, {4, 9914}, {6, 17409}, {22, 69}, {23, 20079}, {25, 66}, {54, 32357}, {67, 10117}, {68, 1503}, {71, 21034}, {72, 3556}, {73, 10831}, {248, 1609}, {265, 9919}, {511, 9908}, {895, 32262}, {924, 2435}, {1176, 19153}, {1177, 13171}, {1351, 15317}, {1593, 14542}, {1853, 15321}, {2393, 6391}, {2781, 5504}, {3357, 19137}, {3519, 9920}, {3580, 18124}, {3589, 4846}, {3827, 28787}, {5020, 6697}, {5093, 15002}, {5486, 19459}, {5621, 11744}, {6000, 23044}, {7484, 31267}, {7503, 15740}, {17040, 19119}, {18125, 26284}

The trilinear polar of X(34207) passes through X(647).

X(34207) is the perspector of ABC and the reflection of the anticevian triangle of X(6) (i.e., the tangential triangle) in X(206). (Let A'BC' = anticevian triangle of X(6); then X(206) is the centroid of {X(6), A', B', C'}.) (Randy Hutson, October 8, 2019)

X(34207) = isogonal conjugate of X(1370)
X(34207) = isotomic conjugate of X(34206)
X(34207) = isogonal conjugate of the anticomplement of X(25)
X(34207) = isogonal conjugate of the complement of X(7500)
X(34207) = isogonal conjugate of the isotomic conjugate of X(13575)
X(34207) = X(i)-cross conjugate of X(j) for these (i,j): {1974, 6}, {2353, 25}
X(34207) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1370}, {2, 18596}, {6, 21582}, {9, 18629}, {19, 28419}, {75, 159}, {92, 23115}, {304, 3162}, {1930, 8793}
X(34207) = cevapoint of X(669) and X(3269)
X(34207) = crosssum of X(i) and X(j) for these (i,j): {159, 23115}, {455, 3162}
X(34207) = barycentric product X(6)*X(13575)
X(34207) = barycentric quotient X (i)/X(j) for these {i,j}: {1, 21582}, {3, 28419}, {6, 1370}, {31, 18596}, {32, 159}, {56, 18629}, {184, 23115}, {1974, 3162}, {13575, 76}
X(34207) = {X(22),X(5596)}-harmonic conjugate of X(159)


X(34208) =  X(4)X(193)∩X(69)X(8754)

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

X(34208) lies on the circumhyperbola {{A,B,C,X(4),X(93)}}, the cubics K181, K616, K675, and K1046, and on these lines: {4, 193}, {69, 8754}, {225, 1738}, {230, 393}, {254, 631}, {264, 1007}, {376, 1300}, {378, 18852}, {403, 16326}, {419, 33630}, {491, 24244}, {492, 24243}, {847, 3090}, {1093, 6622}, {1249, 6531}, {1594, 18854}, {1826, 4028}, {2970, 16051}, {3068, 8940}, {3069, 8944}, {3541, 18853}, {3563, 7612}, {5254, 17040}, {6526, 6530}, {7714, 32085}, {8801, 9766}, {17907, 17983}, {18560, 18849}

X(34208) = isogonal conjugate of X(3167)
X(34208) = isotomic conjugate of X(6337)
X(34208) = polar conjugate of X(193)
X(34208) = isotomic conjugate of the anticomplement of X(13881)
X(34208) = isotomic conjugate of the complement of X(2996)
X(34208) = isotomic conjugate of the isogonal conjugate of X(14248)
X(34208) = polar conjugate of the isotomic conjugate of X(2996)
X(34208) = polar conjugate of the isogonal conjugate of X(8770)
X(34208) = X(i)-cross conjugate of X(j) for these (i,j): {2, 4}, {5013, 8796}, {5139, 2501}, {5254, 2052}, {8770, 2996}, {13881, 2}
X(34208) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3167}, {3, 1707}, {19, 10607}, {31, 6337}, {48, 193}, {63, 3053}, {184, 18156}, {212, 17081}, {255, 6353}, {326, 19118}, {896, 6091}, {906, 3798}, {1437, 4028}, {1790, 21874}, {3566, 4575}, {3787, 34055}, {4100, 21447}, {4592, 8651}
X(34208) = cevapoint of X(i) and X(j) for these (i,j): {2, 2996}, {523, 8754}, {2501, 5139}, {8770, 14248}
X(34208) = trilinear pole of line {2501, 3566} (the perspectrix of the inner and outer Vecten triangles, and the radical axis of the 1st & 2nd Dao-Vecten circles)
X(34208) = barycentric product X(i)*X(j) for these {i,j}: {4, 2996}, {76, 14248}, {92, 8769}, {264, 8770}, {275, 27364}, {393, 6340}, {671, 5203}, {2052, 6391}, {3565, 14618}
X(34208) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 6337}, {3, 10607}, {4, 193}, {6, 3167}, {19, 1707}, {25, 3053}, {92, 18156}, {111, 6091}, {278, 17081}, {393, 6353}, {468, 32459}, {1093, 21447}, {1824, 21874}, {1826, 4028}, {1843, 3787}, {2207, 19118}, {2489, 8651}, {2501, 3566}, {2996, 69}, {3565, 4558}, {5139, 15525}, {5203, 524}, {6340, 3926}, {6353, 439}, {6391, 394}, {7649, 3798}, {8754, 6388}, {8769, 63}, {8770, 3}, {14248, 6}, {27364, 343}
X(34208) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2996, 14248, 4}, {6391, 27364, 2996}


X(34209) =  X(5)X(523)∩X(30)X(74)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - b^6*c^2 - 3*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + 3*a^2*c^6 - b^2*c^6 - c^8) : :
X(34209) = 2 X[5] - 3 X[21315], 3 X[110] + X[31874], X[265] - 3 X[5627], X[265] + 3 X[14993], X[476] + 3 X[5627], X[476] - 3 X[14993], X[477] - 3 X[15061], X[1553] + 3 X[6070], X[1553] - 3 X[25641], X[9158] - 3 X[15362], 2 X[12079] + X[18319], X[14480] - 3 X[14643], X[14508] - 3 X[20126], 3 X[14643] - 2 X[33505], 3 X[14644] - X[20957], X[14731] - 5 X[15081], 2 X[31379] - 3 X[34128]

X(34209) lies on the cubics K187 and K741 and on these lines: {4, 21316}, {5, 523}, {13, 11549}, {14, 11537}, {30, 74}, {94, 10688}, {110, 31874}, {125, 16168}, {140, 14934}, {403, 6344}, {477, 15061}, {550, 21317}, {1141, 10096}, {1511, 22104}, {1522, 1523}, {1539, 32417}, {1989, 16303}, {2070, 31676}, {3258, 20304}, {3845, 14583}, {5961, 15646}, {7471, 32423}, {7575, 11657}, {9158, 15362}, {10272, 14611}, {14140, 14980}, {14480, 14643}, {14644, 20957}, {14731, 15081}, {18300, 18323}, {18478, 18572}, {31379, 34128}

X(34209) = midpoint of X(i) and X(j) for these {i,j}: {265, 476}, {5627, 14993}, {6070, 25641}, {10264, 18319}, {14989, 20127}
X(34209) = reflection of X(i) in X(j) for these {i,j}: {4, 21316}, {1511, 22104}, {3258, 20304}, {7575, 11657}, {10264, 12079}, {14480, 33505}, {14611, 10272}, {14934, 140}, {16340, 125}, {21269, 34150}, {21317, 550}
X(34209) = X(i)-isoconjugate of X(j) for these (i,j): {163, 2411}, {477, 6149}, {662, 2436}, {2624, 30528}
X(34209) = cevapoint of X(30) and X(33505)
X(34209) = crossdifference of every pair of points on line {50, 2436}
X(34209) = barycentric product X(i)*X(j) for these {i,j}: {94, 5663}, {523, 2410}, {850, 2437}, {7480, 14592}
X(34209) = barycentric quotient X(i)/X(j) for these {i,j}: {476, 30528}, {512, 2436}, {523, 2411}, {1989, 477}, {2410, 99}, {2437, 110}, {5663, 323}, {7480, 14590}, {11251, 14920}, {14582, 14220}
X(34209) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {265, 14993, 476}, {476, 5627, 265}, {14480, 14643, 33505}, {14670, 18279, 5}


X(34210) =  X(30)X(110)∩X(54)X(14220)

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

X(34210) lies on the cubic K187 and these lines: {30, 110}, {54, 14220}, {74, 16186}, {526, 14385}, {2411, 14355}, {6148, 10411}, {14910, 15291}, {15035, 15396}, {15462, 30528}

X(34210) = X(7740)-cross conjugate of X(186)
X(34210) = X(i)-isoconjugate of X(j) for these (i,j): {661, 2410}, {1577, 2437}, {2166, 5663}
X(34210) = crosssum of X(30) and X(33505)
X(34210) = trilinear pole of line {50, 2436}
X(34210) = barycentric product X(i)*X(j) for these {i,j}: {99, 2436}, {110, 2411}, {323, 477}, {340, 32663}, {526, 30528}, {14220, 14590}
X(34210) = barycentric quotient X(i)/X(j) for these {i,j}: {50, 5663}, {110, 2410}, {477, 94}, {1576, 2437}, {2088, 6070}, {2411, 850}, {2436, 523}, {14220, 14592}, {14591, 7480}, {32663, 265}


X(34211) =  X(2)X(6)∩X(107)X(110)

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

X(34211) lies on the cubic K231 and these lines: {2, 6}, {20, 18338}, {107, 110}, {525, 2420}, {671, 10733}, {877, 17932}, {879, 2966}, {2409, 2445}, {4558, 4576}, {5889, 6179}, {6793, 15595}, {7760, 34148}, {8779, 30737}, {10330, 14570}

X(34211) = reflection of X(4235) in X(2420)
X(34211) = isotomic conjugate of the polar conjugate of X(2409)
X(34211) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {163, 14721}, {685, 21294}, {32696, 21221}
X(34211) = X(i)-Ceva conjugate of X(j) for these (i,j): {877, 4226}, {17932, 110}
X(34211) = X(i)-isoconjugate of X(j) for these (i,j): {19, 2435}, {647, 8767}, {661, 1297}, {810, 6330}, {1973, 2419}, {2632, 32687}, {20902, 32649}
X(34211) = crosspoint of X(648) and X(2966)
X(34211) = crosssum of X(647) and X(3569)
X(34211) = trilinear pole of line {441, 1503}
X(34211) = crossdifference of every pair of points on line {512, 3269}
X(34211) = barycentric product X(i)*X(j) for these {i,j}: {69, 2409}, {99, 1503}, {110, 30737}, {132, 17932}, {305, 2445}, {326, 24024}, {441, 648}, {799, 2312}, {811, 8766}, {877, 34156}, {2966, 15595}, {3926, 23977}, {4563, 16318}, {4576, 21458}, {6331, 8779}
X(34211) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 2435}, {69, 2419}, {110, 1297}, {132, 16230}, {162, 8767}, {441, 525}, {648, 6330}, {1503, 523}, {2312, 661}, {2409, 4}, {2445, 25}, {2966, 9476}, {6793, 1637}, {8766, 656}, {8779, 647}, {9475, 3569}, {15595, 2799}, {15639, 16318}, {16318, 2501}, {23964, 32687}, {23977, 393}, {24024, 158}, {28343, 2492}, {30737, 850}, {34156, 879}
X(34211) = {X(2407),X(14999)}-harmonic conjugate of X(5468)


X(34212) =  X(2)X(2419)∩X(6)X(520)

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

X(34212) lies on the circumconic {{A,B,C,X(2),X(6)}}, the cubic K231, and these lines: {2, 2419}, {6, 520}, {25, 647}, {111, 1297}, {251, 16040}, {263, 9210}, {393, 523}, {868, 2395}, {1304, 32687}, {1637, 8791}, {1976, 3569}, {2492, 8749}, {2501, 13854}, {6330, 16081}, {8882, 23286}, {23347, 32649}

X(34212) = polar conjugate of the isotomic conjugate of X(2435)
X(34212) = X(i)-cross conjugate of X(j) for these (i,j): {2508, 18105}, {17994, 523}
X(34212) = X(i)-isoconjugate of X(j) for these (i,j): {63, 2409}, {99, 2312}, {162, 441}, {163, 30737}, {304, 2445}, {326, 23977}, {394, 24024}, {648, 8766}, {662, 1503}, {811, 8779}, {2715, 17875}, {4592, 16318}
X(34212) = cevapoint of X(647) and X(3569)
X(34212) = trilinear pole of line {512, 3269}
X(34212) = crossdifference of every pair of points on line {441, 1503}
X(34212) = barycentric product X(i)*X(j) for these {i,j}: {4, 2435}, {25, 2419}, {339, 32649}, {523, 1297}, {647, 6330}, {656, 8767}, {3569, 9476}, {15407, 16230}, {15526, 32687}
X(34212) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 2409}, {512, 1503}, {523, 30737}, {647, 441}, {798, 2312}, {810, 8766}, {878, 34156}, {1096, 24024}, {1297, 99}, {1974, 2445}, {2207, 23977}, {2419, 305}, {2435, 69}, {2489, 16318}, {2491, 9475}, {3049, 8779}, {3569, 15595}, {6330, 6331}, {8767, 811}, {14398, 6793}, {15407, 17932}, {17994, 132}, {18105, 21458}, {32649, 250}, {32687, 23582}


X(34213) =  ISOTOMIC CONJUGATE OF X(6031)

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

X(34213) lies on the cubic K287 and these lines: {599, 8705}, {1383, 6325}, {3734, 7492}

X(34213) = isotomic conjugate of X(6031)
X(34213) = isotomic conjugate of the anticomplement of X(6032)
X(34213) = isotomic conjugate of the isogonal conjugate of X(30488)
X(34213) = X(6032)-cross conjugate of X (2)
X(34213) = X(31)-isoconjugate of X(6031)
X(34213) = barycentric product X(76)*X(30488)
X(34213) = barycentric quotient X (i)/X(j) for these {i,j}: {2, 6031}, {30488, 6}


X(34214) =  ISOGONAL CONJUGATE OF X(5989)

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

X(34214) lies on the cubics K789, K1000, K1131, and these lines: {2, 9467}, {147, 511}, {237, 2076}, {1691, 19556}, {2211, 32748}, {2698, 9862}, {3094, 14251}, {3098, 34157}, {5207, 14603}, {20022, 25332}

X(34214) = isogonal conjugate of X(5989)
X(34214) = anticomplement of X(9467)
X(34214) = anticomplement of the isogonal conjugate of X(9469)
X(34214) = isotomic conjugate of the anticomplement of X(9468)
X(34214) = antigonal conjugate of X(34238)
X(34214) = X(9469)-anticomplementary conjugate of X(8)
X(34214) = X(9468)-cross conjugate of X(2)
X(34214) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5989}, {75, 3506}, {1966, 9467}
X(34214) = cevapoint of X(512) and X(2679)
X(34214) = crosssum of X(i) and X(j) for these (i,j): {147, 8782}, {4027, 8784}
X(34214) = trilinear pole of line {2491, 5113}
X(34214) = barycentric product X(694)*X(9469)
X(34214) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5989}, {32, 3506}, {9468, 9467}, {9469, 3978}


X(34215) =  X(4)X(175)∩X(8)X(492)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Anthrakitis120111 and Peter Moses Hyacinthos 29466.

X(34215) lies on the Feuerbach circumhyperbola and these lines: {2, 13426}, {4, 175}, {8, 492}, {9, 3084}, {79, 481}, {80, 31539}, {176, 3296}, {354, 7269}, {482, 5557}, {1372, 5561}, {5551, 21169}, {5556, 31602}, {5558, 17805}, {5560, 17803}, {13388, 16441}, {13435, 13454}

X(34215) = X(i)-isoconjugate of X(j) for these (i,j): {55, 481}, {10253, 13427}, {30336, 32083}
X(34215) = cevapoint of X(1) and X(13388)
X(34215) = trilinear pole of line {650, 6365}
X(34215) = barycentric product X(7)*X(15889)
X(34215) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 481}, {493, 26495}, {7969, 19030}, {13388, 31534}, {15889, 8}
X(34215) = {X(354),X(7269)}-harmonic conjugate of X(34216)


X(34216) =  X(4)X(176)∩X(8)X(491)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Anthrakitis120111 and Peter Moses Hyacinthos 29466.

X(34216) lies on the Feuerbach circumhyperbola and these lines: {2, 13454}, {4, 176}, {8, 491}, {9, 3083}, {79, 482}, {80, 31538}, {175, 3296}, {354, 7269}, {481, 5557}, {1371, 5561}, {5556, 31601}, {5558, 17802}, {5560, 17806}, {7133, 13389}, {13424, 13426}

X(34216) = isotomic conjugate of the anticomplement of X(8965)
X(34216) = X(i)-cross conjugate of X(j) for these (i,j): {8965, 2}, {8978, 81}
X(34216) = X(i)-isoconjugate of X(j) for these (i,j): {55, 482}, {10252, 13456}, {30335, 32082}
X(34216) = cevapoint of X(1) and X(13389)
X(34216) = trilinear pole of line {650, 6364}
X(34216) = barycentric product X(7)*X(15890)
X(34216) = barycentric quotient X(i)/X(j) for these {i,j}: {57, 482}, {494, 26504}, {7968, 19029}, {13389, 31535}, {15890, 8}
X(34216) = {X(354),X(7269)}-harmonic conjugate of X(34215)


X(34217) =  SINGULAR FOCUS OF THE CUBIC K1129

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

X(34217) = 3 X[3] + X[11641], 2 X[3] + X[15562], X[112] - 3 X[14649], 2 X[11641] - 3 X[15562], X[11641] - 3 X[19165], X[12413] - 5 X[16195]

X(34217) lies on these lines: {3, 114}, {24, 132}, {39, 11610}, {112, 186}, {206, 1511}, {511, 14574}, {549, 23320}, {827, 1297}, {1485, 5961}, {1658, 5171}, {2070, 12918}, {2799, 14270}, {3520, 10735}, {6636, 7711}, {7512, 19164}, {7556, 12253}, {8723, 9517}, {9530, 14070}, {9590, 12784}, {9682, 13923}, {10298, 13219}, {12413, 16195}, {13200, 21844}, {14900, 32534}, {18570, 19163}

X(34217) = singular focus of the cubic K1129
X(34217) = midpoint of X(3) and X(19165)
X(34217) = reflection of X(15562) in X(19165)
X(34217) = circumcircle-inverse of X(6033)
X(34217) = circumperp conjugate of X(38741)


X(34218) =  MIDPOINT OF X(3) AND X(7669)

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 + 3*a^6*b^2*c^2 - 2*a^2*b^6*c^2 + 2*b^8*c^2 + 4*a^6*c^4 - b^6*c^4 - 4*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(34218) lies on these lines: {3, 67}, {5, 15562}, {24, 8754}, {50, 11649}, {110, 14652}, {157, 3818}, {182, 2871}, {231, 1989}, {511, 1976}, {1576, 18114}, {2393, 22463}, {2872, 9126}, {3148, 5476}, {3425, 11187}, {5191, 19140}, {5961, 13289}, {6036, 19165}, {7575, 34010}, {9142, 9976}, {10282, 32734}, {11616, 25644}, {13321, 33886}, {13335, 14649}, {14060, 15462}

X(34218) = midpoint of X(3) and X(7669)
X(34218) = reflection of X(32734) in X(10282)
X(34218) = circumcircle-inverse of X(15545)
X(34218) = crosssum of X(i) and X(j) for these (i,j): {526, 868}, {2072, 3564}
X(34218) = crossdifference of every pair of points on line {2492, 8562}
X(34218) = {X(1576),X(18114)}-harmonic conjugate of X(34155)


X(34219) =  ISOGONAL CONJUGATE OF X(32627)

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

X(34219) lies on the cubics K440 and K1132b and on these lines: {15, 5617}, {61, 15609}, {511, 11600}, {635, 10409}, {3448, 13483}

X(34219) = reflection of X(i) in X(j) for these {i,j}: {61, 15609}, {10409, 635}
X(34219) = isogonal conjugate of X(32627)
X(34219) = antigonal image of X(61)
X(34219) = symgonal image of X(635)


X(34220) =  ISOGONAL CONJUGATE OF X(32628)

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

X(34220) lies on the cubics K440 and K1132a and on these lines: {16, 5613}, {62, 15610}, {511, 11601}, {636, 10410}, {3448, 13484}

X(34220) = reflection of X(i) in X(j) for these {i,j}: {62, 15610}, {10410, 636}
X(34220) = isogonal conjugate of X(32628)
X(34220) = antigonal image of X(62)
X(34220) = symgonal image of X(636)


X(34221) =  ISOGONAL CONJUGATE OF X(5004)

Barycentrics    a^2/(a*b*c*S - Sqrt[2]*a^2*SA*Sqrt[SW]) : :

X(34221) lies on the Jerabek circumhyperbola, the cubics K442 and K570, and on these lines: {51, 125}, {1176, 5005}

X(34221) = isogonal conjugate of X(5004)
X(34221) = reflection of X(34222) in X(125)
X(34221) = antipode in Jerabek hyperbola of X(34222)
X(34221) = {X(51),X(5480)}-harmonic conjugate of X(34222)


X(34222) =  ISOGONAL CONJUGATE OF X(5005)

Barycentrics    a^2/(a*b*c*S + Sqrt[2]*a^2*SA*Sqrt[SW]) : :

X(34222) lies on the Jerabek circumhyperbola, the cubics K442 and K570, and on these lines: {51, 125}, {1176, 5004}

X(34222) = isogonal conjugate of X(5005)
X(34222) = reflection of X(34221) in X(125)
X(34222) = antipode in Jerabek hyperbola of X(34221)
X(34222) = {X(51),X(5480)}-harmonic conjugate of X(34221)


X(34223) =  (name pending)

Barycentrics    (a^8 - 3*a^6*b^2 + 4*a^4*b^4 - 3*a^2*b^6 + b^8 - 4*a^6*c^2 - 6*a^4*b^2*c^2 - 6*a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 13*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 3*a^6*c^2 - 6*a^4*b^2*c^2 + 13*a^2*b^4*c^2 - 4*b^6*c^2 + 4*a^4*c^4 - 6*a^2*b^2*c^4 + 6*b^4*c^4 - 3*a^2*c^6 - 4*b^2*c^6 + c^8) : :
Barycentrics    1/((a^2*SA*(S^2 + 5*SA^2) - (3*S^2 - SA^2)*SB*SC)) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29469.

X(34223) lies on the circumconic {{A,B,C,X(4),X(1656)}} and this line: {1656,18451}


X(34224) =  X(3)X(70)∩X(4)X(6)

Barycentrics    2*a^10-5*(b^2+c^2)*a^8+2*(2*b^4+b^2*c^2+2*c^4)*a^6-2*(b^4-c^4)*(b^2-c^2)*a^4+2*(b^6-c^6)*(b^2-c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    (SB+SC)*S^2+2*(R^2-SW)*SB*SC : :
X(34224) = 2*X(4)-3*X(12022), 7*X(4)-12*X(12024), 3*X(4)-4*X(12241), 5*X(4)-4*X(16621), 7*X(4)-6*X(16654), 3*X(4)-2*X(16655), 9*X(4)-8*X(16656), 5*X(4)-6*X(16657), 4*X(4)-3*X(16658), 3*X(51)-2*X(13419), 4*X(143)-3*X(7540), 4*X(6146)-3*X(12022), 7*X(6146)-6*X(12024), 3*X(6146)-2*X(12241), 5*X(6146)-2*X(16621), 7*X(6146)-3*X(16654), 3*X(6146)-X(16655), 9*X(6146)-4*X(16656), 5*X(6146)-3*X(16657), 8*X(6146)-3*X(16658)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29470.

Let A'B'C' be the reflection triangle. Let AB, AC be the orthogonal projections of A' on CA, AB, resp. Let A" = CAAC∩ABBA, and define B" and C" cyclically. Triangle A"B"C" is homothetic to ABC at X(6). X(34224) = X(4)-of-A"B"C". (Randy Hutson, March 29, 2020)

X(34224) lies on these lines: {2, 9707}, {3, 70}, {4, 6}, {5, 1614}, {11, 9638}, {20, 11411}, {22, 68}, {24, 1899}, {25, 18912}, {26, 3580}, {30, 5889}, {49, 13371}, {51, 13419}, {52, 10116}, {54, 427}, {74, 550}, {96, 98}, {110, 11585}, {125, 10018}, {140, 11464}, {143, 7540}, {154, 7505}, {156, 2072}, {182, 14788}, {184, 1594}, {185, 6152}, {235, 14157}, {265, 15761}, {343, 7512}, {378, 14216}, {382, 12174}, {389, 7576}, {403, 6759}, {428, 9781}, {468, 26882}, {539, 10625}, {542, 1205}, {546, 18394}, {548, 11468}, {568, 11819}, {569, 5133}, {578, 11550}, {858, 1147}, {1204, 10295}, {1209, 7495}, {1352, 7509}, {1370, 6193}, {1595, 15033}, {1656, 26864}, {1853, 19357}, {1885, 12290}, {1993, 14790}, {2888, 6636}, {2918, 2931}, {3043, 23315}, {3047, 23306}, {3060, 7553}, {3100, 12428}, {3147, 23291}, {3153, 22660}, {3515, 26944}, {3517, 26869}, {3518, 13567}, {3520, 6247}, {3541, 18925}, {3542, 11206}, {3564, 11412}, {3567, 11245}, {3575, 5890}, {3796, 7558}, {4296, 18970}, {5064, 11426}, {5422, 7528}, {5576, 14389}, {5663, 18563}, {5876, 13470}, {5944, 13561}, {6000, 18560}, {6143, 23332}, {6403, 26926}, {6515, 31305}, {6642, 18911}, {7383, 25406}, {7395, 18440}, {7403, 13434}, {7487, 18916}, {7502, 18356}, {7506, 18952}, {7507, 19347}, {7542, 23293}, {7583, 11462}, {7584, 11463}, {7667, 33523}, {7687, 14862}, {9544, 9820}, {9545, 31074}, {9730, 18128}, {9825, 15045}, {9862, 17401}, {9908, 26283}, {9920, 21284}, {10112, 29012}, {10114, 13417}, {10127, 15028}, {10192, 14940}, {10257, 11449}, {10263, 11264}, {10264, 15331}, {10574, 31833}, {10575, 17702}, {10594, 31383}, {10605, 17845}, {10619, 11430}, {11381, 13403}, {11413, 12118}, {11414, 12429}, {11441, 18531}, {11444, 31831}, {11455, 13488}, {11459, 12362}, {11461, 15171}, {11466, 11542}, {11467, 11543}, {11572, 18388}, {11645, 13598}, {11649, 11660}, {11750, 12225}, {12082, 32599}, {12111, 12605}, {12161, 31723}, {12278, 15072}, {13160, 18474}, {13171, 13564}, {13198, 32379}, {13367, 20299}, {13383, 26881}, {13491, 30522}, {13568, 18559}, {13619, 17846}, {15133, 20302}, {16238, 26913}, {16252, 16868}, {18390, 26883}, {18404, 32139}, {18445, 18569}, {18533, 18909}, {18583, 19123}, {18990, 19368}, {23335, 34148}, {26937, 32534}

X(34224) = midpoint of X(6241) and X(12289)
X(34224) = reflection of X(i) in X(j) for these (i,j): (4, 6146), (52, 10116), (382, 12370), (3575, 18914), (5876, 13470), (6152, 32377), (6240, 185), (6243, 32358), (6403, 26926), (7553, 13292), (10263, 11264), (11381, 13403), (12111, 12605), (12225, 11750), (12290, 1885), (13417, 10114), (14516, 3), (16654, 12024), (16655, 12241), (16658, 12022), (16659, 4), (18560, 21659)
X(34224) = isogonal conjugate of X(34225)
X(34224) = anticomplement of X(12134)
X(34224) = X(6146)-of-anti-Euler triangle
X(34224) = X(14516)-of-ABC-X3 reflections triangle
X(34224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1498, 32111), (4, 6146, 12022), (4, 6776, 7592), (4, 15032, 12233), (4, 16659, 16658), (1498, 18396, 4), (6146, 16655, 12241), (12022, 16659, 4), (12241, 16655, 4), (16621, 16657, 4)


X(34225) =  ISOGONAL CONJUGATE OF X(34224)

Barycentrics    a^2*(a^10-(2*b^2+3*c^2)*a^8+2*(b^4+b^2*c^2+c^4)*a^6-2*(2*b^6+b^4*c^2-c^6)*a^4+(b^4-c^4)*(5*b^4-2*b^2*c^2+3*c^4)*a^2-(2*b^4+b^2*c^2+c^4)*(b^2-c^2)^3)*(a^10-(3*b^2+2*c^2)*a^8+2*(b^4+b^2*c^2+c^4)*a^6+2*(b^6-b^2*c^4-2*c^6)*a^4-(b^4-c^4)*(3*b^4-2*b^2*c^2+5*c^4)*a^2+(b^4+b^2*c^2+2*c^4)*(b^2-c^2)^3) : :
Barycentrics    ((SA+SC)*S^2+2*(R^2-SW)*SA*SC)*((SA+SB)*S^2+2*(R^2-SW)*SA*SB) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29470.

X(34225) lies on these lines: {3, 8746}, {26, 394}, {52, 17974}, {97, 7512}, {1073, 7506}, {3518, 14919}, {3926, 31305}, {7528, 14376}

X(34225) = isogonal conjugate of X(34224)


X(34226) =  MIDPOINT OF X(12149) AND X(15534)

Barycentrics    a^2*(2*a^8*b^4 + 2*a^6*b^6 - 2*a^4*b^8 - 2*a^2*b^10 + 6*a^8*b^2*c^2 - 31*a^6*b^4*c^2 - 16*a^4*b^6*c^2 + 21*a^2*b^8*c^2 + 2*a^8*c^4 - 31*a^6*b^2*c^4 + 114*a^4*b^4*c^4 - 30*a^2*b^6*c^4 - 2*b^8*c^4 + 2*a^6*c^6 - 16*a^4*b^2*c^6 - 30*a^2*b^4*c^6 - 4*b^6*c^6 - 2*a^4*c^8 + 21*a^2*b^2*c^8 - 2*b^4*c^8 - 2*a^2*c^10) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29472.

X(34226) lies on these lines: {2, 2854}, {12149, 15534}

X(34226) = midpoint of X(12149) and X(15534)


X(34227) =  X(111)X(15271)∩X(126)X(3258)

Barycentrics    2*a^8 - 5*a^6*b^2 + 2*a^4*b^4 - 3*a^2*b^6 - 5*a^6*c^2 + 2*a^4*b^2*c^2 + 8*a^2*b^4*c^2 + 4*b^6*c^2 + 2*a^4*c^4 + 8*a^2*b^2*c^4 - 16*b^4*c^4 - 3*a^2*c^6 + 4*b^2*c^6 : :
X(34227) = X[11162] - 3 X[21358]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29472.

X(34227) lies on these lines: {111, 15271}, {126, 3258}, {141, 543}, {2793, 5026}, {3734, 33962}, {7761, 32424}, {7778, 10717}, {11162, 21358}


X(34228) =  REFLECTION OF X(1367) IN X(1)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29472.

X(34228) lies on the incircle and these lines: {1, 1367}, {11, 4989}, {55, 26702}, {243, 3326}, {1358, 4292}, {1364, 10391}, {1365, 1836}, {1858, 3022}, {3057, 6020}, {5137, 14027}

X(34228) = reflection of X(1367) in X(1)
X(34228) = X(7)-Ceva conjugate of X (1375)
X(34228) = crosspoint of X(7) and X (1375)
X(34248) = complement of anticomplementary conjugate of X(21223)


X(34229) =  X(2)X(6)∩X(4)X(1078)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Anthrakitis250112 and Peter Moses Hyacinthos 29474.

X(34229) lies on the circumconics {{A,B,C,X(4),X(3815)}} and {{A,B,C,X(6),X(5050)}} and these lines: {2, 6}, {3, 32815}, {4, 1078}, {5, 3785}, {20, 9756}, {30, 32885}, {32, 32968}, {39, 32978}, {76, 631}, {83, 32957}, {95, 6340}, {98, 25406}, {99, 3524}, {115, 32986}, {140, 3926}, {148, 33008}, {187, 14033}, {194, 33001}, {264, 6353}, {274, 17567}, {305, 1232}, {311, 7494}, {315, 3090}, {316, 3545}, {317, 8889}, {350, 5218}, {376, 7771}, {547, 14929}, {550, 32826}, {626, 32969}, {632, 32839}, {754, 31415}, {1235, 3147}, {1272, 7664}, {1444, 16434}, {1506, 14023}, {1656, 7767}, {1799, 7392}, {1909, 7288}, {1975, 3523}, {1995, 15574}, {2548, 7780}, {2549, 32457}, {2896, 32961}, {3053, 32971}, {3091, 7750}, {3096, 32951}, {3522, 32819}, {3525, 7763}, {3526, 3933}, {3530, 32886}, {3533, 7769}, {3628, 7776}, {3734, 21843}, {3767, 4045}, {3788, 32977}, {3793, 15484}, {3934, 14001}, {3964, 16419}, {4396, 31497}, {5013, 6392}, {5023, 32981}, {5054, 6390}, {5056, 7773}, {5067, 7752}, {5071, 7811}, {5077, 16509}, {5206, 33239}, {5254, 32990}, {5286, 11285}, {5319, 6683}, {5569, 32456}, {6148, 30786}, {6194, 7616}, {6292, 33221}, {6376, 30478}, {6722, 7865}, {6781, 8182}, {6811, 12323}, {6813, 12322}, {6857, 18140}, {6910, 18135}, {7410, 10449}, {7484, 9723}, {7493, 26235}, {7622, 14148}, {7737, 32983}, {7738, 7824}, {7739, 15482}, {7745, 32987}, {7746, 7800}, {7748, 33226}, {7749, 7795}, {7751, 31401}, {7754, 31400}, {7758, 31455}, {7761, 16041}, {7762, 31404}, {7768, 32823}, {7782, 10299}, {7783, 33012}, {7784, 32972}, {7785, 32999}, {7789, 32989}, {7793, 16924}, {7799, 15709}, {7803, 32960}, {7810, 32984}, {7820, 33224}, {7823, 32962}, {7828, 32956}, {7830, 33238}, {7831, 33190}, {7832, 18840}, {7836, 33000}, {7844, 33223}, {7851, 33202}, {7854, 32976}, {7857, 14069}, {7864, 33258}, {7879, 33249}, {7885, 32963}, {7891, 33206}, {7893, 16922}, {7898, 33006}, {7899, 32958}, {7904, 14063}, {7911, 33292}, {7912, 32998}, {7919, 33230}, {7928, 33283}, {7930, 33195}, {7937, 33196}, {7938, 33248}, {7942, 33194}, {7944, 32953}, {9466, 33216}, {9769, 11061}, {10303, 32830}, {10565, 20477}, {11056, 16051}, {11167, 11172}, {11812, 32892}, {12108, 32888}, {13881, 32974}, {14061, 33285}, {14712, 33016}, {15692, 32893}, {15694, 32837}, {15702, 32833}, {15708, 32874}, {15717, 32872}, {15720, 32824}, {15721, 32869}, {16239, 32884}, {16589, 33044}, {16921, 20065}, {16925, 31276}, {17128, 32964}, {17129, 33015}, {18906, 22712}, {27269, 33055}, {31450, 32450}, {32821, 32835}

X(34229) = isotomic conjugate of X(14494)
X(34229) = isotomic conjugate of the isogonal conjugate of X(5050)
X(34229) = anticomplement of X(31489)
X(34229) = X(31)-isoconjugate of X(14494)
X(34229) = barycentric product X(76)*X(5050)
X(34229) = barycentric quotient X (i)/X(j) for these {i,j}: {2, 14494}, {5050, 6}
X(34229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 69, 1007}, {2, 183, 69}, {2, 193, 3815}, {2, 385, 7736}, {2, 3620, 7778}, {2, 5232, 30761}, {2, 5304, 11174}, {2, 7610, 23055}, {2, 7735, 3618}, {2, 9740, 11163}, {2, 11160, 11184}, {2, 15589, 325}, {2, 17008, 7735}, {2, 26243, 14555}, {5, 3785, 32006}, {76, 631, 6337}, {183, 325, 15589}, {230, 11168, 15271}, {230, 15271, 2}, {325, 15589, 69}, {385, 7736, 1992}, {491, 492, 3620}, {491, 32785, 32806}, {492, 32786, 32805}, {599, 15597, 2}, {1078, 32832, 4}, {1656, 7767, 32816}, {3054, 7778, 2}, {3314, 17006, 2}, {3523, 32834, 1975}, {3526, 3933, 32829}, {3533, 32818, 7769}, {3734, 21843, 32985}, {3767, 7815, 16043}, {3785, 32838, 5}, {3815, 8667, 193}, {7610, 11168, 2}, {7610, 15271, 230}, {7746, 7800, 14064}, {7749, 7795, 32970}, {7771, 11185, 376}, {10299, 32822, 7782}, {11174, 22329, 5304}, {18840, 33189, 7832}, {21356, 23053, 2}, {32805, 32806, 3619}, {32816, 32867, 1656}


X(34230) =  X(1)X(513)∩X(6)X(101)

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

See Antreas Hatzipolakis, Paul Yiu and Angel Montesdeoca, HG040919.

X(34230) lies on these lines: {1,513}, {2,16506}, {3,3446}, {6,101}, {7,528}, {36,23344}, {55,840}, {56,59}, {88,1002}, {518,1026}, {551,16494}, {672,2284}, {997,16504}, {1001,3257}, {1037,1417}, {1149,16501}, {1159,4792}, {1168,15934}, {1193,17109}, {1201,16493}, {1318,4638}, {1319,24029}, {1458,2283}, {1797,3423}, {3304,18771}, {3433,8069}, {3616,16500}, {4080,11330}, {4674,5902}, {4997,30947}, {10247,29349}, {22769,32719}, {24841,31061}, {24870,31139}


X(34231) =  X(1)X(4)∩X(6)X(281)

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

See Antreas Hatzipolakis, Paul Yiu and Angel Montesdeoca, HG040919.

X(34231) lies on these lines: {1,4}, {2,5081}, {6,281}, {8,7078}, {20,17080}, {24,8071}, {28,4267}, {29,81}, {37,3087}, {45,6749}, {53,16884}, {56,7412}, {65,11436}, {108,999}, {109,14647}, {145,318}, {158,31503}, {186,14793}, {196,11529}, {204,461}, {208,3333}, {212,5657}, {222,5768}, {273,3945}, {297,26626}, {317,17321}, {354,1875}, {378,8069}, {381,15252}, {393,1100}, {406,1104}, {412,4313}, {451,499}, {458,17316}, {475,3085}, {519,7046}, {580,1771}, {860,17056}, {1035,12114}, {1038,6865}, {1040,6916}, {1060,6827}, {1062,6850}, {1119,3664}, {1210,1453}, {1212,17916}, {1214,6987}, {1249,1449}, {1394,6245}, {1426,16193}, {1427,4293}, {1451,3075}, {1593,4339}, {1594,10523}, {1697,1753}, {1735,3474}, {1828,28076}, {1835,30274}, {1845,5902}, {1851,1884}, {1861,31397}, {1863,12138}, {1872,9957}, {1876,4307}, {1887,3057}, {1890,23052}, {1892,4310}, {1895,5342}, {1897,3241}, {1905,24476}, {1993,3562}, {2096,7004}, {2202,2280}, {2956,9948}, {3088,5266}, {3100,6925}, {3176,6738}, {3535,5405}, {3536,5393}, {3541,10321}, {3616,17555}, {3671,15005}, {3672,7282}, {3879,32000}, {4185,11406}, {4194,14986}, {4200,5174}, {4207,5155}, {4222,11399}, {4294,15852}, {4296,6836}, {4357,32001}, {4644,18389}, {5125,5703}, {5226,7541}, {5308,26003}, {5665,7149}, {5722,15524}, {6353,24239}, {6748,16777}, {6864,19372}, {6923,18455}, {6928,18447}, {6939,9817}, {7050,11546}, {7079,16572}, {7414,22766}, {7510,15934}, {7524,12433}, {7577,8068}, {8070,16868}, {9630,18961}, {9799,34035}, {10247,21664}, {10580,14004}, {11023,17054}, {13411,20201}, {14792,21844}, {17257,27377}


X(34232) =  X(1)X(5)∩X(514)X(4667)

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

See Antreas Hatzipolakis, Paul Yiu and Angel Montesdeoca, HG040919.

X(34232) lies on these lines: {1,5}, {514,4667}, {655,11041}, {999,2222}, {1168,15934}, {3025,5902}, {6740,26860}


X(34233) =  ISOGONAL CONJUGATE OF X(23291)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Anthrakitis290112 and Peter Moses Hyacinthos 29475.

X(34233) lies on these lines: {20, 6530}, {232, 9909}, {325, 6353}, {511, 3515}, {523, 30549}, {9306, 15394}, {14575, 22263}

X(34233) = isogonal conjugate of X(23291)
X(34233) = cevapoint of X(i) and X(j) for these (i,j): {3, 8780}, {25, 8778}, {154, 3053}, {577, 1974}
X(34233) = barycentric quotient X(6)/X(23291)

X(34234) =  X(2)X(222)∩X(3)X(8)

Barycentrics    (a^3 - a^2*b - a*b^2 + b^3 + 2*a*b*c - a*c^2 - b*c^2)*(a^3 - a*b^2 - a^2*c + 2*a*b*c - b^2*c - a*c^2 + c^3) : :
Barycentrics    1/((a - b + c) (a + b - c) (b + c) - 2 a b c) : :
Trilinears    1/((a - b) cos C + (a - c) cos B) : :

X(34234) is the trilinear product of the circumcircle intercepts of line X(2)X(905). As the trilinear product of circumcircle-X(2)-antipodes, X(34234) lies on the circumellipse with center X(9) and perspector X(1). (Randy Hutson, October 8, 2019)

X(34234) lies on the curves K311 and Q034 and these lines: {2, 222}, {3, 8}, {11, 33650}, {27, 823}, {29, 58}, {46, 20220}, {57, 92}, {63, 190}, {84, 412}, {85, 658}, {88, 2401}, {103, 1309}, {144, 30578}, {189, 5435}, {241, 1952}, {257, 24627}, {295, 660}, {320, 908}, {329, 6557}, {333, 662}, {603, 11109}, {655, 3218}, {664, 16586}, {799, 17206}, {914, 32851}, {936, 7572}, {1013, 3423}, {1155, 14198}, {1220, 24982}, {1311, 2720}, {1748, 21370}, {1768, 24026}, {1796, 4102}, {1803, 32008}, {1897, 7004}, {1936, 2342}, {2067, 7090}, {2316, 14554}, {2399, 3904}, {2423, 20332}, {3562, 27506}, {3762, 24618}, {3911, 5053}, {4181, 20751}, {4598, 27424}, {4604, 17277}, {4606, 5372}, {5174, 6245}, {5176, 10428}, {5361, 30711}, {5777, 7567}, {5906, 6943}, {6502, 14121}, {9364, 26013}, {10265, 13532}, {11679, 20881}, {14986, 15501}, {17923, 34050}, {17972, 30993}, {18031, 34085}, {18151, 18750}, {19861, 31359}, {20348, 30083}, {27003, 30690}

X(34234) = isogonal conjugate of X(2183)
X(34234) = isotomic conjugate of X(908)
X(34234) = polar conjugate of X(1785)
X(34234) = isotomic conjugate of the anticomplement of X(3911)
X(34234) = isotomic conjugate of the complement of X(3218)
X(34234) = isotomic conjugate of the isogonal conjugate of X(909)
X(34234) = polar conjugate of the isogonal conjugate of X(1795)
X(34234) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2316, 153}, {2342, 30578}, {10428, 7}
X(34234) = X(13136)-Ceva conjugate of X(2401)
X(34234) = X(i)-cross conjugate of X(j) for these (i,j): {36, 86}, {80, 903}, {515, 7}, {1737, 75}, {2183, 1}, {2250, 104}, {2316, 1120}, {2401, 13136}, {3738, 664}, {3762, 190}, {3911, 2}, {5053, 1220}, {10265, 18815}, {24618, 673}
X(34234) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2183}, {3, 14571}, {6, 517}, {9, 1457}, {19, 22350}, {31, 908}, {32, 3262}, {37, 859}, {41, 22464}, {44, 14260}, {48, 1785}, {55, 1465}, {58, 21801}, {100, 3310}, {101, 1769}, {104, 23980}, {119, 32655}, {213, 17139}, {219, 1875}, {292, 15507}, {513, 2427}, {604, 6735}, {647, 4246}, {650, 23981}, {652, 23706}, {663, 24029}, {667, 2397}, {692, 10015}, {909, 24028}, {1145, 9456}, {1333, 17757}, {1415, 2804}, {1783, 8677}, {2423, 15632}, {6187, 16586}, {6335, 23220}, {14578, 21664}, {23757, 32665}
X(34234) = cevapoint of X(i) and X(j) for these (i,j): {1, 2183}, {2, 3218}, {9, 519}, {57, 34050}, {63, 914}, {649, 1647}, {900, 1146}, {909, 1795}, {4466, 4707}
X(34234) = trilinear pole of line {1, 522}
X(34234) = barycentric product X(i)*X(j) for these {i,j}: {1, 18816}, {63, 16082}, {75, 104}, {76, 909}, {190, 2401}, {264, 1795}, {273, 1809}, {274, 2250}, {309, 15501}, {312, 34051}, {514, 13136}, {1309, 4025}, {1969, 14578}, {1978, 2423}, {2342, 6063}, {3261, 32641}, {3264, 10428}, {7035, 15635}
X(34234) = barycentric quotient X (i)/X(j) for these {i,j}: {1, 517}, {2, 908}, {3, 22350}, {4, 1785}, {6, 2183}, {7, 22464}, {8, 6735}, {10, 17757}, {19, 14571}, {34, 1875}, {37, 21801}, {56, 1457}, {57, 1465}, {58, 859}, {75, 3262}, {86, 17139}, {101, 2427}, {104, 1}, {106, 14260}, {108, 23706}, {109, 23981}, {162, 4246}, {190, 2397}, {238, 15507}, {513, 1769}, {514, 10015}, {517, 24028}, {519, 1145}, {522, 2804}, {649, 3310}, {651, 24029}, {900, 23757}, {908, 26611}, {909, 6}, {1210, 1532}, {1309, 1897}, {1457, 1361}, {1459, 8677}, {1647, 3259}, {1737, 119}, {1785, 21664}, {1795, 3}, {1809, 78}, {1870, 1845}, {1877, 1846}, {2183, 23980}, {2250, 37}, {2342, 55}, {2401, 514}, {2423, 649}, {2720, 109}, {3086, 1519}, {3218, 16586}, {3582, 12611}, {3943, 21942}, {7192, 23788}, {10428, 106}, {13136, 190}, {14266, 1737}, {14578, 48}, {14776, 8750}, {15501, 40}, {15635, 244}, {16082, 92}, {18391, 1512}, {18816, 75}, {24028, 23101}, {30117, 15906}, {32641, 101}, {32669, 1415}, {32702, 32674}, {34051, 57}, {34063, 25305}
X(34234) = {X(3075),X(14058)}-harmonic conjugate of X(29)


X(34235) =  X(2)X(647)∩X(4)X(39)

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

X(34235) lies on the orthocentroidal circle, the cubic K305, and these lines: {2, 647}, {4, 39}, {6, 6785}, {32, 33695}, {182, 2715}, {316, 14961}, {353, 1495}, {574, 842}, {575, 8779}, {1971, 5038}, {2492, 31953}, {2493, 11005}, {3143, 15048}, {3818, 15355}, {5968, 6787}, {6033, 19163}, {6054, 11672}, {6792, 9465}, {7769, 28407}, {9970, 10766}, {14966, 15915}

X(34235) = Brocard-circle-inverse of X(2715)
X(34235) = polar-circle-inverse of X(232)
X(34235) = orthoptic-circle-of-Steiner-inellipse-inverse of X(647)
X(34235) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(31296)
X(34235) = Moses-radical-circle-inverse of X(2)
X(34235) = psi-transform of X (3569)


X(34236) =  X(1)X(7077)∩X(2)X(51)

Barycentrics    a^2*(a^4*b^4 - a^2*b^6 + 4*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 + 4*b^4*c^4 - a^2*c^6 + b^2*c^6) : :
X(34236) = 3 X[2] + X[263]

X(34236) lies on the cubic K423 and these lines: {1, 7077}, {2, 51}, {6, 3229}, {39, 694}, {83, 3491}, {182, 8841}, {211, 6704}, {237, 5092}, {384, 14135}, {420, 1843}, {512, 7804}, {1431, 17799}, {1613, 5039}, {3398, 3506}, {3972, 32442}, {6310, 7770}, {6784, 7792}, {7711, 15018}, {8675, 12039}, {14096, 14810}, {18553, 20021}, {19130, 21531}, {20985, 23578}

X(34236) = X(3329)-Ceva conjugate of X(39)
X(34236) = crosssum of X(2) and X(10335)
X(34236) = crossdifference of every pair of points on line {3288, 25423}
X(34236) = {X(2),X(11673)}-harmonic conjugate of X(5650)


X(34237) =  X(4)X(66)∩X(98)X(15407)

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

X(34237) lies on the cubics K288 and K1134 and on these lines: {4, 66}, {98, 15407}, {1289, 1503}, {1899, 18018}, {3267, 8673}, {6776, 14376}, {10547, 26926}, {13854, 23291}, {17407, 32064}

X(34237) = polar-circle-inverse of X(27373)
X(34237) = antigonal image of X(34137)
X(34237) = symgonal image of X(34138)
X(34237) = X(290)-Ceva conjugate of X(18018)
X(34237) = X(2172)-isoconjugate of X(34129)
X(34237) = barycentric product X(18018)*X(34137)
X(34237) = barycentric quotient X (i)/X(j) for these {i,j}: {66, 34129}, {34137, 22}


X(34238) =  X(4)X(6071)∩X(6)X(14251)

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

X(34238) lies on the cubics K380, K532, K1134, and these lines: {4, 6071}, {6, 14251}, {98, 385}, {182, 2065}, {237, 694}, {384, 8870}, {733, 2715}, {878, 881}, {1431, 30648}, {1692, 9468}, {2211, 8789}, {2679, 34214}, {2698, 12176}, {3398, 3493}, {5034, 18872}, {5207, 20021}, {5360, 7077}, {14382, 18906}

X(34238) = reflection of X(i) in X(j) for these {i,j}: {805, 9467}, {34214, 2679}
X(34238) = isogonal conjugate of X(5976)
X(34238) = antigonal image of X(34214)
X(34238) = symgonal image of X(9467)
X(34238) = isogonal conjugate of the anticomplement of X(2023)
X(34238) = isogonal conjugate of the complement of X(1916)
X(34238) = cevapoint of circumcircle intercepts of circle {{X(1687),X(1688),PU(1),PU(2)}}
X(34238) = polar conjugate of the isotomic conjugate of X(15391)
X(34238) = X(i)-cross conjugate of X(j) for these (i,j): {6, 1976}, {32, 733}, {512, 805}, {878, 18858}, {1084, 2395}, {14318, 32716}, {32540, 98}
X(34238) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5976}, {237, 1926}, {240, 12215}, {325, 1580}, {385, 1959}, {511, 1966}, {732, 3405}, {1755, 3978}, {2236, 20022}, {2679, 24037}, {9417, 14603}, {14295, 23997}, {14382, 23996}, {16591, 27958}
X(34238) = cevapoint of X(i) and X(j) for these (i,j): {6, 694}, {98, 8870}, {512, 15630}
X(34238) = trilinear pole of line {882, 2422}
X(34238) = barycentric product X(i)*X(j) for these {i,j}: {4, 15391}, {98, 694}, {287, 17980}, {290, 9468}, {733, 20021}, {805, 2395}, {882, 2966}, {1581, 1910}, {1821, 1967}, {1916, 1976}, {2422, 18829}, {3569, 18858}, {8789, 18024}, {9154, 18872}, {14601, 18896}, {16081, 17970}
X(34238) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 5976}, {98, 3978}, {248, 12215}, {290, 14603}, {694, 325}, {733, 20022}, {805, 2396}, {878, 24284}, {881, 3569}, {882, 2799}, {1084, 2679}, {1821, 1926}, {1910, 1966}, {1927, 1755}, {1967, 1959}, {1976, 385}, {2395, 14295}, {2422, 804}, {2715, 17941}, {2966, 880}, {6531, 17984}, {8789, 237}, {9468, 511}, {14601, 1691}, {14604, 9418}, {15391, 69}, {17938, 2421}, {17980, 297}, {18024, 18901}
X(34238) = {X(16068),X(17970)}-harmonic conjugate of X(805)


X(34239) =  X(4)X(32618)∩X(98)X(5001)

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

X(34239) lies on the cubics K025 and K1134 and these lines: {4, 32618}, {98, 5001}, {287, 297}, {290, 34135}, {5002, 30737}

X(34239) = antigonal image of X (5003)
X(34239) = symgonal image of X (5001)
X(34239) = cevapoint of X(1503) and X (5001)


X(34240) =  X(4)X(32619)∩X(98)X(5000)

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

X(34240) lies on the cubics K025 and K1134 and on these lines: {4, 32619}, {98, 5000}, {287, 297}, {290, 34136}, {5003, 30737}

X(34240) = antigonal image of X(5002)
X(34240) = symgonal image of X(5000)
X(34240) = cevapoint of X(1503) and X(5000)


X(34241) =  X(4)X(543)∩X(111)X(352)

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

X(34241) lies on the cubics K289, K305, K474, and on these lines: {4, 543}, {111, 352}, {574, 1296}, {5512, 11185}, {11477, 33900}

X(34241) = reflection of X(i) in X(j) for these {i,j}: {1296, 574}, {11185, 5512}
X(34241) = antigonal image of X(11185)
X(34241) = symgonal image of X(574)
X(34241) = X(13608)-isoconjugate of X(17959)
X(34241) = perspector of ABC and Artzt triangle of 4th anti-Brocard triangle
X(34241) = barycentric product X(1995)*X(5503)
X(34241) = barycentric quotient X(i)/X(j) for these {i,j}: {1995, 22329}, {13493, 18775}, {14262, 17952}, {19136, 2030}


X(34242) =  X(4)X(80)∩X(58)X(65)

Barycentrics    a*(a + b - c)*(a - b + c)*(a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2*b - b^3 + a^2*c - a*b*c - c^3) : :
X(34242) = 3 X[3877] - 4 X[11734], 3 X[5902] - 2 X[11700]

X(34242) lies on the cubic K306 and these lines: {1, 3417}, {4, 80}, {52, 2818}, {58, 65}, {102, 517}, {124, 3869}, {181, 994}, {758, 13532}, {959, 2006}, {2099, 11334}, {2835, 9309}, {2841, 14584}, {3340, 6187}, {3827, 10764}, {3877, 11734}, {5011, 32675}, {5902, 11700}, {6001, 10732}, {10696, 24474}, {10747, 14988}, {18815, 20028}

X(34242) = reflection of X(i) in X(j) for these {i,j}: {109, 65}, {3869, 124}, {10696, 24474}
X(34242) = antigonal image of X(3869)
X(34242) = symgonal image of X(65)
X(34242) = X(14616)-Ceva conjugate of X(2006)
X(34242) = X(i)-isoconjugate of X(j) for these (i,j): {36, 10570}, {2217, 4511}, {2245, 19607}, {2323, 13478}, {2361, 2995}, {3904, 32653}
X(34242) = barycentric product X(i)*X(j) for these {i,j}: {80, 17080}, {573, 18815}, {655, 21189}, {1411, 4417}, {2006, 3869}, {10571, 18359}
X(34242) = barycentric quotient X(i)/X(j) for these {i,j}: {573, 4511}, {759, 19607}, {1411, 13478}, {2006, 2995}, {2161, 10570}, {3185, 2323}, {3869, 32851}, {6589, 3738}, {10571, 3218}, {17080, 320}, {21189, 3904}


X(34243) =  X(3)X(759)∩X(4)X(758)

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

X(34243) lies on the cubics K028 and K306 and on these lines: {3, 759}, {4, 758}

X(34243) = X(30212)-cross conjugate of X(6011)
X(34243) = barycentric quotient X(5086)/X(33116)


X(34244) =  X(2)X(40)∩X(1434)X(16714)

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

X(34244) lies on the cubics K295, K308, and K926, and these lines: {2, 40}, {1434, 16714}

X(34244) = X(4866)-isoconjugate of X(34046)
X(34244) = barycentric quotient X(3616)/X(34255)


X(34245) =  X(2)X(187)∩X(99)X(110)

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

X(34245) lies on the cubic K408 and these lines: {2, 187}, {99, 110}, {183, 1316}, {669, 11634}, {691, 9080}, {877, 7473}, {892, 2395}, {1003, 5108}, {1641, 8598}, {2418, 31614}, {2421, 3288}, {4235, 9170}, {5191, 5939}, {5467, 9182}, {7750, 15000}, {9999, 14360}

X(34245) = X(32694)-anticomplementary conjugate of X(21221)
X(34245) = X(2793)-cross conjugate of X(22329)
X(34245) = X(i)-isoconjugate of X(j) for these (i,j): {798, 5503}, {2643, 2709}
X(34245) = cevapoint of X(2793) and X(22329)
X(34245) = crosssum of X(512) and X(9208)
X(34245) = trilinear pole of line {2030, 18800}
X(34245) = crossdifference of every pair of points on line {3124, 17414}
X(34245) = barycentric product X(i)*X(j) for these {i,j}: {99, 22329}, {670, 2030}, {892, 18800}, {1992, 17937}, {2793, 4590}
X(34245) = barycentric quotient X(i)/X(j) for these {i,j}: {99, 5503}, {249, 2709}, {2030, 512}, {2793, 115}, {9135, 3124}, {17937, 5485}, {18800, 690}, {22329, 523}
X(34245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 5468, 2396}, {99, 17941, 5468}, {4226, 5468, 99}, {4226, 17941, 2396}


X(34246) =  X(2)X(8599)∩X(476)X(2709)

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

X(34246) lies on the cubic K408 and these lines: {2, 8599}, {476, 2709}, {523, 599}, {685, 4235}, {690, 2395}, {850, 9464}, {868, 23288}, {892, 2396}, {1499, 11159}, {2501, 5094}, {2793, 11161}, {2799, 5466}, {9168, 10130}

X(34246) = crosssum of X(i) and X(j) for these (i,j): {2030, 9135}, {5107, 9208}
X(34246) = trilinear pole of line {115, 3906}
X(34246) = X(i)-isoconjugate of X(j) for these (i,j): {163, 22329}, {662, 2030}, {1101, 2793}, {9135, 24041}
X(34246) = barycentric product X(i)*X(j) for these {i,j}: {338, 2709}, {523, 5503}
X(34246) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 2793}, {512, 2030}, {523, 22329}, {690, 18800}, {2709, 249}, {3124, 9135}, {5485, 17937}, {5503, 99}


X(34247) =  X(3)X(984)∩X(6)X(292)

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

X(34247) lies on the cubic K999 and these lines: {3, 984}, {6, 292}, {7, 21320}, {9, 2223}, {12, 5125}, {19, 25}, {31, 3217}, {32, 21830}, {38, 4191}, {41, 19133}, {42, 2277}, {43, 27633}, {44, 3941}, {45, 8053}, {48, 2330}, {56, 78}, {69, 4447}, {71, 4517}, {75, 183}, {100, 192}, {101, 2175}, {171, 20760}, {172, 15370}, {181, 3190}, {200, 1402}, {210, 2352}, {335, 11329}, {344, 8299}, {404, 18048}, {405, 3842}, {474, 24325}, {537, 16371}, {560, 2053}, {573, 3688}, {579, 20683}, {664, 31604}, {726, 25440}, {740, 5687}, {756, 1011}, {976, 13738}, {982, 16059}, {983, 1582}, {1001, 4687}, {1215, 11358}, {1253, 9310}, {1260, 1460}, {1284, 2550}, {1400, 2340}, {1469, 1818}, {1613, 18900}, {1621, 27268}, {1631, 19297}, {1918, 2176}, {2178, 12329}, {2183, 3056}, {2209, 3009}, {2212, 7084}, {2260, 4878}, {2283, 6180}, {2664, 27623}, {3085, 3144}, {3149, 29054}, {3242, 4022}, {3286, 3786}, {3303, 15569}, {3550, 20676}, {3678, 19762}, {3711, 22271}, {3724, 3728}, {3730, 7064}, {3739, 4413}, {3741, 24736}, {3938, 21330}, {3952, 11322}, {3961, 23853}, {3993, 8715}, {4032, 23067}, {4043, 5695}, {4068, 16672}, {4188, 31302}, {4203, 27538}, {4210, 7226}, {4383, 16687}, {4421, 4664}, {4423, 4698}, {4433, 17314}, {4436, 17262}, {4497, 21773}, {4650, 22149}, {4812, 26263}, {4860, 13476}, {5205, 20923}, {5269, 20967}, {6600, 19557}, {7191, 27639}, {8609, 21867}, {9335, 16057}, {11248, 20430}, {11499, 29010}, {11500, 30273}, {12338, 32453}, {13588, 32937}, {16056, 33144}, {16367, 31323}, {16405, 32931}, {16409, 17063}, {16417, 31178}, {16684, 17259}, {17157, 32927}, {17260, 23407}, {17686, 27298}, {17755, 21477}, {17780, 25277}, {18278, 21787}, {18758, 20996}, {19763, 30142}, {22016, 32929}, {24552, 27261}, {25124, 29670}, {26228, 27628}, {28748, 30959}

X(34247) = isogonal conjugate of the isotomic conjugate of X(32937)
X(34247) = X(i)-Ceva conjugate of X(j) for these (i,j): {983, 6}, {7084, 55}, {13588, 3501}
X(34247) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3500}, {7289, 30688}
X(34247) = crosspoint of X(i) and X(j) for these (i,j): {101, 4998}, {7115, 8685}
X(34247) = crosssum of X(i) and X(j) for these (i,j): {513, 21138}, {514, 3271}, {3810, 26932}
X(34247) = trilinear pole of line {22229, 23655}
X(34247) = crossdifference of every pair of points on line {812, 905}
X(34247) = barycentric product X(i)*X(j) for these {i,j}: {1, 3501}, {6, 32937}, {31, 17786}, {37, 13588}, {99, 22229}, {100, 21348}, {101, 17072}, {110, 21958}, {190, 23655}, {692, 21438}, {1252, 23772}, {1897, 22443}, {3508, 8927}, {4551, 21388}, {4552, 23864}, {4559, 21300}
X(34247) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3500}, {3501, 75}, {13588, 274}, {17072, 3261}, {17786, 561}, {21348, 693}, {21388, 18155}, {21958, 850}, {22229, 523}, {22443, 4025}, {23655, 514}, {23772, 23989}, {23864, 4560}, {30689, 17170}, {32937, 76}
X(34247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 20990, 21010}, {9, 2223, 20992}, {37, 15624, 55}, {55, 198, 23868}, {228, 612, 55}, {869, 20964, 6}, {1400, 2340, 3779}, {2178, 12329, 17798}, {2209, 3009, 21769}, {4557, 20990, 6}


X(34248) =  X(31)X(1582)∩X(48)X(1927)

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

X(34248) lies on the cubics K991, K999, and K1031, and on these lines: {31, 1582}, {48, 1927}, {82, 18832}, {560, 1932}, {715, 3222}, {1918, 2176}, {2205, 2209}

X(34248) = isogonal conjugate of X(17149)
X(34248) = isotomic conjugate of X(18837)
X(34248) = isogonal conjugate of the anticomplement of X(16606)
X(34248) = isogonal conjugate of the complement of X(21223)
X(34248) = isogonal conjugate of the isotomic conjugate of X(3223)
X(34248) = X(i)-cross conjugate of X(j) for these (i,j): {1, 31}, {21759, 6}
X(34248) = X(i)-isoconjugate of X(j) for these (i,j): {1, 17149}, {2, 194}, {6, 6374}, {8, 17082}, {31, 18837}, {69, 3186}, {75, 1740}, {76, 1613}, {81, 22028}, {85, 7075}, {86, 21080}, {99, 23301}, {100, 23807}, {190, 21191}, {264, 20794}, {274, 21877}, {305, 11325}, {312, 1424}, {662, 20910}, {664, 25128}, {670, 3221}, {1978, 23572}, {2524, 6331}, {4600, 21144}, {4602, 23503}, {4609, 9491}, {4610, 21056}, {15968, 32747}
X(34248) = cevapoint of X(i) and X(j) for these (i,j): {1, 3223}, {6, 21787}
X(34248) = crosssum of X(21080) and X(22028)
X(34248) = trilinear pole of line {1924, 8640}
X(34248) = crossdifference of every pair of points on line {20910, 21191}
X(34248) = trilinear product of PU(148)
X(34248) = barycentric product X(i)*X(j) for these {i,j}: {1, 3224}, {6, 3223}, {19, 3504}, {31, 2998}, {32, 18832}, {92, 15389}, {798, 3222}, {3112, 19606}
X(34248) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 6374}, {2, 18837}, {6, 17149}, {31, 194}, {32, 1740}, {42, 22028}, {213, 21080}, {512, 20910}, {560, 1613}, {604, 17082}, {649, 23807}, {667, 21191}, {798, 23301}, {1397, 1424}, {1918, 21877}, {1924, 3221}, {1973, 3186}, {1980, 23572}, {2175, 7075}, {2998, 561}, {3063, 25128}, {3121, 21144}, {3222, 4602}, {3223, 76}, {3224, 75}, {3504, 304}, {9247, 20794}, {9426, 23503}, {15389, 63}, {18832, 1502}, {19606, 38}


X(34249) =  X(6)X(43)∩X(141)X(4598)

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

X(34249) lies on the cubic K999 and these lines: {6, 43}, {141, 4598}, {560, 2053}, {19133, 23493}

X(34249) = cevapoint of X(3494) and X(3961)
X(34249) = X(i)-isoconjugate of X(j) for these (i,j): {2, 3502}, {192, 7194}
X(34249) = barycentric product X(i)*X(j) for these {i,j}: {1, 3494}, {87, 3961}, {2162, 17280}, {7121, 33938}, {23493, 33954}
X(34249) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 3502}, {3494, 75}, {3961, 6376}, {7121, 7194}, {17280, 6382}


X(34250) =  X(3)X(984)∩X(48)X(2276)

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

X(34250) lies on the cubics K999 and K1001 and these lines: {3, 984}, {48, 2276}, {75, 17797}, {171, 18207}, {256, 1582}, {560, 8852}, {603, 1469}, {1326, 23850}, {1437, 3736}, {1444, 7224}, {1460, 7053}, {1472, 2212}, {1631, 4492}, {2196, 3862}, {4497, 7241}, {5329, 23086}

X(34250) = isogonal conjugate of X(4388)
X(34250) = isotomic conjugate of X(18835)
X(34250) = isogonal conjugate of the anticomplement of X(171)
X(34250) = isogonal conjugate of the complement of X(20101)
X(34250) = isogonal conjugate of the isotomic conjugate of X(7224)
X(34250) = X(i)-cross conjugate of X(j) for these (i,j): {5322, 56}, {7122, 6}
X(34250) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4388}, {2, 3496}, {6, 17788}, {9, 17086}, {31, 18835}, {75, 23868}, {81, 4109}, {92, 23150}, {100, 4142}, {256, 17797}
X(34250) = crosssum of X(i) and X(j) for these (i,j): {72, 21083}, {23150, 23868}
X(34250) = trilinear pole of line {3250, 22383}
X(34250) = barycentric product X(i)*X(j) for these {i,j}: {1, 3497}, {6, 7224}, {32, 18836}
X(34250) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 17788}, {2, 18835}, {6, 4388}, {31, 3496}, {32, 23868}, {42, 4109}, {56, 17086}, {172, 17797}, {184, 23150}, {649, 4142}, {3497, 75}, {7224, 76}, {18836, 1502}
X(34250) = {X(1582),X(6660)}-harmonic conjugate of X(23868)


X(34251) =  X(6)X(75)∩X(48)X(1613)

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

X(34251) lies on the cubic K999 and these lines: {6, 75}, {48, 1613}, {560, 30634}, {893, 1964}, {1582, 7104}, {1740, 19580}, {1918, 2176}, {1979, 3941}, {3496, 3499}, {4386, 21779}, {9288, 23868}, {12194, 33786}, {14602, 33772}, {16685, 21790}, {18194, 21387}, {21010, 21787}, {21773, 21783}, {23546, 27644}

X(34251) = X(i)-Ceva conjugate of X(j) for these (i,j): {1582, 23868}, {7104, 6}
X(34251) = crossdifference of every pair of points on line {788, 17072}
X(34251) = X(2)-isoconjugate of X(7346)
X(34251) = polar conjugate of the isotomic conjugate of X(23192)
X(34251) = barycentric product X(i)*X(j) for these {i,j}: {1, 6196}, {4, 23192}, {31, 24732}
X(34251) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 7346}, {6196, 75}, {23192, 69}, {24732, 561}
X(34251) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 21776, 18278}, {239, 21751, 6}


X(34252) =  X(1)X(2053)∩X(6)X(43)

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

X(34252) lies on the cubics K673, K983, and K989, and on these lines: {1, 2053}, {6, 43}, {31, 983}, {81, 1178}, {183, 17123}, {238, 385}, {291, 8851}, {330, 985}, {350, 3253}, {726, 727}, {740, 14199}, {757, 7304}, {765, 5383}, {932, 2382}, {982, 18194}, {1428, 8300}, {1580, 1914}, {2109, 3509}, {2280, 25834}, {3329, 17122}, {3494, 5255}, {3733, 4782}, {5247, 27424}, {17598, 18170}, {27458, 32949}

X(34252) = X(i)-Ceva conjugate of X(j) for these (i,j): {87, 8843}, {2319, 8848}
X(34252) = X(239)-cross conjugate of X(238)
X(34252) = X(i)-isoconjugate of X(j) for these (i,j): {43, 291}, {192, 292}, {334, 2209}, {335, 2176}, {660, 4083}, {694, 17752}, {741, 3971}, {813, 3835}, {1403, 4518}, {1423, 4876}, {1911, 6376}, {1922, 6382}, {3009, 33680}, {3123, 5378}, {3212, 7077}, {3572, 4595}, {4562, 20979}, {4583, 8640}, {4584, 21834}, {20906, 34067}
X(34252) = cevapoint of X(3808) and X (27846)
X(34252) = trilinear pole of line {4164, 8632}
X(34252) = crossdifference of every pair of points on line {4083, 20691}
X(34252) = barycentric product X(i)*X(j) for these {i,j}: {87, 239}, {238, 330}, {350, 2162}, {659, 4598}, {812, 932}, {1428, 27424}, {1429, 7155}, {1447, 2319}, {1580, 27447}, {1914, 6384}, {1921, 7121}, {2053, 10030}, {2210, 6383}, {3500, 14199}, {3685, 7153}, {3766, 34071}, {5383, 27846}, {6650, 8843}, {8632, 18830}, {16606, 33295}, {23493, 30940}
X(34252) = barycentric quotient X (i)/X(j) for these {i,j}: {87, 335}, {238, 192}, {239, 6376}, {330, 334}, {350, 6382}, {659, 3835}, {812, 20906}, {932, 4562}, {1428, 1423}, {1429, 3212}, {1447, 30545}, {1580, 17752}, {1914, 43}, {2053, 4876}, {2162, 291}, {2210, 2176}, {2238, 3971}, {2319, 4518}, {3573, 4595}, {3684, 27538}, {3685, 4110}, {3747, 20691}, {4435, 4147}, {4455, 21834}, {4598, 4583}, {5009, 27644}, {6384, 18895}, {7121, 292}, {7153, 7233}, {7193, 22370}, {8632, 4083}, {8843, 6542}, {14199, 17786}, {14599, 2209}, {15373, 295}, {20332, 33680}, {21832, 21051}, {22384, 25098}, {27447, 1934}, {27846, 21138}, {33295, 31008}, {34071, 660}


X(34253) =  X(6)X(7)∩X(43)X(57)

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

X(34253) lies on the cubics K673 and K981 and these lines: {1, 2114}, {6, 7}, {43, 57}, {56, 2110}, {63, 23988}, {77, 16973}, {141, 17077}, {238, 1284}, {239, 10030}, {241, 518}, {511, 20367}, {613, 24248}, {664, 9263}, {812, 4107}, {940, 3475}, {1366, 3323}, {1400, 27633}, {1407, 2991}, {1423, 1743}, {1431, 2665}, {1447, 2238}, {1475, 28391}, {1742, 3056}, {2113, 9501}, {2284, 16593}, {2330, 9440}, {2870, 17463}, {3212, 3780}, {3242, 14151}, {3618, 26125}, {3664, 24237}, {3666, 7004}, {3674, 20963}, {3739, 15984}, {4552, 9055}, {7179, 24512}, {7190, 16972}, {8299, 20778}, {15988, 26806}, {17027, 30545}

X(34253) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 1284}, {57, 241}, {4998, 2283}
X(34253) = crosspoint of X(57) and X(1429)
X(34253) = crosssum of X(9) and X(4876)
X(34253) = crossdifference of every pair of points on line {926, 1024}
X(34253) = X(i)-isoconjugate of X(j) for these (i,j): {6, 33676}, {105, 4876}, {291, 294}, {292, 14942}, {335, 2195}, {660, 1024}, {673, 7077}, {813, 885}, {884, 4562}, {1438, 4518}, {2311, 13576}, {18031, 18265}
X(34253) = barycentric product X(i)*X(j) for these {i,j}: {7, 8299}, {57, 17755}, {238, 9436}, {239, 241}, {273, 20778}, {350, 1458}, {518, 1447}, {659, 883}, {672, 10030}, {812, 1025}, {1284, 30941}, {1428, 3263}, {1429, 3912}, {2223, 18033}, {2283, 3766}, {5236, 20769}, {16609, 18206}
X(34253) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 33676}, {238, 14942}, {241, 335}, {518, 4518}, {659, 885}, {672, 4876}, {883, 4583}, {1025, 4562}, {1284, 13576}, {1362, 22116}, {1428, 105}, {1429, 673}, {1447, 2481}, {1458, 291}, {1914, 294}, {2210, 2195}, {2223, 7077}, {2283, 660}, {3684, 6559}, {4435, 28132}, {8299, 8}, {8632, 1024}, {9436, 334}, {9455, 18265}, {10030, 18031}, {17755, 312}, {20778, 78}, {22384, 23696}, {27919, 3975}
X(34253) = {X(3751),X(4334)}-harmonic conjugate of X(1469)


X(34254) =  X(22)X(315)∩X(25)X(317)

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

See Antreas Hatzipolakis and Francisco Javier García Capitán, Anthrakitis160312 and Peter Moses Hyacinthos 29477.

X(34254) lies on these lines: {2, 39}, {4, 16276}, {20, 16275}, {22, 315}, {25, 317}, {69, 184}, {75, 23556}, {99, 1370}, {183, 7499}, {316, 7500}, {394, 6393}, {427, 1975}, {428, 7773}, {468, 32821}, {648, 8879}, {1007, 7392}, {1184, 7807}, {1236, 18018}, {1368, 6390}, {1369, 7492}, {1899, 12215}, {2548, 16950}, {3933, 6676}, {4159, 7737}, {4176, 4563}, {4232, 32825}, {5064, 32819}, {5094, 32820}, {5133, 11185}, {5207, 31383}, {6337, 7386}, {6340, 30786}, {6353, 32818}, {6504, 8781}, {6636, 14907}, {6995, 32816}, {6997, 7752}, {7378, 32815}, {7408, 32827}, {7409, 32826}, {7493, 7796}, {7714, 32823}, {7738, 30785}, {7776, 9909}, {7789, 11324}, {7791, 21248}, {8889, 32817}, {9723, 23195}, {10330, 33796}, {10565, 23608}, {14023, 14602}, {14033, 15437}, {14961, 28427}, {16051, 19583}, {17076, 20641}, {18138, 28738}

X(34254) = isotomic conjugate of X(13854)
X(34254) = isotomic conjugate of the isogonal conjugate of X(20806)
X(34254) = isotomic conjugate of the polar conjugate of X(315)
X(34254) = X(i)-Ceva conjugate of X(j) for these (i,j): {4590, 4611}, {18020, 4563}
X(34254) = X(i)-cross conjugate of X(j) for these (i,j): {10316, 69}, {20806, 315}, {28405, 17907}
X(34254) = X(i)-isoconjugate of X(j) for these (i,j): {19, 2353}, {25, 2156}, {31, 13854}, {66, 1973}, {798, 1289}, {2643, 15388}
X(34254) = cevapoint of X(i) and X(j) for these (i,j): {69, 28696}, {3926, 28419}
X(34254) = barycentric product X(i)*X(j) for these {i,j}: {22, 305}, {63, 20641}, {69, 315}, {76, 20806}, {127, 4590}, {304, 1760}, {345, 17076}, {670, 8673}, {1502, 10316}, {3267, 4611}, {3718, 7210}, {3926, 17907}, {4123, 7182}, {4150, 17206}, {4561, 21178}, {4563, 33294}, {4601, 18187}, {6393, 31636}
X(34254) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 13854}, {3, 2353}, {22, 25}, {63, 2156}, {69, 66}, {99, 1289}, {127, 115}, {206, 1974}, {249, 15388}, {305, 18018}, {315, 4}, {1370, 17407}, {1760, 19}, {1799, 16277}, {2172, 1973}, {2485, 2489}, {3313, 1843}, {3926, 14376}, {4123, 33}, {4150, 1826}, {4456, 2333}, {4463, 1824}, {4611, 112}, {5562, 27372}, {6393, 34138}, {7210, 34}, {8673, 512}, {8743, 2207}, {10316, 32}, {14396, 14398}, {16757, 6591}, {17076, 278}, {17907, 393}, {18187, 3125}, {20641, 92}, {20806, 6}, {21178, 7649}, {22075, 1501}, {23208, 27369}, {28405, 3767}, {31636, 6531}, {33294, 2501}
X(34254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1180, 7803}, {2, 3926, 305}, {69, 7494, 1799}, {184, 4121, 69}, {1196, 3788, 2}


X(34255) =  X(1)X(2)∩X(7)X(321)

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

X(34255) lies on these lines: {1, 2}, {7, 321}, {9, 14552}, {57, 2321}, {63, 346}, {69, 189}, {75, 9776}, {81, 5749}, {92, 1229}, {100, 11350}, {103, 835}, {144, 4001}, {193, 27064}, {226, 17296}, {319, 14555}, {333, 344}, {345, 5744}, {388, 3714}, {391, 3305}, {443, 5295}, {497, 3416}, {518, 3974}, {553, 4659}, {599, 4415}, {940, 2345}, {944, 16435}, {952, 19517}, {962, 3702}, {1043, 1817}, {1150, 5273}, {1231, 14256}, {1788, 3704}, {1836, 4519}, {1997, 5233}, {2094, 32939}, {2324, 20205}, {2325, 3929}, {2550, 3706}, {2551, 10371}, {2895, 8055}, {2994, 6557}, {3161, 3219}, {3175, 4419}, {3434, 33078}, {3474, 5695}, {3475, 4966}, {3619, 19786}, {3620, 27184}, {3662, 30699}, {3666, 17314}, {3681, 5423}, {3686, 7308}, {3695, 21483}, {3696, 26040}, {3701, 5815}, {3703, 24477}, {3713, 17811}, {3729, 9965}, {3752, 17299}, {3772, 17231}, {3886, 17784}, {3928, 4873}, {3936, 5226}, {3966, 26105}, {3969, 5435}, {4007, 5437}, {4035, 5219}, {4046, 4413}, {4082, 5223}, {4313, 16368}, {4344, 24552}, {4358, 5739}, {4359, 32087}, {4383, 5839}, {4387, 5698}, {4416, 30568}, {4417, 5748}, {4445, 5743}, {4450, 30332}, {4461, 21454}, {4488, 20078}, {4648, 31993}, {4656, 17272}, {4671, 5905}, {4702, 10385}, {4851, 5712}, {4869, 5249}, {4886, 30829}, {4942, 5852}, {4968, 11037}, {5080, 7381}, {5084, 5814}, {5175, 7270}, {5257, 25430}, {5278, 18230}, {5328, 5741}, {5372, 32849}, {5737, 17243}, {5932, 18632}, {6327, 9812}, {6703, 17293}, {9778, 32929}, {9799, 19645}, {9800, 11469}, {10445, 12555}, {16284, 20921}, {16439, 32862}, {17056, 17311}, {17151, 24177}, {17169, 30599}, {17240, 33116}, {17276, 22034}, {17280, 26065}, {17321, 34064}, {17360, 20942}, {17742, 21370}, {18623, 28739}, {19714, 26035}, {19785, 33172}, {19825, 26627}, {20234, 31598}, {21255, 23681}, {24248, 33085}, {24597, 33157}, {26034, 32915}, {26098, 32846}, {28605, 31995}, {28610, 32933}, {32919, 33163}, {33087, 33144}

X(34255) = reflection of X(20043) in X(2999)
X(34255) = complement of X(20043)
X(34255) = anticomplement of X(2999)
X(34255) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1219, 69}, {2297, 8}, {6574, 513}, {7050, 2}, {7091, 7}
X(34255) = X(4673)-Ceva conjugate of X(4461)
X(34255) = X(2255)-isoconjugate of X(14550)
X(34255) = crosssum of X(1015) and X(8662)
X(34255) = barycentric product X(i)*X(j) for these {i,j}: {1219, 28616}, {3596, 34046}
X(34255) = barycentric quotient X(i)/X(j) for these {i,j}: {936, 14550}, {3616, 34244}, {14551, 937}, {28616, 3672}, {34046, 56}
X(34255) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 145, 5256}, {2, 3187, 5222}, {2, 17019, 3616}, {2, 20043, 2999}, {2, 29616, 306}, {10, 17022, 2}, {69, 312, 329}, {75, 18141, 9776}, {319, 18743, 14555}, {345, 14829, 5744}, {1150, 17776, 5273}, {3687, 30567, 2}, {3912, 11679, 2}, {4358, 5739, 18228}, {4417, 28808, 5748}, {4671, 32863, 5905}, {5905, 32863, 21296}, {10327, 17135, 8}, {14829, 17233, 345}, {17292, 29841, 2}, {17294, 30567, 3687}, {18228, 32099, 5739}


X(34256) =  ISOGONAL CONJUGATE OF X(13528)

Barycentrics    a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+a^5 c+5 a^4 b c-6 a^3 b^2 c-6 a^2 b^3 c+5 a b^4 c+b^5 c-4 a^4 c^2+4 a^3 b c^2+8 a^2 b^2 c^2+4 a b^3 c^2-4 b^4 c^2-2 a^3 c^3-6 a^2 b c^3-6 a b^2 c^3-2 b^3 c^3+5 a^2 c^4-2 a b c^4+5 b^2 c^4+a c^5+b c^5-2 c^6) (a^6+a^5 b-4 a^4 b^2-2 a^3 b^3+5 a^2 b^4+a b^5-2 b^6-2 a^5 c+5 a^4 b c+4 a^3 b^2 c-6 a^2 b^3 c-2 a b^4 c+b^5 c-a^4 c^2-6 a^3 b c^2+8 a^2 b^2 c^2-6 a b^3 c^2+5 b^4 c^2+4 a^3 c^3-6 a^2 b c^3+4 a b^2 c^3-2 b^3 c^3-a^2 c^4+5 a b c^4-4 b^2 c^4-2 a c^5+b c^5+c^6) : :
X(34256) = 2*X[11]-X[10309]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29480.

X(34256) lies on the Feuerbach circumhyperbola and these lines: {4,24465}, {8,2829}, {11,10309}, {515,12641}, {1000,6938}, {1320,6001}, {2800,3680}, {6850,33898}

X(34256) = isogonal conjugate of X(13528)
X(34256) = antigonal image of X(10309)


X(34257) =  X(40)X(10692)∩X(1158)X(5552)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 29488.

X(34257) lies on thee lines: {40,10692}, {1158,5552}


X(34258) =  ISOGONAL CONJUGATE OF X(5019)

Barycentrics    b*c*(a*b + b^2 + 2*a*c + b*c)*(2*a*b + a*c + b*c + c^2) : :
Barycentrics    1/(a^2 + 4 R r) : :

For a construction of X(34258), see Dasari Naga Vijay Krishna, "On A Simple Construction of Triangle Centers X(8), X(197), X(K) & X(594)", Scientific Inquiry and Review, Vol. 2, Issue 3, July 2018. The point X(K) in the title is X(34258).

X(34258) lies on the Kiepert circumhyperbola, the cubic K1135, and these lines: {2, 314}, {4, 970}, {5, 3597}, {6, 7058}, {8, 181}, {10, 312}, {43, 2258}, {75, 226}, {76, 1211}, {83, 4383}, {98, 931}, {274, 5712}, {309, 8808}, {321, 3596}, {333, 573}, {386, 1010}, {1446, 6063}, {1751, 17277}, {2051, 5233}, {3030, 3038}, {3772, 20174}, {3944, 4647}, {4023, 10406}, {4049, 4823}, {4080, 28605}, {4104, 4385}, {4359, 30588}, {4671, 6539}, {5278, 24624}, {10471, 10478}, {17748, 28612}, {17758, 18134}, {18816, 27339}, {19701, 32014}, {19792, 30832}, {30116, 34064}, {30599, 31034}

X(34258) = isogonal conjugate of X(5019)
X(34258) = isotomic conjugate of X(940)
X(34258) = polar conjugate of X(4185)
X(34258) = isotomic conjugate of the anticomplement of X(5743)
X(34258) = isotomic conjugate of the complement of X(5739)
X(34258) = isotomic conjugate of the isogonal conjugate of X(941)
X(34258) = X(i)-cross conjugate of X(j) for these (i,j): {2476, 264}, {2517, 668}, {5743, 2}
X(34258) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5019}, {6, 1468}, {31, 940}, {32, 10436}, {48, 4185}, {56, 2268}, {163, 8672}, {184, 5307}, {604, 958}, {662, 8639}, {1106, 3713}, {1397, 11679}, {1415, 17418}, {2206, 31993}, {3714, 16947}
X(34258) = cevapoint of X(i) and X(j) for these (i,j): {2, 5739}, {6, 11337}, {8, 2345}, {386, 573}, {1086, 4801}
X(34258) = trilinear pole of line {523, 4391}
X(34258) = barycentric product X(i)*X(j) for these {i,j}: {75, 31359}, {76, 941}, {313, 5331}, {561, 2258}, {850, 931}, {959, 3596}, {4391, 32038}
X(34258) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 1468}, {2, 940}, {4, 4185}, {6, 5019}, {8, 958}, {9, 2268}, {75, 10436}, {92, 5307}, {312, 11679}, {321, 31993}, {346, 3713}, {512, 8639}, {522, 17418}, {523, 8672}, {931, 110}, {941, 6}, {959, 56}, {2258, 31}, {3701, 3714}, {4391, 23880}, {5331, 58}, {9534, 19283}, {31359, 1}, {32038, 651}, {32693, 1415}
X(34258) = {X(9534),X(9535)}-harmonic conjugate of X(14555)


X(34259) =  ISOGONAL CONJUGATE OF X(4185)

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

X(34259) lies on the Jerabek circumhyperbola, the cubics K506 and K1135, and these lines: on lines {2, 65}, {3, 1812}, {4, 970}, {6, 21}, {8, 15232}, {54, 6875}, {63, 73}, {64, 411}, {69, 22076}, {71, 78}, {72, 345}, {74, 931}, {184, 1798}, {219, 1791}, {280, 1903}, {348, 1439}, {386, 1245}, {1242, 6904}, {1243, 6824}, {1246, 17139}, {1265, 3690}, {1899, 18123}, {2276, 21874}, {2345, 3876}, {2476, 5743}, {3426, 6985}, {3486, 20018}, {3527, 3560}, {3878, 5292}, {5730, 16455}, {5739, 20029}, {5799, 6828}, {7131, 16574}, {10381, 30828}, {10974, 13725}, {11415, 15320}, {26703, 32693}

X(34259) = isogonal conjugate of X(4185)
X(34259) = isotomic conjugate of the polar conjugate of X(941)
X(34259) = X(i)-cross conjugate of X(j) for these (i,j): {2522, 1332}, {4047, 63}
X(34259) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4185}, {4, 1468}, {6, 5307}, {19, 940}, {25, 10436}, {34, 958}, {58, 1867}, {92, 5019}, {108, 17418}, {162, 8672}, {278, 2268}, {608, 11679}, {811, 8639}, {1435, 3713}, {1474, 31993}, {23880, 32674}
X(34259) = cevapoint of X(219) and X(7085)
X(34259) = trilinear pole of line {521, 647}
X(34259) = barycentric product X(i)*X(j) for these {i,j}: {63, 31359}, {69, 941}, {304, 2258}, {306, 5331}, {345, 959}, {521, 32038}, {525, 931}
X(34259) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 5307}, {3, 940}, {6, 4185}, {37, 1867}, {48, 1468}, {63, 10436}, {72, 31993}, {78, 11679}, {184, 5019}, {212, 2268}, {219, 958}, {521, 23880}, {647, 8672}, {652, 17418}, {931, 648}, {941, 4}, {959, 278}, {1260, 3713}, {2258, 19}, {3049, 8639}, {3694, 3714}, {5331, 27}, {31359, 92}, {32038, 18026}, {32693, 108}


X(34260) =  X(386)X(1245)∩X(940)X(1310)

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

X(34260) lies on the cubic K1135 and these lines: {386, 1245}, {940, 1310}, {941, 5739}, {959, 1036}, {2221, 27174}, {2354, 5256}, {3666, 14258}

X(34260) = isogonal conjugate of X(34261)
X(34260) = isogonal conjugate of the complement of X(30479)
X(34260) = X(i)-cross conjugate of X(j) for these (i,j): {6, 959}, {513, 1310}, {6589, 32691}
X(34260) = X(i)-isoconjugate of X(j) for these (i,j): {388, 2268}, {612, 940}, {958, 2285}, {1460, 11679}, {1468, 2345}, {3713, 4320}, {4185, 5227}, {4385, 5019}, {5307, 7085}
X(34260) = cevapoint of X(6) and X(1036)
X(34260) = barycentric product X(959)*X(30479)
X(34260) = barycentric quotient X(i)/X(j) for these {i,j}: {941, 2345}, {959, 388}, {1036, 958}, {1472, 1468}, {2221, 940}, {2258, 612}, {2339, 11679}, {5331, 1010}, {31359, 4385}


X(34261) =  X(3)X(37)∩X(6)X(10)

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

X(34261) lies on the Feuerbach circumhyperbola of the medial triangle, the cubic K321, and these lines: {1, 20227}, {2, 2221}, {3, 37}, {6, 10}, {9, 171}, {12, 478}, {45, 2305}, {55, 14749}, {100, 941}, {119, 5517}, {142, 3772}, {172, 19533}, {198, 5277}, {213, 965}, {214, 5110}, {219, 2295}, {332, 17316}, {427, 608}, {474, 2277}, {572, 30116}, {604, 10459}, {612, 1460}, {750, 1400}, {894, 3718}, {940, 3713}, {958, 5019}, {992, 16466}, {1010, 2303}, {1038, 8898}, {1107, 5120}, {1333, 19259}, {1376, 2092}, {1449, 11530}, {1573, 5042}, {1740, 19584}, {2268, 4185}, {2276, 11358}, {2300, 5710}, {2640, 19557}, {4254, 4386}, {4503, 6180}, {4648, 17060}, {5257, 32916}, {5276, 5749}, {5294, 19725}, {5712, 23600}, {5746, 26939}, {7522, 19730}, {9709, 21857}, {16408, 28244}, {17053, 25524}, {17056, 18642}, {17281, 19276}, {17733, 24325}, {24512, 33137}

X(34261) = isogonal conjugate of X(34260)
X(34261) = complement of X(30479)
X(34261) = complement of the isogonal conjugate of X(1460)
X(34261) = complement of the isotomic conjugate of X(388)
X(34261) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 958}, {109, 8678}, {388, 2887}, {604, 4657}, {612, 1329}, {1038, 1368}, {1402, 4205}, {1460, 10}, {2285, 141}, {2286, 18589}, {2303, 21246}, {2345, 21244}, {2484, 26932}, {4320, 2886}, {5323, 3741}, {7365, 17046}, {8646, 1146}, {8678, 124}, {8898, 17052}, {10375, 23332}, {14594, 21260}
X(34261) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 958}, {100, 8678}
X(34261) = X(i)-isoconjugate of X(j) for these (i,j): {959, 2339}, {2221, 31359}
X(34261) = crosspoint of X(2) and X(388)
X(34261) = crosssum of X(6) and X(1036)
X(34261) = barycentric product X(i)*X(j) for these {i,j}: {388, 958}, {612, 10436}, {940, 2345}, {1468, 4385}, {2285, 11679}, {2303, 31993}, {3713, 7365}, {3714, 5323}, {5227, 5307}, {14594, 17418}
X(34261) = barycentric quotient X(i)/X(j) for these {i,j}: {612, 31359}, {958, 30479}, {1460, 959}, {2268, 2339}, {5019, 2221}
X(34261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {975, 1766, 37}, {5711, 5783, 6}


X(34262) =  X(2)X(573)∩X(8)X(14973)

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

X(34262) lies on the cubic K1135 and these lines: {2,573}, {6,34267}, {8, 14973}, {941,34270}, {2345,34273}, {5739,34271}, {11337,34275}, {34258,34264}, {34259,34265}, {34260,34263}, {34272,34276}

X(34262) = barycentric product X(20028)*X(26115)
X(34262) = barycentric quotient X(i)/X(j) for these {i,j}: {4264, 572}, {26115, 17751}


X(34263) =  X(2)X(2995)∩X(4)X(941)

Barycentrics    a*(a + b - c)*(a - b + c)*(a*b + b^2 + 2*a*c + b*c)*(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(34263) lies on the cubic K1135 and these lines: {2, 2995}, {4, 941}, {6, 959}, {386, 2285}, {478, 16049}, {1295, 32693}

X(34263) = X(i)-isoconjugate of X(j) for these (i,j): {2268, 8048}, {3435, 11679}
X(34263) = barycentric product X(i)*X(j) for these {i,j}: {959, 3436}, {6588, 32038}, {21147, 31359}
X(34263) = barycentric quotient X(i)/X(j) for these {i,j}: {197, 958}, {205, 2268}, {478, 940}, {959, 8048}, {1766, 11679}, {6588, 23880}, {17408, 4185}, {21147, 10436}


X(34264) =  X(386)X(941)∩X(959)X(2345)

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

X(34264) lies on the cubic K1135 and these lines: {386,941}, {959,2345}, {34258,34262}, {34259,34269}, {34263,34274}


X(34265) =  X(386)X(1010)∩X(388)X(959)

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

X(34265) lies on the cubic K1135 and these lines: {386, 1010}, {388, 959}, {835, 958}, {941, 2345}, {4385, 31359}

X(34265) = X(i)-cross conjugate of X(j) for these (i,j): {10, 31359}, {522, 835}
X(34265) = X(i)-isoconjugate of X(j) for these (i,j): {386, 1468}, {5019, 28606}
X(34265) = barycentric quotient X(i)/X(j) for these {i,j}: {941, 386}, {2214, 1468}, {31359, 28606}


X(34266) =  X(2)X(17903)∩X(4)X(6)

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

X(34266) lies on the cubic K1135 and these lines: {2, 17903}, {4, 6}, {19, 5292}, {314, 17907}, {406, 941}, {2331, 5747}, {16318, 19544}


X(34267) =  BARYCENTRIC PRODUCT X(2051)*X(23512)

Barycentrics    (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)*(a^6 + a^5*b - a^2*b^4 - a*b^5 + a^5*c + 3*a^4*b*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c - b^5*c - 2*a^3*b*c^2 + 2*a^2*b^2*c^2 - 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(34267) lies on the cubic K1135 and these lines: {2, 34269}, {4, 386}, {6,34262}, {8,34274}, {2345,34277}

X(34267) = barycentric product X(2051)*X(23512)
X(34267) = barycentric quotient X(i)/X(j) for these {i,j}: {1610, 2975}, {15267, 20617}, {23512, 14829}


X(34268) =  X(4)X(959)∩X(6)X(19607)

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

X(34268) lies on the cubic K1135 and these lines: {4,959}, {6,19607}, {941,34277}, {34260,34269}, {34265,34274}


X(34269) =  (name pending)

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

X(34269) lies on the cubic K1135 and these lines: {2,34267}, {8,573}, {5739,34273}, {11337,34271}, {34258,34270}, {34259,34264}, {34260,34268}, {34263,34265}, {34266,34275}


X(34270) =  (name pending)

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

X(34270) lies on the cubic K1135 and these lines: {386,959}, {941,34262}, {34258,34269}, {34259,34274}


X(34271) =  (name pending)

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

X(34271) lies on the cubic K1135 and these lines: {386,2345}, {5739,34262}, {11337,34269}, {34266,34274}


X(34272) =  (name pending)

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

X(34272) lies on the cubic K1135 and these lines: {4,17869}, {6,34277}, {34262,34276}, {34274,34275}


X(34273) =  (name pending)

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

X(34273) lies on the cubic K1135 and these lines: {2345,34262}, {5739,34269}, {11337,34274}


X(34274) =  (name pending)

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

X(34274) lies on the cubic K1135 and on these lines: {8,34267}, {573,23361}, {11337,34273}, {34259,34270}, {34263,34264}, {34265,34268}, {34266,34271}, {34272,34275}


X(34275) =  (name pending)

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

X(34275) lies on the cubic K1135 and on these lines: {386,5739}, {11337,34262}, {34266,34269}, {34272,34274}


X(34276) =  X(6)-CROSS CONJUGATE OF X(2345)

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

X(34276) lies on the cubic K1135 and these lines: {386,34266}, {2345,11337}, {34262,34272}

X(34276) = X(6)-cross conjugate of X(2345)


X(34277) =  ISOGONAL CONJUGATE OF X(478)

Barycentrics    (a - b - c)*(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(34277) lies on the cubics K555 and K1135 and these lines: {2, 17903}, {8, 197}, {25, 2968}, {63, 573}, {78, 27379}, {280, 4194}, {345, 27540}, {348, 17080}, {1812, 28921}, {1993, 23122}, {2417, 4391}, {17880, 20266}, {30680, 33168}

X(34277) = isogonal conjugate of X(478)
X(34277) = polar conjugate of X(14257)
X(34277) = isogonal conjugate of the complement of X(8048)
X(34277) = X(i)-cross conjugate of X(j) for these (i,j): {6, 8}, {1854, 7}, {1858, 314}, {23983, 4391}
X(34277) = X(i)-isoconjugate of X(j) for these (i,j): {1, 478}, {6, 21147}, {7, 205}, {34, 22132}, {48, 14257}, {56, 1766}, {57, 197}, {63, 17408}, {109, 6588}, {604, 3436}, {1397, 20928}, {1400, 16049}, {1408, 21074}, {1415, 21186}
X(34277) = cevapoint of X(i) and X(j) for these (i,j): {6, 3435}, {650, 2968}
X(34277) = cevapoint of circumcircle intercepts of excircles radical circle
X(34277) = trilinear pole of line {521, 14312}
X(34277) = barycentric product X(i)*X(j) for these {i,j}: {8, 8048}, {3435, 3596}
X(34277) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 21147}, {4, 14257}, {6, 478}, {8, 3436}, {9, 1766}, {21, 16049}, {25, 17408}, {41, 205}, {55, 197}, {219, 22132}, {312, 20928}, {522, 21186}, {650, 6588}, {2321, 21074}, {2968, 123}, {3435, 56}, {8048, 7}


X(34278) =  (name pending)

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

X(34278) lies on the cubic K321 and these lines: {2,34280}, {6,34281}, {56,34261}, {13478,16435}


X(34279) =  X(2)X(478)∩X(3)X(960)

Barycentrics    a*(a - b - c)*(a^2 + a*b + a*c + 2*b*c)*(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(34279) lies on the cubic K321 and these lines: {2, 478}, {3, 960}, {6, 19608}, {1812, 28921}

X(34279) = X(5019)-cross conjugate of X(958)
X(34279) = X(i)-isoconjugate of X(j) for these (i,j): {478, 31359}, {941, 21147}, {959, 1766}, {21186, 32693}
X(34279) = barycentric product X(958)*X(8048)
X(34279) = barycentric quotient X(i)/X(j) for these {i,j}: {958, 3436}, {1468, 21147}, {2268, 1766}, {3435, 959}, {4185, 14257}, {5019, 478}, {11679, 20928}, {17418, 21186}


X(34280) =  X(2)X(2051)∩X(478)X(2221)

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

X(34280) lies on the cubic K321 and these lines: {3, 2051}, {478, 2221}, {19684, 20028}

X(34280) = X(4264)-complementary conjugate of X(15267)


X(34281) =  X(2)X(58)∩X(3)X(31)

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

X(34281) lies on the cubic K321 and these lines: {2, 58}, {3, 31}, {109, 959}, {117, 5230}, {171, 16454}, {213, 22345}, {238, 16342}, {404, 5156}, {405, 10457}, {478, 603}, {750, 16458}, {940, 958}, {992, 4275}, {1150, 5247}, {1191, 23404}, {1408, 5035}, {3736, 16452}, {3915, 4428}, {4225, 4257}, {4252, 13738}, {5398, 19513}, {5711, 16357}, {10458, 16289}, {16347, 17127}, {16456, 17124}, {16457, 17125}, {16468, 27663}, {17123, 19334}, {17126, 19284}, {17187, 19762}, {18792, 19769}, {19518, 19734}, {27623, 27627}

X(34281) = X(i)-Ceva conjugate of X(j) for these (i,j): {58, 1468}, {109, 834}
X(34281) = barycentric product X(i)*X(j) for these {i,j}: {386, 940}, {1468, 28606}, {5019, 5224}


X(34282) =  X(2)X(4263)∩X(4)X(69)

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

X(34282) lies on these lines: {2, 4263}, {4, 69}, {7, 17143}, {8, 30092}, {75, 519}, {86, 386}, {274, 3945}, {312, 4416}, {313, 17360}, {319, 3596}, {321, 17364}, {350, 17272}, {394, 7058}, {516, 4673}, {573, 14829}, {668, 32099}, {991, 1043}, {1269, 17361}, {1444, 7782}, {1654, 30830}, {1742, 3886}, {1909, 10447}, {2979, 17135}, {3631, 18144}, {3663, 17144}, {3686, 20923}, {3718, 33939}, {3902, 17579}, {3948, 17343}, {3963, 17373}, {4043, 17347}, {4358, 17331}, {4417, 24220}, {4441, 21296}, {4445, 17790}, {4675, 20174}, {4888, 32104}, {5224, 30939}, {5232, 18140}, {5933, 31643}, {10453, 21746}, {10472, 26110}, {17178, 24598}, {17202, 32782}, {17233, 29712}, {17234, 29446}, {17251, 25660}, {17271, 18146}, {17294, 17787}, {17333, 22016}, {17346, 18137}, {17363, 20891}, {17375, 20913}, {17861, 20955}, {17863, 33934}, {20892, 29617}, {23659, 30942}

X(34282) = anticomplement of X(4263)
X(34282) = isotomic conjugate of the isogonal conjugate of X(4189)
X(34282) = barycentric product X(76)*X(4189)
X(34282) = barycentric quotient X(4189)/X(6)
X(34282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 314, 76}


X(34283) =  X(6)X(76)∩X(9)X(1909)

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

X(34283) lies on these lines: {6, 76}, {9, 1909}, {75, 527}, {86, 30830}, {183, 5120}, {193, 314}, {194, 2092}, {274, 966}, {305, 5276}, {312, 3879}, {313, 3758}, {321, 17363}, {329, 30710}, {330, 17053}, {350, 1449}, {384, 16946}, {385, 5019}, {538, 4263}, {668, 2345}, {894, 3596}, {1015, 26107}, {1030, 7782}, {1269, 3759}, {1333, 6179}, {1743, 3761}, {1975, 4254}, {2220, 3972}, {2286, 14612}, {2300, 24514}, {2321, 24524}, {3060, 17165}, {3169, 3729}, {3247, 25303}, {3589, 18144}, {3664, 20923}, {3688, 32937}, {3760, 16667}, {3883, 12527}, {3945, 28809}, {3948, 17379}, {3963, 17350}, {3975, 10436}, {4043, 17377}, {4261, 7757}, {4270, 33296}, {4358, 17391}, {4385, 5847}, {4410, 17348}, {4713, 21785}, {4721, 20228}, {4754, 30092}, {5042, 7751}, {5069, 7786}, {5124, 7771}, {5257, 31997}, {5275, 8033}, {5283, 26110}, {5750, 6376}, {5839, 17143}, {12263, 18194}, {17049, 24349}, {17157, 23659}, {17349, 20913}, {17352, 18143}, {17353, 20917}, {17354, 18040}, {17355, 17786}, {17364, 20891}, {17369, 30473}, {17378, 18137}, {17381, 18133}, {17389, 22016}, {20146, 27269}, {20227, 25994}, {24598, 26772}, {27111, 31234}

X(34283) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {941, 21289}, {2258, 2896}, {31359, 1369}
X(34283) = crosssum of X(1084) and X(8639)
X(34283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 3770, 76}, {894, 3765, 3596}


X(34284) =  ISOTOMIC CONJUGATE OF X(941)

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

X(34284) lies on these lines: {1, 4441}, {2, 39}, {4, 16747}, {6, 4754}, {7, 8}, {10, 3761}, {21, 1975}, {32, 16919}, {81, 19281}, {86, 964}, {99, 4189}, {141, 33840}, {142, 29966}, {145, 17143}, {148, 33824}, {183, 404}, {264, 4200}, {279, 6063}, {286, 4198}, {304, 321}, {308, 27005}, {312, 5308}, {313, 19874}, {314, 3945}, {315, 2475}, {325, 2476}, {330, 26801}, {332, 5736}, {333, 379}, {339, 34120}, {346, 27253}, {348, 349}, {350, 3616}, {384, 16998}, {385, 16915}, {442, 3933}, {475, 1235}, {519, 32104}, {536, 24656}, {668, 3617}, {894, 17033}, {941, 1218}, {966, 3770}, {1007, 6933}, {1042, 9312}, {1043, 14828}, {1078, 4188}, {1089, 33942}, {1111, 20345}, {1125, 3760}, {1150, 17206}, {1219, 34018}, {1269, 17321}, {1334, 3729}, {1434, 14829}, {1446, 7182}, {1475, 17026}, {1509, 14996}, {1654, 17680}, {1698, 6381}, {1920, 28659}, {1921, 4699}, {2176, 24330}, {2275, 21264}, {2295, 4363}, {2296, 17018}, {2896, 33823}, {3208, 4659}, {3241, 17144}, {3263, 4385}, {3314, 33841}, {3403, 17116}, {3552, 16996}, {3619, 18143}, {3664, 10447}, {3673, 13725}, {3679, 25278}, {3691, 4384}, {3701, 30758}, {3702, 18156}, {3734, 16920}, {3739, 21615}, {3741, 24215}, {3765, 4359}, {3780, 4361}, {3785, 4190}, {3924, 24291}, {3975, 19804}, {3980, 4039}, {4000, 26965}, {4044, 16831}, {4202, 5224}, {4253, 29433}, {4357, 23536}, {4386, 4400}, {4410, 4643}, {4470, 17790}, {4474, 20907}, {4647, 33936}, {4648, 18157}, {4671, 29569}, {4692, 33937}, {4772, 10009}, {5046, 11185}, {5141, 7752}, {5254, 17550}, {5276, 7754}, {5277, 7751}, {5361, 24587}, {5550, 30963}, {5712, 18138}, {5839, 20174}, {6175, 7788}, {6337, 6910}, {6376, 9780}, {6390, 7483}, {6542, 20432}, {6856, 32818}, {6857, 32817}, {6871, 32816}, {6872, 32815}, {6904, 15589}, {6921, 34229}, {7200, 20446}, {7229, 17787}, {7750, 17579}, {7766, 16913}, {7767, 11112}, {7770, 33854}, {7773, 17577}, {7776, 17532}, {7779, 33030}, {7782, 17548}, {7783, 17684}, {7793, 17693}, {7906, 33045}, {9317, 16822}, {9902, 21443}, {10446, 15971}, {10449, 10471}, {10479, 16887}, {11111, 32822}, {11114, 32819}, {11329, 26243}, {14376, 28426}, {15419, 20948}, {16708, 18141}, {16709, 18147}, {16713, 26961}, {16827, 24514}, {16906, 31090}, {16916, 17000}, {16917, 16997}, {16991, 17673}, {17048, 24629}, {17050, 30036}, {17103, 19271}, {17140, 20247}, {17152, 17753}, {17164, 24282}, {17169, 30962}, {17234, 17672}, {17271, 17679}, {17343, 26079}, {17379, 30940}, {17676, 33940}, {17756, 27020}, {18031, 27304}, {18895, 30669}, {19767, 33296}, {19818, 29833}, {19877, 20943}, {20518, 21132}, {20568, 30590}, {20569, 30589}, {20893, 32847}, {20917, 29611}, {21024, 30945}, {21301, 23807}, {21384, 24592}, {21416, 30660}, {23115, 28718}, {23682, 31330}, {24199, 30030}, {24275, 25497}, {24790, 30107}, {25264, 27255}, {26244, 33830}, {26540, 26550}, {26840, 33934}, {27801, 31025}, {28653, 30596}, {29960, 30949}, {31130, 33933}, {33931, 33943}

X(34284) = isotomic conjugate of X(941)
X(34284) = anticomplement of X(5283)
X(34284) = anticomplement of the isotomic conjugate of X(1218)
X(34284) = isotomic conjugate of the anticomplement of X(10472)
X(34284) = isotomic conjugate of the isogonal conjugate of X(940)
X(34284) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {785, 514}, {1218, 6327}, {2296, 69}
X(34284) = X(1218)-Ceva conjugate of X(2)
X(34284) = X(i)-cross conjugate of X(j) for these (i,j): {10472, 2}, {31993, 10436}
X(34284) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2258}, {31, 941}, {32, 31359}, {41, 959}, {213, 5331}, {663, 32693}, {798, 931}
X(34284) = cevapoint of X(10436) and X(11679)
X(34284) = crosssum of X(2978) and X(3271)
X(34284) = crossdifference of every pair of points on line {669, 3063}
X(34284) = barycentric product X(i)*X(j) for these {i,j}: {75, 10436}, {76, 940}, {85, 11679}, {274, 31993}, {304, 5307}, {305, 4185}, {561, 1468}, {670, 8672}, {958, 6063}, {1218, 10472}, {1502, 5019}, {2268, 20567}, {4554, 23880}, {4572, 17418}, {4609, 8639}
X(34284) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2258}, {2, 941}, {7, 959}, {75, 31359}, {86, 5331}, {99, 931}, {651, 32693}, {940, 6}, {958, 55}, {1468, 31}, {1867, 1824}, {2268, 41}, {3713, 220}, {3714, 210}, {4185, 25}, {4554, 32038}, {5019, 32}, {5307, 19}, {8639, 669}, {8672, 512}, {10436, 1}, {10472, 5283}, {11679, 9}, {17418, 663}, {18078, 3875}, {23880, 650}, {31993, 37}
X(34284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 20888, 4441}, {2, 76, 18135}, {2, 4352, 16705}, {2, 20081, 1655}, {7, 8, 17137}, {75, 85, 20911}, {75, 1909, 8}, {76, 274, 2}, {76, 305, 1228}, {76, 30022, 18152}, {76, 30092, 28660}, {274, 310, 16705}, {350, 31997, 3616}, {1654, 17680, 26085}, {1975, 16992, 21}, {3761, 32092, 10}, {3926, 26541, 18135}, {4968, 20880, 75}, {5277, 7751, 17001}, {7754, 11321, 5276}, {16749, 30599, 274}, {16917, 17129, 16997}, {16919, 17002, 32}, {17000, 17128, 16916}, {17144, 25303, 3241}, {26035, 26978, 2}, {27318, 31276, 2}, {33933, 33941, 31130}


X(34285) =  ISOGONAL CONJUGATE OF X(33586)

Barycentrics    1/(a^4+2a^2(b^2+c^2)-3b^4+2b^2c^2-3c^4) : :
Barycentrics    1/(SA^2-2 SB SC) : :

Let A'B'C' and A"B"C" be the cevian and circumcevian triangles of the orthocenter. Let La be the radical axis of circles with segments BC and A'A" as diameters, and define Lb and Lc cyclically. The triangle formed by lines La, Lb, Lc is perspective to ABC, and the perspector is X(34285). (Angel Montesdeoca, September 15, 2019)

See Angel Montesdeoca, HG140919.

