ENCYCLOPEDIA OF TRIANGLE CENTERS Part18

Encyclopedia of Triangle Centers - ETC

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)	PART 21:	Centers X(40001) - X(42000)
PART 22:	Centers X(42001) - X(44000)	PART 23:	Centers X(44001) - X(46000)	PART 24:	Centers X(46001) - X(48000)
PART 25:	Centers X(48001) - X(50000)	PART 26:	Centers X(50001) - X(52000)	PART 27:	Centers X(52001) - X(54000)
PART 28:	Centers X(54001) - X(56000)	PART 29:	Centers X(56001) - X(58000)	PART 30:	Centers X(58001) - X(60000)
PART 31:	Centers X(60001) - X(62000)	PART 32:	Centers X(62001) - X(64000)	PART 33:	Centers X(64001) - X(66000)
PART 34:	Centers X(66001) - X(68000)	PART 35:	Centers X(68001) - X(70000)	PART 36:	Centers X(70001) - X(72000)

X(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}

Perspectors involving Talitha triangles: X(34016)-X(34088)

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
TT(2,1)	excentral	17277
TT(2,1)	Steiner	100
TT(2,1)	tangential of excentral	75
TT(2,1)	Gemini 2	333
TT(2,1)	Gemini 18	100
TT(2,1)	Gemini 23	86
TT(2,1)	Gemini 90	34016
TT(2,1)	Gemini 92	34017
TT(2,6)	tangential	1078
TT(2,6)	circummedial	1799
TT(2,6)	Ara	5594
TT(2,6)	Gemini 44	1799
TT(2,7)	1st circumperp	927
TT(2,7)	Gemini 7	34018
TT(2,7)	Gemini 30 (inner Conway)	658
TT(2,7)	Gemini 85	34019
TT(2,7)	Soddy	348
TT(2,75)	excentral	34020
TT(2,75)	Gemini 7	274
TT(2,75)	Gemini 17	1978
TT(2,75)	Gemini 19	34021
TT(2,75)	Gemini 90	34022
TT(2,75)	Gemini 92	34023
TT(2,75)	Gemini 104	31008
TT(2,514)	medial	190
TT(2,514)	Gemini 110	34024
TT(6,1)	excentral	572
TT(6,1)	tangential of excentral	48
TT(6,6)	anticomplementary	19121
TT(6,6)	tangential of anticomplementary	206
TT(6,6)	anti-Honsberger	1176
TT(6,6)	Gemini 43	1176
TT(6,7)	incentral	1419
TT(6,7)	excentral	222
TT(6,7)	intouch	222
TT(6,7)	1st circumperp	109
TT(6,7)	2nd circumperp	10571
TT(6,7)	hexyl	1394
TT(6,7)	2nd Conway	18623
TT(6,7)	infinite altitude	221
TT(6,7)	inner Conway	651
TT(6,7)	Gemini 30	651
TT(6,7)	inner tangential mid-arc	34025
TT(6,7)	Yff central	34026

Talitha triangle	2nd triangle	perspector
TT(6,7)	1st Sharygin	34027
TT(6,7)	Honsberger	34028
TT(6,7)	3rd Euler	34029
TT(6,7)	4th Euler	34030
TT(6,7)	2nd Pamfilos-Zhou	34031
TT(6,7)	2nd extouch	34032
TT(6,7)	6th mixtilinear	34033
TT(6,7)	outer tangential mid-arc	34034
TT(6,7)	1st Conway	34035
TT(6,7)	incircle-inverse of ABC	34036
TT(6,7)	outer Hutson	34038
TT(6,7)	T(-1,3)	34039
TT(6,7)	Hutson intouch	34040
TT(6,7)	Atik	34041
TT(6,7)	Ascella	34042
TT(6,7)	reflection of X(1) in BC,CA,AB	34043
TT(6,7)	3rd Conway	34044
TT(6,7)	/Conway-circle-inverse of ABC	34045
TT(6,7)	incircle-circles	34046
TT(6,7)	1st excosine	34047
TT(6,7)	2nd Zaniah	34048
TT(6,7)	Ursa-Major	34049
TT(6,7)	Wasat	34050
TT(6,7)	Gemini 7	34051
TT(6,7)	Gemini 15	34052
TT(6,75)	excentral	32911
TT(6,7)	Steiner	190
TT(6,7)	Wasat	24624
TT(6,7)	Gemini 2	81
TT(6,7)	Gemini 7	81
TT(6,7)	Gemini 17	190
TT(6,7)	Gemini 54	86
TT(6,7)	Gemini 68	27644
TT(6,7)	Gemini 2	81
TT(6,7)	Gemini 2	81
TT(1,1)	anticomplementary	2975
TT(1,1)	tangential	1621
TT(1,1)	Feuerbach	2
TT(1,1)	2nd circumperp	21
TT(1,1)	1st Sharygin	21
TT(1,1)	Steiner	100
TT(1,1)	1st Conway	21
TT(1,1)	2nd Conway	3
TT(1,1)	Garcia reflection	21

Talitha triangle	2nd triangle	perspector
TT(1,1)	Gemini 1	21
TT(1,1)	Gemini 8	21
TT(1,1)	Gemini 18	100
TT(1,1)	Gemini 19	34053
TT(1,1)	Gemini 19	34053
TT(1,6)	cevian of X(1580)	34054
TT(1,6)	cevian of X(3112)	34055
TT(1,7)	medial	3160
TT(1,6)	anticomplementary	347
TT(1,6)	1st circumperp	934
TT(1,6)	inner Conway	651
TT(1,6)	Gemini 1	77
TT(1,6)	Gemini 2	17080
TT(1,6)	Gemini 7	340560
TT(1,6)	Gemini 13	31600
TT(1,6)	Gemini 30	651
TT(1,6)	Gemini 33	31526
TT(1,6)	Gemini 34	34057
TT(1,6)	Gemini 60	34058
TT(1,6)	Gemini 61	34059
TT(1,6)	Gemini 65	34060
TT(1,6)	Gemini 75	34061
TT(1,6)	Gemini 75	34061
TT(1,6)	Gemini 100	34062
TT(1,6)	anticevian of X(31)	18042
TT(1,6)	anticevian of X(88)	9456
TT(1,6)	anticevian of X(100)	692
TT(1,6)	anticevian of X(162)	32676
TT(1,6)	anticevian of X(190)	101
TT(1,6)	anticevian of X(651)	1415
TT(1,6)	anticevian of X(653)	32674
TT(1,6)	anticevian of X(655)	32675
TT(1,6)	anticevian of X(658)	1461
TT(1,6)	anticevian of X(662)	163
TT(1,6)	anticevian of X(673)	1438
TT(1,6)	anticevian of X(799)	662
TT(1,6)	anticevian of X(823)	24019
TT(1,6)	anticevian of X(897)	923
TT(1,6)	anticevian of X(1821)	1910
TT(1,6)	anticevian of X(2349)	2159
TT(1,6)	anticevian of X(2580)	2576
TT(1,6)	anticevian of X(2581)	2577
TT(1,6)	anticevian of X(3257)	32665
TT(1,6)	anticevian of X(32680)	32678

Talitha triangle	2nd triangle	perspector
TT(1,6)	anticevian of X(33760)	1
TT(1,6)	anticevian of X(75)	34065
TT(1,6)	anticevian of X(82)	34066
TT(1,6)	anticevian of X(660)	34067
TT(1,6)	anticevian of X(1156)	34068
TT(1,6)	anticevian of X(1492)	34069
TT(1,6)	anticevian of X(2236)	34070
TT(1,6)	anticevian of X(4598)	34071
TT(1,6)	anticevian of X(4599)	34072
TT(1,6)	anticevian of X(4604)	34073
TT(1,6)	anticevian of X(4606)	34074
TT(1,6)	anticevian of X(4607)	34075
TT(1,6)	anticevian of X(8052)	34076
TT(1,6)	anticevian of X(20332)	34077
TT(1,6)	anticevian of X(20332)	34077
TT(1,6)	anticevian of X(23707)	34078
TT(1,6)	anticevian of X(24624)	34079
TT(1,6)	anticevian of X(27834)	34080
TT(1,6)	anticevian of X(33295)	34081
TT(1,6)	anticevian of X(33764)	34082
TT(1,75)	Soddy	3160
TT(1,75)	tangential	34063
TT(1,75)	Steiner	190
TT(1,75)	Wasat	671
TT(1,75)	Gemini 1	86
TT(1,75)	Gemini 17	190
TT(1,75)	Gemini 24	33296
TT(1,75)	Gemini 105	34064
TT(1,514)	incentral	100
TT(75,1)	anticomplementary	17143
TT(75,1)	2nd Conway	76
TT(75,1)	/Conway-circle-inverse of ABC	314
TT(75,1)	Garcia reflection	314
TT(75,1)	Gemini 8	314
TT(75,1)	Gemini 18	668
TT(75,6)	excentral	33764
TT(75,7)	1st circumperp	36083
TT(75,7)	Gemini 7	36084
TT(75,7)	Gemini 30 (inner Conway)	36085
TT(75,75)	anticomplementary	36086
TT(75,75)	Wasat	36087
TT(75,75)	Gemini 17	1978
TT(75,75)	Gemini 105	36088
TT(6,7)	Hutson intouch	34040

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.

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) : :

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 L_A, L_B, L_C be lines through A', B', C', respectively, parallel to the Euler line. Let L'_A be the reflection of L_A 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)

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

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.

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 A_B, A_C be the orthogonal projections of A' on CA, AB, resp. Let A" = C_AA_C∩A_BB_A, 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 L_a be the radical axis of circles with segments BC and A'A" as diameters, and define L_b and L_c cyclically. The triangle formed by lines L_a, L_b, L_c 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}

Frégier points: X(34341)-X(34344)

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)

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) lies on these lines: {2, 30228}, {6, 522}, {9, 24411}, {37, 5662}, {190, 6184}, {527, 599}, {666, 17350}, {1438, 24840}, {2786, 24279}, {3923, 24274}, {17435, 24410}

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)

Vertex conjugates: X(34426)-X(34449)

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 ,

which passes through X(i) for these i: 99, 1576, 1634, 3455, 9468, 34067. This is the circumconic with perspector X(3051) and center X(34452).

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)

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(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)

Points on the Apollonius circle: X(34453)-X(34466)

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

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}

Bevan-antipodal triangle and related centers: X(34488)-X(34500)

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 V_a, V_b, V_c the antipodes of V on the circumcircles of VBC, VCA and VAB, respectively. The triangle V_aV_bV_c 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)

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(6212) and A", B", C" be as at X(6213). The lines A'A", B'B", C'C" concur in X(34494). (Randy Hutson, November 17, 2019)

Let (O_A), (O_B), (O_C) be the companion incircles used in Elkies' construction of X(176). Let L_A be the external tangent, other than BC, to (O_B) and (O_C); define L_B and L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. 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}

Points of the Moses-Steiner osculatory triangle: X(34505)-X(34511)

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))

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 N_A be the reflection of X(5) in the perpendicular bisector of BC, and define N_B, N_C cyclically. N_AN_BN_C is also the X(140)-anti-altimedial triangle, and X(34507) = X(6)-of N_AN_BN_C. (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}

Vu points: X(34520)-X(34544)

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 T_A and T_A' be the tangents to (PBC) at P and A', respectively.
Let A₁ = T_A∩BC and A₂ = T_A'∩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)

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}

Bankoff equilateral triangles: X(34551)-X(34562)

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 A_b be the rotation of A toward B by an angle |π/6| and let A_c be the rotation of A toward C by the same angle |π/6| (triangle OA_bA_c is an equilateral triangle). Build (B_c, B_a) and (C_a, C_b) cyclically and let A_m, B_m, C_m be the midpoints of B_cC_b, C_aA_c, A_bB_a, respectively. Then, by the precedent theorem, A_mB_mC_m is equilateral.

The triangle A_mB_mC_m 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).

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)

Centers related to the Moses-Jerabek and Moses-Lemoine conics: X(34565)-X(34574)

The Moses-Jerabek and Moses-Lemoine conics are introduced in the preamble before X(34426).

Contributed by Randy Hutson, October 31, 2019.

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)

Cyclologic centers: X(34578)-X(34581)

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)

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).

Let DEF be the intouch triangle and I_aI_bI_c the excentral-triangle. Let D' be the trisector nearest D of segment DI_a. Let L_a be the radical axis of (DEF) and (BCD'), and define L_b and L_c cyclically. The triangle A'B'C' formed by the lines L_a, L_b, L_c is perspective to ABC, and the perspector is X(34578). (Angel Montesdeoca, December 31, 2020)
In general, if D' is a point such that DD' / D'I_a = t (a constant real number) then the perspector of ABC and A'B'C', given by barycentrics

a/(a^3-a^2 (b+c) (1+2 t)-a (b^2 (1-4 t)+c^2 (1-4 t)-2 b c (1-2 t+2 t^2))-(b-c)^2 (b+c) (-1+2 t)) : : ,

lies on the circumhyperbola that is the isogonal conjugate of line X(1)X(6). See also X(39980). (Angel Montesdeoca, December 31, 2020)

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}

Points related to the circumellipse of the medial and incentral triangles: X(34585)-X(34593)

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

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)

HR-ellipses: X(34603)-X(34752)

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)

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 L_A be the polar of X(3) wrt the A-power circle, and define L_B, L_C cyclically. Let A' = L_B∩L_C, 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 B_A, C_A be the anticomplementary circle intercepts of line BC. Let B'_A, C'_A be the {B_A,C_A}-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 lines: {6,25}, {20,2979}, {26,21651}, {155,32063}, {160,26907}, {373,10192}, {511,11206}, {1352,3819}, {1503,3917}, {1853,5650}, {1885,16621}, {1906,3574}, {1993,6759}, {2781,24981}, {3292,15581}, {4319,11189}, {4320,32065}, {5012,10282}, {5446,14530}, {5891,11750}, {6353,12283}, {6391,16199}, {7494,14913}, {7499,15585}, {9909,34382}, {10565,12272}, {11202,17928}, {11381,17845}, {11574,14826}, {12294,31383}, {12324,13348}, {15030,18400}, {15577,22352}, {23208,32078}, {23332,30739}, {26881,27365}

X(34750) = reflection of X(i) in X(j) for these (i,j): (51, 154), (32064, 3819)
X(34750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (161, 1660, 1495), (2203, 9969, 34417), (13366, 15004, 34566)

X(34751) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: TANGENTIAL TO ORTHIC

Barycentrics a^2*((b^2+c^2)*a^8-(2*b^4+3*b^2*c^2+2*c^4)*a^6+2*(b^2+c^2)*b^2*c^2*a^4+(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3) : :
X(34751) = 3*X(51)-X(34750) = 4*X(52)-X(6293) = 4*X(143)-X(9833) = 3*X(154)-2*X(34750) = 4*X(389)-X(17845) = X(1498)-4*X(5446) = 5*X(3522)-8*X(32184) = 8*X(5462)-5*X(17821) = 3*X(5640)-2*X(10192) = X(5895)-4*X(13598) = 4*X(6153)-X(17846) = X(6243)+2*X(18381) = 7*X(9781)-4*X(16252) = X(9924)-4*X(9969) = 4*X(12235)-X(17834)

The center of the reciprocal HR-ellipse of these triangles is X(34750).

Let A'B'C' be the reflection triangle. Let A_B and A_C be the orthogonal projections of A' on lines CA and AB, resp. Define B_C, B_A, C_A, C_B cyclically. Let A" = B_CB_A∩C_AC_B, and define B" and C" cyclically. Triangle A"B"C" is perspective to ABC at X(6), and X(34751) = X(2)-of-A"B"C". (Randy Hutson, March 29, 2020)

X(34751) lies on these lines: {6,25}, {30,7729}, {50,2351}, {52,382}, {68,1154}, {143,9833}, {343,858}, {389,17845}, {394,27365}, {511,1853}, {566,23195}, {568,11225}, {569,2917}, {973,11431}, {1112,31383}, {1498,5446}, {1503,3060}, {1993,15139}, {2781,3448}, {3357,32608}, {3522,32184}, {3819,9967}, {5422,15577}, {5462,17821}, {5640,10192}, {5890,6146}, {6152,18912}, {6153,17846}, {6243,18381}, {6353,11746}, {6403,13567}, {6746,19467}, {6759,15087}, {7494,17710}, {7689,10606}, {8538,9306}, {8549,33586}, {9629,11189}, {9781,16252}, {10117,11800}, {10250,19131}, {10263,14216}, {11002,11206}, {11204,18859}, {11262,32354}, {11442,18382}, {11459,23324}, {11750,18565}, {11819,13292}, {13340,23329}, {13423,26917}, {13754,18405}, {14984,15131}, {15019,15582}, {15073,23292}, {18376,18403}, {18383,18436}, {18438,21243}, {21969,34146}, {23039,23325}

X(34751) = reflection of X(i) in X(j) for these (i,j): (154, 51), (2979, 23332), (11459, 23324), (13340, 23329), (18435, 18376), (23039, 23325)
X(34751) = {X(51), X(20987)}-harmonic conjugate of X(1474)

X(34752) = CENTER OF THE HR-ELLIPSE OF THESE TRIANGLES: SCHROETER TO STEINER

Barycentrics (7*a^4-7*(b^2+c^2)*a^2+5*b^4-3*b^2*c^2+5*c^4)*(b^2-c^2) : :
X(34752) = 13*X(2)-15*X(1649) = 19*X(2)-15*X(5466) = 7*X(2)-5*X(8029) = 17*X(2)-15*X(8371) = 11*X(2)-15*X(9168) = 11*X(2)-10*X(10189) = 4*X(2)-5*X(10190) = 6*X(2)-5*X(10278) = 3*X(2)-5*X(11123) = 19*X(1649)-13*X(5466) = 21*X(1649)-13*X(8029) = 17*X(1649)-13*X(8371) = 11*X(1649)-13*X(9168) = 12*X(1649)-13*X(10190) = 18*X(1649)-13*X(10278) = 9*X(1649)-13*X(11123) = 21*X(5466)-19*X(8029) = 17*X(5466)-19*X(8371) = 11*X(5466)-19*X(9168) = 12*X(5466)-19*X(10190)

X(34752) lies on these lines: {2,523}, {546,8151}, {14869,32204}, {15700,16220}

X(34752) = reflection of X(10278) in X(11123)
X(34752) = {X(10278), X(11123)}-harmonic conjugate of X(10190)

X(34753) = X(1)X(549)∩X(5)X(57)

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

See Antreas Hatzipolakis and Angel Montesdeoca, Euclid 82 .

X(34753) lies on these lines: {1,549}, {2,3927}, {3,938}, {4,13226}, {5,57}, {7,1656}, {8,16417}, {11,3336}, {12,3337}, {30,1210}, {46,496}, {56,952}, {63,17527}, {65,5901}, {78,17564}, {79,7173}, {88,24883}, {140,942}, {165,10386}, {226,3628}, {355,3361}, {381,5704}, {442,27003}, {495,3338}, {498,4860}, {499,5221}, {546,4292}, {547,553}, {548,950}, {550,5722}, {595,3756}, {631,15934}, {632,11374}, {758,6691}, {999,1788}, {1125,4757}, {1155,15171}, {1158,7956}, {1159,3616}, {1387,5903}, {1420,1483}, {1445,5762}, {1466,6914}, {1479,28178}, {1482,6961}, {1532,26877}, {1617,32141}, {1737,18357}, {1836,10593}, {1837,28186}, {1876,21841}, {2093,11373}, {2095,5763}, {3035,3874}, {3086,22791}, {3090,21454}, {3091,18541}, {3218,4187}, {3219,17575}, {3306,8728}, {3333,26446}, {3339,5886}, {3340,10283}, {3419,17563}, {3474,9669}, {3487,3526}, {3525,11036}, {3530,24929}, {3579,11019}, {3586,15704}, {3600,5790}, {3601,15712}, {3627,9581}, {3671,11230}, {3825,17768}, {3845,9579}, {3868,13747}, {3940,17567}, {4188,9945}, {4193,23958}, {4205,24627}, {4298,9956}, {4299,28190}, {4304,33923}, {4308,12645}, {4311,28224}, {4654,15699}, {4848,5844}, {4999,5883}, {5030,21049}, {5044, 6692}, {5045,6684}, {5054,5703}, {5055,5714}, {5070,5226}, {5131,15338}, {5265,10246}, {5273,16853}, {5427,14792}, {5432,18398}, {5433,5902}, {5434,18395}, {5437,5791}, {5439,6675}, {5445,15888}, {5482,12109}, {5657,7373}, {5709,12875}, {5744,11108}, {5770,6918}, {5779,6964}, {5789,6864}, {5806,6705}, {6583,12432}, {6738,13624}, {6825,8732}, {6978,12848}, {6979,13257}, {7354,12019}, {7504,26842}, {7741,11246}, {9709,24477}, {9843,31445}, {10122,11277}, {10360,19347}, {10404,10592}, {10980,31423}, {11231,21620}, {11518,14869}, {11544,17605}, {12053,28212}, {12512,31795}, {12649,16371}, {12702,14986}, {14450,31272}, {15693,15933}, {15932,26470}, {17061,24167}, {18253,19862}, {18527,31730}, {18583,24471}, {19925,31776}, {20060,34122}, {22753,33899}, {23070,32911}, {24473,27385}, {26725,31260}, {28581,30315}, {31792,33575}

X(34753) = midpoint of X(i) and X(j) for these {i, j}: {46,496}, {4848,24928}

X(34754) = ISOGONAL CONJUGATE OF X(33607)

Barycentrics a^2 (9 (-a^2 + b^2 + c^2) + 10 Sqrt[3] S) : :

See Kadir Altintas and Peter Moses, Euclid 90 .

X(34754) lies on these lines: {2,33606}, {3,6}, {4,16960}, {13,3543}, {14,547}, {17,3850}, {18,3533}, {37,11791}, {140,16961}, {193,5463}, {203,10638}, {302,22496}, {395,11812}, {396,3845}, {398,16967}, {533,34540}, {2502,14704}, {2903,10658}, {3171,16021}, {3412,3853}, {3545,10654}, {3589,5464}, {3832,16964}, {5056,5334}, {5059,5335}, {5067,18581}, {5362,16858}, {7005,7051}, {7848,11297}, {7908,11301}, {8740,13596}, {10632,34484}, {11489,15702}, {11539,16241}, {11543,16239}, {12101,33607}, {14483,32585}, {15708,16242}, {16529,22512}, {19106,33703}, {22510,23013}

X(34754) = isogonal conjugate of X(33607)
X(34754) = isogonal conjugate of the complement of X(33608)
X(34754) = Schoutte circle inverse of X(5238)
X(34754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 34755}, {6, 15, 10645}, {6, 10645, 16}, {15, 16, 5238}, {15, 61, 16}, {15, 62, 11480}, {15, 11485, 61}, {61, 10645, 6}, {3389, 3390, 5351}, {3412, 19107, 11542}, {5351, 11486, 16}, {10654, 11488, 16809}, {11485, 22236, 15}, {11543, 16772, 33417}, {16809, 16962, 11488}

X(34755) = ISOGONAL CONJUGATE OF X(33606)

Barycentrics a^2 (9 (-a^2 + b^2 + c^2) - 10 Sqrt[3] S) : :

See Kadir Altintas and Peter Moses, Euclid 90 .

X(34755) lies on these lines: {2,33607}, {3,6}, {4,16961}, {13,547}, {14,3543}, {17,3533}, {18,3850}, {37,11790}, {140,16960}, {193,5464}, {202,1250}, {303,22495}, {395,3845}, {396,11812}, {397,16966}, {532,34541}, {2502,14705}, {2902,10657}, {3170,16022}, {3411,3853}, {3545,10653}, {3589,5463}, {3832,16965}, {5056,5335}, {5059,5334}, {5067,18582}, {5367,16858}, {7006,19373}, {7848,11298}, {7908,11302}, {8739,13596}, {10633,34484}, {11488,15702}, {11539,16242}, {11542,16239}, {12101,33606}, {14483,32586}, {15708,16241}, {16530,22513}, {19107,33703}, {22511,23006}

X(34755) = isogonal conjugate of X(33606)
X(34755) = isogonal conjugate of the complement of X(33609)
X(34755) = Schoutte circle inverse of X(5237)
X(34755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 34754}, {6, 16, 10646}, {6, 10646, 15}, {15, 16, 5237}, {16, 61, 11481}, {16, 62, 15}, {16, 11486, 62}, {62, 10646, 6}, {3364, 3365, 5352}, {3411, 19106, 11543}, {5352, 11485, 15}, {10653, 11489, 16808}, {11486, 22238, 16}, {11542, 16773, 33416}, {16808, 16963, 11489}

X(34756) = X(3)X(1299)∩X(4)X(155)

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

X(34756) lies on the cubics K028 and K1140 and on these lines: {3, 1299}, {4, 155}, {24, 12095}, {54, 14518}, {1594, 32132}, {16266, 20422}

X(34756) = isogonal conjugate of X(34853)
X(34756) = isogonal conjugate of the complement of X(254)
X(34756) = X(i)-cross conjugate of X(j) for these (i,j): {6, 1993}, {924, 13398}, {1147, 24}
X(34756) = X(i)-isoconjugate of X(j) for these (i,j): {68, 920}, {91, 155}, {1820, 6515}, {2351, 33808}
X(34756) = cevapoint of X(924) and X(34338)
X(34756) = polar conjugate of X(39116)
X(34756) = barycentric product X(i)*X(j) for these {i,j}: {24, 6504}, {254, 1993}, {921, 1748}, {11547, 15316}
X(34756) = barycentric quotient X(i)/X(j) for these {i,j}: {24, 6515}, {254, 5392}, {571, 155}, {1147, 6503}, {1748, 33808}, {6504, 20563}, {8745, 3542}
X(34756) = {X(4),X(254)}-harmonic conjugate of X(16172)

X(34757) = X(4)X(52)∩X(925)X(1147)

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

X(34757) lies on these lines: {4, 52}, {925, 1147}

X(34757) = reflection of X(68) in X(847)

X(34758) = X(3)X(8)∩X(21)X(3816)

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

X(34758) lies on these lines: {3, 8}, {21, 3816}, {36, 3878}, {56, 5330}, {191, 997}, {404, 1329}, {758, 14800}, {993, 5445}, {1466, 17097}, {1470, 3616}, {1737, 5267}, {2077, 4861}, {2320, 11507}, {3058, 17549}, {3086, 4189}, {3612, 31660}, {3869, 32612}, {3916, 23961}, {4193, 13273}, {4511, 5884}, {4973, 14804}, {5046, 10090}, {5253, 19525}, {5440, 26201}, {5704, 20846}, {6909, 11826}, {6940, 27529}, {10529, 17548}, {10916, 14794}, {11571, 30144}, {13587, 34606}

X(34758) = {X(3),X(8)}-harmonic conjugate of X(17100)

X(34759) = (name pending)

Barycentrics (b+c)*a^15-(b^2+4*b*c+c^2)*a^14-(b+c)*(5*b^2-2*b*c+5*c^2)*a^13+4*(b^4+c^4+3*(b^2+c^2)*b*c)*a^12+(b+c)*(11*b^4+11*c^4-3*(2*b^2-9*b*c+2*c^2)*b*c)*a^11-(5*b^6+5*c^6+(7*b^4+7*c^4-2*(4*b^2-3*b*c+4*c^2)*b*c)*b*c)*a^10-(b+c)*(15*b^6+15*c^6-(8*b^4+8*c^4-(37*b^2-18*b*c+37*c^2)*b*c)*b*c)*a^9-(13*b^6+13*c^6+(12*b^4+12*c^4+(19*b^2+52*b*c+19*c^2)*b*c)*b*c)*b*c*a^8+(b+c)*(15*b^8+15*c^8-2*(6*b^6+6*c^6+(2*b^4+2*c^4+(8*b^2-5*b*c+8*c^2)*b*c)*b*c)*b*c)*a^7+(b^2-c^2)^2*(5*b^6+5*c^6+(22*b^4+22*c^4+(11*b^2+30*b*c+11*c^2)*b*c)*b*c)*a^6-(b^2-c^2)^2*(b+c)*(11*b^6+11*c^6-(18*b^4+18*c^4+(17*b^2+18*b*c+17*c^2)*b*c)*b*c)*a^5-2*(b^2-c^2)^2*(2*b^8+2*c^8+(7*b^6+7*c^6-(b^2+11*b*c+c^2)*(b+c)^2*b*c)*b*c)*a^4+(b^2-c^2)^4*(b+c)*(5*b^4+5*c^4-(14*b^2+3*b*c+14*c^2)*b*c)*a^3+(b^2-c^2)^4*(b^6+c^6+(5*b^4+5*c^4-4*(b^2+3*b*c+c^2)*b*c)*b*c)*a^2-(b^2-c^2)^5*(b-c)*(b^4+c^4-2*(b+c)^2*b*c)*a-(b^2-c^2)^6*(b-c)^2*b*c : :
Trilinears 64*q*p^7*(p^2-2)+4*(16*q^4-8*q^2+29)*q*p^5-2*(16*q^4-22*q^2+41)*q*p^3-6*(4*q^4+2*q^2-5)*q*p-32*(2*q^2+1)*p^8-64*(q^4-2*q^2-1)*p^6+2*(56*q^4-50*q^2-13)*p^4-(16*q^6+36*q^4-32*q^2+15)*p^2-(q^2-1)*(8*q^2+9) : : where p=sin(A/2), q=cos((B-C)/2)

See Antreas Hatzipolakis and César Lozada, Euclid 109 .

X(34759) lies on this line: {5441, 5842}

X(34760) = X(2)X(99)∩X(691)X(9080)

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

X(34760) lies on the cubics K015 and K052 and these lines: {2, 99}, {691, 9080}, {892, 5466}, {1641, 17948}, {2396, 9187}, {8371, 9182}, {9168, 9170}, {17932, 17937}

X(34760) = isotomic conjugate of X(34763)
X(34760) = isotomic conjugate of the isogonal conjugate of X(23348)
X(34760) = X(i)-cross conjugate of X(j) for these (i,j): {18007, 17948}, {33921, 543}
X(34760) = X(i)-isoconjugate of X(j) for these (i,j): {843, 2642}, {922, 9180}
X(34760) = cevapoint of X(i) and X(j) for these (i,j): {543, 33921}, {1641, 8371}, {17948, 18007}
X(34760) = trilinear pole of line {543, 9182}
X(34760) = barycentric product X(i)*X(j) for these {i,j}: {76, 23348}, {99, 17948}, {543, 892}, {670, 17964}, {671, 9182}, {799, 17955}, {4590, 18007}, {9181, 18023}, {17993, 34537}
X(34760) = barycentric quotient X(i)/X(j) for these {i,j}: {543, 690}, {671, 9180}, {691, 843}, {892, 18823}, {1641, 1649}, {2502, 351}, {8371, 1648}, {9171, 21906}, {9181, 187}, {9182, 524}, {17948, 523}, {17955, 661}, {17964, 512}, {17993, 3124}, {18007, 115}, {23348, 6}, {33921, 23992}
X(34760) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5466, 5468, 892}

X(34761) = X(2)X(98)∩X(476)X(2395)

Barycentrics (a^2 - b^2)*(a^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 - 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(34761) lies on the cubic K015 and these lines: {2, 98}, {476, 2395}, {685, 4240}, {878, 15329}, {879, 2966}, {1640, 7473}, {5468, 17932}, {6037, 32694}, {16092, 34369}

X(34761) = isogonal conjugate of X(23350)
X(34761) = isotomic conjugate of X(34765)
X(34761) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23350}, {1755, 14223}, {1959, 14998}, {32679, 34370}
X(34761) = trilinear pole of line {542, 23967}
X(34761) = barycentric product X(i)*X(j) for these {i,j}: {98, 14999}, {99, 34369}, {287, 7473}, {542, 2966}, {6103, 17932}
X(34761) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 23350}, {98, 14223}, {542, 2799}, {1640, 868}, {1976, 14998}, {2715, 842}, {2966, 5641}, {5191, 3569}, {6103, 16230}, {7473, 297}, {14560, 34370}, {14999, 325}, {23968, 14356}, {34369, 523}
X(34761) = {X(98),X(287)}-harmonic conjugate of X(9140)

X(34762) = X(2)X(45)∩X(901)X(9089)

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

X(34762) lies on the cubic K015 and these lines: {2, 45}, {901, 9089}, {4555, 4618}, {4615, 17934}, {6633, 14475}

X(34762) = X(33920)-cross conjugate of X(545)
X(34762) = X(1635)-isoconjugate of X(2384)
X(34762) = cevapoint of X(i) and X(j) for these (i,j): {545, 33920}, {1644, 14475}
X(34762) = trilinear pole of line {545, 6633}
X(34762) = barycentric product X(i)*X(j) for these {i,j}: {545, 4555}, {903, 6633}
X(34762) = barycentric quotient X(i)/X(j) for these {i,j}: {545, 900}, {901, 2384}, {1644, 6544}, {6633, 519}, {8649, 1960}, {14421, 2087}, {14475, 1647}, {27921, 4448}
X(34762) = {X(6548),X(17780)}-harmonic conjugate of X(4555)

X(34763) = X(2)X(690)∩X(523)X(1648)

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

X(34763) lies on the cubics K015 and K979 and these lines: {2, 690}, {523, 1648}, {524, 1649}, {843, 2770}, {1499, 14694}, {1641, 9164}, {4590, 5468}, {5466, 33919}, {5967, 9125}, {9168, 18823}, {16230, 18012}

X(34763) = isogonal conjugate of X(23348)
X(34763) = isotomic conjugate of X(34760)
X(34763) = X(33921)-cross conjugate of X(690)
X(34763) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23348}, {110, 17955}, {163, 17948}, {662, 17964}, {897, 9181}, {923, 9182}, {1101, 18007}, {17993, 24041}
X(34763) = cevapoint of X(i) and X(j) for these (i,j): {523, 9183}, {690, 33921}
X(34763) = crosssum of X(17964) and X(17993)
X(34763) = trilinear pole of line {690, 23992}
X(34763) = crossdifference of every pair of points on line {2502, 9181}
X(34763) = barycentric product X(i)*X(j) for these {i,j}: {524, 9180}, {690, 18823}, {1648, 9170}
X(34763) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 23348}, {115, 18007}, {187, 9181}, {351, 2502}, {512, 17964}, {523, 17948}, {524, 9182}, {661, 17955}, {690, 543}, {843, 691}, {1648, 8371}, {1649, 1641}, {3124, 17993}, {9180, 671}, {18823, 892}, {21906, 9171}, {23992, 33921}

X(34764) = X(2)X(900)∩X(514)X(1647)

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

X(34764) lies on the cubic K015 and these lines: {2, 900}, {514, 1647}, {519, 6544}, {1016, 17780}, {2384, 2726}, {3766, 20569}, {6548, 6550}, {6630, 20042}, {30580, 31992}

X(34764) = X(33920)-cross conjugate of X(900)
X(34764) = X(i)-isoconjugate of X(j) for these (i,j): {545, 32665}, {3257, 8649}, {6633, 9456}, {9268, 14421}
X(34764) = cevapoint of X(900) and X(33920)
X(34764) = barycentric quotient X(i)/X(j) for these {i,j}: {519, 6633}, {900, 545}, {1647, 14475}, {1960, 8649}, {2087, 14421}, {2384, 901}, {4448, 27921}, {6544, 1644}

X(34765) = X(2)X(1637)∩X(325)X(23350)

Barycentrics (b^2 - 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(34765) lies on the cubic K015 and these lines: {2, 1637}, {325, 23350}, {511, 9141}, {842, 2857}, {877, 4240}

X(34765) = isotomic conjugate of X(34761)
X(34765) = isotomic conjugate of the isogonal conjugate of X(23350)
X(34765) = X(i)-isoconjugate of X(j) for these (i,j): {163, 34369}, {2247, 2715}
X(34765) = barycentric product X(i)*X(j) for these {i,j}: {76, 23350}, {325, 14223}, {868, 6035}, {2799, 5641}
X(34765) = barycentric quotient X(i)/X(j) for these {i,j}: {297, 7473}, {325, 14999}, {523, 34369}, {842, 2715}, {868, 1640}, {2799, 542}, {3569, 5191}, {5641, 2966}, {14223, 98}, {14356, 23968}, {14998, 1976}, {16230, 6103}, {23350, 6}, {34370, 14560}

X(34766) = X(2)X(846)∩X(3570)X(18014)

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

X(34766) lies on the cubic K015 and these lines: {2, 846}, {3570, 18014}, {4615, 5468}, {5466, 17780}

X(34766) = X(2712)-isoconjugate of X(9508)
X(34766) = barycentric quotient X(i)/X(j) for these {i,j}: {2702, 2712}, {2796, 2786}, {5168, 5029}

X(34767) = X(2)X(525)∩X(69)X(3265)

Barycentrics (b^2 - c^2)*(-a^2 + b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 - 2*c^4)*(-a^4 - a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 - c^4) : :
X(34767) = X[2] + 2 X[23616], X[14401] + 3 X[23616]

X(34767 lies on the cubic K015 and these lines: {2, 525}, {69, 3265}, {74, 2373}, {95, 5888}, {253, 6563}, {264, 850}, {287, 14417}, {520, 7998}, {1304, 2867}, {1494, 3268}, {4240, 16077}, {5466, 34765}, {5468, 17932}, {6330, 9979}, {6333, 30786}, {16261, 30209}, {17708, 34211}

X(34767) = isogonal conjugate of X(23347)
X(34767) = isotomic conjugate of X(4240)
X(34767) = anticomplement of X(14401)
X(34767) = anticomplement of the isogonal conjugate of X(34568)
X(34767) = isotomic conjugate of the anticomplement of X(1650)
X(34767) = isotomic conjugate of the isogonal conjugate of X(14380)
X(34767) = isotomic conjugate of the polar conjugate of X(2394)
X(34767) = X(34568)-anticomplementary conjugate of X(8)
X(34767) = X(i)-Ceva conjugate of X(j) for these (i,j): {16077, 1494}, {31621, 15526}
X(34767) = X(i)-cross conjugate of X(j) for these (i,j): {1650, 2}, {6334, 3267}, {9033, 525}, {14380, 2394}, {15526, 31621}
X(34767) = X(i)-isoconjugate of X(j) for these (i,j): {1, 23347}, {19, 2420}, {30, 32676}, {31, 4240}, {32, 24001}, {112, 2173}, {162, 1495}, {163, 1990}, {648, 9406}, {662, 14581}, {811, 9407}, {1099, 32715}, {1576, 1784}, {1973, 2407}, {2631, 23964}, {3284, 24019}, {9409, 24000}
X(34767) = cevapoint of X(i) and X(j) for these (i,j): {30, 23583}, {520, 8552}, {525, 9033}, {1650, 23616}
X(34767) = crosspoint of X(1494) and X(16077)
X(34767) = crosssum of X(i) and X(j) for these (i,j): {1495, 9409}, {3284, 14396}, {9407, 14398}
X(34767) = trilinear pole of line {525, 15526}
X(34767) = crossdifference of every pair of points on line {1495, 9408}
X(34767) = barycentric product X(i)*X(j) for these {i,j}: {69, 2394}, {74, 3267}, {76, 14380}, {305, 2433}, {525, 1494}, {656, 33805}, {850, 14919}, {2349, 14208}, {3265, 16080}, {3926, 18808}, {4563, 12079}, {9033, 31621}, {10152, 14638}, {15526, 16077}
X(34767) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4240}, {3, 2420}, {6, 23347}, {69, 2407}, {74, 112}, {75, 24001}, {122, 14345}, {125, 1637}, {512, 14581}, {520, 3284}, {523, 1990}, {525, 30}, {647, 1495}, {656, 2173}, {810, 9406}, {1304, 23964}, {1494, 648}, {1577, 1784}, {1637, 16240}, {1650, 14401}, {2159, 32676}, {2349, 162}, {2394, 4}, {2433, 25}, {2632, 2631}, {2972, 1636}, {3049, 9407}, {3265, 11064}, {3267, 3260}, {3268, 14920}, {3269, 9409}, {4025, 18653}, {4466, 11125}, {6334, 113}, {8552, 1511}, {8749, 32713}, {9033, 3163}, {9409, 9408}, {10419, 32708}, {11064, 3233}, {11079, 14560}, {12079, 2501}, {14208, 14206}, {14380, 6}, {14385, 14591}, {14401, 3081}, {14417, 5642}, {14582, 14583}, {14592, 14254}, {14919, 110}, {14977, 9214}, {15421, 15454}, {15459, 32230}, {15526, 9033}, {16077, 23582}, {16080, 107}, {17094, 6357}, {18210, 14399}, {18808, 393}, {18877, 1576}, {20975, 14398}, {23616, 1650}, {23870, 6110}, {23871, 6111}, {31621, 16077}, {32112, 232}, {33805, 811}

X(34768) = CENTER OF ANNA'S CIRCLE

Barycentrics (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^12 - 8*a^10*b^2 + 22*a^8*b^4 - 28*a^6*b^6 + 17*a^4*b^8 - 4*a^2*b^10 - 8*a^10*c^2 + 26*a^8*b^2*c^2 - 20*a^6*b^4*c^2 - 9*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - 3*b^10*c^2 + 22*a^8*c^4 - 20*a^6*b^2*c^4 - 7*a^4*b^4*c^4 - 10*a^2*b^6*c^4 + 12*b^8*c^4 - 28*a^6*c^6 - 9*a^4*b^2*c^6 - 10*a^2*b^4*c^6 - 18*b^6*c^6 + 17*a^4*c^8 + 14*a^2*b^2*c^8 + 12*b^4*c^8 - 4*a^2*c^10 - 3*b^2*c^10) : :
X(34768) = 3 X[5] + X[15345], 3 X[547] - X[32551], 9 X[5055] - X[25043], X[5501] - 3 X[34597], 3 X[13856] - X[15345], X[13856] + 2 X[34599], X[15345] + 6 X[34599], X[18807] + 3 X[23516]

Let N = X(5), the center of the nine-point circle. Let A' = nine-point center of NBC, and define B' and C' cyclically. Let A'' = nine-point center of NB'C', and define B'' and C'' cyclically. The points N, A'', B'', C'' lie on a circle, named Anna's circle in Antreas Hatzipolakis and Peter Moses, Euclid 129 .

X(34768) lies on these lines: {3, 7604}, {5, 128}, {30, 15425}, {546, 18016}, {547, 32551}, {1154, 3628}, {5055, 25043}, {5501, 34597}, {7550, 34292}, {10126, 31879}, {10205, 32536}, {10289, 32744}, {22051, 23280}, {27868, 33545}

X(34768) = midpoint of X(i) and X(j) for these {i,j}: {5, 13856}, {546, 18016}, {10126, 31879}, {10205, 32536}, {27868, 33545}
X(34768) = reflection of X(i) in X(j) for these {i,j}: {5, 34599}, {34598, 15425}

X(34769) = 58TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics 2*a^20*b^2 - 15*a^18*b^4 + 46*a^16*b^6 - 70*a^14*b^8 + 42*a^12*b^10 + 28*a^10*b^12 - 70*a^8*b^14 + 54*a^6*b^16 - 20*a^4*b^18 + 3*a^2*b^20 + 2*a^20*c^2 - 26*a^18*b^2*c^2 + 96*a^16*b^4*c^2 - 160*a^14*b^6*c^2 + 142*a^12*b^8*c^2 - 105*a^10*b^10*c^2 + 131*a^8*b^12*c^2 - 146*a^6*b^14*c^2 + 90*a^4*b^16*c^2 - 27*a^2*b^18*c^2 + 3*b^20*c^2 - 15*a^18*c^4 + 96*a^16*b^2*c^4 - 196*a^14*b^4*c^4 + 166*a^12*b^6*c^4 - 47*a^10*b^8*c^4 - 43*a^8*b^10*c^4 + 116*a^6*b^12*c^4 - 142*a^4*b^14*c^4 + 86*a^2*b^16*c^4 - 21*b^18*c^4 + 46*a^16*c^6 - 160*a^14*b^2*c^6 + 166*a^12*b^4*c^6 - 64*a^10*b^6*c^6 - 8*a^6*b^10*c^6 + 86*a^4*b^12*c^6 - 126*a^2*b^14*c^6 + 60*b^16*c^6 - 70*a^14*c^8 + 142*a^12*b^2*c^8 - 47*a^10*b^4*c^8 - 32*a^6*b^8*c^8 - 14*a^4*b^10*c^8 + 87*a^2*b^12*c^8 - 84*b^14*c^8 + 42*a^12*c^10 - 105*a^10*b^2*c^10 - 43*a^8*b^4*c^10 - 8*a^6*b^6*c^10 - 14*a^4*b^8*c^10 - 46*a^2*b^10*c^10 + 42*b^12*c^10 + 28*a^10*c^12 + 131*a^8*b^2*c^12 + 116*a^6*b^4*c^12 + 86*a^4*b^6*c^12 + 87*a^2*b^8*c^12 + 42*b^10*c^12 - 70*a^8*c^14 - 146*a^6*b^2*c^14 - 142*a^4*b^4*c^14 - 126*a^2*b^6*c^14 - 84*b^8*c^14 + 54*a^6*c^16 + 90*a^4*b^2*c^16 + 86*a^2*b^4*c^16 + 60*b^6*c^16 - 20*a^4*c^18 - 27*a^2*b^2*c^18 - 21*b^4*c^18 + 3*a^2*c^20 + 3*b^2*c^20 : :

See Antreas Hatzipolakis and Peter Moses, Euclid 134 .

X(34769) lies on Anna's circle and these lines: {2,3}, {523,13856}, {1291,7604}

X(34770) = X(5)X(49)∩X(17702)X(34598)

Barycentrics 2*a^20*b^2 - 17*a^18*b^4 + 60*a^16*b^6 - 112*a^14*b^8 + 112*a^12*b^10 - 42*a^10*b^12 - 28*a^8*b^14 + 40*a^6*b^16 - 18*a^4*b^18 + 3*a^2*b^20 + 2*a^20*c^2 - 22*a^18*b^2*c^2 + 82*a^16*b^4*c^2 - 140*a^14*b^6*c^2 + 114*a^12*b^8*c^2 - 55*a^10*b^10*c^2 + 77*a^8*b^12*c^2 - 122*a^6*b^14*c^2 + 90*a^4*b^16*c^2 - 29*a^2*b^18*c^2 + 3*b^20*c^2 - 17*a^18*c^4 + 82*a^16*b^2*c^4 - 152*a^14*b^4*c^4 + 124*a^12*b^6*c^4 - 35*a^10*b^8*c^4 - 29*a^8*b^10*c^4 + 112*a^6*b^12*c^4 - 164*a^4*b^14*c^4 + 100*a^2*b^16*c^4 - 21*b^18*c^4 + 60*a^16*c^6 - 140*a^14*b^2*c^6 + 124*a^12*b^4*c^6 - 48*a^10*b^6*c^6 - 2*a^8*b^8*c^6 - 16*a^6*b^10*c^6 + 132*a^4*b^12*c^6 - 170*a^2*b^14*c^6 + 60*b^16*c^6 - 112*a^14*c^8 + 114*a^12*b^2*c^8 - 35*a^10*b^4*c^8 - 2*a^8*b^6*c^8 - 28*a^6*b^8*c^8 - 40*a^4*b^10*c^8 + 169*a^2*b^12*c^8 - 84*b^14*c^8 + 112*a^12*c^10 - 55*a^10*b^2*c^10 - 29*a^8*b^4*c^10 - 16*a^6*b^6*c^10 - 40*a^4*b^8*c^10 - 146*a^2*b^10*c^10 + 42*b^12*c^10 - 42*a^10*c^12 + 77*a^8*b^2*c^12 + 112*a^6*b^4*c^12 + 132*a^4*b^6*c^12 + 169*a^2*b^8*c^12 + 42*b^10*c^12 - 28*a^8*c^14 - 122*a^6*b^2*c^14 - 164*a^4*b^4*c^14 - 170*a^2*b^6*c^14 - 84*b^8*c^14 + 40*a^6*c^16 + 90*a^4*b^2*c^16 + 100*a^2*b^4*c^16 + 60*b^6*c^16 - 18*a^4*c^18 - 29*a^2*b^2*c^18 - 21*b^4*c^18 + 3*a^2*c^20 + 3*b^2*c^20 : :

See Antreas Hatzipolakis and Peter Moses, Euclid 134 .

X(34770) lies on Anna's circle and these lines: {5,49}, {17702,34598}, {20304,23281}

X(34771) = X(5)X(523)∩X(30)X(13856)

Barycentrics 5*a^18*b^4 - 30*a^16*b^6 + 70*a^14*b^8 - 70*a^12*b^10 + 70*a^8*b^14 - 70*a^6*b^16 + 30*a^4*b^18 - 5*a^2*b^20 + 6*a^18*b^2*c^2 - 34*a^16*b^4*c^2 + 66*a^14*b^6*c^2 - 52*a^12*b^8*c^2 + 43*a^10*b^10*c^2 - 121*a^8*b^12*c^2 + 184*a^6*b^14*c^2 - 126*a^4*b^16*c^2 + 37*a^2*b^18*c^2 - 3*b^20*c^2 + 5*a^18*c^4 - 34*a^16*b^2*c^4 + 80*a^14*b^4*c^4 - 70*a^12*b^6*c^4 + 5*a^10*b^8*c^4 + 59*a^8*b^10*c^4 - 154*a^6*b^12*c^4 + 200*a^4*b^14*c^4 - 112*a^2*b^16*c^4 + 21*b^18*c^4 - 30*a^16*c^6 + 66*a^14*b^2*c^6 - 70*a^12*b^4*c^6 + 48*a^10*b^6*c^6 - 8*a^8*b^8*c^6 + 16*a^6*b^10*c^6 - 144*a^4*b^12*c^6 + 182*a^2*b^14*c^6 - 60*b^16*c^6 + 70*a^14*c^8 - 52*a^12*b^2*c^8 + 5*a^10*b^4*c^8 - 8*a^8*b^6*c^8 + 48*a^6*b^8*c^8 + 40*a^4*b^10*c^8 - 187*a^2*b^12*c^8 + 84*b^14*c^8 - 70*a^12*c^10 + 43*a^10*b^2*c^10 + 59*a^8*b^4*c^10 + 16*a^6*b^6*c^10 + 40*a^4*b^8*c^10 + 170*a^2*b^10*c^10 - 42*b^12*c^10 - 121*a^8*b^2*c^12 - 154*a^6*b^4*c^12 - 144*a^4*b^6*c^12 - 187*a^2*b^8*c^12 - 42*b^10*c^12 + 70*a^8*c^14 + 184*a^6*b^2*c^14 + 200*a^4*b^4*c^14 + 182*a^2*b^6*c^14 + 84*b^8*c^14 - 70*a^6*c^16 - 126*a^4*b^2*c^16 - 112*a^2*b^4*c^16 - 60*b^6*c^16 + 30*a^4*c^18 + 37*a^2*b^2*c^18 + 21*b^4*c^18 - 5*a^2*c^20 - 3*b^2*c^20 : :

See Antreas Hatzipolakis and Peter Moses, Euclid 134 .

X(34771) lies on Anna's circle and these lines: {5,523}, {30,13856}, {7604,14979}

X(34772) = X(1)X(2)∩X(21)X(72)

Barycentrics a (a^2 (b+c)+a (b^2+b c+c^2)-a^3-b^3-c^3) : :
X(34772) = 3 X[2] - 4 X[13411], 2 X[10] - 3 X[3584], 2 X[3916] - 3 X[17549], X[3916] - 3 X[33595]

See Antreas Hatzipolakis and Angel Montesdeoca, Euclid 137 .

X(34772) lies on these lines: {1, 2}, {3, 3218}, {4, 5761}, {6, 26690}, {7, 224}, {9, 3984}, {12, 5086}, {20, 5758}, {21, 72}, {29, 1807}, {30, 11015}, {33, 7518}, {35, 758}, {36, 3874}, {37, 2287}, {41, 17522}, {46, 12559}, {55, 3869}, {56, 3873}, {57, 4188}, {58, 1331}, {63, 3601}, {65, 100}, {73, 3152}, {75, 5736}, {81, 1257}, {101, 3970}, {144, 5766}, {149, 946}, {171, 2650}, {191, 4067}, {192, 3100}, {210, 5260}, {214, 3881}, {218, 25082}, {226, 2475}, {228, 4225}, {229, 662}, {269, 32093}, {284, 5279}, {307, 1442}, {321, 1043}, {326, 3945}, {329, 4313}, {346, 3553}, {354, 5253}, {355, 6828}, {377, 3487}, {404, 942}, {405, 3876}, {411, 517}, {442, 5719}, {443, 27186}, {452, 31018}, {474, 15934}, {484, 4084}, {515, 6895}, {516, 20066}, {518, 2330}, {546, 12690}, {581, 31034}, {644, 3991}, {664, 1446}, {740, 31880}, {894, 5764}, {908, 950}, {912, 6906}, {944, 6836}, {952, 6831}, {956, 3897}, {958, 3681}, {960, 1621}, {962, 6261}, {965, 16777}, {984, 10448}, {988, 4392}, {991, 17364}, {993, 5904}, {999, 3889}, {1001, 30628}, {1006, 14054}, {1012, 12528}, {1062, 27407}, {1071, 6909}, {1086, 26729}, {1104, 32911}, {1170, 1280}, {1260, 11020}, {1265, 17776}, {1320, 1389}, {1385, 3555}, {1420, 1445}, {1447, 20247}, {1476, 14151}, {1482, 3149}, {1483, 6922}, {1490, 3146}, {1697, 11682}, {1750, 17578}, {1759, 4262}, {1770, 14450}, {1818, 17300}, {1829, 7466}, {1834, 33133}, {1837, 11681}, {1870, 5125}, {1895, 15500}, {1993, 7078}, {2077, 5884}, {2099, 3913}, {2136, 4917}, {2280, 3061}, {2320, 7162}, {2329, 3930}, {2476, 3419}, {2478, 3488}, {2802, 11009}, {2886, 5178}, {2895, 4101}, {2900, 5175}, {3091, 5720}, {3101, 18673}, {3158, 3340}, {3159, 5497}, {3189, 3434}, {3191, 3995}, {3262, 4360}, {3295, 3877}, {3303, 3890}, {3305, 5436}, {3306, 5438}, {3337, 15015}, {3436, 3486}, {3465, 31294}, {3522, 6282}, {3523, 18443}, {3576, 4430}, {3579, 4018}, {3647, 4127}, {3649, 20292}, {3670, 4256}, {3678, 5251}, {3684, 17451}, {3685, 25253}, {3689, 5836}, {3710, 32849}, {3721, 18755}, {3726, 21008}, {3746, 3878}, {3754, 5425}, {3813, 15950}, {3841, 26725}, {3875, 7269}, {3880, 11011}, {3894, 7280}, {3895, 7982}, {3901, 5010}, {3916, 17549}, {3925, 11281}, {3927, 16370}, {3936, 7270}, {3951, 31424}, {3962, 4640}, {4005, 5302}, {4134, 5426}, {4187, 12433}, {4193, 5722}, {4195, 26223}, {4201, 17184}, {4233, 11363}, {4251, 33950}, {4255, 4850}, {4260, 27678}, {4265, 9021}, {4292, 17483}, {4294, 11415}, {4297, 5538}, {4300, 20101}, {4304, 15680}, {4318, 10571}, {4328, 4373}, {4417, 5016}, {4424, 33771}, {4442, 11553}, {4452, 7190}, {4533, 32635}, {4641, 16948}, {4652, 17548}, {4720, 5295}, {4857, 11813}, {5044, 5047}, {5045, 17614}, {5057, 6284}, {5080, 10572}, {5141, 5219}, {5154, 9581}, {5172, 7098}, {5176, 10950}, {5187, 5748}, {5221, 9352}, {5248, 5692}, {5259, 10176}, {5267, 6763}, {5284, 25917}, {5325, 15672}, {5330, 9957}, {5435, 34489}, {5439, 17531}, {5443, 24387}, {5496, 23555}, {5531, 20085}, {5603, 6835}, {5697, 25439}, {5708, 16371}, {5731, 20076}, {5732, 20059}, {5742, 17362}, {5768, 6890}, {5770, 6977}, {5777, 6912}, {5794, 17718}, {5804, 6953}, {5853, 21617}, {5886, 6991}, {5902, 25440}, {5903, 8715}, {6147, 11112}, {6583, 22935}, {6600, 7672}, {6605, 15853}, {6690, 21677}, {6762, 13384}, {6767, 16293}, {6769, 20070}, {6864, 10595}, {6865, 7967}, {6870, 17857}, {6875, 26921}, {6904, 11036}, {6905, 24474}, {6918, 10247}, {6940, 10202}, {6950, 24467}, {6988, 12245}, {7100, 31155}, {7176, 17136}, {7179, 21285}, {7308, 17570}, {7373, 16410}, {7411, 31793}, {7428, 22458}, {7489, 31835}, {7515, 18455}, {7677, 15185}, {8543, 15733}, {8726, 15717}, {9335, 11512}, {9538, 27402}, {9579, 31164}, {9778, 12520}, {9803, 12616}, {9840, 21319}, {9945, 24470}, {10025, 25237}, {10058, 12532}, {10389, 15829}, {10524, 10590}, {10609, 18990}, {10698, 13278}, {10711, 12738}, {10902, 31806}, {10944, 15844}, {11014, 28234}, {11319, 27064}, {11375, 11680}, {11552, 34600}, {11567, 12737}, {11570, 17100}, {11849, 14988}, {12513, 34471}, {12531, 32537}, {12572, 26792}, {12645, 19920}, {12699, 34629}, {12701, 34611}, {12740, 33176}, {13138, 14919}, {13161, 33153}, {13407, 17647}, {13587, 24473}, {14020, 27776}, {14803, 34758}, {14997, 16485}, {15338, 17768}, {15556, 27086}, {15600, 15839}, {15650, 16418}, {15677, 17781}, {15803, 23958}, {15935, 17527}, {16132, 31730}, {16133, 17668}, {16143, 20084}, {16749, 33296}, {17164, 32932}, {17390, 18635}, {17479, 20222}, {17676, 27184}, {17719, 21935}, {17745, 24036}, {18447, 18641}, {19318, 21808}, {19765, 28606}, {19907, 25416}, {20760, 28348}, {21153, 30389}, {23536, 33148}, {24005, 27524}, {25716, 34059}, {26117, 26580}, {27398, 34064}, {28628, 33108}, {33649, 34340}

X(34772) = anticomplement of X(6734)
X(34772) = reflection of X(i) in X(j) for these {i,j}: {8, 10039}, {411, 33597}, {2975, 2646}, {4861, 1}, {5086, 12}, {6734, 13411}, {6763, 5267}, {6906, 33596}, {17549, 33595}
X(34772) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {56, 2894}, {943, 3436}, {1175, 3869}, {2259, 329}, {2982, 69}, {15439, 513}, {32651, 693}
X(34772) = X(1441)-Ceva conjugate of X(3219)
X(34772) = X(15556)-cross conjugate of X(5174)
X(34772) = X(i)-isoconjugate of X(j) for these (i,j): {56, 6598}, {513, 6011}
X(34772) = crosspoint of X(664) and X(4567)
X(34772) = crosssum of X(663) and X(3125)
X(34772) = barycentric product X(i)*X(j) for these {i,j}: {1, 33116}, {63, 5174}, {190, 6003}, {306, 13739}, {333, 15556}, {644, 31603}, {4567, 8286}, {18359, 27086}, {21961, 24041}
X(34772) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 6598}, {101, 6011}, {5174, 92}, {6003, 514}, {8286, 16732}, {13739, 27}, {15556, 226}, {21961, 1109}, {27086, 3218}, {31603, 24002}, {33116, 75}
X(34772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 42, 17016}, {1, 43, 3924}, {1, 78, 2}, {1, 200, 19860}, {1, 386, 5262}, {1, 976, 3920}, {1, 978, 28082}, {1, 997, 3616}, {1, 1193, 7191}, {1, 1698, 30143}, {1, 3216, 30117}, {1, 3616, 29817}, {1, 3633, 22837}, {1, 3679, 30147}, {1, 3811, 8}, {1, 3870, 145}, {1, 3961, 10459}, {1, 6765, 3872}, {1, 8583, 4666}, {1, 19767, 17011}, {1, 19861, 3622}, {1, 22836, 4511}, {2, 145, 12649}, {2, 20013, 8}, {3, 3868, 3218}, {3, 24475, 26877}, {8, 3616, 19843}, {8, 3811, 3935}, {8, 5703, 2}, {8, 10321, 25005}, {21, 72, 3219}, {55, 12635, 3869}, {57, 4855, 4188}, {63, 3601, 4189}, {72, 24929, 21}, {100, 34195, 65}, {145, 10528, 8}, {200, 19860, 3617}, {214, 3881, 5563}, {284, 22021, 5279}, {329, 4313, 6872}, {377, 3487, 31019}, {386, 5262, 17012}, {404, 942, 27003}, {405, 3876, 27065}, {405, 3940, 3876}, {484, 16126, 4084}, {908, 950, 5046}, {938, 27383, 2}, {942, 5440, 404}, {975, 15954, 2000}, {978, 28082, 7292}, {1210, 27385, 2}, {2099, 3913, 14923}, {3189, 3485, 3434}, {3241, 27383, 938}, {3295, 5730, 3877}, {3303, 5289, 3890}, {3305, 5436, 16859}, {3306, 5438, 17572}, {3419, 11374, 2476}, {3486, 25568, 3436}, {3601, 11523, 63}, {3746, 4867, 3878}, {3872, 6765, 3621}, {3962, 4640, 11684}, {4652, 30282, 17548}, {4855, 11520, 57}, {5175, 5226, 6871}, {5226, 12536, 5175}, {5438, 11518, 3306}, {5552, 18391, 25005}, {6282, 10884, 3522}, {6734, 13411, 2}, {6737, 13405, 24987}, {6738, 6745, 24982}, {9581, 30852, 5154}, {10572, 21077, 5080}, {10950, 12607, 5176}, {24299, 31837, 1006}

Ellipsologic centers: X(34773)-X(34795)

This preamble and centers X(34773)-X(34795) were contributed by César Eliud Lozada, November 14, 2019.

Let T'=A'B'C' and T"=A"B"C" be two triangles. Denote as E'_a the ellipse passing through A' and having foci B" and C". Define E'_b and E'_c cyclically. If theses three ellipses intersect in a unique point Q', then Q' is here named the ellipsologic center of T' to T".

Note: The existence of the ellipsologic center of T' to T" does not imply the existence of the ellipsologic center of T" to T'.

It is known that "if three conics are such that each pair has one common focus and two and only two real points of intersection, the three chords of visible intersection are concurrent" (E. Neville, "A focus-sharing set of three conics", Mathematical Gazette 20(239) (1936) 182-183, and B. Lawrence, "Note on focus sharing conics", Mathematical Gazette 21(243) (1937) 160-161.

The point of intersection, Q_i, of the common chords mentioned above does not lie necessarily on any of the conics, but if it lies on one of the conics then it is obviously the intersection of the three conics. Lawrence gives the following interesting alternative construction of Q_i:

Assume the major axes of the ellipses E'_a, E'_b, E'_c are λ_a = A'B"+A'C", λ_b = B'C"+B'A" and λ_c = C'A"+C'B" and suppose λ_c is the longest axis. If λ'_a = λ_c - λ_b ≥ 0 and λ'_b = λ_c - λ_a ≥ 0 then Q_i is the point at distances λ_a*λ'_a/(2*B"C") and λ_b*λ'_b/(2*C"A") from the perpendicular bisectors of B"C" and C"A", respectively.

The appearance of ( T', T", i, j ) in the following list means that the ellipsologic centers (T' to T") and (T" to T') are X(i) and X(j), respectively. A double-dash "--" means that the ellipsologic center does not exist. It must be clear, under certains conditions of the sidelengths and angles of ABC, that a pair of ellipses may be disjoint, so that the indicated center may not exist. Therefore, items in the following list are valid when the ellipses intersect in pairs:

(ABC, ABC-X3 reflections, 20, 4), (ABC, circumorthic, --, 4), (ABC, 2nd circumperp, 1, --), (ABC, 1st Ehrmann, 6, --), (ABC, Fuhrmann, 80, --), (ABC, inner-Garcia, 100, 1), (ABC, outer-Garcia, 8, 1), (ABC, Gossard, 4240, 1650), (ABC, Johnson, 4, 3), (ABC, Kosnita, 3, --), (ABC, 5th mixtilinear, 145, 8), (ABC, inner-Napoleon, 14, --), (ABC, outer-Napoleon, 13, --), (ABC-X3 reflections, 1st anti-circumperp, --, 20), (ABC-X3 reflections, 1st circumperp, 40, --), (ABC-X3 reflections, inner-Garcia, 104, 1), (ABC-X3 reflections, Trinh, 3, --), (anti-Aquila, Ehrmann-mid, 31673, 34773), (anti-Aquila, intouch, --, 1), (anti-Aquila, medial, 10, 1), (2nd anti-circumperp-tangential, intouch, 65, --), (2nd anti-circumperp-tangential, Mandart-incircle, 6284, 7354), (1st anti-circumperp, circumorthic, 5889, 11412), (anti-Euler, anticomplementary, 4, 20), (3rd anti-Euler, 4th anti-Euler, 5889, 11412), (anti-excenters-reflections, orthic, 185, 11381), (anti-Honsberger, anti-Ursa minor, 66, 34774), (anti-Honsberger, Ehrmann-vertex, 34775, 34776), (anti-Honsberger, 2nd Ehrmann, 34777, 159), (anti-Honsberger, Trinh, 34778, 34779), (anti-Hutson intouch, tangential, 1498, 64), (anti-incircle-circles, anti-inverse-in-incircle, 34780, 34781), (anti-incircle-circles, Ara, 3, --), (anti-incircle-circles, 1st excosine, 3, 1498), (anti-Mandart-incircle, 2nd circumperp tangential, 12513, 3913), (anti-tangential-midarc, intangents, 6285, 7355), (anti-Ursa minor, Kosnita, 34782, 18381), (anti-Wasat, circumorthic, 185, 4), (anticomplementary, Aquila, 8, 1), (anticomplementary, X3-ABC reflections, 4, 3), (Aquila, excentral, 1, --), (1st Auriga, 2nd Auriga, 12455, 12454), (2nd Brocard, McCay, 2, --), (circumorthic, Ehrmann-side, 12111, 34783), (1st circumperp, 2nd circumperp, 1, 40), (2nd circumperp, Wasat, 4, 4297), (inner-Conway, outer-Garcia, 8, --), (inner-Conway, Honsberger, 30628, 34784), (Ehrmann-mid, Euler, 5, 4), (Ehrmann-side, Johnson, 3, --), (Ehrmann-vertex, Johnson, --, 4), (Ehrmann-vertex, Kosnita, 34785, 34786), (2nd Ehrmann, Kosnita, 34787, 34788), (Euler, 3rd Euler, 946, --), (Euler, medial, 3, 4), (Euler, orthic, --, 4), (2nd Euler, medial, 3, --), (2nd Euler, orthic, 52, 5562), (excenters-midpoints, Hutson intouch, --, 8), (excenters-reflections, excentral, 7991, 11531), (excentral, hexyl, 1, 40), (inner-Garcia, X3-ABC reflections, 1, 12331), (Garcia-reflection, 2nd Schiffler, 1768, 34789), (Honsberger, Ursa minor, 7, 14100), (Hutson intouch, intouch, 65, 3057), (Hutson intouch, Mandart-incircle, --, 3057), (Hutson intouch, 2nd midarc, 8422, --), (1st Hyacinth, 2nd Hyacinth, --, 10111), (incircle-circles, inverse-in-incircle, 942, 1), (incircle-circles, 2nd Zaniah, 34790, 34791), (intouch, midarc, 177, --), (1st Johnson-Yff, inner-Yff, 55, 1478), (2nd Johnson-Yff, outer-Yff, 56, 1479), (K798e, Wasat, 5, --), (K798i, 2nd Zaniah, 5, 3035), (Kosnita, Trinh, 3357, 6759), (midarc, 2nd midarc, 8422, 177), (inner-Napoleon, outer-Napoleon, 13, 14), (1st Parry, 2nd Parry, 9979, 9131), (reflection, X3-ABC reflections, 4, --), (reflection, Yiu, 1157, --), (tangential, X3-ABC reflections, --, 3), (tangential-midarc, 2nd tangential-midarc, 8084, 10967)

For definitions of these triangles see here. All pairs of triangles in this index were compared and results are showed in the above list.

Added by Vu Thanh Tung - November 14, 2019:

Let triangle T"= A"B"C" be the reflection of triangle T' = A'B'C' about a point P.
Then T" and T' are symmetrically ellipsologic, i.e there exist Q' = ellipsologic center of T' to T" and Q" = ellipsologic center of T" to T'.
Q' = reflection of A' about midpoint of B"C" = reflection of B' about midpoint of C"A" = reflection of C' about midpoint of A"B".
Q" = reflection of A" about midpoint of B'C' = reflection of B" about midpoint of C'A' = reflection of C" about midpoint of A'B'.
The midpoint of Q'Q" is P.
Let consider a point P and a triangle T' = A'B'C'.
Let A" = reflection of P about the midpoint of B'C', B" = reflection of P about the midpoint of C'A', C" = reflection of P about the midpoint of A'B' and let triangle T" = A"B"C".
Then T" and T' are symmetrically ellipsologic, i.e there exist Q' = ellipsologic center of T' to T" and Q" = ellipsologic center of T" to T'.
Q" = P and Q' = the point such that PQ' = 3 PG' (in vector) where G = centroid of T'.
Let consider a point P and a triangle T' = A'B'C'.
Let A" = reflection of P about the perpendicular bisector of B'C', B" = reflection of P about the perpendicular bisector of C'A', C" = reflection of P about the perpendicular bisector of A'B' and let triangle T" = A"B"C".
Then T" and T' are similar and symmetrically ellipsologic, i.e there exist Q' = ellipsologic center of T' to T" and Q" = ellipsologic center of T" to T'.
Q" = P and Q' = the point such that the two quandrangles A'B'C'Q' and A"B"C"P are similar.

X(34773) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-MID TO ANTI-AQUILA

Barycentrics 4*a^4-2*(b+c)*a^3-(3*b^2-4*b*c+3*c^2)*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(34773) = X(1)-3*X(3655) = 5*X(1)-3*X(3656) = 3*X(1)-X(12699) = 7*X(1)-3*X(31162) = 5*X(1)+3*X(34628) = 2*X(79)-3*X(33668) = X(79)-3*X(33858) = 5*X(3655)-X(3656) = 9*X(3655)-X(12699) = 3*X(3655)+X(18481) = 6*X(3655)-X(22791) = 7*X(3655)-X(31162) = 5*X(3655)+X(34628) = 9*X(3656)-5*X(12699) = 3*X(3656)+5*X(18481) = 6*X(3656)-5*X(22791) = 7*X(3656)-5*X(31162) = X(12699)+3*X(18481) = 2*X(12699)-3*X(22791) = 7*X(12699)-9*X(31162) = 5*X(12699)+9*X(34628)

The reciprocal ellipsologic center of these triangles is X(31673)

X(34773) lies on these lines: {1,30}, {2,18357}, {3,8}, {4,3622}, {5,515}, {10,549}, {11,21842}, {20,1482}, {21,26321}, {35,10944}, {36,10950}, {40,548}, {48,7359}, {65,21578}, {74,12898}, {80,5433}, {140,355}, {145,376}, {153,6902}, {165,33923}, {214,1329}, {378,12135}, {381,3616}, {382,5603}, {388,5719}, {404,18524}, {405,18519}, {411,22765}, {442,3897}, {474,18518}, {495,2646}, {496,1319}, {499,12019}, {516,13607}, {517,550}, {519,3579}, {528,22837}, {529,22836}, {546,5691}, {547,3624}, {551,3845}, {572,17369}, {573,4969}, {631,5790}, {632,9956}, {730,32521}, {912,31786}, {942,4311}, {946,3627}, {950,24928}, {962,1657}, {993,5428}, {999,3486}, {1001,18761}, {1012,16202}, {1056,6851}, {1071,14988}, {1210,5126}, {1317,5697}, {1320,20066}, {1386,21850}, {1387,1388}, {1420,5722}, {1478,34471}, {1484,11715}, {1539,11723}, {1596,11363}, {1597,7718}, {1621,13743}, {1697,7171}, {1699,3853}, {1770,11011}, {1837,15325}, {2098,4302}, {2099,4299}, {2771,3878}, {2779,23157}, {2801,5694}, {2807,13491}, {2829,19907}, {2886,5499}, {2894,6916}, {3098,5846}, {3146,10595}, {3149,16203}, {3241,3534}, {3295,3476}, {3296,3600}, {3485,9655}, {3488,4308}, {3522,12245}, {3524,3617}, {3526,5818}, {3528,20054}, {3529,28182}, {3530,5881}, {3585,15950}, {3586,11373}, {3612,5252}, {3621,10304}, {3626,17504}, {3628,5587}, {3632,3654}, {3634,11539}, {3635,11278}, {3636,15687}, {3650,3869}, {3671,31776}, {3679,12100}, {3753,17563}, {3817,3858}, {3828,15713}, {3850,8227}, {3861,9624}, {3881,31651}, {4189,18515}, {4192,29824}, {4193,34123}, {4293,24470}, {4301,28146}, {4304,9957}, {4313,6767}, {4314,11035}, {4315,5045}, {4316,11009}, {4325,11246}, {4504,30231}, {4511,28459}, {4668,14891}, {4669,15711}, {4677,15759}, {4678,15692}, {4691,6684}, {4701,15714}, {4745,19711}, {4848,5122}, {4861,28458}, {4881,13747}, {5046,10742}, {5054,9780}, {5055,5550}, {5057,7491}, {5066,18492}, {5073,9812}, {5088,32007}, {5180,16116}, {5204,10573}, {5217,12647}, {5248,31649}, {5288,7688}, {5330,12248}, {5445,9897}, {5450,32613}, {5554,16371}, {5560,7741}, {5734,17800}, {5777,31838}, {5880,31657}, {5887,12680}, {5892,23841}, {5903,15326}, {6221,19066}, {6256,11729}, {6398,19065}, {6644,9798}, {6763,16139}, {6823,24301}, {6836,10805}, {6863,10785}, {6903,20060}, {6907,10943}, {6909,11849}, {6914,10267}, {6922,10942}, {6923,12116}, {6924,10269}, {6925,10806}, {6926,27525}, {6928,12115}, {6929,12667}, {6932,18549}, {6958,10786}, {6960,12747}, {6996,29569}, {7508,10902}, {7525,15177}, {7530,11365}, {7580,10680}, {7686,13373}, {7966,9841}, {7972,11010}, {7973,20427}, {7976,9821}, {7977,8725}, {7978,20127}, {7982,12103}, {7984,12121}, {7988,12811}, {8185,12106}, {8583,18528}, {8727,24299}, {8728,13151}, {9613,11374}, {9626,12107}, {9730,16980}, {9778,15696}, {9845,26921}, {9884,14830}, {9933,12163}, {10106,24929}, {10107,34339}, {10113,11735}, {10175,31662}, {10264,11709}, {10272,12368}, {10327,21487}, {10431,10597}, {10701,23240}, {10884,31775}, {11014,11826}, {11038,31671}, {11231,14869}, {11280,15228}, {11362,31663}, {11396,18533}, {11540,19876}, {11545,24914}, {11684,28460}, {11724,22505}, {11725,22515}, {11812,19875}, {12053,25405}, {12054,12195}, {12108,31423}, {12266,20424}, {12512,28234}, {12514,19919}, {12629,34701}, {12650,18443}, {12738,34606}, {12782,32516}, {13464,22793}, {13665,13902}, {13785,13959}, {14093,20053}, {14872,31835}, {15681,20057}, {15688,20050}, {15689,34632}, {15694,19877}, {15699,19862}, {15808,34648}, {16200,28216}, {17057,31260}, {17527,17614}, {17532,18544}, {17538,20070}, {17556,18542}, {17564,24982}, {17566,34122}, {17647,31419}, {18391,34753}, {18446,31789}, {18491,25524}, {19540,30947}, {19542,29833}, {19708,31145}, {19710,28198}, {26492,34126}

X(34773) = midpoint of X(i) and X(j) for these {i,j}: {1, 18481}, {3, 944}, {8, 18526}, {20, 1482}, {74, 12898}, {145, 12702}, {550, 1483}, {962, 1657}, {3241, 3534}, {3244, 31730}, {3656, 34628}, {4297, 5882}, {5887, 12680}, {6224, 12773}, {6361, 8148}, {7972, 12515}, {7973, 20427}, {7976, 9821}, {7977, 8725}, {7978, 20127}, {7984, 12121}, {9884, 14830}, {9933, 12163}, {10701, 23240}, {12119, 12737}
X(34773) = reflection of X(i) in X(j) for these (i,j): (4, 5901), (5, 1385), (10, 13624), (40, 548), (355, 140), (550, 4297), (946, 15178), (1483, 5882), (1484, 11715), (1539, 11723), (3244, 32900), (3627, 946), (3654, 34200), (3679, 12100), (3845, 551), (4301, 33179), (5690, 3), (5691, 546), (5777, 31838), (5884, 26201), (7686, 13373), (10113, 11735), (10175, 31662), (10264, 11709), (11278, 3635), (11362, 31663), (11698, 214), (12368, 10272), (12782, 32516), (14872, 31835), (18480, 1125), (18483, 3636), (18525, 18357), (20424, 12266), (21850, 1386), (22505, 11724), (22515, 11725), (22791, 1), (22793, 13464), (22799, 11729), (22938, 1387), (24475, 12675), (24680, 13607), (31673, 9955), (33668, 33858), (33697, 18483), (33899, 5450)
X(34773) = anticomplement of X(18357)
X(34773) = complement of X(18525)
X(34773) = anticomplement of X(140) with respect to these triangles: 2nd circumperp, hexyl
X(34773) = anticomplement of X(3853) with respect to these triangles: 2nd Fuhrmann, Garcia-reflection
X(34773) = anticomplement of X(9955) with respect to anti-Aquila triangle
X(34773) = anticomplement of X(18357) with respect to these triangles: 1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34773) = anticomplement of X(19925) with respect to K798e triangle
X(34773) = anticomplement of X(31673) with respect to Ehrmann-mid triangle
X(34773) = complement of X(355) with respect to these triangles: 2nd circumperp, hexyl
X(34773) = complement of X(5073) with respect to 2nd Conway triangle
X(34773) = complement of X(8148) with respect to 5th mixtilinear triangle
X(34773) = complement of X(12702) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34773) = complement of X(18525) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34773) = X(3579)-of-inner-Garcia triangle
X(34773) = X(5690)-of-ABC-X3 reflections triangle
X(34773) = X(5876)-of-2nd circumperp triangle
X(34773) = X(5901)-of-anti-Euler triangle
X(34773) = X(10733)-of-K798i triangle
X(34773) = X(13491)-of-1st circumperp triangle
X(34773) = X(13630)-of-hexyl triangle
X(34773) = X(18481)-of-anti-Aquila triangle
X(34773) = X(22791)-of-5th mixtilinear triangle
X(34773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5441, 3058), (1, 10404, 16137), (1, 18990, 6147), (2, 18525, 18357), (3, 12645, 5657), (3, 12773, 2975), (3, 18526, 8), (3, 32141, 33814), (4, 3622, 18493), (4, 10246, 5901), (8, 944, 18526), (944, 5731, 3), (1125, 18480, 5), (1385, 18480, 1125), (3622, 18493, 5901), (3655, 18481, 1), (10246, 18493, 3622), (10404, 16137, 6147), (11230, 19925, 5), (16137, 18990, 10404), (18481, 33858, 7354)

X(34774) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA MINOR TO ANTI-HONSBERGER

Barycentrics 4*a^8-(b^2+c^2)*a^6-(3*b^4-2*b^2*c^2+3*c^4)*a^4+(b^4-c^4)*(b^2-c^2)*a^2-(b^4-c^4)^2 : :
X(34774) = 3*X(2)-5*X(19132) = 4*X(6)-3*X(23326) = X(64)-3*X(25406) = X(66)-3*X(19153) = 2*X(66)-3*X(23332) = X(69)-3*X(154) = 2*X(140)-3*X(23042) = 2*X(141)-3*X(10192) = 3*X(154)-2*X(15585) = 4*X(206)-3*X(10192) = 2*X(3589)-3*X(19153) = 4*X(3589)-3*X(23332) = 3*X(5480)-2*X(18382) = 2*X(5596)+X(15583) = 4*X(5596)+3*X(23326) = 3*X(14912)+X(34781) = 2*X(15583)-3*X(23326) = X(18382)-3*X(34117)

The reciprocal ellipsologic center of these triangles is X(66)

X(34774) lies on these lines: {2,19132}, {4,6}, {30,34776}, {64,25406}, {66,3589}, {69,154}, {110,32264}, {140,23042}, {141,206}, {159,524}, {182,6247}, {193,9924}, {343,19121}, {427,21637}, {511,34782}, {597,23300}, {858,19122}, {1351,9833}, {1352,16252}, {1594,19123}, {1619,19459}, {1853,3618}, {1899,19118}, {1974,13567}, {2393,3629}, {2781,3313}, {3564,6759}, {3630,15647}, {5050,14216}, {5085,6696}, {5092,23328}, {5894,11574}, {5895,14927}, {6329,23327}, {6593,23315}, {8721,15905}, {9715,15577}, {9968,11511}, {10117,11061}, {10519,17821}, {11898,14530}, {12167,31383}, {12359,19154}, {13371,19155}, {14683,32276}, {18381,18583}, {18400,21850}, {19125,23292}, {19130,23324}, {31267,34573}, {32455,34777}

X(34774) = midpoint of X(i) and X(j) for these {i,j}: {6, 5596}, {110, 32264}, {193, 9924}, {1351, 9833}, {1498, 6776}, {5895, 14927}, {10117, 11061}, {14683, 32276}, {34776, 34779}
X(34774) = reflection of X(i) in X(j) for these (i,j): (66, 3589), (69, 15585), (141, 206), (1352, 16252), (2883, 19149), (5480, 34117), (6247, 182), (8549, 12007), (12359, 19154), (13371, 19155), (15583, 6), (18381, 18583), (23315, 6593), (23332, 19153), (34777, 32455)
X(34774) = anticomplement of X(66) with respect to anti-Ursa minor triangle
X(34774) = anticomplement of X(3589) with respect to anti-Honsberger triangle
X(34774) = anticomplement of X(15585) with respect to these triangles: 1st excosine, tangential
X(34774) = complement of X(66) with respect to anti-Honsberger triangle
X(34774) = complement of X(69) with respect to these triangles: 1st excosine, tangential
X(34774) = complement of X(9924) with respect to Aries triangle
X(34774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 15583, 23326), (66, 3589, 23332), (66, 19153, 3589), (69, 154, 15585), (141, 206, 10192), (193, 11206, 9924), (1974, 26926, 13567), (3618, 20079, 1853), (13562, 19126, 141)

X(34775) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO EHRMANN-VERTEX

Barycentrics 3*a^12-3*(b^2+c^2)*a^10-4*(b^4+c^4)*a^8+2*(b^2+c^2)*(b^4+c^4)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^4+(b^4-c^4)^2*(b^2+c^2)*a^2-2*(b^4-c^4)^2*(b^2-c^2)^2 : :
X(34775) = 3*X(3)-4*X(6697) = 3*X(4)-X(5596) = 4*X(5)-3*X(23041) = 6*X(5)-5*X(31267) = X(6)-3*X(18405) = 2*X(6)-3*X(23049) = 2*X(206)-3*X(381) = 3*X(3543)+X(20079) = 2*X(3589)-3*X(23324) = 5*X(5076)-2*X(9968) = 2*X(5596)-3*X(19149) = 3*X(8549)-4*X(15583) = 2*X(18382)-3*X(18405) = 4*X(18382)-3*X(23049) = 9*X(23041)-10*X(31267)

The reciprocal ellipsologic center of these triangles is X(34776)

X(34775) lies on these lines: {3,6697}, {4,6}, {5,23041}, {22,1853}, {30,66}, {64,18124}, {154,5133}, {159,3818}, {182,18383}, {206,381}, {265,9919}, {382,34146}, {511,12293}, {542,34777}, {568,34780}, {1176,18434}, {1177,10113}, {1350,12225}, {1351,32402}, {1352,12605}, {1974,13851}, {2393,18435}, {2781,10733}, {3153,20806}, {3357,29323}, {3543,20079}, {3580,7500}, {3589,23324}, {3845,31166}, {5076,9968}, {5085,13160}, {5092,23325}, {6247,31305}, {7387,14852}, {7403,9833}, {7404,34782}, {7494,23332}, {7503,10516}, {7512,15578}, {7553,14216}, {7566,32395}, {9969,10938}, {10117,26284}, {10249,15760}, {11206,14389}, {11441,32346}, {11572,19124}, {12088,15579}, {12121,15116}, {12140,19457}, {12173,19161}, {12272,15069}, {15141,19506}, {17508,32767}, {18376,19130}, {18377,19139}, {18379,19154}, {18386,19125}, {18392,19121}, {18394,19128}, {18430,19129}, {24206,34785}, {32344,32369}

X(34775) = reflection of X(i) in X(j) for these (i,j): (6, 18382), (159, 3818), (182, 18383), (1177, 10113), (1350, 34118), (12121, 15116), (15141, 19506), (17845, 15577), (19139, 18377), (19149, 4), (19153, 18376), (19154, 18379), (23049, 18405), (31166, 3845), (32344, 32369), (34776, 19130), (34778, 66), (34785, 24206), (34787, 1352)
X(34775) = anticomplement of X(206) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34775) = anticomplement of X(19130) with respect to Ehrmann-vertex triangle
X(34775) = anticomplement of X(34776) with respect to anti-Honsberger triangle
X(34775) = complement of X(34776) with respect to Ehrmann-vertex triangle
X(34775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 18382, 23049), (6, 18405, 18382), (10516, 17845, 15577), (18376, 34776, 19130), (19130, 34776, 19153)

X(34776) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO ANTI-HONSBERGER

Barycentrics 3*a^12-5*(b^2+c^2)*a^10-(b^4+c^4)*a^8+2*(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^6-(b^2-c^2)^2*(b^4+c^4)*a^4+(b^8-c^8)*(b^2-c^2)*a^2-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(34776) = 2*X(5)-3*X(23042) = 4*X(6)-3*X(23048) = 2*X(66)-3*X(23329) = 2*X(141)-3*X(11202) = 3*X(154)-X(18440) = 3*X(182)-2*X(23300) = 3*X(381)-5*X(19132) = 3*X(1853)-5*X(12017) = 4*X(3589)-3*X(23325) = 5*X(3763)-6*X(10182) = 3*X(5085)-2*X(20299) = 4*X(5092)-3*X(23329) = 3*X(5476)-2*X(18382) = 3*X(18381)-4*X(23300)

The reciprocal ellipsologic center of these triangles is X(34775)

X(34776) lies on these lines: {4,19123}, {5,182}, {6,18400}, {30,34774}, {66,5092}, {113,32063}, {141,11202}, {154,18440}, {159,542}, {381,19132}, {389,1843}, {511,12118}, {1351,17845}, {1352,10282}, {1353,34788}, {1495,1899}, {1531,5656}, {1853,12017}, {1974,18390}, {2777,11820}, {3153,19122}, {3517,20987}, {3564,34782}, {3763,10182}, {5085,20299}, {5476,18382}, {5480,34786}, {5596,6000}, {5878,14927}, {5965,34787}, {5972,15142}, {6403,32352}, {6593,19506}, {7398,11451}, {9924,32326}, {9927,19154}, {10192,18358}, {10250,15583}, {11438,26926}, {11645,31166}, {12121,32264}, {12897,34117}, {14216,25406}, {14561,18383}, {14641,34146}, {15030,34781}, {15462,32743}, {15577,23358}, {17821,32348}, {18376,19130}, {18377,19155}, {18388,19125}, {18396,19118}, {18474,19129}, {18533,21851}, {19149,22802}, {23041,24206}

X(34776) = midpoint of X(i) and X(j) for these {i,j}: {1351, 17845}, {5878, 14927}, {6776, 9833}, {12121, 32264}
X(34776) = reflection of X(i) in X(j) for these (i,j): (66, 5092), (1352, 10282), (3818, 206), (9927, 19154), (18376, 19153), (18377, 19155), (18381, 182), (19506, 6593), (22802, 19149), (34507, 15577), (34775, 19130), (34779, 34774), (34786, 5480), (34788, 1353)
X(34776) = anticomplement of X(141) with respect to Kosnita triangle
X(34776) = anticomplement of X(19130) with respect to anti-Honsberger triangle
X(34776) = anticomplement of X(23300) with respect to 1st Ehrmann triangle
X(34776) = anticomplement of X(34775) with respect to Ehrmann-vertex triangle
X(34776) = complement of X(18440) with respect to these triangles: 1st excosine, tangential
X(34776) = complement of X(34775) with respect to anti-Honsberger triangle
X(34776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (66, 5092, 23329), (19130, 34775, 18376), (19153, 34775, 19130)

X(34777) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO 2nd EHRMANN

Barycentrics a^2*(a^6-3*(b^2+c^2)*a^4-(-4*b^2*c^2+(b^2-c^2)^2)*a^2+3*(b^4-c^4)*(b^2-c^2)) : :
X(34777) = 5*X(6)-3*X(154) = 3*X(6)-2*X(206) = 3*X(6)-X(9924) = 2*X(6)-3*X(11216) = X(6)-3*X(17813) = 7*X(6)-5*X(19132) = 4*X(6)-3*X(19153) = 6*X(154)-5*X(159) = 9*X(154)-10*X(206) = 9*X(154)-5*X(9924) = 2*X(154)-5*X(11216) = X(154)-5*X(17813) = 4*X(154)-5*X(19153) = 3*X(159)-4*X(206) = 3*X(159)-2*X(9924) = X(159)-3*X(11216) = X(159)-6*X(17813) = 7*X(159)-10*X(19132) = 2*X(159)-3*X(19153)

The reciprocal ellipsologic center of these triangles is X(159)

X(34777) lies on the cubic K916 and these lines: {3,9972}, {6,25}, {64,14531}, {66,524}, {69,858}, {141,23327}, {157,15905}, {182,34787}, {193,7391}, {382,1351}, {511,3357}, {542,34775}, {575,23041}, {576,19149}, {597,15585}, {599,6697}, {895,32262}, {1353,11819}, {1498,5102}, {1531,18405}, {1576,33582}, {1609,20975}, {1619,3060}, {1992,5596}, {2854,13248}, {3001,3964}, {3098,10249}, {3124,10836}, {3527,16252}, {3564,18569}, {3589,5544}, {3618,10169}, {3631,23332}, {3751,3827}, {3818,23049}, {5050,15577}, {5092,10250}, {5093,34117}, {5097,6759}, {5965,18381}, {6144,25335}, {6329,10192}, {8537,12283}, {8584,31166}, {9605,15270}, {9921,11266}, {9922,11265}, {9934,18534}, {11008,32064}, {11188,26206}, {11255,19139}, {11416,12272}, {11432,12007}, {11443,19121}, {11458,19128}, {11477,34146}, {11482,15581}, {11511,14913}, {11898,34118}, {12161,32368}, {12165,13202}, {12175,32377}, {12421,14790}, {14530,15580}, {14914,29181}, {15516,23042}, {17835,21649}, {18382,18403}, {18859,33878}, {19130,23048}, {19154,32155}, {22530,22658}, {32046,32344}, {32455,34774}

X(34777) = reflection of X(i) in X(j) for these (i,j): (66, 15583), (69, 23300), (159, 6), (6759, 5097), (9924, 206), (11216, 17813), (11898, 34118), (15141, 13248), (18440, 18382), (19139, 11255), (19149, 576), (19153, 11216), (19154, 32155), (31166, 8584), (34774, 32455), (34782, 12007), (34787, 182)
X(34777) = anticomplement of X(6) with respect to 2nd Ehrmann triangle
X(34777) = anticomplement of X(159) with respect to anti-Honsberger triangle
X(34777) = anticomplement of X(3631) with respect to anti-Ursa minor triangle
X(34777) = complement of X(159) with respect to 2nd Ehrmann triangle
X(34777) = X(9973)-of-Ara triangle
X(34777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 159, 19153), (6, 6467, 32621), (6, 7716, 19136), (6, 9924, 206), (6, 9973, 25), (6, 20987, 19118), (159, 11216, 6), (206, 9924, 159), (597, 15585, 31267), (1974, 21639, 6), (6467, 8541, 6), (10602, 12167, 6), (11405, 19125, 6), (11416, 12272, 20806)

X(34778) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER TO TRINH

Barycentrics a^2*(a^10+(b^2+c^2)*a^8-2*(3*b^4-2*b^2*c^2+3*c^4)*a^6+2*(b^2+c^2)*(b^4+c^4)*a^4+(5*b^4+6*b^2*c^2+5*c^4)*(b^2-c^2)^2*a^2-3*(b^4-c^4)^2*(b^2+c^2)) : :
X(34778) = 3*X(3)-2*X(206) = 5*X(3)-2*X(9968) = 4*X(3)-3*X(23041) = 2*X(6)-3*X(10249) = X(6)-3*X(10606) = 3*X(20)+X(20079) = 3*X(64)+X(9924) = 2*X(64)+X(34787) = 5*X(206)-3*X(9968) = 4*X(206)-3*X(19149) = 8*X(206)-9*X(23041) = 3*X(1350)-X(9924) = 2*X(9924)-3*X(34787) = 4*X(9968)-5*X(19149) = 8*X(9968)-15*X(23041) = 2*X(19149)-3*X(23041)

The reciprocal ellipsologic center of these triangles is X(34779)

X(34778) lies on these lines: {3,206}, {6,74}, {20,64}, {30,66}, {141,15311}, {154,6636}, {159,3098}, {376,5596}, {381,6697}, {511,3357}, {549,31267}, {1177,12041}, {1204,12294}, {1352,20427}, {1498,2916}, {1593,19161}, {1597,9969}, {1619,3917}, {1853,3580}, {1974,21663}, {2071,20806}, {2393,33878}, {2777,3818}, {2778,7986}, {2883,7400}, {2892,12244}, {3047,17847}, {3066,5169}, {3088,5480}, {3313,21312}, {3556,3781}, {3589,23328}, {3784,7169}, {5085,8567}, {5092,11204}, {5893,33537}, {5895,10516}, {5907,9914}, {5925,6240}, {6247,17834}, {6759,14810}, {7378,17810}, {7728,15116}, {8703,31166}, {9934,12358}, {10519,12250}, {11250,19139}, {11410,19125}, {11454,19121}, {11468,19128}, {11477,12086}, {12220,13445}, {13293,15141}, {15559,20300}, {15582,16661}, {16252,21167}, {16836,31521}, {18381,29317}, {18440,20127}, {18532,34436}, {19130,23329}, {19140,25564}, {19154,32210}, {21850,23327}, {22241,30270}, {22802,24206}, {23049,23300}

X(34778) = midpoint of X(i) and X(j) for these {i,j}: {64, 1350}, {1352, 20427}, {2892, 12244}
X(34778) = reflection of X(i) in X(j) for these (i,j): (159, 3098), (1177, 12041), (1498, 15577), (5480, 6696), (6759, 14810), (7728, 15116), (10249, 10606), (15141, 13293), (19139, 11250), (19140, 25564), (19149, 3), (19153, 11204), (19154, 32210), (22802, 24206), (31166, 8703), (31670, 23300), (34117, 15578), (34775, 66), (34779, 5092), (34787, 1350)
X(34778) = anticomplement of X(5092) with respect to Trinh triangle
X(34778) = anticomplement of X(34779) with respect to anti-Honsberger triangle
X(34778) = complement of X(6) with respect to anti-Hutson intouch triangle
X(34778) = complement of X(5596) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34778) = complement of X(34779) with respect to Trinh triangle
X(34778) = X(19149)-of-ABC-X3 reflections triangle
X(34778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19149, 23041), (1498, 31884, 15577), (5085, 8567, 15578), (5092, 34779, 19153), (11204, 34779, 5092), (15578, 34117, 5085), (23300, 31670, 23049)

X(34779) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: TRINH TO ANTI-HONSBERGER

Barycentrics a^2*(a^10-4*(b^2+c^2)*a^8+2*(2*b^4-b^2*c^2+2*c^4)*a^6+2*(b^6+c^6)*a^4-(b^2-c^2)^2*(5*b^4+4*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(2*b^4+2*b^2*c^2+2*c^4)) : :
X(34779) = 3*X(3)-5*X(19132) = 2*X(3)-3*X(23042) = 4*X(6)-3*X(10250) = X(64)-3*X(5050) = 2*X(66)-3*X(23325) = 3*X(154)-X(33878) = 2*X(159)-3*X(6759) = X(159)-3*X(19149) = 5*X(159)-3*X(34787) = X(193)+3*X(5656) = X(3357)-4*X(34117) = 5*X(6759)-2*X(34787) = 4*X(19130)-3*X(23325) = 10*X(19132)-9*X(23042) = 5*X(19149)-X(34787)

The reciprocal ellipsologic center of these triangles is X(34778)

X(34779) lies on these lines: {3,19132}, {6,1597}, {25,21851}, {30,34774}, {64,5050}, {66,19130}, {69,6053}, {154,33878}, {155,159}, {182,3357}, {193,1533}, {206,1511}, {378,21637}, {568,32326}, {576,1353}, {578,12294}, {1204,19128}, {1350,10282}, {1351,1498}, {1974,11438}, {2071,19122}, {2777,9970}, {2883,3564}, {3520,19123}, {3589,23329}, {3818,9813}, {5028,32445}, {5092,11204}, {5093,12315}, {5097,8549}, {5476,23300}, {5480,16198}, {5596,18400}, {5878,6776}, {5895,8538}, {6225,14912}, {6247,18583}, {6293,19131}, {6403,26883}, {6467,11456}, {6593,13293}, {7689,19154}, {7728,32264}, {9919,18438}, {9924,32063}, {9934,10752}, {10605,19118}, {10606,12017}, {11250,19155}, {11430,19125}, {12308,32276}, {13346,19139}, {14216,14853}, {14561,20299}, {14810,23041}, {14913,18451}, {15004,32064}, {15083,34380}, {15462,25564}, {15583,23048}, {17854,34470}, {18390,26926}, {19506,32271}, {19924,31166}, {20427,25406}, {29181,34785}, {32139,34382}

X(34779) = midpoint of X(i) and X(j) for these {i,j}: {1351, 1498}, {5596, 31670}, {5878, 6776}, {7728, 32264}, {9934, 10752}, {12308, 32276}
X(34779) = reflection of X(i) in X(j) for these (i,j): (66, 19130), (182, 34117), (1350, 10282), (3098, 206), (3357, 182), (6247, 18583), (6759, 19149), (7689, 19154), (8549, 5097), (11204, 19153), (11250, 19155), (13293, 6593), (13346, 19139), (18381, 5480), (19506, 32271), (34776, 34774), (34778, 5092), (34788, 1351)
X(34779) = anticomplement of X(3098) with respect to Kosnita triangle
X(34779) = anticomplement of X(5092) with respect to anti-Honsberger triangle
X(34779) = anticomplement of X(34778) with respect to Trinh triangle
X(34779) = complement of X(9924) with respect to anti-incircle-circles triangle
X(34779) = complement of X(33878) with respect to these triangles: 1st excosine, tangential
X(34779) = complement of X(34778) with respect to anti-Honsberger triangle
X(34779) = X(21851)-of-Ara triangle
X(34779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (66, 19130, 23325), (206, 3098, 11202), (5092, 34778, 11204), (19153, 34778, 5092)

X(34780) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES TO ANTI-INVERSE-IN-INCIRCLE

Barycentrics 3*a^10-8*(b^2+c^2)*a^8+4*(2*b^4-b^2*c^2+2*c^4)*a^6-6*(b^4-c^4)*(b^2-c^2)*a^4+5*(b^4-c^4)^2*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(34780) = 6*X(2)-5*X(14530) = 7*X(3)-8*X(6696) = 3*X(3)-2*X(9833) = 11*X(3)-12*X(23328) = 5*X(3)-4*X(34782) = 4*X(5)-3*X(32063) = 2*X(5)-3*X(32064) = 3*X(382)-2*X(5895) = 2*X(6247)-3*X(14216) = 12*X(6696)-7*X(9833) = 11*X(9833)-18*X(23328) = 2*X(12250)-3*X(13093) = 3*X(32063)-2*X(34781) = 3*X(32064)-X(34781)

The reciprocal ellipsologic center of these triangles is X(34781)

X(34780) lies on these lines: {2,14530}, {3,66}, {4,3527}, {5,32063}, {25,11457}, {30,11411}, {52,382}, {64,1657}, {140,11206}, {154,3526}, {155,34609}, {161,13564}, {185,18494}, {378,12254}, {381,1498}, {427,19347}, {428,18916}, {546,5656}, {568,34775}, {1181,3574}, {1593,34224}, {1595,6776}, {1597,6146}, {1598,1899}, {1614,5094}, {1619,7506}, {1620,20417}, {1656,1853}, {1885,3426}, {2070,32321}, {2883,3843}, {2888,33524}, {2892,32254}, {3088,31804}, {3167,23335}, {3357,3534}, {3448,9919}, {3517,31383}, {3548,8780}, {3564,12320}, {3627,6225}, {3830,5878}, {5050,5596}, {5054,10282}, {5055,16252}, {5064,7592}, {5070,23332}, {5072,23325}, {5073,15311}, {5076,18405}, {5093,15583}, {5198,16658}, {5447,34750}, {5893,14269}, {5894,15681}, {5907,18536}, {6001,18525}, {6241,7730}, {6243,34146}, {6285,9668}, {6407,8991}, {6408,13980}, {6756,18909}, {7355,9655}, {7387,32140}, {7391,12160}, {7507,11456}, {7553,18917}, {8567,15688}, {9786,13419}, {9899,28146}, {9909,12359}, {9920,21230}, {9968,23049}, {10539,30771}, {10540,15126}, {10594,26869}, {10606,15696}, {10625,11898}, {10691,11487}, {11241,31487}, {11381,18396}, {11402,15559}, {11403,12022}, {11414,11442}, {11645,34726}, {11704,14157}, {12017,14786}, {12085,12166}, {12164,14790}, {12293,14915}, {12964,13665}, {12970,13785}, {13347,18553}, {13488,18945}, {13903,17819}, {13961,17820}, {14516,21312}, {14848,34117}, {15069,15644}, {15484,32445}, {15693,25563}, {15720,17821}, {15811,18390}, {16621,18535}, {17800,20427}, {17834,29012}, {18534,25738}, {21841,23291}, {26888,31479}

X(34780) = reflection of X(i) in X(j) for these (i,j): (3, 14216), (1498, 18381), (1657, 64), (5895, 34786), (6225, 3627), (6759, 14864), (7387, 32140), (9833, 6247), (9919, 3448), (9920, 32337), (12164, 14790), (12315, 4), (13093, 12324), (17800, 20427), (17845, 3357), (32063, 32064), (32254, 2892), (34781, 5)
X(34780) = anticomplement of X(5) with respect to anti-inverse-in-incircle triangle
X(34780) = anticomplement of X(1498) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34780) = anticomplement of X(34781) with respect to anti-incircle-circles triangle
X(34780) = anticomplement of X(34785) with respect to anti-Hutson intouch triangle
X(34780) = complement of X(34781) with respect to anti-inverse-in-incircle triangle
X(34780) = X(14216)-of-X3-ABC reflections triangle
X(34780) = X(16659)-of-Ara triangle
X(34780) = X(34781)-of-Johnson triangle
X(34780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 11245, 3527), (4, 18914, 11432), (5, 34781, 32063), (25, 11457, 26944), (154, 20299, 3526), (1498, 18381, 381), (1595, 6776, 11426), (1853, 6759, 1656), (1899, 16655, 1598), (3357, 17845, 3534), (5895, 34786, 382), (6247, 9833, 3), (6759, 14864, 1853), (9833, 14216, 6247), (10606, 34785, 15696), (11457, 16659, 25), (16658, 18912, 5198), (17821, 23329, 15720), (18405, 22802, 5076), (32064, 34781, 5)

X(34781) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE TO ANTI-INCIRCLE-CIRCLES

Barycentrics 3*a^10-9*(b^2+c^2)*a^8+2*(5*b^4-2*b^2*c^2+5*c^4)*a^6-6*(b^4-c^4)*(b^2-c^2)*a^4+(b^2-c^2)^2*(3*b^2+c^2)*(b^2+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(34781) = 9*X(2)-8*X(20299) = 2*X(4)-3*X(5656) = 2*X(5)-3*X(32063) = 4*X(5)-3*X(32064) = 3*X(20)-2*X(20427) = 3*X(20)-4*X(34785) = 3*X(376)-4*X(34782) = 3*X(9833)-X(20427) = 3*X(9833)-2*X(34785) = 3*X(12250)-4*X(20427) = 2*X(15644)-3*X(34750)

The reciprocal ellipsologic center of these triangles is X(34780)

X(34781) lies on these lines: {2,6759}, {3,11206}, {4,6}, {5,32063}, {20,2979}, {24,1619}, {25,18909}, {30,6193}, {51,11431}, {64,376}, {66,7383}, {69,11414}, {140,14530}, {154,631}, {159,10323}, {161,12088}, {184,3088}, {185,7487}, {221,1056}, {382,31802}, {389,6995}, {399,13203}, {511,12271}, {542,34621}, {550,13093}, {944,3057}, {1058,2192}, {1075,6525}, {1209,3547}, {1352,7400}, {1370,11441}, {1495,26937}, {1568,32605}, {1593,18925}, {1595,11427}, {1597,31804}, {1598,11433}, {1614,3541}, {1853,3090}, {1899,3089}, {1907,11402}, {2060,23239}, {2777,5059}, {3085,26888}, {3086,10535}, {3091,18381}, {3146,5878}, {3357,3522}, {3424,13599}, {3431,16620}, {3448,9934}, {3515,18931}, {3523,10282}, {3524,6696}, {3525,10192}, {3528,10606}, {3529,15311}, {3538,17811}, {3542,11457}, {3543,22802}, {3546,10539}, {3548,10540}, {3564,12311}, {3575,12174}, {3839,18383}, {4293,7355}, {5056,14864}, {5067,23332}, {5068,14862}, {5198,11245}, {5889,7500}, {5894,17538}, {5895,33703}, {5925,11001}, {6241,18533}, {6526,6761}, {6643,18451}, {6823,18440}, {6926,14925}, {7386,17814}, {7387,11411}, {7395,25406}, {7486,32767}, {7488,32321}, {7517,18917}, {7530,18951}, {8408,9838}, {8420,9839}, {8567,21735}, {8718,11180}, {8780,16196}, {9540,10533}, {9899,31730}, {9914,12082}, {10117,12317}, {10299,23328}, {10303,23329}, {10534,13935}, {10594,18916}, {11202,15717}, {11204,21734}, {11270,16623}, {11381,19467}, {11398,18922}, {11399,18915}, {11412,34146}, {11459,11821}, {12118,14915}, {12254,15739}, {12363,18439}, {12420,17702}, {13754,31305}, {14790,32139}, {15030,34776}, {15125,25739}, {15595,28425}, {15697,32903}, {17578,34786}, {17834,34608}, {20125,23315}, {20424,32346}, {21841,26944}

X(34781) = reflection of X(i) in X(j) for these (i,j): (4, 1498), (20, 9833), (64, 34782), (3146, 5878), (3448, 9934), (3529, 17845), (6225, 12315), (6776, 5596), (9899, 31730), (11411, 7387), (12250, 20), (12254, 32359), (12317, 10117), (12324, 3), (13093, 550), (13203, 399), (14216, 6759), (14790, 32139), (20079, 1352), (20427, 34785), (32064, 32063), (33703, 5895), (34780, 5)
X(34781) = anticomplement of X(14216)
X(34781) = anticomplement of X(3) with respect to Aries triangle
X(34781) = anticomplement of X(5) with respect to anti-incircle-circles triangle
X(34781) = anticomplement of X(64) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34781) = anticomplement of X(6247) with respect to these triangles: 1st excosine, tangential
X(34781) = anticomplement of X(14216) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34781) = anticomplement of X(34780) with respect to anti-inverse-in-incircle triangle
X(34781) = complement of X(12324) with respect to Aries triangle
X(34781) = complement of X(34780) with respect to anti-incircle-circles triangle
X(34781) = X(1498)-of-anti-Euler triangle
X(34781) = X(12324)-of-ABC-X3 reflections triangle
X(34781) = X(34780)-of-Johnson triangle
X(34781) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 1498, 5656), (4, 7592, 14853), (4, 14912, 10982), (4, 34224, 18945), (5, 34780, 32064), (6, 16621, 4), (64, 34782, 376), (154, 6247, 631), (185, 31383, 7487), (1181, 16655, 4), (5893, 18405, 4), (6759, 14216, 2), (7592, 16658, 4), (9833, 20427, 34785), (10982, 16654, 4), (11206, 12324, 3), (11456, 16659, 4), (12241, 15811, 4), (20427, 34785, 20), (32063, 34780, 5)

X(34782) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: ANTI-URSA MINOR TO KOSNITA

Barycentrics 4*a^10-9*(b^2+c^2)*a^8+4*(b^2+c^2)^2*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4-(b^4-c^4)*(b^2-c^2)^3 : :
X(34782) = 3*X(2)-5*X(17821) = 3*X(3)-2*X(6696) = 3*X(3)-X(14216) = 4*X(3)-3*X(23328) = 5*X(3)-X(34780) = 3*X(141)-2*X(34118) = 3*X(6247)-4*X(6696) = X(6247)+2*X(9833) = 3*X(6247)-2*X(14216) = 2*X(6247)-3*X(23328) = 5*X(6247)-2*X(34780) = 2*X(6696)+3*X(9833) = 8*X(6696)-9*X(23328) = 10*X(6696)-3*X(34780) = 3*X(9833)+X(14216) = 4*X(9833)+3*X(23328) = 5*X(9833)+X(34780)

The reciprocal ellipsologic center of these triangles is X(18381)

X(34782) lies on these lines: {2,17821}, {3,66}, {4,154}, {5,5944}, {6,7487}, {20,394}, {24,161}, {25,12241}, {26,19908}, {30,156}, {49,32379}, {51,10619}, {52,3629}, {53,1970}, {54,7576}, {64,376}, {68,14070}, {110,12225}, {140,11202}, {182,9825}, {184,3575}, {185,34750}, {186,34224}, {206,578}, {221,4293}, {235,1495}, {343,1601}, {378,16655}, {382,5893}, {389,2393}, {397,11243}, {398,11244}, {403,12289}, {427,13367}, {428,11424}, {511,34774}, {524,6193}, {546,34786}, {548,3357}, {549,20299}, {550,1216}, {569,597}, {575,23326}, {631,1853}, {632,10182}, {858,11449}, {993,20306}, {1181,13568}, {1192,18909}, {1350,5596}, {1353,16625}, {1511,23315}, {1576,2055}, {1587,17819}, {1588,17820}, {1593,16621}, {1594,11464}, {1595,11430}, {1596,13403}, {1597,16656}, {1614,6240}, {1619,11414}, {1620,18931}, {1657,5878}, {1658,12359}, {1854,4305}, {1885,26883}, {1899,3515}, {1971,5254}, {1993,31304}, {2070,9920}, {2192,4294}, {2777,5609}, {2781,10625}, {2854,21651}, {3070,10533}, {3071,10534}, {3079,6523}, {3091,18405}, {3135,15653}, {3518,12022}, {3520,16659}, {3522,10606}, {3523,32064}, {3528,8567}, {3529,5656}, {3530,23329}, {3534,12315}, {3542,15448}, {3556,12114}, {3564,34776}, {3567,34751}, {3589,7401}, {3628,23325}, {3796,6815}, {3827,12675}, {3850,18376}, {3878,4297}, {5050,9815}, {5085,6803}, {5318,30402}, {5321,30403}, {5334,17827}, {5335,17826}, {5449,34351}, {5663,16775}, {5799,7511}, {5842,10537}, {6200,8991}, {6241,10295}, {6253,10536}, {6284,10535}, {6285,15338}, {6293,11412}, {6353,18945}, {6361,7973}, {6396,13980}, {6776,9786}, {7354,26888}, {7355,15326}, {7387,12118}, {7404,34775}, {7405,20300}, {7528,18382}, {7542,18474}, {7553,13352}, {7568,34513}, {7691,32359}, {7715,10110}, {9306,12362}, {9541,19088}, {9820,18569}, {9927,13383}, {9934,12121}, {9971,11387}, {10018,25739}, {10115,32196}, {10117,12088}, {10272,19506}, {10274,22051}, {10539,12605}, {10540,18563}, {10594,16657}, {11204,33923}, {11426,19153}, {11432,12007}, {11438,18914}, {11440,15138}, {11457,32534}, {11496,18621}, {11585,11750}, {12038,23335}, {12107,13289}, {12173,26864}, {12250,17538}, {12278,26881}, {12279,16386}, {12301,23044}, {12429,16195}, {13093,15696}, {13160,13394}, {13371,32171}, {13561,34477}, {13754,32145}, {14157,18560}, {14389,32346}, {14585,27376}, {14683,17835}, {14853,19132}, {14864,15712}, {14865,16658}, {14925,31789}, {14927,28419}, {15034,15131}, {15258,34286}, {15508,22261}, {15644,34146}, {15647,17702}, {15750,26937}, {15761,30522}, {18324,32140}, {18390,21841}, {18583,23042}, {19149,29181}, {19205,26887}, {20307,25440}, {31166,34726}, {31830,32046}, {32734,34579}

X(34782) = midpoint of X(i) and X(j) for these {i,j}: {3, 9833}, {4, 17845}, {20, 1498}, {64, 34781}, {1350, 5596}, {1657, 5878}, {2917, 32354}, {3529, 5895}, {5925, 6225}, {6193, 17834}, {6293, 11412}, {6361, 7973}, {6759, 34785}, {6776, 9924}, {7387, 12118}, {7691, 32359}, {9934, 12121}, {10117, 12383}, {12315, 20427}, {14683, 17835}
X(34782) = reflection of X(i) in X(j) for these (i,j): (4, 16252), (5, 10282), (141, 15577), (382, 5893), (1352, 15585), (2883, 6759), (3357, 548), (5480, 206), (5894, 550), (6247, 3), (9927, 13383), (12359, 1658), (13371, 32171), (14216, 6696), (14864, 25563), (15105, 5894), (15583, 182), (18381, 140), (18569, 9820), (19506, 10272), (22660, 156), (23315, 1511), (23324, 10192), (23332, 11202), (23335, 12038), (32351, 32391), (34777, 12007), (34786, 546)
X(34782) = anticomplement of X(140) with respect to Kosnita triangle
X(34782) = anticomplement of X(16252) with respect to these triangles: 1st excosine, tangential
X(34782) = anticomplement of X(18381) with respect to anti-Ursa minor triangle
X(34782) = complement of X(4) with respect to these triangles: 1st excosine, tangential
X(34782) = complement of X(64) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34782) = complement of X(1498) with respect to Aries triangle
X(34782) = complement of X(5878) with respect to anti-incircle-circles triangle
X(34782) = complement of X(12324) with respect to anti-Hutson intouch triangle
X(34782) = complement of X(18381) with respect to Kosnita triangle
X(34782) = X(6247)-of-ABC-X3 reflections triangle
X(34782) = X(12241)-of-Ara triangle
X(34782) = X(16252)-of-anti-Euler triangle
X(34782) = X(17845)-of-Euler triangle
X(34782) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6247, 23328), (3, 14216, 6696), (4, 154, 16252), (4, 19357, 23292), (6, 7487, 11745), (20, 6225, 5925), (20, 11206, 1498), (24, 6146, 13567), (25, 12241, 15873), (25, 19467, 12241), (140, 18381, 23332), (154, 17845, 4), (184, 3575, 12233), (376, 34781, 64), (1495, 21659, 235), (1498, 5925, 6225), (6696, 14216, 6247), (7487, 18925, 6), (7488, 14516, 343), (11202, 18381, 140)

X(34783) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE TO CIRCUMORTHIC

Barycentrics a^2*((b^2+c^2)*a^4-(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(-a^2+b^2+c^2) : :
X(34783) = 3*X(2)-4*X(13630) = 9*X(2)-8*X(14128) = 5*X(3)-4*X(1216) = 7*X(3)-6*X(3917) = 9*X(3)-8*X(5447) = 3*X(3)-2*X(5562) = 4*X(3)-3*X(23039) = 5*X(185)-2*X(1216) = 7*X(185)-3*X(3917) = 9*X(185)-4*X(5447) = 3*X(185)-X(5562) = 4*X(185)-X(18436) = 8*X(185)-3*X(23039) = 14*X(1216)-15*X(3917) = 9*X(1216)-10*X(5447) = 6*X(1216)-5*X(5562) = 8*X(1216)-5*X(18436) = 16*X(1216)-15*X(23039) = 9*X(3917)-7*X(5562) = 12*X(3917)-7*X(18436) = 8*X(3917)-7*X(23039) = 3*X(5876)-4*X(14128) = 3*X(13630)-2*X(14128)

The reciprocal ellipsologic center of these triangles is X(12111)

X(34783) lies on these lines: {2,5876}, {3,49}, {4,94}, {5,5890}, {20,1154}, {24,10540}, {26,3581}, {30,5889}, {32,22146}, {51,3843}, {52,382}, {54,11440}, {64,15317}, {68,3521}, {69,14861}, {74,11250}, {98,14719}, {125,5448}, {140,10574}, {156,186}, {195,2935}, {355,31728}, {376,6101}, {378,12161}, {381,389}, {399,2929}, {511,1657}, {541,12897}, {546,3567}, {547,15028}, {548,2979}, {549,11444}, {550,11412}, {567,7526}, {576,16010}, {578,14130}, {631,11591}, {974,5654}, {1199,7527}, {1351,8549}, {1482,2807}, {1498,7517}, {1614,1658}, {1656,5907}, {1885,13292}, {1899,18404}, {1993,12084}, {1994,14865}, {2070,6759}, {2072,22660}, {2772,31825}, {2777,10111}, {2781,20427}, {2883,11799}, {2937,32608}, {3060,3627}, {3090,12006}, {3091,5946}, {3146,10263}, {3519,4846}, {3520,32138}, {3522,10627}, {3523,15067}, {3524,32142}, {3526,5891}, {3529,13391}, {3530,7999}, {3534,10625}, {3544,11017}, {3545,15026}, {3548,18913}, {3580,15761}, {3628,15045}, {3629,15105}, {3830,5446}, {3832,10095}, {3845,9781}, {3850,5640}, {3851,5462}, {3853,11455}, {3855,13364}, {3858,16261}, {4550,15037}, {5054,11793}, {5056,13363}, {5070,5892}, {5071,32205}, {5072,5943}, {5073,14915}, {5076,13474}, {5079,15012}, {5093,12294}, {5449,10254}, {5576,12233}, {5655,25711}, {5878,31725}, {5944,10298}, {6146,10938}, {6193,22647}, {6238,18447}, {6288,11442}, {6644,11441}, {6776,15074}, {6971,34462}, {7352,18455}, {7387,12174}, {7391,31815}, {7503,13353}, {7506,9786}, {7509,13339}, {7525,7691}, {7540,16655}, {7575,26882}, {7577,13561}, {7998,15712}, {9544,21844}, {9545,32210}, {9707,18324}, {9714,32063}, {9971,25335}, {10024,12359}, {10110,13321}, {10116,18562}, {10224,10264}, {10226,11468}, {10282,32110}, {10610,11003}, {10619,22815}, {10982,11472}, {11270,16665}, {11413,16266}, {11424,14627}, {11449,15646}, {11457,18569}, {11464,15331}, {11559,15002}, {11806,21650}, {11819,16659}, {11935,11999}, {12041,12219}, {12083,17834}, {12085,12160}, {12106,14094}, {12107,26881}, {12118,12121}, {12134,13568}, {12239,13665}, {12240,13785}, {12270,12278}, {12273,34153}, {12315,18534}, {12358,18931}, {12370,18560}, {12605,18914}, {12699,31732}, {12825,14643}, {12902,21649}, {13201,14677}, {13346,18859}, {13348,15688}, {13445,15801}, {14216,31723}, {14531,17800}, {14641,15681}, {14805,15032}, {14855,15644}, {14862,32223}, {15027,15738}, {15033,15062}, {15041,17855}, {15068,17928}, {15684,21969}, {15693,17704}, {15720,16836}, {15740,34483}, {16003,20299}, {16226,19709}, {17505,32533}, {17702,18565}, {17854,20127}, {18281,20126}, {18377,25739}, {18378,26883}, {18381,31724}, {18394,18567}, {18440,19161}, {18531,18909}, {20478,32617}, {20479,32616}, {21735,33884}, {22802,31726}

X(34783) = midpoint of X(i) and X(j) for these {i,j}: {5889, 6241}, {12270, 12284}
X(34783) = reflection of X(i) in X(j) for these (i,j): (3, 185), (4, 6102), (20, 13491), (355, 31728), (382, 52), (399, 11562), (1657, 10575), (1885, 13292), (3146, 10263), (3830, 14831), (5876, 13630), (5907, 13382), (6243, 5889), (7723, 974), (7728, 1986), (11381, 5446), (11412, 550), (12111, 5), (12134, 13568), (12162, 389), (12219, 12041), (12273, 34153), (12281, 10264), (12290, 3627), (12292, 12236), (12605, 18914), (12699, 31732), (12825, 14708), (12902, 21649), (13201, 14677), (13340, 15072), (13474, 16625), (15684, 21969), (16659, 11819), (18435, 5890), (18436, 3), (18438, 6776), (18439, 4), (18440, 19161), (18560, 12370), (18562, 21659), (18563, 6146), (20127, 17854), (21650, 11806), (21659, 10116), (22584, 125), (22815, 10619)
X(34783) = anticomplement of X(5876)
X(34783) = anticomplement of X(5) with respect to these triangles: 4th anti-Euler, circumorthic
X(34783) = anticomplement of X(3627) with respect to reflection triangle
X(34783) = anticomplement of X(5876) with respect to these triangles: 1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34783) = anticomplement of X(6101) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34783) = anticomplement of X(12111) with respect to Ehrmann-side triangle
X(34783) = anticomplement of X(12162) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34783) = complement of X(12111) with respect to these triangles: 4th anti-Euler, circumorthic
X(34783) = complement of X(12290) with respect to reflection triangle
X(34783) = X(185)-of-X3-ABC reflections triangle
X(34783) = X(6102)-of-anti-Euler triangle
X(34783) = X(12111)-of-Johnson triangle
X(34783) = X(18436)-of-ABC-X3 reflections triangle
X(34783) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 155, 22115), (3, 9703, 12038), (3, 9704, 13367), (3, 18436, 23039), (3, 18445, 49), (4, 6102, 568), (4, 18917, 25738), (4, 25738, 265), (5, 12111, 18435), (155, 10605, 3), (184, 7689, 3), (568, 18439, 4), (1147, 1204, 3), (1181, 12163, 3), (5876, 13630, 2), (5890, 12111, 5), (5890, 15058, 15043), (12038, 21663, 3), (12111, 15043, 15058), (15043, 15058, 5)

X(34784) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO INNER-CONWAY

Barycentrics a*((b+c)*a^3-3*(b^2+b*c+c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-(b^3-c^3)*(b-c)) : :
X(34784) = 6*X(2)-5*X(11025) = 3*X(7)-4*X(15587) = 2*X(9)-3*X(3681) = 4*X(9)-3*X(7671) = 4*X(142)-3*X(3873) = X(144)-3*X(4661) = 3*X(210)-2*X(5572) = 6*X(210)-5*X(18230) = 3*X(3059)-2*X(15587) = 3*X(3681)-X(30628) = 4*X(5572)-5*X(18230) = 3*X(7671)-2*X(30628) = 5*X(11025)-4*X(15185)

The reciprocal ellipsologic center of these triangles is X(30628)

X(34784) lies on these lines: {1,21039}, {2,11025}, {7,8}, {9,1174}, {40,12669}, {63,3174}, {72,390}, {78,7677}, {142,3873}, {144,4661}, {200,1445}, {210,5572}, {306,22312}, {516,5904}, {519,21084}, {527,25722}, {528,12532}, {908,7678}, {956,30284}, {960,8236}, {971,6361}, {984,4343}, {1001,3876}, {1145,12755}, {1698,20116}, {2801,2951}, {3057,12630}, {3243,19860}, {3555,11038}, {3679,30329}, {3729,12718}, {3869,5853}, {3935,6600}, {4326,5223}, {4847,21617}, {4853,11526}, {5686,5728}, {5692,30331}, {5696,5850}, {5809,5815}, {6172,10385}, {6734,7679}, {6743,12573}, {8270,34028}, {9846,20007}, {11220,11495}, {12526,12706}, {12848,17620}, {16133,31938}, {17668,20059}, {18412,31397}, {24393,24987}

X(34784) = reflection of X(i) in X(j) for these (i,j): (7, 3059), (390, 72), (3868, 2550), (5728, 34790), (7671, 3681), (7672, 8), (7673, 3869), (9846, 20007), (10394, 5223), (12573, 6743), (12630, 3057), (12669, 40), (12706, 12526), (12718, 3729), (12755, 1145), (12848, 17658), (16133, 31938), (20059, 17668), (30628, 9)
X(34784) = anticomplement of X(15185)
X(34784) = (inner-Conway)-isogonal conjugate of X(3434)
X(34784) = anticomplement of X(9) with respect to inner-Conway triangle
X(34784) = anticomplement of X(5572) with respect to extouch triangle
X(34784) = anticomplement of X(15185) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34784) = anticomplement of X(30329) with respect to these triangles: Aquila, outer-Garcia
X(34784) = anticomplement of X(30331) with respect to inner-Garcia triangle
X(34784) = anticomplement of X(30628) with respect to Honsberger triangle
X(34784) = complement of X(30628) with respect to inner-Conway triangle
X(34784) = X(66)-of-inner-Conway triangle
X(34784) = X(5596)-of-Honsberger triangle
X(34784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 15185, 11025), (9, 3870, 2346), (9, 30628, 7671), (63, 3174, 7676), (144, 20015, 7674), (210, 5572, 18230), (908, 24389, 7678), (3681, 30628, 9), (5728, 34790, 5686), (7672, 10865, 7)

X(34785) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: EHRMANN-VERTEX TO KOSNITA

Barycentrics 3*a^10-6*(b^2+c^2)*a^8+(b^4+8*b^2*c^2+c^4)*a^6+3*(b^4-c^4)*(b^2-c^2)*a^4-2*(b^2-c^2)^2*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
X(34785) = 5*X(3)-3*X(1853) = 7*X(3)-6*X(10193) = 3*X(3)-2*X(20299) = 4*X(3)-3*X(23329) = 5*X(3)-4*X(25563) = 2*X(1209)-3*X(23358) = 7*X(1853)-10*X(10193) = 3*X(1853)+5*X(17845) = 6*X(1853)-5*X(18381) = 9*X(1853)-10*X(20299) = 4*X(1853)-5*X(23329) = 3*X(1853)-4*X(25563) = 6*X(10193)+7*X(17845) = 12*X(10193)-7*X(18381) = 9*X(10193)-7*X(20299) = 8*X(10193)-7*X(23329) = 15*X(10193)-14*X(25563)

The reciprocal ellipsologic center of these triangles is X(34786)

X(34785) lies on these lines: {2,18383}, {3,161}, {4,1495}, {5,11202}, {20,2979}, {24,18390}, {25,13403}, {26,9938}, {30,156}, {54,18559}, {64,3534}, {66,14810}, {125,32534}, {140,23325}, {154,382}, {159,12085}, {182,31833}, {184,6240}, {186,12289}, {206,11819}, {376,14216}, {381,17821}, {389,18533}, {399,1498}, {511,12118}, {542,12163}, {546,10192}, {548,6247}, {550,1503}, {575,23048}, {578,3575}, {631,32767}, {1092,12225}, {1181,15135}, {1204,10295}, {1511,19506}, {1593,13419}, {1656,10182}, {1658,9927}, {1971,7748}, {3153,11449}, {3292,3529}, {3515,18396}, {3520,11550}, {3522,14864}, {3528,32064}, {3530,23332}, {3627,16252}, {3628,23324}, {3818,7526}, {4316,7355}, {4324,6285}, {4325,32065}, {4330,11189}, {5059,5656}, {5073,14530}, {5449,18324}, {5480,23042}, {5890,12254}, {5894,12103}, {5895,17800}, {5925,12315}, {6146,11438}, {6225,11001}, {6241,13619}, {6696,8703}, {6776,13382}, {7387,22658}, {7487,10110}, {7488,12278}, {7491,14925}, {7505,13851}, {7530,12897}, {7576,11424}, {7592,10619}, {7706,32046}, {8550,34788}, {8907,11413}, {9306,12605}, {9935,12380}, {10274,20424}, {10483,26888}, {10539,18563}, {10540,18562}, {10606,15696}, {10628,11412}, {11216,33749}, {11243,16965}, {11244,16964}, {11414,32321}, {11425,18494}, {11457,21663}, {11820,22951}, {12038,18569}, {12173,18388}, {12293,14070}, {12324,17538}, {12429,32263}, {12584,15581}, {13474,31383}, {13568,31804}, {14915,34350}, {14940,18394}, {15035,32743}, {15311,15704}, {15332,32138}, {16003,16219}, {17506,23294}, {17508,23300}, {18350,18564}, {18377,32171}, {18560,26883}, {18882,32379}, {19106,30402}, {19107,30403}, {19130,23041}, {19149,29317}, {20190,23327}, {21844,25739}, {23049,25555}, {23328,33923}, {24206,34775}, {25738,32110}, {29181,34779}, {32365,32391}

X(34785) = midpoint of X(i) and X(j) for these {i,j}: {3, 17845}, {20, 9833}, {1498, 1657}, {3529, 5878}, {5895, 17800}, {5925, 12315}, {20427, 34781}
X(34785) = reflection of X(i) in X(j) for these (i,j): (4, 10282), (66, 14810), (3357, 550), (3627, 16252), (3818, 15577), (5894, 12103), (6247, 548), (6759, 34782), (9927, 1658), (13293, 16163), (18376, 11202), (18377, 32171), (18381, 3), (18405, 10182), (18569, 12038), (19506, 1511), (22802, 6759), (32138, 15332), (32365, 32391), (32402, 14076), (34775, 24206), (34786, 5), (34788, 8550)
X(34785) = anticomplement of X(18383)
X(34785) = anticomplement of X(5) with respect to Kosnita triangle
X(34785) = anticomplement of X(6247) with respect to Trinh triangle
X(34785) = anticomplement of X(18383) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34785) = anticomplement of X(34786) with respect to Ehrmann-vertex triangle
X(34785) = complement of X(382) with respect to these triangles: 1st excosine, tangential
X(34785) = complement of X(5878) with respect to Aries triangle
X(34785) = complement of X(5895) with respect to anti-incircle-circles triangle
X(34785) = complement of X(14216) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34785) = complement of X(34780) with respect to anti-Hutson intouch triangle
X(34785) = complement of X(34786) with respect to Kosnita triangle
X(34785) = X(10282)-of-anti-Euler triangle
X(34785) = X(13403)-of-Ara triangle
X(34785) = X(18381)-of-ABC-X3 reflections triangle
X(34785) = X(34786)-of-Johnson triangle
X(34785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1853, 25563), (5, 34786, 18376), (20, 34781, 20427), (24, 21659, 18390), (548, 6247, 11204), (3529, 11206, 5878), (9833, 20427, 34781), (10295, 34224, 1204), (11202, 34786, 5), (12173, 19357, 18388), (12315, 15681, 5925), (15696, 34780, 10606), (17800, 32063, 5895), (18533, 19467, 389)

X(34786) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO EHRMANN-VERTEX

Barycentrics 3*a^10-5*(b^2+c^2)*a^8-(b^4-8*b^2*c^2+c^4)*a^6+3*(b^4-c^4)*(b^2-c^2)*a^4+2*(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :
X(34786) = X(3)-3*X(18405) = 2*X(3)-3*X(23325) = 3*X(3)-4*X(32767) = 5*X(3)-4*X(32903) = 3*X(4)-X(9833) = 7*X(4)-3*X(11206) = 7*X(4)-4*X(14862) = 4*X(3574)-3*X(10274) = 2*X(3574)-3*X(32365) = 3*X(6759)-2*X(9833) = 7*X(6759)-6*X(11206) = 7*X(6759)-8*X(14862) = 7*X(9833)-9*X(11206) = 7*X(9833)-12*X(14862) = 3*X(11206)-4*X(14862) = 2*X(18383)-3*X(18405) = 4*X(18383)-3*X(23325) = 3*X(18383)-2*X(32767) = 5*X(18383)-2*X(32903) = 9*X(18405)-4*X(32767) = 15*X(18405)-4*X(32903) = 9*X(23325)-8*X(32767) = 15*X(23325)-8*X(32903) = 5*X(32767)-3*X(32903)

The reciprocal ellipsologic center of these triangles is X(34785)

X(34786) lies on these lines: {3,18383}, {4,54}, {5,11202}, {20,11204}, {24,13851}, {30,3357}, {52,382}, {64,5073}, {66,29317}, {140,23324}, {154,3843}, {156,18567}, {182,18382}, {186,11704}, {376,25563}, {378,11572}, {381,10282}, {389,12173}, {511,12293}, {542,34788}, {546,34782}, {548,23332}, {550,23329}, {575,23049}, {576,1353}, {1092,3153}, {1147,18377}, {1204,25739}, {1498,3830}, {1531,11441}, {1657,1853}, {1658,18379}, {2777,3146}, {2883,3853}, {2937,32345}, {3090,10182}, {3522,10193}, {3542,7687}, {3543,5878}, {3575,18390}, {3845,16252}, {3850,10192}, {3851,17821}, {5448,18568}, {5480,34776}, {5893,15687}, {6001,33697}, {6240,11438}, {6243,10628}, {6288,18564}, {6696,15704}, {7488,18392}, {7502,32401}, {7507,11430}, {7547,13367}, {8549,11645}, {8550,23048}, {8567,15681}, {9306,18404}, {10110,18494}, {10113,13289}, {10296,12111}, {10535,18514}, {10539,18403}, {10606,17800}, {10721,14448}, {11550,18560}, {12084,15126}, {12103,23328}, {12121,32743}, {12262,28168}, {12295,31725}, {12324,15682}, {13093,15684}, {13346,15132}, {13352,31724}, {13565,23358}, {13619,23294}, {14269,14530}, {14805,32395}, {15063,23043}, {15083,32423}, {15580,32367}, {16219,20379}, {17508,20300}, {17578,34781}, {17822,33534}, {18386,19357}, {18474,18563}, {18513,26888}, {18534,32321}, {18553,34787}, {19130,23042}, {20427,32064}

X(34786) = midpoint of X(i) and X(j) for these {i,j}: {64, 5073}, {3146, 14216}, {5895, 34780}, {20427, 33703}
X(34786) = reflection of X(i) in X(j) for these (i,j): (3, 18383), (20, 20299), (64, 14864), (156, 18567), (182, 18382), (1147, 18377), (1658, 18379), (2883, 3853), (3357, 18381), (6759, 4), (10274, 32365), (11202, 18376), (12121, 32743), (13289, 10113), (13346, 18569), (15704, 6696), (17845, 10282), (22802, 3627), (23325, 18405), (23358, 32369), (34776, 5480), (34782, 546), (34785, 5), (34787, 18553)
X(34786) = anticomplement of X(5) with respect to Ehrmann-vertex triangle
X(34786) = anticomplement of X(20) with respect to Trinh triangle
X(34786) = anticomplement of X(548) with respect to anti-Ursa minor triangle
X(34786) = anticomplement of X(10282) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34786) = anticomplement of X(16252) with respect to Ehrmann-mid triangle
X(34786) = anticomplement of X(34785) with respect to Kosnita triangle
X(34786) = complement of X(17800) with respect to anti-Hutson intouch triangle
X(34786) = complement of X(17845) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34786) = complement of X(20427) with respect to anti-inverse-in-incircle triangle
X(34786) = complement of X(34785) with respect to Ehrmann-vertex triangle
X(34786) = X(17845)-of-Ehrmann-mid triangle
X(34786) = X(18383)-of-X3-ABC reflections triangle
X(34786) = X(34785)-of-Johnson triangle
X(34786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 18383, 23325), (3, 18405, 18383), (4, 19467, 18388), (5, 34785, 11202), (20, 20299, 11204), (381, 17845, 10282), (382, 34780, 5895), (578, 6759, 10274), (3153, 12278, 1092), (11206, 14862, 6759), (12173, 18396, 389), (18376, 34785, 5), (32064, 33703, 20427)

X(34787) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: 2nd EHRMANN TO KOSNITA

Barycentrics a^2*(a^10+(b^2+c^2)*a^8-6*(b^2+c^2)^2*a^6+2*(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^4+(5*b^4+6*b^2*c^2+5*c^4)*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)*(-3*b^4+2*b^2*c^2-3*c^4)) : :
X(34787) = 4*X(3)-3*X(10249) = 4*X(5)-3*X(23049) = 3*X(6)-5*X(17821) = 2*X(6)-3*X(23041) = X(64)-3*X(1350) = X(64)+3*X(9924) = 2*X(64)-3*X(34778) = 3*X(154)-X(11477) = 3*X(154)-4*X(15582) = 3*X(154)-2*X(34117) = 2*X(8549)-3*X(10249) = 2*X(9924)+X(34778) = X(11477)-4*X(15582) = 6*X(15577)-5*X(17821) = 4*X(15577)-3*X(23041) = 10*X(17821)-9*X(23041)

The reciprocal ellipsologic center of these triangles is X(34788)

X(34787) lies on these lines: {3,2393}, {5,23049}, {6,24}, {20,64}, {23,154}, {26,1177}, {49,206}, {140,23327}, {141,6643}, {155,159}, {161,394}, {182,34777}, {195,32367}, {389,32621}, {524,6193}, {542,12163}, {575,11202}, {576,10282}, {599,34118}, {1154,9925}, {1352,12605}, {1495,11470}, {1498,2781}, {1511,13248}, {1619,34750}, {1658,8548}, {1660,9909}, {1843,11424}, {1853,16063}, {2883,34621}, {3066,14389}, {3089,5480}, {3098,33543}, {3515,10602}, {3517,19136}, {3527,9969}, {3763,20300}, {4232,10192}, {5085,22467}, {5097,23042}, {5102,19132}, {5622,32534}, {5965,34776}, {6644,15074}, {7395,29959}, {7503,11188}, {8263,12362}, {8541,13367}, {8550,9786}, {8553,10608}, {8567,15579}, {9935,12363}, {9968,32063}, {9971,10982}, {10250,20190}, {10295,10605}, {10516,18382}, {10601,34751}, {11178,18383}, {11255,32171}, {11416,11449}, {11425,12061}, {11898,12307}, {12087,15580}, {12315,18436}, {12584,15141}, {18350,18438}, {18396,32113}, {18400,34507}, {18553,34786}, {18583,31267}, {19161,19459}, {19588,32392}, {19908,34382}, {19924,22802}, {23048,25555}, {24206,33540}

X(34787) = midpoint of X(i) and X(j) for these {i,j}: {1350, 9924}, {15069, 17845}
X(34787) = reflection of X(i) in X(j) for these (i,j): (6, 15577), (195, 32367), (576, 10282), (1351, 206), (1498, 15581), (5480, 15585), (8548, 1658), (8549, 3), (11216, 11202), (11255, 32171), (11477, 34117), (13248, 1511), (15141, 12584), (19149, 159), (32368, 32391), (34117, 15582), (34775, 1352), (34777, 182), (34778, 1350), (34786, 18553), (34788, 575)
X(34787) = anticomplement of X(575) with respect to Kosnita triangle
X(34787) = anticomplement of X(576) with respect to anti-Honsberger triangle
X(34787) = anticomplement of X(9968) with respect to anti-incircle-circles triangle
X(34787) = anticomplement of X(34117) with respect to these triangles: 1st excosine, tangential
X(34787) = anticomplement of X(34788) with respect to 2nd Ehrmann triangle
X(34787) = complement of X(11477) with respect to these triangles: 1st excosine, tangential
X(34787) = complement of X(34788) with respect to Kosnita triangle
X(34787) = X(8549)-of-ABC-X3 reflections triangle
X(34787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 8549, 10249), (6, 15577, 23041), (24, 15073, 6), (69, 14516, 15069), (154, 11477, 34117), (575, 34788, 11216), (576, 10282, 19153), (11202, 34788, 575), (15582, 34117, 154)

X(34788) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: KOSNITA TO 2nd EHRMANN

Barycentrics a^2*(a^10-6*(b^2+c^2)*a^8+2*(4*b^4+9*b^2*c^2+4*c^4)*a^6+2*(b^2+c^2)*(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*a^4-(b^2-c^2)^2*(9*b^4+8*b^2*c^2+9*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(4*b^4-6*b^2*c^2+4*c^4)) : :
X(34788) = 2*X(3)-3*X(10250) = X(3)-3*X(17813) = 2*X(5)-3*X(23048) = 3*X(6)-2*X(10282) = 4*X(6)-3*X(23042) = 2*X(140)-3*X(23326) = 3*X(154)-5*X(11482) = 2*X(206)-3*X(15520) = 4*X(575)-3*X(11202) = 2*X(575)-3*X(11216) = X(3357)-6*X(34777) = 7*X(3357)-6*X(34778) = X(8549)-3*X(34777) = 7*X(8549)-3*X(34778) = 8*X(10282)-9*X(23042) = 3*X(11202)-2*X(34787) = 3*X(11216)-X(34787) = 7*X(34777)-X(34778)

The reciprocal ellipsologic center of these triangles is X(34787)

X(34788) lies on these lines: {3,10250}, {5,23048}, {6,3517}, {24,21639}, {69,15605}, {140,23326}, {154,11482}, {159,5097}, {182,9977}, {184,8537}, {186,11458}, {206,15520}, {389,10602}, {511,3357}, {524,18381}, {542,34786}, {575,11202}, {576,2393}, {578,8541}, {599,32767}, {1092,11416}, {1147,11255}, {1351,1498}, {1353,34776}, {1658,32155}, {1992,9833}, {5093,9924}, {5893,21850}, {6000,11477}, {6243,32276}, {6467,7592}, {7488,11443}, {8538,9306}, {8550,34785}, {8567,33878}, {10274,32368}, {10539,18449}, {10619,14912}, {10982,12167}, {11405,19357}, {12061,19136}, {13346,14984}, {15069,18383}, {15516,23041}, {15583,34380}, {16266,34382}, {18553,23049}, {19153,22330}, {23325,34507}

X(34788) = reflection of X(i) in X(j) for these (i,j): (159, 5097), (1147, 11255), (1658, 32155), (3357, 8549), (6759, 576), (10250, 17813), (10274, 32368), (11202, 11216), (15069, 18383), (34776, 1353), (34779, 1351), (34785, 8550), (34787, 575)
X(34788) = anticomplement of X(575) with respect to 2nd Ehrmann triangle
X(34788) = anticomplement of X(34787) with respect to Kosnita triangle
X(34788) = complement of X(34787) with respect to 2nd Ehrmann triangle
X(34788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (575, 34787, 11202), (8541, 15073, 578), (11216, 34787, 575)

X(34789) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: 2nd SCHIFFLER TO GARCIA-REFLECTION

Barycentrics a^6-(b+c)*a^5-(b^2-3*b*c+c^2)*a^4+(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2*(b-c)^2 : :
X(34789) = 3*X(4)-2*X(6246) = 3*X(4)-X(12247) = 2*X(11)-3*X(1699) = 4*X(11)-3*X(11219) = 3*X(11)-2*X(13226) = 3*X(80)-4*X(6246) = 3*X(80)-2*X(12247) = 3*X(1699)-X(1768) = 9*X(1699)-4*X(13226) = 2*X(1768)-3*X(11219) = 3*X(1768)-4*X(13226) = 9*X(11219)-8*X(13226)

The reciprocal ellipsologic center of these triangles is X(1768)

X(34789) lies on these lines: {1,1537}, {3,12611}, {4,80}, {5,12515}, {7,18240}, {11,57}, {20,214}, {30,6265}, {35,12608}, {36,1519}, {40,119}, {46,2950}, {65,12764}, {79,104}, {84,5533}, {100,516}, {124,24410}, {149,152}, {153,962}, {165,3035}, {191,5771}, {226,11218}, {329,14740}, {355,22799}, {381,12619}, {388,15558}, {484,1532}, {497,5083}, {499,11665}, {515,7972}, {517,10742}, {518,13271}, {528,1750}, {944,25485}, {952,3627}, {971,13274}, {1012,18393}, {1071,4857}, {1145,7991}, {1155,1538}, {1156,15909}, {1158,7741}, {1317,6259}, {1320,4301}, {1387,11522}, {1478,12758}, {1479,5768}, {1484,16159}, {1512,3245}, {1593,9912}, {1697,10956}, {1770,10090}, {2096,10072}, {2448,14503}, {2449,14504}, {2771,7728}, {2951,10427}, {3057,12763}, {3062,3254}, {3070,19078}, {3071,19077}, {3091,6702}, {3146,6224}, {3149,12332}, {3219,5659}, {3336,7681}, {3359,32554}, {3576,11729}, {3582,22835}, {3583,6001}, {3585,10057}, {3624,21154}, {3632,12700}, {3754,13729}, {3817,20292}, {3830,12747}, {4292,13370}, {4295,12736}, {4330,33597}, {4388,20237}, {4679,31235}, {5087,17613}, {5121,25580}, {5400,24715}, {5441,21740}, {5443,6906}, {5444,6950}, {5445,6941}, {5536,17768}, {5541,5812}, {5603,11715}, {5693,10525}, {5697,6256}, {5779,31140}, {5840,6326}, {5854,11531}, {5880,6667}, {5902,26333}, {5904,12665}, {6173,27869}, {6253,12738}, {6260,10087}, {6264,31162}, {6284,12739}, {6713,8227}, {6851,9946}, {6905,15228}, {6909,11813}, {6982,17057}, {7354,12740}, {7686,17654}, {7701,26470}, {7987,34123}, {7989,34122}, {8068,12705}, {8988,31412}, {9580,12831}, {9581,12832}, {9612,10075}, {9614,10085}, {9616,13922}, {9856,13273}, {9964,12540}, {10058,12047}, {10074,30384}, {10265,18483}, {10531,15528}, {10589,14646}, {10707,13243}, {10711,28194}, {10896,20118}, {11010,18242}, {11224,25416}, {11256,13463}, {11698,28174}, {12005,13129}, {12116,16127}, {12137,12173}, {12532,31803}, {12641,32049}, {12678,12735}, {12737,22791}, {12743,12953}, {12943,18976}, {15015,24466}, {15064,33110}, {16209,25522}, {17501,23959}, {17578,20085}, {18480,19914}, {18481,19907}, {21077,25438}, {22935,28146}, {24026,33650}, {26725,33594}, {28164,33337}, {31730,34474}

X(34789) = midpoint of X(i) and X(j) for these {i,j}: {149, 9809}, {153, 962}, {3146, 6224}, {5541, 9589}, {5691, 13253}, {10698, 10728}, {12699, 16128}
X(34789) = reflection of X(i) in X(j) for these (i,j): (1, 1537), (3, 12611), (20, 214), (36, 1519), (40, 119), (80, 4), (100, 21635), (104, 946), (355, 22799), (484, 1532), (944, 25485), (1155, 1538), (1320, 4301), (1768, 11), (2448, 14503), (2449, 14504), (2951, 10427), (3245, 1512), (5531, 13257), (5537, 908), (5904, 12665), (6909, 11813), (7972, 10698), (7991, 1145), (10265, 18483), (10738, 22793), (11219, 1699), (11256, 13463), (12119, 6265), (12247, 6246), (12248, 11715), (12515, 5), (12532, 31803), (12641, 32049), (12737, 22791), (12751, 10742), (14217, 12699), (15071, 11570), (15228, 6905), (17613, 5087), (17638, 9856), (17654, 7686), (18481, 19907), (19914, 18480), (25438, 21077), (26726, 7982), (33898, 22792)
X(34789) = (outer-Yff)-isogonal conjugate of X(36)
X(34789) = incircle-inverse of X(10271)
X(34789) = anticomplement of X(11) with respect to these triangles: 2nd Fuhrmann, Garcia-reflection
X(34789) = anticomplement of X(1768) with respect to 2nd Schiffler triangle
X(34789) = anticomplement of X(12619) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34789) = anticomplement of X(25416) with respect to excenters-reflections triangle
X(34789) = complement of X(149) with respect to 2nd Conway triangle
X(34789) = complement of X(1768) with respect to these triangles: 2nd Fuhrmann, Garcia-reflection
X(34789) = X(1537)-of-Aquila triangle
X(34789) = X(5972)-of-2nd Conway triangle
X(34789) = X(12515)-of-Johnson triangle
X(34789) = X(12611)-of-X3-ABC reflections triangle
X(34789) = X(12825)-of-excenters-reflections triangle
X(34789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12247, 6246), (11, 1768, 11219), (100, 21635, 5660), (104, 946, 16173), (165, 15017, 3035), (1699, 1768, 11), (3583, 11571, 10073), (5603, 12248, 11715), (6246, 12247, 80), (9809, 9812, 149), (12672, 12761, 10057), (12679, 12699, 5691)

X(34790) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO 2nd ZANIAH

Barycentrics a*((b+c)*a^2+2*b*c*a-(b+c)^3) : :
X(34790) = X(1)-3*X(210) = 3*X(1)-5*X(25917) = 3*X(2)-5*X(3697) = 9*X(2)-5*X(3889) = X(4)-3*X(18908) = X(8)+3*X(3681) = 3*X(8)+X(3869) = 3*X(8)-X(10914) = 5*X(8)-X(14923) = X(72)-3*X(3681) = 3*X(72)-X(3869) = 3*X(72)+X(10914) = 5*X(72)+X(14923) = 3*X(210)-2*X(5044) = 9*X(210)-5*X(25917) = X(962)-3*X(5927) = X(3555)-5*X(3697) = 3*X(3555)-5*X(3889) = 9*X(3681)-X(3869) = 9*X(3681)+X(10914) = 15*X(3681)+X(14923) = 3*X(3697)-X(3889) = 5*X(3697)-2*X(5045) = 5*X(3869)+3*X(14923) = 5*X(3889)-6*X(5045) = 6*X(5044)-5*X(25917) = 3*X(5777)-2*X(31937)

The reciprocal ellipsologic center of these triangles is X(34791)

X(34790) lies on these lines: {1,210}, {2,3555}, {3,200}, {4,8}, {5,4847}, {9,3295}, {10,141}, {12,5173}, {21,3935}, {30,12527}, {35,3689}, {37,4270}, {38,3214}, {40,971}, {42,16850}, {44,595}, {55,31445}, {56,3711}, {57,9709}, {63,3579}, {65,3679}, {78,956}, {84,6244}, {100,3916}, {140,6745}, {144,6361}, {145,392}, {165,12680}, {219,7079}, {226,31419}, {239,17681}, {306,30810}, {319,21276}, {341,10449}, {354,1698}, {376,12125}, {386,4849}, {390,9844}, {405,3870}, {442,25006}, {452,20015}, {480,31658}, {496,3452}, {498,16193}, {515,6743}, {519,960}, {529,17647}, {536,24068}, {612,16849}, {726,18252}, {756,6051}, {758,3626}, {762,20963}, {908,9955}, {912,5690}, {928,4528}, {936,999}, {944,20007}, {946,10157}, {952,6737}, {958,3811}, {982,6048}, {984,1716}, {997,12513}, {1018,17746}, {1058,6764}, {1071,5657}, {1089,3706}, {1100,1126}, {1107,20693}, {1125,3740}, {1145,2771}, {1150,5482}, {1155,6763}, {1193,21805}, {1210,3820}, {1259,32613}, {1260,10267}, {1319,5288}, {1329,10916}, {1330,32850}, {1376,9858}, {1386,30145}, {1475,25068}, {1479,4863}, {1482,4853}, {1483,31838}, {1656,5231}, {1697,1864}, {1706,5784}, {1724,3744}, {1737,21031}, {1757,5255}, {1770,34612}, {1788,17625}, {1998,16293}, {2057,10269}, {2262,4034}, {2348,17744}, {2478,18527}, {2551,5722}, {2646,5258}, {2800,32159}, {2801,9943}, {2802,3988}, {2810,11573}, {2886,21077}, {2901,28581}, {2975,4420}, {2999,21529}, {3036,6797}, {3057,3632}, {3085,5791}, {3086,16215}, {3158,31424}, {3219,3871}, {3244,10176}, {3293,3666}, {3303,3715}, {3333,8580}, {3338,4413}, {3339,8581}, {3361,9850}, {3416,34381}, {3428,17857}, {3475,19855}, {3523,33574}, {3617,3753}, {3621,3877}, {3624,17609}, {3625,3878}, {3633,5919}, {3634,3742}, {3635,4547}, {3660,24914}, {3661,33838}, {3670,31855}, {3682,5399}, {3683,3746}, {3686,9052}, {3691,3930}, {3693,4006}, {3695,3717}, {3698,5902}, {3701,17135}, {3702,3952}, {3730,4515}, {3748,5259}, {3751,5711}, {3754,4691}, {3813,7743}, {3824,3925}, {3828,4540}, {3872,3984}, {3873,3921}, {3885,31145}, {3890,20050}, {3892,19862}, {3893,4677}, {3894,4731}, {3899,4816}, {3900,11247}, {3902,25253}, {3913,5220}, {3918,4745}, {3953,16610}, {3957,5047}, {3961,5247}, {3962,4668}, {3965,21061}, {3968,33815}, {3976,16569}, {3995,20051}, {3996,7283}, {4002,24473}, {4007,21871}, {4018,4678}, {4065,4681}, {4067,4669}, {4084,10107}, {4109,4119}, {4113,4692}, {4127,4746}, {4187,26015}, {4301,15064}, {4308,17644}, {4416,15310}, {4500,14077}, {4511,15178}, {4537,34641}, {4640,8715}, {4641,5264}, {4651,4968}, {4666,16842}, {4690,34377}, {4723,17751}, {4855,17502}, {4861,33179}, {4867,11011}, {4900,30294}, {4915,7982}, {4981,26115}, {5087,24387}, {5122,25440}, {5126,8666}, {5219,31493}, {5227,7719}, {5234,16418}, {5236,26942}, {5248,5302}, {5261,7672}, {5268,19319}, {5312,21870}, {5493,15726}, {5552,11231}, {5570,18395}, {5587,5806}, {5686,5728}, {5691,7957}, {5704,17626}, {5705,31479}, {5708,10855}, {5720,22770}, {5779,12705}, {5790,24474}, {5844,13600}, {5850,15587}, {5853,12572}, {5854,18254}, {5881,14110}, {6001,11362}, {6675,12439}, {6684,11227}, {6700,15325}, {6734,9956}, {6735,34339}, {6767,31435}, {7080,9940}, {7081,21554}, {7082,26358}, {7330,10306}, {7373,8583}, {7987,33575}, {7991,12688}, {8270,9370}, {9037,31737}, {9581,17642}, {9614,31142}, {9623,11523}, {9669,24392}, {9710,12609}, {9798,12329}, {9819,9848}, {10039,21677}, {10072,24954}, {10156,31423}, {10175,13374}, {10371,30615}, {10527,11230}, {10528,16465}, {10572,34606}, {10578,16845}, {10580,17559}, {10582,16853}, {10593,24386}, {10826,31141}, {10860,12684}, {10944,34689}, {11019,17527}, {11037,17582}, {11112,31776}, {11113,31795}, {11260,25405}, {11278,11682}, {11373,18236}, {11374,19843}, {11379,11525}, {12526,12702}, {12528,31797}, {12629,15829}, {12710,18249}, {12722,17355}, {13359,31595}, {13360,31594}, {14054,24987}, {15008,30628}, {15481,25439}, {17536,29817}, {17606,18839}, {17612,26062}, {17718,19854}, {17781,28198}, {18398,19875}, {18732,33091}, {20017,31049}, {20117,28234}, {20588,26921}, {20653,33162}, {21342,24046}, {21384,25066}, {26593,33839}, {27065,32635}, {27368,32927}, {28228,31871}, {31429,31461}, {31438,31474}, {31442,31477}, {31446,31480}

X(34790) = midpoint of X(i) and X(j) for these {i,j}: {8, 72}, {40, 14872}, {65, 5904}, {3057, 3632}, {3059, 5223}, {3421, 17658}, {3625, 3878}, {3753, 4661}, {3869, 10914}, {3893, 5697}, {3962, 5903}, {4677, 31165}, {4692, 22275}, {5691, 7957}, {5728, 34784}, {5881, 14110}, {7160, 12692}, {7991, 12688}, {12245, 12672}
X(34790) = reflection of X(i) in X(j) for these (i,j): (1, 5044), (4, 9947), (10, 4662), (145, 31792), (942, 10), (960, 3678), (1071, 31787), (1125, 4015), (1483, 31838), (3555, 5045), (3742, 3956), (3754, 4691), (3868, 31794), (3874, 3812), (3881, 3634), (4084, 10107), (5049, 3740), (5836, 3626), (6797, 3036), (9856, 5777), (9957, 960), (12128, 936), (12672, 31821), (12675, 6684), (12680, 31805), (12710, 18249), (12722, 17355), (12915, 3820), (15171, 12572), (30628, 15008), (31786, 31837), (31788, 5690), (31798, 11362), (34791, 1125)
X(34790) = anticomplement of X(5045)
X(34790) = complement of X(3555)
X(34790) = anticomplement of X(1125) with respect to 2nd Zaniah triangle
X(34790) = anticomplement of X(3881) with respect to 1st Zaniah triangle
X(34790) = anticomplement of X(5044) with respect to extouch triangle
X(34790) = anticomplement of X(5045) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34790) = anticomplement of X(5806) with respect to Fuhrmann triangle
X(34790) = anticomplement of X(12675) with respect to Ascella triangle
X(34790) = anticomplement of X(13373) with respect to K798i triangle
X(34790) = anticomplement of X(13374) with respect to 4th Euler triangle
X(34790) = anticomplement of X(31805) with respect to these triangles: 1st circumperp, 6th mixtilinear, excentral
X(34790) = anticomplement of X(34791) with respect to incircle-circles triangle
X(34790) = complement of X(1) with respect to extouch triangle
X(34790) = complement of X(65) with respect to these triangles: Aquila, outer-Garcia
X(34790) = complement of X(72) with respect to inner-Conway triangle
X(34790) = complement of X(962) with respect to these triangles: 2nd extouch, Atik, Ursa-major
X(34790) = complement of X(3057) with respect to inner-Garcia triangle
X(34790) = complement of X(3555) with respect to these triangles: 1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34790) = complement of X(3633) with respect to Hutson intouch triangle
X(34790) = complement of X(12680) with respect to these triangles: 1st circumperp, 6th mixtilinear, excentral
X(34790) = complement of X(34791) with respect to 2nd Zaniah triangle
X(34790) = X(140)-of-inner-Conway triangle
X(34790) = X(942)-of-outer-Garcia triangle
X(34790) = X(1387)-of-inner-Garcia triangle
X(34790) = X(1595)-of-2nd extouch triangle
X(34790) = X(1598)-of-Atik triangle
X(34790) = X(5044)-of-Aquila triangle
X(34790) = X(5173)-of-outer-Johnson triangle
X(34790) = X(9947)-of-anti-Euler triangle
X(34790) = X(14216)-of-2nd Zaniah triangle
X(34790) = X(16655)-of-Wasat triangle
X(34790) = X(16659)-of-4th Euler triangle
X(34790) = X(18914)-of-excentral triangle
X(34790) = X(21841)-of-Ursa-major triangle
X(34790) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 210, 5044), (1, 30393, 3646), (2, 3555, 5045), (4, 18908, 9947), (8, 329, 5082), (8, 3421, 355), (8, 3436, 3419), (8, 3681, 72), (8, 3869, 10914), (8, 4385, 5295), (8, 4388, 5100), (8, 5080, 5178), (8, 5815, 4), (8, 9954, 9856), (72, 10914, 3869), (329, 5082, 12699), (3419, 3436, 18480), (3555, 3697, 2), (4847, 21075, 5), (10914, 17615, 31937)

X(34791) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO INCIRCLE-CIRCLES

Barycentrics a*((b+c)*a^2+6*b*c*a-(b^2+c^2)*(b+c)) : :
X(34791) = 3*X(1)-X(72) = 5*X(1)-3*X(392) = 7*X(1)-3*X(5692) = 5*X(1)-X(5904) = 4*X(1)-3*X(10179) = 5*X(72)-9*X(392) = 2*X(72)-3*X(960) = X(72)+3*X(3555) = 7*X(72)-9*X(5692) = 5*X(72)-3*X(5904) = 4*X(72)-9*X(10179) = 6*X(392)-5*X(960) = 3*X(392)+5*X(3555) = 7*X(392)-5*X(5692) = 3*X(392)-X(5904) = 4*X(392)-5*X(10179) = X(960)+2*X(3555) = 7*X(960)-6*X(5692) = 5*X(960)-2*X(5904) = 2*X(960)-3*X(10179)

The reciprocal ellipsologic center of these triangles is X(34790)

X(34791) lies on these lines: {1,6}, {2,3983}, {3,12333}, {7,9797}, {8,354}, {10,3742}, {12,26015}, {21,3748}, {40,10178}, {42,4719}, {55,4652}, {56,3870}, {57,3913}, {63,3303}, {65,145}, {78,3304}, {100,32636}, {142,9710}, {200,25524}, {210,3616}, {226,3813}, {241,4322}, {355,13374}, {377,4863}, {388,5175}, {404,3689}, {495,10916}, {496,5087}, {517,550}, {519,942}, {527,12575}, {528,4292}, {529,950}, {551,5044}, {726,12722}, {758,3635}, {912,15083}, {952,7686}, {959,6553}, {961,1280}, {962,12680}, {971,4301}, {978,4849}, {982,4646}, {986,21342}, {997,7373}, {999,3811}, {1002,1219}, {1056,5794}, {1058,24703}, {1071,7982}, {1125,3740}, {1155,3871}, {1210,5123}, {1259,33925}, {1320,17660}, {1329,11019}, {1376,3333}, {1468,3744}, {1475,3693}, {1482,6001}, {1698,3848}, {1706,8168}, {1858,5048}, {1864,34647}, {1887,1897}, {2099,11520}, {2136,3339}, {2334,5256}, {2550,6764}, {2551,10580}, {2800,13600}, {2801,9856}, {2886,21620}, {2975,3957}, {3036,18240}, {3041,14760}, {3057,3241}, {3059,11038}, {3158,3361}, {3174,12854}, {3214,16610}, {3290,3780}, {3293,4694}, {3295,4640}, {3338,5687}, {3340,10912}, {3434,10404}, {3452,18247}, {3475,28628}, {3486,17642}, {3487,34625}, {3579,25439}, {3601,11194}, {3622,3681}, {3623,3869}, {3624,3697}, {3625,5883}, {3632,3753}, {3633,5902}, {3636,3678}, {3660,4848}, {3671,15733}, {3679,5439}, {3680,12127}, {3706,4968}, {3714,10453}, {3746,3916}, {3749,4252}, {3752,3976}, {3816,21075}, {3817,9947}, {3833,4691}, {3838,13407}, {3877,3962}, {3878,31792}, {3879,24471}, {3890,31165}, {3893,20050}, {3894,4018}, {3898,4067}, {3915,4641}, {3918,4701}, {3922,20053}, {3930,17474}, {3935,5253}, {3947,24386}, {3956,19878}, {3968,4746}, {3991,4253}, {3999,24443}, {4002,4668}, {4005,4661}, {4006,25068}, {4298,5853}, {4308,7672}, {4315,12437}, {4345,10866}, {4353,18252}, {4383,28011}, {4421,15803}, {4428,31424}, {4511,20323}, {4515,17754}, {4540,31253}, {4642,17449}, {4673,24349}, {4678,4731}, {4688,31327}, {4696,29824}, {4702,7283}, {4847,25466}, {4853,11518}, {4875,21808}, {4882,5437}, {4955,20244}, {4973,31663}, {4999,13405}, {5082,5880}, {5083,5854}, {5173,10106}, {5208,18178}, {5252,12649}, {5260,29817}, {5290,24392}, {5440,5563}, {5534,22753}, {5542,9953}, {5552,17728}, {5570,33956}, {5603,14872}, {5690,13373}, {5731,7957}, {5732,6766}, {5734,12528}, {5777,13464}, {5784,11036}, {5795,6744}, {5815,26105}, {5837,16201}, {5844,34339}, {5887,10247}, {5903,24473}, {5905,12701}, {5918,20070}, {5927,11522}, {6048,16602}, {6691,6745}, {6734,15888}, {6738,12915}, {6767,12514}, {7962,12711}, {7967,14110}, {7991,10167}, {8094,12646}, {8158,12520}, {8227,18908}, {8580,30343}, {8581,30628}, {8666,24929}, {9026,29958}, {9049,17390}, {9581,11236}, {9612,11235}, {9711,9843}, {9785,14100}, {9844,28609}, {9848,10394}, {9940,11362}, {9942,24474}, {10461,18185}, {10527,17718}, {10528,24914}, {10529,11375}, {10569,11519}, {10573,32537}, {10578,30478}, {10586,24954}, {10624,17768}, {10950,18839}, {11011,34195}, {11018,24391}, {11224,15071}, {11240,11376}, {11281,18241}, {11415,28645}, {11529,12629}, {11570,25416}, {11571,17652}, {12005,28234}, {12410,22769}, {12445,12644}, {12447,20116}, {12563,18251}, {12625,17644}, {12672,16200}, {12699,16127}, {13476,28581}, {13528,26877}, {13607,31786}, {14986,25568}, {15104,30389}, {15178,31837}, {16193,31397}, {16616,18525}, {18391,32049}, {21805,28352}, {21896,24174}, {22836,24928}, {24477,26066}, {25144,33087}, {27304,27475}

X(34791) = midpoint of X(i) and X(j) for these {i,j}: {1, 3555}, {65, 145}, {962, 12680}, {1071, 7982}, {1320, 17660}, {2099, 16465}, {3057, 3868}, {3243, 15185}, {3244, 3874}, {3633, 10914}, {3893, 20050}, {4018, 5697}, {4430, 5919}, {8581, 30628}, {9797, 9850}, {11570, 25416}, {11571, 17652}
X(34791) = reflection of X(i) in X(j) for these (i,j): (8, 3812), (10, 5045), (355, 13374), (942, 3881), (960, 1), (1329, 16215), (3036, 18240), (3041, 14760), (3678, 3636), (3740, 5049), (3742, 3892), (3878, 31792), (4701, 3918), (4711, 3742), (5493, 31805), (5690, 13373), (5777, 13464), (5795, 6744), (5836, 942), (5837, 16201), (9943, 12675), (9957, 3635), (10914, 10107), (11362, 9940), (12448, 10912), (15587, 5542), (18247, 21625), (18251, 12563), (18252, 4353), (18525, 16616), (31786, 13607), (31788, 12005), (31837, 15178), (34790, 1125)
X(34791) = anticomplement of X(4662)
X(34791) = anticomplement of X(10) with respect to 1st Zaniah triangle
X(34791) = anticomplement of X(1125) with respect to incircle-circles triangle
X(34791) = anticomplement of X(3812) with respect to these triangles: intouch, inverse-in-incircle, Ursa-minor
X(34791) = anticomplement of X(4662) with respect to these triangles: 1st anti-Brocard, 1st Brocard-reflected, 1st Brocard, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34791) = anticomplement of X(5044) with respect to anti-Aquila triangle
X(34791) = anticomplement of X(9947) with respect to these triangles: 3rd Euler, Wasat
X(34791) = anticomplement of X(34790) with respect to 2nd Zaniah triangle
X(34791) = complement of X(8) with respect to these triangles: intouch, inverse-in-incircle, Ursa-minor
X(34791) = complement of X(3057) with respect to 5th mixtilinear triangle
X(34791) = complement of X(3869) with respect to Hutson intouch triangle
X(34791) = complement of X(20070) with respect to Hutson extouch triangle
X(34791) = complement of X(34790) with respect to incircle-circles triangle
X(34791) = X(960)-of-5th mixtilinear triangle
X(34791) = X(1498)-of-inverse-in-incircle triangle
X(34791) = X(3555)-of-anti-Aquila triangle
X(34791) = X(6247)-of-intouch triangle
X(34791) = X(12241)-of-excenters-reflections triangle
X(34791) = X(14216)-of-incircle-circles triangle
X(34791) = X(16105)-of-Fuhrmann triangle
X(34791) = X(16252)-of-Ursa-minor triangle
X(34791) = X(16621)-of-excentral triangle
X(34791) = X(16655)-of-2nd circumperp triangle
X(34791) = X(34781)-of-2nd Zaniah triangle
X(34791) = X(34782)-of-Hutson intouch triangle
X(34791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 960, 10179), (1, 3751, 1191), (1, 5247, 1279), (1, 5904, 392), (1, 6762, 958), (1, 11523, 5289), (8, 354, 3812), (8, 3889, 354), (10, 3892, 5045), (10, 5045, 3742), (145, 3600, 3189), (145, 3873, 65), (145, 21454, 12632), (496, 21077, 5087), (942, 12128, 12577), (1125, 34790, 3740), (1210, 12607, 5123), (3333, 6765, 1376), (5049, 34790, 1125), (6764, 11037, 2550)

X(34792) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: (CIRCUMCEVIAN-OF-X(2)) TO (REFLECTION-IN-X(3) OF CIRCUMCEVIAN-OF-X(2))

Barycentrics 8*a^10-21*(b^2+c^2)*a^8-(5*b^4-43*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(23*b^4-32*b^2*c^2+23*c^4)*a^4-3*(b^2-c^2)^2*(b^4+7*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4) : :
X(34792) = 3*X(2)-4*X(31762) = 4*X(3)-3*X(9829) = 3*X(3)-2*X(31744) = 2*X(4)-3*X(6032) = 3*X(4)-4*X(31749) = 3*X(165)-2*X(31746) = 3*X(353)-2*X(31748) = 3*X(376)-2*X(31729) = 3*X(381)-2*X(31824) = 3*X(381)-4*X(32156) = 3*X(6032)-4*X(12506) = 9*X(6032)-8*X(31749) = 3*X(9829)-2*X(12505) = 9*X(9829)-8*X(31744) = 3*X(12505)-4*X(31744) = 3*X(12506)-2*X(31749)

The reciprocal ellipsologic center of these triangles is X(12505).
See Abdilkadir Altintas and César Lozada, Euclid 118.

X(34792) lies on these lines: {2,14866}, {3,9829}, {4,6032}, {20,3849}, {30,31961}, {165,31746}, {353,31748}, {376,31729}, {381,31824}, {631,31606}, {1499,31731}, {1699,31755}, {2979,31736}, {3060,31743}, {3091,10162}, {3522,6031}, {3523,10163}, {3576,31747}, {3830,31840}, {5056,10173}, {5587,31758}, {5692,31823}, {5890,31745}, {8704,11257}, {11227,31818}, {11402,31809}, {11459,31753}, {15055,32311}

X(34792) = reflection of X(i) in X(j) for these (i,j): (4, 12506), (3060, 31743), (3830, 31840), (12505, 3), (14866, 31762), (31824, 32156)
X(34792) = anticomplement of X(14866)
X(34792) = anticomplement of X(4) with respect to 4th Brocard triangle
X(34792) = anticomplement of X(12505) with respect to circummedial triangle
X(34792) = anticomplement of X(14866) with respect to these triangles: 1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34792) = anticomplement of X(31729) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34792) = anticomplement of X(31736) with respect to these triangles: 1st anti-circumperp, 3rd anti-Euler
X(34792) = anticomplement of X(31745) with respect to these triangles: 4th anti-Euler, circumorthic
X(34792) = anticomplement of X(31746) with respect to these triangles: 1st circumperp, 6th mixtilinear, excentral
X(34792) = anticomplement of X(31747) with respect to these triangles: 2nd circumperp, hexyl
X(34792) = anticomplement of X(31748) with respect to circumsymmedial triangle
X(34792) = anticomplement of X(31809) with respect to these triangles: 2nd anti-extouch, anti-Ascella, anti-Conway
X(34792) = anticomplement of X(31818) with respect to Ascella triangle
X(34792) = anticomplement of X(31823) with respect to inner-Garcia triangle
X(34792) = anticomplement of X(31824) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34792) = X(20)-of-circummedial triangle
X(34792) = X(12505)-of-ABC-X3 reflections triangle
X(34792) = X(12506)-of-anti-Euler triangle
X(34792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 12505, 9829), (4, 12506, 6032), (14866, 31762, 2), (31824, 32156, 381)

X(34793) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: (CIRCUMCEVIAN-OF-X(5)) TO (REFLECTION-IN-X(3) OF CIRCUMCEVIAN-OF-X(5))

Barycentrics 8*(4*R^2-SW)*S^4+2*((5*SA-SW)*R^4+(31*SA^2-33*SW*SA+SW^2)*R^2-8*(SA-SW)*SA*SW)*S^2+(R^4-2*SW*R^2-2*SW^2)*R^2*SB*SC : :
X(34793) = 5*X(631)-4*X(31607)

The reciprocal ellipsologic center of these triangles is X(34794).
See Abdilkadir Altintas and César Lozada, Euclid 118.

X(34793) lies on these lines: {3,34794}, {20,32744}, {631,31607}

X(34793) = reflection of X(34794) in X(3)

X(34794) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: (REFLECTION-IN-X(3) OF CIRCUMCEVIAN-OF-X(5)) TO (CIRCUMCEVIAN-OF-X(5))

Barycentrics 8*(4*R^2-SW)*S^4-(R^6-2*(5*SA-2*SW)*R^4-2*(17*SA^2-19*SW*SA+SW^2)*R^2-8*(SB+SC)*SA*SW)*S^2+2*R^2*SC*SB*(R^4-SW^2) : :
X(34794) = 3*X(2)-4*X(31607)

The reciprocal ellipsologic center of these triangles is X(34793).
See Abdilkadir Altintas and César Lozada, Euclid 118.

Let A' be the reflection of X(4) in the Euler line of triangle BCX(4), and define B' and C' cyclically. Then X(34794) = X(20)-of-A'B'C'. (Randy Hutson, March 29, 2020)

X(34794) lies on these lines: {2,31607}, {3,34793}, {4,32744}, {5,10227}, {3153,14097}

X(34794) = reflection of X(34793) in X(3)
X(34794) = anticomplement of the anticomplement of X(31607)
X(34794) = X(4)-of-circumcevian-triangle-of-X(5)
X(34794) = X(34793)-of-ABC-X3 reflections triangle

X(34795) = ELLIPSOLOGIC CENTER OF THESE TRIANGLES: (CIRCUMCEVIAN-OF-X(6)) TO (REFLECTION-IN-X(3) OF CIRCUMCEVIAN-OF-X(6))

Barycentrics (SB+SC)*(45*S^4+3*(21*SA^2-72*R^2*SA-12*SB*SC-2*SW^2)*S^2+8*SA*SW^3) : :
X(34795) = 4*X(3)-3*X(353) = 3*X(3)-2*X(31727) = 3*X(165)-2*X(31740) = 9*X(353)-8*X(31727) = 3*X(353)-2*X(31962) = 3*X(376)-2*X(31731) = 3*X(381)-2*X(31827) = 5*X(631)-4*X(31608) = 3*X(2979)-2*X(31739) = 5*X(3091)-6*X(34512) = 7*X(3523)-6*X(10166) = 3*X(3576)-2*X(31741) = 5*X(3620)-4*X(31742) = 3*X(5692)-2*X(31826) = 3*X(5890)-2*X(31733) = 3*X(9829)-2*X(31748) = 12*X(10160)-13*X(10303) = 4*X(31727)-3*X(31962)

The reciprocal ellipsologic center of these triangles is X(31962).
See Abdilkadir Altintas and César Lozada, Euclid 118.

X(34795) lies on these lines: {2,14867}, {3,352}, {20,9830}, {165,31740}, {376,31731}, {381,31827}, {631,31608}, {2979,31739}, {3091,34512}, {3523,10166}, {3576,31741}, {3620,31742}, {5171,6233}, {5692,31826}, {5890,31733}, {6323,30270}, {9829,31748}, {10160,10303}, {10519,31959}, {11227,31820}, {11402,31813}, {13192,14262}

X(34795) = reflection of X(31962) in X(3)
X(34795) = anticomplement of X(14867)
X(34795) = anticomplement of X(14867) with respect to these triangles: 1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten
X(34795) = anticomplement of X(31731) with respect to these triangles: ABC-X3 reflections, anti-Euler
X(34795) = anticomplement of X(31733) with respect to these triangles: 4th anti-Euler, circumorthic
X(34795) = anticomplement of X(31739) with respect to these triangles: 1st anti-circumperp, 3rd anti-Euler
X(34795) = anticomplement of X(31740) with respect to these triangles: 1st circumperp, 6th mixtilinear, excentral
X(34795) = anticomplement of X(31741) with respect to these triangles: 2nd circumperp, hexyl
X(34795) = anticomplement of X(31748) with respect to circummedial triangle
X(34795) = anticomplement of X(31813) with respect to these triangles: 2nd anti-extouch, anti-Ascella, anti-Conway
X(34795) = anticomplement of X(31820) with respect to Ascella triangle
X(34795) = anticomplement of X(31826) with respect to inner-Garcia triangle
X(34795) = anticomplement of X(31827) with respect to these triangles: Euler, Johnson, X3-ABC reflections
X(34795) = anticomplement of X(31962) with respect to circumsymmedial triangle
X(34795) = X(20)-of-circumsymmedial triangle
X(34795) = X(15305)-of-4th anti-Brocard triangle
X(34795) = X(31962)-of-ABC-X3 reflections triangle
X(34795) = {X(3), X(31962)}-harmonic conjugate of X(353)

X(34796) = X(20)X(1181)∩X(25)X(146)

Barycentrics 2*a^10-(b^2+c^2)*a^8-(8*b^4-9*b^2*c^2+8*c^4)*a^6+2*(b^2+c^2)*(5*b^4-9*b^2*c^2+5*c^4)*a^4-(2*b^4+7*b^2*c^2+2*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics (SA-7*R^2+SW)*S^2+2*(9*R^2-2*SW)*SB*SC : :
X(34796) = 5*X(3091)-8*X(13568)

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34796) lies on these lines: {2, 32620}, {3, 15806}, {20, 1181}, {25, 146}, {74, 31074}, {541, 11455}, {550, 9545}, {1531, 26913}, {1899, 10296}, {2777, 3060}, {3091, 13568}, {3153, 10605}, {3543, 15311}, {4549, 15246}, {4846, 6636}, {5663, 18559}, {9140, 18376}, {9544, 10295}, {9704, 15332}, {11270, 23336}, {11454, 18388}, {11550, 15054}, {13619, 18445}, {15531, 19924}, {15534, 29181}

X(34797) = EULER LINE INTERCEPT OF X(64)X(12244)

Barycentrics (2*a^6-3*(b^2+c^2)*a^4+3*c^2*a^2*b^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics (SA+11*R^2-3*SW)*SB*SC : :
X(34797) = 11*X(4)-12*X(428), 5*X(4)-4*X(1885), 3*X(4)-4*X(3575), 7*X(4)-8*X(6756), 5*X(4)-6*X(7576), 9*X(4)-8*X(13488), 2*X(4)-3*X(18559), 3*X(4)-2*X(18560), 4*X(140)-3*X(18564), 3*X(376)-2*X(12225), 15*X(381)-16*X(13163), 15*X(428)-11*X(1885), 9*X(428)-11*X(3575), 6*X(428)-11*X(6240), 10*X(428)-11*X(7576), 8*X(428)-11*X(18559), 18*X(428)-11*X(18560)

As a point on the Euler line, X(34797) has Shinagawa coefficients (-4*F, E+12*F).

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34797) lies on these lines: {2, 3}, {64, 12244}, {74, 16000}, {93, 16263}, {125, 18394}, {155, 12383}, {185, 12063}, {962, 31948}, {1181, 12254}, {1204, 25739}, {1235, 7802}, {1300, 12092}, {1829, 28168}, {1843, 29323}, {1870, 10483}, {1902, 28154}, {1986, 10263}, {2777, 12281}, {3043, 12121}, {3567, 13403}, {5448, 11449}, {5449, 18392}, {5878, 12112}, {5889, 7722}, {5890, 21659}, {6241, 6242}, {6403, 29012}, {7706, 13434}, {7748, 10312}, {10632, 19106}, {10633, 19107}, {11270, 18434}, {11440, 18474}, {11455, 13419}, {11456, 17845}, {11468, 20299}, {11750, 15072}, {11801, 21400}, {12022, 13568}, {12135, 28178}, {12165, 12316}, {12278, 13754}, {12292, 34584}, {13561, 18430}, {13851, 26917}, {14076, 32340}, {14157, 22802}, {14528, 18363}, {15032, 19467}, {15311, 16659}, {15424, 18848}, {18383, 21663}

X(34797) = reflection of X(i) in X(j) for these (i,j): (4, 6240), (12289, 185), (18560, 3575), (18562, 5)
X(34797) = anticomplement of X(18563)
X(34797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 13619, 3), (4, 21844, 5), (5, 15332, 3), (20, 3547, 376), (20, 34007, 3), (24, 382, 4), (378, 12173, 4), (403, 3627, 4), (452, 34664, 24), (454, 6839, 25), (1885, 7576, 4), (3146, 18533, 4), (3542, 3543, 4), (3575, 18560, 4), (9909, 27723, 25), (15706, 18534, 5), (17586, 33191, 21), (27301, 27865, 30), (27591, 33239, 30), (28020, 33289, 377)

X(34798) = X(4)X(13561)∩X(5)X(20191)

Barycentrics 2*a^10-2*(b^2+c^2)*a^8-(5*b^4-8*b^2*c^2+5*c^4)*a^6+(b^2+c^2)*(7*b^4-13*b^2*c^2+7*c^4)*a^4-(b^2-c^2)^2*(b^4+5*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics (SA-11*R^2+2*SW)*S^2+(29*R^2-7*SW)*SB*SC : :
X(34798) = 5*X(3567)-4*X(15807), 3*X(5890)-X(18562), 3*X(5946)-4*X(13568), 3*X(6102)-2*X(12370), 5*X(6102)-4*X(13292), 3*X(7576)-2*X(32137), 5*X(10574)-3*X(18564), 5*X(12370)-6*X(13292), 3*X(15060)-4*X(31833), X(18439)-3*X(18559)

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34798) lies on these lines: {4, 13561}, {5, 20191}, {30, 52}, {49, 13619}, {74, 31724}, {125, 18567}, {143, 18560}, {235, 1539}, {382, 25739}, {541, 13419}, {550, 18475}, {1204, 18377}, {1353, 29317}, {1493, 15704}, {1511, 22660}, {1594, 32210}, {1657, 12161}, {2777, 3627}, {3518, 7728}, {3521, 7488}, {3567, 15807}, {5448, 15646}, {5663, 6240}, {5889, 18565}, {5890, 18562}, {5894, 32351}, {5895, 7530}, {5946, 13568}, {7576, 32137}, {10224, 21663}, {10295, 32171}, {10574, 18564}, {11819, 15311}, {12041, 13371}, {12086, 20127}, {12107, 34563}, {12173, 34514}, {12236, 34584}, {12585, 29012}, {13367, 15332}, {13406, 32110}, {13630, 18563}, {15060, 31833}, {18323, 26879}, {18439, 18559}, {18568, 26937}, {18951, 33703}

X(34798) = midpoint of X(5889) and X(18565)
X(34798) = reflection of X(i) in X(j) for these (i,j): (18560, 143), (18563, 13630)
X(34798) = {X(3627), X(10264)}-harmonic conjugate of X(18383)

X(34799) = X(2)X(6146)∩X(3)X(2888)

Barycentrics 2*a^10-5*(b^2+c^2)*a^8+(4*b^4+5*b^2*c^2+4*c^4)*a^6-2*(b^6+c^6)*a^4+(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :
Barycentrics (SB+SC-R^2)*S^2+2*(2*R^2-SW)*SB*SC : :
X(34799) = 3*X(2)-4*X(6146), 3*X(4)-4*X(12370), 3*X(568)-4*X(11264), 3*X(3060)-4*X(10112), 5*X(3091)-6*X(12022), 5*X(3091)-4*X(12134), 3*X(3543)-2*X(16659), 7*X(3832)-8*X(12241), 3*X(5890)-4*X(10116), 4*X(6102)-3*X(18559), 3*X(6403)-4*X(12585), 8*X(6756)-9*X(11002), 3*X(7576)-4*X(13292), 3*X(12022)-2*X(12134)

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34799) lies on these lines: {2, 6146}, {3, 2888}, {4, 1994}, {5, 9544}, {20, 11411}, {22, 12429}, {23, 9833}, {49, 7577}, {54, 18474}, {68, 7488}, {94, 847}, {125, 11449}, {155, 3153}, {156, 265}, {161, 32354}, {185, 12278}, {186, 25738}, {193, 1503}, {287, 26170}, {323, 6193}, {539, 11412}, {542, 12111}, {568, 11264}, {578, 5169}, {1092, 31101}, {1147, 25739}, {1594, 9545}, {1614, 9927}, {1899, 22467}, {1993, 32346}, {2071, 11457}, {3060, 10112}, {3091, 12022}, {3410, 7503}, {3520, 32140}, {3543, 16659}, {3564, 12225}, {3574, 11422}, {3832, 12241}, {5448, 18394}, {5449, 11464}, {5889, 18400}, {5890, 10116}, {6102, 6152}, {6241, 12284}, {6288, 32046}, {6403, 12585}, {6515, 31304}, {6756, 11002}, {7401, 15018}, {7544, 34545}, {7566, 11426}, {7576, 13292}, {7687, 18504}, {9538, 12428}, {9703, 10224}, {9707, 14852}, {10298, 12359}, {10619, 21243}, {10733, 22802}, {11003, 13160}, {11441, 14683}, {11442, 19467}, {11455, 12897}, {11456, 12293}, {11468, 16003}, {11565, 15067}, {12024, 15022}, {12038, 23294}, {12086, 14216}, {12112, 31725}, {12289, 13754}, {13142, 34603}, {13353, 14789}, {13367, 23293}, {13403, 15305}, {13470, 23039}, {15078, 26944}, {16655, 17578}, {18381, 31074}, {22804, 32136}

X(34799) = reflection of X(i) in X(j) for these (i,j): (20, 34224), (11412, 11750), (12111, 21659), (12278, 185), (14516, 6146)
X(34799) = anticomplement of X(14516)
X(34799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (156, 265, 16868), (6146, 14516, 2), (11457, 12118, 2071), (12022, 12134, 3091), (13160, 31804, 11003), (18381, 34148, 31074)

X(34800) = ISOGONAL CONJUGATE OF X(7414)

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

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34800) lies on the Jerabek circumhyperbola and these lines: {6, 6985}, {21, 74}, {30, 65}, {54, 411}, {64, 3560}, {71, 2315}, {72, 13754}, {513, 15328}, {521, 14380}, {542, 10101}, {1071, 17702}, {1437, 5504}, {1903, 15947}, {2850, 14220}, {3431, 6876}, {3657, 6003}, {5663, 5887}, {6875, 11270}, {6912, 16835}, {8674, 15453}, {15232, 18480}

X(34800) = isogonal conjugate of X(7414)
X(34800) = trilinear pole of the line {647, 14395}

X(34801) = ISOGONAL CONJUGATE OF X(18533)

Barycentrics a^2*(a^6-(b^2-c^2)*a^4-(b^4-2*b^2*c^2+5*c^4)*a^2+(b^2+3*c^2)*(b^2-c^2)^2)*(-a^2+b^2+c^2)*(a^6+(b^2-c^2)*a^4-(5*b^4-2*b^2*c^2+c^4)*a^2+(3*b^2+c^2)*(b^2-c^2)^2) : :
X(34801) = X(155)-3*X(32620)

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34801) lies on the Jerabek circumhyperbola and these lines: {3, 10938}, {4, 3580}, {5, 14542}, {6, 4550}, {22, 74}, {30, 66}, {54, 155}, {64, 7387}, {65, 7986}, {67, 1350}, {68, 4549}, {70, 12225}, {154, 12412}, {394, 5504}, {525, 15328}, {539, 13622}, {541, 11744}, {567, 12164}, {568, 3527}, {1147, 14528}, {1177, 5663}, {1216, 12301}, {1498, 34438}, {2435, 30213}, {3167, 14805}, {3426, 3581}, {3431, 15066}, {3519, 12429}, {3532, 8717}, {3547, 15740}, {3564, 5486}, {4846, 10605}, {5085, 19151}, {5448, 33537}, {5449, 7706}, {5505, 14984}, {5562, 15316}, {5907, 9908}, {6000, 23044}, {6145, 9927}, {6391, 18438}, {6415, 18457}, {6416, 18459}, {7404, 22660}, {7512, 11270}, {8673, 14380}, {9517, 15453}, {10293, 10620}, {11411, 12022}, {11564, 15085}, {12088, 13452}, {15317, 18436}, {17814, 19908}, {18451, 18532}

X(34801) = isogonal conjugate of X(18533)
X(34801) = midpoint of X(i) and X(j) for these {i,j}: {68, 4549}, {11472, 12163}
X(34801) = reflection of X(7706) in X(5449)
X(34801) = anticomplement of the complementary conjugate of X(18531)
X(34801) = trilinear pole of the line {647, 14396}

X(34802) = ISOGONAL CONJUGATE OF X(10295)

Barycentrics a^2*(a^6-(b^2-2*c^2)*a^4-(b^4-6*b^2*c^2+7*c^4)*a^2+(b^2+4*c^2)*(b^2-c^2)^2)*(-a^2+b^2+c^2)*(a^6+(2*b^2-c^2)*a^4-(7*b^4-6*b^2*c^2+c^4)*a^2+(4*b^2+c^2)*(b^2-c^2)^2) : :
X(34802) = X(155)-3*X(32620)

See Antreas Hatzipolakis and César Lozada, Euclid 145 .

X(34802) lies on the Jerabek circumhyperbola, the cubics K298, K537 and these lines: {3, 15738}, {4, 541}, {6, 5663}, {23, 74}, {26, 3532}, {30, 67}, {54, 7527}, {64, 7530}, {66, 2777}, {69, 4549}, {70, 3146}, {110, 3431}, {125, 4846}, {265, 18323}, {511, 5505}, {512, 15453}, {517, 10101}, {525, 14220}, {542, 5486}, {686, 10097}, {690, 15328}, {895, 13754}, {1177, 6000}, {1511, 32620}, {2775, 3657}, {3292, 5504}, {3426, 10620}, {3519, 18563}, {3521, 15027}, {3627, 6145}, {5169, 10706}, {5609, 7526}, {6593, 9818}, {7552, 15057}, {7556, 11270}, {7575, 15138}, {7706, 14644}, {8717, 15055}, {9517, 14380}, {10024, 14861}, {10113, 18434}, {10264, 11744}, {10293, 11799}, {10296, 11564}, {10733, 33565}, {10938, 13198}, {11806, 14483}, {11820, 15041}, {12292, 18532}, {13622, 32423}, {13623, 15061}, {15462, 19151}, {15740, 20397}

X(34802) = isogonal conjugate of X(10295)
X(34802) = midpoint of X(3426) and X(10620)
X(34802) = reflection of X(i) in X(j) for these (i,j): (110, 4550), (4846, 125), (9970, 31861)
X(34802) = anticomplement of the complementary conjugate of X(10297)
X(34802) = complement of the anticomplementary conjugate of X(10296)
X(34802) = antigonal conjugate of X(4846)
X(34802) = lies on the circumconic with center X(4550)
X(34802) = Jerabek hyperbola-antipode of X(4846)
X(34802) = trilinear pole of the line {647, 5158}

X(34803) = X(315)X(3525)∩X(316)X(3524)

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

See Kadir Altintas and César Lozada, Euclid 147 .

X(34803) lies on these lines: {2, 6}, {3, 32827}, {4, 7769}, {5, 6337}, {20, 32871}, {32, 32977}, {39, 32969}, {76, 5067}, {83, 33189}, {99, 3545}, {140, 32816}, {194, 32998}, {315, 3525}, {316, 3524}, {328, 34336}, {547, 32837}, {574, 16041}, {620, 14033}, {625, 32986}, {626, 32978}, {631, 7752}, {632, 7776}, {1078, 3533}, {1272, 30786}, {1506, 14001}, {1656, 3926}, {1975, 5056}, {2548, 32970}, {2549, 32984}, {2896, 33003}, {3090, 7763}, {3523, 7773}, {3526, 3785}, {3628, 32828}, {3767, 32976}, {3788, 32968}, {3851, 32826}, {3933, 5070}, {4045, 7862}, {4558, 10314}, {5013, 32972}, {5020, 9723}, {5055, 6390}, {5068, 32819}, {5071, 11185}, {5079, 32887}, {5254, 32988}, {5286, 33249}, {5475, 32985}, {5866, 9818}, {6036, 14912}, {6340, 8797}, {6683, 33221}, {6721, 8781}, {6722, 7739}, {7486, 32831}, {7603, 32983}, {7617, 14148}, {7737, 33216}, {7738, 32961}, {7745, 32989}, {7750, 10303}, {7771, 15702}, {7775, 21843}, {7783, 32963}, {7785, 33000}, {7786, 32951}, {7789, 32987}, {7795, 32975}, {7802, 10299}, {7803, 32955}, {7804, 33224}, {7807, 31404}, {7809, 15709}, {7823, 33206}, {7825, 33226}, {7828, 32958}, {7832, 32957}, {7836, 32999}, {7846, 33195}, {7847, 33292}, {7853, 16043}, {7859, 32953}, {7861, 31450}, {7864, 33277}, {7885, 33012}, {7887, 31400}, {7891, 32962}, {7899, 32956}, {7904, 33188}, {7909, 18840}, {7912, 33001}, {7940, 14069}, {8176, 11147}, {8361, 31467}, {9738, 12322}, {9739, 12323}, {9742, 15069}, {10008, 18583}, {10011, 14853}, {10124, 14929}, {15022, 32873}, {15515, 33247}, {15699, 32836}, {15815, 32982}, {16589, 33048}, {16923, 20065}, {17128, 33009}, {32818, 32832}, {32821, 32834}, {32825, 32867}

X(34803) = reflection of X(23053) in X(2)
X(34803) = isotomic conjugate of the isogonal conjugate of X(5093)
X(34803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 325, 34229), (2, 1007, 69), (2, 3815, 3618), (2, 7777, 7735), (2, 7778, 3619), (2, 9770, 23055), (2, 11160, 15597), (2, 11184, 1992), (2, 22110, 21356), (5, 32829, 6337), (183, 325, 10513), (325, 34229, 69), (491, 492, 20080), (620, 31415, 14033), (631, 7752, 32006), (1007, 34229, 325), (3055, 7778, 2), (3533, 32823, 1078), (32805, 32806, 6), (32810, 32811, 15533)

X(34804) = X(5)X(195)∩X(30)X(24573)

Barycentrics (19*R^4-20*(SB+SC)*R^2-8*SA^2+16*SB*SC+4*SW^2)*S^2-(R^2*(R^2-8*SW)+4*SW^2)*SB*SC : :
X(34804) = 3*X(547)-2*X(23280), 3*X(547)-4*X(23281)

See Antreas Hatzipolakis and César Lozada, Euclid 163 .

X(34804) lies on these lines: {5, 195}, {30, 24573}, {128, 31879}, {140, 6150}, {252, 3628}, {546, 25150}, {547, 23280}, {1209, 15425}, {6288, 15335}, {6592, 20414}, {10126, 19552}, {13163, 20189}, {14072, 15307}, {15345, 24306}, {21975, 27090}

X(34804) = reflection of X(i) in X(j) for these (i,j): (140, 31376), (252, 3628), (23280, 23281)
X(34804) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 24385, 20413), (14072, 15307, 20030), (23280, 23281, 547)

X(34805) = X(4)X(9)∩X(101)X(514)

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

See Kadir Altintas, Antreas Hatzipolakis and César Lozada, Euclid 171 .

X(34800) lies these lines: {4, 9}, {101, 514}, {929, 28847}, {5845, 14505}, {15378, 26705}, {24045, 31851}

X(34805) = midpoint of X(927) and X(3732)
X(34805) = reflection of X(101) in X(3234)
X(34805) = reflection of X(101) in its Simson line (line X(118)X(516))
X(34805) = X(522)-he conjugate of X(1768)
X(34805) = lies on the circumconic with center X(33331)
X(34805) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(927)}} and {{A, B, C, X(281), X(666)}}

X(34806) = X(4)X(1499)∩X(111)X(524)

Barycentrics (2*a^8-6*(b^2+c^2)*a^6+(19*b^4-20*b^2*c^2+19*c^4)*a^4-4*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^4-c^4)^2)*(a^2-2*b^2+c^2)*(a^2+b^2-2*c^2) : :
Barycentrics (3*SB-SW)*(3*SC-SW)*((9*SA^2+54*(SB+SC)*R^2-5*SW^2)*S^2-2*(SB+SC)*SW^3) : :

See Kadir Altintas, Antreas Hatzipolakis and César Lozada, Euclid 171 .

X(34806) lies these lines: {4, 1499}, {111, 524}, {2374, 15387}, {16317, 32583}

X(34806) = reflection of X(111) in X(15638)
X(34806) = lies on the circumconic with center X(6092))

Points associated with perspeconics: X(34807)-X(34821)

This preamble and centers X(34807)-X(34821) were contributed by Clark Kimberling and Peter Moses, November 18, 2019.

Suppose that A'B'C' and A''B''C'' are perspective central triangles, so that the points

B'C'∩A''B'', B'C'∩A''C'', C'A'∩B''C'', C'A'∩B''A'', A'B'∩C''B'', A'B'∩C''A''

are distinct and lie on a conic. In the preamble just before X(15254) the conic is named the perspeconic of A'B'C' and A''B''C''.

The perspeconic of ABC and the circumcevian triangle of a point P = p : q : r (barycentrics) is given by

a^2 q^2 r^2 (c^2 q+b^2 r) x^2-p q r (2 b^2 c^2 p^2+a^2 c^2 p q+a^2 b^2 p r+a^4 q r) y z + (cyclic) = 0,

center = p (2 b^2 c^2 p q+2 a^2 c^2 q^2+2 b^2 c^2 p r-a^4 q r+a^2 b^2 q r+a^2 c^2 q r+2 a^2 b^2 r^2) : :

perspector = p (2 b^2 c^2 p+2 a^2 c^2 q+a^2 b^2 r) (2 b^2 c^2 p+a^2 c^2 q+2 a^2 b^2 r) : :

If P lies inside ABC, the conic is an ellipse; otherwise, a hyperbola. As a degenerate case, if P lies on the circumcircle, then the conic is the line tangent to the circumcircle at P. If P is on the line X(187)X(237), the conic consists of two lines that pass through X(6).

The appearance of (i,j) in the following list means that if P = X(i), then the center of the conic is X(j):
(1,1001), (2,15271), (3,3), (6,5024), (8,34807), (20,34808), (25,34809), (30,3810), (32,34811), (37,34812), (40,34813), (55,15288), (56,15287), (58,34814), (64),34815), (110,34291), (513,999), (523,381), (525,20208), (1499,11165), (2574,3), (2575,3), (3309,6600)

The appearance of (i,j) in the following list means that that if P = X(i), then the perspector of the conic is X(j):
(1,10013), (2,34816), (3,348917), (25,34818), (31,34819), (55,34820), (56,34821), (523,13481)

X(34807) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(8)

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

X(34807 lies on these lines: {1, 6}, {1788, 26685}, {3452, 4655}

X(34808) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(20)

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

X(34808) lies on these lines: {3, 30549}, {4, 6}, {64, 1105}, {98, 8770}, {114, 30771}, {154, 15466}, {1192, 3186}, {6337, 6527}, {6389, 16196}, {6677, 20207}, {6696, 32000}

X(34808) = reflection of X(30549) in X(3)

X(34809) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(25)

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

X(34809) lies on these lines: {2, 3964}, {3, 230}, {6, 1196}, {25, 393}, {32, 1598}, {115, 1597}, {577, 1611}, {800, 34481}, {1184, 10311}, {1384, 18534}, {1609, 9909}, {1990, 21313}, {1995, 5304}, {2548, 11484}, {3517, 7755}, {5305, 6642}, {7529, 30435}, {7736, 11284}, {7737, 18535}, {11479, 13881}, {14930, 16042}, {15574, 22329}

X(34809) = {X(25),X(7735)}-harmonic conjugate of X(8573)

X(34810) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(30)

Barycentrics (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(a^8 - 2*a^6*b^2 + a^4*b^4 - 2*a^6*c^2 + 3*a^4*b^2*c^2 - a^2*b^4*c^2 + 2*b^6*c^2 + a^4*c^4 - a^2*b^2*c^4 - 4*b^4*c^4 + 2*b^2*c^6) : :
X(34810) = 3 X[381] - 4 X[14356], 3 X[5050] - 2 X[5967], 2 X[14356] - 3 X[14995]

X(34810) lies on these lines: {2, 9717}, {3, 523}, {4, 12066}, {6, 13}, {30, 2407}, {98, 5968}, {378, 9308}, {385, 7418}, {868, 3564}, {1995, 9755}, {3534, 15919}, {5050, 5967}, {5054, 31378}, {5094, 14920}, {6644, 17423}, {14480, 33927}, {14666, 34106}, {14694, 16317}, {16177, 30771}

X(34810) = reflection of X(i) in X(j) for these {i,j}: {381, 14995}, {399, 14559}
X(34810) = complement of X(36875)
X(34810) = X(32654)-complementary conjugate of X(18593)
X(34810) = X(98)-Ceva conjugate of X(30)
X(34810) = crossdifference of every pair of points on line {526, 3003}
X(34810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14611, 9717}, {6, 15928, 381}

X(34811) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(32)

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

X(34811) lies on these lines: {2, 9462}, {6, 694}, {32, 18374}, {39, 11165}, {599, 3229}, {1196, 3003}, {1613, 5201}, {3001, 3981}, {3051, 34097}, {3124, 13240}, {3291, 7610}

X(34811) = crossdifference of every pair of points on line {804, 9489}
X(34811) = barycentric product X(32)*X(7934)
X(34811) = barycentric quotient X(7934)/X(1502)
X(34811) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1084, 3117, 6}, {1084, 8265, 3117}

X(34812) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(37)

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

X(34812) lies on these lines: {6, 4658}, {405, 27040}, {2229, 5024}, {2238, 9708}, {2551, 4205}, {3295, 21024}, {16302, 26035}, {19259, 26244}

X(34813) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(40)

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

X(34813) lies on these lines: {6, 57}, {198, 347}, {516, 5812}, {3663, 4254}, {4222, 7952}, {9798, 33580}, {16593, 30809}

X(34813) = X(i)-complementary conjugate of X(j) for these (i,j): {7084, 57}, {7123, 946}

X(34814) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(58)

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

X(34814) lies on these lines: {6, 41}, {58, 21009}, {1125, 4363}

X(34814) = crosssum of X(10) and X(4663)

X(34815) = CENTER OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(64)

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

X(34815) lies on the Cubic K004 (the Darboux cubic) and these lines: {3, 6}, {4, 20208}, {20, 32001}, {25, 1073}, {64, 2130}, {393, 15312}, {1593, 11397}, {1657, 18437}, {2883, 15594}, {3089, 33546}, {3516, 23635}, {6389, 29181}, {6617, 33586}, {7390, 18643}, {13611, 26958}, {14379, 33578}, {26906, 33522}, {30517, 33553}, {31860, 33924}, {33582, 34146}

Let A'B'C' be the circumcevian triangle of X(64). Let A"=AO∩B'C', and define B'' and C'' cyclically. The points A'A", B'B", C'C" concur in X(34815). More generally, suppose that a point P lies on the Darboux cubic, and let A'B'C' be the circumcevian triangle of P. Define B'' and C'' cyclically. The points A'A", B'B", C'C" concur in a point, Q(P). The appearance of (i,j) in the following list means that X(i) is on the Darboux cubic and Q(X(i)) = X(j): (1,991), (,3), (4,5890), (64,34815). (Angel Montesdeoca, August 3, 2021). This list can be extended to include (20,44073), (40,44074), (84,44075). (Peter Moses, August 5, 2021)

See X(34815)

X(34815) = midpoint of X(20) and X(32001)
X(34815) = X(i)-complementary conjugate of X(j) for these (i,j): {3424, 20308}, {19614, 7710}
X(34815) = crosssum of X(20) and X(5921)
X(34815) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 15851, 5085}, {216, 31884, 3}

X(34816) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(2)

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

X(34816) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {6, 3934}, {25, 8266}, {42, 21022}, {141, 263}, {183, 251}, {694, 3763}, {3108, 11174}, {3589, 11175}, {14614, 34572}

X(34816) = isotomic conjugate of X(7786)
X(34816) = isotomic conjugate of the anticomplement of X(31239)
X(34816) = isotomic conjugate of the complement of X(31276)
X(34816) = isotomic conjugate of the isogonal conjugate of X(10014)
X(34816) = X(31239)-cross conjugate of X(2)
X(34816) = X(31)-isoconjugate of X(7786)
X(34816) = cevapoint of X(i) and X(j) for these (i,j): {2, 31276}, {115, 3806}
X(34816) = barycentric product X(76)*X(10014)
X(34816) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 7786}, {10014, 6}

X(34817) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(3)

Barycentrics    a^2*(a^2 - b^2 - c^2)*(a^2 + 3*b^2 + c^2)*(a^2 + b^2 + 3*c^2) : :
Trilinears    1/(2 sec A - csc A tan ω) : :
Trilinears    1/(csc A - 2 sec A cot ω) : :
X(34817) = 3 X[10519] + X[11821], X[16936] - 3 X[31884]

The trilinear polar of X(34817) passes through X(647).