X(34285) lies on these lines: {32,8801}, {141,631}, {427,3087}, {1502,3785}, {3515,33582}, {8024,15589}

X(34285) = isogonal conjugate of X(33586)
X(34285) = isotomic conjugate of X(32816)


X(34286) =  X(3)X(10002)∩X(4)X(1192)

Barycentrics    (a^4 - (b^2 - c^2)^2)^2 (9 a^8 - 24 a^6 (b^2 + c^2) + 18 a^4 (b^2 + c^2)^2 - 16 a^2 b^2 c^2 (b^2 + c^2) - (b^2 - c^2)^2 (3 b^4 + 2 b^2 c^2 + 3 c^4)) : :

See Angel Montesdeoca, HG140919.

X(34286) lies on thee lines: {3,10002}, {4,1192}, {20,6525}, {30,6523}, {107,3146}, {381,33531}, {393,3053}, {1503,3183}, {2060,31377}, {2883,3079}, {3346,9530}, {5667,12250}, {5893,6621}, {5895,6616}, {6193,8057}, {6618,13568}, {13346,32713}, {13450,18533}, {14361,17845}


X(34287) =  X(3)X(1075)∩X(97)X(3164)

Barycentrics    (a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 - a^10*c^2 - a^8*b^2*c^2 + 2*a^6*b^4*c^2 + 2*a^4*b^6*c^2 - a^2*b^8*c^2 - b^10*c^2 + 4*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 + 4*b^8*c^4 - 6*a^6*c^6 + 2*a^4*b^2*c^6 + 2*a^2*b^4*c^6 - 6*b^6*c^6 + 4*a^4*c^8 + a^2*b^2*c^8 + 4*b^4*c^8 - a^2*c^10 - b^2*c^10)*(a^10*b^2 - 4*a^8*b^4 + 6*a^6*b^6 - 4*a^4*b^8 + a^2*b^10 - a^10*c^2 + a^8*b^2*c^2 + 2*a^6*b^4*c^2 - 2*a^4*b^6*c^2 - a^2*b^8*c^2 + b^10*c^2 + 4*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 - 4*b^8*c^4 - 6*a^6*c^6 - 2*a^4*b^2*c^6 + 2*a^2*b^4*c^6 + 6*b^6*c^6 + 4*a^4*c^8 + a^2*b^2*c^8 - 4*b^4*c^8 - a^2*c^10 + b^2*c^10) : :
Barycentrics    S^4 + (64 R^4-8 R^2 SB-8 R^2 SC-2 SB SC-16 R^2 SW+2 SB SW+2 SC SW)S^2 -256 R^8 -128 R^6 SB-128 R^6 SC-96 R^4 SB SC+256 R^6 SW+96 R^4 SB SW+96 R^4 SC SW+32 R^2 SB SC SW-96 R^4 SW^2-24 R^2 SB SW^2-24 R^2 SC SW^2-2 SB SC SW^2+16 R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4 : :
Barycentrics    1/(sec^2 A - sec^2 B - sec^2 C) : :

See Antreas Hatzipolakis, Ercole Suppa and Peter Moses, Hyacinthos 29498 and Hyacinthos 29499.

X(34287) lies on the circumconics {{A,B,C,X(2),X(3)}}, {{A,B,C,X(4),X(8613)}}, {{A,B,C,X(64),X(8794)}}, {{A,B,C,X(92),X(6360)}}, {{A,B,C,X(253),X(8795)}}, {{A,B,C,X(275),X(1294)}}, {{A,B,C,X(324),X(3164)}}, {{A,B,C,X(393),X(17849)}}, {{A,B,C,X(459),X(15412)}}, the curve Q124 and these lines: {3, 1075}, {97, 3164}, {394, 8613}, {14941, 34186}

X(34287) = polar conjugate of X(1075)
X(34287) = cyclocevian conjugate of X(68)
X(34287) = isogonal conjugate of tangential isotomic conjugate of X(3)
X(34287) = isotomic conjugate of the anticomplement of X(2052)
X(34287) = polar conjugate of the isogonal conjugate of X(13855)
X(34287) = X(13855)-anticomplementary conjugate of X(21270)
X(34287) = X(i)-cross conjugate of X(j) for these (i,j): {2052, 2}, {15318, 253}
X(34287) = cevapoint of X(216) and X(13322)
X(34287) = barycentric product X(264)*X(13855)
X(34287) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 1075}, {13855, 3}


X(34288) =  ISOGONAL CONJUGATE OF X(15066)

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

X(34288) lies on the circumconics {{A,B,C,X(2),X(6)}}, {{A,B,C,X(4),X(30)}}, the cubic K055, and these lines: {2, 3003}, {4, 6128}, {6, 30}, {25, 1990}, {32, 3163}, {37, 10056}, {111, 1302}, {230, 21448}, {251, 33872}, {263, 2393}, {376, 5063}, {393, 33885}, {477, 32681}, {493, 32787}, {494, 32788}, {523, 2433}, {543, 6096}, {566, 30537}, {588, 19054}, {589, 19053}, {800, 2165}, {1249, 8882}, {1370, 3108}, {1383, 5304}, {1976, 19136}, {1989, 3767}, {1992, 2987}, {2790, 6034}, {2793, 10103}, {2998, 19570}, {3471, 12106}, {3815, 32216}, {5112, 5486}, {5158, 7753}, {5309, 18487}, {5319, 7530}, {8770, 16310}, {9300, 31152}, {15262, 18559}, {18573, 31401}, {33630, 33631}

X(34288) = isogonal conjugate of X(15066)
X(34288) = isotomic conjugate of X(32833)
X(34288) = isotomic conjugate of the anticomplement of X(5309)
X(34288) = polar conjugate of the isotomic conjugate of X(4846)
X(34288) = X(i)-cross conjugate of X(j) for these (i,j): {5309, 2}, {18487, 1989}
X(34288) = X(i)-isoconjugate of X(j) for these (i,j): {1, 15066}, {31, 32833}, {63, 378}, {75, 5063}, {163, 30474}, {662, 8675}, {1959, 11653}, {2167, 5891}, {2349, 10564}
X(34288) = trilinear pole of line {512, 1637}
X(34288) = crossdifference of every pair of points on line {8675, 10564}
X(34288) = barycentric product X(i)*X(j) for these {i,j}: {4, 4846}, {523, 1302}, {850, 32738}
X(34288) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 32833}, {6, 15066}, {25, 378}, {32, 5063}, {51, 5891}, {512, 8675}, {523, 30474}, {1302, 99}, {1495, 10564}, {1976, 11653}, {4846, 69}, {32738, 110}


X(34289) =  ISOGONAL CONJUGATE OF X(5063)

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

X(34289) lies on the Kiepert circumhyperbola, the circumconic {{A,B,C,X(6),X(2501)}}, and these lines: {2, 3003}, {4, 4846}, {6, 2986}, {76, 3580}, {96, 6642}, {98, 1302}, {262, 858}, {264, 16080}, {275, 5422}, {324, 459}, {801, 1993}, {850, 2394}, {5392, 13567}, {5466, 30735}, {6504, 11433}, {7464, 9159}, {7578, 15018}, {8781, 11059}, {14484, 31099}, {14492, 31133}, {14494, 16051}

X(34289) = isogonal conjugate of X(5063)
X(34289) = isotomic conjugate of X(15066)
X(34289) = polar conjugate of X(378)
X(34289) = polar conjugate of the isogonal conjugate of X(4846)
X(34289) = X(381)-cross conjugate of X(264)
X(34289) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5063}, {31, 15066}, {48, 378}, {163, 8675}, {560, 32833}, {1755, 11653}, {2148, 5891}, {2159, 10564}
X(34289) = cevapoint of X(6) and X(6644)
X(34289) = trilinear pole of line {523, 11799}
X(34289) = barycentric product X(i)*X(j) for these {i,j}: {264, 4846}, {850, 1302}
X(34289) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 15066}, {4, 378}, {5, 5891}, {6, 5063}, {30, 10564}, {76, 32833}, {98, 11653}, {381, 4550}, {523, 8675}, {850, 30474}, {1302, 110}, {4846, 3}, {32681, 32640}, {32738, 1576}


X(34290) =  X(2)X(512)∩X(30)X(21733)

Barycentrics    (b^2 - c^2)*(-a^6 + 2*a^4*b^2 - 3*a^2*b^4 + 2*a^4*c^2 + a^2*b^2*c^2 + b^4*c^2 - 3*a^2*c^4 + b^2*c^4) : :
X(34290) = X[69] + 2 X[22260], X[5652] - 4 X[11182], 3 X[5652] - 4 X[11183], 3 X[11182] - X[11183]

X(34290) lies on these lines: {2, 512}, {30, 21733}, {69, 22260}, {351, 14084}, {381, 1499}, {523, 599}, {524, 9178}, {525, 8029}, {542, 5653}, {669, 11328}, {671, 690}, {694, 804}, {850, 20023}, {868, 879}, {882, 2396}, {924, 23327}, {1640, 1853}, {1641, 2444}, {1648, 3143}, {1992, 9171}, {2408, 33921}, {2780, 20126}, {2793, 19905}, {3800, 11123}, {9135, 11176}, {9147, 9208}, {9168, 12073}, {9175, 11179}, {11186, 25423}, {14977, 33919}

X(34290) = anticomplement of X(11183)
X(34290) = reflection of X(i) in X(j) for these {i,j}: {2, 11182}, {1640, 10278}, {1992, 9171}, {5652, 2}, {9135, 11176}, {9147, 9208}, {11179, 9175}
X(34290) = crosspoint of X(i) and X(j) for these (i,j): {98, 892}, {671, 18829}
X(34290) = crosssum of X(i) and X(j) for these (i,j): {187, 5027}, {351, 511}
X(34290) = crossdifference of every pair of points on line {2030, 3231}
X(34290) = barycentric product X(523)*X(10754)
X(34290) = barycentric quotient X(10754)/X(99)


X(34291) =  X(2)X(523)∩X(3)X(512)

Barycentrics    a^2*(b^2 - c^2)*(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 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 - 2*c^6) : :
X(34291) = X[684] + 2 X[6132], 5 X[1656] - 2 X[23105], 2 X[8552] + X[21731]

X(34291) lies on on Thomson-Gibert-Moses hyperbola and these lines: {2, 523}, {3, 512}, {6, 647}, {110, 351}, {154, 924}, {160, 669}, {520, 3167}, {525, 5654}, {574, 10097}, {690, 5655}, {804, 6054}, {842, 7418}, {878, 14355}, {879, 15000}, {1510, 6030}, {1656, 23105}, {2395, 3815}, {2493, 14998}, {2780, 8552}, {2799, 11622}, {3005, 5888}, {3566, 5656}, {3906, 32447}, {5544, 9171}, {5648, 9003}, {5653, 9155}, {7777, 31296}, {8574, 9605}, {9126, 9517}, {11184, 23878}, {11186, 34099}, {14417, 15131}, {14687, 33752}, {15421, 15760}, {20580, 32605}

X(34291) = midpoint of X(351) and X(684)
X(34291) = reflection of X(351) in X(6132)
X(34291) = Thomson isogonal conjugate of X(7422)
X(34291) = X(5649)-Ceva conjugate of X(6)
X(34291) = crosspoint of X(110) and X(842)
X(34291) = crosssum of X(i) and X(j) for these (i,j): {512, 2493}, {523, 542}
X(34291) = crossdifference of every pair of points on line {30, 115} (the Newton line of trapezoid X(13)X(15)X(14)X(16))
X(34291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {647, 10567, 6041}, {2421, 15329, 5467}, {5968, 9213, 9178}, {6041, 10567, 6}, {9717, 32112, 14380}


X(34292) =  MIDPOINT OF X(26) AND X(23709)

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

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 29515.

X(34292) lies on these lines: {2,13467}, {3,54}, {26,23709}, {30,31867}, {186,14978}, {1658,32428}, {2070,32551}, {6368,14809}, {7502,25043}, {7512,14652}, {11264,23195}

X(34292) = midpoint of X(26) and X(23709)


X(34293) =  X(8)X(153)∩X(11)X(18239)

Barycentrics    a (a^11 b-3 a^10 b^2-a^9 b^3+11 a^8 b^4-6 a^7 b^5-14 a^6 b^6+14 a^5 b^7+6 a^4 b^8-11 a^3 b^9+a^2 b^10+3 a b^11-b^12+a^11 c-2 a^10 b c+6 a^9 b^2 c-9 a^8 b^3 c-18 a^7 b^4 c+40 a^6 b^5 c+8 a^5 b^6 c-46 a^4 b^7 c+9 a^3 b^8 c+18 a^2 b^9 c-6 a b^10 c-b^11 c-3 a^10 c^2+6 a^9 b c^2+8 a^8 b^2 c^2-14 a^6 b^4 c^2-36 a^5 b^5 c^2+24 a^4 b^6 c^2+48 a^3 b^7 c^2-23 a^2 b^8 c^2-18 a b^9 c^2+8 b^10 c^2-a^9 c^3-9 a^8 b c^3+24 a^6 b^3 c^3+6 a^5 b^4 c^3+22 a^4 b^5 c^3-32 a^3 b^6 c^3-40 a^2 b^7 c^3+27 a b^8 c^3+3 b^9 c^3+11 a^8 c^4-18 a^7 b c^4-14 a^6 b^2 c^4+6 a^5 b^3 c^4-12 a^4 b^4 c^4-14 a^3 b^5 c^4+22 a^2 b^6 c^4+42 a b^7 c^4-23 b^8 c^4-6 a^7 c^5+40 a^6 b c^5-36 a^5 b^2 c^5+22 a^4 b^3 c^5-14 a^3 b^4 c^5+44 a^2 b^5 c^5-48 a b^6 c^5-2 b^7 c^5-14 a^6 c^6+8 a^5 b c^6+24 a^4 b^2 c^6-32 a^3 b^3 c^6+22 a^2 b^4 c^6-48 a b^5 c^6+32 b^6 c^6+14 a^5 c^7-46 a^4 b c^7+48 a^3 b^2 c^7-40 a^2 b^3 c^7+42 a b^4 c^7-2 b^5 c^7+6 a^4 c^8+9 a^3 b c^8-23 a^2 b^2 c^8+27 a b^3 c^8-23 b^4 c^8-11 a^3 c^9+18 a^2 b c^9-18 a b^2 c^9+3 b^3 c^9+a^2 c^10-6 a b c^10+8 b^2 c^10+3 a c^11-b c^11-c^12) : :
X(34293) = X[11]+X[18239], X[5083]-2*X[12608], 2*X[6667]-X[18238], X[14740]-2*X[32159]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29517.

X(34293) lies on these lines: {8,153}, {11,18239}, {515,15558}, {960,2829}, {971,6713}, {1490,10058}, {2801,24389}, {3036,6001}, {5083,12608}, {6260,10265}, {6261,11715}, {6667,18238}, {14740,32159}

X(34293) = reflection of X(14740) in X(32159)


X(34294) =  X(6)X(17500)∩X(53)X(6531)

Barycentrics    (a^2+b^2) (b-c)^2 (b+c)^2 (a^2+c^2) : :
Barycentrics    (6 R^2 SB+6 R^2 SC+2 SB SC+4 R^2 SW-SB SW-SC SW)S^2 + 2 SB SC SW^2-SB SW^3-SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 29518.

X(34294) lies on these lines: {6,17500}, {53,6531}, {76,25322}, {83,597}, {115,804}, {141,308}, {251,1989}, {338,3124}, {524,20022}, {1086,4374}, {1211,18096}, {1799,13468}, {1990,21459}, {2872,6784}, {3051,30505}, {3589,18092}, {3613,8265}, {6543,18082}, {6748,10550}, {10130,11168}, {16889,23897}, {17056,18703}, {18088,23903}, {18091,23905}, {18104,23917}, {18105,31644}, {27376,32713}

X(34294) = X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {115,1084,7668}, {308,16890,141}, {10549,32085,53}


X(34295) =  X(30)X(299)∩X(265)X(11118)

Barycentrics    Sqrt[3]*(2*a^10 - 5*a^8*b^2 + 6*a^6*b^4 - 8*a^4*b^6 + 8*a^2*b^8 - 3*b^10 - 5*a^8*c^2 - 22*a^6*b^2*c^2 + 8*a^4*b^4*c^2 + 10*a^2*b^6*c^2 + 9*b^8*c^2 + 6*a^6*c^4 + 8*a^4*b^2*c^4 - 36*a^2*b^4*c^4 - 6*b^6*c^4 - 8*a^4*c^6 + 10*a^2*b^2*c^6 - 6*b^4*c^6 + 8*a^2*c^8 + 9*b^2*c^8 - 3*c^10) + 2*(2*a^8 + 7*a^6*b^2 - 3*a^4*b^4 - 5*a^2*b^6 - b^8 + 7*a^6*c^2 + 8*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 14*b^6*c^2 - 3*a^4*c^4 + 5*a^2*b^2*c^4 + 30*b^4*c^4 - 5*a^2*c^6 - 14*b^2*c^6 - c^8)*S : :
X(34295) = 4 X[619] - 3 X[15769]

X(34295) lies on the cubics X046b and K060 and these lines: {30, 299}, {265, 11118}, {619, 15769}, {1525, 22797}, {18331, 18781}

X(34295) = antigonal image of X(8492)
X(34295) = X(8015)-cross conjugate of X(395)
X(34295) = X(6151)-isoconjugate of X(19299)
X(34295) = barycentric product X(395)*X(19777)
X(34295) = barycentric quotient X(i)/X(j) for these {i,j}: {395, 617}, {3441, 6151}


X(34296) =  X(30)X(298)∩X(265)X(11117)

Barycentrics    Sqrt[3]*(2*a^10 - 5*a^8*b^2 + 6*a^6*b^4 - 8*a^4*b^6 + 8*a^2*b^8 - 3*b^10 - 5*a^8*c^2 - 22*a^6*b^2*c^2 + 8*a^4*b^4*c^2 + 10*a^2*b^6*c^2 + 9*b^8*c^2 + 6*a^6*c^4 + 8*a^4*b^2*c^4 - 36*a^2*b^4*c^4 - 6*b^6*c^4 - 8*a^4*c^6 + 10*a^2*b^2*c^6 - 6*b^4*c^6 + 8*a^2*c^8 + 9*b^2*c^8 - 3*c^10) - 2*(2*a^8 + 7*a^6*b^2 - 3*a^4*b^4 - 5*a^2*b^6 - b^8 + 7*a^6*c^2 + 8*a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 14*b^6*c^2 - 3*a^4*c^4 + 5*a^2*b^2*c^4 + 30*b^4*c^4 - 5*a^2*c^6 - 14*b^2*c^6 - c^8)*S : :
X(34296) = 4 X[618] - 3 X[15768]

X(34296) lies on the cubics K046a and K060 and these lines: {30, 298}, {265, 11117}, {618, 15768}, {1524, 22796}, {18331, 18781}

X(34296) = antigonal image of X(8491)
X(34296) = X(8014)-cross conjugate of X(396)
X(34296) = X(2981)-isoconjugate of X(19298)
X(34296) = crosssum of X(15) and X(24303)
X(34296) = barycentric product X(396)*X(19776)
X(34296) = barycentric quotient X (i)/X(j) for these {i,j}: {396, 616}, {3440, 2981}


X(34297) =  X(5)X(20123)∩X(30)X(5667)

Barycentrics    (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 + a^10*b^2 - 11*a^8*b^4 + 14*a^6*b^6 - a^4*b^8 - 7*a^2*b^10 + 3*b^12 - 3*a^10*c^2 + 9*a^8*b^2*c^2 - 2*a^6*b^4*c^2 - 18*a^4*b^6*c^2 + 21*a^2*b^8*c^2 - 7*b^10*c^2 + 3*a^8*c^4 - 10*a^6*b^2*c^4 + 26*a^4*b^4*c^4 - 18*a^2*b^6*c^4 - b^8*c^4 - 2*a^6*c^6 - 10*a^4*b^2*c^6 - 2*a^2*b^4*c^6 + 14*b^6*c^6 + 3*a^4*c^8 + 9*a^2*b^2*c^8 - 11*b^4*c^8 - 3*a^2*c^10 + b^2*c^10 + c^12)*(a^12 - 3*a^10*b^2 + 3*a^8*b^4 - 2*a^6*b^6 + 3*a^4*b^8 - 3*a^2*b^10 + b^12 + a^10*c^2 + 9*a^8*b^2*c^2 - 10*a^6*b^4*c^2 - 10*a^4*b^6*c^2 + 9*a^2*b^8*c^2 + b^10*c^2 - 11*a^8*c^4 - 2*a^6*b^2*c^4 + 26*a^4*b^4*c^4 - 2*a^2*b^6*c^4 - 11*b^8*c^4 + 14*a^6*c^6 - 18*a^4*b^2*c^6 - 18*a^2*b^4*c^6 + 14*b^6*c^6 - a^4*c^8 + 21*a^2*b^2*c^8 - b^4*c^8 - 7*a^2*c^10 - 7*b^2*c^10 + 3*c^12) : :

X(34297) lies on the cubics K060 and K543 and these lines: {5, 20123}, {30, 5667}, {133, 3163}, {1294, 16080}, {1494, 10745}, {7687, 9033}, {14847, 15774}

X(34297) = antigonal image of X(2133)
X(34297) = X(4)-cross conjugate of X(30)
X(34297) = barycentric quotient X(i)/X(j) for these {i,j}: {1990, 5667}, {3163, 15774}, {8431, 14919}


X(34298) =  X(4)X(5627)∩X(30)X(2132)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^4 - 2*a^2*b^2 + b^4 + 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)*(3*a^12 - 7*a^10*b^2 - a^8*b^4 + 14*a^6*b^6 - 11*a^4*b^8 + a^2*b^10 + b^12 - 7*a^10*c^2 + 21*a^8*b^2*c^2 - 18*a^6*b^4*c^2 - 2*a^4*b^6*c^2 + 9*a^2*b^8*c^2 - 3*b^10*c^2 - a^8*c^4 - 18*a^6*b^2*c^4 + 26*a^4*b^4*c^4 - 10*a^2*b^6*c^4 + 3*b^8*c^4 + 14*a^6*c^6 - 2*a^4*b^2*c^6 - 10*a^2*b^4*c^6 - 2*b^6*c^6 - 11*a^4*c^8 + 9*a^2*b^2*c^8 + 3*b^4*c^8 + a^2*c^10 - 3*b^2*c^10 + c^12) : :

X(34298) lies on the cubic K060 and these lines: {4, 5627}, {30, 2132}, {1141, 14451}, {6749, 11079}, {15063, 15395}

X(34298) = antigonal image of X(5667)
X(34298) = symgonal image of X(10745)
X(34298) = X(265)-Ceva conjugate of X(5627)
X(34298) = barycentric quotient X(i)/X(j) for these {i,j}: {5667, 14920}, {11079, 8431}


X(34299) =  X(30)X(1807)∩X(80)X(1784)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + 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)*(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(34299) lies on the cubic K060 and these lines: {30, 1807}, {80, 1784}, {908, 1793}, {1785, 2341}, {7359, 17757}

X(34299) = antigonal image of X(7164)
X(34299) = X(4)-cross conjugate of X(80)
X(34299) = X(i)-isoconjugate of X(j) for these (i,j): {36, 3465}, {1464, 15776}
X(34299) = barycentric product X(3466)*X(18359)
X(34299) = barycentric quotient X(i)/X(j) for these {i,j}: {2161, 3465}, {2341, 15776}, {3466, 3218}


X(34300) =  X(4)X(80)∩X(5)X(2595)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 - b^2 - a*c + 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(34300) lies on the cubic K060 and these lines: {4, 80}, {5, 2595}, {13, 17406}, {14, 17405}, {30, 1807}, {79, 109}, {1901, 2161}, {3145, 10260}, {5627, 19658}, {11584, 14452}, {15065, 33650}

X(34300) = antigonal image of X(3483)
X(34300) = X(265)-Ceva conjugate of X(80)
X(34300) = X(36)-isoconjugate of X(3469)
X(34300) = cevapoint of X(3460) and X(3465)
X(34300) = barycentric product X(3468)*X(18359)
X(34300) = barycentric quotient X(i)/X(j) for these {i,j}: {2161, 3469}, {3468, 3218}, {11069, 7165}


X(34301) =  X(1)X(14844)∩X(4)X(79)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + 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(34301) liews on the cubic K060 and these lines: {1, 14844}, {4, 79}, {5, 3468}, {30, 3464}, {80, 5627}, {515, 26700}, {909, 32678}, {1006, 8606}, {1141, 14452}, {5176, 6742}, {7424, 13486}, {11581, 17405}, {11582, 17406}, {14451, 19658}

X(34301) = polar circle inverse of X(1844)
X(34301) = antigonal image of X(3465)
X(34301) = X(265)-Ceva conjugate of X(79)
X(34301) = X(35)-isoconjugate of X(3466)
X(34301) = cevapoint of X(3464) and X(3468)
X(34301) = barycentric product X(3465)*X(30690)
X(34301) = barycentric quotient X(i)/X(j) for these {i,j}: {2160, 3466}, {3465, 3219}


X(34302) =  X(4)X(1117)∩X(30)X(1141)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 3*a^4*c^2 + a^2*b^2*c^2 - 3*b^4*c^2 + 3*a^2*c^4 + 3*b^2*c^4 - c^6)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - a^4*c^2 + a^2*b^2*c^2 + 3*b^4*c^2 - a^2*c^4 - 3*b^2*c^4 + c^6)*(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) : :
X(34302) = 4 X[10615] - 3 X[15770]

X(34302) lies on the cubic K060 and these lines: {4, 1117}, {30, 1141}, {80, 19658}, {265, 11584}, {1154, 13582}, {10615, 14367}, {11071, 11600}

X(34302) = antigonal image of X(8494)
X(34302) = X(265)-Ceva conjugate of X(1117)
X(34302) = X(6149)-isoconjugate of X(11584)
X(34302) = cevapoint of X(195) and X(14367)
X(34302) = barycentric product X(94)*X(14367)
X(34302) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 11584}, {11071, 3459}, {14367, 323}


X(34303) =  X(5)X(3468)∩X(30)X(3483)

Barycentrics    (a^2 + a*b + b^2 - c^2)*(a^2 - b^2 + a*c + 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)*(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(34303) lies on the cubic K060 and these lines: {5, 3468}, {30, 3483}, {79, 6761}, {1789, 15777}

X(34303) = antigonal image of X(3466)
X(34303) = X(4)-cross conjugate of X(79)
X(34303) = X(i)-isoconjugate of X(j) for these (i,j): {35, 3468}, {2594, 15777}
X(34303) = cevapoint of X(3469) and X(7165)
X(34303) = barycentric product X(3469)*X(30690)
X(34303) = barycentric quotient X(i)/X(j) for these {i,j}: {2160, 3468}, {3469, 3219}


X(34304) =  X(4)X(137)∩X(5)X(2120)

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

X(34304) lies on the cubic K060 and these lines: {4, 137}, {5, 2120}, {30, 3484}, {265, 33664}, {275, 11587}, {5627, 11584}, {6748, 11077}, {14980, 28342}, {17405, 17406}

X(34304) = antigonal image of X(3482)
X(34304) = X(265)-Ceva conjugate of X(1141)
X(34304) = X(6149)-isoconjugate of X(33664)
X(34304) = cevapoint of X(2120) and X(3484)
X(34304) = barycentric quotient X(i)/X(j) for these {i,j}: {1989, 33664}, {3462, 14918}, {11077, 3463}


X(34305) =  X(4)X(19658)∩X(30)X(3483)

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

X(34305) lies on the cubic K060 and these lines: {4, 19658}, {30, 3483}, {79, 11584}, {80, 1141}, {11600, 17406}, {11601, 17405}

X(34305) = antigonal image of X(8480)
X(34305) = X(265)-Ceva conjugate of X(19658)


X(34306) =  X(5)X(399)∩X(110)X(13582)

Barycentrics    (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 3*a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + 3*a^2*c^4 + b^2*c^4 - c^6)*(a^12 - 6*a^10*b^2 + 13*a^8*b^4 - 12*a^6*b^6 + 3*a^4*b^8 + 2*a^2*b^10 - b^12 - 6*a^10*c^2 + 4*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 4*a^4*b^6*c^2 - 6*a^2*b^8*c^2 + 6*b^10*c^2 + 13*a^8*c^4 - 2*a^6*b^2*c^4 - 5*a^4*b^4*c^4 + 4*a^2*b^6*c^4 - 15*b^8*c^4 - 12*a^6*c^6 + 4*a^4*b^2*c^6 + 4*a^2*b^4*c^6 + 20*b^6*c^6 + 3*a^4*c^8 - 6*a^2*b^2*c^8 - 15*b^4*c^8 + 2*a^2*c^10 + 6*b^2*c^10 - c^12) : :
X(34306) = X[399] + 2 X[10277], X[3448] - 4 X[10276]

X(34306) lies on the curve Q004 and these lines: {5, 399}, {110, 13582}, {10272, 14354}


X(34307) =  (name pending)

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

X(34307) lies on the curve Q004 and these lines: {14452, 15767}


X(34308) =  X(5)X(49)∩X(125)X(930)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^10 - 4*a^8*b^2 + 8*a^6*b^4 - 10*a^4*b^6 + 7*a^2*b^8 - 2*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 + 8*a^6*c^4 - 2*a^4*b^2*c^4 + a^2*b^4*c^4 - b^6*c^4 - 10*a^4*c^6 - 3*a^2*b^2*c^6 - b^4*c^6 + 7*a^2*c^8 + 3*b^2*c^8 - 2*c^10) : :
X(34308) = 4 X[125] - X[930], 2 X[128] - 5 X[15081], 2 X[137] + X[3448], 2 X[265] + X[1141], 4 X[11801] - X[31656]

X(34308) lies on the curve Q004 and these lines: {5, 49}, {125, 930}, {128, 15081}, {137, 3448}, {1117, 1263}, {9140, 11117}


X(34309) =  X(80)X(3465)∩X(226)X(2222)

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

X(34309) lies on the curve Q004 and these lines: {80, 3465}, {226, 2222}, {908, 1793}


X(34310) =  X(3)X(125)∩X(136)X(18384)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^8 - a^6*b^2 - 2*a^4*b^4 + 3*a^2*b^6 - b^8 - a^6*c^2 + 5*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + b^6*c^2 - 2*a^4*c^4 - 3*a^2*b^2*c^4 + 3*a^2*c^6 + b^2*c^6 - c^8) : :
X(34310) = 2 X[125] + X[13558], X[265] + 2 X[5961]

X(34310) lies on the curve Q004 and these lines: {3, 125}, {136, 18384}, {468, 14560}, {476, 11657}, {1300, 16080}, {1637, 1989}, {6388, 11060}, {14583, 14847}

X(34310) = midpoint of X(5961) and X(14854)
X(34310) = reflection of X(265) in X(14854)
X(34310) = Dao-Moses-Telv-circle-inverse of X(1989)
X(34310) = barycentric quotient X(16319)/X(14920)


X(34311) =  X(10)X(21)∩X(11)X(2607)

Barycentrics    (-a^2 + a*b - b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^5 - 3*a^3*b^2 + a^2*b^3 + 2*a*b^4 - b^5 + 4*a^3*b*c - a^2*b^2*c - 2*a*b^3*c + 2*b^4*c - 3*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 + 2*a*c^4 + 2*b*c^4 - c^5) : :
X(34311) = 2 X[11] + X[21381], X[80] + 2 X[759], 2 X[12619] + X[14663]

X(34311) lies on the curve Q004 and these lines: {10, 21}, {11, 2607}, {88, 5620}, {12619, 14663}


X(34312) =  REFLECTION OF X(476) IN X(2)

Barycentrics    a^12 - 2*a^10*b^2 + 5*a^8*b^4 - 11*a^6*b^6 + 8*a^4*b^8 + a^2*b^10 - 2*b^12 - 2*a^10*c^2 - 4*a^8*b^2*c^2 + 9*a^6*b^4*c^2 - 10*a^2*b^8*c^2 + 7*b^10*c^2 + 5*a^8*c^4 + 9*a^6*b^2*c^4 - 15*a^4*b^4*c^4 + 9*a^2*b^6*c^4 - 10*b^8*c^4 - 11*a^6*c^6 + 9*a^2*b^4*c^6 + 10*b^6*c^6 + 8*a^4*c^8 - 10*a^2*b^2*c^8 - 10*b^4*c^8 + a^2*c^10 + 7*b^2*c^10 - 2*c^12 : :
X(34312) = 5 X[2] - 4 X[22104], X[476] - 4 X[3258], X[476] + 2 X[14731], 5 X[476] - 8 X[22104], X[477] + 2 X[20957], 2 X[3258] + X[14731], 5 X[3258] - 2 X[22104], 3 X[3524] - 4 X[31379], 3 X[3545] - 2 X[25641], 3 X[3839] - X[34193], X[14480] + 2 X[17511], 5 X[14731] + 4 X[22104]

X(34312) lies on the cubic K1138 and these lines: {2, 476}, {30, 110}, {381, 15111}, {523, 9140}, {541, 14508}, {542, 14480}, {671, 9213}, {858, 10717}, {3268, 5641}, {3524, 31379}, {3545, 25641}, {3830, 9717}, {3839, 34193}, {3845, 11749}, {5066, 18319}, {5640, 16279}, {5968, 9159}, {6054, 10989}, {9829, 9832}, {11237, 33964}, {11238, 33965}, {16171, 34291}, {16186, 18867}, {16340, 20126}

X(34312) = midpoint of X(i) and X(j) for these {i,j}: {2, 14731}, {3845, 11749}
X(34312) = reflection of X(i) in X(j) for these {i,j}: {2, 3258}, {476, 2}, {14989, 3830}, {18319, 5066}, {20126, 16340}
X(34312) = reflection of X(9140) in the Euler line
X(34312) = {X(3258),X(14731)}-harmonic conjugate of X(476)


X(34313) =  X(2)X(25226)∩X(14)X(16)

Barycentrics    Sqrt[3]*(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 + 23*a^6*b^2*c^2 - 13*a^4*b^4*c^2 - 4*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 13*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 - 4*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) + 2*(a^8 - 6*a^6*b^2 - 2*a^4*b^4 + 6*a^2*b^6 + b^8 - 6*a^6*c^2 + 7*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 6*a^2*c^6 + c^8)*S : :

X(34313) lies on the cubic K1138 and these lines: {2, 25226}, {14, 16}, {511, 20126}, {530, 10989}, {531, 691}, {6108, 11580}, {9213, 27551}, {11630, 13084}, {15360, 33958}, {20423, 30440}

X(34313) = Thomson-isogonal conjugate of X(13858)


X(34314) =  X(2)X(25225)∩X(13)X(15)

Barycentrics    Sqrt[3]*(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 + 23*a^6*b^2*c^2 - 13*a^4*b^4*c^2 - 4*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 13*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 2*b^6*c^4 + 2*a^4*c^6 - 4*a^2*b^2*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10) - 2*(a^8 - 6*a^6*b^2 - 2*a^4*b^4 + 6*a^2*b^6 + b^8 - 6*a^6*c^2 + 7*a^4*b^2*c^2 - 2*a^2*b^4*c^2 - 2*a^4*c^4 - 2*a^2*b^2*c^4 - 2*b^4*c^4 + 6*a^2*c^6 + c^8)*S : :

X(34314) lies on the cubic K1138 and these lines: {2, 25225}, {13, 15}, {511, 20126}, {530, 691}, {531, 10989}, {6109, 11580}, {9213, 27550}, {11629, 13083}, {15360, 33957}, {20423, 30439}

X(34314) = Thomson isogonal conjugate of X(13859)


X(34315) =  X(2)X(25225)∩X(14)X(476)

Barycentrics    Sqrt[3]*(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 + 5*a^4*b^4*c^2 + 14*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 5*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) - 2*(a^8 + 6*a^6*b^2 - 2*a^4*b^4 - 6*a^2*b^6 + b^8 + 6*a^6*c^2 - 5*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 2*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - 6*a^2*c^6 + c^8)*S : :
X(34315) = 2 X[16092] - 3 X[22511], 4 X[18579] - 3 X[21158]

X(34315) lies on the cubic K1138 and these lines: {2, 25225}, {14, 476}, {15, 7426}, {23, 531}, {30, 5463}, {511, 5648}, {523, 9162}, {623, 10989}, {2770, 5464}, {16092, 22511}, {18579, 21158}

X(34315) = reflection of X(i) in X(j) for these {i,j}: {15, 7426}, {10989, 623}


X(34316) =  X(2)X(25226)∩X(13)X(476)

Barycentrics    Sqrt[3]*(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 + 5*a^4*b^4*c^2 + 14*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 5*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) + 2*(a^8 + 6*a^6*b^2 - 2*a^4*b^4 - 6*a^2*b^6 + b^8 + 6*a^6*c^2 - 5*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 2*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - 6*a^2*c^6 + c^8)*S : :
X(34316) = 2 X[16092] - 3 X[22510], 4 X[18579] - 3 X[21159]

X(34316) lies on the cubic K1138 and these lines: {2, 25226}, {13, 476}, {16, 7426}, {23, 530}, {30, 5464}, {511, 5648}, {523, 9163}, {624, 10989}, {2770, 5463}, {16092, 22510}, {18579, 21159}

X(34316) = reflection of X(i) in X(j) for these {i,j}: {16, 7426}, {10989, 624}


X(34317) =  REFLECTION OF X(15) IN X(14170)

Barycentrics    a^2*(Sqrt[3]*(2*a^8 - 5*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 - b^8 - 5*a^6*c^2 - 7*a^4*b^2*c^2 + 8*a^2*b^4*c^2 + 4*b^6*c^2 + 3*a^4*c^4 + 8*a^2*b^2*c^4 - 6*b^4*c^4 + a^2*c^6 + 4*b^2*c^6 - c^8) - 2*(7*a^4*b^2 - 6*a^2*b^4 - b^6 + 7*a^4*c^2 - 7*a^2*b^2*c^2 - b^4*c^2 - 6*a^2*c^4 - b^2*c^4 - c^6)*S) : :
X(34317) = X[14538] + 2 X[30485]

X(34317) lies on the cubic K912 and these lines: {15, 23}, {511, 14173}, {5663, 13859}, {9202, 13858}

X(34317) = reflection of X(15) in X(14170)
X(34317) = Thomson-isogonal conjugate of X(13)


X(34318) =  REFLECTION OF X(16) IN X(14169)

Barycentrics    a^2*(Sqrt[3]*(2*a^8 - 5*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 - b^8 - 5*a^6*c^2 - 7*a^4*b^2*c^2 + 8*a^2*b^4*c^2 + 4*b^6*c^2 + 3*a^4*c^4 + 8*a^2*b^2*c^4 - 6*b^4*c^4 + a^2*c^6 + 4*b^2*c^6 - c^8) + 2*(7*a^4*b^2 - 6*a^2*b^4 - b^6 + 7*a^4*c^2 - 7*a^2*b^2*c^2 - b^4*c^2 - 6*a^2*c^4 - b^2*c^4 - c^6)*S) : :
X(34318) = X[14539] + 2 X[30486]

X(34318) lies on the cubic K912 and these lines: {16, 23}, {511, 14179}, {5663, 13858}, {9203, 13859}

X(34318) = reflection of X(16) in X(14169)
X(34318) = Thomson-isogonal conjugate of X(14)


X(34319) =  X(2)X(67)∩X(6)X(13)

Barycentrics    5*a^8 - 3*a^6*b^2 - 4*a^4*b^4 + 3*a^2*b^6 - b^8 - 3*a^6*c^2 + 5*a^4*b^2*c^2 - a^2*b^4*c^2 - 4*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 + 3*a^2*c^6 - c^8 : :
X(34319) = 5 X[2] - 4 X[6698], X[67] - 4 X[6593], 5 X[67] - 8 X[6698], X[67] + 2 X[11061], X[110] + 2 X[25329], 2 X[125] + X[25336], X[265] - 4 X[25556], 2 X[549] - 3 X[15462], 2 X[576] + X[23236], X[599] + 3 X[25331], X[1992] - 3 X[25321], X[2930] + 2 X[5095], 3 X[3524] - X[32247], 3 X[3545] - 2 X[32274], 3 X[5032] + X[14683], 3 X[5055] - X[32306], 2 X[5181] + X[16176], 2 X[5642] + 3 X[25331], X[5648] + 4 X[25329], 4 X[5972] - 3 X[21358], X[6144] + 2 X[32114], 5 X[6593] - 2 X[6698], 2 X[6593] + X[11061], 4 X[6698] + 5 X[11061], 2 X[7426] - 3 X[18374], 2 X[8550] + X[14094], X[9143] + 3 X[25321], 2 X[9970] + X[32233], 4 X[10168] - 3 X[15061], 7 X[10541] - 4 X[20417], X[10989] - 3 X[22151], 2 X[11178] - 3 X[14643], X[11477] + 2 X[30714], X[14982] - 4 X[19140], 5 X[15027] - 8 X[25555], X[15069] - 4 X[16534], 4 X[15118] - X[25335], 2 X[15141] + X[32264], 3 X[19875] - X[32261], 3 X[25055] - 2 X[32238], 2 X[32278] + X[32298]}

X(34319) lies on the cubic K065 and these lines: {2, 67}, {6, 13}, {25, 2930}, {30, 9970}, {110, 524}, {125, 25336}, {141, 13169}, {182, 20126}, {187, 14653}, {230, 9759}, {376, 2781}, {403, 15471}, {519, 32278}, {523, 32313}, {549, 15462}, {576, 23236}, {590, 13643}, {597, 9140}, {599, 5642}, {615, 13762}, {895, 8584}, {1503, 10706}, {1691, 14605}, {1992, 2854}, {2070, 12584}, {2836, 24473}, {3058, 32290}, {3524, 32247}, {3545, 32274}, {3582, 32308}, {3584, 32307}, {3830, 32271}, {5026, 11006}, {5032, 14683}, {5055, 32306}, {5064, 32239}, {5133, 25328}, {5181, 15533}, {5434, 32289}, {5467, 8724}, {5663, 11179}, {5972, 21358}, {6144, 32114}, {6322, 7761}, {7394, 32255}, {7728, 11645}, {7737, 32424}, {7865, 32268}, {8550, 14094}, {8593, 15342}, {9144, 9830}, {9769, 11168}, {9818, 16010}, {10168, 15061}, {10254, 32272}, {10541, 20417}, {10989, 19379}, {11178, 14643}, {11237, 32243}, {11238, 32297}, {11477, 30714}, {11511, 18564}, {12121, 19924}, {15027, 25555}, {15069, 16534}, {15118, 25335}, {15140, 31133}, {15141, 31152}, {18386, 32250}, {19875, 32261}, {22165, 32244}, {23287, 32228}, {25055, 32238}, {32225, 32227}, {32252, 32788}, {32253, 32787}

X(34319) = midpoint of X(i) and X(j) for these {i,j}: {2, 11061}, {1992, 9143}, {2930, 15534}, {8593, 15342}, {15533, 16176}
X(34319) = reflection of X(i) in X(j) for these {i,j}: {2, 6593}, {6, 15303}, {67, 2}, {265, 5476}, {599, 5642}, {895, 8584}, {3818, 25566}, {3830, 32271}, {5476, 25556}, {5648, 110}, {5655, 19140}, {9140, 597}, {9971, 12824}, {11006, 5026}, {11646, 5465}, {13169, 141}, {14982, 5655}, {15360, 32217}, {15533, 5181}, {15534, 5095}, {20126, 182}, {32244, 22165}
X(34319) = psi-transform of X (11628)
X(34319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6593, 11061, 67}, {9143, 25321, 1992}


X(34320) =  X(30)X(111)∩X(67)X(524)

Barycentrics    (a^2 + b^2 - 2*c^2)*(a^2 - 2*b^2 + c^2)*(a^8 + 3*a^6*b^2 - 2*a^4*b^4 - 3*a^2*b^6 + b^8 + 3*a^6*c^2 - 11*a^4*b^2*c^2 + 7*a^2*b^4*c^2 - 2*a^4*c^4 + 7*a^2*b^2*c^4 - 2*b^4*c^4 - 3*a^2*c^6 + c^8) : :
X(34320) = X[2770] - 4 X[31655]

X(34320) lies on the Hutson-Parry circle, the cubics K065 and K479, and on these lines: {2, 691}, {3, 11628}, {23, 11643}, {30, 111}, {67, 524}, {352, 19905}, {523, 10717}, {542, 32583}, {671, 10989}, {1368, 8877}, {1499, 1551}, {5189, 15398}, {5640, 9169}, {5642, 32729}, {5968, 9159}, {5971, 7809}, {5996, 6054}, {6091, 7426}, {6792, 16188}, {7698, 32525}, {10416, 30745}, {19570, 31125}

X(34320) = reflection of X(i) in X(j) for these {i,j}: {2, 31655}, {2770, 2}
X(34320) = reflection of X(10717) in the Euler line
X(34320) = barycentric product X(671)*X(5648)
X(34320) = barycentric quotient X(5648)/X(524)
X(34320) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {858, 15899, 10415}, {5913, 34169, 111}


X(34321) =  ISOGONAL CONJUGATE OF X(6671)

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

See Telv Cohl and Peter Moses, Hyacinthos 29538.

X(34321) lies on these lines: {6, 2380}, {13, 11600}, {16, 2981}, {17, 299}, {61, 1337}, {8603, 11081}, {8742, 9112}

X(34321) = isogonal conjugate of X(6671)
X(34321) = isogonal conjugate of the complement of X(623)
X(34321) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6671}, {532, 3376}
X(34321) = cevapoint of X(8603) and X(21461)
X(34321) = crosssum of X(396) and X(15802)
X(34321) = barycentric product X(i)*X(j) for these {i,j}: {17, 2981}, {2380, 19779}, {8603, 11119}
X(34321) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6671}, {2380, 16771}, {2981, 302}, {8603, 618}, {16459, 8838}, {21461, 396}


X(34322) =  ISOGONAL CONJUGATE OF X(6672)

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

See Telv Cohl and Peter Moses, Hyacinthos 29538.

X(34322) lies on these lines: {6, 2381}, {14, 11601}, {15, 6151}, {18, 298}, {62, 1338}, {8604, 11086}, {8741, 9113}

X(34322) = isogonal conjugate of X(6672)
X(34322) = isogonal conjugate of the complement of X(624)
X(34322) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6672}, {533, 3383}
X(34322) = cevapoint of X(8604) and X(21462)
X(34322) = crosssum of X(395) and X(15778)
X(34322) = barycentric product X(i)*X(j) for these {i,j}: {18, 6151}, {2381, 19778}, {8604, 11120}
X(34322) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6672}, {2381, 16770}, {6151, 303}, {8604, 619}, {16460, 8836}, {21462, 395}


X(34323) =  X(5)X(252)∩X(195)X(10615)

Barycentrics    (a^4 - a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4)*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*(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)*(3*a^12 - 12*a^10*b^2 + 19*a^8*b^4 - 16*a^6*b^6 + 9*a^4*b^8 - 4*a^2*b^10 + b^12 - 12*a^10*c^2 + 20*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 10*a^4*b^6*c^2 + 12*a^2*b^8*c^2 - 6*b^10*c^2 + 19*a^8*c^4 - 4*a^6*b^2*c^4 + 5*a^4*b^4*c^4 - 8*a^2*b^6*c^4 + 15*b^8*c^4 - 16*a^6*c^6 - 10*a^4*b^2*c^6 - 8*a^2*b^4*c^6 - 20*b^6*c^6 + 9*a^4*c^8 + 12*a^2*b^2*c^8 + 15*b^4*c^8 - 4*a^2*c^10 - 6*b^2*c^10 + c^12) : :

See Telv Cohl and Peter Moses, Hyacinthos 29538.

X(34323) lies on these lines: {5, 252}, {195, 10615}, {930, 24385}, {3459, 6150}, {3519, 27246}, {19268, 31675}


X(34324) =  X(1)X(15038)∩X(12)X(79)

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

See Telv Cohl and Peter Moses, Hyacinthos 29538.

X(34324) lies on these lines: {1, 15038}, {12, 79}, {2975, 10176}, {3336, 17483}


X(34325) =  MIDPOINT OF X(13) AND X(11581)

Barycentrics    (2*a^4 - 7*a^2*b^2 + 5*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 5*c^4 - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S)) : :

See Kadir Altintas and Peter Moses, Hyacinthos 29540.

X(34325) lies on these lines: {13, 15}, {140, 8929}, {298, 16770}, {395, 8014}, {397, 11555}, {5478, 11624}, {11078, 11119}, {11142, 11146}, {16645, 21466}, {22797, 25163}

X(34325) = midpoint of X(13) and X(11581)
X(34325) = barycentric product X(13)*X(6669)
X(34325) = barycentric quotient X(6669)/X(298)
X(34325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 11080, 396}, {396, 11080, 11537}


X(34326) =  MIDPOINT OF X(14) AND X(11582)

Barycentrics    (2*a^4 - 7*a^2*b^2 + 5*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 5*c^4 + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S)*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*(Sqrt[3]*a^2 - S)*S) : :

See Kadir Altintas and Peter Moses, Hyacinthos 29540.

X(34326) lies on these lines: {14, 16}, {140, 8930}, {299, 16771}, {396, 8015}, {398, 11556}, {5479, 11626}, {11092, 11120}, {11141, 11145}, {16644, 21467}, {22796, 25153}

X(34326) = midpoint of X(14) and X(11582)
X(34326) = barycentric product X(14)*X(6670)
X(34326) = barycentric quotient X(6670)/X(299)
X(34326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 11085, 395}, {395, 11085, 11549}


X(34327) =  X(15)X(1154)∩X(378)X(10632)

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

See Kadir Altintas and Peter Moses, Hyacinthos 29540.

X(34327) lies on these lines: {15, 1154}, {378, 10632}, {396, 13350}, {11137, 11146}, {11624, 21158}

X(34327) = crosspoint of X(15) and X(11146)
X(34327) = crosssum of X(13) and X(11139)
X(34327) = barycentric product X(i)*X(j) for these {i,j}: {15, 629}, {11146, 23302}
X(34327) = barycentric quotient X(629)/X(300)


X(34328) =  X(16)X(1154)∩X(378)X(10633)

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

See Kadir Altintas and Peter Moses, Hyacinthos 29540.

X(34328) lies on these lines: {16, 1154}, {378, 10633}, {395, 13349}, {11134, 11145}, {11626, 21159}

X(34328) = crosspoint of X(16) and X(11145)
X(34328) = crosssum of X(14) and X(11138)
X(34328) = barycentric product X(i)*X(j) for these {i,j}: {16, 630}, {11145, 23303}
X(34328) = barycentric quotient X(630)/X(301)


X(34329) =  X(74)X(186)∩X(125)X(32417)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(a^4 + a^2*b^2 - 2*b^4 - 2*a^2*c^2 + b^2*c^2 + c^4)*(2*a^10 - 2*a^8*b^2 - 3*a^6*b^4 + a^4*b^6 + 5*a^2*b^8 - 3*b^10 - 2*a^8*c^2 + 8*a^6*b^2*c^2 - a^4*b^4*c^2 - 14*a^2*b^6*c^2 + 9*b^8*c^2 - 3*a^6*c^4 - a^4*b^2*c^4 + 18*a^2*b^4*c^4 - 6*b^6*c^4 + a^4*c^6 - 14*a^2*b^2*c^6 - 6*b^4*c^6 + 5*a^2*c^8 + 9*b^2*c^8 - 3*c^10) : :

See Kadir Altintas and Peter Moses, Hyacinthos 29540.

X(34329) lies on these lines: {74, 186}, {125, 32417}, {184, 14264}, {520, 8431}, {3284, 11079}, {3470, 13367}, {5627, 13851}, {17986, 18400}

X(34329) = X(110)-Ceva conjugate of X(14380)
X(34329) = crosssum of X(1990) and X(3081)
X(34329) = barycentric product X(7687)*X(14919)
X(34329) = {X(39377),X(39378)}-harmonic conjugate of X(3284)


X(34330) =  49TH HATZIPOLAKIS-MOSES-EULER POINT

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 + 6*a^6*b^2*c^2 - 3*a^4*b^4*c^2 + 9*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 3*a^4*b^2*c^4 - 6*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 9*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(34330) = 3 X[2] + X[10201], X[5] + 2 X[10125], 2 X[5] + X[15331], X[26] + 11 X[5070], 4 X[140] - X[10226], 2 X[140] + X[13406], 2 X[546] + X[15332], 2 X[547] + X[15330], 5 X[632] - 2 X[5498], 5 X[632] + 4 X[12010], 5 X[1656] + X[1658], 7 X[3526] - X[11250], 2 X[3628] + X[10020], 4 X[3628] - X[10224], 8 X[3628] + X[12107], X[5498] + 2 X[12010], 2 X[10020] + X[10224], 4 X[10020] - X[12107], 4 X[10125] - X[15331], 4 X[10212] - 7 X[14869], 2 X[10224] + X[12107], X[10226] + 2 X[13406], X[14070] + 7 X[15703]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29543.

X(34330) lies on these lines: {2, 3}, {141, 34155}, {5891, 16223}, {10182, 30522}, {18475, 20304}

X(34330) = midpoint of X(10154) and X(13371)
X(34330) = reflection of X(34331) in X(2)


X(34331) =  50TH HATZIPOLAKIS-MOSES-EULER POINT

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 + 10*a^6*b^2*c^2 - 5*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 5*a^4*b^2*c^4 - 2*a^2*b^4*c^4 + 4*b^6*c^4 + 4*a^4*c^6 + 7*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(34331) = 5 X[2] - X[10201], X[5] + 2 X[5498], 2 X[5] + X[10226], 2 X[140] + X[10224], 4 X[140] - X[15331], X[550] - 4 X[10212], 5 X[632] - 2 X[10125], 5 X[632] - 8 X[12043], 10 X[632] - X[12107], 5 X[632] + X[13371], 5 X[1656] + X[11250], X[1658] - 7 X[3526], 4 X[3530] - X[15332], 4 X[3628] - X[13406], 11 X[5070] + X[12084], 4 X[5498] - X[10226], 4 X[10124] - X[15330], X[10125] - 4 X[12043], 4 X[10125] - X[12107], 2 X[10125] + X[13371], 2 X[10224] + X[15331], 16 X[12043] - X[12107], 8 X[12043] + X[13371], X[12107] + 2 X[13371], X[32171] + 2 X[32767]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29543.

X(34331) lies on these lines: {2, 3}, {49, 9140}, {541, 25563}, {542, 13561}, {567, 15059}, {575, 6698}, {6101, 13857}, {9730, 34128}, {10263, 32225}, {11430, 20304}, {11564, 15027}, {32171, 32767}

X(34331) = reflection of X(34330) in X(2)


X(34332) =  X(3)X(8)∩X(5)X(2969)

Barycentrics    (a^2-b^2-c^2) (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)^2 : :
X(34332) = 2*X[5]-X[2969]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29547.

X(34332) lies on the MacBeath inconic and these lines: {3,8}, {5,2969}, {117,131}, {121,20315}, {339,1234}, {442,2970}, {912,914}, {1060,1411}, {1441,2973}, {1782,12134}, {2834,10743}, {2971,30444}, {2972,21530}, {3549,15253}, {6842,21666}

X(34332) = reflection of X(2969) in X(5)
X(34332) = MacBeath inconic antipode of X(2969)


X(34333) =  X(3)X(74)∩X(4)X(13556)

Barycentrics    a^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)^2 : :
Barycentrics    S^4 + (12 R^4-8 R^2 SB-8 R^2 SC-SB SC+2 SB SW+2 SC SW-SW^2)S^2 + 36 R^4 SB SC-12 R^2 SB SC SW+SB SC SW^2 : :
X(34333) = 2*X[5]-X[2970]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29547.

X(34333) lies on the MacBeath inconic and these lines: {3,74}, {4,13556}, {5,2970}, {25,1299}, {30,16933}, {113,131}, {157,18451}, {216,3016}, {265,20975}, {311,339}, {381,2971}, {454,12164}, {1147,13496}, {1624,25711}, {1986,15329}, {2790,6033}, {2969,6841}, {5158,11060}, {5502,20771}, {5562,23217}, {7687,18114}, {8154,13557}, {10257,31945}, {12091,18403}, {12134,31976}, {14575,18445}, {16163,16186}

X(34333) = reflection of X(2970) in X(5)
X(34333) = MacBeath inconic antipode of X(2970)


X(34334) =  X(3)X(107)∩X(4)X(94)

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)^2 : :
Barycentrics    (12 R^4+4 R^2 SB+4 R^2 SC+SB SC-7 R^2 SW-SB SW-SC SW+SW^2)S^2 36 R^4 SB SC-9 R^2 SB SC SW : :
Barycentrics    (tan A) (cos A - 2 cos B cos A)^2 : :
X(34334) = 2*X[5]-X[2972]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29547.

X(34334) lies on the MacBeath inconic and these lines: {2,18317}, {3,107}, {4,94}, {5,2972}, {25,1300}, {113,133}, {131,132}, {264,339}, {290,3531}, {324,3845}, {338,7687}, {382,1093}, {399,648}, {546,14978}, {1511,4240}, {1596,2971}, {1941,18350}, {2052,3830}, {2968,6841}, {2969,15763}, {2973,15762}, {3534,15466}, {3548,6523}, {3627,13450}, {5667,20127}, {6525,18850}, {6530,11799}, {7480,16168}, {10272,14920}, {10575,14363}, {11563,24977}, {12188,18535}, {12918,18531}, {13202,18507}, {14934,31510}, {15454,20771}, {16163,16240}, {18279,32743}, {18402,23290}

X(34334) = reflection of X(2972) in X(5)
X(34334) = isotomic conjugate of isogonal conjugate of X(16240)
X(34334) = MacBeath inconic antipode of X(2972)
X(34334) = polar conjugate of isotomic conjugate of X(36789)
X(34334) = pole wrt polar circle of line X(74)X(526) (the tangent to the circumcircle at X(74))
X(34334) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {113,133,11251}


X(34335) =  X(3)X(101)∩X(5)X(2973)

Barycentrics    a^2 (a^2-b^2-c^2) (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 : :
X(34335) = 2*X[5]-X[2973]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29547.

X(34335) lies on the MacBeath inconic and these lines: {3,101}, {5,2973}, {321,2968}, {440,2972}, {916,2253}, {2969,8226}, {2970,3136}

X(34335) = reflection of X(2973) in X(5)
X(34335) = MacBeath inconic antipode of X(2973)
X(34335) = pole wrt polar circle of tangent to circumcircle at X(917)


X(34336) =  ISOTOMIC CONJUGATE OF X(15398)

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

X(34336) lies on the MacBeath inconic and these lines: {2, 339}, {3, 2373}, {4, 10748}, {25, 99}, {126, 1560}, {186, 5971}, {264, 2970}, {427, 2971}, {468, 3266}, {1112, 4576}, {1312, 22339}, {1313, 22340}, {2967, 3268}, {2972, 30739}, {4235, 31128}, {4563, 19504}, {6337, 10603}, {7386, 13219}, {13416, 25053}

X(34336) = isotomic conjugate of X(15398)
X(34336) = polar conjugate of X(10630)
X(34336) = isotomic conjugate of the isogonal conjugate of X(5095)
X(34336) = polar conjugate of the isogonal conjugate of X(2482)
X(34336) = X(i)-isoconjugate of X(j) for these (i,j): {31, 15398}, {48, 10630}, {895, 923}, {897, 14908}
X(34336) = cevapoint of X(2482) and X(5095)
X(34336) = crosssum of X(184) and X(14908)
X(34336) = pole wrt polar circle of trilinear polar of X(10630) (line X(111)X(351), the tangent to the circumcircle at X(111))
X(34336) = barycentric product X(i)*X(j) for these {i,j}: {76, 5095}, {92, 24038}, {264, 2482}, {331, 7067}, {468, 3266}, {1366, 7017}, {1649, 6331}, {17983, 23106}
X(34336) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 15398}, {4, 10630}, {187, 14908}, {468, 111}, {524, 895}, {690, 10097}, {1366, 222}, {1649, 647}, {2482, 3}, {3266, 30786}, {4235, 691}, {5095, 6}, {7067, 219}, {8030, 3292}, {14273, 9178}, {16733, 1444}, {23106, 6390}, {23992, 20975}, {24038, 63}, {32459, 6091}


X(34337) =  X(2)X(1897)∩X(25)X(100)

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

X(34337) lies on the MacBeath inconic and these lines: {2, 1897}, {3, 26703}, {4, 10743}, {25, 100}, {92, 427}, {118, 3239}, {120, 20621}, {132, 5513}, {339, 442}, {344, 7071}, {429, 2971}, {468, 5205}, {612, 25968}, {1260, 10327}, {1861, 3693}, {1876, 3717}, {3006, 26020}, {4242, 31073}, {4422, 8750}, {5094, 30741}, {7386, 34188}, {17279, 23050}, {21666, 25985}, {25091, 25882}

X(34337) = polar conjugate of X(6185)
X(34337) = polar conjugate of the isotomic conjugate of X(4437)
X(34337) = polar conjugate of the isogonal conjugate of X(6184)
X(34337) = X(6184)-cross conjugate of X(4437)
X(34337) = X(i)-isoconjugate of X(j) for these (i,j): {48, 6185}, {673, 32658}, {1438, 1814}, {23696, 32735}
X(34337) = crosssum of X(184) and X(32658)
X(34337) = pole wrt polar circle of trilinear polar of X(6185) (line X(105)X(659), the tangent to the circumcircle at X(105))
X(34337) = barycentric product X(i)*X(j) for these {i,j}: {4, 4437}, {92, 4712}, {264, 6184}, {1362, 7017}, {1861, 3912}, {3126, 6335}, {3263, 5089}, {3717, 5236}, {3932, 15149}, {18027, 20776}
X(34337) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 6185}, {518, 1814}, {1362, 222}, {1861, 673}, {1876, 1462}, {2223, 32658}, {2356, 1438}, {3126, 905}, {3912, 31637}, {4437, 69}, {4712, 63}, {5089, 105}, {6184, 3}, {16728, 1444}, {17060, 17170}, {20776, 577}, {23102, 25083}, {23612, 20752}, {24290, 10099}


X(34338) =  X(3)X(1299)∩X(25)X(100)

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

X(34338) lies on the MacBeath inconic and these lines: {3, 1299}, {4, 13556}, {25, 110}, {135, 136}, {235, 34334}, {427, 2974}, {429, 34332}, {430, 34335}, {431, 21664}, {2970, 8901}

X(34338) = isotomic conjugate of the isogonal conjugate of X(6754)
X(34338) = X(1993)-Ceva conjugate of X(6753)
X(34338) = crosssum of X(155) and X(23181)
X(34338) = pole wrt polar circle of line X(925)X(6380) (the tangent to the circumcircle at X(925))
X(34338) = barycentric product X(i)*X(j) for these {i,j}: {76, 6754}, {136, 1993}, {523, 15423}, {6563, 6753}
X(34338) = barycentric quotient X(i)/X(j) for these {i,j}: {136, 5392}, {6753, 925}, {6754, 6}, {15423, 99}


X(34339) =  COMPLEMENT OF X(5887)

Barycentrics    a*(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 - 2*a^2*b^3*c - 3*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^3*b*c^2 - 4*a^2*b^2*c^2 + 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 - 3*a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6) : :
X(34339) = X[1] - 3 X[10202], X[1] - 5 X[15016], 3 X[1] - X[23340], 3 X[3] - X[14110], 3 X[4] + X[9961], X[40] + 3 X[5902], 3 X[65] + X[14110], X[72] - 3 X[26446], 3 X[354] - X[1482], X[355] - 3 X[3753], 3 X[381] - X[12688], 2 X[548] - 3 X[10178], 5 X[631] - X[3869], 3 X[942] - 2 X[6583], X[942] + 2 X[13145], 3 X[942] + X[31798], X[946] - 3 X[5883], X[1071] + 3 X[3753], X[1657] - 3 X[5918], 5 X[1698] - X[5693], X[3057] - 3 X[10246], 7 X[3526] - 5 X[25917], 3 X[3576] + X[5903], X[3579] + 2 X[31794], 5 X[3698] - 3 X[5790], 5 X[3698] - X[14872], 3 X[3740] - 2 X[31835], 3 X[3742] - 2 X[5901], 2 X[3754] + X[13369], 4 X[3812] - X[31937], X[3868] + 3 X[5657], 3 X[3873] + X[12245], X[3878] - 3 X[10165], X[3901] + 7 X[9588], 3 X[3919] + X[4297], 7 X[3922] + X[12680], 7 X[3922] - X[18525], 7 X[4002] - 3 X[18908], 5 X[4004] + 3 X[10167], 5 X[4004] + X[18481], X[4084] + 3 X[10164], 2 X[5044] - 3 X[11231], 3 X[5049] - X[13600], 3 X[5049] - 2 X[33179], 3 X[5054] - X[31165], 5 X[5439] - 3 X[5886], 5 X[5439] - X[12672], 3 X[5587] + X[15071], 3 X[5692] - 7 X[31423], X[5694] - 3 X[11231], 3 X[5790] - X[14872], 5 X[5818] - X[12528], 3 X[5885] - X[6583], 2 X[5885] + X[31788], 6 X[5885] + X[31798], 3 X[5886] - X[12672], 3 X[5902] - X[24474], X[5903] - 3 X[10273], X[6583] + 3 X[13145], 2 X[6583] + 3 X[31788], 2 X[6583] + X[31798], X[7672] + 3 X[21151], 3 X[7967] + X[14923], X[7982] - 5 X[18398], 3 X[10164] - X[31806], 3 X[10165] - 2 X[31838], 3 X[10167] - X[18481], 3 X[10175] - X[31803], 3 X[10202] - 2 X[13373], 3 X[10202] - 5 X[15016], 9 X[10202] - X[23340], 3 X[10246] + X[25413], 3 X[10247] - 5 X[17609], X[10693] - 3 X[15061], 3 X[11227] - 2 X[13624], 3 X[11227] - X[31786], 3 X[11246] + X[11827], 6 X[13145] - X[31798], 2 X[13373] - 5 X[15016], 6 X[13373] - X[23340], 15 X[15016] - X[23340], 3 X[15064] - 5 X[31399], X[16004]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29549.

X(34339) lies on these lines: {1, 3}, {2, 5887}, {4, 9961}, {5, 3812}, {8, 6897}, {10, 912}, {30, 7686}, {72, 5552}, {84, 18761}, {119, 125}, {140, 960}, {182, 3827}, {224, 5687}, {355, 377}, {381, 12688}, {392, 6910}, {404, 21740}, {515, 3754}, {516, 31870}, {518, 5690}, {519, 12005}, {548, 10178}, {601, 3924}, {631, 3869}, {758, 6684}, {944, 4190}, {946, 5883}, {950, 5840}, {952, 5836}, {958, 24467}, {962, 6899}, {971, 5880}, {1064, 24443}, {1125, 2800}, {1158, 3560}, {1483, 3880}, {1490, 18491}, {1519, 6831}, {1657, 5918}, {1698, 5693}, {1706, 5534}, {1737, 1858}, {1770, 7491}, {1788, 6825}, {1836, 6928}, {1837, 6923}, {1871, 14018}, {1888, 7510}, {1898, 10826}, {1902, 7414}, {2182, 16547}, {2390, 5892}, {2392, 31760}, {2649, 9356}, {2778, 12041}, {2801, 3918}, {2802, 13607}, {2818, 9729}, {2975, 26877}, {3474, 6868}, {3485, 6891}, {3486, 6948}, {3526, 25917}, {3555, 12648}, {3556, 6642}, {3577, 9841}, {3616, 6977}, {3627, 15726}, {3634, 20117}, {3654, 11239}, {3698, 5790}, {3740, 31835}, {3742, 5901}, {3868, 5657}, {3873, 12245}, {3874, 11362}, {3878, 10165}, {3881, 28234}, {3901, 9588}, {3919, 4297}, {3922, 12680}, {4002, 18908}, {4004, 6934}, {4084, 10164}, {4292, 5841}, {4295, 6827}, {4511, 6940}, {4848, 18389}, {5044, 5694}, {5054, 31165}, {5057, 6902}, {5086, 6951}, {5437, 7971}, {5439, 5886}, {5450, 30147}, {5553, 30513}, {5587, 15071}, {5603, 6890}, {5692, 31423}, {5721, 23604}, {5722, 10525}, {5770, 19843}, {5784, 5833}, {5787, 18517}, {5806, 22793}, {5818, 12528}, {5927, 6984}, {6259, 17649}, {6261, 6911}, {6265, 17614}, {6824, 14647}, {6836, 10531}, {6850, 18391}, {6862, 10200}, {6863, 24914}, {6882, 12047}, {6883, 12514}, {6913, 12686}, {6942, 9352}, {6947, 11415}, {6955, 10914}, {6958, 11375}, {6971, 17605}, {6980, 17606}, {6985, 12520}, {7672, 21151}, {7967, 14923}, {7992, 18540}, {8094, 8130}, {8129, 12445}, {9947, 17528}, {10039, 10956}, {10044, 10404}, {10085, 18519}, {10175, 31803}, {10265, 25639}, {10391, 31775}, {10543, 24466}, {10609, 32900}, {10693, 15061}, {10935, 11047}, {11112, 26201}, {11246, 11827}, {11374, 12709}, {11491, 18444}, {11499, 18446}, {11684, 26878}, {12433, 12710}, {12515, 12775}, {12564, 16004}, {12594, 24476}, {12664, 31828}, {12749, 17660}, {12778, 13217}, {12905, 13211}, {13374, 22791}, {14054, 20612}, {15064, 31399}, {15952, 18165}, {16049, 18180}, {16154, 17637}, {19860, 22758}, {31752, 31836}, {33597, 33858}

X(34339) = complement of X(5887)
X(34339) = midpoint of X(i) and X(j) for these {i,j}: {3, 65}, {10, 5884}, {40, 24474}, {355, 1071}, {942, 31788}, {1770, 7491}, {3057, 25413}, {3576, 10273}, {3654, 24473}, {3874, 11362}, {4084, 31806}, {5690, 24475}, {5836, 12675}, {5885, 13145}, {6259, 17649}, {6265, 17654}, {7686, 9943}, {12680, 18525}, {17660, 19914}, {31787, 31794}
X(34339) = reflection of X(i) in X(j) for these {i,j}: {1, 13373}, {5, 3812}, {942, 5885}, {960, 140}, {1385, 9940}, {3579, 31787}, {3627, 16616}, {3878, 31838}, {5694, 5044}, {5777, 9956}, {9856, 9955}, {9957, 15178}, {10222, 5045}, {10284, 31792}, {13600, 33179}, {18857, 18856}, {20117, 3634}, {22791, 13374}, {22793, 5806}, {31786, 13624}, {31788, 13145}, {31793, 31663}, {31836, 31752}, {31837, 6684}, {31870, 33815}, {31937, 5}
X(34339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 40, 10679}, {1, 46, 11509}, {1, 2077, 33596}, {1, 3359, 11248}, {1, 5119, 10965}, {1, 10202, 13373}, {1, 10269, 1385}, {1, 12703, 12000}, {1, 14803, 2646}, {1, 15016, 10202}, {1, 18838, 942}, {1, 24927, 15178}, {3, 10246, 3612}, {3, 11507, 26285}, {3, 16203, 22768}, {40, 3576, 16208}, {40, 5902, 24474}, {40, 18443, 10267}, {65, 13750, 942}, {119, 24982, 9956}, {1071, 3753, 355}, {1385, 3579, 32613}, {1385, 10225, 33862}, {1385, 23961, 13624}, {1385, 32612, 18857}, {1385, 33281, 15178}, {3339, 30503, 5709}, {3359, 11248, 3579}, {3698, 14872, 5790}, {3878, 10165, 31838}, {4084, 10164, 31806}, {5049, 13600, 33179}, {5439, 12672, 5886}, {5554, 10940, 377}, {5554, 12115, 355}, {5690, 32213, 10915}, {5694, 11231, 5044}, {6862, 28628, 11230}, {10225, 33862, 31663}, {10246, 25413, 3057}, {10267, 18443, 1385}, {11227, 31786, 13624}, {12000, 12702, 12703}, {12609, 12616, 5}, {14647, 28629, 6824}


X(34340) =  (name pending)

Barycentrics    a (2 a^9-3 a^8 (b+c)-2 a^7 (b^2-b c+c^2)+6 a^6 (b^3+c^3)+a^4 b c (8 b^3-11 b^2 c-11 b c^2+8 c^3)-2 a (b^2-c^2)^2 (2 b^4-b^3 c+5 b^2 c^2-b c^3+2 c^4)+(b-c)^2 (b+c)^3 (3 b^4-4 b^3 c+8 b^2 c^2-4 b c^3+3 c^4)-2 a^5 (3 b^4+b^3 c-7 b^2 c^2+b c^3+3 c^4)-a^2 (b-c)^2 (6 b^5+16 b^4 c+13 b^3 c^2+13 b^2 c^3+16 b c^4+6 c^5)+2 a^3 (5 b^6-b^5 c-5 b^4 c^2+3 b^3 c^3-5 b^2 c^4-b c^5+5 c^6)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 29551.

X(34340) lies on this line: {758,3579}




leftri  Frégier points: X(34341)-X(34344)  rightri

This preamble and centers X(34341)-X(34342) were contributed by Clark Kimberling and Peter Moses, September 30, 2019.

Suppose that P = p : q : r is a point in the plane of a triangle ABC. Suppose further that p,q,r are distinct homogeneous functions of a,b,c. The permutation ellipse of P, denoted by E(P), is the ellipse that passes through the six points p : q : r,    q : r : p,    r : p : q,    p : r : q,    q : p : r,   r : q : p, given by

(q r + r p + p q)(x^2 + y^2 + z^2) - (p^2 + q^2 + r^2)(y z + z x + x y) = 0.

The centroid, X(2), is the center and the perspector of E(P).

E(X(99)) = Steiner circumellipse
E(X(115)) = Steiner inellipse
E(X(148)) = Steiner circumellipse of anticomplementary triangle.

E(X(1)) passes through X(i) for i = 1, 3679, 24338, 24345, 24411.
E(X(6)) passes through X(i) for i = 6, 599, 24281, 24289, 30229.
E(X(10)) passes through X(i) for i = 10, 551, 24348, 25382.
E(X(98)) passes through X(i) for i = 98, 6039, 6040, 6054.
E(X(69)) passes through X(i) for i = 69, 1992, 30225, 30226, 30227, 30228, 34341, 34342, 34343, 34344.
E(X(125)) passes through X(i) for i = 125, 5642, 16278.
E(X(620)) passes through X(i) for i = 620, 4422, 5461, 17044, 23583, 27076

Further discussion of permutation ellipses is given in the preamble just before X(35025).

The Frégier point of P is the point P' shown at Frégier's Theorem. Suppose that P = p : q : r lies on the Steiner circumellipse. Then the Frégier point of P is given by

F(P) = (-5 a^2 + b^2 + c^2) q^2 r^2 + p^2 (r^2 (a^2 + b^2 - 5 c^2) + q^2 (a^2 - 5 b^2 + c^2)) + p q r ((-7 a^2 + 5 b^2 + 5 c^2) p + (5 a^2 + 5 b^2 - 7 c^2) q + (5 a^2 - 7 b^2 + 5 c^2) r) : : .

The locus of F(P) for P on the Steiner circumellipse is the permutation ellipse E(X(69)). Selected points on the Steiner circumellipse and the associated Frégier points on E(X(69)) are shown here:

X(99) -> X(69), X(671) -> X(1992), X(190) -> X(30225), X(290) -> X(30226),
X(648) -> X(30227), X(664) -> X(30228), X(670) -> X(32341), X(903) -> X(34342),
X(3227) -> X(34343), X(3228) -> X(34344); some of these are noted at X(30225).

An equation for the ellipse E(X(69)), that is, Frégier image of the Steiner circumellipse, is

(a-b-c) (a+b-c) (a-b+c) (a+b+c) x^2+(3 a^4-2 a^2 b^2+3 b^4-2 a^2 c^2-2 b^2 c^2+3 c^4) y z + (cyclic) = 0.

An equation for the ellipse E(X(6)), which is the Frégier image of the Steiner inellipse, is

(a^2 b^2+a^2 c^2+b^2 c^2) x^2 - (a^4+b^4+c^4) y z + (cyclic) = 0.

In general, the Frégier image of an ellipse is an ellipse; specifically, it is the dilation of the first ellipse by the following factor:
(semiMajor^2-semiMinor^2)/(semiMajor^2+semiMinor^2). (Peter Moses, September 30, 2019)

underbar



X(34341) =  FRÉGIER POINT OF X(670)

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

X(34341) lies on these lines: {2, 30229}, {69, 512}, {538, 1992}, {543, 25332}, {886, 20023}, {1352, 16084}, {1502, 9428}, {1975, 2421}, {3978, 11185}, {4576, 8591}, {18829, 32833}, {30226, 32815}

X(34341) = anticomplement of X(30229)


X(34342) =  FRÉGIER POINT OF X(903)

Barycentrics    5*a^4 - 5*a^3*b - 2*a^2*b^2 + 7*a*b^3 - b^4 - 5*a^3*c + 7*a^2*b*c - 5*a*b^2*c - 5*b^3*c - 2*a^2*c^2 - 5*a*b*c^2 + 10*b^2*c^2 + 7*a*c^3 - 5*b*c^3 - c^4 : :
X(34342) = 4 X[24281] - X[30225]

X(34342) lies on these lines: {2, 24262}, {69, 519}, {145, 4555}, {514, 1992}, {545, 24807}, {903, 952}, {4561, 12035}, {4597, 30577}, {6542, 30991}, {17953, 31034}

X(34342) = reflection of X(i) in X(j) for these {i,j}: {2, 24281}, {30225, 2}
X(34342) = anticomplement of X(34362)
X(34342) = barycentric product X(190)*X (30190)
X(34342) = barycentric quotient X (30190)/X(514)


X(34343) =  FRÉGIER POINT OF X(3227)

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

X(34343) lies on these lines: {2, 24289}, {69, 536}, {513, 1992}, {3227, 29349}, {24248, 30225}

X(34343) = reflection of X(2) in X (24289)
X(34343) = anticomplement of X(34363)


X(34344) =  FRÉGIER POINT OF X(3228)

Barycentrics    a^6*b^4 - 5*a^4*b^6 - 7*a^6*b^2*c^2 + 11*a^4*b^4*c^2 + 5*a^2*b^6*c^2 + a^6*c^4 + 11*a^4*b^2*c^4 - 19*a^2*b^4*c^4 + b^6*c^4 - 5*a^4*c^6 + 5*a^2*b^2*c^6 + b^4*c^6 : :

X(34344) lies on these lines: {2, 30229}, {69, 538}, {148, 6792}, {512, 1992}, {1899, 30227}, {3229, 7618}, {6787, 7757}, {14608, 14700}

X(34344) = reflection of X(2) in X (30229)
X(34344) = anticomplement of X(34364)


X(34345) =  X(1)X(3)∩X(227)X(18340)

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

See Tran Quang Hung and César Lozada, Hyacinthos 29554.

X(34345) lies on these lines: {1, 3}, {227, 18340}, {676, 2804}, {1465, 1538}, {12515, 15501}

X(34345) = {X(3035), X(10271)}-harmonic conjugate of X(15252)


X(34346) =  MIDPOINT OF X(56) AND X(23981)

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

See Tran Quang Hung and César Lozada, Hyacinthos 29556.

X(34346) lies on this line: {1, 3}

X(34346) = midpoint of X(56) and X(23981)


X(34347) =  X(39)X(512)∩X(115)X(2971)

Barycentrics    a^4 (b-c) (b+c) (a^4 b^4-2 a^2 b^6+b^8-2 b^6 c^2+a^4 c^4+4 b^4 c^4-2 a^2 c^6-2 b^2 c^6+c^8) : :
Barycentrics    (2 R^2+SB-SC)S^4 + (6 R^2 SB SC+12 R^2 SC^2+2 SB SC^2-14 R^2 SB SW+2 R^2 SC SW-2 SB SC SW-2 SC^2 SW-2 R^2 SW^2+2 SB SW^2)S^2 + 2 R^2 SB SC SW^2+4 R^2 SC^2 SW^2+2 SB SC^2 SW^2-2 R^2 SB SW^3-2 R^2 SC SW^3-2 SB SC SW^3-2 SC^2 SW^3+SB SW^4+SC SW^4 : :

See Tran Quang Hung and Ercole Suppa, Hyacinthos 29557.

X(34347) lies on these lines: {39,512}, {115,2971}, {523,7697}, {3095,3566}


X(34348) =  X(30)X(5889)∩X(523)X(23294)

Barycentrics    a^18*b^4 - 6*a^16*b^6 + 14*a^14*b^8 - 14*a^12*b^10 + 14*a^8*b^14 - 14*a^6*b^16 + 6*a^4*b^18 - a^2*b^20 - 2*a^18*b^2*c^2 + 6*a^16*b^4*c^2 - 4*a^14*b^6*c^2 - 7*a^12*b^8*c^2 + 24*a^10*b^10*c^2 - 43*a^8*b^12*c^2 + 44*a^6*b^14*c^2 - 21*a^4*b^16*c^2 + 2*a^2*b^18*c^2 + b^20*c^2 + a^18*c^4 + 6*a^16*b^2*c^4 - 19*a^14*b^4*c^4 + 20*a^12*b^6*c^4 - 23*a^10*b^8*c^4 + 42*a^8*b^10*c^4 - 50*a^6*b^12*c^4 + 23*a^4*b^14*c^4 + 7*a^2*b^16*c^4 - 7*b^18*c^4 - 6*a^16*c^6 - 4*a^14*b^2*c^6 + 20*a^12*b^4*c^6 - a^10*b^6*c^6 - 13*a^8*b^8*c^6 + 19*a^6*b^10*c^6 - 3*a^4*b^12*c^6 - 32*a^2*b^14*c^6 + 20*b^16*c^6 + 14*a^14*c^8 - 7*a^12*b^2*c^8 - 23*a^10*b^4*c^8 - 13*a^8*b^6*c^8 + 2*a^6*b^8*c^8 - 5*a^4*b^10*c^8 + 58*a^2*b^12*c^8 - 28*b^14*c^8 - 14*a^12*c^10 + 24*a^10*b^2*c^10 + 42*a^8*b^4*c^10 + 19*a^6*b^6*c^10 - 5*a^4*b^8*c^10 - 68*a^2*b^10*c^10 + 14*b^12*c^10 - 43*a^8*b^2*c^12 - 50*a^6*b^4*c^12 - 3*a^4*b^6*c^12 + 58*a^2*b^8*c^12 + 14*b^10*c^12 + 14*a^8*c^14 + 44*a^6*b^2*c^14 + 23*a^4*b^4*c^14 - 32*a^2*b^6*c^14 - 28*b^8*c^14 - 14*a^6*c^16 - 21*a^4*b^2*c^16 + 7*a^2*b^4*c^16 + 20*b^6*c^16 + 6*a^4*c^18 + 2*a^2*b^2*c^18 - 7*b^4*c^18 - a^2*c^20 + b^2*c^20 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29559.

X(34348) lies on these lines: {30, 5889}, {523, 23294}, {10255, 14670}


X(34349) =  X(3)X(6)∩X(232)X(14356)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 29561.

X(34349) lies on these lines: {3,6}, {232,14356}, {620,2492}, {4045,18122}, {7669,22146}, {8574,21203}


X(34350) =  51ST HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    3*a^10 - 5*a^8*b^2 - 2*a^6*b^4 + 6*a^4*b^6 - a^2*b^8 - b^10 - 5*a^8*c^2 + 16*a^6*b^2*c^2 - 8*a^4*b^4*c^2 - 6*a^2*b^6*c^2 + 3*b^8*c^2 - 2*a^6*c^4 - 8*a^4*b^2*c^4 + 14*a^2*b^4*c^4 - 2*b^6*c^4 + 6*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(34350) = 3 X[2] - 4 X[10226], 5 X[3] - 4 X[10020], 4 X[3] - 3 X[10201], 3 X[4] - 4 X[10224], 3 X[376] - 2 X[1658], 15 X[631] - 16 X[10212], 5 X[631] - 4 X[13406], 7 X[3090] - 8 X[5498], 5 X[3522] - 4 X[15331], 9 X[3524] - 8 X[10125], 3 X[3534] - X[7387], 2 X[5449] - 3 X[11204], 5 X[8567] - 3 X[14852], 3 X[8703] - 2 X[13383], 16 X[10020] - 15 X[10201], 4 X[10212] - 3 X[13406], 2 X[10224] - 3 X[11250], 3 X[10606] - X[12293], 3 X[14070] - 5 X[15696]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29563.

X(34350) lies on these lines: {2, 3}, {68, 32138}, {74, 25738}, {155, 5925}, {156, 5878}, {1147, 2777}, {1204, 16111}, {3357, 17702}, {4299, 8144}, {4302, 9627}, {4549, 10627}, {4846, 32046}, {5449, 11204}, {5654, 11744}, {5663, 12118}, {6146, 20725}, {7728, 25487}, {8567, 14852}, {10539, 16163}, {10605, 12370}, {10606, 12293}, {10733, 23294}, {11438, 12897}, {12038, 22802}, {12041, 26937}, {12121, 12825}, {12289, 13445}, {13340, 18442}, {13403, 18952}, {13491, 19467}, {14216, 30522}, {14677, 18917}, {15055, 26917}, {15311, 32139}, {17854, 20127}, {18931, 22979}

X(34350) = midpoint of X(i) and X(j) for these {i,j}: {155, 5925}, {1657, 12085}, {3529, 14790}, {12118, 20427}
X(34350) = reflection of X(i) in X(j) for these {i,j}: {4, 11250}, {26, 550}, {68, 32138}, {382, 13371}, {5878, 156}, {7728, 25487}, {22802, 12038}, {32140, 3357}


X(34351) =  52ND HATZIPOLAKIS-MOSES-EULER POINT

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 + 8*a^6*b^2*c^2 - 4*a^4*b^4*c^2 + 8*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 + 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(34351) = X[3] + 2 X[13383], X[5] + 2 X[1658], X[5] - 4 X[10020], X[26] + 2 X[140], X[68] + 5 X[17821], X[549] - 4 X[15330], X[550] - 4 X[15331], 5 X[631] + X[7387], 5 X[632] - 8 X[10125], 35 X[632] - 32 X[12043], 5 X[632] + 4 X[12107], 5 X[632] - 2 X[13371], X[1658] + 2 X[10020], 7 X[3523] - X[12085], 7 X[3526] - X[14790], 4 X[3530] - X[12084], X[3627] - 4 X[13406], 7 X[3857] - 16 X[12010], 3 X[5054] + X[9909], X[6247] - 4 X[20191], X[7689] + 2 X[16252], 7 X[10125] - 4 X[12043], 2 X[10125] + X[12107], 4 X[10125] - X[13371], X[10154] + 4 X[15330], 3 X[10245] + 5 X[15694], X[10264] + 2 X[20773], 2 X[10282] + X[12359], 8 X[12043] + 7 X[12107], 16 X[12043] - 7 X[13371], 2 X[12107] + X[13371]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29563.

X(34351) lies on these lines: {2, 3}, {68, 17821}, {511, 10182}, {524, 1147}, {539, 32391}, {541, 2883}, {542, 10282}, {597, 5462}, {1177, 15132}, {3564, 23041}, {3580, 11464}, {5063, 31406}, {5562, 5642}, {5944, 31804}, {6102, 15361}, {6247, 20191}, {7689, 16252}, {8262, 8550}, {8981, 11266}, {9140, 34224}, {9730, 13394}, {10168, 11695}, {10169, 11649}, {10192, 13754}, {10264, 20773}, {10575, 15738}, {11265, 13966}, {11430, 32223}, {11454, 32111}, {11645, 20299}, {13292, 19357}, {13352, 32269}, {13367, 32225}, {13567, 18475}, {13884, 18459}, {13937, 18457}, {14915, 23328}, {15360, 34148}, {15462, 16789}, {18917, 26864}, {20302, 23358}, {20397, 32274}, {28708, 33878}

X(34351) = midpoint of X(i) and X(j) for these {i,j}: {2, 14070}, {549, 10154}


X(34352) =  X(1)X(140)∩X(5)X(3884)

Barycentrics    2*a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 6*a^3*b^4 - 3*a*b^6 + b^7 + 2*a^6*c - 12*a^5*b*c + 11*a^4*b^2*c + 7*a^3*b^3*c - 12*a^2*b^4*c + 5*a*b^5*c - b^6*c - 3*a^5*c^2 + 11*a^4*b*c^2 - 20*a^3*b^2*c^2 + 12*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 + 7*a^3*b*c^3 + 12*a^2*b^2*c^3 - 10*a*b^3*c^3 + 3*b^4*c^3 + 6*a^3*c^4 - 12*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + 5*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7 : :
X(34352) = 2 X[140] + X[5559], X[1483] + 2 X[15862]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29569.

X(34352) lies on these lines: {1, 140}, {5, 3884}, {10, 1484}, {21, 952}, {495, 13375}, {515, 26202}, {517, 5499}, {519, 31650}, {632, 8256}, {1145, 34126}, {1385, 32905}, {1389, 5901}, {1482, 6853}, {1483, 15862}, {4187, 7705}, {5450, 32613}, {6583, 11362}, {6842, 10129}, {6940, 22765}, {7508, 10944}, {11231, 33895}, {16159, 28212}, {19524, 32141}, {19907, 31659}

X(34352) = reflection of X(1389) in X(5901)


X(34353) =  X(30)X(17643)∩X(65)X(5844)

Barycentrics    a*(a^8*b - 2*a^7*b^2 - 2*a^6*b^3 + 6*a^5*b^4 - 6*a^3*b^6 + 2*a^2*b^7 + 2*a*b^8 - b^9 + a^8*c - 6*a^7*b*c + 9*a^6*b^2*c + 2*a^5*b^3*c - 17*a^4*b^4*c + 14*a^3*b^5*c + 3*a^2*b^6*c - 10*a*b^7*c + 4*b^8*c - 2*a^7*c^2 + 9*a^6*b*c^2 - 24*a^5*b^2*c^2 + 19*a^4*b^3*c^2 + 15*a^3*b^4*c^2 - 27*a^2*b^5*c^2 + 11*a*b^6*c^2 - b^7*c^2 - 2*a^6*c^3 + 2*a^5*b*c^3 + 19*a^4*b^2*c^3 - 40*a^3*b^3*c^3 + 22*a^2*b^4*c^3 + 10*a*b^5*c^3 - 11*b^6*c^3 + 6*a^5*c^4 - 17*a^4*b*c^4 + 15*a^3*b^2*c^4 + 22*a^2*b^3*c^4 - 26*a*b^4*c^4 + 9*b^5*c^4 + 14*a^3*b*c^5 - 27*a^2*b^2*c^5 + 10*a*b^3*c^5 + 9*b^4*c^5 - 6*a^3*c^6 + 3*a^2*b*c^6 + 11*a*b^2*c^6 - 11*b^3*c^6 + 2*a^2*c^7 - 10*a*b*c^7 - b^2*c^7 + 2*a*c^8 + 4*b*c^8 - c^9) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29569.

X(34353) lies on these lines: {30, 17643}, {65, 5844}, {517, 5499}, {1385, 3754}, {1389, 5253}, {1483, 5885}, {5690, 15844}, {5883, 33657}


X(34354) =  8th HUNG-LOZADA-EULER POINT

Barycentrics    (SB+SC)*(5*S^4-(72*R^4-2*(2*SA+23*SW)*R^2-5*SA^2+7*SA*SW+4*SB*SC+7*SW^2)*S^2+(4*R^2-SW)*(72*R^4-2*(8*SW+9*SA)*R^2-7*SB*SC+7*SA^2-4*SW^2)*SA) : :

See Tran Quang Hung and César Lozada, Hyacinthos 29571.

X(34354) lies on these lines: {2, 3}, {8718, 30210}


X(34355) =  X(1)X(12619)∩X(106)X(18976)

Barycentrics    (a-b+c) (a+b-c) (a^7 (b+c)+(b^2-c^2)^4+a^6 (b^2-12 b c+c^2)-a^2 (b^2-c^2)^2 (b^2-b c+c^2)-a (b-c)^2 (b+c)^3 (3 b^2-5 b c+3 c^2)+a^5 (-5 b^3+13 b^2 c+13 b c^2-5 c^3)-a^4 (b^4-11 b^3 c+30 b^2 c^2-11 b c^3+c^4)+a^3 (7 b^5-16 b^4 c+10 b^3 c^2+10 b^2 c^3-16 b c^4+7 c^5)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 29575 and HG031019.

Let Na, Nb, Nc denote the nine-point centers of the triangles IBC, ICA, IAB, and let A"B"C" be the reflection of intouch triangle in the line X(1)X(3). Then NaNbNc is similar to A''B''C'', and the center of similitude is X(34355). (Angel Montesdeoca, October 7, 2019)

X(34355) lies on these lines: {1,12619}, {106,18976}, {11011,23869}


X(34356) =  X(5)X(25043)∩X(930)X(30480)

Barycentrics    b^2 c^2 (a^4-a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) (a^4-2 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^2-2 b^2 c^2+c^4) (-2 a^12+7 a^10 b^2-8 a^8 b^4+2 a^6 b^6+2 a^4 b^8-a^2 b^10+7 a^10 c^2-8 a^8 b^2 c^2-3 a^6 b^4 c^2+3 a^4 b^6 c^2+b^10 c^2-8 a^8 c^4-3 a^6 b^2 c^4-4 a^4 b^4 c^4+a^2 b^6 c^4-4 b^8 c^4+2 a^6 c^6+3 a^4 b^2 c^6+a^2 b^4 c^6+6 b^6 c^6+2 a^4 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : :
Barycentrics    (SC^2-3 S^2) (SW-SB) (SW-SC) (S^2+SB SC) (-4 S^2-SB SC+SB SW-SC^2+SC SW) (-2 R^4+10 R^2 SB+10 R^2 SC-2 R^2 SW+S^2-4 SB SW-4 SC SW+SW^2) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 29576.

X(34356) lies on the cubic K054 and these lines: {5,25043}, {930,30480}


X(34357) =  ISOGONAL CONJUGATE OF X(34358)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29577.

X(34357) lies on these lines: {}

X(34357) = isogonal conjugate of X(34358)
X(34357) = trilinear pole of the line {21, 33668}
X(34357) = barycentric quotient X(110)/X(11849)
X(34357) = trilinear quotient X(662)/X(11849)


X(34358) =  ISOGONAL CONJUGATE OF X(34357)

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

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29577.

X(34358) lies on these lines: {37, 12077}, {647, 661}, {2081, 2245}

X(34358) = isogonal conjugate of X(34357)
X(34358) = crossdifference of every pair of points on line X(21)X(33668)
X(34358) = trilinear product X(661)*X(11849)
X(34358) = trilinear quotient X(11849)/X(662)
X(34358) = barycentric product X(523)*X(11849)
X(34358) = barycentric quotient X(11849)/X(99)


X(34359) =  COMPLEMENT OF X(30226)

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

X(34359) lies on these lines: {2, 30226}, {3, 6}, {99, 287}, {115, 694}, {599, 23878}, {2549, 30229}, {3289, 15920}, {8861, 22416}

X(34359) = reflection of X(6) in X(5661)
X(34359) = complement of X(30226)
X(34359) = X(290)-of-1st-Brocard-triangle
X(34359) = 1st-Brocard-isogonal conjugate of X(36213)
X(34359) = 1st-Brocard-isotomic conjugate of X(2782)


X(34360) =  COMPLEMENT OF X(30227)

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

X(34360) lies on these lines: {2, 30227}, {6, 525}, {30, 599}, {297, 11185}, {378, 18304}, {401, 15066}, {458, 16077}, {543, 15595}, {1003, 2966}, {1975, 2421}, {2482, 5465}, {2777, 15526}, {4048, 15013}, {5118, 9306}, {5641, 11317}, {9308, 23582}, {10718, 10721}, {12037, 13202}, {14417, 14685}, {15014, 18906}

X(34360) = complement of X(30227)


X(34361) =  COMPLEMENT OF X(30228)

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

X(34361) = complement of X(30228)
X(34361) = X(664)-of-1st-Brocard-triangle
X(34361) = 1st-Brocard-isotomic conjugate of X(2785)


X(34362) =  COMPLEMENT OF X(34342)

Barycentrics    a^4 - a^3*b - 4*a^2*b^2 + 5*a*b^3 - 2*b^4 - a^3*c + 5*a^2*b*c - a*b^2*c - b^3*c - 4*a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + 5*a*c^3 - b*c^3 - 2*c^4 : :
X(34362) = X[24281] + 2 X[30225], 3 X[24281] - 2 X[34342], 3 X[30225] + X[34342]

X(34362) lies on these lines: {2, 24262}, {6, 519}, {514, 599}, {952, 4370}, {1121, 13466}, {1146, 12035}, {3735, 24289}, {3912, 30824}, {3943, 24864}, {4234, 5170}, {4555, 17230}, {6633, 17269}, {16672, 25031}, {17310, 31179}, {17953, 31017}

X(34362) = midpoint of X(2) and X(30225)
X(34362) = reflection of X(24281) in X(2)
X(34362) = complement of X(34342)
X(34362) = X(903)-of-1st-Brocard-triangle
X(34362) = 1st-Brocard-isotomic conjugate of X(2796)


X(34363) =  COMPLEMENT OF X(34343)

Barycentrics    2*a^4*b^2 - a^2*b^4 - 5*a^4*b*c + a^3*b^2*c + a^2*b^3*c + a*b^4*c + 2*a^4*c^2 + a^3*b*c^2 + 3*a^2*b^2*c^2 - 5*a*b^3*c^2 + 2*b^4*c^2 + a^2*b*c^3 - 5*a*b^2*c^3 - a^2*c^4 + a*b*c^4 + 2*b^2*c^4 : :
X(34363) = 3 X[24289] - 2 X[34343]

X(34363) lies on these lines: {2, 24289}, {6, 536}, {513, 599}, {3923, 24281}, {4370, 32041}, {4688, 16482}, {4740, 18822}, {13466, 29349}, {17738, 24277}

X(34363) = reflection of X(24289) in X(2)
X(34363) = complement of X(34343)


X(34364) =  COMPLEMENT OF X(34344)

Barycentrics    2*a^6*b^4 - a^4*b^6 - 5*a^6*b^2*c^2 + 4*a^4*b^4*c^2 + a^2*b^6*c^2 + 2*a^6*c^4 + 4*a^4*b^2*c^4 - 11*a^2*b^4*c^4 + 2*b^6*c^4 - a^4*c^6 + a^2*b^2*c^6 + 2*b^4*c^6 : :
X(34364) = X[30229] + 2 X[34341], 3 X[30229] - 2 X[34344], 3 X[34341] + X[34344]

X(34364) lies on these lines: {2, 30229}, {6, 538}, {76, 886}, {99, 2502}, {512, 599}, {690, 14606}, {3111, 9466}, {5118, 9306}, {24254, 24289}, {24259, 24281}

X(34364) = reflection of X(30229) in X(2)
X(34364) = midpoint of X(2) and X(34341)
X(34364) = complement of X(34344)


X(34365) =  X(2)X(15536)∩X(3)X(3447)

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

X(34365) lies on the Lester circle, the curve Q153, and these lines: {2, 15536}, {3, 3447}, {98, 265}, {5664, 14850}, {14560, 24975}, {14993, 15475}


X(34366) =  X(2)X(525)∩X(98)X(230)

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

X(34366) lies on the curve Q153 and these lines: {2, 525}, {98, 230}, {111, 468}, {1529, 10735}, {1551, 23967}, {6103, 17986}, {11580, 13509}, {15341, 24855}

X(34366) = orthoptic-circle-of-Steiner-inellipse-inverse of X(1640)


X(34367) =  X(2)X(249)∩X(14559)X(24875)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-2*a^2 + b^2 + c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^16 - 6*a^14*b^2 + 11*a^12*b^4 - 6*a^10*b^6 - 3*a^8*b^8 + 6*a^6*b^10 - 7*a^4*b^12 + 6*a^2*b^14 - 2*b^16 - 6*a^14*c^2 + 29*a^12*b^2*c^2 - 49*a^10*b^4*c^2 + 41*a^8*b^6*c^2 - 25*a^6*b^8*c^2 + 20*a^4*b^10*c^2 - 16*a^2*b^12*c^2 + 6*b^14*c^2 + 11*a^12*c^4 - 49*a^10*b^2*c^4 + 59*a^8*b^4*c^4 - 29*a^6*b^6*c^4 + 11*a^4*b^8*c^4 + 2*a^2*b^10*c^4 - 7*b^12*c^4 - 6*a^10*c^6 + 41*a^8*b^2*c^6 - 29*a^6*b^4*c^6 - 11*a^4*b^6*c^6 + 5*a^2*b^8*c^6 + 12*b^10*c^6 - 3*a^8*c^8 - 25*a^6*b^2*c^8 + 11*a^4*b^4*c^8 + 5*a^2*b^6*c^8 - 18*b^8*c^8 + 6*a^6*c^10 + 20*a^4*b^2*c^10 + 2*a^2*b^4*c^10 + 12*b^6*c^10 - 7*a^4*c^12 - 16*a^2*b^2*c^12 - 7*b^4*c^12 + 6*a^2*c^14 + 6*b^2*c^14 - 2*c^16) : :

X(34367) lies on the curve Q153 and these lines: {2, 249}, {14559, 24975}


X(34368) =  (name pending)

Barycentrics    (a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(a^2 - b^2 - a*c + c^2)*(a^2 - b^2 + a*c + c^2)*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*(a^18 - 3*a^16*b^2 + 6*a^14*b^4 - 13*a^12*b^6 + 20*a^10*b^8 - 21*a^8*b^10 + 18*a^6*b^12 - 11*a^4*b^14 + 3*a^2*b^16 - 3*a^16*c^2 + 3*a^14*b^2*c^2 + 3*a^12*b^4*c^2 - 8*a^10*b^6*c^2 + 15*a^8*b^8*c^2 - 24*a^6*b^10*c^2 + 24*a^4*b^12*c^2 - 11*a^2*b^14*c^2 + b^16*c^2 + 6*a^14*c^4 + 3*a^12*b^2*c^4 - 9*a^10*b^4*c^4 + 3*a^8*b^6*c^4 + 3*a^6*b^8*c^4 - 18*a^4*b^10*c^4 + 18*a^2*b^12*c^4 - 2*b^14*c^4 - 13*a^12*c^6 - 8*a^10*b^2*c^6 + 3*a^8*b^4*c^6 + 7*a^6*b^6*c^6 + 5*a^4*b^8*c^6 - 18*a^2*b^10*c^6 + 20*a^10*c^8 + 15*a^8*b^2*c^8 + 3*a^6*b^4*c^8 + 5*a^4*b^6*c^8 + 16*a^2*b^8*c^8 + b^10*c^8 - 21*a^8*c^10 - 24*a^6*b^2*c^10 - 18*a^4*b^4*c^10 - 18*a^2*b^6*c^10 + b^8*c^10 + 18*a^6*c^12 + 24*a^4*b^2*c^12 + 18*a^2*b^4*c^12 - 11*a^4*c^14 - 11*a^2*b^2*c^14 - 2*b^4*c^14 + 3*a^2*c^16 + b^2*c^16) : :

X(34368) lies on the curve Q153 and this line: {2, 15536}


X(34369) =  X(6)X(523)∩X(50)X(67)

Barycentrics    (a^4 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - b^2*c^2 + c^4)*(2*a^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(34369) lies on the cubics K381 and K396 and these lines: {6, 523}, {50, 67}, {98, 230}, {287, 524}, {542, 23967}, {685, 1990}, {1976, 1989}, {6034, 34175}, {8550, 32545}

X(34369) = midpoint of X(287) and X(2966)
X(34369) = X(i)-isoconjugate of X(j) for these (i,j): {662, 23350}, {842, 1959}, {1755, 5641}, {14223, 23997}
X(34369) = trilinear pole of line {1640, 5191}
X(34369) = crossdifference of every pair of points on line {511, 23350}
X(34369) = barycentric product X(i)*X(j) for these {i,j}: {98, 542}, {287, 6103}, {290, 5191}, {879, 7473}, {1640, 2966}, {1821, 2247}, {2395, 14999}, {2715, 18312}, {5967, 16092}
X(34369) = barycentric quotient X(i)/X(j) for these {i,j}: {98, 5641}, {512, 23350}, {542, 325}, {1640, 2799}, {1976, 842}, {2247, 1959}, {2395, 14223}, {2422, 14998}, {2715, 5649}, {2966, 6035}, {5191, 511}, {6041, 3569}, {6103, 297}, {7473, 877}, {14999, 2396}
X(34369) = {X(685),X(6531)}-harmonic conjugate of X(1990)


X(34370) =  X(6)X(14560)∩X(523)X(1989)

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

X(34370) lies on the cubic K396 and these lines: {6, 14560}, {523, 1989}, {842, 2493}, {9139, 11079}

X(34370) = barycentric product X(i)*X(j) for these {i,j}: {476, 23350}, {842, 14356}, {2799, 23969}
X(34370) = barycentric quotient X(i)/X(j) for these {i,j}: {23350, 3268}, {23969, 2966}


X(34371) =  X(6)X(57)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34371) lies on these lines: {1, 24328}, {2, 374}, {6, 57}, {7, 2262}, {9, 18725}, {19, 6180}, {30, 511}, {37, 18161}, {44, 16560}, {65, 4644}, {69, 189}, {77, 198}, {101, 6510}, {141, 3452}, {144, 21871}, {193, 3210}, {241, 2183}, {322, 20348}, {599, 31142}, {651, 2182}, {942, 4667}, {960, 4643}, {999, 1386}, {1100, 18162}, {1108, 1423}, {1122, 4000}, {1229, 20248}, {1350, 6282}, {1351, 2095}, {1436, 7013}, {1443, 11349}, {1486, 30621}, {1604, 7053}, {1814, 32677}, {1829, 23154}, {1901, 5929}, {1905, 24476}, {1944, 3732}, {1992, 2094}, {2093, 3751}, {2096, 6776}, {3056, 17642}, {3057, 4419}, {3242, 7962}, {3416, 3421}, {3589, 6692}, {3698, 4470}, {3740, 17251}, {3763, 20196}, {3812, 4670}, {3820, 3844}, {3882, 25083}, {4259, 7960}, {4363, 5836}, {4454, 14923}, {4641, 26934}, {4659, 10914}, {4662, 4690}, {4708, 24317}, {4748, 25917}, {5011, 10756}, {5060, 16702}, {5085, 21164}, {5480, 7682}, {5908, 6260}, {5909, 6245}, {5942, 21279}, {6244, 12329}, {7011, 34052}, {7202, 8609}, {9432, 26273}, {9943, 24683}, {9954, 10859}, {10387, 10388}, {10391, 17441}, {11677, 30620}, {15587, 21867}, {16284, 20719}, {18675, 28369}, {20080, 20214}, {20262, 21239}, {21370, 34048}, {21785, 28022}, {22129, 24611}, {25274, 30082}

X(34371) = psi-transform of X(2)
X(34371) = crossdifference of every pair of points on line {6, 3900}
X(34371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 2097, 57}, {651, 7291, 2182}, {2183, 3942, 241}, {17441, 26892, 10391}


X(34372) =  X(6)X(906)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34372) lies on these lines: {6, 906}, {30, 511}, {52, 3811}, {55, 1331}, {56, 1813}, {1216, 10916}, {2979, 24477}, {3060, 25568}, {3189, 5889}, {3908, 14686}, {5446, 21077}, {12437, 31732}, {16980, 32049}, {24391, 31737}


X(34373) =  X(6)X(2981)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34373) lies on these lines: {2, 11624}, {6, 2981}, {30, 511}, {51, 33459}, {69, 300}, {141, 16536}, {373, 33474}, {2979, 5859}, {3060, 5858}, {3917, 33458}, {5463, 30439}, {5615, 9145}, {5640, 9761}, {5650, 33475}, {7998, 9763}, {9115, 15544}

X(34373) = isogonal conjugate of X(34374)
X(34373) = {X(11126),X(17403)}-harmonic conjugate of X(19294)


X(34374) =  ISOGONAL CONJUGATE OF X(34373)

Barycentrics    1/((Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) + 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34374) lies on these lines: {2, 10409}, {14, 9202}, {99, 11146}, {110, 396}, {112, 463}, {476, 11141}, {2380, 20579}, {5618, 16463}, {5995, 8014}, {6779, 9203}, {9112, 16806}

X(34374) = isogonal conjugate of X(34373)
X(34374) = orthoptic circle of the Steiner inellipse inverse of X(15609)
X(34374) = X(5472)-cross conjugate of X(11085)


X(34375) =  X(6)X(6151)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34375) lies on these lines: {2, 11626}, {6, 6151}, {30, 511}, {51, 33458}, {69, 301}, {141, 16537}, {373, 33475}, {2979, 5858}, {3060, 5859}, {3917, 33459}, {5464, 30440}, {5611, 9145}, {5640, 9763}, {5650, 33474}, {7998, 9761}, {9117, 15544}

X(34375) = isogonal conjugate of X(34376)
X(34375) = {X(11127),X(17402)}-harmonic conjugate of X(19295)


X(34376) =  ISOGONAL CONJUGATE OF X(34375)

Barycentrics    1/((Sqrt[3]*b^2*c^2*(2*a^2 - b^2 - c^2) - 2*(a^2*b^2 - b^4 + a^2*c^2 - c^4)*S)) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34376) lies on these lines: {2, 10410}, {13, 9203}, {99, 11145}, {110, 395}, {112, 462}, {476, 11142}, {2381, 20578}, {5619, 16464}, {5994, 8015}, {6780, 9202}, {9113, 16807}

X(34376) = isogonal conjugate of X(34375)
X(34376) = orthoptic circle of the Steiner inellipse inverse of X(15610)
X(34376) = X(5471)-cross conjugate of X(11080)


X(34377) =  X(6)X(63)∩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) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34377) lies on these lines: {6, 63}, {30, 511}, {65, 4363}, {69, 321}, {72, 4259}, {141, 226}, {193, 17147}, {210, 17251}, {320, 17789}, {326, 2178}, {599, 31164}, {942, 4670}, {960, 4364}, {993, 1386}, {1122, 7232}, {1155, 17977}, {1350, 18446}, {1478, 3416}, {1764, 7289}, {1959, 8609}, {2245, 25083}, {2262, 4361}, {3057, 17318}, {3589, 5745}, {3683, 16792}, {3729, 21853}, {3751, 4424}, {3763, 31266}, {3779, 24326}, {3812, 4472}, {3822, 3844}, {3868, 4644}, {3869, 4419}, {3874, 4667}, {3876, 4748}, {3916, 5135}, {3954, 4503}, {4659, 5903}, {4665, 5836}, {4708, 5044}, {4795, 24473}, {4798, 5439}, {5006, 16702}, {5085, 21165}, {5440, 33844}, {7262, 16793}, {10477, 24476}, {10754, 11611}, {12635, 24328}, {17262, 21871}, {18252, 20713}, {18611, 23075}, {18726, 21061}, {20715, 24699}, {22277, 22325}, {24333, 25368}, {24424, 24705}, {24441, 31165}

X(34377) = crossdifference of every pair of points on line {6, 8678}


X(34378) =  X(6)X(3874)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34378) lies on these lines: {6, 3874}, {10, 24476}, {30, 511}, {69, 1930}, {141, 3678}, {182, 12005}, {193, 17489}, {611, 18389}, {1386, 3881}, {1428, 5083}, {1469, 15556}, {3242, 3878}, {3313, 23156}, {3618, 18398}, {3663, 4523}, {3751, 3868}, {3811, 7289}, {3844, 4015}, {3869, 16496}, {3873, 16475}, {3889, 16491}, {4973, 5096}, {6583, 18583}, {10516, 15064}, {12432, 24471}, {20455, 20715}, {22769, 22836}, {25050, 32846}, {32118, 32935}


X(34379) =  X(10)X(69)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34379) lies on these lines: {1, 193}, {6, 1125}, {8, 17116}, {10, 69}, {30, 511}, {42, 4001}, {44, 4966}, {63, 4028}, {141, 3634}, {226, 32853}, {238, 4684}, {239, 24231}, {306, 32912}, {320, 1738}, {355, 11898}, {551, 1992}, {599, 3828}, {611, 13405}, {895, 13605}, {908, 32919}, {940, 4104}, {946, 1351}, {984, 3879}, {991, 3811}, {1054, 5212}, {1266, 4716}, {1326, 6629}, {1350, 12512}, {1352, 19925}, {1353, 1385}, {1386, 3629}, {1468, 4101}, {1469, 4298}, {1698, 3620}, {1742, 6765}, {1757, 3912}, {2321, 32935}, {3011, 16704}, {3056, 12575}, {3242, 3635}, {3244, 11008}, {3416, 3626}, {3555, 21746}, {3576, 14912}, {3589, 19878}, {3616, 17331}, {3618, 19862}, {3630, 4691}, {3631, 3844}, {3679, 11160}, {3685, 20072}, {3686, 24325}, {3687, 32913}, {3696, 17365}, {3717, 32846}, {3729, 4133}, {3755, 4655}, {3763, 31253}, {3775, 5750}, {3790, 17373}, {3817, 14853}, {3826, 17376}, {3836, 4753}, {3886, 24695}, {3914, 32859}, {3920, 20086}, {3932, 17374}, {3977, 4062}, {3980, 4061}, {4026, 17344}, {4035, 4438}, {4078, 4851}, {4138, 33137}, {4260, 12436}, {4297, 6776}, {4353, 4856}, {4357, 4649}, {4385, 34282}, {4429, 17361}, {4700, 4974}, {4722, 5294}, {4745, 15533}, {4780, 24248}, {4847, 32946}, {4899, 32847}, {4938, 32848}, {5032, 25055}, {5050, 10165}, {5052, 12263}, {5093, 5886}, {5095, 11720}, {5249, 32864}, {5477, 11711}, {5480, 12571}, {5542, 16825}, {5691, 5921}, {5713, 10916}, {5788, 21077}, {5905, 17156}, {6210, 6762}, {9798, 19588}, {9967, 31738}, {10164, 10519}, {10171, 14561}, {10477, 12572}, {10753, 21636}, {10754, 11599}, {10755, 21630}, {10759, 21635}, {10761, 11814}, {12513, 31394}, {13211, 32244}, {15481, 17243}, {15569, 17332}, {16830, 20090}, {17023, 28650}, {17348, 25557}, {17353, 33087}, {17363, 24349}, {17781, 32915}, {18440, 31673}, {18483, 21850}, {19868, 33682}, {21060, 29649}, {24210, 33066}, {25006, 32949}, {26015, 32843}, {26227, 31303}, {26723, 33069}, {29639, 31034}, {30768, 31017}, {31730, 33878}

X(34379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {69, 3751, 10}, {4722, 33081, 5294}, {4851, 5220, 4078}


X(34380) =  X(6)X(140)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34380) lies on these lines: {2, 5093}, {3, 193}, {4, 11898}, {5, 69}, {6, 140}, {30, 511}, {49, 19121}, {51, 10128}, {52, 9825}, {66, 18356}, {141, 576}, {159, 9925}, {182, 3530}, {195, 34002}, {230, 1570}, {265, 32244}, {323, 468}, {325, 10011}, {340, 6530}, {381, 11160}, {382, 5921}, {394, 6677}, {487, 12314}, {488, 12313}, {546, 1352}, {547, 599}, {548, 1350}, {549, 1992}, {550, 6776}, {575, 12108}, {597, 10124}, {613, 28369}, {632, 3618}, {895, 10264}, {1147, 19154}, {1368, 6515}, {1484, 10755}, {1511, 5095}, {1513, 7779}, {1595, 12167}, {1656, 3620}, {1843, 10263}, {1993, 6676}, {1994, 7499}, {2080, 6390}, {2979, 10691}, {3056, 15172}, {3095, 7767}, {3098, 8550}, {3167, 10154}, {3292, 32269}, {3313, 15074}, {3524, 33748}, {3580, 5159}, {3589, 5097}, {3627, 18440}, {3630, 3850}, {3631, 24206}, {3751, 5690}, {3785, 10983}, {3818, 3861}, {3853, 15069}, {3917, 7734}, {4220, 20086}, {5028, 5305}, {5032, 5054}, {5066, 10516}, {5085, 12100}, {5092, 12007}, {5111, 15993}, {5181, 10272}, {5188, 7890}, {5446, 14913}, {5476, 10109}, {5477, 33813}, {5562, 13142}, {5609, 32114}, {5656, 12164}, {5774, 15973}, {5876, 12294}, {5889, 31829}, {5943, 13361}, {6101, 9967}, {6194, 7837}, {6243, 6403}, {6329, 22330}, {6391, 11411}, {6467, 10625}, {6593, 13392}, {6661, 22521}, {6675, 15988}, {6823, 12160}, {6998, 20090}, {7380, 17343}, {7387, 19588}, {7495, 11004}, {7575, 32220}, {7762, 12251}, {7788, 9753}, {8359, 32447}, {8584, 11812}, {8703, 25406}, {9300, 15819}, {9822, 10095}, {9924, 9936}, {9969, 23411}, {9974, 13925}, {9975, 13993}, {10112, 12024}, {10113, 32275}, {10168, 20583}, {10300, 18911}, {10627, 11574}, {10733, 32272}, {10759, 11698}, {11178, 11737}, {11179, 31884}, {11180, 15687}, {11255, 12359}, {11412, 12022}, {11694, 15303}, {12017, 15712}, {12107, 15577}, {12121, 32234}, {12161, 16197}, {12272, 16658}, {12322, 12602}, {12323, 12601}, {12325, 15559}, {12584, 25329}, {12811, 19130}, {13331, 22677}, {13340, 15531}, {13346, 23328}, {13383, 19139}, {13451, 29959}, {13488, 18436}, {13562, 14449}, {14848, 15699}, {14891, 17508}, {15073, 26926}, {15122, 19348}, {15462, 16531}, {15557, 25043}, {15812, 18952}, {16238, 20806}, {16789, 25337}, {18934, 23335}, {19126, 32046}, {19128, 22115}, {19129, 34148}, {19697, 32134}, {20304, 32257}, {25321, 32609}, {25338, 32113}, {32448, 32451}

X(34380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 193, 1353}, {4, 20080, 11898}, {69, 1351, 5}, {141, 576, 18583}, {141, 18583, 3628}, {599, 5102, 14561}, {1352, 11477, 21850}, {1352, 21850, 546}, {1992, 10519, 5050}, {2979, 11245, 10691}, {3527, 11487, 5}, {5050, 10519, 549}, {5480, 18358, 3850}, {6776, 33878, 550}, {14848, 21356, 15699}


X(34381) =  ISOGONAL CONJUGATE OF X(15344)

Barycentrics    a*(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) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34381) lies on these lines: {1, 7083}, {3, 7289}, {6, 169}, {10, 25365}, {30, 511}, {63, 17441}, {65, 3751}, {68, 12587}, {69, 72}, {105, 15382}, {116, 31897}, {141, 5044}, {182, 9940}, {193, 1829}, {228, 18607}, {238, 20601}, {242, 3732}, {354, 3167}, {651, 1876}, {1069, 12595}, {1071, 6776}, {1147, 13373}, {1214, 20760}, {1282, 5018}, {1350, 31793}, {1351, 24474}, {1352, 5777}, {1353, 24475}, {1385, 22769}, {1386, 5045}, {1428, 3660}, {1439, 23603}, {1736, 21362}, {1738, 20455}, {1814, 7193}, {1818, 3942}, {1824, 5905}, {1828, 12649}, {1843, 14054}, {1902, 5921}, {1992, 24473}, {2262, 5781}, {3033, 9436}, {3057, 16496}, {3173, 5173}, {3242, 9957}, {3579, 12329}, {3618, 5439}, {3620, 3876}, {3927, 4047}, {4259, 7352}, {4260, 10481}, {4298, 13572}, {4463, 32859}, {4523, 4655}, {4663, 31794}, {5050, 10202}, {5085, 11227}, {5096, 5122}, {5138, 11018}, {5480, 5806}, {5504, 10100}, {5776, 10441}, {5880, 21867}, {5885, 8548}, {6147, 9895}, {6467, 23154}, {6510, 17976}, {6583, 19139}, {6708, 20256}, {9925, 15178}, {9928, 34339}, {10157, 10516}, {10167, 25406}, {10477, 30625}, {11573, 11574}, {12429, 14872}, {12586, 31937}, {12723, 24695}, {13605, 23296}, {14913, 29957}, {15076, 17635}, {16465, 26892}, {16491, 17609}, {16560, 23693}, {17102, 20805}, {17615, 22321}, {17975, 22148}, {18651, 21015}, {18734, 23167}, {20078, 20243}, {20254, 22149}, {20752, 20811}, {21167, 33575}, {32126, 32263}

X(34381) = isogonal conjugate of X(15344)
X(34381) = isotomic conjugate of the polar conjugate of X(3290)
X(34381) = crossdifference of every pair of points on line {6, 15313}
X(34381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 24476, 942}, {4523, 4655, 18252}


X(34382) =  X(6)X(1147)∩X(30)X(511)

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34382) lies on these lines: {3, 6391}, {4, 12271}, {5, 14913}, {6, 1147}, {20, 12282}, {30, 511}, {49, 21637}, {51, 3167}, {52, 193}, {67, 19477}, {68, 69}, {141, 5449}, {155, 1351}, {159, 32048}, {182, 8548}, {265, 32260}, {389, 1353}, {487, 12604}, {488, 12603}, {575, 22829}, {576, 9925}, {895, 5504}, {1112, 21847}, {1350, 7689}, {1352, 9927}, {1469, 19471}, {1495, 12310}, {1692, 32661}, {1993, 27365}, {2931, 32127}, {3056, 9931}, {3060, 7714}, {3095, 19597}, {3098, 9938}, {3292, 18449}, {3448, 32249}, {3580, 32263}, {3629, 21852}, {3751, 9928}, {3779, 12417}, {3818, 22661}, {5028, 23128}, {5050, 5892}, {5095, 11557}, {5102, 9971}, {5447, 11574}, {5448, 5480}, {5562, 11898}, {5596, 12420}, {5654, 11188}, {5921, 12162}, {6102, 21851}, {6239, 12222}, {6291, 12602}, {6400, 12221}, {6406, 12601}, {6504, 14593}, {6776, 12118}, {6803, 17040}, {7387, 9924}, {8538, 20806}, {9544, 19123}, {9730, 14912}, {9820, 9822}, {9973, 11477}, {10111, 32285}, {10116, 12421}, {10170, 14852}, {10282, 19154}, {10625, 11411}, {11412, 20080}, {11562, 32234}, {11579, 12901}, {12163, 33878}, {12166, 12167}, {12223, 12510}, {12224, 12509}, {12293, 12294}, {12383, 32248}, {12412, 32276}, {12419, 32264}, {12584, 19138}, {12590, 19486}, {12591, 19487}, {13137, 17932}, {13367, 19129}, {13383, 15585}, {14561, 29959}, {15045, 33748}, {15118, 19509}, {15123, 23296}, {15583, 23335}, {18475, 19131}, {18934, 18935}, {19196, 19197}, {19458, 19459}, {20302, 24206}, {20794, 30258}, {21639, 22115}, {21650, 32272}, {21850, 22660}, {32114, 32123}, {32191, 32455}

X(34382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {{6, 9937, 19141}, {69, 9967, 1216}, {69, 15073, 9967}, {193, 6403, 52}, {1147, 12235, 5462}, {1351, 1843, 5446}, {1351, 19588, 155}, {3167, 14914, 5093}, {9926, 19141, 6}, {9937, 15316, 1147}, {10607, 10608, 3}, {11898, 18438, 5562}


X(34383) =  X(6)X(694)∩X(30)X(511)

Barycentrics    a^2*(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) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29595.

X(34383) lies on these lines: {2, 6784}, {6, 694}, {30, 511}, {69, 290}, {76, 4173}, {141, 7668}, {182, 9145}, {193, 8264}, {211, 7805}, {263, 1992}, {373, 12093}, {385, 11673}, {597, 34236}, {671, 6787}, {805, 14931}, {895, 9513}, {1355, 9413}, {1356, 7170}, {1576, 3506}, {1843, 27377}, {1976, 4558}, {1987, 6391}, {2421, 9149}, {2482, 3111}, {3056, 24437}, {3060, 7837}, {3098, 9142}, {3491, 5254}, {3571, 7063}, {3618, 31639}, {3629, 25326}, {4590, 34238}, {5943, 9300}, {5989, 17970}, {6054, 6785}, {6071, 12833}, {6072, 13137}, {6128, 29959}, {7062, 9414}, {7760, 27374}, {7813, 14962}, {7838, 27375}, {7840, 13207}, {8598, 32442}, {10602, 16098}, {10765, 14948}, {11184, 13240}, {12157, 18823}, {14609, 30495}, {14981, 31850}, {15630, 15631}, {15991, 19581}, {21320, 28369}, {24206, 33548}, {30534, 30535}

X(34383) = isotomic conjugate of the isogonal conjugate of X(21444)
X(34383) = crossdifference of every pair of points on line {6, 804}
X(34383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 694, 1084}, {69, 25051, 20021}, {69, 25332, 670}, {141, 25324, 7668}, {6784, 6786, 2}, {7840, 13207, 33873}, {15630, 15631, 22103}


X(34384) =  ISOTOMIC CONJUGATE OF X(51)

Barycentrics    1/(a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)) : :

X(34384) lies on these lines: {69, 8795}, {76, 275}, {95, 183}, {97, 8024}, {290, 3917}, {343, 6331}, {1241, 8882}, {4176, 20023}, {8842, 8901}, {15108, 23962}

X(34384) = isotomic conjugate of X(51)
X(34384) = polar conjugate of X(3199)
X(34384) = isotomic conjugate of the anticomplement of X(3819)
X(34384) = isotomic conjugate of the complement of X(2979)
X(34384) = isotomic conjugate of the isogonal conjugate of X(95)
X(34384) = isotomic conjugate of the polar conjugate of X(276)
X(34384) = X(i)-cross conjugate of X(j) for these (i,j): {95, 276}, {1232, 76}, {3265, 6331}, {3819, 2}, {21403, 75}
X(34384) = X(i)-isoconjugate of X(j) for these (i,j): {5, 560}, {6, 2179}, {19, 217}, {31, 51}, {32, 1953}, {48, 3199}, {53, 9247}, {82, 27374}, {184, 2181}, {216, 1973}, {311, 1917}, {418, 1096}, {669, 2617}, {798, 1625}, {1087, 14573}, {1393, 2175}, {1397, 7069}, {1501, 14213}, {1918, 18180}, {1923, 17500}, {1924, 14570}, {2205, 17167}, {2206, 21807}, {2212, 30493}, {2290, 11060}, {2618, 14574}, {15451, 32676}
X(34384) = cevapoint of X(i) and X(j) for these (i,j): {2, 2979}, {69, 76}, {75, 21579}
X(34384) = trilinear pole of line {3267, 7799}
X(34384) = barycentric product X(i)*X(j) for these {i,j}: {54, 1502}, {69, 276}, {76, 95}, {97, 18022}, {275, 305}, {561, 2167}, {670, 15412}, {1232, 31617}, {1928, 2148}, {2616, 4602}, {2623, 4609}, {3267, 18831}, {3926, 8795}, {4176, 8794}, {6528, 15414}
X(34384) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2179}, {2, 51}, {3, 217}, {4, 3199}, {39, 27374}, {54, 32}, {69, 216}, {75, 1953}, {76, 5}, {85, 1393}, {92, 2181}, {95, 6}, {97, 184}, {99, 1625}, {264, 53}, {274, 18180}, {275, 25}, {276, 4}, {305, 343}, {308, 17500}, {310, 17167}, {312, 7069}, {313, 21011}, {317, 14576}, {321, 21807}, {340, 11062}, {348, 30493}, {394, 418}, {525, 15451}, {561, 14213}, {670, 14570}, {799, 2617}, {850, 12077}, {1141, 11060}, {1232, 233}, {1235, 27371}, {1502, 311}, {2052, 14569}, {2148, 560}, {2167, 31}, {2169, 9247}, {2190, 1973}, {2616, 798}, {2623, 669}, {3261, 21102}, {3265, 17434}, {3267, 6368}, {3268, 2081}, {3926, 5562}, {4563, 23181}, {6389, 6751}, {7763, 52}, {7769, 143}, {7799, 1154}, {8794, 6524}, {8795, 393}, {8882, 1974}, {8884, 2207}, {8901, 3124}, {9291, 27359}, {14376, 27372}, {14533, 14575}, {14573, 9233}, {14586, 14574}, {14587, 23963}, {15412, 512}, {15414, 520}, {16030, 3051}, {16032, 8577}, {16037, 8576}, {16813, 32713}, {18022, 324}, {18027, 13450}, {18315, 1576}, {18831, 112}, {19166, 800}, {19188, 17810}, {19189, 2211}, {19210, 14585}, {20948, 2618}, {23286, 3049}, {26166, 3574}, {31617, 1173}, {32002, 14577}, {32831, 14531}, {32833, 5891}, {32834, 27355}


X(34385) =  ISOTOMIC CONJUGATE OF X(52)

Barycentrics    1/(a^2*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4)) : :

X(34385) lies on these lines: {54, 311}, {68, 317}, {76, 95}, {264, 275}, {327, 2165}, {2367, 32692}, {5408, 16037}, {5409, 16032}

X(34385) = isotomic conjugate of X(52)
X(34385) = polar conjugate of X(14576)
X(34385) = isotomic conjugate of the anticomplement of X(1216)
X(34385) = isotomic conjugate of the complement of X(11412)
X(34385) = isotomic conjugate of the isogonal conjugate of X(96)
X(34385) = X(i)-cross conjugate of X(j) for these (i,j): {69, 95}, {97, 276}, {1216, 2}
X(34385) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2180}, {31, 52}, {47, 51}, {48, 14576}, {53, 563}, {217, 1748}, {467, 9247}, {571, 1953}, {1147, 2181}, {1993, 2179}
X(34385) = cevapoint of X(i) and X(j) for these (i,j): {2, 11412}, {68, 5392}, {69, 20563}, {525, 8901}
X(34385) = trilinear pole of line {850, 15412}
X(34385) = barycentric product X(i)*X(j) for these {i,j}: {68, 276}, {76, 96}, {95, 5392}, {275, 20563}, {561, 2168}, {2167, 20571}
X(34385) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2180}, {2, 52}, {4, 14576}, {54, 571}, {68, 216}, {91, 1953}, {95, 1993}, {96, 6}, {97, 1147}, {264, 467}, {275, 24}, {276, 317}, {847, 53}, {925, 1625}, {1993, 3133}, {2165, 51}, {2167, 47}, {2168, 31}, {2169, 563}, {2351, 217}, {5392, 5}, {5962, 11062}, {8795, 11547}, {8884, 8745}, {14593, 3199}, {15412, 924}, {16032, 372}, {16037, 371}, {20563, 343}, {20571, 14213}, {23286, 30451}, {32692, 1576}


X(34386) =  ISOTOMIC CONJUGATE OF X(53)

Barycentrics    1/((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)) : :
Barycentrics    SA/(S^2 + SB*SC) : :

X(34386) lies on these lines: {54, 69}, {76, 275}, {83, 3289}, {96, 8781}, {97, 3926}, {99, 1298}, {288, 343}, {315, 8884}, {316, 19169}, {323, 26166}, {325, 19179}, {933, 2366}, {1231, 20924}, {1568, 15031}, {3933, 6394}, {4563, 28706}, {4993, 32828}, {6393, 14533}, {7768, 18831}, {7776, 19176}, {8795, 14615}, {15318, 20477}, {19188, 32832}

X(34386) = isogonal conjugate of X(3199)
X(34386) = isotomic conjugate of X(53)
X(34386) = polar conjugate of X(14569)
X(34386) = anticomplement of the isogonal conjugate of X(2984)
X(34386) = isotomic conjugate of the complement of X(20477)
X(34386) = isotomic conjugate of the isogonal conjugate of X(97)
X(34386) = isotomic conjugate of the polar conjugate of X(95)
X(34386) = X(2984)-anticomplementary conjugate of X(8)
X(34386) = X(i)-cross conjugate of X(j) for these (i,j): {97, 95}, {1238, 305}, {3267, 4563}
X(34386) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3199}, {4, 2179}, {5, 1973}, {6, 2181}, {19, 51}, {25, 1953}, {31, 53}, {48, 14569}, {158, 217}, {216, 1096}, {324, 560}, {418, 6520}, {607, 1393}, {608, 7069}, {1474, 21807}, {1974, 14213}, {2180, 14593}, {2203, 21011}, {2290, 18384}, {2333, 18180}, {2489, 2617}, {9247, 13450}, {12077, 32676}, {15451, 24019}
X(34386) = cevapoint of X(i) and X(j) for these (i,j): {2, 20477}, {6, 11206}, {69, 394}
X(34386) = trilinear pole of line {3265, 15414}
X(34386) = barycentric product X(i)*X(j) for these {i,j}: {54, 305}, {69, 95}, {76, 97}, {275, 3926}, {276, 394}, {304, 2167}, {561, 2169}, {648, 15414}, {670, 23286}, {1502, 14533}, {3265, 18831}, {3267, 18315}, {3964, 8795}, {4143, 16813}, {4176, 8884}, {4563, 15412}, {18022, 19210}
X(34386) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2181}, {2, 53}, {3, 51}, {4, 14569}, {6, 3199}, {48, 2179}, {54, 25}, {63, 1953}, {69, 5}, {72, 21807}, {76, 324}, {77, 1393}, {78, 7069}, {95, 4}, {96, 14593}, {97, 6}, {141, 27371}, {160, 15897}, {264, 13450}, {275, 393}, {276, 2052}, {288, 33631}, {298, 6117}, {299, 6116}, {304, 14213}, {305, 311}, {306, 21011}, {323, 11062}, {394, 216}, {426, 6751}, {520, 15451}, {525, 12077}, {577, 217}, {631, 6755}, {850, 23290}, {933, 32713}, {1078, 30506}, {1092, 418}, {1141, 18384}, {1232, 14978}, {1238, 1209}, {1444, 18180}, {1799, 17500}, {1804, 30493}, {1993, 14576}, {1994, 14577}, {2148, 1973}, {2167, 19}, {2169, 31}, {2190, 1096}, {2623, 2489}, {3265, 6368}, {3267, 18314}, {3926, 343}, {3964, 5562}, {4025, 21102}, {4558, 1625}, {4563, 14570}, {4592, 2617}, {6340, 27364}, {7763, 467}, {7769, 14129}, {7799, 14918}, {8552, 2081}, {8795, 1093}, {8882, 2207}, {8884, 6524}, {8901, 8754}, {9723, 52}, {11077, 11060}, {14208, 2618}, {14371, 32319}, {14533, 32}, {15394, 8798}, {15412, 2501}, {15414, 525}, {15958, 1576}, {16030, 1843}, {16813, 6529}, {17206, 17167}, {18315, 112}, {18695, 1087}, {18831, 107}, {19166, 235}, {19180, 800}, {19210, 184}, {20477, 14363}, {20775, 27374}, {23286, 512}, {33629, 3172}
X(34386) = {X(69),X(1092)}-harmonic conjugate of X(1078)


X(34387) =  ISOTOMIC CONJUGATE OF X(59)

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

X(34387) lies on these lines: {1, 26541}, {2, 13006}, {7, 264}, {8, 76}, {75, 311}, {78, 3760}, {95, 7279}, {99, 4996}, {116, 1577}, {312, 28931}, {313, 20895}, {314, 11604}, {318, 1235}, {319, 1232}, {320, 3260}, {338, 1086}, {339, 2968}, {349, 20880}, {350, 4511}, {693, 1565}, {784, 23646}, {850, 17886}, {1111, 3120}, {1146, 4391}, {1226, 1441}, {1227, 1234}, {1228, 21422}, {1230, 3687}, {1447, 10538}, {1847, 7020}, {1909, 4861}, {2973, 20901}, {2980, 7087}, {3123, 23676}, {3271, 18101}, {3705, 8024}, {3761, 3872}, {4359, 28940}, {4384, 26592}, {4560, 7117}, {4858, 21044}, {5552, 18135}, {6376, 28975}, {6381, 6735}, {6734, 20888}, {8735, 26932}, {10527, 34284}, {11998, 23880}, {14615, 21296}, {17860, 32776}, {18140, 27529}, {20237, 28654}, {20913, 28797}, {20937, 30596}, {21208, 24186}, {21420, 27801}, {23528, 26563}, {24548, 26094}, {26539, 26959}, {26666, 27091}

X(34387) = anticomplement of X(13006)
X(34387) = isotomic conjugate of X(59)
X(34387) = polar conjugate of X(7115)
X(34387) = isotomic conjugate of the isogonal conjugate of X(11)
X(34387) = polar conjugate of the isogonal conjugate of X(26932)
X(34387) = X(i)-Ceva conjugate of X(j) for these (i,j): {75, 850}, {76, 4391}, {264, 693}, {290, 3766}, {6063, 3261}
X(34387) = X(i)-cross conjugate of X(j) for these (i,j): {4858, 23989}, {24026, 23978}
X(34387) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2149}, {9, 23979}, {12, 23995}, {31, 59}, {32, 4564}, {41, 1262}, {48, 7115}, {55, 24027}, {56, 1110}, {57, 23990}, {101, 1415}, {108, 32656}, {109, 692}, {163, 4559}, {181, 1101}, {184, 7012}, {269, 6066}, {560, 4998}, {604, 1252}, {651, 32739}, {765, 1397}, {906, 32674}, {1106, 6065}, {1253, 7339}, {1275, 9447}, {1402, 4570}, {1576, 4551}, {1783, 32660}, {1919, 31615}, {1983, 32675}, {2171, 23357}, {2175, 7045}, {2205, 4620}, {2283, 32666}, {2289, 23985}, {2427, 32669}, {3063, 4619}, {3185, 15386}, {6056, 24033}, {6358, 23963}, {6602, 23971}, {23067, 32676}, {23703, 32719}
X(34387) = cevapoint of X(i) and X(j) for these (i,j): {11, 26932}, {4858, 24026}
X(34387) = crosspoint of X(i) and X(j) for these (i,j): {75, 18155}, {3261, 6063}
X(34387) = crosssum of X(i) and X(j) for these (i,j): {692, 20986}, {2175, 32739}
X(34387) = complement of polar conjugate of isogonal conjugate of X(23161)
X(34387) = barycentric product X(i)*X(j) for these {i,j}: {7, 23978}, {8, 23989}, {11, 76}, {60, 23962}, {75, 4858}, {85, 24026}, {92, 17880}, {115, 18021}, {244, 28659}, {261, 338}, {264, 26932}, {305, 8735}, {310, 21044}, {312, 1111}, {313, 17197}, {314, 16732}, {331, 2968}, {333, 21207}, {345, 2973}, {348, 21666}, {522, 3261}, {561, 2170}, {658, 23104}, {693, 4391}, {850, 4560}, {1086, 3596}, {1146, 6063}, {1364, 18027}, {1502, 3271}, {1565, 7017}, {1577, 18155}, {1969, 7004}, {1978, 21132}, {2185, 23994}, {2310, 20567}, {2997, 17878}, {3120, 28660}, {3699, 23100}, {3701, 16727}, {3737, 20948}, {4086, 7199}, {4124, 18895}, {4397, 24002}, {4516, 6385}, {7117, 18022}, {7336, 31625}, {8024, 18101}, {17205, 30713}, {18191, 27801}
X(34387) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2149}, {2, 59}, {4, 7115}, {7, 1262}, {8, 1252}, {9, 1110}, {11, 6}, {55, 23990}, {56, 23979}, {57, 24027}, {60, 23357}, {75, 4564}, {76, 4998}, {85, 7045}, {92, 7012}, {115, 181}, {123, 22132}, {124, 573}, {125, 2197}, {220, 6066}, {244, 604}, {261, 249}, {273, 7128}, {279, 7339}, {310, 4620}, {312, 765}, {314, 4567}, {333, 4570}, {338, 12}, {339, 26942}, {346, 6065}, {479, 23971}, {513, 1415}, {514, 109}, {521, 906}, {522, 101}, {523, 4559}, {525, 23067}, {650, 692}, {652, 32656}, {663, 32739}, {664, 4619}, {668, 31615}, {693, 651}, {850, 4552}, {885, 919}, {918, 2283}, {1015, 1397}, {1024, 32666}, {1086, 56}, {1090, 2170}, {1109, 2171}, {1111, 57}, {1118, 23985}, {1146, 55}, {1358, 1407}, {1364, 577}, {1459, 32660}, {1565, 222}, {1577, 4551}, {1638, 23346}, {1639, 23344}, {1647, 1404}, {2150, 23995}, {2170, 31}, {2185, 1101}, {2310, 41}, {2401, 2720}, {2611, 21741}, {2804, 2427}, {2968, 219}, {2969, 608}, {2970, 8736}, {2973, 278}, {3022, 14827}, {3064, 8750}, {3119, 1253}, {3120, 1400}, {3125, 1402}, {3239, 3939}, {3261, 664}, {3271, 32}, {3326, 23980}, {3596, 1016}, {3676, 1461}, {3700, 4557}, {3737, 163}, {3738, 1983}, {3762, 23703}, {3942, 603}, {4025, 1813}, {4036, 21859}, {4077, 1020}, {4081, 220}, {4086, 1018}, {4092, 1500}, {4124, 1914}, {4391, 100}, {4397, 644}, {4459, 172}, {4466, 73}, {4516, 213}, {4530, 902}, {4534, 3052}, {4542, 1017}, {4560, 110}, {4581, 8687}, {4582, 6551}, {4768, 1023}, {4858, 1}, {4904, 1617}, {4939, 1743}, {4957, 2099}, {4965, 7031}, {4997, 9268}, {5514, 7074}, {5532, 14936}, {6063, 1275}, {6332, 1331}, {7004, 48}, {7017, 15742}, {7117, 184}, {7192, 4565}, {7199, 1414}, {7202, 1399}, {7252, 1576}, {7253, 5546}, {7336, 1015}, {7649, 32674}, {8287, 2594}, {8735, 25}, {10015, 23981}, {11998, 20986}, {13478, 15386}, {14936, 2175}, {15413, 6516}, {15416, 4571}, {15526, 7066}, {15633, 15629}, {16596, 7078}, {16726, 1408}, {16727, 1014}, {16732, 65}, {17059, 4253}, {17197, 58}, {17205, 1412}, {17219, 1790}, {17435, 2223}, {17877, 1708}, {17878, 3868}, {17880, 63}, {17886, 16577}, {17924, 108}, {18021, 4590}, {18101, 251}, {18155, 662}, {18191, 1333}, {18210, 1409}, {20902, 201}, {21044, 42}, {21054, 21794}, {21120, 23845}, {21132, 649}, {21138, 1403}, {21139, 9316}, {21207, 226}, {21666, 281}, {21946, 22277}, {23062, 24013}, {23100, 3676}, {23104, 3239}, {23189, 32661}, {23615, 657}, {23838, 32665}, {23970, 480}, {23978, 8}, {23983, 1259}, {23989, 7}, {23994, 6358}, {24002, 934}, {24010, 6602}, {24026, 9}, {24031, 2289}, {26856, 60}, {26932, 3}, {26933, 2286}, {27010, 5012}, {27918, 1428}, {28659, 7035}, {28660, 4600}, {30805, 6517}, {31611, 18771}, {31623, 5379}
X(34387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1111, 21207, 23989}, {1111, 24026, 17880}, {4560, 27010, 7117}, {17886, 23994, 850}


X(34388) =  ISOTOMIC CONJUGATE OF X(60)

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

X(34388) lies on these lines: {7, 76}, {8, 264}, {9, 26592}, {12, 313}, {69, 10522}, {75, 311}, {77, 3761}, {85, 30596}, {95, 4996}, {99, 7279}, {226, 1230}, {273, 1235}, {286, 5080}, {319, 3260}, {320, 1232}, {321, 8736}, {338, 594}, {339, 6356}, {342, 7101}, {350, 7269}, {651, 3770}, {1226, 20880}, {1442, 1909}, {1631, 2980}, {1826, 22006}, {1837, 2997}, {1893, 5295}, {2197, 3963}, {3760, 7190}, {6358, 28654}, {7081, 30737}, {7179, 8024}, {10436, 26541}, {14615, 32099}, {17077, 20913}, {17791, 20565}, {20236, 23978}, {20923, 28931}, {26125, 31060}

X(34388) = isotomic conjugate of X(60)
X(34388) = polar conjugate of X(2189)
X(34388) = isotomic conjugate of the isogonal conjugate of X(12)
X(34388) = polar conjugate of the isogonal conjugate of X(26942)
X(34388) = X(349)-Ceva conjugate of X(6358)
X(34388) = X(i)-cross conjugate of X(j) for these (i,j): {1089, 28654}, {21692, 10}, {23994, 850}
X(34388) = isogonal conjugate of Lozada perspector of X(58)
X(34388) = complement of polar conjugate of isogonal conjugate of X(23162)
X(34388) = anticomplement of polar conjugate of isogonal conjugate of X(23199)
X(34388) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2150}, {11, 23995}, {21, 2206}, {31, 60}, {32, 2185}, {41, 593}, {48, 2189}, {55, 849}, {58, 2194}, {163, 7252}, {184, 270}, {200, 7342}, {261, 560}, {283, 2203}, {284, 1333}, {604, 7054}, {654, 32671}, {667, 4636}, {757, 2175}, {873, 9448}, {1042, 23609}, {1098, 1397}, {1101, 3271}, {1106, 6061}, {1253, 7341}, {1408, 2328}, {1437, 2299}, {1474, 2193}, {1509, 9447}, {1576, 3737}, {1790, 2204}, {1917, 18021}, {1919, 4612}, {2170, 23357}, {2287, 16947}, {3063, 4556}, {4282, 34079}, {4858, 23963}, {14574, 18155}, {23189, 32676}
X(34388) = cevapoint of X(i) and X(j) for these (i,j): {12, 26942}, {338, 4036}, {1089, 6358}, {16732, 21121}
X(34388) = barycentric product X(i)*X(j) for these {i,j}: {7, 28654}, {10, 349}, {12, 76}, {59, 23962}, {65, 27801}, {75, 6358}, {85, 1089}, {181, 1502}, {201, 1969}, {226, 313}, {264, 26942}, {305, 8736}, {321, 1441}, {331, 3695}, {338, 4998}, {348, 7141}, {561, 2171}, {594, 6063}, {756, 20567}, {850, 4552}, {1254, 28659}, {1365, 31625}, {1446, 3701}, {2197, 18022}, {3596, 6354}, {3668, 30713}, {4024, 4572}, {4033, 4077}, {4036, 4554}, {4551, 20948}, {4564, 23994}, {6356, 7017}, {7066, 18027}, {7178, 27808}, {7235, 18895}
X(34388) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2150}, {2, 60}, {4, 2189}, {7, 593}, {8, 7054}, {10, 284}, {12, 6}, {37, 2194}, {57, 849}, {59, 23357}, {65, 1333}, {72, 2193}, {75, 2185}, {76, 261}, {85, 757}, {92, 270}, {115, 3271}, {125, 7117}, {181, 32}, {190, 4636}, {201, 48}, {225, 1474}, {226, 58}, {279, 7341}, {306, 283}, {307, 1790}, {312, 1098}, {313, 333}, {318, 2326}, {321, 21}, {338, 11}, {339, 26932}, {346, 6061}, {349, 86}, {523, 7252}, {525, 23189}, {594, 55}, {664, 4556}, {668, 4612}, {756, 41}, {758, 4282}, {850, 4560}, {872, 9447}, {1042, 16947}, {1089, 9}, {1091, 2171}, {1109, 2170}, {1211, 4267}, {1214, 1437}, {1231, 1444}, {1237, 27958}, {1254, 604}, {1365, 1015}, {1400, 2206}, {1407, 7342}, {1427, 1408}, {1441, 81}, {1446, 1014}, {1500, 2175}, {1502, 18021}, {1577, 3737}, {1824, 2204}, {1826, 2299}, {1880, 2203}, {2149, 23995}, {2171, 31}, {2197, 184}, {2222, 32671}, {2287, 23609}, {2321, 2328}, {2610, 8648}, {2970, 8735}, {3264, 30606}, {3596, 7058}, {3668, 1412}, {3695, 219}, {3700, 21789}, {3701, 2287}, {3710, 2327}, {3949, 212}, {3952, 5546}, {4013, 2316}, {4024, 663}, {4033, 643}, {4036, 650}, {4053, 2361}, {4064, 652}, {4066, 4877}, {4077, 1019}, {4086, 1021}, {4092, 14936}, {4103, 3939}, {4551, 163}, {4552, 110}, {4559, 1576}, {4564, 1101}, {4566, 4565}, {4572, 4610}, {4605, 109}, {4705, 3063}, {4848, 33628}, {4998, 249}, {6046, 1407}, {6057, 220}, {6058, 1500}, {6063, 1509}, {6354, 56}, {6356, 222}, {6358, 1}, {6370, 654}, {6386, 4631}, {6535, 1334}, {6538, 33635}, {7064, 14827}, {7066, 577}, {7109, 9448}, {7140, 607}, {7141, 281}, {7147, 1106}, {7178, 3733}, {7211, 172}, {7235, 1914}, {7237, 20665}, {7276, 609}, {8736, 25}, {10408, 4264}, {13853, 1413}, {15065, 2341}, {15526, 1364}, {16577, 17104}, {16603, 3736}, {16609, 5009}, {16732, 18191}, {16886, 3056}, {17094, 7254}, {18697, 17185}, {20234, 3794}, {20336, 1812}, {20567, 873}, {20618, 7053}, {20653, 2269}, {20902, 7004}, {20948, 18155}, {21015, 7124}, {21021, 2330}, {21023, 16721}, {21057, 8540}, {21207, 17197}, {21674, 21748}, {21675, 14547}, {21810, 20967}, {21859, 692}, {21958, 23864}, {23067, 32661}, {23994, 4858}, {26942, 3}, {27801, 314}, {27808, 645}, {28654, 8}, {30713, 1043}, {31612, 18772}, {31625, 6064}
X(34388) = {X(313),X(349)}-harmonic conjugate of X(1441)


X(34389) =  ISOTOMIC CONJUGATE OF X(61)

Barycentrics    1/(a^2*(a^2 - b^2 - c^2 - 2*Sqrt[3]*S)) : :

X(34389) lies on these lines: {2, 1225}, {17, 76}, {69, 300}, {264, 298}, {290, 32036}, {299, 1232}, {301, 302}, {308, 21461}, {633, 3519}, {2367, 16806}, {2980, 34008}, {10640, 32037}, {11146, 18354}, {19713, 25043}, {20572, 33529}

X(34389) = isotomic conjugate of X(61)
X(34389) = polar conjugate of X(10642)
X(34389) = isotomic conjugate of the anticomplement of X(635)
X(34389) = isotomic conjugate of the complement of X(633)
X(34389) = isotomic conjugate of the isogonal conjugate of X(17)
X(34389) = X(i)-cross conjugate of X(j) for these (i,j): {299, 301}, {635, 2}
X(34389) = X(i)-isoconjugate of X(j) for these (i,j): {31, 61}, {48, 10642}, {302, 560}, {473, 9247}, {1094, 16463}, {2151, 11083}, {2152, 11141}, {2153, 11137}, {2154, 11135}, {2964, 21462}
X(34389) = cevapoint of X(2) and X(633)
X(34389) = trilinear pole of line {850, 23873}
X(34389) = barycentric product X(i)*X(j) for these {i,j}: {17, 76}, {301, 19779}, {303, 11140}, {305, 8741}, {850, 32036}, {1502, 21461}, {18022, 32585}
X(34389) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 61}, {4, 10642}, {13, 11083}, {14, 11141}, {15, 11137}, {16, 11135}, {17, 6}, {62, 2965}, {76, 302}, {93, 8742}, {264, 473}, {298, 11146}, {299, 11126}, {300, 8838}, {301, 16771}, {303, 1994}, {471, 10632}, {472, 3518}, {633, 10640}, {850, 23872}, {930, 16807}, {2963, 21462}, {3375, 2152}, {3519, 32586}, {8174, 11244}, {8741, 25}, {10677, 11136}, {11078, 6104}, {11080, 16463}, {11087, 3458}, {11130, 3201}, {11139, 3457}, {11140, 18}, {11144, 62}, {11600, 11086}, {16806, 1576}, {19775, 8471}, {19779, 16}, {21461, 32}, {23873, 1510}, {32036, 110}, {32585, 184}
X(34389) = {X(302),X(311)}-harmonic conjugate of X(301)


X(34390) =  ISOTOMIC CONJUGATE OF X(62)

Barycentrics    1/(a^2*(a^2 - b^2 - c^2 + 2*Sqrt[3]*S)) : :

X(34390) lies on these lines: {2, 1225}, {18, 76}, {69, 301}, {264, 299}, {290, 32037}, {298, 1232}, {300, 303}, {308, 21462}, {634, 3519}, {2367, 16807}, {2980, 34009}, {10210, 15321}, {10639, 32036}, {11145, 18354}, {19712, 25043}, {20572, 33530}

X(34390) = isotomic conjugate of X(62)
X(34390) = polar conjugate of X(10641)
X(34390) = isotomic conjugate of the anticomplement of X(636)
X(34390) = isotomic conjugate of the complement of X(634)
X(34390) = isotomic conjugate of the isogonal conjugate of X(18)
X(34390) = X(i)-cross conjugate of X(j) for these (i,j): {298, 300}, {636, 2}
X(34390) = X(i)-isoconjugate of X(j) for these (i,j): {31, 62}, {48, 10641}, {303, 560}, {472, 9247}, {1095, 16464}, {2151, 11142}, {2152, 11088}, {2153, 11136}, {2154, 11134}, {2964, 21461}
X(34390) = cevapoint of X(2) and X(634)
X(34390) = trilinear pole of line {850, 23872}
X(34390) = barycentric product X(i)*X(j) for these {i,j}: {18, 76}, {300, 19778}, {302, 11140}, {305, 8742}, {850, 32037}, {1502, 21462}, {18022, 32586}
X(34390) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 62}, {4, 10641}, {13, 11142}, {14, 11088}, {15, 11136}, {16, 11134}, {18, 6}, {61, 2965}, {76, 303}, {93, 8741}, {264, 472}, {298, 11127}, {299, 11145}, {300, 16770}, {301, 8836}, {302, 1994}, {470, 10633}, {473, 3518}, {634, 10639}, {850, 23873}, {930, 16806}, {2963, 21461}, {3384, 2151}, {3519, 32585}, {8175, 11243}, {8742, 25}, {10678, 11135}, {11082, 3457}, {11085, 16464}, {11092, 6105}, {11131, 3200}, {11138, 3458}, {11140, 17}, {11143, 61}, {11601, 11081}, {16807, 1576}, {19774, 8479}, {19778, 15}, {21462, 32}, {23872, 1510}, {32037, 110}, {32586, 184}
X(34390) = {X(303),X(311)}-harmonic conjugate of X(300)


X(34391) =  ISOTOMIC CONJUGATE OF X(371)

Barycentrics    1/(a^2*(a^2 - b^2 - c^2 - 2*S)) : :

X(34391) lies on these lines: {2, 311}, {68, 637}, {69, 13439}, {76, 485}, {264, 492}, {290, 6413}, {308, 8577}, {847, 24243}, {1232, 1271}, {1235, 13440}, {1270, 3260}, {5409, 16032}, {8035, 18819}, {14615, 32808}

X(34391) = isotomic conjugate of X(371)
X(34391) = polar conjugate of X(5413)
X(34391) = isotomic conjugate of the anticomplement of X(639)
X(34391) = isotomic conjugate of the complement of X(637)
X(34391) = isotomic conjugate of the isogonal conjugate of X(485)
X(34391) = polar conjugate of the isogonal conjugate of X(11090)
X(34391) = X(639)-cross conjugate of X(2)
X(34391) = X(i)-isoconjugate of X(j) for these (i,j): {19, 8911}, {31, 371}, {47, 8576}, {48, 5413}, {492, 560}, {1585, 9247}, {1973, 5408}
X(34391) = cevapoint of X(i) and X(j) for these (i,j): {2, 637}, {69, 13441}, {485, 11090}
X(34391) = barycentric product X(i)*X(j) for these {i,j}: {76, 485}, {264, 11090}, {311, 16032}, {491, 5392}, {1502, 8577}, {1586, 20563}, {6413, 18022}, {13455, 20567}
X(34391) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 371}, {3, 8911}, {4, 5413}, {68, 6414}, {69, 5408}, {76, 492}, {264, 1585}, {372, 571}, {485, 6}, {491, 1993}, {492, 1599}, {493, 8950}, {637, 10962}, {1321, 5411}, {1586, 24}, {2165, 8576}, {5392, 486}, {5409, 1147}, {6413, 184}, {8035, 5062}, {8577, 32}, {8944, 3053}, {10665, 26920}, {11090, 3}, {11091, 10666}, {13439, 372}, {13440, 5412}, {13441, 5409}, {13455, 41}, {16032, 54}, {20563, 11091}, {24246, 10132}


X(34392) =  ISOTOMIC CONJUGATE OF X(372)

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

X(34392) lies on these lines: {2, 311}, {68, 638}, {69, 13428}, {76, 486}, {264, 491}, {276, 26922}, {290, 6414}, {308, 8576}, {847, 24244}, {1232, 1270}, {1235, 13429}, {1271, 3260}, {5408, 16037}, {8036, 18820}, {14615, 32809}

X(34392) = isotomic conjugate of X(372)
X(34392) = polar conjugate of X(5412)
X(34392) = isotomic conjugate of the anticomplement of X(640)
X(34392) = isotomic conjugate of the complement of X(638)
X(34392) = isotomic conjugate of the isogonal conjugate of X(486)
X(34392) = polar conjugate of the isogonal conjugate of X(11091)
X(34392) = X(640)-cross conjugate of X(2)
X(34392) = X(i)-isoconjugate of X(j) for these (i,j): {19, 26920}, {31, 372}, {47, 8577}, {48, 5412}, {491, 560}, {1586, 9247}, {1973, 5409}
X(34392) = cevapoint of X(i) and X(j) for these (i,j): {2, 638}, {69, 13430}, {486, 11091}, {5408, 26922}
X(34392) = barycentric product X(i)*X(j) for these {i,j}: {76, 486}, {264, 11091}, {311, 16037}, {492, 5392}, {1502, 8576}, {1585, 20563}, {6414, 18022}, {18027, 26922}
X(34392) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 372}, {3, 26920}, {4, 5412}, {68, 6413}, {69, 5409}, {76, 491}, {264, 1586}, {371, 571}, {486, 6}, {491, 1600}, {492, 1993}, {637, 26875}, {638, 10960}, {1322, 5410}, {1585, 24}, {2165, 8577}, {3596, 13461}, {5392, 485}, {5408, 1147}, {6414, 184}, {8036, 5058}, {8576, 32}, {8940, 3053}, {10666, 8911}, {11090, 10665}, {11091, 3}, {13428, 371}, {13429, 5413}, {13430, 5408}, {16037, 54}, {20563, 11090}, {24245, 10133}, {26922, 577}


X(34393) =  ISOTOMIC CONJUGATE OF X(515)

Barycentrics    1/(a^3*SA - (b + c)*SB*SC) : :

X(34393) lies on the Steiner circumellipse and these lines: {2, 23986}, {69, 347}, {75, 342}, {99, 102}, {190, 329}, {322, 668}, {333, 648}, {666, 15629}, {1121, 2399}, {4569, 7182}, {4586, 32677}, {5081, 22464}, {17346, 32040}

X(34393) = isotomic conjugate of X(515)
X(34393) = anticomplement of X(23986)
X(34393) = polar conjugate of X(8755)
X(34393) = isotomic conjugate of the isogonal conjugate of X(102)
X(34393) = X(i)-cross conjugate of X(j) for these (i,j): {515, 2}, {5081, 333}, {13532, 34234}, {15633, 2399}, {22464, 75}
X(34393) = X(i)-isoconjugate of X(j) for these (i,j): {6, 2182}, {31, 515}, {41, 34050}, {48, 8755}, {55, 1455}, {650, 2425}, {810, 7452}, {1946, 23987}, {2406, 3063}, {6087, 32652}, {6187, 11700}, {23986, 32677}
X(34393) = cevapoint of X(i) and X(j) for these (i,j): {2, 515}, {8, 908}, {63, 4511}, {2399, 15633}
X(34393) = trilinear pole of line {2, 2399}
X(34393) = barycentric product X(i)*X(j) for these {i,j}: {76, 102}, {561, 32677}, {664, 2399}, {1275, 15633}, {2432, 4572}, {6063, 15629}
X(34393) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2182}, {2, 515}, {4, 8755}, {7, 34050}, {57, 1455}, {102, 6}, {109, 2425}, {515, 23986}, {648, 7452}, {653, 23987}, {664, 2406}, {2399, 522}, {2432, 663}, {3218, 11700}, {4391, 14304}, {14837, 6087}, {15629, 55}, {15633, 1146}, {18026, 24035}, {32677, 31}, {34050, 1359}


X(34394) =  ISOGONAL CONJUGATE OF X(300)

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

X(34394) lies on these lines: {6, 3129}, {13, 14181}, {14, 7578}, {15, 323}, {16, 15080}, {23, 62}, {25, 21461}, {32, 184}, {50, 11136}, {61, 11422}, {470, 30465}, {570, 32586}, {800, 21647}, {1495, 3457}, {3104, 11126}, {3105, 34009}, {3107, 14169}, {5994, 32730}, {6782, 33529}, {8553, 32585}, {8573, 19363}, {9112, 16806}, {9220, 11087}, {11134, 13338}, {11485, 16021}, {13366, 21462}, {14187, 32465}

X(34394) = isogonal conjugate of X(300)
X(34394) = isogonal conjugate of the isotomic conjugate of X(15)
X(34394) = isogonal conjugate of the polar conjugate of X(8739)
X(34394) = X(2380)-Ceva conjugate of X(11135)
X(34394) = X(i)-isoconjugate of X(j) for these (i,j): {1, 300}, {13, 75}, {76, 2153}, {299, 2166}, {304, 8737}, {561, 3457}, {799, 20578}, {1577, 23895}, {2152, 20573}, {5995, 20948}, {23871, 32680}, {24037, 30452}
X(34394) = crosspoint of X(i) and X(j) for these (i,j): {6, 3438}, {15, 8739}, {3457, 21462}
X(34394) = crosssum of X(i) and X(j) for these (i,j): {2, 621}, {298, 303}
X(34394) = crossdifference of every pair of points on line {850, 20578}
X(34394) = barycentric product X(i)*X(j) for these {i,j}: {1, 2151}, {3, 8739}, {6, 15}, {14, 50}, {16, 11086}, {17, 11137}, {18, 11136}, {32, 298}, {61, 8603}, {110, 6137}, {184, 470}, {301, 19627}, {323, 3458}, {512, 17402}, {526, 5994}, {1094, 2153}, {1250, 19373}, {1576, 23870}, {2154, 6149}, {2380, 19294}, {3200, 11082}, {3457, 11131}, {5616, 14579}, {6105, 8604}, {6110, 18877}, {6117, 14533}, {6782, 32654}, {8738, 22115}, {9204, 32729}, {10633, 32586}, {11127, 21462}, {11135, 11600}, {11146, 21461}, {14270, 23896}, {16460, 19295}, {23357, 30465}
X(34394) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 300}, {14, 20573}, {15, 76}, {32, 13}, {50, 299}, {298, 1502}, {470, 18022}, {560, 2153}, {669, 20578}, {1084, 30452}, {1501, 3457}, {1576, 23895}, {1974, 8737}, {2151, 75}, {3200, 11133}, {3458, 94}, {6137, 850}, {8738, 18817}, {8739, 264}, {11086, 301}, {11136, 303}, {11137, 302}, {14270, 23871}, {14574, 5995}, {17402, 670}, {19627, 16}, {30465, 23962}
X(34394) = {X(15),X(3170)}-harmonic conjugate of X(323)


X(34395) =  ISOGONAL CONJUGATE OF X(301)

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

X(34395) lies on these lines: {6, 3130}, {13, 7578}, {14, 14177}, {15, 15080}, {16, 323}, {23, 61}, {25, 21462}, {32, 184}, {50, 11134}, {62, 11422}, {471, 30468}, {570, 32585}, {800, 21648}, {1495, 3458}, {3104, 34008}, {3105, 11127}, {3106, 14170}, {5995, 32730}, {6783, 33530}, {8553, 32586}, {8573, 19364}, {9113, 16807}, {9220, 11082}, {11137, 13338}, {11486, 16022}, {13366, 21461}, {14185, 32466}

X(34395) = isogonal conjugate of X(301)
X(34395) = isogonal conjugate of the isotomic conjugate of X(16)
X(34395) = isogonal conjugate of the polar conjugate of X(8740)
X(34395) = X(2381)-Ceva conjugate of X(11136)
X(34395) = X(i)-isoconjugate of X(j) for these (i,j): {1, 301}, {14, 75}, {76, 2154}, {298, 2166}, {304, 8738}, {561, 3458}, {799, 20579}, {1577, 23896}, {2151, 20573}, {5994, 20948}, {23870, 32680}, {24037, 30453}
X(34395) = crosspoint of X(i) and X(j) for these (i,j): {6, 3439}, {16, 8740}, {3458, 21461}
X(34395) = crosssum of X(i) and X(j) for these (i,j): {2, 622}, {299, 302}
X(34395) = crossdifference of every pair of points on line {850, 20579}
X(34395) = barycentric product X(i)*X(j) for these {i,j}: {1, 2152}, {3, 8740}, {6, 16}, {13, 50}, {15, 11081}, {17, 11135}, {18, 11134}, {32, 299}, {62, 8604}, {110, 6138}, {184, 471}, {300, 19627}, {323, 3457}, {512, 17403}, {526, 5995}, {1095, 2154}, {1576, 23871}, {2153, 6149}, {2381, 19295}, {3201, 11087}, {3458, 11130}, {5612, 14579}, {6104, 8603}, {6111, 18877}, {6116, 14533}, {6783, 32654}, {7051, 10638}, {8737, 22115}, {9205, 32729}, {10632, 32585}, {11126, 21461}, {11136, 11601}, {11145, 21462}, {14270, 23895}, {16459, 19294}, {23357, 30468}
X(34395) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 301}, {13, 20573}, {16, 76}, {32, 14}, {50, 298}, {299, 1502}, {471, 18022}, {560, 2154}, {669, 20579}, {1084, 30453}, {1501, 3458}, {1576, 23896}, {1974, 8738}, {2152, 75}, {3201, 11132}, {3457, 94}, {6138, 850}, {8737, 18817}, {8740, 264}, {11081, 300}, {11134, 303}, {11135, 302}, {14270, 23870}, {14574, 5994}, {17403, 670}, {19627, 15}, {30468, 23962}
X(34395) = {X(16),X(3171)}-harmonic conjugate of X(323)


X(34396) =  ISOGONAL CONJUGATE OF X(327)

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

X(34396) lies on these lines: {2, 3398}, {3, 54}, {6, 157}, {25, 251}, {32, 184}, {39, 13366}, {51, 5007}, {110, 11328}, {154, 15257}, {160, 2965}, {182, 14096}, {206, 13345}, {263, 12212}, {323, 26316}, {418, 10316}, {441, 11245}, {458, 9755}, {460, 5305}, {571, 20775}, {669, 34347}, {800, 21637}, {1316, 14265}, {1495, 5008}, {1627, 20885}, {1899, 14003}, {1994, 3095}, {2080, 11003}, {2210, 9448}, {3053, 17809}, {3060, 6660}, {4159, 18806}, {5063, 23200}, {5157, 22062}, {5188, 22352}, {5201, 19127}, {5304, 6620}, {6636, 9821}, {7499, 7767}, {7766, 19222}, {7854, 16893}, {8573, 19125}, {9301, 15080}, {9407, 19136}, {10350, 33734}, {11451, 21513}, {12176, 22735}, {13338, 18374}, {14880, 14957}, {15033, 32444}, {15531, 22143}, {16257, 16258}, {20854, 26881}, {21309, 26864}, {23163, 26206}

X(34396) = isogonal conjugate of X(327)
X(34396) = isogonal conjugate of the isotomic conjugate of X(182)
X(34396) = isogonal conjugate of the polar conjugate of X(10311)
X(34396) = X(18898)-Ceva conjugate of X(1501)
X(34396) = X(i)-isoconjugate of X(j) for these (i,j): {1, 327}, {75, 262}, {76, 2186}, {263, 561}, {1502, 3402}, {20948, 26714}
X(34396) = crosspoint of X(i) and X(j) for these (i,j): {6, 3425}, {182, 10311}, {249, 33514}
X(34396) = crosssum of X(2) and X(1352)
X(34396) = crossdifference of every pair of points on line {850, 2525}
X(34396) = barycentric product X(i)*X(j) for these {i,j}: {3, 10311}, {6, 182}, {32, 183}, {110, 3288}, {184, 458}, {249, 6784}, {251, 14096}, {560, 3403}, {577, 33971}, {1501, 20023}, {1576, 23878}, {2966, 9420}, {8842, 14602}
X(34396) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 327}, {32, 262}, {182, 76}, {183, 1502}, {458, 18022}, {560, 2186}, {1501, 263}, {1917, 3402}, {3288, 850}, {3403, 1928}, {6784, 338}, {9420, 2799}, {10311, 264}, {14096, 8024}, {14574, 26714}, {33971, 18027}
X(34396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 184, 237}, {32, 14602, 1501}


X(34397) =  ISOGONAL CONJUGATE OF X(328)

Barycentrics    a^2/(b^2*c^2*(a^2 - a*b + b^2 - c^2)*(a^2 + a*b + b^2 - c^2)*(-a^2 + b^2 - a*c - c^2)*(-a^2 + b^2 + a*c - c^2)*(-a^2 + b^2 + c^2)) : :
X(34397) = 2 X[1495] + 3 X[11402]

X(34397) lies on these lines: {2, 19129}, {3, 18532}, {4, 567}, {6, 25}, {22, 18438}, {23, 1112}, {24, 49}, {30, 12228}, {54, 3575}, {110, 468}, {112, 32730}, {125, 15139}, {155, 34116}, {156, 3542}, {182, 5094}, {186, 323}, {235, 1614}, {237, 10317}, {265, 403}, {378, 14805}, {389, 10274}, {421, 2970}, {427, 5012}, {460, 5523}, {511, 19504}, {512, 2623}, {569, 7507}, {578, 12173}, {1092, 15750}, {1147, 3515}, {1199, 6746}, {1503, 13198}, {1514, 9934}, {1593, 14915}, {1594, 13353}, {1692, 14580}, {1976, 8791}, {1990, 14560}, {1993, 21213}, {2211, 14567}, {2904, 6243}, {3092, 9677}, {3131, 18468}, {3132, 18470}, {3135, 23606}, {3155, 18457}, {3156, 18459}, {3516, 10984}, {3517, 9704}, {3518, 32136}, {5064, 11645}, {6000, 19457}, {6146, 32379}, {6353, 9544}, {6403, 11422}, {6759, 18396}, {7488, 31807}, {7493, 19154}, {7505, 18350}, {9027, 32240}, {9407, 32715}, {10151, 14157}, {10295, 15463}, {10539, 14852}, {10938, 11456}, {11004, 18882}, {11064, 15462}, {11363, 31811}, {11413, 18466}, {11572, 32395}, {12112, 12133}, {12165, 12168}, {12828, 32223}, {13171, 34146}, {13367, 32333}, {13434, 23047}, {14165, 14355}, {14975, 21741}, {15135, 19161}, {15136, 32110}, {15138, 32607}, {15471, 32217}, {17835, 21663}, {18382, 31383}, {18405, 26883}, {18569, 34114}, {21844, 33884}, {31802, 34148}

X(34397) = midpoint of X(19504) and X(21284)
X(34397) = isogonal conjugate of X(328)
X(34397) = polar conjugate of X(20573)
X(34397) = isogonal conjugate of the isotomic conjugate of X(186)
X(34397) = polar conjugate of the isotomic conjugate of X(50)
X(34397) = polar conjugate of the isogonal conjugate of X(19627)
X(34397) = X(i)-Ceva conjugate of X(j) for these (i,j): {186, 50}, {1299, 571}, {6344, 2965}, {8749, 32}
X(34397) = X(19627)-cross conjugate of X(50)
X(34397) = crosspoint of X(250) and X(32708)
X(34397) = crosssum of X(i) and X(j) for these (i,j): {2, 3153}, {125, 6334}, {311, 3260}, {525, 16186}
X(34397) = crossdifference of every pair of points on line {343, 525}
X(34397) = X(i)-isoconjugate of X(j) for these (i,j): {1, 328}, {48, 20573}, {63, 94}, {69, 2166}, {75, 265}, {255, 18817}, {304, 1989}, {326, 6344}, {336, 14356}, {476, 14208}, {525, 32680}, {662, 14592}, {799, 14582}, {1141, 18695}, {1807, 20565}, {3267, 32678}, {4592, 10412}, {5961, 20571}, {7100, 20566}, {20948, 32662}
X(34397) = barycentric product X(i)*X(j) for these {i,j}: {4, 50}, {6, 186}, {15, 8740}, {16, 8739}, {19, 6149}, {25, 323}, {32, 340}, {54, 11062}, {112, 526}, {162, 2624}, {184, 14165}, {232, 14355}, {250, 2088}, {264, 19627}, {393, 22115}, {512, 14590}, {523, 14591}, {562, 2965}, {571, 5962}, {648, 14270}, {933, 2081}, {1154, 8882}, {1511, 8749}, {1825, 4282}, {1870, 2174}, {1974, 7799}, {1986, 14910}, {1989, 3043}, {1990, 14385}, {2190, 2290}, {2436, 7480}, {2489, 10411}, {2914, 14579}, {3218, 14975}, {5664, 32715}, {6198, 7113}, {8552, 32713}, {8603, 10632}, {8604, 10633}, {16186, 23964}, {17515, 21741}, {32676, 32679}
X(34397) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 20573}, {6, 328}, {25, 94}, {32, 265}, {50, 69}, {186, 76}, {323, 305}, {340, 1502}, {393, 18817}, {512, 14592}, {526, 3267}, {669, 14582}, {1154, 28706}, {1973, 2166}, {1974, 1989}, {2088, 339}, {2207, 6344}, {2211, 14356}, {2290, 18695}, {2489, 10412}, {2624, 14208}, {3043, 7799}, {6149, 304}, {8739, 301}, {8740, 300}, {9409, 18557}, {11062, 311}, {14165, 18022}, {14270, 525}, {14573, 11077}, {14574, 32662}, {14581, 14254}, {14590, 670}, {14591, 99}, {14975, 18359}, {19627, 3}, {22115, 3926}, {32676, 32680}
X(34397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {110, 19128, 468}, {110, 19138, 32227}, {184, 206, 26864}, {186, 1986, 3581}, {3581, 11597, 22115}, {11402, 26864, 19459}, {19118, 26864, 25}


X(34398) =  X(4)X(7056)∩X(7)X(1857)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34398) lies on these lines: {4, 7056}, {7, 1857}, {33, 77}, {63, 7079}, {69, 7046}, {286, 10400}, {1444, 4183}, {1836, 13149}

X(34398) = cevapoint of X(4) and X(7)
X(34398) = polar conjugate of the complement of X(348)
X(34398) = polar conjugate of the complementary conjugate of X(18639)


X(34399) =  ISOTOMIC CONJUGATE OF X(1837)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34399) lies on these lines: {7, 1264}, {57, 3719}, {69, 1119}, {269, 320}, {479, 7055}, {1396, 1812}, {1462, 3662}, {13425, 13459}, {13437, 13458}

X(34399) = isotomic conjugate of X(1837)
X(34399) = trilinear pole of the line {3669, 3904}


X(34400) =  X(4)X(7215)∩X(7)X(309)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34400) lies on these lines: {4, 7215}, {7, 309}, {69, 271}, {85, 189}, {86, 285}, {280, 6604}, {282, 30705}, {304, 7055}, {1433, 31637}, {7177, 26871}

X(34400) = isotomic conjugate of the polar conjugate of X(1440)


X(34401) =  X(7)X(5552)∩X(279)X(5905)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34401) lies on these lines: {7, 5552}, {279, 5905}, {329, 7183}, {1434, 31631}, {3559, 17139}


X(34402) =  X(7)X(6060)∩X(1088)X(1097)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34402) lies on the circumhyperbola dual of Yff parabola and on these lines: {7, 6060}, {1088, 1097}, {1440, 4872}, {2400, 4131}, {7008, 7013}


X(34403) =  ISOGONAL CONJUGATE OF X(3172)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34403) lies on the cubics K041, K184, K1010 and on these lines: {3, 16096}, {4, 14944}, {20, 64}, {76, 459}, {99, 27082}, {279, 23983}, {312, 1231}, {394, 32840}, {441, 1073}, {1105, 32000}, {1265, 19611}, {1301, 2366}, {3343, 18928}, {3619, 26166}, {6337, 6394}, {10008, 14952}, {11064, 32841}, {13157, 32836}, {14572, 32834}, {33583, 34168}

X(34403) = isogonal conjugate of X(3172)
X(34403) = isotomic conjugate of X(1249)
X(34403) = polar conjugate of X(6525)
X(34403) = trilinear pole of the line {3265, 8057}
X(34403) = {X(253), X(15394)}-harmonic conjugate of X(69)


X(34404) =  ISOGONAL CONJUGATE OF X(2199)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34404) lies on the cubic K184 and on these lines: {63, 7101}, {69, 189}, {75, 7017}, {76, 7182}, {84, 20368}, {271, 333}, {280, 341}, {282, 332}, {1422, 32017}, {1440, 28808}, {7003, 30479}

X(34404) = isogonal conjugate of X(2199)
X(34404) = isotomic conjugate of X(223)
X(34404) = polar conjugate of X(208)
X(34404) = trilinear pole of the line {4397, 6332}


X(34405) =  ISOTOMIC CONJUGATE OF X(1899)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34405) lies on these lines: {4, 4176}, {25, 317}, {69, 6524}, {264, 14593}, {275, 7752}, {297, 315}, {850, 11442}, {1352, 18022}, {1899, 6331}

X(34405) = isotomic conjugate of X(1899)
X(34405) = polar conjugate of X(3767)
X(34405) = trilinear pole of the line {2489, 2799}


X(34406) =  X(4)X(1265)∩X(8)X(1118)

Barycentrics    (a^3-b*a^2-b^2*a+b^3+c^3)*(a^2+b^2-c^2)*(a^3-c*a^2-c^2*a+b^3+c^3)*(a^2-b^2+c^2) : :
Trilinears    1/(sec B csc^2(C/2) + sec C csc^2(B/2)) : :

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

The trilinear polar of X(34406) passes through X(6591) and the isotomic conjugate of X(13149).

X(34406) lies on these lines: {4, 1265}, {8, 1118}, {19, 3692}, {28, 1792}, {34, 78}, {69, 1119}, {286, 5130}, {404, 1809}, {1837, 6335}, {2287, 5016}, {5086, 7017}

X(34406) = polar conjugate of X(3772)
X(34406) = cevapoint of X(4) and X(8)
X(34406) = pole wrt polar circle of trilinear polar of X(3772) (the polar, wrt the Fuhrmann circle, of the perspector of the Fuhrmann circle)


X(34407) =  X(69)X(3079)∩X(253)X(6525)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34407) lies on these lines: {69, 3079}, {253, 6525}, {264, 16096}, {305, 1529}, {1073, 6330}


X(34408) =  (name pending)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34408) lies on this line: {1119, 16091}


X(34409) =  ISOTOMIC CONJUGATE OF X(1836)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

The trilinear polar of X(34409) passes through X(4130).

X(34409) lies on these lines: {8, 7055}, {69, 7046}, {100, 14716}, {200, 319}, {728, 3719}, {1264, 5423}, {1812, 33297}

X(34409) = isotomic conjugate of X(1836)
X(34409) = cevapoint of X(8) and X(69)
X(34409) = trilinear pole of the line {4130, 28898}


X(34410) =  ISOTOMIC CONJUGATE OF X(5895)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34410) lies on this line: {69, 18848}

X(34410) = isotomic conjugate of X(5895)


X(34411) =  (name pending)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34411) lies on this line: {7183, 33066}


X(34412) =  ISOTOMIC CONJUGATE OF X(1853)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34412) lies on these lines: {69, 3079}, {10565, 23608}

X(34412) = isotomic conjugate of X(1853)


X(34413) =  X(189)X(3719)∩X(309)X(1264)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34413) lies on these lines: {189, 3719}, {309, 1264}, {322, 7055}, {326, 1440}

X(34413) = isotomic conjugate of the anticomplement of X(1158)


X(34414) =  X(8)X(7338)∩X(7080)X(7270)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34414) lies on these lines: {8, 7338}, {7080, 7270}


X(34415) =  X(253)X(23608)∩X(441)X(459)

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

See Thanos Kalogerakis and César Lozada, Hyacinthos 29602.

X(34415) lies on these lines: {253, 23608}, {441, 459}, {3079, 6527}


X(34416) =  X(6)X(20879)∩X(25)X(8749)

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

X(34416) lies on these lines: {6, 20897}, {25, 8749}, {32, 682}, {206, 33872}, {237, 5063}, {1084, 1501}, {3618, 12054}, {13338, 18374}, {19137, 22062}, {20775, 33871}

X(34416) = X(561)-isoconjugate of X(3431)
X(34416) = crosssum of X(69) and X(32833)
X(34416) = barycentric product X(i)*X(j) for these {i,j}: {25, 5158}, {32, 381}, {1973, 18477}, {3581, 11060}, {32225, 32740}
X(34416) = barycentric quotient X(i)/X(j) for these {i,j}: {381, 1502}, {1501, 3431}, {5158, 305}
X(34416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {32, 1974, 9407}, {32, 9407, 14575}


X(34417) =  X(4)X(74)∩X(6)X(25)

Barycentrics    a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) : :
X(34417) = 3 X[1995] - X[15066], 3 X[5651] - 2 X[15066]

X(34417) lies on these lines: {2, 3098}, {3, 373}, {4, 74}, {5, 32269}, {6, 25}, {15, 3130}, {16, 3129}, {22, 5092}, {23, 182}, {24, 10110}, {30, 20192}, {32, 3124}, {39, 20897}, {52, 13861}, {54, 14491}, {64, 14490}, {110, 576}, {111, 263}, {143, 10539}, {185, 1598}, {187, 3148}, {235, 11745}, {237, 574}, {323, 3060}, {343, 18358}, {381, 1531}, {389, 10594}, {394, 21969}, {399, 568}, {418, 10979}, {428, 11550}, {462, 5318}, {463, 5321}, {468, 5480}, {511, 1995}, {569, 10095}, {575, 6800}, {578, 3518}, {582, 20840}, {842, 6785}, {1092, 5446}, {1112, 11470}, {1147, 12310}, {1192, 11403}, {1350, 5650}, {1351, 3292}, {1383, 1976}, {1503, 10301}, {1511, 12106}, {1514, 1596}, {1533, 4846}, {1597, 21663}, {1629, 3168}, {1864, 2355}, {1899, 6995}, {1906, 13568}, {1968, 9412}, {1993, 21849}, {2070, 14805}, {2914, 7730}, {3131, 10646}, {3132, 10645}, {3135, 14806}, {3155, 6396}, {3156, 6200}, {3291, 5017}, {3426, 10605}, {3431, 11202}, {3517, 10982}, {3527, 19357}, {3542, 3574}, {3543, 15053}, {3567, 6759}, {3575, 15873}, {3580, 3818}, {3619, 7392}, {3620, 7398}, {3796, 20850}, {3843, 7689}, {3917, 5020}, {4232, 14853}, {4256, 28348}, {4653, 28382}, {5012, 7712}, {5028, 20977}, {5033, 8627}, {5034, 13410}, {5039, 9465}, {5064, 26958}, {5068, 7691}, {5103, 30747}, {5104, 8585}, {5198, 9786}, {5200, 23259}, {5462, 7517}, {5476, 7426}, {5562, 7529}, {5642, 20423}, {5889, 15052}, {5890, 12112}, {5892, 12083}, {6090, 11477}, {6146, 7715}, {6199, 10132}, {6241, 26863}, {6243, 18369}, {6388, 7747}, {6395, 10133}, {6452, 21097}, {6636, 11451}, {6641, 22052}, {6644, 10564}, {6660, 26316}, {6688, 7485}, {6784, 9142}, {6791, 7737}, {7394, 21243}, {7395, 27355}, {7408, 23291}, {7487, 21659}, {7492, 17508}, {7493, 14561}, {7502, 13364}, {7514, 14845}, {7519, 18911}, {7530, 9730}, {7576, 18390}, {7706, 11799}, {7714, 11433}, {7720, 10814}, {7721, 10815}, {7729, 17812}, {7812, 13210}, {7998, 16042}, {8644, 9171}, {9821, 21513}, {9909, 10601}, {9970, 32235}, {10224, 20193}, {10323, 11695}, {10382, 15496}, {10540, 13321}, {11003, 15019}, {11064, 21850}, {11074, 11080}, {11178, 15360}, {11204, 13596}, {11365, 16980}, {11422, 15520}, {11441, 16625}, {11738, 13603}, {12082, 16836}, {12088, 15024}, {12824, 19140}, {13330, 20998}, {13336, 15026}, {13347, 15028}, {13394, 18583}, {13417, 15106}, {13419, 18912}, {13490, 18474}, {13598, 17928}, {13615, 22080}, {13851, 18494}, {14134, 33269}, {14487, 20421}, {14531, 17814}, {14685, 16186}, {14826, 20080}, {14831, 18451}, {15037, 18378}, {15059, 31857}, {15082, 21766}, {16063, 29317}, {16981, 23061}, {17704, 33524}, {19219, 23267}, {21308, 23039}, {23635, 33578}, {29181, 30739}, {31861, 32110}

X(34417) = midpoint of X(7519) and X(18911)
X(34417) = reflection of X(5651) in X(1995)
X(34417) = isogonal conjugate of the isotomic conjugate of X(381)
X(34417) = polar conjugate of the isotomic conjugate of X(5158)
X(34417) = X(i)-Ceva conjugate of X(j) for these (i,j): {381, 5158}, {9064, 647}, {14483, 6}, {32738, 512}
X(34417) = X(34417) = X(i)-isoconjugate of X(j) for these (i,j): {75, 3431}, {326, 16263}
X(34417) = crosspoint of X(i) and X(j) for these (i,j): {4, 34288}, {6, 3426}
X(34417) = crosssum of X(i) and X(j) for these (i,j): {2, 376}, {3, 15066}
X(34417) = crossdifference of every pair of points on line {525, 1636}
X(34417) = barycentric product X(i)*X(j) for these {i,j}: {4, 5158}, {6, 381}, {19, 18477}, {51, 4993}, {74, 18487}, {111, 32225}, {1531, 8749}, {1989, 3581}, {2159, 18486}, {4550, 34288}, {8770, 21970}, {14165, 18479}
X(34417) = barycentric quotient X(i)/X(j) for these {i,j}: {32, 3431}, {381, 76}, {2207, 16263}, {3581, 7799}, {4550, 32833}, {4993, 34384}, {5158, 69}, {11060, 18316}, {18477, 304}, {18487, 3260}, {32225, 3266}
X(34417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 15107, 3098}, {3, 373, 22112}, {3, 3066, 373}, {6, 25, 1495}, {6, 1495, 184}, {6, 31860, 25}, {23, 5640, 182}, {23, 15018, 15080}, {24, 10110, 11424}, {25, 51, 184}, {25, 9777, 154}, {25, 17810, 51}, {51, 184, 15004}, {51, 1495, 6}, {51, 13366, 9777}, {110, 11002, 576}, {154, 9777, 13366}, {154, 13366, 184}, {323, 10546, 9306}, {323, 13595, 10546}, {381, 3581, 4550}, {389, 10594, 26883}, {428, 13567, 11550}, {575, 32237, 6800}, {1350, 11284, 5650}, {3060, 10546, 323}, {3060, 13595, 9306}, {3457, 3458, 32}, {3517, 10982, 13367}, {3518, 9781, 578}, {5020, 33586, 3917}, {5198, 9786, 11381}, {5446, 7506, 1092}, {5462, 7517, 10984}, {5640, 15080, 15018}, {7687, 15473, 13202}, {7714, 11433, 31383}, {7998, 16042, 16187}, {9909, 10601, 22352}, {9971, 19136, 8541}, {10545, 15107, 2}, {10605, 18535, 32062}, {11002, 14002, 110}, {15018, 15080, 182}, {15026, 17714, 13336}, {17810, 31860, 6}, {19130, 32223, 2}, {20423, 26255, 5642}


X(34418) =  ISOGONAL CONJUGATE OF X(19552)

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

See Kadir Altintas and César Lozada, Hyacinthos 29607.

X(34418) lies on the cubics K039, K466 and on these lines: {3, 2888}, {4, 3432}, {26, 11671}, {49, 15345}, {137, 3518}, {184, 13505}, {186, 1141}, {378, 15960}, {570, 14586}, {930, 7512}, {1147, 13504}, {1263, 2070}, {2914, 8157}, {3520, 15620}, {5944, 11273}, {6143, 23319}, {7488, 25150}, {7502, 13512}, {13367, 20574}, {21394, 24144}

X(34418) = anticomplement of the complementary conjugate of X(6150)
X(34418) = isogonal conjugate of X(19552)
X(34418) = antigonal conjugate of the isogonal conjugate of X(21394)


X(34419) =  X(3)X(3648)∩X(46)X(16553)

Barycentrics    a*(a^6-(b-2*c)*a^5-(2*b^2+c^2)*a^4+(2*b^3-4*c^3-(2*b-c)*b*c)*a^3+(b-c)*(b^3+2*b^2*c+c^3)*a^2-(b^2-c^2)*(b^3+2*c^3)*a-(b^2-c^2)^2*(b-c)*c)*(a^6+(2*b-c)*a^5-(b^2+2*c^2)*a^4-(4*b^3-2*c^3-(b-2*c)*b*c)*a^3-(b-c)*(b^3+2*b*c^2+c^3)*a^2+(b^2-c^2)*(2*b^3+c^3)*a+(b^2-c^2)^2*(b-c)*b) : :
Trilinears    (-1+cos(B-C)+cos(2*A)+cos(2*B)+2*cos(2*B+C)-cos(B+2*C))*(-1+cos(B-C)+cos(2*A)+cos(2*C)+2*cos(B+2*C)-cos(2*B+C)) : :

See Kadir Altintas and César Lozada, Hyacinthos 29607.

X(34419) lies on these lines: {3, 3648}, {48, 16553}


X(34420) =  53RD HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^16 + 3*a^14*b^2 - 69*a^12*b^4 + 231*a^10*b^6 - 365*a^8*b^8 + 313*a^6*b^10 - 143*a^4*b^12 + 29*a^2*b^14 - b^16 + 3*a^14*c^2 - 98*a^12*b^2*c^2 + 287*a^10*b^4*c^2 - 192*a^8*b^6*c^2 - 243*a^6*b^8*c^2 + 418*a^4*b^10*c^2 - 207*a^2*b^12*c^2 + 32*b^14*c^2 - 69*a^12*c^4 + 287*a^10*b^2*c^4 - 188*a^8*b^4*c^4 - 79*a^6*b^6*c^4 - 226*a^4*b^8*c^4 + 447*a^2*b^10*c^4 - 172*b^12*c^4 + 231*a^10*c^6 - 192*a^8*b^2*c^6 - 79*a^6*b^4*c^6 - 98*a^4*b^6*c^6 - 269*a^2*b^8*c^6 + 416*b^10*c^6 - 365*a^8*c^8 - 243*a^6*b^2*c^8 - 226*a^4*b^4*c^8 - 269*a^2*b^6*c^8 - 550*b^8*c^8 + 313*a^6*c^10 + 418*a^4*b^2*c^10 + 447*a^2*b^4*c^10 + 416*b^6*c^10 - 143*a^4*c^12 - 207*a^2*b^2*c^12 - 172*b^4*c^12 + 29*a^2*c^14 + 32*b^2*c^14 - c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29611.

X(34420) lies on this line: {2, 3}


X(34421) =  54TH HATZIPOLAKIS-MOSES-EULER POINT

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 + 18*a^6*b^2*c^2 - 9*a^4*b^4*c^2 + 7*a^2*b^6*c^2 - 6*b^8*c^2 + 4*a^6*c^4 - 9*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 4*b^6*c^4 + 8*a^4*c^6 + 7*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    (49*R^2-12*SW)*S^2-(19*R^2-4*SW)*SB*SC : :

X(34421) = 3 (2 J^2 - 3) X[2] + (2 J^2 + 1) X[3], 3*X(2)+X(10226), 15*X(2)+X(34350), 3*X(3)+X(18567), X(3)+3*X(34331)

As a point on the Euler line, X(34421) has Shinagawa coefficients (E-48*F, -3*E+16*F).

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29613 and César Lozada, Hyacinthos 29618.

X(34421) lies on these lines: {2, 3}, {15311, 32415}

X(34421) = midpoint of X(i) and X(j) for these {i,j}: {140, 5498}, {461, 11343}, {10125, 23336}, {18420, 25647}
X(34421) = reflection of X(i) in X(j) for these (i,j): (3536, 33001), (3628, 12043)
X(34421) = complement of the complement of X(10226)


X(34422) =  X(2)X(31744)∩X(3)X(5476)

Barycentrics    3*a^10+2*(b^2+c^2)*a^8-(5*b^4+47*b^2*c^2+5*c^4)*a^6-3*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^4+2*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*(b^2+c^2)^2*a^2+(b^4-c^4)^2*(b^2+c^2) : :

See Kadir Altintas and César Lozada, Hyacinthos 29614.

X(34422) lies on these lines: {2, 31744}, {3, 5476}, {14876, 26613}


X(34423) =  X(110)X(6093)∩X(5189)X(10748)

Barycentrics    9*((SB+SC)*(81*R^4*SA+2*SW^3)+3*(SA-5*SW)*SW^2*R^2)*S^4+(81*(SB+SC)*R^2+2*(9*SA-7*SW)*SW)*SA*SW^3*S^2+4*SB*SC*SW^6 : :

See Kadir Altintas and César Lozada, Hyacinthos 29614.

X(34423) lies on these lines: {110, 6093}, {5189, 10748}


X(34424) =  X(3)X(54)∩X(15)X(1337)

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

See Kadir Altintas and César Lozada, Hyacinthos 29617.

X(34424) lies on these lines: {2, 16626}, {3, 54}, {15, 1337}, {23, 21401}, {61, 21461}, {110, 11146}, {396, 11139}, {3060, 22236}, {3129, 14170}, {3132, 12834}, {3171, 10645}, {5238, 6030}, {5888, 11130}, {8929, 16962}, {11480, 15080}, {14704, 31940}

X(34424) = {X(3),X(5012)}-harmonic conjugate of X(34425)


X(34425) =  X(3)X(54)∩X(16)X(1338)

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

See Kadir Altintas and César Lozada, Hyacinthos 29617.

X(34425) lies on these lines: {2, 16627}, {3, 54}, {16, 1338}, {23, 21402}, {62, 21462}, {110, 11145}, {395, 11138}, {3060, 22238}, {3130, 14169}, {3131, 12834}, {3170, 10646}, {5237, 6030}, {5888, 11131}, {8930, 16963}, {11481, 15080}, {14705, 31939}

X(34425) = {X(3),X(5012)}-harmonic conjugate of X(34424)






leftri  Vertex conjugates: X(34426)-X(34449)  rightri

This preamble and centers X(34426)-X(34449) were contributed by Clark Kimberling and Peter Moses, October 11, 2019.

If P = p : q : r (barycentrics), then

(P-vertex conjugate of P) = isogonal conjugate of anticomplement of isogonal conjugate of P, denoted and given by

V(P) = a^2 / ( -a^2 q r + b^2 r p + c^2 p q) : : .

If P is on the circumcircle, then V(P) = P.

If P is on the Steiner circumellipse, then V(P) lies on the circumconic given by

a^2 c^4 x y+b^2 c^4 x y+a^2 b^4 x z+b^4 c^2 x z+a^4 b^2 y z+a^4 c^2 y z = 0 ,

If P is on the Kiepert hyperbola, then V(P) lies on the circumconic given by

a^4 c^4 x y-b^4 c^4 x y-a^2 c^6 x y+b^2 c^6 x y-a^4 b^4 x z+a^2 b^6 x z-b^6 c^2 x z+b^4 c^4 x z-a^6 b^2 y z+a^4 b^4 y z+a^6 c^2 y z-a^4 c^4 y z = 0 ,

which passes through X(i) for these i: 3, 25 , 32, 98, 184, 228, 878, 1402, 1410, 1799, 2200, 2351, 2353, 3425, 3437, 3438, 3439, 3442, 3443, 3455, 3456, 3504, 6401, 6402, 8825, 8858, 8884, 10547, 14600, 14908, 17970, 22381, 22455, 23716, 23717, 33581. This is the circumconic with perspector X(3049) and center X(17423).

If P is on the Jerabek hyperbola, then V(P) also lies of the Jerabek hyperbola, as in X(i) for i = 34435-34440.

If P is on the Feuerbach hyperbola, then V(P) is on the circumconic given by

a^3 c^3 x y-a^2 b c^3 x y+a b^2 c^3 x y-b^3 c^3 x y-a c^5 x y+b c^5 x y-a^3 b^3 x z+a b^5 x z+a^2 b^3 c x z-b^5 c x z-a b^3 c^2 x z+b^3 c^3 x z-a^5 b y z+a^3 b^3 y z+a^5 c y z-a^3 b^2 c y z+a^3 b c^2 y z-a^3 c^3 y z = 0,

which passes through X(i) for these i: 3, 28, 48, 56, 104, 603, 911, 963, 1333, 1436, 1437, 1444, 1472, 1791, 1811, 2196, 2217, 3417, 3418, 3420, 3433, 3435, 7053, 10623, 15617, 17971, 20779, 23086, 32658, 34121, 34125, 34250. This is the circumconic with perspector X(22383) and center X(34467).

In general, if P is on the circumconic with perspector u : v : w, then V(P) is on the circumconic with perspector a^4 (c^2 v+b^2 w) : : . If CC is a circumconic on the Lemoine line (which passes through X(i) for i = 187, 237, 351, 352, 512, 647, 649, ...}, then V maps CC to CC.

The appearance of (i,j) in the following list means that V(X(i)) = X(j):

(1,6), (2,25), (3,64), (4,3), (5,3432), (6,6), (7,3433), (8,3435), (9,1436), (10,3437), (13,3438), (14,3439), (15,3440), (16,3441), (17,3442), (18,3443), (19,2164), (20,34426), (21,2217), (22, 34427), (23,22258), (24,34428), (25,8770), (27,34429), (28,2218), (30,34178), (31,2162), (32,3224), (34,34430), (36,34431), (37,3444), (39,14370), (40,34432), (42,2248), (48,8761), (53,34433), (54,4), (55,11051), (56,3445), (57,55), (58,1), (59,513), (60,34434), (61,3489), (62,3490), (63,7169), (64,3532), (65,34435), (66,34436), (67,34437), (68,34438), (69,34207), (70,34439), (71,44440), (74,74), (75,7087), (76,2353), (79,34441), (80,34442), (81,31), (82,34443), (83,32), (84,963), (86,34444), (87,34445), (88,6187), (89,34446), (90,34447), (94,34448), (95,2980), (96,34449), (97,32319), (98,98), (99,99), (100,10), (101,101), (102,102), (103,103), (104,104), (105,105), (106,106), (107,107), (108,108), (109,109, (110,110), (111,111), (112,112), (511,34130), (512,(9217), (513,3446), (514,34179), (515,34180), (516,34181), (517,34182), (518,34183), (519,34184), (520,34185), (521,34187), (522,34189), (523,3447), (524,22259), (525,34190)

Although V maps circumconics to circumconics, V does not in general map lines to lines. For example, V maps the Euler line to the circumquartic given by

a^4 c^6 x^2 y^2-b^4 c^6 x^2 y^2-a^2 c^8 x^2 y^2+b^2 c^8 x^2 y^2-2 a^4 b^4 c^2 x^2 y z+2 a^2 b^6 c^2 x^2 y z+2 a^4 b^2 c^4 x^2 y z-2 b^6 c^4 x^2 y z-2 a^2 b^2 c^6 x^2 y z+2 b^4 c^6 x^2 y z-2 a^6 b^2 c^2 x y^2 z+2 a^4 b^4 c^2 x y^2 z+2 a^6 c^4 x y^2 z-2 a^2 b^4 c^4 x y^2 z-2 a^4 c^6 x y^2 z+2 a^2 b^2 c^6 x y^2 z-a^4 b^6 x^2 z^2+a^2 b^8 x^2 z^2-b^8 c^2 x^2 z^2+b^6 c^4 x^2 z^2-2 a^6 b^4 x y z^2+2 a^4 b^6 x y z^2+2 a^6 b^2 c^2 x y z^2-2 a^2 b^6 c^2 x y z^2-2 a^4 b^2 c^4 x y z^2+2 a^2 b^4 c^4 x y z^2-a^8 b^2 y^2 z^2+a^6 b^4 y^2 z^2+a^8 c^2 y^2 z^2-a^6 c^4 y^2 z^2 = 0,

which passes through the following 17 points: A, B, C, vertices of the tangential triangle, and X(i) for these i: 3, 25, 64, 1113, 1114, 2217, 2218, 3432, 8770, 22258, 34178.

Suppose P is on the Jerabek hyperbola, JH, so that V(P) is a point Q also on the Jerabek hyperbola. Let Tp be the tangent to JH at P and let Tq be the tangent to JH at Q. Let T = Tp∩Tq. The locus of T and P traces JH is a conic that is tangent to jJH at X(6) and X(74). An equation for the conic, here named the Moses-Jerabek conic, MJC, is

b^2*c^2*(b^2*c^2*(a^2 - b^2 - c^2)^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)^2*x^2 - a^4*(a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^8 - 2*a^6*b^2 - 5*a^4*b^4 + 8*a^2*b^6 - 3*b^8 - 2*a^6*c^2 + 12*a^4*b^2*c^2 - 8*a^2*b^4*c^2 - 2*b^6*c^2 - 5*a^4*c^4 - 8*a^2*b^2*c^4 + 10*b^4*c^4 + 8*a^2*c^6 - 2*b^2*c^6 - 3*c^8)*y*z) + (cyclic) = 0.

The MJC passes through X(i) for these i: 6, 74, 185, 389, 1199, 1204, 1205, 34469, 34470. The center of the MJC is X(34468). If |OH|/R < 2*sqrt(2), then the MJC is an ellipse. (Based on notes from Peter Moses, October 12, 2019)

Suppose P is on the circumhyperbola LH = {{A, B, C, X(2), X(6)}}. Then V(P) is a point Q also on LH. Let Tp be the tangent to LH at P and let Tq be the tangent to LH at Q. Let T = Tp∩Tq. The locus of T and P traces LH is a conic. An equation for the conic, here named the Moses-Lemoine conic, MLC, is

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

The MLC passes through X(i) for these i: 6, 111, 1194, 1196, 34481, 34482. The center of the MLC is X(34480). (Based on notes from Peter Moses, October 13, 2019)

underbar



X(34426) =  X(20)-VERTEX CONJUGATE OF X(20)

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 + 3*a^8*c^2 + 12*a^6*b^2*c^2 - 30*a^4*b^4*c^2 + 12*a^2*b^6*c^2 + 3*b^8*c^2 - 14*a^6*c^4 + 14*a^4*b^2*c^4 + 14*a^2*b^4*c^4 - 14*b^6*c^4 + 14*a^4*c^6 - 20*a^2*b^2*c^6 + 14*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 - c^10)*(a^10 + 3*a^8*b^2 - 14*a^6*b^4 + 14*a^4*b^6 - 3*a^2*b^8 - b^10 - 3*a^8*c^2 + 12*a^6*b^2*c^2 + 14*a^4*b^4*c^2 - 20*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 - 30*a^4*b^2*c^4 + 14*a^2*b^4*c^4 + 14*b^6*c^4 + 2*a^4*c^6 + 12*a^2*b^2*c^6 - 14*b^4*c^6 - 3*a^2*c^8 + 3*b^2*c^8 + c^10) : :
Barycentrics    a^2/((2 (8 R^2 - SW)) SA^2 + (S^2 - 2 SW (8 R^2 - SW)) SA + 2 S^2 (6 R^2 - SW)) : :

X(34426) lies on the cubic K236 and these lines: {3, 6523}, {20, 1661}, {393, 28783}, {577, 1033}, {1092, 1498}, {1593, 14379}, {3964, 6527}

X(34426) = isogonal conjugate of X(6225)


X(34427) =  X(22)-VERTEX CONJUGATE OF X(22)

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

X(34427) lies on the cubic K174 and these lines: {3, 3162}, {22, 8793}, {159, 394}, {427, 14376}, {1073, 33584}, {1370, 3926}, {18876, 21213}

X(34427) = isogonal conjugate of X(5596)
X(34427) = X(92)-isoconjugate of X(22135)


X(34428) =  X(24)-VERTEX CONJUGATE OF X(24)

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

X(34428) lies on the cubic K044 and these lines: {4, 8905}, {5, 8906}, {24, 8883}, {25, 52}, {26, 15478}, {1609, 2207}, {1824, 11499}, {3542, 6524}, {3563, 11412}, {7395, 14248}, {9714, 13557}, {12362, 15591}, {15316, 34338}

X(34428) = isogonal conjugate of X(6193)


X(34429) =  X(27)-VERTEX CONJUGATE OF X(27)

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

X(34429) lies on these lines: {1, 7416}, {27, 23383}, {33, 2352}, {200, 15624}, {220, 1011}, {1043, 2975}, {2328, 8053}, {4219, 15622}, {7411, 14942}

X(34429) = isogonal conjugate of X(17220)


X(34430) =  X(34)-VERTEX CONJUGATE OF X(34)

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

X(34430) lies on the circumconic {{A,B,C,X(1),X(6)}} and these lines: {1, 11517}, {56, 3215}, {58, 3157}, {106, 11249}, {998, 1104}, {1068, 8747}, {1220, 17526}, {1411, 11510}, {1474, 2178}, {2191, 26357}, {3445, 10966}, {7129, 8609}

X(34430) = isogonal conjugate of X(12649)


X(34431) =  X(36)-VERTEX CONJUGATE OF X(36)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^3*c + a^2*b*c + a*b^2*c - 2*b^3*c + 2*a^2*c^2 - 3*a*b*c^2 + 2*b^2*c^2 + 2*a*c^3 + 2*b*c^3 - 3*c^4)*(a^4 - 2*a^3*b + 2*a^2*b^2 + 2*a*b^3 - 3*b^4 + a^2*b*c - 3*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(34431) lies on the cubics K312 and K685 and these lines: {36, 19619}, {55, 14260}, {517, 3689}, {859, 5127}, {902, 1457}, {953, 22775}, {2183, 3196}, {3025, 10428}, {5180, 17139}

X(34431) = isogonal conjugate of X(6224)


X(34432) =  X(40)-VERTEX CONJUGATE OF X(40)

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

X(34432) lies on the cubic K179 and these lines: {34, 7037}, {55, 207}, {212, 1035}, {1260, 1490}, {1466, 2188}, {1802, 3197}, {8273, 8606}

X(34432) = isogonal conjugate of X(6223)


X(34433) =  X(53)-VERTEX CONJUGATE OF X(53)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 - 4*a^6*c^2 + a^4*b^2*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 + 6*a^4*c^4 + 5*a^2*b^2*c^4 + 6*b^4*c^4 - 4*a^2*c^6 - 4*b^2*c^6 + c^8)*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 2*a^6*c^2 + a^4*b^2*c^2 + 5*a^2*b^4*c^2 - 4*b^6*c^2 + 2*a^4*c^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(34433) lies on these lines: {6, 25044}, {49, 216}, {51, 2965}, {53, 1601}, {2917, 8823}, {2963, 3432}, {15109, 18212}

X(34433) = isogonal conjugate of anticomplement of X(97)


X(34434) =  X(60)-VERTEX CONJUGATE OF X(60)

Barycentrics    a*(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(34434) lies on the cubics K054 and K901 and these lines: {1, 859}, {5, 10}, {8, 14973}, {19, 21770}, {28, 2190}, {31, 2217}, {37, 1953}, {48, 2214}, {51, 10950}, {60, 1610}, {65, 1193}, {72, 4692}, {75, 3869}, {143, 952}, {145, 25048}, {197, 1036}, {214, 5482}, {225, 1829}, {312, 14923}, {354, 31503}, {375, 5795}, {392, 19863}, {513, 7354}, {529, 29958}, {595, 759}, {596, 758}, {674, 6737}, {945, 22753}, {953, 31849}, {957, 3086}, {959, 34267}, {961, 3450}, {997, 31778}, {1376, 31785}, {1460, 3435}, {2176, 18785}, {2218, 3915}, {2390, 4292}, {2650, 13476}, {2933, 5264}, {3216, 4674}, {3668, 3827}, {3714, 10914}, {3754, 20108}, {3877, 31359}, {5289, 10441}, {5434, 23154}, {8679, 10106}, {10571, 20617}, {10944, 16980}, {12435, 15829}, {18827, 34063}

X(34434) = isogonal conjugate of X(2975)
X(34434) = X(92)-isoconjugate of X(22118)


X(34435) =  X(65)-VERTEX CONJUGATE OF X(65)

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

X(34435) lies on the Jerabek circumhyperbola and these lines: {1, 22586}, {3, 501}, {21, 18123}, {35, 72}, {65, 11363}, {68, 6914}, {69, 4189}, {71, 1030}, {73, 1399}, {265, 759}, {484, 2915}, {1176, 5096}, {2160, 3444}, {3145, 14882}, {5358, 16160}, {24929, 28787}

X(34435) = isogonal conjugate of X(2475)


X(34436) =  X(66)-VERTEX CONJUGATE OF X(66)

Barycentrics    a^2*(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(34436) lies on the Jerabek circumhyperbola and these lines: {4, 20300}, {22, 18124}, {25, 15321}, {66, 20987}, {67, 159}, {69, 6636}, {70, 1503}, {248, 8553}, {265, 5621}, {511, 15317}, {895, 12220}, {1176, 19122}, {1350, 15316}, {1351, 15002}, {4846, 7514}, {12087, 15749}, {13622, 19459}, {15750, 16623}, {18374, 34207}, {19125, 19151}

X(34436) = isogonal conjugate of X(7391)
X(34436) = X(92)-isoconjugate of X(22120)


X(34437) =  X(67)-VERTEX CONJUGATE OF X(67)

Barycentrics    a^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)*(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(34437) lies on the Jerabek circumhyperbola and these lines: {4, 5621}, {6, 1205}, {22, 25335}, {23, 18125}, {54, 2781}, {66, 10117}, {67, 19596}, {68, 16010}, {69, 2916}, {125, 15321}, {248, 11063}, {265, 29012}, {542, 3519}, {576, 15002}, {895, 9019}, {1173, 5622}, {1176, 6593}, {1503, 33565}, {5486, 16176}, {11477, 15317}, {12412, 31884}, {13623, 32600}, {15118, 22336}, {15141, 19151}, {20063, 25328}

X(34437) = isogonal conjugate of X(5189)
X(34437) = X(92)-isoconjugate of X(22121)


X(34438) =  X(68)-VERTEX CONJUGATE OF X(68)

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 - 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(34438) lies on the Jerabek circumhyperbola, the cubic K388, and these lines: {3, 6293}, {4, 20303}, {23, 15077}, {24, 70}, {25, 6145}, {26, 68}, {52, 15317}, {54, 2904}, {66, 26937}, {67, 3515}, {69, 7488}, {72, 15177}, {73, 9659}, {265, 7517}, {1176, 34117}, {1503, 18124}, {2917, 3519}, {3521, 5895}, {4846, 7526}, {5504, 16266}, {6644, 34115}, {7527, 31371}, {7530, 32533}, {9914, 11744}, {9937, 12412}, {11559, 13093}, {14118, 15740}, {14490, 32602}, {15068, 32379}, {15316, 17834}, {32316, 33565}

X(34438) = isogonal conjugate of X(37444)
X(34438) = isogonal conjugate of anticomplement of X(24)
X(34438) = isogonal conjugate of complement of X(31304)


X(34439) =  X(70)-VERTEX CONJUGATE OF X(70)

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

X(34439) lies on the Jerabek circumhyperbola and these lines: {22, 68}, {24, 66}, {69, 7512}, {70, 32321}, {265, 7387}, {378, 14542}, {1176, 19123}, {3521, 9818}, {4846, 7503}, {5504, 19908}, {6243, 15317}, {11457, 18124}, {12088, 15077}, {13622, 19468}, {18534, 21400}

X(34439) = isogonal conjugate of X(14790)


X(34440) =  X(71)-VERTEX CONJUGATE OF X(71)

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

X(34440) lies on the Jerabek circumhyperbola and these lines: {27, 8044}, {65, 7120}, {69, 7560}, {71, 199}, {72, 1761}, {73, 1950}, {1246, 2905}, {1439, 9724}, {1942, 2249}

X(34440) = isogonal conjugate of X(3151)


X(34441) =  X(71)-VERTEX CONJUGATE OF X(79)

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

X(34441) lies on these lines: {3, 5904}, {28, 7354}, {39, 32658}, {48, 5124}, {79, 20988}, {104, 3520}, {603, 5172}, {1437, 4278}, {1791, 5303}, {3433, 9672}

X(34441) = isogonal conjugate of anticomplement of X(35)
X(34441) = isogonal conjugate of isotomic conjugate of isogonal conjugate of X(20988)
X(34441) = isogonal conjugate of polar conjugate of isogonal conjugate of X(22122)
X(34441) = isogonal conjugate of complement of X(20066)
X(34441) = X(92)-isoconjugate of X(22122)


X(34442) =  X(80)-VERTEX CONJUGATE OF X(80)

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

X(34442) lies on the cubics K039 and K340 and these lines: {3, 191}, {11, 28}, {48, 1030}, {55, 9912}, {56, 11383}, {80, 20989}, {100, 1791}, {104, 186}, {187, 32658}, {215, 501}, {407, 13273}, {603, 8614}, {1325, 11604}, {1333, 7117}, {1415, 2092}, {1444, 4996}, {1470, 7053}, {1884, 12764}, {2217, 20832}, {2829, 7414}, {3417, 22775}, {3420, 10832}, {3435, 9672}, {3724, 5172}

X(34442) = isogonal conjugate of X(5080)
X(34442) = X(92)-isoconjugate of X(22123)


X(34443) =  X(82)-VERTEX CONJUGATE OF X(82)

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

X(34443) lies on these lines: {35, 238}, {82, 20990}, {983, 4557}, {985, 16679}, {1399, 1428}, {1914, 2174}, {3613, 18101}, {3733, 7032}, {5009, 17104}, {17598, 18170}

X(34443) = isogonal conjugate of X(17165)
X(34443) = X(92)-isoconjugate of X(22164)


X(34444) =  X(86)-VERTEX CONJUGATE OF X(86)

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

X(34444) lies on the circumconic {{A,B,C,X(2),X(6)}} and these lines: {1, 5132}, {55, 10013}, {58, 23383}, {86, 1621}, {87, 16690}, {106, 6577}, {269, 16878}, {292, 16685}, {870, 16684}, {996, 9708}, {1001, 22006}, {1027, 4057}, {1220, 5047}, {1438, 2220}, {2191, 2352}, {5331, 23361}, {15668, 23853}, {16682, 23854}, {17135, 29437}, {17962, 21773}

X(34444) = isogonal conjugate of X(17135)


X(34445) =  X(87)-VERTEX CONJUGATE OF X(87)

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

X(34445) lies on the circumconic {{A,B,C,X(2),X(6)}} and these lines: {1, 34247}, {6, 18613}, {55, 86}, {56, 1918}, {58, 2175}, {87, 8616}, {238, 979}, {269, 1402}, {292, 21769}, {1001, 1220}, {1120, 20037}, {1222, 17277}, {1438, 16946}, {1621, 20146}, {2178, 17962}, {2279, 2300}, {3445, 20470}, {6059, 8747}, {7166, 20676}

X(34445) = isogonal conjugate of X(10453)
X(34445) = X(63)-isoconjugate of X(17920)
X(34445) = X(92)-isoconjugate of X(22127)


X(34446) =  X(89)-VERTEX CONJUGATE OF X(89)

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

X(34446) lies on these lines: {21, 145}, {31, 1404}, {35, 1036}, {41, 902}, {44, 55}, {56, 15854}, {89, 999}, {1260, 1261}, {1460, 6186}, {2078, 3423}, {2194, 3052}, {2226, 14260}, {3251, 8641}, {4217, 5687}, {6187, 7083}, {8851, 11345}

X(34446) = isogonal conjugate of anticomplement of X(45)
X(34446) = isogonal conjugate of isotomic conjugate of X(1000)
X(34446) = isogonal conjugate of complement of X(20073)


X(34447) =  X(90)-VERTEX CONJUGATE OF X(90)

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

X(34447) lies on these lines: {3, 210}, {8, 1444}, {28, 1857}, {42, 603}, {48, 1334}, {55, 1437}, {56, 1824}, {65, 7053}, {90, 15494}, {378, 3417}, {607, 1333}, {975, 22768}

X(34447) = isogonal conjugate of X(11415)


X(34448) =  X(94)-VERTEX CONJUGATE OF X(94)

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

X(34448) lies on these lines: {3, 2888}, {94, 2070}, {98, 9381}, {184, 566}, {2970, 8884}, {3425, 5094}, {5191, 10547}

X(34448) = isogonal conjugate of anticomplement of X(50)
X(34448) = isogonal conjugate of isotomic conjugate of X(33565)


X(34449) =  X(96)-VERTEX CONJUGATE OF X(96)

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

X(34449) lies on the circumconic {{A,B,C, X(4), X(5)} and these lines: {3, 311}, {5, 156}, {25, 13450}, {26, 2351}, {30, 8800}, {32, 53}, {54, 16837}, {96, 3135}, {228, 32141}, {327, 19179}, {578, 3613}, {1503, 27352}, {2165, 9833}, {2200, 21011}, {2980, 13419}, {6193, 32816}, {6243, 9512}, {7502, 25043}, {7528, 10547}, {8797, 18925}, {11819, 27361}, {12362, 27356}, {18400, 22261}, {21841, 33581}

X(34449) = isogonal conjugate of X(11412)


X(34450) =  MIDPOINT OF X(3) AND X(15948)

Barycentrics    9*S^4+(16*R^2*(80*R^2-31*SW)-11*SB*SC+44*SW^2)*S^2-4*(4*R^2-SW)*(112*R^2-17*SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29618.