X(34817) lies on the Jerabek circumhyperbola and these lines: {4, 141}, {6, 3917}, {54, 5085}, {64, 159}, {65, 3242}, {66, 599}, {67, 32114}, {68, 15812}, {71, 1473}, {72, 7289}, {74, 907}, {154, 34207}, {193, 21766}, {248, 22085}, {265, 18536}, {394, 1176}, {511, 3527}, {524, 17040}, {1173, 11477}, {1177, 17847}, {1191, 1245}, {1407, 12329}, {3098, 3426}, {3531, 9822}, {3619, 33586}, {3631, 16774}, {3763, 17810}, {5447, 15316}, {6391, 11574}, {6415, 11513}, {6416, 11514}, {9815, 17834}, {11898, 34483}, {13622, 15533}, {15585, 34778}

X(34817) = midpoint of X(1350) and X(33537)
X(34817) = reflection of X(i) in X(j) for these {i,j}: {6, 31521}, {15435, 141}
X(34817) = isogonal conjugate of X(6995)
X(34817) = isogonal conjugate of the anticomplement of X(7386)
X(34817) = isogonal conjugate of the polar conjugate of X(18840)
X(34817) = X(19459)-cross conjugate of X(6)
X(34817) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6995}, {19, 3618}, {82, 3867}, {92, 30435}, {158, 3796}, {162, 3800}, {811, 3804}, {1096, 3785}, {1897, 3803}
X(34817) = barycentric product X(i)*X(j) for these {i,j}: {3, 18840}, {63, 23051}, {394, 8801}, {525, 907}
X(34817) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 3618}, {6, 6995}, {39, 3867}, {184, 30435}, {394, 3785}, {577, 3796}, {647, 3800}, {907, 648}, {3049, 3804}, {3292, 3793}, {3917, 8362}, {8801, 2052}, {18840, 264}, {22383, 3803}, {23051, 92}
X(34817) = {X(141),X(1350)}-harmonic conjugate of X(7716)

X(34818) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(25)

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

X(34818) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {2, 53}, {6, 1598}, {216, 11484}, {263, 7716}, {1595, 5013}, {1609, 8770}, {2165, 21841}, {2207, 8882}, {6748, 11282}

X(34818) = isogonal conjugate of the anticomplement of X(11433)
X(34818) = isogonal conjugate of the isotomic conjugate of X(8796)
X(34818) = polar conjugate of the isotomic conjugate of X(3527)
X(34818) = X(8796)-Ceva conjugate of X(3527)
X(34818) = X(8573)-cross conjugate of X(6)
X(34818) = X(i)-isoconjugate of X(j) for these (i,j): {63, 631}, {304, 11402}, {326, 3087}
X(34818) = trilinear pole of line {512, 30442}
X(34818) = barycentric product X(i)*X(j) for these {i,j}: {4, 3527}, {6, 8796}, {25, 8797}
X(34818) = barycentric quotient X(i)/X(j) for these {i,j}: {25, 631}, {1974, 11402}, {2207, 3087}, {3527, 69}, {8796, 76}, {8797, 305}

X(34819) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(31)

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

X(34819) lies on these ines: {6, 35}, {31, 2174}, {32, 28615}, {45, 2298}, {81, 16884}, {604, 1399}, {739, 8652}, {1333, 17104}, {2162, 16685}, {2214, 5248}, {2300, 28607}, {5019, 9456}, {14621, 15668}, {18825, 32042}

X(34819) = isogonal conjugate of X(28605)
X(34819) = isogonal conjugate of the anticomplement of X(28606)
X(34819) = isogonal conjugate of the isotomic conjugate of X(25417)
X(34819) = X(i)-isoconjugate of X(j) for these (i,j): {1, 28605}, {2, 1698}, {6, 30596}, {7, 4007}, {8, 4654}, {10, 5333}, {75, 16777}, {81, 4066}, {85, 3715}, {92, 3927}, {99, 4838}, {100, 4823}, {190, 4802}, {306, 31902}, {312, 5221}, {321, 4658}, {335, 4716}, {514, 4756}, {664, 4820}, {668, 4813}, {670, 4826}, {671, 4938}, {903, 4727}, {1441, 4877}, {1978, 4834}, {3679, 30589}, {3903, 4842}, {3952, 4960}, {4033, 4840}, {4373, 4898}, {4555, 4958}, {4562, 4810}, {4803, 30587}, {4880, 18359}, {4942, 9311}, {6742, 23883}
X(34819) = cevapoint of X(1977) and X(8637)
X(34819) = crosssum of X(i) and X(j) for these (i,j): {1698, 4007}, {3927, 16777}
X(34819) = crossdifference of every pair of points on line {4802, 4823}
X(34819) = barycentric product X(i)*X(j) for these {i,j}: {6, 25417}, {31, 30598}, {81, 28625}, {513, 8652}, {667, 32042}, {28607, 30590}
X(34819) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 30596}, {6, 28605}, {31, 1698}, {32, 16777}, {41, 4007}, {42, 4066}, {184, 3927}, {604, 4654}, {649, 4823}, {667, 4802}, {692, 4756}, {798, 4838}, {922, 4938}, {1333, 5333}, {1397, 5221}, {1919, 4813}, {1924, 4826}, {1980, 4834}, {2175, 3715}, {2203, 31902}, {2206, 4658}, {2210, 4716}, {2251, 4727}, {3063, 4820}, {8652, 668}, {20981, 4842}, {25417, 76}, {28607, 30589}, {28625, 321}, {30598, 561}, {32042, 6386}

X(34820) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(55)

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

X(34820) lies on these lines: {6, 1334}, {9, 3913}, {19, 45}, {37, 57}, {55, 3217}, {198, 11051}, {219, 2364}, {220, 284}, {333, 346}, {673, 2345}, {893, 31477}, {909, 4287}, {1436, 3207}, {1751, 17281}, {2160, 16675}, {2256, 3451}, {2291, 8694}, {2299, 7071}, {2339, 3693}, {3247, 5022}, {4254, 33635}, {7077, 10387}, {15279, 16777}, {16672, 21809}

X(34820) = isogonal conjugate of X(21454)
X(34820) = isogonal conjugate of the anticomplement of X(18228)
X(34820) = X(25430)-Ceva conjugate of X(2334)
X(34820) = X(i)-isoconjugate of X(j) for these (i,j): {1, 21454}, {2, 3361}, {7, 1449}, {56, 19804}, {57, 3616}, {81, 3671}, {100, 30723}, {109, 4801}, {222, 5342}, {269, 391}, {278, 4652}, {279, 4512}, {348, 5338}, {461, 7177}, {651, 4778}, {664, 4790}, {934, 4765}, {1014, 5257}, {1088, 4258}, {1214, 31903}, {1396, 4101}, {1407, 4673}, {1414, 4841}, {1461, 4811}, {1462, 4684}, {4565, 4815}, {4573, 4822}, {4625, 4832}, {4626, 4827}, {4635, 8653}, {4637, 4843}, {4734, 7153}, {4835, 7175}, {5586, 25417}
X(34820) = crosspoint of X(4866) and X(25430)
X(34820) = crosssum of X(1449) and X(3361)
X(34820) = trilinear pole of line {663, 4524}
X(34820) = crossdifference of every pair of points on line {4778, 16533}
X(34820) = barycentric product X(i)*X(j) for these {i,j}: {1, 4866}, {8, 2334}, {9, 25430}, {55, 5936}, {522, 8694}, {650, 4606}, {657, 4624}, {3700, 4627}, {3709, 4633}, {4041, 4614}, {4391, 34074}, {14626, 14942}
X(34820) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 21454}, {9, 19804}, {31, 3361}, {33, 5342}, {41, 1449}, {42, 3671}, {55, 3616}, {200, 4673}, {212, 4652}, {220, 391}, {649, 30723}, {650, 4801}, {657, 4765}, {663, 4778}, {1253, 4512}, {1334, 5257}, {2212, 5338}, {2299, 31903}, {2318, 4101}, {2334, 7}, {2340, 4684}, {3063, 4790}, {3689, 4742}, {3709, 4841}, {3900, 4811}, {4041, 4815}, {4524, 4843}, {4606, 4554}, {4614, 4625}, {4627, 4573}, {4866, 75}, {5545, 4616}, {5936, 6063}, {7071, 461}, {8694, 664}, {14626, 9436}, {14827, 4258}, {20967, 4719}, {25430, 85}, {34074, 651}

X(34821) = PERSPECTOR OF PERSPECONIC OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(56)

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

X(34821) lies on the conic {{A,B,C,X(1),X(6)}} and these lines: {1, 1418}, {1458, 2334}, {3445, 21002}

X(34821) = isogonal conjugate of the anticomplement of X(10580)
X(34821) = X(i)-isoconjugate of X(j) for these (i,j): {8, 10389}, {9, 18230}
X(34821) = barycentric product X(57)*X(10390)
X(34821) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 18230}, {604, 10389}, {10390, 312}

Centers of perspeconics of cevian and anticevian triangles: X(34822)-X(34859)

This preamble and centers X(34822)-X(34859) were contributed by Clark Kimberling and Peter Moses, November 20, 2019.

Suppose that A'B'C' and A''B''C'' are perspective central triangles, so that the points

B'C'∩A''B'', B'C'∩A''C'', C'A'∩B''C'', C'A'∩B''A'', A'B'∩C''B'', A'B'∩C''A''

are distinct and lie on a conic. In the preamble just before X(15254) the conic is named the perspeconic of A'B'C' and A''B''C''.

Suppose that P = p : q : r and U = u : v : w. The cevian triangle of P is perspective to the anticevian triangle of U, the perspector being, by definition, the P-Ceva conjugate of U. The perspeconic of the two triangles is given by

q^2 r^2 ((q r u + p r v - p q w) (q r u - p r v + p q w) x^2 + 2 p^2 u (q r u + p r v + p q w) y z) + (cyclic) = 0,

center = (pending) : :

perspector = (pending) : :

The appearance of (i,j,k) in the following list means that X(k) is the center of the perspeconic of the cevian triangle of X(i) and the anticevian triangle of (X(j):

(1,1,15492), (1,6,3185), (1,36,15586), (1,44,902), (1,513,649), (2,1,10), (2,2,2), (2,3,5), (2,4,3), (2,5,140), (2,6,141), (2,7,9), (2,8,1), (2,9,142), (2,10,1125), (2,11,3035), (2,12,4999), (2,13,618), (2,14,619), (2,15,623), (2,16,624), (2,17,629), (2,18,630), (2,19,18589), (2,20,4), (2,21,442), (2,22,427), (2,23,858), (2,24,11585), (2,25,1368), (2,26,13371), (2,27,440), (2,28,21530), (2,29,18641), (2,31,2887), (2,32,626), (2,35,25639), (2,36,3814), (2,37,3739), (2,38,1215), (2,39,3934), (2,40,946), (2,41,17046), (2,42,3741), (2,43,3840), (2,44,3834), (2,46,21616), (2,48,20305), (2,51,3819), (2,52,1216), (2,54,1209), (2,55,2886), (2,56,1329), (2,57,3452), (2,58,3454), (2,61,635), (2,62,636), (2,63,226), (2,64,2883), (2,65,960), (2,66,206,{2,67,6593), (2,68,1147), (2,69,6), (2,70,34116), (2,72,942), (2,74,113), (2,75,37), (2,76,39), (2,77,20262), (2,78,1210), (2,79,3647), ,(2,80,214), (2,81,1211), (2,82,21249), (2,83,6292), (2,84,6260), (2,85,1212), (2,86,1213), (2,88,16594), (2,92,1214), (2,95,233), (2,98,114), (2,99,115), (2,100,11), (2,101,116), (2,102,117), (2,103,118), (2,104,119), (2,105,120), (2,106,121), (2,107,122), (2,108,123), (2,109,124), (2,110,125), (2,111,126), (2,112,127), (2,113,6699), (2,114,6036), (2,115,620), (2,116,6710), (2,117,6711), (2,118,6712), (2,119,6713), (2,120,6714), (2,121,6715), (2,122,6716), (2,123,6717), (2,124,6718), (2,125,5972), (2,126,6719), (2,127,6720), (2,137,13372), (2,140,3628), (2,141,3589), (2,142,6666), (2,143,32142), (2,144,7), (2,145,8), (2,146,74), (2,147,98), (2,148,99), (2,149,100), (2,150,101), (2,151,102), (2,152,103), (2,153,104), (2,154,2333), (2,155,12359), (2,156,13561), (2,157,23333), (2,159,23300), (2,163,21253), (2,164,21633), (2,165,3817), (2,171,3846), (2,174,2090), (2,175,14121), (2,176,7090), (2,177,18258), (2,182,24206), (2,183,3815), (2,184,21243), (2,185,5907), (2,186,2072), (2,187,625), (2,188,178), (2,189,223), (2,190,1086), (2,191,11263), (2,192,75), (2,193,69), (2,194,76), (2,195,21230), (2,197,23304), (2,198,21239), (2,199,34119), (2,200,11019), (2,204,20309), (2,206,6697), (2,210,3742), (2,213,21240), (2,214,6702), (2,216,14767), (2,219,16608), (2,220,21258), (2,221,20306), (2,223,20205), (2,224,10395), (2,226,5745), (2,233,6709), (2,235,16196), (2,237,21531), (2,238,3836), (2,239,3912), (2,244,24003), (2,251,21248), (2,252,31376,(2,253,1249), (2,262,15819), (2,264,216), (2,265,1511), (2,271,20264), (2,274,16589), (2,279,6554), (2,280,7952), (2,281,17073), (2,282,20206), (2,284,17052), (2,286,18591), (2,287,15595), (2,290,11672), (2,291,17793), (2,292,20333), (2,294,17060), (2,297,441), (2,298,396), (2,299,395), (2,302,23302), (2,303,23303), (2,304,16583), (2,305,1196), (2,310,21838), (2,311,570), (2,312,3752), (2,314,2092), (2,315,32), (2,316,187), (2,317,577), (2,318,17102), (2,319,1100), (2,320,44), (2,321,3666), (2,322,1108), (2,323,3580), (2,325,230), (2,326,24005), (2,329,57), (2,330,6376), (2,333,17056), (2,335,17755), (2,340,3284), (2,343,23292), (2,344,17278,(2,345,3772), (2,346,4000), (2,347,281), (2,350,1575), (2,353,34512), (2,354,3740), (2,355,1385), (2,360,1115), (2,365,20334), (2,366,20527), (2,371,639), (2,372,640), (2,373,15082), (2,376,381), (2,377,405), (2,378,15760), (2,379,30810), (2,381,549), (2,382,550), (2,384,6656), (2,385,325), (2,388,958), (2,389,11793), (2,390,2550), (2,391,4648), (2,393,6389), (2,394,13567), (2,399,10264), (2,401,297), (2,402,15184), (2,403,10257), (2,404,4187), (2,405,8728), (2,406,34120), (2,411,6831), (2,427,6676), (2,428,10691), (2,439,32972), (2,440,6678), (2,442,6675), (2,443,11108), (2,452,443), (2,464,7522), (2,468,5159), (2,469,7536), (2,472,466), (2,473,465), (2,474,17527), (2,476,3258), (2,477,25641), (2,479,5574), (2,481,31594), (2,482,31595), (2,484,11813), (2,485,641), (2,486,642), (2,487,486), (2,488,485), (2,489,3071), (2,490,3070), (2,491,615), (2,492,590), (2,497,1376), (2,508,14218), (2,546,3530), (2,547,10124), (2,548,3850), (2,549,547), (2,550,546);

(3,520,32320), (4,2,15591), (4,6,15508), (4,230,460), (4,403,1990), (4,523,2501), (6,6,15513), (6,187,14567), (6,237,9418), (6,512,669), (7,1,15493), (7,514,3676), (8,522,3239), (10,523,4024), (13,523,20578), (14,523,20579), (21,521,23090), (27,514,17925), (29,522,17926), (30,30,3163), (45,381,2226), (69,525,3265), (75,514,693), (76,523,850), (81,238,5009), (81,513,3733), (85,522,693), (86,239,33295), (86,514,7192), (88,1,106), (88,513,23345), (92,240,6530), (94,523,10412), (98,6,1976), (98,523,2395), (99,2,99), (99,6,110), (99,69,4563), (99,86,4610), (99,141,4576), (99,230,4226), (99,323,10411), (99,325,2396), (99,385,17941), (99,524,5468), (100,1,101), (100,6,692), (100,9,3939), (100,37,4557), (100,44,23344), (100,72,4574), (100,518,2284), (101,6,32739), (107,4,6529), (107,6,32713), (110,3,32661), (110,6,1576), (110,32,14574), (110,511,14966), (111,6,32740), (112,184,14574), (162,1,112), (162,19,32713), (162,48,1576), (163,31,14574), (190,1,100), (190,2,190), (190,8,3699), (190,10,3952), (190,42,4557), (190,78,4571), (190,200,4578), (190,239,3570), (190,519,17780), (264,525,850), (274,513,7192), (287,3,17974), (290,2,290,(291,513,3572), (312,522,4397), (321,523,4036), (333,522,7253), (335,514,4444), (336,63,6394), (348,521,4131), (476,6,14560), (476,115,15475), (476,542,23968), (491,217,12322), (492,280,34466), (511,511,11672), (512,512,1084), (513,244,21143), (513,512,6377), (513,513,1015), (514,11,21132), (514,244,764), (514,513,3756), (514,514,1086), (514,523,11), (515,515,23986), (516,516,23972), (517,517,23980), (518,518,6184), (519,519,4370), (522,522,1146), (522,523,6506), (523,115,8029), (523,512,6388), (523,523,115), (524,524,2482), (525,125,5489), (525,520,122), (525,523,13611), (525,525,15526), (526,526,18334), (536,536,13466), (542,542,23967)

X(34822) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(33)

Barycentrics (a^2 - b^2 - c^2)*(a^3*b - a^2*b^2 + a*b^3 - b^4 + a^3*c + 2*a^2*b*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4) : :
Barycentrics sin B + tan B + sin C + tan C : :

X(34822) lies on these lines: {1, 20266), {2, 33), {3, 10), {8, 1038), {19, 26118), {55, 25907), {57, 5800), {118, 123), {141, 20307), {142, 18642), {214, 24301), {216, 1575), {225, 23661), {306, 1818), {497, 30674), {519, 1060), {551, 18455), {577, 4386), {860, 1074), {910, 20262), {960, 2883), {1062, 1125), {1107, 22401), {1210, 1834), {1211, 5784), {1214, 2968), {1329, 6823), {1368, 2886), {1370, 24611), {1377, 1579), {1378, 1578), {1589, 6348), {1590, 6347), {1864, 26005), {2550, 7386), {2551, 10996), {3035, 6676), {3244, 18447), {3434, 30675), {3546, 26363), {3547, 26364), {3741, 6389), {3742, 18214), {3814, 15760), {3946, 11019), {4200, 19372), {4413, 25947), {4999, 16196), {6247, 9942), {6349, 26015), {6350, 25006), {6708, 8727), {6712, 34589), {7004, 21912), {7289, 18935), {9943, 20306), {10167, 26932), {10391, 13567), {11018, 16608), {11574, 17792), {11585, 25639), {17441, 26933), {18608, 22410), {18652, 20277), {20254, 29655), {21629, 24703), {22057, 29639), {22066, 23899), {24411, 33099), {24982, 25876), {25915, 26105), {26153, 27143), {28426, 30104), {30975, 31623}

X(34822) = complement of X(33)
X(34822) = complementary conjugate of X(20262)

X(34823) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(34)

Barycentrics (a - b - c)*(a^2 - b^2 - c^2)*(a^3*b + a^2*b^2 + a*b^3 + b^4 + a^3*c - 2*a^2*b*c - a*b^2*c + a^2*c^2 - a*b*c^2 - 2*b^2*c^2 + a*c^3 + c^4) : :
Barycentrics sin B - tan B + sin C - tan C : :

X(34823) lies on these lines: {2, 34), {3, 10), {8, 1040), {9, 18596), {20, 1861), {33, 27505), {56, 25947), {72, 26932), {121, 123), {124, 21616), {141, 960), {216, 1107), {225, 24984), {227, 25882), {281, 443), {440, 30847), {519, 1062), {551, 18447), {577, 4426), {860, 1076), {1060, 1125), {1104, 1210), {1146, 16605), {1212, 1213), {1329, 1368), {1377, 1578), {1378, 1579), {1575, 22401), {1589, 6347), {1590, 6348), {1891, 4224), {2299, 27405), {2550, 10996), {2551, 7386), {2885, 5123), {2886, 6823), {2968, 6736), {3035, 16196), {3244, 18455), {3452, 3454), {3546, 26364), {3547, 26363), {3634, 34120), {3814, 11585), {4194, 9817), {4314, 24388), {4999, 6676), {6349, 24564), {6350, 24987), {6708, 8728), {6734, 27407), {7400, 19843), {7494, 30478), {9367, 18905), {10916, 34589), {11513, 31453), {15760, 25639), {16968, 24005), {17102, 31397), {20727, 34591), {22341, 28265), {25932, 26001), {26153, 27093), {27385, 28810}

X(34823) = complement of X(34)
X(34823) = complementary conjugate of X(1210)

X(34824) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(45)

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

X(34824) lies on these lines: {1, 4395), {2, 45), {7, 17259), {8, 17313), {10, 141), {37, 1266), {75, 3943), {86, 17366), {145, 4361), {239, 17392), {313, 30044), {320, 16815), {344, 17118), {519, 4405), {524, 4384), {527, 31211), {528, 24331), {536, 4029), {537, 25352), {594, 4699), {597, 3008), {894, 17337), {952, 24262), {966, 7232), {1125, 17067), {1211, 27186), {1213, 3662), {1269, 29982), {2321, 4739), {2345, 17265), {2486, 3816), {3589, 10436), {3616, 4000), {3624, 4657), {3629, 3664), {3630, 3686), {3631, 17275), {3632, 4399), {3635, 4852), {3636, 3946), {3663, 4698), {3707, 4715), {3758, 29628), {3828, 4407), {3912, 4665), {3925, 33120), {3950, 4726), {4357, 31238), {4371, 20053), {4398, 27268), {4402, 20057), {4443, 17063), {4445, 4678), {4478, 4668), {4643, 6173), {4664, 29581), {4667, 8584), {4687, 17246), {4690, 22165), {4701, 17372), {4727, 29601), {4733, 33087), {4740, 29599), {4772, 17233), {4795, 16670), {4798, 29598), {4850, 24184), {4871, 21264), {4967, 17231), {4969, 16816), {4971, 17119), {5241, 31019), {5249, 5743), {5257, 17235), {5296, 17255), {5308, 17318), {5564, 17312), {5718, 24589), {5737, 9776), {5750, 17356), {6666, 17351), {6703, 24789), {7222, 18230), {7227, 17279), {7277, 17349), {7321, 17260), {9955, 12610), {14442, 24110), {14743, 14767), {16593, 24357), {16706, 17398), {16724, 17180), {16826, 17395), {16831, 17301), {17051, 21242), {17056, 19804), {17116, 17263), {17117, 17317), {17133, 29606), {17160, 29569), {17227, 29576), {17237, 24603), {17262, 31995), {17264, 29626), {17269, 29627), {17277, 17365), {17282, 17303), {17283, 28604), {17291, 28653), {17300, 17362), {17306, 33159), {17308, 20582), {17309, 32087), {17320, 29578), {17327, 19877), {17387, 29617), {17399, 29612), {17450, 21027), {19878, 24295), {20257, 25130), {20530, 25382), {20943, 30045), {22791, 24220), {24902, 25454), {26149, 27111), {27017, 27154), {27042, 27311), {27164, 33947), {28640, 31312), {29596, 31243}

X(34824) = isotomic conjugate of X(32013)
X(34824) = complement of X(45)
X(34824) = anticomplement of X(31285)
X(34824) = polar conjugate of isogonal conjugate of X(22068)
X(34824) = complementary conjugate of complement of X(89)
X(34824) = complementary conjugate of isotomic conjugate of isogonal conjugate of X(20973)
X(34824) = complementary conjugate of polar conjugate of isogonal conjugate of X(22083)

X(34825) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(47)

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

X(34825) lies on these lines: {2, 47), {10, 12), {124, 21616), {3814, 31847), {3936, 23518), {19843, 25958), {25760, 26363), {25957, 26364}

X(34825) = complement of X(47)
X(34825) = complementary conjugate of complement of X(91)

X(34826) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(49)

Barycentrics a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10 - 3*a^2*b^6*c^2 + 3*b^8*c^2 + a^6*c^4 - 2*b^6*c^4 - 3*a^4*c^6 - 3*a^2*b^2*c^6 - 2*b^4*c^6 + 3*a^2*c^8 + 3*b^2*c^8 - c^10 : :
Barycentrics b cos 3B + c cos 3C : :

X(34826) lies on these lines: {2, 49), {3, 12278), {4, 3581), {5, 389), {26, 34514), {30, 11572), {54, 11264), {69, 11255), {125, 128), {141, 15074), {143, 3580), {156, 6639), {184, 18356), {186, 6288), {265, 14118), {323, 3519), {343, 6101), {394, 31283), {427, 10263), {550, 20299), {858, 10627), {1154, 1594), {1216, 14076), {1493, 23292), {1495, 18282), {1568, 31834), {1656, 7592), {1658, 18474), {2072, 11591), {2888, 6143), {3410, 14940), {3526, 26913), {3549, 32140), {3574, 13413), {3575, 22804), {3628, 15806), {3819, 21357), {3853, 18488), {5094, 16266), {5133, 10095), {5446, 33332), {5562, 10224), {5609, 12827), {5663, 10024), {5944, 7542), {6689, 15872), {7502, 18381), {7526, 14852), {7527, 15807), {7550, 15027), {7574, 7691), {7577, 18436), {7689, 34798), {8254, 13366), {8550, 24206), {9927, 18570), {10020, 12134), {10254, 12111), {10255, 11459), {10296, 18442), {11412, 23330), {11454, 18565), {11550, 17714), {11585, 15067), {11704, 15056), {11799, 32137), {12006, 13565), {12118, 18580), {12162, 13406), {12363, 32142), {12902, 18364), {13160, 13630), {13363, 14788), {13491, 15760), {14128, 20304), {14516, 32171), {15062, 31726), {15124, 15605), {15646, 20191), {16619, 16621), {16868, 18435), {18379, 18563), {18392, 18562), {18394, 18564), {22052, 34520), {22352, 34004), {25563, 34152}

X(34826) = complement of X(49)
X(34826) = complementary conjugate of X(34833)

X(34827) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(50)

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

X(34827) lies on these lines: {2, 50), {5, 141), {127, 10297), {140, 22463), {297, 11062), {323, 18883), {325, 2493), {468, 14052), {524, 16310), {620, 7575), {868, 3001), {924, 30476), {1995, 7778), {2450, 9019), {3003, 18122), {3284, 24975), {3788, 12106), {6389, 18404), {7542, 31376), {7800, 14787), {9220, 18375), {17710, 23333}

X(34827) = complement of X(50)
X(34827) = complementary conjugate of X(34834)

X(34828) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(53)

Barycentrics (a^2 - b^2 - c^2)*(2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 - b^6 - 3*a^4*c^2 - 4*a^2*b^2*c^2 + b^4*c^2 + 2*a^2*c^4 + b^2*c^4 - c^6) : :
Barycentrics b tan B cos(C - A) + c tan C cos(A - B) : :

X(34828) lies on these lines: {2, 53), {3, 66), {69, 10607), {95, 19212), {97, 1238), {127, 128), {140, 6709), {216, 441), {401, 6748), {418, 8266), {426, 26905), {524, 577), {570, 23292), {620, 2790), {631, 33971), {1368, 23333), {1609, 13567), {1990, 3164), {2871, 11574), {2968, 4399), {3284, 32455), {3523, 20792), {3629, 15905), {3631, 15526), {3917, 6751), {3934, 16197), {5085, 26870), {5158, 6329), {5743, 21482), {7386, 7778), {7494, 15271), {7495, 26895), {7499, 26907), {7745, 28723), {10979, 34573), {12362, 23702), {17045, 17102), {17245, 21940), {17265, 25932), {17327, 25876), {17811, 17849), {18380, 18531), {19126, 19156}

X(34828) = complement of X(53)
X(34828) = complementary conjugate of X(34836)
X(34828) = X(6) of polar triangle of complement of polar circle

X(34829) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(60)

Barycentrics (b + c)^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^2*b^2*c + 2*a*b^3*c + b^4*c + a^3*c^2 + a^2*b*c^2 + a^2*c^3 + 2*a*b*c^3 - a*c^4 + b*c^4 - c^5) : :
Barycentrics b sec^2 (C/2 - A/2) + c sec^2 (A/2 - B/2) : :

X(34829) lies on these lines: {2, 60), {10, 125), {115, 3721), {442, 1737), {908, 14873), {960, 30447), {997, 27685), {1098, 25671), {1109, 21682), {1211, 5044), {2185, 25463), {2392, 3142), {2503, 6537), {3336, 24251), {4187, 8287), {4205, 19869), {4511, 27716), {8286, 24390), {10479, 21243), {11415, 27583), {12514, 27553), {15443, 26942), {21616, 27555}

X(34829) = complement of X(60)
X(34829) = complementary conjugate of X(4999)

X(34830) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(71)

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

X(34830) lies on these lines: {2, 71), {3, 142), {4, 15669), {5, 916), {6, 226), {7, 2260), {11, 18635), {19, 24315), {27, 86), {37, 24424), {48, 379), {69, 30985), {75, 29967), {85, 18161), {113, 116), {141, 674), {206, 17068), {238, 1780), {306, 19792), {307, 24317), {313, 30059), {321, 29964), {497, 2293), {517, 21231), {673, 2259), {908, 17277), {960, 3739), {966, 30961), {1086, 11672), {1111, 18726), {1193, 3485), {1441, 1953), {1740, 17889), {1826, 25935), {1836, 20992), {1918, 3011), {1964, 23682), {2209, 33127), {2274, 23536), {2294, 17863), {2309, 3120), {2325, 22019), {2643, 23688), {2883, 21239), {3006, 17142), {3136, 14053), {3262, 17868), {3286, 4292), {3452, 17259), {3664, 17197), {3687, 20174), {3736, 23537), {3946, 17761), {4001, 29767), {4032, 8609), {4269, 4357), {4279, 24160), {4359, 29985), {4361, 12635), {4363, 20258), {4511, 24435), {4649, 13407), {4858, 18692), {4869, 29824), {5132, 13411), {5443, 24780), {5750, 22006), {5799, 25466), {5901, 17043), {5905, 27317), {7560, 20291), {8227, 18634), {9599, 28078), {10436, 25363), {10446, 25521), {10478, 25525), {10960, 30380), {10962, 30381), {11235, 17313), {11263, 33682), {11281, 17045), {14953, 22054), {15485, 18393), {16470, 17189), {16484, 30384), {16696, 24214), {16713, 20347), {16732, 17443), {16738, 17173), {17049, 20544), {17052, 25639), {17062, 31936), {17138, 26237), {17139, 28287), {17182, 25508), {17298, 29769), {17306, 19863), {17349, 31053), {17379, 29833), {18134, 19803), {18137, 29988), {19804, 30007), {20244, 27514), {20271, 28087), {20883, 23581), {20892, 29979), {21191, 29328), {21195, 31947), {21801, 25001), {22088, 27171), {24596, 27381), {24789, 27623), {24919, 27646), {25512, 31435), {25631, 30970), {25660, 29968), {27644, 33129), {30019, 30044}

X(34830) = complement of X(71)
X(34830) = barycentric product X(1)*X(17866)
X(34830) = complementary conjugate of X(440)

X(34831) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(73)

Barycentrics    (a - b - c)*(a^4*b^2 + a^3*b^3 - a^2*b^4 - a*b^5 + a^2*b^3*c - b^5*c + a^4*c^2 + a*b^3*c^2 + a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 - a*c^5 - b*c^5) : :
Barycentrics    (cos C + cos A) sin 2B + (cos A + cos B) sin 2C : :
Barycentrics    b (sec C + sec A) + c (sec A + sec B) : :

X(34831) lies on these lines: {1, 25490), {2, 73), {3, 10), {6, 1210), {8, 26091), {9, 10479), {29, 270), {65, 26013), {113, 124), {141, 1329), {212, 27378), {255, 5136), {318, 24430), {412, 1861), {442, 26932), {603, 24537), {942, 6708), {950, 4267), {960, 3741), {1107, 1146), {1125, 25493), {1150, 27410), {1209, 3814), {1220, 24982), {1737, 3075), {1935, 11109), {2883, 2886), {3142, 14055), {3670, 4858), {3812, 20617), {3840, 25681), {7004, 23661), {10960, 31453), {11813, 18330), {13567, 15844), {21044, 23640), {21384, 23058), {22072, 27506), {25131, 30983), {30478, 31339), {30943, 31330}

X(34831) = complement of X(73)
X(34831) = complementary conjugate of X(18641)

X(34832) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(87)

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

X(34832) lies on these lines: {1, 25311), {2, 87), {9, 20370), {10, 75), {37, 20532), {42, 26756), {121, 3822), {141, 3816), {142, 20333), {1086, 24182), {1125, 14823), {1329, 3836), {1918, 29379), {2309, 27095), {2885, 3826), {3551, 24451), {3662, 20340), {3741, 17238), {4110, 4941), {4687, 25113), {4699, 25624), {4871, 17232), {17237, 21238), {17239, 24688), {17356, 24742), {17459, 21040), {17786, 24456), {17792, 17793), {18170, 25534), {20540, 20547}

X(34832) = complement of X(87)
X(34832) = isotomic conjugate of isogonal conjugate of X(20971)
X(34832) = polar conjugate of isogonal conjugate of X(22081)
X(34832) = complementary conjugate of X(3840)

X(34833) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(93)

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

X(34833) lies on these lines: {2, 93), {3, 54), {131, 137), {216, 231), {2072, 10600), {6389, 6640), {14106, 34101), {15869, 32423}

X(34833) = complement of X(93)
X(34833) = complementary conjugate of X(34826)

X(34834) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(94)

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

X(34834) lies on these lines: {2, 94), {3, 74), {39, 14389), {50, 323), {99, 2986), {113, 18781), {114, 858), {128, 136), {140, 15869), {1216, 15848), {1649, 6132), {1993, 4558), {2482, 18334), {3003, 3580), {3268, 5664), {3292, 22463), {4576, 6337), {5745, 16585), {6626, 22377), {7495, 15819), {7710, 16063), {12824, 15329), {20772, 30510}

X(34834) = complement of X(94)
X(34834) = complementary conjugate of X(34827)

X(34835) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(96)

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

X(34835) lies on these lines: {2, 54), {3, 2934), {4, 15827), {5, 8800), {24, 14111), {39, 233), {114, 12134), {128, 136), {3147, 14918), {6337, 7401), {6503, 6642), {6509, 10600), {7488, 15848), {7542, 12095), {14788, 31376}

X(34835) = complement of X(96)
X(34835) = complementary conjugate of X(1216)

X(34836) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(97)

Barycentrics (a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 - 3*a^2*b^4*c^2 + 2*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 - 2*b^4*c^4 + 3*a^2*c^6 + 2*b^2*c^6 - c^8) : :
Barycentrics cos B sec(C - A) + cos C sec(A - B) : :

X(34836) lies on these lines: {2, 95), {4, 10600), {5, 51), {125, 129), {127, 138), {132, 5133), {216, 467), {539, 19176), {908, 6708), {2052, 9290), {3078, 11197), {3091, 6523), {3767, 11433), {6503, 6642), {6747, 30258), {8613, 32002), {9722, 13567), {23583, 34545}

X(34836) = complement of X(97)
X(34836) = complementary conjugate of X(34828)

X(34837) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(128)

Barycentrics 2*a^16 - 10*a^14*b^2 + 22*a^12*b^4 - 31*a^10*b^6 + 35*a^8*b^8 - 32*a^6*b^10 + 20*a^4*b^12 - 7*a^2*b^14 + b^16 - 10*a^14*c^2 + 32*a^12*b^2*c^2 - 37*a^10*b^4*c^2 + 12*a^8*b^6*c^2 + 23*a^6*b^8*c^2 - 41*a^4*b^10*c^2 + 28*a^2*b^12*c^2 - 7*b^14*c^2 + 22*a^12*c^4 - 37*a^10*b^2*c^4 + 20*a^8*b^4*c^4 - 9*a^6*b^6*c^4 + 24*a^4*b^8*c^4 - 42*a^2*b^10*c^4 + 22*b^12*c^4 - 31*a^10*c^6 + 12*a^8*b^2*c^6 - 9*a^6*b^4*c^6 - 6*a^4*b^6*c^6 + 21*a^2*b^8*c^6 - 41*b^10*c^6 + 35*a^8*c^8 + 23*a^6*b^2*c^8 + 24*a^4*b^4*c^8 + 21*a^2*b^6*c^8 + 50*b^8*c^8 - 32*a^6*c^10 - 41*a^4*b^2*c^10 - 42*a^2*b^4*c^10 - 41*b^6*c^10 + 20*a^4*c^12 + 28*a^2*b^2*c^12 + 22*b^4*c^12 - 7*a^2*c^14 - 7*b^2*c^14 + c^16 : :
Barycentrics (tan B)(cos 2C + cos 2A)(1 + 2 cos 2B)(cos 2B + 2 cos 2C cos 2A) + (tan C)(cos 2A + cos 2B)(1 + 2 cos 2C)(cos 2C + 2 cos 2A cos 2B) : :

X(34837) lies on these lines: {2, 128), {3, 137), {4, 23516), {5, 11701), {30, 10615), {54, 31376), {140, 6592), {549, 1263), {550, 25147), {620, 6709), {623, 33526), {624, 33527), {631, 930), {632, 14072), {1209, 27196), {1656, 31656), {2072, 12095), {3327, 5433), {3523, 11671), {3628, 5972), {3850, 25339), {5020, 15960), {5054, 13512), {5432, 7159), {6642, 15959), {6644, 23320), {6701, 6718), {6720, 10127), {7542, 15366), {8254, 34768), {11539, 14073), {12383, 34308), {13504, 15043), {13505, 15045), {14140, 24147), {14769, 14788}

X(34837) = complement of X(128)
X(34837) = complementary conjugate of complement of X(15401)

X(34838) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(129)

Barycentrics a^2*(a^2 - b^2 - c^2)*(a^16*b^6 - 6*a^14*b^8 + 15*a^12*b^10 - 20*a^10*b^12 + 15*a^8*b^14 - 6*a^6*b^16 + a^4*b^18 + 2*a^18*b^2*c^2 - 9*a^16*b^4*c^2 + 14*a^14*b^6*c^2 - 8*a^12*b^8*c^2 + 4*a^10*b^10*c^2 - 14*a^8*b^12*c^2 + 22*a^6*b^14*c^2 - 16*a^4*b^16*c^2 + 6*a^2*b^18*c^2 - b^20*c^2 - 9*a^16*b^2*c^4 + 28*a^14*b^4*c^4 - 31*a^12*b^6*c^4 + 16*a^10*b^8*c^4 - 24*a^6*b^12*c^4 + 41*a^4*b^14*c^4 - 28*a^2*b^16*c^4 + 7*b^18*c^4 + a^16*c^6 + 14*a^14*b^2*c^6 - 31*a^12*b^4*c^6 + 18*a^10*b^6*c^6 - a^8*b^8*c^6 + 6*a^6*b^10*c^6 - 37*a^4*b^12*c^6 + 50*a^2*b^14*c^6 - 20*b^16*c^6 - 6*a^14*c^8 - 8*a^12*b^2*c^8 + 16*a^10*b^4*c^8 - a^8*b^6*c^8 + 4*a^6*b^8*c^8 + 11*a^4*b^10*c^8 - 44*a^2*b^12*c^8 + 28*b^14*c^8 + 15*a^12*c^10 + 4*a^10*b^2*c^10 + 6*a^6*b^6*c^10 + 11*a^4*b^8*c^10 + 32*a^2*b^10*c^10 - 14*b^12*c^10 - 20*a^10*c^12 - 14*a^8*b^2*c^12 - 24*a^6*b^4*c^12 - 37*a^4*b^6*c^12 - 44*a^2*b^8*c^12 - 14*b^10*c^12 + 15*a^8*c^14 + 22*a^6*b^2*c^14 + 41*a^4*b^4*c^14 + 50*a^2*b^6*c^14 + 28*b^8*c^14 - 6*a^6*c^16 - 16*a^4*b^2*c^16 - 28*a^2*b^4*c^16 - 20*b^6*c^16 + a^4*c^18 + 6*a^2*b^2*c^18 + 7*b^4*c^18 - b^2*c^20) : :

X(34838) lies on these lines: {2, 129), {3, 130), {620, 11793), {631, 1303), {3819, 13372), {5020, 22551), {5972, 32438), {6688, 6716), {6689, 6720}

X(34838) = complement of X(129)

X(34839) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(130)

Barycentrics a^2*(-(a^14*b^6) + 5*a^12*b^8 - 10*a^10*b^10 + 10*a^8*b^12 - 5*a^6*b^14 + a^4*b^16 + 2*a^16*b^2*c^2 - 7*a^14*b^4*c^2 + 6*a^12*b^6*c^2 + 5*a^10*b^8*c^2 - 9*a^8*b^10*c^2 - a^6*b^12*c^2 + 8*a^4*b^14*c^2 - 5*a^2*b^16*c^2 + b^18*c^2 - 7*a^14*b^2*c^4 + 22*a^12*b^4*c^4 - 19*a^10*b^6*c^4 - 6*a^8*b^8*c^4 + 21*a^6*b^10*c^4 - 20*a^4*b^12*c^4 + 13*a^2*b^14*c^4 - 4*b^16*c^4 - a^14*c^6 + 6*a^12*b^2*c^6 - 19*a^10*b^4*c^6 + 28*a^8*b^6*c^6 - 15*a^6*b^8*c^6 + 4*a^4*b^10*c^6 - 9*a^2*b^12*c^6 + 6*b^14*c^6 + 5*a^12*c^8 + 5*a^10*b^2*c^8 - 6*a^8*b^4*c^8 - 15*a^6*b^6*c^8 + 14*a^4*b^8*c^8 + a^2*b^10*c^8 - 4*b^12*c^8 - 10*a^10*c^10 - 9*a^8*b^2*c^10 + 21*a^6*b^4*c^10 + 4*a^4*b^6*c^10 + a^2*b^8*c^10 + 2*b^10*c^10 + 10*a^8*c^12 - a^6*b^2*c^12 - 20*a^4*b^4*c^12 - 9*a^2*b^6*c^12 - 4*b^8*c^12 - 5*a^6*c^14 + 8*a^4*b^2*c^14 + 13*a^2*b^4*c^14 + 6*b^6*c^14 + a^4*c^16 - 5*a^2*b^2*c^16 - 4*b^4*c^16 + b^2*c^18) : :
Barycentrics (sin^2 B) (sin 2C + sin 2A) (sin 2C - sin 2A)^2 (sin^2(2B) + sin 2C sin 2A) + (sin^2 C) (sin 2A + sin 2B) (sin 2A - sin 2B)^2 (sin^2(2C) + sin 2A sin 2B) : :

X(34839) lies on these lines: {2, 130), {3, 129), {631, 1298), {3917, 21661), {6036, 9729), {6699, 32438), {17811, 22552}

X(34839) = complement of X(130)

X(34840) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(131)

Barycentrics 2*a^16 - 8*a^14*b^2 + 11*a^12*b^4 - 6*a^10*b^6 + 5*a^8*b^8 - 12*a^6*b^10 + 13*a^4*b^12 - 6*a^2*b^14 + b^16 - 8*a^14*c^2 + 28*a^12*b^2*c^2 - 32*a^10*b^4*c^2 + 6*a^8*b^6*c^2 + 20*a^6*b^8*c^2 - 28*a^4*b^10*c^2 + 20*a^2*b^12*c^2 - 6*b^14*c^2 + 11*a^12*c^4 - 32*a^10*b^2*c^4 + 34*a^8*b^4*c^4 - 16*a^6*b^6*c^4 + 11*a^4*b^8*c^4 - 24*a^2*b^10*c^4 + 16*b^12*c^4 - 6*a^10*c^6 + 6*a^8*b^2*c^6 - 16*a^6*b^4*c^6 + 8*a^4*b^6*c^6 + 10*a^2*b^8*c^6 - 26*b^10*c^6 + 5*a^8*c^8 + 20*a^6*b^2*c^8 + 11*a^4*b^4*c^8 + 10*a^2*b^6*c^8 + 30*b^8*c^8 - 12*a^6*c^10 - 28*a^4*b^2*c^10 - 24*a^2*b^4*c^10 - 26*b^6*c^10 + 13*a^4*c^12 + 20*a^2*b^2*c^12 + 16*b^4*c^12 - 6*a^2*c^14 - 6*b^2*c^14 + c^16 : :

X(34840) lies on these lines: {2, 131), {3, 136), {5, 1511), {620, 14767), {631, 925), {1485, 5961), {6642, 13558), {6716, 16238), {7526, 13496), {9826, 23583), {10257, 13553), {15240, 23336}

X(34840) = complement of X(131)

X(34841) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(132)

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

X(34841) lies on these lines: {2, 107), {3, 114), {5, 19160), {6, 13918), {10, 12265), {98, 15526), {112, 631), {140, 6720), {339, 11623), {376, 10735), {468, 12145), {498, 13117), {499, 13116), {550, 19163), {577, 11610), {1001, 12340), {1078, 12207), {1656, 12918), {1698, 12784), {2781, 3819), {2799, 6036), {2806, 6713), {2825, 6710), {2831, 3035), {2853, 6711), {2967, 23583), {3068, 19093), {3069, 19094), {3090, 12253), {3320, 5432), {3523, 13219), {3524, 10718), {3526, 13115), {3576, 13280), {3616, 13099), {3624, 12408), {5020, 12413), {5054, 13310), {5159, 22104), {5188, 28697), {5204, 13296), {5217, 13297), {5418, 13923), {5420, 13992), {5433, 6020), {5657, 10705), {6677, 20203), {6699, 9517), {6712, 9518), {6714, 9523), {6715, 9527), {6718, 9532), {7494, 19164), {7846, 12503), {9540, 19115), {9756, 20208), {10165, 11722), {10519, 10766), {10691, 13372), {10780, 34474), {12796, 15183), {13935, 19114), {14489, 34579), {14676, 18876), {19159, 25524), {28405, 30270}

X(34841) = complement of X(132)
X(34841) = complementary conjugate of complement of X(15407)

X(34842) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(133)

Barycentrics (a^2 - b^2 - c^2)*(2*a^14 - 2*a^12*b^2 - 13*a^10*b^4 + 31*a^8*b^6 - 24*a^6*b^8 + 4*a^4*b^10 + 3*a^2*b^12 - b^14 - 2*a^12*c^2 + 28*a^10*b^2*c^2 - 31*a^8*b^4*c^2 - 12*a^6*b^6*c^2 + 20*a^4*b^8*c^2 - 8*a^2*b^10*c^2 + 5*b^12*c^2 - 13*a^10*c^4 - 31*a^8*b^2*c^4 + 72*a^6*b^4*c^4 - 24*a^4*b^6*c^4 + 5*a^2*b^8*c^4 - 9*b^10*c^4 + 31*a^8*c^6 - 12*a^6*b^2*c^6 - 24*a^4*b^4*c^6 + 5*b^8*c^6 - 24*a^6*c^8 + 20*a^4*b^2*c^8 + 5*a^2*b^4*c^8 + 5*b^6*c^8 + 4*a^4*c^10 - 8*a^2*b^2*c^10 - 9*b^4*c^10 + 3*a^2*c^12 + 5*b^2*c^12 - c^14) : :

X(34842) lies on these lines: {2, 133), {3, 113), {107, 631), {140, 6716), {376, 10152), {517, 11732), {549, 6720), {550, 33531), {620, 2790), {1650, 7687), {1656, 22337), {2797, 6036), {2803, 6713), {2811, 6712), {2816, 6718), {2822, 6710), {2828, 3035), {2846, 6711), {2972, 20417), {3324, 5432), {3523, 23239), {3524, 5667), {5433, 7158), {5657, 10701), {6699, 8552), {6714, 9520), {6715, 9524), {6717, 9528), {6719, 9529), {6759, 31377), {10165, 11718), {10257, 22104), {10519, 10762), {10775, 34474), {12096, 18400), {14379, 20299), {20208, 23329), {21734, 23241}

X(34842) = complement of X(133)
X(34842) = complementary conjugate of complement of X(15404)

X(34843) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(135)

Barycentrics (a^2 - b^2 - c^2)*(2*a^16 - 8*a^14*b^2 + 11*a^12*b^4 - 4*a^10*b^6 - 5*a^8*b^8 + 8*a^6*b^10 - 7*a^4*b^12 + 4*a^2*b^14 - b^16 - 8*a^14*c^2 + 28*a^12*b^2*c^2 - 34*a^10*b^4*c^2 + 22*a^8*b^6*c^2 - 20*a^6*b^8*c^2 + 24*a^4*b^10*c^2 - 18*a^2*b^12*c^2 + 6*b^14*c^2 + 11*a^12*c^4 - 34*a^10*b^2*c^4 + 22*a^8*b^4*c^4 + 4*a^6*b^6*c^4 - 33*a^4*b^8*c^4 + 30*a^2*b^10*c^4 - 16*b^12*c^4 - 4*a^10*c^6 + 22*a^8*b^2*c^6 + 4*a^6*b^4*c^6 + 32*a^4*b^6*c^6 - 16*a^2*b^8*c^6 + 26*b^10*c^6 - 5*a^8*c^8 - 20*a^6*b^2*c^8 - 33*a^4*b^4*c^8 - 16*a^2*b^6*c^8 - 30*b^8*c^8 + 8*a^6*c^10 + 24*a^4*b^2*c^10 + 30*a^2*b^4*c^10 + 26*b^6*c^10 - 7*a^4*c^12 - 18*a^2*b^2*c^12 - 16*b^4*c^12 + 4*a^2*c^14 + 6*b^2*c^14 - c^16) : :

X(34843) lies on these lines: {2, 135), {3, 136), {631, 1299), {1216, 6699), {1368, 6036), {2986, 3546), {9720, 11064), {16760, 16977), {16976, 31379}

X(34843) = complement of X(135)

X(34844) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(136)

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

X(34844) lies on these lines: {2, 136), {3, 125), {539, 12095), {631, 1300), {1656, 13556), {3526, 21667), {4558, 32263), {5159, 16760), {5972, 6132), {6036, 6676), {6388, 32654), {7386, 30789), {7542, 15366), {10257, 31379), {13754, 27087), {16534, 34333), {18531, 22823}

X(34844) = complement of X(136)

X(34845) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(160)

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

X(34845) lies on these lines: {2, 160), {3, 31867), {5, 182), {6, 3613), {115, 8265), {125, 6752), {141, 21531), {157, 458), {230, 427), {264, 523), {311, 3001), {338, 23635), {403, 16264), {1316, 30715), {2393, 14767), {2476, 4429), {3143, 18424), {3767, 16285), {5025, 16890), {5133, 7792), {5169, 7806), {7703, 23301), {7875, 21458), {8266, 14957), {11818, 24270), {15648, 23292), {17004, 31074}

X(34845) = complement of X(160)
X(34845) = complementary conjugate of X(2)-Ceva conjugate of X(51)
X(34845) = complementary conjugate of complement of isogonal conjugate of X(160)
X(34845) = complementary conjugate of complement of isotomic conjugate of X(2979)

X(34846) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(162)

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

X(34846) lies on these lines: {2, 162), {3, 23860), {10, 23998), {11, 127), {122, 124), {123, 125), {226, 20309), {339, 1111), {440, 16593), {525, 7068), {656, 24031), {1086, 2968), {1214, 2887), {2193, 30984), {2972, 34588), {3454, 6700), {4138, 18588), {6349, 25958), {6350, 25959), {10427, 18642), {13609, 13611), {16573, 16592), {16594, 21530), {16597, 18589), {17073, 25760}

X(34846) = complement of X(162)
X(34846) = complementary conjugate of X(8062)

X(34847) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(169)

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

X(34847) lies on these lines: {1, 26101), {2, 169), {3, 142), {5, 6706), {10, 116), {11, 24774), {21, 27006), {55, 20269), {85, 5179), {141, 5044), {226, 241), {277, 497), {481, 30380), {482, 30381), {498, 30742), {517, 21258), {908, 25729), {975, 3487), {1058, 4000), {1212, 1565), {1385, 17044), {3002, 28278), {3011, 20267), {3057, 4904), {3646, 17306), {3730, 9436), {3739, 31419), {3741, 30956), {3879, 30130), {3912, 33942), {3914, 24790), {3946, 30148), {4357, 30110), {4872, 17682), {5195, 27000), {5249, 14021), {5432, 24784), {7683, 9843), {8726, 16388), {9317, 10572), {9560, 17058), {10974, 18635), {12047, 30949), {12053, 17761), {12618, 25365), {16580, 17245), {16593, 25066), {17050, 24331), {17077, 28410), {17171, 30733), {17213, 23649), {17278, 31584), {17742, 28740), {17753, 27129), {17867, 20905), {20335, 21616), {20880, 21073), {23305, 24388), {24580, 29603), {27147, 27301), {28742, 33864), {29596, 30858}

X(34847) = complement of X(169)
X(34847) = complementary conjugate of complement of isotomic conjugate of X(20927)

X(34848) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(170)

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

X(34848) lies on these lines: {2, 170), {4, 9), {946, 2808), {1446, 2310), {1699, 17753), {1996, 11019), {2140, 3817), {3840, 8727), {3991, 28850), {6706, 15726), {9442, 10481), {10939, 24009}

X(34848) = complement of X(170)

X(34849) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(173)

Barycentrics Cos[B] + Cos[C] - 2*(1 + Sin[B/2] + Sin[C/2]) : :

X(34849) lies on these lines: {2, 173), {10, 12614), {142, 178), {226, 2091), {2090, 3452), {4859, 10234), {7048, 21623}

X(34849) = complement of X(173)
X(34849) = complementary conjugate of complement of X(258)

X(34850) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(217)

Barycentrics a^6*b^4 - 2*a^4*b^6 + a^2*b^8 - a^4*b^4*c^2 + b^8*c^2 + a^6*c^4 - a^4*b^2*c^4 - 2*a^2*b^4*c^4 - b^6*c^4 - 2*a^4*c^6 - b^4*c^6 + a^2*c^8 + b^2*c^8 : :
Barycentrics (sin^4 B) cos B cos(C - A) + (sin^4 C) cos C cos(A - B) : :

X(34850) lies on these lines: {2, 217), {3, 66), {127, 129), {290, 6656), {315, 343), {389, 14767), {401, 2896), {511, 27370), {626, 2387), {1235, 22416), {1594, 11674), {2548, 13567), {3269, 26166), {3580, 7785), {3934, 5907), {5103, 18322), {6292, 15595), {7815, 9306), {7865, 27366), {15271, 17814), {17128, 34360}

X(34850) = complement of X(217)
X(34850) = complementary conjugate of complement of X(276)

X(34851) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(225)

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

X(34851) lies on these lines: {2, 225), {3, 10), {33, 27379), {34, 24538), {37, 216), {78, 345), {123, 21530), {124, 131), {281, 6857), {411, 1861), {1038, 6350), {1060, 30147), {1062, 22836), {1125, 16579), {1211, 15823), {1842, 8229), {1944, 25650), {2193, 3686), {2385, 6389), {2968, 3704), {3739, 4999), {3916, 26932), {4640, 20306), {6675, 6708), {7386, 29857), {7494, 29828), {10538, 27531), {16596, 16597), {20222, 23710), {20262, 32561), {22341, 26013), {22447, 34591), {24245, 30557), {24246, 30556), {27410, 33113), {34587, 34588}

X(34851) = complement of X(225)
X(34851) = complementary conjugate of complement of X(283)

X(34852) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(241)

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

X(34852) lies on these lines: {2, 85), {5, 10), {6, 27384), {9, 3739), {44, 1944), {141, 20262), {219, 17348), {220, 4384), {226, 21258), {239, 6603), {281, 17279), {312, 4515), {322, 28778), {514, 4521), {536, 4858), {908, 26001), {910, 6996), {1146, 3912), {1215, 18227), {1229, 3965), {1441, 25067), {1997, 18156), {2324, 4361), {3061, 17284), {3198, 6818), {3673, 3752), {3729, 34524), {3772, 27539), {3834, 26932), {4000, 27508), {4383, 27413), {4641, 28951), {4751, 26059), {4875, 28797), {5328, 29611), {5723, 26006), {5745, 31211), {6687, 7359), {10472, 15479), {14942, 28058), {16706, 27547), {16831, 34522), {17277, 27420), {17278, 27509), {17353, 25971), {17720, 28794), {18228, 28657), {21617, 25964), {24455, 30618), {24612, 26265), {24774, 31183), {24789, 27540), {24993, 27058}

X(34852) = complement of X(241)
X(34852) = complementary conjugate of X(17060)

X(34853) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(2) AND ANTICEVIAN OF X(254)

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

X(34853) lies on these lines: {2, 254), {3, 68), {4, 131), {5, 8906), {20, 5962), {91, 499), {155, 27087), {216, 2165), {454, 1609), {485, 24245), {486, 24246), {3547, 5392), {5654, 34757), {5963, 10298), {6193, 12095), {6389, 11585), {6515, 8883), {9936, 13557), {10257, 31377), {10539, 32734), {10600, 18531), {14376, 34157), {14379, 15454), {15242, 18126), {34030, 34332}

X(34853) = isogonal conjugate of X(34756)
X(34853) = complement of X(254)
X(34853) = crosssum of X(924) and X(34338)

X(34854) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(4) AND ANTICEVIAN OF X(232)

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

X(34854) lies on these lines: {4, 51), {25, 32), {53, 9969), {107, 419), {132, 2450), {184, 8743), {206, 8746), {232, 237), {263, 393), {264, 9822), {297, 511), {420, 14165), {458, 5943), {460, 512), {648, 8681), {1249, 6467), {1495, 8744), {1968, 3148), {1974, 8745), {1990, 2393), {2967, 15143), {3186, 21447), {3819, 11331), {4230, 34157), {6525, 6620), {7745, 27359), {8753, 32695), {9308, 14913), {10002, 12294), {11574, 17907), {14569, 14715), {15262, 21639), {15630, 20031), {16240, 20410}

X(34854) = isogonal conjugate of X(6394)
X(34854) = polar conjugate of isogonal conjugate of X(2211)
X(34854) = polar conjugate of isotomic conjugate of X(232)
X(34854) = barycentric product X(4)*X(232)
X(34854) = X(63)-isoconjugate of X(287)
X(34854) = trilinear pole of line X(2491)X(17994) (the tangent to the inellipse that is the barycentric square of the orthic axis, at the barycentric square of X(232))

X(34855) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(7) AND ANTICEVIAN OF X(241)

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

X(34855) lies on these lines: {1, 738), {6, 57), {7, 354), {55, 77), {56, 3423), {59, 1155), {65, 279), {85, 3812), {105, 1456), {241, 672), {347, 17642), {348, 960), {513, 676), {517, 1323), {518, 1362), {651, 2348), {658, 1447), {664, 3880), {840, 934), {942, 10481), {971, 1541), {1014, 2194), {1122, 3598), {1358, 18838), {1420, 17106), {1428, 15382), {1429, 9500), {1435, 2201), {1442, 3748), {1446, 4059), {1463, 2113), {1565, 6001), {1617, 3433), {2078, 3446), {2264, 34028), {2272, 3942), {2720, 15728), {3025, 3321), {3057, 3160), {3449, 9455), {3663, 12915), {3664, 11018), {3668, 5173), {3674, 7143), {3693, 6168), {3698, 31994), {3889, 32098), {3893, 25718), {4298, 28845), {4513, 6167), {4662, 33298), {5836, 9312), {5902, 21314), {6604, 34791), {7131, 30618), {7195, 14256), {7271, 10980), {7274, 30350), {8581, 10004), {9446, 14828), {9943, 17170), {10269, 14878), {18839, 22464), {23603, 24476), {30946, 31627}

X(34855) = isogonal conjugate of X(28071)
X(34855) = X(230)-of-intouch-triangle

X(34856) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(27) AND ANTICEVIAN OF X(242)

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

X(34856) lies on these lines: {4, 6), {82, 158), {107, 2382), {238, 14024), {242, 1914), {983, 1896), {3733, 17925), {5009, 31905}

X(34857) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(80) AND ANTICEVIAN OF X(37)

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

X(34857) lies on these lines: {8, 80), {42, 4516), {45, 55), {65, 3120), {210, 6535), {660, 17763), {758, 4080), {759, 5293), {762, 1334), {1002, 2006), {1215, 11680), {1411, 2334), {2222, 28471), {2341, 32736), {2611, 4551), {4067, 21093), {4096, 34611), {18398, 33148), {21801, 21805), {21864, 22288}

X(34858) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(104) AND ANTICEVIAN OF X(6)

Barycentrics a^3*(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(34858) lies on these lines: {21, 104), {25, 1397), {41, 9247), {55, 184), {56, 3937), {105, 2720), {154, 16686), {422, 685), {517, 2990), {884, 2423), {1036, 1795), {1404, 6187), {1492, 34234), {1576, 2194), {1976, 5040), {2175, 34446), {2203, 32713), {2218, 14529), {4577, 18816), {7113, 32719), {11402, 20958), {26890, 32736), {32641, 32718), {32669, 32728), {32702, 32727}

X(34858) = isogonal conjugate of X(3262)
X(34858) = polar conjugate of isotomic conjugate of X(14578)
X(34858) = cevapoint of X(2175) and X(2251)
X(34858) = crosssum of X(2) and X(153)
X(34858) = crosspoint of X(6) and X(34182)
X(34858) = trilinear pole of line X(32)X(3063)
X(34858) = barycentric product X(6)*X(104)
X(34858) = barycentric product of circumcircle intercepts of line X(6)X(650)

X(34859) = CENTER OF PERSPECONIC OF THESE TRIANGLES: CEVIAN OF X(112) AND ANTICEVIAN OF X(232)

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

X(34859) lies on these lines: {25, 32), {107, 112), {232, 21525), {1316, 1968), {2395, 23977), {4230, 14966}

X(34860) = ISOGONAL CONJUGATE OF X(3915)

Barycentrics ((b-2*c)*a+b^2+b*c)*((2*b-c)*a-b*c-c^2) : :
X(34860) = 2*X(10)-3*X(24174), 5*X(3616)-3*X(19582)

See Antreas Hatzipolakis and César Lozada, Euclid 195 .

X(34860) lies on these lines: {1, 4234}, {8, 21342}, {10, 982}, {19, 31903}, {37, 2275}, {38, 31359}, {65, 145}, {75, 33947}, {78, 24841}, {82, 1468}, {190, 28011}, {192, 17609}, {225, 26015}, {244, 341}, {312, 3976}, {523, 23835}, {537, 21214}, {759, 8666}, {897, 27368}, {969, 4360}, {979, 17477}, {994, 3874}, {1120, 3680}, {1215, 3624}, {1220, 3677}, {1266, 3668}, {1434, 3875}, {2218, 2975}, {3333, 32926}, {3632, 4674}, {3699, 11512}, {3868, 34434}, {3869, 17154}, {3889, 13476}, {3953, 4385}, {3967, 26093}, {4373, 24797}, {4440, 12701}, {4662, 24620}, {4673, 17155}, {4906, 17697}, {5263, 23051}, {8720, 17715}, {9369, 17054}, {10404, 29840}, {13610, 32921}, {17038, 24325}, {17103, 17393}, {17144, 18827}, {17495, 20047}, {18785, 21384}, {20057, 31503}, {24946, 33122}, {25917, 31302}

X(34860) = isogonal conjugate of X(3915)
X(34860) = isotomic conjugate of X(3875)
X(34860) = Cevapoint of X(i) and X(j) for these (i,j): (244, 522), (512, 16614), (522, 244)
X(34860) = X(522)-Beth conjugate of X(23835)
X(34860) = X(2321)-cross conjugate of X(2)
X(34860) = X(i)-isoconjugate-of X(j) for these {i,j}: {1, 3915}, {2, 16946}, {3, 4186}
X(34860) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 4383), (2, 3875), (6, 3915)
X(34860) = X(1)-Zayin conjugate of X(3915)
X(34860) = lies on the circumconic with center X(i) for i in {244, 3756, 16614}
X(34860) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(10)}} and {{A, B, C, X(2), X(1434)}}
X(34860) = trilinear pole of the line {661, 4521}
X(34860) = barycentric product X(1577)*X(8690)
X(34860) = barycentric quotient X(i)/X(j) for these (i, j): (1, 4383), (2, 3875), (9, 3913), (10, 3175)
X(34860) = trilinear product X(523)*X(8690)
X(34860) = trilinear quotient X(i)/X(j) for these (i, j): (2, 4383), (4, 4186), (6, 16946), (8, 3913)

X(34861) = (name pending)

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

See Antreas Hatzipolakis and César Lozada, Euclid 195 .

X(34861) lies on this line: {3164, 3522}

X(34861) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3522)}} and {{A, B, C, X(4), X(15258)}}

X(34862) = MIDPOINT OF X(3) AND X(84)

Barycentrics a*(2*a^6-(b+c)*a^5-(5*b^2-8*b*c+5*c^2)*a^4+2*(b^3+c^3)*a^3+2*(2*b^2+b*c+2*c^2)*(b-c)^2*a^2-(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*(b+c)^2) : :
X(34862) = 3*X(2)+X(12246), 3*X(3)-X(1490), 3*X(3)+X(12684), 3*X(84)+X(1490), 3*X(84)-X(12684), 3*X(165)+X(10864), X(355)-3*X(14647), 3*X(376)+X(9799), 5*X(631)-X(6223), X(962)+3*X(14646), 4*X(6705)-X(22792)

See Antreas Hatzipolakis and César Lozada, Euclid 195 .

X(34862) lies on these lines: {1, 24645}, {2, 6259}, {3, 9}, {4, 5435}, {5, 6692}, {8, 17613}, {20, 3419}, {21, 10167}, {24, 12136}, {30, 6245}, {35, 12680}, {36, 12688}, {40, 4915}, {55, 10085}, {56, 1709}, {57, 5806}, {63, 31793}, {65, 1768}, {72, 6909}, {104, 1476}, {140, 6260}, {165, 10864}, {355, 6948}, {376, 9799}, {404, 5927}, {405, 11227}, {474, 10157}, {498, 12678}, {499, 1538}, {515, 550}, {516, 3813}, {517, 1158}, {527, 5763}, {631, 6223}, {912, 18238}, {942, 1012}, {946, 20418}, {952, 12640}, {956, 31798}, {958, 31787}, {962, 14646}, {990, 4252}, {993, 9943}, {997, 31821}, {999, 12705}, {1071, 6906}, {1125, 31657}, {1155, 5691}, {1210, 13226}, {1376, 9947}, {1385, 5248}, {1697, 30283}, {1699, 32636}, {2077, 14872}, {2080, 12196}, {2096, 6847}, {2646, 15071}, {2800, 3881}, {2829, 12616}, {2950, 12773}, {3149, 5122}, {3311, 19067}, {3312, 19068}, {3361, 11372}, {3468, 17102}, {3523, 5658}, {3560, 9940}, {3576, 7992}, {3601, 30304}, {3683, 7987}, {3824, 6824}, {3927, 6282}, {3962, 5538}, {4189, 11220}, {4292, 8727}, {4297, 4640}, {4413, 16209}, {4652, 7580}, {4847, 31777}, {5045, 11496}, {5302, 10164}, {5428, 17502}, {5439, 6912}, {5440, 12528}, {5715, 18541}, {5791, 6916}, {5817, 17580}, {5881, 13528}, {5887, 17649}, {6256, 9956}, {6257, 26348}, {6258, 26341}, {6261, 13624}, {6642, 9910}, {6763, 7957}, {6875, 9960}, {6918, 18540}, {6935, 11374}, {6950, 33597}, {7320, 7967}, {7416, 22345}, {7583, 8987}, {7584, 13974}, {7682, 34753}, {7743, 10785}, {7971, 10246}, {8720, 28850}, {8726, 16418}, {9709, 10270}, {9851, 31508}, {10156, 11108}, {10202, 13743}, {10267, 12330}, {10269, 18237}, {10310, 34790}, {10395, 20420}, {10884, 16370}, {11009, 12767}, {11230, 12608}, {11231, 18242}, {11500, 31663}, {12496, 26316}, {12650, 12702}, {12667, 26446}, {12668, 26451}, {12676, 26492}, {12677, 26487}, {12686, 16203}, {12687, 16202}, {13243, 34772}, {13253, 33176}, {13257, 27385}, {13730, 26927}, {15251, 24171}, {15803, 19541}, {16408, 21164}, {17100, 17661}, {17558, 21151}, {21669, 26877}, {22758, 31788}, {23961, 31828}, {26332, 31776}, {31937, 32612}, {32198, 32537}

X(34862) = midpoint of X(i) and X(j) for these {i,j}: {3, 84}, {20, 5787}, {1158, 12114}, {1490, 12684}, {2950, 12773}, {5887, 17649}, {6259, 12246}, {12650, 12702}
X(34862) = reflection of X(i) in X(j) for these (i,j): (5, 6705), (1385, 5450), (6256, 9956), (6260, 140), (6261, 13624), (11500, 31663), (18480, 12616), (22792, 5)
X(34862) = complement of X(6259)
X(34862) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 12246, 6259), (3, 5779, 936), (3, 7171, 31805), (3, 7330, 5044), (3, 12684, 1490), (3, 31445, 31658), (56, 1709, 9856), (84, 1490, 12684), (104, 12672, 24928), (499, 12679, 1538), (1071, 6906, 24929), (6914, 13369, 1385), (9841, 31424, 3)

X(34863) = (name pending)

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

See Antreas Hatzipolakis and César Lozada, Euclid 195 .

X(34863) lies on this line: {3177, 20211}

X(34863) = intersection, other than A,B,C of conics {{A, B, C, X(85), X(3177)}} and {{A, B, C, X(189), X(20211)}}

Circumcevian-inversion perspectors: X(34864)-X(34889)

This preamble and centers X(34864)-X(34889), based on notes by Suren, (Circumcevian Inversion Perspector), were contributed by Clark Kimberling and Peter Moses, November 20, 2019.

Suppose that P = p : q : r (barycentrics) is a point in the plane of a triangle ABC. Let A*B*C* be the circumcevian triangle of P, and let A' be the inverse-in-circumcircle of the reflection of P in A*; define B' and C' cyclically. The triangle A'B'C', here named the circumcevian-inversion triangle of P, is perspective to ABC in the point

P' = p U + a² (b² + c² - a²) V : : , where U = a² b² c² (p+q+r) and V = a² q r + b² r p + c² p q.

If P lies on the circumcircle, then P' = P; if P lies on the line at infinity, then P' = X(3).

If P lies on the Stammler circle, then P' lies on the circumcircle.

If P on the Euler line is given by P = g*X(2) + h*X(3), then P' = 9 a^2 b^2 c^2 g (g + h) X(2) + (9 a^2 b^2 c^2 h (3 g + 2 h) + 4 (a^2 + b^2 + c^2) g^2 S^2) X(3) .

If P on the line X(1)X(3) is given by P = g*X(1) + h*X(3), then P' = g (g + h) R s X(1) + (h (3 g + 2 h) R s + g^2 S) X(3).

The appearance of (i,j) in the following list means that the circumcevian-inversion perspector of X(i) is X(j):

(1,35), (2,7496), (4,3520), (5,34864), (6,574), (7,34865), (8,34755), (9,34867), (10,34868), (15,10645), (16,10646), (20,7488), (23,7492), (31,34869), (32,34870), (35, 34871), (36,7280), (37,34872), (39,34873), (40,11012), (41,34874), (42,34875), (43,34876), (44, 34877), (45,34878), (54,25042), (55,34879), (56, 34880), (57,34881), (58,34882), (64,14379), (69,34883), (75, 34884), (76, 34885), (81, 34886), (82,34887), (83,34888), (86,34889), (115,34866), (186,21844), (187,5210), (351,9130), (399,74), (1319,11510), (1350,8722}, {1657,21844}, {2070,7488), (2080,8722), (3534,7492), (6760,14379), (8008,13594), (8009,13593), (10620,110), (11258,1296), (12188,99), (12331,104), (12702,7280}, {12773,100), (13115,112), (13188,98), (13310,1297), (13512,1141), (14663,6011), (15054,15020), (15154,1114), (15155,1113), (18859,3520), (22765,11012), (32595,25424), (32609,15020}, {33878,5210), (34147,33924)

See the preamble just before X(35000).

The A-vertex, X', of the triangle X'Y'Z' is given by 1st, 2nd, 3rd barycentrics, as follows:

1st: a^2*(c^4*(2*a^2 - b^2 - 2*c^2)*p^2*q^2 + (a - c)*c^4*(a + c)*p*q^3 + b^2*c^2*(5*a^2 - 3*b^2 - 3*c^2)*p^2*q*r + c^2*(3*a^4 + 2*a^2*b^2 - b^4 - 2*a^2*c^2 - b^2*c^2 - c^4)*p*q^2*r + a^2*c^2*(a^2 + b^2 - c^2)*q^3*r + b^4*(2*a^2 - 2*b^2 - c^2)*p^2*r^2 + b^2*(3*a^4 - 2*a^2*b^2 - b^4 + 2*a^2*c^2 - b^2*c^2 - c^4)*p*q*r^2 + a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*q^2*r^2 + (a - b)*b^4*(a + b)*p*r^3 + a^2*b^2*(a^2 - b^2 + c^2)*q*r^3)

2nd: b^2*(-(c^2*(2*a^2 - b^2 + c^2)*p*q) - a^2*c^2*q^2 - b^2*(a^2 - b^2 + c^2)*p*r - a^2*(a^2 - b^2 + 2*c^2)*q*r)*(2*c^2*p*q + c^2*q^2 + 2*b^2*p*r + (a^2 + b^2 + c^2)*q*r + b^2*r^2)

3rd: c^2*(2*c^2*p*q + c^2*q^2 + 2*b^2*p*r + (a^2 + b^2 + c^2)*q*r + b^2*r^2)*(-(c^2*(a^2 + b^2 - c^2)*p*q) - b^2*(2*a^2 + b^2 - c^2)*p*r - a^2*(a^2 + 2*b^2 - c^2)*q*r - a^2*b^2*r^2)

X(34864) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(5)

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

X(34864) lies on these lines: {2, 3}, {49, 5891}, {52, 12307}, {54, 11591}, {110, 10610}, {143, 7691}, {156, 15056}, {182, 34783}, {195, 567}, {265, 1209}, {389, 15047}, {399, 5907}, {569, 15087}, {578, 23039}, {827, 29316}, {1141, 20189}, {1147, 14805}, {1154, 13434}, {1181, 32620}, {1568, 6689}, {1614, 15060}, {1879, 7756}, {1994, 12316}, {2917, 18383}, {3519, 10112}, {3581, 5462}, {4550, 10984}, {5012, 5876}, {5092, 10575}, {5621, 20190}, {5663, 27866}, {5893, 9919}, {6101, 15033}, {6102, 15037}, {8718, 32137}, {9672, 31479}, {9704, 15068}, {10170, 13367}, {10282, 14926}, {10619, 23236}, {10620, 13339}, {11016, 23181}, {11412, 33533}, {11459, 32046}, {11695, 32110}, {11793, 22115}, {12022, 32333}, {13353, 13754}, {14861, 34437}, {15041, 16836}, {15067, 34148}, {15177, 18493}, {15578, 20427}, {15579, 31166}, {18350, 18475}, {18442, 32600}, {22352, 33539}, {23237, 34418}, {26882, 34513}

X(34865) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(7)

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

X(34865) lies on these lines: {3, 7}, {3160, 26357}, {10481, 14794}

X(34866) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(115)

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

X(34866) lies on these lines: {3, 115}, {6, 74}, {39, 14130}, {99, 338}, {111, 7485}, {187, 18859}, {230, 2071}, {399, 9696}, {427, 8428}, {1562, 32607}, {1609, 6128}, {2493, 7503}, {2549, 18570}, {3053, 12084}, {3455, 11641}, {3516, 6103}, {3520, 5254}, {3565, 14060}, {3767, 11250}, {3815, 7527}, {3830, 9699}, {5013, 7526}, {5023, 11413}, {7464, 11063}, {7482, 30715}, {7484, 10418}, {7745, 14865}, {8553, 21312}, {9603, 12162}, {9604, 11430}, {9818, 31489}, {12302, 32661}, {13468, 34883}, {14120, 14729}, {15041, 15544}, {15061, 15538}, {21890, 22586}, {22146, 32761}

X(34867) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(9)

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 + 5 a^3 b c + 3 a^2 b^2 c - 9 a b^3 c + 2 b^4 c - 2 a^3 c^2 + 3 a^2 b c^2 - b^3 c^2 + 2 a^2 c^3 - 9 a b c^3 - b^2 c^3 + a c^4 + 2 b c^4 - c^5) : :

X(34867) lies on these lines: {3, 9}, {21, 8568}, {71, 5030}, {2267, 4262}, {5316, 11349}, {5537, 34522}, {5563, 21872}, {11012, 24047}

X(34868) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(10)

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

X(34868) lies on these lines: {3, 10}, {21, 4972}, {35, 3987}, {36, 5293}, {56, 30115}, {58, 10822}, {759, 17524}, {1011, 33138}, {2944, 2954}, {2975, 4218}, {3145, 5251}, {3520, 17927}, {4413, 20842}, {5258, 20999}, {6211, 11012}, {7081, 7485}, {7484, 8071}, {13732, 26363}, {13733, 19854}, {15618, 19297}, {16287, 24880}, {16374, 17734}, {16452, 25446}, {16453, 24931}, {19879, 28348}

X(34869) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(31)

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

X(34869) lies on these lines: {3, 31}, {849, 5867}, {1621, 3670}, {1914, 4286}

X(34870) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(32)

Barycentrics a^2 (a^6 - a^4 b^2 + 2 a^2 b^4 - a^4 c^2 + a^2 b^2 c^2 + b^4 c^2 + 2 a^2 c^4 + b^2 c^4) : :
Trilinears e^2 sin(A - ω) + sin(A + ω) : :

X(34870) lies on these lines: {3, 6}, {83, 3815}, {98, 5254}, {230, 1078}, {232, 11380}, {237, 1915}, {248, 682}, {325, 10350}, {384, 5976}, {733, 9481}, {1506, 6721}, {1513, 7745}, {1971, 15270}, {2023, 12176}, {2056, 3117}, {2275, 12835}, {2276, 10799}, {2548, 10796}, {2549, 14880}, {3054, 8363}, {3148, 3981}, {3203, 9604}, {3407, 3552}, {3767, 10104}, {3934, 11356}, {4027, 7783}, {5304, 32965}, {5306, 7827}, {5475, 18502}, {6393, 7789}, {7735, 7791}, {7736, 7787}, {7763, 10349}, {7767, 15993}, {7808, 31489}, {7815, 7866}, {7817, 11287}, {7832, 10347}, {7836, 10333}, {7891, 10334}, {7892, 10345}, {9300, 12150}, {9575, 10789}, {9596, 10797}, {9599, 10798}, {10311, 11325}, {10346, 33225}, {10359, 31400}, {10801, 16502}, {12151, 34511}, {12191, 33013}, {14153, 34396}, {15484, 18501}

X(34870) = circumcircle-inverse of X(2458)
X(34870) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(13335)
X(34870) = {X(32),X(39)}-harmonic conjugate of X(3398)

X(34871) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(35)

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

X(34871) lies on these lines: {1, 3}, {80, 5428}, {2475, 4324}, {2476, 18514}, {3583, 6853}, {5267, 6224}, {5499, 15338}, {6876, 18393}, {15712, 26475}

X(34872) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(37)

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

X(34872) lies on these lines: {3, 37}, {35, 16583}, {1030, 5257}, {2163, 16785}, {16367, 32779}

X(34873) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(39)

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

X(34873) lies on these lines: {3, 6}, {3815, 7470}, {4048, 16043}, {5989, 7789}, {6292, 11646}, {6337, 16990}, {7770, 7847}, {7835, 11285}, {8290, 33021}, {8362, 24273}, {10329, 14096}

X(34874) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(41)

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

X(34874) lies on these lines: {3, 41}

X(34875) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(42)

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

X(34875) lies on these lines: {3, 42}

X(34876) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(43)

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

X(34876) lies on these lines: {3, 43}

X(34877) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(44)

Barycentrics a^2 (a^5 - 3 a^4 b - 4 a^3 b^2 + 4 a^2 b^3 + 3 a b^4 - b^5 - 3 a^4 c + 11 a^3 b c + 6 a^2 b^2 c - 21 a b^3 c + 5 b^4 c - 4 a^3 c^2 + 6 a^2 b c^2 + 6 a b^2 c^2 + 2 b^3 c^2 + 4 a^2 c^3 - 21 a b c^3 + 2 b^2 c^3 + 3 a c^4 + 5 b c^4 - c^5) : :

X(34877) lies on these lines: {3, 44}

X(34878) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(45)

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

X(34878) lies on these lines: {3, 45}

X(34879) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(55)

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

X(34879) lies on these lines: {1, 3}, {105, 7465}, {228, 1626}, {672, 2278}, {943, 10404}, {991, 2361}, {993, 10609}, {1001, 1004}, {1005, 4679}, {1621, 5880}, {1836, 7411}, {1837, 6986}, {2194, 4184}, {2245, 2280}, {3185, 16064}, {3651, 11375}, {3683, 5784}, {4316, 15175}, {5248, 11112}, {5259, 12953}, {5260, 5794}, {5338, 20832}, {6067, 6154}, {6187, 30944}, {6836, 10588}, {6889, 10591}, {6910, 26040}, {7580, 17605}, {7702, 16142}, {15624, 20999}, {24953, 25440}

X(34880) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(56)

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

X(34880) lies on these lines: {1, 3}, {11, 5450}, {12, 13747}, {104, 1837}, {221, 8572}, {355, 10090}, {388, 6921}, {404, 5252}, {474, 22759}, {499, 6929}, {604, 4271}, {993, 4187}, {995, 1399}, {1106, 1464}, {1122, 1804}, {1145, 10944}, {1158, 17638}, {1317, 8715}, {1400, 4268}, {1404, 22357}, {1408, 4225}, {1411, 24046}, {1415, 2275}, {1476, 13587}, {1478, 6959}, {1532, 7354}, {1737, 32153}, {2285, 21773}, {2478, 7288}, {2829, 26476}, {2975, 24914}, {3035, 26482}, {3086, 6938}, {3476, 4188}, {3485, 26842}, {3582, 15446}, {3877, 34758}, {3885, 17100}, {4293, 6834}, {4317, 18962}, {5253, 11375}, {5265, 6872}, {5298, 10199}, {5840, 10948}, {5886, 7702}, {5887, 10094}, {5901, 24465}, {6516, 30617}, {6713, 10523}, {6906, 11376}, {6909, 12701}, {7741, 13273}, {10058, 11373}, {10826, 26321}, {10950, 12832}, {10957, 11112}, {11501, 16371}, {12740, 18861}, {13724, 28385}, {13743, 23708}, {17606, 22758}, {20418, 26475}, {23850, 34139}, {25524, 25875}, {28077, 28096}

X(34881) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(57)

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

X(34881) lies on these lines: {1, 3}, {1005, 3911}

X(34882) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(58)

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

X(34882) lies on these lines: {3, 6}, {21, 17596}, {409, 1054}, {986, 4653}, {1125, 24227}, {3821, 11110}, {16948, 17779}

X(34883) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(69)

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

X(34883) lies on these lines: {3, 69}, {183, 2071}, {186, 14907}, {378, 34229}, {1078, 3520}, {3053, 26206}, {3518, 7802}, {3917, 6091}, {5210, 15066}, {7464, 11185}, {7750, 22467}, {12084, 32828}, {13468, 34866}, {14865, 32832}, {15078, 15574}, {17928, 32006}

X(34884) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(75)

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

X(34884) lies on these lines: {3, 75}, {350, 4218}, {19308, 32779}

X(34885) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(76)

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

X(34885) lies on these lines: {3, 76}, {316, 384}, {574, 8150}, {1003, 7883}, {2896, 3552}, {3520, 17984}, {3767, 33004}, {3972, 10350}, {4045, 7749}, {5149, 7836}, {7770, 7899}, {7774, 18769}, {7788, 34682}, {7810, 13586}, {7816, 10997}, {7863, 8290}, {7942, 11285}, {7944, 11356}, {10998, 18860}, {11676, 32152}, {13086, 33008}, {14568, 33273}

X(34886) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(81)

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

X(34886) lies on these lines: {3, 81}, {21, 28612}

X(34887) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(82)

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

X(34887) lies on these lines: {3, 82}

X(34888) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(83)

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

X(34888) lies on these lines: {3, 83}, {827, 12054}

X(34889) = CIRCUMCEVIAN-INVERSION PERSPECTOR OF X(86)

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

X(34889) lies on these lines: {3, 86}, {16374, 17731}

X(34890) = X(1)X(3)∩X(1749)X(26201)

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

See Antreas Hatzipolakis and César Lozada, Euclid 203 .

X(34890) lies on these lines: {1, 3}, {1749, 26201}, {3651, 16173}, {3822, 25542}, {4317, 15175}, {5288, 33337}, {5442, 11491}

X(34890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (46, 23961, 57), (940, 2662, 1467), (1082, 26380, 1), (1402, 31498, 940), (1482, 11883, 2078), (1758, 5711, 1460), (2283, 10902, 1715), (3075, 10383, 1715), (5048, 33178, 1319), (5048, 34593, 982), (5329, 32636, 3072), (5563, 15931, 35), (7982, 8071, 1482), (10202, 11529, 988), (10473, 16201, 942), (10966, 16215, 260), (11278, 34560, 517), (11879, 32613, 1622), (17437, 18838, 354), (18857, 23961, 55)

X(34891) = (name pending)

Barycentrics a*(a^12+4*(b+c)*a^9*b*c-2*(3*b^2-2*b*c+3*c^2)*a^10-4*(b+c)*(3*b^2-2*b*c+3*c^2)*a^7*b*c+(15*b^4+15*c^4-8*(b-c)^2*b*c)*a^8+4*(b+c)*(3*b^4+3*c^4-(4*b^2-5*b*c+4*c^2)*b*c)*a^5*b*c-2*(10*b^6+10*c^6+(7*b^2-6*b*c+7*c^2)*b^2*c^2)*a^6-4*(b^6-c^6)*(b-c)*a^3*b*c+(15*b^8+15*c^8+(8*b^6+8*c^6-(2*b^4+2*c^4+(8*b^2-7*b*c+8*c^2)*b*c)*b*c)*b*c)*a^4-2*(b^2-c^2)^2*(b+c)^2*(3*b^4+3*c^4-(4*b^2-5*b*c+4*c^2)*b*c)*a^2+(b^2-c^2)^6) : :
Trilinears 16*p^5*(p^3-2*p*q^2+q)+2*(8*q^4+24*q^2-17)*p^4-16*q*p^3-2*(8*q^4+11*q^2-12)*p^2+3*q*(2*q+p)-63/16 : : where p=sin(A/2), q=cos((B-C)/2)

See Antreas Hatzipolakis and César Lozada, Euclid 203 .

X(34891) lies on this line: {1, 3}

Vu circlecevian points: X(34892)-X(34902)

This preamble and centers X(34892)-X(34902) were contributed by Vu Thanh Tung and Vu Quoc My, November 20, 2019, with extensions and editing by Clark Kimberling and Peter Moses, November 22, 2019.

Let P be a point in the plane of a triangle ABC, and let A' be the point, other than P, in which the line AP meets the circle (PBC). Define B' and C' cyclically, so that A'B'C' is the circlecevian triangle of P, as in the preamble just before X(3420).

Now let A'B'C' and A''B''C'' be, respectively, the circlecevian triangles of P = p : q : r and U = u : v : w (barycentrics). Let A1 = BC∩A'A'', and define B1 and C1 cyclically. Then AA1, BB1, CC1 concur in a point V(P,U), here named the Vu circlecevian point of P and U, given by

V(P,U) = p*u / (p*(p+q+r)*(a^2*v*w + b^2*w*u + c^2*u*v) - u*(u+v+w)*(a^2*q*r + b^2*r*p + c^2*p*q)) : : .

The appearance of (i,j,k) in the following list means that V(X(i),X(j)) = X(k):

(1,2,34892), (1,3,1807), (1,4,80), (1,6,34893), (1,7,3254), (1,8,12641), (1,9,34894), (1,10,34895), (1,11,34896), (2,3,34897), (2,4,671), (2,6,34898), (2,10,34849), (3,5,34900), (3,4,265), (3,5,34900), (3,6,895), (3,8,34901), (3,4,902), (4,5,1263), (4,6,316), (4,7,1156), (4,8,1320), (4,9,3245), (4,10,11599), (13,14,2), (15,16,323), (485,486,8781)

The Vu circlecevian point of P and U lies on the conic {{A,B,C,P,U}}. (Randy Hutson, November 26, 2019)

Given three points P,Q,R, then 6 points A,B,C, V(P,Q), V(Q,R), V(R,P) lie on a conic. (Tran Quang Hung, Euclid #225)

X(34892) = VU CIRCLECEVIAN POINT V(X(1),X(2))

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

X(34892) lies on the hyperbola {{A,B,C,X(1),X(2)}} and these lines: {1, 597}, {2, 4986}, {28, 12437}, {57, 1018}, {81, 644}, {88, 3912}, {89, 17316}, {105, 519}, {274, 646}, {279, 4099}, {291, 4694}, {346, 7208}, {524, 5525}, {536, 34578}, {551, 1390}, {918, 1022}, {1002, 3892}, {1432, 3970}, {3227, 17264}, {4904, 8056}, {5376, 5387}

X(34892) = isogonal conjugate of X(16784)
X(34892) = isotomic conjugate of X(37756)

X(34893) = VU CIRCLECEVIAN POINT V(X(1),X(6))

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

X(34993) lies on the hyperbola {{A,B,C,X(1),X(6)}} and these lines: {1, 597}, {6, 3722}, {44, 1438}, {56, 4557}, {58, 3939}, {86, 3699}, {106, 518}, {269, 4551}, {524, 5524}, {674, 9432}, {900, 1027}, {998, 5854}, {1431, 22277}, {2163, 3751}, {2191, 3756}, {2254, 23345}, {3214, 32259}, {5378, 5387}, {7194, 17779}, {22108, 23892}

X(34893) = isogonal conjugate of X(7292)

X(34894) = VU CIRCLECEVIAN POINT V(X(1),X(9))

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

X(34894) lies on the Feuerbach hyperbola and these lines: {1, 3939}, {4, 528}, {7, 100}, {8, 4578}, {9, 14740}, {11, 480}, {55, 6068}, {80, 5853}, {84, 2801}, {104, 518}, {149, 7674}, {314, 7256}, {390, 13278}, {516, 25438}, {527, 5537}, {885, 2804}, {952, 3427}, {954, 1145}, {971, 34256}, {1000, 1001}, {1156, 3935}, {1476, 14151}, {2346, 10177}, {2481, 3262}, {2802, 3577}, {3062, 3174}, {3158, 5528}, {3254, 6745}, {3880, 24297}, {5223, 10058}, {5553, 11491}, {5555, 10528}, {5660, 15909}, {5851, 10307}, {6598, 6736}, {6603, 18889}, {7320, 25875}, {10305, 10310}, {10306, 10309}, {10755, 14947}, {11604, 20119}, {18490, 22560}

X(34894) = isogonal conjugate of X(3660)
X(34894) = isotomic conjugate of X(38468)
X(34894) = Vu circlecevian point V(X(7),X(8))

X(34895) = VU CIRCLECEVIAN POINT V(X(1),X(10))

Barycentrics (b + 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) : :

X(34895) lies on the hyperbola {{A,B,C,X(1),X(10)}} and these lines: {1, 8258}, {8, 1247}, {19, 3169}, {37, 16613}, {65, 3178}, {267, 5541}, {519, 759}, {643, 2363}, {740, 2652}, {876, 2785}, {897, 32850}, {994, 29671}, {4730, 23835}

X(34896) = VU CIRCLECEVIAN POINT V(X(1),X(11))

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

X(34896 lies on the hyperbola {{A,B,C,X(1),X(11)}} and these lines: {1, 33562}, {6604, 8047}

X(34897) = VU CIRCLECEVIAN POINT V(X(2),X(3))

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

X(34897) lies on the hyperbola {{A,B,C,X(2),X(3)}} and these lines: {2, 339}, {3, 67}, {30, 935}, {69, 18472}, {127, 543}, {394, 4175}, {441, 14919}, {524, 10317}, {549, 5481}, {907, 9076}, {1368, 10717}, {2072, 22110}, {3933, 4558}, {5055, 14489}, {5159, 10415}, {7540, 34225}, {7728, 34360}, {7813, 22121}, {7818, 18564}, {7840, 15013}, {10748, 34517}, {11165, 20208}, {12037, 17702}, {14376, 34511}, {17974, 22115}, {18403, 31173}, {22366, 31144}, {34427, 34609}

X(34897) = isogonal conjugate of X(8744)
X(34897) = isotomic conjugate of X(37765)
X(34897) = X(19)-isoconjugate of X(23)
X(34897) = isotomic conjugate of polar conjugate of X(67)

X(34898) = VU CIRCLECEVIAN POINT V(X(2),X(6))

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) : :

X(34898) lies on the hyperbola {{A,B,C,X(2),X(6)}} and these lines: {6, 2482}, {25, 2930}, {111, 524}, {251, 8584}, {525, 10103}, {538, 6094}, {542, 11258}, {597, 9146}, {599, 6791}, {690, 9178}, {1383, 1992}, {3457, 30454}, {3458, 30455}, {5969, 14948}, {8030, 32740}, {8770, 15533}, {11054, 18818}, {14916, 28662}

X(34898) = isogonal conjugate of X(11580)

X(34899) = VU CIRCLECEVIAN POINT V(X(2),X(10))

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

X(34899) lies on the Kiepert hyperbola and these lines: {2, 15903}, {4, 2796}, {98, 519}, {115, 4052}, {321, 27733}, {542, 3429}, {671, 17132}, {2784, 3424}, {2799, 4049}

X(34900) = VU CIRCLECEVIAN POINT V(X(3),X(5))

Barycentrics (a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*(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(34900) lies on the conic {{A,B,C,X(3),X(5)}} and these lines: {3, 2888}, {30, 18401}, {216, 34520}, {9381, 13599}, {14140, 18403}, {20625, 25150}

X(34901) = VU CIRCLECEVIAN POINT V(X(3),X(8))

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

X(34901) lies on the on hyperbola {{A,B,C,X(3),X(8)}} and these lines: {59, 1145}, {517, 10742}, {952, 2745}

X(34902) = VU CIRCLECEVIAN POINT V(X(3),X(9))

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

X(34902 lies on the hyperbola {{A,B,C,X(3),X(9)}} and these lines: {40, 6068}, {7078, 23113}

X(34903) = (name pending)

Barycentrics (b+c)*a^9-2*(b^2+3*b*c+c^2)*a^8-(b+c)*(3*b^2-16*b*c+3*c^2)*a^7+2*(3*b^4+3*c^4+(3*b^2-20*b*c+3*c^2)*b*c)*a^6+(b+c)*(3*b^4+3*c^4-(39*b^2-71*b*c+39*c^2)*b*c)*a^5-(6*b^6+6*c^6-(5*b^4+5*c^4+2*(25*b^2-44*b*c+25*c^2)*b*c)*b*c)*a^4-(b+c)*(b^6+c^6-4*(7*b^2-6*b*c+7*c^2)*(b-c)^2*b*c)*a^3-(b^2-c^2)^2*(b+c)*(5*b^2-12*b*c+5*c^2)*a*b*c+2*(b+c)*(b^2-c^2)*(b^3-c^3)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^4*b*c : :
Trilinears 32*q*p^7-16*(6*q^2+1)*p^6+32*(2*q^2+3)*q*p^5-16*(4*q^2+3)*p^4-6*(12*q^2-17)*q*p^3+(48*q^4-22*q^2-19)*p^2-2*(4*q^4+6*q^2-7)*q*p+(4*q^2-3)*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

See Tran Quang Hung and César Lozada, Euclid 218 .

X(34903) lies on this line: {5, 10}

X(34904) = (name pending)

Barycentrics (b+c)*a^9-2*(2*b^2+b*c+2*c^2)*a^8+2*(b+c)*(b^2+3*b*c+c^2)*a^7+2*(5*b^4+5*c^4-6*(2*b^2-b*c+2*c^2)*b*c)*a^6-(b+c)*(12*b^4+12*c^4-(14*b^2-5*b*c+14*c^2)*b*c)*a^5-2*(3*b^6+3*c^6-(25*b^4+25*c^4-(29*b^2-19*b*c+29*c^2)*b*c)*b*c)*a^4+(b+c)*(14*b^6+14*c^6-(46*b^4+46*c^4-31*(b^2+c^2)*b*c)*b*c)*a^3-2*(b^2-c^2)^2*(b^4+c^4+2*(5*b^2-12*b*c+5*c^2)*b*c)*a^2-(b^2-c^2)^3*(b-c)*(5*b^2-16*b*c+5*c^2)*a+2*(b^2-c^2)^4*(b-c)^2 : :
Trilinears 32*q*p^7-16*(10*q^2-3)*p^6+32*(9*q^2-4)*q*p^5-16*(14*q^4-3*q^2-4)*p^4+2*(32*q^4+40*q^2-57)*q*p^3-(48*q^4-34*q^2-21)*p^2+2*(4*q^2-7)*q*p+q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

See Kadir Altintas and César Lozada, Euclid 218 .

X(34904) lies on this line: {1, 5123}

X(34905) = X(514)X(9436)∩X(663)X(672)

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

X(34905) lis on the cubic K721 and these lines: {514, 9436}, {663, 672}, {665, 1362}, {926, 6184}, {2170, 2254}, {2820, 9442}, {3887, 14947}

X(34905) = X(666)-isoconjugate of X(5091)
X(34905) = barycentric product X(i)*X(j) for these {i,j}: {918, 9319}, {2254, 14947}
X(34905) = barycentric quotient X(i)/X(j) for these {i,j}: {665, 9318}, {9319, 666}

X(34906) = X(101)X(514)∩X(238)X(516)

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

X(34906) lies on the cubics K721 and K1074 and on these lines: {101, 514}, {238, 516}

X(34906) = midpoint of X(666) and X(927)
X(34906) = X(i)-isoconjugate of X(j) for these (i,j): {665, 14947}, {2254, 9319}
X(34906) = barycentric product X(666)*X(9318)
X(34906) = barycentric quotient X(i)/X(j) for these {i,j}: {919, 9319}, {5091, 2254}, {9318, 918}

X(34907) = X(40)X(30557)∩X(175)X(13387)

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

X(34907) lies on the cubic K199 and these lines: {40, 30557}, {175, 13387}

X(34907) = X(1)-cross conjugate of X(30557)
X(34907) = X(16232)-isoconjugate of X(32556)
X(34907) = barycentric quotient X(5414)/X(32556)

X(34908) = X(40)X(30556)∩X(176)X(13386)

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

X(34908) lies on the cubic K199 and these lines: {40, 30556}, {176, 13386}

X(34908) = X(1)-cross conjugate of X(30556)
X(34908) = X(2362)-isoconjugate of X(32555)
X(34908) = barycentric quotient X(2066)/X(32555)

X(34909) = X(1)X(1123)∩X(175)X13387)

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

X(34909) lies on the cubic K199 and these lines: {1, 1123}, {175, 13387}, {280, 30557}, {7046, 14121}

X(34909) = X(8)-Ceva conjugate of X(7090)
X(34909) = barycentric quotient X(32555)/X(13389)
X(34909) = {X(1),X(13454)}-harmonic conjugate of X(7090)

X(34910) = X(1)X(1336)∩X(176)X(13386)

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

X(34910) lies on the cubic K199 and these lines: {1, 1336}, {176, 13386}, {280, 30556}, {7046, 7090}

X(34910) = X(8)-Ceva conjugate of X(14121)
X(34910) = barycentric quotient X(32556)/X(13388)
X(34910) = {X(1),X(13426)}-harmonic conjugate of X(14121)

X(34911) = X(8)X(175)∩X(40)X(971)

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

X(34911) lies on the cubic K199 and these lines: {8, 175}, {40, 971}, {200, 15891}, {4853, 16214}, {7046, 7090}

X(34911) = X(i)-isoconjugate of X(j) for these (i,j): {6, 16662}, {56, 175}, {1407, 30413}
X(34911) = barycentric product X(i)*X(j) for these {i,j}: {8, 15891}, {312, 30336}
X(34911) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16662}, {9, 175}, {200, 30413}, {15891, 7}, {30336, 57}

X(34912) = X(8)X(176)∩X(40)X(971)

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

X(34912) lies on the cubic K199 and these lines: {8, 176}, {40, 971}, {200, 15892}, {4853, 16213}, {7046, 14121}

X(34912) = X(i)-isoconjugate of X(j) for these (i,j): {6, 16663}, {56, 176}, {1407, 30412}
X(34912) = barycentric product X(i)*X(j) for these {i,j}: {8, 15892}, {312, 30335}, {5451, 6731}
X(34912) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 16663}, {9, 176}, {200, 30412}, {5451, 555}, {15892, 7}, {30335, 57}

X(34913) = X(1)X(4)∩X(56)X(31825)

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

See Tran Quang Hung and Peter Moses, Euclid 220 .

X(34913) lies on these lines: {1, 4}, {56, 31825}, {214, 3738}, {6001, 34345}, {7335, 34880}, {22350, 23703}

X(34913) = X(80)-isoconjugate of X(2716)
X(34913) = crossdifference of every pair of points on line {652, 2161}
X(34913) = barycentric product X(2800)*X(3218)
X(34913) = barycentric quotient X(i)/X(j) for these {i,j}: {2800, 18359}, {7113, 2716}
X(34913) = {X(1),X(1745)}-harmonic conjugate of X(18340)

Images of Vu T-transform: X(34914)-X(34921)

This preamble and centers X(34914)-X(34921) were contributed by Vu Thanh Tung and Vu Quoc My, November 23, 2019, with extensions and editing by Clark Kimberling and Peter Moses, November 23, 2019.

Let P be a point in the plane of a triangle ABC, and let A' be the point, other than P, in which the line AP meets the circle (PBC). 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 I be the incenter of triangle ABC, and let A0 = BC∩IA'; define B0 and C0 cyclically. Then AA0, BB0, CC0 concur in a point, T(P), here named the Vu T-transform of P. Barycentrics are given by

T(P) = p / (b*c*p^2 + b*c*p* q + c^2*p*q + b^2*p*r + b*c*p*r + a^2*q*r) .

The appearance of (i,j) in the following list means that T(X(i)) = X(j):

(1,1), (2,34914), (3,7100), (4,79), (5,34915), (6,34916), (7,34917), (8,34918), (9,34919), (10,34920)

X(34914) = VU T-TRANSFORM OF X(2)

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

X(34914) lies on these lines: {1, 524}, {2, 14210}, {28, 28619}, {37, 34892}, {57, 7181}, {81, 6629}, {88, 17023}, {89, 26626}, {105, 551}, {291, 4424}, {519, 1390}, {758, 1002}, {961, 3671}, {1211, 25430}, {1255, 29574}, {2224, 16503}, {2895, 27789}, {3227, 17320}, {6703, 8056}, {7132, 16783}, {7208, 17302}, {7313, 17045}, {8682, 30571}, {17382, 34578}

X(34914) = isogonal conjugate of X(16785)

X(34915) = VU T-TRANSFORM OF X(5)

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

X(34915) lies on these lines: {1, 32423}

X(34916) = VU T-TRANSFORM OF X(6)

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

X(34916) lies on these lines: {1, 524}, {6, 896}, {42, 34893}, {56, 4471}, {58, 4719}, {86, 7292}, {106, 1386}, {597, 17779}, {740, 996}, {998, 17768}, {1027, 28209}, {1126, 3931}, {1411, 32259}, {1438, 16666}, {2163, 16475}, {2245, 2279}, {2292, 2334}, {4890, 9027}, {5297, 31144}, {7073, 32286}, {7312, 29821}

X(34916) = isogonal conjugate of X(5297)

X(34917) = VU T-TRANSFORM OF X(7)

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

X(34917) lies on these lines: {4, 12564}, {9, 17718}, {21, 527}, {79, 15726}, {80, 5728}, {84, 6147}, {104, 5542}, {140, 15901}, {226, 1156}, {354, 3254}, {523, 23893}, {954, 15175}, {1172, 23710}, {2078, 2346}, {2320, 11038}, {3065, 5851}, {4866, 12607}, {5045, 34485}, {5572, 15909}, {5665, 6284}, {5856, 6596}, {6598, 15733}, {20116, 24298}

X(34918) = VU T-TRANSFORM OF X(8)

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

X(34918) lies on these lines: {1, 1329}, {7, 5554}, {10, 104}, {21, 5795}, {79, 10107}, {80, 10914}, {84, 355}, {90, 3679}, {474, 17665}, {943, 10915}, {958, 15446}, {960, 5559}, {1000, 2551}, {1156, 5086}, {1320, 12053}, {1389, 11813}, {1392, 6919}, {1476, 5176}, {1837, 3680}, {2320, 5552}, {2550, 10307}, {3036, 3065}, {3057, 12641}, {3436, 5555}, {3577, 22791}, {4863, 31509}, {5434, 7091}, {5794, 16209}, {5837, 32635}, {6256, 10309}, {6597, 21677}, {6842, 9623}, {7284, 9613}, {7320, 12648}, {9578, 25962}, {10305, 12115}, {10742, 22792}, {17556, 21398}, {18236, 32537}

X(34919) = VU T-TRANSFORM OF X(9)

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

X(34919) lies on these lines: {1, 527}, {2, 1156}, {4, 3812}, {7, 3660}, {8, 10394}, {9, 1776}, {55, 6068}, {80, 2550}, {84, 1125}, {90, 6857}, {104, 1001}, {142, 3062}, {144, 2346}, {390, 1320}, {497, 3254}, {516, 3577}, {518, 1000}, {522, 23893}, {528, 24297}, {631, 15297}, {971, 3427}, {1005, 12848}, {1476, 3622}, {3485, 7091}, {3486, 3680}, {3869, 7320}, {4679, 17603}, {4900, 5853}, {5220, 34619}, {5558, 11415}, {5732, 30500}, {6601, 14100}, {6872, 17097}, {7080, 32635}, {7160, 12514}, {8582, 33576}, {9365, 24498}, {9814, 12047}, {9940, 10309}, {12867, 18253}, {31657, 34256}

X(34919) = isogonal conjugate of X(37541)
X(34919) = Kirikami-Euler image of X(7)

X(34920) = VU T-TRANSFORM OF X(10)

Barycentrics T(X(10); (b + 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) : :

X(34920) lies on these lines: {1, 1330}, {2, 1247}, {19, 3144}, {37, 3178}, {75, 18835}, {267, 24169}, {409, 759}, {442, 2652}, {596, 29655}, {876, 29118}, {897, 8052}, {1910, 34076}, {2217, 25361}, {2292, 34895}, {2363, 19786}, {3821, 13610}, {18827, 24214}

X(34921) = X(1)X(2687)∩X(36)X(74)

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

See Antreas Hatzipolakis, César Lozada and Angel Montesdeoca, Euclid 222 .

X(34921) lies on these lines: {1,2687}, {36,74}, {59,8701}, {101,9404}, {102,22765}, {104,3065}, {106,1464}, {109,2605}, {354,2717}, {477,5441}, {484,5951}, {513,26700}, {759,1319}, {953,5563}, {972,5536}, {1141,14452}, {1311,21739}, {1385,2716}, {2078,28471}, {2291,19302}, {2688,3058}, {2695,34773}, {2745,11012}, {5143,29300}, {7113,11075}, {19628,26743}

X(34922) = X(108)X(476)∩X(1309)X(26700)

Barycentrics 1/((a-b-c) (b-c)^2 (a^2-b^2-c^2) (a^2-b^2-b c-c^2)) : :

See Antreas Hatzipolakis, César Lozada and Angel Montesdeoca, Euclid 222 .

X(34922) lies on these lines: {108,476}, {1309,26700}

X(34923) = (name pending)

Barycentrics a*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8 - 6*a^7*b*c + 7*a^6*b^2*c + 12*a^5*b^3*c - 15*a^4*b^4*c - 6*a^3*b^5*c + 9*a^2*b^6*c - b^8*c + a^7*c^2 + 7*a^6*b*c^2 - 28*a^5*b^2*c^2 + 12*a^4*b^3*c^2 + 25*a^3*b^4*c^2 - 20*a^2*b^5*c^2 + 2*a*b^6*c^2 + b^7*c^2 + 12*a^5*b*c^3 + 12*a^4*b^2*c^3 - 38*a^3*b^3*c^3 + 11*a^2*b^4*c^3 + 3*b^6*c^3 - 3*a^5*c^4 - 15*a^4*b*c^4 + 25*a^3*b^2*c^4 + 11*a^2*b^3*c^4 - 2*a*b^4*c^4 - 3*b^5*c^4 - 6*a^3*b*c^5 - 20*a^2*b^2*c^5 - 3*b^4*c^5 + 3*a^3*c^6 + 9*a^2*b*c^6 + 2*a*b^2*c^6 + 3*b^3*c^6 + b^2*c^7 - a*c^8 - b*c^8) : :

See Antreas Hatzipolakis and Peter Moses, Euclid 230 .

X(34923) lies on this line: {1, 3}

Orthology centers associated with osculatory triangle of K721: X(34924)-X(34935)

This preamble and centers X(34924)-X(34935) were contributed by Peter Moses, November 24, 2019.

Suppose that a curve K passes through A, B, C and has osculating circle Oa at A. Let A' be the center of Oa, and define B' and C' cyclically. The triangle A'B'C' is here named the osculatory triangle of K.

X(34924) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO YFF CENTRAL

Barycentrics a*(a + b - c)*(a - b + c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2)*(a^2 - 2*a*b + b^2 - 2*a*c + b*c + c^2) + 2*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - b^2*c - a*c^2 - b*c^2 + c^3)*(a^4 - a^3*b - a*b^3 + b^4 - a^3*c + a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4)*Sin[A/2] : :

X(34924) lies on these lines: {174, 15730}, {2801, 8351}

X(34925) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO EXCENTRAL

Barycentrics a*(a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c - 3*a^4*b*c + 6*a^3*b^2*c - 2*a^2*b^3*c - 3*a*b^4*c + 3*b^5*c - a^4*c^2 + 6*a^3*b*c^2 - 4*a^2*b^2*c^2 + 2*a*b^3*c^2 - 3*b^4*c^2 - 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 - 3*b^2*c^4 + a*c^5 + 3*b*c^5 - c^6) : :
X(34925) = 3 X[101] - 2 X[15730]

X(34925) lies on these lines: {9, 10708}, {40, 2801}, {57, 101}, {63, 544}, {103, 3587}, {116, 3305}, {150, 3219}, {169, 8545}, {484, 1282}, {527, 5011}, {758, 9451}, {2809, 5119}, {3295, 7202}, {3730, 16560}, {10695, 31393}, {17739, 33937}

X(34925) = Bevan-circle-inverse of X(5528)
X(34925) = excentral-isogonal conjugate of X(5527)

X(34926) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO HONSBERGER

Barycentrics (a + b - c)*(a - b + c)*(3*a^7 - 11*a^6*b + 15*a^5*b^2 - 9*a^4*b^3 + a^3*b^4 + 3*a^2*b^5 - 3*a*b^6 + b^7 - 11*a^6*c + 27*a^5*b*c - 21*a^4*b^2*c + 7*a^3*b^3*c - 6*a^2*b^4*c + 6*a*b^5*c - 2*b^6*c + 15*a^5*c^2 - 21*a^4*b*c^2 + 6*a^3*b^2*c^2 + 3*a^2*b^3*c^2 - 3*a*b^4*c^2 - 9*a^4*c^3 + 7*a^3*b*c^3 + 3*a^2*b^2*c^3 + b^4*c^3 + a^3*c^4 - 6*a^2*b*c^4 - 3*a*b^2*c^4 + b^3*c^4 + 3*a^2*c^5 + 6*a*b*c^5 - 3*a*c^6 - 2*b*c^6 + c^7) : :
X(34926) = 3 X[8236] - 2 X[34930], 5 X[18230] - 4 X[34933]

X(34926) lies on these lines: {7, 15730}, {9, 34932}, {101, 30379}, {226, 673}, {390, 2801}, {516, 34931}, {527, 664}, {544, 8545}, {1445, 34925}, {7676, 34927}, {7677, 34928}, {8236, 34930}, {8389, 34924}, {8544, 33520}, {18230, 34933}, {21617, 34934}

X(34926) = reflection of X(i) in X(j) for these {i,j}: {7, 15730}, {7, 34926}, {34932, 9}

X(34927) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO 1ST CIRCUMPERP

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

X(34927) lies on these lines: {3, 2801}, {55, 15730}, {101, 8012}, {103, 2077}, {515, 5144}, {2078, 4845}, {2808, 32613}, {2809, 3428}, {10269, 11714}, {15735, 34486}

X(34928) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO 2ND CIRCUMPERP

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

X(34928) lies on these lines: {3, 2801}, {36, 101}, {55, 2809}, {56, 15730}, {58, 1066}, {103, 7688}, {527, 5144}, {544, 993}, {758, 23398}, {999, 11712}, {1282, 5010}, {5251, 10708}, {5450, 33520}, {12511, 33521}

X(34929) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO HALF-ALTITUDE

Barycentrics (a + b - c)*(a - b + c)*(2*a^5 - 2*a^4*b + a^2*b^3 - 4*a*b^4 + 3*b^5 - 2*a^4*c + 2*a^3*b*c - a^2*b^2*c + a*b^3*c - a^2*b*c^2 + 6*a*b^2*c^2 - 3*b^3*c^2 + a^2*c^3 + a*b*c^3 - 3*b^2*c^3 - 4*a*c^4 + 3*c^5) : :
X(34929) = 3 X[226] - X[15730]

X(34929) lies on these lines: {7, 10708}, {57, 116}, {101, 5226}, {169, 8545}, {226, 544}, {527, 5199}, {942, 2801}, {1319, 11726}, {5122, 6712}, {9579, 33521}, {10739, 15934}, {10770, 14151}, {11374, 33520}

X(34930) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO HUTSON-INTOUCH

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

X(34930) lies on these lines: {1, 651}, {36, 11714}, {55, 2809}, {101, 13384}, {103, 2093}, {515, 1323}, {999, 11028}, {3022, 5048}, {5727, 10708}, {7962, 10695}, {18412, 18473}

X(34930) = reflection of X(15730) in X(1)

X(34931) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO REFLECTION-OF-X(1)-IN-SIDELINES(BC,CA,AB)

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

X(34931) lies on these lines: {1, 651}, {36, 101}, {55, 2808}, {103, 5010}, {150, 10590}, {544, 1478}, {999, 1362}, {1282, 2093}, {2809, 25415}, {3022, 6767}, {3583, 10710}, {4299, 33520}, {5723, 12831}, {7951, 10708}, {11529, 18413}

X(34931) = reflection of X(1) in X(15730)

X(34932) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO INNER CONWAY

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

X(34932) lies on these lines: {2, 15730}, {8, 2801}, {63, 544}, {116, 30852}, {150, 329}, {527, 1121}, {908, 10708}, {2808, 3419}, {3912, 34234}, {4652, 33520}, {4845, 26015}, {4858, 6604}

X(34932) = anticomplement of X(15730)
X(34932) = anticomplement of the isogonal conjugate of X(15734)
X(34932) = X(15734)-anticomplementary conjugate of X(8)

X(34933) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO 2ND ZANIAH

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

X(34933) lies on these lines: {2, 15730}, {9, 10708}, {10, 1071}, {116, 3452}, {150, 5273}, {527, 5199}, {544, 5745}, {3041, 5123}, {4845, 5231}

X(34933) = complement of X(15730)
X(34933) = complement of the isogonal conjugate of X(15734)
X(34933) = X(i)-complementary conjugate of X(j) for these (i,j): {2291, 6594}, {3254, 31844}, {15734, 10}

X(34934) = ORTHOLOGY CENTER OF THESE TRIANGLES: OSCULATORY OF K721 TO WASAT

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

X(34934) lies on these lines: {101, 5249}, {116, 3452}, {150, 31053}, {226, 544}, {527, 5074}, {908, 10708}, {946, 2801}, {21621, 22000}, {24237, 34847}

X(34935) = ORTHOLOGY CENTER OF THESE TRIANGLES: HALF-ALTITUDE TO OSCULATORY OF K721

Barycentrics a^2*(a^8 - a^6*b^2 - 3*a^4*b^4 + 5*a^2*b^6 - 2*b^8 - 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 + 8*a^4*b^2*c^2 - 2*a^3*b^3*c^2 - 5*a^2*b^4*c^2 + 3*a*b^5*c^2 - 2*b^6*c^2 + a^4*b*c^3 - 2*a^3*b^2*c^3 + 4*a^2*b^3*c^3 - 2*a*b^4*c^3 - b^5*c^3 - 3*a^4*c^4 + 2*a^3*b*c^4 - 5*a^2*b^2*c^4 - 2*a*b^3*c^4 + 8*b^4*c^4 - 2*a^2*b*c^5 + 3*a*b^2*c^5 - b^3*c^5 + 5*a^2*c^6 - a*b*c^6 - 2*b^2*c^6 + b*c^7 - 2*c^8) : :
X(34935) = 3 X[64] + X[7959], X[221] - 5 X[8567], 3 X[3556] - X[7959]

X(34935) lies on these lines: {3, 14529}, {64, 71}, {65, 1204}, {74, 30250}, {221, 5217}, {517, 7689}, {2390, 10310}, {2818, 26285}, {3357, 3579}, {3827, 12262}, {3869, 11440}, {5894, 20306}, {6908, 12930}, {7686, 11438}, {8273, 13095}, {9961, 13445}, {11204, 33862}, {14988, 32138}, {28163, 33771}

X(34935) = midpoint of X(i) and X(j) for these {i,j}: {64, 3556}, {5894, 20306}
X(34935) = reflection of X(14529) in X(3)

Polar reciprocal conics: X(34936)-X(34991)

This preamble and centers X(34936)-X(34991) were contributed by César Eliud Lozada, November 24, 2019.

Let K₁, K₂ be two conics (circles included) with centers O₁ and O₂, respectively. The locus of the poles with respect to K₁ of the tangents to K₂ is another conic Φ₂₁ named the reciprocal polar conic of K₂ with respect to K₁. By swapping K₁ and K₂, the reciprocal polar conic Φ₁₂ of K₂ with respect to K₁ is found.

The center of Φ₂₁ is the pole, with respect to K₁, of the polar of O₁ wrt K₂ and, similarly, the center of Φ₁₂ is the pole, with respect to K₂, of the polar of O₂ wrt K₁. (Reference: Eagles, T.H.: Constructive Geometry of Plane Curves, McMillan and Co, London, 1885. pp 201-202.)

The appearance of (K₁, K₂, i, j) in the following list means that the centers of the reciprocal polar conics (K₂ wrt K₁) and (K₁ wrt to K₂) are X(i) and X(j), respectively:

(anticomplementary circle, circumcircle, 14790, 7512), (anticomplementary circle, incircle, 34936, 34937), (anticomplementary circle, nine-point circle, 34938, 34939), (anticomplementary circle, Feuerbach hyperbola, 521, 34940), (anticomplementary circle, Jerabek hyperbola, 520, 34941), (anticomplementary circle, Johnson circumconic, 14216, 34942), (anticomplementary circle, Kiepert hyperbola, 525, 34943), (anticomplementary circle, MacBeath circumconic, 34944, 34945), (anticomplementary circle, Stammler hyperbola, 34946, 34947), (circumcircle, incircle, 7742, 942), (circumcircle, nine-point circle, 26, 5576), (circumcircle, Feuerbach hyperbola, 34948, 34949), (circumcircle, Jerabek hyperbola, 924, 34950), (circumcircle, Johnson circumconic, 10282, 34951), (circumcircle, Kiepert hyperbola, 34952, 34953), (circumcircle, MacBeath circumconic, 154, 3167), (circumcircle, Stammler hyperbola, 523, 523), (incircle, nine-point circle, 496, 7741), (incircle, Feuerbach hyperbola, 513, 513), (incircle, Jerabek hyperbola, 34954, 34955), (incircle, Johnson circumconic, 34956, 34957), (incircle, Kiepert hyperbola, 34958, 34959), (incircle, MacBeath circumconic, 10391, 34960), (incircle, Stammler hyperbola, 6003, 34961), (nine-point circle, Feuerbach hyperbola, 34962, 513), (nine-point circle, Jerabek hyperbola, 34963, 523), (nine-point circle, Johnson circumconic, 5, 5), (nine-point circle, Kiepert hyperbola, 34964, 512), (nine-point circle, MacBeath circumconic, 34965, 34966), (nine-point circle, Stammler hyperbola, 34967, 34968), (Feuerbach hyperbola, Jerabek hyperbola, 34969, 34970), (Feuerbach hyperbola, Johnson circumconic, 34971, 34972), (Feuerbach hyperbola, Kiepert hyperbola, 2310, 34973), (Feuerbach hyperbola, MacBeath circumconic, 34974, 34975), (Feuerbach hyperbola, Stammler hyperbola, 34976, 34977), (Jerabek hyperbola, Johnson circumconic, 34978, 34979), (Jerabek hyperbola, Kiepert hyperbola, 34980, 34981), (Jerabek hyperbola, MacBeath circumconic, 34982, 8673), (Jerabek hyperbola, Stammler hyperbola, 13198, 5972), (Johnson circumconic, Kiepert hyperbola, 34983, 34984), (Johnson circumconic, MacBeath circumconic, 34985, 34986), (Johnson circumconic, Stammler hyperbola, 34987, 6368), (Kiepert hyperbola, MacBeath circumconic, 34988, 32320), (Kiepert hyperbola, Stammler hyperbola, 34989, 34990), (MacBeath circumconic, Stammler hyperbola, 525, 525)

X(34936) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt ANTICOMPLEMENTARY CIRCLE

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

X(34936) lies on these lines: {1,4}, {280,1370}

X(34936) = {X(388), X(497)}-harmonic conjugate of X(34937)

X(34937) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt INCIRCLE

Barycentrics 2*a^4+(b+c)*a^3+(b+c)^2*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a+(b^2-c^2)^2 : :
X(34937) = X(1)+3*X(33152) = 7*X(3624)-3*X(33167) = X(13161)-3*X(33152)

X(34937) lies on these lines: {1,4}, {2,3710}, {3,3663}, {10,3772}, {37,39}, {58,527}, {65,17602}, {78,19785}, {142,975}, {201,3670}, {329,1453}, {386,3946}, {387,11523}, {405,4656}, {443,23681}, {474,24177}, {516,5266}, {519,1834}, {551,3445}, {908,5262}, {936,4000}, {939,954}, {960,17061}, {964,4054}, {976,3914}, {986,6684}, {988,10165}, {1086,12436}, {1089,19869}, {1104,4415}, {1193,3191}, {1210,17720}, {1738,5293}, {2285,3333}, {2292,3011}, {2318,3216}, {2782,15903}, {2911,4989}, {3008,5044}, {3086,3677}, {3187,4101}, {3290,25088}, {3624,33167}, {3646,16020}, {3649,3745}, {3664,4920}, {3666,13411}, {3671,5711}, {3672,5703}, {3695,20106}, {3701,25904}, {3731,16845}, {3744,10624}, {3752,6700}, {3755,3811}, {3782,4292}, {3821,8669}, {3876,26723}, {3931,13405}, {3947,5725}, {3951,24597}, {3986,16844}, {4003,5433}, {4009,25992}, {4021,5719}, {4252,17276}, {4255,17301}, {4295,5269}, {4298,6354}, {4340,4654}, {4419,31424}, {4850,27385}, {4859,17582}, {4887,24470}, {5250,26228}, {5255,28194}, {5292,24391}, {5314,19850}, {5530,17719}, {6692,24046}, {6734,33133}, {7174,19843}, {7322,19855}, {9843,17054}, {11375,17599}, {12609,30142}, {13742,30568}, {14986,28739}, {16408,24175}, {17016,33153}, {20009,26132}, {23536,33143}, {23537,30115}, {24171,25524}, {24178,33147}, {24248,31730}, {24850,28526}, {24851,28150}, {25087,26242}, {25253,26230}, {28038,31393}, {33155,34772}

X(34937) = midpoint of X(1) and X(13161)
X(34937) = crossdifference of every pair of points on line {X(652), X(4057)}
X(34937) = X(13161)-of-anti-Aquila triangle
X(34937) = X(14152)-of-incircle-circles triangle
X(34937) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 226, 5717), (1, 9612, 5716), (1, 33144, 21620), (1, 33152, 13161), (388, 497, 34936), (551, 4052, 11354), (975, 24159, 142), (1104, 4415, 12572), (1125, 17355, 17698)

X(34938) = CENTER OF THIS POLAR RECIPROCAL CONIC: NINE-POINT CIRCLE wrt ANTICOMPLEMENTARY CIRCLE

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

As a point on the Euler line, X(34938) has Shinagawa coefficients (E, -3*E-2*F)

X(34938) lies on these lines: {2,3}, {52,18909}, {68,32064}, {69,10625}, {155,34781}, {389,31670}, {393,23115}, {394,16655}, {511,11411}, {569,25406}, {1056,32047}, {1058,8144}, {1092,31383}, {1147,11206}, {1249,22120}, {1351,18914}, {1352,15644}, {1503,6193}, {2777,12319}, {3060,18916}, {3086,9645}, {3564,12320}, {3618,13336}, {3818,13348}, {3917,11487}, {4846,14641}, {5266,10629}, {5446,11433}, {5656,22660}, {6225,14915}, {6243,18917}, {6247,17834}, {6515,11457}, {6696,18382}, {9815,16836}, {9833,13346}, {10263,18951}, {11381,15438}, {11432,21850}, {12295,18933}, {12318,12324}, {13203,17702}, {13347,19130}, {13352,14927}, {13391,32140}, {13491,31815}, {14855,15740}, {16621,17814}, {22121,33630}, {24680,34643}, {31810,34783}

X(34938) = reflection of X(i) in X(j) for these (i,j): (4, 14790), (20, 12085), (7387, 23335), (9833, 13346), (11411, 14216), (17834, 6247), (31305, 3), (34781, 155)
X(34938) = anticomplement of X(7387)
X(34938) = intersection, other than A,B,C, of conics {{A, B, C, X(68), X(11414)}} and {{A, B, C, X(69), X(10323)}}
X(34938) = anticomplementary circle-inverse of X(403)
X(34938) = X(14790)-of-anti-Euler triangle
X(34938) = X(31305)-of-ABC-X(3) reflections triangle
X(34938) = X(3)-of-polar-triangle-of-anticomplementary-circle
X(34938) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 7403, 2), (4, 6804, 381), (4, 7386, 5), (20, 3088, 3), (20, 7391, 4), (382, 18531, 4), (425, 29950, 377), (631, 34608, 26), (855, 5200, 25), (3538, 7392, 140), (3854, 4226, 25), (7378, 7400, 5), (7387, 23335, 2), (10323, 15559, 2), (11300, 26534, 30), (14008, 15971, 26), (14807, 14808, 403), (16976, 34582, 186), (27206, 31913, 5), (27611, 31102, 30)

X(34939) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt NINE-POINT CIRCLE

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

As a point on the Euler line, X(34939) has Shinagawa coefficients (4*E+2*F, 3*E+2*F)

X(34939) lies on these lines: {2,3}, {141,9781}, {373,26879}, {524,1173}, {597,11423}, {1614,3589}, {5480,7999}, {9722,31404}, {10516,18912}, {11451,12359}, {11457,17825}, {31831,34545}

X(34940) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt FEUERBACH HYPERBOLA

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

X(34940) lies on the line {11,521}

X(34941) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt JERABEK HYPERBOLA

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

X(34941) lies on the line {125,520}

X(34942) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt JOHNSON CIRCUMCONIC

Barycentrics a^2*((b^2+c^2)*a^12-(5*b^4+2*b^2*c^2+5*c^4)*a^10+(b^2+c^2)*(10*b^4-7*b^2*c^2+10*c^4)*a^8-2*(5*b^4+2*b^2*c^2+5*c^4)*(b^4+c^4)*a^6+(b^4-c^4)*(b^2-c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)^3*b^2*c^2-(b^2-c^2)^2*(b^8+c^8-2*(2*b^4+13*b^2*c^2+2*c^4)*b^2*c^2)*a^2)*(-a^2+b^2+c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

X(34942) lies on the line {5,389}

X(34943) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt KIEPERT HYPERBOLA

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

X(34943) lies on the line {115,525}

X(34944) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt ANTICOMPLEMENTARY CIRCLE

Barycentrics a^12-3*(b^2-c^2)^2*a^8-8*(b^4-c^4)*(b^2-c^2)*a^2*b^2*c^2+3*(b^4-c^4)^2*a^4-(b^4-c^4)^2*(b^2-c^2)^2 : :
X(34944) = 3*X(1853)-2*X(13567) = X(6515)-3*X(32064)

X(34944) lies on these lines: {2,32125}, {4,64}, {20,161}, {66,1843}, {122,33581}, {154,7386}, {159,7667}, {235,9914}, {343,34778}, {394,1370}, {541,19479}, {1368,1619}, {1498,6643}, {1899,34146}, {2781,3448}, {2883,6816}, {3357,21243}, {3548,32321}, {3556,21015}, {5064,23300}, {5943,22802}, {6000,18531}, {6242,32337}, {6293,18909}, {6353,10117}, {6696,6815}, {6997,23332}, {7169,26933}, {7528,20299}, {7689,20427}, {8567,10996}, {11206,15139}, {11744,15151}, {12324,22555}, {13093,18404}, {13598,18381}, {13754,14216}, {17825,18537}

X(34944) = reflection of X(i) in X(j) for these (i,j): (1619, 1368), (10605, 6247)
X(34944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1853, 5895, 17810), (13203, 32064, 7391)

X(34945) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt MACBEATH CIRCUMCONIC

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

X(34945) lies on these lines: {2,6}, {23,1915}, {32,2979}, {39,5012}, {110,1194}, {184,1180}, {251,511}, {698,10328}, {1186,10345}, {1501,3094}, {1627,3917}, {1691,8041}, {1976,3108}, {2001,7772}, {3060,5028}, {3148,9605}, {3981,13595}, {4576,16951}, {5007,23061}, {8039,9230}, {9306,9465}, {10329,14567}, {10542,33586}, {10546,34481}, {11205,14153}, {13509,15048}, {14886,16985}, {16932,18906}, {16950,33798}

X(34945) = isogonal conjugate of the isotomic conjugate of X(7832)
X(34945) = barycentric product X(6)*X(7832)
X(34945) = trilinear product X(31)*X(7832)
X(34945) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(7832)}} and {{A, B, C, X(39), X(16893)}}
X(34945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 394, 5359), (6, 3051, 34482), (6, 20965, 34545), (323, 34482, 3051), (394, 5359, 9463), (1501, 3094, 6636), (1691, 8041, 15246), (1915, 20859, 23), (11205, 20976, 14153)

X(34946) = CENTER OF THIS POLAR RECIPROCAL CONIC: STAMMLER HYPERBOLA wrt ANTICOMPLEMENTARY CIRCLE

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

X(34946) lies on these lines: {4,924}, {523,15099}

X(34947) = CENTER OF THIS POLAR RECIPROCAL CONIC: ANTICOMPLEMENTARY CIRCLE wrt STAMMLER HYPERBOLA

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

X(34947) lies on these lines: {110,924}, {184,34950}, {250,8675}, {520,1576}, {647,23963}

X(34948) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt CIRCUMCIRCLE

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

X(34948) lies on these lines: {3,15313}, {11,34467}, {36,238}, {663,23226}, {924,12095}, {1459,8648}, {1946,2605}, {9001,23187}

X(34948) = midpoint of X(667) and X(23224)
X(34948) = barycentric product X(i)*X(j) for these {i, j}: {24, 905}, {47, 514}, {81, 924}, {274, 34952}
X(34948) = barycentric quotient X(i)/X(j) for these (i, j): (24, 6335), (28, 30450), (47, 190), (513, 5392)
X(34948) = trilinear product X(i)*X(j) for these {i, j}: {24, 1459}, {27, 30451}, {47, 513}, {58, 924}
X(34948) = trilinear quotient X(i)/X(j) for these (i, j): (24, 1897), (27, 30450), (47, 100), (58, 925)
X(34948) = intersection, other than A,B,C, of conics {{A, B, C, X(24), X(859)}} and {{A, B, C, X(56), X(1147)}}

X(34949) = CENTER OF THIS POLAR RECIPROCAL CONIC: CIRCUMCIRCLE wrt FEUERBACH HYPERBOLA

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

X(34949) lies on these lines: {11,15313}, {521,3270}, {912,18455}, {1459,7004}, {34971,34974}

X(34950) = CENTER OF THIS POLAR RECIPROCAL CONIC: CIRCUMCIRCLE wrt JERABEK HYPERBOLA

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

X(34950) lies on these lines: {50,8779}, {125,924}, {184,34947}, {520,15526}, {3049,3269}, {5621,34146}, {13754,15781}, {34978,34982}

X(34950) = barycentric product X(394)*X(34978)
X(34950) = trilinear product X(i)*X(j) for these {i, j}: {255, 34978}, {326, 34982}

X(34951) = CENTER OF THIS POLAR RECIPROCAL CONIC: CIRCUMCIRCLE wrt JOHNSON CIRCUMCONIC

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

X(34951) lies on these lines: {5,389}, {129,27370}, {195,520}, {511,15780}

X(34952) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt CIRCUMCIRCLE

Barycentrics a^4*(a^4+b^4+c^4-2*(b^2+c^2)*a^2)*(b^2-c^2) : :
X(34952) = 3*X(351)-X(15451) = X(669)+2*X(14270) = 4*X(8651)-X(21731)

X(34952) lies on these lines: {3,3566}, {32,34347}, {115,17423}, {187,237}, {523,2070}, {525,5926}, {690,22089}, {878,2353}, {924,12095}, {3049,19627}, {25644,32472}

X(34952) = midpoint of X(9420) and X(9494)
X(34952) = Gibert-circumtangential conjugate of X(925)
X(34952) = isogonal conjugate of the isotomic conjugate of X(924)
X(34952) = isogonal conjugate of the polar conjugate of X(6753)
X(34952) = anticomplement of complementary conjugate of X(39013)
X(34952) = polar conjugate of the isotomic conjugate of X(30451)
X(34952) = crossdifference of every pair of points on line X(2)X(311)
X(34952) = X(i)-isoconjugate of X(j) for these {i,j}: {68, 811}, {75, 925}, {91, 99}
X(34952) = barycentric product X(i)*X(j) for these {i, j}: {4, 30451}, {6, 924}, {24, 647}, {32, 6563}
X(34952) = barycentric quotient X(i)/X(j) for these (i, j): (24, 6331), (25, 30450), (32, 925), (47, 799)
X(34952) = trilinear product X(i)*X(j) for these {i, j}: {19, 30451}, {24, 810}, {31, 924}, {42, 34948}
X(34952) = trilinear quotient X(i)/X(j) for these (i, j): (19, 30450), (24, 811), (31, 925), (47, 99)
X(34952) = intersection, other than A,B,C, of conics {{A, B, C, X(24), X(237)}} and {{A, B, C, X(32), X(1147)}}
X(34952) = pole of the trilinear polar of X(925) wrt circumcircle

X(34953) = CENTER OF THIS POLAR RECIPROCAL CONIC: CIRCUMCIRCLE wrt KIEPERT HYPERBOLA

Barycentrics (b^2-c^2)^2*(-a^2+b^2+c^2)*(a^4+2*b^4-3*b^2*c^2+2*c^4-(b^2+c^2)*a^2) : :
X(34953) = 5*X(14061)-X(33803) = X(14120)-4*X(15359)

X(34953) lies on these lines: {115,3566}, {125,525}, {512,14120}, {2072,3564}, {2450,13509}, {6390,19599}, {7668,34964}, {9517,34982}, {14061,33803}, {34984,34988}

X(34953) = isotomic conjugate of the polar conjugate of X(31644)
X(34953) = barycentric product X(i)*X(j) for these {i, j}: {69, 31644}, {125, 14061}, {338, 14060}
X(34953) = barycentric quotient X(115)/X(14052)
X(34953) = trilinear product X(i)*X(j) for these {i, j}: {63, 31644}, {1109, 14060}
X(34953) = trilinear quotient X(1109)/X(14052)

X(34954) = CENTER OF THIS POLAR RECIPROCAL CONIC: JERABEK HYPERBOLA wrt INCIRCLE

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

X(34954) lies on these lines: {513,34958}, {523,942}, {928,6129}, {999,2605}, {3333,3737}, {5214,11021}, {6003,7250}, {8675,31947}, {8676,14353}

X(34954) = X(23290)-of-incircle-circles triangle

X(34955) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt JERABEK HYPERBOLA

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

X(34955) lies on these lines: {}

X(34956) = CENTER OF THIS POLAR RECIPROCAL CONIC: JOHNSON CIRCUMCONIC wrt INCIRCLE

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

X(34956) lies on these lines: {1,1361}, {30,553}, {222,6759}, {389,17102}, {991,20764}, {999,4306}, {1214,15644}, {1425,6906}, {1465,10110}, {1598,34042}, {2810,5399}, {2829,20617}

X(34956) = {X(17102), X(20122)}-harmonic conjugate of X(389)

X(34957) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt JOHNSON CIRCUMCONIC

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

X(34957) lies on the line {5,29958}

X(34958) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt INCIRCLE

Barycentrics (b-c)*(2*a^3+2*c*a*b-(b+c)*a^2-(b^2-c^2)*(b-c)) : :
X(34958) = 3*X(1638)-X(1734) = X(4905)-3*X(30724) = 3*X(14413)+X(21118)

X(34958) lies on these lines: {1,7178}, {495,21051}, {512,942}, {513,34954}, {514,676}, {522,14353}, {523,8043}, {525,4458}, {667,23770}, {900,23815}, {999,4367}, {1019,3333}, {1638,1734}, {2775,5583}, {2826,3669}, {2977,31288}, {3309,3676}, {3800,4369}, {3900,21188}, {3910,20517}, {3960,6362}, {4040,21104}, {4129,21620}, {4151,17069}, {4170,4897}, {4401,6084}, {4449,10015}, {4504,28533}, {4823,29278}, {4843,21192}, {4874,29288}, {4905,30724}, {4990,23875}, {5216,11021}, {14077,14837}, {14205,23599}, {14413,21118}

X(34958) = midpoint of X(i) and X(j) for these {i,j}: {1, 7178}, {667, 23770}, {3669, 21185}, {4040, 21104}, {4170, 4897}, {4449, 10015}
X(34958) = reflection of X(2977) in X(31288)
X(34958) = X(7178)-of-anti-Aquila triangle
X(34958) = X(15451)-of-incircle-circles triangle
X(34958) = X(18314)-of-inverse-in-incircle triangle

X(34959) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt KIEPERT HYPERBOLA

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

X(34959) lies on the line {115,28473}

X(34960) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt MACBEATH CIRCUMCONIC

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

X(34960) lies on these lines: {222,22131}, {2194,7083}, {5452,34381}

X(34961) = CENTER OF THIS POLAR RECIPROCAL CONIC: INCIRCLE wrt STAMMLER HYPERBOLA

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

X(34961) lies on the line {110,6003}

X(34961) = X(2)-Ceva conjugate of X(5546)
X(34961) = X(31)-complementary conjugate of X(5546)
X(34961) = center of the conic {{A, B, C, X(4570), X(5379), X(6064), X(6083)}}

X(34962) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt NINE-POINT CIRCLE

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

X(34962) lies on these lines: {5,513}, {1510,34963}

X(34963) = CENTER OF THIS POLAR RECIPROCAL CONIC: JERABEK HYPERBOLA wrt NINE-POINT CIRCLE

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

X(34963) lies on these lines: {5,523}, {924,34967}, {1510,34962}, {13406,14809}

X(34964) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt NINE-POINT CIRCLE

Barycentrics ((b^2+c^2)*a^6-2*(b^4+b^2*c^2+c^4)*a^4-(b^2-c^2)^2*b^2*c^2+(b^6+c^6)*a^2)*(b^2-c^2) : :
X(34964) = X(22089)-5*X(31279)

X(34964) lies on these lines: {5,512}, {427,16229}, {523,2072}, {1510,34962}, {1594,14618}, {2450,3566}, {2524,5254}, {3613,8430}, {3767,22159}, {5133,5996}, {7668,34953}, {10224,23105}, {22089,31279}

X(34964) = complementary conjugate of X(17423)
X(34964) = pole of the trilinear polar of X(2623) wrt Kiepert hyperbola

X(34965) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt NINE-POINT CIRCLE

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

X(34965) lies on these lines: {2,3}, {53,26905}, {216,6747}, {511,34836}, {1853,17849}, {10184,14769}

X(34965) = orthocentroidal circle-inverse of X(6641)
X(34965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 6641), (404, 33198, 297), (425, 29413, 376), (856, 17553, 2), (3128, 19702, 29), (4187, 27534, 413), (6678, 11292, 409), (6940, 33840, 413), (6981, 33197, 30), (6984, 11288, 24), (6998, 15221, 404), (7528, 23336, 2), (10989, 19315, 29), (11101, 15214, 2), (11286, 33005, 404), (11320, 26602, 377), (15158, 26662, 30), (17555, 27177, 404), (25401, 25766, 21), (27600, 32991, 410), (28799, 30880, 402)

X(34966) = CENTER OF THIS POLAR RECIPROCAL CONIC: NINE-POINT CIRCLE wrt MACBEATH CIRCUMCONIC

Barycentrics a^2*(a^8+8*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4+c^4)*(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
X(34966) = X(12429)-3*X(16072)

X(34966) lies on these lines: {2,8548}, {3,19460}, {5,14457}, {6,6387}, {25,110}, {30,155}, {140,19458}, {323,1370}, {389,1147}, {394,1368}, {511,1660}, {2063,30771}, {3292,8538}, {5504,25487}, {5876,12301}, {6101,9908}, {6193,18531}, {6241,12164}, {6391,26206}, {9306,34382}, {10602,19588}, {12421,18952}, {12429,16072}, {15066,26913}, {26958,32282}

X(34966) = midpoint of X(i) and X(j) for these {i,j}: {6193, 18531}, {10602, 19588}, {12164, 21312}
X(34966) = reflection of X(6644) in X(1147)
X(34966) = pole of the trilinear polar of X(9307) wrt MacBeath circumconic
X(34966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (155, 12118, 32139), (1993, 3167, 19139)

X(34967) = CENTER OF THIS POLAR RECIPROCAL CONIC: STAMMLER HYPERBOLA wrt NINE-POINT CIRCLE

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

X(34967) lies on these lines: {5,1510}, {523,2072}, {924,34963}, {1594,23290}

X(34968) = CENTER OF THIS POLAR RECIPROCAL CONIC: NINE-POINT CIRCLE wrt STAMMLER HYPERBOLA

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

X(34968) lies on these lines: {110,1291}, {3050,32661}, {5663,34191}

X(34969) = CENTER OF THIS POLAR RECIPROCAL CONIC: JERABEK HYPERBOLA wrt FEUERBACH HYPERBOLA

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

X(34969) lies on these lines: {5,24025}, {11,244}, {496,12005}, {1479,20803}, {6284,22342}, {8226,24014}, {14010,24026}

X(34969) = barycentric product X(i)*X(j) for these {i, j}: {1146, 17220}, {1730, 24026}
X(34969) = barycentric quotient X(1730)/X(7045)
X(34969) = trilinear product X(1146)*X(1730)
X(34969) = trilinear quotient X(1730)/X(1262)

X(34970) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt JERABEK HYPERBOLA

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

X(34970) lies on the line {125,526}

X(34971) = CENTER OF THIS POLAR RECIPROCAL CONIC: JOHNSON CIRCUMCONIC wrt FEUERBACH HYPERBOLA

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

X(34971) lies on these lines: {11,6129}, {34949,34974}

X(34972) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt JOHNSON CIRCUMCONIC

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

X(34972) lies on these lines: {5,521}, {6368,18314}, {8677,8819}

X(34973) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt KIEPERT HYPERBOLA

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

X(34973) lies on these lines: {12,21891}, {115,804}, {868,16732}

X(34973) = barycentric product X(1109)*X(18714)
X(34973) = trilinear product X(115)*X(18714)

X(34974) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt FEUERBACH HYPERBOLA

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

X(34974) lies on the line {34949,34971}

X(34975) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt MACBEATH CIRCUMCONIC

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

X(34975) lies on these lines: {3569,6753}, {20796,22160}

X(34975) = pole of the trilinear polar of X(1783) wrt MacBeath circumconic

X(34976) = CENTER OF THIS POLAR RECIPROCAL CONIC: STAMMLER HYPERBOLA wrt FEUERBACH HYPERBOLA

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

X(34976) lies on these lines: {11,16586}, {10391,15569}

X(34977) = CENTER OF THIS POLAR RECIPROCAL CONIC: FEUERBACH HYPERBOLA wrt STAMMLER HYPERBOLA

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

X(34977) lies on these lines: {1,6596}, {3,2778}, {11,16586}, {37,17719}, {109,1214}, {191,6126}, {244,1962}, {405,1718}, {518,8758}, {676,2804}, {942,5496}, {960,34586}, {1001,1421}, {1411,1793}, {1465,5087}, {1772,3812}, {2836,18210}, {3185,20254}, {3740,25091}, {3743,11281}, {3752,33135}, {4458,6370}, {4551,24433}, {4552,26095}, {4646,26727}, {4858,25493}, {5468,16734}, {5836,24028}, {5972,6132}, {6690,16577}, {8143,11263}, {9640,25440}, {17080,24703}, {17768,18593}, {21098,21891}, {28606,33148}

X(34977) = {X(16578), X(24025)}-harmonic conjugate of X(3035)

X(34978) = CENTER OF THIS POLAR RECIPROCAL CONIC: JOHNSON CIRCUMCONIC wrt JERABEK HYPERBOLA

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

X(34978) lies on these lines: {5,34810}, {98,250}, {115,2489}, {125,520}, {338,523}, {385,858}, {403,6761}, {2072,3564}, {2452,9308}, {8901,12079}, {34950,34982}

X(34978) = crosssum of X(15460) and X(15461)
X(34978) = isotomic conjugate of the isogonal conjugate of X(34982)
X(34978) = crosspoint of X(1312) and X(1313)
X(34978) = barycentric product X(i)*X(j) for these {i, j}: {76, 34982}, {338, 13198}
X(34978) = trilinear product X(i)*X(j) for these {i, j}: {75, 34982}, {158, 34950}, {1109, 13198}

X(34979) = CENTER OF THIS POLAR RECIPROCAL CONIC: JERABEK HYPERBOLA wrt JOHNSON CIRCUMCONIC

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

X(34979) lies on these lines: {5,520}, {523,34987}, {6368,18314}

X(34979) = barycentric product X(525)*X(6663)
X(34979) = trilinear product X(656)*X(6663)

X(34980) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt JERABEK HYPERBOLA

Barycentrics    a^4*(-a^2+b^2+c^2)^3*(b^2-c^2)^2 : :
Barycentrics    (tan A) (sec^2 B - sec^2 C)^2 : :
Trilinears    (cos^3 A) (b^2 - c^2)^2 : :

X(34980) lies on the orthic inconic and these lines: {3,1942}, {4,15352}, {6,1987}, {51,6749}, {115,130}, {125,526}, {184,1576}, {185,1205}, {287,34383}, {520,15526}, {822,2638}, {1298,16813}, {1562,8754}, {1974,14642}, {2871,6467}, {2878,3270}, {2882,10602}, {3269,9409}, {4558,17974}, {5167,15341}, {6524,32319}, {9512,18338}, {10762,11596}

X(34980) = isogonal conjugate of the isotomic conjugate of X(2972)
X(34980) = isogonal conjugate of the polar conjugate of X(3269)
X(34980) = barycentric product X(i)*X(j) for these {i, j}: {3, 3269}, {6, 2972}, {48, 2632}, {115, 1092}
X(34980) = barycentric quotient X(i)/X(j) for these (i, j): (25, 34538), (32, 32230), (48, 23999), (125, 18027)
X(34980) = trilinear product X(i)*X(j) for these {i, j}: {31, 2972}, {33, 1363}, {34, 7065}, {48, 3269}
X(34980) = trilinear quotient X(i)/X(j) for these (i, j): (3, 23999), (19, 34538), (25, 24021), (31, 32230), (48, 23582)
X(34980) = intersection, other than A,B,C, of conics {{A, B, C, X(418), X(3134)}} and {{A, B, C, X(577), X(15526)}}
X(34980) = center of the conic {{A, B, C, X(19209), X(23606), X(32319)}}
X(34980) = pole of the trilinear polar of X(4) wrt Jerabek hyperbola
X(34980) = orthic-isogonal conjugate of X(647)
X(34980) = orthic-isotomic conjugate of X(512)
X(34980) = polar conjugate of isotomic conjugate of X(35071)
X(34980) = polar conjugate of barycentric square of X(6528)
X(34980) = X(4)-Ceva conjugate of X(647)
X(34980) = perspector of orthic triangle and tangential triangle of Jerabek hyperbola
X(34980) = trilinear pole, wrt orthic triangle, of line X(4)X(51)
X(34980) = X(i)-isoconjugate of X(j) for these {i,j}: {63, 34538}, {69 ,24021}, {75, 32230}, {92, 23582}
X(34980) = crossdifference of every pair of points on line X(648)X(1625)
X(34980) = X(190)-of-orthic-triangle if ABC is acute
X(34980) = crosssum of MacBeath circumconic intercepts of Euler line

X(34981) = CENTER OF THIS POLAR RECIPROCAL CONIC: JERABEK HYPERBOLA wrt KIEPERT HYPERBOLA

Barycentrics (b^2-c^2)^2*((b^2+c^2)*a^2-b^4+b^2*c^2-c^4) : :
X(34981) = 3*X(1989)-X(9512)

X(34981) lies on these lines: {4,1576}, {5,34990}, {30,22463}, {115,804}, {125,137}, {136,6132}, {247,526}, {338,868}, {427,2493}, {523,8754}, {546,6593}, {1879,34845}, {1989,9512}, {2165,32654}, {2980,14601}, {3014,25051}, {3163,15358}, {3566,34984}, {3613,11672}, {22515,24975}, {23327,32740}

X(34981) = barycentric product X(i)*X(j) for these {i, j}: {115, 7752}, {338, 3060}, {1109, 18041}
X(34981) = trilinear product X(i)*X(j) for these {i, j}: {115, 18041}, {1109, 3060}
X(34981) = pole of the trilinear polar of X(2165) wrt Kiepert hyperbola

X(34982) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt JERABEK HYPERBOLA

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

X(34982) lies on these lines: {115,512}, {125,8673}, {1691,2393}, {9517,34953}, {14965,21639}, {34950,34978}

X(34982) = isogonal conjugate of the isotomic conjugate of X(34978)
X(34982) = barycentric product X(i)*X(j) for these {i, j}: {6, 34978}, {115, 13198}, {393, 34950}
X(34982) = trilinear product X(i)*X(j) for these {i, j}: {31, 34978}, {1096, 34950}

X(34983) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt JOHNSON CIRCUMCONIC

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

X(34983) lies on these lines: {5,525}, {520,34987}, {5899,30210}, {6368,18314}, {15451,17434}

X(34983) = midpoint of X(15451) and X(17434)
X(34983) = barycentric product X(i)*X(j) for these {i, j}: {5, 17434}, {216, 6368}, {343, 15451}, {418, 18314}
X(34983) = barycentric quotient X(i)/X(j) for these (i, j): (51, 16813), (216, 18831), (217, 933), (418, 18315)
X(34983) = trilinear product X(418)*X(2618)
X(34983) = trilinear quotient X(i)/X(j) for these (i, j): (1087, 6528), (1953, 16813)
X(34983) = pole of the trilinear polar of X(110) wrt Johnson circumconic

X(34984) = CENTER OF THIS POLAR RECIPROCAL CONIC: JOHNSON CIRCUMCONIC wrt KIEPERT HYPERBOLA

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

X(34984) lies on these lines: {3566,34981}, {34953,34988}

X(34985) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt JOHNSON CIRCUMCONIC

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

X(34985) lies on these lines: {5,389}, {51,3078}, {418,1495}, {11793,15780}

X(34986) = CENTER OF THIS POLAR RECIPROCAL CONIC: JOHNSON CIRCUMCONIC wrt MACBEATH CIRCUMCONIC

Barycentrics a^2*(2*a^4-3*(b^2+c^2)*a^2+b^4+c^4) : :
X(34986) = X(22)-3*X(184) = X(22)+3*X(1993) = 5*X(22)-9*X(6800) = 3*X(51)-X(27365) = 5*X(184)-3*X(6800) = 5*X(1993)+3*X(6800) = 3*X(11430)-2*X(18570)

X(34986) lies on these lines: {1,20959}, {2,575}, {3,13382}, {5,539}, {6,1196}, {22,184}, {23,21969}, {24,16625}, {25,576}, {39,14153}, {43,20958}, {49,52}, {51,110}, {54,5562}, {97,26907}, {101,23165}, {140,32136}, {154,1351}, {155,578}, {156,5446}, {182,394}, {185,2071}, {186,14831}, {193,21637}, {323,3917}, {343,5965}, {373,34545}, {389,1147}, {399,16194}, {419,7760}, {427,542}, {436,648}, {524,6676}, {567,5891}, {568,9703}, {569,11793}, {572,22139}, {631,11423}, {970,1437}, {1092,7592}, {1154,18475}, {1181,13346}, {1216,32046}, {1352,11427}, {1353,5972}, {1368,8550}, {1495,3060}, {1568,12022}, {1570,3981}, {1613,1692}, {1619,34779}, {1660,34117}, {1691,3787}, {1915,5052}, {1992,6353}, {1995,15004}, {2003,7193}, {2323,3955}, {2979,11003}, {3043,21649}, {3047,13417}, {3051,20976}, {3098,3796}, {3193,15488}, {3270,9637}, {3284,6638}, {3518,9705}, {3564,21243}, {3574,14516}, {3629,10192}, {3845,5609}, {5007,11328}, {5050,15082}, {5065,23163}, {5067,13472}, {5093,8780}, {5133,18553}, {5406,12974}, {5407,12975}, {5422,5651}, {5448,12370}, {5449,32358}, {5476,6997}, {5504,12227}, {5640,34565}, {5654,18390}, {5889,9545}, {6000,13352}, {6090,10601}, {6102,12038}, {6243,9704}, {6467,11416}, {6593,8584}, {6636,23061}, {6759,13598}, {6776,7396}, {6995,20423}, {7386,11179}, {7391,11645}, {7483,18646}, {7485,20190}, {7488,9706}, {7526,15083}, {7539,11178}, {9730,15087}, {9738,10133}, {9739,10132}, {9777,15520}, {9820,13292}, {9909,11477}, {10110,10539}, {10111,32743}, {10114,23306}, {10116,13371}, {10619,12225}, {10984,13348}, {11064,11245}, {11206,31670}, {11284,22234}, {11424,11441}, {11425,12164}, {11426,17814}, {11430,13754}, {11451,34566}, {11511,32621}, {11572,34799}, {11935,32110}, {12160,19357}, {12834,16042}, {13142,16252}, {13335,34396}, {13403,22660}, {13474,32139}, {13482,13596}, {13857,31101}, {14561,14826}, {14627,18350}, {15012,17928}, {15030,15033}, {15448,19155}, {15531,21639}, {15644,16266}, {15872,31807}, {16187,17825}, {16473,23841}, {18917,23329}, {22128,26889}, {25738,32767}, {26864,33586}, {30739,33749}, {31802,34782}

X(34986) = midpoint of X(i) and X(j) for these {i,j}: {184, 1993}, {13352, 18445}
X(34986) = reflection of X(21243) in X(23292)
X(34986) = isogonal conjugate of the polar conjugate of X(27377)
X(34986) = barycentric product X(3)*X(27377)
X(34986) = trilinear product X(48)*X(27377)
X(34986) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11422, 13366), (2, 13366, 575), (6, 2056, 1196), (6, 3167, 9306), (6, 9306, 5943), (25, 576, 21849), (49, 52, 10282), (49, 195, 52), (51, 1994, 5097), (110, 1994, 51), (155, 578, 5907), (182, 394, 3819), (323, 5012, 3917), (394, 11402, 182), (1092, 7592, 9729), (1147, 12161, 389), (3292, 11422, 575), (3292, 13366, 2), (3917, 5012, 5092), (9716, 11422, 3292)

X(34987) = CENTER OF THIS POLAR RECIPROCAL CONIC: STAMMLER HYPERBOLA wrt JOHNSON CIRCUMCONIC

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

X(34987) lies on these lines: {5,6368}, {520,34983}, {523,34979}, {2070,30210}

X(34987) = pole of the trilinear polar of X(18315) wrt Johnson circumconic

X(34988) = CENTER OF THIS POLAR RECIPROCAL CONIC: MACBEATH CIRCUMCONIC wrt KIEPERT HYPERBOLA

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

X(34988) lies on these lines: {115,6587}, {34953,34984}

X(34989) = CENTER OF THIS POLAR RECIPROCAL CONIC: STAMMLER HYPERBOLA wrt KIEPERT HYPERBOLA

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

X(34989) lies on these lines: {2,13582}, {5,542}, {110,10276}, {115,34990}, {141,9722}, {338,18122}, {662,5949}, {9512,25329}, {10264,10277}, {14061,24975}, {15833,17045}

X(34989) = {X(5949), X(7332)}-harmonic conjugate of X(662)

X(34990) = CENTER OF THIS POLAR RECIPROCAL CONIC: KIEPERT HYPERBOLA wrt STAMMLER HYPERBOLA

Barycentrics a^2*((b^2+c^2)*a^4-2*(b^4+c^4)*a^2+b^6+c^6) : :
X(34990) = 3*X(2)+X(14570) = X(237)+3*X(22087) = X(1634)-3*X(9155) = X(3001)-3*X(22087) = 3*X(9155)+X(20975)

X(34990) lies on these lines: {2,94}, {3,1177}, {5,34981}, {6,2987}, {39,597}, {50,22151}, {110,7669}, {115,34989}, {141,216}, {160,17710}, {237,3001}, {297,11062}, {524,3003}, {570,3589}, {599,18573}, {620,2492}, {800,3629}, {1112,23217}, {1511,5961}, {1634,2854}, {1994,13338}, {2023,34294}, {2491,14417}, {2782,7668}, {3053,28710}, {3447,30717}, {3618,13351}, {4422,13006}, {4577,5152}, {5013,32740}, {5024,28662}, {5026,5661}, {5421,6329}, {5467,22085}, {5972,6132}, {7778,22240}, {7784,10718}, {7795,34897}, {8266,20819}, {8553,20806}, {9145,22143}, {9737,19136}, {10717,15302}, {11077,18315}, {11258,11637}, {11328,16776}, {11746,23181}, {14687,33851}, {14838,16578}, {14919,17811}, {14961,32459}, {14981,25328}, {14984,18114}, {15345,32205}, {16308,22110}, {17467,21825}, {18860,32217}

X(34990) = midpoint of X(i) and X(j) for these {i,j}: {237, 3001}, {338, 14570}, {1634, 20975}
X(34990) = isotomic conjugate of the polar conjugate of X(1112)
X(34990) = polar conjugate of the isogonal conjugate of X(23217)
X(34990) = complement of X(338)
X(34990) = barycentric product X(i)*X(j) for these {i, j}: {37, 16734}, {69, 1112}, {264, 23217}
X(34990) = barycentric quotient X(1112)/X(4)
X(34990) = trilinear product X(i)*X(j) for these {i, j}: {42, 16734}, {63, 1112}, {92, 23217}
X(34990) = trilinear quotient X(1112)/X(19)
X(34990) = intersection, other than A,B,C, of conics {{A, B, C, X(94), X(1112)}} and {{A, B, C, X(249), X(338)}}
X(34990) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 14570, 338), (110, 14060, 7669), (237, 22087, 3001)

X(34991) = X(960)X(1706) ∩ X(3687)X(4461)

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

Let (A_b, A_c) be the centers of the polar reciprocal conics of the B-excircle and C-excircle wrt to the A-excircle and define (B_c, B_a), (C_a, C_b) cyclically. The triangle bounded by the lines (A_b, A_c), (B_c, B_a), (C_a, C_b) is perspective to ABC with perspector X(34991). Also, these six points lie on a conic. (César Lozada, Nov. 24, 2019)

X(34991) lies on these lines: {960,1706}, {3687,4461}, {17185,24557}

X(34991) = lies on the circumconic with center X(17419))
X(34991) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(312)}} and {{A, B, C, X(2), X(3062)}}

Perspectors involving circlecevian triangles: X(34992)-X(34999)

This preamble and centers X(34992)-X(34999) were by Clark Kimberling and Peter Moses, November 24, 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, in which the line AP meets the circle (PBC). 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); see the preambles just before X(34892) and X(34914).

The A-vertex of the circlecevian triangle of P is given by A' = -a^2 q r (p + q + r) : q (c^2 p q + b^2 p r + a^2 q r) : r (c^2 p q + b^2 p r + a^2 q r).

Examples:
If P = X(1), then A'B'C' = excentral triangle.
If P = X(4), then A' = reflection of A in BC, and B' and C' are defined cyclically.
If P = X(2), then A' = -3 a^2 : a^2 + b^2 + c^2 : a^2 + b^2 + c^2.
If P = X(3), then A' = (a^2 + b^2 - c^2)(a^2 - b^2 + c^2) : b^2 (-a^2 + b^2 - c^2), -c^2 (a^2 + b^2 - c^2).
If P = X(6), then A' = a^2 + b^2 + c^2 : -3 b^2 : -3 c^2.
If P = X(31), then A' = -b*c*(a^3 + b^3 + c^3) : b^3*(a*b + a*c + b*c), c^3*(a*b + a*c + b*c).
If P = X(75), then A' = -a^2*(a*b + a*c + b*c) : c*(a^3 + b^3 + c^3) : b*(a^3 + b^3 + c^3).
If P = X(76), then A' = -a^2*(a^2*b^2 + a^2*c^2 + b^2*c^2) : c^2*(a^4 + b^4 + c^4) : b^2*(a^4 + b^4 + c^4).

The circlecevian triangle of an arbitrary point P = p : q : r is perspective to the excentral triangle. The perspector, P', is given by

P' = a ( a^3 q^2 r^2 - b^3 p^2 r^2 -c^3 p^2 q^2 + a b c p^2 q r + a b c p q^2 r + a b c p q r^2 - b^2 c p^2 q r - b c^2 p^2 q r - a^2 b p q r^2 - a^2 c p q^2 r + a c^2 p q^2 r + a b^2 p q r^2 ) : :
(Vu Thanh Tung, November 25, 2019)

X(34992) = PERSPECTOR OF THESE TRIANGLES: ARA AND CIRCLECEVIAN OF X(2)

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+a^6 b^2 c^2+6 a^4 b^4 c^2+3 a^2 b^6 c^2-5 b^8 c^2+2 a^6 c^4+6 a^4 b^2 c^4-24 a^2 b^4 c^4-10 b^6 c^4-2 a^4 c^6+3 a^2 b^2 c^6-10 b^4 c^6-3 a^2 c^8-5 b^2 c^8-c^10) : :

X(34992) lies on these lines: {3,9465}, {23,7767}, {25,599}, {7822,11284}

X(34993) = PERSPECTOR OF THESE TRIANGLES: 2ND EXCOSINE AND CIRCLECEVIAN OF X(3)

Barycentrics a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^16+4 a^14 b^2-40 a^12 b^4+100 a^10 b^6-110 a^8 b^8+44 a^6 b^10+16 a^4 b^12-20 a^2 b^14+5 b^16+4 a^14 c^2+40 a^12 b^2 c^2-84 a^10 b^4 c^2-128 a^8 b^6 c^2+412 a^6 b^8 c^2-312 a^4 b^10 c^2+52 a^2 b^12 c^2+16 b^14 c^2-40 a^12 c^4-84 a^10 b^2 c^4+476 a^8 b^4 c^4-456 a^6 b^6 c^4-48 a^4 b^8 c^4+220 a^2 b^10 c^4-68 b^12 c^4+100 a^10 c^6-128 a^8 b^2 c^6-456 a^6 b^4 c^6+688 a^4 b^6 c^6-252 a^2 b^8 c^6+48 b^10 c^6-110 a^8 c^8+412 a^6 b^2 c^8-48 a^4 b^4 c^8-252 a^2 b^6 c^8-2 b^8 c^8+44 a^6 c^10-312 a^4 b^2 c^10+220 a^2 b^4 c^10+48 b^6 c^10+16 a^4 c^12+52 a^2 b^2 c^12-68 b^4 c^12-20 a^2 c^14+16 b^2 c^14+5 c^16) : :

X(34993) lies on these lines: {20,3183}, {25,64}, {207,4319}, {577,1033}, {1885,6525}, {4320,7007}

X(34994) = PERSPECTOR OF THESE TRIANGLES: LUCAS (1)HOMOTHETIC AND CIRCLECEVIAN OF X(6)

Barycentrics a^2 (5 a^8-12 a^6 b^2+2 a^4 b^4+12 a^2 b^6-7 b^8-12 a^6 c^2-132 a^4 b^2 c^2+196 a^2 b^4 c^2-20 b^6 c^2+2 a^4 c^4+196 a^2 b^2 c^4-26 b^4 c^4+12 a^2 c^6-20 b^2 c^6-7 c^8-8 (a^2+b^2+c^2) (3 a^4-2 a^2 b^2+b^4-2 a^2 c^2-10 b^2 c^2+c^4) S) : :

X(34994) lies on these lines: {493,574}, {1992,6462}, {6449,6465}

X(34995) = PERSPECTOR OF THESE TRIANGLES: LUCAS (-1)HOMOTHETIC AND CIRCLECEVIAN OF X(6)

Barycentrics a^2 ((5 a^8-12 a^6 b^2+2 a^4 b^4+12 a^2 b^6-7 b^8-12 a^6 c^2-132 a^4 b^2 c^2+196 a^2 b^4 c^2-20 b^6 c^2+2 a^4 c^4+196 a^2 b^2 c^4-26 b^4 c^4+12 a^2 c^6-20 b^2 c^6-7 c^8)+8 (a^2+b^2+c^2) (3 a^4-2 a^2 b^2+b^4-2 a^2 c^2-10 b^2 c^2+c^4) S) : :

X(34995) lies on these lines: {494,574}, {1992,6463}, {6450,6466}

X(34996) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND CIRCLECEVIAN OF X(31)

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

X(34996) lies on these lines: {1,75}, {99,4093}, {662,8625}, {718,7257}, {758,5539}, {2388,7170}, {3747,7315}

X(34997) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND CIRCLECEVIAN OF X(75)

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

X(34997) lies on these lines: {1,21}, {9,29856}, {57,29858}, {518,1283}, {740,21381}, {1281,3006}, {2245,3509}, {3145,5904}, {3218,29632}, {3219,4425}, {5196,13174}, {5536,8229}, {7075,21366}, {8235,26921}, {9791,29829}, {24697,29861}, {25557,29860}

X(34997) = Gibert-Burek-Moses concurrent circles image of X(31)

X(34998) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND CIRCLECEVIAN OF X(76)

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

X(34998) lies on this line: {1,32}

X(34999) = PERSPECTOR OF THESE TRIANGLES: ANTI-1ST-BROCARD AND CIRCLECEVIAN OF X(76)

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

X(34999) lies on these lines: {2,694}, {4,19910}, {76,8178}, {99,8149}, {385,2458}, {3407,8290}, {5152,9983}, {5989,9865}, {16984,18806}, {17004,32189}

Secondary pre-circumcevian-inversion points: X(35000)-X(35002)

This preamble and centers X(35000)-X(35002) were contributed by Peter Moses, November 25, 2019.

Suppose that a point X is the circumcevian-inversion perspector of a point P = p : q : r, as in the preamble just before X(34864). There is a second point, Q, of which X is the circumcevian-inversion perspector of Q. The point Q, here named the secondary pre-circumcevian-inversion point of P, is given by

Q = a^2 (b^2 c^2 (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) p^2+c^2 (a^6+a^4 b^2+a^2 b^4-3 b^6-3 a^4 c^2+5 b^4 c^2+3 a^2 c^4-b^2 c^4-c^6) p q+3 a^2 b^2 c^2 (a^2-b^2-c^2) q^2+b^2 (a^6-3 a^4 b^2+3 a^2 b^4-b^6+a^4 c^2-b^4 c^2+a^2 c^4+5 b^2 c^4-3 c^6) p r+a^2 (a^2-b^2-c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2+4 b^2 c^2+c^4) q r+3 a^2 b^2 c^2 (a^2-b^2-c^2) r^2) : : .

Let Cip(P) denote the circumcevian-inversion perspector of a point P.

Example 1: Cip(X(1)) = X(35), the circumcevian-inversion perspector of X(1); also, Cip(X(35000)) = X(35), so that X(35000) is the secondary pre-circumcevian-inversion point of X(1).

Example 2: Cip(X(2)) = X(7496), the circumcevian-inversion perspector of X(2); also, Cip(X(35001)) = X(7496), so that X(35001) is the secondary pre-circumcevian-inversion point of X(2).

Example 3: Cip(X(4)) = X(3520), the circumcevian-inversion perspector of X(4); also, Cip(X(18859)) = X(3520), so that X(18859) is the secondary pre-circumcevian-inversion point of X(4).

Example 4: Cip(X(6)) = X(574), the circumcevian-inversion perspector of X(6); also, Cip(X(35002)) = X(574), so that X(35002) is the secondary pre-circumcevian-inversion point of X(6).

Example 5: Cip(X(20)) = X(7488), the circumcevian-inversion perspector of X(20); also, Cip(X(2070)) = X(788), so that X(2070) is the secondary pre-circumcevian-inversion point of X(20).

The appearance of (i,j,k) in the following list means that for the circumcevian-inversion perspector X(i), the two "source points" are

X(j) = pre-circumcevian-inversion perspector of X(i)
X(k) = secondary pre-circumcevian-inversion perspector of X(i).

(35,1,35000), (7496,2,35001), (3520,4,18859), (574,6,35002), (7488,20,2070), (7492,23,3534), (7280,36,12702), (11012,40,22765), (14379,64,6760), (21844,186,1657), (5210,187,33878), (8722,1350,2080), (15020,15054,32609), (351,9130,35447), (1319,11510,35448), (54,25042,35449), (34147,33924,35450), (8,34758,35451), (5,34864,35452), (115,34866,35453), (9,34867,35454), (32,34870,35456), (35,34871,35457), (39,34873,35458), (55,34879,35459), (56.34880,35460) (57,34881,35461), (58,34882,35462), (69,34883,35463), (76,34885,35464), (35372,35373,35465)

X(35000) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(1)

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 + 2*a^2*b^2*c - 3*a*b^3*c - b^4*c - 2*a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 - 3*a*b*c^3 + 2*b^2*c^3 + a*c^4 - b*c^4 - c^5) : :
X(35000) = 3 X[3] - 2 X[36], X[3] + 2 X[5537], 5 X[3] - 4 X[23961], X[36] - 3 X[2077], X[36] + 3 X[5537], 4 X[36] - 3 X[22765], 5 X[36] - 6 X[23961], 3 X[165] - X[5535], 3 X[165] - 2 X[10225], 3 X[376] - X[20067], 3 X[381] - 4 X[3814], 3 X[1482] - 4 X[5048], 4 X[2077] - X[22765], 5 X[2077] - 2 X[23961], 9 X[5054] - 8 X[6681], 9 X[5055] - 10 X[31263], X[5183] - 3 X[13528], 4 X[5537] + X[22765], 5 X[5537] + 2 X[23961], 9 X[10246] - 8 X[25405], 5 X[22765] - 8 X[23961]

X(35000) lies on the cubic K905 and these lines: {1, 3}, {8, 26321}, {10, 13743}, {20, 32141}, {30, 100}, {72, 13465}, {74, 901}, {104, 5844}, {109, 23071}, {140, 5284}, {197, 12083}, {376, 20067}, {381, 1376}, {382, 11499}, {399, 12327}, {404, 22791}, {474, 18493}, {495, 28458}, {500, 33771}, {515, 12331}, {519, 12773}, {535, 3534}, {547, 9342}, {549, 1621}, {550, 11491}, {692, 22115}, {758, 12515}, {759, 2743}, {912, 17613}, {952, 6909}, {953, 28218}, {956, 18515}, {962, 6924}, {993, 3654}, {1001, 5054}, {1012, 5790}, {1259, 18518}, {1290, 5951}, {1324, 12778}, {1436, 22147}, {1532, 12775}, {1537, 6905}, {1597, 1878}, {1618, 17976}, {1656, 3841}, {1657, 11500}, {1770, 16150}, {2070, 20872}, {2699, 29151}, {2703, 29300}, {2742, 28471}, {2932, 4511}, {3158, 7171}, {3526, 8167}, {3652, 3678}, {3655, 25439}, {3830, 18491}, {3871, 34773}, {3913, 18526}, {4413, 5055}, {4423, 15694}, {4428, 15693}, {5057, 6985}, {5123, 9709}, {5176, 5687}, {5180, 6361}, {5251, 28453}, {5260, 31649}, {5274, 6891}, {5293, 5492}, {5326, 15908}, {5657, 6914}, {5690, 6906}, {5762, 30295}, {5841, 24466}, {5842, 10993}, {5899, 20989}, {5901, 6940}, {6199, 19000}, {6211, 24436}, {6395, 18999}, {6840, 13199}, {6842, 31777}, {6850, 8164}, {6882, 10738}, {6903, 20066}, {6923, 10590}, {6942, 20070}, {6958, 10589}, {6971, 10525}, {7489, 26446}, {8168, 12114}, {8299, 13633}, {8715, 18481}, {9037, 12329}, {9655, 11501}, {9668, 11502}, {9708, 28444}, {10620, 13204}, {11813, 12699}, {12178, 13188}, {12188, 13173}, {12245, 32153}, {12307, 12341}, {12315, 12335}, {12340, 13310}, {13115, 13206}, {13411, 16004}, {16117, 21077}, {16548, 26744}, {17796, 19302}, {18357, 21669}, {19549, 27639}, {20075, 34745}, {28212, 34474}

X(35000) = midpoint of X(i) and X(j) for these {i,j}: {40, 5538}, {2077, 5537}, {5180, 6361}, {6840, 13199}
X(35000) = reflection of X(i) in X(j) for these {i,j}: {3, 2077}, {484, 3579}, {3830, 31160}, {5535, 10225}, {6905, 33814}, {10738, 6882}, {10742, 17757}, {12699, 11813}, {18524, 100}, {18525, 5176}, {22765, 3}
X(35000) = circumcircle-inverse of X(3579)
X(35000) = Bevan-circle-inverse of X(24468)
X(35000) = Stammler-circle-inverse of X(12702)
X(35000) = X(17484)-Ceva conjugate of X(17796)
X(35000) = X(2070)-of-1st-circumperp-triangle
X(35000) = X(18859)-of-2nd-circumperp-triangle
X(35000) = X(37938)-of-excentral-triangle
X(35000) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 8148, 56}, {3, 10247, 10269}, {3, 10306, 1482}, {3, 10679, 10246}, {3, 11248, 11849}, {35, 484, 5172}, {35, 3256, 24929}, {35, 3579, 3}, {36, 5048, 999}, {40, 26285, 3}, {165, 5535, 10225}, {165, 32613, 3}, {1381, 1382, 3579}, {2448, 2449, 24468}, {10310, 11248, 3}, {10310, 14882, 3579}, {10902, 31663, 3}, {11012, 26086, 3}, {18515, 34718, 956}

X(35001) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(2)

Barycentrics a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 17*a^4*b^2*c^2 - 9*a^2*b^4*c^2 - 6*b^6*c^2 - 9*a^2*b^2*c^4 + 14*b^4*c^4 + 2*a^2*c^6 - 6*b^2*c^6 - c^8) : :
X(35001) = 3 X[3] - 2 X[23], 7 X[3] - 6 X[186], 4 X[3] - 3 X[2070], 5 X[3] - 6 X[2071], 5 X[3] - 3 X[5899], 7 X[23] - 9 X[186], 8 X[23] - 9 X[2070], 5 X[23] - 9 X[2071], 10 X[23] - 9 X[5899], 8 X[186] - 7 X[2070], 5 X[186] - 7 X[2071], 10 X[186] - 7 X[5899], 3 X[381] - 4 X[858], 3 X[399] - 4 X[3292], 12 X[403] - 13 X[5079], 8 X[468] - 9 X[5054], 3 X[1351] - 4 X[15826], 4 X[1495] - 5 X[15040], 2 X[1533] - 3 X[14643], 15 X[1656] - 16 X[5159], 5 X[2070] - 8 X[2071], 5 X[2070] - 4 X[5899], 12 X[2072] - 11 X[5072], 3 X[3153] - 2 X[3627], 2 X[3581] - 3 X[15041], 6 X[10540] - 7 X[15039], 4 X[10564] - 3 X[32609], 5 X[12017] - 4 X[32217]

X(35001) lies on these lines: {2, 3}, {195, 10575}, {323, 12308}, {399, 3292}, {511, 10620}, {575, 14855}, {691, 9301}, {999, 5160}, {1154, 15054}, {1351, 15826}, {1495, 15040}, {1503, 32254}, {1533, 14643}, {2930, 11645}, {3295, 7286}, {3527, 16227}, {3581, 15041}, {5024, 16308}, {5562, 33541}, {5650, 14926}, {5663, 23061}, {6033, 34013}, {6781, 11063}, {8705, 33878}, {8717, 14805}, {10540, 15039}, {10564, 32609}, {11793, 33539}, {11820, 11935}, {12017, 32217}, {12041, 15107}, {12121, 12584}, {13233, 14830}, {13391, 13445}, {16111, 29317}, {19924, 32305}

X(35001) = reflection of X(i) in X(j) for these {i,j}: {5899, 2071}, {9301, 691}, {12308, 323}, {15107, 12041}
X(35001) = reflection of X(18325) in the De Longchamps axis
X(35001) = circumcircle-inverse of X(8703)
X(35001) = Stammler-circle-inverse of X(3534)
X(35001) = Thomson-isogonal conjugate of antipode of X(5643) in Thomson-Gibert-Moses hyperbola
X(35001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 23, 37958}, {3, 382, 7545}, {3, 3627, 18369}, {3, 3830, 1995}, {3, 5073, 7530}, {3, 5899, 7575}, {3, 7464, 18859}, {3, 7527, 34864}, {3, 11403, 5079}, {3, 12082, 2937}, {3, 15681, 33532}, {3, 17800, 12082}, {23, 858, 11284}, {378, 33532, 3}, {548, 7550, 3}, {550, 7527, 3}, {858, 18325, 381}, {1113, 1114, 8703}, {2071, 7575, 3}, {3520, 7555, 3}, {3830, 32216, 381}, {7492, 18570, 3}, {7496, 8703, 3}, {7530, 11413, 3}, {7556, 11250, 3}, {12082, 12084, 3}, {12084, 17800, 2937}, {12086, 15704, 3}, {12103, 14865, 3}, {15154, 15155, 3534}, {15156, 15157, 15704}

X(35002) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(6)

Barycentrics    a^2*(a^6 - a^4*b^2 + 2*a^2*b^4 - 2*b^6 - a^4*c^2 + 3*a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4 - 2*c^6) : :
Barycentrics    a^2 (3 SA - S Tan[2 ω]) : :
Barycentrics    Sin[A]^2 (3 Cot[A] - Tan[2 ω]) :
X(35002) = 3 X[3] - 2 X[187], 3 X[3] - X[9301], 2 X[6] - 3 X[2456], 4 X[187] - 3 X[2080], 3 X[376] - X[14712], 3 X[381] - 4 X[625], 5 X[631] - 4 X[14693], 2 X[1513] - 3 X[15561], 3 X[1691] - 4 X[5092], 3 X[2080] - 2 X[9301], 9 X[5055] - 10 X[31275], 6 X[5215] - 7 X[15701], 4 X[6390] - 3 X[8724], 3 X[7809] - X[10722], 4 X[12100] - 3 X[26613], 3 X[13188] - 4 X[15301], 3 X[14041] - 2 X[22515]

X(35002) lies on the cubic K905 and these lines: {3, 6}, {4, 7836}, {5, 7832}, {20, 7900}, {23, 7711}, {30, 99}, {74, 805}, {98, 10290}, {111, 32526}, {140, 7859}, {194, 14880}, {237, 15107}, {323, 5191}, {352, 14660}, {376, 7774}, {381, 625}, {382, 13449}, {384, 14881}, {385, 12042}, {399, 5938}, {524, 9142}, {538, 8178}, {542, 7813}, {548, 12122}, {549, 7792}, {631, 14693}, {691, 14388}, {999, 5148}, {1078, 32521}, {1296, 9831}, {1495, 6660}, {1511, 19576}, {1513, 15561}, {1597, 5140}, {1634, 12367}, {1656, 7915}, {2482, 19924}, {2698, 25424}, {2705, 28563}, {2709, 6323}, {2710, 26714}, {2712, 28469}, {2782, 5999}, {2967, 8744}, {3148, 15066}, {3231, 8569}, {3295, 5194}, {3314, 9996}, {3524, 16989}, {3534, 3849}, {3552, 10349}, {3579, 5184}, {3818, 7801}, {3830, 31173}, {3926, 5207}, {4550, 5167}, {5031, 7795}, {5054, 15482}, {5055, 31275}, {5099, 18325}, {5103, 7789}, {5215, 15701}, {5965, 10991}, {5970, 30254}, {6287, 18358}, {6321, 15980}, {7470, 7783}, {7697, 13860}, {7768, 32151}, {7771, 33706}, {7779, 9862}, {7796, 9873}, {7820, 19130}, {7835, 9993}, {7869, 10356}, {7881, 18500}, {7895, 18503}, {8177, 13085}, {8290, 11676}, {8369, 21850}, {8705, 9145}, {9486, 13192}, {10104, 12251}, {10357, 33021}, {11004, 34396}, {11328, 34417}, {11673, 14096}, {12100, 26613}, {12203, 32448}, {13115, 18859}, {13188, 15301}, {14041, 22515}, {14981, 29012}, {15483, 32459}, {18321, 31952}, {22564, 33273}

X(35002) = midpoint of X(7779) and X(9862)
X(35002) = reflection of X(i) in X(j) for these {i,j}: {382, 13449}, {385, 12042}, {2080, 3}, {3830, 31173}, {5104, 3098}, {5184, 3579}, {6033, 325}, {6321, 15980}, {9301, 187}, {11676, 33813}, {18325, 5099}
X(35002) = circumcircle-inverse of X(3098)
X(35002) = Brocard-circle-inverse of X(26316)
X(35002) = Schoutte-circle-inverse of X(1691)
X(35002) = Stammler-circle-inverse of X(33878)
X(35002) = crossdifference of every pair of points on line {523, 5306}
X(35002) = orthocentroidal-to-ABC similarity image of X(6033)
X(35002) = X(512)-vertex conjugate of X(3098)
X(35002) = reflection of X(2080) in X(3)
X(35002) = reflection of X(3) in X(18860)
X(35002) = X(6033)-of-anti-4th-Brocard-triangle
X(35002) = X(6750)-of-Thomson-triangle
X(35002) = X(18859)-of-Grebe-triangle
X(35002) = X(6761)-of-medial-triangle-of-Thomson-triangle
X(35002) = X(6033)-of-anti-orthocentroidal-triangle; see X(10293)
X(35002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 6, 26316}, {3, 39, 12054}, {3, 3095, 3398}, {3, 9301, 187}, {3, 32447, 182}, {6, 26316, 3398}, {15, 16, 1691}, {39, 574, 12055}, {39, 5162, 1691}, {187, 2031, 1384}, {187, 5107, 2031}, {187, 9301, 2080}, {384, 14881, 18502}, {574, 3098, 3}, {574, 30270, 3098}, {1379, 1380, 3098}, {2076, 5013, 2021}, {2080, 2456, 3398}, {2080, 12054, 1691}, {3095, 26316, 6}, {3098, 9737, 574}, {5104, 12055, 1691}, {8722, 9734, 3}, {9737, 30270, 3}, {14810, 21163, 3}

X(35003) = HUNG-LOZADA-HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics a^2*(a^24*b^2 - 7*a^22*b^4 + 19*a^20*b^6 - 21*a^18*b^8 - 6*a^16*b^10 + 42*a^14*b^12 - 42*a^12*b^14 + 6*a^10*b^16 + 21*a^8*b^18 - 19*a^6*b^20 + 7*a^4*b^22 - a^2*b^24 + a^24*c^2 - 6*a^22*b^2*c^2 + 18*a^20*b^4*c^2 - 42*a^18*b^6*c^2 + 79*a^16*b^8*c^2 - 92*a^14*b^10*c^2 + 28*a^12*b^12*c^2 + 76*a^10*b^14*c^2 - 113*a^8*b^16*c^2 + 66*a^6*b^18*c^2 - 14*a^4*b^20*c^2 - 2*a^2*b^22*c^2 + b^24*c^2 - 7*a^22*c^4 + 18*a^20*b^2*c^4 - 24*a^18*b^4*c^4 + 23*a^16*b^6*c^4 + 28*a^14*b^8*c^4 - 42*a^12*b^10*c^4 - 130*a^10*b^12*c^4 + 244*a^8*b^14*c^4 - 93*a^6*b^16*c^4 - 48*a^4*b^18*c^4 + 34*a^2*b^20*c^4 - 3*b^22*c^4 + 19*a^20*c^6 - 42*a^18*b^2*c^6 + 23*a^16*b^4*c^6 - 114*a^14*b^6*c^6 + 94*a^12*b^8*c^6 + 396*a^10*b^10*c^6 - 508*a^8*b^12*c^6 - 66*a^6*b^14*c^6 + 283*a^4*b^16*c^6 - 78*a^2*b^18*c^6 - 7*b^20*c^6 - 21*a^18*c^8 + 79*a^16*b^2*c^8 + 28*a^14*b^4*c^8 + 94*a^12*b^6*c^8 - 712*a^10*b^8*c^8 + 356*a^8*b^10*c^8 + 664*a^6*b^12*c^8 - 526*a^4*b^14*c^8 - 7*a^2*b^16*c^8 + 45*b^18*c^8 - 6*a^16*c^10 - 92*a^14*b^2*c^10 - 42*a^12*b^4*c^10 + 396*a^10*b^6*c^10 + 356*a^8*b^8*c^10 - 1104*a^6*b^10*c^10 + 298*a^4*b^12*c^10 + 272*a^2*b^14*c^10 - 78*b^16*c^10 + 42*a^14*c^12 + 28*a^12*b^2*c^12 - 130*a^10*b^4*c^12 - 508*a^8*b^6*c^12 + 664*a^6*b^8*c^12 + 298*a^4*b^10*c^12 - 436*a^2*b^12*c^12 + 42*b^14*c^12 - 42*a^12*c^14 + 76*a^10*b^2*c^14 + 244*a^8*b^4*c^14 - 66*a^6*b^6*c^14 - 526*a^4*b^8*c^14 + 272*a^2*b^10*c^14 + 42*b^12*c^14 + 6*a^10*c^16 - 113*a^8*b^2*c^16 - 93*a^6*b^4*c^16 + 283*a^4*b^6*c^16 - 7*a^2*b^8*c^16 - 78*b^10*c^16 + 21*a^8*c^18 + 66*a^6*b^2*c^18 - 48*a^4*b^4*c^18 - 78*a^2*b^6*c^18 + 45*b^8*c^18 - 19*a^6*c^20 - 14*a^4*b^2*c^20 + 34*a^2*b^4*c^20 - 7*b^6*c^20 + 7*a^4*c^22 - 2*a^2*b^2*c^22 - 3*b^4*c^22 - a^2*c^24 + b^2*c^24) : :

See Tran Quang Hung, César Lozada, Antreas Hatzipolakis and Peter Moses, Euclid 239 .

X(35003) lies on this line: {2, 3}

X(35004) = X(1)X(3)∩X(5)X(2800)

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^4*b*c + 3*a^3*b^2*c - 4*a*b^4*c + 2*b^5*c - a^4*c^2 + 3*a^3*b*c^2 - 6*a^2*b^2*c^2 + 3*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + 3*a*b^2*c^3 - 4*b^3*c^3 + 2*a^2*c^4 - 4*a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6) : :
X(35004) = 5 X[1] - 8 X[12009], 3 X[10] - 2 X[31835], X[65] - 3 X[10273], 3 X[65] - X[24474], 3 X[354] - X[23340], 3 X[355] - X[12528], 3 X[942] - X[13600], X[946] - 3 X[3919], 2 X[960] - 3 X[11231], 3 X[1385] - 4 X[9940], X[1482] - 3 X[5902], X[3057] - 3 X[10202], 3 X[3753] - X[5887], 3 X[3753] - 2 X[9956], 4 X[3812] - 3 X[11230], X[3869] - 3 X[26446], 3 X[3899] - 7 X[31423], 5 X[4004] - 2 X[9955], 5 X[4004] - X[12672], 3 X[5603] - 2 X[26200], X[5693] - 3 X[5790], 3 X[5694] - 4 X[31835], X[5697] - 3 X[10246], X[5697] - 5 X[15016], 3 X[5883] - 2 X[5901], 4 X[5885] - X[10284], 5 X[5885] - 4 X[12009], 2 X[5885] + X[25413], 3 X[5902] - 2 X[6583], X[5903] + 2 X[13145], 4 X[10107] - X[18480], 8 X[10107] - X[31828], 3 X[10202] - 2 X[15178], 3 X[10222] - 2 X[13600], 3 X[10246] - 5 X[15016], 3 X[10247] - 5 X[18398], 9 X[10273] - X[24474], 5 X[10284] - 16 X[12009], X[10284] + 2 X[25413], 5 X[11522] - 4 X[26088], 8 X[12009] + 5 X[25413]

See Antreas Hatzipolakis and Peter Moses, Euclid 243 .

Let A'B'C' be the Fuhrmann triangle of the orthic triangle of the Fuhrmann triangle. A'B'C' is homothetic to the excentral triangle (and perspective to ABC) at X(1), and X(35004) = X(3)-of-A'B'C'. (Randy Hutson, January 17, 2020)

X(35004) lies on these lines: {1, 3}, {5, 2800}, {8, 6951}, {10, 5694}, {72, 6937}, {140, 3878}, {153, 355}, {404, 6265}, {496, 12736}, {519, 24475}, {573, 21863}, {758, 5499}, {912, 5836}, {944, 26201}, {946, 3919}, {952, 5884}, {960, 11231}, {962, 6903}, {1071, 17579}, {1389, 6909}, {1483, 2802}, {1768, 26321}, {1837, 6797}, {2476, 3753}, {2778, 34935}, {2889, 3868}, {3698, 17057}, {3812, 11230}, {3869, 6853}, {3874, 5844}, {3899, 31423}, {3918, 20117}, {4004, 6830}, {4084, 11362}, {4295, 10526}, {4642, 5396}, {4861, 26877}, {5036, 21853}, {5180, 6902}, {5253, 10698}, {5399, 24028}, {5434, 13375}, {5603, 6972}, {5657, 26487}, {5691, 12409}, {5693, 5790}, {5777, 17532}, {5883, 5901}, {5886, 6952}, {6001, 10107}, {6224, 14923}, {6691, 11729}, {6840, 12699}, {6906, 12515}, {6907, 15556}, {6914, 30147}, {6923, 10573}, {6971, 18393}, {6980, 18395}, {7680, 33592}, {7686, 7706}, {7701, 12767}, {7741, 17638}, {8256, 10942}, {9946, 17563}, {10525, 18391}, {10944, 11570}, {10950, 15530}, {11491, 33858}, {11522, 26088}, {11571, 12763}, {12331, 34600}, {12524, 26921}, {13089, 13465}, {13463, 32214}, {13464, 33815}, {13743, 33856}, {15071, 18525}, {18357, 31803}, {18389, 31775}, {19860, 22936}, {22791, 31870}, {22935, 25440}, {31659, 31806}

X(35004) = midpoint of X(i) and X(j) for these {i,j}: {1, 25413}, {3, 5903}, {4084, 11362}, {11571, 19914}, {15071, 18525}
X(35004) = reflection of X(i) in X(j) for these (i,j): {1, 5885}, {3, 13145}, {5, 3754}, {944, 26201}, {1482, 6583}, {1483, 12005}, {3057, 15178}, {3878, 140}, {5694, 10}, {5887, 9956}, {9957, 13373}, {10222, 942}, {10284, 1}, {12672, 9955}, {13464, 33815}, {20117, 3918}, {22791, 31870}, {22793, 7686}, {23960, 5570}, {31803, 18357}, {31828, 18480}
X(35004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3, 26287}, {1, 40, 11849}, {1, 32612, 1385}, {3, 14882, 26285}, {354, 23340, 33179}, {1385, 3579, 33862}, {1385, 10225, 3}, {1482, 5902, 6583}, {2475, 12247, 355}, {3057, 10202, 15178}, {3057, 13751, 1}, {3304, 5902, 942}, {3336, 11014, 22765}, {3753, 5887, 9956}, {5697, 15016, 10246}, {5885, 25413, 10284}, {5885, 33658, 13751}, {9957, 18856, 1385}, {15178, 18857, 1385}, {15178, 33658, 1}, {23961, 33281, 1385}

Centers related to Vu circlecevian points: X(35005)-X(35009)

Vu circlecevian points are introduced in the preamble before X(34892).

Contributed by Randy Hutson, November 26, 2019.

X(35005) = VU CIRCLECEVIAN POINT V(X(17),X(18))

Barycentrics 1/(3 a^4 + b^4 + c^4 - a^2 b^2 - a^2 c^2 - 3 b^2 c^2) : :

X(35005) lies on the Kiepert hyperbola and these lines: {4, 13188}, {13, 5982}, {14, 5983}, {83, 620}, {98, 5965}, {114, 14492}, {147, 14458}, {262, 8782}, {325, 11606}, {543, 33698}, {598, 19686}, {3406, 10357}, {3407, 10353}, {5962, 6484}, {6721, 7608}, {10159, 12815}, {10302, 14971}, {18840, 33248}, {18842, 33255}

X(35005) = isogonal conjugate of X(35006)

X(35006) = ISOGONAL CONJUGATE OF X(35005)

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

X(35006) lies on these lines: {3, 6}, {141, 33245}, {147, 230}, {183, 10334}, {373, 1915}, {385, 10353}, {597, 7924}, {627, 10617}, {628, 10616}, {732, 5182}, {1501, 5640}, {1503, 14651}, {2715, 5966}, {3589, 5207}, {3618, 5103}, {3629, 13571}, {5031, 33218}, {5254, 12252}, {5965, 15561}, {6090, 21001}, {6329, 14712}, {7379, 11188}, {14567, 20998}

X(35006) = isogonal conjugate of X(35005)
X(35006) = circumcircle-inverse of X(35007)
X(35006) = 1st-Lemoine-circle inverse of X(12212)
X(35006) = radical trace of circumcircle and circle O(61,62)
X(35006) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37517)
X(35006) = {X(371),X(372)}-harmonic conjugate of X(37517)
X(35006) = {X(1691),X(1692)}-harmonic conjugate of X(6)
X(35006) = {X(1379),X(1380)}-harmonic conjugate of X(35007)
X(35006) = {X(1662),X(1663)}-harmonic conjugate of X(12212)

X(35007) = CIRCUMCIRCLE-INVERSE OF X(35006)

Barycentrics    a^2 (4 a^2 - b^2 - c^2) : :
Trilinears    3 sin A - 5 cos A tan ω : :
Trilinears    5 cos A - 3 sin A cot ω : :
Trilinears    5 cos A sin ω - 3 sin A cos ω : :

X(35007) lies on these lines: {1, 9341}, {2, 7843}, {3, 6}, {5, 12815}, {20, 5309}, {23, 1196}, {24, 14581}, {30, 7755}, {99, 7805}, {112, 3199}, {115, 3627}, {140, 7753}, {172, 3746}, {194, 32456}, {230, 546}, {248, 16835}, {251, 7496}, {315, 7874}, {316, 7886}, {376, 5319}, {384, 7780}, {385, 7816}, {439, 34511}, {524, 7863}, {538, 3552}, {543, 33250}, {550, 5306}, {598, 33002}, {599, 33242}, {609, 1500}, {620, 7762}, {625, 7823}, {629, 10617}, {630, 10616}, {632, 1506}, {754, 7807}, {1003, 7751}, {1015, 7031}, {1078, 7804}, {1194, 7492}, {1657, 11648}, {1914, 5563}, {1968, 10594}, {2241, 3304}, {2242, 3303}, {2300, 22357}, {2549, 5346}, {2896, 7915}, {3090, 5475}, {3091, 7737}, {3146, 3767}, {3291, 14002}, {3456, 14908}, {3470, 32640}, {3522, 7739}, {3529, 7735}, {3530, 9300}, {3628, 7603}, {3785, 7822}, {3788, 7845}, {3793, 7789}, {3815, 14869}, {3843, 18362}, {3849, 5025}, {3915, 8649}, {3934, 3972}, {5010, 7296}, {5047, 5277}, {5198, 8778}, {5215, 7775}, {5254, 6781}, {5305, 7756}, {5332, 7280}, {5337, 21516}, {5349, 5479}, {5350, 5478}, {5368, 15048}, {5569, 33001}, {6103, 6240}, {6390, 7890}, {6655, 7817}, {6658, 14568}, {6680, 7750}, {6683, 7787}, {7757, 33014}, {7759, 16925}, {7760, 13586}, {7761, 7852}, {7766, 7782}, {7767, 7820}, {7768, 7880}, {7769, 20088}, {7773, 31275}, {7781, 14614}, {7792, 7830}, {7794, 8369}, {7796, 33246}, {7801, 11160}, {7802, 7806}, {7809, 33245}, {7810, 7819}, {7811, 7849}, {7812, 7907}, {7818, 32954}, {7824, 12150}, {7827, 33260}, {7828, 7842}, {7829, 8356}, {7831, 10583}, {7832, 7848}, {7833, 7856}, {7834, 14907}, {7835, 7893}, {7846, 7904}, {7850, 7945}, {7854, 14001}, {7858, 33259}, {7865, 33217}, {7869, 33220}, {7870, 7946}, {7877, 7891}, {7878, 33004}, {7883, 14043}, {7888, 11288}, {7898, 7942}, {7900, 7940}, {7902, 33234}, {7910, 7932}, {7922, 9939}, {7929, 7930}, {7933, 11057}, {8176, 32998}, {8182, 32990}, {8362, 15810}, {8703, 9607}, {8859, 14042}, {9166, 14044}, {9606, 15712}, {10303, 31455}, {10312, 14865}, {12082, 19220}, {12110, 21445}, {16589, 16865}, {19687, 22329}, {20576, 32152}, {32992, 34506}, {33254, 34504}

X(35007) = complement of X(7860)
X(35007) = circumcircle-inverse of X(35006)
X(35007) = {X(61),X(62)}-harmonic conjugate of X(5097)
X(35007) = {X(1379),X(1380)}-harmonic conjugate of X(35006)
X(35007) = pole wrt circumcircle of line X(512)X(35006)
X(35007) = barycentric product X(6)*X(3629)
X(35007) = barycentric quotient X(3629)/X(76)

X(35008) = VU CIRCLECEVIAN POINT V(PU(6))

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

X(35008) lies on hyperbola {{A,B,C,X(86),X(190),X(789),PU(6)}} and these lines: {86, 22172}, {190, 3250}, {513, 789}, {874, 30665}, {3570, 35009}, {4499, 4623}

X(35008) = barycentric product X(75)*X(35009)
X(35008) = barycentric quotient X(35009)/X(1)

X(35009) = VU CIRCLECEVIAN POINT V(PU(8))

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

X(35009) lies on hyperbola {{A,B,C,X(81),X(100),PU(8)}} and these lines: {81, 1908}, {100, 788}, {649, 4586}, {3570, 35008}, {4610, 25577}

X(35009) = barycentric product X(1)*X(35008)
X(35009) = barycentric quotient X(35008)/X(75)

X(35010) = X(1)X(3)∩X(84)X(7988)

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

See Antreas Hatzipolakis and César Lozada, Euclid 253 .

X(35010) lies on these lines: {1, 3}, {84, 7988}, {191, 10165}, {442, 11219}, {631, 6763}, {1125, 1768}, {1158, 25055}, {2801, 17531}, {3306, 5691}, {4297, 27003}, {4973, 6986}, {5433, 31657}, {5437, 7989}, {5531, 12675}, {5541, 13607}, {5836, 7993}, {6256, 6900}, {6884, 10200}, {6940, 12005}, {7330, 34595}, {7701, 11230}, {7997, 10157}, {10167, 16143}, {10883, 31249}, {15064, 17535}, {15071, 25524}

X(35010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 3337, 5536), (2077, 13373, 1), (5049, 26286, 11510), (5437, 10085, 7989), (10165, 26877, 191), (10269, 15016, 1)

X(35011) = X(106)X(1795)∩X(900)X(1309)

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

Let A', B', C' be the intersections of line X(3)X(8) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(35011). (Randy Hutson, January 17, 2020)

X(35011) lies on the circumcircle and these lines: {106, 1795}, {900, 1309}, {901, 8677}, {952, 2734}, {953, 2818}, {2222, 3738}, {2716, 2800}, {2745, 17100}

X(35011) = isogonal conjugate of X(35013)
X(35011) = reflection of X(1309) in line X(3)X(8)
X(35011) = Collings transform of X(i) for these i: {2800, 3738}
X(35011) = X(i)-isoconjugate of X(j) for these (i,j): {952, 1769}, {2265, 10015}
X(35011) = cevapoint of X(2800) and X(3738)
X(35011) = trilinear pole of line {6, 32641}
X(35011) = barycentric product X(953)*X(13136)
X(35011) = barycentric quotient X(i)/X(j) for these {i,j}: {953, 10015}, {2423, 6075}, {2427, 6073}, {32641, 952}

Centers on the Sherman line: X(35012)-X(35015)

See Paul Yiu, Sherman's Fourth Side of a Triangle .

Contributed by Peter Moses, November 29, 2019.