X(34450) lies on this line: {2, 3}

X(34450) = midpoint of X(3) and X(15948)
X(34450) = reflection of X(25450) in X(10691)


X(34451) =  MIDPOINT OF X(3530) AND X(15949)

Barycentrics    540*S^4+3*(25*R^2*(805*R^2-284*SW)-156*SB*SC+520*SW^2)*S^2-5*(5*R^2*(2125*R^2-764*SW)+296*SW^2)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29618.

X(34451) lies on this line: {2, 3}

X(34451) = midpoint of X(3530) and X(15949)
X(34451) = reflection of X(26028) in X(21518)


X(34452) =  X(2)X(160)∩X(39)X(51)

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

X(34452) lies on the Kiepert circumpyperbola of the medial triangle and these lines: {2, 160}, {3, 31355}, {39, 51}, {114, 7467}, {3051, 20775}, {5976, 8024}, {6292, 14096}, {6337, 33522}, {6636, 8290}, {7782, 19562}, {8041, 23210}, {8891, 15819}, {15270, 32064}

X(34452) = complement of the isogonal conjugate of X(8266)
X(34452) = medial-isogonal conjugate of X(3613)
X(34452) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3613}, {31, 3051}, {8266, 10}, {18051, 626}
X(34452) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 3051}, {160, 23208}
X(34452) = barycentric product X(i)*X(j) for these {i,j}: {39, 8266}, {1923, 18051}
X(34452) = barycentric quotient X(8266)/X(308)






leftri  Points on the Apollonius circle: X(34453)-X(34466)  rightri

This preamble and centers X(34453)-X(34466) are ased on notes by Peter Moses, October, 2019.

The excircles are tangent to the Apollonius circle and the nine-point circle. The centers of the excircles, therefore, lie on an ellipse, here named the Moses-Apollonius ellipse, that is the locus of the center of a (dynamic) circle tangent to the Apollonius and nine-point circles. An equation for this ellipse follows:

b^2*c^2*(a + b - c)*(a - b + c)*(b + c)^2*x^2 - 2*a^2*b*c*(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)*y*z + (cyclic) = 0

The ellipse passes through the vertices of the vertices of the excentral triangle and X(i) for these i: 5400, 34460, 34461, 34462, 34463, 34464, 34465 and has

center X(34466)
perspector X(2051)
foci X(5) and X(970)
eccentricity Sqrt[1-S/(R s)] = OI/R = |X(1)X(3)|/R

underbar

X(34453) =  X(10)X(113)∩X(43)X(9904)

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

X(34453) lies on the Apollonius circle and these lines: {10, 113}, {43, 9904}, {74, 386}, {110, 573}, {125, 2051}, {146, 9534}, {181, 3024}, {399, 9566}, {542, 3029}, {970, 3031}, {1682, 3028}, {1695, 2948}, {2771, 3032}, {2772, 3033}, {2775, 3034}, {2776, 3030}, {2781, 4260}, {2931, 9571}, {3043, 9563}, {3047, 9562}, {3448, 9535}, {4279, 12192}, {7978, 30116}, {9549, 33535}, {9550, 12888}, {9551, 10118}, {9552, 12373}, {9553, 12903}, {9554, 12904}, {9555, 12374}, {9556, 10819}, {9559, 10817}, {9567, 10620}, {9568, 15063}, {9569, 16003}, {9570, 10117}

X(34453) = reflection of X(3031) in X(970)
X(34453) = Apollonius circle antipode of X(3031)


X(34454) =  X(10)X(114)∩X(43)X(9860)

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

X(34454) lies on the Apollonius circle and these lines: {10, 114}, {43, 9860}, {98, 386}, {99, 573}, {115, 2051}, {147, 9534}, {148, 9535}, {181, 3023}, {542, 3031}, {970, 2782}, {1569, 9560}, {1682, 3027}, {1695, 13174}, {2023, 9547}, {2037, 3414}, {2038, 3413}, {2783, 3032}, {2784, 3033}, {2788, 3034}, {2789, 3030}, {3044, 9562}, {4279, 12176}, {7970, 30116}, {9552, 12184}, {9553, 13182}, {9554, 13183}, {9555, 12185}, {9566, 13188}, {9567, 12188}, {9568, 14981}

X(34454) = reflection of X(3029) in X(970)
X(34454) = Apollonius circle antipode of X(3029)


X(34455) =  X(10)X(117)∩X(102)X(386)

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

X(34455) lies on the Apollonius circle and these lines: {10, 117}, {102, 386}, {109, 478}, {124, 2051}, {151, 9534}, {181, 1364}, {970, 2818}, {1361, 1682}, {1845, 5530}, {2779, 3031}, {2792, 3029}, {2800, 3032}, {2807, 3033}, {2814, 3034}, {2815, 3030}, {3040, 9565}, {3042, 9564}, {9535, 33650}, {10696, 30116}

X(34455) = Spieker-radical-circle-inverse of X(123)
X(34455) = reflection of X(34459) in X(970)
X(34455) = Apollonius circle antipode of X(34459)


X(34456) =  X(10)X(117)∩X(108)X(386)

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

X(34456) lies on the Apollonius circle and these lines: {10, 117}, {108, 386}, {181, 1359}, {573, 1295}, {1682, 3318}, {1903, 2823}, {2051, 25640}, {2798, 3029}, {2804, 3032}, {2812, 3033}, {2834, 3034}, {2840, 3030}, {2850, 3031}, {9534, 34188}, {10702, 30116}

X(34456) = Spieker-radical-circle inverse of X(117)


X(34457) =  X(10)X(118)∩X(33)X(181)

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

Let A' be the Spieker-radical-circle-inverse of the midpoint of BC, and define B', C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle at X(43). X(34457) = X(99)-of-A'B'C'. (Randy Hutson, November 17, 2019)

X(34457) lies on the Apollonius circle and these lines: {10, 118}, {33, 181}, {41, 185}, {64, 10822}, {73, 1362}, {101, 102}, {103, 386}, {116, 2051}, {150, 9535}, {152, 9534}, {970, 1490}, {1282, 1695}, {1364, 1415}, {1903, 2823}, {2092, 3269}, {2772, 3031}, {2784, 3029}, {2801, 3032}, {2820, 3034}, {2821, 3030}, {2825, 10974}, {3041, 9565}, {3046, 9562}, {10697, 30116}, {12114, 23630}

X(34457) = reflection of X(3033) in X(970)
X(34457) = Spieker-radical-circle-inverse of X(5514)
X(34457) = Apollonius circle antipode of X(3033)


X(34458) =  X(10)X(119)∩X(11)X(181)

Barycentrics    a*(a^3*b - a^2*b^2 - a*b^3 + b^4 + a^3*c - a^2*b*c + 2*a*b^2*c - b^2*c^2 - a*c^3)*(a^3*b - a*b^3 + a^3*c - a^2*b*c - a^2*c^2 + 2*a*b*c^2 - b^2*c^2 - a*c^3 + c^4) : :
X(34458) = 5 X[1698] - X[12551]

Let A'B'C' be as at X(34457). X(34458) = X(110)-of-A'B'C'. X(34458) is also the perspector of the Apollonius triangle and the polar triangle of the Spieker radical circle. (Randy Hutson, November 17, 2019)

X(34458) lies on the Apollonius circle and these lines: {10, 119}, {11, 181}, {43, 1768}, {100, 573}, {104, 386}, {149, 9535}, {153, 9534}, {952, 970}, {1317, 1682}, {1695, 5541}, {1698, 12551}, {2771, 3031}, {2783, 3029}, {2801, 3033}, {2826, 3034}, {2827, 3030}, {3035, 9564}, {3036, 9565}, {3045, 9562}, {4260, 13226}, {4279, 12199}, {5530, 11570}, {6048, 12767}, {6264, 9549}, {6326, 9548}, {9552, 12763}, {9553, 13273}, {9554, 13274}, {9555, 12764}, {9566, 12331}, {9567, 12773}, {10698, 30116}, {13244, 17156}

X(34458) = reflection of X(3032) in X(970)
X(34458) = complement of X(35649)
X(34458) = Spieker-radical-circle-inverse of X(124)
X(34458) = Apollonius circle antipode of X(3032)


X(34459) =  X(10)X(119)∩X(52)X(2217)

Barycentrics    a^2*(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)*(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(34459) lies on the Apollonius circle and these lines: {10, 119}, {52, 2217}, {101, 102}, {109, 386}, {117, 2051}, {151, 9535}, {155, 3435}, {181, 994}, {184, 3422}, {215, 501}, {407, 1829}, {692, 9912}, {759, 3271}, {928, 3033}, {970, 2818}, {1807, 2875}, {2773, 3031}, {2785, 3029}, {2835, 3034}, {2841, 3030}, {3032, 3738}, {3040, 9564}, {3042, 5810}, {9534, 33650}, {10703, 30116}

X(34459) = reflection of X(34455) in X(970)
X(34459) = Apollonius circle antipode of X(34455)
X(34459) = Spieker-radical-circle-inverse of X(119)


X(34460) =  X(1)X(21859)∩X(5)X(39)

Barycentrics    a*(a^4*b^2 - a^3*b^3 - a^2*b^4 + a*b^5 - 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 - 2*a^2*b^2*c^2 - a^3*c^3 + 3*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(34460) lies on these lines: {1, 21859}, {5, 39}, {6, 6911}, {11, 13006}, {32, 6924}, {37, 11230}, {80, 11998}, {355, 2275}, {517, 1575}, {572, 34079}, {573, 19550}, {574, 6914}, {583, 3013}, {650, 5540}, {905, 9317}, {952, 1015}, {970, 5213}, {992, 5755}, {1018, 32486}, {1107, 9956}, {1385, 16604}, {1415, 10090}, {1500, 5901}, {1574, 5690}, {1656, 5283}, {2229, 19546}, {2241, 32141}, {2276, 5886}, {2548, 6917}, {2549, 6929}, {3560, 5013}, {3628, 16589}, {3767, 6959}, {4426, 26286}, {5024, 6913}, {5069, 5816}, {5276, 6946}, {5286, 6944}, {5291, 22765}, {5396, 24512}, {5400, 21894}, {5443, 20616}, {5603, 17756}, {5790, 16975}, {6184, 15251}, {6824, 31400}, {6826, 7736}, {6862, 31401}, {6867, 31404}, {6893, 7738}, {6905, 33854}, {6918, 9605}, {6970, 7735}, {7377, 24598}, {10222, 20691}, {11499, 16502}, {13731, 28282}, {19513, 28245}, {31649, 31652}


X(34461) =  X(3)X(5400)∩X(5)X(25652)

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

X(34461) lies on these lines: {3, 5400}, {5, 25652}, {517, 3689}, {572, 34079}, {970, 3031}, {1385, 28083}, {2944, 3579}, {10222, 13541}, {14641, 15489}, {17502, 19516}, {29349, 33814}


X(34462) =  X(5)X(113)∩X(11)X(2807)

Barycentrics    a*(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^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 + 4*a^4*b^2*c^2 - 4*a^3*b^3*c^2 - 3*a^2*b^4*c^2 + 5*a*b^5*c^2 - 2*b^6*c^2 - a^5*c^3 + a^4*b*c^3 - 4*a^3*b^2*c^3 + 8*a^2*b^3*c^3 - 3*a*b^4*c^3 - b^5*c^3 - 2*a^4*c^4 + 2*a^3*b*c^4 - 3*a^2*b^2*c^4 - 3*a*b^3*c^4 + 4*b^4*c^4 + 2*a^3*c^5 - 2*a^2*b*c^5 + 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(34462) lies on these lines: {5, 113}, {11, 2807}, {51, 8727}, {149, 517}, {152, 5927}, {162, 4219}, {389, 6831}, {442, 9729}, {513, 1768}, {908, 916}, {970, 6044}, {1532, 6000}, {2476, 10574}, {2771, 18341}, {2772, 21635}, {2779, 10265}, {2800, 15906}, {2808, 13257}, {3042, 26932}, {3270, 15252}, {3567, 6845}, {3937, 13226}, {4187, 5907}, {4193, 12111}, {4551, 15626}, {5132, 7416}, {5396, 13734}, {5400, 33811}, {5462, 6841}, {5482, 6972}, {5562, 6922}, {5640, 10883}, {5752, 6836}, {5810, 6815}, {5889, 6943}, {5890, 6830}, {5892, 6881}, {5909, 18732}, {5943, 8226}, {6241, 6941}, {6828, 15043}, {6829, 15045}, {6882, 13754}, {6932, 15072}, {6945, 15305}, {6963, 11459}, {6975, 15058}, {6990, 15024}, {6991, 15028}, {8757, 26927}, {10095, 16160}, {13744, 33810}, {15310, 17613}

X(34462) = reflection of X(i) in X(j) for these {i,j}: {3937, 13226}, {31849, 10265}
X(34462) = {X(389),X(6831)}-harmonic conjugate of X(18180)


X(34463) =  X(5)X(121)∩X(40)X(5400)

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

X(34463) lies on these lines: {5, 121}, {40, 5400}, {517, 4358}, {573, 19550}, {952, 970}, {21363, 32486}


X(34464) =  X(1)X(953)∩X(5)X(5520)

Barycentrics    a*(a^8 - 2*a^7*b + 4*a^5*b^3 - 4*a^4*b^4 - 2*a^3*b^5 + 4*a^2*b^6 - b^8 - 2*a^7*c + 6*a^6*b*c - 6*a^5*b^2*c - 4*a^4*b^3*c + 14*a^3*b^4*c - 6*a^2*b^5*c - 6*a*b^6*c + 4*b^7*c - 6*a^5*b*c^2 + 17*a^4*b^2*c^2 - 12*a^3*b^3*c^2 - 11*a^2*b^4*c^2 + 16*a*b^5*c^2 - 4*b^6*c^2 + 4*a^5*c^3 - 4*a^4*b*c^3 - 12*a^3*b^2*c^3 + 26*a^2*b^3*c^3 - 10*a*b^4*c^3 - 4*b^5*c^3 - 4*a^4*c^4 + 14*a^3*b*c^4 - 11*a^2*b^2*c^4 - 10*a*b^3*c^4 + 10*b^4*c^4 - 2*a^3*c^5 - 6*a^2*b*c^5 + 16*a*b^2*c^5 - 4*b^3*c^5 + 4*a^2*c^6 - 6*a*b*c^6 - 4*b^2*c^6 + 4*b*c^7 - c^8) : :
X(34464) = 3 X[165] - 2 X[901], 5 X[1698] - 4 X[31841], 3 X[1699] - 4 X[3259]

See Anopolis #403 (June 13, 2013, Antreas Hatzipolakis)

X(34464) lies on the Bevan circle and these lines: {1, 953}, {5, 5520}, {36, 1455}, {55, 23152}, {57, 3025}, {165, 901}, {191, 31847}, {484, 6127}, {513, 1768}, {517, 3689}, {650, 5540}, {910, 16554}, {1054, 21173}, {1282, 5536}, {1697, 13756}, {1698, 31841}, {1699, 3259}, {2077, 18862}, {2948, 5535}, {2957, 5400}, {3336, 31825}, {3339, 33645}, {3464, 5131}, {3632, 8279}, {5119, 15737}, {5531, 14513}, {5538, 34196}, {7993, 14511}, {9904, 33811}, {20989, 22765}

X(34464) = reflection of X(i) in X(j) for these {i,j}: {1, 953}, {5531, 14513}, {7993, 14511}
X(34464) = reflection of X(1768) in the line X(1)X(3)
X(34464) = Stevanovic-circle-inverse of X(5540)
X(34464) = excentral isogonal conjugate of X(900)
X(34464) = X(3738)-Ceva conjugate of X(1)
X(34464) = X(476)-of-excentral-triangle
X(34464) = X(3025)-of-tangential-of-excentral-triangle
X(34464) = X(1)-anti-altimedial-to-ABC similarity image of X(15071)


X(34465) =  X(1)X(5)∩X(6)X(6911)

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

X(34465) lies on the Feuerbach circumhyperbola of the tangential triangle and these lines: {1, 5}, {6, 6911}, {30, 33810}, {52, 27622}, {155, 3149}, {386, 6917}, {474, 15805}, {581, 6862}, {610, 5755}, {912, 1465}, {970, 2818}, {1216, 19513}, {1498, 6985}, {1745, 24467}, {2915, 2918}, {2948, 5535}, {3216, 5398}, {5462, 28258}, {5663, 33811}, {6867, 19767}, {6970, 24597}, {8053, 14723}, {10539, 28077}, {12331, 22141}, {12691, 16586}, {23361, 26286}

X(34465) = X(6905)-Ceva conjugate of X(3)


X(34466) =  X(1)X(5754)∩X(5)X(10)

Barycentrics    a*(a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 + 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 + a^3*c^3 - 3*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(34466) = 3 X[2] + X[5752], 3 X[5] - X[15488], 3 X[549] - 2 X[14131], X[946] + 3 X[10440], 3 X[970] + X[15488], 5 X[1656] - X[10441], 5 X[1698] - X[31778]

X(34466) lies on these lines: {1, 5754}, {2, 5752}, {3, 1724}, {5, 10}, {6, 19547}, {30, 15489}, {40, 5400}, {51, 7483}, {60, 32911}, {72, 5741}, {140, 143}, {392, 5797}, {549, 14131}, {573, 19543}, {582, 19548}, {899, 3579}, {940, 10108}, {942, 5718}, {965, 6918}, {1006, 6045}, {1193, 1385}, {1506, 5164}, {1656, 10441}, {1698, 31778}, {2183, 5755}, {2979, 17566}, {3576, 22392}, {3911, 11573}, {3917, 13747}, {4187, 22076}, {5719, 12109}, {5721, 31786}, {5777, 19542}, {5810, 14555}, {5943, 6675}, {6176, 15178}, {6922, 14554}, {9566, 19540}, {10021, 13364}, {10222, 10459}, {10902, 33656}, {11230, 19863}, {15310, 31663}, {17749, 19550}, {19513, 28273}, {21361, 22458}, {21530, 26005}, {23361, 26286}, {25142, 28475}, {26030, 26446}

X(34466) = midpoint of X(5) and X(970)
X(34466) = reflection of X(5482) in X(140)
X(34466) = complement of X(37536)
X(34466) = complement of complement of X(5752)
X(34466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {40, 5400, 19648}, {5396, 13731, 1385}, {9569, 10440, 970}


X(34467) =  X(2)-CEVA CONJUGATE OF X(22383)

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

X(34467) lies on these lines: {2972, 22096}, {3937, 23224}, {7117, 20982}, {18188, 18210}

X(34467) = X(31)-complementary conjugate of X(22383)
X(34467) = X(2)-Ceva conjugate of X(22383)


X(34468) =  CENTER OF MOSES-JERABEK CONIC

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

The Moses-Jerabek conic is introduced in the preamble just before X(34426).

X(34468) lies on these lines: {51, 13171}, {74, 1199}, {125, 468}, {1511, 13393}, {3448, 22352}, {5092, 14683}, {5621, 13366}, {5622, 13417}

X(34468) = midpoint of X(74) and X(1199)
X(34468) = center of the Moses-Jerabek conic


X(34469) =  X(3)X(74)∩X(25)X(64)

Barycentrics    a^2*(3*a^8 - 2*a^6*b^2 - 12*a^4*b^4 + 18*a^2*b^6 - 7*b^8 - 2*a^6*c^2 + 24*a^4*b^2*c^2 - 18*a^2*b^4*c^2 - 4*b^6*c^2 - 12*a^4*c^4 - 18*a^2*b^2*c^4 + 22*b^4*c^4 + 18*a^2*c^6 - 4*b^2*c^6 - 7*c^8) : :

X(34469) lies on the Moses-Jerabek conic and these lines: {3, 74}, {20, 12429}, {24, 13093}, {25, 64}, {125, 5895}, {154, 3532}, {184, 8567}, {185, 3516}, {186, 12315}, {235, 12250}, {378, 1199}, {389, 1593}, {468, 6225}, {1181, 11410}, {1205, 10602}, {1351, 12086}, {1495, 1620}, {1498, 15750}, {1597, 9781}, {1657, 11457}, {1885, 18913}, {1899, 5894}, {2071, 12164}, {2935, 14448}, {2937, 11999}, {3091, 32601}, {3172, 3269}, {3426, 10594}, {3515, 6000}, {3517, 12290}, {3527, 13596}, {3534, 34224}, {3564, 30552}, {3843, 23294}, {5064, 13568}, {5073, 25739}, {5094, 6696}, {5198, 11438}, {5925, 10990}, {6243, 12085}, {6247, 12173}, {6407, 11462}, {6408, 11463}, {7484, 17704}, {7499, 15740}, {7689, 11414}, {9715, 10575}, {9786, 11403}, {9899, 11363}, {9909, 12279}, {10625, 12163}, {11204, 19357}, {11396, 12262}, {11432, 14865}, {11479, 15028}, {11598, 19504}, {11820, 12088}, {12084, 12160}, {12165, 13293}, {12289, 15681}, {12293, 16003}, {14530, 21844}, {15057, 18504}, {15311, 26937}, {18386, 20299}, {18560, 26944}, {19118, 34146}, {20417, 22802}, {22334, 31860}, {22497, 34207}, {32063, 32534}

X(34469) = crosspoint of X(64) and X(3532)
X(34469) = crosssum of X(20) and X(3146)
X(34469) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12111, 6090}, {3, 12174, 26864}, {64, 1192, 11381}, {64, 1204, 25}, {74, 17854, 15041}, {185, 3516, 11402}, {185, 10606, 3516}, {1192, 11381, 25}, {1204, 11381, 1192}, {1498, 21663, 15750}, {1885, 18913, 26869}, {3357, 10605, 1593}, {10620, 15041, 22584}, {11456, 11468, 3}, {12041, 32139, 3}, {12250, 18931, 235}


X(34470) =  X(6)X(67)∩X(110)X(6467)

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

X(34470) lies on the Moses-Jerabek conic and these lines: {6, 67}, {110, 6467}, {155, 32272}, {182, 32607}, {184, 6593}, {185, 575}, {542, 11441}, {569, 25711}, {895, 3292}, {1092, 8538}, {1177, 1205}, {1181, 15063}, {1204, 2781}, {1425, 32289}, {1843, 13248}, {1899, 11061}, {1992, 32241}, {2930, 10602}, {3047, 19122}, {3270, 32290}, {5050, 19457}, {5972, 26206}, {6776, 19140}, {8541, 32246}, {8548, 32275}, {8549, 32250}, {9967, 19138}, {10117, 19118}, {11064, 23296}, {11550, 32239}, {11579, 21650}, {12227, 14912}, {12584, 15073}, {12893, 18438}, {13198, 21637}, {13289, 19128}, {13367, 15462}, {15136, 18449}, {18919, 32255}, {21659, 32233}, {25329, 26926}, {26937, 32247}

X(34470) = crosspoint of X(895) and X(1177)
X(34470) = crosssum of X(468) and X(858)
X(34470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 15141, 5095}, {67, 15128, 125}, {125, 5095, 32285}, {5622, 9970, 185}, {9967, 19138, 22109}, {21639, 32260, 895}


X(34471) =  MIDPOINT OF X(1) AND X(3612)

Barycentrics    a*(-a+b+c)*(3*a^2+(b+c)*a-2*(b-c)^2) : :
Trilinears    3 cos A + 2 cos B + 2 cos C - 1 : :
X(34471) = 2*X(1)+X(5217)

See Kadir Altintas and César Lozada, Hyacinthos 29622.

X(34471) lies on these lines: {1, 3}, {2, 10950}, {4, 15950}, {6, 17440}, {8, 4999}, {11, 2476}, {12, 944}, {21, 2320}, {33, 4214}, {37, 2261}, {45, 22356}, {78, 3711}, {80, 1656}, {140, 10573}, {145, 5218}, {214, 474}, {226, 9657}, {244, 8572}, {381, 5443}, {382, 18393}, {388, 6840}, {390, 25557}, {392, 1858}, {405, 30144}, {442, 26475}, {497, 2475}, {498, 952}, {515, 10895}, {551, 950}, {632, 11545}, {946, 12953}, {956, 22836}, {958, 3715}, {962, 15338}, {993, 5730}, {1001, 10394}, {1056, 6903}, {1058, 6951}, {1125, 1837}, {1201, 14547}, {1317, 12247}, {1329, 10955}, {1387, 13274}, {1389, 6942}, {1437, 4653}, {1468, 2361}, {1479, 5901}, {1483, 12647}, {1486, 18614}, {1831, 17523}, {1836, 4297}, {1854, 10535}, {1864, 5436}, {2170, 4258}, {2256, 17438}, {2268, 4287}, {2269, 5036}, {2293, 19945}, {2330, 3242}, {2886, 10959}, {2975, 12635}, {3035, 5554}, {3058, 4313}, {3085, 6952}, {3086, 6853}, {3158, 3893}, {3207, 17451}, {3241, 4995}, {3243, 15837}, {3474, 4323}, {3475, 4308}, {3476, 5703}, {3485, 5731}, {3487, 5434}, {3488, 6937}, {3526, 5444}, {3560, 6265}, {3583, 18493}, {3586, 9624}, {3623, 5281}, {3624, 5727}, {3636, 12053}, {3649, 4293}, {3655, 11237}, {3683, 15829}, {3689, 4853}, {3698, 5438}, {3754, 16371}, {3816, 10958}, {3868, 11194}, {3870, 11260}, {3871, 10912}, {3878, 16370}, {3890, 4428}, {3895, 33895}, {3913, 4861}, {3927, 4867}, {3940, 5258}, {4294, 10595}, {4295, 15326}, {4302, 22791}, {4304, 12701}, {4305, 5603}, {4311, 10404}, {4317, 6147}, {4325, 18541}, {4413, 19860}, {4421, 14923}, {4423, 19861}, {4640, 11682}, {4855, 5836}, {4863, 12437}, {4870, 9612}, {5054, 5445}, {5252, 5882}, {5283, 11998}, {5326, 9780}, {5426, 17637}, {5433, 18391}, {5441, 9668}, {5499, 15174}, {5592, 23761}, {5691, 17605}, {5736, 17221}, {5794, 24541}, {5818, 20400}, {5886, 10572}, {6049, 10578}, {6738, 17728}, {6827, 18962}, {6882, 10954}, {7052, 22236}, {7082, 31435}, {7221, 11997}, {7770, 30140}, {7866, 30120}, {7951, 18525}, {7968, 19038}, {7969, 19037}, {7983, 15452}, {9581, 25055}, {9670, 30384}, {9673, 11365}, {9844, 10393}, {10058, 19907}, {10072, 12433}, {10106, 17718}, {10165, 24914}, {10200, 34123}, {10283, 15171}, {10592, 28224}, {10826, 11230}, {10827, 28204}, {11285, 30136}, {11502, 25524}, {11715, 12739}, {11723, 12374}, {11724, 12185}, {11725, 13183}, {11729, 12764}, {11735, 12904}, {12047, 12943}, {12114, 21740}, {12743, 16173}, {13463, 20075}, {13607, 31397}, {13901, 19066}, {13902, 19030}, {13958, 19065}, {13959, 19029}, {15015, 17636}, {15228, 15696}, {17044, 26101}, {17662, 31480}, {18526, 31479}, {21031, 27383}, {21677, 30478}, {22238, 33655}, {23846, 28348}, {24558, 26105}, {28922, 30847}, {28924, 30826}, {30124, 32954}, {31165, 31424}

X(34471) = midpoint of X(1) and X(3612)
X(34471) = reflection of X(i) in X(j) for these (i,j): (5217, 3612), (10895, 11375)
X(34471) = X(3612)-of-anti-Aquila triangle
X(34471) = X(5217)-of-Mandart-incircle triangle
X(34471) = X(7547)-of-2nd circumperp triangle
X(34471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 3304, 10966), (65, 3576, 5204), (999, 7742, 56), (3057, 17609, 18839), (3338, 5126, 56), (3340, 7987, 1155), (3576, 16193, 56), (5010, 11009, 12702), (8071, 16203, 56), (10267, 22766, 5172), (16193, 31786, 65)


X(34472) =  X(5)X(11202)∩X(6)X(3515)

Barycentrics    a^2*(3*a^6-4*(b^2+c^2)*a^4-(b^2-c^2)^2*a^2+2*(b^4-c^4)*(b^2-c^2))*(3*a^8+6*a^4*b^2*c^2-6*(b^2+c^2)*a^6+2*(b^2+c^2)*(3*b^4-4*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) : :
Barycentrics    (SB+SC)*(16*R^2+SA-5*SW)*(S^2+(16*R^2+SA-6*SW)*SA) : :

See Kadir Altintas and César Lozada, Hyacinthos 29622.

X(34472) lies on these lines: {5, 11202}, {6, 3515}, {1147, 18324}, {1493, 23358}, {3292, 22333}, {4550, 10282}, {11821, 15035}, {17821, 33537}

X(34472) = complement of X(38443)


X(34473) =  X(2)X(2794)∩X(3)X(76)

Barycentrics    3*a^8-5*(b^2+c^2)*a^6+(5*b^4+b^2*c^2+5*c^4)*a^4-(b^2-c^2)^2*b^2*c^2-(b^2+c^2)*(3*b^4-4*b^2*c^2+3*c^4)*a^2 : :
X(34473) = 2*X(3)+X(98), 4*X(3)-X(99), X(3)+2*X(12042), 5*X(3)+X(12188), 7*X(3)-X(13188), 10*X(3)-X(23235), 5*X(3)-2*X(33813), X(4)-4*X(6036), 2*X(4)-5*X(14061), 2*X(98)+X(99), X(98)-4*X(12042), 5*X(98)-2*X(12188), 7*X(98)+2*X(13188), 5*X(98)+X(23235), 5*X(98)+4*X(33813), X(99)+8*X(12042), 5*X(99)+4*X(12188), 7*X(99)-4*X(13188), 5*X(99)-2*X(23235), 8*X(6036)-5*X(14061), 5*X(14061)-4*X(23514)

See Kadir Altintas and César Lozada, Hyacinthos 29622.

X(34473) lies on these lines: {2, 2794}, {3, 76}, {4, 6036}, {5, 10722}, {20, 115}, {30, 9166}, {35, 10069}, {36, 10053}, {40, 7983}, {74, 15342}, {83, 13335}, {114, 631}, {140, 6033}, {147, 620}, {148, 3522}, {182, 10753}, {186, 30716}, {187, 5999}, {262, 12150}, {315, 8781}, {371, 19055}, {372, 19056}, {376, 671}, {381, 34127}, {385, 18860}, {511, 21445}, {542, 3524}, {543, 10304}, {549, 6054}, {550, 6321}, {648, 14060}, {690, 15055}, {962, 11725}, {1003, 9756}, {1092, 3044}, {1151, 19109}, {1152, 19108}, {1350, 10754}, {1352, 7835}, {1385, 7970}, {1569, 15515}, {1587, 8980}, {1588, 13967}, {1656, 22505}, {1657, 22515}, {1916, 5188}, {2023, 3053}, {2077, 13189}, {2407, 13479}, {2482, 11177}, {2784, 10164}, {2790, 20792}, {3023, 5204}, {3027, 5217}, {3091, 6722}, {3515, 12131}, {3516, 5186}, {3525, 6721}, {3528, 13172}, {3529, 20398}, {3543, 5461}, {3564, 7799}, {3785, 32458}, {3788, 9863}, {3839, 14971}, {3843, 15092}, {3972, 13860}, {4027, 33004}, {4188, 5985}, {4297, 13178}, {5010, 10086}, {5013, 12829}, {5054, 23234}, {5055, 26614}, {5085, 5182}, {5092, 12177}, {5171, 12176}, {5432, 12184}, {5433, 12185}, {5569, 9877}, {5584, 22514}, {5969, 31884}, {5984, 14981}, {5986, 15246}, {6034, 29181}, {6194, 9888}, {6308, 8295}, {6684, 9864}, {6699, 11005}, {6713, 10768}, {7280, 10089}, {7603, 10486}, {7709, 9734}, {7710, 33216}, {7757, 9755}, {7760, 9737}, {7793, 30270}, {7824, 10333}, {7894, 10983}, {7911, 32152}, {7987, 9860}, {8703, 11632}, {8721, 32964}, {8724, 12100}, {9167, 15708}, {9747, 15078}, {9880, 11001}, {10267, 12190}, {10269, 12189}, {10303, 31274}, {10347, 26316}, {10516, 33220}, {10733, 15359}, {10769, 24466}, {10992, 21735}, {11012, 13190}, {11599, 12512}, {12041, 18332}, {12121, 15535}, {12243, 19708}, {12355, 15695}, {13174, 16192}, {13182, 15326}, {13183, 15338}, {14223, 18556}, {14532, 15655}, {15694, 22566}, {15721, 22247}, {15803, 24472}, {16111, 33511}, {20094, 21734}, {21163, 33273}

X(34473) = midpoint of X(i) and X(j) for these {i,j}: {98, 21166}, {376, 14651}, {14830, 15561}
X(34473) = reflection of X(i) in X(j) for these (i,j): (4, 23514), (99, 21166), (381, 34127), (671, 14651), (3839, 14971), (5055, 26614), (5182, 5085), (6054, 15561), (14651, 6055), (15561, 549), (21166, 3), (23234, 5054), (23514, 6036)
X(34473) = anticomplement of X(36519)
X(34473) = circumperp conjugate of X(23235)
X(34473) = X(5085)-of-1st anti-Brocard triangle
X(34473) = X(21166)-of-ABC-X3 reflections triangle
X(34473) = X(23514)-of-anti-Euler triangle
X(34473) = X(98)-Gibert-Moses-centroid
X(34473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 98, 99), (3, 12042, 98), (3, 12188, 33813), (4, 6036, 14061), (20, 115, 10723), (40, 11710, 7983), (98, 23235, 12188), (631, 9862, 114), (12188, 33813, 23235), (23235, 33813, 99)


X(34474) =  MIDPOINT OF X(165) AND X(15015)

Barycentrics    a*(3*a^6-3*(b+c)*a^5-(6*b^2-7*b*c+6*c^2)*a^4+2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^3+(3*b^4+3*c^4-2*(4*b^2-b*c+4*c^2)*b*c)*a^2+(b^2-c^2)^2*b*c-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a) : :
X(34474) = 2*X(3)+X(100), 4*X(3)-X(104), 5*X(3)+X(12331), 7*X(3)-X(12773), X(3)+2*X(33814), X(4)-4*X(3035), X(4)+2*X(24466), 4*X(5)-X(10724), 2*X(10)+X(12119), 2*X(100)+X(104), 5*X(100)-2*X(12331), 7*X(100)+2*X(12773), X(100)-4*X(33814), 5*X(104)+4*X(12331), 7*X(104)-4*X(12773), X(104)+8*X(33814), X(944)+2*X(1145), 2*X(3035)+X(24466), 2*X(4996)+X(11491), 4*X(5690)-X(12531)

See Kadir Altintas and César Lozada, Hyacinthos 29622.

X(34474) lies on these lines: {2, 5840}, {3, 8}, {4, 3035}, {5, 10724}, {10, 12119}, {11, 631}, {20, 119}, {21, 11231}, {35, 6940}, {36, 10087}, {40, 214}, {80, 6684}, {140, 10738}, {149, 3523}, {153, 3522}, {165, 2800}, {182, 10755}, {371, 19112}, {372, 19113}, {376, 2829}, {404, 5886}, {516, 1519}, {517, 4881}, {528, 3524}, {548, 11698}, {549, 10707}, {550, 10742}, {620, 10768}, {962, 11729}, {1006, 3586}, {1125, 14217}, {1151, 19082}, {1152, 19081}, {1155, 12739}, {1156, 31658}, {1317, 5204}, {1320, 1385}, {1350, 10759}, {1376, 6950}, {1387, 9785}, {1484, 3530}, {1537, 6361}, {1587, 13922}, {1588, 13991}, {1656, 22938}, {1657, 22799}, {1698, 6246}, {1768, 16192}, {1783, 22055}, {1811, 3417}, {1862, 3515}, {2771, 15055}, {2787, 21166}, {2801, 21165}, {2802, 3576}, {2803, 23239}, {3090, 31235}, {3487, 24465}, {3516, 12138}, {3525, 6667}, {3528, 12248}, {3529, 20400}, {3579, 6265}, {3601, 12736}, {3624, 16174}, {3651, 5660}, {3654, 10031}, {3871, 32612}, {4188, 11248}, {4293, 10956}, {4297, 12751}, {4302, 6963}, {4421, 5854}, {5054, 34126}, {5083, 15803}, {5085, 9024}, {5122, 14151}, {5171, 13194}, {5218, 6955}, {5253, 10283}, {5432, 6951}, {5433, 13274}, {5541, 7987}, {5584, 22775}, {5587, 6906}, {5603, 16371}, {5732, 6594}, {5759, 10427}, {5848, 10519}, {5851, 21168}, {5856, 21151}, {6036, 10769}, {6049, 12735}, {6154, 10299}, {6326, 12520}, {6699, 10778}, {6702, 31423}, {6710, 10772}, {6711, 10777}, {6712, 10770}, {6718, 10771}, {6868, 32554}, {6876, 12332}, {6902, 12764}, {6909, 28160}, {6911, 9779}, {6920, 10172}, {6937, 8068}, {6942, 10310}, {6946, 7988}, {6949, 11826}, {6986, 33862}, {7280, 10074}, {7489, 9342}, {7972, 11362}, {7991, 25485}, {8104, 8127}, {8128, 13267}, {8674, 15035}, {9588, 9897}, {10164, 21161}, {10175, 28461}, {10267, 13279}, {10269, 13278}, {10306, 19537}, {10525, 17566}, {11012, 12776}, {11571, 31806}, {12333, 12868}, {12512, 21635}, {12515, 22935}, {12532, 31837}, {12653, 30389}, {12702, 19907}, {12737, 13624}, {12738, 13243}, {12743, 24914}, {12749, 21578}, {12763, 15326}, {13334, 32454}, {13607, 26726}, {13912, 19078}, {13975, 19077}, {15528, 16209}, {15717, 20095}, {16370, 34122}, {17549, 26446}, {17654, 31787}, {24042, 31263}

X(34474) = midpoint of X(165) and X(15015)
X(34474) = reflection of X(i) in X(j) for these (i,j): (5603, 34123), (16173, 10165), (38693, 3)
X(34474) = anticomplement of X(23513)
X(34474) = X(15035)-of-1st circumperp triangle
X(34474) = X(15055)-of-2nd circumperp triangle
X(34474) = X(23515)-of-excentral triangle
X(34474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 100, 104), (3, 5690, 5303), (3, 33814, 100), (20, 119, 10728), (40, 214, 10698), (100, 5303, 12531), (631, 13199, 11), (3035, 24466, 4), (5541, 7987, 11715), (22935, 31663, 12515)


X(34475) =  MIDPOINT OF X(7985) AND X(9902)

Barycentrics    (b+c) (a b+2 b^2-a c+b c) (-a b+a c+b c+2 c^2) : :
X(34475) = X[7985]+X[9902]

See Kadir Altintas and Ercole Suppa, Hyacinthos 29626.

X(34475) lies on the Kiepert circumhyperbola and these lines: {2,726}, {10,22036}, {76,4066}, {226,4135}, {516,14458}, {519,598}, {2321,11599}, {3906,4049}, {3993,21101}, {3994,30588}, {4134,14839}, {4444,30519}, {4709,13576}, {6625,17760}, {7985,9902}, {11167,17132}

X(34475) = isogonal conjugate of X(34476)
X(34475) = midpoint of X(7985) and X(9902)


X(34476) =  ISOGONAL CONJUGATE OF X(34475)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 29626.

X(34476) lies on these lines: {3,6}, {81,10789}, {106,11636}, {110,727}, {385,24267}, {560,595}, {741,30554}, {1357,1412}, {2712,32694}, {4653,11364}, {4658,12194}, {6233,17222}, {7787,25526}, {21793,23095}

X(34476) = isogonal conjugate of X(34475)
X(34476) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {58,33628,1326}, {1333,5009,58}


X(34477) =  55TH HATZIPOLAKIS-MOSES-EULER POINT

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

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29631.

X(34477) lies on these lines: {2, 3}, {343, 1511}, {2883, 32210}, {3917, 16223}, {5642, 11562}, {5663, 10192}, {6697, 11645}, {6699, 20773}, {10182, 13754}, {10193, 14915}, {10282, 20191}, {12359, 32171}, {16226, 32352}, {16252, 32138}, {17821, 32140}

X(34477) = midpoint of X(i) and X(j) for these {i,j}: {2,18324}, {3,10201}, {549,34351}, {14070,18281}, {15331,34330}
X(34477) = reflection of X(i) in X(j) for these {i,j}: {5,34330}, {10201,10020}, {15761,10201}, {18566,5066}, {34330,10125}, {34351,15330}


X(34478) =  56TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    6*a^10 - 15*a^8*b^2 + 6*a^6*b^4 + 12*a^4*b^6 - 12*a^2*b^8 + 3*b^10 - 15*a^8*c^2 + 26*a^6*b^2*c^2 - 13*a^4*b^4*c^2 + 11*a^2*b^6*c^2 - 9*b^8*c^2 + 6*a^6*c^4 - 13*a^4*b^2*c^4 + 2*a^2*b^4*c^4 + 6*b^6*c^4 + 12*a^4*c^6 + 11*a^2*b^2*c^6 + 6*b^4*c^6 - 12*a^2*c^8 - 9*b^2*c^8 + 3*c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29631.

X(34478) lies on these lines: {2, 3}, {3410, 22251}

X(34478) = midpoint of X(i) and X(j) for these {i,j}: {549,34331}, {15330,18281}, {15332,18568}
X(34478) = reflection of X(25401) in X(34331)


X(34479) =  57TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    2*a^16 - 17*a^14*b^2 + 65*a^12*b^4 - 139*a^10*b^6 + 175*a^8*b^8 - 127*a^6*b^10 + 47*a^4*b^12 - 5*a^2*b^14 - b^16 - 17*a^14*c^2 + 90*a^12*b^2*c^2 - 171*a^10*b^4*c^2 + 98*a^8*b^6*c^2 + 89*a^6*b^8*c^2 - 138*a^4*b^10*c^2 + 51*a^2*b^12*c^2 - 2*b^14*c^2 + 65*a^12*c^4 - 171*a^10*b^2*c^4 + 96*a^8*b^4*c^4 + 29*a^6*b^6*c^4 + 72*a^4*b^8*c^4 - 123*a^2*b^10*c^4 + 32*b^12*c^4 - 139*a^10*c^6 + 98*a^8*b^2*c^6 + 29*a^6*b^4*c^6 + 38*a^4*b^6*c^6 + 77*a^2*b^8*c^6 - 94*b^10*c^6 + 175*a^8*c^8 + 89*a^6*b^2*c^8 + 72*a^4*b^4*c^8 + 77*a^2*b^6*c^8 + 130*b^8*c^8 - 127*a^6*c^10 - 138*a^4*b^2*c^10 - 123*a^2*b^4*c^10 - 94*b^6*c^10 + 47*a^4*c^12 + 51*a^2*b^2*c^12 + 32*b^4*c^12 - 5*a^2*c^14 - 2*b^2*c^14 - c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29631.

X(34479) lies on this line: {2, 3}


X(34480) =  X(187)X(1084)∩X(5041)X(32740)

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

X(34480) lies on these lines: {187,1084}, {5041,32740}

X(34480) = center of the Moses-Lemoine conic, MLC, which is introduced in the preamble just before X(34426)


X(34481) =  X(2)X(99)∩X(25)X(32)

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

X(34481) lies on the Moses-Lemoine conic and these lines: {2, 99}, {3, 8770}, {6, 8780}, {22, 3291}, {25, 32}, {39, 5020}, {154, 1692}, {184, 3124}, {187, 1611}, {230, 10154}, {251, 14002}, {576, 2056}, {1194, 1995}, {1368, 7748}, {1504, 8854}, {1505, 8855}, {1506, 7392}, {1570, 3167}, {1899, 6388}, {2548, 7398}, {3051, 34417}, {3053, 20850}, {3098, 21001}, {3229, 30270}, {3767, 6353}, {3787, 33586}, {3815, 10128}, {3926, 18287}, {3981, 5028}, {4232, 7755}, {5034, 5943}, {5052, 17810}, {5064, 15820}, {5254, 6677}, {5268, 31451}, {5651, 20859}, {6387, 18935}, {6676, 7746}, {6995, 7747}, {7386, 7756}, {7484, 15515}, {7494, 7749}, {7667, 16317}, {7714, 7737}, {7735, 33630}, {7751, 33651}, {7888, 34254}, {8024, 16055}, {9465, 13595}

X(34481) = crosspoint of X(i) and X(j) for these (i,j): {25, 8770}, {393, 15591}
X(34481) = crosssum of X(69) and X(193)
X(34481) = crossdifference of every pair of points on line {351, 3265}
X(34481) = barycentric product X(25)*X(30771)
X(34481) = barycentric quotient X(30771)/X(305)
X(34481) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {25, 1196, 32}, {1611, 9909, 187}, {3981, 9306, 5028}, {3981, 20998, 9306}


X(34482) =  X(2)X(6)∩X(23)X(251)

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

X(34482) lies on the Moses-Lemoine conic and these lines: {2, 6}, {22, 30435}, {23, 251}, {32, 1180}, {39, 1627}, {194, 16949}, {305, 7894}, {384, 8267}, {428, 8744}, {1199, 1513}, {1369, 6656}, {1691, 11205}, {1975, 16953}, {2207, 7408}, {3060, 5039}, {3108, 5041}, {3162, 7378}, {3920, 5280}, {5133, 5305}, {5169, 5319}, {5286, 7391}, {5299, 7191}, {6995, 8743}, {7386, 22120}, {7485, 9605}, {7496, 7772}, {7570, 7755}, {7760, 8024}, {7787, 16932}, {7797, 8878}, {7805, 8891}, {7827, 16275}, {7829, 21248}, {7839, 16951}, {8793, 9969}, {9464, 11324}, {9465, 13595}, {10336, 34137}, {10691, 22121}, {12150, 16276}, {12212, 20859}, {16502, 17024}

X(34482) = isogonal conjugate of the isotomic conjugate of X(7859)
X(34482) = crosspoint of X(251) and X(3108)
X(34482) = crosssum of X(141) and X(3589)
X(34482) = crossdifference of every pair of points on line {512, 2528}
X(34482) = barycentric product X(i)*X(j) for these {i,j}: {6, 7859}, {2483, 33951}, {3920, 7191}, {5280, 16706}, {5299, 17289}
X(34482) = barycentric quotient X(7859)/X(76)
X(34482) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 5359, 5354}, {6, 5359, 2}, {32, 1180, 6636}, {39, 1627, 15246}, {251, 1194, 23}, {428, 8792, 8744}, {1194, 5007, 251}, {5276, 33854, 1213}


X(34483) =  X(6)X(3411)∩X(54)X(549)

Barycentrics    (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^4-3 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : :
Barycentrics    (5 S^2+SC^2) (SB+SC-SW) (-4 S^2+SB SC-SB SW+SC^2-SC SW) : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 29636.

X(34483) lies on the Jerabek circumhyperbola and these lines: {4,2889}, {5,14483}, {6,3411}, {20,11738}, {49,1176}, {54,549}, {64,3534}, {74,548}, {185,13623}, {265,1216}, {382,14490}, {1173,3628}, {3426,17800}, {3431,15717}, {3519,3917}, {3521,5562}, {3527,5055}, {3856,14487}, {4846,18436}, {5072,11850}, {7486,14491}, {10303,13472}, {10304,11270}, {10627,15108}, {11559,12121}, {13754,14861}, {15644,16620}, {15704,16659}, {15749,18531}, {15750,18532}

X(34483) = isogonal conjugate of X(34484)


X(34484) =  ISOGONAL CONJUGATE OF X(34483)

Barycentrics    a^2 (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^2-3 b^2 c^2+c^4) : :
Barycentrics SB SC (SB+SC) (-4 S^2-SB SC+SB SW+SC SW-SW^2) : :
Trilinears    4 cos A - 5 sec A : :

As a point on the Euler line, X(34484) has Shinagawa coefficients {-4 f, 5 e + 4 f}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 29636.

X(34484) lies on these lines: {2,3}, {6,15580}, {32,33885}, {51,1199}, {54,1495}, {64,11738}, {74,13474}, {93,32085}, {107,13597}, {110,5446}, {143,10540}, {155,15110}, {156,1994}, {184,9781}, {185,12112}, {232,5041}, {323,10263}, {389,14157}, {569,26881}, {578,26882}, {1056,10046}, {1058,10037}, {1112,2914}, {1173,13366}, {1179,6344}, {1204,11455}, {1216,15107}, {1629,11816}, {1829,33179}, {1831,6198}, {1843,5097}, {1968,10986}, {3060,10539}, {3085,9673}, {3086,9658}, {3199,5008}, {3431,17821}, {3527,26864}, {3563,7953}, {3567,6759}, {3817,9626}, {5102,7716}, {5603,8185}, {5890,26883}, {6242,22750}, {6403,11470}, {7592,17810}, {7687,32340}, {7689,11439}, {7713,16200}, {7967,11365}, {8718,9729}, {8884,11815}, {9590,18483}, {9591,10175}, {9609,31404}, {9625,19925}, {9700,31415}, {9707,10982}, {9713,31418}, {9777,14530}, {9798,10595}, {10095,11817}, {10117,15081}, {10282,15033}, {10575,15053}, {10596,26309}, {10597,26308}, {10984,15024}, {11002,12161}, {11270,14490}, {11278,31948}, {11423,15004}, {11424,11464}, {11438,12290}, {11440,16194}, {11451,13336}, {11456,31860}, {11491,20988}, {11550,26917}, {11572,14644}, {12022,15873}, {12241,12254}, {12310,20125}, {12325,31831}, {13339,32205}, {13353,13364}, {13419,25739}, {13451,14627}, {13472,17809}, {13567,16659}, {13568,32111}, {14683,32358}, {14853,20987}, {15052,18436}, {15062,32110}, {15513,33880}, {15749,18532}, {16534,25714}, {16655,26879}, {18912,31383}

X(34484) = isogonal conjugate of X(34483)
X(34484) = polar conjugate of isotomic conjugate of X(34545)
X(34484) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7517,12088}, {3,4,13596}, {4,24,3520}, {4,25,3518}, {4,186,14865}, {4,3517,17506}, {4,3518,186}, {4,3542,7577}, {4,6143,15559}, {4,7487,18559}, {4,13619,1885}, {4,14940,427}, {4,21844,1593}, {5,23,7512}, {5,7512,7550}, {5,18378,23}, {20,14002,7506}, {22,7529,3090}, {24,378,15750}, {24,1593,21844}, {24,1598,4}, {24,3520,186}, {24,10594,1598}, {25,1598,24}, {25,5198,3517}, {25,10301,23}, {25,10594,4}, {51,1614,1199}, {186,26863,4}, {235,7576,4}, {235,7715,7576}, {378,5198,4}, {378,15750,23040}, {378,23040,3520}, {382,12106,22467}, {382,22467,7464}, {403,6756,4}, {428,1594,4}, {428,21841,1594}, {468,15559,6143}, {546,2070,14118}, {1495,10110,54}, {1593,21844,3520}, {1596,6240,4}, {1656,17714,6636}, {1658,3843,7527}, {1906,18560,4}, {1995,7387,631}, {3199,10312,8744}, {3199,10985,10312}, {3517,5198,378}, {3517,15750,24}, {3518,3520,24}, {3518,10594,26863}, {3518,26863,14865}, {3520,17506,23040}, {3542,6995,4}, {3567,6759,15032}, {3628,13564,15246}, {3855,7556,7503}, {3861,7575,14130}, {5020,10323,3525}, {5899,18369,140}, {7486,7492,7516}, {7503,9714,7556}, {7506,7530,20}, {7517,13861,2}, {7545,18378,5}, {10263,18350,323}, {11799,31830,34007}, {13564,21308,3628}, {15750,23040,17506}, {17928,18534,3529}


X(34485) =  X(8)X(31870)∩X(9)X(5886)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 29637.

X(34485) lies on the Feuerbach circumhyperbola and these lines: {8,31870}, {9,5886}, {21,13464}, {79,12675}, {90,11522}, {946,1156}, {1537,3065}, {4866,30315}, {5551,10806}, {5665,18990}, {5715,33576}, {5882,17097}, {6596,19907}, {7317,10597}, {7319,26332}, {11496,15446}, {11604,12757}

X(34485) = isogonal conjugate of X(34486)


X(34486) =  ISOGONAL CONJUGATE OF X(34485)

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

See Kadir Altintas and Ercole Suppa, Hyacinthos 29637.

X(34486) lies on these lines: {1,3}, {21,5882}, {100,10165}, {104,33812}, {140,6174}, {355,5259}, {392,6326}, {405,5881}, {411,13464}, {498,6978}, {515,1621}, {519,1006}, {551,6905}, {581,3915}, {631,8715}, {632,10943}, {944,5248}, {952,5251}, {993,7967}, {1001,5587}, {1012,4428}, {1125,6946}, {1283,30285}, {1479,6982}, {1483,5288}, {1953,5011}, {2267,2323}, {2302,22356}, {2772,14094}, {2800,18444}, {2975,13607}, {3058,6907}, {3090,3825}, {3146,10587}, {3149,9624}, {3523,11240}, {3525,10806}, {3529,10532}, {3584,6882}, {3616,6796}, {3624,11499}, {3628,26470}, {3651,4301}, {3655,6914}, {3679,6883}, {3871,6684}, {3884,21740}, {4187,20400}, {4304,12119}, {4309,6850}, {4853,11517}, {4857,6842}, {4863,26446}, {5047,24987}, {5079,18544}, {5231,5687}, {5250,5693}, {5270,7491}, {5284,10175}, {5315,5396}, {5398,16474}, {5657,25439}, {6419,26458}, {6420,26464}, {6827,10056}, {6830,10197}, {6853,24387}, {6875,8666}, {6891,31452}, {6911,25055}, {6916,10385}, {6954,10072}, {6985,11522}, {6986,11362}, {6992,11239}, {7411,28194}, {7412,23710}, {7420,18613}, {7489,28204}, {7580,31162}, {7701,12680}, {7741,26487}, {7988,18491}, {7989,18518}, {8227,11500}, {9956,25542}, {10303,10527}, {10386,11826}, {10541,12595}, {10597,17538}, {11024,17572}, {11230,18524}, {11231,12331}, {12672,16132}, {13218,15020}, {14217,33593}, {14869,32214}, {15172,15908}, {15254,18908}, {15325,21155}, {15888,31789}, {16842,17619}, {17531,24541}, {17536,31399}, {17857,31435}, {19546,29640}, {21628,21669}

X(34486) = isogonal conjugate of X(34485)
X(34486) = reflection of X(7688) in X(15931)
X(34486) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1,10267,10902}, {1,10268,12704}, {1,10902,11012}, {1,16208,5709}, {3,3303,7982}, {3,15178,5563}, {55,3576,2077}, {3303,11510,33925}, {3303,33925,1}, {3428,6767,16200}, {3576,12703,30503}, {5010,30392,10269}, {10246,32613,36}, {10267,16202,1}, {10267,24299,14798}


X(34487) =  (name pending)

Barycentrics    5*a^12 - 20*a^10*b^2 + 31*a^8*b^4 - 24*a^6*b^6 + 11*a^4*b^8 - 4*a^2*b^10 + b^12 - 20*a^10*c^2 + 32*a^8*b^2*c^2 - 4*a^6*b^4*c^2 - 16*a^4*b^6*c^2 + 16*a^2*b^8*c^2 - 8*b^10*c^2 + 31*a^8*c^4 - 4*a^6*b^2*c^4 + 7*a^4*b^4*c^4 - 12*a^2*b^6*c^4 + 23*b^8*c^4 - 24*a^6*c^6 - 16*a^4*b^2*c^6 - 12*a^2*b^4*c^6 - 32*b^6*c^6 + 11*a^4*c^8 + 16*a^2*b^2*c^8 + 23*b^4*c^8 - 4*a^2*c^10 - 8*b^2*c^10 + c^12 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 29640.

X(34487) lies on this line: {2, 6}






leftri  Bevan-antipodal triangle and related centers: X(34488) - X(34500)  rightri

This preamble and centers X(34488)-X(34500) were contributed by César Eliud Lozada, October 15, 2019.

Let V be the Bevan point X(40) of ABC and Va, Vb, Vc the antipodes of V on the circumcircles of VBC, VCA and VAB, respectively. The triangle VaVbVc will be named here the Bevan-antipodal triangle of ABC. (see Angel Montesdeoca, HG141019 and Antreas Hatzipolakis Hyacinthos #29638.

Barycentric A-vertex coordinates of the Bevan antipodal triangle are: -a/(-a+b+c) : b/(a-b+c) : c/(a+b-c)

The Bevan-antipodal triangle is perspective to the following triangles with the given perspectors:
(ABC, 57), (ABC-X3 reflections, 3182), (anti-tangential-midarc, 1), (2nd circumperp, 1394), (3rd Conway, 1), (extouch, 34488), (3rd extouch, 223), (Garcia-reflection, 1), (2nd Hatzipolakis, 34489), (hexyl, 1), (Hutson intouch, 1), (incentral, 1419), (incircle-circles, 1), (intangents, 1), (intouch, 1), (Lemoine, 34490), (Macbeath, 34491), (medial, 223), (mixtilinear, 1697), (3rd mixtilinear, 1420), (4th mixtilinear, 8830), (6th mixtilinear, 1), (7th mixtilinear, 2124), (orthic, 34492), (2nd Pamfilos-Zhou, 34493), (2nd inner-Soddy, 34494), (2nd outer-Soddy, 34495), (Steiner, 34496), (symmedial, 34497), (Yff contact, 30719)

The Bevan-antipodal triangle is orthologic to the following triangles with the given centers:
(ABC, 1, 40), (ABC-X3 reflections, 1, 1), (anti-Aquila, 1, 3), (anti-Ara, 1, 1902), (5th anti-Brocard, 1, 12197), (2nd anti-circumperp-tangential, 1, 3057), (anti-Euler, 1, 6361), (anti-inner-Grebe, 1, 1703), (anti-outer-Grebe, 1, 1702), (anti-Mandart-incircle, 1, 10306), (anticomplementary, 1, 962), (Aquila, 1, 7991), (Ara, 1, 9911), (1st Auriga, 1, 12458), (2nd Auriga, 1, 12459), (5th Brocard, 1, 12497), (2nd circumperp tangential, 1, 22770), (Ehrmann-mid, 1, 22793), (Euler, 1, 10), (3rd extouch, 34498, 31965), (outer-Garcia, 1, 4), (Gossard, 1, 12696), (inner-Grebe, 1, 12697), (outer-Grebe, 1, 12698), (Johnson, 1, 12699), (inner-Johnson, 1, 12700), (outer-Johnson, 1, 5812), (1st Johnson-Yff, 1, 1836), (2nd Johnson-Yff, 1, 12701), (Lucas homothetic, 1, 22841), (Lucas(-1) homothetic, 1, 22842), (Mandart-incircle, 1, 65), (medial, 1, 946), (5th mixtilinear, 1, 7982), (3rd tri-squares-central, 1, 13912), (4th tri-squares-central, 1, 13975), (X3-ABC reflections, 1, 12702), (inner-Yff, 1, 5119), (outer-Yff, 1, 46), (inner-Yff tangents, 1, 12703), (outer-Yff tangents, 1, 12704)

The Bevan-antipodal triangle is parallelogic to the following triangles with the given centers: (1st Parry, 1, 9811), (2nd Parry, 1, 9810)

This construction can be generalized for an arbitrary point P (instead of V), and is, in fact, the antipedal triangle of P. Therefore, the Bevan-antipodal triangle is the antipedal triangle of X(40). (Randy Hutson, November 17, 2019)

The Bevan-antipodal triangle is also the anticevian triangle of X(57). (Randy Hutson, January 17, 2020)

underbar

X(34488) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND EXTOUCH

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

X(34488) lies on the cubic K308 and these lines: {1,971}, {2,77}, {57,7023}, {144,7955}, {200,34041}, {222,1449}, {269,4000}, {610,1461}, {651,2324}, {738,14524}, {2270,7053}, {3554,6610}, {4318,34039}, {4853,21147}, {6001,34047}, {6510,34032}

X(34488) = perspector of pedal and antipedal triangles of X(40)
X(34488) = {X(1419), X(34492)}-harmonic conjugate of X(1)


X(34489) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND 2nd HATZIPOLAKIS

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

X(34489) lies on these lines: {1,3}, {7,6872}, {9,25875}, {11,5787}, {34,4306}, {78,3911}, {142,10106}, {145,8732}, {222,1104}, {224,2900}, {225,24159}, {226,2478}, {443,3476}, {497,12520}, {604,17451}, {610,3554}, {614,10571}, {912,1728}, {920,11570}, {936,13747}, {938,6838}, {950,6925}, {958,17625}, {990,2654}, {997,7288}, {1001,12709}, {1042,28082}, {1079,1718}, {1119,4341}, {1145,6765}, {1210,6834}, {1279,34040}, {1423,13724}, {1426,4186}, {1445,11520}, {1453,2003}, {1458,3924}, {1465,17054}, {1471,2650}, {1490,1532}, {1708,3868}, {1722,4551}, {1723,3211}, {1737,17857}, {1788,3811}, {1858,15299}, {1943,19851}, {2647,4334}, {3011,34030}, {3086,5768}, {3487,6947}, {3869,7677}, {3870,4848}, {4187,5219}, {4292,6938}, {4298,30143}, {4299,18223}, {4308,9776}, {4315,30147}, {4318,28079}, {4321,25557}, {4323,29817}, {4511,5265}, {4654,11113}, {4861,6904}, {4999,8583}, {5252,8728}, {5433,5791}, {5720,6959}, {5744,18467}, {5745,19861}, {5805,7354}, {6049,12541}, {6245,10785}, {6841,23708}, {6851,30384}, {6869,21578}, {6881,10827}, {6929,9612}, {6967,13411}, {6989,10039}, {6992,11036}, {8727,11376}, {9578,25466}, {9580,12565}, {9945,20586}, {9946,10074}, {10085,22760}, {10582,11281}, {11319,28968}, {12740,13226}, {15071,30223}, {16859,29007}

X(34489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1467, 57), (1, 8726, 3601), (1, 30503, 1697), (35, 5482, 14795), (40, 30503, 11822), (56, 942, 57), (65, 1319, 11510), (65, 11510, 5119), (165, 559, 10273), (171, 10267, 11883), (260, 16218, 19758), (942, 16678, 26297), (1062, 16541, 14800), (1082, 21164, 5048), (1319, 18838, 32760), (1388, 1466, 24929), (1697, 10310, 30389), (1771, 5584, 5010), (2061, 30337, 9630), (2077, 9441, 18788), (2446, 2572, 30323), (2446, 16193, 24929), (3295, 18443, 10480), (3931, 12555, 10269), (3953, 5885, 14800), (4689, 16209, 10202), (4883, 14794, 26351), (5091, 16204, 11407), (5119, 26423, 1697), (5217, 6767, 26425)


X(34490) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND LEMOINE

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

X(34490) lies on the line {223,5219}


X(34491) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND MACBEATH

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

X(34491) lies on these lines: {1,1537}, {34,34052}, {77,24540}, {222,3670}, {223,936}, {227,34049}, {1394,3468}, {3872,4968}, {10571,27184}, {24046,34051}


X(34492) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND ORTHIC

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

X(34492) lies on these lines: {1,971}, {19,1461}, {34,18328}, {42,34041}, {57,1422}, {77,142}, {222,1100}, {223,5219}, {269,1086}, {278,4341}, {1442,8232}, {1854,34047}, {2910,8757}, {10459,21147}

X(34492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 34488, 1419), (2262, 7053, 57)


X(34493) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND 2nd PAMFILOS-ZHOU

Barycentrics    a*((4*a^9-12*(b+c)*a^8+4*(2*b^2+13*b*c+2*c^2)*a^7+4*(b+c)*(2*b^2-17*b*c+2*c^2)*a^6-4*(4*b-c)*(b-4*c)*(b+c)^2*a^5+4*(b+c)*(4*b^4+4*c^4-(5*b^2+22*b*c+5*c^2)*b*c)*a^4-4*(2*b^6+2*c^6+(17*b^4+17*c^4-2*(11*b^2+13*b*c+11*c^2)*b*c)*b*c)*a^3-4*(b^2-c^2)*(b-c)*(2*b^4+2*c^4-(17*b^2+18*b*c+17*c^2)*b*c)*a^2+4*(b^2-c^2)^2*(3*b^4+3*c^4-(5*b^2+12*b*c+5*c^2)*b*c)*a+(b^2-c^2)^2*(b+c)^3*(-4*b^2+12*b*c-4*c^2))*S+(-a+b+c)*(a^10-3*(b^2+c^2)*a^8-6*(b+c)*a^7*b*c+2*(b^4+c^4+(9*b^2+8*b*c+9*c^2)*b*c)*a^6+2*(5*b-c)*(b-5*c)*(b+c)*b*c*a^5+2*(b^6+c^6-(15*b^4+15*c^4-(11*b^2+38*b*c+11*c^2)*b*c)*b*c)*a^4-2*(b^2-c^2)*(b-c)*(b^2-26*b*c+c^2)*b*c*a^3-(3*b^6+3*c^6+(29*b^2+96*b*c+29*c^2)*b^2*c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*(b^4+c^4+2*(2*b^2-13*b*c+2*c^2)*b*c)*b*c*a+(b^2+6*b*c+c^2)*(b^2-c^2)^4))*(a-b+c)*(a+b-c) : :

X(34493) lies on these lines: {269,8243}, {1394,8234}, {1420,8225}, {1659,12610}


X(34494) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND 2nd INNER-SODDY

Barycentrics    a*(8*S*b*c+(-a+b+c)*(a^3+(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b^2-c^2)*(b-c)))*(a-b+c)*(a+b-c) : :
Trilinears    (a - b + c) (a + b - c) (b c + r s) + 2 r a b c : :

Let A', B', C' be as at X(6213) and A", B", C" be as at X(6212). The lines A'A", B'B", C'C" concur in X(34494). (Randy Hutson, November 17, 2019)

Let (OA), (OB), (OC) be the companion incircles used in Elkies' construction of X(176). Let LA be the external tangent, other than BC, to (OB) and (OC); define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. Triangle A'B'C' is perspective to the cevian triangle of X(176) at X(34494). (Randy Hutson, November 17, 2019)

X(34494) lies on the cubic K414 and these lines: {1,16213}, {7,84}, {9,31535}, {40,176}, {46,1371}, {57,482}, {223,13389}, {1373,3338}, {1394,31546}, {1697,31538}, {1768,15995}, {2124,6212}, {3182,31529}, {5119,17806}, {6204,16572}, {8916,31537}, {10860,31552}, {17805,31393}, {31545,32057}

X(34494) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 3333, 34495), (7177, 11372, 34495)


X(34495) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND 2nd OUTER-SODDY

Barycentrics    a*(-8*S*b*c+(-a+b+c)*(a^3+(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b^2-c^2)*(b-c)))*(a-b+c)*(a+b-c) : :
Trilinears    (a - b + c) (a + b - c) (b c - r s) - 2 r a b c : :

X(34495) lies on the cubic K414 and these lines: {1,16214}, {7,84}, {9,31534}, {40,175}, {46,1372}, {57,481}, {174,3645}, {223,13388}, {1374,3338}, {1394,8225}, {1697,31539}, {1768,15996}, {2124,6213}, {3182,31528}, {5119,17803}, {6203,16572}, {7091,7595}, {8916,31536}, {10860,31551}, {17802,31393}, {31544,32058}

X(34495) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 3333, 34494), (7177, 11372, 34494)


X(34496) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND STEINER

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

X(34496) lies on these lines: {57,7250}, {513,2078}, {522,4318}, {765,4551}, {3287,7180}, {3340,4139}, {3669,21173}, {3676,17418}, {3737,4017}, {4077,7199}


X(34497) = PERSPECTOR OF THESE TRIANGLES: BEVAN-ANTIPODAL AND SYMMEDIAL

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

X(34497) lies on these lines: {1,1362}, {7,17474}, {9,26658}, {41,934}, {57,279}, {77,27626}, {85,17048}, {145,6168}, {222,1424}, {348,21384}, {604,34028}, {651,9310}, {664,3501}, {672,3160}, {1002,3340}, {1025,3208}, {1201,1419}, {1323,4253}, {1416,7121}, {1697,11200}, {1742,20793}, {3177,31526}, {3304,6180}, {3496,7183}, {9312,17754}, {21010,28391}

X(34497) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (279, 1475, 57), (479, 1202, 57), (1025, 25716, 3208), (1200, 9533, 57), (2082, 7177, 57)


X(34498) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BEVAN-ANTIPODAL TO 3rd EXTOUCH

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

The reciprocal orthologic center of these triangles is X(31965)

X(34498) lies on these lines: {1,963}, {19,16572}, {40,221}, {46,10899}, {57,946}, {516,3182}, {962,1440}, {6766,7955}, {15803,37519}

X(34498) = X(4)-of-Bevan antipodal triangle


X(34499) = X(2)-OF- BEVAN-ANTIPODAL TRIANGLE

Barycentrics    a*(a+b-c)*(a-b+c)*(3*a^6+2*(b+c)*a^5-(7*b^2-6*b*c+7*c^2)*a^4-4*(b^2-c^2)*(b-c)*a^3+(5*b^2+6*b*c+5*c^2)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*(b+c)^2) : :
X(34499) = X(57)-4*X(6609) = X(34498)+2*X(34500)

X(34499) lies on these lines: {57,1422}, {223,11212}, {1420,1455}, {3182,3601}, {4654,5603}, {7011,34488}, {34498,34500}

X(34499) = {X(1422), X(6611)}-harmonic conjugate of X(57)


X(34500) = X(3)-OF- BEVAN-ANTIPODAL TRIANGLE

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*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^5+2*(b+c)*(b^2+c^2)*(3*b^2-8*b*c+3*c^2)*a^4-4*(b^4+c^4+3*(b+c)^2*b*c)*(b-c)^2*a^3-4*(b^2-c^2)*(b-c)*(b^4+c^4)*a^2+(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*(b^2-c^2)^2*a+(b^2-c^2)^4*(b+c)) : :
X(34500) = X(34498)-3*X(34499)

X(34500) lies on these lines: {1,4}, {40,1804}, {517,6609}, {1158,1422}, {12705,34492}, {34498,34499}

X(34500) = {X(5307), X(5717)}-harmonic conjugate of X(1072)


X(34501) =  MIDPOINT OF X(9782) AND X(32635)

Barycentrics    (b^2+6*b*c+c^2)*a^2+12*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34501) = 8*X(5)-3*X(13865), 5*X(1698)-X(5506), X(5557)+9*X(19875), 7*X(9780)+X(9782), 7*X(9780)-X(32635), 3*X(10172)-X(34198)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29647.

X(34501) lies these lines: {2, 3303}, {5, 40}, {7, 12}, {10, 354}, {11, 3634}, {20, 26040}, {55, 17552}, {142, 3983}, {442, 3828}, {443, 9657}, {474, 31157}, {495, 11034}, {496, 19872}, {528, 17536}, {548, 5251}, {631, 4413}, {993, 17583}, {1210, 12620}, {1329, 18231}, {1574, 31462}, {2550, 9670}, {2551, 9656}, {2886, 19877}, {3058, 16842}, {3214, 17245}, {3526, 19854}, {3614, 3841}, {3649, 3740}, {3698, 11362}, {4208, 31141}, {4301, 25917}, {4309, 11108}, {4317, 9708}, {4421, 31259}, {4866, 5557}, {4999, 9342}, {5067, 31245}, {5084, 9671}, {5259, 6154}, {5260, 15326}, {5298, 17531}, {5657, 7958}, {5787, 10857}, {6057, 28612}, {6067, 7080}, {6174, 6675}, {7173, 33108}, {7486, 31246}, {8582, 10177}, {8715, 17590}, {9709, 31452}, {9844, 10395}, {9940, 12619}, {9956, 22798}, {10172, 34198}, {12616, 12671}, {12623, 12866}, {16239, 31235}, {16408, 31494}, {17527, 19876}, {17559, 31140}

X(34501) = midpoint of X(9782) and X(32635)
X(34501) = X(1173)-of-4th Euler triangle
X(34501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 17529, 15888), (3826, 9711, 4197), (3826, 9780, 12), (4197, 9711, 12), (4197, 9780, 9711), (4413, 19855, 24953), (5084, 31420, 9671), (8728, 19875, 21031)


X(34502) =  X(1)X(550)∩X(7)X(12)

Barycentrics    (2*a+b+c)*(2*a+3*b+3*c)*(a-b+c)*(a+b-c) : :
X(34502) = X(1)-3*X(5557), 7*X(9780)-15*X(9782), 7*X(9780)-5*X(32635), 3*X(9782)-X(32635)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29647.

X(34502) lies these lines: {1, 550}, {7, 12}, {11, 11544}, {57, 5506}, {65, 3625}, {145, 5434}, {553, 1125}, {1071, 31673}, {1317, 4298}, {3337, 3652}, {3634, 3982}, {3679, 5586}, {4031, 19862}, {4355, 10944}, {4860, 7965}, {5433, 21454}, {5708, 7173}, {6797, 11570}, {9776, 28647}, {11495, 30340}, {11551, 13624}, {11684, 26842}, {12690, 33667}, {14100, 15009}, {16006, 18483}, {17718, 31425}, {23958, 31260}

X(34502) = X(1173)-of-intouch triangle
X(34502) = X(2889)-of-inverse-in-incircle triangle


X(34503) =  X(11)X(3634)∩X(119)X(12811)

Barycentrics    (b^2-10*b*c+c^2)*a^5-(b+c)^3*a^4-(2*b^4+2*c^4+(7*b^2-58*b*c+7*c^2)*b*c)*a^3+2*(b+c)*(b^4-8*b^2*c^2+c^4)*a^2+(b^2+17*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 29647.

X(34503) lies on these lines: {11, 3634}, {119, 12811}, {1156, 5852}, {1484, 5535}




leftri  Points of the Moses-Steiner osculatory triangle: X(34505)-X(34511)  rightri
This preamble and centers X(34505)-X(34511) are based on notes contributed by Peter Moses, October 19-22, 2019.

Suppose that Γ is a curve that circumscribes a triangle ABC and that A', B' C' are the centers of the osculating circles at A,B,C, respectively. The triangle A'B'C' is here named the Moses-Γ osculatory triangle.

If Γ = Steiner circumellipse, then A'B'C' is the Moses-Steiner osculatory triangle, for which

A' = -3 a^4+2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-c^4 : a^2 (a^2+b^2-c^2) : a^2 (a^2-b^2+c^2)
B' = b^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-c^4 : b^2 (-a^2+b^2+c^2)
C' = c^2 (a^2-b^2+c^2) : c^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,

and the three osculating circles meet in the Steiner point, X(99). If you have GeoGebra, you can view Moses-Steiner osculatory triangle.

Let D be the point other than X(99) in which the osculating circles and B and C meet, so that D is the reflection of X(99) in the A-sideline of the Moses-Steiner osculatory triangle, and D is given by

D = a^2-b^2-c^2 : -a^2+b^2+2 c^2 : -a^2+2 b^2+c^2,

Define E and F cyclically. The triangle DEF is here named the Moses-Steiner reflection triangle. See X(34512)-X(34514).

For the Jerabek hyperbola, the center of the A-osculating circle is given by u : v : w, where

u = (b^2-c^2) (-a^2+b^2+c^2) (-2 a^12+7 a^10 b^2-7 a^8 b^4-a^6 b^6+5 a^4 b^8-2 a^2 b^10+7 a^10 c^2-16 a^8 b^2 c^2+14 a^6 b^4 c^2-7 a^4 b^6 c^2+a^2 b^8 c^2+b^10 c^2-7 a^8 c^4+14 a^6 b^2 c^4-4 a^4 b^4 c^4+a^2 b^6 c^4-4 b^8 c^4-a^6 c^6-7 a^4 b^2 c^6+a^2 b^4 c^6+6 b^6 c^6+5 a^4 c^8+a^2 b^2 c^8-4 b^4 c^8-2 a^2 c^10+b^2 c^10)

v = b^2 (a^6-2 a^4 b^2+a^2 b^4-3 a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4-2 b^2 c^4+c^6) (-a^8+3 a^6 b^2-2 a^4 b^4-a^2 b^6+b^8+3 a^6 c^2-5 a^4 b^2 c^2+3 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+3 a^2 b^2 c^4-a^2 c^6-b^2 c^6+c^8)

w = -c^2 (a^6-3 a^4 b^2+a^2 b^4+b^6-2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2+a^2 c^4+b^2 c^4) (-a^8+3 a^6 b^2-2 a^4 b^4-a^2 b^6+b^8+3 a^6 c^2-5 a^4 b^2 c^2+3 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+3 a^2 b^2 c^4-a^2 c^6-b^2 c^6+c^8)

squared radius = -((a^2 b^2 c^2 (-a^8+3 a^6 b^2-2 a^4 b^4-a^2 b^6+b^8+3 a^6 c^2-5 a^4 b^2 c^2+3 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+3 a^2 b^2 c^4-a^2 c^6-b^2 c^6+c^8)^3)/((a-b)^2 (a+b)^2 (-a+c)^2 (a+c)^2 (-b+c)^2 (-a-b+c) (a-b+c) (b+c)^2 (-a+b+c) (a+b+c) (-a^2-b^2+c^2)^2 (a^2-b^2+c^2)^2 (-a^2+b^2+c^2)^2))

For the Kiepert hyperbola, the center of the A-osculating circle is given by u : v : w, where

u = (b^2-c^2) (2 a^8-7 a^6 b^2+9 a^4 b^4-5 a^2 b^6+b^8-7 a^6 c^2+12 a^4 b^2 c^2-8 a^2 b^4 c^2+3 b^6 c^2+9 a^4 c^4-8 a^2 b^2 c^4-5 a^2 c^6+3 b^2 c^6+c^8)

v = (a^4-3 a^2 b^2+2 b^4-2 a^2 c^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)

w = (-a^4+2 a^2 b^2-b^4+3 a^2 c^2-b^2 c^2-2 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)

squared radius = (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)^3/((a-b)^2 (a+b)^2 (a-c)^2 (a-b-c) (b-c)^2 (a+b-c) (a+c)^2 (a-b+c) (b+c)^2 (a+b+c))

For the Feuerbach hyperbola, the center of the A-osculating circle is given by u : v : w, where
u = (b-c) (2 a^7-6 a^5 b^2+6 a^3 b^4-2 a b^6+4 a^5 b c+2 a^4 b^2 c-9 a^3 b^3 c-3 a^2 b^4 c+5 a b^5 c+b^6 c-6 a^5 c^2+2 a^4 b c^2+12 a^3 b^2 c^2+a^2 b^3 c^2-6 a b^4 c^2-3 b^5 c^2-9 a^3 b c^3+a^2 b^2 c^3+6 a b^3 c^3+2 b^4 c^3+6 a^3 c^4-3 a^2 b c^4-6 a b^2 c^4+2 b^3 c^4+5 a b c^5-3 b^2 c^5-2 a c^6+b c^6)

v = b (-a^3+a b^2-a^2 c+b^2 c+a c^2-2 b c^2+c^3) (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)

w = -c (-a^3-a^2 b+a b^2+b^3-2 b^2 c+a c^2+b c^2) (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)

squared radius = -((a 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)^3)/((a-b)^2 (-a+c)^2 (-b+c)^2 (-a-b+c)^3 (a-b+c)^3 (a+b+c)))

For the Steiner circumellipse, the center of the A-osculating circle is given by
-3 a^4+2 a^2 b^2-b^4+2 a^2 c^2+2 b^2 c^2-c^4 : a^2 (a^2+b^2-c^2) : a^2 (a^2-b^2+c^2)

squared radius = a^6/((a+b+c) (-a+b+c) (-b+c+a) (-c+a+b))

Let CE be the circumellipse with perspector X(1). An equation for CE is ayz + bzx + cxy = 0, and CE passes through X(i) for these i: 88, 100, 162, 190, 651, 653, 655, 658, 660, 662, 673, 771, 799, 823, 897, 1156, 1492, 1821, 2349, 2580, 2581, 3257, 4598, 4599, 4604, 4606, 4607, 8052, 20332, 23707, 24624, 27834, 32680, 34085, 34234.
The center of the A-osculating circle is

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

squared radius = -((b (a+b-c)^2 c (a-b+c)^2)/(a^2 (a-b-c) (a+b+c)))

In general, suppose that CCP is the circumconic with perspector P = p : q : r.
The center of the A-osculating circle of CCP is given by u : v : w, where

u = -c^2 (a^2-b^2+c^2) q^3-(a-b-c) (a+b-c) (a-b+c) (a+b+c) p q r-(a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) q^2 r-(a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) q r^2-b^2 (a^2+b^2-c^2) r^3

v = ((a^2-b^2-c^2) q+2 b^2 r) (c^2 q^2+(a^2-b^2-c^2) q r+b^2 r^2)

w =: (-2 c^2 q+(-a^2+b^2+c^2) r) (-c^2 q^2+(-a^2+b^2+c^2) q r-b^2 r^2)

squared radius = ((c^2 q^2+a^2 q r-b^2 q r-c^2 q r+b^2 r^2)^3/((-a+b+c) (a+b-c) (a-b+c) (a+b+c) p^2 q^2 r^2))

Next, suppose that ICP is the inconic with perspector P = p : q : r. Let A'B'C' be the cevian triangle of P.
The center of the A-osculating circle tangent to ICP at A' is given by u : v : w, where

u = 8 a^4 q^2 r^2

v = -q (a^4 p q^2-2 a^2 b^2 p q^2+b^4 p q^2-2 a^2 c^2 p q^2-2 b^2 c^2 p q^2+c^4 p q^2+2 a^4 p q r-4 a^2 b^2 p q r+2 b^4 p q r-4 a^2 c^2 p q r-4 b^2 c^2 p q r+2 c^4 p q r+a^4 p r^2-2 a^2 b^2 p r^2+b^4 p r^2-2 a^2 c^2 p r^2-2 b^2 c^2 p r^2+c^4 p r^2+4 a^4 q r^2+4 a^2 b^2 q r^2-4 a^2 c^2 q r^2)

w = -r (a^4 p q^2-2 a^2 b^2 p q^2+b^4 p q^2-2 a^2 c^2 p q^2-2 b^2 c^2 p q^2+c^4 p q^2+2 a^4 p q r-4 a^2 b^2 p q r+2 b^4 p q r-4 a^2 c^2 p q r-4 b^2 c^2 p q r+2 c^4 p q r+4 a^4 q^2 r-4 a^2 b^2 q^2 r+4 a^2 c^2 q^2 r+a^4 p r^2-2 a^2 b^2 p r^2+b^4 p r^2-2 a^2 c^2 p r^2-2 b^2 c^2 p r^2+c^4 p r^2)

squared radius = -16 a^6 q^4 r^4/((a-b-c) (a+b-c) (a-b+c) (a+b+c) p^2 (q+r)^6))

The center of the osculating circle tangent to the hexyl ellipse at the A-vertex of the excentral triangle is given by u : v : w, where
u = a (a^5 b+2 a^4 b^2-2 a^2 b^4-a b^5+a^5 c+8 a^4 b c+7 a^3 b^2 c-7 a^2 b^3 c-10 a b^4 c-3 b^5 c+2 a^4 c^2+7 a^3 b c^2+10 a^2 b^2 c^2+a b^3 c^2-2 b^4 c^2-7 a^2 b c^3+a b^2 c^3+2 b^3 c^3-2 a^2 c^4-10 a b c^4-2 b^2 c^4-a c^5-3 b c^5)

v = b (a+c) (-a^4 b-2 a^3 b^2+2 a b^4+b^5-3 a^4 c-3 a^3 b c+a^2 b^2 c+a b^3 c-a^3 c^2-4 a^2 b c^2-3 a b^2 c^2-2 b^3 c^2+5 a^2 c^3+a b c^3+3 a c^4+b c^4)

w = (-a-b) c (3 a^4 b+a^3 b^2-5 a^2 b^3-3 a b^4+a^4 c+3 a^3 b c+4 a^2 b^2 c-a b^3 c-b^4 c+2 a^3 c^2-a^2 b c^2+3 a b^2 c^2-a b c^3+2 b^2 c^3-2 a c^4-c^5)

squared radius = (4 a^2 b^2 c^2 (-a^3+2 a b^2+b^3-a b c+2 a c^2+c^3)^3)/((a+b)^2 (-a+b-c)^2 (a+b-c)^2 (a+c)^2 (b+c)^2 (-a+b+c)^2 (a+b+c))

The center of the osculating circle tangent to the hexyl ellipse at the A-vertex of the hexyl triangle is given by u : v : w, where
u = a (a^5 b-2 a^3 b^3+a b^5+a^5 c-4 a^4 b c-7 a^3 b^2 c+3 a^2 b^3 c+8 a b^4 c+3 b^5 c-7 a^3 b c^2-14 a^2 b^2 c^2-3 a b^3 c^2+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+8 a b c^4+2 b^2 c^4+a c^5+3 b c^5)

v = b (a+c) (a^4 b-2 a^2 b^3+b^5+3 a^4 c+a^3 b c-3 a^2 b^2 c+a b^3 c+2 b^4 c+a^3 c^2+4 a^2 b c^2+a b^2 c^2-5 a^2 c^3-3 a b c^3-2 b^2 c^3-3 a c^4-b c^4)

w = (a+b) c (3 a^4 b+a^3 b^2-5 a^2 b^3-3 a b^4+a^4 c+a^3 b c+4 a^2 b^2 c-3 a b^3 c-b^4 c-3 a^2 b c^2+a b^2 c^2-2 b^3 c^2-2 a^2 c^3+a b c^3+2 b c^4+c^5)}.

squared radius = (4 a^2 b^2 c^2 (-a^3+2 a b^2+b^3-a b c+2 a c^2+c^3)^3)/((a+b)^2 (-a+b-c)^2 (a+b-c)^2 (a+c)^2 (b+c)^2 (-a+b+c)^2 (a+b+c))

underbar



X(34504) =  X(3)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    5*a^4 - 5*a^2*b^2 - b^4 - 5*a^2*c^2 + 4*b^2*c^2 - c^4 : :
X(34504) = 4 X[3] - 3 X[5569], 5 X[3] - 3 X[7610], X[4] - 3 X[7618], 2 X[4] - 3 X[8176], 2 X[5] - 3 X[7622], 2 X[20] + X[7759], 4 X[140] - 3 X[7617], X[382] - 3 X[11184], 2 X[546] - 3 X[9771], 4 X[548] - X[7751], 2 X[550] + X[7781], 5 X[631] - 3 X[7615], 6 X[1153] - 7 X[3523], 5 X[1656] - 6 X[7619], X[1657] + 2 X[7764], X[1657] + 3 X[11165], 5 X[3522] - 2 X[7780], 5 X[3522] - 3 X[8182], 7 X[3523] - 3 X[7620], X[3529] + 2 X[7843], X[3529] + 3 X[9770], 4 X[3530] - 3 X[15597], X[3627] - 3 X[12040], 4 X[3628] - 3 X[20112], X[5059] + 3 X[23334], 3 X[5485] - 11 X[21735], 5 X[5569] - 4 X[7610], X[7758] - 3 X[9741], X[7758] + 5 X[17538], 2 X[7764] - 3 X[11165], 2 X[7780] - 3 X[8182], 2 X[7843] - 3 X[9770], X[8667] - 3 X[15688], 3 X[9741] + 5 X[17538], 5 X[15712] - 3 X[16509]

X(34504) lies on these lines: {2, 7748}, {3, 543}, {4, 7618}, {5, 7622}, {20, 3849}, {30, 7775}, {32, 8598}, {39, 33007}, {69, 15301}, {76, 8591}, {99, 3314}, {140, 7617}, {187, 33208}, {376, 538}, {382, 11184}, {524, 550}, {530, 5873}, {531, 5872}, {546, 9771}, {548, 7751}, {549, 18546}, {574, 8370}, {598, 6658}, {599, 7830}, {620, 11318}, {626, 5077}, {631, 7615}, {671, 7746}, {754, 3534}, {1153, 3523}, {1506, 11317}, {1656, 7619}, {1657, 7764}, {1975, 7810}, {1992, 32450}, {2482, 3788}, {2549, 7817}, {3522, 7780}, {3529, 7843}, {3530, 15597}, {3552, 7827}, {3627, 12040}, {3628, 20112}, {3734, 8359}, {3934, 33215}, {4045, 33237}, {5007, 33244}, {5013, 11159}, {5059, 23334}, {5206, 22329}, {5215, 32964}, {5254, 27088}, {5309, 13586}, {5485, 21735}, {6337, 7842}, {6655, 7870}, {6781, 31859}, {7747, 11163}, {7752, 8597}, {7757, 33265}, {7758, 9741}, {7760, 33268}, {7763, 31173}, {7765, 33235}, {7770, 11164}, {7771, 20094}, {7772, 33250}, {7783, 7812}, {7793, 11054}, {7796, 33267}, {7799, 33264}, {7802, 7840}, {7814, 19691}, {7818, 8353}, {7821, 32997}, {7825, 22110}, {7834, 8369}, {7844, 32459}, {7848, 32817}, {7849, 33023}, {7854, 33260}, {7855, 9939}, {7863, 33234}, {7865, 8354}, {7872, 8360}, {7873, 33253}, {7880, 32986}, {7887, 9167}, {7888, 19695}, {7907, 9166}, {7915, 33230}, {8367, 15482}, {8589, 11185}, {8667, 15688}, {8703, 14880}, {8859, 33276}, {9466, 33008}, {9761, 16964}, {9763, 16965}, {9766, 15681}, {9885, 11303}, {9886, 11304}, {9888, 12117}, {10131, 12150}, {10150, 32972}, {11147, 32970}, {13468, 34200}, {14035, 31652}, {14537, 33193}, {14762, 32971}, {14971, 33233}, {15515, 32819}, {15712, 16509}, {15810, 32965}, {16044, 31457}, {21356, 33226}, {21843, 32457}, {26613, 33014}, {31455, 33013}, {32833, 33207}

X(34504) = midpoint of X(i) and X(j) for these {i,j}: {3534, 8716}, {9766, 15681}, {9888, 12117}
X(34504) = reflection of X(i) in X(j) for these {i,j}: {7620, 1153}, {8176, 7618}, {13468, 34200}, {18546, 549}, {34505, 34506}
X(34504) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 7833, 7801}, {671, 33274, 7746}, {2482, 7756, 7841}, {2482, 7841, 3788}, {2549, 32985, 7817}, {3552, 32480, 7827}, {7763, 33192, 31173}, {7783, 9855, 7812}, {7801, 7833, 7761}, {7810, 15300, 1975}, {7817, 32456, 32985}


X(34505) =  X(4)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    a^4 - a^2*b^2 - 2*b^4 - a^2*c^2 + 8*b^2*c^2 - 2*c^4 : :
X(34505) = 5 X[3] - 6 X[5569], 2 X[3] - 3 X[7610], X[4] + 3 X[5485], X[4] - 3 X[7620], 5 X[4] - 3 X[23334], 2 X[5] - 3 X[7615], 4 X[5] - 3 X[11184], 4 X[140] - 3 X[7618], 2 X[140] - 3 X[16509], 3 X[381] - 2 X[7775], X[382] + 2 X[7751], 4 X[546] - X[7758], 2 X[550] - 3 X[8182], 5 X[631] - 6 X[15597], 12 X[1153] - 11 X[15720], 5 X[1656] - 6 X[7617], 5 X[1656] - 2 X[7781], 5 X[1656] - 3 X[11165], X[1657] - 4 X[7780], 7 X[3090] - 3 X[9741], 7 X[3090] - 6 X[9771], 5 X[3091] - 3 X[9770], 5 X[3091] - 6 X[20112], X[3146] + 3 X[9740], 7 X[3526] - 6 X[7622], 2 X[3627] + X[14023], 4 X[3628] - 3 X[12040], 5 X[3843] - 2 X[7759], 7 X[3851] - 4 X[7764], 7 X[3851] - 6 X[8176], 11 X[5056] - 3 X[11148], 5 X[5485] + X[23334], 4 X[5569] - 5 X[7610], 3 X[7617] - X[7781], 5 X[7620] - X[23334], 2 X[7764] - 3 X[8176], 4 X[7775] - 3 X[9766], X[7775] - 3 X[18546], 2 X[7781] - 3 X[11165], X[9766] - 4 X[18546], 3 X[9877] - X[23235]

X(34505) lies on these lines: {2, 1975}, {3, 543}, {4, 524}, {5, 7615}, {6, 8370}, {30, 8667}, {32, 11159}, {76, 338}, {115, 7778}, {126, 21448}, {140, 7618}, {141, 33190}, {148, 183}, {194, 11163}, {230, 32815}, {315, 8352}, {325, 33006}, {376, 13468}, {381, 538}, {382, 3849}, {525, 14852}, {536, 11236}, {546, 7758}, {550, 8182}, {597, 5286}, {598, 7760}, {627, 9761}, {628, 9763}, {631, 15597}, {698, 10516}, {754, 3830}, {1003, 14568}, {1153, 15720}, {1656, 7617}, {1657, 7780}, {1992, 6392}, {2482, 7746}, {2548, 3363}, {2549, 8359}, {2782, 13085}, {3053, 22329}, {3090, 9741}, {3091, 9770}, {3146, 9740}, {3291, 11336}, {3526, 7622}, {3543, 9863}, {3552, 8859}, {3627, 14023}, {3628, 12040}, {3734, 7817}, {3763, 7790}, {3767, 8369}, {3788, 5461}, {3843, 7759}, {3851, 7764}, {3926, 22110}, {4396, 12943}, {4400, 12953}, {5023, 8598}, {5032, 32979}, {5056, 11148}, {5077, 7748}, {5210, 17008}, {5306, 14033}, {5309, 11286}, {5475, 22253}, {6656, 21358}, {7749, 15300}, {7750, 33192}, {7754, 7812}, {7770, 7827}, {7773, 7840}, {7776, 31173}, {7777, 18584}, {7788, 14041}, {7793, 9855}, {7795, 8360}, {7798, 15484}, {7813, 18424}, {7818, 14711}, {7824, 32480}, {7828, 8366}, {7863, 14971}, {7866, 17130}, {7869, 33241}, {7870, 7887}, {7880, 33240}, {7946, 14044}, {8356, 8556}, {8367, 15048}, {8591, 8860}, {8597, 9939}, {9300, 32983}, {9462, 16098}, {9466, 11287}, {9606, 32987}, {9607, 32968}, {9745, 9870}, {9877, 23235}, {9880, 14645}, {10358, 14848}, {11160, 32006}, {11168, 32828}, {11361, 14614}, {11898, 13449}, {12243, 12252}, {14360, 20481}, {15815, 32832}, {16041, 32836}, {16628, 22575}, {16629, 22576}, {17004, 20094}, {17251, 17677}, {19687, 22331}, {20582, 33230}, {21356, 32974}, {22332, 32992}, {26613, 33235}, {31489, 31859}, {32820, 32961}, {32821, 32966}, {32824, 32969}, {32833, 33228}, {32874, 33210}

X(34505) = midpoint of X(5485) and X(7620)
X(34505) = reflection of X(i) in X(j) for these {i,j}: {376, 13468}, {381, 18546}, {7618, 16509}, {8716, 2}, {9741, 9771}, {9766, 381}, {9770, 20112}, {11165, 7617}, {11184, 7615}, {34504, 34506}
X(34505) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {76, 671, 7841}, {76, 7841, 599}, {115, 7801, 11318}, {194, 33013, 11163}, {599, 7841, 7784}, {3734, 7817, 33237}, {3926, 32984, 22110}, {7748, 7810, 5077}, {7754, 7812, 15534}, {7754, 11317, 7812}, {7801, 11318, 7778}, {7812, 11054, 7754}, {7870, 9166, 7887}, {8597, 17129, 9939}, {9466, 11648, 11287}, {11054, 11317, 15534}, {11361, 19570, 14614}, {22329, 32819, 33007}, {22329, 33007, 3053}, {25191, 25195, 13108}, {32828, 33215, 11168}


X(34506) =  X(5)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    4*a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 - 4*b^2*c^2 + c^4 : :
X(34506) = X[3] - 3 X[5569], X[3] + 3 X[7610], X[4] - 3 X[7617], X[4] + 3 X[8182], X[5] - 3 X[15597], X[20] + 3 X[7615], 2 X[140] - 3 X[1153], 4 X[140] - 3 X[7619], 4 X[140] - X[7764], 2 X[140] + X[7780], X[550] + 3 X[16509], 5 X[631] - 3 X[7622], 5 X[631] + X[7751], 5 X[632] - 3 X[9771], 6 X[1153] - X[7764], 3 X[1153] + X[7780], 5 X[1656] - 3 X[8176], X[2896] + 5 X[8150], X[2896] - 5 X[13086], 7 X[2896] + 5 X[18769], 5 X[3522] + 3 X[7620], 7 X[3523] - 3 X[7618], 7 X[3523] - X[7781], 35 X[3523] - 3 X[11148], 11 X[3525] - 3 X[9770], 11 X[3525] + X[14023], 7 X[3526] - X[7759], 7 X[3526] - 3 X[11184], X[3627] - 3 X[20112], 4 X[3628] - X[7843], 3 X[5054] + X[8667], 11 X[5056] - 3 X[23334], 3 X[5485] + 13 X[10299], 3 X[7606] - 2 X[25555], 3 X[7618] - X[7781], 5 X[7618] - X[11148], 3 X[7619] - X[7764], 3 X[7619] + 2 X[7780], 3 X[7622] + X[7751], X[7758] + 3 X[9740], X[7758] - 13 X[10303], X[7759] - 3 X[11184], X[7764] + 2 X[7780], 5 X[7781] - 3 X[11148], 7 X[8150] - X[18769], X[8716] - 5 X[15693], 3 X[9740] + 13 X[10303], X[9766] - 5 X[15694], 3 X[9770] + X[14023], 3 X[11165] - 11 X[15720], 3 X[12040] - 7 X[14869], X[13085] + 3 X[22712], 7 X[13086] + X[18769]

X(34506) lies on these lines: {2, 32}, {3, 543}, {4, 7617}, {5, 3849}, {20, 7615}, {39, 22329}, {76, 2482}, {115, 7771}, {140, 524}, {141, 2030}, {183, 620}, {187, 8370}, {230, 4045}, {316, 17006}, {376, 18546}, {381, 32152}, {384, 26613}, {538, 549}, {542, 10104}, {550, 16509}, {574, 17008}, {597, 6683}, {598, 16921}, {599, 3788}, {625, 3054}, {631, 7622}, {632, 9771}, {635, 33474}, {636, 33475}, {671, 7756}, {736, 15819}, {1656, 8176}, {1992, 31401}, {3055, 3793}, {3314, 31274}, {3522, 7620}, {3523, 7616}, {3525, 9770}, {3526, 7759}, {3627, 20112}, {3628, 7843}, {3734, 21843}, {3767, 23055}, {3934, 8369}, {5023, 11159}, {5025, 14971}, {5054, 8667}, {5056, 23334}, {5077, 13881}, {5206, 32832}, {5215, 7807}, {5461, 7746}, {5485, 10299}, {5913, 10163}, {5969, 32189}, {6179, 9698}, {6655, 9166}, {6656, 15810}, {6719, 20481}, {6722, 7761}, {7606, 25555}, {7735, 15482}, {7747, 33013}, {7750, 31173}, {7755, 7824}, {7758, 9740}, {7765, 33004}, {7767, 22110}, {7768, 16923}, {7769, 7826}, {7772, 33001}, {7782, 15300}, {7783, 11054}, {7794, 7870}, {7804, 8367}, {7816, 27088}, {7822, 8366}, {7825, 23053}, {7829, 11285}, {7838, 11163}, {7844, 33190}, {7854, 33233}, {7863, 33259}, {7869, 21356}, {7873, 33249}, {7886, 8360}, {7888, 33000}, {7895, 22165}, {7902, 32990}, {7908, 15589}, {7915, 8365}, {7916, 11160}, {8556, 11288}, {8584, 31406}, {8588, 11185}, {8598, 15513}, {8716, 15693}, {9172, 9829}, {9761, 33386}, {9763, 33387}, {9766, 15694}, {9890, 12203}, {11165, 15720}, {11303, 33477}, {11304, 33476}, {11648, 33008}, {12040, 14869}, {12815, 32966}, {13083, 22911}, {13084, 22866}, {13085, 22712}, {13087, 13088}, {14148, 17131}, {14568, 33273}, {14693, 24206}, {14762, 19661}, {14907, 33006}, {15271, 33237}, {17130, 32964}, {18362, 33017}, {19569, 32994}, {21358, 32954}, {22566, 32151}, {32480, 33022}

X(34506) = complement of X(7775)
X(34506) = midpoint of X(i) and X(j) for these {i,j}: {376, 18546}, {549, 13468}, {5569, 7610}, {7617, 8182}, {8150, 13086}, {34504, 34505}
X(34506) = reflection of X(7619) in X(1153)
X(34506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 1078, 7810}, {2, 7793, 7812}, {2, 7810, 626}, {2, 7812, 1506}, {2, 9939, 7752}, {76, 33274, 2482}, {140, 7780, 7764}, {230, 8359, 7817}, {1078, 7749, 626}, {5461, 7830, 7841}, {6179, 33015, 9698}, {7746, 7841, 5461}, {7749, 7810, 2}, {7771, 17004, 115}, {7794, 9167, 7870}, {7817, 8359, 4045}, {7824, 8859, 7827}, {7827, 8859, 7755}, {7841, 8860, 7746}, {7870, 7907, 9167}, {8365, 20582, 7915}, {8369, 11168, 3934}, {21843, 34229, 3734}, {23055, 33215, 3767}