X(35012) = X(3)X(901)∩X(855)X(1870)

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

X(35012) lies on these lines: {3, 901}, {855, 1870}, {1361, 1457}, {1459, 3937}, {1532, 21664}, {2967, 8229}, {2972, 34588}, {3259, 3326}

X(35012) = X(i)-Ceva conjugate of X(j) for these (i,j): {264, 10015}, {14260, 8677}
X(35012) = crosspoint of X(264) and X(10015)
X(35012) = crosssum of X(184) and X(32641)
X(35012) = barycentric product X(i)*X(j) for these {i,j}: {222, 3326}, {1361, 26932}, {1565, 23980}, {3937, 26611}, {3942, 24028}, {8677, 10015}
X(35012) = point of tangency of the Sherman line and the MacBeath inconic
X(35012) = barycentric quotient X(i)/X(j) for these {i,j}: {3310, 1309}, {3326, 7017}, {8677, 13136}, {23220, 32641}, {23980, 15742}

X(35013) = X(30)X(511)∩X(901)X(1309)

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

X(35013) lies on these lines: {30, 511}, {901, 1309}, {953, 2734}, {2222, 23981}, {3259, 3326}

X(35013) = isogonal conjugate of X(35011)
X(35013) = crosssum of X(2800) and X(3738)
X(35013) = crosspoint of X(2222) and X(2716)
X(35013) = crossdifference of every pair of points on line {6, 32641}
X(35013) = barycentric quotient X(i)/X(j) for these {i,j}: {8620, 31562}, {19897, 30432}

X(35014) = X(1)X(3)∩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 - 2*a*b*c + b^2*c + b*c^2 - c^3) : :

X(35014) lies on these lines: {1, 3}, {11, 123}, {104, 10702}, {216, 17452}, {1320, 1809}, {1364, 3270}, {1465, 23706}, {1870, 14127}, {3100, 13589}, {3137, 24006}, {3259, 3326}, {10746, 12764}, {12053, 31680}

X(35014) = X(i)-complementary conjugate of X(j) for these (i,j): {104, 20316}, {1459, 119}, {1795, 513}, {2401, 20305}, {2423, 226}, {14578, 514}, {34858, 3239}
X(35014) = X(i)-Ceva conjugate of X(j) for these (i,j): {517, 8677}, {1295, 513}, {1320, 521}, {2968, 10017}, {3218, 14395}
X(35014) = X(i)-isoconjugate of X(j) for these (i,j): {104, 7012}, {109, 1309}, {190, 32702}, {653, 32641}, {664, 14776}, {1809, 24033}, {1897, 2720}, {2149, 16082}, {6335, 32669}, {7115, 34234}, {13136, 32674}
X(35014) = crosspoint of X(i) and X(j) for these (i,j): {517, 2804}, {521, 1807}
X(35014) = crosssum of X(i) and X(j) for these (i,j): {104, 2720}, {108, 1870}
X(35014) = crossdifference of every pair of points on line {650, 1415}
X(35014) = barycentric product X(i)*X(j) for these {i,j}: {517, 26932}, {521, 10015}, {905, 2804}, {908, 7004}, {1214, 14010}, {1465, 2968}, {1769, 6332}, {1875, 23983}, {2183, 17880}, {3262, 7117}, {3942, 6735}, {4391, 8677}, {4858, 22350}, {8611, 23788}, {17219, 21801}, {22464, 34591}
X(35014) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 16082}, {521, 13136}, {650, 1309}, {667, 32702}, {1457, 7128}, {1769, 653}, {1875, 23984}, {1946, 32641}, {2183, 7012}, {2804, 6335}, {3063, 14776}, {3310, 108}, {3937, 34051}, {7004, 34234}, {7117, 104}, {8677, 651}, {10015, 18026}, {14010, 31623}, {22350, 4564}, {22383, 2720}, {23220, 1415}, {26932, 18816}

X(35015) = X(1)X(4)∩X(11)X(244)

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

In the plane of a triangle ABC, let
DEF = orthic triangle;
F' = X(11) = Feuerbach point;
D' = reflection of D in AF;
Da = EF∩D'F';
Oa = circumcenter of ADDa, and define Ob and Oc cyclically;
The finite fixed point of the affine transformation that maps ABC onto OaOObOc is X(35015). (Angel Montesdeoca, May 26, 2022)

X(35015) lies on these lines: {1, 4}, {2, 24410}, {11, 244}, {36, 14127}, {80, 10703}, {109, 34789}, {119, 24028}, {124, 20620}, {201, 15908}, {212, 24703}, {516, 23703}, {650, 33573}, {656, 3139}, {1393, 7681}, {1411, 12764}, {1421, 10776}, {1465, 1538}, {1776, 33140}, {1807, 10738}, {1836, 9316}, {1854, 10896}, {1936, 5057}, {2006, 2342}, {2886, 7069}, {2920, 20988}, {3259, 3326}, {3782, 15845}, {3938, 10947}, {4551, 21635}, {4642, 10958}, {5087, 9371}, {5219, 15737}, {6788, 9581}, {7082, 24892}, {7962, 24864}, {8735, 34591}, {10589, 17889}, {11502, 33094}, {11680, 24430}, {12611, 34586}, {15558, 24222}, {17777, 27542}, {18839, 32856}, {21132, 23615}, {21616, 22072}, {22793, 33649}, {24431, 29690}, {24443, 26476}

X(35015) = X(i)-Ceva conjugate of X(j) for these (i,j): {106, 21119}, {1785, 1769}, {2006, 650}, {6735, 2804}, {17923, 14400}, {22464, 10015}, {23838, 21132}
X(35015) = X(3259)-cross conjugate of X(11)
X(35015) = X(i)-isoconjugate of X(j) for these (i,j): {59, 104}, {100, 2720}, {190, 32669}, {651, 32641}, {909, 4564}, {1252, 34051}, {1332, 32702}, {1415, 13136}, {1795, 7012}, {2149, 34234}, {2342, 7045}, {2423, 31615}, {4998, 34858}, {6516, 14776}
X(35015) = crosspoint of X(i) and X(j) for these (i,j): {80, 522}, {2804, 6735}, {10015, 22464}, {15633, 24026}
X(35015) = crosssum of X(i) and X(j) for these (i,j): {36, 109}, {2342, 32641}
X(35015) = crossdifference of every pair of points on line {101, 652}
X(35015) = polar conjugate of X(39294)
X(35015) = pole wrt polar circle of trilinear polar of X(39294) (line X(109)X(522))
X(35015) = barycentric product X(i)*X(j) for these {i,j}: {11, 908}, {226, 14010}, {514, 2804}, {517, 4858}, {522, 10015}, {1086, 6735}, {1146, 22464}, {1457, 23978}, {1465, 24026}, {1769, 4391}, {1785, 26932}, {2170, 3262}, {2183, 34387}, {2397, 21132}, {3259, 4997}, {3326, 34234}, {3700, 23788}, {14571, 17880}, {17139, 21044}, {17197, 17757}
X(35015) = barycentric quotient X(i)/X(j) for these {i,j}: {11, 34234}, {244, 34051}, {517, 4564}, {522, 13136}, {649, 2720}, {663, 32641}, {667, 32669}, {908, 4998}, {1457, 1262}, {1465, 7045}, {1769, 651}, {1875, 7128}, {2170, 104}, {2183, 59}, {2804, 190}, {3064, 1309}, {3259, 3911}, {3271, 909}, {3310, 109}, {3326, 908}, {4516, 2250}, {4858, 18816}, {6735, 1016}, {7117, 1795}, {8677, 1813}, {10015, 664}, {14010, 333}, {14571, 7012}, {14936, 2342}, {17139, 4620}, {21132, 2401}, {22464, 1275}, {23220, 32660}, {23788, 4573}, {23981, 4619}, {34591, 1809}
X(35015) = {X(1519),X(1785)}-harmonic conjugate of X(1457)

X(35016) = MIDPOINT OF X(1) AND X(21)

Barycentrics a*(2*a^3-(b+c)*a^2-2*(b^2+b*c+c^2)*a+(b+c)*(b^2-3*b*c+c^2)) : :
X(35016) = 3*X(1)+X(191), 2*X(1)+X(3647), X(1)+3*X(5426), 5*X(1)+X(11684), 5*X(1)-X(16126), 3*X(1)-X(34195), 3*X(21)-X(191), X(21)-3*X(5426), 5*X(21)-X(11684), 5*X(21)+X(16126), 3*X(21)+X(34195), 2*X(191)-3*X(3647), X(191)-9*X(5426), 5*X(191)-3*X(11684), 5*X(191)+3*X(16126), X(3647)-6*X(5426), 5*X(3647)-2*X(11684), 5*X(3647)+2*X(16126), 3*X(3647)+2*X(34195)

See Antreas Hatzipolakis and César Lozada, Euclid 257 .

X(35016) lies on these lines: {1, 21}, {2, 15079}, {3, 5883}, {8, 5424}, {10, 6675}, {11, 214}, {30, 551}, {35, 3754}, {40, 21161}, {55, 30147}, {65, 5427}, {79, 2320}, {100, 3918}, {145, 15676}, {284, 25081}, {355, 10197}, {404, 3833}, {405, 10176}, {409, 15792}, {515, 6841}, {517, 5428}, {519, 15670}, {535, 13407}, {942, 4973}, {943, 6596}, {952, 10021}, {958, 12260}, {997, 3646}, {1001, 30144}, {1155, 33815}, {1319, 3636}, {1479, 2475}, {1482, 4428}, {1844, 17515}, {2136, 11525}, {2605, 24457}, {2771, 5609}, {2795, 11711}, {2802, 3746}, {3219, 4127}, {3241, 15672}, {3244, 5837}, {3295, 10912}, {3303, 22837}, {3336, 17549}, {3337, 5303}, {3486, 10198}, {3488, 6598}, {3560, 31803}, {3576, 3651}, {3579, 3919}, {3601, 25440}, {3624, 31254}, {3628, 22935}, {3634, 5440}, {3635, 3748}, {3652, 28453}, {3656, 28460}, {3678, 5251}, {3679, 15671}, {3689, 4691}, {3742, 13624}, {3750, 15955}, {3814, 13411}, {3822, 10572}, {3828, 33595}, {3956, 4420}, {3957, 5288}, {4015, 5260}, {4067, 31445}, {4084, 4640}, {4134, 5302}, {4189, 5902}, {4304, 12609}, {4321, 16133}, {4324, 20292}, {4367, 6161}, {4511, 5259}, {4537, 15481}, {4757, 5425}, {4999, 12433}, {5221, 19535}, {5499, 26287}, {5506, 16861}, {5538, 6986}, {5542, 5625}, {5554, 31452}, {5690, 31650}, {5692, 16865}, {5882, 16617}, {5884, 6914}, {6175, 25055}, {6326, 6920}, {6668, 12019}, {6684, 28465}, {6912, 31871}, {6950, 15016}, {7489, 20117}, {7701, 28461}, {7743, 20288}, {8715, 19860}, {9528, 11718}, {10039, 15863}, {10129, 18514}, {10246, 12114}, {10483, 31019}, {10595, 16113}, {11260, 15673}, {11274, 13607}, {11369, 16124}, {11372, 16132}, {11379, 16143}, {11604, 16173}, {11813, 33961}, {12104, 22937}, {12245, 31669}, {12635, 16418}, {13750, 17010}, {14450, 15677}, {15015, 17531}, {15172, 21630}, {15932, 17097}, {16060, 30131}, {16160, 34773}, {16430, 19765}, {16611, 18755}, {16916, 30135}, {17009, 34339}, {17136, 17201}, {17525, 20323}, {17550, 30123}, {17606, 20104}, {17637, 30538}, {17684, 30139}, {19925, 33597}, {24387, 24541}, {25485, 33281}, {28463, 33179}, {30127, 33821}, {30389, 33557}, {33857, 34471}

X(35016) = midpoint of X(i) and X(j) for these {i,j}: {1, 21}, {79, 15680}, {191, 34195}, {442, 10543}, {1482, 16139}, {2475, 5441}, {3656, 28460}, {6675, 15174}, {11684, 16126}, {13743, 33858}, {16132, 21669}, {16160, 34773}, {22937, 24680}
X(35016) = reflection of X(i) in X(j) for these (i,j): (10, 6675), (442, 1125), (2475, 6701), (3647, 21), (11263, 11281), (16125, 33592), (22937, 12104), (33592, 5901)
X(35016) = X(21)-Beth conjugate of X(2650)
X(35016) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(34195)}} and {{A, B, C, X(58), X(5424)}}
X(35016) = center of the circle {{X(1), X(21), X(3109), X(31525)}}
X(35016) = (anti-Aquila)-isogonal conjugate of X(942)
X(35016) = (anti-Aquila)-anticomplement of X(11281)
X(35016) = (anti-Aquila)-complement of X(11263)
X(35016) = (2nd circumperp)-complement of X(3651)
X(35016) = hexyl-complement of X(3651)
X(35016) = X(21)-of-anti-Aquila-triangle
X(35016) = X(2917)-of-incircle-circles-triangle
X(35016) = X(3574)-of-2nd-circumperp-triangle
X(35016) = X(23358)-of-intouch-triangle
X(35016) = X(32348)-of-hexyl-triangle
X(35016) = X(32401)-of-Hutson-intouch-triangle
X(35016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 191, 34195), (1, 993, 3874), (1, 1621, 3884), (1, 2975, 3881), (1, 4653, 3743), (1, 5248, 3878), (1, 5426, 21), (1, 5429, 4658), (1, 8666, 3892), (1, 31424, 12559), (3, 30143, 5883), (21, 34195, 191), (405, 22836, 10176), (551, 11263, 11281), (942, 5267, 4973), (950, 1125, 25639), (2320, 3622, 21842), (2475, 3616, 26725), (2475, 26725, 6701), (3616, 5441, 6701), (5248, 10122, 3647), (5441, 26725, 2475)

X(35017) = X(1656)X(34793)∩X(10109)X(32744)

Barycentrics 16*(39*R^2-11*SW)*S^4-(15*R^6-2*(10*SA-17*SW)*R^4-4*(66*SA^2-68*SA*SW+SW^2)*R^2-72*(SB+SC)*SA*SW)*S^2-(R^2*(3*R^2+14*SW)+4*SW^2)*R^2*SB*SC : :
X(35017) = 5*X(12812)+X(31607)

See Antreas Hatzipolakis and César Lozada, Euclid 267 .

X(35017) lies on these lines: {1656, 34793}, {10109, 32744}, {12812, 31607}

X(35018) = X(15)X(10188)∩X(16)X(10187)

Barycentrics 2*a^4-9*(b^2+c^2)*a^2+7*(b^2-c^2)^2 : :
X(35018) = 21*X(2)-5*X(3), 3*X(2)+5*X(5), 9*X(2)-5*X(140), 11*X(2)+5*X(381), 15*X(2)+X(382), 3*X(2)+X(546), X(2)-5*X(547), 13*X(2)-5*X(549), 9*X(2)-X(550), 3*X(2)-5*X(3628), 9*X(2)+5*X(3850), 9*X(2)+7*X(3851), 12*X(2)+5*X(3856), 13*X(2)+5*X(3860), 31*X(2)-15*X(5054), X(2)+15*X(5055), 7*X(2)+5*X(5066), 3*X(2)+13*X(5079), X(2)+5*X(10109), 7*X(2)-5*X(10124), 23*X(2)-15*X(11539), 8*X(2)-5*X(11540), 11*X(2)-5*X(11812), 17*X(2)-5*X(12100), 12*X(2)-5*X(12108), 6*X(2)+5*X(12811), 13*X(2)+3*X(14269), 15*X(2)-7*X(14869), 29*X(2)-15*X(14890), 19*X(2)-5*X(14891), 17*X(2)-X(15681), 7*X(2)+X(15687), 19*X(2)-3*X(15688), 7*X(2)-15*X(15699), 23*X(2)-7*X(15700), 25*X(2)-9*X(15707), 27*X(2)-11*X(15720), 23*X(2)-5*X(15759), 6*X(2)-5*X(16239), 11*X(2)-3*X(17504), 5*X(2)-X(34200), 9*X(3)+7*X(4)

As a point on the Euler line, X(35018) has Shinagawa coefficients (9, 5).

See Antreas Hatzipolakis and César Lozada, Euclid 267 .

X(35018) lies on these lines: {2, 3}, {15, 10188}, {16, 10187}, {17, 11543}, {18, 11542}, {125, 13393}, {143, 10170}, {230, 12815}, {233, 1990}, {373, 6102}, {397, 16967}, {398, 16966}, {485, 13993}, {486, 13925}, {496, 8162}, {511, 18874}, {952, 3636}, {1125, 28224}, {1154, 12046}, {1209, 11803}, {1216, 13364}, {3244, 5901}, {3316, 18510}, {3317, 18512}, {3519, 22051}, {3564, 6329}, {3589, 18553}, {3614, 5270}, {3626, 5844}, {3629, 15520}, {3631, 24206}, {3632, 5886}, {3634, 28174}, {3968, 26200}, {4031, 34753}, {4857, 7173}, {5400, 5453}, {5418, 6468}, {5420, 6469}, {5443, 11545}, {5447, 15003}, {5462, 14128}, {5493, 11231}, {5663, 11695}, {5690, 11522}, {5790, 20050}, {5818, 10283}, {5882, 11230}, {5891, 15026}, {5907, 13363}, {5943, 11591}, {5946, 31834}, {6000, 11017}, {6101, 13451}, {6154, 23513}, {6592, 23516}, {6673, 33561}, {6674, 33560}, {6684, 28216}, {6688, 12006}, {6689, 20584}, {6721, 15092}, {7584, 8960}, {7603, 7755}, {7607, 32151}, {7741, 15172}, {7781, 9771}, {7988, 22791}, {7989, 34773}, {8227, 11224}, {8550, 18358}, {8981, 10195}, {9607, 18362}, {9624, 34747}, {9955, 10172}, {10095, 11793}, {10110, 32142}, {10194, 13966}, {10247, 20054}, {10263, 14845}, {10272, 23515}, {10576, 18762}, {10577, 18538}, {10990, 34128}, {10991, 34127}, {11245, 11704}, {11272, 32450}, {11451, 18436}, {11465, 34783}, {11801, 13392}, {12002, 13391}, {12242, 13565}, {12571, 28178}, {12900, 15088}, {13754, 32205}, {14449, 15067}, {14862, 32767}, {14864, 16252}, {15028, 18435}, {15029, 20126}, {15105, 23329}, {15606, 16982}, {16534, 20304}, {16836, 32137}, {17810, 33540}, {17825, 32139}, {19552, 34596}, {19878, 28160}, {20193, 32269}, {23338, 34597}, {24680, 34641}, {25150, 25339}, {31253, 31663}, {31666, 34648}, {31831, 32165}, {32824, 34803}, {32825, 32868}, {32904, 34598}

X(35018) = midpoint of X(i) and X(j) for these {i,j}: {2, 11737}, {3, 3861}, {4, 33923}, {5, 3628}, {140, 3850}, {381, 11812}, {546, 3530}, {547, 10109}, {548, 12102}, {549, 3860}, {3845, 14891}, {3856, 12108}, {5066, 10124}, {5462, 14128}, {6689, 20584}, {6721, 15092}, {10095, 11793}, {10110, 32142}, {10289, 15957}, {10297, 22249}, {11591, 16881}, {11801, 13392}, {12811, 16239}, {12900, 15088}, {13406, 32144}, {14893, 15759}, {15327, 15335}, {15334, 25404}, {15606, 16982}, {31831, 32165}, {32904, 34598}
X(35018) = reflection of X(i) in X(j) for these (i,j): (3856, 12811), (12056, 34420), (12108, 16239), (12811, 5), (16239, 3628)
X(35018) = complement of X(3530)
X(35018) = intersection, other than A,B,C, of conics {{A, B, C, X(264), X(15720)}} and {{A, B, C, X(265), X(12811)}}
X(35018) = X(12102)-of-Ehrmann-mid-triangle
X(35018) = X(12811)-of-Johnson-triangle
X(35018) = X(18874)-of-Wasat- triangle
X(35018) = X(33923)-of-Euler-triangle
X(35018) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3855, 3), (2, 5079, 5), (3, 5071, 5), (5, 15699, 3), (381, 15022, 5), (547, 12812, 5), (631, 15683, 3), (1656, 3851, 2), (1656, 5056, 5), (2072, 34939, 5), (3090, 5055, 5), (3529, 5067, 2), (3530, 3628, 2), (3628, 10109, 5), (3850, 33923, 4), (3851, 15720, 4), (3861, 10124, 3), (5071, 7486, 3), (15687, 15699, 2), (15688, 15703, 2), (15709, 17578, 3), (15717, 16417, 2)

X(35019) = MIDPOINT OF X(13) AND X(6669)

Barycentrics 2*sqrt(3)*(6*a^2+b^2+c^2)*S+2*a^4+11*(b^2+c^2)*a^2-13*(b^2-c^2)^2 : :
X(35019) = 3*X(2)+5*X(13), 21*X(2)-5*X(616), 9*X(2)-5*X(618), X(2)-5*X(5459), 13*X(2)-5*X(5463), 3*X(2)-5*X(6669), 7*X(2)-15*X(22489), 7*X(13)+X(616), 3*X(13)+X(618), X(13)+3*X(5459), 13*X(13)+3*X(5463), 7*X(13)+9*X(22489), 3*X(616)-7*X(618), X(616)-7*X(6669), X(616)-9*X(22489), X(618)-9*X(5459), 13*X(618)-9*X(5463), X(618)-3*X(6669), 13*X(5459)-X(5463), 3*X(5459)-X(6669), 7*X(5459)-3*X(22489)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35019) lies on these lines: {2, 13}, {382, 5478}, {533, 11542}, {542, 6329}, {546, 20252}, {3181, 33413}, {3244, 11705}, {3528, 21156}, {5079, 5617}, {5470, 32552}, {5472, 22847}, {5473, 10299}, {10611, 20377}, {11121, 22845}, {13103, 15720}, {14869, 16001}, {15688, 25154}, {22846, 33464}

X(35019) = midpoint of X(i) and X(j) for these {i,j}: {13, 6669}, {10611, 20377}, {11542, 33560}, {20252, 20415}
X(35019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 5459, 6669), (13, 22489, 616), (6329, 11737, 35020)

X(35020) = MIDPOINT OF X(14) AND X(6670)

Barycentrics -2*sqrt(3)*(6*a^2+b^2+c^2)*S+2*a^4+11*(b^2+c^2)*a^2-13*(b^2-c^2)^2 : :
X(35020) = 3*X(2)+5*X(14), 21*X(2)-5*X(617), 9*X(2)-5*X(619), X(2)-5*X(5460), 13*X(2)-5*X(5464), 3*X(2)-5*X(6670), 7*X(2)-15*X(22490), 7*X(14)+X(617), 3*X(14)+X(619), X(14)+3*X(5460), 13*X(14)+3*X(5464), 7*X(14)+9*X(22490), 3*X(617)-7*X(619), X(617)-7*X(6670), X(617)-9*X(22490), X(619)-9*X(5460), 13*X(619)-9*X(5464), X(619)-3*X(6670), 13*X(5460)-X(5464), 3*X(5460)-X(6670), 7*X(5460)-3*X(22490)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35020) lies on these lines: {2, 14}, {382, 5479}, {532, 11543}, {542, 6329}, {546, 20253}, {3180, 33412}, {3244, 11706}, {3528, 21157}, {3855, 6773}, {5079, 5613}, {5469, 32553}, {5471, 22893}, {5474, 10299}, {10612, 20378}, {11122, 22844}, {13102, 15720}, {14869, 16002}, {15688, 25164}, {22891, 33465}

X(35020) = midpoint of X(i) and X(j) for these {i,j}: {14, 6670}, {11543, 33561}, {20253, 20416}
X(35020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (14, 5460, 6670), (14, 22490, 617), (6329, 11737, 35019)

X(35021) = X(2)X(98)∩X(115)X(382)

Barycentrics 6*a^8-8*(b^2+c^2)*a^6+(9*b^4-4*b^2*c^2+9*c^4)*a^4-2*(b^2+c^2)*(4*b^4-7*b^2*c^2+4*c^4)*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b^2-c^2)^2 : :
X(35021) = 3*X(2)+5*X(98), 9*X(2)-5*X(114), 21*X(2)-5*X(147), 3*X(2)-5*X(6036), 13*X(2)-5*X(6054), X(2)-5*X(6055), 6*X(2)-5*X(6721), 11*X(2)+5*X(11177), 23*X(2)-15*X(23234), 3*X(98)+X(114), 7*X(98)+X(147), 9*X(98)-X(5984), 13*X(98)+3*X(6054), X(98)+3*X(6055), 2*X(98)+X(6721), 11*X(98)-3*X(11177), 7*X(114)-3*X(147), 3*X(114)+X(5984), X(114)-3*X(6036), 13*X(114)-9*X(6054), X(114)-9*X(6055), 2*X(114)-3*X(6721), 11*X(114)+9*X(11177)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35021) lies on these lines: {2, 98}, {99, 10299}, {115, 382}, {230, 29012}, {543, 34200}, {546, 2794}, {550, 11623}, {620, 14869}, {2482, 15707}, {2782, 3530}, {3244, 11710}, {3528, 13172}, {3529, 10723}, {3544, 14061}, {3815, 33749}, {3851, 10991}, {3855, 9862}, {5054, 14692}, {5079, 6033}, {5461, 22505}, {6321, 15681}, {8587, 14488}, {9756, 19130}, {11632, 15688}, {11724, 15808}, {12188, 15720}, {12243, 15715}, {13188, 15700}, {13468, 14810}, {14269, 14830}, {17504, 33813}, {17508, 34229}

X(35021) = midpoint of X(i) and X(j) for these {i,j}: {98, 6036}, {11623, 12042}
X(35021) = reflection of X(6721) in X(6036)
X(35021) = midpoint of the complement and anticomplement of X(114)
X(35021) = reflection of X(35022) in X(3530)
X(35021) = X(3530) of 1st anti-Brocard triangle
X(35021) = {X(98), X(6055)}-harmonic conjugate of X(6036)

X(35022) = X(2)X(99)∩X(114)X(382)

Barycentrics 6*a^4-6*(b^2+c^2)*a^2+b^4+4*b^2*c^2+c^4 : :
X(35022) = 3*X(2)+5*X(99), 9*X(2)-5*X(115), 21*X(2)-5*X(148), 3*X(2)-5*X(620), 13*X(2)-5*X(671), X(2)-5*X(2482), 7*X(2)-5*X(5461), 6*X(2)-5*X(6722), 11*X(2)+5*X(8591), 23*X(2)-15*X(9166), 11*X(2)-15*X(9167), 19*X(2)-15*X(14971), 7*X(2)+5*X(15300), 4*X(2)-5*X(22247), 3*X(99)+X(115), 7*X(99)+X(148), 13*X(99)+3*X(671), X(99)+3*X(2482), 7*X(99)+3*X(5461), 2*X(99)+X(6722), 11*X(99)-3*X(8591), 11*X(99)+9*X(9167), 11*X(99)+5*X(14061), 7*X(99)-3*X(15300), 9*X(99)-X(20094), 4*X(99)+3*X(22247), 7*X(99)+5*X(31274)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35022) lies on these lines: {2, 99}, {3, 14692}, {98, 10299}, {114, 382}, {187, 14148}, {230, 15301}, {315, 33252}, {376, 7908}, {538, 32459}, {542, 3631}, {546, 20399}, {548, 7895}, {550, 2794}, {626, 33234}, {754, 6390}, {1285, 34511}, {1916, 14038}, {2782, 3530}, {3244, 11711}, {3314, 7782}, {3522, 7896}, {3528, 9862}, {3529, 10722}, {3552, 7838}, {3629, 5026}, {3815, 7816}, {3851, 10992}, {3855, 13172}, {5079, 6321}, {5355, 33246}, {5471, 30472}, {5472, 30471}, {5477, 11008}, {5969, 6329}, {6033, 15681}, {6036, 14869}, {6055, 15707}, {6337, 7737}, {6680, 15048}, {6704, 31652}, {6721, 15092}, {6781, 7799}, {7735, 7781}, {7756, 7891}, {7763, 33280}, {7765, 16984}, {7778, 34504}, {7783, 7829}, {7798, 32985}, {7813, 13586}, {7826, 33014}, {7845, 8598}, {7848, 8703}, {7871, 33268}, {7903, 33244}, {7998, 27779}, {8588, 32833}, {8724, 15688}, {11623, 13188}, {11725, 15808}, {12042, 15598}, {12188, 15700}, {13187, 34752}, {13357, 32450}, {14588, 23992}, {15513, 32820}, {32458, 33254}

X(35022) = midpoint of X(i) and X(j) for these {i,j}: {99, 620}, {187, 14148}, {230, 15301}, {5461, 15300}, {6390, 32456}, {11623, 13188} X(35022) = reflection of X(6722) in X(620)
X(35022) = midpoint of the complement and anticomplement of X(115)
X(35022) = reflection of X(35021) in X(3530)
X(35022) = center of the circle {{X(99), X(620), X(907)}}
X(35022) = X(6329)-of-1st anti-Brocard triangle
X(35022) = X(11737)-of-anti-McCay triangle
X(35022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 148, 15300), (99, 2482, 620), (99, 14061, 8591), (99, 15483, 8178), (148, 31274, 5461), (620, 5461, 31274), (620, 6722, 22247), (7782, 7863, 7830), (15300, 31274, 148)

X(35023) = X(2)X(11)∩X(119)X(382)

Barycentrics 6*a^3-6*(b+c)*a^2-(b^2-8*b*c+c^2)*a+(b^2-c^2)*(b-c) : :
X(35023) = 9*X(2)-5*X(11), 3*X(2)+5*X(100), 21*X(2)-5*X(149), 3*X(2)-5*X(3035), 3*X(2)+X(6154), X(2)-5*X(6174), 6*X(2)-5*X(6667), 13*X(2)-5*X(10707), X(11)+3*X(100), 7*X(11)-3*X(149), X(11)-3*X(3035), 5*X(11)+3*X(6154), X(11)-9*X(6174), 2*X(11)-3*X(6667), 13*X(11)-9*X(10707), 3*X(11)+X(20095), 7*X(11)-15*X(31235), 11*X(11)-15*X(31272), 7*X(100)+X(149), 5*X(100)-X(6154), X(100)+3*X(6174), 2*X(100)+X(6667), 13*X(100)+3*X(10707), 9*X(100)-X(20095), 7*X(100)+5*X(31235), 11*X(100)+5*X(31272)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35023) lies on these lines: {2, 11}, {10, 9945}, {104, 10299}, {119, 382}, {165, 13257}, {214, 3244}, {495, 17563}, {518, 11575}, {546, 5840}, {550, 2829}, {952, 3530}, {1145, 3632}, {1155, 5852}, {1317, 1788}, {1319, 32426}, {1329, 4302}, {1387, 15808}, {1698, 12690}, {2802, 3636}, {2932, 19535}, {3036, 9897}, {3528, 12248}, {3529, 10728}, {3631, 5848}, {3634, 31795}, {3681, 27778}, {3851, 10993}, {3855, 13199}, {3982, 24465}, {4293, 12607}, {4316, 17757}, {4855, 8256}, {5079, 10738}, {5217, 9711}, {5220, 13243}, {5440, 5855}, {5541, 34123}, {5552, 12943}, {5851, 6594}, {6329, 9024}, {6691, 8715}, {6713, 14869}, {6745, 17768}, {8683, 15507}, {9324, 17719}, {9668, 26364}, {9780, 9963}, {10058, 19526}, {10164, 13226}, {10742, 15681}, {12331, 15720}, {12732, 16173}, {12773, 15700}, {13996, 20057}, {17564, 25439}, {33337, 34641}, {33559, 34773}

X(35023) = midpoint of X(i) and X(j) for these {i,j}: {10, 9945}, {100, 3035}, {3036, 10609}, {12331, 20418}
X(35023) = reflection of X(6667) in X(3035)
X(35023) = midpoint of the complement and anticomplement of X(11)
X(35023) = complement of the complement of X(6154)
X(35023) = (excenters-midpoints)-complement of X(13226)
X(35023) = X(15151)-of-2nd circumperp triangle
X(35023) = X(15153)-of-Fuhrmann triangle
X(35023) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1376, 5218, 3826), (3826, 5218, 6690)

X(35024) = X(2)X(101)∩X(118)X(382)

Barycentrics 6*a^4-6*(b+c)*a^3+6*b*c*a^2-(b^2-c^2)*(b-c)*a+(b^3-c^3)*(b-c) : :
X(35024) = 3*X(2)+5*X(101), 9*X(2)-5*X(116), 21*X(2)-5*X(150), 3*X(2)-5*X(6710), 13*X(2)-5*X(10708), 3*X(101)+X(116), 7*X(101)+X(150), 13*X(101)+3*X(10708), 9*X(101)-X(20096), 11*X(101)+5*X(31273), 7*X(116)-3*X(150), X(116)-3*X(6710), 13*X(116)-9*X(10708), 3*X(116)+X(20096), 11*X(116)-15*X(31273), X(150)-7*X(6710), 9*X(150)+7*X(20096), 13*X(6710)-3*X(10708), 9*X(6710)+X(20096), 11*X(6710)-5*X(31273)

See Antreas Hatzipolakis and César Lozada, Euclid 268 .

X(35024) lies on these lines: {2, 101}, {103, 10299}, {118, 382}, {546, 20401}, {2808, 3530}, {2809, 3636}, {2810, 6329}, {3244, 11712}, {3529, 10727}, {3626, 28346}, {3851, 33520}, {5079, 10739}, {6712, 14869}, {10741, 15681}, {11726, 15808}

X(35024) = midpoint of X(101) and X(6710)
X(35024) = midpoint of the complement and anticomplement of X(116)

Points on the permutation ellipse of X(1): X(35025)-X(35048)

Contributed by Clark Kimberling and Peter Moses, December 1, 2019.

Suppose that P = p : q : r (barycentrics) is a point other than X(2) = 1 : 1 : 1 in the plane of a triangle ABC. Let T denote the triangle with vertices

p : q : r
q : r : p
r : p : q

Let T' denote the obverse of T, defined in the preamble just before X(24307) by vertices

p : r : q
q : p : r
r : q : p

The six points, corresponding to the permutations pqr, qrp, rpq, prq, qpr, rqp, lie on the permutation ellipse of P, as defined in the preamble just before X(34341), 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 center and perspector of E(P) are both the centroid, X(2). Let u = length of semi-major axis of E(P) and v = length of semi-minor axis of E(P). Then

u^2 = 1/(9 (p + q + r)^2) (p^2 + q^2 + r^2 - q r - r p - p q)((a^2 + b^2 + c^2) + Sqrt[((a^2 + b^2 + c^2)^2 - 12 S^2)])
v^2 = 1/(9 (p + q + r)^2) (p^2 + q^2 + r^2 - q r - r p - p q)((a^2 + b^2 + c^2) - Sqrt[((a^2 + b^2 + c^2)^2 - 12 S^2)])

radius of orthoptic circle = (2 (a^2 + b^2 + c^2) (p^2 + q^2 + r^2 - q r - r p - p q))/(9 ( p + q + r)^2)

If P' lies on E(P), then E(P') = E(P). Moreover, if U = u : v : w is a point, other than P, then the point, other than P', in which the line UP' meets E(P), is here named the E(P,U)-antipode of P', denoted by E(P,P',U) and given by

E(P,P',U) = f(p,q,r,u,v,w) : f(q,r,p,v,w,u) : f(r,p,q,u,v,w), where

f(p,q,r,u,v,w) = (q^3 + q^2 r + q r^2 + r^3 - p q r) u^2 + p (q r + r p + p q) (v^2 + w^2)
- p (p^2 + q^2 + r^2)v w + (p (p q - 2 r^2 - 2 q r) + q (q^2 - r^2)) w u + (p (p r - 2 q^2 - 2 q r) + r (r^2 - q^2)) u v .

Centers X(35025)-X(35048) lie on the permutation ellipse of X(1) = a : b : c. The appearance if (i,j) in the following list means that E(X(1),X(i),X(1)) = X(j):

(2,3679), (6,35026), (31,35028), (37,35032), (69,35042), (75,35025), (76,35027), (82,35038), (87,35032), (100,35033), (141,35037), (175,35031), (190,35030), (256,35044), (320,35034), (321,35036), (512,35040), (517,35046), (536,35043), (561,35029), (662,35039), (693,35035), (824,35047), (894,35044), (1577,35041), (3759,35048), (4361,35045)

X(35025) = E(X(1),X(75))-ANTIPODE OF X(1)

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

X(35025) lies on these lines: {1, 75}, {10, 24505}, {141, 9505}, {190, 2640}, {1574, 33159}, {3679, 4761}, {3826, 23897}, {4363, 24345}, {4364, 24338}, {4418, 27926}, {4436, 8301}, {4440, 17794}, {4986, 5539}, {6653, 20351}, {24336, 24411}

X(35026) = E(X(1),X(6))-ANTIPODE OF X(1)

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

X(35026) lies on these lines: {1, 6}, {75, 33676}, {190, 2310}, {291, 1086}, {1282, 4557}, {3679, 4762}, {4346, 19945}, {4363, 24341}, {4419, 4443}, {4429, 10030}, {8932, 20468}, {17277, 33674}, {24318, 24396}, {24345, 24450}, {24418, 24737}, {24430, 28053}, {25280, 33677}

X(35027) = E(X(1),X(76))-ANTIPODE OF X(1)

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

X(35027) lies on these lines: {1, 76}, {24338, 24357}

X(35028) = E(X(1),X(31))-ANTIPODE OF X(1)

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

X(35028) lies on these lines: {1, 21}, {99, 4736}, {101, 2948}, {984, 24345}, {2607, 24433}, {3571, 4424}, {3679, 4041}, {4493, 24464}, {24342, 24411}

X(35028) = E(X(1),X(21))-antipode of X(1)

X(35029) = E(X(1),X(561))-ANTIPODE OF X(1)

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

X(35029) lies on this line: {1, 561}

X(35030) = E(X(1),X(190))-ANTIPODE OF X(1)

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

X(35030) lies on these lines: {1, 190}, {291, 545}, {551, 24508}, {812, 3679}

X(35031) = E(X(1),X(175))-ANTIPODE OF X(1)

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

X(35031) lies on these lines: {{1,7}, {3959,17435}, {4419,24411}, {4569,24014}, {17596,35047}, {24338,24445}

X(35031) = E(X(1),X(7))-antipode of X(1)

X(35032) = E(X(1),X(87))-ANTIPODE OF X(1)

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

X(35032) lies on these lines: {1,87}, {75,33678}, {190,2108}, {874,24524}, {984,24343}, {3501,34361}, {3679,4785}, {4363,24338}, {24334,24411}, {24335,24345}

X(35033) = E(X(1),X(100))-ANTIPODE OF X(1)

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

X(35033) lies on these lines: {1, 88}, {2254, 3679}, {4424, 24419}, {24338, 24715}, {24411, 24821}

X(35033) = E(X(1),X(88))-antipode of X(1)

X(35034) = E(X(1),X(320))-ANTIPODE OF X(1)

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

X(35034) lies on these lines: {1,320}, {824,3679}

X(35035) = E(X(1),X(693))-ANTIPODE OF X(1)

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

X(35035) lies on this line: {1, 693}, {75, 1734}

X(35036) = E(X(1),X(321))-ANTIPODE OF X(1)

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

X(35036) lies on this line: {1, 321}, {75, 3670}

X(35037) = E(X(1),X(141))-ANTIPODE OF X(1)

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

X(35037) lies on this line: on lines {1, 141}

X(35038) = E(X(1),X(82))-ANTIPODE OF X(1)

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

X(35038) lies on this line: {1, 82}

X(35039) = E(X(1),X(662))-ANTIPODE OF X(1)

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

X(35039) lies on this line: {1, 662}

X(35040) = E(X(1),X(512))-ANTIPODE OF X(1)

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

X(35040) lies on these lines: {1,512}, {38,17143}, {292,5539}, {538,3679}, {846,3229}, {4493,24464}, {5283,30229}, {8868,24576}, {24338,24404}, {24345,24423}, {24426,24455}

X(35041) = E(X(1),X(1577))-ANTIPODE OF X(1)

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

X(35041) lies on these lines: {1, 810}, {2886, 23897}, {9860, 16559}, {20935, 24524}

X(35042) = E(X(1),X(69))-ANTIPODE OF X(1)

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

X(35042) lies on this line: {1, 69}

X(35043) = E(X(1),X(536))-ANTIPODE OF X(1)

Barycentrics 2*a^3*b^2 - a^2*b^3 - 5*a^3*b*c + 4*a^2*b^2*c + a*b^3*c + 2*a^3*c^2 + 4*a^2*b*c^2 - 11*a*b^2*c^2 + 2*b^3*c^2 - a^2*c^3 + a*b*c^3 + 2*b^2*c^3 : :

X(35043) lies on these lines: {1,536}, {2,19945}, {75,889}, {190,9458}, {513,3679}, {1376,23343}, {4418,24345}, {4506,24452}, {4688,16482}, {16494,17118}, {16505,17119}, {17780,24344}, {24722,33908}

X(35044) = E(X(1),X(256))-ANTIPODE OF X(1)

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

X(35044) lies on these lines: {1,256}, {984,35041}, {3216,5661}, {3679,23878}, {4443,24315}, {24248,35040}, {24338,24424}, {24411,24463}

X(35045) = E(X(1),X(4361))-ANTIPODE OF X(1)

Barycentrics 2*a^5*b^2 - 7*a^4*b^3 + 4*a^3*b^4 - 3*a^2*b^5 + 5*a^5*b*c - 14*a^4*b^2*c + 2*a^3*b^3*c + 4*a^2*b^4*c - 9*a*b^5*c + 2*a^5*c^2 - 14*a^4*b*c^2 + a^3*b^2*c^2 + 23*a^2*b^3*c^2 - 3*a*b^4*c^2 - 6*b^5*c^2 - 7*a^4*c^3 + 2*a^3*b*c^3 + 23*a^2*b^2*c^3 + 12*a*b^3*c^3 - 2*b^4*c^3 + 4*a^3*c^4 + 4*a^2*b*c^4 - 3*a*b^2*c^4 - 2*b^3*c^4 - 3*a^2*c^5 - 9*a*b*c^5 - 6*b^2*c^5 : :

X(35045) lies on this line: {1, 3696}

X(35045) = E(X(1),X(3696))-antipode of X(1)

X(35046) = E(X(1),X(517))-ANTIPODE OF X(1)

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

X(35046) lies on these lines: {1,3}, {664,24028}, {984,24411}, {1734,3679}, {24248,24338}

X(35046) = E(X(1),X(3))-antipode of X(1)

X(35047) = E(X(1),X(824))-ANTIPODE OF X(1)

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

X(35047) lies on these lines: {1,824}, {752,1757}, {4418,27926}, {24338,24346}

X(35048) = E(X(1),X(3759))-ANTIPODE OF X(1)

Barycentrics 2*a^5*b^2 - 7*a^4*b^3 + 10*a^3*b^4 - 6*a^2*b^5 + 5*a^5*b*c - 17*a^4*b^2*c + 11*a^3*b^3*c + 4*a^2*b^4*c - 9*a*b^5*c + 2*a^5*c^2 - 17*a^4*b*c^2 + 4*a^3*b^2*c^2 + 8*a^2*b^3*c^2 - 3*b^5*c^2 - 7*a^4*c^3 + 11*a^3*b*c^3 + 8*a^2*b^2*c^3 + 3*a*b^3*c^3 + b^4*c^3 + 10*a^3*c^4 + 4*a^2*b*c^4 + b^3*c^4 - 6*a^2*c^5 - 9*a*b*c^5 - 3*b^2*c^5 : :

X(35048) lies on this line: {1, 872}

X(35048) = E(X(1),X(872))-antipode of X(1)

X(35049) = TRILINEAR POLE OF THE ALTINTAS-LOZADA-NEUBERG LINE

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

Let I = X(1), the incenter. Let P be a point on the Neuberg Cubic of ABC and P* its isogonal conjugate. The locus of the circumcenter of IPP* is a line, named Altintas-Lozada-Neuberg line in Kadir Altintas and César Lozada, Euclid 272 .

X(35049) lies on these lines: {57, 9273}, {249, 18593}, {250, 1835}, {691, 26700}, {1019, 32678}, {1262, 6354}, {1414, 14838}, {4565, 7178}

X(35049) = Cevapoint of X(57) and X(4565)
X(35049) = trilinear pole of the Altintas-Lozada-Neuberg line
X(35049) = X(i)-cross conjugate of X(j) for these (i,j): (1, 1414), (1786, 653)
X(35049) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 6741}, {8, 20982}, {9, 2611}
X(35049) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 6741), (7, 17886), (56, 2611)
X(35049) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(14838)}} and {{A, B, C, X(57), X(1835)}}
X(35049) = center of the conic {{A, B, C, X(5618), X(7339), X(23588), X(23966), X(23971)}}
X(35049) = barycentric product X(i)*X(j) for these {i, j}: {99, 26700}, {664, 13486}, {1414, 6742}, {1444, 34922}
X(35049) = barycentric quotient X(i)/X(j) for these (i, j): (1, 6741), (7, 17886), (56, 2611), (57, 8287)
X(35049) = trilinear product X(i)*X(j) for these {i, j}: {651, 13486}, {662, 26700}, {1262, 3615}, {1789, 7128}
X(35049) = trilinear quotient X(i)/X(j) for these (i, j): (2, 6741), (7, 8287), (56, 20982), (57, 2611)

X(35050) = X(5)X(523)∩X(513)X(1385)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35050) lies on these lines: {5, 523}, {56, 23189}, {110, 9811}, {513, 1385}, {522, 946}, {656, 5903}, {764, 4778}, {953, 2716}, {1001, 4833}, {3737, 11101}, {3738, 6265}, {4017, 10571}, {4086, 11681}

X(35050) = crosssum of X(2265) and X(4041)
X(35050) = center of circle {{X(1),X(3),X(4)}}

X(35051) = X(110)X(9811)∩X(396)X(523)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35051) lies on these lines: {110, 9811}, {396, 523}, {663, 11707}

X(35052) = X(110)X(9811)∩X(395)X(523)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35052) lies on these lines: {110, 9811}, {395, 523}, {663, 11708}

X(35053) = INFINITY POINT OF THE ALTINTAS-LOZADA-NEUBERG LINE

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35053) lies on these lines: {30, 511}, {110, 9811}, {1354, 3028}, {1769, 6126}, {2948, 3464}, {9138, 9810}

X(35053) = isogonal conjugate of X(35056)
X(35053) = crosspoint of X(1414) and X(2349)
X(35053) = crosssum of X(2173) and X(4041)
X(35053) = X(758)-Zayin conjugate of X(110)

X(35054) = X(110)X(9811)∩X(523)X(18285)

Barycentrics a*(b-c)*(2*a^12-3*(3*b^2+b*c+3*c^2)*a^10+4*(b+c)*b*c*a^9+(15*b^4+15*c^4+(7*b^2+17*b*c+7*c^2)*b*c)*a^8-(b+c)*(10*b^2-3*b*c+10*c^2)*b*c*a^7-(10*b^6+10*c^6+(4*b^4+4*c^4+(14*b^2+11*b*c+14*c^2)*b*c)*b*c)*a^6+(b+c)*(6*b^4+6*c^4-(3*b^2-16*b*c+3*c^2)*b*c)*b*c*a^5+(15*b^4+15*c^4+(4*b^2-13*b*c+4*c^2)*b*c)*b^2*c^2*a^4+(b^3+c^3)*(b+2*c)*(2*b+c)*(b^2-3*b*c+c^2)*b*c*a^3+(3*b^6+3*c^6-(b^4+c^4+(5*b^2-b*c+5*c^2)*b*c)*b*c)*(b^2-c^2)^2*a^2-(b^2-c^2)^3*(b-c)*(2*b^2+b*c+2*c^2)*b*c*a-(b^4-c^4)*(b^2-c^2)^3*(b^2-b*c+c^2)) : :

See Kadir Altintas and César Lozada, Euclid 272 .

X(35054) lies on these lines: {110, 9811}, {523, 18285}, {8674, 11699}

X(35055) = X(1)X(526)∩X(21)X(3738)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35055) lies on the cubic K130 and these lines: {1, 526}, {21, 3738}, {36, 238}, {58, 1769}, {74, 759}, {110, 9811}, {191, 6370}, {333, 4768}, {523, 1749}, {900, 3065}, {4516, 18191}, {6089, 12078}

X(35055) = crossdifference of every pair of points on line {X(37), X(3013)}
X(35055) = crosspoint of X(i) and X(j) for these (i,j): (86, 32680), (1414, 24624)
X(35055) = crosssum of X(42) and X(2624)
X(35055) = X(476)-Ceva conjugate of X(1)
X(35055) = X(10)-isoconjugate-of X(34921)
X(35055) = X(i)-reciprocal conjugate of X(j) for these (i,j): (484, 4552), (1333, 34921)
X(35055) = X(1637)-Zayin conjugate of X(9)
X(35055) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(24436)}} and {{A, B, C, X(36), X(74)}}
X(35055) = center of circles: {{X(1), X(484), X(3065), X(3464), X(21381)}}, {{X(5441), X(7727), X(19470)}}
X(35055) = barycentric product X(484)*X(4560)
X(35055) = barycentric quotient X(i)/X(j) for these (i, j): (484, 4552), (1333, 34921)
X(35055) = trilinear product X(484)*X(3737)
X(35055) = trilinear quotient X(i)/X(j) for these (i, j): (58, 34921), (484, 4551)

X(35056) = ISOGONAL CONJUGATE OF X(35053)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35056) lies on these lines: {523, 26700}, {2687, 21669}, {2717, 5520}

X(35056) = isogonal conjugate of X(35053)
X(35056) = cevapoint of X(2173) and X(4041)

X(35057) = ISOGONAL CONJUGATE OF X(26700)

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

See Kadir Altintas and César Lozada, Euclid 272 .

X(35057) lies on these lines: {1, 656}, {8, 4086}, {9, 4171}, {30, 511}, {35, 23226}, {55, 23189}, {80, 14224}, {100, 34921}, {319, 18160}, {1021, 8611}, {1125, 33508}, {1459, 1734}, {1577, 34301}, {1635, 13256}, {2321, 4529}, {2605, 14838}, {3024, 6741}, {3737, 4041}, {3799, 3939}, {4449, 23800}, {4814, 17418}, {4895, 17420}, {4985, 20293}, {9131, 16156}, {9979, 16157}, {10015, 21179}, {14399, 16612}, {21055, 24506}, {24680, 34340}

X(35057) = isogonal conjugate of X(26700)
X(35057) = Cevapoint of X(1) and X(9904)
X(35057) = crossdifference of every pair of points on line {X(6), X(1406)}
X(35057) = crosspoint of X(i) and X(j) for these (i,j): (8, 643), (100, 32635)
X(35057) = crosssum of X(i) and X(j) for these (i,j): (56, 4017), (513, 32636), (523, 27555)
X(35057) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (79, 33650), (109, 3648), (1464, 14731)
X(35057) = X(1043)-Beth conjugate of X(4086)
X(35057) = X(i)-Ceva conjugate of X(j) for these (i,j): (8, 6741), (100, 3647), (662, 9)
X(35057) = X(i)-complementary conjugate of X(j) for these (i,j): (56, 6741), (79, 124), (109, 3647)
X(35057) = X(i)-isoconjugate-of X(j) for these {i,j}: {56, 6742}, {65, 13486}, {79, 109}
X(35057) = X(i)-reciprocal conjugate of X(j) for these (i,j): (6, 26700), (8, 15455), (9, 6742)
X(35057) = X(8)-Waw conjugate of X(14740)
X(35057) = intersection, other than A,B,C, of conics {{A, B, C, X(8), X(758)}} and {{A, B, C, X(9), X(17768)}}
X(35057) = center of circles: {{X(1), X(484), X(3065), X(3464), X(21381)}}, {{X(5441), X(7727), X(19470)}}
X(35057) = barycentric product X(i)*X(j) for these {i, j}: {8, 14838}, {21, 7265}, {35, 4391}, {55, 18160}
X(35057) = barycentric quotient X(i)/X(j) for these (i, j): (8, 15455), (9, 6742), (35, 651), (284, 13486)
X(35057) = trilinear product X(i)*X(j) for these {i, j}: {2, 9404}, {8, 2605}, {9, 14838}, {35, 522}
X(35057) = trilinear quotient X(i)/X(j) for these (i, j): (8, 6742), (21, 13486), (35, 109), (312, 15455)

X(35058) = ISOGONAL CONJUGATE OF X(16685)

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

See Tran Quang Hung and César Lozada, Euclid 270 .

X(35058) lies on the curve Q124 and these lines: {1, 3159}, {2, 18040}, {57, 3187}, {75, 26819}, {81, 4360}, {88, 14829}, {89, 3210}, {149, 16100}, {192, 25417}, {274, 27163}, {279, 19789}, {291, 17135}, {330, 28605}, {959, 3476}, {985, 17150}, {1002, 30614}, {1015, 28654}, {1022, 17496}, {1255, 31035}, {1390, 31111}, {1929, 32914}, {2895, 9263}, {15409, 20222}, {17490, 26745}, {18152, 32020}

X(35058) = isogonal conjugate of X(16685)
X(35058) = isotomic conjugate of X(17147)
X(35058) = cyclocevian conjugate of the isogonal conjugate of X(2915)
X(35058) = cyclocevian conjugate of the isotomic conjugate of X(21287)
X(35058) = Cevapoint of X(523) and X(1015)
X(35058) = X(i)-cross conjugate of X(j) for these (i,j): (321, 2), (596, 75)
X(35058) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 3216}, {19, 22458}, {31, 17147}
X(35058) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 3216), (2, 17147), (3, 22458)
X(35058) = lies on the circumconic with center X(1015))
X(35058) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(4), X(17587)}}
X(35058) = trilinear pole of the line {513, 3814}
X(35058) = barycentric quotient X(i)/X(j) for these (i, j): (1, 3216), (2, 17147), (3, 22458), (10, 3159)
X(35058) = trilinear quotient X(i)/X(j) for these (i, j): (2, 3216), (10, 21858), (63, 22458), (75, 17147)

X(35059) = X(1)X(3)∩X(513)X(3814)

Barycentrics a*((b-c)^2*a^4+(b^2-c^2)*(b-c)*a^3-(b^4+c^4-3*(b^2+c^2)*b*c)*a^2-(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a-(b^2-c^2)^2*b*c) : :
X(35059) = X(36)-3*X(34583)

See Tran Quang Hung and César Lozada, Euclid 270 .

X(35059) lies on these lines: {1, 3}, {513, 3814}, {1739, 18191}, {1878, 5136}, {2392, 8258}, {3784, 26446}, {3937, 17757}, {5044, 32918}, {6679, 6681}, {6684, 11573}, {6699, 22102}, {19335, 34586}

X(35059) = midpoint of X(i) and X(j) for these {i,j}: {2077, 31849}, {3937, 17757}
X(35059) = crossdifference of every pair of points on line {X(650), X(16685)}
X(35059) = {X(2446), X(2447)}-harmonic conjugate of X(3953)

X(35060) = X(3)X(6)∩X(51)X(3972)

Barycentrics a^2*((b^2-c^2)^2*a^4-(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4+c^4)*b^2*c^2) : :
X(35060) = 3*X(3)+X(18322), X(187)-3*X(3111), 5*X(631)-X(11674), 3*X(3111)+X(14962), X(14712)-3*X(32442), X(14712)+3*X(33873)

See Tran Quang Hung and César Lozada, Euclid 270 .

X(35060) lies on these lines: {2, 5167}, {3, 6}, {51, 3972}, {290, 886}, {458, 5140}, {512, 625}, {620, 2387}, {631, 11674}, {2393, 5026}, {3491, 3788}, {3917, 7771}, {5031, 6697}, {5943, 7804}, {6310, 7746}, {6390, 34383}, {6699, 22103}, {7748, 14135}, {7912, 32547}, {9292, 32816}, {12042, 13754}, {14712, 32442}

X(35060) = midpoint of X(i) and X(j) for these {i,j}: {187, 14962}, {18860, 31850}, {32442, 33873}
X(35060) = complement of X(5167)
X(35060) = crossdifference of every pair of points on line {X(523), X(1613)}
X(35060) = crosssum of X(6) and X(864)
X(35060) = {X(3111), X(14962)}-harmonic conjugate of X(187)

X(35061) = X(3)X(3168)∩X(394)X(3164)

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

See Tran Quang Hung and César Lozada, Euclid 270 .

X(35061) lies on these lines: {3, 3168}, {394, 3164}

X(35061) = isotomic conjugate of the anticomplement of X(15466)
X(35061) = polar conjugate of X(64)-Ceva conjugate of X(4)

X(35062) = X(3)X(64)∩X(1352)X(6761)

Barycentrics (SB+SC)*(2*S^4+(8*R^2*(10*R^2-2*SA-3*SW)+3*SA^2+2*SA*SW-SB*SC+SW^2)*S^2-(4*R^2-SW)*(4*R^2*(24*R^2-10*SA-3*SW)+2*SA^2-2*SB*SC+SW^2)*SA) : :

See Tran Quang Hung and César Lozada, Euclid 270 .

X(35062) lies on these lines: {3, 64}, {1352, 6761}

X(35063) = (name pending)

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

See Antreas Hatzipolakis, Angel Montesdeoca and Peter Moses, Euclid 286 and Euclid 287 .

X(35063) lies on this line: {1125, 5663}

X(35064) = (name pending)

Barycentrics 2*a^34 - 24*a^32*b^2 + 137*a^30*b^4 - 499*a^28*b^6 + 1301*a^26*b^8 - 2535*a^24*b^10 + 3653*a^22*b^12 - 3575*a^20*b^14 + 1573*a^18*b^16 + 1573*a^16*b^18 - 3861*a^14*b^20 + 3991*a^12*b^22 - 2561*a^10*b^24 + 1051*a^8*b^26 - 249*a^6*b^28 + 19*a^4*b^30 + 5*a^2*b^32 - b^34 - 24*a^32*c^2 + 244*a^30*b^2*c^2 - 1153*a^28*b^4*c^2 + 3332*a^26*b^6*c^2 - 6393*a^24*b^8*c^2 + 8002*a^22*b^10*c^2 - 5407*a^20*b^12*c^2 - 310*a^18*b^14*c^2 + 3225*a^16*b^16*c^2 + 600*a^14*b^18*c^2 - 7063*a^12*b^20*c^2 + 9256*a^10*b^22*c^2 - 6287*a^8*b^24*c^2 + 2418*a^6*b^26*c^2 - 457*a^4*b^28*c^2 + 10*a^2*b^30*c^2 + 7*b^32*c^2 + 137*a^30*c^4 - 1153*a^28*b^2*c^4 + 4330*a^26*b^4*c^4 - 9268*a^24*b^6*c^4 + 11579*a^22*b^8*c^4 - 6919*a^20*b^10*c^4 - 598*a^18*b^12*c^4 + 1776*a^16*b^14*c^4 + 1951*a^14*b^16*c^4 + 465*a^12*b^18*c^4 - 9338*a^10*b^20*c^4 + 13236*a^8*b^22*c^4 - 8691*a^6*b^24*c^4 + 2887*a^4*b^26*c^4 - 394*a^2*b^28*c^4 - 499*a^28*c^6 + 3332*a^26*b^2*c^6 - 9268*a^24*b^4*c^6 + 12976*a^22*b^6*c^6 - 7871*a^20*b^8*c^6 - 420*a^18*b^10*c^6 + 990*a^16*b^12*c^6 + 1428*a^14*b^14*c^6 + 1115*a^12*b^16*c^6 + 602*a^10*b^18*c^6 - 10486*a^8*b^20*c^6 + 14608*a^6*b^22*c^6 - 8525*a^4*b^24*c^6 + 2194*a^2*b^26*c^6 - 176*b^28*c^6 + 1301*a^26*c^8 - 6393*a^24*b^2*c^8 + 11579*a^22*b^4*c^8 - 7871*a^20*b^6*c^8 - 328*a^18*b^8*c^8 + 752*a^16*b^10*c^8 + 965*a^14*b^12*c^8 + 943*a^12*b^14*c^8 + 874*a^10*b^16*c^8 + 644*a^8*b^18*c^8 - 10563*a^6*b^20*c^8 + 13237*a^4*b^22*c^8 - 6064*a^2*b^24*c^8 + 924*b^26*c^8 - 2535*a^24*c^10 + 8002*a^22*b^2*c^10 - 6919*a^20*b^4*c^10 - 420*a^18*b^6*c^10 + 752*a^16*b^8*c^10 + 912*a^14*b^10*c^10 + 549*a^12*b^12*c^10 + 876*a^10*b^14*c^10 + 858*a^8*b^16*c^10 + 682*a^6*b^18*c^10 - 9759*a^4*b^20*c^10 + 9550*a^2*b^22*c^10 - 2548*b^24*c^10 + 3653*a^22*c^12 - 5407*a^20*b^2*c^12 - 598*a^18*b^4*c^12 + 990*a^16*b^6*c^12 + 965*a^14*b^8*c^12 + 549*a^12*b^10*c^12 + 582*a^10*b^12*c^12 + 984*a^8*b^14*c^12 + 1039*a^6*b^16*c^12 + 865*a^4*b^18*c^12 - 7990*a^2*b^20*c^12 + 4368*b^22*c^12 - 3575*a^20*c^14 - 310*a^18*b^2*c^14 + 1776*a^16*b^4*c^14 + 1428*a^14*b^6*c^14 + 943*a^12*b^8*c^14 + 876*a^10*b^10*c^14 + 984*a^8*b^12*c^14 + 1512*a^6*b^14*c^14 + 1733*a^4*b^16*c^14 + 1558*a^2*b^18*c^14 - 4576*b^20*c^14 + 1573*a^18*c^16 + 3225*a^16*b^2*c^16 + 1951*a^14*b^4*c^16 + 1115*a^12*b^6*c^16 + 874*a^10*b^8*c^16 + 858*a^8*b^10*c^16 + 1039*a^6*b^12*c^16 + 1733*a^4*b^14*c^16 + 2262*a^2*b^16*c^16 + 2002*b^18*c^16 + 1573*a^16*c^18 + 600*a^14*b^2*c^18 + 465*a^12*b^4*c^18 + 602*a^10*b^6*c^18 + 644*a^8*b^8*c^18 + 682*a^6*b^10*c^18 + 865*a^4*b^12*c^18 + 1558*a^2*b^14*c^18 + 2002*b^16*c^18 - 3861*a^14*c^20 - 7063*a^12*b^2*c^20 - 9338*a^10*b^4*c^20 - 10486*a^8*b^6*c^20 - 10563*a^6*b^8*c^20 - 9759*a^4*b^10*c^20 - 7990*a^2*b^12*c^20 - 4576*b^14*c^20 + 3991*a^12*c^22 + 9256*a^10*b^2*c^22 + 13236*a^8*b^4*c^22 + 14608*a^6*b^6*c^22 + 13237*a^4*b^8*c^22 + 9550*a^2*b^10*c^22 + 4368*b^12*c^22 - 2561*a^10*c^24 - 6287*a^8*b^2*c^24 - 8691*a^6*b^4*c^24 - 8525*a^4*b^6*c^24 - 6064*a^2*b^8*c^24 - 2548*b^10*c^24 + 1051*a^8*c^26 + 2418*a^6*b^2*c^26 + 2887*a^4*b^4*c^26 + 2194*a^2*b^6*c^26 + 924*b^8*c^26 - 249*a^6*c^28 - 457*a^4*b^2*c^28 - 394*a^2*b^4*c^28 - 176*b^6*c^28 + 19*a^4*c^30 + 10*a^2*b^2*c^30 + 5*a^2*c^32 + 7*b^2*c^32 - c^34 : :

See Antreas Hatzipolakis and Peter Moses, Euclid 289 .

X(35064) lies on this line: {31607,32904}

X(35065) = CROSSSUM OF X(1) AND X(2720)

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

X(35065) lies on the Sherman line, the de Longchamps ellipse, and these lines: {1, 2716}, {36, 11719}, {55, 10016}, {244, 6129}, {652, 2170}, {1455, 5048}, {2077, 15500}, {3259, 3326}, {7004, 15635}, {10535, 12081}.

X(35065) = crosspoint of X(1) and X(2804)
X(35065) = crosssum of X(1) and X(2720)
.

Points on the Steiner inellipse (the permutation ellipse E(X(115)): X(35066)-X(35095)

Contributed by Clark Kimberling and Peter Moses, December 4, 2019.

This section extends from preambles just before X(34341) and X(35025). The points X(35066)-X(35095) are of the form E(X(115),X(k),X(115)), where E(X(115)) is the Steiner inellipse.

X(35066) = E(X(115),X(1))-ANTIPODE OF X(115)

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

X(35066) lies on the Steiner inellipse and these lines: {1, 115}, {239, 25593}, {1084, 17053}, {1086, 1100}, {1125, 1146}, {2482, 4707}, {3015, 34522}, {4988, 5642}, {15449, 17055}, {15526, 17073}, {17014, 24617}

X(35066) = X(2)-Ceva conjugate of X(17768)
X(35066) = barycentric square of X(17768)

X(35067) = E(X(115),X(3))-ANTIPODE OF X(115)

Barycentrics (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)^2 : :
Barycentrics (sin^2 2A) (b^2 cos^2 C + c^2 cos^2 B - b c cos A)^2 : :

X(35067) lies on the Steiner inellipse and these lines: {3, 115}, {6, 15525}, {32, 34853}, {69, 248}, {131, 187}, {216, 1084}, {1146, 34851}, {2482, 6334}, {3163, 9475}, {3284, 23992}, {6132, 11672}, {6388, 32654}, {14961, 18334}, {15449, 22052}

X(35067) = isogonal conjugate of polar conjugate of X(2974)
X(35067) = complement of X(35142)
X(35067) = X(2)-Ceva conjugate of X(3564)
X(35067) = barycentric square of X(3564)

X(35068) = E(X(115),X(10))-ANTIPODE OF X(115)

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

X(35068) lies on the Steiner inellipse and these lines: {2, 18827}, {6, 645}, {10, 115}, {37, 1084}, {76, 21431}, {99, 9509}, {220, 15628}, {239, 350}, {594, 4092}, {799, 25820}, {960, 1146}, {1015, 1107}, {1086, 1213}, {1500, 4075}, {1509, 17499}, {2482, 27929}, {3124, 3952}, {3661, 20538}, {3912, 10026}, {4010, 4839}, {4037, 4829}, {4039, 16369}, {4847, 23918}, {4969, 7067}, {5969, 24505}, {5976, 6626}, {6376, 21604}, {15449, 21249}, {15526, 18589}, {16592, 24003}, {16597, 23992}, {18268, 25819}, {20340, 20461}, {20683, 20723}, {23917, 25006}, {29593, 31057}

X(35068) = complement of X(18827)
X(35068) = X(2)-Ceva conjugate of X(740)
X(35068) = barycentric square of X(740)

X(35069) = E(X(115),X(37))-ANTIPODE OF X(115)

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

X(35069) lies on the Steiner inellipse and these lines: {2, 14616}, {6, 5546}, {9, 3013}, {36, 2245}, {37, 115}, {125, 25379}, {214, 1015}, {220, 15627}, {338, 4552}, {594, 2197}, {662, 34990}, {1030, 7054}, {1084, 16584}, {1086, 3666}, {1146, 1213}, {1187, 2277}, {1214, 15526}, {2503, 3196}, {2610, 21828}, {2615, 16598}, {2643, 21825}, {2854, 24500}, {3733, 9155}, {4092, 21891}, {4286, 8041}, {4370, 21894}, {4557, 20975}, {8287, 16578}, {8679, 20733}, {15449, 16587}, {17398, 34460}

X(35069) = X(2)-Ceva conjugate of X(758)
X(35069) = barycentric square of X(758)

X(35070) = E(X(115),X(75))-ANTIPODE OF X(115)

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

X(35070) lies on the Steiner inellipse and these lines: {75, 115}, {1015, 4357}, {1084, 4364}, {1086, 3846}, {2482, 4455}, {15526, 20254}, {25530, 26143}

X(35070) = complement of X(35143)
X(35070) = X(2)-Ceva conjugate of X(35101)
X(35070) = barycentric square of X(35101)

X(35071) = E(X(115),X(122))-ANTIPODE OF X(115)

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

Let A'B'C' be the orthic triangle. Let L_A be the van Aubel line of triangle AB'C', and define L_B and L_C cyclically. Let A" = L_B∩L_C, and define B" and C" cyclically. Triangle A"B"C" is inversely similar to ABC, with similicenter X(35071). (Randy Hutson, December 8, 2019)

X(35071) lies on the Steiner inellipse and these lines: {2, 6528}, {3, 1625}, {32, 14379}, {39, 23976}, {115, 122}, {187, 12096}, {216, 549}, {393, 18890}, {577, 18877}, {647, 1562}, {1086, 16595}, {1146, 16573}, {1511, 22052}, {1636, 2972}, {1971, 6760}, {2482, 6509}, {3199, 20329}, {5254, 9243}, {8552, 15526}, {14961, 23967}, {18591, 23980}

X(35071) = midpoint of X(3) and X(14941)
X(35071) = isogonal conjugate of X(34538)
X(35071) = isotomic conjugate of polar conjugate of X(34980)
X(35071) = complement of X(6528)
X(35071) = X(92)-isoconjugate of X(32230)
X(35071) = X(2)-Ceva conjugate of X(520)
X(35071) = perspector of circumparabola centered at X(520)
X(35071) = center of hyperbola {{A,B,C,X(2),X(3)}} (the isogonal conjugate of the van Aubel line, and the locus of trilinear poles of lines passing through X(520))
X(35071) = crossdifference of every pair of points on line X(107)X(1624) (the tangent to the circumcircle at X(107))
X(35071) = crosssum of X(i) and X(j) for these {i,j}: {6, 107}, {112, 1614}, {648, 15466}, {1093, 6529}, {2052, 15352}, {23590, 32230}
X(35071) = crosssum of circumcircle intercepts of van Aubel line
X(35071) = crosspoint of X(i) and X(j) for these {i,j}: {2, 520}, {525, 6662}, {647, 14642}, {577, 32320}
X(35071) = trilinear pole of line X(23613)X(33571)
X(35071) = barycentric square of X(520)

X(35072) = E(X(115),X(123))-ANTIPODE OF X(115)

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

X(35072) lies on the Steiner inellipse and these lines: {2, 18026}, {3, 101}, {6, 268}, {9, 216}, {37, 23986}, {63, 6509}, {115, 123}, {122, 26933}, {219, 577}, {394, 6517}, {418, 3690}, {644, 1809}, {650, 5514}, {663, 11918}, {867, 5190}, {960, 11672}, {1015, 34588}, {1040, 34526}, {1073, 1407}, {1086, 16596}, {1146, 2968}, {1212, 17102}, {1436, 30457}, {2052, 7361}, {2170, 35014}, {2193, 8559}, {2323, 3284}, {2972, 3937}, {3002, 8558}, {3040, 35046}, {3163, 15670}, {4091, 7215}, {5745, 18592}, {6389, 27509}, {7004, 7117}, {7085, 26880}, {7169, 33581}, {14379, 26927}, {15526, 16595}, {16731, 23983}, {20532, 34823}, {26867, 26898}, {26874, 26911}, {26876, 26915}, {26884, 34147}

X(35072) = isogonal conjugate of X(23984)
X(35072) = isotomic conjugate of polar conjugate of X(3270)
X(35072) = complement of X(18026)
X(35072) = X(2)-Ceva conjugate of X(521)
X(35072) = crosspoint of X(2) and X(521)
X(35072) = perspector of circumparabola centered at X(521)
X(35072) = center of hyperbola {{A,B,C,X(2),X(21)}} (the locus of trilinear poles of lines passing through X(521))
X(35072) = crossdifference of every pair of points on line X(108)X(676) (the tangent to the circumcircle at X(108))
X(35072) = crosssum of X(6) and X(108)
X(35072) = barycentric square of X(521)

X(35073) = E(X(115),X(141))-ANTIPODE OF X(115)

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

X(35073) lies on the Steiner inellipse and these lines: {2, 670}, {115, 141}, {126, 325}, {538, 30736}, {599, 34383}, {694, 21358}, {804, 2482}, {888, 6786}, {1015, 3739}, {1086, 3741}, {1146, 21246}, {1368, 15526}, {1502, 32746}, {2882, 7818}, {3163, 9035}, {3314, 17416}, {6374, 7757}, {11160, 25319}, {12036, 31174}, {15449, 21248}, {16098, 32216}, {21356, 25332}, {22165, 25327}

X(35073) = reflection of X(1084) in X(2)
X(35073) = complement of X(3228)
X(35073) = Steiner inellipse antipode of X(1084)
X(35073) = X(2)-Ceva conjugate of X(538)
X(35073) = barycentric square of X(538)

X(35074) = E(X(115),X(142))-ANTIPODE OF X(115)

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

X(35074) lies on the Steiner inellipse and these lines: {115, 142}, {1015, 3946}, {1084, 3752}, {1086, 3742}, {1146, 3739}, {15526, 34822}

X(35074) = complement of X(35144)
X(35074) = X(2)-Ceva conjugate of X(35102)
X(35074) = barycentric square of X(35102)

X(35075) = E(X(115),X(226))-ANTIPODE OF X(115)

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

X(35075) lies on the Steiner inellipse and these lines: {10, 15526}, {115, 226}, {942, 1086}, {1084, 16583}, {1146, 6708}, {1500, 4605}, {1944, 1948}, {3269, 4566}, {5745, 18592}

X(35075) = complement of X(35145)
X(35075) = X(2)-Ceva conjugate of X(8680)
X(35075) = barycentric square of X(8680)

X(35076) = E(X(115),X(244))-ANTIPODE OF X(115)

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

X(35076) lies on the Steiner inellipse and these lines: {2, 6540}, {115, 244}, {3125, 4988}, {4370, 5750}, {4984, 30592}, {6532, 6537}, {6547, 24185}

X(35076) = complement of X(6540)
X(35076) = X(2)-Ceva conjugate of X(4977)
X(35076) = barycentric square of X(4977)

X(35077) = E(X(115),X(325))-ANTIPODE OF X(115)

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

X(35077) lies on the Steiner inellipse and these lines: {2, 18829}, {115, 325}, {141, 23992}, {512, 2482}, {524, 1084}, {5106, 14607}, {5641, 7818}, {7801, 34364}, {7810, 18334}, {34344, 34511}

X(35077) = complement of X(35146)
X(35077) = X(2)-Ceva conjugate of X(5969)
X(35077) = barycentric square of X(5969)

X(35078) = E(X(115),X(512))-ANTIPODE OF X(115)

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

X(35078) lies on the Steiner inellipse and these lines: {2, 18829}, {32, 2966}, {115, 512}, {230, 3229}, {385, 3978}, {523, 1084}, {538, 1569}, {2086, 11183}, {5306, 23967}, {5309, 30229}, {7668, 15449}, {7792, 10161}, {10026, 20532}, {15527, 34294}

X(35078) = X(2)-Ceva conjugate of X(804)
X(35078) = barycentric square of X(804)

X(35079) = E(X(115),X(513))-ANTIPODE OF X(115)

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

X(35079) lies on the Steiner inellipse and these lines: {115, 513}, {230, 23980}, {350, 30996}, {523, 1015}, {524, 13466}, {536, 2482}, {650, 1084}, {1086, 4369}, {3985, 4370}, {5209, 5291}, {8609, 11672}, {24345, 24512}

X(35079) = complement of X(35147)
X(35079) = X(2)-Ceva conjugate of X(2787)
X(35079) = barycentric square of X(2787)

X(35080) = E(X(115),X(514))-ANTIPODE OF X(115)

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

X(35080) lies on the Steiner inellipse and these lines: {115, 514}, {230, 23972}, {239, 24636}, {325, 20532}, {519, 2482}, {523, 1086}, {524, 4370}, {594, 4562}, {1015, 14838}, {1084, 6586}, {1931, 6157}, {3017, 24281}, {3912, 10026}, {6547, 24185}, {8608, 11672}, {15526, 20315}, {17392, 24345}, {25536, 31998}

X(35080) = complement of X(35148)
X(35080) = X(2)-Ceva conjugate of X(2786)
X(35080) = barycentric square of X(2786)

X(35081) = E(X(115),X(515))-ANTIPODE OF X(115)

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

X(35081) lies on the Steiner inellipse and these lines: {115, 515}, {230, 1146}, {523, 23986}, {1084, 8607}, {1086, 1429}, {6589, 11672}

X(35081) = complement of X(35149)
X(35081) = X(2)-Ceva conjugate of X(2792)
X(35081) = barycentric square of X(2792)

X(35082) = E(X(115),X(516))-ANTIPODE OF X(115)

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

X(35082) lies on the Steiner inellipse and these lines: {115, 516}, {230, 1086}, {238, 1146}, {523, 23972}, {1084, 8608}, {6586, 11672}, {10026, 15526}

X(35082) = complement of X(35150)
X(35082) = X(2)-Ceva conjugate of X(2784)
X(35082) = barycentric square of X(2784)

X(35083) = E(X(115),X(517))-ANTIPODE OF X(115)

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

X(35083) lies on the Steiner inellipse and these lines: {115, 517}, {230, 1015}, {523, 23980}, {650, 11672}, {1084, 8609}, {1086, 16609}, {4370, 4529}

X(35083) = complement of X(35151)
X(35083) = X(2)-Ceva conjugate of X(2783)
X(35083) = barycentric square of X(2783)

X(35084) = E(X(115),X(518))-ANTIPODE OF X(115)

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

X(35084) lies on the Steiner inellipse and these lines: {115, 518}, {523, 6184}, {1084, 3290}, {1146, 4966}, {2482, 4762}, {24345, 24512}

X(35084) = complement of X(35152)
X(35084) = X(2)-Ceva conjugate of X(2795)
X(35084) = barycentric square of X(2795)

X(35085) = E(X(115),X(519))-ANTIPODE OF X(115)

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

X(35085) lies on the Steiner inellipse and these lines: {115, 519}, {239, 320}, {514, 2482}, {523, 4370}, {1015, 16611}, {1084, 8610}, {1213, 23992}, {12035, 20532}, {17953, 17969}

X(35085) = complement of X(35153)
X(35085) = X(2)-Ceva conjugate of X(2796)
X(35085) = barycentric square of X(2796)

X(35086) = E(X(115),X(522))-ANTIPODE OF X(115)

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

X(35086) lies on the Steiner inellipse and these lines: {115, 522}, {230, 23986}, {523, 1146}, {527, 2482}, {1084, 6589}, {1086, 17069}, {8607, 11672}, {17950, 17966}

X(35086) = complement of X(35154)
X(35086) = X(2)-Ceva conjugate of X(2785)
X(35086) = barycentric square of X(2785)

X(35087) = E(X(115),X(524))-ANTIPODE OF X(115)

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

X(35087) lies on the Steiner inellipse and these lines: {2, 892}, {115, 524}, {325, 17416}, {523, 2482}, {543, 9182}, {599, 14995}, {671, 31998}, {1084, 3291}, {1641, 8371}, {4590, 8591}, {5159, 15526}, {5461, 23991}, {7840, 15449}, {9165, 14971}, {10717, 15899}, {12036, 31174}, {16092, 31655}

X(35087) = reflection of X(23992) in X(2)
X(35087) = complement of X(18823)
X(35087) = crosssum of circumcircle intercepts of Parry-isodynamic circle
X(35087) = crosssum of circumcircle intercepts of line PU(62) (line X(6)X(351))
X(35087) = Steiner-inellipse antipode of X(23992)
X(35087) = X(2)-Ceva conjugate of X(543)
X(35087) = barycentric square of X(543)

X(35088) = E(X(115),X(525))-ANTIPODE OF X(115)

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

X(35088) lies on the Steiner inellipse and these lines: {2, 2966}, {30, 114}, {115, 525}, {125, 31174}, {230, 23976}, {232, 297}, {338, 15449}, {339, 18314}, {381, 34360}, {401, 7925}, {523, 15526}, {524, 3163}, {599, 14995}, {1084, 2485}, {1086, 21187}, {3258, 14417}, {7752, 10684}, {7888, 34157}, {9473, 9476}, {10026, 23972}, {13485, 23357}, {16041, 30227}, {18311, 23992}

X(35088) = midpoint of X(2) and X(5641)
X(35088) = reflection of X(23967) in X(2)
X(35088) = complement of X(2966)
X(35088) = Steiner inellipse antipode of X(23967)
X(35088) = center of hyperbola {{A,B,C,X(2),X(325)}} (the isotomic conjugate of line X(2)X(98))
X(35088) = X(2)-Ceva conjugate of X(2799)
X(35088) = barycentric square of X(2799)

X(35089) = E(X(115),X(536))-ANTIPODE OF X(115)

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

X(35089) lies on the Steiner inellipse and these lines: {115, 536}, {513, 2482}, {523, 13466}, {524, 1015}, {1086, 4892}, {1211, 23992}

X(35089) = complement of X(35155)
X(35089) = X(2)-Ceva conjugate of X(35103)
X(35089) = barycentric square of X(35103)

X(35090) = E(X(115),X(650))-ANTIPODE OF X(115)

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

X(35090) lies on the Steiner inellipse and these lines: {115, 650}, {187, 1017}, {647, 1015}, {1086, 14838}, {2482, 25083}, {3003, 23980}, {3163, 8609}, {5127, 17796}

X(35090) = complement of X(35156)
X(35090) = X(2)-Ceva conjugate of X(8674)
X(35090) = barycentric square of X(8674)

X(35091) = E(X(115),X(656))-ANTIPODE OF X(115)

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

X(35091) lies on the Steiner inellipse and these lines: {44, 23972}, {115, 656}, {522, 1146}, {527, 1323}, {900, 1566}, {1015, 6129}, {1086, 7658}, {1944, 2482}, {2310, 6608}, {3163, 8558}, {6184, 10427}, {17435, 23757}

X(35091) = complement of X(35157)
X(35091) = X(2)-Ceva conjugate of X(6366)
X(35091) = crosssum of X(109) and X(2291)
X(35091) = crosspoint of X(522) and X(527)
X(35091) = barycentric square of X(6366)

X(35092) = E(X(115),X(661))-ANTIPODE OF X(115)

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

X(35092) lies on the Steiner inellipse and these lines: {2, 4555}, {6, 34232}, {8, 34362}, {32, 32641}, {44, 519}, {45, 24864}, {115, 661}, {121, 20532}, {239, 2482}, {514, 1086}, {650, 1015}, {1016, 4473}, {1017, 1317}, {1145, 6184}, {1146, 4521}, {1573, 5701}, {1647, 2087}, {2226, 8046}, {3008, 31201}, {3125, 4988}, {3163, 30117}, {3912, 13466}, {4422, 6633}, {4542, 33922}, {5222, 24281}, {6550, 14442}, {8609, 23980}, {17310, 32045}, {17435, 23757}

X(35092) = complement of X(4555)
X(35092) = X(2)-Ceva conjugate of X(900)
X(35092) = barycentric square of X(900)
X(35092) = crosssum of X(101) and X(106)
X(35092) = crosspoint of X(514) and X(519)

X(35093) = E(X(115),X(857))-ANTIPODE OF X(115)

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

X(35093) lies on the Steiner inellipse and these lines: {115, 857}, {241, 1015}, {514, 23972}, {516, 1086}, {905, 6184}, {1146, 3008}, {5222, 24281}

X(35093) = complement of X(35158)
X(35093) = X(2)-Ceva conjugate of X(5845)
X(35093) = barycentric square of X(5845)

X(35094) = E(X(115),X(1111))-ANTIPODE OF X(115)

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

X(35094) lies on the Steiner inellipse and these lines: {2, 666}, {7, 34361}, {36, 2482}, {115, 1111}, {116, 514}, {120, 13466}, {239, 25593}, {241, 3693}, {522, 1086}, {527, 4370}, {905, 1015}, {1084, 17058}, {1252, 8047}, {3008, 23972}, {3126, 3675}, {3259, 4728}, {4379, 6075}, {6173, 24411}, {7794, 22116}, {10025, 17266}, {13577, 14827}, {16588, 18214}, {18391, 24281}, {20532, 20540}, {20924, 32458}

X(35094) = reflection of X(35113) in X(2)
X(35094) = complement of X(666)
X(35094) = isotomic conjugate of isogonal conjugate of X(35505)
X(35094) = Steiner-inellipse-antipode of X(35113)
X(35094) = X(2)-Ceva conjugate of X(918)
X(35094) = barycentric square of X(918)

X(35095) = E(X(115),X(1329))-ANTIPODE OF X(115)

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

X(35095) lies on the Steiner inellipse and these lines: {9, 1084}, {115, 1329}, {220, 2311}, {960, 1015}, {1086, 21246}, {1146, 3704}, {3037, 35040}

X(35095) = complement of X(35159)
X(35095) = X(2)-Ceva conjugate of X(35104)
X(35095) = barycentric square of X(35104)

4th intersection of named circumconics: X(35096)-X(35098)

This preamble and centers X(35096)-X(35098) were contributed by César Eliud Lozada, December 4, 2019.

If K1, K2 are two circumconics of ABC then the 4th intersection of K1 and K2 (other than A, B, C) is the trilinear pole wrt ABC of the line joining the perspectors of K1 and K2.

This is a list of the 4th points of intersection of some named conics:
X(4) = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Kiepert hyperbola
X(110) = intersection, other than A, B, C, of these conics: Johnson circumconic and MacBeath circumconic
X(265) = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Johnson circumconic
X(290) = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and Steiner circumellipse
X(648) = intersection, other than A, B, C, of these conics: MacBeath circumconic and Steiner circumellipse
X(671) = intersection, other than A, B, C, of these conics: Kiepert hyperbola and Steiner circumellipse
X(895) = intersection, other than A, B, C, of these conics: Jerabek circumhyperbola and MacBeath circumconic
X(903) = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Steiner circumellipse
X(1246) = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Jerabek hyperbola.
X(2481) = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and Steiner circumellipse
X(2986) = intersection, other than A, B, C, of these conics: Kiepert hyperbola and MacBeath circumconic
X(2989) = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and MacBeath circumconic
X(6528) = intersection, other than A, B, C, of these conics: Johnson circumconic and Steiner circumellipse
X(8759) = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and MacBeath circumconic
X(35096) = intersection, other than A, B, C, of these conics: circumhyperbola dual of Yff parabola and Johnson circumconic
X(35097) = intersection, other than A, B, C, of these conics: Feuerbach hyperbola and Johnson circumconic
X(35098) = intersection, other than A, B, C, of these conics: Johnson circumconic and Kiepert hyperbola

X(35096) = 4TH INTERSECTION OF THESE CIRCUMCONICS: JOHNSON CIRCUMCONIC AND CIRCUMHYPERBOLA DUAL OF YFF PARABOLA

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

X(35096) lies on Johnson circumconic, circumhyperbola dual of Yff parabola, and these lines: {27,1625}, {86,23181}

X(35096) = intersection, other than A,B,C, of circumhyperbola dual of Yff parabola and Johnson circumconic
X(35096) = trilinear pole of the line {216, 514}

X(35097) = 4TH INTERSECTION OF THESE CIRCUMCONICS: FEUERBACH HYPERBOLA AND JOHNSON CIRCUMCONIC

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

X(35097) lies on Feuerbach hyperbola, Johnson circumconic, and these lines: {21,23181}, {943,7567}, {1172,1625}

X(35097) = antigonal conjugate of the isogonal conjugate of X(31849)
X(35097) = Cevapoint of X(i) and X(j) for these (i,j): (5, 517), (11, 8677)
X(35097) = intersection, other than A,B,C, of Feuerbach hyperbola and Johnson circumconic
X(35097) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(10), X(946)}}
X(35097) = trilinear pole of the line {216, 650}

X(35098) = 4TH INTERSECTION OF THESE CIRCUMCONICS: JOHNSON CIRCUMCONIC AND KIEPERT HYPERBOLA

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

X(35098) lies on Johnson circumconic, Kiepert hyperbola, and these lines: {2,23181}, {4,1625}, {76,11444}, {83,13434}, {98,10313}, {110,275}, {262,15355}, {418,8901}, {2052,3060}, {5480,30505}, {13599,26216}, {16277,19164}

X(35098) = isotomic conjugate of the anticomplement of X(3289)
X(35098) = antigonal conjugate of the isogonal conjugate of X(31850)
X(35098) = antigonal conjugate of the antitomic conjugate of X(35098)
X(35098) = antitomic conjugate of the isogonal conjugate of X(31850)
X(35098) = antitomic conjugate of the antigonal conjugate of X(35098)
X(35098) = Cevapoint of X(i) and X(j) for these (i,j): (5, 511), (520, 3150)
X(35098) = lies on the circumconic with center X(3150))
X(35098) = intersection, other than A,B,C, of Johnson circumconic and Kiepert hyperbola
X(35098) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(5), X(30506)}}
X(35098) = trilinear pole of the line {216, 523}

X(35099) = X(1)X(147)∩X(10)X(4220)

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

See César Lozada, Euclid 291 .

X(35099) lies on these lines: {1, 147}, {10, 4220}, {20, 2938}, {185, 2772}, {516, 4647}, {740, 13442}, {1733, 4292}, {3178, 7413}, {3430, 21085}, {9840, 29040}, {17733, 26118}

X(35099) = center of the cevian circle of X(75)

X(35100) = X(1)X(650)∩X(3)X(11124)

Barycentrics a*(a^5-(b+c)*a^4-(2*b^2+b*c+2*c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+(b^4+c^4-(b^2+6*b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c))*(b-c) : :
X(35100) = X(3)-3*X(11124)

See César Lozada, Euclid 291 .

X(35100) lies on these lines: {1, 650}, {3, 11124}, {55, 11247}, {155, 521}, {496, 10006}, {513, 3579}, {514, 6684}, {693, 27529}, {905, 21105}, {1938, 35004}, {3126, 9709}, {3295, 32195}, {3309, 11500}, {3746, 11193}, {8760, 26285}, {9366, 10284}, {9373, 32612}, {25005, 26641}

X(35100) = center of the cevian circle of X(100)

Barycentric positive square roots of points on the Steiner inellipse: X(35101)-X(35104)

Contributed by Clark Kimberling and Peter Moses, December 4, 2019.

This section extends from the introduction to permutation ellipses in the preamble just before X(35025). The points X(35101)-X(35104) are positive square roots of points the form E(X(115),X(k),X(115)), where E(X(115)) is the Steiner inellipse.

Suppose that a point f : g : h lies on the line at infinity (i.e., f + g + h = 0). Then the point f² : g² : h² lies on the Steiner inellipse, E(X(115)). Conversely, if F² : G² : H² lie on the ellipse, then the following six points lie on the line at infinity:

f : g : h

g : h : f

h : f : g

f : h : g

g : f : h

h : g : f

where f, g, h are the positive square roots of F,G,H.

The appearance of (i,j) in the following list means that the positive square root of the point X(i) on E(X(115)) is X(j):

(35066,17768), (35067,3564), (35068,740), (35069,758), (35070,35101), (35071,520), (35072,521), (35073,538), (35074,35102), (35075,8680), (35076,4977},
{35077,5969), (35078,804), (35079,2787), (35080,2786), (35081,2792), (35082,2784), (35083,2783), (35084,2795), (35085,2796), (35086,2785},
{35087,543), (35088,2799), (35089,35103), (35090,8674), (35091,6366), (35092,900), (35093,5845), (35094,918), (35095,35104)

The isogonal conjugates of the points X(35101)-X(35104), all on the circumcircle, are indexed as X(35105)-X(35108).

X(35101) = BARYCENTRIC SQUARE ROOT OF X(35070)

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

X(35101) lies on these lines: {6, 24282}, {30, 511}, {81, 330}, {141, 3735}, {230, 5977}, {257, 312}, {304, 3959}, {350, 21138}, {1086, 20924}, {2176, 21216}, {2238, 17497}, {2895, 20055}, {3125, 14210}, {3589, 24254}, {3727, 20911}, {3752, 6703}, {3912, 21331}, {4424, 25349}, {4595, 33889}, {4642, 16720}, {4754, 17164}, {5291, 33952}, {14951, 22034}, {15985, 18697}, {17144, 33890}, {17759, 33946}, {17760, 20691}, {17946, 32020}, {18156, 20271}, {21025, 33939}

X(35101) = isogonal conjugate of X(35105)
X(35101) = X(2)-Ceva conjugate of X(35070)

X(35102) = BARYCENTRIC SQUARE ROOT OF X(35074)

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

X(35102) lies on these lines: {6, 24249}, {30, 511}, {36, 24685}, {57, 85}, {69, 24247}, {101, 6647}, {329, 1655}, {495, 25353}, {672, 21232}, {956, 24333}, {999, 24331}, {1212, 3452}, {1475, 17048}, {1478, 24694}, {2170, 20347}, {2238, 7200}, {2316, 4510}, {3294, 7278}, {3421, 7960}, {3501, 16284}, {3509, 3732}, {3684, 5088}, {3820, 25352}, {3985, 14210}, {4032, 4416}, {4051, 17753}, {4059, 4875}, {4667, 25371}, {5080, 24712}, {6692, 6706}, {9965, 20089}, {15325, 25342}, {16583, 24215}, {16611, 17205}, {17026, 20925}, {17169, 21921}, {17755, 20924}, {17757, 24318}, {20196, 31269}, {31142, 31169}

X(35102) = isogonal conjugate of X(35106)
X(35102) = X(2)-Ceva conjugate of X(35074)

X(35103) = BARYCENTRIC SQUARE ROOT OF X(35089)

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

X(35103) lies on these lines: {1, 4760}, {2, 3125}, {10, 25383}, {30, 511}, {81, 99}, {98, 28474}, {115, 1211}, {148, 2895}, {291, 4424}, {321, 668}, {551, 9507}, {599, 33936}, {620, 6703}, {1015, 2482}, {1281, 7983}, {1572, 16834}, {2051, 34899}, {3023, 6021}, {4568, 21888}, {4945, 9166}, {5461, 27076}, {8591, 9263}, {8596, 31298}, {8671, 9888}, {10800, 32115}, {12258, 17793}, {14061, 31247}, {17461, 24357}, {17497, 21839}, {17738, 24261}, {20086, 20094}, {21950, 25683}

X(35103) = isogonal conjugate of X(35107)
X(35103) = X(2)-Ceva conjugate of X(35089)

X(35104) = BARYCENTRIC SQUARE ROOT OF X(35095)

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

X(35104) lies on these lines: {1, 2092}, {6, 24265}, {8, 314}, {30, 511}, {51, 32915}, {65, 3879}, {100, 5061}, {181, 1999}, {256, 3680}, {312, 23638}, {958, 4877}, {960, 3686}, {970, 17733}, {1043, 10544}, {1045, 2136}, {1266, 1463}, {1284, 3882}, {1320, 11609}, {1401, 3210}, {1469, 3875}, {3030, 5205}, {3056, 3886}, {3057, 3883}, {3271, 3685}, {3687, 21334}, {3690, 32864}, {3704, 18178}, {3712, 18191}, {3736, 3913}, {3755, 17792}, {3789, 24392}, {3792, 4716}, {3869, 17363}, {3909, 4442}, {3917, 32860}, {3937, 32845}, {4684, 20358}, {7075, 16588}, {9564, 11679}, {12782, 32468}, {17156, 26893}, {18163, 18235}, {22076, 27368}

X(35104) = isogonal conjugate of X(35108)
X(35104) = X(2)-Ceva conjugate of X(35095)

X(35105) = ISOGONAL CONJUGATE OF X(35101)

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

X(35105) lies on these lines: {37, 932}, {99, 192}, {100, 172}, {101, 7122}, {110, 2176}, {604, 29055}, {805, 18268}, {2703, 21760}, {5291, 8709}, {8691, 16468}

X(35106) = ISOGONAL CONJUGATE OF X(35102)

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

X(35106) lies on these lines: {9, 99}, {41, 110}, {100, 1334}, {108, 2333}, {109, 213}, {112, 2212}, {927, 18785}, {934, 1400}, {4649, 14074}

X(35107) = ISOGONAL CONJUGATE OF X(35103)

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

X(35107 lies on these lines: {6, 2703}, {37, 898}, {98, 28475}, {99, 536}, {100, 187}, {101, 922}, {110, 3230}, {111, 667}, {238, 8691}, {511, 28474}, {512, 739}, {572, 2705}, {691, 1333}, {813, 16785}, {1083, 2753}, {2030, 32722}

X(35108) = ISOGONAL CONJUGATE OF X(35104)

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

X(35108) lies on these lines: {1, 6010}, {56, 99}, {65, 29151}, {100, 1402}, {101, 5247}, {107, 7337}, {110, 1397}, {171, 1293}, {901, 5061}, {1001, 8690}, {1319, 2703}, {1388, 29177}, {1420, 29055}, {2743, 5143}, {3600, 26138}

X(35109) = X(1)X(860)∩X(758)X(5494)

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

See Antreas Hatzipolakis and Peter Moses, Euclid 302 .

X(35109) lies on these lines: {1, 860}, {758, 5494}

Points on the Steiner inellipse (the permutation ellipse E(X(115)): X(35110)-X(35135)

Contributed by Clark Kimberling and Peter Moses, December 5, 2019.

This section extends from preambles just before X(34341) and X(35025). The points X(35110)-X(35135) are of the form E(X(115),X(k),X(h)), where E(X(115)) is the Steiner inellipse and h = 1086 or 2482.

X(35110) = E(X(115),X(1))-ANTIPODE OF X(1086)

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

X(35110) lies on the Steiner inellipse and these lines: {1, 528}, {2, 664}, {88, 17014}, {115, 15903}, {220, 651}, {223, 31142}, {241, 16586}, {519, 35094}, {524, 35086}, {527, 1323}, {544, 1565}, {551, 28850}, {918, 1642}, {952, 10708}, {1015, 3752}, {1212, 16578}, {1214, 35072}, {1358, 17439}, {1407, 2094}, {1638, 6174}, {2482, 2785}, {3008, 31201}, {3756, 24281}, {3960, 6184}, {4530, 31192}, {6354, 6505}, {6706, 25723}, {9502, 30725}, {10015, 23972}, {12035, 13466}, {15526, 18641}, {16585, 25091}

X(35110) = reflection of X(1146) in X(2)
X(35110) = complement of X(1121)
X(35110) = X(2)-Ceva conjugate of X(527)
X(35110) = barycentric square of X(527)
X(35110) = Steiner inellipse antipode of X(1146)

X(35111) = E(X(115),X(9))-ANTIPODE OF X(1086)

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

X(35111) lies on the Steiner inellipse and these lines: {1, 3039}, {8, 220}, {9, 1086}, {115, 13540}, {1015, 1212}, {1642, 23972}, {4521, 12035}, {6552, 6558}, {6603, 35092}, {9502, 30725}, {17107, 24796}

X(35111) = complement of X(35160)
X(35111) = X(2)-Ceva conjugate of X(5853)
X(35111) = barycentric square of X(5853)

X(35112) = E(X(115),X(30))-ANTIPODE OF X(1086)

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

X(35112) lies on the Steiner inellipse and these lines: {30, 1086}, {514, 3163}, {519, 15526}, {525, 4370}, {1015, 18593}, {3017, 24281}, {11346, 34360}, {16052, 35088}

X(35112) = complement of X(35161)
X(35112) = X(2)-Ceva conjugate of isogonal conjugate of Moses-radical-circle-inverse of X(106)
X(35112) = barycentric square of isogonal conjugate of Moses-radical-circle-inverse of X(106)

X(35113) = E(X(115),X(44))-ANTIPODE OF X(1086)

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

X(35113) lies on the Steiner inellipse and these lines: {1, 35092}, {2, 666}, {9, 35091}, {44, 527}, {115, 2238}, {294, 5375}, {519, 1146}, {522, 4370}, {650, 6174}, {1015, 3290}, {1642, 1643}, {3509, 35076}, {5222, 24407}, {5452, 31140}, {9502, 35066}, {16601, 35090}

X(35113) = reflection of X(35094) in X(2)
X(35113) = complement of X(18821)
X(35113) = X(2)-Ceva conjugate of X(528)
X(35113) = barycentric square of X(528)
X(35113) = Steiner inellipse antipode of X(35094)

X(35114) = E(X(115),X(86))-ANTIPODE OF X(1086)

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

X(35114) lies on the Steiner inellipse and these lines: {86, 1086}, {115, 1125}, {239, 24636}, {594, 24374}, {1146, 4999}, {4969, 23992}, {7113, 35086}, {17362, 25469}

X(35114) = complement of X(35162)
X(35114) = X(2)-Ceva conjugate of X(17770)
X(35114) = barycentric square of X(17770)

X(35115) = E(X(115),X(238))-ANTIPODE OF X(1086)

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

X(35115) lies on the Steiner inellipse and these lines: {1, 35080}, {115, 3008}, {238, 1086}, {239, 1146}, {514, 35066}, {1125, 35094}, {4364, 35086}, {6184, 14838}

X(35115) = complement of X(35163)
X(35115) = X(2)-Ceva conjugate of isotomic conjugate of X(35163)
X(35115) = barycentric square of isotomic conjugate of X(35163)

X(35116) = E(X(115),X(241))-ANTIPODE OF X(1086)

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

X(35116) lies on the Steiner inellipse and these lines: {37, 35091}, {220, 32641}, {241, 1086}, {650, 23972}, {656, 35069}, {1015, 8608}, {1146, 1737}, {4130, 4370}, {5526, 6149}, {6586, 23980}, {6603, 35072}

X(35116) = complement of X(35164)
X(35116) = X(2)-Ceva conjugate of X(2801)
X(35116) = barycentric square of X(2801)

X(35117) = E(X(115),X(325))-ANTIPODE OF X(1086)

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

X(35117) lies on the Steiner inellipse and these lines: {115, 726}, {141, 35080}, {325, 1086}, {523, 20532}, {1015, 18904}, {1084, 10026}, {1213, 35078}, {2482, 4785}, {24348, 35079}

X(35117) = complement of X(35165)
X(35117) = X(2)-Ceva conjugate of isogonal conjugate of circle-{X(1687),X(1688),PU(1),PU(2)}-inverse of X(101)
X(35117) = barycentric square of isogonal conjugate of circle-{X(1687),X(1688),PU(1),PU(2)}-inverse of X(101)

X(35118) = E(X(115),X(350))-ANTIPODE OF X(1086)

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

X(35118) lies on the Steiner inellipse and these lines: {115, 3836}, {350, 1086}, {513, 35068}, {594, 4562}, {740, 1015}, {1084, 1575}, {13466, 28840}, {24289, 25660}, {25382, 35079}

X(35118) = complement of X(35166)
X(35118) = X(2)-Ceva conjugate of isotomic conjugate of X(35166)
X(35118) = barycentric square of isotomic conjugate of X(35166)

X(35119) = E(X(115),X(513))-ANTIPODE OF X(1086)

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

X(35119) lies on the Steiner inellipse and these lines: {2, 4562}, {6, 666}, {115, 4129}, {239, 350}, {244, 31148}, {513, 1086}, {514, 1015}, {519, 13466}, {536, 4370}, {995, 24281}, {1100, 35084}, {1146, 20317}, {1575, 3008}, {3912, 20333}, {4000, 24289}, {4025, 24134}, {4124, 4448}, {4688, 25382}, {4932, 24191}, {6682, 35077}, {8287, 15449}, {17205, 35076}, {17301, 24338}, {17759, 29590}, {27929, 35092}, {30117, 35075}

X(35119) = reflection of X(35123) in X(2)
X(35119) = complement of X(4562)
X(35119) = complementary conjugate of X(21261)
X(35119) = X(2)-Ceva conjugate of X(812)
X(35119) = barycentric square of X(812)
X(35119) = Steiner inellipse antipode of X(35123)

X(35120) = E(X(115),X(518))-ANTIPODE OF X(1086)

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

X(35120) lies on the Steiner inellipse and these lines: {10, 35094}, {220, 666}, {514, 6184}, {518, 1086}, {1015, 3008}, {1146, 3912}, {1577, 35068}, {1966, 33677}, {4370, 4762}, {29571, 35092}

X(35120) = complement of X(35167)
X(35120) = X(2)-Ceva conjugate of isotomic conjugate of X(35167)
X(35120) = barycentric square of isotomic conjugate of X(35167)

X(35121) = E(X(115),X(519))-ANTIPODE OF X(1086)

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

X(35121) lies on the Steiner inellipse and these lines: {2, 4555}, {115, 3936}, {190, 9460}, {514, 4370}, {519, 1086}, {545, 6633}, {903, 6631}, {1015, 16610}, {1016, 17487}, {1644, 14410}, {2482, 31148}, {3241, 24281}, {3679, 35094}, {3828, 35080}, {3912, 27751}, {24870, 31139}

X(35121) = reflection of X(35092) in X(2)
X(35121) = complement of X(35168)
X(35121) = X(2)-Ceva conjugate of X(545)
X(35121) = barycentric square of X(545)
X(35121) = Steiner inellipse antipode of X(35092)

X(35122) = E(X(115),X(525))-ANTIPODE OF X(1086)

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

X(35122) lies on the Steiner inellipse and these lines: {30, 4370}, {115, 3239}, {297, 26611}, {379, 34360}, {447, 16086}, {514, 15526}, {519, 3163}, {525, 1086}, {1015, 16612}, {3912, 35075}, {5179, 35068}, {24275, 35082}, {34362, 35085}

X(35122) = complement of X(35169)
X(35122) = X(2)-Ceva conjugate of isogonal conjugate of Moses-radical-circle-inverse of X(101)
X(35122) = barycentric square of isogonal conjugate of Moses-radical-circle-inverse of X(101)

X(35123) = E(X(115),X(536))-ANTIPODE OF X(1086)

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

X(35123) lies on the Steiner inellipse and these lines: {2, 4562}, {10, 35092}, {513, 4370}, {514, 13466}, {519, 1015}, {536, 1086}, {599, 18821}, {661, 35068}, {1146, 5123}, {3679, 24496}, {17281, 35043}, {19584, 31178}, {29594, 35094}

X(35123) = reflection of X(35119) in X(2)
X(35123) = complement of X(18822)
X(35123) = X(2)-Ceva conjugate of X(537)
X(35123) = barycentric square of X(537)
X(35123) = Steiner inellipse antipode of X(35119)

X(35124) = E(X(115),X(551))-ANTIPODE OF X(1086)

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

X(35124) lies on the Steiner inellipse and these lines: {2, 4597}, {214, 31138}, {551, 1086}, {1015, 4850}, {1644, 13466}, {4715, 25398}, {8649, 24441}, {14409, 14422}, {32043, 35092}

X(35124) = complement of X(35170)
X(35124) = X(2)-Ceva conjugate of X(4715)
X(35124) = barycentric square of X(4715)

X(35125) = E(X(115),X(650))-ANTIPODE OF X(1086)

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

X(35125) lies on the Steiner inellipse and these lines: {44, 6184}, {650, 1086}, {1015, 6586}, {2238, 35069}, {3693, 4370}, {5526, 19624}, {8608, 23980}, {8609, 23972}

X(35125) = complement of X(35171)
X(35125) = X(2)-Ceva conjugate of X(3887)
X(35125) = barycentric square of X(3887)

X(35126) = E(X(115),X(726))-ANTIPODE OF X(1086)

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

X(35126) lies on the Steiner inellipse and these lines: {115, 3948}, {514, 20532}, {726, 1086}, {1015, 3912}, {3661, 35092}, {4129, 35068}, {4370, 4785}, {7794, 22116}, {17354, 24502}

X(35126) = complement of X(35172)
X(35126) = X(2)-Ceva conjugate of X(9055)
X(35126) = barycentric square of X(9055)

X(35127) = E(X(115),X(740))-ANTIPODE OF X(1086)

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

X(35127) lies on the Steiner inellipse and these lines: {10, 35080}, {115, 3912}, {239, 1015}, {514, 35068}, {740, 1086}, {1500, 4562}, {4370, 28840}, {4472, 35079}, {24603, 35092}

X(35127) = complement of X(35173)
X(35127) = X(2)-Ceva conjugate of isotomic conjugate of X(35173)
X(35127) = barycentric square of isotomic conjugate of X(35173)

X(35128) = E(X(115),X(905))-ANTIPODE OF X(1086)

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

X(35128) lies on the Steiner inellipse and these lines: {6, 32641}, {36, 2245}, {44, 8607}, {214, 6184}, {220, 5548}, {241, 16586}, {650, 1146}, {905, 1086}, {1015, 6589}, {1107, 35083}, {8609, 23986}

X(35128) = complement of X(35174)
X(35128) = X(2)-Ceva conjugate of X(3738)
X(35128) = barycentric square of X(3738)

X(35129) = E(X(115),X(908))-ANTIPODE OF X(1086)

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

X(35129) lies on the Steiner inellipse and these lines: {6, 5548}, {37, 35092}, {44, 1015}, {650, 4370}, {661, 35069}, {908, 1086}, {1146, 3943}, {1252, 3196}, {4791, 13466}, {16814, 35090}, {17237, 35094}

X(35129) = complement of X(35175)
X(35129) = X(2)-Ceva conjugate of X(2802)
X(35129) = barycentric square of X(2802)

X(35130) = E(X(115),X(960))-ANTIPODE OF X(1086)

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

X(35130) lies on the Steiner inellipse and these lines: {115, 3452}, {220, 645}, {960, 1086}, {1015, 5745}, {1084, 1212}, {1146, 3687}, {6603, 35079}, {15526, 34823}

X(35130) = complement of X(35176)
X(35130) = X(2)-Ceva conjugate of isotomic conjugate of X(35176)
X(35130) = barycentric square of isotomic conjugate of X(35176)

X(35131) = E(X(115),X(1))-ANTIPODE OF X(2482)

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

X(35131) lies on the Steiner inellipse and these lines: {1, 2482}, {115, 7200}, {3008, 35085}, {5222, 35066}, {17058, 23992}, {29571, 35068}

X(35131) = complement of X(35177)
X(35131) = X(2)-Ceva conjugate of isotomic conjugate of X(35177)
X(35131) = barycentric square of isotomic conjugate of X(35177)

X(35132) = E(X(115),X(5))-ANTIPODE OF X(2482)

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

X(35132) lies on the Steiner inellipse and these lines: {5, 2482}, {297, 23967}, {338, 17416}, {3163, 8584}, {7777, 11672}, {15526, 34981}

X(35132) = complement of X(35178)
X(35132) = X(2)-Ceva conjugate of isotomic conjugate of X(35178)
X(35132) = barycentric square of isotomic conjugate of X(35178)

X(35133) = E(X(115),X(6))-ANTIPODE OF X(2482)

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

X(35133) lies on the Steiner inellipse and these lines: {6, 2482}, {32, 13608}, {115, 2793}, {230, 35087}, {1648, 15526}, {3163, 7735}, {3815, 35073}, {5158, 35067}, {6388, 23992}, {6791, 9125}, {14653, 21309}, {15993, 35077}

X(35133) = complement of X(35179)
X(35133) = X(2)-Ceva conjugate of X(1499)
X(35133) = barycentric square of X(1499)

X(35134) = E(X(115),X(10))-ANTIPODE OF X(2482)

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

X(35134) lies on the Steiner inellipse and these lines: {10, 2482}, {1086, 4092}, {3661, 20538}, {3912, 35085}, {8287, 23992}, {29571, 35066}

X(35134) = complement of X(35180)
X(35134) = X(2)-Ceva conjugate of isotomic conjugate of X(35180)
X(35134) = barycentric square of isotomic conjugate of X(35180)

X(35135) = E(X(115),X(37))-ANTIPODE OF X(2482)

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

X(35135) lies on the Steiner inellipse and these lines: {6, 5547}, {37, 2482}, {1575, 35089}, {2276, 35069}, {16592, 23992}, {17369, 35068}

X(35135) = complement of X(35181)
X(35135) = X(2)-Ceva conjugate of X(4160)
X(35135) = barycentric square of X(4160)

Points on the Steiner circumellipse: X(35136)-X(35180)

Contributed by Clark Kimberling and Peter Moses, December 5, 2019.

The Steiner circumellipse, E(X(99)), is the anticomplement of the Steiner inellipse, E(X(115)), and the points in this section are presented as anticomplements of points on the Steiner inellipse. As described in the preamble just before X(34341), the two ellipses are permutation ellipses; that is, if p : q : r is on the ellipse then all six of the permutation points, abbreviated by pqr,qrp,rpq,prq,qpr,rqp, are also on the ellipse.

X(35136) = ANTICOMPLEMENT OF X(15525)

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

X(35136) lies on the Steiner circumellipse and these lines: {2, 15525}, {69, 8754}, {76, 14248}, {99, 3565}, {290, 6391}, {315, 671}, {316, 5203}, {1494, 6340}, {2966, 31998}, {3228, 8667}, {4563, 9134}, {8769, 18827}, {14221, 16077}, {18823, 22110}

X(35136) = isogonal conjugate of X(8651)
X(35136) = isotomic conjugate of X(3566)
X(35136) = anticomplement of X(15525)
X(35136) = trilinear pole of line X(2)X(1975)

X(35137) = ANTICOMPLEMENT OF X(15527)

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

X(35137) lies on the Steiner circumellipse and these lines: {2, 15527}, {76, 14250}, {99, 7953}, {290, 1232}, {671, 6656}, {877, 33513}, {892, 31065}, {1494, 7667}, {3108, 3228}, {4576, 4577}, {4590, 31067}, {14970, 18092}, {18823, 31068}, {18829, 33799}

X(35137) = isogonal conjugate of X(8664)
X(35137) = isotomic conjugate of X(7927)
X(35137) = anticomplement of X(15527)
X(35137) = trilinear pole of line X(2)X(3108)

X(35138) = ANTICOMPLEMENT OF X(17416)

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

X(35138) lies on the Steiner circumellipse and these lines: {2, 17416}, {6, 598}, {99, 5467}, {110, 892}, {183, 1494}, {290, 5967}, {308, 10512}, {316, 34319}, {385, 18823}, {670, 5468}, {1383, 3228}, {3329, 23297}, {6593, 18023}, {9182, 18829}, {14970, 30489}

X(35138) = isogonal conjugate of X(17414)
X(35138) = isotomic conjugate of X(3906)
X(35138) = anticomplement of X(17416)
X(35138) = trilinear pole of line X(2)X(187)
X(35138) = Steiner-circumellipse-X(671)-antipode of X(6)

X(35139) = ANTICOMPLEMENT OF X(18334)

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

X(35139) lies on the Steiner circumellipse and these lines: {2, 18334}, {76, 5641}, {94, 671}, {99, 476}, {190, 32680}, {249, 18314}, {265, 290}, {325, 34209}, {328, 1494}, {340, 18817}, {648, 14618}, {670, 14221}, {892, 10412}, {1989, 3228}, {2166, 18827}, {2966, 14592}, {3225, 11060}, {4577, 14560}, {4586, 32678}, {7799, 14993}, {10411, 18878}, {15475, 18829}, {16188, 18333}, {18020, 18831}

X(35139) = isogonal conjugate of X(14270)
X(35139) = isotomic conjugate of X(526)
X(35139) = anticomplement of X(18334)
X(35139) = polar conjugate of PU(4)-harmonic conjugate of X(2493)
X(35139) = trilinear pole of line X(2)X(94) (the isotomic conjugate of the isogonal conjugate of the Fermat axis, and the radical axis of the Parry and orthocentroidal circles)
X(35139) = barycentric product X(76)*X(476)

X(35140) = ANTICOMPLEMENT OF X(23976)

Barycentrics (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) : :
Barycentrics 1/(S^2 a^2 - 2 SB SC SW) : :

X(35140) lies on the Steiner circumellipse and these lines: {2, 20232}, {4, 14944}, {20, 99}, {69, 648}, {76, 6528}, {154, 34412}, {190, 4150}, {290, 2435}, {316, 10152}, {325, 441}, {664, 5930}, {670, 14615}, {1231, 18026}, {1494, 2419}, {2366, 32687}, {3228, 34212}, {3265, 10714}, {4577, 34774}, {7768, 18831}, {23974, 34186}, {32006, 33893}, {33514, 33651}

X(35140) = isotomic conjugate of X(1503)
X(35140) = anticomplement of X(23976)
X(35140) = polar conjugate of X(16318)
X(35140) = trilinear pole of line X(2)X(2419) (the tangent to hyperbola {{A,B,C,X(2),X(107)}} at X(2))

X(35141) = ANTICOMPLEMENT OF X(35066)

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

X(35141) lies on the Steiner circumellipse and these lines: {2, 35066}, {8, 99}, {10, 664}, {85, 6757}, {190, 319}, {281, 648}, {666, 6542}, {668, 3701}, {670, 3596}, {1441, 4569}, {2966, 15628}, {3912, 30856}, {4573, 6741}, {4608, 9140}, {6648, 14624}, {16284, 20951}, {17294, 20538}, {29574, 32040}, {29615, 32041}

X(35141) = isotomic conjugate of X(17768)
X(35141) = anticomplement of X(35066)
X(35141) = trilinear pole of line X(2)X(3700)

X(35142) = ANTICOMPLEMENT OF X(35067)

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

X(35142) lies on the Steiner circumellipse and these lines: {2, 35067}, {4, 99}, {69, 8754}, {76, 847}, {93, 1235}, {136, 4563}, {190, 1826}, {193, 317}, {225, 664}, {230, 297}, {254, 315}, {264, 670}, {316, 1300}, {340, 892}, {868, 6394}, {1093, 6528}, {1494, 18808}, {4577, 32002}, {8737, 23895}, {8738, 23896}, {8741, 32036}, {8742, 32037}, {8884, 18831}

X(35142) = isotomic conjugate of X(3564)
X(35142) = anticomplement of X(35067)
X(35142) = polar conjugate of X(230)
X(35142) = cevapoint of X(4) and X(297)
X(35142) = trilinear pole of line X(2)X(2501)

X(35143) = ANTICOMPLEMENT OF X(35070)

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

X(35143) lies on the Steiner circumellipse and these lines: {2, 35070}, {87, 4128}, {99, 192}, {171, 190}, {321, 18830}, {664, 7175}, {668, 894}, {670, 4363}, {4562, 18787}

X(35143) = anticomplement of X(35070)
X(35143) = trilinear pole of line X(2)X(4367)

X(35144) = ANTICOMPLEMENT OF X(35074)

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

X(35144) lies on the Steiner circumellipse and these lines: {2, 35074}, {9, 99}, {33, 648}, {37, 664}, {190, 210}, {226, 4569}, {312, 670}, {668, 2321}, {1826, 18026}

X(35144) = anticomplement of X(35074)
X(35144) = trilinear pole of line X(2)X(4041)

X(35145) = ANTICOMPLEMENT OF X(35075)

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

X(35145) lies on the Steiner circumellipse and these lines: {1, 648}, {2, 35075}, {63, 99}, {72, 190}, {92, 6528}, {226, 1947}, {274, 4569}, {293, 2966}, {304, 670}, {306, 668}, {333, 664}, {811, 34591}, {1937, 32038}, {2167, 18831}, {2349, 16077}, {2582, 15165}, {2583, 15164}, {4562, 16086}, {4577, 34055}

X(35145) = isotomic conjugate of X(8680)
X(35145) = anticomplement of X(35075)
X(35145) = trilinear pole of line X(2)X(656)
X(35145) = areal center of cevian triangles of PU(20)

X(35146) = ANTICOMPLEMENT OF X(35077)

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

X(35146) lies on the Steiner circumellipse and these lines: {2, 18829}, {6, 892}, {76, 886}, {99, 187}, {190, 4039}, {419, 648}, {512, 671}, {523, 3228}, {524, 670}, {668, 21839}, {1215, 4562}, {2966, 14614}, {3229, 8859}, {4369, 18827}

X(35146) = isogonal conjugate of X(5106)
X(35146) = isotomic conjugate of X(5969)
X(35146) = anticomplement of X(35077)
X(35146) = polar conjugate of PU(4)-harmonic conjugate of X(11176)
X(35146) = trilinear pole of line X(2)X(351)

X(35147) = ANTICOMPLEMENT OF X(35079)

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

X(35147) lies on the Steiner circumellipse and these lines: {2, 35079}, {99, 513}, {190, 661}, {290, 3262}, {320, 18827}, {325, 18816}, {350, 14616}, {385, 17961}, {523, 668}, {524, 3227}, {536, 671}, {648, 6591}, {664, 4017}, {670, 693}, {903, 31177}, {2481, 11609}, {3226, 17954}, {4364, 24505}, {4577, 18108}, {4998, 6648}, {7779, 31118}, {17320, 18822}, {18823, 31143}, {24348, 30966}

X(35147) = reflection of X(35155) in X(2)
X(35147) = isogonal conjugate of X(5040)
X(35147) = isotomic conjugate of X(2787)
X(35147) = anticomplement of X(35079)
X(35147) = trilinear pole of line X(2)X(3125)
X(35147) = Steiner circumellipse antipode of X(35155)

X(35148) = ANTICOMPLEMENT OF X(35080)

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

X(35148) lies on the Steiner circumellipse and these lines: {2, 35080}, {10, 19936}, {99, 514}, {190, 523}, {239, 9278}, {325, 18025}, {385, 3226}, {519, 671}, {524, 903}, {648, 7649}, {662, 31998}, {664, 7178}, {668, 1577}, {670, 3261}, {1016, 4115}, {1929, 3227}, {2054, 3228}, {2481, 18032}, {3570, 18014}, {3912, 30856}, {4049, 4555}, {4577, 10566}, {4584, 21832}, {4600, 4988}, {4615, 4750}, {6542, 6543}, {7779, 31129}, {9505, 32922}, {18822, 29584}, {18823, 24348}, {20016, 20349}, {24074, 32028}, {32042, 33952}

X(35148) = reflection of X(35153) in X(2)
X(35148) = isogonal conjugate of X(5029)
X(35148) = isotomic conjugate of X(2786)
X(35148) = complement of X(39368)
X(35148) = anticomplement of X(35080)
X(35148) = trilinear pole of line X(2)X(846)
X(35148) = Steiner circumellipse antipode of X(35153)

X(35149) = ANTICOMPLEMENT OF X(35081)

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

X(35149) lies on the Steiner circumellipse and these lines: {2, 35081}, {99, 515}, {190, 21277}, {325, 664}, {333, 2966}, {523, 34393}, {648, 8755}, {668, 21587}

X(35149) = isotomic conjugate of X(2792)
X(35149) = anticomplement of X(35081)

X(35150) = ANTICOMPLEMENT OF X(35082)

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

X(35150) lies on the Steiner circumellipse and these lines: {2, 35082}, {86, 2966}, {99, 516}, {190, 325}, {290, 3261}, {523, 18025}, {524, 32040}, {648, 1886}, {664, 4645}, {666, 4416}

X(35150) = isotomic conjugate of X(2784)
X(35150) = anticomplement of X(35082)

X(35151) = ANTICOMPLEMENT OF X(35083)

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

X(35151) lies on the Steiner circumellipse and these lines: {2, 35083}, {81, 2966}, {99, 517}, {190, 1959}, {290, 693}, {325, 668}, {523, 18816}, {648, 14571}, {670, 3262}

X(35151) = isotomic conjugate of X(2783)
X(35151) = anticomplement of X(35083)

X(35152) = ANTICOMPLEMENT OF X(35084)

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

X(35152) lies on the Steiner circumellipse and these lines: {2, 35084}, {37, 666}, {99, 518}, {190, 3930}, {319, 4562}, {523, 2481}, {524, 32041}, {648, 5089}, {668, 3932}, {670, 3263}, {671, 4762}, {14828, 24345}, {16755, 18827}

X(35152) = isotomic conjugate of X(2795)
X(35152) = anticomplement of X(35084)
X(35152) = trilinear pole of line X(2)X(24285)

X(35153) = ANTICOMPLEMENT OF X(35085)

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

X(35153) lies on the Steiner circumellipse and these lines: {2, 35080}, {10, 4555}, {86, 892}, {99, 519}, {190, 524}, {423, 648}, {514, 671}, {523, 903}, {664, 7181}, {666, 29574}, {668, 3992}, {670, 3264}, {4562, 29615}, {6540, 31013}, {22329, 32040}

X(35153) = reflection of X(35148) in X(2)
X(35153) = isogonal conjugate of X(5168)
X(35153) = isotomic conjugate of X(2796)
X(35153) = anticomplement of X(35085)
X(35153) = trilinear pole of line X(2)X(2786)
X(35153) = Steiner circumellipse antipode of X(35148)

X(35154) = ANTICOMPLEMENT OF X(35086)

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

X(35154) lies on the Steiner circumellipse and these lines: {2, 35086}, {99, 522}, {190, 3700}, {325, 34393}, {385, 17963}, {523, 664}, {524, 1121}, {527, 671}, {648, 3064}, {668, 4086}, {2481, 2648}, {2652, 18827}, {4077, 4569}, {4592, 31998}, {10001, 18026}, {14616, 17731}, {14828, 24345}

X(35154) = isogonal conjugate of X(5075)
X(35154) = isotomic conjugate of X(2785)
X(35154) = anticomplement of X(35086)
X(35154) = trilinear pole of line X(2)X(9317)

X(35155) = ANTICOMPLEMENT OF X(35089)

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

X(35155) lies on the Steiner circumellipse and these lines: {2, 35079}, {81, 892}, {99, 536}, {190, 896}, {321, 889}, {422, 648}, {513, 671}, {523, 3227}, {524, 668}, {670, 16741}, {903, 31148}, {3228, 31150}, {4555, 4670}, {4562, 25382}

X(35155) = reflection of X(35147) in X(2)
X(35155) = isogonal conjugate of X(5163)
X(35155) = anticomplement of X(35089)
X(35155) = trilinear pole of line X(2)X(2787)
X(35155) = Steiner circumellipse antipode of X(35147)

X(35156) = ANTICOMPLEMENT OF X(35090)

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

X(35156) lies on the Steiner circumellipse and these lines: {2, 35090}, {99, 693}, {190, 1577}, {316, 2481}, {320, 14616}, {648, 17924}, {664, 4077}, {668, 850}, {1494, 3262}, {3227, 21907}, {3260, 18816}, {5620, 18827}, {18821, 20880}

X(35156) = isotomic conjugate of X(8674)
X(35156) = anticomplement of X(35090)

X(35157) = ANTICOMPLEMENT OF X(35091)

Barycentrics (a - b)*(a - c)*(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(35157) lies on the Steiner circumellipse and these lines: {2, 35091}, {7, 18821}, {99, 4620}, {190, 3239}, {320, 18025}, {522, 664}, {527, 1121}, {648, 17926}, {666, 28132}, {668, 4397}, {671, 17950}, {693, 4569}, {883, 4555}, {892, 17933}, {900, 927}, {903, 9436}, {1156, 2481}, {1494, 16091}, {3227, 34056}, {4577, 32728}, {14727, 23351}, {15633, 34393}

X(35157) = isogonal conjugate of X(6139)
X(35157) = isotomic conjugate of X(6366)
X(35157) = anticomplement of X(35091)
X(35157) = trilinear pole of line X(2)X(664)

X(35158) = ANTICOMPLEMENT OF X(35093)

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

X(35158) lies on the Steiner circumellipse and these lines: {2, 35093}, {8, 666}, {99, 14953}, {190, 516}, {279, 35094}, {514, 18025}, {519, 32040}, {664, 3912}, {668, 30807}, {1146, 6185}, {2481, 4391}, {4555, 29616}

X(35158) = isotomic conjugate of X(5845)
X(35158) = anticomplement of X(35093)
X(35158) = polar conjugate of PU(4)-harmonic conjugate of X(646)
X(35158) = trilinear pole of line X(2)X(676)

X(35159) = ANTICOMPLEMENT OF X(35095)

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

X(35159) lies on the Steiner circumellipse and these lines: {2, 35095}, {7, 670}, {56, 99}, {65, 668}, {190, 1400}, {608, 648}, {664, 1042}, {892, 7316}, {1118, 6528}, {1426, 18026}, {3175, 32041}

X(35159) = anticomplement of X(35095)
X(35159) = trilinear pole of line X(2)X(7180)

X(35160) = ANTICOMPLEMENT OF X(35111)

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

X(35160) lies on the Steiner circumellipse and these lines: {2, 35111}, {7, 190}, {8, 1358}, {85, 668}, {99, 1434}, {145, 279}, {666, 3008}, {1847, 18026}, {2369, 6078}, {3912, 24803}, {4308, 32098}, {4562, 7233}, {4569, 23062}, {6606, 10509}, {7209, 18830}, {16823, 24798}, {17089, 32003}, {21267, 27829}, {29573, 32041}

X(35160) = isogonal conjugate of X(8647)
X(35160) = isotomic conjugate of X(5853)
X(35160) = anticomplement of X(35111)
X(35160) = trilinear pole of line X(2)X(3676)

X(35161) = ANTICOMPLEMENT OF X(35112)

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

X(35161) lies on the Steiner circumellipse and these lines: {2, 35112}, {27, 16077}, {30, 190}, {99, 3977}, {306, 4555}, {447, 519}, {514, 1494}, {525, 903}, {664, 6357}, {668, 14206}, {2966, 4234}, {11351, 34360}

X(35161) = anticomplement of X(35112)
X(35161) = isotomic conjugate of isogonal conjugate of Moses-radical-circle-inverse of X(106)
X(35161) = trilinear pole of line X(2)X(11125)

X(35162) = ANTICOMPLEMENT OF X(35114)

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

X(35162) lies on the Steiner circumellipse and these lines: {2, 35114}, {10, 99}, {12, 664}, {86, 21043}, {190, 594}, {313, 670}, {319, 21089}, {502, 32004}, {648, 1826}, {668, 1089}, {671, 22037}, {1268, 23934}, {4013, 4555}, {4577, 18082}, {6538, 6540}, {6542, 6543}, {21047, 32101}

X(35162) = isotomic conjugate of X(17770)
X(35162) = anticomplement of X(35114)
X(35162) = trilinear pole of line X(2)X(4024)

X(35163) = ANTICOMPLEMENT OF X(35115)

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

X(35163) lies on the Steiner circumellipse and these lines: {2, 35115}, {10, 666}, {99, 3912}, {190, 3932}, {648, 1861}, {664, 6542}, {668, 17789}, {1577, 2481}

X(35163) = anticomplement of X(35115)
X(35163) = isotomic conjugate of X(2)-Ceva conjugate of X(35115)
X(35163) = trilinear pole of line X(2)X(4088)

X(35164) = ANTICOMPLEMENT OF X(35116)

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

X(35164) lies on the Steiner circumellipse and these lines: {2, 35116}, {99, 2717}, {190, 3262}, {320, 4569}, {664, 4511}, {693, 18025}, {1121, 4391}, {3261, 18816}, {4417, 4562}, {4555, 16284}, {5081, 18026}, {6606, 17791}, {7253, 14616}

X(35164) = isotomic conjugate of X(2801)
X(35164) = anticomplement of X(35116)
X(35164) = trilinear pole of line X(2)X(36038)

X(35165) = ANTICOMPLEMENT OF X(35117)

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

X(35165) lies on the Steiner circumellipse and these lines: {2, 35117}, {86, 18829}, {99, 726}, {190, 385}, {523, 3226}, {668, 1757}, {670, 17731}, {671, 4785}, {894, 4562}, {10566, 14970}, {17212, 18827}, {24275, 24502}

X(35165) = anticomplement of X(35117)
X(35165) = isotomic conjugate of isogonal conjugate of circle-{X(1687),X(1688),PU(1),PU(2)}-inverse of X(101)
X(35165) = trilinear pole of line X(2)X(4107)

X(35166) = ANTICOMPLEMENT OF X(35118)

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

X(35166) lies on the Steiner circumellipse and these lines: {2, 35118}, {37, 4562}, {99, 238}, {190, 2238}, {256, 18829}, {350, 670}, {513, 18827}, {648, 2201}, {664, 1284}, {668, 740}, {889, 25382}, {1573, 24505}, {1874, 18026}, {3227, 28840}

X(35166) = anticomplement of X(35118)
X(35166) = isotomic conjugate of X(2)-Ceva conjugate of X(35118)

X(35167) = ANTICOMPLEMENT OF X(35120)

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

X(35167) lies on the Steiner circumellipse and these lines: {1, 666}, {2, 35120}, {8, 4562}, {99, 18206}, {190, 518}, {239, 241}, {514, 2481}, {519, 32041}, {648, 14024}, {668, 3912}, {670, 18157}, {903, 4762}, {4384, 4555}, {4560, 18827}, {4569, 10030}, {5236, 18026}, {9311, 14727}, {30663, 33676}

X(35167) = anticomplement of X(35120)
X(35167) = isotomic conjugate of X(2)-Ceva conjugate of X(35120)
X(35167) = trilinear pole of line X(2)X(2254)

X(35168) = ANTICOMPLEMENT OF X(35121)

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

X(35168) lies on the Steiner circumellipse and these lines: {2, 4555}, {99, 2384}, {190, 519}, {239, 4597}, {514, 903}, {664, 3911}, {666, 3241}, {668, 4358}, {1016, 4370}, {3227, 31150}, {3679, 4562}, {6540, 31011}, {6630, 17487}, {9460, 27191}, {20569, 31139}, {30225, 31145}, {30580, 31992}

X(35168) = reflection of X(4555) in X(2)
X(35168) = isogonal conjugate of X(8649)
X(35168) = isotomic conjugate of X(545)
X(35168) = anticomplement of X(35121)
X(35168) = trilinear pole of line X(2)X(900)
X(35168) = Steiner circumellipse antipode of X(4555)

X(35169) = ANTICOMPLEMENT OF X(35122)

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

X(35169) lies on the Steiner circumellipse and these lines: {2, 35112}, {30, 903}, {99, 4025}, {190, 525}, {447, 30117}, {514, 648}, {519, 1494}, {664, 17094}, {668, 14208}, {2701, 4458}, {2966, 4237}, {4077, 18026}, {5088, 18827}, {5641, 17677}, {11320, 34360}, {30227, 31015}

X(35169) = anticomplement of X(35122)
X(35169) = isotomic conjugate of isogonal conjugate of Moses-radical-circle-inverse of X(101)
X(35169) = trilinear pole of line X(2)X(1762)

X(35170) = ANTICOMPLEMENT OF X(35124)

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

X(35170) lies on the Steiner circumellipse and these lines: {2, 4597}, {99, 5235}, {190, 3679}, {664, 5219}, {668, 4671}, {889, 4506}, {903, 23598}, {3241, 32631}, {4555, 4945}, {20568, 31138}, {30590, 32042}

X(35170) = refection of X(4597) in X(2)
X(35170) = isotomic conjugate of X(4715)
X(35170) = anticomplement of X(35124)
X(35170) = trilinear pole of line X(2)X(4777)
X(35170) = Steiner circumellipse antipode of X(4597)
X(35170) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(80)}}

X(35171) = ANTICOMPLEMENT OF X(35125)

Barycentrics (a - b)*b*(a - c)*c*(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(35171) lies on the Steiner circumellipse and these lines: {2, 35125}, {75, 18821}, {99, 1308}, {190, 693}, {320, 2481}, {664, 24002}, {666, 4585}, {668, 3261}, {1121, 17297}, {3227, 34578}, {3262, 18025}, {4998, 6606}, {14616, 30941}, {17378, 18822}

X(35171) = isogonal conjugate of X(8645)
X(35171) = isotomic conjugate of X(3887)
X(35171) = anticomplement of X(35125)
X(35171) = trilinear pole of line X(2)X(1111)

X(35172) = ANTICOMPLEMENT OF X(35126)

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

X(35172) lies on the Steiner circumellipse and these lines: {1, 4562}, {2, 35126}, {76, 35119}, {99, 9111}, {190, 238}, {239, 668}, {514, 3226}, {664, 1429}, {666, 7760}, {670, 30940}, {889, 4363}, {903, 4785}, {1016, 17475}, {1019, 18827}, {3227, 4378}, {4393, 4555}, {30667, 32028}

X(35172) = isotomic conjugate of X(9055)
X(35172) = anticomplement of X(35126)
X(35172) = trilinear pole of line X(2)X(659)

X(35173) = ANTICOMPLEMENT OF X(35127)

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

X(35173) lies on the Steiner circumellipse and these lines: {2, 35127}, {10, 4562}, {99, 239}, {148, 9510}, {190, 740}, {242, 648}, {257, 18829}, {514, 18827}, {664, 16609}, {668, 3948}, {670, 1921}, {903, 28840}, {4364, 24505}, {4555, 16826}

X(35173) = anticomplement of X(35127)
X(35173) = isotomic conjugate of X(2)-Ceva conjugate of X(35127)
X(35173) = trilinear pole of line X(2)X(4010)

X(35174) = ANTICOMPLEMENT OF X(35128)

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

X(35174) lies on the Steiner circumellipse and these lines: {2, 35128}, {80, 2481}, {85, 18821}, {99, 2222}, {190, 655}, {320, 18816}, {664, 693}, {811, 18831}, {903, 18815}, {1121, 18359}, {1411, 3226}, {1494, 17791}, {1577, 4564}, {2006, 3227}, {3262, 34393}, {4554, 4597}, {4586, 32675}, {5080, 14616}, {14584, 18822}, {18025, 30806}

X(35174) = isogonal conjugate of X(8648)
X(35174) = isotomic conjugate of X(3738)
X(35174) = anticomplement of X(35128)
X(35174) = trilinear pole of line X(2)X(2006)

X(35175) = ANTICOMPLEMENT OF X(35129)

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

X(35175) lies on the Steiner circumellipse and these lines: {2, 35129}, {75, 4555}, {99, 2718}, {190, 3218}, {320, 668}, {664, 1443}, {666, 3758}, {679, 1111}, {693, 903}, {1227, 6635}, {2481, 4406}, {7192, 14616}

X(35175) = isotomic conjugate of X(2802)
X(35175) = anticomplement of X(35129)
X(35175) = trilinear pole of line X(2)X(3762)

X(35176) = ANTICOMPLEMENT OF X(35130)

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

X(35176) lies on the Steiner circumellipse and these lines: {2, 35130}, {34, 648}, {57, 99}, {65, 190}, {85, 670}, {226, 668}, {664, 1427}

X(35176) = anticomplement of X(35130)
X(35176) = isotomic conjugate of X(2)-Ceva conjugate of X(35130)
X(35176) = trilinear pole of line X(2)X(4017)

X(35177) = ANTICOMPLEMENT OF X(35131)

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

X(35177) lies on the Steiner circumellipse and these lines: {2, 35131}, {8, 671}, {645, 892}, {3227, 16823}, {4384, 18827}

X(35177) = anticomplement of X(35131)
X(35177) = isotomic conjugate of X(2)-Ceva conjugate of X(35131)
X(35177) = trilinear pole of line X(2)X(3712)

X(35178) = ANTICOMPLEMENT OF X(35132)

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

X(35178) lies on the Steiner circumellipse and these lines: {2, 35132}, {3, 671}, {112, 33513}, {401, 5641}, {648, 5467}, {892, 4558}, {1494, 15533}, {4235, 6528}

X(35178) = anticomplement of X(35132)
X(35178) = isotomic conjugate of X(2)-Ceva conjugate of X(35132)
X(35178) = trilinear pole of line X(2)X(575)

X(35179) = ANTICOMPLEMENT OF X(35133)

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

X(35179) lies on the Steiner circumellipse and these lines: {2, 35133}, {69, 671}, {76, 14262}, {99, 1296}, {183, 3228}, {315, 34165}, {325, 18823}, {385, 17968}, {524, 9487}, {648, 5468}, {670, 34203}, {892, 4563}, {2966, 9182}, {6082, 8599}, {9146, 18012}

X(35179) = isogonal conjugate of X(8644)
X(35179) = isotomic conjugate of X(1499)
X(35179) = anticomplement of X(35133)
X(35179) = trilinear pole of line X(2)X(2418)

X(35180) = ANTICOMPLEMENT OF X(35134)

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

X(35180) lies on the Steiner circumellipse and these lines: {1, 671}, {2, 35134}, {99, 23889}, {662, 892}, {670, 24039}, {903, 29584}, {1121, 16823}, {4393, 18827}, {6540, 18047}, {32042, 33946}

X(35180) = anticomplement of X(35134)
X(35180) = isotomic conjugate of X(2)-Ceva conjugate of X(35134)
X(35180) = trilinear pole of line X(2)X(896)

X(35181) = ANTICOMPLEMENT OF X(35135)

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

X(35181) lies on the Steiner circumellipse and these lines: {2, 35135}, {75, 671}, {99, 8691}, {799, 892}, {903, 26234}, {3226, 34916}, {3227, 17320}, {4389, 18827}, {4441, 14616}

X(35181) = isotomic conjugate of X(4160)
X(35181) = anticomplement of X(35135)
X(35181) = trilinear pole of line X(2)X(14210)

Haedus transforms: X(35182)-X(35191)

This preamble and centers X(35182)-X(35191) were contributed by César Eliud Lozada, December 5, 2019.

Let ABC be an acute triangle with circumcircle ω. Let t be a tangent line to ω and denote t_a, t_b, t_c the lines obtained by reflecting t in the lines BC, CA and AB, respectively. Then the circumcircle ω′ of the triangle determined by the lines t_a, t_b, t_c is tangent to the circle ω. References: 2011 IMO Shortlist G8 (JPN) problem 6, IMOgeometry and AOPS.

If P = u:v:w (trilinears) is the touchpoint of t and ω and H(P) is the touchpoint of ω and ω′, then

H(P) = u*SA*((a^2*u^2 + 3*b^2*v^2 + 3*c^2*w^2)*SA^2*SB*SC*a^3*u
+ ((2*SA - SB)*S^2 - 3*(SB + SC)*SA^2)*SA*a^4*b*u^2*v
+ ((2*SA - SC)*S^2 - 3*(SB + SC)*SA^2)*SA*a^4*c*u^2*w
+ (S^2 - 3*SA*SB)*SB*SC^2*b^2*c*v^2*w
+ (S^2 - 3*SA*SC)*SC*SB^2*c^2*b*w^2*v
- (S^2 + SA*SC)*SB^2*SC*b^3*v^3
- (S^2 + SA*SB)*SC^2*SB*c^3*w^3
+ (5*S^2 + 6*SA^2 - 24*R^2*SA - 4*SB*SC)*S^2*a^3*b*c*u*v*w) : :

H(P) = IsotomicConjugate( PolarConjugate( BarycentricProduct( P, X₄-antipode-of-P) ) )
where X₄-antipode-of-P means the intersection, other than P, of ω and the line X(4)P, as defined in the preamble just before X(26700).

The point H(P) is here named here the Haedus transform of P.

The appearance of (i, j) in the following list means that H(X(i)) = X(j):

(74, 1304), (98, 2715), (99, 10425), (100, 6099), (101, 35182), (102, 35183), (103, 35184), (104, 2720), (105, 35185), (106, 35186), (107, 1304), (108, 2720), (109, 35187), (110, 10420), (111, 35188), (112, 2715), (476, 35189), (675, 35190), (691, 35191), (915, 6099), (917, 35182), (953, 35011), (1113, 110), (1114, 110), (1300, 10420), (1309, 35011), (3563, 10425), (26704, 35183), (26705, 35184), (26706, 35185), (30247, 35188), (32704, 35186), (32706, 35187), (32710, 35189)

X(35182) = HAEDUS TRANSFORM OF X(101)

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

X(35182) lies on the circumcircle and these lines: {101,8676}, {103,916}, {108,2149}, {112,32699}, {514,1305}, {516,917}, {675,2989}, {915,5011}, {1300,5134}, {2426,32684}, {3234,9057}, {5179,32706}, {15378,26705}

X(35182) = isotomic conjugate of the polar conjugate of X(32699)
X(35182) = reflection of X(i) in the line X(j)X(k) for these (i, j, k): (101, 3, 48), (103, 3, 8676)
X(35182) = barycentric product X(i)*X(j) for these {i, j}: {69, 32699}, {101, 2989}, {917, 1331}
X(35182) = barycentric quotient X(692)/X(1736)
X(35182) = trilinear product X(i)*X(j) for these {i, j}: {63, 32699}, {692, 2989}, {906, 917}
X(35182) = trilinear quotient X(i)/X(j) for these (i, j): (101, 1736), (692, 8608), (906, 916), (917, 17924)
X(35182) = trilinear pole of the line {6, 32656}
X(35182) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(514), X(7252)}}
X(35182) = X(516)-cross conjugate of X(15378)
X(35182) = X(i)-isoconjugate-of X(j) for these {i,j}: {514, 1736}, {693, 8608}, {916, 17924}
X(35182) = X(692)-reciprocal conjugate of X(1736)

X(35183) = HAEDUS TRANSFORM OF X(102)

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

Let A', B', C' be the intersections of line X(3)X(10) and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X(35183). (Randy Hutson, December 8, 2019)

X(35183) lies on the circumcircle and these lines: {108,21189}, {109,15386}, {112,32700}, {522,26704}, {1311,15633}, {2217,2716}, {2432,32689}, {2695,15232}, {2723,13478}, {2734,10570}, {15379,32706}, {26715,32653}, {29068,32677}

X(35183) = isotomic conjugate of the polar conjugate of X(32700)
X(35183) = reflection of X(109) in the line X(3)X(14529)
X(35183) = reflection of X(26704) in line X(3)X(10)
X(35183) = barycentric product X(69)*X(32700)
X(35183) = trilinear product X(63)*X(32700)
X(35183) = trilinear quotient X(102)/X(21189)
X(35183) = trilinear pole of the line {6, 32653}
X(35183) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(522), X(21189)}}
X(35183) = X(522)-cross conjugate of X(15379)
X(35183) = X(515)-isoconjugate-of X(21189)

X(35184) = HAEDUS TRANSFORM OF X(103)

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

X(35184) lies on the circumcircle and these lines: {101,15378}, {112,32701}, {514,26705}, {675,15634}, {677,8701}, {917,15380}, {2424,32682}, {2688,15320}, {2724,14377}, {6577,32642}

X(35184) = isotomic conjugate of the polar conjugate of X(32701)
X(35184) = reflection of X(103) in the line X(3)X(4091)
X(35184) = barycentric product X(i)*X(j) for these {i, j}: {69, 32701}, {677, 14377}, {1815, 26705}
X(35184) = barycentric quotient X(i)/X(j) for these (i, j): (103, 25259), (677, 17233), (911, 1734)
X(35184) = trilinear product X(63)*X(32701)
X(35184) = trilinear quotient X(i)/X(j) for these (i, j): (103, 1734), (677, 3681), (911, 6586)
X(35184) = trilinear pole of the line {6, 22084}
X(35184) = X(514)-cross conjugate of X(15380)
X(35184) = X(i)-isoconjugate-of X(j) for these {i,j}: {516, 1734}, {676, 3681}, {910, 25259}
X(35184) = X(i)-reciprocal conjugate of X(j) for these (i,j): (103, 25259), (677, 17233), (911, 1734)

X(35185) = HAEDUS TRANSFORM OF X(105)

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

X(35185) lies on the circumcircle and these lines: {100,26641}, {105,3827}, {108,32735}, {112,32703}, {518,26703}, {840,3433}, {1292,15402}, {1300,34173}, {2175,3323}, {2862,13577}, {3309,26706}, {9061,15636}, {10099,10100}, {15344,15382}

X(35185) = isotomic conjugate of the polar conjugate of X(32703)
X(35185) = reflection of X(105) in the line X(3)X(905)
X(35185) = barycentric product X(i)*X(j) for these {i, j}: {69, 32703}, {666, 3433}, {919, 13577}
X(35185) = barycentric quotient X(i)/X(j) for these (i, j): (105, 26546), (919, 3434), (1438, 21185)
X(35185) = trilinear product X(63)*X(32703)
X(35185) = trilinear quotient X(i)/X(j) for these (i, j): (105, 21185), (666, 20927), (673, 26546), (919, 169)
X(35185) = trilinear pole of the line {6, 3433}
X(35185) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(518), X(3827)}}
X(35185) = Cevapoint of X(665) and X(2175)
X(35185) = X(i)-isoconjugate-of X(j) for these {i,j}: {169, 918}, {518, 21185}, {665, 20927}
X(35185) = X(i)-reciprocal conjugate of X(j) for these (i,j): (105, 26546), (919, 3434), (1438, 21185)

X(35186) = HAEDUS TRANSFORM OF X(106)

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

X(35186) lies on the circumcircle and these lines: {106,2390}, {112,32705}, {519,2370}, {1293,15403}, {1311,5121}, {3667,32704}, {9083,15637}

X(35186) = isotomic conjugate of the polar conjugate of X(32705)
X(35186) = reflection of X(106) in the line X(3)X(1459)
X(35186) = barycentric product X(69)*X(32705)
X(35186) = trilinear product X(63)*X(32705)
X(35186) = trilinear quotient X(901)/X(14923)
X(35186) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(519), X(2390)}}
X(35186) = X(900)-isoconjugate-of X(14923)

X(35187) = HAEDUS TRANSFORM OF X(109)

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

X(35187) lies on the circumcircle and these lines: {102,15379}, {108,24027}, {112,32707}, {515,32706}, {953,10571}, {1311,2988}, {2425,32683}, {15386,26704}

X(35187) = isotomic conjugate of the polar conjugate of X(32707)
X(35187) = reflection of X(109) in the line X(3)X(73)
X(35187) = barycentric product X(i)*X(j) for these {i, j}: {69, 32707}, {109, 2988}
X(35187) = barycentric quotient X(1415)/X(1735)
X(35187) = trilinear product X(i)*X(j) for these {i, j}: {63, 32707}, {1415, 2988}
X(35187) = trilinear quotient X(i)/X(j) for these (i, j): (109, 1735), (1415, 8607)
X(35187) = trilinear pole of the line {6, 32660}
X(35187) = X(515)-cross conjugate of X(15386)
X(35187) = X(i)-isoconjugate-of X(j) for these {i,j}: {522, 1735}, {1735, 522}
X(35187) = X(1415)-reciprocal conjugate of X(1735)

X(35188) = HAEDUS TRANSFORM OF X(111)

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

X(35188) lies on the circumcircle and these lines: {3,23701}, {6,10102}, {74,352}, {98,5913}, {111,2393}, {112,32709}, {353,32425}, {524,2373}, {1296,15406}, {1300,34169}, {1499,30247}, {2374,15387}, {2407,9080}, {2697,34320}, {2770,5486}, {2868,5108}, {6093,13608}, {9084,15638}, {10097,10098}

X(35188) = reflection of X(23701) in X(3)
X(35188) = circumnormal isogonal conjugate of X(23699)
X(35188) = circumperp conjugate of X(23701)
X(35188) = isotomic conjugate of the polar conjugate of X(32709)
X(35188) = reflection of X(111) in the line X(3)X(647)
X(35188) = barycentric product X(i)*X(j) for these {i, j}: {69, 32709}, {691, 5486}, {895, 30247}
X(35188) = barycentric quotient X(691)/X(11185)
X(35188) = trilinear product X(63)*X(32709)
X(35188) = trilinear quotient X(895)/X(14209)
X(35188) = trilinear pole of the line {6, 14908}
X(35188) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(351), X(20382)}}
X(35188) = circumcircle-antipode of X(23701)
X(35188) = X(468)-isoconjugate-of X(14209)
X(35188) = X(691)-reciprocal conjugate of X(11185)
X(35188) = X(10102)-of-circumsymmedial triangle
X(35188) = X(23701)-of-ABC-X3 reflections triangle

X(35189) = HAEDUS TRANSFORM OF X(476)

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

X(35189) lies on the circumcircle and these lines: {74,5961}, {98,34310}, {110,15453}, {112,32711}, {265,1300}, {477,15396}, {526,10420}, {1302,14559}, {1304,15395}, {2437,32690}, {5663,32710}, {9060,14560}, {13530,14854}

X(35189) = isotomic conjugate of the polar conjugate of X(32711)
X(35189) = barycentric product X(i)*X(j) for these {i, j}: {69, 32711}, {2410, 15396}
X(35189) = barycentric quotient X(2437)/X(25641)
X(35189) = trilinear product X(63)*X(32711)

X(35190) = HAEDUS TRANSFORM OF X(675)

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

X(35190) lies on the circumcircle and the line {9085,15397}

X(35191) = HAEDUS TRANSFORM OF X(691)

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

X(35191) lies on the circumcircle and these lines: {74,6091}, {476,32729}, {477,23698}, {526,10425}, {671,1300}, {842,14984}, {895,3563}, {5663,23700}, {15342,30247}

X(35191) = intersection, other than A,B,C, of conic {{A, B, C, X(67), X(9132)}} and circumcircle

Points associated with the cubic pK(X(35192,X(21)): X(35192)-X(35196)

Contributed by Clark Kimberling and Peter Moses, December 8, 2019.

Suppose that A'B'C' is the circumcevian-inversion triangle of X(1), as defined in the preamble just before X(34864). The locus of a point Q such that the cevian triangle of Q is perspective to A'B'C' is the cubic pK(X(35192),X(21)), which passes through A, B, C, and X(i) for these i: 1, 3, 21, 35, 3467, 11107, 35193, 35194, 35195, 35196.

X(35192) = X(3)X(6)∩X(48)X(163)

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

X(35192) lies on these lines: {3,6}, {9,1793}, {41,2150}, {48,163}, {86,25582}, {112,2331}, {501,3157}, {1172,2074}, {1419,4565}, {1576,15624}, {2174,17104}, {2328,7072}, {2911,5127}, {4612,27958}, {7110,13746}, {11101,16547}, {14794,22058}

X(35192) = isogonal conjugate of polar conjugate of X(11107)

X(35193) = X(1)X(21)∩X(3)X(74)

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

X(35193) lies on the cubics K259 and pK(X(35192),X(21)) and these lines: {1, 21}, {2, 582}, {3, 74}, {8, 643}, {10, 13746}, {28, 37584}, {35, 17104}, {40, 1325}, {46, 229}, {55, 60}, {86, 25581}, {162, 7952}, {163, 3730}, {165, 37294}, {171, 27577}, {184, 16452}, {212, 3876}, {219, 7054}, {220, 5546}, {323, 500}, {333, 17584}, {355, 7424}, {394, 37285}, {501, 5010}, {517, 11101}, {580, 5047}, {759, 5697}, {1259, 6061}, {1399, 1442}, {1414, 3160}, {1437, 4184}, {1479, 24624}, {1495, 35203}, {1724, 3017}, {1754, 4197}, {1793, 5692}, {1816, 23692}, {1993, 37284}, {2074, 26921}, {2478, 24883}, {3109, 5690}, {3145, 22139}, {3219, 6198}, {3258, 36154}, {3579, 37405}, {3648, 18625}, {3654, 7478}, {3746, 9275}, {4193, 24880}, {4648, 6910}, {5012, 16287}, {5060, 9310}, {5398, 16865}, {6043, 31156}, {6097, 22115}, {6919, 24898}, {9306, 16451}, {11699, 16164}, {12699, 37369}, {13329, 17531}, {15107, 20840}, {17566, 25533}, {17574, 37469}, {17671, 24619}, {18653, 31730}, {22076, 37311}, {22136, 37286}, {24892, 37373}, {26446, 37158}, {35976, 37659}

X(35193) = X(i)-Ceva conjugate of X(j) for these (i,j): {21, 35195}, {249, 5546}, {643, 35057}, {6083, 526}, {18315, 23090}
X(35193) = X(i)-cross conjugate of X(j) for these (i,j): {35, 11107}, {3024, 35057}
X(35193) = X(i)-isoconjugate of X(j) for these (i,j): {56, 6757}, {57, 8818}, {65, 79}, {115, 35049}, {225, 7100}, {226, 2160}, {265, 1835}, {523, 26700}, {1254, 3615}, {1400, 30690}, {1402, 20565}, {1427, 7110}, {1441, 6186}, {1464, 2166}, {1989, 18593}, {3668, 7073}, {4017, 6742}, {7180, 15455}, {18210, 34922}, {36035, 36064}
X(35193) = cevapoint of X(1511) and X(6149)
X(35193) = crosssum of X(1365) and X(4017)
X(35193) = crossdifference of every pair of points on line {661, 1637}
X(35193) = barycentric product X(i)*X(j) for these {i,j}: {21, 3219}, {35, 333}, {55, 34016}, {60, 3969}, {63, 11107}, {75, 35192}, {81, 4420}, {99, 9404}, {249, 6741}, {284, 319}, {312, 17104}, {314, 2174}, {323, 6740}, {643, 14838}, {645, 2605}, {662, 35057}, {1043, 2003}, {1098, 16577}, {1442, 2287}, {1812, 6198}, {2185, 3678}, {2194, 33939}, {2327, 7282}, {2328, 17095}, {2594, 7058}, {3939, 16755}, {4467, 5546}, {4636, 7265}, {6064, 20982}, {17190, 32635}
X(35193) = barycentric quotient X(i)/X(j) for these {i,j}: {9, 6757}, {21, 30690}, {35, 226}, {50, 1464}, {55, 8818}, {163, 26700}, {284, 79}, {319, 349}, {333, 20565}, {643, 15455}, {1101, 35049}, {1399, 1427}, {1442, 1446}, {2003, 3668}, {2174, 65}, {2193, 7100}, {2194, 2160}, {2328, 7110}, {2341, 2166}, {2594, 6354}, {2605, 7178}, {3024, 8287}, {3219, 1441}, {3678, 6358}, {3969, 34388}, {4420, 321}, {5546, 6742}, {6149, 18593}, {6740, 94}, {6741, 338}, {7054, 3615}, {7186, 16888}, {9404, 523}, {11107, 92}, {14838, 4077}, {14975, 1880}, {17104, 57}, {17454, 3649}, {20982, 1365}, {21741, 1254}, {32640, 36064}, {34016, 6063}, {35057, 1577}, {35192, 1}, {35195, 17483}
X(35193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {283, 2328, 21}, {643, 1098, 8}

X(35194) = X(1)X(195)∩X(3)X(1393)

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

X(35194) lies on the cubic on pK(X(35192), X(21)) and these lines: {1, 195}, {3, 3460}, {5, 1393}, {9, 1062}, {10, 522}, {12, 1725}, {21, 1807}, {30, 201}, {33, 26921}, {38, 496}, {40, 15430}, {65, 5492}, {109, 3652}, {140, 7004}, {212, 8144}, {381, 37591}, {484, 33642}, {495, 774}, {500, 16577}, {651, 7100}, {942, 1736}, {943, 33761}, {984, 3295}, {1058, 7226}, {1060, 1394}, {1125, 24434}, {1154, 2599}, {1710, 20989}, {1735, 9956}, {1935, 18447}, {2161, 5358}, {2310, 15171}, {2771, 37558}, {2915, 21368}, {3074, 18455}, {3100, 26878}, {3219, 6198}, {3615, 24149}, {4642, 11545}, {5777, 37565}, {5779, 8757}, {5891, 30493}, {6097, 22342}, {7070, 9644}, {7085, 9645}, {9817, 37532}, {16720, 24454}, {18180, 21807}, {20117, 34586}, {22350, 31835}, {22936, 33649}

X(35194) = X(21)-Ceva conjugate of X(35)
X(35194) = X(i)-isoconjugate of X(j) for these (i,j): {36, 1141}, {54, 79}, {95, 6186}, {2148, 30690}, {2160, 2167}, {2190, 7100}, {2616, 13486}, {11077, 17923}
X(35194) = barycentric product X(i)*X(j) for these {i,j}: {5, 3219}, {35, 14213}, {51, 33939}, {311, 2174}, {319, 1953}, {333, 2599}, {343, 6198}, {1154, 18359}, {1273, 2161}, {1807, 14918}, {2290, 20566}, {2617, 7265}, {3678, 17167}, {3969, 18180}, {7069, 17095}, {14975, 28706}, {21807, 34016}
X(35194) = barycentric quotient X(i)/X(j) for these {i,j}: {5, 30690}, {35, 2167}, {51, 2160}, {216, 7100}, {1154, 3218}, {1273, 20924}, {1625, 13486}, {1953, 79}, {2161, 1141}, {2174, 54}, {2179, 6186}, {2290, 36}, {2599, 226}, {3219, 95}, {6198, 275}, {7069, 7110}, {11062, 1870}, {14213, 20565}, {14975, 8882}, {21011, 6757}, {21807, 8818}, {21824, 8901}, {33939, 34384}, {35192, 35196}

X(35195) = X(3)X(54)∩X(21)X(3467)

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

X(35195) lies on the cubic pK(X(35192),X(21)) and these lines: {3, 54}, {21, 3467}, {35, 17104}, {110, 11849}, {501, 11010}, {517, 37405}, {662, 3615}, {1325, 2360}, {3193, 37294}, {3336, 15767}, {3871, 35057}

X(35195) = X(21)-Ceva conjugate of X(35193)
X(35195) = barycentric product X(i)*X(j) for these {i,j}: {333, 35197}, {17483, 35193}
X(35195) = barycentric quotient X(i)/X(j) for these {i,j}: {35192, 3467}, {35197, 226}

X(35196) = (name pending)

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

X(35196) lies on the cubic pK(X(35193),X(21)), the conic {{A,B,C,X(1),X(3)}} and these lines: {1, 1748}, {3, 54}, {21, 1807}, {102, 933}, {275, 5125}, {3469, 11107}, {7100, 15777}, {8764, 16813}

X(35196) = X(i)-cross conjugate of X(j) for these (i,j): {35, 21}, {522, 4636}
X(35196) = X(i)-isoconjugate of X(j) for these (i,j): {5, 65}, {7, 21807}, {10, 1393}, {12, 18180}, {51, 1441}, {53, 1214}, {57, 21011}, {79, 2599}, {108, 6368}, {109, 2618}, {226, 1953}, {307, 2181}, {311, 1402}, {324, 1409}, {343, 1880}, {349, 2179}, {651, 12077}, {1231, 3199}, {1400, 14213}, {1415, 18314}, {2171, 17167}, {3668, 7069}, {4551, 21102}, {8736, 16697}, {13157, 30456}, {13450, 22341}, {15451, 18026}
X(35196) = cevapoint of X(54) and X(2169)
X(35196) = barycentric product X(i)*X(j) for these {i,j}: {21, 2167}, {29, 97}, {54, 333}, {95, 284}, {275, 283}, {314, 2148}, {332, 8882}, {522, 18315}, {652, 18831}, {933, 6332}, {1812, 2190}, {2169, 31623}, {2299, 34386}, {2616, 4612}, {4636, 15412}, {5931, 33629}, {6514, 8884}
X(35196) = barycentric quotient X(i)/X(j) for these {i,j}: {21, 14213}, {29, 324}, {41, 21807}, {54, 226}, {55, 21011}, {60, 17167}, {95, 349}, {97, 307}, {283, 343}, {284, 5}, {332, 28706}, {333, 311}, {522, 18314}, {650, 2618}, {652, 6368}, {663, 12077}, {933, 653}, {1333, 1393}, {1812, 18695}, {2148, 65}, {2150, 18180}, {2167, 1441}, {2169, 1214}, {2174, 2599}, {2194, 1953}, {2204, 2181}, {2299, 53}, {3064, 23290}, {4636, 14570}, {7252, 21102}, {8748, 13450}, {8882, 225}, {14533, 73}, {14586, 109}, {15958, 1813}, {18315, 664}, {33629, 5930}, {35192, 35194}

Points associated with the cubic pK(X(50),X(1)): X(35197)-X(35201)

Contributed by Clark Kimberling and Peter Moses, December 8, 2019.

Suppose that A'B'C' is the circumcevian-inversion triangle of X(1), as defined in the preamble just before X(34864). The locus of a point Q such that the anticevian triangle of Q is perspective to A'B'C' is the cubic pK(X(50),X(1)), which passes through A, B, C, the vertices of the incentral triangle, and X(i) for these i: 1, 35, 36, 1094, 1095, 2169, 5353, 5357, 6126, 6149, 7343, 35197, 35198, 35199, 35200, 35201.

X(35197) = X(1)-CEVA CONJUGATE OF X(35)

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

X(35197) lies on the cubic pK(X(50),X(1)) and these lines: {1, 195}, {6, 18398}, {11, 22051}, {35, 500}, {36, 54}, {46, 1419}, {47, 14795}, {55, 12316}, {65, 6126}, {79, 651}, {221, 5903}, {222, 37524}, {323, 3678}, {484, 8614}, {498, 12325}, {1094, 2307}, {1199, 12005}, {1201, 1203}, {1480, 5697}, {1493, 5563}, {1993, 5904}, {1994, 3874}, {2293, 3746}, {2650, 5425}, {2888, 7951}, {2914, 6198}, {2964, 5399}, {3157, 5902}, {3293, 21173}, {3299, 8953}, {3301, 12965}, {3336, 23070}, {3337, 37509}, {3583, 20424}, {3585, 14101}, {4857, 11803}, {5010, 12307}, {5012, 23156}, {5353, 10677}, {5357, 10678}, {5442, 17074}, {5965, 28369}, {6583, 14627}, {6757, 24149}, {7078, 37571}, {7354, 36966}, {9502, 17745}, {10483, 12254}, {11398, 12175}, {14049, 18968}

X(35197) = X(1)-Ceva conjugate of X(35)
X(35197) = X(i)-isoconjugate of X(j) for these (i,j): {79, 3467}, {3065, 19658}, {3461, 34305}, {3483, 34303}
X(35197) = crosspoint of X(1) and X(3336)
X(35197) = crosssum of X(1) and X(3467)
X(35197) = barycentric product X(i)*X(j) for these {i,j}: {35, 17483}, {226, 35195}, {319, 21773}, {2003, 27529}, {3219, 3336}
X(35197) = barycentric quotient X(i)/X(j) for these {i,j}: {2174, 3467}, {3336, 30690}, {11069, 2166}, {14102, 24148}, {17483, 20565}, {19297, 19658}, {21773, 79}, {21863, 6757}, {35195, 333}
X(35197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {54, 7356, 36}, {2594, 6149, 35}, {3157, 16473, 5902}, {6286, 10066, 3746}, {10066, 15801, 6286}

X(35198) = (name pending)

Barycentrics a^3 (Sqrt[3] (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4)+2 (a^2-b^2-c^2) S) : :
Barycentrics Sin[A] Sin[2 A - Pi/3] : :

X(35198) lies on the cubic pK(X(50), X(1)) and these lines: {1, 3384}, {15, 35}, {31, 1095}, {36, 3165}, {47, 48}, {1094, 2151}, {3170, 5357}, {6126, 19373}

X(35198) = X(1)-Ceva conjugate of X(1094)
X(35198) = crosspoint of X(1) and X(3383)
X(35198) = crosssum of X(1) and X(3384)
X(35198) = X(i)-isoconjugate of X(j) for these (i,j): {2, 11082}, {13, 18}, {14, 11601}, {94, 8604}, {300, 21462}, {2963, 8838}, {3457, 34390}, {11078, 11138}, {11080, 19778}, {11083, 11140}, {11118, 36305}, {11139, 11143}, {20578, 32037}
X(35198) = barycentric product X(i)*X(j) for these {i,j}: {1, 11127}, {31, 11133}, {63, 10633}, {75, 11136}, {303, 2151}, {1094, 16770}, {3383, 11131}, {6149, 8836}
X(35198) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 11082}, {1094, 19778}, {2151, 18}, {2152, 11601}, {2964, 8838}, {8603, 2962}, {10633, 92}, {11088, 2166}, {11127, 75}, {11133, 561}, {11136, 1}
X(35198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {47, 48, 35199}, {2151, 6149, 1094}

X(35199) = (name pending)

Barycentrics a^3 (Sqrt[3] (a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4)-2 (a^2-b^2-c^2) S) : :
Barycentrics Sin[A] Sin[2 A + Pi/3] : :

X(35199) lies on the cubic pK(X(50), X(1)) and these lines: {1, 3375}, {16, 35}, {31, 1094}, {36, 3166}, {47, 48}, {1095, 2152}, {3171, 5353}, {6126, 7051}

X(35199) = X(1)-Ceva conjugate of X(1095)
X(35199) = crosspoint of X(1) and X(3376)
X(35199) = crosssum of X(1) and X(3375)
X(35199) = X(i)-isoconjugate of X(j) for these (i,j): {2, 11087}, {13, 11600}, {14, 17}, {94, 8603}, {301, 21461}, {2963, 8836}, {3458, 34389}, {11085, 19779}, {11088, 11140}, {11092, 11139}, {11117, 36304}, {11138, 11144}, {20579, 32036}
X(35199) = barycentric product X(i)*X(j) for these {i,j}: {1, 11126}, {31, 11132}, {63, 10632}, {75, 11135}, {302, 2152}, {1095, 16771}, {3376, 11130}, {6149, 8838}
X(35199) = barycentric quotient X(i)/X(j) for these {i,j}: {31, 11087}, {1095, 19779}, {2151, 11600}, {2152, 17}, {2964, 8836}, {8604, 2962}, {10632, 92}, {11083, 2166}, {11126, 75}, {11132, 561}, {11135, 1}
X(35199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {47, 48, 35198}, {2152, 6149, 1095}

X(35200) = TRILINEAR PRODUCT X(3)*X(74)

Barycentrics a^3 (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) : :
Trilinears (cos A)/(cos A - 2 cos B cos C) : :

X(35200) is the trilinear product of the circumcircle intercepts of line X(3)X(520). As the trilinear product of circumcircle antipodes, X(35200) lies on the conic {{A,B,C,X(109),X(162),X(163)}} with perspector X(48). (Randy Hutson, December 13, 2019)

Let A'B'C' be the circumcevian triangle of X(6000). Let A" be the trilinear product B'*C', and define B", C" cyclically. A", B", C" are collinear. The lines AA", BB", CC" concur in X(35200). (Randy Hutson, December 13, 2019)

X(35200) lies on the cubic pK(X(50),X(1)) and these lines: {1, 162}, {35, 73}, {47, 19614}, {48, 163}, {255, 4575}, {326, 4592}, {336, 14210}, {820, 2169}, {906, 3990}, {1304, 26701}, {1331, 3682}, {15291, 15627}

X(35200) = isogonal conjugate of X(1784)
X(35200) = isotomic conjugate of polar conjugate of X(2159)
X(35200) = trilinear pole of line X(48)X(822)
X(35200) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 1784}, {2, 1990}, {4, 30}, {19, 14206}, {25, 3260}, {76, 14581}, {92, 2173}, {107, 9033}, {264, 1495}, {281, 6357}, {393, 11064}, {523, 4240}, {648, 1637}, {823, 2631}, {1636, 15352}, {1969, 9406}, {2052, 3284}, {2407, 2501}, {2420, 14618}, {3163, 16080}, {6528, 9409}, {9407, 18022}
X(35200) = trilinear product X(i)*X(j) for these {i,j}: {2, 18877}, {3, 74}, {6, 14919}, {48, 2349}, {63, 2159}, {110, 14380}, {184, 1494}, {222, 15627}, {394, 8749}, {577, 16080}, {1073, 15291}, {1304, 520}, {2394, 32661}, {2433, 4558}, {15459, 32320}
X(35200) = trilinear quotient X(i)/X(j) for these (i,j): (1, 1784), (3, 30), (6, 1990), (32, 14581), (48, 2173), (63, 14206), (69, 3260), (74, 4), (110, 4240), (184, 1495), (222, 6357), (394, 11064), (520, 9033), (577, 3284), (647, 1637), (822, 2631), (1304, 107), (1494, 264), (2159, 19), (2349, 92), (2394, 14618), (2433, 2501), (3284, 3163), (4558, 2407), (8749, 393), (9247, 9406), (14380, 523), (14575, 9407), (14919, 2), (15291, 1249), (15459, 15352), (15627, 281), (16077, 6528), (16080, 2052), (18877, 6), (32320, 1636), (32661, 2420), (32695, 6529)
X(35200) = barycentric product X(i)*X(j) for these {i,j}: {1, 14919}, {3, 2349}, {48, 1494}, {63, 74}, {69, 2159}, {75, 18877}, {77, 15627}, {255, 16080}, {326, 8749}, {822, 16077}, {1304, 24018}, {2394, 4575}, {2433, 4592}, {15291, 19611}
X(35200) = barycentric quotient X(i)/X(j) for these (i,j): (3, 14206), (6, 1784), (31, 1990), (48, 30), (63, 3260), (74, 92), (110, 24001), (163, 4240), (184, 2173), (255, 11064), (822, 9033), (1304, 823), (1494, 1969), (2159, 4), (2349, 264), (2433, 24006), (4575, 2407), (6149, 14920), (8749, 158), (9247, 1495), (14919, 75), (15291, 1895), (15627, 318), (18877, 1)

X(35201) = (name pending)

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

Let A'B'C' be the anti-orthocentroidal triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically. Let A* be the trilinear product B"*C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(35201). (Randy Hutson, December 13, 2019)

X(35201) lies on the cubic pK(X(50),X(1)) and these lines: {1, 162}, {19, 163}, {35, 11587}, {36, 186}, {92, 2166}, {1099, 1784}, {2914, 6198}, {4246, 11720}, {24000, 36062}

X(35201) = X(i)-Ceva conjugate of X(j) for these (i,j): {92, 2173}, {11107, 186}
X(35201) = X(i)-isoconjugate of X(j) for these (i,j): {2, 11079}, {3, 5627}, {74, 265}, {94, 18877}, {125, 15395}, {476, 14380}, {1989, 14919}, {2166, 35200}, {2394, 32662}, {3470, 15392}, {8431, 34298}, {12028, 14264}, {14560, 34767}, {14592, 32640}, {18478, 22455}, {18558, 34568}, {36296, 36311}, {36297, 36308}
X(35201) = barycentric product X(i)*X(j) for these {i,j}: {1, 14920}, {19, 6148}, {92, 1511}, {162, 5664}, {186, 14206}, {323, 1784}, {340, 2173}, {526, 24001}, {4240, 32679}, {14590, 36035}
X(35201) = barycentric quotient X(i)/X(j) for these {i,j}: {19, 5627}, {31, 11079}, {50, 35200}, {186, 2349}, {340, 33805}, {1511, 63}, {1784, 94}, {1990, 2166}, {2173, 265}, {2420, 36061}, {2624, 14380}, {3258, 20902}, {4240, 32680}, {5664, 14208}, {6148, 304}, {6149, 14919}, {14206, 328}, {14591, 36034}, {14920, 75}, {23347, 32678}, {24001, 35139}, {32679, 34767}, {34397, 2159}, {36035, 14592}
{X(1),X(162)}-harmonic conjugate of X(36063)

Points associated with the circumcevian-inversion triangle of X(1): X(35202)-X(35210)

Contributed by Clark Kimberling and Peter Moses, December 8, 2019.

Let T = circumcevian-inversion triangle of X(1), denoted by A'B'C' in the preamble just before X(34864). The triangle T is perspective to each triangle in Column 2 of the following table. These triangles are described in César Lozada's Index of Triangles Referenced in ETC.

Perspector	2nd triangle	Reference
X(3)	excentral	TCCT 6.7
X(3)	hexyl	TCCT 6.36
X(3)	Apus	ETC X(5584)
X(30)	Feuerbach	TCCT 6.10
X(35)	ABC	Euclid
X(36)	incentral	TCCT 6.7
X(7668)	extangents	TCCT 6.17
X(7668)	Trinh	ETC X(7688)
X(7955)	4th mixtilinear	ETC X(7955), X(15931)
X(5251)	Gemini 26	ETC, Index of Triangles
X(25440)	Gemini 39	ETC, Index of Triangle
X(35202)	outer mixtilinear	ETC X(7955)
X(35203)	Apollonius	Index of Triangles
X(35204)	T(-2,1)	TCCT 6.41
X(35205)	Montesceoca-Hung	ETC X(6042)
X(35206)	Gemini 80	ETC X(35206)
X(35207)	1st Przybylowski-Bollin	ETC X(11752)
X(35208)	2nd Przybylowski-Bollin	ETC X(11752)
X(35209)	3rd Przybylowski-Bollin	ETC X(11752)
X(35210)	4th Przybylowski-Bollin	ETC X(11752)

For points associated with the circumcevian-inversion triangle of X(1), see X(35237)-X(35257).

X(35202) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND OUTER MIXTILINEAR

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

X(35202) lies on these lines: {1,3}, {4,25542}, {20,5259}, {30,7958}, {191,10167}, {572,17745}, {954,4355}, {990,27785}, {991,1203}, {1125,7411}, {1621,12512}, {2975,6743}, {3522,5248}, {3523,19855}, {3524,6796}, {3616,12511}, {3624,7580}, {3651,10165}, {3683,31805}, {3817,33557}, {4297,5251}, {4300,5315}, {4314,7677}, {5047,28164}, {5258,5731}, {5267,12447}, {5506,5927}, {5692,10884}, {6764,8715}, {6865,7951}, {6908,7741}, {6987,10483}, {7676,12575}, {7959,17821}, {8001,12333}, {8583,20835}, {9589,11495}, {9899,18621}, {11230,16117}, {12331,31447}, {12432,30284}, {12680,31658}, {13727,25512}, {15717,25440}, {19541,34595}, {22053,34043}

X(35203) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND APOLLONIUS

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

X(35203) lies on these lines: {1,14636}, {3,6}, {10,30}, {21,22080}, {35,181}, {36,1682}, {40,846}, {51,16452}, {140,2051}, {165,6048}, {199,283}, {376,9534}, {498,9553}, {499,9554}, {517,3743}, {540,17748}, {548,9568}, {631,9535}, {946,25354}, {988,28369}, {993,9565}, {1155,5530}, {1478,31496}, {1511,34453}, {1695,3576}, {1764,13731}, {2328,20836}, {2360,22139}, {3029,12042}, {3031,12041}, {3530,9569}, {3556,6000}, {3687,3916}, {3819,16453}, {3917,16451}, {4225,22076}, {4299,9552}, {4302,9555}, {5453,13624}, {5907,7420}, {5943,16287}, {6176,10441}, {6361,19853}, {6644,9571}, {6684,15973}, {6688,16286}, {7688,10822}, {7987,9549}, {9563,22115}, {9564,25440}, {10219,16291}, {10443,31658}, {10824,15931}, {12699,19858}, {12702,30116}, {15082,16297}, {15908,15974}, {15979,31424}, {19513,21363}, {33813,34454}, {33814,34458}

X(35204) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND T(-2,1)

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

Let P₁ be the K798e-to-K798i orthologic center. Let P₂ be the K798i-to-K798e orthologic center. Then X(35204) is the midpoint of P₁ and P₂. (Randy Hutson, January 17, 2020)

X(35204) lies on these lines: {1,6596}, {2,11604}, {3,191}, {8,14795}, {9,1030}, {10,21}, {11,5259}, {30,119}, {36,214}, {55,5426}, {56,16126}, {79,27385}, {104,15931}, {142,10090}, {149,5248}, {165,12332}, {187,1017}, {404,11263}, {411,21635}, {442,3035}, {528,15670}, {906,1333}, {952,5258}, {958,9897}, {960,14794}, {993,6224}, {1006,10265}, {1145,32760}, {1259,5904}, {1317,2078}, {1320,5424}, {1376,17057}, {1484,31650}, {1621,21630}, {1749,5440}, {2092,13006}, {2475,7951}, {2802,3746}, {2975,33337}, {3149,15017}, {3295,12653}, {3428,13253}, {3647,17100}, {3649,24465}, {3651,5660}, {3679,12331}, {3738,32679}, {4188,14450}, {5172,33667}, {5284,33709}, {5288,7972}, {5310,29872}, {5563,34195}, {5587,13743}, {5720,7701}, {5840,6841}, {6264,10267}, {6265,11012}, {6598,10073}, {6600,8069}, {6713,28465}, {6763,17660}, {6924,16159}, {7688,12515}, {7966,7993}, {8071,22754}, {8674,16164}, {10087,12640}, {10427,17768}, {10472,16430}, {10609,18253}, {10707,15671}, {10738,31159}, {10993,16617}, {11281,34123}, {11508,15347}, {11517,18397}, {11530,13205}, {12639,14526}, {12691,26878}, {12737,34486}, {13089,16761}, {13996,15174}, {14804,22836}, {15299,15348}, {15676,20095}, {16160,18406}, {16173,24541}, {16788,19584}, {17060,18637}, {17757,33961}, {19619,34586}, {20104,31254}, {22936,26086}, {25542,31272}, {33856,33862}

X(35205) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND MONTESDEOCA-HUNG

Barycentrics a^2 (a^8+2 a^7 b+a^6 b^2-a^2 b^6-2 a b^7-b^8+2 a^7 c+5 a^6 b c+4 a^5 b^2 c-4 a^3 b^4 c-7 a^2 b^5 c-6 a b^6 c-2 b^7 c+a^6 c^2+4 a^5 b c^2+3 a^4 b^2 c^2-10 a^3 b^3 c^2-17 a^2 b^4 c^2-10 a b^5 c^2-3 b^6 c^2-10 a^3 b^2 c^3-21 a^2 b^3 c^3-14 a b^4 c^3-3 b^5 c^3-4 a^3 b c^4-17 a^2 b^2 c^4-14 a b^3 c^4-2 b^4 c^4-7 a^2 b c^5-10 a b^2 c^5-3 b^3 c^5-a^2 c^6-6 a b c^6-3 b^2 c^6-2 a c^7-2 b c^7-c^8) : :

X(35205) lies on these lines: {3,6043}, {30,11992}, {35,6042}, {36,11993}, {5010,5974}, {7688,11991}, {11995,15931}

X(35206) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND GEMINI 80

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

X(35206) lies on these lines: {1,228}, {2,35}, {3,238}, {31,16451}, {32,2277}, {36,58}, {47,5197}, {100,3831}, {171,16453}, {386,3778}, {404,32772}, {748,16452}, {986,3185}, {1001,19760}, {1030,28244}, {1046,20967}, {1125,4276}, {1283,27628}, {1400,4251}, {1453,16778}, {1722,10434}, {2077,19513}, {2352,16478}, {3142,3583}, {3724,5262}, {3811,21371}, {4184,28247}, {4267,18166}, {4278,27644}, {5010,27639}, {5132,23383}, {5230,14798}, {5247,16678}, {5251,31339}, {5314,28242}, {8666,20036}, {10448,19245}, {10902,13731}, {12047,27653}, {14636,28265}, {16287,17123}, {16299,32784}, {16414,17122}, {16415,33111}, {16468,19762}, {17647,30366}, {19549,26285}, {21321,28258}, {23537,30362}, {27666,28257}, {28238,32760}

X(35207) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND 1ST PRZYBYLOWSKI-BOLLIN

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

X(35207) lies on these lines: {3,11754}, {15,36}, {30,11755}, {35,11753}, {56,11763}, {5563,11762}, {7688,11756}, {11760,15931}, {11764,15325}

X(35208) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND 2ND PRZYBYLOWSKI-BOLLIN

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

X(35208) lies on these lines: {3,11763}, {15,36}, {30,11764}, {35,11762}, {56,11754}, {5563,11753}, {7688,11765}, {11755,15325}, {11769,15931}

X(35209) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND 3RD PRZYBYLOWSKI-BOLLIN

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

X(35209) lies on these lines: {3,11772}, {16,36}, {30,11773}, {35,11771}, {56,11781}, {5563,11780}, {7688,11774}, {11778,15931}, {11782,15325}

X(35210) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(1) AND 4TH PRZYBYLOWSKI-BOLLIN

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

X(35210) lies on these lines: {3,11781}, {16,36}, {30,11782}, {35,11780}, {56,11772}, {5563,11771}, {7688,11783}, {11773,15325}, {11787,15931}

Perspectors of tangential triangle and other triangles: X(35211)-X(35226)

Contributed by Clark Kimberling and Peter Moses, December 10, 2019.

X(35211) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(22)

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

X(35211) lies on these lines: {2, 2138}, {6, 18018}, {22, 8793}, {1370, 5596}

X(35211) = TCC-perspector of X(22)

X(35212) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(37)

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

X(35212) lies on these lines: {3, 31872}, {6, 60}, {35, 37}, {186, 15946}, {199, 20989}, {1325, 5949}, {1631, 7669}, {2916, 9509}, {5124, 21004}, {8053, 20877}, {11340, 19744}, {18755, 32758}, {20472, 21773}

X(35212) = tangential-isogonal conjugate of X(1621)

X(35213) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(38)

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

X(35213) lies on these lines: {22, 15588}, {38, 20994}, {39, 1915}, {141, 2916}

X(35214) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(39)

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

X(35214) lies on these lines: {6, 14247}, {39, 1915}, {76, 15588}, {2076, 2916}, {9482, 14885}, {16689, 20994}, {16877, 23384}

X(35215) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(41)

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

X(35215) lies on these lines: {6, 279}, {41, 9316}, {1615, 21779}, {2176, 3010}

X(35216) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(42)

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

X(35216) lies on these lines: {2, 9509}, {6, 593}, {31, 20675}, {42, 172}, {55, 5147}, {284, 20461}, {1030, 17735}, {1961, 5277}, {5132, 10329}, {16064, 21004}

X(35217) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(66)

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

X(35217) lies on these lines: {6, 22}, {66, 20987}, {159, 21284}, {1614, 15577}

X(35218) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(67)

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

X(35218) lies on these lines: {6, 1112}, {22, 2930}, {23, 25328}, {24, 5621}, {26, 16010}, {67, 19596}, {110, 2916}, {542, 2937}, {1205, 18374}, {1614, 2781}, {2070, 32305}, {2836, 9591}, {5899, 32273}, {12367, 32260}, {12584, 13564}, {13171, 20987}, {15582, 32247}, {16176, 32240}, {18378, 20301}

X(35219) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(69)

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

X(35219) lies on these lines: {22, 69}, {24, 1503}, {66, 26156}, {206, 6800}, {1352, 32321}, {1498, 2916}, {1614, 19149}, {1660, 3313}, {1995, 23300}, {2854, 9924}, {7387, 12318}, {11441, 34146}, {16682, 18619}

X(35220) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(79)

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

X(35220) lies on these lines: {3, 2886}, {26, 1602}, {79, 20988}, {1603, 6644}, {1610, 34773}, {1626, 2937}, {2070, 23850}, {3220, 26201}, {5303, 20833}, {7354, 14667}

X(35221) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(80)

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

X(35221) lies on these lines: {3, 119}, {22, 2932}, {26, 1603}, {80, 20989}, {100, 2915}, {197, 9912}, {952, 1610}, {1324, 2070}, {1602, 6644}, {2933, 2937}, {9798, 12773}, {10058, 20831}, {17100, 20833}

X(35222) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(82)

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

X(35222) lies on these lines: {2, 3613}, {3, 7889}, {6, 694}, {25, 33801}, {82, 20990}, {141, 5201}, {157, 1624}, {160, 3618}, {237, 3589}, {570, 5943}, {571, 19137}, {597, 20775}, {759, 4245}, {800, 29959}, {852, 34828}, {1486, 16375}, {1609, 5020}, {2916, 20854}, {3001, 9969}, {3003, 9822}, {5640, 23181}, {6636, 16987}, {6748, 15143}, {7784, 10790}, {7804, 21177}, {7808, 11360}, {9306, 13345}, {10545, 15329}, {11063, 21513}, {13595, 16984}, {15534, 22152}, {16042, 17006}, {16776, 23635}, {22062, 34573}, {24273, 24729}

X(35223) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(87)

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

X(35223) lies on these lines: {3, 142}, {6, 23433}, {87, 8616}, {1616, 3286}, {5263, 28383}, {7373, 18166}, {20878, 23857}, {23404, 28365}

X(35224) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(87)

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

X(35224) lies on these lines: {3, 3622}, {22, 3007}, {89, 999}, {105, 1995}, {1486, 1623}, {26239, 26241}

X(35225) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(92)

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

Let P₁ and P₂ be the two points with tripolars |sin 2A| : : (or equivalently, |S^2 - SB SC| : :, or sin A' : :, where A'B'C' is the orthic triangle). Then X(35225) is the {P₁,P₂}-harmonic conbjugate of X(3). (Randy Hutson, January 17, 2020)

X(35225) lies on these lines: {3, 161}, {22, 157}, {24, 13450}, {25, 132}, {92, 23843}, {1576, 3060}, {1619, 33582}, {1629, 1632}, {2351, 21213}, {6641, 15577}, {11334, 17188}, {15781, 18405}

X(35225) = X(31)-of-tangential-triangle if ABC is acute
X(35225) = tangential isogonal conjugate of tangential isotomic conjugate of X(3)

X(35226) = PERSPECTOR OF THESE TRIANGLES: TANGENTIAL AND CIRCUMCEVIAN OF X(95)

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

X(35226) lies on these lines: {95, 160}, {99, 33801}, {159, 1632}, {11257, 20987}, {15428, 20200}

X(35227) = MIDPOINT OF X(15600) AND X(15601)

Barycentrics a*(5*a^2-4*(b+c)*a+3*(b-c)^2) : :
X(35227) = 3*X(1)+X(3973), 3*X(1)+2*X(8692), 2*X(1)+X(15601), 2*X(3973)+3*X(15600), 2*X(3973)-3*X(15601), 4*X(8692)+3*X(15600), 4*X(8692)-3*X(15601)

See César Lozada, Euclid 338 .

X(35227) lies on these lines: {1, 6}, {55, 5573}, {57, 902}, {106, 1292}, {145, 17338}, {269, 1319}, {528, 4859}, {551, 4648}, {614, 2177}, {968, 29818}, {999, 21002}, {1149, 2293}, {1253, 7962}, {1388, 1456}, {1418, 13462}, {1420, 2263}, {1458, 33633}, {1471, 3340}, {1480, 18443}, {1621, 3677}, {1697, 28082}, {2646, 15839}, {2999, 3748}, {3052, 10980}, {3058, 23681}, {3158, 5272}, {3332, 13464}, {3333, 4257}, {3445, 30389}, {3601, 28011}, {3616, 17282}, {3620, 3883}, {3622, 3662}, {3636, 4349}, {3679, 17337}, {3744, 10582}, {3749, 5437}, {3755, 8236}, {3886, 17117}, {3915, 11518}, {3928, 8616}, {3938, 7308}, {3945, 17274}, {3957, 14997}, {4000, 30331}, {4421, 8056}, {4428, 4906}, {4512, 17597}, {4666, 5269}, {4684, 20080}, {4702, 17151}, {4779, 28557}, {4902, 28534}, {5284, 7322}, {5853, 16020}, {7982, 13329}, {9581, 28027}, {10385, 24177}, {11529, 21059}, {14996, 29817}, {17245, 25055}, {19624, 25415}, {24175, 34607}, {24841, 25728}, {26728, 31162}, {30117, 31393}

X(35227) = midpoint of X(15600) and X(15601)
X(35227) = reflection of X(i) in X(j) for these (i,j): (3973, 8692), (15600, 1)
X(35227) = X(21)-Beth conjugate of X(3243)
X(35227) = intersection, other than A,B,C, of conics {{A, B, C, X(44), X(2191)}} and {{A, B, C, X(105), X(3243)}}
X(35227) = X(10002)-of-2nd circumperp triangle
X(35227) = X(15600)-of-5th mixtilinear triangle
X(35227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 238, 3243), (1, 1001, 7174), (1, 1279, 7290), (1, 5223, 4864), (3749, 29820, 5437), (3973, 8692, 15601), (5272, 17715, 3158)

X(35228) = X(3)X(66)∩X(6)X(186)

Barycentrics a^2*(a^10-(b^2+c^2)*a^8-2*(b^4+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)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4)) : :
Barycentrics (SB+SC)*((2*R^2-SW)*S^2-(12*R^2+SA-4*SW)*SA*SW) : :
X(35228) = 3*X(3)+X(159), 4*X(3)-X(15579), 4*X(3)+X(15580), 5*X(3)+X(15581), 2*X(3)+X(15582), X(159)-3*X(15577), 2*X(159)+3*X(15578), 4*X(159)+3*X(15579), 4*X(159)-3*X(15580), 5*X(159)-3*X(15581), 2*X(159)-3*X(15582), 2*X(15577)+X(15578), 4*X(15577)+X(15579), 4*X(15577)-X(15580), 5*X(15577)-X(15581), 2*X(15578)+X(15580), 5*X(15578)+2*X(15581)

See César Lozada, Euclid 338 .

X(35228) lies on these lines: {2, 18382}, {3, 66}, {6, 186}, {20, 28408}, {22, 10192}, {24, 5480}, {26, 29181}, {69, 10298}, {140, 20300}, {154, 6636}, {161, 7485}, {182, 9977}, {206, 1511}, {297, 18380}, {376, 2916}, {378, 20987}, {511, 1658}, {524, 18324}, {549, 23300}, {578, 32191}, {631, 2917}, {1092, 32391}, {1350, 7488}, {1853, 15246}, {2070, 31267}, {2393, 5092}, {2854, 12893}, {2883, 10323}, {3564, 15331}, {3589, 6644}, {3763, 34775}, {3818, 18570}, {3827, 13624}, {5085, 22467}, {5447, 7525}, {5893, 11414}, {5895, 16661}, {6000, 33533}, {6697, 18400}, {6759, 15067}, {6776, 21844}, {7492, 10117}, {7512, 17821}, {7514, 34573}, {7550, 18405}, {7575, 21850}, {7998, 15139}, {8546, 18571}, {8550, 32534}, {9682, 13910}, {9821, 15257}, {9924, 10249}, {9969, 11430}, {10182, 12106}, {10201, 23306}, {11178, 32600}, {11250, 29012}, {12007, 15750}, {12017, 34777}, {13347, 32184}, {13367, 19161}, {14649, 18472}, {15040, 15141}, {15462, 18438}, {16063, 23315}, {17714, 29317}, {17847, 33884}, {17907, 18121}, {18916, 19468}, {19153, 33878}, {21213, 23292}, {22109, 33851}, {29323, 32903}, {32274, 32607}, {32598, 32609}

X(35228) = midpoint of X(i) and X(j) for these {i,j}: {3, 15577}, {206, 3098}, {1350, 34117}, {10282, 14810}, {15578, 15582}, {15579, 15580}
X(35228) = reflection of X(i) in X(j) for these (i,j): (15578, 3), (15579, 15578), (15580, 15582), (15582, 15577), (20300, 140)
X(35228) = complement of X(18382)
X(35228) = X(15578)-of-ABC-X3 reflections triangle
X(35228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15582, 15579), (161, 7485, 23332), (1350, 23041, 34117), (3098, 11202, 206), (15577, 15578, 15580), (17821, 31884, 19149)

X(35229) = X(3)X(6)∩X(18)X(531)

Barycentrics a^2*(2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : :
X(35229) = 3*X(15)+X(5237), X(5237)-3*X(30560)

See César Lozada, Euclid 338 .

X(35229) lies on these lines: {3, 6}, {18, 531}, {6109, 16964}, {6781, 16001}, {7749, 16002}, {8594, 33274}, {22510, 31709}

X(35229) = midpoint of X(15) and X(30560)
X(35229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22236, 3104), (50, 13333, 15797), (500, 15796, 4280), (572, 13331, 22052), (576, 5206, 35230), (1380, 11916, 9690), (1805, 8410, 9786), (2673, 4263, 9994), (3312, 32447, 11482), (3368, 15883, 1687), (3386, 5024, 1684), (3386, 32110, 4286), (3394, 22425, 10542), (4266, 8410, 3581), (4271, 18114, 15905), (6441, 15167, 18994), (22052, 35006, 3284)

X(35230) = X(3)X(6)∩X(17)X(530)

Barycentrics a^2*(-2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : :
X(35230) = 3*X(16)+X(5238), X(5238)-3*X(30559)

See César Lozada, Euclid 338 .

X(35230) lies on these lines: {3, 6}, {17, 530}, {6108, 16965}, {6781, 16002}, {7749, 16001}, {8595, 33274}, {22511, 31710}

X(35230) = midpoint of X(16) and X(30559)
X(35230) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22238, 3105), (576, 5206, 35229), (2019, 33871, 12055), (15012, 15851, 22811)

X(35231) = MIDPOINT OF X(3) AND X(1113)

Barycentrics a*((4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b*c-2*(2*a^8-4*(b^2+c^2)*a^6+12*a^4*b^2*c^2+(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2))*S*OH*a) : :
Barycentrics 2*OH*R*(3*S^2-5*SB*SC)+a^2*(S^2-(42*R^2-SA-8*SW)*SA) : :
X(35231) = 3*X(2)+X(15160), 3*X(3)-X(1114), 3*X(3)+X(15154), 5*X(3)-X(15155), 7*X(3)-X(15156), 5*X(3)+X(15157), X(3)+3*X(28447), 7*X(3)-3*X(28448), 2*X(3)+X(30524), 4*X(3)-X(30525), 3*X(376)+X(14808), 3*X(381)-X(10736), 2*X(548)+X(20408), 5*X(631)-X(14807), 3*X(1113)+X(1114), 3*X(1113)-X(15154), 5*X(1113)+X(15155), 7*X(1113)+X(15156), 5*X(1113)-X(15157), X(1113)-3*X(28447), 7*X(1113)+3*X(28448), 4*X(1113)+X(30525)

See César Lozada, Euclid 338 .

X(35231) lies on these lines: {2, 3}, {1511, 2574}, {2100, 3576}, {2102, 10246}, {2103, 12702}, {2104, 5050}, {2105, 33878}, {2575, 12041}, {5085, 15162}, {7740, 10288}, {14500, 16111}

X(35231) = midpoint of X(i) and X(j) for these {i,j}: {3, 1113}, {20, 10751}, {1114, 15154}, {1657, 10737}, {2103, 12702}, {2105, 33878}, {3534, 10720}, {10750, 15160}, {14500, 16111}, {15155, 15157}
X(35231) = reflection of X(i) in X(j) for these (i,j): (1312, 31681), (1313, 140), (13627, 549), (20409, 31682), (30524, 1113), (31682, 3530), (35232, 3)
X(35231) = complement of X(10750)
X(35231) = circumperp conjugate of X(15155)
X(35231) = X(523)-vertex conjugate of X(15154)
X(35231) = circumcircle-inverse of X(15154)
X(35231) = X(10736)-of-Ehrmann-mid triangle
X(35231) = {X(i),X(j)}-harmonic conjugate of X(35232) for these {i,j}: {2, 18571}, {5, 15646}, {23, 12100}, {26, 16976}, {140, 186}, {548, 2071}, {549, 7575}, {550, 34152}, {2070, 3530}, {2072, 15331}

X(35232) = MIDPOINT OF X(3) AND X(1114)

Barycentrics a*((4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b*c+2*(2*a^8-4*(b^2+c^2)*a^6+12*a^4*b^2*c^2+(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2))*S*OH*a) : :
Barycentrics -2*OH*R*(3*S^2-5*SB*SC)+a^2*(S^2-(42*R^2-SA-8*SW)*SA) : :
X(35232) = 3*X(2)+X(15161), 3*X(3)-X(1113), 5*X(3)-X(15154), 3*X(3)+X(15155), 5*X(3)+X(15156), 7*X(3)-X(15157), 7*X(3)-3*X(28447), X(3)+3*X(28448), 4*X(3)-X(30524), 2*X(3)+X(30525), 3*X(376)+X(14807), 3*X(381)-X(10737), 2*X(548)+X(20409), 5*X(631)-X(14808)

See César Lozada, Euclid 338 .

X(35232) lies on these lines: {2, 3}, {1511, 2575}, {2101, 3576}, {2102, 12702}, {2103, 10246}, {2104, 33878}, {2105, 5050}, {2574, 12041}, {5085, 15163}, {7740, 10287}, {14499, 16111}

X(35232) = midpoint of X(i) and X(j) for these {i,j}: {3, 1114}, {20, 10750}, {1113, 15155}, {1657, 10736}, {2102, 12702}, {2104, 33878}, {3534, 10719}, {10751, 15161}, {14499, 16111}, {15154, 15156}
X(35197) = reflection of X(i) in X(j) for these (i,j): (1312, 140), (1313, 31682), (13626, 549), (20408, 31681), (30525, 1114), (31681, 3530), (35231, 3)
X(35232) = complement of X(10751)
X(35232) = circumperp conjugate of X(15154)
X(35232) = X(523)-vertex conjugate of X(15155)
X(35232) = circumcircle-inverse of X(15155)
X(35232) = {X(i),X(j)}-harmonic conjugate of X(35231) for these {i,j}: {2, 18571}, {5, 15646}, {23, 12100}, {26, 16976}, {140, 186}, {548, 2071}, {549, 7575}, {550, 34152}, {2070, 3530}, {2072, 15331}

X(35233) = X(3743)X(15604)∩X(4653)X(15569)

Barycentrics a*(a^6+4*(b+c)*a^5+2*(b+c)^2*a^4-(b+c)*(4*b^2+b*c+4*c^2)*a^3-(2*b^4+2*c^4+(8*b^2+13*b*c+8*c^2)*b*c)*a^2+(b+c)*(2*b^4+2*c^4-(2*b^2+7*b*c+2*c^2)*b*c)*a+(b^2+3*b*c+c^2)*(b^2-c^2)^2) : :

See César Lozada, Euclid 339 .

X(35233) lies on these lines: {3743, 15604}, {4653, 15569}

X(35234) = (name pending)

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

See Antreas P. Hatzipolakis, Chris van Tienhoven and Peter Moses, Euclid 342 .

X(35234) lies on this line: {2, 6}

X(35234) = barycentric product X(11054)*X (35133)
X(35234) = barycentric quotient X (35133)/X(34898)

X(35235) = X(2)X(3)∩X(122)X(135)

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

See Antreas P. Hatzipolakis, Chris van Tienhoven and Peter Moses, Euclid 342 .

X(35235) lies on these lines: {2, 3}, {122, 135}, {125, 136}, {247, 14697}, {265, 1300}, {1648, 2501}, {2972, 34338}, {3258, 16186}, {3563, 30789}, {5502, 16319}, {5962, 22115}, {6531, 8791}, {8287, 22094}, {12079, 18808}, {12893, 22823}, {13567, 18121}, {14165, 14355}, {16080, 18384}, {16177, 16178}, {16933, 31945}, {20957, 32710}, {30786, 35142}

X(35235) = polar circle inverse of X(7471)
X(35235) = polar conjugate of the isogonal conjugate of X(2088)
X(35235) = X(1299)-complementary conjugate of X(8062)
X(35235) = X(i)-Ceva conjugate of X(j) for these (i,j): {1300, 523}, {5962, 526}, {16080, 2501}, {18817, 14618}
X(35235) = X(16221)-cross conjugate of X(2970)
X(35235) = X(i)-isoconjugate of X(j) for these (i,j): {265, 1101}, {328, 23995}, {476, 4575}, {662, 32662}, {4558, 32678}, {4592, 14560}, {32661, 32680}
X(35235) = crosspoint of X(i) and X(j) for these (i,j): {4, 18808}, {850, 2986}, {1300, 14222}, {8901, 12079}, {14618, 18817} X(35235) = crosssum of X(1576) and X(3003)
X(35235) = polar conjugate of X(39295)
X(35235) = crossdifference of every pair of points on line {647, 32661}
X(35235) = orthic-isogonal conjugate of orthocenter of X(4)X(52)X(185)
X(35235) = barycentric product X(i)*X(j) for these {i,j}: {115, 340}, {125, 14165}, {186, 338}, {264, 2088}, {323, 2970}, {470, 30468}, {471, 30465}, {526, 14618}, {860, 8287}, {2052, 16186}, {2501, 3268}, {2986, 16221}, {3258, 16080}, {5664, 18808}, {6334, 14222}, {7799, 8754}, {8901, 14918}, {12079, 14920}, {14590, 23105}, {17923, 21054}, {18334, 18817}, {18384, 23965}, {23962, 34397}, {24006, 32679}
X(35235) = barycentric quotient X(i)/X(j) for these {i,j}: {115, 265}, {136, 18883}, {186, 249}, {338, 328}, {340, 4590}, {512, 32662}, {526, 4558}, {1835, 35049}, {2081, 23181}, {2088, 3}, {2489, 14560}, {2501, 476}, {2624, 4575}, {2970, 94}, {2971, 11060}, {3258, 11064}, {3268, 4563}, {8029, 14582}, {8749, 15395}, {8754, 1989}, {14165, 18020}, {14222, 687}, {14270, 32661}, {14273, 14559}, {14618, 35139}, {16186, 394}, {16221, 3580}, {18334, 22115}, {18384, 23588}, {21824, 1807}, {23105, 14592}, {24006, 32680}, {32679, 4592}, {34397, 23357}
X(35235) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 403, 11251}, {125, 136, 2970}, {868, 1650, 3134}, {3134, 3154, 1650}, {39240, 39241, 23970}

X(35236) = X(2)X(3)∩X(3269)X(9409)

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

See Antreas P. Hatzipolakis, Chris van Tienhoven and Peter Moses, Euclid 347 .

X(35236) lies on these lines: {2, 3}, {3269, 9409}

X(35236) = X(1942)-Ceva conjugate of X(647)
X(35236) = X(i)-isoconjugate of X(j) for these (i,j): {811, 2713}, {1942, 23999}
X(35236) = crosspoint of X(647) and X(1942)
X(35236) = crosssum of X(450) and X(648)
X(35236) = crossdifference of every pair of points on line {647, 648}
X(35236) = barycentric product X(i)*X(j) for these {i,j}: {450, 3269}, {647, 2797}, {851, 16573}
X(35236) = barycentric quotient X (i)/X(j) for these {i,j}: {2797, 6331}, {3049, 2713}

Points associated with the circumcevian-inversion triangle of X(3): X(35237)-X(35257)

Contributed by Clark Kimberling and Peter Moses, December 11, 2019.

Perspector	2nd triangle	Reference
X(3)	ABC	Euclid
X(3)	ABC reflected about X(3)	TCCT 6.12
X(3)	X(3) reflected in ABC	TCCT 6.13
X(3)	Lucas central	MathWorld
X(3)	Lucas antipodal	ETC X(6457)
X(3)	Lucas (-1)antipodal	ETC X(6457)
X(3)	Lucas (-1)central	ETC X(6457)
X(20)	Carnot: reflection of ABC in X(5)	Mathworld
X(30)	medial	TCCT 6.2)
X(30)	Ehrmann mid-triangle	Index of Triangles
X(35)	2nd anti-circumperp-tangential	Index of Triangles
X(36)	Mandart-incircle	ETC X(6018))
X(140)	Euler	TCCT 6.11)
X(376)	anticomplementary	TCCT 6.2)
X(378)	intouch-of-orthic (anti-Ara)	ETC X(11363)
X(381)	Gemini 109	ETC, Index of Triangles
X(549)	Gemini 110	ETC, Index of Triangles
X(550)	infinite altitude	Index of Triangles
X(3098)	Trihn	ETC X(7688)
X(3522)	anti-1st-Euler	Index of Triangles
X(3522)	anti-1st-Euler	Index of Triangles
X(3534)	Gemini 107	ETC, Index of Triangles
X(4299)	1st Johnson-Yff	ETC X(10523)
X(4302)	2nd Johnson-Yff	ETC X(10523)
X(5204)	outer Yff	ETC X(10037)
X(5217)	inner Yff	ETC X(10037)
X(6221)	anti-outer-Grebe	Index of Triangles
X(6398)	anti-inner-Grebe	Index of Triangles
X(6759)	Kosnita	ETC X(1659)
X(12054)	anti-5th-Brocard	Index of Triangles
X(12702)	Caelum (5th mixtilinear)	ETC X(5603)
X(13624)	anti-Aquila	ETC X(10363)
X(18481)	outer Garcia	ETC X(5587)
X(24929)	orthic-of-intouch	TCCT
X(33543)	anti-Hutson intouch	ETC X(11363)
X(35237)	tangential	TCCT 6.5
X(35238)	tangential of 1st circumperp	TCCTETC X(11363)
X(35239)	tangential of 2nd circumperp	ETC X(11363)
X(35240)	2nd Euler	ETC X(3758)
X(35241)	Gossard	ETC X(402)
X(35242)	Aquilla	ETC X(5586), T(1,2) in TCCT 6.40
X(35243)	Ara	ETC X(5594)
X(35244)	1st Auriga	ETC X(5597)
X(35245)	2nd Auriga	ETC X(5597)
X(35246)	inner Grebe	ETC X(1160)
X(35247)	outer Grebe	ETC X(1161)
X(35248)	5th Brocard	ETC X(32)
X(35249)	inner Johnson	ETC X(10522)
X(35250)	outer Johnson	ETC X(10522)
X(35251)	inner Yff tangents	ETC X(10527)
X(35252)	outer Yff tangents	ETC X(10527)
X(35253)	anti-incircle-circles	ETC X(11363)
X(35254)	orthic-of-medial	ETC X(11363)
X(35255)	3rd tri-squares central	ETC X(13637)
X(35256)	4th tri-squares central	ETC X(13637)
X(35257)	Ehrmann side-triangle	Index of Triangles

For points associated with the circumcevian-inversion triangle of X(1), see X(35202)-X(35210).

X(35237) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND TANGENTIAL

Barycentrics a^2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 - 4*a^6*c^2 - 20*a^4*b^2*c^2 + 16*a^2*b^4*c^2 + 8*b^6*c^2 + 6*a^4*c^4 + 16*a^2*b^2*c^4 - 18*b^4*c^4 - 4*a^2*c^6 + 8*b^2*c^6 + c^8) : :

X(35237) lies on these lines: {1, 1406}, {2, 18489}, {3, 1495}, {4, 15805}, {6, 30}, {20, 155}, {22, 74}, {25, 14855}, {26, 2929}, {64, 2917}, {113, 31152}, {140, 15811}, {154, 1511}, {159, 3098}, {195, 1181}, {376, 12112}, {378, 15080}, {381, 1533}, {382, 7706}, {394, 399}, {512, 15919}, {548, 16936}, {550, 1498}, {610, 3587}, {1192, 17714}, {1216, 12315}, {1350, 2930}, {1499, 8719}, {1514, 18531}, {1597, 1974}, {1620, 12107}, {1993, 11001}, {2071, 7712}, {2777, 15141}, {2916, 33533}, {2918, 10323}, {2941, 2948}, {3216, 6985}, {3431, 6800}, {3522, 15052}, {3529, 15032}, {3543, 15018}, {3581, 10605}, {3796, 14805}, {3830, 10601}, {3845, 17825}, {5059, 7592}, {5073, 10982}, {5085, 31861}, {5092, 9818}, {5422, 15682}, {5892, 18535}, {6200, 8939}, {6241, 33524}, {6396, 8943}, {7387, 11438}, {7393, 13474}, {7484, 16194}, {7485, 11455}, {7502, 10606}, {7530, 31860}, {7580, 34465}, {8703, 17811}, {8718, 11413}, {9937, 10575}, {10293, 12121}, {10545, 20791}, {10564, 21312}, {10625, 12174}, {11004, 15683}, {11403, 13336}, {11430, 12085}, {11441, 17538}, {12041, 34802}, {12082, 15072}, {12103, 32139}, {12900, 30771}, {13491, 17834}, {13754, 19588}, {15051, 20771}, {15085, 17854}, {15106, 16111}, {15640, 34545}, {15681, 18445}, {15812, 18358}, {16063, 32111}, {17845, 18431}, {18534, 34417}, {32137, 33537}

X(35237) = tangential-isogonal conjugate of X(6644)
X(35237) = X(1000)-of-tangential-triangle if ABC is acute

X(35238) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND TANGENTIAL OF 1ST CIRCUMPERP

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

X(35238) lies on these lines: {1, 3}, {10, 18761}, {20, 11499}, {30, 1376}, {100, 376}, {119, 6925}, {140, 7956}, {198, 16553}, {200, 7171}, {329, 3648}, {378, 11383}, {381, 4413}, {404, 6361}, {405, 25011}, {474, 12699}, {516, 6911}, {548, 32141}, {549, 1001}, {550, 11500}, {573, 2164}, {601, 2308}, {631, 5284}, {936, 31937}, {946, 16004}, {952, 8168}, {956, 3654}, {962, 6940}, {995, 1480}, {1012, 1512}, {1158, 31837}, {1478, 28458}, {1511, 12327}, {1621, 3524}, {1796, 7416}, {2096, 7411}, {2771, 3940}, {2810, 3098}, {2932, 12515}, {3452, 6985}, {3522, 11491}, {3534, 18524}, {3545, 9342}, {3560, 6684}, {3652, 15650}, {3679, 18519}, {3811, 13369}, {3913, 34773}, {3916, 17658}, {4299, 11501}, {4302, 11502}, {4421, 8703}, {4423, 5054}, {4428, 12100}, {5248, 6692}, {5251, 28444}, {5274, 6926}, {5326, 6863}, {5358, 15952}, {5534, 9841}, {5657, 6909}, {5687, 18481}, {5690, 12114}, {5720, 10860}, {5759, 30295}, {5840, 6827}, {5841, 6948}, {6221, 19000}, {6398, 18999}, {6644, 20872}, {6759, 12335}, {6796, 12512}, {6850, 10590}, {6880, 12775}, {6883, 7682}, {6890, 26470}, {6891, 10589}, {6905, 9778}, {6908, 31659}, {6913, 11231}, {6916, 8164}, {6918, 22793}, {6922, 10525}, {6928, 11826}, {6946, 9812}, {6958, 15908}, {6990, 26060}, {7986, 30115}, {8580, 18540}, {8666, 8668}, {9709, 18480}, {9780, 21669}, {9955, 16408}, {9958, 15979}, {10526, 31775}, {10742, 31141}, {11194, 13205}, {11202, 18621}, {11230, 20195}, {11490, 12054}, {11495, 12332}, {12041, 13204}, {12042, 13173}, {12083, 20989}, {12178, 33813}, {12334, 16163}, {12773, 34718}, {13635, 26241}, {15622, 33543}, {16417, 28198}, {17564, 25893}, {19513, 27639}, {19541, 28146}, {22753, 28174}, {22791, 25524}, {24047, 32561}

X(35238) = X(999)-of-tangential-of-1st-circumperp-triangle
X(35238) = X(6644)-of-1st-circumperp-triangle

X(35239) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND TANGENTIAL OF 2ND CIRCUMPERP

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

X(35239) lies on these lines: {1, 3}, {4, 5260}, {8, 3651}, {9, 31937}, {10, 6985}, {20, 22758}, {21, 6361}, {30, 958}, {100, 6876}, {104, 3522}, {119, 6838}, {140, 22753}, {355, 7580}, {376, 2975}, {378, 22479}, {405, 12699}, {411, 5657}, {474, 24564}, {515, 12511}, {516, 3560}, {548, 32153}, {549, 25524}, {550, 11495}, {573, 2911}, {581, 1126}, {582, 16466}, {595, 1480}, {912, 12520}, {916, 12163}, {944, 7411}, {946, 6883}, {956, 18481}, {962, 1006}, {993, 16004}, {1001, 22791}, {1259, 16139}, {1479, 28459}, {1511, 22583}, {1593, 1871}, {2550, 6869}, {2771, 3927}, {3098, 9052}, {3149, 26446}, {3524, 5253}, {3534, 26321}, {3556, 6000}, {3654, 5687}, {3679, 18518}, {3753, 11344}, {3869, 11517}, {4299, 22759}, {4302, 22760}, {4423, 18493}, {5011, 32561}, {5234, 18540}, {5248, 28194}, {5258, 18519}, {5259, 31162}, {5261, 6908}, {5450, 12512}, {5603, 6986}, {5690, 11500}, {5759, 8543}, {5771, 33899}, {5779, 31828}, {5837, 25440}, {5840, 6868}, {5841, 6850}, {5853, 8666}, {5904, 16132}, {6001, 26921}, {6221, 19014}, {6261, 31837}, {6398, 19013}, {6684, 6911}, {6713, 6926}, {6759, 22778}, {6825, 10588}, {6827, 10591}, {6836, 26470}, {6841, 19854}, {6849, 19855}, {6851, 19843}, {6899, 10527}, {6903, 11680}, {6906, 9778}, {6907, 10526}, {6913, 22793}, {6918, 11231}, {6920, 9812}, {6923, 11827}, {6928, 15908}, {6958, 7294}, {6988, 31659}, {6992, 10531}, {7330, 12565}, {7414, 26377}, {7420, 26935}, {8703, 11194}, {9708, 18480}, {9818, 9895}, {9943, 24467}, {9955, 11108}, {9956, 19541}, {10525, 31789}, {11491, 20013}, {11496, 28174}, {12041, 22586}, {12042, 22514}, {12054, 22520}, {12513, 34773}, {16117, 18525}, {16163, 19478}, {16418, 28198}, {16435, 26723}, {21161, 34632}, {22504, 33813}, {22775, 33814}, {24953, 34618}

X(35239) = {X(1),X(40)}-harmonic conjugate of X(37585)

X(35240) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 2ND EULER

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

X(35240) lies on these lines: {3, 1568}, {20, 1352}, {30, 1209}, {68, 21659}, {125, 12605}, {185, 9967}, {376, 6225}, {511, 34005}, {548, 1511}, {550, 3917}, {1092, 4549}, {1176, 15740}, {1216, 7723}, {1531, 7542}, {1533, 2937}, {2777, 7512}, {3522, 6030}, {3534, 33541}, {3574, 14118}, {5449, 18564}, {6368, 14329}, {7503, 32600}, {7873, 12037}, {10295, 11793}, {10619, 12606}, {11424, 20423}, {11440, 13399}, {11559, 12121}, {12362, 21663}, {13348, 16386}, {13470, 16003}, {15067, 15332}, {15644, 22948}, {16835, 17538}, {18563, 24572}, {30714, 31834}, {32348, 34007}

X(35241) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND GOSSARD

Barycentrics (2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)*(2*a^12 - 3*a^10*b^2 - 7*a^8*b^4 + 18*a^6*b^6 - 12*a^4*b^8 + a^2*b^10 + b^12 - 3*a^10*c^2 + 20*a^8*b^2*c^2 - 19*a^6*b^4*c^2 - 11*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - b^10*c^2 - 7*a^8*c^4 - 19*a^6*b^2*c^4 + 46*a^4*b^4*c^4 - 15*a^2*b^6*c^4 - 5*b^8*c^4 + 18*a^6*c^6 - 11*a^4*b^2*c^6 - 15*a^2*b^4*c^6 + 10*b^6*c^6 - 12*a^4*c^8 + 14*a^2*b^2*c^8 - 5*b^4*c^8 + a^2*c^10 - b^2*c^10 + c^12) : :

X(35241) lies on these lines: {2, 3}, {35, 18958}, {36, 11909}, {1294, 20127}, {1385, 12696}, {1511, 12369}, {3098, 12583}, {3184, 34334}, {3579, 12438}, {4299, 11905}, {4302, 11906}, {5204, 11913}, {5217, 11912}, {6221, 19018}, {6398, 19017}, {6759, 12791}, {9033, 16111}, {11831, 13624}, {11839, 12054}, {11900, 18481}, {11910, 12702}, {12041, 13212}, {12042, 13179}, {12181, 33813}, {12626, 34773}, {12752, 33814}, {12790, 16163}

X(35242) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND AQUILLA

Barycentrics    a*(5*a^3 + a^2*b - 5*a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c - 5*a*c^2 + b*c^2 - c^3) : :
Trilinears    5 cos A - cos B - cos C + 1 : :
Trilinears    r - 6 R cos A : :
X(35242) = X(1) - 6 X(3)

Let A"B"C" be as at X(12512). Then A"B"C" is homothetic to the hexyl triangle at X(5217). (Randy Hutson, January 17, 2020)

X(35242) lies on these lines: {1, 3}, {2, 18483}, {4, 3634}, {5, 19872}, {6, 9582}, {8, 10304}, {9, 2173}, {10, 376}, {20, 5587}, {30, 1698}, {32, 9574}, {63, 4420}, {78, 11684}, {84, 1796}, {100, 4652}, {104, 31509}, {140, 1699}, {172, 31426}, {187, 1571}, {200, 3916}, {355, 548}, {371, 9618}, {372, 9616}, {378, 7713}, {382, 7989}, {392, 19537}, {404, 31435}, {411, 10860}, {474, 3646}, {498, 9579}, {499, 9580}, {500, 5312}, {515, 3522}, {516, 631}, {519, 19708}, {549, 3624}, {550, 5691}, {551, 15698}, {573, 16670}, {574, 9575}, {601, 13329}, {632, 28178}, {936, 4640}, {944, 3625}, {946, 3523}, {952, 4816}, {962, 9624}, {993, 1706}, {1071, 10178}, {1092, 9621}, {1125, 3524}, {1151, 1703}, {1152, 1702}, {1350, 4663}, {1376, 5302}, {1479, 31231}, {1490, 15650}, {1511, 9904}, {1656, 28146}, {1657, 9956}, {1766, 16676}, {1768, 12738}, {1770, 5219}, {1788, 4304}, {1829, 11410}, {1902, 15750}, {2136, 8666}, {2272, 3730}, {2948, 12041}, {2951, 31658}, {2979, 31728}, {3053, 9593}, {3091, 28150}, {3097, 9821}, {3098, 3751}, {3146, 10175}, {3149, 11372}, {3244, 15710}, {3474, 13411}, {3525, 3817}, {3526, 7988}, {3528, 3626}, {3529, 19925}, {3530, 5886}, {3533, 10171}, {3534, 18480}, {3543, 19877}, {3545, 34638}, {3586, 15338}, {3614, 6907}, {3616, 15692}, {3621, 5731}, {3622, 15705}, {3632, 3654}, {3633, 3655}, {3636, 15715}, {3653, 14891}, {3656, 17504}, {3679, 8703}, {3753, 19535}, {3811, 3928}, {3828, 11001}, {3830, 19876}, {3832, 10172}, {3841, 6845}, {3843, 28154}, {3872, 5303}, {3911, 4294}, {4188, 5250}, {4221, 5358}, {4278, 18163}, {4292, 5218}, {4299, 9578}, {4302, 9581}, {4305, 4848}, {4312, 11374}, {4316, 10827}, {4324, 10826}, {4333, 7951}, {4386, 31429}, {4421, 6765}, {4668, 14093}, {4677, 18526}, {4691, 34627}, {4995, 10404}, {5054, 9955}, {5067, 12571}, {5071, 31253}, {5073, 30315}, {5092, 16475}, {5206, 9620}, {5220, 5732}, {5225, 6865}, {5229, 6916}, {5234, 9709}, {5248, 5437}, {5272, 21487}, {5281, 21620}, {5432, 9612}, {5433, 9614}, {5438, 6876}, {5440, 12526}, {5493, 5603}, {5690, 33923}, {5693, 9943}, {5698, 6700}, {5726, 9655}, {5759, 30424}, {5777, 5918}, {5818, 17538}, {5882, 20050}, {5890, 31737}, {5904, 13369}, {6001, 8567}, {6200, 18991}, {6210, 15601}, {6221, 19004}, {6241, 31752}, {6326, 12520}, {6396, 18992}, {6398, 19003}, {6409, 9583}, {6411, 7969}, {6412, 7968}, {6423, 31427}, {6445, 9585}, {6449, 9617}, {6459, 13975}, {6460, 13912}, {6560, 13893}, {6561, 13947}, {6644, 9591}, {6759, 9899}, {6762, 8715}, {6851, 18406}, {6905, 12511}, {6921, 25522}, {6922, 7173}, {6985, 7308}, {6999, 29608}, {7288, 10624}, {7354, 31434}, {7486, 10248}, {7501, 11471}, {7549, 21160}, {7745, 31428}, {7747, 31441}, {7998, 31751}, {8580, 31445}, {8582, 11111}, {8583, 16371}, {8616, 11512}, {8722, 12197}, {9324, 16528}, {9541, 13936}, {9587, 22115}, {9592, 15815}, {9613, 15326}, {9619, 15515}, {9625, 17928}, {9626, 10323}, {9746, 21554}, {9812, 10303}, {9841, 21165}, {9856, 33575}, {9860, 33813}, {9961, 20117}, {10445, 26039}, {10483, 28458}, {10593, 15908}, {10595, 28228}, {10789, 12054}, {10864, 11500}, {10993, 11219}, {11101, 33538}, {11194, 12629}, {11230, 15720}, {11522, 15712}, {11720, 15036}, {12019, 24466}, {12042, 13174}, {12100, 22791}, {12407, 16163}, {12515, 15015}, {12565, 17613}, {12717, 24309}, {12767, 22935}, {13464, 20070}, {13587, 19861}, {14217, 21154}, {14872, 31805}, {15045, 31757}, {15055, 33535}, {15071, 31837}, {15681, 33697}, {15688, 18525}, {15693, 18493}, {15694, 28202}, {15695, 28208}, {15696, 28160}, {15702, 19878}, {15719, 19883}, {16117, 18540}, {16127, 31018}, {16143, 22937}, {16485, 24443}, {16616, 19526}, {17286, 29040}, {17549, 19860}, {17576, 26062}, {20084, 31053}, {22266, 33703}, {25011, 31156}, {30435, 31430}, {32900, 34747}

X(35242) = homothetic center of hexyl triangle and medial triangle of 1st circumperp triangle

X(35243) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND ARA

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

X(35243) lies on these lines: {2, 3}, {6, 9687}, {35, 18954}, {36, 10833}, {64, 14641}, {99, 15574}, {155, 15644}, {159, 3098}, {161, 10606}, {511, 32621}, {517, 22769}, {577, 14581}, {1038, 9645}, {1154, 19459}, {1181, 10625}, {1216, 1498}, {1350, 8717}, {1351, 13391}, {1385, 9911}, {1511, 9919}, {1609, 2549}, {2979, 11456}, {3220, 3587}, {3426, 33533}, {3579, 9798}, {3796, 13352}, {3917, 18451}, {4299, 10831}, {4302, 10832}, {4549, 15311}, {4550, 20987}, {5204, 10046}, {5217, 10037}, {5285, 7171}, {5447, 17814}, {5621, 17702}, {5876, 12315}, {5892, 17810}, {6090, 10540}, {6101, 12164}, {6221, 19006}, {6398, 19005}, {6409, 8276}, {6410, 8277}, {6759, 9914}, {7585, 9695}, {7689, 9937}, {8192, 12702}, {8193, 18481}, {8573, 15048}, {8718, 11441}, {9529, 25644}, {9730, 33586}, {9861, 33813}, {9913, 33814}, {9927, 17712}, {10110, 13347}, {10263, 11432}, {10605, 14855}, {10627, 32139}, {10790, 12054}, {10979, 33843}, {10982, 13336}, {10984, 13366}, {11365, 13624}, {11472, 14810}, {11742, 34866}, {12041, 12310}, {12042, 13175}, {12121, 13171}, {12166, 13491}, {12168, 20127}, {12174, 18436}, {12410, 34773}, {12412, 16163}, {13340, 18445}, {14157, 15066}, {14915, 19596}, {15068, 32063}, {15085, 17855}, {15107, 20791}, {16266, 19347}, {19130, 31521}, {21006, 30230}, {22115, 26864}, {31831, 34781}

X(35244) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 1ST AURIGA

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

X(35244) lies on these lines: {1, 3}, {10, 18497}, {20, 8200}, {30, 5599}, {140, 8196}, {376, 5601}, {378, 11384}, {548, 32146}, {550, 9834}, {1511, 12365}, {3098, 12452}, {3522, 11843}, {3654, 8204}, {4299, 11869}, {4302, 11871}, {5657, 8207}, {5690, 9835}, {6221, 19008}, {6398, 19007}, {6759, 12468}, {8197, 18481}, {8203, 28174}, {8703, 11207}, {11837, 12054}, {12041, 13208}, {12042, 13176}, {12179, 33813}, {12454, 34773}, {12462, 33814}, {12466, 16163}

X(35245) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 2ND AURIGA

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

X(35245) lies on these lines: {1, 3}, {10, 18495}, {20, 8207}, {30, 5600}, {140, 8203}, {376, 5602}, {378, 11385}, {548, 32147}, {550, 9835}, {1511, 12366}, {3098, 12453}, {3522, 11844}, {3654, 8197}, {4299, 11870}, {4302, 11872}, {5657, 8200}, {5690, 9834}, {6221, 19010}, {6398, 19009}, {6759, 12469}, {8196, 28174}, {8204, 18481}, {8703, 11208}, {11838, 12054}, {12041, 13209}, {12042, 13177}, {12180, 33813}, {12455, 34773}, {12463, 33814}, {12467, 16163}

X(35246) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND INNER GREBE

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

X(35246) lies on these lines: {3, 6}, {20, 6215}, {30, 5591}, {35, 18959}, {36, 10927}, {140, 6202}, {323, 13616}, {376, 1271}, {378, 11388}, {382, 10514}, {548, 5875}, {550, 5871}, {1385, 12697}, {1511, 7725}, {1584, 15107}, {3522, 10783}, {3534, 13810}, {3579, 3641}, {4299, 10923}, {4302, 10925}, {5204, 10048}, {5217, 10040}, {5605, 12702}, {5689, 18481}, {5861, 8703}, {6227, 33813}, {6267, 6759}, {6281, 15696}, {6319, 12042}, {7732, 12041}, {9541, 13763}, {11370, 13624}, {12627, 34773}, {12753, 33814}, {12803, 16163}, {18538, 21737}, {26339, 34200}

X(35247) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND OUTER GREBE

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

X(35247) lies on these lines: {3, 6}, {20, 6214}, {30, 5590}, {35, 18960}, {36, 10928}, {140, 6201}, {323, 13617}, {376, 1270}, {378, 11389}, {382, 10515}, {548, 5874}, {550, 5870}, {1385, 12698}, {1511, 7726}, {1583, 15107}, {3522, 10784}, {3534, 13691}, {3579, 3640}, {4299, 10924}, {4302, 10926}, {5204, 10049}, {5217, 10041}, {5604, 12702}, {5688, 18481}, {5860, 8703}, {6226, 33813}, {6266, 6759}, {6278, 15696}, {6320, 12042}, {7733, 12041}, {11371, 13624}, {12628, 34773}, {12754, 33814}, {12804, 16163}, {26340, 34200}

X(35248) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 5TH BROCARD

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

X(35248) lies on these lines: {3, 6}, {20, 9996}, {30, 3096}, {35, 18957}, {36, 10877}, {140, 9993}, {376, 2896}, {378, 11386}, {381, 7914}, {382, 10356}, {548, 32151}, {549, 7846}, {550, 9873}, {1385, 12497}, {1511, 9984}, {3522, 9862}, {3524, 10583}, {3534, 7816}, {3579, 9941}, {3785, 14830}, {3933, 7782}, {4299, 10873}, {4302, 10874}, {5204, 10047}, {5217, 10038}, {6759, 12502}, {7470, 17128}, {7871, 9774}, {7889, 19924}, {8782, 12042}, {9857, 18481}, {9997, 12702}, {11368, 13624}, {12041, 13210}, {12495, 34773}, {12499, 33814}, {12501, 16163}, {14880, 17129}

X(35249) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND INNER JOHNSON

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

X(35249) lies on these lines: {3, 11}, {8, 12248}, {20, 355}, {30, 1376}, {35, 18961}, {36, 10947}, {40, 4316}, {55, 28458}, {140, 10893}, {376, 3434}, {378, 11390}, {390, 1385}, {495, 11248}, {517, 3474}, {548, 10943}, {550, 11495}, {944, 17654}, {1478, 35000}, {1511, 12371}, {1709, 3587}, {2077, 6923}, {3098, 12586}, {3359, 5727}, {3486, 13145}, {3522, 10785}, {3523, 10598}, {3524, 10584}, {3534, 18519}, {3560, 3826}, {3600, 11278}, {3655, 20075}, {3925, 28444}, {4190, 12699}, {4294, 11373}, {4297, 16004}, {4299, 10944}, {4317, 8148}, {4324, 10826}, {5101, 18533}, {5204, 10948}, {5217, 10523}, {5218, 6850}, {5698, 6869}, {5731, 9802}, {5837, 17647}, {5841, 6244}, {5886, 6955}, {6221, 19024}, {6256, 11698}, {6398, 19023}, {6759, 12920}, {6825, 26086}, {6868, 31663}, {6872, 17619}, {6885, 22793}, {6904, 9955}, {6906, 33108}, {6908, 33862}, {6916, 32613}, {6930, 11231}, {6932, 34474}, {6934, 12672}, {6938, 26446}, {6963, 10724}, {8703, 11235}, {10310, 10526}, {10794, 12054}, {10912, 34773}, {10914, 18481}, {11041, 35004}, {11249, 31777}, {12041, 13213}, {12042, 13180}, {12182, 33813}, {12512, 12616}, {12761, 33814}, {12889, 16163}, {15681, 34697}, {16418, 25973}, {17784, 28204}, {18242, 35023}

X(35249) = X(9668)-of-inner-Johnson-triangle
X(35249) = {X(20),X(3579)}-harmonic conjugate of X(35250)

X(35250) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND OUTER JOHNSON

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

X(35250) lies on these lines: {3, 12}, {8, 16139}, {20, 355}, {30, 958}, {35, 18962}, {36, 10953}, {40, 1727}, {56, 28459}, {72, 18481}, {140, 10894}, {376, 3436}, {378, 11391}, {381, 24953}, {390, 11278}, {452, 9955}, {496, 11249}, {515, 26921}, {517, 3486}, {548, 10942}, {550, 11500}, {1385, 3487}, {1511, 12372}, {1657, 6253}, {3098, 12587}, {3428, 7491}, {3474, 13145}, {3522, 10786}, {3523, 10599}, {3524, 10585}, {3534, 18518}, {3655, 20076}, {3841, 6917}, {4293, 11374}, {4302, 10950}, {4309, 8148}, {4316, 10827}, {4355, 18443}, {5080, 6876}, {5130, 18533}, {5204, 10523}, {5217, 10954}, {5428, 10198}, {5791, 6869}, {5886, 6936}, {5903, 16113}, {5905, 33858}, {6221, 19026}, {6361, 15680}, {6398, 19025}, {6759, 12930}, {6827, 7288}, {6865, 32612}, {6872, 12699}, {6885, 11231}, {6899, 10522}, {6926, 23961}, {6928, 7741}, {6930, 22793}, {6934, 26446}, {6948, 31663}, {6985, 18516}, {7688, 10483}, {8703, 11236}, {10056, 28460}, {10404, 13151}, {10795, 12054}, {11248, 31799}, {12041, 13214}, {12042, 13181}, {12183, 33813}, {12248, 12738}, {12635, 34773}, {12762, 33814}, {12890, 16163}, {15681, 34746}, {18407, 19843}, {18525, 21677}, {25466, 28466}

X(35250) = X(9655)-of-outer-Johnson-triangle
X(35250) = {X(20),X(3579)}-harmonic conjugate of X(35249)

X(35251) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND INNER YFF TANGENTS

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

X(35251) lies on these lines: {1, 3}, {10, 28444}, {20, 10942}, {30, 5552}, {44, 2164}, {100, 18518}, {104, 20050}, {119, 382}, {140, 10531}, {376, 10528}, {378, 11400}, {381, 26364}, {528, 18543}, {548, 32213}, {550, 12115}, {962, 11729}, {1012, 18357}, {1511, 12381}, {1597, 26378}, {1656, 26333}, {1657, 6256}, {3085, 28458}, {3098, 12594}, {3149, 12775}, {3522, 10805}, {3523, 10596}, {3524, 10586}, {3534, 18545}, {3560, 9780}, {3614, 11826}, {3617, 6906}, {3625, 5450}, {3626, 22758}, {3651, 20084}, {3715, 22936}, {4299, 10956}, {4302, 10958}, {4421, 10915}, {5054, 10200}, {5225, 6882}, {5229, 11929}, {5554, 16370}, {5687, 18519}, {5748, 6985}, {6199, 26465}, {6221, 19048}, {6395, 26459}, {6398, 19047}, {6735, 18525}, {6759, 13094}, {6863, 31777}, {6923, 10592}, {6943, 13199}, {6958, 10593}, {6966, 10943}, {7173, 10525}, {8703, 11239}, {9655, 26482}, {9668, 26476}, {9709, 13743}, {10526, 24466}, {10803, 12054}, {11499, 31673}, {12041, 13217}, {12042, 13189}, {12083, 26309}, {12189, 33813}, {12332, 12738}, {12515, 12635}, {12648, 34773}, {12773, 25438}, {12905, 16163}, {16371, 22791}, {16417, 18493}, {16418, 24982}, {18483, 25440}

X(35251) = {X(3),X(12702)}-harmonic conjugate of X(35252)

X(35252) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND OUTER YFF TANGENTS

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

X(35252) lies on these lines: {1, 3}, {20, 10943}, {30, 10527}, {140, 10532}, {376, 10529}, {378, 11401}, {381, 26363}, {382, 26470}, {529, 18545}, {548, 32214}, {550, 12116}, {956, 18518}, {993, 18483}, {1511, 12382}, {1597, 26377}, {1656, 26332}, {2975, 6985}, {3086, 28459}, {3098, 12595}, {3149, 18357}, {3522, 10806}, {3523, 10597}, {3524, 10587}, {3534, 18543}, {3614, 10526}, {3616, 28466}, {3617, 6905}, {3622, 21161}, {3625, 6796}, {3626, 11499}, {4299, 10957}, {4302, 10959}, {5054, 10198}, {5225, 7491}, {5229, 6842}, {5258, 18491}, {5289, 16139}, {5303, 6361}, {5550, 6883}, {6199, 26464}, {6221, 19050}, {6395, 26458}, {6398, 19049}, {6734, 18525}, {6759, 13095}, {6841, 30478}, {6863, 10592}, {6911, 9780}, {6928, 10593}, {6958, 31799}, {6962, 10942}, {7173, 11827}, {7580, 32153}, {8703, 11240}, {9655, 26481}, {9668, 26475}, {10525, 30264}, {10804, 12054}, {10916, 11194}, {11491, 20050}, {12041, 13218}, {12042, 13190}, {12083, 26308}, {12190, 33813}, {12649, 34773}, {12699, 28444}, {12738, 22775}, {12776, 33814}, {12906, 16163}, {16370, 22791}, {16417, 24987}, {16418, 18493}, {18499, 24390}, {19843, 28452}, {21487, 26228}, {22758, 31673}

X(35252) = {X(3),X(12702)}-harmonic conjugate of X(35251)

X(35253) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND ANTI-INCIRCLE-CIRCLES

Barycentrics a^2*(a^8 - 6*a^6*b^2 + 12*a^4*b^4 - 10*a^2*b^6 + 3*b^8 - 6*a^6*c^2 - 108*a^4*b^2*c^2 + 82*a^2*b^4*c^2 + 32*b^6*c^2 + 12*a^4*c^4 + 82*a^2*b^2*c^4 - 70*b^4*c^4 - 10*a^2*c^6 + 32*b^2*c^6 + 3*c^8) : :

X(35253) lies on these lines: {3, 32062}, {20, 1351}, {64, 3098}, {376, 11469}, {382, 5544}, {550, 11206}, {1482, 13369}, {1657, 3527}, {3534, 12164}, {5085, 16936}, {10619, 12316}, {11414, 22528}, {11820, 15696}, {12308, 16163}, {12310, 15041}, {14641, 33543}, {15054, 32254}

X(35254) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND ORTHIC-OF-MEDIAL

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

X(35254) lies on these lines: {3, 4549}, {4, 32620}, {5, 32223}, {20, 11472}, {30, 141}, {69, 34801}, {140, 7706}, {146, 6636}, {265, 343}, {376, 12112}, {541, 1511}, {548, 5894}, {549, 18388}, {550, 5447}, {1154, 3629}, {1368, 6699}, {1531, 15760}, {2777, 14810}, {2883, 7525}, {3426, 3534}, {3589, 7514}, {3763, 18420}, {3878, 31730}, {3917, 12358}, {5562, 10619}, {6643, 22661}, {6995, 18489}, {7492, 32111}, {7516, 13568}, {7998, 10295}, {9818, 31670}, {10192, 10272}, {10938, 23039}, {11574, 13754}, {11591, 34782}, {11820, 15696}, {11850, 12118}, {12121, 34802}, {12359, 12362}, {13419, 15704}, {17702, 32257}, {19376, 19378}, {29181, 31861}, {30739, 32110}

X(35255) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 3RD TRI-SQUARES CENTRAL

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

X(35255) lies on these lines: {2, 6221}, {3, 1587}, {4, 6449}, {5, 1151}, {6, 549}, {11, 31500}, {20, 6455}, {30, 590}, {35, 18965}, {36, 13901}, {140, 371}, {355, 9615}, {372, 3530}, {376, 6451}, {378, 13884}, {381, 6445}, {485, 550}, {486, 632}, {487, 11315}, {489, 32491}, {546, 10576}, {547, 6480}, {548, 3070}, {569, 9686}, {631, 3311}, {642, 7914}, {1125, 31439}, {1131, 17538}, {1152, 15712}, {1327, 19710}, {1328, 6439}, {1384, 31403}, {1385, 13912}, {1511, 8994}, {1588, 3526}, {1656, 6407}, {1657, 31412}, {1995, 9695}, {2066, 15325}, {2067, 31499}, {3069, 5054}, {3071, 3628}, {3090, 6519}, {3098, 13910}, {3146, 3316}, {3312, 3523}, {3364, 16773}, {3389, 16772}, {3522, 6496}, {3524, 6398}, {3525, 6447}, {3534, 23249}, {3579, 8983}, {3592, 5420}, {3595, 26619}, {3614, 9649}, {3815, 9675}, {3820, 9678}, {3845, 6433}, {3850, 6484}, {3853, 6486}, {3857, 10147}, {3858, 10195}, {4299, 13897}, {4302, 13898}, {5010, 19030}, {5023, 31411}, {5055, 9690}, {5056, 9543}, {5067, 9692}, {5070, 9691}, {5072, 23263}, {5204, 13905}, {5217, 13904}, {5254, 9674}, {5407, 15236}, {5432, 9663}, {5433, 9648}, {5886, 9616}, {6101, 12239}, {6395, 15693}, {6396, 12100}, {6411, 6560}, {6412, 17504}, {6417, 13935}, {6419, 12108}, {6424, 31406}, {6437, 8252}, {6438, 19711}, {6446, 15700}, {6450, 7581}, {6452, 15692}, {6456, 10299}, {6468, 15699}, {6759, 8991}, {7173, 9662}, {7280, 19028}, {7288, 31474}, {7486, 9693}, {7502, 9682}, {7509, 9694}, {7582, 10303}, {7988, 9584}, {8960, 33923}, {8980, 33813}, {8997, 12042}, {8998, 12041}, {9306, 9687}, {9582, 12699}, {9583, 26446}, {9600, 15048}, {9601, 13881}, {9602, 31489}, {9646, 18990}, {9661, 15171}, {9679, 31419}, {10124, 32790}, {10257, 18457}, {10264, 10819}, {10304, 23267}, {10577, 16239}, {10897, 16196}, {11473, 21841}, {11542, 34552}, {11543, 34551}, {11812, 32788}, {12054, 13885}, {12376, 13392}, {12702, 13902}, {13624, 13883}, {13847, 15713}, {13893, 18481}, {13911, 34773}, {13913, 33814}, {13915, 16163}, {13941, 15702}, {14241, 15697}, {15051, 19052}, {15694, 18510}, {15701, 19053}, {15704, 23251}, {17564, 31473}, {17800, 23253}, {18907, 31463}, {26289, 26361}

X(35255) = {X(6),X(549)}-harmonic conjugate of X(35256)

X(35256) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND 4TH TRI-SQUARES CENTRAL

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

X(35256) lies on these lines: {2, 6398}, {3, 1588}, {4, 6450}, {5, 1152}, {6, 549}, {20, 6456}, {30, 615}, {35, 18966}, {36, 13958}, {140, 372}, {371, 3530}, {376, 6452}, {378, 13937}, {381, 6446}, {485, 632}, {486, 550}, {488, 11316}, {490, 32490}, {546, 10577}, {547, 6481}, {548, 3071}, {631, 3312}, {641, 7914}, {1132, 17538}, {1151, 15712}, {1327, 6440}, {1328, 19710}, {1385, 13975}, {1511, 13969}, {1587, 3526}, {1656, 6408}, {3068, 5054}, {3070, 3628}, {3090, 6522}, {3098, 13972}, {3146, 3317}, {3311, 3523}, {3365, 16773}, {3390, 16772}, {3522, 6497}, {3524, 6221}, {3525, 6448}, {3534, 23259}, {3579, 13971}, {3593, 26620}, {3594, 5418}, {3845, 6434}, {3850, 6485}, {3853, 6487}, {3857, 10148}, {3858, 10194}, {4299, 13954}, {4302, 13955}, {5010, 19029}, {5055, 23249}, {5070, 31412}, {5072, 23253}, {5204, 13963}, {5217, 13962}, {5406, 15235}, {5414, 15325}, {6101, 12240}, {6199, 15693}, {6200, 12100}, {6411, 17504}, {6412, 6561}, {6418, 9540}, {6420, 12108}, {6423, 31406}, {6431, 9680}, {6437, 19711}, {6438, 8253}, {6445, 15700}, {6449, 7582}, {6451, 15692}, {6455, 10299}, {6469, 15699}, {6475, 31414}, {6759, 13980}, {7280, 19027}, {7486, 23269}, {7581, 10303}, {8376, 31463}, {8972, 15702}, {10124, 32789}, {10257, 18459}, {10264, 10820}, {10304, 23273}, {10576, 16239}, {10898, 16196}, {11474, 21841}, {11542, 34551}, {11543, 34552}, {11812, 32787}, {12041, 13990}, {12042, 13989}, {12054, 13938}, {12375, 13392}, {12702, 13959}, {13624, 13936}, {13846, 15713}, {13947, 18481}, {13967, 33813}, {13973, 34773}, {13977, 33814}, {13979, 16163}, {14226, 15697}, {15051, 19051}, {15694, 18512}, {15701, 19054}, {15703, 17851}, {15704, 23261}, {17800, 23263}, {26288, 26362}

X(35256) = {X(6),X(549)}-harmonic conjugate of X(35255)

X(35257) = PERSPECTOR OF THESE TRIANGLES: CIRCUMCEVIAN-INVERSION OF X(3) AND EHRMANN SIDE-TRIANGLE

Barycentrics (a^2 - b^2 - c^2)*(8*a^8 - 5*a^6*b^2 - 12*a^4*b^4 + 7*a^2*b^6 + 2*b^8 - 5*a^6*c^2 + 17*a^4*b^2*c^2 - 7*a^2*b^4*c^2 - 8*b^6*c^2 - 12*a^4*c^4 - 7*a^2*b^2*c^4 + 12*b^4*c^4 + 7*a^2*c^6 - 8*b^2*c^6 + 2*c^8) : :

X(35257) lies on these lines: {3, 1531}, {20, 6288}, {30, 7703}, {69, 12121}, {146, 376}, {382, 15432}, {550, 11459}, {599, 3098}, {1539, 10298}, {6699, 18564}, {6759, 15696}, {10619, 22815}, {12270, 14677}, {16163, 22584}, {33543, 33887}

Centroids of three points on the circumcircle: X(35238)-X(35306)

Contributed by Peter Moses, December 11, 2019.

According to a theorem by Bernard Gibert, every cubic of the type nK0(X6,R), where R = u : v : w, meets the circumcircle in three points, other than A,B,C, and the centroid of those three points is given by

(-3a² + b² + c²)u + 2a²v + 2a²w : :

For centers X(35238)-X(35306), the type is written as nK0(X(6),X(k)), and the circumcircle as Γ .

See Table 4. nK0(X6,R).

X(35258) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(1))

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

X(35258) lies on these lines: {1, 89}, {2, 165}, {3, 392}, {6, 4689}, {8, 3977}, {9, 100}, {10, 4302}, {19, 1013}, {20, 10268}, {21, 40}, {31, 5256}, {33, 1748}, {35, 78}, {38, 3749}, {42, 1707}, {46, 5248}, {55, 63}, {56, 10179}, {57, 1621}, {77, 109}, {145, 8275}, {149, 5231}, {162, 2331}, {171, 968}, {191, 3811}, {200, 3219}, {210, 4421}, {238, 17601}, {283, 601}, {329, 5281}, {354, 4428}, {377, 31730}, {390, 5744}, {404, 31435}, {405, 3579}, {411, 12705}, {452, 24982}, {474, 31663}, {517, 16370}, {612, 846}, {614, 8616}, {896, 2177}, {901, 2725}, {908, 5218}, {946, 6910}, {956, 3895}, {960, 4855}, {962, 24541}, {988, 3915}, {993, 2802}, {997, 5010}, {1001, 1155}, {1004, 11495}, {1006, 3359}, {1054, 15485}, {1150, 3886}, {1158, 10884}, {1253, 1331}, {1279, 17595}, {1293, 9061}, {1376, 3305}, {1385, 19535}, {1420, 3890}, {1445, 10177}, {1512, 6930}, {1519, 6954}, {1697, 2975}, {1698, 5046}, {1706, 5260}, {1708, 3256}, {1743, 3240}, {1758, 2263}, {1770, 10198}, {1790, 2187}, {1801, 2328}, {1836, 6690}, {1979, 16525}, {1998, 4326}, {2000, 4319}, {2094, 11038}, {2339, 5314}, {2478, 6684}, {2646, 11682}, {2771, 18446}, {2800, 3576}, {2999, 17127}, {3011, 24248}, {3035, 4679}, {3052, 3666}, {3158, 3681}, {3207, 4520}, {3247, 9347}, {3295, 3916}, {3361, 3622}, {3416, 3712}, {3434, 5745}, {3474, 5249}, {3523, 10270}, {3601, 3869}, {3612, 3878}, {3616, 15803}, {3617, 5234}, {3634, 5187}, {3646, 17531}, {3647, 8715}, {3654, 17525}, {3663, 26228}, {3679, 15677}, {3689, 5220}, {3699, 17336}, {3711, 15481}, {3722, 16496}, {3729, 4427}, {3731, 5297}, {3744, 21000}, {3750, 4650}, {3753, 16418}, {3755, 24597}, {3816, 31224}, {3821, 29855}, {3873, 3928}, {3883, 17740}, {3897, 7982}, {3923, 29828}, {3935, 5223}, {3952, 25728}, {4054, 24280}, {4141, 4677}, {4184, 10434}, {4188, 8583}, {4190, 12512}, {4193, 31423}, {4199, 22080}, {4229, 26638}, {4294, 6734}, {4297, 16208}, {4298, 10587}, {4309, 10916}, {4312, 31019}, {4314, 12649}, {4338, 11263}, {4362, 28522}, {4384, 4781}, {4413, 15254}, {4450, 33113}, {4511, 30282}, {4660, 29857}, {4847, 20075}, {4850, 7290}, {4862, 33148}, {4999, 12701}, {5057, 5219}, {5080, 31434}, {5128, 5436}, {5129, 25011}, {5131, 25055}, {5175, 18231}, {5269, 28606}, {5271, 32932}, {5273, 17784}, {5284, 5437}, {5291, 31433}, {5338, 14004}, {5432, 24703}, {5552, 12572}, {5587, 11114}, {5657, 11111}, {5687, 31445}, {5691, 15680}, {5794, 15338}, {5840, 11113}, {5905, 13405}, {5918, 17616}, {5919, 11194}, {5985, 13174}, {6244, 13615}, {6284, 26066}, {6361, 6857}, {6745, 31018}, {6904, 24564}, {6933, 18483}, {7322, 33761}, {7330, 11491}, {7411, 10860}, {7465, 24309}, {7483, 12699}, {7987, 17548}, {8053, 23845}, {8171, 10569}, {8580, 27065}, {9441, 16367}, {9574, 33854}, {9580, 11680}, {9588, 25005}, {9965, 10578}, {10303, 26129}, {10310, 11344}, {10385, 24477}, {10527, 10624}, {10528, 12527}, {10529, 12575}, {10582, 27003}, {10980, 23958}, {10987, 16973}, {11231, 17556}, {11415, 13411}, {11523, 11684}, {11679, 32929}, {11849, 26921}, {12446, 25440}, {12526, 34772}, {12530, 21811}, {12702, 17571}, {16469, 17012}, {16482, 23832}, {16570, 32912}, {16832, 24344}, {17011, 30652}, {17064, 33094}, {17272, 33175}, {17274, 33122}, {17276, 17724}, {17282, 24542}, {17284, 33086}, {17298, 29830}, {17304, 26230}, {17338, 26073}, {17502, 19704}, {17532, 28146}, {17566, 25522}, {17718, 17768}, {17781, 25568}, {20292, 25525}, {21077, 31452}, {21511, 25930}, {22060, 23853}, {23681, 29681}, {24392, 34611}, {25527, 32950}, {25734, 32937}, {27811, 29597}, {28516, 32934}, {29665, 33100}, {29675, 32857}, {30944, 31394}

X(35259) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(3))

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

X(35259) lies on these lines: {2, 154}, {3, 1495}, {4, 11064}, {5, 18396}, {6, 110}, {22, 7998}, {23, 1350}, {24, 11459}, {25, 394}, {26, 15067}, {51, 3167}, {64, 22467}, {69, 4232}, {107, 9308}, {125, 18440}, {141, 7493}, {155, 568}, {156, 13363}, {182, 11284}, {184, 373}, {283, 13737}, {323, 11477}, {343, 6353}, {378, 15035}, {381, 5642}, {399, 32235}, {450, 33971}, {468, 1352}, {524, 26255}, {542, 26869}, {575, 30734}, {599, 7426}, {940, 4228}, {1092, 1598}, {1147, 7529}, {1181, 6642}, {1184, 1915}, {1192, 12111}, {1216, 9714}, {1301, 15394}, {1304, 2710}, {1316, 3233}, {1344, 13415}, {1345, 13414}, {1351, 3292}, {1368, 31383}, {1498, 15072}, {1501, 1611}, {1511, 31861}, {1568, 18494}, {1620, 11440}, {1754, 16438}, {1790, 13615}, {1899, 6677}, {1992, 20192}, {1993, 5102}, {2001, 35006}, {2030, 21448}, {2328, 11350}, {2360, 11344}, {2453, 7471}, {3053, 3231}, {3090, 9707}, {3091, 11425}, {3098, 32237}, {3148, 9155}, {3155, 5407}, {3156, 5406}, {3515, 5907}, {3517, 5562}, {3518, 17834}, {3546, 16655}, {3580, 15069}, {3763, 7495}, {3818, 5094}, {3830, 13857}, {3867, 28708}, {3917, 9909}, {4226, 5108}, {4230, 6787}, {4248, 10449}, {4252, 7419}, {5012, 17825}, {5017, 9225}, {5023, 8627}, {5068, 14528}, {5092, 16187}, {5198, 13346}, {5422, 9544}, {5646, 7496}, {5663, 6644}, {5891, 14070}, {5943, 11402}, {6759, 16836}, {6786, 10607}, {6815, 16252}, {6816, 34782}, {6997, 23292}, {7395, 10282}, {7398, 11427}, {7465, 25878}, {7473, 34360}, {7474, 15668}, {7484, 15082}, {7485, 26881}, {7492, 21766}, {7503, 17821}, {7509, 26882}, {7514, 34513}, {7517, 13340}, {7528, 9820}, {7706, 16534}, {7716, 20806}, {8644, 21733}, {9027, 19136}, {9703, 21308}, {9715, 11793}, {9717, 14687}, {9777, 15520}, {9786, 11441}, {9822, 19125}, {9924, 26206}, {10117, 34778}, {10301, 31670}, {10441, 17523}, {10606, 15055}, {10643, 19364}, {10644, 19363}, {10961, 19356}, {10963, 19355}, {10984, 14530}, {11003, 16042}, {11159, 23699}, {11413, 15811}, {11424, 32604}, {11442, 26958}, {11479, 13367}, {11550, 30771}, {11580, 33979}, {11645, 32216}, {11898, 21970}, {11935, 15039}, {12017, 22112}, {12106, 15068}, {12525, 21177}, {14118, 33537}, {14467, 20065}, {14913, 19118}, {15058, 32534}, {15060, 18324}, {15139, 19149}, {15360, 15533}, {16072, 18400}, {16223, 18445}, {16419, 22352}, {17522, 26625}, {18474, 23515}, {19137, 19459}, {19153, 29959}, {19457, 20771}, {20987, 26283}, {22129, 24320}, {26256, 26543}, {29012, 31152}, {32223, 34507}

X(35260) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(4))

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

X(35260) lies on these lines: {2, 154}, {3, 5656}, {4, 1495}, {6, 4232}, {20, 11064}, {23, 15577}, {25, 11427}, {64, 15717}, {69, 110}, {100, 18621}, {107, 1249}, {140, 14530}, {156, 11411}, {159, 1995}, {160, 1624}, {161, 13595}, {184, 6353}, {193, 15585}, {221, 5265}, {323, 34117}, {376, 2777}, {387, 4248}, {394, 10565}, {468, 6776}, {549, 32063}, {631, 5651}, {632, 34780}, {858, 14927}, {924, 15724}, {1353, 21970}, {1370, 26881}, {1498, 3523}, {1614, 3147}, {1619, 7485}, {1649, 3566}, {1660, 5012}, {1971, 7736}, {1992, 7426}, {2192, 5281}, {2360, 27407}, {2393, 5640}, {2409, 6794}, {2781, 33884}, {2883, 3522}, {3066, 9924}, {3068, 10534}, {3069, 10533}, {3088, 16654}, {3089, 16657}, {3090, 9833}, {3091, 18405}, {3167, 10154}, {3231, 32445}, {3357, 10299}, {3524, 5650}, {3525, 14216}, {3528, 5878}, {3530, 12315}, {3533, 20299}, {3541, 16658}, {3542, 9707}, {3545, 18400}, {3556, 5253}, {3619, 5596}, {3620, 34774}, {3763, 20079}, {3832, 17845}, {4549, 16534}, {5059, 5893}, {5067, 18381}, {5218, 10535}, {5894, 21734}, {5943, 34750}, {5972, 16051}, {6090, 10519}, {6193, 13383}, {6247, 10303}, {6515, 9544}, {6618, 11547}, {6622, 19467}, {6676, 8780}, {6988, 14925}, {6995, 23292}, {7288, 26888}, {7473, 30227}, {7492, 10117}, {7494, 9306}, {7528, 32346}, {7533, 18382}, {7585, 17820}, {7586, 17819}, {7712, 13203}, {7998, 34146}, {8889, 31383}, {9123, 14932}, {9820, 31305}, {9919, 13392}, {10174, 11189}, {10181, 11190}, {10193, 15719}, {10201, 32423}, {10304, 15311}, {10546, 31267}, {10606, 15692}, {11001, 13857}, {11204, 15698}, {11241, 19053}, {11242, 19054}, {11456, 18931}, {11487, 18350}, {11488, 30403}, {11489, 30402}, {11645, 30775}, {12317, 32235}, {12319, 20773}, {13093, 15712}, {13289, 20125}, {14002, 15582}, {14389, 23049}, {14643, 18531}, {14644, 18918}, {14862, 20427}, {15066, 19149}, {15077, 34799}, {15581, 16042}, {15702, 23329}, {15740, 22467}, {17538, 22802}, {17811, 21167}, {17813, 20192}, {18475, 18537}, {20423, 32267}, {21766, 34778}, {26283, 28708}, {29317, 34608}, {31670, 32237}

X(35261) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(8))

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

X(35261) lies on these lines: {2, 165}, {8, 902}, {55, 26065}, {100, 7083}, {145, 3977}, {345, 3052}, {1707, 34379}, {3011, 24280}, {3241, 5429}, {3579, 13742}, {3616, 4414}, {3618, 4689}, {3729, 26245}, {3771, 28508}, {4188, 25879}, {4427, 26228}, {4676, 5218}, {5281, 27064}, {5657, 13735}, {10385, 33121}, {17126, 17316}, {17767, 33144}, {26658, 32973}, {27129, 33181}, {28557, 30699}, {33118, 34607}

X(35262) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(9))

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

X(35262) lies on these lines: {1, 88}, {2, 515}, {3, 392}, {8, 1420}, {9, 1055}, {10, 6921}, {21, 3062}, {36, 63}, {40, 4188}, {46, 11682}, {55, 10179}, {56, 78}, {57, 4511}, {104, 5720}, {142, 390}, {145, 26062}, {165, 3877}, {200, 13462}, {210, 11194}, {224, 22766}, {271, 8059}, {355, 6713}, {377, 1125}, {388, 27385}, {405, 13624}, {443, 12116}, {474, 1385}, {499, 17647}, {517, 16371}, {631, 24987}, {908, 4293}, {936, 2975}, {944, 17567}, {946, 4190}, {952, 17564}, {956, 5126}, {960, 4652}, {962, 24558}, {993, 3305}, {999, 3870}, {1004, 22753}, {1149, 3749}, {1155, 5289}, {1193, 7032}, {1319, 1376}, {1388, 5836}, {1450, 1818}, {1476, 2057}, {1478, 30852}, {1482, 17573}, {1512, 6970}, {1519, 6948}, {1621, 30282}, {1698, 17566}, {1699, 17579}, {1706, 4861}, {1737, 31224}, {1837, 6691}, {2093, 9352}, {2475, 8227}, {2476, 3624}, {2478, 4297}, {2646, 25524}, {2751, 14733}, {2771, 32612}, {3035, 5252}, {3146, 26129}, {3158, 3241}, {3244, 4917}, {3333, 34772}, {3337, 12559}, {3338, 11520}, {3361, 3868}, {3419, 15325}, {3436, 4311}, {3476, 6735}, {3579, 19537}, {3600, 27383}, {3623, 14563}, {3646, 16865}, {3753, 10246}, {3811, 5563}, {3812, 34471}, {3869, 15803}, {3876, 12059}, {3889, 30329}, {3897, 17531}, {3899, 5131}, {3913, 20323}, {3924, 11512}, {4187, 18481}, {4189, 31435}, {4193, 5691}, {4299, 21616}, {4308, 7080}, {4315, 6745}, {4317, 21077}, {4420, 6762}, {4421, 5919}, {4512, 17549}, {4666, 24929}, {5046, 25522}, {5080, 30827}, {5087, 12943}, {5154, 18492}, {5177, 5436}, {5187, 31673}, {5276, 9592}, {5277, 9619}, {5303, 31424}, {5330, 7991}, {5426, 6175}, {5433, 5794}, {5437, 13384}, {5450, 34293}, {5552, 10106}, {5554, 5882}, {5687, 24928}, {5722, 10609}, {5727, 6224}, {5840, 5886}, {5853, 11240}, {5880, 15950}, {5881, 10265}, {5901, 17563}, {6734, 7288}, {6857, 24564}, {6931, 19925}, {6933, 19862}, {7280, 12514}, {7292, 16485}, {7354, 25681}, {7415, 24556}, {7504, 34595}, {7675, 10177}, {7701, 17009}, {7966, 17572}, {7988, 17577}, {8256, 12832}, {8273, 11344}, {9061, 28291}, {9310, 14439}, {9578, 27529}, {9613, 11681}, {10200, 10572}, {10269, 18446}, {10444, 24540}, {10470, 16454}, {10882, 16451}, {10884, 18238}, {11230, 17532}, {11246, 34647}, {11329, 24590}, {11529, 27003}, {12114, 25875}, {12635, 32636}, {13742, 25881}, {15326, 24703}, {15679, 30308}, {16056, 24551}, {16370, 17502}, {16408, 18518}, {16410, 33597}, {16842, 31666}, {17136, 26229}, {17556, 28160}, {17619, 18525}, {18483, 31295}, {18857, 22758}, {20067, 27131}, {20076, 21075}, {21214, 28375}, {21477, 25930}, {26050, 26093}

X(35263) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(10))

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

X(35263) lies on these lines: {1, 3977}, {2, 165}, {4, 25982}, {10, 902}, {31, 306}, {40, 17526}, {55, 5294}, {100, 17353}, {142, 24542}, {404, 25881}, {519, 33161}, {527, 33122}, {551, 27811}, {908, 4676}, {1125, 4414}, {1386, 3712}, {1707, 4001}, {2308, 4028}, {2887, 28494}, {3011, 3923}, {3052, 32777}, {3589, 4689}, {3663, 4427}, {3664, 29830}, {3687, 17127}, {3710, 5266}, {3729, 26228}, {3744, 9053}, {3749, 33163}, {3870, 10460}, {3883, 32779}, {3886, 24597}, {3912, 17126}, {3914, 6679}, {4061, 19742}, {4062, 21747}, {4138, 29865}, {4195, 24987}, {4202, 31730}, {4217, 5587}, {4297, 17539}, {4353, 29831}, {4356, 29833}, {4416, 33175}, {4660, 30768}, {4781, 31191}, {4981, 5325}, {5192, 6684}, {5212, 14997}, {5269, 17776}, {5297, 25101}, {5745, 24552}, {5853, 33114}, {6327, 20106}, {7290, 17740}, {11104, 18653}, {13405, 26223}, {13742, 25967}, {16050, 25935}, {17061, 28556}, {17351, 17724}, {17355, 26227}, {17697, 24982}, {17781, 33126}, {20103, 26688}, {23536, 24850}, {24231, 29638}, {24248, 29855}, {25019, 25968}, {26723, 32932}, {28526, 33143}, {28580, 33128}, {29596, 33086}, {29835, 30331}, {29860, 32857}, {29871, 33102}, {29874, 33100}, {30652, 32858}, {30653, 33077}

X(35264) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(22))

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

X(35264) lies on these lines: {2, 154}, {3, 6030}, {4, 9820}, {5, 9707}, {6, 9544}, {22, 1495}, {23, 394}, {24, 10539}, {25, 110}, {26, 18350}, {49, 13861}, {51, 15520}, {54, 7529}, {155, 3518}, {156, 5946}, {159, 26206}, {184, 575}, {186, 18451}, {206, 29959}, {323, 33586}, {343, 15448}, {378, 16194}, {381, 30522}, {468, 11442}, {858, 31383}, {1147, 10594}, {1498, 22467}, {1501, 2502}, {1593, 11449}, {1598, 34148}, {1613, 9998}, {1614, 6642}, {1915, 5359}, {1974, 8681}, {1994, 14002}, {2070, 15068}, {2328, 11340}, {2360, 20846}, {2979, 6090}, {3066, 17809}, {3091, 19357}, {3193, 17562}, {3426, 15051}, {3448, 26958}, {3515, 12111}, {3516, 11439}, {3517, 5889}, {3542, 14516}, {3548, 16659}, {3580, 6353}, {3818, 31236}, {3830, 32609}, {3832, 11425}, {3843, 18504}, {3854, 14528}, {4232, 6515}, {5012, 5020}, {5050, 11451}, {5410, 11448}, {5411, 11447}, {5596, 26156}, {5640, 11402}, {5642, 31133}, {5651, 7485}, {5654, 7576}, {5972, 11550}, {6000, 15078}, {6636, 17811}, {6644, 10540}, {6677, 18911}, {6759, 17928}, {6997, 14389}, {7391, 11064}, {7394, 23292}, {7484, 15080}, {7488, 17814}, {7493, 14826}, {7503, 10282}, {7505, 12134}, {7517, 13391}, {7530, 22115}, {7545, 9703}, {7691, 16195}, {7712, 15246}, {9019, 20806}, {9545, 10982}, {9704, 18369}, {9705, 9781}, {9706, 11426}, {9714, 11412}, {9715, 11444}, {9777, 11422}, {9818, 11464}, {10298, 15052}, {10601, 11003}, {11004, 31860}, {11188, 19153}, {11202, 15030}, {11408, 11453}, {11409, 11452}, {11413, 26883}, {11424, 13570}, {11440, 15750}, {11455, 15035}, {11457, 16238}, {11459, 14070}, {11469, 21734}, {12086, 15811}, {12106, 18445}, {12161, 13321}, {12162, 32534}, {12167, 19122}, {12272, 19118}, {14118, 17821}, {15043, 19347}, {15072, 32063}, {15107, 20850}, {15534, 25321}, {15748, 22334}, {16042, 17825}, {16266, 18378}, {18324, 18435}, {18440, 23293}, {21451, 34799}, {34417, 34986}

X(35265) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(23))

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

X(35265) lies on these lines: {2, 154}, {3, 7712}, {6, 14002}, {20, 32111}, {22, 6090}, {23, 110}, {25, 1994}, {30, 32609}, {49, 34484}, {111, 1692}, {113, 10296}, {146, 10295}, {156, 568}, {182, 10546}, {184, 5640}, {186, 5663}, {206, 11188}, {352, 2076}, {373, 5012}, {399, 7575}, {468, 3448}, {524, 25331}, {575, 10545}, {1092, 12087}, {1199, 13621}, {1351, 9716}, {1499, 13586}, {1511, 7464}, {1514, 3146}, {1614, 9730}, {1691, 2502}, {1915, 34482}, {1974, 15531}, {1993, 16981}, {1995, 5050}, {2071, 14157}, {2854, 18374}, {2888, 13383}, {2930, 32217}, {3047, 34397}, {3091, 9707}, {3431, 31861}, {3542, 34799}, {3564, 7426}, {3580, 14683}, {3581, 5609}, {3819, 6030}, {3832, 19357}, {5102, 11004}, {5111, 13192}, {5189, 11064}, {5446, 9705}, {5502, 33927}, {5642, 10989}, {5650, 15246}, {5651, 7496}, {5965, 15360}, {5971, 12215}, {5972, 30745}, {6000, 15055}, {6146, 21451}, {6593, 12367}, {6636, 7998}, {6759, 15072}, {7488, 10539}, {7492, 15066}, {7512, 15067}, {7527, 11464}, {7533, 14389}, {7556, 15068}, {7574, 10272}, {7693, 18583}, {8115, 15162}, {8116, 15163}, {8627, 9225}, {8705, 19596}, {9545, 10594}, {9706, 10110}, {10282, 14118}, {10298, 18451}, {10564, 15034}, {10620, 18571}, {11061, 32113}, {11134, 11626}, {11137, 11624}, {11145, 14169}, {11146, 14170}, {11202, 15305}, {11422, 15520}, {11449, 12086}, {11799, 12383}, {12088, 13340}, {12106, 15032}, {13857, 29323}, {14094, 32110}, {14530, 17928}, {14567, 20998}, {14865, 32171}, {14912, 26255}, {15041, 15646}, {15078, 32063}, {15082, 22352}, {15139, 15647}, {16252, 34007}, {16319, 17511}, {17506, 18439}, {18325, 34153}, {18504, 34786}, {23515, 25739}, {24981, 32223}, {30551, 32165}, {31074, 31383}

X(35266) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(30))

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

X(35266) lies on these lines: {2, 154}, {6, 20192}, {25, 20423}, {30, 113}, {110, 524}, {230, 2502}, {343, 8780}, {376, 20725}, {419, 12243}, {460, 9880}, {462, 25164}, {463, 25154}, {468, 542}, {511, 32267}, {523, 14697}, {549, 5651}, {597, 1995}, {599, 7493}, {1499, 4786}, {1641, 4226}, {1990, 4240}, {1992, 4232}, {2030, 16317}, {2071, 15152}, {3524, 12112}, {3564, 32225}, {3580, 9143}, {3589, 10546}, {5108, 32459}, {5562, 33591}, {5609, 15361}, {5650, 12100}, {5663, 18579}, {5967, 14694}, {5972, 11645}, {6329, 10545}, {7471, 9129}, {7473, 9144}, {7495, 20582}, {7540, 9820}, {7575, 32315}, {10282, 34664}, {10295, 10706}, {10539, 34351}, {10540, 20126}, {11179, 26864}, {11422, 20583}, {15362, 23236}, {16655, 18281}, {19924, 32237}

X(35267) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(31))

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

X(35267) lies on these lines: {2, 28845}, {31, 36}, {41, 17798}, {48, 674}, {55, 1055}, {165, 15735}, {199, 2187}, {604, 23868}, {902, 15655}, {910, 28125}, {968, 30282}, {1471, 22390}, {1486, 22054}, {2340, 3207}, {4512, 17549}

X(35268) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(32))

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

X(35268) lies on these lines: {2, 6030}, {3, 1495}, {22, 184}, {23, 182}, {24, 16836}, {25, 373}, {26, 9730}, {30, 13394}, {32, 8627}, {51, 3796}, {69, 24981}, {74, 32235}, {110, 3098}, {125, 7493}, {154, 3917}, {160, 6786}, {185, 9715}, {206, 2916}, {376, 2777}, {386, 20851}, {550, 11064}, {568, 2937}, {569, 17714}, {576, 11003}, {578, 12088}, {669, 8723}, {1112, 17710}, {1147, 13340}, {1204, 7488}, {1350, 3292}, {1469, 5370}, {1790, 20841}, {1899, 10565}, {1995, 5092}, {2502, 8588}, {3056, 7302}, {3060, 15520}, {3066, 12017}, {3111, 21525}, {3124, 5033}, {3131, 21158}, {3132, 21159}, {3148, 21163}, {3231, 5206}, {3357, 8718}, {3534, 13857}, {3574, 31305}, {3589, 10301}, {3818, 7495}, {3981, 35006}, {4549, 15063}, {5012, 11002}, {5028, 14567}, {5093, 13366}, {5102, 11402}, {5157, 16776}, {5191, 8722}, {5320, 5347}, {5663, 7502}, {5972, 16063}, {6636, 7998}, {6640, 17712}, {6660, 11171}, {6676, 11550}, {6759, 7512}, {7387, 11424}, {7485, 15082}, {7494, 31383}, {7496, 10546}, {7519, 19130}, {7525, 10539}, {7552, 23325}, {7556, 11438}, {7558, 13419}, {7667, 10192}, {8541, 8705}, {8542, 12367}, {9707, 15644}, {9735, 14170}, {9736, 14169}, {10201, 23515}, {10282, 10323}, {10298, 15055}, {10329, 13331}, {10330, 26233}, {10541, 31860}, {10564, 33532}, {10601, 20850}, {11188, 19126}, {11414, 13367}, {11430, 12082}, {11449, 16661}, {11470, 34397}, {12083, 18475}, {13198, 19121}, {13336, 13363}, {13417, 18438}, {14531, 19347}, {14810, 15066}, {14855, 18324}, {15246, 33879}, {15448, 30739}, {18474, 25337}, {18911, 32223}, {23039, 34006}, {26869, 32225}, {29323, 31133}

X(35268) = intersection of tangents to Thomson-Gibert-Moses hyperbola at X(3) and X(154)
X(35268) = pole of line X(3)X(64) wrt Thomson-Gibert-Moses hyperbola
X(35268) = crosspoint of X(3) and X(154) wrt Thomson triangle

X(35269) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(38))

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

X(35269) lies on these lines: {2, 10186}, {165, 3877}, {171, 4511}, {404, 18788}, {1055, 4640}, {1376, 2170}, {1742, 10868}, {1962, 29597}, {4447, 33299}, {9020, 20990}, {17078, 33126}, {17439, 24264}

X(35270) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(42))

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

X(35270) lies on these lines: {2, 165}, {6, 31}, {43, 31508}, {57, 968}, {103, 110}, {154, 19346}, {210, 14439}, {228, 8012}, {354, 1962}, {579, 4343}, {649, 33570}, {899, 15601}, {910, 3683}, {916, 3167}, {1002, 10389}, {1155, 30944}, {1200, 20967}, {1201, 3747}, {1282, 3219}, {1334, 2223}, {1839, 4207}, {2272, 3185}, {2340, 3730}, {3136, 7965}, {3474, 30949}, {3977, 17135}, {4191, 5646}, {4210, 5888}, {4335, 27624}, {4427, 26237}, {4448, 29328}, {4640, 8299}, {4893, 11124}, {5179, 28133}, {5273, 31330}, {5537, 5643}, {5544, 6244}, {5698, 30961}, {6818, 26040}, {10980, 29814}, {12723, 21811}, {13588, 24557}, {14924, 16373}, {18650, 33171}, {24592, 32932}, {30943, 30970}

X(35271) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(44))

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

X(35271) lies on these lines: {1, 19537}, {2, 17502}, {3, 392}, {20, 1538}, {35, 10179}, {36, 518}, {72, 5204}, {100, 5126}, {165, 19705}, {214, 1155}, {354, 33595}, {404, 13624}, {474, 7987}, {515, 21154}, {516, 34123}, {517, 4881}, {1055, 14439}, {1125, 15338}, {1319, 2802}, {1385, 4188}, {2078, 2932}, {2163, 4663}, {2646, 5883}, {2771, 18861}, {3035, 21578}, {3309, 8643}, {3530, 24987}, {3555, 4855}, {3576, 3753}, {3612, 5439}, {3697, 5438}, {3838, 5444}, {3848, 5426}, {3897, 31666}, {3911, 10609}, {3916, 5692}, {4004, 34471}, {4018, 15803}, {4297, 13747}, {4316, 5087}, {4511, 5122}, {4512, 19704}, {5044, 5303}, {5154, 33697}, {6921, 17619}, {6938, 17618}, {7743, 32558}, {8583, 19535}, {9945, 26015}, {10107, 24926}, {10165, 11112}, {11230, 17579}, {15726, 16370}, {17533, 28164}, {17538, 26129}, {17563, 24541}, {17566, 18480}, {17573, 19860}, {21495, 26672}, {32612, 33597}

X(35272) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(45))

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

X(35272) lies on these lines: {1, 3689}, {2, 952}, {3, 392}, {9, 5126}, {40, 17573}, {45, 8649}, {55, 15015}, {56, 3927}, {78, 7373}, {104, 5779}, {106, 3242}, {214, 1001}, {377, 18493}, {390, 9945}, {404, 12702}, {405, 26321}, {443, 5901}, {474, 1482}, {517, 16417}, {518, 997}, {546, 26129}, {551, 5853}, {936, 24928}, {944, 17527}, {962, 17563}, {971, 3576}, {996, 6789}, {1012, 10861}, {1125, 5722}, {1159, 3306}, {1319, 9708}, {1376, 2802}, {1385, 5720}, {1387, 2550}, {1388, 1698}, {1420, 5044}, {1537, 6955}, {2771, 10269}, {3295, 10179}, {3476, 3820}, {3526, 24987}, {3616, 8728}, {3622, 17582}, {3624, 5727}, {3646, 16860}, {3753, 10247}, {3877, 16371}, {3897, 16842}, {3898, 4421}, {4187, 18525}, {4511, 15934}, {4512, 17502}, {4679, 21578}, {4860, 4867}, {4881, 16370}, {4930, 5902}, {5084, 34773}, {5253, 5708}, {5330, 17572}, {5438, 9957}, {5440, 6767}, {5657, 17564}, {5690, 17567}, {5768, 5789}, {5883, 25524}, {5886, 17528}, {6904, 22791}, {7171, 13624}, {9623, 25405}, {9654, 25681}, {9655, 21616}, {9669, 17647}, {10176, 11194}, {10595, 17580}, {10680, 16410}, {11680, 32558}, {12645, 24982}, {16173, 31140}, {16863, 19860}, {18480, 25522}, {19276, 29327}, {21527, 26672}, {21542, 25930}, {22758, 25893}, {24926, 34595}

X(35273) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(55))

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

X(35273) lies on these lines: {31, 999}, {55, 101}, {154, 392}, {219, 674}, {859, 2328}, {968, 24929}, {1083, 1376}, {2187, 13615}, {2195, 2810}, {2293, 20818}, {3052, 3230}, {3573, 26241}, {5250, 21982}, {8647, 9310}, {11716, 17597}, {16688, 37519}, {21983, 25930}, {23073, 23855}, {24309, 25878}

X(35274) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(75))

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

X(35274) lies on these lines: {1, 17755}, {2, 392}, {8, 26689}, {40, 33828}, {75, 3230}, {213, 18156}, {239, 16483}, {304, 742}, {960, 27248}, {1083, 4676}, {1385, 17696}, {3057, 27299}, {3290, 24282}, {3730, 25918}, {3869, 27097}, {3876, 26759}, {3878, 30110}, {3884, 30107}, {3890, 26965}, {4713, 20925}, {4737, 10027}, {5248, 30132}, {5250, 16060}, {6656, 27129}, {12699, 17680}, {14001, 26658}, {16050, 18465}, {16061, 19861}, {16412, 25899}, {16821, 20154}, {17541, 26653}, {18743, 30114}, {21853, 25504}, {27274, 31359}

X(35275) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(76))

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

X(35275) lies on these lines: {2, 51}, {6, 4563}, {69, 30749}, {76, 3231}, {193, 30793}, {305, 732}, {384, 5651}, {1495, 3552}, {3098, 26257}, {3266, 9463}, {3291, 18906}, {3763, 30785}, {3933, 14467}, {3972, 5108}, {4048, 9225}, {4074, 21001}, {4576, 9465}, {5642, 33246}, {7807, 11064}, {8617, 26235}, {9306, 16951}, {10166, 11149}, {11324, 17811}, {11336, 26869}, {15107, 16055}

X(35276) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(81))

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

X(35276) lies on these lines: {2, 3}, {35, 29574}, {81, 187}, {99, 26243}, {100, 29615}, {239, 5303}, {524, 1030}, {574, 32911}, {597, 5124}, {620, 30760}, {940, 5210}, {980, 5206}, {1211, 32459}, {2223, 29580}, {2482, 31143}, {2895, 6390}, {2975, 29617}, {3793, 20086}, {4254, 5032}, {4421, 4433}, {5010, 29573}, {5204, 26626}, {5217, 17316}, {5337, 15513}, {13624, 26639}, {14996, 15655}, {16815, 32633}, {24271, 32456}

X(35277) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(83))

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

X(35277) lies on these lines: {2, 6030}, {83, 8627}, {732, 1799}, {1495, 7824}, {5092, 26257}, {5650, 33273}, {5651, 33004}, {8356, 13394}, {10130, 10330}, {12215, 15822}

X(35278) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(98))

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

X(35278) lies on these lines: {2, 2794}, {6, 13479}, {22, 22676}, {51, 22521}, {98, 1316}, {99, 110}, {107, 112}, {125, 9862}, {154, 8719}, {184, 7709}, {250, 523}, {262, 3148}, {376, 2777}, {419, 21445}, {458, 9756}, {476, 8599}, {691, 1302}, {827, 925}, {868, 10722}, {930, 7954}, {933, 1289}, {1285, 6791}, {1304, 30247}, {1495, 11676}, {1624, 4230}, {1995, 3972}, {2847, 3163}, {3018, 17983}, {3233, 7472}, {3288, 6037}, {4232, 10418}, {4235, 9125}, {4240, 9189}, {5640, 12150}, {5967, 10753}, {5987, 9999}, {6194, 8922}, {6620, 9752}, {7468, 15724}, {7493, 14907}, {7495, 7831}, {8356, 13394}, {9155, 21166}, {9747, 9755}, {9751, 14096}, {9873, 14003}, {10295, 16319}, {10788, 34417}, {11206, 15428}, {11634, 34519}, {14830, 34094}

X(35279) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(99))

Barycentrics 3*a^6 - 4*a^4*b^2 - a^2*b^4 - 4*a^4*c^2 + 9*a^2*b^2*c^2 - b^4*c^2 - a^2*c^4 - b^2*c^4 : :

X(35279) lies on these lines: {2, 98}, {69, 10418}, {99, 2502}, {111, 5468}, {193, 6791}, {352, 26276}, {524, 9225}, {671, 1641}, {729, 3231}, {1495, 13586}, {1992, 9172}, {1995, 22486}, {2709, 9084}, {4563, 5969}, {5477, 6719}, {5650, 33273}, {6531, 18020}, {8591, 12036}, {9129, 15342}, {11053, 11646}, {11064, 33228}, {13857, 14041}, {17005, 30516}

X(35279) = inverse-in-Thomson-Gibert-Moses-hyperbola of X(5182)
X(35279) = {X(2),X(110)}-harmonic conjugate of X(5182)

X(35280) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(105))

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

X(35280) lies on these lines: {48, 7676}, {59, 513}, {100, 101}, {105, 5091}, {109, 934}, {110, 1292}, {154, 9778}, {165, 15735}, {294, 20672}, {901, 14074}, {999, 17126}, {1002, 3423}, {1156, 2265}, {1308, 4588}, {1429, 8647}, {1768, 11714}, {1814, 20468}, {2187, 7411}, {2283, 8638}, {2346, 18162}, {2398, 3732}, {2742, 9058}, {2800, 3576}, {3315, 11716}, {3428, 4216}, {4427, 25272}, {5773, 14942}, {7677, 20780}, {9441, 11349}, {16384, 34123}, {33139, 33302}

X(35281) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(106))

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

X(35281) lies on these lines: {58, 3987}, {89, 4792}, {100, 109}, {101, 649}, {110, 1293}, {171, 3914}, {283, 3579}, {394, 6244}, {595, 4188}, {664, 4025}, {692, 23832}, {1777, 5552}, {1795, 24028}, {1797, 2810}, {1813, 23981}, {2057, 2956}, {2340, 34931}, {2342, 16586}, {2932, 34586}, {3870, 9316}, {4190, 5264}, {4242, 8750}, {4427, 25268}, {4588, 6014}, {5222, 17126}, {8694, 8697}, {8698, 28192}, {8699, 8701}, {9364, 26015}, {10940, 24159}, {21511, 25930}, {28162, 28218}, {28222, 28226}

X(35282) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(125))

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

X(35282) lies on these lines: {2, 2794}, {3, 113}, {32, 6388}, {110, 14981}, {115, 1316}, {125, 5191}, {126, 15566}, {132, 2409}, {187, 468}, {351, 690}, {441, 1503}, {523, 3163}, {647, 32120}, {1561, 7422}, {1576, 15526}, {1632, 2847}, {2080, 32223}, {2453, 3018}, {3292, 7813}, {5112, 6781}, {5181, 5467}, {5477, 5967}, {5651, 7820}, {6055, 34094}, {6090, 7801}, {6132, 11672}, {6660, 29317}, {6793, 9475}, {7493, 8722}, {9142, 15118}, {9717, 14357}, {10979, 31267}, {11064, 18860}, {13394, 21163}, {13611, 15113}, {17941, 32458}

X(35283) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(140))

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

X(35283) lies on these lines: {2, 154}, {3, 16654}, {5, 1092}, {22, 21167}, {30, 5650}, {51, 10128}, {69, 3066}, {110, 3589}, {125, 18358}, {140, 1495}, {141, 1995}, {343, 5020}, {373, 3564}, {394, 7392}, {428, 3819}, {468, 24206}, {524, 5640}, {547, 5642}, {599, 20192}, {631, 16658}, {1352, 11284}, {1656, 6146}, {2502, 3055}, {3090, 12022}, {3098, 10301}, {3231, 7745}, {3292, 18583}, {3522, 16656}, {3523, 16621}, {3533, 16659}, {3580, 16042}, {3619, 4232}, {3628, 12134}, {3631, 10545}, {3763, 7493}, {3818, 16187}, {5056, 12241}, {5066, 13857}, {5480, 15066}, {5656, 6803}, {5891, 10127}, {5943, 5965}, {6090, 14561}, {6329, 11422}, {6688, 11245}, {6816, 18405}, {6997, 17811}, {7398, 33586}, {7426, 20582}, {7474, 17245}, {7486, 12024}, {7495, 10546}, {7519, 21766}, {7667, 29323}, {7785, 14467}, {7998, 29181}, {10272, 32235}, {10601, 14826}, {10625, 23411}, {11444, 11745}, {12007, 15018}, {12370, 12812}, {13142, 27355}, {13568, 15056}, {15019, 32455}, {15067, 23410}, {15082, 29012}, {16419, 31383}, {17928, 23328}, {20772, 34128}, {21358, 26255}, {23326, 26206}

X(35284) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(145))

Barycentrics 13*a^3 - 7*a^2*b - 3*a*b^2 + b^3 - 7*a^2*c + b^2*c - 3*a*c^2 + b*c^2 + c^3 : :

X(35284) lies on these lines: {2, 165}, {145, 902}, {3621, 3977}, {3622, 4414}, {4427, 26245}, {9053, 21000}, {16192, 25879}, {17126, 29585}

X(35285) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(171))

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

X(35285) lies on these lines: {2, 28845}, {171, 8626}, {1055, 1621}, {2174, 9054}, {9025, 23868}, {16370, 17502}, {16792, 21009}, {19861, 21982}

X(35286) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(192))

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

X(35286) lies on these lines: {2, 392}, {192, 3230}, {1001, 20149}, {3884, 27299}, {3890, 26689}, {4393, 16483}, {4520, 25918}, {5250, 22267}, {9055, 16969}, {10027, 27538}, {16486, 32029}, {25248, 28370}

X(35287) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(193))

Barycentrics 13*a^4 - 10*a^2*b^2 + b^4 - 10*a^2*c^2 + 2*b^2*c^2 + c^4 : :

X(35287) lies on these lines: {2, 3}, {32, 5032}, {69, 5210}, {141, 5585}, {187, 193}, {315, 25486}, {524, 5023}, {538, 32477}, {543, 2996}, {575, 11151}, {597, 15815}, {598, 31404}, {754, 32837}, {1992, 3053}, {2482, 3926}, {2548, 7622}, {3620, 7810}, {3767, 34504}, {3785, 7801}, {3849, 32816}, {5171, 5182}, {5215, 7748}, {5395, 31401}, {5569, 7816}, {5866, 8553}, {6390, 15655}, {6392, 22329}, {6462, 12968}, {6463, 12963}, {6781, 32827}, {7615, 7749}, {7747, 32839}, {7752, 23334}, {7754, 9741}, {7769, 11149}, {7775, 32829}, {7780, 32824}, {7782, 26613}, {7789, 21356}, {7793, 9740}, {7823, 32835}, {7825, 22247}, {7840, 32831}, {7870, 14907}, {7893, 32841}, {7916, 32876}, {8584, 22331}, {8860, 32819}, {9605, 19661}, {9734, 14853}, {11152, 20081}, {13334, 22486}, {21843, 32456}, {22110, 32006}, {23055, 34505}, {32828, 34506}, {33684, 34380}

X(35287) = {X(37172),X(37173)}-harmonic conjugate of X(5)

X(35288) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(194))

Barycentrics 3*a^4*b^2 - a^2*b^4 + 3*a^4*c^2 - 9*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 : :

X(35288) lies on these lines: {2, 51}, {69, 30793}, {193, 11059}, {194, 3231}, {698, 21001}, {1495, 33014}, {3552, 5651}, {3620, 30749}, {4576, 8617}, {7906, 14467}, {7907, 11064}, {11052, 21843}, {13394, 33274}, {21766, 26257}

X(35289) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(213))

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

X(35289) lies on these lines: {3, 392}, {199, 15931}, {228, 518}, {1631, 34879}, {3185, 22060}, {3748, 23391}, {4191, 10434}, {4512, 19346}, {16475, 16778}, {17502, 19254}, {22080, 25941}

X(35290) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(226))

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

X(35290) lies on these lines: {1, 24580}, {2, 515}, {3, 26006}, {10, 24581}, {48, 307}, {77, 22063}, {226, 1055}, {306, 5440}, {379, 1125}, {572, 25019}, {857, 4297}, {946, 14953}, {1214, 7117}, {1375, 1385}, {1790, 17219}, {2360, 26647}, {3616, 24604}, {4209, 17397}, {4466, 22357}, {5222, 5265}, {5253, 11349}, {7987, 14021}, {10444, 24553}, {13624, 30810}, {16054, 24541}, {17073, 18650}, {18481, 30808}, {18589, 22054}, {19861, 24609}, {24590, 26626}, {24608, 25055}, {27000, 29586}, {31014, 31673}, {31186, 34773}

X(35291) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(238))

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

X(35291) lies on these lines: {2, 28845}, {3, 392}, {100, 1055}, {238, 8626}, {518, 17798}, {3576, 19326}, {7113, 9024}, {9441, 19308}

X(35292) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(239))

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

X(35292) lies on these lines: {2, 165}, {40, 17696}, {239, 902}, {439, 26658}, {742, 17735}, {3579, 33821}, {3977, 6542}, {4414, 16826}, {4427, 26247}, {4786, 8643}, {5205, 6651}, {16925, 27129}, {17680, 31730}, {19308, 25899}, {29615, 33161}, {31663, 33828}

X(35293) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(293))

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

X(35293) lies on these lines: {1, 88}, {2, 10186}, {63, 34931}, {165, 15735}, {241, 518}, {656, 35069}, {665, 1642}, {896, 5526}, {1026, 4712}, {1055, 1155}, {1212, 3119}, {1323, 6745}, {1376, 28125}, {1638, 6174}, {1742, 26669}, {2293, 10177}, {2310, 16578}, {3008, 24582}, {3035, 5723}, {3942, 4557}, {4413, 34522}, {5091, 17439}, {5308, 20533}, {9441, 19308}, {11349, 18788}, {21511, 25930}, {21914, 26932}

X(35294) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(294))

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

X(35294) lies on these lines: {2, 51}, {76, 11333}, {305, 3231}, {1501, 5108}, {1613, 32451}, {3051, 11059}, {3167, 5182}, {5651, 16951}

X(35295) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(316))

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

X(35295) lies on these lines: {2, 1495}, {2030, 7665}, {3231, 26613}, {3580, 8593}, {5182, 7426}, {5642, 13586}, {5651, 33274}, {8370, 13394}, {8598, 11064}, {13857, 33265}, {18800, 32223}

X(35296) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(323))

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

X(35296) lies on these lines: {2, 3}, {32, 1994}, {36, 26639}, {39, 588}, {50, 22151}, {69, 8553}, {110, 23700}, {187, 249}, {193, 1609}, {232, 32697}, {239, 4996}, {394, 5023}, {524, 11063}, {574, 15018}, {669, 5940}, {894, 7279}, {1030, 15988}, {1384, 11004}, {1692, 2987}, {1993, 3053}, {2393, 14060}, {2979, 5171}, {3003, 4558}, {3060, 9737}, {3580, 5866}, {3589, 15109}, {3964, 20080}, {4576, 19599}, {5012, 13335}, {5013, 5422}, {5210, 15066}, {5640, 9734}, {6091, 32729}, {6337, 6515}, {7767, 15108}, {8069, 29585}, {10601, 15815}, {11162, 34010}, {13357, 34482}, {14792, 17023}, {14793, 26626}, {15462, 22463}, {17100, 26594}, {18800, 32599}, {26156, 34828}, {34870, 34945}

X(35296) = {X(11145),X(11146)}-harmonic conjugate of X(23)

X(35297) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(325))

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

X(35297) lies on these lines: {2, 3}, {35, 26686}, {36, 26629}, {76, 13468}, {99, 230}, {110, 14605}, {115, 32456}, {141, 7771}, {183, 21843}, {187, 325}, {249, 524}, {315, 5023}, {385, 6390}, {538, 1569}, {543, 5215}, {574, 7792}, {591, 12963}, {597, 3094}, {598, 9771}, {625, 6781}, {626, 15513}, {691, 16320}, {1078, 7789}, {1384, 7774}, {1499, 15724}, {1503, 34473}, {1692, 14645}, {1991, 12968}, {2080, 10352}, {2387, 3111}, {2459, 32421}, {2460, 32419}, {3053, 7762}, {3564, 21445}, {3785, 7881}, {3788, 5206}, {3793, 7779}, {3815, 3972}, {3849, 9167}, {3933, 7793}, {4045, 8589}, {4366, 15325}, {5010, 26590}, {5024, 16989}, {5026, 15993}, {5171, 10349}, {5210, 7778}, {5254, 7782}, {5267, 26558}, {5305, 7783}, {5306, 7757}, {5913, 7664}, {5989, 9890}, {6337, 7754}, {6721, 13449}, {7280, 26561}, {7615, 11164}, {7618, 7739}, {7665, 16317}, {7735, 31859}, {7745, 7769}, {7746, 18546}, {7749, 7816}, {7751, 32820}, {7756, 7886}, {7761, 8588}, {7764, 35007}, {7767, 7836}, {7777, 18907}, {7780, 7863}, {7787, 31406}, {7802, 7940}, {7803, 15815}, {7806, 15048}, {7809, 22110}, {7810, 7880}, {7811, 7870}, {7829, 31652}, {7830, 7874}, {7832, 31168}, {7834, 15515}, {7856, 9607}, {7858, 12156}, {7878, 9606}, {7897, 14929}, {7904, 7945}, {7912, 14976}, {7925, 14712}, {8667, 32833}, {8859, 19570}, {9137, 32231}, {9300, 12150}, {9466, 34506}, {9494, 25423}, {11648, 34504}, {12040, 19661}, {12154, 16268}, {12155, 16267}, {14023, 32821}, {14614, 34511}, {14693, 33813}, {14971, 32479}, {16829, 31157}, {22247, 31173}

X(35297) = isogonal conjugate of X(14498)
X(35297) = circumcircle-inverse of X(37915)
X(35297) = {X(35305),X(35306)}-harmonic conjugate of X(3)

X(35298) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(352))

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

X(35298) lies on these lines: {2, 3}, {32, 11422}, {39, 15019}, {50, 6593}, {99, 33972}, {110, 187}, {230, 33900}, {323, 2080}, {574, 5640}, {895, 3003}, {2482, 15360}, {2930, 11063}, {3053, 14567}, {5013, 20977}, {5866, 23181}, {7998, 8722}, {8588, 10546}, {8589, 10545}, {9486, 11580}, {9734, 34417}, {10510, 34990}, {11171, 15018}, {11186, 32231}, {12177, 32599}, {14602, 35007}, {15107, 18860}, {19911, 34013}, {32269, 32459}

X(35299) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(371))

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

X(35299) lies on these lines: {2, 9758}, {3, 1495}, {110, 6200}, {372, 5640}, {394, 21097}, {511, 3155}, {1583, 5085}, {1599, 6800}, {1995, 6396}, {3066, 6398}, {5050, 10133}, {5409, 6090}, {6481, 10545}, {11188, 11514}, {11824, 33884}

X(35300) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(372))

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

X(35300) lies on these lines: {2, 9757}, {3, 1495}, {110, 6396}, {371, 5640}, {511, 3156}, {1584, 5085}, {1600, 6800}, {1995, 6200}, {3066, 6221}, {5050, 10132}, {5408, 6090}, {6480, 10545}, {11188, 11513}, {11825, 33884}

X(35301) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(384))

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

X(35301) lies on these lines: {2, 154}, {110, 26257}, {384, 1495}, {5642, 7924}, {5650, 33273}, {5651, 7824}, {5972, 30777}, {6655, 11064}, {11003, 16055}, {16951, 26881}

X(35302) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(394))

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

X(35302) lies on these lines: {2, 3}, {187, 394}, {524, 1609}, {574, 10601}, {599, 8553}, {800, 10607}, {1384, 1993}, {1992, 8573}, {1994, 21309}, {2482, 6503}, {3167, 9155}, {5024, 5422}, {5191, 8780}, {5210, 17811}, {5943, 9734}, {6390, 6515}, {8069, 29574}, {8939, 13663}, {8943, 13783}, {9167, 9699}, {9737, 21849}, {11063, 15533}, {13567, 32459}, {15066, 15655}, {18860, 33586}

X(35303) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(395))

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

X(35303) lies on these lines: {2, 3}, {13, 9885}, {15, 597}, {16, 524}, {62, 8584}, {141, 5463}, {187, 395}, {230, 6775}, {299, 6390}, {396, 574}, {530, 619}, {542, 13349}, {543, 6108}, {599, 11481}, {618, 15810}, {1153, 33477}, {1503, 21159}, {1992, 11486}, {2080, 8594}, {3054, 22576}, {3181, 3793}, {3589, 10645}, {3629, 34755}, {3642, 13084}, {5210, 16645}, {5237, 22165}, {5321, 5460}, {5459, 8589}, {5461, 31710}, {5476, 9735}, {5613, 11154}, {5859, 34511}, {5981, 22329}, {6329, 34754}, {6337, 33618}, {7618, 9763}, {7870, 9989}, {8182, 9761}, {8588, 23303}, {8595, 8724}, {9114, 22570}, {9762, 22513}, {10168, 13350}, {11153, 25154}, {11489, 15655}, {14830, 14905}, {15534, 22238}, {16242, 33474}, {16773, 34508}, {16940, 19661}, {16963, 22496}, {22489, 23005}, {22493, 33459}, {22495, 33458}, {31709, 32479}

X(35303) = reflection of X(35304) in X(27088)
X(35303) = {X(2),X(3)}-harmonic conjugate of X(35304)

X(35304) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(396))

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

X(35304) lies on these lines: {2, 3}, {14, 9886}, {15, 524}, {16, 597}, {61, 8584}, {141, 5464}, {187, 396}, {230, 6772}, {298, 6390}, {395, 574}, {531, 618}, {542, 13350}, {543, 6109}, {599, 11480}, {619, 15810}, {1153, 33476}, {1503, 21158}, {1992, 11485}, {2080, 8595}, {3054, 22575}, {3180, 3793}, {3589, 10646}, {3629, 34754}, {3643, 13083}, {5210, 16644}, {5238, 22165}, {5318, 5459}, {5460, 8589}, {5461, 31709}, {5476, 9736}, {5617, 11153}, {5858, 34511}, {5980, 22329}, {6329, 34755}, {6337, 33619}, {7618, 9761}, {7870, 9988}, {8182, 9763}, {8588, 23302}, {8594, 8724}, {9116, 22568}, {9760, 22512}, {10168, 13349}, {11154, 25164}, {11488, 15655}, {14830, 14904}, {15534, 22236}, {16241, 33475}, {16772, 34509}, {16941, 19661}, {16962, 22495}, {22490, 23004}, {22494, 33458}, {22496, 33459}, {31710, 32479}

X(35304) = reflection of X(35303) in X(27088)
X(35304) = {X(2),X(3)}-harmonic conjugate of X(35303)

X(35305) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(491))

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

X(35305) lies on these lines: {2, 3}, {99, 13758}, {187, 491}, {486, 13789}, {524, 12963}, {615, 32459}, {1991, 3053}, {1992, 6424}, {2482, 13757}, {3102, 11157}, {3926, 5860}, {7801, 32808}, {11147, 13934}, {13637, 31411}

X(35305) = {X(2),X(32985)}-harmonic conjugate of X(35306)
X(35305) = {X(3),X(35297)}-harmonic conjugate of X(35306)

X(35306) = CENTROID OF THREE POINTS IN Γ∩nK0(X(6),X(492))

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

X(35306) lies on these lines: {2, 3}, {99, 13638}, {187, 492}, {485, 13669}, {524, 12968}, {590, 32459}, {591, 3053}, {1992, 6423}, {2482, 13637}, {3103, 11158}, {3618, 9600}, {3926, 5861}, {7801, 32809}, {11147, 13882}

X(35306) = {X(2),X(32985)}-harmonic conjugate of X(35305)
X(35306) = {X(3),X(35297)}-harmonic conjugate of X(35305)

Areal centers: X(35307)-X(35368)

This preamble and centers X(35307)-X(35368) were contributed by César Eliud Lozada, December 12, 2019.

Let A'B'C', A"B"C" be two triangles inscribed in ABC. The areal center of the two triangles is a point S such that the triangles SA'A", SB'B", SC'C" have the same area. (Reference: The Triangles Web)

Construction of the areal center of two inscribed triangles

Let A'B'C', A"B"C" be the two inscribed triangles, L_a the line through A parallel to the line joining the midpoints of B'C" and B"C'.
Construct analogously L_b, L_c.
The lines L_a, L_b, L_c intersect at the areal center S.
(Reference: The Triangles Web)

The appearance of (T₁, T₂, n) in the following list means that the areal center of triangles T₁, T₂ is X(n):

(extouch, incentral, 1018), (extouch, intouch, 100), (extouch, Lemoine, 35368), (extouch, Macbeath, 35320), (extouch, medial, 100), (extouch, orthic, 101), (extouch, Steiner, 35347), (extouch, symmedial, 35334), (extouch, Yff contact, 35348), (2nd Hatzipolakis, intouch, 35349), (2nd Hatzipolakis, orthic, 35350), (incentral, intouch, 4551), (incentral, Lemoine, 35351), (incentral, medial, 3952), (incentral, orthic, 4559), (incentral, Steiner, 35352), (incentral, symmedial, 35309), (incentral, Yff contact, 35353), (intouch, Macbeath, 35321), (intouch, medial, 100), (intouch, orthic, 109), (intouch, Steiner, 35354), (intouch, symmedial, 35333), (intouch, Yff contact, 35355), (Lemoine, medial, 35356), (Lemoine, orthic, 35357), (Lemoine, Steiner, 35358), (Lemoine, symmedial, 35359), (Macbeath, medial, 35360), (Macbeath, orthic, 23181), (Macbeath, Steiner, 35361), (Macbeath, symmedial, 35362), (Macbeath, Yff contact, 35363), (medial, orthic, 110), (medial, Steiner, 5466), (medial, symmedial, 4576), (medial, Yff contact, 6548), (orthic, Steiner, 35364), (orthic, symmedial, 35325), (orthic, Yff contact, 35365), (Steiner, symmedial, 35366), (Steiner, Yff contact, 3120), (symmedial, Yff contact, 35367)

X(35307) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND CEVIAN-OF-X(3)

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

X(35307) lies on these lines: {12,1146}, {101,108}, {163,2222}, {2171,8818}, {2197,23980}

X(35307) = barycentric product X(i)*X(j) for these {i, j}: {5, 4551}, {12, 2617}, {59, 2618}, {442, 35320}
X(35307) = barycentric quotient X(i)/X(j) for these (i, j): (5, 18155), (51, 3737), (181, 2616), (1625, 2185)
X(35307) = trilinear product X(i)*X(j) for these {i, j}: {5, 4559}, {12, 1625}, {51, 4552}, {53, 23067}
X(35307) = trilinear quotient X(i)/X(j) for these (i, j): (5, 4560), (12, 15412), (51, 7252), (59, 18315)
X(35307) = X(2149)-Ceva conjugate of X(2171)
X(35307) = X(i)-isoconjugate-of X(j) for these {i,j}: {11, 18315}, {54, 4560}, {60, 15412}
X(35307) = X(i)-reciprocal conjugate of X(j) for these (i,j): (5, 18155), (51, 3737), (181, 2616)

X(35308) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND CEVIAN-OF-X(5)

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

X(35308) lies on the line {1783,3064}

X(35308) = barycentric product X(655)*X(21012)
X(35308) = trilinear product X(2222)*X(21012)

X(35309) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND SYMMEDIAL

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

X(35309) lies on these lines: {2,27800}, {11,594}, {38,8041}, {100,101}, {244,25748}, {765,5389}, {799,4562}, {1500,6377}, {1978,3807}, {2321,22032}, {3952,7239}, {3963,21404}, {4876,33115}, {7035,27805}, {15523,21037}, {20706,21026}, {29527,33798}

X(35309) = barycentric product X(i)*X(j) for these {i, j}: {10, 4553}, {37, 4568}, {38, 3952}, {39, 4033}
X(35309) = barycentric quotient X(i)/X(j) for these (i, j): (37, 10566), (38, 7192), (39, 1019), (42, 18108)
X(35309) = trilinear product X(i)*X(j) for these {i, j}: {37, 4553}, {38, 1018}, {39, 3952}, {42, 4568}
X(35309) = trilinear quotient X(i)/X(j) for these (i, j): (10, 10566), (37, 18108), (38, 1019), (39, 3733)
X(35309) = trilinear pole of the line {3954, 20969}
X(35309) = intersection, other than A,B,C, of conics {{A, B, C, X(38), X(3573)}} and {{A, B, C, X(100), X(4553)}}
X(35309) = Cevapoint of X(2084) and X(21035)
X(35309) = crosspoint of X(1018) and X(4033)
X(35309) = X(1110)-Ceva conjugate of X(3949)
X(35309) = X(2084)-cross conjugate of X(21035)
X(35309) = X(i)-isoconjugate-of X(j) for these {i,j}: {58, 10566}, {81, 18108}, {82, 1019}
X(35309) = X(i)-reciprocal conjugate of X(j) for these (i,j): (37, 10566), (38, 7192), (39, 1019)
X(35309) = {X(3952), X(7239)}-harmonic conjugate of X(661)

X(35310) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND CEVIAN-OF-X(9)

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

X(35310) lies on these lines: {11,6184}, {37,3120}, {100,919}, {149,5701}, {210,21856}, {528,23988}, {594,6741}, {693,26795}, {1018,4551}, {1500,16592}, {2276,17723}, {3693,4071}, {3925,21795}, {4554,32041}, {4885,27134}, {5249,14746}, {6154,14936}, {16588,34612}, {20691,21870}, {25925,26692}, {28743,31250}, {35326,35338}

X(35310) = barycentric product X(i)*X(j) for these {i, j}: {10, 35338}, {100, 3925}, {142, 1018}, {190, 21808}
X(35310) = barycentric quotient X(i)/X(j) for these (i, j): (142, 7199), (354, 7192), (1018, 32008), (1020, 10509)
X(35310) = trilinear product X(i)*X(j) for these {i, j}: {10, 35326}, {37, 35338}, {65, 35341}, {100, 21808}
X(35310) = trilinear quotient X(i)/X(j) for these (i, j): (142, 7192), (354, 1019), (1018, 2346), (1212, 3737)
X(35310) = trilinear pole of the line {21039, 21808}
X(35310) = crossdifference of every pair of points on line {X(3675), X(18184)}
X(35310) = crosspoint of X(1018) and X(4552)
X(35310) = crosssum of X(i) and X(j) for these (i,j): (513, 21007), (1019, 7252)
X(35310) = X(i)-isoconjugate-of X(j) for these {i,j}: {1019, 2346}, {1170, 3737}, {1174, 7192}
X(35310) = X(i)-reciprocal conjugate of X(j) for these (i,j): (142, 7199), (354, 7192), (1018, 32008)
X(35310) = {X(35338), X(35341)}-harmonic conjugate of X(35326)

X(35311) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(5)

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

X(35311) lies on these lines: {4,195}, {107,110}, {125,14920}, {250,14480}, {264,11422}, {324,34986}, {394,15576}, {685,31065}, {877,10330}, {930,933}, {1493,14978}, {1993,19174}, {1994,30506}, {2407,23181}, {2888,3462}, {5468,6331}, {5609,34334}, {5965,14918}, {7480,14611}, {8884,15801}, {11562,15454}, {13398,30247}, {15958,18831}, {35318,35324}

X(35311) = isotomic conjugate of anticomplement of X(35441)
X(35311) = anticomplement of X(35442)
X(35311) = polar conjugate of the isogonal conjugate of X(35324)
X(35311) = barycentric product X(i)*X(j) for these {i, j}: {95, 35318}, {112, 1232}, {140, 648}, {162, 20879}
X(35311) = barycentric quotient X(i)/X(j) for these (i, j): (110, 31626), (112, 1173), (140, 525), (233, 6368)
X(35311) = trilinear product X(i)*X(j) for these {i, j}: {92, 35324}, {112, 20879}, {140, 162}, {648, 17438}
X(35311) = trilinear quotient X(i)/X(j) for these (i, j): (140, 656), (162, 1173), (662, 31626), (1232, 14208)
X(35311) = trilinear pole of the line {140, 233}
X(35311) = lies on the circumconics with center X(i) for i in {140, 233, 1493, 3628, 6709, 12242, 33549}
X(35311) = intersection, other than A,B,C, of conics {{A, B, C, X(110), X(35324)}} and {{A, B, C, X(140), X(3471)}}
X(35311) = Cevapoint of X(i) and X(j) for these (i,j): (523, 12242), (525, 6709)
X(35311) = crosspoint of X(648) and X(18831)
X(35311) = crosssum of X(647) and X(15451)
X(35311) = X(648)-Ceva conjugate of X(35318)
X(35311) = X(i)-isoconjugate-of X(j) for these {i,j}: {656, 1173}, {661, 31626}, {1173, 656}
X(35311) = X(i)-reciprocal conjugate of X(j) for these (i,j): (110, 31626), (112, 1173), (140, 525)

X(35312) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(9)

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

X(35312) lies on these lines: {2,3119}, {7,149}, {100,658}, {110,927}, {223,26265}, {226,31058}, {279,20247}, {329,31527}, {347,21273}, {651,2428}, {883,3952}, {1025,4552}, {1088,3873}, {1111,18240}, {2898,5905}, {3160,5744}, {3218,14189}, {3306,9312}, {3321,6154}, {3434,7056}, {3616,25586}, {3676,4551}, {3681,31627}, {3957,9446}, {4569,30704}, {4576,4625}, {5932,21286}, {6063,17140}, {6167,26653}, {6183,13397}, {6327,7055}, {7182,17135}, {7205,25295}, {8059,9086}, {9533,17784}, {20245,33673}, {21104,35326}, {23062,30628}

X(35312) = isotomic conjugate of the anticomplement of X(6608)
X(35312) = anticomplement of X(3119)
X(35312) = barycentric product X(i)*X(j) for these {i, j}: {85, 35338}, {109, 1233}, {142, 664}, {190, 10481}
X(35312) = barycentric quotient X(i)/X(j) for these (i, j): (100, 6605), (101, 10482), (109, 1174), (142, 522)
X(35312) = trilinear product X(i)*X(j) for these {i, j}: {7, 35338}, {85, 35326}, {100, 10481}, {109, 20880}
X(35312) = trilinear quotient X(i)/X(j) for these (i, j): (100, 10482), (142, 650), (190, 6605), (354, 663)
X(35312) = trilinear pole of the line {142, 1212}
X(35312) = lies on the circumconics with center X(i) for i in {142, 1212, 5572, 6666, 6706}
X(35312) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(3254)}} and {{A, B, C, X(110), X(354)}}
X(35312) = Cevapoint of X(i) and X(j) for these (i,j): (142, 6362), (354, 21104), (514, 5572)
X(35312) = crosspoint of X(664) and X(4569)
X(35312) = crosssum of X(663) and X(8641)
X(35312) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (59, 30695), (658, 33650), (1262, 144)
X(35312) = X(664)-Ceva conjugate of X(35338)
X(35312) = X(i)-isoconjugate-of X(j) for these {i,j}: {513, 10482}, {649, 6605}, {650, 1174}
X(35312) = X(i)-reciprocal conjugate of X(j) for these (i,j): (100, 6605), (101, 10482), (109, 1174)
X(35312) = X(664)-Waw conjugate of X(4552)
X(35312) = X(218)-Zayin conjugate of X(657)
X(35312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (658, 664, 100), (664, 934, 17136), (664, 4566, 21272), (883, 4554, 3952)

X(35313) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(11)

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

X(35313) lies on these lines: {144,6163}, {145,6630}, {666,885}, {12531,14942}

X(35313) = barycentric product X(666)*X(3035)
X(35313) = barycentric quotient X(i)/X(j) for these (i, j): (885, 31611), (919, 18771)
X(35313) = trilinear product X(i)*X(j) for these {i, j}: {666, 17439}, {673, 14589}, {919, 20881}
X(35313) = trilinear pole of the line {3035, 14589}
X(35313) = lies on the circumconics with center X(i) for i in {3035, 6667}
X(35313) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(35340)}} and {{A, B, C, X(110), X(35328)}}
X(35313) = X(i)-reciprocal conjugate of X(j) for these (i,j): (885, 31611), (919, 18771)
X(35313) = {X(666), X(5377)}-harmonic conjugate of X(885)

X(35314) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(13)

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

X(35314) lies on these lines: {2,14}, {99,110}, {476,10409}, {616,9143}, {618,16256}, {2407,17402}, {3180,21466}, {5980,14170}, {6138,35336}, {14570,23896}, {30468,32552}, {35329,35343}

X(35314) = reflection of X(i) in X(j) for these (i,j): (3180, 30454), (16256, 618)
X(35314) = isotomic conjugate of the isogonal conjugate of X(35329)
X(35314) = isotomic conjugate of anticomplement of X(35443)
X(35314) = anticomplement of X(30465)
X(35314) = barycentric product X(i)*X(j) for these {i, j}: {76, 35329}, {99, 396}, {463, 4563}, {476, 14922}
X(35314) = barycentric quotient X(i)/X(j) for these (i, j): (110, 2981), (249, 10409), (396, 523), (463, 2501)
X(35314) = trilinear product X(i)*X(j) for these {i, j}: {75, 35329}, {396, 662}, {463, 4592}
X(35314) = trilinear quotient X(i)/X(j) for these (i, j): (396, 661), (662, 2981)
X(35314) = trilinear pole of the line {396, 532}
X(35314) = lies on the circumconics with center X(i) for i in {396, 618, 6669, 33526}
X(35314) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(32036)}} and {{A, B, C, X(14), X(99)}}
X(35314) = Cevapoint of X(523) and X(6669)
X(35314) = crossdifference of every pair of points on line {X(3124), X(6138)}
X(35314) = crosspoint of X(99) and X(23895)
X(35314) = crosssum of X(512) and X(6137)
X(35314) = X(1101)-anticomplementary conjugate of X(616)
X(35314) = X(476)-Ceva conjugate of X(35315)
X(35314) = X(99)-Daleth conjugate of X(35315)
X(35314) = X(661)-isoconjugate-of X(2981)
X(35314) = X(i)-reciprocal conjugate of X(j) for these (i,j): (110, 2981), (249, 10409), (396, 523)
X(35314) = X(1634)-Vertex conjugate of X(35315)
X(35314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 110, 35315), (110, 14185, 4226), (4226, 5468, 35315), (17402, 23895, 2407), (30508, 30509, 35315)

X(35315) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(14)

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

X(35315) lies on these lines: {2,13}, {99,110}, {476,10410}, {617,9143}, {619,16255}, {2407,17403}, {3181,21467}, {5981,14169}, {6137,35337}, {14570,23895}, {30465,32553}, {35330,35344}

X(35315) = reflection of X(i) in X(j) for these (i,j): (3181, 30455), (16255, 619)
X(35315) = isotomic conjugate of the isogonal conjugate of X(35330)
X(35315) = isotomic conjugate of anticomplement of X(35444)
X(35315) = anticomplement of X(30468)
X(35315) = barycentric product X(i)*X(j) for these {i, j}: {76, 35330}, {99, 395}, {462, 4563}, {476, 14921}
X(35315) = barycentric quotient X(i)/X(j) for these (i, j): (110, 6151), (249, 10410), (395, 523), (462, 2501)
X(35315) = trilinear product X(i)*X(j) for these {i, j}: {75, 35330}, {395, 662}, {462, 4592}
X(35315) = trilinear quotient X(i)/X(j) for these (i, j): (395, 661), (662, 6151)
X(35315) = trilinear pole of the line {395, 533}
X(35315) = lies on the circumconics with center X(i) for i in {395, 619, 6670, 33527}
X(35315) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(32037)}} and {{A, B, C, X(13), X(99)}}
X(35315) = Cevapoint of X(523) and X(6670)
X(35315) = crossdifference of every pair of points on line {X(3124), X(6137)}
X(35315) = crosspoint of X(99) and X(23896)
X(35315) = crosssum of X(512) and X(6138)
X(35315) = X(1101)-anticomplementary conjugate of X(617)
X(35315) = X(476)-Ceva conjugate of X(35314)
X(35315) = X(99)-Daleth conjugate of X(35314)
X(35315) = X(661)-isoconjugate-of X(6151)
X(35315) = X(i)-reciprocal conjugate of X(j) for these (i,j): (110, 6151), (249, 10410), (395, 523)
X(35315) = X(1634)-Vertex conjugate of X(35314)
X(35315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 110, 35314), (110, 14187, 4226), (4226, 5468, 35314), (17403, 23896, 2407), (30508, 30509, 35314)

X(35316) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(15)

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

X(35316) lies on these lines: {476,10409}, {622,11582}, {2407,17403}, {3180,21467}, {35139,35317}

X(35316) = barycentric product X(623)*X(23896)
X(35316) = barycentric quotient X(623)/X(23871)
X(35316) = lies on the circumconics with center X(i) for i in {623, 6671}
X(35316) = Cevapoint of X(526) and X(6671)
X(35316) = X(623)-reciprocal conjugate of X(23871)

X(35317) = AREAL CENTER OF THESE TRIANGLES: MEDIAL AND CEVIAN-OF-X(16)

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

X(35317) lies on these lines: {476,10410}, {621,11581}, {2407,17402}, {3181,21466}, {35139,35316}

X(35317) = barycentric product X(624)*X(23895)
X(35317) = barycentric quotient X(624)/X(23870)
X(35317) = lies on the circumconics with center X(i) for i in {624, 6672}
X(35317) = Cevapoint of X(526) and X(6672)
X(35317) = X(624)-reciprocal conjugate of X(23870)

X(35318) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND CEVIAN-OF-X(5)

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

X(35318) lies on these lines: {52,27376}, {53,1263}, {112,930}, {216,549}, {648,16813}, {1625,14391}, {23582,33513}, {35311,35324}

X(35318) = barycentric product X(i)*X(j) for these {i, j}: {5, 35311}, {110, 14978}, {233, 648}, {324, 35324}
X(35318) = barycentric quotient X(i)/X(j) for these (i, j): (112, 288), (233, 525), (648, 31617), (1576, 20574)
X(35318) = trilinear product X(i)*X(j) for these {i, j}: {162, 233}, {163, 14978}, {823, 32078}, {1953, 35311}
X(35318) = trilinear quotient X(i)/X(j) for these (i, j): (162, 288), (163, 20574), (233, 656), (811, 31617)
X(35318) = trilinear pole of the line {233, 3078}
X(35318) = lies on the circumconic with center X(140))
X(35318) = crosssum of X(647) and X(23286)
X(35318) = X(648)-Ceva conjugate of X(35311)
X(35318) = X(i)-isoconjugate-of X(j) for these {i,j}: {288, 656}, {656, 288}, {810, 31617}
X(35318) = X(i)-reciprocal conjugate of X(j) for these (i,j): (112, 288), (233, 525), (648, 31617)

X(35319) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND SYMMEDIAL

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

X(35319) lies on these lines: {110,647}, {125,11672}, {542,23584}, {850,11794}, {1625,2081}, {1634,35325}, {3448,5661}, {5133,14773}, {6331,31174}, {35322,35323}, {35336,35337}

X(35319) = barycentric product X(i)*X(j) for these {i, j}: {5, 1634}, {38, 2617}, {39, 14570}, {51, 4576}
X(35319) = barycentric quotient X(i)/X(j) for these (i, j): (39, 15412), (216, 4580), (1625, 83), (1634, 95)
X(35319) = trilinear product X(i)*X(j) for these {i, j}: {38, 1625}, {39, 2617}, {799, 27374}, {1634, 1953}
X(35319) = trilinear quotient X(i)/X(j) for these (i, j): (5, 18070), (38, 15412), (39, 2616), (1625, 82)
X(35319) = crossdifference of every pair of points on line {X(868), X(8901)}
X(35319) = crosspoint of X(i) and X(j) for these (i,j): (110, 11794), (1625, 14570)
X(35319) = crosssum of X(523) and X(3050)
X(35319) = X(i)-isoconjugate-of X(j) for these {i,j}: {54, 18070}, {82, 15412}, {83, 2616}
X(35319) = X(i)-reciprocal conjugate of X(j) for these (i,j): (39, 15412), (216, 4580), (1625, 83)

X(35320) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND INTOUCH

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

X(35320) lies on these lines: {162,4551}, {1618,2222}

X(35320) = trilinear product X(5)*X(15439)
X(35320) = trilinear pole of the line {1953, 2599}
X(35320) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(35321)}} and {{A, B, C, X(101), X(23181)}}

X(35321) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND EXTOUCH

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

X(35321) lies on these lines: {163,1021}, {1983,23703}, {2250,16548}, {3882,13136}, {20367,34234}

X(35321) = barycentric product X(i)*X(j) for these {i, j}: {1953, 13136}, {2250, 14570}
X(35321) = barycentric quotient X(i)/X(j) for these (i, j): (51, 1769), (1953, 10015), (2179, 3310), (2250, 15412)
X(35321) = trilinear product X(i)*X(j) for these {i, j}: {5, 32641}, {216, 1309}, {343, 14776}, {2250, 2617}
X(35321) = trilinear quotient X(i)/X(j) for these (i, j): (5, 10015), (51, 3310), (216, 8677), (217, 23220)
X(35321) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(35320)}} and {{A, B, C, X(101), X(1625)}}
X(35321) = Cevapoint of X(1953) and X(2600)
X(35321) = X(i)-isoconjugate-of X(j) for these {i,j}: {54, 10015}, {95, 3310}, {275, 8677}
X(35321) = X(i)-reciprocal conjugate of X(j) for these (i,j): (51, 1769), (1953, 10015), (2179, 3310)

X(35322) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND CEVIAN-OF-X(15)

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

X(35322) lies on these lines: {35319,35323}, {35336,35346}

X(35322) = barycentric product X(623)*X(1625)

X(35323) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(3) AND CEVIAN-OF-X(16)

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

X(35323) lies on these lines: {35319,35322}, {35337,35346}

X(35323) = barycentric product X(624)*X(1625)

X(35324) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(5)

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

X(35324) lies on these lines: {6,3200}, {99,35178}, {110,112}, {115,30714}, {154,11641}, {1147,10311}, {1511,3269}, {1970,18350}, {1971,22115}, {2439,14586}, {2715,7953}, {2888,13527}, {3044,9149}, {7954,26714}, {14590,18315}, {22146,32609}, {22416,32171}, {35311,35318}

X(35324) = isogonal conjugate of the polar conjugate of X(35311)
X(35324) = barycentric product X(i)*X(j) for these {i, j}: {3, 35311}, {97, 35318}, {99, 13366}, {101, 17168}
X(35324) = barycentric quotient X(i)/X(j) for these (i, j): (140, 850), (233, 18314), (250, 33513), (1576, 1173)
X(35324) = trilinear product X(i)*X(j) for these {i, j}: {48, 35311}, {110, 17438}, {140, 163}, {162, 22052}
X(35324) = trilinear quotient X(i)/X(j) for these (i, j): (140, 1577), (163, 1173), (233, 2618), (1232, 20948)
X(35324) = trilinear pole of the line {13366, 22052}
X(35324) = lies on the circumconic with center X(1493))
X(35324) = intersection, other than A,B,C, of conics {{A, B, C, X(110), X(35311)}} and {{A, B, C, X(112), X(14579)}}
X(35324) = crossdifference of every pair of points on line {X(125), X(137)}
X(35324) = crosspoint of X(i) and X(j) for these (i,j): (110, 18315), (648, 930)
X(35324) = crosssum of X(i) and X(j) for these (i,j): (523, 12077), (647, 1510)
X(35324) = X(i)-isoconjugate-of X(j) for these {i,j}: {288, 2618}, {1173, 1577}, {1577, 1173}
X(35324) = X(i)-reciprocal conjugate of X(j) for these (i,j): (140, 850), (233, 18314), (250, 33513)
X(35324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 32661, 1625), (1625, 32661, 2420)

X(35325) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND SYMMEDIAL

Barycentrics a^2*(b^2+c^2)*(a^2-b^2+c^2)*(a^2-c^2)*(a^2+b^2-c^2)*(a^2-b^2) : :
Trilinears sin(A + ω)/(tan B - tan C) : :
X(35325) = 2 X[38356] - 3 X[46128]

X(35325) lies on the circumconic with center X(141), and on these lines: {2, 25053}, {3, 23584}, {4, 9463}, {6, 67}, {22, 38663}, {25, 694}, {39, 46147}, {107, 26714}, {110, 112}, {141, 46165}, {162, 660}, {323, 41363}, {352, 8744}, {378, 38653}, {394, 3162}, {427, 3051}, {468, 2211}, {511, 14580}, {599, 36879}, {647, 1624}, {648, 670}, {858, 14965}, {933, 2623}, {1112, 3124}, {1576, 2445}, {1634, 35319}, {1691, 34397}, {1783, 8050}, {1843, 46154}, {1968, 44080}, {2076, 21284}, {2501, 35360}, {2781, 38356}, {2972, 9475}, {3172, 6090}, {3229, 44896}, {3289, 16318}, {3291, 44084}, {3917, 40938}, {4235, 10330}, {4576, 41676}, {6388, 12828}, {6792, 18947}, {7879, 8743}, {7998, 39575}, {8778, 9412}, {8879, 37669}, {9210, 37937}, {9306, 17409}, {10117, 35901}, {12294, 40130}, {13203, 35902}, {14389, 41334}, {14768, 40684}, {14966, 23181}, {15639, 32320}, {17171, 46158}, {17442, 46160}, {18020, 41209}, {20976, 46130}, {21001, 37453}, {21779, 44097}, {27369, 46156}, {32676, 35326}, {36824, 38303}, {37119, 43843}, {40805, 45141}

X(35325) = reflection of X(46165) in X(141)
X(35325) = isogonal conjugate of X(4580)
X(35325) = complement of X(25053)
X(35325) = polar conjugate of the isotomic conjugate of X(1634)
X(35325) = complement of X(25053)
X(35325) = anticomplement of the complementary conjugate of X(23285)
X(35325) = X(i)-Ceva conjugate of X(j) for these (i,j): {648, 41676}, {18020, 25}, {41676, 1634}, {44183, 3}
X(35325) = X(i)-cross conjugate of X(j) for these (i,j): {826, 6}, {3005, 427}
X(35325) = cevapoint of X(i) and X(j) for these (i,j): {3005, 3051}, {3569, 40601}
X(35325) = crosspoint of X(i) and X(j) for these (i,j): {110, 44766}, {112, 648}, {41676, 46151}
X(35325) = crosssum of X(i) and X(j) for these (i,j): {523, 2485}, {525, 647}
X(35325) = trilinear pole of line {39, 1843}
X(35325) = crossdifference of every pair of points on line {125, 127}
X(35325) = trilinear pole of the line {39, 1843}
X(35325) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(4630)}} and {{A, B, C, X(39), X(2420)}}
X(35325) = crossdifference of every pair of points on line {X(125), X(127)}
X(35325) = crosspoint of X(112) and X(648)
X(35325) = crosssum of X(i) and X(j) for these (i,j): (523, 2485), (525, 647)
X(35325) = X(826)-cross conjugate of X(6)
X(35325) = X(112)-Daleth conjugate of X(4230)
X(35325) = X(i)-isoconjugate-of X(j) for these {i,j}: {3, 18070}, {72, 10566}, {82, 525}, {827, 20902}
X(35325) = X(i)-line conjugate of X(j) for these (i,j): (6, 125), (67, 125), (110, 9517)
X(35325) = X(i)-reciprocal conjugate of X(j) for these (i,j): (6, 4580), (19, 18070), (38, 14208)
X(35325) = pole wrt polar circle of line X(338)X(3124)
X(35325) = perspector of unary cofactor triangles of 1st and 2nd orthosymmedial triangles
X(35325) = X(i)-isoconjugate of X(j) for these (i,j): {1, 4580}, {3, 18070}, {72, 10566}, {82, 525}, {83, 656}, {125, 4599}, {251, 14208}, {304, 18105}, {306, 18108}, {308, 810}, {339, 34072}, {521, 18097}, {523, 34055}, {647, 3112}, {661, 1799}, {822, 46104}, {827, 20902}, {879, 3405}, {905, 18082}, {1176, 1577}, {2632, 42396}, {3049, 18833}, {3695, 39179}, {3708, 4577}, {4025, 18098}, {4592, 34294}, {4593, 20975}, {9517, 37221}, {10547, 20948}, {24006, 28724}, {24018, 32085}, {24284, 43763}, {39182, 44706}
X(35325) = barycentric product X(i)*X(j) for these {i,j}: {3, 46151}, {4, 1634}, {6, 41676}, {21, 46152}, {25, 4576}, {27, 46148}, {28, 4553}, {29, 46153}, {38, 162}, {39, 648}, {99, 1843}, {101, 17171}, {107, 3917}, {110, 427}, {112, 141}, {163, 20883}, {186, 46155}, {250, 826}, {275, 35319}, {419, 46161}, {468, 36827}, {662, 17442}, {670, 27369}, {692, 16747}, {811, 1964}, {823, 4020}, {907, 3867}, {935, 9019}, {1113, 46167}, {1114, 46166}, {1235, 1576}, {1289, 3313}, {1296, 41585}, {1401, 36797}, {1474, 4568}, {1783, 16696}, {1897, 17187}, {1930, 32676}, {2409, 46164}, {2525, 23964}, {2530, 5379}, {3005, 18020}, {3051, 6331}, {3565, 41584}, {3933, 32713}, {4230, 20021}, {4235, 46154}, {4238, 46149}, {4240, 46147}, {4242, 46160}, {4249, 46158}, {4556, 21016}, {4558, 27376}, {4570, 21108}, {6528, 20775}, {7473, 46157}, {7482, 36824}, {7953, 46026}, {8041, 42396}, {8750, 16887}, {10098, 41583}, {15149, 46163}, {15388, 23881}, {16030, 35360}, {16789, 39382}, {18315, 27371}, {19174, 23181}, {19189, 35362}, {26706, 41582}, {29959, 30247}, {37168, 46162}, {40938, 44766}
X(35325) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 4580}, {19, 18070}, {38, 14208}, {39, 525}, {107, 46104}, {110, 1799}, {112, 83}, {141, 3267}, {162, 3112}, {163, 34055}, {250, 4577}, {427, 850}, {648, 308}, {688, 20975}, {811, 18833}, {826, 339}, {933, 39287}, {1235, 44173}, {1401, 17094}, {1474, 10566}, {1576, 1176}, {1634, 69}, {1843, 523}, {1923, 810}, {1964, 656}, {1974, 18105}, {2084, 3708}, {2203, 18108}, {2445, 21458}, {2489, 34294}, {2525, 36793}, {3005, 125}, {3051, 647}, {3917, 3265}, {4020, 24018}, {4230, 20022}, {4553, 20336}, {4568, 40071}, {4576, 305}, {6331, 40016}, {7813, 45807}, {8041, 2525}, {8061, 20902}, {8623, 24284}, {8750, 18082}, {8882, 39182}, {14574, 10547}, {16696, 15413}, {16747, 40495}, {17171, 3261}, {17187, 4025}, {17442, 1577}, {18020, 689}, {18831, 41488}, {20775, 520}, {20883, 20948}, {21035, 4064}, {21108, 21207}, {21123, 4466}, {23208, 8673}, {23964, 42396}, {27369, 512}, {27371, 18314}, {27374, 15451}, {27376, 14618}, {32661, 28724}, {32674, 18097}, {32676, 82}, {32713, 32085}, {35319, 343}, {36827, 30786}, {40938, 33294}, {40972, 8611}, {41272, 10097}, {41331, 3049}, {41676, 76}, {44090, 18010}, {44102, 22105}, {46147, 34767}, {46148, 306}, {46151, 264}, {46152, 1441}, {46153, 307}, {46154, 14977}, {46155, 328}, {46161, 40708}, {46164, 2419}, {46166, 22340}, {46167, 22339}
X(35325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1112, 44467, 3124}, {2211, 3231, 468}, {14966, 45215, 23181}

X(35326) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(9)

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

X(35326) lies on these lines: {6,244}, {55,20995}, {100,2284}, {101,109}, {110,919}, {222,5452}, {354,20229}, {651,658}, {666,4573}, {692,2428}, {910,20752}, {1357,20662}, {1615,7074}, {1759,22126}, {1979,2176}, {2248,21779}, {2426,32656}, {3732,20507}, {4574,35342}, {7004,9502}, {8012,22053}, {8652,28899}, {17435,17660}, {18191,18785}, {21104,35312}, {23832,25577}, {26007,34253}, {26716,28148}, {32676,35325}, {35310,35338}

X(35326) = barycentric product X(i)*X(j) for these {i, j}: {1, 35338}, {55, 35312}, {57, 35341}, {59, 6362}
X(35326) = barycentric quotient X(i)/X(j) for these (i, j): (59, 6606), (101, 32008), (109, 21453), (142, 3261)
X(35326) = trilinear product X(i)*X(j) for these {i, j}: {6, 35338}, {41, 35312}, {56, 35341}, {58, 35310}
X(35326) = trilinear quotient X(i)/X(j) for these (i, j): (100, 32008), (101, 2346), (109, 1170), (142, 693)
X(35326) = trilinear pole of the line {1475, 2293}
X(35326) = intersection, other than A,B,C, of conics {{A, B, C, X(101), X(658)}} and {{A, B, C, X(109), X(4626)}}
X(35326) = pole of the trilinear polar of X(1110) wrt circumcircle
X(35326) = Cevapoint of X(650) and X(14746)
X(35326) = crossdifference of every pair of points on line {X(11), X(116)}
X(35326) = crosspoint of X(101) and X(651)
X(35326) = crosssum of X(i) and X(j) for these (i,j): (513, 6586), (514, 650)
X(35326) = X(1110)-Ceva conjugate of X(6)
X(35326) = X(101)-Daleth conjugate of X(2283)
X(35326) = X(i)-isoconjugate-of X(j) for these {i,j}: {513, 32008}, {514, 2346}, {522, 1170}
X(35326) = X(i)-reciprocal conjugate of X(j) for these (i,j): (59, 6606), (101, 32008), (109, 21453)
X(35326) = X(101)-Waw conjugate of X(4557)
X(35326) = X(7)-Zayin conjugate of X(650)
X(35326) = {X(35338), X(35341)}-harmonic conjugate of X(35310)

X(35327) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(10)

Barycentrics a^2*(2*a+b+c)*(a-c)*(a-b) : :
X(35327) = 2*X(22356)+X(23854)

X(35327) lies on these lines: {3,2772}, {6,2054}, {48,8053}, {55,19302}, {100,8652}, {101,692}, {109,28148}, {110,351}, {219,1631}, {284,4068}, {523,7479}, {560,16685}, {651,14315}, {662,3573}, {674,22356}, {901,28180}, {922,3285}, {1001,23095}, {1486,20818}, {2175,20990}, {2187,15621}, {2194,20326}, {2245,21009}, {2426,32656}, {2784,21045}, {2911,4497}, {3009,9459}, {3271,17455}, {3683,23201}, {3733,23363}, {4427,35343}, {4570,4629}, {4579,23343}, {4588,28152}, {5548,32719}, {7193,20470}, {8693,28895}, {8701,28176}, {11712,17463}, {16681,22126}, {16684,18042}, {17796,17798}, {17976,20872}, {21784,23861}, {23073,23855}, {28156,28162}, {28196,28200}, {29022,29048}, {29055,29117}, {29067,29161}, {29143,29289}, {29165,29261}

X(35327) = isogonal conjugate of X(4608)
X(35327) = anticomplement of the complementary conjugate of X(4988)
X(35327) = complement of the anticomplementary conjugate of X(14779)
X(35327) = barycentric product X(i)*X(j) for these {i, j}: {1, 35342}, {6, 4427}, {56, 30729}, {58, 4115}
X(35327) = barycentric quotient X(i)/X(j) for these (i, j): (100, 32018), (101, 1268), (110, 32014), (430, 14618)
X(35327) = trilinear product X(i)*X(j) for these {i, j}: {6, 35342}, {31, 4427}, {100, 2308}, {101, 1100}
X(35327) = trilinear quotient X(i)/X(j) for these (i, j): (100, 1268), (101, 1255), (163, 1171), (190, 32018)
X(35327) = trilinear pole of the line {2308, 20970}
X(35327) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(4629)}} and {{A, B, C, X(101), X(4556)}}
X(35327) = pole of the trilinear polar of X(4570) wrt circumcircle
X(35327) = crossdifference of every pair of points on line {X(115), X(116)}
X(35327) = crosspoint of X(101) and X(110)
X(35327) = crosssum of X(i) and X(j) for these (i,j): (10, 24076), (512, 6586), (513, 14838)
X(35327) = X(i)-isoconjugate-of X(j) for these {i,j}: {244, 6540}, {513, 1268}, {514, 1255}
X(35327) = X(i)-reciprocal conjugate of X(j) for these (i,j): (100, 32018), (101, 1268), (110, 32014)
X(35327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (101, 692, 4557), (662, 3573, 4436), (692, 4557, 23344), (922, 3747, 3285), (35343, 35344, 4427)

X(35328) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(11)

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

X(35328) lies on these lines: {919,2423}, {2195,34858}

X(35328) = intersection, other than A,B,C, of conics {{A, B, C, X(109), X(35340)}} and {{A, B, C, X(110), X(35313)}}

X(35329) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(13)

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

X(35329) lies on these lines: {6,3129}, {110,351}, {249,14183}, {827,10410}, {1625,5994}, {2070,5615}, {2420,5995}, {35314,35343}

X(35329) = isogonal conjugate of the isotomic conjugate of X(35314)
X(35329) = isogonal conjugate of anticomplement of X(35443)
X(35329) = barycentric product X(i)*X(j) for these {i, j}: {6, 35314}, {110, 396}, {463, 4558}, {476, 19294}
X(35329) = barycentric quotient X(i)/X(j) for these (i, j): (396, 850), (463, 14618), (1576, 2981)
X(35329) = trilinear product X(i)*X(j) for these {i, j}: {31, 35314}, {163, 396}, {463, 4575}
X(35329) = trilinear quotient X(i)/X(j) for these (i, j): (163, 2981), (396, 1577), (463, 24006), (1101, 10409)
X(35329) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(16806)}} and {{A, B, C, X(396), X(2421)}}
X(35329) = crossdifference of every pair of points on line {X(115), X(23871)}
X(35329) = crosspoint of X(110) and X(5995)
X(35329) = crosssum of X(523) and X(23870)
X(35329) = X(110)-Daleth conjugate of X(35330)
X(35329) = X(i)-isoconjugate-of X(j) for these {i,j}: {1109, 10409}, {1577, 2981}
X(35329) = X(i)-reciprocal conjugate of X(j) for these (i,j): (396, 850), (463, 14618), (1576, 2981)
X(35329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 1576, 35330), (110, 17403, 1634)

X(35330) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(14)

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

X(35330) lies on these lines: {6,3130}, {110,351}, {249,14184}, {827,10409}, {1625,5995}, {2070,5611}, {2420,5994}, {35315,35344}

X(35330) = isogonal conjugate of the isotomic conjugate of X(35315)
X(35330) = isogonal conjugate of anticomplement of X(35444)
X(35330) = barycentric product X(i)*X(j) for these {i, j}: {6, 35315}, {110, 395}, {462, 4558}, {533, 5995}
X(35330) = barycentric quotient X(i)/X(j) for these (i, j): (395, 850), (462, 14618), (1576, 6151)
X(35330) = trilinear product X(i)*X(j) for these {i, j}: {31, 35315}, {163, 395}, {462, 4575}
X(35330) = trilinear quotient X(i)/X(j) for these (i, j): (163, 6151), (395, 1577), (462, 24006), (1101, 10410)
X(35330) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(16807)}} and {{A, B, C, X(110), X(3457)}}
X(35330) = crossdifference of every pair of points on line {X(115), X(23870)}
X(35330) = crosspoint of X(110) and X(5994)
X(35330) = crosssum of X(523) and X(23871)
X(35330) = X(110)-Daleth conjugate of X(35329)
X(35330) = X(i)-isoconjugate-of X(j) for these {i,j}: {1109, 10410}, {1577, 6151}
X(35330) = X(i)-reciprocal conjugate of X(j) for these (i,j): (395, 850), (462, 14618), (1576, 6151)
X(35330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 1576, 35329), (110, 17402, 1634)

X(35331) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(15)

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

X(35331) lies on these lines: {6,11087}, {1625,5994}, {2421,32036}, {4630,32737}

X(35331) = barycentric product X(623)*X(16806)

X(35332) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND CEVIAN-OF-X(16)

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

X(35332) lies on these lines: {6,11082}, {1625,5995}, {2421,32037}, {4630,32737}

X(35332) = barycentric product X(624)*X(16807)

X(35333) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND INTOUCH

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

X(35333) lies on these lines: {101,4040}, {662,3737}, {666,1026}, {831,919}, {927,29052}, {1438,33854}, {1581,2664}, {4602,7257}

X(35333) = barycentric product X(i)*X(j) for these {i, j}: {38, 666}, {105, 4568}, {673, 4553}, {919, 1930}
X(35333) = barycentric quotient X(i)/X(j) for these (i, j): (38, 918), (39, 2254), (105, 10566), (666, 3112)
X(35333) = trilinear product X(i)*X(j) for these {i, j}: {39, 666}, {105, 4553}, {141, 919}, {927, 3688}
X(35333) = trilinear quotient X(i)/X(j) for these (i, j): (38, 2254), (39, 665), (105, 18108), (141, 918)
X(35333) = trilinear pole of the line {38, 17456}
X(35333) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(4576)}} and {{A, B, C, X(101), X(1634)}}
X(35333) = X(i)-isoconjugate-of X(j) for these {i,j}: {82, 2254}, {83, 665}, {251, 918}
X(35333) = X(i)-reciprocal conjugate of X(j) for these (i,j): (38, 918), (39, 2254), (105, 10566)
X(35333) = X(31)-Zayin conjugate of X(2254)

X(35334) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND EXTOUCH

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

X(35334) lies on these lines: {163,643}, {813,8707}, {1025,6648}, {2298,4284}

X(35334) = barycentric product X(i)*X(j) for these {i, j}: {38, 8707}, {1220, 4553}
X(35334) = barycentric quotient X(i)/X(j) for these (i, j): (141, 4509), (1018, 27067)
X(35334) = trilinear product X(i)*X(j) for these {i, j}: {39, 8707}, {141, 32736}, {1634, 14624}
X(35334) = trilinear quotient X(i)/X(j) for these (i, j): (39, 6371), (141, 3004), (1220, 10566)
X(35334) = trilinear pole of the line {1964, 33299}
X(35334) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(4576)}} and {{A, B, C, X(101), X(35325)}}
X(35334) = X(i)-isoconjugate-of X(j) for these {i,j}: {83, 6371}, {251, 3004}, {1193, 10566}
X(35334) = X(i)-reciprocal conjugate of X(j) for these (i,j): (141, 4509), (1018, 27067)

X(35335) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND CEVIAN-OF-X(9)

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

X(35335) lies on the line {4041,21272}

X(35335) = barycentric product X(i)*X(j) for these {i, j}: {141, 35338}, {142, 4553}, {354, 4568}
X(35335) = barycentric quotient X(i)/X(j) for these (i, j): (354, 10566), (1475, 18108)
X(35335) = trilinear product X(i)*X(j) for these {i, j}: {38, 35338}, {141, 35326}, {354, 4553}, {1475, 4568}
X(35335) = trilinear quotient X(i)/X(j) for these (i, j): (142, 10566), (354, 18108)
X(35335) = X(1174)-isoconjugate-of X(10566)
X(35335) = X(i)-reciprocal conjugate of X(j) for these (i,j): (354, 10566), (1475, 18108)

X(35336) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND CEVIAN-OF-X(15)

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

X(35336) lies on these lines: {110,16807}, {6138,35314}, {35319,35337}, {35322,35346}

X(35336) = barycentric product X(623)*X(1634)
X(35336) = trilinear quotient X(623)/X(18070)

X(35337) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND CEVIAN-OF-X(16)

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

X(35337) lies on these lines: {110,16806}, {6137,35315}, {35319,35336}, {35323,35346}

X(35337) = barycentric product X(624)*X(1634)
X(35337) = trilinear quotient X(624)/X(18070)

X(35338) = AREAL CENTER OF THESE TRIANGLES: INTOUCH AND CEVIAN-OF-X(9)

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

X(35338) lies on these lines: {1,528}, {9,1742}, {37,17668}, {40,2807}, {42,4667}, {43,4274}, {46,34372}, {48,24309}, {99,29052}, {100,109}, {101,1292}, {112,831}, {142,2293}, {165,16554}, {190,1026}, {214,32486}, {219,11495}, {241,15733}, {269,3174}, {326,4149}, {516,1818}, {522,4552}, {527,2340}, {601,5150}, {662,3737}, {664,4569}, {674,20367}, {906,35342}, {934,6575}, {991,2550}, {1018,2284}, {1019,1634}, {1020,2283}, {1042,12437}, {1044,11523}, {1045,13610}, {1275,6606}, {1308,20219}, {1418,15185}, {1458,5853}, {1740,18794}, {1770,3191}, {1777,11517}, {2310,16578}, {2323,9441}, {2324,2951}, {2809,3942}, {2947,10860}, {3035,5400}, {3189,4306}, {3190,3474}, {3216,16400}, {3243,4334}, {3247,4335}, {3293,4663}, {3682,31730}, {3925,17194}, {4014,21320}, {4322,21627}, {4447,6007}, {4557,21362}, {4847,22053}, {4858,28850}, {6180,6600}, {7202,21889}, {9945,34586}, {12016,24028}, {13576,17197}, {15507,29349}, {16548,18788}, {16553,18790}, {16688,27626}, {17092,30628}, {24466,33810}, {33814,34465}, {35310,35326}

X(35338) = midpoint of X(2340) and X(3000)
X(35338) = reflection of X(i) in X(j) for these (i,j): (2310, 16578), (21362, 4557)
X(35338) = barycentric product X(i)*X(j) for these {i, j}: {7, 35341}, {9, 35312}, {75, 35326}, {83, 35335}
X(35338) = barycentric quotient X(i)/X(j) for these (i, j): (100, 32008), (101, 2346), (109, 1170), (142, 693)
X(35338) = trilinear product X(i)*X(j) for these {i, j}: {2, 35326}, {55, 35312}, {57, 35341}, {59, 6362}
X(35338) = trilinear quotient X(i)/X(j) for these (i, j): (100, 2346), (101, 1174), (142, 514), (190, 32008)
X(35338) = trilinear pole of the line {354, 1212}
X(35338) = lies on the circumconics with center X(i) for i in {142, 3925}
X(35338) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(20219)}} and {{A, B, C, X(9), X(6606)}}
X(35338) = crossdifference of every pair of points on line {X(2170), X(17463)}
X(35338) = crosspoint of X(100) and X(664)
X(35338) = crosssum of X(513) and X(663)
X(35338) = X(i)-Aleph conjugate of X(j) for these (i,j): (9, 2958), (100, 9), (101, 43)
X(35338) = X(i)-Ceva conjugate of X(j) for these (i,j): (664, 35312), (1252, 1), (1275, 9)
X(35338) = X(100)-Daleth conjugate of X(1025)
X(35338) = X(i)-isoconjugate-of X(j) for these {i,j}: {513, 2346}, {514, 1174}, {649, 32008}
X(35338) = X(i)-reciprocal conjugate of X(j) for these (i,j): (100, 32008), (101, 2346), (109, 1170)
X(35338) = X(100)-Waw conjugate of X(1018)
X(35338) = X(i)-Zayin conjugate of X(j) for these (i,j): (6, 4040), (31, 3737), (55, 513)
X(35338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100, 651, 3939), (100, 3888, 3882), (190, 1026, 4069), (269, 3174, 8271), (4436, 4553, 1018), (35310, 35326, 35341)

X(35339) = AREAL CENTER OF THESE TRIANGLES: INTOUCH AND CEVIAN-OF-X(10)

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

X(35339) lies on these lines: {100,4069}, {643,4614}, {1018,34074}, {1929,17594}, {2334,5264}, {3065,4866}

X(35339) = barycentric product X(i)*X(j) for these {i, j}: {1125, 4606}, {1213, 4614}, {1269, 34074}, {1962, 4633}
X(35339) = barycentric quotient X(i)/X(j) for these (i, j): (1100, 4778), (1125, 4801), (1213, 4815)
X(35339) = trilinear product X(i)*X(j) for these {i, j}: {1100, 4606}, {1125, 8694}, {1213, 4627}, {1962, 4614}
X(35339) = trilinear quotient X(i)/X(j) for these (i, j): (553, 30723), (1125, 4778), (1213, 4841), (1962, 4822)
X(35339) = lies on the circumconic with center X(3647))
X(35339) = X(i)-isoconjugate-of X(j) for these {i,j}: {1126, 4778}, {1171, 4841}, {1255, 4790}
X(35339) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1100, 4778), (1125, 4801), (1213, 4815)
X(35339) = X(37)-Zayin conjugate of X(4790)
X(35339) = {X(100), X(4606)}-harmonic conjugate of X(8694)

X(35340) = AREAL CENTER OF THESE TRIANGLES: INTOUCH AND CEVIAN-OF-X(11)

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

X(35340) lies on these lines: {1,9357}, {165,29374}, {1707,9282}, {2743,14733}

X(35340) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(35313)}} and {{A, B, C, X(109), X(35328)}}

X(35341) = AREAL CENTER OF THESE TRIANGLES: EXTOUCH AND CEVIAN-OF-X(9)

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

X(35341) lies on these lines: {1,23988}, {9,11}, {57,16593}, {100,101}, {190,658}, {200,14943}, {244,25096}, {643,1021}, {894,27256}, {2284,4551}, {3119,14740}, {3161,30947}, {3174,5580}, {3219,31058}, {3239,3952}, {3306,16549}, {3501,4936}, {3729,21404}, {3730,5744}, {4025,25266}, {4712,11028}, {4728,25737}, {4750,22003}, {4847,8012}, {6011,6078}, {17194,21795}, {18240,24036}, {20588,28070}, {21061,22032}, {21363,26265}, {21371,29969}, {35310,35326}

X(35341) = barycentric product X(i)*X(j) for these {i, j}: {8, 35338}, {99, 21039}, {100, 4847}, {101, 1229}
X(35341) = barycentric quotient X(i)/X(j) for these (i, j): (100, 21453), (101, 1170), (142, 24002), (190, 31618)
X(35341) = trilinear product X(i)*X(j) for these {i, j}: {8, 35326}, {9, 35338}, {21, 35310}, {99, 21795}
X(35341) = trilinear quotient X(i)/X(j) for these (i, j): (100, 1170), (142, 3676), (190, 21453), (354, 3669)
X(35341) = trilinear pole of the line {1212, 2293}
X(35341) = lies on the circumconics with center X(i) for i in {142, 6067}
X(35341) = intersection, other than A,B,C, of conics {{A, B, C, X(11), X(3887)}} and {{A, B, C, X(100), X(3254)}}
X(35341) = Cevapoint of X(1212) and X(21127)
X(35341) = crosspoint of X(190) and X(644)
X(35341) = crosssum of X(649) and X(3669)
X(35341) = X(644)-Daleth conjugate of X(1026)
X(35341) = X(i)-isoconjugate-of X(j) for these {i,j}: {513, 1170}, {649, 21453}, {663, 10509}
X(35341) = X(i)-reciprocal conjugate of X(j) for these (i,j): (100, 21453), (101, 1170), (142, 24002)
X(35341) = X(644)-Waw conjugate of X(4069)
X(35341) = X(i)-Zayin conjugate of X(j) for these (i,j): (7, 649), (218, 513)
X(35341) = {X(35310), X(35326)}-harmonic conjugate of X(35338)

X(35342) = AREAL CENTER OF THESE TRIANGLES: EXTOUCH AND CEVIAN-OF-X(10)

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

X(35342) lies on these lines: {1,1929}, {2,4262}, {3,16552}, {6,16371}, {9,1030}, {32,3216}, {35,3294}, {36,3684}, {41,16549}, {43,609}, {78,1759}, {99,3570}, {100,101}, {110,15322}, {116,24582}, {163,662}, {169,4855}, {172,3293}, {187,2238}, {190,32042}, {199,200}, {214,2170}, {224,1729}, {391,15692}, {404,4251}, {474,4258}, {514,17136}, {519,1055}, {664,23890}, {673,25532}, {813,8708}, {831,919}, {906,35338}, {910,5440}, {931,6010}, {978,7031}, {1078,29433}, {1111,24685}, {1146,10609}, {1213,15670}, {1292,28879}, {1308,28903}, {1310,28847}, {1376,16788}, {1384,4383}, {1415,4551}, {1609,1713}, {1724,3053}, {1743,2305}, {1761,16553}, {1764,28920}, {1983,2613}, {2246,24036}, {2802,17439}, {2915,17742}, {3207,5687}, {3550,9431}, {3686,22054}, {3691,5267}, {3693,20918}, {3811,17736}, {3916,17746}, {4051,21842}, {4103,17780}, {4115,4427}, {4188,4253}, {4256,5276}, {4511,5011}, {4557,21003}, {4559,23703}, {4561,33952}, {4567,4596}, {4574,35326}, {4875,13624}, {4919,5541}, {5022,19537}, {5030,13587}, {5275,19322}, {5291,31855}, {5540,15015}, {5816,6974}, {6012,28852}, {6574,28226}, {7267,8682}, {7280,21384}, {7719,20832}, {8300,16479}, {8678,22280}, {8715,9310}, {13396,28887}, {14543,22003}, {16917,17175}, {17499,17693}, {17729,20347}, {18047,23891}, {18206,19308}, {18786,34996}, {20367,20769}, {21921,35016}, {24398,28111}, {27524,32431}, {33828,33953}

X(35342) = barycentric product X(i)*X(j) for these {i, j}: {1, 4427}, {57, 30729}, {59, 4985}, {75, 35327}
X(35342) = barycentric quotient X(i)/X(j) for these (i, j): (1, 4608), (100, 1268), (101, 1255), (163, 1171)
X(35342) = trilinear product X(i)*X(j) for these {i, j}: {2, 35327}, {6, 4427}, {56, 30729}, {58, 4115}
X(35342) = trilinear quotient X(i)/X(j) for these (i, j): (99, 32014), (100, 1255), (101, 1126), (110, 1171)
X(35342) = trilinear pole of the line {1100, 1962}
X(35342) = lies on the circumconic with center X(3647))
X(35342) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(4596)}} and {{A, B, C, X(100), X(1929)}}
X(35342) = Cevapoint of X(i) and X(j) for these (i,j): (649, 4272), (661, 3723), (1213, 4976)
X(35342) = crossdifference of every pair of points on line {X(244), X(2611)}
X(35342) = crosspoint of X(i) and X(j) for these (i,j): (100, 662), (190, 6742)
X(35342) = crosssum of X(i) and X(j) for these (i,j): (513, 661), (514, 4129), (649, 2605)
X(35342) = X(i)-Aleph conjugate of X(j) for these (i,j): (99, 16552), (100, 846), (190, 1761)
X(35342) = X(100)-Daleth conjugate of X(3573)
X(35342) = X(i)-isoconjugate-of X(j) for these {i,j}: {115, 6578}, {512, 32014}, {513, 1255}
X(35342) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 4608), (100, 1268), (101, 1255)
X(35342) = X(1018)-Vertex conjugate of X(4557)
X(35342) = X(100)-Waw conjugate of X(4436)
X(35342) = X(i)-Zayin conjugate of X(j) for these (i,j): (21, 1019), (37, 513), (72, 4063)
X(35342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (41, 25440, 16549), (100, 101, 1018), (101, 1018, 1023), (474, 4258, 16783), (4427, 30729, 4115), (5277, 18755, 1)

X(35343) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(10) AND CEVIAN-OF-X(13)

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

X(35343) lies on these lines: {4427,35327}, {35314,35329}

X(35343) = barycentric product X(i)*X(j) for these {i, j}: {396, 4427}, {1213, 35314}, {1230, 35329}
X(35343) = trilinear product X(i)*X(j) for these {i, j}: {396, 35342}, {1962, 35314}
X(35343) = {X(4427), X(35327)}-harmonic conjugate of X(35344)

X(35344) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(10) AND CEVIAN-OF-X(14)

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

X(35344) lies on these lines: {4427,35327}, {35315,35330}

X(35344) = barycentric product X(i)*X(j) for these {i, j}: {395, 4427}, {1213, 35315}, {1230, 35330}
X(35344) = trilinear product X(i)*X(j) for these {i, j}: {395, 35342}, {1962, 35315}
X(35344) = {X(4427), X(35327)}-harmonic conjugate of X(35343)

X(35345) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(13) AND CEVIAN-OF-X(14)

Barycentrics (SA-SC)*(SA-SB)*(4*S^2-3*(SB+SC)^2) : :
X(35345) = X(2407)+3*X(4226) = X(2407)-3*X(5467)

X(35345) lies on these lines: {30,14356}, {99,827}, {395,8015}, {396,8014}, {476,1291}, {512,35346}, {523,2407}, {597,5092}, {3233,15329}, {3329,6636}, {4235,23347}, {5468,10190}, {6593,33813}, {7473,14590}, {7668,12042}, {14588,17941}, {15358,18122}, {23342,34245}, {35314,35329}, {35315,35330}

X(35345) = midpoint of X(4226) and X(5467)
X(35345) = barycentric product X(i)*X(j) for these {i, j}: {395, 35314}, {396, 35315}
X(35345) = crosssum of X(2088) and X(22260)

X(35346) = AREAL CENTER OF THESE TRIANGLES: CEVIAN-OF-X(15) AND CEVIAN-OF-X(16)

Barycentrics (SA-SB)*(SB+SC)*(SA-SC)*(2*S^4+(3*SA^2-14*SB*SC+SW^2)*S^2-3*(SB+SC)^2*SA^2) : :

X(35346) lies on these lines: {512,35345}, {526,2407}, {1154,14356}, {1510,5467}, {3613,21243}, {4226,20188}, {15060,19130}, {35322,35336}, {35323,35337}

X(35347) = AREAL CENTER OF THESE TRIANGLES: EXTOUCH AND STEINER

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

X(35347) lies on these lines: {1,4041}, {10,7265}, {65,4705}, {75,4086}, {759,28471}, {5620,30574}, {9278,24290}, {18827,35141}

X(35347) = barycentric product X(i)*X(j) for these {i, j}: {661, 35141}, {1577, 28471}
X(35347) = barycentric quotient X(661)/X(17768)
X(35347) = trilinear product X(i)*X(j) for these {i, j}: {512, 35141}, {523, 28471}
X(35347) = trilinear quotient X(523)/X(17768)
X(35347) = trilinear pole of the line {661, 2611}
X(35347) = lies on the circumconic with center X(244))
X(35347) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(10)}} and {{A, B, C, X(100), X(5466)}}
X(35347) = X(110)-isoconjugate-of X(17768)
X(35347) = X(661)-reciprocal conjugate of X(17768)

X(35348) = AREAL CENTER OF THESE TRIANGLES: EXTOUCH AND YFF CONTACT

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

X(35348) lies on these lines: {1,1643}, {2,522}, {57,23351}, {81,3737}, {88,1156}, {89,4724}, {105,1024}, {165,11193}, {274,18155}, {277,7658}, {278,7649}, {279,3676}, {676,2006}, {885,3911}, {957,29350}, {1022,3675}, {1026,5376}, {1121,3227}, {1638,2826}, {2222,14733}, {2224,34068}, {3887,4845}, {5219,10006}, {8056,21189}, {10389,11124}, {11247,15803}, {14392,23057}, {14413,34056}, {23703,35340}, {28095,28096}

X(35348) = barycentric product X(i)*X(j) for these {i, j}: {7, 23893}, {85, 23351}, {513, 1121}, {514, 1156}
X(35348) = barycentric quotient X(i)/X(j) for these (i, j): (56, 23890), (244, 1638), (513, 527), (514, 30806)
X(35348) = trilinear product X(i)*X(j) for these {i, j}: {7, 23351}, {11, 14733}, {57, 23893}, {513, 1156}
X(35348) = trilinear quotient X(i)/X(j) for these (i, j): (56, 23346), (57, 23890), (244, 14413), (513, 1155)
X(35348) = trilinear pole of the line {513, 2170}
X(35348) = lies on the circumconics with center X(i) for i in {1015, 6615, 14413}
X(35348) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(100), X(6548)}}
X(35348) = Cevapoint of X(i) and X(j) for these (i,j): (513, 14413), (663, 22108)
X(35348) = crossdifference of every pair of points on line {X(1055), X(6603)}
X(35348) = crosssum of X(1155) and X(14413)
X(35348) = X(i)-isoconjugate-of X(j) for these {i,j}: {8, 23346}, {9, 23890}, {59, 6366}
X(35348) = X(i)-reciprocal conjugate of X(j) for these (i,j): (56, 23890), (244, 1638), (513, 527)
X(35348) = X(i)-Zayin conjugate of X(j) for these (i,j): (354, 23890), (513, 1155)

X(35349) = AREAL CENTER OF THESE TRIANGLES: 2nd HATZIPOLAKIS AND INTOUCH

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

X(35349) lies on these lines: {101,2222}, {109,1783}, {644,31343}, {1743,13539}, {10703,34591}, {17494,26693}

X(35349) = barycentric product X(i)*X(j) for these {i, j}: {100, 9581}, {318, 35350}, {644, 23681}
X(35349) = trilinear product X(i)*X(j) for these {i, j}: {101, 9581}, {281, 35350}, {644, 17054}
X(35349) = X(1435)-Zayin conjugate of X(652)

X(35350) = AREAL CENTER OF THESE TRIANGLES: 2nd HATZIPOLAKIS AND ORTHIC

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

X(35350) lies on these lines: {100,905}, {109,8059}, {1331,23113}

X(35350) = barycentric product X(i)*X(j) for these {i, j}: {77, 35349}, {1331, 23681}, {1332, 17054}, {1813, 9581}
X(35350) = trilinear product X(i)*X(j) for these {i, j}: {222, 35349}, {906, 23681}, {1331, 17054}
X(35350) = crossdifference of every pair of points on line {X(5514), X(5521)}

X(35351) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND LEMOINE

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

X(35351) lies on the line {190,3908}

X(35352) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND STEINER

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

X(35352) lies on these lines: {10,514}, {12,7178}, {80,291}, {313,3261}, {523,594}, {741,2372}, {813,2690}, {826,1089}, {894,10566}, {1734,29070}, {1826,7649}, {3572,6133}, {3837,20507}, {4010,24290}, {4013,4049}, {4088,21053}, {4562,35148}, {4589,5380}, {6538,31010}, {6543,18004}, {6545,15523}, {15232,29324}, {18827,35162}, {21204,29653}

X(35352) = barycentric product X(i)*X(j) for these {i, j}: {10, 4444}, {115, 4589}, {291, 1577}, {292, 850}
X(35352) = barycentric quotient X(i)/X(j) for these (i, j): (10, 3570), (37, 3573), (115, 4010), (125, 24459)
X(35352) = trilinear product X(i)*X(j) for these {i, j}: {10, 876}, {37, 4444}, {115, 4584}, {291, 523}
X(35352) = trilinear quotient X(i)/X(j) for these (i, j): (10, 3573), (115, 21832), (291, 110), (292, 163)
X(35352) = trilinear pole of the line {3120, 4024}
X(35352) = lies on the circumconics with center X(i) for i in {4988, 18004}
X(35352) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(12)}} and {{A, B, C, X(37), X(5377)}}
X(35352) = Cevapoint of X(523) and X(18004)
X(35352) = crossdifference of every pair of points on line {X(1914), X(5009)}
X(35352) = crosspoint of X(335) and X(4589)
X(35352) = crosssum of X(1914) and X(4455)
X(35352) = X(i)-isoconjugate-of X(j) for these {i,j}: {58, 3573}, {99, 2210}, {100, 5009}
X(35352) = X(i)-reciprocal conjugate of X(j) for these (i,j): (10, 3570), (37, 3573), (115, 4010)

X(35353) = AREAL CENTER OF THESE TRIANGLES: INCENTRAL AND YFF CONTACT

Barycentrics (-b*c+(2*b-c)*a)*(b*c+(b-2*c)*a)*(b^2-c^2) : :
X(35353) = X(321)+2*X(8034)

X(35353) lies on the Kiepert hyperbola and these lines: {2,4448}, {4,6591}, {10,661}, {76,693}, {83,18108}, {98,739}, {226,4017}, {321,523}, {649,4672}, {671,2787}, {690,11611}, {876,4358}, {889,35147}, {898,1290}, {1019,32944}, {1647,4444}, {2051,3667}, {2254,14554}, {3835,17758}, {3840,8042}, {4010,4080}, {4806,30588}, {4824,27797}, {4931,34475}, {30992,30993}

X(35353) = barycentric product X(i)*X(j) for these {i, j}: {313, 23892}, {523, 3227}, {661, 31002}, {739, 850}
X(35353) = barycentric quotient X(i)/X(j) for these (i, j): (37, 23343), (115, 14431), (512, 3230), (523, 536)
X(35353) = trilinear product X(i)*X(j) for these {i, j}: {313, 23349}, {321, 23892}, {512, 31002}, {661, 3227}
X(35353) = trilinear quotient X(i)/X(j) for these (i, j): (10, 23343), (321, 23891), (523, 899), (661, 3230)
X(35353) = trilinear pole of the line {523, 3125}
X(35353) = lies on the circumconic with center X(14431))
X(35353) = intersection, other than A,B,C, of Kiepert hyperbola and conic {{A, B, C, X(37), X(16482)}}
X(35353) = Cevapoint of X(523) and X(14431)
X(35353) = X(i)-isoconjugate-of X(j) for these {i,j}: {58, 23343}, {110, 899}, {163, 536}
X(35353) = X(i)-reciprocal conjugate of X(j) for these (i,j): (37, 23343), (115, 14431), (512, 3230)

X(35354) = AREAL CENTER OF THESE TRIANGLES: INTOUCH AND STEINER

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

X(35354) lies on these lines: {9,661}, {210,4705}, {312,1577}, {2321,4024}, {8818,14321}

X(35354) = barycentric product X(1109)*X(6083)
X(35354) = trilinear product X(115)*X(6083)
X(35354) = trilinear quotient X(i)/X(j) for these (i, j): (115, 6089), (2501, 1884)
X(35354) = trilinear pole of the line {2643, 4041}
X(35354) = intersection, other than A,B,C, of conics {{A, B, C, X(9), X(33)}} and {{A, B, C, X(100), X(5466)}}
X(35354) = Cevapoint of X(661) and X(2610)
X(35354) = X(i)-isoconjugate-of X(j) for these {i,j}: {249, 6089}, {1884, 4558}

X(35355) = AREAL CENTER OF THESE TRIANGLES: INTOUCH AND YFF CONTACT

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

X(35355) lies on the Feuerbach hyperbola and these lines: {1,3309}, {4,28591}, {7,3667}, {8,514}, {9,3126}, {21,1019}, {80,2826}, {104,1477}, {294,1027}, {314,5214}, {522,6601}, {764,3680}, {876,4876}, {900,3254}, {1000,28292}, {1022,1280}, {1023,1308}, {1025,5377}, {1156,2827}, {2346,6003}, {2481,35160}, {3062,30198}, {3255,28217}, {3551,24802}, {3577,30199}, {3738,34894}, {4900,14077}, {5559,28473}, {6006,34919}, {6362,6598}, {6366,12641}

X(35355) = barycentric product X(i)*X(j) for these {i, j}: {514, 1280}, {650, 35160}, {1111, 6078}, {1477, 4391}
X(35355) = barycentric quotient X(i)/X(j) for these (i, j): (55, 23704), (244, 6084), (513, 3008), (649, 1279)
X(35355) = trilinear product X(i)*X(j) for these {i, j}: {513, 1280}, {522, 1477}, {663, 35160}, {1086, 6078}
X(35355) = trilinear quotient X(i)/X(j) for these (i, j): (9, 23704), (513, 1279), (514, 3008), (522, 5853)
X(35355) = trilinear pole of the line {244, 650}
X(35355) = lies on the circumconic with center X(661))
X(35355) = intersection, other than A,B,C, of Feuerbach hyperbola and conic {{A, B, C, X(100), X(6548)}}
X(35355) = Cevapoint of X(513) and X(2254)
X(35355) = crosssum of X(2254) and X(4864)
X(35355) = X(1)-Hirst inverse of X(14201)
X(35355) = X(i)-isoconjugate-of X(j) for these {i,j}: {57, 23704}, {100, 1279}, {101, 3008}
X(35355) = X(i)-reciprocal conjugate of X(j) for these (i,j): (55, 23704), (244, 6084), (513, 3008)
X(35355) = X(650)-Zayin conjugate of X(2348)
X(35355) = orthocenter of X(1)X(7)X(8)
X(35355) = orthocenter of X(4)X(8)X(9)

X(35356) = AREAL CENTER OF THESE TRIANGLES: LEMOINE AND MEDIAL

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

X(35356) lies on these lines: {2,353}, {69,5648}, {99,110}, {111,5182}, {183,6800}, {184,33798}, {352,13586}, {512,34205}, {542,7664}, {669,1634}, {892,4577}, {1383,1992}, {1495,26276}, {2715,9069}, {5108,31128}, {5642,14928}, {5971,12215}, {6082,12074}, {6233,9100}, {6593,25052}, {6791,18800}, {6792,7665}, {7417,12177}, {7771,15080}, {9999,10553}, {10552,14645}, {11206,33796}, {12073,35359}, {14570,14999}, {20976,25047}, {23297,30516}

X(35356) = reflection of X(1992) in X(20380)
X(35356) = isotomic conjugate of the anticomplement of X(17436)
X(35356) = isotomic conjugate of the isogonal conjugate of X(35357)
X(35356) = anticomplement of X(8288)
X(35356) = barycentric product X(i)*X(j) for these {i, j}: {76, 35357}, {99, 597}, {110, 26235}, {670, 5008}
X(35356) = barycentric quotient X(i)/X(j) for these (i, j): (99, 10302), (249, 12074), (597, 523)
X(35356) = trilinear product X(i)*X(j) for these {i, j}: {75, 35357}, {163, 26235}, {597, 662}, {799, 5008}
X(35356) = trilinear quotient X(i)/X(j) for these (i, j): (597, 661), (799, 10302)
X(35356) = trilinear pole of the line {597, 5008}
X(35356) = lies on the circumconics with center X(i) for i in {597, 14762, 15810, 20582}
X(35356) = Cevapoint of X(i) and X(j) for these (i,j): (523, 14762), (597, 12073)
X(35356) = crossdifference of every pair of points on line {X(3124), X(9208)}
X(35356) = crosspoint of X(99) and X(35138)
X(35356) = crosssum of X(512) and X(17414)
X(35356) = X(99)-Daleth conjugate of X(34245)
X(35356) = X(798)-isoconjugate-of X(10302)
X(35356) = X(i)-reciprocal conjugate of X(j) for these (i,j): (99, 10302), (249, 12074), (597, 523)
X(35356) = X(1634)-Vertex conjugate of X(5468)
X(35356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 110, 5468), (99, 5468, 4576), (2502, 5026, 2), (5468, 10330, 99), (30508, 30509, 34245)

X(35357) = AREAL CENTER OF THESE TRIANGLES: LEMOINE AND ORTHIC

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

X(35357) lies on these lines: {3,19140}, {110,351}, {691,827}, {1625,3050}, {1995,15364}, {2080,18374}, {5191,6593}, {5201,9407}, {15919,32609}, {19596,23164}, {31951,32694}

X(35357) = isogonal conjugate of the anticomplement of X(17436)
X(35357) = isogonal conjugate of the isotomic conjugate of X(35356)
X(35357) = anticomplement of the complementary conjugate of X(17436)
X(35357) = barycentric product X(i)*X(j) for these {i, j}: {6, 35356}, {99, 5008}, {110, 597}, {249, 12073}
X(35357) = barycentric quotient X(i)/X(j) for these (i, j): (110, 10302), (597, 850)
X(35357) = trilinear product X(i)*X(j) for these {i, j}: {31, 35356}, {163, 597}, {662, 5008}, {1101, 12073}
X(35357) = trilinear quotient X(i)/X(j) for these (i, j): (597, 1577), (662, 10302), (1101, 12074)
X(35357) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(31951)}} and {{A, B, C, X(110), X(35356)}}
X(35357) = crossdifference of every pair of points on line {X(115), X(17416)}
X(35357) = crosspoint of X(110) and X(11636)
X(35357) = crosssum of X(523) and X(3906)
X(35357) = X(i)-isoconjugate-of X(j) for these {i,j}: {661, 10302}, {1109, 12074}
X(35357) = X(i)-reciprocal conjugate of X(j) for these (i,j): (110, 10302), (597, 850)
X(35357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110, 1576, 5467), (110, 5467, 1634)

X(35358) = AREAL CENTER OF THESE TRIANGLES: LEMOINE AND STEINER

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

X(35358) lies on these lines: {523,599}, {598,1499}, {12073,15810}

X(35358) = barycentric product X(597)*X(34246)
X(35358) = barycentric quotient X(597)/X(34245)
X(35358) = lies on the circumconic with center X(8288))
X(35358) = X(597)-reciprocal conjugate of X(34245)

X(35359) = AREAL CENTER OF THESE TRIANGLES: LEMOINE AND SYMMEDIAL

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

X(35359) lies on these lines: {691,10330}, {732,31125}, {826,4576}, {892,5466}, {12073,35356}

X(35359) = barycentric quotient X(597)/X(22105)
X(35359) = X(597)-reciprocal conjugate of X(22105)

X(35360) = AREAL CENTER OF THESE TRIANGLES: MACBEATH AND MEDIAL

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

The trilinear polar of X(35360), line X(5)X(53), is the Brocard axis of the orthic triangle, and the polar-circle-inverse of circle {{X(4),X(15),X(16),X(186),X(3484)}}. (Randy Hutson, January 17, 2020)

Let P₁ and P₂ be the intersections, other than X(3) and X(4), of the Jerebek hyperbola and circle O(3,4). Then X(35360) is the crosssum of P₁ and P₂. (Randy Hutson, January 17, 2020)

X(35360) lies on the Johnson circumconic and these lines: {2,1972}, {3,6663}, {4,94}, {20,1075}, {27,13243}, {51,324}, {52,13450}, {107,110}, {112,925}, {133,15063}, {136,12828}, {162,655}, {186,14934}, {250,476}, {264,5640}, {323,450}, {338,11746}, {427,31127}, {436,1994}, {467,14569}, {523,1624}, {653,3658}, {685,4630}, {811,21272}, {877,4576}, {1093,5889}, {1249,7493}, {1301,1302}, {1304,14480}, {1370,12384}, {1625,14391}, {1896,35097}, {1897,4246}, {1941,22467}, {1990,32269}, {1995,9308}, {2052,3060}, {2322,7474}, {2501,35325}, {2979,15466}, {3078,31610}, {3153,6761}, {3183,30552}, {3186,4232}, {3580,6530}, {4230,11794}, {5562,14363}, {5972,14920}, {5984,6995}, {6515,6524}, {7049,9538}, {10095,14978}, {12111,14249}, {14570,23181}, {14611,31510}, {15059,16080}, {15958,16813}, {16240,24981}, {18022,33798}, {18831,33513}

X(35360) = reflection of X(4) in X(34334)
X(35360) = isogonal conjugate of X(23286)
X(35360) = polar conjugate of X(15412)
X(35360) = isotomic conjugate of the anticomplement of X(17434)
X(35360) = anticomplementary conjugate of the anticomplement of X(32230)
X(35360) = anticomplement of X(2972)
X(35360) = trilinear pole of the line {5, 53}
X(35360) = lies on the circumconics with center X(i) for i in {140, 216, 389, 6663, 14363, 14767, 18402}
X(35360) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(933)}} and {{A, B, C, X(5), X(4240)}}
X(35360) = Cevapoint of X(i) and X(j) for these (i,j): (5, 6368), (51, 12077), (140, 520)
X(35360) = crosspoint of X(i) and X(j) for these (i,j): (99, 30441), (648, 6528)
X(35360) = crosssum of X(i) and X(j) for these (i,j): (54, 19208), (512, 30442)
X(35360) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (162, 34186), (823, 13219)
X(35360) = X(i)-Ceva conjugate of X(j) for these (i,j): (250, 4), (476, 4240), (648, 1625)
X(35360) = X(i)-cross conjugate of X(j) for these (i,j): (143, 250), (1625, 14570)
X(35360) = X(i)-isoconjugate-of X(j) for these {i,j}: {3, 2616}, {48, 15412}, {54, 656}
X(35360) = X(i)-reciprocal conjugate of X(j) for these (i,j): (4, 15412), (5, 525), (6, 23286)
X(35360) = X(1624)-Vertex conjugate of X(4240)
X(35360) = Kiepert image of X(4)
X(35360) = pole wrt polar circle of trilinear polar of X(15412) (line X(125)X(526))
X(35360) = barycentric product X(i)*X(j) for these {i, j}: {4, 14570}, {5, 648}, {51, 6331}, {52, 30450}, {107, 343}
X(35360) = barycentric quotient X(i)/X(j) for these (i, j): (4, 15412), (5, 525), (19, 2616), (25, 2623)
X(35360) = trilinear product X(i)*X(j) for these {i, j}: {4, 2617}, {5, 162}, {19, 14570}, {51, 811}
X(35360) = trilinear quotient X(i)/X(j) for these (i, j): (4, 2616), (5, 656), (19, 2623), (51, 810)
X(35360) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 324, 30506), (107, 110, 4240), (107, 648, 110), (110, 648, 35311), (877, 6331, 4576), (1112, 2970, 4), (4240, 35311, 110)

X(35361) = AREAL CENTER OF THESE TRIANGLES: MACBEATH AND STEINER

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

X(35361) lies on these lines: {6,2501}, {216,12077}, {343,18314}, {2088,14582}, {2351,8029}, {11077,14910}

X(35361) = barycentric product X(i)*X(j) for these {i, j}: {5, 15328}, {53, 15421}, {1300, 6368}
X(35361) = barycentric quotient X(i)/X(j) for these (i, j): (51, 15329), (53, 16237), (1300, 18831)
X(35361) = lies on the circumconic with center X(15450))
X(35361) = X(1725)-isoconjugate-of X(18315)
X(35361) = X(i)-reciprocal conjugate of X(j) for these (i,j): (51, 15329), (53, 16237), (1300, 18831)

X(35362) = AREAL CENTER OF THESE TRIANGLES: MACBEATH AND SYMMEDIAL

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

X(35362) lies on these lines: {879,2966}, {6368,14570}, {9019,20021}

X(35362) = barycentric product X(290)*X(35319)
X(35362) = trilinear product X(1821)*X(35319)

X(35363) = AREAL CENTER OF THESE TRIANGLES: MACBEATH AND YFF CONTACT

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

X(35363) lies on these lines: {58,17925}, {216,21102}

X(35364) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND STEINER

Barycentrics a^2*(a^4+b^4-b^2*c^2+2*c^4-(2*b^2+c^2)*a^2)*(a^4+2*b^4-b^2*c^2+c^4-(b^2+2*c^2)*a^2)*(b^2-c^2) : :
X(35364) = X(879)-3*X(34290)

X(35364) lies on the Jerabek hyperbola and these lines: {3,512}, {4,3566}, {6,924}, {68,525}, {69,523}, {71,4079}, {72,4705}, {74,3563}, {248,2422}, {265,690}, {290,35142}, {520,6391}, {526,895}, {691,10411}, {826,3519}, {868,879}, {876,8773}, {1176,1510}, {1499,4846}, {2065,23350}, {2088,10097}, {2780,34802}, {3309,34800}, {3426,20186}, {3521,32478}, {4230,32696}, {5467,14560}, {5504,9517}, {7927,34483}, {8673,15316}, {15321,20184}, {30209,34801}

X(35364) = isogonal conjugate of X(4226)
X(35364) = antigonal conjugate of the isogonal conjugate of X(7468)
X(35364) = anticomplement of the complementary conjugate of X(868)
X(35364) = barycentric product X(i)*X(j) for these {i, j}: {115, 10425}, {125, 32697}, {512, 8781}, {523, 2987}
X(35364) = barycentric quotient X(i)/X(j) for these (i, j): (351, 5477), (512, 230), (647, 3564), (661, 1733)
X(35364) = trilinear product X(i)*X(j) for these {i, j}: {512, 8773}, {656, 3563}, {661, 2987}, {798, 8781}
X(35364) = trilinear quotient X(i)/X(j) for these (i, j): (512, 8772), (523, 1733), (656, 3564), (661, 230)
X(35364) = trilinear pole of the line {647, 3124}
X(35364) = lies on the circumconics with center X(i) for i in {3005, 3569}
X(35364) = intersection, other than A,B,C, of Jerabek hyperbola and conic {{A, B, C, X(98), X(5968)}}
X(35364) = Cevapoint of X(512) and X(3569)
X(35364) = crossdifference of every pair of points on line {X(114), X(230)}
X(35364) = crosssum of X(i) and X(j) for these (i,j): (511, 6132), (523, 6036)
X(35364) = X(i)-cross conjugate of X(j) for these (i,j): (684, 523), (878, 34212)
X(35364) = X(i)-isoconjugate-of X(j) for these {i,j}: {99, 8772}, {110, 1733}, {162, 3564}
X(35364) = X(i)-reciprocal conjugate of X(j) for these (i,j): (6, 4226), (351, 5477), (512, 230)
X(35364) = orthocenter of X(3)X(4)X(69)
X(35364) = orthocenter of X(4)X(6)X(68)
X(35364) = pole wrt polar circle of line X(460)X(3564)

X(35365) = AREAL CENTER OF THESE TRIANGLES: ORTHIC AND YFF CONTACT

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

X(35365) lies on these lines: {3,14825}, {57,23800}, {103,9085}, {295,3572}, {901,29241}, {1790,3733}, {1797,23345}, {3798,14377}, {7192,17206}

X(35365) = barycentric product X(1086)*X(29241)
X(35365) = barycentric quotient X(i)/X(j) for these (i, j): (58, 4237), (649, 3011), (1015, 29240), (1459, 9028)
X(35365) = trilinear product X(i)*X(j) for these {i, j}: {244, 29241}, {905, 9085}
X(35365) = trilinear quotient X(i)/X(j) for these (i, j): (81, 4237), (244, 29240), (513, 3011), (905, 9028)
X(35365) = trilinear pole of the line {1015, 1459}
X(35365) = lies on the circumconic with center X(513))
X(35365) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(27)}} and {{A, B, C, X(101), X(35348)}}
X(35365) = crossdifference of every pair of points on line {X(3011), X(5513)}
X(35365) = crosssum of X(514) and X(31380)
X(35365) = X(i)-isoconjugate-of X(j) for these {i,j}: {37, 4237}, {100, 3011}, {765, 29240}
X(35365) = X(i)-reciprocal conjugate of X(j) for these (i,j): (58, 4237), (649, 3011), (1015, 29240)

X(35366) = AREAL CENTER OF THESE TRIANGLES: STEINER AND SYMMEDIAL

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

X(35366) lies on these lines: {2,512}, {141,3005}, {729,9076}, {826,8024}, {850,1502}, {882,3266}, {3228,9140}, {3613,23301}

X(35366) = barycentric product X(i)*X(j) for these {i, j}: {729, 23285}, {826, 3228}
X(35366) = barycentric quotient X(i)/X(j) for these (i, j): (39, 5118), (141, 23342), (688, 33875), (729, 827)
X(35366) = trilinear quotient X(i)/X(j) for these (i, j): (38, 5118), (729, 34072), (826, 2234)
X(35366) = lies on the circumconic with center X(15449))
X(35366) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(66)}} and {{A, B, C, X(39), X(3111)}}
X(35366) = X(i)-isoconjugate-of X(j) for these {i,j}: {82, 5118}, {538, 34072}, {827, 2234}
X(35366) = X(i)-reciprocal conjugate of X(j) for these (i,j): (39, 5118), (141, 23342), (688, 33875)

X(35367) = AREAL CENTER OF THESE TRIANGLES: SYMMEDIAL AND YFF CONTACT

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

X(35367) lies on these lines: {86,3253}, {141,21123}, {334,876}, {1930,2530}

X(35367) = barycentric quotient X(i)/X(j) for these (i, j): (141, 23354), (727, 4628)
X(35367) = X(1575)-isoconjugate-of X(4628)
X(35367) = X(i)-reciprocal conjugate of X(j) for these (i,j): (141, 23354), (727, 4628)

X(35368) = AREAL CENTER OF THESE TRIANGLES: EXTOUCH AND LEMOINE

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

X(35368) lies on these lines: {662,4069}, {2748,35342}

X(35368) = intersection, other than A,B,C, of conics {{A, B, C, X(100), X(35356)}} and {{A, B, C, X(101), X(35357)}}

X(35369) = ANTICOMPLEMENT OF X(20094)

Barycentrics 5*a^4 - 5*a^2*b^2 - 3*b^4 - 5*a^2*c^2 + 11*b^2*c^2 - 3*c^4 : :
X(35369) = 9 X[2] - 8 X[99], 15 X[2] - 16 X[115], 3 X[2] - 4 X[148], 33 X[2] - 32 X[620], 7 X[2] - 8 X[671], 17 X[2] - 16 X[2482], 31 X[2] - 32 X[5461], 63 X[2] - 64 X[6722], 5 X[2] - 4 X[8591], 23 X[2] - 24 X[9166], 49 X[2] - 48 X[9167], 39 X[2] - 40 X[14061], 47 X[2] - 48 X[14971], 19 X[2] - 16 X[15300]

X(35369) lies on these lines: {2, 99}, {20, 12188}, {114, 3854}, {147, 17578}, {183, 33260}, {1007, 32993}, {1916, 14068}, {2782, 3146}, {2796, 31145}, {2996, 33014}, {3091, 13188}, {3329, 9607}, {3522, 12042}, {3617, 13174}, {3622, 11599}, {3832, 6321}, {3839, 12355}, {4678, 13178}, {5059, 5984}, {5186, 7408}, {5286, 19693}, {5939, 33244}, {5969, 20080}, {6655, 7879}, {6658, 7766}, {7748, 7922}, {7783, 33024}, {7798, 20088}, {7933, 32822}, {8781, 32881}, {8782, 32997}, {9862, 15683}, {14651, 15717}, {14929, 33256}, {16044, 31406}, {19691, 20081}, {22577, 32552}, {22578, 32553}

X(35369) = reflection of X(i) in X(j) for these {i,j}: {2, 8596}, {5059, 5984}, {20094, 148}
X(35369) = anticomplement of X(20094)
X(35369) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(6719)
X(35369) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {99, 671, 6722}, {115, 22247, 14061}, {148, 8591, 115}, {148, 20094, 2}, {8596, 20094, 148}

X(35370) = MIDPOINT OF X(468) AND X(32246)

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

See Antreas P. Hatzipolakis and Angel Montesdeoca, Euclid 358 .

X(35370) lies on these lines: {5,141}, {66,12290}, {468,2393}, {1503,16270}, {1514,15738}, {1995,8542}, {5094,9971}, {5159,9019}, {6000,32274}, {6593,8681}, {7505,15073}, {10510,11284}, {18374,32251}

X(35370) = midpoint of X(468) and X(32246)

X(35371) = X(3)X(6)∩X(468)X(2393)

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

See Antreas P. Hatzipolakis and Angel Montesdeoca, Euclid 358 .

X(35371) lies on these lines: {3,6}, {468,2393}, {1177,1205}, {2781,15471}, {3147,18919}, {3292,32245}, {5181,8681}, {8262,13567}, {8705,11746}, {12061,23326}, {15074,34351}, {16511,19510}

X(35372) = X(25)X(14264)∩X(30)X(1272)

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

X(35372) lies on these lines: {25,14264},{30,1272},{381,14583},{399,1495},{512,34329},{3003,14581},{3447,10628},{10421,16263},{12281,14979}

X(35372) = isogonal conjugate of X(12383)
X(35372) = isogonal conjugate of the anticomplement of X(265)
X(35372) = X(21650)-cross conjugate of X(4)
X(35372) = X(1)-isoconjugate of X(12383)
X(35372) = crosssum of X(399) and X(2931)
X(35372) = trilinear pole of line {686,14398}
X(35372) = barycentric quotient X (i)/X(j) for these {i,j}: {6,12383}, {8749,10421}

X(35373) = X(69)X(15454)∩X(265)X(18780)

X(35373) lies on the Jerabek circumhyperbola and these lines: {69,15454},{265,18780},{1300,33565}

X(35373) = X(19457)-cross conjugate of X(12028)
X(35373) = X(1725)-isoconjugate of X(12383)
X(35373) = barycentric quotient X(14910)/X(12383)

Centers of circles through couples of bicentric pairs: X(35374)-X(35440)

This preamble and centers X(35374)-X(35440) were contributed by César Eliud Lozada, December 14, 2019.