X(34507) =  X(6)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    a^6 - 2*a^4*b^2 + 2*a^2*b^4 - b^6 - 2*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(34507) = X[3] - 3 X[599], X[4] + 3 X[69], X[4] - 3 X[1352], 2 X[4] - 3 X[3818], 5 X[4] - 3 X[31670], 4 X[5] - 3 X[5476], 2 X[5] - 3 X[11178], 3 X[6] - 5 X[1656], 3 X[6] - 4 X[25555], X[20] + 3 X[11180], 2 X[39] - 3 X[11261], X[52] - 3 X[29959], 3 X[66] - 2 X[14864], 2 X[69] + X[3818], 3 X[69] + 2 X[18553], 5 X[69] + X[31670], 2 X[140] - 3 X[141], 4 X[140] - 3 X[182], 3 X[141] - X[8550], 2 X[143] - 3 X[16776], 3 X[182] - 2 X[8550], 3 X[193] - 11 X[5056], X[193] - 3 X[14561], 3 X[381] - X[11477], 2 X[550] - 3 X[3098], X[550] - 6 X[3631], 2 X[576] - 3 X[5476], X[576] - 3 X[11178], 3 X[597] - 4 X[3628], 6 X[597] - 5 X[22234], 3 X[599] + X[15069], 5 X[631] - 3 X[11179], 5 X[631] - 4 X[20190], 5 X[631] - 9 X[21356], 5 X[632] - 6 X[20582], 3 X[1350] - X[1657], 3 X[1351] - 7 X[3851], X[1351] - 3 X[10516], 3 X[1352] - 2 X[18553], 5 X[1352] - X[31670], 5 X[1656] + 3 X[11898], 5 X[1656] - 6 X[24206], 5 X[1656] - 4 X[25555], X[1657] + 3 X[18440], 3 X[1992] - 7 X[3090], 3 X[1992] - 4 X[22330], 7 X[3090] - 4 X[22330], 5 X[3091] + 3 X[11160], 5 X[3091] - 3 X[20423], 5 X[3091] - 6 X[25561], X[3098] - 4 X[3631], 5 X[3522] + 3 X[5921], 5 X[3522] - 9 X[10519], 5 X[3522] - 6 X[14810], 7 X[3523] - 15 X[3620], 7 X[3523] - 6 X[5092], 7 X[3523] - 3 X[6776], 7 X[3526] - 6 X[10168], 7 X[3526] - 9 X[21358], 7 X[3526] - 4 X[33749], 17 X[3533] - 21 X[3619], 17 X[3533] - 9 X[14912], 5 X[3618] - 4 X[15516], 7 X[3619] - 3 X[14912], 5 X[3620] - 2 X[5092], 5 X[3620] - X[6776], 8 X[3628] - 5 X[22234], 2 X[3629] - 3 X[15520], 3 X[3630] + 4 X[3850], X[3630] + 2 X[18358], 5 X[3763] - 3 X[5050], 3 X[3818] - 4 X[18553], 5 X[3818] - 2 X[31670], 4 X[3850] - 3 X[5480], 2 X[3850] - 3 X[18358], 7 X[3851] - 9 X[10516], 7 X[3851] - 6 X[19130], 5 X[3858] - 3 X[21850], 9 X[5032] - 17 X[7486], 9 X[5054] - 7 X[10541], 9 X[5055] - 5 X[11482], 3 X[5055] - X[15534], 11 X[5056] - 6 X[5097], 11 X[5056] - 9 X[14561], 13 X[5068] - 9 X[14853], 13 X[5068] + 3 X[20080], X[5073] + 3 X[33878], 13 X[5079] - 9 X[14848], 13 X[5079] - 12 X[25565], 9 X[5085] - 11 X[15720], 3 X[5093] - X[6144], 2 X[5095] - 3 X[34155], 2 X[5097] - 3 X[14561], 3 X[5181] - X[30714], 3 X[5648] - X[23236], X[5921] + 3 X[10519], X[5921] + 2 X[14810], X[6243] - 3 X[9971], 3 X[7697] - X[13330], 5 X[7925] - 3 X[22525], 2 X[10168] - 3 X[21358], 3 X[10168] - 2 X[33749], 3 X[10170] - X[32284], 3 X[10249] - 4 X[25563], 13 X[10299] - 9 X[25406], 3 X[10516] - 2 X[19130], 3 X[10519] - 2 X[14810], X[11160] + 2 X[25561], 3 X[11161] + X[23235], 3 X[11179] - 4 X[20190], X[11179] - 3 X[21356], 3 X[11188] + X[11412], X[11257] - 3 X[22677], 5 X[11444] - X[15073], X[11477] + 3 X[15533], 5 X[11482] - 3 X[15534], X[11898] + 2 X[24206], 3 X[11898] + 4 X[25555], 4 X[12242] - 3 X[19150], 3 X[12584] - 2 X[30714], X[12584] + 2 X[32275], 3 X[13169] + X[14094], 3 X[14643] - 2 X[25556], 3 X[14848] - 4 X[25565], 3 X[14853] + X[20080], 4 X[14862] - 3 X[19149], 3 X[15067] - X[15074], 3 X[15462] - X[32234], 3 X[15520] - 4 X[18583], 3 X[15561] - 2 X[32135], 10 X[15712] - 9 X[17508], 4 X[16534] - 3 X[19140], 4 X[18282] - 3 X[19154], 10 X[18553] - 3 X[31670], 3 X[19662] - 2 X[20398], 4 X[20190] - 9 X[21356], X[20417] - 3 X[32257], 4 X[20417] - 3 X[32305], 9 X[21358] - 4 X[33749], 3 X[23327] - 4 X[32767], 3 X[24206] - 2 X[25555], X[30714] + 3 X[32275], 5 X[31276] - 3 X[31958], 4 X[32257] - X[32305]

Let NA be the reflection of X(5) in the perpendicular bisector of BC, and define NB, NC cyclically. NANBNC is also the X(140)-anti-altimedial triangle, and X(34507) = X(6)-of NANBNC. (Randy Hutson, November 17, 2019)

X(34507) lies on the cubic K481 and these lines: {2, 575}, {3, 67}, {4, 69}, {5, 524}, {6, 17}, {20, 11180}, {30, 22165}, {32, 15993}, {39, 11261}, {52, 29959}, {66, 14864}, {68, 5486}, {98, 3314}, {114, 183}, {125, 9976}, {140, 141}, {143, 16776}, {147, 22712}, {156, 19127}, {159, 2918}, {184, 7495}, {193, 5056}, {262, 7779}, {298, 5613}, {299, 5617}, {343, 468}, {381, 11477}, {382, 19924}, {394, 5094}, {403, 11470}, {520, 18312}, {539, 7514}, {547, 8584}, {550, 1503}, {597, 3628}, {626, 11623}, {631, 7870}, {632, 20582}, {635, 5873}, {636, 5872}, {895, 20301}, {1078, 12177}, {1216, 2393}, {1350, 1657}, {1351, 3851}, {1353, 3589}, {1469, 5270}, {1594, 8541}, {1899, 3819}, {1975, 10992}, {1992, 3090}, {1994, 7570}, {2072, 8538}, {2781, 5876}, {2782, 7761}, {2783, 4655}, {2810, 12587}, {2836, 5694}, {2854, 15067}, {2888, 11444}, {2896, 11257}, {2904, 5095}, {2937, 19596}, {2979, 3410}, {3056, 4857}, {3060, 7533}, {3091, 7946}, {3095, 7855}, {3398, 7822}, {3406, 10159}, {3448, 7998}, {3522, 5921}, {3523, 3620}, {3526, 10168}, {3533, 3619}, {3567, 12325}, {3580, 5651}, {3618, 15516}, {3629, 15520}, {3630, 3850}, {3642, 22736}, {3643, 22737}, {3763, 5050}, {3858, 21850}, {3917, 11442}, {3933, 9737}, {4232, 14826}, {4260, 5820}, {4663, 9956}, {5032, 7486}, {5038, 31455}, {5054, 10541}, {5055, 11482}, {5059, 29323}, {5068, 14853}, {5073, 29317}, {5079, 14848}, {5085, 7666}, {5093, 6144}, {5169, 23061}, {5171, 7767}, {5182, 7907}, {5447, 32140}, {5449, 8548}, {5475, 7697}, {5477, 7749}, {5609, 31744}, {5650, 18911}, {5847, 13464}, {5891, 18390}, {5943, 6515}, {5972, 6090}, {5980, 22509}, {5981, 22507}, {6036, 7778}, {6101, 9019}, {6243, 9971}, {6390, 9734}, {6393, 32820}, {6688, 11433}, {6759, 16618}, {6800, 24981}, {6803, 15012}, {6811, 32809}, {6813, 32808}, {6997, 21849}, {6998, 17271}, {7380, 17378}, {7393, 32621}, {7394, 21969}, {7395, 10112}, {7401, 16625}, {7502, 15582}, {7516, 10116}, {7544, 14531}, {7552, 11061}, {7574, 11649}, {7577, 8537}, {7608, 7777}, {7610, 25486}, {7611, 24697}, {7709, 7831}, {7741, 8540}, {7748, 11646}, {7758, 13085}, {7762, 10358}, {7788, 13860}, {7795, 13335}, {7800, 13334}, {7811, 11676}, {7818, 15980}, {7833, 11161}, {7835, 21445}, {7868, 9755}, {7876, 32467}, {7883, 19905}, {7893, 12110}, {7925, 22525}, {7934, 14651}, {7951, 19369}, {7999, 32255}, {8262, 12106}, {8549, 20299}, {8586, 18424}, {8593, 33274}, {8675, 23105}, {9003, 14314}, {9027, 10170}, {9052, 12586}, {9729, 11411}, {9744, 15819}, {9926, 20302}, {9927, 11591}, {9938, 32600}, {9970, 32244}, {9996, 32515}, {10008, 32824}, {10011, 13468}, {10219, 18928}, {10222, 28538}, {10224, 11255}, {10249, 25563}, {10255, 18449}, {10272, 25329}, {10274, 21230}, {10299, 25406}, {10601, 11225}, {11305, 20415}, {11306, 20416}, {11318, 19662}, {11331, 15595}, {11511, 11585}, {11695, 18951}, {12111, 32247}, {12134, 16789}, {12161, 12585}, {12203, 32027}, {12596, 33547}, {13169, 14094}, {13248, 32743}, {13347, 18914}, {13348, 14216}, {13562, 21841}, {13571, 32451}, {14002, 15360}, {14160, 32827}, {14265, 20021}, {14643, 25556}, {14683, 15080}, {14790, 15606}, {14862, 19149}, {15068, 16534}, {15122, 23329}, {15462, 32234}, {15561, 32135}, {15577, 23358}, {15712, 17508}, {16044, 22486}, {16836, 18917}, {17297, 21554}, {17704, 18909}, {17825, 32068}, {18281, 19510}, {18282, 19154}, {18325, 18435}, {18350, 18374}, {18356, 32142}, {18376, 18572}, {19126, 34002}, {19357, 32348}, {23293, 30745}, {23327, 32767}, {24728, 29093}, {31276, 31958}, {31394, 33082}, {31395, 32846}, {32423, 33533}, {33384, 33385}

X(34507) = midpoint of X(i) and X(j) for these {i,j}: {3, 15069}, {6, 11898}, {69, 1352}, {381, 15533}, {1350, 18440}, {2930, 32306}, {3630, 5480}, {5181, 32275}, {9970, 32244}, {11160, 20423}, {25335, 32254}, {32233, 32272}
X(34507) = reflection of X(i) in X(j) for these {i,j}: {4, 18553}, {6, 24206}, {182, 141}, {193, 5097}, {576, 5}, {895, 20301}, {1351, 19130}, {1353, 3589}, {3629, 18583}, {3818, 1352}, {4663, 9956}, {5476, 11178}, {5480, 18358}, {6776, 5092}, {8548, 5449}, {8549, 20299}, {8550, 140}, {8584, 547}, {9926, 20302}, {9976, 125}, {9977, 1209}, {11255, 10224}, {12584, 5181}, {12596, 33547}, {13248, 32743}, {18381, 34118}, {20423, 25561}, {25329, 10272}, {32273, 32274}
X(34507) = anticomplement of X(575)
X(34507) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1, 7616}, {7608, 8}
X(34507) = crossdifference of every pair of points on line {1510, 2492}
X(34507) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 1352, 18553}, {4, 18553, 3818}, {5, 576, 5476}, {6, 1656, 25555}, {17, 18, 7746}, {140, 8550, 182}, {141, 8550, 140}, {193, 14561, 5097}, {576, 11178, 5}, {599, 15069, 3}, {631, 11179, 20190}, {633, 634, 76}, {637, 638, 11185}, {1351, 10516, 19130}, {2979, 3410, 11550}, {3410, 15108, 2979}, {3629, 18583, 15520}, {6177, 6178, 140}, {7810, 14981, 3}, {9744, 16990, 15819}, {24206, 25555, 1656}


X(34508) =  X(13)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    a^6 - 4*a^4*b^2 + 2*a^2*b^4 + b^6 - 4*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 + 2*Sqrt[3]*(a^4 + a^2*b^2 - b^4 + a^2*c^2 - c^4)*S : :
X(34508) = X[3] - 3 X[9761], 2 X[3] - 3 X[13084], X[4] - 3 X[22491], 4 X[140] - 3 X[13083], 2 X[140] - 3 X[33474], 5 X[1656] - 3 X[9763], 5 X[3091] - 3 X[22492], 3 X[3545] + X[5862], 4 X[3628] - 3 X[33475], 3 X[5055] - X[5859], 5 X[5071] - X[5863], 2 X[20415] - 3 X[33477], 2 X[22114] + X[22871]

X(34508) lies on these lines: {2, 18}, {3, 531}, {4, 530}, {5, 524}, {6, 623}, {13, 3181}, {14, 76}, {15, 302}, {16, 621}, {30, 33459}, {32, 395}, {62, 7812}, {69, 624}, {140, 13083}, {141, 11543}, {183, 6114}, {193, 18582}, {299, 7752}, {303, 16967}, {317, 6117}, {371, 33393}, {372, 33395}, {381, 532}, {396, 7746}, {397, 31693}, {398, 7801}, {511, 33482}, {542, 5872}, {543, 16002}, {547, 33458}, {591, 18587}, {599, 636}, {617, 16242}, {618, 6337}, {619, 16645}, {622, 16809}, {627, 5463}, {629, 22236}, {1351, 7684}, {1352, 7685}, {1656, 9763}, {1991, 18586}, {1992, 5459}, {3091, 22492}, {3095, 5617}, {3545, 5862}, {3628, 33475}, {3629, 11542}, {5055, 5859}, {5071, 5863}, {5339, 11295}, {5470, 16630}, {5471, 22689}, {5487, 12817}, {5615, 20428}, {5980, 6777}, {6115, 7774}, {6116, 9308}, {6295, 6774}, {6297, 22714}, {6299, 10104}, {6671, 11485}, {6672, 11489}, {6773, 12252}, {6782, 22687}, {7747, 9115}, {7749, 9117}, {7870, 11307}, {7878, 11289}, {7922, 22490}, {8594, 33274}, {8838, 11004}, {11131, 16771}, {11133, 22856}, {11296, 22238}, {11300, 16963}, {11312, 21358}, {12355, 16628}, {15067, 34375}, {15765, 32419}, {18585, 32421}, {20415, 33477}, {21159, 33960}, {22512, 32553}, {22693, 25192}

X(34508) = midpoint of X(i) and X(j) for these {i,j}: {381, 5858}, {3104, 25191}
X(34508) = reflection of X(i) in X(j) for these {i,j}: {6295, 6774}, {6298, 5617}, {13083, 33474}, {13084, 9761}, {22866, 18}, {33458, 547}, {34509, 5}
X(34508) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {14, 298, 3643}, {69, 18581, 624}, {599, 11306, 636}, {636, 5460, 11306}


X(34509) =  X(14)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    a^6 - 4*a^4*b^2 + 2*a^2*b^4 + b^6 - 4*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 - 2*Sqrt[3]*(a^4 + a^2*b^2 - b^4 + a^2*c^2 - c^4)*S : :
X(34509) = X[3] - 3 X[9763], 2 X[3] - 3 X[13083], X[4] - 3 X[22492], 4 X[140] - 3 X[13084], 2 X[140] - 3 X[33475], 5 X[1656] - 3 X[9761], 5 X[3091] - 3 X[22491], 3 X[3545] + X[5863], 4 X[3628] - 3 X[33474], 3 X[5055] - X[5858], 5 X[5071] - X[5862], 2 X[20416] - 3 X[33476], 2 X[22113] + X[22916]

X(34509) lies on these lines: {2, 17}, {3, 530}, {4, 531}, {5, 524}, {6, 624}, {13, 76}, {14, 3180}, {15, 622}, {16, 303}, {30, 33458}, {32, 396}, {61, 7812}, {69, 623}, {140, 13084}, {141, 11542}, {183, 6115}, {193, 18581}, {298, 7752}, {302, 16966}, {317, 6116}, {371, 33394}, {372, 33392}, {381, 533}, {395, 7746}, {397, 7801}, {398, 31694}, {511, 33483}, {542, 5873}, {543, 16001}, {547, 33459}, {591, 18586}, {599, 635}, {616, 16241}, {618, 16644}, {619, 6337}, {621, 16808}, {628, 5464}, {630, 22238}, {1351, 7685}, {1352, 7684}, {1656, 9761}, {1991, 18587}, {1992, 5460}, {3091, 22491}, {3095, 5613}, {3545, 5863}, {3628, 33474}, {3629, 11543}, {5055, 5858}, {5071, 5862}, {5340, 11296}, {5469, 16631}, {5472, 22687}, {5488, 12816}, {5611, 20429}, {5981, 6778}, {6114, 7774}, {6117, 9308}, {6296, 22715}, {6298, 10104}, {6582, 6771}, {6671, 11488}, {6672, 11486}, {6770, 12252}, {6783, 22689}, {7747, 9117}, {7749, 9115}, {7870, 11308}, {7878, 11290}, {7922, 22489}, {8595, 33274}, {8836, 11004}, {11130, 16770}, {11132, 22900}, {11295, 22236}, {11299, 16962}, {11311, 21358}, {12355, 16629}, {15067, 34373}, {15765, 32421}, {18585, 32419}, {20416, 33476}, {21158, 33959}, {22513, 32552}, {22694, 25196}

X(34509) = midpoint of X(i) and X(j) for these {i,j}: {381, 5859}, {3105, 25195}
X(34509) = reflection of X(i) in X(j) for these {i,j}: {6299, 5613}, {6582, 6771}, {13083, 9763}, {13084, 33475}, {22911, 17}, {33459, 547}, {34508, 5}
X(34509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {13, 299, 3642}, {69, 18582, 623}, {599, 11305, 635}, {635, 5459, 11305}


X(34510) =  X(39)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    4*a^8 - 9*a^6*b^2 + 7*a^4*b^4 - 2*b^8 - 9*a^6*c^2 + 4*a^4*b^2*c^2 + 7*a^2*b^4*c^2 + 3*b^6*c^2 + 7*a^4*c^4 + 7*a^2*b^2*c^4 - 2*b^4*c^4 + 3*b^2*c^6 - 2*c^8 : :
X(34510) = 5 X[3] - 3 X[9774], 2 X[140] - 3 X[15810], X[382] - 3 X[10033], 3 X[598] - 5 X[1656], 3 X[3545] + X[14976], 5 X[5071] - X[19569], 2 X[7750] + X[14881], 2 X[7830] + X[32151]

X(34510) lies on these lines: {3, 6054}, {4, 7616}, {5, 3849}, {30, 5188}, {140, 15810}, {183, 22515}, {376, 2896}, {381, 1078}, {382, 10033}, {512, 15067}, {542, 7830}, {547, 7749}, {549, 626}, {550, 11645}, {598, 1656}, {2782, 7833}, {3095, 9939}, {3545, 14976}, {5071, 19569}, {6655, 11632}, {7750, 14881}, {7761, 12042}, {7801, 33813}, {7812, 11272}, {7841, 10104}, {7899, 15694}, {9996, 14907}, {11318, 34127}, {13464, 28562}, {14458, 15681}, {15092, 17004}

X(34510) = midpoint of X(i) and X(j) for these {i,j}: {381, 11057}, {3095, 9939}, {7810, 32152}, {14458, 15681}
X(34510) = reflection of X(i) in X(j) for these {i,j}: {7812, 11272}, {14537, 547}


X(34511) =  X(98)-OF-MOSES-STEINER-OSCULATORY-TRIANGLE

Barycentrics    a^4 - 4*a^2*b^2 + b^4 - 4*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(34511) = 2 X[3] - 3 X[7618], 2 X[3] + X[7758], 4 X[3] - 3 X[8182], X[3] - 3 X[11165], 4 X[3] - X[14023], X[4] - 4 X[7764], X[4] + 2 X[7781], X[4] + 3 X[9741], X[4] - 3 X[9770], 4 X[5] - 3 X[7615], 2 X[5] - 3 X[11184], X[20] + 2 X[7759], 2 X[39] + X[6309], 4 X[39] - X[31981], 4 X[140] - 3 X[7610], 2 X[140] - 3 X[12040], X[194] + 2 X[8149], 2 X[194] + X[18768], 5 X[631] - 6 X[7622], 5 X[631] - 2 X[7751], 12 X[1153] - 13 X[10303], 5 X[1656] - 6 X[9771], 7 X[3090] - 3 X[5485], 7 X[3090] - 6 X[7617], 5 X[3091] - 3 X[7620], 5 X[3091] - 6 X[8176], 5 X[3091] + 3 X[11148], X[3146] - 4 X[7843], X[3146] - 3 X[23334], 7 X[3523] - 6 X[5569], 7 X[3523] - 4 X[7780], 7 X[3523] - 3 X[9740], 11 X[3525] - 12 X[7619], 7 X[3526] - 6 X[15597], 3 X[3545] - 2 X[18546], 4 X[3628] - 3 X[16509], 7 X[3851] - 6 X[20112], 3 X[5054] - 2 X[13468], 3 X[5503] + X[23235], 3 X[5569] - 2 X[7780], 2 X[6309] + X[31981], 3 X[7618] + X[7758], 6 X[7618] - X[14023], 3 X[7622] - X[7751], 2 X[7758] + 3 X[8182], X[7758] + 6 X[11165], 2 X[7758] + X[14023], 2 X[7764] + X[7781], 4 X[7764] + 3 X[9741], 4 X[7764] - 3 X[9770], 2 X[7775] + 3 X[9741], 2 X[7775] - 3 X[9770], 4 X[7780] - 3 X[9740], 2 X[7781] - 3 X[9741], 2 X[7781] + 3 X[9770], 4 X[7843] - 3 X[23334], 4 X[8149] - X[18768], 2 X[8176] + X[11148], X[8182] - 4 X[11165], 3 X[8182] - X[14023], 12 X[11165] - X[14023]

X(34511) lies on the cubic K527 and these lines: {2, 39}, {3, 524}, {4, 543}, {5, 7615}, {6, 6390}, {20, 3849}, {30, 8716}, {32, 1992}, {69, 574}, {99, 7737}, {115, 1007}, {140, 7610}, {141, 5024}, {187, 193}, {230, 22253}, {315, 7783}, {316, 33192}, {325, 2549}, {376, 754}, {381, 9767}, {385, 21843}, {525, 5654}, {542, 9737}, {549, 8667}, {576, 19911}, {597, 7789}, {598, 7858}, {599, 3933}, {620, 7735}, {626, 7738}, {627, 9885}, {628, 9886}, {631, 7622}, {671, 7752}, {698, 14561}, {732, 11171}, {736, 7709}, {1153, 10303}, {1384, 3629}, {1656, 9771}, {1975, 2548}, {2021, 32451}, {3053, 15534}, {3090, 5485}, {3091, 7620}, {3095, 5969}, {3146, 7843}, {3523, 5569}, {3525, 7619}, {3526, 15597}, {3545, 18546}, {3552, 13571}, {3618, 7820}, {3628, 16509}, {3734, 7736}, {3761, 31497}, {3785, 7855}, {3793, 5210}, {3851, 20112}, {4045, 7908}, {5007, 5032}, {5054, 13468}, {5077, 7776}, {5206, 7890}, {5215, 32989}, {5254, 11318}, {5306, 11288}, {5319, 7807}, {5461, 7862}, {5475, 32815}, {5965, 9734}, {6179, 26613}, {6329, 22246}, {6389, 14961}, {6655, 32480}, {6656, 32821}, {6680, 33197}, {6776, 18860}, {6792, 14515}, {7745, 11159}, {7747, 15300}, {7748, 31173}, {7749, 23055}, {7753, 14033}, {7754, 22329}, {7755, 9167}, {7756, 7903}, {7760, 16925}, {7762, 8598}, {7765, 7888}, {7767, 15533}, {7768, 32965}, {7770, 32820}, {7772, 7863}, {7773, 8352}, {7777, 11185}, {7778, 8360}, {7779, 9939}, {7782, 7905}, {7785, 8591}, {7788, 8356}, {7791, 7796}, {7792, 8366}, {7794, 16043}, {7809, 33017}, {7811, 33008}, {7814, 14063}, {7818, 32986}, {7821, 32825}, {7823, 9855}, {7825, 32823}, {7826, 15515}, {7829, 14069}, {7830, 7916}, {7835, 16989}, {7837, 13586}, {7839, 7891}, {7847, 7871}, {7848, 10513}, {7849, 33202}, {7854, 15810}, {7860, 32997}, {7864, 7947}, {7866, 9607}, {7869, 32956}, {7873, 33023}, {7878, 14037}, {7902, 32951}, {7907, 8859}, {7941, 8597}, {7946, 33260}, {8362, 21358}, {8367, 31406}, {8584, 30435}, {8588, 11008}, {8589, 20080}, {9166, 32961}, {9300, 11286}, {9698, 17130}, {10519, 21163}, {11057, 33207}, {11164, 19687}, {11285, 31450}, {11317, 32819}, {11648, 16041}, {12150, 33255}, {13745, 19758}, {14357, 17708}, {14971, 32969}, {16044, 31417}, {16628, 22491}, {16629, 22492}, {17131, 34229}, {17346, 21937}, {21309, 32455}, {22562, 32492}, {22563, 32495}, {32824, 32971}

X(34511) = midpoint of X(i) and X(j) for these {i,j}: {6309, 13085}, {7620, 11148}, {7757, 9764}, {7775, 7781}, {8716, 9766}, {9741, 9770}
X(34511) = reflection of X(i) in X(j) for these {i,j}: {4, 7775}, {5485, 7617}, {7610, 12040}, {7615, 11184}, {7618, 11165}, {7620, 8176}, {7775, 7764}, {8182, 7618}, {8667, 549}, {9740, 5569}, {9890, 8724}, {13085, 39}, {31981, 13085}
X(34511) = barycentric product X(3266)*X(15268)
X(34511) = barycentric quotient X(15268)/X(111)
X(34511) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 3926, 7801}, {2, 5286, 7817}, {2, 7757, 7739}, {2, 7801, 7795}, {2, 32836, 9466}, {3, 7758, 14023}, {4, 9770, 7775}, {32, 2482, 32985}, {39, 3926, 7795}, {39, 6309, 31981}, {39, 7801, 2}, {69, 33215, 7810}, {99, 7774, 7737}, {99, 7812, 33007}, {194, 7763, 3767}, {194, 8149, 18768}, {325, 31859, 2549}, {574, 7810, 33215}, {574, 7813, 69}, {597, 7789, 33237}, {599, 5013, 8359}, {599, 8359, 7800}, {620, 7798, 7735}, {671, 7752, 33006}, {1975, 11163, 8370}, {1992, 6337, 32985}, {1992, 32985, 32}, {3629, 32459, 1384}, {3734, 14148, 32817}, {3788, 7817, 2}, {3788, 32450, 5286}, {3933, 5013, 7800}, {3933, 8359, 599}, {5210, 6144, 3793}, {5254, 22110, 11318}, {5286, 32831, 3788}, {6294, 6581, 194}, {6337, 32985, 2482}, {6392, 32829, 7746}, {7736, 32817, 3734}, {7738, 32818, 626}, {7756, 7903, 32006}, {7757, 7799, 2}, {7757, 7870, 7827}, {7764, 7775, 9770}, {7764, 7781, 4}, {7765, 7888, 14064}, {7772, 7863, 14001}, {7774, 33007, 7812}, {7777, 11185, 31415}, {7782, 7905, 20065}, {7783, 7840, 7833}, {7783, 7906, 315}, {7799, 7827, 7870}, {7812, 33007, 7737}, {7827, 7870, 2}, {7833, 7840, 315}, {7833, 7906, 7840}, {7854, 31652, 32990}, {8370, 11163, 2548}, {9605, 33237, 597}, {9698, 17130, 32968}, {31400, 32830, 3934}, {32825, 32974, 7821}


X(34512) =  X(2)-OF-MOSES-STEINER-REFLECTION TRIANGLE

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

X(34512) lies on these lines: {2, 353}, {5, 14867}, {6, 23297}, {125, 13234}, {126, 12494}, {140, 31731}, {141, 858}, {468, 9128}, {524, 6032}, {625, 5167}, {1656, 31608}, {3090, 31962}, {3094, 31125}, {3628, 31727}, {3634, 31740}, {3849, 13378}, {3850, 31827}, {5108, 11178}, {5133, 11226}, {5462, 31733}, {7617, 9169}, {7697, 9148}, {10163, 11645}, {10173, 30516}, {11793, 31739}, {15271, 33973}


X(34513) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: MCKAY TO MOSES-STEINER-REFLECTION

Barycentrics    a^2*(3*a^8 - 7*a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 - 2*b^8 - 7*a^6*c^2 + 4*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 + 3*a^2*c^6 + b^2*c^6 - 2*c^8) : :
X(34513) = 5 X[3] + X[11456], 5 X[6800] - X[11456], X[7502] + 2 X[18475], 2 X[7555] + X[13352]

X(34513) lies on these lines: {3, 74}, {5, 15448}, {23, 14805}, {24, 13363}, {26, 10610}, {30, 13394}, {154, 15060}, {182, 7575}, {373, 12106}, {511, 7502}, {549, 1503}, {567, 7556}, {568, 1199}, {569, 12107}, {1493, 17834}, {1658, 9730}, {2070, 5640}, {3431, 7492}, {3534, 6030}, {3581, 11003}, {3796, 18324}, {5050, 5946}, {5085, 6644}, {6101, 19357}, {6639, 13470}, {7512, 13340}, {7516, 17821}, {7525, 13367}, {7526, 15811}, {7555, 13352}, {7691, 9704}, {9715, 10263}, {10170, 10282}, {10182, 15113}, {10984, 15331}, {11179, 15361}, {11935, 23061}, {12105, 34417}, {12236, 15074}, {12279, 18364}, {14915, 18570}, {16261, 26881}, {22115, 33884}

X(34513) = midpoint of X(3) and X(6800)
X(34513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 110, 33533}, {3, 9707, 11591}, {3, 32609, 7998}, {7998, 11464, 32609}


X(34514) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO MOSES-STEINER-REFLECTION

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 - a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - 2*a^2*b^4*c^4 - 2*b^6*c^4 - a^4*c^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(34514) = 3 X[381] - X[11456], 5 X[1656] - 3 X[6800], 4 X[3628] - 3 X[13394]

X(34514) lies on these lines: {3, 18432}, {4, 94}, {5, 182}, {20, 6288}, {24, 13561}, {30, 343}, {52, 18356}, {68, 10263}, {69, 6101}, {125, 12106}, {156, 1594}, {316, 18321}, {378, 30522}, {381, 5422}, {394, 31181}, {546, 18914}, {567, 5169}, {578, 33332}, {621, 34389}, {622, 34390}, {1154, 11442}, {1181, 7564}, {1209, 7525}, {1352, 14791}, {1510, 14592}, {1511, 18281}, {1531, 11572}, {1595, 12370}, {1656, 6800}, {1853, 6644}, {1899, 5946}, {2070, 23293}, {3146, 18387}, {3153, 18435}, {3357, 14677}, {3410, 23039}, {3580, 7540}, {3627, 9927}, {3628, 13394}, {3629, 32358}, {3763, 7516}, {3845, 18390}, {4846, 6145}, {5097, 11232}, {5189, 13340}, {5448, 6053}, {5449, 13419}, {5576, 14389}, {5609, 5654}, {5876, 18569}, {5944, 9833}, {6240, 32138}, {6699, 20299}, {7391, 13391}, {7394, 13364}, {7502, 21243}, {7503, 13470}, {7507, 32139}, {7528, 15026}, {7530, 14852}, {7544, 12006}, {7558, 13565}, {7565, 15032}, {7574, 11459}, {7577, 10540}, {7703, 11464}, {7768, 14387}, {8705, 34507}, {10024, 16659}, {10095, 18912}, {10170, 18553}, {10201, 31383}, {10224, 10272}, {10254, 14157}, {10255, 27866}, {10264, 11438}, {10610, 32354}, {11064, 12134}, {11250, 16163}, {11411, 31815}, {11439, 18394}, {11455, 18392}, {11457, 13630}, {11470, 13292}, {11472, 18405}, {11799, 16658}, {11819, 12359}, {12111, 31724}, {12289, 14130}, {13339, 14789}, {13352, 32423}, {13363, 18911}, {13406, 26883}, {13490, 13567}, {13621, 26917}, {13851, 16194}, {14002, 15027}, {15060, 18531}, {15062, 18562}, {15068, 18440}, {15305, 18403}, {15646, 23329}, {15761, 16655}, {16000, 18550}, {18027, 32002}, {18396, 31861}, {18400, 18570}, {18420, 32064}, {18488, 21659}, {22115, 31074}, {23236, 31857}, {23323, 23324}

X(34514) = midpoint of X(i) and X(j) for these {i,j}: {{11442, 31723}, {11550, 18474}
X(34514) = reflection of X(7502) in X(21243)
X(34514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 3448, 568}, {4, 25738, 143}, {4, 32140, 6102}, {1352, 14791, 15067}, {1899, 11818, 5946}, {5576, 34224, 32046}, {7528, 18952, 15026}, {11455, 18392, 31726}, {11572, 12162, 18377}, {18379, 32137, 4}


X(34515) =  X(2)X(3591)∩X(6)X(25)

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

X(34515) lies on these lines: {2, 3591}, {6, 25}, {22, 3594}, {39, 21097}, {371, 16419}, {486, 11548}, {1151, 7485}, {1152, 6636}, {1588, 8889}, {3070, 7408}, {3071, 7378}, {3311, 8855}, {5020, 6419}, {6417, 8854}, {6420, 9909}, {6425, 7484}, {6428, 20850}, {7392, 32787}, {7398, 19054}, {7494, 32788}, {7499, 13847}, {7584, 18289}, {8280, 13785}, {8281, 18510}, {10565, 19053}, {18290, 19116}

X(34515) = {X(6),X(25)}-harmonic conjugate of X(34516)


X(34516) =  X(2)X(3590)∩X(6)X(25)

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

X(34516) lies on these lines: {2, 3590}, {6, 25}, {22, 3592}, {32, 21097}, {372, 16419}, {485, 11548}, {1151, 6636}, {1152, 7485}, {1587, 8889}, {3070, 7378}, {3071, 7408}, {3312, 8854}, {5020, 6420}, {6418, 8855}, {6419, 9909}, {6426, 7484}, {6427, 20850}, {7392, 32788}, {7398, 19053}, {7494, 32787}, {7499, 13846}, {7583, 18290}, {8280, 18512}, {8281, 13665}, {10565, 19054}, {18289, 19117}

X(34516) = {X(6),X(25)}-harmonic conjugate of X(34515)


X(34517) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: MEDIAL TO ORTHIC INCONIC ORTHOPTIC

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

X(34517) lies on these lines: {2, 14908}, {5, 5181}, {25, 7664}, {30, 114}, {126, 127}, {625, 2393}, {626, 2882}, {1370, 31128}, {7775, 19136}, {9775, 10718}, {10602, 11318}

X(34517) = complement of X(14908)
X(34517) = medial-isogonal conjugate of X(14961)
X(34517) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 14961}, {4, 16611}, {19, 3291}, {75, 5159}, {92, 524}, {162, 2492}, {264, 4892}, {273, 17070}, {468, 37}, {524, 1214}, {690, 16573}, {811, 690}, {896, 216}, {1969, 625}, {3266, 18589}, {3292, 828}, {4062, 18591}, {4235, 14838}, {5203, 16605}, {7181, 17102}, {14210, 3}, {14273, 16592}, {14417, 16595}, {18022, 21256}, {24006, 1648}, {34336, 16597}


X(34518) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ORTHIC INCONIC ORTHOPTIC

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

X(34518) lies on these lines: {2, 14908}, {4, 14984}, {25, 7665}, {30, 147}, {315, 670}, {316, 2393}, {1370, 13219}, {7812, 19136}, {7841, 10602}, {8263, 8370}, {9517, 34163}, {25052, 32006}

X(34518) = anticomplement of X(14908)
X(34518) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {4, 17497}, {75, 858}, {92, 524}, {264, 17491}, {273, 4442}, {286, 17162}, {468, 192}, {524, 6360}, {811, 690}, {823, 9979}, {896, 3164}, {1969, 316}, {3112, 11416}, {3266, 4329}, {4062, 18666}, {4235, 4560}, {6629, 20222}, {14210, 20}, {14273, 21220}, {16741, 17134}, {18022, 21298}, {24039, 6563}


X(34519) =  PARALOGIC CENTER OF THESE TRIANGLES: PARRY TO ORTHIC INCONIC ORTHOPTIC

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

X(34519) lies on these lines: {3, 9125}, {22, 9123}, {23, 385}, {25, 2793}, {351, 6091}, {690, 2936}, {8644, 30209}, {9156, 9157}

X(34519) = crosssum of X(i) and X(j) for these (i,j): {512, 24855}, {523, 19510}, {690, 30739}, {5159, 30209}




leftri  Vu points: X(34520)-X(34544)  rightri

This preamble and centers X(34520)-X(34544) are based on notes contributed by Vu Thanh Tung and Vu Quoc My, October 24, 2019.

Let P = p:q:r (barycentrics) be a point in the plane of a triangle ABC. Let A' be the point, other than P, in which the line AP meets the circle (PBC), and define B' and C' cyclically; the triangle A'B'C' is called the circlecevian triangle of P with respect to triangle ABC by Floor van Lamoen ( Hyacinthos # 10039).

Let TA and TA' be the tangents to (PBC) at P and A', respectively.
Let A1 = TA∩BC and A2 = TA'∩BC.
Define B1 and C1 cyclically, and define B2 and C2 cyclically.
Then A1,B1,C1 are collinear and A2,B2,C2 are collinear.

The trilinear pole of the line B1C1 is given by Q1(P) = 1/(-a^2 q r + c^2 q (q + r) + b^2 r (q + r)) : : , here named the 1st Vu point of P.

The trilinear pole of the line B2C2 is given by Q2(P) = p^2 (-a^2 q r + c^2 q (q + r) + b^2 r (q + r)) : : , here named the 2nd Vu point of P.

Examples:

Q1(X(1)) = X(57) and Q2(X(1)) = X(9)
Q1(X(2)) = X(598) and Q2(X(2)) = X(599)
Q1(X(3)) = X(2) and Q2(X(3)) = X(32)
Q1X(4)) = X(2052) and Q2(X(4)) = X(6)

If P lies on the circumcircle, then Q1(P) is the trilinear pole of the tangent to the circumcircle at P, and Q2(P) = X(6). (Randy Hutson, October 24, 2019)

Q2(P) is the X(2)-Ceva conjugate of the Dao image of P. (Randy Hutson, November 17, 2019)

underbar



X(34520) =  2ND VU POINT OF X(5)

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

X(34520) lies on these lines: {6, 17}, {577, 2120}

X(34520) = X(2)-Ceva conjugate of X(15345)


X(34521) =  1ST VU POINT OF X(7)

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

X(34521) lies on these lines: {1088, 1323}


X(34522) =  2ND VU POINT OF X(7)

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

X(34522) lies on these lines: {1, 6}, {2, 664}, {3, 5011}, {20, 5829}, {41, 34471}, {55, 2170}, {56, 17451}, {57, 15855}, {65, 5022}, {78, 4875}, {101, 10246}, {142, 1323}, {169, 1385}, {241, 3306}, {281, 17398}, {294, 2320}, {348, 21258}, {583, 15830}, {672, 2099}, {910, 3576}, {999, 15288}, {1015, 16588}, {1125, 5199}, {1334, 2098}, {1388, 9310}, {1442, 25878}, {1482, 3730}, {1572, 3052}, {2082, 2646}, {2275, 17054}, {2278, 2301}, {2340, 3711}, {2550, 11200}, {2886, 24247}, {3008, 24281}, {3086, 21049}, {3208, 10912}, {3218, 5228}, {3304, 21808}, {3509, 11194}, {3616, 6554}, {3624, 23058}, {3689, 28043}, {3693, 3872}, {3735, 16579}, {3752, 9592}, {3816, 6506}, {3897, 33950}, {3913, 4051}, {3959, 5013}, {4360, 26059}, {4423, 20277}, {4513, 4861}, {4515, 4853}, {4534, 5432}, {4860, 32578}, {5179, 5886}, {5273, 20182}, {5275, 21332}, {5308, 5328}, {5527, 11495}, {5603, 17747}, {5731, 5819}, {5781, 18444}, {6173, 21314}, {6180, 24554}, {6706, 9312}, {7223, 30949}, {7960, 17365}, {7982, 21872}, {8074, 10165}, {8273, 20838}, {8555, 11108}, {8558, 16418}, {9619, 16583}, {10025, 31169}, {11051, 11227}, {12702, 24047}, {15817, 21773}, {16826, 30854}, {17014, 24597}, {17045, 27509}, {17095, 26531}, {17136, 24596}, {17325, 26932}, {17394, 27420}, {17732, 22791}, {18230, 31721}, {18493, 24045}, {19722, 28950}, {24203, 24352}, {26526, 27187}, {26659, 27340}, {27384, 28639}, {29571, 30826}

X(34522) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15346}, {14074, 17072} X(34522) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 15346}, {6173, 4860}
X(34522) = X(41)-isoconjugate of X(18810)
X(34522) = crosspoint of X(5231) and X(6173)
X(34522) = crossdifference of every pair of points on line {513, 6139}
X(34522) = barycentric product X(i)*X(j) for these {i,j}: {1, 5231}, {8, 4860}, {9, 6173}, {85, 32578}, {200, 21314}, {4554, 17425}
X(34522) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 18810}, {4860, 7}, {5231, 75}, {6173, 85}, {17425, 650}, {21314, 1088}, {32578, 9}
X(34522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 9, 6603}, {1, 1212, 220}, {9, 6603, 220}, {169, 1385, 3207}, {1212, 6603, 9}, {2082, 2646, 4258}, {4861, 25082, 4513}, {5239, 5240, 1001}, {30556, 30557, 15254}


X(34523) =  1ST VU POINT OF X(8)

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

X(34523) lies on these lines: {312, 3911}, {341, 519}, {4358, 27130}

X(34523) = isogonal conjugate of X(34543)
X(34523) = cevapoint of X(312) and X(18743)
X(34523) = trilinear pole of line X(900)X(4397)


X(34524) =  2ND VU POINT OF X(8)

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

X(34524) lies on these lines: {1, 6}, {281, 17340}, {346, 1146}, {480, 2310}, {1329, 24248}, {3161, 6554}, {4012, 4953}, {4422, 27509}, {4862, 25580}, {5328, 17595}, {6180, 26669}, {8169, 17596}, {8545, 25067}, {17261, 30854}, {17267, 26932}, {17281, 20262}, {17336, 27420}, {17351, 27384}, {17354, 27547}, {17594, 18227}, {24231, 25681}, {25878, 29007}, {26635, 27065}, {27522, 27525}

X(34524) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15347}, {30236, 17072}
X(34524) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 15347}, {30827, 2098}, {31343, 3900}
X(34524) = X(604)-isoconjugate of X(18811)
X(34524) = barycentric product X(i)*X(j) for these {i,j}: {8, 2098}, {9, 30827}, {200, 4862}, {341, 32577}, {646, 17424}
X(34524) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 18811}, {2098, 7}, {4862, 1088}, {17424, 3669}, {30827, 85}, {32577, 269}
X(34524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {9, 219, 16885}, {9, 2324, 44}, {9, 3731, 1212}, {37, 44, 3554}, {30556, 30557, 11260}


X(34525) =  1ST VU POINT OF X(9)

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

X(34525) lies on these lines: {9, 2078}, {200, 5526}, {346, 3935}


X(34526) =  2ND VU POINT OF X(9)

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

X(34526) lies on these lines: {1, 6}, {78, 6554}, {200, 1146}, {223, 31142}, {241, 25934}, {326, 27420}, {908, 948}, {910, 6282}, {1467, 5022}, {1572, 16283}, {1802, 2082}, {2170, 6602}, {3057, 7368}, {3452, 24249}, {3601, 32561}, {5179, 5720}, {6506, 30827}, {7079, 33299}, {15288, 24929}

X(34526) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15348}, {30237, 17072}
X(34526) = X(2)-Ceva conjugate of X(15348)
X(34526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2324, 6603}, {1212, 6603, 6}


X(34527) =  1ST VU POINT OF X(10)

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

X(34527) lies on these lines: {2, 18654}, {8, 6535}, {29, 7140}, {312, 17299}, {333, 594}, {2994, 17350}, {4997, 34064}, {6542, 28654}


X(34528) =  2ND VU POINT OF X(10)

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

X(34528) lies on these lines: {2, 6}, {12, 594}, {37, 3178}, {75, 20337}, {115, 2321}, {313, 338}, {1330, 2305}, {2092, 3454}, {2887, 21857}, {3943, 27558}, {3948, 17788}, {5257, 6537}, {8287, 17231}, {8818, 17281}, {9560, 15349}, {15526, 18589}, {16777, 23905}, {17058, 21255}, {17233, 23947}, {17275, 28628}, {17314, 23903}, {17362, 27368}, {20461, 33064}, {21033, 23921}, {21711, 21873}, {33156, 33329}

X(34528) = isotomic conjugate of the isogonal conjugate of X(9560)
X(34528) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 15349}, {6010, 4369}
X(34528) = X(2)-Ceva conjugate of X(15349)
X(34528) = X(1333)-isoconjugate of X(18812)
X(34528) = barycentric product X(i)*X(j) for these {i,j}: {76, 9560}, {594, 26840}, {1978, 17411}
X(34528) = barycentric quotient X(i)/X(j) for these {i,j}: {10, 18812}, {9560, 6}, {17411, 649}, {26840, 1509}
X(34528) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 24935, 17398}, {594, 5949, 23897}, {1213, 10026, 6}, {3936, 27707, 17300}, {3949, 16886, 594}, {4053, 20654, 594}


X(34529) =  1ST VU POINT OF X(11)

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

X(34529) lies on these lines: {149, 4564}, {484, 516}, {17484, 30807}


X(34530) =  2ND VU POINT OF X(11)

Barycentrics    (b - c)^4*(-a + b + c)^3*(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(34530) lies on this line: {9, 2957}


X(34531) =  1ST VU POINT OF X(12)

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

X(34531) lies on this lines: {7161, 10039}


X(34532) =  2ND VU POINT OF X(12)

Barycentrics    (b + c)^4*(a^3 + a^2*b - a*b^2 - b^3 + a^2*c + 3*a*b*c + b^2*c - a*c^2 + b*c^2 - c^3)*(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(34532) lies on these lines: {6, 24880}, {3336, 24251}, {5949, 11263}


X(34533) =  1ST VU POINT OF X(61)

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

X(34533) lies on these lines: {2963, 18813}, {2981, 19781}, {3130, 21461}

X(34533) = isogonal conjugate of X(34540)


X(34534) =  1ST VU POINT OF X(62)

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

X(34534) lies on these lines: {2963, 18814}, {3129, 21462}, {6151, 19780}

X(34534) = isogonal conjugate of X(34541)


X(34535) =  1ST VU POINT OF X(80)

Barycentrics    b*(-a + b - c)*(a + b - c)*c*(a^2 - a*b + b^2 - c^2)^2*(-a^2 + b^2 + a*c - c^2)^2 : :
Barycentrics    (Cos[A]-1) / (1-2 Cos[A])^2 : :

X(34535) lies on these lines: {80, 517}, {655, 16548}, {908, 18359}, {1465, 2006}, {2222, 3724}, {3911, 18815}, {5219, 14628}, {20920, 21587}

X(34535) = isogonal conjugate of X(34544)
X(34535) = X(i)-cross conjugate of X(j) for these (i,j): {6, 2166}, {661, 2222}, {1953, 759}, {14584, 18815}
X(34535) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34544}, {2, 215}, {6, 4996}, {36, 2323}, {758, 4282}, {1252, 3025}, {1983, 3738}, {2150, 4736}, {2361, 3218}, {3028, 7054}, {4511, 7113}, {4585, 8648}
X(34535) = barycentric product X(i)*X(j) for these {i,j}: {80, 18815}, {1411, 20566}, {2006, 18359}, {4858, 23592}
X(34535) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4996}, {6, 34544}, {12, 4736}, {31, 215}, {80, 4511}, {244, 3025}, {655, 4585}, {1254, 3028}, {1411, 36}, {2006, 3218}, {2161, 2323}, {6187, 2361}, {14584, 214}, {18359, 32851}, {18815, 320}, {23592, 4564}, {32675, 1983}, {34079, 4282}


X(34536) =  1ST VU POINT OF X(98)

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

X(34536) lies on the cubic K380 and these lines: {2, 14382}, {98, 237}, {248, 290}, {287, 3978}, {297, 22456}, {419, 685}, {850, 9476}, {1316, 14265}, {1821, 16609}, {20021, 20026}

X(34536) = isogonal conjugate of X(11672)
X(34536) = isotomic conjugate of X(36790)
X(34536) = polar conjugate of X(2967)
X(34536) = trilinear pole of line X(98)X(804) (the tangent to the circumcircle at X(98))
X(34536) = barycentric square of X(1821)
X(34536) = X(i)-cross conjugate of X(j) for these (i,j): {6, 98}, {523, 22456}, {2422, 18858}, {3050, 2715}, {3288, 6037}, {14265, 290}
X(34536) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11672}, {6, 23996}, {9, 1355}, {42, 16725}, {48, 2967}, {57, 7062}, {75, 9419}, {237, 1959}, {240, 3289}, {325, 9417}, {511, 1755}, {560, 32458}, {798, 15631}, {1821, 23611}, {1910, 23098}, {3569, 23997}, {5360, 17209}, {17462, 34157}
X(34536) = cevapoint of X(i) and X(j) for these (i,j): {6, 98}, {290, 14382}
X(34536) = crossdifference of every pair of points on line {23611, 33569}
X(34536) = barycentric product X(i)*X(j) for these {i,j}: {98, 290}, {287, 16081}, {879, 22456}, {1821, 1821}, {1976, 18024}, {14295, 18858}
X(34536) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 23996}, {4, 2967}, {6, 11672}, {32, 9419}, {55, 7062}, {56, 1355}, {76, 32458}, {81, 16725}, {98, 511}, {99, 15631}, {237, 23611}, {248, 3289}, {290, 325}, {511, 23098}, {685, 4230}, {879, 684}, {1821, 1959}, {1910, 1755}, {1976, 237}, {2065, 34157}, {2395, 3569}, {2422, 2491}, {2715, 14966}, {2782, 6072}, {2966, 2421}, {3288, 33569}, {5967, 9155}, {6531, 232}, {9154, 5968}, {14265, 114}, {14382, 5976}, {14601, 9418}, {16081, 297}, {18858, 805}, {22456, 877}, {34238, 14251}


X(34537) =  1ST VU POINT OF X(99)

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

Line X(99)X(670) (the trilinear polar of X(34537)) is the locus of the trilinear pole of the tangent at P to hyperbola {{A,B,C,X(6),P}}, as P moves on line X(2)X(6). The line X(99)X(670) is also the tangent to the circumcircle at X(99), and the perspectrix of any pair of {3rd, 5th and 6th Brocard triangles}. (Randy Hutson, October 24, 2019)

X(34537) lies on these lines: {76, 5108}, {99, 669}, {305, 31614}, {385, 3266}, {419, 18020}, {670, 850}, {689, 805}, {799, 4369}, {873, 1215}, {880, 5468}, {2106, 4601}, {3005, 18829}, {3231, 3978}, {4039, 4600}, {4563, 4609}, {5970, 31128}, {6064, 18021}, {8024, 31632}, {8033, 25819}, {10754, 34087}

X(34537) = isogonal conjugate of X(1084)
X(34537) = isotomic conjugate of X(3124)
X(34537) = polar conjugate of X(2971)
X(34537) = crosssum of X(9427) and X(23216)
X(34537) = trilinear pole of line {99, 670}
X(34537) = crossdifference of every pair of points on line {1645, 23099}
X(34537) = cevapoint of circumcircle-intercepts of line X(2)X(6)
X(34537) = X(2143)-complementary conjugate of X(8287)
X(34537) = X(i)-cross conjugate of X(j) for these (i,j): {2, 689}, {6, 99}, {76, 670}, {183, 9063}, {230, 22456}, {305, 4609}, {732, 18829}, {1509, 4623}, {1611, 107}, {1613, 110}, {3231, 9150}, {3763, 6572}, {3770, 668}, {4074, 4576}, {4563, 31614}, {4601, 24037}, {7304, 4610}, {7754, 648}, {7760, 4577}, {8033, 799}, {17499, 190}, {21001, 3222}, {21431, 75}, {21779, 100}
X(34537) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1084}, {2, 4117}, {9, 1356}, {31, 3124}, {32, 2643}, {42, 3121}, {48, 2971}, {57, 7063}, {75, 9427}, {92, 23216}, {115, 560}, {163, 22260}, {213, 3122}, {244, 7109}, {338, 1917}, {512, 798}, {523, 1924}, {661, 669}, {662, 23099}, {667, 4079}, {756, 1977}, {799, 23610}, {810, 2489}, {872, 1015}, {904, 21823}, {923, 21906}, {1109, 1501}, {1365, 9447}, {1500, 3248}, {1577, 9426}, {1755, 15630}, {1918, 3125}, {1919, 4705}, {1923, 34294}, {1967, 2086}, {1973, 20975}, {1974, 3708}, {1980, 4024}, {2084, 18105}, {2205, 3120}, {2206, 21833}, {3402, 6784}, {7104, 21725}, {7148, 21762}, {8754, 9247}, {9233, 23994}, {9494, 18070}, {21835, 23493}
X(34537) = cevapoint of X(i) and X(j) for these (i,j): {2, 4576}, {6, 99}, {75, 21604}, {76, 670}, {110, 1627}, {305, 4563}, {799, 873}, {1509, 4623}, {2421, 5976}, {4631, 18021}, {5468, 31128}
X(34537) = barycentric product X(i)*X(j) for these {i,j}: {75, 24037}, {76, 4590}, {99, 670}, {110, 4609}, {249, 1502}, {274, 4601}, {305, 18020}, {310, 4600}, {561, 24041}, {662, 4602}, {668, 4623}, {689, 4576}, {799, 799}, {850, 31614}, {873, 7035}, {880, 18829}, {886, 23342}, {1101, 1928}, {1509, 31625}, {1978, 4610}, {3596, 7340}, {4554, 4631}, {4563, 6331}, {4567, 6385}, {4620, 28660}, {4625, 7257}, {4635, 7258}, {4998, 18021}, {6063, 6064}
X(34537) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 3124}, {4, 2971}, {6, 1084}, {31, 4117}, {32, 9427}, {55, 7063}, {56, 1356}, {69, 20975}, {75, 2643}, {76, 115}, {81, 3121}, {86, 3122}, {98, 15630}, {99, 512}, {110, 669}, {163, 1924}, {183, 6784}, {184, 23216}, {190, 4079}, {249, 32}, {250, 1974}, {261, 3271}, {264, 8754}, {274, 3125}, {300, 30452}, {301, 30453}, {304, 3708}, {305, 125}, {308, 34294}, {310, 3120}, {313, 21043}, {314, 4516}, {321, 21833}, {385, 2086}, {512, 23099}, {523, 22260}, {524, 21906}, {552, 1357}, {561, 1109}, {593, 1977}, {645, 3709}, {648, 2489}, {662, 798}, {668, 4705}, {669, 23610}, {670, 523}, {757, 3248}, {765, 872}, {799, 661}, {805, 881}, {850, 8029}, {873, 244}, {874, 4155}, {877, 17994}, {880, 804}, {887, 33918}, {892, 9178}, {894, 21823}, {1016, 1500}, {1101, 560}, {1252, 7109}, {1502, 338}, {1509, 1015}, {1576, 9426}, {1634, 688}, {1909, 21725}, {1928, 23994}, {1978, 4024}, {2396, 3569}, {2407, 14398}, {2421, 2491}, {2782, 6071}, {2966, 2422}, {3231, 1645}, {3261, 21131}, {3266, 1648}, {3596, 4092}, {3926, 3269}, {4076, 7064}, {4176, 2972}, {4427, 8663}, {4556, 1919}, {4558, 3049}, {4563, 647}, {4567, 213}, {4570, 1918}, {4573, 7180}, {4576, 3005}, {4577, 18105}, {4590, 6}, {4592, 810}, {4600, 42}, {4601, 37}, {4602, 1577}, {4609, 850}, {4610, 649}, {4612, 3063}, {4616, 7250}, {4620, 1400}, {4623, 513}, {4625, 4017}, {4631, 650}, {4635, 7216}, {4998, 181}, {5118, 887}, {5383, 6378}, {5468, 351}, {5976, 2679}, {6035, 14998}, {6063, 1365}, {6064, 55}, {6331, 2501}, {6385, 16732}, {6386, 4036}, {7035, 756}, {7058, 14936}, {7192, 8034}, {7256, 4524}, {7257, 4041}, {7258, 4171}, {7304, 6377}, {7340, 56}, {7799, 2088}, {8033, 16592}, {9146, 17414}, {9182, 9171}, {9464, 8288}, {10330, 8664}, {10411, 14270}, {11059, 6791}, {12833, 34347}, {14089, 14090}, {14587, 14573}, {14999, 6041}, {17103, 4128}, {17929, 18002}, {17930, 18001}, {17931, 18000}, {17932, 878}, {17933, 17992}, {17934, 17990}, {17935, 17989}, {17937, 17999}, {17941, 5027}, {18020, 25}, {18021, 11}, {18022, 2970}, {18829, 882}, {23106, 14444}, {23342, 888}, {23357, 1501}, {23582, 2207}, {23963, 9233}, {23995, 1917}, {23999, 1096}, {24037, 1}, {24039, 2642}, {24041, 31}, {27644, 21835}, {28660, 21044}, {31614, 110}, {31625, 594}, {31632, 20998}, {33769, 7668}, {33939, 21824}, {34016, 20982}, {34245, 9135}, {34384, 8901}
X(34537) = barycentric square of X(799)


X(34538) =  1ST VU POINT OF X(107)

Barycentrics    (a - b)^2*b^2*(a + b)^2*(a - c)^2*c^2*(a + c)^2*(a^2 + b^2 - c^2)^4*(a^2 - b^2 + c^2)^4 : :
Barycentrics    a^2/(sec^2 B - sec^2 C)^2 : :

Line X(107)X(1624) (the trilinear polar of X(34538)) is the locus of the trilinear pole of the tangent at P to hyperbola {{A,B,C,X(6),P}}, as P moves on the van Aubel line. (Randy Hutson, October 24, 2019)

X(34538) lies on these lines: {401, 23582}, {1093, 32545}, {14157, 32230}, {14249, 18338}, {14618, 15352}

X(34538) = isogonal conjugate of X(35071)
X(34538) = isogonal conjugate of the complement of X(6528)
X(34538) = isotomic conjugate of the isogonal conjugate of X(23590)
X(34538) = polar conjugate of X(2972)
X(34538) = trilinear pole of line X(107)X(1624) (the tangent to the circumcircle at X(107))
X(34538) = barycentric square of X(823)
X(34538) = X(i)-cross conjugate of X(j) for these (i,j): {6, 107}, {1498, 99}, {1503, 22456}, {2052, 15352}, {5596, 689}, {14249, 6528}, {17849, 925}, {32445, 112}
X(34538) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1363}, {32, 24020}, {42, 16730}, {48, 2972}, {57, 7065}, {125, 4100}, {255, 3269}, {520, 822}, {560, 23974}, {577, 2632}, {656, 32320}, {823, 23613}, {1092, 3708}, {6507, 20975}, {14585, 17879}, {20902, 23606}, {23103, 24019}
X(34538) = cevapoint of X(i) and X(j) for these (i,j): {6, 107}, {112, 1614}, {648, 15466}, {1093, 6529}, {2052, 15352}, {23590, 32230}
X(34538) = cevapoint of circumcircle intercepts of van Aubel line
X(34538) = crossdifference of every pair of points on line {23613, 33571}
X(34538) = barycentric product X(i)*X(j) for these {i,j}: {75, 24021}, {76, 23590}, {107, 6528}, {158, 23999}, {264, 32230}, {561, 24022}, {648, 15352}, {823, 823}, {1093, 18020}, {1502, 23975}, {2052, 23582}, {6331, 6529}, {18027, 23964}
X(34538) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 2972}, {55, 7065}, {56, 1363}, {75, 24020}, {76, 23974}, {81, 16730}, {107, 520}, {112, 32320}, {158, 2632}, {250, 1092}, {393, 3269}, {520, 23103}, {823, 24018}, {1093, 125}, {2052, 15526}, {3331, 33571}, {6331, 4143}, {6520, 3708}, {6521, 20902}, {6524, 20975}, {6528, 3265}, {6529, 647}, {14249, 122}, {14618, 23616}, {15352, 525}, {15384, 14379}, {15742, 4158}, {18020, 3964}, {23582, 394}, {23590, 6}, {23964, 577}, {23975, 32}, {23999, 326}, {24000, 255}, {24019, 822}, {24021, 1}, {24022, 31}, {24033, 7138}, {32230, 3}, {32646, 2430}


X(34539) =  1ST VU POINT OF X(691)

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

X(34539) lies on these lines: {110, 9171}, {249, 2502}, {351, 691}, {671, 1641}, {892, 9979}, {9129, 14246}

X(34539) = isogonal conjugate of X(23992)
X(34539) = isogonal conjugate of the complement of X(892)
X(34539) = cevapoint of X(i) and X(j) for these (i,j): {6, 691}, {110, 111}
X(34539) = trilinear pole of line {691, 5467} (the tangent to circumcircle at X(691)
X(34539) = crossdifference of every pair of points on line {14443, 14444}
X(34539) = X(i)-cross conjugate of X(j) for these (i,j): {6, 691}, {2930, 99}, {14246, 892}, {16175, 670}, {20998, 111}
X(34539) = barycentric square of isogonal conjugate of X(2642)
X(34539) = barycentric square of X(36085)
X(34539) = barycentric product X(i)*X(j) for these {i,j}: {691, 892}, {4590, 10630}, {15398, 18020}
X(34539) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 23992}, {110, 1649}, {111, 1648}, {187, 14444}, {249, 2482}, {250, 5095}, {512, 14443}, {691, 690}, {1383, 20382}, {5467, 33915}, {9178, 33919}, {10630, 115}, {14246, 5099}, {15398, 125}, {17993, 14423}, {18020, 34336}, {23348, 33921}, {24041, 24038}, {32729, 351}, {32740, 21906}
X(34539) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23992}, {661, 1649}, {662, 14443}, {690, 2642}, {896, 1648}, {897, 14444}, {2482, 2643}, {3124, 24038}, {3708, 5095}, {14210, 21906}, {23889, 33919}, {23894, 33915}


X(34540) =  2ND VU POINT OF X(17)

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

X(34540) lies on these lines: {2, 6}, {15, 633}, {16, 627}, {115, 11122}, {148, 14904}, {338, 34389}, {383, 18358}, {470, 11409}, {574, 11133}, {616, 7898}, {617, 618}, {622, 623}, {629, 22901}, {631, 5872}, {634, 7912}, {636, 16967}, {1080, 33878}, {2993, 11130}, {3091, 5864}, {3098, 5617}, {3104, 18581}, {3522, 5868}, {3642, 10646}, {3643, 16809}, {5092, 6773}, {5318, 7885}, {5321, 17128}, {7879, 11289}, {7881, 11307}, {7891, 11480}, {7904, 11481}, {7908, 11129}, {10640, 11131}, {11290, 11543}, {11542, 22113}, {19781, 33225}, {22906, 33518}, {29579, 30414}

X(34540) = isogonal conjugate of X(34533)
X(34540) = crosspoint of X(2) and X(5487)
X(34540) = complement of the isotomic conjugate of X(5487)
X(34540) = X(i)-complementary conjugate of X(j) for these (i,j): {5487, 2887}, {30252, 4369}
X(34540) = X(1)-isoconjugate of X(34533)
X(34540) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34533}, {17, 18813}
X(34540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 298, 3181}, {2, 3620, 34541}, {6, 3314, 34541}, {141, 302, 2}, {3619, 11489, 2}, {3619, 16990, 34541}, {3631, 7925, 34541}, {3631, 23302, 299}


X(34541) =  2ND VU POINT OF X(18)

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

X(34541) lies on these lines: {2, 6}, {15, 628}, {16, 634}, {115, 11121}, {148, 14905}, {338, 34390}, {383, 33878}, {471, 11408}, {574, 11132}, {616, 619}, {617, 7898}, {621, 624}, {630, 22855}, {631, 5873}, {633, 7912}, {635, 16966}, {1080, 18358}, {2992, 11131}, {3091, 5865}, {3098, 5613}, {3105, 18582}, {3522, 5869}, {3642, 16808}, {3643, 10645}, {5092, 6770}, {5318, 17128}, {5321, 7885}, {7879, 11290}, {7881, 11308}, {7891, 11481}, {7904, 11480}, {7908, 11128}, {10639, 11130}, {11289, 11542}, {11543, 22114}, {19780, 33225}, {22862, 33517}, {29579, 30415}

X(34541) = isogonal conjugate of X(34534)
X(34541) = complement of the isotomic conjugate of X(5488)
X(34541) = X(i)-complementary conjugate of X(j) for these (i,j): {5488, 2887}, {30253, 4369}
X(34541) = crosspoint of X(2) and X(5488)
X(34541) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34534}, {18, 18814}
X(34541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 299, 3180}, {2, 3620, 34540}, {6, 3314, 34540}, {141, 303, 2}, {3619, 11488, 2}, {3619, 16990, 34540}, {3631, 7925, 34540}, {3631, 23303, 298}


X(34542) =  2ND VU POINT OF X(75)

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

X(34542) lies on these lines: {1, 16886}, {6, 10}, {32, 7270}, {76, 20234}, {85, 7794}, {115, 312}, {304, 626}, {345, 2549}, {495, 594}, {574, 32851}, {1015, 3705}, {1107, 30172}, {1573, 29641}, {1909, 30149}, {1914, 4680}, {2176, 4153}, {2241, 5015}, {2275, 30171}, {2345, 31409}, {2887, 3735}, {3006, 16975}, {3125, 4165}, {3314, 20924}, {3661, 3936}, {3679, 20483}, {3685, 9664}, {3695, 5254}, {3797, 7790}, {3912, 17720}, {4372, 30104}, {4799, 33952}, {4872, 7818}, {4950, 33953}, {5025, 33939}, {5088, 7801}, {5949, 21024}, {6376, 20444}, {7187, 7796}, {7283, 7748}, {7764, 25918}, {7821, 17181}, {7888, 17095}, {7951, 21057}, {9346, 33121}, {16971, 33120}, {20541, 33936}, {21044, 29687}, {24275, 32777}, {25280, 30153}, {29593, 31034}, {33841, 33935}

X(34542) = barycentric product X(10)*X(30984)
X(34542) = barycentric quotient X(30984)/X(86)
X(34542) = {X(4165),X(25957)}-harmonic conjugate of X(3125)


X(34543) =  2ND VU POINT OF X(56)

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

X(34543) lies on these lines: {1, 24265}, {6, 101}, {9, 9351}, {32, 604}, {87, 24264}, {478, 16502}, {572, 2241}, {574, 2269}, {1100, 5114}, {2175, 3248}, {2300, 5042}, {5053, 21769}, {7113, 16946}

X(34543) = isogonal conjugate of X(34523)
X(34543) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34523}, {9, 18811}
X(34543) = crosssum of X(312) and X(18743)
X(34543) = crossdifference of every pair of points on line {900, 4397}
X(34543) = barycentric product X(i)*X(j) for these {i,j}: {1, 32577}, {31, 4862}, {56, 2098}, {604, 30827}, {651, 17424}, {1407, 34524}
X(34543) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34523}, {56, 18811}, {2098, 3596}, {4862, 561}, {17424, 4391}, {30827, 28659}, {32577, 75}
X(34543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {572, 21785, 2241}, {604, 20228, 32}


X(34544) =  2ND VU POINT OF X(36)

Barycentrics    a^3*(a - b - c)*(a^2 - b^2 + b*c - c^2)^2 : :
Barycentrics    1 + Cos[3 A] : :

X(34544) lies on these lines: {9, 48}, {36, 2245}, {41, 2316}, {57, 1813}, {219, 5124}, {284, 2170}, {662, 4858}, {1800, 33178}, {2161, 2278}, {2167, 6358}, {2171, 17439}, {3928, 6507}, {5949, 15833}

X(34544) = isogonal conjugate of X(34535)
X(34544) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 6149}, {2361, 3647}, {6186, 1737}, {7073, 3814}, {7110, 21237}, {8648, 6741}
X(34544) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 6149}, {662, 3738}, {2167, 758}
X(34544) = X(i)-isoconjugate of X(j) for these (i,j): {1, 34535}, {11, 23592}, {80, 2006}, {1168, 14628}, {1411, 18359}, {2161, 18815}
X(34544) = barycentric product X(i)*X(j) for these {i,j}: {1, 4996}, {36, 4511}, {60, 4736}, {75, 215}, {320, 2361}, {654, 4585}, {765, 3025}, {1098, 3028}, {1983, 3904}, {2323, 3218}, {3936, 4282}, {7113, 32851}
X(34544) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 34535}, {36, 18815}, {215, 1}, {1983, 655}, {2149, 23592}, {2323, 18359}, {2361, 80}, {3025, 1111}, {4282, 24624}, {4511, 20566}, {4736, 34388}, {4996, 75}, {7113, 2006}, {17455, 14628}
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {101, 572, 2265}, {101, 909, 16554}


X(34545) =  MIDPOINT OF X(15037) AND X(15038)

Barycentrics    a^2*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 3*b^2*c^2 + c^4) : :
Barycentrics    1 + 4 Sin^2[A] : :
Barycentrics    a^2 (b^2 c^2 + 4 S^2) : :
Barycentrics    a^2 + R^2 : :
Trilinears    4 sin A + csc A : :

X(34545) lies on these lines: {2, 6}, {3, 14449}, {5, 1199}, {8, 16472}, {22, 5050}, {23, 51}, {25, 11003}, {30, 15037}, {39, 588}, {49, 15026}, {54, 5462}, {61, 11145}, {62, 11146}, {83, 11140}, {97, 3284}, {110, 5943}, {125, 32068}, {140, 2889}, {143, 7512}, {154, 14002}, {155, 5056}, {182, 3060}, {184, 5640}, {186, 567}, {195, 3628}, {251, 1692}, {288, 18315}, {381, 15032}, {389, 13434}, {399, 5066}, {511, 15246}, {569, 3567}, {576, 2979}, {578, 15043}, {611, 17024}, {613, 29815}, {623, 11143}, {624, 11144}, {651, 26842}, {1092, 15028}, {1147, 15024}, {1154, 7550}, {1173, 5446}, {1180, 5034}, {1181, 3832}, {1351, 7485}, {1493, 32205}, {1583, 6418}, {1584, 6417}, {1591, 19116}, {1592, 19117}, {1599, 3312}, {1600, 3311}, {1625, 31610}, {1627, 5052}, {1899, 5169}, {1915, 13410}, {1995, 8780}, {2003, 27003}, {2004, 34395}, {2005, 34394}, {2052, 7578}, {2071, 9730}, {2323, 27065}, {2888, 13292}, {2914, 20304}, {2987, 3108}, {3047, 20772}, {3066, 17809}, {3090, 12161}, {3091, 7592}, {3124, 14153}, {3146, 10982}, {3167, 5645}, {3218, 26740}, {3219, 16579}, {3292, 5643}, {3448, 5133}, {3518, 32046}, {3521, 15807}, {3525, 16266}, {3545, 15052}, {3564, 7605}, {3616, 16473}, {3819, 22330}, {3839, 11456}, {3845, 12112}, {3855, 32139}, {3917, 5097}, {4858, 24149}, {5038, 20859}, {5068, 11441}, {5071, 15068}, {5085, 16981}, {5092, 21969}, {5093, 7484}, {5111, 8041}, {5154, 5707}, {5280, 26639}, {5396, 27086}, {5406, 6432}, {5407, 6431}, {5410, 15188}, {5411, 15187}, {5476, 11550}, {5483, 25065}, {5644, 9716}, {5661, 14773}, {5890, 7527}, {5899, 13451}, {6358, 24148}, {6509, 15860}, {6642, 9545}, {6776, 7394}, {6800, 17810}, {6995, 33748}, {6997, 14912}, {7391, 14853}, {7492, 33586}, {7502, 13321}, {7503, 11432}, {7533, 8550}, {7565, 25739}, {7570, 11225}, {7583, 15233}, {7584, 15234}, {7691, 16625}, {7998, 15520}, {9140, 34155}, {9306, 11422}, {9696, 15563}, {9813, 15531}, {10095, 11817}, {10264, 10821}, {10303, 15805}, {10328, 13196}, {10329, 20977}, {10539, 11423}, {10540, 13364}, {10564, 13482}, {10574, 11424}, {10880, 15192}, {10881, 15191}, {10984, 12087}, {11289, 19778}, {11290, 19779}, {11426, 17928}, {11430, 15053}, {11442, 14561}, {11793, 15801}, {11800, 27866}, {12007, 14683}, {12215, 33798}, {12241, 34007}, {12254, 31830}, {13339, 13391}, {13352, 15045}, {13363, 22115}, {13382, 15062}, {13616, 26341}, {13617, 26348}, {13630, 14865}, {14763, 34470}, {14848, 31133}, {15022, 17814}, {15107, 21849}, {17536, 22136}, {18350, 32136}, {18911, 31074}, {18914, 19361}, {20062, 25406}, {22052, 31626}, {25488, 32255}, {26869, 31236}, {26881, 34417}, {32064, 34117}

X(34545) = complement of X(15108)
X(34545) = midpoint of X(15037) and X(15038)
X(34545) = complement of the isotomic conjugate of X(11538)
X(34545) = isotomic conjugate of the polar conjugate of X(34484)
X(34545) = X(11538)-complementary conjugate of X(2887)
X(34545) = X(i)-isoconjugate of X(j) for these (i,j): {19, 34483}, {661, 20189}, {17438, 34110}
X(34545) = cevapoint of X(6) and X(15047)
X(34545) = crosspoint of X(2) and X(11538)
X(34545) = crosssum of X(6) and X(15109)
X(34545) = barycentric product X(i)*X(j) for these {i,j}: {69, 34484}, {95, 10095}, {99, 20188}, {340, 31676}
X(34545) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 34483}, {110, 20189}, {1173, 34110}, {10095, 5}, {11817, 15559}, {20188, 523}, {31676, 265}, {34484, 4}
X(34545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 1994}, {2, 1994, 323}, {2, 5422, 15018}, {2, 11004, 394}, {6, 597, 22151}, {6, 5422, 2}, {6, 10601, 1993}, {6, 15018, 323}, {22, 9777, 11002}, {51, 575, 5012}, {51, 5012, 23}, {110, 12834, 5943}, {143, 13353, 7512}, {182, 3060, 6636}, {182, 15004, 3060}, {184, 5640, 13595}, {389, 13434, 14118}, {567, 5946, 186}, {569, 3567, 7488}, {575, 15019, 23}, {578, 15043, 22467}, {588, 589, 39}, {1993, 5422, 10601}, {1993, 10601, 2}, {1994, 15018, 2}, {1995, 11402, 9544}, {3545, 18445, 15052}, {3618, 6515, 2}, {5012, 15019, 51}, {5050, 9777, 22}, {5133, 11245, 3448}, {5943, 13366, 110}, {5943, 15516, 13366}, {9306, 11451, 16042}, {9730, 15033, 2071}, {10574, 11424, 12086}, {11245, 18583, 5133}, {11422, 11451, 9306}, {11430, 16226, 15053}, {13292, 14788, 2888}, {13567, 14389, 2}, {14627, 15047, 140}, {15066, 17825, 2}, {21849, 22352, 15107}


X(34546) =  ISOGONAL CONJUGATE OF X(1604)

Barycentrics    (a^6-a^4 b^2-a^2 b^4+b^6+4 a^4 b c-4 a^3 b^2 c-4 a^2 b^3 c+4 a b^4 c-3 a^4 c^2+4 a^3 b c^2+6 a^2 b^2 c^2+4 a b^3 c^2-3 b^4 c^2-4 a^2 b c^3-4 a b^2 c^3+3 a^2 c^4-4 a b c^4+3 b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6+4 a^4 b c+4 a^3 b^2 c-4 a^2 b^3 c-4 a b^4 c-a^4 c^2-4 a^3 b c^2+6 a^2 b^2 c^2-4 a b^3 c^2+3 b^4 c^2-4 a^2 b c^3+4 a b^2 c^3-a^2 c^4+4 a b c^4-3 b^2 c^4+c^6) : :
X(34546) = 5*X(20196)-3*X(34499)

See Kadir Altintas, Ercole Suppa and César Lozada, Euclid 10 and Euclid 16 .

X(34546) lies these lines: {2,6609}, {40,2123}, {223,3452}, {5514,6612}, {14256,26563}, {20196, 34499}

X(34546) = isogonal conjugate of X(1604)
X(34546) = isotomic conjugate of anticomplement of X(1407)
X(34546) = cyclocevian conjugate of X(6553)
X(34546) = anticomplement of X(6609)
X(34546) = antigonal image of isogonal conjugate of X(17112)


X(34547) =  MIDPOINT OF X(3146) AND X(20097)

Barycentrics    a^7-3 a^6 b+5 a^5 b^2-7 a^4 b^3+7 a^3 b^4-5 a^2 b^5+3 a b^6-b^7-3 a^6 c+3 a^5 b c+a^4 b^2 c-6 a^3 b^3 c+7 a^2 b^4 c-5 a b^5 c+3 b^6 c+5 a^5 c^2+a^4 b c^2+2 a^3 b^2 c^2-2 a^2 b^3 c^2-3 a b^4 c^2-3 b^5 c^2-7 a^4 c^3-6 a^3 b c^3-2 a^2 b^2 c^3+10 a b^3 c^3+b^4 c^3+7 a^3 c^4+7 a^2 b c^4-3 a b^2 c^4+b^3 c^4-5 a^2 c^5-5 a b c^5-3 b^2 c^5+3 a c^6+3 b c^6-c^7 : :
X(34547) = X[3146]+X[20097]

See Kadir Altintas and Ercole Suppa, Euclid 10 .

X(34547) lies on the anticomplementary circle and these lines: {2,1292}, {4,10743}, {20,105}, {120,3091}, {146,2836}, {147,2795}, {148,2788}, {149,2826}, {150,2820}, {151,2835}, {152,962}, {153,528}, {388,3021}, {497,1358}, {516,5540}, {2550,3039}, {2775,3448}, {2814,33650}, {2838,12384}, {3034,9535}, {3146,20097}, {3523,6714}, {3839,10712}, {5731,11716}, {9519,21290}, {9522,14360}, {9523,13219}

X(34547) = midpoint of X(3146) and X(20097)
X(34547) = reflection of X(i) in X(j) for these {i,j}: {4,15521}, {20,105}, {1292,5511}
X(34547) = anticomplement of X(1292)
X(34547) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1292,5511,2}


X(34548) =  MIDPOINT OF X(3146) AND X(20098)

Barycentrics    a^6-3 a^5 b-a^4 b^2+a^2 b^4+3 a b^5-b^6-3 a^5 c+15 a^4 b c-6 a^3 b^2 c+6 a^2 b^3 c-15 a b^4 c+3 b^5 c-a^4 c^2-6 a^3 b c^2-10 a^2 b^2 c^2+12 a b^3 c^2+b^4 c^2+6 a^2 b c^3+12 a b^2 c^3-6 b^3 c^3+a^2 c^4-15 a b c^4+b^2 c^4+3 a c^5+3 b c^5-c^6 : :
X(34548) = X[3146]+X[20098]

See Kadir Altintas and Ercole Suppa, Euclid 10 .

X(34548) lies on the anticomplementary circle and these lines: {2,1293}, {4,10744}, {20,106}, {121,3091}, {146,2842}, {147,2796}, {148,2789}, {149,2827}, {150,2821}, {151,2841}, {152,2810}, {153,962}, {388,6018}, {497,1357}, {516,1054}, {1699,11814}, {2550,3038}, {2776,3448}, {2815,33650}, {2844,12384}, {3030,9535}, {3146,20098}, {3523,6715}, {3543,10730}, {3839,10713}, {4301,13541}, {5731,11717}, {8055,9519}, {9526,14360}, {9527,13219}, {9778,14664}

X(34548) = midpoint of X(3146) and X(20098)
X(34548) = reflection of X(i) in X(j) for these {i,j}: {4,15522}, {20,106}, {1293,5510}, {13541,4301}, {21290,4}
X(34548) = anticomplement of X(1293)
X(34548) ={X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1293,5510,2}


X(34549) =  X(2)X(133)∩X(4)X(2972)

Barycentrics    a^16+a^14 b^2-16 a^12 b^4+31 a^10 b^6-20 a^8 b^8-a^6 b^10+4 a^4 b^12+a^2 b^14-b^16+a^14 c^2+25 a^12 b^2 c^2-29 a^10 b^4 c^2-57 a^8 b^6 c^2+79 a^6 b^8 c^2-a^4 b^10 c^2-19 a^2 b^12 c^2+b^14 c^2-16 a^12 c^4-29 a^10 b^2 c^4+154 a^8 b^4 c^4-78 a^6 b^6 c^4-96 a^4 b^8 c^4+51 a^2 b^10 c^4+14 b^12 c^4+31 a^10 c^6-57 a^8 b^2 c^6-78 a^6 b^4 c^6+186 a^4 b^6 c^6-33 a^2 b^8 c^6-49 b^10 c^6-20 a^8 c^8+79 a^6 b^2 c^8-96 a^4 b^4 c^8-33 a^2 b^6 c^8+70 b^8 c^8-a^6 c^10-a^4 b^2 c^10+51 a^2 b^4 c^10-49 b^6 c^10+4 a^4 c^12-19 a^2 b^2 c^12+14 b^4 c^12+a^2 c^14+b^2 c^14-c^16 : :

See Kadir Altintas and Ercole Suppa, Euclid 10 .

X(34549) lies on the anticomplementary circle and these lines: {2,133}, {4,2972}, {20,107}, {23,14703}, {30,5667}, {122,3091}, {146,9033}, {147,2797}, {148,2790}, {149,2828}, {150,2822}, {151,2846}, {152,2811}, {153,2803}, {253,317}, {388,7158}, {497,3324}, {2777,3146}, {2816,33650}, {2848,12384}, {3522,23239}, {3523,6716}, {3529,23240}, {3839,10714}, {5731,11718}, {9520,20344}, {9524,21290}, {9529,14360}, {15055,24930}

X(34549) = reflection of X(i) in X(j) for these {i,j}: {4,22337}, {20,107}, {1294,133}, {3529,23240}, {34186,4}
X(34549) = anticomplement of X(1294)
X(34549) = anticomplementary-circle-antipode of X(34186)
X(34549) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {133,1294,2}


X(34550) =  X(2)X(1295)∩X(4)X(280)

Barycentrics    a^13-a^12 b-2 a^11 b^2+2 a^10 b^3-a^9 b^4+a^8 b^5+4 a^7 b^6-4 a^6 b^7-a^5 b^8+a^4 b^9-2 a^3 b^10+2 a^2 b^11+a b^12-b^13-a^12 c+9 a^11 b c-6 a^10 b^2 c-11 a^9 b^3 c+19 a^8 b^4 c-14 a^7 b^5 c-12 a^6 b^6 c+18 a^5 b^7 c-3 a^4 b^8 c+5 a^3 b^9 c+2 a^2 b^10 c-7 a b^11 c+b^12 c-2 a^11 c^2-6 a^10 b c^2+28 a^9 b^2 c^2-20 a^8 b^3 c^2-28 a^7 b^4 c^2+44 a^6 b^5 c^2-16 a^5 b^6 c^2+14 a^3 b^8 c^2-22 a^2 b^9 c^2+4 a b^10 c^2+4 b^11 c^2+2 a^10 c^3-11 a^9 b c^3-20 a^8 b^2 c^3+76 a^7 b^3 c^3-28 a^6 b^4 c^3-34 a^5 b^5 c^3+40 a^4 b^6 c^3-52 a^3 b^7 c^3+10 a^2 b^8 c^3+21 a b^9 c^3-4 b^10 c^3-a^9 c^4+19 a^8 b c^4-28 a^7 b^2 c^4-28 a^6 b^3 c^4+66 a^5 b^4 c^4-38 a^4 b^5 c^4-12 a^3 b^6 c^4+52 a^2 b^7 c^4-25 a b^8 c^4-5 b^9 c^4+a^8 c^5-14 a^7 b c^5+44 a^6 b^2 c^5-34 a^5 b^3 c^5-38 a^4 b^4 c^5+94 a^3 b^5 c^5-44 a^2 b^6 c^5-14 a b^7 c^5+5 b^8 c^5+4 a^7 c^6-12 a^6 b c^6-16 a^5 b^2 c^6+40 a^4 b^3 c^6-12 a^3 b^4 c^6-44 a^2 b^5 c^6+40 a b^6 c^6-4 a^6 c^7+18 a^5 b c^7-52 a^3 b^3 c^7+52 a^2 b^4 c^7-14 a b^5 c^7-a^5 c^8-3 a^4 b c^8+14 a^3 b^2 c^8+10 a^2 b^3 c^8-25 a b^4 c^8+5 b^5 c^8+a^4 c^9+5 a^3 b c^9-22 a^2 b^2 c^9+21 a b^3 c^9-5 b^4 c^9-2 a^3 c^10+2 a^2 b c^10+4 a b^2 c^10-4 b^3 c^10+2 a^2 c^11-7 a b c^11+4 b^2 c^11+a c^12+b c^12-c^13 : :

See Kadir Altintas and Ercole Suppa, Euclid 10 .

X(34550) lies on the anticomplementary circle and these lines: {2,1295}, {4,280}, {20,108}, {123,3091}, {146,2850}, {147,2798}, {148,2791}, {149,2829}, {150,2823}, {151,2849}, {152,2812}, {153,2804}, {347,6925}, {388,3318}, {497,1359}, {962,2817}, {2475,9528}, {2778,3448}, {3523,6717}, {3543,10731}, {3839,10715}, {5731,11719}, {9521,20344}, {9525,21290}, {9531,14360}

X(34550) = reflection of X(i) in X(j) for these {i,j}: {4,33566}, {20,108}, {1295,25640}, {34188,4}
X(34550) = anticomplement of X(1295)
X(34550) = anticomplementary-circle-antipode of X(34188)
X(34550) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {1295,25640,2}




leftri  Bankoff equilateral triangles: X(34551) - X(34562)  rightri

This preamble and centers X(34551)-X(34562) were contributed by César Eliud Lozada, October 29, 2019.

Let OA1A2, OB1B2, OC1C2 be three congruent equilateral triangles with the same orientation. Then the midpoints of A2B1, B2C1 and C2A1 are vertices of an equilateral triangle. (Martin Gardner, The Asymmetric Propeller, The College Mathematics Journal, Vol. 30, No. 1, Jan. 1999, pp. 18-22, based on the article by Leon Bankoff, Paul Erdös, and Murray Klamkin, The asymmetric propeller, Mathematics Magazine, 46: 5, 1973, pp 270-272)1 2.

Application: Let ABC be a triangle with circumcenter O. Centering at O, let Ab be the rotation of A toward B by an angle |π/6| and let Ac be the rotation of A toward C by the same angle |π/6| (triangle OAbAc is an equilateral triangle). Build (Bc, Ba) and (Ca, Cb) cyclically and let Am, Bm, Cm be the midpoints of BcCb, CaAc, AbBa, respectively. Then, by the precedent theorem, AmBmCm is equilateral.

The triangle AmBmCm will be named here the (ABC)-Bankoff equilateral triangle.

1 The original work by Bankoff, Erdös, and Klamkin refers to three equilateral triangles, not necessarily congruent, but with the same result, i.e, the indicated midpoints are vertices of another equilateral triangle. Martin Gardner added:

...The original propeller theorem goes back at least to the early 1930's and is of unknown origin. It concerns three congruent equilateral triangles with corners meeting at a point as shown shaded in Figure 1. The triangles resemble the blades of a propeller.

...They showed that the three equilateral triangles need not be congruent. They can be of any size, as shown in Figure 2, and the theorem still holds.

... Later, Bankoff made three further generalizations. As far as I know they have not been published.

  • Second generalization: The propeller triangles need not meet at a point. They may meet at the corners of any equilateral triangle, as shown in Figure 3.
  • Third generalization: The propeller triangles need not be equilateral! They need only be similar triangles of any sizes that meet at a point. The midpoints of the three added lines will then form a triangle similar to each of the propellers, as shown in Figure 4.
  • Fourth generalization: The similar triangles need not meet at a point! If the propellers meet at the corners of a fourth triangle of any size, provided it is similar to each propeller, the midpoints of the added lines will form a triangle similar to each propeller. Vertices of the interior triangle must touch corresponding corners of the propellers.

See the above mentioned figures here.

2 The cited Martin Gardner's article was referenced by Richard Guy and Antreas Hatzipolakis in Hyacinthos #26 & #28 (December 28, 1999).

underbar

X(34551) = CENTER OF THE (ABC)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    -((b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+4*a^4-5*(b^2+c^2)*a^2+(b^2-c^2)^2 : :
X(34551) = (1-2*sqrt(3))*X(3)-X(4)

As a point on the Euler line, X(34551) has Shinagawa coefficients (-1+2*sqrt(3), -2*sqrt(3)+3)

X(34551) lies on these lines: {2,3}, {395,3364}, {396,3390}, {511,34553}, {515,34557}, {516,34556}, {618,640}, {619,639}, {1151,10654}, {1152,10653}, {3366,16242}, {3392,16241}, {3849,34554}, {5418,16645}, {5420,16644}, {6560,11481}, {6561,11480}, {6774,33444}, {9830,34558}, {13754,34555}

X(34551) = midpoint of X(i) and X(j) for these {i,j}: {3, 2044}, {20, 18587}, {14813, 15765}
X(34551) = reflection of X(i) in X(j) for these (i,j): (4, 34562), (860, 16917), (15765, 140), (16908, 11315), (18586, 34559), (34552, 3)
X(34551) = anticomplement of X(34559)
X(34551) = complement of X(18586)
X(34551) = X(34552)-of-ABC-X3 reflections triangle
X(34551) = X(34553)-of-1st circumperp triangle
X(34551) = X(34555)-of-2nd circumperp triangle
X(34551) = X(34558)-of-circumsymmedial triangle
X(34551) = X(34562)-of-anti-Euler triangle
X(34551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 140, 34552), (2, 2042, 15765), (2, 2044, 14813), (2, 15765, 5), (2, 18586, 34559), (2, 18587, 3628), (3, 2042, 140), (3, 2045, 3530), (3, 7419, 7527), (3, 11293, 7459), (3, 14813, 5), (3, 15765, 549), (3, 18586, 18585), (3, 29890, 415), (3, 31913, 34004), (26, 4241, 29902), (26, 10691, 34552), (26, 14955, 7493), (26, 20855, 2071), (26, 23250, 17682), (26, 27404, 21493), (26, 27510, 5117), (26, 27583, 10201), (27, 15200, 17517), (27, 30769, 15700), (27, 32964, 26833), (27, 33192, 6823), (27, 33532, 7000), (28, 5133, 31913), (28, 6901, 7463)


X(34552) = CENTER OF THE (ABC-X3 REFLECTIONS)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+4*a^4-5*(b^2+c^2)*a^2+(b^2-c^2)^2 : :
X(34552) = (2*sqrt(3)+1)*X(3)-X(4)

As a point on the Euler line, X(34552) has Shinagawa coefficients (2*sqrt(3)+1, -2*sqrt(3)-3)

X(34552) lies on these lines: {2,3}, {395,3365}, {396,3389}, {511,34555}, {515,34556}, {516,34557}, {618,639}, {619,640}, {1151,10653}, {1152,10654}, {3367,16242}, {3391,16241}, {5418,16644}, {6560,11480}, {6561,11481}, {6771,33446}, {13754,34553}

X(34552) = midpoint of X(i) and X(j) for these {i,j}: {3, 2043}, {20, 18586}, {14814, 18585}
X(34552) = reflection of X(i) in X(j) for these (i,j): (4, 34559), (15764, 34200), (18585, 140), (18587, 34562), (34551, 3)
X(34552) = anticomplement of X(34562)
X(34552) = complement of X(18587)
X(34552) = X(34551)-of-ABC-X3 reflections triangle
X(34552) = X(34553)-of-2nd circumperp triangle
X(34552) = X(34555)-of-1st circumperp triangle
X(34552) = X(34559)-of-anti-Euler triangle
X(34552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 22, 4198), (2, 25, 23260), (2, 140, 34551), (2, 405, 19532), (2, 405, 24538), (2, 407, 6893), (2, 447, 33203), (2, 471, 26600), (2, 472, 1316), (2, 1325, 6918), (2, 1557, 864), (2, 2041, 18585), (2, 2043, 14814), (2, 2047, 7819), (2, 2476, 11311), (26, 415, 27656), (26, 416, 11484), (26, 1651, 21482), (26, 2060, 7575), (26, 3523, 25463), (26, 3832, 26), (26, 3832, 3832), (26, 3861, 18585), (26, 6856, 27859), (26, 6906, 6979), (26, 6936, 16052), (26, 6939, 4236), (26, 7519, 18585), (26, 7532, 13742), (26, 7539, 4212)


X(34553) = CENTER OF THE (1st ANTI-CIRCUMPERP)-BANKOFF EQUILATERAL TRIANGLE

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

X(34553) lies on these lines: {3,54}, {143,2042}, {511,34551}, {2041,11591}, {2043,5663}, {2044,13391}, {2045,12006}, {2046,32142}, {5876,14814}, {5946,15765}, {10263,14813}, {13754,34552}, {15060,18587}, {15067,18585}

X(34553) = reflection of X(34555) in X(3)
X(34553) = X(34551)-of-1st anti-circumperp triangle
X(34553) = X(34552)-of-circumorthic triangle
X(34553) = X(34555)-of-ABC-X3 reflections triangle
X(34553) = X(34559)-of-3rd anti-Euler triangle
X(34553) = X(34562)-of-4th anti-Euler triangle
X(34553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2979, 10627, 34555), (5890, 13630, 34555)


X(34554) = CENTER OF THE (CIRCUMMEDIAL)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    (2*(b^2+c^2)*a^8+10*(b^4-5*b^2*c^2+c^4)*a^6-6*((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2)*a^4-2*(5*b^8+5*c^8-b^2*c^2*(19*b^4-12*b^2*c^2+19*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(4*b^4-16*b^2*c^2+4*c^4))*sqrt(3)+44*a^10-109*(b^2+c^2)*a^8+(5*b^4+74*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(107*b^4-59*b^2*c^2+107*c^4)*a^4-(49*b^8+49*c^8-b^2*c^2*(41*b^4+120*b^2*c^2+41*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4-8*b^2*c^2+2*c^4) : :

X(34554) lies on these lines: {3,9829}, {2042,32156}, {3849,34551}, {14814,31824}

X(34554) = X(34551)-of-circummedial triangle


X(34555) = CENTER OF THE (CIRCUMORTHIC)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    a^2*(3*a^2*(b^6+c^6)-2*a^2*b^2*c^2*(-a^2+b^2+c^2)*sqrt(3)+(b^2+c^2)*a^6-(3*b^4+2*b^2*c^2+3*c^4)*a^4-(b^6-c^6)*(b^2-c^2)) : :
X(34555) = ((5+2*sqrt(3))*R^2-2*SW)*X(3)-(5*R^2-2*SW)*X(54)

X(34555) lies on these lines: {3,54}, {143,2041}, {511,34552}, {2042,11591}, {2043,13391}, {2044,5663}, {2045,32142}, {2046,12006}, {5876,14813}, {5946,18585}, {10263,14814}, {13754,34551}, {15060,18586}, {15067,15765}

X(34555) = reflection of X(34553) in X(3)
X(34555) = X(34551)-of-circumorthic triangle
X(34555) = X(34552)-of-1st anti-circumperp triangle
X(34555) = X(34553)-of-ABC-X3 reflections triangle
X(34555) = X(34559)-of-4th anti-Euler triangle
X(34555) = X(34562)-of-3rd anti-Euler triangle
X(34555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2979, 10627, 34553), (5890, 13630, 34553)


X(34556) = CENTER OF THE (1st CIRCUMPERP)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    a*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)-2*sqrt(3)*(-a^2+b^2+c^2)*a) : :
X(34556) = X(1)-(2*sqrt(3)+1)*X(3)

X(34556) lies on these lines: {1,3}, {515,34552}, {516,34551}, {2041,9956}, {2042,9955}, {2043,28160}, {2044,28146}, {11230,15765}, {11231,18585}, {14813,22793}, {14814,18480}

X(34556) = reflection of X(34557) in X(3)
X(34556) = X(34551)-of-1st circumperp triangle
X(34556) = X(34552)-of-2nd circumperp triangle
X(34556) = X(34557)-of-ABC-X3 reflections triangle
X(34556) = X(34559)-of-excentral triangle
X(34556) = X(34561)-of-anti-Mandart-incircle triangle
X(34556) = X(34562)-of-hexyl triangle
X(34556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10832, 13750), (1, 17798, 10253), (35, 5706, 3303), (36, 10383, 1617), (36, 26425, 31794), (46, 982, 32760), (46, 17102, 56), (46, 31792, 18967), (55, 18443, 3075), (56, 9364, 6583), (57, 10473, 12000), (57, 11367, 18398), (57, 11518, 5707), (165, 31663, 34557), (171, 10252, 11021), (171, 10474, 1735), (171, 26296, 11529), (241, 16203, 1), (260, 982, 17603), (260, 11521, 5363), (260, 31797, 8193), (484, 24927, 7146), (942, 23703, 3339), (982, 11875, 11521), (986, 11882, 11018), (986, 14801, 7373), (986, 30392, 18330), (988, 11529, 11018), (999, 2099, 11011), (999, 3295, 20790), (999, 17601, 3931), (999, 26437, 5708), (1038, 5570, 12001), (1040, 2352, 30337), (1060, 3931, 15804), (1060, 10389, 11011), (1082, 1159, 982), (1082, 18453, 940)


X(34557) = CENTER OF THE (2nd CIRCUMPERP)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    a*((b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)+2*sqrt(3)*(-a^2+b^2+c^2)*a) : :
X(34557) = X(1)+(2*sqrt(3)-1)*X(3)

X(34557) lies on these lines: {1,3}, {515,34551}, {516,34552}, {2041,9955}, {2042,9956}, {2043,28146}, {2044,28160}, {11230,18585}, {11231,15765}, {14813,18480}, {14814,22793}

X(34557) = midpoint of X(1) and X(34560)
X(34557) = reflection of X(34556) in X(3)
X(34557) = X(34551)-of-2nd circumperp triangle
X(34557) = X(34552)-of-1st circumperp triangle
X(34557) = X(34556)-of-ABC-X3 reflections triangle
X(34557) = X(34559)-of-hexyl triangle
X(34557) = X(34560)-of-anti-Aquila triangle
X(34557) = X(34561)-of-2nd circumperp tangential triangle
X(34557) = X(34562)-of-excentral triangle
X(34557) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1388, 12000), (3, 17715, 14798), (40, 1460, 31781), (40, 5697, 988), (40, 10247, 7280), (46, 12915, 16687), (165, 31663, 34556), (171, 9659, 17102), (559, 23890, 25405), (986, 26393, 11881), (1062, 7982, 5425), (1155, 10500, 1429), (1214, 3075, 10508), (1385, 17502, 34556), (1402, 18967, 33179), (1402, 23981, 10247), (1420, 13462, 5706), (1482, 17603, 26296), (1617, 17601, 4038), (1622, 11884, 17716), (1622, 26352, 1082), (1715, 14804, 17600), (1735, 18421, 3338), (1758, 18330, 26297), (1771, 16877, 10832), (2061, 4689, 2449), (2093, 2283, 12001), (2099, 3072, 1467), (2446, 20789, 13373), (2447, 11873, 11881)


X(34558) = CENTER OF THE (CIRCUMSYMMEDIAL)-BANKOFF EQUILATERAL TRIANGLE

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

X(34558) lies on these lines: {3,352}, {9830,34551}, {14814,31827}

X(34558) = X(34551)-of-circumsymmedial triangle


X(34559) = CENTER OF THE (MEDIAL)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    -2*((b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(34559) = 3*X(2)+X(18586) = (2*sqrt(3)+1)*X(3)+(2*sqrt(3)-1)*X(4) = 3*X(5)-X(18585)

As a point on the Euler line, X(34559) has Shinagawa coefficients (2*sqrt(3)+1, -3+2*sqrt(3))

X(34559) lies on these lines: {2,3}, {395,3392}, {396,3366}, {3642,23311}, {3643,23312}

X(34559) = midpoint of X(i) and X(j) for these {i,j}: {4, 34552}, {5, 15765}, {860, 17685}, {18586, 34551}
X(34559) = reflection of X(i) in X(j) for these (i,j): (1592, 17558), (34562, 5)
X(34559) = complement of X(34551)
X(34559) = X(34552)-of-Euler triangle
X(34559) = X(34553)-of-3rd Euler triangle
X(34559) = X(34555)-of-4th Euler triangle
X(34559) = X(34562)-of-Johnson triangle
X(34559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3628, 34562), (2, 18586, 34551), (2, 29929, 26835), (2, 34552, 16239), (3, 423, 444), (3, 10109, 34562), (3, 14790, 29726), (3, 23408, 25404), (3, 27553, 32963), (3, 30773, 7465), (3, 31214, 2937), (4, 446, 8229), (4, 4188, 12811), (4, 8703, 6971), (4, 11737, 34562)


X(34560) = CENTER OF THE (EXCENTRAL)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    a*(2*(b+c)*a^2-4*b*c*a-2*(b^2-c^2)*(b-c)-2*sqrt(3)*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))) : :
X(34560) = (1-sqrt(3))*X(1)-(1-2*sqrt(3))*X(3)

X(34560) lies on these lines: {1,3}, {516,18586}, {962,2045}, {2042,6361}, {2043,5657}, {2044,9778}, {15765,28174}, {18587,26446}

X(34560) = reflection of X(1) in X(34557)
X(34560) = X(18586)-of-1st circumperp triangle
X(34560) = X(34551)-of-excentral triangle
X(34560) = X(34557)-of-Aquila triangle
X(34560) = X(34559)-of-6th mixtilinear triangle
X(34560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1429, 5228), (1, 7373, 5711), (1, 10225, 3256), (1, 10269, 7962), (1, 11518, 13750), (1, 24927, 3601), (1, 30389, 10965), (3, 8187, 10857), (35, 36, 34561), (35, 1467, 4689), (36, 21334, 999), (46, 1470, 31511), (55, 12009, 13384), (55, 15178, 26357), (56, 3660, 5706), (260, 2283, 19761), (260, 5048, 31515), (354, 12915, 26417), (484, 9630, 11011), (484, 16217, 5172), (559, 31781, 16206), (940, 5329, 3361), (940, 11849, 10474), (940, 17593, 26393), (942, 1402, 3340), (942, 17102, 26086), (942, 24464, 17600), (980, 3333, 5269), (980, 10225, 13389), (980, 31393, 5919), (982, 3550, 4883), (982, 5183, 1060), (988, 8187, 5662), (999, 5137, 3337), (999, 5143, 5363), (999, 6583, 26437), (999, 11021, 3333), (999, 13601, 3338), (999, 16687, 1159), (999, 18421, 31794), (1038, 7962, 33657), (1038, 23703, 33177)


X(34561) = CENTER OF THE (INTOUCH)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    a*(4*b*c*a*sqrt(3)+(b+c)*a^2-2*b*c*a-(b^2-c^2)*(b-c)) : :
X(34561) = (2*sqrt(3)*R+r)*X(1)-r*X(3)

X(34561) lies on the line {1,3}

X(34561) = X(2044)-of-incircle-circles triangle
X(34561) = X(18586)-of-inverse-in-incircle triangle
X(34561) = X(34551)-of-intouch triangle
X(34561) = X(34552)-of-Hutson intouch triangle
X(34561) = X(34556)-of-Mandart-incircle triangle
X(34561) = X(34557)-of-2nd anti-circumperp-tangential triangle
X(34561) = X(34559)-of-Ursa-minor triangle
X(34561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1062, 31511, 6766), (1082, 7011, 1381), (1715, 31797, 18856), (2061, 11248, 7991), (2077, 5709, 7994), (2093, 23961, 5119), (2447, 32612, 24474), (2448, 26286, 484), (2557, 33176, 1764), (3337, 13462, 3339), (3359, 11823, 3579), (3675, 26908, 13151), (3999, 15016, 30392), (5348, 5535, 26285), (5482, 8726, 12458), (7070, 15932, 15803), (7146, 24310, 11009), (8162, 17642, 9627), (8186, 9441, 12702), (9371, 31778, 10856), (9630, 16678, 8726), (10857, 26286, 2572), (11280, 11849, 5347), (11531, 14802, 11822), (14793, 22770, 23171), (14800, 26358, 55), (16204, 31508, 165), (17595, 31797, 2098)


X(34562) = CENTER OF THE (EULER)-BANKOFF EQUILATERAL TRIANGLE

Barycentrics    2*((b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2 : :
X(34562) = 3*X(2)+X(18587) = (1-2*sqrt(3))*X(3)-(1+2*sqrt(3))*X(4) = 3*X(5)-X(15765)

As a point on the Euler line, X(34562) has Shinagawa coefficients (-1+2*sqrt(3), 2*sqrt(3)+3)

X(34562) lies on these lines: {2,3}, {395,3391}, {396,3367}, {3642,23312}, {3643,23311}

X(34562) = midpoint of X(i) and X(j) for these {i,j}: {4, 34551}, {5, 18585}, {15687, 15764}, {18587, 34552}
X(34562) = reflection of X(i) in X(j) for these (i,j): (7512, 21491), (34559, 5)
X(34562) = complement of X(34552)
X(34562) = X(34551)-of-Euler triangle
X(34562) = X(34553)-of-4th Euler triangle
X(34562) = X(34555)-of-3rd Euler triangle
X(34562) = X(34559)-of-Johnson triangle
X(34562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 430, 7572), (2, 466, 11007), (2, 1344, 11289), (2, 1984, 26906), (2, 2046, 16396), (2, 3628, 34559), (2, 5117, 6143), (2, 5501, 20408), (2, 7396, 7523), (2, 7924, 26170), (2, 13723, 15673), (2, 14011, 25933), (2, 16061, 964), (2, 16353, 25802), (2, 16858, 21520)


X(34563) =  X(3)X(1568)∩X(30)X(1493)

Barycentrics    (a^2-b^2-c^2) (4 a^8+2 a^6 b^2-13 a^4 b^4+4 a^2 b^6+3 b^8+2 a^6 c^2+26 a^4 b^2 c^2-4 a^2 b^4 c^2-12 b^6 c^2-13 a^4 c^4-4 a^2 b^2 c^4+18 b^4 c^4+4 a^2 c^6-12 b^2 c^6+3 c^8) : :
Barycentrics    (SB+SC-SW) (-80 R^2 SB-80 R^2 SC+3 S^2-15 SB SC+17 SB SW+17 SC SW) : :

See Kadir Altintas and Ercole Suppa, Euclid 15 .

X(34563) lies on these lines: {3,1568}, {4,13399}, {30,1493}, {74,18363}, {125,546}, {143,185}, {184,3529}, {389,13202}, {576,3146}, {1112,22538}, {1173,10721}, {1176,31371}, {1204,3091}, {1562,5007}, {1594,10990}, {2777,3574}, {3628,21663}, {5895,11403}, {6000,11808}, {7728,18369}, {10594,22802}, {10605,33541}, {11381,19161}, {12102,13851}, {12103,13367}, {15054,18428}


X(34564) =  X(4)X(1173)∩X(52)X(550)

Barycentrics    (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2-8 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6) : :
Barycentrics    (3 S^2-SB SC) (8 R^2+3 SB+3 SC-2 SW) : :

See Kadir Altintas and Ercole Suppa, Euclid 15 .

X(34564) lies on these lines: {4,1173}, {6,17823}, {52,550}, {54,13420}, {113,3850}, {125,1199}, {140,1493}, {155,1656}, {182,193}, {185,15105}, {389,6152}, {1843,6746}, {1986,10990}, {3515,32621}, {3519,15037}, {3574,11245}, {5059,15741}, {5068,11442}, {5073,11820}, {5462,24981}, {5890,16880}, {7488,33749}, {12007,13367}, {12241,13202}, {13358,30714}, {13597,26862}, {14862,15032}

X(34564) = reflection of X(125) in X(10821)




leftri  Centers related to the Moses-Jerabek and Moses-Lemoine conics: X(34565) - X(34574)  rightri

The Moses-Jerabek and Moses-Lemoine conics are introduced in the preamble before X(34426).

Contributed by Randy Hutson, October 31, 2019.

underbar



X(34565) = POLE OF BROCARD AXIS WRT MOSES-JERABEK CONIC

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

X(34565) lies on these lines: {2, 5097}, {6, 25}, {54, 13433}, {155, 27355}, {182, 21969}, {237, 5041}, {323, 6688}, {373, 1993}, {389, 3520}, {394, 11482}, {428, 12007}, {511, 15246}, {575, 3060}, {576, 3917}, {578, 32534}, {1173, 1199}, {1570, 20965}, {1994, 3292}, {3527, 26883}, {3567, 13367}, {3819, 15018}, {5012, 15516}, {5032, 14826}, {5093, 5650}, {5102, 7484}, {5107, 8041}, {5133, 11225}, {5158, 26907}, {5447, 15047}, {5462, 14627}, {5476, 11442}, {5622, 13417}, {5643, 10219}, {6329, 7499}, {6427, 10132}, {6428, 10133}, {6749, 14569}, {7409, 11550}, {10605, 11424}, {10619, 11745}, {10982, 11381}, {11004, 11451}, {11431, 26937}, {11746, 32226}, {12039, 15534}, {13321, 18475}, {13352, 16226}, {13434, 16625}, {13472, 26882}, {13754, 15038}, {20958, 20961}, {20959, 20962}

X(34565) = isogonal conjugate of isotomic conjugate of X(3628)
X(34565) = crosssum of X(2) and X(140)
X(34565) = crosspoint of X(6) and X(1173)
X(34565) = intersection of tangents to Jerabek hyperbola at X(6) and X(1173)
X(34565) = intersection of tangents to Moses-Jerabek conic at X(6) and X(389)
X(34565) = pole of Brocard axis wrt Moses-Jerabek conic
X(34565) = barycentric product X(6)*X(3628)
X(34565) = barycentric quotient X(3628)/X(76)
X(34565) = {X(6),X(13366)}-harmonic conjugate of X(34566)


X(34566) = POLE OF VAN AUBEL LINE WRT MOSES-JERABEK CONIC

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

X(34566) lies on these lines: {6, 25}, {389, 17506}, {575, 2979}, {1199, 6000}, {1993, 22234}, {1994, 3819}, {2781, 34468}, {3284, 32078}, {3292, 5643}, {5012, 22330}, {5097, 22352}, {5422, 16187}, {5644, 5651}, {7592, 32062}, {10282, 13472}, {11424, 13093}, {12099, 32226}, {15520, 21969}, {15860, 23606}

X(34566) = isogonal conjugate of isotomic conjugate of X(16239)
X(34566) = crosssum of X(2) and X(3628)
X(34566) = crosspoint of X(6) and X(34567)
X(34566) = intersection of tangents to Jerabek hyperbola at X(6) and X(34567)
X(34566) = intersection of tangents to Moses-Jerabek conic at X(6) and X(1199)
X(34566) = pole of van Aubel line wrt Moses-Jerabek conic
X(34566) = barycentric product X(6)*X(16239)
X(34566) = barycentric quotient X(16239)/X(76)
X(34566) = {X(6),X(13366)}-harmonic conjugate of X(34565)


X(34567) = X(34566)-CROSS CONJUGATE OF X(6)

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

The trilinear polar of X(34567) passes through X(647).

X(34567) lies on the Jerabek hyperbola and these lines: {6, 26882}, {64, 15033}, {68, 5056}, {69, 3533}, {74, 1199}, {140, 34483}, {248, 5041}, {265, 3850}, {550, 13623}, {578, 11270}, {1173, 13366}, {1176, 5097}, {1181, 14490}, {1614, 3527}, {3426, 7592}, {3519, 8254}, {3532, 5890}, {3545, 15077}, {3832, 32533}, {3845, 17505}, {4846, 5059}, {8550, 15321}, {10018, 13622}, {12242, 33565}, {13596, 16835}, {13603, 15032}, {16774, 23294}, {31371, 33703}

X(34567) = isogonal conjugate of X(3628)
X(34567) = X(34566)-cross conjugate of X(6)


X(34568) = TRILINEAR POLE OF LINE X(74)X(186)

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

Line X(74)X(186) is the tangent to the Jerabek hyperbola and the Moses-Jerabek conic at X(74).

X(34568) lies on these lines: {6, 11079}, {1304, 5502}, {1597, 9139}, {2394, 2404}, {2420, 14590}, {16077, 16237}

X(34568) = isogonal conjugate of X(14401)
X(34568) = X(647)-cross conjugate of X(74)
X(34568) = cevapoint of X(i) and X(j) for these {i,j}: {74, 647}, {112, 1304}
X(34568) = trilinear pole of line X(74)X(186)
X(34568) = barycentric product X(i)*X(j) for these {i,j}: {74, 16077}, {112, 31621}, {1304, 1494}
X(34568) = barycentric quotient X(i)/X(j) for these (i,j): (6, 14401), (74, 9033), (107, 34334), (110, 16163), (112, 3163), (1304, 30), (16077, 3260), (31621, 3267)


X(34569) = POLE OF FERMAT AXIS WRT MOSES-LEMOINE CONIC

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

Let U = X(1151)-Ceva conjugate of X(1152) and V = X(1152)-Ceva conjugate of X(1151). Then X(34569) is the Brocard axis intercept of line UV.

X(34569) lies on these lines: {3, 6}, {2549, 5702}, {14581, 15262}

X(34569) = crosssum of X(2) and complement of X(11064)
X(34569) = crosspoint of X(6) and X(34570)
X(34569) = crossdifference of PU(163)
X(34569) = crossdifference of every pair of points on line X(523)X(3146)
X(34569) = X(2)-Ceva conjugate of X(39084)
X(34569) = perspector of conic {{A,B,C,X(110),X(3532)}}
X(34569) = intersection of tangents to hyperbola {{A,B,C,X(2),X(6)}} at X(6) and X(34570)


X(34570) = X(34569)-CROSS CONJUGATE OF X(6)

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

X(34570) lies on hyperbola {{A,B,C,X(2),X(6)}} and these lines: {6, 15051}, {112, 5896}, {393, 3163}, {1636, 2433}, {1976, 21639}, {2071, 3284}, {2165, 6128}, {2881, 9178}

X(34570) = isogonal conjugate of complement of X(11064)
X(34570) = X(34569)-cross conjugate of X(6)
X(34570) = trilinear pole of line X(154)X(512)


X(34571) = POLE OF LINE X(2)X(6) WRT MOSES-LEMOINE CONIC

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

X(34571) lies on these lines: {3, 6}, {115, 3856}, {597, 7826}, {1992, 7822}, {3291, 34482}, {3629, 7889}, {3854, 5475}, {5032, 7795}, {5286, 14537}, {5346, 7603}, {5368, 10109}, {6292, 6329}, {7745, 14893}, {7747, 33699}, {7794, 32455}, {7804, 7894}, {7805, 7878}, {7817, 7921}, {7821, 16989}, {7829, 7845}, {7837, 7915}, {7838, 7852}, {7843, 7920}, {7858, 31275}, {7875, 7882}, {7890, 8584}, {12150, 32450}, {14614, 31239}

X(34571) = isogonal conjugate of isotomic conjugate of complement of X(34573)
X(34571) = crosssum of X(2) and X(34573)
X(34571) = crosspoint of X(6) and X(34572)
X(34571) = intersection of tangents to hyperbola {{A,B,C,X(2),X(6)}} at X(6) and X(34572)
X(34571) = intersection of tangents to Moses-Lemoine conic at X(6) and X(34482)
X(34571) = pole of line X(2)X(6) wrt Moses-Lemoine conic


X(34572) = X(34571)-CROSS CONJUGATE OF X(6)

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

The trilinear polar of X(34572) meets the line at infinity at X(512).

X(34572) lies on hyperbola {{A,B,C,X(2),X(6)}} and these lines: {2, 7826}, {111, 34482}, {2963, 5306}, {3108, 5007}, {5359, 21448}

X(34572) = isogonal conjugate of X(34573)
X(34572) = X(34571)-cross conjugate of X(6)


X(34573) = ISOGONAL CONJUGATE OF X(34572)

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

X(34573) lies on these lines: {2, 6}, {4, 21167}, {5, 3098}, {10, 9053}, {37, 29596}, {53, 11331}, {66, 10192}, {140, 1503}, {142, 4472}, {159, 16419}, {182, 632}, {187, 5031}, {239, 4478}, {308, 30736}, {344, 16677}, {346, 26104}, {441, 22052}, {442, 33844}, {487, 6437}, {488, 6438}, {511, 3628}, {518, 3634}, {542, 10124}, {545, 17235}, {546, 14810}, {547, 19130}, {549, 3818}, {574, 7789}, {594, 4395}, {625, 8367}, {631, 10516}, {635, 11543}, {636, 11542}, {639, 18762}, {640, 18538}, {698, 3934}, {732, 6683}, {742, 4698}, {858, 5888}, {894, 7238}, {902, 32781}, {1030, 21516}, {1086, 7227}, {1125, 3844}, {1216, 32191}, {1350, 3090}, {1352, 3526}, {1384, 7800}, {1386, 19862}, {1428, 7294}, {1495, 7499}, {1656, 5480}, {2330, 5326}, {2345, 7263}, {2781, 12900}, {2854, 6723}, {2916, 5596}, {3008, 17239}, {3091, 31884}, {3094, 8363}, {3096, 7745}, {3242, 9780}, {3247, 4657}, {3313, 5650}, {3416, 3624}, {3525, 5085}, {3533, 6776}, {3564, 16239}, {3661, 4399}, {3662, 7228}, {3723, 3912}, {3731, 4364}, {3739, 9055}, {3751, 19872}, {3819, 9969}, {3828, 9041}, {3834, 5750}, {3867, 5094}, {3943, 17302}, {3946, 4971}, {3973, 4643}, {4000, 4665}, {4048, 11285}, {4257, 17698}, {4265, 5047}, {4357, 4422}, {4361, 29611}, {4370, 17258}, {4445, 5222}, {4550, 15311}, {4670, 21255}, {4708, 5845}, {4755, 17225}, {5008, 7767}, {5024, 7795}, {5026, 31274}, {5044, 9021}, {5055, 31670}, {5067, 10519}, {5070, 14561}, {5096, 17531}, {5103, 5104}, {5157, 5651}, {5159, 8705}, {5210, 14001}, {5257, 31285}, {5318, 11289}, {5321, 11290}, {5585, 32973}, {5646, 16051}, {5748, 7232}, {5847, 19878}, {5921, 10541}, {5969, 6722}, {5972, 6698}, {6411, 11292}, {6412, 11291}, {6661, 7831}, {6667, 9024}, {6704, 7849}, {7392, 31860}, {7393, 15577}, {7405, 11745}, {7485, 20987}, {7495, 10546}, {7716, 8889}, {7784, 16045}, {7786, 32449}, {7815, 33185}, {7820, 8359}, {7824, 24273}, {7830, 19697}, {7832, 12055}, {7859, 10159}, {7865, 18907}, {7869, 31406}, {7881, 9606}, {7784, 16045}, {7786, 32449}, {7815, 33185}, {7820, 8359}, {7824, 24273}, {7830, 19697}, {7832, 12055}, {7859, 10159}, {7865, 18907}, {7869, 31406}, {7881, 9606}, {7937, 8370}, {8358, 32456}, {8361, 24256}, {8369, 8588}, {8998, 32304}, {9015, 31287}, {9019, 9822}, {9022, 20106}, {9167, 14928}, {9478, 14931}, {9830, 22247}, {10109, 19924}, {10387, 10589}, {10545, 32269}, {11178, 11539}, {11311, 18582}, {11312, 18581}, {11313, 23312}, {11314, 23311}, {11645, 11812}, {11742, 33023}, {12007, 34507}, {12100, 25561}, {13154, 15582}, {13881, 33221}, {14065, 18906}, {14869, 17508}, {15059, 25328}, {15485, 32784}, {15492, 17237}, {15585, 23300}, {15655, 33237}, {15694, 18440}, {15699, 21850}, {16674, 17321}, {17023, 17231}, {17227, 17368}, {17228, 17367}, {17348, 31191}, {17710, 29959}, {17768, 24295}, {18424, 33184}, {18584, 32968}, {23515, 33851}, {25555, 34380}

X(34573) = midpoint of X(i) and X(j) for these {i,j}: {2, 20582}, {6, 3631}, {69, 32455}, {140, 24206}, {141, 3589}
X(34573) = isogonal conjugate of X(34572)
X(34573) = complement of X(3589)
X(34573) = anticomplement of isotomic conjugate of isogonal conjugate of X(34571)
X(34573) = center of conic {{X(13),X(14),X(15),X(16),X(141)}}


X(34574) = TRILINEAR POLE OF LINE X(23)X(111)

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

Line X(23)X(111) is the tangent to hyperbola {{A,B,C,X(2),X(6)}} and the Moses-Lemoine conic at X(111), and to hyperbola {{A,B,C,X(691),PU(2)}} at X(691).

X(34574) lies on these lines: {6, 10630}, {110, 9171}, {671, 34171}, {691, 5467}, {892, 5466}, {5967, 9154}, {5968, 9139}

X(34574) = isogonal conjugate of X(1649)
X(34574) = crossdifference of every pair of points on line X(14444)X(23992)
X(34574) = trilinear pole of line X(23)X(111)
X(34574) = barycentric product X(i)*X(j) for these {i,j}: {99, 10630}, {111, 892}, {112, 5095}, {523, 34539}, {671, 691}, {18023, 32729}
X(34574) = barycentric quotient X(i)/X(j) for these (i,j): (6, 1649), (110, 2482), (111, 690), (351, 14444), (512, 23992), (691, 524), (892, 3266), (5095, 3267), (5467, 8030), (10630, 523), (32729, 187), (34539, 99)


X(34575) =  (name pending)

Barycentrics    (35*a^6 + 27*a^4*b^2 - 15*a^2*b^4 - 7*b^6 + 27*a^4*c^2 - 3*a^2*b^2*c^2 - 21*b^4*c^2 - 15*a^2*c^4 - 21*b^2*c^4 - 7*c^6)*(a^6 + 9*a^4*b^2 + 9*a^2*b^4 + b^6 + a^4*c^2 + 17*a^2*b^2*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 + 9*a^4*c^2 + 17*a^2*b^2*c^2 - b^4*c^2 + 9*a^2*c^4 + b^2*c^4 + c^6) : :

See Antreas Hatzipolakis and Peter Moses, Euclid 19 .

X(34575) lies on this line: {6,8937}

X(34575) = second Lemoine circle inverse of X(8937)


X(34576) =  X(3)X(33664)∩X(216)X(3463)

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

See Antreas Hatzipolakis and Peter Moses, Euclid 19 .

X(34576) lies on these lines: {3, 33664}, {216, 3463}

X(34576) = circumcircle-inverse of X(33664)


X(34577) =  MIDPOINT OF X(5) AND X(7488)

Barycentrics    2 a^10-5 a^8 (b^2+c^2)+(b^2-c^2)^4 (b^2+c^2)+2 a^6 (b^4+b^2 c^2+c^4)-a^2 (b^2-c^2)^2 (4 b^4+3 b^2 c^2+4 c^4)+a^4 (4 b^6+b^4 c^2+b^2 c^4+4 c^6) : :
Barycentrics    19 R^2 S^2-R^2 SB SC-6 S^2 SW+2 SB SC SW : :

As a point on the Euler line, X(34577) has Shinagawa coefficients [-5e-24f,7e+8f].

See Tran Quang Hung, Angel Montesdeoca and Ercole Suppa, Euclid 21 and Euclid 23 .

X(34577) lies on these lines: {2,3}, {52,22051}, {110,10203}, {143,8254}, {1154,15806}, {1291,15641}, {3470,16243}, {3519,9705}, {3580,32165}, {5944,32423}, {5972,32142}, {6689,10095}, {10272,10628}, {11743,18874}, {11808,15426}, {13364,20193}, {14128,32348}, {14449,23292}, {20299,20391}, {20379,23060}

X(34577) = midpoint of X(5) and X(7488)
X(34577) = reflection of X(i) in X(j) for these {i, j}: {140,7542}, {1594,3628}, {20299,20391}
X(34577) = center of pedal circle of X(5) wrt anticevian triangle of X(5)




leftri  Cyclologic centers: X(34578)-X(34581)  rightri

This preamble and centers X(34578)-X(34581) are based on notes contributed by Vu Thanh Tung and Vu Quoc My, October 31, 2019.

Let P = p:q:r (barycentrics) be a point in the plane of a triangle ABC, and let A' be the point, other than P, where the line AP meets the circle (PBC). Let U = u:v:w be a point (as a function of a, b, c), and let A'' = U-of-A'BC. Define B'' and C'' cyclically. Let T* = A''B''C''. The triangles ABC and T* are cyclologic; i.e., the circumcircles of A''BC, AB''C, ABC'' concur in a single point, called the ABC-to-T* cyclologic center, denoted by S(P,U).

S(P,X(2)) = (c^2 p^2 q^2 + a^2 p^2 q r + 2 b^2 p^2 q r - c^2 p^2 q r + 2 a^2 p q^2 r + b^2 p q^2 r - c^2 p q^2 r - 2 b^2 p^2 r^2 - a^2 p q r^2 - b^2 p q r^2 + c^2 p q r^2 - 2 a^2 q^2 r^2) (2 c^2 p^2 q^2 - a^2 p^2 q r + b^2 p^2 q r - 2 c^2 p^2 q r + a^2 p q^2 r - b^2 p q^2 r + c^2 p q^2 r - b^2 p^2 r^2 - 2 a^2 p q r^2 + b^2 p q r^2 - c^2 p q r^2 + 2 a^2 q^2 r^2) : :

S(P,X(3)) = p (-a^2 p^2 + b^2 p^2 + c^2 p^2 - a^2 p q + b^2 p q - c^2 p q - a^2 p r - b^2 p r + c^2 p r - 2 a^2 q r) (-a c p q + c^2 p q - a c q^2 + b^2 p r + a^2 q r - a c q r) (a c p q + c^2 p q + a c q^2 + b^2 p r + a^2 q r + a c q r) (c^2 p q - a b p r + b^2 p r + a^2 q r - a b q r - a b r^2) (c^2 p q + a b p r + b^2 p r + a^2 q r + a b q r + a b r^2) : :

S(P,X(4)) = p (a^2 p q - b^2 p q + a^2 q^2 - b^2 q^2 + c^2 q^2 - b^2 p r - b^2 q r + c^2 q r) (-c^2 p q + a^2 p r - c^2 p r + b^2 q r - c^2 q r + a^2 r^2 + b^2 r^2 - c^2 r^2) : :

The circumcircles of AB"C", A"BC", A"B"C also concur in a single point, called the T*-to-ABC cyclologic center, denoted by T(P,U).

T(P,X(2)) = a^2 (p - q) q (p - r) r : :

T(P,X(3)) = a^2 (2 b^2 c^2 p^2 q + a^2 c^2 p q^2 + b^2 c^2 p q^2 - c^4 p q^2 - a^2 b^2 p^2 r + b^4 p^2 r + b^2 c^2 p^2 r + a^4 q^2 r - a^2 b^2 q^2 r - a^2 c^2 q^2 r - a^2 b^2 p r^2 - b^4 p r^2 + b^2 c^2 p r^2 - 2 a^2 b^2 q r^2) (-a^2 c^2 p^2 q + b^2 c^2 p^2 q + c^4 p^2 q - a^2 c^2 p q^2 + b^2 c^2 p q^2 - c^4 p q^2 + 2 b^2 c^2 p^2 r - 2 a^2 c^2 q^2 r + a^2 b^2 p r^2 - b^4 p r^2 + b^2 c^2 p r^2 + a^4 q r^2 - a^2 b^2 q r^2 - a^2 c^2 q r^2) : :

T(P,X(4)) = a^2 (a^2 p - b^2 p - c^2 p + a^2 q - b^2 q + c^2 q) (a^2 p - b^2 p - c^2 p + a^2 r + b^2 r - c^2 r) (a^2 p q - b^2 p q - c^2 p q + a^2 q^2 - b^2 q^2 + c^2 q^2 - 2 b^2 p r - a^2 q r - b^2 q r + c^2 q r) (2 c^2 p q - a^2 p r + b^2 p r + c^2 p r + a^2 q r - b^2 q r + c^2 q r - a^2 r^2 - b^2 r^2 + c^2 r^2) : :

The appearance of (i,j,k) in the following list means that S(X(i),X(j)) = X(k):

1,2,34578
3,2,34579
5,2,34580
2,3,34581
4,2,671
6,2,6094
3,3,5961
4,3,265
5,3,24772
6,3,13493
1,4,80
2,4,671
3,4,265
6,4,67

The appearance of (i,j,k) in the following list means that T(X(i),X(j)) = X(k):

1,2,100
3,2,107
4,2,110
5,2,933
6,2,99
2,3,1296
3,3,110
4,3,110
5,3,1291
6,3,1296
1,4,100
2,4,1296
4,3,925
4,4,930
6,4,30247

S(P,X(2)) lies on the circumconic {{A,B,C,X(2),P}}. (Peter Moses, November 4, 2019)

S(P,X(4)) lies on the circumconic {{A,B,C,X(4),P}}. (Vu Thanh Tung, November 5, 2019)

underbar



X(34578) =  CYCLOLOGIC CENTER S(X(1),X(2))

Barycentrics    (a^2 + a*b + b^2 - 2*a*c - 2*b*c + c^2)*(a^2 - 2*a*b + b^2 + a*c - 2*b*c + c^2) : :
X(34578) = 2 X[1358] + X[5540]

If you have GeoGebra, you can view X(34578).

X(34578) lies on the circumconic {{A,B,C,X(1),X(2)}}, the cubics K086 and K949, and these lines: {1, 528}, {2, 1111}, {7, 15730}, {36, 105}, {57, 1358}, {80, 10708}, {81, 17205}, {88, 3008}, {89, 5222}, {241, 2006}, {291, 1739}, {519, 1280}, {527, 5526}, {544, 9317}, {553, 2982}, {651, 1170}, {673, 4089}, {903, 5376}, {955, 30274}, {957, 2835}, {1002, 2809}, {1022, 6084}, {1255, 5249}, {1292, 15931}, {1323, 15727}, {1390, 10712}, {1638, 2826}, {1930, 30701}, {2795, 26725}, {3021, 13384}, {3570, 32097}, {3576, 28915}, {4000, 7208}, {5425, 10699}, {8056, 24795}, {17078, 34018}, {32019, 33943}

X(34578) = isogonal conjugate of X(5526)
X(34578) = isotomic conjugate of X(17264)
X(34578) = X(i)-cross conjugate of X(j) for these (i,j): {1155, 7}, {5011, 79}, {5030, 5557}
X(34578) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5526}, {2, 19624}, {6, 3935}, {9, 2078}, {31, 17264}, {100, 22108}, {101, 3887}, {103, 28345}, {190, 8645}, {692, 30565}, {2291, 6594}, {4845, 15730}
X(34578) = cevapoint of X(i) and X(j) for these (i,j): {1086, 1638}, {1323, 10481}
X(34578) = trilinear pole of line {354, 513}
X(34578) = crossdifference of every pair of points on line {8645, 22108}
X(34578) = Gibert-Burek-Moses concurrent circles image of X(5528)
X(34578) = barycentric product X(i)*X(j) for these {i,j}: {7, 3254}, {693, 1308}
X(34578) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3935}, {2, 17264}, {6, 5526}, {31, 19624}, {56, 2078}, {513, 3887}, {514, 30565}, {649, 22108}, {667, 8645}, {910, 28345}, {1155, 6594}, {1308, 100}, {3254, 8}, {6610, 15730}


X(34579) =  CYCLOLOGIC CENTER S(X(3),X(2))

Barycentrics    (a^8 + a^6*b^2 - 4*a^4*b^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 + 2*a^4*c^4 - a^2*b^2*c^4 + 2*b^4*c^4 - 2*a^2*c^6 - 2*b^2*c^6 + c^8)*(a^8 - 2*a^6*b^2 + 2*a^4*b^4 - 2*a^2*b^6 + b^8 + a^6*c^2 + 2*a^4*b^2*c^2 - a^2*b^4*c^2 - 2*b^6*c^2 - 4*a^4*c^4 + 2*a^2*b^2*c^4 + 2*b^4*c^4 + a^2*c^6 - 2*b^2*c^6 + c^8) : :
Trilinears    1/(b^2 cos(C - A) (cos C) (sin 2A - sin 2B) - c^2 cos(A - B) (cos B) (sin 2C - sin 2A)) : :

X(34579) lies on the circumconic {{A,B,C,X(2),X(3)}}, the cubic K811 and these lines: {3, 9530}, {30, 17974}, {97, 112}, {127, 1073}, {276, 15352}, {297, 10718}, {394, 1625}, {2794, 32063}, {3926, 14570}

X(34579) = isogonal conjugate of X(13509)
X(34579) = trilinear pole of line {51, 520}
X(34579) = barycentric quotient X(6)/X(13509)


X(34580) =  CYCLOLOGIC CENTER S(X(5),X(2))

Barycentrics    (2*a^14 - 7*a^12*b^2 + 9*a^10*b^4 - 4*a^8*b^6 - 4*a^6*b^8 + 9*a^4*b^10 - 7*a^2*b^12 + 2*b^14 - 7*a^12*c^2 + 15*a^10*b^2*c^2 - 8*a^8*b^4*c^2 - 8*a^4*b^8*c^2 + 15*a^2*b^10*c^2 - 7*b^12*c^2 + 6*a^10*c^4 - 11*a^8*b^2*c^4 + 5*a^6*b^4*c^4 + 5*a^4*b^6*c^4 - 11*a^2*b^8*c^4 + 6*b^10*c^4 + 5*a^8*c^6 + 9*a^6*b^2*c^6 - a^4*b^4*c^6 + 9*a^2*b^6*c^6 + 5*b^8*c^6 - 10*a^6*c^8 - 8*a^4*b^2*c^8 - 8*a^2*b^4*c^8 - 10*b^6*c^8 + 3*a^4*c^10 + 3*b^4*c^10 + 2*a^2*c^12 + 2*b^2*c^12 - c^14)*(2*a^14 - 7*a^12*b^2 + 6*a^10*b^4 + 5*a^8*b^6 - 10*a^6*b^8 + 3*a^4*b^10 + 2*a^2*b^12 - b^14 - 7*a^12*c^2 + 15*a^10*b^2*c^2 - 11*a^8*b^4*c^2 + 9*a^6*b^6*c^2 - 8*a^4*b^8*c^2 + 2*b^12*c^2 + 9*a^10*c^4 - 8*a^8*b^2*c^4 + 5*a^6*b^4*c^4 - a^4*b^6*c^4 - 8*a^2*b^8*c^4 + 3*b^10*c^4 - 4*a^8*c^6 + 5*a^4*b^4*c^6 + 9*a^2*b^6*c^6 - 10*b^8*c^6 - 4*a^6*c^8 - 8*a^4*b^2*c^8 - 11*a^2*b^4*c^8 + 5*b^6*c^8 + 9*a^4*c^10 + 15*a^2*b^2*c^10 + 6*b^4*c^10 - 7*a^2*c^12 - 7*b^2*c^12 + 2*c^14) : :

X(34580) lies on the circumconic {{A,B,C,X(2),X(5)}} these lines: {}


X(34581) =  CYCLOLOGIC CENTER S(X(2),X(3))

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

X(34581) lies on the circumconic {{A,B,C,X(2),X(3)}}, the curves K043 and Q053, and these lines: {2, 10354}, {3, 13493}, {6, 2482}, {468, 8753}, {597, 34422}, {2444, 9125}, {5461, 5512}, {6593, 13608}

X(34581) = isogonal conjugate of X(13492)
X(34581) = complement of X(34164)
X(34581) = circumcircle-inverse of X(13493)
X(34581) = antigonal image of X(25409)
X(34581) = symgonal image of X(16939)
X(34581) = X(187)-cross conjugate of X(13608)
X(34581) = X(i)-isoconjugate of X(j) for these (i,j): {1, 13492}, {897, 10354}
X(34581) = crosssum of X(6) and X(10355)
X(34581) = crossdifference of every pair of points on line {6088, 10354}
X(34581) = complementary conjugate of complement of X(38533)
X(34581) = barycentric product X(i)*X(j) for these {i,j}: {1499, 6082}, {13608, 34166}
X(34581) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 13492}, {187, 10354}, {1384, 11580}, {1992, 11054}, {8644, 6088}, {13493, 14262}, {18775, 17952}


X(34582) =  X(2)X(3)∩X(4669)X(11900)

Barycentrics    (2 a^4-(b^2-c^2)^2-a^2 (b^2+c^2)) (5 a^8-5 a^6 (b^2+c^2)+7 a^2 (b^2-c^2)^2 (b^2+c^2)+a^4 (-6 b^4+17 b^2 c^2-6 c^4)-(b^2-c^2)^2 (b^4+7 b^2 c^2+c^4)) : :

See Tran Quang Hung and Angel Montesdeoca, Euclid 29 .

X(34582) lies on these lines: {2,3}, {4669,11900}, {4677,12438}, {11055,12794}, {12583,15533}, {13392,20123}, {14583,16319}, {22698,33706}

X(34582) = midpoint of X(i) and X(j) for these {i, j}: {2,3081}, {1651,4240}, {2870,29343}, {11251,20128}
X(34582) = reflection of X(i) in X(j) for these {i, j}: {402,1651}, {11049,402}, {11050,15184}
X(34582) = X(2)-of-anticevian-triangle-of-X(30)


X(34583) =  X(1)X(3)∩X(2)X(513)

Barycentrics    a*(a^3*b^2 - a*b^4 - 4*a^3*b*c + 2*a^2*b^2*c + 3*a*b^3*c - b^4*c + a^3*c^2 + 2*a^2*b*c^2 - 6*a*b^2*c^2 + b^3*c^2 + 3*a*b*c^3 + b^2*c^3 - a*c^4 - b*c^4) : :
X(34583) = 4 X[140] - X[31847], X[3025] + 2 X[22102], 2 X[3035] + X[3937], X[3878] + 2 X[13752]

See Antreas Hatzipolakis and Peter Moses, Euclid 33 .

X(34583) lies on these lines: {1, 3}, {2, 513}, {11, 29349}, {59, 17074}, {63, 23343}, {140, 31847}, {497, 33646}, {614, 16501}, {649, 5701}, {650, 24484}, {901, 1621}, {1054, 18191}, {1357, 17724}, {1376, 15635}, {1464, 19335}, {2810, 6174}, {2818, 21154}, {2841, 34123}, {2886, 6075}, {3025, 5432}, {3035, 3937}, {3109, 3833}, {3259, 3816}, {3306, 16494}, {3681, 17780}, {3740, 9458}, {3752, 16507}, {3848, 32772}, {3878, 13752}, {3911, 29353}, {4499, 4997}, {5218, 33647}, {5442, 34466}, {6789, 10176}, {9004, 16504}, {16467, 32911}, {16495, 16610}, {20718, 33852}, {25048, 30577}, {26910, 34151}

X(34583) = incircle-inverse of X(3999)
X(34583) = crossdifference of every pair of points on line {650, 3230}
X(34583) = intersection, other than X(3), of line X(1)X(3) and circle O(2,3)
X(34583) = intersection, other than X(2), of circles O(2,3) and O(2,165)
X(34583) = X(100)-of-X(2)-Brocard-triangle
X(34583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2446, 2447, 3999}


X(34584) =  X(30)X(511)∩X(74)X(382)

Barycentrics    4 a^10-5 a^8 (b^2+c^2)-2 (b^2-c^2)^4 (b^2+c^2)+a^6 (-7 b^4+22 b^2 c^2-7 c^4)-a^2 (b^2-c^2)^2 (b^4+13 b^2 c^2+c^4)+a^4 (11 b^6-12 b^4 c^2-12 b^2 c^4+11 c^6) : :

See Kadir Altintas and Angel Montesdeoca, Euclid 44 .

X(34584) lies on these lines: {3,1539}, {4,12041}, {5,13202}, {20,1511}, {26,2935}, {30,511}, {74,382}, {110,1657}, {113,550}, {125,3627}, {143,974}, {146,3529}, {155,17812}, {156,5895}, {265,3146}, {376,14643}, {381,15055}, {399,17800}, {546,6699}, {548,5972}, {1112,13630}, {1658,13293}, {1885,10095}, {1986,11565}, {3024,10483}, {3357,18379}, {3448,33703}, {3530,12900}, {3534,15035}, {3830,14644}, {3843,15059}, {3845,23515}, {3850,6723}, {3853,7687}, {4299,12374}, {4302,12373}, {5059,12383}, {5073,10620}, {5076,15021}, {5446,17855}, {5642,15686}, {5654,11744}, {5655,11001}, {5894,13561}, {5925,14852}, {6070,21269}, {6101,12825}, {6102,17854}, {6240,12133}, {7699,18550}, {7723,18565}, {7731,12279}, {8144,19505}, {9140,15684}, {9919,12085}, {10065,12943}, {10081,12953}, {10117,12084}, {10118,32047}, {10264,10990}, {10272,12103}, {10575,13417}, {10706,15681}, {10817,13903}, {10818,13961}, {11250,13289}, {11541,12317}, {11561,18563}, {11598,19506}, {11709,22793}, {11801,20417}, {11806,13598}, {12102,20397}, {12219,18439}, {12290,13201}, {12824,18564}, {12893,17714}, {12902,15054}, {13406,32743}, {13416,14128}, {13488,15473}, {13915,23251}, {13979,23261}, {14708,16105}, {14855,16223}, {15051,15696}, {15063,34153}, {15081,17578}, {15122,20725}, {15331,25564}, {15472,32046}, {15647,22802}, {15682,20126}, {15704,16163}, {15761,23315}, {31824,32311}




leftri  Points related to the circumellipse of the medial and incentral triangles: X(34585)-X(34593)  rightri

This preamble is based on notes from Dasari Naga Vijay Krishna and Peter Moses, November, 2019.

The circumellipse of the medial and incentral triangles (the CEMIT), given by the equation

b*c*x^2 - a*c*x*y - b*c*x*y + a*c*y^2 - a*b*x*z - b*c*x*z - a*b*y*z - a*c*y*z + a*b*z^2 = 0,

has center X(1125) and perpsector X(34585). The axes are parallel to the asymptotes of the Feuerbach hyperbola, and

Major axis has length = (b + c) (c + a) (a + b) (R + |OI|) / (32 R s), in the line X(1125)X(3307)
Minor axis has length = (b + c) (c + a) (a + b) (R - |OI|) / (32 R s), in the line X(1125)X(3308)

The CEMIT passes through the following points:
vertices of the medial triangle
vertices of the incentral triangle
vertices of the anti-Aquila triangle
X(i) for these i: 11, 214, 244, 1015, 8054, 8299, 10494, 14714, 17417, 17419, 17421, 17761, 17793, 34586, 34587, 34588, 34589, 34590, 34591, 34592, 34593.

If P = p:q;r lies on the circumcircle then the point f(P) = a (q c (a + c) + r b (a + b)) : : lies on the CEMIT. Inversely, if U lies on the CEMIT, then the point g(U) = (a (a + b) (a + c) (-u + v + w) : : lies on the circumcircle.

The appearance of (i,j) in the following list means that f(X(i)) = X(j): (99,1015), (100,244), (101,17761), (104,34586), (105,8299), (106,34587), (109,34589), (110,11), (741,17793), (759,214), (901,34590), (934,345191), (1113,34592), (1114,34593), (34594,8054).

The CEMIT is the bicevian conic of X(1) and X(2). For a discussion of general bicevian conics, see Bernard Gibert, Bicevian Conics and CPCC Cubics

underbar



X(34585) =  PERSPECTOR OF THE CIRCUMELLIPSE OF THE MEDIAL AND INCENTRAL TRIANGLES

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

X(34585) lies on these lines: {238, 18166}, {350, 16709}, {740, 1125}, {757, 5284}, {1100, 2238}, {1284, 32636}, {3616, 25660}, {16696, 30571}, {16826, 18082}

X(34585) = cevapoint of X(244) and X(4983)
X(34585) = trilinear pole of line {4979, 6372}
X(34585) = X(16726)-cross conjugate of X(513)
X(34585) = X(i)-isoconjugate of X(j) for these (i,j): {10, 33774}, {37, 33766}, {42, 33770}, {101, 31290}, {213, 33779}, {1252, 24185}
X(34585) = barycentric quotient X(i)/X(j) for these {i,j}: {58, 33766}, {81, 33770}, {86, 33779}, {244, 24185}, {513, 31290}, {1333, 33774}


X(34586) =  X(1)X(5)∩X(3)X(102)

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

X(34586) lies on the Jerabek circumhyperbola of the medial triangle, the circumellipse of the medial and incentral triangles, and these lines: {1, 5}, {3, 102}, {6, 101}, {36, 1464}, {43, 26727}, {55, 1480}, {56, 215}, {59, 953}, {65, 1772}, {73, 1385}, {104, 651}, {113, 10017}, {141, 997}, {214, 3738}, {222, 10269}, {244, 942}, {255, 26286}, {500, 2646}, {517, 1457}, {603, 32612}, {860, 1870}, {1062, 2883}, {1064, 2293}, {1066, 1201}, {1125, 25493}, {1149, 25405}, {1209, 18447}, {1318, 1391}, {1361, 23981}, {1419, 3576}, {1458, 5126}, {1493, 5563}, {1735, 14988}, {1745, 18481}, {1769, 14299}, {1771, 6924}, {2605, 11720}, {2617, 3109}, {2635, 28160}, {2654, 9955}, {2771, 7004}, {2800, 24025}, {2812, 11712}, {2841, 23845}, {2964, 14804}, {3216, 24885}, {3240, 11041}, {3295, 33537}, {3487, 33148}, {3579, 22072}, {3939, 22141}, {3946, 17761}, {4303, 13624}, {4306, 23072}, {4559, 13006}, {5204, 33556}, {5313, 11529}, {5755, 22134}, {5887, 17102}, {5902, 26742}, {6923, 34029}, {6958, 34030}, {7078, 11249}, {8757, 12114}, {8776, 9502}, {9946, 12016}, {10742, 18340}, {11248, 34040}, {16203, 34046}, {17055, 30115}, {17421, 19861}, {17502, 22053}, {18254, 24433}, {21664, 23706}, {22758, 34048}, {22765, 23071}, {22935, 33649}, {23703, 33814}, {24806, 26446}, {30116, 30858}

X(34586) = midpoint of X(i) and X(j) for these {i,j}: {1, 4551}, {1457, 22350}
X(34586) = complement of X(38955)
X(34586) = complement of the isogonal conjugate of X(859)
X(34586) = complement of the isotomic conjugate of X(17139)
X(34586) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 2245}, {58, 517}, {162, 8677}, {517, 3454}, {603, 856}, {849, 15325}, {859, 10}, {908, 21245}, {1333, 3911}, {1457, 442}, {1465, 17052}, {1474, 26011}, {1769, 125}, {2183, 1211}, {2206, 8609}, {2427, 4129}, {3310, 8287}, {4246, 20316}, {10015, 21253}, {17139, 2887}, {22350, 21530}, {23220, 16573}, {23788, 21252}
X(34586) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 517}, {2, 2245}, {59, 23981}, {110, 8677}, {651, 654}, {1262, 1983}, {6742, 2804}
X(34586) = X(i)-isoconjugate of X(j) for these (i,j): {80, 104}, {909, 18359}, {2161, 34234}, {2250, 24624}, {2342, 18815}, {6187, 18816}
X(34586) = crosspoint of X(i) and X(j) for these (i,j): {1, 36}, {2, 17139}, {859, 14260}
X(34586) = crosssum of X(1) and X(80)
X(34586) = crossdifference of every pair of points on line {654, 900}
X(34586) = barycentric product X(i)*X(j) for these {i,j}: {1, 16586}, {36, 908}, {63, 1845}, {320, 2183}, {517, 3218}, {859, 3936}, {1457, 32851}, {1465, 4511}, {1769, 4585}, {1785, 22128}, {2245, 17139}, {2323, 22464}, {2427, 4453}, {3262, 7113}, {3738, 24029}, {3904, 23981}, {17923, 22350}
X(34586) = barycentric quotient X(i)/X(j) for these {i,j}: {36, 34234}, {517, 18359}, {859, 24624}, {908, 20566}, {1457, 2006}, {1465, 18815}, {1845, 92}, {1870, 16082}, {2183, 80}, {3218, 18816}, {3724, 2250}, {7113, 104}, {16586, 75}, {21801, 15065}, {23981, 655}
X(34586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1718, 1411}, {1, 6127, 80}, {1, 6326, 1807}, {1, 32486, 1387}, {202, 203, 17455}, {1066, 1201, 24928}


X(34587) =  X(10)X(11)∩X(106)X(190)

Barycentrics    (2*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(34587) = 3 X[392] + X[22306], 5 X[3616] - X[17154], X[22313] - 5 X[25917]

X(34587) lies on the circumellipse of the medial and incentral triangles and these lines: {1, 3952}, {2, 4674}, {10, 11}, {37, 537}, {80, 24709}, {100, 6789}, {106, 190}, {192, 995}, {214, 900}, {244, 1125}, {514, 4465}, {519, 3992}, {596, 25253}, {891, 17793}, {997, 4319}, {1193, 4065}, {1201, 3159}, {1227, 17195}, {1387, 4422}, {2835, 18589}, {3230, 17475}, {3259, 11813}, {3616, 17154}, {3739, 17761}, {3878, 25652}, {3899, 30957}, {4013, 24222}, {4368, 14432}, {4511, 5497}, {4574, 16685}, {4687, 18061}, {4717, 20891}, {5541, 9458}, {6532, 17164}, {6544, 21832}, {6788, 26139}, {8683, 25440}, {11700, 11796}, {16173, 33115}, {16489, 32927}, {19582, 24068}, {21290, 24864}, {24176, 28352}

X(34587) = midpoint of X(i) and X(j) for these {i,j}: {1, 3952}, {4738, 17460}
X(34587) = reflection of X(i) in X(j) for these {i,j}: {10, 24003}, {244, 1125}, {4065, 14752}
X(34587) = complement of X(4674)
X(34587) = complement of the isotomic conjugate of X(30939)
X(34587) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 3936}, {21, 5123}, {44, 1211}, {58, 519}, {81, 3834}, {86, 21241}, {110, 900}, {163, 3960}, {214, 31845}, {519, 3454}, {593, 4395}, {662, 4928}, {692, 21894}, {741, 25351}, {759, 6702}, {900, 125}, {902, 1213}, {1023, 4129}, {1319, 442}, {1333, 16610}, {1404, 17056}, {1412, 17067}, {1576, 3310}, {1635, 8287}, {1960, 115}, {2206, 8610}, {2251, 16589}, {3285, 2}, {3733, 1647}, {3762, 21253}, {3911, 17052}, {4273, 27751}, {4358, 21245}, {5440, 21530}, {9459, 21838}, {14407, 6627}, {16704, 141}, {17780, 31946}, {22086, 15526}, {22356, 440}, {23202, 18591}, {23344, 661}, {30576, 3739}, {30606, 21246}, {30939, 2887}
X(34587) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 519}, {3952, 900}
X(34587) = crosspoint of X(2) and X(30939)
X(34587) = crossdifference of every pair of points on line {4491, 21786}
X(34587) = barycentric product X(519)*X(17495)
X(34587) = barycentric quotient X(i)/X(j) for these {i,j}: {17495, 903}, {23169, 1797}
X(34587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {121, 1145, 10}, {1145, 16594, 121}, {3884, 25079, 10}, {4013, 25031, 24222}, {24222, 30566, 4013}


X(34588) =  X(3)X(214)∩X(11)X(123)

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

X(34588) lies on the circumellipse of the medial and incentral triangles and these lines: {3, 214}, {11, 123}, {219, 23137}, {392, 17102}, {521, 24031}, {1001, 16579}, {1214, 25941}, {2170, 17417}, {3452, 22027}, {3771, 17793}, {17073, 21233}, {17421, 18210}

X(34588) = isotomic conjugate of polar conjugate of X(38345)
X(34588) = complement of the isogonal conjugate of X(23189)
X(34588) = X(i)-complementary conjugate of X(j) for these (i,j): {21, 20316}, {48, 1577}, {58, 521}, {60, 8062}, {78, 31946}, {162, 3042}, {212, 661}, {219, 4129}, {261, 21259}, {270, 520}, {283, 513}, {332, 21260}, {521, 3454}, {603, 656}, {652, 1211}, {905, 17052}, {1019, 16608}, {1333, 14837}, {1408, 21172}, {1437, 522}, {1444, 17072}, {1459, 442}, {1790, 4885}, {1808, 3837}, {1812, 3835}, {1946, 1213}, {2150, 525}, {2185, 30476}, {2193, 514}, {2194, 3239}, {2206, 6588}, {2299, 14298}, {2327, 20317}, {3733, 1210}, {3737, 5}, {3937, 8286}, {4091, 18642}, {4558, 21232}, {4560, 20305}, {4575, 3035}, {6332, 21245}, {7004, 125}, {7117, 8287}, {7252, 226}, {7254, 142}, {15411, 21244}, {15419, 17046}, {17219, 21252}, {18155, 21243}, {21789, 20262}, {22096, 16613}, {22379, 6739}, {22383, 17056}, {23090, 3452}, {23092, 20528}, {23189, 10}, {23224, 18641}, {26932, 21253}, {32661, 16578}
X(34588) = X(1)-Ceva conjugate of X(521)
X(34588) = complement of polar conjugate of X(4560)
X(34588) = X(i)-isoconjugate of X(j) for these (i,j): {4, 15386}, {109, 26704}, {653, 32653}, {2217, 7012}, {2406, 32700}, {7115, 13478}
X(34588) = crosspoint of X(i) and X(j) for these (i,j): {1, 21189}, {4560, 8048}
X(34588) = crosssum of X(197) and X(4559)
X(34588) = crossdifference of every pair of points on line {1415, 32653}
X(34588) = barycentric product X(i)*X(j) for these {i,j}: {63, 124}, {573, 17880}, {2968, 17080}, {3869, 26932}, {4417, 7004}, {6332, 21189}, {17219, 21078}, {22134, 34387}
X(34588) = barycentric quotient X(i)/X(j) for these {i,j}: {48, 15386}, {124, 92}, {573, 7012}, {650, 26704}, {1946, 32653}, {3185, 7115}, {6589, 108}, {7004, 13478}, {7117, 2217}, {10571, 7128}, {21189, 653}, {22134, 59}, {26932, 2995}
X(34588) = {X(2968),X(16596)}-harmonic conjugate of X(123)


X(34589) =  X(10)X(140)∩X(11)X(124)

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(34589) lies on the circumellipse of the medial and incentral triangles and these lines: {1, 26095}, {2, 4551}, {9, 30942}, {10, 140}, {11, 124}, {56, 10570}, {109, 34234}, {116, 123}, {125, 6075}, {244, 1109}, {496, 20306}, {522, 7004}, {946, 2818}, {958, 16286}, {1015, 1146}, {1125, 25493}, {1386, 11019}, {1647, 8054}, {1768, 24410}, {2170, 17419}, {2310, 4939}, {2323, 32919}, {2810, 3038}, {2886, 24251}, {3119, 4521}, {3660, 26011}, {3741, 4154}, {3742, 6708}, {3831, 5795}, {3911, 26013}, {4025, 20901}, {4030, 4847}, {5267, 6097}, {6245, 12262}, {6590, 21339}, {6682, 16579}, {6718, 25968}, {10479, 30478}, {11813, 12261}, {12259, 21616}, {17219, 34387}, {17417, 21044}, {17761, 24285}, {26871, 34029}, {30827, 30957}

X(34589) = midpoint of X(7004) and X(24026)
X(34589) = complement of X(4551)
X(34589) = complement of the isogonal conjugate of X(3737)
X(34589) = complement of the isotomic conjugate of X(18155)
X(34589) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 1577}, {8, 31946}, {9, 4129}, {11, 125}, {21, 513}, {28, 521}, {29, 20316}, {55, 661}, {56, 656}, {58, 522}, {60, 523}, {81, 4885}, {86, 17072}, {107, 3042}, {110, 3035}, {163, 16578}, {244, 8286}, {261, 512}, {270, 8062}, {272, 8676}, {283, 20315}, {284, 514}, {314, 21260}, {333, 3835}, {513, 442}, {514, 17052}, {521, 21530}, {522, 3454}, {593, 17069}, {643, 24003}, {645, 27076}, {649, 17056}, {650, 1211}, {652, 440}, {662, 21232}, {663, 1213}, {667, 2092}, {741, 25380}, {759, 3738}, {884, 2238}, {905, 18642}, {1014, 3900}, {1015, 17058}, {1019, 142}, {1021, 3452}, {1169, 3910}, {1178, 3907}, {1333, 905}, {1364, 122}, {1408, 6129}, {1412, 7658}, {1413, 17898}, {1436, 24018}, {1459, 18641}, {1474, 14837}, {1509, 17066}, {1576, 13006}, {1817, 20314}, {1946, 18591}, {2053, 798}, {2150, 14838}, {2170, 8287}, {2185, 4369}, {2189, 525}, {2194, 650}, {2203, 6588}, {2204, 2509}, {2206, 6589}, {2287, 20317}, {2299, 3239}, {2311, 812}, {2328, 4521}, {3063, 16589}, {3248, 16613}, {3271, 115}, {3286, 3126}, {3669, 18635}, {3709, 6537}, {3733, 1}, {3737, 10}, {3738, 31845}, {4391, 21245}, {4560, 141}, {4565, 17044}, {4833, 17057}, {4858, 21253}, {5324, 17115}, {5546, 4422}, {7117, 15526}, {7192, 2886}, {7199, 17046}, {7203, 11019}, {7252, 2}, {7253, 1329}, {7254, 17073}, {7255, 17792}, {7256, 3038}, {15419, 18639}, {16726, 4904}, {17096, 21258}, {17197, 116}, {17205, 17059}, {17217, 20338}, {17925, 16608}, {18021, 23301}, {18101, 7668}, {18155, 2887}, {18191, 11}, {18197, 20528}, {19302, 32679}, {21789, 9}, {22383, 18592}, {23189, 3}, {26932, 127}, {27527, 21250}, {28660, 21262}, {34079, 10015}
X(34589) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 522}, {2995, 514}, {11109, 21173}
X(34589) = X(11998)-cross conjugate of X(24237)
X(34589) = X(i)-isoconjugate of X(j) for these (i,j): {59, 34434}, {2051, 2149}
X(34589) = crosspoint of X(i) and X(j) for these (i,j): {1, 21173}, {2, 18155}
X(34589) = crossdifference of every pair of points on line {4559, 23845}
X(34589) = polar conjugate of isogonal conjugate of X(38344)
X(34589) = barycentric product X(i)*X(j) for these {i,j}: {8, 24237}, {11, 14829}, {75, 11998}, {522, 17496}, {572, 34387}, {2975, 4858}, {4391, 21173}, {11109, 26932}, {17074, 24026}, {17197, 17751}
X(34589) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 2051}, {572, 59}, {2170, 34434}, {2975, 4564}, {11998, 1}, {14829, 4998}, {17074, 7045}, {17197, 20028}, {17496, 664}, {20986, 2149}, {21173, 651}, {23187, 1813}, {24237, 7}
X(34589) = {X(11),X(26932)}-harmonic conjugate of X(124)


X(34590) =  X(11)X(513)∩X(244)X(523)

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

X(34590) lies on the circumellipse of the medial and incentral triangles and these lines: {2, 16506}, {11, 513}, {121, 17757}, {214, 519}, {244, 523}, {650, 1015}, {672, 4370}, {1086, 4379}, {1459, 3756}, {1647, 4448}, {3712, 8299}, {4124, 21129}, {4369, 17761}, {4871, 16594}, {6163, 25531}, {7004, 14284}, {21104, 21139}, {23758, 23772}

X(34590) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 21894}, {44, 4129}, {58, 900}, {81, 4928}, {519, 31946}, {649, 3936}, {900, 3454}, {902, 661}, {1019, 3834}, {1333, 3960}, {1404, 1577}, {1635, 1211}, {1647, 125}, {1960, 1213}, {2087, 8287}, {2206, 3310}, {3285, 514}, {3733, 519}, {3737, 5123}, {3762, 21245}, {4570, 6550}, {7192, 21241}, {14407, 6537}, {16704, 3835}, {22086, 440}, {30576, 4369}, {30725, 17052}, {30939, 21260}, {34079, 21198}
X(34590) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 900}, {2, 21894}, {4581, 6550}
X(34590) = crossdifference of every pair of points on line {2427, 23832}
X(34590) = barycentric product X(900)*X(21222)
X(34590) = barycentric quotient X(i)/X(j) for these {i,j}: {1647, 14554}, {5053, 9268}, {21222, 4555}, {21786, 901}


X(34591) =  X(9)X(48)∩X(19)X(102)

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

X(34591) lies on the circumellipse of the medial and incentral triangles and these lines: {1, 1783}, {2, 1952}, {9, 48}, {11, 1146}, {19, 102}, {36, 8558}, {37, 22063}, {56, 7367}, {63, 6516}, {73, 1212}, {77, 25915}, {78, 1802}, {123, 31653}, {124, 5190}, {201, 22070}, {219, 1807}, {220, 7124}, {238, 8766}, {244, 2632}, {281, 1953}, {656, 3269}, {946, 1855}, {960, 8299}, {1015, 17435}, {1071, 22088}, {1435, 2184}, {1457, 5089}, {1565, 3942}, {1826, 20263}, {1984, 2310}, {2188, 30223}, {2260, 9119}, {2272, 6001}, {3061, 6554}, {3721, 9367}, {3900, 24010}, {3930, 6603}, {4423, 20277}, {4858, 17761}, {6332, 17880}, {6507, 6513}, {7004, 7117}, {8227, 23058}, {11375, 17451}, {12739, 17439}, {16502, 20280}, {16550, 33587}, {17421, 20974}, {21246, 30017}, {28070, 34526}, {30827, 30858}

X(34591) = isogonal conjugate of X(7128)
X(34591) = complement of X(4566)
X(34591) = complement of the isogonal conjugate of X(21789)
X(34591) = complement of the isotomic conjugate of X(7253)
X(34591) = isotomic conjugate of the polar conjugate of X(2310)
X(34591) = isogonal conjugate of the polar conjugate of X(24026)
X(34591) = polar conjugate of X(24032)
X(34591) = polar conjugate of the isotomic conjugate of X(24031)
X(34591) = polar conjugate of the isogonal conjugate of X(2638)
X(34591) = crosssum of circumcircle intercepts of line X(1)X(19)
X(34591) = X(i)-complementary conjugate of X(j) for these (i,j): {21, 17072}, {31, 656}, {41, 1577}, {58, 3900}, {163, 17044}, {200, 31946}, {220, 4129}, {284, 4885}, {649, 18635}, {650, 17052}, {652, 18642}, {657, 1211}, {663, 442}, {667, 1834}, {1019, 21258}, {1021, 141}, {1043, 21260}, {1098, 512}, {1146, 21253}, {1253, 661}, {1333, 7658}, {1576, 24025}, {1946, 18641}, {2150, 17069}, {2185, 17066}, {2194, 522}, {2203, 21172}, {2204, 14837}, {2206, 6129}, {2208, 17898}, {2287, 3835}, {2299, 521}, {2310, 125}, {2326, 30476}, {2328, 513}, {2638, 122}, {3063, 17056}, {3239, 21245}, {3271, 8286}, {3733, 11019}, {3737, 2886}, {3900, 3454}, {4183, 20316}, {4560, 17046}, {5546, 21232}, {7054, 4369}, {7118, 24018}, {7121, 17478}, {7252, 142}, {7253, 2887}, {7259, 27076}, {8641, 1213}, {8648, 6739}, {14936, 8287}, {17926, 20305}, {18155, 17047}, {18191, 17059}, {21789, 10}, {22383, 18643}, {23090, 18589}, {27527, 20547}, {32676, 15252}, {32713, 24030}
X(34591) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 3900}, {2, 656}, {9, 652}, {63, 521}, {92, 522}, {219, 8611}, {282, 650}, {2184, 513}, {2297, 2522}, {2326, 1021}, {2349, 3738}, {7097, 649}, {10570, 4041}, {24026, 2310}, {26932, 7004}
X(34591) = X(i)-cross conjugate of X(j) for these (i,j): {2638, 24031}, {3270, 7004}
X(34591) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7128}, {3, 23984}, {4, 1262}, {7, 7115}, {19, 7045}, {25, 1275}, {34, 4564}, {48, 24032}, {57, 7012}, {59, 278}, {63, 24033}, {69, 23985}, {92, 24027}, {100, 32714}, {108, 651}, {109, 653}, {112, 4566}, {162, 1020}, {250, 6354}, {264, 23979}, {273, 2149}, {281, 7339}, {608, 4998}, {658, 8750}, {664, 32674}, {692, 13149}, {765, 1435}, {934, 1783}, {1016, 1398}, {1110, 1847}, {1119, 1252}, {1407, 15742}, {1415, 18026}, {1425, 23582}, {1426, 4567}, {1427, 5379}, {1461, 1897}, {4619, 7649}, {6356, 23964}, {7046, 23971}, {7071, 23586}, {7079, 24013}
X(34591) = crosspoint of X(i) and X(j) for these (i,j): {1, 905}, {2, 7253}, {9, 3239}, {63, 521}, {78, 6332}, {84, 514}, {92, 522}, {219, 23090}, {1021, 2326}, {1265, 15411}, {2968, 26932}, {7003, 17926}
X(34591) = crosssum of X(i) and X(j) for these (i,j): {1, 1783}, {19, 108}, {34, 32674}, {40, 101}, {48, 109}, {57, 1461}, {651, 17080}, {1435, 32714}
X(34591) = crossdifference of every pair of points on line {108, 109}
X(34591) = center of hyperbola {{A,B,C,X(1),X(63)}} (the X(6)-isoconjugate of the Euler line)
X(34591) = perspector of hyperbola {{A,B,C,X(521),X(522)}} (the circumconic centered at X(656), and the isogonal conjugate of line X(108)X(109))
X(34591) = barycentric product X(i)*X(j) for these {i,j}: {1, 2968}, {3, 24026}, {4, 24031}, {8, 7004}, {9, 26932}, {11, 78}, {19, 23983}, {48, 23978}, {55, 17880}, {63, 1146}, {69, 2310}, {75, 3270}, {77, 4081}, {84, 7358}, {125, 1098}, {200, 1565}, {210, 17219}, {212, 34387}, {219, 4858}, {244, 1265}, {255, 21666}, {264, 2638}, {270, 7068}, {282, 16596}, {304, 14936}, {312, 7117}, {318, 1364}, {332, 4516}, {341, 3937}, {345, 2170}, {346, 3942}, {348, 3119}, {521, 522}, {525, 1021}, {649, 15416}, {650, 6332}, {652, 4391}, {656, 7253}, {657, 15413}, {661, 15411}, {905, 3239}, {1043, 18210}, {1086, 3692}, {1111, 1260}, {1459, 4397}, {1577, 23090}, {1789, 6741}, {1792, 3120}, {1802, 23989}, {1812, 21044}, {1826, 16731}, {2287, 4466}, {2326, 15526}, {2327, 16732}, {3022, 7182}, {3271, 3718}, {3694, 17197}, {3708, 7058}, {3710, 18191}, {3719, 8735}, {3900, 4025}, {3949, 26856}, {4086, 23189}, {4171, 15419}, {4183, 17216}, {4560, 8611}, {4571, 21132}, {6506, 6513}, {6516, 23615}, {7054, 20902}, {7056, 24010}, {7177, 23970}, {14208, 21789}, {17926, 24018}
X(34591) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 7045}, {4, 24032}, {6, 7128}, {11, 273}, {19, 23984}, {25, 24033}, {41, 7115}, {48, 1262}, {55, 7012}, {63, 1275}, {78, 4998}, {184, 24027}, {200, 15742}, {212, 59}, {219, 4564}, {244, 1119}, {514, 13149}, {521, 664}, {522, 18026}, {603, 7339}, {647, 1020}, {649, 32714}, {650, 653}, {652, 651}, {656, 4566}, {657, 1783}, {663, 108}, {905, 658}, {906, 4619}, {1015, 1435}, {1021, 648}, {1086, 1847}, {1090, 2973}, {1098, 18020}, {1146, 92}, {1260, 765}, {1265, 7035}, {1364, 77}, {1459, 934}, {1565, 1088}, {1792, 4600}, {1802, 1252}, {1812, 4620}, {1946, 109}, {1973, 23985}, {1984, 15146}, {2170, 278}, {2310, 4}, {2326, 23582}, {2327, 4567}, {2328, 5379}, {2632, 6356}, {2638, 3}, {2968, 75}, {3022, 33}, {3063, 32674}, {3119, 281}, {3122, 1426}, {3239, 6335}, {3248, 1398}, {3270, 1}, {3271, 34}, {3692, 1016}, {3708, 6354}, {3900, 1897}, {3937, 269}, {3942, 279}, {4025, 4569}, {4081, 318}, {4466, 1446}, {4516, 225}, {4587, 31615}, {4858, 331}, {6332, 4554}, {7004, 7}, {7053, 24013}, {7056, 24011}, {7099, 23971}, {7117, 57}, {7177, 23586}, {7253, 811}, {7254, 4637}, {7358, 322}, {8611, 4552}, {8641, 8750}, {9247, 23979}, {14714, 4219}, {14935, 1041}, {14936, 19}, {15411, 799}, {15416, 1978}, {15419, 4635}, {16731, 17206}, {17435, 5236}, {17880, 6063}, {17926, 823}, {18210, 3668}, {20975, 1254}, {21789, 162}, {22096, 1106}, {22383, 1461}, {23090, 662}, {23189, 1414}, {23696, 927}, {23970, 7101}, {23978, 1969}, {23983, 304}, {24010, 7046}, {24012, 7071}, {24026, 264}, {24031, 69}, {26932, 85}
X(34591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1146, 6506, 21044}, {1146, 13609, 5514}, {2170, 3119, 1146}, {3269, 16573, 656}, {16596, 26932, 4466}, {21044, 33573, 6506}


X(34592) =  X(1)X(3)∩X(1113)X(3100)

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

X(34592) lies on the circumellipse of the medial and incentral triangles and these lines: {1, 3}, {11, 1313}, {33, 1344}, {73, 14375}, {243, 2586}, {1015, 15166}, {1113, 3100}, {1114, 1870}, {1822, 2361}, {2463, 12943}, {2574, 7004}, {3583, 10750}, {6198, 14709}, {8144, 20478}

X(i)-complementary conjugate of X(j) for these (i,j): {58, 2574}, {1114, 20316}, {1459, 1312}, {1823, 513}, {2206, 8105}, {2574, 3454}, {2578, 1211}, {2582, 21245}, {2584, 21530}, {8116, 3835}
X(34592) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2574}, {1114, 513}
X(34592) = {X(1),X(36)}-harmonic conjugate of X(34593)
X(34592) = {X(3),X(18455)}-harmonic conjugate of X(34593)


X(34593) =  X(1)X(3)∩X(1114)X(3100)

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

X(34593) lies on the circumellipse of the medial and incentral triangles and these lines: {1, 3}, {11, 1312}, {33, 1345}, {73, 14374}, {243, 2587}, {1015, 15167}, {1113, 1870}, {1114, 3100}, {1823, 2361}, {2464, 12943}, {2575, 7004}, {3583, 10751}, {6198, 14710}, {8144, 20479}

X(34593) = X(i)-complementary conjugate of X(j) for these (i,j): {58, 2575}, {1113, 20316}, {1459, 1313}, {1822, 513}, {2206, 8106}, {2575, 3454}, {2579, 1211}, {2583, 21245}, {2585, 21530}, {8115, 3835}
X(34593) = X(i)-Ceva conjugate of X(j) for these (i,j): {1, 2575}, {1113, 513}
X(34593) = {X(1),X(36)}-harmonic conjugate of X(34592)
X(34593) = {X(3),X(18455)}-harmonic conjugate of X(34592)


X(34594) =  ISOGONAL CONJUGATE OF X(4132)

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

X(34594) lies on these lines: {21, 106}, {81, 727}, {98, 19649}, {100, 1634}, {104, 15952}, {596, 759}, {643, 901}, {644, 29303}, {662, 8701}, {675, 1444}, {689, 4623}, {741, 1621}, {799, 8709}, {1325, 2758}, {1414, 29279}, {2370, 16049}, {3733, 3952}, {4228, 9083}, {4236, 6012}, {4575, 32682}, {4592, 29233}

X(34594) = isogonal conjugate of X(4132)
X(34594) = Collings transform of X(i) for these i: {3670, 4187}
X(34594) = X(i)-cross conjugate of X(j) for these (i,j): {1018, 662}, {4427, 100}, {5124, 249}, {20833, 250}
X(34594) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4132}, {6, 4129}, {10, 4057}, {37, 4063}, {42, 20295}, {71, 17922}, {213, 20949}, {512, 4360}, {513, 3293}, {523, 595}, {649, 3995}, {656, 4222}, {661, 32911}, {798, 18140}, {1577, 2220}, {1826, 22154}, {3733, 4075}, {3871, 4017}, {3952, 8054}, {4557, 21208}
X(34594) = cevapoint of X(i) and X(j) for these (i,j): {1, 3733}, {513, 3670}, {523, 4187}, {667, 1100}, {905, 18733}, {1019, 16696}, {3737, 18178}
X(34594) = trilinear pole of line {6, 474}
X(34594) = barycentric product X(i)*X(j) for these {i,j}: {81, 8050}, {596, 662}, {645, 20615}
X(34594) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4129}, {6, 4132}, {28, 17922}, {58, 4063}, {81, 20295}, {86, 20949}, {99, 18140}, {100, 3995}, {101, 3293}, {110, 32911}, {112, 4222}, {163, 595}, {596, 1577}, {662, 4360}, {1018, 4075}, {1019, 21208}, {1333, 4057}, {1437, 22154}, {1576, 2220}, {5546, 3871}, {8050, 321}, {20615, 7178}


X(34595) =  COMPLEMENT OF X(19877)

Barycentrics    5*a + 4*b + 4*c : :
X(34595) = X[1] + 12 X[2], 9 X[1] + 4 X[8], 5 X[1] + 8 X[10], 17 X[1] - 4 X[145], 11 X[1] - 24 X[551], 3 X[1] - 16 X[1125], 3 X[1] + 10 X[1698], 25 X[1] - 12 X[3241], 21 X[1] - 8 X[3244], 7 X[1] - 20 X[3616], 19 X[1] + 20 X[3617], 35 X[1] + 4 X[3621], 15 X[1] - 28 X[3622], 33 X[1] - 20 X[3623], X[1] - 14 X[3624], 31 X[1] + 8 X[3625], 23 X[1] + 16 X[3626], 11 X[1] + 2 X[3632], 15 X[1] - 2 X[3633], 7 X[1] + 32 X[3634], 29 X[1] - 16 X[3635], 19 X[1] - 32 X[3636], 7 X[1] + 6 X[3679], 17 X[1] + 48 X[3828], 8 X[1] + 5 X[4668], 41 X[1] + 24 X[4669], 10 X[1] + 3 X[4677], 37 X[1] + 28 X[4678], 33 X[1] + 32 X[4691], 49 X[1] + 16 X[4701], 43 X[1] + 48 X[4745], 59 X[1] + 32 X[4746], 29 X[1] + 10 X[4816], 5 X[1] - 44 X[5550], 11 X[1] + 28 X[9780], 17 X[1] - 56 X[15808], X[1] - 40 X[19862], 5 X[1] + 34 X[19872], 4 X[1] + 9 X[19875], 5 X[1] + 21 X[19876], X[1] + 4 X[19877], X[1] + 64 X[19878], 7 X[1] - 72 X[19883], 69 X[1] - 4 X[20014], 103 X[1] - 12 X[20049], 43 X[1] - 4 X[20050], 71 X[1] + 20 X[20052], 61 X[1] + 4 X[20053], 87 X[1] + 4 X[20054], 41 X[1] - 28 X[20057], 43 X[1] + 152 X[22266], 27 X[2] - X[8], 15 X[2] - 2 X[10]

X(34595) lies on these lines: {1, 2}, {3, 7988}, {5, 7987}, {9, 3337}, {12, 13462}, {20, 10171}, {35, 4423}, {36, 11108}, {40, 3526}, {46, 3646}, {55, 16863}, {56, 16853}, {58, 17125}, {63, 5506}, {72, 3848}, {79, 4679}, {140, 165}, {182, 9587}, {191, 3306}, {238, 31243}, {312, 6533}, {320, 31289}, {333, 28618}, {355, 30315}, {377, 18514}, {392, 10107}, {405, 7280}, {442, 3847}, {443, 3583}, {452, 4316}, {474, 5010}, {484, 31435}, {515, 5067}, {516, 10303}, {547, 18481}, {549, 30308}, {590, 19003}, {595, 17124}, {615, 19004}, {631, 1699}, {632, 5886}, {944, 10172}, {946, 3525}, {958, 16854}, {993, 17536}, {1001, 16862}, {1089, 30829}, {1213, 16667}, {1376, 16864}, {1385, 5070}, {1420, 5726}, {1478, 17559}, {1479, 17582}, {1573, 9336}, {1574, 9331}, {1656, 3576}, {1743, 17398}, {1750, 6832}, {1757, 6687}, {2093, 5443}, {2478, 18513}, {2951, 6890}, {2975, 17546}, {3035, 12732}, {3068, 13942}, {3069, 13888}, {3090, 5691}, {3097, 6683}, {3305, 6763}, {3336, 5437}, {3338, 7308}, {3339, 7294}, {3361, 5219}, {3467, 15297}, {3522, 12571}, {3523, 3817}, {3524, 18483}, {3533, 6684}, {3579, 15694}, {3585, 5084}, {3628, 5587}, {3653, 18357}, {3666, 31318}, {3731, 17369}, {3742, 4539}, {3746, 4413}, {3752, 27785}, {3763, 16475}, {3816, 17529}, {3825, 4197}, {3833, 3869}, {3844, 16491}, {3851, 17502}, {3868, 4536}, {3876, 3894}, {3885, 3968}, {3889, 4015}, {3890, 3918}, {3901, 10176}, {3928, 28645}, {3947, 5265}, {3973, 5257}, {3983, 5049}, {4002, 10179}, {4297, 5056}, {4299, 5129}, {4302, 17580}, {4324, 6904}, {4355, 5226}, {4398, 17322}, {4653, 17551}, {4731, 31792}, {4748, 4758}, {4798, 25358}, {4850, 27784}, {4857, 26105}, {4873, 16673}, {4902, 10436}, {4999, 5234}, {5044, 18398}, {5047, 5303}, {5054, 9955}, {5055, 13624}, {5057, 5131}, {5068, 28164}, {5071, 31673}, {5079, 28160}, {5204, 16857}, {5235, 28620}, {5248, 17531}, {5251, 16842}, {5253, 17534}, {5258, 16856}, {5267, 16859}, {5284, 17535}, {5290, 7288}, {5316, 13407}, {5326, 9819}, {5426, 31254}, {5436, 31262}, {5439, 5692}, {5444, 10826}, {5541, 32557}, {5563, 16855}, {5603, 9588}, {5690, 16189}, {5715, 6878}, {5732, 6884}, {5790, 32900}, {5880, 7483}, {5901, 11224}, {5902, 25917}, {6051, 16602}, {6326, 34126}, {6361, 15709}, {6532, 17155}, {6667, 10609}, {6691, 25525}, {6701, 15671}, {6707, 17306}, {6713, 15017}, {6824, 10857}, {6861, 8726}, {6918, 15931}, {6958, 30503}, {6972, 12565}, {7082, 13089}, {7485, 9591}, {7486, 19925}, {7516, 9625}, {7741, 8728}, {7746, 9592}, {7815, 10789}, {7951, 17527}, {7967, 31399}, {7982, 11231}, {7998, 31757}, {8040, 11263}, {8056, 24161}, {8185, 11284}, {8252, 18991}, {8253, 18992}, {8715, 9342}, {8983, 32786}, {9581, 10543}, {9583, 10577}, {9589, 10164}, {9593, 31455}, {9612, 16845}, {9614, 31508}, {9621, 13353}, {9624, 11531}, {9779, 12512}, {9897, 34123}, {9956, 18526}, {9963, 15015}, {10124, 22791}, {10156, 12688}, {10299, 28150}, {10434, 19549}, {10980, 11374}, {11373, 30337}, {11465, 31760}, {11539, 31162}, {12245, 16191}, {13384, 17606}, {13893, 32789}, {13947, 32790}, {13971, 32785}, {15024, 31738}, {15045, 31751}, {15668, 16468}, {15670, 16118}, {15720, 22793}, {16173, 31235}, {16291, 16678}, {16296, 20470}, {16457, 17123}, {16469, 17245}, {16472, 17811}, {16473, 17825}, {16844, 19749}, {17151, 28653}, {17259, 28650}, {17290, 25498}, {17296, 28640}, {17304, 24295}, {17360, 30598}, {17384, 31244}, {17400, 24342}, {17557, 31280}, {17575, 25466}, {18253, 25681}, {18421, 24914}, {19249, 23383}, {19265, 23361}, {19334, 32944}, {19709, 33697}, {20003, 27918}, {20196, 24953}, {21734, 28158}, {24719, 31288}, {24954, 31260}, {25089, 26242}, {25354, 27147}, {28174, 31425}, {28628, 31190}, {30963, 32092}

X(34595) = complement of X(19877)
X(34595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 10, 4677}, {1, 19875, 4668}, {1, 19876, 10}, {2, 10, 19872}, {2, 1125, 1698}, {2, 3616, 3634}, {2, 3624, 1}, {2, 5550, 10}, {2, 9780, 31253}, {2, 19862, 3624}, {2, 19883, 3679}, {2, 25055, 19876}, {2, 25492, 19863}, {2, 26094, 19858}, {2, 27148, 30107}, {2, 29603, 16832}, {2, 29609, 4384}, {2, 29612, 17308}, {10, 1125, 3622}, {10, 3621, 3679}, {10, 3622, 3633}, {10, 5550, 25055}, {10, 19872, 19876}, {10, 25055, 1}, {140, 8227, 165}, {474, 5259, 5010}, {474, 8167, 5259}, {551, 3632, 1}, {551, 4691, 3623}, {551, 9780, 3632}, {551, 31253, 9780}, {631, 1699, 16192}, {632, 5886, 31423}, {1125, 1698, 1}, {1125, 3244, 3616}, {1125, 3622, 25055}, {1125, 3634, 3244}, {1125, 4691, 551}, {1125, 31253, 4691}, {1656, 3576, 7989}, {1698, 3624, 1125}, {1698, 3633, 10}, {1698, 25055, 3633}, {3090, 10165, 5691}, {3216, 26102, 1}, {3244, 3621, 3633}, {3244, 19883, 1125}, {3526, 11230, 40}, {3616, 3634, 3679}, {3616, 3679, 1}, {3622, 3633, 1}, {3622, 5550, 1125}, {3623, 4691, 3632}, {3623, 9780, 4691}, {3623, 31253, 1698}, {3624, 19872, 25055}, {3624, 25055, 5550}, {3633, 19872, 1698}, {3633, 25055, 3622}, {3634, 19883, 3616}, {3720, 5312, 1}, {3720, 17749, 5312}, {3828, 15808, 145}, {4423, 16408, 35}, {4677, 19876, 19875}, {4999, 30827, 5234}, {5055, 13624, 18492}, {5219, 5433, 3361}, {5284, 17535, 25440}, {5550, 19872, 1}, {5886, 31423, 7991}, {6707, 17306, 31312}, {7294, 11375, 31231}, {9624, 26446, 11531}, {11375, 31231, 3339}, {16842, 25524, 5251}, {17284, 29646, 1}, {19858, 29825, 6048}, {19862, 19878, 2}, {19872, 25055, 10}, {19876, 25055, 4677}, {28257, 30950, 386}, {29598, 29637, 1}, {30315, 30392, 355}


X(34596) =  X(5)X(49)∩X(547)X(15307)

Barycentrics    a^16 - 3*a^14*b^2 - 7*a^12*b^4 + 46*a^10*b^6 - 90*a^8*b^8 + 89*a^6*b^10 - 47*a^4*b^12 + 12*a^2*b^14 - b^16 - 3*a^14*c^2 - 12*a^12*b^2*c^2 + 60*a^10*b^4*c^2 - 48*a^8*b^6*c^2 - 64*a^6*b^8*c^2 + 123*a^4*b^10*c^2 - 69*a^2*b^12*c^2 + 13*b^14*c^2 - 7*a^12*c^4 + 60*a^10*b^2*c^4 - 51*a^8*b^4*c^4 - 16*a^6*b^6*c^4 - 63*a^4*b^8*c^4 + 135*a^2*b^10*c^4 - 58*b^12*c^4 + 46*a^10*c^6 - 48*a^8*b^2*c^6 - 16*a^6*b^4*c^6 - 26*a^4*b^6*c^6 - 78*a^2*b^8*c^6 + 131*b^10*c^6 - 90*a^8*c^8 - 64*a^6*b^2*c^8 - 63*a^4*b^4*c^8 - 78*a^2*b^6*c^8 - 170*b^8*c^8 + 89*a^6*c^10 + 123*a^4*b^2*c^10 + 135*a^2*b^4*c^10 + 131*b^6*c^10 - 47*a^4*c^12 - 69*a^2*b^2*c^12 - 58*b^4*c^12 + 12*a^2*c^14 + 13*b^2*c^14 - c^16 : :

See Kadir Altintas and Peter Moses, Euclid 55 .

X(34596) lies on these lines: {5, 49}, {547, 15307}, {930, 15957}, {1487, 22051}, {3090, 24573}, {13469, 14140}

X(34596) = {X(5),X(15425)}-harmonic conjugate of X(1141)


X(34597) =  X(5)X(49)∩X(547)X(32744)

Barycentrics    2*a^16 - 7*a^14*b^2 - 5*a^12*b^4 + 61*a^10*b^6 - 125*a^8*b^8 + 123*a^6*b^10 - 63*a^4*b^12 + 15*a^2*b^14 - b^16 - 7*a^14*c^2 - 10*a^12*b^2*c^2 + 81*a^10*b^4*c^2 - 70*a^8*b^6*c^2 - 87*a^6*b^8*c^2 + 168*a^4*b^10*c^2 - 91*a^2*b^12*c^2 + 16*b^14*c^2 - 5*a^12*c^4 + 81*a^10*b^2*c^4 - 72*a^8*b^4*c^4 - 27*a^6*b^6*c^4 - 84*a^4*b^8*c^4 + 183*a^2*b^10*c^4 - 76*b^12*c^4 + 61*a^10*c^6 - 70*a^8*b^2*c^6 - 27*a^6*b^4*c^6 - 42*a^4*b^6*c^6 - 107*a^2*b^8*c^6 + 176*b^10*c^6 - 125*a^8*c^8 - 87*a^6*b^2*c^8 - 84*a^4*b^4*c^8 - 107*a^2*b^6*c^8 - 230*b^8*c^8 + 123*a^6*c^10 + 168*a^4*b^2*c^10 + 183*a^2*b^4*c^10 + 176*b^6*c^10 - 63*a^4*c^12 - 91*a^2*b^2*c^12 - 76*b^4*c^12 + 15*a^2*c^14 + 16*b^2*c^14 - c^16 : :
X(34597) = X[5] + 2 X[15425], 7 X[3090] - X[14143], 2 X[3628] + X[31879], 2 X[10289] + X[20414], 4 X[13469] - X[32551], X[13856] + 2 X[15957], X[18016] + 2 X[19940]

See Kadir Altintas and Peter Moses, Euclid 55 .

X(34597) lies on these lines: {5, 49}, {547, 32744}, {1503, 17727}, {3090, 14143}, {3628, 31879}, {10289, 20414}, {13469, 32551}, {13856, 15957}, {18016, 19940}

X(34597) = {X(5),X(7604)}-harmonic conjugate of X(8254)


X(34598) =  X(4)X(7604)∩X(5)X(11701)

Barycentrics    2*a^16 - 8*a^14*b^2 + 8*a^12*b^4 + 11*a^10*b^6 - 35*a^8*b^8 + 38*a^6*b^10 - 22*a^4*b^12 + 7*a^2*b^14 - b^16 - 8*a^14*c^2 + 12*a^12*b^2*c^2 + 13*a^10*b^4*c^2 - 20*a^8*b^6*c^2 - 27*a^6*b^8*c^2 + 59*a^4*b^10*c^2 - 38*a^2*b^12*c^2 + 9*b^14*c^2 + 8*a^12*c^4 + 13*a^10*b^2*c^4 - 16*a^8*b^4*c^4 - 11*a^6*b^6*c^4 - 32*a^4*b^8*c^4 + 72*a^2*b^10*c^4 - 34*b^12*c^4 + 11*a^10*c^6 - 20*a^8*b^2*c^6 - 11*a^6*b^4*c^6 - 10*a^4*b^6*c^6 - 41*a^2*b^8*c^6 + 71*b^10*c^6 - 35*a^8*c^8 - 27*a^6*b^2*c^8 - 32*a^4*b^4*c^8 - 41*a^2*b^6*c^8 - 90*b^8*c^8 + 38*a^6*c^10 + 59*a^4*b^2*c^10 + 72*a^2*b^4*c^10 + 71*b^6*c^10 - 22*a^4*c^12 - 38*a^2*b^2*c^12 - 34*b^4*c^12 + 7*a^2*c^14 + 9*b^2*c^14 - c^16 : :
X(34598) = 5 X[1656] - X[30484]

See Kadir Altintas and Peter Moses, Euclid 55 .

X(34598) lies on these lines: {3, 32536}, {4, 7604}, {5, 11701}, {30, 15425}, {137, 24144}, {140, 20414}, {1154, 23338}, {1510, 32205}, {1656, 30484}, {3850, 18400}, {5501, 25150}, {10285, 13856}, {13372, 21975}, {15957, 32423}

X(34598) = midpoint of X(i) and X(j) for these {i,j}: {3, 32536}, {5, 33545}, {140, 20414}, {10285, 13856}


X(34599) =  X(5)X(128)∩X(1154)X(12046)

Barycentrics    (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 12*a^10*b^2 + 38*a^8*b^4 - 52*a^6*b^6 + 33*a^4*b^8 - 8*a^2*b^10 - 12*a^10*c^2 + 42*a^8*b^2*c^2 - 32*a^6*b^4*c^2 - 21*a^4*b^6*c^2 + 30*a^2*b^8*c^2 - 7*b^10*c^2 + 38*a^8*c^4 - 32*a^6*b^2*c^4 - 15*a^4*b^4*c^4 - 22*a^2*b^6*c^4 + 28*b^8*c^4 - 52*a^6*c^6 - 21*a^4*b^2*c^6 - 22*a^2*b^4*c^6 - 42*b^6*c^6 + 33*a^4*c^8 + 30*a^2*b^2*c^8 + 28*b^4*c^8 - 8*a^2*c^10 - 7*b^2*c^10) : :
X(34599) = 3 X[5] + X[13856], 7 X[5] + X[15345], 3 X[5066] + X[18016], 5 X[12812] - X[32551], 7 X[13856] - 3 X[15345]

See Kadir Altintas and Peter Moses, Euclid 55 .

X(34599) lies on these lines: {5, 128}, {1154, 12046}, {5066, 18016}, {12812, 32551}, {23338, 32904}

X(34599) = midpoint of X(23338) and X(32904)


X(34600) =  X(1)X(149)∩X(3)X(191)

Barycentrics    a*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c + 2*a*b^4*c - a^4*c^2 + 3*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 - 2*a*b^2*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + c^6) : :
X(34600) = 2 X[11] - 3 X[26725], X[79] + 2 X[10609], X[191] - 3 X[15015], 4 X[214] - X[3065], 3 X[1699] - 4 X[33594], 4 X[6702] - 5 X[31254], 5 X[7987] - 4 X[17009], 4 X[11281] - 3 X[16173], 3 X[28443] - 2 X[33856]

See Antreas Hatzipolakis and Peter Moses, Euclid 57 .

X(34600) lies on the cubic K853 and these lines: {1, 149}, {3, 191}, {11, 26725}, {21, 214}, {30, 6265}, {36, 33598}, {56, 33667}, {79, 6596}, {80, 442}, {100, 484}, {104, 15910}, {952, 5499}, {1001, 5426}, {1699, 33594}, {1749, 27086}, {2099, 12653}, {2800, 3651}, {2802, 34195}, {3647, 32633}, {3649, 12739}, {3679, 5531}, {3913, 5541}, {4316, 4511}, {5267, 18259}, {6261, 16143}, {6264, 31140}, {6702, 31254}, {6830, 15017}, {6840, 21635}, {6853, 10265}, {7987, 17009}, {9897, 33108}, {10543, 12740}, {10738, 33592}, {11277, 12738}, {11281, 16173}, {12047, 12444}, {12520, 12767}, {12524, 12927}, {12769, 12845}, {12786, 23016}, {12919, 26202}, {13743, 26287}, {14450, 22836}, {15680, 30144}, {16139, 33814}, {20085, 30147}, {28443, 33856}

X(34600) = midpoint of X(i) and X(j) for these {i,j}: {1, 13146}, {2475, 6224}, {5541, 16126}, {6326, 16132}
X(34600) = reflection of X(i) in X(j) for these {i,j}: {21, 214}, {80, 442}, {1749, 27086}, {3065, 21}, {10738, 33592}, {11604, 11263}, {16139, 33814}


X(34601) =  X(107)X(1651)∩X(122)X(125)

Barycentrics    (b - c)^2*(b + c)^2*(a^2 - b^2 - c^2)^2*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(3*a^12 - 7*a^10*b^2 - a^8*b^4 + 14*a^6*b^6 - 11*a^4*b^8 + a^2*b^10 + b^12 - 7*a^10*c^2 + 21*a^8*b^2*c^2 - 18*a^6*b^4*c^2 - 2*a^4*b^6*c^2 + 9*a^2*b^8*c^2 - 3*b^10*c^2 - a^8*c^4 - 18*a^6*b^2*c^4 + 26*a^4*b^4*c^4 - 10*a^2*b^6*c^4 + 3*b^8*c^4 + 14*a^6*c^6 - 2*a^4*b^2*c^6 - 10*a^2*b^4*c^6 - 2*b^6*c^6 - 11*a^4*c^8 + 9*a^2*b^2*c^8 + 3*b^4*c^8 + a^2*c^10 - 3*b^2*c^10 + c^12) : :
X(34601) = 2 X[107] - 3 X[1651], 4 X[122] - 3 X[1650], 3 X[12113] - 2 X[23240]

See Antreas Hatzipolakis and Peter Moses, Euclid 57 .

X(34601) lies on these lines: {30, 34186}, {107, 1651}, {110, 12113}, {122, 125}, {2132, 10733}, {3081, 9530}, {5667, 15774}

X(34601) = reflection of X(5667) in X(15774)
X(34601) = crosspoint of X(1294) and X(9033)
X(34601) = crosssum of X(i) and X(j) for these (i,j): {1304, 6000}, {1495, 32695}
X(34601) = barycentric product X(15526)*X(15774)
X(34601) = barycentric quotient X(15774)/X(23582)


X(34602) =  X(15)X(115)∩X(18)X(542)

Barycentrics    36*S^4+3*(3*SA^2-5*SW^2)*S^2+6*SB*SC*SW^2+S*sqrt(3)*(3*(5*SA+SW)*(SB+SC)*SW-(6*SA+14*SW)*S^2) : :

See Tran Quang Hung and César Lozada, Euclid 68 .

X(34602) lies on these lines: {13, 15561}, {15, 115}, {18, 542}, {114, 6108}, {620, 14145}, {628, 33460}, {630, 2482}, {5470, 12355}, {9698, 20415}, {12815, 20190}, {22511, 25559}

X(34602) = reflection of X(i) in X(j) for these (i,j): (115, 22846), (9115, 14139), (14145, 620)




leftri  HR-ellipses: X(34603) - X(34752)  rightri

This preamble and centers X(34603)-X(34752) were contributed by César Eliud Lozada, November 7, 2019.

Let T'=A'B'C' and T"=A"B"C" be two homothetic triangles. Denote (B'a, C'a) the reflections of B' and C' in A", respectively, and define (C'b, A'b) and (A'c, B'c) cyclically. Then these six points lie on an ellipse here named the HR-ellipse of T' to T" (letters H and R stands for homothetic and reflections).

By swapping T' and T", the HR-ellipse T" to T' is found.

Suppose T' and T" are homothetic to the reference triangle ABC, with H' = homothetic center (ABC, T') and H"=homothetic center (ABC, T"). Denote λ'=AA'/AH' and λ"=AA"/AH". Then, if H', H" have normalized barycentrics coordinates H' = x' : y' : z' and H" = x" : y" : z", the centers O', O" of the HR-ellipses T' to T" and T" to T' are:

  O' = (3*x' - 1)*λ' - 2*(3*x" - 1)*λ" - 1 : :

  O" = (3*x" - 1)*λ" - 2*(3*x' - 1)*λ' - 1 : :

In general, { H', H", O' , O"} are aligned, O' = complement-of-O" with respect to T' and O" = complement-of-O' with respect to T".

The appearance of (T', T", i, j) in the following lists means that the centers of HR-ellipses (T' to T") and (T" to T') are X(i) and X(j), respectively:

(ABC, ABC-X3 reflections, 20, 4), (ABC, anti-Aquila, 1, 10), (ABC, anti-Euler, 20, 4), (ABC, Aquila, 8, 1), (ABC, 5th Brocard, 9939, 7812), (ABC, Euler, 4, 3), (ABC, outer-Garcia, 8, 1), (ABC, Johnson, 4, 3), (ABC, 5th mixtilinear, 145, 8), (ABC, X3-ABC reflections, 4, 3), (ABC-X3 reflections, anticomplementary, 4, 20), (ABC-X3 reflections, Johnson, 3543, 15681), (ABC-X3 reflections, medial, 4, 20), (anti-Aquila, anticomplementary, 10, 1), (anti-Aquila, medial, 10, 1), (2nd anti-circumperp-tangential, Mandart-incircle, 6284, 7354), (anti-Euler, anticomplementary, 4, 20), (anti-Euler, Euler, 3543, 15681), (anti-Euler, medial, 4, 20), (anti-Euler, X3-ABC reflections, 3543, 15681), (anti-Mandart-incircle, 2nd circumperp tangential, 12513, 3913), (anticomplementary, Aquila, 8, 1), (anticomplementary, 5th Brocard, 9939, 7812), (anticomplementary, Euler, 4, 3), (anticomplementary, outer-Garcia, 8, 1), (anticomplementary, Johnson, 4, 3), (anticomplementary, medial, 2, 2), (anticomplementary, 5th mixtilinear, 145, 8), (anticomplementary, X3-ABC reflections, 4, 3), (Aquila, medial, 1, 8), (5th Brocard, medial, 7812, 9939), (Euler, medial, 3, 4), (Euler, X3-ABC reflections, 381, 381), (outer-Garcia, medial, 1, 8), (Johnson, medial, 3, 4), (medial, 5th mixtilinear, 145, 8), (medial, X3-ABC reflections, 4, 3)

underbar

X(34603) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO ANTI-ARA

Barycentrics    2*a^6+(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(34603) = 9*X(2)-8*X(7734) = 7*X(2)-8*X(10128) = 5*X(2)-4*X(10691) = 15*X(2)-16*X(13361) = X(4)+2*X(7553) = 4*X(4)-X(12225) = 5*X(4)-2*X(12605) = X(20)-4*X(6756) = 5*X(51)-4*X(32068) = 2*X(382)+X(6240)

As a point on the Euler line, X(34603) has Shinagawa coefficients (-E-2*F, 6*E+6*F)
The center of the reciprocal HR-ellipse of these triangles is X(7667)

X(34603) lies on these lines: {2,3}, {51,29012}, {52,16659}, {53,10313}, {230,18353}, {251,5254}, {316,16276}, {324,16264}, {343,15107}, {519,34633}, {524,9973}, {528,34655}, {529,34653}, {542,13417}, {543,19568}, {597,1176}, {614,10483}, {671,16277}, {754,34651}, {1180,7745}, {1194,7747}, {1503,3060}, {1992,5596}, {1993,31383}, {1994,21850}, {2979,29181}, {3058,9629}, {3108,9607}, {3580,11550}, {3583,5322}, {3585,5310}, {3867,19121}, {3917,29317}, {3920,6284}, {5012,5480}, {5032,19119}, {5345,18514}, {5359,7737}, {5392,14458}, {5446,34224}, {5523,17409}, {5889,16655}, {5943,29323}, {5986,6321}, {6748,22240}, {7191,7354}, {7298,18513}, {7762,8267}, {7842,21248}, {8024,32819}, {10574,11745}, {11002,11245}, {11439,16656}, {11442,33586}, {11645,21849}, {12111,16621}, {12279,13568}, {13419,14516}, {13754,16658}, {15171,29815}, {15305,16654}, {17024,18990}, {17810,18911}, {18907,34482}, {19130,22352}, {23292,26881}, {23293,32269}

X(34603) = midpoint of X(i) and X(j) for these {i,j}: {7576, 34613}, {15682, 18559}
X(34603) = reflection of X(i) in X(j) for these (i,j): (2, 428), (3, 13490), (7576, 7540), (15305, 16654)
X(34603) = anticomplement of X(7667)
X(34603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5, 5020), (2, 20, 31074), (2, 23, 10154), (2, 186, 28431), (2, 444, 7463), (2, 453, 21545), (2, 474, 29877), (2, 549, 6955), (2, 852, 26207), (2, 858, 7484), (2, 858, 31856), (2, 1312, 15248), (2, 1347, 15245), (2, 1368, 16419), (2, 1370, 7499)


X(34604) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO 5th ANTI-BROCARD

Barycentrics    5*a^4+(b^2+c^2)*a^2-b^4+b^2*c^2-c^4 : :
X(34604) = X(2896)-4*X(12206) = 3*X(3839)-2*X(34681) = 4*X(5007)-X(6655) = X(6658)+2*X(7760) = X(7883)-3*X(12150) = 2*X(10350)+X(20065) = X(34615)+2*X(34682) = 2*X(34635)+X(34674) = 4*X(34635)-X(34686) = 2*X(34674)+X(34686)

The center of the reciprocal HR-ellipse of these triangles is X(7883)

X(34604) lies on these lines: {2,32}, {4,11177}, {5,8859}, {6,7833}, {20,576}, {30,34615}, {61,530}, {62,531}, {147,10788}, {148,7737}, {194,1992}, {316,5008}, {376,3095}, {381,9863}, {384,524}, {385,8370}, {519,34635}, {528,34672}, {529,34670}, {538,19686}, {542,12110}, {543,6658}, {597,7750}, {598,6179}, {599,7893}, {671,7747}, {1003,7837}, {1285,7774}, {1383,7665}, {1384,7777}, {1975,15534}, {2482,7838}, {3053,7921}, {3314,33237}, {3329,8359}, {3411,34508}, {3412,34509}, {3552,13571}, {3793,8367}, {3839,34681}, {3849,5007}, {3972,7779}, {5077,7864}, {5254,8597}, {5286,33192}, {5305,8352}, {5306,14041}, {5319,33019}, {7610,16921}, {7615,10807}, {7617,33024}, {7618,33014}, {7620,14068}, {7735,33006}, {7739,33264}, {7745,22329}, {7754,11159}, {7755,9166}, {7757,33265}, {7759,7870}, {7762,7836}, {7765,19691}, {7768,19689}, {7772,33260}, {7776,8366}, {7780,33020}, {7783,8598}, {7788,14036}, {7791,9731}, {7794,19692}, {7797,7823}, {7798,20094}, {7805,11054}, {7806,11318}, {7807,19661}, {7828,31173}, {7829,19690}, {7839,8584}, {7858,26613}, {7878,33021}, {7885,8360}, {7898,16989}, {7907,11184}, {7932,32006}, {7941,22110}, {7946,14001}, {8182,33004}, {9300,33273}, {9607,33267}, {9737,10304}, {9740,32971}, {9741,33239}, {9766,33246}, {9770,16925}, {9771,16923}, {11165,33235}, {11171,22679}, {11361,14614}, {14063,23334}, {14537,14568}, {14830,14881}, {14976,32986}, {15484,17004}, {15597,16922}, {16895,21358}, {16898,21356}, {19569,33017}, {31407,33188}

X(34604) = reflection of X(i) in X(j) for these (i,j): (2, 12150), (6655, 7827), (7827, 5007)
X(34604) = anticomplement of X(7883)
X(34604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7812, 7785), (2, 9939, 2896), (2, 20065, 9939), (2, 20088, 7812), (32, 7812, 2), (32, 20088, 7785), (83, 7810, 2), (7787, 9939, 2), (7787, 20065, 2896), (10350, 12206, 7787), (14466, 28802, 33454), (25779, 28802, 27964)


X(34605) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    2*a^4-(b^2-5*b*c+c^2)*a^2-(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34605) = X(8)-4*X(18990) = X(145)+2*X(7354) = 2*X(1770)+X(3885) = X(3869)-4*X(10106) = 4*X(4292)-X(14923) = 3*X(5434)-X(34606) = X(17579)-4*X(34637) = X(17579)+2*X(34690) = X(34617)+2*X(34698) = 2*X(34637)+X(34690)

The center of the reciprocal HR-ellipse of these triangles is X(34606)

X(34605) lies on these lines: {1,535}, {2,12}, {4,10707}, {8,2094}, {30,944}, {55,20067}, {57,5176}, {79,22837}, {100,4293}, {145,528}, {149,12943}, {153,22753}, {376,10805}, {404,4317}, {515,3873}, {517,23155}, {519,3868}, {527,3869}, {551,3897}, {754,34669}, {908,4315}, {956,17528}, {999,5080}, {1056,1621}, {1317,34741}, {1319,31053}, {1478,11680}, {1770,3885}, {1836,34640}, {2099,17483}, {2320,5719}, {2475,9657}, {2476,5270}, {3057,28534}, {3085,5303}, {3218,5252}, {3227,7812}, {3244,10483}, {3303,15680}, {3304,5046}, {3336,3679}, {3474,12648}, {3476,5905}, {3487,11113}, {3621,34720}, {3623,6284}, {3635,34649}, {3655,5812}, {3839,10893}, {3870,34701}, {3871,4299}, {3872,20292}, {3889,10572}, {4188,6174}, {4189,15888}, {4193,5563}, {4197,5258}, {4257,24222}, {4316,25439}, {4325,8715}, {4392,5724}, {4428,15677}, {4654,34716}, {5086,9613}, {5178,6762}, {5229,10529}, {5289,17484}, {5691,31146}, {5735,11520}, {6173,19860}, {6735,9352}, {6933,31410}, {9965,34742}, {10056,17549}, {10197,14804}, {10385,10965}, {10914,31776}, {11238,34739}, {11280,12559}, {11691,31734}, {13462,30852}, {15171,20057}, {16370,34740}, {17016,17301}, {17564,17757}, {17572,21031}, {19861,31142}, {20070,34630}, {24473,28204}, {25005,32636}, {28452,34627}, {31145,34612}, {31254,31458}

X(34605) = midpoint of X(i) and X(j) for these {i,j}: {7354, 34749}, {10483, 34719}
X(34605) = reflection of X(i) in X(j) for these (i,j): (2, 5434), (8, 11112), (145, 34749), (3621, 34720), (11112, 18990), (11114, 1), (20070, 34630), (31145, 34612), (34611, 3241), (34627, 28452), (34629, 1482), (34649, 3635), (34719, 3244), (34745, 1483)
X(34605) = anticomplement of X(34606)
X(34605) = pole of the line {1638, 3910} wrt Steiner circumellipse
X(34605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11236, 11681), (2, 20060, 11236), (2, 20076, 34610), (2, 34610, 2975), (56, 11236, 2), (56, 20060, 11681), (388, 20076, 2975), (388, 34610, 2), (3436, 3600, 5253), (11194, 11237, 2)


X(34606) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ABC

Barycentrics    2*a^4-(b-c)^2*a^2-4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34606) = 2*X(8)+X(6284) = 4*X(10)-X(7354) = X(65)-4*X(5795) = X(65)+2*X(12527) = 4*X(355)-X(6253) = 2*X(355)+X(11827) = 3*X(5434)-2*X(34605) = 4*X(5690)-X(11826) = X(5691)+2*X(31799) = 2*X(5795)+X(12527) = X(6253)+2*X(11827) = 2*X(11114)+X(34720)

The center of the reciprocal HR-ellipse of these triangles is X(34605)

X(34606) lies on these lines: {1,4679}, {2,12}, {3,6174}, {4,31140}, {5,5258}, {8,190}, {9,5252}, {10,535}, {11,956}, {21,12607}, {30,40}, {36,3820}, {55,3421}, {63,34742}, {65,527}, {72,519}, {99,12349}, {200,34701}, {210,515}, {257,17333}, {329,2099}, {376,11500}, {381,10526}, {404,9711}, {405,15888}, {428,5130}, {443,9657}, {452,3303}, {495,5251}, {496,5288}, {549,10942}, {551,20323}, {599,12587}, {671,13181}, {754,34670}, {855,15621}, {903,24835}, {908,15950}, {952,5692}, {960,10944}, {984,5724}, {993,5432}, {1056,4423}, {1146,5282}, {1259,4421}, {1317,5289}, {1319,3452}, {1376,15326}, {1420,24954}, {1478,3925}, {1698,18990}, {1836,9623}, {2094,5221}, {2183,3691}, {2475,9710}, {2478,11240}, {2550,12943}, {2646,21075}, {2829,5657}, {2886,5080}, {3219,5176}, {3241,5330}, {3304,5084}, {3434,34706}, {3476,18228}, {3486,5815}, {3524,10786}, {3534,18518}, {3545,10894}, {3582,10523}, {3584,10954}, {3585,31419}, {3586,4863}, {3614,26363}, {3625,34649}, {3632,15171}, {3633,15172}, {3649,19860}, {3683,31397}, {3689,4304}, {3697,17647}, {3698,4292}, {3753,11246}, {3811,10543}, {3813,5046}, {3830,18517}, {3845,31159}, {3872,24703}, {3893,10624}, {3899,5844}, {3913,6872}, {3927,10573}, {4005,6737}, {4030,4737}, {4126,16086}, {4187,8666}, {4293,4413}, {4299,9709}, {4302,6154}, {4315,5316}, {4317,16408}, {4325,17563}, {4390,17747}, {4428,11239}, {4640,6735}, {4853,12701}, {4915,9580}, {4995,10955}, {5054,26487}, {5055,11929}, {5064,11391}, {5071,10599}, {5082,12953}, {5177,9656}, {5217,7080}, {5223,5727}, {5234,9578}, {5250,32049}, {5270,8728}, {5302,24987}, {5445,5791}, {5463,12932}, {5464,12931}, {5563,17527}, {5584,12667}, {5687,15338}, {5793,17251}, {5812,31162}, {5836,28534}, {5881,31789}, {6054,12183}, {6173,10404}, {6762,31146}, {6963,20418}, {7173,10527}, {7750,25280}, {7757,12933}, {7810,13466}, {7865,10872}, {7958,10532}, {8168,20075}, {8370,16829}, {8582,32636}, {9140,13214}, {9596,31490}, {9639,10149}, {9654,19854}, {9678,13901}, {10039,31445}, {10056,16418}, {10106,18250}, {10197,15670}, {10371,17294}, {10522,11235}, {10590,31245}, {10706,12372}, {10711,12762}, {10718,13295}, {10895,19843}, {10953,11238}, {11374,25055}, {11499,30264}, {12245,34629}, {12645,34745}, {12678,30503}, {12934,31168}, {13462,20196}, {13694,13712}, {13814,13835}, {16417,34740}, {16980,22299}, {17271,21277}, {17274,30617}, {17313,26101}, {17571,31452}, {19025,32788}, {19026,32787}, {19028,31453}, {22704,22712}, {28204,28459}, {31145,34611}, {31456,31460}, {32288,34319}

X(34606) = midpoint of X(i) and X(j) for these {i,j}: {8, 11114}, {3058, 34689}, {3625, 34649}, {3632, 34719}, {6284, 34720}, {11827, 34746}, {12245, 34629}, {12645, 34745}, {31145, 34611}
X(34606) = reflection of X(i) in X(j) for these (i,j): (3058, 11113), (5434, 2), (6253, 34746), (6284, 11114), (7354, 11112), (11112, 10), (11246, 3753), (34612, 3679), (34630, 40), (34699, 3058), (34719, 15171), (34720, 8), (34746, 355), (34749, 1)
X(34606) = complement of X(34605)
X(34606) = pole of the line {3910, 30565} wrt Steiner circumellipse
X(34606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3436, 11236), (2, 11194, 5298), (2, 11236, 12), (2, 34610, 56), (12, 958, 24953), (958, 3436, 12), (958, 11236, 2), (1329, 2975, 5433), (2551, 34610, 2), (5260, 20060, 25466), (7288, 8165, 31246)


X(34607) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO ANTI-MANDART-INCIRCLE

Barycentrics    5*a^3-5*(b+c)*a^2+(b+c)^2*a-(b^2-c^2)*(b-c) : :
X(34607) = 5*X(2)-4*X(3829) = X(4)-4*X(8715) = X(20)+2*X(3913) = 2*X(40)+X(3189) = X(2136)+2*X(4297) = 2*X(3829)-5*X(4421) = 6*X(3829)-5*X(11235) = 3*X(4421)-X(11235) = 2*X(13205)+X(20095) = X(34619)+2*X(34707)

The center of the reciprocal HR-ellipse of these triangles is X(11235)

X(34607) lies on these lines: {2,11}, {3,34625}, {4,8715}, {8,4640}, {19,7714}, {20,529}, {30,10306}, {35,5082}, {40,376}, {56,34699}, {65,3241}, {71,4685}, {165,5853}, {200,5698}, {329,3689}, {388,3871}, {428,11406}, {443,3746}, {498,31159}, {516,3158}, {518,5918}, {527,2951}, {535,11001}, {550,34740}, {551,28629}, {553,4321}, {678,33094}, {754,34671}, {962,34647}, {1056,25439}, {1058,25440}, {1706,4314}, {1992,3779}, {2177,5712}, {2551,4294}, {2801,14646}, {2802,7967}, {2900,5759}, {3059,6172}, {3085,17532}, {3146,12607}, {3175,3198}, {3303,6904}, {3421,4302}, {3436,20066}, {3474,3870}, {3476,3895}, {3509,17314}, {3522,12513}, {3523,3813}, {3524,10806}, {3525,24387}, {3528,8666}, {3543,6253}, {3617,32157}, {3622,13463}, {3655,17648}, {3679,5234}, {3742,8236}, {3748,9776}, {3749,4000}, {3750,4648}, {3811,6361}, {3839,34706}, {3880,5731}, {3929,20588}, {3961,4419}, {3974,32929}, {4299,34690}, {4305,10914}, {4309,5084}, {4313,5836}, {4339,4646}, {4863,5744}, {5217,31157}, {5225,5552}, {5229,10528}, {5415,19054}, {5416,19053}, {5437,30331}, {5438,12575}, {5440,30305}, {5493,11523}, {5584,10304}, {5695,7172}, {5745,31508}, {5768,13528}, {5880,10578}, {6261,6769}, {6284,7080}, {6601,15931}, {6745,9580}, {6765,31730}, {6767,19706}, {6856,31452}, {6919,9670}, {6934,34617}, {6948,10993}, {7674,11495}, {7688,19708}, {7735,10988}, {7957,34632}, {7987,21627}, {7991,12437}, {9337,24217}, {9709,10386}, {9710,17558}, {10056,10629}, {10072,34719}, {10164,24392}, {10679,28452}, {11190,11206}, {11240,13587}, {11260,12541}, {12115,13199}, {12635,20070}, {12701,27383}, {15170,16417}, {15338,34689}, {15677,21677}, {20085,32198}, {24703,30332}, {31405,31451}

X(34607) = midpoint of X(i) and X(j) for these {i,j}: {2136, 34716}, {3189, 34744}, {3913, 34626}
X(34607) = reflection of X(i) in X(j) for these (i,j): (2, 4421), (20, 34626), (962, 34647), (3146, 34739), (3543, 11236), (6762, 34646), (24392, 10164), (24477, 165), (25568, 3158), (34610, 376), (34625, 3), (34646, 12512), (34716, 4297), (34739, 12607), (34740, 550), (34744, 40)
X(34607) = anticomplement of X(11235)
X(34607) = intangents-to-extangents similarity image of X(2)
X(34607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 17784, 34612), (2, 20075, 34611), (2, 34611, 497), (2, 34612, 2550), (55, 6154, 17784), (55, 17784, 2550), (55, 34612, 2), (100, 20075, 497), (100, 34611, 2), (390, 1376, 26105), (4995, 31140, 2), (6174, 11238, 2)


X(34608) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO ARA

Barycentrics    5*a^6+(b^2+c^2)*a^4-(5*b^4-2*b^2*c^2+5*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :
X(34608) = 3*X(2)-4*X(10154) = 10*X(2)-9*X(30775) = X(4)-4*X(7387) = X(4)+2*X(31305) = 8*X(26)-5*X(631) = 7*X(26)-4*X(23336) = 2*X(34642)+X(34712) = 4*X(34642)-X(34730) = 2*X(34712)+X(34730)

As a point on the Euler line, X(34608) has Shinagawa coefficients (-E-2*F, 3*E+3*F)
The center of the reciprocal HR-ellipse of these triangles is X(34609)

X(34608) lies on these lines: {2,3}, {51,25406}, {69,16276}, {154,29181}, {388,5310}, {497,5322}, {511,11206}, {519,34642}, {524,5596}, {528,34702}, {529,34691}, {534,4319}, {754,34675}, {1058,9645}, {1249,10313}, {1285,5359}, {1350,14826}, {1352,33522}, {1478,7298}, {1479,5345}, {1899,14927}, {1992,6467}, {3060,14912}, {3618,22352}, {3796,14853}, {5225,5370}, {5229,7302}, {5485,16277}, {6515,15107}, {7747,15437}, {8024,32822}, {8144,29815}, {8280,22644}, {8281,22615}, {9530,19164}, {9921,12320}, {9922,12321}, {11179,21849}, {11427,31670}, {12162,33523}, {17024,32047}, {18928,34417}, {23291,32269}, {29012,32064}, {32006,34254}

X(34608) = reflection of X(i) in X(j) for these (i,j): (2, 9909), (10201, 17714), (14790, 10201), (34609, 10154)
X(34608) = anticomplement of X(34609)
X(34608) = pole of the line {185, 3618} wrt Jerabek hyperbola
X(34608) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 20, 7667), (2, 297, 26205), (2, 377, 409), (2, 384, 2047), (2, 403, 28964), (2, 416, 32956), (2, 427, 31074), (2, 442, 29875), (2, 451, 32978), (2, 453, 33205), (2, 470, 7410), (2, 470, 12043), (2, 632, 33029), (2, 964, 18369), (2, 1368, 3525)


X(34609) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO ABC

Barycentrics    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) : :
X(34609) = 7*X(2)-9*X(30775) = X(3)+2*X(14790) = X(3)-4*X(23335) = X(6391)-4*X(15583) = 8*X(9820)-5*X(14530) = X(9919)-4*X(23306) = X(12164)+2*X(14216) = 2*X(34643)+X(34713) = 4*X(34643)-X(34729) = 2*X(34713)+X(34729)

As a point on the Euler line, X(34609) has Shinagawa coefficients (E-F, -3*E-3*F)

The center of the reciprocal HR-ellipse of these triangles is X(34608).