The appearance of (i, j, n) in the following list means that the bicentric pairs PU(i) and PU(j) are concyclic on a circle with center X(n):

(1, 2, 1691), (1, 39, 32), (1, 182, 3), (1, 183, 3398), (1, 189, 3), (1, 190, 6), (1, 191, 6), (1, 192, 32), (2, 39, 35374), (2, 182, 35375), (2, 183, 35376), (2, 188, 35433), (2, 189, 35383), (2, 190, 35377), (2, 191, 35378), (2, 192, 35379), (5, 61, 35380), (5, 162, 35434), (5, 163, 35384), (5, 164, 35435), (5, 165, 35381), (5, 166, 35382), (5, 174, 4), (5, 175, 3), (39, 182, 35385), (39, 183, 35386), (39, 188, 35436), (39, 189, 35387), (39, 190, 35388), (39, 191, 35389), (39, 192, 32), (40, 186, 11632), (55, 111, 35390), (61, 162, 35391), (61, 163, 35392), (61, 164, 35393), (61, 165, 35394), (61, 166, 35395), (61, 173, 35396), (61, 174, 35397), (61, 175, 35398), (105, 181, 35399), (120, 121, 1), (120, 122, 1), (121, 122, 1), (162, 163, 35400), (162, 164, 15685), (162, 165, 35401), (162, 166, 35402), (162, 173, 35403), (162, 174, 35404), (162, 175, 14893), (163, 164, 381), (163, 165, 35405), (163, 166, 35406), (163, 173, 35407), (163, 174, 35408), (163, 175, 35409), (164, 165, 35410), (164, 166, 35411), (164, 173, 35412), (164, 174, 35413), (164, 175, 35414), (165, 166, 35415), (165, 173, 35416), (165, 174, 35417), (165, 175, 35418), (166, 173, 35419), (166, 174, 35420), (166, 175, 35421), (173, 174, 2), (173, 175, 381), (174, 175, 5), (182, 183, 35422), (182, 188, 21163), (182, 189, 3), (182, 190, 182), (182, 191, 35423), (182, 192, 35424), (183, 188, 35437), (183, 189, 35425), (183, 190, 35426), (183, 191, 35427), (183, 192, 35428), (188, 189, 35438), (188, 190, 5052), (188, 191, 35439), (188, 192, 35440), (189, 190, 35429), (189, 191, 13354), (189, 192, 35430), (190, 191, 6), (190, 192, 35431), (191, 192, 35432)

Some notes: (apply for PU(1) to PU(192), except PU(3), PU(4), PU(117), PU(118) and PU(119))

PU(n) lie on the circumcircle of ABC for n ∈ {25,42,87,89,91,97,99,101,103,105,107,109,133,134}
PU(n) lie in the infinity for n ∈ {24, 41, 176, 177, 178, 179, 180}
If m, n ∈ {5, 61, 162, 163, 164, 165, 166, 173, 174, 175}, then PU(m) and PU(n) are concyclic on a circle whose center lies on the Euler line of ABC. (excepting {PU(5), PU(173)} which are collinear on the tripolar of X(94))
If m, n ∈ {2, 39, 182, 183, 189, 190, 191, 192}, then PU(m) and PU(n) are concyclic on a circle whose center lies on the Brocard axis of ABC. Also, the calculated centers O* of circles related to these PU's have all the same relative positions with respect to ABC and the circumsymmedial triangle, i.e., O*-of-ABC = O*-of-circumsymmedial triangle.
PU(1), PU(182), PU(189) lie on the 2nd Brocard circle; PU(120), PU(121), PU(122) lie on the Adams circle.
The following sets of PU's are concyclic: {PU(1), PU(39), PU(192)}, {PU(1), PU(190), PU(191)}, {PU(120), P(121), P(122)}

X(35374) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(39)

Barycentrics a^2*(a^8+(b^2+c^2)*a^6-(b^4+3*b^2*c^2+c^4)*a^4+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4)*a^2-(b^4+c^4)^2) : :
X(35374) = 2*X(5017)+X(5162)

X(35374) lies on these lines: {3,6}, {5031,5475}, {5103,8361}, {5149,6393}, {5207,7737}, {5477,12215}, {10008,20065}, {13586,14645}

X(35374) = midpoint of X(2076) and X(5017)
X(35374) = reflection of X(i) in X(j) for these (i,j): (6, 16385), (1570, 2032), (2456, 13335), (2458, 187), (5028, 1692), (5162, 2076), (35388, 32)
X(35374) = X(35374)-of-circumsymmedial triangle
X(35374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1691, 2076, 35375), (1691, 35379, 32), (2459, 2460, 5171), (4289, 8554, 584), (5017, 35424, 32), (35387, 35424, 35385)

X(35375) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(182)

Barycentrics a^2*(a^8-(b^2+c^2)*a^6-(b^4+5*b^2*c^2+c^4)*a^4+(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^2-b^8-c^8) : :
Trilinears cos A (cot ω - 4 sin 2ω) + sin A : :
X(35375) = 2*X(187)+X(3098) = X(1351)-4*X(8590) = X(1351)-3*X(35006) = X(2080)+2*X(14810) = 3*X(5050)-X(15514) = 2*X(5092)+X(5104) = 4*X(8590)-3*X(35006) = X(35377)+2*X(35383)

The circle {{PU(2),PU(182)}} is the inverse-in-circumcircle of line PU(1) (line X(39)X(512)), and also passes through X(3) and X(2076). (Randy Hutson, January 17, 2020)

X(35375) lies on these lines: {3,6}, {98,10997}, {140,5103}, {384,24206}, {542,13586}, {732,33813}, {1003,11178}, {1352,3552}, {4048,34507}, {5031,7761}, {5152,21445}, {5207,14907}, {5965,12215}, {5999,6036}, {6636,11673}, {6776,33014}, {7488,32529}, {7770,13449}, {9751,22503}, {10168,33273}, {10998,12110}, {11676,29012}, {15246,33873}, {19924,26613}

X(35375) = midpoint of X(i) and X(j) for these {i,j}: {3, 2076}, {1691, 35383}, {2456, 5104}, {2458, 35387}
X(35375) = reflection of X(i) in X(j) for these (i,j): (576, 1692), (2456, 5092), (5103, 140), (5111, 575), (35377, 1691), (35431, 16385)
X(35375) = reflection of X(182) in the line X(512)X(25644)
X(35375) = circumtangential isogonal conjugate of X(8295)
X(35375) = circumperp conjugate of X(9821)
X(35375) = isogonal conjugate of the antigonal conjugate of X(3406)
X(35375) = isogonal conjugate of the antitomic conjugate of X(3406)
X(35375) = circumcircle-inverse of X(3095)
X(35375) = X(6)-Hirst inverse of X(3095)
X(35375) = X(512)-vertex conjugate of X(3095)
X(35375) = X(35375)-of-circumsymmedial triangle
X(35375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35424, 182), (187, 5188, 2080), (1379, 1380, 3095), (1691, 2076, 35374), (1691, 35379, 35376), (2459, 2460, 32), (5188, 14810, 3098), (12974, 12975, 5171), (13349, 13350, 2080)

X(35376) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(183)

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

X(35376) lies on these lines: {3,6}, {12176,29012}

X(35376) = X(35376)-of-circumsymmedial triangle
X(35376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1691, 35377, 182), (1691, 35379, 35375), (2674, 8407, 1970), (3398, 35422, 182), (3398, 35428, 35426)

X(35377) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(190)

Barycentrics a^2*(a^8-3*(b^2+c^2)*a^6+(3*b^4-b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2+b^8+c^8-2*(b^2-c^2)^2*b^2*c^2) : :
X(35377) = X(3)-3*X(35006) = 5*X(6)+X(9301) = 5*X(182)-2*X(18860) = X(1350)-4*X(8590) = 2*X(1570)-3*X(15520) = 3*X(1691)-X(35383) = 5*X(1692)-X(18860) = 4*X(2030)-X(3098) = X(2080)+2*X(5097) = 3*X(5093)-X(15514) = 3*X(35375)-2*X(35383)

X(35377) lies on these lines: {3,6}, {114,385}, {542,14568}, {1994,11673}, {3788,14693}, {5103,18583}, {5149,13196}, {5207,14561}, {7766,9744}, {7901,25555}, {8177,34507}, {11675,19150}, {12215,32135}, {33873,34545}

X(35377) = midpoint of X(i) and X(j) for these {i,j}: {1351, 2076}, {2080, 5111}, {2458, 35389}
X(35377) = reflection of X(i) in X(j) for these (i,j): (182, 1692), (2456, 575), (5103, 18583), (5111, 5097), (12215, 32135), (35375, 1691), (35424, 16385)
X(35377) = 2nd Lemoine circle (cosine circle)-inverse of X(13330)
X(35377) = X(35377)-of-circumsymmedial triangle
X(35377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 35431, 576), (182, 35376, 1691), (575, 13354, 182), (1570, 1692, 2024), (1666, 1667, 13330), (2459, 2460, 5206)

X(35378) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(191)

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

X(35378) lies on these lines: {3,6}, {5149,12151}, {8593,11054}, {10753,15993}

X(35378) = X(35378)-of-circumsymmedial triangle

X(35379) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(192)

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

X(35379) lies on these lines: {3,6}, {5031,10796}, {5103,20576}, {5207,10788}

X(35379) = reflection of X(i) in X(j) for these (i,j): (182, 16385), (5103, 20576)
X(35379) = X(35379)-of-circumsymmedial triangle
X(35379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 35374, 1691), (35375, 35376, 1691)

X(35380) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(61)

Barycentrics 3*S^4-(R^2*(R^2-4*SW)+9*SB*SC+SW^2)*S^2+3*(3*R^2-SW)*(5*R^2-SW)*SB*SC : :

X(35380) lies on the line {2,3}

X(35381) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(165)

Barycentrics 73*a^4-119*(b^2+c^2)*a^2+46*(b^2-c^2)^2 : :
X(35381) = 7*X(2)-18*X(632) = 20*X(2)-9*X(1656) = X(2)+9*X(3525) = 29*X(2)-9*X(5056) = 14*X(2)-9*X(5070) = 8*X(2)+3*X(15693) = 2*X(2)+9*X(15694) = 10*X(2)+X(15695) = 9*X(2)+2*X(15711) = 5*X(2)+6*X(15713) = 4*X(2)+X(15716) = 7*X(2)+3*X(15719) = 11*X(2)+9*X(15721) = 4*X(2)-9*X(15723) = 14*X(2)-3*X(19709) = 11*X(3)-6*X(35418) = 2*X(4)-17*X(5070) = 4*X(4)+7*X(14093) = 3*X(4)+17*X(15719) = 6*X(4)-17*X(19709) = 22*X(4)-17*X(35401) = 11*X(4)-16*X(35417) = 28*X(4)-17*X(35434)

X(35381) lies on the line {2,3}

X(35381) = midpoint of X(5071) and X(15717)
X(35381) = reflection of X(i) in X(j) for these (i,j): (15694, 3525), (15718, 631), (35400, 19515)

X(35382) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(166)

Barycentrics 7*a^4-101*(b^2+c^2)*a^2+94*(b^2-c^2)^2 : :
X(35382) = 13*X(3)-8*X(35421) = 13*X(4)-18*X(35420) = 22*X(5)-7*X(5068) = 6*X(5)+7*X(5071) = 8*X(5)+7*X(5079) = 14*X(5)+X(10303) = 12*X(5)+X(15694) = 20*X(5)-7*X(19709) = 4*X(20)-17*X(15693) = 4*X(381)+9*X(1656) = 31*X(381)-18*X(3843) = X(381)+12*X(5071) = X(381)+9*X(5079) = 10*X(381)+3*X(14093) = 9*X(381)+4*X(15692) = 7*X(381)+6*X(15694) = 5*X(381)-18*X(19709) = 13*X(381)-3*X(35402) = 19*X(381)-6*X(35403) = 16*X(381)-3*X(35434)

X(35382) lies on the line {2,3}

X(35383) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(189)

Barycentrics a^2*(a^8-(3*b^4+7*b^2*c^2+3*c^4)*a^4+4*(b^2+c^2)*(b^4+c^4)*a^2-2*b^8-2*c^8+(b^2-c^2)^2*b^2*c^2) : :
X(35383) = 2*X(187)+X(33878) = 2*X(1350)+X(2080) = 2*X(1570)-3*X(5050) = 3*X(1691)-2*X(35377) = 2*X(3098)+X(5104) = 4*X(3098)-X(35002) = 3*X(5085)-X(15514) = 4*X(5092)-X(8586) = 2*X(5104)+X(35002) = 2*X(5107)-5*X(12017) = X(11477)-3*X(35006) = 3*X(35375)-X(35377)

X(35383) lies on these lines: {3,6}, {99,3564}, {489,9991}, {490,9992}, {805,3563}, {5207,10519}, {7467,11673}, {7784,13449}, {10104,18906}, {11674,32529}, {12215,33813}, {14971,19924}, {15980,29181}

X(35383) = midpoint of X(i) and X(j) for these {i,j}: {1350, 2076}, {11674, 32529}
X(35383) = reflection of X(i) in X(j) for these (i,j): (1351, 1692), (1691, 35375), (2080, 2076), (2456, 3), (5111, 182), (12215, 33813), (35388, 13335), (35389, 16385)
X(35383) = reflection of X(6) in the line X(512)X(25644)
X(35383) = circumperp conjugate of X(5171)
X(35383) = intersection, other than A,B,C, of conics {{A, B, C, X(32), X(23700)}} and {{A, B, C, X(249), X(9737)}}
X(35383) = circumcircle-inverse of X(9737)
X(35383) = X(6)-Hirst inverse of X(9737)
X(35383) = X(512)-vertex conjugate of X(9737)
X(35383) = X(2456)-of-ABC-X3 reflections triangle
X(35383) = X(35383)-of-circumsymmedial triangle
X(35383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1379, 1380, 9737), (2459, 2460, 3053), (3094, 35424, 3398), (3098, 5104, 35002), (3098, 35387, 3), (13354, 14810, 3)

X(35384) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(163)

Barycentrics 133*a^4-59*(b^2+c^2)*a^2-74*(b^2-c^2)^2 : :
X(35384) = X(4)+4*X(35408) = 22*X(5)-19*X(3522) = 34*X(5)-19*X(3529) = 24*X(5)-19*X(3534) = 14*X(5)-19*X(3543) = 4*X(5)-19*X(5073) = 16*X(5)-19*X(5076) = 8*X(5)-7*X(14093) = 12*X(5)-7*X(15685) = 20*X(5)-19*X(15694) = 21*X(5)-19*X(15711) = 8*X(20)-13*X(35402) = 11*X(381)-16*X(382) = 17*X(381)-16*X(1656) = 13*X(381)-8*X(1657) = 21*X(381)-16*X(3534) = 7*X(381)-8*X(5076) = 19*X(381)-16*X(14093) = 9*X(381)-8*X(15693) = 5*X(381)-4*X(15696) = 12*X(381)-11*X(35381) = X(381)-11*X(35405) = 13*X(381)-16*X(35434) = 28*X(381)-13*X(35435)

X(35384) lies on the line {2,3}

X(35384) = reflection of X(i) in X(j) for these (i,j): (1656, 15684), (1657, 35434), (5059, 15714), (15695, 15682), (17538, 35404), (17800, 5071), (35403, 3146)
X(35384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3839, 15684, 382), (5159, 27988, 2), (6807, 35298, 237), (6997, 26654, 24), (7393, 24939, 140), (10245, 16044, 379), (11345, 27301, 2), (11818, 27403, 24), (12107, 33821, 30), (16066, 19284, 24), (16066, 27938, 22), (16298, 21737, 186), (17527, 33260, 140), (23265, 33289, 186), (23276, 26645, 377), (25870, 33207, 2), (27207, 33741, 377), (27212, 31186, 23), (27592, 33002, 377), (28664, 30774, 29)

X(35385) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(182)

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

X(35385) lies on these lines: {3,6}, {83,10998}, {98,18806}, {384,2794}, {3552,8721}, {5999,7828}, {7768,34885}, {7886,13860}, {9863,10000}

X(35385) = reflection of X(i) in X(j) for these (i,j): (3398, 13335), (35386, 32)
X(35385) = X(35385)-of-circumsymmedial triangle
X(35385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2076, 5188), (3, 35424, 32), (32, 32452, 35388), (13335, 35430, 32), (35387, 35424, 35374)

X(35386) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(183)

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

X(35386) lies on these lines: {3,6}, {736,12110}, {9753,14023}

X(35386) = reflection of X(35385) in X(32)
X(35386) = X(35386)-of-circumsymmedial triangle
X(35386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3398, 35428, 32), (35389, 35431, 35388)

X(35387) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(189)

Barycentrics a^2*(a^8+(b^2+c^2)*a^6-3*(b^2+c^2)^2*a^4+(b^2+c^2)*(3*b^4-2*b^2*c^2+3*c^4)*a^2-2*(b^4+c^4)^2) : :
X(35387) = 3*X(32)-2*X(35431) = X(315)-3*X(10519) = 4*X(6680)-3*X(14561) = X(35389)-4*X(35424) = 3*X(35389)-4*X(35431) = 3*X(35424)-X(35431)

X(35387) lies on these lines: {3,6}, {98,18906}, {99,6776}, {315,10519}, {694,34481}, {1352,2794}, {6680,14561}, {7844,31670}, {9744,32458}, {11179,14645}, {20576,21850}

X(35387) = midpoint of X(1350) and X(5017)
X(35387) = reflection of X(i) in X(j) for these (i,j): (6, 13335), (32, 35424), (2458, 35375), (5028, 182), (13355, 3), (21850, 20576), (30270, 3098), (35389, 32)
X(35387) = X(13355)-of-ABC-X3 reflections triangle
X(35387) = X(35387)-of-circumsymmedial triangle
X(35387) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35383, 3098), (3, 35429, 5092), (3, 35430, 32), (182, 9734, 5116), (1664, 1665, 13354), (35374, 35385, 35424)

X(35388) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(190)

Barycentrics a^2*(a^8-(b^2+c^2)*a^6+(b^4-3*b^2*c^2+c^4)*a^4-(b^2+c^2)*b^2*c^2*a^2+(b^4+c^4)^2) : :
X(35388) = 2*X(2031)+X(5107)

X(35388) lies on these lines: {3,6}, {323,32526}, {3767,5207}, {5031,7746}, {7799,14645}, {11673,34945}

X(35388) = midpoint of X(5017) and X(15514)
X(35388) = reflection of X(i) in X(j) for these (i,j): (187, 2032), (2076, 16385), (2458, 1692), (5028, 1570), (5162, 1691), (35374, 32), (35383, 13335)
X(35388) = X(35388)-of-circumsymmedial triangle
X(35388) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 35431, 32), (32, 32452, 35385), (1504, 1505, 32452), (1570, 1692, 2025), (1692, 2025, 5034), (35389, 35431, 35386)

X(35389) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(191)

Barycentrics a^2*(a^8-3*(b^2+c^2)*a^6+((b^2-c^2)^2-4*b^2*c^2)*a^4-(b^2+c^2)^3*a^2+2*(b^4+c^4)^2) : :
X(35389) = 3*X(32)-2*X(35424) = X(69)-3*X(9753) = X(315)-3*X(14853) = 2*X(626)-3*X(14561) = 3*X(35387)-4*X(35424) = X(35387)-4*X(35431) = X(35424)-3*X(35431)

X(35389) lies on these lines: {3,6}, {69,9753}, {147,193}, {315,14853}, {626,14561}, {754,20423}, {1352,7751}, {1974,2967}, {2001,3060}, {2794,31670}, {5476,7818}, {6776,7760}, {8364,18583}, {10358,24256}, {12110,18906}, {12252,14912}, {20576,33185}, {25406,32467}

X(35389) = midpoint of X(5017) and X(11477)
X(35389) = reflection of X(i) in X(j) for these (i,j): (32, 35431), (1350, 13335), (2458, 35377), (5028, 576), (7818, 5476), (13355, 6), (30270, 182), (35383, 16385), (35387, 32)
X(35389) = X(35389)-of-circumsymmedial triangle
X(35389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 35423, 575), (6, 35427, 576), (6, 35432, 32), (3371, 13356, 4284), (35386, 35388, 35431)

X(35390) = CENTER OF THE CIRCLE THROUGH PU(55) AND PU(111)

Barycentrics a*(2*(b+c)*a^11+(5*b^2+8*b*c+5*c^2)*a^10-7*(b+c)*(b^2+c^2)*a^9-(20*b^4+20*c^4+b*c*(17*b^2+8*b*c+17*c^2))*a^8+(b+c)*(8*b^4+8*c^4-b*c*(8*b^2+31*b*c+8*c^2))*a^7+(30*b^6+30*c^6+(2*b^4+2*c^4+3*b*c*(b^2+16*b*c+c^2))*b*c)*a^6-2*(b+c)*(b^6+c^6-4*(3*b^4+3*c^4+8*b*c*(b^2-b*c+c^2))*b*c)*a^5-2*(10*b^8+10*c^8-(8*b^6+8*c^6+(2*b^4+2*c^4+5*b*c*(b^2+12*b*c+c^2))*b*c)*b*c)*a^4-(b^2-c^2)*(b-c)*(2*b^6+2*c^6+(28*b^4+28*c^4+b*c*(61*b^2+22*b*c+61*c^2))*b*c)*a^3+(b^2-c^2)^2*(5*b^6+5*c^6-2*b*c*(5*b^4+29*b^2*c^2+5*c^4))*a^2+(b^4-c^4)*(b^2-c^2)^2*(b-c)*(b^2+10*b*c+c^2)*a+(b^2-c^2)^4*b*c*(b^2+6*b*c+c^2)) : :

X(35390) lies on the line {1,3}

X(35391) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(162)

Barycentrics 9*S^4-(2*R^2*(173*R^2-104*SW)+27*SB*SC+31*SW^2)*S^2+3*(2*R^2*(439*R^2-250*SW)+71*SW^2)*SB*SC : :

X(35391) lies on the line {2,3}

X(35392) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(163)

Barycentrics S^4-(4*R^2*(241*R^2-138*SW)+3*SB*SC+79*SW^2)*S^2+(4*R^2*(849*R^2-485*SW)+277*SW^2)*SB*SC : :

X(35392) lies on the line {2,3}

X(35392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1599, 6617, 21), (3133, 15183, 24), (6658, 18378, 28), (6855, 6975, 30), (6933, 16294, 377), (7382, 32996, 377), (7549, 26833, 5), (11345, 27301, 26), (13595, 33204, 186), (14017, 26968, 2), (15014, 32958, 377), (17566, 29942, 24), (17674, 27504, 186), (18420, 34093, 297), (19545, 28121, 3), (21479, 33826, 30), (21869, 31074, 29), (21898, 28733, 297), (23277, 33053, 21), (28136, 34331, 186)

X(35393) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(164)

Barycentrics 9*S^4-(8*R^2*(1256*R^2-719*SW)+27*SB*SC+823*SW^2)*S^2+3*(4*R^2*(2204*R^2-1259*SW)+719*SW^2)*SB*SC : :

X(35393) lies on the line {2,3}

X(35393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381, 25012, 5), (4194, 34559, 140), (4201, 33746, 2), (5066, 27121, 26), (6622, 17570, 30), (6657, 16179, 30), (6818, 7561, 5), (6853, 30920, 30), (6888, 21515, 24), (7376, 7553, 27), (10125, 28947, 140), (11251, 21555, 5), (11345, 27301, 21), (15201, 16445, 24), (16066, 27938, 2), (17226, 18869, 23), (18868, 26650, 23), (19317, 21527, 2), (21487, 26968, 3), (22366, 25742, 2), (27212, 31186, 2)

X(35394) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(165)

Barycentrics 121*S^4-(8*R^2*(248*R^2-159*SW)+363*SB*SC+199*SW^2)*S^2+(4*R^2*(564*R^2-305*SW)+157*SW^2)*SB*SC : :

X(35394) lies on the line {2,3}

X(35394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (21, 3153, 3109), (436, 7575, 1004), (1316, 26623, 1658), (3141, 28450, 1889), (3559, 6810, 3133), (6913, 15213, 468), (6966, 27744, 1012), (7391, 19263, 1345), (7866, 16180, 2045), (11145, 14096, 1008), (11317, 26644, 2554), (11318, 16901, 405), (15183, 25018, 1008), (16050, 23408, 2074), (17226, 25701, 1590), (19685, 26554, 404), (23260, 33838, 447), (25032, 30815, 401), (27099, 30776, 3148), (27171, 28035, 3080)

X(35395) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(166)

Barycentrics 169*S^4-(2*R^2*(853*R^2-584*SW)+507*SB*SC+191*SW^2)*S^2+(2*R^2*(557*R^2-270*SW)+53*SW^2)*SB*SC : :

X(35395) lies on the line {2,3}

X(35395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1006, 21542, 4193), (3155, 32973, 3090), (6849, 24879, 419), (6892, 21976, 384), (7554, 32976, 24), (10126, 33218, 866), (11300, 33275, 3575), (11352, 27940, 468), (13746, 28800, 3130), (17529, 18323, 3850), (19227, 30803, 458), (19526, 27407, 1599), (19532, 33286, 2566), (19676, 27203, 1584), (21844, 34354, 407), (25484, 25910, 3560), (26618, 29409, 2050), (28433, 33196, 441), (29912, 33742, 404), (29915, 33187, 442)

X(35396) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(173)

Barycentrics S^4-(R^2*(33*R^2-20*SW)+3*SB*SC+3*SW^2)*S^2+(R^2*(113*R^2-64*SW)+9*SW^2)*SB*SC : :

X(35396) lies on the line {2,3}

X(35397) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(174)

Barycentrics 3*S^4-(R^2*(R^2-4*SW)+9*SB*SC+SW^2)*S^2+3*(R^2*(113*R^2-64*SW)+9*SW^2)*SB*SC : :

X(35397) lies on the line {2,3}

X(35398) = CENTER OF THE CIRCLE THROUGH PU(61) AND PU(175)

Barycentrics 3*S^4-(4*R^2*(37*R^2-22*SW)+9*SB*SC+13*SW^2)*S^2+3*(16*R^2-5*SW)*(4*R^2-SW)*SB*SC : :

X(35398) lies on the line {2,3}

X(35399) = CENTER OF THE CIRCLE THROUGH PU(105) AND PU(181)

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

X(35399) lies on these lines: {3,5106}, {110,384}, {182,2782}

X(35400) = CENTER OF THE CIRCLE THROUGH PU(162) AND PU(163)

Barycentrics 37*a^4-17*(b^2+c^2)*a^2-20*(b^2-c^2)^2 : :
X(35400) = 20*X(2)-19*X(3) = 17*X(2)-19*X(4) = 23*X(2)-19*X(20) = 21*X(2)-19*X(376) = 18*X(2)-19*X(381) = 14*X(2)-19*X(382) = 26*X(2)-19*X(1657) = 11*X(2)-19*X(3146) = 29*X(2)-19*X(3529) = 22*X(2)-19*X(3534) = 15*X(2)-19*X(3543) = 16*X(2)-19*X(3830) = 19*X(2)-20*X(3859) = 35*X(2)-19*X(5059) = 8*X(2)-19*X(5073) = 25*X(2)-19*X(11001) = 7*X(2)-8*X(12102) = 7*X(2)-19*X(15640) = 24*X(2)-19*X(15681) = 13*X(2)-19*X(15682) = 27*X(2)-19*X(15683) = 12*X(2)-19*X(15684) = 28*X(2)-19*X(15685) = 32*X(2)-19*X(17800) = 5*X(2)-19*X(33703)

X(35400) lies on these lines: {2,3}, {485,10137}, {486,10138}, {1131,6472}, {1132,6473}, {1327,6407}, {1328,6408}, {5102,29323}, {10247,28172}, {11531,28208}, {16200,28168}, {28150,34718}, {28160,34748}, {28182,34627}

X(35400) = midpoint of X(11541) and X(15640)
X(35400) = reflection of X(i) in X(j) for these (i,j): (382, 15640), (1657, 15682), (3529, 33699), (3534, 3146), (3830, 5073), (5059, 3845), (15681, 15684), (15683, 35404), (15685, 382), (17800, 3830), (35381, 19515), (35412, 35402)
X(35400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (376, 15723, 3), (381, 15686, 3), (382, 5059, 3), (1657, 3853, 3), (1982, 33821, 2), (3534, 3545, 3), (8964, 24933, 2), (10018, 33021, 2), (11345, 27301, 3), (11812, 15688, 3), (14461, 27054, 2), (16066, 27938, 4), (16067, 27173, 2), (17550, 19274, 2), (17568, 21898, 4), (21559, 25990, 3), (21979, 33188, 2), (24879, 27172, 2), (25041, 25824, 2), (26028, 29722, 4), (27212, 31186, 2)

X(35401) = CENTER OF THE CIRCLE THROUGH PU(162) AND PU(165)

Barycentrics 47*a^4-7*(b^2+c^2)*a^2-40*(b^2-c^2)^2 : :
X(35401) = 9*X(3)-20*X(381) = 31*X(3)-20*X(3534) = 3*X(3)+8*X(3543) = X(3)+10*X(3830) = 5*X(3)-16*X(3845) = 5*X(3)-8*X(5056) = 7*X(3)-10*X(5070) = 11*X(3)-20*X(5072) = 19*X(3)-8*X(11001) = 4*X(3)-15*X(14269) = 21*X(3)-10*X(15681) = 6*X(3)+5*X(15684) = 16*X(3)-5*X(15685) = 27*X(3)-16*X(15686) = 19*X(3)-20*X(15716) = 9*X(3)-10*X(15718) = 7*X(3)-8*X(15719) = 17*X(3)-20*X(15720) = 3*X(3)-4*X(15723) = 9*X(3)+2*X(35400) = 11*X(3)+4*X(35405) = 22*X(3)-13*X(35415)

X(35401) lies on the line {2,3}

X(35401) = midpoint of X(15682) and X(15717)
X(35401) = reflection of X(i) in X(j) for these (i,j): (3534, 3525), (5056, 3845), (15716, 3855), (15718, 381)
X(35401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 3543, 547), (4, 35404, 381), (381, 35400, 3), (464, 24887, 1591), (547, 15681, 3), (3523, 3627, 382), (3832, 5073, 3), (3845, 15702, 381), (4203, 25829, 424), (5055, 11001, 3), (7435, 19691, 416), (7525, 30972, 1370), (11113, 31681, 447), (11325, 16055, 860), (13731, 15202, 858), (14893, 15683, 381), (15695, 15708, 3), (16409, 30844, 1313), (17522, 31257, 426), (21521, 29519, 1347)

X(35402) = CENTER OF THE CIRCLE THROUGH PU(162) AND PU(166)

Barycentrics 61*a^4-11*(b^2+c^2)*a^2-50*(b^2-c^2)^2 : :
X(35402) = 3*X(3)+10*X(3543) = 7*X(3)-20*X(3845) = 7*X(3)-10*X(5067) = 23*X(3)-10*X(11001) = 33*X(3)-20*X(15686) = 13*X(3)+5*X(35406) = 21*X(3)-5*X(35412) = 24*X(4)-11*X(381) = 2*X(4)+11*X(3830) = 29*X(4)-11*X(5068) = 21*X(4)-8*X(11737) = 17*X(4)-4*X(12100) = 14*X(4)-X(15685) = 15*X(4)-2*X(15686) = 16*X(4)-3*X(15688) = 18*X(4)-5*X(15694) = 19*X(4)-6*X(15699) = 8*X(20)+5*X(35384) = 20*X(140)-7*X(11001) = 6*X(140)+7*X(35404)

X(35402) lies on the line {2,3}

X(35402) = midpoint of X(i) and X(j) for these {i,j}: {10303, 15682}, {35400, 35412}
X(35402) = reflection of X(5067) in X(3845)
X(35402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (382, 35403, 381), (1585, 6992, 461), (3534, 14893, 381), (3543, 35400, 382), (3545, 15721, 547), (3845, 15723, 381), (4188, 7948, 410), (6676, 23335, 237), (6756, 21036, 413), (8364, 15779, 1592), (11355, 23246, 1656), (12100, 34199, 1006), (13727, 14120, 384), (14093, 14269, 381), (15144, 33275, 441), (15685, 15694, 376), (15687, 35434, 381), (17555, 19340, 401), (25798, 28451, 868), (27051, 32958, 1658)

X(35403) = CENTER OF THE CIRCLE THROUGH PU(162) AND PU(173)

Barycentrics 17*a^4-(b^2+c^2)*a^2-16*(b^2-c^2)^2 : :
X(35403) = 16*X(2)-11*X(3) = X(2)-11*X(4) = 31*X(2)-11*X(20) = 21*X(2)-11*X(376) = 6*X(2)-11*X(381) = 13*X(2)-11*X(631) = 10*X(2)-11*X(1656) = 7*X(2)-11*X(3091) = 19*X(2)-11*X(3522) = 26*X(2)-11*X(3534) = 9*X(2)+11*X(3543) = 4*X(2)+11*X(3830) = 4*X(2)-11*X(3843) = X(2)+4*X(3853) = 9*X(2)-11*X(5071) = 4*X(2)+X(5073) = 2*X(2)+11*X(5076) = 11*X(2)-16*X(12811) = 18*X(2)-11*X(14093) = 9*X(2)-4*X(15691) = 15*X(2)-11*X(15692) = 14*X(2)-11*X(15693) = 12*X(2)-11*X(15694) = 20*X(2)-11*X(15695) = 23*X(2)-11*X(15697) = 7*X(2)-2*X(15704) = 25*X(2)-11*X(17538) = 5*X(2)+11*X(17578) = 17*X(2)-11*X(19708) = 8*X(2)-11*X(19709) = 13*X(2)-8*X(33923) = 3*X(2)+2*X(35404) = 9*X(2)-X(35414) = 6*X(2)+11*X(35434)

X(35403) lies on these lines: {2,3}, {3531,18550}, {4701,12699}, {5790,28232}, {6472,8976}, {6473,13951}, {6500,23251}, {6501,23261}, {6767,18513}, {7373,18514}, {7753,22246}, {10248,12645}, {11455,13321}, {12355,22505}, {14627,15811}, {18439,21849}, {18492,28202}, {28228,34718}, {28234,34648}, {28236,34748}

X(35403) = midpoint of X(i) and X(j) for these {i,j}: {381, 35434}, {382, 15693}, {632, 33699}, {3522, 15682}, {3543, 5071}, {3830, 3843}, {15685, 35407}, {15714, 35404}
X(35403) = reflection of X(i) in X(j) for these (i,j): (2, 3858), (3, 19709), (20, 15711), (1657, 15697), (3091, 3845), (3534, 631), (3830, 5076), (8703, 12812), (14093, 5071), (15681, 14093), (15693, 3091), (15694, 381), (15695, 1656), (15696, 2), (15697, 632), (15712, 5066), (15713, 3859), (17538, 15713), (19708, 5), (19709, 3843), (35384, 3146)
X(35403) = anticomplement of X(15714)
X(35403) = X(14093)-of-Ehrmann-mid triangle
X(35403) = X(19708)-of-Johnson triangle
X(35403) = X(19709)-of X3-ABC reflections triangle
X(35403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12102, 382), (4, 15687, 381), (376, 3845, 381), (381, 382, 376), (381, 15684, 3), (381, 15700, 5), (381, 35402, 382), (547, 3839, 381), (3091, 35407, 3), (3543, 14893, 381), (3627, 5059, 382), (3830, 5055, 382), (3830, 14269, 3), (3843, 15694, 381), (5055, 15685, 3), (5055, 35400, 376), (14269, 15684, 381), (15681, 15703, 3), (15683, 15708, 376), (15686, 15759, 376), (15688, 15722, 3)

X(35404) = CENTER OF THE CIRCLE THROUGH PU(162) AND PU(174)

Barycentrics 20*a^4-7*(b^2+c^2)*a^2-13*(b^2-c^2)^2 : :
X(35404) = 13*X(2)-11*X(3) = 7*X(2)-11*X(4) = 10*X(2)-11*X(5) = 19*X(2)-11*X(20) = 15*X(2)-11*X(376) = 9*X(2)-11*X(381) = X(2)-11*X(382) = 12*X(2)-11*X(549) = 16*X(2)-11*X(550) = 25*X(2)-11*X(1657) = 5*X(2)+11*X(3146) = 31*X(2)-11*X(3529) = 17*X(2)-11*X(3534) = 3*X(2)-11*X(3543) = 4*X(2)-11*X(3627) = 5*X(2)-11*X(3830) = 8*X(2)-11*X(3845) = 4*X(2)-5*X(3858) = 14*X(2)-11*X(8703) = 23*X(2)-11*X(11001) = 7*X(2)-8*X(12811) = 21*X(2)-11*X(15681) = X(2)+11*X(15682) = 27*X(2)-11*X(15683) = 3*X(2)+11*X(15684) = 29*X(2)-11*X(15685) = 18*X(2)-11*X(15686) = 6*X(2)-11*X(15687) = 7*X(2)-5*X(15696) = 6*X(2)-5*X(15714) = 20*X(2)-11*X(19710) = 2*X(2)-11*X(33699) = 5*X(2)-4*X(33923) = 3*X(2)-5*X(35403) = 21*X(2)-5*X(35414)

X(35404) lies on these lines: {2,3}, {551,28168}, {597,29323}, {1353,11645}, {1483,28208}, {3655,28190}, {3679,28178}, {4701,28194}, {5349,16963}, {5350,16962}, {6429,12818}, {6430,12819}, {6441,6561}, {6442,6560}, {10283,28164}, {12943,15170}, {13491,21849}, {15361,23324}, {28146,34648}, {28186,31162}, {28202,31673}, {28212,34627}, {28216,34718}

X(35404) = midpoint of X(i) and X(j) for these {i,j}: {2, 5073}, {3, 15640}, {382, 15682}, {3146, 3830}, {3534, 33703}, {3543, 15684}, {15683, 35400}, {15697, 35407}, {17538, 35384}
X(35404) = reflection of X(i) in X(j) for these (i,j): (2, 3853), (3, 12101), (5, 3830), (20, 5066), (376, 14893), (549, 15687), (550, 3845), (1657, 12100), (3529, 15690), (3534, 546), (3627, 33699), (3845, 3627), (8703, 4), (11001, 140), (12100, 12102), (12103, 3860), (13491, 21849), (15681, 547), (15683, 34200), (15685, 548), (15686, 381), (15687, 3543), (15690, 3861), (15704, 2), (15713, 5076), (15714, 35403), (19710, 5), (33699, 382)
X(35404) = anticomplement of X(15691)
X(35404) = anticomplement of X(549) with respect to Ehrmann-mid triangle
X(35404) = anticomplement of X(15691) with respect to these triangles: {1st anti-Brocard, 1st Brocard, 1st Brocard-reflected, 1st half-diamonds, 1st half-squares, 1st Neuberg, 2nd half-diamonds, 2nd half-squares, 2nd Neuberg, anti-Artzt, anti-McCay, anticomplementary, Artzt, inner-Fermat, inner-Vecten, McCay, medial, outer-Fermat, outer-Vecten}
X(35404) = anticomplement of X(34200) with respect to these triangles: {Euler, Johnson, X3-ABC reflections}
X(35404) = complement of X(15683) with respect to these triangles: {Euler, Johnson, X3-ABC reflections}
X(35404) = X(12101)-of X3-ABC reflections triangle
X(35404) = X(15683)-of-Ehrmann-mid triangle
X(35404) = X(19710)-of-Johnson triangle
X(35404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 12103, 5), (376, 14893, 5), (381, 10124, 5), (381, 35401, 4), (546, 3523, 5), (549, 19710, 376), (1657, 12102, 5), (1657, 15703, 376), (3146, 3543, 376), (3544, 19549, 3), (3830, 5054, 4), (3839, 12100, 5), (3853, 12811, 4), (3860, 5054, 5), (3860, 35421, 2), (5054, 15681, 376), (6940, 19647, 3), (6977, 19649, 21), (11001, 14269, 140), (15681, 35434, 4), (15691, 33923, 376)

X(35405) = CENTER OF THE CIRCLE THROUGH PU(163) AND PU(165)

Barycentrics 89*a^4-39*(b^2+c^2)*a^2-50*(b^2-c^2)^2 : :
X(35405) = 9*X(3)-10*X(5056) = 21*X(3)-10*X(5059) = 14*X(3)-15*X(15723) = X(3)+10*X(33703) = 11*X(3)-15*X(35401) = 24*X(4)-13*X(1657) = 15*X(4)-13*X(5056) = 2*X(4)-13*X(5073) = 16*X(4)-13*X(15720) = 11*X(4)-9*X(15721) = 29*X(4)-18*X(19710) = 17*X(4)-13*X(21735) = 20*X(140)-9*X(5059) = 6*X(376)-7*X(5070) = 27*X(381)-16*X(1657) = 9*X(381)-8*X(15720) = 11*X(381)-10*X(35381) = X(381)+10*X(35384) = 18*X(382)-7*X(1657) = 31*X(382)-20*X(3091) = 29*X(382)-18*X(5055) = 15*X(382)-4*X(5059) = 23*X(382)-14*X(5070) = 11*X(382)-7*X(5072) = 3*X(382)-14*X(5073) = 7*X(382)+4*X(11541) = X(382)-12*X(15640) = 17*X(382)-6*X(15685) = 26*X(382)-15*X(15693) = 19*X(382)-8*X(15704) = 7*X(382)-4*X(15717) = 12*X(382)-7*X(15720) = 5*X(382)-3*X(15723) = 5*X(382)+6*X(35400) = X(382)+10*X(35407) = 11*X(382)-3*X(35410) = 11*X(382)-10*X(35416)

X(35405) lies on the line {2,3}

X(35405) = midpoint of X(11541) and X(15717)
X(35405) = reflection of X(i) in X(j) for these (i,j): (15718, 15682), (17800, 3525)
X(35405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1600, 17678, 377), (2048, 7532, 379), (3091, 15684, 382), (3146, 15685, 382), (4186, 6756, 25), (4244, 10303, 377), (6914, 21574, 140), (7452, 25028, 24), (7498, 27769, 186), (7515, 14958, 235), (11345, 27301, 2), (15640, 35407, 382), (16239, 17574, 21), (16429, 18569, 2), (16897, 19704, 186), (18404, 28263, 237), (19235, 33275, 140), (26057, 27510, 186), (27258, 32977, 2), (27530, 28032, 377), (28445, 30922, 24)

X(35406) = CENTER OF THE CIRCLE THROUGH PU(163) AND PU(166)

Barycentrics 107*a^4-47*(b^2+c^2)*a^2-60*(b^2-c^2)^2 : :
X(35406) = 7*X(3)-20*X(3146) = 33*X(3)-20*X(3529) = 25*X(3)-12*X(5059) = 11*X(3)-12*X(5067) = 2*X(3)-15*X(5073) = 9*X(3)-10*X(5079) = 19*X(3)-20*X(10303) = 28*X(3)-15*X(17800) = X(3)+12*X(33703) = 4*X(3)+9*X(35400) = 13*X(3)-18*X(35402) = 17*X(3)-9*X(35412) = 13*X(3)-11*X(35415) = 27*X(381)-14*X(3529) = 15*X(381)-14*X(5067) = X(381)-14*X(15640) = 24*X(381)-11*X(17800) = 26*X(381)-11*X(35411) = 33*X(382)-20*X(632) = 31*X(382)-18*X(3524) = 4*X(382)-17*X(5073) = 27*X(382)-17*X(5079) = 25*X(382)-12*X(15686) = 28*X(382)-15*X(15695) = 29*X(382)-16*X(33923) = 10*X(382)-3*X(35412)

X(35406) lies on the line {2,3}

X(35406) = midpoint of X(10303) and X(11541)
X(35406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (550, 17578, 381), (2567, 6846, 4), (4221, 6925, 376), (5079, 6850, 21), (5079, 11251, 28), (6821, 21520, 377), (6829, 24567, 377), (6842, 29961, 3), (7506, 25702, 24), (16050, 32989, 384), (17528, 19250, 297), (17537, 25803, 2), (19239, 27209, 2), (19518, 28819, 140), (19674, 29467, 21), (25953, 26254, 377), (26058, 27333, 2), (26256, 30773, 186), (26863, 33849, 235), (33208, 33739, 26)

X(35407) = CENTER OF THE CIRCLE THROUGH PU(163) AND PU(173)

Barycentrics 43*a^4-19*(b^2+c^2)*a^2-24*(b^2-c^2)^2 : :
X(35407) = X(2)+4*X(35408) = 7*X(3)-12*X(382) = 15*X(3)-16*X(632) = 11*X(3)-12*X(1656) = 17*X(3)-12*X(1657) = 7*X(3)-8*X(3091) = 3*X(3)-8*X(3146) = 13*X(3)-8*X(3529) = 11*X(3)-16*X(3627) = 13*X(3)-18*X(3830) = 5*X(3)-6*X(3843) = X(3)-6*X(5073) = 3*X(3)-4*X(5076) = 7*X(3)+8*X(11541) = 23*X(3)-18*X(15681) = 4*X(3)-9*X(15684) = 14*X(3)-9*X(15685) = 17*X(3)-18*X(15694) = 19*X(3)-18*X(15695) = 13*X(3)-12*X(15696) = 21*X(3)-16*X(15704) = 9*X(3)-8*X(17538) = 11*X(3)-6*X(17800) = 8*X(3)-9*X(19709) = 7*X(3)-9*X(35403) = 7*X(3)-11*X(35416) = 7*X(3)-13*X(35419)

X(35407) lies on these lines: {2,3}, {11482,29323}

X(35407) = midpoint of X(3091) and X(11541)
X(35407) = reflection of X(i) in X(j) for these (i,j): (1657, 17578), (5076, 3146), (14093, 15682), (15685, 35403), (15697, 35404), (17800, 1656), (19709, 15684)
X(35407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3529, 3830, 3), (3568, 25484, 26), (3627, 17800, 3), (3628, 15689, 3), (3851, 12103, 3), (8727, 28790, 25), (11109, 26578, 2), (11345, 27301, 2), (14015, 31216, 25), (15708, 27328, 21), (15711, 32967, 29), (16066, 27938, 4), (16349, 16862, 2), (22248, 26032, 24), (26607, 28121, 21), (27123, 33181, 21), (27204, 33022, 21), (27212, 31186, 2), (28102, 34562, 25), (33703, 35406, 3)

X(35408) = CENTER OF THE CIRCLE THROUGH PU(163) AND PU(174)

Barycentrics 130*a^4-59*(b^2+c^2)*a^2-71*(b^2-c^2)^2 : :
X(35408) = X(2)-5*X(35407) = X(4)-5*X(35384) = 15*X(5)-13*X(376) = 19*X(5)-13*X(1657) = 7*X(5)-13*X(3146) = 11*X(5)-13*X(3830) = 7*X(5)-5*X(11001) = 14*X(5)-13*X(12100) = 16*X(5)-13*X(12103) = 12*X(5)-13*X(14893) = 3*X(5)-5*X(15684) = 17*X(5)-15*X(15688) = 6*X(5)-5*X(15691) = 11*X(5)-10*X(15759) = 17*X(5)-13*X(19710) = 9*X(5)-13*X(35404) = 3*X(140)-4*X(3543) = 14*X(140)-17*X(3853) = 16*X(140)-17*X(5066) = 20*X(140)-17*X(12103) = 15*X(140)-17*X(14893) = 13*X(140)-12*X(15688) = 19*X(140)-17*X(15690) = 21*X(140)-20*X(15714) = 23*X(140)-20*X(17538) = 7*X(140)-4*X(17800) = 19*X(140)-20*X(19709) = 5*X(140)-4*X(19710) = 11*X(140)-12*X(23046) = 18*X(140)-17*X(34200)

X(35408) lies on the line {2,3}

X(35408) = reflection of X(i) in X(j) for these (i,j): (5059, 10109), (12100, 3146), (15691, 15684)
X(35408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1984, 25912, 21), (3839, 15701, 5), (4201, 15766, 24), (6678, 17537, 24), (6805, 27532, 21), (7503, 28820, 24), (7519, 14093, 24), (10021, 25647, 21), (11099, 14789, 24), (11345, 27301, 2), (12105, 32991, 5), (14960, 16953, 24), (14960, 18859, 3), (16066, 27938, 4), (21532, 33729, 24), (25396, 25696, 21), (25482, 25911, 3), (25802, 28136, 21), (25989, 31681, 24), (27212, 31186, 2)

X(35409) = CENTER OF THE CIRCLE THROUGH PU(163) AND PU(175)

Barycentrics 65*a^4-28*(b^2+c^2)*a^2-37*(b^2-c^2)^2 : :
X(35409) = 14*X(2)-17*X(4) = 23*X(2)-17*X(20) = 20*X(2)-17*X(376) = 5*X(2)-17*X(3146) = 18*X(2)-17*X(3524) = 32*X(2)-17*X(3529) = 11*X(2)-17*X(3543) = 16*X(2)-17*X(3545) = 15*X(2)-17*X(3839) = 19*X(2)-17*X(10304) = 26*X(2)-17*X(11001) = X(2)+17*X(15640) = 8*X(2)-17*X(15682) = 29*X(2)-17*X(15683) = 7*X(2)-10*X(35434) = X(3)+5*X(35384) = 23*X(4)-14*X(20) = 10*X(4)-7*X(376) = 19*X(4)-16*X(547) = 5*X(4)-14*X(3146) = 9*X(4)-7*X(3524) = 16*X(4)-7*X(3529) = 11*X(4)-14*X(3543) = 8*X(4)-7*X(3545) = 15*X(4)-14*X(3839) = 5*X(4)-4*X(5054) = 11*X(4)-8*X(8703) = 19*X(4)-14*X(10304) = 13*X(4)-7*X(11001) = 11*X(4)+7*X(11541) = 25*X(4)-16*X(12103) = X(4)+14*X(15640) = 7*X(4)-4*X(15681) = 4*X(4)-7*X(15682) = 29*X(4)-14*X(15683) = 13*X(4)-10*X(15692) = 29*X(4)-20*X(15696) = 4*X(4)-3*X(15710) = 14*X(4)-11*X(15719) = 23*X(4)-20*X(19709) = 2*X(4)+7*X(33703) = 5*X(4)-8*X(35404) = 17*X(4)-20*X(35434)

X(35409) lies on the line {2,3}

X(35409) = reflection of X(i) in X(j) for these (i,j): (3529, 3545), (3545, 15682), (5054, 35404), (5059, 15689), (10304, 382), (15683, 14269), (15685, 23046), (15689, 33699), (17800, 17504)
X(35409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35434, 4), (440, 2676, 21), (1009, 27125, 297), (3543, 8703, 4), (5079, 17578, 4), (7537, 33264, 2), (7567, 28744, 186), (11334, 25989, 24), (14893, 15759, 5), (14958, 27883, 235), (15970, 35382, 2), (16843, 21898, 297), (19230, 26096, 186), (19337, 19548, 140), (19674, 27308, 297), (19697, 26802, 24), (21527, 33313, 186), (21937, 27505, 140), (22377, 28141, 5), (25911, 28721, 2), (30917, 33050, 2)

X(35410) = CENTER OF THE CIRCLE THROUGH PU(164) AND PU(165)

Barycentrics 349*a^4-191*(b^2+c^2)*a^2-158*(b^2-c^2)^2 : :
X(35410) = 2*X(2)-13*X(17800) = 31*X(20)-20*X(12101) = 8*X(140)-19*X(15685) = 8*X(376)-7*X(15723) = 18*X(376)-7*X(35400) = 2*X(376)-13*X(35412) = 22*X(376)-15*X(35416) = 17*X(381)-18*X(5070) = 29*X(381)-18*X(5073) = 9*X(381)-20*X(15683) = 7*X(381)-18*X(15685) = 8*X(381)-9*X(15716) = 9*X(381)-10*X(15718) = 11*X(381)-12*X(15721) = 14*X(381)-15*X(15723) = 21*X(381)-10*X(35400) = 11*X(381)-10*X(35401) = 11*X(382)-14*X(5072) = 5*X(382)-16*X(15685) = 5*X(382)-7*X(15716) = 3*X(382)-4*X(15723) = 27*X(382)-16*X(35400) = 11*X(382)-8*X(35405)

X(35410) lies on the line {2,3}

X(35410) = reflection of X(15718) in X(15683)
X(35410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (450, 6922, 29), (3138, 25749, 21), (3153, 16025, 5), (4194, 20420, 25), (6756, 14063, 24), (6945, 19248, 30), (6953, 27209, 5), (7435, 19234, 24), (7533, 28261, 23), (11345, 27301, 2), (14022, 24605, 3), (16066, 27938, 22), (16356, 28696, 2), (16400, 19711, 21), (19669, 25518, 2), (23252, 25442, 29), (24568, 25012, 2), (26643, 32960, 2), (27212, 31186, 22), (28732, 30924, 21)

X(35411) = CENTER OF THE CIRCLE THROUGH PU(164) AND PU(166)

Barycentrics 407*a^4-223*(b^2+c^2)*a^2-184*(b^2-c^2)^2 : :
X(35411) = 24*X(5)-11*X(35400) = 32*X(376)-19*X(15684) = 26*X(376)-19*X(35402) = 2*X(376)-19*X(35435) = 2*X(381)-15*X(17800) = 26*X(381)-15*X(35406) = 20*X(548)-17*X(5068) = 29*X(1657)-16*X(11812) = 31*X(1657)-18*X(15710)

X(35411) lies on the line {2,3}

X(35411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3081, 13629, 2), (4237, 20422, 29), (5084, 16296, 21), (6939, 15215, 24), (6989, 28719, 21), (7450, 25795, 2), (11345, 27301, 2), (15672, 16424, 2), (15681, 35400, 5), (16066, 27938, 4), (17513, 33191, 29), (17677, 30033, 3), (19245, 28770, 3), (19674, 34478, 21), (21497, 29779, 21), (21545, 31914, 5), (25910, 32964, 24), (26257, 30918, 5), (27212, 31186, 4), (28066, 33256, 29), (28378, 33025, 24)

X(35412) = CENTER OF THE CIRCLE THROUGH PU(164) AND PU(173)

Barycentrics 359*a^4-199*(b^2+c^2)*a^2-160*(b^2-c^2)^2 : :
X(35412) = 3*X(2)-16*X(35413) = 33*X(3)-20*X(15684) = 7*X(3)-20*X(15685) = 21*X(3)-8*X(35400) = 21*X(3)-16*X(35402) = 17*X(3)-8*X(35406) = 24*X(20)-11*X(15684) = 20*X(376)-7*X(35400) = 10*X(376)-7*X(35402) = 2*X(376)+11*X(35410) = 28*X(381)-15*X(35407) = 3*X(381)+10*X(35414) = 4*X(382)-17*X(15685) = 30*X(382)-17*X(35400) = 15*X(382)-17*X(35402) = 10*X(382)-7*X(35406) = 28*X(547)-15*X(15640) = 25*X(1657)-12*X(11539)

X(35412) lies on the line {2,3}

X(35412) = reflection of X(35400) in X(35402)
X(35412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (423, 20851, 23), (3559, 30100, 140), (6658, 27961, 5), (6865, 26058, 29), (6945, 19250, 5), (11101, 26535, 5), (11285, 33274, 28), (11345, 27301, 2), (13617, 26183, 3), (15769, 19524, 24), (16066, 27938, 4), (16349, 19290, 2), (16861, 25404, 25), (18369, 19333, 2), (24865, 29496, 23), (25674, 28702, 186), (25696, 34725, 186), (25905, 33280, 30), (26782, 35232, 29), (27212, 31186, 21), (27985, 33034, 4), (29892, 33229, 186)

X(35413) = CENTER OF THE CIRCLE THROUGH PU(164) AND PU(174)

Barycentrics 362*a^4-205*(b^2+c^2)*a^2-157*(b^2-c^2)^2 : :
X(35413) = 3*X(2)+13*X(35412) = 3*X(4)+13*X(35435) = 3*X(5)+5*X(35414) = 17*X(20)-13*X(35421) = X(140)-13*X(3529) = 17*X(140)-13*X(3830) = 9*X(140)-13*X(15686) = 19*X(376)-15*X(12812) = 7*X(381)-3*X(35408) = 7*X(382)-11*X(12100) = 8*X(382)-11*X(12811) = 19*X(547)-15*X(17578) = 19*X(548)-15*X(3545) = X(548)-5*X(15685) = 9*X(548)-5*X(35404) = 17*X(549)-15*X(3850) = 7*X(549)-15*X(11001) = 19*X(549)-15*X(12101) = 11*X(549)-15*X(12103) = 3*X(549)-11*X(15683) = 19*X(549)-11*X(15684)

X(35413) lies on the line {2,3}

X(35413) = reflection of X(11540) in X(15704)
X(35413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (413, 6873, 404), (856, 28066, 401), (3854, 5187, 404), (4188, 33249, 377), (5084, 26992, 140), (6832, 33035, 377), (7407, 16293, 384), (7472, 29928, 405), (7509, 27223, 402), (10245, 28062, 237), (11277, 33722, 402), (12056, 26057, 384), (13745, 30449, 30), (14161, 16053, 410), (14787, 17685, 24), (17532, 32981, 24), (21979, 33286, 2), (26173, 28102, 24), (27212, 31186, 235), (28384, 33824, 409)

X(35414) = CENTER OF THE CIRCLE THROUGH PU(164) AND PU(175)

Barycentrics 181*a^4-98*(b^2+c^2)*a^2-83*(b^2-c^2)^2 : :
X(35414) = 16*X(2)-11*X(3146) = 10*X(2)-11*X(3522) = 17*X(2)-16*X(3858) = X(2)-11*X(5059) = 13*X(2)-8*X(5073) = 6*X(2)-11*X(15683) = 7*X(2)-8*X(15696) = 11*X(2)-16*X(15704) = 15*X(2)-16*X(15714) = 13*X(2)-11*X(17578) = 9*X(2)-8*X(35403) = 21*X(2)-16*X(35404) = 3*X(3)-13*X(35435) = 9*X(4)-14*X(15681) = 6*X(4)-7*X(15692) = 11*X(4)-14*X(15696) = 13*X(4)-14*X(19709) = 15*X(4)-14*X(35434) = 3*X(5)-8*X(35413) = 21*X(20)-16*X(547) = 23*X(20)-17*X(3091) = 32*X(20)-17*X(3146) = 20*X(20)-17*X(3522) = 27*X(20)-17*X(3543) = 11*X(20)-8*X(3858) = 2*X(20)-17*X(5059) = 17*X(20)-12*X(14269) = 37*X(20)-17*X(15640) = 12*X(20)-17*X(15683) = 21*X(20)-17*X(15692) = 5*X(20)-4*X(15693) = 19*X(20)-17*X(15697) = 23*X(20)-18*X(15709) = 26*X(20)-17*X(17578) = 13*X(20)-8*X(33699)

X(35414) lies on the line {2,3}

X(35414) = reflection of X(i) in X(j) for these (i,j): (631, 15685), (15640, 17538), (15697, 3529), (33703, 15695)
X(35414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (860, 25450, 24), (2455, 9818, 25), (2479, 28101, 2), (3543, 15691, 2), (7887, 28031, 24), (10151, 21490, 25), (11322, 16868, 5), (11345, 27301, 2), (11634, 33740, 5), (14008, 24939, 24), (14014, 25865, 25), (14142, 24904, 5), (15013, 24951, 2), (15184, 26204, 2), (15766, 19282, 24), (16043, 33747, 2), (16066, 27938, 4), (16358, 28818, 21), (16902, 26255, 24), (18572, 21495, 25), (24939, 31905, 2), (25877, 32962, 2), (27212, 31186, 2)

X(35415) = CENTER OF THE CIRCLE THROUGH PU(165) AND PU(166)

Barycentrics 383*a^4-263*(b^2+c^2)*a^2-120*(b^2-c^2)^2 : :
X(35415) = 33*X(3)-20*X(5072) = 31*X(3)-20*X(5079) = 22*X(3)-9*X(35401) = 13*X(3)-2*X(35406) = 18*X(376)-7*X(5079)

X(35415) lies on the line {2,3}

X(35415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (408, 33207, 1344), (4246, 19231, 3363), (7426, 8964, 3517), (7576, 11359, 438), (9909, 19247, 379), (11818, 33831, 2676), (12101, 16426, 381), (12105, 20478, 30), (13728, 28941, 445), (15722, 31014, 415), (16415, 18562, 1564), (16863, 32979, 3147), (20420, 28035, 1012), (20845, 21487, 868), (21511, 27023, 632), (25544, 35306, 2675), (25798, 29411, 438), (26170, 26662, 3134), (27120, 35417, 381), (33029, 34608, 1556)

X(35416) = CENTER OF THE CIRCLE THROUGH PU(165) AND PU(173)

Barycentrics 137*a^4-41*(b^2+c^2)*a^2-96*(b^2-c^2)^2 : :
X(35416) = 11*X(2)-16*X(35417) = 11*X(3)-16*X(5072) = 5*X(3)-16*X(5076) = 7*X(3)-18*X(35403) = 7*X(3)+4*X(35407) = 22*X(376)-7*X(35410) = 11*X(381)-6*X(35418) = 9*X(382)+2*X(3091) = 13*X(382)+2*X(15717) = 17*X(382)+3*X(15723) = 8*X(382)+3*X(35403) = 11*X(382)-X(35405) = 12*X(382)-X(35407) = 24*X(382)-13*X(35419) = 20*X(546)-9*X(15697) = 8*X(550)-19*X(3843) = 8*X(550)-13*X(5070) = 14*X(550)-19*X(15717) = 2*X(550)-13*X(17578) = 4*X(631)-15*X(3830) = 6*X(631)-17*X(5076) = 21*X(631)-10*X(15704) = 14*X(631)-15*X(15723)

X(35416) lies on the line {2,3}

X(35416) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (472, 28442, 1003), (4223, 21534, 3078), (6829, 25991, 1584), (6878, 15717, 1006), (6917, 14065, 1586), (6979, 33307, 436), (7401, 25450, 410), (11316, 17547, 474), (11317, 15247, 852), (12084, 28382, 21), (13726, 30952, 861), (14106, 24904, 631), (16848, 21903, 1010), (16860, 19250, 2073), (17517, 21032, 2676), (17575, 28030, 2676), (17585, 28943, 1011), (21980, 26024, 549), (28105, 32978, 1344), (35403, 35407, 3)

X(35417) = CENTER OF THE CIRCLE THROUGH PU(165) AND PU(174)

Barycentrics 290*a^4+119*(b^2+c^2)*a^2-409*(b^2-c^2)^2 : :
X(35417) = 11*X(2)+5*X(35416) = 11*X(4)+5*X(35381) = 33*X(4)-17*X(35401) = 13*X(5)-5*X(15716) = 11*X(5)-3*X(35418) = 14*X(381)-3*X(11540) = 19*X(381)-3*X(15717) = 13*X(546)-X(5056) = 11*X(547)-7*X(15721) = 11*X(547)+9*X(35401)

X(35417) lies on the line {2,3}

X(35418) = CENTER OF THE CIRCLE THROUGH PU(165) AND PU(175)

Barycentrics 145*a^4-122*(b^2+c^2)*a^2-23*(b^2-c^2)^2 : :
X(35418) = 25*X(2)-14*X(3839) = 17*X(2)-14*X(5056) = 11*X(2)-8*X(5072) = 3*X(2)-14*X(10304) = 3*X(2)+8*X(15689) = 4*X(2)-7*X(15717) = 5*X(2)-8*X(15718) = 11*X(2)-14*X(15721) = X(2)-4*X(21735) = 27*X(2)-16*X(23046) = 11*X(3)-5*X(35381) = 4*X(5)-15*X(15710) = 2*X(5)-5*X(15716) = 11*X(5)-8*X(35417) = X(20)+2*X(15719) = 3*X(20)+8*X(17504) = 5*X(376)+X(3525) = 10*X(376)+X(3839) = 9*X(376)+2*X(5054) = 6*X(376)+5*X(10304) = X(376)+10*X(15688) = 21*X(376)-10*X(15689) = 8*X(376)+3*X(15705) = 13*X(376)+5*X(15715) = 7*X(376)+2*X(15718) = 7*X(376)+5*X(21735)

X(35418) lies on the line {2,3}

X(35418) = reflection of X(i) in X(j) for these (i,j): (3839, 3525), (5056, 3524)

X(35419) = CENTER OF THE CIRCLE THROUGH PU(166) AND PU(173)

Barycentrics 223*a^4-79*(b^2+c^2)*a^2-144*(b^2-c^2)^2 : :
X(35419) = 13*X(2)-18*X(35420) = 7*X(3)+6*X(35407) = 6*X(20)-19*X(5076) = 6*X(20)-11*X(5079) = 13*X(381)-8*X(35421) = 15*X(382)-2*X(3091) = 17*X(382)-2*X(5067) = 13*X(382)-3*X(35402) = 16*X(382)-3*X(35403) = 12*X(382)+X(35407) = 24*X(382)-11*X(35416) = 7*X(1656)+6*X(15640)

X(35419) lies on the line {2,3}

X(35419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (28, 17560, 436), (410, 27119, 443), (466, 7397, 432), (3148, 21561, 297), (3568, 16956, 434), (3628, 5020, 447), (5077, 8021, 401), (6175, 17692, 464), (6871, 13726, 441), (15672, 33739, 409), (16856, 21504, 443), (19698, 24903, 447), (21494, 27656, 439), (21567, 29938, 140), (25014, 27125, 442), (26004, 27122, 5), (26124, 32491, 442), (26622, 32490, 186), (27087, 32953, 468), (33059, 33239, 404)

X(35420) = CENTER OF THE CIRCLE THROUGH PU(166) AND PU(174)

Barycentrics 500*a^4+101*(b^2+c^2)*a^2-601*(b^2-c^2)^2 : :
X(35420) = 13*X(2)+5*X(35419) = 13*X(4)+5*X(35382) = 23*X(5)-10*X(15705) = 13*X(5)-4*X(35421)

X(35420) lies on the line {2,3}

X(35421) = CENTER OF THE CIRCLE THROUGH PU(166) AND PU(175)

Barycentrics 250*a^4-203*(b^2+c^2)*a^2-47*(b^2-c^2)^2 : :
X(35421) = 22*X(2)-9*X(3853) = 24*X(2)-11*X(12101) = 2*X(2)+11*X(15690) = 4*X(2)+9*X(15691) = 5*X(2)-18*X(33923) = 13*X(3)-5*X(35382) = 13*X(5)-9*X(35420) = 17*X(20)-4*X(35413) = 3*X(140)+10*X(15697) = 8*X(376)+5*X(548) = 12*X(376)+X(12100) = 14*X(376)-X(12103) = 18*X(376)-5*X(15690) = 7*X(376)+X(21734) = 11*X(376)+2*X(33923) = 13*X(381)-5*X(35419) = 9*X(546)+4*X(15685) = 7*X(546)-20*X(15692) = X(547)-14*X(548) = 17*X(547)-16*X(5079) = 3*X(547)-16*X(8703) = 27*X(547)-14*X(12101) = 5*X(547)+8*X(12103) = 5*X(547)-16*X(21734)

X(35421) lies on the line {2,3}

X(35421) = midpoint of X(10303) and X(15686)
X(35421) = reflection of X(5067) in X(14891)

X(35422) = CENTER OF THE CIRCLE THROUGH PU(182) AND PU(183)

Barycentrics a^2*(a^8+(b^2+c^2)*a^6-(b^4+3*b^2*c^2+c^4)*a^4-3*(b^2+c^2)*b^2*c^2*a^2-b^8-c^8-2*b^2*c^2*(b^2+c^2)^2) : :
X(35422) = 3*X(3398)-X(35427) = X(12212)+2*X(14810) = 3*X(35426)-2*X(35427)

X(35422) lies on these lines: {3,6}, {98,22498}, {114,7931}, {141,5149}, {384,29012}, {5999,19130}, {6660,34236}, {7470,29317}, {7897,9744}, {9751,10352}, {10292,24206}, {12215,34885}, {18553,24273}, {33014,33750}

X(35422) = reflection of X(35426) in X(3398)
X(35422) = X(35422)-of-circumsymmedial triangle
X(35422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 2076, 14810), (3, 35424, 3098), (182, 35376, 3398), (1692, 20190, 182), (5092, 13354, 182)

X(35423) = CENTER OF THE CIRCLE THROUGH PU(182) AND PU(191)

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

X(35423) lies on these lines: {3,6}, {114,3763}, {141,9744}, {384,25406}, {1503,7770}, {3589,13860}, {3618,5999}, {5480,7803}, {7470,14853}, {8177,9755}, {8925,31521}, {12215,32830}

X(35423) = X(35423)-of-circumsymmedial triangle
X(35423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (182, 13354, 6), (216, 9737, 5096), (575, 35389, 6), (5092, 35424, 3)

X(35424) = CENTER OF THE CIRCLE THROUGH PU(182) AND PU(192)

Barycentrics a^2*(a^8-2*(b^4+3*b^2*c^2+c^4)*a^4+2*(b^6+c^6)*a^2-(b^4+c^4)^2) : :
Trilinears cos(A + ω) (sin A sin(A - ω) + sin B sin(B - ω) + sin C sin(C - ω)) + sin(A - ω) (sin A cos(A - ω) + sin B cos(B - ω) + sin C cos(C - ω)) : :
X(35424) = 3*X(32)-X(35389) = 3*X(9753)-X(31670) = 3*X(10519)+X(20065) = 3*X(35387)+X(35389) = 2*X(35387)+X(35431) = 2*X(35389)-3*X(35431)

X(35424) lies on these lines {3,6}, {384,1352}, {542,1003}, {2794,3818}, {2882,6644}, {3425,17970}, {3552,6776}, {3564,4048}, {3852,15577}, {5480,20576}, {5989,9755}, {5999,7806}, {6036,13860}, {7770,24206}, {7787,10998}, {10104,24256}, {10323,14133}, {10519,20065}, {11178,11286}, {11179,13586}, {21445,31958}

X(35424) = midpoint of X(i) and X(j) for these {i,j}: {3, 5017}, {32, 35387}
X(35424) = reflection of X(i) in X(j) for these (i,j): (182, 13335), (5028, 575), (5480, 20576), (13355, 5092), (30270, 14810), (35377, 16385), (35431, 32)
X(35424) = X(35424)-of-circumsymmedial triangle
X(35424) = center of circle {{X(3),X(5017),PU(39)}}
X(35424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5050, 5116), (3, 35423, 5092), (32, 35374, 5017), (32, 35385, 3), (182, 35375, 3), (3098, 35422, 3), (3398, 35383, 3094), (13343, 13344, 39), (13349, 13350, 8722), (35374, 35385, 35387)

X(35425) = CENTER OF THE CIRCLE THROUGH PU(183) AND PU(189)

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

X(35425) lies on the line {3,6}

X(35425) = X(35425)-of-circumsymmedial triangle
X(35425) = {X(12049), X(14075)}-harmonic conjugate of X(13357)

X(35426) = CENTER OF THE CIRCLE THROUGH PU(183) AND PU(190)

Barycentrics a^2*(a^8-(b^2+c^2)*a^6-(b^4+7*b^2*c^2+c^4)*a^4-5*(b^2+c^2)*b^2*c^2*a^2+b^8+c^8) : :
X(35426) = X(576)+2*X(5007) = 2*X(5097)+X(12212) = X(7768)-4*X(25555) = X(35422)+2*X(35427)

X(35426) lies on these lines: {3,6}, {51,3506}, {385,24206}, {698,32134}, {1352,7766}, {7759,8177}, {7768,25555}, {11178,14614}

X(35426) = midpoint of X(3398) and X(35427)
X(35426) = reflection of X(35422) in X(3398)
X(35426) = X(35426)-of-circumsymmedial triangle
X(35426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 35431, 182), (3398, 35428, 35376), (5007, 5052, 12212), (5052, 5097, 576)

X(35427) = CENTER OF THE CIRCLE THROUGH PU(183) AND PU(191)

Barycentrics a^2*(a^8-2*(b^2+c^2)*a^6-(b^4+9*b^2*c^2+c^4)*a^4-6*(b^2+c^2)*b^2*c^2*a^2+2*b^8+2*c^8+b^2*c^2*(b^2+c^2)^2) : :
X(35427) = 2*X(1351)+X(12212) = 3*X(3398)-2*X(35422) = 2*X(5007)+X(11477) = X(35422)-3*X(35426)

X(35427) lies on these lines: {3,6}, {262,8177}, {698,12110}, {1503,7760}, {3629,10753}, {5480,7762}, {7751,10516}

X(35427) = reflection of X(3398) in X(35426)
X(35427) = X(35427)-of-circumsymmedial triangle
X(35427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (576, 1351, 5111), (576, 35389, 6), (3095, 35431, 1691), (5097, 13354, 6)

X(35428) = CENTER OF THE CIRCLE THROUGH PU(183) AND PU(192)

Barycentrics (SB+SC)*(S^6-(3*SA-2*SW)*SW*S^4+(6*SA-7*SW)*SW^3*S^2+SA*SW^5) : :

X(35428) lies on the line {3,6}

X(35428) = X(35428)-of-circumsymmedial triangle
X(35428) = X(35431)-of-5th anti-Brocard triangle
X(35428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 35386, 3398), (35376, 35426, 3398)

X(35429) = CENTER OF THE CIRCLE THROUGH PU(189) AND PU(190)

Barycentrics a^2*(a^8-6*(b^2+c^2)*a^6+2*(3*b^4+b^2*c^2+3*c^4)*a^4-2*((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2)*a^2+b^8+c^8-2*b^2*c^2*(b^4-5*b^2*c^2+c^4)) : :
X(35429) = 2*X(574)-5*X(12017) = X(1351)+2*X(8722) = 2*X(5034)-3*X(5050)

X(35429) lies on these lines: {3,6}, {114,15271}, {183,3564}, {385,14912}, {1993,34095}, {3060,30535}, {5422,7467}, {6776,32828}, {7766,33748}, {7803,18583}, {8177,8550}, {14561,15980}

X(35429) = X(35429)-of-circumsymmedial triangle
X(35429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5050, 2456), (182, 1692, 5050), (182, 13354, 3), (575, 35431, 6), (5092, 35387, 3)

X(35430) = CENTER OF THE CIRCLE THROUGH PU(189) AND PU(192)

Barycentrics (SB+SC)*((SA-2*SW)*S^4+2*SA*SW^2*S^2-SA*SW^4) : :
X(35430) = X(315)-3*X(22712) = 2*X(626)-3*X(15819) = 3*X(6194)+X(20065)

X(35430) lies on these lines: {3,6}, {315,22712}, {626,15819}, {2794,6248}, {6194,20065}, {7828,9753}, {14881,20576}

X(35430) = reflection of X(i) in X(j) for these (i,j): (39, 13335), (3095, 13357), (6248, 18806), (14881, 20576), (32452, 13334)
X(35430) = X(35430)-of-circumsymmedial triangle
X(35430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 35385, 13335), (32, 35387, 3), (4274, 6441, 22425), (5188, 13354, 3)

X(35431) = CENTER OF THE CIRCLE THROUGH PU(190) AND PU(192)

Barycentrics a^2*(a^8-2*(b^2+c^2)*a^6-6*b^2*c^2*a^4-2*(b^2+c^2)*b^2*c^2*a^2+(b^4+c^4)^2) : :
X(35431) = 3*X(32)-X(35387) = X(315)-3*X(14561) = X(1352)-3*X(9753) = 3*X(14853)+X(20065) = X(35387)+3*X(35389) = 2*X(35387)-3*X(35424) = 2*X(35389)+X(35424)

X(35431) lies on these lines: {3,6}, {5,8177}, {25,3506}, {141,20576}, {183,24206}, {315,14561}, {385,1352}, {542,14614}, {754,5476}, {1976,3060}, {3852,34117}, {4048,32515}, {6776,7766}, {8667,11178}, {10788,18906}, {10796,24256}, {12177,32451}, {14853,20065}, {22521,31958}

X(35431) = midpoint of X(i) and X(j) for these {i,j}: {32, 35389}, {1351, 5017}
X(35431) = reflection of X(i) in X(j) for these (i,j): (141, 20576), (3098, 13335), (5028, 5097), (13355, 575), (30270, 5092), (35375, 16385), (35424, 32)
X(35431) = X(35428)-of-5th Brocard triangle
X(35431) = X(35431)-of-circumsymmedial triangle
X(35431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 35429, 575), (32, 35388, 6), (182, 35426, 6), (576, 35377, 6), (1691, 35427, 3095), (35386, 35388, 35389)

X(35432) = CENTER OF THE CIRCLE THROUGH PU(191) AND PU(192)

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

X(35432) lies on these lines: {3,6}, {193,9865}, {263,8569}, {7745,24256}, {14994,18806}

X(35432) = midpoint of X(5017) and X(13330)
X(35432) = reflection of X(i) in X(j) for these (i,j): (3094, 13357), (14994, 18806)
X(35432) = X(35432)-of-circumsymmedial triangle
X(35432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 35389, 6), (5052, 13354, 6)

X(35433) = CENTER OF THE CIRCLE THROUGH PU(2) AND PU(188)

Barycentrics (SB+SC)*(2*S^6+(13*SA+11*SW)*SW*S^4+2*(SA-4*SW)*SW^3*S^2+(5*SA-SW)*SW^5) : :

X(35433) lies on the line {3,6}

X(35433) = X(35433)-of-circumsymmedial triangle

X(35434) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(162)

Barycentrics 29*a^4-7*(b^2+c^2)*a^2-22*(b^2-c^2)^2 : :
X(35434) = 22*X(2)-17*X(3) = 7*X(2)-17*X(4) = 37*X(2)-17*X(20) = 27*X(2)-17*X(376) = 12*X(2)-17*X(381) = 19*X(2)-17*X(631) = 16*X(2)-17*X(1656) = 13*X(2)-17*X(3091) = 25*X(2)-17*X(3522) = 32*X(2)-17*X(3534) = 3*X(2)+17*X(3543) = 2*X(2)-17*X(3830) = 10*X(2)-17*X(3843) = 11*X(2)-16*X(3856) = 15*X(2)-17*X(5071) = 4*X(2)-17*X(5076) = 24*X(2)-17*X(14093) = 21*X(2)-17*X(15692) = 20*X(2)-17*X(15693) = 18*X(2)-17*X(15694) = 26*X(2)-17*X(15695) = 28*X(2)-17*X(15696) = 29*X(2)-17*X(15697) = 31*X(2)-17*X(17538) = X(2)-17*X(17578) = 23*X(2)-17*X(19708) = 14*X(2)-17*X(19709) = 6*X(2)-17*X(35403)

X(35434) lies on these lines: {2,3}, {6498,22615}, {6499,22644}, {9812,34748}, {11648,34571}, {28232,34648}

X(35434) = midpoint of X(i) and X(j) for these {i,j}: {1657, 35384}, {3091, 15682}, {3146, 19708}, {15684, 15694}
X(35434) = reflection of X(i) in X(j) for these (i,j): (20, 15713), (381, 35403), (631, 3845), (1657, 15695), (3534, 1656), (3830, 17578), (3858, 12101), (5076, 3830), (8703, 3859), (14093, 381), (15681, 15692), (15685, 17538), (15693, 3843), (15695, 3091), (15696, 19709), (15697, 5), (15711, 546), (19708, 3858), (19709, 4)
X(35434) = anticomplement-of X(15694) with respect to these triangles: {Euler, Johnson, X3-ABC reflections}
X(35434) = X(15697)-of-Johnson triangle
X(35434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 14893, 381), (4, 15681, 381), (381, 3543, 382), (547, 15687, 4), (549, 14269, 381), (3543, 3830, 381), (3843, 5071, 381), (3845, 15703, 381), (5076, 15696, 4), (6813, 26151, 427), (7539, 26642, 384), (15681, 35401, 4), (15684, 15687, 381), (17678, 33266, 384), (18568, 35297, 403), (21485, 28097, 425), (27123, 31156, 413), (27537, 33211, 406), (28431, 33182, 405), (35401, 35404, 381)

X(35435) = CENTER OF THE CIRCLE THROUGH PU(5) AND PU(164)

Barycentrics 371*a^4-205*(b^2+c^2)*a^2-166*(b^2-c^2)^2 : :
X(35435) = 3*X(3)+10*X(35414) = 3*X(4)-16*X(35413) = 11*X(376)-9*X(5068) = 30*X(376)-17*X(15684) = 24*X(376)-17*X(35402) = 2*X(376)+17*X(35411) = 28*X(381)-15*X(35384) = 9*X(381)-8*X(35402) = 24*X(546)-11*X(35400) = 28*X(549)-15*X(5073) = 16*X(549)-15*X(5079) = 23*X(549)-10*X(35408) = 21*X(1657)-8*X(3543) = 25*X(1657)-12*X(5055) = 17*X(1657)-8*X(5068) = 23*X(1657)-11*X(5079) = 29*X(1657)-16*X(8703) = 27*X(1657)-14*X(15700) = 21*X(1657)-10*X(35382) = 27*X(1657)-11*X(35402)

X(35435) lies on the line {2,3}

X(35435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (377, 13615, 2), (446, 19692, 20), (3148, 16452, 2), (6932, 21984, 3), (7453, 23410, 30), (11306, 29725, 24), (11345, 27301, 4), (12108, 27051, 21), (13413, 19518, 140), (13734, 26624, 26), (14892, 15775, 30), (15713, 28262, 24), (16066, 27938, 3), (19265, 19323, 2), (19324, 24538, 2), (20405, 27330, 186), (25742, 31220, 21), (27212, 31186, 2), (27650, 32964, 25), (27656, 30776, 24), (28721, 33312, 186)

X(35436) = CENTER OF THE CIRCLE THROUGH PU(39) AND PU(188)

Barycentrics a^2*((b^2+c^2)*a^8-4*(b^2+c^2)^2*a^6+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4-4*b^4*c^4*a^2+(b^2+c^2)*(b^4+c^4)^2) : :
X(35436) = X(76)-3*X(9753) = 3*X(262)-X(315) = 4*X(6680)-3*X(15819)

X(35436) lies on these lines: {3,6}, {76,9753}, {262,315}, {736,6248}, {2967,27369}, {6680,15819}, {7819,20576}, {7843,32189}, {8362,11272}, {9983,13862}, {12251,14001}, {13085,20423}

X(35436) = reflection of X(i) in X(j) for these (i,j): (3, 13357), (5188, 13335), (30270, 13334), (35430, 32), (35432, 35431)
X(35436) = Gallatly circle-inverse of X(2024)
X(35436) = X(13357)-of X3-ABC reflections triangle
X(35436) = X(35436)-of-circumsymmedial triangle
X(35436) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32, 30270, 35424), (39, 35438, 11171), (39, 35439, 3095), (39, 35440, 32), (2026, 2027, 2024), (3095, 11171, 10983)

X(35437) = CENTER OF THE CIRCLE THROUGH PU(183) AND PU(188)

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

X(35437) lies on these lines: {3,6}, {262,7759}, {6292,11272}, {8149,9753}, {14853,31981}

X(35437) = reflection of X(35385) in X(13357)
X(35437) = X(35437)-of-circumsymmedial triangle
X(35437) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3398, 35426, 5007), (3398, 35427, 35386)

X(35438) = CENTER OF THE CIRCLE THROUGH PU(188) AND PU(189)

Barycentrics a^2*((b^2+c^2)*a^8-12*(b^2+c^2)^2*a^6+2*(b^2+c^2)*(7*b^4+2*b^2*c^2+7*c^4)*a^4-4*(b^8+c^8-4*b^2*c^2*(b^2+c^2)^2)*a^2+(b^2+c^2)*(b^8+c^8-2*b^2*c^2*(2*b^4-7*b^2*c^2+2*c^4))) : :
X(35438) = X(10983)-3*X(11171)

X(35438) lies on these lines: {3,6}, {6337,22712}, {7789,15819}

X(35438) = reflection of X(5013) in X(13334)
X(35438) = X(35438)-of-circumsymmedial triangle
X(35438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35429, 5171), (8409, 12232, 2037), (11171, 35436, 39)

X(35439) = CENTER OF THE CIRCLE THROUGH PU(188) AND PU(191)

Barycentrics a^2*((b^2+c^2)*a^6-5*(b^2+c^2)^2*a^4+(b^2-3*c^2)*(3*b^2-c^2)*(b^2+c^2)*a^2+b^8+c^8+2*b^2*c^2*(2*b^4-b^2*c^2+2*c^4)) : :
X(35439) = X(69)-3*X(262) = X(76)-3*X(14853) = X(1350)-3*X(13331) = 2*X(3098)-3*X(21163) = 4*X(3589)-3*X(15819) = 3*X(5050)-X(9821) = 3*X(5102)-X(13330) = 3*X(11171)-X(33878) = 3*X(13331)-2*X(13334)

X(35439) lies on these lines: {3,6}, {4,32451}, {5,14994}, {69,262}, {76,14853}, {538,20423}, {732,5480}, {1352,7759}, {1843,2967}, {1916,10753}, {2782,21850}, {3564,14881}, {3589,15819}, {3618,22712}, {3818,22682}, {3934,14561}, {5476,9466}, {7786,10519}, {7821,24206}, {10759,32454}, {12110,12215}, {12251,16045}, {14039,22486}

X(35439) = midpoint of X(i) and X(j) for these {i,j}: {4, 32451}, {1351, 3095}, {1916, 10753}, {3094, 11477}, {10759, 32454}
X(35439) = reflection of X(i) in X(j) for these (i,j): (1350, 13334), (5052, 576), (5188, 182), (6248, 5480), (9466, 5476), (13354, 6), (14994, 5), (35387, 13357)
X(35439) = X(14994)-of-Johnson triangle
X(35439) = X(32451)-of-Euler triangle
X(35439) = X(35439)-of-circumsymmedial triangle
X(35439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1351, 35389), (1350, 13331, 13334), (3094, 13331, 22332), (3095, 35436, 39), (5118, 18466, 568)

X(35440) = CENTER OF THE CIRCLE THROUGH PU(188) AND PU(192)

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

X(35440) lies on the line {3,6}

X(35440) = reflection of X(574) in X(13357)
X(35440) = {X(32), X(35436)}-harmonic conjugate of X(39)

X(35441) = X(323)X(401)∩X(523)X(14460)

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

See César Lozada, Euclid 371 .

X(35441) lies on these lines: {323, 401}, {523, 14460}, {2081, 6368}

X(35441) = complement of the isotomic conjugate of X(35311)
X(35441) = crossdifference of every pair of points on line {X(51), X(54)}
X(35441) = crosspoint of X(i) and X(j) for these (i,j): (2, 35311), (233, 35318), (525, 6368)
X(35441) = crosssum of X(i) and X(j) for these (i,j): (112, 933), (647, 1199)
X(35441) = X(i)-isoconjugate-of X(j) for these {i,j}: {162, 288}, {288, 162}, {823, 20574}
X(35441) = X(i)-reciprocal conjugate of X(j) for these (i,j): (5, 33513), (140, 18831), (233, 648)
X(35441) = intersection, other than A,B,C, of conics {{A, B, C, X(323), X(22052)}} and {{A, B, C, X(401), X(3078)}}
X(35441) = center of the circumconic {{ A, B, C, X(525), X(6368), X(18314) }}
X(35441) = barycentric product X(i)*X(j) for these {i, j}: {140, 6368}, {233, 525}, {520, 14978}, {850, 32078}
X(35441) = barycentric quotient X(i)/X(j) for these (i, j): (5, 33513), (140, 18831), (233, 648), (525, 31617)
X(35441) = trilinear product X(i)*X(j) for these {i, j}: {233, 656}, {822, 14978}, {1577, 32078}
X(35441) = trilinear quotient X(i)/X(j) for these (i, j): (233, 162), (656, 288), (822, 20574)

X(35442) = COMPLEMENT OF X(35311)

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

See César Lozada, Euclid 371 .

X(35442) lies on these lines: {2, 35311}, {3, 2888}, {4, 19774}, {5, 35360}, {74, 23240}, {122, 125}, {127, 5522}, {135, 14672}, {137, 20625}, {216, 35319}, {324, 23607}, {339, 868}, {343, 418}, {417, 12359}, {852, 3580}, {1368, 31127}, {3078, 11197}, {3519, 19210}, {5064, 12384}, {5984, 7494}, {7897, 16051}, {8884, 15319}, {10600, 31388}, {10745, 14644}, {12041, 12113}, {13157, 14569}, {13409, 21243}, {14059, 26917}, {14918, 32428}, {14919, 15059}, {18403, 21316}, {18870, 34186}, {20208, 26869}

X(35442) = complement of X(35311)
X(35442) = crossdifference of every pair of points on line {X(112), X(933)}
X(35442) = crosspoint of X(i) and X(j) for these (i,j): (5, 6368), (95, 525), (324, 850)
X(35442) = crosssum of X(i) and X(j) for these (i,j): (51, 112), (54, 933), (107, 8884)
X(35442) = X(i)-Ceva conjugate of X(j) for these (i,j): (5, 6368), (95, 525), (343, 17434)
X(35442) = X(i)-isoconjugate-of X(j) for these {i,j}: {54, 24000}, {158, 14587}, {162, 933}
X(35442) = X(i)-reciprocal conjugate of X(j) for these (i,j): (5, 23582), (51, 23964), (53, 32230)
X(35442) = intersection, other than A,B,C, of conics {{A, B, C, X(5), X(1650)}} and {{A, B, C, X(125), X(33565)}}
X(35442) = center of the circumconic {{ A, B, C, X(5), X(95), X(140), X(233), X(3519) }}
X(35442) = pole of the trilinear polar of X(15319) wrt Kiepert hyperbola
X(35442) = barycentric product X(i)*X(j) for these {i, j}: {5, 15526}, {122, 13157}, {125, 343}, {216, 339}
X(35442) = barycentric quotient X(i)/X(j) for these (i, j): (5, 23582), (51, 23964), (53, 32230), (115, 8884)
X(35442) = trilinear product X(i)*X(j) for these {i, j}: {5, 2632}, {51, 17879}, {216, 20902}, {343, 3708}
X(35442) = trilinear quotient X(i)/X(j) for these (i, j): (5, 24000), (125, 2190), (255, 14587), (311, 23999)
X(35442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (125, 2972, 1650), (125, 15526, 2972), (11197, 34836, 3078)

X(35443) = X(2)X(20579)∩X(395)X(523)

Barycentrics (2*S+(-a^2+b^2+c^2)*sqrt(3))*(sqrt(3)*a^2+2*S)*(b^2-c^2) : :
X(35443) = 3*X(9200)-2*X(20578)

See César Lozada, Euclid 371 .

X(35443) lies on these lines: {2, 20579}, {395, 523}, {826, 1640}, {1649, 9201}, {3258, 15609}, {3268, 6137}, {5027, 13305}, {8562, 23284}, {9115, 13304}, {11126, 23871}

X(35443) = reflection of X(14447) in X(6137)
X(35443) = complement of the isogonal conjugate of X(35329)
X(35443) = complement of the isotomic conjugate of X(35314)
X(35443) = crossdifference of every pair of points on line {X(15), X(1337)}
X(35443) = crosspoint of X(i) and X(j) for these (i,j): (2, 35314), (523, 23870)
X(35443) = crosssum of X(110) and X(5995)
X(35443) = X(2)-Ceva conjugate of X(30465)
X(35443) = X(i)-complementary conjugate of X(j) for these (i,j): (31, 30465), (396, 21253)
X(35443) = X(i)-isoconjugate-of X(j) for these {i,j}: {163, 11119}, {662, 16459}
X(35443) = X(i)-reciprocal conjugate of X(j) for these (i,j): (15, 10409), (396, 23895), (512, 16459)
X(35443) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(14922)}} and {{A, B, C, X(396), X(11537)}}
X(35443) = center of the circumconic {{ A, B, C, X(523), X(23284), X(23870), X(23896)}}
X(35443) = barycentric product X(i)*X(j) for these {i, j}: {396, 23870}, {523, 618}, {532, 23284}
X(35443) = barycentric quotient X(i)/X(j) for these (i, j): (15, 10409), (396, 23895), (512, 16459), (523, 11119)
X(35443) = trilinear product X(618)*X(661)
X(35443) = trilinear quotient X(i)/X(j) for these (i, j): (618, 662), (661, 16459), (1577, 11119)
X(35443) = {X(1640),X(11123)}-harmonic conjugate of X(35444)

X(35444) = X(2)X(20578)∩X(396)X(523)

Barycentrics (-2*S+(-a^2+b^2+c^2)*sqrt(3))*(sqrt(3)*a^2-2*S)*(b^2-c^2) : :
X(35444) = 3*X(9201)-2*X(20579)

See César Lozada, Euclid 371 .

X(35444) lies on these lines: {2, 20578}, {396, 523}, {826, 1640}, {1649, 9200}, {3258, 15610}, {3268, 6138}, {5027, 13304}, {8562, 23283}, {9117, 13305}, {11127, 23870}

X(35444) = reflection of X(14446) in X(6138)
X(35444) = complement of the isogonal conjugate of X(35330)
X(35444) = complement of the isotomic conjugate of X(35315)
X(35444) = crossdifference of every pair of points on line {X(16), X(1338)}
X(35444) = crosspoint of X(i) and X(j) for these (i,j): (2, 35315), (523, 23871)
X(35444) = crosssum of X(110) and X(5994)
X(35444) = X(2)-Ceva conjugate of X(30468)
X(35444) = X(i)-complementary conjugate of X(j) for these (i,j): (31, 30468), (395, 21253)
X(35444) = X(i)-isoconjugate-of X(j) for these {i,j}: {163, 11120}, {662, 16460}, {2154, 10410}
X(35444) = X(i)-reciprocal conjugate of X(j) for these (i,j): (16, 10410), (395, 23896), (512, 16460)
X(35444) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(14921)}} and {{A, B, C, X(395), X(11549)}}
X(35444) = center of the circumconic {{ A, B, C, X(523), X(23283), X(23871), X(23895)}}
X(35444) = barycentric product X(i)*X(j) for these {i, j}: {395, 23871}, {523, 619}, {533, 23283}
X(35444) = barycentric quotient X(i)/X(j) for these (i, j): (16, 10410), (395, 23896), (512, 16460), (523, 11120)
X(35444) = trilinear product X(619)*X(661)
X(35444) = trilinear quotient X(i)/X(j) for these (i, j): (619, 662), (661, 16460), (1577, 11120)
X(35444) = {X(1640),X(11123)}-harmonic conjugate of X(35443)

X(35445) = X(1)X(3)∩X(9)X(100)

Barycentrics a*(5*a^2 - 4*a*b - b^2 - 4*a*c + 2*b*c - c^2) : :
X(35445) = 2 X[1] - 3 X[13384], 3 X[354] - 5 X[17603]

Starting with P = X(5537), let A'B'C' be the circumcevian-inversion triangle defined in the preamble just before X(34864). Then X(35445) is the perspector of A'B'C' and the excentral triangle.

Let B_c be the reflection of B in the internal angle bisector of angle C, and let C_b be the reflection of C in the internal angle bisector of angle B. Let Γ_a be the circle that passes through A, B_c, C_b. Let A' be the pole of BC, with respect to Γ_a and define B 'and C' cyclically. Then X(1156) is the perspector of A'B'C' and ABC, and X(35445) is the perspector of A'B'C' and the excentral triangle. (Angel Montesdeoca, January 13, 2021)

X(35445) lies on the cubic K716 and these lines: {1, 3}, {2, 9580}, {8, 34701}, {9, 100}, {10, 11111}, {20, 9578}, {21, 1706}, {30, 31434}, {32, 31426}, {45, 910}, {63, 3158}, {84, 11491}, {104, 7966}, {105, 6014}, {109, 1253}, {140, 9614}, {187, 31433}, {200, 3929}, {210, 5696}, {220, 15855}, {226, 5281}, {244, 35227}, {376, 31397}, {388, 12512}, {390, 3911}, {452, 9780}, {497, 10164}, {516, 5218}, {528, 5231}, {548, 31436}, {550, 9613}, {553, 10578}, {631, 10624}, {672, 3240}, {759, 33538}, {846, 7322}, {899, 15601}, {902, 7290}, {968, 25430}, {1152, 31432}, {1266, 26245}, {1293, 15728}, {1317, 8275}, {1335, 9582}, {1376, 4512}, {1449, 17126}, {1479, 31423}, {1571, 16780}, {1621, 5437}, {1698, 6284}, {1699, 5432}, {1742, 4551}, {1768, 27778}, {1770, 31452}, {1788, 4314}, {1836, 4995}, {1837, 9588}, {1914, 9574}, {2003, 7074}, {2136, 2975}, {2241, 31422}, {2280, 5053}, {2717, 15737}, {2951, 15837}, {2999, 3052}, {3085, 9579}, {3218, 3243}, {3474, 4654}, {3485, 5493}, {3522, 10106}, {3523, 12053}, {3528, 4311}, {3530, 11373}, {3586, 10993}, {3614, 7965}, {3617, 5273}, {3621, 12640}, {3624, 12701}, {3634, 5084}, {3680, 17548}, {3683, 8580}, {3689, 5223}, {3699, 25728}, {3752, 21000}, {3870, 3928}, {3871, 4652}, {3872, 17549}, {3916, 6765}, {3977, 4901}, {4002, 19526}, {4031, 11038}, {4184, 18163}, {4222, 5338}, {4294, 6684}, {4302, 5587}, {4304, 5657}, {4305, 11362}, {4308, 21734}, {4309, 31425}, {4312, 17718}, {4313, 4848}, {4324, 10827}, {4330, 10826}, {4414, 7174}, {4428, 10582}, {4659, 4781}, {4666, 9352}, {4679, 6174}, {4847, 34607}, {4855, 15829}, {4862, 17724}, {5054, 7743}, {5250, 5438}, {5297, 16548}, {5414, 9616}, {5541, 11525}, {5687, 31424}, {5691, 15338}, {5698, 6745}, {5726, 12943}, {5729, 10382}, {5732, 17613}, {5744, 5853}, {5745, 17784}, {6361, 13411}, {6796, 12705}, {6848, 18483}, {6970, 8227}, {7173, 17527}, {7285, 32635}, {7288, 12575}, {7330, 32141}, {7411, 8544}, {7989, 12953}, {8053, 8683}, {9554, 29825}, {9589, 11375}, {9623, 16370}, {9668, 11231}, {9785, 15717}, {9860, 15452}, {9899, 26888}, {10165, 30305}, {10178, 17625}, {10384, 21153}, {10385, 11019}, {10590, 28150}, {10914, 19535}, {11495, 30353}, {12648, 34716}, {12912, 19547}, {15600, 17449}, {15624, 23845}, {16502, 31421}, {16572, 24047}, {17353, 35261}, {17567, 19862}, {18524, 18540}, {20075, 24392}, {25440, 31435}, {26685, 35284}, {28146, 31479}, {31447, 31795}

X(35445) = reflection of X(5219) in X(5218)
X(35445) = crosssum of X(650) and X(23056)
X(35445) = crossdifference of every pair of points on line {650, 14413}
X(35445) = barycentric product X(i)*X(j) for these {i,j}: {1, 6172}, {1275, 23056}
X(35445) = barycentric quotient X(i)/X(j) for these {i,j}: {6172, 75}, {23056, 1146}
X(35445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 165, 1155}, {1, 1155, 57}, {1, 2093, 1159}, {1, 3579, 5128}, {3, 55, 2078}, {3, 1697, 1420}, {35, 40, 3601}, {35, 5537, 55}, {40, 3601, 3340}, {55, 57, 10389}, {55, 165, 57}, {55, 1155, 1}, {55, 6244, 3256}, {57, 2078, 1420}, {100, 35258, 9}, {165, 5537, 40}, {165, 31508, 55}, {200, 4640, 3929}, {497, 10164, 31231}, {1159, 24929, 1}, {1376, 4512, 7308}, {1697, 2078, 10389}, {3085, 31730, 9579}, {3295, 31663, 15803}, {3359, 32613, 3576}, {3474, 13405, 4654}, {3513, 3514, 13462}, {3550, 17594, 5269}, {3576, 5119, 7962}, {3612, 11010, 7982}, {3749, 17596, 3677}, {3871, 4652, 6762}, {4294, 6684, 9581}, {4304, 5657, 5727}, {4421, 4640, 200}, {5010, 5119, 3576}, {5281, 9778, 226}, {5698, 6745, 31142}, {6244, 11248, 5537}, {8186, 8187, 36}, {9957, 11575, 354}, {32622, 32623, 3576}

X(35446) = EULER LINE INTERCEPT OF X(74)X(16936)

Barycentrics a^2*(3*a^8 - 6*a^6*b^2 + 6*a^2*b^6 - 3*b^8 - 6*a^6*c^2 - 30*a^4*b^2*c^2 + 26*a^2*b^4*c^2 + 10*b^6*c^2 + 26*a^2*b^2*c^4 - 14*b^4*c^4 + 6*a^2*c^6 + 10*b^2*c^6 - 3*c^8) : :
X(35446) = 30 X[2] + (3 J^2 - 55) X[3]

Starting with P = X(631), let A'B'C' be the circumcevian-inversion triangle defined in the preamble just before X(34864). Then X(35446) is the perspector of A'B'C' and the anti-incircle-circles triangle; see X(11363).

X(35446) lies on these lines: {2, 3}, {74, 16936}, {110, 33543}, {11456, 15606}, {12290, 31884}

X(35446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {16661, 35243, 10323}

Source-points for a pair of circumcevian-inversion points: X(35447)-X(35465)

This preamble and centers X(35447)-X(35465) were contributed by Peter Moses, December 16, 2019.

The secondary pre-circumcevian-inversion point of a point P is defined just before X(35000). The [primary] point is defined just before X(34864). For a list of points P and associated primary and secondary points, see the preamble just before X(35000).

X(35447) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(351)

Barycentrics a^2*(a^14 - 5*a^12*b^2 - 3*a^10*b^4 + 21*a^8*b^6 - 9*a^6*b^8 - 15*a^4*b^10 + 11*a^2*b^12 - b^14 - 5*a^12*c^2 + 44*a^10*b^2*c^2 - 57*a^8*b^4*c^2 - 80*a^6*b^6*c^2 + 149*a^4*b^8*c^2 - 48*a^2*b^10*c^2 - 3*b^12*c^2 - 3*a^10*c^4 - 57*a^8*b^2*c^4 + 251*a^6*b^4*c^4 - 153*a^4*b^6*c^4 - 75*a^2*b^8*c^4 + 41*b^10*c^4 + 21*a^8*c^6 - 80*a^6*b^2*c^6 - 153*a^4*b^4*c^6 + 232*a^2*b^6*c^6 - 37*b^8*c^6 - 9*a^6*c^8 + 149*a^4*b^2*c^8 - 75*a^2*b^4*c^8 - 37*b^6*c^8 - 15*a^4*c^10 - 48*a^2*b^2*c^10 + 41*b^4*c^10 + 11*a^2*c^12 - 3*b^2*c^12 - c^14) : :

X(35447) lies on these lines: {3, 351}, {74, 33962}, {111, 12041}, {125, 22338}, {126, 7728}, {1296, 5663}, {2777, 10748}, {2854, 10620}, {3325, 10065}, {5512, 15061}, {6019, 10081}, {9970, 14688}, {10734, 34584}, {11258, 15041}, {12244, 14360}, {14532, 32424}, {14650, 15055}, {20127, 23699}

X(35448) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(1319)

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

X(35448) lies on these lines: {1, 3}, {8, 18519}, {30, 3436}, {100, 6361}, {226, 16004}, {376, 3871}, {381, 1329}, {388, 28458}, {474, 22791}, {516, 11499}, {529, 3534}, {582, 3052}, {958, 3654}, {962, 6911}, {976, 7986}, {998, 4646}, {1012, 5690}, {1193, 1480}, {1260, 18524}, {1351, 22277}, {1376, 12699}, {1597, 1828}, {1598, 2355}, {1656, 3826}, {1657, 2829}, {1770, 11501}, {2550, 6841}, {3149, 28174}, {3434, 18544}, {3560, 5260}, {3617, 21669}, {3652, 5220}, {3656, 25524}, {3679, 18761}, {3830, 31141}, {3870, 13369}, {3913, 18481}, {4294, 28459}, {4413, 9955}, {5054, 6691}, {5055, 31246}, {5261, 6850}, {5493, 6796}, {5528, 16009}, {5534, 10860}, {5804, 6883}, {5854, 12773}, {6259, 11500}, {6600, 16117}, {6765, 7171}, {6825, 8543}, {6842, 10588}, {6845, 33110}, {6851, 17784}, {6882, 10591}, {6890, 10943}, {6899, 20075}, {6905, 20070}, {6909, 12245}, {6923, 11929}, {6924, 28212}, {6925, 10942}, {7354, 35249}, {7580, 32141}, {8256, 9708}, {8679, 33878}, {8715, 31730}, {9778, 11491}, {9798, 23845}, {10526, 11826}, {10738, 32554}, {11108, 25011}, {11362, 22758}, {11496, 26446}, {12327, 12778}, {12515, 13205}, {12607, 18545}, {15338, 35250}, {16408, 18493}, {17613, 24467}, {17757, 18542}, {18516, 21031}, {18517, 34612}, {19000, 31439}, {19549, 27628}, {25440, 28194}, {26321, 34718}

X(35449) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(54)

Barycentrics a^2*(a^14 - 3*a^12*b^2 + 3*a^10*b^4 - 5*a^8*b^6 + 15*a^6*b^8 - 21*a^4*b^10 + 13*a^2*b^12 - 3*b^14 - 3*a^12*c^2 + 8*a^10*b^2*c^2 - 13*a^8*b^4*c^2 + 14*a^6*b^6*c^2 + a^4*b^8*c^2 - 14*a^2*b^10*c^2 + 7*b^12*c^2 + 3*a^10*c^4 - 13*a^8*b^2*c^4 + 23*a^6*b^4*c^4 - 7*a^4*b^6*c^4 - 3*a^2*b^8*c^4 - 3*b^10*c^4 - 5*a^8*c^6 + 14*a^6*b^2*c^6 - 7*a^4*b^4*c^6 + 8*a^2*b^6*c^6 - b^8*c^6 + 15*a^6*c^8 + a^4*b^2*c^8 - 3*a^2*b^4*c^8 - b^6*c^8 - 21*a^4*c^10 - 14*a^2*b^2*c^10 - 3*b^4*c^10 + 13*a^2*c^12 + 7*b^2*c^12 - 3*c^14) : :

X(35449) lies on these lines: {3, 54}, {4, 14140}, {20, 32744}, {30, 13512}, {381, 16336}, {382, 19552}, {1656, 16337}, {5054, 10615}, {5217, 14102}, {5899, 14979}, {14531, 15787}

X(35450) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(34147)

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 + 28*a^4*b^2*c^2 - 18*a^2*b^4*c^2 - 8*b^6*c^2 - 12*a^4*c^4 - 18*a^2*b^2*c^4 + 30*b^4*c^4 + 18*a^2*c^6 - 8*b^2*c^6 - 7*c^8) : :

X(35450) lies on these lines: {3, 64}, {4, 34469}, {5, 12250}, {20, 34780}, {22, 11820}, {25, 74}, {30, 32064}, {51, 1597}, {140, 6225}, {146, 30744}, {185, 11426}, {378, 11402}, {381, 15311}, {382, 6247}, {399, 11598}, {548, 34781}, {549, 5656}, {550, 12324}, {999, 10060}, {1154, 12085}, {1192, 13474}, {1204, 1598}, {1351, 2781}, {1385, 9899}, {1482, 12262}, {1503, 3534}, {1593, 5890}, {1596, 18931}, {1656, 5878}, {1657, 5894}, {1853, 2777}, {1885, 26944}, {1971, 15655}, {1993, 15054}, {2393, 33878}, {2883, 3526}, {2979, 13445}, {3167, 5663}, {3295, 10076}, {3515, 12290}, {3516, 6241}, {3517, 11381}, {3520, 12174}, {3527, 13452}, {3843, 5895}, {3851, 22802}, {4550, 16419}, {5020, 11472}, {5050, 10249}, {5054, 23328}, {5055, 23329}, {5072, 5893}, {5073, 5925}, {5093, 10250}, {5094, 18431}, {5644, 9730}, {5892, 11479}, {6221, 11241}, {6398, 11242}, {6417, 19087}, {6418, 19088}, {6445, 17819}, {6446, 17820}, {6449, 12964}, {6450, 12970}, {6451, 10533}, {6452, 10534}, {7387, 32138}, {7395, 15062}, {7506, 9914}, {8263, 18440}, {8703, 11206}, {9715, 12279}, {9777, 13596}, {9833, 15696}, {9909, 14915}, {10117, 16219}, {10182, 15707}, {10192, 15693}, {10193, 15701}, {10254, 32125}, {10990, 11550}, {11410, 11456}, {11414, 11440}, {11438, 18535}, {11468, 15750}, {12017, 19153}, {12084, 12164}, {12086, 12160}, {12100, 35260}, {12308, 13293}, {12940, 31479}, {13399, 18396}, {13488, 18913}, {14269, 23325}, {15055, 35264}, {15681, 18400}, {15684, 18405}, {15720, 16252}, {17853, 19457}, {20186, 21733}, {20417, 26958}, {20850, 32110}, {31133, 34796}, {31978, 34783}

X(35451) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(8)

Barycentrics a^2*(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 2*a^7*c + 8*a^6*b*c - 4*a^5*b^2*c - 14*a^4*b^3*c + 14*a^3*b^4*c + 4*a^2*b^5*c - 8*a*b^6*c + 2*b^7*c - 2*a^6*c^2 - 4*a^5*b*c^2 + 21*a^4*b^2*c^2 - 8*a^3*b^3*c^2 - 17*a^2*b^4*c^2 + 12*a*b^5*c^2 - 2*b^6*c^2 + 6*a^5*c^3 - 14*a^4*b*c^3 - 8*a^3*b^2*c^3 + 24*a^2*b^3*c^3 - 6*a*b^4*c^3 - 2*b^5*c^3 + 14*a^3*b*c^4 - 17*a^2*b^2*c^4 - 6*a*b^3*c^4 + 6*b^4*c^4 - 6*a^3*c^5 + 4*a^2*b*c^5 + 12*a*b^2*c^5 - 2*b^3*c^5 + 2*a^2*c^6 - 8*a*b*c^6 - 2*b^2*c^6 + 2*a*c^7 + 2*b*c^7 - c^8) : :

X(35451) lies on these lines: {3, 8}, {11, 13743}, {36, 1749}, {56, 11571}, {79, 12611}, {80, 26321}, {153, 6924}, {214, 33858}, {381, 1470}, {404, 11698}, {1482, 12332}, {1484, 6906}, {1768, 26286}, {2800, 4973}, {2802, 35000}, {3058, 10058}, {3816, 7489}, {5253, 33668}, {5441, 12743}, {5533, 8071}, {5884, 6265}, {6264, 26285}, {6326, 32612}, {7354, 10090}, {11849, 12737}, {12114, 12747}, {12702, 22560}, {13369, 33598}, {13624, 35204}, {14793, 18515}, {15654, 35221}, {22935, 26201}

X(35452) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(5)

Barycentrics a^2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8 - 2*a^6*c^2 + 13*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 + 10*b^4*c^4 + 2*a^2*c^6 - 4*b^2*c^6 - c^8) : :

X(35452) lies on these lines: {2, 3}, {74, 13391}, {195, 13491}, {477, 20189}, {567, 14855}, {691, 29316}, {1154, 10620}, {1533, 14156}, {2935, 5898}, {5621, 19924}, {5876, 15103}, {6000, 12308}, {6781, 34866}, {7373, 10149}, {10540, 10564}, {10575, 34986}, {10627, 15062}, {11440, 12280}, {11649, 33878}, {12316, 34783}, {14128, 33539}, {14157, 32609}, {14915, 22115}, {15072, 15087}, {16030, 19651}

X(35453) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(115)

Barycentrics a^2*(a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12 - 4*a^10*c^2 + 23*a^8*b^2*c^2 - 35*a^6*b^4*c^2 + 27*a^4*b^6*c^2 - 9*a^2*b^8*c^2 - 2*b^10*c^2 + 5*a^8*c^4 - 35*a^6*b^2*c^4 + 24*a^4*b^4*c^4 - 7*a^2*b^6*c^4 + 17*b^8*c^4 + 27*a^4*b^2*c^6 - 7*a^2*b^4*c^6 - 28*b^6*c^6 - 5*a^4*c^8 - 9*a^2*b^2*c^8 + 17*b^4*c^8 + 4*a^2*c^10 - 2*b^2*c^10 - c^12) : :

X(35453) lies on these lines: {3, 115}, {30, 5866}, {74, 3565}, {1593, 3563}, {1597, 5139}, {7393, 15565}

X(35453) = circumcircle-inverse of X(38736)
X(35453) = Stammler-circle-inverse of X(38730)

X(35454) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(9)

Barycentrics a^2*(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 + 7*a^7*b*c - 4*a^6*b^2*c - 5*a^5*b^3*c - 10*a^4*b^4*c + 13*a^3*b^5*c + 12*a^2*b^6*c - 15*a*b^7*c + 3*b^8*c - 4*a^7*c^2 - 4*a^6*b*c^2 + 16*a^5*b^2*c^2 + 8*a^4*b^3*c^2 - 20*a^3*b^4*c^2 - 4*a^2*b^5*c^2 + 8*a*b^6*c^2 + 4*a^6*c^3 - 5*a^5*b*c^3 + 8*a^4*b^2*c^3 + 22*a^3*b^3*c^3 - 12*a^2*b^4*c^3 - 9*a*b^5*c^3 - 8*b^6*c^3 + 6*a^5*c^4 - 10*a^4*b*c^4 - 20*a^3*b^2*c^4 - 12*a^2*b^3*c^4 + 30*a*b^4*c^4 + 6*b^5*c^4 - 6*a^4*c^5 + 13*a^3*b*c^5 - 4*a^2*b^2*c^5 - 9*a*b^3*c^5 + 6*b^4*c^5 - 4*a^3*c^6 + 12*a^2*b*c^6 + 8*a*b^2*c^6 - 8*b^3*c^6 + 4*a^2*c^7 - 15*a*b*c^7 + a*c^8 + 3*b*c^8 - c^9) : :

X(35454) lies on these lines: {3, 9}, {18859, 34928}

X(35455) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(10)

Barycentrics a^2*(a^8 - a^7*b - 2*a^6*b^2 + 3*a^5*b^3 - 3*a^3*b^5 + 2*a^2*b^6 + a*b^7 - b^8 - a^7*c + 3*a^6*b*c - a^5*b^2*c - 5*a^4*b^3*c + 5*a^3*b^4*c + a^2*b^5*c - 3*a*b^6*c + b^7*c - 2*a^6*c^2 - a^5*b*c^2 + 16*a^4*b^2*c^2 - 6*a^3*b^3*c^2 - 10*a^2*b^4*c^2 + 7*a*b^5*c^2 - 4*b^6*c^2 + 3*a^5*c^3 - 5*a^4*b*c^3 - 6*a^3*b^2*c^3 + 14*a^2*b^3*c^3 - 5*a*b^4*c^3 - b^5*c^3 + 5*a^3*b*c^4 - 10*a^2*b^2*c^4 - 5*a*b^3*c^4 + 10*b^4*c^4 - 3*a^3*c^5 + a^2*b*c^5 + 7*a*b^2*c^5 - b^3*c^5 + 2*a^2*c^6 - 3*a*b*c^6 - 4*b^2*c^6 + a*c^7 + b*c^7 - c^8) : :

X(35455) lies on these lines: {3, 10}, {56, 3465}, {1597, 1785}, {10538, 21312}

X(35456) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(32)

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

X(35456) lies on these lines: {3, 6}, {30, 5207}, {69, 32151}, {381, 5031}, {698, 12188}, {1352, 18503}, {1503, 13188}, {1513, 7925}, {1975, 9873}, {3564, 9862}, {5103, 31670}, {5999, 8782}, {6656, 10357}, {7807, 21850}, {10583, 18583}, {12501, 32306}, {17128, 18358}, {20885, 33873}

X(35457) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(35)

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

X(35457) lies on these lines: {1, 3}, {30, 5180}, {140, 1389}, {145, 6903}, {381, 5289}, {535, 4930}, {758, 12773}, {952, 6840}, {2475, 5330}, {2476, 18493}, {3869, 13465}, {3878, 13743}, {3940, 5176}, {4511, 18524}, {4867, 28204}, {5055, 17057}, {5080, 5730}, {5690, 6952}, {5762, 14151}, {5790, 6830}, {5844, 12247}, {5901, 6853}, {5905, 34698}, {6882, 11545}, {6972, 12245}, {7354, 16150}, {10698, 28174}, {12635, 18526}, {20067, 34773}

X(35458) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(39)

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

X(35458) lies on these lines: {3, 6}, {25, 33873}, {30, 12215}, {381, 5103}, {384, 21850}, {732, 12188}, {1003, 34615}, {3314, 13860}, {3564, 5984}, {4048, 31670}, {5207, 7776}, {7470, 7762}, {7484, 11673}, {8925, 34130}, {14880, 32451}, {19130, 24273}, {21312, 32529}

X(35459) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(55)

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

X(35459) lies on these lines: {1, 3}, {30, 4511}, {60, 15952}, {63, 18515}, {72, 26321}, {78, 18525}, {381, 997}, {515, 12738}, {516, 6265}, {518, 12773}, {535, 18481}, {550, 21740}, {908, 10742}, {952, 3935}, {960, 13743}, {1455, 23071}, {1657, 6261}, {2771, 4867}, {3534, 28534}, {3617, 6833}, {3621, 6890}, {3625, 12616}, {3656, 5880}, {3683, 28453}, {3811, 18526}, {3814, 5794}, {3872, 34718}, {3940, 17615}, {4420, 5176}, {5160, 31524}, {5220, 22758}, {5267, 16139}, {5440, 18524}, {5550, 5804}, {5762, 18450}, {6326, 28160}, {6831, 18357}, {6862, 9780}, {6882, 12019}, {6909, 14988}, {6966, 12245}, {7489, 15254}, {11112, 22791}, {11813, 17647}, {12248, 17484}, {12520, 15696}, {12699, 30144}, {12737, 28234}, {12739, 21578}, {17528, 8493}, {18480, 31160}, {19907, 28212}, {28460, 33857}, {34772, 34773}

X(35460) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(56)

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

X(35460) lies on these lines: {1, 3}, {30, 1145}, {63, 34718}, {381, 5123}, {519, 12515}, {962, 6959}, {1158, 12645}, {1532, 28174}, {1656, 22835}, {3219, 5690}, {3626, 3652}, {3814, 12699}, {3872, 18515}, {3880, 12773}, {5080, 6361}, {5180, 5748}, {5541, 28204}, {5657, 6929}, {5836, 13743}, {6001, 12331}, {6735, 10742}, {6834, 20070}, {9955, 31263}, {10039, 16140}, {10914, 26321}, {11813, 28194}, {12647, 35249}, {13465, 34790}, {13747, 22791}, {17615, 18518}, {18526, 33956}, {23958, 34631}, {28198, 31160}

X(35461) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(56)

Barycentrics a^2*(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 2*a^7*c + 3*a^6*b*c - a^5*b^2*c - 2*a^4*b^3*c + 8*a^3*b^4*c - 5*a^2*b^5*c - 5*a*b^6*c + 4*b^7*c - 2*a^6*c^2 - a^5*b*c^2 + 10*a^4*b^2*c^2 - 20*a^3*b^3*c^2 - 4*a^2*b^4*c^2 + 21*a*b^5*c^2 - 4*b^6*c^2 + 6*a^5*c^3 - 2*a^4*b*c^3 - 20*a^3*b^2*c^3 + 38*a^2*b^3*c^3 - 18*a*b^4*c^3 - 4*b^5*c^3 + 8*a^3*b*c^4 - 4*a^2*b^2*c^4 - 18*a*b^3*c^4 + 10*b^4*c^4 - 6*a^3*c^5 - 5*a^2*b*c^5 + 21*a*b^2*c^5 - 4*b^3*c^5 + 2*a^2*c^6 - 5*a*b*c^6 - 4*b^2*c^6 + 2*a*c^7 + 4*b*c^7 - c^8) : :

X(35461) lies on these lines: {1, 3}, {5082, 26321}, {5853, 12773}

X(35462) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(58)

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

X(35462) lies on these lines: {3, 6}, {30, 20558}, {381, 20546}, {726, 12188}, {2959, 3579}

X(35463) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(69)

Barycentrics a^2*(a^2 - b^2 - c^2)*(a^10 - 3*a^8*b^2 + 2*a^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10 - 3*a^8*c^2 + 13*a^6*b^2*c^2 - 6*a^4*b^4*c^2 + 9*a^2*b^6*c^2 - b^8*c^2 + 2*a^6*c^4 - 6*a^4*b^2*c^4 - 16*a^2*b^4*c^4 + 2*a^4*c^6 + 9*a^2*b^2*c^6 - 3*a^2*c^8 - b^2*c^8 + c^10) : :

X(35463) lies on these lines: {3, 69}, {187, 399}, {1597, 5203}, {2070, 11641}, {12188, 18859}, {34106, 35002}

X(35464) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(76)

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

X(35464) lies on these lines: {3, 76}, {30, 32528}, {542, 2076}, {1003, 34682}, {1597, 32527}, {5149, 8724}, {6033, 7747}, {7820, 15561}, {19910, 30270}

X(35465) = SECONDARY PRE-CIRCUMCEVIAN-INVERSION POINT OF X(35372)

Barycentrics a^2*(a^6 - a^4*b^2 - a^2*b^4 + b^6 - 2*a^4*c^2 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 + b^2*c^4)*(a^6 - 2*a^4*b^2 + a^2*b^4 - a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 - 2*b^2*c^4 + c^6)*(a^20 - 6*a^18*b^2 + 17*a^16*b^4 - 34*a^14*b^6 + 56*a^12*b^8 - 70*a^10*b^10 + 56*a^8*b^12 - 22*a^6*b^14 - a^4*b^16 + 4*a^2*b^18 - b^20 - 6*a^18*c^2 + 28*a^16*b^2*c^2 - 52*a^14*b^4*c^2 + 46*a^12*b^6*c^2 - 14*a^10*b^8*c^2 - 4*a^8*b^10*c^2 - 8*a^6*b^12*c^2 + 22*a^4*b^14*c^2 - 16*a^2*b^16*c^2 + 4*b^18*c^2 + 17*a^16*c^4 - 52*a^14*b^2*c^4 + 69*a^12*b^4*c^4 - 40*a^10*b^6*c^4 - 22*a^8*b^8*c^4 + 48*a^6*b^10*c^4 - 31*a^4*b^12*c^4 + 20*a^2*b^14*c^4 - 9*b^16*c^4 - 34*a^14*c^6 + 46*a^12*b^2*c^6 - 40*a^10*b^4*c^6 + 60*a^8*b^6*c^6 - 30*a^6*b^8*c^6 - 22*a^4*b^10*c^6 - 4*a^2*b^12*c^6 + 24*b^14*c^6 + 56*a^12*c^8 - 14*a^10*b^2*c^8 - 22*a^8*b^4*c^8 - 30*a^6*b^6*c^8 + 64*a^4*b^8*c^8 - 4*a^2*b^10*c^8 - 54*b^12*c^8 - 70*a^10*c^10 - 4*a^8*b^2*c^10 + 48*a^6*b^4*c^10 - 22*a^4*b^6*c^10 - 4*a^2*b^8*c^10 + 72*b^10*c^10 + 56*a^8*c^12 - 8*a^6*b^2*c^12 - 31*a^4*b^4*c^12 - 4*a^2*b^6*c^12 - 54*b^8*c^12 - 22*a^6*c^14 + 22*a^4*b^2*c^14 + 20*a^2*b^4*c^14 + 24*b^6*c^14 - a^4*c^16 - 16*a^2*b^2*c^16 - 9*b^4*c^16 + 4*a^2*c^18 + 4*b^2*c^18 - c^20) : :

X(35465) lies on these lines: {3, 35372}, {14910, 15550}

X(35466) = CROSSDIFFERENCE OF EVERY PAIR OF POINTS ON X(55)X(512)

Barycentrics 2*a^3 - a*b^2 + b^3 - b^2*c - a*c^2 - b*c^2 + c^3 : :
X(35466) = 3 X[2] + X[16704], X[896] + 2 X[17070], X[4831] + 2 X[4892]

X(35466) lies on these lines: {1, 5791}, {2, 6}, {3, 1714}, {5, 1724}, {9, 17720}, {10, 4434}, {11, 238}, {12, 5247}, {21, 1834}, {27, 1865}, {31, 2886}, {37, 27492}, {38, 17061}, {42, 6690}, {43, 5432}, {44, 908}, {55, 8731}, {56, 28258}, {57, 1723}, {58, 442}, {63, 3772}, {88, 21907}, {100, 33139}, {140, 3216}, {171, 3925}, {226, 4641}, {239, 26629}, {241, 514}, {257, 17367}, {306, 17372}, {312, 17339}, {377, 4252}, {386, 7483}, {387, 6857}, {405, 5292}, {440, 2193}, {511, 18191}, {518, 3011}, {528, 902}, {536, 3977}, {580, 6831}, {594, 32779}, {595, 24390}, {673, 17963}, {698, 19683}, {740, 3712}, {748, 3816}, {750, 3826}, {752, 21241}, {756, 29683}, {846, 4854}, {851, 3286}, {857, 26145}, {896, 3120}, {899, 1818}, {958, 5230}, {978, 5433}, {984, 17602}, {1001, 11269}, {1010, 25446}, {1038, 1722}, {1046, 3649}, {1086, 3218}, {1104, 6734}, {1108, 25939}, {1155, 1738}, {1191, 10527}, {1193, 4999}, {1279, 26015}, {1329, 28273}, {1386, 17726}, {1411, 26727}, {1451, 15844}, {1453, 5705}, {1468, 25466}, {1503, 8229}, {1616, 10529}, {1621, 33142}, {1698, 5725}, {1707, 1836}, {1736, 15252}, {1743, 5219}, {1751, 13478}, {1754, 8727}, {1757, 17719}, {1764, 5755}, {1778, 1901}, {1780, 6841}, {1999, 17315}, {2161, 21368}, {2292, 18253}, {2308, 33105}, {2475, 16948}, {2650, 11281}, {2915, 5358}, {3006, 5846}, {3017, 4653}, {3052, 3434}, {3058, 8616}, {3073, 15908}, {3142, 27660}, {3187, 33113}, {3219, 4415}, {3242, 26228}, {3306, 17278}, {3416, 29857}, {3550, 32865}, {3663, 33996}, {3666, 3946}, {3681, 29665}, {3683, 24210}, {3703, 4362}, {3704, 27368}, {3741, 6679}, {3744, 4847}, {3749, 4863}, {3751, 17718}, {3752, 16585}, {3755, 4689}, {3756, 7292}, {3757, 33121}, {3769, 29641}, {3771, 32853}, {3791, 29671}, {3813, 3915}, {3831, 25992}, {3844, 30768}, {3873, 29681}, {3891, 4884}, {3914, 4640}, {3916, 23537}, {3928, 23681}, {3932, 17763}, {3943, 32849}, {3944, 7262}, {3973, 31142}, {4000, 5744}, {4026, 29631}, {4030, 29673}, {4046, 33160}, {4054, 17351}, {4205, 25441}, {4220, 5324}, {4224, 5347}, {4255, 6910}, {4257, 11112}, {4265, 7465}, {4358, 4422}, {4361, 17740}, {4395, 17495}, {4414, 33128}, {4427, 4442}, {4437, 26247}, {4649, 29640}, {4650, 11246}, {4656, 5325}, {4671, 17340}, {4672, 25385}, {4831, 4892}, {4850, 17366}, {4871, 31289}, {4966, 29632}, {4969, 27757}, {5121, 5972}, {5132, 30944}, {5137, 7193}, {5222, 34522}, {5231, 7290}, {5264, 31419}, {5271, 28634}, {5272, 17728}, {5294, 5830}, {5305, 16552}, {5336, 11679}, {5435, 7365}, {5483, 17020}, {5706, 6824}, {5707, 6861}, {5710, 19843}, {5711, 19854}, {5831, 18229}, {5852, 32856}, {6057, 33164}, {6542, 34024}, {6682, 29654}, {6691, 27627}, {7081, 33118}, {7227, 31025}, {7683, 13442}, {7819, 29433}, {9053, 20045}, {9362, 26048}, {10479, 17698}, {10974, 18180}, {11240, 16486}, {11680, 17127}, {13728, 20083}, {13747, 17749}, {14206, 16732}, {15048, 24296}, {15325, 34586}, {15485, 24217}, {15509, 28260}, {16455, 19763}, {16466, 26363}, {16468, 17717}, {16475, 17723}, {16702, 18653}, {16706, 24627}, {16816, 24384}, {16827, 26686}, {16885, 31018}, {17045, 29833}, {17126, 33108}, {17160, 26070}, {17246, 33155}, {17323, 19823}, {17324, 19786}, {17326, 19832}, {17329, 27184}, {17332, 26580}, {17334, 33151}, {17338, 30829}, {17353, 30818}, {17362, 33077}, {17365, 31019}, {17469, 29690}, {17540, 29455}, {17596, 33132}, {17597, 24477}, {17734, 17757}, {17735, 21956}, {17737, 17747}, {17775, 31266}, {17783, 25568}, {18178, 22076}, {18792, 24885}, {19804, 20912}, {20072, 25529}, {20075, 21000}, {20470, 27628}, {21997, 24366}, {23511, 31231}, {24178, 32636}, {24473, 26728}, {24542, 29824}, {25453, 32916}, {25459, 26117}, {25968, 26013}, {26098, 31245}, {26131, 31254}, {26227, 33114}, {26627, 34824}, {26685, 28808}, {26724, 27003}, {28027, 34791}, {29846, 32864}, {29850, 32918}, {29856, 32784}, {29858, 33087}, {29861, 33076}, {29862, 32846}, {29865, 33081}, {29867, 32781}, {29872, 33075}, {29873, 33078}, {31146, 35227}, {31192, 31210}, {31237, 33080}, {32912, 33127}, {32913, 33130}, {32914, 33119}

X(35466) = complement of X(3936)
X(35466) = midpoint of X(i) and X(j) for these {i,j}: {896, 3120}, {902, 33136}, {3936, 16704}, {4427, 4442}
X(35466) = X(35466) = reflection of X(i) in X(j) for these {i,j}: {3120, 17070}, {17724, 3011}
X(35466) = complement of the isogonal conjugate of X(34079)
X(35466) = complement of the isotomic conjugate of X(24624)
X(35466) = isotomic conjugate of the polar conjugate of X(1884)
X(35466) = X(i)-complementary conjugate of X(j) for these (i,j): {6, 31845}, {32, 35069}, {80, 21245}, {604, 6739}, {759, 141}, {1333, 214}, {1411, 17052}, {2161, 3454}, {2206, 16586}, {2341, 1329}, {6187, 1211}, {6740, 21244}, {9273, 21254}, {9274, 620}, {11060, 5949}, {14616, 626}, {24624, 2887}, {32671, 523}, {34079, 10}
X(35466) = X(7)-Ceva conjugate of X(34194)
X(35466) = X(34194)-cross conjugate of X(7)
X(35466) = X(i)-isoconjugate of X(j) for these (i,j): {110, 35354}, {661, 6083}
X(35466) = crosspoint of X(i) and X(j) for these (i,j): {2, 24624}, {655, 1016}
X(35466) = crosssum of X(i) and X(j) for these (i,j): {6, 2245}, {654, 1015}, {2088, 9404}
X(35466) = trilinear pole of line {6089, 34194}
X(35466) = crossdifference of every pair of points on line {55, 512}
X(35466) = barycentric product X(i)*X(j) for these {i,j}: {69, 1884}, {99, 6089}, {257, 27970}
X(35466) = barycentric quotient X(i)/X(j) for these {i,j}: {110, 6083}, {661, 35354}, {1884, 4}, {6089, 523}, {27970, 894}
X(35466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 6, 5718}, {2, 69, 30811}, {2, 81, 17056}, {2, 193, 30828}, {2, 333, 1211}, {2, 1150, 141}, {2, 1654, 30832}, {2, 2895, 30831}, {2, 5235, 1213}, {2, 5278, 5743}, {2, 5361, 32782}, {2, 5372, 33172}, {2, 16704, 3936}, {2, 17277, 5241}, {2, 17349, 5233}, {2, 19742, 5741}, {2, 24597, 6}, {2, 31034, 30834}, {6, 31187, 2}, {21, 24883, 1834}, {31, 24892, 2886}, {58, 24880, 442}, {63, 3772, 3782}, {69, 31232, 2}, {81, 31204, 2}, {86, 31205, 2}, {171, 33138, 3925}, {238, 33140, 11}, {333, 30832, 1654}, {387, 6857, 19765}, {748, 29662, 3816}, {846, 33135, 4854}, {851, 3286, 15447}, {984, 29658, 17602}, {1046, 24161, 3649}, {1150, 31229, 2}, {1386, 29639, 17726}, {1654, 30832, 1211}, {1707, 17064, 1836}, {3008, 3911, 16610}, {3218, 33129, 1086}, {3219, 33133, 4415}, {3550, 32865, 34612}, {3911, 34050, 241}, {4000, 5744, 17595}, {4362, 4438, 3703}, {4650, 17889, 11246}, {4921, 30831, 2895}, {5231, 7290, 17721}, {8616, 33141, 3058}, {17763, 33115, 3932}, {24597, 31187, 5718}, {29631, 32917, 4026}, {29632, 32919, 4966}

X(35467) = X(3)X(161)∩X(5)X(23320)

Barycentrics a^2*(a^20-6*(b^2+c^2)*a^18+(15*b^4+26*b^2*c^2+15*c^4)*a^16-4*(b^2+c^2)*(5*b^4+6*b^2*c^2+5*c^4)*a^14+(14*b^8+14*c^8+(35*b^4+47*b^2*c^2+35*c^4)*b^2*c^2)*a^12-10*(b^4+b^2*c^2+c^4)*(b^2+c^2)*b^2*c^2*a^10-(14*b^12+14*c^12-(b^8+c^8+(b^4+6*b^2*c^2+c^4)*b^2*c^2)*b^2*c^2)*a^8+(b^4-c^4)*(b^2-c^2)*(20*b^8+20*c^8+(4*b^4+25*b^2*c^2+4*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(15*b^12+15*c^12+(b^8+c^8-(3*b^4+4*b^2*c^2+3*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^3*(6*b^8+6*c^8-(2*b^4-b^2*c^2+2*c^4)*b^2*c^2)*a^2-(b^6+c^6)*(b^2+c^2)*(b^2-c^2)^6) : :

See Kadir Altintas and César Lozada, Euclid 374 .

X(35467) lies on these lines: {3, 161}, {5, 23320}, {54, 34418}, {1157, 6242}, {2070, 21394}, {3432, 20414}, {6150, 13368}, {6799, 10214}, {18016, 32171}

X(35468) = X(3)X(31817)∩X(4)X(9)

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

See Kadir Altintas and César Lozada, Euclid 375 .

X(35468) lies on these lines: {3, 31817}, {4, 9}, {30, 21672}, {35, 9275}, {63, 21081}, {191, 2127}, {484, 21674}, {502, 21696}, {2245, 3743}, {3336, 27577}, {4039, 24068}, {4062, 6763}, {4570, 14366}, {13323, 22276}, {22299, 35004}

X(35469) = X(2)X(3)∩X(15)X(74)

Barycentrics a^2*(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) - 8*(a^2 - b^2 - c^2)*S^3) : :
Barycentrics a^2*(8*(c^2-a^2+b^2)*S^3+3^(1/2)*(a^2+b^2-c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+4*b^2*c^2+c^4)*(a^2-b^2+c^2)) : :

X(35469) lies on these lines: {2, 3}, {15, 74}, {61, 5890}, {62, 15033}, {621, 14369}, {1503, 14173}, {4550, 11130}, {5318, 11142}, {5352, 8837}, {10564, 11131}, {10605, 22236}, {11126, 13352}, {11127, 13754}, {13445, 34424}, {14170, 14915}, {14538, 18863}

X(35469) = {X(3),X(378)}-harmonic conjugate of X(35470)

X(35470) = X(2)X(3)∩X(16)X(74)

Barycentrics a^2*(Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 4*b^2*c^2 + c^4) + 8*(a^2 - b^2 - c^2)*S^3) : :
Barycentrics a^2*(8*(c^2-a^2+b^2)*S^3-3^(1/2)*(a^2+b^2-c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+4*b^2*c^2+c^4)*(a^2-b^2+c^2)) : :

X(35470) lies on these lines: {2, 3}, {16, 74}, {61, 15033}, {62, 5890}, {622, 14368}, {1503, 14179}, {4550, 11131}, {5321, 11141}, {5351, 8839}, {10564, 11130}, {10605, 22238}, {11126, 13754}, {11127, 13352}, {13445, 34425}, {14169, 14915}, {14539, 18864}

X(35470) = {X(3),X(378)}-harmonic conjugate of X(35469)

X(35471) = X(2)X(3)∩X(70)X(74)

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

X(35471) lies on these lines: {2, 3}, {33, 4324}, {34, 4316}, {50, 393}, {64, 16659}, {67, 13452}, {68, 12278}, {70, 74}, {125, 34786}, {146, 15132}, {156, 34798}, {185, 34785}, {566, 3087}, {1192, 18396}, {1204, 11457}, {1235, 14907}, {1495, 22802}, {1620, 18405}, {1870, 4299}, {1899, 12289}, {1968, 6781}, {1974, 29317}, {1986, 6243}, {2549, 10312}, {2777, 26883}, {3053, 5523}, {3580, 12293}, {4293, 9630}, {4302, 6198}, {4549, 11444}, {5654, 11449}, {5878, 14157}, {5889, 12118}, {5890, 19467}, {5894, 16655}, {5895, 32111}, {6145, 11270}, {6193, 7722}, {6225, 12112}, {6241, 9833}, {6242, 6776}, {6403, 11663}, {6560, 10880}, {6561, 10881}, {7592, 13568}, {7689, 11442}, {7756, 10311}, {8537, 11179}, {9786, 12022}, {9927, 32110}, {10313, 15075}, {10605, 17845}, {11363, 28146}, {11430, 32903}, {11438, 18912}, {11456, 34782}, {11470, 19924}, {11572, 23329}, {12140, 16111}, {12163, 14516}, {12244, 12250}, {12290, 20427}, {12292, 20127}, {13346, 15463}, {15032, 18925}, {15083, 30714}, {15424, 18850}, {15472, 27866}, {18381, 21663}, {18917, 34799}, {19128, 31670}, {20125, 32605}, {25738, 30522}, {25739, 26937}, {31815, 34397}, {32210, 34514}, {34469, 34780}

X(35471) = anticomplement of X(18404)
X(35471) = {X(3),X(4)}-harmonic conjugate of X(37119)

X(35472) = X(2)X(3)∩X(74)X(154)

Barycentrics a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^4 - 10*a^2*b^2 + 5*b^4 - 10*a^2*c^2 + 8*b^2*c^2 + 5*c^4) : :
Barycentrics 5 Sin[2 A]-Tan[A] : :

X(35472) lies on these lines: {2, 3}, {54, 1192}, {64, 26882}, {74, 154}, {112, 5210}, {232, 8588}, {394, 15035}, {1204, 9707}, {1350, 15036}, {1495, 11204}, {1498, 11468}, {1620, 19357}, {1986, 2979}, {1993, 32110}, {2351, 22455}, {3431, 11402}, {5092, 8541}, {5206, 8743}, {5410, 6452}, {5411, 6451}, {5523, 21843}, {5621, 15577}, {5890, 17809}, {5963, 16391}, {6241, 17821}, {6409, 10881}, {6410, 10880}, {7722, 15040}, {8537, 10541}, {8567, 12290}, {8589, 10311}, {8739, 10645}, {8740, 10646}, {10193, 11550}, {10312, 15815}, {10605, 11464}, {10606, 14157}, {11202, 11456}, {11270, 34469}, {11405, 12017}, {11423, 14528}, {11430, 15004}, {11438, 13366}, {11449, 12163}, {11454, 18451}, {12254, 26944}, {14581, 15513}, {15055, 26881}, {15533, 32234}, {17845, 23294}, {19128, 31884}

X(35473) = X(2)X(3)∩X(74)X(184)

Barycentrics a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - 4*a^2*b^2 + 2*b^4 - 4*a^2*c^2 + 5*b^2*c^2 + 2*c^4) : :
X(35473) = 4*(4*R^2-SW)*X(3)-R^2*X(4)

X(35473) lies on these lines: {2, 3}, {6, 20421}, {49, 32138}, {54, 1204}, {64, 9707}, {74, 184}, {112, 574}, {125, 10193}, {154, 12112}, {182, 32901}, {185, 11468}, {232, 8589}, {393, 15109}, {576, 13482}, {1068, 14794}, {1119, 7279}, {1147, 11440}, {1199, 11425}, {1235, 7782}, {1272, 14558}, {1495, 11455}, {1511, 18435}, {1614, 3357}, {1620, 10982}, {1870, 5010}, {1899, 32607}, {1968, 15515}, {1986, 14805}, {2781, 19379}, {2979, 10564}, {3060, 32110}, {3098, 8541}, {3455, 9862}, {3563, 33638}, {4550, 15051}, {4996, 7046}, {5012, 15055}, {5063, 8749}, {5206, 10312}, {5410, 6451}, {5411, 6452}, {5621, 6776}, {5651, 15036}, {5663, 9544}, {5890, 11430}, {6000, 11464}, {6030, 8717}, {6198, 7280}, {6241, 13367}, {6325, 30247}, {6403, 14810}, {6696, 34224}, {6699, 26913}, {7689, 34148}, {7722, 11003}, {7952, 14792}, {8567, 19357}, {8588, 10311}, {8739, 10646}, {8740, 10645}, {8743, 15815}, {9306, 15035}, {9545, 32210}, {9609, 21397}, {9744, 13200}, {10192, 32111}, {10246, 31948}, {10282, 12290}, {10539, 15062}, {10605, 15032}, {10606, 11456}, {10986, 33843}, {11058, 14634}, {11202, 14157}, {11381, 26882}, {11405, 33878}, {11423, 13382}, {11438, 15004}, {11442, 12383}, {11449, 12162}, {11454, 13754}, {11457, 12254}, {11472, 35264}, {12038, 12111}, {12219, 25487}, {12244, 13293}, {12289, 20299}, {12292, 15060}, {12834, 15053}, {13403, 26917}, {14855, 15080}, {14915, 26881}, {15072, 18475}, {16163, 21243}, {17508, 19128}, {17702, 23293}, {18394, 32767}, {18439, 32171}, {19457, 23291}, {19596, 35228}, {21659, 23294}, {23329, 25739}

X(35473) = {X(3),X(4)}-harmonic conjugate of X(21844)

X(35474) = X(2)X(3)∩X(74)X(290)

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

X(35474) lies on these lines: {2, 3}, {74, 290}, {141, 16264}, {187, 6531}, {264, 3098}, {275, 22352}, {316, 16163}, {340, 542}, {511, 648}, {805, 1300}, {1350, 33971}, {1629, 3917}, {1986, 18322}, {2966, 21445}, {3168, 33586}, {5523, 5667}, {6530, 29181}, {7760, 14831}, {8884, 15644}, {9308, 33878}, {16080, 32225}, {17907, 31670}

X(35474) = {X(3),X(4)}-harmonic conjugate of X(37124)

X(35475) = X(2)X(3)∩X(74)X(578)

Barycentrics a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^4 - 4*a^2*b^2 + 2*b^4 - 4*a^2*c^2 + 7*b^2*c^2 + 2*c^4) : :
Barycentrics 4 Sin[2 A]+3 Tan[A] : :

X(35475) lies on these lines: {2, 3}, {52, 11454}, {54, 3357}, {74, 578}, {112, 7772}, {185, 11423}, {389, 11468}, {568, 32210}, {1147, 15062}, {1173, 11204}, {1199, 10605}, {1204, 15033}, {1493, 34783}, {1614, 3431}, {2914, 10620}, {3043, 14094}, {3087, 11063}, {3567, 21663}, {5013, 8744}, {5237, 11475}, {5238, 11476}, {5286, 34866}, {5351, 10633}, {5352, 10632}, {5410, 6519}, {5411, 6522}, {5663, 9545}, {5876, 22948}, {5890, 34566}, {6241, 11430}, {6453, 11474}, {6454, 11473}, {6696, 12022}, {6776, 15579}, {7592, 10606}, {7722, 32136}, {8743, 22332}, {9544, 18439}, {9707, 12112}, {10282, 11455}, {10564, 11444}, {10986, 15513}, {11003, 12300}, {11381, 11464}, {11425, 15032}, {11440, 13352}, {12038, 15305}, {12244, 32607}, {12254, 14216}, {12279, 18475}, {12290, 13367}, {12294, 20190}, {13293, 18912}, {13403, 23294}, {13474, 26882}, {14641, 15080}, {14644, 25564}, {14853, 15578}, {15021, 15472}, {15054, 15463}, {15083, 34148}, {17821, 22334}, {18388, 34563}, {18913, 19457}, {23328, 26879}, {25563, 26917}

X(35476) = X(2)X(3)∩X(74)X(695)

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

X(35476) lies on these lines: {2, 3}, {74, 695}, {112, 34870}, {682, 9862}, {689, 1300}, {1078, 17984}, {1235, 5976}, {1691, 10312}, {2211, 3094}, {3186, 22062}

X(35477) = X(2)X(3)∩X(74)X(1181)

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 - 6*a^2*c^2 + 8*b^2*c^2 + 3*c^4) : :
Barycentrics    a^2*(3*a^4-6*(b^2+c^2)*a^2+3*b^4+8*b^2*c^2+3*c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    3 Sin[2 A]+Tan[A] : :
Trilinears    6 cos A + sec A : :

X(35477) lies on these lines: {2, 3}, {6, 32329}, {54, 3532}, {64, 1614}, {74, 1181}, {112, 5013}, {154, 12290}, {155, 11440}, {185, 11204}, {232, 15515}, {574, 8743}, {578, 21663}, {1192, 3567}, {1204, 7592}, {1498, 11464}, {1853, 12289}, {1870, 5217}, {1902, 17502}, {1986, 10574}, {1993, 7689}, {3043, 9704}, {3092, 6412}, {3093, 6411}, {3357, 11456}, {5023, 10312}, {5024, 8778}, {5204, 6198}, {5351, 8739}, {5352, 8740}, {5410, 6455}, {5411, 6456}, {5493, 15177}, {5585, 10986}, {5621, 32234}, {5890, 11425}, {6000, 9707}, {6146, 23328}, {6241, 10606}, {6403, 31884}, {6409, 10880}, {6410, 10881}, {6696, 11457}, {6749, 8553}, {6800, 10575}, {7722, 15041}, {7999, 15103}, {8550, 15578}, {8745, 14806}, {9638, 10060}, {9786, 15033}, {10114, 20417}, {10193, 13403}, {10249, 15073}, {10311, 15513}, {10619, 32401}, {10990, 13293}, {11202, 11381}, {11441, 12038}, {11449, 15062}, {11454, 12163}, {11462, 19087}, {11463, 19088}, {11470, 20190}, {12022, 26937}, {12041, 13148}, {12112, 14530}, {12292, 15035}, {12293, 23293}, {12294, 17508}, {12901, 30714}, {13093, 26864}, {14157, 17821}, {14448, 16219}, {14810, 19124}, {15036, 33537}, {18396, 23294}, {18445, 32138}, {19347, 34469}, {21659, 23329}

X(35477) = {X(3),X(4)}-harmonic conjugate of X(32534)

X(35478) = X(2)X(3)∩X(74)X(1199)

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 - 6*a^2*c^2 + 11*b^2*c^2 + 3*c^4) : :
Barycentrics 6 Sin[2 A]+5 Tan[A] : :

X(35478) lies on these lines: {2, 3}, {74, 1199}, {112, 5041}, {1498, 3431}, {1902, 31662}, {1994, 32138}, {3092, 6434}, {3093, 6433}, {3357, 15032}, {3567, 11204}, {5462, 15055}, {6484, 10880}, {6485, 10881}, {9786, 11270}, {10990, 12242}, {11424, 11468}, {11456, 11738}, {12002, 32110}, {12112, 13367}, {15061, 15807}, {15515, 33885}, {15739, 17854}, {31948, 33179}, {32137, 35265}

X(35479) = X(2)X(3)∩X(74)X(1620)

Barycentrics a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^4 - 10*a^2*b^2 + 5*b^4 - 10*a^2*c^2 + 4*b^2*c^2 + 5*c^4) : :
Barycentrics 5 Sin[2 A]-3 Tan[A] : :

X(35479) lies on these lines: {2, 3}, {74, 1620}, {1192, 1614}, {1829, 31666}, {1986, 15034}, {3431, 11426}, {3592, 10881}, {3594, 10880}, {5013, 10986}, {5351, 10641}, {5352, 10642}, {5410, 6448}, {5411, 6447}, {5890, 17821}, {6152, 15043}, {7592, 11202}, {7689, 35264}, {7999, 12300}, {8567, 11455}, {8743, 35007}, {9707, 11438}, {9786, 11464}, {10311, 31652}, {10605, 26882}, {10632, 22238}, {10633, 22236}, {10985, 15515}, {11423, 19357}, {11432, 13472}, {11441, 32110}, {11449, 15801}, {11468, 16835}, {11477, 19128}, {11704, 18405}, {12024, 18912}, {12112, 34469}, {12254, 26869}, {12289, 26958}, {12292, 15021}, {13093, 13452}, {15576, 19189}, {15579, 20987}, {17845, 26917}

X(35480) = X(2)X(3)∩X(74)X(1853)

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

X(35480) lies on these lines: {2, 3}, {64, 16000}, {74, 1853}, {113, 35264}, {125, 16219}, {155, 12278}, {185, 34786}, {1179, 18848}, {1181, 12289}, {1192, 26917}, {1204, 18383}, {1289, 14388}, {1300, 1629}, {1531, 9306}, {1614, 17845}, {1829, 33697}, {1870, 12943}, {1986, 3060}, {1993, 17702}, {2777, 11550}, {2781, 6403}, {2794, 14983}, {3167, 12383}, {3357, 11572}, {3426, 33565}, {3448, 34796}, {5422, 7706}, {5523, 7737}, {5622, 23049}, {5878, 16659}, {5889, 12293}, {5890, 18396}, {5893, 15152}, {5895, 6152}, {6198, 12953}, {6746, 13491}, {6776, 11216}, {7592, 21659}, {7722, 12902}, {7747, 8743}, {8541, 11645}, {9707, 34785}, {10311, 19220}, {10421, 14264}, {10605, 18405}, {10723, 20774}, {10880, 23251}, {10881, 23261}, {11438, 13851}, {11456, 18400}, {12140, 13202}, {12254, 19347}, {12295, 12828}, {12300, 32338}, {13166, 16983}, {13399, 18381}, {13568, 18912}, {13754, 27365}, {14644, 26958}, {14852, 18392}, {15534, 32234}, {18445, 30522}, {19124, 29323}, {21663, 23325}, {31383, 32111}

X(35481) = X(2)X(3)∩X(74)X(1899)

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

X(35481) lies on these lines: {2, 3}, {33, 4316}, {34, 4324}, {64, 34224}, {68, 11440}, {74, 1899}, {112, 2549}, {125, 11204}, {146, 3043}, {154, 32111}, {184, 2777}, {340, 11057}, {1181, 5925}, {1204, 13403}, {1514, 10192}, {1614, 5878}, {1870, 4302}, {1968, 7756}, {1986, 20127}, {1994, 34796}, {2781, 6776}, {2883, 9707}, {2979, 4549}, {3168, 5667}, {3357, 11457}, {3431, 11744}, {4294, 9627}, {4299, 6198}, {4550, 12140}, {4846, 5012}, {5702, 14836}, {5894, 6146}, {5895, 19357}, {6103, 11648}, {6241, 19467}, {6344, 18850}, {6696, 15153}, {6781, 10311}, {7687, 10193}, {7967, 31948}, {8541, 19924}, {9306, 16163}, {9638, 12950}, {9833, 12290}, {10605, 12022}, {10606, 18396}, {10733, 23293}, {11206, 12112}, {11270, 22466}, {11381, 34785}, {11442, 17702}, {11455, 31383}, {11456, 15311}, {11468, 26937}, {11577, 12250}, {11738, 13622}, {11821, 22951}, {12111, 12118}, {12121, 12292}, {12133, 15060}, {12278, 15062}, {12289, 14216}, {12383, 12825}, {12828, 16111}, {13367, 22802}, {13851, 23329}, {15055, 26913}, {16237, 18880}, {16659, 17845}, {18390, 21663}, {18442, 22948}, {19124, 29317}, {25738, 32138}

X(35482) = X(2)X(3)∩X(74)X(3574)

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

X(35482) lies on these lines: {2, 3}, {74, 3574}, {110, 18488}, {185, 22949}, {389, 7731}, {933, 13597}, {3043, 24981}, {3431, 9833}, {3448, 11264}, {3567, 23329}, {6153, 10625}, {6247, 15032}, {6696, 17835}, {7699, 22802}, {7703, 9927}, {7722, 12300}, {9820, 20125}, {10095, 15061}, {10112, 13482}, {10264, 14627}, {11424, 23294}, {11430, 12254}, {11592, 11817}, {12121, 22804}, {12242, 13399}, {15033, 20299}, {15081, 32767}, {31455, 33885}

X(35483) = X(2)X(3)∩X(74)X(5095)

Barycentrics (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(13*a^6 - 25*a^4*b^2 + 11*a^2*b^4 + b^6 - 25*a^4*c^2 + 42*a^2*b^2*c^2 - b^4*c^2 + 11*a^2*c^4 - b^2*c^4 + c^6) : :
Barycentrics (13*a^6-25*(b^2+c^2)*a^4+(11*b^4+42*b^2*c^2+11*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(35483) lies on these lines: {2, 3}, {74, 5095}, {112, 5702}, {8550, 10606}, {11204, 11433}, {14528, 15105}, {17983, 18852}

X(35484) = X(2)X(3)∩X(74)X(5480)

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

X(35484) lies on these lines: {2, 3}, {51, 20417}, {74, 5480}, {112, 6749}, {1300, 12074}, {1353, 12317}, {1514, 7699}, {1986, 12294}, {1990, 13337}, {2935, 20300}, {6103, 33843}, {6696, 9781}, {8550, 15033}, {10294, 12292}, {11064, 16261}, {11430, 16658}, {11455, 23292}, {11464, 16654}, {11468, 11745}, {14389, 14915}, {15472, 32234}, {16194, 16534}, {16656, 26882}

X(35485) = X(2)X(3)∩X(74)X(5486)

Barycentrics (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(7*a^6 - 13*a^4*b^2 + 5*a^2*b^4 + b^6 - 13*a^4*c^2 + 18*a^2*b^2*c^2 - b^4*c^2 + 5*a^2*c^4 - b^2*c^4 + c^6) : :
Barycentrics (7*a^6-13*(b^2+c^2)*a^4+(5*b^4+18*b^2*c^2+5*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(35485) lies on these lines: {2, 3}, {74, 5486}, {112, 5063}, {182, 15472}, {184, 10990}, {340, 14907}, {477, 10098}, {841, 935}, {1296, 1300}, {1299, 20187}, {1352, 16163}, {1614, 12250}, {1899, 11204}, {2549, 6103}, {2696, 32710}, {3431, 10293}, {3563, 30256}, {4549, 10564}, {4846, 16111}, {5095, 11179}, {5181, 11180}, {5656, 11464}, {5894, 19357}, {5897, 30251}, {5921, 12383}, {6146, 8567}, {6193, 11440}, {6225, 9707}, {6451, 13884}, {6452, 13937}, {8550, 10605}, {11003, 15463}, {11468, 18913}, {12022, 18931}, {13367, 20427}, {13394, 20725}, {15055, 18911}, {15578, 19457}, {18396, 23328}, {19128, 33750}, {31804, 34469}, {32111, 35260}

X(35486) = X(2)X(3)∩X(74)X(5656)

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

X(35486) lies on these lines: {2, 3}, {54, 5486}, {68, 30714}, {74, 5656}, {232, 21843}, {254, 17983}, {340, 7763}, {1092, 5095}, {1112, 13340}, {1177, 32247}, {1614, 18913}, {1899, 11202}, {3087, 10986}, {5063, 10312}, {5181, 15034}, {5410, 35256}, {5411, 35255}, {5654, 32110}, {5878, 10990}, {6193, 11449}, {6221, 13937}, {6398, 13884}, {6759, 20417}, {6776, 11464}, {7689, 16534}, {8550, 18916}, {9540, 10881}, {9707, 18909}, {10182, 11438}, {10192, 10605}, {10282, 26937}, {10293, 11270}, {10519, 19128}, {10880, 13935}, {11456, 18931}, {11468, 12250}, {12165, 13392}, {13148, 18436}, {14649, 14900}, {15073, 35371}, {15448, 23328}, {18925, 26879}, {18945, 26917}, {23329, 31383}, {26882, 34781}

X(35487) = X(2)X(3)∩X(74)X(5893)

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

X(35487) lies on these lines: {2, 3}, {74, 5893}, {1112, 14128}, {1870, 7173}, {1986, 5907}, {2883, 23294}, {2904, 17814}, {3580, 5448}, {3614, 6198}, {6146, 14644}, {6746, 18874}, {7687, 13367}, {10192, 12289}, {12290, 23332}, {12292, 23515}, {13148, 34826}, {13382, 26879}, {15105, 32125}, {16252, 25739}, {16659, 23325}, {18394, 34782}, {18504, 23293}, {20299, 32111}

X(35488) = X(2)X(3)∩X(74)X(5895)

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

X(35488) lies on these lines: {2, 3}, {64, 23294}, {74, 5895}, {110, 12293}, {112, 13881}, {113, 2904}, {115, 8743}, {125, 22802}, {154, 12289}, {156, 10113}, {264, 15031}, {265, 32139}, {847, 14249}, {974, 6241}, {1112, 5876}, {1300, 33640}, {1495, 34786}, {1498, 25739}, {1514, 6247}, {1539, 32138}, {1614, 18396}, {1853, 12290}, {1870, 10896}, {1879, 8745}, {1986, 12111}, {1993, 5448}, {2883, 11457}, {3531, 13418}, {5925, 10721}, {6152, 11743}, {6198, 10895}, {6759, 13851}, {7592, 18390}, {7687, 11456}, {7699, 22750}, {8746, 18353}, {9707, 12140}, {10605, 26917}, {10733, 11449}, {10880, 23261}, {10881, 23251}, {11264, 18445}, {11381, 23325}, {11470, 18553}, {12112, 34780}, {12133, 13491}, {12143, 22681}, {12254, 26864}, {12645, 31948}, {14157, 18394}, {14216, 32111}, {15024, 22948}, {16655, 23324}, {17845, 26882}, {18383, 26883}, {18430, 34397}, {32364, 32377}

X(35489) = X(2)X(3)∩X(74)X(5900)

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

X(35489) lies on these lines: {2, 3}, {74, 5900}, {539, 12383}, {935, 29316}, {1138, 1157}, {6242, 13630}, {11270, 14216}, {11649, 25406}, {11692, 15043}, {12244, 14157}, {13376, 15024}, {14861, 23060}, {19128, 19924}, {20189, 32710}, {32608, 34153}

X(35490) = X(2)X(3)∩X(74)X(5925)

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

X(35490) lies on these lines: {2, 3}, {64, 25739}, {74, 5925}, {146, 34799}, {156, 1539}, {254, 18846}, {1112, 13491}, {1300, 18848}, {1498, 12289}, {1514, 34782}, {1531, 13346}, {1614, 15472}, {1853, 18394}, {1870, 12953}, {1902, 33697}, {1974, 29323}, {1986, 5895}, {2904, 11456}, {3357, 13851}, {5876, 12133}, {5878, 34224}, {6198, 12943}, {7592, 13403}, {7728, 12419}, {7748, 8743}, {9833, 32111}, {9927, 12295}, {10113, 32138}, {10152, 14264}, {10606, 23294}, {10733, 12111}, {11363, 28168}, {11381, 34786}, {11440, 14852}, {11441, 17702}, {11449, 20771}, {11457, 15311}, {11468, 14644}, {11470, 11645}, {12244, 34469}, {12278, 18451}, {12290, 22535}, {14157, 17845}, {15062, 18392}, {18526, 31948}

X(35491) = X(2)X(3)∩X(74)X(6146)

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

X(35491) lies on these lines: {2, 3}, {74, 6146}, {1204, 12022}, {1614, 15311}, {1870, 15338}, {1986, 16111}, {1990, 13489}, {2777, 13367}, {2883, 11464}, {3357, 34224}, {5523, 7756}, {5878, 9707}, {5893, 10721}, {5894, 6241}, {5907, 12292}, {5925, 19357}, {6152, 16622}, {6198, 15326}, {6247, 12289}, {6696, 25739}, {7722, 14677}, {8567, 18396}, {10282, 32111}, {10606, 11457}, {10619, 10628}, {11454, 12359}, {11456, 20427}, {12133, 14128}, {12134, 15062}, {12290, 34782}, {12300, 14641}, {12825, 30714}, {12897, 32110}, {13403, 21663}, {13474, 32903}, {13568, 15033}, {13851, 25563}, {15463, 20127}, {15644, 22948}, {16659, 34785}, {18394, 23332}, {23294, 23328}

X(35492) = X(2)X(3)∩X(74)X(8550)

Barycentrics (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(10*a^6 - 19*a^4*b^2 + 8*a^2*b^4 + b^6 - 19*a^4*c^2 + 30*a^2*b^2*c^2 - b^4*c^2 + 8*a^2*c^4 - b^2*c^4 + c^6) : :
Barycentrics (10*a^6-19*(b^2+c^2)*a^4+2*(4*b^4+15*b^2*c^2+4*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

X(35492) lies on these lines: {2, 3}, {74, 8550}, {541, 10294}, {10990, 11430}, {12022, 20417}, {14389, 16111}, {16163, 18553}

X(35493) = X(2)X(3)∩X(74)X(9544)

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

X(35493) lies on these lines: {2, 3}, {74, 9544}, {110, 11204}, {146, 25564}, {184, 15055}, {1204, 9545}, {3098, 11416}, {3431, 12219}, {5921, 15578}, {7280, 9538}, {8588, 10313}, {8589, 22240}, {9306, 15051}, {10193, 16163}, {11202, 13445}, {11270, 16665}, {11468, 12038}, {12278, 25563}, {21663, 34986}, {23291, 32607}

X(35494) = X(2)X(3)∩X(74)X(9545)

Barycentrics a^2*(4*a^8 - 8*a^6*b^2 + 8*a^2*b^6 - 4*b^8 - 8*a^6*c^2 + 21*a^4*b^2*c^2 - 10*a^2*b^4*c^2 - 3*b^6*c^2 - 10*a^2*b^2*c^4 + 14*b^4*c^4 + 8*a^2*c^6 - 3*b^2*c^6 - 4*c^8) : :
Barycentrics a^2*(4*a^8-8*(b^2+c^2)*a^6+21*b^2*c^2*a^4+2*(b^2+c^2)*(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^2-(4*b^4+11*b^2*c^2+4*c^4)*(b^2-c^2)^2) : :

X(35494) lies on these lines: {2, 3}, {74, 9545}, {193, 15578}, {578, 15055}, {3047, 15054}, {3357, 9544}, {3431, 13491}, {3448, 25564}, {6102, 11270}, {6696, 34799}, {7280, 9539}, {11204, 34148}, {12041, 32136}, {12901, 14683}, {13198, 15021}, {13452, 16665}, {18913, 32607}

X(35495) = X(2)X(3)∩X(74)X(9703)

Barycentrics a^2*(4*a^8 - 8*a^6*b^2 + 8*a^2*b^6 - 4*b^8 - 8*a^6*c^2 + 21*a^4*b^2*c^2 - 11*a^2*b^4*c^2 - 2*b^6*c^2 - 11*a^2*b^2*c^4 + 12*b^4*c^4 + 8*a^2*c^6 - 2*b^2*c^6 - 4*c^8) : :
Barycentrics a^2*(4*a^8-8*(b^2+c^2)*a^6+21*b^2*c^2*a^4+(b^2+c^2)*(8*b^4-19*b^2*c^2+8*c^4)*a^2-2*(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :

X(35495) lies on these lines: {2, 3}, {74, 9703}, {399, 10606}, {7280, 9641}, {11204, 22115}, {11468, 11999}, {11898, 15578}, {11935, 12041}, {12121, 23329}, {15040, 18451}, {15041, 18445}

X(35496) = X(2)X(3)∩X(74)X(9704)

Barycentrics a^2*(4*a^8 - 8*a^6*b^2 + 8*a^2*b^6 - 4*b^8 - 8*a^6*c^2 + 19*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 + 12*b^4*c^4 + 8*a^2*c^6 - 2*b^2*c^6 - 4*c^8) : :
Barycentrics a^2*(4*a^8-8*(b^2+c^2)*a^6+19*b^2*c^2*a^4+(b^2+c^2)*(8*b^4-17*b^2*c^2+8*c^4)*a^2-2*(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2)) : :

X(35496) lies on these lines: {2, 3}, {49, 11204}, {74, 9704}, {1181, 15041}, {7280, 9642}, {8567, 10620}, {9703, 32138}, {11935, 11999}, {15055, 32046}, {26944, 32607}

X(35497) = X(2)X(3)∩X(74)X(9705)

Barycentrics a^2*(3*a^8 - 6*a^6*b^2 + 6*a^2*b^6 - 3*b^8 - 6*a^6*c^2 + 15*a^4*b^2*c^2 - 8*a^2*b^4*c^2 - b^6*c^2 - 8*a^2*b^2*c^4 + 8*b^4*c^4 + 6*a^2*c^6 - b^2*c^6 - 3*c^8) : :
Barycentrics a^2*(3*a^8-6*(b^2+c^2)*a^6+15*b^2*c^2*a^4+2*(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^2-(3*b^4+7*b^2*c^2+3*c^4)*(b^2-c^2)^2) : :

X(35497) lies on these lines: {2, 3}, {49, 12041}, {74, 9705}, {146, 5894}, {185, 9706}, {323, 7689}, {1092, 11454}, {1147, 11468}, {3357, 11449}, {5204, 9538}, {7280, 9643}, {8567, 11441}, {9545, 10605}, {10282, 13445}, {10313, 15513}, {10539, 33556}, {11202, 12279}, {11204, 12111}, {12121, 13561}, {12162, 15035}, {12278, 23329}, {12290, 35265}, {14516, 23328}, {15051, 15052}, {15057, 32607}, {15063, 25564}, {15069, 15578}, {15515, 22240}, {16163, 25563}, {21663, 34148}, {22115, 32210}

X(35498) = X(2)X(3)∩X(74)X(9706)

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

X(35498) lies on these lines: {2, 3}, {49, 10620}, {54, 12041}, {74, 9706}, {185, 15041}, {265, 25563}, {399, 12038}, {1192, 13321}, {1204, 15087}, {1511, 15062}, {3521, 16111}, {5204, 9642}, {5663, 9705}, {5944, 13445}, {6288, 16163}, {7280, 9644}, {8567, 18445}, {10116, 20126}, {10540, 33541}, {10564, 15606}, {11204, 34783}, {12162, 32609}, {12901, 23236}, {12902, 13561}, {13403, 15061}, {13630, 15055}, {14531, 32608}, {14677, 15806}, {15040, 18350}, {16003, 25564}, {32210, 34148}

X(35499) = X(2)X(3)∩X(74)X(9716)

Barycentrics a^2*(10*a^8 - 20*a^6*b^2 + 20*a^2*b^6 - 10*b^8 - 20*a^6*c^2 + 59*a^4*b^2*c^2 - 30*a^2*b^4*c^2 - 9*b^6*c^2 - 30*a^2*b^2*c^4 + 38*b^4*c^4 + 20*a^2*c^6 - 9*b^2*c^6 - 10*c^8) : :
Barycentrics a^2*(10*a^8-20*(b^2+c^2)*a^6+59*b^2*c^2*a^4+10*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^2*(5*b^2+2*c^2)*(2*b^2+5*c^2)) : :

X(35499) lies on these lines: {2, 3}, {74, 9716}, {576, 15055}, {9737, 14388}, {11004, 12041}, {11204, 23061}

X(35500) = X(2)X(3)∩X(74)X(9729)

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

X(35500) lies on these lines: {2, 3}, {49, 15060}, {54, 5907}, {74, 9729}, {113, 6689}, {156, 14805}, {182, 6241}, {184, 15058}, {195, 31834}, {323, 11591}, {567, 5876}, {569, 4550}, {578, 11459}, {1147, 15056}, {1173, 16625}, {1176, 16835}, {1199, 13434}, {1204, 15045}, {1614, 15030}, {1994, 18436}, {2888, 12370}, {2914, 7723}, {3047, 5609}, {3581, 10095}, {5012, 12162}, {5422, 12163}, {5446, 7691}, {5562, 15033}, {5663, 13353}, {5866, 7769}, {5891, 34148}, {5925, 15578}, {5944, 35265}, {6102, 34545}, {7689, 15043}, {7999, 13346}, {8718, 13474}, {9545, 15068}, {9625, 12571}, {9659, 10591}, {9672, 10590}, {9723, 32822}, {9730, 11440}, {10113, 13565}, {10540, 10610}, {10541, 15579}, {10821, 11806}, {10984, 12290}, {11003, 32139}, {11411, 32166}, {11412, 11424}, {11422, 15083}, {11438, 15024}, {11444, 13352}, {11454, 15028}, {11562, 15054}, {11587, 20625}, {11695, 21663}, {12134, 12254}, {12307, 14449}, {12317, 18914}, {13198, 14094}, {13336, 15072}, {13472, 34801}, {14128, 22115}, {14389, 22660}, {15426, 30531}, {15582, 34775}, {16252, 32345}, {16261, 26883}, {16881, 32608}, {18435, 32046}, {19357, 33537}, {22330, 32599}, {31371, 34439}

X(35501) = X(2)X(3)∩X(74)X(9777)

Barycentrics    a^2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(5*a^4 - 10*a^2*b^2 + 5*b^4 - 10*a^2*c^2 + 26*b^2*c^2 + 5*c^4) : :
Barycentrics    a^2*(5*a^4-10*(b^2+c^2)*a^2+(b^2+5*c^2)*(5*b^2+c^2))*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    5 Cos[A] Sin[A]+4 Tan[A] : :

X(35501) lies on these lines: {2, 3}, {64, 11426}, {74, 9777}, {184, 3426}, {578, 13093}, {1204, 3527}, {1384, 33843}, {3167, 11472}, {3357, 11432}, {3531, 11204}, {3796, 11820}, {5093, 19124}, {5412, 6445}, {5413, 6446}, {6000, 17809}, {6417, 11474}, {6418, 11473}, {8567, 10110}, {8780, 16194}, {9605, 14581}, {9691, 10880}, {10605, 15004}, {11425, 12315}, {11430, 32063}, {11455, 26864}, {12017, 14855}, {12160, 15062}, {13474, 14530}

X(35502) = X(2)X(3)∩X(74)X(9781)

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 + 8*b^2*c^2 + c^4) : :
Barycentrics    a^2*(a^4-2*(b^2+c^2)*a^2+b^4+8*b^2*c^2+c^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    Sin[2 A]+3 Tan[A] : :
Trilinears    2 cos A + 3 sec A : :

X(35502) lies on these lines: {2, 3}, {6, 6241}, {33, 5563}, {34, 3746}, {51, 3357}, {54, 1498}, {61, 11475}, {62, 11476}, {64, 1173}, {74, 9781}, {155, 15305}, {184, 13474}, {185, 34565}, {394, 15058}, {569, 14915}, {575, 19124}, {576, 12294}, {578, 11381}, {1147, 16194}, {1181, 11423}, {1192, 11468}, {1204, 10110}, {1493, 18445}, {1614, 11425}, {1870, 3303}, {1902, 10222}, {1968, 5007}, {1974, 20190}, {1986, 15054}, {1993, 12162}, {2914, 12308}, {2935, 14644}, {3060, 12163}, {3092, 3594}, {3093, 3592}, {3199, 31652}, {3304, 6198}, {3426, 11426}, {3527, 13452}, {3567, 10605}, {3574, 22802}, {3796, 8718}, {3964, 32822}, {4550, 10625}, {4994, 19172}, {5023, 10986}, {5206, 10985}, {5237, 10642}, {5238, 10641}, {5412, 6453}, {5413, 6454}, {5480, 15579}, {5609, 12133}, {6000, 7592}, {6247, 16657}, {6419, 11473}, {6420, 11474}, {6425, 10880}, {6426, 10881}, {6759, 32062}, {7772, 8743}, {7802, 32002}, {8744, 9605}, {8778, 15433}, {8976, 9694}, {9608, 34866}, {9707, 11430}, {9833, 16658}, {9925, 18440}, {9927, 18488}, {10311, 35007}, {10312, 22331}, {10541, 19128}, {10721, 19457}, {11387, 11817}, {11402, 12315}, {11439, 18451}, {11441, 13352}, {11457, 12241}, {11470, 22330}, {11472, 12111}, {11550, 13403}, {12022, 14216}, {12038, 35264}, {12161, 18439}, {12168, 25714}, {12250, 14853}, {12279, 13434}, {12292, 14094}, {12897, 18474}, {13035, 13036}, {13336, 14641}, {13346, 15030}, {13382, 15004}, {13445, 15043}, {14157, 15811}, {15805, 20791}, {16266, 18435}, {16654, 34782}, {16659, 19467}, {19005, 23275}, {19006, 23269}, {31521, 33750}

X(35502) = {X(3),X(4)}-harmonic conjugate of X(10594)

X(35503) = X(2)X(3)∩X(74)X(9833)

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

X(35503) lies on these lines: {2, 3}, {50, 1249}, {70, 11270}, {74, 9833}, {1192, 12022}, {1620, 18396}, {5023, 5523}, {5878, 26882}, {5925, 32111}, {6145, 20421}, {6225, 9934}, {10606, 16659}, {11411, 12383}, {11457, 21663}, {11468, 14216}, {12112, 12250}, {12254, 18909}, {12289, 26937}, {14157, 20427}, {18912, 32903}, {19124, 33751}, {20070, 31948}

X(35504) = X(11)X(18635)∩X(55)X(2249)

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

See César Lozada, Euclid 380 .

X(35504) lies on the incircle and these lines: {11, 18635}, {55, 2249}, {1364, 18734}, {1365, 21746}, {2361, 7062}

X(35504) = crosspoint of X(7) and X(3002)
X(35504) = X(7)-Ceva conjugate of X(3002)
X(35504) = isogonal conjugate of X(8680) wrt these triangles: intouch, Mandart-incircle, 2nd midarc
X(35504) = X(2249)-of-Mandart-incircle triangle

X(35505) = X(6)X(840)∩X(220)X(6078)

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

See César Lozada, Euclid 380 .

X(35505) lies on the Brocard inellipse and these lines: {6, 840}, {220, 6078}, {244, 3124}, {649, 3937}, {663, 1015}, {813, 24484}, {1017, 1055}, {1146, 4462}, {1362, 1458}, {1407, 24016}, {1566, 3323}, {1977, 22383}, {3125, 23755}, {3218, 34253}, {3446, 23990}, {3726, 17747}, {14991, 20982}, {17966, 35128}, {20671, 20860}

X(35505) = isogonal conjugate of the isotomic conjugate of X(35094)
X(35505) = crossdifference of every pair of points on line {X(666), X(885)}
X(35505) = crosspoint of X(i) and X(j) for these (i,j): (6, 665), (241, 513), (918, 22116)
X(35505) = crosssum of X(i) and X(j) for these (i,j): (2, 666), (100, 294), (1252, 5377)
X(35505) = X(i)-Ceva conjugate of X(j) for these (i,j): (6, 665), (1086, 3675)
X(35505) = X(i)-isoconjugate-of X(j) for these {i,j}: {673, 5377}, {765, 6185}
X(35505) = X(i)-reciprocal conjugate of X(j) for these (i,j): (665, 666), (1015, 6185), (1362, 4998)
X(35505) = barycentric product X(i)*X(j) for these {i, j}: {6, 35094}, {11, 1362}, {55, 3323}, {241, 17435}
X(35505) = barycentric quotient X(i)/X(j) for these (i, j): (665, 666), (1015, 6185), (1362, 4998)
X(35505) = trilinear product X(i)*X(j) for these {i, j}: {31, 35094}, {41, 3323}, {85, 15615}, {244, 6184}
X(35505) = trilinear quotient X(i)/X(j) for these (i, j): (244, 6185), (672, 5377), (1362, 4564)

X(35506) = X(6)X(8687)∩X(1977)X(7117)

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

See César Lozada, Euclid 380 .

X(35506) lies on the Brocard inellipse and these lines: {6, 8687}, {1977, 7117}, {2170, 3124}, {17419, 17420}

X(35506) = crosssum of X(2) and X(6648)
X(35506) = X(1682)-reciprocal conjugate of X(4998)
X(35506) = barycentric product X(11)*X(1682)
X(35506) = barycentric quotient X(1682)/X(4998)
X(35506) = trilinear product X(1682)*X(2170)
X(35506) = trilinear quotient X(1682)/X(4564)

X(35507) = CROSSPOINT OF X(351) AND X(598)

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

See César Lozada, Euclid 380 .

X(35507) lies on the Lemoine inellipse and this line: {7668, 8288}

X(35507) = crosspoint of X(351) and X(598)
X(35507) = crosssum of X(574) and X(892)
X(35507) = X(598)-Ceva conjugate of X(351)

X(35508) = ISOGONAL CONJUGATE OF X(23586)

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

See César Lozada, Euclid 380 .

X(35508) lies on the Steiner inellipse and these lines: {2, 4569}, {6, 2338}, {9, 1742}, {32, 7367}, {37, 23972}, {45, 16588}, {115, 5514}, {220, 3939}, {657, 3270}, {1015, 17435}, {1086, 13609}, {1212, 16578}, {1500, 7079}, {2310, 3119}, {3041, 35026}, {4081, 4130}, {5745, 35074}, {6554, 13466}, {16589, 35075}, {20310, 23986}, {26932, 35094}

X(35508) = midpoint of X(9) and X(14943)
X(35508) = complement of X(4569)
X(35508) = isogonal conjugate of X(23586)
X(35508) = (medial)-isotomic conjugate of X(3900)
X(35508) = crossdifference of every pair of points on line {X(934), X(1633)}
X(35508) = crosspoint of X(i) and X(j) for these (i,j): (220, 4105), (480, 4130), (657, 7071)
X(35508) = crosssum of X(i) and X(j) for these (i,j): (6, 934), (279, 4626), (479, 4617)
X(35508) = X(9)-Beth conjugate of X(23972)
X(35508) = X(2)-Ceva conjugate of X(3900)
X(35508) = X(i)-complementary conjugate of X(j) for these (i,j): (32, 7658), (41, 4885), (55, 17072)
X(35508) = X(i)-isoconjugate-of X(j) for these {i,j}: {2, 24013}, {6, 24011}, {59, 23062}
X(35508) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 24011), (6, 23586), (31, 24013)
X(35508) = center of the circumconic {{ A, B, C, X(2), X(9), X(200), X(281), X(282) }}
X(35508) = barycentric square of X(3900)
X(35508) = barycentric product X(i)*X(j) for these {i, j}: {1, 24010}, {6, 23970}, {8, 3022}, {9, 3119}
X(35508) = barycentric quotient X(i)/X(j) for these (i, j): (1, 24011), (31, 24013), (32, 23971), (220, 1275)
X(35508) = trilinear product X(i)*X(j) for these {i, j}: {2, 24012}, {6, 24010}, {9, 3022}, {11, 6602}
X(35508) = trilinear quotient X(i)/X(j) for these (i, j): (2, 24011), (6, 24013), (11, 23062), (31, 23971)

X(35509) = X(1086)X(21202)∩X(1566)X(35092)

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

See César Lozada, Euclid 380 .

X(35509) lies on the Steiner inellipse and these lines: {1086, 21202}, {1566, 35092}, {6547, 35093}

X(35509) = X(i)-complementary conjugate of X(j) for these (i,j): (244, 926), (665, 21232), (667, 24980)
X(35509) = center of the circumconic {{ A, B, C, X(2), X(11), X(23989), X(31611), X(31619) }}

X(35510) = ISOTOMIC CONJUGATE OF X(3146)

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

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

X(35510) lies on these lines: {2, 15851}, {69, 3522}, {144, 306}, {253, 3146}, {264, 3832}, {287, 20080}, {297, 20218}, {305, 10513}, {307, 3160}, {1494, 6527}, {4035, 27382}, {5068, 32000}, {6330, 17037}, {8797, 15022}, {9229, 33025}, {11090, 32814}, {16284, 20336}

X(35510) = isotomic conjugate of X(3146)
X(35510) = polar conjugate of X(33630)
X(35510) = cyclocevian conjugate of the isogonal conjugate of X(1661)
X(35510) = cyclocevian conjugate of the isotomic conjugate of X(6225)
X(35510) = Cevapoint of X(i) and X(j) for these (i,j): (525, 13611), (1270, 1271)
X(35510) = X(20)-cross conjugate of X(2)
X(35510) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 18594}, {41, 18624}, {48, 33630}
X(35510) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 18594), (2, 3146), (4, 33630)
X(35510) = lies on the circumconics with center X(i) for i in {13609, 13611, 15526, 15613}
X(35510) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(69)}} and {{A, B, C, X(3), X(3832)}}
X(35510) = trilinear pole of the line {525, 7658}
X(35510) = barycentric product X(76)*X(3532)
X(35510) = barycentric quotient X(i)/X(j) for these (i, j): (1, 18594), (2, 3146), (4, 33630), (7, 18624)
X(35510) = trilinear product X(i)*X(j) for these {i, j}: {75, 3532}, {1895, 15400}
X(35510) = trilinear quotient X(i)/X(j) for these (i, j): (2, 18594), (75, 3146), (85, 18624), (92, 33630)

X(35511) = ISOTOMIC CONJUGATE OF X(148)

Barycentrics (a^4+(b^2-3*c^2)*a^2-b^4+b^2*c^2+c^4)*(a^4-(3*b^2-c^2)*a^2+b^4+b^2*c^2-c^4) : :
X(35511) = X(8591)-4*X(18823), 3*X(8591)-4*X(33799), 3*X(18823)-X(33799)

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

X(35511) lies on these lines: {2, 31372}, {69, 19610}, {148, 523}, {193, 5967}, {385, 468}, {524, 2076}, {892, 23991}, {2770, 7665}, {3266, 7779}, {4062, 9395}, {4580, 25054}, {4590, 23992}, {8591, 18823}, {14712, 16103}, {26081, 35148}, {26147, 35080}

X(35511) = reflection of X(i) in X(j) for these (i,j): (892, 23991), (4590, 23992), (31372, 31998)
X(35511) = anticomplement of X(31998)
X(35511) = complement of X(31372)
X(35511) = isogonal conjugate of X(20998)
X(35511) = isotomic conjugate of X(148)
X(35511) = cyclocevian conjugate of X(670)
X(35511) = antitomic conjugate of X(4590)
X(35511) = Cevapoint of X(i) and X(j) for these (i,j): (2, 20094), (523, 23991), (524, 23992)
X(35511) = crosspoint of X(2) and X(31373)
X(35511) = X(798)-anticomplementary conjugate of X(31372)
X(35511) = X(99)-cross conjugate of X(2)
X(35511) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 2640}, {19, 22143}, {31, 148}
X(35511) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 2640), (2, 148), (3, 22143)
X(35511) = lies on the circumconics with center X(i) for i in {9151, 23991, 23992}
X(35511) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(468)}} and {{A, B, C, X(4), X(13586)}}
X(35511) = pole of the trilinear polar of X(9293) wrt Steiner circumellipse
X(35511) = trilinear pole of the line {620, 690} (the orthic axis of the 1st Brocard triangle)
X(35511) = barycentric product X(i)*X(j) for these {i, j}: {75, 9395}, {99, 9293}, {799, 9396}
X(35511) = barycentric quotient X(i)/X(j) for these (i, j): (1, 2640), (2, 148), (3, 22143), (7, 17085)
X(35511) = trilinear product X(i)*X(j) for these {i, j}: {2, 9395}, {75, 9217}, {99, 9396}, {662, 9293}
X(35511) = trilinear quotient X(i)/X(j) for these (i, j): (2, 2640), (10, 21899), (63, 22143), (75, 148)
X(35511) = perspector of conic through X(2), X(8), and the extraversion triangle of X(8)
X(35511) = {X(2), X(31372)}-harmonic conjugate of X(31998)

X(35512) = ISOGONAL CONJUGATE OF X(21312)

Barycentrics (2*S^2-12*R^2*SB+2*SB^2-SA*SC)*(2*S^2-12*R^2*SC+2*SC^2-SA*SB) : :

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

X(35512) lies on the Jerabek circumhyperbola and these lines: {3, 5656}, {4, 30443}, {6, 15311}, {30, 6391}, {64, 3089}, {68, 12324}, {69, 6000}, {74, 6353}, {265, 32064}, {541, 5504}, {879, 20186}, {895, 2777}, {1499, 2435}, {1596, 3426}, {1853, 14490}, {3521, 18431}, {3527, 13488}, {3532, 15105}, {3566, 14380}, {4846, 5892}, {5486, 34146}, {5878, 6804}, {6247, 22334}, {6677, 35450}, {11206, 35237}, {11744, 15151}, {11820, 34781}, {14216, 15077}, {14457, 18909}, {15585, 34778}, {15749, 18381}, {18296, 18383}, {22802, 31371}

X(35512) = reflection of X(34781) in X(11820)
X(35512) = anticomplement of the complementary conjugate of X(1596)
X(35512) = isogonal conjugate of X(21312)
X(35512) = X(6)-reciprocal conjugate of X(21312)
X(35512) = X(3)-vertex conjugate of X(69)
X(35512) = intersection, other than A,B,C, of Jerabek hyperbola and conic {{A, B, C, X(20), X(3089)}}

X(35513) = EULER LINE INTERCEPT OF X(68)X(14641)

Barycentrics 4*R^2*S^2-(12*R^2-SW)*SB*SC : :
X(35513) = 7*X(4)-8*X(11818), 3*X(4)-4*X(18420), 3*X(376)-4*X(35243), 5*X(631)-4*X(9818)

As a point on the Euler line, X(35513) has Shinagawa coefficients (E, -2*E+F).

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

X(35513) lies on these lines: {2, 3}, {68, 14641}, {69, 6000}, {389, 15740}, {394, 5656}, {800, 2549}, {1056, 3100}, {1058, 4296}, {1285, 10313}, {1350, 15311}, {3426, 3620}, {3587, 26939}, {4846, 12220}, {5065, 7737}, {5562, 6225}, {5878, 15644}, {5921, 11820}, {6515, 15072}, {7171, 26929}, {10575, 11411}, {11459, 12058}, {14855, 18950}, {14927, 18400}, {15438, 25712}, {16836, 18928}, {20477, 32815}

X(35513) = reflection of X(3146) in X(18494)
X(35513) = anticomplement of X(1597)
X(35513) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(21312)}} and {{A, B, C, X(378), X(17040)}}
X(35513) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1596, 2), (4, 3537, 2), (20, 22, 376), (376, 6353, 3), (382, 7401, 4), (3146, 6815, 4), (3543, 6997, 4), (6040, 25909, 235), (6883, 26121, 186), (7409, 25950, 407), (7465, 31907, 24), (10298, 21550, 406), (11096, 33277, 384), (11339, 27744, 413), (14022, 24950, 405), (15333, 16054, 413), (16414, 25763, 24), (17506, 33010, 406), (17566, 26210, 410), (19263, 35236, 406), (33210, 33306, 401)

X(35514) = MIDPOINT OF X(4312) AND X(7991)

Barycentrics a^6-2*(b+c)*a^5+(b^2+14*b*c+c^2)*a^4-8*(b+c)*b*c*a^3-(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)^3*a-(b^2-c^2)^2*(b-c)^2 : :
X(35514) = 2*X(1)-3*X(21151), 2*X(9)-3*X(5657), 4*X(10)-3*X(5817), 4*X(142)-3*X(5603), 3*X(376)-4*X(11495), 5*X(631)-4*X(1001), 4*X(1385)-3*X(8236), 2*X(1482)-3*X(11038), 2*X(5698)-3*X(21168), 3*X(5817)-2*X(11372), 3*X(11038)-4*X(31657)

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

Let P_A be the parabola with focus A and directrix BC. Let L_A be the polar of X(1) wrt P_A. Define L_B and L_C cyclically. Let A' = L_B∩ L_C, B' = L_C∩L_A, C' = L_A∩L_B. The lines AA', BB', CC' concur in X(1000), and X(35514) is the isotomic conjugate of X(1000) wrt A'B'C'. (Randy Hutson, January 17, 2020)

X(35514) lies on these lines: {1, 21151}, {2, 6244}, {3, 390}, {4, 9}, {7, 517}, {8, 971}, {20, 956}, {63, 14646}, {104, 376}, {142, 5603}, {144, 153}, {165, 497}, {200, 5658}, {226, 7994}, {355, 31797}, {377, 20070}, {388, 4312}, {392, 443}, {515, 2951}, {518, 12245}, {527, 12115}, {631, 1001}, {673, 7397}, {912, 34784}, {938, 31787}, {944, 5732}, {946, 17582}, {954, 6908}, {1210, 10384}, {1385, 8236}, {1445, 3359}, {1482, 11038}, {1538, 5328}, {1698, 24644}, {1699, 5316}, {1738, 12652}, {1788, 15299}, {2077, 3524}, {2256, 3332}, {2346, 10679}, {3059, 6001}, {3062, 3679}, {3090, 3826}, {3146, 31799}, {3174, 18446}, {3189, 12520}, {3434, 5744}, {3487, 6769}, {3488, 4326}, {3528, 10806}, {3529, 11826}, {3576, 30331}, {3579, 5704}, {3600, 8158}, {3654, 6172}, {4294, 5584}, {4295, 7957}, {4298, 6766}, {4847, 10860}, {4848, 10398}, {4863, 5918}, {5084, 31658}, {5218, 5537}, {5223, 11362}, {5252, 31391}, {5542, 7982}, {5686, 5690}, {5709, 16004}, {5728, 31788}, {5735, 10532}, {5762, 6850}, {5771, 5789}, {5785, 5837}, {5806, 11024}, {5815, 6259}, {5840, 20119}, {5880, 6897}, {6223, 34790}, {6684, 17559}, {6713, 35238}, {6825, 8543}, {6826, 28174}, {6827, 10738}, {6842, 7679}, {6864, 12699}, {6882, 7678}, {6891, 30312}, {6898, 15254}, {6907, 8164}, {6939, 18230}, {6947, 13528}, {6948, 30295}, {6954, 35000}, {6982, 30311}, {6988, 11248}, {7411, 20075}, {7580, 17784}, {8148, 30340}, {9812, 18482}, {9911, 17562}, {10164, 26105}, {10202, 11025}, {10385, 15931}, {10427, 10698}, {10431, 33110}, {10531, 21153}, {10580, 11227}, {10629, 11010}, {11496, 16845}, {11827, 33703}, {12120, 15998}, {14100, 18391}, {15016, 20116}

X(35514) = midpoint of X(4312) and X(7991)
X(35514) = reflection of X(i) in X(j) for these (i,j): (4, 2550), (390, 3), (944, 5732), (962, 5805), (1056, 6916), (1482, 31657), (3488, 30503), (5223, 11362), (5728, 31788), (5759, 40), (5779, 5690), (6172, 3654), (7982, 5542), (10698, 10427), (11372, 10)
X(35514) = intersection, other than A,B,C, of conics {{A, B, C, X(19), X(10307)}} and {{A, B, C, X(104), X(7719)}}
X(35514) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 11372, 5817), (1482, 31657, 11038), (5690, 5779, 5686), (5744, 9778, 17613)

X(35515) = X(20)X(20208)∩X(1249)X(1853)

Barycentrics SB*SC*(5*S^2-16*(SB-2*SW)*R^2+4*SB^2-2*SA*SC-8*SW^2)*(5*S^2-16*(SC-2*SW)*R^2+4*SC^2-2*SA*SB-8*SW^2) : :

See Antreas P. Hatzipolakis and César Lozada, Euclid 388 .

X(35515) lies on these lines: {20, 20208}, {1249, 1853}

X(35515) = polar conjugate of the anticomplement of X(20200)
X(35515) = lies on the circumconic with center X(122)
X(35515) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(20)}} and {{A, B, C, X(66), X(459)}}

Points on De Longchamps line: X(35516)-X(35575)

This preamble and centers X(35516)-X(35575) were contributed by César Eliud Lozada, December 19, 2019.

The De Longchamps line X(325)X(523) is the isotomic conjugate of the circumcircle.

The appearance of (i, j) in the following list means that the isotomic conjugate of X(i) is X(j), whereb (X(i) is on the circumcircle and X(j) is on the De Longchamps line):

(74, 3260), (98, 325), (99, 523), (100, 693), (101, 3261), (102, 35516), (103, 35517), (104, 3262), (105, 3263), (106, 3264), (107, 3265), (108, 35518), (109, 35519), (110, 850), (111, 3266), (112, 3267), (476, 3268), (477, 35520), (675, 3006), (681, 35521), (689, 3005), (691, 35522), (697, 35523), (699, 35524), (701, 35525), (703, 35526), (705, 35527), (707, 35528), (709, 35529), (711, 35530), (713, 35531), (715, 35532), (717, 35533), (719, 35534), (721, 35535), (723, 35536), (725, 35537), (727, 35538), (729, 30736), (731, 35539), (733, 35540), (735, 35541), (737, 35542), (739, 35543), (741, 35544), (743, 35545), (745, 35546), (747, 35547), (753, 35548), (755, 35549), (759, 35550), (761, 35551), (767, 35552), (769, 35553), (773, 35554), (777, 35555), (779, 35556), (781, 35557), (783, 35558), (785, 35559), (787, 35560), (789, 1491), (791, 35561), (793, 35562), (795, 35563), (797, 35564), (805, 14295), (827, 23285), (925, 6563), (932, 20906), (934, 4397), (1113, 22339), (1114, 22340), (1141, 1273), (1297, 30737), (1302, 30474), (1305, 20294), (1310, 2517), (1311, 33864), (2367, 3001), (2370, 3007), (2373, 858), (3222, 23301), (6079, 4927), (8707, 3004), (8709, 3837), (9063, 17415), (9066, 5996), (9080, 9191), (9150, 9148), (22456, 684), (35565, 2512), (35566, 2513), (35567, 2514), (35568, 3014), (35569, 14277), (35570, 14279), (35571, 14343), (35572, 21128), (35573, 22226), (35574, 23770), (35575, 30735)

X(35516) = ISOTOMIC CONJUGATE OF X(102)

Barycentrics b^2*c^2*(2*a^4-(b+c)*a^3-(b-c)^2*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : :
Trilinears (sec^2 A) (sin B (sec A - sec B) + sin C (sec A - sec C)) : :

X(35516) lies on these lines {2,8607}, {7,1226}, {69,313}, {75,225}, {99,2695}, {183,26236}, {290,35154}, {304,309}, {311,1269}, {320,18816}, {325,523}, {670,35149}, {4858,26012}, {7017,18750}, {14829,19810}, {17895,23989}, {17902,28435}, {19799,20928}, {21207,24209}, {23978,30807}

X(35516) = isotomic conjugate of X(102)
X(35516) = anticomplementary conjugate of the anticomplement of X(2988)
X(35516) = anticomplement of X(8607)
X(35516) = barycentric product X(i)*X(j) for these {i, j}: {76, 515}, {305, 8755}, {561, 2182}, {1455, 28659}
X(35516) = barycentric quotient X(i)/X(j) for these (i, j): (1, 32677), (2, 102), (8, 15629), (76, 34393)
X(35516) = trilinear product X(i)*X(j) for these {i, j}: {75, 515}, {76, 2182}, {304, 8755}, {312, 34050}
X(35516) = trilinear quotient X(i)/X(j) for these (i, j): (2, 32677), (75, 102), (515, 31), (561, 34393)
X(35516) = intersection, other than A,B,C, of conics {{A, B, C, X(75), X(35518)}} and {{A, B, C, X(102), X(117)}}
X(35516) = Cevapoint of X(2) and X(151)
X(35516) = X(314)-Beth conjugate of X(22464)
X(35516) = X(117)-cross conjugate of X(2)
X(35516) = X(i)-isoconjugate-of X(j) for these {i,j}: (6, 32677), (31, 102), (560, 34393)
X(35516) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 32677), (2, 102), (8, 15629)
X(35516) = pole wrt polar circle of line X(25)X(663)
X(35516) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3260, 3262, 35517), (3264, 35517, 3262), (3596, 14615, 322)

X(35517) = ISOTOMIC CONJUGATE OF X(103)

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

X(35517) lies on these lines: {2,8608}, {8,349}, {10,18738}, {69,674}, {75,1088}, {76,4301}, {92,264}, {99,2688}, {183,18613}, {290,35148}, {314,15909}, {325,523}, {338,10026}, {350,14942}, {519,21207}, {668,35164}, {670,35150}, {942,17867}, {1226,20930}, {1231,23528}, {1441,2886}, {1886,26006}, {4087,20446}, {4875,25002}, {5564,20565}, {6684,29477}, {7360,16090}, {17747,30807}, {17861,33141}, {17866,24390}, {17904,28404}, {17905,28409}, {17911,28410}, {18816,35171}, {21075,21579}, {21271,22276}, {21403,34790}, {30806,34387}

X(35517) = isotomic conjugate of X(103)
X(35517) = anticomplementary conjugate of the anticomplement of X(2989)
X(35517) = polar conjugate of the isogonal conjugate of X(26006)
X(35517) = anticomplement of X(8608)
X(35517) = barycentric product X(i)*X(j) for these {i, j}: {75, 30807}, {76, 516}, {264, 26006}, {305, 1886}
X(35517) = barycentric quotient X(i)/X(j) for these (i, j): (1, 911), (2, 103), (3, 32657), (8, 2338)
X(35517) = trilinear product X(i)*X(j) for these {i, j}: {2, 30807}, {75, 516}, {76, 910}, {92, 26006}
X(35517) = trilinear quotient X(i)/X(j) for these (i, j): (2, 911), (63, 32657), (75, 103), (100, 32642)
X(35517) = lies on the circumconic with center X(26006))
X(35517) = intersection, other than A,B,C, of conics {{A, B, C, X(69), X(17233)}} and {{A, B, C, X(75), X(4397)}}
X(35517) = cevapoint of X(i) and X(j) for these (i,j): (2, 152), (516, 26006)
X(35517) = crosssum of X(2175) and X(9454)
X(35517) = pole wrt polar circle of line X(25)X(649)
X(35517) = X(917)-anticomplementary conjugate of X(5905)
X(35517) = X(i)-Beth conjugate of X(j) for these (i,j): (314, 9436), (668, 6735)
X(35517) = X(118)-cross conjugate of X(2)
X(35517) = X(i)-isoconjugate-of X(j) for these {i,j}: (6, 911), (19, 32657), (31, 103)
X(35517) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 911), (2, 103), (3, 32657)
X(35517) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (264, 322, 313), (1233, 4441, 1269), (3260, 3262, 35516), (3262, 35516, 3264)

X(35518) = ISOTOMIC CONJUGATE OF X(108)

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

X(35518) lies on these lines: {2,6588}, {10,23799}, {69,4131}, {75,17896}, {99,2766}, {325,523}, {650,3975}, {656,4025}, {668,4998}, {905,15420}, {1459,24560}, {2533,22318}, {3676,4163}, {3699,25724}, {3762,11068}, {4077,20907}, {6332,8611}, {6589,25902}, {7192,20293}, {10436,23724}, {14298,14555}, {17899,17924}, {20316,25008}, {20952,30910}, {24459,25098}

X(35518) = reflection of X(15420) in X(905)
X(35518) = isotomic conjugate of X(108)
X(35518) = anticomplement of X(6588)
X(35518) = barycentric product X(i)*X(j) for these {i, j}: {7, 15416}, {8, 15413}, {21, 3267}, {63, 35519}
X(35518) = barycentric quotient X(i)/X(j) for these (i, j): (1, 32674), (2, 108), (3, 1415), (7, 32714)
X(35518) = trilinear product X(i)*X(j) for these {i, j}: {2, 6332}, {3, 35519}, {8, 4025}, {9, 15413}
X(35518) = trilinear quotient X(i)/X(j) for these (i, j): (2, 32674), (8, 8750), (21, 32676), (29, 32713)
X(35518) = trilinear pole of the line {17880, 23983}
X(35518) = lies on the circumconic with center X(905))
X(35518) = intersection, other than A,B,C, of conics {{A, B, C, X(11), X(23770)}} and {{A, B, C, X(21), X(858)}}
X(35518) = Cevapoint of X(2) and X(34188)
X(35518) = crossdifference of every pair of points on line {X(32), X(1395)}
X(35518) = crosspoint of X(i) and X(j) for these (i,j): (332, 4561), (668, 3596)
X(35518) = crosssum of X(667) and X(1397)
X(35518) = X(i)-Beth conjugate of X(j) for these (i,j): (314, 17896), (333, 16757)
X(35518) = X(i)-Ceva conjugate of X(j) for these (i,j): (76, 23983), (668, 69)
X(35518) = X(521)-cross conjugate of X(4391)
X(35518) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 32674}, {19, 1415}, {25, 109}
X(35518) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 32674), (2, 108), (3, 1415)
X(35518) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (693, 4397, 35519), (4025, 14208, 15413)

X(35519) = ISOTOMIC CONJUGATE OF X(109)

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

X(35519) lies on these lines: {2,6589}, {10,23801}, {69,8999}, {75,4025}, {76,23104}, {99,2689}, {105,2866}, {290,35149}, {312,3239}, {321,25259}, {325,523}, {514,17789}, {522,4087}, {525,23685}, {668,35174}, {670,35154}, {834,20293}, {905,17899}, {918,21438}, {1459,24353}, {1577,4500}, {1769,25008}, {1920,20518}, {1978,3699}, {3064,6332}, {3310,30864}, {3676,20907}, {3700,3910}, {3762,28468}, {3766,4083}, {3776,20908}, {3996,4105}, {4077,23877}, {4086,4522}, {4406,7192}, {4408,23813}, {4468,21611}, {4885,24622}, {6586,25258}, {6588,25981}, {7252,28960}, {7253,23289}, {7658,19804}, {8676,21300}, {9029,25301}, {14208,18076}, {15413,17896}, {16892,20909}, {20512,23768}, {20950,29739}, {21123,30094}, {21132,28659}, {25511,30476}, {26146,28423}

X(35519) = midpoint of X(18071) and X(20952)
X(35519) = reflection of X(3261) in X(35559)
X(35519) = isotomic conjugate of X(109)
X(35519) = polar conjugate of X(32674)
X(35519) = anticomplement of X(6589)
X(35519) = barycentric product X(i)*X(j) for these {i, j}: {8, 3261}, {11, 1978}, {21, 20948}, {29, 3267}
X(35519) = barycentric quotient X(i)/X(j) for these (i, j): (1, 1415), (2, 109), (3, 32660), (4, 32674)
X(35519) = trilinear product X(i)*X(j) for these {i, j}: {2, 4391}, {4, 35518}, {7, 4397}, {8, 693}
X(35519) = trilinear quotient X(i)/X(j) for these (i, j): (2, 1415), (8, 692), (9, 32739), (11, 667)
X(35519) = trilinear pole of the line {4858, 21044}
X(35519) = lies on the circumconics with center X(i) for i in {1577, 6332, 21186}
X(35519) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(33864)}} and {{A, B, C, X(7), X(3007)}}
X(35519) = Cevapoint of X(i) and X(j) for these (i,j): (2, 33650), (514, 21186), (522, 6332)
X(35519) = crossdifference of every pair of points on line {X(32), X(1397)}
X(35519) = crosspoint of X(i) and X(j) for these (i,j): (76, 4572), (314, 668), (561, 1978)
X(35519) = crosssum of X(i) and X(j) for these (i,j): (560, 1919), (667, 1402), (810, 3725)
X(35519) = X(314)-Beth conjugate of X(4025)
X(35519) = X(i)-Ceva conjugate of X(j) for these (i,j): (76, 23978), (668, 313)
X(35519) = X(124)-cross conjugate of X(2)
X(35519) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 1415}, {19, 32660}, {31, 109}
X(35519) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 1415), (2, 109), (3, 32660)
X(35519) = pole wrt polar circle of trilinear polar of X(32674) (line X(25)X(31))
X(35519) = trilinear product of Feuerbach hyperbola intercepts of de Longchamps line
X(35519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (693, 850, 3261), (693, 4397, 35518), (693, 20906, 3004), (4025, 17894, 75), (25258, 26049, 6586)

X(35520) = ISOTOMIC CONJUGATE OF X(477)

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

X(35520) lies on these lines: {2,3018}, {69,74}, {183,7664}, {264,328}, {302,18777}, {303,18776}, {311,339}, {325,523}, {338,18122}, {1007,9214}, {6527,13219}, {7493,9769}, {11061,35278}, {13479,25320}, {14570,15526}, {18354,22468}

X(35520) = isotomic conjugate of X(477)
X(35520) = anticomplement of X(3018)
X(35520) = barycentric product X(i)*X(j) for these {i, j}: {76, 5663}, {1553, 31621}
X(35520) = barycentric quotient X(i)/X(j) for these (i, j): (2, 477), (3, 32663), (99, 30528), (323, 34210)
X(35520) = trilinear product X(75)*X(5663)
X(35520) = trilinear quotient X(i)/X(j) for these (i, j): (63, 32663), (75, 477), (799, 30528)
X(35520) = intersection, other than A,B,C, of conics {{A, B, C, X(74), X(523)}} and {{A, B, C, X(99), X(3260)}}
X(35520) = crossdifference of every pair of points on line {X(32), X(14398)}
X(35520) = X(i)-isoconjugate-of X(j) for these {i,j}: (19, 32663), (31, 477), (798, 30528)
X(35520) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 477), (3, 32663), (99, 30528)
X(35520) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69, 99, 6148), (69, 1272, 99), (99, 1494, 69), (1272, 1494, 6148), (22339, 22340, 3260)

X(35521) = ISOTOMIC CONJUGATE OF X(681)

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

X(35521) lies on these lines: {325,523}, {4091,4131}

X(35521) = isotomic conjugate of X(681)
X(35521) = barycentric product X(76)*X(680)
X(35521) = barycentric quotient X(i)/X(j) for these (i, j): (2, 681), (680, 6)
X(35521) = trilinear product X(75)*X(680)
X(35521) = trilinear quotient X(i)/X(j) for these (i, j): (75, 681), (680, 31)
X(35521) = crossdifference of every pair of points on line {X(32), X(1096)}
X(35521) = X(31)-isoconjugate-of X(681)
X(35521) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 681), (680, 6)

X(35522) = ISOTOMIC CONJUGATE OF X(691)

Barycentrics    b^2*c^2*(2*a^2-b^2-c^2)*(b^2-c^2) : :
Barycentrics    (csc^2 A) (cot B - cot C) (2 cot A - cot B - cot C) : :
Barycentrics    csc A sin(B - C) (2 a^2 - b^2 - c^2) : :
X(35522) = 3*X(351)-2*X(22105) = X(850)-3*X(3267) = 2*X(850)-3*X(23285) = 3*X(1649)-X(14272) = 3*X(2489)-5*X(31277) = 4*X(3589)-3*X(14398) = 3*X(4108)-4*X(6134) = 3*X(9191)-2*X(14279) = 2*X(14272)-3*X(23287) = X(14273)-3*X(14417) = 2*X(14273)-3*X(18311)

The trilinear polar of X(35522) passes through X(1648).