Let LA be the polar of X(3) wrt the A-power circle, and define LB, LC cyclically. Let A' = LB∩LC, and define B', C' cyclically. Triangle A'B'C' is homothetic to ABC at X(5094), and X(34609) is the centroid of A'B'C'. (Randy Hutson, November 17, 2019)

Let BA, CA be the anticomplementary circle intercepts of line BC. Let B'A, C'A be the {BA,CA}-harmonic conjugates of B, C resp. Define C'B, A'B, A'C, B'C cyclically. X(34609) is the centroid of B'AC'AC'BA'BA'CB'C. (Randy Hutson, November 17, 2019)

X(34609) lies on these lines: {2,3}, {52,26944}, {66,524}, {115,1611}, {125,33586}, {154,29012}, {305,7776}, {343,33878}, {394,11550}, {511,1853}, {519,34643}, {528,34703}, {529,34692}, {542,17847}, {599,3313}, {612,9654}, {614,9669}, {754,34676}, {801,14458}, {1151,8280}, {1152,8281}, {1350,21243}, {1351,1899}, {1503,3167}, {1992,26926}, {3060,26869}, {3162,22120}, {3564,32064}, {3818,17811}, {3819,10516}, {3867,15812}, {3920,32047}, {5093,11245}, {5102,11225}, {5268,10895}, {5272,10896}, {5654,32063}, {6101,33523}, {7191,8144}, {7788,14615}, {8780,11064}, {8854,23251}, {8855,23261}, {9140,13201}, {9629,11238}, {9777,18911}, {9820,14530}, {9919,23306}, {11265,13903}, {11266,13961}, {11433,21850}, {11442,11898}, {11457,12160}, {12164,14216}, {12310,23315}, {12315,22660}, {12429,18381}, {13567,31670}, {14806,31489}, {17825,19130}, {17834,20299}, {18909,31802}, {19161,21849}, {19583,32816}, {21970,26958}, {23332,29181}, {25055,34712}

X(34609) = reflection of X(i) in X(j) for these (i,j): (7387, 10201), (9909, 2), (10201, 13371), (14070, 18281), (17714, 34330), (32063, 5654), (34608, 10154), (34726, 14070)
X(34609) = anticomplement of X(10154)
X(34609) = complement of X(34608)
X(34609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 23, 5), (2, 23, 20843), (2, 25, 30734), (2, 405, 19532), (2, 423, 1584), (2, 1370, 7667), (2, 1375, 23262), (2, 2047, 7819), (2, 2476, 11311), (2, 3134, 33025), (2, 3138, 33840), (2, 3542, 16057), (2, 3543, 7714), (2, 3628, 19285), (2, 3857, 7539)


X(34610) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    5*a^4-4*(b-c)^2*a^2-4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34610) = 10*X(1)-X(28647) = X(4)-4*X(8666) = 5*X(4)-8*X(24387) = X(20)+2*X(12513) = X(2136)-4*X(12512) = X(3189)-4*X(4297) = X(3189)+2*X(6762) = 5*X(8666)-2*X(24387) = 3*X(11194)-X(11236) = X(34625)+2*X(34740)

The center of the reciprocal HR-ellipse of these triangles is X(11236)

X(34610) lies on these lines: {1,527}, {2,12}, {3,34619}, {4,535}, {8,1155}, {9,4315}, {20,528}, {21,33925}, {30,22770}, {36,3421}, {40,376}, {55,34749}, {63,3476}, {65,2094}, {144,5289}, {329,1319}, {443,4317}, {452,3304}, {497,11114}, {499,31160}, {515,24477}, {518,5731}, {550,34707}, {551,3487}, {754,34679}, {758,7967}, {950,31146}, {956,2550}, {960,4308}, {962,11260}, {993,1056}, {999,26105}, {1420,12527}, {2099,9965}, {2183,21384}, {3057,3241}, {3086,17556}, {3146,3813}, {3296,30143}, {3303,17576}, {3361,5795}, {3434,20067}, {3452,13462}, {3474,3872}, {3485,31164}, {3522,3913}, {3523,12607}, {3524,10805}, {3528,8715}, {3543,11235}, {3555,4305}, {3576,25568}, {3616,4679}, {3621,13996}, {3655,31786}, {3679,15803}, {3680,5493}, {3829,3839}, {3880,9778}, {3897,20323}, {4298,6173}, {4299,5082}, {4302,34719}, {4421,10304}, {5080,10589}, {5084,5563}, {5177,9657}, {5204,6174}, {5225,10529}, {5229,10527}, {5252,5744}, {5270,6856}, {5303,10528}, {6834,10711}, {6938,34629}, {7354,31140}, {9655,31418}, {10032,15677}, {10056,34690}, {10072,10629}, {10912,20070}, {11239,17549}, {11249,12667}, {12629,31730}, {13407,25055}, {15326,17784}, {15680,34741}, {17528,18990}, {20057,28646}, {24392,28164}

X(34610) = midpoint of X(i) and X(j) for these {i,j}: {3241, 28610}, {3928, 34716}, {6762, 34701}, {12513, 34620}
X(34610) = reflection of X(i) in X(j) for these (i,j): (2, 11194), (20, 34620), (962, 34640), (2136, 34639), (3146, 34706), (3189, 34701), (3543, 11235), (3928, 34646), (25568, 3576), (28609, 551), (34607, 376), (34619, 3), (34639, 12512), (34640, 11260), (34701, 4297), (34706, 3813), (34707, 550), (34711, 40), (34739, 3829), (34744, 3928)
X(34610) = anticomplement of X(11236)
X(34610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 20076, 34605), (2, 34605, 388), (2, 34606, 2551), (56, 34606, 2), (388, 2975, 30478), (961, 29524, 30979), (2975, 20076, 388), (2975, 34605, 2), (5298, 31141, 2), (5484, 25914, 27283), (8169, 26686, 26558), (8169, 30001, 26259), (11237, 31157, 2), (15844, 27410, 5260), (27994, 31157, 2975), (31039, 31246, 958)


X(34611) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC TO MANDART-INCIRCLE

Barycentrics    2*a^3-2*(b+c)*a^2+(b^2-b*c+c^2)*a-(b^2-c^2)*(b-c) : :
X(34611) = X(8)-4*X(15171) = X(145)+2*X(6284) = 4*X(950)-X(14923) = 3*X(3058)-X(34612) = X(3869)-4*X(10624) = X(3885)+2*X(10572) = X(11114)-4*X(34649) = X(11114)+2*X(34719) = X(34629)+2*X(34745) = 2*X(34649)+X(34719)

The center of the reciprocal HR-ellipse of these triangles is X(34612)

X(34611) lies on these lines: {1,11015}, {2,11}, {4,11239}, {8,11113}, {21,4309}, {30,944}, {56,20066}, {145,529}, {333,21283}, {376,10806}, {516,3873}, {517,5890}, {519,3869}, {527,30628}, {754,34683}, {902,33141}, {950,14923}, {1058,5253}, {1279,33131}, {1479,3871}, {1697,5086}, {1770,3889}, {1836,3957}, {2177,33106}, {2475,3303}, {2476,3746}, {2975,4294}, {3052,33142}, {3120,17715}, {3175,4463}, {3219,4863}, {3242,33100}, {3244,34690}, {3295,17532}, {3543,12667}, {3583,25439}, {3586,3895}, {3616,15172}, {3621,34689}, {3623,7354}, {3635,10483}, {3655,12700}, {3679,5178}, {3681,5853}, {3685,5014}, {3689,27131}, {3722,3944}, {3744,33134}, {3748,31019}, {3749,33133}, {3750,33104}, {3813,4189}, {3839,10894}, {3870,5057}, {3886,33075}, {3890,12575}, {3913,5046}, {3935,24703}, {3938,33095}, {3979,24725}, {4030,4671}, {4193,4857}, {4330,8666}, {4387,33091}, {4430,17768}, {4432,33117}, {4450,10453}, {4514,32929}, {4660,32943}, {4689,29680}, {4693,32854}, {4702,32858}, {4854,29815}, {5080,9668}, {5082,5260}, {5225,10528}, {5249,30331}, {5303,10529}, {5441,22837}, {5536,31146}, {5603,28452}, {5658,9812}, {5695,33090}, {5840,7967}, {5880,29817}, {6261,31162}, {7504,31452}, {8616,33136}, {9352,11019}, {9589,11520}, {9669,27529}, {9709,26127}, {9710,16859}, {10056,17577}, {10072,13587}, {10129,13405}, {10269,13199}, {10284,12672}, {10386,24390}, {10543,13463}, {10724,12115}, {10914,31795}, {11112,15170}, {11237,34706}, {11355,19738}, {11451,22278}, {11691,31770}, {12513,15680}, {12649,34744}, {12701,34647}, {12953,20060}, {15310,23155}, {16371,34707}, {17155,17764}, {17220,17378}, {17233,28599}, {17597,33102}, {17765,32925}, {17766,32915}, {18134,21282}, {18990,20057}, {19538,31494}, {20070,34618}, {24473,28198}, {28610,30332}, {29818,33149}, {29844,32845}, {31145,34606}, {32782,32941}

X(34611) = midpoint of X(6284) and X(34699)
X(34611) = reflection of X(i) in X(j) for these (i,j): (2, 3058), (8, 11113), (145, 34699), (3621, 34689), (10483, 34637), (11112, 15170), (11113, 15171), (17579, 1), (20070, 34618), (31145, 34606), (34605, 3241), (34617, 1482), (34637, 3635), (34690, 3244), (34698, 1483)
X(34611) = anticomplement of X(34612)
X(34611) = pole of the line {518, 11681} wrt Feuerbach hyperbola
X(34611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 33094, 33146), (2, 149, 11235), (2, 11235, 11680), (2, 20075, 34607), (2, 34607, 100), (55, 149, 11680), (55, 11235, 2), (390, 3434, 1621), (497, 20075, 100), (497, 34607, 2), (1621, 3434, 33108), (3829, 4995, 2), (4421, 11238, 2), (4428, 31140, 2)


X(34612) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO ABC

Barycentrics    2*a^3-2*(b+c)*a^2+(b+c)^2*a-(b^2-c^2)*(b-c) : :
X(34612) = 2*X(8)+X(7354) = 4*X(10)-X(6284) = 2*X(40)+X(6253) = 2*X(355)+X(11826) = 3*X(3058)-2*X(34611) = X(3893)+2*X(10106) = 4*X(5690)-X(11827) = X(5691)+2*X(31777) = 2*X(10914)+X(10944) = X(10914)+2*X(17647) = 2*X(17579)+X(34689)

The center of the reciprocal HR-ellipse of these triangles is X(34611).

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. A'B'C' is homothetic to ABC at X(55), and X(34612) is the centroid of A'B'C'. (Randy Hutson, November 17, 2019)

X(34612) lies on these lines: {1,34699}, {2,11}, {3,31157}, {4,21031}, {5,31159}, {8,529}, {10,3683}, {12,5687}, {19,428}, {21,9710}, {30,40}, {35,24953}, {51,22278}, {56,5082}, {57,4863}, {65,519}, {71,17330}, {72,17646}, {75,4030}, {78,34647}, {99,12348}, {141,32945}, {142,3748}, {165,5659}, {190,4126}, {200,1836}, {209,752}, {210,516}, {226,3689}, {230,10988}, {329,3711}, {354,5853}, {376,5584}, {377,3913}, {381,10306}, {395,10637}, {396,10636}, {404,3813}, {442,8715}, {443,3303}, {515,5918}, {517,5891}, {518,11246}, {524,3779}, {527,3059}, {535,4669}, {549,10902}, {550,5258}, {551,17614}, {594,5282}, {597,19133}, {599,12586}, {612,4854}, {671,13180}, {754,34684}, {851,15621}, {903,24834}, {936,12701}, {950,3698}, {956,15326}, {958,15338}, {999,19706}, {1086,3938}, {1155,4847}, {1211,4660}, {1479,9709}, {1503,11190}, {1698,15171}, {1706,1837}, {1708,3419}, {1738,3744}, {1776,5086}, {2093,4677}, {2177,17056}, {2334,4340}, {2475,12607}, {2551,12953}, {2829,14646}, {3011,21949}, {3158,17718}, {3174,6173}, {3175,28580}, {3189,3241}, {3304,6904}, {3416,4046}, {3421,12943}, {3436,34739}, {3524,10785}, {3534,18519}, {3545,10893}, {3550,32865}, {3582,10948}, {3583,3820}, {3584,10523}, {3614,5552}, {3624,15172}, {3625,34637}, {3632,18990}, {3649,3811}, {3681,17768}, {3703,32850}, {3704,5300}, {3712,29641}, {3715,5698}, {3729,30615}, {3745,3755}, {3746,8728}, {3749,24789}, {3782,3961}, {3828,17619}, {3830,18516}, {3845,18406}, {3870,5880}, {3871,25466}, {3899,28212}, {3914,17602}, {3922,6738}, {3932,32929}, {3935,20292}, {3957,25557}, {3962,6743}, {3971,17764}, {3983,12572}, {3996,4645}, {4023,4388}, {4113,4416}, {4190,12513}, {4299,34740}, {4302,9708}, {4309,11108}, {4386,21956}, {4415,33094}, {4450,4651}, {4534,24247}, {4640,25006}, {4661,5852}, {4668,10483}, {4679,8580}, {4853,34716}, {4857,17527}, {4882,9579}, {4884,32845}, {5046,9711}, {5054,26492}, {5055,11928}, {5064,11390}, {5071,10598}, {5084,9670}, {5160,9639}, {5217,19843}, {5259,10386}, {5260,20066}, {5298,10949}, {5306,10315}, {5415,19024}, {5416,19023}, {5433,24390}, {5438,11376}, {5440,15950}, {5463,12922}, {5464,12921}, {5524,33096}, {5537,8727}, {5563,17563}, {5657,5842}, {5692,28174}, {5695,6057}, {5718,33109}, {5741,21282}, {5743,32947}, {5762,15104}, {5790,5840}, {5794,34687}, {5836,10391}, {5846,32860}, {5881,30304}, {6054,12182}, {6172,16112}, {6197,7576}, {6745,17605}, {6765,10404}, {6769,12700}, {6856,31420}, {6914,10993}, {6919,9671}, {7080,10895}, {7173,26364}, {7263,32923}, {7688,8703}, {7757,12923}, {7767,32104}, {7865,10871}, {7957,12672}, {7964,17613}, {7991,20420}, {7994,31142}, {8356,16829}, {8539,8584}, {9053,17155}, {9140,13213}, {9312,30623}, {9614,24954}, {9679,18965}, {9816,10128}, {10056,17528}, {10072,16417}, {10319,10691}, {10522,11236}, {10543,19860}, {10591,31246}, {10624,25917}, {10706,12371}, {10711,12761}, {10718,13294}, {10826,19875}, {11237,18961}, {11355,19723}, {11373,15170}, {11499,15908}, {12245,34617}, {12436,17609}, {12645,34698}, {12647,13996}, {12924,31168}, {13693,13712}, {13747,24387}, {13813,13835}, {13901,31484}, {15177,28454}, {15624,21926}, {15625,28349}, {16418,34707}, {17061,33131}, {17070,29665}, {17366,17469}, {17395,29816}, {17615,17781}, {17724,17889}, {17728,24392}, {17765,24165}, {17782,29661}, {19038,31413}, {19535,31458}, {20323,21627}, {21927,34247}, {22313,23638}, {22703,22712}, {22758,24466}, {24693,29651}, {24712,25355}, {25280,32819}, {25351,29672}, {28198,31937}, {28204,28458}, {28465,32613}, {28530,32925}, {31145,34605}, {31260,31493}, {31416,31448}, {32287,34319}

X(34612) = midpoint of X(i) and X(j) for these {i,j}: {8, 17579}, {3625, 34637}, {3632, 34690}, {5434, 34720}, {6253, 34618}, {7354, 34689}, {11826, 34697}, {12245, 34617}, {12645, 34698}, {31145, 34605}
X(34612) = reflection of X(i) in X(j) for these (i,j): (51, 22278), (3058, 2), (5434, 11112), (6284, 11113), (7354, 17579), (11113, 10), (34606, 3679), (34618, 40), (34689, 8), (34690, 18990), (34697, 355), (34699, 1), (34719, 15170), (34749, 5434)
X(34612) = complement of X(34611)
X(34612) = homothetic center of extangents triangle and reflection of intangents triangle in X(2)
X(34612) = X(354)-of-extangents-triangle if ABC is acute
X(34612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3434, 11235), (2, 4421, 4995), (2, 11235, 11), (2, 17784, 34607), (2, 34607, 55), (55, 2550, 3925), (55, 17784, 6154), (100, 2886, 5432), (100, 33110, 2886), (390, 26040, 4423), (1376, 3434, 11), (1376, 11235, 2), (2550, 17784, 55), (2550, 34607, 2), (3925, 6154, 55)


X(34613) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ANTI-ARA

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4+7*b^2*c^2+c^4)*a^6+4*(b^2+c^2)*(b^4+c^4)*a^4+10*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    3*(2*R^2+SW)*SB*SC-SW*S^2 : :
X(34613) = 3*X(3)-4*X(23410) = 7*X(4)-4*X(12362) = 3*X(5603)-2*X(34634) = X(11412)-4*X(16621) = 4*X(13598)-X(34224) = 5*X(15058)-8*X(16656)

As a point on the Euler line, X(34613) has Shinagawa coefficients (-2*E-2*F, 9*E+6*F)
The center of the reciprocal HR-ellipse of these triangles is X(34614)

X(34613) lies on these lines: {2,3}, {511,16658}, {515,34657}, {516,34633}, {517,34668}, {524,16655}, {528,34663}, {529,34655}, {542,16659}, {2781,11660}, {3058,9628}, {5254,13338}, {5476,10984}, {5562,19924}, {5603,34634}, {7592,20423}, {7745,13337}, {11412,16621}, {11456,31670}, {11459,16654}, {11645,13598}, {12022,29012}, {12359,15360}, {15030,29317}, {15032,21850}, {15058,16656}, {20299,32225}

X(34613) = reflection of X(i) in X(j) for these (i,j): (376, 428), (3534, 13490), (7576, 34603), (11459, 16654)
X(34613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 297, 19257), (2, 417, 33291), (2, 422, 7380), (2, 441, 16843), (2, 441, 19536), (2, 441, 27280), (2, 447, 32970), (2, 456, 26121), (2, 457, 6893), (2, 631, 23266), (2, 632, 21529), (2, 852, 442), (2, 861, 19292), (2, 868, 30802), (2, 964, 18369)


X(34614) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO ABC-X3 REFLECTIONS

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4+22*b^2*c^2+c^4)*a^6+4*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^4+16*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics    3*(8*R^2+SW)*SB*SC-(12*R^2+SW)*S^2 : :
X(34614) = 11*X(3)-8*X(23411) = 4*X(20)-X(3575) = 5*X(20)-2*X(31829) = 3*X(9778)-X(34668)

As a point on the Euler line, X(34614) has Shinagawa coefficients (4*E+F, -9*E-3*F)
The center of the reciprocal HR-ellipse of these triangles is X(34613)

X(34614) lies on these lines: {2,3}, {515,34656}, {516,34634}, {517,34667}, {528,34662}, {529,34654}, {542,34660}, {9778,34668}, {13857,16252}, {19924,34650}

X(34614) = reflection of X(i) in X(j) for these (i,j): (428, 376), (3543, 10691), (13490, 15690)
X(34614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24, 16405), (2, 29, 34478), (2, 140, 31106), (2, 235, 4249), (2, 297, 15236), (2, 377, 16376), (2, 383, 6676), (2, 383, 7519), (2, 384, 16398), (2, 405, 33730), (2, 409, 15330), (2, 410, 11286), (2, 411, 7470), (2, 414, 31050), (2, 416, 25828)


X(34615) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 5th ANTI-BROCARD

Barycentrics    a^8-11*(b^2+c^2)*a^6+(2*b^4-17*b^2*c^2+2*c^4)*a^4+(b^2+c^2)*(7*b^4-6*b^2*c^2+7*c^4)*a^2+(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2 : :
Barycentrics    8*S^4+(6*SA^2-9*SB*SC-2*SW^2)*S^2-9*SB*SC*SW^2 : :
X(34615) = 3*X(4)-2*X(34681) = 3*X(376)-2*X(34616) = 3*X(3545)-2*X(7883) = 3*X(5603)-2*X(34636) = 3*X(12150)-X(34616) = 3*X(34604)-2*X(34682)

The center of the reciprocal HR-ellipse of these triangles is X(34616)

X(34615) lies on these lines: {2,9821}, {4,754}, {30,34604}, {182,376}, {515,34674}, {516,34635}, {517,34686}, {528,34680}, {529,34672}, {542,34678}, {576,34624}, {3543,5984}, {3545,7883}, {3830,7823}, {5066,7885}, {5603,34636}, {5969,32474}, {7739,7756}, {7810,14492}, {7818,9993}, {8356,21850}, {9862,31670}, {10131,33265}, {10350,14033}, {15031,32006}, {19130,31168}, {22521,29181}

X(34615) = reflection of X(376) in X(12150)
X(34615) = pole of the line {5650, 12054} wrt Stammler hyperbola


X(34616) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO ABC-X3 REFLECTIONS

Barycentrics    7*a^8+7*(b^2+c^2)*a^6-(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^4-11*(b^2+c^2)*(b^4+c^4)*a^2-(2*b^4+3*b^2*c^2+2*c^4)*(b^2-c^2)^2 : :
Barycentrics    S^4+(3*SA^2+8*SW^2)*S^2-18*SB*SC*SW^2 : :
X(34616) = 3*X(376)-X(34615) = 3*X(3534)-X(34682) = 3*X(7883)-2*X(34681) = 3*X(9778)-X(34686) = 3*X(12150)-2*X(34615)

The center of the reciprocal HR-ellipse of these triangles is X(34615)

X(34616) lies on these lines: {20,754}, {30,7883}, {182,376}, {515,34673}, {516,34636}, {517,34685}, {528,34679}, {529,34671}, {542,34677}, {3098,31168}, {3534,7757}, {3830,7911}, {7802,11001}, {9778,34686}, {10350,33265}, {12117,13188}

X(34616) = reflection of X(12150) in X(376)


X(34617) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    2*a^7-2*(b+c)*a^6-(3*b^2-5*b*c+3*c^2)*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4-4*(b^2-4*b*c+c^2)*b*c*a^3+8*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :
X(34617) = 3*X(4)-2*X(34697) = 3*X(3545)-2*X(34606) = 3*X(5603)-2*X(11113) = 5*X(5734)-2*X(7491) = X(6361)-4*X(18990) = 5*X(10595)-2*X(11827)

The center of the reciprocal HR-ellipse of these triangles is X(34618)

X(34617) lies on these lines: {2,10532}, {4,529}, {8,28452}, {30,944}, {55,376}, {149,3543}, {381,956}, {515,34690}, {516,34637}, {517,2979}, {528,34631}, {535,31162}, {542,34694}, {1012,34740}, {1770,5697}, {2095,12247}, {3304,6903}, {3545,10894}, {3555,28204}, {3656,11114}, {3679,5536}, {3957,18481}, {4345,15170}, {5258,6990}, {5603,11113}, {5734,7491}, {5842,34699}, {6361,18990}, {6845,8666}, {6876,15888}, {6934,34607}, {7680,31157}, {10197,11012}, {10284,28198}, {10595,11827}, {11239,11491}, {12245,34612}, {17532,22770}, {19924,34669}, {22753,31141}, {28458,34632}

X(34617) = reflection of X(i) in X(j) for these (i,j): (8, 28452), (376, 5434), (11114, 3656), (12245, 34612), (34611, 1482), (34632, 28458)


X(34618) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ABC-X3 REFLECTIONS

Barycentrics    2*a^7-2*(b+c)*a^6-(b+3*c)*(3*b+c)*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4+8*(b+c)^2*b*c*a^3+8*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :
X(34618) = 4*X(40)-X(6253) = 3*X(3534)-X(34698) = X(6284)+2*X(6361) = X(7354)-4*X(31730) = 3*X(9778)-X(17579)

The center of the reciprocal HR-ellipse of these triangles is X(34617)

X(34618) lies on these lines: {2,5584}, {20,529}, {30,40}, {55,376}, {65,3058}, {381,3925}, {428,11471}, {515,34689}, {516,3753}, {517,14855}, {519,7957}, {528,9803}, {535,34638}, {542,34693}, {549,7688}, {2550,3543}, {2951,12678}, {3428,31157}, {3534,10306}, {3579,28452}, {6154,15681}, {6284,6361}, {6836,11235}, {6849,34501}, {6925,34739}, {7354,31397}, {7994,34628}, {8703,10902}, {9778,17579}, {10197,12511}, {11190,15311}, {11529,15170}, {12565,28609}, {12607,33557}, {15683,17784}, {15687,18406}, {19924,34670}, {20070,34611}, {28198,28459}, {30503,31162}

X(34618) = midpoint of X(20070) and X(34611)
X(34618) = reflection of X(i) in X(j) for these (i,j): (5434, 376), (6253, 34612), (28452, 3579), (34612, 40), (34746, 3654)
X(34618) = {X(1762), X(2100)}-harmonic conjugate of X(7330)


X(34619) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ANTI-MANDART-INCIRCLE

Barycentrics    a^4-2*(b^2+4*b*c+c^2)*a^2+4*(b+c)*b*c*a+(b^2-c^2)^2 : :
X(34619) = X(4)+2*X(3913) = X(4)-4*X(12607) = 3*X(4)-2*X(34706) = X(8)+2*X(3811) = X(8)+2*X(3811) = X(8)-4*X(10915) = X(8)-4*X(10915) = 2*X(10)+X(6765) = 2*X(10)+X(6765) = X(145)-4*X(22836) = X(145)-4*X(22836) = 4*X(1125)-X(12629) = 4*X(1125)-X(12629) = X(3913)+2*X(12607) = 3*X(3913)+X(34706) = 3*X(11236)-X(34706) = 6*X(12607)-X(34706)

The center of the reciprocal HR-ellipse of these triangles is X(34620)

X(34619) lies on these lines: {1,2}, {3,34610}, {4,528}, {12,5082}, {20,535}, {30,10306}, {40,527}, {46,2094}, {55,3421}, {56,6174}, {100,4293}, {153,25438}, {329,5119}, {341,17264}, {345,4737}, {355,3189}, {376,529}, {388,5687}, {390,25439}, {443,15888}, {452,3746}, {484,9965}, {495,2550}, {497,17556}, {515,3158}, {516,34639}, {517,25568}, {518,5657}, {542,34705}, {631,12513}, {758,15104}, {908,3895}, {943,4428}, {944,32049}, {946,2136}, {956,5218}, {962,21077}, {993,5281}, {999,17564}, {1000,5289}, {1056,1376}, {1058,1329}, {1466,5434}, {1478,17784}, {1479,31160}, {1697,21075}, {1706,6173}, {1788,3555}, {1792,4234}, {2096,13528}, {2475,31410}, {2551,3295}, {2802,5660}, {2886,8164}, {3058,31141}, {3090,3813}, {3091,12632}, {3262,3673}, {3303,5084}, {3304,17567}, {3338,26062}, {3434,10590}, {3436,3871}, {3452,31393}, {3475,3753}, {3476,5440}, {3485,10914}, {3487,5836}, {3523,8666}, {3524,11194}, {3545,10598}, {3600,25440}, {3654,31788}, {3656,5761}, {3672,4868}, {3680,6964}, {3689,5252}, {3754,11036}, {3814,5274}, {3820,6767}, {3829,5071}, {3880,5603}, {3893,11375}, {3926,24524}, {3950,5199}, {3991,6554}, {4295,31164}, {4452,17885}, {4487,33113}, {4515,17281}, {4646,17301}, {4723,17776}, {4995,34689}, {5056,24387}, {5080,20075}, {5175,10827}, {5177,31420}, {5258,31452}, {5286,20691}, {5587,5853}, {5726,8232}, {5734,6953}, {5748,30384}, {5758,6260}, {5766,12572}, {5787,28204}, {5815,6172}, {5854,34710}, {5881,6847}, {5882,6926}, {5883,11038}, {6154,12943}, {6361,28534}, {6675,31480}, {6684,6762}, {6833,32537}, {6837,12536}, {6846,12625}, {6848,7982}, {6908,11362}, {6916,34742}, {6928,34745}, {6944,24680}, {6983,33895}, {8227,21627}, {8703,34740}, {9588,11407}, {9711,17559}, {9785,21616}, {10156,24477}, {10175,24392}, {10385,11113}, {10588,24390}, {10591,10707}, {10595,10912}, {10786,12245}, {11001,34626}, {11237,18961}, {11238,34699}, {11500,34687}, {12631,15170}, {12641,25485}, {14740,18397}, {15621,19262}, {15677,31660}, {15682,34739}, {15733,18908}, {17144,32828}, {17313,21258}, {17448,31400}, {19924,34671}, {21096,23058}, {33908,34511}

X(34619) = midpoint of X(3913) and X(11236)
X(34619) = reflection of X(i) in X(j) for these (i,j): (4, 11236), (376, 4421), (11001, 34626), (11236, 12607), (15682, 34739), (24392, 10175), (24477, 26446), (34610, 3), (34625, 2), (34625, 2), (34740, 8703), (34744, 3654)
X(34619) = pole of the line {2826, 7649} wrt polar circle
X(34619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 145, 11240), (2, 11240, 3086), (2, 14986, 10199), (2, 29675, 33120), (2, 31241, 26660), (8, 3085, 19843), (8, 10528, 3085), (8, 30163, 27148), (10, 25761, 33117), (42, 3840, 29867), (42, 25761, 30145), (42, 28371, 1722), (43, 17026, 30143), (78, 28742, 26030), (145, 5552, 3086), (306, 17734, 30121), (306, 29643, 26626), (386, 31339, 612), (387, 29577, 25352), (387, 33175, 29571), (551, 29606, 26981), (612, 24603, 26807), (612, 29639, 6686), (612, 29861, 29598), (869, 3741, 15865), (869, 29699, 29680), (938, 21718, 29873), (938, 26181, 29580), (997, 26169, 29637), (997, 26597, 19876), (997, 29613, 26658), (997, 30164, 6347), (1103, 17022, 29661), (1103, 26964, 30143)


X(34620) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ABC-X3 REFLECTIONS

Barycentrics    7*a^4-(5*b^2-4*b*c+5*c^2)*a^2-2*(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34620) = 2*X(20)+X(12513) = 3*X(376)-X(34619) = 4*X(550)-X(3913) = X(1657)+2*X(8666) = 5*X(3522)-2*X(12607) = X(3529)+2*X(3813) = 3*X(3534)-X(34707) = 4*X(4297)-X(12635) = 3*X(4421)-2*X(34619) = X(5073)-4*X(24387) = 3*X(11194)-X(34706) = 3*X(11235)-2*X(34706) = 3*X(34626)-2*X(34707) = X(34626)+2*X(34740) = X(34707)+3*X(34740)

The center of the reciprocal HR-ellipse of these triangles is X(34619)

X(34620) lies on these lines: {1,28534}, {2,3614}, {3,535}, {20,528}, {30,10525}, {36,17556}, {55,20067}, {56,11114}, {376,529}, {405,4325}, {515,34700}, {516,34640}, {517,34710}, {518,34701}, {519,3534}, {527,4297}, {542,34704}, {550,3913}, {956,4316}, {958,4299}, {993,17528}, {1001,4293}, {1376,15326}, {1657,8666}, {2094,3486}, {2646,31164}, {2975,31140}, {3304,15680}, {3436,6174}, {3522,12607}, {3529,3813}, {3543,3829}, {3584,19704}, {3649,15678}, {3654,34717}, {3679,3916}, {3828,19706}, {3928,10085}, {4189,9657}, {4324,34719}, {4428,5434}, {5073,24387}, {5267,9655}, {5270,19535}, {5289,21578}, {5298,18961}, {5731,17768}, {6284,11240}, {6934,34746}, {6942,10711}, {7280,31160}, {8715,15696}, {9688,19028}, {9778,34711}, {10707,12953}, {11001,34625}, {11237,17549}, {12512,32049}, {12943,17577}, {13587,31141}, {15338,20076}, {17648,34716}, {19924,34672}, {24467,28204}

X(34620) = midpoint of X(i) and X(j) for these {i,j}: {20, 34610}, {3534, 34740}, {3928, 34628}, {11001, 34625}
X(34620) = reflection of X(i) in X(j) for these (i,j): (3543, 3829), (4421, 376), (11235, 11194), (11236, 3), (12513, 34610), (34626, 3534), (34717, 3654), (34739, 2)


X(34621) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO ARA

Barycentrics    a^10-3*(b^2+c^2)*a^8+2*(b^2-4*b*c+c^2)*(b^2+4*b*c+c^2)*a^6+2*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^4-(b^2-c^2)^2*(3*b^4-14*b^2*c^2+3*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^3 : :
X(34621) = 7*X(2)-8*X(10201) = 3*X(5603)-2*X(34643) = X(6225)+2*X(17834)

As a point on the Euler line, X(34621) has Shinagawa coefficients (E+F, -3*E)
The center of the reciprocal HR-ellipse of these triangles is X(34622)

X(34621) lies on these lines: {2,3}, {193,11456}, {511,5656}, {515,34712}, {516,34642}, {517,34730}, {524,1498}, {528,34724}, {529,34702}, {542,34722}, {599,15811}, {1181,1992}, {1514,28419}, {3600,9645}, {5032,7592}, {5603,34643}, {6225,17834}, {10519,15030}, {12112,20080}, {12279,15360}, {13598,20423}, {14927,18396}, {16657,25406}, {18931,32269}, {19924,34675}, {26937,32225}

X(34621) = reflection of X(i) in X(j) for these (i,j): (376, 9909), (12085, 34351), (21036, 29912)
X(34621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 20, 865), (2, 21, 26154), (2, 29, 28065), (2, 297, 29949), (2, 401, 26654), (2, 403, 24906), (2, 406, 26648), (2, 409, 19324), (2, 414, 31107), (2, 415, 31258), (2, 415, 33732), (2, 419, 26002), (2, 440, 868), (2, 441, 16843), (2, 441, 19536)


X(34622) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO ABC-X3 REFLECTIONS

Barycentrics    7*a^10-12*(b^2+c^2)*a^8-4*(b^2-3*b*c-c^2)*(b^2+3*b*c-c^2)*a^6+2*(b^2+c^2)*(7*b^4-18*b^2*c^2+7*c^4)*a^4-(b^2-c^2)^2*(3*b^4+22*b^2*c^2+3*c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(34622) = 11*X(3)-8*X(13383) = 4*X(3357)-X(12429) = X(5925)+2*X(13346) = 3*X(9778)-X(34730) = 2*X(12118)+X(13093) = X(12164)+2*X(20427)

As a point on the Euler line, X(34622) has Shinagawa coefficients (E-5*F, -3*E+9*F)
The center of the reciprocal HR-ellipse of these triangles is X(34621)

X(34622) lies on these lines: {2,3}, {64,542}, {155,541}, {185,32284}, {511,7729}, {515,34713}, {516,34643}, {517,34729}, {524,5894}, {528,34723}, {529,34703}, {1351,20725}, {3167,15311}, {3357,12429}, {5925,13346}, {9778,34730}, {11424,14848}, {11645,17845}, {12118,13093}, {12164,20427}, {13568,20423}, {19456,20127}, {19924,34676}, {23326,29181}

X(34622) = reflection of X(9909) in X(376)
X(34622) = pole of the line {6103, 8770} wrt Dao-Moses-Telv circle
X(34622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 21, 11348), (2, 23, 19537), (2, 27, 28103), (2, 235, 30506), (2, 377, 16376), (2, 381, 6846), (2, 405, 33730), (2, 410, 27592), (2, 413, 10124), (2, 413, 11585), (2, 415, 16405), (2, 416, 25828), (2, 416, 26556), (2, 419, 24904), (2, 420, 34725)


X(34623) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 5th BROCARD

Barycentrics    4*a^8-5*(b^2+c^2)*a^6+(2*b^2-c^2)*(b^2-2*c^2)*a^4+(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^2-(2*b^4+3*b^2*c^2+2*c^4)*(b^2-c^2)^2 : :
Barycentrics    8*S^4+(6*SA^2-9*SB*SC-2*SW^2)*S^2+3*SB*SC*SW^2 : :
X(34623) = 3*X(4)-2*X(34733) = X(194)-4*X(32151) = 3*X(376)-2*X(34624) = 3*X(3524)-4*X(7810) = 3*X(3545)-2*X(7812) = 5*X(5071)-4*X(7753) = 3*X(5603)-2*X(34645) = 3*X(7709)-4*X(8356) = 2*X(7826)+X(9873) = 2*X(9863)+X(12251) = 3*X(9939)-2*X(34734)

The center of the reciprocal HR-ellipse of these triangles is X(34624)

X(34623) lies on these lines: {2,3398}, {4,754}, {30,9863}, {69,74}, {98,7818}, {182,31168}, {194,32151}, {381,385}, {515,34714}, {516,34644}, {517,34738}, {528,34732}, {529,34704}, {547,7806}, {549,3314}, {631,7909}, {732,22678}, {1352,10788}, {2782,33264}, {3186,7576}, {3524,7810}, {3545,7812}, {3564,7709}, {5071,7735}, {5603,34645}, {7766,9996}, {7826,9873}, {7897,12042}, {7898,12188}, {7929,14880}, {7931,15694}, {10358,12156}, {10516,22521}, {10722,17131}, {11178,12150}, {11288,21445}, {11676,15069}, {12243,33017}, {14651,16041}, {15703,16984}, {19924,34677}

X(34623) = reflection of X(376) in X(7811)
X(34623) = pole of the line {1495, 3095} wrt Stammler hyperbola


X(34624) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO ABC-X3 REFLECTIONS

Barycentrics    2*a^8+5*(b^2+c^2)*a^6-(5*b^4+7*b^2*c^2+5*c^4)*a^4-(b^2+c^2)*(b^4+c^4)*a^2-(b^4+3*b^2*c^2+c^4)*(b^2-c^2)^2 : :
Barycentrics    S^4+(3*SA^2-4*SW^2)*S^2+6*SB*SC*SW^2 : :
X(34624) = 3*X(376)-X(34623) = 3*X(3534)-X(34734) = 2*X(7810)-3*X(10304) = 3*X(7811)-2*X(34623) = 3*X(7812)-2*X(34733) = 3*X(9778)-X(34738)

The center of the reciprocal HR-ellipse of these triangles is X(34623)

X(34624) lies on these lines: {2,8721}, {20,754}, {30,3095}, {69,74}, {147,7818}, {381,7790}, {515,34715}, {516,34645}, {517,34737}, {528,34731}, {529,34705}, {549,7835}, {576,34615}, {1352,31168}, {1503,8356}, {2387,15072}, {2549,3543}, {2794,33264}, {3534,34734}, {3564,22676}, {5071,7919}, {5984,8722}, {7470,7796}, {7709,29012}, {7710,16041}, {7810,10304}, {7831,18440}, {7857,14880}, {9778,34738}, {9890,11177}, {11179,12150}, {14039,15428}, {19924,34678}

X(34624) = reflection of X(i) in X(j) for these (i,j): (3543, 7753), (7811, 376)
X(34624) = pole of the line {1495, 5188} wrt Stammler hyperbola


X(34625) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    a^4-2*(b^2-4*b*c+c^2)*a^2-4*(b+c)*b*c*a+(b^2-c^2)^2 : :
X(34625) = X(4)-4*X(3813) = X(4)+2*X(12513) = 3*X(4)-2*X(34739) = X(8)-4*X(10916) = 2*X(10)+X(12629) = X(20)-4*X(8666) = X(145)-4*X(22837) = 4*X(1125)-X(6765) = 5*X(1698)+X(11519) = 5*X(3616)-2*X(3811) = 5*X(3616)+X(6764) = 2*X(3813)+X(12513) = 6*X(3813)-X(34739) = 3*X(11235)-X(34739) = 3*X(12513)+X(34739)

The center of the reciprocal HR-ellipse of these triangles is X(34626)

X(34625) lies on these lines: {1,2}, {3,34607}, {4,529}, {11,3421}, {20,8666}, {30,22770}, {36,17784}, {40,21627}, {55,31157}, {56,5082}, {63,30305}, {69,24203}, {104,376}, {144,21630}, {329,30384}, {346,24036}, {377,31420}, {381,7956}, {388,17532}, {390,993}, {443,3304}, {452,5258}, {496,2551}, {497,956}, {515,24392}, {516,34646}, {517,5770}, {518,5603}, {527,11372}, {535,3543}, {542,34731}, {631,3913}, {740,10186}, {752,3332}, {944,11260}, {946,5811}, {958,1058}, {962,1709}, {999,2550}, {1056,2886}, {1108,17281}, {1158,3928}, {1319,4863}, {1385,3189}, {1387,3940}, {1478,31159}, {1479,5288}, {1482,5789}, {1617,16371}, {1788,10914}, {2136,6684}, {2256,17330}, {2802,11219}, {2975,4294}, {3035,8168}, {3058,11111}, {3090,12607}, {3091,24387}, {3158,10165}, {3295,30478}, {3303,6857}, {3419,3476}, {3434,4293}, {3436,10591}, {3485,3555}, {3523,8715}, {3524,4421}, {3545,3829}, {3576,5853}, {3654,31786}, {3656,5887}, {3680,6926}, {3825,8165}, {3880,5657}, {3881,11036}, {3892,11038}, {3893,24914}, {3902,17740}, {3926,17144}, {4308,17647}, {4309,17576}, {4323,12559}, {4361,17044}, {4461,4717}, {4479,32836}, {4653,16713}, {4737,28808}, {4742,17776}, {5045,28629}, {5119,5744}, {5173,24473}, {5180,20078}, {5223,5825}, {5261,25639}, {5265,25440}, {5281,25439}, {5286,17448}, {5298,34720}, {5434,31140}, {5563,6904}, {5587,24386}, {5686,10176}, {5687,7288}, {5734,6837}, {5745,31393}, {5748,23708}, {5791,31792}, {5815,21616}, {5818,32049}, {5855,34743}, {5881,6848}, {5882,6908}, {5886,25568}, {6824,24680}, {6833,33895}, {6846,11523}, {6847,7982}, {6856,15888}, {6865,34687}, {6871,31410}, {6923,34698}, {6983,32537}, {6989,15178}, {7373,31419}, {7742,13587}, {8703,34707}, {8732,13462}, {9614,12527}, {9708,26105}, {9710,17582}, {9785,12514}, {10385,16370}, {10589,17757}, {10590,11680}, {10595,12635}, {10680,28452}, {10785,10912}, {10953,11238}, {11001,34620}, {11037,12609}, {11237,34749}, {11373,18236}, {11522,30326}, {12114,34742}, {12248,13271}, {12575,31424}, {12777,16417}, {15170,16418}, {15682,34706}, {15692,15931}, {16202,28465}, {17558,31458}, {17642,31165}, {19924,34679}, {20691,31400}, {24524,32828}

X(34625) = midpoint of X(i) and X(j) for these {i,j}: {962, 28610}, {6762, 28609}, {11235, 12513}
X(34625) = reflection of X(i) in X(j) for these (i,j): (4, 11235), (376, 11194), (3158, 10165), (5587, 24386), (11001, 34620), (11235, 3813), (11236, 3829), (15682, 34706), (25568, 5886), (28609, 946), (34607, 3), (34619, 2), (34707, 8703), (34711, 3654)
X(34625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 25755, 12647), (1, 25864, 29880), (1, 30884, 6738), (2, 145, 11239), (2, 11239, 3085), (2, 26814, 19875), (2, 29631, 30124), (8, 5704, 10), (8, 10529, 3086), (10, 24551, 19875), (10, 29581, 29670), (42, 5231, 19854), (42, 27968, 24541), (43, 24902, 17032), (43, 26774, 20037), (499, 3632, 7080), (499, 20107, 17795), (499, 24641, 26111), (551, 26610, 29644), (551, 29603, 21674), (551, 29662, 29819), (612, 25935, 30106), (614, 30915, 30145), (869, 3085, 17016), (869, 16833, 11679), (869, 29616, 26757), (936, 24641, 17016), (936, 27700, 29611), (936, 29595, 19856), (975, 16828, 29598), (975, 29609, 31191), (976, 31520, 29659), (997, 27148, 30159), (1103, 15523, 29611), (1103, 29819, 30121)


X(34626) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ABC-X3 REFLECTIONS

Barycentrics    7*a^4-(5*b^2+4*b*c+5*c^2)*a^2+2*(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34626) = 7*X(3)-4*X(24387) = 2*X(20)+X(3913) = 3*X(376)-X(34625) = 4*X(550)-X(12513) = X(1657)+2*X(8715) = 5*X(3522)-2*X(3813) = 3*X(3534)-X(34740) = 3*X(4421)-X(34739) = 3*X(11194)-2*X(34625) = 7*X(11235)-8*X(24387) = 3*X(11236)-2*X(34739) = X(22560)-4*X(24466) = X(34620)+2*X(34707) = 3*X(34620)-2*X(34740) = 3*X(34707)+X(34740)

The center of the reciprocal HR-ellipse of these triangles is X(34625)

X(34626) lies on these lines: {2,3847}, {3,11235}, {20,529}, {30,4421}, {35,17532}, {55,17579}, {56,20066}, {100,31141}, {104,376}, {381,24042}, {474,4330}, {515,34717}, {516,34647}, {517,34743}, {519,3534}, {527,34638}, {535,15681}, {542,34732}, {545,24683}, {550,12513}, {551,5880}, {958,15338}, {1125,19706}, {1158,28204}, {1376,4302}, {1657,8715}, {3058,22768}, {3189,28610}, {3241,3474}, {3434,31157}, {3522,3813}, {3524,3829}, {3529,12607}, {3582,19705}, {3654,34700}, {3679,4640}, {3880,34716}, {4188,9670}, {4294,25524}, {4297,10912}, {4316,34690}, {4324,5687}, {4428,11112}, {4857,19537}, {4995,10953}, {5010,31159}, {5560,19875}, {5853,34646}, {5884,34710}, {6154,8168}, {6925,13272}, {6938,34697}, {7354,11239}, {8666,15696}, {9671,17566}, {9689,19030}, {9710,17576}, {9778,34744}, {10860,34628}, {11001,34619}, {11238,13587}, {11496,28452}, {13996,31145}, {15326,20075}, {15678,18253}, {17549,31140}, {19924,34680}, {25439,34637}

X(34626) = midpoint of X(i) and X(j) for these {i,j}: {20, 34607}, {3189, 28610}, {3534, 34707}, {11001, 34619}
X(34626) = reflection of X(i) in X(j) for these (i,j): (3913, 34607), (11194, 376), (11235, 3), (11236, 4421), (34620, 3534), (34700, 3654), (34706, 2)


X(34627) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO OUTER-GARCIA

Barycentrics    7*a^4-6*(b+c)*a^3-2*(b^2-6*b*c+c^2)*a^2+6*(b-c)*(b^2-c^2)*a-5*(b^2-c^2)^2 : :
X(34627) = 2*X(1)-3*X(3545) = 5*X(2)-4*X(1385) = 7*X(4)-4*X(4301) = X(4)+2*X(5881) = 5*X(4)-2*X(7982) = 4*X(8)-X(6361) = 5*X(8)-2*X(12702) = X(8)+2*X(18525) = 4*X(10)-3*X(3524) = 3*X(40)-2*X(34638) = X(145)-3*X(3839) = X(145)-4*X(18480) = 6*X(165)-5*X(376) = 3*X(165)-5*X(3679) = 4*X(165)-5*X(5657)

The center of the reciprocal HR-ellipse of these triangles is X(34628)

X(34627) lies on these lines: {1,3545}, {2,355}, {4,519}, {8,30}, {10,3524}, {20,3654}, {40,4669}, {55,28461}, {80,3476}, {100,18519}, {104,16371}, {114,9884}, {119,10031}, {145,3656}, {149,18516}, {150,17079}, {165,376}, {381,952}, {517,3543}, {518,11180}, {528,16112}, {529,34700}, {535,34744}, {542,34715}, {546,5734}, {547,10246}, {549,5731}, {551,5071}, {553,9613}, {631,19875}, {958,21161}, {962,3830}, {1000,3586}, {1056,5727}, {1478,9897}, {1482,3845}, {1483,5066}, {1512,34716}, {1698,15709}, {1699,16191}, {2096,12247}, {2975,18518}, {3090,5882}, {3146,28198}, {3244,18492}, {3486,10056}, {3487,10950}, {3488,3748}, {3529,11362}, {3533,30389}, {3534,5690}, {3544,9624}, {3576,3828}, {3579,4678}, {3616,5055}, {3617,10304}, {3621,12699}, {3623,9955}, {3632,31673}, {3633,18483}, {3829,6941}, {3832,24680}, {3855,13464}, {3871,18761}, {3872,18528}, {3885,31937}, {3895,18540}, {3913,21669}, {4297,4745}, {4305,4995}, {4421,6906}, {4668,31730}, {4677,5691}, {4860,5434}, {4870,10590}, {5054,9780}, {5056,15178}, {5067,19883}, {5258,6876}, {5435,11545}, {5550,15699}, {5658,31140}, {5844,9812}, {5901,19709}, {6684,15698}, {6739,27721}, {6797,18419}, {6845,12607}, {6905,11194}, {6990,15888}, {7319,9669}, {7983,9880}, {7987,15719}, {7989,13607}, {7991,33703}, {8148,20053}, {9588,21735}, {9778,15681}, {9803,34698}, {9881,13172}, {10164,15715}, {10165,19876}, {10172,30392}, {10248,12101}, {10267,16858}, {10283,11737}, {10385,10572}, {10595,19925}, {10944,11238}, {11231,15721}, {11278,20014}, {11491,16370}, {11499,13587}, {11539,19877}, {12243,13178}, {12248,15863}, {13624,15708}, {14269,20050}, {15623,19251}, {15640,20070}, {15679,16116}, {15683,28160}, {15684,28174}, {15692,26446}, {16226,23841}, {17549,22758}, {17561,24987}, {18493,20057}, {18517,20060}, {19924,34673}, {20052,33697}, {26006,30844}, {28452,34605}

X(34627) = midpoint of X(i) and X(j) for these {i,j}: {3543, 31145}, {3830, 12645}, {4677, 5691}, {12245, 15682}, {15640, 20070}
X(34627) = reflection of X(i) in X(j) for these (i,j): (2, 355), (20, 3654), (40, 4669), (145, 3656), (376, 3679), (944, 2), (962, 3830), (1482, 3845), (1483, 5066), (3241, 381), (3534, 5690), (3656, 18480), (4297, 4745), (5731, 5790), (7967, 5587), (7983, 9880), (9884, 114), (10031, 119), (11001, 40), (12243, 13178), (12245, 4677), (13172, 9881), (15682, 5691), (16116, 15679), (31162, 34648), (34605, 28452), (34631, 31162), (34632, 34718)
X(34627) = anticomplement of X(3655)
X(34627) = pole of the line {4926, 26275} wrt orthoptic circle of Steiner inellipse
X(34627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 34631, 31162), (8, 34632, 34718), (8, 34668, 34730), (145, 3839, 3656), (355, 944, 5818), (376, 3679, 5657), (381, 3241, 5603), (551, 5587, 5071), (3576, 3828, 15702), (3653, 9956, 2), (3656, 18480, 3839), (5071, 7967, 551), (9779, 10247, 5603), (30389, 31399, 3533), (31162, 34648, 4)


X(34628) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: OUTER-GARCIA TO ABC-X3 REFLECTIONS

Barycentrics    11*a^4-3*(b+c)*a^3-(7*b^2-6*b*c+7*c^2)*a^2+3*(b-c)*(b^2-c^2)*a-4*(b^2-c^2)^2 : :
X(34628) = 3*X(1)-4*X(3655) = 5*X(1)-4*X(3656) = 7*X(1)-4*X(12699) = X(1)-4*X(18481) = 4*X(2)-5*X(7987) = 8*X(2)-7*X(7989) = 5*X(2)-4*X(19925) = 4*X(3)-3*X(19875) = 2*X(4)-3*X(25055) = 4*X(4)-5*X(30308) = 4*X(4)-7*X(30389) = 2*X(10)-3*X(10304) = 6*X(25055)-5*X(30308) = 6*X(25055)-7*X(30389) = 5*X(30308)-7*X(30389)

The center of the reciprocal HR-ellipse of these triangles is X(34627)

X(34628) lies on these lines: {1,30}, {2,4297}, {3,19875}, {4,25055}, {10,10304}, {20,519}, {40,3534}, {98,9875}, {165,376}, {355,8703}, {381,3576}, {390,9814}, {484,7171}, {516,3241}, {517,15681}, {528,2951}, {529,34701}, {535,5538}, {542,34714}, {549,5587}, {550,3654}, {551,1699}, {553,3486}, {631,30315}, {944,11001}, {946,15682}, {952,15686}, {971,31165}, {1125,3839}, {1155,30286}, {1385,3830}, {1420,11238}, {1482,15685}, {1657,7982}, {1698,3524}, {1750,11113}, {2093,4316}, {2094,24645}, {2784,8591}, {3091,19883}, {3146,11522}, {3339,4299}, {3361,10572}, {3488,30350}, {3522,9588}, {3529,5882}, {3545,3624}, {3579,4668}, {3584,5726}, {3586,10072}, {3600,30343}, {3601,11237}, {3627,9624}, {3632,31730}, {3633,6361}, {3634,15708}, {3653,3845}, {3828,15692}, {3851,31666}, {3897,15679}, {3928,10085}, {3929,10864}, {4293,10980}, {4294,30337}, {4298,15933}, {4301,5059}, {4302,9819}, {4305,5290}, {4325,6869}, {4512,15677}, {4669,12512}, {4870,12943}, {4995,9578}, {5054,18480}, {5055,13624}, {5071,10165}, {5073,15178}, {5234,17647}, {5250,15678}, {5252,31508}, {5270,6851}, {5298,9581}, {5531,6282}, {5603,28172}, {5690,15690}, {5727,15326}, {5732,17579}, {5790,14093}, {5818,15698}, {5886,15687}, {5901,33699}, {6684,19708}, {6987,30326}, {7580,11194}, {7688,18519}, {7753,9592}, {7967,28150}, {7994,34618}, {8273,16857}, {8666,33557}, {8726,28452}, {9590,15078}, {9613,10056}, {9615,13846}, {9619,14537}, {9626,18324}, {9778,28236}, {9780,15705}, {9902,33706}, {9956,15693}, {10031,13253}, {10106,10385}, {10175,15702}, {10246,15684}, {10247,28154}, {10299,31399}, {10442,17378}, {10711,15015}, {10723,12258}, {10860,34626}, {10902,28444}, {11231,15700}, {11362,17538}, {12100,31423}, {12117,13174}, {12407,20126}, {13464,33703}, {14269,33697}, {15622,19254}, {15688,18525}, {15691,28224}, {15694,17502}, {15695,31663}, {15709,19872}, {15931,16418}, {16191,28178}, {16200,28146}, {17504,18357}, {17613,34717}, {17800,24680}, {19924,34674}, {26446,34200}, {28458,30503}, {31425,33923}

X(34628) = midpoint of X(i) and X(j) for these {i,j}: {944, 11001}, {1482, 15685}, {3241, 15683}
X(34628) = reflection of X(i) in X(j) for these (i,j): (2, 4297), (40, 3534), (355, 8703), (1699, 5731), (3543, 551), (3654, 550), (3679, 376), (3830, 1385), (3928, 34620), (4669, 12512), (4677, 40), (5690, 15690), (5691, 2), (5881, 3654), (9875, 98), (9902, 33706), (10723, 12258), (12407, 20126), (13174, 12117), (13253, 10031), (15682, 946), (31162, 3655), (33699, 5901), (34632, 34638)
X(34628) = anticomplement of X(34648)
X(34628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 25055, 30308), (79, 15174, 10910), (1081, 3649, 11544), (3655, 31162, 1), (4297, 5691, 7987), (4654, 32047, 18990), (5434, 9579, 13408), (5691, 7987, 7989), (7354, 13995, 11544), (9611, 31162, 79), (12696, 16155, 18506), (13408, 15171, 1), (22791, 34657, 31162), (30308, 30389, 25055), (34634, 34712, 1)


X(34629) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO MANDART-INCIRCLE

Barycentrics    2*a^7-2*(b+c)*a^6-3*(b^2-b*c+c^2)*a^5+(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-8*b^2*c^2*a^3-4*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(34629) = 3*X(4)-2*X(34746) = 3*X(3545)-2*X(34612) = 3*X(5603)-2*X(11112) = X(6361)-4*X(15171) = 5*X(10595)-2*X(11826)

The center of the reciprocal HR-ellipse of these triangles is X(34630)

X(34629) lies on these lines: {2,10531}, {4,528}, {30,944}, {56,376}, {104,11240}, {381,5687}, {515,34719}, {516,34649}, {517,3060}, {519,5693}, {529,34631}, {535,7982}, {542,34735}, {938,6361}, {950,5903}, {2077,10199}, {2829,34749}, {3149,34707}, {3543,20060}, {3545,10893}, {3651,4309}, {3656,17579}, {4313,15170}, {5071,31418}, {5537,6963}, {5603,11112}, {6174,7681}, {6903,9670}, {6938,34610}, {10306,17556}, {10385,11508}, {10525,17577}, {10595,11826}, {10609,22791}, {11496,31140}, {12245,34606}, {13199,22753}, {18491,20095}, {19924,34683}, {28459,34632}, {31162,34701}

X(34629) = reflection of X(i) in X(j) for these (i,j): (376, 3058), (12245, 34606), (17579, 3656), (34605, 1482), (34632, 28459)
X(34629) = {X(4), X(34619)}-harmonic conjugate of X(10711)


X(34630) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO ABC-X3 REFLECTIONS

Barycentrics    2*a^7-2*(b+c)*a^6-3*(b^2-6*b*c+c^2)*a^5+(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-4*(3*b^2+2*b*c+3*c^2)*b*c*a^3-4*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(34630) = 3*X(3534)-X(34745) = X(6284)-4*X(31730) = 2*X(6361)+X(7354) = 3*X(9778)-X(11114)

The center of the reciprocal HR-ellipse of these triangles is X(34629)

X(34630) lies on these lines: {20,528}, {30,40}, {56,376}, {392,516}, {515,34720}, {517,34749}, {519,12680}, {527,7957}, {529,34632}, {535,5493}, {542,34736}, {549,5259}, {1155,1210}, {2551,3543}, {3057,4292}, {3534,22770}, {3651,27197}, {5584,11111}, {6173,12651}, {6174,10310}, {6361,7354}, {6836,34706}, {6925,11236}, {7994,12678}, {9778,11114}, {9841,31146}, {10157,28146}, {12246,34689}, {12565,34701}, {12679,31142}, {12702,13996}, {19924,34684}, {20070,34605}, {26200,28198}, {28202,28452}

X(34630) = midpoint of X(20070) and X(34605)
X(34630) = reflection of X(i) in X(j) for these (i,j): (3058, 376), (34606, 40), (34697, 3654)


X(34631) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ABC-X3 REFLECTIONS TO 5th MIXTILINEAR

Barycentrics    5*a^4-12*(b+c)*a^3-4*(b^2-6*b*c+c^2)*a^2+12*(b-c)*(b^2-c^2)*a-(b^2-c^2)^2 : :
X(34631) = 4*X(1)-3*X(3524) = 5*X(2)-4*X(5690) = 7*X(2)-8*X(5901) = 4*X(2)-5*X(10595) = 5*X(4)-8*X(4301) = 7*X(4)-4*X(5881) = X(4)-4*X(7982) = 3*X(4)-4*X(31162) = 3*X(4)-2*X(34627) = 9*X(4)-8*X(34648) = 2*X(8)-3*X(3545) = 5*X(8)-8*X(9955) = X(8)-4*X(11278) = 3*X(3545)-4*X(3656) = 15*X(3545)-16*X(9955) = 3*X(3545)-8*X(11278) = 14*X(4301)-5*X(5881)

The center of the reciprocal HR-ellipse of these triangles is X(34632)

X(34631) lies on these lines: {1,3524}, {2,1482}, {4,519}, {8,3545}, {30,145}, {40,19708}, {376,517}, {381,5844}, {388,11280}, {528,34617}, {529,34629}, {542,34737}, {549,10247}, {551,5657}, {631,3654}, {944,11001}, {946,4677}, {952,3543}, {956,28461}, {962,15682}, {1000,2099}, {1056,11551}, {1058,30323}, {1385,15698}, {1483,3534}, {3090,5734}, {3244,6361}, {3295,21161}, {3525,11362}, {3528,7991}, {3529,28198}, {3544,11522}, {3576,15715}, {3579,15710}, {3616,15709}, {3617,5055}, {3621,3839}, {3622,5054}, {3623,10304}, {3653,15719}, {3679,5071}, {3845,12645}, {4421,6942}, {4669,5818}, {4678,18493}, {4745,8227}, {4870,8164}, {4930,10698}, {5067,13464}, {5697,10385}, {5846,11180}, {5854,10711}, {5882,17538}, {6950,11194}, {7962,11041}, {7983,12243}, {9589,11541}, {9884,13172}, {9957,15933}, {10031,13199}, {10056,11009}, {10246,15692}, {10283,15694}, {10299,15178}, {10679,17549}, {10680,13587}, {11111,23340}, {12000,28466}, {12248,25416}, {12699,20050}, {14269,20054}, {15681,28212}, {15683,28174}, {15684,28224}, {18357,20052}, {18480,20053}, {18525,20014}, {19924,34685}, {26006,31218}, {28208,33703}

X(34631) = midpoint of X(3543) and X(20049)
X(34631) = reflection of X(i) in X(j) for these (i,j): (2, 1482), (8, 3656), (376, 3241), (3534, 1483), (3654, 24680), (3656, 11278), (4677, 946), (5603, 11224), (5657, 16200), (11001, 944), (12243, 7983), (12245, 2), (12645, 3845), (13172, 9884), (13199, 10031), (15682, 962), (20070, 3534), (31145, 381), (34627, 31162), (34632, 3655)
X(34631) = anticomplement of X(34718)
X(34631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 3656, 3545), (376, 3241, 7967), (551, 5657, 15702), (1482, 12245, 10595), (3241, 34632, 3655), (3655, 34632, 376), (3679, 5603, 5071), (31162, 34627, 4)


X(34632) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO ABC-X3 REFLECTIONS

Barycentrics    5*a^4+6*(b+c)*a^3-4*(b^2+3*b*c+c^2)*a^2-6*(b-c)*(b^2-c^2)*a-(b^2-c^2)^2 : :
X(34632) = 2*X(1)-3*X(10304) = 5*X(2)-4*X(946) = 7*X(2)-8*X(6684) = 11*X(2)-10*X(8227) = 8*X(3)-5*X(5734) = X(8)+2*X(6361) = X(8)-4*X(12702) = 7*X(8)-4*X(18525) = 3*X(8)-2*X(34627) = 4*X(10)-3*X(3839) = X(20)-4*X(5493) = X(20)+2*X(7991) = 3*X(20)-2*X(34628) = X(145)-4*X(31730) = 3*X(165)-2*X(551) = 6*X(165)-5*X(15692) = 4*X(551)-5*X(15692)

The center of the reciprocal HR-ellipse of these triangles is X(34631)

X(34632) lies on these lines: {1,10304}, {2,40}, {3,5734}, {4,3654}, {7,5119}, {8,30}, {10,3839}, {20,519}, {35,4323}, {36,4345}, {46,9785}, {65,10385}, {125,27582}, {145,31730}, {147,9881}, {165,551}, {355,15682}, {376,517}, {381,5657}, {390,2093}, {411,4421}, {484,5435}, {497,5183}, {515,15683}, {516,3543}, {528,9803}, {529,34630}, {542,34738}, {549,5603}, {553,1697}, {938,3058}, {944,3534}, {952,15681}, {1125,15708}, {1385,19708}, {1482,8703}, {1483,15690}, {1699,3828}, {1788,11238}, {1902,7714}, {2796,9860}, {3091,9589}, {3146,11362}, {3245,18391}, {3428,17549}, {3434,5775}, {3474,5434}, {3485,4995}, {3522,7982}, {3523,4301}, {3524,3579}, {3528,24680}, {3529,28208}, {3533,31447}, {3545,9780}, {3584,5226}, {3587,9776}, {3622,15705}, {3653,10595}, {3817,19876}, {3829,6943}, {3830,5690}, {3845,5818}, {3854,31399}, {3895,9965}, {3913,33557}, {4295,10056}, {4297,15697}, {4308,5697}, {4313,5903}, {4344,4424}, {4428,5584}, {4488,4737}, {4669,5691}, {4678,31673}, {4740,29054}, {4870,5218}, {5054,5550}, {5055,19877}, {5056,9588}, {5059,5881}, {5071,9779}, {5128,14986}, {5180,5748}, {5556,9654}, {5587,28232}, {5704,12701}, {5709,11240}, {5759,11113}, {5790,15687}, {5815,17781}, {5844,15686}, {5886,15702}, {5901,15693}, {5902,8236}, {6049,30323}, {6244,16371}, {6909,11194}, {6925,28534}, {7957,34607}, {8148,15688}, {9800,34746}, {9802,12515}, {10031,24466}, {10164,15721}, {10246,34200}, {10247,14093}, {10283,14891}, {10303,11522}, {10310,13587}, {10580,15170}, {11001,12245}, {11111,31798}, {11496,16858}, {11531,12512}, {11539,18493}, {12565,34639}, {12645,15685}, {13464,15717}, {13624,15710}, {15178,21735}, {15684,28178}, {18481,20050}, {19924,34686}, {20049,28234}, {21454,31393}, {28458,34617}, {28459,34629}

X(34632) = midpoint of X(i) and X(j) for these {i,j}: {2, 20070}, {11001, 12245}, {12645, 15685}, {15683, 31145}
X(34632) = reflection of X(i) in X(j) for these (i,j): (2, 40), (4, 3654), (147, 9881), (944, 3534), (962, 2), (1482, 8703), (1483, 15690), (3241, 376), (3543, 3679), (3656, 3579), (3830, 5690), (5691, 4669), (5731, 9778), (9812, 5657), (10031, 24466), (15640, 5691), (15682, 355), (34617, 28458), (34627, 34718), (34628, 34638), (34629, 28459), (34631, 3655)
X(34632) = anticomplement of X(31162)
X(34632) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 20070, 962), (65, 10385, 15933), (165, 551, 15692), (376, 3241, 5731), (376, 34631, 3655), (484, 30305, 5435), (3241, 9778, 376), (3524, 3656, 3616), (3579, 3656, 3524), (3655, 34631, 3241), (5493, 7991, 20), (6361, 12702, 8), (26129, 31435, 5250), (34627, 34718, 8), (34628, 34638, 20), (34656, 34730, 8)


X(34633) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO ANTI-ARA

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4)*a^3-(b+c)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^4-c^4)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34633) = 3*X(3244)-2*X(34667) = 3*X(4669)-2*X(34656) = 3*X(10165)-4*X(23410) = 8*X(13163)-5*X(31666) = 3*X(34603)-X(34657)

The center of the reciprocal HR-ellipse of these triangles is X(34634)

X(34633) lies on these lines: {10,30}, {428,551}, {515,7540}, {516,34613}, {519,34603}, {527,34663}, {535,34653}, {3244,34667}, {3830,9798}, {4669,34656}, {7713,18559}, {8185,31133}, {10165,23410}, {13163,31666}, {19924,31737}

X(34633) = midpoint of X(34657) and X(34668)
X(34633) = reflection of X(551) in X(428)
X(34633) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (34603, 34668, 34657), (34642, 34648, 10)


X(34634) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO ANTI-AQUILA

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(b^4+6*b^2*c^2+c^4)*a^3-2*(b^4+c^4)*(b+c)*a^2-(b^4-c^4)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34634) = 3*X(1)-X(34657) = 3*X(3653)-2*X(10127) = 3*X(5603)-X(34613) = X(7540)-3*X(10246) = X(7553)-4*X(15178) = 3*X(7667)-X(34656) = 4*X(7734)-3*X(19875) = 4*X(9825)-7*X(30389)

The center of the reciprocal HR-ellipse of these triangles is X(34633)

X(34634) lies on these lines: {1,30}, {2,9798}, {376,8193}, {428,551}, {515,34664}, {516,34614}, {519,7667}, {527,34662}, {535,34652}, {1479,28037}, {3534,12410}, {3653,10127}, {3679,10691}, {5603,34613}, {7540,10246}, {7553,15178}, {7734,19875}, {8192,31152}, {9626,33591}, {9825,30389}

X(34634) = midpoint of X(34656) and X(34667)
X(34634) = reflection of X(i) in X(j) for these (i,j): (428, 551), (3679, 10691)
X(34634) = complement of X(34668)
X(34634) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 34628, 34712), (3655, 34643, 1), (7667, 34667, 34656)


X(34635) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO 5th ANTI-BROCARD

Barycentrics    8*a^5+11*(b+c)*a^4+3*(b+c)*(b^2+c^2)*a^2-4*(b^4+c^4)*a-(b^4-3*b^2*c^2+c^4)*(b+c) : :
X(34635) = 3*X(551)-2*X(34636) = 3*X(3244)-2*X(34685) = 3*X(4669)-2*X(34673) = 3*X(34604)-X(34674) = 3*X(34604)+X(34686)

The center of the reciprocal HR-ellipse of these triangles is X(34636)

X(34635) lies on these lines: {10,754}, {515,34682}, {516,34615}, {519,34604}, {527,34680}, {535,34670}, {551,11364}, {3244,34685}, {4669,34673}, {19924,34638}

X(34635) = midpoint of X(34674) and X(34686)
X(34635) = reflection of X(551) in X(12150)
X(34635) = {X(34604), X(34686)}-harmonic conjugate of X(34674)


X(34636) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-AQUILA

Barycentrics    a^5-2*(b+c)*a^4+3*(b^2+c^2)*a^3+(4*b^4+3*b^2*c^2+4*c^4)*a+(b^4+c^4)*(b+c) : :
X(34636) = 3*X(1)-X(34674) = 3*X(551)-X(34635) = 3*X(5603)-X(34615) = 3*X(7883)-X(34673) = 3*X(7883)+X(34685) = 3*X(10246)-X(34682) = 3*X(12150)-2*X(34635)

The center of the reciprocal HR-ellipse of these triangles is X(34635)

X(34636) lies on these lines: {1,754}, {2,9941}, {515,34681}, {516,34616}, {519,7883}, {527,34679}, {535,34669}, {551,11364}, {5603,34615}, {7818,9997}, {10246,34682}, {19924,31162}

X(34636) = midpoint of X(34673) and X(34685)
X(34636) = reflection of X(12150) in X(551)
X(34636) = complement of X(34686)
X(34636) = {X(7883), X(34685)}-harmonic conjugate of X(34673)


X(34637) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    4*a^4-2*(b^2-4*b*c+c^2)*a^2-(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34637) = X(10)-4*X(18990) = 3*X(551)-2*X(11113) = X(3244)+2*X(7354) = 3*X(3244)-2*X(34699) = 2*X(3635)+X(10483) = 3*X(4669)-2*X(34689) = 3*X(5434)-X(11113) = 3*X(7354)+X(34699) = 3*X(11112)-X(34689) = X(17579)+3*X(34605) = 3*X(34605)-X(34690)

The center of the reciprocal HR-ellipse of these triangles is X(11113)

X(34637) lies on these lines: {2,4317}, {10,529}, {30,4301}, {226,535}, {388,5267}, {515,34698}, {516,34617}, {519,3868}, {527,34696}, {952,4744}, {993,34740}, {999,34739}, {3218,3679}, {3241,17483}, {3244,7354}, {3625,34612}, {3635,10483}, {3822,31157}, {3892,28160}, {4297,33596}, {4299,11239}, {4669,11112}, {6246,7682}, {8666,9657}, {9655,11235}, {17660,24473}, {24475,28204}, {25439,34626}, {28609,30144}

X(34637) = midpoint of X(i) and X(j) for these {i,j}: {10483, 34611}, {17579, 34690}
X(34637) = reflection of X(i) in X(j) for these (i,j): (551, 5434), (3625, 34612), (4669, 11112), (34611, 3635)
X(34637) = {X(17579), X(34605)}-harmonic conjugate of X(34690)


X(34638) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO ANTI-EULER

Barycentrics    16*a^4+3*(b+c)*a^3-(11*b^2+6*b*c+11*c^2)*a^2-3*(b-c)*(b^2-c^2)*a-5*(b^2-c^2)^2 : :
X(34638) = 11*X(2)-7*X(10248) = 5*X(2)-4*X(12571) = 4*X(3)-3*X(19883) = 7*X(4)-13*X(31425) = 5*X(10)-8*X(3579) = 19*X(10)-16*X(18357) = 11*X(10)-8*X(18480) = 2*X(20)+X(5493) = 5*X(20)+X(7991) = 3*X(20)-X(34628) = 3*X(40)-X(34627) = 3*X(165)-X(3543) = 3*X(165)-2*X(3828) = 5*X(376)-3*X(3576) = 7*X(376)-3*X(5603)

The center of the reciprocal HR-ellipse of these triangles is X(31162)

X(34638) lies on these lines: {2,10248}, {3,19883}, {4,31425}, {10,30}, {20,519}, {40,4669}, {165,3543}, {355,15685}, {376,516}, {381,10164}, {515,15681}, {517,15686}, {527,34626}, {528,34646}, {529,34639}, {535,34618}, {542,34644}, {547,28182}, {549,3817}, {550,4301}, {553,4314}, {946,8703}, {962,15697}, {1125,10304}, {1385,15690}, {1482,3534}, {1657,3654}, {1699,15692}, {3058,21625}, {3146,19875}, {3241,28228}, {3244,6361}, {3522,25055}, {3523,30308}, {3524,19862}, {3624,15705}, {3634,3839}, {3653,15695}, {3656,15689}, {3671,15338}, {3679,9778}, {3830,6684}, {3845,31663}, {3947,4995}, {4298,10385}, {4304,5425}, {4333,10056}, {4342,15326}, {4745,5691}, {5054,18483}, {5882,12103}, {5886,14093}, {5918,24473}, {7688,28461}, {7988,15721}, {8169,25440}, {8227,15698}, {9955,17504}, {9956,33699}, {10165,28178}, {10171,15702}, {10175,15687}, {11230,14891}, {11231,14893}, {11362,15704}, {11495,16418}, {12100,22793}, {12511,16370}, {12699,15688}, {13464,15696}, {14537,31396}, {15682,19925}, {15684,26446}, {15691,28174}, {15708,19878}, {18481,34748}, {19710,28204}, {19924,34635}

X(34638) = midpoint of X(i) and X(j) for these {i,j}: {40, 11001}, {355, 15685}, {1657, 3654}, {3679, 15683}, {34628, 34632}
X(34638) = reflection of X(i) in X(j) for these (i,j): (2, 12512), (551, 376), (946, 8703), (1385, 15690), (3543, 3828), (3830, 6684), (3845, 31663), (4297, 3534), (4669, 40), (5691, 4745), (15682, 19925), (22793, 12100), (33699, 9956)
X(34638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 34632, 34628), (165, 3543, 3828), (9778, 15683, 3679), (12512, 12571, 16192), (18480, 34633, 34648)


X(34639) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO ANTI-MANDART-INCIRCLE

Barycentrics    8*a^4+3*(b+c)*a^3-(7*b^2+22*b*c+7*c^2)*a^2-(b+c)*(3*b^2-14*b*c+3*c^2)*a-(b^2-c^2)^2 : :
X(34639) = 5*X(10)-8*X(32157) = 3*X(551)-2*X(34640) = X(2136)+2*X(12512) = 3*X(3244)-2*X(34710) = 2*X(3913)+X(5493) = X(4301)-4*X(8715) = 3*X(4421)-X(34640) = 3*X(4669)-2*X(34700) = X(12541)-7*X(16192) = 4*X(13463)-7*X(15808) = 3*X(34607)-X(34701)

The center of the reciprocal HR-ellipse of these triangles is X(34640)

X(34639) lies on these lines: {2,12575}, {10,528}, {40,376}, {100,4342}, {515,34707}, {516,34619}, {527,3913}, {529,34638}, {535,34687}, {551,3295}, {1210,34719}, {3149,4301}, {3158,28228}, {3241,3339}, {3244,34710}, {3671,3871}, {3679,4294}, {3895,4315}, {3950,5011}, {4304,5541}, {4669,34700}, {4882,6172}, {5542,25439}, {6174,12053}, {6736,11114}, {6940,34486}, {9948,28204}, {11024,25055}, {11500,28194}, {12541,16192}, {12565,34632}, {13463,15808}, {21628,34648}

X(34639) = midpoint of X(i) and X(j) for these {i,j}: {2136, 34610}, {34701, 34711}
X(34639) = reflection of X(i) in X(j) for these (i,j): (551, 4421), (34610, 12512)
X(34639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (21628, 34746, 34648), (34607, 34711, 34701)


X(34640) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ANTI-AQUILA

Barycentrics    a^4-3*(b+c)*a^3-2*(b^2-5*b*c+c^2)*a^2+(b+c)*(3*b^2-8*b*c+3*c^2)*a+(b^2-c^2)^2 : :
X(34640) = X(1)+2*X(13463) = 3*X(1)-X(34701) = X(4)+2*X(33895) = 3*X(551)-X(34639) = 2*X(946)+X(10912) = 4*X(946)-X(32049) = X(962)+2*X(11260) = 2*X(1537)+X(11256) = 5*X(3091)-2*X(32537) = 7*X(3624)-4*X(32157) = 2*X(4301)+X(12513) = 3*X(4421)-2*X(34639) = 2*X(10912)+X(32049) = X(12699)+2*X(22837) = 6*X(13463)+X(34701)

The center of the reciprocal HR-ellipse of these triangles is X(34639)

X(34640) lies on these lines: {1,528}, {2,3057}, {4,33895}, {8,5087}, {65,11240}, {78,34720}, {355,381}, {388,3241}, {515,34706}, {516,34620}, {527,4301}, {529,31162}, {535,12699}, {551,3295}, {903,7185}, {962,11260}, {1001,4342}, {1145,23708}, {1320,5252}, {1329,3679}, {1387,17564}, {1537,11256}, {1836,34605}, {1837,10707}, {2098,5794}, {2550,4345}, {2802,5886}, {2886,7962}, {3061,17281}, {3091,32537}, {3340,31146}, {3434,5048}, {3545,7704}, {3624,32157}, {3655,34708}, {3680,11522}, {3813,6831}, {3828,31493}, {3872,24703}, {3880,5603}, {3885,11375}, {3895,15950}, {3913,6918}, {3922,10586}, {4479,20449}, {4853,31142}, {4861,11114}, {4870,11239}, {5434,7702}, {5450,11194}, {5587,5854}, {5697,26066}, {5722,21630}, {5734,6835}, {5855,11224}, {6256,28204}, {6690,9819}, {6917,24680}, {6971,24387}, {7705,26129}, {7951,12653}, {8148,10916}, {9957,28628}, {10107,14986}, {10199,11373}, {10246,34707}, {10914,25681}, {10915,18493}, {11111,30305}, {11508,16371}, {12436,19706}, {12625,16189}, {12648,17605}, {17556,30384}, {19860,34687}, {19875,25522}, {30960,31136}, {31164,34749}

X(34640) = midpoint of X(i) and X(j) for these {i,j}: {962, 34610}, {10912, 11236}, {11224, 24392}, {34700, 34710}
X(34640) = reflection of X(i) in X(j) for these (i,j): (3679, 3829), (4421, 551), (11236, 946), (32049, 11236), (34610, 11260), (34647, 3656)
X(34640) = complement of X(34711)
X(34640) = pole of the line {145, 18839} wrt Feuerbach hyperbola
X(34640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (946, 10912, 32049), (3680, 11522, 12607), (11235, 34710, 34700)


X(34641) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO AQUILA

Barycentrics    8*a-7*b-7*c : :
X(34641) = 5*X(1)-7*X(2) = X(1)-7*X(8) = 4*X(1)-7*X(10) = 13*X(1)-7*X(145) = 6*X(1)-7*X(551) = 11*X(1)-14*X(1125) = 9*X(1)-7*X(3241) = 10*X(1)-7*X(3244) = 11*X(1)+7*X(3621) = 2*X(1)+7*X(3625) = 5*X(1)-14*X(3626) = 5*X(1)+7*X(3632) = 19*X(1)-7*X(3633) = 17*X(1)-14*X(3635) = 3*X(1)-7*X(3679) = 9*X(1)-14*X(3828) = 2*X(1)-7*X(4669) = X(1)+7*X(4677) = X(1)+14*X(4701) = 4*X(8)-X(10)

The center of the reciprocal HR-ellipse of these triangles is X(3241)

X(34641) lies on these lines: {1,2}, {319,903}, {355,14269}, {376,28236}, {381,28234}, {382,28194}, {515,15681}, {516,34627}, {517,4525}, {518,4744}, {527,34717}, {535,34689}, {537,4686}, {546,4301}, {549,31662}, {550,11362}, {944,15710}, {952,34200}, {993,8168}, {2321,4370}, {2802,4134}, {3303,17545}, {3529,5493}, {3530,5882}, {3543,28228}, {3629,28538}, {3631,9041}, {3654,4297}, {3655,10164}, {3656,3851}, {3678,3893}, {3686,4072}, {3707,4908}, {3817,5844}, {3839,11531}, {3855,7982}, {3902,4125}, {3913,17571}, {3919,24473}, {3950,17330}, {3956,5919}, {3982,5252}, {4029,16590}, {4031,5434}, {4058,16671}, {4060,17281}, {4067,10914}, {4098,4545}, {4399,21255}, {4416,17487}, {4421,5267}, {4428,16866}, {4478,17382}, {4665,4796}, {4690,28309}, {4692,4793}, {4711,10176}, {4717,4737}, {4725,10022}, {5054,13607}, {5066,11278}, {5071,16200}, {5079,13464}, {5258,17574}, {5288,13587}, {5587,34631}, {5657,15715}, {5690,17504}, {5846,20583}, {6684,15707}, {8666,19537}, {8715,19535}, {12513,17573}, {15684,28232}, {15699,33179}, {15721,30392}, {16418,25439}, {17133,24441}, {17563,24391}, {17678,19820}, {26446,34748}, {28164,34632}

X(34641) = midpoint of X(i) and X(j) for these {i,j}: {2, 3632}, {8, 4677}, {3625, 4669}, {3654, 12645}, {3679, 31145}
X(34641) = reflection of X(i) in X(j) for these (i,j): (1, 4745), (2, 3626), (10, 4669), (551, 3679), (3241, 3828), (3244, 2), (3625, 4677), (4297, 3654), (4669, 8), (4677, 4701), (4745, 4746), (5919, 3956), (10176, 4711), (11278, 5066)
X(34641) = pole of the line {3057, 4746} wrt Feuerbach hyperbola
X(34641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8, 4746), (1, 3632, 20054), (2, 15808, 19883), (8, 3621, 4668), (8, 3625, 10), (8, 3632, 3626), (8, 4701, 3625), (8, 4816, 4701), (8, 20052, 1), (8, 31145, 3679), (10, 1125, 22166), (10, 3244, 15808), (145, 4691, 19862), (551, 3679, 10), (551, 4669, 3679), (975, 27097, 26676), (1961, 29720, 25377), (2664, 17018, 30117), (2999, 27020, 1961), (3008, 26111, 9797), (3241, 3679, 3828), (3241, 3828, 551), (3244, 3625, 3632), (3244, 3626, 10), (3616, 29604, 8582), (3617, 29594, 4678), (3621, 4668, 1125), (3626, 3632, 3244), (3632, 4668, 20057), (3633, 4678, 3634), (3633, 11679, 4668), (3633, 30106, 1201), (3636, 20050, 3244), (3679, 4677, 31145)


X(34642) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO ARA

Barycentrics    8*a^7+11*(b+c)*a^6+4*(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-8*(b^4+c^4)*a^3-(b+c)*(11*b^4-6*b^2*c^2+11*c^4)*a^2-4*(b^4-c^4)*(b^2-c^2)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34642) = 3*X(3244)-2*X(34729) = X(4301)-4*X(7387) = 3*X(4669)-2*X(34713) = 4*X(10154)-3*X(19883) = 9*X(30775)-10*X(31253) = 3*X(34608)-X(34712) = 3*X(34608)+X(34730)

The center of the reciprocal HR-ellipse of these triangles is X(34643)

X(34642) lies on these lines: {10,30}, {376,7713}, {515,34726}, {516,34621}, {519,34608}, {527,34724}, {535,34691}, {551,9909}, {3244,34729}, {4301,7387}, {4669,34713}, {9798,28194}, {10154,19883}, {10165,33591}, {30775,31253}

X(34642) = midpoint of X(34712) and X(34730)
X(34642) = reflection of X(551) in X(9909)
X(34642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 34633, 34648), (34608, 34730, 34712)


X(34643) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO ANTI-AQUILA

Barycentrics    a^7-2*(b+c)*a^6-4*(b^2+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*a^3+2*(b+c)*(b^4+c^4)*a^2+4*(b^4-c^4)*(b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34643) = 3*X(1)-X(34712) = 3*X(3653)-2*X(14070) = 3*X(5603)-X(34621) = 7*X(9780)-9*X(30775) = 2*X(10154)-3*X(25055) = 3*X(10246)-X(34726) = 4*X(15178)-X(31305) = 4*X(18281)-3*X(26446) = 3*X(34609)-X(34713)

The center of the reciprocal HR-ellipse of these triangles is X(34642)

X(34643) lies on these lines: {1,30}, {2,1829}, {515,34725}, {516,34622}, {519,34609}, {527,34723}, {535,34692}, {551,9909}, {3543,7718}, {3653,14070}, {5090,31133}, {5603,34621}, {9780,30775}, {10154,25055}, {10246,34726}, {11396,31152}, {12410,28194}, {15178,31305}, {18281,26446}

X(34643) = midpoint of X(34713) and X(34729)
X(34643) = reflection of X(9909) in X(551)
X(34643) = complement of X(34730)
X(34643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 34634, 3655), (79, 7100, 13408), (11809, 18506, 13408), (34609, 34729, 34713)


X(34644) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO 5th BROCARD

Barycentrics    8*a^5+8*(b+c)*a^4-3*(b+c)*(b^2+c^2)*a^2-4*(b^4+c^4)*a-(b+c)*(4*b^4+3*b^2*c^2+4*c^4) : :
X(34644) = 3*X(551)-2*X(34645) = 3*X(3244)-2*X(34737) = 3*X(4669)-2*X(34715) = 4*X(7810)-3*X(19883) = 3*X(7811)-X(34645) = 3*X(9939)-X(34714) = 3*X(9939)+X(34738)

The center of the reciprocal HR-ellipse of these triangles is X(34645)

X(34644) lies on these lines: {10,754}, {515,34734}, {516,34623}, {519,9939}, {527,34732}, {535,34693}, {542,34638}, {551,7811}, {3244,34737}, {4669,34715}, {7810,19883}, {9902,14976}

X(34644) = midpoint of X(i) and X(j) for these {i,j}: {9902, 14976}, {34714, 34738}
X(34644) = reflection of X(551) in X(7811)
X(34644) = {X(9939), X(34738)}-harmonic conjugate of X(34714)


X(34645) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO ANTI-AQUILA

Barycentrics    2*a^5+2*(b+c)*a^4+3*(b^2+c^2)*a^3-(b^4-3*b^2*c^2+c^4)*a-(b+c)*(b^4+c^4) : :
X(34645) = 3*X(1)-X(34714) = 3*X(551)-X(34644) = 3*X(5603)-X(34623) = 2*X(7810)-3*X(25055) = 3*X(7811)-2*X(34644) = 3*X(7812)-X(34715) = 3*X(7812)+X(34737) = 3*X(10246)-X(34734)

The center of the reciprocal HR-ellipse of these triangles is X(34644)

X(34645) lies on these lines: {1,754}, {2,12194}, {515,34733}, {516,34624}, {519,7812}, {527,34731}, {535,34694}, {542,31162}, {551,7811}, {730,11361}, {1572,3679}, {5603,34623}, {7810,25055}, {7818,10800}, {10246,34734}, {12156,12195}

X(34645) = midpoint of X(34715) and X(34737)
X(34645) = reflection of X(i) in X(j) for these (i,j): (3679, 7753), (7811, 551)
X(34645) = complement of X(34738)
X(34645) = {X(7812), X(34737)}-harmonic conjugate of X(34715)


X(34646) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    8*a^4+3*(b+c)*a^3-(7*b^2-10*b*c+7*c^2)*a^2-(b+c)*(3*b^2+2*b*c+3*c^2)*a-(b^2-c^2)^2 : :
X(34646) = 3*X(551)-2*X(34647) = 3*X(3244)-2*X(34743) = 3*X(3928)+X(34716) = 3*X(3928)-X(34744) = X(4301)-4*X(8666) = 3*X(4669)-2*X(34717) = X(5493)+2*X(12513) = X(6762)+2*X(12512) = 3*X(11194)-X(34647)

The center of the reciprocal HR-ellipse of these triangles is X(34647)

X(34646) lies on these lines: {1,28610}, {2,3361}, {10,529}, {36,21060}, {40,376}, {63,4315}, {144,13462}, {226,31157}, {515,34740}, {516,34625}, {527,551}, {528,34638}, {535,34648}, {1012,4301}, {1125,28609}, {1155,34689}, {2975,3671}, {3241,9819}, {3244,34743}, {3576,5850}, {3600,18249}, {3679,4293}, {3874,9957}, {3911,31141}, {4311,6763}, {4321,6172}, {4652,11239}, {4669,17647}, {4847,17579}, {5493,12513}, {5853,34626}, {8158,12114}, {10164,21164}, {11019,11113}, {12577,31424}, {12675,31806}, {17625,31165}, {24392,28158}, {24477,28164}, {31397,34690}

X(34646) = midpoint of X(i) and X(j) for these {i,j}: {1, 28610}, {3928, 34610}, {6762, 34607}, {34716, 34744}
X(34646) = reflection of X(i) in X(j) for these (i,j): (551, 11194), (28609, 1125), (34607, 12512)
X(34646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (551, 24405, 33682), (3928, 34716, 34744), (34610, 34744, 34716)


X(34647) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-AQUILA

Barycentrics    a^4-3*(b+c)*a^3-2*(b^2-b*c+c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a+(b^2-c^2)^2 : :
X(34647) = 3*X(1)-X(34716) = 3*X(551)-X(34646) = 2*X(946)+X(12635) = X(1482)+2*X(21077) = 2*X(1482)+X(32049) = 5*X(3616)-X(28610) = 10*X(3616)-X(28646) = 7*X(3622)+2*X(28645) = X(3811)+2*X(22791) = 2*X(3813)-5*X(11522) = 2*X(3813)+X(11523) = X(3928)-3*X(25055) = 3*X(5603)-X(34625) = X(7982)+2*X(12607) = 3*X(11194)-2*X(34646) = 5*X(11522)+X(11523) = 3*X(28609)+X(34716)

The center of the reciprocal HR-ellipse of these triangles is X(34646)

X(34647) lies on these lines: {1,529}, {2,65}, {8,17605}, {12,11682}, {30,6261}, {63,15950}, {78,34612}, {191,3338}, {226,5289}, {355,381}, {376,28534}, {496,12559}, {497,3241}, {499,4018}, {515,34739}, {516,34626}, {518,5603}, {527,551}, {528,1537}, {535,3655}, {758,5886}, {908,2099}, {962,34607}, {997,5880}, {1125,5708}, {1319,5905}, {1329,3340}, {1532,7982}, {1836,4511}, {1848,5155}, {2093,3035}, {2551,4323}, {2646,11415}, {2886,3679}, {3057,11239}, {3339,6691}, {3419,4867}, {3436,11011}, {3475,10179}, {3576,17768}, {3616,3683}, {3622,28645}, {3649,19861}, {3653,28443}, {3654,25413}, {3671,25524}, {3811,22791}, {3813,8226}, {3816,11529}, {3868,11376}, {3870,34699}, {3872,34689}, {3874,11373}, {3877,17718}, {3878,10197}, {3880,25568}, {3894,16173}, {3897,20323}, {3913,4301}, {3962,10527}, {4234,5327}, {4421,6796}, {4715,24316}, {4999,12526}, {5087,18391}, {5123,5748}, {5252,31053}, {5434,31164}, {5570,10072}, {5587,5855}, {5722,11813}, {5730,5794}, {5734,6957}, {5854,11224}, {6765,13463}, {6913,12513}, {6929,24680}, {6968,32537}, {7489,8666}, {8148,10915}, {9897,25416}, {10200,31794}, {10246,34740}, {10595,11260}, {10916,18493}, {12699,22836}, {12701,34611}, {15829,25466}, {16370,22766}, {17313,18589}, {17781,18967}, {18421,30827}, {26729,28370}

X(34647) = midpoint of X(i) and X(j) for these {i,j}: {1, 28609}, {381, 4930}, {962, 34607}, {11235, 12635}, {34717, 34743}
X(34647) = reflection of X(i) in X(j) for these (i,j): (11194, 551), (11235, 946), (28646, 28610), (34640, 3656)
X(34647) = complement of X(34744)
X(34647) = pole of the line {3486, 11015} wrt Feuerbach hyperbola
X(34647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (960, 3485, 28628), (1482, 21077, 32049), (3869, 11375, 26066), (4867, 18393, 3419), (4870, 31165, 2), (5730, 12047, 5794), (11236, 34743, 34717), (11522, 11523, 3813)


X(34648) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO EULER

Barycentrics    8*a^4-3*(b+c)*a^3-(b^2-6*b*c+c^2)*a^2+3*(b-c)*(b^2-c^2)*a-7*(b^2-c^2)^2 : :
X(34648) = X(1)-3*X(3839) = 7*X(2)-5*X(7987) = 5*X(2)-7*X(7989) = 4*X(4)-X(4301) = 5*X(4)+X(5881) = 7*X(4)-X(7982) = 4*X(5)-3*X(19883) = 7*X(10)-4*X(3579) = 5*X(10)-8*X(18357) = X(10)-4*X(18480) = X(20)-3*X(19875) = 3*X(165)-X(15683) = 3*X(355)-X(34718) = X(376)-3*X(5587) = 2*X(376)-3*X(10164) = 2*X(4745)+X(15682)

The center of the reciprocal HR-ellipse of these triangles is X(3655)

X(34648) lies on these lines: {1,3839}, {2,4297}, {4,519}, {5,19883}, {10,30}, {20,19875}, {40,4745}, {147,9875}, {165,15683}, {355,3830}, {376,3828}, {381,515}, {382,3654}, {516,3543}, {517,4525}, {527,34739}, {535,34646}, {546,5882}, {547,10165}, {549,10175}, {550,31399}, {553,1837}, {671,2784}, {944,12571}, {946,1483}, {950,11237}, {952,14893}, {962,4677}, {997,18529}, {1125,3545}, {1385,5066}, {1478,5542}, {1698,10304}, {1699,3241}, {2771,4744}, {2796,9864}, {2801,24473}, {3091,25055}, {3244,3656}, {3524,3634}, {3534,6684}, {3576,5071}, {3582,4311}, {3583,4342}, {3584,4304}, {3585,3671}, {3586,8232}, {3625,12699}, {3627,11362}, {3653,19709}, {3668,20289}, {3832,30308}, {3843,13464}, {3858,15178}, {3860,5901}, {3861,24680}, {3874,16616}, {3947,10572}, {3986,32431}, {4314,10056}, {4315,10072}, {4654,5229}, {4691,6361}, {5055,18481}, {5059,9588}, {5068,30389}, {5080,21060}, {5086,17781}, {5267,18761}, {5290,15933}, {5434,11019}, {5657,28158}, {5690,28202}, {5731,10171}, {5790,15684}, {5818,11001}, {5927,31165}, {6245,28452}, {6246,7682}, {7991,17578}, {8273,19536}, {8703,9956}, {9578,10385}, {9812,31145}, {9880,11599}, {9881,10723}, {9955,23046}, {10106,11238}, {10124,17502}, {10172,15694}, {10248,11531}, {10443,17330}, {11114,12617}, {11180,34379}, {11194,19541}, {11224,20049}, {11230,11737}, {11231,28190}, {12101,22793}, {13624,15699}, {15681,26446}, {15686,28168}, {15692,19876}, {15697,16192}, {15705,19877}, {15709,31253}, {15717,30315}, {15931,16861}, {18391,30424}, {19708,31423}, {19710,31663}, {21628,34639}, {21849,31732}, {26006,31048}

X(34648) = midpoint of X(i) and X(j) for these {i,j}: {2, 5691}, {40, 15682}, {147, 9875}, {355, 3830}, {382, 3654}, {962, 4677}, {3543, 3679}, {3656, 18525}, {5690, 33699}, {9881, 10723}, {31162, 34627}
X(34648) = reflection of X(i) in X(j) for these (i,j): (2, 19925), (40, 4745), (376, 3828), (551, 381), (946, 3845), (1385, 5066), (3244, 3656), (3534, 6684), (3656, 18483), (4297, 2), (4669, 355), (5493, 3654), (5731, 10171), (5901, 3860), (8703, 9956), (10164, 5587), (11001, 12512), (11599, 9880), (19710, 31663), (22793, 12101), (31732, 21849)
X(34648) = complement of X(34628)
X(34648) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 34627, 31162), (10, 34633, 34642), (376, 3828, 10164), (376, 5587, 3828), (381, 551, 3817), (3656, 14269, 18483), (5691, 19925, 4297), (14269, 18525, 3656), (18357, 31730, 10), (18357, 33697, 31730), (18480, 31673, 10), (18480, 33697, 18357), (18483, 18525, 3244), (21628, 34746, 34639), (31673, 31730, 33697)


X(34649) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-AQUILA TO MANDART-INCIRCLE

Barycentrics    4*a^4-2*(b^2+4*b*c+c^2)*a^2+(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34649) = X(10)-4*X(15171) = 3*X(551)-2*X(11112) = 3*X(3058)-X(11112) = X(3244)+2*X(6284) = 3*X(3244)-2*X(34749) = 3*X(4669)-2*X(34720) = 3*X(6284)+X(34749) = 3*X(11113)-X(34720) = X(11114)+3*X(34611) = 3*X(34611)-X(34719)

The center of the reciprocal HR-ellipse of these triangles is X(11112)

X(34649) lies on these lines: {2,4309}, {10,528}, {30,4301}, {35,10707}, {497,10199}, {515,34745}, {516,34629}, {519,3869}, {527,34741}, {535,3244}, {551,2646}, {3295,34706}, {3625,34606}, {3635,34605}, {3746,17577}, {3825,6174}, {3871,31160}, {3874,28534}, {3892,28146}, {4294,5267}, {4302,11240}, {4304,21630}, {4342,33337}, {4669,11113}, {4744,28174}, {5248,31140}, {8715,9670}, {9668,11236}, {10197,10385}, {10386,25639}, {21740,31162}, {25440,34707}, {26201,28202}, {30144,34701}

X(34649) = midpoint of X(11114) and X(34719)
X(34649) = reflection of X(i) in X(j) for these (i,j): (551, 3058), (3625, 34606), (4669, 11113), (34605, 3635)
X(34649) = pole of the line {3874, 17606} wrt Feuerbach hyperbola
X(34649) = {X(11114), X(34611)}-harmonic conjugate of X(34719)


X(34650) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO 5th ANTI-BROCARD

Barycentrics    2*a^10+3*(b^2+c^2)*a^8+(b^4-16*b^2*c^2+c^4)*a^6-(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4-(b^4+c^4)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4) : :
X(34650) = 3*X(428)-2*X(34651) = 3*X(12150)-X(34651)

The center of the reciprocal HR-ellipse of these triangles is X(34651)

X(34650) lies on these lines: {30,34604}, {428,11380}, {754,7667}, {19924,34614}

X(34650) = reflection of X(428) in X(12150)
X(34650) = {X(34676), X(34682)}-harmonic conjugate of X(34604)


X(34651) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-ARA

Barycentrics    2*a^10+3*(b^2+c^2)*a^8+(b^2+c^2)^2*a^6-(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+c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4) : :
X(34651) = 3*X(428)-X(34650)

The center of the reciprocal HR-ellipse of these triangles is X(34650)

X(34651) lies on these lines: {30,7883}, {428,11380}, {754,34603}, {5562,19924}

X(34651) = reflection of X(12150) in X(428)
X(34651) = {X(34675), X(34681)}-harmonic conjugate of X(7883)


X(34652) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34653)

X(34652) lies on these lines: {30,944}, {34,428}, {528,34667}, {529,7667}, {535,34634}, {5262,18990}

X(34652) = reflection of X(428) in X(5434)
X(34652) = {X(34692), X(34698)}-harmonic conjugate of X(34605)


X(34653) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ANTI-ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34652)

X(34653) lies on these lines: {30,40}, {34,428}, {528,34668}, {529,34603}, {535,34633}

X(34653) = reflection of X(5434) in X(428)
X(34653) = {X(34691), X(34697)}-harmonic conjugate of X(34606)


X(34654) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO ANTI-MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34655)

X(34654) lies on these lines: {30,10306}, {428,4421}, {519,34662}, {528,7667}, {529,34614}

X(34654) = reflection of X(428) in X(4421)
X(34654) = {X(34703), X(34707)}-harmonic conjugate of X(34607)


X(34655) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ANTI-ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34654)

X(34655) lies on these lines: {2,9673}, {30,10525}, {428,4421}, {519,34663}, {528,34603}, {529,34613}

X(34655) = reflection of X(4421) in X(428)
X(34655) = {X(34702), X(34706)}-harmonic conjugate of X(11235)


X(34656) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO AQUILA

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(-4*b^2*c^2+(b^2-c^2)^2)*a^3-2*(b+c)*(b^4+6*b^2*c^2+c^4)*a^2-(b^2-c^2)*(b^4-c^4)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34656) = 3*X(8)-X(34668) = 3*X(428)-2*X(34657) = X(1885)+2*X(7991) = X(3575)-4*X(11362) = 3*X(4669)-X(34633) = 3*X(7667)-2*X(34634) = 3*X(7667)-X(34667)

The center of the reciprocal HR-ellipse of these triangles is X(34657)

X(34656) lies on these lines: {2,12410}, {8,30}, {376,8192}, {428,3679}, {515,34614}, {517,34664}, {519,7667}, {1829,28194}, {1885,7991}, {3241,10691}, {3575,11362}, {4669,34633}, {5262,15170}

X(34656) = reflection of X(i) in X(j) for these (i,j): (428, 3679), (3241, 10691), (34667, 34634)
X(34656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 34632, 34730), (7667, 34667, 34634), (34713, 34718, 8)


X(34657) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO ANTI-ARA

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(b^4-3*b^2*c^2+c^4)*a^3-2*(b^4+c^4)*(b+c)*a^2-(b^2-c^2)*(b^4-c^4)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34657) = 3*X(1)-2*X(34634) = 3*X(1699)-2*X(34664) = 2*X(3575)+X(9589) = 4*X(6756)-X(7991) = 2*X(7553)+X(7982) = 2*X(7667)-3*X(25055) = 8*X(10128)-7*X(19876) = 4*X(23410)-3*X(26446) = 3*X(34603)-2*X(34633)

The center of the reciprocal HR-ellipse of these triangles is X(34656)

X(34657) lies on these lines: {1,30}, {381,8193}, {428,3679}, {515,34613}, {517,7540}, {519,34603}, {1699,34664}, {3575,9589}, {3654,13490}, {3830,12410}, {6756,7991}, {7553,7982}, {7576,28194}, {7667,25055}, {7718,18559}, {9625,34351}, {10128,19876}, {11179,16472}, {11365,31152}, {16473,20423}, {23410,26446}

X(34657) = reflection of X(i) in X(j) for these (i,j): (3654, 13490), (3679, 428), (34668, 34633)
X(34657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9580, 10910, 16154), (10543, 11544, 16152), (16152, 33668, 33154), (31162, 34712, 1), (34603, 34668, 34633)


X(34658) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO ARA

Barycentrics    2*a^12-(b^2+c^2)*a^10-(5*b^4+18*b^2*c^2+5*c^4)*a^8+2*(b^4-c^4)*(b^2-c^2)*a^6+4*(b^8+c^8+2*(2*b^4-b^2*c^2+2*c^4)*b^2*c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :

The center of the reciprocal HR-ellipse of these triangles is X(34659)

X(34658) lies on the line {2,3}

X(34658) = reflection of X(428) in X(9909)
X(34658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 384, 27201), (2, 406, 33190), (2, 409, 11321), (2, 443, 33745), (2, 458, 405), (2, 1011, 8356), (2, 1583, 16861), (2, 3540, 27612), (2, 3560, 21545), (2, 3859, 28454), (2, 4187, 15671), (2, 4193, 1982), (2, 4197, 19337), (2, 4208, 27616), (2, 4214, 6990), (26, 406, 1589), (26, 1344, 28034), (26, 1817, 29889), (26, 4233, 18871), (26, 4244, 444), (26, 6657, 28034), (26, 6868, 21308), (26, 6875, 26182), (26, 6940, 29775), (26, 6960, 27847), (26, 7445, 32995), (26, 7557, 18377), (26, 7807, 31923), (26, 9825, 6957), (26, 11323, 19277)


X(34659) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO ANTI-ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34658)

X(34659) lies on the line {2,3}

X(34659) = midpoint of X(7394) and X(7473)
X(34659) = reflection of X(i) in X(j) for these (i,j): (9909, 428), (15146, 420), (17696, 28117), (33313, 3147)
X(34659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2555, 28821), (2, 6882, 14021), (2, 7525, 21475), (2, 7575, 11358), (2, 11110, 25766), (2, 15207, 4204), (2, 15246, 15709), (2, 17669, 24951), (2, 18420, 25949), (2, 19248, 19523), (2, 19698, 15670), (2, 21993, 26123), (2, 24904, 3109), (2, 25017, 14007), (2, 25018, 16407), (26, 6840, 7534), (26, 19705, 19345), (26, 24932, 16380), (26, 28044, 33268), (27, 17573, 11097), (27, 26527, 6834), (28, 11403, 34562), (29, 3078, 4199), (29, 3136, 27203), (29, 3515, 140), (29, 3524, 21498), (29, 6853, 21508), (29, 11305, 7887), (29, 14006, 7538), (29, 14011, 27534)


X(34660) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO 5th BROCARD

Barycentrics    2*(b^2+c^2)*a^8+(b^4+20*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^4+c^4)*(b^4+10*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(34660) = 3*X(428)-2*X(34661)

The center of the reciprocal HR-ellipse of these triangles is X(34661)

X(34660) lies on these lines: {30,9863}, {428,7811}, {542,34614}, {754,7667}

X(34660) = reflection of X(428) in X(7811)
X(34660) = {X(34721), X(34734)}-harmonic conjugate of X(9939)


X(34661) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO ANTI-ARA

Barycentrics    2*(b^2+c^2)*a^8+(b^4+8*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^8+c^8+4*(b^4-b^2*c^2+c^4)*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(34661) = 3*X(428)-X(34660)

The center of the reciprocal HR-ellipse of these triangles is X(34660)

X(34661) lies on these lines: {30,3095}, {428,7811}, {542,16659}, {754,34603}

X(34661) = reflection of X(7811) in X(428)
X(34661) = {X(34722), X(34733)}-harmonic conjugate of X(7812)


X(34662) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4+14*b^2*c^2+c^4)*a^6+4*(b^6+c^6+(5*b^2-8*b*c+5*c^2)*b^2*c^2)*a^4+16*(b+c)*b^3*c^3*a^3+8*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

The center of the reciprocal HR-ellipse of these triangles is X(34663)

X(34662) lies on these lines: {30,22770}, {428,11194}, {519,34654}, {527,34634}, {528,34614}, {529,7667}, {535,34664}

X(34662) = reflection of X(428) in X(11194)
X(34662) = {X(34723), X(34740)}-harmonic conjugate of X(34610)


X(34663) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ANTI-ARA

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^4+5*b^2*c^2+c^4)*a^6+2*(2*b^6+2*c^6+(b^2-4*b*c+c^2)*b^2*c^2)*a^4+4*(b+c)*b^3*c^3*a^3+8*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(34663) = 3*X(428)-X(34662)

The center of the reciprocal HR-ellipse of these triangles is X(34662)

X(34663) lies on these lines: {2,9658}, {30,4421}, {428,11194}, {519,34655}, {527,34633}, {528,34613}, {529,34603}, {535,7540}

X(34663) = reflection of X(11194) in X(428)
X(34663) = {X(34724), X(34739)}-harmonic conjugate of X(11236)


X(34664) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO EULER

Barycentrics    2*a^10-3*(b^2+c^2)*a^8-2*(b^2-c^2)^2*a^6+4*(b^2+c^2)*(b^4+c^4)*a^4-8*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(34664) = 2*X(3)+X(1885) = 3*X(1699)-X(34657) = 2*X(5447)+X(12897) = X(5562)+2*X(12241) = 2*X(5907)+X(6146) = X(10627)+2*X(15807) = X(11412)+2*X(13142) = 2*X(11591)+X(12370) = 2*X(11793)+X(13403) = X(12111)+2*X(18914) = 2*X(13292)+X(18436) = X(14516)-7*X(15056) = 5*X(15058)+X(34224) = 3*X(16261)-X(16658) = 2*X(31834)+X(32358)

As a point on the Euler line, X(34664) has Shinagawa coefficients (E+F, -3*F)
The center of the reciprocal HR-ellipse of these triangles is X(7540)

X(34664) lies on these lines: {2,3}, {343,18390}, {511,16657}, {515,34634}, {517,34656}, {524,5562}, {535,34662}, {541,25711}, {542,5907}, {569,22660}, {597,12233}, {598,13599}, {952,34667}, {973,21849}, {1181,11179}, {1352,18396}, {1498,31166}, {1503,15030}, {1514,5092}, {1531,3589}, {1568,23292}, {1699,34657}, {1992,12160}, {2777,16836}, {2883,10984}, {3003,7745}, {3564,11459}, {5063,5254}, {5181,32274}, {5447,12897}, {5486,15069}, {5642,13367}, {5656,25406}, {6593,15063}, {7691,15360}, {7728,13339}, {10170,17702}, {10516,18405}, {10627,15807}, {10982,20423}, {11064,11430}, {11180,18945}, {11245,13754}, {11412,13142}, {11441,31804}, {11456,26206}, {11591,12370}, {11645,16655}, {11793,13403}, {12111,18914}, {12168,18933}, {13292,18436}, {13598,19924}, {14516,15056}, {14852,32620}, {15058,34224}, {16261,16658}, {16654,29012}, {16776,29181}, {17814,19467}, {18842,31363}, {22261,27356}, {31834,32358}

X(34664) = midpoint of X(11459) and X(12022)
X(34664) = reflection of X(i) in X(j) for these (i,j): (376, 10691), (428, 381), (13490, 5066), (20841, 11634)
X(34664) = pole of the line {6, 18281} wrt Evans conic
X(34664) = pole of the line {468, 10414} wrt Lester circle
X(34664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3, 14890), (2, 4221, 28049), (2, 7375, 11311), (2, 8362, 25039), (2, 10210, 21567), (2, 12084, 15970), (2, 12108, 11539), (2, 13371, 28426), (2, 15646, 19238), (2, 15706, 140), (2, 15776, 11318), (2, 16245, 4227), (2, 16661, 15970), (2, 16860, 33048), (2, 17692, 28791), (26, 6890, 19671), (26, 11317, 33832), (26, 17516, 27234), (26, 21547, 14869), (26, 21552, 31651), (26, 27915, 6822), (26, 30776, 1985), (27, 419, 33008), (27, 10298, 7527), (27, 16956, 6999), (27, 18641, 7567), (27, 27847, 25986), (27, 28029, 28430), (27, 28666, 1564), (27, 28822, 25547), (2043, 2044, 25)


X(34665) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO MANDART-INCIRCLE

Barycentrics    2*a^6+(b^2+8*b*c+c^2)*a^4-8*(b+c)*b*c*a^3-2*(b^2+c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*b*c*a-(b^4-c^4)*(b^2-c^2) : :
X(34665) = 3*X(428)-2*X(34666)

The center of the reciprocal HR-ellipse of these triangles is X(34666)

X(34665) lies on these lines: {30,944}, {33,428}, {528,7667}, {529,34667}

X(34665) = reflection of X(428) in X(3058)
X(34665) = {X(34727), X(34745)}-harmonic conjugate of X(34611)


X(34666) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO ANTI-ARA

Barycentrics    2*a^6+(b+c)^2*a^4-2*(b+c)*b*c*a^3-(2*b^2+3*b*c+2*c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*b*c*a-(b^4-c^4)*(b^2-c^2) : :
X(34666) = 3*X(428)-X(34665) = 3*X(3058)-2*X(34665)

The center of the reciprocal HR-ellipse of these triangles is X(34665)

X(34666) lies on these lines: {2,20989}, {30,40}, {33,428}, {210,29024}, {528,34603}, {529,34668}, {5434,21147}

X(34666) = reflection of X(3058) in X(428)
X(34666) = {X(34728), X(34746)}-harmonic conjugate of X(34612)


X(34667) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-ARA TO 5th MIXTILINEAR

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(b^4+18*b^2*c^2+c^4)*a^3-2*(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*a^2-(b^2-c^2)*(b^4-c^4)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34667) = 3*X(428)-2*X(34668) = 3*X(3244)-X(34633) = 3*X(7667)-4*X(34634) = 3*X(7667)-2*X(34656)

The center of the reciprocal HR-ellipse of these triangles is X(34668)

X(34667) lies on these lines: {30,145}, {428,3241}, {517,34614}, {519,7667}, {528,34652}, {529,34665}, {952,34664}, {3244,34633}, {10691,31145}, {12135,28204}

X(34667) = reflection of X(i) in X(j) for these (i,j): (428, 3241), (31145, 10691), (34656, 34634)
X(34667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (34634, 34656, 7667), (34729, 34748, 145)


X(34668) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO ANTI-ARA

Barycentrics    2*a^7+2*(b+c)*a^6+(b^2+c^2)*a^5+(b+c)*(b^2+c^2)*a^4-2*(b^4+3*b^2*c^2+c^4)*a^3-2*(b^4-3*b^2*c^2+c^4)*(b+c)*a^2-(b^2-c^2)*(b^4-c^4)*a-(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34668) = 3*X(8)-2*X(34656) = 3*X(9778)-2*X(34614) = 3*X(10246)-4*X(23410) = 3*X(34603)-4*X(34633)

The center of the reciprocal HR-ellipse of these triangles is X(34667)

X(34668) lies on these lines: {2,9798}, {8,30}, {381,8192}, {428,3241}, {517,34613}, {519,34603}, {528,34653}, {529,34666}, {542,16980}, {952,7540}, {1829,7576}, {5262,5434}, {7291,34746}, {7426,8185}, {9778,34614}, {10246,23410}

X(34668) = reflection of X(i) in X(j) for these (i,j): (3241, 428), (34657, 34633)
X(34668) = anticomplement of X(34634)
X(34668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (34627, 34730, 8), (34633, 34657, 34603)


X(34669) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    2*a^8+(b^2+3*b*c+c^2)*a^6-3*(b+c)*b*c*a^5+(5*b^2+2*b*c+5*c^2)*b*c*a^4-(b+c)*(b^2+c^2)*b*c*a^3-(2*b^6+2*c^6-(b^2+b*c+c^2)*(6*b^2-7*b*c+6*c^2)*b*c)*a^2-(b+c)*b^3*c^3*a-(b^6-c^6)*(b^2-c^2) : :
X(34669) = 3*X(5434)-X(34670)

The center of the reciprocal HR-ellipse of these triangles is X(34670)

X(34669) lies on these lines: {30,34683}, {528,34685}, {529,7883}, {535,34636}, {754,34605}, {5434,12150}, {19924,34617}

X(34669) = reflection of X(12150) in X(5434)


X(34670) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 5th ANTI-BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34669)

X(34670) lies on these lines: {30,34684}, {528,34686}, {529,34604}, {535,34635}, {754,34606}, {5434,12150}, {19924,34618}

X(34670) = reflection of X(5434) in X(12150)


X(34671) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO ANTI-MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34672)

X(34671) lies on these lines: {519,34679}, {528,7883}, {529,34616}, {754,34607}, {4421,11490}, {19924,34619}

X(34671) = reflection of X(12150) in X(4421)


X(34672) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 5th ANTI-BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34671)

X(34672) lies on these lines: {519,34680}, {528,34604}, {529,34615}, {754,11235}, {4421,11490}, {19924,34620}

X(34672) = reflection of X(4421) in X(12150)


X(34673) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO AQUILA

Barycentrics    5*a^5-(b+c)*a^4+3*(b^2+c^2)*a^3-3*(b+c)*(b^2+c^2)*a^2+(2*b^4+3*b^2*c^2+2*c^4)*a-(b+c)*(4*b^4+3*b^2*c^2+4*c^4) : :
X(34673) = 3*X(8)-X(34686) = 3*X(3679)-X(34674) = 3*X(4669)-X(34635) = 3*X(7883)-2*X(34636) = 3*X(7883)-X(34685)

The center of the reciprocal HR-ellipse of these triangles is X(34674)

X(34673) lies on these lines: {2,12495}, {8,754}, {515,34616}, {517,34681}, {519,7883}, {3679,10789}, {4669,34635}, {9857,31168}, {19924,34627}

X(34673) = reflection of X(i) in X(j) for these (i,j): (12150, 3679), (34685, 34636)
X(34673) = {X(7883), X(34685)}-harmonic conjugate of X(34636)


X(34674) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO 5th ANTI-BROCARD

Barycentrics    7*a^5+4*(b+c)*a^4+3*(b^2+c^2)*a^3+(b^4+3*b^2*c^2+c^4)*a-2*(b+c)*(b^4+c^4) : :
X(34674) = 3*X(1)-2*X(34636) = 3*X(1699)-2*X(34681) = 3*X(3679)-2*X(34673) = 2*X(7883)-3*X(25055) = 3*X(12150)-X(34673) = 3*X(34604)-2*X(34635) = 3*X(34604)-X(34686)

The center of the reciprocal HR-ellipse of these triangles is X(34673)

X(34674) lies on these lines: {1,754}, {515,34615}, {517,34682}, {519,34604}, {1699,34681}, {3679,10789}, {7818,11368}, {7883,25055}, {19924,34628}

X(34674) = reflection of X(i) in X(j) for these (i,j): (3679, 12150), (34686, 34635)
X(34674) = {X(34604), X(34686)}-harmonic conjugate of X(34635)


X(34675) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO ARA

Barycentrics    3*a^10+8*(b^2+c^2)*a^8+(4*b^4+5*b^2*c^2+4*c^4)*a^6-(b^2+c^2)*(8*b^4-9*b^2*c^2+8*c^4)*a^4-(b^4+b^2*c^2+c^4)*(7*b^4-6*b^2*c^2+7*c^4)*a^2-(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(34675) = 3*X(9909)-X(34676)

The center of the reciprocal HR-ellipse of these triangles is X(34676)

X(34675) lies on these lines: {30,7883}, {754,34608}, {9909,10790}, {19924,34621}

X(34675) = reflection of X(12150) in X(9909)
X(34675) = {X(7883), X(34651)}-harmonic conjugate of X(34681)


X(34676) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO 5th ANTI-BROCARD

Barycentrics    3*a^10-7*(b^2+c^2)*a^8-(8*b^4-5*b^2*c^2+8*c^4)*a^6+(b^2+c^2)*(7*b^4-12*b^2*c^2+7*c^4)*a^4+(b^4+b^2*c^2+c^4)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+2*(b^4-c^4)*(b^2-c^2)*b^2*c^2 : :
X(34676) = 3*X(9909)-2*X(34675)

The center of the reciprocal HR-ellipse of these triangles is X(34675)

X(34676) lies on these lines: {30,34604}, {754,34609}, {9909,10790}, {19924,34622}

X(34676) = reflection of X(9909) in X(12150)
X(34676) = {X(34604), X(34650)}-harmonic conjugate of X(34682)


X(34677) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO 5th BROCARD

Barycentrics    4*a^8+5*(b^2+c^2)*a^6+(b^4+3*b^2*c^2+c^4)*a^4-2*(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^2-2*b^8-(4*b^4+5*b^2*c^2+4*c^4)*b^2*c^2-2*c^8 : :
X(34677) = 3*X(7811)-X(34678) = 3*X(12150)-2*X(34678)

The center of the reciprocal HR-ellipse of these triangles is X(34678)

X(34677) lies on these lines: {2,32}, {542,34616}, {19924,34623}

X(34677) = reflection of X(12150) in X(7811)
X(34677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3785, 29419, 30954), (7787, 24889, 29568), (7815, 27662, 10337), (7883, 9481, 28704), (10292, 28415, 28436)


X(34678) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO 5th ANTI-BROCARD

Barycentrics    2*a^8+7*(b^2+c^2)*a^6+(2*b^4+9*b^2*c^2+2*c^4)*a^4-2*(b^6+c^6)*a^2-b^8-(2*b^4+b^2*c^2+2*c^4)*b^2*c^2-c^8 : :
X(34678) = 3*X(7811)-2*X(34677) = 3*X(12150)-X(34677)

The center of the reciprocal HR-ellipse of these triangles is X(34677)

X(34678) lies on these lines: {2,32}, {542,34615}, {19924,34624}

X(34678) = reflection of X(7811) in X(12150)
X(34678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3785, 26664, 7867), (6292, 31124, 28415), (7753, 7810, 27602), (7787, 21248, 26586), (7914, 29445, 25779), (15821, 25752, 7867), (15870, 28802, 25886), (22870, 25779, 1799), (27260, 29792, 31024), (27602, 29445, 30817), (28724, 29944, 24546)


X(34679) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO 2nd CIRCUMPERP TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34680)

X(34679) lies on these lines: {519,34671}, {527,34636}, {528,34616}, {529,7883}, {535,34681}, {754,34610}, {11194,12150}, {19924,34625}

X(34679) = reflection of X(12150) in X(11194)


X(34680) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 5th ANTI-BROCARD

Barycentrics    3*a^8-2*(5*b^2+2*b*c+5*c^2)*a^6+2*(b+c)*b*c*a^5+(2*b^4+2*c^4-(4*b^2+15*b*c+4*c^2)*b*c)*a^4+2*(b+c)*(b^2+c^2)*b*c*a^3+(5*b^6+5*c^6-4*(b^4+b^2*c^2+c^4)*b*c)*a^2+2*(b^3+c^3)*(b^2+b*c+c^2)*b*c*a+2*(b^2-c^2)^2*b^2*c^2 : :

The center of the reciprocal HR-ellipse of these triangles is X(34679)

X(34680) lies on these lines: {519,34672}, {527,34635}, {528,34615}, {529,34604}, {535,34682}, {754,11236}, {11194,12150}, {19924,34626}

X(34680) = reflection of X(11194) in X(12150)


X(34681) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO EULER

Barycentrics    5*a^8-(b^2+c^2)*a^6+(b^4+5*b^2*c^2+c^4)*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^2-(4*b^4+3*b^2*c^2+4*c^4)*(b^2-c^2)^2 : :
X(34681) = 3*X(4)-X(34615) = 3*X(381)-X(34682) = 3*X(1699)-X(34674) = 3*X(3839)-X(34604) = 3*X(7883)-X(34616) = 3*X(12150)-2*X(34682)

The center of the reciprocal HR-ellipse of these triangles is X(34682)

X(34681) lies on these lines: {2,9873}, {4,754}, {30,7883}, {76,3830}, {83,18500}, {98,381}, {515,34636}, {517,34673}, {535,34679}, {952,34685}, {1078,18503}, {1352,3543}, {1699,34674}, {3818,10722}, {3839,34604}, {3845,14568}, {5066,7828}, {5152,22566}, {6054,9888}, {7795,11001}, {7812,14492}, {7832,8703}, {7841,14458}, {7924,11645}, {7930,15693}, {9996,31168}, {10350,14041}

X(34681) = reflection of X(12150) in X(381)
X(34681) = {X(7883), X(34651)}-harmonic conjugate of X(34675)


X(34682) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO 5th ANTI-BROCARD

Barycentrics    7*a^8-8*(b^2+c^2)*a^6+(2*b^4-11*b^2*c^2+2*c^4)*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-2*(b^4+c^4)*(b^2-c^2)^2 : :
X(34682) = 3*X(381)-2*X(34681) = 3*X(3534)-2*X(34616) = 3*X(5054)-2*X(7883) = 3*X(10246)-2*X(34636) = 3*X(34604)-X(34615)

The center of the reciprocal HR-ellipse of these triangles is X(34681)

X(34682) lies on these lines: {2,32151}, {3,754}, {30,34604}, {98,381}, {515,34635}, {517,34674}, {535,34680}, {952,34686}, {1351,15681}, {3534,7757}, {3830,5309}, {5054,7883}, {7753,14830}, {7762,8703}, {7776,15693}, {7787,18503}, {7818,26316}, {10246,34636}, {10350,11287}, {11361,12188}, {11645,12212}, {12156,14881}, {12206,18501}

X(34682) = reflection of X(381) in X(12150)
X(34682) = {X(34604), X(34650)}-harmonic conjugate of X(34676)


X(34683) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34684)

X(34683) lies on these lines: {30,34669}, {528,7883}, {529,34685}, {754,34611}, {3058,10799}, {19924,34629}

X(34683) = reflection of X(12150) in X(3058)


X(34684) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO 5th ANTI-BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34683)

X(34684) lies on these lines: {30,34670}, {528,34604}, {529,34686}, {754,34612}, {3058,10799}, {19924,34630}

X(34684) = reflection of X(3058) in X(12150)


X(34685) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th ANTI-BROCARD TO 5th MIXTILINEAR

Barycentrics    7*a^5-5*(b+c)*a^4+9*(b^2+c^2)*a^3-3*(b+c)*(b^2+c^2)*a^2+(10*b^4+9*b^2*c^2+10*c^4)*a-(b+c)*(2*b^4+3*b^2*c^2+2*c^4) : :
X(34685) = 3*X(3244)-X(34635) = 3*X(7883)-4*X(34636)

The center of the reciprocal HR-ellipse of these triangles is X(34686)

X(34685) lies on these lines: {145,754}, {517,34616}, {519,7883}, {528,34669}, {529,34683}, {952,34681}, {3241,10800}, {3244,34635}, {9997,31168}, {19924,34631}

X(34685) = reflection of X(i) in X(j) for these (i,j): (12150, 3241), (34673, 34636)
X(34685) = {X(34636), X(34673)}-harmonic conjugate of X(7883)


X(34686) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO 5th ANTI-BROCARD

Barycentrics    a^5+7*(b+c)*a^4-3*(b^2+c^2)*a^3+3*(b+c)*(b^2+c^2)*a^2-(5*b^4+3*b^2*c^2+5*c^4)*a+(b+c)*(b^4+3*b^2*c^2+c^4) : :
X(34686) = 3*X(8)-2*X(34673) = 3*X(3241)-2*X(34685) = 3*X(9778)-2*X(34616) = 3*X(12150)-X(34685) = 3*X(34604)-4*X(34635) = 3*X(34604)-2*X(34674)

The center of the reciprocal HR-ellipse of these triangles is X(34685)

X(34686) lies on these lines: {2,9941}, {8,754}, {517,34615}, {519,34604}, {528,34670}, {529,34684}, {952,34682}, {3241,10800}, {9778,34616}, {19924,34632}

X(34686) = reflection of X(i) in X(j) for these (i,j): (3241, 12150), (34674, 34635)
X(34686) = anticomplement of X(34636)
X(34686) = {X(34635), X(34674)}-harmonic conjugate of X(34604)


X(34687) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ANTI-MANDART-INCIRCLE

Barycentrics    2*a^7-2*(b+c)*a^6-(3*b^2+2*b*c+3*c^2)*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4+4*(b^2+8*b*c+c^2)*b*c*a^3+8*(b+c)*(b^2-3*b*c+c^2)*b*c*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(34687) = X(2136)+2*X(31799)

The center of the reciprocal HR-ellipse of these triangles is X(34688)

X(34687) lies on these lines: {8,190}, {20,529}, {30,12686}, {40,34742}, {519,14110}, {535,34639}, {1004,15888}, {2136,31799}, {2478,11235}, {3870,34749}, {3885,34699}, {4421,5434}, {5433,13279}, {5794,34612}, {5795,10624}, {6765,18481}, {6865,34625}, {10953,31140}, {11112,31397}, {11500,34619}, {12625,31789}, {19860,34640}, {24466,25438}

X(34687) = reflection of X(i) in X(j) for these (i,j): (5434, 4421), (34742, 40)


X(34688) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34687)

X(34688) lies on these lines: {4,529}, {30,34708}, {119,11236}, {145,528}, {519,34696}, {535,12699}, {1482,34741}, {3555,34690}, {4188,15888}, {4293,13205}, {4421,5434}, {7173,10527}, {10944,17579}, {11112,12647}, {12943,13271}, {28452,34717}

X(34688) = reflection of X(i) in X(j) for these (i,j): (4421, 5434), (34717, 28452), (34741, 1482)


X(34689) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO AQUILA

Barycentrics    2*a^4-(b^2-10*b*c+c^2)*a^2-8*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34689) = 4*X(8)-X(7354) = 3*X(8)-X(17579) = 3*X(3058)-4*X(11113) = 3*X(3058)-2*X(34699) = 2*X(3632)+X(6284) = 3*X(3679)-X(34690) = X(3893)+2*X(12527) = 5*X(4668)-2*X(18990) = 3*X(4669)-X(34637) = 3*X(5434)-2*X(34690) = 3*X(7354)-4*X(17579) = 3*X(11112)-2*X(34637) = 2*X(11113)-3*X(34606) = 2*X(17579)-3*X(34612) = 3*X(34606)-X(34699)

The center of the reciprocal HR-ellipse of these triangles is X(34690)

X(34689) lies on these lines: {2,3304}, {8,529}, {11,3421}, {12,31493}, {30,4677}, {57,3679}, {72,519}, {144,528}, {200,34716}, {515,34618}, {517,34697}, {551,5316}, {956,5432}, {958,11239}, {1155,34646}, {1317,3940}, {1836,4915}, {2098,5815}, {3241,5289}, {3434,34739}, {3436,11235}, {3476,3711}, {3621,34611}, {3632,6284}, {3654,24467}, {3872,34647}, {3893,12527}, {4317,19706}, {4661,5855}, {4668,18990}, {4669,11112}, {4853,28609}, {4995,34619}, {5048,21060}, {5220,12648}, {5288,5433}, {6154,8168}, {6174,11194}, {6765,10543}, {8163,11238}, {10197,24953}, {11827,12645}, {12246,34630}, {15326,34740}, {15338,34607}, {21677,32049}, {28204,31793}

X(34689) = midpoint of X(3621) and X(34611)
X(34689) = reflection of X(i) in X(j) for these (i,j): (3058, 34606), (5434, 3679), (7354, 34612), (11112, 4669), (34612, 8), (34699, 11113), (34720, 4677), (34749, 2)
X(34689) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3421, 34625, 31141), (11113, 34699, 3058), (31141, 34625, 11), (34606, 34699, 11113)


X(34690) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    2*a^4-(b^2-7*b*c+c^2)*a^2-2*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34690) = 3*X(1)-2*X(11113) = 2*X(145)+X(10483) = 3*X(1699)-2*X(34697) = X(3632)-4*X(18990) = X(3633)+2*X(7354) = 3*X(3679)-2*X(34689) = X(3893)-4*X(31776) = X(3901)+2*X(10944) = 3*X(5434)-X(34689) = X(5904)-4*X(10106) = X(17579)-3*X(34605) = 3*X(34605)-2*X(34637) = 4*X(34699)-3*X(34719) = X(34699)-3*X(34749) = X(34719)-4*X(34749)

The center of the reciprocal HR-ellipse of these triangles is X(34689)

X(34690) lies on these lines: {1,529}, {2,5258}, {30,7982}, {35,11239}, {55,34740}, {57,3679}, {145,10483}, {376,5537}, {388,5288}, {495,31157}, {515,34617}, {517,34698}, {519,3868}, {528,26726}, {535,3241}, {551,908}, {952,3894}, {999,31141}, {1056,5251}, {1478,31159}, {1699,34697}, {2829,11224}, {2975,10197}, {3244,34611}, {3245,12648}, {3336,32049}, {3476,4867}, {3555,34688}, {3582,11236}, {3583,34739}, {3584,11194}, {3585,11235}, {3632,18990}, {3633,7354}, {3680,4338}, {3893,31776}, {3895,15228}, {3901,10944}, {3913,4325}, {4299,34607}, {4316,34626}, {4677,11112}, {4880,12647}, {5261,31262}, {5270,12513}, {5881,28452}, {5904,10106}, {6264,31162}, {7741,20060}, {10056,34610}, {10072,31160}, {11237,26437}, {11374,25055}, {16418,33925}, {19876,31190}, {20067,25439}, {24473,34696}, {24474,28204}, {31397,34646}

X(34690) = reflection of X(i) in X(j) for these (i,j): (3632, 34612), (3679, 5434), (4677, 11112), (5881, 28452), (17579, 34637), (34611, 3244), (34612, 18990)
X(34690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1478, 34625, 31159), (17579, 34605, 34637)


X(34691) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34692)

X(34691) lies on these lines: {30,40}, {528,34730}, {529,34608}, {535,34642}, {5434,9909}

X(34691) = reflection of X(5434) in X(9909)
X(34691) = {X(34606), X(34653)}-harmonic conjugate of X(34697)


X(34692) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34691)

X(34692) lies on these lines: {30,944}, {528,34729}, {529,34609}, {535,34643}, {5434,9909}

X(34692) = reflection of X(9909) in X(5434)
X(34692) = {X(34605), X(34652)}-harmonic conjugate of X(34698)


X(34693) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 5th BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34694)

X(34693) lies on these lines: {30,9902}, {528,34738}, {529,9939}, {535,34644}, {542,34618}, {754,34606}, {5434,7811}

X(34693) = reflection of X(5434) in X(7811)


X(34694) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34693)

X(34694) lies on these lines: {30,7976}, {528,34737}, {529,7812}, {535,34645}, {542,34617}, {754,34605}, {5434,7811}, {7753,16975}

X(34694) = reflection of X(7811) in X(5434)


X(34695) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 2nd CIRCUMPERP TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34696)

X(34695) lies on these lines: {2,12}, {30,84}, {40,34709}, {519,14110}, {527,5728}, {528,9803}, {535,34646}, {3655,34716}, {3813,6895}, {5705,18990}, {5857,5902}, {6284,12649}, {6734,7354}, {6831,8666}, {6836,12513}, {6986,12607}, {11114,28610}, {12527,17625}

X(34695) = midpoint of X(11114) and X(28610)
X(34695) = reflection of X(i) in X(j) for these (i,j): (5434, 11194), (34709, 40), (34742, 3928)
X(34695) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12, 3600, 15844), (1220, 26561, 5484), (2551, 26686, 27283), (5253, 30778, 2), (5484, 30826, 26561), (15843, 31157, 26561), (20076, 30001, 29730), (25524, 27944, 5434), (25524, 31221, 31141), (30778, 30979, 26561)


X(34696) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34695)

X(34696) lies on these lines: {2,12}, {30,34741}, {519,34688}, {527,34637}, {528,34617}, {535,3655}, {1482,34708}, {3913,6934}, {6917,12513}, {6942,12607}, {7967,33961}, {24473,34690}, {28443,34740}, {28452,34700}

X(34696) = reflection of X(i) in X(j) for these (i,j): (11194, 5434), (34700, 28452), (34708, 1482)
X(34696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (958, 3600, 6645), (5484, 29468, 24583), (27283, 31246, 26558)


X(34697) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO EULER

Barycentrics    2*a^7-2*(b+c)*a^6-(3*b^2-14*b*c+3*c^2)*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4-4*(b^2-4*b*c+c^2)*b*c*a^3+8*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-10*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(34697) = 3*X(4)-X(34617) = 4*X(355)-X(11826) = 3*X(1699)-X(34690) = 3*X(3839)-X(34605) = X(6284)+2*X(18525) = X(7354)-4*X(18480) = X(10944)-4*X(31937) = 5*X(18492)-2*X(18990)

The center of the reciprocal HR-ellipse of these triangles is X(34698)

X(34697) lies on these lines: {2,12114}, {4,529}, {11,381}, {12,18761}, {30,40}, {153,7680}, {376,1376}, {392,515}, {517,34689}, {519,12672}, {528,16112}, {535,34646}, {549,5251}, {952,34699}, {1699,34690}, {1737,7354}, {2829,14647}, {3058,9957}, {3361,10826}, {3434,3543}, {3545,10785}, {3614,18542}, {3656,34749}, {3830,8158}, {3839,10893}, {3845,10943}, {4847,31673}, {4995,28444}, {5055,26492}, {5101,15942}, {5252,18540}, {5290,11373}, {5433,26321}, {5587,21164}, {5655,12889}, {6054,12348}, {6256,17532}, {6284,12647}, {6849,9657}, {6866,9656}, {6938,34626}, {6945,20418}, {8185,28454}, {10629,11238}, {10914,12527}, {11112,12616}, {11239,11496}, {11928,14269}, {12607,21669}, {15326,18491}, {15338,18518}, {17647,18236}, {18492,18990}, {19541,34740}, {22758,31157}, {28208,28459}, {30326,31789}

X(34697) = reflection of X(i) in X(j) for these (i,j): (5434, 381), (7354, 28452), (11826, 34612), (28452, 18480), (34612, 355), (34630, 3654), (34749, 3656)
X(34697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (18516, 18519, 11), (34606, 34653, 34691)


X(34698) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO 2nd ANTI-CIRCUMPERP-TANGENTIAL

Barycentrics    2*a^7-2*(b+c)*a^6-(3*b^2-11*b*c+3*c^2)*a^5+(b-3*c)*(3*b-c)*(b+c)*a^4-(7*b^2-16*b*c+7*c^2)*b*c*a^3+8*(b^2-c^2)*(b-c)*b*c*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(34698) = 3*X(3534)-2*X(34618) = 3*X(5054)-2*X(34606) = 2*X(7354)+X(18526) = 3*X(10246)-2*X(11113) = X(18525)-4*X(18990)

The center of the reciprocal HR-ellipse of these triangles is X(34697)

X(34698) lies on these lines: {2,32153}, {3,529}, {11,381}, {30,944}, {65,28204}, {376,20067}, {388,26321}, {495,18515}, {515,34637}, {517,34690}, {519,25413}, {528,34748}, {535,3655}, {549,17757}, {952,17579}, {2829,10247}, {3534,10306}, {3653,25681}, {3654,32049}, {3830,12001}, {4293,18524}, {5054,26487}, {5708,18391}, {5714,18493}, {5770,5790}, {5840,34699}, {6923,34625}, {7354,18526}, {9803,34627}, {10246,11113}, {10269,31141}, {11237,22766}, {11239,11849}, {12115,22765}, {12645,34612}, {12648,12702}, {19914,34717}, {28458,34718}

X(34698) = reflection of X(i) in X(j) for these (i,j): (381, 5434), (12645, 34612), (18525, 28452), (28452, 18990), (34611, 1483), (34718, 28458)
X(34698) = {X(34605), X(34652)}-harmonic conjugate of X(34692)


X(34699) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL TO 5th MIXTILINEAR

Barycentrics    2*a^4-(b^2+14*b*c+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34699) = 2*X(145)+X(6284) = 4*X(1482)-X(6253) = 4*X(1483)-X(11826) = 3*X(3058)-2*X(11113) = 3*X(3058)-X(34689) = 3*X(3241)-X(17579) = 4*X(3244)-X(7354) = 3*X(3244)-X(34637) = 3*X(5434)-2*X(17579) = 3*X(7354)-4*X(34637) = 4*X(11113)-3*X(34606) = 3*X(34606)-2*X(34689) = X(34690)+3*X(34719) = 2*X(34690)-3*X(34749) = 2*X(34719)+X(34749)

The center of the reciprocal HR-ellipse of these triangles is X(17579)

X(34699) lies on these lines: {1,34612}, {2,3303}, {7,528}, {12,11235}, {30,7982}, {55,31157}, {56,34607}, {72,519}, {145,529}, {495,31159}, {497,31141}, {517,14855}, {549,34486}, {551,3748}, {943,15998}, {952,34697}, {999,6154}, {1071,28194}, {1259,4428}, {1482,6253}, {1483,11826}, {1750,31162}, {2550,8162}, {3244,7354}, {3295,24953}, {3632,15172}, {3633,15171}, {3656,34746}, {3679,4863}, {3870,34647}, {3871,5433}, {3885,34687}, {3894,28212}, {3902,20336}, {3925,6767}, {4302,34740}, {4421,5298}, {4677,4866}, {4917,25681}, {5288,10386}, {5432,25439}, {5840,34698}, {5842,34617}, {5853,5919}, {6174,10072}, {7173,10528}, {9797,28610}, {10197,24390}, {11238,34619}, {12348,13189}, {12356,13180}, {12648,34717}, {12701,28609}, {13600,28204}, {13996,18391}, {15326,20075}, {15888,17532}, {16189,20420}, {16371,33925}, {24680,28452}

X(34699) = midpoint of X(145) and X(34611)
X(34699) = reflection of X(i) in X(j) for these (i,j): (3679, 15170), (5434, 3241), (6284, 34611), (28452, 24680), (34606, 3058), (34612, 1), (34689, 11113), (34720, 2), (34746, 3656)
X(34699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 34625, 31157), (3058, 34689, 11113), (4421, 11240, 5298), (11113, 34689, 34606), (11235, 11239, 12)


X(34700) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO AQUILA

Barycentrics    5*a^4-6*(b+c)*a^3-(b^2-8*b*c+c^2)*a^2+2*(b+c)*(3*b^2-5*b*c+3*c^2)*a-4*(b^2-c^2)^2 : :
X(34700) = 3*X(8)-X(34711) = 4*X(355)-X(12635) = 5*X(355)-2*X(21077) = X(3621)+2*X(13463) = 3*X(3679)-X(34701) = 3*X(4421)-2*X(34701) = 3*X(4669)-X(34639) = 7*X(4678)-4*X(32157) = 2*X(5881)+X(12513) = X(10912)+2*X(12645) = 3*X(11235)-2*X(34640) = 3*X(11235)-X(34710) = X(12625)+2*X(32537) = X(13205)-4*X(15863)

The center of the reciprocal HR-ellipse of these triangles is X(34701)

X(34700) lies on these lines: {2,10950}, {8,190}, {12,3241}, {35,958}, {72,4677}, {80,5289}, {355,381}, {515,34620}, {517,34706}, {529,34627}, {535,18525}, {956,9897}, {1001,5727}, {1388,10031}, {2098,10707}, {2099,17577}, {3149,5881}, {3436,31145}, {3560,3913}, {3621,13463}, {3654,34626}, {3813,6941}, {3880,18908}, {3893,17615}, {4669,34639}, {4678,32157}, {4685,15232}, {4745,5791}, {5086,31140}, {5252,21617}, {5691,28534}, {5693,34739}, {5730,31160}, {6253,34744}, {6828,12607}, {9041,12587}, {10525,12762}, {10573,11112}, {10944,11240}, {10954,17530}, {11194,11500}, {11207,11868}, {11208,11867}, {11545,17564}, {12625,32537}, {12649,34749}, {12738,22837}, {13205,15863}, {15079,25055}, {15677,21677}, {24392,33956}, {28452,34696}

X(34700) = reflection of X(i) in X(j) for these (i,j): (3241, 3829), (4421, 3679), (11236, 355), (12635, 11236), (34626, 3654), (34696, 28452), (34710, 34640), (34743, 3656)
X(34700) = {X(11235), X(34710)}-harmonic conjugate of X(34640)


X(34701) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO ANTI-MANDART-INCIRCLE

Barycentrics    7*a^4-3*(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-2*(b^2-c^2)^2 : :
X(34701) = 7*X(1)-4*X(13463) = 3*X(1)-2*X(34640) = 4*X(3)-X(12625) = 2*X(20)+X(11523) = X(20)+2*X(12437) = 2*X(944)+X(2136) = 3*X(1699)-2*X(34706) = 3*X(3158)-2*X(34619) = X(3189)+2*X(4297) = 2*X(3189)+X(6762) = 5*X(3522)+X(12536) = 5*X(3522)-2*X(24391) = 3*X(3679)-2*X(34700) = 3*X(4421)-X(34700) = X(11523)-4*X(12437) = X(12536)+2*X(24391) = 6*X(13463)-7*X(34640)

The center of the reciprocal HR-ellipse of these triangles is X(34700)

X(34701) lies on these lines: {1,528}, {2,950}, {3,12625}, {9,4304}, {20,527}, {30,1490}, {35,958}, {40,376}, {56,31146}, {78,11015}, {100,5727}, {145,2094}, {200,34606}, {381,33596}, {515,3158}, {517,34707}, {518,34620}, {529,34628}, {535,3811}, {551,1058}, {553,3241}, {960,5696}, {1043,17294}, {1420,11240}, {1699,34706}, {1706,3486}, {1837,6174}, {2646,31140}, {2900,6282}, {3243,4293}, {3245,3633}, {3419,30282}, {3434,13384}, {3488,5437}, {3522,12536}, {3576,24392}, {3586,5440}, {3632,3916}, {3655,34709}, {3680,5882}, {3813,30389}, {3870,34605}, {3880,10167}, {3885,10031}, {3895,6224}, {3913,5537}, {3984,15680}, {4190,11518}, {4294,15829}, {4511,9580}, {4654,17579}, {4668,32157}, {4853,34720}, {4930,28198}, {5219,9963}, {5691,11236}, {5722,9945}, {5731,5853}, {5732,34742}, {5881,6906}, {5903,24473}, {6172,20007}, {6765,18481}, {6901,9624}, {6934,7982}, {7962,20075}, {9579,31164}, {11235,17614}, {11248,28204}, {11682,20066}, {12526,15338}, {12565,34630}, {12635,28534}, {15670,19875}, {15678,31938}, {16548,18596}, {17528,24929}, {17532,33595}, {17571,31446}, {25568,28164}, {30144,34649}, {31162,34629}

X(34701) = midpoint of X(3189) and X(34610)
X(34701) = reflection of X(i) in X(j) for these (i,j): (3679, 4421), (3928, 376), (5691, 11236), (6762, 34610), (24392, 3576), (34610, 4297), (34711, 34639)
X(34701) = pole of the line {3962, 18839} wrt Feuerbach hyperbola
X(34701) = pole of the line {2826, 3737} wrt hexyl circle
X(34701) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 11112, 6173), (20, 12437, 11523), (78, 11114, 31142), (3189, 4297, 6762), (3522, 12536, 24391), (3586, 5440, 30827), (34607, 34711, 34639)


X(34702) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34703)

X(34702) lies on these lines: {30,10525}, {197,4421}, {519,9911}, {528,34608}, {529,34621}, {3913,7387}, {9645,25524}

X(34702) = reflection of X(4421) in X(9909)
X(34702) = {X(11235), X(34655)}-harmonic conjugate of X(34706)


X(34703) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO ANTI-MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34702)

X(34703) lies on these lines: {2,7071}, {30,10306}, {197,4421}, {519,34723}, {528,34609}, {529,34622}, {15940,16418}

X(34703) = reflection of X(9909) in X(4421)
X(34703) = {X(34607), X(34654)}-harmonic conjugate of X(34707)


X(34704) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 5th BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34705)

X(34704) lies on these lines: {519,34732}, {528,9939}, {529,34623}, {542,34620}, {754,11235}, {4421,7811}

X(34704) = reflection of X(4421) in X(7811)


X(34705) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO ANTI-MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34704)

X(34705) lies on these lines: {519,34731}, {528,7812}, {529,34624}, {542,34619}, {754,34607}, {4421,7811}

X(34705) = reflection of X(7811) in X(4421)


X(34706) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO EULER

Barycentrics    5*a^4-(b^2+4*b*c+c^2)*a^2+2*(b+c)*b*c*a-4*(b^2-c^2)^2 : :
X(34706) = 4*X(4)-X(3913) = 5*X(4)-2*X(12607) = 3*X(4)-X(34619) = 3*X(381)-X(34707) = 2*X(382)+X(12513) = X(1657)-4*X(24387) = 3*X(1699)-X(34701) = 5*X(3913)-8*X(12607) = 3*X(3913)-4*X(34619) = 3*X(4421)-2*X(34707) = 3*X(11194)-2*X(34620) = 3*X(11235)-X(34620) = 5*X(11236)-4*X(12607) = 3*X(11236)-2*X(34619) = 6*X(12607)-5*X(34619)

The center of the reciprocal HR-ellipse of these triangles is X(34707)

X(34706) lies on these lines: {2,3847}, {4,528}, {30,10525}, {55,17577}, {56,10707}, {149,12943}, {376,3829}, {381,4421}, {382,535}, {404,9671}, {515,34640}, {517,34700}, {519,3830}, {529,3543}, {952,34710}, {958,11114}, {1001,9668}, {1376,3583}, {1479,11112}, {1657,24387}, {1699,34701}, {2475,9670}, {2886,11111}, {3146,3813}, {3189,10248}, {3295,34649}, {3434,34606}, {3436,34720}, {3585,34719}, {3839,34607}, {3843,8715}, {4428,17532}, {5073,8666}, {5080,8168}, {5687,18514}, {5691,10912}, {6836,34630}, {7354,11240}, {9579,31146}, {9669,10199}, {9689,13898}, {10431,34742}, {10543,15679}, {10724,22560}, {10738,22753}, {11237,34611}, {11238,17579}, {12635,22793}, {12672,34717}, {13205,18491}, {14794,16370}, {15682,34625}, {22792,28204}, {24386,28158}

X(34706) = midpoint of X(i) and X(j) for these {i,j}: {3146, 34610}, {15682, 34625}
X(34706) = reflection of X(i) in X(j) for these (i,j): (376, 3829), (3913, 11236), (4421, 381), (11194, 11235), (11236, 4), (34610, 3813), (34626, 2), (34739, 3830)
X(34706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11114, 31140, 958), (11235, 34655, 34702), (12953, 31140, 11114)


X(34707) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO ANTI-MANDART-INCIRCLE

Barycentrics    7*a^4-(5*b^2+8*b*c+5*c^2)*a^2+4*(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34707) = 7*X(3)-4*X(3813) = 3*X(381)-2*X(34706) = X(382)-4*X(8715) = X(1657)+2*X(3913) = 3*X(3534)-2*X(34620) = 5*X(3811)-2*X(28645) = 4*X(3829)-5*X(15694) = 3*X(4421)-X(34706) = 3*X(5054)-2*X(11235) = X(5073)-4*X(12607) = 3*X(10246)-2*X(34640) = 3*X(34607)-X(34619) = X(34620)-3*X(34626) = 4*X(34620)-3*X(34740) = 4*X(34626)-X(34740)

The center of the reciprocal HR-ellipse of these triangles is X(34706)

X(34707) lies on these lines: {2,9669}, {3,528}, {30,10306}, {35,31140}, {55,17528}, {56,34719}, {84,28204}, {100,9668}, {381,4421}, {382,8715}, {497,17564}, {515,34639}, {517,34701}, {519,3534}, {529,15681}, {535,1657}, {550,34610}, {551,19706}, {943,17532}, {952,34711}, {956,20095}, {999,20075}, {1479,6174}, {2475,31480}, {3058,16417}, {3149,34629}, {3158,28146}, {3295,11112}, {3679,31445}, {3811,28534}, {3829,15694}, {3830,18542}, {3871,9655}, {4294,9709}, {4299,34749}, {4302,6154}, {4309,16408}, {4930,28194}, {5054,11235}, {5073,12607}, {5687,11114}, {7957,15071}, {8703,34625}, {9708,11111}, {9945,30305}, {10246,34640}, {11194,15688}, {12513,15696}, {12632,17538}, {12675,25413}, {12953,31160}, {14377,17313}, {15338,34720}, {15720,24387}, {16371,34611}, {16418,34612}, {17571,31494}, {17577,31479}, {24466,30283}, {25440,34649}, {25568,28178}, {28202,28609}

X(34707) = reflection of X(i) in X(j) for these (i,j): (381, 4421), (382, 11236), (3534, 34626), (11236, 8715), (34610, 550), (34625, 8703), (34740, 3534)
X(34707) = {X(34607), X(34654)}-harmonic conjugate of X(34703)


X(34708) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34709)

X(34708) lies on these lines: {2,11}, {30,34688}, {519,34741}, {529,34629}, {1482,34696}, {3655,34640}, {3813,6950}, {3913,6929}, {6938,12513}

X(34708) = reflection of X(i) in X(j) for these (i,j): (4421, 3058), (34696, 1482)


X(34709) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO ANTI-MANDART-INCIRCLE

Barycentrics    2*a^6-4*(b+c)*a^5+(b^2+18*b*c+c^2)*a^4+2*(b+c)*(b^2-8*b*c+c^2)*a^3-2*(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2+2*(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*(b-c)^2 : :
X(34709) = X(2136)+2*X(31777)

The center of the reciprocal HR-ellipse of these triangles is X(34708)

X(34709) lies on these lines: {2,11}, {30,12686}, {40,34695}, {519,1071}, {529,34630}, {1532,8715}, {2136,31777}, {3655,34701}, {3913,6925}, {5853,10167}, {6284,6735}, {7354,12648}, {12120,34716}, {21628,34639}

X(34709) = reflection of X(i) in X(j) for these (i,j): (3058, 4421), (34695, 40)


X(34710) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE TO 5th MIXTILINEAR

Barycentrics    7*a^4-12*(b+c)*a^3-(5*b^2-28*b*c+5*c^2)*a^2+2*(3*b-2*c)*(2*b-3*c)*(b+c)*a-2*(b^2-c^2)^2 : :
X(34710) = 3*X(3241)-X(34711) = 3*X(3244)-X(34639) = 3*X(4421)-2*X(34711) = 3*X(11235)-4*X(34640) = 3*X(11235)-2*X(34700) = X(13205)-4*X(25416) = 2*X(13463)+X(20050) = 5*X(16189)-2*X(32537) = 7*X(20057)-4*X(32157)

The center of the reciprocal HR-ellipse of these triangles is X(34711)

X(34710) lies on these lines: {56,3241}, {145,528}, {355,381}, {517,34620}, {529,34629}, {535,8148}, {952,34706}, {3244,34639}, {3829,11681}, {3913,6924}, {4428,10965}, {5854,34619}, {5884,34626}, {5903,24473}, {6906,12513}, {10074,13205}, {11194,11248}, {11224,33956}, {11531,28534}, {13463,20050}, {16189,32537}, {20049,20060}, {20057,32157}

X(34710) = reflection of X(i) in X(j) for these (i,j): (4421, 3241), (11236, 1482), (31145, 3829), (34700, 34640), (34717, 3656)
X(34710) = {X(34640), X(34700)}-harmonic conjugate of X(11235)


X(34711) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO ANTI-MANDART-INCIRCLE

Barycentrics    a^4+6*(b+c)*a^3-2*(b^2+10*b*c+c^2)*a^2-2*(b+c)*(3*b^2-8*b*c+3*c^2)*a+(b^2-c^2)^2 : :
X(34711) = 3*X(8)-2*X(34700) = X(3146)-4*X(32537) = X(3189)+2*X(12245) = 3*X(3241)-2*X(34710) = 7*X(3523)-4*X(33895) = 5*X(3616)-8*X(32157) = 3*X(4421)-X(34710) = X(7991)+2*X(12640) = 3*X(9778)-2*X(34620) = 7*X(9780)-4*X(13463) = X(9802)-4*X(32198) = X(20070)+2*X(32049) = 3*X(25568)-4*X(34619) = 3*X(34607)-4*X(34639) = 3*X(34607)-2*X(34701)

The center of the reciprocal HR-ellipse of these triangles is X(34710)

X(34711) lies on these lines: {2,3057}, {8,190}, {40,376}, {56,3241}, {145,1155}, {411,3913}, {517,25568}, {527,7991}, {529,34630}, {535,6361}, {952,34707}, {962,11236}, {1145,17556}, {1445,3895}, {1479,2551}, {1788,3885}, {2094,34749}, {2098,6174}, {2183,3208}, {2802,5657}, {3035,4345}, {3146,32537}, {3474,12648}, {3523,33895}, {3616,32157}, {3617,4679}, {3654,31786}, {3880,24477}, {4669,12572}, {4848,31146}, {5119,11111}, {5731,5854}, {6256,28194}, {6736,31142}, {6865,11362}, {6927,7982}, {6942,8715}, {8256,9785}, {8668,11194}, {9778,34620}, {9780,13463}, {9802,32198}, {9819,26105}, {10573,34719}, {11041,25439}, {11235,27870}, {18221,20323}, {20066,31145}, {20070,28534}, {22770,32141}, {28459,34718}

X(34711) = reflection of X(i) in X(j) for these (i,j): (962, 11236), (3241, 4421), (34610, 40), (34625, 3654), (34701, 34639)
X(34711) = anticomplement of X(34640)
X(34711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6068, 30332, 5698), (34639, 34701, 34607)


X(34712) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO ARA

Barycentrics    7*a^7+4*(b+c)*a^6-(b^2+c^2)*a^5+2*(b+c)*(b^2+c^2)*a^4-(7*b^4-6*b^2*c^2+7*c^4)*a^3-4*(b+c)*(b^4+c^4)*a^2+(b^4-c^4)*(b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(-2*b-2*c) : :
X(34712) = 4*X(10154)-3*X(19875) = 10*X(19862)-9*X(30775) = 3*X(25055)-2*X(34609)

The center of the reciprocal HR-ellipse of these triangles is X(34713)

X(34712) lies on these lines: {1,30}, {376,7718}, {515,34621}, {517,34726}, {519,34608}, {1658,31425}, {1699,34725}, {3679,8185}, {5881,7387}, {7982,31305}, {9588,16195}, {9624,14790}, {10154,19875}, {11363,31152}, {12410,28204}, {16548,18596}, {19862,30775}, {25055,34609}, {26208,31048}, {26446,33591}, {31423,34351}

X(34712) = reflection of X(i) in X(j) for these (i,j): (3679, 9909), (34730, 34642)
X(34712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 34628, 34634), (1, 34657, 31162), (1081, 10149, 10910), (13408, 13995, 4654), (15172, 16137, 13995), (34608, 34730, 34642)


X(34713) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO AQUILA

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

The center of the reciprocal HR-ellipse of these triangles is X(34712)

X(34713) lies on these lines: {2,12135}, {8,30}, {381,5090}, {515,34622}, {517,34725}, {519,34609}, {1829,3830}, {3622,30775}, {3679,8185}, {4669,34642}, {10246,18281}, {11396,31133}

X(34713) = reflection of X(i) in X(j) for these (i,j): (9909, 3679), (34729, 34643)
X(34713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 34656, 34718), (34609, 34729, 34643)


X(34714) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO 5th BROCARD

Barycentrics    4*a^5+4*(b+c)*a^4-3*(b^2+c^2)*a^3-(2*b^4+3*b^2*c^2+2*c^4)*a-2*(b+c)*(b^4+c^4) : :
X(34714) = 3*X(1)-2*X(34645) = 3*X(1699)-2*X(34733) = 3*X(3097)-4*X(8356) = 3*X(3679)-2*X(34715) = 4*X(7810)-3*X(19875) = 3*X(7811)-X(34715) = 2*X(7812)-3*X(25055) = 3*X(9939)-2*X(34644) = 3*X(9939)-X(34738)

The center of the reciprocal HR-ellipse of these triangles is X(34715)

X(34714) lies on these lines: {1,754}, {515,34623}, {517,34734}, {519,9939}, {542,34628}, {726,33264}, {1699,34733}, {3097,8356}, {3099,3679}, {7810,19875}, {7812,25055}, {7818,11364}, {10791,31168}

X(34714) = reflection of X(i) in X(j) for these (i,j): (3679, 7811), (34738, 34644)
X(34714) = {X(9939), X(34738)}-harmonic conjugate of X(34644)


X(34715) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO AQUILA

Barycentrics    2*a^5+2*(b+c)*a^4-3*(b^2+c^2)*a^3+3*(b+c)*(b^2+c^2)*a^2-(b^4+3*b^2*c^2+c^4)*a-(b^4-3*b^2*c^2+c^4)*(b+c) : :
X(34715) = 3*X(8)-X(34738) = 3*X(3679)-X(34714) = 3*X(4669)-X(34644) = 3*X(7811)-2*X(34714) = 3*X(7812)-2*X(34645) = 3*X(7812)-X(34737)

The center of the reciprocal HR-ellipse of these triangles is X(34714)

X(34715) lies on these lines: {2,12195}, {8,754}, {515,34624}, {517,34733}, {519,7812}, {542,34627}, {3099,3679}, {3241,7753}, {4669,34644}, {11361,14839}

X(34715) = reflection of X(i) in X(j) for these (i,j): (3241, 7753), (7811, 3679), (34737, 34645)
X(34715) = {X(7812), X(34737)}-harmonic conjugate of X(34645)


X(34716) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    7*a^4-3*(b+c)*a^3-(5*b^2-14*b*c+5*c^2)*a^2+(b-3*c)*(3*b-c)*(b+c)*a-2*(b^2-c^2)^2 : :
X(34716) = 3*X(1)-2*X(34647) = 2*X(944)+X(6762) = 3*X(1699)-2*X(34739) = X(2136)-4*X(4297) = 5*X(3522)-2*X(12640) = 3*X(3679)-2*X(34717) = X(5691)-4*X(11260) = X(5881)-4*X(8666) = 4*X(5882)-X(11523) = 5*X(7987)-2*X(32049) = 7*X(9588)-4*X(32537) = 3*X(11194)-X(34717) = 4*X(12513)-X(12625) = 3*X(24392)-4*X(34625) = 3*X(28609)-4*X(34647)

The center of the reciprocal HR-ellipse of these triangles is X(34717)

X(34716) lies on these lines: {1,529}, {2,1420}, {9,3476}, {20,3680}, {30,12650}, {36,956}, {40,376}, {145,28610}, {200,34689}, {390,527}, {515,24392}, {517,34740}, {528,2951}, {535,31162}, {551,1056}, {960,17644}, {1319,30827}, {1512,34627}, {1699,34739}, {1706,4311}, {3158,5731}, {3340,20076}, {3522,12640}, {3555,3901}, {3601,11239}, {3655,34695}, {3872,8544}, {3880,34626}, {4315,5437}, {4321,5434}, {4654,34605}, {4853,34612}, {4861,9579}, {5176,31231}, {5252,31157}, {5691,11235}, {5881,6905}, {5882,6987}, {6904,11530}, {6938,7982}, {6965,9624}, {7580,12513}, {7987,32049}, {9588,32537}, {9589,33895}, {9613,17532}, {11224,17768}, {11236,25055}, {11249,28204}, {12120,34709}, {12607,30389}, {12629,18481}, {13462,31190}, {17648,34620}, {19535,31436}, {24477,28236}

X(34716) = midpoint of X(145) and X(28610)
X(34716) = reflection of X(i) in X(j) for these (i,j): (2136, 34607), (3158, 5731), (3679, 11194), (3928, 34610), (5691, 11235), (11235, 11260), (28609, 1), (34607, 4297), (34744, 34646)
X(34716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (34610, 34744, 34646), (34646, 34744, 3928)


X(34717) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO AQUILA

Barycentrics    5*a^4-6*(b+c)*a^3-(b^2-16*b*c+c^2)*a^2+2*(b+c)*(3*b^2-7*b*c+3*c^2)*a-4*(b^2-c^2)^2 : :
X(34717) = 5*X(8)-X(28610) = 3*X(8)-X(34744) = 4*X(355)-X(10912) = 3*X(3679)-X(34716) = X(3913)+2*X(5881) = X(3913)-4*X(32537) = 3*X(4669)-X(34646) = X(5881)+2*X(32537) = 3*X(11194)-2*X(34716) = 3*X(28610)-5*X(34744)

The center of the reciprocal HR-ellipse of these triangles is X(34716)

X(34717) lies on these lines: {2,1388}, {8,529}, {11,3241}, {36,956}, {355,381}, {515,34626}, {517,34739}, {528,16112}, {535,34718}, {551,31479}, {999,15863}, {1012,3913}, {2099,12531}, {3036,3476}, {3434,31145}, {3632,28609}, {3654,34620}, {3812,17644}, {3813,6945}, {3880,5927}, {4421,12114}, {4669,17647}, {4677,5904}, {5176,5289}, {5249,5252}, {5587,33956}, {6830,12607}, {6911,12513}, {9041,12586}, {10950,11239}, {11113,12647}, {11207,11866}, {11208,11865}, {12648,34699}, {12672,34706}, {12737,34748}, {13205,18519}, {13271,18516}, {17613,34628}, {17614,19875}, {17615,31165}, {17619,25055}, {19914,34698}, {28452,34688}

X(34717) = midpoint of X(3632) and X(28609)
X(34717) = reflection of X(i) in X(j) for these (i,j): (10912, 11235), (11194, 3679), (11235, 355), (34620, 3654), (34688, 28452), (34710, 3656), (34743, 34647)
X(34717) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5881, 32537, 3913), (11236, 34743, 34647)


X(34718) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO AQUILA

Barycentrics    a^4-6*(b+c)*a^3+(b^2+12*b*c+c^2)*a^2+6*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(34718) = 2*X(1)-3*X(5054) = 5*X(2)-4*X(5901) = 7*X(2)-5*X(10595) = 3*X(3)-2*X(3655) = 7*X(3)-4*X(5882) = X(3)-4*X(11362) = 3*X(3)-X(34748) = 5*X(8)+X(6361) = 2*X(8)+X(12702) = 4*X(8)-X(18525) = 3*X(8)-X(34627) = 3*X(8)+X(34632) = 4*X(10)-3*X(5055) = 4*X(10)-X(8148) = 8*X(10)-5*X(18493) = 2*X(3656)-3*X(5055) = 2*X(6361)-5*X(12702)

The center of the reciprocal HR-ellipse of these triangles is X(31162)

X(34718) lies on these lines: {1,5054}, {2,1482}, {3,519}, {8,30}, {10,3656}, {40,3534}, {55,28443}, {145,3524}, {165,14093}, {210,381}, {355,3830}, {376,952}, {382,7991}, {515,15681}, {516,15684}, {528,19914}, {535,34717}, {547,5603}, {549,3241}, {551,10247}, {944,8703}, {946,4745}, {956,18515}, {958,28453}, {962,3845}, {1145,3940}, {1159,31397}, {1385,15693}, {1483,12100}, {1656,7982}, {1657,5881}, {1698,11278}, {2098,3582}, {2099,3584}, {2102,13627}, {2103,13626}, {3058,10573}, {3244,15707}, {3304,5559}, {3421,10742}, {3428,12331}, {3526,24680}, {3543,28174}, {3545,3617}, {3576,15700}, {3579,3632}, {3616,11539}, {3621,10304}, {3622,15709}, {3623,15708}, {3625,15689}, {3626,12699}, {3628,5734}, {3633,13624}, {3653,6684}, {3828,5886}, {3829,6971}, {3839,4678}, {3851,4301}, {3871,21161}, {3962,18545}, {4297,15695}, {4668,18480}, {4701,31730}, {4746,31673}, {4848,7373}, {4870,25415}, {4921,15952}, {5066,5818}, {5070,13464}, {5072,30308}, {5076,9589}, {5079,11522}, {5252,18541}, {5285,28456}, {5434,12647}, {5493,17800}, {5691,28202}, {5697,11238}, {5731,34200}, {5846,11179}, {5903,11237}, {7967,15692}, {7987,15716}, {9588,15178}, {9778,15686}, {9780,15699}, {9812,14893}, {9881,13188}, {9884,33813}, {9956,11531}, {10031,33814}, {10124,10283}, {10306,28444}, {10679,16418}, {10680,16417}, {11224,11230}, {11231,15723}, {11235,31806}, {11545,30305}, {11849,16370}, {11900,20128}, {12355,13178}, {15170,18391}, {15679,16150}, {15682,20070}, {15683,28186}, {15687,28212}, {15705,20014}, {15710,20054}, {16371,22765}, {17504,20050}, {18527,30286}, {28458,34698}, {28459,34711}, {28610,31775}, {30389,31447}, {31423,33179}, {31425,31666}

X(34718) = midpoint of X(i) and X(j) for these {i,j}: {2, 12245}, {40, 4677}, {376, 31145}, {3534, 12645}, {15682, 20070}, {34627, 34632}
X(34718) = reflection of X(i) in X(j) for these (i,j): (2, 5690), (3, 3654), (355, 4669), (381, 3679), (944, 8703), (946, 4745), (962, 3845), (1482, 2), (1483, 12100), (2102, 13627), (2103, 13626), (3241, 549), (3534, 40), (3654, 11362), (3656, 10), (3830, 355), (8148, 3656), (9884, 33813), (10031, 33814), (10246, 5657), (10247, 26446), (11224, 11230), (12355, 13178), (12645, 4677), (13188, 9881), (16150, 15679), (16200, 11231), (34698, 28458), (34745, 28459), (34748, 3655)
X(34718) = complement of X(34631)
X(34718) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 34748, 3655), (8, 12702, 18525), (8, 34632, 34627), (8, 34656, 34713), (10, 3656, 5055), (10, 8148, 18493), (381, 3679, 5790), (549, 3241, 10246), (551, 26446, 15694), (3241, 5657, 549), (3579, 3632, 18526), (3656, 5055, 18493), (5055, 8148, 3656), (5690, 12245, 1482), (10247, 15694, 551)


X(34719) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: AQUILA TO MANDART-INCIRCLE

Barycentrics    2*a^4-(b^2+7*b*c+c^2)*a^2+2*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34719) = 3*X(1)-2*X(11112) = 5*X(1698)-8*X(15172) = 3*X(1699)-2*X(34746) = 3*X(3058)-X(34720) = 4*X(3244)-X(10483) = X(3632)-4*X(15171) = X(3633)+2*X(6284) = 3*X(3679)-2*X(34720) = X(3893)-4*X(31795) = X(5904)-4*X(10624) = X(11114)-3*X(34611) = 3*X(34611)-2*X(34649) = X(34690)-4*X(34699) = 3*X(34690)-4*X(34749) = 3*X(34699)-X(34749)

The center of the reciprocal HR-ellipse of these triangles is X(34720)

X(34719) lies on these lines: {1,528}, {2,3746}, {30,7982}, {36,11240}, {46,31146}, {56,34707}, {80,3895}, {100,10199}, {145,535}, {149,7951}, {390,5251}, {496,6174}, {515,34629}, {517,34745}, {519,3869}, {1210,34639}, {1479,31160}, {1697,1837}, {1698,15172}, {1699,34746}, {3241,4295}, {3244,10483}, {3295,31140}, {3303,17528}, {3555,28534}, {3582,4421}, {3583,11236}, {3584,11235}, {3585,34706}, {3632,15171}, {3633,6284}, {3655,11014}, {3871,7741}, {3893,31795}, {3894,28174}, {3913,4857}, {4294,5288}, {4302,34610}, {4309,5258}, {4324,34620}, {4330,12513}, {4360,33866}, {4677,11113}, {4867,30305}, {5082,5259}, {5100,17264}, {5274,31263}, {5541,5722}, {5692,5853}, {5842,11224}, {10056,31159}, {10072,34607}, {10389,26725}, {10573,34711}, {11238,26358}, {11373,15170}, {12688,23340}

X(34719) = reflection of X(i) in X(j) for these (i,j): (3632, 34606), (3679, 3058), (4677, 11113), (10483, 34605), (11114, 34649), (34605, 3244), (34606, 15171), (34612, 15170)
X(34719) = pole of the line {17606, 18839} wrt Feuerbach hyperbola
X(34719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (149, 25439, 7951), (1479, 34619, 31160), (11114, 34611, 34649), (15170, 34612, 25055)


X(34720) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO AQUILA

Barycentrics    2*a^4-(b^2+10*b*c+c^2)*a^2+8*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34720) = 4*X(8)-X(6284) = 3*X(8)-X(11114) = 3*X(3058)-2*X(34719) = 2*X(3632)+X(7354) = 3*X(3679)-X(34719) = 2*X(3893)+X(10944) = 7*X(3983)-4*X(12575) = 5*X(4668)-2*X(15171) = 3*X(4669)-X(34649) = 3*X(5434)-4*X(11112) = 3*X(5434)-2*X(34749) = 3*X(6284)-4*X(11114) = 2*X(11112)-3*X(34612) = 3*X(11113)-2*X(34649) = 2*X(11114)-3*X(34606) = 3*X(34612)-X(34749)

The center of the reciprocal HR-ellipse of these triangles is X(34719)

X(34720) lies on these lines: {2,3303}, {8,190}, {12,5082}, {30,4677}, {65,519}, {78,34640}, {515,34630}, {517,16194}, {529,31145}, {535,3625}, {956,6154}, {1329,10707}, {1334,17330}, {1376,11240}, {1697,1837}, {1706,31146}, {2082,17281}, {3434,8168}, {3436,34706}, {3621,34605}, {3632,7354}, {3649,6765}, {3689,15950}, {3711,30305}, {3871,24953}, {3983,12575}, {4420,13463}, {4421,31157}, {4668,15171}, {4669,11113}, {4853,34701}, {4882,12701}, {4917,28628}, {5221,6764}, {5298,34625}, {5433,5687}, {5853,10177}, {6253,12245}, {7080,7173}, {8162,26040}, {11518,11524}, {11826,12645}, {12607,17577}, {12688,28194}, {15170,19875}, {15326,17784}, {15338,34707}, {15888,17528}, {17556,21031}, {28204,31798}

X(34720) = midpoint of X(3621) and X(34605)
X(34720) = reflection of X(i) in X(j) for these (i,j): (3058, 3679), (5434, 34612), (6284, 34606), (11113, 4669), (34606, 8), (34689, 4677), (34699, 2), (34749, 11112)
X(34720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5082, 34619, 31140), (11112, 34749, 5434), (31140, 34619, 12), (34612, 34749, 11112)


X(34721) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO 5th BROCARD

Barycentrics    4*a^10-3*(b^2+c^2)*a^8-(4*b^4-13*b^2*c^2+4*c^4)*a^6+(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^4-(5*b^4-6*b^2*c^2+5*c^4)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4-2*b^2*c^2-2*c^4) : :
X(34721) = 3*X(7811)-X(34722)

The center of the reciprocal HR-ellipse of these triangles is X(34722)

X(34721) lies on these lines: {30,9863}, {64,542}, {754,34609}, {7811,9909}

X(34721) = reflection of X(9909) in X(7811)
X(34721) = {X(9939), X(34660)}-harmonic conjugate of X(34734)


X(34722) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO ARA

Barycentrics    2*a^10+3*(b^2+c^2)*a^8-(2*b^4-11*b^2*c^2+2*c^4)*a^6-(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4-(7*b^4-6*b^2*c^2+7*c^4)*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4) : :
X(34722) = 3*X(7811)-2*X(34721)

The center of the reciprocal HR-ellipse of these triangles is X(34721)

X(34722) lies on these lines: {30,3095}, {542,34621}, {754,34608}, {7811,9909}

X(34722) = reflection of X(7811) in X(9909)
X(34722) = {X(7812), X(34661)}-harmonic conjugate of X(34733)


X(34723) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO 2nd CIRCUMPERP TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34724)

X(34723) lies on these lines: {2,1398}, {30,22770}, {519,34703}, {527,34643}, {528,34622}, {529,34609}, {535,34725}, {9909,11194}

X(34723) = reflection of X(9909) in X(11194)
X(34723) = {X(34610), X(34662)}-harmonic conjugate of X(34740)


X(34724) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34723)

X(34724) lies on these lines: {30,4421}, {519,9911}, {527,34642}, {528,34621}, {529,34608}, {535,34726}, {7387,12513}, {9909,11194}

X(34724) = reflection of X(11194) in X(9909)
X(34724) = {X(11236), X(34663)}-harmonic conjugate of X(34739)


X(34725) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO EULER

Barycentrics    5*a^10-6*(b^2+c^2)*a^8-4*(2*b^4-b^2*c^2+2*c^4)*a^6+2*(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^4+(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^2-4*(b^4-c^4)*(b^2-c^2)^3 : :
X(34725) = 5*X(3)-8*X(13371) = 3*X(1699)-X(34712) = 8*X(5448)-5*X(14530) = X(17834)-4*X(18383)

As a point on the Euler line, X(34725) has Shinagawa coefficients (E+F, -3*E-9*F)
The center of the reciprocal HR-ellipse of these triangles is X(34726)

X(34725) lies on these lines: {2,3}, {511,18405}, {515,34643}, {517,34713}, {524,12429}, {535,34723}, {539,32402}, {541,13093}, {542,12164}, {952,34729}, {1351,18396}, {1498,11645}, {1568,8780}, {1699,34712}, {1992,18945}, {2883,31166}, {3167,18400}, {5093,12022}, {5448,14530}, {9919,19479}, {10112,15534}, {10733,15106}, {11179,12233}, {11750,19347}, {11821,21356}, {12241,20423}, {12310,19506}, {13346,15139}, {13851,33586}, {17834,18383}

X(34725) = reflection of X(i) in X(j) for these (i,j): (9909, 381), (17506, 30772), (19239, 19337), (25494, 28721)
X(34725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3135, 21491), (3, 6924, 20063), (3, 13726, 11295), (3, 15246, 6954), (3, 18536, 3137), (3, 26254, 6918), (3, 27029, 14017), (3, 27652, 33011), (3, 30445, 26803), (3, 32976, 28465), (3, 33284, 6883), (4, 15769, 6839), (4, 16423, 8021), (4, 21501, 5999), (4, 21977, 25674), (26, 6810, 30071), (26, 14785, 29521), (26, 25867, 29464), (27, 10125, 11106), (27, 11845, 21312), (27, 16300, 15717), (27, 28735, 6756), (28, 6894, 25868), (28, 17589, 26256), (28, 31926, 26761), (29, 3127, 7667), (29, 5500, 29961), (29, 14865, 28464), (29, 24949, 27124), (140, 429, 6948)


X(34726) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO ARA

Barycentrics    7*a^10-12*(b^2+c^2)*a^8-4*(b^4+b^2*c^2+c^4)*a^6+2*(b^2+c^2)*(7*b^4-6*b^2*c^2+7*c^4)*a^4-(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(34726) = 2*X(2)-3*X(10245) = 3*X(10246)-2*X(34643)

As a point on the Euler line, X(34726) has Shinagawa coefficients (-2*E-5*F, 6*E+9*F)
The center of the reciprocal HR-ellipse of these triangles is X(34725)

X(34726) lies on these lines: {2,3}, {515,34642}, {517,34712}, {524,9833}, {535,34724}, {542,17834}, {569,14848}, {952,34730}, {1879,5210}, {1992,31804}, {10246,34643}, {11179,11432}, {11426,20423}

X(34726) = reflection of X(i) in X(j) for these (i,j): (381, 9909), (14790, 34351), (34609, 14070)
X(34726) = anticomplement of the anticomplement of X(33591)
X(34726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 402, 29894), (2, 402, 33747), (2, 442, 17526), (2, 461, 470), (2, 857, 15719), (2, 857, 21844), (2, 859, 6896), (2, 861, 140), (2, 864, 11484), (2, 1316, 14040), (2, 1600, 19650), (2, 1656, 25491), (2, 1658, 19324), (2, 2045, 29952), (2, 2045, 32986), (26, 235, 28382), (26, 379, 10997), (26, 381, 13737), (26, 446, 26195), (26, 464, 28076), (26, 550, 27983), (26, 3860, 28382), (26, 4194, 28102), (26, 5142, 6820), (26, 6656, 6837), (26, 6836, 16238), (26, 6851, 15764), (26, 6904, 21292), (26, 7391, 18569), (26, 7419, 26552)


X(34727) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34728)

X(34727) lies on these lines: {30,944}, {528,34609}, {529,34729}, {3058,9909}

X(34727) = reflection of X(9909) in X(3058)
X(34727) = {X(34611), X(34665)}-harmonic conjugate of X(34745)


X(34728) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO ARA

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

The center of the reciprocal HR-ellipse of these triangles is X(34727)

X(34728) lies on these lines: {30,40}, {528,34608}, {529,34730}, {3058,9909}

X(34728) = reflection of X(3058) in X(9909)
X(34728) = {X(34612), X(34666)}-harmonic conjugate of X(34746)


X(34729) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ARA TO 5th MIXTILINEAR

Barycentrics    7*a^7-5*(b+c)*a^6-10*(b^2+c^2)*a^5+2*(b+c)*(b^2+c^2)*a^4-(7*b^4-18*b^2*c^2+7*c^4)*a^3+(b+c)*(5*b^4-6*b^2*c^2+5*c^4)*a^2+10*(b^4-c^4)*(b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(-2*b-2*c) : :

The center of the reciprocal HR-ellipse of these triangles is X(34730)

X(34729) lies on these lines: {30,145}, {381,11396}, {517,34622}, {519,34609}, {528,34692}, {529,34727}, {952,34725}, {3241,8192}, {3244,34642}, {3830,12135}, {4678,30775}

X(34729) = reflection of X(i) in X(j) for these (i,j): (9909, 3241), (34713, 34643)
X(34729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (145, 34667, 34748), (34643, 34713, 34609)


X(34730) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO ARA

Barycentrics    a^7+7*(b+c)*a^6+5*(b^2+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(b^4+6*b^2*c^2+c^4)*a^3-(b+c)*(7*b^4-6*b^2*c^2+7*c^4)*a^2-5*(b^4-c^4)*(b^2-c^2)*a+(b^4-c^4)*(b^2-c^2)*(b+c) : :
X(34730) = 10*X(1698)-9*X(30775) = 3*X(10246)-4*X(33591)

The center of the reciprocal HR-ellipse of these triangles is X(34729)

X(34730) lies on these lines: {2,1829}, {8,30}, {517,34621}, {519,34608}, {528,34691}, {529,34728}, {952,34726}, {1698,30775}, {3241,8192}, {3543,5090}, {9778,34622}, {10246,33591}, {24608,26203}

X(34730) = reflection of X(i) in X(j) for these (i,j): (3241, 9909), (34712, 34642)
X(34730) = anticomplement of X(34643)
X(34730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 34632, 34656), (8, 34668, 34627), (34642, 34712, 34608)


X(34731) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO 2nd CIRCUMPERP TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34732)

X(34731) lies on these lines: {519,34705}, {527,34645}, {528,34624}, {529,7812}, {535,34733}, {542,34625}, {754,34610}, {7811,11194}

X(34731) = reflection of X(7811) in X(11194)


X(34732) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 5th BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34731)

X(34732) lies on these lines: {519,34704}, {527,34644}, {528,34623}, {529,9939}, {535,34734}, {542,34626}, {754,11236}, {7811,11194}

X(34732) = reflection of X(11194) in X(7811)


X(34733) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO EULER

Barycentrics    2*a^8-7*(b^2+c^2)*a^6+(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^4+(b^2+c^2)*(5*b^4-6*b^2*c^2+5*c^4)*a^2-(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2 : :
X(34733) = 3*X(4)-X(34623) = 3*X(262)-2*X(8356) = 3*X(381)-X(34734) = 3*X(1699)-X(34714) = 3*X(3545)-2*X(7810) = 3*X(3839)-X(9939) = X(7802)-4*X(14881) = 3*X(7812)-X(34624) = 2*X(7823)+X(9873)

The center of the reciprocal HR-ellipse of these triangles is X(34734)

X(34733) lies on these lines: {2,5171}, {4,754}, {5,7936}, {30,3095}, {146,148}, {183,316}, {262,8356}, {376,574}, {511,11361}, {515,34645}, {517,34715}, {535,34731}, {547,7934}, {549,3972}, {952,34737}, {1699,34714}, {3096,18502}, {3545,7810}, {3830,11054}, {3839,9939}, {5476,7924}, {6054,9890}, {7802,14881}, {7823,9873}, {7846,18501}, {7898,19130}, {8370,33706}, {9737,33265}, {9753,16041}, {12117,33193}, {14853,33210}, {14976,32152}, {20423,33017}, {23698,32469}

X(34733) = reflection of X(i) in X(j) for these (i,j): (376, 7753), (7811, 381), (14976, 32152), (33706, 8370)
X(34733) = {X(7812), X(34661)}-harmonic conjugate of X(34722)


X(34734) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO 5th BROCARD

Barycentrics    4*a^8-11*(b^2+c^2)*a^6+5*(b^4-b^2*c^2+c^4)*a^4+(b^2+c^2)*(4*b^4+3*b^2*c^2+4*c^4)*a^2-2*(b^4+c^4)*(b^2-c^2)^2 : :
X(34734) = 3*X(381)-2*X(34733) = 3*X(3534)-2*X(34624) = 3*X(5054)-2*X(7812) = 3*X(5055)-4*X(7810) = 4*X(7753)-5*X(15694) = 4*X(8356)-3*X(32447) = 3*X(9939)-X(34623) = 3*X(10246)-2*X(34645)

The center of the reciprocal HR-ellipse of these triangles is X(34733)

X(34734) lies on these lines: {2,32134}, {3,754}, {30,9863}, {183,316}, {376,7779}, {515,34644}, {517,34714}, {535,34732}, {542,15681}, {549,7777}, {952,34738}, {1384,7753}, {2080,7818}, {2896,18501}, {3534,34624}, {5054,7812}, {5055,7810}, {8356,32447}, {10246,34645}, {10796,31168}, {32515,33264}

X(34734) = midpoint of X(12251) and X(14976)
X(34734) = reflection of X(381) in X(7811)
X(34734) = {X(9939), X(34660)}-harmonic conjugate of X(34721)


X(34735) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34736)

X(34735) lies on these lines: {30,7976}, {528,7812}, {529,34737}, {542,34629}, {754,34611}, {3058,7811}

X(34735) = reflection of X(7811) in X(3058)


X(34736) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO 5th BROCARD

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

The center of the reciprocal HR-ellipse of these triangles is X(34735)

X(34736) lies on these lines: {30,9902}, {528,9939}, {529,34738}, {542,34630}, {754,34612}, {3058,7811}

X(34736) = reflection of X(3058) in X(7811)


X(34737) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th BROCARD TO 5th MIXTILINEAR

Barycentrics    2*a^5+2*(b+c)*a^4+9*(b^2+c^2)*a^3-3*(b+c)*(b^2+c^2)*a^2-(b^4-9*b^2*c^2+c^4)*a-(b+c)*(b^4+3*b^2*c^2+c^4) : :
X(34737) = 3*X(3241)-X(34738) = 3*X(3244)-X(34644) = 3*X(7811)-2*X(34738) = 3*X(7812)-4*X(34645) = 3*X(7812)-2*X(34715)

The center of the reciprocal HR-ellipse of these triangles is X(34738)

X(34737) lies on these lines: {145,754}, {517,34624}, {519,7812}, {528,34694}, {529,34735}, {542,34631}, {952,34733}, {3241,7811}, {3244,34644}, {7753,31145}

X(34737) = reflection of X(i) in X(j) for these (i,j): (7811, 3241), (31145, 7753), (34715, 34645)
X(34737) = {X(34645), X(34715)}-harmonic conjugate of X(7812)


X(34738) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO 5th BROCARD

Barycentrics    4*a^5+4*(b+c)*a^4+3*(b^2+c^2)*a^3-3*(b+c)*(b^2+c^2)*a^2-(2*b^4-3*b^2*c^2+2*c^4)*a-(b+c)*(2*b^4+3*b^2*c^2+2*c^4) : :
X(34738) = 3*X(8)-2*X(34715) = 3*X(3241)-2*X(34737) = 3*X(9778)-2*X(34624) = 3*X(9939)-4*X(34644) = 3*X(9939)-2*X(34714)

The center of the reciprocal HR-ellipse of these triangles is X(34737)

X(34738) lies on these lines: {2,12194}, {8,754}, {517,34623}, {519,9939}, {528,34693}, {529,34736}, {542,34632}, {730,33264}, {952,34734}, {3241,7811}, {9778,34624}, {10800,31168}

X(34738) = reflection of X(i) in X(j) for these (i,j): (3241, 7811), (34714, 34644)
X(34738) = anticomplement of X(34645)
X(34738) = {X(34644), X(34714)}-harmonic conjugate of X(9939)


X(34739) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO EULER

Barycentrics    5*a^4-(b^2-4*b*c+c^2)*a^2-2*(b+c)*b*c*a-4*(b^2-c^2)^2 : :
X(34739) = 5*X(4)-2*X(3813) = 4*X(4)-X(12513) = 3*X(4)-X(34625) = 3*X(381)-X(34740) = 2*X(382)+X(3913) = 3*X(1699)-X(34716) = X(3146)+2*X(12607) = X(3811)+2*X(33697) = 4*X(3813)-5*X(11235) = 8*X(3813)-5*X(12513) = 6*X(3813)-5*X(34625) = 3*X(4421)-2*X(34626) = 3*X(11194)-2*X(34740) = 3*X(11235)-2*X(34625) = 3*X(11236)-X(34626)

The center of the reciprocal HR-ellipse of these triangles is X(34740)

X(34739) lies on these lines: {2,3614}, {4,529}, {21,9656}, {30,4421}, {153,528}, {377,34501}, {381,535}, {382,3913}, {515,34647}, {517,34717}, {519,3830}, {527,34648}, {952,34743}, {956,18513}, {958,3585}, {999,34637}, {1001,1478}, {1376,5080}, {1699,34716}, {2095,6246}, {3146,12607}, {3245,3679}, {3434,34689}, {3436,34612}, {3583,34690}, {3648,15679}, {3811,33697}, {3829,3839}, {3843,8666}, {4428,11114}, {4720,21291}, {5046,9657}, {5073,8715}, {5302,19875}, {5691,12635}, {5693,34700}, {5789,19925}, {6284,11239}, {6925,34618}, {9589,32537}, {9654,10197}, {9655,25524}, {9688,13897}, {10728,13205}, {10912,22793}, {11238,34605}, {12953,20060}, {15682,34619}, {16371,31160}, {17768,34744}, {21031,31295}, {22560,22799}

X(34739) = midpoint of X(i) and X(j) for these {i,j}: {3146, 34607}, {5691, 28609}, {15682, 34619}
X(34739) = reflection of X(i) in X(j) for these (i,j): (4421, 11236), (11194, 381), (11235, 4), (12513, 11235), (12635, 28609), (34607, 12607), (34610, 3829), (34620, 2), (34706, 3830)
X(34739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3839, 34610, 3829), (5080, 12943, 1376), (5080, 17579, 31141), (11114, 11237, 4428), (11236, 34663, 34724), (12943, 31141, 17579), (17579, 31141, 1376)


X(34740) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    7*a^4-(5*b^2-8*b*c+5*c^2)*a^2-4*(b+c)*b*c*a-2*(b^2-c^2)^2 : :
X(34740) = 7*X(3)-4*X(12607) = 3*X(381)-2*X(34739) = X(382)-4*X(8666) = X(1657)+2*X(12513) = 3*X(3534)-2*X(34626) = 4*X(3813)-X(5073) = 4*X(3829)-3*X(14269) = 2*X(3913)-5*X(15696) = 2*X(4421)-3*X(15688) = 3*X(5054)-2*X(11236) = 3*X(11194)-X(34739) = 3*X(34610)-X(34625) = 3*X(34620)-X(34626) = 4*X(34620)-X(34707) = 4*X(34626)-3*X(34707)

The center of the reciprocal HR-ellipse of these triangles is X(34739)

X(34740) lies on these lines: {2,9654}, {3,529}, {10,19706}, {30,22770}, {36,31141}, {55,34690}, {376,6244}, {377,31494}, {381,535}, {382,8666}, {515,34646}, {517,34716}, {519,3534}, {527,3655}, {528,15681}, {550,34607}, {551,24703}, {944,28610}, {952,34744}, {956,17579}, {993,34637}, {999,11113}, {1012,34617}, {1155,3679}, {1385,28609}, {1478,31157}, {1617,5434}, {1657,12513}, {2975,9655}, {3295,20076}, {3813,5073}, {3829,14269}, {3830,18544}, {3913,15696}, {3928,28204}, {3940,21578}, {4293,9708}, {4299,34612}, {4302,34699}, {4317,11108}, {4421,15688}, {5054,11236}, {5076,24387}, {8703,34619}, {10246,34647}, {10247,17768}, {12943,31159}, {15174,15678}, {15326,34689}, {16370,34605}, {16417,34606}, {19535,31480}, {19541,34697}, {24392,28168}, {24477,28186}, {28443,34696}

X(34740) = midpoint of X(944) and X(28610)
X(34740) = reflection of X(i) in X(j) for these (i,j): (381, 11194), (382, 11235), (3534, 34620), (4930, 3655), (11235, 8666), (28609, 1385), (34607, 550), (34619, 8703), (34707, 3534)
X(34740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2975, 9655, 31493), (34610, 34662, 34723)


X(34741) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO MANDART-INCIRCLE

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

The center of the reciprocal HR-ellipse of these triangles is X(34742)

X(34741) lies on these lines: {2,14882}, {4,528}, {30,34696}, {145,529}, {519,34708}, {527,34649}, {535,34745}, {1071,28534}, {1259,31140}, {1317,34605}, {1482,34688}, {3058,10966}, {3614,5552}, {5440,18393}, {10573,11113}, {10950,11114}, {11235,11496}, {12699,22836}, {15680,34610}

X(34741) = reflection of X(i) in X(j) for these (i,j): (11194, 3058), (34688, 1482)


X(34742) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO 2nd CIRCUMPERP TANGENTIAL

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

The center of the reciprocal HR-ellipse of these triangles is X(34741)

X(34742) lies on these lines: {2,1466}, {7,5289}, {8,529}, {20,528}, {30,84}, {40,34687}, {63,34606}, {72,527}, {377,11236}, {519,1071}, {535,34746}, {936,28609}, {1210,3916}, {1259,6174}, {3058,10966}, {3649,19861}, {3868,34749}, {5302,24982}, {5732,34701}, {5777,28452}, {6284,26015}, {6916,34619}, {9965,34605}, {10431,34706}, {12114,34625}, {12629,18481}, {18961,31141}

X(34742) = midpoint of X(17579) and X(28610)
X(34742) = reflection of X(i) in X(j) for these (i,j): (3058, 11194), (34687, 40), (34695, 3928)


X(34743) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL TO 5th MIXTILINEAR

Barycentrics    7*a^4-12*(b+c)*a^3-5*(b^2-4*b*c+c^2)*a^2+2*(b+c)*(6*b^2-11*b*c+6*c^2)*a-2*(b^2-c^2)^2 : :
X(34743) = 3*X(3241)-X(34744) = 3*X(3244)-X(34646) = 3*X(11194)-2*X(34744) = 3*X(11236)-4*X(34647) = 3*X(11236)-2*X(34717)

The center of the reciprocal HR-ellipse of these triangles is X(34744)

X(34743) lies on these lines: {55,3241}, {145,529}, {149,20049}, {355,381}, {517,34626}, {528,34617}, {535,34748}, {952,34739}, {2099,31019}, {3244,34646}, {3555,3901}, {3633,28609}, {3913,6905}, {4421,11249}, {5855,34625}, {6914,12513}, {11009,17532}, {11680,31145}

X(34743) = midpoint of X(3633) and X(28609)
X(34743) = reflection of X(i) in X(j) for these (i,j): (11194, 3241), (11235, 1482), (34700, 3656), (34717, 34647)
X(34743) = {X(34647), X(34717)}-harmonic conjugate of X(11236)


X(34744) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: 5th MIXTILINEAR TO 2nd CIRCUMPERP TANGENTIAL

Barycentrics    a^4+6*(b+c)*a^3-2*(b+c)^2*a^2-2*(b+c)*(3*b^2-4*b*c+3*c^2)*a+(b^2-c^2)^2 : :
X(34744) = 3*X(8)-2*X(34717) = 4*X(40)-X(3189) = 3*X(3241)-2*X(34743) = 10*X(3617)-X(28647) = 3*X(3928)-2*X(34646) = 7*X(4678)+2*X(28646) = 2*X(5493)+X(12625) = X(7991)+2*X(24391) = 3*X(9778)-2*X(34626) = 3*X(11194)-X(34743) = 3*X(24477)-2*X(34625) = 3*X(28610)+2*X(34717)

The center of the reciprocal HR-ellipse of these triangles is X(34743)

X(34744) lies on these lines: {2,65}, {8,529}, {10,5714}, {40,376}, {55,3241}, {329,31141}, {517,5770}, {527,1478}, {528,9803}, {535,34627}, {549,4930}, {551,11529}, {758,5657}, {920,11111}, {952,34740}, {962,11235}, {993,11041}, {2094,5434}, {2099,5744}, {2551,4848}, {2886,5775}, {3085,4018}, {3218,3476}, {3339,5837}, {3340,30478}, {3487,4084}, {3543,28534}, {3617,20292}, {3654,31788}, {3868,11239}, {3913,7411}, {3962,7080}, {4004,19855}, {4295,17532}, {4323,4999}, {4421,5584}, {4428,15933}, {4678,28646}, {4757,10198}, {4880,12647}, {5128,6737}, {5176,20078}, {5183,17784}, {5252,9965}, {5289,5435}, {5493,12625}, {5554,11684}, {5698,5729}, {5731,5855}, {5745,18421}, {5815,8256}, {6175,11236}, {6253,34700}, {6909,12513}, {6916,11362}, {6935,7982}, {6950,8666}, {7288,11682}, {7991,24391}, {8168,30295}, {9778,34626}, {10306,32153}, {12649,34611}, {16126,31452}, {17281,21866}, {17768,34739}, {24392,28228}, {28458,34698}

X(34744) = midpoint of X(8) and X(28610)
X(34744) = reflection of X(i) in X(j) for these (i,j): (962, 11235), (3189, 34607), (3241, 11194), (4930, 549), (25568, 5657), (28609, 10), (34607, 40), (34610, 3928), (34619, 3654), (34716, 34646)
X(34744) = anticomplement of X(34647)
X(34744) = pole of the line {3486, 10179} wrt Feuerbach hyperbola
X(34744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3485, 24954, 28629), (3928, 34716, 34646), (4848, 12526, 2551), (34646, 34716, 34610)


X(34745) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: EULER TO MANDART-INCIRCLE

Barycentrics    2*a^7-2*(b+c)*a^6-3*(b^2+b*c+c^2)*a^5+(b+c)*(3*b^2+2*b*c+3*c^2)*a^4+(3*b^2-8*b*c+3*c^2)*b*c*a^3-4*(b^2-c^2)*(b-c)*b*c*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :
X(34745) = 3*X(3534)-2*X(34630) = 3*X(5054)-2*X(34612) = 2*X(6284)+X(18526) = 3*X(10246)-2*X(11112) = X(10483)-4*X(32900) = 4*X(15171)-X(18525) = 8*X(15172)-5*X(18493)

The center of the reciprocal HR-ellipse of these triangles is X(34746)

X(34745) lies on these lines: {2,32141}, {3,528}, {12,381}, {30,944}, {376,20066}, {497,18524}, {515,34649}, {517,34719}, {529,34748}, {535,34741}, {549,24390}, {952,11114}, {3057,28204}, {3534,22770}, {3655,34640}, {3830,6256}, {4084,28194}, {4294,26321}, {4302,12773}, {4309,13743}, {5054,26492}, {5703,15172}, {5842,10247}, {6284,18526}, {6767,18499}, {6928,34619}, {9668,10742}, {10246,11112}, {10267,31140}, {10483,32900}, {10707,11491}, {11238,11508}, {11240,22765}, {11849,12116}, {12645,34606}, {12649,12702}, {15170,28452}, {15171,18525}, {15694,31493}, {16202,17528}, {28459,34711}

X(34745) = reflection of X(i) in X(j) for these (i,j): (381, 3058), (12645, 34606), (28452, 15170), (34605, 1483), (34718, 28459)
X(34745) = {X(34611), X(34665)}-harmonic conjugate of X(34727)


X(34746) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO EULER

Barycentrics    2*a^7-2*(b+c)*a^6-3*(b+c)^2*a^5+(b+c)*(3*b^2+2*b*c+3*c^2)*a^4-8*b^2*c^2*a^3-4*(b^2-c^2)*(b-c)*b*c*a^2+(b^2+6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :
X(34746) = 3*X(4)-X(34629) = 2*X(355)+X(6253) = 3*X(1699)-X(34719) = 3*X(3839)-X(34611) = X(5881)+2*X(20420) = X(6284)-4*X(18480) = X(7354)+2*X(18525) = 2*X(15171)-5*X(18492)

The center of the reciprocal HR-ellipse of these triangles is X(34745)

X(34746) lies on these lines: {2,11500}, {4,528}, {5,34486}, {11,18491}, {12,381}, {30,40}, {72,28194}, {376,958}, {495,18406}, {515,3753}, {516,18908}, {517,16194}, {527,14872}, {529,34627}, {535,34742}, {549,24953}, {551,33597}, {942,5434}, {952,34749}, {1699,34719}, {1836,18528}, {3303,6849}, {3436,3543}, {3545,10786}, {3651,9710}, {3656,34699}, {3830,10526}, {3839,10894}, {3845,10942}, {5055,26487}, {5220,6361}, {5432,18524}, {5655,12890}, {5842,11114}, {5881,20420}, {6054,12349}, {6174,11499}, {6284,10039}, {6736,31673}, {6903,9711}, {6934,34620}, {7173,18544}, {7291,34668}, {7354,10573}, {7681,10707}, {7958,16202}, {7965,10679}, {9580,18529}, {9799,17579}, {9800,34632}, {10629,11237}, {11240,22753}, {11374,15170}, {11929,14269}, {12658,12858}, {12738,22791}, {15104,28174}, {15171,18492}, {15326,18519}, {15338,18761}, {15908,31140}, {17577,18242}, {17857,31162}, {18499,18516}, {21628,34639}, {28208,28458}

X(34746) = midpoint of X(6253) and X(34606)
X(34746) = reflection of X(i) in X(j) for these (i,j): (3058, 381), (5434, 28452), (11827, 34606), (34606, 355), (34618, 3654), (34699, 3656)
X(34746) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355, 6253, 11827), (18517, 18518, 12), (34612, 34666, 34728), (34639, 34648, 21628)


X(34747) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: OUTER-GARCIA TO 5th MIXTILINEAR

Barycentrics    11*a-4*b-4*c : :
X(34747) = 5*X(1)-4*X(2) = 7*X(1)-4*X(8) = 11*X(1)-8*X(10) = X(1)-4*X(145) = 9*X(1)-8*X(551) = 19*X(1)-16*X(1125) = 13*X(1)-10*X(1698) = 3*X(1)-4*X(3241) = 5*X(1)-8*X(3244) = 23*X(1)-20*X(3616) = 29*X(1)-20*X(3617) = 13*X(1)-4*X(3621) = 17*X(1)-20*X(3623) = 17*X(1)-14*X(3624) = 17*X(1)-8*X(3625) = 25*X(1)-16*X(3626) = 5*X(1)-2*X(3632) = X(1)+2*X(3633) = 13*X(1)-16*X(3635) = 3*X(1)-2*X(3679)

The center of the reciprocal HR-ellipse of these triangles is X(31145)

X(34747) lies on these lines: {1,2}, {30,11531}, {40,15688}, {165,3655}, {355,30308}, {376,28234}, {381,16200}, {382,7982}, {515,34631}, {517,15681}, {528,26726}, {529,34719}, {537,3644}, {546,3656}, {549,30392}, {550,7991}, {903,3875}, {952,11224}, {1317,13462}, {1385,15707}, {1482,14269}, {1483,3654}, {1699,16191}, {1743,4370}, {2093,6154}, {2136,3336}, {2802,3894}, {3174,13144}, {3303,16866}, {3476,4031}, {3524,13607}, {3528,5882}, {3529,28194}, {3530,30389}, {3543,28236}, {3544,13464}, {3576,15700}, {3629,9041}, {3653,14869}, {3746,17571}, {3830,11278}, {3851,24680}, {3855,11522}, {3880,24473}, {3885,3901}, {3893,18398}, {3913,19537}, {3928,11010}, {3973,4898}, {4312,12630}, {4317,12632}, {4399,31312}, {4421,7280}, {4428,5258}, {4460,4862}, {4654,10944}, {4659,4796}, {4725,24441}, {4795,4971}, {4859,4916}, {4870,5726}, {4889,17313}, {4908,16670}, {4910,17296}, {5010,11194}, {5055,33179}, {5119,16558}, {5288,16370}, {5434,18421}, {5541,10031}, {5563,17573}, {5587,11737}, {7967,15715}, {7983,9875}, {7988,10247}, {7989,12645}, {8148,28208}, {9053,20583}, {9884,13174}, {9897,25416}, {10299,11362}, {11034,11041}, {11260,33595}, {11274,15015}, {11280,34710}, {11357,19751}, {12245,15710}, {12513,19535}, {15170,34689}, {15683,28228}, {15718,31662}, {16667,17281}, {16673,17330}, {17079,20121}, {17151,17378}, {17274,17377}, {17549,25439}, {17678,19830}, {18526,28198}, {28581,31178}, {31160,33956}

X(34747) = midpoint of X(i) and X(j) for these {i,j}: {2, 20050}, {3241, 20049}
X(34747) = reflection of X(i) in X(j) for these (i,j): (2, 3244), (3621, 4669), (3632, 2), (3654, 1483), (3679, 3241), (3830, 11278), (4669, 3635), (4677, 1), (5541, 10031), (5881, 3656), (9875, 7983), (13174, 9884), (31145, 551), (34689, 15170)
X(34747) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4677, 19875), (1, 19876, 551), (2, 3636, 25055), (8, 3828, 3679), (145, 3633, 1), (145, 20049, 3241), (145, 20050, 3244), (239, 23600, 12629), (306, 25980, 27272), (551, 3679, 19876), (551, 10578, 29680), (551, 31145, 3679), (612, 26801, 29853), (997, 1193, 29599), (997, 27255, 29610), (1125, 20053, 4816), (1698, 3635, 1), (1722, 20014, 145), (1999, 20050, 3216), (2535, 31199, 25879), (3017, 26759, 30166), (3178, 29853, 28961), (3241, 3679, 1), (3241, 31145, 551), (3244, 3626, 20057), (3244, 3632, 1), (3244, 15808, 3635), (3244, 20050, 3632), (3582, 33175, 26805), (3584, 19868, 30147), (3584, 30116, 27577), (3616, 3622, 5268), (3617, 17023, 19861), (3617, 30177, 26973), (3621, 3622, 31183), (3621, 3635, 1698), (3621, 29582, 145), (3622, 29631, 30135), (3623, 3624, 1)


X(34748) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: JOHNSON TO 5th MIXTILINEAR

Barycentrics    11*a^4-12*(b+c)*a^3-(7*b^2-24*b*c+7*c^2)*a^2+12*(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2 : :
X(34748) = 4*X(1)-3*X(5055) = 5*X(3)-4*X(3654) = 3*X(3)-4*X(3655) = 5*X(3)-8*X(5882) = 11*X(3)-8*X(11362) = 3*X(3)-2*X(34718) = 2*X(8)-3*X(5054) = 4*X(40)-5*X(15695) = 4*X(145)-X(8148) = 2*X(145)+X(18526) = 3*X(145)-X(34631) = 4*X(1483)-X(12645) = 3*X(3654)-5*X(3655) = X(8148)+2*X(18526) = 3*X(8148)-4*X(34631)

The center of the reciprocal HR-ellipse of these triangles is X(34627)

X(34748) lies on these lines: {1,5055}, {2,1483}, {3,519}, {8,5054}, {30,145}, {40,15695}, {355,19709}, {376,5844}, {381,952}, {515,15684}, {517,15681}, {528,34698}, {529,34745}, {535,34743}, {549,7967}, {551,5790}, {944,3534}, {956,28443}, {999,7972}, {1159,5434}, {1317,10072}, {1385,4677}, {1482,3830}, {3244,3656}, {3295,28453}, {3524,3621}, {3543,28224}, {3545,3623}, {3576,15718}, {3617,11539}, {3622,15699}, {3632,15707}, {3633,4880}, {3635,18493}, {3653,4669}, {3679,10246}, {3843,24680}, {3851,5881}, {4678,15709}, {5066,10595}, {5070,25055}, {5071,10283}, {5073,7982}, {5657,15700}, {5690,15693}, {5731,14093}, {6684,15722}, {7983,12355}, {8703,12245}, {9053,11179}, {9884,13188}, {10031,12331}, {10304,20014}, {11531,28202}, {11910,20128}, {12737,34717}, {15178,19875}, {15683,28212}, {15685,28194}, {15688,20050}, {15706,20053}, {15708,20052}, {17504,20054}, {17530,32213}, {17632,31792}, {17800,28198}, {18357,20057}, {18481,34638}, {18515,25439}, {19710,20070}

X(34748) = midpoint of X(376) and X(20049)
X(34748) = reflection of X(i) in X(j) for these (i,j): (2, 1483), (381, 3241), (3534, 944), (3654, 5882), (3656, 3244), (3830, 1482), (4669, 13607), (4677, 1385), (12245, 8703), (12331, 10031), (12355, 7983), (12645, 2), (13188, 9884), (18525, 3656), (20070, 19710), (31145, 549), (34718, 3655)
X(34748) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (145, 18526, 8148), (145, 34667, 34729), (381, 3241, 10247), (551, 5790, 15703), (3655, 34718, 3), (3656, 18525, 14269), (3679, 10246, 15694), (4669, 13607, 3653), (7967, 31145, 549)


X(34749) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: MANDART-INCIRCLE TO 5th MIXTILINEAR

Barycentrics    2*a^4-(b^2-14*b*c+c^2)*a^2-4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(34749) = 2*X(145)+X(7354) = 3*X(3058)-2*X(11114) = 3*X(3241)-X(11114) = 4*X(3244)-X(6284) = 3*X(3244)-X(34649) = 2*X(3555)+X(10944) = X(3893)-4*X(4298) = 3*X(5434)-2*X(11112) = 3*X(5434)-X(34720) = 3*X(6284)-4*X(34649) = 4*X(11112)-3*X(34612) = 3*X(34612)-2*X(34720) = 2*X(34690)+X(34699) = 3*X(34690)+X(34719) = 3*X(34699)-2*X(34719)

The center of the reciprocal HR-ellipse of these triangles is X(11114)

X(34749) lies on these lines: {1,4679}, {2,3304}, {8,4860}, {11,11236}, {30,7982}, {55,34610}, {56,6174}, {65,519}, {145,528}, {388,31140}, {496,31160}, {517,34630}, {518,27492}, {527,3057}, {529,2098}, {535,3244}, {551,21075}, {952,34746}, {1056,3925}, {1222,17297}, {1483,11827}, {1837,31146}, {2094,34711}, {2829,34629}, {3303,11111}, {3333,3679}, {3614,10529}, {3633,18990}, {3656,34697}, {3689,4315}, {3698,12577}, {3813,17577}, {3868,34742}, {3870,34687}, {3880,11246}, {3894,5844}, {4293,6154}, {4299,34707}, {4430,5855}, {4853,6173}, {4995,11194}, {5288,24953}, {5563,17564}, {6762,21677}, {6769,34618}, {7681,10711}, {9369,17264}, {9579,12127}, {10056,31157}, {10199,17757}, {10404,12629}, {10459,17392}, {10707,20060}, {10950,18839}, {11237,34625}, {12349,13190}, {12357,13181}, {12536,17579}, {12649,34700}, {12679,31162}, {15338,20076}, {24954,25055}, {31164,34640}

X(34749) = midpoint of X(145) and X(34605)
X(34749) = reflection of X(i) in X(j) for these (i,j): (3058, 3241), (7354, 34605), (34606, 1), (34612, 5434), (34689, 2), (34697, 3656), (34720, 11112)
X(34749) = pole of the line {3812, 12053} wrt Feuerbach hyperbola
X(34749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56, 34619, 6174), (5434, 34720, 11112), (11112, 34720, 34612), (11194, 11239, 4995), (11236, 11240, 11)


X(34750) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: ORTHIC TO TANGENTIAL

Barycentrics    a^2*((b^2+c^2)*a^8-2*(b^4+6*b^2*c^2+c^4)*a^6+8*(b^2+c^2)*b^2*c^2*a^4+2*(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(34750) = 4*X(20)-X(30443) = 4*X(26)-X(21651) = 3*X(51)-2*X(34751) = 3*X(154)-X(34751) = 4*X(159)-X(1843) = 3*X(373)-4*X(10192) = 2*X(1853)-3*X(5650) = 2*X(5446)-5*X(14530) = X(5562)+2*X(9833) = X(6467)+2*X(9924) = X(11381)+2*X(17845) = X(12324)-4*X(13348)

The center of the reciprocal HR-ellipse of these triangles is X(34751)

X(34750) lies on these l