X(35522) lies on these lines {2,2492}, {6,9035}, {67,9517}, {69,526}, {76,9180}, {99,670}, {115,127}, {126,1560}, {141,3569}, {264,23350}, {290,1494}, {316,20403}, {325,523}, {328,10412}, {351,7664}, {625,8430}, {690,5181}, {1649,14272}, {2489,31277}, {2780,14982}, {2793,14278}, {2881,13219}, {3288,9030}, {3589,14398}, {4108,6134}, {4768,18698}, {5468,14559}, {6088,14360}, {6370,18697}, {9178,30786}, {9293,9479}, {9464,34206}, {9979,18310}, {11672,35073}, {14977,18019}, {16235,21732}, {18023,23288}

X(35522) = midpoint of X(22339) and X(22340)
X(35522) = reflection of X(i) in X(j) for these (i,j): (6, 24284), (3569, 141), (8430, 625), (9979, 18310), (18311, 14417), (21732, 16235), (23285, 3267), (23287, 1649)
X(35522) = isogonal conjugate of X(32729)
X(35522) = isotomic conjugate of X(691)
X(35522) = anticomplementary conjugate of the anticomplement of X(17708)
X(35522) = polar conjugate of the isogonal conjugate of X(14417)
X(35522) = anticomplement of X(2492)
X(35522) = barycentric product X(i)*X(j) for these {i, j}: {76, 690}, {264, 14417}, {300, 9204}, {301, 9205}
X(35522) = barycentric quotient X(i)/X(j) for these (i, j): (2, 691), (13, 9206), (14, 9207), (76, 892)
X(35522) = trilinear product X(i)*X(j) for these {i, j}: {75, 690}, {76, 2642}, {92, 14417}, {115, 24039}
X(35522) = trilinear quotient X(i)/X(j) for these (i, j): (75, 691), (313, 5380), (338, 23894), (351, 560)
X(35522) = lies on the circumconics with center X(i) for i in {1649, 14417, 18310}
X(35522) = intersection, other than A,B,C, of conics {{A, B, C, X(67), X(468)}} and {{A, B, C, X(99), X(115)}}
X(35522) = Cevapoint of X(i) and X(j) for these (i,j): (523, 18310), (690, 14417)
X(35522) = crossdifference of every pair of points on line {X(32), X(1084)}
X(35522) = crosspoint of X(i) and X(j) for these (i,j): (99, 2373), (670, 18023)
X(35522) = crosssum of X(i) and X(j) for these (i,j): (512, 2393), (669, 14567)
X(35522) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (67, 21221), (662, 11061), (935, 5905)
X(35522) = X(339)-Hirst inverse of X(18312)
X(35522) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 691}, {110, 923}, {111, 163}
X(35522) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 691), (6, 32729), (13, 9206)
X(35522) = pole wrt polar circle of line X(25)X(111)

X(35523) = ISOTOMIC CONJUGATE OF X(697)

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

X(35523) lies on these lines: {325,523}, {3721,17184}, {19562,21352}

X(35523) = isotomic conjugate of X(697)
X(35523) = barycentric product X(i)*X(j) for these {i, j}: {76, 696}, {1502, 8619}
X(35523) = barycentric quotient X(i)/X(j) for these (i, j): (2, 697), (76, 18824), (696, 6)
X(35523) = trilinear product X(i)*X(j) for these {i, j}: {75, 696}, {561, 8619}
X(35523) = trilinear quotient X(i)/X(j) for these (i, j): (75, 697), (561, 18824), (696, 31)
X(35523) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 697}, {560, 18824}, {697, 31}
X(35523) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 697), (76, 18824), (696, 6)

X(35524) = ISOTOMIC CONJUGATE OF X(699)

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

X(35524) lies on these lines: {2,2998}, {76,7849}, {141,6664}, {148,16084}, {305,7897}, {325,523}, {385,670}, {1502,3314}, {1916,14603}, {1978,33889}, {3978,7779}, {16986,26235}, {32547,32548}

X(35524) = isotomic conjugate of X(699)
X(35524) = barycentric product X(i)*X(j) for these {i, j}: {76, 698}, {561, 2227}, {1502, 3229}
X(35524) = barycentric quotient X(i)/X(j) for these (i, j): (2, 699), (76, 3225), (305, 8858), (698, 6)
X(35524) = trilinear product X(i)*X(j) for these {i, j}: {75, 698}, {76, 2227}, {561, 3229}, {1928, 32748}
X(35524) = trilinear quotient X(i)/X(j) for these (i, j): (75, 699), (561, 3225), (698, 31), (1926, 32544)
X(35524) = crossdifference of every pair of points on line {X(32), X(9491)}
X(35524) = crosspoint of X(1502) and X(18896)
X(35524) = crosssum of X(i) and X(j) for these (i,j): (32, 32748), (1501, 14602)
X(35524) = X(1916)-Ceva conjugate of X(8024)
X(35524) = X(1502)-Daleth conjugate of X(8024)
X(35524) = X(i)-isoconjugate-of X(j) for these {i,j}: (31, 699), (560, 3225), (1927, 32544)
X(35524) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 699), (76, 3225), (305, 8858)
X(35524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (325, 30736, 35540), (325, 35540, 3266), (1502, 3314, 8024)

X(35525) = ISOTOMIC CONJUGATE OF X(701)

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

X(35525) lies on these lines: {321,20859}, {325,523}, {561,2887}, {1978,3783}, {6382,19562}

X(35525) = isotomic conjugate of X(701)
X(35525) = barycentric product X(76)*X(700)
X(35525) = barycentric quotient X(i)/X(j) for these (i, j): (2, 701), (700, 6)
X(35525) = trilinear product X(75)*X(700)
X(35525) = trilinear quotient X(i)/X(j) for these (i, j): (75, 701), (700, 31)
X(35525) = X(31)-isoconjugate-of X(701)
X(35525) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 701), (700, 6)

X(35526) = ISOTOMIC CONJUGATE OF X(703)

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

X(35526) lies on these lines: {76,19562}, {325,523}, {689,8627}, {3124,14603}

X(35526) = isotomic conjugate of X(703)
X(35526) = barycentric product X(76)*X(702)
X(35526) = barycentric quotient X(i)/X(j) for these (i, j): (2, 703), (702, 6)
X(35526) = trilinear product X(75)*X(702)
X(35526) = trilinear quotient X(i)/X(j) for these (i, j): (75, 703), (702, 31)
X(35526) = X(31)-isoconjugate-of X(703)
X(35526) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 703), (702, 6)

X(35527) = ISOTOMIC CONJUGATE OF X(705)

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

X(35527) lies on these lines: {313,20859}, {325,523}, {3596,19562}

X(35527) = isotomic conjugate of X(705)
X(35527) = barycentric product X(76)*X(704)
X(35527) = barycentric quotient X(i)/X(j) for these (i, j): (2, 705), (704, 6)
X(35527) = trilinear product X(75)*X(704)
X(35527) = trilinear quotient X(i)/X(j) for these (i, j): (75, 705), (704, 31)
X(35527) = X(31)-isoconjugate-of X(705)
X(35527) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 705), (704, 6)

X(35528) = ISOTOMIC CONJUGATE OF X(707)

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

X(35528) lies on these lines: {2,14603}, {305,19562}, {325,523}, {689,10997}, {698,8024}, {736,3978}, {4609,9865}

X(35528) = isotomic conjugate of X(707)
X(35528) = barycentric product X(76)*X(706)
X(35528) = barycentric quotient X(i)/X(j) for these (i, j): (2, 707), (706, 6)
X(35528) = trilinear product X(75)*X(706)
X(35528) = trilinear quotient X(i)/X(j) for these (i, j): (75, 707), (706, 31)
X(35528) = X(31)-isoconjugate-of X(707)
X(35528) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 707), (706, 6)

X(35529) = ISOTOMIC CONJUGATE OF X(709)

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

X(35529) lies on the line {325,523}

X(35529) = isotomic conjugate of X(709)
X(35529) = barycentric product X(76)*X(708)
X(35529) = barycentric quotient X(i)/X(j) for these (i, j): (2, 709), (708, 6)
X(35529) = trilinear product X(75)*X(708)
X(35529) = trilinear quotient X(i)/X(j) for these (i, j): (75, 709), (708, 31)
X(35529) = X(31)-isoconjugate-of X(709)
X(35529) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 709), (708, 6)

X(35530) = ISOTOMIC CONJUGATE OF X(711)

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

X(35530) lies on these lines: {325,523}, {4074,9230}, {14603,19590}, {16890,20859}

X(35530) = isotomic conjugate of X(711)
X(35530) = barycentric product X(76)*X(710)
X(35530) = barycentric quotient X(i)/X(j) for these (i, j): (2, 711), (710, 6)
X(35530) = trilinear product X(75)*X(710)
X(35530) = trilinear quotient X(i)/X(j) for these (i, j): (75, 711), (710, 31)
X(35530) = X(31)-isoconjugate-of X(711)
X(35530) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 711), (710, 6)

X(35531) = ISOTOMIC CONJUGATE OF X(713)

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

X(35531) lies on these lines: {141,3721}, {325,523}, {670,19623}, {1502,1928}, {3739,6374}

X(35531) = isotomic conjugate of X(713)
X(35531) = barycentric product X(i)*X(j) for these {i, j}: {76, 712}, {561, 2228}, {1502, 8620}
X(35531) = barycentric quotient X(i)/X(j) for these (i, j): (2, 713), (76, 18825), (321, 27810), (712, 6)
X(35531) = trilinear product X(i)*X(j) for these {i, j}: {75, 712}, {76, 2228}, {561, 8620}
X(35531) = trilinear quotient X(i)/X(j) for these (i, j): (75, 713), (313, 27810), (561, 18825), (712, 31)
X(35531) = X(i)-isoconjugate-of X(j) for these {i,j}: (31, 713), (560, 18825), (2206, 27810)
X(35531) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 713), (76, 18825), (321, 27810)

X(35532) = ISOTOMIC CONJUGATE OF X(715)

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

X(35532) lies on these lines: {321,1237}, {325,523}, {561,1928}, {740,1978}, {1502,30632}, {3728,6374}, {3994,27808}, {6382,21020}

X(35532) = isotomic conjugate of X(715)
X(35532) = barycentric product X(i)*X(j) for these {i, j}: {76, 714}, {561, 2229}
X(35532) = barycentric quotient X(i)/X(j) for these (i, j): (2, 715), (76, 18826), (714, 6)
X(35532) = trilinear product X(i)*X(j) for these {i, j}: {75, 714}, {76, 2229}
X(35532) = trilinear quotient X(i)/X(j) for these (i, j): (75, 715), (561, 18826), (714, 31)
X(35532) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 715}, {560, 18826}
X(35532) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 715), (76, 18826), (714, 6)

X(35533) = ISOTOMIC CONJUGATE OF X(717)

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

X(35533) lies on these lines: {76,1928}, {325,523}, {1925,3727}, {1926,3726}

X(35533) = isotomic conjugate of X(717)
X(35533) = barycentric product X(i)*X(j) for these {i, j}: {1, 30875}, {76, 716}, {561, 2230}, {1502, 8621}
X(35533) = barycentric quotient X(i)/X(j) for these (i, j): (2, 717), (716, 6)
X(35533) = trilinear product X(i)*X(j) for these {i, j}: {6, 30875}, {75, 716}, {76, 2230}, {561, 8621}
X(35533) = trilinear quotient X(i)/X(j) for these (i, j): (75, 717), (716, 31)
X(35533) = X(31)-isoconjugate-of X(717)
X(35533) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 717), (716, 6)

X(35534) = ISOTOMIC CONJUGATE OF X(719)

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

X(35534) lies on these lines: {75,1928}, {313,3721}, {325,523}

X(35534) = isotomic conjugate of X(719)
X(35534) = barycentric product X(i)*X(j) for these {i, j}: {76, 718}, {561, 2231}
X(35534) = barycentric quotient X(i)/X(j) for these (i, j): (2, 719), (718, 6)
X(35534) = trilinear product X(i)*X(j) for these {i, j}: {75, 718}, {76, 2231}
X(35534) = trilinear quotient X(i)/X(j) for these (i, j): (75, 719), (718, 31)
X(35534) = X(31)-isoconjugate-of X(719)
X(35534) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 719), (718, 6)

X(35535) = ISOTOMIC CONJUGATE OF X(721)

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

X(35535) lies on these lines: {2,1928}, {325,523}, {3721,8024}

X(35535) = isotomic conjugate of X(721)
X(35535) = barycentric product X(i)*X(j) for these {i, j}: {76, 720}, {561, 2232}
X(35535) = barycentric quotient X(i)/X(j) for these (i, j): (2, 721), (720, 6)
X(35535) = trilinear product X(i)*X(j) for these {i, j}: {75, 720}, {76, 2232}
X(35535) = trilinear quotient X(i)/X(j) for these (i, j): (75, 721), (720, 31)
X(35535) = X(31)-isoconjugate-of X(721)
X(35535) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 721), (720, 6)

X(35536) = ISOTOMIC CONJUGATE OF X(723)

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

X(35536) lies on these lines: {1,1925}, {325,523}

X(35536) = isotomic conjugate of X(723)
X(35536) = barycentric product X(i)*X(j) for these {i, j}: {76, 722}, {561, 2233}
X(35536) = barycentric quotient X(i)/X(j) for these (i, j): (2, 723), (722, 6)
X(35536) = trilinear product X(i)*X(j) for these {i, j}: {75, 722}, {76, 2233}
X(35536) = trilinear quotient X(i)/X(j) for these (i, j): (75, 723), (722, 31)
X(35536) = X(31)-isoconjugate-of X(723)
X(35536) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 723), (722, 6)

X(35537) = ISOTOMIC CONJUGATE OF X(725)

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

X(35537) lies on these lines: {6,1928}, {325,523}

X(35537) = isotomic conjugate of X(725)
X(35537) = barycentric product X(76)*X(724)
X(35537) = barycentric quotient X(i)/X(j) for these (i, j): (2, 725), (724, 6)
X(35537) = trilinear product X(75)*X(724)
X(35537) = trilinear quotient X(i)/X(j) for these (i, j): (75, 725), (724, 31)
X(35537) = X(31)-isoconjugate-of X(725)
X(35537) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 725), (724, 6)

X(35538) = ISOTOMIC CONJUGATE OF X(727)

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

X(35538) lies on these lines: {75,982}, {76,4066}, {141,321}, {313,561}, {325,523}, {334,18891}, {350,1978}, {670,17731}, {1086,20440}, {1237,20911}, {1921,20446}, {1930,27801}, {3314,30713}, {3971,18133}, {6381,27808}, {18040,21101}, {18050,29960}, {20255,21435}, {20335,20453}, {20928,30078}, {20936,27424}, {21240,21412}, {21331,30026}, {28659,33930}, {31119,31130}

X(35538) = isotomic conjugate of X(727)
X(35538) = barycentric product X(i)*X(j) for these {i, j}: {76, 726}, {561, 1575}, {668, 20908}, {670, 21053}
X(35538) = barycentric quotient X(i)/X(j) for these (i, j): (1, 34077), (2, 727), (75, 20332), (76, 3226)
X(35538) = trilinear product X(i)*X(j) for these {i, j}: {75, 726}, {76, 1575}, {190, 20908}, {313, 18792}
X(35538) = trilinear quotient X(i)/X(j) for these (i, j): (2, 34077), (75, 727), (76, 20332), (313, 18793)
X(35538) = trilinear pole of the line {20908, 21053}
X(35538) = intersection, other than A,B,C, of conics {{A, B, C, X(75), X(20906)}} and {{A, B, C, X(313), X(523)}}
X(35538) = crosspoint of X(561) and X(18895)
X(35538) = crosssum of X(560) and X(14599)
X(35538) = X(334)-Ceva conjugate of X(313)
X(35538) = X(561)-Daleth conjugate of X(313)
X(35538) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 34077}, {31, 727}, {32, 20332}
X(35538) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 34077), (2, 727), (75, 20332)
X(35538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (561, 33931, 313), (3263, 35543, 35544), (3263, 35544, 3264)

X(35539) = ISOTOMIC CONJUGATE OF X(731)

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

X(35539) lies on these lines: {75,700}, {141,313}, {325,523}, {668,3792}, {1978,32848}, {3596,6374}, {4439,27808}

X(35539) = isotomic conjugate of X(731)
X(35539) = barycentric product X(i)*X(j) for these {i, j}: {76, 730}, {1502, 8622}
X(35539) = barycentric quotient X(i)/X(j) for these (i, j): (2, 731), (730, 6)
X(35539) = trilinear product X(i)*X(j) for these {i, j}: {75, 730}, {76, 2235}, {561, 8622}
X(35539) = trilinear quotient X(i)/X(j) for these (i, j): (75, 731), (730, 31)
X(35539) = X(31)-isoconjugate-of X(731)
X(35539) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 731), (730, 6)

X(35540) = ISOTOMIC CONJUGATE OF X(733)

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

X(35540) lies on the cubic K1024 and these lines: {2,308}, {76,4045}, {141,6665}, {305,3314}, {325,523}, {385,3978}, {670,7779}, {706,32748}, {1916,19573}, {1920,1926}, {3329,9230}, {8290,8783}, {9464,31124}, {9477,18896}, {9865,14603}, {18277,18891}, {26192,31268}

X(35540) = isotomic conjugate of X(733)
X(35540) = barycentric product X(i)*X(j) for these {i, j}: {38, 1926}, {39, 14603}, {76, 732}, {141, 3978}
X(35540) = barycentric quotient X(i)/X(j) for these (i, j): (2, 733), (38, 1967), (39, 9468), (76, 14970)
X(35540) = trilinear product X(i)*X(j) for these {i, j}: {38, 3978}, {39, 1926}, {75, 732}, {76, 2236}
X(35540) = trilinear quotient X(i)/X(j) for these (i, j): (38, 9468), (39, 1927), (75, 733), (141, 1967)
X(35540) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(3005)}} and {{A, B, C, X(39), X(9865)}}
X(35540) = crossdifference of every pair of points on line {X(32), X(881)}
X(35540) = X(699)-complementary conjugate of X(1215)
X(35540) = X(141)-Hirst inverse of X(8024)
X(35540) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 733}, {82, 9468}, {83, 1927}
X(35540) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 733), (38, 1967), (39, 9468)
X(35540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (305, 6374, 3314), (325, 30736, 35524), (3266, 35524, 325)

X(35541) = ISOTOMIC CONJUGATE OF X(735)

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

X(35541) lies on these lines: {1,704}, {325,523}

X(35541) = isotomic conjugate of X(735)
X(35541) = barycentric product X(i)*X(j) for these {i, j}: {76, 734}, {561, 2237}
X(35541) = barycentric quotient X(i)/X(j) for these (i, j): (2, 735), (734, 6)
X(35541) = trilinear product X(i)*X(j) for these {i, j}: {75, 734}, {76, 2237}
X(35541) = trilinear quotient X(i)/X(j) for these (i, j): (75, 735), (734, 31)
X(35541) = X(31)-isoconjugate-of X(735)
X(35541) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 735), (734, 6)

X(35542) = ISOTOMIC CONJUGATE OF X(737)

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

X(35542) lies on these lines: {6,706}, {325,523}, {3051,8039}

X(35542) = isotomic conjugate of X(737)
X(35542) = barycentric product X(76)*X(736)
X(35542) = barycentric quotient X(i)/X(j) for these (i, j): (2, 737), (736, 6)
X(35542) = trilinear product X(75)*X(736)
X(35542) = trilinear quotient X(i)/X(j) for these (i, j): (75, 737), (736, 31)
X(35542) = crossdifference of every pair of points on line {X(32), X(9006)}
X(35542) = X(31)-isoconjugate-of X(737)
X(35542) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 737), (736, 6)

X(35543) = ISOTOMIC CONJUGATE OF X(739)

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

X(35543) lies on these lines: {75,4392}, {76,321}, {310,4980}, {325,523}, {670,16741}, {716,2229}, {1109,20629}, {1150,3403}, {1920,4359}, {1921,1978}, {3681,25278}, {3741,20889}, {3891,18056}, {3978,25298}, {3994,6381}, {4485,26234}, {4495,17763}, {4671,21615}, {4766,20496}, {5057,20345}, {6374,20892}, {7196,19809}, {7244,32914}, {8024,30713}, {10009,24589}, {18059,25303}, {21435,26562}, {21596,21605}, {27104,28596}, {28605,30637}, {30635,33931}, {30660,33075}

X(35543) = isotomic conjugate of X(739)
X(35543) = barycentric product X(i)*X(j) for these {i, j}: {75, 6381}, {76, 536}, {310, 3994}, {561, 899}
X(35543) = barycentric quotient X(i)/X(j) for these (i, j): (2, 739), (76, 3227), (100, 32718), (190, 34075)
X(35543) = trilinear product X(i)*X(j) for these {i, j}: {2, 6381}, {75, 536}, {76, 899}, {85, 4009}
X(35543) = trilinear quotient X(i)/X(j) for these (i, j): (75, 739), (190, 32718), (514, 23349), (536, 31)
X(35543) = intersection, other than A,B,C, of conics {{A, B, C, X(76), X(693)}} and {{A, B, C, X(321), X(523)}}
X(35543) = crossdifference of every pair of points on line {X(32), X(1980)}
X(35543) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 739}, {101, 23349}, {560, 3227}
X(35543) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 739), (76, 3227), (100, 32718)
X(35543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (561, 6382, 321), (1921, 1978, 4358), (4671, 30638, 21615), (35538, 35544, 3263)

X(35544) = ISOTOMIC CONJUGATE OF X(741)

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

X(35544) lies on these lines: {2,4485}, {10,1237}, {38,75}, {76,4647}, {86,18059}, {313,321}, {325,523}, {334,18275}, {350,740}, {668,758}, {756,28593}, {1920,27798}, {1962,30963}, {1966,33295}, {1978,20947}, {2292,6376}, {2294,17786}, {2650,24524}, {3212,25307}, {3596,6382}, {3721,21435}, {3743,18140}, {3930,4033}, {3948,4037}, {3954,21412}, {3992,27808}, {4016,30473}, {4071,20496}, {4441,17163}, {17762,28660}, {18050,33299}, {18147,32915}, {18148,24443}, {18895,20437}, {20896,31089}, {30758,30801}

X(35544) = isotomic conjugate of X(741)
X(35544) = barycentric product X(i)*X(j) for these {i, j}: {10, 1921}, {37, 18891}, {75, 3948}, {76, 740}
X(35544) = barycentric quotient X(i)/X(j) for these (i, j): (1, 18268), (2, 741), (8, 2311), (10, 292)
X(35544) = trilinear product X(i)*X(j) for these {i, j}: {2, 3948}, {10, 350}, {37, 1921}, {42, 18891}
X(35544) = trilinear quotient X(i)/X(j) for these (i, j): (2, 18268), (10, 1911), (37, 1922), (42, 14598)
X(35544) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(3783)}} and {{A, B, C, X(38), X(756)}}
X(35544) = Cevapoint of X(10) and X(20496)
X(35544) = crossdifference of every pair of points on line {X(32), X(1924)}
X(35544) = crosspoint of X(1921) and X(18891)
X(35544) = crosssum of X(1922) and X(14598)
X(35544) = X(1921)-Ceva conjugate of X(3948)
X(35544) = X(313)-Daleth conjugate of X(1230)
X(35544) = X(i)-Hirst inverse of X(j) for these (i,j): (313, 321), (321, 313), (1921, 18035)
X(35544) = X(i)-isoconjugate-of X(j) for these {i,j}: {6, 18268}, {31, 741}, {58, 1911}
X(35544) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 18268), (2, 741), (8, 2311)
X(35544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 27801, 1237), (1921, 4087, 350), (3263, 35543, 35538), (3264, 35538, 3263), (3596, 6382, 33931), (21435, 22028, 3721)

X(35545) = ISOTOMIC CONJUGATE OF X(743)

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

X(35545) lies on these lines: {2,561}, {305,6382}, {321,8024}, {325,523}, {3920,18059}, {17443,18053}

X(35545) = isotomic conjugate of X(743)
X(35545) = barycentric product X(i)*X(j) for these {i, j}: {76, 742}, {561, 2239}, {1502, 8624}
X(35545) = barycentric quotient X(i)/X(j) for these (i, j): (2, 743), (742, 6)
X(35545) = trilinear product X(i)*X(j) for these {i, j}: {75, 742}, {76, 2239}, {561, 8624}
X(35545) = trilinear quotient X(i)/X(j) for these (i, j): (75, 743), (742, 31)
X(35545) = crossdifference of every pair of points on line {X(32), X(8630)}
X(35545) = X(31)-isoconjugate-of X(743)
X(35545) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 743), (742, 6)

X(35546) = ISOTOMIC CONJUGATE OF X(745)

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

X(35546) lies on these lines: {1,561}, {321,26589}, {325,523}, {799,34997}, {1920,29631}, {1921,29632}, {20896,30179}

X(35546) = isotomic conjugate of X(745)
X(35546) = barycentric product X(i)*X(j) for these {i, j}: {76, 744}, {561, 2240}, {1502, 8625}
X(35546) = barycentric quotient X(i)/X(j) for these (i, j): (2, 745), (744, 6)
X(35546) = trilinear product X(i)*X(j) for these {i, j}: {75, 744}, {76, 2240}, {561, 8625}
X(35546) = trilinear quotient X(i)/X(j) for these (i, j): (75, 745), (744, 31)
X(35546) = X(31)-isoconjugate-of X(745)
X(35546) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 745), (744, 6)

X(35547) = ISOTOMIC CONJUGATE OF X(747)

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

X(35547) lies on these lines: {6,561}, {325,523}

X(35547) = isotomic conjugate of X(747)
X(35547) = barycentric product X(76)*X(746)
X(35547) = barycentric quotient X(i)/X(j) for these (i, j): (2, 747), (746, 6)
X(35547) = trilinear product X(75)*X(746)
X(35547) = trilinear quotient X(i)/X(j) for these (i, j): (75, 747), (746, 31)
X(35547) = crossdifference of every pair of points on line {X(32), X(9008)}
X(35547) = X(31)-isoconjugate-of X(747)
X(35547) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 747), (746, 6)

X(35548) = ISOTOMIC CONJUGATE OF X(753)

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

X(35548) lies on these lines: {1,76}, {75,33120}, {313,4358}, {325,523}, {551,30893}, {4485,31120}, {9464,29832}, {11059,29857}, {20947,29643}, {26230,26235}, {29638,30963}

X(35548) = isotomic conjugate of X(753)
X(35548) = barycentric product X(i)*X(j) for these {i, j}: {1, 30874}, {76, 752}, {310, 4144}, {561, 2243}
X(35548) = barycentric quotient X(i)/X(j) for these (i, j): (2, 753), (668, 5386), (752, 6)
X(35548) = trilinear product X(i)*X(j) for these {i, j}: {6, 30874}, {75, 752}, {76, 2243}, {85, 4070}
X(35548) = trilinear quotient X(i)/X(j) for these (i, j): (75, 753), (752, 31), (1978, 5386)
X(35548) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 753}, {753, 31}, {1919, 5386}
X(35548) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 753), (668, 5386), (752, 6)

X(35549) = ISOTOMIC CONJUGATE OF X(755)

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

X(35549) lies on these lines: {6,76}, {305,9766}, {325,523}, {338,3978}, {566,7763}, {736,33875}, {5201,5976}, {7774,9464}, {7778,11059}, {7792,26235}, {16890,23642}

X(35549) = isotomic conjugate of X(755)
X(35549) = barycentric product X(i)*X(j) for these {i, j}: {76, 754}, {310, 4156}, {561, 2244}, {670, 14420}
X(35549) = barycentric quotient X(i)/X(j) for these (i, j): (2, 755), (668, 5389), (754, 6)
X(35549) = trilinear product X(i)*X(j) for these {i, j}: {75, 754}, {76, 2244}, {85, 4157}, {274, 4156}
X(35549) = trilinear quotient X(i)/X(j) for these (i, j): (75, 755), (754, 31)
X(35549) = intersection, other than A,B,C, of conics {{A, B, C, X(6), X(3005)}} and {{A, B, C, X(76), X(23285)}}
X(35549) = crossdifference of every pair of points on line {X(32), X(688)}
X(35549) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 755}, {755, 31}, {1919, 5389}
X(35549) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 755), (668, 5389), (754, 6)
X(35549) = {X(3260), X(3266)}-harmonic conjugate of X(30736)

X(35550) = ISOTOMIC CONJUGATE OF X(759)

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

X(35550) lies on these lines: {1,75}, {2,20896}, {9,20444}, {12,313}, {37,20234}, {99,12030}, {190,8680}, {306,25361}, {312,31266}, {320,758}, {321,3943}, {325,523}, {333,20929}, {344,25255}, {350,29632}, {536,20432}, {668,17791}, {696,20590}, {742,15994}, {744,3747}, {857,21094}, {1213,27705}, {1266,20893}, {1962,29638}, {2064,25080}, {2292,4389}, {2294,17234}, {2614,18160}, {2650,17378}, {3596,6757}, {3662,4016}, {3702,11281}, {3718,20930}, {3739,21442}, {3743,17320}, {3936,4053}, {3948,16732}, {3958,17347}, {4019,21231}, {4066,6541}, {4137,25957}, {4150,18589}, {4358,17895}, {4359,4395}, {4427,18661}, {4441,29830}, {4812,16777}, {4968,28503}, {4980,28309}, {10180,29860}, {15903,24209}, {16086,16099}, {16586,17923}, {17023,30892}, {17250,33934}, {17263,25081}, {17271,20955}, {17277,17788}, {17776,25254}, {17861,18147}, {18137,20236}, {18805,20985}, {20437,20947}, {21020,33120}, {21421,33942}, {21425,28594}, {21907,32849}, {24315,30882}, {26109,28605}, {26230,26234}, {26772,27727}, {27042,27697}, {27798,29861}, {29643,33931}, {29832,31115}, {29857,30758}, {29858,30963}

X(35550) = reflection of X(3747) in X(21254)
X(35550) = isotomic conjugate of X(759)
X(35550) = barycentric product X(i)*X(j) for these {i, j}: {10, 20924}, {36, 27801}, {75, 3936}, {76, 758}
X(35550) = barycentric quotient X(i)/X(j) for these (i, j): (1, 34079), (2, 759), (8, 2341), (10, 2161)
X(35550) = trilinear product X(i)*X(j) for these {i, j}: {2, 3936}, {10, 320}, {36, 313}, {37, 20924}
X(35550) = trilinear quotient X(i)/X(j) for these (i, j): (10, 6187), (36, 2206), (75, 759), (76, 24624)
X(35550) = trilinear pole of the line {2610, 4707}
X(35550) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(12)}} and {{A, B, C, X(75), X(850)}}
X(35550) = X(75)-Ceva conjugate of X(1227)
X(35550) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 759}, {32, 24624}, {58, 6187}
X(35550) = X(i)-reciprocal conjugate of X(j) for these (i,j): (1, 34079), (2, 759), (8, 2341)
X(35550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75, 20932, 314), (1441, 20336, 313), (3262, 3263, 3264), (3948, 20912, 16732), (18697, 18698, 75)

X(35551) = ISOTOMIC CONJUGATE OF X(761)

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

X(35551) lies on these lines: {6,75}, {42,20627}, {55,20641}, {325,523}, {740,20629}, {3685,20643}, {7774,31130}, {7778,30758}, {7792,26234}, {18138,20890}

X(35551) = isotomic conjugate of X(761)
X(35551) = barycentric product X(i)*X(j) for these {i, j}: {76, 760}, {1502, 8628}
X(35551) = barycentric quotient X(i)/X(j) for these (i, j): (2, 761), (760, 6)
X(35551) = trilinear product X(i)*X(j) for these {i, j}: {75, 760}, {561, 8628}
X(35551) = trilinear quotient X(i)/X(j) for these (i, j): (75, 761), (760, 31)
X(35551) = crossdifference of every pair of points on line {X(32), X(788)}
X(35551) = X(31)-isoconjugate-of X(761)
X(35551) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 761), (760, 6)

X(35552) = ISOTOMIC CONJUGATE OF X(767)

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

X(35552) lies on these lines: {1,6}, {141,17447}, {325,523}, {536,4516}, {674,1959}, {1284,9021}, {1626,4123}, {1631,1760}, {2611,32848}, {3122,20590}, {3675,3834}, {3703,21318}, {3739,21804}, {3778,4118}, {3912,17463}, {4022,7237}, {4073,18161}, {4149,16551}, {4436,25083}, {4463,16678}, {4523,5132}, {4712,21801}, {4884,21333}, {7202,9025}, {7297,8301}, {7774,29832}, {7778,29857}, {7792,26230}, {14839,17444}, {15523,21325}, {16574,20713}, {16732,20544}, {16989,29831}, {17049,17443}, {17265,20275}, {17470,21035}, {18183,28358}, {19898,19929}, {20486,23772}, {21023,21236}, {24326,33120}

X(35552) = reflection of X(i) in X(j) for these (i,j): (4436, 25083), (16732, 20544)
X(35552) = isotomic conjugate of X(767)
X(35552) = barycentric product X(i)*X(j) for these {i, j}: {76, 766}, {1502, 8629}
X(35552) = barycentric quotient X(766)/X(6)
X(35552) = trilinear product X(i)*X(j) for these {i, j}: {75, 766}, {561, 8629}
X(35552) = trilinear quotient X(766)/X(31)
X(35552) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(3261)}} and {{A, B, C, X(6), X(693)}}
X(35552) = X(31)-isoconjugate-of X(767)
X(35552) = X(766)-reciprocal conjugate of X(6)
X(35552) = {X(3778), X(4118)}-harmonic conjugate of X(18179)

X(35553) = ISOTOMIC CONJUGATE OF X(769)

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

X(35553) lies on the line {325,523}

X(35553) = isotomic conjugate of X(769)
X(35553) = barycentric product X(76)*X(768)
X(35553) = barycentric quotient X(i)/X(j) for these (i, j): (2, 769), (768, 6)
X(35553) = trilinear product X(75)*X(768)
X(35553) = trilinear quotient X(i)/X(j) for these (i, j): (75, 769), (768, 31)
X(35553) = X(31)-isoconjugate-of X(769)
X(35553) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 769), (768, 6)

X(35554) = ISOTOMIC CONJUGATE OF X(773)

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

X(35554) lies on the line {325,523}

X(35554) = isotomic conjugate of X(773)
X(35554) = barycentric product X(76)*X(772)
X(35554) = barycentric quotient X(i)/X(j) for these (i, j): (2, 773), (772, 6)
X(35554) = trilinear product X(75)*X(772)
X(35554) = trilinear quotient X(i)/X(j) for these (i, j): (75, 773), (772, 31)
X(35554) = X(31)-isoconjugate-of X(773)
X(35554) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 773), (772, 6)

X(35555) = ISOTOMIC CONJUGATE OF X(777)

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

X(35555) lies on the line {325,523}

X(35555) = isotomic conjugate of X(777)
X(35555) = barycentric product X(76)*X(776)
X(35555) = barycentric quotient X(i)/X(j) for these (i, j): (2, 777), (776, 6)
X(35555) = trilinear product X(75)*X(776)
X(35555) = trilinear quotient X(i)/X(j) for these (i, j): (75, 777), (776, 31)
X(35555) = X(31)-isoconjugate-of X(777)
X(35555) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 777), (776, 6)

X(35556) = ISOTOMIC CONJUGATE OF X(779)

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

X(35556) lies on the line {325,523}

X(35556) = isotomic conjugate of X(779)
X(35556) = barycentric product X(76)*X(778)
X(35556) = barycentric quotient X(i)/X(j) for these (i, j): (2, 779), (778, 6)
X(35556) = trilinear product X(75)*X(778)
X(35556) = trilinear quotient X(i)/X(j) for these (i, j): (75, 779), (778, 31)
X(35556) = X(31)-isoconjugate-of X(779)
X(35556) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 779), (778, 6)

X(35557) = ISOTOMIC CONJUGATE OF X(781)

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

X(35557) lies on the line {325,523}

X(35557) = isotomic conjugate of X(781)
X(35557) = barycentric product X(76)*X(780)
X(35557) = barycentric quotient X(i)/X(j) for these (i, j): (2, 781), (780, 6)
X(35557) = trilinear product X(75)*X(780)
X(35557) = trilinear quotient X(i)/X(j) for these (i, j): (75, 781), (780, 31)
X(35557) = X(31)-isoconjugate-of X(781)
X(35557) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 781), (780, 6)

X(35558) = ISOTOMIC CONJUGATE OF X(783)

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

X(35558) lies on these lines: {325,523}, {689,17995}, {804,8783}

X(35558) = isotomic conjugate of X(783)
X(35558) = barycentric product X(76)*X(782)
X(35558) = barycentric quotient X(i)/X(j) for these (i, j): (2, 783), (76, 18828), (782, 6)
X(35558) = trilinear product X(75)*X(782)
X(35558) = trilinear quotient X(i)/X(j) for these (i, j): (75, 783), (561, 18828), (782, 31)
X(35558) = crossdifference of every pair of points on line {X(32), X(14946)}
X(35558) = crosssum of X(688) and X(3852)
X(35558) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 783}, {560, 18828}
X(35558) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 783), (76, 18828), (782, 6)

X(35559) = ISOTOMIC CONJUGATE OF X(785)

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

X(35559) lies on these lines: {325,523}, {768,20910}, {784,23594}, {834,3766}, {1577,21262}, {4083,4408}, {6589,24622}, {20907,23785}

X(35559) = midpoint of X(3261) and X(35519)
X(35559) = isotomic conjugate of X(785)
X(35559) = barycentric product X(i)*X(j) for these {i, j}: {10, 23594}, {76, 784}, {850, 27164}, {1502, 2978}
X(35559) = barycentric quotient X(i)/X(j) for these (i, j): (2, 785), (784, 6)
X(35559) = trilinear product X(i)*X(j) for these {i, j}: {37, 23594}, {75, 784}, {523, 10471}, {561, 2978}
X(35559) = trilinear quotient X(i)/X(j) for these (i, j): (75, 785), (784, 31)
X(35559) = X(31)-isoconjugate-of X(785)
X(35559) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 785), (784, 6)

X(35560) = ISOTOMIC CONJUGATE OF X(787)

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

X(35560) lies on the line {325,523}

X(35560) = isotomic conjugate of X(787)
X(35560) = barycentric product X(76)*X(786)
X(35560) = barycentric quotient X(i)/X(j) for these (i, j): (2, 787), (786, 6)
X(35560) = trilinear product X(i)*X(j) for these {i, j}: {75, 786}, {693, 27020}
X(35560) = trilinear quotient X(i)/X(j) for these (i, j): (75, 787), (786, 31)
X(35560) = X(31)-isoconjugate-of X(787)
X(35560) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 787), (786, 6)

X(35561) = ISOTOMIC CONJUGATE OF X(791)

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

X(35561) lies on these lines: {325,523}, {2084,20953}

X(35561) = isotomic conjugate of X(791)
X(35561) = barycentric product X(76)*X(790)
X(35561) = barycentric quotient X(i)/X(j) for these (i, j): (2, 791), (790, 6)
X(35561) = trilinear product X(75)*X(790)
X(35561) = trilinear quotient X(i)/X(j) for these (i, j): (75, 791), (790, 31)
X(35561) = X(31)-isoconjugate-of X(791)
X(35561) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 791), (790, 6)

X(35562) = ISOTOMIC CONJUGATE OF X(793)

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

X(35562) lies on these lines: {325,523}, {9008,21305}

X(35562) = isotomic conjugate of X(793)
X(35562) = barycentric product X(76)*X(792)
X(35562) = barycentric quotient X(i)/X(j) for these (i, j): (2, 793), (792, 6)
X(35562) = trilinear product X(75)*X(792)
X(35562) = trilinear quotient X(i)/X(j) for these (i, j): (75, 793), (792, 31)
X(35562) = X(31)-isoconjugate-of X(793)
X(35562) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 793), (792, 6)

X(35563) = ISOTOMIC CONJUGATE OF X(795)

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

X(35563) lies on the line {325,523}

X(35563) = isotomic conjugate of X(795)
X(35563) = barycentric product X(76)*X(794)
X(35563) = barycentric quotient X(i)/X(j) for these (i, j): (2, 795), (794, 6), (1491, 14945)
X(35563) = trilinear product X(75)*X(794)
X(35563) = trilinear quotient X(i)/X(j) for these (i, j): (75, 795), (794, 31), (824, 14945)
X(35563) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 795}, {825, 14945}
X(35563) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 795), (794, 6), (1491, 14945)

X(35564) = ISOTOMIC CONJUGATE OF X(797)

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

X(35564) lies on the line {325,523}

X(35564) = isotomic conjugate of X(797)
X(35564) = barycentric product X(76)*X(796)
X(35564) = barycentric quotient X(i)/X(j) for these (i, j): (2, 797), (796, 6)
X(35564) = trilinear product X(75)*X(796)
X(35564) = trilinear quotient X(i)/X(j) for these (i, j): (75, 797), (796, 31)
X(35564) = X(31)-isoconjugate-of X(797)
X(35564) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 797), (796, 6)

X(35565) = ISOTOMIC CONJUGATE OF X(2512)

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

X(35565) lies on the circumcircle and these lines: {670,931}, {799,8693}, {4554,32693}, {4623,8690}

X(35565) = isotomic conjugate of X(2512)
X(35565) = trilinear quotient X(75)/X(2512)
X(35565) = trilinear pole of the line {6, 4754}
X(35565) = lies on the circumconic with center X(17550))
X(35565) = Cevapoint of X(523) and X(17550)
X(35565) = X(31)-isoconjugate-of X(2512)
X(35565) = X(2)-reciprocal conjugate of X(2512)
X(35565) = Ψ(X(6), X(4754))

X(35566) = ISOTOMIC CONJUGATE OF X(2513)

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

X(35566) lies on the circumcircle and the line {670,26714}

X(35566) = isotomic conjugate of X(2513)
X(35566) = trilinear quotient X(75)/X(2513)
X(35566) = lies on the circumconic with center X(33734))
X(35566) = Cevapoint of X(523) and X(33734)
X(35566) = X(31)-isoconjugate-of X(2513)
X(35566) = X(2)-reciprocal conjugate of X(2513)
X(35566) = trilinear pole of line X(6)X(20023)
X(35566) = Ψ(X(6), X(20023))

X(35567) = ISOTOMIC CONJUGATE OF X(2514)

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

X(35567) lies on the circumcircle and these lines: {111,1241}, {112,670}, {783,18829}

X(35567) = isotomic conjugate of X(2514)
X(35567) = barycentric product X(99)*X(1241)
X(35567) = barycentric quotient X(i)/X(j) for these (i, j): (99, 1194), (670, 6656), (799, 17446), (1241, 523)
X(35567) = trilinear product X(662)*X(1241)
X(35567) = trilinear quotient X(i)/X(j) for these (i, j): (75, 2514), (670, 17446), (799, 1194), (1241, 661)
X(35567) = trilinear pole of the line {6, 305}
X(35567) = Cevapoint of X(1634) and X(4563)
X(35567) = X(i)-cross conjugate of X(j) for these (i,j): (141, 34537), (1799, 4590)
X(35567) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 2514}, {669, 17446}, {798, 1194}
X(35567) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 2514), (99, 1194), (670, 6656)
X(35567) = Ψ(X(6), X(305))

X(35568) = ISOTOMIC CONJUGATE OF X(3014)

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

X(35568) lies on the circumcircle and these lines: {76,476}, {110,7799}, {112,340}, {298,5994}, {299,5995}, {691,7811}, {759,18160}, {2715,7835}, {5152,20404}, {5641,23969}

X(35568) = isotomic conjugate of X(3014)
X(35568) = trilinear quotient X(75)/X(3014)
X(35568) = trilinear pole of the line {6, 3268}
X(35568) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(76), X(298)}}
X(35568) = X(31)-isoconjugate of X(3014)
X(35568) = X(2)-reciprocal conjugate of X(3014)
X(35568) = Ψ(X(6), X(3268))

X(35569) = ISOTOMIC CONJUGATE OF X(14277)

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

X(35569) lies on the circumcircle and these lines: {111,316}, {112,34205}

X(35569) = isotomic conjugate of X(14277)
X(35569) = trilinear quotient X(75)/X(14277)
X(35569) = trilinear pole of the line {6, 7664}
X(35569) = reflection of X(99) in the line X(3)X(3266)
X(35569) = lies on the circumconic with center X(7810))
X(35569) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(316), X(892)}}
X(35569) = Cevapoint of X(690) and X(7810)
X(35569) = X(31)-isoconjugate-of X(14277)
X(35569) = X(2)-reciprocal conjugate of X(14277)
X(35569) = Ψ(X(6), X(7664))

X(35570) = ISOTOMIC CONJUGATE OF X(14279)

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

X(35570) lies on the circumcircle and these lines: {141,843}, {827,9182}

X(35570) = isotomic conjugate of X(14279)
X(35570) = trilinear quotient X(75)/X(14279)
X(35570) = trilinear pole of the line {6, 14360}
X(35570) = intersection, other than A,B,C, of circumcircle and conic {{A, B, C, X(141), X(9182)}}
X(35570) = X(31)-isoconjugate-of X(14279)
X(35570) = X(2)-reciprocal conjugate of X(14279)
X(35570) = Ψ(X(6), X(14360))

X(35571) = ISOTOMIC CONJUGATE OF X(14343)

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

X(35571) lies on the circumcircle and the line {253,1297}

X(35571) = isotomic conjugate of X(14343)
X(35571) = trilinear quotient X(75)/X(14343)
X(35571) = trilinear pole of the line {6, 253}
X(35571) = X(31)-isoconjugate-of X(14343)
X(35571) = X(2)-reciprocal conjugate of X(14343)
X(35571) = Ψ(X(6), X(253))

X(35572) = ISOTOMIC CONJUGATE OF X(21128)

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

X(35572) lies on the circumcircle and the line {727,3993}

X(35572) = isotomic conjugate of X(21128)
X(35572) = barycentric quotient X(i)/X(j) for these (i, j): (100, 17459), (101, 20971), (190, 34832), (668, 20899)
X(35572) = trilinear quotient X(i)/X(j) for these (i, j): (75, 21128), (100, 20971), (190, 17459), (668, 34832)
X(35572) = Cevapoint of X(190) and X(932)
X(35572) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 21128}, {513, 20971}, {649, 17459}
X(35572) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 21128), (100, 17459), (101, 20971)

X(35573) = ISOTOMIC CONJUGATE OF X(22226)

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

X(35573) lies on the circumcircle and the line {699,16998}

X(35573) = isotomic conjugate of X(22226)
X(35573) = barycentric quotient X(81)/X(23464)
X(35573) = trilinear quotient X(i)/X(j) for these (i, j): (75, 22226), (86, 23464)
X(35573) = trilinear pole of the line {6, 34020}
X(35573) = Cevapoint of X(274) and X(16695)
X(35573) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 22226}, {42, 23464}
X(35573) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 22226), (81, 23464)
X(35573) = Ψ(X(6), X(34020))

X(35574) = ISOTOMIC CONJUGATE OF X(23770)

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

X(35574) lies on the circumcircle and these lines: {98,16085}, {101,32094}, {105,3263}, {108,4998}, {111,32849}, {112,4567}, {190,28847}, {668,1292}, {739,2991}, {874,927}, {14665,34159}

X(35574) = isotomic conjugate of X(23770)
X(35574) = barycentric product X(668)*X(2991)
X(35574) = barycentric quotient X(i)/X(j) for these (i, j): (99, 16752), (100, 3290), (190, 1738), (666, 14267)
X(35574) = trilinear product X(190)*X(2991)
X(35574) = trilinear quotient X(i)/X(j) for these (i, j): (75, 23770), (190, 3290), (668, 1738), (799, 16752)
X(35574) = trilinear pole of the line {6, 344}
X(35574) = lies on the circumconic with center X(6547))
X(35574) = X(518)-cross conjugate of X(1016)
X(35574) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 23770}, {649, 3290}, {667, 1738}
X(35574) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 23770), (99, 16752), (100, 3290)
X(35574) = Ψ(X(6), X(344))
X(35574) = Λ(X(3125), X(3271))

X(35575) = ISOTOMIC CONJUGATE OF X(30735)

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

X(35575) lies on the circumcircle and these lines: {69,98}, {107,877}, {111,15066}, {112,2421}, {476,9146}, {670,22456}, {691,12833}, {925,4576}, {1302,5468}, {2715,4558}, {5866,23700}, {10411,11636}

X(35575) = isotomic conjugate of X(30735)
X(35575) = barycentric quotient X(i)/X(j) for these (i, j): (110, 7735), (112, 6620), (249, 35278), (662, 4008)
X(35575) = trilinear quotient X(i)/X(j) for these (i, j): (75, 30735), (99, 4008), (162, 6620), (662, 7735)
X(35575) = trilinear pole of the line {6, 3964}
X(35575) = lies on the circumconic with center X(5028))
X(35575) = intersection, other than A,B,C, of conic {{A, B, C, X(69), X(670)}} and circumcircle
X(35575) = Cevapoint of X(512) and X(5028)
X(35575) = X(1350)-cross conjugate of X(249)
X(35575) = X(i)-isoconjugate-of X(j) for these {i,j}: {31, 30735}, {512, 4008}, {656, 6620}
X(35575) = X(i)-reciprocal conjugate of X(j) for these (i,j): (2, 30735), (110, 7735), (112, 6620)
X(35575) = Ψ(X(6), X(3964))

X(35576) = X(56)X(551)∩X(65)X(4714)

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

See Angel Montesdeoca, Euclid 392 .

X(35576) lies on these lines: {56,551}, {65,4714}, {1400,3649}

X(35577) = X(1)X(15828)∩X(57)X(3623)

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

See Angel Montesdeoca and César Lozada, Euclid 393 .

X(35577) lies on these lines: {1, 15828}, {57, 3623}, {145, 8056}

X(35577) = Cevapoint of X(145) and X(3622)
X(35577) = X(1293)-isoconjugate-of X(14352)
X(35577) = lies on the circumconic with center X(1015))
X(35577) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(2)}} and {{A, B, C, X(4), X(15828)}}
X(35577) = trilinear pole of the line {513, 31182}

X(35578) = X(2)X(7)∩X(8)X(524)

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

See Angel Montesdeoca, Euclid 396 .

X(35578) lies on these lines: {1,4454}, {2,7}, {6,4402}, {8,524}, {37,4488}, {69,7229}, {75,1992}, {145,4659}, {190,5308}, {193,17116}, {239,5032}, {320,21356}, {335,17487}, {346,3664}, {391,25590}, {404,24328}, {519,4307}, {536,3241}, {551,4310}, {594,15533}, {597,4000}, {599,2345}, {1002,6007}, {1266,17014}, {1267,13757}, {1278,4460}, {1449,4452}, {1654,5936}, {2297,7274}, {2325,29621}, {3161,4648}, {3616,4419}, {3618,7321}, {3629,4371}, {3729,3945}, {3758,5222}, {3875,27830}, {3879,4461}, {3946,4373}, {4346,17023}, {4361,7231}, {4364,5550}, {4440,26626}, {4470,4643}, {4472,4748}, {4480,16831}, {4645,4715}, {4672,16020}, {4675,29627}, {4713,26103}, {4796,20050}, {4869,4888}, {4887,29598}, {4896,17284}, {4911,33190}, {5391,13637}, {5543,27340}, {5564,11008}, {5731,29069}, {5764,16393}, {5839,7277}, {7081,9740}, {7227,22165}, {7232,20582}, {9309,17049}, {9843,14269}, {10022,17251}, {11160,17364}, {11238,25729}, {12436,15687}, {13639,32794}, {13759,32793}, {16670,24599}, {16826,20073}, {17092,25099}, {17224,17378}, {17237,26039}, {17261,29622}, {17296,32093}, {17318,20057}, {17347,31144}, {17369,21358}, {17740,31179}, {17951,27818}, {19822,31143}, {24231,25055}, {24315,26062}

X(35578) = reflection of X(17251) in X(10022)

Intersections of Simson lines on the nine-point circle: X(35579)-X(35594)

This preamble and centers X(35579)-X(35594) were contributed by César Eliud Lozada, December 22, 2019.

Let P and Q be antipodal points on the circumcircle of ABC. Then the Simson lines of P and Q intersect at a point on the nine-point circle of ABC.

If X is such point of intersection, and P^-1, Q^-1 are the isogonal conjugates of P and Q, then:

X is crosssum of P and Q; therefore, X is the crosspoint of P^-1 and Q^-1.
X is crossdifference of every pair of points on the polar trilinear of X^-1.
X is center of the rectangular circum-hyperbola {{A, B, C, X(4), P^-1, Q^-1}}.

Moreover, if Y is the intersection, other than A,B,C, of the circum-parabolas {{A, B, C, P^-1}} and {{A, B, C, Q^-1}}, then:

X is the complement of Y.
X is the X(4)-Ceva conjugate-of Y^-1.

The appearance of (i, j, k) in the following list means that X(k) is the intersection of Simson lines of the circumcircle-antipodal centers X(i) and X(j):

(74, 110, 3258), (98, 99, 2679), (100, 104, 3259), (101, 103, 1566), (102, 109, 10017), (105, 1292, 5519), (106, 1293, 5516), (107, 1294, 35579), (108, 1295, 35580), (111, 1296, 31654), (112, 1297, 33504), (476, 477, 35581), (691, 842, 35582), (759, 6011, 35583), (827, 29011, 35584), (840, 2742, 35585), (843, 2709, 35586), (901, 953, 35587), (925, 1300, 35588), (927, 2724, 35589), (929, 2723, 35590), (930, 1141, 35591), (933, 18401, 35592), (934, 972, 35593), (935, 2697, 35594), (1113, 1114, 125), (1379, 1380, 115), (1381, 1382, 11)

X(35579) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(107) AND X(1294)

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

X(35579) lies on the nine-point circle and these lines: {2,6080}, {122,520}, {133,1515}, {10575,25641}

X(35579) = complementary conjugate of X(6086)
X(35579) = complement of X(6080)
X(35579) = center of the circumconic {{ A, B, C, X(4), X(520), X(6000), X(14379) }}
X(35579) = crossdifference of every pair of points on line {X(2430), X(2442)}
X(35579) = crosspoint of X(520) and X(6000)
X(35579) = crosssum of X(107) and X(1294)
X(35579) = X(4)-Ceva conjugate of X(6086)
X(35579) = X(1)-complementary conjugate of X(6086)

X(35580) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(108) AND X(1295)

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

X(35580) lies on the nine-point circle and these lines: {2,6081}, {122,656}, {123,521}, {1528,6001}, {5514,14298}

X(35580) = complementary conjugate of X(6087)
X(35580) = complement of X(6081)
X(35580) = center of the circumconic {{ A, B, C, X(4), X(521), X(1433), X(6001) }}
X(35580) = crosssum of X(108) and X(1295)
X(35580) = crossdifference of every pair of points on line {X(2431), X(2443)}
X(35580) = crosspoint of X(i) and X(j) for these (i,j): (4, 14312), (521, 6001)
X(35580) = X(4)-Ceva conjugate of X(6087)
X(35580) = X(1)-complementary conjugate of X(6087)

X(35581) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(476) AND X(477)

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

X(35581) lies on the nine-point circle and these lines: {2,16170}, {4,16169}, {133,1986}, {526,3258}, {1522,1523}

X(35581) = midpoint of X(4) and X(16169)
X(35581) = complementary conjugate of X(16171)
X(35581) = complement of X(16170)
X(35581) = center of the circumconic {{ A, B, C, X(4), X(526), X(5663), X(14385), X(16169) }}
X(35581) = crossdifference of every pair of points on line {X(2436), X(2437)}
X(35581) = crosspoint of X(526) and X(5663)
X(35581) = X(4)-Ceva conjugate of X(16171)
X(35581) = X(1)-complementary conjugate of X(16171)

X(35582) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(691) AND X(842)

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

X(35582) lies on the nine-point circle and these lines: {2,20404}, {113,22566}, {126,5026}, {132,5095}, {526,2679}, {542,1550}, {690,2682}, {804,3258}, {2782,25641}, {5512,16278}, {5663,33330}, {15359,20389}

X(35582) = reflection of X(2682) in X(14114)
X(35582) = complementary conjugate of X(20403)
X(35582) = complement of X(20404)
X(35582) = center of the circumconic {{ A, B, C, X(4), X(542), X(690), X(1640), X(5967) }}
X(35582) = crosspoint of (X(i) and X(j) for these {i,j}: {526,5663}, {542,690}
X(35582) = crosssum of (X(i) and X(j) for these {i,j}: {691, 842}, {476,499}
X(35582) = X(4)-Ceva conjugate of X(20403)
X(35582) = X(1)-complementary conjugate of X(20403)

X(35583) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(759) AND X(6011)

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

X(35583) lies on the nine-point circle and these lines: {2,6083}, {115,14321}, {758,31845}, {6003,8286}

X(35583) = complementary conjugate of X(6089)
X(35583) = complement of X(6083)
X(35583) = center of the circumconic {{ A, B, C, X(4), X(758), X(6003), X(15556), X(27086) }}
X(35583) = crosspoint of X(758) and X(6003)
X(35583) = crosssum of X(759) and X(6011)
X(35583) = X(4)-Ceva conjugate of X(6089)
X(35583) = X(1)-complementary conjugate of X(6089)

X(35584) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(827) AND X(29011)

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

X(35584) lies on the nine-point circle and these lines: {1556,29012}, {5026,13499}

X(35584) = center of the circumconic {{ A, B, C, X(4), X(826), X(14378), X(29012) }}
X(35584) = crosspoint of X(826) and X(29012)
X(35584) = crosssum of X(827) and X(29011)

X(35585) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(840) AND X(2742)

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

X(35585) lies on the nine-point circle and these lines: {900,5519}, {3259,6084}, {28915,31841}

X(35585) = center of the circumconic {{ A, B, C, X(4), X(528), X(2826) }}
X(35585) = crosspoint of X(528) and X(2826)
X(35585) = crosssum of X(840) and X(2742)

X(35586) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(843) AND X(2709)

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

X(35586) lies on the nine-point circle and these lines: {804,31654}, {2679,6088}, {2782,6092}, {33330,33962}

X(35586) = center of the circumconic {{ A, B, C, X(4), X(543), X(2793) }}
X(35586) = crosspoint of X(543) and X(2793)
X(35586) = crosssum of X(843) and X(2709)

X(35587) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(901) AND X(953)

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

X(35587) lies on the nine-point circle and these lines: {117,25485}, {121,214}, {900,3259}, {952,6073}, {5151,25640}

X(35587) = center of the circumconic {{ A, B, C, X(4), X(900), X(952), X(14584) }}
X(35587) = crosspoint of X(900) and X(952)
X(35587) = crosssum of X(901) and X(953)

X(35588) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(925) AND X(1300)

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

X(35588) lies on the nine-point circle and these lines: {52,25641}, {131,1516}, {136,924}

X(35588) = center of the circumconic {{ A, B, C, X(4), X(924), X(1147), X(13754) }}
X(35588) = crosspoint of X(924) and X(13754)
X(35588) = crosssum of X(925) and X(1300)

X(35589) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(927) AND X(2724)

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

X(35589) lies on the nine-point circle and these lines: {926,1566}, {1517,2808}

X(35589) = center of the circumconic {{ A, B, C, X(4), X(926), X(2808) }}
X(35589) = crosspoint of X(926) and X(2808)
X(35589) = crosssum of X(927) and X(2724)

X(35590) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(929) AND X(2723)

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

X(35590) lies on the nine-point circle and these lines: {926,10017}, {928,15612}, {1521,2807}, {1566,8677}, {2818,33331}

X(35590) = center of the circumconic {{ A, B, C, X(4), X(928), X(2807) }}
X(35590) = crosspoint of X(928) and X(2807)
X(35590) = crosssum of X(929) and X(2723)

X(35591) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(930) AND X(1141)

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

X(35591) lies on the nine-point circle and these lines: {4,15907}, {5,33333}, {113,16337}, {128,1154}, {137,1510}, {143,24772}

X(35591) = midpoint of X(143) and X(24772)
X(35591) = reflection of X(33333) in X(5)
X(35591) = complementary conjugate of X(25149)
X(35591) = complement of the isogonal conjugate of X(25149)
X(35591) = center of the circumconic {{ A, B, C, X(4), X(143), X(1154), X(1510), X(11135) }}
X(35591) = nine-point circle-antipode of X(33333)
X(35591) = crosspoint of X(1154) and X(1510)
X(35591) = crosssum of X(930) and X(1141)
X(35591) = X(4)-Ceva conjugate of X(25149)
X(35591) = X(1)-complementary conjugate of X(25149)
X(35591) = X(476)-of-orthic-triangle

X(35592) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(933) AND X(18401)

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

X(35592) lies on the nine-point circle and these lines: {1540,18400}, {6368,20625}

X(35592) = center of the circumconic {{ A, B, C, X(4), X(6368), X(18400) }}
X(35592) = crosssum of X(933) and X(18401)

X(35593) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(934) AND X(972)

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

X(35593) lies on the nine-point circle and these lines: {11,17427}, {971,1543}, {3160,33331}, {3900,5514}

X(35593) = center of the circumconic {{ A, B, C, X(4), X(971), X(3900), X(7367) }}
X(35593) = crosspoint of X(971) and X(3900)
X(35593) = crosssum of X(934) and X(972)
X(35593) = X(676)-complementary conjugate of X(3817)

X(35594) = INTERSECTION OF SIMSON LINES OF THE CIRCUMCIRCLE-ANTIPODAL CENTERS X(935) AND X(2697)

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

X(35594) lies on the nine-point circle and these lines: {133,20410}, {526,33504}, {1554,2781}, {2881,3258}

X(35594) = center of the circumconic {{ A, B, C, X(4), X(2781), X(9517) }}
X(35594) = crosspoint of X(2781) and X(9517)
X(35594) = crosssum of X(935) and X(2697)

X(35595) = X(1)X(9330)∩X(2)X(7)

Barycentrics a*(a^2 - b^2 - 5*b*c - c^2) : :
Barycentrics 5 + 2 Cos[A] : :
X(35595) = 3 (r + 7 R) X[2] - (r + 4 R) X[7]

X(35595) lies on these lines: {1, 9330}, {2, 7}, {6, 17021}, {10, 149}, {37, 17012}, {40, 5068}, {45, 4850}, {72, 17536}, {78, 16859}, {81, 16669}, {88, 31197}, {100, 15254}, {140, 13257}, {190, 24589}, {210, 3957}, {214, 5251}, {238, 5297}, {239, 31035}, {344, 33077}, {392, 1320}, {632, 26877}, {748, 3920}, {756, 7191}, {936, 16865}, {942, 17534}, {984, 7292}, {997, 2320}, {1001, 3711}, {1100, 17019}, {1150, 17335}, {1621, 3689}, {1656, 26878}, {1698, 5141}, {1743, 14996}, {1757, 30950}, {2323, 15018}, {3008, 33155}, {3220, 7496}, {3247, 17013}, {3294, 16815}, {3336, 31253}, {3523, 7171}, {3533, 24467}, {3555, 32635}, {3587, 3839}, {3617, 3895}, {3622, 3646}, {3678, 25542}, {3681, 4423}, {3690, 6688}, {3691, 29569}, {3715, 3873}, {3752, 33761}, {3781, 5640}, {3826, 5057}, {3842, 32944}, {3846, 29873}, {3868, 16842}, {3870, 30393}, {3876, 11108}, {3916, 17535}, {3927, 16854}, {3936, 17263}, {3937, 15082}, {3940, 17542}, {3952, 16823}, {4011, 26037}, {4054, 30578}, {4078, 32842}, {4096, 32923}, {4104, 33173}, {4358, 17277}, {4383, 16777}, {4384, 4671}, {4415, 26724}, {4420, 5259}, {4422, 5241}, {4430, 10582}, {4640, 9342}, {4656, 33150}, {4661, 4666}, {4679, 33108}, {4687, 27491}, {4703, 25961}, {4861, 14740}, {4867, 10176}, {4997, 31205}, {5011, 29610}, {5044, 5047}, {5067, 26921}, {5187, 9780}, {5233, 27757}, {5235, 30818}, {5253, 5302}, {5256, 16673}, {5260, 12739}, {5268, 17127}, {5272, 7226}, {5278, 18743}, {5287, 16667}, {5314, 13595}, {5361, 30567}, {5439, 17546}, {5440, 16858}, {5525, 17023}, {5708, 16856}, {5709, 7486}, {5743, 33157}, {6763, 19878}, {6945, 26446}, {7262, 17124}, {7291, 29613}, {7322, 29815}, {7330, 10303}, {7411, 10157}, {10304, 18540}, {11231, 12515}, {11451, 26893}, {12514, 19877}, {14555, 32858}, {15601, 30653}, {15650, 16853}, {15674, 27385}, {15934, 19536}, {16484, 21805}, {16552, 19740}, {16558, 19876}, {16610, 16814}, {16675, 17020}, {16831, 30561}, {16861, 24929}, {17231, 31143}, {17244, 31034}, {17261, 17495}, {17266, 31017}, {17278, 33151}, {17289, 26998}, {17292, 21383}, {17337, 33129}, {17531, 31445}, {17572, 31424}, {18249, 25011}, {18250, 20060}, {18259, 20104}, {18359, 26591}, {20073, 24184}, {21242, 24709}, {21384, 29595}, {21516, 25066}, {21565, 32555}, {21568, 32556}, {24003, 32917}, {24723, 24988}, {25101, 32849}, {25502, 32912}, {25960, 29872}, {26038, 32929}, {27121, 31247}, {27747, 31171}, {28605, 30568}, {31289, 32775}

X(35595) = barycentric product X(86)*X(3956)
X(35595) = barycentric quotient X(3956)/X(10)
X(35595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 9, 3218}, {2, 329, 27186}, {2, 3219, 27003}, {2, 3305, 27065}, {2, 17350, 26627}, {2, 17484, 142}, {2, 18228, 31053}, {2, 26792, 5249}, {2, 27065, 3219}, {2, 31018, 31019}, {9, 3218, 3219}, {210, 5284, 3957}, {756, 17123, 7191}, {908, 6666, 2}, {984, 17125, 7292}, {3218, 27065, 9}, {3305, 7308, 2}, {3681, 4423, 29817}, {3715, 8167, 3873}, {4422, 5241, 32779}, {5044, 5047, 34772}, {17335, 30829, 1150}, {18250, 24564, 20060}

X(35596) = X(2)X(7)∩X(30)X(13243)

Barycentrics 5*a^3 + a^2*b - 5*a*b^2 - b^3 + a^2*c + 3*a*b*c + b^2*c - 5*a*c^2 + b*c^2 - c^3 : :
X(35596) = (4 r + 7 R) X[2] - 2 (r + 4 R) X[7] = 5 X[2] - 4 X[908], 7 X[2] - 8 X[3911], 2 X[908] - 5 X[3218], 7 X[908] - 10 X[3911], 8 X[908] - 5 X[17484], 7 X[3218] - 4 X[3911], 4 X[3218] - X[17484], 16 X[3911] - 7 X[17484]

Let E be the circumellipse with center X(9); the perspector of E is X(1), and X(35596) is the E-inverse of X(35595). (Peter Moses, December 24, 2019)

X(35596) lies on these lines: {2, 7}, {30, 13243}, {89, 4419}, {149, 28534}, {239, 20092}, {514, 4984}, {519, 4316}, {535, 4880}, {1121, 21739}, {3241, 3894}, {4980, 20920}, {5211, 31301}, {5852, 6174}, {6542, 30579}, {10707, 17768}, {10916, 20084}, {15677, 24473}, {17297, 32849}, {20049, 28234}, {24593, 30578}, {26070, 31029}, {28542, 32919}

X(35596) = reflection of X(i) in X(j) for these {i,j}: {2, 3218}, {17484, 2}
X(35596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {20078, 23958, 26792}

X(35597) = X(1)X(3)∩X(551)X(5499)

Barycentrics a (-4 a^6+7 a^5 (b+c)+5 a^4 (b^2-4 b c+c^2)+a^3 (-14 b^3+9 b^2 c+9 b c^2-14 c^3)+2 a^2 (b^4+7 b^3 c-13 b^2 c^2+7 b c^3+c^4)+a (b-c)^2 (7 b^3-2 b^2 c-2 b c^2+7 c^3)-3 (b-c)^4 (b+c)^2) : :

See Antreas P. Hatzipolakis, Francisco Javier Garcia Capitan and Angel Montesdeoca, Euclid 414 .

X(35597) lies on these lines: {1,3}, {551,5499}, {3655,6903}, {5901,6701}, {6224,18480}, {6265,21669}, {19907,26200}

X(35598) = X(5)X(79)∩X(952)X(16125)

Barycentrics 6 a^10-6 a^9 (b+c)+a^8 (-17 b^2+12 b c-17 c^2)+2 a^7 (5 b^3-b^2 c-b c^2+5 c^3)+2 a^6 (11 b^4-6 b^3 c+15 b^2 c^2-6 b c^3+11 c^4)+a^5 (6 b^5+4 b^4 c-4 b^3 c^2-4 b^2 c^3+4 b c^4+6 c^5)+a^4 (-24 b^6+2 b^5 c+5 b^4 c^2+16 b^3 c^3+5 b^2 c^4+2 b c^5-24 c^6) -2 a^3 (b-c)^2 (9 b^5+7 b^4 c-b^3 c^2-b^2 c^3+7 b c^4+9 c^5)+a^2 (b^2-c^2)^2 (20 b^4-16 b^3 c+b^2 c^2-16 b c^3+20 c^4)+2 a (b-c)^4 (b+c)^3 (4 b^2-5 b c+4 c^2)-7 (b-c)^6 (b+c)^4 : :

See Antreas P. Hatzipolakis, Francisco Javier Garcia Capitan and Angel Montesdeoca, Euclid 414 .

X(35598) lies on these lines: {5,79}, {952,16125}

X(35599) = X(1)X(6)∩X(35)X(1190)

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

See Angel Montesdeoca, Euclid 416 and HG231219 .

X(35599) lies on these lines: {1,6}, {35,1190}, {644,6764}, {1170,8232}, {1174,3295}, {2348,7957}

X(35600) = (name pending)

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

See Angel Montesdeoca, HG231219 .

X(35600) lies on this line: {728,34784}

X(35601) = X(143)X(32744)∩X(2070)X(14097)

Barycentrics a^26*b^2 - 9*a^24*b^4 + 37*a^22*b^6 - 93*a^20*b^8 + 162*a^18*b^10 - 210*a^16*b^12 + 210*a^14*b^14 - 162*a^12*b^16 + 93*a^10*b^18 - 37*a^8*b^20 + 9*a^6*b^22 - a^4*b^24 + a^26*c^2 - 18*a^24*b^2*c^2 + 95*a^22*b^4*c^2 - 248*a^20*b^6*c^2 + 367*a^18*b^8*c^2 - 287*a^16*b^10*c^2 + 8*a^14*b^12*c^2 + 258*a^12*b^14*c^2 - 311*a^10*b^16*c^2 + 178*a^8*b^18*c^2 - 39*a^6*b^20*c^2 - 10*a^4*b^22*c^2 + 7*a^2*b^24*c^2 - b^26*c^2 - 9*a^24*c^4 + 95*a^22*b^2*c^4 - 322*a^20*b^4*c^4 + 497*a^18*b^6*c^4 - 324*a^16*b^8*c^4 - 33*a^14*b^10*c^4 + 89*a^12*b^12*c^4 + 182*a^10*b^14*c^4 - 307*a^8*b^16*c^4 + 124*a^6*b^18*c^4 + 47*a^4*b^20*c^4 - 49*a^2*b^22*c^4 + 10*b^24*c^4 + 37*a^22*c^6 - 248*a^20*b^2*c^6 + 497*a^18*b^4*c^6 - 338*a^16*b^6*c^6 - 45*a^14*b^8*c^6 + 74*a^12*b^10*c^6 + 37*a^10*b^12*c^6 + 131*a^8*b^14*c^6 - 189*a^6*b^16*c^6 - 57*a^4*b^18*c^6 + 147*a^2*b^20*c^6 - 46*b^22*c^6 - 93*a^20*c^8 + 367*a^18*b^2*c^8 - 324*a^16*b^4*c^8 - 45*a^14*b^6*c^8 + 70*a^12*b^8*c^8 + 44*a^10*b^10*c^8 + 11*a^8*b^12*c^8 + 78*a^6*b^14*c^8 + 7*a^4*b^16*c^8 - 245*a^2*b^18*c^8 + 130*b^20*c^8 + 162*a^18*c^10 - 287*a^16*b^2*c^10 - 33*a^14*b^4*c^10 + 74*a^12*b^6*c^10 + 44*a^10*b^8*c^10 + 48*a^8*b^10*c^10 + 17*a^6*b^12*c^10 + 19*a^4*b^14*c^10 + 238*a^2*b^16*c^10 - 255*b^18*c^10 - 210*a^16*c^12 + 8*a^14*b^2*c^12 + 89*a^12*b^4*c^12 + 37*a^10*b^6*c^12 + 11*a^8*b^8*c^12 + 17*a^6*b^10*c^12 - 10*a^4*b^12*c^12 - 98*a^2*b^14*c^12 + 372*b^16*c^12 + 210*a^14*c^14 + 258*a^12*b^2*c^14 + 182*a^10*b^4*c^14 + 131*a^8*b^6*c^14 + 78*a^6*b^8*c^14 + 19*a^4*b^10*c^14 - 98*a^2*b^12*c^14 - 420*b^14*c^14 - 162*a^12*c^16 - 311*a^10*b^2*c^16 - 307*a^8*b^4*c^16 - 189*a^6*b^6*c^16 + 7*a^4*b^8*c^16 + 238*a^2*b^10*c^16 + 372*b^12*c^16 + 93*a^10*c^18 + 178*a^8*b^2*c^18 + 124*a^6*b^4*c^18 - 57*a^4*b^6*c^18 - 245*a^2*b^8*c^18 - 255*b^10*c^18 - 37*a^8*c^20 - 39*a^6*b^2*c^20 + 47*a^4*b^4*c^20 + 147*a^2*b^6*c^20 + 130*b^8*c^20 + 9*a^6*c^22 - 10*a^4*b^2*c^22 - 49*a^2*b^4*c^22 - 46*b^6*c^22 - a^4*c^24 + 7*a^2*b^2*c^24 + 10*b^4*c^24 - b^2*c^26 : :

See Antreas P. Hatzipolakis and Peter Moses, Euclid 419 .

X(35601) lies on these lines: {143, 32744}, {2070, 14097}, {10264, 30483}, {25150, 32749}

X(35602) = ISOGONAL CONJUGATE OF X(6526)

Barycentrics    a^2*(-a^2+b^2+c^2)^2*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
Barycentrics    a^2 SA^2(S^2 - 2 SB SC) : :
Barycentrics    (cos^2 A) (tan A - tan B - tan C) : :

See Antreas P. Hatzipolakis and César Lozada, Euclid 421 .

Let A'B'C' be the cevian triangle of X(20). Let A" be the circumcircle-inverse of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(35602). (Randy Hutson, January 17, 2020)

X(35602) lies on the conic {{X(3),X(6),X(24),X(60),X(143),X(1511),X(1986)}} and these lines: {2, 11425}, {3, 49}, {4, 11064}, {6, 2929}, {20, 154}, {22, 11449}, {24, 33586}, {25, 13346}, {26, 1511}, {64, 2063}, {68, 10257}, {69, 3523}, {110, 1498}, {125, 12429}, {143, 6644}, {186, 17834}, {255, 1804}, {343, 631}, {376, 9707}, {378, 15058}, {381, 12897}, {382, 1568}, {417, 10607}, {511, 3515}, {549, 32358}, {567, 15805}, {578, 10601}, {1192, 5889}, {1259, 6507}, {1350, 7488}, {1368, 19467}, {1370, 34782}, {1503, 28419}, {1531, 5073}, {1593, 9306}, {1657, 7728}, {1853, 14516}, {1899, 16196}, {1986, 11412}, {1993, 9786}, {2060, 3344}, {2072, 12293}, {2888, 20376}, {3053, 3289}, {3066, 6642}, {3146, 35264}, {3348, 6617}, {3516, 5907}, {3522, 6800}, {3532, 15055}, {3546, 6146}, {3561, 7011}, {3564, 26937}, {3955, 26935}, {3964, 14379}, {5020, 11424}, {5050, 32284}, {5059, 35265}, {5480, 28708}, {5504, 15316}, {5651, 11479}, {5925, 16386}, {6101, 18324}, {6593, 15020}, {6640, 14852}, {6759, 21312}, {6815, 23292}, {7193, 26927}, {7395, 11430}, {7400, 13394}, {7503, 17811}, {7526, 14128}, {7527, 33537}, {7771, 34386}, {8780, 26883}, {8907, 34787}, {9545, 17809}, {9706, 20791}, {9715, 11202}, {9729, 11402}, {9927, 14156}, {9938, 19457}, {10112, 26869}, {10249, 15069}, {10282, 11414}, {10323, 11464}, {10516, 28408}, {10539, 10564}, {10606, 12111}, {10625, 14070}, {10996, 15438}, {11250, 15068}, {11438, 12160}, {11459, 35477}, {11477, 22151}, {11585, 12118}, {12082, 15034}, {12084, 18451}, {12278, 18405}, {12289, 31180}, {12310, 17701}, {13403, 16072}, {13434, 17825}, {13857, 34725}, {15040, 22109}, {15122, 32140}, {15311, 30552}, {15815, 22416}, {16051, 18945}, {19353, 19458}, {23115, 32661}, {23163, 30262}, {26881, 33524}, {34621, 35266}

X(35602) = isogonal conjugate of X(6526)
X(35602) = isotomic conjugate of the polar conjugate of X(15905)
X(35602) = anticomplement of the complementary conjugate of X(31377)
X(35602) = crosssum of X(4) and X(6622)
X(35602) = X(3964)-Ceva conjugate of X(394)
X(35602) = X(i)-isoconjugate-of X(j) for these {i,j}: {19, 459}, {64, 158}, {253, 1096}
X(35602) = X(i)-reciprocal conjugate of X(j) for these (i,j): (3, 459), (6, 6526), (20, 2052)
X(35602) = intersection, other than A,B,C, of conics {{A, B, C, X(3), X(20)}} and {{A, B, C, X(4), X(10605)}}
X(35602) = barycentric product X(i)*X(j) for these {i, j}: {20, 394}, {69, 15905}, {110, 20580}, {122, 249}
X(35602) = barycentric quotient X(i)/X(j) for these (i, j): (3, 459), (6, 6526), (20, 2052), (122, 338)
X(35602) = trilinear product X(i)*X(j) for these {i, j}: {20, 255}, {63, 15905}, {122, 1101}, {154, 326}
X(35602) = trilinear quotient X(i)/X(j) for these (i, j): (20, 158), (63, 459), (122, 1109), (154, 1096)
X(35602) = perspector of ABC and the 2nd pedal triangle of X(20)
X(35602) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 155, 10605), (3, 1092, 394), (3, 1147, 1181), (3, 3167, 185), (3, 12164, 1204), (3, 19357, 3796), (3, 22115, 155), (22, 11449, 17821), (110, 11413, 1498), (1204, 3292, 12164), (1993, 22467, 9786), (2071, 11441, 64), (3516, 6090, 5907), (5889, 15078, 1192), (6642, 10982, 3066), (6642, 13352, 10982), (10539, 10564, 12085), (11202, 15644, 9715), (11412, 15035, 32534), (17928, 34148, 6)

X(35603) = ISOGONAL CONJUGATE OF X(32132)

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

See Antreas P. Hatzipolakis and César Lozada, Euclid 421 .

Let A'B'C' be the cevian triangle of X(24). Let A" be the circumcircle-inverse of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(35603). (Randy Hutson, January 17, 2020)

X(35603) lies on the conic {{X(3),X(6),X(24),X(60),X(143),X(1511),X(1986)}} and these lines: {4, 6}, {24, 52}, {25, 143}, {26, 34397}, {68, 403}, {110, 9937}, {155, 3542}, {186, 17834}, {195, 3517}, {235, 13292}, {343, 7505}, {378, 569}, {394, 3147}, {1112, 7517}, {1511, 3515}, {1593, 13491}, {1594, 5422}, {1598, 15087}, {1609, 8883}, {1994, 7487}, {6243, 21213}, {7509, 19131}, {8538, 12082}, {9777, 32341}, {9967, 10323}, {10018, 15066}, {10539, 12235}, {10984, 11470}, {11412, 19128}, {11432, 19362}, {11750, 35490}, {12085, 15472}, {12160, 19118}, {12897, 35480}, {13198, 32321}, {14627, 18494}, {18474, 35488}, {19154, 31807}

X(35603) = isogonal conjugate of X(32132)
X(35603) = X(i)-isoconjugate-of X(j) for these {i,j}: {68, 921}, {91, 15316}, {1820, 6504}
X(35603) = X(i)-reciprocal conjugate of X(j) for these (i,j): (6, 32132), (24, 6504), (135, 338)
X(35603) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(1993)}} and {{A, B, C, X(6), X(155)}}
X(35603) = barycentric product X(i)*X(j) for these {i, j}: {24, 6515}, {135, 249}, {155, 11547}, {317, 1609}
X(35603) = barycentric quotient X(i)/X(j) for these (i, j): (6, 32132), (24, 6504), (135, 338), (571, 15316)
X(35603) = trilinear product X(i)*X(j) for these {i, j}: {24, 920}, {47, 3542}, {135, 1101}, {1609, 1748}
X(35603) = trilinear quotient X(i)/X(j) for these (i, j): (24, 921), (47, 15316), (135, 1109), (920, 68)
X(35603) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (24, 2904, 1993), (1181, 2883, 11456), (3515, 19504, 16266)

X(35604) = REFLECTION OF X(11) IN X(34949)

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

See Antreas P. Hatzipolakis and César Lozada, Euclid 421 .

X(35604) lies on these lines: {11, 15313}, {55, 1618}, {3024, 3025}, {10703, 13756}

X(35604) = reflection of X(11) in X(34949)
X(35604) = pole of the trilinear polar of X(34529) wrt Feuerbach hyperbola
X(35604) = (Pelletier)-isogonal conjugate of X(11)
X(35604) = X(1618)-of-Mandart-incircle triangle

X(35605) = MIDPOINT OF X(249) AND X(3448)

Barycentrics (b^2-c^2)^2*(2*a^6-3*(b^2+c^2)*a^4+2*(b^4+b^2*c^2+c^4)*a^2-b^6-c^6) : :
X(35605) = 4*X(125)-X(5099), 3*X(9140)+X(9218), 2*X(10264)+X(16188)

See Antreas P. Hatzipolakis and César Lozada, Euclid 421 .

X(35605) lies on these lines: {2, 33803}, {115, 3566}, {125, 512}, {249, 3448}, {427, 15544}, {525, 15357}, {1899, 32761}, {3111, 21243}, {3143, 10413}, {5576, 11554}, {6328, 10279}, {6787, 26913}, {7668, 15609}, {9140, 9218}, {10264, 16188}, {15359, 32478}, {20299, 31850}

X(35605) = midpoint of X(249) and X(3448)
X(35605) = reflection of X(115) in X(34953)
X(35605) = complement of X(33803) wrt these co-centroidal triangles: (ABC, anti-Artzt, 1st anti-Brocard, anti-McCay, anticomplementary, Artzt, 1st Brocard-reflected, 1st Brocard, inner-Fermat, outer-Fermat, 1st half-diamonds, 2nd half-diamonds, 1st half-squares, 2nd half-squares, inverse-in-excircles, McCay, medial, 1st Neuberg, 2nd Neuberg, inner-Vecten, outer-Vecten)
X(35605) = (Schroeter)-isogonal conjugate of X(115)

X(35606) = X(2)X(99)∩X(6)X(523)

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

X(35606) lies on the cubic K1141 and on these lines: {2, 99}, {3, 5915}, {6, 523}, {30, 1648}, {187, 4226}, {385, 34245}, {476, 843}, {538, 5468}, {2502, 2782}, {5254, 15000}, {5466, 17964}, {7735, 23967}, {7766, 35146}, {7804, 14608}, {9169, 11159}, {11052, 15301}, {15048, 31945}

X(35606) = psi-transform of X(5912)
X(35606) = crosssum of X(6) and X(33928)
X(35606) = crossdifference of every pair of points on line {351, 511}

X(35607) = X(2)X(1341)∩X(6)X(1344)

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

X(35607) lies on the cubics K138, K881, K886, K887, K1141, and these lines: {2, 1341}, {6, 1344}, {111, 2470}, {1113, 1379}

X(35607) = reflection of X(35608) in X(9173)
X(35607) = psi-transform of X(14899)
X(35607) = crossdifference of every pair of points on line {2575, 5639}
X(35607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1341, 31863, 14899}, {1344, 13414, 35608}, {5638, 13722, 14899}

X(35608) = X(2)X(1340)∩X(6)X(1344)

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

X(35608) lies on the cubics K138, K881, K886, K887, K1141, and these lines: {2, 1340}, {6, 1344}, {111, 2469}, {1113, 1380}

X(35608) = reflection of X(35607) in X(9173)
X(35608) = psi-transform of X(35609)
X(35608) = crossdifference of every pair of points on line {2575, 5638}
X(35608) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1340, 31862, P3}, {1344, 13414, 35607}, {5639, 13636, 35609}

X(35609) = X(2)X(1340)∩X(6)X(1345)

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

X(35609) lies on the cubics K138, K881, K886, K887, K1141, and these lines: {2, 1340}, {6, 1345}, {111, 2470}, {1114, 1380}

X(35609) = reflection of reflection of X(14899) in X(9174)
X(35609) = psi-transform of X(35608)
X(35609) = crossdifference of every pair of points on line {2574, 5638}
X(35609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1340, 31862, 35608}, {1345, 13415, 14899}, {5639, 13636, 35608}

Miscellaneous triangles and related centers: X(35610)-X(35900)

This preamble and centers X(35610)-X(35900) were contributed by César Eliud Lozada, December 29, 2019.

The following triangles are defined in ETC:

(ABC)-Bankoff equilateral triangle (shortened to Bankoff-equilateral triangle), in the preamble just before X(34551)
Bevan antipodal, in the preamble just before X(34488)
inverse-in-Conway, in X(10439)
inverse-in-excircles, in X(8055)
1st Kenmotu-free-vertices, in X(12375)
2nd Kenmotu-free-vertices, in X(12376)
Moses-Steiner osculatory, in the preamble just before X(34504)
Moses-Steiner reflection, in the preamble just before X(34504)
orthic axes, in X(2501)
Yiu tangents, in X(7495)

A complete list of centers related to these triangles can be seen here.

X(35610) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO BEVAN ANTIPODAL

Barycentrics a*(a^3+2*(b+c)*a^2-2*S*a-(b^2+4*b*c+c^2)*a-2*(b^2-c^2)*(b-c)) : :
X(35610) = 3*X(371)-2*X(7969) = 3*X(371)-4*X(31439) = 4*X(7969)-3*X(35641) = 4*X(13883)-3*X(35822) = 8*X(31439)-3*X(35641)

The reciprocal orthologic center of these triangles is X(1)

X(35610) lies on these lines: {1,6200}, {3,35642}, {4,35788}, {6,12702}, {8,6561}, {10,6565}, {40,372}, {46,35769}, {65,35808}, {145,9541}, {355,35821}, {371,517}, {484,6502}, {485,962}, {486,5657}, {515,35842}, {516,35820}, {590,22791}, {946,10576}, {1151,1482}, {1385,35811}, {1587,20070}, {1702,6419}, {1703,35770}, {1737,35803}, {1836,35800}, {1902,35764}, {2066,5903}, {2067,5697}, {2093,31432}, {2800,35882}, {2802,35856}, {3057,35768}, {3070,28174}, {3071,5690}, {3245,3299}, {3365,34560}, {3579,6396}, {3617,23259}, {3654,13973}, {4301,13912}, {5119,16232}, {5289,9679}, {5414,11010}, {5418,5603}, {5587,35787}, {5734,9680}, {5790,23261}, {5812,35798}, {5840,35852}, {6001,35864}, {6221,8148}, {6361,6560}, {6409,10246}, {6449,10247}, {6453,7982}, {6459,12245}, {6480,11278}, {6482,9584}, {6484,9615}, {6486,33179}, {6564,12699}, {7583,28212}, {7978,10819}, {8253,18493}, {9556,30116}, {9583,11531}, {9647,10944}, {9660,10950}, {9911,35776}, {10039,35801}, {10306,35772}, {10577,26446}, {10704,11835}, {12197,35766}, {12376,12778}, {12458,35778}, {12459,35780}, {12497,35782}, {12515,35857}, {12696,35790}, {12697,35792}, {12698,35794}, {12700,35796}, {12701,35802}, {12703,35816}, {12704,35818}, {13211,35835}, {13883,28194}, {13893,31162}, {13975,35814}, {15950,31499}, {19914,35853}, {22770,35784}, {22793,35786}, {22841,35804}, {22842,35806}, {34339,35817}

X(35610) = reflection of X(i) in X(j) for these (i,j): (7969, 31439), (35641, 371)
X(35610) = X(40)-of-1st Kenmotu-free-vertices triangle
X(35610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35642, 35762), (6, 12702, 35611), (40, 35775, 372), (1151, 1482, 35763), (1702, 7991, 35774), (1702, 35774, 6419), (3071, 5690, 35789), (3579, 7968, 6396), (5119, 16232, 35809), (6361, 19066, 6560), (7969, 31439, 371), (12699, 13911, 6564)

X(35611) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO BEVAN ANTIPODAL

Barycentrics a*(a^3+2*(b+c)*a^2+2*S*a-(b^2+4*b*c+c^2)*a-2*(b^2-c^2)*(b-c)) : :
X(35611) = 3*X(372)-2*X(7968) = 4*X(7968)-3*X(35642) = 4*X(13936)-3*X(35823)

The reciprocal orthologic center of these triangles is X(1)

X(35611) lies on these lines: {1,6396}, {3,35641}, {4,35789}, {6,12702}, {8,6560}, {10,6564}, {40,371}, {46,35768}, {65,35809}, {355,35820}, {372,517}, {484,2067}, {485,5657}, {486,962}, {515,35843}, {516,35821}, {615,22791}, {946,10577}, {1152,1482}, {1385,35810}, {1588,20070}, {1702,35771}, {1703,6420}, {1737,35802}, {1836,35801}, {1902,35765}, {2066,11010}, {2362,5119}, {2800,35883}, {2802,35857}, {3057,35769}, {3070,5690}, {3071,28174}, {3245,3301}, {3389,34560}, {3579,6200}, {3617,23249}, {3654,13911}, {4301,13975}, {5414,5903}, {5420,5603}, {5587,35786}, {5697,6502}, {5790,23251}, {5812,35799}, {5840,35853}, {6001,35865}, {6361,6561}, {6398,8148}, {6410,10246}, {6450,10247}, {6454,7982}, {6460,12245}, {6480,9582}, {6481,11278}, {6485,16200}, {6487,33179}, {6565,12699}, {7584,28212}, {7978,10820}, {8252,18493}, {9911,35777}, {10039,35800}, {10306,35773}, {10576,26446}, {10704,11836}, {12197,35767}, {12375,12778}, {12458,35781}, {12459,35779}, {12497,35783}, {12515,35856}, {12696,35791}, {12697,35795}, {12698,35793}, {12700,35797}, {12701,35803}, {12703,35817}, {12704,35819}, {13211,35834}, {13912,35815}, {13936,28194}, {13947,31162}, {19914,35852}, {22770,35785}, {22793,35787}, {22841,35807}, {22842,35805}, {34339,35816}

X(35611) = reflection of X(35642) in X(372)
X(35611) = X(40)-of-2nd Kenmotu-free-vertices triangle
X(35611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35641, 35763), (6, 12702, 35610), (40, 35774, 371), (1152, 1482, 35762), (1703, 7991, 35775), (1703, 35775, 6420), (2362, 5119, 35808), (3070, 5690, 35788), (3579, 7969, 6200), (6361, 19065, 6561), (12699, 13973, 6565)

X(35612) = HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND INVERSE-IN-CONWAY

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

X(35612) lies on these lines: {1,3}, {2,3786}, {6,18165}, {9,16345}, {72,16343}, {81,184}, {142,3741}, {314,30962}, {386,28258}, {405,10461}, {443,10449}, {511,5712}, {518,5737}, {579,8731}, {971,10888}, {1125,35632}, {1150,3757}, {1368,18635}, {1401,28108}, {1447,5736}, {3305,16355}, {3664,3784}, {3705,29981}, {3720,28274}, {3742,15668}, {3794,17379}, {3819,4648}, {3868,16342}, {3876,19334}, {4228,5320}, {4259,17056}, {5044,16457}, {5272,27623}, {5439,16458}, {5738,26118}, {5744,35614}, {5745,35628}, {5751,8727}, {6675,35637}, {6857,34259}, {8728,10479}, {8729,35627}, {8732,35617}, {8733,35624}, {8734,35625}, {9776,10453}, {9856,12548}, {9858,35629}, {9942,35635}, {9943,12544}, {9944,10167}, {9945,35636}, {9946,35638}, {10381,13725}, {10446,14548}, {10454,20420}, {10855,35613}, {10858,35622}, {11019,24220}, {11020,19645}, {11854,35618}, {11855,35619}, {12385,35630}, {12436,35633}, {12437,35634}, {12439,35639}, {12442,35643}, {12443,35644}, {12444,35646}, {13226,35649}, {14996,15080}, {16351,24473}, {16849,19727}, {17612,35626}, {19767,25059}, {21746,26098}, {26102,27626}, {28082,34281}, {31540,35648}, {31541,35647}, {34042,34045}, {35640,35655}

X(35612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10473, 35645), (2, 5208, 10477), (81, 4224, 5138), (12410, 14799, 3428)

X(35613) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND INVERSE-IN-CONWAY

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

X(35613) lies on these lines: {1,2}, {57,3706}, {69,1699}, {75,10980}, {165,3886}, {312,5223}, {314,3062}, {982,17151}, {1043,7987}, {1150,4512}, {1743,32853}, {1764,10860}, {2321,24477}, {2886,17296}, {3333,5295}, {3416,24392}, {3686,26105}, {3696,5437}, {3702,12526}, {3714,6762}, {3928,5695}, {3929,4387}, {3973,4011}, {4023,20196}, {4042,7308}, {4046,17728}, {4361,5573}, {4417,7988}, {4673,7991}, {4720,35262}, {4891,5737}, {4923,6692}, {4966,25525}, {5208,10861}, {5274,32099}, {5372,35258}, {5774,31393}, {5927,10439}, {8581,10473}, {8951,25079}, {9581,10371}, {9856,10441}, {9947,35631}, {9948,35635}, {9949,12544}, {9950,10444}, {9951,35636}, {9952,35638}, {9953,35639}, {9954,35645}, {10476,10864}, {10478,10863}, {10480,10866}, {10569,11021}, {10855,35612}, {10865,35617}, {10867,35622}, {10868,35623}, {11035,35620}, {11678,35614}, {11856,35618}, {11857,35619}, {11858,35624}, {11859,35625}, {11860,35627}, {12386,35630}, {12446,35632}, {12448,35634}, {12449,35643}, {12450,35644}, {12451,35646}, {13227,35649}, {15587,35892}, {16120,35637}, {16469,32942}, {16667,25496}, {17064,33087}, {17272,24210}, {17286,33121}, {17604,21334}, {18227,35628}, {18743,30393}, {31542,35648}, {31543,35647}, {34041,34045}, {35640,35656}

X(35613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 30567, 8580), (3886, 14829, 165), (5271, 29824, 10582), (10453, 11679, 1), (17156, 30942, 2999)

X(35614) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-CONWAY AND INVERSE-IN-CONWAY

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

X(35614) lies on these lines: {1,21}, {2,181}, {8,10441}, {9,35617}, {10,30007}, {72,35631}, {78,10476}, {100,1764}, {144,35892}, {145,10480}, {197,28920}, {200,35621}, {209,33121}, {228,4511}, {314,17135}, {329,2810}, {511,4388}, {517,32932}, {518,1999}, {651,34045}, {674,4514}, {752,7186}, {908,3741}, {1150,20788}, {1370,21280}, {1400,24550}, {1401,26840}, {1402,3218}, {1460,26625}, {1469,27184}, {1761,22099}, {2300,7191}, {2887,3792}, {2891,3436}, {2979,3888}, {3061,5364}, {3434,10446}, {3496,23623}, {3616,35620}, {3621,35634}, {3681,10439}, {3687,29311}, {3705,26893}, {3781,29641}, {3786,31330}, {3799,32862}, {3917,4645}, {3980,5903}, {4259,32773}, {4551,29472}, {5279,20752}, {5303,10470}, {5730,20760}, {5744,35612}, {5904,17733}, {8125,35625}, {8126,35627}, {8679,33066}, {9961,12547}, {10442,25722}, {10444,12530}, {10478,11680}, {10479,11681}, {10889,30628}, {11678,35613}, {11685,35618}, {11686,35619}, {11687,35622}, {11690,35624}, {11691,35644}, {12125,35629}, {12389,35630}, {12435,14923}, {12527,35633}, {12528,35635}, {12529,12544}, {12531,35636}, {12532,35638}, {12533,35639}, {12534,35643}, {12535,35646}, {14829,22275}, {17137,18659}, {17140,21273}, {17165,30710}, {17597,21769}, {17615,35626}, {20535,21218}, {20718,32939}, {21246,24996}, {22276,32851}, {22744,27950}, {23155,32859}, {24101,30665}, {24551,27659}, {30478,34259}, {31547,35648}, {31548,35647}, {35640,35659}

X(35614) = reflection of X(1999) in X(21334)
X(35614) = isotomic conjugate of the isogonal conjugate of X(16872)
X(35614) = anticomplementary conjugate of the anticomplement of X(261)
X(35614) = anticomplement of X(181)
X(35614) = barycentric product X(i)*X(j) for these {i, j}: {76, 16872}, {274, 22301}
X(35614) = trilinear product X(i)*X(j) for these {i, j}: {75, 16872}, {86, 22301}
X(35614) = pole of the trilinear polar of X(18021) with respect to Steiner circumellipse
X(35614) = crosspoint of X(670) and X(4564)
X(35614) = crosssum of X(669) and X(2170)
X(35614) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (21, 1654), (60, 192), (81, 17778)
X(35614) = X(1629)-of-inner-Conway triangle
X(35614) = X(1629)-of-inverse-in-Conway triangle
X(35614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17185, 1621), (2979, 6327, 3888), (10473, 35628, 2)

X(35615) = PERSPECTOR OF THESE TRIANGLES: FUHRMANN AND INVERSE-IN-CONWAY

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

X(35615) lies on these lines: {1,3896}, {3,35638}, {8,35636}, {191,1150}, {314,5224}, {355,10371}, {940,4647}, {969,10447}, {1089,5220}, {1158,1764}, {2475,10449}, {4714,16458}, {4975,16343}, {5880,35892}, {9782,10453}, {15668,28611}

X(35616) = PERSPECTOR OF THESE TRIANGLES: OUTER-GARCIA AND INVERSE-IN-CONWAY

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

X(35616) lies on these lines: {1,3996}, {8,10441}, {10,314}, {40,11679}, {100,35638}, {321,11684}, {986,17143}, {2550,10449}, {3339,32104}, {3812,20174}, {4385,5223}, {4647,30710}, {7270,35637}, {9534,10480}, {10453,11024}, {10479,32773}, {19874,25058}, {24715,31964}

X(35617) = HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND INVERSE-IN-CONWAY

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

X(35617) lies on these lines: {1,1170}, {7,310}, {9,35614}, {390,10441}, {1156,35649}, {1764,7676}, {1999,15185}, {2285,3873}, {3741,21617}, {4326,35621}, {5572,21334}, {5728,35631}, {7670,35644}, {7671,10439}, {7673,12435}, {7675,10476}, {7678,10478}, {7679,10479}, {8232,10477}, {8236,10480}, {8237,35622}, {8238,35623}, {8385,35618}, {8386,35619}, {8387,35624}, {8388,35625}, {8389,35627}, {8732,35612}, {9846,35629}, {10444,12718}, {10865,35613}, {11038,35620}, {11679,34784}, {12399,35630}, {12544,12706}, {12560,35632}, {12573,35633}, {12630,35634}, {12669,35635}, {12730,35636}, {12755,35638}, {12846,35639}, {12847,35643}, {12848,35645}, {12850,35646}, {16133,35637}, {17620,35626}, {18230,35628}, {31565,35648}, {31566,35647}, {34028,34045}, {35640,35668}

X(35617) = intersection, other than A,B,C, of conics {{A, B, C, X(310), X(2346)}} and {{A, B, C, X(314), X(10482)}}
X(35617) = {X(10473), X(35892)}-harmonic conjugate of X(7)

X(35618) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-HUTSON AND INVERSE-IN-CONWAY

Barycentrics a*(2*(a+b-c)*(a-b+c)*((b^2+b*c+c^2)*a+b^2*c+b*c^2)*sin(A/2)-2*(-a+b+c)*(a+b-c)*(a+c)*sin(B/2)*b^2-2*(a+b)*(-a+b+c)*(a-b+c)*sin(C/2)*c^2-(b+c)*a^4-(3*b^2+b*c+3*c^2)*a^3+(b^3+c^3)*a^2+(b^2-b*c+c^2)*(3*b^2+2*b*c+3*c^2)*a+3*(b^2-c^2)*(b-c)*b*c) : :

X(35618) lies on these lines: {1,289}, {1764,8107}, {3741,21618}, {5208,11886}, {5934,10477}, {6732,35625}, {8111,10476}, {8113,10473}, {8133,35624}, {8140,35619}, {8377,10478}, {8380,10479}, {8385,35617}, {8390,10480}, {8391,35623}, {9783,10453}, {9836,10441}, {9847,35629}, {10439,11222}, {10444,12719}, {11039,35620}, {11685,35614}, {11854,35612}, {11856,35613}, {11922,35622}, {11923,35627}, {12488,35631}, {12544,12707}, {12561,35632}, {12574,35633}, {12633,35634}, {12673,35635}, {12733,35636}, {12759,35638}, {12851,35639}, {12878,35643}, {12879,35644}, {12880,35645}, {12882,35646}, {12886,35630}, {13260,35649}, {16135,35637}, {17607,21334}, {17621,35626}, {22993,35628}, {34037,34045}

X(35619) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-HUTSON AND INVERSE-IN-CONWAY

Barycentrics a*(2*(a+b-c)*(a-b+c)*((b^2+b*c+c^2)*a+b^2*c+b*c^2)*sin(A/2)-2*(-a+b+c)*(a+b-c)*(a+c)*b^2*sin(B/2)-2*(a+b)*(-a+b+c)*(a-b+c)*c^2*sin(C/2)+(b+c)*a^4+(3*b^2+b*c+3*c^2)*a^3-(b^3+c^3)*a^2-(b^2-b*c+c^2)*(3*b^2+2*b*c+3*c^2)*a-3*(b^2-c^2)*(b-c)*b*c) : :

X(35619) lies on these lines: {1,168}, {314,8372}, {1764,8108}, {3741,21619}, {5208,11887}, {5935,10477}, {8112,10476}, {8114,10473}, {8135,35624}, {8138,35625}, {8140,35618}, {8378,10478}, {8381,10479}, {8386,35617}, {8392,10480}, {9787,10453}, {9837,10441}, {9849,35629}, {10439,11223}, {10444,12720}, {11040,35620}, {11686,35614}, {11855,35612}, {11857,35613}, {11925,35622}, {11926,35623}, {12489,35631}, {12544,12708}, {12562,35632}, {12576,35633}, {12634,35634}, {12674,35635}, {12734,35636}, {12760,35638}, {12852,35639}, {12881,35630}, {12883,35643}, {12884,35644}, {12885,35645}, {12887,35646}, {13261,35649}, {16136,35637}, {17608,21334}, {17623,35626}, {22994,35628}, {34038,34045}

X(35719) = X(4)-of-orthic-axes-triangle

X(35620) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND INVERSE-IN-CONWAY

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

X(35620) lies on these lines: {1,3}, {314,3296}, {495,10479}, {496,10478}, {1056,10449}, {1058,10446}, {1125,35628}, {1191,18165}, {1387,35649}, {1999,3889}, {3244,35634}, {3487,10477}, {3488,10465}, {3555,11679}, {3616,35614}, {3741,21620}, {3881,17733}, {4260,19843}, {5083,35638}, {5208,11036}, {5542,35892}, {6797,12550}, {8092,35627}, {8351,35625}, {9856,12547}, {10444,12722}, {10453,11037}, {10454,18990}, {11035,35613}, {11038,35617}, {11039,35618}, {11040,35619}, {11042,35622}, {11043,35623}, {11044,35624}, {11373,17617}, {11374,19863}, {12128,35629}, {12401,35630}, {12544,12710}, {12563,35632}, {12577,35633}, {12675,35635}, {12735,35636}, {12853,35639}, {12907,35643}, {12908,35644}, {12909,35646}, {15825,30329}, {16137,35637}, {17624,35626}, {18229,34790}, {31569,35648}, {31570,35647}, {34045,34046}, {35640,35670}

X(35620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1764, 3295), (1, 3338, 1402), (1, 10473, 10441), (1, 10882, 24929), (1, 12435, 9957), (5045, 35631, 1), (17609, 21334, 1)

X(35621) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND 6th MIXTILINEAR

Barycentrics a*((b+c)*a^4+(3*b^2+b*c+3*c^2)*a^3-(b^3+c^3)*a^2-(b^2-b*c+c^2)*(3*b^2+2*b*c+3*c^2)*a-3*(b^2-c^2)*(b-c)*b*c) : :
X(35621) = 3*X(165)-4*X(20368)

X(35621) lies on these lines: {1,3}, {20,35633}, {43,29311}, {167,35644}, {200,35614}, {314,3062}, {516,10453}, {573,26102}, {1125,1695}, {1699,3741}, {1709,35626}, {1750,10477}, {1766,32913}, {2051,29827}, {2951,35892}, {3840,9535}, {4326,35617}, {4649,16435}, {5208,5732}, {5223,11679}, {5531,35649}, {5573,21769}, {5691,10449}, {6996,32853}, {7988,10478}, {7989,10479}, {7992,35635}, {7993,35636}, {7996,10444}, {8001,35639}, {8089,35624}, {8090,35625}, {8140,35618}, {8244,35622}, {8245,35623}, {8423,35627}, {8580,35628}, {9315,23441}, {9851,35629}, {10862,30291}, {10886,30308}, {10887,30315}, {10888,30326}, {10889,30330}, {10892,30363}, {11519,35634}, {11894,30371}, {11895,30395}, {11896,30394}, {12404,35630}, {12565,35632}, {12767,35638}, {13069,35643}, {13101,35646}, {16143,35637}, {18229,30393}, {19542,33087}, {19645,32919}, {21363,25502}, {31573,35648}, {31574,35647}, {34033,34045}, {35640,35673}

X(35621) = reflection of X(9535) in X(3840)
X(35621) = X(6638)-of-3rd Conway triangle
X(35621) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1764, 165), (1, 11021, 30350), (1, 11521, 16189), (1, 12435, 11531), (40, 35631, 1), (46, 7373, 11493), (57, 21334, 1), (260, 3601, 3953), (1403, 17595, 18398), (1470, 5348, 3072), (1764, 10439, 1), (2078, 9627, 18956), (2448, 18330, 18956), (3340, 26399, 3338), (5173, 22766, 26399), (5363, 7146, 10475), (5570, 33658, 1771), (5597, 20764, 5902), (10441, 10476, 1), (10474, 11509, 5228)

X(35622) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND 2nd PAMFILOS-ZHOU

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

X(35622) lies on these lines: {1,372}, {314,7595}, {1764,8224}, {3741,12610}, {5208,10885}, {7596,10441}, {8228,10478}, {8230,10479}, {8233,10477}, {8234,10476}, {8237,35617}, {8239,10480}, {8243,10473}, {8244,35621}, {8246,35623}, {8247,35624}, {8248,35625}, {9789,10453}, {10439,10891}, {10444,12724}, {10858,35612}, {10867,35613}, {11042,35620}, {11687,35614}, {11922,35618}, {11925,35619}, {11996,35627}, {12129,35629}, {12490,35631}, {12544,12712}, {12566,35632}, {12578,35633}, {12638,35634}, {12681,35635}, {12744,35636}, {12768,35638}, {12865,35639}, {13070,35643}, {13090,35644}, {13096,35645}, {13120,35646}, {13262,35649}, {16144,35637}, {17610,21334}, {17627,35626}, {18234,35628}, {31575,35648}, {34031,34045}, {35640,35674}

X(35623) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND 1st SHARYGIN

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

X(35623) lies on these lines: {1,21}, {2,4476}, {10,19810}, {37,4269}, {42,25060}, {43,3786}, {55,23369}, {72,4281}, {86,982}, {228,4276}, {244,5333}, {256,314}, {274,24165}, {333,984}, {579,8731}, {756,5235}, {986,1010}, {1284,10473}, {1326,27174}, {1764,4220}, {2303,3509}, {3242,18185}, {3666,3736}, {3670,25526}, {3705,17202}, {3821,33730}, {3944,14009}, {3953,28619}, {4003,16700}, {4022,4038}, {4184,4414}, {4199,10477}, {4267,20760}, {4278,22060}, {4392,8025}, {4424,32932}, {4877,5336}, {5051,10479}, {5214,6615}, {5262,27660}, {7174,18163}, {7226,16704}, {8229,10478}, {8235,10476}, {8238,35617}, {8240,10480}, {8245,35621}, {8246,35622}, {8249,35624}, {8250,35625}, {8391,35618}, {8425,35627}, {9791,10453}, {9840,10441}, {9852,35629}, {9959,35631}, {10439,10892}, {10444,12725}, {10449,26117}, {10455,24179}, {10471,33945}, {10868,35613}, {11043,35620}, {11926,35619}, {12405,35630}, {12544,12713}, {12579,35633}, {12642,35634}, {12683,35635}, {12746,35636}, {12770,35638}, {12869,35639}, {13071,35643}, {13091,35644}, {13097,35645}, {13123,35646}, {13265,35649}, {13588,17596}, {14005,24443}, {14007,24174}, {17063,25507}, {17139,26098}, {17155,30599}, {17167,29639}, {17173,29664}, {17174,29680}, {17182,24239}, {17184,30984}, {17591,18792}, {17611,21334}, {17628,35626}, {18173,20984}, {18235,35628}, {21035,32780}, {22024,30710}, {24450,27798}, {27644,29821}, {31576,35648}, {31577,35647}, {34027,34045}, {35640,35675}

X(35623) = barycentric product X(i)*X(j) for these {i, j}: {81, 32778}, {662, 29017}
X(35623) = barycentric quotient X(163)/X(29018)
X(35623) = trilinear product X(i)*X(j) for these {i, j}: {58, 32778}, {110, 29017}
X(35623) = trilinear quotient X(110)/X(29018)
X(35623) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(7018)}} and {{A, B, C, X(31), X(256)}}
X(35623) = X(523)-isoconjugate-of X(29018)
X(35623) = X(163)-reciprocal conjugate of X(29018)
X(35623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3786, 25059, 43), (5208, 25058, 1)

X(35624) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND TANGENTIAL-MIDARC

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

X(35624) lies on these lines: {1,164}, {177,314}, {188,35628}, {1764,8075}, {2089,10473}, {3741,21622}, {5208,11888}, {8079,10477}, {8081,10476}, {8084,11895}, {8085,10478}, {8087,10479}, {8089,35621}, {8091,10441}, {8095,35635}, {8097,35636}, {8099,35631}, {8101,35645}, {8103,35649}, {8133,35618}, {8135,35619}, {8241,10480}, {8247,35622}, {8249,35623}, {8387,35617}, {8733,35612}, {9793,10453}, {9853,35629}, {10439,11192}, {10444,12726}, {10503,21334}, {10967,12554}, {11044,35620}, {11690,35614}, {11858,35613}, {12544,12714}, {12568,35632}, {12580,35633}, {12643,35634}, {12771,35638}, {12870,35639}, {12916,35630}, {13072,35643}, {13124,35646}, {16146,35637}, {17629,35626}, {31578,35648}, {31579,35647}, {34025,34045}

X(35624) = {X(1), X(35644)}-harmonic conjugate of X(35625)

X(35625) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND 2nd TANGENTIAL-MIDARC

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

X(35625) lies on these lines: {1,164}, {174,10473}, {1764,8076}, {3741,21623}, {5208,11889}, {6732,35618}, {7028,35628}, {8080,10477}, {8082,10476}, {8083,11021}, {8084,12554}, {8086,10478}, {8088,10479}, {8090,35621}, {8092,10441}, {8096,35635}, {8098,35636}, {8100,35631}, {8102,35645}, {8104,35649}, {8125,35614}, {8138,35619}, {8242,10480}, {8248,35622}, {8250,35623}, {8351,35620}, {8388,35617}, {8734,35612}, {9795,10453}, {9854,35629}, {10439,11217}, {10444,12727}, {10501,21334}, {10967,11894}, {11859,35613}, {12544,12715}, {12569,35632}, {12581,35633}, {12644,35634}, {12772,35638}, {12871,35639}, {13073,35643}, {13125,35646}, {13475,35630}, {16147,35637}, {17630,35626}, {31580,35648}, {31581,35647}, {34034,34045}

X(35625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35644, 35624), (174, 10473, 35627), (11021, 11896, 8083)

X(35626) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND URSA MAJOR

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

X(35626) lies on these lines: {1,474}, {11,1211}, {354,32921}, {355,10449}, {392,5737}, {517,14829}, {599,11235}, {1709,35621}, {1764,17613}, {3057,32916}, {3434,4645}, {4891,18165}, {5208,17616}, {5927,10439}, {9025,33106}, {10441,12672}, {10444,17651}, {10473,17625}, {10476,12114}, {10478,17618}, {10479,17619}, {10480,17622}, {11679,17658}, {12544,17650}, {17612,35612}, {17615,35614}, {17620,35617}, {17621,35618}, {17623,35619}, {17624,35620}, {17627,35622}, {17628,35623}, {17629,35624}, {17630,35625}, {17631,35627}, {17644,35629}, {17645,35630}, {17646,35632}, {17647,35633}, {17649,35635}, {17652,35636}, {17653,35637}, {17654,35638}, {17655,35639}, {17656,35643}, {17657,35644}, {17659,35646}, {17661,35649}, {17668,35892}, {17792,33141}, {18236,35628}, {20359,32941}, {24239,35104}, {30986,33084}, {31586,35648}, {31587,35647}, {34045,34049}, {35640,35678}

X(35626) = {X(10439), X(35613)}-harmonic conjugate of X(10477)

X(35627) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND YFF CENTRAL

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

X(35627) lies on these lines: {1,168}, {174,10473}, {177,314}, {236,35628}, {1764,7589}, {3741,21624}, {5208,11890}, {7590,10476}, {7593,10477}, {8092,35620}, {8126,35614}, {8351,10441}, {8379,10478}, {8382,10479}, {8389,35617}, {8423,35621}, {8425,35623}, {8729,35612}, {10439,11195}, {10444,12728}, {10453,11891}, {10480,11924}, {10502,21334}, {11021,11033}, {11032,12554}, {11860,35613}, {11923,35618}, {11996,35622}, {12130,35629}, {12406,35630}, {12491,35631}, {12544,12716}, {12570,35632}, {12582,35633}, {12646,35634}, {12685,35635}, {12748,35636}, {12774,35638}, {12873,35639}, {13074,35643}, {13092,35644}, {13098,35645}, {13127,35646}, {13267,35649}, {16151,35637}, {17631,35626}, {31592,35648}, {31593,35647}, {34026,34045}, {35640,35681}

X(35627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (174, 10473, 35625), (11021, 11895, 11033)

X(35628) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY AND 2nd ZANIAH

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

X(35628) lies on these lines: {1,6}, {2,181}, {8,314}, {10,3781}, {21,2175}, {55,17185}, {56,16574}, {63,1402}, {65,10436}, {188,35624}, {210,4042}, {236,35627}, {516,24705}, {760,1944}, {894,3869}, {936,10476}, {940,22275}, {980,2274}, {993,7193}, {1002,5208}, {1125,35620}, {1329,10479}, {1376,1764}, {1463,17274}, {1469,4357}, {1999,3681}, {2550,10446}, {2551,10449}, {2886,10478}, {2979,33083}, {3035,35649}, {3036,35636}, {3056,3883}, {3057,3886}, {3059,10889}, {3452,3741}, {3678,17733}, {3685,3877}, {3690,33163}, {3696,10447}, {3717,4517}, {3740,10439}, {3742,11021}, {3775,21244}, {3792,32784}, {3878,3923}, {3917,26034}, {4026,4259}, {4363,20718}, {4640,10434}, {4643,8679}, {4966,30847}, {5044,35631}, {5061,26625}, {5087,10886}, {5219,30986}, {5737,20788}, {5745,35612}, {5777,35635}, {5784,10444}, {5793,22299}, {5836,10456}, {5837,20258}, {5903,24342}, {7015,10570}, {7028,35625}, {7064,27549}, {7998,33086}, {8580,35621}, {9943,12547}, {10442,10862}, {10453,18228}, {10474,19860}, {10475,19861}, {12544,18251}, {12672,12717}, {14555,23638}, {17018,25058}, {18227,35613}, {18230,35617}, {18234,35622}, {18235,35623}, {18236,35626}, {18247,35629}, {18248,35630}, {18249,35632}, {18250,35633}, {18253,35637}, {18254,35638}, {18255,35639}, {18257,35643}, {18258,35644}, {18259,35646}, {18792,24464}, {19863,25681}, {21233,24325}, {22993,35618}, {22994,35619}, {24987,29967}, {26911,33166}, {27410,27420}, {30710,32937}, {31594,35648}, {31595,35647}, {34045,34048}, {35640,35682}

X(35628) = barycentric product X(i)*X(j) for these {i, j}: {8, 980}, {312, 2274}
X(35628) = barycentric quotient X(i)/X(j) for these (i, j): (55, 981), (980, 7), (2274, 57)
X(35628) = trilinear product X(i)*X(j) for these {i, j}: {8, 2274}, {9, 980}
X(35628) = trilinear quotient X(i)/X(j) for these (i, j): (9, 981), (980, 57), (2274, 56)
X(35628) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(980)}} and {{A, B, C, X(6), X(314)}}
X(35628) = X(57)-isoconjugate-of X(981)
X(35628) = X(i)-reciprocal conjugate of X(j) for these (i,j): (55, 981), (980, 7), (2274, 57)
X(35628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10477, 35892), (2, 35614, 10473), (8, 10480, 35634), (210, 21334, 11679), (3242, 21769, 1), (30556, 30557, 213)

X(35629) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO ANDROMEDA

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

The reciprocal orthologic center of these triangles is X(1)

X(35629) lies on these lines: {1,2}, {740,3339}, {942,17151}, {971,10441}, {1706,28581}, {1764,9841}, {1834,17296}, {5208,9859}, {5234,32853}, {5295,25590}, {8951,18743}, {9842,10478}, {9844,10477}, {9845,10476}, {9846,35617}, {9847,35618}, {9848,10480}, {9849,35619}, {9850,10473}, {9851,35621}, {9852,35623}, {9853,35624}, {9854,35625}, {9858,35612}, {10439,12126}, {10454,12555}, {12125,35614}, {12128,35620}, {12129,35622}, {12130,35627}, {12435,35662}, {12526,32915}, {13740,16667}, {17314,24391}, {17632,21334}, {17644,35626}, {18247,35628}

X(35629) = X(17814)-of-3rd Conway triangle
X(35629) = X(18909)-of-inverse-in-Conway triangle
X(35629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4853, 10587, 10), (6633, 24902, 1125), (8583, 30036, 4668), (10327, 25813, 25698), (16829, 26964, 26965), (16831, 25741, 30132), (18229, 30167, 29697), (19784, 26766, 10586), (20653, 29819, 24541), (25838, 29870, 27525), (26251, 30128, 30970), (26594, 33171, 19874), (26626, 31339, 27255), (26807, 30751, 29576), (28977, 30145, 20069)

X(35630) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO ANTLIA

Barycentrics a^10-7*(b+c)*a^9+2*(b+3*c)*(3*b+c)*a^8-2*(b+c)*(3*b^2+5*b*c+3*c^2)*a^7+2*(8*b^4+8*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a^6+2*(b+c)*(2*b^4+2*c^4-(11*b^2-10*b*c+11*c^2)*b*c)*a^5-2*(11*b^6+11*c^6-(13*b^4+13*c^4+(b+c)^2*b*c)*b*c)*a^4+2*(b^2-c^2)*(b-c)^3*(3*b^2+b*c+3*c^2)*a^3-(b^6+c^6-(8*b^4+8*c^4+(21*b^2+40*b*c+21*c^2)*b*c)*b*c)*(b-c)^2*a^2+(b^4-c^4)*(b-c)^3*(3*b^2+2*b*c+3*c^2)*a+2*(b^4-c^4)*(b^2-c^2)*(b-c)^2*b*c : :
X(35630) = 3*X(10439)-X(12392)

The reciprocal orthologic center of these triangles is X(1)

X(35630) lies on these lines: {1,7056}, {1764,12387}, {3741,21626}, {5208,12390}, {10439,12392}, {10453,12391}, {10473,12402}, {10476,12398}, {10477,12397}, {10478,12393}, {10479,12394}, {10480,12400}, {12385,35612}, {12386,35613}, {12389,35614}, {12399,35617}, {12401,35620}, {12404,35621}, {12405,35623}, {12406,35627}, {12881,35619}, {12886,35618}, {12916,35624}, {13475,35625}, {17633,21334}, {17645,35626}, {18248,35628}

X(35631) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO AYME

Barycentrics a*((b^2+c^2)*a^4+(b+c)*(b^2+b*c+c^2)*a^3-(b^2+c^2)*(b^2-b*c+c^2)*a^2-(b+c)*(b^4+c^4+(b-c)^2*b*c)*a-(b^2-c^2)^2*b*c) : :
X(35631) = X(1)+3*X(10439) = 3*X(1)+X(12435) = X(5752)-3*X(5886) = 3*X(10165)-2*X(15489) = 3*X(10439)-X(10441) = 9*X(10439)-X(12435) = 3*X(10440)-5*X(19862) = 3*X(10441)-X(12435) = 3*X(11230)-2*X(34466)

The reciprocal orthologic center of these triangles is X(10)

X(35631) lies on these lines: {1,3}, {4,10453}, {5,3741}, {19,22127}, {30,12545}, {42,19513}, {43,19549}, {72,35614}, {320,10446}, {355,10449}, {392,11110}, {511,946}, {515,15488}, {518,17733}, {674,3813}, {908,16980}, {970,1125}, {971,35635}, {978,9567}, {1071,5208}, {1872,7009}, {1953,22065}, {1999,3555}, {2771,35637}, {3452,23841}, {3720,13731}, {3757,21554}, {3781,19843}, {3784,4295}, {4281,18178}, {4999,22276}, {5044,35628}, {5728,35617}, {5752,5886}, {5777,10477}, {6001,15486}, {8099,35624}, {8100,35625}, {9548,26102}, {9549,21214}, {9947,35613}, {9955,10478}, {9956,10479}, {9959,35623}, {10165,15489}, {10440,19862}, {10454,28160}, {10465,18481}, {10527,26893}, {11019,12109}, {11230,19863}, {11415,26892}, {11679,34790}, {12488,35618}, {12489,35619}, {12490,35622}, {12491,35627}, {12908,31784}, {15171,31774}, {17617,31937}, {18990,31782}, {19518,19860}, {21853,33863}, {22076,24541}, {25941,28258}, {29309,31730}, {31783,32183}

X(35631) = midpoint of X(i) and X(j) for these {i,j}: {1, 10441}, {942, 31779}, {1482, 31778}, {7982, 31785}, {9957, 31781}, {12908, 31784}, {15171, 31774}, {18990, 31782}, {31780, 31792}, {31783, 32183}, {35638, 35649}
X(35631) = reflection of X(i) in X(j) for these (i,j): (970, 1125), (3579, 5482)
X(35631) = barycentric product X(1)*X(30007)
X(35631) = trilinear product X(6)*X(30007)
X(35631) = Conway circle-inverse of X(484)
X(35631) = X(5)-of-inverse-in-Conway triangle
X(35631) = X(140)-of-3rd Conway triangle
X(35631) = X(10441)-of-anti-Aquila triangle
X(35631) = X(13322)-of-2nd circumperp triangle
X(35631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10439, 10441), (1, 10473, 942), (1, 10475, 24928), (1, 10476, 3), (1, 10480, 9957), (1, 35620, 5045), (1, 35621, 40), (980, 10388, 26424), (1403, 16193, 5563), (7962, 18838, 2556), (10247, 23207, 4689)

X(35632) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 4th CONWAY

Barycentrics a*((b^2+b*c+c^2)*a^4+(b+c)^3*a^3-(b^2-3*b*c+c^2)*(b+c)^2*a^2-(b+c)*(b^4+c^4+(b^2-b*c+c^2)*b*c)*a-(b^3+c^3)*(b+c)*b*c) : :
X(35632) = 3*X(10439)-X(12544) = 3*X(10439)+X(12548)

The reciprocal orthologic center of these triangles is X(1)

X(35632) lies on these lines: {1,21}, {10,3779}, {516,10441}, {1125,35612}, {1764,12511}, {2304,3509}, {3178,26893}, {3671,10473}, {3741,12609}, {3757,5904}, {3771,10974}, {3841,10479}, {4294,20101}, {4295,10453}, {4314,10480}, {4645,10449}, {6001,15486}, {10381,21746}, {10439,12544}, {10476,12520}, {10478,12558}, {10888,31871}, {12446,35613}, {12560,35617}, {12561,35618}, {12562,35619}, {12563,35620}, {12565,35621}, {12566,35622}, {12568,35624}, {12569,35625}, {12570,35627}, {12635,23383}, {12711,21334}, {17646,35626}, {18180,32853}, {18249,35628}

X(35632) = midpoint of X(12544) and X(12548)
X(35632) = X(578)-of-inverse-in-Conway triangle
X(35632) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10461, 993), (10439, 12548, 12544), (10441, 35892, 35633)

X(35633) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 5th CONWAY

Barycentrics 2*(b+c)*a^3+(b^2+4*b*c+c^2)*a^2-(b^3+c^3)*a-(b+c)^2*b*c : :
X(35633) = 5*X(1698)-3*X(4685)

The reciprocal orthologic center of these triangles is X(1)

X(35633) lies on these lines: {1,2}, {6,21071}, {20,35621}, {21,32919}, {76,3879}, {314,3664}, {320,24851}, {405,32853}, {515,15488}, {516,10441}, {726,2901}, {740,942}, {752,15171}, {950,10381}, {960,4891}, {986,4022}, {1010,4038}, {1046,3685}, {1100,21024}, {1479,32946}, {1764,12512}, {1834,4966}, {2260,3501}, {2300,4856}, {2321,17750}, {2650,3702}, {3670,4970}, {3686,16589}, {3730,3950}, {3775,4205}, {3812,28581}, {3868,32915}, {3896,24443}, {3913,20470}, {3946,21240}, {3971,5904}, {4292,5208}, {4297,10476}, {4298,10473}, {4357,33297}, {4649,13740}, {4658,33682}, {4684,13161}, {4709,28612}, {4851,34830}, {4852,20255}, {4889,25102}, {5015,32846}, {5047,32864}, {5051,33081}, {5295,24325}, {5711,32941}, {6376,17377}, {7283,32913}, {8715,23853}, {9345,16454}, {10439,10454}, {10452,31964}, {10477,12572}, {10478,12571}, {10480,12575}, {12109,35104}, {12435,28228}, {12436,35612}, {12527,35614}, {12573,35617}, {12574,35618}, {12576,35619}, {12577,35620}, {12578,35622}, {12579,35623}, {12580,35624}, {12581,35625}, {12582,35627}, {16062,33087}, {17355,21070}, {17363,27269}, {17388,20691}, {17533,33084}, {17647,35626}, {18250,35628}, {18398,24165}, {22189,23659}, {24214,30941}

X(35633) = midpoint of X(2901) and X(3874)
X(35633) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(20012)}} and {{A, B, C, X(291), X(28247)}}
X(35633) = Conway circle-inverse of X(5529)
X(35633) = X(389)-of-inverse-in-Conway triangle
X(35633) = X(11793)-of-3rd Conway triangle
X(35633) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3633, 20037), (1, 3741, 1125), (1, 10449, 10), (145, 29824, 1193), (1210, 4028, 17748), (4701, 5262, 1698), (6764, 26364, 27368), (10441, 35892, 35632), (25761, 29819, 498)

X(35634) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO EXCENTERS-MIDPOINTS

Barycentrics a*(-a+b+c)*((b^2+b*c+c^2)*a^3+(b+c)*(2*b^2-3*b*c+2*c^2)*a^2+(b^4+c^4-(b+c)^2*b*c)*a+(b+c)*(b^2-4*b*c+c^2)*b*c) : :
X(35634) = 3*X(10439)-X(12546)

The reciprocal orthologic center of these triangles is X(1)

X(35634) lies on these lines: {1,474}, {8,314}, {65,3875}, {145,10473}, {517,35635}, {518,10442}, {519,10441}, {528,10454}, {740,31785}, {958,18163}, {1764,12513}, {1999,14923}, {2802,17733}, {3057,11679}, {3244,35620}, {3621,35614}, {3741,21627}, {3813,10479}, {3870,10474}, {3893,21334}, {4421,10470}, {5208,12536}, {5691,9025}, {5853,35892}, {5854,35636}, {10439,12546}, {10453,12541}, {10472,12640}, {10476,12629}, {10477,12625}, {10478,12607}, {10882,11260}, {11519,35621}, {11521,12635}, {12437,35612}, {12448,35613}, {12630,35617}, {12633,35618}, {12634,35619}, {12638,35622}, {12642,35623}, {12643,35624}, {12644,35625}, {12646,35627}

X(35634) = X(64)-of-inverse-in-Conway triangle
X(35634) = X(2883)-of-3rd Conway triangle
X(35634) = {X(8), X(10480)}-harmonic conjugate of X(35628)

X(35635) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO EXTOUCH

Barycentrics a^7+2*(b+c)*a^6+(b^2+6*b*c+c^2)*a^5-2*(b+c)*b*c*a^4-(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^3-2*(b^4-c^4)*(b-c)*a^2-(b^2-c^2)^2*(b+c)^2*a-2*(b^2-c^2)^2*(b+c)*b*c : :
X(35635) = 3*X(10439)-X(12547)

The reciprocal orthologic center of these triangles is X(72)

X(35635) lies on these lines: {1,4}, {3,10472}, {40,11679}, {84,309}, {516,17733}, {517,35634}, {960,5786}, {962,1999}, {971,35631}, {1010,3576}, {1071,10473}, {1158,1764}, {1402,3149}, {2050,3931}, {2800,12435}, {2829,35638}, {3702,19645}, {3741,6245}, {4221,5450}, {4362,35639}, {5208,9960}, {5587,17533}, {5777,35628}, {6001,10441}, {6684,18229}, {6705,10856}, {6796,10434}, {7992,35621}, {8095,35624}, {8096,35625}, {9548,34258}, {9799,10453}, {9942,35612}, {9948,35613}, {10439,12547}, {10477,12664}, {10479,12616}, {10480,12672}, {10889,11372}, {11021,12005}, {11500,19544}, {12514,13478}, {12528,35614}, {12669,35617}, {12673,35618}, {12674,35619}, {12675,35620}, {12681,35622}, {12683,35623}, {12685,35627}, {12688,21334}, {12699,29207}, {17649,35626}

X(35635) = reflection of X(84) in X(15486)
X(35635) = intersection, other than A,B,C, of conics {{A, B, C, X(223), X(17206)}} and {{A, B, C, X(225), X(309)}}
X(35635) = Conway circle-inverse of X(1785)
X(35635) = X(68)-of-inverse-in-Conway triangle
X(35635) = X(1147)-of-3rd Conway triangle
X(35635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (223, 1058, 1870), (946, 5290, 8862), (1056, 3585, 5236), (1072, 1079, 34231), (1870, 13607, 3488), (1877, 3586, 10629), (3465, 3485, 3586), (7009, 10597, 11522)

X(35636) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO FUHRMANN

Barycentrics a*((b^2+b*c+c^2)*a^4+(b^2-c^2)*(b-c)*a^3-(b^4+c^4+(b^2+b*c+c^2)*b*c)*a^2-(b+c)*(b^4+c^4-(b^2-b*c+c^2)*b*c)*a-(b^2-3*b*c+c^2)*(b+c)^2*b*c) : :
X(35636) = 3*X(10439)-X(12550)

The reciprocal orthologic center of these triangles is X(8)

X(35636) lies on these lines: {1,88}, {2,3032}, {8,35615}, {11,10479}, {80,313}, {104,1764}, {119,10478}, {149,10449}, {153,10446}, {517,35638}, {528,35892}, {952,10441}, {1145,10472}, {1317,10473}, {2800,12435}, {2801,10442}, {3036,35628}, {3741,21630}, {3753,28639}, {4360,5902}, {5208,9963}, {5840,10454}, {5854,35634}, {6224,31061}, {6264,10476}, {7993,35621}, {8097,35624}, {8098,35625}, {9024,10477}, {9802,10453}, {9945,35612}, {9951,35613}, {10439,12550}, {10470,34474}, {10474,12739}, {10475,20586}, {10698,11521}, {10882,11715}, {10886,16174}, {12531,35614}, {12730,35617}, {12733,35618}, {12734,35619}, {12735,35620}, {12744,35622}, {12746,35623}, {12748,35627}, {16173,19863}, {17636,21334}, {17652,35626}

X(35636) = midpoint of X(12435) and X(13244)
X(35636) = anticomplement of X(3032)
X(35636) = Conway circle-inverse of X(106)
X(35636) = X(74)-of-inverse-in-Conway triangle
X(35636) = X(113)-of-3rd Conway triangle

X(35637) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 2nd FUHRMANN

Barycentrics a*((b^2+b*c+c^2)*a^2+(b+c)*b*c*a-(b^3+c^3)*(b+c))*(a+c)*(a+b) : :
X(35637) = 3*X(10439)-X(16124)

The reciprocal orthologic center of these triangles is X(4)

X(35637) lies on these lines: {1,21}, {30,10441}, {72,18165}, {79,314}, {86,18398}, {141,442}, {284,1759}, {333,5904}, {354,28619}, {518,18180}, {859,12635}, {942,25526}, {978,18173}, {1010,5902}, {1043,5903}, {1479,17139}, {1698,3786}, {1764,3651}, {2475,10449}, {2771,35631}, {3218,4278}, {3336,13588}, {3555,18178}, {3649,10473}, {3670,3736}, {3678,5235}, {3741,11263}, {3742,28618}, {3833,17551}, {4221,5884}, {4225,22836}, {4269,22021}, {4276,34772}, {5312,25059}, {5358,20602}, {5563,18465}, {5692,11110}, {5883,14005}, {6675,35612}, {6841,10478}, {7270,35616}, {9275,17521}, {10176,17557}, {10439,12547}, {10453,14450}, {10470,21161}, {10471,33930}, {10476,16132}, {10480,10543}, {10916,17167}, {10974,25645}, {11679,31938}, {16120,35613}, {16133,35617}, {16135,35618}, {16136,35619}, {16137,35620}, {16143,35621}, {16144,35622}, {16146,35624}, {16147,35625}, {16151,35627}, {17637,21334}, {17653,35626}, {17768,35892}, {18253,35628}, {19863,26725}, {27660,30117}, {31737,32949}, {31757,32843}

X(35637) = barycentric product X(81)*X(30172)
X(35637) = trilinear product X(58)*X(30172)
X(35637) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(20565)}} and {{A, B, C, X(31), X(79)}}
X(35637) = Conway circle-inverse of X(5127)
X(35637) = X(54)-of-inverse-in-Conway triangle
X(35637) = X(1209)-of-3rd Conway triangle

X(35638) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO INNER-GARCIA

Barycentrics a^7-(b^2-5*b*c+c^2)*a^5+(b+c)*(b^2-5*b*c+c^2)*a^4-2*(2*b^2-3*b*c+2*c^2)*b*c*a^3-(b+c)*(b^4+c^4-2*(3*b^2-4*b*c+3*c^2)*b*c)*a^2-(b^2-c^2)^2*b*c*a-(b^2-c^2)^2*(b+c)*b*c : :
X(35638) = 3*X(1)-X(12550) = 3*X(10439)-X(12551) = X(12550)+3*X(13244)

The reciprocal orthologic center of these triangles is X(3869)

X(35638) lies on these lines: {1,5}, {3,35615}, {99,104}, {100,35616}, {214,10472}, {517,35636}, {1145,11679}, {1320,1999}, {1402,10090}, {1764,12515}, {1768,10476}, {2771,35631}, {2800,10441}, {2801,35892}, {2802,17733}, {2829,35635}, {3741,10265}, {5083,35620}, {5208,9964}, {9803,10453}, {9946,35612}, {9952,35613}, {10074,10475}, {10434,33814}, {10439,12551}, {10449,12247}, {10465,12248}, {10473,11570}, {10477,12691}, {10478,12611}, {10479,12619}, {10480,12758}, {12532,35614}, {12755,35617}, {12759,35618}, {12760,35619}, {12767,35621}, {12768,35622}, {12770,35623}, {12771,35624}, {12772,35625}, {12774,35627}, {17638,21334}, {17654,35626}, {18254,35628}

X(35638) = midpoint of X(1) and X(13244)
X(35638) = reflection of X(35649) in X(35631)
X(35638) = Conway circle-inverse of X(80)
X(35638) = X(265)-of-inverse-in-Conway triangle
X(35638) = X(1511)-of-3rd Conway triangle
X(35638) = X(13244)-of-anti-Aquila triangle

X(35639) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO HUTSON EXTOUCH

Barycentrics a^7+2*(b+c)*a^6+(b^2-10*b*c+c^2)*a^5-18*(b+c)*b*c*a^4-(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^3-2*(b^2-4*b*c+c^2)*(b+c)^3*a^2-(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2)*(b+c)^2*a-2*(b^2-c^2)^2*(b+c)*b*c : :
X(35639) = 3*X(10439)-X(12552)

The reciprocal orthologic center of these triangles is X(3555)

X(35639) lies on these lines: {1,12521}, {314,7160}, {1764,12516}, {3741,21631}, {4362,35635}, {5208,12537}, {5920,10480}, {8001,35621}, {9804,10453}, {9953,35613}, {10439,12552}, {10472,12631}, {10473,12854}, {10476,12842}, {10477,12692}, {10478,12612}, {10479,12620}, {12439,35612}, {12533,35614}, {12846,35617}, {12851,35618}, {12852,35619}, {12853,35620}, {12865,35622}, {12869,35623}, {12870,35624}, {12871,35625}, {12873,35627}, {17639,21334}, {17655,35626}, {18255,35628}

X(35640) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO INVERSE-IN-EXCIRCLES

Barycentrics a*((2*b^2+b*c+2*c^2)*a^7+4*(b+c)*b*c*a^6-(6*b^4+6*c^4-(11*b^2+10*b*c+11*c^2)*b*c)*a^5+6*(b+c)*(b^2+c^2)*b*c*a^4+(6*b^4+6*c^4+(3*b^2-2*b*c+3*c^2)*b*c)*(b-c)^2*a^3-8*(b+c)*(b^4+c^4)*b*c*a^2-(b^2-c^2)^2*(2*b^4+2*c^4+(3*b^2+14*b*c+3*c^2)*b*c)*a-2*(b^4-c^4)*(b^2-c^2)*(b+c)*b*c) : :
X(35640) = 2*X(9943)-3*X(35672) = 3*X(10439)-X(35662)

The reciprocal orthologic center of these triangles is X(1)

X(35640) lies on these lines: {1,1407}, {962,10439}, {1764,35657}, {3741,35680}, {5208,35660}, {10453,35661}, {10473,35671}, {10476,35667}, {10477,35666}, {10478,35663}, {10479,35664}, {10480,35669}, {21334,35679}, {35612,35655}, {35613,35656}, {35614,35659}, {35617,35668}, {35620,35670}, {35621,35673}, {35622,35674}, {35623,35675}, {35626,35678}, {35627,35681}, {35628,35682}

X(35640) = X(1217)-of-inverse-in-Conway triangle

X(35641) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO INVERSE-IN-CONWAY

Barycentrics a*(a^3-2*(b+c)*a^2-2*S*a-(b^2-4*b*c+c^2)*a+2*(b^2-c^2)*(b-c)) : :
X(35641) = 5*X(371)-4*X(31439) = 5*X(7969)-2*X(31439) = 4*X(7969)-X(35610) = 8*X(31439)-5*X(35610) = 3*X(35822)-X(35842)

The reciprocal orthologic center of these triangles is X(10441)

X(35641) lies on these lines: {1,372}, {3,35611}, {5,35789}, {6,1482}, {8,485}, {10,10576}, {30,35854}, {40,6200}, {55,35778}, {65,35768}, {145,1587}, {355,6564}, {371,517}, {486,5603}, {515,35820}, {518,35840}, {519,35822}, {551,13975}, {590,5690}, {615,5901}, {730,35866}, {944,6560}, {946,6565}, {952,3070}, {962,6561}, {1124,2098}, {1151,12702}, {1152,10246}, {1327,34627}, {1335,2099}, {1377,5289}, {1385,6396}, {1686,30116}, {1699,35787}, {1702,11531}, {1829,35764}, {1837,35802}, {2066,5697}, {2067,5903}, {2800,35856}, {2802,35882}, {3057,35808}, {3068,12245}, {3069,10595}, {3071,22791}, {3093,11396}, {3103,14839}, {3301,11009}, {3311,8148}, {3312,10247}, {3616,5420}, {3622,13935}, {3640,35794}, {3641,35792}, {3656,35823}, {3878,31453}, {5252,35800}, {5418,5657}, {5844,7583}, {5886,10577}, {6001,35844}, {6265,35883}, {6419,7982}, {6420,7968}, {6453,7991}, {6454,15178}, {6460,7967}, {6482,9617}, {6484,9582}, {6487,30392}, {8960,13911}, {8983,11362}, {9541,20070}, {9624,13947}, {9798,35776}, {9812,22615}, {9941,35782}, {10283,13966}, {10679,35773}, {10680,35785}, {10738,35853}, {11224,19004}, {11278,35771}, {12047,35801}, {12194,35766}, {12438,35790}, {12440,35804}, {12441,35806}, {12645,13665}, {12647,31472}, {12699,35821}, {12737,35857}, {13464,13936}, {13846,34718}, {13883,28234}, {13971,35814}, {16189,19003}, {16200,18992}, {16232,25415}, {18480,35786}, {18525,23251}, {19054,34631}, {30384,35803}

X(35641) = reflection of X(i) in X(j) for these (i,j): (371, 7969), (35610, 371)
X(35641) = X(1)-of-1st Kenmotu-free-vertices triangle
X(35641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 372, 35762), (1, 2362, 35769), (1, 35774, 372), (6, 1482, 35642), (8, 485, 35788), (372, 35810, 1), (372, 35816, 35809), (372, 35818, 35769), (5603, 19065, 486), (5657, 13902, 5418), (5886, 13973, 10577), (6420, 35811, 7968), (6564, 35843, 355), (7968, 24680, 35811), (7982, 18991, 35775), (18991, 35775, 6419), (35611, 35763, 3), (35772, 35784, 3), (35774, 35810, 35762), (35778, 35780, 55), (35796, 35798, 6564)

X(35642) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO INVERSE-IN-CONWAY

Barycentrics a*(a^3-2*(b+c)*a^2+2*S*a-(b^2-4*b*c+c^2)*a+2*(b^2-c^2)*(b-c)) : :
X(35642) = 4*X(7968)-X(35611) = 3*X(35823)-X(35843)

The reciprocal orthologic center of these triangles is X(10441)

X(35642) lies on these lines: {1,371}, {3,35610}, {5,35788}, {6,1482}, {8,486}, {10,10577}, {30,35855}, {40,6396}, {55,35779}, {65,35769}, {145,1588}, {355,6565}, {372,517}, {485,5603}, {515,35821}, {518,35841}, {519,35823}, {551,13912}, {590,5901}, {615,5690}, {730,35867}, {944,6561}, {946,6564}, {952,3071}, {962,6560}, {1124,2099}, {1151,10246}, {1152,12702}, {1328,34627}, {1335,2098}, {1378,5289}, {1385,6200}, {1685,30116}, {1699,35786}, {1703,11531}, {1829,35765}, {1837,35803}, {2362,25415}, {2800,35857}, {2802,35883}, {3057,35809}, {3068,10595}, {3069,12245}, {3070,22791}, {3092,11396}, {3102,14839}, {3299,11009}, {3311,10247}, {3312,8148}, {3616,5418}, {3622,9540}, {3640,35793}, {3641,35795}, {3656,35822}, {5252,35801}, {5414,5697}, {5420,5657}, {5844,7584}, {5886,10576}, {5903,6502}, {6001,35845}, {6265,35882}, {6419,7969}, {6420,7982}, {6453,15178}, {6454,7991}, {6459,7967}, {6480,9615}, {6486,30392}, {8960,13464}, {8981,10283}, {8983,35815}, {9582,30389}, {9624,13893}, {9646,15950}, {9798,35777}, {9812,22644}, {9941,35783}, {10679,35772}, {10680,35784}, {10738,35852}, {11224,19003}, {11278,35770}, {11362,13971}, {12047,35800}, {12194,35767}, {12438,35791}, {12440,35807}, {12441,35805}, {12645,13785}, {12699,35820}, {12737,35856}, {13847,34718}, {13936,28234}, {16189,19004}, {16200,18991}, {18480,35787}, {18525,23261}, {19053,34631}, {30384,35802}

X(35642) = reflection of X(i) in X(j) for these (i,j): (372, 7968), (35611, 372)
X(35642) = X(1)-of-2nd Kenmotu-free-vertices triangle
X(35642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 371, 35763), (1, 16232, 35768), (1, 35775, 371), (6, 1482, 35641), (8, 486, 35789), (371, 35811, 1), (371, 35817, 35808), (371, 35819, 35768), (5603, 19066, 485), (5657, 13959, 5420), (5886, 13911, 10576), (6419, 35810, 7969), (6565, 35842, 355), (7969, 24680, 35810), (7982, 18992, 35774), (18992, 35774, 6420), (35610, 35762, 3), (35773, 35785, 3), (35775, 35811, 35763), (35779, 35781, 55), (35797, 35799, 6565)

X(35643) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO MANDART-EXCIRCLES

Barycentrics a*((b^2+b*c+c^2)*a^10+(b^2+c^2)*(b+c)*a^9-(2*b^4+2*c^4+(5*b^2+8*b*c+5*c^2)*b*c)*a^8-2*(b+c)*(b^4+c^4+(2*b^2-3*b*c+2*c^2)*b*c)*a^7+2*(b^4+c^4+4*(2*b^2+3*b*c+2*c^2)*b*c)*b*c*a^6+2*(b^2-c^2)*(b-c)*b^2*c^2*a^5+2*(b^8+c^8-(b^6+c^6+(4*b^4+4*c^4+(9*b^2-2*b*c+9*c^2)*b*c)*b*c)*b*c)*a^4+2*(b+c)*(b^8+c^8+(2*b^6+2*c^6-(7*b^4+7*c^4-4*(b^2-b*c+c^2)*b*c)*b*c)*b*c)*a^3-(b^8+c^8-(7*b^6+7*c^6-(14*b^4+14*c^4-3*(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*(b+c)^2*a^2-(b^2-c^2)^2*(b+c)*(b^6+c^6-(3*b^2-8*b*c+3*c^2)*b^2*c^2)*a-(b^4-c^4)*(b^2-c^2)^3*b*c) : :
X(35643) = 3*X(10439)-X(12553)

The reciprocal orthologic center of these triangles is X(3555)

X(35643) lies on these lines: {1,12522}, {1764,12517}, {3741,21632}, {5208,12538}, {10439,12553}, {10453,12542}, {10473,12912}, {10476,12843}, {10477,12693}, {10478,12613}, {10479,12621}, {10480,12876}, {12442,35612}, {12449,35613}, {12534,35614}, {12847,35617}, {12878,35618}, {12883,35619}, {12907,35620}, {13069,35621}, {13070,35622}, {13071,35623}, {13072,35624}, {13073,35625}, {13074,35627}, {17640,21334}, {17656,35626}, {18257,35628}

X(35644) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO MIDARC

Barycentrics a*(((b+c)*b*c+(b^2+b*c+c^2)*a)*sin(A/2)-(a+c)*b^2*sin(B/2)-(a+b)*c^2*sin(C/2)) : :
X(35644) = 3*X(10439)-X(12554)

The reciprocal orthologic center of these triangles is X(1)

X(35644) lies on these lines: {1,164}, {167,35621}, {177,10473}, {1764,12518}, {3741,21633}, {5208,12539}, {7670,35617}, {8422,10480}, {9807,10453}, {10439,12554}, {10441,31783}, {10476,12844}, {10477,12694}, {10478,12614}, {10479,12622}, {11691,35614}, {12443,35612}, {12450,35613}, {12879,35618}, {12884,35619}, {12908,35620}, {13090,35622}, {13091,35623}, {13092,35627}, {17641,21334}, {17657,35626}, {18258,35628}

X(35644) = X(10)-of-3rd Conway triangle
X(35644) = {X(35624), X(35625)}-harmonic conjugate of X(1)

X(35645) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO MIXTILINEAR

Barycentrics a*((b^2+b*c+c^2)*a^3-(b+c)*b*c*a^2-(b^2+c^2)*(b^2-b*c+c^2)*a-(b^2-c^2)*(b-c)*b*c) : :
X(35645) = 3*X(10439)-X(12555)

The reciprocal orthologic center of these triangles is X(1)

X(35645) lies on these lines: {1,3}, {149,2979}, {329,2810}, {388,15488}, {496,5752}, {497,511}, {516,3784}, {527,35892}, {970,3086}, {1191,18178}, {1401,24248}, {1469,24210}, {2478,16980}, {2550,3819}, {3421,10449}, {3434,3917}, {3452,3741}, {3474,29309}, {3781,4847}, {3792,33141}, {3820,10479}, {4551,19540}, {5057,23155}, {5084,23841}, {5208,9965}, {5399,19543}, {5562,12116}, {5943,26105}, {6735,30006}, {7015,29652}, {7074,16434}, {7288,15489}, {7956,10478}, {7998,33110}, {8101,35624}, {8102,35625}, {8679,24703}, {9580,15310}, {9954,35613}, {10241,10862}, {10527,22076}, {11019,29311}, {11415,23154}, {11573,12699}, {11679,17658}, {12513,22299}, {12848,35617}, {12880,35618}, {12885,35619}, {13096,35622}, {13097,35623}, {13098,35627}, {22097,31394}, {22300,25524}, {26015,26893}

X(35645) = barycentric product X(1)*X(29965)
X(35645) = trilinear product X(6)*X(29965)
X(35645) = intersection, other than A,B,C, of conics {{A, B, C, X(57), X(29965)}} and {{A, B, C, X(314), X(23853)}}
X(35645) = Conway circle-inverse of X(5537)
X(35645) = X(25)-of-inverse-in-Conway triangle
X(35645) = X(1368)-of-3rd Conway triangle
X(35645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1764, 23853), (1, 10473, 35612), (3750, 11493, 11021)

X(35646) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 1st SCHIFFLER

Barycentrics a*(-a+b+c)*((b^2+b*c+c^2)*a^9+3*(b+c)*b*c*a^8-4*(b^4+c^4)*a^7-2*(b+c)*(4*b^2-3*b*c+4*c^2)*b*c*a^6+(6*b^6+6*c^6-(4*b^2+5*b*c+4*c^2)*(b^2-b*c+c^2)*b*c)*a^5+(b+c)*(8*b^4+8*c^4-(8*b^2-5*b*c+8*c^2)*b*c)*b*c*a^4-(4*b^8+4*c^8-(4*b^6+4*c^6+(5*b^4+5*c^4-(b^2+10*b*c+c^2)*b*c)*b*c)*b*c)*a^3-(b+c)*(4*b^6+4*c^6-(6*b^4+6*c^4+(b^2-8*b*c+c^2)*b*c)*b*c)*b*c*a^2+(b^2-c^2)^3*(b-c)*(b^3+c^3)*a+(b^4-c^4)*(b^2-c^2)^2*(b-c)*b*c) : :
X(35646) = 3*X(10439)-X(12557)

The reciprocal orthologic center of these triangles is X(21)

X(35646) lies on these lines: {1,6597}, {314,10266}, {1764,12519}, {3741,21634}, {5208,12540}, {10439,12557}, {10453,12543}, {10472,12639}, {10473,12913}, {10476,12845}, {10477,12695}, {10478,12615}, {10479,12623}, {10480,12877}, {12444,35612}, {12451,35613}, {12535,35614}, {12850,35617}, {12882,35618}, {12887,35619}, {12909,35620}, {13101,35621}, {13120,35622}, {13123,35623}, {13124,35624}, {13125,35625}, {13127,35627}, {17643,21334}, {17659,35626}, {18259,35628}

X(35647) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 2nd INNER-SODDY

Barycentrics a*((b^2+b*c+c^2)*a^4+(b+c)*(b^2+c^2)*a^3-(b^4+c^4)*a^2-(b^2-c^2)^2*b*c-(b+c)*(b^4+c^4)*a+(2*(b^2+b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a-2*(b^2+c^2)*b*c)*S) : :
X(35647) = 3*X(10439)-X(31554)

The reciprocal orthologic center of these triangles is X(1)

X(35647) lies on these lines: {1,372}, {482,10473}, {516,10441}, {1764,31545}, {3741,31591}, {5208,31550}, {10439,31554}, {10453,31552}, {10476,31564}, {10477,31562}, {10478,31556}, {10479,31558}, {10480,31568}, {21334,31589}, {31541,35612}, {31543,35613}, {31548,35614}, {31566,35617}, {31570,35620}, {31574,35621}, {31577,35623}, {31579,35624}, {31581,35625}, {31587,35626}, {31593,35627}, {31595,35628}

X(35647) = X(371)-of-inverse-in-Conway triangle
X(35647) = X(639)-of-3rd Conway triangle
X(35647) = {X(10441), X(35892)}-harmonic conjugate of X(35648)

X(35648) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO 2nd OUTER-SODDY

Barycentrics a*((-2*(b^2+b*c+c^2)*a^2+2*(b^2-c^2)*(b-c)*a+2*(b^2+c^2)*b*c)*S+(b^2+b*c+c^2)*a^4+(b+c)*(b^2+c^2)*a^3-(b^4+c^4)*a^2-(b+c)*(b^4+c^4)*a-(b^2-c^2)^2*b*c) : :
X(35648) = 3*X(10439)-X(31553)

The reciprocal orthologic center of these triangles is X(1)

X(35648) lies on these lines: {1,371}, {481,10473}, {516,10441}, {1764,31544}, {3741,31590}, {5208,31549}, {10439,31553}, {10453,31551}, {10476,31563}, {10477,31561}, {10478,31555}, {10479,31557}, {10480,31567}, {21334,31588}, {31540,35612}, {31542,35613}, {31547,35614}, {31565,35617}, {31569,35620}, {31573,35621}, {31575,35622}, {31576,35623}, {31578,35624}, {31580,35625}, {31586,35626}, {31592,35627}, {31594,35628}

X(35648) = X(372)-of-inverse-in-Conway triangle
X(35648) = X(640)-of-3rd Conway triangle
X(35648) = {X(10441), X(35892)}-harmonic conjugate of X(35647)

X(35649) = PARALLELOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-CONWAY TO FUHRMANN

Barycentrics a*((b^2+b*c+c^2)*a^6-(b+c)*(b^2+c^2)*a^5-(2*b^4+2*c^4-(2*b^2-b*c+2*c^2)*b*c)*a^4+(b+c)*(2*b^4+2*c^4-3*(b^2-b*c+c^2)*b*c)*a^3+(b-c)^2*(b^2+c^2)^2*a^2-(b^2-c^2)*(b-c)*(b^4+c^4-(b^2-3*b*c+c^2)*b*c)*a-(b^2-c^2)*(b-c)*(b^3+c^3)*b*c) : :
X(35649) = 3*X(10439)-X(13244)

The reciprocal parallelogic center of these triangles is X(4)

X(35649) lies on these lines: {1,104}, {2,34458}, {11,10473}, {100,1764}, {119,10479}, {149,10446}, {153,10449}, {214,10882}, {952,10441}, {1156,35617}, {1317,10480}, {1320,11521}, {1387,35620}, {2771,35631}, {2801,10439}, {2802,12435}, {2829,10454}, {3035,35628}, {3741,21635}, {5208,13243}, {5531,35621}, {5851,35892}, {6224,10465}, {6265,18417}, {6326,10476}, {6702,10887}, {8103,35624}, {8104,35625}, {9809,10453}, {10475,12740}, {10477,13257}, {11021,18240}, {13226,35612}, {13227,35613}, {13260,35618}, {13261,35619}, {13262,35622}, {13265,35623}, {13267,35627}, {17660,21334}, {17661,35626}

X(35649) = midpoint of X(i) and X(j) for these {i,j}: {1, 12551}, {12435, 12550}
X(35649) = reflection of X(i) in X(j) for these (i,j): (35636, 10441), (35638, 35631)
X(35649) = anticomplement of X(34458)
X(35649) = Conway circle-inverse of X(109)
X(35649) = X(110)-of-inverse-in-Conway triangle
X(35649) = X(125)-of-3rd Conway triangle
X(35649) = X(12551)-of-anti-Aquila triangle
X(35649) = {X(10473), X(17617)}-harmonic conjugate of X(10478)

X(35650) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-EXCIRCLES AND MIDHEIGHT

Barycentrics a*(a^3+(b+c)*a^2+(b^2+b*c+c^2)*a+b^3+c^3)*(b+c)*(a-b+c)*(a+b-c) : :
X(35650) = 3*X(354)-X(10544)

X(35650) lies on these lines: {1,3430}, {4,15314}, {7,1330}, {10,12}, {28,34}, {56,2915}, {169,1046}, {354,10544}, {511,942}, {517,34937}, {540,553}, {573,986}, {938,17170}, {1210,1905}, {1284,3743}, {1400,4456}, {1403,19763}, {1426,3668}, {1431,5883}, {1460,4347}, {1469,3874}, {1730,24443}, {2099,30145}, {2171,28594}, {2392,18838}, {2825,5185}, {3008,6678}, {3212,10449}, {3361,5429}, {3812,5750}, {3911,6693}, {7286,32636}, {10473,28109}, {12564,21746}, {13731,25065}, {15171,29032}, {17450,21471}, {18421,28038}, {20077,21454}, {24391,24476}

X(35650) = barycentric product X(i)*X(j) for these {i, j}: {57, 5051}, {65, 19786}, {226, 5262}
X(35650) = trilinear product X(i)*X(j) for these {i, j}: {56, 5051}, {65, 5262}, {226, 16470}
X(35650) = intersection, other than A,B,C, of conics {{A, B, C, X(10), X(28)}} and {{A, B, C, X(12), X(34)}}
X(35650) = crossdifference of every pair of points on line {X(7252), X(8611)}
X(35650) = crosssum of X(55) and X(21033)
X(35650) = X(6750)-of-intouch triangle
X(35650) = {X(65), X(181)}-harmonic conjugate of X(12432)

X(35651) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-EXCIRCLES AND 7th MIXTILINEAR

Barycentrics a*(a^10+2*(b+c)*a^9-(27*b^2-10*b*c+27*c^2)*a^8+8*(b+c)*(7*b^2-22*b*c+7*c^2)*a^7-2*(7*b^4+7*c^4+2*(54*b^2-163*b*c+54*c^2)*b*c)*a^6-12*(b^2-c^2)*(b-c)*(7*b^2-62*b*c+7*c^2)*a^5+2*(49*b^4+49*c^4-2*(4*b^2+217*b*c+4*c^2)*b*c)*(b-c)^2*a^4-8*(b^2-c^2)*(b-c)*(b^4+c^4+2*(42*b^2-53*b*c+42*c^2)*b*c)*a^3-(51*b^6+51*c^6-(290*b^4+290*c^4+3*(129*b^2-76*b*c+129*c^2)*b*c)*b*c)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)^5*(17*b^2+62*b*c+17*c^2)*a+(b^2-c^2)^2*(b-c)^4*(-7*b^2-2*b*c-7*c^2)) : :

X(35651) lies on these lines: {65,3062}, {3664,9533}

X(35652) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-EXCIRCLES TO ANTI-ARTZT

Barycentrics (b+c)*a^2+(b^2+4*b*c+c^2)*a-2*(b+c)*b*c : :
X(35652) = X(942)+2*X(3159) = X(2901)+2*X(5044) = 2*X(3971)+X(4891) = 4*X(4075)-X(34790) = 2*X(5045)+X(24068)

The reciprocal orthologic center of these triangles is X(35653)

X(35652) lies on these lines: {1,3967}, {2,37}, {6,30568}, {42,4009}, {45,11679}, {57,17262}, {141,4656}, {210,28581}, {226,17243}, {329,4851}, {333,16814}, {354,28582}, {375,35104}, {518,3971}, {519,960}, {545,553}, {612,4387}, {726,3742}, {740,3740}, {756,3706}, {940,17351}, {942,3159}, {975,19276}, {1089,6051}, {1100,27064}, {1104,33309}, {1155,32936}, {1211,17229}, {1215,15569}, {1279,32926}, {1386,4011}, {2176,16834}, {2321,5743}, {2886,4078}, {2901,5044}, {2999,17318}, {3185,4421}, {3241,8834}, {3305,17348}, {3452,3950}, {3679,3714}, {3683,17763}, {3687,3943}, {3715,17156}, {3720,3994}, {3725,14752}, {3731,5737}, {3745,32930}, {3748,32927}, {3782,3834}, {3823,3914}, {3838,29653}, {3844,4425}, {3846,6541}, {3848,24165}, {3912,4415}, {3923,4682}, {3932,24210}, {3948,25102}, {3961,4702}, {3991,34852}, {4003,30957}, {4052,17758}, {4066,27784}, {4135,24325}, {4361,7308}, {4362,15254}, {4363,17022}, {4383,4852}, {4417,17242}, {4439,29655}, {4500,4762}, {4519,31330}, {4640,29649}, {4643,34255}, {4654,17313}, {4670,5287}, {4679,33088}, {4715,17781}, {4719,25079}, {4849,27538}, {4883,17165}, {4884,11019}, {4918,24982}, {4970,24003}, {5045,24068}, {5087,29671}, {5268,5695}, {5302,17733}, {5739,17372}, {5905,17376}, {6687,26723}, {6703,17355}, {9639,22836}, {10157,29016}, {14555,17299}, {14759,33908}, {14829,17261}, {15481,32853}, {15668,25430}, {16676,18229}, {17231,27184}, {17265,23681}, {17267,25527}, {17276,18141}, {17314,18228}, {17354,29841}, {17374,33066}, {17605,29643}, {17777,33073}, {20363,21902}, {20691,25125}, {21071,29594}, {28609,29573}

X(35652) = midpoint of X(i) and X(j) for these {i,j}: {2, 3175}, {210, 32915}, {354, 32925}
X(35652) = reflection of X(24165) in X(3848)
X(35652) = lies on the circumconic with center X(35113))
X(35652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (192, 18743, 3752), (3210, 30829, 16602), (3666, 3995, 4681), (3995, 4358, 3666), (4664, 20942, 2), (4718, 16602, 3210), (20691, 30830, 25125), (27064, 34064, 1100)

X(35653) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-ARTZT TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*(2*b^2-b*c+2*c^2)*a^5+4*(b+c)*(3*b^2-5*b*c+3*c^2)*a^4+2*(2*b^2+11*b*c+2*c^2)*b*c*a^3-2*(b+c)*(6*b^4+6*c^4-(11*b^2-b*c+11*c^2)*b*c)*a^2-(6*b^6+6*c^6-(7*b^4+7*c^4+2*(2*b^2-17*b*c+2*c^2)*b*c)*b*c)*a+2*(b+c)*(3*b^4+3*c^4-(b^2+c^2)*b*c)*b*c) : :

The reciprocal orthologic center of these triangles is X(35652)

X(35653) lies on these lines: {2,35654}, {13637,35676}, {13757,35677}

X(35654) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ARTZT TO INVERSE-IN-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(35652)

X(35654) lies on these lines: {1,3430}, {2,35653}, {9746,10156}, {13638,35676}, {13758,35677}

X(35655) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ASCELLA TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^8+2*(b^2+3*b*c+c^2)*a^7-2*(b^2-c^2)*(b-c)*a^6-2*(3*b^4+3*c^4+(5*b^2-24*b*c+5*c^2)*b*c)*a^5-4*(b+c)^3*b*c*a^4+2*(3*b^6+3*c^6+(b^4+c^4-(23*b^2+10*b*c+23*c^2)*b*c)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(b^4+6*b^2*c^2+c^4)*a^2-2*(b^2-c^2)^2*(b^4+c^4-(b^2-4*b*c+c^2)*b*c)*a-(b^2-c^2)^3*(b-c)^3) : :
X(35655) = 3*X(2)+X(35660)

The reciprocal orthologic center of these triangles is X(1)

X(35655) lies on these lines: {1,35889}, {2,35660}, {3,7290}, {57,8915}, {142,35680}, {942,4301}, {1125,35886}, {3601,35669}, {5744,35659}, {5745,35682}, {8726,35667}, {8727,35663}, {8728,35664}, {8729,35681}, {8731,35675}, {8732,35668}, {9776,35661}, {10855,35656}, {10856,35662}, {10857,35673}, {10858,35674}, {11018,35672}, {11518,35665}, {17603,35679}, {17612,35678}, {35612,35640}

X(35655) = midpoint of X(i) and X(j) for these {i,j}: {1, 35889}, {8915, 35671}, {35660, 35666}
X(35655) = reflection of X(35886) in X(1125)
X(35655) = complement of X(35666)
X(35655) = X(1217)-of-Ascella triangle
X(35655) = X(35889)-of-anti-Aquila triangle
X(35655) = {X(2), X(35660)}-harmonic conjugate of X(35666)

X(35656) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ATIK TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^8-2*(b-c)^2*a^7-2*(b+c)*(b^2-4*b*c+c^2)*a^6+2*(b^2+c^2)*(3*b^2-8*b*c+3*c^2)*a^5-4*(b+c)*(5*b^2-12*b*c+5*c^2)*b*c*a^4-2*(3*b^6+3*c^6-(10*b^4+10*c^4+3*(b^2-12*b*c+c^2)*b*c)*b*c)*a^3+2*(b+c)*(b^6+c^6+(8*b^4+8*c^4-(25*b^2-48*b*c+25*c^2)*b*c)*b*c)*a^2+2*(b^2-c^2)^2*(b^4+c^4-2*(2*b^2+3*b*c+2*c^2)*b*c)*a+(b^2-c^2)^2*(b+c)*(-b^4-c^4-2*(2*b^2-b*c+2*c^2)*b*c)) : :

The reciprocal orthologic center of these triangles is X(1)

X(35656) lies on these lines: {8,12688}, {3062,35673}, {5927,35666}, {8580,8915}, {8581,35671}, {8582,35664}, {8583,35658}, {10855,35655}, {10860,35657}, {10861,35660}, {10862,35662}, {10863,35663}, {10864,35667}, {10865,35668}, {10866,35669}, {10867,35674}, {10868,35675}, {11019,35672}, {11035,35670}, {11519,35665}, {11678,35659}, {11860,35681}, {17604,35679}, {18227,35682}, {35613,35640}

X(35656) = midpoint of X(35666) and X(35678)
X(35656) = reflection of X(35672) in X(35680)
X(35656) = X(1217)-of-Atik triangle

X(35657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st CIRCUMPERP TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*a^6+2*(b+c)*a^5+(b^2+6*b*c+c^2)*a^4+4*(b^3+c^3)*a^3-(3*b^4+3*c^4+2*(4*b^2-3*b*c+4*c^2)*b*c)*a^2-2*(b+c)*(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(35657) = 3*X(165)-X(8915) = 3*X(165)+X(35673) = 3*X(9778)+X(35661)

The reciprocal orthologic center of these triangles is X(1)

X(35657) lies on these lines: {1,3430}, {2,35663}, {3,7290}, {4,35664}, {40,936}, {55,35671}, {56,35669}, {57,35672}, {100,35659}, {165,978}, {516,35680}, {1036,2999}, {1125,9746}, {1155,35679}, {1350,3333}, {1764,35640}, {3295,35670}, {3428,9121}, {5293,7991}, {5584,35889}, {6210,31424}, {7411,35660}, {7580,35666}, {7589,35681}, {7676,35668}, {8224,35674}, {8726,30269}, {9778,35661}, {10434,35662}, {10860,35656}, {17613,35678}

X(35657) = midpoint of X(i) and X(j) for these {i,j}: {40, 35667}, {7991, 35665}, {8915, 35673}
X(35657) = reflection of X(i) in X(j) for these (i,j): (4, 35664), (35658, 3)
X(35657) = anticomplement of X(35663)
X(35657) = X(1217)-of-1st circumperp triangle
X(35657) = X(35658)-of-ABC-X3 reflections triangle
X(35657) = X(35664)-of-anti-Euler triangle
X(35657) = X(35669)-of-2nd circumperp tangential triangle
X(35657) = X(35671)-of-anti-Mandart-incircle triangle
X(35657) = {X(165), X(35673)}-harmonic conjugate of X(8915)

X(35658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*a^6-(5*b^2-14*b*c+5*c^2)*a^4+(b^4+c^4-2*(6*b^2+5*b*c+6*c^2)*b*c)*a^2+(b^2-c^2)^2*(b-c)^2) : :
X(35658) = 3*X(1)-X(35665) = 3*X(3576)-X(35667) = 5*X(3616)-X(35661) = 5*X(7987)-X(35673) = 3*X(8915)+X(35665)

The reciprocal orthologic center of these triangles is X(1)

X(35658) lies on these lines: {1,1407}, {2,35664}, {3,7290}, {4,35663}, {20,5269}, {21,35660}, {40,58}, {55,1394}, {56,35671}, {84,7174}, {109,1496}, {165,1453}, {223,939}, {405,35666}, {940,12651}, {958,35682}, {975,11372}, {991,1066}, {999,35670}, {1001,35886}, {1125,35680}, {2646,35679}, {2975,35659}, {3073,15601}, {3576,15839}, {3601,4300}, {3616,35661}, {5266,5732}, {5438,35338}, {7587,35681}, {7677,35668}, {7987,35673}, {8225,35674}, {8583,35656}, {10388,34046}, {10882,35662}, {17614,35678}

X(35658) = midpoint of X(i) and X(j) for these {i,j}: {1, 8915}, {35669, 35889}
X(35658) = reflection of X(i) in X(j) for these (i,j): (4, 35663), (35657, 3), (35680, 1125)
X(35658) = anticomplement of X(35664)
X(35658) = X(21)-Beth conjugate of X(7091)
X(35658) = X(1217)-of-2nd circumperp triangle
X(35658) = X(8915)-of-anti-Aquila triangle
X(35658) = X(35657)-of-ABC-X3 reflections triangle
X(35658) = X(35663)-of-anti-Euler triangle
X(35658) = X(35669)-of-anti-Mandart-incircle triangle
X(35658) = X(35671)-of-2nd circumperp tangential triangle

X(35659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-CONWAY TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5+(3*b^2-b*c+3*c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3-2*(b^4+c^4-(b^2+8*b*c+c^2)*b*c)*a^2-(b+c)*(3*b^4+3*c^4-2*(2*b^2-9*b*c+2*c^2)*b*c)*a-(b^4+c^4-(b^2-4*b*c+c^2)*b*c)*(b+c)^2) : :
X(35659) = 3*X(2)-4*X(35682) = 5*X(3616)-4*X(35670) = 3*X(3873)-4*X(35672)

The reciprocal orthologic center of these triangles is X(1)

X(35659) lies on these lines: {2,35671}, {9,35668}, {63,8915}, {78,35667}, {100,35657}, {145,35669}, {200,35673}, {329,35661}, {518,35679}, {908,35680}, {2975,35658}, {3616,35670}, {3869,7957}, {3873,35672}, {5744,35655}, {8126,35681}, {11678,35656}, {11679,35662}, {11680,35663}, {11681,35664}, {11682,35665}, {11687,35674}, {11688,35675}, {17615,35678}, {35614,35640}

X(35659) = reflection of X(i) in X(j) for these (i,j): (145, 35669), (35660, 8915), (35661, 35666), (35668, 9), (35671, 35682)
X(35659) = anticomplement of X(35671)
X(35659) = X(1217)-of-inner-Conway triangle
X(35659) = {X(35671), X(35682)}-harmonic conjugate of X(2)

X(35660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: CONWAY TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^8+(2*b^2+3*b*c+2*c^2)*a^7-(b+c)*(2*b^2-3*b*c+2*c^2)*a^6-(2*b-c)*(b-2*c)*(b+3*c)*(3*b+c)*a^5-(b+c)*(5*b^2+6*b*c+5*c^2)*b*c*a^4+(6*b^6+6*c^6+(b^4+c^4-2*(13*b^2+5*b*c+13*c^2)*b*c)*b*c)*a^3+(b+c)*(2*b^6+2*c^6+(b^4+c^4+10*(b^2-b*c+c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(2*b^4+2*c^4-(b^2-18*b*c+c^2)*b*c)*a+(b^2-c^2)^2*(b+c)*(-b^4-c^4+(b^2-4*b*c+c^2)*b*c)) : :
X(35660) = 3*X(2)-4*X(35655) = 5*X(3616)-4*X(35886)

The reciprocal orthologic center of these triangles is X(1)

X(35660) lies on these lines: {2,35655}, {7,35661}, {8,35889}, {21,35658}, {63,8915}, {3616,35886}, {3868,9797}, {4197,35664}, {4313,35669}, {5208,35640}, {5249,35680}, {5273,35682}, {5732,35673}, {7411,35657}, {10391,35679}, {10444,35662}, {10861,35656}, {10883,35663}, {10884,35667}, {10885,35674}, {11020,35672}, {11036,35670}, {11520,35665}, {11890,35681}, {17616,35678}

X(35660) = reflection of X(i) in X(j) for these (i,j): (8, 35889), (35659, 8915), (35661, 35671), (35666, 35655)
X(35660) = anticomplement of X(35666)
X(35660) = X(1217)-of-Conway triangle
X(35660) = {X(35655), X(35666)}-harmonic conjugate of X(2)

X(35661) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd CONWAY TO INVERSE-IN-EXCIRCLES

Barycentrics 3*a^7+3*(b+c)*a^6-9*(b-c)^2*a^5-(b+c)*(9*b^2-34*b*c+9*c^2)*a^4+(9*b^4+9*c^4-2*(2*b^2+29*b*c+2*c^2)*b*c)*a^3+(b+c)*(9*b^4+9*c^4-2*(18*b^2-35*b*c+18*c^2)*b*c)*a^2-(3*b^2+14*b*c+3*c^2)*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c)*(-3*b^2+2*b*c-3*c^2) : :
X(35661) = 3*X(2)-4*X(35680) = 5*X(3616)-4*X(35658) = 3*X(9778)-4*X(35657) = 9*X(9779)-8*X(35663) = 7*X(9780)-8*X(35664)

The reciprocal orthologic center of these triangles is X(1)

X(35661) lies on these lines: {2,8915}, {7,35660}, {8,12688}, {20,35667}, {145,35665}, {329,35659}, {497,35679}, {516,35673}, {962,21296}, {3434,35678}, {3616,35658}, {4295,32098}, {9776,35655}, {9778,35657}, {9779,35663}, {9780,35664}, {9785,35669}, {9789,35674}, {9791,35675}, {10446,35662}, {10453,35640}, {10580,35672}, {11037,35670}, {11891,35681}, {18228,35682}

X(35661) = reflection of X(i) in X(j) for these (i,j): (20, 35667), (145, 35665), (8915, 35680), (35659, 35666), (35660, 35671)
X(35661) = anticomplement of X(8915)
X(35661) = X(1217)-of-2nd Conway triangle
X(35661) = {X(8915), X(35680)}-harmonic conjugate of X(2)

X(35662) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd CONWAY TO INVERSE-IN-EXCIRCLES

Barycentrics a*((2*b^2+b*c+2*c^2)*a^7+4*(b+c)*(2*b^2-b*c+2*c^2)*a^6+(10*b^2-13*b*c+10*c^2)*(b+c)^2*a^5+2*(b+c)*(3*b^2+14*b*c+3*c^2)*b*c*a^4-(10*b^6+10*c^6+(b^4+c^4-2*(11*b^2-3*b*c+11*c^2)*b*c)*b*c)*a^3-8*(b+c)*(b^6+c^6+(3*b^2-4*b*c+3*c^2)*b^2*c^2)*a^2-(2*b^6+2*c^6+3*(b^4+c^4-2*(b^2-3*b*c+c^2)*b*c)*b*c)*(b+c)^2*a-2*(b^2-c^2)^3*(b-c)*b*c) : :
X(35662) = 3*X(10439)-2*X(35640)

The reciprocal orthologic center of these triangles is X(1)

X(35662) lies on these lines: {1,1122}, {962,10439}, {1764,8915}, {10434,35657}, {10444,35660}, {10446,35661}, {10473,35679}, {10478,35680}, {10856,35655}, {10862,35656}, {10882,35658}, {10886,35663}, {10887,35664}, {10888,35666}, {10889,35668}, {10891,35674}, {10892,35675}, {11021,35672}, {11521,35665}, {11679,35659}, {11896,35681}, {12435,35629}, {17617,35678}, {18229,35682}

X(35662) = X(1217)-of-3rd Conway triangle

X(35663) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER TO INVERSE-IN-EXCIRCLES

Barycentrics (b+c)*a^6+2*(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4-4*(b^4+c^4-(3*b^2+4*b*c+3*c^2)*b*c)*a^3-(b^2-c^2)^2*(b+c)*a^2-2*(b^2-c^2)^2*(b^2+4*b*c+c^2)*a+(b^2-c^2)^2*(b+c)*(-3*b^2+2*b*c-3*c^2) : :
X(35663) = 3*X(1699)+X(8915) = 3*X(3817)-X(35680) = 9*X(7988)-X(35673) = 5*X(8227)-X(35667) = 9*X(9779)-X(35661) = 5*X(11522)-X(35665)

The reciprocal orthologic center of these triangles is X(1)

X(35663) lies on these lines: {2,35657}, {4,35658}, {5,35664}, {10,7683}, {11,35671}, {12,35669}, {226,35672}, {496,35670}, {946,3812}, {1699,8915}, {2886,35682}, {3817,35680}, {7678,35668}, {7988,35673}, {8226,35666}, {8227,35667}, {8228,35674}, {8229,35675}, {8379,35681}, {8727,35655}, {9779,35661}, {10478,35640}, {10863,35656}, {10883,35660}, {10886,35662}, {11522,35665}, {11680,35659}, {17605,35679}, {17618,35678}

X(35663) = midpoint of X(4) and X(35658)
X(35663) = reflection of X(35664) in X(5)
X(35663) = complement of X(35657)
X(35663) = X(1217)-of-3rd Euler triangle
X(35663) = X(35658)-of-Euler triangle
X(35663) = X(35664)-of-Johnson triangle

X(35664) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th EULER TO INVERSE-IN-EXCIRCLES

Barycentrics (b+c)*a^6-4*b*c*a^5+(b^2-c^2)*(b-c)*a^4-8*(b^2+c^2)*b*c*a^3-(b+c)*(5*b^4+5*c^4-2*(2*b^2-7*b*c+2*c^2)*b*c)*a^2+12*(b^2-c^2)^2*b*c*a+(b^2-c^2)^2*(b+c)*(3*b^2-2*b*c+3*c^2) : :
X(35664) = 5*X(1698)-X(8915) = 3*X(3679)+X(35665) = 3*X(5587)+X(35667) = 7*X(7989)+X(35673) = 7*X(9780)+X(35661)

The reciprocal orthologic center of these triangles is X(1)

X(35664) lies on these lines: {2,35658}, {4,35657}, {5,35663}, {10,9842}, {11,35669}, {12,35671}, {442,35666}, {495,35670}, {516,5955}, {946,3454}, {1210,35672}, {1329,35682}, {1698,8915}, {2886,35886}, {3679,35665}, {3925,35889}, {4197,35660}, {5051,35675}, {5587,35667}, {7679,35668}, {7989,35673}, {8230,35674}, {8382,35681}, {8582,35656}, {8728,35655}, {9780,35661}, {10479,35640}, {10887,35662}, {11681,35659}, {17606,35679}, {17619,35678}

X(35664) = midpoint of X(i) and X(j) for these {i,j}: {4, 35657}, {10, 35680}
X(35664) = reflection of X(35663) in X(5)
X(35664) = complement of X(35658)
X(35664) = X(1217)-of-4th Euler triangle
X(35664) = X(35657)-of-Euler triangle
X(35664) = X(35663)-of-Johnson triangle

X(35665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: EXCENTERS-REFLECTIONS TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*a^6-6*(b+c)*a^5+(b^2+14*b*c+c^2)*a^4+12*(b+c)*(b^2-4*b*c+c^2)*a^3-(11*b^4+11*c^4+2*(6*b^2-31*b*c+6*c^2)*b*c)*a^2-6*(b^2-c^2)*(b-c)*(b^2-6*b*c+c^2)*a+(7*b^2-2*b*c+7*c^2)*(b^2-c^2)^2) : :
X(35665) = 3*X(1)-2*X(35658) = 3*X(3679)-4*X(35664) = 3*X(8915)-4*X(35658) = 5*X(11522)-4*X(35663)

The reciprocal orthologic center of these triangles is X(1)

X(35665) lies on these lines: {1,1407}, {8,35680}, {145,35661}, {517,35667}, {1743,12672}, {2098,35679}, {3340,35671}, {3679,35664}, {5293,7991}, {7962,35669}, {10912,35678}, {11518,35655}, {11519,35656}, {11520,35660}, {11521,35662}, {11522,35663}, {11523,35666}, {11526,35668}, {11529,35670}, {11531,35673}, {11532,35674}, {11533,35675}, {11535,35681}, {11682,35659}, {15829,35682}

X(35665) = midpoint of X(i) and X(j) for these {i,j}: {145, 35661}, {11531, 35673}
X(35665) = reflection of X(i) in X(j) for these (i,j): (8, 35680), (7991, 35657), (8915, 1), (35889, 35670)
X(35665) = X(1217)-of-excenters-reflections triangle
X(35665) = X(8915)-of-5th mixtilinear triangle

X(35666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH TO INVERSE-IN-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(1)

X(35666) lies on these lines: {1,35886}, {2,35655}, {4,7101}, {9,8915}, {10,35889}, {72,1750}, {226,35671}, {329,35659}, {392,15852}, {405,35658}, {442,35664}, {950,35669}, {1490,35667}, {1864,35679}, {3487,35670}, {4199,35675}, {5728,35672}, {5927,35656}, {7580,35657}, {7593,35681}, {8226,35663}, {8232,35668}, {8233,35674}, {10477,35640}, {10888,35662}, {11523,35665}

X(35666) = midpoint of X(35659) and X(35661)
X(35666) = reflection of X(i) in X(j) for these (i,j): (1, 35886), (8915, 35682), (35660, 35655), (35671, 35680), (35678, 35656), (35889, 10)
X(35666) = anticomplement of X(35655)
X(35666) = complement of X(35660)
X(35666) = X(8)-Beth conjugate of X(35889)
X(35666) = X(1217)-of-2nd extouch triangle
X(35666) = X(35886)-of-Aquila triangle
X(35666) = X(35889)-of-outer-Garcia triangle
X(35666) = {X(2), X(35660)}-harmonic conjugate of X(35655)

X(35667) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HEXYL TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*a^6+3*(b+c)^2*a^4+4*(2*b-c)*(b-2*c)*(b+c)*a^3-(7*b^2+22*b*c+7*c^2)*(b-c)^2*a^2-4*(b+c)*(2*b^4+2*c^4-5*(b-c)^2*b*c)*a+(b^2-c^2)^2*(b+c)^2) : :
X(35667) = 3*X(3576)-2*X(35658) = 3*X(5587)-4*X(35664) = 5*X(8227)-4*X(35663)

The reciprocal orthologic center of these triangles is X(1)

X(35667) lies on these lines: {1,1122}, {3,7963}, {4,35680}, {20,35661}, {40,936}, {56,35679}, {78,35659}, {517,35665}, {1490,35666}, {3333,35672}, {3576,15839}, {5587,35664}, {7590,35681}, {7675,35668}, {8227,35663}, {8234,35674}, {8235,35675}, {8726,35655}, {10476,35640}, {10864,35656}, {10884,35660}, {12114,35678}, {30503,35889}

X(35667) = midpoint of X(i) and X(j) for these {i,j}: {1, 35673}, {20, 35661}
X(35667) = reflection of X(i) in X(j) for these (i,j): (4, 35680), (40, 35657), (8915, 3)
X(35667) = X(1217)-of-hexyl triangle
X(35667) = X(8915)-of-ABC-X3 reflections triangle
X(35667) = X(35673)-of-anti-Aquila triangle
X(35667) = X(35679)-of-2nd circumperp tangential triangle
X(35667) = X(35680)-of-anti-Euler triangle

X(35668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HONSBERGER TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^7+(b^2+b*c+c^2)*a^6-(b+c)*(3*b^2+2*b*c+3*c^2)*a^5-(3*b^4+3*c^4+(9*b^2-40*b*c+9*c^2)*b*c)*a^4+(b+c)*(3*b^4+3*c^4-2*(2*b^2+23*b*c+2*c^2)*b*c)*a^3+(3*b^6+3*c^6+(b^2+3*b*c+c^2)*(7*b^2-12*b*c+7*c^2)*b*c)*a^2-(b^2-c^2)*(b-c)*(b^4+c^4-2*(2*b^2-11*b*c+2*c^2)*b*c)*a-(b^4+c^4-(b^2-4*b*c+c^2)*b*c)*(b^2-c^2)^2) : :
X(35668) = 3*X(8236)-2*X(35669) = 5*X(11025)-4*X(35672) = 3*X(11038)-4*X(35670) = 5*X(18230)-4*X(35682)

The reciprocal orthologic center of these triangles is X(1)

X(35668) lies on these lines: {7,35660}, {9,35659}, {1445,8915}, {4326,35673}, {5572,35679}, {7675,35667}, {7676,35657}, {7677,35658}, {7678,35663}, {7679,35664}, {8232,35666}, {8236,35669}, {8237,35674}, {8238,35675}, {8389,35681}, {8732,35655}, {10865,35656}, {10889,35662}, {11025,35672}, {11038,35670}, {11526,35665}, {17620,35678}, {18230,35682}, {21617,35680}, {35617,35640}

X(35668) = reflection of X(i) in X(j) for these (i,j): (7, 35671), (35659, 9), (35679, 5572)
X(35668) = X(1217)-of-Honsberger triangle

X(35669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5+(3*b-c)*(b-3*c)*a^4+2*(b^2-c^2)*(b-c)*a^3-2*(b^4+6*b^2*c^2+c^4)*a^2-(3*b^2-2*b*c+3*c^2)*(b+c)^3*a-(b^2-c^2)^2*(b-c)^2) : :
X(35669) = 3*X(1)-2*X(35670) = 3*X(8236)-X(35668) = 4*X(35670)-3*X(35671)

The reciprocal orthologic center of these triangles is X(1)

X(35669) lies on these lines: {1,1122}, {8,35682}, {11,35664}, {12,35663}, {20,3057}, {55,1394}, {56,35657}, {65,10544}, {145,35659}, {497,35886}, {950,35666}, {1010,3698}, {1697,8915}, {3601,35655}, {4313,35660}, {4320,10387}, {7962,35665}, {8236,35668}, {8239,35674}, {8240,35675}, {9785,35661}, {10480,35640}, {10866,35656}, {11924,35681}, {12053,35680}, {17622,35678}

X(35669) = midpoint of X(i) and X(j) for these {i,j}: {145, 35659}, {3057, 35679}
X(35669) = reflection of X(i) in X(j) for these (i,j): (8, 35682), (65, 35672), (35671, 1), (35889, 35658)
X(35669) = X(1217)-of-Hutson intouch triangle
X(35669) = X(35657)-of-2nd anti-circumperp-tangential triangle
X(35669) = X(35658)-of-Mandart-incircle triangle
X(35669) = X(35671)-of-5th mixtilinear triangle

X(35670) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5+(3*b^2+8*b*c+3*c^2)*a^4+2*(b+c)*(b^2+b*c+c^2)*a^3-2*(b^4+c^4-3*(b^2+8*b*c+c^2)*b*c)*a^2-(b+c)*(b^2-4*b*c+c^2)*(3*b^2-2*b*c+3*c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(35670) = 3*X(1)-X(35669) = 5*X(3616)-X(35659) = 3*X(11038)+X(35668) = 5*X(17609)-X(35679) = X(35669)+3*X(35671)

The reciprocal orthologic center of these triangles is X(1)

X(35670) lies on these lines: {1,1122}, {495,35664}, {496,35663}, {517,959}, {942,4301}, {999,35658}, {1125,35682}, {3295,35657}, {3333,8915}, {3487,35666}, {3616,35659}, {5045,35672}, {8092,35681}, {11035,35656}, {11036,35660}, {11037,35661}, {11038,35668}, {11042,35674}, {11043,35675}, {11529,35665}, {17609,35679}, {17624,35678}, {21620,35680}, {35620,35640}

X(35670) = midpoint of X(i) and X(j) for these {i,j}: {1, 35671}, {35665, 35889}
X(35670) = reflection of X(i) in X(j) for these (i,j): (35672, 5045), (35682, 1125)
X(35670) = X(1217)-of-incircle-circles triangle
X(35670) = X(35671)-of-anti-Aquila triangle

X(35671) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INTOUCH TO INVERSE-IN-EXCIRCLES

Barycentrics a*(a^2+2*(b+c)*a+(b-c)^2)*((b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2) : :
X(35671) = 3*X(354)-2*X(35672) = 3*X(354)-X(35679) = X(35669)-4*X(35670)

The reciprocal orthologic center of these triangles is X(1)

X(35671) lies on these lines: {1,1122}, {2,35659}, {7,35660}, {11,35663}, {12,35664}, {55,35657}, {56,35658}, {57,8915}, {65,497}, {174,35681}, {226,35666}, {354,17114}, {1284,35675}, {1361,10373}, {3057,4854}, {3340,35665}, {3485,35886}, {8243,35674}, {8581,35656}, {9785,24471}, {10473,35640}, {17625,35678}

X(35671) = midpoint of X(i) and X(j) for these {i,j}: {7, 35668}, {35660, 35661}
X(35671) = reflection of X(i) in X(j) for these (i,j): (1, 35670), (8915, 35655), (35659, 35682), (35666, 35680), (35669, 1), (35679, 35672)
X(35671) = anticomplement of X(35682)
X(35671) = complement of X(35659)
X(35671) = crosspoint of X(7) and X(2999)
X(35671) = crosssum of X(55) and X(2297)
X(35671) = X(1217)-of-intouch triangle
X(35671) = X(35657)-of-Mandart-incircle triangle
X(35671) = X(35658)-of-2nd anti-circumperp-tangential triangle
X(35671) = X(35669)-of-5th mixtilinear triangle
X(35671) = X(35670)-of-Aquila triangle
X(35671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35659, 35682), (354, 35679, 35672)

X(35672) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5-(b^2-6*b*c+c^2)*a^4-2*(b+c)*(b^2-5*b*c+c^2)*a^3+2*(b^4+c^4+(b^2+4*b*c+c^2)*b*c)*a^2+(b+c)*(b^4+c^4-2*(b^2-5*b*c+c^2)*b*c)*a-(b^4-c^4)*(b^2-c^2)) : :
X(35672) = 3*X(354)-X(35671) = 3*X(354)+X(35679) = 3*X(3873)+X(35659) = 2*X(9943)+X(35640) = 5*X(11025)-X(35668)

The reciprocal orthologic center of these triangles is X(1)

X(35672) lies on these lines: {1,1407}, {57,35657}, {65,10544}, {72,171}, {109,12709}, {221,10383}, {226,35663}, {354,17114}, {518,35682}, {601,12672}, {940,12711}, {942,4307}, {1071,5711}, {1193,17612}, {1210,35664}, {1834,17668}, {3333,35667}, {3664,35889}, {3873,35659}, {4646,35338}, {5045,35670}, {5439,26098}, {5710,17625}, {5728,35666}, {8083,35681}, {10580,35661}, {10980,35673}, {11018,35655}, {11019,35656}, {11020,35660}, {11021,35662}, {11025,35668}, {11030,35674}, {11031,35675}, {17626,35678}

X(35672) = midpoint of X(i) and X(j) for these {i,j}: {65, 35669}, {35671, 35679}
X(35672) = reflection of X(i) in X(j) for these (i,j): (35656, 35680), (35670, 5045)
X(35672) = X(1217)-of-inverse-in-incircle triangle
X(35672) = {X(354), X(35679)}-harmonic conjugate of X(35671)

X(35673) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR TO INVERSE-IN-EXCIRCLES

Barycentrics a*(3*a^6+2*(b+c)*a^5+(9*b^2-2*b*c+9*c^2)*a^4+12*(b^2-c^2)*(b-c)*a^3-(11*b^4+11*c^4+2*(2*b^2-23*b*c+2*c^2)*b*c)*a^2-2*(b+c)*(7*b^4+7*c^4-2*(3*b-2*c)*(2*b-3*c)*b*c)*a-(b^2-6*b*c+c^2)*(b^2-c^2)^2) : :
X(35673) = 3*X(165)-2*X(8915) = 3*X(165)-4*X(35657) = 3*X(1699)-4*X(35680) = 5*X(7987)-4*X(35658) = 9*X(7988)-8*X(35663) = 7*X(7989)-8*X(35664)

The reciprocal orthologic center of these triangles is X(1)

X(35673) lies on these lines: {1,1122}, {40,8951}, {57,35679}, {72,1750}, {165,978}, {200,35659}, {516,35661}, {1699,35680}, {1709,35678}, {3062,35656}, {4326,35668}, {5732,35660}, {7987,35658}, {7988,35663}, {7989,35664}, {8244,35674}, {8245,35675}, {8423,35681}, {8580,35682}, {10857,35655}, {10980,35672}, {11531,35665}, {35621,35640}

X(35673) = reflection of X(i) in X(j) for these (i,j): (1, 35667), (8915, 35657), (11531, 35665)
X(35673) = X(1217)-of-6th mixtilinear triangle
X(35673) = X(35667)-of-Aquila triangle
X(35673) = {X(8915), X(35657)}-harmonic conjugate of X(165)

X(35674) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU TO INVERSE-IN-EXCIRCLES

Barycentrics a*(2*((b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(5*b^2-18*b*c+5*c^2)*a^4+4*(b^2+4*b*c+c^2)*(b-c)^2*a^3+(b+c)*(7*b^4+7*c^4-2*(10*b^2-21*b*c+10*c^2)*b*c)*a^2-2*(b^2-c^2)^2*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2*(b+c)*(-3*b^2+2*b*c-3*c^2))*S*b*c-(a+b-c)*(a+c-b)*(c-a+b)*((b+c)*a^7+(3*b^2+2*b*c+3*c^2)*a^6+3*(b^3+c^3)*a^5+(b^4+c^4-(b^2-32*b*c+c^2)*b*c)*a^4-(b+c)*(b^4+c^4-2*(b^2-5*b*c+c^2)*b*c)*a^3-3*(b^4-c^4)*(b^2-c^2)*a^2-(b+c)*(3*b^6+3*c^6-(b^4+c^4-(17*b^2-22*b*c+17*c^2)*b*c)*b*c)*a-(b^2-c^2)^2*(b+c)*(b^3+c^3))) : :

The reciprocal orthologic center of these triangles is X(1)

X(35674) lies on these lines: {8224,35657}, {8225,35658}, {8228,35663}, {8230,35664}, {8231,8915}, {8233,35666}, {8234,35667}, {8237,35668}, {8239,35669}, {8243,35671}, {8244,35673}, {8246,35675}, {9789,35661}, {10858,35655}, {10867,35656}, {10885,35660}, {10891,35662}, {11030,35672}, {11042,35670}, {11532,35665}, {11687,35659}, {11996,35681}, {12610,35680}, {17610,35679}, {17627,35678}, {18234,35682}, {35622,35640}

X(35675) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st SHARYGIN TO INVERSE-IN-EXCIRCLES

Barycentrics a*(2*(b+c)*a^8+(b^2+6*b*c+c^2)*a^7-(b+c)*(3*b^2-14*b*c+3*c^2)*a^6+(b^4+c^4+2*(2*b^2-3*b*c+2*c^2)*b*c)*a^5+(b+c)*(b^4+c^4-2*(6*b^2-5*b*c+6*c^2)*b*c)*a^4-(5*b^6+5*c^6+(10*b^4+10*c^4+(19*b^2+44*b*c+19*c^2)*b*c)*b*c)*a^3-(b+c)*(b^6+c^6+(2*b^4+2*c^4+(23*b^2-4*b*c+23*c^2)*b*c)*b*c)*a^2+(3*b^4+14*b^2*c^2+3*c^4)*(b^2-c^2)^2*a+(b^4-c^4)^2*(b+c)) : :

The reciprocal orthologic center of these triangles is X(1)

X(35675) lies on these lines: {1,3430}, {21,35658}, {846,8915}, {1284,35671}, {4199,35666}, {4425,35680}, {5051,35664}, {8229,35663}, {8235,35667}, {8238,35668}, {8240,35669}, {8245,35673}, {8246,35674}, {8425,35681}, {8731,35655}, {9791,35661}, {10868,35656}, {10892,35662}, {11031,35672}, {11043,35670}, {11533,35665}, {11688,35659}, {17611,35679}, {17628,35678}, {18235,35682}, {35623,35640}

X(35675) = X(1217)-of-1st Sharygin triangle

X(35676) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TRI-SQUARES TO INVERSE-IN-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(35652)

X(35676) lies on these lines: {2,35677}, {13637,35653}, {13638,35654}

X(35677) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TRI-SQUARES TO INVERSE-IN-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(35652)

X(35677) lies on these lines: {2,35676}, {13757,35653}, {13758,35654}

X(35678) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MAJOR TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^8-2*(3*b^2-4*b*c+3*c^2)*a^7-2*(b+c)*(b^2-7*b*c+c^2)*a^6+2*(9*b^4+9*c^4-2*(8*b^2+b*c+8*c^2)*b*c)*a^5-2*(b+c)*(17*b^2-50*b*c+17*c^2)*b*c*a^4-2*(9*b^6+9*c^6-(20*b^4+20*c^4+9*(b^2-8*b*c+c^2)*b*c)*b*c)*a^3+2*(b+c)*(b^6+c^6+(13*b^4+13*c^4-(53*b^2-94*b*c+53*c^2)*b*c)*b*c)*a^2+2*(b^2-c^2)^2*(3*b^4+3*c^4-2*(4*b^2-b*c+4*c^2)*b*c)*a+(b^2-c^2)^2*(b+c)*(-b^4-c^4-6*(b^2-b*c+c^2)*b*c)) : :

The reciprocal orthologic center of these triangles is X(1)

X(35678) lies on these lines: {11,35679}, {1376,8915}, {1709,35673}, {3434,35661}, {5691,10914}, {5927,35656}, {10912,35665}, {12114,35667}, {17612,35655}, {17613,35657}, {17614,35658}, {17615,35659}, {17616,35660}, {17617,35662}, {17618,35663}, {17619,35664}, {17620,35668}, {17622,35669}, {17624,35670}, {17625,35671}, {17626,35672}, {17627,35674}, {17628,35675}, {17631,35681}, {18236,35682}, {35626,35640}

X(35678) = reflection of X(i) in X(j) for these (i,j): (35666, 35656), (35679, 35680)
X(35678) = X(1217)-of-Ursa-major triangle
X(35678) = X(35679)-of-inner-Johnson triangle

X(35679) = ORTHOLOGIC CENTER OF THESE TRIANGLES: URSA MINOR TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5-5*(b-c)^2*a^4-2*(b-3*c)*(3*b-c)*(b+c)*a^3+6*(b^2-c^2)^2*a^2+(b+c)*(5*b^2-2*b*c+c^2)*(b^2-2*b*c+5*c^2)*a-(b^2-c^2)^2*(b+c)^2) : :
X(35679) = 3*X(210)-4*X(35682) = 3*X(354)-2*X(35671) = 3*X(354)-4*X(35672) = 5*X(17609)-4*X(35670)

The reciprocal orthologic center of these triangles is X(1)

X(35679) lies on these lines: {11,35678}, {20,3057}, {55,8915}, {56,35667}, {57,35673}, {210,35682}, {354,17114}, {497,35661}, {518,35659}, {1155,35657}, {1864,35666}, {2098,35665}, {2646,35658}, {5572,35668}, {10374,35889}, {10391,35660}, {10473,35662}, {10502,35681}, {17603,35655}, {17604,35656}, {17605,35663}, {17606,35664}, {17609,35670}, {17610,35674}, {17611,35675}, {21334,35640}

X(35679) = reflection of X(i) in X(j) for these (i,j): (3057, 35669), (35668, 5572), (35671, 35672), (35678, 35680)
X(35679) = X(1217)-of-Ursa-minor triangle
X(35679) = X(8915)-of-Mandart-incircle triangle
X(35679) = X(35667)-of-2nd anti-circumperp-tangential triangle
X(35679) = X(35678)-of-2nd Johnson-Yff triangle
X(35679) = {X(35671), X(35672)}-harmonic conjugate of X(354)

X(35680) = ORTHOLOGIC CENTER OF THESE TRIANGLES: WASAT TO INVERSE-IN-EXCIRCLES

Barycentrics (b+c)*a^6-2*(b-c)^2*a^5-(b+c)*(5*b^2-18*b*c+5*c^2)*a^4+4*(b^2+4*b*c+c^2)*(b-c)^2*a^3+(b+c)*(7*b^4+7*c^4-2*(10*b^2-21*b*c+10*c^2)*b*c)*a^2-2*(b^2-c^2)^2*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2*(b+c)*(-3*b^2+2*b*c-3*c^2) : :
X(35680) = 3*X(2)+X(35661) = 3*X(1699)+X(35673) = 3*X(3817)-2*X(35663)

The reciprocal orthologic center of these triangles is X(1)

X(35680) lies on these lines: {2,8915}, {4,35667}, {8,35665}, {10,9842}, {11,35678}, {142,35655}, {226,35666}, {516,35657}, {908,35659}, {946,21255}, {1125,35658}, {1699,35673}, {3452,35682}, {3741,35640}, {3817,35663}, {4425,35675}, {5249,35660}, {10478,35662}, {11019,35656}, {12053,35669}, {12610,35674}, {21617,35668}, {21620,35670}, {21624,35681}

X(35680) = midpoint of X(i) and X(j) for these {i,j}: {4, 35667}, {8, 35665}, {8915, 35661}, {35656, 35672}, {35666, 35671}, {35678, 35679}
X(35680) = reflection of X(i) in X(j) for these (i,j): (10, 35664), (35658, 1125)
X(35680) = complement of X(8915)
X(35680) = X(1217)-of-Wasat triangle
X(35680) = X(35667)-of-Euler triangle
X(35680) = {X(2), X(35661)}-harmonic conjugate of X(8915)

X(35681) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YFF CENTRAL TO INVERSE-IN-EXCIRCLES

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

The reciprocal orthologic center of these triangles is X(1)

X(35681) lies on these lines: {173,8915}, {174,35671}, {236,35682}, {7587,35658}, {7589,35657}, {7590,35667}, {7593,35666}, {8083,35672}, {8092,35670}, {8126,35659}, {8379,35663}, {8382,35664}, {8389,35668}, {8423,35673}, {8425,35675}, {8729,35655}, {10502,35679}, {11535,35665}, {11860,35656}, {11890,35660}, {11891,35661}, {11896,35662}, {11924,35669}, {11996,35674}, {17631,35678}, {21624,35680}, {35627,35640}

X(35682) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd ZANIAH TO INVERSE-IN-EXCIRCLES

Barycentrics a*((b+c)*a^5+(3*b^2-4*b*c+3*c^2)*a^4+2*(b+c)*(b^2-4*b*c+c^2)*a^3-2*(b^2-c^2)^2*a^2-(b+c)*(3*b^4+26*b^2*c^2+3*c^4)*a-(b^2+c^2)*(b+c)^4) : :
X(35682) = 3*X(2)+X(35659) = 3*X(210)+X(35679) = 5*X(18230)-X(35668)

The reciprocal orthologic center of these triangles is X(1)

X(35682) lies on these lines: {2,35659}, {8,35669}, {9,8915}, {40,936}, {210,35679}, {236,35681}, {518,35672}, {958,35658}, {1125,35670}, {1329,35664}, {2886,35663}, {3452,35680}, {3931,10179}, {5273,35660}, {5711,34791}, {5745,35655}, {8580,35673}, {15829,35665}, {18227,35656}, {18228,35661}, {18229,35662}, {18230,35668}, {18234,35674}, {18235,35675}, {18236,35678}, {35628,35640}

X(35682) = midpoint of X(i) and X(j) for these {i,j}: {8, 35669}, {8915, 35666}, {35659, 35671}
X(35682) = reflection of X(35670) in X(1125)
X(35682) = complement of X(35671)
X(35682) = X(1217)-of-2nd Zaniah triangle
X(35682) = {X(2), X(35659)}-harmonic conjugate of X(35671)

X(35683) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd SHARYGIN TO INVERSE-IN-EXCIRCLES

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

The reciprocal parallelogic center of these triangles is X(1)

X(35683) lies on these lines: {}

X(35683) = X(1217)-of-2nd Sharygin triangle

X(35684) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 3rd ANTI-TRI-SQUARES

Barycentrics (a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2+2*(2*(b^2+c^2)*a^2-b^4-c^4)*S : :
X(35684) = 4*X(140)-3*X(13087) = X(6281)-3*X(9767)

The reciprocal orthologic center of these triangles is X(22562)

X(35684) lies on these lines: {3,591}, {4,33433}, {6,642}, {32,13989}, {76,486}, {140,13087}, {372,7762}, {487,7793}, {543,35703}, {576,7764}, {639,6421}, {3564,6312}, {5480,6311}, {6229,32494}, {6251,6289}, {6281,6813}, {6393,35841}, {6419,7807}, {7774,13926}, {7775,35702}, {8184,13758}, {8782,33340}, {9891,33274}, {12123,32433}, {12158,35305}, {12252,12256}, {16628,22605}, {16629,22606}

X(35684) = {X(576), X(7764)}-harmonic conjugate of X(35685)

X(35685) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 4th ANTI-TRI-SQUARES

Barycentrics (a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2-2*(2*(b^2+c^2)*a^2-b^4-c^4)*S : :
X(35685) = 4*X(140)-3*X(13088) = X(6278)-3*X(9768)

The reciprocal orthologic center of these triangles is X(22563)

X(35685) lies on these lines: {3,1991}, {4,33432}, {6,641}, {32,8997}, {76,485}, {140,13088}, {371,7762}, {488,7793}, {543,35702}, {576,7764}, {640,6422}, {3564,6311}, {5480,6315}, {6228,32497}, {6250,6290}, {6278,6811}, {6393,35840}, {6420,7807}, {7774,13873}, {7775,35703}, {8180,13638}, {8782,33341}, {9893,33274}, {12124,32436}, {12159,35306}, {12252,12257}, {16628,22634}, {16629,22635}

X(35685) = {X(576), X(7764)}-harmonic conjugate of X(35684)

X(35686) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO BANKOFF

Barycentrics 18*S^3-3*(-1+2*sqrt(3))*SW*S^2-sqrt(3)*(3*SA*SW+6*sqrt(3)*SB*SC-SW^2)*S+3*(2*sqrt(3)-3)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(34509)

X(35686) lies on these lines: {3,531}, {524,35748}, {530,35742}, {543,34551}, {1151,22579}, {5464,35739}, {11292,22490}, {22571,35730}, {22573,35731}, {22575,35732}, {22577,35733}, {31695,35740}, {33476,35738}, {35692,35734}, {35693,35735}, {35694,35736}, {35695,35737}

X(35687) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO CIRCUMSYMMEDIAL

Barycentrics a^2*((b^2-2*c^2)*(2*b^2-c^2)*a^4-2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*b^2*c^2) : :
X(35687) = X(20)-4*X(14135) = 4*X(140)-3*X(12525) = 7*X(3523)-4*X(6310)

The reciprocal orthologic center of these triangles is X(99)

X(35687) lies on these lines: {3,3231}, {4,6331}, {20,185}, {140,12525}, {263,33007}, {373,32971}, {512,34511}, {2396,11459}, {2979,33207}, {3060,33193}, {3111,3767}, {3523,6310}, {5640,14035}, {5650,32990}, {5652,5654}, {6337,6786}, {6658,11002}, {6787,7763}, {7998,32965}, {9879,33274}, {11326,35259}, {11673,33208}, {27374,33239}, {33192,33873}, {33258,33879}, {33260,33884}, {34504,35704}

X(35687) = reflection of X(35704) in X(34504)
X(35687) = X(6323)-of-Moses-Steiner osculatory triangle

X(35688) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO INNER-FERMAT

Barycentrics -2*(a^4-6*(b^2+c^2)*a^2+2*b^4+2*b^2*c^2+2*c^4)*sqrt(3)*S+3*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2 : :
X(35688) = 4*X(140)-3*X(22866)

The reciprocal orthologic center of these triangles is X(22568)

X(35688) lies on these lines: {3,533}, {16,7877}, {18,76}, {61,30471}, {62,619}, {140,22866}, {383,22665}, {576,35689}, {627,10654}, {630,16644}, {634,22850}, {5617,16628}, {7755,22848}, {7764,34509}, {7772,8260}, {7781,34508}, {12252,14538}, {16627,16629}, {22567,33274}

X(35689) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO OUTER-FERMAT

Barycentrics 2*(a^4-6*(b^2+c^2)*a^2+2*b^4+2*b^2*c^2+2*c^4)*sqrt(3)*S+3*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2 : :
X(35689) = 4*X(140)-3*X(22911)

The reciprocal orthologic center of these triangles is X(22570)

X(35689) lies on these lines: {3,532}, {15,7877}, {17,76}, {61,618}, {62,30472}, {140,22911}, {576,35688}, {628,10653}, {629,16645}, {633,22894}, {1080,22666}, {5613,16629}, {7755,22892}, {7764,34508}, {7772,8259}, {7781,34509}, {12252,14539}, {16626,16628}, {22569,33274}

X(35690) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics 2*(a^2+c^2+b^2)*S+(7*a^4-6*(b^2+c^2)*a^2-5*b^4+18*b^2*c^2-5*c^4)*sqrt(3) : :
X(35690) = 7*X(2)-6*X(9885) = 11*X(2)-12*X(33477) = 3*X(148)-2*X(35693) = 3*X(616)-4*X(33459) = 11*X(9885)-14*X(33477) = 12*X(9885)-7*X(35691) = 9*X(9885)-7*X(35696) = 3*X(9885)-7*X(35697) = 24*X(33477)-11*X(35691) = 18*X(33477)-11*X(35696) = 6*X(33477)-11*X(35697) = 3*X(35691)-4*X(35696) = X(35691)-4*X(35697) = X(35696)-3*X(35697)

The reciprocal orthologic center of these triangles is X(34508)

X(35690) lies on these lines: {2,99}, {524,35749}, {530,5862}, {616,5321}, {5858,33625}, {5859,33624}, {5863,35752}, {33458,33609}

X(35690) = reflection of X(i) in X(j) for these (i,j): (2, 35697), (5863, 35752), (20094, 22570), (35691, 2), (35750, 5858)
X(35690) = anticomplement of X(35696)
X(35690) = {X(18546), X(35692)}-harmonic conjugate of X(2)

X(35691) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics -2*(a^2+b^2+c^2)*S+(9*a^4-10*(b^2+c^2)*a^2-3*b^4+14*b^2*c^2-3*c^4)*sqrt(3) : :
X(35691) = 5*X(2)-6*X(9885) = 13*X(2)-12*X(33477) = 3*X(616)-2*X(5858) = 3*X(8591)-2*X(35692) = 13*X(9885)-10*X(33477) = 12*X(9885)-5*X(35690) = 3*X(9885)-5*X(35696) = 9*X(9885)-5*X(35697) = 24*X(33477)-13*X(35690) = 6*X(33477)-13*X(35696) = 18*X(33477)-13*X(35697) = X(35690)-4*X(35696) = 3*X(35690)-4*X(35697) = 3*X(35696)-X(35697)

The reciprocal orthologic center of these triangles is X(34508)

X(35691) lies on these lines: {2,99}, {524,35750}, {530,5863}, {616,5858}, {5474,6770}, {5859,33611}, {5862,35751}

X(35691) = reflection of X(i) in X(j) for these (i,j): (2, 35696), (148, 22568), (5862, 35751), (35690, 2), (35749, 5859)
X(35691) = anticomplement of X(35697)

X(35692) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd INNER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics 2*(a^2+c^2+b^2)*S+(5*a^4+4*b^2*c^2-6*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3) : :
X(35692) = 2*X(2)-3*X(9886) = 7*X(2)-6*X(33476) = 3*X(8591)-X(35691) = 7*X(9886)-4*X(33476) = 3*X(9886)-X(35693) = 9*X(9886)-2*X(35694) = 3*X(9886)+2*X(35695) = 3*X(22570)-X(35697) = 12*X(33476)-7*X(35693) = 18*X(33476)-7*X(35694) = 6*X(33476)+7*X(35695) = 3*X(35693)-2*X(35694) = X(35693)+2*X(35695) = X(35694)+3*X(35695)

The reciprocal orthologic center of these triangles is X(34509)

X(35692) lies on these lines: {2,99}, {531,3534}, {617,5863}, {3181,6781}, {3845,9760}, {5066,22575}, {5464,22900}, {5613,13103}, {5862,33611}, {5969,9114}, {8716,11296}, {9830,35751}, {10611,14144}, {35686,35734}

X(35692) = midpoint of X(2) and X(35695)
X(35692) = reflection of X(i) in X(j) for these (i,j): (148, 33460), (22568, 99), (35693, 2), (35696, 15300)
X(35692) = complement of X(35694)
X(35692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35690, 18546), (9886, 35693, 2)

X(35693) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th INNER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics -2*(a^2+c^2+b^2)*S+(3*a^4-2*(b^2+c^2)*a^2-(b^2-3*c^2)*(3*b^2-c^2))*sqrt(3) : :
X(35693) = 4*X(2)-3*X(9886) = 5*X(2)-6*X(33476) = 3*X(148)-X(35690) = 3*X(9885)-2*X(15300) = 5*X(9886)-8*X(33476) = 3*X(9886)-2*X(35692) = 3*X(9886)+4*X(35694) = 9*X(9886)-4*X(35695) = 3*X(22568)-2*X(35696) = 12*X(33476)-5*X(35692) = 6*X(33476)+5*X(35694) = 18*X(33476)-5*X(35695) = X(35692)+2*X(35694) = 3*X(35692)-2*X(35695) = 3*X(35694)+X(35695)

The reciprocal orthologic center of these triangles is X(34509)

X(35693) lies on these lines: {2,99}, {14,33459}, {531,3830}, {3845,22575}, {5066,9760}, {5478,5613}, {5862,33627}, {5863,33625}, {9830,35752}, {11295,34505}, {11898,13102}, {12816,33458}, {22578,22850}, {35686,35735}

X(35693) = midpoint of X(2) and X(35694)
X(35693) = reflection of X(i) in X(j) for these (i,j): (99, 33461), (22570, 115), (35692, 2)
X(35693) = complement of X(35695)
X(35693) = {X(2), X(35692)}-harmonic conjugate of X(9886)

X(35694) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st OUTER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics -2*(a^2+c^2+b^2)*S+(7*a^4-6*(b^2+c^2)*a^2-5*b^4+18*b^2*c^2-5*c^4)*sqrt(3) : :
X(35694) = 7*X(2)-6*X(9886) = 11*X(2)-12*X(33476) = 3*X(148)-2*X(35697) = 3*X(617)-4*X(33458) = 11*X(9886)-14*X(33476) = 9*X(9886)-7*X(35692) = 3*X(9886)-7*X(35693) = 12*X(9886)-7*X(35695) = 18*X(33476)-11*X(35692) = 6*X(33476)-11*X(35693) = 24*X(33476)-11*X(35695) = X(35692)-3*X(35693) = 4*X(35692)-3*X(35695) = 4*X(35693)-X(35695)

The reciprocal orthologic center of these triangles is X(34509)

X(35694) lies on these lines: {2,99}, {531,5863}, {617,5318}, {5858,33622}, {5859,33623}, {9830,35749}, {33459,33608}, {35686,35736}

X(35694) = reflection of X(i) in X(j) for these (i,j): (2, 35693), (20094, 22568), (35695, 2)
X(35694) = anticomplement of X(35692)
X(35694) = {X(18546), X(35696)}-harmonic conjugate of X(2)

X(35695) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd OUTER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics 2*(a^2+c^2+b^2)*S+(9*a^4-10*(b^2+c^2)*a^2-3*b^4+14*b^2*c^2-3*c^4)*sqrt(3) : :
X(35695) = 5*X(2)-6*X(9886) = 13*X(2)-12*X(33476) = 3*X(617)-2*X(5859) = 3*X(8591)-2*X(35696) = 13*X(9886)-10*X(33476) = 3*X(9886)-5*X(35692) = 9*X(9886)-5*X(35693) = 12*X(9886)-5*X(35694) = 6*X(33476)-13*X(35692) = 18*X(33476)-13*X(35693) = 24*X(33476)-13*X(35694) = 3*X(35692)-X(35693) = 4*X(35692)-X(35694) = 4*X(35693)-3*X(35694)

The reciprocal orthologic center of these triangles is X(34509)

X(35695) lies on these lines: {2,99}, {531,5862}, {617,5859}, {5473,6773}, {5858,33610}, {9830,35750}, {35686,35737}

X(35695) = reflection of X(i) in X(j) for these (i,j): (2, 35692), (148, 22570), (35694, 2)
X(35695) = anticomplement of X(35693)

X(35696) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics -2*(a^2+c^2+b^2)*S+(5*a^4+4*b^2*c^2-6*(b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3) : :
X(35696) = 2*X(2)-3*X(9885) = 7*X(2)-6*X(33477) = 3*X(8591)-X(35695) = 7*X(9885)-4*X(33477) = 9*X(9885)-2*X(35690) = 3*X(9885)+2*X(35691) = 3*X(9885)-X(35697) = 3*X(22568)-X(35693) = 18*X(33477)-7*X(35690) = 6*X(33477)+7*X(35691) = 12*X(33477)-7*X(35697) = X(35690)+3*X(35691) = 2*X(35690)-3*X(35697) = 2*X(35691)+X(35697)

The reciprocal orthologic center of these triangles is X(34508)

X(35696) lies on these lines: {2,99}, {524,35751}, {530,3534}, {616,5862}, {3180,6781}, {3845,9762}, {5066,22576}, {5463,22856}, {5617,13102}, {5863,33610}, {5969,9116}, {8716,11295}, {10612,14145}, {33458,35752}

X(35696) = midpoint of X(i) and X(j) for these {i,j}: {2, 35691}, {5863, 35750}
X(35696) = reflection of X(i) in X(j) for these (i,j): (148, 33461), (22570, 99), (35692, 15300), (35697, 2), (35752, 33458)
X(35696) = complement of X(35690)
X(35696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35694, 18546), (9885, 35697, 2)

X(35697) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO MOSES-STEINER OSCULATORY

Barycentrics 2*(a^2+c^2+b^2)*S+(3*a^4-2*(b^2+c^2)*a^2-(b^2-3*c^2)*(3*b^2-c^2))*sqrt(3) : :
X(35697) = 4*X(2)-3*X(9885) = 5*X(2)-6*X(33477) = 3*X(148)-X(35694) = 5*X(9885)-8*X(33477) = 3*X(9885)+4*X(35690) = 9*X(9885)-4*X(35691) = 3*X(9885)-2*X(35696) = 3*X(9886)-2*X(15300) = 3*X(22570)-2*X(35692) = 6*X(33477)+5*X(35690) = 18*X(33477)-5*X(35691) = 12*X(33477)-5*X(35696) = 3*X(35690)+X(35691) = 2*X(35690)+X(35696) = 2*X(35691)-3*X(35696)

The reciprocal orthologic center of these triangles is X(34508)

X(35697) lies on these lines: {2,99}, {13,33458}, {524,35752}, {530,3830}, {3845,22576}, {5066,9762}, {5479,5617}, {5862,33623}, {5863,33626}, {11296,34505}, {11898,13103}, {12817,33459}, {22577,22894}

X(35697) = midpoint of X(i) and X(j) for these {i,j}: {2, 35690}, {5862, 35749}
X(35697) = reflection of X(i) in X(j) for these (i,j): (99, 33460), (22568, 115), (35696, 2), (35751, 33459)
X(35697) = complement of X(35691)
X(35697) = {X(2), X(35696)}-harmonic conjugate of X(9885)

X(35698) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO MOSES-STEINER OSCULATORY

Barycentrics 3*S^4+(12*SA^2+33*SB*SC-5*SW^2)*S^2-3*SB*SC*SW^2+3*S*(3*S^2-SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4)

X(35698) lies on these lines: {6,12355}, {30,35824}, {372,671}, {381,35879}, {485,8591}, {530,35850}, {531,35753}, {542,35820}, {543,35822}, {1587,8596}, {2482,10576}, {2782,35838}, {5969,35866}, {6200,12117}, {6321,35823}, {6419,19058}, {6560,12243}, {6564,8724}, {6565,9880}, {7841,35830}, {9830,35840}, {9875,35774}, {9876,35776}, {9878,35782}, {9881,35788}, {9882,35792}, {9883,35794}, {9884,35810}, {9974,10488}, {10054,35809}, {10070,35769}, {10992,13908}, {12132,35764}, {12191,35766}, {12258,35762}, {12326,35772}, {12345,35778}, {12346,35780}, {12347,35790}, {12348,35796}, {12349,35798}, {12350,35800}, {12351,35802}, {12352,35804}, {12353,35806}, {12354,35808}, {12356,35816}, {12357,35818}, {13968,35814}, {18969,35768}, {19057,35770}, {22565,35784}, {22566,35786}

X(35698) = reflection of X(35878) in X(35822)
X(35698) = {X(6), X(12355)}-harmonic conjugate of X(35699)

X(35699) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO MOSES-STEINER OSCULATORY

Barycentrics 3*S^4+(12*SA^2+33*SB*SC-5*SW^2)*S^2-3*SB*SC*SW^2-3*S*(3*S^2-SW^2)*(SB+SC) : :

The reciprocal orthologic center of these triangles is X(4)

X(35699) lies on these lines: {6,12355}, {30,35825}, {371,671}, {381,35878}, {486,8591}, {530,35851}, {531,35754}, {542,35821}, {543,35823}, {1588,8596}, {2482,10577}, {2782,35839}, {5969,35867}, {6321,35822}, {6396,12117}, {6420,19057}, {6561,12243}, {6564,9880}, {6565,8724}, {7841,35831}, {9830,35841}, {9875,35775}, {9876,35777}, {9878,35783}, {9881,35789}, {9882,35795}, {9883,35793}, {9884,35811}, {9975,10488}, {10054,35808}, {10070,35768}, {10992,13968}, {12132,35765}, {12191,35767}, {12258,35763}, {12326,35773}, {12345,35781}, {12346,35779}, {12347,35791}, {12348,35797}, {12349,35799}, {12350,35801}, {12351,35803}, {12352,35807}, {12353,35805}, {12354,35809}, {12356,35817}, {12357,35819}, {13908,35815}, {18969,35769}, {19058,35771}, {22565,35785}, {22566,35787}

X(35699) = reflection of X(35879) in X(35823)
X(35699) = {X(6), X(12355)}-harmonic conjugate of X(35698)

X(35700) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 1st NEUBERG

Barycentrics (b^4-b^2*c^2+c^4)*a^4+(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-2*(b^4+c^4)*b^2*c^2 : :
X(35700) = 4*X(140)-3*X(13085) = 3*X(3095)-4*X(7764) = 2*X(7764)-3*X(8149)

The reciprocal orthologic center of these triangles is X(9888)

X(35700) lies on these lines: {3,538}, {4,5969}, {39,32954}, {76,141}, {114,3095}, {140,13085}, {194,5976}, {576,35701}, {732,6776}, {736,9821}, {1513,9764}, {1691,7754}, {1916,32966}, {1975,34870}, {2023,32969}, {2782,18768}, {5182,7760}, {6309,32515}, {6658,8782}, {7757,7807}, {7821,32452}, {7866,9466}, {9887,33274}, {11055,35297}, {11178,34505}, {11287,14711}, {11356,17130}, {15589,20081}, {16045,24256}, {16628,25191}, {16629,25195}, {18906,32971}, {19568,20885}

X(35700) = reflection of X(3095) in X(8149)

X(35701) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 2nd NEUBERG

Barycentrics a^8-2*(b^2+c^2)*a^6-(2*b^4+7*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 : :
X(35701) = 4*X(140)-3*X(13086) = 2*X(7780)-3*X(8150) = 2*X(7781)+3*X(13111) = X(7781)+3*X(18548)

The reciprocal orthologic center of these triangles is X(9890)

X(35701) lies on these lines: {3,754}, {6,76}, {140,13086}, {315,5116}, {325,2896}, {384,13571}, {576,35700}, {1003,12156}, {2076,7762}, {3095,7781}, {3398,7780}, {3552,6337}, {5149,7838}, {5476,34505}, {5989,7785}, {6292,7778}, {6309,10796}, {7470,12252}, {7745,12215}, {7750,34873}, {7772,11356}, {7922,11285}, {8177,10359}, {9478,32961}, {9765,13860}, {9889,33274}, {11606,32993}, {16628,25192}, {16629,25196}

X(35701) = reflection of X(13111) in X(18548)

X(35702) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 1st TRI-SQUARES-CENTRAL

Barycentrics 3*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2-2*(4*a^4+2*(b^2+c^2)*a^2-5*b^4+8*b^2*c^2-5*c^4)*S : :
X(35702) = 4*X(140)-3*X(13700)

The reciprocal orthologic center of these triangles is X(13676)

X(35702) lies on these lines: {32,13908}, {76,1327}, {140,13700}, {543,35685}, {576,35703}, {6337,32495}, {7775,35684}, {12252,13674}, {13677,33274}, {13687,32419}, {16628,25193}, {16629,25197}

X(35703) = ORTHOLOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 3*(a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4)*a^2+2*(4*a^4+2*(b^2+c^2)*a^2-5*b^4+8*b^2*c^2-5*c^4)*S : :
X(35703) = 4*X(140)-3*X(13820)

The reciprocal orthologic center of these triangles is X(13796)

X(35703) lies on these lines: {32,13968}, {76,1328}, {140,13820}, {543,35684}, {576,35702}, {6337,32492}, {7775,35685}, {12252,13794}, {13797,33274}, {13807,32421}, {16628,25194}, {16629,25198}

X(35704) = PARALLELOGIC CENTER OF THESE TRIANGLES: MOSES-STEINER OSCULATORY TO CIRCUMSYMMEDIAL

Barycentrics a^2*((2*b^2+c^2)*(b^2+2*c^2)*a^4-2*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^2-c^2)^2*b^2*c^2) : :
X(35704) = 4*X(140)-3*X(13240)

The reciprocal parallelogic center of these triangles is X(98)

X(35704) lies on these lines: {20,512}, {39,263}, {76,34095}, {140,13240}, {211,7738}, {376,2387}, {511,34511}, {2871,13340}, {2979,14917}, {3111,35287}, {6787,33192}, {7763,33873}, {13207,33274}, {27375,31400}, {34504,35687}

X(35704) = reflection of X(35687) in X(34504)

X(35705) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND MOSES-STEINER REFLECTION

Barycentrics a^8+2*(b^2+c^2)*a^6-3*(b^4+b^2*c^2+c^4)*a^4+(b^6+c^6)*a^2-b^8+2*(b^4+c^4)*b^2*c^2-c^8 : :
X(35705) = X(148)-3*X(33016)

X(35705) lies on these lines: {2,353}, {30,99}, {69,147}, {114,3818}, {115,11174}, {148,7736}, {160,35894}, {183,542}, {262,6321}, {543,5475}, {620,7868}, {671,3363}, {1513,12177}, {1975,14981}, {2482,7761}, {2679,33755}, {2782,9744}, {3329,6034}, {3589,10352}, {3629,12830}, {4027,7806}, {5149,7820}, {5152,8291}, {5182,7792}, {5463,9760}, {5464,9762}, {5477,14614}, {5969,7774}, {6114,22687}, {6115,22689}, {6287,7769}, {6787,12093}, {7610,10488}, {7664,9832}, {7737,9890}, {7771,9774}, {7777,14931}, {7931,8290}, {8356,15483}, {8591,9770}, {8593,9877}, {8598,19911}, {11177,34229}, {15300,25486}, {22561,33013}, {23235,32819}

X(35705) = reflection of X(i) in X(j) for these (i,j): (671, 3363), (13860, 114)
X(35705) = crossdifference of every pair of points on line {X(6041), X(9208)}
X(35705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99, 6054, 325), (5978, 5979, 8724), (8593, 9877, 22329)

X(35706) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ANTI-MCCAY TO MOSES-STEINER REFLECTION

Barycentrics 3*(6*R^2+SA-3*SW)*S^2-3*R^2*(SW^2+9*SB*SC)+SW*(SW^2+15*SB*SC) : :

The reciprocal orthologic center of these triangles is X(99)

X(35706) lies on these lines: {2,34513}, {1503,8598}, {5663,9855}, {6800,11317}, {8593,8705}, {33274,34514}

X(35707) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st EHRMANN TO MOSES-STEINER REFLECTION

Barycentrics a^2*(a^6+(b^2+c^2)*a^4-(b^2+c^2)^2*a^2-b^6-c^6) : :
X(35707) = X(6)-3*X(6800) = 2*X(3589)-3*X(13394) = 2*X(5092)-3*X(34513)

The reciprocal orthologic center of these triangles is X(69)

X(35707) lies on these lines: {2,19596}, {3,66}, {6,23}, {22,524}, {25,597}, {26,8550}, {69,2916}, {182,12106}, {184,9019}, {206,6593}, {542,7502}, {575,9969}, {576,11536}, {599,6636}, {895,19121}, {1176,9973}, {1350,11456}, {1658,8262}, {1995,3589}, {2070,11179}, {2393,19127}, {2854,35268}, {3098,5663}, {3292,3313}, {3455,9830}, {3518,7716}, {3525,15435}, {3564,7555}, {3618,14002}, {3629,19459}, {3763,7496}, {5012,9971}, {5092,34513}, {5157,32154}, {5480,7530}, {5621,10298}, {5652,21006}, {5899,20423}, {6776,7556}, {7485,20582}, {7493,25328}, {7512,15069}, {7525,34507}, {7545,14561}, {7550,10516}, {8177,9149}, {8542,19126}, {8584,9909}, {9822,20190}, {9966,14655}, {9968,32367}, {9970,18438}, {10117,25329}, {10510,12220}, {11061,35218}, {11188,12367}, {11477,12088}, {11645,18570}, {12082,29181}, {12893,32305}, {13452,34817}, {14173,34009}, {14179,34008}, {15246,21358}, {18374,26881}, {18911,32218}, {20775,33801}, {22352,29959}, {24206,34514}, {31959,35890}

X(35707) = midpoint of X(i) and X(j) for these {i,j}: {1350, 11456}, {31959, 35890}
X(35707) = reflection of X(34514) in X(24206)
X(35707) = crossdifference of every pair of points on line {X(2485), X(3906)}
X(35707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (206, 11511, 6593), (6593, 17710, 11511)

X(35708) = PERSPECTOR OF THESE TRIANGLES: 4th ANTI-EULER AND ORTHIC AXES

Barycentrics (SB+SC)*(7*S^4+(-R^2*(52*R^2+36*SA-25*SW)+9*SA^2-2*SB*SC-3*SW^2)*S^2+(R^2+SW)*(4*R^2-SW)*(4*R^2+SA-2*SW)*SA) : :

X(35708) lies on these lines: {1885,11455}, {3567,6748}, {6241,35717}

X(35708) = X(4)-Waw conjugate of X(9781)

X(35709) = PERSPECTOR OF THESE TRIANGLES: ANTI-WASAT AND ORTHIC AXES

Barycentrics ((b^2-c^2)^2*a^8-(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4)*a^6+(3*b^4+5*b^2*c^2+3*c^4)*(b^2-c^2)^2*a^4-(b^4-c^4)*(b^2-c^2)*(b^4+3*b^2*c^2+c^4)*a^2+(b^2-c^2)^4*b^2*c^2)*a^2 : :
X(35709) = 3*X(51)-4*X(6748) = 3*X(51)-2*X(6752) = 2*X(16264)-3*X(32062)

X(35709) lies on these lines: {25,31382}, {51,6748}, {53,34980}, {160,1495}, {185,35717}, {1503,1885}, {4173,16655}, {16264,32062}

X(35709) = crosspoint of X(4) and X(1988)
X(35709) = crosssum of X(3) and X(3164)
X(35709) = X(4)-Waw conjugate of X(51)

X(35710) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND ORTHIC AXES

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

X(35710) lies on these lines: {2,35884}, {4,8719}, {262,7378}, {427,9752}, {1853,14853}, {6748,7612}

X(35711) = PERSPECTOR OF THESE TRIANGLES: 2nd EXCOSINE AND ORTHIC AXES

Barycentrics SB*SC*(8*R^2*(16*R^2+SA-5*SW)-2*SA^2+SB*SC+2*SW^2) : :

X(35711) lies on these lines: {4,64}, {5,20199}, {25,17830}, {53,22334}, {393,11381}, {1033,3087}, {1073,20329}, {1515,3090}, {1888,17832}, {1906,34993}, {3079,8567}, {3346,8798}, {3462,5656}, {3529,16253}, {6225,10002}, {6564,22838}, {6565,22839}, {6616,6696}, {6995,18288}, {12134,18850}, {13155,20208}, {31685,35714}, {31686,35715}, {33893,35515}

X(35711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 64, 6525), (4, 6247, 6526)

X(35712) = PERSPECTOR OF THESE TRIANGLES: 15th FERMAT-DAO AND ORTHIC AXES

Barycentrics SB*SC*(sqrt(3)*(50*R^2+SA-4*SW)*S^2+(20*S^2+2*(12*R^2-SA-SW)*R^2+5*SA^2+4*SB*SC-SW^2)*S+sqrt(3)*(2*R^2-SW)*SB*SC) : :

X(35712) lies on these lines: {}

X(35713) = PERSPECTOR OF THESE TRIANGLES: 16th FERMAT-DAO AND ORTHIC AXES

Barycentrics SB*SC*(sqrt(3)*(50*R^2+SA-4*SW)*S^2-(20*S^2+2*(12*R^2-SA-SW)*R^2+5*SA^2+4*SB*SC-SW^2)*S+sqrt(3)*(2*R^2-SW)*SB*SC) : :

X(35713) lies on these lines: {}

X(35714) = PERSPECTOR OF THESE TRIANGLES: 3rd ISODYNAMIC-DAO AND ORTHIC AXES

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

X(35714) lies on these lines: {4,13}, {25,1605}, {53,115}, {403,6117}, {427,6115}, {463,6111}, {470,6669}, {471,618}, {472,530}, {473,5459}, {542,31688}, {1112,20412}, {1300,8741}, {1585,6306}, {1586,6302}, {1597,6772}, {1785,1832}, {2914,10657}, {3087,9112}, {5318,18400}, {5472,6748}, {6564,31689}, {6565,31692}, {6778,20774}, {18494,22513}, {31685,35711}

X(35714) = polar circle-inverse of X(5669)
X(35714) = X(4)-Waw conjugate of X(6117)
X(35714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 13, 31687), (4, 8737, 6110), (53, 1596, 35715)

X(35715) = PERSPECTOR OF THESE TRIANGLES: 4th ISODYNAMIC-DAO AND ORTHIC AXES

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

X(35715) lies on these lines: {4,14}, {25,1606}, {53,115}, {403,6116}, {427,6114}, {462,6110}, {470,619}, {471,6670}, {472,5460}, {473,531}, {542,31687}, {1112,20411}, {1300,8742}, {1585,6307}, {1586,6303}, {1597,6775}, {1785,1833}, {2914,10658}, {3087,9113}, {5321,18400}, {5471,6748}, {6564,31691}, {6565,31690}, {6777,20774}, {18494,22512}, {31686,35711}

X(35715) = polar circle-inverse of X(5668)
X(35715) = X(4)-Waw conjugate of X(6116)
X(35715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 14, 31688), (4, 8738, 6111), (53, 1596, 35714)

X(35716) = PERSPECTOR OF THESE TRIANGLES: ORTHIC AXES AND X3-ABC REFLECTIONS

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

X(35716) lies on these lines: {3,53}, {4,31831}, {1656,6747}, {5064,35719}, {11671,35502}

X(35716) = X(4)-Waw conjugate of X(1598)

X(35717) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC AXES TO MACBEATH

Barycentrics SB*SC*(4*S^2-4*R^2*(8*R^2+SA-5*SW)+SA^2-2*SB*SC-3*SW^2) : :
X(35717) = 2*X(4)-3*X(35884) = 3*X(31867)-X(35728)

The reciprocal orthologic center of these triangles is X(14978)

X(35717) lies on these lines: {3,34836}, {4,54}, {20,17035}, {30,35887}, {185,35709}, {324,10112}, {378,14111}, {403,34598}, {427,31867}, {539,14978}, {570,3575}, {1852,1877}, {1879,22261}, {6241,35708}, {6755,34782}, {7507,31976}, {8565,8882}, {8887,12022}, {14531,27377}, {14918,32348}, {32423,35719}

X(35717) = {X(4), X(275)}-harmonic conjugate of X(3574)

X(35718) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC AXES TO YIU

Barycentrics SB*SC*(4*S^2-R^2*(6*R^2-SA-9*SW)-SA^2-3*SW^2) : :

The reciprocal orthologic center of these triangles is X(5)

X(35718) lies on these lines: {4,93}, {427,35729}, {1157,8882}, {1510,14618}, {1594,10615}, {7576,16336}

X(35718) = polar circle-inverse of X(6243)
X(35718) = {X(4), X(14111)}-harmonic conjugate of X(6152)

X(35719) = ORTHOLOGIC CENTER OF THESE TRIANGLES: ORTHIC AXES TO YIU TANGENTS

Barycentrics SB*SC*(S^2-(R^2-SW)*SA)*(S^2+SB*SC) : :
X(35719) = X(3)-3*X(11197) = 2*X(140)-3*X(10184) = 5*X(1656)-3*X(32078) = 4*X(3628)-3*X(12012) = 4*X(3850)-3*X(14635) = 4*X(15557)-3*X(35884) = 3*X(35884)-2*X(35887)

The reciprocal orthologic center of these triangles is X(35720)

X(35719) lies on these lines: {3,11197}, {4,93}, {5,53}, {24,34292}, {140,10184}, {143,324}, {195,4994}, {264,6101}, {275,1493}, {382,31388}, {403,31607}, {427,35720}, {1075,5946}, {1594,13856}, {1656,32078}, {2052,15026}, {3518,32551}, {3628,12012}, {3850,14635}, {5064,35716}, {5392,31810}, {6644,23709}, {6748,32358}, {8146,18400}, {8884,10610}, {10018,13467}, {10095,30506}, {13364,13450}, {15557,35884}, {15559,25150}, {32423,35717}

X(35719) = midpoint of X(i) and X(j) for these {i,j}: {4, 14978}, {382, 31388}
X(35719) = reflection of X(35887) in X(15557)
X(35719) = barycentric product X(324)*X(13353)
X(35719) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(14577)}} and {{A, B, C, X(5), X(11140)}}
X(35719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 93, 6152), (15557, 35887, 35884)

X(35720) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YIU TANGENTS TO ORTHIC AXES

Barycentrics (3*S^2-2*R^2*(5*R^2+3*SA-4*SW)-4*SB*SC-SW^2)*(S^2+SB*SC) : :
X(35720) = 3*X(5)-4*X(13856) = 15*X(5)-16*X(34599) = 7*X(5)-8*X(34768) = 3*X(549)-4*X(18016) = 5*X(632)-4*X(32551) = 4*X(10205)-3*X(35885) = 2*X(13856)-3*X(15345) = 5*X(13856)-4*X(34599) = 7*X(13856)-6*X(34768) = 15*X(15345)-8*X(34599) = 7*X(15345)-4*X(34768) = 14*X(34599)-15*X(34768) = 3*X(35885)-2*X(35888)

The reciprocal orthologic center of these triangles is X(35719)

X(35720) lies on these lines: {3,19553}, {5,128}, {30,23337}, {54,930}, {140,25043}, {185,550}, {427,35719}, {539,30484}, {548,24305}, {549,18016}, {632,32551}, {6368,12038}, {8254,33992}, {10205,35885}, {11273,21230}, {32423,35728}

X(35720) = reflection of X(i) in X(j) for these (i,j): (5, 15345), (14072, 18807), (25043, 140), (35888, 10205)
X(35720) = circumnormal isogonal conjugate of X(7691)
X(35720) = {X(10205), X(35888)}-harmonic conjugate of X(35885)

X(35721) = PERSPECTOR OF THESE TRIANGLES: ABC-X3 REFLECTIONS AND YIU TANGENTS

Barycentrics S^4-(64*R^4+2*R^2*(5*SA-27*SW)-4*SA^2-SB*SC+11*SW^2)*S^2+(32*R^2*(5*R^2-4*SW)+25*SW^2)*SB*SC : :
X(35721) = 9*X(549)-8*X(32904) = 3*X(550)-2*X(30484) = 4*X(550)-3*X(35885) = 3*X(3627)-4*X(20414) = 3*X(15687)-4*X(33545) = 4*X(30484)-3*X(35728) = 8*X(30484)-9*X(35885) = 4*X(32536)-3*X(33699) = 2*X(35728)-3*X(35885)

X(35721) lies on these lines: {3,15619}, {30,23337}, {549,32904}, {550,6247}, {3574,33992}, {3627,20414}, {14073,15704}, {15687,33545}, {32536,33699}

X(35721) = reflection of X(i) in X(j) for these (i,j): (15619, 3), (35728, 550)
X(35721) = {X(550), X(35728)}-harmonic conjugate of X(35885)

X(35722) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND YIU TANGENTS

Barycentrics 5*(40*R^2-3*SW)*S^4-(2*R^2*(25*SA-7*SW)-20*SA^2+95*SB*SC+11*SW^2)*SW*S^2-5*(8*R^2-5*SW)*SB*SC*SW^2 : :

X(35722) lies on these lines: {2,35885}, {9754,35727}, {16652,35725}, {16653,35726}

X(35723) = PERSPECTOR OF THESE TRIANGLES: CIRCUMMEDIAL AND YIU TANGENTS

Barycentrics 2*(20*R^2*(5*R^2-4*SW)+11*SW^2)*S^4-(2*R^2*(50*SA^2-45*SA*SW+SW^2)-SW*(15*SA^2-11*SA*SW-SW^2))*SW*S^2+3*SB*SC*SW^4 : :

X(35723) lies on the line {7485,35896}

X(35724) = PERSPECTOR OF THESE TRIANGLES: CIRCUMORTHIC AND YIU TANGENTS

Barycentrics 21*S^4-3*(2*(8*R^2+5*SA-9*SW)*R^2-4*SA^2+9*SB*SC+5*SW^2)*S^2+(4*R^2*(10*R^2-11*SW)+13*SW^2)*SB*SC : :

X(35724) lies on these lines: {4,35728}, {54,35885}, {140,35888}, {185,550}, {12242,30484}

X(35725) = PERSPECTOR OF THESE TRIANGLES: INNER-FERMAT AND YIU TANGENTS

Barycentrics -2*(12*a^10-20*(b^2+c^2)*a^8-(3*b^4+14*b^2*c^2+3*c^4)*a^6+2*(b^2+c^2)*(11*b^4+2*b^2*c^2+11*c^4)*a^4-(b^2-c^2)^2*(9*b^4+13*b^2*c^2+9*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4+b^2*c^2-2*c^4))*S+sqrt(3)*(4*a^12-24*(b^2+c^2)*a^10+(49*b^4+58*b^2*c^2+49*c^4)*a^8-(b^2+c^2)*(41*b^4-14*b^2*c^2+41*c^4)*a^6+3*(3*b^8+3*c^8-(5*b^4+4*b^2*c^2+5*c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^2-(2*b^4-b^2*c^2+2*c^4)*(b^2-c^2)^4) : :

X(35725) lies on these lines: {3,35726}, {15,35727}, {16652,35722}

X(35726) = PERSPECTOR OF THESE TRIANGLES: OUTER-FERMAT AND YIU TANGENTS

Barycentrics 2*(12*a^10-20*(b^2+c^2)*a^8-(3*b^4+14*b^2*c^2+3*c^4)*a^6+2*(b^2+c^2)*(11*b^4+2*b^2*c^2+11*c^4)*a^4-(b^2-c^2)^2*(9*b^4+13*b^2*c^2+9*c^4)*a^2+(b^4-c^4)*(b^2-c^2)*(-2*b^4+b^2*c^2-2*c^4))*S+sqrt(3)*(4*a^12-24*(b^2+c^2)*a^10+(49*b^4+58*b^2*c^2+49*c^4)*a^8-(b^2+c^2)*(41*b^4-14*b^2*c^2+41*c^4)*a^6+3*(3*b^8+3*c^8-(5*b^4+4*b^2*c^2+5*c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(5*b^4+4*b^2*c^2+5*c^4)*a^2-(2*b^4-b^2*c^2+2*c^4)*(b^2-c^2)^4) : :

X(35726) lies on these lines: {3,35725}, {16,35727}, {16653,35722}

X(35727) = PERSPECTOR OF THESE TRIANGLES: REFLECTION AND YIU TANGENTS

Barycentrics 27*S^4-((6*(5*SA-3*SW))*R^2-12*SA^2+37*SB*SC+9*SW^2)*S^2+3*SB*SC*SW^2 : :

X(35727) lies on these lines: {15,35725}, {16,35726}, {511,550}, {7495,26276}, {9754,35722}, {11063,15358}

X(35728) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YIU TANGENTS TO MACBEATH

Barycentrics 7*S^4-(2*R^2*(8*R^2+5*SA-9*SW)-4*SA^2+9*SB*SC+5*SW^2)*S^2+(4*R^2*(20*R^2-17*SW)+15*SW^2)*SB*SC : :
X(35728) = 5*X(5)-4*X(33545) = 9*X(5)-8*X(34598) = 2*X(550)-3*X(35885) = 5*X(3858)-4*X(20414) = 4*X(30484)-X(35721) = 4*X(30484)-3*X(35885) = 3*X(31867)-2*X(35717) = 9*X(33545)-10*X(34598) = X(35721)-3*X(35885)

The reciprocal orthologic center of these triangles is X(14978)

X(35728) lies on these lines: {4,35724}, {5,11701}, {30,15619}, {54,33992}, {427,31867}, {550,6247}, {1510,12162}, {3858,20414}, {6240,31976}, {21975,31656}, {32423,35720}

X(35728) = reflection of X(i) in X(j) for these (i,j): (550, 30484), (35721, 550)
X(35728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (550, 30484, 35885), (35721, 35885, 550)

X(35729) = ORTHOLOGIC CENTER OF THESE TRIANGLES: YIU TANGENTS TO YIU

Barycentrics 10*S^4-(R^2*(17*R^2+10*SA-20*SW)-4*SA^2+14*SB*SC+6*SW^2)*S^2+3*(R^2*(5*R^2-6*SW)+2*SW^2)*SB*SC : :
X(35729) = 3*X(5)-4*X(10615) = 3*X(376)-X(35449) = 3*X(549)-2*X(16336) = 3*X(6150)-2*X(10615) = 3*X(8703)-X(14141)

The reciprocal orthologic center of these triangles is X(5)

X(35729) lies on these lines: {3,14140}, {5,6150}, {30,1141}, {54,10205}, {140,19552}, {185,550}, {252,10285}, {376,35449}, {427,35718}, {539,14073}, {549,16336}, {930,6343}, {1510,30481}, {3627,16337}, {7354,14102}, {7488,18212}, {7604,10289}, {8703,14141}, {21230,35888}

X(35729) = reflection of X(i) in X(j) for these (i,j): (5, 6150), (3627, 16337), (14140, 3), (19552, 140)

X(35730) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 3rd FERMAT-DAO

Barycentrics 13*S^2+3*(4-sqrt(3))*(SB+SC)*S-(25-16*sqrt(3))*SB*SC : :

X(35730) lies on these lines: {3,13}, {14,3592}, {15,35740}, {16,32785}, {61,1588}, {62,3366}, {371,16809}, {381,6425}, {1151,16808}, {2044,16962}, {3364,16966}, {3389,19106}, {3390,16960}, {3411,8960}, {3412,35823}, {5470,35748}, {6419,18586}, {8976,22238}, {11542,35739}, {14813,32788}, {16267,34551}, {16529,35759}, {22489,35741}, {22510,35742}, {22571,35686}, {22602,35743}, {22631,35744}, {22688,35745}, {22845,33352}, {22846,35746}, {25151,35761}, {25157,35755}, {25158,35756}, {25159,35757}, {25160,35758}, {25217,35760}, {33351,33414}

X(35730) = {X(16960), X(35733)}-harmonic conjugate of X(3390)

X(35731) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 7th FERMAT-DAO

Barycentrics 11*sqrt(3)*S^2-3*(-2*sqrt(3)+1)*(SB+SC)*S+3*(8-5*sqrt(3))*SB*SC : :
X(35731) = X(3389)+2*X(35740)

X(35731) lies on these lines: {2,3364}, {3,13}, {14,371}, {15,2044}, {16,590}, {18,35812}, {30,3389}, {61,14813}, {62,2042}, {115,35748}, {381,1151}, {395,35738}, {396,3390}, {617,33352}, {2043,3391}, {2045,5237}, {3365,6460}, {5318,34552}, {5464,6304}, {6109,35742}, {6200,16808}, {6221,16809}, {6306,11303}, {6564,19106}, {6565,16966}, {8253,16967}, {8960,22238}, {9117,35759}, {9739,20425}, {10654,35732}, {13846,16963}, {13876,22511}, {13929,35743}, {16267,35739}, {16962,35733}, {18762,23302}, {22573,35686}, {22691,35745}, {22847,35746}, {25178,35761}, {25183,35755}, {25184,35756}, {25185,35757}, {25186,35758}, {25220,35760}, {33440,34509}

X(35731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371, 18586, 14), (396, 34551, 3390)

X(35732) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 11th FERMAT-DAO

Barycentrics S^2+(sqrt(3)-1)*SB*SC : :
X(35732) = 3*X(2)-4*X(14813) = 9*X(2)-8*X(35738) = 5*X(20)-4*X(14814)

As a point on the Euler line, X(35732) has Shinagawa coefficients (2-sqrt(3), 3*sqrt(3)-5)

X(35732) lies on these lines: {2,3}, {6,35740}, {14,35742}, {15,23259}, {16,23249}, {61,1588}, {62,1587}, {371,5334}, {372,5335}, {397,3594}, {398,3592}, {487,621}, {488,622}, {1151,5321}, {1152,5318}, {3070,22238}, {3071,22236}, {3364,18581}, {3366,9540}, {3390,18582}, {3392,13935}, {5237,23253}, {5238,23263}, {5339,6425}, {5340,6426}, {5343,6453}, {5344,6454}, {5613,35759}, {5870,33352}, {5871,33353}, {10654,35731}, {11485,23273}, {11486,23267}, {11489,31412}, {11542,13939}, {11543,13886}, {16626,35747}, {16628,35746}, {16808,35739}, {16809,35733}, {18435,34555}, {22491,35741}, {22575,35686}, {22605,35743}, {22634,35744}, {22693,35745}, {25164,35748}, {25180,35761}, {25191,35755}, {25192,35756}, {25193,35757}, {25194,35758}, {25224,35760}

X(35732) = reflection of X(2042) in X(14813)
X(35732) = anticomplement of X(2042)

X(35733) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 15th FERMAT-DAO

Barycentrics 61*S^2+(32*sqrt(3)-49)*SB*SC+(8+sqrt(3))*(SB+SC)*S : :

X(35733) lies on these lines: {3,16808}, {13,34551}, {16,23267}, {18,590}, {1151,16964}, {2044,3389}, {3390,16960}, {6409,19107}, {6410,16965}, {6777,35742}, {10576,16242}, {16241,35821}, {16809,35732}, {16962,35731}, {22493,35741}, {22577,35686}, {22607,35743}, {22636,35744}, {22695,35745}, {22849,35746}, {22894,35747}, {25166,35748}, {25182,35761}, {25199,35755}, {25200,35756}, {25201,35757}, {25202,35758}, {25228,35760}, {25236,35759}, {33416,35738}

X(35733) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3390, 35730, 16960), (34551, 35740, 35739), (35739, 35740, 13)

X(35734) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 3rd INNER-FERMAT-DAO-NHI

Barycentrics 83*sqrt(3)*S^2+9*(-7*sqrt(3)+8)*SB*SC : :
X(35734) = 3*X(381)+8*X(15764)

As a point on the Euler line, X(35734) has Shinagawa coefficients (-4*sqrt(3)+31, 12*sqrt(3)-27)

X(35734) lies on these lines: {2,3}, {5859,35741}, {33624,35746}, {35686,35692}

X(35735) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 4th INNER-FERMAT-DAO-NHI

Barycentrics 107*S^2+3*(-33+16*sqrt(3))*SB*SC : :

As a point on the Euler line, X(35735) has Shinagawa coefficients (-8*sqrt(3)+37, 24*sqrt(3)-45)

X(35735) lies on these lines: {2,3}, {5858,35741}, {33627,35746}, {35686,35693}

X(35736) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 1st OUTER-FERMAT-DAO-NHI

Barycentrics 52*S^2+3*(7*sqrt(3)-15)*SB*SC : :

As a point on the Euler line, X(35736) has Shinagawa coefficients (-14*sqrt(3)+74, 42*sqrt(3)-81)

X(35736) lies on these lines: {2,3}, {5862,35741}, {33622,35747}, {35686,35694}

X(35737) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 2nd OUTER-FERMAT-DAO-NHI

Barycentrics 46*S^2+3*(-12+5*sqrt(3))*SB*SC : :

As a point on the Euler line, X(35737) has Shinagawa coefficients (-10*sqrt(3)+68, 30*sqrt(3)-63)

X(35737) lies on these lines: {2,3}, {5863,35741}, {33626,35747}, {35686,35695}

X(35738) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 1st HALF-DIAMONDS-CENTRAL

Barycentrics ((b^2+c^2)*a^2-(b^2-c^2)^2)*sqrt(3)+2*a^2*(-a^2+b^2+c^2) : :
X(35738) = 3*X(2)+X(2042) = 9*X(2)-X(35732) = X(3)-3*X(15765) = X(3)+3*X(18586) = 5*X(4)-X(2041) = 5*X(631)-3*X(15764) = 5*X(632)-3*X(34551)

As a point on the Euler line, X(35738) has Shinagawa coefficients (1+sqrt(3), 5-3*sqrt(3))

X(35738) lies on these lines: {2,3}, {15,18762}, {16,18538}, {18,35746}, {51,34553}, {61,3392}, {62,3366}, {371,11543}, {372,11542}, {395,35731}, {485,22238}, {486,22236}, {619,35759}, {629,35747}, {1151,18581}, {1152,18582}, {3364,23303}, {3365,5318}, {3367,5352}, {3389,5321}, {3390,23302}, {3391,5351}, {3817,34556}, {5460,35748}, {5891,34555}, {6118,6672}, {6119,6671}, {6774,35742}, {10162,34554}, {10175,34557}, {16966,35739}, {28216,34560}, {33416,35733}, {33444,35744}, {33445,35743}, {33474,35741}, {33476,35686}, {33478,35745}, {33481,35761}, {33482,35755}, {33484,35756}, {33486,35757}, {33488,35758}, {33491,35760}, {34512,34558}

X(35738) = midpoint of X(i) and X(j) for these {i,j}: {382, 14814}, {2042, 14813}, {15765, 18586}
X(35738) = complement of X(14813)

X(35739) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 1st ISODYNAMIC-DAO

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

X(35739) lies on these lines: {2,33360}, {3,6}, {13,34551}, {30,3367}, {140,3366}, {530,33392}, {2044,3392}, {2046,3391}, {5464,35686}, {6778,35742}, {7599,14705}, {11542,35730}, {16267,35731}, {16808,35732}, {16966,35738}, {22495,35741}, {22609,35743}, {22638,35744}, {22701,35745}, {22855,35746}, {22900,35747}, {22997,35748}, {22999,35761}, {23000,35755}, {23001,35756}, {23002,35757}, {23003,35758}, {23004,35759}, {23007,35760}

X(35739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 372, 3364), (3, 1152, 16), (3, 3365, 10646), (3, 3390, 15), (3, 6396, 3365), (3, 9739, 14538), (3, 11485, 6409), (15, 5237, 3365), (16, 3365, 62), (16, 6396, 15), (371, 5238, 15), (372, 3364, 62), (2271, 13332, 2025), (2278, 10542, 11485), (3284, 8961, 5110), (5050, 20970, 2022), (6427, 22236, 61), (13327, 16836, 6200), (14631, 19011, 7772), (22385, 35388, 6421)

X(35740) = HOMOTHETIC CENTER OF THESE TRIANGLES: BANKOFF AND 3rd ISODYNAMIC-DAO

Barycentrics 2*S^2+(SB+SC)*S+2*(sqrt(3)-1)*SB*SC : :
X(35740) = X(3389)-3*X(35731)

X(35740) lies on these lines: {3,5318}, {4,590}, {5,3364}, {6,35732}, {13,34551}, {15,35730}, {16,18538}, {30,3389}, {115,35742}, {371,5321}, {372,397}, {395,485}, {396,2044}, {398,3311}, {640,6306}, {1152,5335}, {2042,3070}, {3390,11542}, {3592,5334}, {5340,6410}, {5350,14814}, {6783,35759}, {10576,15765}, {11481,32789}, {18585,35821}, {31693,35741}, {31695,35686}, {31697,35743}, {31699,35744}, {31701,35745}, {31703,35746}, {31705,35747}, {31707,35761}, {31709,35748}, {31711,35755}, {31713,35756}, {31715,35757}, {31717,35758}, {31719,35760}

X(35740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13, 35733, 35739), (35733, 35739, 34551)

X(35741) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO ANTI-ARTZT

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

The reciprocal orthologic center of these triangles is X(12155)

X(35741) lies on these lines: {2,3364}, {3,530}, {30,9738}, {524,34551}, {531,2044}, {543,35748}, {1151,11295}, {3642,6306}, {5459,6300}, {5858,35735}, {5859,35734}, {5862,35736}, {5863,35737}, {6307,31709}, {13084,15765}, {14813,34508}, {15764,32421}, {16241,33440}, {22489,35730}, {22491,35732}, {22493,35733}, {22495,35739}, {31693,35740}, {33474,35738}

X(35742) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 1st ANTI-BROCARD

Barycentrics 33*S^3+3*(-9+4*sqrt(3))*S*SB*SC+6*(-5+sqrt(3))*SB*SC*SW-(6+sqrt(3))*(3*SA-3*SW+2*sqrt(3)*SW)*S^2 : :

The reciprocal orthologic center of these triangles is X(5979)

X(35742) lies on these lines: {3,619}, {4,6307}, {14,35732}, {20,33443}, {30,35748}, {115,35740}, {530,35686}, {531,2044}, {542,34551}, {640,2794}, {1151,22512}, {3364,6114}, {3390,6783}, {5460,13663}, {6109,35731}, {6230,32553}, {6774,35738}, {6777,35733}, {6778,35739}, {14813,35744}, {15765,35757}, {22510,35730}, {35747,35759}

X(35743) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 3rd ANTI-TRI-SQUARES

Barycentrics 39*S^3-3*(2*sqrt(3)+5)*(-sqrt(3)*SW+SA+2*SW)*S^2+(8*sqrt(3)-45)*SB*SC*S-(7*sqrt(3)-15)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22601)

X(35743) lies on these lines: {3,6300}, {4,33367}, {487,633}, {530,35758}, {623,642}, {1151,22611}, {12123,14539}, {13929,35731}, {22602,35730}, {22605,35732}, {22607,35733}, {22609,35739}, {31697,35740}, {32419,34551}, {33445,35738}

X(35744) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 4th ANTI-TRI-SQUARES

Barycentrics 69*S^3-(9-2*sqrt(3))*(3*SA-(6+sqrt(3))*SW)*S^2-3*(13-8*sqrt(3))*SB*SC*S+3*(5*sqrt(3)-11)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22630)

X(35744) lies on these lines: {3,6304}, {5,6118}, {16,485}, {140,33446}, {530,35757}, {636,641}, {1151,22640}, {2045,22635}, {12257,33353}, {13088,34509}, {13876,22511}, {14813,35742}, {22631,35730}, {22634,35732}, {22636,35733}, {22638,35739}, {31699,35740}, {32421,34551}, {33444,35738}

X(35745) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 1st BROCARD-REFLECTED

Barycentrics 3*(-2*S+sqrt(3)*SA)*S^3-(2*sqrt(3)-3)*(-3*SA^2+2*(3+sqrt(3))*SA*SW+(9+4*sqrt(3))*SW^2)*S^2-sqrt(3)*(3*S+2*sqrt(3)*SW-6*SW)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(22687)

X(35745) lies on these lines: {3,22687}, {511,34551}, {1151,22707}, {2782,35748}, {22688,35730}, {22691,35731}, {22693,35732}, {22695,35733}, {22701,35739}, {31701,35740}, {33478,35738}

X(35746) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO INNER-FERMAT

Barycentrics 97*S^3-(sqrt(3)+10)*(6*SA+2*sqrt(3)*SW-3*SW)*S^2+(-91+20*sqrt(3))*SB*SC*S+(-24+17*sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(616)

X(35746) lies on these lines: {3,299}, {18,35738}, {533,34551}, {630,639}, {1151,22861}, {16628,35732}, {22846,35730}, {22847,35731}, {22849,35733}, {22855,35739}, {31703,35740}, {33624,35734}, {33627,35735}

X(35747) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO OUTER-FERMAT

Barycentrics 44*S^3+(5+sqrt(3))*(-3*SA+2*sqrt(3)*SW)*S^2+4*(5*sqrt(3)-8)*SB*SC*S-(-9+7*sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13)

X(35747) lies on these lines: {3,13}, {5,33473}, {532,34551}, {629,35738}, {1151,22906}, {16626,35732}, {22894,35733}, {22900,35739}, {31705,35740}, {33622,35736}, {33626,35737}, {35742,35759}

X(35748) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 8th FERMAT-DAO

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

The reciprocal orthologic center of these triangles is X(115)

X(35748) lies on these lines: {3,67}, {14,1151}, {30,35742}, {115,35731}, {511,35760}, {512,35761}, {524,35686}, {531,34551}, {543,35741}, {2042,20416}, {2044,9738}, {2782,35745}, {3390,9117}, {5460,35738}, {5470,35730}, {5474,12306}, {5969,35755}, {6307,6774}, {14813,16002}, {18587,22797}, {22997,35739}, {25164,35732}, {25166,35733}, {31709,35740}

X(35748) = reflection of X(35759) in X(34551)

X(35749) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st INNER-FERMAT-DAO-NHI TO BANKOFF

Barycentrics 2*(13*a^2-5*b^2-5*c^2)*S+(11*a^4-4*(b^2+c^2)*a^2-7*(b^2-c^2)^2)*sqrt(3) : :
X(35749) = 5*X(2)-6*X(13) = 4*X(2)-3*X(616) = 13*X(2)-12*X(618) = 11*X(2)-12*X(5459) = 7*X(2)-6*X(5463) = 17*X(2)-18*X(22489) = 8*X(13)-5*X(616) = 13*X(13)-10*X(618) = 11*X(13)-10*X(5459) = 7*X(13)-5*X(5463) = 23*X(13)-20*X(6669) = 17*X(13)-15*X(22489) = 12*X(13)-5*X(35750) = 9*X(13)-5*X(35751) = 3*X(13)-5*X(35752) = 13*X(616)-16*X(618) = 11*X(616)-16*X(5459) = 7*X(616)-8*X(5463) = 3*X(616)-2*X(35750) = 9*X(616)-8*X(35751) = 3*X(616)-8*X(35752)

The reciprocal orthologic center of these triangles is X(35748)

X(35749) lies on these lines: {2,13}, {148,33625}, {524,35690}, {542,10721}, {543,5863}, {634,11296}, {671,33603}, {2482,33615}, {3524,32907}, {3534,6770}, {3545,16001}, {3830,33627}, {3845,13103}, {4669,9901}, {5859,33611}, {5862,33623}, {6771,15719}, {6778,8591}, {9830,35694}, {11001,33626}, {15702,20415}, {33602,33608}

X(35749) = reflection of X(i) in X(j) for these (i,j): (2, 35752), (5862, 35697), (35691, 5859), (35750, 2)
X(35749) = anticomplement of X(35751)
X(35749) = {X(2), X(35750)}-harmonic conjugate of X(616)

X(35750) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd INNER-FERMAT-DAO-NHI TO BANKOFF

Barycentrics 2*(11*a^2-7*b^2-7*c^2)*S+(13*a^4-8*(b^2+c^2)*a^2-5*(b^2-c^2)^2)*sqrt(3) : :
X(35750) = 7*X(2)-6*X(13) = 2*X(2)-3*X(616) = 11*X(2)-12*X(618) = 13*X(2)-12*X(5459) = 5*X(2)-6*X(5463) = 19*X(2)-18*X(22489) = 4*X(13)-7*X(616) = 11*X(13)-14*X(618) = 13*X(13)-14*X(5459) = 5*X(13)-7*X(5463) = 12*X(13)-7*X(35749) = 3*X(13)-7*X(35751) = 9*X(13)-7*X(35752) = 11*X(616)-8*X(618) = 13*X(616)-8*X(5459) = 5*X(616)-4*X(5463) = 25*X(616)-16*X(6669) = 19*X(616)-12*X(22489) = 3*X(616)-X(35749) = 3*X(616)-4*X(35751) = 9*X(616)-4*X(35752)

The reciprocal orthologic center of these triangles is X(35748)

X(35750) lies on these lines: {2,13}, {524,35691}, {542,11001}, {543,5862}, {617,15300}, {628,22494}, {631,32907}, {671,33605}, {2482,33617}, {3534,33624}, {4745,9901}, {5066,13103}, {5071,16001}, {5473,15697}, {5858,33625}, {5863,33610}, {6770,8703}, {6777,8596}, {8591,33611}, {9830,35695}, {12816,33613}, {15682,33622}, {15709,20415}, {31694,33412}

X(35750) = reflection of X(i) in X(j) for these (i,j): (2, 35751), (5863, 35696), (35690, 5858), (35749, 2)
X(35750) = anticomplement of X(35752)
X(35750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35751, 616), (616, 35749, 2)

X(35751) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd OUTER-FERMAT-DAO-NHI TO BANKOFF

Barycentrics 2*(5*a^2-4*b^2-4*c^2)*S+(7*a^4-5*(b^2+c^2)*a^2-2*(b^2-c^2)^2)*sqrt(3) : :
X(35751) = 4*X(2)-3*X(13) = X(2)-3*X(616) = 5*X(2)-6*X(618) = 7*X(2)-6*X(5459) = 2*X(2)-3*X(5463) = 13*X(2)-12*X(6669) = 10*X(2)-9*X(22489) = X(13)-4*X(616) = 5*X(13)-8*X(618) = 7*X(13)-8*X(5459) = 13*X(13)-16*X(6669) = 5*X(13)-6*X(22489) = 9*X(13)-4*X(35749) = 3*X(13)+4*X(35750) = 3*X(13)-2*X(35752) = 5*X(616)-2*X(618) = 7*X(616)-2*X(5459) = 13*X(616)-4*X(6669) = 10*X(616)-3*X(22489) = 9*X(616)-X(35749) = 3*X(616)+X(35750) = 6*X(616)-X(35752)

The reciprocal orthologic center of these triangles is X(35748)

X(35751) lies on these lines: {2,13}, {3,22494}, {14,22577}, {18,31694}, {22,13859}, {30,22493}, {394,10657}, {524,35696}, {542,1350}, {543,5858}, {671,32553}, {1384,9112}, {2482,6778}, {3411,11304}, {3845,5617}, {4669,12781}, {5054,32907}, {5055,16001}, {5066,25154}, {5862,35691}, {6770,19708}, {6771,15701}, {8591,33610}, {9761,16809}, {9830,35692}, {9885,33458}, {12793,34582}, {12817,33459}, {13103,19709}, {15534,25235}, {15693,21156}, {15694,20415}, {16962,22495}, {16964,22496}, {22490,31695}, {33607,33619}, {33608,33623}

X(35751) = midpoint of X(i) and X(j) for these {i,j}: {2, 35750}, {5862, 35691}
X(35751) = reflection of X(i) in X(j) for these (i,j): (13, 5463), (671, 32553), (5463, 616), (9114, 9116), (22495, 35304), (22577, 14), (35697, 33459), (35752, 2)
X(35751) = complement of X(35749)
X(35751) = anticomplement of anticomplement of X(36768)
X(35751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35752, 13), (616, 35750, 2), (5463, 22489, 618), (5463, 35752, 2), (22495, 35304, 16962), (33440, 33441, 6669)

X(35752) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 4th OUTER-FERMAT-DAO-NHI TO BANKOFF

Barycentrics 2*(7*a^2-2*b^2-2*c^2)*S+(5*a^4-(b^2+c^2)*a^2-4*(b^2-c^2)^2)*sqrt(3) : :
X(35752) = 2*X(2)-3*X(13) = 5*X(2)-3*X(616) = 7*X(2)-6*X(618) = 5*X(2)-6*X(5459) = 4*X(2)-3*X(5463) = 11*X(2)-12*X(6669) = 8*X(2)-9*X(22489) = 5*X(13)-2*X(616) = 7*X(13)-4*X(618) = 5*X(13)-4*X(5459) = 11*X(13)-8*X(6669) = 4*X(13)-3*X(22489) = 19*X(13)-16*X(35019) = 3*X(13)+2*X(35749) = 9*X(13)-2*X(35750) = 3*X(13)-X(35751)

The reciprocal orthologic center of these triangles is X(35748)

X(35752) lies on these lines: {2,13}, {3,32907}, {4,22496}, {14,31695}, {25,13859}, {30,22495}, {62,31694}, {148,33623}, {381,16001}, {524,35697}, {531,19106}, {532,22493}, {542,1351}, {543,5859}, {671,6777}, {1993,10657}, {2482,33620}, {3845,25154}, {4677,9901}, {4745,12781}, {5054,20415}, {5066,5617}, {5464,15300}, {5469,14137}, {5473,8703}, {5858,12816}, {5863,35690}, {6109,22571}, {6770,11001}, {6771,15693}, {6772,9112}, {8584,22580}, {8591,32552}, {9166,32553}, {9761,16808}, {9830,35693}, {10646,33475}, {11296,16965}, {11486,22490}, {11602,17503}, {12100,21156}, {16267,35304}, {16962,35229}, {21359,31693}, {22238,33415}, {31683,33602}, {33458,35696}

X(35752) = midpoint of X(i) and X(j) for these {i,j}: {2, 35749}, {5863, 35690}
X(35752) = reflection of X(i) in X(j) for these (i,j): (3, 32907), (14, 31695), (381, 16001), (616, 5459), (5463, 13), (8591, 32552), (35696, 33458), (35751, 2)
X(35752) = complement of X(35750)
X(35752) = anticomplement of X(36769)
X(35752) = inner-Napoleon circle-inverse of X(22489)
X(35752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 35751, 5463), (13, 5463, 22489), (13, 35751, 2), (6779, 9762, 5463), (10653, 33517, 13), (16965, 22494, 11296)

X(35753) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO BANKOFF

Barycentrics 8*S^3-(2*sqrt(3)+3)*(SW-sqrt(3)*SW+SA)*S^2-(SB+SC)*(9*SA-sqrt(3)*SW)*S+sqrt(3)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(3)

X(35753) lies on these lines: {6,13103}, {13,372}, {371,35848}, {485,616}, {530,35822}, {531,35698}, {542,35834}, {615,20252}, {618,10576}, {3070,35846}, {5473,6200}, {5478,6565}, {5613,35879}, {5617,6564}, {6268,35794}, {6270,35792}, {6321,35851}, {6396,6771}, {6419,19074}, {6420,16001}, {6454,20415}, {6560,6770}, {7975,35810}, {9901,35774}, {9916,35776}, {9982,35782}, {10062,35809}, {10078,35769}, {11304,33441}, {11705,35762}, {12142,35764}, {12205,35766}, {12337,35772}, {12472,35778}, {12473,35780}, {12781,35788}, {12793,35790}, {12922,35796}, {12932,35798}, {12942,35800}, {12952,35802}, {12990,35804}, {12991,35806}, {13076,35808}, {13105,35816}, {13107,35818}, {13917,35812}, {13982,35814}, {18974,35768}, {19073,35770}, {22773,35784}, {22796,35786}, {25154,35823}

X(35753) = {X(6), X(13103)}-harmonic conjugate of X(35754)

X(35754) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO BANKOFF

Barycentrics 8*S^3-(2*sqrt(3)-3)*(SA+SW+sqrt(3)*SW)*S^2-(9*SA+sqrt(3)*SW)*(SB+SC)*S+sqrt(3)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(3)

X(35754) lies on these lines: {6,13103}, {13,371}, {372,35847}, {486,616}, {530,35823}, {531,35699}, {542,35835}, {590,20252}, {618,10577}, {3071,35849}, {5473,6396}, {5478,6564}, {5613,35878}, {5617,6565}, {6200,6771}, {6268,35793}, {6270,35795}, {6321,35850}, {6419,16001}, {6420,19073}, {6453,20415}, {6561,6770}, {7975,35811}, {9901,35775}, {9916,35777}, {9982,35783}, {10062,35808}, {10078,35768}, {11304,33440}, {11705,35763}, {12142,35765}, {12205,35767}, {12337,35773}, {12472,35781}, {12473,35779}, {12781,35789}, {12793,35791}, {12922,35797}, {12932,35799}, {12942,35801}, {12952,35803}, {12990,35807}, {12991,35805}, {13076,35809}, {13105,35817}, {13107,35819}, {13917,35815}, {13982,35813}, {18974,35769}, {19074,35771}, {22773,35785}, {22796,35787}, {25154,35822}

X(35754) = {X(6), X(13103)}-harmonic conjugate of X(35753)

X(35755) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 1st NEUBERG

Barycentrics 2*SW*S^3-(2*(SB+SC)*SA+1/3*sqrt(3)*(2*sqrt(3)*SW-3*SA)*SW)*S^2+(3*SA^2-SW^2)*SW*S-(sqrt(3)-2)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6582)

X(35755) lies on these lines: {3,6581}, {538,34551}, {1151,23018}, {3107,11291}, {5969,35748}, {11292,25167}, {23000,35739}, {25157,35730}, {25183,35731}, {25191,35732}, {25199,35733}, {31711,35740}, {33482,35738}

X(35756) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 2nd NEUBERG

Barycentrics 6*S^4-12*SW*S^3-sqrt(3)*(-3*SA*SW+2*sqrt(3)*SB*SC+6*SW^2-10*sqrt(3)*SW^2)*S^2-3*(3*SA^2-6*SB*SC-SW^2)*SW*S+15*(sqrt(3)-2)*SB*SC*SW^2 : :

The reciprocal orthologic center of these triangles is X(6298)

X(35756) lies on these lines: {3,6296}, {754,34551}, {1151,23019}, {23001,35739}, {25158,35730}, {25184,35731}, {25192,35732}, {25200,35733}, {31713,35740}, {33484,35738}

X(35757) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 1st TRI-SQUARES-CENTRAL

Barycentrics 219*S^3+(-9+10*sqrt(3))*(3*SA+2*SW+5*sqrt(3)*SW)*S^2+3*(-51+8*sqrt(3))*SB*SC*S+3*(19*sqrt(3)-39)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13705)

X(35757) lies on these lines: {3,13706}, {30,33444}, {530,35744}, {1151,23020}, {15765,35742}, {23002,35739}, {25159,35730}, {25185,35731}, {25193,35732}, {25201,35733}, {31715,35740}, {33486,35738}

X(35758) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 363*S^3-(15+14*sqrt(3))*(3*SA-10*SW+7*sqrt(3)*SW)*S^2+3*(-147+8*sqrt(3))*SB*SC*S-3*(-51+25*sqrt(3))*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(13825)

X(35758) lies on these lines: {3,13826}, {530,35743}, {1151,23021}, {2044,25198}, {23003,35739}, {25160,35730}, {25186,35731}, {25194,35732}, {25202,35733}, {31717,35740}, {33488,35738}

X(35759) = PARALLELOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 1st INNER-FERMAT-DAO-NHI

Barycentrics 12*S^3-(3+sqrt(3))*(-3*SA+2*sqrt(3)*SW)*S^2+12*sqrt(3)*SB*SC*S+3*(sqrt(3)-1)*SB*SC*SW : :

The reciprocal parallelogic center of these triangles is X(35750)

X(35759) lies on these lines: {3,14}, {5,33446}, {115,3390}, {486,9739}, {511,35761}, {512,35760}, {531,34551}, {542,2044}, {619,35738}, {1151,23013}, {2042,33442}, {3364,5471}, {5613,35732}, {6303,14813}, {6783,35740}, {9117,35731}, {16529,35730}, {23004,35739}, {25236,35733}, {35742,35747}

X(35759) = reflection of X(35748) in X(34551)

X(35760) = PARALLELOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 2nd FERMAT-DAO

Barycentrics (SB+SC)*(11*sqrt(3)*S^4-33*(SW+3*R^2)*S^3-11*(sqrt(3)*(-2*SW+SA)*SW+3*(sqrt(3)*SW-6*SA)*R^2)*S^2+(-9+4*sqrt(3))*(-33*R^2+9*SA+4*sqrt(3)*SA)*SA*SW*S-11*(sqrt(3)-2)*(3*R^2+3*SA+2*sqrt(3)*SA-2*sqrt(3)*SW-3*SW)*SA*SW^2) : :

The reciprocal parallelogic center of these triangles is X(25216)

X(35760) lies on these lines: {3,14188}, {511,35748}, {512,35759}, {1151,23022}, {3390,25178}, {23007,35739}, {25217,35730}, {25220,35731}, {25224,35732}, {25228,35733}, {31719,35740}, {33491,35738}, {34551,35761}

X(35760) = reflection of X(35761) in X(34551)

X(35761) = ORTHOLOGIC CENTER OF THESE TRIANGLES: BANKOFF TO 2nd FERMAT-DAO

Barycentrics a^2*((12*(a^2-b^2-c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)*sqrt(3)*a^2*b^2*c^2-24*(b^2-c^2)^2*S^2*(a^2-b^2)*(a^2-c^2))*S+sqrt(3)*(-3*(b^2+c^2)*a^4*b^4*c^4+(b^2+c^2)^2*a^10+(7*b^4+10*b^2*c^2+7*c^4)*b^2*c^2*a^6-2*(b^2+c^2)^3*a^8-2*(a^2+c^2+b^2)*(a^2-b^2-c^2)*(a^4-(b^2+c^2)*a^2+b^4-b^2*c^2+c^4)*sqrt(3)*a^2*b^2*c^2+(b^2+c^2)*(2*b^8+2*c^8-9*(b^4-b^2*c^2+c^4)*b^2*c^2)*a^4-(b^8-c^8)*(b^2-c^2)*b^2*c^2-(b^12+c^12-(5*b^8+5*c^8-(5*b^4-6*b^2*c^2+5*c^4)*b^2*c^2)*b^2*c^2)*a^2)) : :

The reciprocal orthologic center of these triangles is X(25207)

X(35761) lies on these lines: {3,14182}, {511,35759}, {512,35748}, {1151,23017}, {3390,25220}, {22999,35739}, {25151,35730}, {25178,35731}, {25180,35732}, {25182,35733}, {31707,35740}, {33481,35738}, {34551,35760}

X(35761) = reflection of X(35760) in X(34551)

X(35762) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 1st KENMOTU-FREE-VERTICES

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

X(35762) lies on these lines: {1,372}, {2,35788}, {3,35610}, {6,10246}, {8,5420}, {10,35842}, {11,35852}, {145,13935}, {214,35882}, {355,10577}, {371,1385}, {485,3616}, {486,944}, {515,6565}, {517,6396}, {551,35822}, {615,952}, {946,35820}, {999,35784}, {1124,34471}, {1125,10576}, {1152,1482}, {1317,13958}, {1319,35768}, {1335,1388}, {1386,35840}, {1483,13966}, {1587,3622}, {1702,6453}, {2067,21842}, {2646,35808}, {3069,7967}, {3070,5901}, {3071,34773}, {3242,19146}, {3244,13975}, {3295,35772}, {3299,24926}, {3576,6200}, {3649,35854}, {3655,35823}, {5418,19066}, {5603,6560}, {5691,35787}, {5731,6561}, {5790,8252}, {5844,35256}, {5882,13971}, {5886,6564}, {6265,35857}, {6398,10247}, {6410,12702}, {6419,18992}, {6420,7969}, {6450,8148}, {6454,24680}, {6460,10595}, {6480,31662}, {6481,16200}, {6482,9618}, {6484,9584}, {6485,11278}, {6487,11531}, {7984,10820}, {8192,8277}, {9583,30392}, {9955,35786}, {10572,35803}, {11363,35764}, {11364,35766}, {11365,35776}, {11366,35778}, {11367,35780}, {11368,35782}, {11370,35792}, {11371,35794}, {11373,35796}, {11374,35798}, {11375,35800}, {11376,35802}, {11377,35804}, {11378,35806}, {11513,24301}, {11705,35753}, {11706,35850}, {11709,35826}, {11710,35824}, {11711,35878}, {11715,35856}, {11720,12375}, {11722,35880}, {11739,35848}, {11740,35846}, {11831,35790}, {12114,35844}, {12258,35698}, {12259,35836}, {12260,35862}, {12261,35834}, {12262,35864}, {12263,35866}, {12264,35868}, {12265,35828}, {12266,12965}, {12267,35870}, {12268,35830}, {12269,35832}, {12737,35883}, {13607,13936}, {13667,35872}, {13787,35874}, {13883,35812}, {13951,18526}, {13973,35813}, {18481,35821}, {18493,23251}, {18762,28224}, {18991,35770}, {22475,35838}, {22476,35860}, {32238,35876}, {32331,35858}

X(35762) = midpoint of X(6396) and X(35811)
X(35762) = reflection of X(35789) in X(615)
X(35762) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 372, 35641), (1, 35774, 35810), (3, 35642, 35610), (6, 10246, 35763), (372, 35810, 35774), (944, 13959, 486), (1152, 1482, 35611), (1385, 7968, 371), (3576, 35775, 6200), (35774, 35810, 35641), (35813, 35843, 13973)

X(35763) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-AQUILA AND 2nd KENMOTU-FREE-VERTICES

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

X(35763) lies on these lines: {1,371}, {2,35789}, {3,35611}, {6,10246}, {8,5418}, {10,35843}, {11,35853}, {145,9540}, {214,35883}, {355,10576}, {372,1385}, {485,944}, {486,3616}, {515,6564}, {517,6200}, {551,35823}, {590,952}, {946,35821}, {999,35785}, {1124,1388}, {1125,10577}, {1151,1482}, {1317,13901}, {1319,35769}, {1335,34471}, {1386,35841}, {1483,8981}, {1588,3622}, {1703,6454}, {2646,35809}, {3068,7967}, {3070,34773}, {3071,5901}, {3242,19145}, {3244,13912}, {3295,35773}, {3301,24926}, {3576,6396}, {3649,35855}, {3655,35822}, {5289,9678}, {5420,19065}, {5603,6561}, {5691,35786}, {5731,6560}, {5734,9681}, {5790,8253}, {5844,35255}, {5882,8960}, {5886,6565}, {6221,10247}, {6265,35856}, {6409,12702}, {6419,7968}, {6420,18991}, {6449,8148}, {6453,24680}, {6459,10595}, {6480,9616}, {6481,31662}, {6484,11278}, {6486,9582}, {6502,21842}, {7982,9615}, {7984,10819}, {8192,8276}, {8976,18526}, {9557,30116}, {9584,16191}, {9646,10944}, {9661,10950}, {9955,35787}, {10572,35802}, {11363,35765}, {11364,35767}, {11365,35777}, {11366,35781}, {11367,35779}, {11368,35783}, {11370,35795}, {11371,35793}, {11373,35797}, {11374,35799}, {11375,35801}, {11376,35803}, {11377,35807}, {11378,35805}, {11514,24301}, {11705,35754}, {11706,35851}, {11709,35827}, {11710,35825}, {11711,35879}, {11715,35857}, {11720,12376}, {11722,35881}, {11739,35847}, {11740,35849}, {11831,35791}, {12114,35845}, {12258,35699}, {12259,35837}, {12260,35863}, {12261,35835}, {12262,35865}, {12263,35867}, {12264,35869}, {12265,35829}, {12266,12971}, {12267,35871}, {12268,35833}, {12269,35831}, {12737,35882}, {13607,13883}, {13667,35873}, {13787,35875}, {13911,35812}, {13936,35813}, {18481,35820}, {18493,23261}, {18538,28224}, {18992,35771}, {22475,35839}, {22476,35861}, {30144,31453}, {31439,33179}, {32238,35877}, {32331,35859}

X(35763) = midpoint of X(6200) and X(35810)
X(35763) = reflection of X(35788) in X(590)
X(35763) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 371, 35642), (1, 9583, 35775), (1, 35775, 35811), (3, 35641, 35611), (6, 10246, 35762), (371, 35811, 35775), (944, 13902, 485), (1151, 1482, 35610), (1385, 7969, 372), (3576, 35774, 6396), (9583, 35775, 371), (35775, 35811, 35642), (35812, 35842, 13911)

X(35764) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 1st KENMOTU-FREE-VERTICES

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

X(35764) lies on these lines: {3,10961}, {4,371}, {5,11513}, {6,1598}, {24,6396}, {25,372}, {33,35808}, {34,35768}, {235,6565}, {381,10897}, {389,12964}, {427,10576}, {428,35822}, {486,3089}, {546,11265}, {578,10533}, {590,1595}, {615,21841}, {640,1586}, {1112,12375}, {1151,1597}, {1152,3517}, {1351,12238}, {1578,5020}, {1587,6995}, {1593,6200}, {1596,3071}, {1829,35641}, {1843,35840}, {1862,35882}, {1902,35610}, {1907,13884}, {3070,6756}, {3088,5418}, {3091,11417}, {3092,5198}, {3311,18535}, {3518,6454}, {3542,10577}, {3575,35820}, {3843,18457}, {4232,13935}, {5058,33842}, {5090,35788}, {5186,35878}, {5408,15187}, {5411,35770}, {5413,5419}, {5420,6353}, {5446,10665}, {6289,12961}, {6290,12960}, {6453,11403}, {6455,35501}, {6486,13596}, {6560,7487}, {7387,11514}, {7517,10898}, {7529,10963}, {7687,13287}, {7713,35774}, {8280,15809}, {8948,12167}, {9733,12979}, {10605,35865}, {10881,34484}, {10982,19355}, {11363,35762}, {11380,35766}, {11381,35864}, {11383,35772}, {11384,35778}, {11385,35780}, {11386,35782}, {11388,35792}, {11389,35794}, {11390,35796}, {11391,35798}, {11392,35800}, {11393,35802}, {11394,35804}, {11396,35810}, {11398,35809}, {11399,35769}, {11400,35816}, {11401,35818}, {11576,12965}, {11832,35790}, {12131,35824}, {12132,35698}, {12133,35826}, {12134,35836}, {12135,35842}, {12136,35844}, {12137,35852}, {12138,35856}, {12139,35862}, {12140,35834}, {12141,35850}, {12142,35753}, {12143,35866}, {12144,35868}, {12145,35828}, {12146,35870}, {12147,35830}, {12148,35832}, {12424,22660}, {13166,35880}, {13668,35872}, {13788,35874}, {13937,35814}, {16114,35854}, {16198,18538}, {18378,18459}, {18494,23251}, {22479,35784}, {22480,35838}, {22481,35846}, {22482,35848}, {22483,35860}, {32239,35876}, {32332,35858}

X(35764) = barycentric product X(485)*X(15188)
X(35764) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(15188)}} and {{A, B, C, X(485), X(494)}}
X(35764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5412, 371), (4, 10880, 11473), (6, 1598, 35765), (24, 11474, 6396), (25, 3093, 372), (3092, 5410, 6419), (5198, 5410, 3092), (5412, 11473, 10880), (10880, 11473, 371)

X(35765) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ARA AND 2nd KENMOTU-FREE-VERTICES

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

X(35765) lies on these lines: {3,10963}, {4,372}, {5,11514}, {6,1598}, {24,6200}, {25,371}, {33,35809}, {34,35769}, {235,6564}, {381,10898}, {389,12970}, {427,10577}, {428,35823}, {485,3089}, {546,11266}, {578,10534}, {590,21841}, {615,1595}, {639,1585}, {1112,12376}, {1151,3517}, {1152,1597}, {1351,12237}, {1579,5020}, {1588,6995}, {1593,6396}, {1596,3070}, {1829,35642}, {1843,35841}, {1862,35883}, {1902,35611}, {1907,13937}, {3071,6756}, {3088,5420}, {3091,11418}, {3093,5198}, {3312,18535}, {3518,6453}, {3542,10576}, {3575,35821}, {3843,18459}, {4232,9540}, {5062,33842}, {5090,35789}, {5186,35879}, {5409,15188}, {5410,35771}, {5412,5417}, {5418,6353}, {5446,10666}, {6289,12967}, {6290,12966}, {6454,11403}, {6456,35501}, {6487,13596}, {6561,7487}, {7387,11513}, {7517,10897}, {7529,10961}, {7687,13288}, {7713,35775}, {8281,15809}, {8946,12167}, {9732,12978}, {10605,35864}, {10880,34484}, {10982,19356}, {11363,35763}, {11380,35767}, {11381,35865}, {11383,35773}, {11384,35781}, {11385,35779}, {11386,35783}, {11388,35795}, {11389,35793}, {11390,35797}, {11391,35799}, {11392,35801}, {11393,35803}, {11395,35805}, {11396,35811}, {11398,35808}, {11399,35768}, {11400,35817}, {11401,35819}, {11576,12971}, {11832,35791}, {12131,35825}, {12132,35699}, {12133,35827}, {12134,35837}, {12135,35843}, {12136,35845}, {12137,35853}, {12138,35857}, {12139,35863}, {12140,35835}, {12141,35851}, {12142,35754}, {12143,35867}, {12144,35869}, {12145,35829}, {12146,35871}, {12147,35833}, {12148,35831}, {12425,22660}, {13166,35881}, {13668,35873}, {13788,35875}, {13884,35815}, {16114,35855}, {16198,18762}, {18378,18457}, {18494,23261}, {22479,35785}, {22480,35839}, {22481,35849}, {22482,35847}, {22483,35861}, {32239,35877}, {32332,35859}

X(35765) = barycentric product X(486)*X(15187)
X(35765) = intersection, other than A,B,C, of conics {{A, B, C, X(4), X(15187)}} and {{A, B, C, X(486), X(493)}}
X(35765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5413, 372), (4, 10881, 11474), (6, 1598, 35764), (24, 11473, 6200), (25, 3092, 371), (3093, 5411, 6420), (5198, 5411, 3093), (5413, 11474, 10881), (10881, 11474, 372)

X(35766) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 1st KENMOTU-FREE-VERTICES

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

X(35766) lies on these lines: {3,6}, {83,10576}, {98,6565}, {384,35866}, {485,7787}, {3070,32134}, {3552,32471}, {4027,35878}, {5418,10359}, {5420,7793}, {6560,10788}, {6564,10796}, {10104,10577}, {10789,35774}, {10790,35776}, {10791,35788}, {10794,35796}, {10795,35798}, {10797,35800}, {10798,35802}, {10799,35808}, {10800,35810}, {10801,35809}, {10802,35769}, {10803,35816}, {10804,35818}, {11364,35762}, {11380,35764}, {11490,35772}, {11837,35778}, {11838,35780}, {11839,35790}, {11840,35804}, {11841,35806}, {12110,35820}, {12150,35822}, {12176,35824}, {12191,35698}, {12192,35826}, {12193,35836}, {12194,35641}, {12195,35842}, {12196,35844}, {12197,35610}, {12198,35852}, {12199,35856}, {12200,35862}, {12201,35834}, {12202,35864}, {12204,35850}, {12205,35753}, {12206,35868}, {12207,35828}, {12208,12965}, {12209,35870}, {12210,35830}, {12211,35832}, {12375,13193}, {12835,35768}, {13194,35882}, {13195,35880}, {13672,35872}, {13792,35874}, {13885,35812}, {13938,35814}, {14880,35821}, {16115,35854}, {18501,23251}, {18502,35786}, {22520,35784}, {22521,35838}, {22522,35846}, {22523,35848}, {22524,35860}, {32242,35876}, {32335,35858}

X(35766) = {X(6), X(11842)}-harmonic conjugate of X(35767)

X(35767) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th ANTI-BROCARD AND 2nd KENMOTU-FREE-VERTICES

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

X(35767) lies on these lines: {3,6}, {83,10577}, {98,6564}, {384,35867}, {486,7787}, {3071,32134}, {3552,32470}, {4027,35879}, {5418,7793}, {5420,10359}, {6561,10788}, {6565,10796}, {10104,10576}, {10789,35775}, {10790,35777}, {10791,35789}, {10794,35797}, {10795,35799}, {10797,35801}, {10798,35803}, {10799,35809}, {10800,35811}, {10801,35808}, {10802,35768}, {10803,35817}, {10804,35819}, {11364,35763}, {11380,35765}, {11490,35773}, {11837,35781}, {11838,35779}, {11839,35791}, {11840,35807}, {11841,35805}, {12110,35821}, {12150,35823}, {12176,35825}, {12191,35699}, {12192,35827}, {12193,35837}, {12194,35642}, {12195,35843}, {12196,35845}, {12197,35611}, {12198,35853}, {12199,35857}, {12200,35863}, {12201,35835}, {12202,35865}, {12204,35851}, {12205,35754}, {12206,35869}, {12207,35829}, {12208,12971}, {12209,35871}, {12210,35833}, {12211,35831}, {12376,13193}, {12835,35769}, {13194,35883}, {13195,35881}, {13672,35873}, {13792,35875}, {13885,35815}, {13938,35813}, {14880,35820}, {16115,35855}, {18501,23261}, {18502,35787}, {22520,35785}, {22521,35839}, {22522,35849}, {22523,35847}, {22524,35861}, {32242,35877}, {32335,35859}

X(35767) = {X(6), X(11842)}-harmonic conjugate of X(35766)

X(35768) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 1st KENMOTU-FREE-VERTICES

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

X(35768) lies on these lines: {1,371}, {3,3298}, {4,35802}, {5,35801}, {6,101}, {11,6565}, {12,9661}, {34,35764}, {36,5414}, {46,35611}, {55,6200}, {56,372}, {57,35774}, {65,35641}, {388,485}, {390,9541}, {486,3086}, {495,590}, {496,3071}, {497,6561}, {499,10577}, {613,35841}, {614,8855}, {615,15325}, {942,7969}, {1001,9678}, {1056,3068}, {1058,6459}, {1060,11513}, {1124,3304}, {1125,31453}, {1151,3295}, {1317,35882}, {1319,35762}, {1377,25524}, {1378,12513}, {1398,3093}, {1469,35840}, {1478,6564}, {1479,35821}, {1587,3600}, {1588,14986}, {1703,3361}, {1737,35789}, {1870,5412}, {2099,35810}, {2241,12963}, {2362,3338}, {3023,35824}, {3024,35826}, {3027,35878}, {3028,12375}, {3057,35610}, {3058,9660}, {3070,18990}, {3085,5418}, {3297,3311}, {3299,35771}, {3301,5563}, {3303,6453}, {3320,35880}, {3333,18991}, {3487,13902}, {3585,35786}, {3592,31474}, {3913,9679}, {4293,6560}, {5225,22615}, {5252,35788}, {5265,13935}, {5274,23259}, {5298,13958}, {5415,11529}, {5420,7288}, {5434,19030}, {6020,35828}, {6198,11473}, {6221,6767}, {6284,9647}, {6285,35864}, {6424,16781}, {7133,7284}, {7354,35820}, {7968,24928}, {8164,32785}, {8253,31479}, {8960,13904}, {8983,21620}, {9600,31477}, {9616,31393}, {9646,15888}, {9655,23251}, {9669,23261}, {9841,31564}, {10046,35777}, {10047,35783}, {10048,35795}, {10049,35793}, {10069,35825}, {10070,35699}, {10071,35837}, {10072,35823}, {10073,35853}, {10074,35857}, {10075,35863}, {10076,35865}, {10077,35851}, {10078,35754}, {10079,35867}, {10080,35869}, {10081,35827}, {10082,12971}, {10083,35833}, {10084,35831}, {10085,35845}, {10089,35879}, {10090,35883}, {10091,12376}, {10523,35799}, {10573,35843}, {10802,35767}, {10896,35787}, {10897,18447}, {10944,35842}, {10948,35797}, {11237,13898}, {11399,35765}, {11508,35773}, {11509,35772}, {11879,35781}, {11880,35779}, {11913,35791}, {11953,35807}, {11954,35805}, {12688,35844}, {12835,35766}, {12904,35835}, {12962,31471}, {12965,18984}, {13117,35829}, {13129,35871}, {13312,35881}, {13715,35873}, {13838,35875}, {13905,35815}, {13963,35813}, {16153,35855}, {18954,35776}, {18955,35778}, {18956,35780}, {18957,35782}, {18958,35790}, {18959,35792}, {18960,35794}, {18961,35796}, {18962,35798}, {18963,35804}, {18964,35806}, {18966,35814}, {18967,35818}, {18968,35834}, {18969,35698}, {18970,35836}, {18971,35838}, {18972,35846}, {18973,35848}, {18974,35753}, {18975,35850}, {18976,35852}, {18977,35854}, {18978,35860}, {18979,35862}, {18982,35866}, {18983,35868}, {18985,35870}, {18986,35872}, {18987,35874}, {18988,35832}, {18989,35830}, {18995,35770}, {22730,35839}, {22767,35785}, {22885,35849}, {22930,35847}, {22981,35861}, {31409,31463}, {31439,31792}, {32243,35876}, {32308,35877}, {32336,35858}, {32404,35859}

X(35768) = intersection, other than A,B,C, of conics {{A, B, C, X(89), X(35769)}} and {{A, B, C, X(106), X(16232)}}
X(35768) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 371, 35808), (1, 2067, 371), (1, 16232, 35642), (3, 3298, 35809), (6, 999, 35769), (12, 9661, 10576), (36, 5414, 6396), (56, 1335, 372), (371, 35819, 35642), (388, 485, 35800), (496, 3071, 35803), (1124, 18996, 6419), (3301, 5563, 6502), (3301, 6502, 6420), (3304, 18996, 1124), (3311, 7373, 3297), (9646, 18965, 35812), (13904, 31472, 8960), (15888, 18965, 9646)

X(35769) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CIRCUMPERP-TANGENTIAL AND 2nd KENMOTU-FREE-VERTICES

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

X(35769) lies on these lines: {1,372}, {3,3297}, {4,35803}, {5,35800}, {6,101}, {11,6564}, {12,10577}, {34,35765}, {36,2066}, {46,35610}, {55,6396}, {56,371}, {57,35775}, {65,35642}, {388,486}, {485,3086}, {495,615}, {496,3070}, {497,6560}, {499,10576}, {590,15325}, {613,35840}, {614,8854}, {942,7968}, {956,31473}, {1056,3069}, {1058,6460}, {1060,11514}, {1151,31474}, {1152,3295}, {1317,35883}, {1319,35763}, {1335,3304}, {1377,12513}, {1378,25524}, {1398,3092}, {1469,35841}, {1478,6565}, {1479,35820}, {1587,14986}, {1588,3600}, {1702,3361}, {1737,35788}, {1870,5413}, {2067,3299}, {2099,35811}, {2241,12968}, {3023,35825}, {3024,35827}, {3027,35879}, {3028,12376}, {3057,35611}, {3071,18990}, {3085,5420}, {3298,3312}, {3301,35770}, {3303,6454}, {3320,35881}, {3333,18992}, {3338,16232}, {3487,13959}, {3585,35787}, {4293,6561}, {5225,22644}, {5252,35789}, {5265,9540}, {5274,23249}, {5298,13901}, {5416,11529}, {5418,7288}, {5433,9646}, {5434,19029}, {6020,35829}, {6198,11474}, {6285,35865}, {6398,6767}, {6423,16781}, {7354,35821}, {7969,24928}, {8164,32786}, {8252,31479}, {8666,31453}, {8953,30556}, {8960,9661}, {9583,13462}, {9655,23261}, {9660,15326}, {9669,23251}, {9678,11194}, {9841,31563}, {10046,35776}, {10047,35782}, {10048,35792}, {10049,35794}, {10069,35824}, {10070,35698}, {10071,35836}, {10072,35822}, {10073,35852}, {10074,35856}, {10075,35862}, {10076,35864}, {10077,35850}, {10078,35753}, {10079,35866}, {10080,35868}, {10081,35826}, {10082,12965}, {10083,35830}, {10084,35832}, {10085,35844}, {10089,35878}, {10090,35882}, {10091,12375}, {10523,35798}, {10573,35842}, {10802,35766}, {10896,35786}, {10898,18447}, {10944,35843}, {10948,35796}, {11237,13955}, {11399,35764}, {11508,35772}, {11509,35773}, {11879,35778}, {11880,35780}, {11913,35790}, {11953,35804}, {11954,35806}, {12688,35845}, {12835,35767}, {12904,35834}, {12971,18984}, {13117,35828}, {13129,35870}, {13312,35880}, {13715,35872}, {13838,35874}, {13905,31475}, {13963,35814}, {13971,21620}, {15803,31432}, {15888,18966}, {16153,35854}, {18954,35777}, {18955,35781}, {18956,35779}, {18957,35783}, {18958,35791}, {18959,35795}, {18960,35793}, {18961,35797}, {18962,35799}, {18963,35807}, {18964,35805}, {18965,35815}, {18967,35819}, {18968,35835}, {18969,35699}, {18970,35837}, {18971,35839}, {18972,35849}, {18973,35847}, {18974,35754}, {18975,35851}, {18976,35853}, {18977,35855}, {18978,35861}, {18979,35863}, {18982,35867}, {18983,35869}, {18985,35871}, {18986,35873}, {18987,35875}, {18988,35831}, {18989,35833}, {18996,35771}, {22730,35838}, {22767,35784}, {22885,35846}, {22930,35848}, {22981,35860}, {31413,34625}, {32243,35877}, {32308,35876}, {32336,35859}, {32404,35858}

X(35769) = intersection, other than A,B,C, of conics {{A, B, C, X(89), X(35768)}} and {{A, B, C, X(106), X(2362)}}
X(35769) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 372, 35809), (1, 2362, 35641), (1, 6502, 372), (3, 3297, 35808), (6, 999, 35768), (36, 2066, 6200), (56, 1124, 371), (372, 35818, 35641), (388, 486, 35801), (496, 3070, 35802), (499, 31472, 10576), (1335, 18995, 6420), (2067, 3299, 6419), (3086, 31408, 485), (3299, 5563, 2067), (3304, 18995, 1335), (3312, 7373, 3298), (9661, 19028, 8960)

X(35770) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INNER-GREBE AND 1st KENMOTU-FREE-VERTICES

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

X(35770) lies on these lines: {2,35814}, {3,6}, {4,12819}, {81,21555}, {140,35815}, {485,3591}, {486,3545}, {487,5032}, {547,7583}, {590,16239}, {615,8960}, {940,21551}, {1328,23253}, {1587,3832}, {1588,3543}, {1703,35610}, {3068,3533}, {3069,5067}, {3070,3845}, {3071,3853}, {3155,34565}, {3299,35809}, {3301,35769}, {3850,6564}, {4383,21544}, {5059,6561}, {5408,34545}, {5411,35764}, {5413,34484}, {5418,15702}, {5420,7585}, {6413,34567}, {6460,11001}, {6560,7582}, {7375,26340}, {7736,19103}, {7838,11313}, {7968,33179}, {8253,13961}, {8976,13847}, {9540,15708}, {11278,35642}, {11474,13596}, {11531,19003}, {11539,13966}, {12375,19110}, {12965,19095}, {13785,35786}, {13846,15723}, {13925,32790}, {13936,35788}, {16200,18992}, {16477,31546}, {16854,31473}, {18510,23251}, {18991,35762}, {18995,35768}, {18999,35772}, {19004,30392}, {19005,35776}, {19007,35778}, {19009,35780}, {19013,35784}, {19017,35790}, {19023,35796}, {19025,35798}, {19027,35800}, {19029,35802}, {19031,35804}, {19033,35806}, {19037,35808}, {19047,35816}, {19049,35818}, {19051,35834}, {19055,35824}, {19057,35698}, {19059,35826}, {19061,35836}, {19063,35838}, {19065,35842}, {19067,35844}, {19069,35846}, {19071,35848}, {19073,35753}, {19075,35850}, {19077,35852}, {19079,35854}, {19081,35856}, {19083,35860}, {19085,35862}, {19087,35864}, {19089,35866}, {19091,35868}, {19093,35828}, {19097,35870}, {19099,35872}, {19101,35874}, {19102,35832}, {19104,35830}, {19108,35878}, {19112,35882}, {19114,35880}, {21552,32911}, {22644,23273}, {31454,35256}, {32252,35876}, {32342,35858}

X(35770) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(35771)}} and {{A, B, C, X(3), X(12819)}}
X(35770) = Brocard circle-inverse of X(35771)
X(35770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6431, 371), (6, 3312, 371), (6, 6420, 372), (6, 6432, 3), (371, 3312, 372), (1151, 6434, 3), (1152, 6417, 371), (1152, 6429, 3), (3311, 6409, 371), (3312, 6501, 6), (3592, 6407, 371), (3594, 6396, 372), (6395, 6454, 372), (6396, 6486, 3), (6418, 6428, 6), (6430, 6437, 3), (6432, 35771, 372), (6446, 10137, 3), (6449, 6470, 371), (6480, 6485, 3), (6484, 6487, 3)

X(35771) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-OUTER-GREBE AND 2nd KENMOTU-FREE-VERTICES

Barycentrics a^2*(a^2-b^2-c^2-10*S) : :

X(35771) lies on these lines: {2,35815}, {3,6}, {4,12818}, {81,21552}, {140,35814}, {485,3545}, {486,3590}, {488,5032}, {547,7584}, {590,19116}, {615,16239}, {940,21544}, {1327,23263}, {1587,3543}, {1588,3832}, {1702,35611}, {3068,5067}, {3069,3533}, {3070,3853}, {3071,3845}, {3156,34565}, {3299,35768}, {3301,35808}, {3850,6565}, {4383,21551}, {5059,6560}, {5409,34545}, {5410,35765}, {5412,34484}, {5418,7586}, {5420,15702}, {6414,34567}, {6459,11001}, {6561,7581}, {7376,26339}, {7736,19104}, {7838,11314}, {7969,33179}, {8225,16477}, {8252,13903}, {8253,31487}, {8981,11539}, {11278,35641}, {11473,13596}, {11531,19004}, {12376,19111}, {12971,19096}, {13665,35787}, {13846,13951}, {13847,15723}, {13883,35789}, {13935,15708}, {13966,31454}, {13993,32789}, {16200,18991}, {16864,31473}, {18512,23261}, {18992,35763}, {18996,35769}, {19000,35773}, {19003,30392}, {19006,35777}, {19008,35781}, {19010,35779}, {19014,35785}, {19018,35791}, {19024,35797}, {19026,35799}, {19028,35801}, {19030,35803}, {19032,35807}, {19034,35805}, {19038,35809}, {19048,35817}, {19050,35819}, {19052,35835}, {19056,35825}, {19058,35699}, {19060,35827}, {19062,35837}, {19064,35839}, {19066,35843}, {19068,35845}, {19070,35847}, {19072,35849}, {19074,35754}, {19076,35851}, {19078,35853}, {19080,35855}, {19082,35857}, {19084,35861}, {19086,35863}, {19088,35865}, {19090,35867}, {19092,35869}, {19094,35829}, {19098,35871}, {19100,35875}, {19103,35831}, {19105,35833}, {19109,35879}, {19113,35883}, {19115,35881}, {21555,32911}, {22541,35873}, {22615,23267}, {23275,31414}, {32253,35877}, {32343,35859}

X(35771) = intersection, other than A,B,C, of conics {{A, B, C, X(2), X(35770)}} and {{A, B, C, X(3), X(12818)}}
X(35771) = Brocard circle-inverse of X(35770)
X(35771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6432, 372), (6, 3311, 372), (6, 6419, 371), (6, 6431, 3), (372, 3311, 371), (1151, 6418, 372), (1151, 6430, 3), (1152, 6433, 3), (3311, 6500, 6), (3312, 6410, 372), (3592, 6200, 371), (3594, 6408, 372), (6199, 6453, 371), (6200, 6487, 3), (6417, 6427, 6), (6429, 6438, 3), (6431, 35770, 371), (6445, 10138, 3), (6450, 6471, 372), (6481, 6484, 3), (6485, 6486, 3)

X(35772) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 1st KENMOTU-FREE-VERTICES

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

X(35772) lies on these lines: {3,35611}, {6,11849}, {35,35774}, {55,372}, {56,35810}, {100,485}, {197,35776}, {371,11248}, {1151,35000}, {1335,14882}, {1376,10576}, {1482,35785}, {1621,5420}, {2362,32760}, {3070,32141}, {3295,35762}, {3560,35789}, {3913,35842}, {4421,35822}, {5687,35788}, {6200,10310}, {6396,10267}, {6419,19000}, {6560,11491}, {6564,11499}, {6565,11496}, {7969,26285}, {10306,35610}, {10679,35642}, {11383,35764}, {11490,35766}, {11492,35778}, {11493,35780}, {11494,35782}, {11497,35792}, {11498,35794}, {11500,35798}, {11501,35800}, {11502,35802}, {11503,35804}, {11504,35806}, {11507,35809}, {11508,35769}, {11509,35768}, {11510,35818}, {11848,35790}, {12178,35824}, {12326,35698}, {12327,35826}, {12328,35836}, {12329,35840}, {12330,35844}, {12331,35852}, {12332,35856}, {12333,35862}, {12334,35834}, {12335,35864}, {12336,35850}, {12337,35753}, {12338,35866}, {12339,35868}, {12340,35828}, {12341,12965}, {12342,35870}, {12343,35830}, {12344,35832}, {12375,13204}, {13173,35878}, {13205,13909}, {13206,35880}, {13675,35872}, {13795,35874}, {13887,35812}, {13940,35814}, {16117,35854}, {18491,35786}, {18524,23251}, {18999,35770}, {22556,35838}, {22557,35846}, {22558,35848}, {22559,35860}, {22758,35843}, {32256,35876}, {32347,35858}

X(35772) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35641, 35784), (6, 11849, 35773)

X(35773) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-MANDART-INCIRCLE AND 2nd KENMOTU-FREE-VERTICES

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

X(35773) lies on these lines: {3,35610}, {6,11849}, {35,35775}, {55,371}, {56,35811}, {100,486}, {197,35777}, {372,11248}, {1124,14882}, {1152,35000}, {1376,10577}, {1482,35784}, {1621,5418}, {3071,32141}, {3295,35763}, {3560,35788}, {3913,35843}, {4421,35823}, {5687,35789}, {6200,10267}, {6396,10310}, {6420,18999}, {6561,11491}, {6564,11496}, {6565,11499}, {7968,26285}, {9677,20986}, {10306,35611}, {10679,35641}, {11383,35765}, {11490,35767}, {11492,35781}, {11493,35779}, {11494,35783}, {11497,35795}, {11498,35793}, {11500,35799}, {11501,35801}, {11502,35803}, {11503,35807}, {11504,35805}, {11507,35808}, {11508,35768}, {11509,35769}, {11510,35819}, {11848,35791}, {12178,35825}, {12326,35699}, {12327,35827}, {12328,35837}, {12329,35841}, {12330,35845}, {12331,35853}, {12332,35857}, {12333,35863}, {12334,35835}, {12335,35865}, {12336,35851}, {12337,35754}, {12338,35867}, {12339,35869}, {12340,35829}, {12341,12971}, {12342,35871}, {12343,35833}, {12344,35831}, {12376,13204}, {13173,35879}, {13205,35883}, {13206,35881}, {13675,35873}, {13795,35875}, {13887,35815}, {13940,35813}, {16117,35855}, {16232,32760}, {18491,35787}, {18524,23261}, {19000,35771}, {22556,35839}, {22557,35849}, {22558,35847}, {22559,35861}, {22758,35842}, {32256,35877}, {32347,35859}

X(35773) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35642, 35785), (6, 11849, 35772)

X(35774) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 1st KENMOTU-FREE-VERTICES

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

X(35774) lies on these lines: {1,372}, {3,7969}, {4,19065}, {5,13973}, {6,517}, {8,1587}, {10,485}, {35,35772}, {36,35784}, {40,371}, {46,2067}, {57,35768}, {65,1335}, {80,30432}, {165,6200}, {355,3070}, {392,31473}, {486,946}, {515,6560}, {516,6561}, {590,26446}, {615,5886}, {631,13902}, {942,3298}, {944,6460}, {950,31561}, {960,1377}, {962,1588}, {1124,3057}, {1125,5420}, {1151,3579}, {1152,1385}, {1320,19081}, {1378,5836}, {1386,19146}, {1387,13977}, {1482,3312}, {1697,35808}, {1698,10576}, {1699,6565}, {1702,6419}, {1768,35856}, {1829,3093}, {1902,3092}, {2066,5119}, {2098,18995}, {2099,19037}, {2948,12375}, {3068,5657}, {3069,5603}, {3071,12699}, {3099,35782}, {3103,12782}, {3297,9957}, {3299,5697}, {3301,5903}, {3311,12702}, {3428,19000}, {3576,6396}, {3592,31439}, {3594,24680}, {3616,13935}, {3632,35842}, {3654,32787}, {3656,32788}, {3679,35788}, {3751,35840}, {3878,30556}, {3911,5393}, {4640,9678}, {4663,9974}, {5418,6684}, {5541,35882}, {5587,6564}, {5588,35794}, {5589,35792}, {5604,8416}, {5690,7583}, {5691,35820}, {5790,13665}, {5818,31412}, {5881,35843}, {5901,13966}, {6264,35857}, {6326,35883}, {6361,6459}, {6395,10247}, {6398,10246}, {6409,31663}, {6410,13624}, {6412,17502}, {6418,8148}, {6420,7982}, {6426,15178}, {6432,11278}, {6434,31662}, {6453,9582}, {6481,30392}, {7581,12245}, {7584,22791}, {7713,35764}, {7970,19108}, {7978,19110}, {7983,19055}, {7984,19059}, {7992,35844}, {8185,35776}, {8186,35778}, {8187,35780}, {8188,35804}, {8189,35806}, {8227,10577}, {8252,11230}, {8253,11231}, {8277,11365}, {8960,13893}, {9541,9778}, {9578,35800}, {9581,35802}, {9588,13888}, {9600,31443}, {9612,35801}, {9614,35803}, {9615,35242}, {9624,35813}, {9661,24914}, {9812,23259}, {9860,35824}, {9875,35698}, {9896,35836}, {9897,35852}, {9898,35862}, {9899,35864}, {9900,35850}, {9901,35753}, {9902,35866}, {9903,35868}, {9904,35826}, {9905,12965}, {9906,35830}, {9907,35832}, {9928,10665}, {10039,31472}, {10310,19014}, {10595,13959}, {10698,19112}, {10705,19093}, {10789,35766}, {10820,11720}, {10826,35796}, {10827,35798}, {10864,35845}, {11362,13883}, {11480,34556}, {11481,34557}, {11531,19003}, {11723,13990}, {11724,13989}, {11725,13967}, {11729,13991}, {11735,13969}, {11822,19010}, {11823,19008}, {11852,35790}, {12407,35834}, {12408,35828}, {12409,35870}, {12514,31453}, {12650,19067}, {12962,31437}, {13099,19114}, {13174,35878}, {13221,35880}, {13464,13971}, {13679,35872}, {13799,35874}, {13942,35814}, {13951,18493}, {13958,15950}, {13976,16174}, {14217,19077}, {16118,35854}, {16200,35811}, {18480,23251}, {18492,35786}, {19047,23340}, {19049,24474}, {22644,31673}, {22650,35838}, {22651,35846}, {22652,35848}, {22653,35860}, {22793,23261}, {31162,35823}, {31398,31463}, {32261,35876}, {32356,35858}, {33535,35827}

X(35774) = reflection of X(35775) in X(6)
X(35774) = intersection, other than A,B,C, of conics {{A, B, C, X(80), X(6213)}} and {{A, B, C, X(485), X(998)}}
X(35774) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1703, 372), (40, 18991, 371), (165, 9583, 6200), (371, 35611, 40), (372, 35641, 1), (372, 35810, 35762), (946, 13936, 486), (1125, 13975, 5420), (1482, 3312, 7968), (1702, 7991, 35610), (1702, 19004, 6419), (3301, 5903, 16232), (5690, 7583, 13911), (6396, 35763, 3576), (6419, 35610, 1702), (6564, 35789, 5587), (6684, 8983, 5418), (7991, 19004, 1702), (35641, 35762, 35810), (35762, 35810, 1)

X(35775) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 2nd KENMOTU-FREE-VERTICES

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

X(35775) lies on these lines: {1,371}, {3,7968}, {4,19066}, {5,13911}, {6,517}, {8,1588}, {10,486}, {35,35773}, {36,35785}, {40,372}, {46,6502}, {57,35769}, {65,1124}, {80,7133}, {165,6396}, {355,3071}, {485,946}, {515,6561}, {516,6560}, {590,5886}, {615,26446}, {631,13959}, {942,3297}, {944,6459}, {950,31562}, {960,1378}, {962,1587}, {1125,5418}, {1151,1385}, {1152,3579}, {1320,19082}, {1335,3057}, {1377,5836}, {1386,19145}, {1387,13913}, {1482,3311}, {1697,35809}, {1698,10577}, {1699,6564}, {1703,6420}, {1768,35857}, {1829,3092}, {1902,3093}, {2098,18996}, {2099,19038}, {2362,3299}, {2948,12376}, {3068,5603}, {3069,5657}, {3070,12699}, {3099,35783}, {3102,12782}, {3298,9957}, {3301,5697}, {3312,12702}, {3428,18999}, {3576,6200}, {3592,24680}, {3616,9540}, {3632,35843}, {3654,32788}, {3656,32787}, {3679,35789}, {3751,35841}, {3753,31473}, {3878,30557}, {3911,5405}, {4295,31408}, {4663,9975}, {5119,5414}, {5420,6684}, {5541,35883}, {5587,6565}, {5588,35793}, {5589,35795}, {5605,8396}, {5690,7584}, {5691,35821}, {5731,9541}, {5790,13785}, {5881,35842}, {5901,8981}, {6199,10247}, {6221,10246}, {6264,35856}, {6326,35882}, {6361,6460}, {6409,13624}, {6410,31663}, {6411,17502}, {6417,8148}, {6419,7982}, {6425,15178}, {6431,11278}, {6433,31662}, {6453,9615}, {6480,30392}, {7582,12245}, {7583,22791}, {7713,35765}, {7970,19109}, {7978,19111}, {7983,19056}, {7984,19060}, {7987,9582}, {7992,35845}, {8185,35777}, {8186,35781}, {8187,35779}, {8188,35807}, {8189,35805}, {8227,10576}, {8252,11231}, {8253,11230}, {8276,11365}, {8960,11522}, {8976,18493}, {8980,11725}, {8983,13464}, {8988,16174}, {8994,11735}, {8997,11724}, {8998,11723}, {9578,35801}, {9581,35803}, {9588,13942}, {9592,31427}, {9612,35800}, {9614,35802}, {9619,31437}, {9623,31438}, {9624,31440}, {9646,11375}, {9661,11376}, {9812,23249}, {9860,35825}, {9875,35699}, {9896,35837}, {9897,35853}, {9898,35863}, {9899,35865}, {9900,35851}, {9901,35754}, {9902,35867}, {9903,35869}, {9904,35827}, {9905,12971}, {9906,35833}, {9907,35831}, {9928,10666}, {10310,19013}, {10595,13902}, {10698,19113}, {10705,19094}, {10789,35767}, {10819,11720}, {10826,35797}, {10827,35799}, {10864,35844}, {11362,13936}, {11480,34557}, {11481,34556}, {11531,19004}, {11729,13922}, {11822,19009}, {11823,19007}, {11852,35791}, {12047,31472}, {12407,35835}, {12408,35829}, {12409,35871}, {12650,19068}, {13099,19115}, {13174,35879}, {13221,35881}, {13679,35873}, {13799,35875}, {13888,35815}, {13901,15950}, {14217,19078}, {16118,35855}, {16200,35810}, {18480,23261}, {18492,35787}, {19048,23340}, {19050,24474}, {22615,31673}, {22650,35839}, {22651,35849}, {22652,35847}, {22653,35861}, {22793,23251}, {31162,35822}, {32261,35877}, {32356,35859}, {33535,35826}

X(35775) = reflection of X(35774) in X(6)
X(35775) = intersection, other than A,B,C, of conics {{A, B, C, X(80), X(6212)}} and {{A, B, C, X(486), X(998)}}
X(35775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1702, 371), (1, 9583, 35763), (1, 31432, 35808), (40, 18992, 372), (371, 35642, 1), (371, 35763, 9583), (371, 35811, 35763), (372, 35610, 40), (946, 13883, 485), (1385, 31439, 1151), (1482, 3311, 7969), (1703, 7991, 35611), (1703, 19003, 6420), (3299, 5903, 2362), (3576, 9616, 6200), (6200, 35762, 3576), (6420, 35611, 1703), (7991, 19003, 1703), (35642, 35763, 35811), (35763, 35811, 1)

X(35776) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 1st KENMOTU-FREE-VERTICES

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

X(35776) lies on these lines: {3,3366}, {6,7517}, {20,9682}, {22,485}, {23,1587}, {24,6560}, {25,372}, {26,3070}, {159,35840}, {197,35772}, {371,7387}, {378,22644}, {486,10594}, {615,13861}, {1124,9673}, {1151,12083}, {1152,7506}, {1335,9658}, {1598,6565}, {1995,5420}, {2937,13665}, {3068,9683}, {3069,34484}, {3071,7530}, {3311,5899}, {3312,18378}, {3518,6460}, {5198,35787}, {5418,10323}, {5594,35794}, {5595,35792}, {6200,8276}, {6396,6642}, {6419,19006}, {7488,23249}, {7512,31412}, {7525,18538}, {7529,10577}, {7545,13951}, {7556,23269}, {7583,17714}, {8185,35774}, {8190,35778}, {8191,35780}, {8192,35810}, {8193,35788}, {8194,35804}, {8976,13564}, {9541,12087}, {9798,35641}, {9818,35786}, {9861,35824}, {9876,35698}, {9908,35836}, {9909,35822}, {9910,35844}, {9911,35610}, {9912,35852}, {9913,35856}, {9914,35864}, {9915,35850}, {9916,35753}, {9917,35866}, {9918,35868}, {9919,35826}, {9920,12965}, {9921,35830}, {9922,35832}, {10037,35809}, {10046,35769}, {10665,32048}, {10790,35766}, {10828,35782}, {10829,35796}, {10830,35798}, {10831,35800}, {10832,35802}, {10833,35808}, {10834,35816}, {10835,35818}, {11365,35762}, {11641,35880}, {11853,35790}, {12310,12375}, {12410,35842}, {12411,35862}, {12412,35834}, {12413,35828}, {12414,35870}, {13175,35878}, {13222,35882}, {13595,13935}, {13680,35872}, {13800,35874}, {13889,35812}, {13943,35814}, {16119,35854}, {18534,35821}, {18954,35768}, {19005,35770}, {22654,35784}, {22655,35838}, {22656,35846}, {22657,35848}, {22658,35860}, {32262,35876}, {32357,35858}

X(35776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7517, 35777), (3068, 12088, 9683), (8276, 11414, 6200)

X(35777) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2nd KENMOTU-FREE-VERTICES

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

X(35777) lies on these lines: {2,9683}, {3,3367}, {6,7517}, {22,486}, {23,1588}, {24,6561}, {25,371}, {26,3071}, {159,35841}, {197,35773}, {372,7387}, {378,22615}, {485,10594}, {590,13861}, {1124,9658}, {1151,7506}, {1152,12083}, {1335,9673}, {1598,6564}, {1995,5418}, {2937,13785}, {3068,34484}, {3069,12088}, {3070,7530}, {3311,18378}, {3312,5899}, {3518,6459}, {5198,35786}, {5420,10323}, {5594,35793}, {5595,35795}, {6200,6642}, {6396,8277}, {6420,19005}, {7488,23259}, {7525,18762}, {7529,10576}, {7545,8976}, {7556,23275}, {7584,17714}, {8185,35775}, {8190,35781}, {8191,35779}, {8192,35811}, {8193,35789}, {8195,35805}, {9540,13595}, {9680,9695}, {9798,35642}, {9818,35787}, {9861,35825}, {9876,35699}, {9908,35837}, {9909,35823}, {9910,35845}, {9911,35611}, {9912,35853}, {9913,35857}, {9914,35865}, {9915,35851}, {9916,35754}, {9917,35867}, {9918,35869}, {9919,35827}, {9920,12971}, {9921,35833}, {9922,35831}, {10037,35808}, {10046,35768}, {10666,32048}, {10790,35767}, {10828,35783}, {10829,35797}, {10830,35799}, {10831,35801}, {10832,35803}, {10833,35809}, {10834,35817}, {10835,35819}, {11365,35763}, {11641,35881}, {11853,35791}, {12310,12376}, {12410,35843}, {12411,35863}, {12412,35835}, {12413,35829}, {12414,35871}, {13175,35879}, {13222,35883}, {13564,13951}, {13680,35873}, {13800,35875}, {13889,35815}, {13943,35813}, {16119,35855}, {18534,35820}, {18954,35769}, {19006,35771}, {22654,35785}, {22655,35839}, {22656,35849}, {22657,35847}, {22658,35861}, {32262,35877}, {32357,35859}

X(35777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 7517, 35776), (3518, 6459, 9682), (8277, 11414, 6396)

X(35778) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 1st KENMOTU-FREE-VERTICES

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

X(35778) lies on these lines: {6,11875}, {55,35641}, {371,11252}, {372,5597}, {485,5601}, {1482,35779}, {3070,32146}, {5598,35810}, {5599,10576}, {6200,11822}, {6419,19008}, {6560,11843}, {6564,8200}, {6565,8196}, {8186,35774}, {8190,35776}, {8197,35788}, {8198,35792}, {8199,35794}, {8201,35804}, {8202,35806}, {8207,35843}, {9834,35820}, {11207,35822}, {11366,35762}, {11384,35764}, {11492,35772}, {11493,35784}, {11837,35766}, {11861,35782}, {11863,35790}, {11865,35796}, {11867,35798}, {11869,35800}, {11871,35802}, {11873,35808}, {11877,35809}, {11879,35769}, {11881,35816}, {11883,35818}, {12179,35824}, {12345,35698}, {12365,35826}, {12375,13208}, {12415,35836}, {12452,35840}, {12454,35842}, {12456,35844}, {12458,35610}, {12460,35852}, {12462,35856}, {12464,35862}, {12466,35834}, {12468,35864}, {12470,35850}, {12472,35753}, {12474,35866}, {12476,35868}, {12478,35828}, {12480,12965}, {12482,35870}, {12484,35830}, {12486,35832}, {13176,35878}, {13228,35882}, {13229,35880}, {13682,35872}, {13802,35874}, {13890,35812}, {13944,35814}, {16121,35854}, {18495,35786}, {18955,35768}, {19007,35770}, {22668,35838}, {22669,35846}, {22670,35848}, {22671,35860}, {32265,35876}, {32360,35858}

X(35778) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11875, 35781), (55, 35641, 35780)

X(35779) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND 2nd KENMOTU-FREE-VERTICES

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

X(35779) lies on these lines: {6,11876}, {55,35642}, {371,5598}, {372,11253}, {486,5602}, {1482,35778}, {3071,32147}, {5597,35811}, {5600,10577}, {6396,11823}, {6420,19009}, {6561,11844}, {6564,8203}, {6565,8207}, {8187,35775}, {8191,35777}, {8200,35842}, {8204,35789}, {8205,35795}, {8206,35793}, {8208,35807}, {8209,35805}, {9835,35821}, {11208,35823}, {11367,35763}, {11385,35765}, {11492,35785}, {11493,35773}, {11838,35767}, {11862,35783}, {11864,35791}, {11866,35797}, {11868,35799}, {11870,35801}, {11872,35803}, {11874,35809}, {11878,35808}, {11880,35768}, {11882,35817}, {11884,35819}, {12180,35825}, {12346,35699}, {12366,35827}, {12376,13209}, {12416,35837}, {12453,35841}, {12455,35843}, {12457,35845}, {12459,35611}, {12461,35853}, {12463,35857}, {12465,35863}, {12467,35835}, {12469,35865}, {12471,35851}, {12473,35754}, {12475,35867}, {12477,35869}, {12479,35829}, {12481,12971}, {12483,35871}, {12485,35833}, {12487,35831}, {13177,35879}, {13230,35883}, {13231,35881}, {13683,35873}, {13803,35875}, {13891,35815}, {13945,35813}, {16122,35855}, {18497,35787}, {18956,35769}, {19010,35771}, {22672,35839}, {22673,35849}, {22674,35847}, {22675,35861}, {32266,35877}, {32361,35859}

X(35779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11876, 35780), (55, 35642, 35781)

X(35780) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 1st KENMOTU-FREE-VERTICES

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

X(35780) lies on these lines: {6,11876}, {55,35641}, {371,11253}, {372,5598}, {485,5602}, {1482,35781}, {3070,32147}, {5597,35810}, {5600,10576}, {6200,11823}, {6419,19010}, {6560,11844}, {6564,8207}, {6565,8203}, {8187,35774}, {8191,35776}, {8200,35843}, {8204,35788}, {8205,35792}, {8206,35794}, {8208,35804}, {8209,35806}, {9835,35820}, {11208,35822}, {11367,35762}, {11385,35764}, {11492,35784}, {11493,35772}, {11838,35766}, {11862,35782}, {11864,35790}, {11866,35796}, {11868,35798}, {11870,35800}, {11872,35802}, {11874,35808}, {11878,35809}, {11880,35769}, {11882,35816}, {11884,35818}, {12180,35824}, {12346,35698}, {12366,35826}, {12375,13209}, {12416,35836}, {12453,35840}, {12455,35842}, {12457,35844}, {12459,35610}, {12461,35852}, {12463,35856}, {12465,35862}, {12467,35834}, {12469,35864}, {12471,35850}, {12473,35753}, {12475,35866}, {12477,35868}, {12479,35828}, {12481,12965}, {12483,35870}, {12485,35830}, {12487,35832}, {13177,35878}, {13230,35882}, {13231,35880}, {13683,35872}, {13803,35874}, {13891,35812}, {13945,35814}, {16122,35854}, {18497,35786}, {18956,35768}, {19009,35770}, {22672,35838}, {22673,35846}, {22674,35848}, {22675,35860}, {32266,35876}, {32361,35858}

X(35780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11876, 35779), (55, 35641, 35778)

X(35781) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND 2nd KENMOTU-FREE-VERTICES

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

X(35781) lies on these lines: {6,11875}, {55,35642}, {371,5597}, {372,11252}, {486,5601}, {1482,35780}, {3071,32146}, {5598,35811}, {5599,10577}, {6396,11822}, {6420,19007}, {6561,11843}, {6564,8196}, {6565,8200}, {8186,35775}, {8190,35777}, {8197,35789}, {8198,35795}, {8199,35793}, {8201,35807}, {8202,35805}, {8207,35842}, {9834,35821}, {11207,35823}, {11366,35763}, {11384,35765}, {11492,35773}, {11493,35785}, {11837,35767}, {11861,35783}, {11863,35791}, {11865,35797}, {11867,35799}, {11869,35801}, {11871,35803}, {11873,35809}, {11877,35808}, {11879,35768}, {11881,35817}, {11883,35819}, {12179,35825}, {12345,35699}, {12365,35827}, {12376,13208}, {12415,35837}, {12452,35841}, {12454,35843}, {12456,35845}, {12458,35611}, {12460,35853}, {12462,35857}, {12464,35863}, {12466,35835}, {12468,35865}, {12470,35851}, {12472,35754}, {12474,35867}, {12476,35869}, {12478,35829}, {12480,12971}, {12482,35871}, {12484,35833}, {12486,35831}, {13176,35879}, {13228,35883}, {13229,35881}, {13682,35873}, {13802,35875}, {13890,35815}, {13944,35813}, {16121,35855}, {18495,35787}, {18955,35769}, {19008,35771}, {22668,35839}, {22669,35849}, {22670,35847}, {22671,35861}, {32265,35877}, {32360,35859}

X(35781) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11875, 35778), (55, 35642, 35779)

X(35782) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 1st KENMOTU-FREE-VERTICES

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

X(35782) lies on these lines: {3,6}, {485,2896}, {3070,32151}, {3096,10576}, {3099,35774}, {5418,10357}, {5420,10583}, {6560,9862}, {6564,9996}, {6565,9993}, {7811,35822}, {8782,35878}, {9857,35788}, {9873,35820}, {9878,35698}, {9923,35836}, {9941,35641}, {9981,35850}, {9982,35753}, {9983,35866}, {9984,35826}, {9985,12965}, {9986,35830}, {9987,35832}, {9997,35810}, {10038,35809}, {10047,35769}, {10828,35776}, {10871,35796}, {10872,35798}, {10873,35800}, {10874,35802}, {10875,35804}, {10876,35806}, {10877,35808}, {10878,35816}, {10879,35818}, {11368,35762}, {11386,35764}, {11494,35772}, {11861,35778}, {11862,35780}, {11885,35790}, {12375,13210}, {12495,35842}, {12496,35844}, {12497,35610}, {12498,35852}, {12499,35856}, {12500,35862}, {12501,35834}, {12502,35864}, {12503,35828}, {12504,35870}, {13235,35882}, {13236,35880}, {13685,35872}, {13805,35874}, {13892,35812}, {13946,35814}, {16123,35854}, {18500,35786}, {18503,23251}, {18957,35768}, {22678,35838}, {22744,35784}, {22745,35846}, {22746,35848}, {22747,35860}, {32268,35876}, {32362,35858}

X(35782) = {X(6), X(9301)}-harmonic conjugate of X(35783)

X(35783) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5th BROCARD AND 2nd KENMOTU-FREE-VERTICES

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

X(35783) lies on these lines: {3,6}, {486,2896}, {3071,32151}, {3096,10577}, {3099,35775}, {5418,10583}, {5420,10357}, {6561,9862}, {6564,9993}, {6565,9996}, {7811,35823}, {8782,35879}, {9857,35789}, {9873,35821}, {9878,35699}, {9923,35837}, {9941,35642}, {9981,35851}, {9982,35754}, {9983,35867}, {9984,35827}, {9985,12971}, {9986,35833}, {9987,35831}, {9997,35811}, {10038,35808}, {10047,35768}, {10828,35777}, {10871,35797}, {10872,35799}, {10873,35801}, {10874,35803}, {10875,35807}, {10876,35805}, {10877,35809}, {10878,35817}, {10879,35819}, {11368,35763}, {11386,35765}, {11494,35773}, {11861,35781}, {11862,35779}, {11885,35791}, {12376,13210}, {12495,35843}, {12496,35845}, {12497,35611}, {12498,35853}, {12499,35857}, {12500,35863}, {12501,35835}, {12502,35865}, {12503,35829}, {12504,35871}, {13235,35883}, {13236,35881}, {13685,35873}, {13805,35875}, {13892,35815}, {13946,35813}, {16123,35855}, {18500,35787}, {18503,23261}, {18957,35769}, {22678,35839}, {22744,35785}, {22745,35849}, {22746,35847}, {22747,35861}, {32268,35877}, {32362,35859}

X(35783) = {X(6), X(9301)}-harmonic conjugate of X(35782)

X(35784) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 1st KENMOTU-FREE-VERTICES

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

X(35784) lies on these lines: {3,35611}, {6,22765}, {36,35774}, {55,35810}, {56,372}, {104,6560}, {371,11249}, {485,2975}, {956,35788}, {958,10576}, {999,35762}, {1482,35773}, {3070,32153}, {3428,6200}, {5253,5420}, {6396,10269}, {6419,19014}, {6564,22758}, {6565,22753}, {6911,35789}, {7969,26286}, {10680,35642}, {10966,35808}, {11194,35822}, {11492,35780}, {11493,35778}, {11499,35843}, {12114,35796}, {12375,22586}, {12513,35842}, {12773,35852}, {12965,22781}, {13743,35854}, {18237,35844}, {18761,35786}, {19013,35770}, {19159,35828}, {19162,35880}, {19478,35834}, {22479,35764}, {22504,35824}, {22514,35878}, {22520,35766}, {22560,35882}, {22565,35698}, {22583,35826}, {22595,35830}, {22624,35832}, {22654,35776}, {22659,35836}, {22680,35838}, {22744,35782}, {22755,35790}, {22756,35792}, {22757,35794}, {22759,35800}, {22760,35802}, {22761,35804}, {22762,35806}, {22763,35812}, {22764,35814}, {22766,35809}, {22767,35769}, {22768,35816}, {22769,35840}, {22770,35610}, {22771,35846}, {22772,35848}, {22773,35753}, {22774,35850}, {22775,35856}, {22776,35860}, {22777,35862}, {22778,35864}, {22779,35866}, {22780,35868}, {22782,35870}, {22783,35872}, {22784,35874}, {23251,26321}, {32270,35876}, {32363,35858}

X(35784) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35641, 35772), (6, 22765, 35785)

X(35785) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CIRCUMPERP TANGENTIAL AND 2nd KENMOTU-FREE-VERTICES

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

X(35785) lies on these lines: {3,35610}, {6,22765}, {36,35775}, {55,35811}, {56,371}, {104,6561}, {372,11249}, {486,2975}, {956,35789}, {958,10577}, {999,35763}, {1482,35772}, {3071,32153}, {3428,6396}, {5253,5418}, {6200,10269}, {6420,19013}, {6564,22753}, {6565,22758}, {6911,35788}, {7968,26286}, {10680,35641}, {10966,35809}, {11194,35823}, {11492,35779}, {11493,35781}, {11499,35842}, {12114,35797}, {12376,22586}, {12513,35843}, {12773,35853}, {12971,22781}, {13743,35855}, {18237,35845}, {18761,35787}, {19014,35771}, {19159,35829}, {19162,35881}, {19478,35835}, {22479,35765}, {22504,35825}, {22514,35879}, {22520,35767}, {22560,35883}, {22565,35699}, {22583,35827}, {22595,35833}, {22624,35831}, {22654,35777}, {22659,35837}, {22680,35839}, {22744,35783}, {22755,35791}, {22756,35795}, {22757,35793}, {22759,35801}, {22760,35803}, {22761,35807}, {22762,35805}, {22763,35815}, {22764,35813}, {22766,35808}, {22767,35768}, {22768,35817}, {22769,35841}, {22770,35611}, {22771,35849}, {22772,35847}, {22773,35754}, {22774,35851}, {22775,35857}, {22776,35861}, {22777,35863}, {22778,35865}, {22779,35867}, {22780,35869}, {22782,35871}, {22783,35873}, {22784,35875}, {23261,26321}, {32270,35877}, {32363,35859}

X(35785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 35642, 35773), (6, 22765, 35784)

X(35786) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-MID AND 1st KENMOTU-FREE-VERTICES

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

X(35786) lies on these lines: {2,22644}, {4,371}, {5,6396}, {6,3843}, {30,10576}, {113,35834}, {372,381}, {382,6200}, {428,8280}, {486,3832}, {546,3070}, {548,32789}, {590,3627}, {615,3850}, {946,35811}, {1131,6435}, {1132,1327}, {1151,3830}, {1152,3851}, {1328,7582}, {1478,35802}, {1479,35800}, {1539,35826}, {1657,8253}, {1699,35642}, {1853,35865}, {2066,18514}, {2067,18513}, {3071,3845}, {3091,6454}, {3093,18386}, {3102,22682}, {3146,5418}, {3311,14269}, {3316,9680}, {3317,3855}, {3366,16808}, {3391,16809}, {3543,6486}, {3545,5420}, {3583,35808}, {3585,35768}, {3818,35840}, {3853,6480}, {3856,18762}, {3857,13966}, {3861,7583}, {5055,6410}, {5064,8854}, {5070,6412}, {5072,8252}, {5073,6409}, {5076,6453}, {5198,35777}, {5413,35488}, {5587,35611}, {5691,35763}, {6202,35833}, {6411,17800}, {6419,13665}, {6450,19709}, {6455,15684}, {6476,8972}, {6497,15703}, {7394,8855}, {7408,18289}, {7547,11474}, {8276,11403}, {8981,15687}, {9540,17578}, {9677,26883}, {9682,35502}, {9733,22810}, {9818,35776}, {9955,35762}, {10113,12375}, {10533,34786}, {10895,35809}, {10896,35769}, {10897,18403}, {11001,34089}, {11265,18567}, {11513,18404}, {12101,13925}, {12239,16194}, {12611,35852}, {12699,35788}, {12811,35256}, {12964,18383}, {12965,22804}, {13785,35770}, {14639,35825}, {14644,35827}, {14881,35866}, {14893,32787}, {18480,35641}, {18491,35772}, {18492,35774}, {18495,35778}, {18497,35780}, {18500,35782}, {18502,35766}, {18507,35790}, {18509,35792}, {18511,35794}, {18516,35796}, {18517,35798}, {18520,35804}, {18522,35806}, {18525,35810}, {18542,35816}, {18544,35818}, {18761,35784}, {19090,35839}, {19160,35828}, {19163,35880}, {19925,35789}, {22505,35824}, {22515,35878}, {22566,35698}, {22596,35830}, {22625,35832}, {22660,35836}, {22681,35838}, {22791,35842}, {22792,35844}, {22793,35610}, {22794,35846}, {22795,35848}, {22796,35753}, {22797,35850}, {22798,35854}, {22799,35856}, {22800,35860}, {22801,35862}, {22802,35864}, {22803,35868}, {22805,35870}, {22806,35872}, {22807,35874}, {22938,35882}, {23046,32788}, {23273,31414}, {26332,35819}, {26333,35817}, {32271,35876}, {32364,35858}, {32785,33703}

X(35786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 485, 35821), (4, 3068, 22615), (4, 6564, 371), (4, 31412, 6561), (5, 35820, 6396), (6, 3843, 35787), (381, 23251, 372), (485, 6459, 35815), (485, 35821, 371), (546, 3070, 6565), (1327, 3839, 35823), (3070, 6565, 6420), (3071, 35822, 35771), (3091, 6560, 10577), (3091, 23253, 6560), (6459, 35815, 371), (6560, 10577, 6454), (6561, 8960, 371), (6561, 31412, 8960), (6564, 8960, 31412), (6564, 35821, 485), (35815, 35821, 6459)

X(35787) = HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-MID AND 2nd KENMOTU-FREE-VERTICES

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

X(35787) lies on these lines: {2,22615}, {4,372}, {5,6200}, {6,3843}, {30,10577}, {113,35835}, {371,381}, {382,6396}, {428,8281}, {485,3832}, {546,3071}, {548,32790}, {590,3850}, {615,3627}, {946,35810}, {1131,1328}, {1132,6436}, {1151,3851}, {1152,3830}, {1327,7581}, {1478,35803}, {1479,35801}, {1539,35827}, {1657,8252}, {1699,35641}, {1853,35864}, {3070,3845}, {3091,6453}, {3092,18386}, {3103,22682}, {3146,5420}, {3312,14269}, {3316,3855}, {3367,16808}, {3392,16809}, {3543,6487}, {3545,5418}, {3583,35809}, {3585,35769}, {3614,9660}, {3818,35841}, {3853,6481}, {3856,18538}, {3857,8981}, {3858,8960}, {3859,31454}, {3861,7584}, {5055,6409}, {5064,8855}, {5068,9541}, {5070,6411}, {5072,8253}, {5073,6410}, {5076,6454}, {5198,35776}, {5412,35488}, {5414,18514}, {5587,35610}, {5691,35762}, {6201,35832}, {6412,17800}, {6420,13785}, {6449,19709}, {6456,15684}, {6477,13941}, {6496,15703}, {6502,18513}, {7173,9647}, {7394,8854}, {7408,18290}, {7547,11473}, {8277,11403}, {9681,32785}, {9732,22809}, {9818,35777}, {9955,35763}, {10113,12376}, {10534,34786}, {10895,35808}, {10896,35768}, {10898,18403}, {11001,34091}, {11266,18567}, {11514,18404}, {12101,13993}, {12240,16194}, {12611,35853}, {12699,35789}, {12811,35255}, {12970,18383}, {12971,22804}, {13665,35771}, {13935,17578}, {13966,15687}, {14639,35824}, {14644,35826}, {14881,35867}, {14893,32788}, {18480,35642}, {18491,35773}, {18492,35775}, {18495,35781}, {18497,35779}, {18500,35783}, {18502,35767}, {18507,35791}, {18509,35795}, {18511,35793}, {18516,35797}, {18517,35799}, {18520,35807}, {18522,35805}, {18525,35811}, {18542,35817}, {18544,35819}, {18761,35785}, {19089,35838}, {19160,35829}, {19163,35881}, {19925,35788}, {22505,35825}, {22515,35879}, {22566,35699}, {22596,35833}, {22625,35831}, {22660,35837}, {22681,35839}, {22791,35843}, {22792,35845}, {22793,35611}, {22794,35849}, {22795,35847}, {22796,35754}, {22797,35851}, {22798,35855}, {22799,35857}, {22800,35861}, {22801,35863}, {22802,35865}, {22803,35869}, {22805,35871}, {22806,35873}, {22807,35875}, {22938,35883}, {23046,32787}, {23275,31412}, {26332,35818}, {26333,35816}, {32271,35877}, {32364,35859}, {32786,33703}

X(35787) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 486, 35820), (4, 3069, 22644), (4, 6565, 372), (5, 35821, 6200), (6, 3843, 35786), (381, 23261, 371), (486, 6460, 35814), (486, 35820, 372), (546, 3071, 6564), (1328, 3839, 35822), (3070, 35823, 35770), (3071, 6564, 6419), (3091, 6561, 10576), (3091, 23263, 6561), (3832, 23259, 485), (6460, 35814, 372), (6561, 10576, 6453), (6565, 35820, 486), (13785, 23251, 6420), (35814, 35820, 6460)

X(35788) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND 1st KENMOTU-FREE-VERTICES

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

X(35788) lies on these lines: {1,10576}, {2,35762}, {4,35610}, {5,35642}, {6,5790}, {8,485}, {10,372}, {40,35820}, {65,35800}, {72,35798}, {80,2066}, {100,35852}, {355,371}, {486,5818}, {515,6200}, {517,6564}, {519,35810}, {590,952}, {944,5418}, {956,35784}, {1151,18525}, {1587,3617}, {1737,35769}, {1837,35808}, {3057,35802}, {3070,5690}, {3071,18357}, {3416,35840}, {3560,35773}, {3679,35774}, {5090,35764}, {5252,35768}, {5420,9780}, {5587,6565}, {5657,6560}, {5687,35772}, {5688,35794}, {5689,35792}, {5844,18538}, {5881,13893}, {5886,35811}, {6361,22644}, {6396,26446}, {6419,13883}, {6420,13973}, {6453,13912}, {6502,18395}, {6911,35785}, {7967,32785}, {7968,9956}, {7969,8960}, {8193,35776}, {8197,35778}, {8204,35780}, {8214,35804}, {8215,35806}, {8253,10246}, {8976,12645}, {9646,10950}, {9661,10944}, {9857,35782}, {9864,35824}, {9881,35698}, {9928,35836}, {10039,35809}, {10573,31472}, {10791,35766}, {10826,35803}, {10827,16232}, {10914,35796}, {10915,35816}, {10916,35818}, {11684,35854}, {11900,35790}, {12245,31412}, {12368,35826}, {12375,13211}, {12619,35857}, {12667,35844}, {12699,35786}, {12702,23251}, {12751,35856}, {12777,35862}, {12778,35834}, {12779,35864}, {12780,35850}, {12781,35753}, {12782,35866}, {12783,35868}, {12784,35828}, {12785,12965}, {12786,35870}, {12787,35830}, {12788,35832}, {13178,35878}, {13280,35880}, {13688,35872}, {13808,35874}, {13936,35770}, {13947,35814}, {13971,31399}, {18480,35821}, {19925,35787}, {20070,23253}, {22697,35838}, {22851,35846}, {22896,35848}, {22941,35860}, {28224,35255}, {32278,35876}, {32371,35858}, {33899,35845}

X(35788) = reflection of X(35763) in X(590)
X(35788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5790, 35789), (8, 485, 35641), (355, 13911, 371), (3070, 5690, 35611), (5587, 35775, 6565), (5818, 19066, 486), (7968, 9956, 10577), (8960, 35843, 7969), (10827, 16232, 35801)

X(35789) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND 2nd KENMOTU-FREE-VERTICES

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

X(35789) lies on these lines: {1,10577}, {2,35763}, {4,35611}, {5,35641}, {6,5790}, {8,486}, {10,371}, {40,35821}, {65,35801}, {72,35799}, {80,5414}, {100,35853}, {355,372}, {485,5818}, {515,6396}, {517,6565}, {519,35811}, {615,952}, {944,5420}, {956,35785}, {1152,18525}, {1588,3617}, {1737,35768}, {1837,35809}, {2067,18395}, {2362,10827}, {3057,35803}, {3070,18357}, {3071,5690}, {3416,35841}, {3560,35772}, {3679,35775}, {5090,35765}, {5252,35769}, {5418,9780}, {5587,6564}, {5657,6561}, {5687,35773}, {5688,35793}, {5689,35795}, {5844,18762}, {5881,13947}, {5886,35810}, {6200,26446}, {6361,22615}, {6419,13911}, {6420,13936}, {6454,13975}, {6911,35784}, {7967,32786}, {7968,35842}, {7969,9956}, {8193,35777}, {8197,35781}, {8204,35779}, {8214,35807}, {8215,35805}, {8252,10246}, {8960,18991}, {8983,31399}, {9583,19875}, {9857,35783}, {9864,35825}, {9881,35699}, {9928,35837}, {10039,35808}, {10791,35767}, {10826,35802}, {10914,35797}, {10915,35817}, {10916,35819}, {11684,35855}, {11900,35791}, {12368,35827}, {12376,13211}, {12619,35856}, {12645,13951}, {12667,35845}, {12699,35787}, {12702,23261}, {12751,35857}, {12777,35863}, {12778,35835}, {12779,35865}, {12780,35851}, {12781,35754}, {12782,35867}, {12783,35869}, {12784,35829}, {12785,12971}, {12786,35871}, {12787,35833}, {12788,35831}, {13178,35879}, {13280,35881}, {13688,35873}, {13808,35875}, {13883,35771}, {13893,35815}, {18480,35820}, {19925,35786}, {20070,23263}, {22697,35839}, {22851,35849}, {22896,35847}, {22941,35861}, {28224,35256}, {32278,35877}, {32371,35859}, {33899,35844}

X(35789) = reflection of X(35762) in X(615)
X(35789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 5790, 35788), (8, 486, 35642), (355, 13973, 372), (2362, 10827, 35800), (3071, 5690, 35610), (5587, 35774, 6564), (5818, 19065, 485), (7969, 9956, 10576), (10577, 35843, 1)

X(35790) = HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 1st KENMOTU-FREE-VERTICES

Barycentrics (3*SB*SC*(6*R^2-SW)+(SW^2+2*R^2*(36*R^2-11*SW-18*SA)+9*SA^2-6*SB*SC)*S+(2*SW-12*R^2+SA)*S^2+8*S^3)*(S^2-3*SB*SC) : :

X(35790) lies on these lines: {6,11911}, {30,590}, {371,11251}, {372,402}, {485,4240}, {1650,10576}, {1651,35822}, {3070,32162}, {6396,26451}, {6419,19018}, {6560,11845}, {6565,11897}, {11831,35762}, {11832,35764}, {11839,35766}, {11848,35772}, {11852,35774}, {11853,35776}, {11863,35778}, {11864,35780}, {11885,35782}, {11900,35788}, {11901,35792}, {11902,35794}, {11903,35796}, {11904,35798}, {11905,35800}, {11906,35802}, {11907,35804}, {11908,35806}, {11909,35808}, {11910,35810}, {11912,35809}, {11913,35769}, {11914,35816}, {11915,35818}, {12113,35820}, {12181,35824}, {12347,35698}, {12369,35826}, {12375,13212}, {12418,35836}, {12438,35641}, {12583,35840}, {12626,35842}, {12668,35844}, {12696,35610}, {12729,35852}, {12752,35856}, {12789,35862}, {12790,35834}, {12791,35864}, {12792,35850}, {12793,35753}, {12794,35866}, {12795,35868}, {12796,35828}, {12797,12965}, {12798,35870}, {12799,35830}, {12800,35832}, {13179,35878}, {13268,35882}, {13281,35880}, {13689,35872}, {13809,35874}, {13894,35812}, {13948,35814}, {16129,35854}, {18507,35786}, {18508,23251}, {18958,35768}, {19017,35770}, {22698,35838}, {22755,35784}, {22852,35846}, {22897,35848}, {22943,35860}, {32279,35876}, {32372,35858}

X(35790) = {X(6), X(11911)}-harmonic conjugate of X(35791)

X(35791) = HOMOTHETIC CENTER OF THESE TRIANGLES: GOSSARD AND 2nd KENMOTU-FREE-VERTICES

Barycentrics (3*SB*SC*(6*R^2-SW)-(SW^2+2*R^2*(36*R^2-11*SW-18*SA)+9*SA^2-6*SB*SC)*S+(2*SW-12*R^2+SA)*S^2-8*S^3)*(S^2-3*SB*SC) : :

X(35791) lies on these lines: {6,11911}, {30,615}, {371,402}, {372,11251}, {486,4240}, {1650,10577}, {1651,35823}, {3071,32162}, {6200,26451}, {6420,19017}, {6561,11845}, {6564,11897}, {11831,35763}, {11832,35765}, {11839,35767}, {11848,35773}, {11852,35775}, {11853,35777}, {11863,35781}, {11864,35779}, {11885,35783}, {11900,35789}, {11901,35795}, {11902,35793}, {11903,35797}, {11904,35799}, {11905,35801}, {11906,35803}, {11907,35807}, {11908,35805}, {11909,35809}, {11910,35811}, {11912,35808}, {11913,35768}, {11914,35817}, {11915,35819}, {12113,35821}, {12181,35825}, {12347,35699}, {12369,35827}, {12376,13212}, {12418,35837}, {12438,35642}, {12583,35841}, {12626,35843}, {12668,35845}, {12696,35611}, {12729,35853}, {12752,35857}, {12789,35863}, {12790,35835}, {12791,35865}, {12792,35851}, {12793,35754}, {12794,35867}, {12795,35869}, {12796,35829}, {12797,12971}, {12798,35871}, {12799,35833}, {12800,35831}, {13179,35879}, {13268,35883}, {13281,35881}, {13689,35873}, {13809,35875}, {13894,35815}, {13948,35813}, {16129,35855}, {18507,35787}, {18508,23261}, {18958,35769}, {19018,35771}, {22698,35839}, {22755,35785}, {22852,35849}, {22897,35847}, {22943,35861}, {32279,35877}, {32372,35859}

X(35791) = {X(6), X(11911)}-harmonic conjugate of X(35790)

X(35792) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 1st KENMOTU-FREE-VERTICES

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

X(35792) lies on these lines: {3,6}, {485,1271}, {3070,5875}, {3641,35641}, {5418,10517}, {5589,35774}, {5591,10576}, {5595,35776}, {5605,35810}, {5689,35788}, {5861,35822}, {5871,35820}, {6202,6565}, {6215,6564}, {6227,35824}, {6258,35844}, {6263,35852}, {6267,35864}, {6270,35753}, {6271,35850}, {6273,35866}, {6275,35868}, {6277,12965}, {6279,35832}, {6281,35830}, {6319,35878}, {6560,10783}, {7725,35826}, {7732,12375}, {8198,35778}, {8205,35780}, {8216,35804}, {8217,35806}, {8974,35812}, {9882,35698}, {9929,35836}, {10040,35809}, {10048,35769}, {10919,35796}, {10921,35798}, {10923,35800}, {10925,35802}, {10927,35808}, {10929,35816}, {10931,35818}, {11370,35762}, {11388,35764}, {11497,35772}, {11901,35790}, {12627,35842}, {12697,35610}, {12753,35856}, {12801,35862}, {12803,35834}, {12805,35828}, {12807,35870}, {13269,35882}, {13282,35880}, {13690,35872}, {13810,35874}, {13949,35814}, {16130,35854}, {18509,35786}, {18959,35768}, {22699,35838}, {22756,35784}, {22853,35846}, {22898,35848}, {22945,35860}, {23251,26336}, {32280,35876}, {32373,35858}

X(35792) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1161, 371), (6, 11916, 35795), (371, 6420, 30435), (372, 35840, 35794), (3103, 35840, 372), (3371, 3372, 26348), (9605, 11477, 35793)

X(35793) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 2nd KENMOTU-FREE-VERTICES

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

X(35793) lies on these lines: {3,6}, {486,1270}, {3071,5874}, {3640,35642}, {5420,10518}, {5588,35775}, {5590,10577}, {5594,35777}, {5604,35811}, {5688,35789}, {5860,35823}, {5870,35821}, {6201,6564}, {6214,6565}, {6226,35825}, {6257,35845}, {6262,35853}, {6266,35865}, {6268,35754}, {6269,35851}, {6272,35867}, {6274,35869}, {6276,12971}, {6278,35831}, {6280,35833}, {6320,35879}, {6561,10784}, {7726,35827}, {7733,12376}, {8199,35781}, {8206,35779}, {8218,35807}, {8219,35805}, {8975,35815}, {9883,35699}, {9930,35837}, {10041,35808}, {10049,35768}, {10920,35797}, {10922,35799}, {10924,35801}, {10926,35803}, {10928,35809}, {10930,35817}, {10932,35819}, {11371,35763}, {11389,35765}, {11498,35773}, {11902,35791}, {12628,35843}, {12698,35611}, {12754,35857}, {12802,35863}, {12804,35835}, {12806,35829}, {12808,35871}, {13270,35883}, {13283,35881}, {13691,35873}, {13811,35875}, {13950,35813}, {16131,35855}, {18511,35787}, {18960,35769}, {22700,35839}, {22757,35785}, {22854,35849}, {22899,35847}, {22947,35861}, {23261,26346}, {32281,35877}, {32374,35859}

X(35793) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1160, 372), (6, 11917, 35794), (371, 35841, 35795), (372, 6419, 30435), (3102, 35841, 371), (3385, 3386, 26341), (9605, 11477, 35792)

X(35794) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 1st KENMOTU-FREE-VERTICES

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

X(35794) lies on these lines: {3,6}, {485,1270}, {3070,5874}, {3640,35641}, {5418,10518}, {5588,35774}, {5590,10576}, {5594,35776}, {5604,35810}, {5688,35788}, {5860,35822}, {5870,35820}, {6201,6565}, {6214,6564}, {6226,35824}, {6257,35844}, {6262,35852}, {6266,35864}, {6268,35753}, {6269,35850}, {6272,35866}, {6274,35868}, {6276,12965}, {6278,32499}, {6280,35830}, {6320,35878}, {6560,10784}, {7726,35826}, {7733,12375}, {8199,35778}, {8206,35780}, {8218,35804}, {8219,35806}, {8975,35812}, {9883,35698}, {9930,35836}, {10041,35809}, {10049,35769}, {10515,26469}, {10920,35796}, {10922,35798}, {10924,35800}, {10926,35802}, {10928,35808}, {10930,35816}, {10932,35818}, {11371,35762}, {11389,35764}, {11498,35772}, {11902,35790}, {12628,35842}, {12698,35610}, {12754,35856}, {12802,35862}, {12804,35834}, {12806,35828}, {12808,35870}, {13270,35882}, {13283,35880}, {13691,35872}, {13811,35874}, {13950,35814}, {16131,35854}, {18511,35786}, {18960,35768}, {22700,35838}, {22757,35784}, {22854,35846}, {22899,35848}, {22947,35860}, {23251,26346}, {32281,35876}, {32374,35858}

X(35794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1160, 371), (6, 11917, 35793), (372, 6453, 2459), (372, 35840, 35792), (5102, 30435, 35795)

X(35795) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 2nd KENMOTU-FREE-VERTICES

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

X(35795) lies on these lines: {3,6}, {486,1271}, {3071,5875}, {3641,35642}, {5420,10517}, {5589,35775}, {5591,10577}, {5595,35777}, {5605,35811}, {5689,35789}, {5861,35823}, {5871,35821}, {6202,6564}, {6215,6565}, {6227,35825}, {6258,35845}, {6263,35853}, {6267,35865}, {6270,35754}, {6271,35851}, {6273,35867}, {6275,35869}, {6277,12971}, {6279,35831}, {6281,32498}, {6319,35879}, {6561,10783}, {7725,35827}, {7732,12376}, {8198,35781}, {8205,35779}, {8216,35807}, {8217,35805}, {8974,35815}, {9882,35699}, {9929,35837}, {10040,35808}, {10048,35768}, {10514,26468}, {10919,35797}, {10921,35799}, {10923,35801}, {10925,35803}, {10927,35809}, {10929,35817}, {10931,35819}, {11370,35763}, {11388,35765}, {11497,35773}, {11901,35791}, {12627,35843}, {12697,35611}, {12753,35857}, {12801,35863}, {12803,35835}, {12805,35829}, {12807,35871}, {13269,35883}, {13282,35881}, {13690,35873}, {13810,35875}, {13949,35813}, {16130,35855}, {18509,35787}, {18959,35769}, {22699,35839}, {22756,35785}, {22853,35849}, {22898,35847}, {22945,35861}, {23261,26336}, {32280,35877}, {32373,35859}

X(35795) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 1161, 372), (6, 11916, 35792), (371, 6454, 2460), (371, 35841, 35793), (5102, 30435, 35794)

X(35796) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 1st KENMOTU-FREE-VERTICES

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

X(35796) lies on these lines: {6,11928}, {11,372}, {12,35816}, {355,6564}, {371,10525}, {485,3434}, {486,10598}, {1376,10576}, {3070,10943}, {5420,10584}, {6200,11826}, {6396,26492}, {6419,19024}, {6560,10785}, {6565,10893}, {10523,35809}, {10794,35766}, {10826,35774}, {10829,35776}, {10871,35782}, {10912,35842}, {10914,35788}, {10919,35792}, {10920,35794}, {10944,35800}, {10945,35804}, {10946,35806}, {10947,35808}, {10948,35769}, {10949,35818}, {11235,35822}, {11373,35762}, {11390,35764}, {11865,35778}, {11866,35780}, {11903,35790}, {12114,35784}, {12182,35824}, {12348,35698}, {12371,35826}, {12375,13213}, {12422,35836}, {12586,35840}, {12676,35844}, {12700,35610}, {12737,35852}, {12761,35856}, {12857,35862}, {12889,35834}, {12920,35864}, {12921,35850}, {12922,35753}, {12923,35866}, {12924,35868}, {12925,35828}, {12926,12965}, {12927,35870}, {12928,35830}, {12929,35832}, {13180,35878}, {13271,35882}, {13294,35880}, {13693,35872}, {13813,35874}, {13895,35812}, {13952,35814}, {16138,35854}, {18516,35786}, {18519,23251}, {18961,35768}, {19023,35770}, {22703,35838}, {22857,35846}, {22902,35848}, {22956,35860}, {32287,35876}, {32380,35858}

X(35796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11928, 35797), (6564, 35641, 35798)

X(35797) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 2nd KENMOTU-FREE-VERTICES

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

X(35797) lies on these lines: {6,11928}, {11,371}, {12,35817}, {355,6565}, {372,10525}, {485,10598}, {486,3434}, {1376,10577}, {3071,10943}, {5418,10584}, {6200,26492}, {6396,11826}, {6420,19023}, {6561,10785}, {6564,10893}, {10523,35808}, {10794,35767}, {10826,35775}, {10829,35777}, {10871,35783}, {10912,35843}, {10914,35789}, {10919,35795}, {10920,35793}, {10944,35801}, {10945,35807}, {10946,35805}, {10947,35809}, {10948,35768}, {10949,35819}, {11235,35823}, {11373,35763}, {11390,35765}, {11865,35781}, {11866,35779}, {11903,35791}, {12114,35785}, {12182,35825}, {12348,35699}, {12371,35827}, {12376,13213}, {12422,35837}, {12586,35841}, {12676,35845}, {12700,35611}, {12737,35853}, {12761,35857}, {12857,35863}, {12889,35835}, {12920,35865}, {12921,35851}, {12922,35754}, {12923,35867}, {12924,35869}, {12925,35829}, {12926,12971}, {12927,35871}, {12928,35833}, {12929,35831}, {13180,35879}, {13271,35883}, {13294,35881}, {13693,35873}, {13813,35875}, {13895,35815}, {13952,35813}, {16138,35855}, {18516,35787}, {18519,23261}, {18961,35769}, {19024,35771}, {22703,35839}, {22857,35849}, {22902,35847}, {22956,35861}, {32287,35877}, {32380,35859}

X(35797) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11928, 35796), (6565, 35642, 35799)

X(35798) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 1st KENMOTU-FREE-VERTICES

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

X(35798) lies on these lines: {6,11929}, {11,35818}, {12,372}, {72,35788}, {355,6564}, {371,10526}, {485,3436}, {486,10599}, {958,10576}, {3070,10942}, {5420,10585}, {5812,35610}, {6200,11827}, {6396,26487}, {6419,19026}, {6560,10786}, {6565,10894}, {10523,35769}, {10795,35766}, {10827,35774}, {10830,35776}, {10872,35782}, {10921,35792}, {10922,35794}, {10950,35802}, {10951,35804}, {10952,35806}, {10953,35808}, {10954,35809}, {10955,35816}, {11236,35822}, {11374,35762}, {11391,35764}, {11500,35772}, {11867,35778}, {11868,35780}, {11904,35790}, {12183,35824}, {12349,35698}, {12372,35826}, {12375,13214}, {12423,35836}, {12587,35840}, {12635,35842}, {12677,35844}, {12738,35852}, {12762,35856}, {12858,35862}, {12890,35834}, {12930,35864}, {12931,35850}, {12932,35753}, {12933,35866}, {12934,35868}, {12935,35828}, {12936,12965}, {12937,35870}, {12938,35830}, {12939,35832}, {13181,35878}, {13272,35882}, {13295,35880}, {13694,35872}, {13814,35874}, {13896,35812}, {13953,35814}, {16139,35854}, {18517,35786}, {18518,23251}, {18962,35768}, {19025,35770}, {22704,35838}, {22858,35846}, {22903,35848}, {22957,35860}, {32288,35876}, {32381,35858}

X(35798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11929, 35799), (6564, 35641, 35796)

X(35799) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 2nd KENMOTU-FREE-VERTICES

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

X(35799) lies on these lines: {6,11929}, {11,35819}, {12,371}, {72,35789}, {355,6565}, {372,10526}, {485,10599}, {486,3436}, {958,10577}, {3071,10942}, {5418,10585}, {5812,35611}, {6200,26487}, {6396,11827}, {6420,19025}, {6561,10786}, {6564,10894}, {10523,35768}, {10795,35767}, {10827,35775}, {10830,35777}, {10872,35783}, {10921,35795}, {10922,35793}, {10950,35803}, {10951,35807}, {10952,35805}, {10953,35809}, {10954,35808}, {10955,35817}, {11236,35823}, {11374,35763}, {11391,35765}, {11500,35773}, {11867,35781}, {11868,35779}, {11904,35791}, {12183,35825}, {12349,35699}, {12372,35827}, {12376,13214}, {12423,35837}, {12587,35841}, {12635,35843}, {12677,35845}, {12738,35853}, {12762,35857}, {12858,35863}, {12890,35835}, {12930,35865}, {12931,35851}, {12932,35754}, {12933,35867}, {12934,35869}, {12935,35829}, {12936,12971}, {12937,35871}, {12938,35833}, {12939,35831}, {13181,35879}, {13272,35883}, {13295,35881}, {13694,35873}, {13814,35875}, {13896,35815}, {13953,35813}, {16139,35855}, {18517,35787}, {18518,23261}, {18962,35769}, {19026,35771}, {22704,35839}, {22858,35849}, {22903,35847}, {22957,35861}, {32288,35877}, {32381,35859}

X(35799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 11929, 35798), (6565, 35642, 35797)

X(35800) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND 1st KENMOTU-FREE-VERTICES

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

X(35800) lies on these lines: {1,6564}, {4,35808}, {5,35769}, {6,9650}, {12,372}, {55,35820}, {56,10576}, {65,35788}, {371,1478}, {381,3297}, {388,485}, {390,23253}, {486,10590}, {495,3070}, {498,6396}, {590,18990}, {615,10592}, {1056,31412}, {1124,6565}, {1151,9655}, {1152,31479}, {1335,11237}, {1377,11236}, {1479,35786}, {1587,5261}, {1703,5726}, {1836,35610}, {2066,3585}, {2067,5270}, {2099,35842}, {2362,10827}, {3085,6560}, {3157,35836}, {3295,23251}, {3298,13665}, {3299,35823}, {4293,5418}, {4294,22644}, {5229,6561}, {5252,35641}, {5420,10588}, {5434,9661}, {5714,19066}, {6200,7354}, {6419,19028}, {6453,9647}, {6460,8164}, {6502,7951}, {8068,35857}, {9578,35774}, {9612,35775}, {9656,19038}, {9657,13897}, {10039,35611}, {10088,35834}, {10797,35766}, {10831,35776}, {10873,35782}, {10923,35792}, {10924,35794}, {10944,35796}, {10956,35816}, {10957,35818}, {11375,35762}, {11392,35764}, {11501,35772}, {11869,35778}, {11870,35780}, {11905,35790}, {11930,35804}, {11931,35806}, {12047,35642}, {12184,35824}, {12350,35698}, {12373,35826}, {12375,12903}, {12588,35840}, {12678,35844}, {12739,35852}, {12763,35856}, {12837,35866}, {12859,35862}, {12940,35864}, {12941,35850}, {12942,35753}, {12944,35868}, {12945,35828}, {12946,12965}, {12947,35870}, {12948,35830}, {12949,35832}, {13182,35878}, {13273,35882}, {13296,35880}, {13695,35872}, {13815,35874}, {13954,35814}, {15326,31499}, {16140,35854}, {19027,35770}, {22705,35838}, {22759,35784}, {22859,35846}, {22904,35848}, {22958,35860}, {23261,31474}, {32289,35876}, {32382,35858}

X(35800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6564, 35802), (381, 3297, 35803), (495, 3070, 35809), (1124, 10895, 6565), (1478, 31472, 371), (2066, 3585, 35821), (2362, 10827, 35789), (6502, 7951, 10577), (7354, 9646, 6200), (9647, 13901, 6453), (10590, 31408, 486)

X(35801) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st JOHNSON-YFF AND 2nd KENMOTU-FREE-VERTICES

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

X(35801) lies on these lines: {1,6565}, {4,35809}, {5,35768}, {6,9650}, {12,371}, {55,35821}, {56,10577}, {65,35789}, {372,1478}, {381,3298}, {388,486}, {390,23263}, {485,10590}, {495,3071}, {498,6200}, {590,10592}, {615,18990}, {1124,11237}, {1151,31479}, {1152,9655}, {1335,6564}, {1378,11236}, {1479,35787}, {1588,5261}, {1702,5726}, {1836,35611}, {2067,7951}, {2099,35843}, {3085,6561}, {3157,35837}, {3295,23261}, {3297,13785}, {3301,35822}, {3585,5414}, {3614,9661}, {3822,31453}, {4293,5420}, {4294,22615}, {5229,6560}, {5252,35642}, {5270,6502}, {5418,10588}, {5432,9647}, {5714,19065}, {6396,7354}, {6419,31472}, {6420,19027}, {6454,13958}, {6459,8164}, {8068,35856}, {8960,18996}, {9578,35775}, {9612,35774}, {9656,19037}, {9657,13954}, {10039,35610}, {10088,35835}, {10797,35767}, {10827,16232}, {10831,35777}, {10873,35783}, {10923,35795}, {10924,35793}, {10944,35797}, {10956,35817}, {10957,35819}, {11375,35763}, {11392,35765}, {11501,35773}, {11869,35781}, {11870,35779}, {11905,35791}, {11930,35807}, {11931,35805}, {12047,35641}, {12184,35825}, {12350,35699}, {12373,35827}, {12376,12903}, {12588,35841}, {12678,35845}, {12739,35853}, {12763,35857}, {12837,35867}, {12859,35863}, {12940,35865}, {12941,35851}, {12942,35754}, {12944,35869}, {12945,35829}, {12946,12971}, {12947,35871}, {12948,35833}, {12949,35831}, {13182,35879}, {13273,35883}, {13296,35881}, {13695,35873}, {13815,35875}, {13897,35815}, {16140,35855}, {19028,35771}, {22705,35839}, {22759,35785}, {22859,35849}, {22904,35847}, {22958,35861}, {31408,31410}, {32289,35877}, {32382,35859}

X(35801) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6565, 35803), (381, 3298, 35802), (388, 486, 35769), (495, 3071, 35808), (1335, 10895, 6564), (3585, 5414, 35820), (10827, 16232, 35788)

X(35802) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND 1st KENMOTU-FREE-VERTICES

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

X(35802) lies on these lines: {1,6564}, {4,35768}, {5,35809}, {6,9665}, {11,372}, {55,10576}, {56,35820}, {80,35843}, {371,1479}, {381,3298}, {485,497}, {486,10591}, {496,3070}, {499,6396}, {590,15171}, {615,10593}, {999,23251}, {1058,31412}, {1069,35836}, {1124,11238}, {1151,9668}, {1335,6565}, {1378,11235}, {1478,35786}, {1587,5274}, {1737,35611}, {1837,35641}, {2066,4857}, {2067,3583}, {2098,35842}, {3057,35788}, {3058,9646}, {3086,6560}, {3297,13665}, {3301,23315}, {3600,23253}, {4293,22644}, {4294,5418}, {5225,6561}, {5414,7741}, {5420,10589}, {5533,35857}, {6200,6284}, {6419,19030}, {6453,9660}, {7743,7968}, {9581,35774}, {9614,35775}, {9670,13898}, {9671,18996}, {10091,35834}, {10572,35763}, {10798,35766}, {10826,35789}, {10832,35776}, {10874,35782}, {10925,35792}, {10926,35794}, {10950,35798}, {10958,35816}, {10959,35818}, {11376,35762}, {11393,35764}, {11502,35772}, {11871,35778}, {11872,35780}, {11906,35790}, {11932,35804}, {11933,35806}, {12185,35824}, {12351,35698}, {12374,35826}, {12375,12904}, {12589,35840}, {12679,35844}, {12701,35610}, {12740,35852}, {12764,35856}, {12836,35866}, {12860,35862}, {12950,35864}, {12951,35850}, {12952,35753}, {12954,35868}, {12955,35828}, {12956,12965}, {12957,35870}, {12958,35830}, {12959,35832}, {13183,35878}, {13274,35882}, {13297,35880}, {13696,35872}, {13816,35874}, {13955,35814}, {14986,23249}, {15172,18538}, {16141,35854}, {19029,35770}, {22706,35838}, {22760,35784}, {22860,35846}, {22905,35848}, {22959,35860}, {30384,35642}, {32290,35876}, {32383,35858}

X(35802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6564, 35800), (6, 9669, 35803), (381, 3298, 35801), (485, 497, 35808), (496, 3070, 35769), (1335, 10896, 6565), (2067, 3583, 35821), (5414, 7741, 10577), (6284, 9661, 6200), (9660, 18965, 6453)

X(35803) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd JOHNSON-YFF AND 2nd KENMOTU-FREE-VERTICES

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

X(35803) lies on these lines: {1,6565}, {4,35769}, {5,35808}, {6,9665}, {11,371}, {55,10577}, {56,35821}, {80,35842}, {372,1479}, {381,3297}, {485,10591}, {486,497}, {496,3071}, {499,6200}, {590,10593}, {615,15171}, {999,23261}, {1069,35837}, {1124,6564}, {1152,9668}, {1335,11238}, {1377,11235}, {1478,35787}, {1588,5274}, {1737,35610}, {1837,35642}, {2066,7741}, {2098,35843}, {3057,35789}, {3086,6561}, {3298,13785}, {3299,35822}, {3583,6502}, {3600,23263}, {4293,22615}, {4294,5420}, {4857,5414}, {5225,6560}, {5418,10589}, {5433,9660}, {5533,35856}, {6284,6396}, {6420,19029}, {6454,18966}, {7173,9646}, {7743,7969}, {8960,19038}, {9581,35775}, {9614,35774}, {9670,13955}, {9671,18995}, {10091,35835}, {10572,35762}, {10798,35767}, {10826,35788}, {10832,35777}, {10874,35783}, {10925,35795}, {10926,35793}, {10950,35799}, {10958,35817}, {10959,35819}, {11376,35763}, {11393,35765}, {11502,35773}, {11871,35781}, {11872,35779}, {11906,35791}, {11932,35807}, {11933,35805}, {12185,35825}, {12351,35699}, {12374,35827}, {12376,12904}, {12589,35841}, {12679,35845}, {12701,35611}, {12740,35853}, {12764,35857}, {12836,35867}, {12860,35863}, {12950,35865}, {12951,35851}, {12952,35754}, {12954,35869}, {12955,35829}, {12956,12971}, {12957,35871}, {12958,35833}, {12959,35831}, {13183,35879}, {13274,35883}, {13297,35881}, {13696,35873}, {13816,35875}, {13898,35815}, {14986,23259}, {15172,18762}, {16141,35855}, {19030,35771}, {22706,35839}, {22760,35785}, {22860,35849}, {22905,35847}, {22959,35861}, {24387,31453}, {30384,35641}, {32290,35877}, {32383,35859}

X(35803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6565, 35801), (6, 9669, 35802), (381, 3297, 35800), (486, 497, 35809), (496, 3071, 35768), (1124, 10896, 6564), (2066, 7741, 10576), (3583, 6502, 35820)

X(35804) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND LUCAS HOMOTHETIC

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

X(35804) lies on these lines: {6,11949}, {371,10669}, {372,493}, {485,487}, {3070,32177}, {6200,11828}, {6419,19032}, {6461,35806}, {6560,11846}, {6564,8220}, {6565,8212}, {8188,35774}, {8194,35776}, {8201,35778}, {8208,35780}, {8210,35810}, {8214,35788}, {8216,35792}, {8218,35794}, {8222,10576}, {9838,35820}, {10875,35782}, {10945,35796}, {10951,35798}, {11377,35762}, {11394,35764}, {11503,35772}, {11840,35766}, {11907,35790}, {11930,35800}, {11932,35802}, {11947,35808}, {11951,35809}, {11953,35769}, {11955,35816}, {11957,35818}, {12152,35822}, {12186,35824}, {12352,35698}, {12375,13215}, {12377,35826}, {12426,35836}, {12440,35641}, {12590,35840}, {12636,35842}, {12741,35852}, {12765,35856}, {12861,35862}, {12894,35834}, {12965,12998}, {12986,35864}, {12988,35850}, {12990,35753}, {12992,35866}, {12994,35868}, {12996,35828}, {13000,35870}, {13004,35832}, {13184,35878}, {13275,35882}, {13298,35880}, {13697,35872}, {13817,35874}, {13899,35812}, {13956,35814}, {16161,35854}, {18245,35844}, {18520,35786}, {18963,35768}, {19031,35770}, {22709,35838}, {22761,35784}, {22841,35610}, {22863,35846}, {22908,35848}, {22963,35860}, {32295,35876}, {32388,35858}

X(35804) = {X(6), X(11949)}-harmonic conjugate of X(35807)

X(35805) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND LUCAS HOMOTHETIC

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

X(35805) lies on these lines: {6,11950}, {371,494}, {372,10673}, {486,488}, {3071,32178}, {6396,11829}, {6420,19033}, {6461,35807}, {6561,11847}, {6564,8213}, {6565,8221}, {8189,35775}, {8195,35777}, {8202,35781}, {8209,35779}, {8211,35811}, {8215,35789}, {8217,35795}, {8219,35793}, {8223,10577}, {9839,35821}, {10876,35783}, {10946,35797}, {10952,35799}, {11378,35763}, {11395,35765}, {11504,35773}, {11841,35767}, {11908,35791}, {11931,35801}, {11933,35803}, {11948,35809}, {11952,35808}, {11954,35768}, {11956,35817}, {11958,35819}, {12153,35823}, {12187,35825}, {12353,35699}, {12376,13216}, {12378,35827}, {12427,35837}, {12441,35642}, {12591,35841}, {12637,35843}, {12742,35853}, {12766,35857}, {12862,35863}, {12895,35835}, {12971,12999}, {12987,35865}, {12989,35851}, {12991,35754}, {12993,35867}, {12995,35869}, {12997,35829}, {13001,35871}, {13003,35833}, {13185,35879}, {13276,35883}, {13299,35881}, {13698,35873}, {13818,35875}, {13900,35815}, {13957,35813}, {16162,35855}, {18246,35845}, {18522,35787}, {18964,35769}, {19034,35771}, {22710,35839}, {22762,35785}, {22842,35611}, {22864,35849}, {22909,35847}, {22964,35861}, {32296,35877}, {32389,35859}

X(35805) = {X(6), X(11950)}-harmonic conjugate of X(35806)

X(35806) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND LUCAS(-1) HOMOTHETIC

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

X(35806) lies on these lines: {6,11950}, {25,372}, {371,10673}, {485,6463}, {3070,32178}, {6200,11829}, {6419,19034}, {6461,35804}, {6560,11847}, {6564,8221}, {6565,8213}, {8189,35774}, {8202,35778}, {8209,35780}, {8211,35810}, {8215,35788}, {8217,35792}, {8219,35794}, {8223,10576}, {9839,35820}, {10876,35782}, {10946,35796}, {10952,35798}, {11378,35762}, {11504,35772}, {11841,35766}, {11908,35790}, {11931,35800}, {11933,35802}, {11948,35808}, {11952,35809}, {11954,35769}, {11956,35816}, {11958,35818}, {12153,35822}, {12187,35824}, {12353,35698}, {12375,13216}, {12378,35826}, {12427,35836}, {12441,35641}, {12591,35840}, {12637,35842}, {12742,35852}, {12766,35856}, {12862,35862}, {12895,35834}, {12965,12999}, {12987,35864}, {12989,35850}, {12991,35753}, {12993,35866}, {12995,35868}, {12997,35828}, {13001,35870}, {13003,35830}, {13005,35832}, {13185,35878}, {13276,35882}, {13299,35880}, {13698,35872}, {13818,35874}, {13900,35812}, {13957,35814}, {16162,35854}, {18246,35844}, {18522,35786}, {18964,35768}, {19033,35770}, {22710,35838}, {22762,35784}, {22842,35610}, {22864,35846}, {22909,35848}, {22964,35860}, {32296,35876}, {32389,35858}

X(35806) = {X(6), X(11950)}-harmonic conjugate of X(35805)

X(35807) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND LUCAS(-1) HOMOTHETIC

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

X(35807) lies on these lines: {6,11949}, {25,371}, {372,10669}, {486,6462}, {3071,32177}, {6396,11828}, {6420,19031}, {6461,35805}, {6561,11846}, {6564,8212}, {6565,8220}, {8188,35775}, {8201,35781}, {8208,35779}, {8210,35811}, {8214,35789}, {8216,35795}, {8218,35793}, {8222,10577}, {9838,35821}, {10875,35783}, {10945,35797}, {10951,35799}, {11377,35763}, {11503,35773}, {11840,35767}, {11907,35791}, {11930,35801}, {11932,35803}, {11947,35809}, {11951,35808}, {11953,35768}, {11955,35817}, {11957,35819}, {12152,35823}, {12186,35825}, {12352,35699}, {12376,13215}, {12377,35827}, {12426,35837}, {12440,35642}, {12590,35841}, {12636,35843}, {12741,35853}, {12765,35857}, {12861,35863}, {12894,35835}, {12971,12998}, {12986,35865}, {12988,35851}, {12990,35754}, {12992,35867}, {12994,35869}, {12996,35829}, {13000,35871}, {13002,35833}, {13004,35831}, {13184,35879}, {13275,35883}, {13298,35881}, {13697,35873}, {13817,35875}, {13899,35815}, {13956,35813}, {16161,35855}, {18245,35845}, {18520,35787}, {18963,35769}, {19032,35771}, {22709,35839}, {22761,35785}, {22841,35611}, {22863,35849}, {22908,35847}, {22963,35861}, {32295,35877}, {32388,35859}

X(35807) = {X(6), X(11949)}-harmonic conjugate of X(35804)

X(35808) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND MANDART-INCIRCLE

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

X(35808) lies on these lines: {1,371}, {3,3297}, {4,35800}, {5,35803}, {6,595}, {11,9646}, {12,6565}, {33,35764}, {35,6396}, {55,372}, {56,6200}, {65,35610}, {388,6561}, {390,1587}, {485,497}, {486,3085}, {495,3071}, {496,590}, {498,10577}, {519,31453}, {611,35841}, {612,8855}, {999,1151}, {1001,1378}, {1056,6459}, {1058,3068}, {1062,11513}, {1317,35856}, {1335,3303}, {1377,3913}, {1478,35821}, {1479,6564}, {1697,35774}, {1709,35845}, {1837,35788}, {1870,11473}, {2098,35810}, {2242,12963}, {2362,5119}, {2646,35762}, {3023,35878}, {3024,12375}, {3027,35824}, {3028,35826}, {3056,35840}, {3057,35641}, {3058,19028}, {3070,15171}, {3086,5418}, {3093,7071}, {3298,3311}, {3299,3746}, {3301,35771}, {3304,6453}, {3320,35828}, {3333,9616}, {3361,9582}, {3488,19066}, {3583,35786}, {3600,9541}, {3811,30556}, {4294,6560}, {4309,31475}, {4995,18966}, {5045,31439}, {5218,5420}, {5229,22615}, {5261,23259}, {5281,13935}, {5412,6198}, {5415,18992}, {5416,26464}, {5433,31499}, {5434,9647}, {5687,31473}, {5722,13911}, {6020,35880}, {6221,7373}, {6284,35820}, {6421,31477}, {6422,16781}, {6765,31438}, {7133,7162}, {7354,9660}, {7355,35864}, {7583,15172}, {7968,24929}, {7969,9957}, {8831,31508}, {8960,13905}, {9538,11417}, {9540,14986}, {9654,23261}, {9661,13901}, {9668,23251}, {9678,12513}, {9679,25524}, {10037,35777}, {10038,35783}, {10039,35789}, {10040,35795}, {10041,35793}, {10053,35825}, {10054,35699}, {10055,35837}, {10056,35823}, {10057,35853}, {10058,35857}, {10059,35863}, {10060,35865}, {10061,35851}, {10062,35754}, {10063,35867}, {10064,35869}, {10065,35827}, {10066,12971}, {10067,35833}, {10068,35831}, {10086,35879}, {10087,35883}, {10088,12376}, {10482,30335}, {10523,35797}, {10799,35766}, {10801,35767}, {10833,35776}, {10877,35782}, {10895,35787}, {10897,18455}, {10927,35792}, {10928,35794}, {10947,35796}, {10950,35842}, {10953,35798}, {10954,35799}, {10965,35816}, {10966,35784}, {11019,13912}, {11238,13897}, {11398,35765}, {11507,35773}, {11873,35778}, {11874,35780}, {11877,35781}, {11878,35779}, {11909,35790}, {11912,35791}, {11947,35804}, {11948,35806}, {11951,35807}, {11952,35805}, {12354,35698}, {12428,35836}, {12647,35843}, {12680,35844}, {12743,35852}, {12863,35862}, {12896,35834}, {12903,35835}, {12965,13079}, {13075,35850}, {13076,35753}, {13077,35866}, {13078,35868}, {13080,35870}, {13081,35830}, {13082,35832}, {13116,35829}, {13128,35871}, {13311,35881}, {13699,35872}, {13714,35873}, {13819,35874}, {13837,35875}, {13904,35815}, {13958,35814}, {13962,31452}, {15170,32787}, {16142,35854}, {16152,35855}, {18991,31393}, {19037,35770}, {22711,35838}, {22729,35839}, {22766,35785}, {22865,35846}, {22884,35849}, {22910,35848}, {22929,35847}, {22965,35860}, {22980,35861}, {31561,31568}, {32297,35876}, {32307,35877}, {32390,35858}, {32403,35859}

X(35808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 371, 35768), (1, 6212, 8953), (1, 31432, 35775), (3, 3297, 35769), (6, 3295, 35809), (11, 9646, 10576), (35, 6502, 6396), (55, 1124, 372), (371, 35817, 35642), (485, 497, 35802), (495, 3071, 35801), (1335, 19038, 6419), (1479, 31472, 6564), (2241, 31471, 6), (2362, 5119, 35611), (3295, 31474, 6), (3299, 3746, 5414), (3299, 5414, 6420), (3303, 19038, 1335), (4294, 31408, 6560)

X(35809) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND MANDART-INCIRCLE

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

X(35809) lies on these lines: {1,372}, {3,3298}, {4,35801}, {5,35802}, {6,595}, {11,10577}, {12,6564}, {33,35765}, {35,2067}, {55,371}, {56,6396}, {65,35611}, {388,6560}, {390,1588}, {485,3085}, {486,497}, {495,3070}, {496,615}, {498,10576}, {611,35840}, {612,8854}, {999,1152}, {1001,1377}, {1056,6460}, {1058,3069}, {1062,11514}, {1124,3303}, {1317,35857}, {1378,3913}, {1478,35820}, {1479,6565}, {1697,35775}, {1709,35844}, {1837,35789}, {1870,11474}, {2066,3301}, {2098,35811}, {2242,12968}, {2646,35763}, {3023,35879}, {3024,12376}, {3027,35825}, {3028,35827}, {3056,35841}, {3057,35642}, {3058,19027}, {3071,15171}, {3086,5420}, {3092,7071}, {3297,3312}, {3299,35770}, {3304,6454}, {3320,35829}, {3488,19065}, {3583,35787}, {3811,30557}, {4294,6561}, {4421,9679}, {4995,18965}, {5119,16232}, {5218,5418}, {5229,22644}, {5248,31453}, {5261,23249}, {5281,9540}, {5413,6198}, {5415,26458}, {5416,18991}, {5432,9661}, {5722,13973}, {6020,35881}, {6284,35821}, {6398,7373}, {6421,16781}, {6422,31477}, {7355,35865}, {7584,15172}, {7968,9957}, {7969,24929}, {8164,31412}, {8833,9582}, {8960,9646}, {9538,11418}, {9647,15338}, {9654,23251}, {9668,23261}, {10037,35776}, {10038,35782}, {10039,35788}, {10040,35792}, {10041,35794}, {10053,35824}, {10054,35698}, {10055,35836}, {10056,31472}, {10057,35852}, {10058,35856}, {10059,35862}, {10060,35864}, {10061,35850}, {10062,35753}, {10063,35866}, {10064,35868}, {10065,35826}, {10066,12965}, {10067,35830}, {10068,35832}, {10086,35878}, {10087,35882}, {10088,12375}, {10198,31484}, {10482,30336}, {10523,35796}, {10799,35767}, {10801,35766}, {10833,35777}, {10877,35783}, {10895,35786}, {10898,18455}, {10927,35795}, {10928,35793}, {10947,35797}, {10950,35843}, {10953,35799}, {10954,35798}, {10965,35817}, {10966,35785}, {11019,13975}, {11238,13954}, {11398,35764}, {11507,35772}, {11873,35781}, {11874,35779}, {11877,35778}, {11878,35780}, {11909,35791}, {11912,35790}, {11947,35807}, {11948,35805}, {11951,35804}, {11952,35806}, {12354,35699}, {12428,35837}, {12647,35842}, {12680,35845}, {12743,35853}, {12863,35863}, {12896,35835}, {12903,35834}, {12971,13079}, {13075,35851}, {13076,35754}, {13077,35867}, {13078,35869}, {13080,35871}, {13081,35833}, {13082,35831}, {13116,35828}, {13128,35870}, {13311,35880}, {13699,35873}, {13714,35872}, {13819,35875}, {13837,35874}, {13901,35815}, {13904,31452}, {13935,14986}, {13958,35813}, {13962,35814}, {15170,32788}, {16142,35855}, {16152,35854}, {18992,31393}, {19038,35771}, {22711,35839}, {22729,35838}, {22766,35784}, {22865,35849}, {22884,35846}, {22910,35847}, {22929,35848}, {22965,35861}, {22980,35860}, {31562,31567}, {32297,35877}, {32307,35876}, {32390,35859}, {32403,35858}

X(35809) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 372, 35769), (1, 5414, 372), (3, 3298, 35768), (6, 3295, 35808), (35, 2067, 6200), (55, 1335, 371), (372, 35816, 35641), (486, 497, 35803), (495, 3070, 35800), (1124, 19037, 6420), (2066, 3301, 6419), (3301, 3746, 2066), (3303, 19037, 1124), (3312, 6767, 3297), (4995, 18965, 31499), (5119, 16232, 35610), (9646, 19030, 8960)

X(35810) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND 5th MIXTILINEAR

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

X(35810) lies on these lines: {1,372}, {5,35843}, {6,10247}, {8,10576}, {55,35784}, {56,35772}, {145,485}, {371,1482}, {486,10595}, {517,6200}, {519,35788}, {590,5844}, {615,10283}, {944,35820}, {946,35787}, {952,6564}, {1151,8148}, {1320,35882}, {1385,35611}, {1483,3070}, {1587,3623}, {1702,16189}, {2067,11009}, {2098,35808}, {2099,35768}, {3241,35822}, {3242,35840}, {3622,5420}, {5418,12245}, {5441,35854}, {5597,35780}, {5598,35778}, {5603,6565}, {5604,35794}, {5605,35792}, {5886,35789}, {5901,10577}, {6396,10246}, {6419,7969}, {6453,7982}, {6484,11531}, {6560,7967}, {7968,33179}, {7970,35824}, {7971,35844}, {7972,35852}, {7973,35864}, {7974,35850}, {7975,35753}, {7976,35866}, {7977,35868}, {7978,35826}, {7979,12965}, {7980,35830}, {7981,35832}, {7983,35878}, {7984,12375}, {8000,35862}, {8192,35776}, {8210,35804}, {8211,35806}, {9583,11224}, {9884,35698}, {9933,35836}, {9997,35782}, {10698,35856}, {10705,35880}, {10800,35766}, {10944,35796}, {10950,35798}, {11396,35764}, {11910,35790}, {12898,35834}, {13099,35828}, {13100,35870}, {13702,35872}, {13822,35874}, {13902,35812}, {13959,35814}, {16200,35775}, {18525,35786}, {18526,23251}, {19907,35883}, {22713,35838}, {22791,35821}, {22867,35846}, {22912,35848}, {22969,35860}, {32298,35876}, {32394,35858}

X(35810) = reflection of X(6200) in X(35763)
X(35810) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35641, 372), (1, 35774, 35762), (6, 10247, 35811), (145, 485, 35842), (7969, 24680, 35642), (7969, 35642, 6419), (35641, 35762, 35774), (35762, 35774, 372), (35816, 35818, 372)

X(35811) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND 5th MIXTILINEAR

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

X(35811) lies on these lines: {1,371}, {5,35842}, {6,10247}, {8,10577}, {55,35785}, {56,35773}, {145,486}, {372,1482}, {485,10595}, {517,6396}, {519,35789}, {590,10283}, {615,5844}, {944,35821}, {946,35786}, {952,6565}, {1152,8148}, {1320,35883}, {1385,35610}, {1483,3071}, {1588,3623}, {1703,16189}, {2098,35809}, {2099,35769}, {3241,35823}, {3242,35841}, {3622,5418}, {5420,12245}, {5441,35855}, {5597,35779}, {5598,35781}, {5603,6564}, {5604,35793}, {5605,35795}, {5886,35788}, {5901,10576}, {6200,10246}, {6420,7968}, {6454,7982}, {6485,11531}, {6502,11009}, {6561,7967}, {7969,33179}, {7970,35825}, {7971,35845}, {7972,35853}, {7973,35865}, {7974,35851}, {7975,35754}, {7976,35867}, {7977,35869}, {7978,35827}, {7979,12971}, {7980,35833}, {7981,35831}, {7983,35879}, {7984,12376}, {8000,35863}, {8192,35777}, {8210,35807}, {8211,35805}, {8960,19066}, {9884,35699}, {9933,35837}, {9997,35783}, {10698,35857}, {10705,35881}, {10800,35767}, {10944,35797}, {10950,35799}, {11396,35765}, {11910,35791}, {12898,35835}, {13099,35829}, {13100,35871}, {13702,35873}, {13822,35875}, {13902,35815}, {13959,35813}, {16200,35774}, {18525,35787}, {18526,23261}, {19907,35882}, {22713,35839}, {22791,35820}, {22867,35849}, {22912,35847}, {22969,35861}, {32298,35877}, {32394,35859}

X(35811) = reflection of X(6396) in X(35762)
X(35811) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35642, 371), (1, 35775, 35763), (6, 10247, 35810), (145, 486, 35843), (7968, 24680, 35641), (7968, 35641, 6420), (35642, 35763, 35775), (35763, 35775, 371), (35817, 35819, 371)

X(35812) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND 3rd TRI-SQUARES-CENTRAL

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

X(35812) lies on these lines: {2,6419}, {3,8960}, {4,6453}, {5,371}, {6,3411}, {18,35731}, {20,485}, {140,6420}, {372,631}, {381,6425}, {382,1151}, {486,5067}, {548,3070}, {549,6454}, {615,16239}, {632,32788}, {637,6118}, {640,20065}, {858,8280}, {1327,3529}, {1328,5068}, {1587,15717}, {1588,7486}, {1656,3592}, {1907,13884}, {1991,7780}, {3090,10195}, {3298,31480}, {3311,5070}, {3316,3855}, {3525,19054}, {3528,6560}, {3530,6396}, {3533,19053}, {3543,3590}, {3594,5054}, {3830,6519}, {3832,6561}, {3843,6221}, {3851,6447}, {3853,6480}, {4301,13912}, {4317,31472}, {5188,22720}, {5319,31465}, {5412,15559}, {5420,7585}, {5881,13893}, {6228,7793}, {6229,7828}, {6409,13665}, {6410,18512}, {6417,8252}, {6421,31492}, {6426,15720}, {6427,13847}, {6428,15694}, {6431,13951}, {6435,19116}, {6448,15701}, {6449,17800}, {6470,18510}, {6484,31412}, {7493,8854}, {7584,32789}, {7746,12962}, {7775,11314}, {7817,11313}, {8276,9715}, {8974,35792}, {8975,35794}, {8980,14981}, {8983,11362}, {8987,35844}, {8988,35852}, {8991,35864}, {8992,35866}, {8993,35868}, {8994,15063}, {8995,12965}, {8997,35878}, {8998,12375}, {9541,9692}, {9588,13888}, {9624,31440}, {9646,15888}, {9647,9663}, {9648,9660}, {9657,13897}, {9661,13901}, {9670,13898}, {10992,13908}, {11241,20299}, {12963,31481}, {13848,35874}, {13879,35832}, {13883,35762}, {13885,35766}, {13887,35772}, {13889,35776}, {13890,35778}, {13891,35780}, {13892,35782}, {13894,35790}, {13895,35796}, {13896,35798}, {13899,35804}, {13900,35806}, {13902,35810}, {13904,31452}, {13905,31475}, {13906,35816}, {13907,35818}, {13909,35836}, {13910,35840}, {13911,35763}, {13913,35856}, {13914,35862}, {13915,35834}, {13916,35850}, {13917,35753}, {13918,35828}, {13919,35870}, {13920,35872}, {13921,35830}, {13922,35882}, {13923,35880}, {15069,19145}, {16148,35854}, {19030,31499}, {19105,31400}, {22763,35784}, {22876,35846}, {22921,35848}, {22976,35860}, {32303,35876}, {32399,35858}

X(35812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 8960, 35822), (3, 13846, 8960), (5, 8981, 31454), (5, 31454, 371), (6, 3526, 35813), (6, 13903, 35815), (20, 9540, 9680), (20, 9680, 6200), (140, 32787, 6420), (371, 10576, 6565), (485, 6200, 35820), (485, 9540, 6200), (485, 9680, 20), (590, 8981, 371), (590, 31454, 5), (1151, 8976, 6564), (3068, 5418, 372), (3526, 13903, 31487), (3526, 31487, 6), (8972, 9540, 485)

X(35813) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND 3rd TRI-SQUARES-CENTRAL

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

X(35813) lies on these lines: {2,6420}, {3,13847}, {4,6454}, {5,372}, {6,3411}, {20,486}, {140,6419}, {371,631}, {381,6426}, {382,1152}, {485,5067}, {498,31475}, {548,3071}, {549,6453}, {590,16239}, {591,7780}, {632,32787}, {638,6119}, {639,20065}, {858,8281}, {1327,5068}, {1328,3529}, {1587,7486}, {1588,9681}, {1656,3594}, {1907,13937}, {3090,10194}, {3297,31480}, {3312,5070}, {3317,3855}, {3525,19053}, {3528,6561}, {3530,6200}, {3533,19054}, {3543,3591}, {3592,5054}, {3830,6522}, {3832,6560}, {3843,6398}, {3851,6448}, {3853,6481}, {4301,13975}, {5076,17852}, {5188,22721}, {5413,15559}, {5418,7586}, {5881,13947}, {6228,7828}, {6229,7793}, {6409,18510}, {6410,13785}, {6418,8253}, {6422,31492}, {6425,15720}, {6427,15694}, {6428,13846}, {6432,8976}, {6436,19117}, {6447,15701}, {6450,17800}, {6471,18512}, {6485,33703}, {7493,8855}, {7583,32790}, {7746,12969}, {7775,11313}, {7817,11314}, {8277,9715}, {9588,13942}, {9624,35774}, {9657,13954}, {9670,13955}, {10992,13968}, {11242,20299}, {11362,13971}, {12376,13990}, {12971,13986}, {13849,35875}, {13880,35831}, {13933,35833}, {13936,35763}, {13938,35767}, {13940,35773}, {13943,35777}, {13944,35781}, {13945,35779}, {13946,35783}, {13948,35791}, {13949,35795}, {13950,35793}, {13952,35797}, {13953,35799}, {13956,35807}, {13957,35805}, {13958,35809}, {13959,35811}, {13962,31452}, {13963,35768}, {13964,35817}, {13965,35819}, {13967,14981}, {13969,15063}, {13970,35837}, {13972,35841}, {13973,35762}, {13974,35845}, {13976,35853}, {13977,35857}, {13978,35863}, {13979,35835}, {13980,35865}, {13981,35851}, {13982,35754}, {13983,35867}, {13984,35869}, {13985,35829}, {13987,35871}, {13988,35873}, {13989,35879}, {13991,35883}, {13992,35881}, {15069,19146}, {15888,18966}, {16149,35855}, {19102,31400}, {22764,35785}, {22877,35849}, {22922,35847}, {22977,35861}, {26363,31486}, {31401,31465}, {31423,31440}, {32304,35877}, {32400,35859}

X(35813) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 6420, 8960), (6, 3526, 35812), (6, 13961, 35814), (140, 32788, 6419), (372, 615, 10577), (372, 10577, 6564), (486, 6396, 35821), (486, 13935, 6396), (615, 13966, 372), (1152, 13951, 6565), (1588, 15717, 9681), (1656, 3594, 35822), (3069, 5420, 371), (3312, 8252, 10576), (5418, 7586, 35771), (13935, 13941, 486), (13973, 35762, 35843), (13993, 35256, 3071)

X(35814) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND 4th TRI-SQUARES-CENTRAL

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

X(35814) lies on these lines: {2,35770}, {4,372}, {6,3411}, {20,6481}, {140,35771}, {371,549}, {376,6487}, {382,6438}, {485,7486}, {548,6396}, {550,6485}, {615,3628}, {639,13757}, {1152,3534}, {1587,15022}, {1588,10304}, {1656,6432}, {1657,6430}, {3070,5066}, {3071,6454}, {3312,5055}, {3316,6436}, {3524,6484}, {3530,6480}, {3594,5072}, {3832,12818}, {3856,18762}, {5054,6431}, {5418,15709}, {5420,6419}, {6200,9543}, {6398,17800}, {6409,15706}, {6410,18510}, {6418,8252}, {6426,13785}, {6428,8253}, {6429,15693}, {6434,15696}, {6435,31454}, {6453,7582}, {6471,13665}, {6483,12103}, {6486,15712}, {6501,13846}, {12375,13990}, {12965,13986}, {13607,13936}, {13849,35874}, {13880,35832}, {13886,34091}, {13937,35764}, {13938,35766}, {13940,35772}, {13942,35774}, {13943,35776}, {13944,35778}, {13945,35780}, {13946,35782}, {13947,35788}, {13948,35790}, {13949,35792}, {13950,35794}, {13952,35796}, {13953,35798}, {13954,35800}, {13955,35802}, {13956,35804}, {13957,35806}, {13958,35808}, {13959,35810}, {13962,35809}, {13963,35769}, {13964,35816}, {13965,35818}, {13967,35824}, {13968,35698}, {13969,35826}, {13970,35836}, {13971,35641}, {13972,35840}, {13973,35842}, {13974,35844}, {13975,35610}, {13976,35852}, {13977,35856}, {13978,35862}, {13979,35834}, {13980,35864}, {13981,35850}, {13982,35753}, {13983,35866}, {13984,35868}, {13985,35828}, {13987,35870}, {13988,35872}, {13989,35878}, {13991,35882}, {13992,35880}, {14692,35825}, {15640,22615}, {15684,23261}, {16149,35854}, {18966,35768}, {19117,32790}, {22721,35838}, {22764,35784}, {22877,35846}, {22922,35848}, {22977,35860}, {32304,35876}, {32400,35858}

X(35814) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3526, 35815), (6, 13961, 35813), (372, 486, 35820), (372, 35787, 6460), (486, 6460, 35787), (486, 35820, 6565), (615, 6420, 10576), (3312, 10577, 35822), (3312, 13847, 10577), (3594, 13951, 6564), (5420, 7586, 6419), (6418, 8252, 8960), (6460, 35787, 35820), (13966, 32788, 371)

X(35815) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND 4th TRI-SQUARES-CENTRAL

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

X(35815) lies on these lines: {2,35771}, {4,371}, {6,3411}, {20,6480}, {140,35770}, {372,549}, {376,6486}, {382,6437}, {486,7486}, {548,6200}, {550,6484}, {590,3628}, {640,13637}, {1151,3534}, {1587,10304}, {1588,15022}, {1656,6431}, {1657,6429}, {3070,6453}, {3071,5066}, {3311,5055}, {3317,6435}, {3524,6485}, {3530,6481}, {3592,5072}, {3832,12819}, {3856,18538}, {5054,6432}, {5418,6420}, {5420,15709}, {6221,17800}, {6396,9540}, {6409,18512}, {6410,15706}, {6417,8253}, {6425,13665}, {6427,8252}, {6430,15693}, {6433,15696}, {6454,7581}, {6460,9680}, {6470,13785}, {6476,31414}, {6482,12103}, {6487,15712}, {6500,13847}, {8974,35795}, {8975,35793}, {8980,35825}, {8983,35642}, {8987,35845}, {8988,35853}, {8991,35865}, {8992,35867}, {8993,35869}, {8994,35827}, {8995,12971}, {8997,35879}, {8998,12376}, {9543,15683}, {9681,23249}, {9682,35479}, {13607,13883}, {13848,35875}, {13884,35765}, {13885,35767}, {13887,35773}, {13888,35775}, {13889,35777}, {13890,35781}, {13891,35779}, {13892,35783}, {13893,35789}, {13894,35791}, {13895,35797}, {13896,35799}, {13897,35801}, {13898,35803}, {13899,35807}, {13900,35805}, {13901,35809}, {13902,35811}, {13904,35808}, {13905,35768}, {13906,35817}, {13907,35819}, {13908,35699}, {13909,35837}, {13910,35841}, {13911,35843}, {13912,35611}, {13913,35857}, {13914,35863}, {13915,35835}, {13916,35851}, {13917,35754}, {13918,35829}, {13919,35871}, {13920,35873}, {13921,35833}, {13922,35883}, {13923,35881}, {13939,34089}, {14692,35824}, {15640,22644}, {15684,23251}, {16148,35855}, {18965,35769}, {19116,32789}, {22720,35839}, {22763,35785}, {22876,35849}, {22921,35847}, {22976,35861}, {32303,35877}, {32399,35859}

X(35815) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 3526, 35814), (6, 13903, 35812), (371, 485, 35821), (371, 3068, 8960), (371, 8960, 6564), (371, 35786, 6459), (485, 6459, 35786), (485, 35821, 6564), (590, 6419, 10577), (3311, 10576, 35823), (3592, 8976, 6565), (5418, 7585, 6420), (6459, 35786, 35821), (7583, 31454, 6200), (8960, 35821, 485), (8981, 32787, 372), (13903, 31487, 6)

X(35816) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND INNER-YFF TANGENTS

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

X(35816) lies on these lines: {1,372}, {6,12000}, {12,35796}, {371,10679}, {485,10528}, {486,10596}, {3070,32213}, {5420,10586}, {5552,10576}, {6200,11248}, {6396,16203}, {6419,19048}, {6560,10805}, {6564,10942}, {6565,10531}, {10803,35766}, {10834,35776}, {10878,35782}, {10915,35788}, {10929,35792}, {10930,35794}, {10955,35798}, {10956,35800}, {10958,35802}, {10965,35808}, {11239,35822}, {11400,35764}, {11509,35768}, {11729,35883}, {11881,35778}, {11882,35780}, {11914,35790}, {11955,35804}, {11956,35806}, {12115,35820}, {12189,35824}, {12356,35698}, {12375,13217}, {12381,35826}, {12430,35836}, {12594,35840}, {12648,35842}, {12686,35844}, {12703,35610}, {12749,35852}, {12775,35856}, {12874,35862}, {12905,35834}, {12965,13121}, {13094,35864}, {13104,35850}, {13105,35753}, {13109,35866}, {13112,35868}, {13118,35828}, {13130,35870}, {13132,35830}, {13134,35832}, {13189,35878}, {13278,35882}, {13313,35880}, {13716,35872}, {13839,35874}, {13906,35812}, {13964,35814}, {16154,35854}, {18542,35786}, {18545,23251}, {19047,35770}, {22731,35838}, {22768,35784}, {22886,35846}, {22931,35848}, {22982,35860}, {26333,35787}, {32309,35876}, {32405,35858}, {34339,35611}

X(35816) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12000, 35817), (372, 35810, 35818), (35641, 35809, 372)

X(35817) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND INNER-YFF TANGENTS

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

X(35817) lies on these lines: {1,371}, {6,12000}, {12,35797}, {372,10679}, {485,10596}, {486,10528}, {3071,32213}, {5418,10586}, {5552,10577}, {6200,16203}, {6396,11248}, {6420,19047}, {6561,10805}, {6564,10531}, {6565,10942}, {10803,35767}, {10834,35777}, {10878,35783}, {10915,35789}, {10929,35795}, {10930,35793}, {10955,35799}, {10956,35801}, {10958,35803}, {10965,35809}, {11239,35823}, {11400,35765}, {11509,35769}, {11729,35882}, {11881,35781}, {11882,35779}, {11914,35791}, {11955,35807}, {11956,35805}, {12115,35821}, {12189,35825}, {12356,35699}, {12376,13217}, {12381,35827}, {12430,35837}, {12594,35841}, {12648,35843}, {12686,35845}, {12703,35611}, {12749,35853}, {12775,35857}, {12874,35863}, {12905,35835}, {12971,13121}, {13094,35865}, {13104,35851}, {13105,35754}, {13109,35867}, {13112,35869}, {13118,35829}, {13130,35871}, {13132,35833}, {13134,35831}, {13189,35879}, {13278,35883}, {13313,35881}, {13716,35873}, {13839,35875}, {13906,35815}, {13964,35813}, {16154,35855}, {18542,35787}, {18545,23261}, {19048,35771}, {22731,35839}, {22768,35785}, {22886,35849}, {22931,35847}, {22982,35861}, {26333,35786}, {32309,35877}, {32405,35859}, {34339,35610}

X(35817) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12000, 35816), (371, 35811, 35819), (35642, 35808, 371)

X(35818) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES AND OUTER-YFF TANGENTS

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

X(35818) lies on these lines: {1,372}, {6,12001}, {11,35798}, {371,10680}, {485,10529}, {486,10597}, {3070,32214}, {5420,10587}, {6200,11249}, {6396,16202}, {6419,19050}, {6560,10806}, {6564,10943}, {6565,10532}, {10527,10576}, {10804,35766}, {10835,35776}, {10879,35782}, {10916,35788}, {10931,35792}, {10932,35794}, {10949,35796}, {10957,35800}, {10959,35802}, {10966,35784}, {11240,35822}, {11401,35764}, {11510,35772}, {11883,35778}, {11884,35780}, {11915,35790}, {11957,35804}, {11958,35806}, {12116,35820}, {12190,35824}, {12357,35698}, {12375,13218}, {12382,35826}, {12431,35836}, {12595,35840}, {12649,35842}, {12687,35844}, {12704,35610}, {12750,35852}, {12776,35856}, {12875,35862}, {12906,35834}, {12965,13122}, {13095,35864}, {13106,35850}, {13107,35753}, {13110,35866}, {13113,35868}, {13119,35828}, {13131,35870}, {13133,35830}, {13135,35832}, {13190,35878}, {13279,35882}, {13314,35880}, {13717,35872}, {13840,35874}, {13907,35812}, {13965,35814}, {15868,31472}, {16155,35854}, {18543,23251}, {18544,35786}, {18967,35768}, {19049,35770}, {22732,35838}, {22887,35846}, {22932,35848}, {22983,35860}, {26332,35787}, {32310,35876}, {32406,35858}

X(35818) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12001, 35819), (372, 35810, 35816), (35641, 35769, 372)

X(35819) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES AND OUTER-YFF TANGENTS

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

X(35819) lies on these lines: {1,371}, {6,12001}, {11,35799}, {372,10680}, {485,10597}, {486,10529}, {3071,32214}, {5418,10587}, {6200,16202}, {6396,11249}, {6420,19049}, {6561,10806}, {6564,10532}, {6565,10943}, {10527,10577}, {10804,35767}, {10835,35777}, {10879,35783}, {10916,35789}, {10931,35795}, {10932,35793}, {10949,35797}, {10957,35801}, {10959,35803}, {10966,35785}, {11240,35823}, {11401,35765}, {11510,35773}, {11883,35781}, {11884,35779}, {11915,35791}, {11957,35807}, {11958,35805}, {12116,35821}, {12190,35825}, {12357,35699}, {12376,13218}, {12382,35827}, {12431,35837}, {12595,35841}, {12649,35843}, {12687,33417}, {12704,35611}, {12750,35853}, {12776,35857}, {12875,35863}, {12906,35835}, {12971,13122}, {13095,35865}, {13106,35851}, {13107,35754}, {13110,35867}, {13113,35869}, {13119,35829}, {13131,35871}, {13133,35833}, {13135,35831}, {13190,35879}, {13279,35883}, {13314,35881}, {13717,35873}, {13840,35875}, {13907,35815}, {13965,35813}, {16155,35855}, {18543,23261}, {18544,35787}, {18967,35769}, {19050,35771}, {22732,35839}, {22887,35849}, {22932,35847}, {22983,35861}, {26332,35786}, {32310,35877}, {32406,35859}

X(35819) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12001, 35818), (371, 35811, 35817), (35642, 35768, 371)

X(35820) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO ABC

Barycentrics 3*a^4+2*S*a^2-(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(35820) = 3*X(371)-4*X(7583) = 5*X(371)-6*X(32787) = 2*X(371)-3*X(35822) = 2*X(486)-3*X(35830) = 3*X(3070)-2*X(7583) = 5*X(3070)-3*X(32787) = 4*X(3070)-3*X(35822) = 2*X(3103)-3*X(35838) = 10*X(7583)-9*X(32787) = 8*X(7583)-9*X(35822) = 4*X(32787)-5*X(35822)

The reciprocal orthologic center of these triangles is X(4)

X(35820) lies on these lines: {2,12818}, {3,3366}, {4,372}, {5,6396}, {6,382}, {20,485}, {22,8280}, {30,371}, {40,35788}, {55,35800}, {56,35802}, {115,12968}, {265,35827}, {355,35611}, {376,1327}, {381,1152}, {428,8855}, {487,33456}, {490,639}, {511,35866}, {515,35641}, {516,35610}, {517,35842}, {542,35698}, {546,615}, {548,18538}, {550,590}, {637,32421}, {640,11293}, {641,32489}, {944,35810}, {946,35762}, {1124,12953}, {1151,1657}, {1335,12943}, {1351,22809}, {1370,8854}, {1478,35809}, {1479,35769}, {1503,35840}, {1587,3146}, {1588,3543}, {1656,6410}, {2067,10483}, {2777,35826}, {2781,35876}, {2794,35824}, {2800,35852}, {2829,35856}, {3068,3529}, {3071,3627}, {3091,5420}, {3093,12173}, {3103,13749}, {3297,9668}, {3298,9655}, {3311,5073}, {3312,3830}, {3317,6477}, {3365,18586}, {3390,18587}, {3526,6412}, {3528,32785}, {3530,32789}, {3534,6409}, {3545,6487}, {3575,35764}, {3583,6502}, {3585,5414}, {3592,18512}, {3594,5076}, {3832,6481}, {3843,6398}, {3845,13966}, {3850,6485}, {3851,6450}, {3853,7584}, {3854,10194}, {3855,32786}, {3861,18762}, {4296,9632}, {4302,31472}, {5055,6456}, {5059,9541}, {5064,8281}, {5070,6452}, {5198,8277}, {5412,6240}, {5663,35834}, {5691,35774}, {5787,35845}, {5840,35882}, {5870,35794}, {5871,35792}, {6033,35879}, {6221,17800}, {6250,6811}, {6284,35808}, {6289,13449}, {6321,35825}, {6407,15685}, {6411,15696}, {6417,15684}, {6426,13951}, {6432,18510}, {6436,23273}, {6437,31487}, {6438,13961}, {6449,13846}, {6459,23267}, {6480,31454}, {6482,9542}, {6484,11001}, {6486,13925}, {6496,15689}, {7354,35768}, {7378,18290}, {7581,15682}, {7586,23263}, {7728,12376}, {7968,22793}, {7969,28160}, {8276,21312}, {8981,15704}, {9646,15338}, {9647,19030}, {9660,19028}, {9661,15326}, {9676,13346}, {9681,31414}, {9682,11413}, {9683,12082}, {9732,12602}, {9834,35778}, {9835,35780}, {9838,35804}, {9839,35806}, {9873,35782}, {10195,21735}, {10533,34785}, {10575,12239}, {10665,12375}, {10733,35835}, {10738,35857}, {10742,35883}, {10749,35829}, {10880,34797}, {10897,18563}, {10961,31833}, {11473,18560}, {11500,35772}, {11513,12605}, {12103,35255}, {12110,35766}, {12113,35790}, {12114,35784}, {12115,35816}, {12116,35818}, {12124,32499}, {12257,22646}, {12293,35837}, {12509,22592}, {12699,35642}, {12918,35881}, {12964,12965}, {12970,22802}, {13712,33364}, {13754,35836}, {13847,14269}, {13883,28150}, {13980,23324}, {13993,14893}, {14216,35865}, {14880,35767}, {15311,35864}, {15687,32788}, {17578,23259}, {18289,34608}, {18405,19087}, {18457,18562}, {18480,35789}, {18481,35763}, {18525,35843}, {18534,35777}, {19053,23275}, {22485,26289}, {22791,35811}, {22810,33878}, {28154,31439}, {31670,35841}, {32495,35832}

X(35820) = reflection of X(i) in X(j) for these (i,j): (371, 3070), (490, 639), (35821, 7748), (35832, 32495)
X(35820) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6564, 10576), (3, 23251, 6564), (4, 372, 6565), (4, 486, 35787), (4, 6460, 486), (4, 6560, 372), (6, 382, 35821), (20, 485, 6200), (20, 1131, 9540), (20, 23249, 485), (372, 486, 35814), (372, 35787, 486), (485, 6200, 35812), (485, 9680, 8972), (486, 6460, 372), (486, 6560, 6460), (486, 35787, 6565), (1131, 9540, 485), (3366, 3391, 6564), (6396, 35786, 5), (6460, 35787, 35814), (6560, 22644, 4), (6565, 35814, 486), (9540, 23249, 1131), (16964, 19107, 35821), (16965, 19106, 35821)

X(35821) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO ABC

Barycentrics 3*a^4-2*S*a^2-(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(35821) = 3*X(372)-4*X(7584) = 5*X(372)-6*X(32788) = 2*X(372)-3*X(35823) = 2*X(485)-3*X(35831) = 3*X(3071)-2*X(7584) = 5*X(3071)-3*X(32788) = 4*X(3071)-3*X(35823) = 2*X(3102)-3*X(35839) = 10*X(7584)-9*X(32788) = 8*X(7584)-9*X(35823) = 4*X(32788)-5*X(35823)

The reciprocal orthologic center of these triangles is X(4)

X(35821) lies on these lines: {2,12819}, {3,3367}, {4,371}, {5,6200}, {6,382}, {11,9647}, {12,9660}, {20,486}, {22,8281}, {30,372}, {40,35789}, {55,35801}, {56,35803}, {115,12963}, {265,35826}, {355,35610}, {376,1328}, {381,1151}, {428,8854}, {488,33457}, {489,640}, {511,35867}, {515,35642}, {516,35611}, {517,35843}, {542,35699}, {546,590}, {548,18762}, {550,615}, {578,9677}, {638,32419}, {639,11294}, {642,32488}, {944,35811}, {946,35763}, {1124,12943}, {1152,1657}, {1335,12953}, {1351,22810}, {1367,5413}, {1370,8855}, {1478,35808}, {1479,35768}, {1503,35841}, {1587,3543}, {1588,3146}, {1656,6409}, {2066,3585}, {2067,3583}, {2777,35827}, {2781,35877}, {2794,35825}, {2800,35853}, {2829,35857}, {3069,3529}, {3070,3627}, {3091,5418}, {3092,12173}, {3102,13748}, {3297,9655}, {3298,9668}, {3311,3830}, {3312,5073}, {3316,6476}, {3364,18587}, {3389,18586}, {3526,6411}, {3528,32786}, {3530,32790}, {3534,6410}, {3545,6486}, {3575,35765}, {3592,5076}, {3594,18510}, {3614,31499}, {3832,6480}, {3843,6221}, {3845,8981}, {3850,6484}, {3851,6449}, {3853,7583}, {3854,10195}, {3855,9680}, {3861,18538}, {5055,6455}, {5064,8280}, {5070,6451}, {5198,8276}, {5663,35835}, {5691,35775}, {5787,35844}, {5840,35883}, {5870,35793}, {5871,35795}, {6033,35878}, {6251,6813}, {6284,35809}, {6290,13449}, {6321,35824}, {6398,17800}, {6408,15685}, {6412,15696}, {6418,15684}, {6425,8976}, {6431,18512}, {6435,23267}, {6437,13903}, {6450,13847}, {6460,23273}, {6485,11001}, {6487,13993}, {6497,15689}, {6502,10483}, {7354,35769}, {7378,18289}, {7503,9683}, {7582,15682}, {7585,23253}, {7728,12375}, {7968,28160}, {7969,22793}, {7989,9582}, {8277,21312}, {8991,23324}, {9616,18492}, {9676,10539}, {9682,10594}, {9733,12601}, {9834,35781}, {9835,35779}, {9838,35807}, {9839,35805}, {9873,35783}, {10194,21735}, {10534,34785}, {10575,12240}, {10666,12376}, {10733,35834}, {10738,35856}, {10742,35882}, {10749,35828}, {10881,34797}, {10898,18563}, {10963,31833}, {11474,18560}, {11500,35773}, {11514,12605}, {12103,35256}, {12110,35767}, {12113,35791}, {12114,35785}, {12115,35817}, {12116,35819}, {12123,32498}, {12256,22617}, {12293,35836}, {12510,22591}, {12699,35641}, {12918,35880}, {12964,22802}, {12970,12971}, {13754,35837}, {13835,33365}, {13846,14269}, {13925,14893}, {13936,28150}, {13966,15704}, {14216,35864}, {14880,35766}, {15311,35865}, {15687,32787}, {16241,35733}, {17578,23249}, {18290,34608}, {18405,19088}, {18459,18562}, {18480,35788}, {18481,35762}, {18525,35842}, {18534,35776}, {18585,35740}, {19054,23269}, {22484,26288}, {22791,35810}, {22809,33878}, {31670,35840}, {32492,35833}

X(35821) = reflection of X(i) in X(j) for these (i,j): (372, 3071), (489, 640), (35820, 7748), (35833, 32492)
X(35821) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6565, 10577), (3, 23261, 6565), (4, 371, 6564), (4, 485, 35786), (4, 6459, 485), (4, 6561, 371), (6, 382, 35820), (371, 485, 35815), (371, 6564, 8960), (371, 35786, 485), (485, 6459, 371), (485, 6561, 6459), (485, 35786, 6564), (485, 35815, 8960), (3367, 3392, 6565), (6200, 35787, 5), (6459, 35786, 35815), (6561, 22615, 4), (6564, 35815, 485), (16964, 19107, 35820), (16965, 19106, 35820)

X(35822) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO ANTI-ARTZT

Barycentrics a^4+6*S*a^2+(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(35822) = X(371)+2*X(3070) = X(371)-4*X(7583) = 2*X(371)+X(35820) = 2*X(485)+X(35832) = X(3070)+2*X(7583) = 4*X(3070)-X(35820) = 2*X(3103)+X(35866) = 8*X(7583)+X(35820) = 2*X(10665)+X(35836) = 4*X(32787)+X(35820) = 2*X(35641)+X(35842)

The reciprocal orthologic center of these triangles is X(2)

X(35822) lies on these lines: {2,372}, {3,8960}, {4,1327}, {5,6420}, {6,13}, {20,6453}, {30,371}, {61,18587}, {62,18586}, {140,6454}, {376,3068}, {382,3592}, {397,15765}, {398,18585}, {428,35764}, {486,3545}, {491,32833}, {511,35838}, {519,35641}, {524,35840}, {528,35882}, {530,35753}, {531,35850}, {532,35848}, {533,35846}, {538,1991}, {539,10665}, {541,35826}, {543,35698}, {547,615}, {549,590}, {550,31454}, {551,35762}, {591,7775}, {639,32808}, {754,35868}, {1124,11238}, {1131,1328}, {1151,3534}, {1152,5054}, {1335,11237}, {1504,11648}, {1506,12969}, {1651,35790}, {1656,3594}, {1657,6425}, {1703,19875}, {2459,13663}, {3058,19028}, {3069,5071}, {3071,3845}, {3093,5064}, {3241,35810}, {3299,35803}, {3301,35801}, {3311,3830}, {3312,5055}, {3316,15709}, {3365,16267}, {3387,12822}, {3390,16268}, {3522,9680}, {3524,5418}, {3525,10195}, {3526,6426}, {3529,9681}, {3543,6561}, {3582,6502}, {3584,5414}, {3654,13911}, {3655,35763}, {3656,35642}, {3679,35774}, {3843,6427}, {3851,6428}, {3855,14226}, {4421,35772}, {4995,9646}, {5058,14537}, {5066,7584}, {5298,9661}, {5412,7576}, {5434,19030}, {5860,35794}, {5861,35792}, {6054,19058}, {6055,13908}, {6199,15684}, {6221,15681}, {6280,26468}, {6321,35699}, {6395,8252}, {6398,8253}, {6409,13903}, {6410,15693}, {6411,14093}, {6412,15700}, {6417,14269}, {6418,19709}, {6432,13951}, {6435,14893}, {6436,7586}, {6438,15723}, {6447,17800}, {6449,15689}, {6450,15701}, {6452,15718}, {6455,15695}, {6456,15707}, {6459,15682}, {6471,13961}, {6479,11540}, {6480,15686}, {6481,15702}, {6485,11812}, {6486,15690}, {6487,15708}, {7389,32809}, {7486,10194}, {7739,31411}, {7748,12962}, {7811,35782}, {7849,11313}, {7969,28204}, {8703,8981}, {8724,35879}, {8972,15692}, {9166,19055}, {9530,35828}, {9540,10304}, {9541,15683}, {9583,34628}, {9682,15078}, {9741,26620}, {9909,35776}, {9974,15534}, {10056,31472}, {10072,35769}, {10124,32789}, {10127,10961}, {10784,26330}, {10880,18559}, {11184,13926}, {11194,35784}, {11207,35778}, {11208,35780}, {11235,35796}, {11236,35798}, {11239,35816}, {11240,35818}, {11241,18400}, {11299,33441}, {11300,33443}, {11737,18762}, {12100,13925}, {12150,35766}, {12152,35804}, {12153,35806}, {13639,26289}, {13692,23002}, {13713,25201}, {13786,13833}, {13794,13832}, {13830,32471}, {13883,28194}, {13966,15699}, {13967,14971}, {13968,23514}, {14639,19057}, {18457,18564}, {19003,30308}, {19024,34697}, {19026,34746}, {19103,32499}, {19108,23234}, {19111,35835}, {20126,35827}, {20423,35841}, {21163,22720}, {22615,23253}, {22630,22640}, {22632,22641}, {25154,35754}, {25164,35851}, {28202,31439}, {31162,35775}, {32419,35830}, {34200,35255}

X(35822) = midpoint of X(i) and X(j) for these {i,j}: {3070, 32787}, {35698, 35878}
X(35822) = reflection of X(i) in X(j) for these (i,j): (371, 32787), (32787, 7583), (32808, 639), (35823, 5309)
X(35822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 8960, 35812), (6, 381, 35823), (6, 6564, 6565), (6, 13665, 6564), (13, 14, 6564), (371, 3070, 35820), (372, 485, 10576), (381, 35823, 6565), (485, 1587, 372), (3068, 6560, 6200), (3068, 23267, 6560), (3070, 7583, 371), (3163, 18390, 35823), (3545, 7581, 19053), (3545, 14241, 31412), (3545, 19053, 486), (5476, 7753, 35823), (6564, 35823, 381), (7581, 14241, 3545), (7581, 31412, 486), (13665, 18512, 6), (19053, 31412, 3545), (31715, 31718, 1327)

X(35823) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO ANTI-ARTZT

Barycentrics a^4-6*S*a^2+(b^2+c^2)*a^2-2*(b^2-c^2)^2 : :
X(35823) = X(372)+2*X(3071) = X(372)-4*X(7584) = 2*X(372)+X(35821) = 2*X(486)+X(35833) = X(3071)+2*X(7584) = 4*X(3071)-X(35821) = 2*X(3102)+X(35867) = 8*X(7584)+X(35821) = 2*X(10666)+X(35837) = 4*X(32788)+X(35821) = 2*X(35642)+X(35843)

The reciprocal orthologic center of these triangles is X(2)

X(35823) lies on these lines: {2,371}, {3,13847}, {4,1328}, {5,6419}, {6,13}, {20,6454}, {30,372}, {61,18586}, {62,18587}, {140,6453}, {376,3069}, {382,3594}, {397,18585}, {398,15765}, {428,35765}, {485,3545}, {492,32833}, {511,35839}, {519,35642}, {524,35841}, {528,35883}, {530,35754}, {531,35851}, {532,35847}, {533,35849}, {538,591}, {539,10666}, {541,35827}, {543,35699}, {547,590}, {549,615}, {551,35763}, {640,32809}, {754,35869}, {1124,11237}, {1132,1327}, {1151,5054}, {1152,3534}, {1335,11238}, {1505,11648}, {1506,12962}, {1651,35791}, {1656,3592}, {1657,6426}, {1702,19875}, {1991,7775}, {2066,3584}, {2067,3582}, {2460,13783}, {3058,19027}, {3068,5071}, {3070,3845}, {3092,5064}, {3241,35811}, {3299,35800}, {3301,23315}, {3311,5055}, {3312,3830}, {3317,15709}, {3364,16267}, {3374,12823}, {3389,16268}, {3412,35730}, {3523,9681}, {3524,5420}, {3525,9680}, {3526,6425}, {3543,6560}, {3628,31454}, {3654,13973}, {3655,35762}, {3656,35641}, {3679,35775}, {3843,6428}, {3851,6427}, {3855,14241}, {4421,35773}, {5062,14537}, {5066,7583}, {5079,31487}, {5413,7576}, {5434,19029}, {5860,35793}, {5861,35795}, {6054,19057}, {6055,13968}, {6199,8253}, {6221,8252}, {6279,26469}, {6321,35698}, {6395,15684}, {6398,15681}, {6409,15693}, {6410,13961}, {6411,15700}, {6412,14093}, {6417,19709}, {6418,14269}, {6431,8976}, {6435,7585}, {6436,14893}, {6437,15723}, {6448,17800}, {6449,15701}, {6450,15689}, {6451,15718}, {6455,15707}, {6456,15695}, {6460,15682}, {6470,13903}, {6478,11540}, {6480,15702}, {6481,15686}, {6484,11812}, {6486,15708}, {6487,15690}, {7388,32808}, {7486,10195}, {7748,12969}, {7811,35783}, {7849,11314}, {7968,28204}, {8703,13966}, {8724,35878}, {8980,14971}, {8981,15699}, {9166,19056}, {9530,35829}, {9541,13941}, {9647,18966}, {9660,13958}, {9741,26619}, {9909,35777}, {9975,15534}, {10056,35808}, {10072,35768}, {10124,32790}, {10127,10963}, {10304,13935}, {10783,26331}, {10881,18559}, {11184,13873}, {11194,35785}, {11207,35781}, {11208,35779}, {11235,35797}, {11236,35799}, {11239,35817}, {11240,35819}, {11242,18400}, {11299,33440}, {11300,33442}, {11737,18538}, {12100,13993}, {12150,35767}, {12152,35807}, {12153,35805}, {13666,13769}, {13674,13831}, {13710,32470}, {13759,26288}, {13812,23003}, {13836,25202}, {13908,23514}, {13936,28194}, {14639,19058}, {16962,35731}, {18459,18564}, {19004,30308}, {19023,34697}, {19025,34746}, {19104,32498}, {19109,23234}, {19110,35834}, {20126,35826}, {20423,35840}, {21163,22721}, {22601,22611}, {22603,22612}, {22644,23263}, {25154,35753}, {25164,35850}, {31162,35774}, {32421,35831}, {34200,35256}

X(35823) = midpoint of X(i) and X(j) for these {i,j}: {3071, 32788}, {35699, 35879}
X(35823) = reflection of X(i) in X(j) for these (i,j): (372, 32788), (32788, 7584), (32809, 640), (35822, 5309)
X(35823) = intersection, other than A,B,C, of conics {{A, B, C, X(265), X(1328)}} and {{A, B, C, X(486), X(1989)}}
X(35823) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 6419, 8960), (6, 381, 35822), (6, 6565, 6564), (6, 13785, 6565), (13, 14, 6565), (371, 486, 10577), (372, 3071, 35821), (381, 35822, 6564), (485, 7582, 35771), (486, 1588, 371), (3069, 6561, 6396), (3069, 23273, 6561), (3071, 7584, 372), (3163, 18390, 35822), (3545, 7582, 19054), (3545, 19054, 485), (5476, 7753, 35822), (6565, 35822, 381), (7582, 14226, 3545), (13785, 18510, 6), (31716, 31717, 1328)

X(35824) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999)

X(35824) lies on these lines: {3,35879}, {6,13}, {30,35698}, {98,372}, {99,6200}, {114,10576}, {147,485}, {148,6561}, {371,2782}, {385,32421}, {486,14651}, {690,35826}, {1151,13188}, {1587,5984}, {2783,35882}, {2787,35856}, {2794,35820}, {2799,35828}, {3023,35768}, {3027,35808}, {3070,35868}, {6055,13989}, {6226,35794}, {6227,35792}, {6251,22501}, {6321,35821}, {6396,12042}, {6419,19056}, {6453,23235}, {6560,9862}, {7970,35810}, {8980,14981}, {9541,20094}, {9860,35774}, {9861,35776}, {9864,35788}, {10053,35809}, {10069,35769}, {11710,35762}, {12131,35764}, {12176,35766}, {12178,35772}, {12179,35778}, {12180,35780}, {12181,35790}, {12182,35796}, {12183,35798}, {12184,35800}, {12185,35802}, {12186,35804}, {12187,35806}, {12189,35816}, {12190,35818}, {13967,35814}, {14639,35787}, {14692,35815}, {19055,35770}, {22504,35784}, {22505,35786}, {35838,35840}

X(35824) = reflection of X(35878) in X(371)
X(35824) = {X(6), X(12188)}-harmonic conjugate of X(35825)

X(35825) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st ANTI-BROCARD

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

The reciprocal orthologic center of these triangles is X(5999)

X(35825) lies on these lines: {3,35878}, {6,13}, {30,35699}, {98,371}, {99,6396}, {114,10577}, {147,486}, {148,6560}, {372,2782}, {385,32419}, {485,14651}, {690,35827}, {1152,13188}, {1588,5984}, {2783,35883}, {2787,35857}, {2794,35821}, {2799,35829}, {3023,35769}, {3027,35809}, {3071,35869}, {6055,8997}, {6200,12042}, {6226,35793}, {6227,35795}, {6250,22502}, {6321,35820}, {6420,19055}, {6454,23235}, {6561,9862}, {7970,35811}, {8960,11623}, {8980,35815}, {9860,35775}, {9861,35777}, {9864,35789}, {10053,35808}, {10069,35768}, {11710,35763}, {12131,35765}, {12176,35767}, {12178,35773}, {12179,35781}, {12180,35779}, {12181,35791}, {12182,35797}, {12183,35799}, {12184,35801}, {12185,35803}, {12186,35807}, {12187,35805}, {12189,35817}, {12190,35819}, {13967,14981}, {14639,35786}, {14692,35814}, {19056,35771}, {22504,35785}, {22505,35787}, {35839,35841}

X(35825) = reflection of X(35879) in X(372)
X(35825) = {X(6), X(12188)}-harmonic conjugate of X(35824)

X(35826) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO ANTI-ORTHOCENTROIDAL

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

The reciprocal orthologic center of these triangles is X(12112)

X(35826) lies on these lines: {3,12376}, {6,10620}, {30,35834}, {74,372}, {110,6200}, {113,10576}, {125,6565}, {146,485}, {265,35821}, {371,5663}, {399,1151}, {541,35822}, {542,35878}, {690,35824}, {1152,15041}, {1503,35876}, {1539,35786}, {1986,11473}, {2066,19470}, {2067,7727}, {2771,35882}, {2777,35820}, {2781,35840}, {3024,35768}, {3028,35808}, {3071,10264}, {3098,32292}, {3448,6561}, {5412,12292}, {5621,19146}, {6000,13287}, {6221,12308}, {6396,12041}, {6409,32609}, {6411,15040}, {6419,15054}, {6453,10819}, {6459,12317}, {6560,12244}, {6564,7728}, {7689,12892}, {7725,35792}, {7726,35794}, {7978,35810}, {8674,35856}, {8994,15063}, {9517,35828}, {9541,14683}, {9904,35774}, {9919,35776}, {9984,35782}, {10065,35809}, {10081,35769}, {10577,15061}, {10628,12965}, {10666,12901}, {10820,15055}, {11242,16219}, {11579,35841}, {11709,35762}, {12133,35764}, {12192,35766}, {12327,35772}, {12365,35778}, {12366,35780}, {12368,35788}, {12369,35790}, {12371,35796}, {12372,35798}, {12373,35800}, {12374,35802}, {12377,35804}, {12378,35806}, {12381,35816}, {12382,35818}, {12963,14901}, {12970,13289}, {13969,35814}, {14644,35787}, {17702,35836}, {17835,19088}, {19059,35770}, {20126,35823}, {22583,35784}, {33535,35775}

X(35826) = reflection of X(12375) in X(371)
X(35826) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 10620, 35827), (3448, 6561, 35835)

X(35827) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO ANTI-ORTHOCENTROIDAL

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

The reciprocal orthologic center of these triangles is X(12112)

X(35827) lies on these lines: {3,12375}, {6,10620}, {30,35835}, {74,371}, {110,6396}, {113,10577}, {125,6564}, {146,486}, {265,35820}, {372,5663}, {399,1152}, {541,35823}, {542,35879}, {690,35825}, {1151,15041}, {1503,35877}, {1539,35787}, {1986,11474}, {2771,35883}, {2777,35821}, {2781,35841}, {3024,35769}, {3028,35809}, {3070,10264}, {3098,32291}, {3448,6560}, {5413,12292}, {5414,19470}, {5621,19145}, {6000,13288}, {6200,12041}, {6398,12308}, {6410,32609}, {6412,15040}, {6420,15054}, {6454,10820}, {6460,12317}, {6502,7727}, {6561,12244}, {6565,7728}, {7689,12891}, {7725,35795}, {7726,35793}, {7978,35811}, {8674,35857}, {8960,20417}, {8994,35815}, {9517,35829}, {9904,35775}, {9919,35777}, {9984,35783}, {10065,35808}, {10081,35768}, {10576,15061}, {10628,12971}, {10665,12901}, {10819,15055}, {11241,16219}, {11579,35840}, {11709,35763}, {12133,35765}, {12192,35767}, {12327,35773}, {12365,35781}, {12366,35779}, {12368,35789}, {12369,35791}, {12371,35797}, {12372,35799}, {12373,35801}, {12374,35803}, {12377,35807}, {12378,35805}, {12381,35817}, {12382,35819}, {12964,13289}, {12968,14901}, {13969,15063}, {14644,35786}, {17702,35837}, {17835,19087}, {19060,35771}, {20126,35822}, {22583,35785}, {33535,35774}

X(35827) = reflection of X(12376) in X(372)
X(35827) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 10620, 35826), (3448, 6560, 35834)

X(35828) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st ANTI-ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(19158)

X(35828) lies on these lines: {3,35881}, {6,13115}, {112,6200}, {127,6565}, {132,10576}, {371,35880}, {372,1297}, {485,12384}, {1151,13310}, {2781,12375}, {2794,35878}, {2799,35824}, {2806,35856}, {2831,35882}, {3320,35808}, {6020,35768}, {6419,19094}, {6560,12253}, {6561,13219}, {6564,12918}, {9517,35826}, {9530,35822}, {10749,35821}, {12145,35764}, {12207,35766}, {12265,35762}, {12340,35772}, {12408,35774}, {12413,35776}, {12478,35778}, {12479,35780}, {12503,35782}, {12784,35788}, {12796,35790}, {12805,35792}, {12806,35794}, {12925,35796}, {12935,35798}, {12945,35800}, {12955,35802}, {12996,35804}, {12997,35806}, {13099,35810}, {13116,35809}, {13117,35769}, {13118,35816}, {13119,35818}, {13918,35812}, {13985,35814}, {19093,35770}, {19159,35784}, {19160,35786}

X(35828) = reflection of X(35880) in X(371)
X(35828) = {X(6), X(13115)}-harmonic conjugate of X(35829)

X(35829) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st ANTI-ORTHOSYMMEDIAL

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

The reciprocal orthologic center of these triangles is X(19158)

X(35829) lies on these lines: {3,35880}, {6,13115}, {112,6396}, {127,6564}, {132,10577}, {371,1297}, {372,35881}, {486,12384}, {1152,13310}, {2781,12376}, {2794,35879}, {2799,35825}, {2806,35857}, {2831,35883}, {3320,35809}, {6020,35769}, {6420,19093}, {6560,13219}, {6561,12253}, {6565,12918}, {9517,35827}, {9530,35823}, {10749,35820}, {12145,35765}, {12207,35767}, {12265,35763}, {12340,35773}, {12408,35775}, {12413,35777}, {12478,35781}, {12479,35779}, {12503,35783}, {12784,35789}, {12796,35791}, {12805,35795}, {12806,35793}, {12925,35797}, {12935,35799}, {12945,35801}, {12955,35803}, {12996,35807}, {12997,35805}, {13099,35811}, {13116,35808}, {13117,35768}, {13118,35817}, {13119,35819}, {13918,35815}, {13985,35813}, {19094,35771}, {19159,35785}, {19160,35787}

X(35829) = reflection of X(35881) in X(372)
X(35829) = {X(6), X(13115)}-harmonic conjugate of X(35828)

X(35830) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 3rd ANTI-TRI-SQUARES

Barycentrics -(a^2+c^2+b^2)*(a^2-b^2-c^2)*a^2+(2*a^4-2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S : :
X(35830) = 2*X(486)+X(35820)

The reciprocal orthologic center of these triangles is X(486)

X(35830) lies on these lines: {3,115}, {4,372}, {5,32494}, {6,12601}, {30,35874}, {76,6229}, {194,9867}, {371,5254}, {385,9991}, {485,487}, {488,32982}, {642,7389}, {671,13087}, {1587,12221}, {1588,22617}, {3068,12509}, {3070,3564}, {3103,6248}, {3311,22809}, {3734,11313}, {3818,23251}, {5025,33340}, {5286,6419}, {6119,7388}, {6200,12123}, {6280,35794}, {6281,35792}, {6300,11303}, {6301,11304}, {6420,7745}, {6561,12296}, {7816,11316}, {7841,35698}, {7844,11315}, {7861,11314}, {7980,35810}, {8281,32587}, {9906,35774}, {9921,35776}, {9986,35782}, {10067,35809}, {10083,35769}, {10577,13934}, {12147,35764}, {12210,35766}, {12268,35762}, {12343,35772}, {12484,35778}, {12485,35780}, {12787,35788}, {12799,35790}, {12928,35796}, {12938,35798}, {12948,35800}, {12958,35802}, {13003,35806}, {13081,35808}, {13132,35816}, {13133,35818}, {13921,35812}, {13928,31709}, {13929,31710}, {18989,35768}, {19104,35770}, {19146,23261}, {22595,35784}, {22596,35786}, {32419,35822}, {33349,33433}

X(35830) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12601, 35833), (486, 6251, 6565), (486, 6560, 12256), (3070, 35840, 35832)

X(35831) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 3rd ANTI-TRI-SQUARES

Barycentrics -(a^2+c^2+b^2)*(a^2-b^2-c^2)*a^2-(2*a^4-2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S : :
X(35831) = 2*X(485)+X(35821)

The reciprocal orthologic center of these triangles is X(486)

X(35831) lies on these lines: {3,115}, {4,371}, {5,32497}, {6,12602}, {30,35873}, {76,6228}, {194,9868}, {372,5254}, {385,9992}, {486,488}, {487,32982}, {641,7388}, {671,13088}, {1587,22646}, {1588,12222}, {3069,12510}, {3071,3564}, {3102,6248}, {3312,22810}, {3734,11314}, {3818,23261}, {5025,33341}, {5286,6420}, {6118,7389}, {6278,35793}, {6279,35795}, {6304,11303}, {6305,11304}, {6396,12124}, {6419,7745}, {6560,12297}, {7816,11315}, {7841,35699}, {7844,11316}, {7861,11313}, {7981,35811}, {8280,32588}, {9907,35775}, {9922,35777}, {9987,35783}, {10068,35808}, {10084,35768}, {10576,13882}, {12148,35765}, {12211,35767}, {12269,35763}, {12344,35773}, {12486,35781}, {12487,35779}, {12788,35789}, {12800,35791}, {12929,35797}, {12939,35799}, {12949,35801}, {12959,35803}, {13004,35807}, {13082,35809}, {13134,35817}, {13135,35819}, {13875,31709}, {13876,31710}, {13880,35813}, {18988,35769}, {19103,35771}, {19145,23251}, {22624,35785}, {22625,35787}, {32421,35823}, {33348,33432}

X(35831) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12602, 35832), (485, 6250, 6564), (485, 6561, 12257), (3071, 35841, 35833)

X(35832) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 4th ANTI-TRI-SQUARES

Barycentrics (3*a^4-8*(b^2+c^2)*a^2+5*b^4-6*b^2*c^2+5*c^4)*a^2-(6*a^4+2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S : :
X(35832) = 2*X(485)-3*X(35822)

The reciprocal orthologic center of these triangles is X(485)

X(35832) lies on these lines: {2,372}, {4,22646}, {6,12602}, {30,35872}, {371,12961}, {542,9974}, {2459,8960}, {3068,12510}, {3070,3564}, {3311,22810}, {3312,13881}, {5093,35833}, {5309,6418}, {6200,12124}, {6201,35787}, {6250,6565}, {6278,32499}, {6279,35792}, {6289,6564}, {6419,19103}, {6423,13873}, {6560,12257}, {6561,12297}, {7583,32497}, {7981,35810}, {9907,35774}, {9922,35776}, {9987,35782}, {10068,35809}, {10084,35769}, {12148,35764}, {12211,35766}, {12269,35762}, {12344,35772}, {12486,35778}, {12487,35780}, {12788,35788}, {12800,35790}, {12929,35796}, {12939,35798}, {12949,35800}, {12959,35802}, {13004,35804}, {13005,35806}, {13082,35808}, {13134,35816}, {13135,35818}, {13879,35812}, {13880,35814}, {18988,35768}, {19102,35770}, {22624,35784}, {22625,35786}, {32495,35820}

X(35832) = reflection of X(i) in X(j) for these (i,j): (32497, 7583), (35820, 32495)
X(35832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12602, 35831), (485, 641, 10576), (1587, 12222, 485), (3070, 35840, 35830), (22638, 22639, 6564)

X(35833) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 4th ANTI-TRI-SQUARES

Barycentrics (3*a^4-8*(b^2+c^2)*a^2+5*b^4-6*b^2*c^2+5*c^4)*a^2+(6*a^4+2*(b^2+c^2)*a^2-4*(b^2-c^2)^2)*S : :
X(35833) = 2*X(486)-3*X(35823)

The reciprocal orthologic center of these triangles is X(485)

X(35833) lies on these lines: {2,371}, {4,22617}, {6,12601}, {30,35875}, {372,12966}, {542,9975}, {2460,13934}, {3069,12509}, {3071,3564}, {3311,13881}, {3312,22809}, {5093,35832}, {5309,6417}, {6202,35786}, {6251,6564}, {6280,35793}, {6281,32498}, {6290,6565}, {6396,12123}, {6420,19104}, {6424,13926}, {6560,12296}, {6561,12256}, {7584,32494}, {7980,35811}, {9906,35775}, {9921,35777}, {9986,35783}, {10067,35808}, {10083,35768}, {12147,35765}, {12210,35767}, {12268,35763}, {12343,35773}, {12484,35781}, {12485,35779}, {12787,35789}, {12799,35791}, {12928,35797}, {12938,35799}, {12948,35801}, {12958,35803}, {13002,35807}, {13003,35805}, {13081,35809}, {13132,35817}, {13133,35819}, {13921,35815}, {13933,35813}, {18989,35769}, {19105,35771}, {22595,35785}, {22596,35787}, {32492,35821}

X(35833) = reflection of X(i) in X(j) for these (i,j): (32494, 7584), (35821, 32492)
X(35833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12601, 35830), (1588, 12221, 486), (3071, 35841, 35831), (22609, 22610, 6565)

X(35834) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO AAOA

Barycentrics (3*R^2+2*SA-2*SW)*S^2-(9*R^2-2*SW)*(SB+SC)*S-(27*R^2-8*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(7574)

X(35834) lies on these lines: {4,12376}, {6,12902}, {30,35826}, {110,6564}, {113,35786}, {125,6396}, {146,22644}, {265,372}, {371,12891}, {399,23251}, {485,12383}, {511,35876}, {542,35753}, {590,34153}, {615,11801}, {1327,9143}, {1511,10576}, {2771,35852}, {2777,35864}, {3070,12375}, {3448,6560}, {3818,32292}, {5420,15081}, {5663,35820}, {6200,12121}, {6419,19052}, {6565,10113}, {8253,15040}, {8960,10819}, {9927,12892}, {10088,35800}, {10091,35802}, {10577,10820}, {10628,35858}, {10666,19479}, {10733,35821}, {10778,35857}, {11005,35879}, {12140,35764}, {12201,35766}, {12261,35762}, {12334,35772}, {12407,35774}, {12412,35776}, {12466,35778}, {12467,35780}, {12501,35782}, {12778,35788}, {12790,35790}, {12803,35792}, {12804,35794}, {12889,35796}, {12890,35798}, {12894,35804}, {12895,35806}, {12896,35808}, {12898,35810}, {12903,35809}, {12904,35769}, {12905,35816}, {12906,35818}, {13211,35611}, {13287,18400}, {13915,35812}, {13979,35814}, {14683,23249}, {18968,35768}, {19051,35770}, {19110,35823}, {19478,35784}

X(35834) = reflection of X(12375) in X(3070)
X(35834) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12902, 35835), (3448, 6560, 35827), (10820, 14644, 10577)

X(35835) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO AAOA

Barycentrics (3*R^2+2*SA-2*SW)*S^2+(9*R^2-2*SW)*(SB+SC)*S-(27*R^2-8*SW)*SB*SC : :

The reciprocal orthologic center of these triangles is X(7574)

X(35835) lies on these lines: {4,12375}, {6,12902}, {30,35827}, {110,6565}, {113,35787}, {125,6200}, {146,22615}, {265,371}, {372,12892}, {399,23261}, {486,12383}, {511,35877}, {542,35754}, {590,11801}, {615,34153}, {1328,9143}, {1511,10577}, {2771,35853}, {2777,35865}, {3071,12376}, {3448,6561}, {3818,32291}, {5418,15081}, {5663,35821}, {6396,12121}, {6420,19051}, {6564,10113}, {8252,15040}, {9927,12891}, {10088,35801}, {10091,35803}, {10576,10819}, {10628,35859}, {10665,19479}, {10733,35820}, {10778,35856}, {11005,35878}, {12140,35765}, {12201,35767}, {12261,35763}, {12334,35773}, {12407,35775}, {12412,35777}, {12466,35781}, {12467,35779}, {12501,35783}, {12778,35789}, {12790,35791}, {12803,35795}, {12804,35793}, {12889,35797}, {12890,35799}, {12894,35807}, {12895,35805}, {12896,35809}, {12898,35811}, {12903,35808}, {12904,35768}, {12905,35817}, {12906,35819}, {13211,35610}, {13288,18400}, {13915,35815}, {13979,35813}, {14683,23259}, {18968,35769}, {19052,35771}, {19111,35822}, {19478,35785}

X(35835) = reflection of X(12376) in X(3071)
X(35835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12902, 35834), (3448, 6561, 35826), (10819, 14644, 10576)

X(35836) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO ARIES

Barycentrics (3*a^8-5*(b^2+c^2)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*a^4+2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S*a^2-3*(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
X(35836) = 2*X(10665)-3*X(35822)

The reciprocal orthologic center of these triangles is X(9833)

X(35836) lies on these lines: {6,10112}, {30,35864}, {68,372}, {155,6564}, {371,12424}, {485,6193}, {539,10665}, {542,12964}, {1069,35802}, {1147,10576}, {1154,35858}, {3070,3564}, {3157,35800}, {5412,14516}, {6146,11513}, {6200,12118}, {6396,12359}, {6419,19062}, {6560,11411}, {6565,9927}, {8281,12232}, {8909,8960}, {9896,35774}, {9908,35776}, {9923,35782}, {9928,35788}, {9929,35792}, {9930,35794}, {9933,35810}, {10055,35809}, {10071,35769}, {10577,14852}, {11265,32423}, {11417,34799}, {11442,11474}, {12134,35764}, {12164,23251}, {12193,35766}, {12259,35762}, {12293,35821}, {12328,35772}, {12415,35778}, {12416,35780}, {12418,35790}, {12422,35796}, {12423,35798}, {12426,35804}, {12427,35806}, {12428,35808}, {12430,35816}, {12431,35818}, {13754,35820}, {13909,35812}, {13970,35814}, {14984,35876}, {17702,35826}, {18970,35768}, {19061,35770}, {22659,35784}, {22660,35786}

X(35836) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12429, 35837), (9927, 10666, 6565)

X(35837) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO ARIES

Barycentrics (3*a^8-5*(b^2+c^2)*a^6+(3*b^4+2*b^2*c^2+3*c^4)*a^4-2*(a^2+b^2-c^2)*(a^2-b^2+c^2)*S*a^2-3*(b^4-c^4)*(b^2-c^2)*a^2+2*(b^2-c^2)^4)*(-a^2+b^2+c^2) : :
X(35837) = 2*X(10666)-3*X(35823)

The reciprocal orthologic center of these triangles is X(9833)

X(35837) lies on these lines: {6,10112}, {30,35865}, {68,371}, {155,6565}, {372,12425}, {486,6193}, {488,8910}, {539,10666}, {542,12970}, {1069,35803}, {1147,10577}, {1154,35859}, {3071,3564}, {3157,35801}, {5413,14516}, {6146,11514}, {6200,12359}, {6396,12118}, {6420,19061}, {6561,11411}, {6564,9927}, {8280,12231}, {8909,10576}, {9896,35775}, {9908,35777}, {9923,35783}, {9928,35789}, {9929,35795}, {9930,35793}, {9933,35811}, {10055,35808}, {10071,35768}, {11266,32423}, {11418,34799}, {11442,11473}, {12134,35765}, {12164,23261}, {12193,35767}, {12259,35763}, {12293,35820}, {12328,35773}, {12415,35781}, {12416,35779}, {12418,35791}, {12422,35797}, {12423,35799}, {12426,35807}, {12427,35805}, {12428,35809}, {12430,35817}, {12431,35819}, {13754,35821}, {13909,35815}, {13970,35813}, {14984,35877}, {17702,35827}, {18970,35769}, {19062,35771}, {22659,35785}, {22660,35787}

X(35837) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 12429, 35836), (8909, 14852, 10576), (9927, 10665, 6564)

X(35838) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st BROCARD-REFLECTED

Barycentrics 3*(b^2+c^2)*a^6+(b^4+7*b^2*c^2+c^4)*a^4-2*(b^2-c^2)^2*b^2*c^2+6*((b^2+c^2)*a^2+b^2*c^2)*S*a^2-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2 : :
X(35838) = 4*X(3070)-X(35866) = 2*X(3103)+X(35820)

The reciprocal orthologic center of these triangles is X(3)

X(35838) lies on these lines: {6,22728}, {262,372}, {371,35868}, {485,6194}, {511,35822}, {2782,35698}, {3070,32515}, {3103,13749}, {5188,22720}, {6200,22676}, {6419,19064}, {6420,14881}, {6454,11272}, {6560,7709}, {6564,7697}, {6565,22682}, {8960,9821}, {10576,15819}, {18971,35768}, {19063,35770}, {19089,35787}, {22475,35762}, {22480,35764}, {22521,35766}, {22556,35772}, {22622,22642}, {22650,35774}, {22655,35776}, {22668,35778}, {22672,35780}, {22678,35782}, {22680,35784}, {22681,35786}, {22697,35788}, {22698,35790}, {22699,35792}, {22700,35794}, {22703,35796}, {22704,35798}, {22705,35800}, {22706,35802}, {22709,35804}, {22710,35806}, {22711,35808}, {22713,35810}, {22721,35814}, {22729,35809}, {22730,35769}, {22731,35816}, {22732,35818}, {23251,35867}, {35824,35840}

X(35838) = {X(6), X(22728)}-harmonic conjugate of X(35839)

X(35839) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st BROCARD-REFLECTED

Barycentrics 3*(b^2+c^2)*a^6+(b^4+7*b^2*c^2+c^4)*a^4-2*(b^2-c^2)^2*b^2*c^2-6*((b^2+c^2)*a^2+b^2*c^2)*S*a^2-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2 : :
X(35839) = 4*X(3071)-X(35867) = 2*X(3102)+X(35821)

The reciprocal orthologic center of these triangles is X(3)

X(35839) lies on these lines: {6,22728}, {262,371}, {372,35869}, {486,6194}, {511,35823}, {2782,35699}, {3071,32515}, {3102,13748}, {5188,22721}, {6396,22676}, {6419,14881}, {6420,19063}, {6453,11272}, {6561,7709}, {6564,22682}, {6565,7697}, {10577,15819}, {18971,35769}, {19064,35771}, {19090,35786}, {22475,35763}, {22480,35765}, {22521,35767}, {22556,35773}, {22593,22613}, {22650,35775}, {22655,35777}, {22668,35781}, {22672,35779}, {22678,35783}, {22680,35785}, {22681,35787}, {22697,35789}, {22698,35791}, {22699,35795}, {22700,35793}, {22703,35797}, {22704,35799}, {22705,35801}, {22706,35803}, {22709,35807}, {22710,35805}, {22711,35809}, {22713,35811}, {22720,35815}, {22729,35808}, {22730,35768}, {22731,35817}, {22732,35819}, {23261,35866}, {35825,35841}

X(35839) = {X(6), X(22728)}-harmonic conjugate of X(35838)

X(35840) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st EHRMANN

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

The reciprocal orthologic center of these triangles is X(3)

X(35840) lies on these lines: {3,6}, {51,8855}, {69,485}, {141,10576}, {159,35776}, {193,637}, {265,35877}, {343,8280}, {394,8854}, {486,14853}, {518,35641}, {524,35822}, {542,35753}, {611,35809}, {613,35769}, {615,18583}, {732,35866}, {1327,11180}, {1352,6564}, {1386,35762}, {1469,35768}, {1503,35820}, {1589,8944}, {1843,35764}, {1991,13926}, {1992,32419}, {2781,35826}, {2854,12375}, {2987,5409}, {3056,35808}, {3070,3564}, {3071,21850}, {3093,12167}, {3242,35810}, {3416,35788}, {3618,5420}, {3751,35774}, {3818,35786}, {5032,26618}, {5412,6403}, {5418,10519}, {5480,6565}, {5846,35842}, {5921,23249}, {5965,35846}, {5969,35878}, {6393,35685}, {6460,14912}, {6560,6776}, {7583,34380}, {9024,35882}, {9830,35698}, {9970,12376}, {10577,14561}, {10665,34382}, {11427,18290}, {11579,35827}, {11898,13665}, {12177,35879}, {12222,32982}, {12329,35772}, {12452,35778}, {12453,35780}, {12583,35790}, {12586,35796}, {12587,35798}, {12588,35800}, {12589,35802}, {12590,35804}, {12591,35806}, {12594,35816}, {12595,35818}, {12970,34779}, {13847,14848}, {13910,35812}, {13972,35814}, {18440,23251}, {19140,32292}, {20423,35823}, {22769,35784}, {31670,35821}, {34146,35864}, {35824,35838}, {35872,35874}

X(35840) = midpoint of X(193) and X(637)
X(35840) = reflection of X(i) in X(j) for these (i,j): (69, 639), (371, 6), (11824, 13355), (35841, 5028)
X(35840) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1504, 371), (3, 3311, 8414), (3, 6566, 6396), (6, 1152, 5050), (6, 1350, 19145), (6, 1351, 35841), (6, 5102, 9975), (6, 12968, 1692), (182, 5107, 35841), (372, 6200, 2459), (372, 35792, 3103), (576, 5052, 35841), (1350, 19145, 6200), (1570, 5062, 6), (3106, 12231, 2037), (3312, 5093, 6), (3385, 3386, 6423), (3594, 15884, 372), (6398, 8589, 6396), (9974, 15884, 11482), (35792, 35794, 372)

X(35841) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st EHRMANN

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

The reciprocal orthologic center of these triangles is X(3)

X(35841) lies on these lines: {3,6}, {51,8854}, {69,486}, {141,10577}, {159,35777}, {193,638}, {265,35876}, {343,8281}, {394,8855}, {485,14853}, {518,35642}, {524,35823}, {542,35754}, {590,18583}, {591,13873}, {611,35808}, {613,35768}, {732,35867}, {1328,11180}, {1352,6565}, {1386,35763}, {1469,35769}, {1503,35821}, {1590,8940}, {1843,35765}, {1992,32421}, {2781,35827}, {2854,12376}, {2987,5408}, {3056,35809}, {3070,21850}, {3071,3564}, {3092,12167}, {3242,35811}, {3416,35789}, {3618,5418}, {3751,35775}, {3818,35787}, {5032,26617}, {5413,6403}, {5420,10519}, {5480,6564}, {5846,35843}, {5921,23259}, {5965,35847}, {5969,35879}, {6393,35684}, {6459,8982}, {6561,6776}, {7584,34380}, {8996,19459}, {9024,35883}, {9830,35699}, {9970,12375}, {10576,14561}, {10666,34382}, {11427,18289}, {11579,35826}, {11898,13785}, {12177,35878}, {12221,32982}, {12329,35773}, {12452,35781}, {12453,35779}, {12583,35791}, {12586,35797}, {12587,35799}, {12588,35801}, {12589,35803}, {12590,35807}, {12591,35805}, {12594,35817}, {12595,35819}, {12964,34779}, {13846,14848}, {13910,35815}, {13972,35813}, {18440,23261}, {19140,32291}, {20423,35822}, {22769,35785}, {31670,35820}, {34146,35865}, {35825,35839}, {35873,35875}

X(35841) = midpoint of X(193) and X(638)
X(35841) = reflection of X(i) in X(j) for these (i,j): (69, 640), (372, 6), (11825, 13355), (35840, 5028)
X(35841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1505, 372), (3, 3312, 8406), (6, 1151, 5050), (6, 1350, 19146), (6, 1351, 35840), (6, 5102, 9974), (6, 12963, 1692), (182, 5107, 35840), (371, 6396, 2460), (371, 35793, 3102), (576, 5052, 35840), (1350, 19146, 6396), (1570, 5058, 6), (3311, 5093, 6), (3371, 3372, 6424), (3592, 15883, 371), (6221, 8589, 6200), (6398, 33872, 2027), (9975, 15883, 11482), (35793, 35795, 371)

X(35842) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO EXCENTERS-MIDPOINTS

Barycentrics 3*a^4+2*S*a^2-4*(b+c)*a^3-(b^2-8*b*c+c^2)*a^2+4*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(35842) = 2*X(35641)-3*X(35822)

The reciprocal orthologic center of these triangles is X(10)

X(35842) lies on these lines: {1,10576}, {5,35811}, {6,12645}, {8,372}, {10,35762}, {80,35803}, {145,485}, {355,6565}, {371,952}, {515,35610}, {517,35820}, {519,35641}, {590,1483}, {758,35854}, {944,6200}, {1151,18526}, {1317,9661}, {1327,34631}, {1482,6564}, {1587,3621}, {1703,4677}, {2098,35802}, {2099,35800}, {2802,35852}, {3070,5844}, {3617,5420}, {3632,35774}, {3913,35772}, {4669,13975}, {4678,13935}, {5418,7967}, {5690,6396}, {5790,10577}, {5846,35840}, {5881,35775}, {6419,19066}, {6560,12245}, {7968,35789}, {8148,23251}, {8200,35779}, {8207,35781}, {10573,35769}, {10912,35796}, {10944,35768}, {10950,35808}, {11499,35785}, {12135,35764}, {12195,35766}, {12410,35776}, {12454,35778}, {12455,35780}, {12495,35782}, {12513,35784}, {12626,35790}, {12627,35792}, {12628,35794}, {12635,35798}, {12636,35804}, {12637,35806}, {12647,35809}, {12648,35816}, {12649,35818}, {13846,34748}, {13911,35763}, {13973,35814}, {14839,35866}, {18525,35821}, {19065,35770}, {19914,35857}, {22758,35773}, {22791,35786}

X(35842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35788, 10576), (6, 12645, 35843), (145, 485, 35810), (355, 35642, 6565), (13911, 35763, 35812)

X(35843) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO EXCENTERS-MIDPOINTS

Barycentrics 3*a^4-2*S*a^2-4*(b+c)*a^3-(b^2-8*b*c+c^2)*a^2+4*(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(35843) = 2*X(35642)-3*X(35823)

The reciprocal orthologic center of these triangles is X(10)

X(35843) lies on these lines: {1,10577}, {5,35810}, {6,12645}, {8,371}, {10,35763}, {80,35802}, {145,486}, {355,6564}, {372,952}, {515,35611}, {517,35821}, {519,35642}, {615,1483}, {758,35855}, {944,6396}, {1152,18526}, {1328,34631}, {1482,6565}, {1588,3621}, {1702,4677}, {2098,35803}, {2099,35801}, {2802,35853}, {3071,5844}, {3617,5418}, {3632,35775}, {3913,35773}, {4668,9583}, {4669,13912}, {4678,9540}, {5420,7967}, {5690,6200}, {5790,10576}, {5846,35841}, {5881,35774}, {6420,19065}, {6561,12245}, {7969,8960}, {8148,23261}, {8200,35780}, {8207,35778}, {10573,35768}, {10912,35797}, {10944,35769}, {10950,35809}, {11499,35784}, {12135,35765}, {12195,35767}, {12410,35777}, {12454,35781}, {12455,35779}, {12495,35783}, {12513,35785}, {12626,35791}, {12627,35795}, {12628,35793}, {12635,35799}, {12636,35807}, {12637,35805}, {12647,35808}, {12648,35817}, {12649,35819}, {13847,34748}, {13911,35815}, {13973,35762}, {14839,35867}, {18525,35820}, {19066,35771}, {19914,35856}, {22758,35772}, {22791,35787}

X(35843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 35789, 10577), (6, 12645, 35842), (145, 486, 35811), (355, 35641, 6564), (7969, 35788, 8960), (13973, 35762, 35813)

X(35844) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40)

X(35844) lies on these lines: {6,12684}, {84,372}, {371,971}, {485,6223}, {515,35610}, {1490,6200}, {1709,35809}, {2829,35852}, {5418,5658}, {5787,35821}, {6001,35641}, {6245,6565}, {6257,35794}, {6258,35792}, {6259,6564}, {6260,10576}, {6396,34862}, {6419,19068}, {6560,12246}, {6561,9799}, {7971,35810}, {7992,35774}, {8987,35812}, {9910,35776}, {10085,35769}, {10864,35775}, {12114,35762}, {12136,35764}, {12196,35766}, {12330,35772}, {12456,35778}, {12457,35780}, {12496,35782}, {12667,35788}, {12668,35790}, {12676,35796}, {12677,35798}, {12678,35800}, {12679,35802}, {12680,35808}, {12686,35816}, {12687,35818}, {12688,35768}, {13974,35814}, {18237,35784}, {18245,35804}, {18246,35806}, {19067,35770}, {22792,35786}, {33899,35789}

X(35844) = {X(6), X(12684)}-harmonic conjugate of X(35845)

X(35845) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO EXTOUCH

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

The reciprocal orthologic center of these triangles is X(40)

X(35845) lies on these lines: {6,12684}, {84,371}, {372,971}, {486,6223}, {515,35611}, {1490,6396}, {1709,35808}, {2829,35853}, {5420,5658}, {5787,35820}, {6001,35642}, {6200,34862}, {6245,6564}, {6257,35793}, {6258,35795}, {6259,6565}, {6260,10577}, {6420,19067}, {6560,9799}, {6561,12246}, {7971,35811}, {7992,35775}, {8987,35815}, {9910,35777}, {10085,35768}, {10864,35774}, {12114,35763}, {12136,35765}, {12196,35767}, {12330,35773}, {12456,35781}, {12457,35779}, {12496,35783}, {12667,35789}, {12668,35791}, {12676,35797}, {12677,35799}, {12678,35801}, {12679,35803}, {12680,35809}, {12686,35817}, {12687,33417}, {12688,35769}, {13974,35813}, {18237,35785}, {18245,35807}, {18246,35805}, {19068,35771}, {22792,35787}, {33899,35788}

X(35845) = {X(6), X(12684)}-harmonic conjugate of X(35844)

X(35846) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO INNER-FERMAT

Barycentrics 2*(3*a^4+(-(b^2+c^2)*a^2+4*b^2*c^2-2*b^4-2*c^4)*sqrt(3)-6*(b^2-c^2)^2)*(-2*sqrt(3)+1)*S+sqrt(3)*(11*a^4+8*b^4-22*b^2*c^2+8*c^4-19*(b^2+c^2)*a^2+(5*(b^2+c^2)*a^2-5*b^4-5*c^4)*sqrt(3))*a^2 : :

The reciprocal orthologic center of these triangles is X(3)

X(35846) lies on these lines: {6,16628}, {18,372}, {371,35850}, {485,628}, {533,35822}, {630,10576}, {642,30471}, {1587,22114}, {3070,35753}, {3365,22846}, {3366,11603}, {5965,35840}, {6200,22843}, {6419,19072}, {6560,22531}, {6564,16627}, {6565,22831}, {11740,35762}, {18972,35768}, {19069,35770}, {22481,35764}, {22522,35766}, {22557,35772}, {22651,35774}, {22656,35776}, {22669,35778}, {22673,35780}, {22745,35782}, {22771,35784}, {22794,35786}, {22851,35788}, {22852,35790}, {22853,35792}, {22854,35794}, {22857,35796}, {22858,35798}, {22859,35800}, {22860,35802}, {22863,35804}, {22864,35806}, {22865,35808}, {22867,35810}, {22876,35812}, {22877,35814}, {22884,35809}, {22885,35769}, {22886,35816}, {22887,35818}

X(35846) = {X(6), X(16628)}-harmonic conjugate of X(35849)

X(35847) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO INNER-FERMAT

Barycentrics -2*(3*a^4+(-(b^2+c^2)*a^2+4*b^2*c^2-2*b^4-2*c^4)*sqrt(3)-6*(b^2-c^2)^2)*(-2*sqrt(3)+1)*S+sqrt(3)*(11*a^4+8*b^4-22*b^2*c^2+8*c^4-19*(b^2+c^2)*a^2+(5*(b^2+c^2)*a^2-5*b^4-5*c^4)*sqrt(3))*a^2 : :

The reciprocal orthologic center of these triangles is X(3)

X(35847) lies on these lines: {6,16629}, {17,371}, {372,35754}, {486,627}, {532,35823}, {629,10577}, {641,30472}, {1588,22113}, {3071,35851}, {3389,22891}, {3392,11602}, {5965,35841}, {6396,22890}, {6420,19071}, {6561,22532}, {6564,22832}, {6565,16626}, {11739,35763}, {18973,35769}, {19070,35771}, {22482,35765}, {22523,35767}, {22558,35773}, {22652,35775}, {22657,35777}, {22670,35781}, {22674,35779}, {22746,35783}, {22772,35785}, {22795,35787}, {22896,35789}, {22897,35791}, {22898,35795}, {22899,35793}, {22902,35797}, {22903,35799}, {22904,35801}, {22905,35803}, {22908,35807}, {22909,35805}, {22910,35809}, {22912,35811}, {22921,35815}, {22922,35813}, {22929,35808}, {22930,35768}, {22931,35817}, {22932,35819}

X(35847) = {X(6), X(16629)}-harmonic conjugate of X(35848)

X(35848) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO OUTER-FERMAT

Barycentrics 2*(3*a^4-(-(b^2+c^2)*a^2+4*b^2*c^2-2*b^4-2*c^4)*sqrt(3)-6*(b^2-c^2)^2)*(2*sqrt(3)+1)*S-sqrt(3)*(11*a^4+8*b^4-22*b^2*c^2+8*c^4-19*(b^2+c^2)*a^2-(5*(b^2+c^2)*a^2-5*b^4-5*c^4)*sqrt(3))*a^2 : :

The reciprocal orthologic center of these triangles is X(3)

X(35848) lies on these lines: {6,16629}, {17,372}, {371,35753}, {485,627}, {532,35822}, {629,10576}, {642,30472}, {1587,22113}, {3070,35850}, {3390,22891}, {3391,11602}, {5965,35840}, {6200,22890}, {6419,19070}, {6560,22532}, {6564,16626}, {6565,22832}, {11739,35762}, {18973,35768}, {19071,35770}, {22482,35764}, {22523,35766}, {22558,35772}, {22652,35774}, {22657,35776}, {22670,35778}, {22674,35780}, {22746,35782}, {22772,35784}, {22795,35786}, {22896,35788}, {22897,35790}, {22898,35792}, {22899,35794}, {22902,35796}, {22903,35798}, {22904,35800}, {22905,35802}, {22908,35804}, {22909,35806}, {22910,35808}, {22912,35810}, {22921,35812}, {22922,35814}, {22929,35809}, {22930,35769}, {22931,35816}, {22932,35818}

X(35848) = {X(6), X(16629)}-harmonic conjugate of X(35847)

X(35849) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO OUTER-FERMAT

Barycentrics -2*(3*a^4-(-(b^2+c^2)*a^2+4*b^2*c^2-2*b^4-2*c^4)*sqrt(3)-6*(b^2-c^2)^2)*(2*sqrt(3)+1)*S-sqrt(3)*(11*a^4+8*b^4-22*b^2*c^2+8*c^4-19*(b^2+c^2)*a^2-(5*(b^2+c^2)*a^2-5*b^4-5*c^4)*sqrt(3))*a^2 : :

The reciprocal orthologic center of these triangles is X(3)

X(35849) lies on these lines: {6,16628}, {18,371}, {372,35851}, {486,628}, {533,35823}, {630,10577}, {641,30471}, {1588,22114}, {3071,35754}, {3364,22846}, {3367,11603}, {5965,35841}, {6396,22843}, {6420,19069}, {6561,22531}, {6564,22831}, {6565,16627}, {11740,35763}, {18972,35769}, {19072,35771}, {22481,35765}, {22522,35767}, {22557,35773}, {22651,35775}, {22656,35777}, {22669,35781}, {22673,35779}, {22745,35783}, {22771,35785}, {22794,35787}, {22851,35789}, {22852,35791}, {22853,35795}, {22854,35793}, {22857,35797}, {22858,35799}, {22859,35801}, {22860,35803}, {22863,35807}, {22864,35805}, {22865,35809}, {22867,35811}, {22876,35815}, {22877,35813}, {22884,35808}, {22885,35768}, {22886,35817}, {22887,35819}

X(35849) = {X(6), X(16628)}-harmonic conjugate of X(35846)

X(35850) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st INNER-FERMAT-DAO-NHI

Barycentrics 8*S^3+(2*sqrt(3)-3)*(SA+SW+sqrt(3)*SW)*S^2-(SB+SC)*(9*SA+sqrt(3)*SW)*S-sqrt(3)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(15682)

X(35850) lies on these lines: {6,13102}, {14,372}, {371,35846}, {485,617}, {530,35698}, {531,35822}, {542,35753}, {615,20253}, {619,10576}, {3070,35848}, {5474,6200}, {5479,6565}, {5613,6564}, {5617,35879}, {6269,35794}, {6271,35792}, {6321,35754}, {6396,6774}, {6419,19076}, {6420,16002}, {6454,20416}, {6560,6773}, {7974,35810}, {9900,35774}, {9915,35776}, {9981,35782}, {10061,35809}, {10077,35769}, {11303,33443}, {11706,35762}, {12141,35764}, {12204,35766}, {12336,35772}, {12470,35778}, {12471,35780}, {12780,35788}, {12792,35790}, {12921,35796}, {12931,35798}, {12941,35800}, {12951,35802}, {12988,35804}, {12989,35806}, {13075,35808}, {13104,35816}, {13106,35818}, {13916,35812}, {13981,35814}, {18975,35768}, {19075,35770}, {22774,35784}, {22797,35786}, {25164,35823}

X(35850) = {X(6), X(13102)}-harmonic conjugate of X(35851)

X(35851) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st INNER-FERMAT-DAO-NHI

Barycentrics 8*S^3-(-2*sqrt(3)-3)*(SA+SW-sqrt(3)*SW)*S^2-(SB+SC)*(9*SA-sqrt(3)*SW)*S-sqrt(3)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(15682)

X(35851) lies on these lines: {6,13102}, {14,371}, {372,35849}, {486,617}, {530,35699}, {531,35823}, {542,35754}, {590,20253}, {619,10577}, {3071,35847}, {5474,6396}, {5479,6564}, {5613,6565}, {5617,35878}, {6200,6774}, {6269,35793}, {6271,35795}, {6321,35753}, {6419,16002}, {6420,19075}, {6453,20416}, {6561,6773}, {7974,35811}, {9900,35775}, {9915,35777}, {9981,35783}, {10061,35808}, {10077,35768}, {11303,33442}, {11706,35763}, {12141,35765}, {12204,35767}, {12336,35773}, {12470,35781}, {12471,35779}, {12780,35789}, {12792,35791}, {12921,35797}, {12931,35799}, {12941,35801}, {12951,35803}, {12988,35807}, {12989,35805}, {13075,35809}, {13104,35817}, {13106,35819}, {13916,35815}, {13981,35813}, {18975,35769}, {19076,35771}, {22774,35785}, {22797,35787}, {25164,35822}

X(35851) = {X(6), X(13102)}-harmonic conjugate of X(35850)

X(35852) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO FUHRMANN

Barycentrics 3*a^7-5*(b+c)*a^6-(2*b^2-13*b*c+2*c^2)*a^5+8*(b^2-c^2)*(b-c)*a^4+2*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*S*a^2-(5*b^4+5*c^4+(5*b^2-18*b*c+5*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(b^2-10*b*c+c^2)*a^2+4*(b^2-c^2)^2*(b-c)^2*a-2*(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(3)

X(35852) lies on these lines: {6,12747}, {11,35762}, {80,372}, {100,35788}, {214,10576}, {355,35883}, {485,6224}, {515,35856}, {952,3070}, {1587,20085}, {2771,35834}, {2800,35820}, {2802,35842}, {2829,35844}, {5840,35610}, {6200,12119}, {6246,6565}, {6262,35794}, {6263,35792}, {6265,6564}, {6396,12619}, {6419,19078}, {6560,12247}, {7972,35810}, {8988,35812}, {9897,35774}, {9912,35776}, {10057,35809}, {10073,35769}, {10738,35642}, {12137,35764}, {12198,35766}, {12331,35772}, {12460,35778}, {12461,35780}, {12498,35782}, {12611,35786}, {12729,35790}, {12737,35796}, {12738,35798}, {12739,35800}, {12740,35802}, {12741,35804}, {12742,35806}, {12743,35808}, {12749,35816}, {12750,35818}, {12773,35784}, {13976,35814}, {18976,35768}, {19077,35770}, {19914,35611}

X(35852) = {X(6), X(12747)}-harmonic conjugate of X(35853)

X(35853) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO FUHRMANN

Barycentrics 3*a^7-5*(b+c)*a^6-(2*b^2-13*b*c+2*c^2)*a^5+8*(b^2-c^2)*(b-c)*a^4-2*(a^3-(b+c)*a^2-(b^2-3*b*c+c^2)*a+(b^2-c^2)*(b-c))*S*a^2-(5*b^4+5*c^4+(5*b^2-18*b*c+5*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(b^2-10*b*c+c^2)*a^2+4*(b^2-c^2)^2*(b-c)^2*a-2*(b^2-c^2)^3*(b-c) : :

The reciprocal orthologic center of these triangles is X(3)

X(35853) lies on these lines: {6,12747}, {11,35763}, {80,371}, {100,35789}, {214,10577}, {355,35882}, {486,6224}, {515,35857}, {952,3071}, {1588,20085}, {2771,35835}, {2800,35821}, {2802,35843}, {2829,35845}, {5840,35611}, {6200,12619}, {6246,6564}, {6262,35793}, {6263,35795}, {6265,6565}, {6396,12119}, {6420,19077}, {6561,12247}, {7972,35811}, {8988,35815}, {9897,35775}, {9912,35777}, {10057,35808}, {10073,35768}, {10738,35641}, {12137,35765}, {12198,35767}, {12331,35773}, {12460,35781}, {12461,35779}, {12498,35783}, {12611,35787}, {12729,35791}, {12737,35797}, {12738,35799}, {12739,35801}, {12740,35803}, {12741,35807}, {12742,35805}, {12743,35809}, {12749,35817}, {12750,35819}, {12773,35785}, {13976,35813}, {18976,35769}, {19078,35771}, {19914,35610}

X(35853) = {X(6), X(12747)}-harmonic conjugate of X(35852)

X(35854) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 2nd FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3)

X(35854) lies on these lines: {6,16150}, {30,35641}, {79,372}, {485,3648}, {758,35842}, {1587,20084}, {2771,35834}, {3647,10576}, {3649,35762}, {3652,6564}, {5441,35810}, {6200,16113}, {6419,19080}, {6560,16116}, {6565,16125}, {11684,35788}, {13743,35784}, {16114,35764}, {16115,35766}, {16117,35772}, {16118,35774}, {16119,35776}, {16121,35778}, {16122,35780}, {16123,35782}, {16129,35790}, {16130,35792}, {16131,35794}, {16138,35796}, {16139,35798}, {16140,35800}, {16141,35802}, {16142,35808}, {16148,35812}, {16149,35814}, {16152,35809}, {16153,35769}, {16154,35816}, {16155,35818}, {16161,35804}, {16162,35806}, {18977,35768}, {19079,35770}, {22798,35786}

X(35854) = {X(6), X(16150)}-harmonic conjugate of X(35855)

X(35855) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 2nd FUHRMANN

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

The reciprocal orthologic center of these triangles is X(3)

X(35855) lies on these lines: {6,16150}, {30,35642}, {79,371}, {486,3648}, {758,35843}, {1588,20084}, {2771,35835}, {3647,10577}, {3649,35763}, {3652,6565}, {5441,35811}, {6396,16113}, {6420,19079}, {6561,16116}, {6564,16125}, {11684,35789}, {13743,35785}, {16114,35765}, {16115,35767}, {16117,35773}, {16118,35775}, {16119,35777}, {16121,35781}, {16122,35779}, {16123,35783}, {16129,35791}, {16130,35795}, {16131,35793}, {16138,35797}, {16139,35799}, {16140,35801}, {16141,35803}, {16142,35809}, {16148,35815}, {16149,35813}, {16152,35808}, {16153,35768}, {16154,35817}, {16155,35819}, {16161,35807}, {16162,35805}, {18977,35769}, {19080,35771}, {22798,35787}

X(35855) = {X(6), X(16150)}-harmonic conjugate of X(35854)

X(35856) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40)

X(35856) lies on these lines: {3,35883}, {6,12773}, {11,6565}, {80,2067}, {100,6200}, {104,372}, {119,10576}, {149,6561}, {153,485}, {214,31453}, {371,952}, {515,35852}, {590,11698}, {1151,12331}, {1317,35808}, {1484,3071}, {1702,7993}, {1768,35774}, {2066,7972}, {2771,7969}, {2783,35878}, {2787,35824}, {2800,35641}, {2802,35610}, {2806,35828}, {2829,35820}, {2831,35880}, {5531,9583}, {5533,35803}, {6264,35775}, {6265,35763}, {6419,19082}, {6560,12248}, {6564,10742}, {8068,35801}, {8674,35826}, {9541,20095}, {9913,35776}, {10058,35809}, {10074,35769}, {10698,35810}, {10738,35821}, {10778,35835}, {11715,35762}, {12138,35764}, {12199,35766}, {12332,35772}, {12462,35778}, {12463,35780}, {12499,35782}, {12515,35611}, {12619,35789}, {12737,35642}, {12751,35788}, {12752,35790}, {12753,35792}, {12754,35794}, {12761,35796}, {12762,35798}, {12763,35800}, {12764,35802}, {12765,35804}, {12766,35806}, {12775,35816}, {12776,35818}, {13913,35812}, {13977,35814}, {19081,35770}, {19914,35843}, {22775,35784}, {22799,35786}

X(35856) = reflection of X(35882) in X(371)
X(35856) = lies on the circumconic with center X(7144))
X(35856) = {X(6), X(12773)}-harmonic conjugate of X(35857)

X(35857) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO INNER-GARCIA

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

The reciprocal orthologic center of these triangles is X(40)

X(35857) lies on these lines: {3,35882}, {6,12773}, {11,6564}, {80,6502}, {100,6396}, {104,371}, {119,10577}, {149,6560}, {153,486}, {372,952}, {515,35853}, {615,11698}, {1152,12331}, {1317,35809}, {1484,3070}, {1703,7993}, {1768,35775}, {2771,7968}, {2783,35879}, {2787,35825}, {2800,35642}, {2802,35611}, {2806,35829}, {2829,35821}, {2831,35881}, {5414,7972}, {5533,35802}, {6264,35774}, {6265,35762}, {6420,19081}, {6561,12248}, {6565,10742}, {8068,35800}, {8674,35827}, {8960,20418}, {9913,35777}, {10058,35808}, {10074,35768}, {10698,35811}, {10738,35820}, {10778,35834}, {11715,35763}, {12138,35765}, {12199,35767}, {12332,35773}, {12462,35781}, {12463,35779}, {12499,35783}, {12515,35610}, {12619,35788}, {12737,35641}, {12751,35789}, {12752,35791}, {12753,35795}, {12754,35793}, {12761,35797}, {12762,35799}, {12763,35801}, {12764,35803}, {12765,35807}, {12766,35805}, {12775,35817}, {12776,35819}, {13913,35815}, {13977,35813}, {19082,35771}, {19914,35842}, {22775,35785}, {22799,35787}

X(35857) = reflection of X(35883) in X(372)
X(35857) = {X(6), X(12773)}-harmonic conjugate of X(35856)

X(35858) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO HATZIPOLAKIS-MOSES

Barycentrics (R^2*(4*R^2+6*SA-7*SW)+2*(SB+SC)*SW)*S^2-(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S-(R^2*(52*R^2-41*SW)+8*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146)

X(35858) lies on these lines: {6,32402}, {371,18400}, {372,6145}, {485,32354}, {1154,35836}, {6200,32330}, {6288,10898}, {6419,32343}, {6560,32337}, {6564,32379}, {6565,32369}, {10576,32391}, {10628,35834}, {32331,35762}, {32332,35764}, {32335,35766}, {32336,35768}, {32342,35770}, {32347,35772}, {32356,35774}, {32357,35776}, {32360,35778}, {32361,35780}, {32362,35782}, {32363,35784}, {32364,35786}, {32371,35788}, {32372,35790}, {32373,35792}, {32374,35794}, {32380,35796}, {32381,35798}, {32382,35800}, {32383,35802}, {32388,35804}, {32389,35806}, {32390,35808}, {32394,35810}, {32399,35812}, {32400,35814}, {32403,35809}, {32404,35769}, {32405,35816}, {32406,35818}

X(35858) = {X(6), X(32402)}-harmonic conjugate of X(35859)

X(35859) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO HATZIPOLAKIS-MOSES

Barycentrics (R^2*(4*R^2+6*SA-7*SW)+2*(SB+SC)*SW)*S^2+(5*R^2-2*SW)*(4*R^2-SW)*(SB+SC)*S-(R^2*(52*R^2-41*SW)+8*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(6146)

X(35859) lies on these lines: {6,32402}, {371,6145}, {372,18400}, {486,32354}, {1154,35837}, {6288,10897}, {6396,32330}, {6420,32342}, {6561,32337}, {6564,32369}, {6565,32379}, {10577,32391}, {10628,35835}, {32331,35763}, {32332,35765}, {32335,35767}, {32336,35769}, {32343,35771}, {32347,35773}, {32356,35775}, {32357,35777}, {32360,35781}, {32361,35779}, {32362,35783}, {32363,35785}, {32364,35787}, {32371,35789}, {32372,35791}, {32373,35795}, {32374,35793}, {32380,35797}, {32381,35799}, {32382,35801}, {32383,35803}, {32388,35807}, {32389,35805}, {32390,35809}, {32394,35811}, {32399,35815}, {32400,35813}, {32403,35808}, {32404,35768}, {32405,35817}, {32406,35819}

X(35859) = {X(6), X(32402)}-harmonic conjugate of X(35858)

X(35860) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 3rd HATZIPOLAKIS

Barycentrics (R^2*(24*R^2+10*SA-15*SW)+2*(SB+SC)*SW)*S^2-(7*R^2*(8*R^2-3*SW)+2*SW^2)*(SB+SC)*S-(R^2*(184*R^2-77*SW)+8*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(12241)

X(35860) lies on these lines: {6,22979}, {371,22960}, {372,22466}, {485,22647}, {6200,22951}, {6419,19084}, {6560,22533}, {6564,22955}, {6565,22833}, {10576,22966}, {18978,35768}, {19083,35770}, {22476,35762}, {22483,35764}, {22524,35766}, {22559,35772}, {22653,35774}, {22658,35776}, {22671,35778}, {22675,35780}, {22747,35782}, {22776,35784}, {22800,35786}, {22941,35788}, {22943,35790}, {22945,35792}, {22947,35794}, {22956,35796}, {22957,35798}, {22958,35800}, {22959,35802}, {22963,35804}, {22964,35806}, {22965,35808}, {22969,35810}, {22976,35812}, {22977,35814}, {22980,35809}, {22981,35769}, {22982,35816}, {22983,35818}

X(35860) = {X(6), X(22979)}-harmonic conjugate of X(35861)

X(35861) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 3rd HATZIPOLAKIS

Barycentrics (R^2*(24*R^2+10*SA-15*SW)+2*(SB+SC)*SW)*S^2+(7*R^2*(8*R^2-3*SW)+2*SW^2)*(SB+SC)*S-(R^2*(184*R^2-77*SW)+8*SW^2)*SB*SC : :

The reciprocal orthologic center of these triangles is X(12241)

X(35861) lies on these lines: {6,22979}, {371,22466}, {372,22961}, {486,22647}, {6396,22951}, {6420,19083}, {6561,22533}, {6564,22833}, {6565,22955}, {10577,22966}, {18978,35769}, {19084,35771}, {22476,35763}, {22483,35765}, {22524,35767}, {22559,35773}, {22653,35775}, {22658,35777}, {22671,35781}, {22675,35779}, {22747,35783}, {22776,35785}, {22800,35787}, {22941,35789}, {22943,35791}, {22945,35795}, {22947,35793}, {22956,35797}, {22957,35799}, {22958,35801}, {22959,35803}, {22963,35807}, {22964,35805}, {22965,35809}, {22969,35811}, {22976,35815}, {22977,35813}, {22980,35808}, {22981,35768}, {22982,35817}, {22983,35819}

X(35861) = {X(6), X(22979)}-harmonic conjugate of X(35860)

X(35862) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO HUTSON EXTOUCH

Barycentrics a*(a^9-6*(b^2+b*c+c^2)*a^7+2*(b+c)*(b^2+4*b*c+c^2)*a^6+2*(3*b^2+b*c+2*c^2)*(2*b^2+b*c+3*c^2)*a^5-2*(b+c)*(3*b^4+3*c^4+2*(4*b^2+13*b*c+4*c^2)*b*c)*a^4-2*(5*b^6+5*c^6+(b^2+12*b*c+c^2)*(b^2-5*b*c+c^2)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(5*b^2+19*b*c+5*c^2)*b*c)*a^2-2*(a^6-2*(b+c)*a^5-(b^2+10*b*c+c^2)*a^4+4*(b+c)^3*a^3-(b^4+c^4-2*(6*b^2+5*b*c+6*c^2)*b*c)*a^2-2*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2*(b-c)^2)*S*a+(b^2-c^2)^2*(b-c)^2*(3*b^2+4*b*c+3*c^2)*a-2*(b^2-c^2)^4*(b+c)) : :

The reciprocal orthologic center of these triangles is X(40)

X(35862) lies on these lines: {6,12872}, {372,7160}, {485,9874}, {6200,12120}, {6419,19086}, {6560,12249}, {6564,12856}, {6565,12599}, {8000,35810}, {9898,35774}, {10059,35809}, {10075,35769}, {10576,12864}, {12139,35764}, {12200,35766}, {12260,35762}, {12333,35772}, {12411,35776}, {12464,35778}, {12465,35780}, {12500,35782}, {12777,35788}, {12789,35790}, {12801,35792}, {12802,35794}, {12857,35796}, {12858,35798}, {12859,35800}, {12860,35802}, {12861,35804}, {12862,35806}, {12863,35808}, {12874,35816}, {12875,35818}, {13914,35812}, {13978,35814}, {18979,35768}, {19085,35770}, {22777,35784}, {22801,35786}

X(35862) = {X(6), X(12872)}-harmonic conjugate of X(35863)

X(35863) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO HUTSON EXTOUCH

Barycentrics a*(a^9-6*(b^2+b*c+c^2)*a^7+2*(b+c)*(b^2+4*b*c+c^2)*a^6+2*(3*b^2+b*c+2*c^2)*(2*b^2+b*c+3*c^2)*a^5-2*(b+c)*(3*b^4+3*c^4+2*(4*b^2+13*b*c+4*c^2)*b*c)*a^4-2*(5*b^6+5*c^6+(b^2+12*b*c+c^2)*(b^2-5*b*c+c^2)*b*c)*a^3+2*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(5*b^2+19*b*c+5*c^2)*b*c)*a^2+2*(a^6-2*(b+c)*a^5-(b^2+10*b*c+c^2)*a^4+4*(b+c)^3*a^3-(b^4+c^4-2*(6*b^2+5*b*c+6*c^2)*b*c)*a^2-2*(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)*a+(b^2-c^2)^2*(b-c)^2)*S*a+(b^2-c^2)^2*(b-c)^2*(3*b^2+4*b*c+3*c^2)*a-2*(b^2-c^2)^4*(b+c)) : :

The reciprocal orthologic center of these triangles is X(40)

X(35863) lies on these lines: {6,12872}, {371,7160}, {486,9874}, {6396,12120}, {6420,19085}, {6561,12249}, {6564,12599}, {6565,12856}, {8000,35811}, {9898,35775}, {10059,35808}, {10075,35768}, {10577,12864}, {12139,35765}, {12200,35767}, {12260,35763}, {12333,35773}, {12411,35777}, {12464,35781}, {12465,35779}, {12500,35783}, {12777,35789}, {12789,35791}, {12801,35795}, {12802,35793}, {12857,35797}, {12858,35799}, {12859,35801}, {12860,35803}, {12861,35807}, {12862,35805}, {12863,35809}, {12874,35817}, {12875,35819}, {13914,35815}, {13978,35813}, {18979,35769}, {19086,35771}, {22777,35785}, {22801,35787}

X(35863) = {X(6), X(12872)}-harmonic conjugate of X(35862)

X(35864) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO MIDHEIGHT

Barycentrics (SB+SC)*(2*S^2-(4*R^2-SW)*S-(28*R^2-4*SA-3*SW)*SA) : :
X(35864) = 7*X(371)-6*X(11241) = 3*X(371)-2*X(12964) = 9*X(11241)-7*X(12964)

The reciprocal orthologic center of these triangles is X(4)

X(35864) lies on these lines: {6,13093}, {30,35836}, {64,372}, {371,6000}, {485,6225}, {1151,12315}, {1152,35450}, {1498,6200}, {1853,35787}, {2777,35834}, {2883,10576}, {3357,6396}, {5412,12290}, {5418,5656}, {5878,6564}, {6001,35610}, {6241,11473}, {6247,6565}, {6266,35794}, {6267,35792}, {6285,35768}, {6409,32063}, {6411,14530}, {6419,19088}, {6560,12250}, {6561,12324}, {7355,35808}, {7973,35810}, {8991,35812}, {9899,35774}, {9914,35776}, {10060,35809}, {10076,35769}, {10575,11513}, {10605,35765}, {11381,35764}, {12202,35766}, {12262,35762}, {12335,35772}, {12468,35778}, {12469,35780}, {12502,35782}, {12779,35788}, {12791,35790}, {12920,35796}, {12930,35798}, {12940,35800}, {12950,35802}, {12986,35804}, {12987,35806}, {13094,35816}, {13095,35818}, {13980,35814}, {14216,35821}, {15311,35820}, {19087,35770}, {22615,32064}, {22778,35784}, {22802,35786}, {34146,35840}

X(35864) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 13093, 35865), (3357, 12970, 6396)

X(35865) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO MIDHEIGHT

Barycentrics (SB+SC)*(2*S^2+(4*R^2-SW)*S-(28*R^2-4*SA-3*SW)*SA) : :
X(35865) = 7*X(372)-6*X(11242) = 3*X(372)-2*X(12970)

The reciprocal orthologic center of these triangles is X(4)

X(35865) lies on these lines: {6,13093}, {30,35837}, {64,371}, {372,6000}, {486,6225}, {1151,35450}, {1152,12315}, {1498,6396}, {1853,35786}, {2777,35835}, {2883,10577}, {3357,6200}, {5413,12290}, {5420,5656}, {5878,6565}, {6001,35611}, {6241,11474}, {6247,6564}, {6266,35793}, {6267,35795}, {6285,35769}, {6410,32063}, {6412,14530}, {6420,19087}, {6560,12324}, {6561,12250}, {7355,35809}, {7973,35811}, {8991,35815}, {9899,35775}, {9914,35777}, {10060,35808}, {10076,35768}, {10575,11514}, {10605,35764}, {11381,35765}, {12202,35767}, {12262,35763}, {12335,35773}, {12468,35781}, {12469,35779}, {12502,35783}, {12779,35789}, {12791,35791}, {12920,35797}, {12930,35799}, {12940,35801}, {12950,35803}, {12986,35807}, {12987,35805}, {13094,35817}, {13095,35819}, {13980,35813}, {14216,35820}, {15311,35821}, {19088,35771}, {22644,32064}, {22778,35785}, {22802,35787}, {34146,35841}

X(35865) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 13093, 35864), (3357, 12964, 6200)

X(35866) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st NEUBERG

Barycentrics (b^2+c^2)*a^6+3*(b^2+c^2)*b^2*c^2*a^2-(b^4+3*b^2*c^2+c^4)*a^4-2*(b^2-c^2)^2*b^2*c^2+2*((b^2+c^2)*a^2+b^2*c^2)*S*a^2 : :
X(35866) = 4*X(3070)-3*X(35838) = 2*X(3103)-3*X(35822)

The reciprocal orthologic center of these triangles is X(3)

X(35866) lies on these lines: {2,32471}, {6,13108}, {39,10576}, {76,372}, {194,485}, {371,2782}, {384,35766}, {511,35820}, {538,1991}, {590,32448}, {730,35641}, {732,35840}, {1587,20081}, {3068,32470}, {3070,32515}, {3095,6564}, {3102,6248}, {5418,7709}, {5420,31276}, {5969,35698}, {6200,11257}, {6272,35794}, {6273,35792}, {6419,19090}, {6560,12251}, {7697,10577}, {7976,35810}, {8976,32519}, {8992,35812}, {9902,35774}, {9917,35776}, {9983,35782}, {10063,35809}, {10079,35769}, {11361,32421}, {12143,35764}, {12263,35762}, {12338,35772}, {12474,35778}, {12475,35780}, {12782,35788}, {12794,35790}, {12836,35802}, {12837,35800}, {12923,35796}, {12933,35798}, {12992,35804}, {12993,35806}, {13077,35808}, {13109,35816}, {13110,35818}, {13665,32520}, {13885,23235}, {13983,35814}, {14839,35842}, {14881,35786}, {18982,35768}, {19089,35770}, {22779,35784}, {23261,35839}

X(35866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 13108, 35867), (3102, 6248, 6565)

X(35867) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st NEUBERG

Barycentrics (b^2+c^2)*a^6+3*(b^2+c^2)*b^2*c^2*a^2-(b^4+3*b^2*c^2+c^4)*a^4-2*(b^2-c^2)^2*b^2*c^2-2*((b^2+c^2)*a^2+b^2*c^2)*S*a^2 : :
X(35867) = 4*X(3071)-3*X(35839) = 2*X(3102)-3*X(35823)

The reciprocal orthologic center of these triangles is X(3)

X(35867) lies on these lines: {2,32470}, {6,13108}, {39,10577}, {76,371}, {194,486}, {372,2782}, {384,35767}, {511,35821}, {538,591}, {615,32448}, {730,35642}, {732,35841}, {1588,20081}, {3069,32471}, {3071,32515}, {3095,6565}, {3103,6248}, {5418,31276}, {5420,7709}, {5969,35699}, {6272,35793}, {6273,35795}, {6396,11257}, {6420,19089}, {6561,12251}, {7697,10576}, {7976,35811}, {8992,35815}, {9902,35775}, {9917,35777}, {9983,35783}, {10063,35808}, {10079,35768}, {11361,32419}, {12143,35765}, {12263,35763}, {12338,35773}, {12474,35781}, {12475,35779}, {12782,35789}, {12794,35791}, {12836,35803}, {12837,35801}, {12923,35797}, {12933,35799}, {12992,35807}, {12993,35805}, {13077,35809}, {13109,35817}, {13110,35819}, {13785,32520}, {13938,23235}, {13951,32519}, {13983,35813}, {14839,35843}, {14881,35787}, {18982,35769}, {19090,35771}, {22779,35785}, {23251,35838}

X(35867) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 13108, 35866), (3103, 6248, 6564)

X(35868) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3)

X(35868) lies on these lines: {6,382}, {83,372}, {371,35838}, {485,2896}, {732,35840}, {754,35822}, {1587,20088}, {3070,35824}, {3103,35878}, {6200,12122}, {6249,6565}, {6274,35794}, {6275,35792}, {6287,6564}, {6292,10576}, {6419,19092}, {6560,12252}, {7977,35810}, {8993,35812}, {9903,35774}, {9918,35776}, {10064,35809}, {10080,35769}, {12144,35764}, {12206,35766}, {12264,35762}, {12339,35772}, {12476,35778}, {12477,35780}, {12783,35788}, {12795,35790}, {12924,35796}, {12934,35798}, {12944,35800}, {12954,35802}, {12994,35804}, {12995,35806}, {13078,35808}, {13112,35816}, {13113,35818}, {13984,35814}, {18983,35768}, {19091,35770}, {22780,35784}, {22803,35786}

X(35868) = {X(6), X(13111)}-harmonic conjugate of X(35869)

X(35869) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 2nd NEUBERG

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

The reciprocal orthologic center of these triangles is X(3)

X(35869) lies on these lines: {6,382}, {83,371}, {372,35839}, {486,2896}, {732,35841}, {754,35823}, {1588,20088}, {3071,35825}, {3102,35879}, {6249,6564}, {6274,35793}, {6275,35795}, {6287,6565}, {6292,10577}, {6396,12122}, {6420,19091}, {6561,12252}, {7977,35811}, {8993,35815}, {9903,35775}, {9918,35777}, {10064,35808}, {10080,35768}, {12144,35765}, {12206,35767}, {12264,35763}, {12339,35773}, {12476,35781}, {12477,35779}, {12783,35789}, {12795,35791}, {12924,35797}, {12934,35799}, {12944,35801}, {12954,35803}, {12994,35807}, {12995,35805}, {13078,35809}, {13112,35817}, {13113,35819}, {13984,35813}, {18983,35769}, {19092,35771}, {22780,35785}, {22803,35787}

X(35869) = {X(6), X(13111)}-harmonic conjugate of X(35868)

X(35870) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st SCHIFFLER

Barycentrics 3*a^10-2*(b+c)*a^9-4*(3*b^2-2*b*c+3*c^2)*a^8+6*(b^3+c^3)*a^7+(20*b^4+20*c^4-9*(2*b^2-3*b*c+2*c^2)*b*c)*a^6-2*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^5-(18*b^6+18*c^6-(16*b^4+16*c^4-3*(b^2+c^2)*b*c)*b*c)*a^4+2*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))^2*S*a^2+2*(b^2-c^2)*(b-c)*(b^4+c^4+3*(b^2+c^2)*b*c)*a^3-4*(b^2-c^2)^3*(b-c)*b*c*a+(9*b^4+9*c^4-10*(b^2+c^2)*b*c)*(b^2-c^2)^2*a^2-2*(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(79)

X(35870) lies on these lines: {6,13126}, {372,10266}, {485,12849}, {6200,12556}, {6419,19098}, {6502,18244}, {6560,12255}, {6564,12919}, {6565,12600}, {10576,13089}, {12146,35764}, {12209,35766}, {12267,35762}, {12342,35772}, {12409,35774}, {12414,35776}, {12482,35778}, {12483,35780}, {12504,35782}, {12786,35788}, {12798,35790}, {12807,35792}, {12808,35794}, {12927,35796}, {12937,35798}, {12947,35800}, {12957,35802}, {13000,35804}, {13001,35806}, {13080,35808}, {13100,35810}, {13128,35809}, {13129,35769}, {13130,35816}, {13131,35818}, {13919,35812}, {13987,35814}, {18985,35768}, {19097,35770}, {22782,35784}, {22805,35786}

X(35870) = {X(6), X(13126)}-harmonic conjugate of X(35871)

X(35871) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st SCHIFFLER

Barycentrics 3*a^10-2*(b+c)*a^9-4*(3*b^2-2*b*c+3*c^2)*a^8+6*(b^3+c^3)*a^7+(20*b^4+20*c^4-9*(2*b^2-3*b*c+2*c^2)*b*c)*a^6-2*(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^5-(18*b^6+18*c^6-(16*b^4+16*c^4-3*(b^2+c^2)*b*c)*b*c)*a^4-2*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b^2-c^2)*(b-c))^2*S*a^2+2*(b^2-c^2)*(b-c)*(b^4+c^4+3*(b^2+c^2)*b*c)*a^3-4*(b^2-c^2)^3*(b-c)*b*c*a+(9*b^4+9*c^4-10*(b^2+c^2)*b*c)*(b^2-c^2)^2*a^2-2*(b^2-c^2)^4*(b-c)^2 : :

The reciprocal orthologic center of these triangles is X(79)

X(35871) lies on these lines: {6,13126}, {371,10266}, {486,12849}, {2067,18244}, {6396,12556}, {6420,19097}, {6561,12255}, {6564,12600}, {6565,12919}, {10577,13089}, {12146,35765}, {12209,35767}, {12267,35763}, {12342,35773}, {12409,35775}, {12414,35777}, {12482,35781}, {12483,35779}, {12504,35783}, {12786,35789}, {12798,35791}, {12807,35795}, {12808,35793}, {12927,35797}, {12937,35799}, {12947,35801}, {12957,35803}, {13000,35807}, {13001,35805}, {13080,35809}, {13100,35811}, {13128,35808}, {13129,35768}, {13130,35817}, {13131,35819}, {13919,35815}, {13987,35813}, {18985,35769}, {19098,35771}, {22782,35785}, {22805,35787}

X(35871) = {X(6), X(13126)}-harmonic conjugate of X(35870)

X(35872) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st TRI-SQUARES-CENTRAL

Barycentrics 3*(3*a^4-8*(b^2+c^2)*a^2+5*b^4-6*b^2*c^2+5*c^4)*a^2-(34*a^4-2*(b^2+c^2)*a^2-20*(b^2-c^2)^2)*S : :

The reciprocal orthologic center of these triangles is X(13665)

X(35872) lies on these lines: {6,13713}, {30,35832}, {372,1327}, {485,13678}, {3839,22616}, {5093,35875}, {6200,13666}, {6419,22541}, {6560,13674}, {6564,13692}, {6565,13687}, {10576,13701}, {13667,35762}, {13668,35764}, {13672,35766}, {13675,35772}, {13679,35774}, {13680,35776}, {13682,35778}, {13683,35780}, {13685,35782}, {13688,35788}, {13689,35790}, {13690,35792}, {13691,35794}, {13693,35796}, {13694,35798}, {13695,35800}, {13696,35802}, {13697,35804}, {13698,35806}, {13699,35808}, {13702,35810}, {13714,35809}, {13715,35769}, {13716,35816}, {13717,35818}, {13920,35812}, {13988,35814}, {18986,35768}, {19099,35770}, {22783,35784}, {22806,35786}, {35840,35874}

X(35872) = {X(6), X(13713)}-harmonic conjugate of X(35873)

X(35873) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st TRI-SQUARES-CENTRAL

Barycentrics 3*(a^2+c^2+b^2)*(-a^2+b^2+c^2)*a^2-(22*a^4-14*(b^2+c^2)*a^2-20*(b^2-c^2)^2)*S : :

The reciprocal orthologic center of these triangles is X(13665)

X(35873) lies on these lines: {3,12815}, {6,13713}, {30,35831}, {371,1131}, {486,13678}, {6396,13666}, {6420,19099}, {6561,13674}, {6564,13687}, {6565,13692}, {10577,13701}, {13667,35763}, {13668,35765}, {13672,35767}, {13675,35773}, {13679,35775}, {13680,35777}, {13682,35781}, {13683,35779}, {13685,35783}, {13688,35789}, {13689,35791}, {13690,35795}, {13691,35793}, {13693,35797}, {13694,35799}, {13695,35801}, {13696,35803}, {13697,35807}, {13698,35805}, {13699,35809}, {13702,35811}, {13714,35808}, {13715,35768}, {13716,35817}, {13717,35819}, {13920,35815}, {13988,35813}, {18986,35769}, {22541,35771}, {22783,35785}, {22806,35787}, {35841,35875}

X(35873) = {X(6), X(13713)}-harmonic conjugate of X(35872)

X(35874) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 3*(c^2+a^2+b^2)*(-a^2+b^2+c^2)*a^2+(22*a^4-14*(b^2+c^2)*a^2-20*(b^2-c^2)^2)*S : :

The reciprocal orthologic center of these triangles is X(13785)

X(35874) lies on these lines: {3,12815}, {6,13836}, {30,35830}, {372,1132}, {485,13798}, {6200,13786}, {6419,19100}, {6560,13794}, {6564,13812}, {6565,13807}, {10576,13821}, {13787,35762}, {13788,35764}, {13792,35766}, {13795,35772}, {13799,35774}, {13800,35776}, {13802,35778}, {13803,35780}, {13805,35782}, {13808,35788}, {13809,35790}, {13810,35792}, {13811,35794}, {13813,35796}, {13814,35798}, {13815,35800}, {13816,35802}, {13817,35804}, {13818,35806}, {13819,35808}, {13822,35810}, {13837,35809}, {13838,35769}, {13839,35816}, {13840,35818}, {13848,35812}, {13849,35814}, {18987,35768}, {19101,35770}, {22784,35784}, {22807,35786}, {35840,35872}

X(35874) = {X(6), X(13836)}-harmonic conjugate of X(35875)

X(35875) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 2nd TRI-SQUARES-CENTRAL

Barycentrics 3*(3*a^4-8*(b^2+c^2)*a^2+5*b^4-6*b^2*c^2+5*c^4)*a^2+(34*a^4-2*(b^2+c^2)*a^2-20*(b^2-c^2)^2)*S : :

The reciprocal orthologic center of these triangles is X(13785)

X(35875) lies on these lines: {6,13836}, {30,35833}, {371,1328}, {486,13798}, {3839,22645}, {5093,35872}, {6396,13786}, {6420,19101}, {6561,13794}, {6564,13807}, {6565,13812}, {10577,13821}, {13787,35763}, {13788,35765}, {13792,35767}, {13795,35773}, {13799,35775}, {13800,35777}, {13802,35781}, {13803,35779}, {13805,35783}, {13808,35789}, {13809,35791}, {13810,35795}, {13811,35793}, {13813,35797}, {13814,35799}, {13815,35801}, {13816,35803}, {13817,35807}, {13818,35805}, {13819,35809}, {13822,35811}, {13837,35808}, {13838,35768}, {13839,35817}, {13840,35819}, {13848,35815}, {13849,35813}, {18987,35769}, {19100,35771}, {22784,35785}, {22807,35787}, {35841,35873}

X(35875) = {X(6), X(13836)}-harmonic conjugate of X(35874)

X(35876) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO WALSMITH

Barycentrics (3*R^2*(6*SA-SW)-2*(3*SA-SW)*SW)*S^2+(9*R^2-2*SW)*(SB+SC)*SW*S+(9*R^2-4*SW)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(125)

X(35876) lies on these lines: {6,7579}, {67,372}, {265,35841}, {371,542}, {485,11061}, {511,35834}, {1352,12376}, {1503,35826}, {2781,35820}, {6200,32233}, {6419,32253}, {6560,32247}, {6564,9970}, {6565,32274}, {6593,10576}, {9974,16176}, {14984,35836}, {32238,35762}, {32239,35764}, {32242,35766}, {32243,35768}, {32252,35770}, {32256,35772}, {32261,35774}, {32262,35776}, {32265,35778}, {32266,35780}, {32268,35782}, {32270,35784}, {32271,35786}, {32278,35788}, {32279,35790}, {32280,35792}, {32281,35794}, {32287,35796}, {32288,35798}, {32289,35800}, {32290,35802}, {32292,34507}, {32295,35804}, {32296,35806}, {32297,35808}, {32298,35810}, {32303,35812}, {32304,35814}, {32307,35809}, {32308,35769}, {32309,35816}, {32310,35818}

X(35876) = {X(6), X(32306)}-harmonic conjugate of X(35877)

X(35877) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO WALSMITH

Barycentrics (3*R^2*(6*SA-SW)-2*(3*SA-SW)*SW)*S^2-(9*R^2-2*SW)*(SB+SC)*SW*S+(9*R^2-4*SW)*SB*SC*SW : :

The reciprocal orthologic center of these triangles is X(125)

X(35877) lies on these lines: {6,7579}, {67,371}, {265,35840}, {372,542}, {486,11061}, {511,35835}, {1352,12375}, {1503,35827}, {2781,35821}, {6396,32233}, {6420,32252}, {6561,32247}, {6564,32274}, {6565,9970}, {6593,10577}, {9975,16176}, {14984,35837}, {32238,35763}, {32239,35765}, {32242,35767}, {32243,35769}, {32253,35771}, {32256,35773}, {32261,35775}, {32262,35777}, {32265,35781}, {32266,35779}, {32268,35783}, {32270,35785}, {32271,35787}, {32278,35789}, {32279,35791}, {32280,35795}, {32281,35793}, {32287,35797}, {32288,35799}, {32289,35801}, {32290,35803}, {32291,34507}, {32295,35807}, {32296,35805}, {32297,35809}, {32298,35811}, {32303,35815}, {32304,35813}, {32307,35808}, {32308,35768}, {32309,35817}, {32310,35819}

X(35877) = {X(6), X(32306)}-harmonic conjugate of X(35876)

X(35878) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st ANTI-BROCARD

Barycentrics 4*S^4-3*(SB+SC)*S^3+(9*SA^2-5*SA*SW-SW^2)*S^2+(SB+SC)*SW^2*S+SB*SC*SW^2 : :

The reciprocal parallelogic center of these triangles is X(385)

X(35878) lies on these lines: {3,35825}, {6,1569}, {98,6200}, {99,372}, {114,6565}, {115,10576}, {147,6561}, {148,485}, {371,2782}, {381,35699}, {542,35826}, {543,35698}, {690,12375}, {1151,12188}, {1587,20094}, {2459,32456}, {2482,13967}, {2783,35856}, {2787,35882}, {2794,35828}, {2799,35880}, {3023,35808}, {3027,35768}, {3103,35868}, {4027,35766}, {5186,35764}, {5418,14651}, {5613,35754}, {5617,35851}, {5969,35840}, {5984,9541}, {6033,35821}, {6319,35792}, {6320,35794}, {6321,6564}, {6396,33813}, {6419,19109}, {6560,13172}, {7983,35810}, {8724,35823}, {8782,35782}, {8997,35812}, {9732,12602}, {10086,35809}, {10089,35769}, {10577,15561}, {10819,22265}, {11005,35835}, {11711,35762}, {12177,35841}, {13173,35772}, {13174,35774}, {13175,35776}, {13176,35778}, {13177,35780}, {13178,35788}, {13179,35790}, {13180,35796}, {13181,35798}, {13182,35800}, {13183,35802}, {13184,35804}, {13185,35806}, {13189,35816}, {13190,35818}, {13989,35814}, {19108,35770}, {22514,35784}, {22515,35786}

X(35878) = reflection of X(i) in X(j) for these (i,j): (35698, 35822), (35824, 371)
X(35878) = {X(6), X(13188)}-harmonic conjugate of X(35879)

X(35879) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st ANTI-BROCARD

Barycentrics 4*S^4+3*(SB+SC)*S^3+(9*SA^2-5*SA*SW-SW^2)*S^2-(SB+SC)*SW^2*S+SB*SC*SW^2 : :

The reciprocal parallelogic center of these triangles is X(385)

X(35879) lies on these lines: {3,35824}, {6,1569}, {98,6396}, {99,371}, {114,6564}, {115,10577}, {147,6560}, {148,486}, {372,2782}, {381,35698}, {542,35827}, {543,35699}, {690,12376}, {1152,12188}, {1588,20094}, {2460,32456}, {2482,8980}, {2783,35857}, {2787,35883}, {2794,35829}, {2799,35881}, {3023,35809}, {3027,35769}, {3102,35869}, {4027,35767}, {5186,35765}, {5420,14651}, {5613,35753}, {5617,35850}, {5969,35841}, {6033,35820}, {6200,33813}, {6319,35795}, {6320,35793}, {6321,6565}, {6420,19108}, {6561,13172}, {7983,35811}, {8724,35822}, {8782,35783}, {8997,35815}, {9733,12601}, {10086,35808}, {10089,35768}, {10576,15561}, {10820,22265}, {11005,35834}, {11711,35763}, {12177,35840}, {13173,35773}, {13174,35775}, {13175,35777}, {13176,35781}, {13177,35779}, {13178,35789}, {13179,35791}, {13180,35797}, {13181,35799}, {13182,35801}, {13183,35803}, {13184,35807}, {13185,35805}, {13189,35817}, {13190,35819}, {13989,35813}, {19109,35771}, {22514,35785}, {22515,35787}

X(35879) = reflection of X(i) in X(j) for these (i,j): (35699, 35823), (35825, 372)
X(35879) = {X(6), X(13188)}-harmonic conjugate of X(35878)

X(35880) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO 1st ANTI-ORTHOSYMMEDIAL

Barycentrics (SB+SC)*(2*S^4-(3*R^2-SW)*S^3-((9*SA-8*SW)*R^2-2*SA^2+SB*SC+2*SW^2)*S^2+SW^2*(4*R^2-SW)*(S-SA)) : :

The reciprocal parallelogic center of these triangles is X(10313)

X(35880) lies on these lines: {3,35829}, {6,13310}, {112,372}, {127,10576}, {132,6565}, {371,35828}, {485,13219}, {1151,13115}, {1297,6200}, {2781,35826}, {2794,35820}, {2799,35878}, {2806,35882}, {2831,35856}, {3320,35768}, {6020,35808}, {6419,19115}, {6560,13200}, {6561,12384}, {6564,10749}, {9517,12375}, {10705,35810}, {11641,35776}, {11722,35762}, {12918,35821}, {13166,35764}, {13195,35766}, {13206,35772}, {13221,35774}, {13229,35778}, {13231,35780}, {13236,35782}, {13280,35788}, {13281,35790}, {13282,35792}, {13283,35794}, {13294,35796}, {13295,35798}, {13296,35800}, {13297,35802}, {13298,35804}, {13299,35806}, {13311,35809}, {13312,35769}, {13313,35816}, {13314,35818}, {13923,35812}, {13992,35814}, {19114,35770}, {19162,35784}, {19163,35786}

X(35880) = reflection of X(35828) in X(371)
X(35880) = {X(6), X(13310)}-harmonic conjugate of X(35881)

X(35881) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO 1st ANTI-ORTHOSYMMEDIAL

Barycentrics (SB+SC)*(2*S^4+(3*R^2-SW)*S^3-((9*SA-8*SW)*R^2-2*SA^2+SB*SC+2*SW^2)*S^2+SW^2*(4*R^2-SW)*(-S-SA)) : :

The reciprocal parallelogic center of these triangles is X(10313)

X(35881) lies on these lines: {3,35828}, {6,13310}, {112,371}, {127,10577}, {132,6564}, {372,35829}, {486,13219}, {1152,13115}, {1297,6396}, {2781,35827}, {2794,35821}, {2799,35879}, {2806,35883}, {2831,35857}, {3320,35769}, {6020,35809}, {6420,19114}, {6560,12384}, {6561,13200}, {6565,10749}, {9517,12376}, {10705,35811}, {11641,35777}, {11722,35763}, {12918,35820}, {13166,35765}, {13195,35767}, {13206,35773}, {13221,35775}, {13229,35781}, {13231,35779}, {13236,35783}, {13280,35789}, {13281,35791}, {13282,35795}, {13283,35793}, {13294,35797}, {13295,35799}, {13296,35801}, {13297,35803}, {13298,35807}, {13299,35805}, {13311,35808}, {13312,35768}, {13313,35817}, {13314,35819}, {13923,35815}, {13992,35813}, {19115,35771}, {19162,35785}, {19163,35787}

X(35881) = reflection of X(35829) in X(372)
X(35881) = {X(6), X(13310)}-harmonic conjugate of X(35880)

X(35882) = PARALLELOGIC CENTER OF THESE TRIANGLES: 1st KENMOTU-FREE-VERTICES TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1)

X(35882) lies on these lines: {3,35857}, {6,12331}, {11,9646}, {80,2066}, {100,372}, {104,6200}, {119,6565}, {149,485}, {153,6561}, {214,35762}, {355,35853}, {371,952}, {528,35822}, {590,1484}, {1151,12773}, {1173,1587}, {1317,35768}, {1320,35810}, {1702,5531}, {1862,35764}, {2067,7972}, {2771,35826}, {2783,35824}, {2787,35878}, {2800,35610}, {2802,35641}, {2806,35880}, {2831,35828}, {3071,11698}, {5415,19078}, {5541,35774}, {5840,35820}, {6174,13977}, {6265,35642}, {6326,35775}, {6396,33814}, {6419,19113}, {6560,13199}, {6564,10738}, {7968,22935}, {7993,9583}, {8674,12375}, {9024,35840}, {10087,35809}, {10090,35769}, {10742,35821}, {11729,35817}, {12737,35763}, {13194,35766}, {13205,13909}, {13222,35776}, {13228,35778}, {13230,35780}, {13235,35782}, {13268,35790}, {13269,35792}, {13270,35794}, {13271,35796}, {13272,35798}, {13273,35800}, {13274,35802}, {13275,35804}, {13276,35806}, {13278,35816}, {13279,35818}, {13922,35812}, {13991,35814}, {15863,31453}, {19112,35770}, {19907,35811}, {22560,35784}, {22938,35786}

X(35882) = reflection of X(35856) in X(371)
X(35882) = {X(6), X(12331)}-harmonic conjugate of X(35883)

X(35883) = PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd KENMOTU-FREE-VERTICES TO INNER-GARCIA

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

The reciprocal parallelogic center of these triangles is X(1)

X(35883) lies on these lines: {3,35856}, {6,12331}, {11,10577}, {80,5414}, {100,371}, {104,6396}, {119,6564}, {149,486}, {153,6560}, {214,35763}, {355,35852}, {372,952}, {528,35823}, {615,1484}, {1152,12773}, {1317,35769}, {1320,35811}, {1588,20095}, {1703,5531}, {1862,35765}, {2771,35827}, {2783,35825}, {2787,35879}, {2800,35611}, {2802,35642}, {2806,35881}, {2831,35829}, {3070,11698}, {5416,19077}, {5541,35775}, {5840,35821}, {6174,13913}, {6200,33814}, {6265,35641}, {6326,35774}, {6420,19112}, {6502,7972}, {6561,13199}, {6565,10738}, {7969,22935}, {8674,12376}, {9024,35841}, {10087,35808}, {10090,35768}, {10742,35820}, {11729,35816}, {12737,35762}, {13194,35767}, {13205,35773}, {13222,35777}, {13228,35781}, {13230,35779}, {13235,35783}, {13268,35791}, {13269,35795}, {13270,35793}, {13271,35797}, {13272,35799}, {13273,35801}, {13274,35803}, {13275,35807}, {13276,35805}, {13278,35817}, {13279,35819}, {13922,35815}, {13991,35813}, {19113,35771}, {19907,35810}, {22560,35785}, {22938,35787}

X(35883) = reflection of X(35857) in X(372)
X(35883) = {X(6), X(12331)}-harmonic conjugate of X(35882)

X(35884) = X(2)-OF-ORTHIC AXES TRIANGLE

Barycentrics (a^2+b^2-c^2)*(a^2-b^2+c^2)*(2*a^8-8*(b^2+c^2)*a^6+(11*b^4+8*b^2*c^2+11*c^4)*a^4-6*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2) : :
X(35884) = 2*X(4)+X(35717) = 4*X(15557)-X(35719) = 2*X(15557)+X(35887) = X(35719)+2*X(35887)

X(35884) lies on these lines: {2,35710}, {4,54}, {30,12012}, {427,3054}, {1907,31867}, {5064,9756}, {6748,11245}, {6755,10192}, {15557,35719}

X(35884) = polar conjugate of the isotomic conjugate of X(6709)
X(35884) = barycentric product X(4)*X(6709)
X(35884) = trilinear product X(19)*X(6709)
X(35884) = crosssum of X(3) and X(32078)
X(35884) = {X(15557), X(35887)}-harmonic conjugate of X(35719)

X(35885) = X(2)-OF-YIU TANGENTS TRIANGLE

Barycentrics 13*S^4+(2*R^2*(16*R^2-5*SA-9*SW)+4*SA^2-19*SB*SC+SW^2)*S^2-(8*R^2-5*SW)*SB*SC*SW : :
X(35885) = 7*X(5)-4*X(32536) = 2*X(20)+X(15619) = X(550)+2*X(30484) = 4*X(550)-X(35721) = 2*X(550)+X(35728) = 5*X(3858)-8*X(32904) = 4*X(10205)-X(35720) = 2*X(10205)+X(35888) = 7*X(14869)-4*X(33545) = 8*X(30484)+X(35721) = 4*X(30484)-X(35728) = X(35720)+2*X(35888) = X(35721)+2*X(35728)

X(35885) lies on these lines: {2,35722}, {5,27684}, {20,15619}, {54,35724}, {427,3054}, {550,6247}, {1510,3917}, {3858,32904}, {10205,35720}, {14869,33545}

X(35885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (550, 35728, 35721), (10205, 35888, 35720)

X(35886) = X(3)-OF-INVERSE-IN-EXCIRCLES TRIANGLE

Barycentrics a*((b+c)*a^8+2*(b^2-3*b*c+c^2)*a^7-2*(b+c)*(b^2-3*b*c+c^2)*a^6-6*(b^3-c^3)*(b-c)*a^5-14*(b^2-c^2)*(b-c)*b*c*a^4+2*(3*b^6+3*c^6+(3*b^4+3*c^4+(9*b^2-14*b*c+9*c^2)*b*c)*b*c)*a^3+2*(b+c)*(b^6+c^6+(5*b^4+5*c^4-(13*b^2-30*b*c+13*c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^2*(b^4+c^4+3*(b^2+4*b*c+c^2)*b*c)*a-(b^4-c^4)*(b^2-c^2)*(b+c)^3) : :
X(35886) = 5*X(3616)-X(35660)

X(35886) lies on these lines: {1,35666}, {2,35889}, {40,936}, {497,35669}, {946,21255}, {1001,35658}, {1125,35655}, {2886,35664}, {3485,35671}, {3616,35660}, {8915,31435}

X(35886) = midpoint of X(1) and X(35666)
X(35886) = reflection of X(35655) in X(1125)
X(35886) = complement of X(35889)
X(35886) = X(3)-of-inverse in incircles triangle

X(35887) = X(3)-OF-ORTHIC AXES TRIANGLE

Barycentrics SB*SC*(3*S^2-SB*SC)*(2*S^2-R^2*SA) : :
X(35887) = 2*X(15557)-3*X(35884) = X(35719)-3*X(35884)

X(35887) lies on these lines: {4,195}, {5,97}, {30,35717}, {140,233}, {143,32438}, {275,8254}, {427,14111}, {549,22268}, {6750,22051}, {15557,35719}, {15559,32744}

X(35887) = reflection of X(35719) in X(15557)
X(35887) = intersection, other than A,B,C, of conics {{A, B, C, X(140), X(13582)}} and {{A, B, C, X(233), X(1263)}}
X(35887) = {X(35719), X(35884)}-harmonic conjugate of X(15557)

X(35888) = X(3)-OF-YIU TANGENTS TRIANGLE

Barycentrics (4*S^2-(SB+SC)*(SA+SB))*(4*S^2-(SA+SC)*(SB+SC))*(S^2+(10*R^2+3*SA-6*SW)*SA) : :
X(35888) = 5*X(5)-4*X(31879) = 5*X(1656)-4*X(23338) = 2*X(10205)-3*X(35885) = 5*X(14143)-2*X(31879) = 8*X(15425)-9*X(15699) = X(35720)-3*X(35885)

X(35888) lies on these lines: {3,15620}, {5,252}, {30,15619}, {74,550}, {93,6240}, {140,35724}, {427,14111}, {549,21975}, {562,3520}, {1154,30484}, {1487,3858}, {1656,23338}, {3627,31392}, {3628,23337}, {10205,35720}, {15331,21394}, {15425,15699}, {19268,31656}, {21230,35729}

X(35888) = reflection of X(i) in X(j) for these (i,j): (5, 14143), (23337, 3628), (35720, 10205)
X(35888) = intersection, other than A,B,C, of conics {{A, B, C, X(30), X(7691)}} and {{A, B, C, X(74), X(5944)}}
X(35888) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (252, 19552, 5), (35720, 35885, 10205)

X(35889) = X(4)-OF-INVERSE-IN-EXCIRCLES TRIANGLE

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

Let A'B'C' be the extangents triangle. Let A" be the {B',C'}-reciprocal conjugate of A', and define B" and C" cyclically (see X(8795)). Let A* be the {B,C}-reciprocal conjugate of A, wrt A'B'C', and define B* and C* cyclically. The lines A"A*, B"B*, C"C* concur in X(35889). (Randy Hutson, January 17, 2020)

X(35889) lies on these lines: {1,35655}, {2,35886}, {8,35660}, {10,35666}, {40,1743}, {55,1394}, {65,497}, {1122,9943}, {3664,35672}, {3925,35664}, {5584,35657}, {10374,35679}, {11529,35665}, {30503,35667}

X(35889) = midpoint of X(8) and X(35660)
X(35889) = reflection of X(i) in X(j) for these (i,j): (1, 35655), (35665, 35670), (35666, 10), (35669, 35658)
X(35889) = anticomplement of X(35886)
X(35889) = X(8)-Beth conjugate of X(35666)

X(35890) = X(4)-OF-MOSES-STEINER REFLECTION TRIANGLE

Barycentrics 9*(3*R^2-SW)*(SA+SW)*S^4-(162*(SA-SW)*SA*R^4-3*(54*SA^2-57*SA*SW-SW^2)*SW*R^2+(36*SA^2-37*SA*SW-3*SW^2)*SW^2)*S^2+6*(R^2-SW)*SB*SC*SW^3 : :
X(35890) = 3*X(6800)-2*X(31608) = 3*X(34512)-2*X(34514)

X(35890) lies on these lines: {568,8550}, {6800,31608}, {31959,35707}, {34512,34514}

X(35890) = reflection of X(31959) in X(35707)

X(35891) = X(6)-OF-BEVAN ANTIPODAL TRIANGLE

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

X(35891) lies on these lines: {223,3452}, {269,6614}

X(35892) = X(6)-OF-INVERSE-IN-CONWAY TRIANGLE

Barycentrics a*((b^2+b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2+c^2)*b*c) : :
X(35892) = 3*X(10439)-X(10442)

X(35892) lies on these lines: {1,6}, {7,310}, {8,17049}, {55,16574}, {65,3886}, {69,21746}, {142,3741}, {144,35614}, {193,3271}, {344,20683}, {354,10436}, {390,10480}, {516,10441}, {527,35645}, {528,35636}, {579,8299}, {674,4851}, {869,21330}, {894,3873}, {971,35631}, {980,4022}, {982,1045}, {1002,5749}, {1071,12717}, {1402,1445}, {1469,4684}, {1764,11495}, {1999,30628}, {2550,10449}, {2801,35638}, {2951,35621}, {3041,27508}, {3056,3879}, {3059,11679}, {3616,3786}, {3681,17260}, {3685,3868}, {3688,17316}, {3757,34784}, {3779,3912}, {3783,17065}, {3789,5257}, {3826,10479}, {3870,21371}, {3874,3923}, {3875,20358}, {3888,17375}, {3938,20964}, {4259,4966}, {4363,13476}, {4430,17350}, {4553,17311}, {4890,17321}, {5542,35620}, {5732,10476}, {5851,35649}, {5853,35634}, {5880,35615}, {6600,23853}, {6666,22312}, {9054,17243}, {10439,10442}, {10443,29311}, {10456,11021}, {10889,14100}, {11019,21246}, {14839,17314}, {15587,35613}, {16112,17617}, {17259,22271}, {17269,21865}, {17279,22277}, {17293,22279}, {17296,17792}, {17312,25279}, {17377,25048}, {17668,35626}, {17768,35637}, {18206,20992}, {18398,24342}, {26015,29967}

X(35892) = reflection of X(22312) in X(6666)
X(35892) = barycentric product X(1)*X(29966)
X(35892) = trilinear product X(6)*X(29966)
X(35892) = intersection, other than A,B,C, of conics {{A, B, C, X(1), X(29966)}} and {{A, B, C, X(7), X(213)}}
X(35892) = Conway circle-inverse of X(5526)
X(35892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10477, 35628), (7, 35617, 10473), (5208, 10453, 10473), (14100, 21334, 10889), (35632, 35633, 10441), (35647, 35648, 10441)

X(35893) = X(6)-OF-INVERSE-IN-EXCIRCLES TRIANGLE

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

X(35893) lies on these lines: {}

X(35893) = (inverse-in-excircles)-isogonal conjugate of X(2)

X(35894) = X(6)-OF-MOSES-STEINER REFLECTION TRIANGLE

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

X(35894) lies on these lines: {69,2871}, {160,35705}, {316,1502}

X(35895) = X(6)-OF-ORTHIC AXES TRIANGLE

Barycentrics SC*SB*((27*R^2-2*SA-10*SW)*S^2-R^2*SB*SC) : :

X(35895) lies on these lines: {115,6748}, {233,546}, {427,35896}, {11062,23047}

X(35896) = X(6)-OF-YIU TANGENTS TRIANGLE

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

X(35896) lies on these lines: {3,35725}, {427,35895}, {3520,19189}, {7485,35723}

X(35897) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-EXCIRCLES TO INNER-HUTSON

Barycentrics a*(-2*(a^2+2*(b+c)*a+(b-c)^2)*((b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2)*sin(A/2)-2*(a^2+2*(b-c)*a+(b+c)^2)*((b-2*c)*a^3+(b+2*c)*(b-c)*a^2-(b+c)*(b^2+3*b*c-2*c^2)*a-(b^2-c^2)*(b^2-b*c+2*c^2))*sin(B/2)+2*(a^2-2*(b-c)*a+(b+c)^2)*((2*b-c)*a^3+(2*b+c)*(b-c)*a^2-(b+c)*(2*b^2-3*b*c-c^2)*a-(b^2-c^2)*(2*b^2-b*c+c^2))*sin(C/2)+3*a^6+2*(b+c)*a^5+(9*b^2-2*b*c+9*c^2)*a^4+12*(b^2-c^2)*(b-c)*a^3-(11*b^4+11*c^4+2*b*c*(2*b^2-23*b*c+2*c^2))*a^2-2*(b+c)*(7*b^4+7*c^4-2*b*c*(3*b-2*c)*(2*b-3*c))*a-(b^2-6*b*c+c^2)*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(35897) lies on these lines: {363,8915}, {5934,35666}, {6732,35900}, {8107,35657}, {8109,35658}, {8111,35667}, {8113,35671}, {8133,35899}, {8140,35673}, {8377,35663}, {8380,35664}, {8385,35668}, {8390,35669}, {8391,35675}, {9783,35661}, {11026,35672}, {11039,35670}, {11527,35665}, {11685,35659}, {11854,35655}, {11856,35656}, {11886,35660}, {11892,35662}, {11922,35674}, {11923,35681}, {17607,35679}, {17621,35678}, {21618,35680}, {22993,35682}, {35618,35640}

X(35897) = reflection of X(35898) in X(35673)

X(35898) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INVERSE-IN-EXCIRCLES TO OUTER-HUTSON

Barycentrics a*(2*(a^2+2*(b+c)*a+(b-c)^2)*((b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2)*sin(A/2)+2*(a^2+2*(b-c)*a+(b+c)^2)*((b-2*c)*a^3+(b+2*c)*(b-c)*a^2-(b+c)*(b^2+3*b*c-2*c^2)*a-(b^2-c^2)*(b^2-b*c+2*c^2))*sin(B/2)-2*(a^2-2*(b-c)*a+(b+c)^2)*((2*b-c)*a^3+(2*b+c)*(b-c)*a^2-(b+c)*(2*b^2-3*b*c-c^2)*a-(b^2-c^2)*(2*b^2-b*c+c^2))*sin(C/2)+3*a^6+2*(b+c)*a^5+(9*b^2-2*b*c+9*c^2)*a^4+12*(b^2-c^2)*(b-c)*a^3-(11*b^4+11*c^4+2*b*c*(2*b^2-23*b*c+2*c^2))*a^2-2*(b+c)*(7*b^4+7*c^4-2*b*c*(3*b-2*c)*(2*b-3*c))*a-(b^2-6*b*c+c^2)*(b^2-c^2)^2) : :

The reciprocal orthologic center of these triangles is X(1)

X(35898) lies on these lines: {168,8915}, {5935,35666}, {7707,35681}, {8108,35657}, {8110,35658}, {8112,35667}, {8114,35671}, {8135,35899}, {8138,35900}, {8140,35673}, {8378,35663}, {8381,35664}, {8386,35668}, {8392,35669}, {9787,35661}, {11027,35672}, {11040,35670}, {11528,35665}, {11686,35659}, {11855,35655}, {11857,35656}, {11887,35660}, {11893,35662}, {11925,35674}, {11926,35675}, {17608,35679}, {17623,35678}, {21619,35680}, {22994,35682}, {35619,35640}

X(35898) = reflection of X(35897) in X(35673)

X(35899) = ORTHOLOGIC CENTER OF THESE TRIANGLES: TANGENTIAL-MIDARC TO INVERSE-IN-EXCIRCLES

Barycentrics a*(2*(-a+b+c)*(a^2+2*(b+c)*a+(b-c)^2)*((b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2)*sin(A/2)+2*(a-b+c)*(a^2+2*(b-c)*a+(b+c)^2)*((b-2*c)*a^3+(b+2*c)*(b-c)*a^2-(b+c)*(b^2+3*b*c-2*c^2)*a-(b^2-c^2)*(b^2-b*c+2*c^2))*sin(B/2)-2*(a+b-c)*(a^2-2*(b-c)*a+(b+c)^2)*((2*b-c)*a^3+(2*b+c)*(b-c)*a^2-(b+c)*(2*b^2-3*b*c-c^2)*a-(b^2-c^2)*(2*b^2-b*c+c^2))*sin(C/2)+4*S^2*(3*a^3+(b+c)*a^2+(b^2+10*b*c+c^2)*a+(b+c)*(3*b^2-2*b*c+3*c^2))) : :

The reciprocal orthologic center of these triangles is X(1).

X(35899) lies on these lines: {1,35900}, {177,35681}, {188,35682}, {2089,35671}, {8075,35657}, {8077,35658}, {8078,8915}, {8079,35666}, {8081,35667}, {8085,35663}, {8087,35664}, {8089,35673}, {8133,35897}, {8135,35898}, {8241,35669}, {8247,35674}, {8249,35675}, {8387,35668}, {8733,35655}, {9793,35661}, {10503,35679}, {11032,35672}, {11044,35670}, {11534,35665}, {11690,35659}, {11858,35656}, {11888,35660}, {11894,35662}, {17629,35678}, {21622,35680}, {35624,35640}

X(35899) = reflection of X(35900) in X(1)

X(35900) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO INVERSE-IN-EXCIRCLES

Barycentrics a*(-2*(-a+b+c)*(a^2+2*(b+c)*a+(b-c)^2)*((b+c)*a^3+(b-c)^2*a^2-(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2)*sin(A/2)-2*(a-b+c)*(a^2+2*(b-c)*a+(b+c)^2)*((b-2*c)*a^3+(b+2*c)*(b-c)*a^2-(b+c)*(b^2+3*b*c-2*c^2)*a-(b^2-c^2)*(b^2-b*c+2*c^2))*sin(B/2)+2*(a+b-c)*(a^2-2*(b-c)*a+(b+c)^2)*((2*b-c)*a^3+(2*b+c)*(b-c)*a^2-(b+c)*(2*b^2-3*b*c-c^2)*a-(b^2-c^2)*(2*b^2-b*c+c^2))*sin(C/2)+4*S^2*(3*a^3+(b+c)*a^2+(b^2+10*b*c+c^2)*a+(b+c)*(3*b^2-2*b*c+3*c^2))) : :

The reciprocal orthologic center of these triangles is X(1)

X(35900) lies on these lines: {1,35899}, {174,35671}, {258,8915}, {6732,35897}, {7028,35682}, {7588,35658}, {8076,35657}, {8080,35666}, {8082,35667}, {8086,35663}, {8088,35664}, {8090,35673}, {8125,35659}, {8138,35898}, {8242,35669}, {8248,35674}, {8250,35675}, {8351,35670}, {8388,35668}, {8734,35655}, {9795,35661}, {10501,35679}, {11033,35672}, {11859,35656}, {11889,35660}, {11895,35662}, {11899,35665}, {17630,35678}, {21623,35680}, {35625,35640}

X(35900) = reflection of X(35899) in X(1)
X(35900) = {X(174), X(35671)}-harmonic conjugate of X(35681)

X(35901) = X(3)X(647)∩X(6)X(25)

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

X(35901 lies on the Brocard circle, the cubic K1142, and these lines: {3, 647}, {6, 25}, {110, 10766}, {111, 5622}, {125, 10418}, {351, 8429}, {574, 14685}, {842, 35188}, {1316, 2549}, {1555, 6232}, {1562, 5642}, {1640, 18338}, {1648, 1899}, {1995, 34235}, {2395, 6795}, {2409, 6794}, {2502, 3269}, {2715, 6800}, {3124, 13198}, {6776, 6792}, {8743, 23964}, {10117, 35325}, {15341, 35266}, {15647, 28343}, {16165, 32661}, {22075, 26881}, {34137, 35265}

X(35901) = circumcircle-inverse of X(647)
X(35901) = 2nd-Lemoine-circle-inverse of X(21639)
X(35901) = Parry-circle-inverse of X(8429)
X(35901) = Moses-radical-circle-inverse of X(3)
X(35901) = psi-transform of X(112)
X(35901) = crossdifference of every pair of points on line {468, 525}
X(35901) = X(2373)-of-1st-Brocard-triangle
X(35901) = 1st-Brocard-isotomic conjugate of X(125)

X(35902) = X(4)X(6)∩X(69)X(525)

Barycentrics (a^2 - b^2 - c^2)*(a^10 - a^8*b^2 - 2*a^6*b^4 + a^2*b^8 + b^10 - a^8*c^2 + 5*a^6*b^2*c^2 - a^2*b^6*c^2 - 3*b^8*c^2 - 2*a^6*c^4 + 2*b^6*c^4 - a^2*b^2*c^6 + 2*b^4*c^6 + a^2*c^8 - 3*b^2*c^8 + c^10) : :
X(35902) = 2 X[6] - 3 X[6794], 4 X[15341] - 3 X[34137]

X(35902) lies on the cubics K955 and K1142, and on these lines: {4, 6}, {69, 525}, {111, 125}, {115, 5622}, {542, 1562}, {1352, 18337}, {1648, 23291}, {1899, 6792}, {2088, 18347}, {2420, 14927}, {2697, 2715}, {3269, 11646}, {3818, 15355}, {5941, 7418}, {7748, 15073}, {8779, 11645}, {13203, 35325}

X(35902) = reflection of X(i) in X(j) for these {i,j}: {6776, 18338}, {10766, 1562}, {18337, 1352}
X(35902) = psi-transform of X(1560)

X(35903) = X(2)X(99)∩X(6)X(1562)

Barycentrics 2*a^10 - 6*a^8*b^2 - a^6*b^4 + 9*a^4*b^6 - a^2*b^8 - 3*b^10 - 6*a^8*c^2 + 20*a^6*b^2*c^2 - 13*a^4*b^4*c^2 - 18*a^2*b^6*c^2 + 9*b^8*c^2 - a^6*c^4 - 13*a^4*b^2*c^4 + 38*a^2*b^4*c^4 - 6*b^6*c^4 + 9*a^4*c^6 - 18*a^2*b^2*c^6 - 6*b^4*c^6 - a^2*c^8 + 9*b^2*c^8 - 3*c^10 : :

X(35903) lies on the cubic K1142 and these lines: {2, 99}, {6, 1562}, {187, 16386}, {232, 10151}, {5254, 6103}, {11410, 34866}

X(35904) = X(2)X(15113)∩X(4)X(5972)

Barycentrics a^2*(3*a^10 - 4*a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 + a^2*b^8 - 2*b^10 - 4*a^8*c^2 + 23*a^6*b^2*c^2 - 15*a^4*b^4*c^2 - 15*a^2*b^6*c^2 + 11*b^8*c^2 - 4*a^6*c^4 - 15*a^4*b^2*c^4 + 36*a^2*b^4*c^4 - 9*b^6*c^4 + 6*a^4*c^6 - 15*a^2*b^2*c^6 - 9*b^4*c^6 + a^2*c^8 + 11*b^2*c^8 - 2*c^10) : :

X(35904) lies on the cubic K1142 and these lines: {2, 15113}, {4, 5972}, {6, 110}, {74, 15030}, {373, 13198}, {542, 18950}, {858, 15448}, {1112, 6090}, {5020, 5622}, {5085, 15647}, {5663, 6642}, {7464, 15036}, {7529, 32609}, {8567, 10117}, {9934, 16836}, {12828, 14826}, {16051, 35268}, {18919, 32300}

X(35904) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1995, 35259, 11188}, {5020, 20772, 5622}

X(35905) = X(6)X(8673)∩X(691)X(10423)

Barycentrics a^2*(b^2 - c^2)*(a^20 - 3*a^16*b^4 - 3*a^14*b^6 + 5*a^12*b^8 + 9*a^10*b^10 - 7*a^8*b^12 - 9*a^6*b^14 + 6*a^4*b^16 + 3*a^2*b^18 - 2*b^20 + a^14*b^4*c^2 + 13*a^12*b^6*c^2 - 21*a^10*b^8*c^2 - 21*a^8*b^10*c^2 + 39*a^6*b^12*c^2 + 3*a^4*b^14*c^2 - 19*a^2*b^16*c^2 + 5*b^18*c^2 - 3*a^16*c^4 + a^14*b^2*c^4 + 5*a^12*b^4*c^4 - 18*a^10*b^6*c^4 + 36*a^8*b^8*c^4 - 3*a^6*b^10*c^4 - 39*a^4*b^12*c^4 + 20*a^2*b^14*c^4 + b^16*c^4 - 3*a^14*c^6 + 13*a^12*b^2*c^6 - 18*a^10*b^4*c^6 + 20*a^8*b^6*c^6 - 31*a^6*b^8*c^6 + 21*a^4*b^10*c^6 + 4*a^2*b^12*c^6 - 6*b^14*c^6 + 5*a^12*c^8 - 21*a^10*b^2*c^8 + 36*a^8*b^4*c^8 - 31*a^6*b^6*c^8 + 18*a^4*b^8*c^8 - 8*a^2*b^10*c^8 + b^12*c^8 + 9*a^10*c^10 - 21*a^8*b^2*c^10 - 3*a^6*b^4*c^10 + 21*a^4*b^6*c^10 - 8*a^2*b^8*c^10 + 2*b^10*c^10 - 7*a^8*c^12 + 39*a^6*b^2*c^12 - 39*a^4*b^4*c^12 + 4*a^2*b^6*c^12 + b^8*c^12 - 9*a^6*c^14 + 3*a^4*b^2*c^14 + 20*a^2*b^4*c^14 - 6*b^6*c^14 + 6*a^4*c^16 - 19*a^2*b^2*c^16 + b^4*c^16 + 3*a^2*c^18 + 5*b^2*c^18 - 2*c^20) : :

X(35905) lies on these lines: {6, 8673}, {691, 10423}, {21733, 34146}

X(35905) = singular focus of the cubic K1142

X(35906) = BARYCENTRIC PRODUCT X(30)*X(98)

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

X(35906) lies on the cubics K055 and K1143 and these lines: {2, 2966}, {4, 32}, {6, 523}, {30, 2420}, {187, 7422}, {230, 868}, {232, 2409}, {287, 1992}, {290, 7766}, {393, 32230}, {477, 2549}, {1007, 6394}, {1138, 7739}, {1384, 5915}, {1976, 19136}, {1990, 9407}, {2407, 3260}, {3163, 9214}, {3815, 15000}, {5007, 14265}, {5304, 9154}, {5309, 34175}, {5319, 33753}, {7774, 31636}, {7787, 14382}, {9476, 11348}, {13352, 17974}, {17932, 32815}

X(35906) = isogonal conjugate of X(35910)
X(35906) = X(6793)-cross conjugate of X(1990)
X(35906) = trilinear pole of line {1495, 1637} (the line through X(1637) parallel to its trilinear polar)
X(35906) = crossdifference of every pair of points on line {511, 684}
X(35906) = polar conjugate of isotomic conjugate of X(35912)
X(35906) = X(i)-isoconjugate of X(j) for these (i,j): {63, 35908}, {74, 1959}, {237, 33805}, {240, 14919}, {297, 35200}, {325, 2159}, {511, 2349}, {662, 32112}, {1494, 1755}, {2394, 23997}
X(35906) = barycentric product X(i)*X(j) for these {i,j}: {30, 98}, {287, 1990}, {290, 1495}, {293, 1784}, {685, 9033}, {879, 4240}, {1637, 2966}, {1821, 2173}, {1910, 14206}, {1976, 3260}, {2395, 2407}, {3284, 16081}, {5642, 9154}, {5967, 9214}, {6357, 15628}, {6531, 11064}, {6793, 9476}, {9407, 18024}, {9409, 22456}, {14254, 14355}
X(35906) = barycentric quotient X(i)/X(j) for these {i,j}: {30, 325}, {98, 1494}, {248, 14919}, {512, 32112}, {685, 16077}, {878, 14380}, {879, 34767}, {1495, 511}, {1637, 2799}, {1821, 33805}, {1910, 2349}, {1976, 74}, {1990, 297}, {2173, 1959}, {2395, 2394}, {2407, 2396}, {2420, 2421}, {2422, 2433}, {4240, 877}, {6531, 16080}, {6793, 15595}, {9033, 6333}, {9406, 1755}, {9407, 237}, {9409, 684}, {11064, 6393}, {14398, 3569}, {14581, 232}, {14583, 14356}, {14600, 18877}, {20031, 15459}, {23347, 4230}, {32696, 1304}
X(35906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 32545, 34156}, {6, 34369, 5967}, {1316, 1640, 35606}

X(35907) = X(4)X(6)∩X(112)X(523)

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

X(35907) lies on the cubics K186 abnd K1143 and on these lines: {4, 6}, {107, 685}, {112, 523}, {525, 648}, {1304, 9209}, {1637, 31510}, {6103, 17986}, {6529, 32230}, {14560, 32713}, {15388, 32708}

X(35907) = reflection of X(5523) in X(1990)
X(35907) = isogonal conjugate of X(35911)
X(35907) = antigonal image of X(17986)
X(35907) = polar conjugate of the isotomic conjugate of X(7473)
X(35907) = X(1640)-cross conjugate of X(6103)
X(35907) = X(i)-isoconjugate of X(j) for these (i,j): {63, 35909}, {255, 14223}, {326, 14998}, {822, 5641}, {842, 24018}, {2632, 5649}
X(35907) = cevapoint of X(1640) and X(6103)
X(35907) = trilinear pole of line {5191, 6103}
X(35907) = barycentric product X(i)*X(j) for these {i,j}: {4, 7473}, {107, 542}, {393, 14999}, {648, 6103}, {823, 2247}, {1640, 23582}, {4240, 17986}, {5191, 6528}, {6530, 34761}, {14165, 23968}, {18312, 23964}
X(35907) = barycentric quotient X(i)/X(j) for these {i,j}: {107, 5641}, {393, 14223}, {542, 3265}, {1640, 15526}, {2207, 14998}, {2247, 24018}, {5191, 520}, {6041, 3269}, {6103, 525}, {6530, 34765}, {7473, 69}, {14999, 3926}, {17986, 34767}, {23582, 6035}, {23964, 5649}, {32713, 842}, {34761, 6394}, {34854, 23350}

X(35908) = X(3)X(250)∩X(4)X(523)

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

X(35908) lies on the cubic K1143 and these lines: {3, 250}, {4, 523}, {6, 74}, {24, 3447}, {25, 842}, {30, 16237}, {186, 23347}, {232, 7418}, {262, 5094}, {264, 339}, {325, 877}, {427, 12079}, {511, 4230}, {648, 34810}, {868, 6530}, {1344, 16071}, {1345, 16070}, {1597, 9139}, {1993, 10689}, {1995, 14919}, {2433, 6785}, {2790, 5186}, {2967, 5968}, {3470, 10594}, {3613, 7577}, {7480, 33927}, {9308, 15928}, {10421, 18494}, {14989, 35480}, {15078, 34233}, {15292, 34212}, {15459, 33971}, {15463, 15468}, {17994, 23350}, {19189, 21525}, {32224, 35474}

X(35908) = isogonal conjugate of X(35912)
X(35908) = crosssum of X(1495) and X(6793)
X(35908) = trilinear pole of line {232, 3569}
X(35908) = crossdifference of every pair of points on line {3284, 9033}
X(35908) = polar conjugate of isotomic conjugate of X(35910)
X(35908) = X(i)-isoconjugate of X(j) for these (i,j): {63, 35906}, {30, 293}, {248, 14206}, {287, 2173}, {336, 1495}, {1784, 17974}, {1821, 3284}, {1910, 11064}, {2631, 2966}
X(35908) = barycentric product X(i)*X(j) for these {i,j}: {74, 297}, {232, 1494}, {240, 2349}, {325, 8749}, {511, 16080}, {648, 32112}, {684, 15459}, {877, 2433}, {1304, 2799}, {2394, 4230}, {2421, 18808}, {3569, 16077}, {6333, 32695}, {6530, 14919}
X(35908) = barycentric quotient X(i)/X(j) for these {i,j}: {74, 287}, {232, 30}, {237, 3284}, {240, 14206}, {297, 3260}, {511, 11064}, {1304, 2966}, {2159, 293}, {2211, 1495}, {2349, 336}, {2433, 879}, {2491, 9409}, {3569, 9033}, {4230, 2407}, {8749, 98}, {14919, 6394}, {15459, 22456}, {16080, 290}, {17994, 1637}, {18877, 17974}, {32112, 525}, {32695, 685}, {32715, 2715}, {34854, 1990}, {34859, 23347}
X(35908) = {X(25),X(9717)}-harmonic conjugate of X(1304)

X(35909) = X(3)X(684)∩X(4)X(690)

Barycentrics a^2*(b^2 - c^2)*(a^2 - b^2 - c^2)*(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(35909) lies on the Jerabek circumhyperbola, the cubic K1143, and these lines: {3, 684}, {4, 690}, {6, 526}, {54, 33695}, {67, 523}, {69, 6333}, {74, 512}, {110, 5649}, {125, 879}, {246, 35364}, {248, 647}, {265, 525}, {290, 850}, {520, 895}, {826, 33565}, {924, 1177}, {1499, 10293}, {1510, 34437}, {2780, 3426}, {3269, 10097}, {3431, 15920}, {3566, 11744}, {3906, 11564}, {5191, 34291}, {5486, 9003}, {5504, 8673}, {5505, 8675}, {5900, 7927}, {6368, 18125}, {7624, 11579}, {9970, 33752}, {14380, 20975}, {15328, 34290}, {30209, 34802}

X(35909) = reflection of X(i) in X(j) for these {i,j}: {879, 125}, {9970, 33752} X(35909) = isogonal conjugate of X(7473)
X(35909) = antigonal image of X(879)
X(35909) = isotomic conjugate of the polar conjugate of X(14998)
X(35909) = isogonal conjugate of the polar conjugate of X(14223)
X(35909) = X(14223)-Ceva conjugate of X(14998)
X(35909) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7473}, {19, 14999}, {162, 542}, {240, 34761}, {648, 2247}, {662, 6103}, {811, 5191}
X(35909) = crosspoint of X(5641) and X(5649)
X(35909) = crosssum of X(1640) and X(5191)
X(35909) = trilinear pole of line {647, 16186}
X(35909) = crossdifference of every pair of points on line {542, 6103}
X(35909) = polar conjugate of isotomic conjugate of X(35911)
X(35909) = Jerabek-hyperbola-antipode of X(879)
X(35909) = orthocenter of X(3)X(4)X(67)
X(35909) = orthocenter of X(3)X(6)X(74)
X(35909) = barycentric product X(i)*X(j) for these {i,j}: {3, 14223}, {69, 14998}, {125, 5649}, {248, 34765}, {287, 23350}, {525, 842}, {647, 5641}, {6035, 20975}
X(35909) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 14999}, {6, 7473}, {125, 18312}, {248, 34761}, {512, 6103}, {647, 542}, {810, 2247}, {842, 648}, {878, 34369}, {2433, 17986}, {3049, 5191}, {5641, 6331}, {5649, 18020}, {10097, 16092}, {14223, 264}, {14998, 4}, {20975, 1640}, {23350, 297}

X(35910) = ISOGONAL CONJUGATE OF X(35906)

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

X(35910) lies on the cubics K022 and K657 and on these lines: {2, 525}, {3, 74}, {6, 5649}, {76, 6331}, {249, 394}, {323, 2420}, {458, 16077}, {511, 4230}, {599, 1494}, {1304, 2710}, {1352, 17986}, {2396, 6393}, {5085, 15407}, {5094, 18304}, {5968, 32112}, {9308, 15459}, {9513, 14380}, {11007, 12079}

X(35910) = isogonal conjugate of X(35906)
X(35910) = isotomic conjugate of the polar conjugate of X(35908)
X(35910) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35906}, {30, 1910}, {98, 2173}, {248, 1784}, {290, 9406}, {293, 1990}, {336, 14581}, {685, 2631}, {878, 24001}, {1495, 1821}, {1976, 14206}
X(35910) = trilinear pole of line {511, 684}
X(35910) = crossdifference of every pair of points on line {1495, 1637}
X(35910) = barycentric product X(i)*X(j) for these {i,j}: {69, 35908}, {74, 325}, {99, 32112}, {297, 14919}, {511, 1494}, {684, 16077}, {877, 14380}, {1304, 6333}, {1755, 33805}, {1959, 2349}, {2394, 2421}, {2396, 2433}, {4230, 34767}, {6393, 8749}
X(35910) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35906}, {74, 98}, {232, 1990}, {237, 1495}, {240, 1784}, {325, 3260}, {511, 30}, {684, 9033}, {1304, 685}, {1494, 290}, {1755, 2173}, {1959, 14206}, {2159, 1910}, {2211, 14581}, {2349, 1821}, {2421, 2407}, {2433, 2395}, {2491, 14398}, {3289, 3284}, {3569, 1637}, {4230, 4240}, {5968, 9214}, {8749, 6531}, {9139, 9154}, {9155, 5642}, {9417, 9406}, {9418, 9407}, {9475, 6793}, {9717, 5967}, {14356, 14254}, {14380, 879}, {14385, 14355}, {14919, 287}, {14966, 2420}, {15627, 15628}, {16077, 22456}, {16080, 16081}, {17209, 18653}, {18877, 248}, {32112, 523}, {32640, 2715}, {32695, 20031}, {32715, 32696}, {35200, 293}, {35908, 4}

X(35911) = ISOGONAL CONJUGATE OF X(35907)

Barycentrics a^2*(b^2 - c^2)*(a^2 - b^2 - c^2)^2*(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(35911) lies on the cubic K022 and these lines: {2, 1637}, {3, 684}, {112, 5649}, {127, 15421}, {394, 1636}, {520, 17974}, {525, 34897}, {647, 10766}, {842, 1297}, {2781, 34291}, {3265, 18558}, {5641, 10718}, {6334, 14376}

X(35911) = isogonal conjugate of X(35907)
X(35911) = isotomic conjugate of the polar conjugate of X(35909)
X(35911) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35907}, {19, 7473}, {107, 2247}, {162, 6103}, {542, 24019}, {823, 5191}, {1096, 14999}, {1640, 24000}, {6041, 23999}
X(35911) = crosssum of X(1640) and X(6103)
X(35911) = crossdifference of every pair of points on line {5191, 6103}
X(35911) = barycentric product X(i)*X(j) for these {i,j}: {69, 35909}, {394, 14223}, {520, 5641}, {842, 3265}, {3269, 6035}, {3926, 14998}, {5649, 15526}, {6394, 23350}, {17974, 34765}
X(35911) = barycentric quotient X(i)/X(j) for these {i,j}: {3, 7473}, {6, 35907}, {394, 14999}, {520, 542}, {647, 6103}, {822, 2247}, {842, 107}, {3269, 1640}, {5641, 6528}, {5649, 23582}, {14223, 2052}, {14380, 17986}, {14998, 393}, {15526, 18312}, {17974, 34761}, {23350, 6530}, {35909, 4}

X(35912) = ISOGONAL CONJUGATE OF X(35908)

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

X(35912) lies on the cubic K022 and these lines: {2, 98}, {3, 525}, {4, 23582}, {20, 9218}, {30, 2420}, {69, 17932}, {248, 4846}, {290, 3431}, {376, 2966}, {511, 34211}, {685, 35260}, {1249, 20031}, {1495, 4240}, {1650, 11064}, {1651, 35266}, {2395, 6795}, {2697, 2715}, {3269, 12042}, {3524, 31621}, {6467, 15630}, {6531, 18850}, {7502, 35362}, {11050, 13857}, {13137, 15073}, {13367, 14265}, {14585, 14880}, {18347, 26937}

X(35912) = isogonal conjugate of X(35908)
X(35912) = isotomic conjugate of the polar conjugate of X(35906)
X(35912) = psi-transform of X(7473)
X(35912) = X(i)-isoconjugate of X(j) for these (i,j): {1, 35908}, {74, 240}, {162, 32112}, {232, 2349}, {297, 2159}, {1755, 16080}, {1959, 8749}, {2211, 33805}, {6530, 35200}, {18808, 23997}
X(35912) = cevapoint of X(1495) and X(6793)
X(35912) = trilinear pole of line {3284, 9033}
X(35912) = crossdifference of every pair of points on line {232, 3569}
X(35912) = barycentric product X(i)*X(j) for these {i,j}: {30, 287}, {69, 35906}, {98, 11064}, {248, 3260}, {290, 3284}, {293, 14206}, {336, 2173}, {879, 2407}, {1636, 22456}, {1637, 17932}, {1990, 6394}, {2966, 9033}
X(35912) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 35908}, {30, 297}, {98, 16080}, {248, 74}, {287, 1494}, {293, 2349}, {336, 33805}, {647, 32112}, {685, 15459}, {878, 2433}, {879, 2394}, {1495, 232}, {1636, 684}, {1637, 16230}, {1976, 8749}, {1990, 6530}, {2173, 240}, {2395, 18808}, {2407, 877}, {2420, 4230}, {2715, 1304}, {2966, 16077}, {3284, 511}, {6793, 132}, {9033, 2799}, {9407, 2211}, {9409, 3569}, {11064, 325}, {14398, 17994}, {14581, 34854}, {17974, 14919}, {32696, 32695}, {34369, 17986}, {35906, 4}
X(35912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 34761, 5967}, {98, 11653, 20021}, {32618, 32619, 5967}

X(35913) = X(2)X(1341)∩X(4)X(6)

Barycentrics (a^2 - b^2 - c^2)*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) - 2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(35913) = 3 X[2] - 4 X[1348]

X(35913) lies on these lines: {2, 1341}, {4, 6}, {20, 1379}, {2028, 3767}, {2039, 22242}, {3091, 14631}, {3146, 3557}, {5024, 19660}, {6039, 7735}, {6040, 7710}

X(35913) = crossdifference of every pair of points on line {520, 5639}
X(35913) = {X(4),X(6776)}-harmonic conjugate of X(35914)

X(35914) = X(2)X(1340)∩X(4)X(6)

Barycentrics (a^2 - b^2 - c^2)*(3*a^4 + b^4 - 2*b^2*c^2 + c^4) + 2*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4] : :
X(35914) = 3 X[2] - 4 X[1349]

X(35914) lies on these lines: {2, 1340}, {4, 6}, {20, 1380}, {2029, 3767}, {2040, 22243}, {3091, 14630}, {3146, 3558}, {5024, 19659}, {6039, 7710}, {6040, 7735}

X(35914) = crossdifference of every pair of points on line {520, 5638}
X(35914) = {X(4),X(6776)}-harmonic conjugate of X(35913)

X(35915) = EULER LINE INTERCEPT OF X(8)X(99)

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

X(35915) lies on these lines: {2, 3}, {8, 99}, {40, 17209}, {662, 5698}, {1326, 24248}, {2185, 3474}, {5744, 19642}, {16020, 24617}, {17169, 33867}, {18653, 35258}, {19785, 19849}, {20558, 27704}, {24280, 27958}

X(35916) = EULER LINE INTERCEPT OF X(10)X(99)

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

X(35916) lies on these lines: {2, 3}, {10, 99}, {86, 1326}, {261, 1738}, {333, 17596}, {662, 24723}, {986, 17206}, {1330, 10026}, {2185, 33068}, {4658, 16712}, {11599, 24851}, {16823, 24617}, {16825, 24378}, {19642, 24627}, {19786, 19849}, {24248, 27958}

X(35917) = EULER LINE INTERCEPT OF X(15)X(99)

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

X(35917) lies on these lines: {2, 3}, {15, 99}, {16, 3972}, {61, 7757}, {62, 12150}, {187, 5980}, {616, 2076}, {617, 4048}, {3106, 12204}, {3642, 9981}, {3734, 5981}, {5149, 5978}, {5152, 6109}, {5162, 5979}, {5615, 10788}, {6295, 25157}, {8291, 9885}, {8716, 22236}, {9735, 22715}, {11485, 31859}, {14904, 35375}, {21158, 33388} X(35917) = {X(2),X(35925)}-harmonic conjugate of X(35918)
X(35917) = {X(3),X(1003)}-harmonic conjugate of X(35918)

X(35918) = EULER LINE INTERCEPT OF X(16)X(99)

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

X(35918) lies on these lines: {2, 3}, {15, 3972}, {16, 99}, {61, 12150}, {62, 7757}, {187, 5981}, {616, 4048}, {617, 2076}, {3107, 12205}, {3643, 9982}, {3734, 5980}, {5149, 5979}, {5152, 6108}, {5162, 5978}, {5611, 10788}, {6582, 25167}, {8292, 9886}, {8716, 22238}, {9736, 22714}, {11486, 31859}, {14905, 35375}, {21159, 33389} X(35918) = {X(2),X(35925)}-harmonic conjugate of X(35917)
X(35918) = {X(3),X(1003)}-harmonic conjugate of X(35917)

X(35919) = EULER LINE INTERCEPT OF X(51)X(99)

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

X(35919) lies on these lines: {2, 3}, {51, 99}, {184, 3972}, {263, 12215}, {1993, 10788}, {1994, 22521}, {5422, 7709}, {7757, 15004}, {8719, 17825}, {9777, 31859}, {12150, 13366}

X(35920) = EULER LINE INTERCEPT OF X(53)X(99)

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

X(35920) lies on these lines: {2, 3}, {53, 99}, {112, 27377}, {393, 31859}, {620, 33842}, {1968, 7762}, {1990, 7757}, {3972, 6748}, {6749, 12150}, {7792, 33843}, {27371, 32819}

X(35921) = EULER LINE INTERCEPT OF X(95)X(99)

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

X(35921) lies on these lines: {2, 3}, {49, 10610}, {52, 7691}, {54, 5562}, {74, 827}, {76, 14558}, {95, 99}, {104, 26712}, {110, 5891}, {112, 216}, {128, 34418}, {141, 12383}, {182, 5890}, {184, 11459}, {323, 14805}, {343, 12022}, {477, 1287}, {511, 15033}, {567, 1154}, {568, 34545}, {569, 1199}, {575, 14831}, {578, 11412}, {691, 14979}, {841, 16166}, {1056, 10832}, {1058, 10831}, {1092, 7999}, {1147, 11444}, {1166, 8883}, {1216, 34148}, {1285, 1609}, {1292, 26707}, {1296, 5966}, {1614, 5907}, {1986, 19129}, {2055, 31388}, {2079, 3054}, {2383, 3565}, {2914, 12219}, {2931, 15081}, {2979, 13352}, {3043, 12358}, {3085, 9672}, {3086, 9659}, {3316, 8276}, {3317, 8277}, {3431, 15066}, {3455, 6054}, {3459, 8800}, {3581, 5946}, {3763, 34775}, {3796, 11456}, {3817, 9625}, {3917, 11430}, {4550, 12112}, {5012, 13754}, {5085, 10605}, {5116, 18371}, {5481, 18876}, {5603, 15177}, {5650, 15035}, {5651, 11202}, {5888, 12901}, {5892, 15053}, {5944, 14128}, {6000, 22352}, {6030, 16194}, {6102, 13353}, {6241, 10984}, {6403, 9813}, {6759, 15058}, {6800, 18451}, {7622, 10870}, {7689, 10574}, {7771, 34883}, {7853, 34217}, {8718, 11381}, {8744, 22240}, {9306, 11464}, {9544, 15068}, {9590, 10175}, {9591, 18483}, {9608, 31404}, {9626, 19925}, {9682, 32785}, {9699, 31415}, {9707, 17814}, {9712, 31418}, {9723, 32817}, {9967, 11416}, {10282, 32401}, {10314, 10986}, {10316, 26216}, {10539, 15056}, {10540, 15052}, {10575, 15062}, {10606, 15578}, {11003, 18445}, {11438, 15045}, {11454, 20791}, {11468, 13347}, {11562, 27866}, {11635, 32710}, {11642, 14649}, {11793, 13367}, {12168, 12317}, {12254, 14516}, {12307, 14627}, {12325, 32333}, {12412, 20125}, {12893, 15059}, {13445, 14855}, {14157, 15030}, {14389, 35254}, {14644, 22109}, {14652, 31656}, {15037, 32608}, {15067, 22115}, {15109, 34866}, {15740, 34439}, {16261, 35268}, {16836, 21663}, {18383, 23358}, {18436, 32046}, {18442, 34798}, {19192, 26907}, {21243, 25739}, {32345, 34782}, {34513, 35265}

X(35922) = EULER LINE INTERCEPT OF X(99)X(125)

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

X(35922) lies on these lines: {2, 3}, {69, 2396}, {99, 125}, {110, 9862}, {147, 9155}, {691, 3258}, {895, 6148}, {1285, 2420}, {1494, 32257}, {3819, 31848}, {5466, 5664}, {5650, 6787}, {5967, 25406}, {5972, 35278}, {7709, 18911}, {7761, 11052}, {9154, 34229}, {9168, 18556}, {14907, 17941}, {26869, 31859}

X(35923) = EULER LINE INTERCEPT OF X(99)X(127)

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

X(35923) lies on these lines: {2, 3}, {69, 525}, {99, 127}, {141, 34360}, {147, 10749}, {148, 339}, {287, 17702}, {315, 15075}, {316, 14961}, {543, 15526}, {2777, 15595}, {2966, 14907}, {3284, 3849}, {6794, 34211}, {7802, 10316}, {7823, 22120}, {7842, 22401}, {8182, 23967}, {8591, 34897}, {10317, 14712}, {10568, 11550}, {11161, 32275}, {17907, 23582}

X(35924) = EULER LINE INTERCEPT OF X(99)X(160)

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

X(35924) lies on these lines: {2, 3}, {76, 15270}, {99, 160}, {112, 19121}, {1634, 32833}, {1691, 19127}, {1975, 23208}, {2916, 34866}, {3094, 9019}, {6393, 16789}, {7762, 9917}, {8266, 14907}, {20775, 31859}

X(35925) = EULER LINE INTERCEPT OF X(99)X(182)

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

X(35925) lies on these lines: {2, 3}, {32, 12251}, {39, 10359}, {69, 35424}, {76, 13335}, {83, 9737}, {98, 3734}, {99, 182}, {114, 7835}, {183, 21445}, {187, 22712}, {194, 3398}, {262, 7804}, {315, 34885}, {511, 3972}, {574, 21166}, {575, 7757}, {576, 12150}, {1075, 8863}, {1285, 5017}, {1351, 22521}, {1352, 9862}, {1916, 8350}, {2076, 10519}, {2080, 3407}, {2549, 13172}, {2782, 8289}, {2794, 7820}, {3095, 7787}, {3096, 32152}, {3098, 10347}, {3111, 13137}, {4048, 6776}, {5050, 31859}, {5149, 9744}, {5152, 11185}, {5162, 7737}, {5207, 14907}, {5651, 35278}, {5989, 32815}, {7697, 12042}, {7766, 11842}, {7771, 15819}, {7781, 32467}, {7782, 13334}, {7790, 23698}, {7795, 35385}, {7816, 11257}, {7844, 14639}, {7919, 10723}, {7934, 13449}, {8721, 12252}, {10104, 31276}, {10796, 35002}, {11261, 14810}, {12054, 32522}, {12110, 30270}, {12215, 14912}, {13349, 22714}, {13350, 22715}, {21163, 32456}, {34507, 34623}

X(35925) = {X(35917),X(35918)}-harmonic conjugate of X(2)
X(35925) = {X(35938),X(35939)}-harmonic conjugate of X(384)

X(35926) = EULER LINE INTERCEPT OF X(99)X(184)

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

X(35926) lies on these lines: {2, 3}, {51, 3972}, {99, 184}, {194, 34396}, {287, 35387}, {1976, 18906}, {3060, 10788}, {3228, 35178}, {5012, 7709}, {7757, 13366}, {8716, 17809}, {9306, 35278}, {9862, 11442}, {10328, 14826}, {11402, 31859}, {12150, 15004}

X(35927) = EULER LINE INTERCEPT OF X(99)X(193)

Barycentrics 11*a^4 - 6*a^2*b^2 - b^4 - 6*a^2*c^2 + 6*b^2*c^2 - c^4 : :

X(35927) lies on these lines: {2, 3}, {99, 193}, {187, 32815}, {620, 32827}, {754, 3926}, {1007, 32459}, {1285, 31859}, {1992, 8716}, {2482, 32837}, {2996, 14568}, {3053, 6392}, {3620, 14907}, {3785, 7816}, {5023, 13468}, {5032, 5052}, {5034, 12150}, {5206, 32828}, {5210, 34229}, {5395, 31400}, {6337, 9766}, {6781, 7818}, {7618, 7753}, {7620, 26613}, {7709, 33748}, {7737, 32456}, {7739, 34504}, {7747, 32829}, {7823, 32831}, {7836, 14976}, {7893, 32840}, {8182, 9466}, {8719, 25406}, {8782, 11148}, {9752, 23698}, {11147, 11184}, {11160, 32833}, {14023, 32824}, {18546, 32826}, {20080, 32817}, {21166, 34733}, {32885, 34506}

X(35928) = EULER LINE INTERCEPT OF X(99)X(216)

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

X(35928) lies on these lines: {2, 3}, {83, 22401}, {99, 216}, {127, 7931}, {577, 3972}, {1060, 4366}, {1062, 6645}, {3284, 12150}, {3329, 14961}, {5158, 7757}, {7787, 23115}, {7864, 15075}, {11511, 22486}

X(35929) = EULER LINE INTERCEPT OF X(99)X(251)

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

X(35929) lies on these lines: {2, 3}, {32, 8267}, {99, 251}, {1180, 3972}, {1369, 14712}, {1627, 16276}, {1691, 33798}, {1799, 31078}, {1915, 4576}, {2056, 35356}, {2076, 10328}, {3051, 10330}, {4074, 8627}, {6781, 21248}, {7816, 8024}, {10546, 35294}, {31068, 32820}

X(35930) = EULER LINE INTERCEPT OF X(99)X(262)

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

Let (O*) be circle that is the locus of crosssums of Brocard circle antipodes, with center X(10796). Then X(35930) is the antipode of X(6) in (O*). (Randy Hutson, January 17, 2020)

X(35930) lies on these lines: {2, 3}, {6, 2782}, {32, 6248}, {39, 10358}, {76, 12110}, {83, 11257}, {98, 3972}, {99, 262}, {114, 5475}, {155, 3499}, {182, 7804}, {183, 2080}, {385, 10788}, {511, 3734}, {538, 576}, {543, 5476}, {598, 6054}, {736, 35431}, {754, 34507}, {1160, 7696}, {1161, 7695}, {1350, 22677}, {1351, 18906}, {1352, 5017}, {1916, 11170}, {1975, 3095}, {2076, 9996}, {2549, 14561}, {2794, 3818}, {3053, 10104}, {3329, 7709}, {3407, 9755}, {3564, 18907}, {3767, 20576}, {3849, 11178}, {3934, 5171}, {4048, 5480}, {5013, 11272}, {5093, 22253}, {5097, 7798}, {5149, 19130}, {5152, 14639}, {5162, 13449}, {5989, 6321}, {6232, 9129}, {7753, 14981}, {7754, 13108}, {7757, 23235}, {7761, 24206}, {7766, 22521}, {7806, 14651}, {7808, 13334}, {7816, 9737}, {7878, 32467}, {8290, 13172}, {8722, 15819}, {8724, 11163}, {9605, 32448}, {9734, 32456}, {9753, 11185}, {9756, 12042}, {9993, 10000}, {10753, 22486}, {10797, 13077}, {10798, 18982}, {11171, 11174}, {11184, 19911}, {11472, 33877}, {11632, 33997}, {12215, 14853}, {12251, 17128}, {14538, 22693}, {14539, 22694}, {15048, 18583}, {15483, 33813}, {18424, 23514}, {18860, 22682}, {20428, 22512}, {20429, 22513}, {22503, 22728}, {30435, 32134}

X(35930) = reflection of X(6) in X(10796)
X(35930) = {X(3552),X(37334)}-harmonic conjugate of X(3)

X(35931) = EULER LINE INTERCEPT OF X(99)X(298)

Barycentrics 7*a^8 - 21*a^6*b^2 + 19*a^4*b^4 - 3*a^2*b^6 - 2*b^8 - 21*a^6*c^2 + 16*a^4*b^2*c^2 + 7*a^2*b^4*c^2 + 6*b^6*c^2 + 19*a^4*c^4 + 7*a^2*b^2*c^4 - 8*b^4*c^4 - 3*a^2*c^6 + 6*b^2*c^6 - 2*c^8 - 6*Sqrt[3]*a^2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2)*S : :

X(35931) lies on these lines: {2, 3}, {14, 13084}, {15, 530}, {99, 298}, {187, 6772}, {299, 14907}, {302, 22491}, {303, 10645}, {511, 8595}, {524, 616}, {532, 35751}, {538, 35696}, {542, 8594}, {543, 5980}, {599, 617}, {622, 9763}, {671, 6109}, {2482, 5978}, {3104, 7757}, {3181, 31859}, {3389, 35741}, {3643, 5464}, {3849, 5979}, {5238, 34509}, {5321, 33474}, {5460, 16242}, {5473, 12155}, {5858, 8716}, {6108, 26613}, {6671, 19106}, {7801, 9988}, {9166, 31709}, {9749, 11153}, {9886, 22512}, {11645, 14904}, {16962, 35229}, {34504, 35692}, {34505, 35694}

X(35932) = EULER LINE INTERCEPT OF X(99)X(299)

Barycentrics 7*a^8 - 21*a^6*b^2 + 19*a^4*b^4 - 3*a^2*b^6 - 2*b^8 - 21*a^6*c^2 + 16*a^4*b^2*c^2 + 7*a^2*b^4*c^2 + 6*b^6*c^2 + 19*a^4*c^4 + 7*a^2*b^2*c^4 - 8*b^4*c^4 - 3*a^2*c^6 + 6*b^2*c^6 - 2*c^8 + 6*Sqrt[3]*a^2*(a^2 - b^2 - c^2)*(a^2 + b^2 + c^2)*S : :

X(35932) lies on these lines: {2, 3}, {13, 13083}, {16, 531}, {99, 299}, {187, 6775}, {298, 14907}, {302, 10646}, {303, 22492}, {511, 8594}, {524, 617}, {538, 35692}, {542, 8595}, {543, 5981}, {599, 616}, {621, 9761}, {628, 35749}, {634, 35750}, {671, 6108}, {2482, 5979}, {3105, 7757}, {3180, 31859}, {3642, 5463}, {3849, 5978}, {5237, 34508}, {5318, 33475}, {5459, 16241}, {5474, 12154}, {5859, 8716}, {6109, 26613}, {6672, 19107}, {7801, 9989}, {9166, 31710}, {9750, 11154}, {9885, 22513}, {11645, 14905}, {16963, 35230}, {34504, 35696}, {34505, 35690}, {34509, 35752}

X(35933) = EULER LINE INTERCEPT OF X(99)X(323)

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

X(35933) lies on these lines: {2, 3}, {99, 323}, {524, 1272}, {1993, 8716}, {1994, 7757}, {3566, 9147}, {3972, 15018}, {6800, 8719}, {11004, 31859}, {11162, 15566}, {12150, 34545}, {32224, 35345}, {35265, 35278}

X(35934) = EULER LINE INTERCEPT OF X(99)X(327)

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

X(35934) lies on these lines: {2, 3}, {39, 13754}, {99, 327}, {160, 3818}, {182, 1576}, {567, 34396}, {574, 3016}, {1154, 3095}, {1352, 1634}, {2080, 6785}, {2421, 30541}, {5191, 14805}, {5201, 20423}, {5663, 11171}, {6000, 13334}, {8266, 31670}, {9821, 13391}, {14915, 21163}, {15270, 18400}, {18438, 23635}, {18440, 20775}, {22062, 33878}, {33533, 35002}

X(35935) = EULER LINE INTERCEPT OF X(99)X(333)

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

X(35935) lies on these lines: {2, 3}, {81, 2094}, {86, 6173}, {99, 333}, {239, 3916}, {284, 527}, {552, 553}, {1043, 17294}, {1333, 17301}, {2287, 6172}, {3663, 33628}, {3794, 10167}, {3928, 9311}, {5222, 16948}, {5327, 28534}, {13624, 24559}, {14552, 32817}, {14907, 18134}, {18653, 26638}, {24271, 26244}, {24473, 29584}, {27398, 31142}, {35278, 35285}

X(35936) = EULER LINE INTERCEPT OF X(99)X(338)

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

X(35936) lies on these lines: {2, 3}, {99, 338}, {287, 19457}, {2079, 14061}, {2930, 11161}, {5182, 5621}, {8593, 16010}, {10568, 22352}, {14608, 22259}, {14907, 18122}, {18800, 32305}

X(35937) = EULER LINE INTERCEPT OF X(99)X(343)

Barycentrics 2*a^8 - 7*a^6*b^2 + 7*a^4*b^4 - a^2*b^6 - b^8 - 7*a^6*c^2 + 4*a^4*b^2*c^2 + a^2*b^4*c^2 + 2*b^6*c^2 + 7*a^4*c^4 + a^2*b^2*c^4 - 2*b^4*c^4 - a^2*c^6 + 2*b^2*c^6 - c^8 : :

X(35937) lies on these lines: {2, 3}, {53, 95}, {97, 112}, {99, 343}, {216, 27377}, {394, 14907}, {1494, 22165}, {1853, 8719}, {6515, 31859}, {7709, 11245}, {10192, 35278}, {13468, 31635}, {14810, 15595}

X(35938) = EULER LINE INTERCEPT OF X(99)X(371)

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

X(35938) lies on these lines: {2, 3}, {99, 371}, {372, 3972}, {3311, 31859}, {3592, 8716}, {6229, 9986}, {6419, 7757}, {6420, 12150}, {6423, 26429}, {7828, 35830}, {9733, 33434}, {9738, 22727}, {18906, 19145}

X(35938) = {X(2),X(35950)}-harmonic conjugate of X(35939)
X(35938) = {X(3),X(1003)}-harmonic conjugate of X(35939)
X(35938) = {X(384),X(35925)}-harmonic conjugate of X(35939)
X(35938) = {X(13586),X(35951)}-harmonic conjugate of X(35939)

X(35939) = EULER LINE INTERCEPT OF X(99)X(372)

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

X(35939) lies on these lines: {2, 3}, {99, 372}, {371, 3972}, {3312, 31859}, {3594, 8716}, {6228, 9987}, {6419, 12150}, {6420, 7757}, {6424, 26430}, {7828, 35831}, {9732, 33435}, {9739, 22726}, {18906, 19146}

X(35939) = {X(2),X(35950)}-harmonic conjugate of X(35938)
X(35939) = {X(3),X(1003)}-harmonic conjugate of X(35938)
X(35939) = {X(384),X(35925)}-harmonic conjugate of X(35938)
X(35939) = {X(13586),X(35951)}-harmonic conjugate of X(35938)

X(35940) = EULER LINE INTERCEPT OF X(99)X(393)

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

X(35940) lies on these lines: {2, 3}, {99, 393}, {112, 193}, {1249, 31859}, {1285, 27377}, {1968, 3926}, {1990, 8716}, {2207, 6337}, {3087, 3972}, {9308, 32817}, {14581, 34511}

X(35941) = EULER LINE INTERCEPT OF X(99)X(394)

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

X(35941) lies on these lines: {2, 3}, {99, 394}, {154, 8719}, {287, 1350}, {290, 8667}, {317, 34828}, {343, 14907}, {577, 9308}, {1494, 15533}, {1993, 31859}, {3164, 15905}, {3972, 10601}, {7709, 11402}, {8716, 9289}, {9777, 10788}, {10607, 14615}

X(35942) = EULER LINE INTERCEPT OF X(99)X(395)

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

X(35942) lies on these lines: {2, 3}, {99, 395}, {396, 3972}, {524, 11128}, {5182, 8595}, {5215, 33476}, {5858, 32833}, {6772, 7792}, {12154, 22496}, {16529, 22494}, {32819, 35697}

X(35942) = {X(2),X(1003)}-harmonic conjugate of X(35943)
X(35942) = {X(32985),X(35955)}-harmonic conjugate of X(35943)

X(35943) = EULER LINE INTERCEPT OF X(99)X(396)

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

X(35943) lies on these lines: {2, 3}, {99, 396}, {395, 3972}, {524, 11129}, {5182, 8594}, {5215, 33477}, {5859, 32833}, {6775, 7792}, {12155, 22495}, {16530, 22493}, {32819, 35693}

X(35943) = {X(2),X(1003)}-harmonic conjugate of X(35942)
X(35943) = {X(32985),X(35955)}-harmonic conjugate of X(35942)

X(35944) = EULER LINE INTERCEPT OF X(99)X(487)

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

X(35944) lies on these lines: {2, 3}, {99, 487}, {182, 9541}, {372, 12123}, {486, 26521}, {488, 14907}, {489, 12256}, {642, 26469}, {1285, 3312}, {1587, 9738}, {5870, 6231}, {6290, 33365}, {6417, 14482}, {6460, 9732}, {6463, 12509}, {7581, 12313}, {12601, 13939}, {12975, 13935}, {13786, 35823}, {22676, 33434}, {33363, 33456}

X(35944) = {X(3),X(376)}-harmonic conjugate of X(35945)

X(35945) = EULER LINE INTERCEPT OF X(99)X(488)

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

X(35945) lies on these lines: {2, 3}, {99, 488}, {371, 12124}, {485, 26516}, {487, 14907}, {490, 12257}, {511, 9541}, {641, 26468}, {1285, 3311}, {1588, 9739}, {5871, 6230}, {6289, 33364}, {6418, 14482}, {6459, 9733}, {6462, 12510}, {7582, 12314}, {9540, 12974}, {12602, 13886}, {13666, 35822}, {15885, 31411}, {22676, 33435}, {33362, 33457}

X(35945) = {X(3),X(376)}-harmonic conjugate of X(35944)

X(35946) = EULER LINE INTERCEPT OF X(99)X(489)

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

X(35946) lies on these lines: {2, 3}, {99, 489}, {485, 6569}, {487, 26295}, {490, 11824}, {638, 12306}, {1588, 15884}, {2460, 6560}, {6421, 6459}, {6424, 6460}, {8982, 9732}, {9541, 25406}, {10723, 13926}, {12124, 22676}, {12296, 13934}

X(35946) = {X(20),X(376)}-harmonic conjugate of X(35947)

X(35947) = EULER LINE INTERCEPT OF X(99)X(490)

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

X(35947) lies on these lines: {2, 3}, {99, 490}, {486, 6568}, {488, 26294}, {489, 11825}, {637, 12305}, {1587, 15883}, {2459, 6561}, {6422, 6460}, {6423, 6459}, {9541, 19145}, {9733, 26441}, {10723, 13873}, {12123, 22676}, {12297, 13882}

X(35947) = {X(20),X(376)}-harmonic conjugate of X(35946)

X(35948) = EULER LINE INTERCEPT OF X(99)X(491)

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

X(35948) lies on these lines: {2, 3}, {69, 26288}, {99, 491}, {187, 13758}, {372, 489}, {487, 5861}, {492, 6396}, {591, 637}, {642, 35820}, {1271, 32817}, {1285, 7586}, {1991, 8716}, {1992, 26289}, {2459, 13757}, {2549, 13638}, {3068, 9600}, {3103, 7757}, {3618, 9541}, {3972, 6561}, {5024, 13644}, {6119, 35787}, {6422, 19054}, {6423, 13798}, {7811, 32808}, {9166, 13873}, {12322, 13935}, {12968, 32788}, {13637, 35822}, {13712, 31168}, {13786, 13789}, {22485, 35685}, {23249, 32806}, {32421, 32809}, {33343, 33435}, {33365, 33456}, {35823, 35875}

X(35948) = reflection of X(35949) in X(1003)
X(35948) = {X(2),X(376)}-harmonic conjugate of X(35949)
X(35948) = {X(3),X(8356)}-harmonic conjugate of X(35949)
X(35948) = {X(20),X(14039)}-harmonic conjugate of X(35949)
X(35948) = {X(549),X(35955)}-harmonic conjugate of X(35949)

X(35949) = EULER LINE INTERCEPT OF X(99)X(492)

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

X(35949) lies on these lines: {2, 3}, {69, 9541}, {99, 492}, {112, 26912}, {187, 13638}, {371, 490}, {488, 5860}, {491, 6200}, {591, 8716}, {638, 1151}, {641, 35821}, {1270, 32817}, {1285, 7585}, {1992, 26288}, {2460, 13637}, {2549, 13758}, {3102, 7757}, {3972, 6560}, {5024, 13763}, {6118, 35786}, {6421, 19053}, {6424, 13678}, {6565, 32807}, {7811, 32809}, {9166, 13926}, {9540, 12323}, {12963, 32787}, {13666, 13669}, {13757, 35823}, {13835, 31168}, {22484, 35684}, {23259, 32805}, {32419, 32808}, {33342, 33434}, {33364, 33457}, {35822, 35872}

X(35949) = reflection of X(35948) in X(1003)
X(35949) = {X(2),X(376)}-harmonic conjugate of X(35948)
X(35949) = {X(3),X(8356)}-harmonic conjugate of X(35948)
X(35949) = {X(20),X(14039)}-harmonic conjugate of X(35948)
X(35949) = {X(549),X(35955)}-harmonic conjugate of X(35948)

X(35950) = EULER LINE INTERCEPT OF X(99)X(575)

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

X(35950) lies on these lines: {2, 3}, {99, 575}, {576, 3972}, {3329, 33813}, {3734, 21445}, {6321, 16984}, {7757, 22234}, {7792, 13172}, {7804, 21166}, {7816, 23235}, {7827, 10992}, {10788, 11477}, {12150, 22330}

X(35950) = {X(3),X(1003)}-harmonic conjugate of X(35951)
X(35950) = {X(35938),X(35939)}-harmonic conjugate of X(2)

X(35951) = EULER LINE INTERCEPT OF X(99)X(576)

Barycentrics 4*a^8 - 11*a^6*b^2 + 8*a^4*b^4 - a^2*b^6 - 11*a^6*c^2 + a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 3*b^6*c^2 + 8*a^4*c^4 - 3*a^2*b^2*c^4 - 6*b^4*c^4 - a^2*c^6 + 3*b^2*c^6 : :

X(35951) lies on these lines: {2, 3}, {32, 23235}, {99, 576}, {262, 32456}, {575, 3972}, {7608, 7622}, {7747, 20399}, {7757, 22330}, {7766, 13188}, {7816, 12251}, {7857, 20398}, {8719, 10541}, {9753, 13172}, {11482, 22521}, {12150, 22234}

X(35951) = {X(3),X(1003)}-harmonic conjugate of X(35950)
X(35951) = {X(35938),X(35939)}-harmonic conjugate of X(13586)

X(35952) = EULER LINE INTERCEPT OF X(99)X(577)

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

X(35952) lies on these lines: {2, 3}, {99, 577}, {112, 3164}, {127, 7898}, {194, 10316}, {216, 3972}, {339, 18472}, {2896, 14376}, {3284, 7757}, {5158, 12150}, {5182, 11511}, {6389, 14907}, {7766, 10317}, {7782, 22401}, {7783, 23115}, {7811, 15526}, {9862, 18437}, {10788, 30258}, {15905, 31859}

X(35953) = EULER LINE INTERCEPT OF X(99)X(590)

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

X(35953) lies on these lines: {2, 3}, {83, 32494}, {99, 590}, {591, 12968}, {615, 3972}, {1991, 32833}, {3068, 31859}, {7757, 19090}, {7774, 13644}, {7782, 32497}, {8716, 13846}, {12150, 32788}

X(35954) = EULER LINE INTERCEPT OF X(99)X(597)

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

X(35954) lies on these lines: {2, 3}, {99, 597}, {385, 19661}, {524, 3972}, {543, 7792}, {598, 7835}, {1285, 11160}, {2482, 7804}, {2549, 11164}, {3734, 22329}, {3849, 7820}, {3933, 34604}, {5032, 32817}, {5306, 11054}, {7603, 22247}, {7618, 11174}, {7745, 7870}, {7762, 7801}, {7789, 7812}, {7816, 15300}, {7817, 32819}, {7840, 18907}, {7875, 32480}, {8584, 12150}, {8591, 15048}, {11168, 26613}, {14061, 20112}, {14711, 18806}, {14907, 21358}, {15534, 32833}

X(35955) = EULER LINE INTERCEPT OF X(99)X(599)

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

X(35955) lies on these lines: {2, 3}, {99, 599}, {115, 5569}, {183, 543}, {316, 11184}, {325, 7618}, {385, 32480}, {524, 14907}, {574, 3849}, {598, 11155}, {671, 7610}, {1078, 34505}, {1296, 11162}, {1975, 7810}, {2482, 7761}, {2549, 8182}, {3053, 7827}, {3734, 11164}, {5013, 7812}, {5023, 7847}, {5024, 14712}, {5206, 7817}, {5210, 7790}, {5215, 7844}, {6781, 11174}, {7620, 34229}, {7622, 8589}, {7748, 34506}, {7750, 34511}, {7757, 15534}, {7773, 15515}, {7782, 7881}, {7783, 9939}, {7784, 7870}, {7801, 7830}, {7802, 15815}, {7806, 15655}, {7811, 8716}, {7831, 21358}, {7840, 11165}, {7851, 15513}, {7868, 32456}, {8667, 11054}, {9605, 34604}, {9741, 11160}, {9766, 11057}, {10788, 14848}, {11168, 11185}, {12117, 22712}, {16989, 19661}, {22165, 32833}, {22486, 22676}

X(35955) = anti-Artzt-isotomic conjugate of X(598)
X(35955) = X(183)-of-anti-Artzt-triangle
X(35955) = {X(35942),X(35943)}-harmonic conjugate of X(32985)
X(35955) = {X(35948),X(35949)}-harmonic conjugate of X(549)

Points on the permutation ellipse of X(75): X(35956)-X(35962)

Contributed by Clark Kimberling and Peter Moses, January 2, 2020.

Suppose that P is a point not in {X(2),A,B,C}. The permutation ellipse of P is denoted by E(P). For details, see the preamble to X(35025)-X(35048).

X(35956) = E(X(75),X(1))-ANTIPODE OF X(75)

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

X(35956) lies on these lines: {1, 75}, {37, 24505}, {99, 4094}, {1573, 31323}, {3571, 4589}, {4664, 28840}, {9263, 33888}

X(35956) = reflection of X(24505) in X(37)

X(35957) = E(X(75),X(514))-ANTIPODE OF X(75)

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

X(35957) lies on these lines: {1, 3570}, {2, 2087}, {8, 14947}, {9, 1016}, {10, 19933}, {75, 514}, {76, 4051}, {190, 2802}, {239, 980}, {244, 3227}, {335, 33908}, {519, 751}, {666, 3872}, {668, 2170}, {1573, 31323}, {1654, 6630}, {1762, 2985}, {3008, 31233}, {3125, 9263}, {3661, 35092}, {3662, 6547}, {3679, 4562}, {3729, 32028}, {3807, 4738}, {3912, 5233}, {4384, 4555}, {4595, 24036}, {4986, 33946}, {5540, 18047}, {6542, 31035}, {6631, 17277}, {6633, 17335}, {10009, 20924}, {17256, 24864}, {17336, 32094}, {17761, 18159}, {29615, 34362}, {30117, 30882}, {30583, 31992}, {33888, 35103}

X(35957) = barycentric product X(75)*X(24482)
X(35957) = barycentric quotient X (24482)/X(1)
X(35957) = {X(668),X(2170)}-harmonic conjugate of X(18061)

X(35958) = E(X(75),X(10))-ANTIPODE OF X(75)

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

X(35958) lies on these lines: {1, 3253}, {10, 75}, {37, 24502}, {335, 1015}, {4583, 24413}, {4664, 4785}, {17261, 33681}, {18140, 33679}, {20366, 32020}, {27481, 35126}

X(35958) = reflection of X(24502) in X (37)

X(35959) = E(X(75),X(693))-ANTIPODE OF X(75)

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

X(35959) lies on these lines: {75, 693}, {4664, 4715}

X(35960) = E(X(75),X(523))-ANTIPODE OF X(75)

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

X(35960) lies on these lines: {11, 20939}, {38, 3571}, {75, 523}, {86, 24345}, {141, 35080}, {190, 20538}, {261, 31998}, {322, 325}, {385, 26242}, {524, 4664}, {799, 2611}, {2486, 18032}, {2643, 18827}, {5224, 24348}, {7779, 31087}, {10026, 17243}, {17271, 35147}, {17277, 35148}, {21220, 21833}, {35151, 35154}

X(35960) = barycentric product X(75)*X(24504)
X(35960) = barycentric quotient X(24504)/X(1)

X(35961) = E(X(75),X(69))-ANTIPODE OF X(75)

Barycentrics b*c*(2*a^5*b - 3*a^4*b^2 + 2*a^3*b^3 - a^2*b^4 + 2*a^5*c - 3*a^4*b*c + a^3*b^2*c + a^2*b^3*c + a*b^4*c - 3*a^4*c^2 + a^3*b*c^2 - a^2*b^2*c^2 - a*b^3*c^2 + b^4*c^2 + 2*a^3*c^3 + a^2*b*c^3 - a*b^2*c^3 - 2*b^3*c^3 - a^2*c^4 + a*b*c^4 + b^2*c^4) : :
X(35961) = 5 X[4687] - 4 X[5701]

X(35961) lies on these lines: {1, 33674}, {7, 8}, {668, 17755}, {874, 3729}, {4664, 4762}, {4687, 5701}, {27475, 35167}

X(35962) = E(X(75),X(519))-ANTIPODE OF X(75)

Barycentrics 2*a^3*b + a^2*b^2 - a*b^3 + 2*a^3*c - 11*a^2*b*c + 4*a*b^2*c + 2*b^3*c + a^2*c^2 + 4*a*b*c^2 - 5*b^2*c^2 - a*c^3 + 2*b*c^3 : :

X(35962) lies on these lines: {1, 4555}, {2, 2087}, {75, 519}, {239, 24594}, {320, 24864}, {514, 4664}, {678, 4597}, {874, 35043}, {903, 2802}, {3241, 34342}, {3758, 6633}, {6548, 14421}, {17244, 35092}, {17250, 25031}, {17310, 31179}, {17460, 20568}, {18059, 31161}, {24281, 29584}, {31349, 33908}

Points on the permutation ellipse of X(6): X(35963)-X(35966)

Contributed by Clark Kimberling and Peter Moses, January 2, 2020.

Suppose that P is a point not in {X(2),A,B,C}. The permutation ellipse of P is denoted by E(P). For details, see the preamble to X(35025)-X(35048).

X(35963) = E(X(6),X(1))-ANTIPODE OF X(6)

Barycentrics a*(a - b - c)*(-(a^2*b^4) + a*b^5 + a^4*b*c - a*b^4*c + b^5*c - a^2*b^2*c^2 + a*b^3*c^2 - 2*b^4*c^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(35963) lies on these lines: {1, 6}, {8, 33676}, {599, 4762}, {668, 1146}, {2170, 7077}, {3094, 24248}, {3662, 10030}, {3735, 24249}, {3923, 24274}, {20468, 23402}

X(35964) = E(X(6),X(75))-ANTIPODE OF X(6)

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

X(35964) lies on these lines: {6, 75}, {24281, 24293}

X(35965) = E(X(6),X(76))-ANTIPODE OF X(6)

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

X(35965) lies on these lines: {6, 76}, {99, 15588}, {148, 25332}, {3493, 7794}, {4045, 30229}, {13519, 14931}

X(35966) = E(X(6),X(31))-ANTIPODE OF X(6)

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

X(35966) lies on these lines: {6, 31}, {3094, 24281}

X(35967) = X(11)X(11193)∩X(115)X(35509)

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

See César Lozada, Euclid 455 .

X(35967) lies on the nine-point circle and these lines: {11, 11193}, {115, 35509}, {15612, 35094}

X(35967) = complement of trilinear pole of line X(6)X(59)
X(35967) = X(i)-complementary conjugate of X(j) for these (i,j): (244, 676), (513, 24980), (918, 21232)

X(35968) = COMPLEMENT OF X(1301)

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

See César Lozada, Euclid 455 .

X(35968) lies on the nine-point circle and these lines: {2, 1301}, {3, 133}, {4, 5897}, {113, 2883}, {114, 30771}, {115, 13611}, {131, 3548}, {132, 1368}, {135, 3134}, {136, 1650}, {440, 20622}, {1249, 1560}, {2072, 18809}, {3150, 5139}, {3154, 16178}, {5514, 34846}, {10257, 25641}, {15526, 15613}, {21530, 25640}

X(35968) = midpoint of X(4) and X(5897)
X(35968) = complement of X(1301)
X(35968) = complementary conjugate of X(8057)
X(35968)= X(i)-complementary conjugate of X(j) for these (i,j): (65, 14302), (122, 34846), (154, 16612)
X(35968) = center of the circumconic {{ A, B, C, X(4), X(5879), X(5897), X(11413) }}
X(35968) = orthoptic circle of Steiner inellipse-inverse of X(34168)
X(35968) = polar circle-inverse of X(30249)
X(35968) = center of the circle {{X(4), X(1294), X(5897)}}
X(35968) = reflection of X(122) in the line X(5)X(34842)
X(35968) = X(5897)-of-Euler triangle
X(35968) = isogonal conjugate of X(8057) wrt these triangles: 3rd Euler, medial, orthic
X(35968) = isogonal conjugate of X(15311) wrt these triangles: Euler, 2nd Euler, 4th Euler

X(35969) = COMPLEMENT OF X(1288)

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

See César Lozada, Euclid 455 .

X(35969) lies on the nine-point circle and these lines: {2, 1288}, {22, 132}, {113, 6759}, {126, 30802}, {133, 15761}, {16188, 16977}, {18809, 23323}

X(35969) = complement of X(1288)
X(35969) = complementary conjugate of the isogonal conjugate of X(1288)
X(35969) = X(i)-complementary conjugate of X(j) for these (i,j): (647, 18590), (656, 13371), (810, 571)
X(35969) = reflection of X(125) in the line X(5)X(15647)

X(35970) = COMPLEMENT OF X(681)

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

See César Lozada, Euclid 458 .

X(35970) lies on the nine-point circle and these lines: {2, 681}, {125, 16595}

X(35970) = complement of X(681)
X(35970) = complementary conjugate of X(680)
X(35970) = crosspoint of X(2) and X(35521)
X(35970) = X(4)-Ceva conjugate of X(680)
X(35970) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 680), (680, 10)

X(35971) = COMPLEMENT OF X(689)

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

See César Lozada, Euclid 458 .

X(35971) is the point of concurrence of the cevian circles of the vertices of the tangential triangle. (Randy Hutson, August 7, 2020)

X(35971) lies on the nine-point circle, the symmedial circle, and these lines: {2, 689}, {11, 6026}, {12, 7334}, {39, 33666}, {116, 6377}, {125, 1084}, {325, 9496}, {1194, 13499}, {2080, 8623}, {2679, 3005}, {3229, 15573}, {3842, 16587}, {9152, 32526}

X(35971) = complement of X(689)
X(35971) = complementary conjugate of X(688)
X(35971) = center of the circumconic {{ A, B, C, X(4), X(384), X(1031), X(2998), X(3114) }}
X(35971) = intersection, other than X(125), of nine-point circle and symmedial circle
X(35971) = orthoptic-circle-of-Steiner-inellipse-inverse of X(733)
X(35971) = barycentric product X(i)*X(j) for these {i, j}: {782, 882}, {881, 35558}
X(35971) = barycentric quotient X(i)/X(j) for these (i, j): (782, 880), (881, 783), (882, 18828)

X(35972) = COMPLEMENT OF X(697)

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

See César Lozada, Euclid 458 .

X(35972) lies on the nine-point circle and these lines: {2, 697}, {11, 6028}

X(35972) = complement of X(697)
X(35972) = complementary conjugate of X(696)
X(35972) = crosspoint of X(2) and X(35523)
X(35972) = X(4)-Ceva conjugate of X(696)
X(35972) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 696), (696, 10)

X(35973) = EULER LINE INTERCEPT OF X(33)X(100)

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

X(35973) lies on these lines: {2, 3}, {8, 11399}, {33, 100}, {34, 5253}, {81, 3192}, {92, 108}, {102, 9107}, {162, 593}, {232, 5276}, {273, 26229}, {281, 26258}, {1061, 1320}, {1452, 3869}, {1621, 30687}, {1876, 27003}, {1892, 31053}, {1905, 4511}, {1974, 15988}, {3195, 17126}, {3199, 5277}, {3616, 11398}, {3871, 6198}, {3920, 15500}, {5090, 25005}, {5347, 26005}, {5362, 10642}, {5367, 10641}, {5554, 7718}, {7282, 33864}, {7713, 19861}, {9058, 32706}, {9308, 17001}, {10311, 33854}, {11392, 11681}, {15344, 33637}, {20988, 25968}

X(35974) = EULER LINE INTERCEPT OF X(34)X(100)

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

X(35974) lies on these lines: {2, 3}, {19, 26690}, {33, 5253}, {34, 100}, {145, 1398}, {1063, 1320}, {1452, 9352}, {1861, 2975}, {1870, 3871}, {1876, 34772}, {1968, 33854}, {3622, 7071}, {4881, 11363}, {5277, 33843}, {5362, 11476}, {5367, 11475}, {15988, 19124}

X(35975) = EULER LINE INTERCEPT OF X(38)X(100)

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

X(35975) lies on these lines: {2, 3}, {35, 29656}, {36, 29655}, {38, 100}, {986, 3871}, {1283, 24169}, {1621, 3821}, {1626, 4429}, {2975, 29673}, {5253, 29820}, {5303, 29861}

X(35976) = EULER LINE INTERCEPT OF X(46)X(100)

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

X(35976) lies on these lines: {2, 3}, {7, 11509}, {35, 5249}, {36, 10916}, {46, 100}, {63, 3678}, {65, 3871}, {651, 3215}, {943, 31019}, {1155, 4420}, {1420, 13279}, {1612, 23604}, {1621, 12609}, {1754, 3193}, {2352, 19850}, {2975, 17647}, {3188, 6516}, {3434, 7742}, {3612, 5253}, {4304, 14803}, {4313, 22768}, {5250, 12511}, {5330, 14110}, {5732, 16209}, {5784, 15481}, {5880, 7676}, {5905, 11517}, {6594, 8544}, {6796, 10884}, {7688, 24987}, {8715, 11520}, {9352, 17700}, {10940, 11248}, {11491, 18444}, {15823, 18227}, {28628, 34879}

X(35976) = {X(404),X(7411)}-harmonic conjugate of X(21)

X(35977) = EULER LINE INTERCEPT OF X(57)X(100)

Barycentrics a*(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 - 4*a*b^2*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(35977) lies on these lines: {2, 3}, {35, 12436}, {36, 4847}, {55, 9776}, {56, 3189}, {57, 100}, {142, 1621}, {200, 4973}, {942, 3871}, {1088, 6516}, {1260, 9965}, {1376, 5744}, {1617, 6601}, {2975, 25006}, {3219, 17616}, {3306, 10383}, {3434, 7677}, {3587, 3877}, {3601, 4666}, {4260, 8849}, {4881, 24929}, {5687, 20015}, {5787, 25005}, {6282, 35262}, {8543, 20292}, {8583, 12511}, {9441, 25941}, {9940, 11491}, {10582, 30282}, {11018, 27003}, {11023, 11508}, {11500, 26062}, {14923, 34489}, {19764, 19769}, {33925, 34607}

X(35977) = {X(404),X(7411)}-harmonic conjugate of X(2)

X(35978) = EULER LINE INTERCEPT OF X(58)X(100)

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

X(35978) lies on these lines: {1, 18601}, {2, 3}, {8, 3286}, {58, 100}, {81, 3871}, {1014, 6604}, {1043, 29766}, {1621, 25526}, {2223, 4968}, {2975, 4278}, {3295, 8025}, {3724, 24850}, {3915, 18792}, {4653, 5253}, {5255, 17187}, {5330, 18465}, {5687, 16704}, {5710, 18166}, {9350, 25440}, {12699, 17174}, {19765, 19769}, {19821, 19850}

X(35979) = EULER LINE INTERCEPT OF X(65)X(100)

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

X(35979) lies on these lines: {2, 3}, {35, 12609}, {36, 6734}, {46, 78}, {55, 11281}, {56, 12649}, {57, 224}, {65, 100}, {79, 27385}, {191, 936}, {938, 22766}, {1155, 11684}, {1210, 10073}, {1259, 5905}, {1376, 21677}, {1446, 6516}, {1454, 9352}, {1617, 10529}, {1621, 12701}, {1792, 3936}, {1998, 3361}, {2245, 2287}, {2646, 3742}, {2975, 5794}, {3612, 35016}, {3647, 16120}, {3649, 10940}, {3652, 17653}, {3916, 17616}, {5124, 5742}, {5441, 14803}, {5554, 11500}, {5687, 20013}, {5703, 11507}, {5705, 7280}, {7742, 10527}, {8261, 33857}, {10427, 12913}, {10543, 22768}, {11263, 13411}, {11604, 17100}, {12511, 35258}, {12775, 33594}, {13279, 24928}, {14450, 27383}, {15803, 31938}, {15931, 24541}, {15932, 20612}, {16143, 16209}, {16586, 33178}, {26285, 33592}, {33597, 33858}

X(35980) = EULER LINE INTERCEPT OF X(69)X(100)

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

X(35980) lies on these lines: {2, 3}, {6, 15447}, {36, 11269}, {42, 46}, {65, 17018}, {69, 100}, {228, 5905}, {1155, 3240}, {1214, 20243}, {1714, 4278}, {1754, 1790}, {2223, 26228}, {2352, 19785}, {2646, 29814}, {3198, 18607}, {3286, 24597}, {3434, 16678}, {3612, 3720}, {3724, 24248}, {3914, 16778}, {4297, 26013}, {5010, 29640}, {7280, 33140}, {13397, 20624}, {17126, 19133}, {17647, 31330}, {19766, 19769}, {20075, 23853}, {20078, 20760}, {20242, 27339}, {20992, 27628}, {22369, 27267}, {26227, 34284}, {31394, 35289}

X(35981) = EULER LINE INTERCEPT OF X(71)X(100)

Barycentrics a*(a + b)*(a + c)*(a^5*b - 2*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 - 2*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(35981) lies on these lines: {2, 3}, {43, 1780}, {55, 5327}, {71, 100}, {81, 1936}, {162, 2193}, {243, 23207}, {1210, 4278}, {1998, 18206}, {2328, 2947}, {2651, 26893}, {2939, 12514}, {3869, 6508}, {15447, 18635}, {15931, 17188}

X(35982) = EULER LINE INTERCEPT OF X(79)X(100)

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

X(35982) lies on these lines: {2, 3}, {79, 100}, {484, 3678}, {1376, 3648}, {1621, 6701}, {2802, 34195}, {3649, 3871}, {3652, 10225}, {3746, 11263}, {5253, 5441}, {11499, 16116}, {14450, 25568}, {14526, 20292}, {16118, 25440}

X(35983) = EULER LINE INTERCEPT OF X(86)X(100)

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

X(35983) lies on these lines: {2, 3}, {31, 27643}, {36, 30970}, {42, 4658}, {55, 5333}, {58, 899}, {81, 1376}, {86, 100}, {110, 29352}, {274, 26227}, {750, 3736}, {999, 4720}, {1043, 5253}, {1213, 15447}, {1621, 25507}, {1698, 4278}, {2229, 5277}, {2303, 17756}, {3218, 3786}, {3286, 4413}, {3724, 24342}, {3744, 16736}, {3746, 28620}, {3913, 17018}, {4276, 29825}, {4653, 30950}, {5208, 27003}, {5563, 31136}, {7081, 30599}, {7236, 30966}, {8666, 31330}, {8715, 28619}, {8849, 31073}, {9534, 19769}, {10458, 17122}, {11221, 16585}, {16752, 26228}, {17126, 27644}, {19726, 19765}, {19818, 19848}, {25440, 25526}, {27666, 32944}, {30965, 33175}

X(35984) = EULER LINE INTERCEPT OF X(100)X(141)

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

X(35984) lies on these lines: {2, 3}, {35, 29632}, {36, 29631}, {39, 2240}, {42, 3670}, {43, 6763}, {55, 18139}, {56, 29829}, {100, 141}, {171, 5161}, {228, 17184}, {238, 34869}, {2223, 26230}, {2238, 4286}, {2352, 32774}, {2975, 33139}, {3185, 32950}, {3218, 4260}, {3240, 17595}, {3285, 24512}, {3583, 30980}, {3589, 15447}, {3724, 3821}, {3909, 4271}, {3936, 5132}, {4276, 30984}, {4972, 16678}, {5010, 29858}, {7280, 29856}, {7761, 30954}, {7831, 30987}, {8053, 24542}, {11688, 33102}, {15624, 33122}, {17596, 34997}, {20913, 26227}, {25557, 29822}, {26228, 26978}

X(35985) = EULER LINE INTERCEPT OF X(100)X(142)

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

X(35985) lies on these lines: {2, 3}, {57, 3681}, {100, 142}, {200, 3874}, {277, 26228}, {942, 3935}, {1376, 3475}, {1706, 3870}, {1998, 3306}, {2550, 33925}, {3303, 28629}, {3957, 10914}, {4413, 5744}, {4571, 7321}, {4847, 5563}, {5178, 5253}, {5745, 9342}, {6745, 12436}, {6763, 8580}, {7677, 33108}, {10106, 25006}

X(35986) = EULER LINE INTERCEPT OF X(100)X(144)

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

X(35986) lies on these lines: {2, 3}, {8, 12511}, {40, 3935}, {100, 144}, {165, 3219}, {200, 3951}, {390, 33925}, {516, 31019}, {962, 34486}, {991, 14996}, {1155, 10394}, {1214, 9539}, {1742, 17126}, {1750, 27065}, {1770, 13405}, {1998, 3218}, {2078, 30332}, {2951, 35258}, {3240, 9441}, {3579, 12528}, {3617, 5584}, {3681, 7964}, {3746, 4295}, {3870, 7991}, {3957, 7982}, {4297, 26015}, {4305, 5563}, {4666, 30389}, {5259, 10248}, {5537, 5905}, {5550, 35202}, {5658, 26792}, {5759, 17484}, {6223, 6796}, {6244, 20214}, {6745, 12512}, {8545, 35445}, {9352, 10178}, {9812, 15931}, {10167, 23958}, {11407, 27003}, {12279, 22076}, {13329, 14997}, {20095, 35514}, {21153, 35595}

X(35987) = EULER LINE INTERCEPT OF X(100)X(189)

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

X(35987) lies on these lines: {2, 3}, {40, 18607}, {55, 77}, {100, 189}, {103, 3190}, {224, 12262}, {962, 1622}, {1804, 7070}, {1813, 2192}, {3100, 7011}, {13243, 23089}, {15494, 15726}

X(35988) = EULER LINE INTERCEPT OF X(100)X(198)

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

X(35988) lies on these lines: {2, 3}, {33, 3101}, {41, 3240}, {81, 33586}, {100, 198}, {108, 347}, {197, 17784}, {208, 4296}, {228, 3995}, {329, 5285}, {498, 9591}, {972, 9058}, {1400, 17126}, {1420, 7191}, {1697, 3920}, {1824, 9536}, {2000, 7291}, {2078, 26228}, {2550, 20989}, {3100, 24611}, {3220, 5744}, {3796, 32911}, {5012, 5320}, {5121, 7280}, {5218, 20872}, {5297, 16548}, {5314, 18228}, {5345, 24239}, {5435, 7293}, {5739, 11206}, {5795, 29667}, {7080, 8193}, {8185, 19843}, {9070, 26702}, {9539, 20243}, {9957, 29815}, {11680, 26259}, {14996, 15107}, {14997, 15080}, {17024, 24928}, {17127, 27659}, {25934, 31884}, {26540, 32269}, {26668, 35260}, {28081, 33148}

X(35989) = EULER LINE INTERCEPT OF X(100)X(210)

Barycentrics a*(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 + 2*a*b^2*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(35989) lies on these lines: {2, 3}, {35, 16154}, {55, 5905}, {63, 2900}, {79, 14798}, {100, 210}, {144, 6600}, {191, 200}, {500, 3193}, {758, 3870}, {1319, 29817}, {1621, 1836}, {1709, 35258}, {3057, 3957}, {3218, 10391}, {3255, 3256}, {3648, 31660}, {3649, 11510}, {3838, 5284}, {3935, 11684}, {3951, 8715}, {4666, 35016}, {5426, 10582}, {5441, 26015}, {5554, 5584}, {7280, 31249}, {7971, 16132}, {7995, 16143}, {10543, 10936}, {10578, 11508}, {10580, 22767}, {11491, 12528}, {12511, 19860}, {13405, 16152}, {15931, 34789}, {19850, 19852}, {19919, 32141}, {24703, 34879}

X(35990) = EULER LINE INTERCEPT OF X(100)X(226)

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

X(35990) lies on these lines: {2, 3}, {7, 8730}, {9, 30295}, {56, 5175}, {79, 6745}, {100, 226}, {191, 8580}, {200, 758}, {329, 1376}, {643, 28979}, {950, 5253}, {1014, 2893}, {1320, 3957}, {1476, 6598}, {1621, 9580}, {1708, 9352}, {2136, 3340}, {2900, 3873}, {3218, 17616}, {3306, 10382}, {3487, 3871}, {3586, 31249}, {3649, 11501}, {4666, 13384}, {4847, 5288}, {5128, 11684}, {5176, 25006}, {5714, 11517}, {5728, 27003}, {5777, 17653}, {7677, 11680}, {7742, 31418}, {9612, 25440}, {10582, 35016}, {11263, 13405}, {14647, 26062}, {26725, 30384}

X(35991) = EULER LINE INTERCEPT OF X(100)X(229)

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

X(35991) lies on these lines: {2, 3}, {46, 2651}, {56, 24378}, {58, 1054}, {65, 662}, {81, 24443}, {100, 229}, {291, 8935}, {501, 3754}, {849, 1739}, {946, 25533}, {1014, 3212}, {1098, 1155}, {1210, 19642}, {1958, 3868}, {2185, 3812}, {2363, 3752}, {5883, 15792}, {18653, 24982}, {20472, 23897}, {20653, 33079}, {23536, 24617}, {24169, 25526}

X(35992) = EULER LINE INTERCEPT OF X(100)X(238)

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

X(35992) lies on these lines: {2, 3}, {36, 4871}, {43, 595}, {56, 30947}, {100, 238}, {993, 29827}, {1284, 17777}, {1621, 25496}, {2223, 5205}, {2229, 33854}, {2352, 18743}, {2975, 30942}, {3240, 3871}, {3741, 5258}, {3877, 30272}, {4432, 5143}, {5248, 29825}, {5253, 30950}, {5260, 30970}, {8666, 31137}, {10453, 12513}, {11688, 32930}, {19726, 19769}, {19801, 19850}, {20470, 25531}, {23407, 29828}

X(35993) = EULER LINE INTERCEPT OF X(100)X(242)

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

X(35993) lies on these lines: {2, 3}, {33, 38}, {55, 1851}, {100, 242}, {108, 840}, {672, 2201}, {1430, 2356}, {1633, 1776}, {1861, 33115}, {5698, 7085}

X(35994) = EULER LINE INTERCEPT OF X(100)X(278)

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

X(35994) lies on these lines: {2, 3}, {33, 3306}, {34, 4855}, {43, 1430}, {55, 17923}, {56, 5174}, {92, 1376}, {100, 278}, {108, 5435}, {162, 4383}, {243, 11502}, {899, 1957}, {1155, 1748}, {1435, 3870}, {1621, 17917}, {1838, 25440}, {1897, 3210}, {7071, 17302}, {7076, 16569}, {11471, 19861}

X(35995) = EULER LINE INTERCEPT OF X(100)X(283)

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

X(35995) lies on these lines: {2, 3}, {81, 1771}, {100, 283}, {947, 1790}, {1437, 11491}, {1444, 33298}, {1788, 3286}, {3193, 3871}, {20222, 23171}, {23207, 23661}

X(35996) = EULER LINE INTERCEPT OF X(100)X(312)

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

X(35996) lies on these lines: {2, 3}, {33, 20243}, {81, 3060}, {100, 312}, {102, 9058}, {105, 29681}, {108, 17080}, {197, 3434}, {198, 26258}, {612, 5119}, {908, 5285}, {940, 33586}, {1145, 33091}, {1319, 7191}, {1331, 21361}, {1376, 15494}, {1610, 5086}, {1629, 31623}, {1763, 2000}, {1791, 5016}, {1859, 3101}, {1875, 4296}, {2194, 5012}, {2886, 20989}, {3011, 14798}, {3057, 3920}, {3193, 5752}, {3452, 5314}, {3796, 4383}, {3911, 7293}, {4268, 33854}, {4271, 5276}, {5310, 32760}, {5322, 24239}, {5432, 20872}, {5552, 8193}, {6690, 20988}, {7069, 21368}, {7085, 31018}, {8185, 26363}, {8192, 10529}, {9082, 29325}, {9798, 10527}, {10327, 26264}, {10528, 12410}, {10530, 10834}, {11206, 14555}, {11510, 26228}, {15107, 18165}, {17074, 20122}, {18141, 33522}

X(35997) = EULER LINE INTERCEPT OF X(100)X(319)

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

X(35997) lies on these lines: {2, 3}, {46, 4658}, {81, 1155}, {100, 319}, {110, 15731}, {165, 1790}, {283, 35242}, {1030, 15447}, {1437, 31663}, {3193, 3579}, {4720, 10609}, {5333, 5880}, {8666, 17156}, {12609, 28618}

X(35998) = EULER LINE INTERCEPT OF X(100)X(341)

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

X(35998) lies on these lines: {2, 3}, {36, 28018}, {100, 341}, {106, 13397}, {950, 7293}, {1319, 4296}, {1473, 12649}, {1633, 3869}, {2975, 4514}, {3007, 17320}, {3057, 3100}, {3434, 22654}, {4316, 9591}, {4329, 4389}, {5119, 20243}, {6012, 26703}, {7354, 20872}, {8192, 20075}, {11510, 28037}, {12410, 20076}, {14798, 28027}, {19860, 24309}, {28017, 34489}

X(35999) = EULER LINE INTERCEPT OF X(100)X(386)

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

X(35999) lies on these lines: {2, 3}, {31, 3216}, {35, 32772}, {100, 386}, {940, 19769}, {1150, 19762}, {1402, 3702}, {2298, 4261}, {2975, 10479}, {3871, 5710}, {19792, 19850}, {20470, 26094}, {32944, 35206}

X(36000) = EULER LINE INTERCEPT OF X(100)X(387)

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

X(36000) lies on these lines: {2, 3}, {8, 2352}, {56, 4966}, {78, 579}, {100, 387}, {386, 4855}, {580, 1801}, {1100, 2275}, {1211, 19759}, {1714, 25440}, {3868, 3998}, {5712, 19769}, {5739, 19762}, {10449, 17740}, {19793, 19850}

This is the end of 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)	PART 21:	Centers X(40001) - X(42000)
PART 22:	Centers X(42001) - X(44000)	PART 23:	Centers X(44001) - X(46000)	PART 24:	Centers X(46001) - X(48000)
PART 25:	Centers X(48001) - X(50000)	PART 26:	Centers X(50001) - X(52000)	PART 27:	Centers X(52001) - X(54000)
PART 28:	Centers X(54001) - X(56000)	PART 29:	Centers X(56001) - X(58000)	PART 30:	Centers X(58001) - X(60000)
PART 31:	Centers X(60001) - X(62000)	PART 32:	Centers X(62001) - X(64000)	PART 33:	Centers X(64001) - X(66000)
PART 34:	Centers X(66001) - X(68000)	PART 35:	Centers X(68001) - X(70000)	PART 36:	Centers X(70001) - X(72000)